Chiral algebras

Contents Introduction

1

Chapter 1. Axiomatic Patterns

11

1.1. Pseudo-tensor categories

11

1.2. Complements

18

1.3. Compound tensor categories

24

1.4. Rudiments of compound geometry

32

Chapter 2. Geometry of D-schemes

53

2.1. D-modules: Recollections and notation

53

2.2. The compound tensor structure

67

2.3. DX -schemes

79

2.4. The spaces of horizontal sections

89

∗

2.5. Lie algebras and algebroids

95

2.6. Coisson algebras

113

2.7. The Tate extension

117

2.8. Tate structures and characteristic classes

129

2.9. The Harish-Chandra setting and the setting of c-stacks

143

Chapter 3. Local Theory: Chiral Basics

157

3.1. Chiral operations

157

3.2. Relation to “classical” operations

163

3.3. Chiral algebras and modules

164

3.4. Factorization

172

3.5. Operator product expansions

194

3.6. From chiral algebras to associative algebras

200

3.7. From Lie∗ algebras to chiral algebras

212

3.8. BRST, alias semi-infinite, homology

227

3.9. Chiral differential operators

238

3.10. Lattice chiral algebras and chiral monoids

257

Chapter 4. Global Theory: Chiral Homology

275

1

2

CONTENTS

4.1. The cookware

275

4.2. The construction and first properties

296

4.3. The BV structure and products

313

4.4. Correlators and coinvariants

325

4.5. Rigidity and flat projective connections

230

4.6. The case of commutative DX algebras

341

4.7. Chiral homology of the de Rham-Chevalley algebras

347

4.8. Chiral homology of chiral envelopes

353

4.9. Chiral homology of lattice chiral algebras

357

Bibliography

363

Index and Notation

369

Introduction V nastowe$i rabote ne soderats svedeni, kotorye mogli by sostavit~ predmet izobreteni ili otkryti. Anon., “Akt kspertizy”, ok. 1956.† 0.0. This book is an exposition of basic (local and global) aspects of chiral algebra theory. Chiral algebras have their origin in mathematical physics; they lie at the heart of conformal field theory1 ; see [BPZ]. Mathematicians, since the pioneering work of Borcherds [B1], usually look at them through the formalism of vertex algebras incorporated into representation theory of infinite-dimensional algebras. We follow a different approach which tastes more of algebraic geometry than of representation theory. In the introduction we outline the principal structures involved and their interrelations. More specific information, together with bibliographical comments and references, can be found in brief introductions to sections. 0.1. Chiral algebras are “quantum” objects. Let us describe first the corresponding “classical” objects; we call them coisson algebras (“coisson” may be considered as an abbreviation for “chiral Poisson” or “compound Poisson”, the word “compound” being related to the notion of compound tensor category which we discuss in 0.4). In fact, “coisson algebra” is a new name for a well-known class of objects. Informally, a coisson algebra is defined by a local Poisson bracket on a space of classical fields. In the simplest case where “classical field” means “a function u(x)” a mathematical physicist would define a local Poisson bracket by a formula like (0.1.1)

{u(x), u(y)} =

n X

ϕi (x, u(x), u0 (x), . . . )δ (i) (x − y),

ϕi ∈ A.

i=1

Here A denotes the algebra of functions of x, u, u0 , . . . , u(k) (k is not fixed). Of course in the left-hand side of (0.1.1) u(x) is understood as a functional u 7→ u(x) on the space of classical fields. The bracket should be skew-symmetric and satisfy the Jacobi identity, so the collection of functions ϕi should satisfy certain differential equations. For f ∈ A and every x we have the functional `f,x on the space of † “This article does not contain any information that could be the subject of an invention or a discovery.”, Anon., “Expert Certification”, circa 1956. (The signing of this Patriot Act was a precondition for publication of a mathematical paper in yonder Russia.) 1 More precisely, of its purely holomorphic sector which is “holomorphic quantum mechanics”.

1

2

INTRODUCTION

classical fields defined by `f,x (u) = f (x, u(x), u0 (x), . . . ). It follows from (0.1.1) that X f,g (0.1.2) {`f,x , `g,y }(u) = ϕi (x, u(x), u0 (x), . . . )δ (i) (x − y) i

for some (0.1.3)

ϕf,g i

∈ A. Formula (0.1.2) has a more symmetric version: X f,g {`f,x , `g,y }(u) = ∂xi ∂yi {ψij (x, u(x), u0 (x), . . . )δ(x − y)},

f,g ψij ∈ A.

i,j f,g Of course the ψij are not uniquely determined by ϕf,g i ; we must take into account the relations

(0.1.4)

(∂x + ∂y )(aδ) = (∂x a) · δ,

a ∈ A,

where δ = δ(x − y). Now from the algebraic point of view the primary object is the algebra A rather than the “space of classical fields”. For instance, one can take A = C[x, u, u0 , u00 , . . . ]. More generally, A can be any commutative differential unital algebra over C[x], and a “classical field” is a homomorphism of differential unital C[x]-algebras A → B where B is, e.g., the algebra of C ∞ -functions of x ∈ R. A choice of a set of generators and defining relations for the differential C[x]-algebra A identifies classical fields with collections of functions satisfying certain systems of differential equations. Let us try to formulate in terms of A what is a “local Poisson bracket on the space of classical fields”. A glance at (0.1.4) and the right-hand side of (0.1.3) shows that such an object is defined by a mapping (0.1.5)

A⊗A → V C

where V is the module over C[∂x , ∂y ] generated by the symbols aδ, a ∈ A, with the defining relations (0.1.4). We prefer another interpretation of V : V is the module over C[x, y, ∂x , ∂y ] generated by the symbols aδ, a ∈ A, with the defining relations (0.1.4) and also the following ones: (0.1.6)

f (x, y) · (aδ) = (f (x, x)a) · δ,

f ∈ C[x, y], a ∈ A.

The mapping (0.1.5) should satisfy certain properties. First of all, it should beh a morphism of C[x, y, ∂x , ∂y ]-modules (notice that A ⊗ A is a module over i ∂ C x, ∂x ⊗ C[x, ∂y ] = C[x, y, ∂x , ∂y ]). Before discussing the other properties, let us rewrite (0.1.5) in geometric terms. An algebra A over C[x] is the same as a quasi-coherent OX -algebra A where X := Spec C[x] is the affine line. Differential C[x]-algebras correspond to DX -algebras (see 0.2). The DX×X -module corresponding to the C[x, y, ∂x , ∂y ]-module V from (0.1.5) is canonically isomorphic to ∆∗ A where ∆: X → X × X is the diagonal embedding. So (0.1.5) is equivalent to a DX×X -module morphism (0.1.7)

A  A → ∆∗ A.

We denote by  the external tensor product, i.e., A  A := p∗1 A ⊗ p∗2 A where p1 and p2 are the projections X × X → X.

INTRODUCTION

3

Notice that (0.1.7) makes sense for an arbitrary smooth variety X. In 0.6 we will define a coisson algebra on X as a commutative (= commutative associative unital) DX -algebra A with a D-module morphism (0.1.7) satisfying certain properties which actually mean that (0.1.3) is a Poisson bracket on the “space(s) of classical fields”. To formulate these properties in a natural way, we need some polylinear algebra in the category of DX -modules (see 0.2 – 0.5). Remarks. (i) The notions of Poisson OX -algebra or Poisson DX -algebra are inadequate for expressing the idea of “Poisson bracket on the space(s) of classical fields”: a Poisson bracket on A is a morphism A ⊗OX A → A while (0.1.7) is an object of a different nature. The reason is that Spec A is the space of “jets of classical fields”, not the space of classical fields. (ii) The reader can compare our notion of coisson algebra with other ways of formalizing the notion of “Poisson bracket on the space of classical fields” (see [DG1], [DG2], [Ku], [KuM], [Ma] and references therein). As far as we understand, our approach is essentially equivalent to that of [KuM] and [Ma]. 0.2. Let X be a smooth complex algebraic variety. Consider the category M` (X) of left D-modules on X (see, e.g., [Ber], [Ba], or [Kas2]). The OX -tensor product ⊗ makes it a tensor category with unit object OX . A commutative algebra in M` (X) (commutative DX -algebra) is just a quasi-coherent commutative OX algebra A` equipped with a flat connection along X. We call an X-scheme equipped with a flat connection along X a DX -scheme; so the spectra of commutative DX algebras are DX -schemes affine over X. A standard example: for any X-scheme Y → X the space J(Y /X) of ∞-jets of sections of Y /X is a DX -scheme. Horizontal sections of J(Y /X) are the same as arbitrary sections of Y /X. One may view the (closed) DX -subschemes of J(Y /X) as systems of differential equations on sections of Y /X (see [G]). What makes the tensor category M` (X) substantially different from, say, the tensor category of OX -modules is the absence of duals. Precisely, for F ∈ M` (X) the dual object in the tensor category sense (see, e.g., [D1]) exists iff F is coherent as an OX -module, i.e., is a vector bundle with an integrable connection. For example, for a group DX -scheme G one has its Lie coalgebra CoLie(G) but we cannot dualize it to get the Lie algebra (unless G is finite dimensional as a usual scheme). We will see that the category M` (X) carries a richer structure (that of compound tensor category) which remedies the above flaw. 0.3. Consider the category M(X) of right D-modules on X. Let I be a finite non-empty set; denote by ∆(I) : X ,→ X I the diagonal embedding. For an I-family of D-modules Li ∈ M(X) and M ∈ M(X) set (0.3.1)

(I)

PI∗ ({Li }, M ) := Hom( Li , ∆∗ M ). (I)

Elements of PI∗ ({Li }, M ) are called ∗ I-operations; they are X-local (since ∆∗ M sits on the diagonal). The ∗ operations compose in a natural way,2 just as polylinear maps between vector spaces do (i.e., for a surjective map π: J → I, a J-family {Kj } of D-modules, and ϕ ∈ PI∗ ({Li }, M ), ψi ∈ Pπ∗−1 (i) ({Kj }, Li ) we have ϕ(ψi ) ∈ PJ∗ ({Kj }, M )). The composition is associative; if |I| = 1, then PI∗ = Hom. We call such data of operations (or “polylinear maps”) between the objects of a category 2 We

switched to right D-modules to make the “sign rule” for the ∗ operations obvious.

4

INTRODUCTION

a pseudo-tensor structure. Thus M(X) carries a canonical pseudo-tensor structure; M(X) equipped with this structure is denoted by M(X)∗ . One may view pseudo-tensor categories as a straightforward generalization of operads: an operad is just a pseudo-tensor category with single object. The notions of algebras, modules over them, etc., make perfect sense in any pseudo-tensor category. For example, a Lie algebra in M(X)∗ (or simply a Lie∗ algebra on X) is a D-module L together with a binary ∗ operation [ ] ∈ P2∗ ({L, L}, L) which is skew-symmetric and satisfies the Jacobi identity. For M ∈ M(X) set h(M ) := M ⊗ OX (the sheaf of middle de Rham cohomolDX

ogy). There is a canonical map PI∗ ({Li }, M ) → Hom(⊗h(Li ), h(M )) compatible with composition of operations. Therefore h sends Lie∗ algebras to the sheaves of usual Lie algebras. In fact, many important Lie algebras (including the Virasoro and affine Kac-Moody algebras) arise naturally from Lie∗ algebras. 0.4. Let us identify the categories of left and right D-modules by the usual ∼ −1 ⊗−1 equivalence M(X) −→ M` (X), M 7→ M ` = M ωX = M ⊗ ωX (here ωX = OX

X ). Therefore M(X) has the tensor product M ⊗! L := (M ` ⊗ L` )ωX . Now Ωdim X the ! tensor product of two ∗ operations is defined according to the following pattern. Let I, J be finite sets, i0 ∈ I, j0 ∈ J; denote by I ∨ J the disjoint union of I, J with i0 , j0 identified. Then there is a natural map ! ∗ ∗ ∗ ! ⊗I,J i0 ,j0 : PI ({Mi } , L) ⊗ PJ ({Nj } , K) → PI∨J ({Mi , Nj , Mi0 ⊗ Nj0 } i6=i0 , L ⊗ K). j6=j0

If I = {i0 }, J = {j0 } this is the usual tensor product of morphisms. A compound tensor structure on a category is the data of a tensor and a pseudotensor structure related as above by the ⊗I,J i0 ,j0 maps which are associative and commutative in the obvious sense. Thus M(X) (and M` (X)) is a compound tensor category. 0.5. The compound tensor structure makes it possible to do basic differential geometry in M(X) implementing a complementarity principle: the “functions” multiply according to ⊗! format, while “operators” (such as infinitesimal symmetries) act in the ∗ sense. In other words, the geometric objects we consider are DX -schemes, the Lie algebras are actually Lie∗ algebras, and one defines easily what the action of a Lie∗ algebra on a DX -scheme means. Note that an action of a Lie∗ algebra L on a DX -scheme Y yields an action on Y of the usual Lie algebra h(L). For example, let G be a group DX -scheme such that CoLie(G) is a locally free DX -module of finite rank. Set Lie(G) := CoLie(G)◦ = HomDX (CoLie(G), DX ) (the usual duality of D-modules theory). Then Lie(G) is a Lie∗ algebra. A G-action on a DX -scheme Y yields a CoLie(G)-coaction on OY (in the ⊗! sense). Dualizing, we get a Lie(G)-action on Y. In fact, for locally free DX -modules of finite rank the duality yields an identi∼ fication PI∗ ({Mi }, L) −→ Hom(L◦ , ⊗Mi◦ ). So, on an appropriate derived category level, the ∗ operations can be recovered from the usual tensor structure and the duality functor ◦ . This is a very awkward thing to do however, when dealing with both ! and ∗ operations simultaneously (as happens in coisson algebras), and it precludes chiral quantization (see 0.8 and 0.9).

INTRODUCTION

5

0.6. Now a coisson algebra is simply a Poisson algebra in the compound setting. Namely, this is a commutative DX -algebra A` together with a Lie∗ bracket on A (coisson bracket) such that the adjoint action is compatible with the multiplicative structure on A` . From this we get an action of h(A) on A` in the usual sense. Derivations of A` that come from h(A) are called hamiltonian vector fields. Coisson brackets localize nicely so one knows what is a coisson structure on any DX -scheme. 0.7. Let Y = Spec A` be an affine DX -scheme. The space of horizontal sections X → Y is an ind-affine ind-scheme hYi = hYi(X) = Spf hAi; if X is compact, this is actually a scheme. This construction also makes sense locally. For example, for x ∈ X the space of horizontal sections of Y over the formal punctured disc at x is an ind-affine scheme Spf Aas x . Suppose that X is compact; set Ux := X r{x}. The evaluation map which assigns to a global section on Ux its restriction to the punctured disc is a closed embedding of the ind-schemes of sections hYi(Ux ) ,→ SpfAas x ; its ideal is generated by the image of a certain canonical map rx : Γ(Ux , h(A)) → Aas x . We also have an embedding Yx ,→ Spf Aas whose image consists of horizontal secx tions that extend to x, so hYi(X) = hYi(Ux ) ∩ Yx . If A is a coisson algebra finitely generated as a DX -algebra, then Aasx is a topological Poisson algebra, Γ(Ux , h(A)) is a Lie algebra, and rx commutes with brackets. Therefore rx is a hamiltonian action of Γ(Ux , h(A)) on Spf Aas x , and hYi(Ux ) is the zero fiber of the momentum map. When we pass to quantization and chiral homology (see 0.9 and 0.10), the zero fiber changes into the Hamiltonian reduction. The algebra hAi is also denoted by H0ch (X, A). The derived version of this construction yields a graded commutative superalgebra H·ch (X, A). 0.8. Now let us pass to chiral algebras. There are two complementary (equivalent) approaches to this notion: Lie algebra style and commutative algebra style.3 We begin with the “Lie algebra” approach; for the “commutative algebra” picture see 0.12. From now on we assume that dim X = 1. Let j (I) : U (I) ,→ X I be the complement to the diagonal divisor. Set (0.8.1)

(I)

(I)

PIch ({Li }, M ) := Hom(j∗ j (I)∗  Li , ∆∗ M ).

These are chiral I-operations. One defines their composition in the obvious manner. We get the chiral pseudo-tensor structure on M(X); the corresponding pseudotensor category is denoted by M(X)ch . Now a chiral algebra on X is simply a Lie algebra in M(X)ch with an additional property (existence of unit). We refer to the corresponding Liech bracket as the chiral product. 0.9. Let us explain why the notion of chiral algebra is a quantization of that of coisson algebra. For every DX -module A we have a canonical exact sequence (0.9.1)

0 → Hom(A`⊗2 , A` ) → P2ch ({A, A}, A) → P2∗ ({A, A}, A).

The right arrow assigns to every chiral product µ on A a Lie∗ bracket [ ]µ . If the latter vanishes (we say then that our chiral algebra is commutative), then µ can be considered as a binary operation on A` with respect to ⊗. This way one 3A

poetically-minded reader may call them “dynamic” and “static” points of view.

6

INTRODUCTION

identifies commutative chiral algebra structures on A with commutative DX -algebra structures on A` . Now assume we have a family At of chiral algebras that
depends on a parameter is a coisson bracket t ∈ C such that A0 is commutative. Then { } := t−1 [ ]µt t=0 on A`0 . Thus chiral algebras are quantizations of coisson algebras as promised. In fact, the whole chiral pseudo-tensor structure can be considered as a quantization of the compound tensor structure (see 3.2). The problem of quantization of coisson algebras is fairly interesting. We do not know how to solve it in general. In the main body of this work we treat the simplest cases of linear brackets. In particular, we construct chiral enveloping algebras of Lie∗ algebras and Lie∗ algebroids (chiral algebras of twisted differential operators). We also discuss (in 3.9.10) quantizations modt2 . The general theory of deformations of chiral algebras was recently developed in [Tam1]. In a sense, in the “chiral world” chiral algebras are parallel to associative algebras while Lie∗ algebras and commutative DX -algebras play, respectively, the roles of Lie and commutative algebras. 0.10. The constructions from 0.7 generalize to arbitrary chiral algebras. For a chiral algebra A one can define its chiral homology Hich (X, A) (see 0.12). Here H0ch (X, A) is what mathematical physicists usually call “the space of conformal blocks”. For any x ∈ X we have an associative algebra Aas x ; the fiber Ax is natuch rally a quotient of Aas x modulo a left ideal. H0 (X, A) identifies canonically with a quotient of Ax modulo a right ideal generated by the image of a Lie algebra morphism rx : Γ(Ux , h(A)) → Aas x . In the classical limit we return to the objects from 0.7. 0.11. Let us sketch now the “commutative algebra” style description of chiral algebras. This approach is essential for certain subjects, in particular, for the definition of chiral homology. Consider the space R(X) of finite non-empty subsets of X; for such a subset S ⊂ X we denote the corresponding point of R(X) by [S]. So R(X) carries a natural filtration R(X)n ; the open stratum R(X)◦n is the space of configurations of n (distinct) points on X. Remarkably enough, R(X) is contractible. Informally, a factorization algebra on X is an O-module A`R(X) on R(X) such that for every two non-intersecting subsets S, T ⊂ X there is a canonical identification of fibers A`[S∪T ] = A`[S] ⊗ A`[T ] which is associative and commutative in the obvious sense. We also demand the existence of a unit section (the definition is left to the reader). Such A`R(X) provides an O-module A` on X = R(X)1 . In fact, A` is automatically a left DX -module, and A`R(X) amounts to a certain structure on A` of local origin (referred to as factorization algebra structure). Now a factorization algebra structure on A` amounts to a chiral algebra structure on A. Namely, the chiral product corresponding to a factorization algebra structure is the composition j∗ j ∗ A  A = j∗ j ∗ AX×X → ∆∗ ∆! AX×X = ∆∗ A. Here AX×X is the pull-back of AR(X) by the obvious map X × X → R(X)2 , ∆ : X ,→ X × X the diagonal, j : U ,→ X × X its complement, the equalities are structure identifications, and the arrow is a canonical morphism. The O-tensor product of factorization algebras is evidently a factorization algebra, so chiral algebras form a tensor category.

INTRODUCTION

7

0.12. Therefore chiral algebras can be considered as geometric objects on R(X). In this vein, the chiral homology of A is defined as the de Rham homology of R(X) with coefficients in A`R(X) . The chiral homology functor has many remarkable properties; e.g., it commutes with tensor products. In particular, higher chiral homology of the unit chiral algebra vanishes (which also follows from contractibility of R(X)). Chiral homology is naturally defined for DG chiral algebras and it is preserved by quasi-isomorphisms. In the exposition we do not pave a road across the morass of homotopy theory of chiral algebras, but we resort to an unworthy (yet solid) path of functorial resolutions. Namely, chiral homology can be realized as homology of certain functorial chiral chain complexes C ch (X, A)PQ which depend on appropriate auxiliary resolutions P, Q of OX (see 4.2.12).4 These complexes resemble Chevalley homology complexes of Lie algebras: for example, they carry a canonical BV (Batalin-Vilkovisky) algebra structure. In the classical limit it becomes an odd Poisson bracket. The comeuppance is the lack of understanding of the structure of the homotopy category of chiral algebras (see Remarks in 3.3.13). By the way, the latter lies outside of Quillen’s model category framework due to the absence of cofibrant objects (e.g., the chiral algebra freely generated by DX ∈ M(X) does not exist). 0.13. Physicists usually describe a chiral algebra structure in terms of operator product expansions, ope for short (see, e.g., [BPZ]). The same approach is common in the literature on vertex algebras; see 0.15. To explain what ope is, one needs to consider some non-quasi-coherent Dˆ ∗ A` be a sheaf of left DX×X -modules modules. Thus, for a left DX -module A` , let ∆ supported on the diagonal X ,→ X × X which is I-adically complete and satisfies ˆ ∗ A` /I ∆ ˆ ∗ A` = A` (here I ⊂ OX×X is the ideal of the diagonal). Such ∆ ˆ ∗ A` ∆ ` exists and is unique. Note that for any local section a ∈ A there is a unique ˆ ∗ A` such that a(1) is horizontal along the second variable and section a(1) ∈ ∆ (1) ` ˆ ∗ A = a. Therefore, if t is a local coordinate on X, one may write any a mod I ∆ ˆ ∗ A` . ˆ ∗ A` as a formal power series P a(1) (t2 −t1 )i , ai = 1 ∂ti ϕ mod I ∆ section ϕ of ∆ i i! 2 i≥0

˜ ∗ A` be the localization of ∆ ˆ ∗ A` with respect to the equation of the Now let ∆ P (1) diagonal. Its section is a Laurent power series ai (t2 − t1 )i ; i.e., we can write i−∞

˜ ∗ A` as A` ((t2 − t1 )). Notice that ∆ ˜ ∗ A` /∆ ˆ ∗ A` = ∆∗ A` . ∆ ˜ ∗ A` . Composing Now an ope is a morphism of DX×X -modules ◦ : A`  A` → ∆ ` it with the above projection to ∆∗ A , we get a binary chiral operation µ = µ◦ on A. One can explain what it means for ◦ to be commutative and associative. In fact, ◦ 7→ µ◦ is a bijective correspondence between the set of commutative and associative ope and that of chiral algebra structures on A. From the point of view of factorization algebras, our ope is the gluing isomorphism that reconstructs A`X×X from its restriction to U (which is A`  A` |U ) and ˆ ∗ A` ). to the formal neighbourhood of the diagonal (which is ∆ ∗ Notice also that the Lie bracket of a chiral algebra is just the the polar part of the ope. 4 A poetically-minded reader may say that P resolves ultraviolet problems of A, while Q takes care of its infrared behavior.

8

INTRODUCTION

0.14. To see an example of a non-commutative chiral algebra, let us describe in geometric terms the chiral enveloping algebra of a Kac-Moody Lie algebra. Let G be an algebraic group. For x ∈ X consider the set of pairs (F, α), where F is a G-bundle on X and α is a trivialization of F on X r {x}. This is the set of points of a formally smooth ind-scheme GRx ; a choice of a parameter tx at x identifies GRx with G(C((t)) )/G(C[[t]]). Our GRx are fibers of an ind-scheme GRX over X called the affine Grassmannian. This is a DX -ind-scheme: when x varies infinitesimally, X r {x}, hence GRx , do not change. The trivialized G-bundle (F, α) on X defines the section e: X → GR (which is the only horizontal section of GR). Denote by A`x the vector space of distributions on GRx supported at e(x); i.e., A`x is the (topological) dual to the formal completion of the local ring OGRx ,e(x) . When x varies, the A`x form a left DX -module A`X . Let us show that A is naturally a chiral algebra. As in 0.11 we have to define a factorization algebra A`R(X) . We have a D-ind-scheme GRR(X) over R(X) with fibers GR[S] equal to the space of pairs (F, α) where F is a G-bundle on X and α is a trivialization of F on X r S. There is a canonical horizontal section e of GRX n , and we set A`[S] to be the vector space of distributions on GR[S] supported at e. The factorization property for A`R(X) follows from the similar property of GRR(X) itself: there is a canonical identification GR[S∪T ] = GR[S] × GR[T ] if S ∩ T = ∅. To get the chiral envelope of a Kac-Moody algebra of a non-zero level, one must twist A` by an appropriate canonical line bundle on GR. 0.15. Chiral algebras are related to (various versions of) vertex algebras as follows. The category of vertex algebras in the sense of [B1] and [K] is equivalent to the category of chiral algebras on X = A1 equivariant with respect to the group T of translations.5 Namely, the vertex algebra VA corresponding to a T -equivariant chiral algebra A is the vector space of T -invariant sections of A` with the vertex algebra structure given by the operator product expansion (see 0.13); of course, VA identifies naturally with the fiber A`x at any point x ∈ A1 . In the same manner the category of VA -modules is identified with that of (weakly) T -equivariant A-modules or, more conveniently, with that of A-modules supported at x.6 Replacing T on the chiral side by the group Af f of affine transformations, we get on the vertex side essentially an object called the “vertex algebra” in [FBZ], “Z-graded vertex algebra” in [GMS2], and “graded vertex algebra” in [K].7 Adding to the structure a Virasoro vector, a.k.a. stress-energy tensor, which is a morphism from the Virasoro Lie∗ algebra (of some central charge) to A compatible with the T - or Af f -action, we get essentially a “conformal vertex algebra” from [K] and [FBZ] or the “vertex operator algebra” of [FLM], [FHL], [DL], [Hu]. Similarly, a chiral algebra on the “coordinate disc” Spec C[[t]] equivariant with respect to the action of the group ind-scheme Aut C[[t]] is the same as the “quasiconformal vertex algebra” from [FBZ]. The equivariance means that such an object yields a chiral algebra on any non-coordinate formal disc, and, in fact, by patching, 5 Here “equivariant” means that the group T of translations acts on A as on the O -module, X and the action of each t ∈ T is compatible with the chiral algebra structure. In particular, A is a weakly T -equivariant D-module. 6 One identifies a weakly T -equivariant A-module M with the A-module (j j ∗ M )/M supx∗ x ported at x; here jx is the embedding A1 r {x} ,→ A1 . 7 Here “essentially” means that we discard the varying finiteness conditions of the references.

INTRODUCTION

9

on any curve (see [FBZ] 18.3.3 and [HL2]). In this way it amounts to a “universal” chiral algebra, i.e., a rule that assigns to any curve a chiral algebra on it, in a way compatible with the ´etale localization When one is interested in local questions, such as the usual representation theory, chiral algebras are essentially equivalent to vertex algebras. In practice, however, chiral algebras are considerably more flexible: for example, twists of vertex algebras (see 3.4.17) and constructions like that of Aas x (see 3.6.4) are painful in the pure vertex algebra setting. Let us mention that all kinds of vertex/chiral algebras can be seen as chiral algebras on appropriate c-stacks (see 2.9, 3.1.16, 3.3.14), the above-mentioned functors and equivalences being mere base change. In the exposition we stick to the usual D-module setting. 0.16. We started this work with the modest intent of understanding the ingenious formal power series manipulations that haunt the books on vertex operator algebras. For a VOA insider, untrammelled by algebro-geometric affections, the mode of the output might resemble though zadopasomy$i vol’s8 features. A challenging problem is to define chiral algebras on higher-dimensional X (see [B2] and [Tam2] in this respect). The definitions from 0.8 and 0.11 formally work also for dim X > 1, but to make them sensible, one must plunge at once into the homotopy setting which we had no courage to do (see 0.12). Notice that coisson algebras live in any dimension, so it may be reasonable to look first for the corresponding quasi-classical objects. Let us also mention that while the general format of the classical setting is covered nicely by the compound tensor category axiomatics (see 1.4), chiral algebras proper defy such treatment. We consider chiral algebras on usual curves, while some applications demand the setting of super curves. Presumably, the rendition should not be difficult. 0.17. The book consists of four chapters. The first one discusses some relevant abstract nonsense. The classical (coisson) story occupies the second chapter. The third chapter deals with the local theory of chiral algebras proper (some readers may find the exposition of the textbooks [K], [FBZ] livelier,9 and we do recommend parallel reading). It divides into two parts: the first considers basic structures and their interrelations while the second deals with elementary methods of constructing chiral algebras. The final chapter treats global theory, i.e., the formalism of chiral homology. It contains an exposition of the general machinery and some results 8 “V barbari e sut~ voly glagolts epistony, zane ne pasutc obyqa$ino kak vol mesta sego, iduwe vpered, no onye pasuts nazad, potomu qto rogi u nih sut~ krkovaty i sklonts napered ... utykatc rogi ih v zeml”, iz “Damaskina arhiere Studita sobrani ot drevnih filosofov o nekih sobstvah estestva ivotnyh”. “In Barbary there are oxen called epistomi for

they do not graze walking forward in the usual manner of the ox of our land, but move backwards, since horns of theirs are crooked and bent forward ... sticking into the earth,” from “A collection (of excerpts) from ancient philosophers on certain properties of the nature of animals” by Damaskin, the high priest of the Studiy monastery. 9 “U nas dvady dva toe qetyre, da vyhodit kak-to bo$ iqee.” M. Gasparov, “Zapiski i Vypiski”; kommentari$i k ponti “Narodnost~”. “At

ours two by two is also four, yet it comes livelier”, from M. Gasparov, “Notes and Excerpts”, a comment on the notion narodnost’ (which has no adequate English analog).

10

INTRODUCTION

showing that the chiral homology functor transforms the constructions of chiral algebras from Chapter 3 to parallel constructions for BV algebras. It is known that distilled axioms are pretty indigestible. We strongly suggest the reader follow the complicated alembics of Chapter 1 simultaneously with respective sections of more wholesome Chapters 2 and 3. 0.18. We were particularly motivated by the observation that chiral algebras provide a natural tool for tackling geometric automorphic forms in the D-module setting. This can be seen already in the oper construction at the critical level (see [BD]). One hopes that general automorphic D-modules come as (higher) chiral homology of twisted chiral Hecke algebras (where the moduli of G-bundles and de Rham G∨ -local systems are parameters of the twists). A related local problem is to describe a “spectral decomposition” of the category of representations of an affine Kac-Moody algebra (at negative integral level) with spectral parameters being moduli of G∨ -local systems on the punctured formal disc (see [Be]). The chiral Hecke algebras arise from the global geometry of the affine Grassmannian; when G is a torus, they amount to lattice Heisenberg algebras. We will return to these subjects elsewhere. A general matter of special interest is the factorization property of chiral homology for degenerating families of curves. At the moment, it is to some extent understood only for H0ch of rational field theories (the Verlinde rules). Presumably, there should be an interesting theory beyond the rational situation with Verlinde’s summation over the finite set of irreducibles replaced by “integration” over an appropriate “compact space”. The (twisted) chiral Hecke algebra may provide an example of such a situation. 0.19. Our sincere gratitude is due to Sasha Belavin and Pierre Deligne. Apart from an obvious influence of their ideas, it was in their apartments in 1992–1995 where the prime part of the work was done. During the (all too long) course of writing, we were greatly helped by many mathematicians; we are very grateful to S. Arkhipov, J. Bernstein, R. Bezrukavnikov, P. Bressler, B. Feigin, E. Frenkel, V. Ginzburg, V. Hinich, Y.-Z. Huang, M. Kapranov, D. Kazhdan, J. Lepowski, Yu. Manin, B. Mazur, V. Schechtman, I. Shapiro, G. Segal, J. Wiennfield, G. Zuckerman, and, especially, D. Gaitsgory, for their interest, inspiration, and correction of mistakes. The first author would also like to thank his wife and children10 for their ability to endure the rewritings of the draft. We are grateful to Dottie Phares of IAS and Richard Lloyd of MIT for the careful typing of the first version of the first part of the manuscript back in 1994–1995, and to Sergei Gelfand and Arlene O’Sean of the AMS for the editing of the manuscript. The authors were partially supported by NSF grant DMS-0100108. 0.20. A few words about notation commonly used in the book. “x ∈ Y ” means that either x is an element of a set Y or x is an object of a category Y or x is a local section of a sheaf Y . For a category M we denote by M◦ the dual category; Sets is the category of all sets. For a smooth variety X we denote by ΘX and DX the tangent sheaf and the algebra of differential operators. 10 “I slonovu e oustraxaet rost~ malaa kviqawaa svinaa porosta”, iz “Pohvaly Bogu o sotvorenii vse$i tvari Georgi Pisidy.” “And the

elefant rage is terrified by tiny squealing swine piggies,” from “A praise unto the Lord for the creation of all living creatures” by George Pisida.

CHAPTER 1

Axiomatic Patterns Da porazit t puwe groma Uasna, sil~na aksioma. A. A. Nahimov, “Pot i Matematik”, Har~kovski$i Demokrit, 1816

†

1.1. Pseudo-tensor categories The notion of pseudo-tensor category is an immediate generalization of that of operad in the same way that the notion of category generalizes that of monoid. On the other hand, the pseudo-tensor structure appears naturally on any (not necessary closed under the tensor product) subcategory of a tensor category,1 hence the adjective “pseudo-tensor”. For example, you can play with Lie algebras and their representations, but the enveloping algebras are out of reach. Pseudo-tensor categories were considered sporadically and under various names for quite a while; see [La] (“multicategories”) or [Li], [B2] (“multilinear categories”). Operads were introduced by P. May [May] in the topological context; today they are very popular: see, e.g., [GK], [H], [Kap1], [KrM], [L], [J], and [MSS]. Pseudo-tensor categories are defined in 1.1.1, pseudo-tensor functors and adjunction are in 1.1.5, examples are in 1.1.6, the linear setting is considered in 1.1.7, the tensor product of pseudo-tensor categories is defined in 1.1.9. A funny lemma in 1.1.10 will be used later to identify commutative chiral algebras with commutative DX -algebras; its explanation in 1.1.11 is due to V. Ginzburg. In 1.1.13 we explain that a pseudo-tensor structure on the category of R-modules, R an associative algebra, can be naturally defined the moment one knows the operad of endooperations of R. In 1.1.16 we fix “super” conventions. 1.1.1. Denote by S the category of finite non-empty sets and surjective maps. For a morphism π: J  I in S and i ∈ I set Ji := π −1 (i) ⊂ J; let · ∈ S be the one element set. Definition. A pseudo-tensor category is a class of objects M together with the following datum: †

Let thee be smitten as by a thunderbolt By a dreadful, potent axiom. A. A. Nakhimov, “A Poet and a Mathematician”, Kharkovski Democritus, 1816 1 In fact, every pseudo-tensor category appears in this manner in a universal way; see Remark in 1.1.6(i).

11

12

1.

AXIOMATIC PATTERNS

(a) For any I ∈ S, an I-family of objects Li ∈ M, i ∈ I, and an object M ∈ M one has the set PIM ({Li }, M ) = PI ({Li }, M ); its elements are called Ioperations. (b) For any map π: J  I in S, families of objects {Li }i∈I , {Kj }j∈J , and an object M one has the composition map Y (1.1.1.1) PI ({Li }, M ) × PJi ({Kj }, Li ) → PJ ({Kj }, M ), (ϕ, (ψi )) 7→ ϕ(ψi ). I

Here in the notation PJi ({Kj }, Li ) we assume that j ∈ Ji . The following properties should hold: (i) The composition is associative: if H  J is another surjective map, {Fh } an H-family of objects, χj ∈ PHj ({Fh }, Kj ), then ϕ(ψi (χj )) = (ϕ(ψi ))(χj ) ∈ PH ({Fh }, M ). (ii) For any M ∈ M there is an element idM ∈ P· ({M }, M ) such that for any ϕ ∈ PI ({Li }, M ) one has idM (ϕ) = ϕ(idLi ) = ϕ. Q We often use notation PJ/I ({Kj }, {Li }) = Pπ ({Kj }, {Li }) for PJi ({Kj }, Li ) I

to write the composition map as PI × PJ/I → PJ . We also write Pn := P{1,...,n} . We denote our pseudo-tensor category simply by M by abuse of notation. 1.1.2. Let M be a pseudo-tensor category. For L, M ∈ M set Hom(L, M ) = P1 ({L}, M ). Composition of morphisms comes from (b), so we have a usual category with the same objects as M; we denote it also by M. The composition with morphisms makes PI a functor (M◦ )I × M → Sets. Therefore, we may consider the pseudo-tensor categories as usual categories equipped with extra pseudo-tensor structure formed by a collection of functors PI , |I| ≥ 2, together with composition morphisms. Every category admits a trivial pseudo-tensor structure with PI = ∅ for |I| > 1. 1.1.3. Assume that a family {Li }i∈I has the property that the functor M 7→ PI ({Li }, M ) is representable. We denote the corresponding object of M as ⊗ Li I

and call it the pseudo-tensor product of Li ’s. If the pseudo-tensor product exists for any family of objects, then we call our pseudo-tensor structure representable. In other words, a representable pseudo-tensor structure on a category M is a rule that assigns to any I ∈ S a functor ⊗: MI → M and to any map π: J  I a natural I

compatibility morphism (1.1.3.1)

επ : ⊗ Kj → ⊗(⊗ Kj ). J

I Ji

The morphisms επ must behave naturally with respect to composition of the maps π, and ⊗ is the identity functor. ·

If all the compatibility morphisms επ are isomorphisms, then we arrive at the usual notion of tensor (= symmetric monoidal) category. 1.1.4. A pseudo-tensor category B with single object is the same as an operad. Namely, such B amounts to a collection of sets Bn = PnB , n ≥ 1, equipped with the action of the symmetric group Σn , together with appropriate “composition of operations” maps between them, and our axioms coincide here with those of an operad.

1.1. PSEUDO-TENSOR CATEGORIES

13

Examples. (i) The operad Σ of linear orders: ΣI = {linear orders on I}; the composition is defined by the lexicographic rule. Notice that an operad over Σ, i.e., an operad B equipped with a morphism of operads B → Σ, is the same as “colored operad”. (ii) The operad of Q projections which assigns to I the same set I; the composition map for J  I is I × Ji → J, (i0 , (ji )) 7→ ji0 . (iii) The operad of trees which assigns to I the set TI of (homeomorphism classes of) trees with I ingoing and one outgoing edge; the composition is the gluing of the corresponding outgoing and ingoing edges. Notice that TI is naturally ordered (one tree is larger than another if the latter can be obtained from the former by contracting some connected subgraphs to points); the composition preserves the order. 1.1.5. Let M, N be pseudo-tensor categories. A pseudo-tensor functor τ : N → M is a law that assigns to any object G ∈ N an object τ (G) ∈ M and to {Fi }i∈I , G ∈ N, a map τI : PIN ({Fi }, G) → PIM ({τ (Fi )}, τ (G)) so that τI are compatible with composition and τ· (idG ) = idτ (G) . One defines a morphism between pseudo-tensor functors in the obvious way. So, if N is equivalent to a small category, pseudo-tensor functors N → M form a category. The composition of pseudo-tensor functors is a pseudo-tensor functor in the obvious way. Any pseudo-tensor functor τ defines an usual functor between the corresponding usual categories. We call τ a pseudo-tensor extension of this usual functor. ν

An adjoint pair is a pair of pseudo-tensor functors M  N together with idenτ

∼

tifications a: PIN ({ν(Mi )}, G) −→ PIM ({Mi }, τ (G)) for Mi ∈ M, G ∈ N, compatible with composition of operations.2 We say that ν is left adjoint to τ and τ is right adjoint to ν. The corresponding usual functors are obviously adjoint. Exercise. Show that for a given pseudo-tensor functor its left (resp. right) adjoint pseudo-tensor functor is determined uniquely up to a unique isomorphism. 1.1.6. Examples. (i) A pseudo-tensor subcategory of a pseudo-tensor category M is given by a subclass of objects T ⊂ M together with subsets PIT ({Li }, M ) ⊂ PIM ({Li }, M ) for Li , M ∈ T such that PIT are closed under the composition of operations, and idM ∈ P.T ({M }, M ) for any M ∈ T. Then (T, P T ) is a pseudo-tensor category, and T ,→ M is a pseudo-tensor functor. If PIT ({Li }, M ) = PIM ({Li }, M ) for any Li , M ∈ P, then we call T a full pseudo-tensor subcategory of M. Any subclass of objects of M may be considered as a full pseudo-tensor subcategory of M. In particular, for any M ∈ M we have the full pseudo-tensor subcategory with single object M . We denote by P (M ) = P M (M ) the corresponding operad. Remark. Any pseudo-tensor category M can be realized as a full pseudotensor subcategory of a tensor category. Here is a universal construction of such an embedding M ,→ M⊗ . By definition, an object of M⊗ is a collection {Mi }i∈I of objects of M labeled by some finite non-empty set I; we denote it by ⊗Mi = ⊗I Mi . A morphism ϕ : ⊗J Nj → ⊗I Mi is a collection (π, {ϕi }i∈I ) where π : J  I and ϕi ∈ PJi ({Nj }, Mi ); we write ϕ = ⊗ϕi = ⊗π ϕi . The composition of morphisms is defined using composition of operations. The tensor structure on M⊗ and the embedding M ,→ M⊗ are the obvious ones. 2 I.e.,

we demand that for ϕ ∈ PIN ({ν(Mi )}, G), ψi ∈ PJM ({Lj }, Mi ) one has a(ϕ(ν(ψi ))) = i

N ({G }, H), ϕ ∈ P N ({ν(M )}, G ) one has a(χ(ϕ )) = τ (χ)(a(ϕ )). a(ϕ)(ψi ) and for χ ∈ PK i k k k k k I k

14

1.

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(ii) Assume that M, N have representable pseudo-tensor structures. Then a pseudo-tensor functor N → M is the same as a usual functor τ : N → M together with natural morphisms νI : ⊗ τ (Ni ) → τ (⊗ Ni ) for any finite families of objects I

I

Ni ∈ N, such that νI commute with the compatibility morphisms επ (i.e., for any π: J  I and a J-family Kj ∈ N the diagram ⊗τ (Kj ) J  M yεπ (1.1.6.1)

ν

−−−J−→

τ (⊗Kj ) J

   N τ (επ ) y

⊗(⊗τ (Kj )) I Ji  ⊗νJi yI ν

⊗τ (⊗Kj ) −−−I−→ τ (⊗(⊗Kj )) I

Ji

I Ji

commutes). In particular, any tensor functor between tensor categories is a pseudotensor functor. (iii) If B is an operad, then we call a pseudo-tensor functor B → M a B algebra in M; we denote the category of B algebras in M as B(M). A B algebra in M amounts to an object L ∈ M together with a morphism of operads B → P M (L). For example, a Σ algebra in M is just a monoid, i.e., an object L equipped with an associative product · ∈ P2 ({L, L}, L) = P2M (L). An algebra for a “trivial” 1-point operad is a commutative monoid. (iv) For any family {Mj } of pseudo-tensor categories one defines their product ΠMj in the obvious way. A pseudo-tensor functor N → ΠMj is the same as a collection of pseudo-tensor functors N → Mj . (v) If A is a tensor category, then the product functor A × A → A is a pseudotensor functor in the obvious way. Let M be a pseudo-tensor category. An A-action on M is a pseudo-tensor functor ⊗: A × M → M together with an identification of ∼ two pseudo-tensor functors A × A × M → M, (A1 ⊗ A2 ) ⊗ M −→ A1 ⊗ (A2 ⊗ M ), that satisfies the obvious compatibilities. In concrete terms, such an action is given by an action of A on M as on a usual category together with natural maps PIM ({Mi }, N ) → PIM ({Ai ⊗ Mi }, (⊗Ai ) ⊗ N ) for Mi , N ∈ M, Ai ∈ A. (vi) We are in situation of (v), so we have a pseudo-tensor functor ⊗ : A⊗M → M. Let P be a commutative monoid in A. Then M → A × M, M 7→ (P, M ), is naturally a pseudo-tensor functor (see (iv) above). Thus M → M, M 7→ P ⊗ M , is a pseudo-tensor functor. Similarly, if F is a commutative monoid in M, then A → M, A 7→ A ⊗ F , is naturally a pseudo-tensor functor. So if a A is a B algebra in A and M is a B algebra in M, then P ⊗ M , A ⊗ F are B algebras in M. 1.1.7. The above definitions make sense when operations PI ({Li }, M ) are not sets but objects of a certain fixed tensor (or even pseudo-tensor!) category A.3 We call such thing a pseudo-tensor A-category. The common example is A = the tensor category of k-modules, where k is a commutative ring; this is the case of pseudo-tensor k-categories. Other examples (super setting, DG categories, etc· ) are discussed in 1.1.16. 3 So

the composition maps are morphisms PI ⊗ (⊗I PJi ) → PJ in A, etc.

1.1. PSEUDO-TENSOR CATEGORIES

15

Explicitly, a pseudo-tensor k-category is a pseudo-tensor category M together with a k-module structure on the sets PI such that the composition map is kpolylinear. For example, if M consists of a single object, then we have a k-operad. A pseudo-tensor k-category M is additive if it is additive as a usual k-category, and M is abelian if it is abelian as a usual k-category and the polylinear functors PI are left exact. A pseudo-tensor functor F : N → M between the pseudo-tensor k-categories is k-linear if the maps FI are k-linear. If B is a k-operad, we get a notion of a B k-algebra in M (we usually call them simply B algebras). For example, we have the favorite k-operads Ass, Com, Lie, Pois which give rise, respectively, to associative, commutative and associative, Lie, and Poisson algebras. Note that Ass, Com are k-envelopes of the corresponding set-theoretic operads: Ass = k [Σ], Com = k. To avoid problems with skew-symmetry, we always assume that 1/2 ∈ k when speaking about Lie or Poisson algebras. 1.1.8. Remarks. (i) Let B be any k-operad. For I ∈ S denote by FB(I) the free B algebra in the tensor category k mod of k-modules with generators ei labeled by elements i ∈ I. Then the morphism of k-modules BI → FB(I), ϕ 7→ ϕ(⊗ ei ), I

is injective, so one may consider BI as the k-submodule of FB(I) generated by “multiplicity free monomials of degree I”. (ii) The operad Pois carries a natural grading such that the product and the Poisson bracket have degrees, respectively, 0 and 1. The operad Ass carries a natural decreasing filtration such that gr Ass = Pois. 1.1.9. Let Nj , j ∈ J, be a finite family of pseudo-tensor k-categories. Their tensor product ⊗Nj is a pseudo-tensor category defined as follows. Its objects are collections {Nj }, Nj ∈ Nj ; we denote this object of ⊗Nj by ⊗Nj . One has N PI ({⊗Nji } , ⊗Kj ) := ⊗ PI j ({Nji }, Kj ); the composition maps are tensor prodkJ

ucts of those in Nj . Note that for any pseudo-tensor k-category M a k-linear pseudo-tensor functor ⊗Nj → M is the same as a k-polylinear pseudo-tensor functor ΠNj → M. 1.1.10. Let M be any pseudo-tensor k-category. Then M ⊗ Lie coincides with M as a usual k-category, and one has PIM⊗Lie ({Li }, M ) = PIM ({Li }, M ) ⊗ LieI . The following funny lemma will be of use: Lemma. Lie(M ⊗ Lie) = Com(M). Proof. We have to show that Lie algebras in M ⊗ Lie are the same as commutative algebras in M. Let M be an object in M. One has P2M⊗Lie (M ) = P2M (M ) ⊗ Lie2 , so a skew-symmetric binary M ⊗ Lie operation [ ]M on M amounts to a symmetric binary M operation ·M . Let us identify Lien with the space of Lie polynomials of variables e1 , . . . , en , so [ ]M = ·M ⊗ [e1 , e2 ]. Since Lie3 is generated by [e1 , [e2 , e3 ]], [e3 , [e1 , e2 ]], [e2 , [e3 , e1 ]] with the only relation [e1 , [e2 , e3 ]] + [e3 , [e1 , e2 ]] + [e2 , [e3 , e1 ]] = 0, we see that [ ]M satisfies the Jacobi relation if and only if ·M is associative.  1.1.11. The material of this subsection will not be used in the sequel. The above lemma is a particular case of a general duality statement for quadratic operads. To formulate it, we need some notions from §2 of [GK]. Assume for simplicity that k is a field. Let A be a quadratic k-operad (i.e., A is a quotient

16

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of the free k-operad generated by a finite-dimensional vector space with Σ2 -action A2 modulo relations in A3 ). Its quadratic dual A! is defined in [GK] 2.1.9 (this is a quadratic operad with A!2 equal to the vector space dual to A2 , the Σ2 -action is twisted by sgn). If B is another quadratic operad, then we have a quadratic operad A • B defined in [GK] 2.2.5 (one has (A • B)2 = A2 ⊗ B2 with the Σ2 -action twisted by sgn) . Notice that for any k-operad P the map Hom(A, P) → Hom(A2 , P2 ) is injec= P2 and relations coming from tive. Let Pquad be a quadratic operad with Pquad 2 ∼ P3 . Then the obvious morphism Pquad → P yields a bijection Hom(A, Pquad ) −→ Hom(A, P). Lemma. For any quadratic operads A, B and a pseudo-tensor k-category M one has (1.1.11.1)

A(M ⊗ B) = (A • B! )(M) = B! (M ⊗ A! ).

Proof. Since • is symmetric, it suffices to define the first identification. We may assume that M has a single object. So we have a k-operad P, and we want to ∼ define a canonical bijection Hom(A, P ⊗ B) −→ Hom(A • B! , P). It comes from the obvious identification Hom(A2 , (P ⊗ B)2 ) = A!2 ⊗ P2 ⊗ B2 = Hom((A • B! )2 , P2 ). To check the relations, we may replace P by Pquad , P ⊗ B by (P ⊗ B)quad = Pquad ◦ B (see [GK] 2.2.3), and we are done by [GK] 2.2.6(b).  If A = B = Com, then A! = B! = Lie (see [GK] 2.1.11) and the above lemma becomes the lemma from 1.1.10. 1.1.12. Remark. Suppose that our M is actually a tensor category with tensor product ⊗. Let us show that the (pseudo) tensor structure of M reconstructs uniquely from that of M ⊗ Lie. Note that the pseudo-tensor structure of M ⊗ Lie is representable; denote by ⊗Lie the corresponding tensor product. One has ⊗Lie = Lie∗I ⊗ ⊗. So one I

I

has a canonical identification M1 ⊗ M2 = M1 ⊗Lie M2 , and the commutativity constraints differ by sign. It remains to recover the associativity constraint. For ∼ M1 , M2 , M3 ∈ M consider the composition ⊗ Lie Mi → M1 ⊗Lie (M2 ⊗Lie M3 ) −→ {1,2,3}

M1 ⊗ (M2 ⊗ M3 ); here the first arrow is the compatibility morphism for ⊗Lie (see 1.1.3), and the second isomorphism comes from the ordering of our indices ∼ (e.g., Lie{2,3} −→ k, [e2 , e3 ] 7→ 1). Let ⊗ Lie Mi → M1 ⊗ (M2 ⊗ M3 ) ⊕ M3 ⊗ {1,2,3}

(M1 ⊗ M2 ) ⊕ M2 ⊗ (M3 ⊗ M1 ) be the sum of these maps for cyclic transposition of indices; denote by K its cokernel. The projection maps each of the 3 terms to K isomorphically, and the associativity constraint for ⊗ coincides with composition ∼ ∼ M1 ⊗ (M2 ⊗ M3 ) −→ K −→ M3 ⊗ (M1 ⊗ M2 ) = (M1 ⊗ M2 ) ⊗ M3 . 1.1.13. One can use operads to define pseudo-tensor structures on some categories of modules and their dual categories. Let us explain how to do this in the k-linear setting. Let R be an associative k-algebra; denote by Rmod, modR, the category of left (right) R-modules. Let B be a strict R-operad, i.e., a k-operad endowed with ∼ an isomorphism of k-algebras R −→ B1 . So for any I ∈ S the k-module BI is actually an (R − R⊗I )-bimodule (here ⊗ is the tensor power over k), and for J 

1.1. PSEUDO-TENSOR CATEGORIES

17

I the composition map may be written as a morphism of (R − R⊗J )-bimodules BI ⊗ BJ/I → BJ . R⊗I

Such B defines the pseudo-tensor structures P ∗ = P ∗B and P ! = P !B on modR and Rmod◦ , respectively. Namely, for Li , M ∈ modR set   (1.1.13.1) PI∗ ({Li }, M ) := HommodR⊗I ⊗Li , M ⊗ BI . R

The composition of ∗ operations comes from the composition law in B. Similarly, for Ci , D ∈ Rmod set   ! (1.1.13.2) PI (D, {Ci }) := HomRmod D, BI ⊗ (⊗Ci ) . R⊗I I

Here we write the operations in Rmod◦ as operations on R-modules with reversed order of arguments, so that P.! (D, {C}) = HomRmod (D, C) (= HomRmod◦ (C, D)). The composition of ! operations comes in an obvious manner from the composition law in B. Remarks. (i) B = P ∗ (R) = P ! (R); here we consider R as right and left R-module, respectively. (ii) The ∗ and ! structures are abelian (see 1.1.7) iff BI are flat as, respectively, R- and R⊗I -modules. 1.1.14. Example. Assume that R has a cocommutative Hopf algebra struc⊗I , and the composition ture. It defines a strict R-operad BR . Namely, BR I := R (i) (i) R R (Ji ) map BI ⊗ BJ/I → BJ is (⊗ri ) ⊗ (⊗(⊗rj )) 7→ ⊗(∆ (ri ) · (⊗rj )) ∈ ⊗R⊗Ji = I

I Ji

I

Ji

I

R⊗J . Here ∆(Ji ) : R → R⊗Ji is the Ji -multiple coproduct. Note that the ! pseudotensor structure coincides with the standard tensor category structure on Rmod. 1.1.15. The above notions easily sheafify. Namely, let X be a topological space (or a Grothendieck topology) and M a sheaf of categories on X (we assume that both morphisms and objects of M have local nature, i.e., satisfy the gluing property). A pseudo-tensor structure on M assigns to any open U a pseudo-tensor structure on M(U ) and to any j: V → U a pseudo-tensor extension of the pullback functor j ∗ : M(U ) → M(V ). We assume that this extension is compatible with the composition of j’s and the pseudo-tensor operations have local nature (i.e., for any Li , M ∈ M(U ), the prehseaf P I ({Li }, M ) on U , P I ({Li }, M )(V ) = PI ({Li }, M ), is actually a sheaf). We call such M a sheaf of pseudo-tensor V V categories. All the notions we discuss in Chapter 1 have straightforward sheaf versions. We will tacitly assume them for the next sections. 1.1.16. Super conventions. Sometimes we need to consider super objects in a given category. Let M be a k-category. Its super (:= Z/2-graded) objects form a k-category Ms . In other words, Ms is the tensor product of M and the category of finitedimensional super vector spaces. Ms is also a “super” category: for M, N ∈ Ms the k-module of all M-morphisms Hom(M, N ) has an obvious Z/2-grading, and the morphisms in Ms are the even part in Hom(M, N ). If M is a tensor category, then Ms is automatically a tensor (super) category according to the Koszul rule of signs. A Z-graded super object M in M is a super object in M equipped with an extra Z-grading. For such M we set M [1] := k[1] ⊗ M where k[1] is an odd copy

18

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of k placed in degree −1; i.e., M [1] is the Z-graded super object obtained from M by simultaneous shift of Z- and Z/2-gradings by 1. The shift is an automorphism of the category of Z-graded super objects, so we have M [a] for every a ∈ Z. We consider M a as a subobject of M . The category MsZ of Z-graded super objects has an obvious “Z-graded super structure”: for M, N ∈ MsZ we have a Z-graded super k-module Hom(M, N ) with components Hom(M, N )i = ΠHom(M a , N a+i ). If M is a tensor category, then so is MsZ (the signs come from the “super” grading, so the forgetting of the Z-grading is a tensor functor). A DG super object, or super complex, in M is a Z-graded super object equipped with an action of the commutative graded Lie super algebra k[−1], which is the same as an odd endomorphism d of degree 1 with square 0. They form a DG super category CMs in the obvious manner. This is a tensor DG super category if M is a tensor category. We identify the category of Z-graded objects in M with a full subcategory in MsZ of those objects for which the Z-degree mod 2 coincides with the super degree. In the same manner the category of complexes CM is a full subcategory of CMs . If M is a tensor category, then these are embeddings of tensor subcategories. If M is a pseudo-tensor k-category, then Ms is automatically a pseudo-tensor super k-category (see 1.1.7). Namely, for Li , M ∈ Ms the k-module PI ({Li }, M ) carries the obvious (Z/2)I × Z/2-grading; hence it is a super k-module with respect to the total Z/2-grading. The composition takes the Koszul sign rule into account. Thus for any Li , M ∈ Ms and free super k-modules Vi , U of finite rank one has a canonical identification of super k-modules PI ({Vi ⊗ Li }, U ⊗ M ) = Hom(⊗ Vi , U ) ⊗ PI ({Li }, M ).4 One can also view Ms as a plain pseudo-tensor I

k-category considering only even operations. Similarly, MsZ is a pseudo-tensor Z-graded super k-category and CMs is a pseudo-tensor DG super k-category. One has PI ({Li [ai ]}, M [b]) = P ({Li }, M )[b − Σai ]. Notation. For L ∈ CM (or CMs ) we set L† := Cone(idL ), or, equivalently, L† = k† ⊗ L. Notice that k† is naturally a comutative DG k-algebra (indeed, we can write k† = k[η], d† = ∂η , for an odd variable η of Z-degree −1). So, by 1.1.6(vi),5 for a pseudo-tensor M the functor L 7→ L† is a DG pseudo-tensor endofunctor of CM or CMs and the canonical morphism L → L† is a morphism of DG pseudo-tensor functors. In particular, if L is a B algebra, then so is L† and L → L† is a morphism of B algebras. 1.2. Complements This section is a continuation of 1.1. We consider inner objects of operations in 1.2.1, augmented pseudo-tensor structures in 1.2.4–1.2.7, augmented pseudotensor functors in 1.2.8, modules over “operadic” algebras in 1.2.11–1.2.15, the corresponding monoids (or associative algebras) in 1.2.16, and the action of an augmentation functor on algebras and modules in 1.2.17–1.2.18. 1.2.1. Inner Hom. Let M be a pseudo-tensor category, I a finite set, {Li }i∈I , and M objects in M. Assume we have an object X ∈ M and an element ε ∈ PIe({Li , X}, M ) where Ie := I t · . Then for any J ∈ S and a J-family of objects 4 And

this identification is compatible with composition of operations. to A = C(k) (the category of k-complexes), P = k† .

5 Applied

1.2. COMPLEMENTS

19

{Aj } we get a natural map (1.2.1.1)

κ = κJ (ε) : PJ ({Aj }, X) → PJtI ({Aj , Li }, M ),

κ(ϕ) := ε(ϕ, idLi ). Here “natural” means that for any H  J, an H-family of objects {Bh } and χj ∈ PHj ({Bh }, Aj ) one has κ(ϕ(χj )) = κ(ϕ)(χj , idLi ). A pair (X, ε) with the property that all the maps κ are isomorphisms is uniquely defined (if its exists); we denote it by (PI ({Li }, M ), εI,{Li },M ) and call it the inner P object. Note that Pφ (M ) = M , and P1 (L, M ) = Hom(L, M ) is the “inner Hom” object. If well defined, inner P objects depend on their arguments in a functorial manner: for a surjection E  I, ϕ ∈ Hom(M, N ), and ψi ∈ PEi ({Ke }, Li ) (as usual Ei := π −1 (i)) there is an obvious morphism PI ({Li }, M ) → PE ({Ke }, N ) which is compatible with the composition of ϕ’s and ψ’s. 1.2.2. Here is an “inner” version of the functoriality property. Let π: E → I be any map of finite sets, {Ke } an E-family of objects. Assume that the objects PI ({Li }, M ), PE ({Ke }, M ), PEi ({Ke }, Li ) exist. Then one has a canonical “composition” element (1.2.2.1)

c ∈ PIe({PI ({Li }, M ), PEi ({Ke }, Li )}, PE ({Ke }, M ))


defined as the image of the composition εI,{Li },M idPI ({Li },M ) , εEi ,{Ke },Li by the canonical isomorphism κ−1 . Note that if E = ∅, then c = ε. One may check that the composition is associative in the obvious sense. Example. Consider L ∈ M such that End L = Hom(L, L) exists. Then the composition c ∈ P2 ({End L, End L}, End L) is associative; hence it makes End L a monoid. In the k-linear setting End L is an associative k-algebra. 1.2.3. Example. Suppose we are in situation 1.1.13, so M = modR and the pseudo-tensor structure is given by the functors P ∗ defined by the operad B. For Li , M as above set X := HomR⊗I (⊗Li , M ⊗ BIe). This is a right R-module in the I

R

obvious way, and we have the canonical operation ε ∈ PIe∗ ({Li , X}, M ). The maps κJ (ε) are isomorphisms for |J| = 1; i.e., X necessarily coincides with PI ({Li }, M ) if the latter object exists. This happens in the following situations: (i) Assume that the composition maps BI ⊗ BJ/I → BJ are isomorphisms. R⊗I

Then X = PI ({Li }, M ) if the Li are projective modules of finite rank. (ii) Suppose (i) holds and all BI are flat R-modules. Then X = PI ({Li }, M ) if the Li are finitely presented R-modules. ˆ the category 1.2.4. Augmented pseudo-tensor structures. Denote by S ˆ If in 1.1.1 of arbitrary finite sets and maps between them, so S is a subcategory of S. ˆ we replace S by S, then we get the definition of augmented pseudo-tensor category. An augmented pseudo-tensor category with single object is called an augmented operad. Examples. (i) A tensor category with a unit object 1 is an augmented tensor category: set PI ({Li }, M ) = Hom(⊗Li , M ). We assume that ⊗ = 1. I

φ

(ii) The linear orders operad Σ (see 1.1.4) extends in the obvious way to an ∧ ∧ augmented operad Σ with Σ∅ = {point}. We also have the “trivial” augmented ∧ ∧ ∧ one-point operad, and their k-linear envelopes Ass = k[Σ ] and Com = k.

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1.2.5. One may consider an augmented pseudo-tensor category as a usual pseudo-tensor category equipped with some extra structure. Let us describe this structure explicitly. We need the following definition. Let M be any pseudo-tensor category. Suppose we have a functor h: M → Sets,

(1.2.5.1) and for any finite set I of order (1.2.5.2)

≥2

and i0 ∈ I we have natural maps

hI,i0 : PI ({Li }, M ) × h(Li0 ) → PIr{i0 } ({Li }, M )

(here, as usual, {Li } is an I-family of objects, M ∈ M). We call these data an augmentation functor if the following compatibilities (i), (ii) hold: (i) Let J be a finite set of order ≥ 2, π: J  I a surjective map, j0 ∈ J. Then for ϕ ∈ PI ({Li }, M ), ψi ∈ PJi ({Kj }, Li ), a ∈ h(Kj0 ) one has hJ,j0 (ϕ(ψi ), a) = ϕ(ψi0 , hJi0 ,j0 (ψi0 , a)) (here i0 = π(j0 ), i0 ∈ I r {i0 }) if |Ji0 | ≥ 2, and hJ,j0 (ϕ(ψi ), a) = hI,i0 (ϕ, ψi0 (a))(ψi0 ) if |Ji0 | = 1. (ii) Assume that |I| ≥ 2; let i0 , i1 ∈ I be two distinct elements. Then for ϕ ∈ PI ({Li }, M ), ai0 ∈ h(Li0 ), ai1 ∈ h(Li1 ), one has hIr{i0 },i1 (hI,i0 (ϕ, a0 ), a1 ) = hIr{i1 },i0 (hI,i1 (ϕ, a1 ), a0 ) ∈ PIr{i0 ,i1 } ({Li }, M ) if |I| > 2, and hI,i0 (ϕ, a0 )a1 = hI,i1 (ϕ, a1 )a0 ∈ h(M ) if |I| = 2. One defines morphisms of augmentation functors in the obvious way. Remark. Any subfunctor of an augmentation functor is itself an augmentation functor. The functor h(M ) ≡ ∅ is an augmentation functor. Definition. We say that our augmentation functor is non-degenerate if all the maps PI ({Li }, M ) → Maps(h(Li0 ), PIr{i0 } ({Li }, M )) coming from (1.2.5.2) are injective. ∧

1.2.6. Now let M be an augmented pseudo-tensor category, M the corresponding usual pseudo-tensor category. Then M carries an augmentation functor h = P∅ ; hI,i0 is the composition map for the embedding I r {i0 } ,→ I. Lemma. The above construction yields an 1-1 correspondence between augmented pseudo-tensor categories and usual pseudo-tensor categories equipped with an augmentation functor.  1.2.7. Note that any augmentation functor h: M → Sets is automatically a pseudo-tensor functor (we consider Sets as a tensor category with ⊗ = Π); i.e., we have natural maps (1.2.7.1)

hI : PI ({Li }, M ) → Hom(⊗h(Li ), h(M )). I

1.2. COMPLEMENTS

21

To see this, we can assume, by the above lemma, that h came from an augmented pseudo-tensor structure. Then hI is the composition map for ∅ → I. Equivalently, for ϕ ∈ Pn ({Li }, M ), ai ∈ h(Li ) one has (here hn : = h{1,...,n} ) hn (ϕ)(a1 ⊗ · · · ⊗ an ) = h{n−1,n},n−1 (. . . (h{1,...,n},1 (ϕ, a1 ) . . . ), an−1 )an . In the situation of 1.2.1 there is a canonical map (1.2.7.2)

¯ : h(PI ({Li }, M )) → PI ({Li }, M ), h

λ 7−→ hI,· e (εI,{Li },M , λ).

¯ arrows intertwine h e(c) and the usual composition map for For c as in 1.2.2 the h I P operations. We say that PI ({Li }, M ) is an inner P object in the augmented sense if (1.2.7.2) is an isomorphism. This amounts to the fact that the canonical morphism (1.2.2.1) (for X = PI ({Li }, M )) is an isomorphism for J = ∅ as well. ∧

∧

∧

1.2.8. One defines an augmented pseudo-tensor functor τ : N → M between augmented pseudo-tensor categories in the obvious way. Such τ ∧ amounts to a usual pseudo-tensor functor τ : N → M together with a morphism of augmentation functors hN → hM τ that are compatible in the obvious manner. For example, given ∧ ∧ an augmented operad B , we have the notion of a B algebra in any augmented ∧ pseudo-tensor category M . ∧

∧

Examples. Σ algebras in M are semigroups; algebras for the trivial aug∧ mented operad are commutative semigroups. In the k-linear setting Ass algebras ∧ are unital associative algebras, Com algebras are unital commutative algebras. If A is a unital associative algebra, then the image 1A ∈ h(A) of 1 ∈ k = Ass∅ is called the unit element in h(A). Remark. 1A is the unit of the associative algebra h(A). So a unital associative ∧ algebra in M is the same as an associative algebra in M such that the associative algebra h(A) is unital and the left and right product with the unit element of h(A) is the identity endomorphism of A. 1.2.9. Following the pattern of 1.1.13, one may use augmented operads to define augmented pseudo-tensor structures on some categories of modules and their duals. ∧ Let R be an associative k-algebra and B an augmented strict R-operad; i.e., ∧ ∧ ∼ B is an augmented k-operad equipped with an isomoprhism R −→ B1 . Such B is the same as a usual strict R-operad B together with a left R-module h = B0 and a system of R⊗Ir{i} -linear morphisms hI,i0 : BI ⊗ B0 → BIr{i0 } (here |I| ≥ 2, i0 ∈ I, R

and BI is considered as a right R-module via the i0 th action) that satisfy obvious compatibilities. According to 1.1.13, B defines the ∗ and ! pseudo-tensor structures on modR and Rmod◦ , respectively. Now h defines the augmentation functors h∗ , h! for these structures. Namely, for M ∈ modR, D ∈ Rmod set (1.2.9.1)

h∗ (M ) := M ⊗ h, R

h! (D) := HomRmod (D, h).

∗B The structure morphisms h∗I,i0 : PI∗B ({Li }, M ) ⊗ h∗ (Li0 ) → PIr{i ({Li }, M ) and 0} ! !B ! ! hI,i0 : PI (D, {Ci })⊗h (Ci0 ) → PIr{i0 } (D, {Ci }) come from the hI,i0 maps. Com∧

patibilities (i) and (ii) in 1.2.5 are clear. Notice that B = P ∗∧ (R) = P !∧ (R).

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1.2.10. Example. Assume that R has a cocommutative Hopf algebra structure with counit ε : R → k. It defines an augmented strict R-operad BR ∧ . Namely, we already defined BR in 1.1.14. Set h = k (the R-module structure is defined by ∼ ε), and let hI,i0 be the obvious isomorphisms R⊗I ⊗ k = R⊗Ir{i0 } ⊗ (R ⊗ k) −→ R

R

R⊗Ir{i0 } . We leave it to the reader to check the compatibilities. Note that k is a unit object for the standard tensor structure on Rmod, and ! augmented structure is the corresponding augmented pseudo-tensor structure (see example (i) in 1.2.4). 1.2.11. Let A be a pseudo-tensor category and P a plain category. An Aaction on P is a pseudo-tensor category C together with an identification of A with a full pseudo-tensor subcategory of C and P with a full subcategory of C such that (a) As a plain category, C is equivalent to a disjoint union of A and P. (b) If PIC ({Ci }, B) is non-empty, then either B and every Ci are in A, or B and exactly one of Ci are in P. To define such a structure, one has to present for every P, Q in P and {Ai }i∈I in A, where I ∈ S, a set PI˜({Ai , P }, Q) which behaves naturally with respect to morphisms of P and Q and operations of Ai ’s. The definition can be rewritten in the k-linear setting in the obvious way. Remark. If A is actually a tensor category, then its action on P as of a tensor category can be considered as an A-action in the above sense: just set PI˜({Ai , P }, Q) := Hom((⊗Ai ) ⊗ P, Q). If both A and P have single object, then we refer to C as an A module operad. The A and P objects of C are denoted, respectively, by • and ∗. 1.2.12. Examples. (i) Let A be any pseudo-tensor category. There is a canonical A-action on A, considered as a plain category. The corresponding C is denoted by Am . It is equipped with a faithful pseudo-tensor functor Am → A m ∼ left inverse to both structure embeddings A ⊂ Am such that P A −→ P A unless prohibited by condition (b). This determines Am uniquely. (ii) Consider operad Σ (see 1.1.4). The Σ module operad Σm contains two Σ m m m module suboperads Σm ` , Σr ⊂ Σ . Namely, ΣI := ΣIt{∗} = set of linear orders m m on I t {∗}, and Σ`I (resp. ΣrI ) is the subset of orders such that ∗ is the maximal (resp. the minimal) element. (iii) Let Bi be operads, and let Ci be Bi module operads, i ∈ I. Denote by Π0 Ci ⊂ Π Ci the full pseudo-tensor subcategory with objects • := (• i ), ∗ := (∗i ). I

I

This is a Π Bi module operad. I

1.2.13. Let B be an operad, C a B module operad, and M a pseudo-tensor category. Let τ : C → M be a pseudo-tensor functor. Then L := τ (• ) is a B algebra in M. We call M := τ (∗) an L-module with respect to C, or simply an L-module (if C is fixed). Explicitly, a structure of an L-module on an object M ∈ M is given by a system of compatible “action” maps CI → PIe({Li , M }, M ). Denote by C(M) the category of pseudo-tensor functors τ : C → M (see 1.1.5); its objects are pairs (L, M ), where L is a B algebra and M is an L-module. If L is a fixed B algebra, then we denote by M(L) = M(L, C) ⊂ C(M) the subcategory of L-modules: its objects are pairs (L, M ); the morphisms are identity on L. Example. Let L be an object of a pseudo-tensor category M such that End L ∈ M exists. Then L is an End L-module with respect to Σm ` .

1.2. COMPLEMENTS

23

Variant. More generally, suppose we have B, C, M as above, and a category P equipped with an M-action D. Denote by C(D) the category of pseudo-tensor functors τ : C → D which map • to M, ∗ to P. Then L := τ (• ) is a B algebra in M and P := τ (∗) is an L-module in P (with respect to C, D). For fixed L we get the category P(L) = P(L, C, D) of L-modules in P; its objects are pairs (L, P ) ∈ C(D), and the morphisms are morphisms in C(D) which are identity on L. The situation described in the beginning of this subsection corresponds to P = M, D = Mm (see 1.2.12(i)). 1.2.14. The above definitions render to the k-linear setting in the obvious way. Lemma. If M is an abelian pseudo-tensor k-category, then M(L, C) is an abelian k-category.  A similar statement for P(L, C, D) is left to the reader. 1.2.15. Examples. (i) B = Lie; C = Bm . Then M(L) is the usual category of modules over the Lie algebra L: its object is M ∈ M equipped with a pairing · ∈ P2 ({L, M }, M ) that satisfies the condition `1 · (`2 · m) − `2 · (`1 · m) = [`1 , `2 ] · m. (ii) B = Ass = k[Σ]. Let L be an associative algebra. If C = Bm = k[Σm ], m then M(L, C) is the category of L-bimodules. If C = k[Σm ` ] (resp. C = k[Σr ]), then M(L, C) is the category of left (resp. right) L-modules. 1.2.16. Let B be an operad, C a B module operad, and L a B algebra in the tensor category Sets (with ⊗ = Π; see 1.2.7). We have the obvious “forgetting of the L-module structure” functor o: Sets(L, C) → Sets. Denote by C(L) the monoid of endormorphisms of this functor. Here is an explicit description of C(L). It is generated by elements ((`i ), ϕ), where I is a finite set, `i ∈ L is an I-family of elements of L, ϕ ∈ CI . The relations for C(L) come from surjective maps π: Je → Ie such that π(·) = · and collections ψi ∈ BJi , i ∈ I. The relation corresponding to π, (ψi ) is (1.2.16.1)

((`j ), ϕ1 (ϕ2 , ψi )) = ((ψi (`j )j∈Ji , ϕ1 ) · ((`j )j∈J. , ϕ2 )).

Here (`j ) is a J-family of elements of L, ϕ1 ∈ CI , ϕ2 ∈ CJ. , where J. := J ∩ π −1 (·). The element 1 = (∅, id.) is the unit in C(L). Denote by C(L)mod the category of sets equipped with C(L)-action. The functor o lifts to the functor o˜: SetsLC → C(L)mod which is an equivalence of categories. More generally, one may define a C-action of L on an object of any category, and this is the same as a C(L)-action. The above construction has its k-linear version. Namely, for a k-operad B, a B module operad C, and a B algebra L in kmod (or in the category of sheaves of k-modules on some space) we get an associative k-algebra with unit C(L) (or a sheaf of k-algebras) such that the category of L-modules coincides with the one of C(L)-modules. We call C(L) the enveloping algebra of L. In situation 1.2.15(i) it coincides with the usual enveloping algebra. ∧

1.2.17. Let now M = (M, h) be an augmented pseudo-tensor category, B, C as above. Let L be a B algebra in M, and let M ∈ M(L, C) be an L-module. Then h(L) is a B algebra in Sets (since h is a pseudo-tensor functor), and M (considered as an object of the usual category M) carries a canonical C(h(L))-action. Explicitly, a generator ((`i ), ϕ) ∈ C(h(L)) acts as hI (aI (ϕ))(`i ) ∈ End M . Therefore we have a canonical “identity” functor (1.2.17.1)

M(L, C) → C(h(L))-modules in M.

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1.2.18. Example. Let M be a pseudo-tensor k-category, and let L be a Lie algebra in M. Let {Mi }i∈I , N be a finite collection of L-modules. We say that an P operation ϕ ∈ PI ({Mi }, N ) is L-compatible if ·N (idL , ϕ) = ϕ(·Mi0 , idMi )i6=i0 ∈ i0 ∈I

PIe({L, Mi }, N ).6 Let PLI ({Mi }, N ) ⊂ PI ({Mi }, N ) be the k-submodule of Lcompatible operations. It is easy to see that composition of L-linear operations is L-linear. Therefore PLI is a pseudo-tensor structure on M(L). If M was an abelian pseudo-tensor category, then M(L) is also an abelian pseudo-tensor category (see 1.2.14). If h is a k-linear augmentation functor on M, then h(L) is a Lie k-algebra, and for every M ∈ M(L), h(M ) is an h(L)-module. The functor hL of h(L)invariants, M 7→ hL (M ) := h(M )h(L) , is an augmentation functor on M(L). h(L) If {Fj }j∈J , G are h(L)-modules in M, then the subspace PJ ({Fj }, G) ⊂ PJ ({Fj }, G) of h(L)-linear operations is defined in the obvious way. Therefore h(L)-modules form a pseudo-tensor k-category. The “identity” functor M(L) → h(L)-modules in M

(1.2.18.1)

is naturally a pseudo-tensor functor. For a k-operad B a B algebra in M(L) is the same as a B algebra A in M on which L acts by derivations. According to (1.2.18.1) such an A carries a canonical h(L)-action by derivations. 1.3. Compound tensor categories The notion of pseudo-tensor category, as opposed to that of tensor category, is not self-dual. Thus “algebraic geometry” in a general pseudo-tensor k-category is very poor: one knows what are commutative algebras, but there are no coalgebras, hence no group schemes, etc. The notion of compound pseudo-tensor category restores self-duality which makes it possible to develop algebro-geometric basics on these grounds as we will see in the next section. We do recommend the reader to consider the basic example of 2.2 below before, or while, reading this section. He may also restrict himself to compound tensor categories (see 1.3.12–1.3.15) instead of general compound pseudo-tensor categories (see 1.3.4–1.3.7), but in this case the self-duality 1.3.8(i) will be lost. The notion of transversality for quotient sets of a given finite set is considered in 1.3.1–1.3.3. We define compound pseudo-tensor structure in 1.3.6–1.3.7, compound pseudo-tensor functors in 1.3.9, augmented version in 1.3.10. We consider duality in 1.3.11 and representable setting in 1.3.12–1.3.13. The notion of a compound category is treated in 1.3.14–1.3.16. An approach to the construction of compound tensor structures parallel to 1.1.13 is in 1.3.18–1.3.19; for an example see 1.3.20. 1.3.1. We have to fix some notation. For a finite non-empty set I denote by Q(I) the set of all equivalence relations on I. We identify an element of Q(I) with the corresponding quotient map πS : I → S and denote it by (S, πS ) or simply S by abuse of notation; as usual for s ∈ S we set Is := πS−1 (s). The set Q(I) is ordered: we write S1 ≤ S2 iff S1 is a quotient of S2 . It is a lattice: for any S, T ∈ Q(I) there exist inf{S, T } ∈ Q(I) and sup{S, T } ∈ Q(I). For any S ∈ Q(I) we have the obvious identifications Q(I) ≤ S := {S 0 ∈ Q(I): S 0 ≤ S} = Q(S), Q(I) ≥ S := {S 00 ∈ Q(I): S 00 ≥ S} = ΠQ(Is ). S

6 Here

·N ∈ P2 ({L, N }, N ), ·Mi ∈ P2 ({L, Mi }, Mi ) are the L-actions.

1.3. COMPOUND TENSOR CATEGORIES

25

We say that S, T ∈ Q(I) are transversal if |S| + |T | = |I| + | inf{S, T }|; S and T are called complementary if they are transveral and | inf{S, T }| = 1. 1.3.2. Proposition. (i) Let V be a non-zero vector space. Let us assign to S ∈ Q(I) the corresponding “diagonal” subspace V S ⊂ V I . Then S, T are transversal iff the subspaces V S , V T ⊂ V I are (i.e., V S + V T = V I ), and S, T are complementary iff, in addition, V S ∩ V T = V · ⊂ V I . (ii) Consider the graph G with vertices labeled by S t T and edges labeled by I, such that an edge i ∈ I connects πS (i) and πT (i). Then S and T are complementary (resp. transversal) iff G is a tree (resp. forest). Proof. (i) Note that |S| = dim V S / dim V , and for any S, T one has V inf{S,T } = V ∩ V T , S = T iff V S = V T . Therefore |S| + |T | ≤ | sup{S, T }| + | inf{S, T }|, and the equality holds iff V sup{S,T } = V S + V T . Hence |S| + |T | ≤ |I| + | inf{S, T }|, and equality holds iff V I = V S + V T . (ii) Note that | inf{S, T }| coincides with the number of components of G, and compute the Euler characteristic of G.  S

1.3.3. Corollary. (a) Suppose S, T ∈ Q(I) are transversal. Then (i) for any α ∈ inf{S, T } the equivalence relations induced by S, T on Iα are complementary; (ii) for any J ∈ Q(I) such that J ≥ S the pair T, J ∈ Q(I) is transversal as well as inf{T, J}, S ∈ Q(J). The map Q(I) ≥ S → Q(T ), J 7→ inf{T, J} is injective. (b) Suppose S, P ∈ Q(I) are such that inf{S, P } = ·. Then P = inf {T } where T is the set of all T ∈ Q(I) such that T

T ∈T

≥P

and T , S are complementary.



1.3.4. Let M be a category. For any functors F : M → Sets, G: M◦ → Sets we denote by hF, Gi the quotient of the disjoint union of sets F (X) × G(X), X ∈ M, modulo the relations (ϕ(f ), g) = (f, t ϕ(g)), where f ∈ F (X), g ∈ G(Y ), ϕ ∈ Hom(X, Y ) (and X, Y are objects in M). We tacitly assume that hF, Gi is a well-defined set; to assure this, it suffices to know that isomorphism classes of objects in M form a set, or that either F or G is a representable functor (if, say, F = Hom(XF , ·), then hF, Gi = G(XF )). The above construction generalizes to the setting of 1.1.7 in the obvious way. In particular, in the k-linear situation (so M is a k-category and F , G are klinear functors with values in k-modules), the k-module hF, Gik is the quotient of ⊕ F (X) ⊗ G(X) modulo the relations ϕ(f ) ⊗ g = f ⊗ t ϕ(g). We have an obvious X∈M

k

map of sets hF, Gi → hF, Gik which is bijective if either F or G is representable. 1.3.5. Now assume that both categories M and M◦ carry the pseudo-tensor structures given by functors PI∗ and PI! , respectively. It will be convenient to write the variables for PI! in reverse order, i.e., as PI! (M, {Li }) so that P.! (M, L) = HomM (M, L). For any I, J ∈ S and I- and J-families of objects {Li }, {Mj }, respectively, set

 ∗! (1.3.5.1) PI,J ({Li }, {Mj }) := PI∗ ({Li }, ·), PJ! (·, {Mj }) . ∗! Our PI,J is a functor; moreover, the ∗ and ! operations act on P ∗! . Precisely, for any surjective maps H  J, G  I and families of objects {Nh }, {Kg } we have the composition map ! ∗! ∗ ∗! PH/J ({Mj }, {Nh }) × PI,J ({Li }, {Mj }) × PG/I ({Kg }, {Li }) → PG,H ({Kg }, {Nh }).

26

1.

AXIOMATIC PATTERNS

∗! ∗! Note that PI,. = PI∗ , P.,J = PJ! .

1.3.6. Assume that for any I ∈ S and a complementary pair S, T ∈ Q(I) we have a natural map (1.3.6.1)

h

I

∗ ! ∗! ({Li }, {Ms }) × PI/T ({Nt }, {Li }) → PT,S iS,T : PI/S ({Nt }, {Ms }), I

((ϕs ), (ψt )) 7→ h(ϕs ), (ψt )iS,T . Here {Li }, {Ms }, {Nt } are arbitrary families of objects parametrized by I, S, T , respectively, and “natural” means that h

I

iS,T is

I iS,T

functorial with respect to Ms and Nt variables. Note that h extends uniquely to the family of maps Y Y I ∗! h iS,T : PI∗!s ,Hs ({Li }, {Bh }) × PG∗!t ,It ({Ag }, {Li }) → PG,H ({Ag }, {Bh }) S

T

labeled by surjecitons H  S, G  T , in a way compatible with composition of ∗ and ! operations (acting on Ag and Bh , respectively). 1.3.7. Definition. The data (P ∗ , P ! , h i) form a compound pseudo-tensor structure on M if the following compatibility axioms hold (here I, S, T , {Li }, {Ms }, {Nt } are as above): (i) Take H ∈ Q(I) such that H ≥ S; set V := inf{H, T }. Then for any H-family of objects {Kh } the following diagram commutes: ! ∗ ! ∗ ∗ PH/S ({Kh }, {Ms })×PI/H ({Li }, {Kh })×PI/T ({Nt }, {Li }) −−→ PI/S ({Li }, {Ms })×PI/T ({Nt }, {Li }) ?Q ? ? h iIv ? ? ?h iI Hv ,Tv S,T yV y ∗ PH/S ({Kh }, {Ms }) ×

Q V

PT∗!v ,Hv ({Nt }, {Kh })

−−→

PT∗!,S ({Nt }, {Ms })

Here the upper horizontal arrow is the product of composition maps for Is  Hs , the lower one is h iH S,V . (ii) Take G ∈ Q(I) such that G ≥ T ; set U := inf{S, G}. Then for any G-family of objects {Rg } the following diagram commutes: ∗ ! ! ∗ ! PI/S ({Li }, {Ms })×PI/G ({Rg }, {Li })×PG/T ({Nt }{Rg }) −−→ PI/S ({Li }, {Ms })×PI/T ({Nt }, {Li }) ?Q ? ? h iIu ? ? ?h iI Su ,Gu S,T yU y

Q U

∗! ! PG ({Rg }, {Ms }) × PG/T ({Nt }, {Rg }) u ,Su

−−→

PT∗!,S ({Nt }, {Ms })

Here the upper horizontal arrow is the product of composition maps for It  Gt , the lower one is h iG U,T . I

(iii) If S = I, T = . , Mi = Li , then h(idLi ), ψiI, . = ψ for any ψ ∈ PI! (N, {Li }). I

(iv) If S = . , T = I, Ni = Li , then hϕ, (idLi )i. ,I = ϕ for any ϕ ∈ PI∗ ({Li }, M ). We call a category M equipped with a compound pseudo-tensor structure a compound pseudo-tensor category. We usually denote it by M∗! , while M∗ and Mo! denote M and Mo considered as plain pseudo-tensor categories (with the operations P ∗ and P ! , respectively).

1.3. COMPOUND TENSOR CATEGORIES

27

1.3.8. Remarks. (i) The axioms (i)–(ii) and (iii)–(iv) are mutually dual. Therefore a compound pseudo-tensor structure on M amounts to that on M◦ . (ii) Any tensor structure on M defines a compound pseudo-tensor structure: set I PI∗ ({Li }, M ) = Hom(⊗Li , M ), PI! (N, {Li }) = Hom(N, ⊗Li ), and h(ϕs ), (ψ)t iS,T = I

I

∗! (⊗ϕs ) ◦ (⊗ψt ) (note that PT,S ({Nt }, {Ms }) = Hom(⊗Nt , ⊗Ms )). S

T

T

S

1.3.9. A compound pseudo-tensor functor τ ∗! : N∗! → M∗! between the compound pseudo-tensor categories is a pair of pseudo-tensor functors τ ∗ : N∗ → M∗ , τ ! : No! → Mo! that give rise to the same usual functor τ : N → M, such that all the diagrams ∗ ! PI/S ({Li }, {Ms }) × PI/T ({Nt }, {Li })   τ ∗ ×τ ! y

< >IS,T

−−−−−→

∗! PT,S ({Nt }, {Ms })   τ ∗! y

< >IS,T

∗ ! ∗! PI/S ({τ Li }, {τ Ms }) × PI/T ({τ Nt }, {τ Li }) −−−−−→ PT,S ({τ Nt }, {τ Ms })

commute. Here the arrow τ ∗! is defined in the obvious way using τ ∗ and τ ! . 1.3.10. Let us explain what an augmented compound pseudo-tensor category is. Let M∗! be a compound tensor category and h an augmentation functor on M∗ . We say that h is compatible with the compound structure, or h is a ∗-augmentation functor on M∗! , if it satisfies the following condition. Let S, T ∈ Q(I) be a complementary pair and i0 ∈ I an element such that πT−1 (t0 ) = {i0 }, |πS−1 (s0 )| ≥ 2 where s0 = πS (i0 ), t0 = πT (i0 ). Set I 0 := I r {i0 }, T 0 := T r {t0 }. We may consider S and T 0 as a complementary pair in Q(I 0 ). Our condition is commutativity of the diagram ∗ ! PI/S ({Li }, {Ms }) × PI/T ({Nt }, {Li }) × h(Nt0 ) −−→ PI∗0 /S ({Li }, Ms ) × PI! 0 /T 0 (Nt , {Li }) ? ? ? ? ?< >I ?< >I 0 S,T y y S,T 0

PT∗!,S ({Nt }, {Ms }) × h(Nt0 )

−−→

PT∗!0 ,S ({Nt }, {Ms })

∗! Here the lower horizontal map is the obvious extension of hT,t0 to PT, . ; the upper one is ((ϕs ), (ψt ), a) 7→ ((ϕs , hIs0 ,i (ϕs0 , ψt0 (a))), ψt ). 0 0 One spells out the compatibility property of a !-augmentation functor in a similar (dual) way. We call a compound pseudo-tensor category endowed with both ∗- and !-augmentation functors an augmented compound pseudo-tensor category.

The above definitions extend to the setting of 1.1.7 in the obvious way. In particular, one has the k-linear version (replace product of sets by tensor product of k-modules, etc.). 1.3.11. Duality. One can use the augmentation functors to relate the pseudotensor categories M∗ and Mo! . For example, consider a !-augmentation functor h! on M∗! . Assume that h! is representable by an object 1, so h! (M ) = HomM (M, 1), and for any M ∈ M the object M ◦ := HomM∗ (M, 1) exists (here HomM∗ means inner Hom with respect to ∗ structure; see 1.2.1).

28

1.

AXIOMATIC PATTERNS

Lemma. The functor M◦ → M, M 7→ M ◦ , admits a canonical pseudo-tensor extension M◦! → M∗ .

(1.3.11.1)

Proof. We need to define the maps PI! (M, {Li }) → PI∗ ({L◦i }, M ◦ ). Let I 1 , I 2 be two copies of I. Set J := I 1 t I 2 , Ie = I t ·, and let πI : J  I, πIe: J  Ie be the projections defined by formulas πI (i1 ) = πI (i2 ) = i, πIe(i1 ) = i, πIe(i2 ) = ·; here i ∈ I, and i1 ∈ I 1 , i2 ∈ I 2 are the copies of i. Then I, Ie ∈ Q(J) is a complementary pair (e.g., by 1.3.2(ii)). Consider a J-family of objects {Aj }, Ai1 = L◦i , Ai2 = Li , e an I-family of objects {1i } (copies of 1), and an I-family of objects which are Li for i ∈ I ⊂ Ie and the · one is M . The desired map is the composition ∼

∗! ◦ ∗ ◦ PI! (M, {Li }) → PI,I → PI∗ ({L◦i }, M ◦ ). e ({Li , M }, {1i }) → PIe ({Li , M }, 1) −

J Here the first arrow sends ψ ∈ PI! (M, {Li }) to (κi ), (ψ, idL◦i ) I,Ie (where κi ∈ P2∗ ({Ai1 , Ai2 }, 1i ) = P2∗ ({L◦i , Li }, 1) is the canonical pairing), the second arrow is ! ∗! λ 7→ h!I (λ)(id⊗I in the obvious manner), and 1 ) (for hI see 1.2.7; we extend it to P the last one is the canonical isomorphism (see 1.2.1). We leave it to the reader to check that the construction is compatible with the composition of operations.  Remark. Assume in addition that we have a ∗-augmentation functor h on M∗! . Then M → h(M ◦ ) is an augmentation functor on M◦! . We have a canonical morphism h(M ◦ ) → h! (M )

(1.3.11.3)

defined as restriction of the structure map P2∗ ({M ◦ , M }, 1) × h(M ◦ ) → Hom(M, 1) (see (1.2.5.2)) to κ × h(M ◦ ) where κ is the canonical pairing. We leave it to the reader to check that (1.3.11.3) is a morphism of augmentation functors on M◦! . 1.3.12. Assume that the ! pseudo-tensor structure is representable (see 1.1.3). I ∗ Then PI,J ({Li }, {Mj }) = PI∗ ({Li }, ⊗! Mj ), and we may rewrite the h iS,T data as J

a collection of canonical morphisms (the compound tensor product maps) (1.3.12.1)

∗ ⊗IS,T : PI/S ({Li }, {Ms }) =

Y

PI∗s ({Li }, Ms ) → PT∗ ({⊗! Li }, ⊗! Ms ), It

S

S

⊗IS,T := h·, (id⊗! Li )iIS,T . The axioms (i), (ii) of 1.3.7 can be rewritten as commuIt

tativity of the following diagrams (where I, S, T, . . . have the same meaning as in 1.3.2):

(i)

∗ ∗ PH/S ({Kh }, {Ms }) × PI/H ({Li }, {Kh })  Q Iv ⊗H y S,V × V ⊗Hv ,Tv

∗ −−−−→ PI/S ({Li }, {Ms })  ⊗I y S,T

PV∗ ({ ⊗ ! Kh }, ⊗! Ms ) × PT∗/V ({⊗! Li }, ⊗ ! Kh ) −−−−→ PT∗ ({⊗! Li }, ⊗! Ms ) Hv

S

It

Hv

It

S

1.3. COMPOUND TENSOR CATEGORIES

29

(the horizontal arrows are composition maps) ⊗IS,T

∗ PI/S ({Li }, {Ms }) Q  ⊗Iu y U Su ,Gu

(ii)

−−−−→

PT∗ ({⊗! Li }, ⊗! Ms ) It S   y

∗ PG/U ({⊗! Li }, {⊗ ! Ms }) −−−−→ PT∗ ({⊗ ! (⊗! Li )}, ⊗! Ms ) Ig

Gt

Su

Ig

S

Here the right vertical arrow is the ∗ composition map with compatibility morphisms ⊗ ! (⊗! Li ) → ⊗! Li ; the lower horizontal arrow is the composition of ⊗G U,T Gt

It

Ig

and the map of the PT∗ arising from ⊗! (⊗ ! Ms ) → ⊗! Ms . U

Su

S

1.3.13. The simplest non-trivial ⊗IS,T operations are those with |S| = 2 (the binary compound tensor products). They look as follows. If S = {1, 2}, then the projection I  S amounts to a non-trivial decomposition I = I1 t I2 . The complementarity of T , S means that the only non-trivial T -equivalence class It has two elements i1 ∈ I1 , i2 ∈ I2 ; i.e., T coincides with I1 ∨ I2 = the union of I1 , I2 with i1 , i2 identified. We call the map ⊗ = ⊗Ii11,i,I22 := ⊗IS,T : PI∗1 ({Li }, M1 ) ⊗ PI∗2 ({Li }, M2 )

i1 ,i2

→ PI∗1 ∨I2 ({Li , Li1 ⊗! Li2 }, M1 ⊗! M2 ), ϕ1 , ϕ2 7→ ϕ1 ⊗ ϕ2 , the binary tensor product at i1 , i2 (here the objects Li under i1 ,i2

the PI∗1 ∨I2 sign are those for i ∈ I r {i1 , i2 }). 1.3.14. Assume that M◦! is actually a tensor category; we call such M∗! a compound tensor category. One may consider ⊗! as a tensor product on M itself; denote this tensor category as M! (so M! is the tensor category dual to Mo! ). If we are in the k-linear situation, then we call M∗! abelian if M is an abelian k-category, the PI∗ are left exact functors, and ⊗! is right exact. A compound tensor functor τ ∗! : N∗! → M∗! between the compound tensor categories is a compound pseudo-tensor functor such that all the canonical morphisms νI : τ (⊗! Ni ) → ⊗ τ (Ni ) from 1.1.6(ii) are isomorphisms; i.e., τ ! : N! → M! is a tensor I

I

functor. Such τ ∗! amounts to a pair (τ ∗ , τ ! ) where τ ∗ : N∗ → M∗ , τ ! : N! → M! are, respectively, the pseudo-tensor and tensor extensions of the same functor τ : N → M which commute with ⊗IS,T maps. Remark. If M is a compound tensor k-categrory, then the category of super objects in M is a compound tensor category in the obvious way, as well as the categories of graded super objects and DG super objects (see 1.1.16 for the terminology). 1.3.15. If M∗! is a compound tensor category, then the right vertical arrow in diagram (ii) in 1.3.12 is an isomorphism. Therefore all the morphisms ⊗IS,T are completely determined by binary ones (use this diagram to compute ⊗IS,T inductively). The binary tensor products are functorial, commutative and associative. Let us spell out these properties explicitly. Functoriality means that the binary products

30

1.

AXIOMATIC PATTERNS

are compatible with composition of ∗ operations in the obvious sense, and ⊗Ii11,i,I22 coincides with the usual ! tensor product of morphisms if I1 = {i1 }, I2 = {i2 }. Commutativity means that ⊗Ii11,i,I22 = ⊗Ii22,i,I11 . To spell out the associativity property, assume that we have three sets I1 , I2 , I3 and the elements i1 ∈ I1 , i12 , i32 ∈ I2 , i3 ∈ I3 ; denote by I1 ∨ I2 ∨ I3 the disjoint union of Ij ’s modulo the relation i1 = i12 , i32 = i3 . Take ϕj ∈ PI∗j ({Li }, Mj ). The associativity says that (ϕ1 ⊗ ϕ2 ) ⊗ ϕ3 = ϕ1 ⊗ (ϕ2 ⊗ ϕ3 ) i1 ,i12

i1 ,i12

i32 ,i3

∈

PI∗1

∨

I2

i32 ,i3

∨

I3 ({Li , Li1

⊗! Li12 , Li32 ⊗! Li3 }, M1 ⊗! M2 ⊗! M3 )

if i12 6= i32 , and (ϕ1 ⊗ ϕ2 ) ⊗ ϕ3 = ϕ1 ⊗ (ϕ2 ⊗ ϕ3 ) = (ϕ1 ⊗ ϕ3 ) ⊗ ϕ2 i1 ,i2

i2 ,i3

i1 ,i2

∈

i2 ,i3

PI∗1 ∨I2 ∨I3 ({Li , Li1

i1 ,i3

!

i1 ,i2

!

⊗ Li2 ⊗ Li3 }, M1 ⊗! M2 ⊗! M3 )

if i12 = i32 = i2 . It is easy to check that for a category M equipped with a tensor structure ⊗! and a pseudo-tensor structure P ∗ the data of functorial, commutative and associative binary tensor products amount to the whole data of compound tensor products, i.e., to a compound tensor structure. Remark. (We use terminology from 1.1.6(v).) The obvious action of the tensor category M! on M extends canonically to an M! -action on M∗ defined by the maps PI∗ ({Mi }, N ) → PI∗ ({Ai ⊗! Mi }, (⊗Ai ) ⊗! N ), ϕ 7→ ϕ ⊗ {idAi }. 1.3.16. Let M∗! be a compound tensor category such that M! has unit object 1 (as a tensor category). Then h! (M ) := HomM (M, 1) is a !-augmentation functor on Mo! . We say that 1 is a unit object in M∗! if h! is compatible with the compound structure (see 1.3.10). Explicitly, this means that for any I-family of objects {Li } and an object M the binary tensor product at i0 ∈ I, End 1 × PI∗ ({Li }, M ) → PI∗ ({Li , 1 ⊗! Li0 }i6=i0 , 1 ⊗! M ) = PI∗ ({Li }, M ), sends id1 × ϕ to ϕ for any ϕ ∈ PI∗ ({Li }, M ). Recall that in any tensor category F = End 1 is an abelian semigroup (a commutative k-algebra in the k-linear situation) that acts canonically on any object M (the action F → End M sends f ∈ F to f ⊗! idM ∈ End(1 ⊗! M ) = End M ). The above compatibility implies that the F -action on PI∗ ({Li }, M ) through M coincides with the action through any Li . If we are in the k-linear situation this means that the PI∗ are canonically F -modules and M∗! is actually a compound tensor F -category. We will call a compound tensor category M∗! which has a unit and is endowed with a ∗-augmentation functor h simply an augmented compound tensor category. We leave the definition of an augmented compound tensor functor between such categories to the reader. 1.3.17. Assume we are in the k-linear situation. We say that our unit object 1 is a strong unit object in M∗! if for any I ∈ S and Li , M ∈ M one has PIe∗ ({Li , 1}, M ) = 0.

1.3. COMPOUND TENSOR CATEGORIES

31

1.3.18. A compound operad is the same as a compound pseudo-tensor category with single object. One may use compound operads to define compound tensor structures on some categories of modules (cf. 1.1.13). As in 1.1.13 we explain this in the k-linear setting. Let R be an associative k-algebra, and let B∗! be a compound strict R-operad, ∼ i.e., a compound k-operad B∗! together with isomorphism of k-algebras R −→ B1 . Explicitly, such B∗! is a pair (B∗ , B! ), where B∗ is a strict R-operad, and B! is a strict Ro -operad (Ro is the opposite algebra, so B∗I is an (R − R⊗I )-bimodule and B!I is an (R⊗I − R)-bimodule), together with the pairings h

(1.3.18.1)

I

iS,T : (⊗ B∗Is ) ⊗ (⊗ B!It ) → B!S ⊗ B∗T S

R⊗I T

R

defined for any complementary pair S, T ∈ Q(I). These maps are morphisms of (R⊗S − R⊗T )-bimodules, and the axioms in 1.3.7 just say that the diagram (here I, S, T, H, V are as in (i) in 1.3.7) (⊗ B∗Hs ) ⊗ (⊗ B∗Ih ) ⊗ (⊗ B!It ) −−−−→ (⊗ B∗I3 ) ⊗ (⊗ B!It ) S S R⊗H H R⊗I T R⊗I T   y   (⊗ B∗HS ) ⊗ (⊗ B!Hv ) ⊗ (⊗ B∗Tv ) y S R⊗H V R⊗V V   y

(1.3.18.2)

B!S ⊗ B∗V ⊗ (⊗ B∗Tv )

B!S ⊗ B∗T

−−−−→

R⊗V V

R

R

I

commutes as well as the corresponding “dual” diagram, and h iS,T coincides with idB∗T , idB!S if | S |= 1, resp. | T |= 1. Now any such B∗! defines a compound pseudo-tensor structure on modR. Namely, set (see 1.1.13) PI∗ ({Li }, M ) = HommodR⊗I (⊗ Li , M ⊗ B∗I ), I R
PI! (N, {Li }) = HommodR N, (⊗ Li ) ⊗ B!I .

(1.3.18.3)

R⊗I

I

The composition of operations comes from composition laws in B∗ and B! . Note !

that the ! pseudo-tensor structure is representable (one has ⊗ Li = (⊗ Li ) ⊗ B!I ), I

I

R⊗I

so we may use the format of 1.3.12. The compound tensor product !

⊗IS,T : ⊗ PI∗s ({Li }, Ms ) → PT∗ ({⊗ Li }, ⊗! Ms )

(1.3.18.4)

S

It

S

sends ⊗ϕs ∈ ⊗ HommodR⊗Is (⊗ Li , Ms ⊗ B∗Is ) to the composition S

Is

R


⊗ϕs ⊗ (⊗ Li ) ⊗ B!It = (⊗ Li ) ⊗ (⊗ B!It ) −−→ (⊗ Ms ) ⊗ (⊗ B∗Is ) ⊗ (⊗ B!It ) T

It

R⊗It

I

R⊗I T

S

h

R⊗S S

R⊗I T

iIS,T

−−−−−→ (⊗ Ms ) ⊗ B!S ⊗ B∗T S

R⊗S

R

32

1.

AXIOMATIC PATTERNS

We leave it to the reader to check the commutativity of diagrams (i) and (ii) in 1.3.12. Note that ⊗! is a tensor product (i.e., we have a compound tensor structure) iff for any J  I the composition map (⊗ B!Ji ) ⊗ B!I − → B!J is an isomorphism. R⊗I

I

1.3.19. Now suppose that B∗! is a ∗-augmented compound operad (see 1.3.10). So we have a left R-module B∗0 and the maps B∗I ⊗ B∗0 → BI\{i0 } , i0 ∈ I, that are R

compatible with h i in the sense explained in 1.3.10. According to 1.2.9, we get an augmentation functor h∗ for the ∗ structure on modR, h∗ (M ): = M ⊗ B∗0 . We R

leave it to the reader to check that h∗ is a ∗-augmentation functor for our compound pseudo-tensor structure (see 1.3.10). Similarly, a !-augmentation for B∗! defines a !-augmentation h! on modR by the formula h! (M ) = HommodR (M, B!0 ). 1.3.20. Example. Assume we have a cocommutative Hopf algebra structure on R. It yields a similar structure on Ro (the coproduct map for Ro coincides with that for R). By 1.1.14 we have the strict R- and Ro -operads B∗ = BR o and B! = BR , respectively. Let us show that B∗ and B! form a compound I strict R-operad. We need to define the maps h iS,T . For S ∈ Q(I) denote by 4IS : R⊗S → R⊗I the morphism of k-algebras ⊗ 4(Is ) (recall that 4(Is ) : R → R⊗Is S

is the Is -multiple coproduct). Let us consider R⊗I as a (R⊗S − R⊗T )-bimodule with respect to the action (⊗rs )(⊗ri )(⊗rt ): = 4IS (⊗rs ) · (⊗ri ) · 4IT (⊗rt ). We have the obvious indentifications of (R⊗S − R⊗T )-bimodules (⊗ B∗Is ) ⊗ (⊗ B!It ) = S

(⊗ R S

⊗Is

) ⊗ (⊗ R

⊗It

) = (R

R⊗I T

sider the morphism of (R

⊗S

⊗I

) ⊗ (R R⊗I ⊗T

−R

⊗I

) = R

⊗I

,

B!S

⊗ B∗T R

= R

⊗S

R⊗I T ⊗T

⊗R

. Con-

R

)-bimodules

ν: R⊗S ⊗ R⊗T → R⊗I , R


ν (⊗rs ) ⊗ (⊗rt ) : = (⊗rs )1(⊗rt ) = 4IS (⊗rs ) · 4IT (⊗rt ). If S, T are complementary, I then ν is an isomorphism (use 1.3.2(ii)), and our h iS,T is ν −1 . We leave it to the reader to check that B∗! = (B∗ , B! , h i) is a compound operad. If our Hopf algebra has counit, then the operads B∗ and B! are augmented in a natural way (see 1.2.10); it is easy to see that (B∗ˆ, B!ˆ, h i) is an augmented compound operad. 1.4. Rudiments of compound geometry This section treats some basic algebro-geometric constructions in compound setting. We consider matrix algebras in 1.4.2, Lie∗ algebras and their cohomology in 1.4.4–1.4.5, commutative! algebras in 1.4.6, reliable augmentation functors in 1.4.7, differential ∗ operations in 1.4.8, Lie∗ algebroids and modules over them in 1.4.11–1.4.12, Lie coalgebroids and the de Rham-Chevalley complexes in 1.4.14, the formal groupoid of a Lie coalgebroid in 1.4.15, the tangent Lie∗ algebroid in 1.4.16, connections in 1.4.17, coisson and odd coisson algebras in 1.4.18, hamiltonian reduction in 1.4.19, coisson modules in 1.4.20, the BRST reduction in 1.4.21–1.4.26. Finally, we explain in 1.4.27 how to encode a compound tensor structure by a single pseudo-tensor structure; the construction looks ad hoc at the moment, but in 3.2 we will see that it represents the “classical limit” of the chiral structure.

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33

The reader is invited to attend to the continuation of the list. For usual Lie algebroids see the textbook [M], written in the differential geometry setting, or a brief exposition in 2.9.1 below. For the BRST in the usual (non-compound) setting see, e.g., [KS]. Statement 1.4.24(iii) is a version of a theorem of Akman [A]. In this section M∗! is an abelian augmented compound tensor k-category (see 1.3.14 and 1.3.16); everything adapts immediately to the sheafified setting of 1.1.15. We abbreviate ⊗! to ⊗. There are no assumptions about exactness of the ∗augmentation functor h. 1.4.1. B∗ and B! algebras. For M ∈ M set Γ(M ) := Hom(1, M ). Then Γ is an augmentation functor on M! ; in particular Γ: M! → kmod is a pseudo-tensor functor (see 1.2.4 and 1.2.7). For a k-operad B we denote by B(M∗ ) the category of B algebras in M∗ ; we call them simply B∗ algebras. In particular we have the category Lie(M∗ ) of Lie∗ ∧ ∧ algebras. If B is an augmented k-operad, then B algebras in the augmented ∧ ∗ ∗ category M are called B algebras. In particular, we have the category of unital associative∗ algebras (see 1.2.8). Similarly, we refer to B algebras in M! as B! algebras. As follows from Remark in 1.3.15 (and 1.1.6(v)), if E is a B∗ algebra and R a 0 B ! algebra, then E ⊗! R is a (B⊗B0 )∗ algebra. In particular, if R is a commutative! algebra, then E ⊗ R is a B∗ algebra. 1.4.2. Matrix algebras and non-degenerate ∗ pairings. Assume we have V, V 0 ∈ M and h i ∈ P2∗ ({V 0 , V }, 1) a ∗ pairing. Then E = V ⊗ V 0 carries a canonical structure of associative∗ algebra which acts on V from the left and on V 0 from the right. Namely, the product operation · ∈ P2∗ ({E, E}, E) is the image of idV ⊗ h i ⊗ idV 0 ∈ End(V ) ⊗ P2∗ ({V 0 , V }, 1) ⊗ End(V 0 ) by the ⊗IS,T map, where I = {1, 2, 3, 4}, and S, T are equivalence relations {2 = 3} and {1 = 2, 3 = 4}, respectively. The E-action on V is the image of idV ⊗ h i by the binary map {1},{2,3} ⊗1,2 : End(V )⊗P2∗ ({V 0 , V }, 1) → P2∗ ({E, V }, V ) (see 1.3.13 for notation). We leave the definition of the right E-action on V 0 , as well as verification of associativity, to the reader. One often wants to know if the E-action on V identifies E with End V . The following condition on h i assures this: Definition. A ∗ pairing h i ∈ P2∗ ({V 0 , V }, 1) is left non-degenerate if for every finite set I and objects Ai , B ∈ M, i ∈ I, the canonical map (1.4.2.1)

PI∗ ({Ai }, B ⊗ V 0 ) → PI˜∗ ({Ai , V }, B),

ϕ 7→ (idB ⊗ h i)(ϕ, idV ),

is an isomorphism.7 We say that h i is right non-degenerate if the transposed pairing is left non-degenerate; h i is non-degenerate if it is both left and right nondegenerate. If h i is left non-degenerate, then for every B ∈ M one has (1.4.2.2)

∼

B ⊗ V 0 −→ Hom(V, B),

∼

h(B ⊗ V 0 ) −→ Hom(V, B)

(see 1.2.1 and 1.2.2, and (1.4.2.1) for I = ∅). In particular, taking B = 1, we see that ∼ V 0 = Hom(V, 1); i.e., V 0 equals the ∗ dual V ◦ to V , and E = V ⊗ V 0 −→ End V , 7 Here

idB ⊗ h i ∈ P2∗ ({B ⊗ V 0 , V }, B).

34

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∼

h(E) −→ End V . Our E is a unital associative∗ algebra: the unit element8 is idV ∈ End V = h(E). Notice that V 0 is ⊗-flat; if h is right exact, then V is a projective object. Exercise. The above definitions make sense if we replace M by the DG category of complexes CM (or CMs ; see 1.1.16). Show that a complex L admits a dual L◦ if and only if every Ln admits a dual (then (L◦ )n = (L−n )◦ , dL◦ = −t dL ). A pairing h i ∈ P2∗ ({L0 , L}, 1) is non-degenerate if and only if L is supported in finitely many degrees and every component ∈ P2∗ ({L0n , L−n }, 1) is non-degenerate. 1.4.3. We will often use the fact that ⊗ behaves naturally with respect to ∗ actions on multiples. Precisely, assume we have a map I → J of finite sets and families of objects {Li }i∈I , {Mj }j∈J , {Nj }j∈J . Then there is a canonical map (1.4.3.1)

⊗ItJ : ⊗ PI˜∗j ({Li , Mj }, Nj ) → PI˜∗ ({Li , ⊗ Mj }, ⊗ Nj ) , J,I˜ J

J

J

⊗ϕj 7→ ⊗ ϕj , J

compatible with composition of operations. Example. Assume we have a family {Bj }j∈J of k-operads together with the corresponding Bj module operads Cj . Let Lj be a B∗j algebra, Mj an Lj -module in M∗ with respect to Cj . Then ⊗ Mj carries canonical Lj -actions that mutually J

commute. Namely, for j0 ∈ J the j0 ’s action map Cj0 I → PI˜∗ ({Lj0 I , ⊗ Mj }, ⊗ Mj ) J

J

sends c ∈ Cj0 I to cMj0 ⊗ ( ⊗ idMj ), where cMj0 ∈ PI˜∗ ({Lj0 I , Mj0 }, Mj0 ) is the j6=j0

c-action on Mj0 . Recall that h(Lj ) is a Bj algebra and Mj is automatically an h(Lj )-module (see 1.2.17). It is easy to see that the action of h(Lj0 ) on ⊗ Mj that comes from J

the above Lj0 -action on ⊗ Mj coincides with the action a 7→ aMj0 ⊗ (⊗idMj )j6=j0 . J

∗

1.4.4. Lie algebras and modules. Let L be a Lie∗ algebra. Denote by M(L) the category of L-modules in M∗ . Lemma. M(L) is naturally an abelian augmented compound tensor k-category. We denote it by M(L)∗! . Proof. According to 1.2.18, M(L) carries a canonical abelian pseudo-tensor structure PL∗ . If {Mj }j∈J is a finite family of L-modules, then L acts on ⊗ Mj via J

the sum of the above J commuting actions (i.e., by Leibniz rule). This is the ! tensor product of L-modules. One easily checks that the maps ⊗IS,T send L-linear operations to L-linear ones; this provides the maps ⊗IS,T for M(L). Compatibility axioms 1.3.12 are clear. The object 1 with trivial L-action is a unit object in M∗! L . For an L-module M the L-action on M yields the morphism of k-modules h(M ) → HomM (L, M ).  Denote by hL (M ) its kernel. hL is a ∗-augmentation functor on M(L)∗! . Remark. A morphism L0 → L of Lie∗ algebras yields the “restriction of action” augmented compound tensor functor M(L)∗! → M(L0 )∗! . 8 See

1.2.8.

1.4. RUDIMENTS OF COMPOUND GEOMETRY

35

1.4.5. The Chevalley complex. The constructions below (except Remark (ii)) use only ∗ pseudo-tensor structure on M. For a Lie∗ algebra L and an L-module M we have the Chevalley complex C(L, M ) which is a complex with terms C n (L, M ) := Pn∗ ({L, . . . , L}, M )sgn (the skew-coinvariants of the action of the symmetric group Σn on operations of n variables) and differential given by the standard formula (see below). Set H · (L, M ) = H · C(L, M ); these are the cohomology of L with coefficients in M . As in the case of usual Lie algebras, the Chevalley complex is naturally defined in the more general super setting (see 1.1.16; below we often drop the adjective “super”). Namely, let L be a super Lie∗ algebra, M a super L-module. Consider the graded super k-module with components Pn∗ ({L[1], . . . , L[1]}, M )Σn (coinvariants of the symmetric group action on super operations of n variables). It carries a canonical odd differential δ of degree 1 coming from the morphism ∗ Pn∗ ({L[1], . . . , L[1]}, M )[−1] → Pn+1 ({L[1], . . . , L[1]}, M ), ϕ 7→ ·M (idL[1] , ϕ) − 1 ϕ([ ], id , . . . , id ) where · ∈ P2 ({L, M }, M ) is the action of L on M . M L[1] L[1] 2 One has δ 2 = 0, so we have a DG super k-module denoted by C(L, M ). Let us describe its natural symmetries. Consider the DG Lie∗ algebra L† (see 1.1.16 for notation). The DG Lie algebra h(L† ) = h(L)† acts on C(L, M ) in a canonical way. Namely, h(L) ⊂ h(L† ) acts via the adjoint action on copies of L[1] and the L-module structure on M , and the action of its complement h(L)[1] comes from the map h(L[1]) ⊗ Pn∗ ({L[1] . . . .L[1]}, M ) → ∗ Pn−1 ({L, . . . , L}, M ) which is minus the sum of the canonical “convolution” maps for each of the n arguments. The above constructions are functorial in the obvious manner. Remarks. (i) If for every n the inner P object P∗n ({L, . . . , L}, M ) is well defined (see 1.2.1), then one has the inner Chevalley complex C(L, M ) defined in the obvious way. It carries a canonical ∗ action of L† . There is a canonical morphism of DG h(L† )-modules h(C(L, M )) → C(L, M ) which is an isomorphism if our P’s are P objects in the augmented sense (see 1.2.7). (ii) Let us assume that the ∗ dual L◦ := Hom∗ (L, 1) is well defined and the canonical pairing h i ∈ P2∗ ({L◦ , L}, 1) is non-degenerate (see 1.4.2). Then for every n ≥ 0 and every N ∈ M one has N ⊗ L◦⊗n = P∗n ({L, . . . , L}, N ) and h(N ⊗ ∼ L◦⊗n ) −→ Pn∗ ({L, .., L}, N ) (see 1.2.1 and 1.2.7). Thus the inner Chevalley complex is well-defined (for an L-module M one has C(L, M )n = M ⊗ Symn (L◦ [−1])), and we have a canonical isomorphism of complexes (1.4.5.1)

∼

h(C(L, M )) −→ C(L, M ).

Consider now the super DG setting (see 1.1.16), so L is a DG Lie∗ super algebra, M a DG super L-module. The Chevalley super Q complex C(L, M ) is defined then as follows. Its components are C a (L, M ) := Pn∗ ({L[1], . . . , L[1]}, M )a (the n

operations of degree a with respect to the Z-gradings). Its differential is the sum of two components: the Chevalley component considered above and that coming from differentials of L, M by transport of structure. As above, the DG Lie super algebra h(L† ) acts on C(L, M ). ∗ Our C(L, M ) carries a decreasing filtration F n = C ≥n (L, M ) formed by P≥n . We equip L† with a decreasing filtration F −1 = L† , F 0 = L, F 1 = 0; then the h(L† )-action on C(L, M ) is compatible with the F -filtrations.

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1.4.6. Commutative! algebras and modules. We can do some commutative algebra in M! as in any abelian tensor k-category. Denote by Comu(M! ) the category of commutative and associative algebras with unit in M! . An object A ∈ Comu(M! ) (we call it a commutative! algebra) is an object A ∈ M together with a commutative and associative product · = ·A : A ⊗ A → A and a morphism 1 = 1A : 1 → A which is a unit for ·. For a finite family {Ai } of commutative! algebras the tensor product ⊗ Ai has an obvious structure of commutative! algebra; I

this is the coproduct of {Ai } in Comu(M! ). Therefore ⊗ defines a tensor category structure on Comu(M! ); the object 1 endowed with an obvious product is a unit in Comu(M! ). For A ∈ Comu(M! ) we denote by M(A) the category of unital A-modules in M! ; this is an abelian k-category. The usual tensor product ⊗ of A-modules is A

right exact, so M(A) is an abelian tensor k-category with unit object A. The “forgetting of A-action” functor M(A) → M admits a left adjoint “induction” functor M → M(A), M 7→ A ⊗ M , which is a tensor functor. Below for an Amodule N we denote by ·N : A ⊗ N → N the action map. Lemma. M(A) is naturally an abelian augmented compound tensor k-category. We denote it by M(A)∗! . Proof. The tensor structure has been discussed already, so it remains to de∗ , the compound tensor product maps ⊗IS,T , and the ∗fine the ∗ operations PAI augmentation functor. Take I ∈ S and objects {Li }i∈I , M of M. Assume that some Li0 , i0 ∈ I, and M are A-modules. We say that an operation ϕ ∈ PI∗ ({Li }, M ) is A-linear with respect to the i0 th variable if the operations ·M (ϕ ⊗i0 idA ), ϕ(idLi , ·Li0 )i6=i0 ∈ PI∗ ({Li , A ⊗ Li0 }i6=i0 , M ) coincide. Here ϕ ⊗i0 idA ∈ PI∗ ({Li , A ⊗ Li0 }i6=i0 , A ⊗ M ). ∗ ({Li }, M ) ⊂ PI∗ ({Li }, M ) the kIf all Li are A-modules, then we denote by PAI submodule of A-polylinear (i.e., A-linear with respect to each variable) operations. ∗ The composition of A-polylinear operations is again A-polylinear, so PAI form a ∗ pseudo-tensor structure on M(A). The functors PAI are left exact. ∗ ∗ ({ ⊗ Li }, ⊗ Ms ), ({Li }, Ms ) → PAT To define the structure maps ⊗IS,T : ⊗ PAI s AIt

S

AS

∗ consider the embedding PAT ({ ⊗ Li }, ⊗ Ms ) ,→ PT∗ ({⊗Li }, ⊗ Ms ) defined by the AIt

AS

It

AS

∗ projections ⊗Li → ⊗ Li . The composition ⊗ PAI ({Lis }, Ms ) → ⊗ PI∗s ({Li }, Ms ) It

AIt

S

S

⊗IS,T

−−−→ PT∗ ({⊗Li }, ⊗Ms ) → PT∗ ({⊗Li }, ⊗ Ms ) takes values in the image of the It

S

It

AS

∗ embedding. Therefore we can consider it as a morphism ⊗ PAI ({Li }, Ms ) → s

PT∗ ({ ⊗ Lit }, ⊗ Ms ); this is the ⊗IS,T map for M(A)∗! . AIt

S

AS

Compatibility axioms 1.3.12 for M(A)∗! follow from those for M∗! . The object A, considered as an A-module, is a unit object in M(A)∗! . Note ∼ that the morphism 1: 1 → A yields a canonical isomorphism EndM(A) (A) −→ Γ(A) of commutative k-algebras. Therefore, by 1.3.16, M(A)∗! is actually a compound tensor Γ(A)-category. We define the ∗-augmentation functor on M∗! A simply as the restriction of h to M(A) (note that hI,i0 maps preserve A-polylinear operations). 

1.4. RUDIMENTS OF COMPOUND GEOMETRY

37

Remarks. (i) The “induction” functor M → M(A) extends naturally to an augmented compound tensor functor M∗! → M(A)∗! . Namely, we already mentioned that the induction is a ! tensor functor. The map ∗ PI∗ ({Fi }, G) → PAI ({A ⊗ Fi }, A ⊗ G)

(1.4.6.1)

∗ is the composition PI∗ ({Fi }, G) → PI∗ ({A ⊗ Fi }, A⊗I ⊗ G) → PAI ({A ⊗ Fi }, A ⊗ G) ItI where the first arrow sends ϕ to ⊗I,I ˜ (ϕ, {idA }i∈I ) and the second arrow is the

composition with ·I ⊗ idG where ·I : A⊗I → A is the I-fold product. We leave it to the reader to check the compatibility with ⊗IS,T maps (see 1.3.14). The arrow h(F ) → h(A ⊗ F ) is h(1A ⊗ idF ). More generally, any morphism of commutative! algebras f : B → A yields a compound tensor functor (base change) f ∗ : M(B)∗! → M(A)∗!

(1.4.6.2)

such that for M ∈ M(B) one has f ∗ M = A ⊗ M . To define f ∗ , you just replace B

M∗! by M(B)∗! in the above considerations. The base change is compatible with the composition of f ’s in the obvious way. (ii) The induction pseudo-tensor functor M∗ → M(A)∗ is left adjoint (see 1.1.5) to the obvious forgetful pseudo-tensor functor M(A)∗ → M∗ . To see this, consider ∗ ∗ ({Fi , Lj }, M ) ({A⊗Fi , Lj }, M ) → PItJ for M ∈ M(A) and Fi , Lj ∈ M the map PItJ sending an operation ψ to its composition with 1A ⊗idFi : Fi → A ⊗Fi . This projection identifies the subspace of operations A-linear with respect to I variables with ∗ ∗ PItJ ({Fi , Lj }, M ). The inverse map sends ϕ ∈ PItJ ({Fi , Lj }, M ) to the compo⊗I ∗ sition of ϕ ⊗I (⊗idA ) ∈ PItJ ({A ⊗ Fi , Lj }, A ⊗ M ) with the product morphism A⊗I ⊗ M → M . If the Lj are A-modules, then our maps preserve operations Alinear with respect to J-variables, so PA∗ ItJ ({A ⊗ Fi , Lj }, M ) identifies with the ∗ ({Fi , Lj }, M ) that consists of operations A-linear with respect to subspace of PItJ the J variables. In the setting of (1.4.6.2) the pseudo-tensor functor f ∗ : M(B)∗ → M(A)∗ is left adjoint to the obvious forgetful pseudo-tensor functor M(A)∗ → M(B)∗ .9 (iii) If a ∗ pairing h i ∈ P2∗ ({L0 , L}, 1) in M is non-degenerate (see 1.4.2), then so is the corresponding pairing h iA ∈ P2∗ ({A ⊗ L0 , A ⊗ L}, A) in M(A). 1.4.7. Reliable augmentation functors. We know that h defines a pseudotensor functor on M∗ with values in the tensor category of k-modules. Usually this pseudo-tensor functor is not fully faithful. Let us formulate a property of h which is an effective remedy for this nuisance. ˜ For M ∈ M let h(M ) : Comu(M! ) → kmod be the functor R 7→ h(R ⊗ M ) (so we consider R as a variable “test” algebra). Since h is a pseudo-tensor functor on each M(R)∗ , we see that every ∗ operation ϕ ∈ PI∗ ({Mi }, N ) yields a morphism of ˜ ˜ i ) → h(N ˜ ) on Comu(M! ). functors10 h(ϕ) : ⊗ h(M I

9 To

see this, replace M by M(B) in the above argument. ˜ the functor ⊗ h(M i ) is R 7→ ⊗ h(R ⊗ Mi ).

10 Here

I

I

38

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Definition. We say that our augmentation functor h is reliable if for ev˜ : P ∗ ({Mi }, N ) → ery non-empty family of objects {Mi } and N ∈ M the map h I 11 ˜ ˜ Mor(⊗ h(Mi ), h(N )) is bijective. I

Therefore if h is reliable, then for a k-operad B and an object M ∈ M a B∗ algebra structure on M amounts to a rule that assigns to every commutative! algebra R a B algebra structure on h(R ⊗ M ) such that for every morphism of commutative! algebras R1 → R2 the corresponding morphism h(R1 ⊗M ) → h(R2 ⊗ M ) is a morphism of B algebras. Remarks. (i) Assume that the category Φ of all functors F : Comu(M! ) → kmod, R 7→ FR , makes sense (which happens if M is equivalent to a small category). ˜ : M∗ → Φ⊗ is a Then Φ is a tensor category (one has (⊗Fi )R := ⊗(FiR )) and h ˜ pseudo-tensor functor. Reliability of h just means that h is a fully faithful pseudotensor embedding. (ii) If we are playing with sheaves of categories, then in the above definition kmod should be replaced by the tensor category of sheaves of k-modules. (iii) If h is reliable, then the corresponding augmentation functor on every M(A)∗ is also reliable. 1.4.8. Differential ∗ operations. Many important ∗ operations between A-modules are not A-polylinear but belong to a larger class of A-differential ∗ operations we are going to define. Let I ⊂ A ⊗ A be the ideal of the diagonal, i.e., the kernel of the product map A ⊗ A → A. For an A-module L consider A ⊗ L as an A ⊗ A-module and set L(n) := A ⊗ L/In+1 (A ⊗ L) = A(n) ⊗ L. Now assume we have M ∈ M(A) and A

Li ∈ M. Pick some i0 ∈ I and identify PI∗ ({Li }, M ) with the subspace of operations in PI∗ ({Li , A ⊗ Li0 }, M ) A-linear with respect to the i0 th variable (see Remark (ii) in 1.4.6). Assume that Li0 is also an A-module. The operations vanishing on In+1 (A ⊗ Li0 ) form a subspace of PI∗ ({Li }, M ); its elements are differential operations of order ≤ n with respect to the i0 th variable. If n = 0, these are the same as A-linear operations. If all the Li are A-modules, then the ∗ operations which are differential operations of some order with respect to each variable are called A-polydifferential S (n) ∗ operations of finite order. The space of such operations is PA∗ I ({Li }, M ) ⊂ n

PA∗ I ({A ⊗ Li }, M ) = PI∗ ({Li }, M ). They are stable with respect to composition, so A-polydifferential ∗ operations form another pseudo-tensor structure on M(A). 1.4.9. Action of Lie∗ algebras on commutative! algebras. If L is a Lie∗ algebra, then an action of L (or an L-action by derivations) on a commutative! algebra A is an L-module structure τ ∈ P2∗ ({L, A}, A) on A (as on an object of M∗ ) such that ·A : A ⊗ A → A, 1A : 1 → A are morphisms of L-modules. Let τ be a fixed action of L on A. Then for M ∈ M(A) a τ -action (or a τ -compatible action) of L on M is an L-module structure on M (as on an object of M∗ ) such that ·M : A ⊗ M → M is a morphism of L-modules. The A-modules equipped with τ -actions form an abelian category M(A, L, τ ). 11 Here

˜ ˜ Mor(⊗ h(M i ), h(N )) denotes the collection of morphisms of functors. I

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39

Lemma. M(A, L, τ ) is naturally an abelian augmented compound tensor kcategory. We denote it by M(A, L, τ )∗! . ∗ One defines the ∗ operations in M(A, L, τ ) as PA,L,I ({Mi }, N ) := ∗ ∗ ∩ PAI ({Mi }, N ) (the intersection is taken in PI ({Mi }, N )). Notice that ⊗ Mi is a quotient L-module of ⊗Mi , so we define the ! tensor product on

Proof.

∗ PLI ({Mi }, N ) AI

I

M(A, L, τ ) as ⊗. The ⊗IS,T operations all come from M(A)∗! . The unit object of A

M(A, L, τ ) is A. The ∗-augmentation functor on M(A, L, τ )∗! is hL (see 1.4.4).  1.4.10. The inner Chevalley complex. Let L be a Lie∗ algebra such that the ∗ dual L◦ := Hom∗ (L, 1) exists and the canonical pairing h i ∈ P2∗ ({L◦ , L}, 1) is non-degenerate (see 1.4.2). Below we give another interpretation of the inner Chevalley complexes from Remark (ii) in 1.4.5. We deal with the general super DG setting (skipping the adjectives). Consider first the graded commutative! algebra Sym(L◦ [−1]). It carries a canonical action of the graded Lie∗ algebra L† (we forget about the differentials at the moment). Namely, L ⊂ L† acts according to the coadjoint action, and its complement L[1] ⊂ L† acts on the generators L◦ [−1] by − h i ∈ P2∗ ({L[1], L◦ [−1]}, 1). Now the graded algebra Sym(L◦ [−1]) admits a unique differential δ (odd of degree 1; see 1.1.16) such that the action of L† is compatible with the differentials. ν Explicitly, the restriction of δ to L◦ [−1] is equal to the composition L◦ [−2] − → −1/2·

L◦ [−1] ⊗ L◦ [−1] −−−→ Sym2 (L◦ [−1]) where ν is the morphism dual to the Lie∗ bracket.12 Our DG commutative! algebra carries a decreasing filtration by the DG ideals ◦ ≥i ◦ Sym≥i (L◦ [−1]). The topological DG algebra lim ←− Sym(L [−1])/Sym (L [−1]) is denoted by C(L); it carries a natural (continuous) L† -action. The above filtration yields a filtration C≥i (L) on C(L), and the L† -action is compatible with the filtrations.13 Remark. Here “topological algebra” means simply the (formal) projective limit of algebras. We pass to the completion in order to assure that the functor C preserves quasi-isomorphisms. Let M be any L-module. Consider M as an L† -module (we forget about the differential on L† at the moment) via the projection L† → L† /L[1] = L. Then L† acts ◦ ≥i ◦ ˆ on the topological C(L)-module M ⊗C(L) := ← lim − M ⊗(Sym(L [−1])/Sym (L [−1])) ˆ via the tensor product of the our actions. As above, M ⊗C(L)) admits a unique differential such that the actions of L† and C(L) are compatible with the differentials. Explicitly, on the C(L)-generators M the differential is the sum of the differential on M and the morphism M → M ⊗ L◦ dual to the L-action. We denote this topological DG C(L)-module equipped with the L† -action by C(L, M ). It carries a natural filtration by DG submodules C≥i (L, M ) compatible with the filtration C≥i (L). We leave it to the reader to check that C(L) coincides with the inner Chevalley complex from Remark (ii) of 1.4.5. ˆ If L acts on a commutative! algebra A by derivations, then L† acts on A⊗C(L) also by derivations, and the Chevalley differential is a derivation. Thus C(L, A) if h i2 ∈ P3∗ ({L◦ ⊗ L◦ , L, L}, 1) is the tensor product of two copies of h i, then ν is uniquely determined by property h i2 (ν, idL , idL ) = h i (idL◦ , [ ]) ∈ P3∗ ({L◦ , L, L}, 1). 13 Here L is equipped with the filtration defined at the end of 1.4.5. † 12 So

40

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is a topological DG commutative! algebra equipped with a DG L† -action. It is a C(L)-deformation of A = C(L, A)/C≥1 (L, A). If M is an A-module equipped with a compatible L-action (see 1.4.9), then C(L, M ) is a topological DG C(L, A)-module equipped with a compatible L† -action. Remark. The morphism ν : L◦ → L◦ ⊗ L◦ is a Lie cobracket. The duality ∼ isomorphism P2∗ ({L, M }, M ) −→ Hom(M, L◦ ⊗ M ) identifies ∗ actions of L on M ◦ with ! coactions of L on M . So the category of L-modules in M∗ coincides with the category of L◦ -comodules in M! . If M is an L-module, then C(L, M ) is the usual Chevalley complex of the Lie coalgebra L◦ with coefficients in L◦ -comodule M in the tensor category M! . 1.4.11. Lie∗ algebroids. Let A be any commutative unital algebra in M! . Definition. A Lie∗ A-algebroid is an A-module L together with a Lie∗ bracket [ ] ∈ P2∗ ({L, L}, L) and an action τ = τL ∈ P2∗ ({L, A}, A) of L (considered as a Lie∗ algebra) on A by derivations. This datum should satisfy the following properties: (i) τ is A-linear with respect to the L-variable; (ii) the adjoint action adL is a τ -action of L (as a Lie∗ algebra) on L (as an A-module). Notice that the structure ∗ operations (a), (b) are A-polydifferential (see 1.4.8). ∗ . Note that Lie(M(A)∗ ) coincides Lie∗ A-algebroids form a category LieAlgA ∗ with the full subcategory of LieAlgA that consists of L with τ = 0. ∗ 1.4.12. For L ∈ LieAlgA an L-module M is an A-module M equipped with a τ -action τM ∈ P2∗ ({L, M }, M ) of L (considered as a plain Lie∗ algebra) such that τM is A-linear with respect to L-variable. L-modules form a full subcategory14 M(A, L) ⊂ M(A, L, τ ) closed under subquotients and ⊗; clearly A ∈ M(A, L). A

∗ Define the ∗ operations in M(A, L) as PA,L ; i.e., let us consider M(A, L) as a full augmented compound tensor subcategory M(A, L)∗! in M(A, L, τ )∗! .

Lemma. M(A, L)∗! is an abelian augmented compound tensor k-category.



Remark. If our compound tensor structure is a mere tensor one (see 1.3.8(ii)), then Lie∗ algebroids are the same as Lie algebroids. The above L-modules are the same as left L-modules for L considered as a usual Lie algebroid (see 2.9.1). There seems to be no adequate compound analog for the notion of right module over a usual Lie algebroid. 1.4.13. Let L be any Lie∗ algebra, τ an action of L on A. Lemma. (i) The induced A-module LA := A ⊗ L has a unique structure of a Lie∗ A-algebroid such that the obvious morphism 1A ⊗ idL : L → LA is a morphism of Lie∗ algebras compatible with their actions on A. The functor L 7→ LA from the category of Lie∗ algebras acting on A to that of Lie∗ A-algebroids is left adjoint to the “forgetting of the A-action” functor. (ii) The functor M(A, LA )∗! → M(A, L, τ )∗! assigning to an LA -module M the same M considered as an L-module is an equivalence of augmented compound tensor categories.  14 In

the notation M(A, L, τ ) we consider L as a plain Lie∗ algebra acting on A.

1.4. RUDIMENTS OF COMPOUND GEOMETRY

41 ∼

For a Lie∗ A-algebroid L we refer to an isomorphism of Lie∗ algebroids LA −→ L as (L, τ )-rigidification of L. 1.4.14. The de Rham-Chevalley complex. As in 1.4.5, 1.4.10, we consider the super setting. (i) Let L be a Lie∗ A-algebroid. For M ∈ M(A, L) the A-polylinear operations form a subcomplex CA (L, M ) ⊂ C(L, M ) closed under the h(L† )-action. This is the de Rham–Chevalley complex of L with coefficients in M . Set CA (L) := CA (L, A). Below we discuss an “inner” version of this complex. (ii) The next definitions make sense in any unital abelian tensor k-category M! (we do not use the compound structure at the moment). Let A be a commutative unital algebra, L◦ an A-module. A Lie A-coalgebroid structure on L◦ is an odd differential d = dL◦ of degree 1 and square 0 on the algebra SymA (L◦ [−1]). We denote this commutative DG algebra by CA (L◦ ); this is the de Rham-Chevalley DG algebra of L◦ . Let L◦ be a Lie A-coalgebroid. For an A-module M a L◦ -action, or L◦ module structure, on M is a dL◦ -action on SymA (L◦ [−1]) ⊗ M , i.e., a differential A

of SymA (L◦ [−1]) ⊗ M that makes it a DG CA (L◦ )-module. Such a differential is A

uniquely determined by its restriction M → L◦ ⊗ M . We denote this DG CA (L◦ )A

module by CA (L◦ , M ). The category of L◦ -modules is denoted by M(A, L◦ ). This is a tensor category: ◦ if L acts on (finitely many) A-modules Mα , then it acts naturally on ⊗Mα = ⊗ Mα so that CA (L◦ , ⊗M ) =

⊗

CA (L◦ )

CA (L◦ , Mα ).

A

(iii) Let us return to the compound tensor category setting. Suppose that an A-module L admits a dual L◦ and the canonical ∗ pairing ∗ h i ∈ PA2 ({L◦ , L}, A) is non-degenerate (in M(A)∗! ; see 1.4.2). Lemma. There is a natural bijective correspondence between Lie∗ A-algebroid structures on L and Lie A-coalgebroid structures on L◦ . Sketch of a proof. Suppose we have a Lie∗ A-algebroid structure on L. There is a unique action ad◦L of the Lie∗ algebra L on L◦ (called the coadjoint action) such that h i ∈ P2∗ ({L◦ , L}, A) is L-invariant (here L acts on L by the adjoint action, on A by τ ). Indeed, the operation τ (idL , h i) − h[ ], idL◦ i ∈ P3∗ ({L, L, L◦ }, A) is A-linear with respect to the second argument L, so, since h i is non-degenerate (use Remark (ii) from 1.4.6) there is a unique ad◦L ∈ P2∗ ({L, L◦ }, L◦ ) such that the above operation equals h i (idL , ad◦L ). By the universality property of h i, our ad◦L is a τ -action. Consider the graded commutative! algebra SymA (L◦ [−1]). The Lie∗ algebra L acts on it (via τ and ad◦L ). This action extends to an action of the graded Lie∗ algebra L† (see 1.1.16); we forget about the differential on L† at the moment. Namely, L[1] ⊂ L† kills A and acts on L◦ [−1] by h i. We leave it to the reader to show that L◦ carries a unique Lie A-coalgebroid structure such that the L† -action on SymA (L◦ [−1]) is compatible with the differentials and to check that this map is, indeed, bijective.  For a Lie∗ A-algebroid L as above we call L◦ the dual Lie coalgebroid and CA (L) = CA (L, A) := CA (L◦ ) the de Rham-Chevalley DG algebra of L.

42

1.

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Remark. This DG algebra can be seen as the algebra of functions on the quotient space of Spec A modulo the action of L. Lemma. For an A-module M there is a natural bijection between L- and L◦ module structures on M . This bijection is compatible with tensor products, so we get a natural equivalence of tensor categories (1.4.14.1)

∼

M(A, L) −→ M(A, L◦ ).

Sketch of a proof. The subspace of operations φ ∈ P2∗ ({L, M }, M ) which are A-linear with respect to the first variable identifies naturally with Hom(M, L◦ ⊗ M ) A

(see 1.4.2). We leave it to the reader to check that φ is an L-module structure on M if and only if the morphism M → L◦ ⊗ M is an L◦ -action on M .  A

For M ∈ M(A, L) we usually denote the DG CA (L)-module CA (L◦ , M ) by CA (L, M ); this is the inner de Rham-Chevalley complex of L with coefficients in ∼ M . One has h(CA (L, M )) −→ CA (L, M ) (cf. (1.4.5.1)). Remarks. (i) For M ∈ M(A, L) consider the graded SymA (L◦ [−1])-module M ⊗ SymA (L◦ [−1]). It carries a natural action of the graded Lie∗ algebra L† : A

namely, L ⊂ L† acts according to the tensor product of the L-actions, and L[1] ⊂ L† acts via the above action on SymA (L◦ [−1]). Now the differential in CA (L, M ) is characterized by the property that the L† -action is compatible with the differentials. (ii) In the situation of 1.4.13 the above constructions amount to that of 1.4.10: one has CA (LA ) = C(L, A) and CA (LA , M ) = C(L, M ). 1.4.15. Lie coalgebroids and formal groupoids. In this subsection the compound tensor structure is not used, so M! can be any unital abelian k-category (⊗ is right exact, and k is of characteristic 0); for simplicity, we assume that M! has many flat objects. We have the category of commutative unital algebras in M! and its dual category of affine schemes in M! . Many notions of affine algebraic geometry generalize in a straightforward manner to such a setting. In particular, we know what are the affine formal schemes in M! . Let A be a commutative unital algebra in M! , Y := Spec A the corresponding affine scheme. Let G = Spf E be a formal groupoid on Y . Thus we have the structure projection G → Y ×Y and the “identity morphism” embedding i : Y ,→ G; let I ⊂ E be the ideal of i. We say that G is flat if E is an I-adic topological algebra, I/I 2 is ∼ P a flat A-module, and SymA (I/I 2 ) −→ I n /I n+1 . We say that a Lie A-coalgebroid P is flat if P is flat as an A-module. Proposition. The category of flat formal groupoids on Spec A is naturally equivalent to the category of flat Lie A-coalgebroids. Sketch of a proof. (a) Let P be a flat Lie A-coalgebroid, C := CA (P ) its de Rham-Chevalley DG algebra. Let us construct the corresponding formal groupoid G = Spf E on Y . Idea of the construction: Spec C is a DG version of the orbit space of G; thus G is fiber square of Y over Spec C with its evident groupoid structure. Of course, the fiber square should be taken in the DG sense and appropriately completed. We consider C as a filtered topological pro-algebra: C = ← lim − Cn where Cn := C/F n+1 and F n := C ≥n is the “stupid” filtration, so grF C = SymA (P [−1]). Set

1.4. RUDIMENTS OF COMPOUND GEOMETRY

43

L

Tn := A ⊗ A; one computes it using a Cn -flat resolution of A. One has the projecCn

tions . . . → C2 → C1 → C0 = A and . . . → T2 → T1 → T0 = A. Set En := H 0 Tn ; these are commutative algebras. The natural morphisms A = A ⊗ Cn → Tn ← Cn

Cn ⊗ A = A show that Tn and En are A ⊗ A-algebras. Cn

Lemma. One has H >0 Tn = 0 for every n. The projections . . . → E2 → E1 → E0 = A are P surjective. Set In := Ker(En → A); then Ker(En → Em ) = Inm+1 for m < n, and Ina /Ina+1 = SymA≤ n P for n ≥ 1. Proof of Lemma. Cn is filtered by the stupid filtration, and we can compute Tn using a filtered resolution of A. One gets a filtration on Tn such that grF Tn = L

≤n A ⊗ A. Then H >0 grF Tn = 0 = H <0 grF≤ n Tn and H 0 grF · Tn = SymA P by a

grF Cn

usual Koszul resolution computation, which implies the lemma.  Set E := ← lim E . By the lemma, this is an I-adic topological A ⊗ A-algebra, n − P n n+1 where I := lim I /I = SymA P . ←− In , and It remains to define on G := Spf E the structure of a formal groupoid on Y . To do this, we consider for every a ≥ 0 a formal Y a+1 -scheme Ga = Spf E (a) defined as the appropriately completed DG fiber product of a + 1 copies of Y over Spec C (i.e., the DG pull-back of the (Spec C)a+1 -scheme Y a+1 by the diagonal embedding Spec C ,→ (Spec C)a+1 ). Thus G0 = Y , G1 = G. Now the G· form a formal simplicial scheme in the obvious way, and each standard projection Ga → G × . . . × G (a times) Y

Y

is an isomorphism. Therefore G is a groupoid on Y (so that G· is the “homotopy quotient” of Y modulo G, i.e., the “classifying space” BG of G). The details are left to the reader. Remark. The formal groupoid G carries a canonical coaction of the Lie A ⊗ Acoalgebroid P1 := (P ⊗A)×(A⊗P ). Namely, for n ≥ 0 let CnF be a copy of Cn conL

sidered as a filtered Cn -algebra (with filtration F · ). Set Φn := CnF ⊗ CnF . This is a Cn

·

filtered DG Cn -algebra, and gr Φn = gr

·

CnF

·

⊗ gr

Cn ⊗2

Therefore ⊕ H a gra Φn = (Sym≤n A (P [−1])) a

L

CnF

=

⊗2 (Sym≤n A (P [−1]))

⊗ Tn .

A⊗A

⊗ En . Being the first term of the

A⊗A

spectral sequence for the filtration on Φn , this graded algebra carries a natural differential. Passing to the projective limit by n, we see that (Sym·A⊗A (P1 [−1])) ⊗ E A⊗A

carries a natural differential, which is the promised P1 -coaction on G. This coaction is compatible with the groupoid structure on G in an evident way. In fact, each Y a+1 -scheme Ga carries a natural action of the Lie coalgebroid Pa := the product of copies of P acting along each of the coordinates, and the structure projections are compatible with these actions. (b) Suppose we have a flat formal groupoid G = Spf E. The corresponding Lie A-coalgebroid is P := I/I 2 . We will identify SymA (P [−1]) with the de Rham algebra of invariant forms on G; the Lie coalgebroid structure on P is given then by the de Rham differential. (1) Let DRG/Y be the de Rham complex of G relative to the first structure pro(1)

jection G → Y . Similarly, let DRG2 /G be the de Rham complex of G2 := G × G Y

44

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relative to the first projection G × G → G. The second projection and the product Y

(1)

(1)

map G × G → G yield the pull-back morphisms of DG algebras DRG/Y → DRG2 /G . Y

(`)

Their equalizer is the DG algebra DRG/Y of left invariant forms on G. Forgetting about the differential, we have an evident morphism of graded A(`) (1) algebras DRG/Y → i∗ DRG/Y = SymA (P [−1]); one checks immediately that this is an isomorphism. We have defined the promised Lie A-coalgebroid structure on P . (r)

Remark. We can also consider the DG algebra DRG/Y of right invariant differential forms (which lies in the relative de Rham complex for the second structure ∼ (`) (r) projection G → Y ). There is a canonical identification DRG/Y −→ DRG/Y coming from the involution g 7→ g −1 of G. (c) It remains to show that the functors defined in (a), (b) are mutually inverse: Suppose we have a flat Lie coalgebroid P ; let G = Spf E be the corresponding formal groupoid as in (a), and P 0 its Lie coalgebroid as in (b). To define a canonical ∼ isomorphism P −→ P 0 , consider the canonical P1 -coaction on G from Remark in (a). Its A⊗P -component is a differential on the graded algebra E ⊗ Sym(P [−1]); denote A (1)

this formal DG algebra by DRG,P . One checks that the canonical morphism of DG (1)

(1)

algebras DRG/Y → DRG,P is an isomorphism. Its inverse identifies Sym(P [−1]) (`)

with DRG/Y . This identification is compatible with the differential; this is the ∼

promised isomorphism of Lie coalgebroids P −→ P 0 . Now suppose we have a flat formal groupoid G = Spf E. Let P be the corresponding Lie A-coalgebroid, C its de Rham-Chevalley DG algebra as in (b), and G0 = Spf E 0 the formal groupoid defined by P as in (a). Let us define a canonical ∼ isomorphism G −→ G0 . The “identity morphism” embedding Y ,→ G yields a morphism of algebras (1) (1) DRG/Y  A. Its kernel is a DG ideal; denote by F n the filtration on DRG/Y by powers of this ideal. By construction, we have an embedding of DG alge(1) bras C ,→ DRG/Y compatible with the F · filtrations. Notice that each projection (1)

(1)

DRG/Y /F n+1  DRG/Y /F 1 = A is a quasi-isomorphism. It yields a morphism of L

(1)

A ⊗ A-algebras En0 := H 0 (A ⊗ A) → (DRG/Y /F n+1 ) ⊗ A = E/I n+1 =: En . One Cn

Cn

checks that this is an isomorphism; passing to the limit, we get an isomorphism ∼ of formal Y × Y -schemes G −→ G0 . We leave it to the reader to check that it is compatible with the groupoid composition law.  Remark. Here is another (equivalent) way to define the Lie coalgebroid of a formal groupoid G. Consider the formal simplicial scheme G· = BG , Ga = G × . . . × G Y

Y

(a times) as in (a). Let Spec Q· ⊂ G· be the maximal simplicial subscheme such that Q1 = E/I 2 . Thus Q· is a cosimplicial algebra; our C is the corresponding normalized complex. This is a DG algebra in the usual way; one has C 0 = A, C 1 = P . We leave it to the reader to check that C is strictly commutative, equals SymA (P [−1]) as a mere graded algebra, and the corresponding Lie coalgebroid structure on P coincides with one from (b).

1.4. RUDIMENTS OF COMPOUND GEOMETRY

45

Exercise. Let P be an A-flat A-coalgebroid and G the corresponding formal groupoid. Show that for any A-module a P -action on it amounts to a G-action. Thus M(A, P ) coincides with the category of G-equivariant A-modules. Returning to the compound tensor category setting, consider a Lie∗ A-algebroid L as in 1.4.14(iii). Its dual Lie A-coalgebroid L◦ is A-flat (see 1.4.2). We get the corresponding formal groupoid G = GL . By Exercise and (1.4.14.1), L-modules are the same as GL -equivariant A-modules. 1.4.16. The tangent algebroid. For M ∈ M(A) we have the k-module of derivations (with respect to the ! tensor structure) Der(A, M ) ⊂ HomM (A, M ). The functor Der(A, ·) is representable: there is a universal derivation d: A → ΩA . More generally, for any finite collection of objects {Ni }i∈I of M we have a k-submodule Der(A, M )({Ni })I ⊂ PI˜∗ ({Ni , A}, M ) of operations ϕ that are derivations with respect to A, i.e., satisfy ϕ(idNi , ·A ) = ·M (ϕ ⊗1 idA ) + ·M (ϕ ⊗2 idA ) ∈ PI˜∗ ({Ni , A ⊗ A}, M ). Here ϕ ⊗1 idA , ϕ ⊗2 idA ∈ PI˜∗ ({Ni , A ⊗ A}, M ) are the operations defined in 1.4.3; the subscripts 1 and 2 show which copy of A in A ⊗ A we insert in ϕ (the transposition of factors symmetry of A ⊗ A interchanges them). Note that for any Ni -valued ∗ operations χi the composition ϕ(χi , idA ) is again a derivation with respect to A, and in case I = ∅ we have our old Der(A, M ). By Remark (ii) in 1.4.6 (and left exactness of P ∗ ), the map PI˜∗ ({Ni , ΩA }, M ) → ∗ PI˜ ({Ni , A}, M ), ψ 7→ ψ(idNi , d), identifies the subspace of A-linear with respect to ΩA operations with Der(A, M )({Ni })I . If the Ni are also A-modules, then this map identifies PA∗ I˜({Ni , ΩA }, M ) with the subspace of derivations A-linear with respect to Ni . Suppose that the object Der(A, M ): = HomM(A)∗ (ΩA , M ) ∈ M(A) exists (see 1.3.12). By the above and Remark (ii) in 1.4.6 for any finite collection {Gi }, i ∈ I,
one has PI∗ {Gi }, Der(A, M ) = Der(A, M )({Gi })I ⊂ PI˜∗ ({Gi , A}, M ). Let us show that ΘA := Der(A, A) (if it exists) is a Lie∗ A-algebroid in a natural way; this is the tangent algebroid of A. Namely, the action τ ∈ P2∗ ({ΘA , A}, A) is the universal derivation (that corresponds to idΘA under the above identification), and [ ] ∈ P2∗ ({ΘA , ΘA }, ΘA ) is uniquely defined by the condition that τ is a Lie∗ action of ΘA on A. Compatibilities (i) and (ii) in 1.4.11 are clear. ∗ there Remarks. (i) ΘA is a universal Lie∗ A-algebroid: for any L ∈ LieAlgA ∗ is a unique morphism of Lie A-algebroids τ : L → ΘA called the anchor morphism. Notice that L♦ := Ker τ is the maximal Lie∗ A-subalgebra of L. (ii) If the canonical ∗ pairing ∈ PA∗ ({ΘA , ΩA }, A) is non-degenerate, then the de Rham-Chevalley DG algebra CA (ΘA ) equals the de Rham complex DRA , DRi = ΩiA := Λi ΩA . (iii) Let M be an A-module. One may look for a universal Lie∗ A-algebroid E(M ) acting on M . If E(M ) exists, then E(M )♦ ⊂ E(M ) is the Lie∗ A-algebra that corresponds to the associative∗ A-algebra End M ; we denote it by gl(M ).

1.4.17. Connections. The notion of connection on an A-module M is clear (it uses only the tensor category structure); we know what the curvature of a connection is (this is a morphism M → Ω2A ⊗ M ), etc. Let us explain what the connections in the context of Lie∗ algebroids are.

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Let L be a Lie∗ A-algebroid as in 1.4.14(ii), so we have the de Rham-Chevalley DG commutative! algebra C = CA (L). Let P be a Lie∗ A-algebra, and E a P extension of L. So E is an A-algebroid, and we have a short exact sequence 0 → i π P − →E− → L → 0 where i and π are morphisms of Lie∗ A-algebroids. An L-connection on E is an A-linear section ∇ : L → E of π. Its curvature is c(∇) := [∇, ∇] − ∇([ ]) ∈ P2∗ ({L, L}, P ) ⊂ P2∗ ({L, L}, E). One checks that c(∇) is skew-symmetric and A-bilinear. Thus, by (1.4.2.1), one can write c(∇) ∈ h(C2 ⊗P ). Consider the Lie∗ C· -algebra C ⊗ P . Then ∇ defines a dC -derivation d∇ of the C-module C⊗P such that d∇ : P → C1 ⊗P corresponds to [∇, idP ] ∈ P2∗ ({L, P }, P ) via (1.4.2.1). If non-empty, the set of L-connections on E is a torsor for Hom(L, P ) = h(C1 ⊗ P ). For ν ∈ h(C1 ⊗ P ) one has 1 c(∇ + ν) = c(∇) + d∇ (ν) + [ν, ν], d∇+ν = d∇ + adν . 2 Remark. We call ΘA -connections simply connections (here π is the canonical projection). If M is an A-module such that E(M ) is well defined (see Remark (iii) in 1.4.16), then a connection on E(M ) is the same as a connection on M ; the identification Hom(M, Ω2A ⊗ M ) = h(Ω2A ⊗ gl(M )) identifies the corresponding curvatures. (1.4.17.1)

The set of Lie∗ algebroids equipped with an L-connection can be described as follows. Namely, for a given Lie∗ A-algebra P consider the following two sets: (i) the set15 F(P ) of pairs (E, ∇) where E is a P -extension of L, ∇ a connection on E; (ii) the set D(P ) of pairs (d, c) where d is a dC -derivation of the Lie∗ C-algebra C ⊗ P ,16 and c ∈ h(C2 ⊗ P ) is an element such that d(c) = 0 and d2 = adc . We call such a pair (d, c) compatible. The group h(C1 ⊗ P ) acts on both sets: for ν ∈ h(C1 ⊗ P ) the ν-translations are (E, ∇) 7→ (E, ∇ + ν), (d, c) 7→ (d + adν , c + d(ν) + 21 [ν, ν]). Lemma. For any (E, ∇) ∈ F(P ) the pair (d∇ , c(∇)) is compatible. The map F(P ) → D(P ), (E, ∇) 7→ (d∇ , c(∇)), is bijective and compatible with the h(C1 ⊗ P )actions.  Remark. Consider a larger set D0 (P ) of pairs (d, c) which satisfy only the second compatibility d2 = adc from (ii). The h(C1 ⊗ P )-action extends to D0 (P ) by the same formula. The function (d, c) 7→ d(c) is constant on the h(C1 ⊗ P )-orbits. 1.4.18. Coisson and odd coisson algebras. A compound Poisson, or simply coisson, bracket on A ∈ Comu(M! ) is a Lie∗ bracket { } ∈ P2∗ ({A, A}, A) such that the adjoint action τ{ } is an action of A (considered as a Lie∗ algebra) on A (as on a commutative! algebra). In other words, { } satisfies the Leibniz rule with respect to either of the variables. Such pair (A, { }) is called a coisson algebra. By 1.4.13 (i) a coisson bracket { } yields a Lie∗ A-algebroid structure on A ⊗ A. One checks easily that the kernel of the projection A ⊗ A → ΩA , a ⊗ b 7→ adb, is an ideal in A ⊗ A, so ΩA inherits the Lie∗ A-algebroid structure. In other words, ΩA carries a unique Lie∗ A-algebroid structure such that d: A → ΩA is a morphism of Lie∗ algebras and τΩA (d, idA ) = { }. 15 Pairs 16 I.e.,

(E, ∇) are rigid, so we make no distinction between (E, ∇) and its isomorphism class. d is a dC -derivation for the C-module structure and a derivation for the Lie∗ bracket.

1.4. RUDIMENTS OF COMPOUND GEOMETRY

47

Conversely, a structure of a Lie∗ A-algebroid on ΩA yields a binary ∗ operation { }: = τΩA (d, idA ) on A. If { } is skew-symmetric and d{ } = [d, d] ∈ P2∗ ({A, A}, ΩA ), then { } is a coisson bracket. Therefore coisson brackets on A are the same as Lie∗ A-algebroid structures on ΩA that satisfy the above conditions. The category of coisson algebras Cois(M∗! ) is a tensor category. Namely, for a finite family (AP i , { }i ), i ∈ I, of coisson algebras their tensor product (A, { }) is ∼ ∗ A = ⊗Ai , { } = { }∼ i where for j ∈ I the bracket { }j ∈ P2 ({A, A}, A) is the i∈I

⊗ Ai -linear extension of { }j (see Remark (ii) in 1.4.6). The algebra 1 with zero Ir{j}

coisson bracket is a unit object for this tensor structure. Remark. The evident morphisms Ai → A are morphisms of coisson algebras which mutually coisson commute in the obvious sense. This datum is universal: for any coisson algebra B a morphism of coisson algebras A → B amounts to a collection (ϕi ) of mutually coisson commuting morphisms ϕi : Ai → B. Examples. (i) If L is a Lie∗ algebra, then the symmetric! algebra Sym L carries a unique coisson bracket { } such that L = Sym1 L ,→ Sym L is a morphism of Lie∗ algebras; this is the Kostant-Kirillov bracket. The functor Lie(M∗ ) → Cois(M∗ ), L 7→ (Sym L, { }), is left adjoint to the “forgetting of commutative! algebra structure” functor Cois(M∗! ) → Lie(M∗ ). (ii) More generally, if A is a commutative! algebra and L is a Lie∗ A-algebroid, then the symmetric! algebra Sym L := SymA L of the A-module L carries a unique coisson bracket { } such that {Symp L, Symq L} ⊂ Symp+q−1 L, and { } coincides with [ ] on Sym1 × Sym1 and with τL on Sym1 × Sym0 . For example, Sym ΘA is the algebra of functions on the cotangent bundle to Spec A and { } is a canonical coisson structure on it. (iii) The above constructions often come in a twisted version. Let L be a Lie∗ algebra and L[ a central extension of L by 1. The twisted symmetric algebra Sym[ L is the quotient of Sym(L[ ) modulo the ideal generated by the image of the difference of embeddings 1 = Sym0 (L[ ) ,→ Sym(L[ ), 1 ⊂ L[ = Sym1 (L[ ) ,→ Sym(L[ ). This quotient algebra inherits the coisson bracket from Sym(L[ ). Similarly, in the setting of (ii), if L[ is a central extension of a Lie∗ algebroid L by A, then the twisted symmetric algebra Sym[A L inherits a coisson bracket from SymA (L[ ). The above definition, of course, makes sense in the super, or DG super, setting (see 1.1.16). For a commutative super, or DG super, algebra A we can also consider n-coisson brackets on A, which are Lie∗ brackets { } ∈ P2∗ ({A[−n], A[−n]}, A[−n]) on A[−n] such that the adjoint action of A[−n] on A satisfies the Leibniz rule. One calls (A, { }) an n-coisson algebra. For n = 0 this is the usual coisson algebra. As above, for n-coisson A the A-module ΩA [−n] is naturally a Lie∗ algebroid. Remarks. (i) For the plain super setting, it is only the parity of n that matters, so we speak of odd and even (= usual) coisson algebras. (ii) In (the particular case of) the mere tensor (non-compound) category setting, we talk of n-Poisson and odd or even Poisson algebras. Example. For any Lie∗ A-algebroid L the symmetric! algebra SymA (L[n]) is naturally an n-coisson algebra whose bracket is characterized by the property that it coincides with [ ] on Sym1 × Sym1 and with τL on Sym1 × Sym0 . In particular, if ΘA is well defined, then SymA (ΘA [1]) is an odd coisson algebra; its odd coisson

48

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bracket is called the Schouten-Nijenhuis bracket. More generally, SymA (ΘA [n]) is an n-coisson algebra. Exercise. Any ν ∈ HomA (L, A) yields a degree 1 odd derivation dν of the algebra SymA (L[1]) which equals ν on Sym1 . Show that it is compatible with the odd coisson bracket if and only if ν, considered as an element of CA (L)1 (see 1.4.14(i)), is closed. In particular, suppose that ΘA is well defined and the canonical ∗ pairing ∈ PA2 ({ΘA , ΩA }, A) is non-degenerate. Then ν ∈ h(Ω1A ) is closed if and only if dν is a derivation of the Schouten-Nijenhuis bracket. Let A be an n-coisson algebra. Suppose ΘA is well defined and the canonical ∗ pairing ∈ PA2 ({ΩA , ΘA }, A) is non-degenerate. Consider the de Rham-Chevalley algebra CA (ΩA [−n]) of the Lie∗ algebroid ΩA [−n] (see 1.4.14). As a mere graded algebra, it equals SymA (ΘA [n − 1]). Since ΘA is a Lie∗ algebroid, the latter is an n − 1-coisson algebra. One checks that the de Rham-Chevalley differential is compatible with the coisson bracket, so CA (ΩA ) is a DG n − 1-coisson algebra. In particular, for an odd coisson A, Ccois (A) is a coisson algebra. The n − 1-coisson bracket is compatible with the stupid filtration CA (ΩA )≥a , so ≥a the completion Ccois (A) := lim ←− CA (ΩA [−n])/CA (ΩA [−n]) is naturally a topologcois ical (n − 1)-coisson algebra. The spectrum of C (A) is “the space of symplectic leaves” of SpecA. 1.4.19. Hamiltonian reduction. The usual operations on Poisson algebras have obvious coisson versions. For example, if (A, { }) is a coisson algebra and I ⊂ A is an ideal such that {I, I} ⊂ I, then I acts on A/I as a Lie∗ algebra. Set B := (A/I)I (:= the maximal I-invariant subobject of A/I; we tacitly assume that it exists). Then B is a ! subalgebra of A/I, and B inherits a coisson bracket { } such that for b1 , b2 ∈ B and ˜bi ∈ A, ˜bi mod I = bi , one has {b1 , b2 } = {˜b1 , ˜b2 }mod I. In particular, for (A, { }) ∈ Cois(M∗! ), L ∈ Lie(M∗ ), and a morphism of Lie∗ algebras α: L → A, the commutative! algebra B: = (A/α(L)A)L carries a canonical coisson bracket { } (here α(L)A is the ideal generated by α(L) ⊂ A, so L acts on A via α and { }, A/α(L)A carries the quotient action, and ( )L is the subobject of L-invariants). We call (B, { }) the hamiltonian reduction of (A, { }). 1.4.20. Coisson modules. Let (A, { }) be a coisson algebra. A coisson Amodule, or (A, { })-module, is an A-module M (here A is considered as a plain commutative! algebra) together with a τ{ } -action of A on M (here A is considered (A) as a Lie∗ algebra) τM ∈ P2∗ ({A, M }, M ) which is a derivation with respect to the A-variable. The coisson modules form a category M(A, { }). In fact M(A,{ }) coincides with the category M(A, ΩA ) where ΩA is the Lie∗ ∼ A-algebroid from 1.4.18. The equivalence M(A, ΩA ) −→ M(A, { }) assigns to an (A) ΩA -module M the same M with τM coming from the morphism of Lie∗ algebras d : A → ΩA . Therefore M(A, { }) carries a canonical augmented compound tensor structure; we denote it M(A, { })∗! . 1.4.21. BRST reduction (coisson version). This is a refined version of the hamiltonian reduction; both are compared in 1.4.26. Below we tacitly deal with DG super objects, i.e., with the category CMs , following the conventions of 1.1.16. For A ∈ CMs , h(A) is a complex of super k-modules. We assume that k ⊃ Q. Suppose we have L, L0 ∈ CMs and a ∗ pairing h i ∈ P2∗ ({L0 , L}, 1). Consider L[1] ⊕ L0 [−1] as a commutative Lie∗ algebra. Let (L[1] ⊕ L0 [−1])[ = L[1] ⊕

1.4. RUDIMENTS OF COMPOUND GEOMETRY

49

L0 [−1] ⊕ 1 be its central extension by 1 with commutator ∈ P2∗ ({L0 [−1], L[1]}, 1) = P2∗ ({L0 , L}, 1) equal to h i. Set Clc := Sym[ (L0 [−1] ⊕ L[1]) (see Example (iii) in (·) 1.4.18).17 This is the Clifford coisson algebra. It carries an extra Z-grading Clc (−1) 0 (1) determined by the conditions L[1] ⊂ Cl , L [−1] ⊂ Cl . One has: (i) There are obvious embeddings Sym(L0 [−1]), Sym(L[1]) ,→ Clc ; the product morphism Sym(L0 [−1] ⊕ L[1]) = Sym(L0 [−1]) ⊗ Sym(L[1]) → Clc is an isomorphism of commutative! algebras. (ii) The coisson bracket vanishes on both L0 [−1] and L[1]; the { } pairing between L0 [−1] and L[1] is equal to h i. (0) (iii) The subobject L ⊗ L0 = L[1] ⊗ L0 [−1] ⊂ Clc is a Lie∗ subalgebra. The coisson bracket on it coincides with the bracket that comes from the associative∗ algebra structure as defined in 1.4.2. This subalgebra normalizes L[1], L0 [−1] ⊂ (±1) Clc , and the corresponding adjoint action coincides with the action from 1.4.2. We will need a simple lemma. For A ∈ CMs consider the coisson Clc -module A ⊗ Clc . Let Fc ⊂ h(A ⊗ Clc ) be the centralizer of L[1] ⊂ Clc . In other words, Fc is the kernel of the morphism ψ : h(A ⊗ Clc ) → Hom(L[1], A ⊗ Clc ) that comes from the adjoint action of L[1] on Clc . It is clear that Fc ⊃ h(A ⊗ Sym(L[1])). 1.4.22. Lemma. If the ∗ pairing h i is non-degenerate (see 1.4.2), then Fc = h(A ⊗ Sym(L[1])). (a)

Proof. We have h(A ⊗ Clc ) = ⊕ h(A ⊗ Symn (L0 [−1]) ⊗ Symn−a (L[1])). Take (a) any ν ∈ h(A ⊗ Clc ); let n be the largest number such that the nth component of ν is non-zero; denote it by ν¯ ∈ h(A ⊗ Symn (L0 [−1]) ⊗ Symn−a (L[1])). The morphism χ := ψ(ν) sends L[1] to the degree a−1 part of A⊗Sym≤n−1 (L0 [−1])⊗Sym(L[1]); let χ ¯ be its composition with the projection to A ⊗ Symn−1 (L0 [−1]) ⊗ Symn−a (L[1]). Thus (see 1.4.2) χ ¯ ∈ Hom(L[1], A ⊗ Symn−1 (L0 [−1]) ⊗ Symn−a (L[1])) = h(A ⊗ 0 n−1 (L [−1] ⊗ Sym (L0 [−1])) ⊗ Symn−a (L[1])). According to the Leibnitz formula the image of χ ¯ by the product map L0 [−1] ⊗ Symn−1 (L0 [−1]) → Symn (L0 [−1]) equals n¯ ν . If ν lies in Fc , i.e., χ = 0, then n¯ ν = 0, hence n = 0.  1.4.23. From now on L is a Lie∗ algebra. We assume that the ∗ dual L◦ := Hom(L, 1) exists and the canonical pairing h i ∈ P2∗ ({L◦ , L}, 1) is non-degenerate (see 1.4.2). Then the adjoint action of L on L, hence on L[1], yields a morphism of (0) Lie∗ algebras β : L → L ⊗ L◦ = L[1] ⊗ L◦ [−1] ⊂ Clc . We can consider β as an (0) ◦ element of h(Clc ⊗ L ). Denote by β˜c ∈ h(Clc [1]) its image by the product map Clc ⊗ L◦ [−1] → Clc . Let A be a coisson algebra, and let α : L → A be a morphism of Lie∗ algebras. Consider the graded coisson algebra A ⊗ Clc· . We have a morphism of mere graded Lie∗ algebras (1.4.23.1)

` : L† → A ⊗ Clc .

Here L† is a contractible DG Lie∗ algebra defined in 1.1.16 (we forget about its differential for the moment), and the components of ` are `(0) = α+β : L → A⊗Cl(0) and `(−1) := the composition L[1] ,→ Cl(−1) → A ⊗ Cl. 17 The

subscript “c” is for “coisson” or “classical.”

50

1.

AXIOMATIC PATTERNS

Let us rewrite α as α ˜ ∈ h(A ⊗ L◦ )even ⊂ h(A ⊗ Clc [1]). Let β˜ ∈ h(A ⊗ Clc [1]) ˜ be the image of βc by the morphism of algebras Clc → A ⊗ Clc . Set (1.4.23.2)

dc := α ˜+

1 ˜ β ∈ h(A ⊗ Clc [1]) = Hom(k[−1], h(A ⊗ Clc ); 2

this is the (classical) BRST charge. Its adjoint action is an odd derivation d of degree 1 of the graded coisson algebra A ⊗ Clc called the BRST differential. 1.4.24. Proposition. (i) One has {dc , dc } = 0, so d2 = 0. We denote the DG coisson algebra (A ⊗ Clc , d) by CBRST (L, A)c . (ii) ` commutes with the differentials, so we have a morphism of DG Lie∗ algebras (1.4.24.1)

` : L† → CBRST (L, A)c .

We refer to this as the BRST property of d or d. It determines d and d uniquely: (iii) dc is a unique even element of degree zero in h(A ⊗ Clc [1]) such that [dc , `(−1) ] equals `(0) , and d is a unique odd derivation of A ⊗ Cl of degree 1 such that d`(−1) = `(0) . Proof. (i), (ii) are straightforward computations. (iii) follows from the next lemma. Let Derc be the graded Lie algebra of derivations of the coisson algebra A ⊗ Clc . (a)

(a−1)

), 1.4.25. Lemma. The maps h(A ⊗ Clc ) → Der ac → Hom(L[1], A ⊗ Clc where the first arrow is the adjoint action and the second one is δ 7→ δ`(−1) , are injective for a > 0. Proof. (a) Let δ be a coisson derivation of A ⊗ Clc of degree a > 0 such that δ`(−1) = 0; i.e., δ kills L[1] ⊂ Clc . Then it sends L◦ [−1] and A to the centralizer of (a) (a+1) and A ⊗ Clc , which is zero by 1.4.22. Thus δ = 0. L[1] in, respectively, A ⊗ Clc (b) The kernel of the first arrow is contained in the centralizer of L[1]; by 1.4.22 it vanishes if a > 0.  Considered as an object of an appropriate homotopy category of coisson algebras, CBRST (L, A)c is called the BRST reduction of A with respect to α. Its · (L, A)c . The reduction is regucohomology is a graded coisson algebra HBRST 6=0 lar if HBRST (L, A)c = 0. Then the BRST reduction is a plain coisson algebra 0 HBRST (L, A)c . 1.4.26. Let us compare the BRST and hamiltonian reductions (see 1.4.19). Set A¯ := A/α(L)A. This is a commutative! algebra; the L-action on A (via ¯ Let I be the DG ideal of α and the coisson bracket) yields an L-action on A. (−1) (−1) CBRST (L, A)c generated by the image of ` : L[1] → A ⊗ Clc . The morphism ◦ of graded algebras A ⊗ Sym(L [−1]) → A ⊗ Clc /I is surjective; by 1.4.24 its kernel is the ideal generated by α(L). Thus CBRST (L, A)c /I = A¯ ⊗ Sym(L◦ [−1]). One ◦ ¯ checks immediately that the BRST differential yields on A⊗Sym(L [−1]) the usual Chevalley differential (see 1.4.10), so we have a canonical surjective morphism of DG commutative! algebras (1.4.26.1)

¯ CBRST (L, A)c  C(L, A).

1.4. RUDIMENTS OF COMPOUND GEOMETRY

51

Passing to zero cohomology, we get a canonical morphism of commutative algebras 0 HBRST (L, A)c → A¯L .

(1.4.26.2)

It is easy to see that it is compatible with the coisson brackets. 1.4.27. The compound tensor structure on M can be encoded in a single pseudo-tensor structure Mc . For the meaning of this construction see 3.2. So let us bring the sundry polylinear operations of a compound tensor category under one roof. For an I-family Li ∈ M, M ∈ M, and S ∈ Q(I) (see 1.3.1) set (1.4.27.1)

PIc ({Li } , M )S := PS∗ ({ ⊗ Li }, M ) ⊗ ( ⊗ LieIs ). i∈Is

s∈S

The k-module of c operations is (1.4.27.2)

PIc ({Li } , M ) :=

⊕ S∈Q(I)

PIc ({Li } , M )S .

The c operations compose in a natural manner. Namely, assume we have J  I and a J-family of objects Kj ∈ M. Take ϕ ∈ PIc ({Li } , M )S , ψi ∈ PJci ({Kj }, M )Ti (where Ti ∈ Q(Ji )). For R ∈ Q(J) the R-component ϕ(ψi )R ∈ PJc ({Kj } , M )R vanishes unless R ≤ T ,18 I and R are transversal in Q(T ), and inf{I, R} = S (see 1.3.1). If these conditions are satisfied, then ϕ(ψi )R = ϕ(γs ) where for s ∈ S one has19 γs := ⊗TIss,Rs ψi ∈ PR∗ s ({ ⊗ Kj }, ⊗ Li ) ⊗ ( ⊗ LieJt ) and the composition j∈Jr

i∈Is

t∈Ts

ϕ(γs ) is the tensor product of compositions of ∗ and Lie operations.20 We leave it to the reader to check its associativity. We have defined an abelian (see 1.1.7) pseudo-tensor structure Mc on M. Set PIc ({Li }, M )n :=

⊕ S∈Q(I,|I|−n)

PIc ({Li } , M )S where Q(I, m) ⊂ Q(I) con-

sists of quotients I  S, |S| = m. This is a natural Z-grading on P c compatible with the composition of c operations. One has PIc ({Li }, M )|I| = Hom(⊗ Li , M )⊗LieI , PIc ({Li }, M )0 = PI∗ ({Li }, M ). I

The corresponding embedding and projection define the maps P ! ⊗Lie → P c → P ∗ which are compatible with the composition of operations, so we have the pseudotensor functors (1.4.27.3)

M! ⊗ Lie → Mc → M∗

that extend the identity functor on M. Remark. By 1.1.12 the compound pseudo-tensor structure M∗! is uniquely determined by the pseudo-tensor structure Mc together with the Z-grading on P c . 1.4.28. Lemma. Lie algebras in Mc are the same as coisson algebras without unit in M∗! . Proof. A binary c operation µ ∈ P2c (M ) is the same as a pair (·, { }) where21 · ∈ Hom(M ⊗ M, M ) and { } ∈ P2∗ (M ). The skew-symmetry of µ amounts to the 18 Here

T := tTi ∈ Q(J). 1.3.12 for notation. 20 The composition of Lie operations is the tensor product of maps Lie Is ⊗ ( ⊗ LieJt ) → 19 See

∼

t∈Tr

LieJr labeled by r ∈ Rs . We use the fact that Tr −→ Is (the transversality condition). ∼ 21 We use the identification k − → Lie2 , 1 7→ [ ].

52

1.

AXIOMATIC PATTERNS

symmetry of · and the skew-symmetry of { }. The Jacobi identity for µ amounts to three identities (corresponding to the graded components): – the Jacobi identity for · ⊗ [ ] which, by 1.1.10, is the same as the associativity of ·, – the Jacobi identity for { }, – the identity which says that { } acts by derivations of ·. 

CHAPTER 2

Geometry of D-schemes . . . inogda e, udals~ v bliz~ leawu rowu, igral na fle$itravere, ostav druga molodago medu grobov odnogo, ko by dl togo, qto b izdali emu pritnee bylo sluxat~ muzyku. M. I. Kovalinski$i, “izn~ Grigori Skovorody. Pisana 1794 goda v drevnem vkuse.” †

2.1. D-modules: Recollections and notation The language of D-modules plays for us about the same role as the language of linear algebra for the usual commutative algebra and algebraic geometry. Good references are [Ber], [Ba], [Kas2] (we will not use the theory of holonomic Dmodules though). This section collects some information that will be of use; we recommend the reader to skip it, returning when necessary. We discuss left and right D-modules in 2.1.1, basic D-module functoriality in 2.1.2, Kashiwara’s lemma in 2.1.3, the exactness of the pull-back functor for locally complete intersection maps in 2.1.4, locally projective D-modules in 2.1.5, the sheafified middle de Rham cohomology functor h in 2.1.6, the induced D-modules in 2.1.8–2.1.10 (we refer to [S1] for all details) and the quasi-induced D-modules in 2.1.11, maximal constant quotients and de Rham homology in 2.1.12, modifications of a D-module at a point and completions of the local de Rham cohomology in 2.1.13–2.1.15, and a variant of the same subject when the point moves in 2.1.16. Finally, in 2.1.17 we prove an auxiliary result from commutative algebra which explains how the formalism of I-adic topology can be rendered to the setting of arbitrary (possibly infinitely generated) modules. All schemes and algebras are defined over a fixed base field k of characteristic zero. The reader may prefer to work all the way with super objects following conventions of 1.1.16. 2.1.1. Left and right D-modules; homotopical flatness. For a scheme X we denote by MO (X) the category of O-modules on X (: = quasi-coherent sheaves of OX -modules). If X is smooth, then Mr (X) (resp. M` (X)) denotes the category †

“. . . sometimes he played flute in a nearby grove, abandoning the young friend betwixt the tombs, as if the music were to be better appreciated from afar.” M. I. Kovalinski, “The life of Gregori Skovoroda, written in 1794 in ancient gusto.”

53

54

2.

GEOMETRY OF D-SCHEMES

of right (resp. left) D-modules on X (: = sheaves of DX -modules quasi-coherent as OX -modules). Notice that left D-modules are the same as O-modules equipped with an integrable connection. We mostly deal with right D-modules and often call them simply D-modules writing M(X) := Mr (X). One has the obvious forgetting of D-action functors or : M(X) → MO (X), o` : M` (X) → MO (X). If G, L are left D-modules, then G ⊗ L = G ⊗ L is a left D-module in a natural OX

way. Therefore M` (X) is a tensor category with a unit object OX , and o` is a tensor functor. If M is a right D-module and L a left D-module, then M ⊗ L is naturally OX

a right D-module (a field τ ∈ ΘX ⊂ DX acts as (m ⊗ `)τ = mτ ⊗ ` − m ⊗ τ `) X denoted by M ⊗ L. The sheaf ωX = Ωdim (we consider it as an even super line) X has a canonical right D-module structure, namely, ντ = −Lieτ (ν) for ν ∈ ωX . The functor M` (X) → M(X), L 7→ Lr := LωX = ωX ⊗ L, is an equivalence of −1 categories. The inverse equivalence is M 7→ M ` := M ⊗ ωX . We refer to complexes of left (right) D-modules as left (right) D-complexes on X; sometimes right D-complexes are called simply D-complexes. We call the identification of derived categories ∼

DM` (X) −→ DMr (X),

(2.1.1.1)

L 7→ LωX [dim X],

the canonical equivalence. So DM(X) carries two standard Mr and M` t-structures, o`

differing by shift by dim X. The composition DM(X) = DM` (X) −→ DMO (X) is denoted simply by o` . For details about the next definitions see [Sp].1 We say that a complex M of O-modules is homotopically OX -flat if for every acyclic complex F of O-modules the complex M ⊗ F is acyclic. A D-complex M ∈ CM(X) is homotopically OX -flat if it is homotopically flat as a complex of OX -modules. Equivalently, this means that for every acyclic L ∈ CM` (X) the complex M ⊗ L is acyclic. Similarly, M ∈ CM(X) is homotopically DX -flat if for every L as above the complex of sheaves of k-vector spaces M ⊗ L is acyclic. A D-complex M is homotopically DX -flat if and only DX

if it is homotopically OX -flat and for every P ∈ CM` (X) the canonical projection DR(M ⊗ P ) → h(M ⊗ P ) (see 2.1.7 below) is a quasi-isomorphism. 2.1.2. Functoriality. A morphism f : X → Y of smooth varieties yields canonical functors f∗

(2.1.2.1)

DM(X)  DM(Y ) f!

that satisfy standard compatibilities (the base change property, adjunction property for proper f , etc.). If f is quasi-finite, then f∗ is left exact, and if f is affine, then f∗ is right exact with respect to the Mr t-structure. 1 Our

“homotopically flat” is Spaltenstein’s “K-flat”.

2.1. D-MODULES: RECOLLECTIONS AND NOTATION

55

The diagram of functors Rf !

DMO (X) ←− ↑ or (2.1.2.2)

DM(X)

DMO (Y ) ↑ or

f!

←−

↓ o`

DM(Y ) ↓ o`

Lf ∗

DMO (X) ←−

DMO (Y )

commutes; here Rf ! , Lf ∗ are standard pull-back functors for O-modules (recall that for any F ∈ DMO (Y ) one has (Lf ∗ F ) ⊗ ωX [dim X] = Rf ! (F ⊗ ωY [dim Y ])). In particular, f ! is right exact with respect to the M` t-structure. We often denote the corresponding functor M` (Y ) → M` (X) by f ∗ ; this is the usual pull-back of an O-module equipped with a connection. If Xi , i ∈ I, is a finite collection of smooth schemes, then have the exterior Q we Q r r tensor product functors for both left and right D-modules M (X Xi ), i) → M ( Q ` Q M (Xi ) → M` ( Xi ), (Mi ) 7→ Mi . The canonical equivalence (2.1.1.1) identifies the corresponding derived functors.2 The functors f∗ ,Q f ! are compatible with : ifQfi : Xi → Yi are morphisms of smooth schemes, then ( fi )∗ (Mi ) = (fi∗ Mi ), ( fi )! (Ni ) = (fi! Ni ). Assume that all Xi coincide with X; let ∆(I) : X → X I be the diagonal embedding. For Li ∈ DM` (X) one has a canonical isomorphism (2.1.2.3)

L

⊗Li = ∆(I)! (Li ).

In particular, for Li ∈ M` (X) one has (2.1.2.4)

⊗Li = ∆(I)∗ (Li ).

2.1.3. If i: X ,→ Y is a closed embedding, then i∗ is exact with respect to the Mr t-structure, and its right adjoint i! is left exact. They define mutually inverse equivalences between M(X) and the full subcategory M(Y )X ⊂ M(Y ) that consists of DY -modules that vanish on Y \ X (Kashiwara’s lemma). Our functors can be described explicitly as follows. Let I ⊂ OY be the ideal of X. If N ∈ M(Y ), then i! N coincides, as an O-module, with the subsheaf of N that consists of sections killed by I. For a L ∈ M` (Y ) one has i∗ L = L/IL. For M ∈ M(X) one has (2.1.3.1)

i∗ M = i· (M ⊗ i∗ DY ). DX

Here we consider DY as a left DY -module, and the right DY -module structure on i∗ M comes from the right action of DY on DY (as on a left D-module). The ∼ identification M −→ i! i∗ M is m 7→ m ⊗ 1. For L ∈ M` (Y ) and M ∈ M(X) there is a canonical isomorphism (2.1.3.2)

∼

i∗ (M ⊗ i∗ L) −→ (i∗ M ) ⊗ L. ∼

2 There is a canonical isomorphism of super lines ω ΠXi [dim ΠXi ] −→ (ωXi [dim Xi ]); in particular, it does not depend on the ordering of I.

56

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GEOMETRY OF D-SCHEMES ∼

The corresponding isomorphism M ⊗i∗ L −→ i! ((i∗ M )⊗L) sends m⊗` to (m⊗1)⊗`. The reader can skip the next remark. Remark. We will use only D-modules on smooth algebraic varieties. However at some places (such as 2.1.4) the smoothness restriction on X is unnatural. Here is a short comment about the notion of D-module on an arbitraly singular scheme. One can assign to any k-scheme X of locally finite type an abelian k-category M(X) whose objects are called “D-modules on X” together with a conservative3 left exact functor o : M(X) → MO (X). If X is smooth, then M(X) is the category of right D-modules on X and o is the functor or from 2.1.1. The important point is that the notion of left D-module on an arbitrary singular X makes no sense. The standard functors from 2.1.2 exist in this setting4 and satisfy the usual properties. According to Kashiwara’s lemma, if one has a closed embedding i : X ,→ Y where Y is smooth, then i∗ identifies M(X) with the full subcategory of usual right Dmodules on Y which vanish on Y r X; for M ∈ M(X) one has o(M ) = i! (or i∗ M ). There exist two (different yet equivalent) ways to define M(X) (see [S2]5 and [BD] 7.10). The method of [S2] is to consider all possible closed embeddings iY : U ,→ Y where U ⊂ X is an open subset and Y is a smooth scheme; a D-module on X is a rule M which assigns to every such datum a D-module iY ∗ M on Y in a compatible manner. [BD] presents a crystalline approach: a D-module M on X is a rule that assigns to any U as above and an infinitesimal thickening6 iZ : U ,→ Z an O-module MZ = o(iZ∗ M ) on Z in a compatible (with respect to morphisms of Z’s) manner. 2.1.4. The pull-back functor. If f is a smooth morphism then f ∗ is exact with respect to the M` t-structure. Left D-modules satisfy the smooth descent property, so the categories M` (U ), U is a smooth X-scheme, form a sheaf of abelian categories on the smooth topology Xsm , which we denote by M` (Xsm ). One defines then the category M` (Z) for any smooth algebraic stack Z in the usual way. Remark. If our schemes are not necessarily smooth (so we are in the setting of Remark in 2.1.3) but f is a smooth morphism, then the functor f † := f ! [− dim X/Y ] : DM(Y ) → DM(X) is t-exact (i.e., f † (M(Y )) ⊂ M(X)), so we have a canonical exact functor M(Y ) → M(X) which we denote also by f † . The smooth descent property holds. Of course, if our schemes are smooth, then f † coincides with the pull-back functor f ∗ : M` (Y ) → M` (X).7 The above picture generalizes to the case of locally complete intersection morphisms. Namely, assume that f : X → Y makes X a locally complete intersection over Y of pure relative dimension d ≥ 0 (i.e., X can be locally represented as a closed subscheme of Y × An defined by n − d equations and the fibers of f have pure dimension d). Set f † := f ! [−d] : DM(Y ) → DM(X). Proposition. The functor f † is t-exact, i.e., f † M is a D-module for every D-module M on Y . We give two proofs. The first one makes sense even if D-modules are replaced, say, by `-adic perverse sheaves. The second one is based on the relation between D-modules and O-modules. 3 Which

means that o(M ) 6= 0 for M 6= 0. assume that our schemes are quasi-compact and separated. 5 M. Saito considers D-modules in the analytic setting; the algebraic picture is similar. 6 I.e., i is a closed embedding i : U ,→ Z defined by a nilpotent ideal. Z Z 7 We identify M with M` as in 2.1.1. 4 We

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57

First proof. For any f : X → Y with fibers of dimension ≤ d one has H i f ! M = 0 for i < −d.8 So it remains to show that H i f ! M = 0 for i > −d. We can assume that f = πi where π : Y˜ → Y is smooth of pure relative dimension n and i : X ,→ Y˜ is a closed embedding such that Y˜ \ i(X) is covered by n − d open subsets Uj affine over Y˜ . Now compute i∗ f ! M = i∗ i! π ∗ M [n] using the Uj ’s. The direct image with respect to an affine open embedding is an exact functor, so H i f ! M = 0 for i > −d, and we are done. Second proof. We can assume that there is a closed embedding X ,→ Y × An such that i(X) ⊂ Y × An is defined by n − d equations. One can find subschemes Xk ⊂ Y × An , k ∈ N, such that Xk ⊂ Xk+1 , (Xk )red = i(Xred ), each Xk is a locally complete intersection over Y , and every subscheme Z ⊂ Y ×An with Zred = i(Xred ) is contained in some Xk . The OY ×An -module corresponding to H p f ! M is the direct limit of the OXk -modules H p fk! MO , where fk is the morphism Xk → Y and MO is the OY -module corresponding to M . But Xk is Cohen-Macaulay over Y , so H p fk! MO = 0 for p 6= −d.  Remark. In the above proposition it is not enough to require X to be CohenMacaulay over Y (a counterexample: Y is a point and X is the scheme of 2 × 3 matrices of rank ≤ 1). 2.1.5. Vector D-bundles. If X is affine, then the projectivity of a DX module is a local property for the Zariski or ´etale topology of X (see 2.3.6 below where this fact is proven in a more general situation). For arbitrary X we call a locally projective right DX -module of finite rank a vector DX -bundle or a vector D-bundle on X. For a vector DX -bundle V its dual V ◦ is HomDX (V ` , DX ). So a section of V ◦ is a morphism of left DX -modules V ` → DX , and the right DX -module structure on V ◦ comes from the right DX -action on DX . It is clear that V ◦ is again a vector DX -bundle, and (V ◦ )◦ = V . For further discussion see 2.2.16. A vector DX -bundle need not be a locally free DX -module as the following example from [CH] (see also [BGK]) shows. Example. Assume that X is an affine curve, and let V ⊂ DX be a Dsubmodule such that V equals DX outside of a point x ∈ X. Then V is a projective DX -module (since the category of D-modules has homological dimension 1).9 By 2.1.13 below, such V ’s are the same as open k-vector subspaces of the formal completion Ox of the local ring Ox . If V is locally free, then (shrinking X if necessary) we can assume that it is generated by a single section φ; i.e., V = φDX ⊂ DX . Since φDX = DX on X r {x}, the differential operator φ has order 0; i.e., V = tn DX for some n ≥ 0 (where t is a parameter at x). Therefore locally free V ’s correspond to open ideals in Ox . 2.1.6. The middle de Rham cohomology sheaf. For M ∈ M(X) set h(M ): = M ⊗ OX = M/M ΘX . For a local section m ∈ M we denote by m its class DX

8 To prove this, choose a stratification of Y such that for each stratum Y α the morphism ! f ! M , ν : X ,→ X. Xα := (f −1 (Yα ))red → Yα is smooth and look at the sheaves H i να α α 9 In fact, it is enough to know that D /V has a projective resolution of length 1, and this X follows from Kashiwara’s lemma.

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in h(M ). We consider h(M ) as a sheaf on the ´etale topology of X. This will be relevant in Chapter 4; for the present chapter Zariski localization works as well. Notice that for any M ∈ M(X), L ∈ M` (X) one has M ⊗ L = h(M ⊗ L). DX

The reader can skip the next technical lemma returning to it when necessary. Lemma. (i) If M is a non-zero coherent DX -module then h(M ) 6= 0. (ii) If M is a coherent DX -module with no OX -torsion then for every DX module L the map Hom(L, M ) → Hom(h(L), h(M )) is injective. Proof. (i) (a) Using induction by the dimension of the support of M , we can assume, by Kashiwara’s lemma (see 2.1.3), that M is non-zero at the generic point of X. Replacing X by its open subset, we can assume that M is a free OX -module.10 (b) Let us show that for any (not necessarily coherent) non-zero DX -module M which is free as an OX -module one has h(M ) 6= 0. We use induction by dim X. Replacing X by its open subset, we can find a smooth hypersurface i : Y ,→ X such that the determinant of the normal bundle to Y is trivial. Let j : U ,→ X be its complement. Set MY := i! (j∗ j ∗ M/M ) = (i ∗ M ` )r ; this is a non-zero D-module on Y which is free as an OY -module by the condition on Y . Hence h(MY ) 6= 0 by the induction hypothesis. Notice that j· j · h(M ) = h(j∗ j ∗ M ) and h(i∗ MY ) = i· h(MY ). Since h is right exact and i∗ MY is a quotient of j∗ j ∗ M , we see that h(M ) 6= 0. (ii) (a) Since h is right exact, it suffices to show that for every non-zero L ⊂ M the map h(L) → h(M ) is non-zero. The condition on M assures that we can replace X by any open subset, so we can assume that L, M , and M/L are free OX -modules. (b) Let us show that for every (not necessarily coherent) non-zero DX -modules L ⊂ M such that L, M , M/L are free OX -modules, the map h(L) → h(M ) is non-zero. We use induction by dim X and follow notation from (i)(b) above. It suffices to show that the map h(j∗ j ∗ L) → h(j∗ j ∗ M ) is non-zero. It admits as a quotient the map h(LY ) → h(MY ) which is non-zero by induction by dim X.  2.1.7. The de Rham complex. Denote by DR(M ) the de Rham complex of M , so DR(M )i : = M ⊗ Λ−i ΘX . One has h(M ) = H 0 DR(M ); in fact, as an object OX

of the derived category of sheaves of k-vector spaces on XZar or X´et , DR(M ) equals L

to M ⊗ OX . In particular, if M is a locally projective DX -module (say, a vector DX

DX -bundle), then the projection DR(M ) → h(M ) is a quasi-isomorphism. Set ΓDR (X, M ) := Γ(X, DR(M )), RΓDR (X, M ) := RΓ(X, DR(M )). For a morphism f : X → Y of smooth schemes there is a canonical isomorphism ∼ Rf· (DR(M )) −→ DR(f∗ M ) in the derived category of sheaves on Y ; here f· is the sheaf-theoretical direct image, and Rf· is its derived functor. For a more precise setting for this quasi-isomorphism; see 2.1.11. We mention two particular cases: (a) The global de Rham cohomology coincides with the push-forward by pro· jection π: X → (point): one has HDR (X, M ) := H · (X, DR(M )) = H · π∗ M . (b) If i: X ,→ Y is a closed embedding, then the above quasi-isomorphism arises as a canonical embedding of complexes i· DR(M ) ,→ DRi∗ (M ) defined by the obvious embedding of OY -modules i· M ,→ i∗ M . Thus i· h(M ) = h(i∗ M ). 10 Indeed, choose a good filtration on M ; then, according to [EGA IV] 6.9.2, grM is O -free X over a non-empty open V ⊂ X.

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59

Notice also that in situation (b) for N ∈ DM(Y ) the obvious morphism DR(Ri! N ) → Ri! DR(N ) is a quasi-isomorphism.11 0 Lemma. If X is an affine curve, then the map HDR (X, M ) → Γ(X, h(M )) is surjective. d

Proof. Consider the de Rham complex M ` − → M ; set K := Ker(d), I := Im(d). Notice that H 2 (X, K) = 0 (since dim X = 1) and H 1 (X, M ` ) = 0 (since X is affine). Thus H 1 (X, I) = 0, which implies that Γ(X, M )  Γ(X, h(M )).  2.1.8. Induced D-modules. See [S1] for all the details. Let X be a smooth scheme. For an O-module B on X we have the induced right D-module BD : = B ⊗ DX ∈ M(X). The de Rham complex DR(BD ) is acyclic in OX

non-zero degrees, and the projection DR(BD )0 = B ⊗ DX → B, b ⊗ ∂ 7→ b∂(1), OX

∼

yields an isomorphism h(BD ) −→ B. In other words, there is a canonical quasiisomorphism νB : DR(BD ) → B.

(2.1.8.1)

If C is another O-module, then for any ϕ ∈ HomM(X) (BD , CD ) the corresponding map h(ϕ): h(BD ) → h(CD ) is a differential operator B → C. In fact, the map h: Hom(BD , CD ) → Diff(B, C),

(2.1.8.2)

where Diff is the vector space of all differential operators, is an isomorphism. The inverse map assigns to a differential operator ψ: B → C the morphism ψD : BD → CD , b ⊗ ∂ 7→ ψb ∂, where ψb ∈ C ⊗ DX = Diff(OX , C) is the differential operator ψb (f ) = ψ(bf ). Let Dif f (X) be the category whose objects are O-modules on X and where morphisms are differential operators. We see that the induction B 7→ BD defines a fully faithful functor Dif f (X) ,→ M(X). For N ∈ M` (X), B ∈ MO (X) there is a canonical isomorphism of D-modules ∼

(B ⊗ N )D −→ BD ⊗ N,

(2.1.8.3)

OX

(b ⊗ n) ⊗ ∂ 7→ (b ⊗ n)∂.

Therefore N, B 7→ B ⊗ N := B ⊗ N is an action of the tensor category M` (X) on OX

Dif f (X); the induction functor ·D commutes with M` (X)-actions.12 The “identity” functor from Dif f (X) to the category Sh(X) of sheaves of kvector spaces on X is faithful, so we know what a Dif f (X)-structure on P ∈ Sh(X) is. Explicitly, this is an equivalence class of quasi-coherent OX -module structures on P , where two OX -module structures P1 , P2 are said to be equivalent if idP : P1 → P2 is a differential operator. Remark. For an O-module B the structure action of OX on B yields an OX action on the D-module BD . Therefore BD is an (OX , DX )-bimodule. This bimod· ule satisfies the following property: the ideal of the diagonal Ker(OX ⊗ OX − → OX ) k

acts on BD in a locally nilpotent way. As follows from Kashiwara’s lemma, the 11 Use

the standard exact triangle i∗ Ri! N → N → Rj∗ j ∗ N → i∗ Ri! N [1] for j : Y ri(X) ,→ Y . M` (X) acts on M(X) by the usual tensor product.

12 Here

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functor B 7→ BD from the category of O-modules (and O-linear maps) to that of (OX , DX )-bimodules satisfying the above property is an equivalence of categories. Examples. (i) Let X ,→ Y be a closed embedding. Suppose that the formal neighborhood Yˆ of X in Y admits a retraction Yˆ → X. Let E be an OY -modules E supported (set-theoretically) on X. Then E, considered as a sheaf of k-vector spaces on X, admits a canonical Dif f (X)-structure. Indeed, every retraction π : Yˆ → X yields an OX -module structure on E, and its equivalence class does not depend on the choice of π. A particular situation we will use is the diagonal embedding X ,→ X n . Notice that for E as above the corresponding DX -module coincides with ∆! (E ⊗ DX n ). OX n

(ii) For a right D-module M its de Rham complex DR(M ) is a complex in Dif f (X). If L is a left D-module, then DR(L ⊗ M ) = L ⊗ DR(M ). (iii) Let i: X ,→ Y be a closed embedding. For B as above consider the canonical embeddings of OY -modules i· B ,→ i· (B ⊗ DX ) ,→ i∗ (BD ). The composition OX

induces a canonical morphism of DY -modules γ: (i· B)D → i∗ (BD ) which is an isomorphism. 2.1.9. For any M ∈ M(X) the de Rham complex DR(M ) is a complex in Dif f (X) (the de Rham differential is a first order differential operator), so we have the corresponding complex DR(M )D of D-modules. It is acyclic in degrees 6= 0, and the projection DR(M )0D = M ⊗ DX → M , m ⊗ ∂ 7→ m∂, defines an isomorphism ∼

OX

H 0 DR(M )D −→ M . In other words we have a canonical quasi-isomorphism (2.1.9.1)

µM : DR(M )D → M.

2.1.10. One may replace modules by complexes of modules to get the functors between the DG categories of complexes (2.1.10.1)

DR: CM(X) → CDif f (X),

·D : CDif f (X) → CM(X)

together with natural morphisms ν, µ as in (2.1.8.1) and (2.1.9.1). The category Dif f (X) is not abelian. Following Saito, a morphism ψ: B → C of complexes in Dif f (X) is called a quasi-isomorphism if ψD is. One defines the derived category DDif f (X) by inverting quasi-isomorphisms. Remark. A morphism ψ : B → C in CDif f (X) is called a naive quasiisomorphism if it is a quasi-isomorphism of complexes of plain sheaves of vector spaces. It is clear that ψ is a quasi-isomorphism if and only if for any O-flat left D-module N the morphism idN ⊗ ψ : N ⊗ B → N ⊗ C is a naive quasiisomorphism. If both B, C are bounded above complexes of coherent O-modules, then every naive quasi-isomorphism ψ is necessarily a quasi-isomorphism.13 For general quasi-coherent B, C this may be false (for example, take B = DR(M ) where M is a constant sheaf equal to the field of fractions of DX , C = 0). Both µ and ν are quasi-isomorphisms. They are compatible in the following sense: for every B · ∈ CDif f (X), M · ∈ CM(X) the morphisms DR(µM ), νDR(M ) : DR(DR(M )D ) → DR(M ) are homotopic; the same is true for the morphisms 13 Let M be the top non-zero cohomology of the cone of ψ . Then M is a coherent D-module D such that h(M ) = 0. We are done by the lemma from 2.1.6.

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(νB )D , µBD : DR(BD )D → BD . Therefore DR and ·D define mutually inverse equivalences between the derived categories: ∼

DM(X)↔DDif f (X).

(2.1.10.2)

For a morphism of schemes f : X → Y there is an obvious sheaf-theoretic pushforward functor Rf· : DDif f (X) → DDif f (Y ). Equivalence (2.1.10.2) identifies it with f∗ : there are canonical isomorphisms (2.1.10.3)

DR(f∗ M ) = Rf· DR(M ),

f∗ (BD ) = (Rf· (B))D .

For particular cases see 2.1.7 and Example (iii) in 2.1.8. 2.1.11. Quasi-induced D-modules and D-complexes. The full subcategory of induced D-modules, i.e., D-modules isomorphic to ND for some O-module N , does not look reasonable. For example, a direct summand of an induced Dmodule need not be induced (as the example from 2.1.5 shows); induced D-modules do not have a local nature. A more friendly category of quasi-induced D-modules is defined as follows. We say that M ∈ M(X) is quasi-induced if for every OX -flat L ∈ M` (X) DX one has T or>0 (M, L) = 0; i.e., the canonical projection DR(M ⊗ L) → h(M ⊗ L) is a quasi-isomorphism.14 Similarly, a DX -complex M ∈ CM(X) is homotopically quasi-induced if for every homotopically OX -flat L ∈ CM` (X) one has L

∼

M ⊗ L −→ M ⊗ L; i.e., the canonical projection DR(M ⊗ L) → h(M ⊗ L) is a DX

DX

quasi-isomorphism. Here is an intrinsic characterization of quasi-induced D-modules. Let η be a (not necessarily closed) point of X, iη : η ,→ X the embedding. For C ∈ DM(X) denote by i!η C the fiber of i!Y C at η, where Y is a non-empty open smooth subvariety of the closure of η and iY is the embedding Y ,→ X. So i!η C belongs to the derived category of modules over the ring of differential operators on η. Let us say that C is strongly non-negative if for every η ∈ X the Tor-dimension of i!η C is nonnegative. Equivalently, C is strongly non-negative if and only if for every η ∈ X the Tor-dimension of the stalk Cη of C at η is ≤ codim η. It is easy to see that strong non-negativity implies the usual one (i.e., H <0 C = 0) and that strong nonnegativity is preserved under direct images with respect to arbitrary morphisms. Lemma. M ∈ M(X) is quasi-induced if and only if it is strongly non-negative. Proof. Suppose M is quasi-induced. We can assume that X is affine. Take any η ∈ X and L ∈ M` (X). Let P be a left resolution of L such that Pa = 0 for a > codim η and P0 , . . . , Pcodim η−1 are projective D-modules. Localizing at L

η, we see that every Paη is a flat Oη -module. Thus (M ⊗ L)η = h(Pη ); hence DX

T oraDX (M, L)η = 0 for a > codim η; q.e.d. Suppose M is strongly non-negative. Then M ⊗ L is strongly non-negative for any OX -flat L ∈ M` (X). So it suffices to check that H a DR(M ) = 0 for every a < 0. Suppose this is not true. Let η be a point of smallest codimension such that DR(M )η has non-zero negative cohomology. Since DR(i!η M ) = i!η DR(M ), one has 14 This local property amounts to the corresponding global property which says that for every ∼ OX -flat L ∈ M` (X) one has RΓDR (X, M ⊗ L) −→ RΓ(X, h(M ⊗ L)).

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H a DR(i!η M ) = H a DR(M )η for a < 0. Therefore i!η M has negative Tor-dimension, and we are done.  Remarks. (i) Assume that k = C. A holonomic complex with regular singularities is strongly non-negative if and only if the complex of (usual, not perverse) sheaves corresponding to its Verdier dual is non-positive. (ii) An induced D-module is quasi-induced. The direct image of a quasi-induced D-module under an affine morphism is quasi-induced. The exterior tensor product of quasi-induced D-modules is quasi-induced. A holonomic D-module is quasiinduced if and only if its support has dimension 0. A tensor product of an OX -flat D-module and a quasi-induced D-module is quasi-induced. (iii) If X is a curve, then a D-module is quasi-induced if and only if its restriction to any open subset does not contain a non-trivial lisse submodule. 2.1.12. Maximal constant quotients and de Rham homology. Suppose that X is connected and dim X = n. −a (X, M [n]); this is the de Rham homolFor M ∈ M(X) set HaDR (X, M ) := HDR ogy of X with coefficients in M . Notice that HaDR (X, M ) vanish unless a ∈ [0, 2n]; if X is affine, the vanishing holds unless a ∈ [n, 2n]. The vector space H0DR (X, M ) vanishes if X is non-proper. We say that M ∈ M(X) is constant if M ` is generated by global horizontal sections. The category of constant D-modules is closed under subquotients. It identifies canonically with the category of vector spaces by means of the functors M 7→ Hom(OX , M ` ) = Γ∇ (X, M ` ), V 7→ V ⊗ ωX .15 Every D-module M admits the maximal constant submodule Γ∇ (X, M ` ) ⊗ ωX ⊂ M . For any M ∈ M(X) all its constant quotients form a directed projective system of constant D-modules Mconst . The functor M 7→ Mconst is right exact. Suppose X is proper. Then M admits the maximal constant quotient; i.e., Mconst is a plain DX -module.16 The adjunction formula for the projection from X to a point (see 2.1.2) yields ∼ ` a canonical identification Mconst −→ H0DR (X, M ) ⊗ OX . For any point s ∈ X πs the composition H 0 Ri!s M [n] = Ms` −→ (Mconst )s = H0DR (X, M ) comes from the ! canonical morphism is∗ Ris M → M . Let S ⊂ X be any finite non-empty subset, jS : U ,→ X its complement. The standard exact triangle ⊕ is∗ Ri!s M → M → RjS∗ jS∗ M → ⊕ is∗ Ri!s M [1] shows that S

S

the kernel of the morphism πS = ⊕πs : ⊕ Ms` → H0DR (X, M [n]) is equal to the S

n−1 image of the “residue” map ResS = ⊕Ress : HDR (U, M ) → ⊕ Ms` . S

2.1.13. Modifications at a point. Let x ∈ X be a (closed) point. Denote by ix the embedding {x} ,→ X; let jx : Ux := X r {x} ,→ X be its complement. For a D-module M on X let Ξx (M ) be the set of all submodules Mξ ⊂ M such that M/Mξ is supported at x. This is a subfilter in the ordered set of all submodules of M , so Ξx (M ) is a topology on M . 15 To prove that these functors are mutually inverse, notice that for a vector space V all sub-D-modules of V ⊗ OX are of the form W ⊗ OX , W ⊂ V ; this follows from the irreducibility of the D-module OX . 16 A direct way to see this: reduce to the case where M ` is coherent; then notice that dim HomD (M ` , OX ) < ∞.

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Consider the vector space h(M )x . For every Mξ ∈ Ξx (M ) the sequence 0 → −1 h(Mξ )x → h(M )x → h(M/Mξ ) → 0 is exact (since HDR (M/Mξ ) = 0). Therefore the map N 7→ h(N )x ⊂ h(M )x is a bijection between the set of D-submodules N ⊂ M containing Mξ and that of vector subspaces of h(M )x containing h(Mξ )x (since D-submodules of M/Mξ are the same as vector subspaces of h(M/Mξ )x ). The subspaces h(Mξ )x , Mξ ∈ Ξx (M ), form a topology on h(M )x called the Ξx topology. Thus Mξ 7→ h(Mξ )x is a bijection between Ξx (M ) and the set of open vector subspaces for the Ξx -topology. ˆ x (M ) the completion of h(M )x with respect to the Ξx -topology, Denote by h i.e., the projective limit of the system of vector spaces i!x (M/Mξ ) = h(M/Mξ )x , ˆ x as a functor from M(X) to the category of complete Mξ ∈ Ξx (M ). We consider h separated topological vector spaces.17 For any discrete vector space F one has18 (2.1.13.1)

ˆ x (M ), F ) = Hom(M, ix∗ F ), Homcont (h

ˆ x is right exact. so the functor h n Remarks. (i) Let Ox := ← lim − Ox /mx be the formal completion of the local ring at x. Set MOx := M ⊗ Ox ; this is a DOx -module. Then the above picture OX

depends only on MOx . Indeed, the map Mξ 7→ MξOx is a bijection between Ξx (M ) and the set of DOx -submodules of MOx with the quotients supported at x.19 So Ξx (M ) can be understood as the Ξx -topology on MOx . The vector space h(MOx ) = M ⊗ Ox carries the Ξx -topology formed by the subspaces h(MξOx ). For every DX

Mξ ∈ Ξx (M ) one has M/Mξ = MOx /MξOx ; hence h(M/Mξ ) = h(MOx /MξOx ) = ˆ x (M ) is the completion of h(MO ) with respect to h(MOx )/h(MξOx ). Therefore h x the Ξx -topology. ˆ x (M ) is a profinite-dimensional vec(ii) If M is a coherent D-module, then h tor space. Indeed, every M/Mξ is a coherent D-module supported at x; hence i!x (M/Mξ ) is a finite-dimensional vector space. We refer to any topology Ξ?x (M ) weaker than Ξx (M ) as a topology at x on M . As above, it amounts to a topology on h(M )x which is weaker than the Ξx -topology. ˆ ? (M ) equals lim i! (M/Mξ ), Mξ ∈ Ξ? (M ). As The corresponding completion h x x ←− x ˆ ? (M ) as the in Remark (i) we can consider Ξ?x (M ) as a topology on MOx and h x completion of h(MOx ). Examples. (i) Assume that M is an induced D-module, M = BD (see 2.1.8), so h(M ) = B. Denote by Ξx (B) the topology on B formed by all quasi-coherent O-submodules P ⊂ B such that B/P is supported set-theoretically at x. Then ˆ x (M ) is the Ξx (M ) is generated by submodules PD , P ∈ Ξx (B).20 Therefore h completion of B with respect to the topology Ξx (B). If B is O-coherent, then this is just the formal completion of B at x. We will prove in 2.1.17 that for arbitrary B the Ξx (B)-completion of B equals B ⊗ Ox . 17 By “topological vector space” we always mean the vector space equipped with a linear topology (there is a base of neighborhoods of 0 formed by linear subspaces). 18 We would denote h ˆ x (M ) by i∗ M but, unfortunately, the notation was already used in 2.1.2. x 19 Since the O-modules supported at x are the same as the O -modules supported at x. x 20 For M ∈ Ξ (M ) set P := M ∩ B. Then P ∈ Ξ (B) and M ⊃ P . x x ξ ξ ξ D

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ˆ x (M ), one can (ii) Assume that M is any coherent D-module. To compute h m n represent M as a cokernel of a morphism ϕ : DX → DX . We have seen that ˆ x (DX ) = Ox , so h ˆ x (M ) is the cokernel of the corresponding matrix of differential h 21 m operators ϕ : Ox → Oxn . 2.1.14. Lemma. If the restriction of M to Ux := X r {x} is a coherent D-module, then the Ξx -topology on h(MOx )22 is complete and separated; i.e., ∼ ˆ h(MOx ) −→ h x (M ).

(2.1.14.1)

ˆ x (M ) Proof. We want to show that the morphism of vector spaces h(MOx ) → h 0 is an isomorphism. Our M is an extension of a D-module M supported at x by a coherent D-module M 00 . We have already know the statement for M 0 (see Remark (i) in 2.1.13) and M 00 (see Example (ii) of 2.1.13). Since the sequences ˆ x (M 00 ) → h ˆ x (M ) → h ˆ x (M 0 ) → 0 → h(MO00 x ) → h(MOx ) → h(MO0 x ) → 0 and 0 → h 0 are exact, we are done.  ˆ x (M ) admits an open Remarks. (i) If M is a D-module as in 2.1.14, then h profinite-dimensional subspace.23 Such topological vector spaces are discussed in 2.7.9 under the name of Tate vector spaces. (ii) If M is an arbitrary D-module, then the canonical map φM : h(MOx ) → ˆ hx (M ) need not be an isomorphism. Here is an example. Assume that X = Spec R 0 is an affine curve, so R = HDR (X, DX ) = Ext1M(X) (ωX , DX ). Let M be the ˆ x (M ) = 0 universal extension of R ⊗ ωX by DX . Then M ⊗ Ox = Ox /R 6= 0 and h DX

ˆ x (M ) = 0. is the completion of Ox /R with respect to the quotient topology; i.e., h However φM is always an isomorphism if M is either an induced D-module (see Example (i) of 2.1.13 and see 2.1.17) or isomorphic to a direct sum of coherent D-modules. In fact, the argument used in the proof of the above lemma shows that it suffices to assume that the restriction of M to Ux satisfies either of these conditions. 2.1.15. Remark. Sometimes it is convenient to consider a finer object than ˆ x (M ). Namely, consider the directed set of all a plain topological vector space h ˜ x (M ) := lim h ˆ D-coherent submodules Mα ⊂ M and set h lim −→ x (Mα ) = − → Mα ⊗ Ox . DX

˜ x (M ) equals M ⊗ Ox as a plain vector space, but we consider it as an indThus h DX

object of the category of profinite-dimensional vector spaces. Recall that such an animal is the same as a left exact contravariant functor on the category of profinite˜ x (M ) “represents” the dimensional vector spaces with values in vector spaces. So h ˆ functor F 7→ lim −→ Hom(F, hx (Mα )), F a profinite-dimensional vector space. Notice ˜ that the dual object hx (M )∗ – which is a pro-object of the category of vector spaces – equals ← lim − Hom(Mα , ix∗ k). 21 We

use the fact that a morphism of profinite-dimensional vector spaces has closed image. Remark (i) in 2.1.13. 23 Namely, the image of h ˆ x (Mξ ) where Mξ ∈ Ξx (M ) is a coherent DX -module (as explained ˆ x (Mξ ) is profinite-dimensional). in 2.1.13, h 22 See

2.1. D-MODULES: RECOLLECTIONS AND NOTATION

65

2.1.16. Moving the point. We will also need a version of 2.1.13 in the case where x depends on parameters. So let Y be a scheme which we assume to be quasi-compact and quasi-separated,24 x ∈ X(Y ), ix : Y ,→ X × Y the graph of x. Denote by M(X × Y /Y ) the category of right DX×Y /Y := DX  OY -modules which are quasi-coherent as OX×Y -modules. Let M(X × Y /Y )x be the full subcategory of DX×Y /Y -modules supported (set-theoretically) at the graph of x. A version of Kashiwara’s lemma (see 2.1.3) says that the functor i!x identifies M(X × Y /Y )x with the category of quasi-coherent OY -modules. For M ∈ M(X × Y /Y ) we denote by Ξx (M ) the set of submodules Mξ ⊂ M such that M/Mξ ∈ M(X × Y /Y )x . This is a subfilter in the ordered set of all submodules of M . We have a Ξx (M )-projective system of quasi-coherent OY modules i!x (M/Mξ ) connected by surjective maps. For N ∈ M(X × Y /Y ) set h(N ) := N ⊗ OX×Y ; this is a p·Y OY -module.25 DX×Y /Y

Since i!x (M/Mξ ) = i·x h(M/Mξ ), the projective limit of i!x (M/Mξ ) equals the completion of i·x h(M ) with respect to the topology defined by the images of the i·x h(Mξ ). ˆ x (M ). We denote it by h If M is an induced module, M = B ⊗ DX×Y /Y where B is an OX×Y -module, ˆ x (M ) is the completion of B with respect to the topology then h(M ) = B and h formed by all quasi-coherent O-submodules P ⊂ B which coincide with B outside ˆ x (M ) is the formal of (the graph of) x. In particular, if B is O-coherent, then h completion of B at x. ˆ x (M ) with the As in 2.1.14 for a coherent DX×Y /Y -module M one identifies h DX×Y /Y -tensor product of M and the formal completion of OX×Y at x. Remarks. (i) The above construction is compatible with the base change by arbitrary morphisms f : Y 0 → Y . Namely, the evident morphism of pro-OY 0 ˆ xf ((f × idX )∗ M ) → f ∗ h ˆ x (M ) is an isomorphism. modules h (ii) If Y is a smooth variety and M is a right DX×Y -module, then the differential operators on Y act on M in a continuous way with respect to the Ξx -topology. ˆ x (M ) is a sheaf of topological right DY -modules. Similarly, if DY acts Therefore h ˆ x (M ) is a sheaf of topological left DY -modules. on M from the left, then h ˆ x (M ) → F is a continuous DY -morphism, (iii) If F is a D-module on Y and ϕ : h ˆ then it vanishes on the image of hx (Mξ ) where Mξ ∈ Ξx (M ) is a sub-DX×Y -module of M . In particular, assume that Y = X, x = idX , so ix = ∆, and M = L  OX for a D-module L on X. Then every morphism ϕ as above is trivial (if dim X 6= 0). 2.1.17. I-topology. In this section we prove a result in commutative algebra we referred to in Example (i) of 2.1.13 and Remark (ii) in 2.1.14. It will not be used in the rest of this work. Let A be a noetherian ring, I ⊂ A an ideal such that A is complete in the I-adic topology. Let C be the category of A-modules and C0 ⊂ C the full subcategory of A-modules M such that each element of M is annihilated by some power of I. The I-topology on an A-module M is the weakest topology such that every A-linear map from M to a module from C0 equipped with the discrete topology 24 The assumption is needed to assure that topological O -modules have a local nature, i.e., Y pro-objects in the category of quasi-coherent O-modules form a sheaf of categories on YZar , and our constructions are compatible with the localization of Y . 25 Here p : X × Y → Y is the projection. Y

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˜ ⊂ M such that M/M ˜ ∈ C0 is a is continuous. The set BM of all submodules M base of neighborhoods of 0 for the I-topology. If M is finitely generated, then the I-topology coincides with the I-adic topology. Theorem. (a) If N is a submodule of an A-module M then the I-topology on M induces the I-topology on N . (b) If A has finite Krull dimension then every A-module M is complete and separated for the I-topology. ˜ ∈ BN there is an M ˜ ∈ Proof. To prove (a), we have to show that for every N ˜ ∩N = N ˜ . Let P be the set of pairs (L, L) ˜ where L is a submodule BM such that M ˜ ∈ BL . Equip P with the following ordering: (L, L) ˜ ≤ (L0 , L ˜ 0 ) if of M and L 0 0 ˜ ˜ ˜ L ⊂ L and L = L ∩ L . By Zorn’s lemma there is a maximal (L, L) ∈ P such that ˜ ≥ (N, N ˜ ). It remains to show that L = M . Indeed, if x ∈ M , x ∈ (L, L) / L, then ˜ 0 ) > (L, L) ˜ by putting L0 := L + Ax, L ˜ 0 := L + U where one can construct (L0 , L ˜ ∩ Ax (U exists because U ⊂ Ax is an open submodule of Ax such that L ∩ U = L the I-topology on L ∩ Ax is induced by the I-topology on Ax). Using (a) we reduce the proof of separatedness of M to the well-known case where M is finitely generated (A need not have finite Krull dimension for this property). ˆ the completion of M with respect to the I-topology. Let C ? ⊂ C Denote by M ˆ is bijective. be the full subcategory of A-modules M such that the map M → M ? We will show that C = C if A has finite Krull dimension. Lemma. C ? has the following properties: (i) Finitely generated modules belong to C ? . (ii) Suppose that the sequence 0 → M 0 → M 00 → M → 0 is exact and M ∈ C ? . Then M 0 ∈ C ? ⇔ M 00 ∈ C ? . L (iii) If Mj ∈ C ? , j ∈ J, then j Mj ∈ C ? . S (iv) Suppose M = i Mi , i = 1, 2, . . . , Mi ⊂ Mi+1 . If Mi ∈ C ? for all i, then M ∈ C ?. Proof of Lemma. (i) is well known. ˆ , which is proved (ii) is a corollary of the left exactness of the functor M 7→ M ˆ2 → M ˆ 3 ) is as follows. If the sequence 0 → M1 → M2 → M3 is exact, then Ker(M ˆ the closure of M1 in M2 , i.e., the completion of M1 with respect to the topology induced from the I-topology on M2 . We have proved that the induced topology is ˆ2 → M ˆ 3) = M ˆ 1. the I-topology on M1 , so Ker(M L To prove (iii) it suffices to show that if M = j Mj , then the image of the ˆ →Q M ˆ j equals L M ˆ j . We only have to prove that natural embedding f : M j L ˆ Q jˆ 0 Imf ⊂ M . Assume that x = (x ) ∈ j j j j Mj and J := {j ∈ J|xj 6= 0} is infinite. Choose open submodules Uj ⊂ Mj so thatQ for every j ∈ J 0 the image of xj in Mj /Uj is non-zero. Then the image of x in j Mj /Uj does not belong to L / Imf . j Mj /Uj , so x ∈ ˆ belongs to the closure of To prove (iv), it suffices to show that every x ∈ M ˆ for some i (this closure equals M ˆ i = Mi ). If this is not true, then for every i Mi ⊂ M there is an open submodule Ui ⊂ M such that the image of x in M/(M S i +Ui ) is nonzero. We can assume that U1 ⊃ U2 ⊃ · · · . The submodule U := i (Mi ∩ Ui ) ⊂ M is open and U + Mi ⊂ Ui + Mi . So for every i the image of x in M/(Mi + U ) is

2.2. THE COMPOUND TENSOR STRUCTURE

67

non-zero; i.e., the image of x in M/U S does not belong to the image of Mi in M/U . This is impossible because M = i Mi .  We assume now that A has finite Krull dimension. It remains to show that if a full subcategory C ? ⊂ C satisfies properties (i)–(iv) from the lemma, then C ? = C. Let Ci ⊂ C be the full subcategory consisting of modules M such that Mp = 0 for every prime ideal p ⊂ A with dim A/p > i. We will prove by induction that Ci ⊂ C ? . Suppose we know that Ci−1 ⊂ C ? . Let M ∈ CQ i . Denote by Pi the set of all primes p ⊂ A such that dim A/p = i. The map M → p∈Pi Mp factors through P P p∈Pi Mp , and the kernel and cokernel of the morphism M → p∈Pi Mp belong to Ci−1 . By (ii),(iii), and the induction assumption, to prove that M ∈ C ? it suffices to show that Mp ∈ C ? for each p ∈ Pi . Let p ∈ Pi . Our Mp is an Ap -module such that each element of Mp is annihilated by a power of pAp . By (ii) and (iv) we are reduced to proving that vector spaces over Ap /pAp belong to C ? . By (iii) it suffices to show that Ap /pAp ∈ C ? . The morphism A/p → Ap /pAp is injective and its cokernel belongs to Ci−1 ⊂ C ? . So Ap /pAp ∈ C ? by (i) and (ii).  Remark. If A is a 1-dimensional local domain, then the proof of statement (b) of the above theorem is very simple. In this case every A-module M has a free submodule N such that M/N ∈ C0 , so we can assume that M is free. Then apply statement (iii) of the lemma. 2.2. The compound tensor structure In this section we describe a canonical augmented compound tensor structure on the category of DX -modules M(X) where X is a smooth scheme. We define the tensor structure ⊗! in 2.2.1, the pseudo-tensor structure P ∗ in 2.2.3, the compound tensor product maps ⊗IS,T in 2.2.6, and the augmentation functor h in 2.2.7; h is non-degenerate and reliable (see 2.2.8). We show in 2.2.9 that the translation functor M → M [− dim X] admits a canonical homotopy pseudo-tensor extension CM(X)! → CM(X)∗ . In 2.2.10 it is shown that DR is naturally a homotopy pseudo-tensor functor; another construction, that works in case dim X = 1, is presented in 2.2.11. We explain in 2.2.12 that our compound tensor structure can be encoded into a compound DX -operad, discuss duality and inner Hom in 2.2.15– 2.2.17, explain how ∗ operations can be seen on the level of polylinear operations between the h sheaves in 2.2.18, and establish their continuity properties in 2.2.19. 2.2.1. The ! tensor structure. The category M` is a tensor category (see ∼ −1 2.1.1). We use the standard equivalence M(X) −→ M` (X), M 7→ M ` : = M ωX , to ! ! ` define the tensor product ⊗ on M(X), so we have M1 ⊗ M2 = M1 ⊗ M2 . The OX

object ωX is a unit object for ⊗! . L

2.2.2. Consider the derived tensor product ⊗! on DM(X).26 According to (2.1.2.3) for Mi ∈ DM(X) one has L

⊗!I Mi = ∆(I)! ((Mi [dim X]))[− dim X] = ∆(I)! (Mi ) ⊗ λI⊗ dim X [(|I| − 1) dim X] 26 Notice

L

that ⊗! is identified with the tensor product on DM` (X) by the above “standard” t-exact equivalence, and not by the canonical equivalence (2.1.1.1).

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where λI := det(k I )[|I|] = (k[1])⊗I [−|I|] (the last equality comes since k[dim X]⊗I dim X = λ⊗ [|I| dim X]). Therefore for Mi ∈ M(X) there is a canonical isomorphism I ⊗!I Mi = H (|I|−1) dim X ∆(I)! (Mi ) ⊗ λI⊗ dim X .

(2.2.2.1)

!

⊗I I Example. Assume that Mi = ωX . One has ωX = ωX and ωX = ωX I ⊗ ⊗ dim X (|I|−1) dim X (I)! λI , so (2.2.2.1) comes from the identification ωX = H ∆ (ωX I ).

2.2.3. The ∗ pseudo-tensor structure. For an I-family Li of D-modules, I ∈ S, and a D-module M set (I)

PI∗ ({Li }, M ) := Hom(Li , ∆∗ M ).

(2.2.3.1)

The elements of PI∗ are called ∗ I-operations. They have ´etale local nature: we have the sheaf P ∗I ({Li }, M ), U 7→ PI∗ ({Li }, M ), on the ´etale topology of X. U U For a surjective map π: J  I the composition map PI∗ ({Li }, M ) ⊗ (⊗I PJ∗i ({Kj }, Li )) −→ PJ∗ ({Kj }, M )

(2.2.3.2)

sends ϕ ⊗ (⊗ψi ) to ϕ(ψi ) defined as the composition ψi

(Ji )

J Kj −−→ I ∆∗

∆(π) (ϕ)

(π)

(π)

(I)

(J)

∗ Li = ∆∗ (I Li ) −−− −−→ ∆∗ ∆∗ M = ∆∗ M

where ∆(π) = Π ∆(Ji ) : X I ,→ X J . The composition is associative, so PI∗ define on I

M(X) an abelian pseudo-tensor structure. We denote this pseudo-tensor category by M(X)∗ . The pseudo-tensor categories M(U )∗ , U ∈ X´et , form a sheaf of pseudotensor categories on the ´etale topology of X. For ϕ ∈ PI∗ ({Li }, M ) we will sometimes denote its image considered (by Kashiwara’s lemma) as a submodule of M by ϕ({Li }). Remarks. (i) The sheaf-theoretic restriction to the diagonal identifies our ∗ (I)· (I) I-operations with D⊗I ∆∗ M . X -module morphisms ⊗ Li → ∆ k

(ii) One has PI∗ ({Li }, M ) = Hom(⊗∗ Li , M ) where ⊗∗ Li := ∆(I)∗ (Li ) is the projective system of all the DX I -module quotients of Li supported on the diagonal, considered as a projective system of DX -modules via the ∆(I)! -equivalence. (iii) The category MO (X) of quasi-coherent OX -modules carries a natural M(X)∗ -action (see 1.2.11). Namely, the vector space of operations PI˜∗ ({Mi , F }, G), where F , G are (quasi-coherent) OX -modules and Mi , i ∈ I, are DX -modules, is ˜ (I)

˜ (I)

defined as PI˜∗ ({Mi , F }, G) := HomDI OX ((Mi )  F, ∆∗ G). Here ∆∗ G := X

˜ (I)

(∆· G) ⊗ (DI X  OX ). The composition of these operations with usual ∗ operO

˜ XI

ations between D-modules and morphisms of O-modules (as needed in 1.2.11) is clear. Consider the induction functor MO (X) → M(X), F 7→ FD (see 2.1.8). One has an obvious natural map PI˜∗ ({Mi , F }, G) ,→ PI˜∗ ({Mi , FD }, GD ). On the other hand, if P, Q are D-modules, then there is an obvious map PI˜∗ ({Mi , P }, Q) ,→ PI˜∗ ({Mi , PO }, QO ), where PO , QO are P, Q considered as O-modules. Therefore both induction and restriction functors MO (X)  M(X) are compatible with the M(X)∗ -actions. Here the M(X)∗ acts on M(X) in the standard way; see 1.2.12(i).

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69

2.2.4. Examples. (i) Let us describe the ∗ pseudo-tensor structure on the subcategory of induced modules (see 2.1.8). For O-modules {Fi }i∈I , G denote by DiffI ({Fi }, G) ⊂ Homk (⊗ Fi , G) the subspace of differential I-operations (those ϕ’s k

for which the maps Fi → G, fi 7→ ϕ(fi ⊗ ( ⊗ fj )), are differential operators for j6=i

any i and any fixed local sections fj ∈ Fj , j 6= i). Equivalently, DiffI ({Fi }, G) = (I) Diff(Fi , ∆· G). The differential operations are stable with respect to composition so they define a pseudo-tensor structure on the category Dif f (X); denote this pseudo-tensor category by Dif f (X)∗ . One has PI∗ ({FiD }, GD ) = Hom((Fi )D , (∆· G)D ) = DiffI ({Fi }, G) (see 2.1.8, in particular Example (iii) there); i.e., ·D extends to a fully faithful pseudo-tensor functor ·D : Dif f (X)∗ ,→ M(X)∗ .

(2.2.4.1)

(ii) Assume that |I| ≥ 2 and one of the D-modules Li is lisse (e.g., equal to ωX ). Then PI∗ ({Li }, M ) = 0. (iii) The de Rham DG algebra DR := (OX → Ω1X → · · · ) is a commutative DG algebra in Dif f (X)∗ , so the corresponding induced complex DRD is a commutative DG algebra in M(X)∗ . Notice that DRD is naturally a resolution of ωX [− dim X] (see, e.g., (2.1.9.1)), so ωX [− dim X] is naturally a homotopy commutative DG algebra in M(X)∗ . Exercise. Show that the cone of the product morphism DRD  DRD → ∆∗ DRD is naturally quasi-isomorphic to j! ωU [−2 dim X + 1] where j! is the usual functor on the derived category of holonomic D-modules. 2.2.5. Let i: X ,→ Y be an embedding of smooth schemes. The left exact functors H 0 i∗ , H 0 i! between M(X) and M(Y ) act on PI∗ in the obvious way; i.e., they are pseudo-tensor functors. If Y is a closed subscheme, then H 0 i! is right adjoint to i∗ = H 0 i∗ as pseudo-tensor functors (which means that PI∗ ({i∗ Ni }, M ) = PI∗ ({Ni }, H 0 i! M ) for any Ni ∈ M(X), M ∈ M(Y )), so i∗ is a fully faithful embedding of pseudo-tensor categories. 2.2.6. The tensor product maps. We have to define the canonical maps (see 1.3.12) (2.2.6.1)

⊗IS,T : ⊗S PI∗s ({Li }, Ms ) → PT∗ ({⊗!It Li }, ⊗!S Ms ).

According to the first formula in 2.2.2, one has canonical isomorphisms in DM(X T ) (here α := k[− dim X], so α−1 = k[dim X]): L

∆(πT )! (I Li ) = T (∆(It )! (It Li )) = T ((⊗ !It Li ) ⊗ α−1 ⊗ α⊗It ) L

= (T (⊗ !It Li )) ⊗ (α−1 )⊗T ⊗ α⊗I , (I )

(πS )

∆(πT )! (S (∆∗ s Ms )) = ∆(πT )! ∆∗

(T )

= ∆∗



(T )

(S Ms ) = ∆∗ ∆(S)! (S Ms )  L ! ⊗ S Ms ⊗ α−1 ⊗ α⊗S . (π )

In the second line we used the base change canonical isomorphism ∆(πT )! ∆∗ S = (T ) ∆∗ ∆(S)! (recall that X S , X T ⊂ X I are transveral and X S ∩X T = X; see 1.3.2(i)).

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Consider the diagonal embeddings ∆(S) : k ,→ k S , ∆(πT ) : k T ,→ k I . The em∼ bedding ∆(πS ) : k S → k I induces an isomorphism between the quotients k S /k −→ k I /k T . One has α−1 ⊗ α⊗S = det((k S /k)[− dim X]) and (α−1 )⊗T ⊗ α⊗I = det((k I /k)T [− dim X]), so we get the identification α−1 ⊗ α⊗S = (α−1 )⊗T ⊗ α⊗I . Now ⊗IS,T is the composition (I )

(T )

⊗S PI∗s ({Li }, Ms ) → Hom(I Li , S ∆∗ s Ms ) → Hom(T (⊗!It Li ), ∆∗ (⊗!S Ms )) where the first arrow is S and the second one is H (|I|−|T |) dim X ∆(πT )! . The commutativity of diagrams (i) and (ii) in 1.3.12 is clear. Therefore we have defined the compound tensor category structure on M(X). This compound tensor category M(X)∗! is abelian (see 1.3.14) with a strong unit object ωX (see 1.3.16, 1.3.17, 2.2.4(ii)). The above constructions are compatible with ´etale localization, so M(U )∗! , U ∈ X´et , form a sheaf of compound tensor categories M(X´et )∗! over X´et . 2.2.7. The augmentation functor. Let us show that the de Rham functor h (see 2.1.6) is a non-degenerate ∗-augmentation functor on M(X´et )∗! (see 1.2.5, 1.3.10). We have to define for a finite set I of order ≥ 2 and i0 ∈ I a canonical morphism of sheaves27 (2.2.7.1)

∗ hI,i0 : PI∗ ({Li }, M ) ⊗ h(Li0 ) → PIr{i ({Li }, M ), 0}

ϕ ⊗ ¯l 7→ ϕ¯l .

Namely, for ϕ ∈ PI∗ ({Li }, M ) and li ∈ Li one has     (2.2.7.2) ϕ¯li0  l i = T r i0 ϕ  l i . Ir{i0 }

(I) T ri0 : prI,i0 · ∆∗ M I Ir{i0 }

I

(Ir{i0 }) ∆∗ M

Here → is the trace morphism for the projection prI,i0 : X → X . It is easy to see that hI,i0 is well-defined by (2.2.7.2). Compatibilities (i), (ii) in 1.2.5 and 1.3.10 are straightforward, so h makes M(X)∗! an augmented compound tensor category. Let Sh(X)⊗ be the tensor category of sheaves of k vector spaces. According to 1.2.7, h extends to a pseudo-tensor functor (2.2.7.3)

h : M(X)∗ → Sh(X)⊗ .

The maps hI : PI∗ ({Li }, M ) → Hom(⊗I h(Li ), h(M )) are hI (ϕ)(⊗¯li ) = ϕ(li ); here (I) we consider ϕ(li ) as a section of ∆(I)· h(∆∗ M ) = h(M ) (see 2.1.7). If i: X ,→ Y is a closed embedding, then i∗ : M(X)∗ ,→ M(Y )∗ is compatible with the augmentation functors; i.e., it is an augmented pseudo-tensor functor (see 1.2.8). Remark. The map hI : PI∗ ({Li }, M ) → Hom(⊗I h(Li ), h(M )) need not be injective. However it is injective if M is either an induced D-module (e.g., if M is supported at a single point) or a coherent DX -module without OX -torsion.28 2.2.8. The next proposition shows that these difficulties disappear if we con˜ from 1.4.7 instead of h: sider h we consider P ∗ as a sheaf on the ´ etale topology of X. In the induced case by left exactness of h we can assume that the Li are also induced (replace Li by LiD  Li if necessary), and then use 2.2.4(i). In the coherent case use statement (ii) of the lemma from 2.1.6. 27 Here

28 Proof:

2.2. THE COMPOUND TENSOR STRUCTURE

71

Proposition. Our h is non-degenerate (see 1.2.5) 29 and reliable (see 1.4.7). Proof. Let us show that h is non-degenerate. Consider the map α : PI∗ ({Li }, M ) ∗ → Hom(h(Li0 ), PIr{i ({Li }, M )) coming from (2.2.7.1). We want to show that α 0} is injective. Take a non-zero ϕ ∈ PI∗ ({Li }, M ). Replacing M by the image of ϕ, we can assume that ϕ is surjective. To check that α(ϕ) 6= 0, we will show that the ∗ sum of images of all operations ϕ¯l ∈ PIr{i ({Li }, M ), l ∈ Li0 , equals M . Take 0} (Ir{i })

(I)

0 m ∈ ∆∗ M . Choose m ˜ ∈ ∆∗ M such ˜ Since ϕ is surjective, P Pthat m = T ri0 m. one can write m ˜ = ϕ(lia ). Then m = ϕ¯lia (  lia ); q.e.d.

a

a

0

i6=i0

Let us show that h is reliable. A commutative unital algebra R ∈ M(X)! is the same as a commutative unital algebra R` in the tensor category M` (X), i.e., a quasicoherent OX -algebra equipped with an integrable connection along X. We call (see 2.3.1) such R` a commutative DX -algebra and denote the corresponding category ˜ by Comu! (X). Every M ∈ M(X) yields a functor h(M ) : Comu! (X) → Sh(X), ` ` ` ˜ R 7→ h(M )R` := h(M ⊗ R ) = M ⊗ R . We want to show that the map (see DX

˜ : P ∗ ({Mi }, N ) → Mor(⊗ h(M ˜ i ), h(N ˜ )) is bijective. 1.4.7) h I I

˜ ˜ i ), h(N ˜ )) → P ∗ ({Mi }, N ) to h. Let us construct the inverse map κ : Mor(⊗h(M I ` ˜ ˜ Let ψ : ⊗h(Mi ) → h(N ) be a morphism of functors. Let A L be the commutative DX -algebra freely generated by I sections ai , so A` = Sym( DX · ai ) = Sym(D`I ) where D` is DX considered as a left DX -module. Notice that End A` ⊃ End(D`I ) = opp I ˜ i )A` → MatI (Dopp (the diagonal matrices), so the morphism ψA` : ⊗h(M X ) ⊃ DX I I 30 ˜ h(N )A` commutes with the right action of DX . The obvious Z -grading on A` ˜ A` ; since it comes from the action of diagonal matrices yields a ZI -grading on h(?) k I ⊂ EndA` , our ψA` is compatible with the grading. The component of degree ˜ )A` equals 1I = (1, . . . , 1) of A` is D⊗I , so the corresponding component of h(N ⊗I (I)· (I) 31 ˜ i )A` conN ⊗ D = ∆ ∆∗ N . The component of the same degree of ⊗h(M DX

tains ⊗ Mi : we identify ⊗mi with ⊗(mi ⊗ai ). Restricting ψ to this subsheaf, we get k

(I)

a morphism κ(ψ) : ⊗ Mi → ∆(I)· ∆∗ N . Notice that for every (∂i ) ∈ DIX ⊂ End A` , k

˜ i )A` and ∆(I)· ∆(I) ˜ the restriction of its action to ⊗ Mi ⊂ ⊗h(M ∗ N ⊂ h(N )A` coink

cides with the obvious action of ⊗∂i ∈ D⊗I X coming from the D-module structure on the Mi and N . Since κ(ψ) commutes with this action, one has κ(ψ) ∈ PI∗ ({Mi }, N ) according to Remark (i) in 2.2.3. ˜ So it remains to show that κ is injective; It is clear that κ is left inverse to h. i.e., ψ is uniquely determined by κ(ψ). Take any commutative DX -algebra R` ˜ i )R` → h(N ˜ )R` . Take a local section and consider the morphism ψR` : ⊗h(M ˜ i )R` ; let us compute ψR` (γ) ∈ h(N ˜ )R` in terms of κ(ψ). γ = ⊗(mi ⊗ ri ) ∈ h(M We can assume that the ri ’s are global sections of R` (replacing R` by j∗ j ∗ R` where j : U ,→ X is an open subset on which all ri ’s are defined).32 Consider 29 See

2.2.15 for a more precise statement. acts as a semigroup with respect to multiplication. latter equality comes from (2.1.3.1) for i = ∆(I) since ∆(I)∗ DX I = ∆(I)∗ D`I = D`⊗I by (2.1.2.4). See also 2.2.9 below. 32 We use the fact that the morphism of D -algebras R` → j j ∗ R` yields a morphism ∗ X ˜ ) ` → h(N ˜ ) ∗ ` which is an isomorphism over U . h(N R j∗ j R 30 DI X 31 The

72

2.

GEOMETRY OF D-SCHEMES

the morphism ν : A` → R` of DX -algebras which sends ai to ri . Then ψR` (γ) = νψA` (⊗(mi ⊗ ai )) = νκ(ψ)(mi ). We are done.  2.2.9. According to Remark in 1.3.15, the tensor category M(X)! acts on M(X)∗ , so the tensor product extends canonically to a pseudo-tensor functor (2.2.9.1)

⊗ : M(X)! ⊗ M(X)∗ → M(X)∗ .

Therefore, by 1.1.6(vi), any commutative algebra F ∈ M(X)∗ yields a pseudo-tensor functor M(X)! → M(X)∗ , M 7→ M ⊗ F . For example, DRD ∈ CM(X)∗ (see 2.2.4(iii)) yields a pseudo-tensor DG functor (2.2.9.2)

CM(X)! → CM(X)∗ ,

M 7→ M ⊗ DRD ,

which is a resolution of the functor M 7→ M [− dim X]. Similarly, by 1.1.6(vi), any commutative (not necessary unital) algebra P ∈ M(X)! yields a pseudo-tensor functor M(X)∗ → M(X)∗ , M 7→ M ⊗ P . This fact can be used as follows. 2.2.10. Enhanced de Rham complexes. The de Rham functor (see 2.1.7) DR : CM(X) → CSh(X) is not a pseudo-tensor functor, as opposed to h (see (2.2.7.3)). It is naturally a homotopy pseudo-tensor functor though. In fact, there is a natural family of mutually homotopically equivalent pseudo-tensor functors which are homotopically equivalent to DR as mere DG functors. Below we suppose that X is quasi-projective. Let P : P → OX be any homotopically DX -flat non-unital commutative! algebra resolution of OX . So P is a commutative DG algebra in the tensor category of left DX -modules, P a quasi-isomorphism of such algebras, and we assume that P is homotopically DX -flat as a mere complex of DX -modules (see 2.1.1). Example. If X is a curve, then a simplest P is a two term resolution of OX with P0 = Sym>0 DX , P sending 1 ∈ DX ⊂ P0 to 1 ∈ OX , and P−1 := KerP . Remark. For an arbitrary X one can take for P any resolution of OX which is isomorphic as a mere graded commutative! algebra to Sym>0 T , where T is an X-locally projective graded DX -module having degrees ≤ 0. For a quasi-projective X, it can be choosen so that T is isomorphic to a direct sum of D-modules of type DX ⊗ L where L is a line bundle (if needed, one can assume that all the L’s are negative powers of a given ample line bundle). By (2.2.7.3) and 2.2.9, we have a pseudo-tensor DG functor (2.2.10.1)

CM(X)∗ → CSh(X)⊗ ,

M 7→ h(M ⊗ P). 

P The canonical quasi-isomorphisms h(M ⊗ P) ← DR(M ⊗ P) −→ DR(M ) for M ∈ CM(X) make (2.2.10.1) a homotopy version of the de Rham functor.

The rest of the section can be skipped by the reader. Consider the category of all P = (P, P ) as above. It is a tensor category (without unit) in the obvious way. Lemma. Every functor from our category to a groupoid is isomorphic to a trivial one. Proof. Denote our functor by ¯. For two objects P, P0 let πP : P ⊗ P0 → P be the morphism πP (p ⊗ p0 ) := P0 (p0 )p.

2.2. THE COMPOUND TENSOR STRUCTURE

73

(i) Let ν, µ : P1 → P2 be two morphisms and let P be any object. Set νP := ν ⊗ idP , and the same for µP . Then ν¯ = µ ¯ if ν¯P = µ¯P . This follows since νπP1 = πP2 νP , µπP1 = πP2 µP . 0 0 (ii) Take any P. We have the morphisms πP , πP : P ⊗ P → P where πP := πP σ where σ is the transposition of multiples. These morphisms produce the same arrow 0 in the groupoid. Indeed, by (i) it suffices to check that the images of (πP )P , (πP )P : 0 (P ⊗ P) ⊗ P → P ⊗ P in the groupoid coincide, which is clear since πP (πP )P = 0 0 πP (πP )P . (iii) Let ζ, η : P1 → P2 be any two morphisms. Then ζ¯ = η¯. This follows from 0 0 (ii) since ζπP1 = πP2 (ζ ⊗ η) and ηπP = πP (ζ ⊗ η). 1 2 (iv) For any finite collection of objects {Pα } one can find P such that for every α there is a morphism φα : P → Pα . Namely, one can take P := ⊗Pα and πα := idPα ⊗ ( ⊗ Pα0 ). α0 6=α

(v) To finish the proof, let us show that for a pair of objects P0 , Pn and a chain ψn χn χ1 ψ1 χ2 ¯n ψ¯n−1 · · · ψ¯1−1 χ ¯1 : of morphisms P0 → P01 ← P1 → · · · → P0n ← Pn the composition χ ¯ ¯ P0 → Pn does not depend on the chain. To see this, choose φi : P → Pi as in (iv). Since χ ¯i φ¯i−1 = ψ¯i φ¯i by (iii), our composition equals φ¯n φ¯−1  0 , and we are done. Remarks. (i) It is clear that one can find P such that P>0 = 0 and each Pa is DX -flat. Since the homological dimension of DX equals dim X, for every such P the truncated DG algebra P− dim X /d(P− dim X−1 ) → P− dim X+1 → · · · → P0 satisfies the same conditions. So one can find P supported in degrees [− dim X, 0] and such that every Pa is DX -flat. (ii) The categories of these two kinds of P satisfy the above lemma as well (the proof does not change). 2.2.11. This section will not be used in the text. The reader can skip it. In case dim X = 1, there is another way to make DR a homotopy pseudo-tensor functor. It is convenient to consider DR as a functor with values in CDif f (X). Let us show that it extends naturally to a pseudo-tensor functor (see 2.2.4(i)) (2.2.11.1)

¯ → CDif f (X)∗ DR : CM(X)∗ ⊗ E

¯ is a certain DG operad homotopically equivalent to the unit k-operad Com. where E Consider the operad of projections (see 1.1.4(ii)) as an operad of groupoids with the contractible groupoid structure. Passing from groupoids to the classifying simplicial sets (see 4.1.1(ii)), we get a simplicial operad E. Let E := Norm k[E] be the ¯ := E/τ≤−1 E; this corresponding DG operad of normalized k-chains. Finally, set E ¯ is a quotient DG operad of E. Notice that both E and E are naturally resolutions of ¯ 0 is the k-linear envelope of the operad of projections (see Com. The operad E0 = E 0 1.1.4), the projection E → Com is E0I → k, ei 7→ 1, where ei are the base vectors ¯ is a length 2 resolution of Com; it is uniquely determined by of E0I := k[I]. Our E ¯ 0 . We define cii0 ∈ E ¯ −1 by the condition dcii0 = ei − ei0 . the above description of E I (I) For M ∈ CM(X), I ∈ S the complexes DR(M ) and DR(∆∗ M ) can be considered naturally as objects CDif f (X) (see 2.1.8, especially Example (i) there). (I) One has a canonical embedding DR(M ) ,→ DR(∆∗ M ) (see 2.1.7(b)) which is a quasi-isomorphism (see 2.1.10). We will define in a moment a natural morphism (I) (I) ¯ I → DR(M ) such that π(m ⊗ ei ) is the of complexes π = πM : DR(∆∗ M ) ⊗ E integral of m along the fibers of the ith projection X I → X. Such π is unique,

74

2.

GEOMETRY OF D-SCHEMES

since the de Rham differential DR(M )−1 → DR(M )0 is an injective morphism in Dif f (X). To construct π, consider first the case M = DX . We need to specify the (I) ¯ −1 → DR(DX )−1 , i.e., ∆∗(I) DX ⊗ E ¯ −1 → DX ⊗ ΘX , component DR(∆∗ DX )0 ⊗ E I I (I) of πDX . Since the differential in DR(DX ) is injective as a usual map of sheaves, we determine it from the formula dπ(m ⊗ c) = π(m ⊗ dc). Notice that DX is actually a DX -bimodule, and the above construction used only the right DX -module structure (I) on it, so πDX just defined is a morphism of left DX -modules. Now for arbitrary (I)

(I)

M one has DR(M ) = M ⊗ DR(DX ), DR(∆∗ M ) = M ⊗ DR(∆∗ DX ), and DX

(I)

DX

(I)

πM := idM ⊗ πDX .

(I)

One defines (2.2.11.1) as follows. For a ∗ operation ϕ : Li → ∆∗ M and ¯ I the operation DR(ϕ ⊗ c) : ⊗DR(Li ) → DR(M ) is ⊗`i 7→ π (I) ϕ(`i ) ⊗ c. c∈E M The compatibility with composition of operations is immediate. The rest of this section contains auxiliary technical material; we suggest the reader skip it, returning when necessary. 2.2.12. We return to arbitrary dim X. Let us show that our augmented compound tensor structure comes from a certain augmented compound strict DX operad B∗! in the way explained in 1.3.18, 1.3.19. Let us define B∗! = (B∗ , B! , < >IS,T ). For a finite set I let B∗I be the tensor product of I copies of DX considered as left D-modules. This is a left D-module on X, and the right DX -action on each copy of DX defines the right D⊗I X -action on B∗I (here D⊗I is the Ith tensor power of D considered as a sheaf of k-algebras). X X ⊗I ! ! ∗ So BI is a (DX − DX )-bimodule. Similarly, let BI be the ⊗ -product of I copies of DX considered as right DX -modules; this is a (D⊗I X − DX )-bimodule. -module, by the subsheaf OX = OX 1⊗I Note that B∗I is generated, as right D⊗I X ⊗I ⊗I ∗ on which OX ⊂ DX acts via the product map O⊗I X → OX . In fact BI = (1−|I|) (1−|I|) ⊗I ⊗I ⊗I ! OX ⊗ DX . Similarly, BI = DX ⊗ (ωX ) as left DX -modules where ωX O⊗I X

O⊗I X

is the OX -tensor power. For a map J → I the composition morphisms for our augmented operads B∗I ⊗ (B∗Ji ) → B∗J , (⊗B!Ji ) ⊗ B!I → B!J are the isomorphisms of, respectively, D⊗I X

D⊗I X

D⊗J X )-

(D⊗J X

(DX − and − DX )-bimodules, uniquely determined by the property (1−|J|) that fI ⊗ (⊗fJi ) 7→ fI ΠfJi ∈ OX ⊂ B∗J , (⊗νJi ) ⊗ νI 7→ (ΠνJi )νI ∈ ωX ⊂ B!J (1−|I|) for fI ∈ OX ⊂ B∗I , νI ∈ ωX ⊂ B!I , etc. I It remains to define < >S,T : (⊗ B∗Is ) ⊗ (⊗ B!It ) → B!S ⊗ B∗T . The arrow is, in S

T D⊗I X

DX

fact, an isomorphism, and we will construct its inverse. One has (⊗ B∗Is ) ⊗ (⊗ B!It ) S

=

O⊗S X

−1 ⊗I ⊗ (DX ωX ) ⊗ O⊗I O⊗I X X

⊗T ωX

where the

O⊗I X -module

structure on

T D⊗I X ⊗S OX ,

⊗T ωX

⊗S ⊗I ⊗T comes from the πS and πT product morphisms O⊗I X → OX , OX → OX . This is −|I| ⊗S ⊗T ⊗StT ⊗S ⊗T a (DX − DX )-bimodule containing an OX -submodule OX ⊗ ωX ⊗ ωX O⊗I X

O⊗I X

2.2. THE COMPOUND TENSOR STRUCTURE −|S|+1

= ωX

75

⊗T . On the other hand, B!S ⊗ B∗T is a (D⊗S X , DX )-bimodule freely generOX

−|S|+1

ated by its O⊗StT -module ωX X

. Therefore we get a morphism β : B!S ⊗ B∗T → OX

−1 ⊗I ⊗T (⊗ B∗Is ) ⊗ (⊗ B!It ). Notice that the action of ΘX on O⊗S ⊗ (DX ωX ) ⊗ ωX X T D⊗I X

S

O⊗I X

O⊗I X

(D⊗S X

by transport of structure coincides with the adjoint action for the − D⊗T X )⊗T bimodule structure of the diagonal embedded ΘX ,→ D⊗S X , DX . In particular, the −|S|+1 latter action of ΘX on ωX ⊂ (⊗ B∗Is ) ⊗ (⊗ B!It ) coincides with the evident T D⊗I X

S

action; hence β factors through

B!S

⊗

DX

B∗T

→ (⊗ B∗Is ) ⊗ (⊗ B!It ). A computation T D⊗I X

S

on the level of symbols shows that this arrow is an isomorphism; our < >IS,T is its inverse. The axioms of augmented compound operad are immediate. Let us check that the augmented compound tensor structure defined on M(X) by B∗! (see 1.3.18 and 1.3.19) coincides with that defined in 2.2.1 – 2.2.7. It is clear that ⊗!I Li = (⊗Li ) ⊗ B!I . The sheaf-theoretic restriction of the DX I -module D⊗I X

(I) ∆∗ M

⊗I to the diagonal is a D⊗I X -module. There is a unique isomorphism of DX modules ∼

(I)

M ⊗ B∗I −→ ∆(I)· ∆∗ M

(2.2.12.1)

DX

which induces the identity isomorphism between the subsheaves M = M ⊗ 1⊗I and (I) M = ∆(I)! ∆∗ M . This yields the isomorphism (2.2.12.2)

∼

HomD⊗I (⊗Li , M ⊗ B∗I ) −→ PI∗ ({Li }, M ). X

DX

We leave it to the reader to identify the maps <

>IS,T .

2.2.13. We finish this section with several remarks about the ∗ pseudo-tensor structure. The operad B∗I from 2.2.12 satisfies conditions (i) and (ii) of 1.2.3. Therefore, for any coherent D-modules Li and an arbitrary D-module M , the object PI ({Li }, M ) ∈ M(X) exists. Explicitly, one has (2.2.13.1)

PI ({Li }, M ) = HomD⊗I (⊗ Li , M ⊗ B∗Ie). X

k

DX

(OX , D⊗I X )-bimodules

Consider the morphism of B∗I → B∗Ie, b 7→ 1· ⊗ b (recall that Ie = { · } t I). The corresponding morphism DX ⊗ B∗I → B∗Ie of (DX , D⊗I X )OX

bimodules is an isomorphism. It yields the identification M ⊗ B∗I = M ⊗ B∗Ie; OX

DX

hence also (2.2.13.2)

PI ({Li }, M ) = HomD⊗I (⊗ Li , M ⊗ B∗I ). X

k

OX

Note that the right DX -action on M ⊗ B∗Ie (used to define the D-module strucOX

ture on the Hom sheaf) identifies with the right D-module structure on M ⊗ B∗I OX

that comes from the right DX -module structure on M and the left DX -module structure on B∗I (see 2.1.1).

76

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Remark. The functor P∗I obviously commutes with inverse images for ´etale morphisms; i.e., it is defined on the sheaf of categories M(X´et ). It also commutes with direct images for closed embeddings (as follows immediately from 2.2.5). 2.2.14. If we deal with induced DX -modules, i.e., Li = FiD , M = GD , where Fi are coherent O-modules, then the corresponding D-module P∗I can be described as follows. Consider the sheaf of differential I-operations DiffI ({Fi }, G)X .33 This is an object of Dif f (X) by Example (i) in 2.1.8, so we have the corresponding ˜ D-module DiffI ({Fi }, G)D = DiffI ({Fi }, G)XD . Now the canonical I-operation ∗ ∈ DiffI˜({DiffI ({Fi }, G)X , Fi }, G) = PI˜ ({DiffI ({Fi }, G)D , FiD }, GD ) yields an isomorphism34 ∼

DiffI ({Fi }, G)D −→ P∗I ({FiD }, GD ).

(2.2.14.1)

2.2.15. According to 2.2.13, for a DX -coherent L and any M ∈ M(X) we have the Hom DX -module Hom∗ (L, M ) = HomDX (L, M ⊗ DX ) ∈ M(X) equipped OX

with a universal “evaluation” ∗ pairing ε ∈ P2∗ ({Hom∗ (L, M ), L}, M ). If K is another coherent D-module, then the composition ∗ pairing (2.2.15.1)

c ∈ P2∗ ({Hom∗ (L, M ), Hom∗ (K, L)}, Hom∗ (K, M ))

is defined; the composition is associative and compatible with the evaluation (see 1.2.2). Here is a more detailed description. Recall that M ⊗ DX := M ⊗ DX carries OX

two commuting right DX -actions: the first one comes from the right DX -action on DX ; the second one comes from the right DX -module structure on M and the left DX -module structure on DX . Since35 M ⊗ DX = M ⊗ B∗{1,2} , one has a canonical DX

automorphism of M ⊗ DX interchanging the two D-module structures. Now a section of Hom∗ (L, M ) is a morphism of DX -modules L → M ⊗ DX with respect to the first D-module structure on M ⊗ DX ; the D-module structure on Hom∗ (L, M ) comes from the second D-module structure on M ⊗ DX . Here ε is just the usual evaluation morphism Hom(L, M ⊗ DX ) ⊗ L → M ⊗ DX . To describe k

c explicitly, it is convenient to understand HomDX (L, M ⊗ DX ) in a way different from that used above: the morphisms are understood with respect to the second Dmodule structure on M ⊗ DX , and the action of DX on HomDX (L, M ⊗ DX ) comes from the first one. Let us identify Hom∗ (L, M ) with HomDX (L, M ⊗ DX ) using the above symmetry of M ⊗ DX interchanging the D-module structures. Then the ∗ pairing c becomes the usual composition Hom(L, M ⊗ DX ) ⊗ Hom(K, L ⊗ DX ) → Hom(K, (M ⊗ DX ) ⊗ DX ) k

= Hom(K, M ⊗ DX ) ⊗ DX . 33 Which is the X-sheafified version of the vector space of differential I-operations from 2.2.4(i). 34 To check that our arrow is an isomorphism, one reduces (using the left exactness of our functors) to the case Fi = OX where the statement is obvious. 35 See 2.2.13 for notation.

2.2. THE COMPOUND TENSOR STRUCTURE

77

2.2.16. We see that for any coherent M ∈ M(X) the D-module End∗ (M ) = Hom∗ (M, M ) is an associative algebra in the ∗ sense acting on M . This action is universal: for any associative aglebra A in M(X)∗ an A-module structure on M is the same as a morphism of associative algebras A → End∗ M . Set M ◦ := Hom∗ (M, ωX ) = HomDX (M, ωX ⊗ DX ); this is again a coherent OX

module. We have a canonical pairing h i ∈ P2∗ ({M ◦ , M }, ωX ). If M is a vector DX -bundle (see 2.1.5), then h i is non-degenerate (see 1.4.2); ∼ in particular, M ⊗ M ◦ −→ End∗ M . Remarks. (i) Notice that ωX ⊗ DX considered as a right DX -module with OX

respect to the second D-module structure (see 2.2.15) equals (DX )r where we consider DX as a left D-module. So, by 2.2.15, one has M ◦ = HomDX (M ` , DX ). (ii) Let F be a vector bundle. It yields the corresponding left and right induced DX -modules D F := DX ⊗ F ∈ M` (X), FD := F ⊗ DX ∈ M(X). We see that36 OX

OX

(2.2.16.1)

◦

∗

(FD ) = (F ⊗ ωX )D ,

(D F )◦ = (F ∗ )D .

2.2.17. Global duality. Suppose X is proper of dimension n. Let V , V ◦ be mutually dual vector DX -bundles. The canonical pairing ∈ P2∗ ({V ◦ , V }, ωX ) together with the trace morphism tr : RΓDR (X, ωX ) → k[−n] yields a canonical pairing (2.2.17.1)

RΓDR (X, V ◦ ) ⊗ RΓDR (X, V ) → k[−n].

Lemma. This pairing is non-degenerate; i.e., RΓDR (X, V ◦ ) is dual to the complex RΓDR (X, V )[n]. Proof. Using appropriate resolutions, one is reduced to the induced situation where the above duality is the usual Serre duality.  If X is not proper, the above lemma holds if one replaces one of the cohomology groups by cohomology with compact supports. We do not need the statement in i (X, V ) = 0 whole generality; consider just the case when X is affine. Then HDR 0 0 for i 6= 0, and HDR (X, V ) = Γ(X, h(V )). Now the morphism HDR (X, V ) → Hom(V ◦ , ωX ), defined by the canonical pairing, is an isomorphism.37 One can rewrite it as a canonical isomorphism (see 2.1.12 for notation)38 (2.2.17.2)

∼

0 (V ◦ )const −→ HDR (X, V )∗ ⊗ ωX .

Remark. We can consider families of vector DX -bundles; i.e., locally projective F ⊗ DX -modules V of finite rank where F is any commutative k-algebra. For such V its dual V ◦ is HomF ⊗DX (V ` , F ⊗ DX ). The above statements (together with the proofs) remain valid with the following modifications: If X is projective, then RΓDR (X, V ) is a perfect F -complex, and the canonical pairing identifies RΓDR (X, V ◦ ) with the F -complex dual to RΓDR (X, V )[dim X]. If X is affine, then i 0 HDR (X, V ) = 0 for i 6= 0, HDR (X, V ) = Γ(X, h(V )) is a projective F -module, and 0 0 (2.2.17.2) holds with HDR (X, V )∗ := HomF (HDR (X, V ), F ). the second equality we identified left and right D-modules in the usual way. suffices to check this statement for V = DX where it is obvious. 38 Here H 0 (X, V )∗ is considered as a profinite-dimensional vector space. DR 36 In

37 It

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2.2.18. According to 1.3.11, the duality functor ◦ : M(X)◦coh → M(X)coh ex∗ tends canonically to a pseudo-tensor functor ◦ : M(X)◦! coh → M(X)coh . It induces an equivalence between the full pseudo-tensor subcategories of vector DX -bundles. ! On M(X)◦! coh we have a standard augmentation functor h (M ) := Hom(M, ωX ) ◦ ! and a canonical morphism h(M ) → h (M ) of augmentation functors; see (1.3.11.3). If M is a vector DX -bundle, then this is an isomorphism.39 ◦ Remark. If Fi , G are vector bundles on X, then the duality Hom(G◦D , ⊗! FiD ) ∗ −→ PI ({FiD }, GD ) is the composition of the following identifications (see 2.2.12) Hom( D G∗ , ⊗( D Fi∗ )) = HomOX (G∗ , B∗I ⊗ (⊗ Fi∗ )) = HomOX I (Fi , G ⊗ B∗I ) = ∼

(I) Hom(FiD , (∆· G)D )

=

OX

O⊗I X (I) Hom(FiD , ∆∗ GD ).

2.2.19. A remark about the augmentation functor. Take I, J ∈ S, the families of objects {Li }, {Aj }, M and consider the composition morphism for I ,→ I t J in our augmented pseudo-tensor structure P ∗ItJ ({Li , Aj }, M ) ⊗ ⊗ h(Aj ) → J

P ∗I ({Li }, M ). We can rewrite it as an embedding40

∗ hIJ : PItJ ({Li , Aj }, M ) ,→ Hom (⊗ h(Aj ), P ∗I ({Li }, M )) .

(2.2.19.1)

J

Lemma. The image of hIJ is the subspace of those ϕ: ⊗ h(Aj ) → PI∗ ({Li }, M ) J

aj )(⊗`i ), for which the map ϕ: e (⊗Li ) ⊗ (⊗Aj ) → M ⊗ B∗I , (⊗`i ) ⊗ (⊗aj ) 7→ ϕ(⊗¯ DX

is a differential operator with respect to any of the variables Aj (see Example (iii) in 2.1.8). In other words, ϕ satisfies the following condition (here I is the kernel of ⊗ItJ OX → OX ): (∗) For any local section s ∈ (⊗Li ) ⊗ (⊗Aj ) one has ϕ(I e N s) = 0 for N  0. ∗ Proof. Let T r: M ⊗ B∗ItJ → M ⊗ B∗ItJ ⊗ O⊗J X = M ⊗ BI be the trace DX

DX

along J-variables. The morphism M ⊗

DX

B∗ItJ

DX

D⊗J X

→

⊗J HomD⊗I (D⊗I X ⊗OX , M X

⊗ B∗I ) of

DX

D⊗ItJ -modules which assigns to b ∈ M ⊗ B∗ItJ the map α 7→ T r(bα) is injective. X DX

⊗J N ∗ ⊂ Its image is the subspace of those ψ: D⊗I X ⊗ OX → M ⊗ BI that kill some I DX

⊗ItJ ⊗I ∗ OX ⊂ DX ⊗ O⊗J X . This morphism maps PItJ ({Li , Aj }, M ) injectively to ⊗I ∗ HomD⊗ItJ ((⊗Li ) ⊗ (⊗Aj ), HomD⊗I (DX ⊗ O⊗J X , M ⊗ BI )) X

DX

X

= HomD⊗I ((⊗Li ) ⊗ (⊗h(Aj )), M ⊗ X

DX

B∗I )

= Hom(⊗h(Aj ), P ∗I ({Li }, M )).

The image coincides with the subspace defined by condition (∗). Since this map coincides with hI,J , we are done.  2.2.20. The following technical lemma will be of use. Let Li , M be D-modules on X, ϕ ∈ PI∗ ({Li }, M ), and let x ∈ X be a closed point. Set Ux := X r {x}. We use the terminology of 2.1.13. 39 It

suffices to check this statement for M = DX where it is obvious. h is non-degenerate by 2.2.8.

40 Our

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79

Lemma. (i) The polylinear map h(ϕ) : ⊗h(Li )x → h(M )x is Ξx -continuous with respect to each variable. (ii) Assume that the restriction of each Li to Ux is a countably generated Dmodule. Then h(ϕ) is Ξx -continuous. (iii) Assume that for certain i0 ∈ I every Li , but, possibly Li0 , is a coherent DX -module. Then for each Mξ ∈ Ξx (M ) there exists Li0 ξ ∈ Ξx (Li0 ) such that (I)

ϕ(Li0 ξ  (  Li )) ⊂ ∆∗ Mξ .

(2.2.20.1)

i6=i0

Proof. (i) Pick i0 ∈ I and l¯i ∈ h(Li )x for i 6= i0 . We have to show that the map h(Li0 )x → h(M )x , ¯li0 7→ h(ϕ)(¯li0 ⊗ (⊗¯li )), is Ξx -continuous. This is clear since this map comes from a morphism of D-modules Li0 → M that comes from ϕ and the ¯li ’s by the augmentation functor structure on h. (ii) An induction by |I| shows (use the augmentation functor structure on h) that to prove continuity of h(ϕ), it suffices to find for every Mξ ∈ Ξx (M ) some Liξ ∈ Ξx (Li ) such that h(ϕ)(⊗Liξ ) ⊂ Mξ . Shrinking X if necessary, we can assume that X is affine and we have functions ta , a = 1, . . . , n, on it which are local coordinates at x and have no other common zero. Let li1 , li2 , . . . be sections of Li that generate the restriction of Li to Ux as a DUx -module. The quotient (I) (I) ∆∗ M/∆∗ Mξ = ∆∗ (M/Mξ ) is a DX I -module supported at ∆(I) (x), so every its section is killed by sufficiently high power of each ta lifted from either copy of X. Therefore, using induction by b = 1, 2, . . . , we can choose integers n(a, b, i) ≥ 0 (I) (I) n(a,b ,i) such that h(ϕ)( ta i libi ) ∈ ∆∗ Mξ ⊂ ∆∗ M for every 1 ≤ bi ≤ b. Our Liξ is I

n(a,b,i)

lib }. the D-submodule of Li generated by {ta (iii) Choose for i 6= i0 a finite set {lij } of local sections of Li at x that generates (I) Li . Now take for Li0 ξ the intersection of the preimages of ∆∗ Mξ by all maps (I) Li0 → ∆∗ M , l 7→ ϕ(l  (lij )).  We see that in situation (ii) the morphism hI (ϕ): ⊗h(Li )x → h(M )x exˆ I (ϕ)x : ⊗ ˆ x (Li ) → h ˆ x (M ). Here ⊗ ˆ x (Li ) := bh bh tends by continuity to the map h lim ⊗h (L /L ). x i i ξi ←− ˜ x from 2.1.15 instead of plain topological If we decide to play with objects h vector spaces, then the countability condition of (ii) becomes irrelevant. 2.3. DX -schemes A commutative algebra in M! (X) can be considered as a “coordinate-free” version of a system of non-linear differential equations (just as D-modules provide a coordinate-free language for systems of linear differential equations). In this section we deal with basic algebro-geometric properties of these objects. The bulk of the literature on foundational subjects of the geometric theory of non-linear differential equations is huge (take [G] and [V] to estimate the span). The Euler-Lagrange equations (the classical calculus of variations) are treated in the nice review articles [DF], [Z]. We consider jet schemes in 2.3.2–2.3.3, quasi-coherent OY [DX ]-modules on a DX -scheme Y in 2.3.5, and prove in 2.3.6–2.3.9 that for affine Y projectivity of such modules is a local property, which is a D-version of a theorem of Raynaud-Gruson [RG]. The ∗ operations between quasi-coherent OY [DX ]-modules are considered

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in 2.3.11–2.3.12, the notion of smoothness for DX -schemes in 2.3.13–2.3.16. We show that smooth DX do not necessary admit local coordinate systems (see 2.3.17– 2.3.18), and discuss the notion of vector DX -scheme in 2.3.19. The setting of the calculus of variations is briefly described in 2.3.20. 2.3.1. A commutative DX -algebra is a commutative unital OX -algebra equipped with an integrable connection along X. Unless stated explicitly otherwise, our DX -algebras are assumed to be OX -quasi-coherent. A DX -scheme is an X-scheme equipped with an integrable connection along X. Denote by ComuD (X), SchD (X) the corresponding categories. We have a fully faithful functor Spec: ComuD (X)◦ ,→ SchD (X); its essential image is the category Af f SchD (X) of DX -schemes affine over X. Replacing schemes by algebraic spaces, we get the notion of an algebraic DX space. For a DX -algebra R` and an algebraic DX -space Y a morphism of algebraic spaces Spec R` → Y is sometimes referred to as a DX -algebra R` -point of Y. Remark. Let xa be a coordinate system on X. Assume we have a system of non-linear differential equations Pα (ui , ∂a ui , . . . ) = 0 where Pα are polynomial functions with coefficients in functions on X. It yields a DX -algebra A` defined as the quotient of the free DX -algebra Sym(⊕DX · ui ) modulo the relations Pα . A solution of our system with values in a (not necessarily quasi-coherent) DX algebra R` is a collection of φi ∈ R` such that Pα (φi , ∂a φi , . . . ) = 0. This is the same as a morphism of DX -algebras A → R, ui 7→ φi , i.e., a DX -algebra R` point of Spec A` . Therefore DX -algebras provide a “coordinate-free” language for non-linear differential equations. Set Comu! (X) := Comu(M! (X)) (we use the notation of 1.4.6). The equivalence ∼ −1 of tensor categories M! (X) −→ M` (X), A 7→ A` = AωX , yields the equivalence (2.3.1.1)

∼

Comu! (X) −→ ComuD (X).

2.3.2. Jet schemes. Denote by Comu(X) the category of commutative OX algebras quasi-coherent as OX -modules and by Sch(X) the category of X-schemes. We have the obvious forgetting of connection functors (2.3.2.1)

o: ComuD (X) → Comu(X),

SchD (X) → Sch(X).

They admit the left, resp. right, adjoint functors (2.3.2.2)

Comu(X) → ComuD (X),

Sch(X) → SchD (X).

We denote them both by J. For R ∈ Comu(X), JR is generated by R.Explicitly, it  is the DX -algebra quotient of the symmetric algebra Sym· DX ⊗ R modulo the  OX  2 ideal generated by elements ∂(1⊗r1 ·1⊗r2 −1⊗r1 r2 ) ∈ Sym DX ⊗ R +DX ⊗ R OX

OX

and ∂(1 ⊗ 1R − 1) ∈ DX ⊗ R + OX where ri ∈ R, ∂ ∈ DX , 1R is the unit of R. This construction is compatible with localization. The functor J: Sch(X) → SchD (X) is Spec R 7→ Spec JR for affine schemes; for arbitrary schemes use gluing. Note that for any Z ∈ Sch(X) the canonical projection JZ → Z induces a bijection ∼ (JZ)D (X) −→ Z(X) where (JZ)D (X) is the set of all horizontal sections X → Z.

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81

2.3.3. Here is a different description of JR. Denote by I the kernel of the ⊗2 n product map O⊗2 X = OX ⊗ OX → OX ; for n ≥ 1 set OX (n) := OX /I . These OX (n) k

form a projective system of algebras; there are two obvious morphisms OX ⇒ OX (n) . Therefore any OX -algebra B yields two OX (n) -algebras: B ⊗ OX (n) and OX

OX (n) ⊗ B. OX

Lemma. For any R, B ∈ Comu(X) one has   (2.3.3.1) Hom(JR, B) = lim ←− Hom OX (n) ⊗ R, B ⊗ OX (n) OX

OX

where the left and right Hom mean, respectively, the morphisms of OX - and OX (n) algebras. Proof. Our statement is X-local, so we can assume that X is affine, X = Spec C. Thus OX -algebras are the same as C-algebras. For a C-algebra B denote by JB n the kernel of the map B ⊗ C → B, b ⊗ c 7→ bc. Set TB := lim ←− B ⊗ C/JB . Let us k

consider TB as a C-algebra via the morphism C → TB, c 7→ 1 ⊗ c. The projection TB → C, b ⊗ c 7→ bc, is a morphism of C-algebras. Note that TB is a DX -algebra in the obvious way (the elments b ⊗ 1 ∈ TB are horizontal). It is easy to see that the functor Comu(X) → ComuD (X), B 7→ TB, is right adjoint to the functor o: ComuD (X) → Comu(X). Therefore we have Hom(oJR, B) = Hom(JR, TB) = Hom(R, oTB). The latter set coincides with ← lim − Hom in the statement of the lemma; we are done.  So for any X-scheme Y the k-points of JY are the same as pairs (x, γ) where x ∈ X and γ is a section of Y on the formal neighborhood of x. We call JY the jet scheme of Y . Remark. The above constructions are compatible with ´etale localization, so we may replace schemes by algebraic DX -spaces. 2.3.4. For A` ∈ ComuD (X) we denote by A` [DX ] the ring of differential operators with coefficients in A` . By definiton, this is a sheaf of associative algebras equipped with a morphism of algebras A` → A` [DX ] and that of Lie algebras ΘX → A` [DX ] which satisfy the relations τ · a − a · τ = τ (a), f · τ = f τ for a ∈ A` , f ∈ OX ⊂ A` , τ ∈ ΘX , and universal with respect to these data in the obvious sense. The embedding of ΘX extends to the morphism of associative algebras DX → A` [DX ]. One checks easily that the morphism of sheaves A` ⊗ DX → A` [DX ], a ⊗ ∂ 7→ a · ∂ is an isomorphism. OX

For any algebraic DX -space Y we have the corresponding sheaf OY [DX ] on Y´et .

2.3.5. An A` -module in the tensor category M` (X) is the same as a left A [DX ]-module quasi-coherent as an OX -module. Similarly, an A-module in M! (X) is the same as a right A` [DX ]-module quasi-coherent as an OX -module. We denote by M` (X, A` ), M(X, A` ) = Mr (X, A` ) the categories of left, resp. right, A` [DX ]modules. These are tensor categories (canonically identified) with unit object equal to A` , resp· A. Every morphism of commutative DX -algebras f : A` → B ` yields an adjoint pair of functors f ∗ : M(X, A` ) → M(X, B ` ), f· : M(X, B ` ) → M(X, A` ) where `

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f ∗ M := B ` ⊗ M , f· N is N considered as an A` [DX ]-module. Notice that f ∗ is a A`

tensor functor. The faithfully flat descent works in any abelian tensor category, so A` [DX ]modules are local objects with respect to the flat topology.41 So for any algebraic DX -space Y we have the corresponding categories M` (Y), Mr (Y) = M(Y). Their objects are sheaves of left, resp. right, OY [DX ]-modules which are quasi-coherent as OY -modules. The categories M` (U), M(U) for U ∈ Y´et form sheaves of categories M` (Y´et ) on Y´et . One has M` (Spec A` ) = M` (X, A), M(Spec A` ) = M(X, A` ). Most of the facts mentioned in 2.1 render easily to the OY [DX ]-modules setting. E.g., if G, L are left OY [DX ]-modules and M is right one, then G ⊗ L is a left OY

OY [DX ]-module in a natural way, and L ⊗ M is a right one. We have the standard OY

equivalence M` (Y) → Mr (Y), L 7→ LωX . Any N ∈ M(Y) admits a canonical finite left resolution (DR(N ))D whose terms are induced OY [DX ]-modules. 2.3.6. Theorem. Assume that Y is a DX -scheme which is affine (as a kscheme). Then projectivity of an OY [DX ]-module is a local property for the Zariski or ´etale topology. Proof. It is found in 2.3.7–2.3.9. 2.3.7. Let us first prove the theorem for a finitely generated OY [DX ]-module P (which is sufficient for the most of applications). Suppose that P is projective over some ´etale covering of Y. We need to show that P is projective. Take N ∈ M(Y). Consider the sheaf Hom(P, N ) on Y´et . It depends on N in an exact way, so projectivity of P follows if we prove that H i (Y´et , Hom(P, N )) = 0 for i > 0. This is true if N is an induced module, N = F ⊗ DX . Indeed, in this situOY

ation Hom(P, N ) admits a structure of a quasi-coherent OY -module. Namely, the OY -module structure on Hom(P, N ) comes from an OY -action on N inherited from F . Quasi-coherence is an ´etale local property, so it suffices to check it under the assumption that P is projective. Then Hom(P, N ) is isomorphic to a direct summand of a sum of several copies of N , which is obviously quasi-coherent. To prove acyclicity of Hom(P, N ) for arbitrary N , use the fact that N admits a finite left resolution by induced modules (see 2.1.9 and 2.3.5). We are done. 2.3.8. Without the finite generatedness assumption the theorem is nontrivial even if X is a point. In this case it was conjectured by Grothendieck (see Remark 9.5.8 from [Gr1]) and proved by Raynaud and Gruson (even for the fpqc topology); see [RG] 3.1.4. We will use the method of [RG]. It is based on the following fact: a flat module M over a ring R is projective if and only if M has the following properties: (i) M is a direct sum of countably generated modules; (ii) M is a Mittag-Leffler module (the implication projectivity ⇒(i) is due to Kaplansky [Ka]; the fact that (i) and (ii) imply projectivity is proved in [RG], p. 74). A flat module M is said to be Mittag-Leffler if for some (or for every) representation of M as a direct limit of finitely generated projective modules Pi (i belongs to a directed ordered set I) the 41 f is faithfully flat as a morphism of commutative algebras in the tensor category M` (X) if and only if it is faithfully flat as a morphism of plain OX -algebras.

2.3. DX -SCHEMES

83

projective system (Pi∗ ) satisfies the Mittag-Leffler condition (i.e., for every i ∈ I there exists j ≥ i such that Im(Pj∗ → Pi∗ ) = Im(Pk∗ → Pi∗ ) for all k ≥ j). Here Pi∗ := HomR (Pi , R). Now the theorem is easily reduced to the following lemma. 2.3.9. Lemma. Assume that X is affine and Y is an affine DX -scheme. Then a flat OY [DX ]-module M is globally flat; i.e., Γ(Y, M ) is flat over Γ(Y, OY [DX ]). Proof. M has a finite left resolution consisting of globally flat OY [DX ]-modules (indeed, M is OY -flat, so the canonical resolution (DR(M ))D has the required property). Therefore it suffices to show that if there is an exact sequence 0 → F1 → F0 → M → 0 with F0 , F1 globally flat and M flat, then M is globally flat; i.e., for every left OY [DX ]-module L the morphism Γ(Y, F1 ) ⊗Γ(Y,OY [DX ]) Γ(Y, L) → Γ(Y, F0 ) ⊗Γ(Y,OY [DX ]) Γ(Y, L) is injective. The map Γ(Y, Fi ) ⊗Γ(Y,OY [DX ]) Γ(Y, L) → Γ(Y, Fi ⊗OY [DX ] L) is an isomorphism (represent Fi as a direct limit of finitely generated free OY [DX ]-modules). Finally, the morphism F1 ⊗OY [DX ] L → F0 ⊗OY [DX ] L is injective because M is flat.  2.3.10. We see that for any algebraic DX -space Y there is a notion of Y-locally projective OY [DX ]-modules. Notice that for OY [DX ]-modules on an affine Y the property of being finitely generated is obviously ´etale local. So we know what Ylocally finitely generated OY [DX ]-modules on any algebraic DX -space Y are. We call a Y-locally finitely generated and projective OY [DX ]-module a vector DX -bundle on Y. 2.3.11. According to 1.4.6 (and 2.2) for a commutative DX -algebra A` the category M(X, A` ) carries a canonical augmented compound tensor structure. As in 1.4.6 for Li , M ∈ M(X, A` ) we denote the corresponding space of A` -polylinear ∗ operations by PA∗ I ({Li }, M ). In fact, this structure comes from an augmented compound strict A` [DX ]∗! operad B∗! A . We leave the explicit construction of BA to the reader (repeat 2.2.12 ` replacing DX by A [DX ] everywhere). The discussion in 2.2.13–2.2.18 remains literally valid for A` [DX ]-modules if we replace the words “coherent DX -module” by “finitely presented A` [DX ]-module”. The lemma from 2.2.19 also remains valid. According to Remark (i) of 1.4.6 for every morphism of commutative DX algebras f : B ` → A` the corresponding tensor functor f ∗ from 2.3.5 extends to a compound tensor functor (base change) (2.3.11.1)

f ∗ : M(X, A` )∗! → M(X, B ` )∗! .

We also have a forgetful pseudo-tensor functor (2.3.11.2)

f· : M(X, B ` )∗ → M(X, A` )∗

right adjoint to f ∗ considered as a ∗ pseudo-tensor functor (see Remark (ii) of 1.4.6). Lemma. M(X, A` )∗! has a local nature with respect to the flat topology of Spec A` . Proof. We already discussed the flat descent of A` [DX ]-modules in 2.3.5, so it remains to check that the A-polylinear ∗ operations satisfy the flat descent property. For Li , M ∈ M(X, A` ) the above adjunction identifies the descent data

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with operations ψ ∈ PA∗ I ({Li }, B ` ⊗ M ) whose composition with the two arrows A` B ` ⊗ M −→ −→B ` ⊗ B ` ⊗ M , b ⊗ m 7→ b ⊗ 1 ⊗ m, 1 ⊗ b ⊗ m, coincide. Now the exact A`

A`

A`

sequence 0 → M → B ` ⊗ M → B ` ⊗ B ` ⊗ M (the latter arrow is the difference of A`

A`

A`

the above two standard arrows) together with the left exactness of PA∗ implies the desired descent property.  2.3.12. For Li , M as above a ∗ operation ϕ ∈ PI∗ ({Li }, M ) is A-polydifferential if for every finitely generated A` [DX ]-submodules L0i ⊂ Li the restriction ϕ|{L0i } ∈ PI∗ ({L0i }, M ) is an A-polydifferential ∗ operation of finite order (see 1.4.8). The composition of A-polydifferential operations is A-polydifferential, so they define another pseudo-tensor structure on M(X, A` ). Lemma. A-polydifferential ∗ operations have a local nature with respect to the ´etale topology of Spec A` . Proof. The case of arbitrary A-polydifferential operations reduces immediately to that of A-polydifferential operations of finite order, and then to that of A(n) (n) polylinear ones (replace Li by Li as in 1.4.8 and notice that the Li have an ´etale local nature). Then use the previous lemma.  Therefore for any algebraic DX -space Y we have the compound tensor category M(Y)∗! and the sheaf of compound tensor categories M(Y´et )∗! on Y´et , the pull-back functors f ∗ are compound tensor functors, etc. The polydifferential ∗ operations also make sense in this setting. Remarks. (i) One has the following immediate analog of 2.2.13. Let {Li } be a finite collection of finitely presented A` [DX ]-modules, M any A` [DX ]-module. Then the A` [DX ]-module P∗A I ({Li }, M ) is well defined. The functor P∗A I is compatible with flat pull-backs. So the functor P∗I makes sense on any algebraic DX -space Y. (ii) In particular, for every finitely presented OY [DX ]-module L its dual L◦ := Hom∗ (L, OY ωX ) is well defined. In particular, if the OY [DX ]-module of differentials ΩY = ΩY/X is finitely presented,42 then ΘY := Ω◦Y is well defined. (iii) If L is a vector DX -bundle on Y, then L◦ is also a vector DX -bundle on Y, and the canonical pairing is non-degenerate (see 1.4.2). 2.3.13. We are going to discuss the notion of smoothness for algebraic DX spaces. First let us recall the definition of formally smooth algebras. Let M be any abelian tensor k-category with unit object (so ⊗ is right exact). A commutative algebra A in M is formally smooth if for any (commutative) algebra R and an ideal I ⊂ R such that I 2 = 0, any morphism f : A → R/I may be lifted to a morphism f˜: A → R. It suffices to check this in the case when f is an e := A × R, isomorphism; i.e., R/I = A, f = id (replace arbitrary R, I, f by R R/I

e → A) = I, f = id). Note that the symmetric algebra Sym V is formally Ie := Ker(R smooth if and only if V is a projective object of M. Assume that M has many projective objects. We may represent A as a quotient B/J, B = Sym V for a projective V . Then A is formally smooth iff the projection B/J 2 → B/J = A has a section σ: A → B/J 2 (if σ exists, then for any R, I, 42 This

happens if Y is (locally) finitely DX -presented or if ΩY is a vector DX -bundle.

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85

f as above f lifts to fˆ: B/J 2 → R and one may set f˜ = f σ). Note that the sections σ correspond bijectively to derivations ∂: B → J/J 2 such that ∂(x) = xmod J 2 for x ∈ J (namely, σ(bmod J) = b − ∂(b)). We may consider instead of ∂ the corresponding morphism of B-modules ϕ: ΩB /J · ΩB → J/J 2 ; the condition on ∂ means that ϕν = idJ/J 2 where ν: J/J 2 → ΩB /J · ΩB is the usual map ν(x mod J 2 ) = dx mod J · ΩB . Therefore A = B/J is formally smooth iff ν is a split (−1) injection. Set ΩA := Ker ν. Since Coker ν = ΩA and ΩB /J · ΩB is a projective A-module, we see (as in [Gr1] 9.5.7) that (−1) A is formally smooth iff ΩA = 0 and ΩA is a projective A-module. (−1)

Remark. The above modules ΩA , ΩA are cohomology of the cotangent complex L ΩA (see [Gr1], [Il], or [H] §7) which is the left derived functor of the functor A 7→ ΩA . To define it, one extends the category of commutative algebras to that of commutative unital DG algebras where the standard homotopy theory formalism is available (see, e.g., [H]). Now for a cofibrant DG algebra R its cotangent complex coincides with the R-module ΩR considered as the object of the derived category D(R) of R-modules. For arbitrary DG algebra R one considers its cofi˜ → R and defines L ΩR as the image of Ω ˜ under the equivalence brant resolution R R ∼ ˜ ˜ If A is a plain algebra, then D(R) −→ D(R); it does not depend on the choice of R. (−1) L ΩA is acyclic in degrees > 0 and H 0 (L ΩA ) = ΩA , H −1 (L ΩA ) = ΩA . 2.3.14. Let us return to our tensor category M` (X) assuming for the moment that X is affine. Let A` be a commutative DX -algebra. One check immediately that ΩA considered as a mere A` -module (we forget about the DX -action) coincides with the module of relative differentials ΩA` /OX . The same is true for the cotangent complex. Proposition. (i) A` is formally smooth iff it is formally smooth as an OX algebra and ΩA` is a projective A` [DX ]-module. (ii) A` is formally smooth as an OX -algebra iff it is formally smooth as a kalgebra. Proof. (i) Use the remark above, together with the fact that every projective A` [DX ]-module is a projective A` -module. (ii) Follows from the next lemma (the morphism ΩA` → ΩX ⊗ A` left inverse OX

to µ corresponds to the canonical derivation ∇: A` → ΩX ⊗ A` ). OX

Lemma. If A is an OX -algebra, then A is a formally smooth OX -algebra iff A is formally smooth as a k-algebra and the morphism of A-modules µ: ΩX ⊗ A → ΩA OX

is a split injection. Proof of Lemma. Since X is smooth over k, the standard long exact sequence (−1) (−1) reduces to 0 → ΩA → ΩA/OX → ΩX ⊗ A → ΩA → ΩA/OX → 0; this implies the OX

lemma. Here is another ad hoc proof. Let us represent A as B/J, B = Sym V , V a free OX -module. The formal smoothness of A over OX means that the morphism of A-modules ν: J/J 2 → ΩB/O /J ·ΩB/O admits a left inverse. The formal smoothness over k means that the morphism ν˜: J/J 2 → ΩB/k /J · ΩB/k admits a left inverse. It remains to apply the following easy sublemma to P = J/J 2 , Q = ΩB/k /J · ΩB/k , R = ΩX ⊗ A. OX

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g

Easy Sublemma. Suppose we have morphisms P −→Q←−R where g is a split injection. Then P → Q/Im g is a split injection iff both f : P → Q and R → Q/Imf are such. Proof of Easy Sublemma. Write Q as R⊕S, so g(r) = (r, 0), f (p) = (ϕ(p), ψ(p)). The morphism R → Q/f (P ) is a split injection iff there is a γ: S → R such that ϕ = γψ, i.e., when ϕ ∈ Hom(S, R)ψ; the morphism f is a split injection iff idP ∈ Hom(R, P )ϕ + Hom(S, P )ψ. We need to show that idP ∈ Hom(S, P )ψ if and only if idP ∈ Hom(R, P )ϕ + Hom(S, P )ψ and ϕ ∈ Hom(S, R)ψ, which is clear.  2.3.15. From now on we do not assume that X is affine. (−1) Let A` be a DX -algebra. Back in 2.3.13 we defined ΩA , ΩA assuming that X is affine. The construction is compatible with the localization of X, and we define these A` [DX ]-modules for arbitrary X by gluing. More generally, we have all H a (L ΩA ) ∈ M(X, A` ).43 Definition. We say that A` is – formally smooth if ΩA is a projective A` [DX ]-module locally on X (see 2.3.6) (−1) and ΩR vanishes; – smooth if, in addition, A` is finitely generated as a DX -algebra; – very smooth if it is smooth and H 6=0 (L ΩA ) = 0. Notice that the H a (L ΩA ) are compatible with the ´etale localization of Spec A` , so the above notions make sense for an arbitrary algebraic DX -space Y. If Y is smooth, then ΩY is a vector DX -bundle on Y. Remarks. (i) Y is formally smooth if and only if for every affine scheme S equipped with a DX -scheme structure and every closed DX -subscheme S0 ⊂ S defined by an ideal I with I 2 = 0, the map MorDX (S, Y) → MorDX (S0 , Y) is surjective.44 (ii) We do not know if there exists a smooth A` which is not very smooth.45 We also do not know if a formally smooth A` is necessary OX -flat and reduced. (iii) Suppose that A` is smooth and, as a mere OX -algebra, it can be represented locally on X as an inductive limit of smooth OX -algebras. Then A` is very smooth (see the remark that opens 2.3.14). In particular, the jet DX -scheme JZ for a smooth X-scheme Z is very smooth. 2.3.16. A morphism ϕ : Y → Z of algebraic DX -spaces is formally smooth if for every (S, S0 ) as in Remark (i) from 2.3.15, any pair of ϕ-compatible morphisms f : S0 → Y, g : S → Z lifts to a morphism f˜ : S → Y. The reformulation of this property in terms of relative differentials is left to the reader. One says that ϕ is formaly ´etale if, in addition, f˜ is unique. 43 In

fact, at least when X is quasi-projective, one has a well-defined object L ΩA of the derived category DM(X, A` ) defined by means of a global semi-free resolution of A` ; see 4.6.1 and 4.6.4. 44 To prove the “if” statement, notice that locally a D -morphism f : S → Y can be 0 X extended to a morphism S → Y and these extensions form a torsor over Hom(f ∗ ΩY , I). Such a torsor is the same as an extension of f ∗ ΩY by I. Now f ∗ ΩY is locally projective and therefore projective (see 2.3.6), so Ext(f ∗ ΩY , I) = 0 and our torsor is trivial. 45 There seems to be no known example of a formally smooth algebra over a field k of characteristic 0 such that H 6=0 (L ΩA ) does not vanish. For char k 6= 0 such an example was communicated to us by O. Gabber.

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2.3.17. Coordinate systems. For a DX -scheme Y a coordinate system on Y is an ´etale morphism of DX -schemes ν: Y → Spec Sym· (DnX ), n ≥ 0. Any Y that admits locally a coordinate system is very smooth.46 The jet scheme JZ for Z/X smooth admits a local coordinate system (for any ´etale ϕ: Z → AnX the morphism Jϕ: JZ → JAnX = Spec Sym· (DnX ) is a coordinate system on JZ). This is not true (even at a generic point!) for arbitrary smooth DX -schemes. Here is the reason. Suppose that our Y = Spec A` is a smooth reduced DX scheme such that ΩA` ' A` [DX ]. Notice that any invertible element of A` [DX ] lies ∼ in A` ⊂ A` [DX ]. Denote by L the image of A` by any isomorhism A` [DX ] −→ ΩA` . This is a canonically defined line subbundle of ΩA . If Y admits locally a coordinate system, then L is an integrable subbundle; i.e., d(L) ⊂ L ∧ ΩA` ⊂ Ω2A` . 2.3.18. Example. Let us construct a smooth A` with non-integrable L. Take X = Spec k[t], B ` = k[t, u, v, u0 , v 0 , . . . , v 0−1 ] (here u0 := ∂t u, etc., so B ` is the localization of Sym· (D2X ) = k[t, u, v, u0 , . . . ] by v 0 ). Let I ⊂ B ` be the DX -ideal generated by ϕ = v + u0 v 0 ; set A` := B ` /I. Consider the morphism of B ` [DX ]modules χ: B ` [DX ] ⊕ B ` [DX ] → ΩB ` , (∂1 , ∂2 ) 7→ ∂1 dϕ + ∂2 ξ, where ξ := du + (u0 /v 0 )dv ∈ ΩB ` . Localizing B ` (and A` ) further by v 0 − u00 v 0 + u0 v 00 , we see that χ becomes an isomorphism. Now A` is smooth, and the map A` [DX ] → ΩA` , ∂ 7→ ∂ξA` , is an isomorphism. Here ξA` is the restriction of ξ to Spec A` ; this is a generator of L. One has ξA` ∧ dξA` = ξA` ∧ d(u0 /v 0 ) ∧ dv = 2 −ξA` ∧ d(v/v 0 ) ∧ dv = αξA` ∧ dv ∧ dv 0 , α 6= 0, where u and v are considered as ` elements of A . It is easy to check that dv = β∂t ξA` , β 6= 0, so ξA` ∧ dξA` 6= 0 and L is non-integrable. 2.3.19. Vector DX -schemes. By definition, these are vector space objects in the category Af f SchD (X) of DX -schemes affine over X. For a left D-module N ` and R` ∈ ComuD (X) morphisms of DX -algebras Sym(N ` ) → R` are the same as morphisms of D-modules N ` → R` , so they form a vector space. Thus V(N ` ) := Spec Sym(N ` ) is a vector DX -scheme.47 It is easy to see that the functor (2.3.19.1)

V : M` (X)◦ → {vector DX -schemes}

is an equivalence of categories. The inverse functor assigns to a vector DX -scheme V the pull-back of ΩV by the zero section morphism X → V . Consider the category Φ = Φ(X) of all functors F : ComuD (X) → Sh(X);48 this is a tensor category in the obvious way. A vector DX -scheme V can be considered as an object of Φ: for an open j : U → X one has V (R` )(U ) := V |U (R` |U ) = V (j∗ j ∗ R` ). It is clear that the category of vector DX -schemes is a full subcategory of Φ, so we have a fully faithful embedding V : M` (X)◦ ,→ Φ. On the other hand, ˜ : M(X) → Φ. according to 2.2.8, we have a fully faithful embedding h Lemma. The intersection of these two subcategories is the category of vector DX -bundles. 46 Use

the remark that opens 2.3.14. a closed point x ∈ X the fiber V(N ` )x is the profinite-dimensional vector space Nx`∗ dual to Nx` . 48 Recall that Sh(X) is the category of sheaves of k-vector spaces on the ´ etale topology of X. 47 For

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Proof. Let M be a vector DX -bundle; M ◦ := HomDX (M ` , DX ) is its dual (see ∼ ◦ ˜ 2.2.16). One has h(M )(R` ) = h(R` ⊗ M ◦ ) = M ◦ ⊗ R` −→ HomDX (M ` , R` ) = DX

◦ ˜ V(M ` ); hence h(M ) = V(M ` ). It remains to show that if for some N ` ∈ M(X)` there exists M ∈ M(X) such ˜ that h(M ) = V(N ` ), then N is a vector DX -bundle. We can assume that X is affine. We need some notation. For L` ∈ M` (X) the multiplicative group k ∗ acts on ` L by homotheties, so for any F ∈ Φ it acts on the vector space F (SymL` ). Denote by F 1 (L` ) ⊂ F (SymL` ) the subsheaf on which k ∗ acts by the standard character. Therefore F yields a functor F 1 : M` (X) → Sh(X). ˜ )1 (L` ) = M ⊗ L` = h(M ⊗ One has V(N ` )1 (L` ) = HomDX (N ` , L` ) and h(M DX

L` ). The latter functor commutes with direct limits and is right exact. If the functors are equal, then N ` is a finitely presented and projective D-module, i.e., a vector DX -bundle. We are done.  2.3.20. The calculus of variations. To end the section let us explain, as a mere illustration, the first notions of the calculus of variations in the present context. Suppose we have a DX -scheme Y = Spec R` such that ΘY is well defined (see Remark (ii) of 2.3.12), e.g., Y is a smooth DX -scheme. Any ν ∈ h(Ω1Y/X ) defines a morphism of R` [DX ]-modules iν : ΘY → R; its image is the Euler-Lagrange ideal Iν . One gets the Euler-Lagrange DX -scheme Yν := Spec R` /Iν` . One usually considers the situation when ν = df¯ for certain f ∈ R called the lagrangian; f¯ ∈ h(R) is called the action. Remark. A more natural object is the DG DX -algebra SymR` (Θ`Y [1]) with the differential dν which equals iν on the degree −1 component Θ`Y [1]. If ν is closed (in particular, if it comes from a lagrangian), then it is an odd coisson DG algebra (see Exercise in 1.4.18). Example. Suppose that Y = JZ for a smooth X-scheme Z. Let π : Y → Z be the canonical morphism. Then dπ : π ∗ Ω1Z/X → Ω1Y/X identifies Ω1Y/X with the left R` [DX ]-module induced from π ∗ Ω1Z/X . Thus h(Ω1Y/X ) = π ∗ Ω1Z/X ⊗ ωX = R ⊗ Ω1Z/X . The de Rham differential h(d) : h(R) → h(Ω1Y/X ) = R ⊗ Ω1Z/X is OZ

OZ

called the variational derivative. By (2.2.16.1) there is a canonical isomorphism (2.3.20.1)

∼

(π ∗ ΘZ/X )D −→ ΘY

of right R` [DX ]-modules.49 Thus for a local coordinate system xi on Z the EulerLagrange ideal is generated, as a DX -ideal, by the df¯-images of ∂xi ∈ ΘZ/X ⊂ ΘY in R. We leave it to the reader to check that these are indeed the usual EulerLagrange equations. Remark. Here is another way to define (2.3.20.1). Consider ΘZ/X as a Lie algebra acting on Z/X. It acts on Y = JZ by transport of structure. Since ΘZ/X is a Lie OZ -algebroid, this action permits us to define a Lie OY -algebroid structure on 49 Explicitly,

it is the composition of the identifications ΘY := HomR` [DX ] (Ω1Y/X , R` [DX ]) =

HomR` (π ∗ Ω1Z/X , R` [DX ]) = HomR` (π ∗ Ω1Z/X , R` ) ⊗ DX = (π ∗ ΘZ/X )D .

2.4. THE SPACES OF HORIZONTAL SECTIONS

89

π ∗ ΘZ/X . Thus (π ∗ ΘZ/X )D is naturally a Lie∗ algebra acting on Y, and (2.3.20.1) is the action morphism. Suppose we have a Lie∗ algebra L acting on Y (see 1.4.9, 2.5.3, 2.5.6(c)). We say ˜ ˜ that our action is L-invariant if the h(L)-action on h(R) leaves f¯ invariant. In other ¯ words, the Lie algebra h(L) fixes f ∈ h(R), and this happens also after any base change, i.e., replacing L by L ⊗ F ` , R` by R` ⊗ F ` where F ` is any commutative DX -algebra. Since h is reliable (see 2.2.8) this happens if and only if the map h(R) → HomDX (L, R) defined by the action map (see (2.2.7.1)) L  R → ∆∗ R sends f¯ to 0. This is essentially Noether’s theorem. d

Remark. The above map is equal to the composition h(R` ) − → h(ΩY/X ) → Hom(ΘY , R) → Hom(L, R) where the last arrow comes from the action morphism L → ΘY . So the property of being L-invariant actually depends on df¯. 2.4. The spaces of horizontal sections In this section we assume that X 6= ∅ is connected of dimension n. We consider the space of global horizontal sections of a DX -scheme Spec, R` affine over X, which is the same as the space of global solutions of the corresponding system of differential equations. This is an ind-affine ind-scheme (see 2.4.1) which is a true affine scheme if X is proper (see 2.4.2); the corresponding algebra of functions is denoted by hRi. The non-proper situation can be reduced to the proper situation by considering all possible extensions of R` to a compactification (see 2.4.3). In 2.4.4–2.4.6 we give some explicit descriptions of hRi which will be of use in Chapter 4. In 2.4.7 we explain how one checks the smoothness of hRi assuming that R` is smooth. In the case of dim X = 1 we consider the ind-scheme of horizontal sections of Spec R` over the formal punctured disc at a point x ∈ X, describe its (topological) algebra of functions Rxas in 2.4.8–2.4.11, and relate the local and global pictures in 2.4.12. For the (more manageable) derived version of the functor R` 7→ hRi, see 4.6. 2.4.1. We will play with ind-schemes in the strict sense calling them simply ind-schemes. So an ind-scheme Y is a functor on the category of commutative algebras, R 7→ Y (R), which can be represented as the inductive limit of a directed family of quasi-compact schemes and closed embeddings. Some basic material about ind-schemes can be found in [BD] 7.11–7.12. When discussing a topological commutative algebra Q, we always tacitly assume that the topology is complete and has a base formed bySopen ideals Iα , so Q=← lim − Q/Iα . For more details, see 3.6.1. We write Spf Q := Spec Q/Iα ; this is an ind-affine ind-scheme. Denote by Comu(k) the category of commutative unital k-algebras. For any F ∈ Comu(k) the constant left D-module F ⊗ OX is a DX -algebra in the obvious way. This functor Comu(k) → ComuD (X) identifies Comu(k) with the 50 full subcategory Comu∇ It has a right adjoint D (X) of constant DX -algebras. ∇ ∇ ` ∇ functor Γ : ComuD (X) → Comu(k), Γ (R ) = Γ (X, R` ) = Hom(OX , R` ) = 0 HDR (X, R[−n]). Let us see if our embedding admits a left adjoint. So for R` ∈ ComuD (X) consider the functor F 7→ Hom(R` , F ⊗ OX ) on Comu(k). 50 I.e.,

those DX -algebras which are constant as a DX -module; see 2.1.12.

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Lemma. (i) This functor is representable by a topological commutative algebra hRi = hRi(X). For any x ∈ X(k), hRi is naturally a completion of Rx` . ˆ α i. (ii) R 7→ hRi is a tensor functor: one has h⊗Rα i = ⊗hR Proof. (i) Consider the set CR of DX -ideals I ` ⊂ R` such that R` /I ` is a constant D-algebra. Since any subquotient of a constant D-module is constant, we see that CR is a subfilter in the ordered set of all ideals in R` . For every I ` ∈ CR we have a commutative algebra hRiI := Γ∇ (X, R` /I ` ), S and for I 0 ⊃ I a surjective ` map hRiI  hRiI 0 . Therefore Hom(R , F ⊗ OX ) = Hom(R` /I ` , F ⊗ OX ) = S Hom(hRiI , F ) = Homcont (hRi, F ) where hRi is the CR -projective limit of the hRiI ’s. For any I ` ∈ CR one has hRiI = Rx` /Ix` , so we have a natural morphism ` Rx → hRi with dense image; q.e.d. (ii) Evident.  S By construction, the ind-affine ind-scheme Spf hRi := SpechRiI is the space of horizontal sections X → Spec R` . It is naturally an ind-subscheme of any fiber Spec Rx` . We denote it also by hSpec R` i. Exercise. If R is formally smooth and X is affine, then hRi is formally smooth. Consider the category hRimod of discrete hRi-modules. Since hRi ⊗ OX is a completion of R` , we have a fully faithful embedding (2.4.1.1)

hRimod → M(X, R` ),

V 7→ V ⊗ ωX .

Exercise. Show that its essential image is equal to the subcategory of R` [DX ]modules which are constant as DX -modules. Notice that hRi(X) is covariantly functorial with respect to the ´etale localization of X. Precisely, for any morphism U 0 → U in Xe´t , where U 0 , U are connected, there is a natural morphism hRi(U 0 )  hRi(U ). It is immediate that U 7→ SpfhRi(U ) is a sheaf of ind-schemes on Xe´t ; we denote it by SpfhRiX = hSpec R` iX . 2.4.2. Suppose that X is proper. Proposition. (i) hRi is a discrete algebra. (ii) If R is finitely generated as a DX -algebra, then hRi is finitely generated. Proof. (i) We want to show that R` admits the maximal constant D-algebra quotient R` /I0` . By 2.1.12, R` admits the maximal constant D-module quotient ` σR : R`  Rconst (see 2.1.12). Then I0` is the ideal generated by the kernel of σR . ∼ (ii) Clear, since the equivalence Comu∇ → Comu(k) identifies constant D (X) − DX -algebras which are finitely generated as DX -algebras with finitely generated algebras.  2.4.3. Let j : U ,→ X be a non-empty Zariski open subset. For R` ∈ ComuD (U ) the left D-module j· R` := H 0 j∗ R` is a DX -algebra in the obvious way. Let Ξc (R) denotes the set of DX -subalgebras Rξ` ⊂ j· R` such that j ∗ Rξ` = R` . The intersection of two subalgebras from Ξc (R) belongs to Ξc (R), so this is a subfilter in the ordered set of all DX -subalgebras of j· R` . We have a Ξc (R)-projective system of topological algebras51 hRξ i(X) connected by epimorphisms. The canonical maps 51 If

X is proper, then the hRξ i(X) are discrete; see 2.4.2.

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91

hRi(U ) → hRξ i(X) yield a morphism (the limit is taken over Ξc (R)) hRi(U ) → lim ←−hRξ i(X).

(2.4.3.1)

Proposition. This is an isomorphism. Proof. It suffices to show that any morphism52 ϕ: R` → F ⊗OU is the restriction to U of some morphism ϕX : Rξ` → F ⊗ OX , Rξ` ∈ Ξc (R). Note that ϕ defines a morphism of DX -algebras j· (ϕ): j· R` → j· (F ⊗OU ), and F ⊗OX is a DX -subalgebra  of j· (F ⊗ OU ). Set Rξ` := j· (ϕ)−1 (F ⊗ OX ), ϕX := j· (ϕ) ` . We are done. Rξ

Corollary. If R is a finitely generated DX -algebra, then SpfhRiX is a sheaf of ind-schemes of ind-finite type.  2.4.4. In 2.4.4–2.4.7 we assume that X is proper. Here are some “explicit” descriptions of hRi. ` We saw in 2.1.12 that Rconst = H0DR (X, R) ⊗ OX , so hRi is a quotient of H0DR (X, R). Let I` ⊂ R` ⊗ R` be the kernel of the multiplication map R` ⊗ R` → R` . ⊗2 Consider the morphism σR : R` ⊗ R` → H0DR (X, R)⊗2 ⊗ OX . Restricting it to I, ∼ passing to the de Rham cohomology, and using tr : H0DR (X, ωX ) −→ k, we get a morphism (2.4.4.1)

ν : H0DR (X, I) → H0DR (X, R)⊗2 .

The projection H0DR (X, R)  hRi together with the product on hRi identifies hRi with a quotient of H0DR (X, R)⊗2 . Lemma. One has hRi = Coker ν. Proof. Consider a commutative diagram R` x  

hRi ⊗ OX x  

−−−−→

R` ⊗ R` −−−−→ H0DR (X, R)⊗2 ⊗ OX According to the proof of 2.4.2(i), hRi ⊗ OX is the quotient of R` modulo the ideal R` · Ker σR , or, equivalently, the quotient of R` ⊗ R` modulo I + Ker σR ⊗ R` + R` ⊗ Ker σR . This is Cokerν ⊗ OX ; q.e.d.  2.4.5. Here is another description of hRi. Let ∆: X ,→ X × X be the diagonal embedding, j: U := X × X r ∆(X) ,→ X × X its complement. Consider the morphism Rj∗ j ∗ R  R → ∆∗ R[−n + 1] in DM(X × X) defined as the composi∼ tion of the canonical “residue” morphism Rj∗ j ∗ (R  R) → ∆∗ ∆! (R  R)[1] −→ L

L

∆∗ (R ⊗ ! R)[−n + 1] and the product R ⊗ ! R → R ⊗! R → R. Passing to de Rham cohomology,53 we get a morphism of vector spaces (2.4.5.1) 52 Here 53 One

0 µ : HDR (U, R  R[2n − 1]) → H0DR (X, R).

F is a commutative algebra. · · · · has HDR (X × X, j∗ j ∗ B) = HDR (U, B), HDR (X × X, ∆∗ C) = HDR (X, C).

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Lemma. One has hRi = Coker µ. Proof. Denote by C the cone of our morphism j∗ j ∗ R  R[−1] → ∆∗ R[−n]; this 2n is an object of DM(X × X).54 One has55 Coker µ = HDR (X × X, C). Consider the obvious morphism R  R → C. Denote by D its cone, so we have the exact triangle R  R → C → D. The complex D is supported on the diagonal; 2n−1 its top non-zero cohomology is H n−1 D = ∆∗ I. Therefore HDR (X × X, D) = n HDR (X, I) and all the higher de Rham cohomologies vanish. So the long exact 2n sequence of de Rham cohomology identifies HDR (X × X, C) with the cokernel of 2n−1 2n the arrow HDR (X × X, D) → HDR (X × X, R  R) which may be rewritten as n n ν 0 : HDR (X, I) → HDR (X, R)⊗2 . So we see that Coker µ = Coker ν 0 . We leave it to the reader to check that our 0 ν equals the map ν from (2.4.4.1). Now use 2.4.4.  2.4.6. Now let {xs } ⊂ X, s ∈ S, be a finite non-empty subset. Q ` We have the n−1 canonical residue map ResS = (Resxs ): HDR (X r S, R) → Rxs (see 2.1.12); denote its image by VS . Let ⊗Rx` s  hRi be the ring morphism that combines the surjections Rx` s  hRi, s ∈ S (see 2.4.1 and 2.4.2). As follows from the definition Q of hRi (and 2.1.12), its kernel contains f (VS ) where the map f : Rx` s → ⊗Rx` s Q ` sends (as ) ∈ Rxs to Σas ⊗ 1⊗Sr{s} . Hence it contains the ideal IS generated by f (VS ). We have defined αS : (⊗Rx` s )/IS  hRi. ∼

Lemma. One has αS : (⊗Rx` s )/IS −→ hRi. ` )xs = H0DR (X, R). Proof. Consider the natural projections πxs : Rx` s  (Rconst Q ` DR According to 2.1.12, the kernel of πS = Σπxs : Rxs  H0 (X, R) equals VS . (a) Assume that |S| = 1, so we have a single point xs = x ∈ X. As follows from the proof of 2.4.2(i), hRi ⊗ OX is the quotient of R` modulo the ideal generated by ` the kernel of the projection π : R`  Rconst . Therefore, passing to the x-fibers, we ` see that hRi is the quotient of Rx modulo the ideal Ix generated by Ker πx = Vx , and we are done. (b) Assume that |S| ≥ 2. Take any 0 ∈ S. Then Rx` 0 ∩ VS = Vx0 (indeed, both subspaces are equal to the kernel of πx0 ). Therefore the morphism Rx` 0 → ⊗ Rx` s Q ` S yields a morphism ζ : Rx` 0 /Ix0 → (⊗ Rx` s )/IS . Since Rx` 0 + VS = Rxs (which S

follows from the surjectivity of πx0 ), we see that ζ is surjective. The composition of ζ with αS is the map αx0 : Rx` 0 /Ix0 → hRi for the 1-point set {x0 } which we know to be an isomorphism by (a). Thus αS is an isomorphism.  Remark. One may also deduce 2.4.6 in the case |S| = 1 (i.e., (a) above) from 2.4.5. Namely, set Ux := X r {x}; denote by ix , ˜ix the embeddings {x} ,→ X, X = {x} × X ,→ X × X. Then ˜i!x (R  R) = (i!s R) ⊗ R, so we have a commutative diagram µ

0 HDR (U, R  R[2n − 1]) −−−−→ H0DR (X, R) x x    

(2.4.6.1)

µx

0 Rx` ⊗ HDR (Ux , R[n − 1]) −−−−→ 54 In

Rx`

the most important case of n = 1 our C is actually a D-module, not merely a complex. 2n (U, B) = 0 for every B ∈ M(U ). U is non-compact, HDR

55 Since

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the vertical arrows come from canonical morphisms ˜ix∗˜i!x (RR) → RR, ix∗ i!x R → R, and µx sends a ⊗ b to a Resx b. The standard long exact sequences show that the vertical arrows are surjective; the kernel of the right one is Im (Resx ). Since Resx (b) = µx (1⊗b), one has Im (Resx ) ⊂ Im(µx ). Therefore, Coker µ = Rx` /Im(µx ) ∼ = Rx` /Ix , so, by 2.4.5, we have Rx` /Ix −→ hRi. One checks immediately that this is the morphism of 2.4.6. 2.4.7. For M ∈ M(X, R` ) the quotient MhRi := M/I0` M is an hRi ⊗ DX module; therefore H0DR (X, MhRi ) is an hRi-module. It follows from 2.1.12 that the functor M 7→ H0DR (X, MhRi ) is left adjoint to the above embedding. ∼

Proposition. (i) One has a canonical isomorphism ΩhRi −→ H0DR (X, (ΩR )hRi ). (ii) Suppose R` is smooth, H1DR (X, (ΩR )hRi ) = 0, and H0DR (X, (ΩR )hRi ) is a projective hRi-module. Then hRi is smooth. The same is true for “smooth” replaced by “formally smooth”. Proof. (i) Follows from the adjunction. (ii) By 2.4.2 it suffices to consider formal smoothness. We want to show that every extension P of hRi by an ideal N of square 0 (which is an hRi-module) ˜ ` be the pull-back of P ⊗ OX by the morphism R` → hRi ⊗ OX ; splits. Let R this is a DX -algebra extension of R` by the ideal N ⊗ OX of square 0. By adjunction, it suffices to show that this extension splits. It happens locally on X since R` is formally smooth. The obstruction to the existence of global splitting lies in Ext1R` [DX ] (ΩR , N ⊗ ωX ) = Ext1hRi⊗DX ((ΩR )hRi , N ⊗ ωX ) (the latter equality comes since ΩR is a locally projective R` [DX ]-module). By duality, it equals HomhRi (RΓDR (X, ΩR )hRi [n − 1], N ) which vanishes by our conditions.  Remark. Suppose dim X = 1. For smooth R the conditions of (ii) amount to the smoothness of hRi = H0ch (X, R) and the vanishing of H1ch (X, R) (see 4.6). 2.4.8. For the rest of the section, X is a curve. Let x ∈ X be a (closed) point, jx : Ux ,→ X the complement to x. For R` ∈ ComuD (Ux ) consider the as topology Ξas x = Ξx (jx∗ R) on jx∗ R at x (see 2.1.13 for terminology) formed by all DX -subalgebras Rξ` such that jx∗ Rξ = R. We have (2.4.8.1)

` Rξx = i!x ((jx∗ R)/Rξ ) = h((jx∗ R)/Rξ )x .

` We see that for Rξ` 0 ⊂ Rξ` the morphism of fibers Rξ` 0 x → Rξx is surjective. Denote as as ` 56 . As a plain topoby Rx the Ξx -projective limit of the commutative algebras Rξx logical vector space it is equal to the Ξas -completion of the vector space h(jx∗ R)x x (see 2.1.13).

Remark. The functor R` 7→ Rxas is not compatible with the localization of R` , so it cannot be extended to non-affine DUx -schemes. Let Q be any topological commutative algebra. Following [BD] 7.11.1, we call an ideal I ⊂ Q (and the quotient Q/I) reasonable if I is open and for every open ideal J ⊂ I the ideal I/J ⊂ Q/J is finitely generated. We say that Q (and the ind-scheme Spf Q) is reasonable if the reasonable ideals form a base of the topology. 56 The

in 3.6.

superscript “as” means “associated” or “associative”; this notation will be clearified

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Example. The topological algebras k[t0 , t1 , · · · ]] := lim ←− k[t0 , t1 , · · · , tn ] and k[· · · , t−1 , t0 , t1 , · · · ]] := ← lim k[· · · , t , t , t , · · · , t ] are reasonable, while the alge−1 0 1 n − n bra k[[t0 , t1 , · · · ]] := lim k[t , t , · · · ]/m is unreasonable (here m ⊂ k[t0 , t1 , · · · ] is 0 1 ←− the ideal generated by all ti ’s). Lemma. For any finitely generated D-algebra R` the topological algebra Rxas is reasonable. More precisely, if Rξ` ∈ Ξas x (X) is a finitely generated DX -algebra, then ` as Rξx is a reasonable quotient of Rx . Proof. It suffices to consider the case of a free finitely generated DUx -algebra; here the statement is obvious.  S as ` 2.4.9. Let us show that the ind-affine ind-scheme Spf Rx := Spec Rξx , ξ ∈ as Ξx , is the space of horizontal sections of Spec R` over the formal punctured disc at n x. Denote by Kx the field of fractions of Ox := lim ←− Ox /mx . For a vector space H n n b Ox := lim H ⊗Ox /mO , H ⊗ b Kx := lim H ⊗Kx /mO . So if t is a parameter set H ⊗ ←− ←− x x b Ox = H[[t]], H ⊗ b Kx = H((t)). Let us consider H ⊗ b Ox ⊂ H ⊗ b Kx as at x, then H ⊗ sheaves on X supported at x; these are non-quasi-coherent sheaves of DX -modules (DX -sheaves in the terminology of 3.5.1) in the obvious way. If H is an algebra, b Ox , H ⊗ b Kx are naturally (non-quasi-coherent) DX -algebras. then H ⊗ Proposition. For a commutative algebra H there is a canonical identification (2.4.9.1)

b Kx ). Hom(Rxas , H) = Hom(jx∗ R` , H ⊗

Here the former Hom is the set of continuous morphisms of commutative algebras; the latter one is that of morphisms of commutative DX -algebras. Proof (cf. the proof of the proposition in 2.4.3). Suppose we have a morphism of b Kx . Since H ⊗ b Ox is a DX -subalgebra commutative DX -algebras φ : jx∗ R` → H ⊗ b b b of H ⊗ Kx and the quotient H ⊗ Kx /H ⊗ Ox is quasi-coherent, we see that Rφ` := ` b Ox ) ⊂ jx∗ R` belongs to Ξas φ−1 (H ⊗ x (jx∗ R) and φx : Rφx → H is a morphism T ` b Kx ) → Hom(Rξx of algebras. We have defined a map Hom(jx∗ R` , H ⊗ , H) = as Hom(Rx , H), φ 7→ φx . The verification of its bijectivity reduces easily to the case when R` is a free DUx -algebra, then to that of R` = Sym(DUx ) where it is clear.  Remark. The above proposition also follows easily from its linear counterpart (3.5.4.2). 2.4.10. Corollary. (i) For N ` ∈ M` (Ux ) one has (2.4.10.1)

ˆ (Sym N ` )as x = Sym hx (jx∗ N )

where the latter Sym is the topological symmetric algebra, i.e., the completion of the plain symmetric algebra with respect to the topology formed by ideals generated ˆ x (jx∗ N ). In particular, (Sym DU )as = Sym(ωK ). by open vector subspaces of h x x x (ii) If R is formally smooth, then the topological algebra Rxas , or the ind-scheme Spf Rxas , is formally smooth. ˆ as (iii) R 7→ Rxas is a tensor functor: one has (⊗Rα )as  x = ⊗Rαx . Remark. For an explicit description of formally smooth topological algebras, see [BD] 7.12.20–7.12.23.

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2.4.11. For a commutative D-algebra R` on X and an R` [DX ]-module N we denote by ΞR x (N ) the topology on N at x (see 2.1.13) formed by all Nη ∈ Ξx (N ) which are R` -submodules of N . Since each Nηx is an Rx` -module, the completion ˆ R (N ) is a topological R` -module. h x x For R` as in 2.4.8 and an R` -module M let ΞR x (jx∗ M ) be the topology at x R on jx∗ M formed by all Mη ⊂ jx∗ M such that Mη ∈ Ξx ξ (jx∗ M ) for some Rξ ∈ as ˆR ˆR Ξas x (jx∗ R). The completion hx (M ) := hx (jx∗ M ) is a topological Rx -module. Remarks. (i) M is generated by finitely many sections {mα }. Then P Suppose ` the submodules Rξ` [DX ] · tn mα , where Rξ` ∈ Ξas x (R ) and n a positive integer, form a base of the topology ΞR x (jx∗ M ). Here t is a parameter at x, and we restricted X to make t invertible on Ux . ˆ R (M ) = lim M ` , we see that there is a natural continuous trans(ii) Since h x ←− ηx R ˆ (⊗Mi ) → ⊗ ˆ R (Mi ). Thus h ˆ R transforms !-coalgebras to topological ˆ formation h h x x x coalgebras. 2.4.12. Let {xs } ⊂ X, s ∈ S, be a finite set, jS : US ,→ X its complement. b Rxass := lim ⊗Rξ` x . Assume now that X For a D-algebra R on US set RSas = ⊗ ←− xs s c is proper. Subalgebras from Ξ (R) (see 2.4.3) are just intersections of subalgebras as from Ξas xs (jS∗ R), s ∈ S, so, by 2.4.3, we have the canonical maps Rxs → hRi(US ). They define a continuous morphism (2.4.12.1)

v: RSas → hRi(US ).

According to 2.4.6, it identifies hRi(US ) with the quotient of RSas modulo the closed ideal generated by the image of the map (2.4.12.2)

R

ξ 0 as as rS = ← lim − ResS : HDR (US , R) → ΠRxs → RS .

0 (US , R). Remark. rS evidently factors thru the quotient Γ(US , h(R)) of HDR

In geometric language, the embedding Spf hRi(US ) ,→ Spf RSas assigns to a horizontal section of Spec R` over US the collection of its restrictions to the formal punctured discs at xs . Notice that Spec hRi(X) is the intersection of Spf hRi(US ) Q and Spec Rx` s in Spf RSas . 2.5. Lie∗ algebras and algebroids The notion (if not the name) of the Lie∗ algebra has been, undoubtedly, well known in mathematical physics for quite a long time. Since the first version of this chapter was available as a preprint in 1995, Lie∗ algebras migrated to mathematical literature. For example, they appear (in the translation equivariant setting; cf. 0.15) in [K], [DK] under the alias of “conformal algebras”, which further mutated into “pseudoalgebras” of [BAK], and in [P], [DLM], [FBZ] as “vertex Lie algebras” (while “conformal algebras” of [FBZ] are translation equivariant Lie∗ algebras equipped with a “Virasoro vector”). We begin with remarks about B∗ algebras for any operad B and the corresponding h-sheaves of B algebras (see 2.5.1 and 2.5.2). Passing to Lie∗ algebras, we describe Lie∗ brackets in terms of the corresponding adjoint actions in 2.5.5. First examples of Lie∗ algebras are in 2.5.6; in 2.5.7 we discuss the relation between Lie! coalgebras and Lie∗ algebras. The examples of Kac-Moody and Virasoro algebras (here X is a curve) are in 2.5.8–2.5.10. In 2.5.12–2.5.15 we play with a natural

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topology on the Lie algebra h(L)x , L a Lie∗ algebra, x ∈ X. The rest of the section deals with Lie∗ algebroids. They are introduced in 2.5.16; the equivariant setting is discussed in 2.5.17. We show that a Lie∗ algebroid yields (topological) Lie algebroids on ind-schemes of sections over formal punctured discs in 2.5.18–2.5.21. For a global theory see 4.6.10 and 4.6.11. Elliptic Lie∗ algebroids are considered in 2.5.22; a geometric example (suggested by D. Gaitsgory) is discussed in 2.5.23. The reader is referred to [K], [DK], [BAK] for solutions to some natural classification problems (e.g. a description of simple translation equivariant Lie∗ algebras on A1 which are coherent D-modules), and to [BKV] for some computations of the Lie∗ algebra cohomology. A certain ∗ version of the Wedderburn theorem can be found in [Re]. 2.5.1. Having at hand the ∗ pseudo-tensor structure, we may consider for any k-operad B the category B∗ (X) of B algebras in M(X)∗ , and for any L ∈ B∗ (X) (and a B-module operad C) the corresponding category M(X, L) = M(X)(L, C) of L-modules. We have the sheaves of categories B∗ (X´et ) and M(X´et , L) on X´et . For a locally closed embedding i: X ,→ Y one has (see 2.2.5) the adjoint functors i∗

(2.5.1.1)

∗ B∗ (X)−→ ←−B (Y ). i!

If X is a closed subscheme, then i∗ identifies B∗ (X) with the full subcategory of B∗ (Y ) that consists of algebras supported on X. 2.5.2. The functor h sends B∗ algebras on X to B algebras in the tensor category Sh⊗ (X) of sheaves of k-vector spaces on X. If R` is a commutative DX -algebra, then a B∗ algebra structure on a Dmodule L yields a B∗ algebra structure on R` ⊗ L; hence a B algebra structure ` ˜ on h(L)(R ) := h(L ⊗ R` ) = L ⊗ R` (see 1.4.6). Since h is a reliable augmentaDX

tion functor (see 1.4.7 and 2.2.8), this gives a bijective correspondence between B∗ algebra structures on L and data which assign to every commutative DX -algebra R` a B algebra structure on h(L ⊗ R` ) in a way compatible with morphisms of DX -algebras. For L ∈ B∗ (X) an L-module structure on a D-module M defines on M the C(h(L))-module structure (see 1.2.17). Since h is non-degenerate (see 2.2.8), we get a fully faithful embedding (2.5.2.1)

M(X, L) ,→ C(h(L))-modules in M(X).

Its image can be described using 2.2.19; we leave this description to the reader. Due to Remark (iii) in 2.2.3 and 1.2.13, we can also consider L-module structures on OX -modules. 2.5.3. We will be mainly interested in Lie∗ algebras. Therefore a Lie∗ algebra is a D-module L equipped with a Lie∗ bracket, i.e., a ∗-pairing [ ] ∈ P2∗ ({L, L}, L) which is skew-symmetric and satisfies the Jacobi identity. Then the sheaf h(L) is ` ˜ ) := h(L ⊗ R` ) a Lie algebra with bracket [`¯1 , `¯2 ] = [ ](`1  `2 ); moreover, h(L)(R is a Lie algebra for every commutative DX -algebra R` . Recall that C(h(L)) is the universal enveloping algebra of h(L) and a C(h(L))-module is the same as an h(L)-module. Consider the category ΦLie of all functors F on ComuD (X) with values in the category of sheaves of Lie k-algebras on X (i.e., ΦLie is the category of Lie algebras

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˜ in the tensor category Φ from 2.3.19). A Lie∗ algebra L yields h(L) ∈ ΦLie ; the functor ˜ : Lie∗ (X) → ΦLie h

(2.5.3.1)

is a fully faithful embedding. For D-submodules M, N ⊂ L, following the notation of 2.2.3, we define [M, N ] ⊂ L by ∆∗ [M, N ] := [ ](M  N ). 2.5.4. An L-action on M ∈ M(X) yields an h(L)-action on M given by ¯ `(m) = T r2 ( · (`  m)).

(2.5.4.1)

Here · ∈ P2∗ ({L, M }, M ) is the L-action and T r2 : ∆· ∆∗ M = pr2· ∆∗ M → M is the trace map for the projection pr2 : X × X → X. According to 2.2.19, this way we get a bijective correspondence between the L-module structures on M ∈ M(X) and such h(L)-actions on M (we call them good ¯ is a differential operator. actions) that for every m ∈ M the map L → M , ` 7→ `m, Remarks. (i) Let M be an L-module. For a section m ¯ of h(M ) its stabilizer ¯ Lm ⊂ L is the kernel of the morphism L → M , l 7→ lm; ¯ this is a Lie∗ subalgebra ¯ of L. For R` ∈ ComuD (X) there is an obvious morphism from h(Lm ⊗ R` ) to the ` ` stabilizer of m ¯ ∈ h(M ⊗ R ) in h(L ⊗ R ); it need not be an isomorphism (see Example in 2.5.7). (ii) Any lisse D-submodule of M is killed by the L-action (see 2.2.4(ii)). Examples. (i) Let V be a coherent D-module. Then the h(End∗ V )-action on V coincides with the map h(Hom(V, V ⊗ DX )) → End V , which sends ϕ, ¯ ϕ∈ O ϕX

Hom(V, V ⊗ DX ) to the composition V −→V ⊗ DX → V ; the second arrow is OX

OX

v ⊗ ∂ 7→ v∂ (ii) The adjoint action of L yields an h(L)-action on L denoted by `¯ → 7 ad`¯ ∈ End L. The next technical proposition may be useful when one wants to check if a given ∗ operation is actually a Lie∗ bracket. The reader can skip it, returning when necessary. 2.5.5. Proposition. For a D-module L Lie∗ brackets on L are in bijective correspondence with the morphisms of sheaves h(L) → EndL, `¯ 7→ ad`¯, which satisfy the  following  properties: (i) ad`¯1 , ad`¯2 = adad`¯ (`¯2 ) for any `¯1 , `¯2 ∈ h(L). 1 (ii) ad`¯(`) = 0 for any ` ∈ L. (iii) For any `0 ∈ L the map L → L, ` 7→ ad`¯(`0 ), is a differential operator (with respect to the O-module structure on L). Proof. We use 2.2.19. The condition (iii) is exactly the condition (∗) of 2.2.19; hence it means that ad comes from a ∗-pairing [ ] ∈ P2∗ ({L, L}, L). The property (i) is equivalent to the Jacobi identity by 2.2.19. The property (ii) means that T r2 [ ]: L⊗2 → L vanishes on the symmetric part L⊗2+ ⊂ L⊗2 . Set N := ∆· ∆∗ L. ⊗2+ Since [ ]: L⊗2 → N is D⊗2 ) is a D⊗2+ X -linear, M := [ ](L X -submodule of N such that T r2 (M ) = 0. So the skew-symmetry of [ ] follows from the next lemma.

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Lemma. If M ⊂ N is a D⊗2+ X -submodule such that T r2 (M ) = 0, then M = 0. Proof. Take m ∈ M . For anyf ∈ OX , θ ∈ ΘX we have T r2 (m·(f ⊗θ +θ ⊗f )) = 0. Since T r2 (n·(θ⊗1)) = 0 and T r2 (n·(1⊗θ)) = T r2 (n)·θ for any n ∈ N , we obtain (T r2 (m · (f ⊗ 1)))θ = 0. Since ΘX DX = DX , we have T r2 (m · (f ⊗ 1)) = 0 for any f ∈ OX . Choose a coordinate system t1 , . . . , tn on X and for α = (α1 , . . . , αn ) ∈ Zk+ αk α1 α 1 k set tα := tα tα 1 . . .P k ∈ OX , ∂ := ∂t1 . . . ∂tk ∈ DX . Since N = L P· (DXα ⊗ 1), we α can write m as `α · (∂ ⊗ 1), `α ∈ L. Then T r2 (m · (f ⊗ 1)) = `α ∂ (f ). The α α P equality `α ∂ α (f ) = 0 for all f ∈ OX implies `α = 0 (set f = tβ ).  α

2.5.6. Basic examples. (a) For any coherent DX -module M we have an associative∗ , hence Lie∗ , algebra End∗ (M ) (see 2.2.15 and 2.2.16). If M is a vector DX -bundle, then End∗ (M ) considered as a Lie∗ algebra is denoted by gl(M ). (b) By 2.2.4(i) for an O-module P a Lie bracket on P which is a bidifferential operator is the same as a Lie∗ bracket on the induced D-module PD . Notice that ˜ )(R` ) = P ⊗ R` , and the bracket on this sheaf is just the extension of the Lie h(P OX

bracket on P defined by the DX -algebra structure on R` . Examples. (i) The Lie bracket of vector fields defines a Lie∗ algebra structure on ΘD := ΘXD . For every O-module F equipped with a ΘX -action such that the action map is a differential operator with respect to ΘX 57 the corresponding ⊗j D-module FD is a ΘD -module. E.g., the (ωX )D are ΘD -modules. (ii) If g is a Lie algebra, then gO = g ⊗ OX is a Lie OX -algebra; hence gD := ˜ D )(R` ) = g ⊗ R` ; the bracket g ⊗ DX = (gO )D is a Lie∗ algebra. Notice that h(g extends the bracket on g by R` -linearity. (iii) If F is a coherent O-module, then the sheaf Diff(F, F )X of differential operators is an associative algebra in Dif f (X)∗ which acts on F . So Diff(F, F )D := Diff(F, F )XD is an associative∗ (hence Lie∗ ) algebra that acts on FD . This action ∼ yields a canonical isomorphism of associative∗ algebras Diff(F, F )D −→ End∗ (FD ) (see 2.2.16 and (2.2.14.1)). Remark. As follows from 2.2.10, any Lie∗ algebra L admits a functorial left ˜ whose terms are induced D-modules locally on X. Namely, Lie algebra resolution L ˜ one can take L = L ⊗ P where P is a resolution of OX from Remark in 2.2.10. ∗

(c) Let L be a Lie∗ algebra. We know what its action on any DX -algebra is (see 1.4.9). Such an action has the ´etale local nature, so we know what the L-action on any algebraic DX -space is. For example, let R` be a DX -algebra such that ΩR is a finitely-presented ` R [DX ]-module, so ΘR is well defined (see Remark (ii) in 2.3.12). Then ΘR is a Lie∗ algebra acting on R` (see 1.4.16). On the other hand, we have the Lie algebra E := DerDX R` of horizontal OX -derivations of R` (experts in non-linear differential equations sometimes call E the Lie-B¨ acklund algebra). We consider E as a sheaf on X. One has the canonical morphism of Lie algebras h(ΘR ) → E. If ΩR` is a projective R` [DX ]-module (which happens if R` is DX -smooth), then this is an isomorphism (see Remark (iii) in 2.3.12). ΘR is the first example of a Lie∗ R-algebroid (see 1.4.11 and 2.5.16 below). 57 We assume that Θ X acts on F as on an O-module, so the action is automatically a differential operator with respect to F .

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Remark. If ΩR is a free R` [DX ]-module, then ΘR has the following descrip∼ tion. Choosing generators ν1 , . . . , νn ∈ ΩR , we get an identification E −→ R`n , τ 7→ (hνi , τ i). It yields, in particular, an OX -module structure on E. With respect to this structure the bracket on E is a bidifferential operator, so we have the Lie∗ algebra ED . Clearly ΘR = ED . (d) If F is a commutative∗ algebra and P is a Lie! algebra, then P ⊗ F is naturally a Lie∗ algebra (see 2.2.9). Taking F = DRD (see 2.2.4(iii)), we find that the complex P ⊗ DRD , which is a resolution of P [− dim X], is naturally a Lie∗ DG algebra. Take P = gO , where g is a Lie k-algebra; we see that g ⊗ F is a Lie∗ algebra. Thus g ⊗ ωX [− dim X] is naturally a homotopy Lie∗ algebra. 2.5.7. Lie! coalgebras versus Lie∗ algebras. Some Lie∗ algebras arise as duals of natural Lie! coalgebras. If the D-modules we play with are vector DX bundles, then the two structures are equivalent. Otherwise, as we will see, they are pretty different.58 We are basically interested in Lie∗ algebras (and not in Lie! algebras) because of their connection with chiral algebras. Let us start with some generalities: (i) A Lie! coalgebra is just a Lie algebra in the tensor category M(X)!◦ . Equivalently, this is a left DX -module N ` equipped with a morphism N ` → N ` ⊗ N ` which satisfies the Lie cobracket property. If R` is a commutative DX -algebra and N ` is a Lie! coalgebra, then V(N ` )(R` ) = HomDX (N ` , R` ) = Hom(SymN ` , R` ) has a natural structure of a Lie k-algebra. Therefore the vector DX -scheme (see 2.3.19) V(N ` ) carries a canonical Lie algebra structure. It is easy to see that the functor (2.5.7.1)

V : {Lie! coalgebras}◦ → {Lie k-algebras in Af f SchD (X)}

is an equivalence of categories. We have a fully faithful embedding V of the category dual to that of Lie! coalgebras to the category ΦLie (see 2.5.3), V(N ` )(R` ) = Hom(L` , R` ). We also ˜ : Lie∗ (X) ,→ ΦLie ; see (2.5.3.1). The next have the fully faithful embedding h lemma describes the intersection of these subcategories: Lemma. (a) A Lie* algebra L corresponds to some Lie! coalgebra if and only if L is a vector DX -bundle. (b) A Lie! coalgebra N ` corresponds to some Lie* coalgebra if and only if N ` is a vector DX -bundle. (c) If a Lie* algebra L and a Lie! coalgebra N ` correspond to each other, then L and N ` are dual D-modules; i.e., N ` = HomDX (L, DX ) and L = HomDX (N ` , DX ). Proof. Use the lemma in 2.3.19.



If F is a vector bundle on X, then FD := F ⊗ DX is dual to N ` := OX

DX ⊗ F ∗ , so a Lie∗ bracket on FD is the same as a Lie cobracket OX ∗

DF

∗

→

DF

∗

DF

∗

:= ⊗

, i.e., a section of F ⊗ Λ2 ( D F ∗ ) satisfying a certain condition. Sometimes it is convenient to write Lie∗ brackets on FD in this format. DF

58 Of course, in the D -coherent situation the difference disappears if one passes to derived X categories.

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Example. Consider the standard Lie∗ algebra structure on ΘD from (b)(i) in P 2.5.6. The corresponding section of 59 ΘX ⊗Λ2 ( D Ω) is ∂i ⊗((1⊗dxj )∧(∂j ⊗dxi )). i,j

Similarly, in the situation of (b)(ii) in 2.5.6 (we assume that dim g < ∞) the corresponding section is the usual cobracket element in g∗ ⊗ Λ2 g∗ ⊂ g∗ ⊗ Λ2 ( D g∗ ). (ii) For a Lie coalgebra N ` let M(X, N ` ) be the category of N ` -comodules. Notice that for M ∈ M(X, N ` ) the Lie algebra V(N ` )(R` ) acts on M ⊗ R` as on an R` [DX ]-module in a way compatible with the morphisms of the R` ’s. Therefore ˜ V(N ` ) ∈ ΦLie acts on h(M ) ∈ Φ. Assume that N is a vector DX -bundle, so its dual N ◦ := HomDX (N ` , DX ) is ˜ ◦ ) ∈ ΦLie . The above h(N ˜ ◦ )-action on h(M ˜ a Lie∗ algebra, and V(N ` ) = h(N ) ◦ 60 amounts to a ∗ action of N on M . Therefore we have an equivalence of tensor categories (2.5.7.2)

∼

M(X, N ` ) −→ M(X, N ◦ ).

(iii) Assume that N ` is a coherent D-module, but not necessarily a vector DX -bundle. The above constructions can be partially extended to this situation as follows. According to 2.2.18, the dual to a Lie cobracket N ` → N ` ⊗ N ` is a ˜ ◦) → Lie∗ bracket on the dual D-module N ◦ . We have a canonical morphism h(N ` ` Lie ◦ ` V(N ) in Φ . Namely, the morphism h(N ⊗ R ) → HomDX (N , R` ) comes from the pairing idR` ⊗ h i ∈ P2∗ ({R` ⊗ N ◦ , N }, R) where h i ∈ P2∗ ({N ◦ , N }, ωX ) is the canonical pairing. In particular, for R` = Ox , x ∈ X,61 we get a canonical ˆ x (N ◦ ) = Γ(X, h(N ◦ ⊗ Ox )) → morphism of profinite-dimensional Lie algebras h ` `∗ HomDX (N , Ox ) = Nx . We see that any N ` -comodule M is automatically an N ◦ -module, so we have a faithful functor M(X, N ` ) → M(X, N ◦ ). Therefore an N ` -coaction on an algebraic DX -space Y induces an action of N ◦ on Y. (iv) If G is a group DX -scheme, then the restriction of ΩG = ΩG/X to the unit section X ⊂ G has a natural structure of Lie! coalgebra. We denote it by CoLie(G). If G is smooth, then CoLie(G) is a vector DX -bundle, so we have the dual Lie∗ algebra Lie(G) called the Lie∗ algebra of G. A G-action on an algebraic DX -space Y yields a CoLie(G)-coaction on Y, hence an action of Lie(G) on Y. (v) The next example illustrates the fact that the world of Lie! coalgebras differs from that of Lie∗ algebras if one considers not only vector DX -bundles. We will show that on a symplectic variety X the Lie algebra OX of hamiltonians comes naturally from a Lie∗ algebra, while the Lie algebra of symplectic vector fields comes from a Lie! coalgebra and not a Lie∗ algebra. Example. Assume that our X is a symplectic variety with symplectic form ν. For every commutative DX -algebra R` one has the Lie algebra Sympl(R` ) := the stabilizer of ν with respect to the action of the Lie algebra R` ⊗ ΘX on R` ⊗ Ω2X (e.g., if R` is the sheaf of holomorphic functions, then Sympl(R` ) is the sheaf of holomorphic symplectic vector fields). Symplectic vector fields on X can be Ω = Ω1X = Θ∗X . can be constructed explicitly as the composition of h i ⊗ idM ∈ P2∗ ({N ◦ , N ⊗ M }, M ) with the coaction morphism M → N ⊗ M . 61 Here O is the formal completion of O considered as a D -algebra (i.e., O = the direct x x x X image of the structure sheaf by the morphism SpecOx → X). 59 Here 60 It

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identified with closed 1-forms, i.e., DX -module morphisms K ` → OX where K ` is the cokernel of the differential D Λ2 Θ → D Θ from the de Rham complex DR(DX ) of the right DX -module DX (see 2.1.7). Moreover, one has a canonical isomorphism HomDX (K ` , R` ) = Sympl(R` ) functorial in the DX -algebra R` . So K ` is a Lie! coalgebra. K ` does not come from a Lie* algebra because the DX -module K ` is not locally projective (DR(DX ) is a locally projective resolution of the left DX -module −1 OX , so if K ` were locally projective, then HDR (M ) = Tor1 (M, OX ) would vanish for all left DX -modules M ). The following realization of K ` may be convenient. We have the Poisson bracket on OX , the Lie bracket on ΘX = ΩX , and the hamiltonian action morphism f : OX → ΘX = ΩX ; quite similarly, for every commutative DX algebra R` one has the Lie algebra structures on R` and R` ⊗OX ΩX as well as the Lie algebra morphism fR : R` → R` ⊗OX ΩX , which factors through Sympl(R` ). Since R` = HomDX (DX , R` ) and R` ⊗OX ΩX = HomDX (D Θ, R` ), we get Lie! coalgebra structures on DX = D O and D Θ as well as the Lie! coalgebra morphisms ϕ : D Θ → K ` , ψ : K ` → D O. It is easy to see that ϕ is the canonical projection 2 D Θ → Coker(D Λ Θ → D Θ) and ψϕ is the differential from the complex DR(DX ). ∼ Since DR(DX ) is a resolution of OX , we see that ψ induces an isomorphism K ` −→ Ker(D O → OX ). Now let us look at the Lie* algebras associated to our symplectic variety (X, ν). The Poisson bracket on OX and the Lie bracket on ΘX induce Lie* structures on the corresponding induced D-modules OD = DX and ΘD . The morphism a : OD → ΘD corresponding to the hamiltonian action morphism OX → ΘX is injective (because a non-zero morphism from DX to a vector DX -bundle is always injective). Denote by Stabν the stabilizer62 of ν ∈ Ω2X = h(Ω2X ⊗OX DX ) with respect to the action of ΘD on Ω2D := Ω2X ⊗OX DX . Then Stabν = Im a. Indeed, α

d

D D Stabν = Ker(ΘD −−→ Ω2D ) = Ker(Ω1D −−→ Ω2D ) where dD , αD are morphisms of 1 induced D-modules corresponding to d : ΩX → Ω2X and α : ΘX → Ω2X , α(τ ) := τ (ν). Thus Stabν = Im a by 2.1.9.

2.5.8. ω-extensions. Let L be a Lie∗ algebra. We will consider Lie∗ algebra extensions L[ of L by ω = ωX . Notice that any such extension is automatically ∼ central (indeed, one has P2∗ ({L, L}, L[ ) −→ P2∗ ({L[ , L[ }, L[ ), see Remark (ii) in [ 63 2.5.4). All L form a Picard groupoid in the usual way; we denote it by P(L). From now until 2.5.10 we assume that dim X = 1. We will define canonical ω-extensions of some natural Lie∗ algebras of the type considered in 2.5.6(b). So these are examples of Lie∗ algebras which do not come from Lie! coalgebras.64 The constructions follow the same pattern. Namely, to define our ω-extension L[ of L we first construct an L-module extension L\ of the adjoint representation by an L-module which equals ωD := ω ⊗ DX as a plain D-module and has the property that the canonical morphism ωD  ω is L-invariant. Now L[ is the pushout of L\ by this arrow. The L-action on L[ yields a binary ∗ operation [ ] on L[ , 62 See

Remark (i) from 2.5.4 for the definition of stabilizer that a Picard groupoid is a tensor category whose morphisms and objects are invertible, see [SGA 4] Exp. XVIII 1.4. 64 Notice that due to the presence of ω the corresponding D-modules are not induced, nor even quasi-induced in the sense of 2.1.11. For the same reason they cannot be represented as duals. 63 Recall

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[a, b] := π(a)b (where π : L[ → L is the projection), which lifts the Lie∗ bracket on L. It satisfies automatically the Jacobi identity; skew-symmetry follows for separate reasons. 2.5.9. The Kac-Moody extension. We are in the situation of Example (ii) in 2.5.6(b). Let κ be an ad-invariant symmetric bilinear form on g. We will define an ω-extension gκD of gD called the (affine) Kac-Moody Lie∗ algebra; if g is commutative, it is the Heisenberg Lie∗ algebra. ˜ b) := κ(da, b) and the corresponding Consider the pairing φ˜ : gO × gO → ω, φ(a, ∗ ˜ ∗ pairing φD ∈ P2 ({gD , gD }, ωD ); let φ ∈ P2∗ ({gD , gD }, ω) be the composition of φ˜D with the canonical morphism ωD = ω⊗DX → ω. Since κ is ad-invariant, one has ˜ ˜ [b, c])− φ(b, ˜ [a, c]). Since κ is symmetric, one has φ(a, ˜ b)+ φ(b, ˜ a) = φ([a, b], c) = φ(a, ∗ dκ(a, b); hence φ is skew-symmetric. Therefore φ is a 2-cocycle of the Lie algebra gD . Our gκD is the extension corresponding to this cocycle. ˜ language (see 2.5.3) the definition is even simpler. Namely, for a test In the h ˜ κ )(R` ) is the extension g(R` )κ of DX -algebra R` the corresponding Lie algebra h(g D ` ` ` g(R ) = g ⊗ R by h(R ) defined by the cocycle a, b 7→ κ(da, b). Notice that gD acts on gD ⊕ ωD so that the only non-zero components of the action ∈ P2∗ ({gD , gD ⊕ωD }, gD ⊕ωD ) are [ , ]gD and φ˜D . Denote this gD -module by Pκ ; this is an extension of the adjoint representation by the trivial gD -module ωD . The adjoint action of gκD factors through gD ; as a gD -module, gκD is the push-out of Pκ by the projection ωD  ω. Remark. The Lie algebra ΘX acts on gO and ω in the obvious way, and φ˜ is invariant with respect to this action. Thus Pκ and gκD carry a canonical action of the Lie∗ algebra ΘD . Suppose that g is the Lie algebra of an algebraic group G and κ is AdG -invariant. We have the corresponding jet group DX -scheme JG = JGX (see 2.3.2); its Lie∗ algebra equals gD . Notice that Pκ , considered as a mere vector DX -bundle, carries a natural JG-action. Namely, for a test DX -algebra R` a DX -scheme R` -point of JG is the same as g ∈ G(R` ). Such g acts on Pκ ⊗ R` so that the corresponding action on h(Pκ ⊗ R` ) = g ⊗ R` ⊕ ω ⊗ R` is (2.5.9.1)

g(a + ν) := Adg (a) + κ((g −1 dg, a) + ν).

Here g −1 dg ∈ g ⊗ ω and we differentiate g using the structure connection on R` . Now the corresponding action of Lie(JG) = gD coincides with the action we defined previously. The above formula can be interpreted as follows. Consider the space of left G-invariant 1-forms on GX = G × X as a vector bundle on X; denote by P∗ the induced vector DX -bundle. For a test DX -algebra R` one can interpret h(P∗ ⊗ R` ) as the space of left G-invariant 1-forms µ on G × Spec R` along the preimage of the horizontal foliation of Spec R` defined by the structure connection on R` . Then G(R` ) acts on this space by right translations. Therefore JG acts on P∗ . Our P∗ is an extension of g∗D (1-forms relative to the projection GX → X) by ωD , and the JG-action preserves the projection. Our κ defines a morphism of vector DX -bundles gD → g∗D . It lifts canonically to a morphism of ωD -extensions of (2.5.9.2)

αP : Pκ → P∗

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103

which commutes with the JG-actions. Namely, our P∗ admits an evident decomposition P∗ = g∗D ⊕ωD , and αP identifies it with the corresponding decomposition of P. The compatibility with the JG-action follows from (2.5.9.1), since the G(R` )-action on P∗ in terms of the above decomposition is g(a∗ + ν) = Ad∗g (a) + a(g −1 dg) + ν. We have seen that G(OX ) acts on our picture, so we can twist it by any Gtorsor F getting the twisted Kac-Moody extension g(F)κD , etc. Notice that G(F) is the group X-scheme Aut(F) of automorphisms of F, JG(F) = JAut(F) is the group DX -scheme Aut(JF) of automorphisms of the JG-torsor JF, and g(F)D its Lie∗ algebra. Our P∗ (F) is the induced vector DX -bundle corresponding to the vector bundle of G-invariant 1-forms on F which is an ωD -extension of g∗ (F)D ; a section µ ∈ h(P∗ (F)) is the same as a G-invariant 1-form on FR (:= the pull-back of F to Spec R` ) along the horizontal foliation on Spec R` . Our αP identifies P(F) with the pull-back of the P∗ (F) by the morphism g(F)D → g∗ (F)D defined by κ. Here is a convenient interpretation of g(F)κD . Consider the DX -scheme Conn(F) of connections on F. By definition, for a test DX -algebra R` a DX -algebra R` -point Spec R` → Conn(F) is the same as a horizontal connection ∇R , i.e., a connection along the horizontal foliation on Spec R` , on the G-torsor FR . Horizontal connections form a torsor with respect to g(F) ⊗ ω ⊗ R` = g(F) ⊗ R, so Conn(F) is a torsor for the vector DX -scheme Spec Sym(g∗ (F)`D ). Thus Conn(F) = Spec A` where the DX -algebra A` = A` (Conn(F)) carries a natural filtration A`· such that grA` = Sym(g∗ (F)`D ). In particular, A1 is an ω-extension of g∗ (F)D . The group DX -scheme Aut(JF) = JAut(F) acts naturally on Conn(F): namely, a DX -scheme R` -point of Aut(JF) is the same as an element of Aut(F)(R` ), and it acts on ∇R by transport of structure. Lemma. A1 identifies canonically with the push-out of the ωD -extension P∗ (F) by the projection ωD  ω. The identification commutes with the Aut(JF)-action. Proof. The promised identification is the same as a morphism of DX -modules ϕ : P∗ (F) → A1 compatible with projections to g∗ (F)D and equal to the projection ωD  ω on ωD ⊂ P∗ (F). Such ϕ amounts to a rule that assigns to a G-invariant 1form µ on FR along the preimage of the horizontal foliation on R` and ∇R as above an element ϕ(µ)∇R ∈ R. Our φ should be R` -linear with respect to µ, functorial with respect to morphisms of R` , and should satisfy the following properties: (i) ϕ(µ)∇R = µ if µ came from Spec R` ; i.e., µ is a 1-form along the horizontal leaves on Spec R` = an element of R` ⊗ ω = R. (ii) ϕ(µ)∇R +ψ = ϕ(µ)∇R + µ(ψ) for any ψ ∈ g(F) ⊗ ω ⊗ R` = g(F) ⊗ R. (iii) ϕ(gµ)g∇R = ϕ(µ)∇R for any g ∈ G(R` ). Set ϕ(ν)∇R := ∇∗R (ν). Here ∇R is considered as a lifting of a horizontal vector field on Spec R` to a G-invariant vector field on FR . The above properties are evident.  Combining the above identification with (2.5.9.2), we get a canonical Aut(JF)equivariant morphism (2.5.9.3)

ακ : g(F)κD → A1

which is the identity on ω and lifts the morphism g(F)D → g∗ (F)D defined by κ. It is an isomorphism if κ is non-degenerate. Notice that every connection ∇R as above defines a splitting of the h(R)extension h(A1 ⊗ R) of g∗ (F) ⊗ R` whose image is the kernel of the retraction

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h(A`1 ⊗ R) → h(R) defined by the “value at ∇” morphism A`1 ⊗ R` → R` . It yields, via (2.5.9.3), a splitting s∇R : g(F) ⊗ R` → h(g(F)κD ⊗ R` ). It is clear that splittings s∇R are functorial with respect to the morphisms of the R` ’s (base change) and satisfy the following properties: (a) For any ψ ∈ g(F) ⊗ R one has s∇R +ψ (a) = s∇R (a) − κ(ψ, a). (b) For g ∈ Aut(F)(R` ) one has g(s∇R ) = sg(∇R ) . Corollary. g(F)κD is a unique ω-extension of g(F)D equipped with an action of Aut(JF) that integrates the adjoint action of g(F)D on g(F)κD and a map s from κ ˜ Conn(F) to splittings of h(g(F)  D ) which satisfies (a), (b). Remarks. (i) The construction of g(F)κ in terms of connections immediately generalizes to the situation when F is a G-torsor on algebraic DX -space Y. Here Conn(F) is the space of horizontal connections ∇h on F, i.e., connections along the horizontal foliation of Y. Notice that any “vertical” (i.e., relative to X) connection ∇v on F provides a connection ∇vConn : Conn(F) → Ω1Y/X ⊗ g(F)ω on the torsor Conn(F), a derivation d∇v of Ω·Y/X ⊗ g(F), and its lifting dκ∇v to Ω·Y/X ⊗ g(F)κ such that dκ∇v s∇h (a) = s∇h d∇v (a) − (∇vConn (∇h ), a). (ii) g(F) is the OX -linear part of the Lie OX -algebroid A(F) of all infinitesimal symmetries of (F, X). Our g(F)κD comes from a canonical ω-extension of A(F)D . For every connection ∇ on F this extension canonically splits over ∇(ΘD ). See [BS] where a more general setting is discussed. 2.5.10. The Virasoro extension. This is an ω-extension of the Lie∗ algebra ΘD . Its construction is similar to that of the Kac-Moody extension with the torsor of connections on F replaced by the torsor of projective connections on X. (a) Let X (n) be the nth infinitesimal neighborhood of the diagonal X ,→ X ×X for some n ≥ 0, so OX (n) := OX×X /In+1 where I is the ideal of the diagonal. The scheme X (n) carries a canonical involution σ (transposition of coordinates); the action of the Lie algebra ΘX on X = X (0) extends in the obvious (diagonal) way to an action on X (n) commuting with σ. The filtration Ia OX (n) is (ΘX , σ)-invariant. ⊗a One has gra OX (n) = ωX (for a ≤ n) with the standard action of ΘX ; σ acts by a multiplication by (−1) . An O-module E on X (n) yields, by Example (i) in 2.1.8, the D-module ED on X. Our E carries a canonical filtration Ia E; if E is a locally free OX (n) -module, ⊗a then gra E = EX ⊗ ωX for a ≤ n,65 so the corresponding filtration on ED has ⊗a successive quotients (EX ⊗ ωX )D . If E is σ-equivariant, then σ acts on ED . We σ have the plain sheaf of σ-invariants E σ and the D-module ED := (ED )σ ; one has σ σ h(ED ) = E . An action of ΘX on E which is a bidifferential operator yields an action of the Lie∗ algebra ΘD on ED (see 2.2.4(i)). If the actions of σ and ΘX on E commute, then the actions of σ and ΘD on ED also commute; thus ΘD acts on σ ED . Remark. The projection O∗X  O∗X (n) has a canonical section s : O∗X → O∗X (n) uniquely defined by the properties that it has an ´etale local nature and s(f 2 ) = f  f |X (n) . It is clear that s is σ-invariant and commutes with the action of ΘX . Explicit formula: s(f )(x, y) = f (x) + 21 f 0 (x)(y − x) + 14 (f 00 (x) − f 0 (x)2 /f (x))(y − x)2 + . . . . So a line bundle L on X yields a σ-equivariant line bundle Ls on X (n) . To construct 65 Here

EX := E/IE.

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105

Ls one should choose (locally) a square root L⊗1/2 ; then Ls = L⊗1/2  L⊗1/2 |X (n) . If ΘX acts on L, then Ls is a (σ, ΘX )-equivariant line bundle. (b) We will use (a) with n = 2 to construct an ωX -extension of the Lie∗ algebra ΘD . We follow the format of 2.5.8, so our Lie∗ algebra extension comes from a D-module extension of Θ\D by ωD equipped with a Lie∗ action of ΘD which equals the usual action on the subquotients. There are two ways to define such Θ\D : (i) Assume we have a (σ, ΘX )-equivariant line bundle ν on X (2) together with σ an equivariant identification νX := ν/Iν = ΘX .66 Our extension is Θ\D (ν) := νD . (2) (ii) Assume we have a (σ, ΘX )-equivariant line bundle λ on X together with an equivariant identification λX = OX .67 Then λσ is a ΘX -equivariant extension ⊗2 ⊗2 of OX by ωX , so the preimage P(λ) of 1 ∈ OX is a ΘX -equivariant ωX -torsor. Tensoring it by ΘX , we get a ΘX -equivariant OX -module extension P(λ) of ΘX by ωX (so P(λ) is the torsor of OX -linear splittings of P(λ)). Set Θ\D (λ) := P(λ)D . Remark. For λ, ν as above the extension Θ\D (λ ⊗ ν) is the Baer sum of extensions Θ\D (λ) and Θ\D (ν). Similarly, Θ\D (λ ⊗ λ0 ) is the Baer sum of Θ\D (λ) and Θ\D (λ0 ). As was explained in 2.5.8, any Θ\D as above yields (by push-out by the canonical morphism ωD → ωX ) a ΘD -module extensions Θ[D of ΘD by ωX ; the ΘD -action can be rewritten as a binary ∗ operation on Θ[D which lifts the Lie∗ bracket on ΘD . If this operation happens to be skew-symmetric, then this is a Lie∗ bracket, so Θ[D is a Lie∗ algebra extension of ΘD . (c) We apply the format of (b) to the following (σ, ΘX )-equivariant line bundles. Our ν is the restriction of OX×X (∆) to X (2) ; σ is the transposition action multiplied s . by −1, and ΘX acts in the obvious way. Set λ := ν ⊗ ωX We will see at the end of (d) below that the bracket on Θ[D (λ) is skewsymmetric, so Θ[D (λ) is a Lie∗ algebra extension of ΘD by ωX . Definition. Θ[D (λ) is called the Virasoro Lie∗ algebra. For c ∈ k the Virasoro extension of central charge c is the c-multiple of the extension Θ[D (λ); we denote it (c) by ΘD . Remark. Since ΘD is a perfect Lie∗ algebra,68 any its ω-extension is rigid. Lemma. Θ[D (ν) coincides with the Virasoro extension of central charge −2. Proof. We will show that the extension Θ\D (ν) is canonically isomorphic to the −2-multiple of Θ\D (λ). Such an identification amounts to a canonical splitting of Θ\D (λ⊗2 ⊗ ν). To define this splitting, notice that λ⊗2 ⊗ ν is the restriction of ωX  ωX (3∆) to X (2) . Let Res1 , Res2 : ωX  ωX (∞∆) → ωX be the residue around the diagonal along the first or the second variable. The restriction of Resi to ωX  ωX (∆) is the obvious projection ωX  ωX (∆) → ωX  ωX (∆)/ωX  ωX = ωX , so the Resi yield ΘX -equivariant retractions of λ⊗2 ⊗ ν to ωX ⊂ λ⊗2 ⊗ ν. The restrictions of the corresponding D-module retractions ResiD : λ⊗2 ⊗ ν → ωD to (λ⊗2 ⊗ ν)σD = Θ\D (λ⊗2 ⊗ ν) coincide; this retraction defines the desired splitting.  66 Here

σ acts on ΘX trivially. σ acts on OX trivially. 68 A proof of this fact is contained in the proof of the sublemma in 2.7.3. 67 Here

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⊗2 (d) Elements of the ωX -torsor P := P(λ) (see (b)(ii), (c)) are called projective connections.

Remarks. (i) A local coordinate t on X yields a projective connection ρt defined as the restriction of (x − y)−1 dx1/2 dy 1/2 to X (2) .69 An immediate calculation 1 000 f (t)dt2 . shows that (f (t)∂t )(ρt ) = 12 ⊗m/2 ⊗m/2 ⊗m (ii) We have λ = (ωX ωX )(m∆)/((m−3)∆). So for m = 3, 4 one has ⊗m/2 ⊗m/2 ⊗m/2 ⊗1−m/2 ⊗m/2 ∗ ⊗m/2 embeddings mP ,→ (j∗ j ωX ωX )/ωX ωX = Diff(ωX , ωX ) ⊗−1/2 70 that identify projective connections with symmetric differential operators ωX ⊗3/2 → ωX of order 2 and principal symbol 1 and, respectively, skew-symmetric dif⊗−1 ⊗2 ferential operators ωX → ωX of order 3 and principal symbol 1. For an interpretation of projective connections as sl2 -opers, see sect. 3.1.6 and 3.1.7 in [BD]. As follows from Remark (i), a coordinate t yields a D-module splitting of the extension Θ\X (λ). In terms of this splitting the ΘD -action is given by the cocycle 1 000 f (t)g(t)dt. Thus the bracket on Θ[X (λ) is given by the classif (t)∂t , g(t)∂t 7→ 12 1 000 cal Virasoro 2-cocycle f (t)∂t , g(t)∂t 7→ 12 f (t)g(t)dt (modulo exact forms). It is obviously skew-symmetric, which fulfills the promise made in (c) above. 2.5.11. Let L be a Lie∗ algebra, x ∈ X a (closed) point, jx : Ux := X r {x} ,→ X. The vector space h(L)x carries the Ξx -topology (see 2.1.13). Lemma. (i) The Lie bracket on h(L)x is Ξx -continuous with respect to each variable. (ii) If jx∗ L is a countably generated DUx -module, then the Lie bracket is Ξx continuous. Proof. See (i) and (ii) in 2.2.20.



˜ x from 2.1.15, then the Remark. If we decide to play with the better objects h countability condition becomes irrelevant. Examples. If we are in the situation of 2.5.6(b) and the restriction of P to ˆ x (PD ) = P ⊗ Ox (see 2.1.14) and the Lie the complement of x is O-coherent, then h bracket comes from the bidifferential operator defining the Lie bracket on P . ˆ x∗ j ∗ gκ ) is the central extension of In the situation of 2.5.9 the Lie algebra h(j x D ∗ ∗ ˆ ˆ g(Kx ) = h(jx∗ jx gD ) by k = h(jx∗ jx ωX ) defined by the 2-cocycle a, b 7→ Resx (da, b), i.e., the usual affine Kac-Moody algebra. 2.5.12. Denote by ΞLie x (L) the topology on L at x (see 2.1.13 for terminology) whose base consists of those Lξ ∈ Ξx (L) which are Lie∗ subalgebras of L. As follows from the Remark in 2.2.7, the latter condition amounts to the fact that h(Lξ )x is a Lie subalgebra of h(L)x . Therefore one can describe the ΞLie x -topology on h(L)x as the topology whose base is formed by all Ξx -open Lie subalgebras of h(L)x . The Lie bracket on h(L)x is automatically continuous with respect to the ΞLie x topology. Indeed, the base of our topology is formed by Lie subalgebras, and for every open Lie subalgebra V the adjoint action of V on h(L)x /V is continuous by ˆ Lie (L) is a topological Lie algebra. 2.5.11(i). Therefore the completion h x 69 Here 70 Here

x, y are coordinates on X × X corresponding to t. we identify P with 3P, 4P by means of multiplications by 3 and 4.

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Remark. As in Remark (i) in 2.1.13 we can also consider ΞLie as a topology x ˆ Lie (L). on the Lie algebra h(LOx ); the corresponding completion coincides with h x ˆ x (L) is a topological Lie algebra if L satisfies the condition of Similarly, h ˆ x (L) → h ˆ Lie (L) of Lie 2.5.11(ii). We have a canonical continuous morphism h x algebras. The two topologies do not differ if L satisfies a coherency condition: 2.5.13. Lemma. (i) Let M be an L-module such that its restriction to Ux is a coherent D-module. Then for any Mξ , Mξ0 ∈ Ξx (M ) such that Mξ0 is a coherent DX -module, there exists Lζ ∈ ΞLie x (L) such that Lζ (Mξ 0 ) ⊂ Mξ . (ii) If the restriction of L to Ux is a coherent D-module, then the topologies Ξx (L) and ΞLie x (L) coincide. Remark. Using the terminology of 2.7.9 the above statements can be reformuˆ x (M ) is a Tate vector space (see 2.1.13 and 2.7.9 lated as follows. Recall that F := h ˆ x (M ) for terminology), and (i) just says that the action morphism h(L)x → End h Lie is continuous with respect to the Ξx -topology and the topology on End F introˆ x (L) is a topological Lie algebra which is duced in 2.7.9. In the situation of (ii), h a Tate vector space. The argument below actually proves that any such object has a base of the topology formed by open Lie subalgebras. Proof. (i) Replacing Mξ by Mξ ∩Mξ0 we may assume that Mξ ⊂ Mξ0 . According to 2.2.20(iii), the maximal sub-D-module Lζ ⊂ L such that Lζ (Mξ0 ) ⊂ Mξ belongs to Ξx (L). It is also a Lie∗ algebra, so we are done. (ii) We want to show that every Lξ ∈ Ξx (L) contains some Lζ ∈ ΞLie x (L). We can assume that Lξ is a coherent D-module. According to the proof of (i) the normalizer N of Lξ is open. Set Lζ := N ∩ Lξ .  can be very 2.5.14. In the non-coherent situation the topologies Ξx and ΞLie x different: Example. Consider the Lie∗ algebra L = gl(DX ). Then h(L) is DX conˆ x (L) is a sidered as a sheaf of Lie algebras. For every x ∈ X the completion h Lie algebra because L satisfies the condition of 2.5.11(ii). Our L is an induced ˆ x (L) is the Lie algebra DX -module, so, by Example (i) of 2.1.13, the completion h D = DOx of differential operators with coefficients in Ox (the formal completion of the local ring at x). Denote by D≤n the subspace of differential operators of order ≤ n; this is a free Ox -module of rank n + 1. A vector subspace of D is open in the Ξx -topology if its intersection with every D≤n is open. The ΞLie x -topology is defined by Ξx -open Lie subalgebras. It is easy to see that for every open subspace P ⊂ D≤2 the Lie subalgebra generated by P is open. Such subalgebras form a Lie base of the topology ΞLie x ; in particular, the topology Ξx , as opposed to Ξx , has a countable base. Let us describe the ΞLie x -topology on D explicitly assuming that dim X = 1. Let t be a local parameter at x, so Ox = k[[t]]. Set ∂ = ∂t . For a ≥ 0, b ≥ 1 let Da,b ⊂ D be the space of differential operators Σφi (t)∂ i , φi (t) ∈ ta+bi Ox . Lemma. The Da,b form a basis of the ΞLie x -topology on D. Proof. Indeed, the Da,b are open in the Ξx -topology and they are associative, hence Lie, subalgebras of D. It remains to check that any open Lie subalgebra

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Q ⊂ D contains some Da,b . Set Dia,b := Da,b ∩ D≤i . A simple calculation shows that the operator Ab := adtb+1 ∂ 2 ∈ End D preserves Da,b . It sends D≤i to D≤i+1 and the corresponding operator gri Da,b → gri+1 Da,b is an isomorphism if a ≥ 1, i ≥ 0. Therefore Da,b = ⊕ Aib (ta Ox ). Now choose a, b ≥ 1 such that ta Ox ⊂ Q i≥o

and tb+1 ∂ 2 ∈ Q. Then Da,b ⊂ Q.



ˆ Lie (L) consists of “differential operators of infinite We see that the completion h x order” Σφi (t)∂ i such that the number of the first non-vanishing coefficient of φi (t) grows faster than any linear function of i. Remarks. (i) We will see in 2.7.13 that the above L carries an important topology which is weaker than ΞLie x . (ii) The above wonders seem to be artificial: they disappear if we consider ˆ x or h ˆ Lie the more sophisticated object h ˜ x (L) from 2.1.15. instead of h x 2.5.15. Let L be any Lie∗ algebra. Notice that for an h(L)x -module V the action map h(L)x ×V → V is continuous with respect to the ΞLie x -topology (and the discrete topology on V ) if and only if it is continuous with respect to the Ξx -topology (since the stabilizer of every v ∈ V is a Lie subalgebra). Such h(L)x -modules are ˆ x (L)mod. called discrete; the corresponding category is denoted by h Let M(X, L)x be the category of L-modules supported at x. For M ∈ M(X, L)x the Lie algebra h(L)x acts canonically on M ; hence i!x M = h(M ) is an h(L)x module. By 2.5.13(i) it is discrete. ˆ x (L)mod is an equivalence of cateLemma. The functor h : M(X, L)x → h gories. Proof. Follows from 2.5.4.



2.5.16. Lie∗ algebroids. Let R` be a commutative DX -algebra. According to 1.4.11, we have the notion of Lie∗ R-algebroid. We denote by M(X, R, L) the category of L-modules (see 1.4.12). As follows from the lemma in 2.3.12, these objects have an ´etale local nature. Therefore we know what a Lie∗ algebroid L on any algebraic DX -space Y is and what are L-modules are. Then h(L) is a sheaf of Lie algebras (on Ye´t ) acting on OY by horizontal OX -derivations. The constructions of 1.4.14 are ´etale local as well (for the same reason). So for a Lie∗ algebroid L on Y which is a vector DX -bundle on Y (see 2.3.10) we have a commutative DG (super)algebra C(L)Y called the de Rham-Chevalley complex of L, and for an L-module M we have a DG C(L)Y -module C(L, M )Y . If we forget about the differential, then C(L)Y = Sym(L◦ [−1]), C(L, M )Y = Sym(L◦ [−1]) ⊗ M where L◦ is the vector DX -bundle on Y dual to L. For example, if ΩY is finitely presented, then the tangent Lie algebroid ΘY is well defined (see 1.4.16 and Remarks in 2.3.12). If ΩY is a vector DY -bundle, which happens if Y is smooth (see 2.3.15), then h(ΘY ) is the Lie algebra of all horizontal OX -derivations of OY . One has C(ΘY )Y = DRY/X (the relative de Rham complex). More generally, if V is a vector DX -bundle on such Y, then the Lie∗ algebroid E(V ) on Y (see Remark (iii) of 1.4.16) is well defined. The morphism τ : E(V ) → ΘY is surjective, so E(V ) is an extension of ΘY by gl(V ). 2.5.17. Equivariant Lie∗ algebroids. The format of equivariant Lie∗ algebroids is very parallel to the one of the usual equivariant Lie algebroids (see,

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e.g., [BB] 1.8). Below we denote by S a test DX -algebra, YS := Y × Spec S; for X

M ∈ M(Y ) set MS := M ⊗ S ∈ M(YS ). Suppose we have a group DX -scheme G acting on Y and a Lie∗ algebroid L on Y. A weak G-action on L is an action of G on L as on an OY [DX ]-module which preserves the Lie∗ bracket and the L-action on OY . In more details, for any test DX -algebra S the OYS [DX ]-module LS is naturally a Lie∗ algebroid on YS (acting along the fibers of the projection YS → Y); the group of DX -scheme points Spec S → G acts on YS . Now a weak G-action on L is a rule which assigns to each S the action of the above group on the Lie∗ OYS -algebroid LS in a way compatible with the base change. Suppose G is smooth; set L := Lie(G). A strong G-action on L is a weak G-action together with a morphism of Lie∗ algebras α : L → L which satisfies the following conditions: (i) α is compatible with the Lie∗ actions on OY and with the G-actions (the G-action on L is the adjoint one). (ii) The L-action on L coming from the G-action coincides with the L-action via α and the adjoint action of L. We refer to a Lie∗ algebroid equipped with a weak or strong G-action as a weakly or strongly G-equivariant Lie∗ algebroid. Example. If V is a G-equivariant vector DX -bundle then E(V ), if well defined, is a strongly G-equivariant Lie∗ algebroid. As in the setting of the usual Lie algebroids, the strong G-action can be naturally interpreted as follows (we give a brief sketch of constructions leaving the details to the reader). If π : Z → Y is a smooth morphism of algebraic DX -spaces which is flat as a morphism of the usual schemes, then the pull-back π † (L) of L is defined. This is a Lie∗ algebroid on Z constructed as follows (see 2.9.3 for a parallel construction in the setting of the usual Lie algebroids). Consider the right OZ [DX ]-module π ∗ L := OZ ⊗ π −1 L. The action of L on OY is a morphism of right OZ [DX ]π −1 OY

modules L → HomOY [DX ] (ΩY , OY [DX ]); pulling it back to Z, we get a morphism π ∗ L → HomOZ [DX ] (π ∗ ΩY , OZ [DX ]). A section of π † L is a pair (`, θ) where ` ∈ π ∗ L and θ ∈ HomOZ [DX ] (ΩZ , OZ [DX ]) a morphism whose restriction to π ∗ ΩY ⊂ ΩZ coincides with the morphism defined by `. As a mere OZ [DX ]-module, π † L is an extension of π ∗ L by ΘZ/Y . Now π † L is naturally a Lie∗ algebroid on Z; for an L-module M its pull-back ∗ π M to Z is naturally a π † L-module. The functors π † are compatible with the composition of π’s and satisfy the descent property. Therefore, if a smooth group DX -scheme G acts on Y, then we know what a G-action on L is. We leave it to the reader to check that this is the same as a strong G-action on L as defined above. 2.5.18. Here is a local version of the above picture (cf. 2.5.11–2.5.13). The following terminology will be of use. Let Q be a topological commutative algebra and P a Q-algebroid equipped with a linear topology. One says that P is a topological Lie Q-algebroid if the bracket on P , the Q-action on P , and the P -action on Q are continuous.71 For the rest of the section we assume that dim X = 1. Let 71 We

assume, as always, that the topologies on P and Q are complete.

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x ∈ X be a point, Ux := X r {x} its complement. Let R` be a commutative D-algebra on Ux , L a Lie∗ R-algebroid. Since L is an R` -module, jx∗ L carries the topology ΞR x at x (see 2.4.11). as Lemma. (i) The action of hx (jx∗ L) on hx (jx∗ R) is ΞR x -Ξx -continuous with R respect to each variable. The Lie bracket on hx (jx∗ L)x is Ξx -continuous with respect to each variable. (ii) If R` is a countably generated DUx -algebra and L is a countably generas R ated R` [DUx ]-module, then the action is ΞR x -Ξx -continuous and the bracket is Ξx continuous.

¯ ¯−1 (r) ∈ Proof. (i) Take Rξ ∈ Ξas x (jx∗ R). For l ∈ hx (jx∗ L) one has {r ∈ Rξ : l as R ¯ Rξ } ∈ Ξx . For r¯ ∈ hx (R) one has {l ∈ L : l(¯ r) ∈ hx (Rξ )} ∈ Ξx . (ii) Take Rξ as above and Lζ ∈ ΞR which is an Rξ -module. It suffices to find x as Lη ∈ ΞR , R ∈ Ξ such that the action map on Lη , Rν takes values in Rξ and the ν x x bracket on Lη takes values in Lζ . Together with (i) this proves the continuity. Let P ⊂ Rξ , Q ⊂ Lζ be countably generated D-submodules such that P |Ux generates R` as a DUx -algebra and Q|Ux generates L as an R` [DUx ]-module. According to 2.2.20(ii), there exists P 0 ∈ Ξx (P ), Q0 ∈ Ξx (Q) such that the action map on Q0 , P 0 takes values in Rξ and the bracket on Q0 takes values in Lζ . Now let Rν be the DX -subalgebra of jx∗ R generated by P 0 , and let Lη be the Rν -submodule of jx∗ L generated by Q0 .  ˆ R (L) := We see that the countability conditions of (ii) in the lemma imply that h x as R ˆ hx (jx∗ L) is a topological Lie Rx -algebroid. ˜x Remark. The countability condition is irrelevant in the setting of objects h from 2.1.15. L be the set of pairs (Rξ , Lξ ) ∈ Ξas 2.5.19. For arbitrary R, L let ΞR x (jx∗ R) × x ∗ Ξx (jx∗ L) such that Lξ is a Lie Rξ -subalgebroid of jx∗ L.72 The corresponding Rξ and Lξ form topologies on jx∗ R and jx∗ L at x which we denote by ΞasL x (R), asL ˆ LieR (jx∗ L) be the completions. ˆ LieR (L) := h ˆ asL (jx∗ R), h := h (L). Let R ΞLieR x x x x x ˆ LieR (L) a topological Lie Then RxasL is a commutative topological algebra and h x asL Rx -algebroid.

Lemma. Suppose that R` is a finitely generated DUx -algebra and L is a finitely generated R` [DUx ]-module. Then the above topologies on jx∗ R and jx∗ L coincide: asL LieR ˆ LieR (L) = h ˆ R (L). one has Ξas and ΞR . Thus RxasL = Rxas , h x = Ξx x = Ξx x x ` Proof. Take Rξ ∈ Ξas x (jx∗ R) and Lξ ∈ Ξx (jx∗ L) which is an Rξ -submodule of RL jx∗ L. We want to find (Rζ , Lζ ) ∈ Ξx such that Rζ ⊂ Rξ , Lζ ⊂ Lξ . Shrinking Rξ , Lξ if necessary, we can assume that Rξ is a finitely generated DX -algebra and Lξ is a finitely generated Rξ` [DX ]-module. Set Rζ = Rξ and define Lζ to be the intersection of Lξ and the normalizers of Rξ and Lξ in jx∗ L. We leave it to the reader to show that (Rζ , Lζ ) ∈ ΞRL  x .

2.5.20. Suppose that R` is a finitely generated DUx -algebra and our L is a projective R` [DUx ]-module of finite rank. By 1.4.14, the dual module L◦ is naturally 72 I.e., L is a Lie∗ subalgebra and an R` -submodule of j L, and the action of L on j R x∗ x∗ ξ ξ ξ preserves Rξ .

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a Lie! coalgebra in M(X, R` ) which yields (see 1.4.15) a formal DX -scheme groupoid ˆ R (L◦ ) is a topological G on Spec R` . Then (see Remark in 2.4.11 and 2.4.10(iii)) h x as as Lie Rx -coalgebroid and Gx is a formal groupoid on Spec Rxas . On the other hand, the duality ∗-pairing between L and L◦ yields a continuous as ˆ R (L) × h ˆ R (L◦ ) → Ras . Rx -bilinear pairing h x x x Exercise. Show that the latter pairing is non-degenerate, i.e., identifies either ˆ R (L), h ˆ R (L◦ ) with the topological dual to another. of the topological Rxas -modules h x x as ˆ R (L◦ ) is dual to the topoThen the topological Lie Rx -coalgebroid structure on h x as R R ◦ ˆ (L), and h ˆ (L ) identifies canonically with logical Lie Rx -algebroid structure on h x x the Lie Rxas -coalgebroid of the formal groupoid Gas x . 2.5.21. Let Q be a reasonable topological commutative algebra (see 2.4.8), ΘQ the Lie Q-algebroid ΘQ of continuous derivations of Q. Then Q carries a natural topology whose base is formed by all subalgebroids ΘQ;I,S ⊂ ΘQ ; here I ⊂ Q is a reasonable ideal, S ⊂ Q/I a finite set, and ΘQ;I,S := {τ ∈ ΘQ : τ (I), τ (S) ⊂ I}. This topology makes ΘQ a topological Lie Q-algebroid. It is a final object in the category of topological Lie Q-algebroids. Suppose R` is smooth (see 2.3.15). Then Rxas is reasonable and formally smooth (see Lemma in 2.4.8 and 2.4.10(ii)), so we have a topological Lie Rxas -algebroid ΘRxas . The Lie∗ R-algebroid ΘR satisfies the conditions of 2.5.19; it defines a topological ˆ R (ΘR ). Lie Rxas -algebroid h x ∼ ˆ R (ΘR ) −→ Proposition. One has h ΘRxas . x

ˆ R (ΘR ) → ΘRas of topological Proof. Consider the canonical morphism ψ : h x x as Lie Rx -algebroids. We want to prove that ψ is an isomorphism. It suffices to show that it is an isomorphism of abstract vector spaces. This implies automatically that ψ is a homeomorphism since ψ is continuous and our topological vector spaces admit a countable base of open linear subspaces of countable codimension. Take any Rξ` ∈ Ξas x (R). By 2.4.9, there is a canonical morphism of DX -algebras ` ` b ) and an identificajx∗ R` → Rξx ⊗ Kx (corresponding to the projection Rxas → Rξx ∼ ` ` ` b ` b as ⊗ Kx )). tion Dercont (Rx , Rξx ) −→DerD (jx∗ R , Rξx ⊗ Kx ) = h(jx∗ ΘR ⊗ (Rξx jx∗ R`

`

If V is any finitely generated projective R [DUx ]-module, then the vector space R ` b h(jx∗ V ⊗ (Rξx ⊗ Kx )) equals the completion of hx (jx∗ V ) with respect to the Ξx ξ jx∗ R`

R

topology (see 2.4.11). Indeed, each submodule N ∈ Ξx ξ (jx∗ V ) yields a projection ` b ` b ` b jx∗ V ⊗ (Rξx ⊗ Kx ) = N ⊗ (Rξx ⊗ Kx )  N ⊗ (Rξx ⊗ Kx /Ox ) = ix∗ Nx` , hence a jx∗ R`

Rξ`

projection h(jx∗ V map h(jx∗ V

⊗ jx∗ R

⊗

` (Rξx `

Rξ`

b Kx ))  ⊗

jx∗ R ` b (Rξx ⊗ Kx )) `

Nx` .

Passing to the limit, we get a linear

R

ˆ x ξ (jx∗ V ). To see that this is an isomorphism, it →h

suffices (since both sides are additive in V ) to consider the case V = R` [DUx ]; here b Kx . both our vector spaces equal Rξ` ⊗ ∼ ˆ Rξ Taking V = ΘR , we get an isomorphism φξ : Dercont (Ras , R` ) −→ h x (jx∗ ΘR ). x

ξx

Passing to the projective limit with respect to the Rξ` ’s, we get an isomorphism of ∼ ˆR vector spaces φ : ΘRxas −→ h  x (ΘR ) which is left inverse to ψ. We are done.

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2.5.22. Elliptic algebroids. As above, we assume that X is a curve. Suppose we have a morphism φ : M → N of vector DX -bundles on an algebraic DX -space Y. We say that φ is elliptic if it is injective and Coker τ is a locally projective OY -module of finite rank. This property is stable under duality: φ is elliptic if and only if φ◦ : N ◦ → M ◦ is. Notice that (Coker φ)` and (Coker φ◦ )` are mutually dual vector bundles with horizontal connections on Y. Suppose Y is smooth, and let L be a Lie∗ algebroid on Y which is a vector DX -bundle. We say that L is elliptic if the anchor morphism τ : L → ΘY is elliptic. Let G be the formal DX -scheme groupoid G on Y that corresponds to L (see 1.4.15). Then L is elliptic if and only if G is formally transitive and the corresponding formal group DX -scheme on Y – the restriction of G to the diagonal – is smooth as a mere formal Y-scheme. The Lie algebra g`L of the latter formal group scheme equals (Coker τ )` as a mere vector bundle with a horizontal connection on Y. The Lie bracket on (Coker τ )` comes from the cobracket on the dual vector bundle (Coker τ ◦ )` which is the quotient of the Lie! coalgebroid L◦ (see 1.4.14). Notice that G acts on g`L by adjoint action, and this action is compatible with the Lie bracket. Thus gL is a Lie algebra in the tensor category of L-modules. The projection ΘY  Coker τ = gL is compatible with the L-actions where L acts on ΘY in the adjoint way. 2.5.23. Some important elliptic algebroids arise in the following way. Let G be an algebraic group with Lie algebra g, Y a smooth algebraic DX -space, FY a Gtorsor on Y equipped with a horizontal connection ∇h ; i.e., (FY , ∇h ) is a DX -scheme G-torsor. Let A be the universal groupoid that acts on (F, Y); i.e., a groupoid on Y whose arrows connecting two points y, y 0 of Y are isomorphisms of G-torsors ∼ FYy −→ FYy0 . The connection on FY yields a horizontal connection on the groupoid, so it is a DX -scheme groupoid. In other words, if y, y 0 were DX -scheme points, then a DX -scheme point of our groupoid connecting them is an isomorphism of G-torsors ∼ FYy −→ FYy0 compatible with the connections. Let Aˆ be the formal completion of A and let L◦ be its Lie OY -coalgebroid. We say that (FY , ∇h ) is non-degenerate if Aˆ is formally smooth over Y in the DX -scheme sense or, equivalently, if L◦ is a vector DX -bundle on Y. If this happens, then the dual Lie∗ algebroid L is defined (see 1.4.15 and 1.4.14). It is automatically elliptic. The corresponding Lie algebra g`L equals g(FY ) (the FY -twist of g with respect to the adjoint action). The non-degeneracy condition can be checked as follows. Our L◦ consists of G-invariant 1-forms on FY relative to X; it is an extension of g∗ (FY ) by Ω1Y/X . Such a left OY [DX ]-module extension amounts to an extension of OY by g(FY ) ⊗ Ω1Y/X ; let α ∈ Γ(Y, h(g(FY ) ⊗ Ω1Y/X )) be its (local) class. To compute it explicitly, choose a vertical connection ∇v on FY and consider the component of the curvature of ∇v + ∇h which is a section of g(FY ) ⊗ Ω1Y/X ⊗ ωX ; our α is the class of this form. Now α amounts to a morphism of OY [DX ]-modules Θ`Y → g(FY ). Then (FY , ∇h ) is non-degenerate if and only if the latter morphism is surjective. Example. We follow the notation of 2.5.9. Let F be a G-torsor on X. Set Y := Conn(F); let FY be the pull-back of F to Y and let ∇h be the universal horizontal connection. Then (FY , ∇h ) is non-degenerate. To see this, consider the gauge action of the group DX -scheme JAut(F) on Y and FY . The corresponding

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groupoid equals A(FY ). Therefore L◦ equals the pull-back of g∗ (F)`D to Y, which is evidently a vector DX -bundle on Y, and we are done. Notice that L is the Lie∗ algebroid defined by the gauge action of the Lie∗ algebra g(F)D on Y (see 1.4.13). For another example see 2.6.8. 2.6. Coisson algebras The coisson algebra structure appeared in the study of Hamiltonian formalism in the calculus of variations at the end of 1970s, see [DG1], [DG2], [Ku], [KuM], [Ma], and the recent book [Di] and references therein. The reader may also look in Chapter 15 of [FBZ] or [DLM] (where the term “vertex Poisson algebra” is used). We discussed coisson algebras in the general setting of compound tensor categories in 1.4. Below we show that the functor R 7→ Rxas transforms coisson algebras to topological Poisson algebras. For a global version of this statement see 4.3.1(iii) and 4.7.3. We consider then elliptic coisson structures and treat briefly two geometric examples, namely, the space of connections on a given G-bundle and the space of opers; for the latter subject see [DS], [BD], [Fr], or [FBZ] 15.6, 15.7. 2.6.1. According to 1.4.18 we have the category Cois(X) = Cois(M∗! (X)) of coisson algebras on X. Explicitly, a coisson algebra is a commutative DX -algebra A` together with a Lie∗ bracket { } on A (the coisson bracket) such that for any local sections a, b, c ∈ A` and ν ∈ ωX one has {abν, cν} = p∗1 a · {bν, cν} + p∗1 b · {aν, cν} ∈ ∆∗ A (we use the left p∗1 A` -module structure on ∆∗ A). Coisson brackets on a DX -algebra A` are A-bidifferential operators, so they have the ´etale local nature (see the lemma in 2.3.12). Thus they make sense on any algebraic DX -space Y. According to 1.4.18 a coisson structure on Y yields a Lie∗ algebroid structure on ΩY (which determines the coisson structure uniquely). 2.6.2. For the rest of the section we assume that X is a curve. Let x ∈ X be a (closed) point, jx : Ux := X r {x} ,→ X its complement. Let A` be a coisson algebra on Ux . Consider the commutative topological algebra Aas x defined in 2.4.8. Recall that Aas x is the completion of the vector space h(jx∗ A)x with respect to the Ξas x -topology. The coisson bracket defines a Lie algebra structure on h(jx∗ A)x . Lemma. (i) The Lie bracket is Ξas x -continuous with respect to each variable. (ii) If A` is a countably generated DUx -algebra,73 then the Lie bracket is Ξas x continuous. Its continuous extension to Aas is a Poisson bracket. x Proof (cf. 2.5.11). (i) We want to show that for every ¯l ∈ h(jx∗ A)x the operator as ad¯l is Ξas x -continuous. By 2.2.20(i) it is Ξx -continuous; hence for every Aξ ∈ Ξx −1 −1 as its preimage ad¯l (Aξ ) is Ξx -open. Since ad¯l is a derivation, ad¯l (Aξ ) ∩ Aξ ∈ Ξx ; q.e.d. as 74 (ii) We will find for every Aξ ∈ Ξas x some Aζ ∈ Ξx such that {Aζ , Aζ } ⊂ Aξ . as Together with (i) it proves the Ξx -continuity of the bracket. Our Aξ is a countably generated D-module. According to 2.2.20(ii) the morphism h(Aξ )x ⊗ h(Aξ )x → h(jx∗ A)x coming from the coisson bracket is Ξx -continuous. So for some L ∈ Ξx (Aξ ) one has {h(L)x , h(L)x } ⊂ h(Aξ )x ; i.e., {L, L} ⊂ Aξ . Our Aζ is the subalgebra generated by L.  this means that A is a countably generated D-module. 2.5.3 for the definition of {Aζ , Aζ }.

73 Equivalently, 74 See

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2.6.3. Assume that A satisfies the countability condition of (ii) in the above lemma. We define the bracket on Aas x extending the bracket on hx (jx∗ A) by continuity. It is automatically a Poisson bracket since h(jx∗ A) acts on jx∗ A by derivations. ˆ A (ΩA ) = lim Ω ` , Aξ ∈ Ξas (A); i.e., it is Remark. It is easy to see that h x x ←− Aξx as ˆ the module ΩAas of the topological Kahler differentials of Ax . Now a coisson x structure on A yields the Lie A-algebroid structure on ΩA , hence, according to ˆA 2.5.18, the topological Lie Aas x -algebroid structure on hx (ΩA ). On the other hand, as ˆ Aas carries the topological Lie A -algebroid structure coming from the Poisson Ω x x bracket on Aas x . The two structures coincide. Remark. The countability condition becomes irrelevant if we consider instead of plain topologies the finer structure from 2.1.15. Let A` be an arbitrary (not necessarily countably generated) coisson algebra. Denote by Ξcois the topology at x on jx∗ A formed by all Aξ ∈ Ξas x x which are coisson subalgebras of jx∗ A. The coisson bracket on hx (jx∗ A) is continuous with respect to this topology,75 so the corresponding completion is a topological Poisson algebra which we denote by Acois x . is the completion of Aas If A` is countably generated, then Acois x with respect x to the topology formed by those open ideals which are Lie subalgebras of Aas x . 2.6.4. Lemma. If A` is a finitely generated DX -algebra, then the topologies cois and Ξas = Aas x on jx∗ A coincide, so Ax x .

Ξcois x

as Proof (cf. 2.5.13). We want to show that every Aξ ∈ Ξas x contains Aζ ∈ Ξx which is a coisson subalgebra. We can assume that Aξ is generated as a DX algebra by a DX -coherent submodule M ⊂ Aξ . Let Aζ ⊂ Aξ be the maximal DX -submodule such that {Aζ , Aξ } ⊂ Aξ . We will show that Aζ ∈ Ξcois x . Our Aζ is the maximal DX -submodule of Aξ such that {Aζ , M } ⊂ Aξ , so, by 2.2.20, Aζ ∈ Ξx . Clearly Aζ is a Lie∗ subalgebra. One has {Aζ · Aζ , Aξ } ⊂ Aζ · {Aζ , Aξ } ⊂ Aξ · Aξ ⊂ Aξ , so Aζ · Aζ ⊂ Aζ . 

and Ξas Remark. If A` is not finitely generated, then the topologies Ξcois x can x ` 76 be different. This happens, e.g., for A = Sym(gl(DX )) (see 2.5.14). 2.6.5. Let S ⊂ X be a finite subset, jS : US ,→ X the complementary open embedding. Let A be a countably generated DUS -algebra equipped with a coisson b as structure, so, by 2.6.2, Aas S = ⊗ As (see 2.4.12) is a topological Poisson algebra. s∈S

The canonical morphism Γ(US , h(A)) → Aas S from 2.4.12 commutes with brackets; i.e., it is a hamiltonian action of the Lie algebra Γ(US , h(A)) on Aas S . If X is proper and S is non-empty, then, according to 2.4.6 and 2.4.12, the zero fiber of Q as the momentum map coincides with Spf hAi(US ) ⊂ Spf Aas = Spf A . s S Remark. The hamiltonian reduction appears when one looks at the chiral homology of a chiral quantization (mod t2 ) of our coisson algebra. 75 Indeed, for every A ∈ Ξcois the action of the Lie algebra h (A ) on A` is continuous x ξ ξ x ξx with respect to the topology on hx (Aξ ) formed by hx (Aζ ) where Aζ ∈ Ξcois (A), Aζ ⊂ Aξ . x 76 Indeed, for any Lie∗ algebra L the topology Ξas on the coisson algebra Sym L induces on x L ⊂ Sym L the topology Ξx , and Ξcois induces on L the topology ΞLie x x .

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2.6.6. Elliptic brackets. Suppose that R` is a smooth DX -algebra, { } a coisson bracket. We say that { } is non-degenerate or symplectic if the anchor morphism τ = τ{ } : ΩR → ΘR for the Lie∗ algebroid structure on ΩR is an isomorphism. Symplectic brackets occur quite rarely though. Here is a more relevant notion: We say that our bracket is elliptic if ΩR is an elliptic Lie∗ algebroid. By 2.5.22, an elliptic { } yields a Lie algebra g{ } := gΩR in the tensor category of coisson R-modules (see 1.4.20) which is a locally projective R` -module. This g{ } vanishes if and only if { } is symplectic. Proposition. g carries a natural non-degenerate ad- and ΩR -invariant symmetric bilinear form, i.e., a morphism of coisson modules ( , ) : Sym2R` g`{ } → R` . Proof. The anchor morphism τ : ΩR → ΘR has property τ ◦ = τ (we use the identifications ΘR = Ω◦R , ΩR = Θ◦R ). Thus we have a non-degenerate skewsymmetric pairing ∈ P2∗ ({Cone(τ ), Cone(τ )}, R) which yields a tensor symmetric ` pairing g`⊗2  { } → R . We leave it to the reader to check the properties. In the rest of the section we discuss briefly two geometric examples of elliptic brackets. 2.6.7. The coisson structure on Conn(F). We follow the notation of 2.5.9. Consider the twisted Kac-Moody extension g(F)κD . Let A be its twisted symmetric algebra; i.e., A` is the quotient of Sym(g(F)κD )` modulo the relation 1κ = 1 where 1κ is the generator of OX ⊂ g(F)κD . According to Example (iii) in 1.4.18, A` is a coisson algebra; the bracket is defined by the condition that g(F)κD ,→ A is a morphism of Lie∗ algebras. The coisson bracket is denoted by { }κ . Suppose that the form κ is non-degenerate. Then, by 2.5.9, one has a canonical identification of the DX -schemes Spec A` = Conn(F). Proposition. The Lie∗ algebroid ΩA defined by { }κ identifies canonically with the Lie∗ algebroid L from Example in 2.5.23. Therefore the bracket { }κ is elliptic and the Lie A` [DX ]-algebra g{ }κ equals g(F)A . Proof. Back in 2.5.23 we identified L with the Lie∗ algebroid A` ⊗g(F)D defined by the gauge action of the Lie∗ algebra g(F)D on Conn(F). It remains to identify the latter Lie∗ algebroid with ΩA . Since Conn(F) is a torsor for the vector DX ∼ scheme of g(F)-valued forms, there is a canonical identification A` ⊗ g∗ (F)D −→ ΩA . ∼ ∼ ∗ ` Using κ : g −→ g , we can rewrite it as an identification A ⊗ g(F)D −→ ΩA . By 2.5.9, the latter is an isomorphism of Lie∗ A-algebroids, and we are done.  τ

Remark. The resolution ΩA − → ΘA of g{ }κ identifies naturally with canonical resolution (2.1.9.1) of g(F)A . The form ( , ) on g{ }κ from 2.6.6 becomes the form κ on g(F)A . 2.6.8. The coisson structure on the moduli of opers. For all details and proofs of the statements below, see the references at the beginning of the section. (a) Let G be a semi-simple group; for simplicity, we assume it to be adjoint. Let N ⊂ B ⊂ G be a Borel subgroup and its nilradical, T = B/N the Cartan torus, n ⊂ b ⊂ g, t = b/n the corresponding Lie algebras, ψ : n → k a non-degenerate character.77 Let FB be a B-torsor on X and FG , FT the induced G- and T -torsors; 77 The vector space n/[n, n] is the direct sum of lines corresponding to simple roots; “nondegenerate” means that ψ : n/[n, n] → k does not vanish on each of the lines.

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the adjoint action yields the corresponding twisted Lie OX -algebras n(F) ⊂ b(F) ⊂ g(F). For a character γ : B → Gm we denote by FγB the corresponding induced O∗X -torsor = line bundle on X. Suppose that for every simple root γ : B → Gm ∼ we are given an identification of the line bundles FγB −→ ωX . Then ψ yields a morphism n(F) → ωX which extends to a morphism of Lie∗ algebras n(F)D → ωX ; we denote it again by ψ. Let κ be an ad-invariant symmetric bilinear form on g, g(F)κD the Kac-Moody extension of gD twisted by FG , and A the coisson algebra from 2.6.7. The restriction of κ to n vanishes, so we have a canonical embedding of Lie∗ algebras α : n(F)D ,→ g(F)κD ⊂ A. Consider the corresponding BRST reduction CBRST (n(F), A)c (see 1.4.21). Proposition. The BRST reduction is regular, i.e., H 6=0 CBRST (n(F), A)c van0 ishes, and the coisson algebra Wcκ := HBRST (n(F)D , A) coincides with the hamiltonian n(F)D -reduction of A (see 1.4.26).  The above hamiltonian reduction was introduced in [DS], and Wcκ is known as the Gelfand-Dikii algebra. It can be interpreted geometrically as follows. (b) For a test DX -algebra R` an g-oper on Spec R` is a triple (FG , ∇, FB ) where FG is a G-torsor on Spec R` , ∇ is a horizontal connection on FG , and FB ⊂ FG is a reduction of FG to B. We demand that: (i) ∇ satisfies the Griffith transversality condition with respect to FB ; i.e., the horizontal form c(∇) := ∇ mod b(FB ) ∈ R` ⊗ ωX ⊗ (g/b)(FB ) takes values in n⊥ /b(FB ); (ii) c(∇) is non-degenerate; i.e., its components with respect to the simple root ∼ Q decomposition n⊥ /b −→ kγ are all invertible. The functor which assigns to R the set of isomorphism classes78 of g-opers is representable by a smooth DX -scheme Opg . (c) Suppose that κ is non-degenerate. Then there is a natural morphism of DX -schemes Spec Wcκ → Opg

(2.6.8.1)

defined as follows. Let I ⊂ A be the ideal generated by α(n(F)) ⊂ A. The canonical connection on the restriction of FG A to Spec A/I ⊂ Conn(FG ) together with the B-structure FB form a g-oper on Spec A/I. As follows from Proposition in 2.6.7, the fibers of the smooth projection Spec A/I → Spec Wcκ are orbits of the gauge action of the group DX -scheme Ker(JAut(FB ) → JAut(FT )) ⊂ JAut(FG ) whose Lie∗ algebra equals n(FB ). This action is free; it lifts to FB preserving the canonical connection. So our g-oper descends to Spec Wcκ ; this is (2.6.8.1). The proposition from 2.6.7 implies that the Lie∗ algebroid ΩWcκ defined by the coisson structure acts naturally on FG preserving ∇. Let L◦ be the Lie coalgebroid on Spec Wcκ defined by the pair (FG , ∇) as in 2.5.23.79 The above action can be rewritten as a canonical morphism of Lie coalgebroids L◦ → Θ`Wcκ .

(2.6.8.2) 78 g-opers 79 In

are rigid, i.e., admit no non-trivial automorphisms. 2.5.23 we used the notation (FY , ∇h ).

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Proposition. Both (2.6.8.1) and (2.6.8.2) are isomorphisms. So the GelfandDikii coisson structure is elliptic and the pair (FG , ∇) on Opg is non-degenerate (see 2.5.23). The form ( , ) from 2.6.6 identifies with the form κ on g(FG ).  2.7. The Tate extension In this section dim X = 1. We will discuss a canonical central extension of a matrix Lie∗ algebra on a curve. A linear algebra version of this ubiquitous extension first appeared (implicitly) in Tate’s remarkable note [T].80 It was rediscovered as a Lie algebra of symmetries of a Heisenberg or Clifford module (metaplectic representation) and studied under various names81 in the beginning of the 1980s; see [Sa], [DJKM], [DJM], [KP], [SW], [PS], and [Kap2]. For this point of view see section 3.8. As was pointed out in [Kap2], the linear algebra objects called Tate vector spaces here were, in fact, introduced by Lefschetz ([Lef], pp. 78–79) under the name of locally linearly compact spaces and were used by Chevalley (these authors did not perceive the Tate extension though). A group-theoretic version of the Tate extension was studied in the L2 setting in [PS];82 for an algebraic variant see [Kap2], [Dr2] and [BBE]. We begin in 2.7.1 with a general format for the construction of central extensions of Lie algebras. The Tate extension in the D-module setting is defined in 2.7.2 and 2.7.3; the situation with extra parameters is considered in 2.7.6. Tate’s original linear algebra construction is presented in 2.7.7–2.7.9; the two constructions are related in 2.7.10–2.7.14. The Kac-Moody and Virasoro extensions can be embedded naturally in the Tate extension; see 2.7.5. The exposition of 2.7.1–2.7.14 essentially follows [BS]; the meaning of the constructions will become clear in the chiral context of 3.8.5–3.8.6. We do not discuss representation theory of the Tate extension; for this subject see [FKRW] and [KR]. 2.7.1. Some concrete nonsense. We work in an abelian pseudo-tensor kcategory. For simplicity of notation formulas are written in terms of “symbolic elements” of our objects. (i) Let L be a Lie algebra. Denote by E(L) the category of pairs (L\ , π) where L is an L-module, π : L\ → L a surjective morphism of L-modules (we consider L as an L-module with respect to the adjoint action). We often write L\ for (L\ , π) and set L\0 := Ker π. Any L\ ∈ E(L) yields a symmetric L-invariant pairing ( ) ∈ P2 ({L\ , L\ }, L\0 ), (a, b) := π(a)b + π(b)a. Notice that the L-action on L\0 is trivial if and only if ( ) ∈ P2 ({L, L}, L\0 ) ⊂ P2 ({L\ , L\ }, L\0 ). We say that L\ is central if ( ) = 0. \

80 Which

was, apparently, the first work on the subject of algebraic conformal field theory. in Moscow they used to call it “the Japanese extension”; cf. A. Brehm’s discourse in “The Life of Animals” on the names of the common cockroach Blatta germanica in different tongues: Russians call the fellow Prussian, in Austrian highlands he is known as Russian, etc. 82 [PS] deals with the “semi-global” setting of function spaces on a fixed circle; a simpler picture for the space of germs of holomorphic functions with (possibly essential) singularities at the origin (i.e., the case of an infinitely small circle) is discussed in a sequel to [BBE]. 81 E.g.,

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Denote by CE(L) the category of central Lie algebra extensions of L. Notice that for every L[ ∈ CE(L) the adjoint action of L[ factors through L. So L[ is an L-module; the structure projection L[ → L makes it an object of E(L). Lemma. The functor CE(L) → E(L) we have defined is fully faithful. It identifies CE(L) with the category of central objects in E(L).  Assume we have L\ ∈ E(L) and an L-invariant morphism tr : L\0 → F (the L-action on F is trivial). We say that tr is central if tr( ) ∈ P2 ({L, L}, F ) equals 0. Denote by L\tr the push-forward of L\ by tr; this is an object of E(L) which is an extension of L by F . It is clear that L\tr ∈ CE(L) if and only if tr is central. Remark. If every L-invariant symmetric F -valued pairing on L is trivial, then every tr as above is automatically central. We suggest that the reader skip the rest of this section, returning to it when necessary. (ii) Here is an example of extensions that arise in the above manner: Lemma. Let L be a Lie algebra, Lc , Ld ⊂ L ideals such that Lc + Ld = L, and tr : Lf := Lc ∩ Ld → F a morphism such that tr([lc , ld ]) = 0 for every lc ∈ Lc , ld ∈ Ld . Then there exists a central extension L[ of L by F together with sections sc : Lc → L[ , sd : Ld → L[ such that the images of sc , sd are ideals in L[83 and for any l ∈ Lf one has (sc − sd )l = tr(l) ∈ F ⊂ L[ . Such (L[ , sc , sd ) is unique (up to a unique isomorphism). Proof. The uniqueness of (L[ , sc , sd ) is clear. Let us construct L[ . Set L\ := α

β

Lc ⊕ Ld , and consider the extension of L-modules 0 → Lf → L\ → L → 0 where α(l) = (l, −l), β(lc , ld ) = lc +ld (the action of L is the adjoint one). Then L\ ∈ E(L) and tr is a central morphism (the L-invariance of tr and the vanishing of tr( ) follow from the property tr([Lc , Ld ]) = 0). Now set L[ := L\tr .  ∼

Remark. In the situation of the above lemma one has Lc /Lf −→ L/Ld . The Lie algebra Lc /Lf has a central extension (Lc /Lf )[ defined as the push-out of the extension 0 → Lf → Lc → Lc /Lf → 0 by tr : Lf → F . Now sc yields an ∼ isomorphism (Lc /Lf )[ −→ L[ /sd (Ld ). ∼ Similarly, sd yields an isomorphism (Ld /Lf )[ −→ L[ /sc (Lc ) where the central extension (Ld /Lf )[ of Ld /Lf is the push-out of the extension 0 → Lf → Ld → Ld /Lf → 0 by −tr : Lf → F . (iii) Let A be an associative algebra. Denote by ALie our A considered as a Lie algebra. If A[ is a central extension of ALie by F , then its bracket comes from a skewsymetric pairing [ ][ ∈ P2 ({A, A}, A[ ). We say that A[ satisfies the cyclic property if for every a, b, c ∈ A one has [ab, c][ + [bc, a][ + [ca, b][ = 0. Denote by E(A) the category of pairs (A\ , π) where A\ is an A-bimodule, π : \ A → A a surjective morphism of A-bimodules. Notice that any A-bimodule is an ALie -module with respect to the commutator action. Thus every object of 83 Or, equivalently, that s , s are morphisms of L-modules (with respect to the adjoint c d actions).

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E(A) can be considered as an object of E(ALie ); i.e., we have a faithful functor E(A) → E(ALie ). Take A\ ∈ E(A) and let tr : A\f := Ker π → F be a central ALie -invariant morphism, so A[ := A\tr is a central Lie algebra extension of ALie by F . Lemma. A[ satisfies the cyclic property. Proof. Let c˜ be a lifting of c. We have a tautological identity [ab, c˜] + [b˜ c, a] + [˜ ca, b] = 0 in A\ (it holds in any A-bimodule).84 Now notice that tr([bc, a ˜]−[b˜ c, a]) = tr(a, bc) = 0 (since b˜ c is a lifting of bc) and similarly tr([ca, ˜b]−[˜ ca, b]) = tr(ca, b) = 0.  For example, if we have two-sided ideals Ac , Ad ⊂ A, Ac + Ad = A, and a morphism tr : Af := Ac ∩ Ad → F which vanishes on [Ac , Ad ] ⊂ Af , then the central extension A[ of ALie defined in (ii) above satisfies the cyclic property. Remark. Assume that our pseudo-tensor category is a tensor unital category (say, the category of k-modules) and A is a commutative unital algebra. Consider the A-module of differentials ΩA , so we have a universal derivation d : A → ΩA . For every object F the map Hom(ΩA , F ) → P2 ({A, A}, F ), which sends ϕ : ΩA → F to the operation a, b 7→ ϕ(adb) is injective, and its image consists of those operations ψ that ψ(a, bc) = ψ(ab, c) + ψ(ac, b). Notice that ϕ vanishes on the image of d if and only if the corresponding ψ is skew-symmetric. Now for (A\ , tr) as above one has [ ][ ∈ P2 ({A, A}, F ) (since A is commutative), and our lemma shows that [ ][ yields a morphism ΩA /d(A) → F . This is one of the principal ideas of Tate’s work [T], a portent of the cyclic homology formalism. (iv) Suppose our pseudo-tensor category is augmented. For A, A\ as above h(A) is an associative algebra and A\ is an h(A)-bimodule (see 1.2.8). The A-actions also yield morphisms A ⊗ h(A\ ) → A\ , h(A\ ) ⊗ A → A\ which are morphisms of, respectively, (A, h(A))- and (h(A), A)-bimodules. If z \ ∈ h(A) is such that z := π(z \ ) ∈ h(A) is an A\ -central element (i.e., left and right multiplications by z on A\ coincide), then the morphism ∂z\ : A → A\0 , a 7→ az \ − z \ a, is a derivation. If tr : A\f → F is an ALie -invariant morphism, then the composition tr ∂z\ : A → F depends only on z, so we denote it by tr ∂z . It is clear that tr ∂z kills the commutant [A, A]. Let us assume that A is a unital associative algebra (see 1.2.8), A\ is a unital bimodule, and 1 = 1A ∈ h(A) lifts to h(A\ ). We get a canonical morphism tr ∂1 : A → F. Lemma. One has tr(a, b) = tr ∂1 (ab). Therefore tr is central if and only if tr ∂1 = 0. Proof. Choose ˜ 1 ∈ h(A\ ). We have two sections A → A\ , a 7→ 1\ a, a1\ . One \ \ \ has (1 a, b1 ) = ab1 − 1\ ab. Applying tr, we get the lemma.  In particular, if ALie is perfect, i.e., [A, A] = A, then every tr is central. 84 This identity can be interpreted as follows. For any A-bimodule M and a ∈ A denote by la , ra the left, resp. right, multiplication by a; set ada := la − ra . The identity says that adab = ada lb +adb ra . Define the A-bimodule structure on End(M ) by af := f ra , f b := f lb . Then the identity says that adab = ada b + aadb ; i.e., the map A → End(M ), a 7→ ada , is a derivation.

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2.

2.7.2. The Tate extension in the D-module setting. Let V , V 0 be Dmodules on X, h i ∈ P2∗ ({V, V 0 }, ωX ) = P2∗ ({V 0 , V }, ωX ) a ∗ pairing. By 1.4.2 the D-module G := V ⊗ V 0 is an associative∗ algebra which acts on V , V 0 . We have the corresponding Lie∗ algebra GLie ; the pairing is GLie -invariant. The Tate extension is a canonical central extension of Lie∗ algebras (2.7.2.1)

π[

0 → ωX → G[ −→GLie → 0.

To define G[ , consider the exact sequence85 of the D-modules on X × X (2.7.2.2)

ε

π

V  V 0 −→j∗ j ∗ (V  V 0 )−→∆∗ (V ⊗ V 0 ) → 0.

Here ∆: X ,→ X × X is the diagonal embedding, j: U := X × X r ∆(X) ,→ X × X is the complementary embedding, and π is the canonical arrow. Namely, according to (2.2.2.1), one has V ⊗ V 0 = H 1 ∆! (V  V 0 ) = Coker ε; explicitly, π sends (t2 − t1 )−1 v  v 0 ∈ j∗ j ∗ V  V 0 to v ⊗ v 0 (dt)−1 ∈ ∆· (V ⊗ V 0 ) ⊂ ∆∗ (V ⊗ V 0 ). Note that h i vanishes on Ker ε.86 So, pushing out the above exact sequence by h i, we get an extension of ∆∗ (V ⊗ V 0 ) by ∆∗ ωX . This extension is supported on the diagonal; applying ∆! , we get the Tate extension G[ . We denote the canonical morphism j∗ j ∗ (V  V 0 ) → ∆∗ G[ by µ = µG . 2.7.3. Let us define now the Lie∗ bracket [ ]G[ on G[ . Let L be a Lie∗ algebra that acts on V and V 0 preserving the pairing. Then the action of L on V ⊗ V 0 = G lifts canonically to G[ . Indeed, the action of the sheaf of Lie algebras h(L) on G lifts canonically to G[ . Namely, global sections of h(L) act on V , V 0 preserving the ∗ pairing; hence they act on j∗ j ∗ (V  V 0 ) and G[ by transport of structure, and everything is compatible with localization of X. One checks that the h(L)-action on G[ is good in the sense of 2.5.4, so we actually have an L-action. In particular, the action of G on V , V 0 yields a canonical action of GLie on G[ . Proposition. There is a unique Lie∗ algebra structure on G[ such that the adjoint action of G[ on G[ is the above action composed with G[ → GLie . The projection G[ → GLie is a morphism of Lie∗ algebras, so G[ is a central extension of GLie . This extension satisfies the cyclic property (see 2.7.1(iii)). Proof. The uniqueness is clear, as well as the latter property (notice that the commutator on G[ comes from the action of GLie on G[ that lifts the adjoint action). Let us show that our picture fits into the format of 2.7.1(iii); this implies the existence statement. The cyclic property follows then from the lemma in 2.7.1(iii). Consider the ordered set of all submodules Nα ⊂ j∗ j ∗ V  V 0 such that j∗ j ∗ V  0 V /Nα is supported on the diagonal. Set G\α := ∆! (j∗ j ∗ V  V 0 /Nα ) and G\ := \ lim ←− Gα . The pro-DX -module G\ carries a canonical structure of the topological Gbimodule. This means that for every coherent DX -submodule Gβ ⊂ G we have mutually commuting ∗ pairings (left and right action) in ·l ∈ P2∗ ({Gβ , G\ }, G\ ) and ·r ∈ P2∗ ({G\ , Gβ }, G\ ) which are compatible with respect to the embeddings of the Gβ ’s and satisfy the left and right G-action properties.87 85 It

is also exact from the left if V , V 0 have no OX -torsion. L

O

−1 Ker ε = ∆∗ H −1 (V ⊗V 0 ) = ∆∗ Tor1 X (V, V 0 ωX ), its sections have finite supports. makes sense since for every two coherent DX -submodules Gβ , Gβ 0 of G their product (i.e., the image in G of Gβ  Gβ 0 by the product morphism) is coherent. 86 Since 87 This

2.7. THE TATE EXTENSION

121

To define ·l , consider the morphism ·V idV 0 : GV V 0 → ∆1=2 (V V 0 ). Here ∗ 1=2 ·V is the ∗ action of G on V and ∆ is the diagonal embedding (x, y) 7→ (x, x, y). Localizing, we get the morphism ˜·l : G  j∗ j ∗ (V  V 0 ) → ∆1=2 j∗ j ∗ (V  V 0 ). It ∗ satisfies the following continuity property: for every Nα , Gβ as above there exists an Nα0 such that ˜·l (Gβ  Nα0 ) ⊂ ∆1=2 (Nα ).88 Now ·l is the completion of ˜·l with ∗ respect to the Nα -topology. One defines ·r in a similar way using the action of G on V 0 instead of V . It is clear that the ·l,r define on G\ a topological G-bimodule structure. The projection π from (2.7.2.2) yields the projection G\  G which we also denote by π. This is a morphism of (topological) G-bimodules. The ∗ pairing h i yields a GLie -invariant morphism G\0 := Ker π → ωX which we denote by tr. One has89 G[ = G\tr , and the h(GLie )-module structure on G[ defined in 2.7.2 comes from the G[ -module structure on G\tr . Therefore, by 2.7.1(i), our proposition follows from the next lemma: Lemma. tr is a central morphism. Proof of Lemma. (a) It suffices to check our lemma in case V = DnX and V = ωDnX is its dual. Indeed, our problem is local, so we can choose a surjective morphism DnX  V . Then h i yields a morphism V ◦ → ωDnX . Our constructions are functorial with respect to the morphisms of the (V, V ◦ , h i)’s. Looking at (V, V ◦ )  (DnX , V ◦ ) → (DnX , ωDnX ), we get the desired reduction. (b) For such V , V ◦ our G is the unital associative algebra and G\ is a unital G-module. One has GLie = gl(DnX ) (see 2.5.6(a)), so, by 2.7.1(iv), it remains to check the following fact: ◦

Sublemma.. The Lie∗ algebra gl(DnX ) is perfect. Proof of Sublemma. It suffices to consider the case n = 1. One has gl(DX ) = Lie DD where DLie is the sheaf D = DX of differential operators considered as a Lie algebra in Dif f (X)∗ ; see Example (iii) in 2.5.6(b). We want to show that Lie the adjoint action of DLie = h(DD ) on DD has trivial coinvariants. In fact, the Lie action of the subalgebra ΘX ⊂ D of vector fields has trivial coinvariants. To see this, notice that this action preserves the filtration by the degree of the differential operator, and gr DD = ⊕ Θ⊗n XD . Let us show that the coinvariants of the ΘX n≥0

⊗n action on ΘD are trivial for n 6= −1. Since we deal with a coherent D-module ⊗n (see (i) in the lemma from 2.1.6), this amounts to the equality [ΘX , Θ⊗n X ] = ΘX . n+1 n+1 n+1 If v ∈ ΘX , η ∈ ωX , then [v, v η] = v [v, η] = v d(v, η). Such sections for v(x) 6= 0 generate the stalk of Θ⊗n at x if n = 6 −1.  X

2.7.4. If V is a vector DX -bundle (see 2.2.16), V ◦ is its dual, and h i is the canonical pairing, then GLie = gl(V ) (see 2.5.6(a)). Therefore we have a canonical central extension gl(V )[ of gl(V ) by ωX . Passing to homology, we get a central ¯ )[ of the sheaf of Lie algebras gl(V ¯ ) = h(gl(V )) (which is the sheaf extension gl(V of endomorphisms of V considered as a Lie algebra) by h(ωX ). 88 Let g be a finite set of generators of G , so we have the correponding classes g ¯i ∈ h(Gβ ) i β and endomorphisms g¯i˜·l of j∗ j ∗ V  V 0 . Our Nα0 is the intersection of preimages of Nα by these endomorphisms. 89 We use the notation of 2.7.1(i); the material of 2.7.1 renders itself to the setting of topological modules in the obvious way.

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The Tate extension satisfies the following properties: (i) Additivity: Let F be a finite flag V1 ⊂ · · · ⊂ Vn = V such that every gri V is a vector DX -bundle. Let gl(V, F) ⊂ gl(V ) be the corresponding parabolic Lie∗ subalgebra, so we have the projections gl(V, F) → gl(gri V ). Let gl(V, F)[ be 0 the restriction of the Tate extension gl(V )[ to gl(V, F), and let gl(V, F)[ be the sum of the pull-backs of the Tate extensions gl(gri V )[ . Now there is a canonical isomorphism of extensions (2.7.4.1)

∼

0

gl(V, F)[ −→ gl(V, F)[ .

To see this, consider the submodule P := ΣVj  (V /Vj−1 )◦ ⊂ V  V ◦ ; we have the obvious projections πi : P → gri V  (gri V )◦ . Now gl(V, F) = ΣVi ⊗ (V /Vi−1 )◦ ⊂ V ⊗ V ◦ ; i.e., gl(V, F) = ∆! P ⊂ ∆! (V  V ◦ ) = V ⊗ V ◦ . So the exact sequence 0 → ∆∗ ωX → gl(V, F)[ → gl(V, F) → 0 is the push-out of 0 → P → j∗ j ∗ P → ∆! P → 0 by f : P → ∆∗ ωX , where f is the restriction of the canonical pairing V  V ◦ → ∆∗ ωX . Since f is the sum of the canonical pairings gri V  (gri V )◦ → ∆∗ ωX composed with πi , we get (2.7.4.1). We leave it to the reader to check compatibility with Lie∗ brackets. (ii) Self-duality: There is a canonical isomorphism of ωX -extensions (2.7.4.2)

∼

gl(V )[ −→ gl(V ◦ )[ ∼

which lifts the standard isomorphism of Lie∗ algebras gl(V ) −→ gl(V ◦ ). It comes from the transposition of the multiples symmetry of (2.7.2.2). 2.7.5. Induced DX -bundles; fitting Kac-Moody and Virasoro into Tate. Consider the case of the induced DX -bundle, so V = FD := F ⊗ DX , OX

¯ ) = Diff(F, F )X where F is a locally free OX -module of finite rank. We have gl(V and gl(V ) = Diff(F, F )D (see Example (iii) in 2.5.6(b)). The dual D-module V ◦ ◦ equals FD where F ◦ := F ∗ ωX , so j∗ j ∗ (V V ◦ ) = (j∗ j ∗ F F ◦ )D and ∆∗ (V ⊗V ◦ ) = ∗ (j∗ j (F  F ◦ )/F  F ◦ )D . Applying h, we get a canonical identification (2.7.5.1)

∼

Diff(F, F )X −→ (j∗ j ∗ F  F ◦ )/F  F ◦ .

This is the usual Grothendieck isomorphism which can be described as follows. The action of Diff(F, F )X on F makes (j∗ j ∗ F  F ◦ )/F  F ◦ a Diff(F, F )X -module, and (2.7.5.1) is a unique morphism of Diff(F, F )X -modules which maps 1 ∈ Diff(F, F ) to 1 ∈ End F = F  F ◦ (∆)/F  F ◦ .90 The inverse isomorphism assigns to a “kernel” k(x, y) ∈ j∗ j ∗ F  F ◦ the differential operator f 7→ k(f ), k(f )(x) := (y) Resy=x hk(x, y), f (y)i. ¯ )[ is the push-forward of the extension We see that ∆· gl(V (2.7.5.2)

π

0 → F  F ◦ → j∗ j ∗ F  F ◦ − → Diff(F, F )X → 0

by the map F  F ◦ → ∆· ωX → h(ωX ), f (x)  g(y) 7→ hf (x), g(x)i. The left and ¯ ) = Diff(F, F )X on ∆· h(j∗ j ∗ (V  V ◦ )) = ∆· j∗ j ∗ (F  F ◦ ) come right actions of gl(V from the usual left and right actions of Diff(F, F )X on F and F o , respectively; ¯ )[ . Therefore gl(V ¯ )[ the adjoint action yields our Lie bracket on the quotient gl(V coincides with the Tate extension of the Lie algebra Diff(F, F )X as defined in [BS]. 90 One can consider the right Diff(F, F ) -module structure defined by the right Diff(F, F ) X X action on F ◦ as well.

2.7. THE TATE EXTENSION

123

¯ ) = DX . Exercises. Assume that F = OX , so F ◦ = ωX and gl(F dy 91 (i) A local coordinate t defines a section δt := y−x ∈ j∗ j ∗ OX  ωX such that π(δt ) = 1 ∈ DX . It yields a splitting st : DX → D[X , A 7→ Ax · δt (here Ax denotes the differential operator A acting along the x variable). Show that in terms of this splitting the Lie bracket on D[X is given by the 2-cocycle (2.7.5.3)

f (t)∂tm , g(t)∂tn 7→ (−1)m+1

m!n! f (m+n+1) (t)g(t)dt. (m + n + 1)!

(ii) Suppose that X is compact of genus g. Show that the image of 1 ∈ DX by the boundary map H 0 (X, DX ) → H 1 (X, h(ω)) = k for the Tate extension equals g − 1.92 Let A(F ) ,→ Diff(F, F )X be the Lie subalgebra of differential operators of first order whose symbol lies in ΘX = ΘX · idF ⊂ ΘX ⊗ End(F ). Thus A(F ) is an extension of ΘX by gl(F ); its elements are pairs (τ, τ˜) where τ is a vector field and τ˜ is an action of τ on F ; i.e., A(F ) is the Lie algebroid of infinitesimal symmetries of (X, F ). The pull-back of the Tate extension by A(F )D ,→ Diff(F, F )D = gl(V ) is an ωX -extension A(F )[D of A(F )D ; we refer to it again as the Tate extension. Consider the two natural ωX -extensions of the Lie∗ algebra gl(F )D : the KacMoody extension gl(F )κD where κ is of the form a, b 7→ −tr(ab) on gl (see 2.5.9) and the restriction gl(F )[D of A(F )[D to gl(F )D ⊂ A(F )D . Proposition. There is a canonical isomorphism of ωX -extensions (2.7.5.4)

∼

−1/2 κ )D

gl(F )[D −→ gl(F ωX ∼

−1/2

which lifts the obvious identification gl(F ) −→ gl(F ωX

).

Proof. According to the corollary in 2.5.9, the promised isomorphism amounts −1/2 to a rule that assigns to each connection ∇ on F ωX a splitting s∇ : gl(F ) → gl(F )[ in a way compatible with the Aut(F )-action and so that for every ψ ∈ gl(F )ωX one has s∇+ψ (a) = s∇ (a) + tr(ψa). Strictly speaking, we have to define s∇ in the presence of the parameters Spec R` , R` ∈ ComuD (X) (see 2.5.9) which brings no complications except notational ones. Let X (1) ⊂ X × X be the first infinitesimal neighborhood of the diagonal (see 2.5.10(a)). By (2.7.5.2) one has the projection of sheaves µ : F  F ◦ (∆)|X (1) → ⊗1/2 ⊗1/2 gl(F )[ . Let 1[ ∈ (ωX  ωX )(∆)|X (1) be the section which lifts 1 ∈ OX = ⊗1/2 ⊗1/2 (ωX  ωX )(∆)|X and is anti-invariant with respect to the transposition of co∼ −1/2 −1/2 ordinates. Consider ∇ as an identification93 (p∗2 F ωX )|X (1) −→ (p∗1 F ωX )|X (1) . 1/2 −1/2 Tensoring it by the identity map for p∗2 (F ∗ ωX ) = p∗2 (F ◦ ωX ), one gets an ∼ −1/2 −1/2 isomorphism s∇ : p∗2 (F ⊗ F ∗ )|X (1) −→ (F ωX )  (F ◦ ωX )|X (1) . We define ⊗1/2 [ [ s∇ : gl(F ) → gl(F ωX ) as s∇ (a) := µ(s∇ (a) · 1 ). The required properties are obvious.  91 Here

x, y are coordinates on X × X defined by t. any non-constant meromorphic function t on X provides a meromorphic lifting δt of 1. It remains to compute the residues of its singular parts. 93 Here p : X × X → X are the projections. i 92 Hint:

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Let us pass to the Virasoro extension. Assume that F is a ΘX -equivariant bundle such that the ΘX -action morphism ΘX → A(F ) is a differential operator.94 Then we can restrict A(F )[D to ΘD ⊂ A(F )D . Thus every such F yields an ωX extension Θ[D (F ) of ΘD which is a Lie∗ subalgebra of A(F )[D . ⊗j Consider the case F = ωX with the usual Lie action of ΘX . Proposition. There is a unique isomorphism of ωX -extensions (2(6j 2 −6j+1)) ∼

(2.7.5.5)

⊗j −→ Θ[D (ωX ).

ΘD

Proof. The uniqueness is clear since ωX -extensions of ΘD are rigid; see Remark in 2.5.10(c). Thus it suffices to define our isomorphism locally in terms of a ⊗j coordinate t. Let us identify ωX with OX by means of the section (dt)⊗j . Then τ = f (t)∂t ∈ ΘX acts on ωX as the first order differential operator f (t)∂t + jf 0 (t). ⊗j So the section st from the Exercise (i) in 2.7.5 yields a splitting of Θ[D (ωX ). As 2 −1 000 follows from (2.7.5.3) its 2-cocycle is f (t)∂t , g(t)∂t 7→ (j − j + 6 )f (t)g(t)dt. As was shown in the end of 2.5.10, t also yields a splitting of the Virasoro extension 1 000 of central charge 1 with the cocycle f (t)∂t , g(t)∂t 7→ 12 f (t)g(t)dt. Now (2.7.5.5) is the isomorphism which identifies the splittings.  2.7.6. The construction of 2.7.2 renders itself easily to the situation with extra parameters, i.e., to the case of A` [DX ]-modules. Namely, suppose we have A` ∈ ∗ ComuD (X), V, V 0 ∈ M(X, A` ) and h i ∈ PA2 ({V, V 0 }, A). This datum yields the associative∗ matrix algebra95 V ⊗ V 0 ∈ M(X, A); denote by GA the corresponding A

`

0 96 Lie∗ algebra. Assume that A is OX -flat and TorA We will define a 1 (V, V ) = 0. ∗ canonical central extension of Lie algebras in M(X, A)

0 → A → G[A → GA → 0.

(2.7.6.1)

Let us construct G[A as a plain A` [DX ]-module. Consider the short exact sequence 0 → V  V 0 → j∗ j ∗ V  V 0 → ∆∗ (V ⊗ V 0 ) → 0 of A` [DX ]2 -modules. Its push-forward by h i : V V 0 → ∆∗ A is an A` [DX ]2 -module extension of ∆∗ (V ⊗V 0 ) ˜ [ → V ⊗V 0 → 0 be the corresponding (A` )⊗2 [DX ]-module by ∆∗ A. Let 0 → A → G A extension. ˜ [ /I ` G ˜ [ where I ` is the ideal of the diagonal, I ` := Ker(A` ⊗ Now set G[A = G A A ` · ` A − → A ). The Tor1 vanishing assures that G[A is, indeed, an extension of GA by A. One defines the bracket on G[A as in 2.7.2 and 2.7.3; again, we refer to 3.8 for a natural Clifford realization of G[A . As in 2.7.3, the Tate extension satisfies the cyclic property (for the associative∗ algebra structure on the matrix algebra GA ). The above construction is compatible with the ´etale localization, so it works in the setting of algebraic DX -spaces. Namely, let Y be an algebraic DX -space and let V, V 0 be (right) OY [DX ]-modules equipped with an OrY -valued ∗ pairing h i; assume that Y is flat over X and Tor1OY (V, V 0 ) vanishes. We have a Lie∗ algebra GY = V ⊗ V 0 ∈ M(Y) and its Tate (central) extension G[Y by OrY . The construction of GY , G[Y is compatible with the base change (with respect to the morphisms of the Y’s). If V , V 0 are equipped with an action of a Lie∗ algebroid E on Y such that 94 I.e., 95 As 96 We

the ΘX -action satisfies the condition from Example (i) in 2.5.6(b). usually, we write the tensor product of right A[DX ]-modules as ⊗ instead of ⊗ ! . A

do not know if one can get rid of this technical condition.

A

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125

this action preserves h i, then E acts on GY as on an associative∗ OY -algebra; as in 2.7.2, this action lifts canonically to an E-action on G[Y . An important particular case of the above situation: Y is an OX -flat algebraic DX -space and V , V 0 are mutually dual vector DX -bundles on Y. Then GY = gl(V ), so we have the Tate extension gl(V )[ . An action of a Lie∗ algebroid E on V yields its action on V 0 compatible with the canonical pairing, so E acts on gl(V )[ . 2.7.7. Tate’s linear algebra. Let us describe the Tate construction in its original linear algebra setting. Let F be a k-vector space equipped with a separated complete linear97 topology. A closed vector subspace L ⊂ F is compact, resp. cocompact, if for any open vector subspace U ⊂ F one has dim L/(L∩U ) < ∞, resp. dim F/(L+U ) < ∞. A c-lattice is an open compact subspace; dually, a d-lattice is a discrete cocompact subspace. It is easy to see that F has a c-lattice if and only if it has a d-lattice. Such an F is called a Tate vector space. For a Tate vector space F its dual98 F ∗ is again a Tate space; the canonical map F → F ∗∗ is an isomorphism. Any discrete vector space99 is a Tate space. Any compact topological vector space100 is a Tate space. Duality interchanges discrete and compact Tate spaces. Any Tate vector space can be represented as a direct sum of a discrete and a compact space. Let A : F → G be a morphism of Tate vector spaces. We say that A is compact (resp. discrete) if it factors through a compact (resp. discrete) vector space. Equivalently, A is compact if the closure of Im A is compact; A is discrete if Ker A is open. The composition of a compact operator with an arbitrary one is compact; the same is true for discrete operators. A is compact if and only if A∗ : G∗ → F ∗ is discrete. We denote by Homc (F, G), Homd (F, G) ⊂ Hom(F, G) the subspaces of compact, resp. discrete, operators, and by Homf (F, G) the intersection of these subspaces. Any operator may be represented as the sum of a compact and a discrete operator. The above vector spaces of operators carry natural topologies. Notation: for subspaces U ⊂ F and V ⊂ G set Hom(F, G)U,V := {A ∈ Hom(F, G) : A(U ) ⊂ V }. Now bases of topologies on Hom(F, G), Homc (F, G), Homd (F, G), and Homf (F, G) are formed, respectively, by subspaces Hom(F, G)U,V , Hom(F, V ), Hom(F/U, G), and Hom(F/U, V ) where U ⊂ F and V ⊂ G are c-lattices. These topologies are complete and separated. The exact sequence (2.7.7.1)

α

β

0 → Homf (F, G) − → Homc (F, G) ⊕ Homd (F, G) − → Hom(F, G) → 0,

where α(A) := (−A, A), β(A, B) := A + B, is strongly compatible with the topologies.101 Notice that Homc , Homd and Homf are two-sided ideals in the obvious sense, and all the composition maps like Homd (G, H) × Homc (F, G) → Homf (F, H), ∼ ∼ etc., are continuous. The maps Hom(F, G) −→ Hom(G∗ , F ∗ ), Hom(F, G)c −→ ∗ ∗ ∗ Hom(G , F )d , etc., which send A to A , are homeomorphisms. 97 Here

“linear” means that the topology has a base formed by vector subspaces of F . definition, F ∗ is the space of all continuous linear functionals F → k; open subspaces in F ∗ are orthogonal complements to compact subspaces of F . 99 I.e., a vector space equipped with a discrete topology. 100 Which is the same as a profinite-dimensional vector space. 101 I.e., α is a homeomorphism onto a closed subspace and β is continuous and open. 98 By

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Exercise. Consider an (abstract) vector space G ⊗ F . It carries four natural topologies with bases of open subspaces formed, respectively, by subspaces V ⊗ F + G ⊗ U , V ⊗ F , G ⊗ U , and V ⊗ U , where V ⊂ G, U ⊂ F are c-lattices. Now replace F by F ∗ and consider the obvious canonical map G ⊗ F ∗ → Homf (F, G). Show that it yields homeomorphisms between the completions of G ⊗ F ∗ with respect to these topologies and, respectively, Hom(F, G), Homc (F, G), Homd (F, G), and Homf (F, G). So for a Tate space F we have an associative algebra End F together with two-sided ideals Endc F, Endd F ⊂ End F such that Endc F + Endd F = End F . An operator A belongs to Endf F := Endc F ∩ Endd F if and only if there are c-lattices V ⊂ U such that A(V ) = 0, A(F ) ⊂ U . For such an A its trace tr A is the trace of the induced operator on U/V ; it does not depend on the auxiliary choice of lattices. For every Ac ∈ Endc F , Ad ∈ Endd F one has tr([Ac , Ad ]) = 0. Since End F = Endc F + Endd F , one has tr[A, B] = 0 for any A ∈ Endf F , B ∈ End F . The functional tr is continuous with respect to our topology on Endf F . In terms of the above exercise, tr is simply the continuous extension of the canonical pairing F ⊗ F ∗ → k. 2.7.8. The Tate extension in the linear algebra setting. For a Tate vector space F let gl(F ) be End F considered as a Lie algebra. By 2.7.7 we have ideals glc (F ), gld (F ) ⊂ gl(F ) and tr : glf (F ) := glc (F ) ∩ gld (F ) → k that fit into the setting of 2.7.1(ii). The corresponding central extension gl(F )[ is called the Tate extension. It satisfies the cyclic property (see 2.7.1(iii)). We have canonical [ F [ sections sc = sF c : glc (F ) → gl(F ) , sd = sd : gld (F ) → gl(F ) . According to 2.7.7 the Lie algebras gl(F ), glc (F ), gld (F ), and glf (F ) carry canonical topologies, and the functional tr is continuous. Our gl(F )[ is also a topological Lie algebra: the topology is defined by the condition that the map sc + sd : glc (F ) × gld (F ) → gl(F )[ is continuous and open. It is complete and separated; the projection gl(F )[ → gl(F ) is continuous and open. ∼

Remark. The identification gl(F ) −→ gl(F ∗ ), A 7→ −A∗ , lifts canonically to ∼ an identification of topological k-extensions gl(F )[ −→ gl(F ∗ )[ which interchanges sc and sd . 2.7.9. The additivity property. Let F be a finite flag F1 ⊂ F2 ⊂ · · · ⊂ Fn = F of closed subspaces of F .102 Then Fi are Tate vector spaces, as well as gri F . Let gl(F, F) ⊂ gl(F ) be the stabilizer of our flag, so one has the projections 0 πi : gl(F, F) → gl(gri F ). There are two natural k-extensions gl(F, F)[ and gl(F, F)[ of gl(F, F): the first one is the restriction of gl(F )[ to gl(F, F) and the second one is the Baer sum of the πi -pull-backs of the extensions gl(gri F )[ . 0

Lemma. The topological central extensions gl(F, F)[ , gl(F, F)[ are canonically isomorphic. Proof. Consider the ideals glc (F, F), gld (F, F) defined as intersection of gl(F, F) with glc (F ), gld (F ). One has glc (F, F) + gld (F, F) = gl(F, F), and the images of these ideals by πi lie in, respectively, glc (gri F ), gld (gri F ). Both gl(F, F)[ and 0 gl(F, F)[ are equipped with sections over our ideals: these are, respectively, sc , sd 102 Any

such flag splits; i.e., it comes from a grading.

2.7. THE TATE EXTENSION

127

iF F 0 ∗ gri F defined as restrictions of sF ), s0d := Σπi∗ (sgr ). Over c , sd , and sc := Σπi (sc d 0 0 glf (F, F) := glc (F, F) ∩ gld (F, F) one has sc − sd = sc − sd = tr. Now use the uniqueness statement of the lemma from 2.7.1(ii). 

Remark. Suppose that every gri F is either compact or discrete. Then the iF iF corresponding sections sgr or sgr (together with the lemma) provide a splitting c d F/P [ of gl(F )F . For example, if F consists of a single c-lattice P ⊂ F , then sP c and sd [ 0 yield a canonical splitting sP of gl(F )P . If P ⊂ P is another c-lattice, then on gl(F )P ∩ gl(F )P 0 one has sP − sP 0 = trP/P 0 . 2.7.10. The local duality. We return to the geometric setting of 2.7.2. Let x ∈ X be a (closed) point, jx : Ux ,→ X the complement to x, Ox ⊂ Kx the completion of the local ring at x and its field of fractions. Let V be a coherent ˆ x (jx∗ V ) (see 2.1.13). This is a Tate vector space D-module on Ux . Set V(x) := h (see Remark (i) in 2.1.14). Applying 2.1.14 to M = jx∗ V , we get a canonical isomorphism of topological vector spaces (2.7.10.1) Here V ⊗ Kx := Γ(Ux , V ) DUx

∼

V ⊗ Kx −→ V(x) . DUx

⊗ Γ(Ux ,D)

Kx = Vη ⊗ Kx (where η is the generic point Dη

of X) and its topology is formed by the images of Vξ ⊗ Ox , Vξ ∈ Ξx (jx∗ V ) (see DX

Remark (i) in 2.1.13). Suppose that V is a vector D-bundle on Ux ;103 let V ◦ be its dual. The ∗ pairing jx∗ V  jx∗ V ◦ → ∆∗ jx∗ ωUx composed with the canonical projection jx∗ ωUx  ix∗ k ◦ yields a canonical k-valued pairing between the Tate vector spaces V(x) and V(x) . ◦ Lemma. This pairing is non-degenerate, so V(x) and V(x) are mutually dual Tate vector spaces.

Proof. It is enough to consider the example of V = DUx . Then V ◦ = ωUx ⊗DUx , ◦ V(x) = Kx , V(x) = ωKx , and our pairing is f, ν 7→ Resx (f ν). It is clearly nondegenerate.  Remark. The above discussion remains true for families of vector D-bundles. Precisely, let F be a commutative algebra, V a projective right F ⊗ DUx -module of finite rank, V ◦ := HomF ⊗DUx (V, F ⊗ (ωUx )D ) the dual F ⊗ DUx -module. We define V(x) as the completion of h(jx∗ V )x with respect to the topology formed by all F ⊗ DX -submodules of jx∗ V which equal V on the complement to x. Then ∼ ◦ ˆ x ) −→ V ⊗ (F ⊗K V(x) and the canonical pairing identifies V(x) with (V(x) )∗ := F ⊗DUx

the module of continuous F -linear morphisms V(x) → F . 2.7.11. Consider now the Lie∗ algebra jx∗ gl(V ). It acts on jx∗ V and jx∗ V ◦ . Suppose we have Vξ ∈ Ξx (jx∗ V ), Vξ◦0 ∈ Ξx (jx∗ V 0 ) such that the canonical ∗ pairing sends Vξ  Vξ◦0 to ∆∗ ωX ⊂ ∆∗ jx∗ ωUx . Then the matrix algebra Vξ ⊗ Vξ◦0 is a Lie∗ subalgebra of jx∗ gl(V ) which equals gl(V ) on Ux . We call such a subalgebra special. Special Lie∗ subalgebras form a base of a topology Ξsp x on jx∗ gl(V ) at x (called the special topology) which is weaker than the topology ΞLie (see 2.5.12).104 x 103 Then V becomes a free D-module after replacing X by a Zariski neighborhood of x. This follows from the fact that every ideal of Dη is principal (here η is the generic point of X). 104 In fact, as follows from 2.5.14 and 2.7.11 the special topology is stricty weaker than ΞLie . x

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It is clear that the action of jx∗ gl(V ) on jx∗ V is continuous with respect to the 105 Ξsp x - and Ξx -topologies. Denote by gl(V )(x) the completion of the Lie algebra hx (jx∗ gl(V )) with respect to the Ξsp x -topology (see 2.5.12 and Remark in 2.5.13). This is a topological Lie algebra106 which acts continuously on the Tate vector space V(x) . 2.7.12. Lemma. This action yields an isomorphism of topological Lie algebras (2.7.12.1)

∼

r : gl(V )(x) −→ gl(V(x) ).

Proof. It suffices to consider the case V = DUx . Then hx (jx∗ gl(V )) is the vector space Dη of all meromorphic differential operators. The special topology has a base formed by the subspaces Dm,n := {∂ : ∂(t−m Ox ) ⊂ tn Ox }; here t is a parameter at x. This is exactly the topology induced from the topology on gl(V(x) ) ∼ (see 2.7.8, 2.7.7). It remains to show that Dη /Dm,n −→ Hom(t−m Ox , Kx /tn Ox ) (where Hom means continuous linear maps); this is a refreshing exercise for the reader.  2.7.13. Now consider the central extension jx∗ gl(V )[ of jx∗ gl(V ) by jx∗ ωUx . Notice that for any special subalgebra G = Vξ ⊗ Vξ◦0 ⊂ jx∗ gl(V ) (see 2.7.11) its Tate extension107 G[ is a Lie∗ subalgebra of jx∗ gl(V )[ . We call such a Lie∗ subalgebra of [ jx∗ gl(V )[ special. Special subalgebras form a base of a topology Ξsp[ x on jx∗ gl(V ) at Lie x (called the special topology) which is weaker than the Ξx -topology (see 2.5.12). Let gl(V )[(x) be the completion of hx (jx∗ gl(V )[ ) with respect to the Ξsp[ x topology. The adjoint action of jx∗ gl(V ) on jx∗ gl(V )[ is continuous with respect to special topologies, so gl(V )[(x) is a topological Lie algebra. For every special subalgebra G[ ⊂ jx∗ gl(V )[ we have G[ ∩ jx∗ ωUx = ωX , so the identification ∼ Resx : jx∗ ωUx /ωX −→ k tells us that gl(V )[(x) is a topological central extension of gl(V )(x) by k. 2.7.14. Proposition. The isomorphism r of 2.7.12 lifts canonically to an isomorphism of topological central extensions (2.7.14.1)

∼

r[ : gl(V )[(x) −→ gl(V(x) )[ .

Notice that such a lifting r[ is unique since the extension gl(V(x) )[ has no automorphisms. We give two constructions of r[ . The one right below is a version of considerations from [BS]. A more conceptual and simple approach uses chiral Clifford algebras; it can be found in 3.8.23. Proof. (i) Let us compare the constructions of Tate extensions: (a) Let η be the generic point of X. Apply the functor h to (2.7.2.2) and ¯ )η consider sections of the resulting sheaves over η × η. We get an extension of gl(V 108 modules (2.7.14.2)

¯ )\ → gl(V ¯ )η → 0. 0 → h(V )η ⊗ h(V ◦ )η → gl(V η

fact, Ξsp x is the weakest topology on jx∗ gl(V ) such that the jx∗ gl(V )-action on jx∗ V equipped with Ξx -topology is continuous. 106 In fact, this is a topological associative algebra. 107 With respect to the ∗ pairing ∈ P ∗ ({V , V ◦ }, ω ). X ξ 2 ξ0 108 We use the notation from 2.7.4. 105 In

2.8. TATE STRUCTURES AND CHARACTERISTIC CLASSES

129 ()

Consider the push-forward of this extension by the map h(V )η ⊗ h(V ◦ )η −→ Resx ¯ )η by k; the bracket h(ω)η −−−→ k (see 2.7.2). This is a central extension of gl(V ¯ )η -module structure on gl(V ¯ )\ . Our gl(V )[ is the completion comes from the gl(V η

(x)

of this Lie algebra with respect to the topology defined by the images of vector spaces h(x,x) (j∗ j ∗ Vξ  Vξ◦0 ) where Vξ ,Vξ◦0 are as in 2.7.11. (b) Look at (2.7.7.1) for F = G = V(x) . This is an extension (2.7.14.3)

α

β

0 → glf (V(x) ) − → gl(V(x) )\ − → gl(V(x) ) → 0

of topological gl(V(x) )-modules; here gl(V(x) )\ := glc (V(x) ) ⊕ gld (V(x) ) and α = (+, −), β = (+, +). Now gl(V(x) )[ is the push-forward of this extension by tr : glf (V(x) ) → k; the bracket comes from the gl(V(x) )-module structure on gl(V(x) )\ . ¯ )η on the completion V(x) of h(V )η yields the morphism (ii) The action of gl(V ¯ )η → gl(V(x) ) of Lie algebras. We will lift it to a continuous morphism of r : gl(V extensions (2.7.14.4)

¯ )\ → gl(V(x) )\ r\ : gl(V η

such that r\ is an r-morphism of gl(V )η -modules and its restriction to h(V )η ⊗ ◦ h(V ◦ )η equals the obvious morphism h(V )η ⊗ h(V ◦ )η → V(x) ⊗ V(x) → gl0 (V(x) ). It 109 \ [ is clear from (a), (b) above that such an r yields the desired r . ¯ )\ = hj∗ j ∗ (V  V ◦ )η×η ; let u ∈ gl(V ¯ )η be the image of (iii) Take u\ ∈ gl(V η \ \ \ \ u . To define r (u ) ∈ gl(V(x) ) = glc (V(x) ) ⊕ gld (V(x) ) we must represent r(u) ∈ gl(V(x) ) as rc (u\ ) + rd (u\ ) where rc (u\ ) ∈ glc (V(x) ), rd (u\ ) ∈ gld (V(x) ). Notice that r(u) : h(V )η → h(V )η equals R∆ ◦ fu\ where fu\ : h(V )η → hj∗ j ∗ (V  ωX )η×η maps s ∈ h(V )η to (idV  s)(u\ ) (here we consider s as a morphism of DX -modules V ◦ → ωX at η) and R∆ : hj∗ j ∗ (V  ωX )η×η → R h(V )η isR the residue at η × x along the second variable. We have R∆ = −Rx + η where η denotes integration over the second variable and Rx is the residue at η × x along the second variable. It remains to prove the following lemma, which is left to the reader: R Lemma. (a) The map η ◦fu\ : h(V )η → h(V )η extends to a compact operator rc\ (u\ ) : V(x) → V(x) . (b) The map −Rx ◦ fu\ : h(V )η → h(V )η extends to a discrete operator rd\ (u\ ) : V(x) → V(x) . ¯ )\ → gl(V(x) )\ defined by r\ := (r\ , r\ ) satisfies the (c) The map r\ : gl(V η c d properties promised in (ii) above.  2.8. Tate structures and characteristic classes The Tate extension replaces the absent trace map for the matrix Lie∗ algebras. In fact, for X of arbitrary dimension n and a vector DX -bundle V on X (or any DX scheme Y) there is a canonical Lie∗ algebra cohomology class τV ∈ H n+1 (gl(V ), O) which is the trace map for n = 0 and the Tate extension class for n = 1. When we are in a situation with extra parameters, i.e., we have a DX -scheme Y, then one can define Chern classes chD a (V ) of a DX -bundle V on Y repeating the usual 109 Use

the interpretation of tr at the end of 2.7.7.

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Weil construction with the trace map replaced by τV . Below we do this for n = 1, leaving the case n > 1 to an interested reader. It would be nice to understand the Riemann-Roch story in this setting. We are mostly interested in the first Chern class chD 1 (V ) which is an obstruction to the existence of a Tate structure on V . We will see in (3.9.20.2) that a Tate structure on the tangent bundle ΘY (a.k.a. the Tate structure on Y) is exactly the datum needed to construct an algebra of chiral differential operators (cdo) on Y. The groupoid of Tate structures Tate(V ) on V is defined in in 2.8.1; if nonempty, this is a torsor over the Picard groupoid Pcl (Θ) (see 2.8.2). The rest of the section consists of two independent parts: (a) Tate structures and chD 1 : We identify Tate(V ) with certain down-to-earth groupoids defined in terms of connections on V in 2.8.3–2.8.9, which helps to determine the obstruction to the existence of a Tate structure. The classes chD a are defined in 2.8.10. If V is induced, V = FD , then chD a (V ) comes from the conven1/2 tional de Rham Chern class cha+1 (F ωX ) (see 2.8.13). (b) Some constructions and concrete examples of Tate structures: We consider weakly equivariant Tate structures in 2.8.14 and identify them explicitly in the case when Y is a torsor in 2.8.15. The descent construction of Tate structures is treated in 2.8.16. As an application, we identify weakly equivariant Tate structures on the jet scheme of a flag space with Miura opers (see 2.8.17). The material of 2.8.3–2.8.9 is inspired by [GMS1], [GMS2]. After the identification of Tate structures on Θ with cdo, 2.8.15 becomes essentially Theorem 4.4 of [AG] or [GMS3] 2.5. The result in 2.8.17 was conjectured by E. Frenkel and D. Gaitsgory; it is a variation on the theme of [GMS3] 4.10 which considered the translation equivariant setting (the original statement of [GMS3] 4.10 is presented as the exercise at the end of 2.8.17). As in the previous section, we assume that dim X = 1. 2.8.1. We return to the setting of 2.7.6, so Y is an algebraic DX -space flat over X, V a vector DX -bundle on Y, V ◦ the dual bundle, so we have the Tate extension gl(V )[ of gl(V ). From now on we assume that ΩY := ΩY/X is a vector DX -bundle (which happens if Y is smooth), so the Lie∗ algebroid E(V ) (see 2.5.16) is well defined. It acts on V , V ◦ preserving the ∗-pairing, so E(V ) acts on gl(V )[ (see 2.7.6); denote this action by ad[ . The usefulness of the following definition will become clear in 3.9.20: Definition. A Tate structure on V is the following datum (i), (ii): (i) An extension E(V )[ of the Lie∗ algebroid E(V ) on Y by OrY . The corresponding Lie∗ OY -algebra E(V )[♦ = Ker(τ ) is an extension of gl(V ) by OrY . (ii) An identification of this extension with gl(V )[ compatible with the E(V )[ actions. Here E(V )[ acts on E(V )[♦ by the adjoint action and on gl(V )[ by ad[ . Tate structures form a groupoid Tate(V ) = Tate(Y, V ). They have the ´etale local nature, so we have a sheaf of groupoids Tate(V )Y´et on Y´et . We refer to a Tate structure on ΘY as the Tate structure on Y and write Tate(Y) := Tate(ΘY ). Remark. We wiil see that Tate(V ) can be empty. We do not know if Tate structures always exist locally on Y (this is true for locally trivial V , which is the case if V is induced).

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131

2.8.2. For a Lie∗ algebroid L on Y let110 Pcl (L) = Pcl (Y, L) be the groupoid of Lie∗ algebroid extensions Lc of L by OrY (we assume that the adjoint action of Lc on OrY ⊂ Lc coincides with the structure action). This is a Picard groupoid, or rather a k-vector space in categories, in the obvious way (the sum is the Baer sum operation). The objects of Pcl (L) have the ´etale local nature, so we have a sheaf Pcl (L)Y´et of Picard groupoids on Y´et . There is a canonical action of Pcl (ΘY ) on Tate(V ). Namely, an object Θc of cl P (ΘY ) amounts, by pull-back, to an extension E(V )c of E(V ) by OrY trivialized over gl(V ). The corresponding automorphism of Tate(V ) sends E(V )[ to its Baer sum with E(V )c . Lemma. If Tate(V ) is non-empty, then it is a Pcl (ΘY )-torsor.



2.8.3. We are going to describe Tate(V ) “explicitly” by means of auxiliary data of connections. Let us deal with Pcl (ΘY ) first. Consider the relative de Rham complex DRY/X . This is a complex of left i OY [DX ]-modules with terms DRY/X = Ωi := ΩiY/X . Passing to the right DX modules (which we skip in the notation), one gets a complex of sheaves h(DRY/X ) on Ye´t . Consider its 2-term subcomplex h(DRY/X )[1,2) := τ≤2 σ≥1 h(DRY/X ) with components h(Ω1 ) and h(Ω2 )closed := Ker(d : h(Ω2 ) → h(Ω3 )). As in [SGA 4] Exp. XVIII 1.4, our complex defines a sheaf of Picard groupoids on Ye´t . Its objects, called h(DRY/X )[1,2) -torsors, are pairs (C, c) where C is an h(Ω1 )-torsor and c is a trivialization of the corresponding induced h(Ω2 )closed torsor; i.e., c : C → h(Ω2 )closed is a map such that for ν ∈ h(Ω1 ), ∇ ∈ C one has c(∇ + ν) = c(∇) + dν. Denote this Picard groupoid by h(DRY/X )[1,2) -tors. Lemma. There is a canonical equivalence of Picard groupoids (2.8.3.1)

∼

Pcl (ΘY ) −→ h(DRY/X )[1,2) -tors.

Proof. The functor is Θc 7→ (C, c) where C is the h(Ω1 )-torsor of connections (every Θc admits a connection locally) on Θc and c is the curvature map. It is an equivalence due to the lemma in 1.4.17.  2.8.4. Now let us pass to Tate structures; we follow the notation of 2.8.1 and 1.4.17. Suppose we have a connection ∇ : ΘY → E(V ) on V (which is the same as a connection on E(V )). It defines a compatible pair (d∇ , c(∇)) ∈ D(gl(V )) (see 1.4.17) where d∇ is a d-derivation of the Lie∗ Ω· -algebra Ω· ⊗ gl(V ) and c(∇) ∈ h(Ω2 ⊗gl(V )). Since E(V ) acts on the Lie∗ OY -algebra gl(V )[ , our ∇ also defines a dderivation d[∇ of the Lie∗ Ω· -algebra Ω· ⊗ gl(V )[ such that d[∇ : gl(V )[ → Ω1 ⊗gl(V )[ corresponds to ad[∇ ∈ P2∗ ({ΘY , gl(V )[ }, gl(V )[ ). One checks that d[∇ lifts d∇ , its restriction to Ω· ⊂ Ω· ⊗ gl(V )[ equals d, and (d[∇ )2 = adc(∇) . Suppose we have a Tate structure E(V )[ on V and a connection ∇[ on the ∗ Lie algebroid E(V )[ . Such a ∇[ yields a connection ∇ on V (we say that ∇[ lifts ∇). It also yields a compatible pair (d∇[ , c(∇[ )) ∈ D(gl(V )[ ). It is clear that d∇[ coincides with d[∇ defined by ∇; hence d[∇ (c(∇[ )) = 0, and c(∇[ ) ∈ h(Ω2 ⊗ gl(V )[ ) lifts c(∇). 110 Here

the superscript “cl” means “classical”; see 3.9.6, 3.9.7 for an explanation.

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We will consider the above objects locally on Ye´t . Denote by T the sheaf of pairs (E(V )[ , ∇[ ) as above.111 Let C be the sheaf of pairs (∇, c(∇)[ )) where ∇ is a connection on V and c(∇)[ ∈ h(Ω2 ⊗ gl(V )[ ) is a lifting of c(∇) such that d[∇ (c(∇)[ ) = 0. One has a morphism (E(V )[ , ∇[ ) 7→ (∇, c(∇[ )).

c : T → C,

(2.8.4.1)

2.8.5. Lemma. c is an isomorphism of sheaves. Proof. Use the lemma in 1.4.17.

 [

[

2.8.6. Our T is naturally a groupoid: a morphism between the (E(V ) , ∇ )’s is a morphism between the corresponding E(V )[ ’s. Since every Tate structure admits a connection locally, the corresponding sheaf of groupoids is canonically equivalent to Tate(V )Ye´t . Our C also carries a groupoid structure. Namely, a morphism (∇, c(∇)[ ) → 0 (∇ , c(∇0 )[ ) is a 1-form ν [ ∈ h(Ω1 ⊗ gl(V )[ ) such that its image in h(Ω1 ⊗ gl(V )) equals ∇0 −∇ and c(∇0 )[ −c(∇)[ = d[∇ (ν [ )+ 12 [ν [ , ν [ ]. The composition of morphisms is the addition of the ν [ ’s (check the relation!). Now (2.8.4.1) lifts naturally to an equivalence of groupoids ∼

c : T −→ C

(2.8.6.1) 0

0

0

that sends φ : (Θc , ∇[ ) → (Θc , ∇ [ ) to a morphism c(φ) : (∇, c(∇[ )) → (∇0 , c(∇ [ )) 0 defined by the form ν = c(φ) := φ−1 (∇ [ ) − ∇[ (the compatibilities follow from (1.4.17.1)). We have proved 2.8.7. Proposition. The formula (2.8.6.1) yields an equivalence between Tate(V )Ye´t and the sheaf of groupoids defined by C.  Remark. Notice that C carries a natural action of the groupoid defined by the 2-term complex h(DRY/X )[1,2) . Namely, the translation by ω ∈ h(Ω2 )cl acts on objects of C as (∇, c(∇)[ ) 7→ (∇, c(∇)[ + ω) and sends a morphism defined by a 1form ν [ to a morphism defined by the same form. For µ ∈ h(Ω1 ) the corresponding morphism (∇, c(∇)[ ) → (∇, c(∇)[ + dµ) in C is defined by µ1[ ∈ h(Ω1 ⊗ gl(V )[ ). Now 2.8.7, 2.8.3 identify the corresponding sheafified action with the Pcl (ΘY )-action on Tate(V ) from 2.8.2. 2.8.8. The above description of Tate(V ) can be made more concrete by introducing auxiliary choices. First, choose an ´etale hypercovering Y· such that ≥a each Yi is a disjoint union of affine schemes. Let DRY/X be the subcomplex a a+1 ˇ Ω → Ω → · · · of DRY/X . Denote by C the complex of Cech cochains with ≥1 coefficients in h(DRY/X ). Remark. Since ΩiY/X is a direct summand of an induced DX -module for i > 0, one has H a (Yb , h(ΩiY/X )) = 0 for every a, i > 0. Thus C computes

≥1 RΓ(Y, h(DRY/X )).

Below, ∂i : Yn → Yn−1 , i = 0, . . . , n, are the (´etale) face morphisms. 111 Our

classes.

pairs are rigid, so we do not distinguish them from the corresponding isomorphism

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133

Now choose a triple (∇, χ, µ) where ∇ is a connection for V on Y0 , χ is a section of h(Ω2 ⊗ gl(V )[ ) on Y0 which lifts c(∇), and µ is a section of h(Ω1 ⊗ gl(V )[ ) on Y1 which lifts ∂0∗ ∇ − ∂1∗ ∇. Set ζ := ζ0 + ζ1 + ζ2 ∈ C 3 where ζ0 := d∇ (χ) ∈ Γ(Y0 , h(Ω3 )) ⊂ Γ(Y0 , h(Ω3 ⊗ gl(V )[ )), ζ1 := ∂0∗ χ − ∂1∗ χ − d∂0∗ ∇ µ + 12 [µ, µ] ∈ Γ(Y1 , h(Ω2 )) ⊂ Γ(Y1 , h(Ω2 ⊗ gl(V )[ )), ζ2 := −∂0∗ µ + ∂1∗ µ − ∂2∗ µ ∈ Γ(Y3 , h(Ω1 )) ⊂ Γ(Y2 , h(Ω1 ⊗ gl(V )[ )). Using the second remark in 1.4.17, one checks that ζ is a cocycle whose class ≥1 ≥1 chD (V ) ∈ H 3 (C) = H 3 (Y, h(DRY/X )) = H 4 (Y, DRX (DRY/X )) does not depend 1 on the auxiliary choices. Let Cζ be the groupoid whose set of objects is {a ∈ C 2 : 0 1 dC (a) = −ζ} and for two objects a, a0 one has Hom(a, a0 ) := d−1 C (a − a) ⊂ C ; 1 the composition of morphisms is the addition in C . If Cζ is non-empty, i.e., if chD 1 (V ) = 0, then Cζ is a τ≤2 C-torsor in the obvious way. 2.8.9. Corollary. (i) chD 1 (V ) vanishes if and only if V admits a Tate structure. (ii) The groupoid Tate(V ) is canonically equivalent to Cζ . (iii) Pcl (ΘY ) is canonically equivalent to the Picard groupoid defined by the 2-term complex τ≤2 C. The action of τ≤2 C on Cζ corresponds via the above equivalences to the Pcl (ΘY )-action on Tate(V ) from 2.8.2. Proof. Take any a ∈ Cζ , so a = (a0 , a1 ), a0 ∈ Γ(Y0 , h(Ω2 )), a1 ∈ Γ(Y1 , h(Ω1 )). Then ca := (∇, χ + a0 ) ∈ C(Y0 ), and γa := µ + a1 is a morphism ∂1∗ ca → ∂0∗ ca in C(Y2 ) that satisfies the cocycle property. The map a 7→ (ca , γa ) is a bijection between Cζ and the set of pairs (c, γ) with the above property. Morphisms in Cζ correspond to γ-compatible morphisms between the c’s. By 2.8.6 we have identified Cζ with the groupoid of pairs (E(V )[ , ∇) where E(V )[ is a Tate structure on V , ∇ is a connection on E(V )[ |Y0 , the morphisms are morphisms between the E(V )[ ’s. Since any E(V )[ admits a connection on Y0 , we are done.  2.8.10. In fact, for a vector DX -bundle V one has a whole sequence of char≥a 2a+1 (Y, h(DRY/X )), a ≥ 0. One constructs them as acteristic classes chD a (V ) ∈ H follows. Assume for a moment that V admits a global connection ∇, so we have c(∇) ∈ h(Ω2 ⊗ gl(V )). Notice that gl(V ) = End(V ), hence (see 1.4.1) Ω· ⊗ gl(V ), is an associative∗ algebra. Thus h(Ω· ⊗ gl(V )) is an associative algebra, so we have 1 a 2a ⊗ gl(V )). Assume further that this section admits a global lifting a! c(∇) ∈ h(Ω a 2a χ ∈ Γ(Y, h(Ω ⊗ gl(V )[ )). Then d∇ (χa ) ∈ h(Ω2a+1 ) ⊂ h(Ω2a+1 ⊗ gl(V )[ ). This ≥a 2a+1 form is closed, and chD (Y, h(DRY/X )). a (V ) is its class in H One extends this definition to a general situation (when global ∇ and χa need not exist) using a Chern-Simons type interpolation procedure. We need some notation. Below ∆n ⊂ An+1 is the affine hyperplane t0 + · · · + tn = 1. For a DX -scheme Z we consider Z × ∆n as a DX -scheme. One has h(Ω·Z×∆n /X ) = h(Ω·Z/X ) ⊗ Ω·∆n , so the integration over the simplex Σti = 1, 0 ≤ ti ≤ 1, yields a map ρn : Γ(Z × ∆n , h(Ω·Z×∆n /X )) → Γ(Z, h(Ω·−n Z )). It satisfies the Stokes formula ρn (dφ) = dρn (φ) + Σ(−1)i ρn−1 (φ|∆in−1 ) where ∆in−1 ⊂ ∆n , i = 0, . . . , n, is given by the equation ti = 0. Choose a hypercovering Y· as in 2.8.8 and a connection ∇ for V on Y0 . Pulling it back by the structure ´etale maps pi : Yn → Y0 , we get n+1 connections ∇0n , . . . , ∇nn on every Yn . Now the pull-back of V to Yn × ∆n gets a canonical connection ∇n :=

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a Σti ∇in , so we have c(∇n )a ∈ h(Ω2a Yn ×∆n /X ⊗gl(V )). Choose any liftings χn ∈ Γ(Yn × 2a+1 1 a a a [ ∆n , h(Ω2a Yn ×∆n /X ⊗ gl(V ) )) of a! c(∇n ) . Set ξn := d∇n (χn ) ∈ h(ΩYn ×∆n /X ) ⊂ a 2a [ a i ∗ a i h(Ω2a+1 Yn ×∆n /X ⊗ gl(V ) ) and ψn := Σ(−1) (∂i χn−1 − χn |∆n−1 ) ∈ h(ΩYn ×∆n−1 /X ) ⊂ [ h(Ω2a Yn ×∆n−1 /X ⊗ gl(V ) ). Our ∇n are flat in ∆n -directions, so we can (and will) choose χan with zero [ a a a 2a+1−n image in h(ΩYn ⊗ Ω>a ) ∆n ⊗ gl(V ) ). Then ζn := ρn (ξn ) + ρn−1 (ψn ) ∈ Γ(Yn , Ω ≥a a a ˇ vanish for n > a + 1, so ζ := Σζn is a Cech cochain with coefficients in h(DRY/X ). This is a cocycle due to the Stokes formula whose cohomology class chD a (V ) does not depend on the auxiliary choices of Y· , ∇, and χan .

Remarks. (i) Using induction by n, the forms χan can be choosen so that χan |∆in−1 = ∂i∗ χan−1 for every n and i = 0, . . . , n. Then ζ a = Σρn (ξna ). ◦ a+1 (ii) One has chD chD a (V ) = (−1) a (V ) (use (2.7.4.2)). Let us show that the above class chD 1 (V ) coincides with the one from 2.8.8. Indeed, for given ∇ any datum (χ, µ) as in 2.8.8 defines χn = χ1n as above. Namely, χ0 := χ and for n ≥ 1 we set χn := p∗0 χ1 + d∇0n µn + 21 [µn , µn ] where µn ∈ Γ(Yn ×∆n , h(Ω1Yn ×∆n /X ⊗gl(V )[ )) is a lifting of ∇n −∇0n , µn := t1 p∗01 µ+· · ·+tn p∗0n µ. The cocycle ζ for these χn ’s coincides with ζ for χ, µ from 2.8.8. 2.8.11. Let us compare the cohomology with coefficients in h(DRY/X ) with the absolute de Rham cohomology of Y. Below, for a given a left DX -module N we denote by DRX (N ) the corresponding de Rham complex placed in degrees [0, 1], so H 1 (DRX (N )) = h(N ). Denote by DRY the absolute de Rham complex OY → Ω1Y → · · · of Y. Then the complex DRX (DRY/X ) coincides with DRY . Indeed, the DX -scheme structure connection provides a decomposition Ω1Y = Ω1Y/X ⊕ π ∗ Ω1X , hence the isomorphisms ∼

∼

ΩnY −→ ⊕ΩiY/X ⊗ π ∗ ΩjX which form the isomorphism of complexes DRY/X −→ DRX (DRY/X ). We refer to the DRY/X and DRX components dY/X , dX of the de Rham differential dY as the “vertical” and “horizontal” differentials. Since ΩiY/X is a direct summand of an induced DX -module for every i > 0, we know that H 0 (DRX (ΩiY/X )) = 0 for i > 0 and H 0 (DRX (OY )) = H 0 (DRY ).112

≥a ≥a So the projection DRX (DRY/X ) → h(DRY/X )[−1] is a quasi-isomorphism for 0 a > 0, and for a = 0 the projection DRY /H (DRY ) → h(DRY/X )[−1] is a quasi∼ ≥a ≥a isomorphism. Therefore H m (Y, DRX (DRY/X )) −→ H m−1 (Y, h(DRY/X )) unless a = m = 0 (when the right-hand side vanishes).

2.8.12. Suppose now that our V is induced, so V = FD for a vector bundle F on Y. Then chD a (V ) can be expressed in terms of conventional characteristic classes of F . Namely, a classical Weil-Chern-Simons construction113 provides (for ≥a any algebraic space Y and vector bundle F ) Chern classes cha (F ) ∈ H 2a (Y, DRY ) ≥a ≥0 where DRY are the subcomplexes ΩaY → Ωa+1 → · · · of DR = DR . The Y Y Y ≥a product of forms makes ⊕H 2a (Y, DRY ) a graded commutative ring. 112 Proof:

Suppose ϕ ∈ OY is killed by dX ; i.e., dY (ϕ) ⊂ Ω1Y/X ⊂ Ω1Y . Then dX (dY (ϕ)) = 0;

H 0 (DR

1 i.e., dY (ϕ) ∈ X (ΩY/X )). Therefore dY (ϕ) = 0 by the first equality. 113 To be recalled in the course of the proof in 2.8.13.

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≥a 2a Remark. The image of cha (F ) by the map H 2a (Y, DRY ) → HDR (Y) is the conventional de Rham cohomology Chern class of F . The classes cha form the ≥∗ theory of Chern classes for the cohomology theory Y 7→ H · (Y, DRY ). ≥a+1 ≥a Since DRY ⊂ DRX (DRY/X ) ⊂ DRY ,114 one has a canonical map (2.8.12.1) ≥a+1 ≥a ≥a H 2a+2 (Y, DRY ) → H 2a+2 (Y, DRX (DRY/X )) = H 2a+1 (Y, h(DRY/X )). −1/2

2.8.13. Theorem. chD ) = cha+1 (F ) − a (V ) equals the image of cha+1 (F ωX −1/2 by (2.8.12.1). In particular, chD (V ) is the image of ch2 (F ωX ). 1

1 2 cha (F )ch1 (ωX )

Proof. First we prove our statement on the level of chains in the case when −1/2 F ωX admits a global connection. The general situation follows then by the Chern-Simons interpolation. −1/2 −1/2 (a) Suppose F ωX has a global connection ∇. Then the classes cha (F ωX ) 1 a 2a are represented by closed forms cha (∇) := a! tr(c(∇) ) ∈ Γ(Y, ΩY ) where c(∇) ∈ Ω2Y ⊗ End(F ω −1/2 ) = Ω2Y ⊗ End(F ) is the curvature of ∇. The structure connection on Y provides a decomposition Ω1Y = Ω1Y/X ⊕ π ∗ Ω1X which yields a decomposition of ∇ into a sum of vertical ∇v and horizontal ∇h parts. Since dim X = 1, one has Ω2Y = Ω2Y/X ⊕ Ω1Y/X ⊗ π ∗ Ω1X . Denote by c(∇v ) and c(∇)1,1 the components of c(∇). Then c(∇v ) is the curvature of the relative connection ∇v , and c(∇)1,1 = ∇vConn (∇h ) in terms of the notation of Remark (i) in 2.5.9. The image of cha+1 (∇) by the map Ω2a+2 → h(Ω2a+1 Y Y/X ) (see 2.8.11) coincides 1 v a with that of the form a! tr(c(∇ ) c(∇)1,1 ). We will show that chD a (V ) is represented by the same form. Since π ∗ ωX is constant along the fibers of π, our ∇v can be considered as a vertical connection on F . One has FD = V and (Ω1Y/X ⊗ F )D = Ω1Y/X ⊗ V ; set ∇V := (∇v )D : V → Ω1Y/X ⊗ V .115 Then ∇V is a D-module connection on V whose curvature is the image of c(∇v ) by the map Ω2Y/X ⊗ gl(F ) → h(Ω2 ⊗ gl(V )). By (2.7.5.4)116 the restriction of gl(V )[ to gl(F )D ⊂ gl(V ) is the Kac-Moody −1/2 and the invariant form κ = ( , ), (a, a) = extension for the vector bundle F ωX −tr(a2 ). So, as in 2.5.9, the horizontal connection ∇h provides a splitting s∇h : 1 [ gl(F )D → gl(V )[ . Set χa := s∇h ( a! c(∇v )a ) ∈ h(Ω2a+1 Y/X ⊗ gl(V ) ). Then, according 2a+1 a to 2.8.10, the class chD a (V ) is represented by the form d∇V (χ ) ∈ h(ΩY/X ) ⊂ [ h(Ω2a+1 Y/X ⊗ gl(V ) ).

The derivative d∇V preserves the subalgebra gl(F )[D ⊂ gl(V )[ and coincides there with the derivative dκ∇v defined by ∇v (see Remark (i) of 2.5.9). Since 1 d∇v (c(∇v )a ) vanishes, the formula in loc. cit. gives d∇V (χa ) = dκ∇v s∇h ( a! c(∇v )a ) = 1 1 h v a v −(∇Conn (∇ ), a! c(∇ ) ) = a! tr(c(∇ )c(∇)1,1 ) = cha+1 (∇); q.e.d. 114 Due 115 So

∇v .

to the fact that dim X = 1. h(V ) = F and h(Ω1Y/X ⊗ V ) = Ω1Y/X ⊗ F , and ∇V is defined by the property h(∇V ) =

116 Or, rather, its immediate generalization to the situation with extra parameters Y; see Remark (i) in 2.5.9.

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(b) Consider now the general situation, so F ωX need not have a global −1/2 ≥a connection. Recall the construction of cha (F ωX ) ∈ H 2a (Y, DRY ). −1/2 Choose a hypercovering Y· as in 2.8.8 and a connection ∇ for F ωX on Y0 . As in 2.8.10 we get connections ∇0n , . . . , ∇nn on Yn and ∇n := Σti ∇in on Yn × ∆n hence 1 forms cha (∇n ) := a! c(∇n )a ∈ Γ(Yn × ∆n , Ω2a Yn ×∆n ). As in 2.8.10 we have the integration map ρn : Γ(Yn × ∆n , Ω·Yn ×∆n ) = Γ(Yn , Ω·Yn ) ⊗ Γ(∆n , Ω·∆n ) → Γ(Yn , Ω·−n Yn ). −1/2 117 ˇ The class cha (F ω ) is represented by a Cech cocycle Σρn (cha (∇n )). X

−1/2

To compare cha+1 (F ωX ) with chD a (V ), one computes the latter class us1 c(∇vn )a ) as in (a). These forms sating connection ∇V and forms χan = s∇hn ( a! isfy the condition from Remark (i) in 2.8.10, so chD a (V ) is represented by cochain −1/2 a Σρn d∇V (χn ). According to (a) it equals cha+1 (F ωX ).  2.8.14. Rigidified and weakly equivariant Tate structures. In the rest of this section we describe two general procedures for constructing Tate structures together with some examples. (a) Let ∇ be an integrable connection on a vector DX -bundle V . As follows from 2.8.4, 2.8.5, there is a unique pair (E(V )[∇ , ∇[ ) where E(V )[ is a Tate structure on V and ∇[ is a lifting of ∇ to an integrable connection on E(V )[ . By 2.8.2, ∼ one can use E(V )[∇ as the base point for an identification Pcl (ΘY ) −→ Tate(V ), ΘcY 7→ E(V )[+c ∇ . An example of this situation: Suppose that ΘY is L-rigidified; i.e., we have an ∼ action τ of a Lie∗ algebra L on Y which yields an isomorphism LOY := L⊗OY −→ ΘY (see 1.4.13). Such a τ yields an integrable connection ∇τ on Θ = ΘY defined by the property ∇τ τ = τΘ , where τΘ : L → E(Θ) is τΘ (ξ)(θ) = [τ (ξ), θ] for ξ ∈ L, θ ∈ Θ. Thus we have E(Θ)[τ := E(Θ)[∇τ ∈ Tate(Θ). Denote by P(L) the Picard groupoid of Lie∗ algebra ωX -extensions of L. Any Lc ∈ P(L) yields the corresponding Lie∗ algebroid extension ΘcY := LcOY ∈ Pcl (ΘY ) (see 1.4.13); hence the Tate structure c := E(Θ)[+c E(Θ)[+c τ ∇τ . We refer to it as the L -rigidified Tate structure on Y (with respect to τ ). One has canonical embeddings of Lie∗ algebras τ c : Lc → ΘcY , c : Lc → E(Θ)τ[+c which lift τ , τΘ . τΘ (b) Let G be a smooth group DX -scheme affine over X acting on Y and let V be a G-equivariant vector DX -bundle. Then E(V ) is a G-equivariant Lie∗ algebroid (see 2.5.17) and G acts on the Lie∗ algebra gl(V )[ by transport of structure. One defines a weakly G-equivariant Tate structure on V demanding E(V )[ to be a weakly G-equivariant Lie∗ algebroid such that the maps gl(V )[ → E(V )[ → E(V ) are compatible with the G-actions. Weakly G-equivariant Tate structures form a torsor Tate(V )G over the groupoid Pcl (Θ)G of weakly G-equivariant OrY -extensions of ΘY . If we have ∇ as in (a) preserved by the G-action, then E(V )[∇ is weakly G∼ equivariant and we have the equivalence Pcl (ΘY )G −→ Tate(V )G , ΘcY 7→ E(V )[+c ∇ . Suppose that we have an L-rigidification τ of ΘY and an action of G on the Lie∗ algebra L such that τ is compatible with the G-actions. Then Θ = ΘY is a G-equivariant vector DX -bundle and the connection ∇τ on it is compatible with the G-action, so the above constructions apply. Let P(L)G be the Picard groupoid of G-equivariant Lie∗ algebra ωX -extensions Lc of L (we assume that G acts on ωX ⊂ Lc trivially). For Lc ∈ P(L)G the extension ΘcY is weakly G-equivariant in 117 One

≥a has ρn (cha (∇n )) = 0 for n > a since ∇n is flat along ∆n , so our cocycle is in DRY .

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137

the evident way, so we get a morphism of Picard groupoids and their torsors (2.8.14.1)

P(L)G → Pcl (ΘY )G ,

P(L)G → Tate(Y)G ,

c Lc 7→ ΘcY , E(Θ)τ[+c . The embeddings τ c , τΘ from (a) are compatible with the Gactions.

Exercise. Define a strongly G-equivariant Tate structure. What do we need to make E(V )cτ strongly G-equivariant? 2.8.15. The case of a bitorsor. Let G, G0 be smooth group DX -schemes affine over X and let Y be a (G, G0 )-bitorsor; i.e., Y is a DX -scheme equipped with a G × G0 -action which makes it both G- and G0 -torsor. In other words, Y is a G-torsor, and G0 is the group DX -scheme of its automorphisms (= the twist of G by the G-torsor Y with respect to the adjoint action of G). Let us 0 0 describe the groupoids Tate(Y)G , Tate(Y)G , Tate(Y)G×G of weakly equivariant Tate structures on Y; these are torsors over the corresponding Picard groupoids 0 0 Pcl (ΘY )G , Pcl (ΘY )G , Pcl (ΘY )G×G . 0 ∗ Let L, L be the Lie algebras of G, G0 , and τ : L → ΘY , τ 0 : L0 → ΘY their actions on Y. Notice that τ is compatible with the G×G0 -actions where the G-action 0 ∼ on L is the adjoint action and the G0 -action is trivial; in fact, τ : L −→ (ΘY )G (the ∗ 0 Lie subalgebra of G -invariants in ΘY ). The same is true for G interchanged with G0 . Since both τ and τ 0 are rigidifications of ΘY , we can use (2.8.14.1) to get the morphisms of the Picard groupoids and their torsors 0

0

(2.8.15.1)

P(L) → Pcl (ΘY )G ,

P(L) → Tate(Y)G ,

(2.8.15.2)

P(L0 ) → Pcl (ΘY )G ,

P(L0 ) → Tate(Y)G ,

0

0

(2.8.15.3) P(L)G → P(ΘY )G×G ← P(L0 )G ,

0

0

P(L)G → Tate(Y)G×G ← P(L0 )G .

Lemma. The above functors are equivalences of groupoids. 0

Proof. Consider, for example, (2.8.15.1). The inverse functor Pcl (ΘY )G → 0 P(L) assigns to ΘcY its Lie∗ subalgebra (ΘcY )G of G0 -invariants. The inverse functor 0 0 Tate(Y)G → P(L) sends E(Θ)c to the pull-back by τΘ : L → E(Θ)G (see (a) in 0 2.8.14) of the Lie∗ subalgebra (E(Θ)c )G of G0 -invariants in E(Θ)c (the latter is 0 an ωX -extension of E(Θ)G ). The inverse functors to (2.8.15.2) and (2.8.15.3) are defined in a similar way.  The Tate extension L[ of L is the pull-back of the Tate extension gl(L)[ by the adjoint action morphism ad : L → gl(L). The group G acts on it by transport of structure. So ad lifts to a morphism of G-equivariant ωX -extensions ad[ : L[ → [ [ [ gl(L)[ . We also have the Tate extension L0 ∈ P(L0 ) and a morphism ad0 : L0 → 0 [ gl(L ) . 0 0 c0 For Lc ∈ P(L)G consider the corresponding ΘcY ∈ Pcl (ΘY )G×G , L0 ∈ P(L0 )G 0 0 Tc and E(Θ)c ∈ Tate(Y)G×G , L0 ∈ P(L0 )G coming from equivalences (2.8.15.3). Proposition. There is a canonical isomorphism of G0 -equivariant extensions (2.8.15.4)

L0

Tc ∼

−→ L0

c0 +[

.

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◦

Proof. Let j 0 = τ 0 ⊗ τ 0 : gl(L0 ) ,→ gl(Θ) be the embedding whose image is the ∼ [ subalgebra gl(Θ)G of G-invariants. We also have the isomorphisms j 0 : gl(L0 )[ −→ ∼ [ G 0 0 0 0 c G (gl(Θ) ) , (τ , j ) : L n gl(L ) −→ (E(Θ) ) . Consider the flat connection ∇τ on Θ compatible with the G × G0 -action (see 0 0 (a) in 2.8.14). Since the images of τΘ , τΘ commute, one has τΘ = ∇τ τ 0 + j 0 ad0 . c0 The extension ΘcY equals the pull-back of E(Θ)c by ∇τ . Therefore L0 is the Tc pull-back of the ωX -extension (E(Θ)c )G of E(Θ)G by ∇τ τ 0 . Now L0 is the pull0 back of (E(Θ)c )G by τΘ . Since the images of j 0 and ∇τ commute, the pull-back by 0 τΘ of any extension of E(Θ)G is the Baer sum of its pull-backs by ∇τ τ 0 and j 0 ad0 , and we are done  We say that Lc ∈ P(L)G is strongly G-equivariant if the L-action adc on Lc coming from the G-action Adc coincides with the adjoint action of Lc (factored through L). One checks immediately that Lc is strongly G-equivariant if and only c0 if the images of τ c and τ 0 in ΘcY mutually commute. If this happens, then the Tc c images of τΘ and τ 0 Θ in E(Θ)c also mutually commute. ∼ Notice that there are two natural equivalences of Picard groupoids P(L)G −→ 0 0 c P(L0 )G . The first one is Lc 7→ L0 considered above. The second equivalence c c 0 L 7→ (L ) is defined as follows. Consider isomorphisms of G×G0 -equivariant vector ∼ ∼ ∼ DX -bundles L ⊗ OY −→ Θ ← L0 ⊗ OY defined by τ , τ 0 ; let α : L ⊗ OY −→ L0 ⊗ OY be minus the composition. Both L ⊗ OY and L0 ⊗ OY are G × G0 -equivariant Lie∗ OY -algebras in the obvious way. A usual computation shows that α is compatible with the Lie∗ brackets.118 There is an obvious identification of P(L)G with the Picard groupoid of the G × G0 -equivariant Lie∗ OY -algebra extensions of L ⊗ OY and the same for L replaced by L0 ; together with α, they produce the promised 0 ∼ equivalence P(L)G −→ P(L0 )G , Lc 7→ (Lc )0 . Lemma. For a strongly G-equivariant Lc there is a canonical identification (2.8.15.5)

∼

(Lc )0 −→ L0

−c0

.

Proof. Consider isomorphisms of G × G0 -equivariant DX -vector bundles Lc ⊗ ∼ ∼ ∼ c0 c0 c0 OY −→ ΘcY ← L0 ⊗ OY defined by τ c , τ 0 ; let αc : Lc ⊗ OY −→ L0 ⊗ OY be minus c0 the composition. Consider Lc ⊗ OY and L0 ⊗ OY as G × G0 -equivariant Lie∗ OY algebras. The computation from the previous footnote with L replaced by Lc , etc., shows that αc is compatible with the Lie∗ brackets (here the strong G-equivariance of Lc is used). Our identification is the restriction of αc to the G-invariants.  Example. The Tate extension L[ is strongly G-equivariant, and there is a [ canonical identification (L[ )0 = L0 (for the corresponding G × G0 -equivariant extension of the Lie∗ OY -algebra L ⊗ OY = L0 ⊗ OY is its Tate extension). Now take any a, a0 ∈ k such that a + a0 = 1. According to the last lemma and the proposition, the weakly G × G0 -equivariant Tate structures that correspond to 0 a0 [ La[ ∈ P(L)G and L0 ∈ P(L0 )G by (2.8.15.3) are canonically identified. Denote this Tate structure by E(Θ)[a,a0 . It is equipped with canonical Lie∗ algebra embeddings La[ ,→ E(Θ)[a,a0 ←- L0

a0 [

whose images mutually commute.

118 Suppose that τ 0 (χ) = Σf τ (ξ ), τ 0 (ψ) = Σg τ (η ); we want to show that τ 0 ([χ, ψ]) = i i j j −Σfi gj τ ([ξi , ηj ]). Indeed, τ 0 ([χ, ψ]) = [τ 0 (χ), τ 0 (ψ)] = −Σfi gj [τ (ξi ), τ (ηj ]) + Σfi [τ (ξi ), τ 0 (ψ)] − Σgj [τ (ηj ), τ 0 (χ)] = −Σfi gj τ ([ξi , ηj ]); q.e.d.

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139

Exercises. (i) Show that the construction of the proposition is symmetric with respect to the interchange of G and G0 . Precisely, show that the composition 0 0 0 00 ∼ ∼ ∼ ∼ LT(Tc) −→ LT(c +[) −→ L(c +[) +[ −→ Lc −[+[ −→ Lc coincides with the evident identification. Here the first arrow is (2.8.15.4) transformed by T : P(L0 ) → P(L), the second arrow is (2.8.15.4) for G, G0 interchanged, the third one comes from the identification [0 = −[ (see Example and the last lemma), and the fourth is the evident identification c00 = c. (ii) Suppose our bitorsor Y is trivialized; i.e., we have a horizontal X-point ∼ e ∈ Y(X). It yields an identification G −→ G0 , g 7→ g 0 , where g 0 e = g −1 e. Show G ∼ 0 G0 that the equivalence P(L) −→ P(L ) , Lc 7→ (Lc )0 , comes from the identification of G = G0 . (iii) We are in situation (ii). Suppose that Lc is strongly G-equivariant, and Tc let LTc ∈ P(L)G be the extension that corresponds to L0 by the identification G = G0 . By (ii) we can rewrite (2.8.15.4), (2.8.15.5) as a canonical isomorphism ∼ Lc+Tc −→ L[ . Show that it can be rephrased as follows. It suffices to describe Tc c its composition Lc+Tc → gl(L)[ with ad[ . Since the images of τΘ , τ 0 Θ commute, 0 Tc c Tc c ∗ c we have a morphism of Lie algebras τΘ + τ Θ L × L → E(Θ) . The Baer sum Lc+Tc is the subquotient of Lc × LTc , and the above morphism yields a morphism c : Lc+Tc → E(Θ)c . Its value at e ∈ Y belongs to gl(L)[ ⊂ E(Θ)e ; this is our τ˜Θ canonical morphism Lc+Tc → gl(L)[ . 2.8.16. The descent of Tate structures. Suppose that a smooth group DX -scheme G affine over X acts freely on a smooth DX -scheme Y; i.e., we have a morphism of smooth DX -schemes π : Y → Z such that Y is a G-torsor over Z. Let L be the Lie∗ algebra of G and let L[ be its Tate extension (see 2.8.15). Suppose we have a weakly G-equivariant Tate structure E(ΘY )[ (see 2.8.14) and a morphism of Lie∗ algebras α : L[ → E(ΘY )[ which sends 1[ to 1[ , commutes with the action of G, and lifts the morphism L → E(ΘY ) corresponding to the L-action on ΘY . Proposition. A pair (E(ΘY )[ , α) as above defines a Tate structure E(ΘZ )[α ∈ Tate(Z) called the descent of E(ΘY )[ with respect to α. Proof. Below for a Lie∗ algebroid L on Y we denote by L/Z the preimage of ΘY/Z ⊂ ΘY by the anchor map τL : L → ΘY ; this is a Lie∗ subalgebroid of L. We will consider Z-families of Tate structure along the fibers of π (which are OrY -extensions of E(ΘY/Z )/Z ) and call them simply Tate structures on Y/Z. The material of 2.8.14 and 2.8.15 immediately generalizes to the setting of families. Consider a filtration B ⊂ A[ ⊂ E(ΘY )[ defined as follows. Let A[ be the preimage of the Lie∗ subalgebroid A ⊂ E(ΘY ) preserving ΘY/Z ⊂ ΘY . Our B is the Lie∗ subalgebra of all endomorphisms of ΘY having image in ΘY/Z ⊂ ΘY , embedded in gl(ΘY )[ as in 2.7.4(i).119 Now A[ is a Lie∗ subalgebroid of E(ΘG )[ , B is an ideal in A[ , and the G-action preserves our datum. Therefore we have a weakly G-equivariant Lie∗ algebroid A/B on Y and its OrY -extension A[ /B. By construction, A/B acts on ΘY/Z and π ∗ ΘZ , and these actions yield an ∼ isomorphism A/B −→ E(ΘY/Z ) × E(π ∗ ΘZ ). By 2.7.4(i) the restriction of A[ /B to ΘY

gl(ΘY/Z ) × gl(π ∗ ΘZ ) is the Baer product of the Tate extensions. 119 Consider the filtration Θ Y/Z ⊂ ΘY ; by (2.7.4.1) the Tate extension of gl(ΘY ) splits canonically over B.

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Now A/B contains E(ΘY/Z )/Z as a Lie∗ subalgebroid. Its preimage E(ΘY/Z )[/Z ⊂ A[ /B is a weakly G-equivariant Tate structures on Y/Z. The morphism α : L[ → E(ΘY )[ has an image in A[ , and, modulo B, in E(ΘY/Z )[/Z . Therefore α identifies E(ΘY/Z )[/Z with the weakly G-equivariant L[ -rigidified Tate structure on Y/Z (see (2.8.14.1)). Equivalently, E(ΘY/Z )[/Z identifies canonically with the Tate structure E(Θ)[1,0 from Example in 2.8.15. So the morphism of Lie∗ OZ -algebras L0Z ,→ π∗ ΘY/Z (whose image is the Lie∗ subalgebra of G-invariant vector fields) lifts naturally to L0Z ,→ π∗ E(ΘY/Z )[/Z whose image is invariant with respect to the G-action. Denote the latter embedding by iα . Set C := (π∗ A/B)G , C [ := (π∗ A[ /B)G . These are Lie∗ algebroids on Z, and [ C is an OrZ -extension of C. Notice that C acts on ΘZ ; the morphism C → E(ΘZ ) is surjective, and its kernel equals (π∗ E(ΘY/Z )[/Z )G . In particular, it contains i(L0Z ). Since all our constructions were natural, i(L0Z ) is an ideal in C [ . Finally, set E(ΘZ )[ := C [ /iα (L0Z ). This is a Tate structure on Z: the identi∼ fication gl(ΘZ )[ −→ Ker(E(ΘZ )[ → ΘZ ) comes from the morphism π ∗ (gl(ΘZ )[ ) = gl(π ∗ ΘZ )[ → A[ /B. We are done.  Denote by φ(L) the vector space of DX -module morphisms φ : L → ωX compatible with the actions of G (the adjoint and the trivial ones). Notice that such a φ is automatically a morphism of Lie∗ algebras. One has a natural morphism of Picard groupoids (2.8.16.1)

φ(L) → Pcl (ΘZ ),

φ 7→ ΘφZ .

Namely, (π∗ ΘY )G is naturally a Lie∗ algebroid on Z which is an extension of ΘZ by L0Z . A morphism φ : L → ωX from φ(L) yields, by twist, a morphism of Lie∗ OZ -algebras φ0Z : L0Z → OrZ . Our ΘφZ is the push-out of (π∗ ΘY )G by φ0Z . It is a Lie∗ OZ -algebroid since φZ is compatible with the (π∗ ΘY )G -actions. Lemma. Suppose we have (E(ΘY )[ , α) as in the proposition. Then for any φ ∈ φ(L) one has a canonical identification (2.8.16.2)

∼

E(ΘZ )[α+φ −→ E(ΘZ )[+φ α .

Here α + φ in the left-hand side is the composition of α and the automorphism idL[ + φ of L[ , and E(ΘZ )[+φ is the Baer sum of E(ΘZ )[α and ΘφZ . α Proof. We use notation from the proof of the proposition. Recall that E(ΘZ )[α = C [ /iα (L0Z ). Replacing α by α + φ, we do not change C [ , and iα+φ = iα − φ0Z . So (2.8.16.2) amounts to a morphism of Lie∗ OZ -algebroids χ : C [ → E(ΘZ )[+φ which α lifts the identity morphism of E(ΘZ ), sends 1[ to 1[ , and satisfies χiα = φ0Z . One constructs it as follows. Denote the projection C [ → E(ΘZ )[α by η. Let  be the composition of morphisms of Lie∗ OZ -algebroids C [ → (π∗ ΘY )G → ΘφZ where the first arrow (η,)

comes from the anchor map for A[ /B. Now our χ is the composition C [ −−−→ E(ΘZ )[α × ΘφZ → E(ΘZ )[+φ where the second arrow is the Baer sum projection.  α ΘZ

2.8.17. Tate structures on the jet scheme of a flag space. We change notation: now G is a semi-simple group. Let N ⊂ B ⊂ G be a Borel subgroup and

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141

its nilradical, H := B/N the Cartan group, n ⊂ b ⊂ g and h the corresponding Lie algebras, Q := G/B the flag space. Set GX := G × X, JG := JGX (see 2.3.2 for notation), etc. So we have smooth group DX -schemes JB ⊂ JG, and ∼ JG/JB −→ JQ. The Lie∗ algebra of JG equals gD , etc. We will consider Tate structures and related objects on JQ locally with respect to X. Denote by Pcl (ΘJQ )X and Tate(JQ)X the corresponding sheaves of groupoids on X. We also have the weakly JG-equivariant counterparts Pcl (ΘJQ )JG X , Tate(JQ)JG . X × Let ρ ∈ h∗ be the half-sum of the positive roots. Let ωX be the O× X -torsor of invertible sections of ωX . Define the Miura h∗ ⊗ ωX -torsor MX as the push-out of × ∗ ωX by ρ ⊗ d log : O× X → h ⊗ ωX . Remark. Sections of MX are sometimes called Miura opers (for the Langlands dual group G∨ ); see, e.g., [DS], [BD], [Fr]. They can be interpreted as connections × ρ on an H ∨ -torsor (ωX ) (here H ∨ is the torus dual to H). Theorem. The groupoid Pcl (ΘJQ )X is discrete, so we can consider it as a sheaf of k-vector spaces on X. There are canonical identifications of sheaves of vector spaces and their torsors Pcl (ΘJQ )JG → h∗ ⊗ ω X , X − ∼

(2.8.17.1)

Tate(JQ)JG → MX . X − ∼

Proof. (a) Consider the complex h(DRJQ/X )[1,2) from 2.8.3. According to (2.8.3.1), Pcl (ΘJQ )X identifies canonically with the Picard groupoid defined by the 2-term complex τ≤1 Rp∗ (JQ, h(DRJQ/X )[1,2) [1]) where p : JQ → X is the projection. To compute it, we use the canonical affine projection π : JQ → QX . The morphism of OJQ -modules π ∗ (Ω1Q ⊗ OX ) → Ω1JQ/X yields an isomorphism of ∼

left OJQ [DX ]-modules DX ⊗ π ∗ (Ω1Q ⊗ OX ) −→ Ω1JQ/X or right OJQ [DX ]-modules ∼

1 1 1 (π ∗ Ω1Q ⊗ ωX )D −→ Ω1r JQ/X . Therefore π∗ h(ΩJQ/X ) = Rπ∗ h(ΩJQ/X ) = (ΩQ ⊗ ωX ) ⊗ π∗ OJQ . The group GX of G-valued functions on X = horizontal sections of JG acts on the above sheaves, and the isomorphisms are compatible with this action. The sheaf (π∗ OJQ )/OQX admits an increasing filtration compatible with the GX -action with successive quotients isomorphic (as GX -modules) to direct sum⊗m , n > 0. Since Γ(Q, (Ω1Q )⊗n ) = 0 for mands of sheaves of type (Ω1Q )⊗n ⊗ ωX n > 0,120 one has p∗ h(Ω1JQ/X ) = 0. In particular, Pcl (ΘJQ )X is discrete; hence Pcl (ΘJQ )X = R2 p∗ h(DRJQ/X )[1,2) . (b) Since H 1 (Q, (Ω1Q )⊗n )G = 0 for n > 1,121 we see that the embedding Ω1Q ⊗ ωX ,→ π∗ h(Ω1JQ ) yields an isomorphism

(2.8.17.2)

∼

H 1 (Q, Ω1Q ) ⊗ ωX −→ (R1 p∗ h(Ω1JQ/X ))GX .

We will see in a moment that the arrow (2.8.17.3)

(R2 p∗ h(DRJQ/X )[1,2) )GX → R1 p∗ h(Ω1JQ/X )GX

120 It suffices to show that for every G-equivariant line bundle subquotient L of (Ω1 )⊗n one Q has Γ(Q, L) = 0, which is clear since the weight of L is the sum of n > 0 negative roots. 121 It suffices to check that for every L as in the previous footnote one has H 1 (Q, L)G = 0. Indeed, if H 1 (Q, L)G 6= 0, then, by Borel-Weil-Bott, its weight is a simple negative root, and the weight of our L is the sum of n > 1 negative roots.

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coming from the projection h(DRJQ/X )[1,2) [1] → h(Ω1JQ/X ) is an isomorphism. We define the first isomorphism in (2.8.17.1) as the composition of (2.8.17.3), the ∼ inverse to (2.8.17.2), and the standard Chern class identification H 1 (Q, Ω1Q ) −→ h∗ . One has p∗ h(Ω2JQ/X ) = 0: indeed, Ω2JQ/X is a direct summand in (Ω1JQ/X )⊗2 = DX ⊗ (π ∗ Ω1Q ⊗ DX ⊗ π ∗ Ω1Q ); hence π∗ h(Ω2JQ/X ) is a direct summand in ((Ω1Q )⊗2 ⊗ ωX ⊗ DX ) ⊗ π∗ OJQ , and the push-forward to X of the latter sheaf vanishes for the same reason that p∗ h(Ω1JQ/X ) did. Therefore the map R2 p∗ h(DRJQ/X )[1,2) → R1 p∗ h(Ω1JQ/X ) is injective. Its surjectivity on GX -invariants follows from the fact ∼

that H 1 (Q, Ω1closed ) −→ H 1 (Q, Ω1Q ). To see this, notice that the arrow (2.8.17.2) Q d

lifts to R2 p∗ h(DRJQ/X )[1,2) since Ω1closed ⊗ ωX ,→ Ker(π∗ h(Ω1JQ ) − → π∗ h(Ω2JQ )). Q ˜ b[D

(c) Now let us pass to Tate structures. Let ∈ P(bD ) be the restriction of the Tate extension g[D of gD to bD . Since the adjoint action of n on g is nilpotent, this extension, as well as the Tate extension b[D , are canonically trivialized on nD ˜ [/2−[

(by 2.7.4(i)). Set bµD := bD ; let hµD ∈ P(hD ) be its descent to hD according to the above trivialization over nD . Denote by TX the sheaf of trivializations of the extension hµD . If hµD is locally trivial, then TX is a Hom(hD , ωX ) = h∗ ⊗ ωX -torsor. We define the second isomorphism in (2.8.17.1) as the composition of natural maps MX → TX → Tate(JQ)JG X

(2.8.17.4)

which commute with the h∗ ⊗ ωX -actions (here the action on Tate(JQ)JG X comes from the first isomorphism in (2.8.17.1)). Since MX is locally non-empty, our maps are isomorphisms of h∗ ⊗ ωX -torsors. (i) The construction of the map MX → TX : By the definition of MX , it amounts × to a morphism ωX → TX , ν 7→ γν , such that γf ν = ρ ⊗ d log f + γν for f ∈ O× X . To define γν , we fix a non-degenerate Ad-invariant bilinear form κ on g. It yields an ∼ ∼ identification g/b −→ n∗ ; hence (g/b ⊗ ωX )D −→ (nD )◦ (see (2.2.16.1)). Thus for ∼ × any ν ∈ ωX we get an isomorphism ικν : gD /bD −→ (nD )◦ . The adjoint action of bD on gD preserves the filtration nD ⊂ bD ⊂ gD and is trivial on the subquotient bD /nD . By (2.7.4.1) we have a canonical identification ˜ ∼ a : b[+] → b[D where b]D is the pull-back of the Tate extension gl(gD /bD )[ by the D − adjoint action map bD → gl(gD /bD ). The isomorphism ικν is compatible with the bD -actions. Together with (2.7.4.2), ∼ it yields an identification σνκ : b[D −→ b]D . It follows from, say, (2.7.5.4) or a direct κ calculation that σν does not depend on κ and it satisfies σfκν = σνκ + 2ρ ⊗ d log f . ∼

˜

Now consider an isomorphism a/2(idb[ + σνκ ) : b[D −→ b[/2 . It is compatible D with the trivializations of our extensions over nD ; hence it can be considered as a trivialization of hµD , i.e., an element of TX . This is our γν . (ii) The construction of the map TX → Tate(JQ)X , γ 7→ E(ΘJQ )[γ : Consider JG as (the trivialized) (JG, JG)-bitorsor with respect to the left and right translations. Let E(ΘJG )[ be the weakly JG × JG-equivariant Tate structure on JG denoted [/2 by E(Θ)[1/2,1/2 in Example from 2.8.15. We have the canonical embedding τ 0 Θ : [/2

gD → E(ΘJG )[ lifting the right translation action τ 0 of gD . Restricting it to bD , ˜ [/2

˜ [/2

we get τ 0 Θ : bD → E(ΘJG )[ .

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˜ [/2

Therefore γ ∈ TX yields a morphism αγ := τ 0 Θ γ : b[D → E(ΘJG )[ . Now the projection JG is a JB-torsor over JQ = JG/JB. The pair (E(ΘJG )[ , αγ ) satisfies the conditions from 2.8.16. So, by the proposition from loc. cit., it yields a Tate structure on JQ which is weakly JG-equivariant (since E(ΘJG )[ is weakly JG-equivariant with respect to left translations). This is our E(ΘJQ )[γ ∈ Tate(JQ)JG X . According to (2.8.16.2), it behaves in a right way under h∗ ⊗ ωX -translations of α.  Exercise. Suppose that X is an affine line, so it carries an action of the group Af f of the affine transformations. It lifts in the obvious way to X-schemes GX , QX , hence, by transport of structure, to the corresponding jet DX -schemes. Show that there is a unique Tate structure on JQ which is (weakly) Af f -equivariant in the obvious sense, and this Tate structure is automatically weakly GX -equivariant.122 2.9. The Harish-Chandra setting and the setting of c-stacks This section is quite isolated and can be skipped by the reader. Let Y be a scheme and let L be a Lie algebroid on Y which is a locally free OY -module of finite rank. We will play with the quotient stack Y /L of Y modulo the action of (the formal groupoid of) L and similar objects called c-stacks (where “c” is for “crystalline”). The key fact is that the category of O-modules on a cstack (which are L-modules on Y in the case of Y /L) is naturally an augmented compound tensor category. The particular cases we already came across are the category of D-modules (were L is the tangent algebroid) and, more generally, the category M(Y ) where Y is a DX -scheme (here L is the horizontal foliation defined by the structure connection). A c-stack covered by a point amounts to a Harish-Chandra pair (g, K) (the notation for this stack is B(g,K) ), and O-modules on B(g,K) are the same as (g, K)-modules. Specialists in non-linear partial differential equations explain that it is important to consider symmetries of differential equations that mix the dependent and independent variables. In the setting of 2.3, this means that for a DX -scheme Y the structure that really matters is the horizontal foliation, and the projection Y → X can be forgotten. Therefore, for an algebraic geometer, non-linear differential equations are c-stacks. For this, see [V] and references therein.123 The c-stack functoriality is quite convenient. For example, suppose a smooth variety X carries a (g, K)-structure, i.e., a K-torsor whose space is equipped with a simple transitive g-action compatible with the K-action. Such a structure yields a morphism of c-stacks X/ΘX → B(g,K) , hence a compound tensor functor (the base change) from (g, K)-modules to D-modules on X. Thus one gets, say, Lie∗ algebras on X from Lie∗ algebras in the category of (g, K)-modules. An important (g, K) corresponds to the group ind-scheme of automorphisms of the formal ball Spec k[[t1 , . . . , tn ]]: according to Gelfand-Kazhdan [GeK], every X of dimension n carries a canonical (g, K)-structure (the space of formal coordinate systems), so (g, K)-modules give rise to “universal” D-modules. 122 Hint: X = A1 carries a unique Af f -invariant Miura oper, which yields an Af f -equivariant f Tate structure. One checks that Pcl (ΘJQ )Af = 0 by an argument similar to part (a) of the proof X of the theorem. 123 [V] considers, under the name of diffieties, c-stacks of type Y /L such that L is a foliation. The rank of L is called in loc. cit. the Dimension of the diffiety.

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In 2.9.1 we recall what Lie algebroids and modules over them are;124 for more information about Lie algerboids in the framework of the usual differential geometry see [M]. The enveloping algebras of Lie algebroids are considered in 2.9.2. The corresponding Poincar´e-Birkhoff-Witt theorem was proved originally by Rinehart [Rin] for Lie R-algebroids which are projective R-modules. For yet another proof (which makes sense also in the chiral setting) see 3.9.13 below. In 2.9.4 we show that for a Lie algebroid L which is a locally free O-module of finite rank, the category of L-modules is naturally an augmented compound tensor category (generalizing thus the construction of 2.2). In 2.9.5 we define the pull-back of a Lie algebroid by a smooth morphism; the reader can skip this section (it will be used in 4.5). In 2.9.7 we define a canonical augmented compound tensor structure on the category of Harish-Chandra modules for a Harish-Chandra pair (g, K) such that dim g/k < ∞. In 2.9.8 we define the notion of the (g, K)-structure on a smooth variety and the corresponding functor from Harish-Chandra modules to D-modules. The GelfandKazhdan structure is considered in 2.9.9 (we follow the exposition of [W]). In the rest of the section we discuss briefly the general setting of c-stacks (for the language of stacks see [LMB]). The definition of c-stack is in 2.9.10; in 2.9.11 we show that the quotient of an algebraic stack modulo the action of appropriate Lie algebroid is a c-stack. In 2.9.11 we formulate (without a proof) the statement that O-modules on a c-stack form naturally an augmented compound tensor category which combines 2.9.4 and 2.9.7. 2.9.1. Lie algebroids and modules over them. Let R be a commutative k-algebra. A Lie R-algebroid is a k-module L equipped with R-module and Lie kalgebra structures and an action τ of the Lie algebra L on R (so τ is a morphism of Lie algebras L → Derk (R) called the anchor map). One demands that for f, g ∈ R, l, l0 ∈ L one has τ (f l)g = f (τ (l)g) and [l, f l0 ] = τ (l)(f )l0 + f [l, l0 ]. Lie R-algebroids are local objects with respect to the Zariski or ´etale topology of Spec R, so we know what Lie OX -algebroids = Lie algebroids on X for any scheme or algebraic space X are. Lie algebroids form a category which we denote by LieAlgR or LieAlgX . Remarks. (i) Derk (R) is a final object of the category of Lie R-algebroids. In the setting of a scheme or an algebraic space, Lie algebroids are assumed to be quasi-coherent OX -modules, so if ΘX := Derk (OX ) is OX -quasi-coherent (which happens if X is of finite type), then it is a final object of LieAlgX . (ii) For a Lie R-algebroid L the kernel L♦ of the anchor map is a Lie R-algebra. In the setting of a scheme or an algebraic space, L♦ is a Lie OX -algebra which is OX -quasi-coherent ΘX is. For a Lie R-algebroid L denote by L− a copy of L considered as a mere Lie k-algebra equipped with an action τ on R (i.e., we forget about the R-module structure on L). An L− -module is an R-module M equipped with an action of L− which is compatible with the L− -action on R; i.e., for every f ∈ R, m ∈ M , and l ∈ L one has l(f m) = l(f )m + f (lm). We say that M is a left (resp. right) L-module if, in addition, one has (f l)m = f (lm) (resp. (f l)n = l(f n)). 124 According to K. Mackenzie, Lie algebroids enjoyed some fourteen different names for a half century of existence.

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The category M(R, L− ) of L− -modules is an abelian k-category; the R-tensor product makes it a tensor category. The full subcategories M` (R, L), Mr (R, L) of left and right L-modules are closed under subquotients. The tensor product of two left L-modules is a left L-module, and that of a left and a right L-module is a right L-module. Therefore M` (R, L) is a tensor category which acts on Mr (R, L). Any right L-module M defines the (homological) de Rham-Chevalley complex CrR (L, M ). As a mere graded module, it equals M ⊗ SymR (L[1]), and the differR

ential is determined by the property that the evident surjective map C(L− , M ) → CrR (L, M ) is a morphism of complexes. Here C(L− , M ) is the usual Lie algebra homology complex (which equals M ⊗ Symk (L[1]) as a mere graded module). Equivak

lently, the differential is characterized by the property that the evident action of the r Lie algebra L− † on M ⊗ SymR (L[1]) (as on a mere graded module) makes CR (L, M ) R

a DG L− † -module (see 1.1.16). Any left L-module P yields the (cohomological) deLRham-Chevalley complex (L[1]), P ) and the C`R (L, P ). As a mere graded module, it equals HomR ( SymiRL differentialLis defined by the property that the inclusion HomR ( SymiR (L[1]), P ) ,→ Homk ( Symik (L[1]), P ) identifies C`R (L, P ) with a subcomplex of the usual Lie algebra cochain complex C(L− , P ). For left L-modules P , P 0 and a right L-module M we have the evident pairings ` CR (L, P ) ⊗C`R (L, P 0 ) → C`R (L, P ⊗ P 0 ), C`R (L, P ) ⊗ CrR (L, M ) → CrR (L, P ⊗ M ) which are morphisms of complexes. In particular, CR (L) = C`R (L, R) is a commutative DG algebra and C`R (L, P ), CrR (L, M ) are CR (L)-modules. The complex C`R (P ) carries a decreasing filtration by subcomplexes F · , where ` CR (P )/F n = HomR (Sym
morphism LR → L is the same as a Lie algebra morphism L → L compatible with ∼ the actions on R. We refer to an isomorphism LR −→ L as an L-rigidification of L. For such a rigidification the restrictions of the obvious “identity” functor M(R, L− ) → M(R, L) (:= the category of R-modules equipped with an L-action compatible with the L-action on R) to the subcategories of left and right L-modules are equivalences of categories: (2.9.1.1)

∼

∼

M` (R, L) −→ M(R, L) ←− Mr (R, L).

The complexes CrR (L, M ), C`R (L, P ) coincide with the usual Chevalley Lie algebra homology, resp. cohomology, complexes of L with coefficients in M , P . (ii) Suppose R is an n-Poisson algebra (see 1.4.18), so ΩR [−n] is a Lie Rpois algebroid. (R) := CR (ΩR [−n]). As a mere graded algebra, Cpois (R) equals L Set iC HomR ( SymR (ΩR [1−n]), R); thus it carries a canonical continuous (n−1)-Poisson bracket which is the usual commutator on Der(R) and the action of Der(R) on R in appropriate degrees. This bracket is compatible with the differential and filtration F · , so Cpois R (R) is naturally a topological filtered (n−1)-Poisson algebra (cf. 1.4.18).

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Left and right L-modules and their tensor products have the ´etale local nature with respect to Spec R, so they make sense on any scheme or algebraic space X; the corresponding categories are denoted by M` (X, R), Mr (X, L). The homological de Rham-Chevalley complexes are naturally sheaves on X. The same is true for cohomological complexes if L is a finitely presented OX -module. Example. If X is a smooth variety and L = ΘX , then left, resp. right, Lmodules are the same as left, resp. right, DX -modules, and the corresponding de Rham-Chevalley complexes are the usual de Rham complexes. 2.9.2. Enveloping algebras. Let L be a Lie algebroid on X. The enveloping algebra of L is a sheaf U (L) = U (OX , L) of associative algebras on X equiped with a morphism of algebras ιO : OX → U (L) and a morphism of Lie algebras ιL : L → U (L) such that ιO is a morphism of Lie algebra L-modules (L acts on U (L) by ιL composed with the adjoint action) and ιL is a morphism of OX -modules (OX acts on U (L) by ιO and left multiplication), and U (L) is universal with respect to this datum. Then left, resp. right, L-modules are the same as left (resp. right) U (L)-modules which are quasi-coherent as OX -modules. U (L) carries a natural ring filtration: U (L)0 := ιO (OX ), U (L)i := (ιO (OX ) + ιL (L))i for i ≥ 1; we refer to it as the standard filtration. The filtration is commutative (i.e., gr U (L) is a commutative algebra), so gr U (L) is naturally a Poisson algebra. Remarks. (i) Considered as an OX -bimodule, U (L) is a quasi-coherent OX×X module supported set-theoretically on the diagonal X ,→ X × X. o`

or

(ii) The evident forgetful functors125 M` (X, L) −→ MO (X) ← Mr (X, L) admit left adjoints sending an OX -module N to, respectively, U (L) N ∈ M` (X, L) and NU (L) ∈ Mr (X, L) (the induced left and right L-modules). One has U (L) N = U (L) ⊗ N , NU (L) := N ⊗ U (L). OX

OX

Notice that Sym L := SymOX L carries a natural Poisson structure characterized by the property that for `, `0 ∈ L ⊂ Sym L and f ∈ OX ⊂ Sym L one has {`, `0 } = [`, `0 ], {`, f } = τ (`)f . The morphisms ιO and ιL yield a surjective morphism of graded Poisson algebras (2.9.2.1)

Sym L  gr· U (L).

One has the following version of the Poincar´e-Birkhoff-Witt theorem: Proposition. If L is OX -flat, then (2.9.2.1) is an isomorphism. Proof. Consider a graded OX -algebra OX [t] with t of degree 1, a graded Lie OX [t]-algebroid L[t], and its subalgebroid Lt := tL[t]. It yields a graded k[t]-algebra (a) Ut := U (OX [t], Lt ); denote by Ut its components. One has Ut /tUt = Sym(L) and Ut /(t − 1)Ut = U (L). So it suffices to show that the OX -flatness of L implies that Ut is k[t]-flat. Our Ut carries the standard filtration, and we will actually prove ∼ ∼ that (Sym L)[t] −→ gr Ut ; i.e., Syma L ⊕ tSyma−1 L ⊕ · · · ⊕ ta OX −→ gr Uta for every a ≥ 0. Denote the latter statement by (∗)a . Consider Ut as a right Lt -module. Let Ct be the corresponding de Rham(a) Chevalley complex; this is a complex of graded k[t]-modules, and we denote by Ct 125 Here

MO (X) denotes the category of quasi-coherent OX -modules.

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its components with respect to the grading. Notice that ∼

OX [t] ⊕ Ut Lt −→ Ut ;

(2.9.2.2) ∼

i.e., OX [t] −→ Ut /Ut Lt = H 0 Ct .126 We will prove (∗)a by induction. (∗)0 is evident. Suppose (∗)i is known for each (a) i < a. We will show in a moment that H −1 Ct = 0. This implies (∗)a . Indeed, 0 (a) (a)0 look at the truncated complex C(a) := Ct /Ct . The above vanishing of H −1 means that the differential in C (2.9.2.3)

0

d

H −1 C(a) − → (Ut Lt )(a) ∼

(a)

is an isomorphism. We have seen that (Ut Lt )(a) −→ Ut /ta OX . The exactness of the Koszul complexes (here the flatness of L is used) together with (∗)i for i < a 0 0 imply that H j gr C(a) = 0 for j 6= −1 and H −1 gr C(a) = L ⊕ · · · ⊕ Syma L. The −1 (a)0 latter group equals H C , which gives (∗)a . (a) It remains to prove that H −1 Ct = 0, i.e., that (2.9.2.3) is an isomorphism. Our Ut is a universal right Lt -module, so to get an inverse to (2.9.2.3), it suffices to define 0 (0) (a−1) on F := Ut ⊕ · · · ⊕ Ut ⊕ (H −1 C(a) ⊕ ta OX ) (the latter summand has degree a) (≥a) a structure of the graded right Lt -module such that on F/F (a) = Ut /Ut it is the (1) (a−1) usual Lt -module structure, and the action of ` ∈ Lt = L on m ∈ F (a−1) = Ut (a)0 −1 −1 (a)0 (a) is the class of m ⊗ ` ∈ C in H C ⊂ F . This formula defines a Lie algebra action of L on F . The multiplication by t operator F (a−1) → F (a) is 0 0 t (a−1) ∼ the composition Ut ← H −1 C(a−1) ⊕ ta−1 OX − → H −1 C(a) ⊕ ta OX where the first arrow comes from isomorphism (2.9.2.3) for a − 1 (known by induction) and (2.9.2.2). The OX -action on F (a) recovers uniquely from the Lie algebroid property. We leave it to the reader to check the compatibilities.  2.9.3. Suppose that L is a locally free OX -module of finite rank. The adjoint action of L on L makes it an L− -module. One checks that the line bundle ωL := det(L∗ )[rkL] is a right L-module. There is a canonical equivalence of categories (2.9.3.1)

∼

M` (X, L) −→ Mr (X, L),

M 7→ M r := M ⊗ ωL .

We denote these categories thus identified by M(X, L). This is a tensor category since M` (X, L) is. Below we denote by ⊗! the corresponding tensor product on Mr (X, L); i.e., −1 ! ⊗ Ni = (⊗(Ni ⊗ ωL )) ⊗ ωL . Remark. Suppose that L is L-rigidified (see Example (i) in 2.9.1). Then the composition of equivalences (2.9.1.1) differs from (2.9.3.1) by the twist by det L. 2.9.4 The compound tensor structure. Let us show that for L as in 2.9.3, M(X, L) is naturally an augmented compound tensor category (we already know that it is a tensor category). The construction is a generalization of the one from 2.2 (the case of L = ΘX ). We present it first in a formal way as in 2.2.12, explaining a more geometric picture similar to 2.2.1–2.2.7 later. 126 Our map is evidently surjective, and it admits a left inverse which sends the class in H 0 C t of u ∈ Ut to u · 1 ∈ OX [t] where · is the left Ut -action on OX [t].

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The structure has a local origin, so we can assume that X is affine. The whole compound tensor structure comes then from a compound augmented U (L)operad B∗! ; see 1.3.18 and 1.3.19 (cf. 1.3.20). We define B∗I as the tensor product of I copies of U (L) considered as a left L-module. This is an (U (L) − U (L)⊗I )bimodule: namely, the right U (L)⊗I -action comes by transport of structure from the right U (L)-action on U (L) and the left U (L)-module structure comes since B∗I is a left L-module. Similarly, B! is the ⊗! tensor product of I copies of U (L) considered as a right U (L)-module. This is a (U (L)⊗I − U (L))-bimodule. The operad structure on B∗ : for J  I the map B∗I ⊗ (⊗ B∗Ji ) → B∗J is the I

projection ( ⊗ U (L)i ) ⊗ (⊗( ⊗ U (L)ji )) → ⊗ (U (L)i ⊗ ( ⊗ U (L)ji )) = B∗J . The OX

I OX

OX

operad structure on B! is defined in a similar way. So for Mi , N ∈ Mr (X, L) one has ⊗! Mi = (⊗Mi )

U (L) OX

⊗ U (L)⊗I

B! and PI∗ ({Mi }, N ) :=

HomU (L)⊗I (⊗Mi , N ⊗U (L) B∗I ). The construction of structure morphisms < >IS,T : (⊗ B∗Is ) S

⊗

(⊗ B!It ) →

U (L)⊗I T

B!S ⊗ B∗T repeats the one we considered in 2.2.12 for the particular case of DU (L)

modules (i.e., L = ΘX ). The axioms of the augmented compound operad are immediate.

Here is a geometric explanation of the picture. Let G be the formal groupoid on X that corresponds to L (see 1.4.15). For I ∈ S one has a formal groupoid GI on X I ; it contains subgroupoids Gi := G × X Ir{i} , i ∈ I. Denote the corresponding Q Ir{i} Lie algebroids by L(I) , Li . So Li := OX  L and L(I) = Li . i∈I

Let T (I) be the restriction of GI , considered as a mere formal X I ×X I -scheme, to I X × X ,→ X I × X I , ((xi ), x) 7→ ((xi ), (x, . . . , x)). This is a formal X I × X-scheme equipped with an action of the groupoid GI × G. The action of the subgroupoid X I × G is free; denote by T (I) the corresponding quotient. This is a formal X I scheme equipped with an L(I) -action; let p(I) : T (I) → X I be the projection and (I) ∆L : X → T (I) the lifting of the diagonal embedding that comes from the “unit” section of the groupoid. We consider OT (I) as a topological OX I -algebra; then (I) OX := ∆· OX = OT (I) /I, and the topology of OT (I) is the I-adic one. Notice also that the L(I) -action identifies I/I2 with the OX -module LI /L (the quotient modulo ∼ the image of the diagonal embedding) and SymOX (I/I2 ) −→ Σ Ia /Ia+1 . Let Mr (T (I) , L(I) ) be the category of right L(I) -modules equipped with a compatible discrete OT (I) -action. For any of its objects the OX -submodule of sections (I)! killed by I is naturally a right L-module; let ∆L : Mr (T (I) , L(I) ) → Mr (X, L) be the corresponding functor. It follows easily from Kashiwara’s lemma that it is ∼ (I) an equivalence of categories; let ∆L∗ : Mr (X, L) −→ Mr (T (I) , L(I) ) be the inverse equivalence. We leave it to the reader to check that for any N ∈ Mr (X, L) the L(I) -module (I) (I) ∆L∗ N equals N ⊗U (L) B∗I . Thus PI∗ ({Mi }, N ) = HomMr (X I ,L(I) ) (Mi , ∆L∗ N ). The constructions of 2.2.3, 2.2.6, 2.2.7 generalize directly to our setting with the (I) (I) ∆∗ functor replaced by ∆L∗ , and we arrive at the same augmented pseudo-tensor structure as above.

2.9. THE HARISH-CHANDRA SETTING AND THE SETTING OF C-STACKS

149

2.9.5. Smooth localization of Lie algebroids. Let π : Y → X is a smooth morphism of schemes, L any Lie algebroid on X. One defines its pull-back π † (L), which is a Lie algebroid on Y , as follows. Let Θπ ⊂ ΘY be the subsheaf of vector fields preserving π −1 OX ⊂ OY . It is a Lie π −1 OX -algebroid on Y (in the obvious sense) which is an extension of π −1 ΘX by ΘY .127 Set π ] L := π −1 L × Θπ ; this π −1 ΘX

is a Lie π −1 OX -algebroid which is an extension of π −1 L by ΘY /X . By construction, it acts on OY , so OY ⊗ π ] L is a Lie OY -algebroid. Finally, our π † (L) is the pushout of OY

π −1 OX

⊗

π −1 OX

π ] L by the product map OY

algebroid quotient of OY

⊗

π −1 OX

⊗

π −1 OX

ΘY /X → ΘY /X . It is a Lie OY -

π ] L. As an OY -module, π † L is an extension of π ∗ L

by ΘY /X ; thus π † L is OY -quasi-coherent if L is OX -quasi-coherent. The projection OY ⊗ π ] L → π † L identifies π ] L with the normalizer of π −1 OX ⊂ OY . π −1 OX

If M is a left L-module, then its O-module pull-back π ∗ M is naturally a left π L-module. Namely, a section f (`, θ) of π † L (here f ∈ OY , (`, θ) ∈ π ] L) acts on π ∗ M as f (`, θ)(gm) = f θ(g)m + f g`(m). Similarly, if N is a right L-module, then π ◦ N := π ∗ M ⊗ ωY /X is a right π † L-module; here ωY /X is the line bundle det ΩY /X on Y . Namely, the action is (nν)(f (`, π)) = (n`)f ν − nLieθ (f ν); here ν ∈ ωX/Y , and Lieθ is the natural (“Lie derivative”) action of θ ∈ Θπ on ωY /X . One easily checks that the functor π † is compatible with the composition of the π’s and satisfies the descent property. The functors π ∗ and π ◦ for the left and right modules also satisfy the smooth descent property. Therefore Lie algebroids and left/right modules over them are objects of a local nature with respect to smooth topology; hence they make sense on any algebraic stack. †

Remarks. (i) The functor π ∗ commutes with tensor products, so the category of left L-modules is a tensor category in the stack setting. (ii) For a Lie algebroid the property of being a locally free O-module of finite rank is local with respect to the smooth topology. Pull-backs π ∗ and π ◦ are naturally identified by (2.9.3.1), so the picture of 2.9.3 makes sense in the stack setting. (iii) For π, L as above there is an evident isomorphism of Lie OY -algebras ∼ ∗ π (L♦ ) −→ π † (L)♦ (we assume that L♦ is OX -quasi-coherent; see Remark (ii) in 2.9.1). Thus a Lie algebroid L on an algebraic stack Y defines a Lie OY -algebra L♦ . (iv) Let π be as above and let ψ : L → L0 be a morphism of Lie algebroids on X, so we have the morphism ψ † : π † (L) → π † (L0 ) of Lie algebroids on Y . There are evident isomorphisms of OY -modules Ker π † (ψ) = π ∗ (Ker ψ) and Coker π † (ψ) = ψ ∗ (Coker ψ). Thus a morphism ψ of Lie algebroids on an algebraic stack Y yields the O-modules Ker ψ, Coker ψ on Y. 2.9.6. Let Y be an algebraic stack. Let Z → Y be any smooth surjective morphism where Z is a scheme, so Y is the quotient stack of Z modulo the action of a smooth groupoid K = Z × Z. Denote by ΘZ/Y the corresponding Lie algebroid on Y

∼

Z. For any smooth π : Z 0 → Z there is a natural identification ΘZ 0 /Y −→ π † ΘZ/Y , so the ΘZ/Y form a Lie algebroid Θ/Y on Y. Lemma. Θ/Y is the initial object of the category of Lie algebroids on Y. 127 We

use the fact that π is smooth.

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Proof. Let L be a Lie algebroid on Y. We want to show that there is a unique morphism of Lie algebroids φ : Θ/Y → L.

(2.9.6.1)

Uniqueness: For Z as above let p1 , p2 : K → Z be the structure projections, (1) e : Z → K the “unit” section of K over the diagonal Z ,→ Z × Z. Denote by ΘK/Z , (2)

ΘK/Z the tangent bundle along the fibers of p1 , p2 . One has canonical embeddings ΘK/Z ,→ p†1 ΘZ/Y = ΘK/Y , ΘK/Z ,→ p†2 ΘZ/Y = (1)

(2)

∼

ΘK/Y , and one checks easily that ΘK/Z ⊕ ΘK/Z −→ ΘK/Y . Since φK = p†1 φZ , the (1)

(2)

restriction of φK to ΘK/Z coincides with the canonical embedding ΘK/Z ,→ p†1 LZ . (1)

(1)

(2)

The same is true for the restriction of φK to ΘK/Z . Thus φK is uniquely defined, hence φ is. (1) Existence: Consider a canonical embedding ΘK/Z ,→ p†2 LZ = LK and a projec-

tion LK = p†1 LZ  p∗1 LZ . Pulling the composition back by e, we get a morphism (1) of OZ -modules φZ : ΘZ/Y := e∗ ΘZ/Y → LZ . We leave it to the reader to check that φZ comprize a morphism φ : Θ/Y → L of Lie algebroids on Y.  Exercise. Suppose that Y is covered by a point ·, so Y = BK where K is an algebraic group. Set k := LieK. Consider the category HCK of triples (g, φ, α) where g is a Lie algebra, φ : k → g is a morphism of Lie algebras, α is a K-action on g (as on a Lie algebra) such that the k-action coming from α equals adφ . One has a functor (2.9.6.2)

LieAlgBK → HCK ,

L 7→ (l, φ, α),

where l is the Lie algebra L· , φ : k → l is the morphism (2.9.6.1) over ·, and α arises since we have a Lie O-algebra L♦ on BK which equals l over the covering · (see Remark (iii) in 2.9.5). Show that (2.9.6.2) is an equivalence of categories. 2.9.7. The setting of Harish-Chandra modules. The compound pseudotensor structure from 2.9.4 is a particular case of a canonical compound structure on the category of O-modules on a c-stack. We will sketch the general picture in 2.9.10–2.9.12 below. Prior to this, let us consider another particular case of the c-stack setting which is especially relevant for applications. There are two equivalent ways to describe the notion of a Harish-Chandra pair: (i) We consider a reasonable ind-affine group formal scheme G. Here the formal scheme is the same as an ind-scheme such that Gred is a scheme; “reasonable” (see 2.4.8) amounts to the property that for any subscheme Gred ⊂ P ⊂ G the ideal of Gred in OP is finitely generated. (ii) We consider a collection (g, K) = (g, K, φ, α) where K is an affine group scheme, g a Lie algebra which is a Tate vector space, φ : k := Lie K ,→ g a continuous embedding of Lie algebras with an open image,128 and α a K-action on g (as on a Lie algebra). We demand that the k-action on g coming from α is equal to adφ . Notice that for any G as in (i) the pair (g, Gred ) with obvious φ and α fits (ii). One can show that this functor from the category of G’s as in (i) to that of (g, K)’s as in (ii) is an equivalence of categories. 128 Notice

that k is naturally a profinite-dimensional Lie algebra.

2.9. THE HARISH-CHANDRA SETTING AND THE SETTING OF C-STACKS

151

For G as in (i) a G-module is a vector space V equipped with a G-action.129 In the setting of (ii), a (g, K)-module is a (discrete) vector space equipped with a K-action and a g-action which are compatible in the obvious manner. We denote the corresponding categories by M(G) and M(g, K); these are tensor categories in the obvious way. One checks that G-modules are the same as (g, Gred )-modules. Proposition. If dim g/k < ∞, then M(g, K) is naturally an abelian augmented compound tensor category. Sketch of a proof. We already have the tensor product. Let us construct the ∗ operations. Set n := dim g/k. The G-module picture is convenient here. Write G = Spf F where F is a ˆ G topological algebra. For I ∈ S set F (I) := (F ⊗I ) , so Spf F (I) = GI /G, where we consider the diagonal right translation action of G on GI . Notice that GI /G is a formally smooth formal scheme whose reduced scheme K I /K has codimension (|I|−1)n (|I| − 1)n. Set E (I) := lim (F (I) /J, F (I) ) where J runs the set of all open −→ ExtF (I) (I) (I) ideals in F . This is a discrete F -module equivariant with respect to the left translation action of GI on GI /G. ˆ ∗ N := (OGI ⊗N ˆ )G be the topological vector space For a G-module V let ∆ I of N -valued functions on G equivariant with respect to the diagonal right Gtranslations on GI . This is a topological F (I) -module equivariant with respect (I) (I) ˆ ˆ ˆ ∗N = ˆ ∗ N = lim ∆ ⊗ ∆ to the GI -action, and ∆ ←− ∗ N/J∆∗ N . Set ∆∗ N := E F (I) (|I|−1)n (I) (I) I (I) ˆ lim (F /J, ∆∗ N ); this is a discrete G -equivariant F -module. The −→ ExtF (I) (I) functor ∆∗ : M(G) → MδGI (GI /G) := the category of discrete GI -equivariant (I)

F -modules is actually an equivalence of categories; the inverse functor P 7→ Tor(|I|−1)n (ke , P ) where ke is the skyscraper sheaf at the distinguished point e ∈ GI /G fixed by the action of G ⊂ GI . Now, for a collection {Mi }, i ∈ I, of G-modules we set (2.9.7.1)

(I)

PI∗ ({Mi }, N ) := HomGI (⊗Mi , ∆∗ N ) ⊗ λ⊗n I

where λI is as in 2.2.2. Equivalently, a ∗ operation is a morphism of GI -equivariant (I) F (I) -modules (⊗Mi )⊗F (I) → ∆∗ N ⊗λ⊗n I . The composition of ∗ operations comes, (J) (J/I) (I) as in the case of D-modules, from natural identifications ∆∗ N = ∆∗ ∆∗ N (J/I) (here J  I is a surjection) where ∆∗ : M(GI ) → M(GJ ) is a functor defined in the same manner as above (using the space GJ /GI ). The compound tensor product maps ⊗IS,T (see 1.3.12) are defined as in the D-module case using restrictions to the diagonal GT /G ⊂ GI /G. The augmentation functor is h(N ) := Extn (k, N ) where k is the trivial Gmodule, and Ext is computed in M(G). The definition of the compatibility morphisms is left to the reader.  2.9.8. (g, K)-structures. One uses the above picture as follows (this is an example of c-functoriality; see 2.9.12). Let X be a smooth variety, (g, K) a Harish-Chandra pair, G the corresponding group formal scheme. A (g, K)-structure on X is a scheme Y equipped with a 129 Which is the linear action of the group valued functor R 7→ G(R) on the functor R 7→ R⊗V and R is a test commutative algebra.

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projection π : Y → X and a (g, K)-action (which is the same as a G-action) such that (i) K acts along the fibers of π and makes Y a K-torsor over X. (ii) The action of g is formally simply transitive. Notice that (ii) can be replaced by (ii)0 The morphism (g/k) ⊗ OY → π ∗ ΘX defined by the g-action and dπ, is an isomorphism. Notice that for any left DX -module its pull-back to Y is a G-equivariant OY module, and this functor is an equivalence of categories between M` (X) and the category of G-equivariant (quasi-coherent) O-modules on Y . Any G-module V yields a G-equivariant OY -module V ⊗ OY , so we have defined an exact faithful tensor functor M(g, K) → M` (X), V 7→ V (Y ) . Here is a more explicit definition of V (I) . Our Y defines a G-torsor Yˆ on X ˆ ⊂ X × X be the formal completion equipped with a flat connection. Namely, let X ˜ ˆ → X at the diagonal. Then Y is the pull-back of Y by the first projection X ˆ considered as an X-scheme via the second projection X → X; the G-action on Yˆ comes from the G-action on Y and the connection is the derivation along the ˆ Now V (Y ) is simply the Yˆ -twist of V . One has Y ⊂ Yˆ ; i.e., second variable in X. as a mere G-torsor (we forget about the connection) our Yˆ is the G-torsor induced from the K-torsor Y . Therefore as a mere OX -module V (Y ) is the twist of V by the K-torsor Y . Now the above functor extends to an augmented compound tensor functor (2.9.8.1)

M(g, K)∗! → M(X)∗! .

Our functor is evidently a tensor functor. Let us explain the how it transforms the ∗ operations; the compatibility with compound tensor products is defined in a similar ˆ (I) be the formal way. For I ∈ S we have the (g, K)I -structure Y I on X I . Let X I (I) I completion of X at the diagonal and Yˆ the pull-back of Y to XˆI . One has the (I) I diagonal embedding Y ,→ Yˆ , and the G -action on Y (I) yields an isomorphism ∼ (GI ×Y )/G −→ Y (I) . Thus we have an affine projection p : Y (I) → GI /G. Therefore (I) for a G-module V we have a GI -equivariant “discrete” O-module p∗ ∆∗ V on the (I) I ˆ ˆ (I) formal scheme Y . Its K -invariant sections form a discrete O-module on X I which is naturally a left D-module, i.e., a left D-module Q on X supported at (I) the diagonal. One checks immediately that Q = (∆∗ (V (Y ) )r )` . Now it is clear ∗ that a ∗ operation ∈ PI ({Mi }, V ) in M(G) yields a morphism of left DX I -modules (Y ) (Y )r Mi → Q ⊗ λ⊗n , i.e., a ∗ operation ∈ PI∗ ({Mi }, V (Y )r ), and we are done. Remark. Instead of a single X equipped with a (g, K)-structure, we can consider smooth families X/S of varieties equipped with a fiberwise (g, K)-structure, so (2.9.10.2) becomes a functor with values in the corresponding fiberwise D-modules. Suppose we have a morphism of smooth families f : X 0 /S 0 → X/S which is fiberwise ´etale. Then for any (g, K)-structure Y on X/S its pull-back Y 0 := Y × X 0 is naturally a (g, K)-structure on X 0 , and for V ∈ M(g, K) one has V (Y

0

X )

= f ∗ V (Y ) .

Exercise. Let G˜be a group scheme which contains K and its formal completion at K is identified with G (so g is the Lie algebra of G˜). Then X := K r G˜ is equipped with an evident (g, K)-structure given by the projection Y = G˜→ X.

2.9. THE HARISH-CHANDRA SETTING AND THE SETTING OF C-STACKS

153

Show that for each (g, K)-module V the corresponding D-module V (Y ) on X is naturally weakly G˜-equivariant130 with respect to the right G˜-action on X, and the functor V 7→ V (Y ) establishes an equivalence between M(g, K) and the augmented compound pseudo-tensor category of weakly G˜-equivariant D-modules on X. For example, for G˜ = Gna , K = 1, we get an equivalence between the augmented compound pseudo-tensor category of k[∂1 , . . . , ∂n ]-modules and that of the D-modules on An weakly equivariant with respect to translations. 2.9.9. The Gelfand-Kazhdan structure. Set On := k[[t1 , . . . , tn ]]. One has a group formal scheme G :=Aut On : for a commutative algebra R an R-point ˆ n = R[[t1 , . . . , tn ]] over R. Then of G is a continuous automorphism g of R⊗O g := LieG = Derk On , and an R-point of G lies in K := Gred if and only if g preserves the ideal generated by t1 , . . . , tn . According to [GeK], each smooth variety X of dimension n admits a canonical (g, K)-structure YGK called the Gelfand-Kazhdan structure. Namely, YGK is the ˆ nspace of all formal coordinate systems on X; i.e., an R-point of YGK is an R⊗O ∼ ∗ point ξ of X such that dξ : ΘR⊗O → ξ ΘX . The group G acts on YGK by ˆ n /R − transport of structure; the axioms of the (g, K)-structure are immediate. As in Remark in 2.9.8, we can consider families of smooth n-dimensional varieties, and the Gelfand-Kazhdan structure is functorial with respect to fiberwise ´etale morphisms of families. A universal left D-module (in dimension n) is a rule M which assigns to every smooth family X/S of fiberwise dimension n an OX -quasi-coherent left DX/S module MX/S on X and to every fiberwise ´etale morphisms of families f : X 0 /S 0 → ∼ X/S an identification MX 0 /S 0 −→ f ∗ MX/S ; the obvious compatibility with composition of the f ’s should hold. Universal D-modules form an abelian augmented compound pseudo-tensor category M∗! n . By 2.9.8, the Gelfand-Kazhdan structure yields an augmented compound pseudo-tensor functor (2.9.9.1)

M(Aut On )∗! = M(g, K)∗! → M∗! n.

Proposition. This is an equivalence of augmented compound pseudo-tensor categories. Proof. Left to the reader.



Exercise. Define the category of universal O-modules in dimension n and show that it is canonically equivalent, as a tensor category, to the category of K-modules where K is as above. One can use (g, K)-modules to deal with “non-universal” D-modules as well. Namely, let X be an affine smooth variety of dimension n. So YGK is also affine. Then A := Γ(YGK , OYGK ) carries a (g, K)-action; i.e., it is a commutative! algebra in (g, K)mod. As in 1.4.6, we have the augmented compound pseudo-tensor category M(A)∗! of A-modules equipped with a compatible (g, K)-action. For a left Dmodule M on X, set ΓGK (M ) := Γ(YGK , π ∗ M ). This is an object of M(A) in the 130 Recall (see, e.g., [BB] or [Kas1] where the term “quasi-equivariant” is used) that a weakly G˜-equivariant D-module on X is a D-module M together with a lifting of the G˜-action on X to M considered as an OX -module so that for every g ∈ G the action of the corresponding translation on M is compatible with the action of DX .

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evident way; since YGK is affine, we get an equivalence of categories ∼

ΓGK : M` (X) −→ M(A).

(2.9.9.2)

Exercises. (i) The composition of (2.9.9.2) with the forgetful functor M(A) → M(g, K) is right adjoint to the functor M(g, K) → M(X), V 7→ V (YGK ) . (ii) Show that (2.9.9.2) extends naturally to an augmented compound pseudotensor equivalence. 2.9.10. C-stacks. The rest of the section is a brief sketch of a general setting which includes the settings of 2.9.4 and 2.9.7. We need a dictionary. (a) We play with sheaves and (1-)stacks on the category of affine k-schemes equipped with the fpqc topology, calling them simply spaces and stacks. Schemes, formal schemes, etc., are identified with the corresponding spaces, and spaces with stacks having trivial “inner symmetries”. A morphism of stacks f : P → Q is said to be space morphism if for every morphism S → Q where S is a space, the fibered product P × S is a space. It Q

suffices to check this for affine schemes S. If P is a space, then f is a space morphism. (b) Suppose one has a class of morphisms of spaces whose target is a scheme which is stable by the base change. We say that a space morphism f : P → Q belongs to our class if for every morphism S → Q where S is a scheme, the morphism P × S → S is there. Q

E.g., we have schematic morphisms coming from the class of morphisms whose source is a scheme, flat schematic morphisms, quasi-compact schematic morphisms, fpqc coverings := surjective flat quasi-compact schematic morphisms, etc. Suppose, in addition, that our class is stable under composition with flat quasicompact schematic morphisms from the right.131 We say that f as above belongs to our class fpqc locally if for every morphism S → Q where S is a scheme, there exists an fpqc covering K → P × S such that the composition K → P × S → S is Q

Q

in the class. (c) A formal scheme F is said to be smooth in formal directions of rank n if there exists an open ideal J ⊂ OF such that the topology of OF is J-adic, J/J2 is a ∼ locally free OF /J-module of finite rank n, and SymOF /J (J/J2 ) −→ ΣJa /Ja+1 . One checks that n does not depend on the choice of J. Consider the class of morphisms F → Z where Z is a scheme, F a formal scheme smooth in formal directions of rank n, and J as above can be chosen so that Spec OF /J is flat and quasi-compact over Z. It satisfies both conditions of (b). We say that a space morphism f of stacks is a c-morphism of c-rank n if it belongs fpqc locally to our class. If, in addition, f is surjective, then it is called a c-covering of c-rank n. Definition. A c-stack of c-rank n is a stack which admits a c-covering of crank n by a scheme. A morphism P → Q of c-stacks is called c-morphism if for any c-covering S → Q where S is a scheme, there exists a surjective flat quasi-compact schematic morphism T → P × S where T is a scheme. Q

g

f

131 I.e., for Y − → Z −→ T such that g is a flat quasi-compact schematic morphism and f in the class, the composition f g is in the class.

2.9. THE HARISH-CHANDRA SETTING AND THE SETTING OF C-STACKS

155

Thus c-stacks are the same as quotient stacks S/G where S is a scheme and G a groupoid on S such that either of the projections G → S is a c-covering. Remarks. (i) In the definition of c-morphism it suffices to consider a single c-covering S → Q. (ii) c-morphisms exist only between c-stacks of the same c-rank. Examples. (o) Algebraic stacks are c-stacks of c-rank 0. (i) In the situation of 2.9.3 the quotient Y /GL of Y modulo the action of the formal groupoid GL defined by L is a c-stack of c-rank equal to the rank of L as an OY -module. (ii) For G as in 2.9.7 with dim g/k < ∞ the classifying stack BG is a c-stack of c-rank equal to dim g/k. Here is a common generalization of Examples (i) and (ii) (for dim G < ∞): 2.9.11. Let Y be an algebraic stack and L a Lie algebroid on Y. Suppose that Ker φ = 0 and Coker φ is a locally free OY -module of rank n; here φ is the canonical morphism (2.9.6.1). Proposition. Such a (Y, L) yields a c-stack Q = Y/L of c-rank n equipped with a morphism Y → Q. Proof. Let Z → Y a smooth covering where Z is a scheme, so Y is the quotient of Z modulo the action of the smooth groupoid K := Z × Z. We have the Lie Y

algebroid LZ on Z; let Gˆbe the corresponding formal groupoid. By our condition, Gˆcontains the formal completion132 Kˆof K (which is the formal groupoid of ΘZ/Y ). We will embed K into a larger formal groupoid G on Y whose formal completion equals Gˆ, and we will define Q as the quotient stack Z/G. Let G` be the quotient of Gˆ× K modulo the relation (gr, k) = (g, rk) where Z

g ∈ Gˆ, r ∈ Kˆ, k ∈ K, and the two products are well defined. Our condition assures that G` is a formal scheme over Z × Z. Denote by Gr the quotient of K × Gˆmodulo Z

a similar relation. We write the points of G` , Gr as, respectively, g · k, k 0 · g 0 . ∼ There is a canonical isomorphism ν : G` −→ Gr of formal Z × Z-schemes defined as follows. Consider the Lie algebroid LK on K; let H be its formal groupoid. Since LK = p†1 LZ = p†2 LZ , H coincides with the formal completion of the pull-back of Gˆ either by p1 or by p2 . Therefore a point of H over a pair of infinitely close points (k, k 0 ) ∈ K×K can be written as either a point g ` ∈ Gˆover (p1 (k), p1 (k 0 )) or g r ∈ Gˆ over (p1 (k), p1 (k 0 )). Now ν identifies g ` · k with k 0 · g r . One easily checks that ν is well defined. Our G equals G` or Gr identified by ν. The groupoid structure on G is defined using ν. The details (such as the independence of the construction of the choice of Z) are left to the reader.  2.9.12. Recall that for a stack Q an O-module on Q is a rule N that assigns to any f ∈ Q(S), S a scheme, a quasi-coherent OS -module f ∗ N , to any arrow f → f 0 ∼ ∗ in Q(S) an isomorphism f ∗ N −→ f 0 N , and to any g : S 0 → S an isomorphism ∼ (f g)∗ N −→ g ∗ f ∗ N , so that the evident compatibilities are satisfied. The category of O-modules on Q is denoted by MO (Q). This is a tensor category. 132 I.e.,

the formal completion at the “identity”, which is a formal groupoid.

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For example, for Q = Y/L as in 2.9.11, one has MO (Q) = M` (Y, L) (see 2.9.5). In the situation of Example (ii) of 2.9.10, this is the category of (g, K)-modules. One has the following generalization of 2.9.4: Proposition. If Q is a c-stack, then M(Q) is naturally an abelian augmented compound pseudo-tensor category. The pull-back functor for a c-morphism of cstacks is naturally an augmented pseudo-tensor functor.  The particular cases corresponding to Examples (i) and (ii) from 2.9.10 were considered in 2.9.4 and 2.9.7. The construction in the general situation is essentially a combination of those from 2.9.4 and 2.9.7, and we omit it. Remark. Functoriality (2.9.8.1) is a particular case of functoriality with respect to c-morphisms of c-stacks. Namely, for our smooth variety X consider the c-stack X − which is the quotient of X modulo the formal groupoid G generated by ΘX ; i.e., G = the formal completion of X × X at the diagonal. The category MO (X − ) is the category of left D-modules M` (X). Now for a (g, K)-structure Y the projection G r Y → X − defined by π is an equivalence of c-stacks, and (2.9.8.1) is ∼ the composition of the pull-back functors for the c-morphisms X − ← G r Y → BG .

CHAPTER 3

Local Theory: Chiral Basics Algebraisty obyqno opredelt gruppy kak mnoestva s operacimi, udovletvorwimi dlinnomu rdu trudnozapominaemyh aksiom. Pont~ takoe opredelenie, na mo$i vzgld, nevozmono. Qlen-korrespondent AN SSSR V. I. Arnol~d, “Matematika s qeloveqeskim licom”, Priroda, }3, 1988, 117–118. †

3.1. Chiral operations From now on we assume that X is a smooth curve. We define chiral operations between D-modules on X in 3.1.1. The action of the tensor category M(X)! on M(X)ch is defined in 3.1.3. The operad P ch (ω) of chiral operations acting on ωX identifies canonically with the Lie operad (see 3.1.5). This fact is (the de Rham version of) a theorem of F. Cohen [C]; an alternative proof is given in 3.1.6–3.1.9. In 3.1.16 we mention that chiral operations can also be defined in the setting of (g, K)-modules with dim g/k = 1 (and, in fact, for O-modules on any c-stack of c-rank 1; see 2.9), and (g, K)-structures provide pseudo-tensor functors between the (g, K)-modules and D-modules. 3.1.1. For a finite non-empty set I put U (I) := {(xi ) ∈ X I : xi1 6= xi2 for every i1 6= i2 }. So U (I) is the complement to the diagonal divisor if |I| ≥ 2. Denote by j (I) : U (I) ,→ X I the open embedding. We write U (n) for U ({1,... ,n}) . Let Li , i ∈ I, and M be (right) D-modules on X. Set (3.1.1.1)

(I)

(I)

PIch ({Li }, M ) := HomM(X I ) (j∗ j (I)∗ ( Li ), ∆∗ M ). I

The elements of this vector space are called chiral I-operations. They have the local nature with respect to the ´etale topology of X. As usual, we write Pnch for ch ch P{1,... ,n} ; the elements of P2 are called chiral pairings. Remark. Let Fi ,G be quasi-coherent OX -modules. By 2.1.8 (see, in par(I) ticular, Example (iii) there) we have PIch ({FiD }, GD ) = Diff(j∗ j (I)∗ (Fi ), ∆· G); elements of this vector space are called chiral polydifferential operators. † “Algebraists define a group as a set equipped with operations, subject to a long series of axioms which are difficult to remember. Understanding such a definition is, in my opinion, impossible.” V. I. Arnold, Corresponding Member of the Academy of Sciences of the USSR, “Mathematics with a Human Face”, Priroda, No. 3, 1988, 117–118.

157

158

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LOCAL THEORY: CHIRAL BASICS

3.1.2. Our PIch are left exact k-polylinear functors on M(X). Let us define the composition of chiral operations. Let π: J  I be a surjective map of finite non-empty sets, {Kj } a J-family of DX -modules. The composition map (3.1.2.1)

PIch ({Li }, M ) ⊗ (⊗I PJch ({Kj }, Li )) −→ PJch ({Kj }, M ) i

sends ϕ ⊗ (⊗ψi ) to ϕ(ψi ) defined as the composition (J)

(π)

(Ji ) (Ji )∗

j∗ j (J)∗ (J Kj ) = j∗ j (π)∗ (I (j∗

j

ψi

∆(π) ∗ (ϕ)

(π) (I)

(π)

(Ji )

Ji Kj )) −−→ j∗ j (π)∗ (I ∆∗ (π)

(I)

Li )

(J)

= ∆∗ j∗ j (I)∗ (I Li ) −−−−−→ ∆∗ ∆∗ M = ∆∗ M. Here j (π) : U (π) := {(xj ) ∈ X J : xj1 6= xj2 if π(j1 ) 6= π(j2 )} ,→ X J ; for the rest of the notation see 2.2.3. The composition is associative, so the PIch define on M(X) an abelian pseudotensor structure on M(X). We call it the chiral structure and denote it by M(X)ch . The pseudo-tensor categories M(U )ch , U ∈ X´et , form a sheaf of pseudo-tensor categories M(X´et )ch on the ´etale topology of X. The de Rham functor h (see 2.1.6) is an augmentation functor on M(X´et )ch (see 1.2.5).1 The structure map (3.1.2.2)

ch hI,i0 : PIch ({Li }, M ) ⊗ h(Li0 ) → PIr{i ({Li }, M ) 0}

sends ϕ ⊗ `¯i0 , where ϕ ∈ PIch ({Li }, M ), `i0 ∈ Li0 , to the chiral I r {i0 }-operation (I) (Ir{i0 }) a 7→ T ri0 ϕ(a  `i0 ) (cf. 2.2.7). Here, as in 2.2.7, T ri0 : prI,i0 · ∆∗ M → ∆∗ M is the trace map for the projection prI,i0 : X I → X Ir{i0 } . Compatibilities (i) and (ii) in 1.2.5 are immediate. 3.1.3. The action of the tensor category M(X)! on M(X) extends naturally to an action of M(X)! on M(X)ch (see 1.1.6(v)). The corresponding morphisms PIch ({Mi }, N ) → PIch ({Mi ⊗Ai }, N ⊗(⊗Ai )) send a chiral operation ϕ to ϕ⊗idAi ; (I) (I) we use the identification (∆∗ N ) ⊗ (Ai ) = ∆∗ (N ⊗ (⊗Ai )) of (2.1.3.2). 3.1.4. Consider the operad P ch (ωX ) (see 1.1.6) of chiral operations on ωX , so (I) (I) I , ∆∗ ωX ). Set λI := (k[1])⊗I [−|I|] = (det(k I ))[|I|]; := Hom(j∗ j (I)∗ ωX this is a line of degree 0 on which Aut I acts by the sgn character. One has a canonical, hence Aut I-equivariant, isomorphism

PIch (ωX )

∼

I (3.1.4.1) εI : ωX ⊗ λI −→ ωX I ,

(νi ) ⊗ (ei1 ∧ · · · ∧ ein ) 7→ pri∗1 νi1 ∧ · · · ∧ pri∗n νin ;

here νi ∈ ωX , {i1 , . . . , in } is an ordering of I, and {ei } is the standard base of k I . ∼ Equivalently, εI [|I|] : (ωX [1])I −→ ωX I [|I|] is the compatibility of Grothendieck’s dualizing complexes with products (see 2.2.2). (I) (I) If |I| = 2, then the residue morphism Res: j∗ j (I)∗ ωX I → ∆∗ ωX yields a canonical map rI : λI → PIch (ωX ). 3.1.5. Theorem. There is a unique isomorphism of operads ∼

κ : Lie −→ P ch (ωX ) which coincides with rI for |I| = 2. 1 Contrary

to the ∗ situation, in the chiral setting h is highly degenerate.

3.1. CHIRAL OPERATIONS

159

Proof. Let C(ω)X I be the Cousin complex for ωX I [|I|] = (ωX [1])I with respect (I/T ) (T ) (T )∗ to the diagonal stratification. It equals ⊕ ∆∗ j∗ j (ωX [1])T as a mere T ∈Q(I)

graded module; here Q(I) is the set of quotients I  T (see 1.3.1), and ∆(I/T ) is the diagonal embedding X T ,→ X I . The non-zero components of the differential are 0 (I/T ) (T ) ∆∗ (ResT 0 ) for X T ⊂ X T , |T 0 | = |T | − 1, where ResT 0 : j∗ j (T )∗ (ωX [1])T → 0 0 0 (T /T ) (T ) (T 0 )∗ ∆∗ j∗ j (ωX [1])T is the residue morphism. The complex CI is a resoluI tion of (ωX [1]) . The existence of κ is easy: the map rI : λI → PIch (ωX ), |I| = 2, is Aut Iequivariant, so it defines a skew-symmetric operation [ ] ∈ P2ch (ωX ), and we have only to check that the operation α := [[1, 2], 3] + [[2, 3]1] + [[3, 1], 2] ∈ P3ch (ωX ) is zero. This is true since α equals the square of the differential in C(ω)X 3 . The uniqueness of κ is clear. We will show that κ is bijective in 3.1.8–3.1.9 after a necessary digression of 3.1.6–3.1.7. 3.1.6. Let M be a DX I -module. We say that a finite increasing filtration W· on M is special if for every l the successive quotient grW l M is a finite sum of copies (I/T ) of ∆∗ ωX T for T ∈ Q(I, −l). Such a filtration is unique if it exists. We call M a special DX I -module if it admits a special filtration. Let M(X I )sp ⊂ M(X I ) be the full subcategory of special D-modules. It is closed under finite direct sums and subquotients, so it is an abelian category. Any morphism in M(X I )sp is strictly compatible with W· ; i.e., grW is an exact functor. Denote by M(X I )sp >n the full subcategory of special modules with Wn = 0. (I)

3.1.7. Lemma. j∗ j (I)∗ ωX I is a special DX I -module. −|I|

Proof of Lemma. We want to show that C(ω)X I is special. By induction −|I|+1

by |I| we can assume that C(ω)X I −|I|

−|I|+1

∈ M(X I )sp >−|I| . Thus such is the image of −|I|

d : C(ω)X I → C(ω)X I . Therefore C(ω)X I is an extension of a special module with vanishing W−|I| by ωX I , and hence it is special; q.e.d.  (I/T ) (T )

j∗ j (T )∗ ωX T is special. 3.1.8. So for any T ∈ Q(I) the DX I -module ∆∗ I sp In fact, this is an injective object of M(X ) . This follows from the exactness of (I/T ) (T ) (T )∗ the functor P 7→ P/W−|T |−1 P since ∆∗ j∗ j ωX T is obviously an injective (I) object of the subcategory M(X I )sp . In particular, ∆∗ ωX is an injective >−|T |−1 object of M(X I )sp . (I) Suppose that |I| > 1, so Hom(ωX I , ∆∗ ωX ) = 0. Since C(ω)X I is a resolution (I) of ωX I [I] in M(X I )sp , the complex CI := Hom(C(ω)X I , ∆∗ ωX ) is acyclic. ch As a mere graded module CI equals ⊕ PT (ωX ) ⊗ λT [−|T |]. The non-zero T ∈Q(I)

components of the differential are PTch0 (ωX ) ⊗ λT 0 [−|T 0 |] → PTch (ωX ) ⊗ λT [−|T |] for T 0 ∈ Q(T ), |T | = |T 0 | + 1. In terms of the operad structure it is the insertion ch of κ([ ])α1 ,α2 ∈ P{α (ωX ); here αi ∈ T are the elements such that T 0 is the 1 ,α2 } quotient of T modulo the relation α1 = α2 . 3.1.9. Now we can finish the proof of 3.1.5. The surjectivity of the top differential in the complexes CI means that for n ≥ 2 every operation in Pnch (ωX ) is a sum of (n − 1)-operations composed with binary operations. Therefore P ch (ωX ) is

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generated by P2ch (ωX ). Acyclicity of CI implies that all the relations in our operad come from the Jacobi relation for [ ] ∈ P2ch (ωX ).  3.1.10. Remarks. (i) For S, T ∈ Q(I) let LieS/T be the vector space ⊗ LieSt T

if S ≥ T and 0 otherwise. Let us define a canonical isomorphism ∼ (I/S) (S) (S)∗ S j∗ j ωX ) −→

(3.1.10.1) α(I, S)l : grW −l (∆∗

⊕ T ∈Q(S,l)

(I/T )

∆∗

T ωX ⊗ Lie∗S/T .

First notice that both parts are supported on X S ⊂ X I , so (applying ∆(I/S)! and replacing S by I) we may assume that S = I. The case of the top quotient l = 1 is just the isomorphism κI of 3.1.5. Assume that l > 1. It suffices to define α(I, I)l = α(I)l over the complement to diagonals of dimension < l. Its T -component α(I)T is determined by the restriction to a neighbourhood of X T . (I) (I ) It I There j∗ j (I)∗ ωX coincides with  j∗ t j (It )∗ ωX , and α(I)T :=  α(It )1 . T

T

(I/T )

T (ii) Let us describe M(X I )sp explicitly. Its irreducible objects are ∆∗ ωX , (I/T ) (T ) (T )∗ T T ∈ Q(I). The object IT := ∆∗ j∗ j ωX is an injective envelope of (I/T ) T 2 ∆∗ ωX . Isomorphism (3.1.10.1) yields a natural identification Hom(IS , IT ) = LieS/T . The composition of morphisms corresponds to (a tensor product) of compositions of Lie operations. So let Q(I) be the k-category whose objects are elements of Q(I) and morphisms HomQ(I) (S, T ) = LieS/T ; the composition of morphisms is the tensor product of compositions of Lie operations. We have defined a fully faithful embedding Q(I) → M(X I )sp , T 7→ IT . A Q(I)-module is a k-linear functor Q(I) → (finite-dimensional vector spaces); Q(I)-modules form an abelian category V(I). There is an obvious functor M(X I )sp → V(I)◦ which sends a D-module M to the Q(I)-module T 7→ Hom(M, IT ). This functor is an equivalence of categories.

3.1.11. Remarks. We assume that k = C. (i) For any special DX I -module P one has H l DRan (P ) = H l DRan (grW l P ); this sheaf is isomorphic to a direct sum of constant sheaves supported on X S , |S| = −l. Here DRan denotes the analytic de Rham complex (used in the Riemann-Hilbert correspondence). If l = −1, then H l DRan (P ) is the constant sheaf on the diagonal (I) X ⊂ X I with fiber V ∗ where V := Hom(P, ∆∗ ωX ). (I) (ii) The filtration W· on j∗ ωU (I) coincides with the weight filtration on the (I) mixed Hodge module j∗ C(|I|)U (I) [|I|]. (I) (I) (iii) By 3.1.4, one has PIch (ωX ) = Hom(j∗ ωU (I) , ∆∗ ωX ) ⊗ λI . So remark (i) above implies that PIch (ωX ) ⊗ λI = H|I|−1 (YI , C) where YI is the configuration space of I-tuples of different points on C = R2 . Up to homotopy the YI ’s are the spaces of the “small disc” operad. The composition law in the operad P ch (ωX ) coincides with the one on the homology operad of this topological operad. So 3.1.5 amounts to Cohen’s theorem [C]. 3.1.12. For M ∈ M(X) we define the unit operation εM ∈ P2ch ({ωX , M }, M ) ∼ as the composition j∗ j ∗ ωX  M → (j∗ j ∗ ωX  M )/ωX  M −→ ∆∗ M where the last ! arrow comes from the canonical isomorphism ωX ⊗ M = M . 2 Indeed,

(I/T )

IT is injective by 3.1.8; it contains ∆∗

T and is indecomposable. ωX

3.1. CHIRAL OPERATIONS

161

3.1.13. Lemma. (i) For Mi , N ∈ M(X) there is a canonical isomorphism 3 X ∼ (3.1,13.1) PIch ({Mi }, N )I −→ PI˜ch ({ωX , Mi }, N ), (ϕi ) 7→ ϕi (εMi , idMi0 )i0 6=i . i∈I

(ii) Chiral operations satisfy the Leibnitz the unit operations: P rule with respect to ch for ϕ ∈ PIch ({Mi }, N ) one has εN ϕ = ϕ(εMi , idMi0 )i0 6=i ∈ PI˜ ({ωX , Mi }, N ). i∈I

˜

Proof. (i) Let ∆i : X I ,→ X I be the ith diagonal section of the projection X → X I . The images of the ∆i ’s are disjoint over U (I) , so one has a short exact sequence (3.1.13.1) I˜

˜ (I)

(I)

˜

(I)

0 → ωX  j∗ j (I)∗ (Mi ) → j∗ j (I)∗ ωX  (Mi ) → ⊕ ∆i∗ j∗ j (I)∗  Mi → 0. Here we use the standard identifications j∗ j ∗ ωX  Mi /ωX  Mi = ∆∗ (ωX ⊗! Mi ) = (I)

˜ (I)

∆∗ Mi . Since Hom(ωX  j∗ j (I)∗ (Mi ), ∆∗ N ) = 0, (3.2.4.1) yields a canonical ∼ isomorphism PIch ({Mi }, N )I −→ PI˜ch ({ωX , Mi }, N ). One checks in a moment that it coincides with the map in (3.1.13.1). (ii) Let t be a local coordinate on X, t0 , ti the corresponding local coordinates ˜ (I) on X I . For i ∈ I set ν i := t0dt−to i . For any m ∈ j∗ j (I)∗  Mi one has εN ϕ(ν i · m) = ˜ (I)

∆i· (ϕ(m)) = ϕ(εMi , idMi0 )i0 6=i (νi · m) ∈ ∆∗ N and ϕ(εMi00 , idMi0 )i0 6=i00 = 0 for i00 6= i. Now use (i).  3.1.14. Variants. (i) Let L be any Lie∗ algebra, Mi , N ∈ M(X, L). A chiral operation ϕ ∈ PIch ({Mi }, N ) is L-compatible if it satisfies the Leibnitz rule with respect to the L-actions. Explicitly, we demand that the composition ·N ϕ ∈ ˜ (I) (I) Hom(L  j∗ j (I)∗  Mi , ∆∗ N ) equals the sum of I operations ϕ(·Mi , idMi0 )i0 6=i , 4 i ∈ I. This amounts to the fact that ϕ satisfies the Leibnitz rule with respect to ch the h(L)-action on our D-modules. We denote the set of such operations by PLI ; they define on M(X, L) a pseudo-tensor structure M(X, L)ch . (ii) Let R` be a commutative DX -algebra and Mi , N are R` [DX ]-modules. (I) (I) Consider the DX I -algebra R`I . Then j∗ j (I)∗  Mi and ∆∗ N are R`I -modules (I) (I)∗ (I) 5 in the obvious way. We say that a chiral operation ϕ : j∗ j  Mi → ∆∗ N is ` `I R -polylinear if it is a morphism of R -modules. R` -polylinear chiral operations are closed under composition, so they define a pseudo-tensor structure on M(X, R` ), which we denote by M(X, R` )ch . According to 3.1.3, the functor M 7→ M ⊗ R` extends naturally to a pseudotensor functor M(X)ch → M(X, R` )ch . More generally, for any morphism of commutative DX -algebras R` → F ` the base change functor N 7→ N ⊗ F ` extends R`

naturally to a pseudo-tensor functor M(X, R` )ch → M(X, F ` )ch . The above constructions make evident sense in the DG super setting. I˜ := I t · (see 1.2.1). ·Mi , ·N are the ∗ actions of L. 5 The R`I -module structure on ∆(I) N comes from the R`⊗I -module structure on N (de∗ (I) fined by the product morphism of algebras R`⊗I → R` ) and the fact that ∆∗ sends R`⊗I = (I)∗ `I `I ∆ R -modules to R -modules (to define the module structure use (2.1.3.2) for i = ∆(I) , L = R`I , M = N ). 3 Here

4 Here

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Exercises. (i) Suppose L is a Lie∗ algebra which is a vector DX -bundle, so L is a Lie! coalgebra (see 2.5.7). Let C(L) be the Chevalley DG DX -algebra; for an L-module M we have a DG C(L)-module C(L, M ) (see 1.4.10). Show that the functor C(L, ·) : M(X, L) → M(X, C(L)) extends naturally to a faithful pseudotensor functor C(L, ·) : M(X, L)ch → M(X, C(L))ch . (ii) Extend the above constructions to the setting of modules over a Lie∗ Ralgebroid (see 2.5.16). ◦

3.1.15. The category MO (X) of quasi-coherent OX -modules carries a natural M(X)ch -action (see 1.2.11). The vector space of operations PI˜ch ({Mi , F }, G), where F , G are (quasi-coherent) OX -modules and Mi , i ∈ I, are DX -modules, is defined ˜ (I)

˜ (I)

˜

˜ (I)

as PI˜ch ({Mi , F }, G) := HomDI OX (j∗ j (I)∗ ((Mi )  F ), ∆∗ G). Here ∆∗ G := X

˜ (I)

(∆· G) ⊗ (DI X  OX ). The composition of these operations with usual chiral O

˜ XI

operations between D-modules and morphisms of O-modules (as needed in 1.2.11) is clear. Consider the induction functor MO (X) → M(X), F 7→ FD (see 2.1.8). One has an obvious natural map PI˜ch ({Mi , F }, G) ,→ PI˜ch ({Mi , FD }, GD ). On the other hand, if P, Q are D-modules, then there is an obvious map PI˜ch ({Mi , P }, Q) ,→ PI˜ch ({Mi , PO }, QO ), where PO , QO are P, Q considered as O-modules. Therefore both induction and restriction functors MO (X)  M(X) are compatible with the M(X)ch -actions. Here the M(X)ch acts on M(X) in the standard way; see 1.2.12(i). 3.1.16. The setting of Harish-Chandra modules. Let us mention briefly that chiral operations are also defined for O-modules on any c-stack of c-rank 1 (see 2.9.10). We will not define it in full generality, but we will just consider the case of the category of (g, K)-modules (see 2.9.7; we use notation from loc. cit.) with dim g/k = 1. For I ∈ S define the diagonal divisor in GI /G as the union of preimages of K by all maps GI /G → G, (gi ) 7→ gi g −1 . Let F˜ (I) be the localization 1

i2

of F (I) with respect to the equations of the diagonal divisor; this is a GI -equivariant F (I) -module. Now for Mi , N ∈ M(G) we set (3.1.16.1)

(I) PIch ({Mi }, N ) := Hom((⊗Mi ) ⊗ F˜ (I) , ∆∗ N ) ⊗ λI

where the morphisms are taken in the category of GI -equivariant F (I) -modules. The composition of chiral operations is defined similarly to the D-module situation. Theorem 3.1.5 remains valid in this context: one has P ch (k) = Lie. For a (g, K)structure Y on X (see 2.9.8) the corresponding functor M(g, K) → M(X), V 7→ V (Y ) , extends naturally to a pseudo-tensor functor (3.1.16.2)

M(g, K)ch → M(X)ch .

Its definition is similar to the definition of the ∗ pseudo-tensor extension (see 2.9.8), and we skip it. For example, for g = k the category of g-modules = that of k[∂]-modules, carries a natural chiral pseudo-tensor structure, and we have a natural pseudotensor functor from it to M(A1 ) (see Exercise in 2.9.8). The functor (3.1.16.2) for the Gelfand-Kazhdan structure is a pseudo-tensor equivalence between the chiral

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pseudo-tensor category of Aut k[[t]]-modules and that of universal D-modules on curves (with chiral operations defined in the obvious way); see 2.9.9. 3.2. Relation to “classical” operations In this section we explain why “classical” operations from 2.2 (or, more precisely, the corresponding c operations as defined in 1.4.27) can be considered as “symbols”, or “classical limits”, of chiral operations. The key reason is Cohen’s theorem (see 3.1.5). (I)

∗

3.2.1. The obvious morphism Li → j∗ j (I)  Li yields a map (3.2.1.1)

βI : PIch ({Li }, M ) → PI∗ ({Li }, M ).

It is clear that the β’s are compatible with the composition of operations, so we have a pseudo-tensor functor β: M(X)ch → M(X)∗

(3.2.1.2)

which extends the identity functor on M(X). It is compatible with the augmentations, so β is an augmented pseudo-tensor functor. Remark. The above pseudo-tensor functor commutes the action of M(X)! on M(X)ch and M(X)∗ (see 2.2.9 and 3.1.3). 3.2.2. Consider the tensor category M(X)! and the pseudo-tensor category M(X)! ⊗Lie (see 1.1.10). So M(X)! ⊗Lie coincides with M(X) as a usual category, and the corresponding operations are PI!Lie ({Li }, M ) := Hom(⊗! Li , M ) ⊗ LieI where Lie is the Lie algebras operad. There is a canonical faithful pseudo-tensor functor (3.2.2.1)

α: M(X)! ⊗ Lie → M(X)ch

which extends the identity functor on M(X). Namely, the map (3.2.2.2)

αI : Hom(⊗! Li , M ) ⊗ LieI ,→ PIch ({Li }, M )

sends φ ⊗ ν ∈ Hom(⊗! Li , M ) ⊗ LieI to a chiral operation equal to the composition (I)

(I)

(I)

(I)

(I)

I j∗ j (I)∗ (Li ) = (L`i ) ⊗ j∗ j (I)∗ ωX → (L`i ) ⊗ ∆∗ ωX = ∆∗ (⊗! Li ) → ∆∗ M.

Here the first arrow is κ(ν) (see 3.1.5) tensored by the identity morphism of L`i , (I) the second one is ∆∗ (φ), and the equalities are standard canonical isomorphisms. We leave it to the reader to check injectivity of αI (notice that the canonical map (I) (I) I j∗ j (I)∗ ωX → ∆∗ ωX ⊗ Lie∗I coming from κ is surjective) and the compatibility of the αI ’s with the composition of operations. 3.2.3. Remark. The composition βI αI : PI!Lie → PI∗ vanishes for |I| ≥ 2. ∼ 7 The sequence 0 → P2!Lie → P2ch → P2∗ is exact. Notice that P2! −→ P2!Lie , φ → φ ⊗ [ ]. The above pseudo-tensor functors α, β are parts of the following general picture:

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(I)

(I)

I 3.2.4. Notice that j∗ j (I)∗ Li = (j∗ j (I)∗ ωX )⊗(L`i ). So the special filtra(I) (I)∗ I (I) tion W· on j∗ j ωX (see 3.1.6 and 3.1.7) yields a finite filtration on j∗ j (I)∗ Li W (I) (I)∗ I which we denote also by W· . The canonical identifications gr−l j∗ j ωX =

⊕ T ∈Q(I,l)

(I/T )

∆∗

(3.2.4.1)

T ωX ⊗ Lie∗I/T of (3.1.10.1) yield the canonical surjections

⊕ T ∈Q(I,l)

(I/T )

∆∗

!

(I)

W (  (⊗ Li ) ⊗ Lie∗It )  gr−l j∗ j (I)∗  Li t∈T It

which are isomorphisms if the Li are OX -flat. The above filtration yields a canonical filtration on k-modules of the chiral operations: (I)

(I)

(I)

(3.2.4.2) PIch ({Li }, M )n := Hom(j∗ j (I)∗  Li /Wn−1−|I| j∗ j (I)∗  Li , ∆∗ M ). The morphisms (3.2.4.1) yield the canonical embeddings (3.2.4.3)

grn PIch ({Li }, M ) ,→

⊕ T ∈Q(I,|I|−n)

PT∗ ({⊗!It Li }, M ) ⊗ LieI/T

which are isomorphisms if the Li are projective DX -modules. Notice that the maps α, β from 3.2.1 and 3.2.2 occur as “boundary” morphisms of (3.2.4.3). Namely, the morphism (3.2.4.3) for n = |I| − 1 is always an isomorch |I|−1 phism. Its inverse (composed with the embedding gr|I|−1 PIch = PI ,→ PIch ) coincides with αI of (3.2.2.2). Simlarly, βI of (3.2.1.1) is the morphism (3.2.4.3) for n = 0 composed with the projection PIch  PIch /PIch 1 = gr0 PIch . 3.2.5. The above filtration is compatible with the composition of chiral operations. Namely, for J  I and a J-family of DX -modules Kj the composition maps the tensor product PIch ({Li }, M )n ⊗ (⊗ PJch ({Kj }, Li )mi ) to PJch ({Kj }, M )n+Σmi . i I

Therefore we can define a new pseudo-tensor structure on M with operations P cl := gr· P ch ; this is the classical limit of the chiral structure. We can rewrite (3.2.4.3) as a canonical embedding PIcl ({Li }, M ) ,→ PIc ({Li }, M ) (recall that c operations were defined in 1.4.27 using the compound pseudo-tensor structure M∗! ) which is an isomorphism if the Li are projective DX -modules. It follows from the definitions that this embedding is compatible with the composition of operations. So we defined a canonical fully faithful embedding of pseudo-tensor categories (3.2.5.1)

M(X)cl ,→ M(X)c

which extends the identity functor on M. 3.2.6. The above constructions remain valid in the setting of (g, K)-modules (see 3.1.16) and pseudo-tensor functors defined by (g, K)-structures are compatible with them. 3.3. Chiral algebras and modules In this section we define the basic objects to play with following the “Lie algebra style” approach (see 0.8). We begin with non-unital chiral algebras which are simply Lie algebras in the pseudo-tensor category M(X)ch . The standard functors relating chiral operations with ! and ∗ operations provide the canonical Lie∗

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bracket on every chiral algebra and identify commutative DX -algebras with commutative chiral algebras (see 3.3.1–3.3.2). Unital chiral algebras (or simply chiral algebras) are considered in 3.3.3. We pass then to modules over chiral algebras and operations between them in 3.3.4–3.3.5, define an induction functor from Lie∗ modules to chiral modules in 3.3.6 (for a more general procedure see 3.7.15), consider various commutativity properties for chiral modules and their sections in 3.3.7– 3.3.8, identify modules over commutative DX -algebras with central modules over the corresponding commutative chiral algebras in 3.3.9, consider families of chiral algebras in 3.3.10, explain why coisson algebras are “classic limits” of chiral algebras in 3.3.11, consider the filtered setting in 3.3.12, and homotopy chiral algebras in 3.3.13. Chiral algebras in the setting of (g, K)-modules are mentioned in 3.3.14. Chiral operations between modules over a chiral algebra (see 3.3.4) are closely related to the notion of fusion product of A-modules (cf. [KL] and [HL1]); due to the lack of our understanding, the latter subject will not be discussed in the book. Below, we deal either with plain DX -modules or tacitly assume to be in the DG super setting. 3.3.1. For a k-operad B a Bch algebra on X is a B algebra in the pseudo-tensor category M(X)ch . The category B(M(X)ch ) of Bch algebras will also be denoted by Bch (X). The canonical pseudo-tensor functors α, β from 3.2 yield the functors between the categories of B algebras (3.3.1.1)

βB

B

B(M(X)! ⊗ Lie)−−α−−→Bch (X)−−−−→B∗ (X).

Our prime objects of interest are Liech algebras. According to 1.1.10, one has Lie(M(X)! ⊗ Lie) = Com! (X) := Com(M(X)! ), so we can rewrite (3.3.1.1) as (3.3.1.2)

β Lie

Lie

α Com! (X)−− −−→Liech (X)−−−−→Lie∗ (X).

Note that β Lie is a faithful functor, and αLie is a fully faithful embedding (since α is a faithful pseudo-tensor functor). 3.3.2. For a Liech algebra A we denote by µ = µA ∈ P2ch ({A, A}, A) the commutator and by [ ] = [ ]A ∈ P2∗ ({A, A}, A) the ∗ Lie bracket β(µ). We usually call µ the chiral product. Consider the sheaf of Lie algebras h(A) := h(β Lie (A)) (see 2.5.3). It acts on the D-module A (the adjoint action) by derivations of µA (see the end of 1.2.18). A Liech algebra A is said to be commutative if [ ]A = 0. Denote by Liech com (X) ⊂ ch ch Lie (X) the full subcategory of commutative Lie algebras. It follows from 3.2.3 that αLie (Com! (X)) = Liech com (X); i.e., we have the equivalence of categories (3.3.2.1)

∼

αLie : Com! (X) −→ Liech com (X).

3.3.3. Let A be a Liech algebra. A unit in A is a morphism of D-modules 1 = 1A : ωX → A (i.e., 1 is a horizontal section of A` ) such that µA (1, idA ) ∈ P2ch ({ωX , A}, A) is the unit operation εA (see 3.1.12). A unit in A is unique if it exists. A Liech algebra with unit is called a unital Liech algebra or simply a chiral algebra. We will also refer to Liech algebras as chiral algebras without unit, or non-unital chiral algebras. Let CA(X) ⊂ Liech (X) be the subcategory of chiral

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algebras and morphisms preserving units; the category Liech (X) will also be denoted by CAnu (X). One has an obvious “adding of unit” functor6 CAnu (X) → CA(X), B 7→ B + := B ⊕ ωX , left adjoint to the embedding CA(X) ,→ CAnu (X). In particular, we have the unit chiral algebra ω = ωX ; for a chiral algebra A the structure morphism of chiral algebras 1A : ω → A is called the unit morphism. An ideal I ⊂ A means Liech ideal; for an ideal I the quotient A/I is a chiral algebra in the evident way. Chiral algebras form a sheaf of categories CA(X´et ) on the ´etale topology of X. For a chiral algebra A we denote the corresponding Lie∗ algebra β Lie (A) by A , so ALie is A considered as a Lie∗ algebra with bracket [ ]A . For a Lie∗ algebra L an L-action on A is a Lie∗ action of L on the D-module A such that µA is L-compatible (see 3.1.14(i)).7 E.g., we have the adjoint ALie -action on A. Lie

Let CA(X)com ⊂ CA(X) be the full subcategory of commutative chiral algebras. The above equivalence (3.3.2.1) identifies the corresponding subcategories of unital algebras, so we have a canonical equivalence (see 2.3.1 for notation) (3.3.3.1)

∼

ComuD (X) = Comu! (X) −→ CA(X)com .

Every chiral algebra A admits the maximal commutative quotient (which is the quotient of A modulo the ideal generated by the image of [ ]A ). Exercise. Suppose A is generated as a chiral algebra by a set of sections {ai } that mutually commute (i.e., µA (ai  aj ) = 0). Then A is commutative. 3.3.4. For a chiral algebra A an A-module (or chiral A-module) always means a unital A-module, i.e., a Liech algebra A-module M such that µM (1A , idM ) ∈ P2ch ({ωX , M }, M ) is the unit operation εM (see 3.1.12); here µM ∈ P2ch ({A, M }, M ) is the A-action on M . We denote the category of A-modules by M(X, A). Remark. Liech algebra A-modules are referred to as non-unital (chiral) Amodules. They make sense for any non-unital chiral algebra A and coincide with A+ -modules. By 1.2.18, M(X, A) carries an abelian augmented pseudo-tensor structure; we denote it by M(X, A)ch . For Mi , N ∈ M(X, A) the corresponding vector space of ch chiral A-operations is denoted by PAI ({Mi }, N ) ⊂ PIch ({Mi }, N ). The augmentation functor on M(X, A)ch is M 7→ hA (M ) := h(M )h(A) . ch Exercise. A chiral operation ϕ ∈ PIch ({Mi }, N ) belongs to PAI ({Mi }, N ) if and only if it satisfies the following condition: For every m ∈ Mi , a ∈ A, and i ∈ I there exists n = n(a, m, i)  0 such that µN (idA , ϕ)(f a  m) = ϕ(µMi , {idMi0 }i0 6=i )(f a  m) for every function f on X × X I which vanishes of order ≥ n at every diagonal x = xi0 , i0 6= i, and may have poles at any other diagonal.

Example. For the unit chiral algebra ω the forgetful functor M(X, ω) → M(X) is an equivalence of augmented pseudo-tensor categories (see 3.1.13(ii)). 6 The 7 The

Jacobi identity for B + follows from 3.1.13(ii). unit is automatically fixed by any L-action (as well as any horizontal section of A` ).

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167

Remarks. (i) If N is a left DX -module, M a chiral A-module, then N ⊗ M is a chiral A-module. The tensor category M` (X) acts on M(X, A)ch (see 1.1.6(v)). (ii) A-modules localize in the obvious manner, so we have a sheaf of augmented pseudo-tensor categories M(X´et , A)ch . (iii) Let a : P ,→ A be a sub-D-module which generates A as a chiral algebra. ch Then for Mi , N as above an operation A-operation if Pϕ ∈ PI ({Mi }, N ) is a chiral ch and only if the operations µN (a, ϕ), ϕ(idMj6=i , µMi (a, idMi )) ∈ PI˜ ({P, Mi }, N ) i∈I

coincide. 3.3.5. Remarks. (i) An A-module structure on a D-module M amounts to a structure of a chiral algebra on A ⊕ M such that A ,→ A ⊕ M and A ⊕ M  A are morphisms of chiral algebras and the restriction of the chiral product to M × M vanishes. (ii) Due to 3.1.15 and 1.2.13, the notion of an A-module structure (= A-action) makes sense not only for DX -modules, but also for OX -modules. The corresponding category of unital A-modules is denoted by MO (X, A). The induction and restriction yield adjoint faithful functors (3.3.5.1)

MO (X, A)  M(X, A).

Exercise. An object of M(X, A) is the same as an O-module M equipped with an A-action and a right D-module structure such that the A-action on M ` is horizontal. 3.3.6. Any A-module is automatically (via the functor β) an ALie -module. The functor (3.3.6.1)

βA : M(X, A) → M(X, ALie )

ch : M(X, A)ch → M(X, ALie )ch . So for extends to a pseudo-tensor functor βA = βA any A-module M the Lie algebra h(A) acts canonically on M (considered as a plain D-module), and chiral A-operations satisfy the Leibnitz rule with respect to this action.

Proposition. The functor βA admits a left adjoint IndA : M(X, ALie ) → ch M(X, A) which extends to a left pseudo-tensor adjoint IndA =Indch A of βA . Proof. An explicit construction of IndA N for an ALie -module N takes 2 steps: (a) First we construct a triple (I, α, µ) where I is an ALie -module, α : N → I a morphism of ALie -modules, and µ ∈ P2ch ({A, I}, I) a chiral pairing compatible with ALie -actions such that the corresponding ∗ pairing equals the ALie -action on I. Our (I, α, µ) is a universal triple with these properties. Notice that [Ators , N tors ], where Ators ⊂ A, N tors ⊂ N are the OX -torsion submodules, is an ALie -submodule of N . Denote by N v the quotient module. The ALie -action map AN v → ∆∗ N v kills all sections supported at the diagonal, so the push-forward of the exact sequence A  N v → j∗ j ∗ (A  N v ) → ∆∗ A ⊗ N v → 0 by the above action map is well defined. Let I(N ) be its pull-back to the diagonal; it is an extension of A⊗N v by N v . By construction, one has a morphism α : N → I(N ) and a chiral operation µ ∈ P2ch ({A, N }, I(N )). It is easy to see that I(N ) admits a unique structure of the ALie -module such that i and µ are compatible with ALie actions. The ∗ operation that corresponds to µ is α composed with the ALie -action map.

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Iterating this construction, we get a sequence of morphisms I n (N ) → I n+1 (N ) of ALie -modules, I 0 (N ) = N . Our I is its inductive limit. The above α and µ define α : N → I(N ) and µ ∈ P2ch ({A, I(N )}, I(N )). This (I, α, µ) is the promised universal triple. (b) The operation µ usually does not satisfy the properties of an A-action. However I admits a maximal quotient on which µ defines a structure of the unital A-module. This is IndA N . We leave it to the reader to check the adjunction property: for every Ni ∈ ch M(X, ALie ), M ∈ M(X, A) the maps PAI ({IndA Ni }, M ) → PAchLie I ({Ni }, M ), φ 7→ φ({iNi }), are isomorphisms.  Remark. For any A-module M and its ALie -submodule N the image I of the chiral operation (µM )|N ∈ P2ch ({A, N }, M ) is an A-submodule of M ; in particular, in the notation of the lemma above, IndA N is a quotient of I(N ) already. This can (3) be seen from the following trick. Let F1 ⊂ F2 ⊂ j∗ j (3)∗ OX 3 be the subalgebras of functions having poles at the diagonals x1 = x2 and x1 = x2 , x2 = x3 , respectively. Since every function on the diagonal x2 = x3 with pole at x1 = x2 is a restriction of some function from F1 , we see that the image of (µM )|I ∈ P2ch ({A, I}, M ), i.e., the image of µM (idA , (µM )|N ) ∈ P3ch ({A, A, N }, M ), coincides already with the ∗ (3) µM (idA , µM )-image of the F2 -submodule generated by A  A  N in j∗ j (3) (A  A  N ). By the Jacobi identity and the condition on N , the latter image equals I. 3.3.7. Let M be an A-module and m ∈ Γ(X, M ) a section. We say that a ∈ A kills m if µM (f a  m) ∈ ∆∗ M vanishes for every f ∈ j∗ j ∗ OX×X . Such a form a subsheaf Ann(m) ⊂ A called the annihilator of m. The centralizer Cent(m) of m consists of sections a ∈ A that commute with m, i.e., µM (a  m) = 0. This is a chiral subalgebra of A. We say that m is an A-central section of M if Cent(m) = A. Both Ann(m) and Cent(m) have ´etale local nature. Exercises. (i) If the D-submodule of M generated by m is lisse, then m is A-central (see 2.2.4(ii)). In particular, every horizontal section of M is A-central. (ii) The map ϕ 7→ ϕ(1) identifies HomM(X,A) (A, M ) with the vector space of all horizontal sections of M ` .8 (iii) The maximal constant DX -submodule of A (see 2.1.12) is a commutative chiral subalgebra of A which centralizes every A-module. (iv) If Ann(m) is a quasi-coherent OX -module, then it is automatically an ideal in A.9 The quasi-coherence always holds if M is OX -flat (since the flatness implies 8 Let

m be a horizontal section of M ` . By (i), µM (·, m) vanishes on A  ωX , so it yields a (3) morphism of D-modules ϕm : A → M . The Jacobi identity (restricted to sections of j∗ j (3)∗ A  A  M which have no pole at the diagonal x1 = x3 ) shows that ϕm is a morphism of A-modules. One checks immediately that m 7→ ϕm is the inverse map to that of (ii). 9 Sketch of a proof (cf. Remark in 3.3.6). Fix some n ≥ 0. Notice that every function on the diagonal x1 = x2 in X × X × X with possible pole at the diagonal X is the restriction of a function on X × X × X with possible pole at x2 = x3 and zero of order ≥ n at x1 = x3 . Now take any a ∈ A and b ∈ Ann(m). We want to show that the µA -image P of j∗ OU · a  b ⊂ j∗ j ∗ A  A lies in Ann(m). The above remark implies that the D-module generated by µA (j∗ OU · P  m) coincides with the D-module generated by the µM (µA , idM )-images of all elements f · a  b  m where the function f can have arbitrary poles at the diagonals x1 = x2 and x2 = x3 and zero of order ≥ n at x1 = x3 . Choose n such that µM (a  m) ∈ ∆∗ M is killed by the nth power of the equation of the diagonal, and apply the Jacobi identity.

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169

that a section of A belongs to Ann(m) iff its restriction to some Zariski open is). If m 6= 0 vanishes on a Zariski open subset, then Ann(m) is not OX -quasi-coherent. The annihilator Ann(M ) = Ann(A, M ) of M is the subsheaf of A whose sections over an affine open U are those a that kill every section m of M over U . Similarly, the centralizer Cent(M ) = Cent(A, M ) of M is the subsheaf of A whose sections over U as above are those a that commute with every m. Both these subsheaves are OX -submodules of A. If we know that Ann(M ) (resp. Cent(M )) is a quasi-coherent OX -module, then it is an ideal (resp. a chiral subalgebra) in A. This always happens if M is OX -flat. T Exercise. If {mα } are A-generators of M , then Ann(M ) = Ann(mα ). We say that M is central (or the A-action on M is central) if Cent(M ) = A, or, equivalently, if every m ∈ M is A-central. The centralizer of A (considered as an A-module) is called the center of A; we denote it by Z(A). If Z(A) is OX -quasi-coherent (see above), then this is a commutative subalgebra of A. ch We denote by M(X, A)ch the full pseudo-tensor subcategory of cent ⊂ M(X, A) central modules. The subcategory M(X, A)cent ⊂ M(X, A) is closed under direct sums and subquotients. Any A-module M has maximal central sub- and quotient modules which we denote, respectively, by M cent and Mcent . Notice that Mcent is the quotient of M modulo the A-submodule generated by the image of the ∗ action of ALie on M . So the functor M 7→ Mcent is compatible with the ´etale localization. Every chiral algebra A admits the maximal commutative quotient algbera Acom . As an A-module it equals the maximal central A-module quotient of A. ch 3.3.8. Lemma. The obvious functor M(X, Acom )ch cent → M(X, A)cent is an equivalence of pseudo-tensor categories.

Proof. Let us prove that the action of A on any central A-module M factors through Acom . Let a, a0 ∈ A and m ∈ M be any sections, and f (x1 , x2 , x3 ) any function on X ×X ×X which may have poles at the diagonal x1 = x3 and is regular elsewhere. We want to show that the operation µM (µA , idM ) ∈ P3ch ({A, A, M }, M ) vanishes on f · a1  a2  m, which is clear by the Jacobi identity.  3.3.9. Let R be a commutative chiral algebra, so R` is a commutative DX algebra. The functor α identifies R` -actions on a DX -module M with central chiral R-actions. One gets a canonical equivalence of categories (3.3.9.1)

∼

M(X, R` ) −→ M(X, R)cent .

We leave it to the reader to check that a chiral operation ϕ between R` [DX ]modules is R` -polylinear (see 3.1.14(ii)) if and only if it satisfies the Leibnitz rule with respect to the chiral actions of R on these modules. So we have a canonical equivalence of pseudo-tensor categories (3.3.9.2)

∼

M(X, R` )ch −→ M(X, R)ch cent .

Remark. The pseudo-tensor category M(X, R` )ch , and hence the category of non-unital chiral R-algebras, makes sense for a non-unital commutative DX -algebra R` (use 3.1.3).

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3.3.10. Everything said in this section remains valid if we replace M(X)ch by M(X, R` )ch . We leave it to the reader to check that (3.3.9.2) identifies chiral algebras in M(X, R` ) with pairs (A, i) where A is a chiral algebra on X and i : R → A is a morphism of chiral algebras such that A is a central R-module; we call such a pair (A, i) a chiral R-algebra. Denote the corresponding category by CA(X, R` ). An A-module in M(X, R` )ch is the same as an A-module M such that the R-action on M (via the structure morphism i) is central; such objects are called R-central A-modules or A-modules on Spec R` . Example. Let A be a chiral algebra and L a Lie∗ algebra that acts on A (see 3.3.3). Suppose that L is a vector DX -bundle, so we have the Chevalley topological ≥i DG DX -algebra C(L) = lim ←− C(L)/C (L) (see 1.4.10). Then, by Exercise (i) in 3.1. 14, C(L, A) is a topological DG chiral C(L)-algebra. It is a C(L)-deformation of A = C(L, A)/C≥1 (L, A). If A is commutative, then C(L, A) is commutative, and the above construction coincides with the one from 1.4.10. Recall that R` [DX ]-modules have the local nature with respect to the ´etale topology of Spec R` (see 2.3.5). R` -polylinear chiral operations also have the local nature. So for every algebraic DX -space Y we have the pseudo-tensor category M(Y)ch of OY [DX ]-modules (see 2.3.5), the category CA(Y) of chiral OY -algebras on Y, and for A ∈ CA(Y) the pseudo-tensor category M(Y, A)ch of A-modules on Y, as well as the corresponding sheaves of categories on Y´et . For example, if Y is constant, i.e., Y = X × Y with a “trivial” connection, then OY [DX ]-modules are the same as Y -families of D-modules on X. A chiral algebra A on Y is the same as Y -families of chiral algebras on X. We also call A a chiral OY -algebra on X. By 3.1.14(ii), the above objects behave naturally with respect to morphisms of Y’s or R` ’s: for any morphism of commutative DX -algebras R` → F ` we have a base change functor CA(X, R` ) → CA(X, F ` ), A 7→ A ⊗ F ` , etc. R`

3.3.11. Let us explain why coisson algebras are “classical limits” of chiral algebras. Let At be a one-parameter flat family of chiral algebras; i.e., At is a chiral k[t]-algebra which is flat as a k[t]-module (see 3.3.10). Assume that A := At=0 := At /tAt is a commutative chiral algebra. This means that the ∗ bracket [ ]t on At is divisible by h. Thus { }t := t−1 [ ]t is a Lie∗ bracket on At ; the corresponding Lie∗ algebra acts on the chiral algebra At (see 3.3.3) by the adjoint action. Reducing this picture modulo t, we see (as in 3.3.3) that A` is a commutative DX -algebra, and { } := { }t=0 a coisson bracket on it. One calls At the quantization of the coisson algebra (A` , { }) with respect to the parameter t. Remark. The above construction is also a direct consequence of 3.2.5 and 1.4.28. Notice the difference between the usual quantization picture and the chiral quantization. The former exploits the fact that the operad Ass is a deformation of the operad Pois (see 1.1.8(ii)), while the tensor structure on the category of vector spaces remains fixed. The latter deals with the fixed operad Lie, and the pseudo-tensor structure on M(X) is being deformed as in 3.2. As in the usual Poisson setting, one can consider quantizations mod tn+1 , n ≥ 0, of a given coisson algebra (A, { }). Namely, these are triples (A(n) , { }(n) , α) where

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A(n) is a chiral k[t]/tn+1 -algebra flat as a k[t]/tn+1 -module, { }(n) a k[t]/tn+1 bilinear Lie∗ bracket on A(n) such that t{ } equals the Lie∗ bracket for the chiral ∼ algebra structure, and α : A(n) /tA(n) −→ A an isomorphism of chiral algebras which (n) sends { } mod t to { }. Quantizations mod tn+1 form a groupoid. For n ≥ 1 the reduction mod tn of a quantization mod tn+1 is a quantization mod tn . A compatible sequence of such quantizations A(n) , A(n) /tn−1 A(n) = A(n−1) , is a formal quantization of (A, { }). 3.3.12. Let A be a not necessarily unital chiral algebra. A filtration S on A is an increasing sequence of D-submodules A0 ⊂ A1 ⊂ · · · ⊂ A such that Ai = A and µ(j∗ j ∗ Ai  Aj ) ⊂ ∆∗ Ai+j . Then gr A is a Liech algebra in the obvious way. If A is unital, then, unless it is stated explicitly otherwise, we assume that our filtration is unital; i.e., 1A ∈ A0 . Then gr· A is a chiral algebra. The filtration is commutative if gr· A is a commutative chiral algebra. Then gr· A` is a coisson algebra: the coisson bracket { } ∈ P2∗ ({gri A, grj A}, Ai+j−1 ) comes from the Lie∗ bracket on A. Consider the Rees algebra At := ⊕Ai . This is a chiral k[t]-algebra10 which is k[t]-flat. One has At=1 = A, At=0 = gr A. If our filtration is commutative, then we are in the situation of 3.3.11, and the above coisson bracket equals the one from 3.3.11. 3.3.13. We denote by HoCA(X) the homotopy category of chiral algebras, defined as the localization of the category of DG super chiral algebras with respect to quasi-isomorphisms.11 One can consider the non-unital setting as well. We will not exploit HoCA(X) to any length. Notice that the category of DG chiral algebras is not a closed model category (e.g., since the coproduct of chiral algebras usually does not exist). The next remarks show the level of our understanding of HoCA(X): Remarks. (i) For any homotopy Lie operad Lie˜ (i.e., a DG operad equipped with a quasi-isomorphism Lie˜ → Lie) one has the corresponding category of Lie˜ algebras in M(X)ch . We do not know if the corresponding homotopy category is equivalent to the homotopy category of (non-unital) chiral algebras. In particular, we do not know if chiral DG algebras satisfy the “homotopy descent” property. If the answer is negative, then, probably, the right homotopy theory should be based on different objects. (ii) A morphism of DG chiral algebras φ : A → B yields an evident exact DG functor between the DG categories of chiral DG modules M(X, B) → M(X, A). Suppose that φ is a quasi-isomorphism. We do not know if the corresponding functor of the derived categories DM(X, B) → DM(X, A) is an equivalence of categories. 3.3.14. The above definitions have immediate counterparts in the setting of (g, K)-modules (see 3.1.16). The functors (3.1.16.2) transform chiral algebras to chiral algebras. In particular, a chiral algebra in the category of Aut k[[t]]-modules 10 The chiral product on A is compatible with the grading and comes from the chiral product t on A; multiplication by t is the sum of embeddings Ai ,→ Ai+1 . 11 To see that HoCA(X) is well defined (no set-theoretic difficulties occur), one shows that any diagram representing a morphism A → B in HoCA(X) can be replaced by a subdiagram all of whose terms (but the last one, B) have cardinality less than or equal to that of A.

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amounts to a universal chiral algebra, i.e., a chiral algebra in the category of universal D-modules on curves; see 2.9.9. 3.4. Factorization In this section we give another definition of chiral algebras in terms of Omodules on the space R(X) of finite non-empty subsets of X which satisfy a certain factorization property for the disjoint union of subsets. Ran’s space R(X) was, in fact, introduced by Borsuk and Ulam [BU]; Ziv Ran [R1], [R2] was, probably, first to consider it in algebraic geometry. Remarkably enough, it is contractible. This result (in fact, a stronger one) is due to Curtis and Nhu (see Lemmas 3.3 and 3.6 from [CN]); simple-connectedness of R(X)3 was established by Bott [Bo] who showed that R(S 1 )3 = S 3 . We give a simple algebraic proof in 3.4.1(iv); a similar proof was found indepently by Jacob Mostovoy. We begin in 3.4.1 with a review of the basic topological properties of R(X). In 3.4.2–3.4.3 we play with O- and D-modules on R(X). Notice that R(X) is not an algebraic variety but merely an ind-scheme in a broad sense, which brings some (essentially notational) complications. Factorization algebras are defined in 3.4.4; equivalent definitions are considered in 3.4.5 and 3.4.6. In 3.4.7 we show that, in fact, every factorization algebra is a D-module in a canonical way. We define a functor from the category of factorization algebras to that of chiral algebras in 3.4.8. The theorem in 3.4.9 says that it is an equivalence of categories. In the course of the proof (see 3.4.10–3.4.12) some constructions that will be used later in Chapter 4 are introduced. First we realize M(X)ch as a full pseudo-tensor subcategory in a certain abelian tensor category (of a non-local nature) and the same for M(X)∗ (see 3.4.10). This permits us to define the Chevalley-Cousin complex of a chiral algebra; the theorem in 3.4.9 amounts to its acyclicity property (see 3.4.12). Having at hand the identification of chiral and factorization algebras, we consider “free” chiral algebras in 3.4.14 (in the setting of vertex algebras this construction was studied in [Ro]), define the tensor product of chiral algebras in 3.4.15, play with chiral Hopf algebras in 3.4.16, explain in 3.4.17 what are twists of chiral algebras, describe chiral modules and chiral operations between them in factorization terms (see 3.4.18–3.4.19), and give a “multijet” geometric construction of the factorization algebra that corresponds to a commutative chiral algebra (see 3.4.22). 3.4.1. A topological digression. This subsection plays an important yet purely motivational role. Let X be a topological space. Its exponential Exp(X) is the set of all its finite subsets; for S ⊂ X the corresponding point of Exp(X) is denoted by [S]. For any finite index set I we have an obvious map rI : X I → Exp(X). We equip Exp(X) with the strongest topology such that all maps rI are continuous.12 Thus Exp(X) is the disjoint union of the base point [∅] and the subspace R(X) called Ran’s space associated to X. For [S] ∈ R(X) let R(X)[S] ⊂ R(X) be the subspace whose points are finite subsets of X that contain S. If X is connected, then so are R(X) and R(X)[S] . Here are some properties of Exp and R: 12 Therefore two points [S], [S 0 ] ∈ Exp(X) are close iff S lies in a small neighborhood of S 0 and vice versa.

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(i) The subspaces Exp(X)n that consists of [S] with |S| ≤ n form an increasing filtration on Exp(X); the projection rn : X n → Exp(X)n is an open map. The same is true for Exp replaced by R. One has R(X)0 = ∅, R(X)1 = X, R(X)2 = Sym2 X. For any n the stratum R(X)on := R(X)n r R(X)n−1 is equal to the complement to the diagonals in Symn X; this is the space of configurations of n points in X. (ii) For any surjection J  I we have the corresponding diagonal embedding ∆(J/I) : X I → X J . One has rJ ∆(J/I) = rI , and Exp(X) is the inductive limit of the topological spaces X I with respect to all diagonal embeddings. The same is true for R(X) (add the condition I 6= ∅). (iii) Exp(X) is a commutative monoid with respect to the operation [S]◦[S 0 ] := [S ∪ S 0 ]; the unit element is [∅]. The map Exp(X1 ) × Exp(X2 ) → Exp(X1 t X2 ), ([S1 ], [S2 ]) 7→ [S1 ] ◦ [S2 ] = [S1 t S2 ], is a homeomorphism. It is clear that R(X) is a subsemigroup of Exp(X), as well as each subspace R(X)[S] , [S] ∈ R(X). Notice that [S] is a unit element in [R(X)[S] . (iv) Proposition. Suppose that X is linearly connected. Then all the homotopy groups of R(X) and R(X)[S] vanish. So if X is a CW complex, then R(X) and R(X)[S] are contractible. Proof. Since R(X) and R(X)[S] are linearly connected, the proposition follows from (iii) above and the next lemma: Lemma. Let R be a linearly connected topological space which admits an associative and commutative product ◦ : R × R → R such that for any v ∈ R one has v ◦ v = v. Then all the homotopy groups of R vanish. Proof of Lemma. Suppose for a moment that ◦ has a neutral element e ∈ R (which happens if R = R(X)[S] ). Then R is an H-space. The space of all continuous maps γ : (S i , ·) → (R, e) carries a natural product ◦ coming from the product on R. Passing to the homotopy classes of the γ’s, we get a product on πi (R, e). It is well known that this product coincides with the usual product on the homotopy group. Since for any γ one has γ ◦ γ = γ, we see that πi (R, e) is trivial; q.e.d. If we do not assume the existence of a neutral element, the above argument should be modified as follows. Below we write the usual group law on the homotopy groups πi , i ≥ 1, multiplicatively. Let v ∈ R be any base point. Consider the maps (3.4.1.1)

δ

◦

R− →R×R− →R

where δ is the diagonal embedding; one has ◦δ = idR . For γ ∈ πi (R(X), v) one has δ(γ) = i1 (γ) · i2 (γ) ∈ πi (R × R, (v, v)) where i1 , i2 : R ,→ R × R are the embeddings r 7→ (r, v), (v, r). Therefore γ = ◦δ(γ) = (v ◦ γ)2 . We see that γ = 1 if v ◦ γ = 1. Notice that v ◦ v ◦ γ = v ◦ γ, so, replacing γ in the previous formula by v ◦ γ, we get v ◦ γ = (v ◦ γ)2 . Thus v ◦ γ = 1, so γ = 1, and we are done.  (v) A factorization algebra is a sheaf B of vector spaces on Exp(X) together ∼ with natural identifications B[S1 ] ⊗ B[S2 ] −→ B[S1 ]◦[S2 ] defined for disjoint S1 , S2 , which are associative and commutative in the obvious manner. Denote by BX the restriction of B to X ⊂ Exp(X). The restriction of B to Exp(X)oi equals to the restriction of Symi (BX ) to Symi Xr(diagonals). A morphism between factorization algebras is completely determined by its restriction to the BX ’s, so one can consider B as a sheaf BX on X equipped with a certain factorization structure. It has the X-local nature, so one can speak of factorization algebras on X.

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3.4.2. O-modules and left D-modules on R(X). We want to play with factorization algebras in the algebro-geometric setting of quasi-coherent sheaves. From now on X is our curve. Notice that the R(X)i are not algebraic varieties for i ≥ 3: Exercise. Let f be a germ of a holomorphic function on R(X)3 at a point x ∈ X = R(X)1 ⊂ R(X)3 ; i.e., f is a germ of a holomorphic function on X 3 at (x, x, x) which is symmetric and satisfies the relation f (y, z, z) = f (y, y, z). Then the restriction of f to X is constant. Nevertheless, we will consider O- and D-modules on R(X); they will be introduced directly using 3.4.1(ii). An O-module on R(X), or an OR(X) -module, is a rule F that assigns to every finite non-empty I a quasi-coherent OX I -module FX I and to every π : J  I an identification (3.4.2.1)

(π)

ν (π) = νF

∼

= ν (J/I) : ∆(π)∗ FX J −→ FX I

compatible with the composition of the π’s. We demand that the FX I have no non-zero local sections supported at the diagonal divisor. OR(X) -modules form a k-category MO (R(X)). This category is exact in the sense of [Q3].13 It is also a tensor k-category in the obvious way. The unit object is OR(X) (:= the O-module with components OX I ). Replacing O-modules in the above definition by left D-modules, we get the notion of the left D-module on R(X). These objects form an exact tensor k-category M` (R(X)) with unit object OR(X) . 3.4.3. Let F be an OR(X) -module. Fix some J ∈ S; let ` : V ,→ X J be the complement to the diagonal strata of codimension ≥ 2. Lemma. (i) The OX J -module FX J is flat along every diagonal X I ,→ X J ; i.e., Tori (FX J , OX I ) = 0 for i > 0. (ii) The morphism FX J → `∗ `∗ FX J is an isomorphism.14 Proof. (i) Our statement amounts to the claim that for every π : J  I the projection L∆(π)∗ FX J → ∆(π)∗ FX J is an isomorphism. Let ν˜(π) : L∆(π)∗ FX J → FX I be its composition with the structure isomorphism ν (π) . Our ν˜(π) ’s are compatible with the composition of the π’s, so (since every diagonal embedding is a composition of diagonal embeddings of codimension 1) it suffices to treat the codimension 1 case. Here our statement amounts to the property that FX J has no sections supported on X I , and we are done. (ii) We know that our map is injective. For every diagonal X I ⊂ X J of codimension ≥ 2 one has Ext1 (OX I , FX J ) = 0 by (i) (indeed, Extj (OX I , FX J ) is locally isomorphic to Tor|J|−|I|−j (OX I , FX J )15 ). This implies the surjectivity of our map. Namely, take the smallest m such that C := Coker(FX J → `∗ `∗ FX J ) vanishes outside of diagonals of dimension ≤ m. Then for a diagonal X I ⊂ X J of dimension m our C is non-zero on X I r (diagonals); hence Hom(OX I , C) 6= 0. The latter sheaf equals Ext1 (OX I , FX J ), and we are done.  13 A short sequence is exact if and only if all the corresponding sequences of O n -modules X are exact. Probably, MO (R(X)) is not an abelian category. 14 Here ` is the naive sheaf-theoretic direct image (no higher derived functors are considered). ∗ 15 Compute them using a Koszul resolution of O XI .

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Exercise. F is flat (as an object of the exact tensor category MO (R(X))) if and only if every FX I is a flat OX I -module. We say that a complex of OR(X) -modules is homotopically flat if each of the complexes of OX I -modules is homotopically OX I -flat (see 2.1.1). 3.4.4. Factorization algebras. For π : J  I let j [J/I] : U [J/I] ,→ X J be the complement to all the diagonals that are transversal to ∆(J/I) : X I ,→ X J . Therefore one has U [J/I] = {(xj ) ∈ X J : xj1 6= xj2 if π(j1 ) 6= π(j2 )}. Let B be an O-module on R(X). A factorization structure on B is a rule that assigns to every J  I an isomorphism of OU [J/I] -modules (3.4.4.1)

∼

c[J/I] : j [J/I]∗ ( BX Ji ) −→ j [J/I]∗ BX J . I

We demand that the c’s are mutually compatible: for K  J the isomorphism c[K/J] coincides with the composition c[K/I] (c[Ki /Ji ] ). And c should be compatible 0 0 0 with ν: for every J  J 0  I one has ν (J/J ) ∆(J/J )∗ (c[J/I] ) = c[J 0 /I] (ν (Ji /Ji ) ). J , so we have a canonical Notice that c[J/J] identifies j (J)∗ BX J with j (J)∗ BX embedding (3.4.4.2)

(J)

J BX J ,→ j∗ j (J)∗ BX .

O-modules on R(X) equipped with the factorization structure form a tensor category in the obvious manner. This category is not additive. As follows from (3.4.4.2), the functor B 7→ BX is faithful. Therefore we can consider an O-module B on R(X) equipped with a factorization structure as an Omodule BX on X equipped with a certain structure which we also call a factorization structure. Notice that the factorization structure has the X-local nature, so Omodules with the factorization structure form a sheaf of tensor categories on the ´etale topology of X. In the above discussion one may replace O-modules everywhere by left Dmodules. It also renders to the super or DG super setting in the obvious way. Remarks. (i) If an OR(X) -module B admits a factorization structure, then B is flat if (and only if) BX is a flat OX -module.16 Similarly, if we play with complexes of O-modules, then B is homotopically flat if (and only if) BX is homotopically OX -flat. (ii) Let ϕ : B → B 0 be a morphism of DG OR(X) -modules equipped with a 0 factorization structure such that ϕX : BX → BX is a quasi-isomorphism. Then 0 every ϕX I : BX I → BX I is a quasi-isomorphism. Definition. Let B be an O-module equipped with a factorization structure. We say that the factorization structure is unital or B is a factorization algebra if there exists a global section 1 = 1B of BX such that for every f ∈ BX one has 2 1  f ∈ BX 2 ⊂ j∗ j ∗ BX and ∆∗ (1  f ) = f . Notice that such a section 1 is uniquely defined; it is called the unit of B. 16 Assume that B X is flat. We want to show that every BX n is flat. It amounts to the vanishing of T or>0 (BX n , K) where Spec K is the generic point of any irreducible subscheme of X n . Using induction by n and 3.4.3(i) we can assume that Spec K ⊂ U (n) . Now BU (n) is flat by factorization, and we are done.

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Factorization algebras form a tensor category (we demand that the morphisms preserve units) denoted by FA(X). They are objects of the X-local nature, so we have a sheaf of categories FA(X´et ) on the ´etale topology of X. The unit object of our tensor category is the “trivial” factorization algebra O. For any B ∈ FA(X) there is a unique morphism O → B. Therefore for a collection Bα of factorization algebras and B := ⊗Bα one has canonical morphisms να : Bα → B (defined as tensor product of idBα and the canonical morphisms O → Bα0 , α0 6= α). Remark. In the category FA(X) infinite tensor products are well defined.17 Namely, if Bα is any collection of factorization algebras then ⊗Bα is the inductive limit of ⊗ Bα where S runs the set of finite subsets of indices α. Here for S ⊂ S 0 α∈S

the corresponding arrow ⊗ Bα → ⊗ Bα is ⊗bα 7→ (⊗bα ) ⊗ ( α∈S 0

α∈S

⊗

α0 ∈S 0 rS

10α ).

3.4.5. Factorization algebras can be viewed from a slightly different perspecˆ of all finite sets and arbitrary maps. For any tive. Consider the category S ˆ morphism π : J → I in S we have the corresponding map ∆(π) : X I → X J , (xi ) 7→ (xπ(j) ). We also have the open subset j [π] : U [π] := {(xj ) ∈ X J : xj1 6= xj2 if π(j1 ) 6= π(j2 )} ,→ X J . We write ∆(J/I) := ∆(π) , etc. ˆ a quasi-coherent OX I Suppose we have a rule B which assigns to every I ∈ S module BX I , and to every π : J → I a morphism of OX I -modules ν (π) : ∆(π)∗ BX J → BX I

(3.4.5.1)

and an isomorphism of OU [π] -modules ∼

c[π] : j [π]∗ ( BX Ji ) −→ j [π]∗ BX J .

(3.4.5.2)

I

We demand that our datum satisfies the following properties: (a) ν (π) are compatible with the composition of π’s; (b) for π surjective ν (π) is an isomorphism; (c) for K → J → I one has c[K/J] = c[K/I] (c[Ki /Ji ] )|U [K/J] ; 0 0 0 (d) for J → J 0 → I one has ν (J/J ) ∆(J/J )∗ (c[J/I] ) = c[J 0 /I] (ν (Ji /Ji ) ); (e) the BX I have no non-zero local sections supported at the diagonal divisor; (f) the vector space BX ∅ is non-zero. ∼ Notice that c yields an isomorphism of the vector spaces BX ∅ ⊗ BX ∅ −→ BX ∅ . This map is a commutative and associative product on BX ∅ . This implies that dim BX ∅ = 1,18 so one has a canonical identification of algebras BX ∅ = k. Lemma. The above objects are the same as factorization algebras. Proof. Any B as above is automatically an O-module on R(X) equipped with a factorization structure. The section 1B of BX is the image of 1 by the morphism ν : BX ∅ ⊗ OX → BX . Conversely, suppose we have a factorization algebra B. We set BX ∅ = k and recover the ν (π) as the unique system of morphisms compatible with the composition of the π’s such that for π surjective ν (π) is the structure isomorphism from 3.4.2, and 17 Just

as in the usual category of unital associative algebras. Since the product map for BX ∅ is injective and is commutative, we see that for any one has a ⊗ b = b ⊗ a ∈ B ⊗2∅ , and we are done by (e).

18 Proof.

a, b ∈ BX ∅

X

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for π injective our ν (π) : ∆(π)∗ BX J = BX J OX IrJ → BX I is bf 7→ c(bf 1IrJ ). Here c is the factorization isomorphism (3.4.4.1); to see that the latter section belongs to BX I (see (3.4.4.2)), use (ii) of the lemma in 3.4.3 and the definition of the unit in B (see 3.4.4). The two constructions are clearly mutually inverse.  3.4.6. Here is still another description of the factorization algebras. Let Z be a test affine scheme. Recall that an effective Cartier divisor in X ×Z/Z proper over Z is the same as a subscheme S ⊂ X × Z which is finite and flat over Z (see [KM] 1.2.3). Denote by C(X)Z the set of equivalence classes of such an S 0 where S 0 is said to be equivalent to S if Sred = Sred . This is an ordered set (where 0 0 S ≤ S iff Sred ⊂ Sred ), so we can consider it as a category. Our C(X)Z form a category C(X) fibered over the category of affine schemes. Consider a pair (B, c) where: (i) B is a morphism from the fibered category C(X) to that of quasi-coherent O-modules. So B assigns to every S ∈ C(X)Z an OZ -module BS = BS,Z , for S 0 ≤ S we have a morphism BS 0 → BS , and everything is compatible with the base change. In particular, for each n ≥ 0 the universal effective divisor of degree n (see [SGA 4] Exp. XVII 6.3.9) yields an OSymn X -module BSymn X . (ii) c is a rule that assigns to every pair of mutually disjoint divisors S1 , S2 ∈ ∼ C(X)Z an identification cS1 ,S2 : BS1 ⊗ BS2 −→ BS1 +S2 . We demand that these identifications are commutative and associative in the obvious manner and that they are compatible with the natural morphisms from (i). Assume that the following conditions hold: (a) The O-modules BSymn X on Symn X have no non-zero local sections supported at the discriminant divisor. (b) One has BSym0 X 6= 0. We will show that such a (B, c) is the same as a factorization algebra. For the moment we call such an object a factorization algebra in the sense of 3.4.6. They form a category which we denote by FA(X)0 . This is a tensor category in the obvious manner (one has (⊗Bi )S = ⊗(BiS ). Remarks. (i) The associativity and commutativity of c show that for any finite family of mutually disjoint divisors Sα there is a canonical identification (3.4.6.1)

∼

c{Sα } : ⊗BSα −→ BΣSα .

(ii) As in 3.4.5, one gets a canonical identification BSym0 X = k. For any S ∈ C(X)Z the embedding ∅ ⊂ S yields a canonical morphism OZ = B∅,Z → BS,Z , i.e., a canonical section of BS,Z . The structure morphisms for B preserve these sections; we refer to all of them as the unit section of B. 0 (iii) If S 0 , S are Cartier divisors such that Sred ⊂ Sred , then S 0 ⊂ nS for some n ≥ 1. Thus C(X)Z is the localization of the category of the Cartier divisors and inclusions with respect to the family of all morphisms S ⊂ nS, n ≥ 1. Therefore in the definition of B we can replace C(X)Z by the ordered set of Cartier divisors in X × Z/Z proper over Z, adding the condition that the canonical morphisms BS → BnS , n ≥ 1, are all isomorphisms. Proposition. There is a canonical equivalence of tensor categories (3.4.6.2)

∼

FA(X)0 −→ FA(X).

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Proof. Let (B, c) be a factorization algebra in the sense of 3.4.6. For any finite I the union of the subschemes x = xi of X × X I , i ∈ I, is a Cartier divisor of degree |I| in X × X I /X I . So we have the corresponding OX I -module BX I ; equivalently, BX I is the pull-back of BSym|I| X by the projection p|I| : X I → Sym|I| X. These O-modules together with the evident ν (π) and c[π] form a datum from 3.4.5 which satisfies properties (a)–(f) from loc. cit. So we have a factorization algebra; we denote it again by B. The construction is obviously functorial and compatible with tensor products. We have defined the promised tensor functor FA(X)0 → FA(X). Let us construct the inverse functor. Let B be a factorization algebra in the sense of 3.4.5. We want to define the corresponding factorization algebra (B, c) in the sense of 3.4.6. First let us construct the O-modules BSymn X . Consider the projection pn : X n → Symn X. The symmetric group Σn acts on X n and on BX n . Set BSymn X := (pn∗ BX n )Σn . By 3.4.2 it satisfies condition (a) in 3.4.6. Lemma. The canonical morphism ψ : p∗n BSymn X → BX n is an isomorphism. Proof of the lemma. (a) Let ` : V ,→ X n be the complement to the diagonals of codimension ≥ 2 and `¯ : V¯ ,→ Symn X be its pn -image. By (ii) in the lemma in 3.4.3 we know that BX n = `∗ `∗ BX n ; thus BSymn X = `¯∗ `¯∗ BSymn X . Since pn is flat, we conclude that it suffices to check that ψ is an isomorphism over V . By factorization it suffices to consider the case n = 2. (b) By ´etale descent our ψ is an isomorphism over the complement to the diagonal divisor. Therefore ψ is injective (since pn is flat). (c) It remains to prove that ψ : p∗2 BSym2 X → BX 2 is surjective. Consider its cokernel C. This is a Σ2 -equivariant quotient of BX 2 such that (p2∗ C)Σ2 = 0. We know that C is supported (set-theoretically) on the diagonal divisor X, so one has C Σ2 = 0. Let f be a (local) equation of the diagonal X ⊂ X × X which is Σ2 -antiinvariant. Since Σ2 acts on C by the non-trivial character sgn multiplication by f kills C, i.e., C is supported on the diagonal scheme-theoretically. Thus C is a quotient of the pull-back of BX 2 to X, i.e., of BX . But Σ2 acts on BX trivially. So C = 0, and we are done.  π

1 Remark. Let G ⊂ Σn be a subgroup, so we have the projections X n −→ n π2 n ∗ G G r X −→ Sym X. The lemma implies that π2 BSymn X = (π1∗ BX n ) .

Now let us construct our B following Remark (iii) above. Let us define BS,Z for any Cartier divisor S in X × Z/Z proper over Z. Our problem is Z-local, so we can assume that the degree n of S over Z is constant. Such an S amounts to a morphism Z → Symn X, and BS,Z is the pull-back of BSymn X . Let us define for S 0 ⊂ S a canonical morphism BS 0 → BS . We have S = 0 S + T for an effective Cartier divisor T . It suffices to construct our morphism in the universal situation when Z = Symn X × Symm X and S 0 , T are the universal divisors of order n and m, respectively. So let q (n) : Symn X × Symm X → Symn X be the projection and let p : Symn X × Symm X → Symn+m X be the addition map. We want to define a canonical morphism q (n)∗ BSymn X → p∗ BSymn+m X . By the above lemma and remark this amounts to a Σn × Σm -equivariant morphism BX n  OX m → BX n+m . This is the structure morphism ν (π) from (3.4.5.1) for π : {1, . . . , n} ,→ {1, . . . , n + m}.

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The condition from Remark (iii) is satisfied,19 so we have defined our B. The factorization isomorphisms c come from the isomorphisms (3.4.5.2) (consider the universal situation and use the above remark). So we have defined the factorization algebra in the sense of 3.4.6; i.e., we have a functor FA(X) → FA(X)0 . It follows from the construction (use the lemma) that it is inverse to the previously defined functor in the opposite direction. We are done.  3.4.7. The canonical connection. Let B be a factorization algebra. Proposition. The O-module B on R(X) admits a left D-module structure compatible with the factorization structure and such that the section 1B of BX is horizontal. Such a D-module structure is unique. We call it the canonical D-module structure (or the canonical connection) on B. Proof. Existence. It is evident from the picture of 3.4.6. Indeed, suppose we 0 have two morphisms of schemes f, g : Z 0 ⇒ Z which coincide on Zred . Then the ∗ ∗ pull-back maps f , g : C(X)Z → C(X)Z 0 coincide, so for any S ∈ C(X)Z one has a canonical identification (3.4.7.1)

∼

f ∗ BS −→ g ∗ BS

of OZ 0 -modules. The identifications are transitive, so BS carries a canonical action of the universal formal groupoid on Z (whose space is the formal completion of X × X at the diagonal). The structure morphisms of B are compatible with this action. If Z is smooth, then, according to Grothendieck [Gr2], such an action amounts to an integrable connection. The compatibility with the base change shows that 1B is a horizontal section. Looking at BX I ’s, we get our canonical connection. Here is an equivalent construction that starts from the picture of 3.4.5. We will define the connection on BX ; the connections on BX I ’s are defined similarly. For I ∈ S let X be the formal completion of X I at the diagonal X ,→ X I ; for π as above let ∆<π> : X → X <J> be the formal completion of ∆(π) . Let BX be the pull-back of BX I to X (= the formal completion of BX I at X ⊂ X I ) and ν <π> : ∆<π>∗ BX <J> → BX be the completion of ν (π) . Key remark: each ν <π> is an isomorphism. This follows from statement (i) of the lemma in 3.4.3 and the fact that the pull-back of ν (π) to the diagonal X ,→ X I is an isomorphism (equal to idBX ). The isomorphisms ν <π> are compatible with the composition of the π’s. A datum of such isomorphisms amounts to an action on BX of the universal formal groupoid on X (the action itself is the identification of the pull-backs of BX by the two projections X <2> ⇒ X). We have defined the canonical connection on BX . Uniqueness. Our condition just says that all morphisms ν (π) from (3.4.5.1) are compatible with the connections. Therefore the above action of the universal formal groupoid on BX is compatible with the connection, which determines the connection uniquely.  Remark. Here is a down-to-earth definition of the canonical D-module structure (which follows directly from the proof of 3.4.7). Let t be a local coordinate on 19 Use the fact that p O n is isomorphic to O n∗ X Symn X [Σn ] as an OSymn X [Σn ]-module locally on Symn X.

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X. Then for b ∈ B one has 1  b − b  1 = (∂t b)  1(t2 − t1 )mod(t2 − t1 )2 BX×X .

(3.4.7.2)

In fact, the restriction of 1  b to the formal neighbourhood of the diagonal X ⊂ P (t2 −t1 )i i X ×X equals (∂t b)1. This defines our D-module structure on BX . And i! (I) I BX I is a D-submodule of j∗ j (I)∗ BX . 3.4.8. From factorization algebras to chiral algebras. Let B r = ωX ⊗ BX be the right DX -module that corresponds to the left DX -module BX (see 2.1.1). Let us show that it carries a canonical structure of a chiral algebra. For each surjection J  I we have a natural isomorphism of left D-modules ∼ ∆(J/I)∗ BX J −→ BX I . By (2.1.3.2), we can rewrite it as an isomorphism of right DX J -modules (J/I)

(3.4.8.1)

∆∗

∼

(J/I)

I ωX ⊗ BX J −→ ∆∗

I (ωX ⊗ BX I ).

In particular, we get a canonical map r P ch (ωX )J −→ P ch (BX )J

(3.4.8.2)

(J)

(J)

J which sends an operation ϕ : j∗ j (J)∗ ωX −→ ∆∗ ωX to the composition

(3.4.8.3)

(J)

(J)

(J)

(J)

J j∗ j (J)∗ B rJ = j∗ j (J)∗ ωX ⊗ BX J −→ ∆∗ ωX ⊗ BX J = ∆∗ B r .

Here the first equality is the factorization isomorphism tensored by idωJ , the arrow X is ϕ tensored by idBX J , and the last equality is (3.4.8.1) (for I = ·). We leave it to the reader to check that (3.4.8.2) is compatible with the composition of chiral operations. By 3.1.5 we can rewrite (3.4.8.2) as a canonical morphism r . of operads Lie → P ch (B r ) which is the same as a Liech algebra structure on BX ch r r The section 1 of BX is a unit in the Lie algebra B . So B is a chiral algebra. 3.4.9. The next theorem is the principal result of this section: Theorem. The functor FA(X) → CA(X), B 7→ B r , is an equivalence of categories. Remarks. (i) The factorization algebra corresponding to a commutative chiral algebra (i.e., to an affine DX -scheme) can be constructed geometrically (see 3.4.21– 3.4.22). But to prove that the construction from 3.4.21 indeed gives a factorization algebra, we compare it in 3.4.22 with the construction from 3.4.11–3.4.12. (ii) The theorem and its proof remain valid in the super and DG super setting. It also generalizes immediately to the case of families of chiral algebras (see 3.3.10). Namely, for a scheme (or algebraic space) Z one defines the notion of Z-family of factorization algebras and the above functor in the evident way. It establishes an equivalence between the category of OZ -flat families of factorization algebras and the category of OZ -flat families of chiral algebras. Proof. It is found in 3.4.10–3.4.12. We will construct the inverse functor. So for every chiral algebra A we want to define on the left DX -module A` a canonical structure of a factorization algebra. The key point is the acyclicity property of the Chevalley-Cousin complex C(A) of A. We define C(A) in 3.4.11 after the necessary preliminaries of 3.4.10; the acyclicity lemma is proven in 3.4.12.

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3.4.10. The category M(X S ) and its two tensor structures. In this subsection we embed M(X) in a larger abelian category M(X S ) so that both ∗ and chiral pseudo-tensor structures on M(X) are induced from some tensor structures on M(X S ). The reader should compare this construction with the universal one from Remark in 1.1.6(i). A right D-module M on X S is a rule that assigns to I ∈ S a right D-module MX I (π) (π) on X I and to π : J  I a morphism of D-modules θ(π) = θM : ∆∗ MX I → MX J . We demand that the θ(π) are compatible with the composition of the π’s (i.e., (π ) θ(π1 π2 ) = θ(π2 ) ∆∗ 2 (θ(π1 ) )) and θ(idI ) = idMX I . These objects form an abelian k-category M(X S ). There is an exact fully (S) (S) (I) faithful embedding ∆∗ : M(X) ,→ M(X S ) defined by ∆∗ MX I := ∆∗ M , θ(π) = id∆(J) M . This embedding is left adjoint to the projection functor M(X S ) → M(X), ∗ M 7→ MX . Remark. For n ≥ 1 let Sn ⊂ S be the subcategory of sets of order ≤ n. The above category has an obvious “truncated” version M(X Sn ). The “restricπn tion” functor M(X S ) −→ M(X Sn ) admits a left adjoint an : M(X Sn ) → M(X S ), (I/S) NX S where the inductive limit is taken over the ordered set an (N )X I = lim ∆∗ −→

Q(I, ≤ n). Since πn an is the identity functor, we see that πn identifies M(X Sn ) with the quotient category of M(X S ) modulo the full subcategory M(X ≥n+1 ) of such M ’s that πn M = 0; i.e., MX I = 0 if |I| ≤ n. In particular, grn M(X S ) := M(X ≥n )/M(X ≥n+1 ) equals the category of Σn -equivariant D-modules on X n . The category M(X S ) carries two tensor products ⊗∗ and ⊗ch defined as follows. Let Mi , i ∈ I, be a finite non-empty family of objects of M(X S ). One has (⊗∗I Mi )X J := ⊕ (Mi )X Ji ,

(3.4.10.1)

JI I

where the arrows θ

(π)

are the obvious ones. Similarly, set [J/I] [J/I]∗

(⊗ch I Mi )X J := ⊕ j∗

(3.4.10.2)

j

JI

 (Mi )X Ji ,

where the arrows θ(π) are obvious ones. Here j [J/I] was defined in 3.4.4. Our tensor products are associative and commutative in the obvious way, so they define on M(X S ) (non-unital) tensor category structures which we denote by M(X S )∗ , M(X S )ch . There is an obvious canonical morphism ⊗∗ Mi → ⊗ch Mi

(3.4.10.3)

compatible with the constraints, so the identity functor for M(X S ) extends to a pseudo-tensor functor β S : M(X S )ch → M(X S )∗ .

(3.4.10.4)

(S)

(J/I)

Examples. For Ni ∈ M(X) one has (⊗∗I ∆∗ Ni )X J = ⊕ ∆∗ JI

particular,

(S) (⊗∗I ∆∗ Ni )X I

(S) (Symm ∗ ∆∗ N )X I

=

⊕

⊕

=

I

ν (Ni ). Similarly, for N ∈ M(X) one has ∗

ν∈Aut I (I/T ) T ∆∗ N .

T ∈Q(I,m) ∗

 Ni . In

Here Symm ∗ , m ≥ 1, is the mth symmet-

ric power with respect to ⊗ , and Q(I, m) ⊂ Q(I) is formed by quotient sets of order m. These formulas are modified in the obvious way in the ⊗ch case.

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The embedding M(X) ,→ M(X S ) extends canonically to fully faithful pseudotensor embeddings (3.4.10.5)

(S)

∆∗

: M(X)∗ ,→ M(X S )∗ , (S)

M(X)ch ,→ M(X S )ch . (S)

∼

Namely, for Mi , N ∈ M(X) our Hom(⊗∗I ∆∗ Mi , ∆∗ N ) −→ PI∗ ({Mi }, N ) assigns (S) (S) to ϕ : ⊗∗I ∆∗ Mi → N the restriction of ϕX I to  MI ⊂ (⊗∗I ∆∗ Mi )X I . One I

(S)

∼

defines Hom(⊗∗I ∆∗ Mi , N ) −→ PI∗ ({Mi }, N ) in a similar way. Notice that the restriction of β S to M(X)∗ is the pseudo-tensor functor β from (3.2.1.2). The category M(X S ) admits a natural action of the tensor category M` (R(X)) (see 3.4.2): the action functor (3.4.10.6)

⊗ : M` (R(X)) × M(X S ) → M(X S )

sends F ∈ M` (R(X)), M ∈ M(X S ) to M ⊗ F with (M ⊗ F )X I := MX I ⊗ FX I and (π) (π)−1 (π) (π) via (2.1.3.2). The θM ⊗F : ∆∗ MX I ⊗ FX I → MX J ⊗ FX J equal to θM ⊗ νF associativity constraint for ⊗ is the evident one. 3.4.11. The Chevalley-Cousin complex. Let A be a (not necessarily uni(S) tal) chiral algebra on X. According to (3.4.10.5) the right D-module ∆∗ A on X S S ch is a Lie algebra in the tensor category M(X ) . We define the Chevalley-Cousin complex C(A) of A as the reduced Chevalley complex of this Lie algebra.20 Let us describe C(A) explicitly. Forget about the differential for a moment. As a plain Z-graded D-module, C(A) is the free commutative (non-unital) algebra in (S) M(X S )ch generated by ∆∗ A[1]. Therefore one has (3.4.11.1)

C(A)·X I =

⊕ T ∈Q(I)

(I/T ) (T ) (T )∗ j∗ j (A[1])T .

∆∗

In other words, C(A)·X I is a Q(I)-graded module, and for T ∈ Q(I) its T -component (I/T ) (T ) (T )∗ T is ∆∗ j∗ j A ⊗ λT sitting in degree −|T |. (I/T ) (T ) (T )∗ The differential looks as follows. Its component dT,T 0 : ∆∗ j∗ j (A[1])T 0 0 (I/T ) (T ) (T 0 )∗ T 0 0 → ∆∗ j∗ j (A[1]) can be non-zero only for T ∈ Q(T, |T | − 1). Then T = T 00 t {α0 , α00 }, T 0 = T 00 t {α} and dT,T 0 is the exterior tensor product of the chiral product map µA [1] : j∗ j ∗ (Aα0 [1]  Aα00 [1]) → ∆∗ Aα [1] and the identity map 00 0 for AT (localized at the diagonals transversal to X T ). The reader can skip the next remark at the moment. Remark. As with any Chevalley complex, C(A) carries a canonical structure of a BV algebra with respect to ⊗ch (see 4.1.7). The product · : C(A)· ⊗ch C(A)· → C(A)· and the bracket { } : C(A)[−1]⊗ch C(A)[−1] → C(A)[−1] can be described as follows. For J = J1 t J2 the morphism ·J1 J2 is the obvious embedding C(A)·X J1  C(A)·X J2 → C(A)·X J localized at appropriate diagonals. For T1 ∈ Q(J1 ), T2 ∈ Q(J2 ), T ∈ Q(J) the corresponding component of { }J1 J2 : ˜j∗ ˜j ∗ C(A)X J1 [−1]  C(A)X J2 [−1] → C(A)X J [−1] (here ˜j is the embedding of the complement to the diagonals xj1 = xj2 , j1 ∈ J1 , j2 ∈ J2 ) can be non-zero only if T is the quotient of T1 t T2 modulo a relation t1 = t2 where t1 ∈ T1 , t2 ∈ T2 . Then this component is 20 We must consider the reduced Chevalley complex since M(X S )ch is a non-unital tensor category. The basic facts about the Chevalley complex are recalled in 4.1.6.

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the chiral product j∗ j ∗ A  A → ∆∗ A tensored by the identity map for (A[1])T where T 0 := T r {t} = T1 r {t1 } t T2 r {t2 } (localized at the other diagonals).

0

Notice that if v : V ,→ X I is an open embedding such that X I r V is a union of diagonal strata, then the canonical map C(A)X I → v∗ v ∗ C(A)X I admits an obvious canonical section (3.4.11.2)

s : v∗ v ∗ C(A)·X I → C(A)·X I

which preserves the Q(I)-grading but does not commute with the differentials. The complex C(A) enjoys the following two properties: (i) For every π : J  I the structure morphism yields an isomorphism of complexes (3.4.11.3)

∼

C(A)X I −→ ∆(π)! C(A)X J ,

and this plain ∆(π)! pull-back equals the derived functor pull-back. In other words, there is a canonical short exact sequence of complexes (3.4.11.4)

(π)

0 → ∆∗ C(A)X I → C(A)X J → v∗ v ∗ C(A)X J → 0.

Here v : X J r ∆(π) (X I ) ,→ X J is the open embedding complementary to ∆(π) ; the naive v∗ coincides here with the derived one. (ii) (Factorization) In the setting of 3.4.4 for J  I the product morphism for the commutative algebra structure on C(A)· yields an isomorphism of complexes (3.4.11.5)

∼

c[J/I] : j [J/I]∗ ( C(A)X Ji ) −→ j [J/I]∗ C(A)X J . I

The identifications c are mutually compatible: for K  J the isomorphism c[K/J] coincides with the composition c[K/I] (c[Ki /Ji ] ). 3.4.12. Lemma. For a chiral algebra A one has H n C(A)X I = 0 for n 6= −|I|. End of the proof of 3.4.9. Let us define the factorization algebra structure on A` . −|I| −|I|+1 ` Set A`X I := H −|I| C(A)`X I = Ker(d : C(A)X I → C(A)X I ) . By the lemma we (I/S)∗ ` can rewrite (3.4.10.3) as a canonical identification ∆ AX I = A`X S (see 2.1.2). These identifications are obviously transitive, so the A`X I ’s form a left D-module, (I) (I)∗ `I hence an O-module, on R(X). So21 A`X I ⊂ C(A)0` A . We leave it X I = j∗ j to the reader to check that this embedding defines a factorization algebra structure on our OR(X) -module (with the unit equal to the unit in A). Notice that our factorization algebra came together with a D-module structure, which obviously coincides with the canonical structure (see 3.4.7). The functor FA(X) → CA(X) we have defined is clearly inverse to that of 3.4.9. We are done.  Proof of Lemma. The statement is clear if |I| = 1 (one has C(A)X = A[1]). For |I| > 1 we use induction on |I|. The proof is in three steps: (i) The module H a C(A)X I for a 6= −|I| is supported on the diagonal X ⊂ X I . This follows from (3.4.11.5) and the induction assumption. 0 Let i : Y = X I ,→ X I be a diagonal of codimension 1, v : X I r Y ,→ X I the complementary embedding. Consider the canonical embedding ξ : i∗ C(A)Y ,→ C(A)X I (see (3.4.11.4)). 21 We

I ⊗ λ ; see 3.1.4. use the identification ωX I = ωX I

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(ii) The morphism H a ξ : i∗ H a C(A)Y = H a i∗ C(A)Y → H a C(A)X I is surjective for a 6= −|I|. Indeed, the complex v∗ v ∗ C(A)X I is acyclic outside degree −|I| by (i) (since v is affine), so our claim follows from the exact sequence 0 → i∗ C(A)Y → C(A)X I → v∗ v ∗ C(A)X I → 0 (see (3.4.11.4)). (iii) It remains to show that H a ξ = 0. If a 6= 1 − |I|, this is clear because in this case H a C(A)Y = 0 by the induction assumption. Now let a = 1 − |I|. Choose a section I 0 → I of I  I 0 . Then we have the 0 decomposition X I = X × X I = X × Y and the embedding i : Y ,→ X × Y . We want to show that ξ(i∗ ZYa ) ⊂ d(C a−1 (A)X I ) where ZYa := Ker(d : C a (A)Y → C a+1 (A)Y ). Let s be a local section of i∗ ZYa = v∗ v ∗ (ωX  ZYa )/ωX  ZYa . Lift it to a local section s˜ of v∗ v ∗ (ωX  ZYa ). Then ξ(s) = d(h(˜ s)) where h is the composition f

v∗ v ∗ (ωX  ZYa )−→v∗ v ∗ (A  C a (A)Y ) ,→ C a−1 (A)X I and f comes from 1A : ωX → A.



3.4.13. Remarks. (i) For n ≥ 1 an n-truncated factorization algebra is defined exactly as the usual factorization algebra but we take into consideration only the X I ’s with |I| ≤ n. Denote the corresponding categories by FA≤n (X). As follows from 3.4.9, for n ≥ 3 the obvious functors FA(X) → FA≤n (X), FA≤n+1 (X) → FA≤n (X) are equivalences of categories.22 (ii) Let B be a factorization algebra, B r the corresponding unital chiral algebra. Then there is a canonical isomorphism of complexes of right D-modules on X S (see (3.4.10.6)) (3.4.13.1)

∼

C(B r ) −→ C(ω) ⊗ B (I)

(I)

I I which identifies the summand j∗ j (I)∗ B rI = (j∗ j (I)∗ ωX ) ⊗ BX in C(B r )X I (I) (I)∗ I with the one (j∗ j ωX )⊗BX I in (C(ω)⊗B)X I via the factorization isomorphism (see (3.4.4.2)).

3.4.14. Free factorization algebras. It is easy to see that the chiral algebra freely generated (in the obvious sense) by a given non-empty set of sections does not exist. The nuisance can be corrected as follows. Suppose we have a pair (N, P ) where N is a quasi-coherent OX -module and P ⊂ j∗ j ∗ N  N is a quasi-coherent OX×X -submodule such that P |U = N  N |U . Consider a functor on the category CA(X) which assigns to a chiral algebra A the set of all OX -linear morphisms φ : N → A such that the chiral product µA kills φ2 (P ) ⊂ j∗ j ∗ A2 . Theorem. This functor is representable. We refer to the corresponding universal chiral algebra as the chiral algebra freely generated by (N, P ). Remarks. (i) From the point of view of chiral operations, P should be thought of as “generalized commutation relations”. For example, if P = N  N , then A is the commutative chiral algebra freely generated by ND . From the point of view of chiral algebras, P is the module of “degree 2 generators” (see below). 22 They

are fully faithful for n = 2 and faithful for n = 1.

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(ii) If C is any chiral algebra and N ⊂ C an OX -submodule that generates C, then C is a quotient of the chiral algebra freely generated by (N, P ) where P is, µA say, the kernel of the map j∗ j ∗ N  N → j∗ j ∗ A −−→ ∆∗ A. Proof of Theorem. It is more convenient to consider the setting of factorization algebras. Here the theorem asserts representability of the functor which assigns to a factorization algebra B the set of all OX -linear morphisms φ : N → B such that φ2 (P ) is contained in BX×X ⊂ j∗ j ∗ BX BX . We call the corresponding universal factorization algebra the factorization algebra freely generated by (N, P ). In the course of construction we use certain auxiliary objects: (i) A quasi-factorization algebra in formed by the datum (BX I , ν (π) , c[π] ) from 3.4.5 subject to axioms (a)–(d) and (f) in loc. cit. (ii) A pre-factorization algebra is defined in the same way as a quasi-factorization algebra except that the morphisms ν (π) are defined only for injective maps π : J ,→ I. Apart from the (obvious modification of) axioms (a)–(d) and (f) in 3.4.5, we demand that for every n ≥ 2 the next property is satisfied: (∗)n The action of the transposition σ = σ1,2 on TX n induces the trivial automorphism of the pull-back of TX n to the diagonal x1 = x2 . (I)

Example. T (N )X I := j∗ j (I)∗ (N ⊕ OX )I together with evident structure morphisms ν and c form a pre-factorization algebra T (N ). Pre- and quasi-factorization algebras form tensor categories which we denote by PFA(X) and QFA(X). One has evident tensor functors (in fact, fully faithful embeddings) FA(X) → QFA(X) → PFA(X). Lemma. These functors admit left adjoints FA(X) → PFA(X) → QFA(X). Granted the lemma, let us construct the factorization algebra freely generated by (N, P ). Let T (N, P ) ⊂ T (N ) be a pre-factorization subalgebra defined as follows. (i) For |I| ≤ 1 one has T (N, P )X I = T (N )X I . (ii) T (N, P )X 2 is the minimal σ-invariant OX 2 -submodule of T (N )X 2 which contains P ⊕ N  OX ⊕ OX  N ⊕ OX  OX , is preserved by the action of the transposition σ, and satisfies property (∗)2 .23 (iii) If |I| > 2, then T (N, P )X I is the maximal OX I -submodule of T (N )X I such that for every i1 6= i2 ∈ I its restriction to the complement of the union of all the diagonals except for xi1 = xi2 is equal to T (N, P )X {i1 ,i2 }  T (N )X Ir{i1 ,i2 } . It is easy to see that T (N, P ) is indeed a pre-factorization subalgebra of T (N ).24 Now for any factorization algebra B a morphism of pre-factorization algebras T (N, P ) → B amounts to a morphism of OX -modules φ : N → B such that φ2 (P ) ⊂ BX×X . The promised factorization algebra freely generated by (N, P ) is the image of T (N, P ) by the composition of the functors from the lemma. Proof of Lemma. (i) Let T be any pre-factorization algebra. Let us construct the corresponding universal quasi-factorization algebra B = B(T ). S T (N, P )X 2 is well defined and is OX -quasi-coherent. Indeed, T (N, P )X 2 = Ti where T0 ⊂ T1 ⊂ · · · are the submodules of j∗ j ∗ T (N )2 X defined by induction as follows. One has T0 := (P + σ(P )) ⊕ N  OX ⊕ OX  N ⊕ OX  OX , and Ti+1 is obtained from Ti by adding all sections (t1 − t2 )−1 a where a ∈ Ti is such that a|∆ ∈ ∆∗ Ti is σ-anti-invariant. 24 To check (∗) , notice that the pull-back of T (N, P ) n to the diagonal x = x has no n 1 2 X non-trivial local sections supported at the diagonal divisor. 23 Our

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Notice that for any surjection π : J  I the natural action of the group Aut (J/I) on the OX I -module ∆(J/I)∗ TX J is trivial.25 Let us define a natural morphism α(π) = α(J/I) : TX I → ∆(J/I)∗ TX J . Choose a section s : I ,→ J of π, so we have the structure morphism ν (s) : ∆(s)∗ TX I → TX J . Now set α(J/I) := ∆(J/I)∗ (ν (s) ). Our α(J/I) does not depend on the auxiliary choice of s. The α(π) are compatible with the composition of π’s in an evident way. Therefore for any I ∈ S we have a functor (S/I)◦ → MO (X I ) which assigns to J/I the OX I -module ∆(J/I)∗ TX J and to an arrow K  J the morphism ∆(J/I)∗ α(K/J) . Let BX I ∈ MO (X I ) be the inductive limit. Our B is a quasi-factorization algebra. Namely, the isomorphisms c for B (see (3.4.5.2)) come from the corresponding isomorphisms of T . For f : I → K the morphism ν (f ) : ∆(f )∗ BX I → BX K (see (3.4.5.1)) comes from the functor S/I → S/K, J/I 7→ Jf /K := J t(K rf (I))/K, and the corresponding morphism of the inductive systems ∆(f )∗ ∆(J/I)∗ TX J = ∆(Jf /K)∗ ∆(J/Jf )∗ TX J → ∆(Jf /K)∗ TX Jf where the last arrow is the ∆(Jf /K) -pull-back of the structure morphism ν (J/Jf ) for T . We leave it to the reader to check axioms (a)–(d) and (f) in 3.4.5. The universality property of B is evident. (ii) Let B be a quasi-factorization algebra B. The corresponding universal ¯ is the maximal factorization algebra quotient of B. To factorization algebra B construct it, consider first the quotient qB of B where qBX I is the quotient of BX I modulo the OX I -submodule generated by all sections of type ν (J/I) (a) where J  I and a ∈ BX J is a local section supported at the diagonal divisor of X J . We leave it to the reader to check that qB is indeed a quasi-factorization algebra. We get a ¯ = lim q i B. system of quotients B  qB  q 2 B := q(qB)  · · · . Now B  −→ 3.4.15. Tensor products. We know that FA(X) is a tensor category (see 3.4.4); hence so is CA(X). Let us explain the meaning of the tensor product of chiral algebras in purely chiral terms. Let Aα be a finite family of chiral algebras. Let A := ⊗Aα ∈ CA(X) be the corresponding tensor product, so A` = ⊗A`α . By 3.4.4 we have canonical morphisms of chiral algebras να : Aα → A, a 7→ a ⊗ ( ⊗ 1Aα0 ). α0 6=α

Let ϕα : Aα → C be morphisms of chiral algebras. We say that ϕα mutually commute if [ϕα0 , ϕα00 ]C ∈ P2∗ ({Aα0 , Aα00 }, C) vanishes for every α0 6= α00 . Denote ({Aα }, C) the set of all mutually commuting (ϕα )’s. by Homcom I Proposition. {να } is a universal family of mutually commuting morphisms: ({Aα }, C), ψ 7→ (ψνα ), is for every C ∈ CA(X) the map Hom(⊗Aα , C) → Homcom I bijective. Proof. Let us translate our proposition back to the setting of factorization algebras. Assume for simplicity of notation that we play with two algebras A1 , A2 . Notice that ϕ1 , ϕ2 as above mutually commute if and only if for every a1 ∈ A`1 , a2 ∈ ` ` lies in CX×X . One has ν1 (a1 )ν2 (a2 ) = A`2 the section ϕ1 (a1 )ϕ2 (a2 ) of j∗ j ∗ CX×X ` ` ` ` (a1  1A1 ) ⊗ (1A2  a2 ) ∈ A1X×X ⊗ A2X×X = (A1 ⊗ A2 )X×X , so our ν1,2 mutually commute. For ϕ1,2 ∈ Homcom ({A1 , A2 }, C) the Ith component of the corresponding I morphism ψ : A`1 ⊗ A`2 → C ` sends m1 ⊗ m2 , where m1 ∈ A`1X I , m2 ∈ A`2X I , to the 25 It suffices to check that the transposition of two elements of some J ⊂ J acts trivially. i By (∗)|J| it acts trivially already on the pull-back of TX J to the corresponding codimension 1 diagonal.

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pull-back of 26 ϕ1 (m1 )  ϕ2 (m2 ) by the diagonal map X I ,→ X I × X I . The details are left to the reader.  Remarks. (i) Let Nα ⊂ Aα be a sub-OX -module that generates Aα as a chiral algebra. Then ϕα0 , ϕα00 mutually commute if (and only if) their restrictions to the N ’s commute, i.e., the image of [ϕα0 , ϕα00 ]C in P2∗ ({Nα0 , Nα00 }, C) vanishes. (ii) Suppose that Aα is freely P generated P by (Nα , Pα ) (see 3.4.14). Then ⊗Aα is freely generated by (⊕Nα , ( Pα ) + Nα0  Nα00 ). α0 6=α00

Let M be a DX -module and µα ∈ P2ch ({Aα , M }, M ) are chiral Aα -actions on M . We say that the µα mutually commute if for every α 6= α0 the restriction of µα (idAα , µα0 ) − µα0 (idAα0 , µα ) ∈ P3ch ({Aα , Aα0 , M }, M ) to the localization of Aα  Aα0  M with respect to the diagonals x1 = x3 and x2 = x3 vanishes. Exercises. (i) Show that for every ⊗Aα -action on M the corresponding Aα actions mutually commute, and this provides 1-1 correspondence between the sets of ⊗Aα -module structures and collections of mutually commuting Aα -actions. (ii) Let Mα be Aα -modules. Then the Aα -actions on ⊗Mα mutually commute, so ⊗Mα is a ⊗Aα -module.27 3.4.16. Hopf algebras. A Hopf chiral algebra is a coassociative coalgebra object in the tensor category CA(X), i.e., a chiral algebra F equipped with a coassociative morphism of chiral algebras δ = δF : F → F ⊗ F . As a part of general tensor category story, we know what a counit F : F → ω is, and what it means for F to be cocommutative. We also know what a coaction of F on any chiral algebra A is (this is a morphism δ = δA : A → F ⊗ A satisfying a certain compatibility property with δF ), when such a coaction is counital and when it is trivial. For F , A as above set AF := {a ∈ A : δ(a) = 1F ⊗ a}. This is the maximal chiral subalgebra of A on which the coaction of F is trivial. We denote by CA(X)F the category of chiral algebras A equipped with a coacF tion of F ; let CA(X)F f l ⊂ CA(X) be the full subcategory of OX -flat algebras. If F F is counital, then CA(X)F cu ⊂ CA(X) is the full subcategory of those A that δA F F is counital and CA(X)cu,f l := CA(X)cu ∩ CA(X)F f l. The tensor product of Hopf chiral algebras is naturally a Hopf chiral algebra. If Fα are Hopf chiral algebras and Aα ∈ CA(X)Fα , then ⊗Aα ∈ CA(X)⊗Fα . Remark. If M , N are F -modules, then M ⊗ N is again an F -module via δF and Exercise (ii) in 3.4.15. Thus M(X, F ) is a monoidal category; it is a tensor category if F is cocommutative. Suppose F is an OX -flat cocommutative Hopf chiral algebra. Then CA(X)F fl F

is a tensor category. Namely, for Aα ∈ CA(X)F f l their tensor product ⊗ Aα is the chiral subalgebra of ⊗Aα that consists of those sections a of ⊗Aα that all δα (a) := (δAα ⊗ ( ⊗ idAα0 ))(a) ∈ F ⊗ (⊗Aα ) coincide. If F is counital, then α0 6=α

F CA(X)F cu,f l is a tensor subcategory of CA(X)f l , and F is naturally a unit object F F in CA(X)cu,f l . Invertible objects of CA(X)cu,f l are called F -cotorsors; they form a Picard groupoid P(F ) = P(X, F ). For examples see 3.7.12, 3.7.20, and 3.10. 26 This 27 This

` section belongs to CX I ×X I : use 3.4.4 and the fact that ϕ1,2 mutually commute. also follows from the interpretation of A-modules in factorization terms, see 3.4.19.

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3.4.17. Twists. In the following constructions we use only tensor products of an arbitrary chiral algebra and a commutative chiral algebra, so they can also be performed in the context of 3.3.10 (see Exercise in 3.4.20 below). Suppose that a Hopf chiral algebra F is commutative (as a chiral algebra). Such F is the same as a semigroup DX -scheme G = Spec F affine over X. We call the F -coaction on A a G-action and write AG := AF . If A itself is commutative, then a G-action on A is the same as a G-action on Spec A` in the category of DX -schemes, i.e., a scheme G-action compatible with connections. If G acts on two chiral algebras, then it acts naturally on their tensor product (via the diagonal embedding G ,→ G×G), so the category of chiral algebras equipped with a G-action is a tensor category. Suppose G is a group DX -scheme. Let P = Spec R` be a DX -scheme Gtorsor (i.e., a G-torsor equipped with a connection compatible with the G-action). For a chiral algebra A equipped with a G-action its P -twist is the chiral algebra A(P ) := (R ⊗ A)G . Remarks. (i) If G acts on A trivially, then A(P ) = A. (ii) As a plain D-module, A(P ) is the P -twist of the D-module A; in particular, ∼ any section s : X → P defines an isomorphism of O-modules is : A −→ A(P ). If s is horizontal, then is is an isomorphism of chiral algebras. (iii) Denote by G∇ the sheaf of horizontal sections of G; this is a sheaf of groups on Xe´t . Suppose that P admits locally a horizontal section; then the sheaf of horizontal sections P ∇ is a G∇ -torsor. Notice that G∇ acts on A, so we have the corresponding P ∇ -twisted chiral algebra A(P ∇ ). The isomorphisms is , s ∈ P ∇ , provide a canonical identification of chiral algebras (3.4.17.1)

∼

A(P ∇ ) −→ A(P ).

(iv) Let H be a group X-scheme, and suppose that G = JH (the group jet DX -scheme; see 2.3.2). There is a canonical equivalence of groupoids (3.4.17.2)

∼

{H-torsors} −→ {DX -scheme G-torsors}

which assigns to an H-torsor Q on X the DX -scheme G-torsor of jets P = JQ; its inverse is the push-out functor for the canonical homomorphism of the group X-schemes G = JH → H. Notice that the canonical projections G → H, P → Q identify the sheaves of horizontal sections of left-hand sides with those of arbitrary sections of right-hand sides, so, by (3.4.17.1), the P -twist can be interpreted as the plain twist by the torsor of sections of Q. (v) If G is commutative, A and B are two chiral algebras equipped with GF

actions, then the tensor product A ⊗ B from 3.4.16 coincides with invariants in A ⊗ B of the action of the anti-diagonal subgroup G ,→ G × G, g 7→ (g, g −1 ). 3.4.18. Factorization modules. Let us describe modules over chiral algee bras in factorization terms. Below for a finite set I we set Ie := I t·, so X I = X ×X I . Let B be a factorization algebra. A factorization B-module is a triple B(M ) = (B(M ), ν˜, c˜) where: (i) B(M ) is a rule that assigns to every finite set I a left D-module B(M )X Ie on X I × X. We demand that B(M )X Ie have no sections supported at the diagonal divisors. Set M := B(M )X .

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(ii) ν˜ assigns to every π : Je  Ie preserving the ·’s an identification (3.4.18.1)

∼

ν˜(π) : ∆(π)∗ B(M )X Je −→ B(M )X Ie.

We demand that the ν˜(π) ’s are compatible with the composition of the π’s. (iii) Consider a surjection Je  Ie preserving the ·’s. Then Je is the disjoint union of the preimage subsets Je· and Ji , i ∈ I. Our c˜ assigns to such a surjection 28 an isomorphism of DU [J/ e Ie] -modules (3.4.18.2)

∼

[J/I]∗ c˜[J/ (( BX Ji ))  B(M )X Je· ) −→ j [J/I]∗ B(M )X J . e I] e : j e e

e e

I

˜  Je the isomorphism We demand that the c˜’s be mutually compatible: for K c˜[K/ ˜[K/ ˜K˜ · /J˜· ) where c are the ˜ J] ˜ coincides with the composition c ˜ I] ˜ ((c[Ki /Ji ] )  c factorization isomorphisms for B. They should also be compatible with the isomorphisms ν˜ (and the isomorphisms ν for B) in the obvious way.29 Finally, for every m ∈ M the section c˜(1  m) ∈ j∗ j ∗ BX  M should belong to B(M )X×X and one should have ν˜((1  m)|∆ ) = m. e (I)

I As in 3.4.4, the above c˜[I/·] define an embedding B(M )X I˜ ,→ j∗ j (I)∗ (BX  e M ), so the functor B(M ) 7→ M is fully faithful. We call B(M ) a factorizatization B-module structure on the DX -module M . e

˜

Remark. Let ` : V ,→ X I be the complement to the diagonal strata of codi∼ mension 2. As in 3.4.3, one shows that MX I˜ −→ `∗ `∗ MX I˜ . 3.4.19. We denote the category of factorization B-modules by M` (X, B). This is an abelian k-category. As in 3.3.4, M(X, B r ) is the category of chiral B r -modules. Proposition. There is a canonical equivalence of categories (3.4.19.1)

∼

M` (X, B) −→ M(X, B r ).

Proof. Our functor assigns to B(M ) ∈ M` (X, B) the right DX -module M r equipped with a chiral operation µM ∈ P2ch ({B r , M r }, M r ) defined as the compo∼ ∼ sition j∗ j ∗ B r  M r −→ j∗ j ∗ B(M )rX×X → j∗ j ∗ B(M )rX×X /B(M )rX×X −→ ∆∗ M r where the first and the last arrow come from the appropriate isomorphisms c˜ and ν˜. As in 3.4.8, one checks immediately that µM is a B r -module structure on M r . Let us construct the functor in inverse direction. According to 3.3.5(i), for a chiral B r -module M r the DX -module B r ⊕ M r is naturally a chiral algebra; so F := B ⊕ M carries a factorization algebra structure. For a finite set I let ˜ ˜ (I) B(M )X I˜ ⊂ FX I˜ be the submodule of sections which belong to j∗ j (I)∗ B I  M ⊂ ˜ (I)

˜

j∗ j (I)∗ FX I˜ . The isomorphisms ν˜ and c˜ come from the corresponding isomorphisms of the factorization structure on F . It is clear that (B(M ), ν˜, c˜) is a factorization B-module structure on M . It is clear (use the remark in 3.4.18 and the construction of F in 3.4.12) that the above functors are mutually inverse, and we are done.  28 See 29 I.e.,

3.4.4 for notation. in the same way as the c are compatible with ν; see 3.4.4.

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` Remark. The definition of 3.4.19 makes sense if we take for an MX an I˜ OX  DX I -module instead of a DX I˜ -module. The above argument shows that the corresponding category is equivalent to MO (X, B r ) (see 3.3.5(ii)).

Let us describe chiral B r -operations (see 3.3.4) in factorization terms. Suppose we have a finite family of chiral B r -modules {Mir }, i ∈ I. Then, as in 3.3.5(i), B r ⊕ (⊕Mir ) is a chiral algebra; let G be the corresponding factorization algebra. For a finite set J let B({Mi })X JtI ⊂ GX JtI be the submodule of sections whose restriction to U (JtI) belongs to B J  (Mi ). Equivalently, B({Mi })X JtI (JtI) (JtI)∗ J is the submodule of j∗ j B  (Mi ) which consists of sections killed by ¯ ¯ ¯ (JtI) (JtI) (JtI)∗ J j (JtI)∗ B J  (Mi ) the morphisms µj,i : j∗ j B  (Mi ) → ∆j=i ∗ j∗ for all j ∈ J and i ∈ I; here J¯ := J r {j} and µj,i is the chiral B r -action on Mi (I) tensored by the identity map at the variables 6= j, i. So B({Mi })X I = j∗ j (I)∗ Mi , etc. This module satisfies the usual factorization properties (inherited from G). In particular, let j [J,I] : U [J,I] ,→ X JtI be the complement to all the diagonals xj = xi , (I) j ∈ J, i ∈ I; then j [J,I]∗ B({Mi })X JtI = j [J,I]∗ (BX J  j∗ j (I)∗  Mi ), so (3.4.19.2)

[J,I] [J,I]∗

B({Mi })X JtI ,→ j∗

j

(I)

(BX J  j∗ j (I)∗  Mi ).

Let N r be another B r -module and ϕ ∈ PIch ({Mir }, N r ) be a chiral operation. (I) (I) For any J set ϕJ := idBX J  ϕ : BX J  j∗ j (I)∗  Mi → BX J  ∆∗ N . Lemma. If ϕ is a chiral B r -operation, then for every finite set J the morphism [J,I] (I) ϕJ sends B({Mi })X JtI to (idX J ×∆(I) )∗ B(N )X J˜ ⊂ j∗ j [J,I]∗ BX J ∆∗ N . Conr versely, if this condition is satisfied for J = ·, then ϕ is a chiral B -operation.  3.4.20. We are going to describe the factorization algebra that corresponds to a commutative chiral algebra in geometric terms (see 3.4.22). We start with some technical preliminaries. A factorization algebra is said to be commutative if the corresponding chiral algebra is commutative. Proposition. (i) The next properties of a factorization algebra B are equivalent: (a) B is commutative; (b) for every J  I the isomorphism (3.4.4.1) extends to a morphism (3.4.20.1)

 BX Ji → BX J ; I

(c) same as (b) but in the particular case J = {1, 2}, I = {1}; i.e., the factor∼ ization isomorphism j ∗ (BX  BX ) −→ j ∗ BX 2 extends to a morphism (3.4.20.2)

BX  BX → BX 2 ;

(d) there exists a morphism of factorization algebras m : B ⊗ B → B whose restrictions to B ⊗1 ⊂ B ⊗B and 1⊗B ⊂ B ⊗B are the identity morphisms. (ii) The morphism m mentioned in (d) is unique: the corresponding morphism mX : BX ⊗BX → BX is the multiplication on BX corresponding to the commutative r chiral structure on BX = BX ⊗ωX . For every I the morphism mX I : BX I ⊗BX I → BX I defines a commutative DX I -algebra structure on BX I .

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(iii) If B is a commutative factorization algebra, then the multiplication morphism BX ⊗ BX → BX corresponding to the DX -algebra structure on BX is the restriction of (3.4.20.2) to the diagonal X ,→ X 2 . Proof. Clearly (b)⇒(c)⇔(a). To see that (c)⇒(b), use 3.4.3(ii). To prove that (d)⇒(c), define the morphism (3.4.20.2) to be the composition of f : BX  BX → BX 2 ⊗ BX 2 and mX 2 : BX 2 ⊗ BX 2 → BX 2 , where f (b1 ⊗ b2 ) := (b1  1) ⊗ (1  b2 ). Statement (iii) follows from the definitions. Now let us show that (a)⇒(d). If B is commutative, then B ⊗ B is commutative (use the equivalence (a)⇔(c)). Applying (iii) to B ⊗ B, we see that the multiplication in BX ⊗ BX = (B ⊗ B)X coming from the factorization structure on B ⊗ B is the usual tensor product multiplication (i.e., (b1 ⊗ b2 )(b3 ⊗ b4 ) = b1 b3 ⊗ b2 b4 ). So we have the DX -algebra morphism BX ⊗ BX → BX , b1 ⊗ b2 7→ b1 b2 . The corresponding morphism of factorization algebras m : B ⊗ B → B has the property mentioned in (d). To prove (ii) consider any m : B ⊗ B → B satisfying the property mentioned in (d). Then mX : BX ⊗ BX → BX is a DX -algebra morphism such that b ⊗ 1 7→ b and 1 ⊗ b 7→ b. So there is only one possibility for mX and therefore for m. The rest of (ii) is obvious.  Exercise. According to 3.3.10 (or, equivalently, by 3.1.3 and (3.3.3.1)) we know what the tensor product of any chiral algebra and a commutative chiral algebra is. Show that it coincides with the tensor product of these chiral algebras in the sense of 3.4.15. 3.4.21. If B is a commutative factorization algebra, then BX I is a commutative DX I -algebra for every I (see (ii) in the proposition in 3.4.20). By 3.4.9, BX I can be uniquely reconstructed from BX . We are going to explain the geometric meaning of the DX I -scheme Spec BX I in terms of the DX -scheme Y := Spec BX . We will show that Spec BX I is the scheme YX I of horizontal infinite multijets of Y . First we have to define YX I . A k-point of YX I is a pair consisting of a point ˆ x → Y , where X ˆ x is the formal x = (xi )i∈I ∈ X I and a horizontal section s : X completion of X at the subset {xi } ⊂ X. One defines R-points of YX I quite ˆ x is the formal completion of X ⊗ R similarly (if x = (xi )i∈I ∈ X I (R), then X along the union of the graphs of xi : Spec R → X). Therefore we have a functor {k-algebras}→{sets}, and we claim that it is representable by a scheme YX I affine over X I . It is enough to prove this if Y = Spec Sym(DS X ⊗ V ) where V is a vector space. In this case YX I = Spec Sym(L ⊗ V ), L := N (π∗ OX×X I /IN )∗ , where π : X × X I → X I is the projection to the second factor and I ⊂ OX×X I is the product of the ideals of the subschemes x = xi of X × X I , i ∈ I (we use the fact that the sheaves π∗ OX×X I /IN are locally free30 ). Remarks. (to be used in 3.4.22) (i) We have shown that if Y = Spec Sym(M ) and M is a free DX -module, then YX I is flat over X I . (ii) Suppose that a group scheme G over k acts on the DX -scheme Y . Then it acts on YX n . Let Y G ⊂ Y be the subscheme of fixed points of G. Applying the above multijet construction to Y G instead of Y , we get a DX n -scheme (Y G )X n . Then the morphism (Y G )X n → YX n induces an isomorphism (Y G )X n → (YX n )G (this follows immediately from the definitions). 30 This is not true if dim X > 1, so it is not clear if our multijet functor is representable in this case.

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(iii) The above construction has ´etale local origin with respect to YX , so it makes sense for an arbitrary (not necessary affine) algebraic DX -space YX . The X I -scheme YX I has a natural structure of the DX I -scheme. Indeed, the fibers of YX I over infinitely close R-points x, x ˜ of X I are naturally identified because ˆ x and X ˆ x˜ coincide (“infinitely close” means that the restrictions the completions X of x, x ˜ : Spec R → X to the reduced part of Spec R coincide). So YX I = Spec AX I for some DX I -algebra AX I . Notice that YX = Y and therefore AX = BX . We will show that AX I = BX I . As a first step, let us show that A has a structure of the quasi-factorization algebra (see 3.4.14). Recall that this structure consists of morphisms ν (π) and c[π] of (3.4.5.1) and (3.4.5.2) subject to axioms (a)–(d) and (f) from 3.4.5. Suppose we have π : J → I. It yields a map ∆(π) : X I → X J and a DX I morphism YX I → YX J ×X J X I (restricting a horizontal section of Y over a formal neighbourhood of a finite subscheme to a formal neighbourhood of a smaller subscheme). We get the morphism ν (π) of (3.4.5.1). We also have a similarly defined Q DX J -morphism YX J → YX Ji which is an isomorphism over U [π] , and therefore I

the isomorphism c[π] of (3.4.5.1). The axioms (a)–(d) and (f) from 3.4.5 are evident, so A is a quasi-factorization algebra. 3.4.22. Theorem. There is a (unique) isomorphism of quasi-factorization al∼ ∼ gebras A −→ B such that the corresponding isomorphism AX −→ BX is the identity. Proof. (i) If we know that A is a factorization algebra (i.e., that AX I has no non-zero sections supported at the diagonal divisor), then A = B. Indeed, A is equipped with a morphism m : A ⊗ A → A satisfying the conditions (i)(d) of the proposition in 3.4.20 and such that mX : AX ⊗ AX → AX is the multiplication in ∼ BX = AX . So, by (ii) in the proposition in 3.4.20, the identity map AX −→ BX is an isomorphism of DX -algebras. Therefore it extends in a unique way to an ∼ isomorphism A −→ B of factorization algebras. (ii) If BX is a free commutative DX -algebra, then AX I has no non-zero sections supported at the diagonal divisor (see Remark (i) from 3.4.21) and therefore A = B. (iii) Everything said above in 3.4.20–3.4.22 renders itself to the super setting and to the DG setting. So if BX is a free commutative differential graded DX algebra, then A = B (as usual, “free” means “free as a Z-graded commutative super DX -algebra”). (iv) The statement of the theorem is local, so we can assume that X is affine. Then BX has a free DG resolution; i.e., there is a free commutative differential ˜X placed in degrees ≤ 0 equipped with a quasi-isomorphism graded DX -algebra B ˜ BX → BX (we consider BX as a DG algebra placed in degree 0). Applying the ˜X extends to a factorization DG DG version of the theorem in 3.4.9, we see that B ˜ ˜ ˜ algebra B = {BX I }I∈S . Applying to BX the multijet construction from 3.4.21, we get a quasi-factorization DG algebra A˜ equipped with a morphism A˜ → A. ˜ For every I the morphism B ˜X I → BX I is a quasiAs explained in (iii), A˜ = B. isomorphism (see Remark (ii) in 3.4.4). So it remains to show that the morphism (3.4.22.1)

H 0 (A˜X I ) → AX I

is an isomorphism. Indeed, according to 1.1.16 a DG algebra is the same as a super algebra with an action of the semi-direct product G of Gm and the super group H

3.4. FACTORIZATION

193

with Lie algebra k[−1] (k[−1] is odd and λ ∈ Gm acts on k[−1] as multiplication by λ); the action of G defines the grading and the action of H defines the differential. Since AX I is placed in non-positive degrees, Spec H 0 (A˜X I ) = (Spec A˜X I )G . So the super version of Remark (ii) from 3.4.21 shows that (3.4.22.1) is indeed an isomorphism.  ` ` 3.4.23. Let MX be a left DX -module. Then BX := Sym MX is a commutative DX -algebra. We will give an explicit description of the corresponding factorization algebra B = {BX I }I∈S . Let ZI ⊂ X × X I be the union of the subschemes x = xi , i ∈ I (notice that ZI is singular if |I| > 1). We have the projections πI : ZI → X and pI : Z → X I . Put

(3.4.23.1)

MX I := pI∗ πI† MX = (pI )∗ πI! MX [1 − n],

n := |I|.

Here we use the notion of a D-module on a singular scheme (see Remark in 2.1.3) and notation from 2.1.4. Notice that since ZI naturally appears as a subscheme of the smooth scheme X × X I , one can easily reformulate the definition of MX I using only D-modules on smooth schemes. Namely, one has (3.4.23.2)

MX I = p∗ (j∗ j ∗ (MX  ωX I )/(MX  ωX I ))

where j : X × X I r ZI ,→ X × X I and p : X × X I → X I is the projection. According to the proposition in 2.1.4, πI† MX is a D-module (not merely a complex of D-modules). Since πI is finite, MX I is also a D-module. On the complement U (I) of the diagonal divisor of X I our MX I is canonically isomorphic P † to pri MX , where pri : X I → X is the ith projection. So on U (I) we have a ` I canonical isomorphism between Sym MX I and BX . Proposition. This isomorphism comes from a unique isomorphism of DX I modules (3.4.23.3)

` BX I = Sym (MX I ).

Proof. We use notation from 3.4.5. For any f : J → I we have ∆(f ) : X I → X J . Then ∆(f )! MX J = (pI )∗ α∗ α! πI! MX [1 − n], n := |J|, where α is the embedding 0 ZI 0 × X I\I ,→ X I and I 0 := f (I). So we get a canonical morphism (3.4.23.4)

` ` ν (f ) : ∆(f )∗ MX J → MX I .

` ` J We also have a canonical morphism Σpri∗ MX → X Ji is Ji → MX J where pri : X [f ] the projection. Its restriction to U (see 3.4.5) is an isomorphism X ∼ ` ` (3.4.23.5) c[f ] : j [f ]∗ pr∗i MX → j [f ]∗ MX Ji − J.

˜X I := Sym M ` I . The (3.4.23.4) and (3.4.23.5) yield morphisms (3.4.5.1) Set B X ˜ which evidently satisfy properties (a)–(f) in 3.4.5.31 So B ˜ and (3.4.5.2) for B is a commutative factorization algebra. By (ii) in the proposition in 3.4.20 the ` corresponding commutative DX -algebra is BX := Sym MX equipped with the usual multiplication.  31 Axiom (b) holds since (3.4.23.4) for surjective f is an isomorphism; axiom (e) holds since ` MX I has no non-zero sections supported at the diagonal divisor.

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3.4.24. Let MX be a DX -module, MX I the DX I -module defined by (3.4.23.1). ` ` Consider the DX -scheme Y := V(MX ) := Spec Sym(MX ) and denote by YX I the corresponding multijet DX I -scheme defined in 3.4.21. ` Proposition. One has YX I = V(MX More precisely, the canonical isoI ). ∼ (I) I (I) ∼ ` (I) morphism YX I ×X I U −→ (YX ) ×X I U −→ V(MX over the comI ) ×X I U (I) I plement U ⊂ X of the diagonal divisor extends (uniquely) to an isomorphism ∼ ` YX I −→ V(MX I ).

Proof. This is an immediate consequence of 3.4.22 and 3.4.23. Here is a direct proof. An R-point of YX I is a pair consisting of x = (xi ) ∈ X(R)I and an R ⊗ DX ` linear morphism R ⊗ MX → lim ←− OX⊗R /OX⊗R (−nDx ), where Dx is the sum of the graphs of xi : Spec R → X considered as divisors on X ⊗ R. Put Lx := (jx )∗ jx∗ (ωX ⊗ R)/(ωX ⊗ R), where jx is the embedding (X ⊗ R) \ Dx ,→ X ⊗ R. ` An OX ⊗ R-morphism f : R ⊗ MX → lim ←− OX⊗R /OX⊗R (−nDx ) is the same as ` an R-morphism ϕ : (pR )· (Lx ⊗OX MX ) → R where pR : X ⊗ R → Spec R is the projection and (pR )· is the sheaf-theoretic direct image (ϕ is the composition ` of the morphism (pR )· (Lx ⊗OX MX ) → (pR )· Lx induced by f and the “sum of ` residues” morphism (pR )· Lx → R). An OX ⊗ R-linear morphism f : R ⊗ MX → lim O /O (−nD ) is D ⊗ R-linear if and only if the corresponding ϕ : X⊗R X⊗R x X ←− ` ` ). ) → R factors through (pR )· (Lx ⊗DX MX (pR )· (Lx ⊗OX MX So R-points of YX I bijectively correspond to R-morphisms Px → R, where ` Px := (pR )· (Lx ⊗DX MX ); i.e., (3.4.24.1)

Px = (pR )· (ωX ⊗DX Nx` ),

` ` Nx` := jx∗ jx∗ (MX ⊗ R)/(MX ⊗ R).

` I On the other hand, an R-point of V(MX I ) is a pair consisting of x = (xi ) ∈ X(R) and an R-linear morphism Mx` → R where Mx` is the x-pull-back of the OX I -module ` ` MX I to Spec R. We claim that Mx = Px . To show this, notice that Dx is the xpull-back of the closed subscheme ZI ⊂ X ×X I from 3.4.23. According to (3.4.23.2) one has ` ` MX = p· (ωX ⊗DX N ` ), I = p∗ N

` ` N ` := j∗ j ∗ (MX  OX I )/(MX  OX I ).

` ` Comparing this description of MX I with (3.4.24.1), we see that Mx = Px . ∼ ` So we have constructed an X-isomorphism YX I −→ V(MX I ). It is easy to see that it is a DX I -isomorphism and that it induces the required isomorphism over the complement of the diagonal divisor. 

3.5. Operator product expansions In this section we identify a chiral algebra structure on a D-module with a commutative and associative ope product. Here ope stands for “operator product expansion”. This shows that our chiral algebras are related to their namesake from mathematical physics and - in the translation equivariant setting - amount to vertex algebras (see 0.15). This approach uses non-quasi-coherent D-modules (referred to as “D-sheaves” below), which requires some care. The ope set-up is convenient for writing formulas and can be considered as a direct generalization of the notion of the DX -algebra (see 2.3). It is also very convenient in the (g, K)-modules setting (see 3.5.15).

3.5. OPERATOR PRODUCT EXPANSIONS

195

3.5.1. Let us begin with some preliminary considerations. For a smooth variety P a DP -sheaf is any (not necessarily quasi-coherent) sheaf of left D-modules on Pe´t . ¯ ` (P ), so M` (P ) is a full subcategory of We denote the category of DP -sheaves by M ` ¯ ¯ ` (P ) a structure of the tensor category. M (P ). The OP -tensor product defines on M Let i : Z ,→ P be a closed smooth subvariety and I ⊂ OP the corresponding ideal. If F is a DP -sheaf, then i∗ F := i· (F/IF ) is a DZ -sheaf in the usual way. ¯ ` (P ) → M ¯ ` (Z) admits a right adjoint 3.5.2. Lemma. (i) The functor i∗ : M ` ` ¯ ¯ ¯ ` (Z) functor iˆ∗ : M (Z) → M (P ) which is exact and fully faithful. It identifies M n with the full subcategory of those DP -sheaves F for which F = ← lim − F/I F . ¯ ` (Z) → (ii) For a DZ -sheaf G set i∗ G := (i∗ ωZ )` ⊗ iˆ∗ G.32 The functor i∗ : M ` ∗ ¯ M (P ) is exact, fully faithful, and right inverse to i . A DP -sheaf belongs to its image if and only if everyone of its local sections is killed by some power of I. (iii) The above i∗ coincides on M` (Z) with the direct image functor from 2.1.3: for N ∈ M(Z) one has i∗ (N ` ) = (i∗ N )` . Sketch of a proof. (i) The existence of iˆ∗ follows from right exactness of i∗ , as well as the fact that iˆ∗ has the local nature. One can describe iˆ∗ G explicitly: for an open U one has Γ(U, iˆ∗ G) = Hom(i∗ DU , GU ) (the morphisms of DU ∩Z -sheaves). This easily implies all the assertions in (i) and also (ii) and (iii) (use Kashiwara’s lemma; see 2.1.3).  Remark. The functor i∗ commutes with tensor products. Therefore iˆ∗ is naturally a pseudo-tensor functor adjoint to i∗ as a pseudo-tensor functor (see 1.1.5). ¯ ` (P ) one has the identification Hom(⊗i∗ Fi , G) = Hom(i∗ ⊗ Namely, for Fi ∈ M Fi , G) = Hom(⊗Fi , iˆ∗ G). So if A is a DZ -algebra, then iˆ∗ A is a DP -algebra, and if N is an A-module, then iˆ∗ N and i∗ N are iˆ∗ A-modules. 3.5.3. Suppose Z is a divisor; let j : U := P r Z ,→ P be the complementary open embedding. For a DZ -sheaf G set (3.5.3.1)

¯ ` (P ). i˜∗ G := (iˆ∗ G) ⊗ j∗ OU ∈ M

It is easy to see that the canonical map iˆ∗ G → i˜∗ G is injective. Since (i∗ ωZ )` = j∗ OU /OU , we get a canonical exact sequence (3.5.3.2)

0 → iˆ∗ G → i˜∗ G → i∗ G → 0.

Remark. Since · ⊗ j∗ OU is a tensor functor, the remark in 3.5.2 shows that i˜∗ is a pseudo-tensor functor. 3.5.4. Lemma. (i) The projection i˜∗ G → i∗ G is a universal morphism from a DP -sheaf which is a j∗ OU -module to i∗ G. The same is true for OP -module morphisms. (ii) For any DP -sheaf F one has (3.5.4.1)

∼

∼

Hom(F, i˜∗ G) ←− Hom(F ⊗ j∗ OU , i˜∗ G) −→ Hom(F ⊗ j∗ OU , i∗ G).

Proof. (ii) follows from (i), and (i) essentially says that one recovers i˜∗ G from i∗ G by the “Tate module” construction. Namely, for a DP - (or OP -) sheaf Q which is a j∗ OU -module one lifts a morphism α : Q → i∗ G to α ˜ : Q → i˜∗ G by the formula n −n α ˜ (f ) = lim q α(q f ). Here f ∈ F ⊗ j O , q ∈ O an equation of Z, so q n α(q −n f ) ∗ U P ←− n is a well-defined element of i˜∗ G/I iˆ∗ G.  32 Here

(i∗ ωZ )` ∈ M` (P ); see 2.1.1 and 2.1.3.

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Remark. We also have the canonical isomorphisms (3.5.4.2)

∼

∼

Hom(F, i˜∗ G) ←− lim Hom(Fξ , iˆ∗ G) −→ lim Hom(i∗ Fξ , G), −→

−→

where Fξ runs the set of DP -subsheaves of F ⊗j∗ OU such that Fξ ·j∗ OU = F ⊗j∗ OU . The first arrow assigns to Fξ → iˆ∗ G its tensor product with j∗ OU ; the second arrow is the adjunction isomorphism. The inverse map to the first arrow sends ϕ : F ⊗ j∗ OU → i˜∗ G to ϕξ : Fξ → iˆ∗ G where Fξ := ϕ−1 (iˆ∗ G), ϕξ is the restriction of ϕ to Fξ . Notice that if both F and G are quasi-coherent, then, as follows from the above lemma, we may assume in (3.5.4.2) that all the Fξ ’s are quasi-coherent. 3.5.5. Let us return to our situation, so X is a curve. For a finite non-empty (I) ˆ (I) I and a DX -sheaf G set ∆ ˆ∗ G and ∗ G := ∆ (3.5.5.1)

(I) I ¯ ˜ (I) ˆ (I) ∆ ∗ G := (∆∗ G) ⊗ j∗ OU (I) ∈ M(X ).

ˆ (I) ˜ (I) ˆ (I) The obvious morphism ∆ ∗ G → ∆∗ G is injective. Note that ∆∗ G has the Ir{0} following explicit description. For 0 ∈ I consider a DX I -sheaf pr0∗ G := GOX . (I) n ∗ (I) ∗ ∗ n ˆ lim Then ∆ ˇpr0 G = G; hence ∆∗ G = ← lim − OX I /I∆ ⊗ G where − pr0 G/I∆ pr0 G = ← OX

I∆ ⊂ OX I is the ideal of the diagonal X ⊂ X I .

ˆ ∗(I) G = Example. Let t be a local coordinate on X, I = {1, 2}. Then ∆ ˜ (I) G1 [[t1 − t2 ]] = G2 [[t1 − t2 ]] and ∆ ∗ G = G1 ((t1 − t2 )) = G2 ((t1 − t2 )) where G1 , resp. G2 , is a copy of G considered as a D-sheaf along the first, resp. second, variable in X × X. More generally, for S ∈ Q(I) let ∆(I/S) : X S ,→ X I be the diagonal embedding and j (I/S) : U (I/S) ,→ X I the complement to the diagonals containing X S . For S ¯ F ∈ M(X ) set (3.5.5.2)

(I/S) I ¯ ˜ (I/S) ˆ (I/S) ∆ F := (∆ F ) ⊗ j∗ OU (I/S) ∈ M(X ). ∗ ∗

(S) (I) ˜ (I/S) Note that if F is a j∗ OU (S) -module, then ∆ F is a j∗ OU (I) -module. ∗ ¯ For an interval I = {I1 < · · · < In = I} ⊂ Q(I) and G ∈ M(X) set

(3.5.5.3)

1) I ¯ ˜ (I) ˜ (In /In−1 ) . · · · ∆ ˜ ∗(I2 /I1 ) ∆ ˜ (I ∆ ∗ G := ∆∗ ∗ G ∈ M(X ). 0

(I) ) ˜ (I ˜ (I) This is a j∗ OU (I) -module. For I0 ⊂ I one has ∆ ∗ G ⊂ ∆∗ G.

˜ ({I}) ˜ (I) Examples. One has ∆ G=∆ ∗ ∗ G. If I = {1, · · · , n} and Ia is its quotient ˜ ∗(I) G = Gi ((t1 −t2 )) · · · ((tn−1 −tn )). modulo the relation a = a+1 = · · · = n, then ∆ Here i is any element in [1, n] and Gi is a copy of G considered as a D-sheaf along the ith variable in X n . 3.5.6. Suppose I is a maximal interval; i.e., |Ia | = a+1. We have a natural pro(I) (I/Ik ) ˜ (I≤k ) ˜ (I) ∆∗ G→ jection πI : ∆ ∗ G → ∆∗ G defined as the composition of maps ∆∗ (I/Ik−1 ) ˜ (I≤k−1 ) (I /I ) (I k k−1 k /Ik−1 ) ˜ G coming from the projection ∆∗ → ∆∗ . It is clear ∆∗ ∆∗ (I) (I) ˜ (I) that any non-trivial j∗ OU (I) -submodule of ∆ ∗ G projects non-trivially to ∆∗ G, ∼ (I) (I) (I) ˜ ∗ G) ← Hom(N ⊗ j∗ OU (I) , ∆ ˜ ∗ G) ,→ so for any DX I -sheaf N we have Hom(N, ∆

3.5. OPERATOR PRODUCT EXPANSIONS (I)

197

(I)

Hom(N ⊗ j∗ OU (I) , ∆∗ G) where the right arrow comes from πI . In particular, for any Li , M ∈ M(X) one has a canonical embedding (here λI comes from I ω X I ⊗ λI = ω X , see 3.1.4, and we use 3.5.2(iii)) ch ˜ (I) Hom(L`i , ∆ ∗ M ) ,→ PI ({Li }, M ) ⊗ λI .

(3.5.6.1)

3.5.7. We will need a technical lemma. Consider the Cousin complex for ˆ ∗(I) G)ω −1I [−|I|], we get a comC(ω)X I from the proof in 3.1.5. Tensoring it by (∆ X (I/T ) ˜ (T ) plex CI (G) of DX I -sheaves with terms CI (G)m := ⊕ ∆∗ ∆∗ G (in T ∈Q(I,|I|−m)

˜ (I) ˆ (I) particular, CI (G)0 = ∆ ∗ G) which is a resolution of ∆∗ G. (I) I ¯ Lemma. Suppose that N ∈ M(X ) is a j∗ OU (I) -module. Then the sequence

0 → Hom(N, CI (G)0 ) → Hom(N, CI (G)1 ) → Hom(N, CI (G)2 ) is exact. Proof. The statement is X-local, so let us choose an equation q = 0 of the n ˆ (I) ˜ (I) ˜ (I) diagonal divisor; then ∆ lim ∗ G=← − ∆∗ G/q ∆∗ G. Now use the fact that for any −n 0

1

d d 0 q n ˆ (I) ˜ (I) −−−→ CI (G)1 −→ CI (G)2 is exact.  n the sequence 0 → ∆ ∗ G/q ∆∗ G → CI −

3.5.8. For an I-family of DX -sheaves set (3.5.8.1)

(I) ˜ (I) ˜ (I) OI ({Fi }, G) := Hom(Fi , ∆ ∗ G) = Hom((Fi ) ⊗ j∗ OU (I) , ∆∗ G).

This is the vector space of I-ope’s. They have a local nature with respect to the ´etale topology of X. For example, in the notation of Example in 3.5.5, a binary ope ◦ : F  F 0 → P (I) ˜ ∆∗ G is a series f, f 0 7→ (t1 − t2 )i (f ◦ f 0 )2 where f ◦ f 0 ∈ G and (f ◦ f 0 )2 is the i

i

i

corresponding section of G2 . Ope’s are not operations in the sense of 1.1 for the composition of ope’s need not be an ope: it belongs to a larger vector space. Namely, for J  I and a J-family {Ej } of DX -sheaves the ope composition map (3.5.8.2)

˜ ({I,J}) OI ({Fi }, G) ⊗ (⊗ OJi ({Ej }, Fi )) −→ Hom(Ej , ∆ G) ∗ I

δi i) ˜ (J sends γ ⊗ (⊗δi ) to γ(δi ) defined as the composition  Ej −−→ ∆ ∗ Fi −→ J

˜ (J/I) ∆ (Fi ) ∗

˜ (J/I) ∆ ((Fi ) ∗

˜ (J/I) (γ) ∆ (I) j∗ OU (I) ) −−∗−−−−→

I

˜ ({I,J}) ∆ G. ∗

−→ ⊗ ˜ (J) We say that ope’s (γ, {δi }) compose nicely if γ(δi ) takes values in ∆ ∗ G ⊂ ({I,J}) ˜∗ ∆ G, so γ(δi ) ∈ OJ ({Ej }, G). (I)

ˆ ∗ G) = Hom(⊗Fi , G) (see Remark. The canonical identification Hom(Fi , ∆ (I) (I) ˆ ˜ 3.5.2(i)) and the embedding ∆∗ G ,→ ∆∗ G yield an embedding (3.5.8.3)

Hom(⊗Fi , G) ⊂ OI ({Fi }, G).

Ope’s from the left-hand side always compose nicely, and their ope composition coincides with the composition in M(X)! . The composition of ope’s is associative in the following sense. Let K  J be another surjection. For a K-family {Ck } of DX -sheaves and ope’s ε ∈ OKj ({Ck }, Ej )

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˜ ({I,J,K}) the double composition morphisms (γ(δi ))(εj ), γ(δi (εj )) ∈ Hom(Ck , ∆ G) ∗ are defined, respectively, as compositions εj

(Kj )

˜∗ Ck −−→ ∆

(K/J)

˜∗ Ej −→ ∆

˜ (K/J) (γ(δi )) ∆

({I,J,K})

∗ ˜∗ (Ej ) −−− −−−−−−→ ∆

G,

˜ (K/J) ˜ (J/I)

∆∗ ∆ (γ) δi (εj ) i ,Ki }) ˜ ({J ˜ (K/J) ˜ ({I,J,K}) ˜ (J/I) Ck −−−−−→ ∆ Fi −→ ∆ −−−−∗−−−−→ ∆ G. ∆ (Fi ) −−− ∗ ∗ ∗ ∗

The associativity property says that they coincide. ˜ ∗ G. We 3.5.9. Let ◦ ∈ O2 ({G, G}, G) be a binary ope, a  b 7→ a ◦ b ∈ ∆ say that ◦ is associative if both (◦, {◦, idG }), (◦, {idG , ◦}) compose nicely and the composition ope’s ∈ O3 ({G, G, G}, G) coincide, and we say ◦ is commutative if it is fixed by an obvious “transposition of coordinates” symmetry of O2 ({G, G}, G). We call ◦ an ope algebra product on G and (G, ◦) an ope algebra. A unit for ◦ is a horizontal section 1 ∈ G such that for every a ∈ G one has ˆ ∗G ⊂ ∆ ˜ ∗ G and modulo I∆ ∆ ˆ ∗ G both a ◦ 1 and 1 ◦ a equal a ∈ G. A a ◦ 1, 1 ◦ a ∈ ∆ unit is unique, if it exists. Remarks. (i) For ◦ ∈ Hom(G ⊗ G, G) ⊂ O2 ({G, G}, G) (see (3.5.8.3)) the above notions of associativity and commutativity are the same as the usual notions in M(X)! . (ii) If ◦ is associative and commutative, then the triple product ◦3 := ◦({◦, idG }) is fixed by the action of the symmetric group of three variables. 3.5.10. According to (3.5.4.1) the canonical embedding (3.5.6.1) is an isomorphism for |I| = 2, so for any A ∈ M(X) one has a canonical bijection (3.5.10.1)

∼

P2ch ({A, A}, A) ⊗ λ{1,2} −→ O2 ({A` , A` }, A` ),

µ 7→ ◦µ .

Now we can state the main result of this section: Theorem. The above bijection identifies the set of chiral Lie brackets with the set of commutative and associative ope algebra products. A horizontal section 1 ∈ A` is a unit for µ if and only if it is a unit for ◦µ . Proof. We will prove the first statement; the second one is left to the reader. Since (3.5.10.1) commutes with transposition of coordinates, skew-commutative µ correspond to commutative ◦µ . Take such µ. Let us show that the Jacobi identity for µ amounts to the associativity of ◦ = ◦µ . (I) Set R := j∗ j (I)∗ (A` I ). By 3.5.4 for every S ∈ Q(I, |I| − 1) we have isomorphisms ∼ ∼ (I/S) ˜ (S) ` ` ˜ (S) ˜ ∗({S,I}) A` ) ←− ˜ (I/S) ∆∗ A ). Hom(R, ∆ ∆ Hom(A` I , ∆ ∗ ∗ A ) −→ Hom(R, ∆∗ ` ` I ˜ (I) ` ˜ (I) Since Hom(R, ∆ , ∆∗ A ), the exact sequence from 3.5.7 ∗ A ) = Hom(A looks like 0

d ` ˜ (I) 0 → Hom(A` I , ∆ ∗ A ) −→

˜ ({S,I}) Hom(A` I , ∆ A` ) ∗

⊕ S∈Q(I,|I|−1)

d1

−→

⊕ T ∈Q(I,|I|−2)

(I/T )

Hom(R, ∆∗

The symmetric group of I indices acts on it in the obvious way.

) ` ˜ (T ∆ ∗ A ).

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199

Now suppose that I = {1, 2, 3}. The third vector space in the above exact (3) (3) sequence is Hom(j∗ j (3)∗ (A` 3 ), ∆∗ A` ) = P3ch ({A, A, A}, A) ⊗ λ{1,2,3} (use the 3 identification ωX = ωX 3 ⊗λ{1,2,3} ). Our ◦ is commutative, so for every S ∈ Q(I, 2) ˜ ({S,I}) we have the corresponding iterated product ¯◦S ∈ Hom(A` 3 , ∆ A` ). For ∗ example, if S identifies 1, 2 ∈ I, then ¯ ◦S = ◦(◦, id). The element ¯◦3 := Σ¯◦S of the middle term of the above exact sequence is invariant with respect to the action of the symmetric group. As follows from the construction, one has d1 (◦(◦, id)) = µ(µ, idA ). Therefore d1 (¯ ◦3 ) is exactly the sum of terms of the Jacobi identity, so d1 (¯◦3 ) = 0 if and only if µ is a Lie bracket. On the other hand, by the very definition, ◦ is associative if and only if ¯ ◦3 lies in the image of d0 . Since our sequence is exact, we are done.  3.5.11. From the point of view of factorization algebras, for a chiral algebra A the corresponding ope ◦A := ◦µA is the glueing datum for A`X×X . Namely, for a, b ∈ A` the ope a ◦A b is the restriction of a  b, considered as a section of A`X 2 on the complement to the diagonal, to the punctured formal neighbourhood of the diagonal. This implies, in particular, that ◦A is compatible with the tensor product of ˜ ∗ A` ) → ∆ ˜ ∗ (⊗A` ) (see the chiral algebras (see 3.4.15). Namely, the morphism ⊗(∆ α α remark in 3.5.3) sends ⊗◦Aα to ◦⊗Aα . 3.5.12. Lemma. Let (A` , ◦) be a commutative and associative ope algebra. Then for any finite non-empty I one has a canonical I-fold ope product operation ◦I ∈ OI ({A` }, A` ). These operations are uniquely determined by the following properties: (i) If |I| = 1, then ◦I = idA` ; if |I| = 2, then ◦I = ◦. (ii) For every J  I the operations (◦I , {◦Ji }) compose nicely; the composition is ◦J . Proof. We define ◦I by induction. We know ◦I if |I| ≤ 3. Now let I be a finite set of order n > 3, and assume that we know ◦I 0 for |I 0 | < n so that (i), (ii) hold. Consider the exact sequence from the proof in 3.5.10. For S ∈ Q(I, n − 1) ˜ (I,S) A` ). Here a, b ∈ I are the two set ¯◦S := ◦S (◦{a,b} , idA`i )i6=a,b ∈ Hom(A`I , ∆ ˜ (I,S) A` ). distinct S-equivalent elements. Set ¯ ◦I := Σ¯◦S ∈ ⊕ Hom(A`I , ∆ S∈Q(I,n−1)

The associativity of composition (see 3.5.8) and the existence of ◦3 imply that d1 (¯◦I ) = 0. Hence ¯ ◦I = d0 (◦I ) for certain ◦I uniquely determined by this condition. Property (ii) follows from associativity of composition.  Remark. Let us describe ◦I from the point of view of factorization algebras. Consider a factorization algebra that corresponds to our ope algebra. In particular, we have a DX I -module A`X I whose pull-back to the diagonal equals A` . Thus its ˆ ∗ A` (see 3.5.2 and 3.5.5). Now ◦I is the formal completion at the diagonal equals ∆ (I) (I)∗ `I ∼ (I) (I)∗ ` `I ˜ ∗(I) A` . composition A → j∗ j A −→ j∗ j AX I → ∆ 3.5.13. Let us translate some of the definitions from 3.3 in the ope language. Let (A, µ) be a chiral algebra, and let (A` , ◦) be the corresponding ope algebra. By definition, the Lie∗ bracket [ ]µ : A  A → ∆∗ A coincides with the compo◦ ˜ ` ` sition A`  A` − →∆ ∗ A → ∆∗ A tensored by idωX 2 ; hence [ ]µ is the singular part of ◦. Therefore our chiral algebra is commutative if and only if ◦ takes values in

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ˆ ∗ A` ⊂ ∆ ˜ ∗ A` ; i.e., ◦ ∈ Hom(A`⊗2 , A` ) ⊂ O2 ({A` , A` }, A` ) (see (3.5.8.3)). Notice ∆ that Remark (i) from 3.5.9 provides another proof of (3.3.2.1), (3.3.3.1). 3.5.14. If (G, ◦) is an ope algebra, then a G-module is a DX -sheaf N together with an ope pairing ◦N ∈ O2 ({G, N }, N ) such that both ◦N (◦, idN ) and ◦N (idG , ◦N ) compose nicely and the compositions ∈ O3 ({G, G, N }, N ) coincide. If G has unit, then one defines a unital G-module in the obvious manner. Let A be a chiral algebra as above, M an A-module, so we have the chiral action µM ∈ P2ch ({A, M }, M ). Let ◦M ∈ O2 ({A` , M ` }, M ` ) be the corresponding ope pairing (see 3.5.10). This is an (A` , ◦)-module structure on M . This identifies the category of (A, µ)-modules with that of (quasi-coherent) (A` , ◦)-modules. This follows directly from 3.5.10 (use 3.3.5(i)). As in 3.5.8 one can write ◦M P in terms of a local coordinate t as a morphism A`  M → M2 ((t1 − t2 )), a, m 7→ (t1 − t2 )i (a ◦ m)2 . For fixed a the map M → i

M2 ((t1 − t2 )), m 7→ a ◦M m, is the vertex operator corresponding to a. 3.5.15. The ope approach is quite convenient in the setting of (g, K)-modules (see 3.1.16; we follow the notation of loc. cit. and 2.9.7). Namely, for Mi N ∈ ˜ (I) ˜ (I) M(g, K) one defines OI ({Mi }, N ) as HomGI (⊗Mi , ∆ ∗ N ) where we set ∆∗ N := ˆ (I) F˜ (I) ⊗ ∆ ∗ N . With this definition the results of this section (in particular, the F (I)

theorem in 3.5.10) remain valid for (g, K)-modules. For g = k an ope algebra is the same as a vertex algebra in the sense of [B1] or [K] 1.3. An ope algebra for G = Aut k[[t]] (see 2.9.9) is the same as a quasi-conformal vertex algebra in the sense of [FBZ] 5.2.4. The functors defined by (g, K)-structures transform ope algebras to ope algebras; this is the same as the old transformation of chiral algebras (see 3.3.14). In particular, quasi-conformal vertex algebras define universal chiral algebras on any curve, which is [FBZ] 18.3.3. 3.6. From chiral algebras to associative algebras In this section we consider some functors which assign to a chiral algebra A on X certain associative “algebras of observables.” In a local situation, fixing a point x ∈ X, we define two associative (closely related) algebras: a topological algebra Aas x which governs the category of A-modules supported at x and a filtered Wick algebra Aw x that encodes the standard rules of thumb for manipulating vertex operators (e.g., the expression of the coefficients of the operator product expansion as normally ordered products). If A is commutative, then Aas x is the algebra of functions on the space of horizontal sections of Spec A` over the formal punctured disc at x. The above associative algebras are omnipresent in the vertex algebra literature as algebras of operators acting on concrete modules. The reader who wants to acquire some computational skills is advised to look into [K], [FBZ]. ~ of topological vector In 3.6.1 we introduce “normally ordered” tensor product ⊗ spaces, which is a general monoidal category context for the topological associative algebras we will consider. The topological algebra Aas x is studied in 3.6.2–3.6.10. We define Aas in 3.6.2, describe it as a certain completion of h(jx∗ jx∗ A)x in 3.6.4–3.6.7, x check compatibility with tensor products in 3.6.8, and compare it with a construction from [FBZ] 4.1.4, 4.1.5 in 3.6.9–3.6.10. The Wick algebra Aw x is defined in 3.6.11; it is mapped into Aas in 3.6.12. The situation when x varies is considered in x

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3.6.13–3.6.14. We describe arbitrary A-modules as modules over a sheaf of topological associative DX -algebras Aas in 3.6.15–3.6.17. If A is commutative, then Aas x coincides with the same-noted algebra from 2.4. We look at the situation when x varies and show in 3.6.19 that the functor A 7→ Aas x preserves formally smooth and formally ´etale morphisms. Finally 3.6.20–3.6.21 contains some remarks about a global version of the Wick algebra. A natural problem is to assign (in the analytic setting) to any oriented loop γ on as X an associative algebra Aas γ ; the algebra Ax should correspond to the infinitely small loop around the punctured point x. The construction of the global Wick algebra has to do with this question. ~ . The reader can skip this subsection at the mo3.6.1. A digression on ⊗ ment, returning when necessary. Below, “topological k-vector space” means a k-vector space equipped with a complete and separated linear topology. The category of topological vector spaces ~ is a monoidal k-category with respect to the “normally ordered” tensor product ⊗ ~ ···⊗ ~ Vn is the completion defined as follows. For V1 , . . . , Vn the tensor product V1 ⊗ of the plain tensor product V1 ⊗ · · · ⊗ Vn with respect to a topology in which a vector subspace U is open if and only if for every i, 1 ≤ i ≤ n, and vectors vi+1 ∈ Vi+1 , . . . , vn ∈ Vn there exists an open subspace P ⊂ Vi such that U ⊃ ~ is associative but not V1 ⊗ · · · ⊗ Vi−1 ⊗ P ⊗ vi+1 ⊗ · · · ⊗ vn . The tensor product ⊗ commutative. The unit object is k. Examples. Consider k((t)) as a topological k-vector space equipped with the usual “t-adic” topology. Then for every topological k-vector space V there is ~ k((t)) = V ((t)). One has (EndV )((t)) ⊂ an obvious canonical identification V ⊗ ~ ~t End(V ⊗k((t))). Thus for every endomorphism g of V the endomorphism 1 − g ⊗ ~ ~ ~ of V ⊗k((t)) is invertible; if g is invertible, then so is g ⊗1 − 1⊗t. In particular, we see that (3.6.1.1)

~ ···⊗ ~ k((tn )) = k((t1 )) · · · ((tn )). k((t1 ))⊗

Any k((t))-vector space V is naturally a topological k-vector space: a subspace U ⊂ V is open if its intersection with every k((t))-line in V is “t-adically” open. Suppose that Vi , i = 1, . . . , n, are k((ti ))-vector spaces. Then (3.6.1.1) yields a canonical map (3.6.1.2)

(V1 ⊗ · · · ⊗ Vn )

⊗

~ ···⊗ ~ Vn k((t1 )) · · · ((tn )) → V1 ⊗

k((t1 ))⊗···⊗k((tn ))

which is an isomorphism if the Vi ’s have finite k((ti ))-dimension. An associative algebra in our monoidal category is the same as a topological associative algebra whose topology has a base formed by left ideals. Below we call such an R simply a topological associative algebra;33 we always assume it to be unital. For a topological vector space V the free topological associative algebra ~2 generated by V is denoted by T~ V , so T~ V = k ⊕ V ⊕ V ⊗ ⊕ · · · (equipped with the direct limit topology). For a k-vector space M considered as a discrete topological vector space a left unital R-action on M in the sense of our monoidal category is the same as a 33 So the ring of Laurent formal power series k((t)) with its usual topology is not a topological algebra in this sense.

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continuous left unital R-action (which means that the annihilator of every m ∈ M is an open ideal). M equipped with such an action is called a discrete R-module. The category of discrete R-modules is denoted by Rmod. Remarks. (i) Our R reconstructs from Rmod and the obvious forgetful functor from Rmod to (discrete) vector spaces as the (topological) endomorphism algebra of this functor. (ii) For a discrete vector space V the algebra End(V ) equipped with the weakest topology such that its action on V is continuous is a topological associative algebra. The category of discrete End(V )-modules is semisimple; each irreducible object is isomorphic to V . (iii) A right R-action on M in our monoidal category is the same as a plain right R-action such that the annihilator of M is open in R; i.e., the action factors through R/I where I is an open two-sided ideal. We will not consider such senseless objects. The category of topological vector spaces also carries a symmetric monoidal ˆ α := lim ⊗(Vα /Vαξ ), structure defined by the usual completed tensor product ⊗V ←− where Vαξ runs the set all open subspaces in Vα . So we have a natural continuous ~ ···⊗ ~ Vn → V 1 ⊗ ˆ · · · ⊗V ˆ n . The tensor products ⊗ ~ and ⊗ ˆ mutually morphism V1 ⊗ commute: for a collection {Vαi } of topological vector spaces bi-indexed by {α} ˆ α1 ⊗ ~ ···⊗ ~ Vαn ) = (⊗V ˆ α1 )⊗ ~ ···⊗ ~ (⊗V ˆ αn ). Therefore if and i = 1, . . . , n one has ⊗(V ˆ α ; if the Vα are the Rα are topological associative (unital) algebras, then so is ⊗R ˆ α -module. discrete (unital) Rα -modules, then ⊗Vα is a discrete (unital) ⊗R Remarks. (i) A topological associative algebra which is commutative (as an ˆ abstract algebra) is the same as a commutative algebra with respect to ⊗. (ii) The topological associative algebra End(V ) (see the previous Remark (ii)) ˆ and the action morphism End(V )⊗ ~ V → V does is not an algebra in the sense of ⊗ ˆ unless dim V < ∞. not extend to End(V )⊗V 3.6.2. The first definition of Aas x . Let x ∈ X be a closed point and ix : {x} ,→ X, jx : Ux ,→ X the complementary embeddings. Denote by Ox the formal completion of the local ring at x, Kx its quotient field. Below, t is a formal parameter at x (we fix it for mere notational convenience), so Ox = k[[t]], Kx = k((t)). Let A be a chiral algebra on X. We will assign to it a topological associative algebra Aas x . Here is a quick definition. Denote by M(X, A)x the category of Amodules supported at x. We have a faithful exact functor i!x = h on M(X, A)x with values in the category of vector spaces. Our Aas x is the algebra of endomorphisms of this functor equipped with the standard topology (its base is formed by annihilators of elements of hM , M ∈ M(X, A)x ). Unfortunately this definition says little about the structure of Aas x (to the extent that it smells of set-theoretic problems). So we will give another “concrete” definition of Aas x comparing it with the above definition later in 3.6.6. Remark. The reader may prefer to consider instead of M(X, A)x the subcategory MO (X, A)x of MO (X, A) (see 3.3.5(ii)) which consists of modules supported scheme-theoretically at x.34 The induction functor (3.3.5.1) identifies it with M(X, A)x . 34 I.e.,

killed by the maximal ideal mx ⊂ Ox .

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3.6.3. Lemma. The functor M(X, jx∗ jx∗ A)x → M(X, A)x defined by the obvious morphism of chiral algebras A → jx∗ jx∗ A is an equivalence of categories. Proof. Let M be a D-module supported at x. We want to show that every A-action on M extends uniquely to a jx∗ jx∗ A-action, and this extension is compatible with the morphisms of the M ’s. Notice that for every finite family of DX ˜ ˜ (I) modules Ai , i ∈ I, the obvious morphism of DX I˜ -modules35 j∗ j (I)∗ (( Ai )M ) →

I ˜ (I)∗ ˜ (I) ∗ ch ∗ j∗ j (( jx∗ jx Ai )  M ) is an isomorphism. Thus PI˜ ({jx∗ jx Ai , M }, M ) I ch PI˜ ({Ai , M }, M ). This implies the desired statement (take Ai = A).

∼

−→ 

Remark. Suppose M is an A-module supported at x. Let us write the Aaction as the ope ◦M (see 3.5.14). By Kashiwara’s lemma we can replace M by i!x M = hM , i.e., consider ◦M as a morphism36 A`Ox → Hom(hM, hM ((t))), a 7→ (m 7→ a ◦M m), compatible with the action of differential operators.37 Then the ope for the jx∗ jx∗ A-action on M is simply the Kx -linear extension A`Kx → Hom(hM, hM ((t))) of ◦M . 3.6.4. The second definition of Aas x . We change the notation: from now till 3.6.13 our A is a chiral algebra on Ux . Set M(X, A)x := M(X, jx∗ A)x . Denote by Ξas x the set of chiral subalgebras Aξ ⊂ jx∗ A which coincide with A as over Ux . If Aξ , Aξ0 are in Ξas x , then so is Aξ ∩ Aξ 0 , so Ξx is a topology on jx∗ A at x (see 2.1.13 for terminology). Remark. As in Remark (i) in 2.1.13 the map Aξ 7→ AξOx identifies Ξas x with the set of chiral subalgebras of AKx = (jx∗ A)Ox = AOx ⊗ Kx with torsion quotient, and we can consider Ξas x as a topology on AKx or on h(AKx ). The fibers A`ξx := i∗x A`ξ = i!x (jx∗ A/Aξ ) form a Ξas x -projective system of vector spaces connected by surjective morphisms. We denote by Aas x its projective limit. Equivalently (see 2.1.13), Aas is the completion of h(j A) or h(AKx ) with respect x∗ x x ` as to the Ξas -topology. So every A is a quotient of A modulo an open submodule x x ξx Iξ . We are going to define on Aas x a canonical structure of the associative algebra such that the Iξ become left ideals. To do this, we need an auxiliary lemma. For as ∗ ` ! ξ ∈ Ξas x let 1ξx ∈ Aξ /Iξ = ix Aξ = ix (jx∗ A/Aξ ) be the value at x of the unit section of A`ξ . 7 ϕ(1ξx ) 3.6.5. Lemma. For every ξ ∈ Ξas x and M ∈ M(X, Aξ ) the map ϕ → identifies Hom(jx∗ A/Aξ , M ) with the space of Aξ -central sections of M supported scheme-theoretically at x. Proof (cf. Exercise (ii) in 3.3.7). It is clear that 1ξx is a central section. Since it generates jx∗ A/Aξ , our map is injective. The inverse map is constructed as follows. Assume we have an Aξ -central section m of M supported scheme-theoretically at x; let us define the corresponding morphism ψm . We can assume that M is generated by m; hence M is supported at x. Thus, by 3.6.3, M is a jx∗ A-module. Consider the chiral action morphism jx∗ A  M → ∆∗ M . Pulling it back by (idX × ix )! , we 35 Here

I˜ := I t ·. that AOx := A ⊗ Ox , AKx := A ⊗ Kx .

36 Recall 37 D

X

Ox

Ox

acts on the right-hand side as on the Laurent series k((t)) = Kx .

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get a morphism of DX -modules ψ : jx∗ A ⊗ i!x M → M . Since m is Aξ -central, the morphism jx∗ A → M , b 7→ ψ(b ⊗ m), vanishes on Aξ , so we have ψm : jx∗ A/Aξ → M . One checks immediately that it is a morphism of Aξ -modules. This is the morphism we want.  3.6.6. Set Φξ := jx∗ A/Aξ . This is an Aξ -module supported at x, hence a jx∗ A-module by 3.6.3. Notice that the projection jx∗ A → Φξ is not a morphism of jx∗ A-modules unless Φξ = 0. For Aξ0 ⊂ Aξ the canonical projection Φξ0  Φξ is a morphism of Aξ0 -modules, hence, by 3.6.3, that of jx∗ A-modules. We have defined a Ξas x -projective system as Φ = {Φξ } in M(X, A)x . One has i!x Φ = Aas . Set 1 := lim 1 x x ←− ξx ∈ Ax . Take any M ∈ M(X, A)x . For every m ∈ i!x M its centralizer (see 3.3.7; here we consider m as a section of M ) belongs to Ξas x . Therefore, by 3.6.5, we have [ ∼ ! (3.6.6.1) Hom(Φ, M ) := Hom(Φξ , M ) −→ ix M = h(M ), ψ 7→ ψ(1x ). as In particular, Aas x = End Φ. We define on Ax the structure of an associative algebra so that this identification becomes an anti-isomorphism of algebras (so Φ as is a right Aas x -module). Therefore Ax is the algebra of endomorphisms of the functor h on M(X, A)x , as was promised in 3.6.2. The element 1x is the unit of our associative algebra. We see that Aas x is a topological associative unital algebra in the sense of 3.6.1 and every i!x M is a discrete unital Aas x -module.

Remarks. (i) Consider A as a Lie∗ algebra. Thus h(AKx ) is a Lie algebra and the map (3.6.6.2)

h(AKx ) → Aas x

is obviously a morphism of Lie algebras. Notice that the Ξas x -topology is weaker -topology (see 2.5.12), so (3.6.6.2) extends by continuity to the ΞLie than the ΞLie x x completion. (ii) Let us spell out the definition of the Aas x product more concretely. Take any ` as ; let us compute (ab)ξ ∈ Aas . Choose any A ∈ Ξ a, b ∈ Aas ξ x /Iξ = Aξx . Consider x x ` ` ` the ope product ◦ξ : Aξ  Aξ → (Aξ )2 ((t1 − t2 )). Restrict ◦ξ to X × {x} and fix the second argument to be bξ ∈ A`ξx ; we get a morphism of D-modules A`ξ → A`ξx ((t)), c 7→ c ◦ξ bξ . Let A`ξ0 ⊂ A`ξ be the preimage of A`ξx [[t]] ⊂ A`ξx ((t)). Then Aξ0 ∈ Ξas x and we have a morphism of D-modules A`ξ0 → A`ξ [[t]], hence the map A`ξ0 → A`ξ of fibers at x. Our (ab)ξ is the image of aξ0 . 3.6.7. Lemma. The functor i!x = h : M(X, A)x → Aas x mod is an equivalence of categories. Proof. Let us define the inverse functor. Recall that by Kashiwara’s lemma (see 2.1.3) the functor i!x is an equivalence between the category of right DX -modules supported at x and that of vector spaces; the inverse functor sends a vector space V to ix∗ V . Now assume that V is a discrete Aas x -module. Then ix∗ V is a chiral A-module. Indeed, by Kashiwara’s lemma, one has ix∗ Aas x = Φ. So, since ix∗ commutes with direct limits, we have ix∗ V = Φ ⊗ V which yields the promised A-module Aas x

! structure. The functor ix∗ : Aas x mod → M(X, A)x is obviously inverse to ix .



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38 3.6.8. The functor A 7→ Aas x is compatible with tensor products:

Lemma. There is a canonical isomorphism of topological associative algebras ∼ ˆ as ⊗A → (⊗Aα )as αx − x .

(3.6.8.1)

Proof. It is clear that any subalgebra (⊗Aα )ξ contains a subalgebra of type ˆ as ⊗(Aαξα ).39 This yields an isomorphism of topological vector spaces ⊗A αx := ∼ ` as lim ⊗A − → (⊗A ) which is obviously compatible with the products.  α x αξα x ←− Corollary. If A is a Hopf chiral algebra, then Aas x is a topological Hopf algebra. 3.6.9. The third definition of Aas x . Here is another, less economic, conˆ struction of Aas x (cf. [FBZ] 4.1.5). Denote by hAKx the completion of hAKx or ˆ K → Aas yields hx jx∗ A with respect to the Ξx -topology (see 2.1.13). The map hA x x a morphism of topological algebras (see 3.6.1) ˆ K → Aas . π : T~ hA x x

(3.6.9.1)

~ˆ It appears that π identifies Aas x with a quotient of T hAKx modulo explicit quadratic relations we are going to define. Consider AKx itself as a topological k-vector space: we declare a subspace P ⊂ AKx to be open if it intersects every Kx -line by a subspace open in the “tˆ K is continuous. adic” topology (see 3.6.1). Then the obvious map ζ : AKx → hA x (2) Denote by Kx the localization of k[[t1 , t2 ]] with respect to t1 , t2 and t1 − t2 ; let (2) (2) σ be an involution of Kx which interchanges ti . We have an embedding Kx ,→ k((t1 ))((t2 )). (2) Now take any a, b ∈ AKx and f = f (t1 , t2 ) ∈ Kx . Consider f as an iterated P P j Laurent power series f = fij ti1 t2 ∈ k((t1 ))((t2 )). The series fij (ti1 a) ⊗ (tj2 b) P ~2 ~2 ˆ K )⊗ converges in A⊗ fij ζ(ti1 a) ⊗ ζ(tj2 b) ∈ (hA . x Kx , so we have ζ(f, a, b) := ~2 ⊗ ˆ ˆ Set r(f, a, b) := ζ(f, a, b) − ζ(σ(f ), b, a) − µ(f, a, b) ∈ (hAKx ) ⊕ hAKx ⊂ ˆ K . Here µ(f, a, b) ∈ hA ˆ K is the image of f a  b by the chiral product map T~ hA x x ˆ K . composed with the projection ∆∗ AKx → h∆∗ AKx = hAKx → hA x as0 ˆ ~ Let Ax be the topological quotient algebra of T hAKx modulo the relations r(f, a, b) = 0 for all f, a, b as above and an extra relation ζ(1A dt/t) = 1. 3.6.10. Lemma. The morphism π from (3.6.9.1) yields an isomorphism of topological algebras (3.6.10.1)

0

∼

Aas → Aas x − x .

~ˆ Proof. Consider the functor Aas x mod → T hAKx mod defined by π. Our statement amounts to the fact that it is a fully faithful embedding, and a discrete ˆ K -module M comes from an Aas -module if and only if the T~ hA ˆ K -action on T~ hA x x x M kills our relations. ˆ K -action on it amounts to a morphism (hA ˆ K )⊗ ~M For a vector space M a T~ hA x x → M which is the same as a morphism of DX -modules jx∗ A ⊗ M → ix∗ M , or, by 38 See

3.4.15 and 3.6.1. for Aαξα the preimage of (⊗Aα )ξ by the morphism Aα → ⊗Aα .

39 Take

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Kashiwara’s lemma applied to X × {x} ⊂ X × X, that of DX 2 -modules jx∗ A  ix∗ M → ∆∗ ix∗ M . Since jx∗ A  ix∗ M = j∗ j ∗ (jx∗ A  ix∗ M ), this is the same as a ˆ K mod chiral operation µ ∈ P2ch ({jx∗ A, ix∗ M }, ix∗ M ). Therefore ix∗ identifies T~ hA x ch with the category of pairs (P, µ) where P ∈ M(X)x and µ ∈ P2 ({jx∗ A, P }, P ). Our relations just mean that µ is a chiral unital action of jx∗ A on ix∗ M . We are done by 3.6.7.  3.6.11. The Wick algebra. A better way to appreciate the construction of 3.6.9 and the like is to consider the Wick algebra Aw x of A. Here is a definition. (n) For n ≥ 1 let Ox be the formal completion of the local ring of (x, . . . , x) ∈ X n (n) (n) and let Kx be the localization of Ox with respect to the equations of the diagonal (n) divisor and the divisors xi = x, i = 1, . . . , n. Set Txw n A := An ⊗ Kx = O(x,... ,x)

A⊗n Kx

⊗ Kx⊗n

(n) Kx

40

and

T¯xw n A := h(Txw n A). Then Txw A := ⊕Txw n A and its quotient

T¯xw A := ⊕T¯xw n A are Z≥0 -graded associative algebras with respect to the exterior tensor product. For every n ≥ 2 the action of the symmetric group Σn on X n and An provides a Σn -action on the nth component of Txw n A and T¯xw n A. For i = 1, . . . , n − 1 let σi ∈ Σn be the transposition of i, i + 1 and let µi : T¯w n A(X) → T¯w n−1 A(X) be the map induced by the chiral product µA at i, i + 1 variables. One has µi σi = −µi . The span of elements r(¯ a) := a ¯ − σi a ¯ − µi a ¯ for a ¯ ∈ T¯xw n A, n, i as above, is a w w ¯ two-sided ideal Ix A. We define the Wick algebra Ax of A at x as the quotient of T¯xw A modulo I¯xw A and an extra (central) relation 1A · ν¯ = Resx ν¯ for ν¯ ∈ h(ωKx ).41 w w w Our Aw x carries a commutative filtration Ax0 ⊂ Ax1 ⊂ · · · , Axn := the image w ≤n ¯ of Tx A. There is a canonical morphism of Lie algebras (3.6.11.1)

hAKx → Aw 1x .

Let us define a canonical morphism of associative algebras (3.6.11.2)

as δ : Aw x → Ax

compatible with the standard Lie algebra maps from h(AKx ). 3.6.12. Lemma. There is a commutative diagram of morphisms of associative algebras δ˜ Txw A −−−−→ T~ AKx    ~ y yT ζ

(3.6.12.1)

δ¯ ˆ K T¯xw A −−−−→ T~ hA x    π y y δ

Aw −−−−→ x 40 As

Aas x

usual, h means coinvariants of the Lie algebra of vector fields acting on our right D

(n)

Kx

-

module; if t is a local coordinate at x, these coinvariants coincide with the coinvariants of the vector fields ∂t1 , . . . , ∂tn . 41 Here 1 · ν ¯w 1 A ¯ ∈ h(AKx ) = Tx A.

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in which the top horizontal arrow δ˜ is a morphism of Z≥0 -graded algebras such that (n) every component δ˜n is Ox -linear and δ˜1 = idAKx . The left vertical arrows are the projections from 3.6.11; for the right ones see 3.6.9. Such a diagram is unique. ~

(n)

n Proof. According to 3.6.1 every A⊗ Kx is an Ox -module and the multiplication by ti − tj , i 6= j, on it is invertible. Therefore the morphism of graded algebras T AKx → T~ AKx which is idAKx in degree 1 extends uniquely to a morphism of (n) graded algebras δ˜ : Txw A → T~ AKx such that every δ˜n is Ox -linear. The left vertical arrows in (3.6.12.1) are surjections, so δ˜ determines the other horizontal arrows uniquely. It remains to show that they are well defined. δ˜ ~n Consider δ¯ first. We need to check that the composition Txw n AKx − → A⊗ Kx → ~n ˆ K )⊗ (hA vanishes on the image of every operator ∂t , 1 ≤ i ≤ n. Notice that ∂t x

i

~

n is a continuous operator on AKx . Define ∂ti ∈ End A⊗ Kx as the tensor product of ∂t at the ith place and idAKx at other places. Then δ˜ commutes with ∂ti . The ~

∂t

~

~n i n n ⊗ ˆ composition A⊗ −→ A⊗ vanishes, and we are done. Kx − Kx → h(AKx ) δ¯ π ˆ K )− To see that δ is well defined, we have to check that T¯xw A − → T~ h(A → Aas x x wn 42 kills r(¯ a) := a ¯ − σi a ¯ − µi a ¯ for every a ∈ Tx A, n ≥ 2, i = 1, . . . , n − 1. For n = 2 this follows from the Jacobi identity for the chiral bracket.43 For n, i arbitrary we (n) have a = f (b ⊗ c ⊗ d) where f is in Ox localized with respect to all the ta − tb ’s ⊗2 −1 except ti − ti+1 , b ∈ A⊗i−1 ], d ∈ A⊗n−i−1 . Consider f as Kx , c ∈ AKx [(ti − ti+1 ) Kx −1 an element of the ring Rn,i := k((t1 )) · · · ((ti−1 ))[[ti , ti+1 ]][t−1 , t ]((t )) · · · ((tn )), so i+2 i i+1 P αn 1 ¯ f= P fα1 ···αn tα · · · t where f ∈ k. Then δr(¯ a ) is the sum of a convergent α1 ···αn n 1 αi−1 αi+2 αi αi+1 αn 1 series fα1 ···αn (tα d). Since π is a contin· · · t b) ⊗ r(t t c) ⊗ (t · · · t n 1 i−1 i i+1 i+2 i αi+1 ¯ a) = 0, and uous morphism of algebras which kills every r(tα t c), one has π δr(¯ i i+1 we are done. 

Remark. By 3.6.10, the map π identifies Aas x with the topological algebra ˆ K modulo the ideal generated by δ(Ker( ¯ quotient of T~ hA T¯xw A → Aw x )). x 3.6.13. The construction of Aas x generalizes immediately to the situation when x depends on parameters. Namely, assume we are in the situation of 2.1.16, so we have a quasi-compact and quasi-separated scheme Y and a Y -point x ∈ X(Y ). Let ix : Y ,→ X × Y be the graph of x and let jx : Ux ,→ X × Y be its complement. For a chiral OY -algebra A on X (see 3.3.10) jx∗ jx∗ A is again a chiral OY -algebra on X. The lemma in 3.6.3 (together with its proof) remains valid in the present setting: we have an equivalence of categories (3.6.13.1)

∼

M(X × Y, jx∗ jx∗ A)x −→ M(X × Y, A)x

where the lower index x means the full subcategory of modules supported (settheoretically) at the graph of x (see 3.3.10 for notation). ∗ Denote by Ξas x the set of chiral OY -subalgebras Aξ ⊂ jx∗ jx A which coincide ∗ with A over Ux . This is a topology on jx∗ jx A at x (we use terminology from 2.1.13 in the version of 2.1.16). The fibers A`ξx := i∗x A`ξ = i!x (jx∗ jx∗ A)/Aξ ) form a Ξas x -projective system of OY -modules connected by surjective morphisms. We 42 The 43 Cf.

extra relation 1A · ν¯ = Resx ν¯ for ν¯ ∈ h(ωKx ) holds by the definition of Aas x . the proof in 3.6.10.

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as denote by Aas x its projective limit. Equivalently (see 2.1.16), Ax is the completion · ∗ as ` of ix h(jx∗ jx A) with respect to the Ξx -topology. So every Aξx is a quotient of Aas x as modulo an open submodule Iξ and Aas = ← lim A /I . ξ − The lemma in 3.6.5 (together with its proof) remains valid, as well as the constructions and statements of 3.6.6. Therefore Aas x is an associative unital topological OY -algebra; every Iξ is an open left ideal in Aas x . The lemma in 3.6.7 together with its proof 44 remain valid, so we have an equivalence of categories

(3.6.13.2)

∼

i!x = hx : M(X × Y, A)x −→ Aas x mod

as where Aas x mod is the category of discrete Ax -modules := the sheaves of discrete left Ax -modules on Y which are quasi-coherent as OY -modules. The lemma in 3.6.8 together with the proof also remains valid.

Exercise. (cf. Remark (i) in 2.1.16). Check that the formation Aas xY is compatible with the base change (i.e., for every f : Y 0 → Y the pro-OY 0 -modules f ∗ Aas xY and Aas xf Y 0 coincide). 3.6.14. Suppose now that Y from 3.6.13 is another copy of X, x = idX (so ix = ∆, jx = j : U ,→ X × X). Let A be a chiral algebra on X. Applying 3.6.13 to A = AY := A  OY , we get an associative topological OX -algebra Aas := Aas x . Let τ be any infinitesimal automorphism of X. It acts on X × X = X × Y along the Y -copy of X preserving U . This action lifts in the obvious way to AY , hence to j∗ j ∗ AY . Therefore45 τ acts on Aas as on a topological OX -algebra in a natural way. Thus Aas carries a canonical flat connection; we denote it by ∇. Let I0 ⊂ Aas be the open left ideal that corresponds to AY ∈ Ξas x . Then I0 is preserved by ∇ and one has a canonical identification of left DX -modules (3.6.14.1)

Aas /I0 = A` .

For a section a of j∗ j ∗ A  OX we denote by V(a) the image of a in Aas . The map V commutes with the action of DX along the second variable of the source, and is a differential operator with respect the OX -action along the first variable. For b ∈ A the section V(b  1) belongs to J0 , is ∇-horizontal, and depends only on the class ¯b ∈ h(A). We denote it by ¯bas . So one has a morphism of sheaves (3.6.14.2)

as ∇ ¯ h(A) → I∇ 7 ¯bas . 0 ⊂ (A ) , b →

3.6.15. Suppose M is an A-module. Then the OX -module M ` carries a canonical Aas -action. Indeed, M  OY is an AY -module, so (j∗ j ∗ M  OX )/M  OX is an AY -module supported at the diagonal. Since M ` = ∆! ((j∗ j ∗ M  OX )/M  OX ) = h((j∗ j ∗ M  OX )/M  OX ), it is an Aas -module by (3.6.13.2); here h is the integration along the X-multiple of X × Y (see 2.1.16). This picture is equivariant with respect to the action of infinitesimal symmetries of X acting along the Y -multiple,46 so the Aas -action on M ` is compatible with the connections.47 44 Use

a version of Kashiwara’s lemma from 2.1.16. Exercise in 3.6.13. 46 Use the h part of the above formula. 47 M ` carries a connection since it is a left D -module. X 45 Use

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Example. If M = A, then the Aas -action on M ` comes from (3.6.14.1). Denote by M` (X, Aas ) the category whose objects are discrete Aas -modules equipped with a connection (as OX -modules) such that the Aas -action is horizontal. To say it differently, the connection ∇ on Aas defines on Aas [DX ] := Aas ⊗ DX OX

the structure of an associative algebra,48 and M` (X, Aas ) is the category of discrete Aas [DX ]-modules, i.e., left unital Aas [DX ]-modules which are discrete Aas -modules. 3.6.16. Proposition. There is a canonical equivalence of categories (3.6.16.1)

∼

M(X, A) −→ M` (X, Aas ).

Proof. We have seen in 3.6.15 that for M ∈ M(X, A) the left DX -module M ` carries a canonical horizontal Aas -action, so we have the functor M(X, A) → M` (X, Aas ). This is an equivalence of categories. Indeed, by Kashiwara’s lemma, the functor M 7→ (j∗ j ∗ M  OY )/M  OY identifies the category of A-modules with the category of AY -modules supported on the diagonal and equivariant with respect to the action of infinitesimal automorphisms of Y (acting along the Y multiple; recall that Y is another copy of X). Now use (3.6.13.2) and the definition of the canonical connection ∇ on Aas .  Variant. Forgetting about ∇, we see that the category of discrete Aas-modules is canonically equivalent to MO (X, A) (see 3.3.5(ii)). Remark. One can describe chiral A-operations in terms of the Aas -module structures using the lemma from 3.4.19. 3.6.17. For an OX -module M the above 1-1 correspondence between discrete Aas -actions and chiral A-actions on M can be rewritten as follows. The (non-quasi-coherent) OX -module Aas ⊗ M carries a topology whose open subsheaves are V ⊂ Aas ⊗ M that satisfy the following property: for every local section m ∈ M there exist an open ideal Iξ ⊂ Aas such that Iξ ⊗ m ⊂ V . It is clear that every open subsheaf contains a smaller open subsheaf V such that V is an O-submodule and Aas ⊗ M/V is a quasi-coherent OX -module. Denote the ˆ . completion by Aas ⊗M ∗ The sheaf j∗ j A  M carries a topology whose open subsheaves are W ⊂ j∗ j ∗ A  M that satisfy the following property: for every local section m ∈ M there exists Aξ ⊂ j∗ j ∗ (A  OX ) in Ξas ∆ such that Aξ ⊗ m ⊂ W . Every open subsheaf contains a smaller open subsheaf W which is a quasi-coherent DX  OX -module. The completion (j∗ j ∗ A  M )ˆis a topological DX  OX -module. The identifications j∗ j ∗ A  OX /Aξ = ∆∗ A`ξ∆ = ∆∗ (Aas /Iξ ) of DX  OX modules yield a canonical isomorphism of topological DX  OX -modules (3.6.17.1)

∼

ˆ ). (j∗ j ∗ A  M )ˆ −→ ∆∗ (Aas ⊗M

Now every chiral A-action on M is automatically continuous in the above topology, so it can be considered as a morphism (j∗ j ∗ A  M )ˆ→ ∆∗ M . Similarly, every ˆ → M . A chiral discrete Aas -action on M can be cosidered as a morphism Aas ⊗M as A-action µM is identified with an A -action ·M if (3.6.17.1) identifies ∆∗ (·M ) with µM . 48 It is characterized by the properties that the maps Aas , D as X → A [DX ], a 7→ a ⊗ 1, ∂ 7→ 1 ⊗ ∂, are morphisms of algebras, and for a ∈ Aas , τ ∈ ΘX one has τ a − aτ = τ (a).

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If M is a DX -module, then the DX -action on Aas ⊗ M extends by continuity ˆ , (j∗ j ∗ A  M )ˆ is a topological DX×X -module, and (3.6.17.1) is an isoto Aas ⊗M morphism of topological DX×X -modules. Therefore µM is horizontal if and only if ·M is. 3.6.18. The commutative case. Suppose now that our A is commutative. as Then the algebras Aas are also commutative. We already met Aas x , A x in the situation when x is a single point, x ∈ X(k) (see 2.4.8). The “geometric” description of Aas x from 2.4.9 generalizes immediately to the general situation. Namely, for x ∈ X(Y ) as in 3.6.13 and a commutative quasi-coherent OY -algebra F we define ˆ as the formal completion of OX  F at a non-quasi-coherent OX×Y -algebra Ox ⊗F ˆ be its localization with respect to an equation of x. (the graph) of x; let Kx ⊗F ˆ ⊂ Kx ⊗F ˆ are DX×Y /Y -algebras; i.e., they carry an obvious connection Our Ox ⊗F along X. Now one has a canonical identification ` ∗ ` ˆ ˆ (3.6.18.1) Hom(Aas x , F ) = Hom(A  OY , Kx ⊗F ) = Hom(jx∗ jx (A  OY ), Kx ⊗F )

where the first Hom means continuous morphisms of OY -algebras49 and the next ones mean morphisms of DX×Y /Y -algebras. For the construction see the proof in 2.4.9. As was mentioned in loc. cit., (3.6.18.1) is the subset of the algebra morphisms in (3.5.4.2). 3.6.19. Lemma. If ϕ : A` → B ` is a formally smooth morphism of commutative DX -algebras, then the corresponding morphism of topological OY -algebras as Aas x → Bx is formally smooth. The same is true if we replace “formally smooth” by “formally ´etale.” 50 Proof. Assume that ϕ is formally smooth. Let F be a commutative k-algebra, as I ⊂ F an ideal such that I 2 = 0. Let f : Spec F → Spf Aas x , g : Spec F/I → Spf Bx as be ϕ-compatible morphisms. We want to find a lifting h : Spec F → Spf Bx of f that extends g. According to (3.6.18.1) we may rewrite f, g as morphisms of DX×Y /Y -algebras ˆ , g 0 : B `  OY → Kx ⊗F/I. ˆ ˆ , Kx ⊗F/I ˆ f 0 : A`  OY → Kx ⊗F Here Kx ⊗F are non-quasi-coherent sheaves of DX×Y /Y -algebras. Our problem is local (see [RG]) so we can assume that X and Y are affine. Denote by Γ the functor of global sections over X × Y . So we have commutative ˆ ), Γ(Kx ⊗F/I) ˆ O(X)-algebras Γ(Kx ⊗F equipped with a connection along X. The second algebra is a quotient of the first one modulo an ideal of square 0. Our f 0 , g 0 amount to maps of commutative O(X)-algebras Γf 0 : A` (X) → ˆ ), Γg 0 : B ` (X) → Γ(Kx ⊗F/I) ˆ compatible with connections along X. Γ(Kx ⊗F Since ϕ is a formally smooth morphism of DX -algebras, we can find a lifting Γh0 : ˆ ) of Γf 0 that extends Γg 0 . As above, Γh0 is the same as a B ` (X) → Γ(Kx ⊗F morphism h : Spec F → Spf Bxas which is the desired lifting. The case of formally ´etale ϕ is left to the reader.  3.6.20. The global Wick algebra. In the rest of this section we make a few remarks about the global version Aw (X) of the Wick algebra; they will not be used elsewhere in the book. We assume that X is an affine curve. 49 F

is assumed to carry the discrete topology. the definition of formally smooth and formally ´ etale morphisms of commutative DX algebras (or, more generally, algebraic DX -spaces) see 2.3.16. 50 For

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First we have a Z≥0 -graded associative algebra T w A(X) = ⊕T w n A(X) where (we use the notation of 3.1.1) T w n A(X) := Γ(U (n) , An ) and the multiplication 0 is the exterior tensor product. Passing to HDR , we get the quotient algebra51 w 0 (n) n ¯ T A(X) := ⊕HDR (U , A ). For every n ≥ 2 the action of the symmetric group Σn on X n and An provides a Σn -action on the nth component of T w n A(X) and T¯w n A(X). For i = 1, . . . , n−1 let σi ∈ Σn be the transposition of i, i + 1 and let µi : T¯w n A(X) → T¯w n−1 A(X) be the map induced by the chiral product µA at i, i + 1 variables. One has µi σi = −µi . Let I¯w A(X) ⊂ T¯w A(X) be the span of (non-homogeneous) elements r(¯ a) := a ¯− σi a ¯ − µi a ¯ for all a ¯ ∈ T¯w n A(X), n, i as above. This is a two-sided ideal in T¯w A(X). Our Aw (X) is the quotient algebra T¯w A(X)/I¯w A(X). It carries an increasing ¯w ≤n A(X) in Aw (X), Aw (X) = S Aw (X). Thus filtration Aw (X) := the image of T n n Aw · is a Z≥0 -filtered associative unital algebra that depends on A in a functorial way. Remarks. (i) The constructions are functorial with respect to the ´etale base change, so, replacing X by ´etale X-schemes, we get a presheaf Aw on Xe´t . There is an evident morphism of Lie algebras (3.6.20.1)

h(A) → Aw 1.

(ii) There is a (more local) variant of the above constructions with T w n A(X) ˜ (n) OX ) (see (3.5.5.1) for notation). replaced by Γ(X n , An ⊗ ∆ The following properties of Aw are immediate: w w w w (i) Aw · is a commutative filtration: one has [Am , An ] ⊂ Am+n−1 . So gr· A is a commutative algebra. (ii) If the morphisms of chiral algebras A, B → C mutually commute (see 3.4.15), then the images of Aw , B w in C w mutually commute. In particular, if R is a commutative chiral algebra, then Rw is commutative; if A is a chiral R-algebra, then Aw is an Rw -algebra. 3.6.21. Let us compute the Wick algebra of the unit chiral algebra ω. Recall 1 (V ). that h(ω) is the sheaf V 7→ HDR Proposition. There is a canonical isomorphism of filtered algebras ∼

Sym h(ω) −→ ω w . Proof. Since ω w is commutative, (3.6.20.1) yields a morphism Sym h(ω) → ω w . We want to prove that this is an isomorphism of filtered algebras or, equivalently, that the corresponding morphism of graded algebras Sym h(ω) → gr ω w is an iso1 morphism. It suffices to show that the maps Symn HDR (X) → grn ω w (X) are isomorphisms. n Surjectivity: Let K be the kernel of HDR (U (n) ) = T¯w n ω(X)  grn ω w (X). We 1 1 want to check that K together with the image of Symn HDR (X) ⊂ HDR (X)⊗n = n n n (n) n (n) HDR (X ) in HDR (U ) span HDR (U ). Notice that the action of the symmetric n group Σn on T¯w n ω(X) is the obvious action on HDR (U (n) ) multiplied by the sign character (see 3.1.4). Look at the Σn -action on the spectral sequence computing (n) · HDR (U (n) ) which corresponds to the filtration W on j∗ ωU (n) (see 3.1.6 and 3.1.7). 51 Recall

0 that the An are right D-modules, so HDR is the “middle” cohomology; see 2.1.7.

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1 n It shows that the map Symn HDR (X) → HDR (U (n) ) is injective and its image is n (n) sgn the subspace HDR (U ) of the elements that transform according to the sign character. As follows from the definition of I¯w , the subspace K contains every n subspace HDR (U (n) )σi of σi -invariants. We are done since for every Σn -module P one has P = P sgn + ΣP σi .52 Injectivity: Here is a topological argument. We can assume that k = C, so the 1 dual vector space to HDR (X) identifies with the singular homology group H1 (X, C). We have proved that Spec gr ω w (X) is a closed subscheme of H1 (X, C), and we want to show that it equals H1 (X, C). It suffices to check that it contains every integral homology class γ. Thus we need to construct a morphism of C-algebras 1 γ w : ω w (X) → C whose composition with the canonical map HDR (X) → ω w (X) is integration along γ. Let us represent γ by a union of several mutually nonintersecting oriented C ∞ -loops γα each of which has no self-intersections (i.e., γα is an image of a regular embedding S 1 ,→ X). Let yα be a normal coordinate function on a tubular neighbourhood of γα . We assume that the orientation of yα , i.e., the sign of dyα on γα = {yα = 0}, is choosen in such a way that if γα is the unit circle S 1 ⊂ C with the standard orientation, then yα (z) = log |z| is well oriented. So for a small  ∈ R we have an oriented loop γα  := {yα = }. For every n ≥ 1 consider a cycle γn = Σγα n on U (n) , γn αR := γα 1 × · · · × γα n where the i are n (U (n) ) → C form a morphism of small and 1 > · · · > n . The functionals : HDR γn

C-algebras T w γ : T¯w ω(X) → C. It kills the ideal I¯w ω(X), so we have defined our γ w : ω w (X) → C.  According to property (ii) in 3.6.20 and 3.6.21 the Wick algebra of any chiral algebra is, in fact, a filtered associative Sym h(ω)-algebra. Remark. For chiral algebras A, B a canonical morphism Aw ⊗B w → (A⊗B)w defined by property (iii) in 3.6.20 comes from Aw ⊗ B w → (A ⊗ B)w . The Sym h(ω)

latter arrow need not be surjective. 3.7. From Lie∗ algebras to chiral algebras The rest of this chapter deals with some methods of constructing chiral algebras. This section treats chiral enveloping algebras. Different proofs of the theorems in 3.7.1 and 3.7.14 in the graded vertex algebra setting (see 0.15) can be found in [FBZ] and in the preprint version of [GMS2]. The observation that the vacuum representation of a Kac-Moody or Virasoro algebra carries a canonical structure of vertex algebra goes back to the first days of the vertex algebra theory and beyond. Needless to say, this fact was always known to mathematical physicists. We begin in 3.7.1 with the existence theorem for chiral envelopes. A more general theorem which deals with arbitrary Jacobi type operads (instead of the Lie operad) is formulated in 3.7.2; the proof takes 3.7.3–3.7.4 (the results of 3.7.2–3.7.4 will not be used elsewhere in the book and the reader can skip them). An explicit factorization algebra construction of chiral envelopes is presented in 3.7.5–3.7.11; 52 To see this, we can assume that P is irreducible and P 6= P sgn . Then P σi 6= 0 (by Young’s table picture), so it suffices to show P s := ΣP σi ⊂ P is preserved by the Σn -action, or, equivalently, P s is preserved by every σi . Now σi preserves P σj for j 6= i ± 1 (since σi , σj commute), and it preserves both P σi + P σi±1 (an exercise in representation theory of Σ3 ).

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the key tool is an auxiliary Lie algebra L\ in the tensor category of the left Dmodules on R(X) (see 3.7.6). This construction implies immediately the Poincar´eBirkhoff-Witt theorem for chiral envelopes, see 3.7.14. We relate L- and U (L)modules in 3.7.15–3.7.18, describe the associative algebra U (L)as x in 3.7.19, and consider the twisted envelopes in 3.7.20–3.7.22. The enveloping algebra construction for a Lie∗ algebra acting on a chiral algebra is treated in 3.7.23–3.7.24. Finally, we discuss in 3.7.25 Virasoro vectors and the Sugawara construction. 3.7.1. Consider the forgetful functor CA(X) → Lie∗ (X), A 7→ ALie . Theorem. This functor admits a left adjoint functor U : Lie∗ (X) → CA(X). For a Lie∗ algebra L we call U (L) the chiral enveloping algebra, or simply the chiral envelope of L. Proof. This is an immediate corollary of the theorem in 3.4.14. Namely, let A be the chiral algebra freely generated by (L, P ) where P is the image of Ker[ ]L ⊂ LL in j∗ j ∗ L  L. Then U (L) is the quotient of A modulo the ideal generated by the relations saying that the morphism L → U (L) is a morphism of Lie∗ algebras. Precisely, let iA : L → A be the universal morphism. Let A0 be the quotient of A modulo the ideal generated by elements i(`τ ) − i(`)τ where ` ∈ L and τ ∈ ΘX , so iA0 : L → A0 is a morphism of DX -modules. Then U (L) is the quotient of A0 0 modulo the ideal generated by the image of iA0 [ ]L − [ ]A0 i2 A0 : L  L → ∆ ∗ A .  3.7.2. The above theorem remains true for algebras over more general operads. Definition. A k-operad B is of Jacobi type if it is generated by B1 and B2 , and for every α, α0 ∈ B2 there exist β, β 0 , γ, γ 0 ∈ B2 such that α(x1 , α0 (x2 , x3 )) = β(x3 , β 0 (x1 , x2 )) + γ(x2 , γ 0 (x3 , x1 )). For example, the operads Lie, Poiss, Com, Ass are of Jacobi type. Theorem. For any operad B of Jacobi type the functor β B of (3.3.1.1) admits a left adjoint functor UB : B∗ (X) → Bch (X). Composing ULie with the “adding of unit” functor (see 3.3.3) one gets the functor U of 3.7.1. Proof. Here is the idea. If the chiral pseudo-tensor structure were representable (see 1.1.3), then 3.7.2 would be immediate for arbitrary B. Indeed, representability implies that for every L the free Bch algebra generated by L is well defined. So, if L is a B∗ algebra, then UB (L) is a quotient of the free Bch algebra modulo the obvious relations. Now the binary chiral pseudo-tensor product is representable up to the non-representability of the ∗ pseudo-tensor product. This shows that the quadratic part of UB (L) is well defined. To get control of all of UB (L), one needs the Jacobi property of B. Let us turn to the actual proof. 3.7.3. We begin with some complements to 3.4.10. Let M(X S )∆ ⊂ M(X S ) be the full subcategory of those M for which every MX I is supported on the diagonal X ⊂ X I ; it is closed under subquotients and extensions. M(X S )∆ is equivalent to the category M(X)S of all functors S◦ → M(X) (one identifies M ∈ M(X S )∆ with the functor I 7→ ∆(I)! MX I ). Notice that (S) ∆∗ identifies M(X) with the full subcategory of M(X S )∆ , and this embedding

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admits a left adjoint functor  : M(X S )∆ → M(X), M 7→ lim ∆(I)! MX I . It extends −→ in the obvious way to pseudo-tensor functors (3.7.3.1)

M(X S )∗∆ → M(X)∗ ,

ch M(X S )ch ∆ → M(X)

which are left adjoint to the pseudo-tensor functors from (3.4.10.5). For any k-operad B denote by B∗ (X S ), Bch (X S ) the categories of B algebras in M(X S )∗ , M(X S )ch (we call them B∗ and Bch algebras on X S ). Notice that by (3.4.10.4) every B algebra in M(X S )ch is automatically a B algebra in M(X S )∗ ; i.e., the pseudo-tensor functor β S yields a functor Bch (X S ) → B∗ (X S ). This functor admits a left adjoint functor (3.7.3.2)

S UB : B∗ (X S ) → Bch (X S ).

S Indeed, for a B∗ algebra L the corresponding Bch algebra UB (L) is the quotient of ch the free B algebra generated by L modulo the relations needed to assure that the S canonical morphism iS : L → UB (L) is a morphism of B∗ algebras. Precisely, let ∗ ch ∗ ch Φ , Φ be the free B and B algebras generated by L. Since Φch is a B∗ algebra, the canonical morphism ich : L → Φch extends to a morphism of B∗ algebras α : Φ∗ → Φch . Since L is a B∗ algebra, the identity morphism idL extends to a S (L) is the quotient of Φch modulo morphism of B∗ algebras β : Φ∗ → L. Now UB ch ch the B ideal I generated by the image of α − i β : Φ∗ → Φch .

3.7.4. We return to the proof of 3.7.2. Let L be a B∗ algebra on X. We want to construct the corresponding universal Bch algebra UB (L) on X equipped with a morphism of B∗ algebras i : L → UB (L). (S) (S) S According to (3.4.10.5), ∆∗ L is a B∗ algebra on X S . Set U := UB (∆∗ L) ∈ ch S S B (X ). In the next lemma we will show that U ∈ M(X )∆ . Assuming this, set UB (L) := (U ), i := (iS ) : L → UB (L). By (3.7.3.1), UB (L) is a B algebra in M(X)ch and i is a morphism of B∗ algebras; by the adjunction property of  the pair (UB (L), i) satisfies the desired universal property. We are done. It remains to prove the following lemma: Lemma. If B is of Jacobi type, then U ∈ M(X S )∆ . Proof of Lemma. (a) For a finite set I and α, β ∈ I, α 6= β, let Bαβ I ⊂ BI be the image of the composition map BI/{α,β} ⊗ B{α,β} → BI . A simple induction shows that B is of Jacobi type if and only if it satisfies the following property: For any finite non-empty sets I1 , I2 the vector space BI1 tI2 is generated by the subspaces Bαβ I1 tI2 , α ∈ I1 , β ∈ I2 . (b) Let Φch be the free Bch algebra on X S generated by L as in 3.7.3. It has ch ⊗ch I a natural gradation Φch = ⊕ Φch ⊗ BI )AutI where I is a n such that Φn = (L n≥1

set of order n. This gradation yields an increasing filtration on U = Φch /I. We are going to show that gr U ∈ M(X S )∆ ; this will prove our lemma. P ⊗ch I Clearly Ker(Φch Kαβ ⊗Bαβ ⊗BI n  grn U ) ⊃ NAut I where N := I ⊂L α6=β∈I

ch

ch

and Kαβ is the image of (L ⊗∗ L) ⊗ch L⊗ (Ir{α,β}) in L⊗ ch show that M := (L⊗ I ⊗ BI )/N belongs to M(X S )∆ .

I

. So it is enough to

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215

Set Uαβ := {(xi ) ∈ X I : xi 6= xj if i 6= j, {i, j} 6= {α, β}}; let jαβ : Uαβ ,→ X I P (I) be the embedding. Set F := j∗ OU (I) ⊗ BI , G := jαβ∗ OUαβ ⊗ Bαβ ⊂ F, I α6=β∈I

(J/I)

H := F/G. By definition, for every J ∈ S one has MX J = ⊕ ∆∗

(LI ⊗ H).

JI I

It remains to show that H is supported on the diagonal X ⊂ X . (c) For I = I1 t I2 where I1 , I2 6= ∅ set UI1 I2 := {(xi ) ∈ X I : xi 6= xj if i ∈ (I) I1 , j ∈ I2 }. On UI1 I2 the sheaves jαβ∗ OUαβ coincide with j∗ OU (I) . So (a) above implies that the restriction of H to UI1 I2 is zero. The union of all the UI1 I2 ’s is the complement to X ⊂ X I , and we are done.  3.7.5. Let us return to the setting of 3.7.1. Let L be a Lie∗ algebra. Below we will give an explicit construction of U (L) as a factorization algebra (together with the canonical D-module structure). Our problem is X-local, so we can assume that X is affine. The first step is to define an auxiliary Lie algebra L\ in the tensor category of the left D-modules on R(X) (see 3.7.6). The chiral envelope is constructed in 3.7.7 (it is denoted by V there). The universality property is checked in 3.7.11. Remark. A slightly unpleasant point of our construction is that the auxiliary Lie algebra L\ is a non-local object (while U (L) is of course local). A remedy would be to consider in (3.7.6.1)Sinstead of global cohomology the one of the formal neighbourhood of the divisor {x = xi }. The price is to deal with non quasicoherent modules; we prefer not to do this. In 3.7.6–3.7.11 we assume that X is affine. 3.7.6. Let I be a finite set. Denote by pI the projection X × X I → X I , and let jI : V = V ˜I ,→ X × X I be the open subset of those (x, (xi )) that x ∈ V(xi ) := X r {xi }. For L ∈ Mr (X)53 consider the de Rham complex DR(L) = Cone(L ⊗ ΘX → L). Set (3.7.6.1)

L\X I := H 0 (pI jI )· jI∗ (DR(L)  OX I ).

0 (V(xi ) , L) = Γ(V(xi ) , h(L)). This is an OX I -module; its fiber at (xi ) ∈ X I equals HDR I · The obvious left pI DX -module structure on DR(L)  OX I shows that L\X I is a left DX I -module. In other words, L\r = H 0 (pI jI )∗ jI∗ (L  ωX I ). XI Our L\ carries a bunch of structures described in (i)–(iii) below that will be used in the construction of the chiral envelope: (i) For every diagonal embedding ∆(I/S) : X S ,→ X I we have an obvious identification of left D-modules

(3.7.6.2)

∼

∆(I/S)∗ L\X I −→ L\X S .

So the L\X I form a left D-module on R(X) (see 3.4.2). More generally, the L\X I behave nicely with respect to arbitrary standard morphisms between X I ’s. Namely, if π : J → I is any map of finite sets, then the pull-back of V J˜ by ∆(π) : X I → X J contains V ˜I . So we have the restriction map (3.7.6.3) 53 We

∆(π)∗ L\X J → L\X I do not assume that L is a Lie∗ algebra at the moment.

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which coincides with (3.7.6.2) if π is surjective. These maps are injective morphisms of D-modules; they are compatible with the composition of the π’s. For example, if J = ∅, then the corresponding vector space L\0 := L\X ∅ is Γ(X, h(L)) (see 2.5.3). We get a canonical embedding of DX I -modules L\0 ⊗ OX I ,→ L\X I .

(3.7.6.4)

(ii) Denote by L`X I the cokernel of (3.7.6.4). Since X is affine, one has L`X I = H pI· F X˜I where F X˜I := (jI· jI∗ DR(L)  OX I )/(DR(L)  OX I ). The corresponding right DX I -module equals pI∗ ΦX˜I where ΦX˜I := jI∗ jI∗ (L  ωX I )/L  ωX I . In particular, L`X = L` . Notice that L`X I , unlike L\X I , depends on L in a purely X-local way; i.e., L`(xi ) depends only on the restriction of L to a neighbourhood of {xi }. For S ∈ Q(I) consider the open subset j [I/S] : U [I/S] ,→ X I (see 3.4.4); for s ∈ S let ps : X I → X Is be the projection and p˜s := idX F X˜I Q× p∗s . Decomposing ∼ [I/S]∗ [I/S]∗ p˜s F X˜Is −→ j F X˜I by connected components of its support, we get j 0

s∈S

hence the isomorphism Y

j [I/S]∗

(3.7.6.5)

∼

p∗s L`X Is −→ j [I/S]∗ L`X I .

s∈S

Q ∼ In particular, one has j (I)∗ p∗i L` −→ j (I)∗ L`X I . Here is another interpretation of (3.7.6.5). For s ∈ S consider the projection qs : I X → X IrIs and the coresponding embedding (3.7.6.3) of D-modules qs∗ L\X IrIs ,→ L\X I . We have a commutative diagram of embeddings of DX I -modules Q ∗ \ −→ (L\X I )S ←− L\X I ps LX Is S

↑

(3.7.6.6) (L\0

↑

⊗ OX I )

S

−→

Q S

qs∗ L\X IrIs

↑ ←−

L\0

⊗ OX I

Q The morphisms between the cokernels of vertical embeddings (p∗s L\X Is /L\0 ⊗ S Q OX Is ) → L\X I /L\X IrIs ← L\X I /L\0 ⊗ OX I being restricted to U [I/S] become isoS

morphisms; the composition is (3.7.6.5). (iii) Assume that L is a Lie∗ algebra. Then L\X I is a Lie algebra in the tensor category M` (X I ). One defines the Lie bracket on L\X I in the same way that we defined the bracket on h(L). Namely, the ∗ bracket on L yields a morphism DR(L) DR(L) → DR(∆∗ L) on X × X. Take the exterior tensor product of this morphism with OX I , restrict it to V × V ⊂ (X × X) × X I , and push forward to X I . We get XI

a morphism of complexes of the left DX I -modules ((pI jI )· jI∗ (DR(L)  OX I ))⊗2 → (pI jI )· jI∗ (∆· DR(∆∗ L)  OX I ). The second term is canonically quasi-isomorphic to (pI jI )· jI∗ (DR(L)  OX I ) (see (b) in 2.1.7). Passing to H 0 , we get a morphism [ ]\ : L\X I ⊗ L\X I → L\X I which is the desired Lie bracket. All the canonical morphisms in (i) and (ii) above are compatible with our Lie algebra structure. In particular, L\ is a Lie algebra in the tensor category of left D-modules on R(X).

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217

3.7.7. Now we are ready to define our factorization algebra. Set VX I := U (L\X I )/U (L\X I )L\0 .

(3.7.7.1)

Here U (L\X I ) is the enveloping associative algebra of our Lie algebra L\X I in the abelian tensor k-category M` (X I ). So VX I is the vacuum representation := the L\X I -module induced from the trivial L\0 ⊗ OX I -module. The construction of the enveloping algebra and induced module is compatible with pull-backs. Therefore U (L\X I ) and VX I form left D-modules U (L\ ) and V on R(X). Our V has a canonical factorization structure. To define factorization isomorphisms c (see 3.4.4.1), look at (3.7.6.6) and consider the corresponding morphisms of induced modules (3.7.7.2)

 VX Is −→ ⊗ U (L\X I )/U (L\X I )qs∗ L\X IrIs ←− VX I . S

S

The following well-known lemma (together with (ii) in 3.7.6) shows that over U [I/S] both arrows in (3.7.7.2) are isomorphisms. This provides the factorization isomorphisms. The compatibilities from 3.4.4 obviously hold. 3.7.8. Lemma. Let φ : L1 → L2 be a morphism of Lie algebras over a commutative ring R, and let N1 ⊂ L1 , N2 ⊂ L2 be Lie subalgebras such that φ(N1 ) ⊂ N2 ∼ and L1 /N1 −→ L2 /N2 . Then the morphism U (L1 )/U (L1 )N1 → U (L2 )/U (L2 )N2 is an isomorphism. Remark. The following stronger statement holds: the map U (L1 ) ⊗ U (N2 ) U (N1 )

→ U (L2 ) is an isomorphism. In fact, this statement follows from the lemma.54 We give two proofs of the lemma. The first one is elementary. The second proof is more “immediate” but it uses a machinery too heavy for the purpose.55 First Proof. Set Pi := U (Li )/U (Li )Ni ; let vi ∈ Pi be the “vacuum vectors” and let π : P1 → P2 be our morphism. (i) π is surjective. Indeed, π is compatible with the standard filtrations on the Pi ’s. Since gr· Pi is a quotient of Sym· (Li /Ni ), gr π is surjective, and we are done. (ii) So to prove our lemma, it suffices to construct a left inverse P2 → P1 of π. To do this, we will define an L2 -module structure on P1 such that: (a) The obvious L1 -action on P1 equals the one coming from the L2 -action via π. (b) The action of N2 ⊂ L2 kills v1 . As follows from (b), there is a unique morphism of L2 -modules P2 → P1 which sends v2 to v1 . By (a) it is left inverse to π. (iii) It remains to construct the L2 -action. We use the following remark: for any L1 -modules Q, R one has HomL1 (Q ⊗ P1 , R) = HomL1 (P1 , Hom(Q, R)) = HomN1 (Q, R). Taking Q = L2 , R = P1 , we see that there is a unique morphism of L1 -modules a : L2 ⊗ P1 → P1 with the property a(l2 ⊗ v1 ) = l1 v1 for every li ∈ Li such that l1 modN1 = l2 mod N2 . 54 It suffices to consider the case L ⊂ L . Then apply the lemma to L ˜ 1 := L1 × N2 , 1 2 ˜ 2 := L2 × L2 , N ˜1 := N1 , N ˜2 := L2 . L 55 It will be of use in the proof of the Poincar´ e-Birkhoff-Witt theorem 3.7.14 though.

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Let us show that a is a Lie algebra action. We need to check that the two maps Λ2 L2 ⊗ P1 → P1 coincide. Both maps are L1 -equivariant, so by the above remark it suffices to show that their restrictions to Λ2 L2 ⊗ v1 are equal. This is clear since both maps vanish on Λ2 N2 ⊗ v1 , and on the image of L1 ⊗ L2 ⊗ v1 they coincide by the L1 -equivariance of a. Property (a) follows from the case Q = L1 , R = P1 in the above remark; property (b) holds by construction. We are done. Second Proof. Consider the derived version of the induced module construction. Namely, for an emedding of Lie algebras N ⊂ L one may find Lie DG algebras L˜, N˜ placed in degrees ≤ 0 whose components are flat (e. g. free) R-modules, together with projections L˜ → L, N˜ → N which are quasi-isomorphisms, and morphism of Lie DG algebras N˜ → L˜ that lifts the embedding N ,→ L. Consider the enveloping associative DG algebra U (L˜) as an (L˜, N˜)-bimodule (with respect to the left and right actions). Denote by C the homological Chevalley complex of N˜ with coefficients in U (L˜) (where N˜acts on U (L˜) by right multiplication). This is a DG L˜-module (L˜acts by left multiplication) of degrees ≤ 0. There is an obvious ∼ isomorphism H 0 C −→ U (L)/U (L)N . As a graded module, C equals U (L˜) ⊗ Sym(N˜[1]). It carries a filtration C0 ⊂ C1 ⊂ · · · equal to the tensor product of the standard filtration U (L˜)· and the filtration Sym≤· (N˜[1]). This filtration is stable with respect to the differential, and there is a canonical isomorphism gri C = Symi (Cone(N˜ → L˜)). Therefore, as an object of the derived category of R-modules, gri C equals LSymi (L/N ). Let us return to the situation of our lemma. Choose resolutions as above for both (L1 , N1 ), (L2 , N2 ) and morphisms of Lie DG algebras φ˜ L : L˜ 1 → L˜ 2, φN ˜ : N˜1 → N˜2 that lift φ and for which the diagram φ˜ L

L˜1 −−−−→ x  

L˜2 x  

φN ˜

N˜1 −−−−→ N˜2 commutes. We get a morphism of filtered complexes φC : C1 → C2 . Notice that gri (φC ) is the ith symmetric power of φ˜ : Cone(N˜1 → L˜1 ) → Cone(N˜2 → L˜2 ). The condition of 3.7.4 assures that this φ˜, hence Sym· φ˜, is a quasi-isomorphism. Therefore φC is a quasi-isomorphism. Passing to H 0 we get our lemma.  Remark. The above lemma also shows that VX has the ´etale local nature. Namely, let π : X 0 → X be an ´etale map, LX 0 the π-pull-back of L = LX , VX 0 the corresponding vacuum module. One has an obvious morphism of Lie algebras π ∗ L\X → L\X 0 . It yields the morphism of vacuum modules π ∗ VX → VX 0 which is an isomorphism. 3.7.9. Lemma. Our V is a factorization algebra. Its D-module structure coincides with the canonical D-module structure defined by the factorization structure (see 3.4.7). Proof. One needs only to check the flatness along codimension 1 diagonals. Indeed, the unit in V is the canonical generator 1 of the induced module VX ; since 1 is horizontal, our D-module structure coincides with the canonical one (see 3.4.7). Assume that I = {1, . . . , n} and the codimension 1 diagonal X n−1 ,→ X n we consider is given by the equation xn−1 = xn . Let j 0 : U 0 ,→ X n be the

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219

complement to the union of diagonals x1 = xn , . . . , xn−1 = xn , and the projections q : X n → X n−1 , p : X n → X are, respectively, (x1 , . . . , xn ) 7→ (x1 , . . . , xn−1 ), (x1 , . . . , xn ) 7→ xn . The embeddings q ∗ L\X n−1 ,→ L\X n ←- p∗ L\X (see (i) in 3.7.6) yield a short exact sequence ∗

0 → q ∗ L\X n−1 → L\X n → j∗0 j 0 p∗ L`X → 0.

(3.7.9.1)

Consider the canonical filtration on the enveloping algebra U (L\X n ). The Poincar´e-Birkhoff-Witt theorem gr U = Sym holds for Lie algebras in any abelian tensor Q-category (see, e.g., [DM] 1.3.7), so gra U (L\X n ) = Syma L\X n . Let us consider (3.7.9.1) as a two-step increasing filtration on L\X n . Notice that the sub- and quotient modules in our filtration are Tor-independent. Therefore it defines on every Syma L\X n an increasing filtration with successive quotients q ∗ (Syma−m L\X n−1 ) ⊗ ∗ ∗ j∗0 j 0 p∗ (Symm L`X ). Set Q := U (L\X n )/q ∗ U (L\X n−1 ); we see that Q = j∗0 j 0 Q. The Lie algebra L\0 ⊂ q ∗ L\X n−1 ⊂ L\X n acts on U (L\X n ) by right multiplications, and VX n is the DX n -module of coinvariants U (L\X n )L\ . The short exact sequence of 0

L\0 -modules 0 → q ∗ U (L\X n−1 ) → U (L\X n ) → Q → 0 yields the short exact sequence of coinvariants 0 → q ∗ VX n−1 → VX n → QL\ → 0.

(3.7.9.2)

0

Indeed, one need only check the exactness from the left which follows from the fact that every morphism from a j∗0 OU 0 -module to q ∗ VX n−1 vanishes. Both sub- and quotient modules in (3.7.9.2) are flat along our diagonal X n−1 ⊂ ∗ n X (recall that QL\ = j∗0 j 0 QL\ ), so the same is true for the middle term; q.e.d.  0

0

3.7.10. Consider the chiral algebra V r that corresponds to our factorization algebra V . So V r is the right D-module corresponding to the left DX -module VX = U (L\X )/U (L\X )L\0 . The chiral product µ : j∗ j ∗ VX  VX → ∆∗ VX may be described explicitly as follows. The exact sequence 0 → p∗2 L\X → L\X×X → j∗ j ∗ p∗1 L`X → 0 (see (3.7.9.1)) shows that the embedding p∗1 L\X ,→ L\X×X yields an isomorphism between the induced modules j ∗ p∗1 VX −→ j ∗ U (L\X×X )/U (L\X×X )p∗1 L\X . In partic∼ ular, L\X×X acts on j∗ j ∗ p∗1 VX ; the same is true for j∗ j ∗ p∗2 VX . Therefore the tensor product j∗ j ∗ VX  VX is an L\X×X -module. There is a canonical isomorphism of L\X×X -modules j∗ j ∗ VX×X −→ j∗ j ∗ VX  VX which sends the generator 1X×X to ∼ 1X  1X . Our chiral product µ is the composition of the inverse to this isomorphism and the projection j∗ j ∗ VX×X → (j∗ j ∗ VX×X )/VX×X = ∆∗ VX where the latter identification is the structure isomorphism ∆∗ VX×X = VX . Notice that µ is a morphism of L\X×X -modules. The morphism L\X → VX , l 7→ l · 1, kills L\0 , so it yields a morphism of right DX -modules i : L → V r . The next proposition finishes the proof of 3.7.1: 3.7.11. Proposition. The above i : L → V r is a morphism of Lie∗ algebras. It satisfies the universality property: for every chiral algebra A the map HomCA(X) (V r , A) → HomLie∗ (X) (L, ALie ), ϕ 7→ ϕi, is bijective. Proof. We begin with a useful general construction.

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Let A be a chiral algebra, M an A-module. Consider the Lie algebra56 A\X := tensor category M` (X) (see 3.7.6). One has H 0 p2∗ j∗ j ∗ A  M =  ωX ) ⊗! M = A\X ⊗ M ; here p2 : X × X → X is the second projection. Therefore we get a canonical morphism ALie\ in the X 0 H (p2∗ j∗ j ∗ A

(3.7.11.1)

• = •M := p2∗ (µM ) : A\X ⊗ M → M.

Exercise: Check that this is a Lie algebra action of A\X on the D-module M . 0 (X, A) = Γ(X, h(A)) ⊂ A\X this action coincides On the Lie subalgebra A\0 = HDR with the usual h(A)-action for the ALie -module structure on M (see 2.5.4). Notice that •M determines µM uniquely (since p2∗ is right exact and for DX×X modules supported on the diagonal it is fully faithful). In particular, we have a canonical action of A\X on A. The morphism A\X → A` , a\ 7→ a\ •1, coincides with the standard projection A\X → A\X /A\0 ⊗OX = A` (see (ii) in 3.7.6). A morphism of chiral algebras A → A0 is the same as a morphism of D-modules ϕ : A → A0 which is compatible with • actions (i.e., ϕ •A = •A0 (ϕ\ ⊗ϕ)) and sends 1A to 1A0 . Let us return to our chiral algebra V r . Let •L : L\X ⊗ VX → VX be the morphism •V r composed with i\ : L\X → VXr\ . Sublemma. •L coincides with our old canonical L\ -action on VX . Proof of Sublemma. We know that the chiral product µ : j∗ j ∗ VX VX → ∆∗ VX is a morphism of L\X×X -modules (see 3.7.10). Restricting this action to p∗2 L\X ⊂ L\X×X and pushing µ forward by p2 , we see that • : VXr\ ⊗ VX → VX is a morphism of L\X -modules. Here L\X acts on VXr\ := H 0 p2∗ j∗ j ∗ p∗1 VX via the embedding p∗2 L\X ,→ L\X×X and the L\X×X -action on j∗ j ∗ p∗1 VX = U (L\X×X )/U (L\X×X )p∗2 L\X (see 3.7.10), and on VX in our old canonical way (as on the vacuum module). One checks easily that i\ : L\X → VXr\ is a morphism of L\X -modules. So for l, l0 ∈ L\X , v ∈ VX one has l0 (l •L v) = [l0 , l] •L v + l •L (l0 v). Since l •L 1 = l1 and 1 generates  VX as an L\X -module, we are done. End of proof of Proposition. Take l ∈ L\0 ⊂ L\X , l0 ∈ L. By the sublemma, l •L i(l0 ) = i(adl l0 ). This shows that i is a morphism of Lie∗ algebras (see 2.5.2). Our sublemma also implies that V r is generated by i(L) as a chiral algebra, so the map HomCA(X) (V r , A) → HomLie∗ (X) (L, ALie ) is injective. Let us define its inverse. Let ψ : L → ALie be a morphism of Lie∗ algebras. It yields a morphism of Lie algebras ψ \ : L\X → A\X ; hence •A defines an L\X -action on A. Since L\0 ⊂ L\X kills 1A , we get a morphism of L\X -modules ψ˜ : VX → A` which sends 1 to 1A . One ˜ = ψ, so it remains to check that ψ˜ is a morphism of chiral algebras. Since it has ψi ˜ V r on the image of is a morphism of L\X -modules, we know that µA (ψ˜2 ) equals ψµ j∗ j ∗ L  V r → j∗ j ∗ V r  V r . Since the image of L generates V r as a chiral algebra, we are done.  3.7.12. Let Lα be a family of Lie∗ algebras. Consider the Lie∗ algebra ⊕Lα . It follows immediately from the universality property of ⊗ and U (see 3.4.15 and 56 Specialists

in vertex algebras sometimes call a similar object “the Lie algebra of local fields.”

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221

3.7.1) that one has a canonical isomorphism of chiral algebras Y ∼ U ( Lα ) −→ ⊗U (Lα ).

(3.7.12.1)

In particular, for any Lie∗ algebra L the diagonal map L → L × L yields a coproduct morphism U (L) → U (L) ⊗ U (L) which makes U (L) a Hopf chiral algebra (see 3.4.16). 3.7.13. We are going to prove a chiral version of the Poincar´e-Birkhoff-Witt theorem. Let A be a chiral algebra equipped with a filtration A0 ⊂ A1 ⊂ · · · ⊂ A (see 3.3.12). Notice that A0 is a chiral subalgebra of A and A1 is an A0 -submodule. We say that our filtration A· is 1-generated if all the arrows j∗ j ∗ Ai  Aj → ∆∗ Ai+j , i, j ≥ 0, are surjective. If R ⊂ A is a chiral subalgebra of A and M ⊃ R is an R-submodule of A that generates A, then there is a unique 1-generated filtration A· such that A0 = R, A1 = M . Namely, Ai+1 := µ(j∗ j ∗ A1  Ai ) for i ≥ 0. We say that this filtration is generated by (R, M ); if R = ωX · 1, then we simply say that our filtration is generated by M . Assume that M is a Lie∗ subalgebra of A,57 R is commutative, and M normalizes R. Then our filtration is commutative (see 3.3.12), and the obvious surjective morphism of Z-graded commutative DX -algebras Sym· M ` → gr· A` is compatible with the coisson brackets. 3.7.14. Let us return to our situation. Denote by Ltor the OX -torsion submodule of L. Since any binary chiral operation on an O-torsion D-module vanishes, the canonical morphism L → U (L) kills the commutator [Ltors , Ltors ] (which is a ∼ Lie ideal in L). Therefore U (L) −→ U (L/[Ltor , Ltor ]). The chiral algebra U (L) is generated by the image of L which is a Lie∗ subalgebra of U (L). The corresponding filtration U (L)· is called the Poincar´e-BirkhoffWitt filtration. We have a canonical surjective morphism of coisson algebras Sym· L`  gr· U (L)` ,

(3.7.14.1) which factors through (3.7.14.2)

Sym· (L/[Ltor , Ltor ])`  gr· U (L)` .

Theorem. The morphism (3.7.14.2) is an isomorphism. In particular, if L is OX -flat, then (3.7.14.1) is an isomorphism. Proof. We can assume that X is affine. Consider the projection U (L\X ) → VX = U (L)` . As follows from the proof of 3.7.11 (see the sublemma), our PBW filtration equals the image of the usual PBW filtration on the enveloping algebra U (L\X ). Consider the filtered OX -complex C from the second proof of 3.7.8 for the pair L\0 ⊂ L\X . Then H 0 C equals U (L)` (as a filtered module) and gri C = LSymiOX (L` ). So one has H 0 gri C = Symi L` , H −1 gri C = Symi−2 (L/Ltor )` ⊗ tor` X TorO , Ltor` )Σ2 = Symi−2 (L/Ltor )` ⊗ ∆! (Ltor  Ltor )` sgn . The differential 1 (L −2,1 −1 d1 : H gr2 C → H 0 gr1 C in the spectral sequence coincides with the Lie∗ bracket 57 The

next conditions hold automatically if R = ωX · 1.

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∆! (Ltor  Ltor )` sgn → Ltor` ⊂ L` .58 Now recall that C is a module over some Lie algebra whose quotient is L\X . This implies that the image of the differential d1 : H −1 gr C → H 0 gr C is the ideal generated by the image of d1−2,1 , and all higher −2,1 dp,q ). We are done.  r ’s for p + q = −1 vanish (since they all vanish on Ker d1 3.7.15. We are going to describe U (L)-modules in more explicit terms (see 3.7.18). Notice that, contrary to the usual (non-chiral) setting, U (L)-modules are not the same as L-modules. Proposition. The obvious functor M(X, U (L)) → M(X, L) admits a left adjoint induction functor (3.7.15.1)

Ind = IndL : M(X, L) → M(X, U (L)).

Proof. For a D-module M an L-action on it amounts to a Lie∗ algebra structure on L ⊕ M such that the bracket on M vanishes and L ,→ L ⊕ M is a morphism of Lie∗ algebras. So an L-module M yields a chiral algebra U (L⊕M ). The Gm -action on M by homotheties defines a Z-grading on this chiral algebra. Its zero component is U (L). The degree 1 component is Ind M . To check the universality property use 3.3.5(i).  The induction functor is compatible with the ´etale localization of X. Let us describe the induced module Ind M explicitly. We can assume that X is affine. Consider the Lie algebra L\X defined in (iii) in 3.7.6. The Lie∗ action of L on M yields an action on M of the Lie subalgebra L\0 ⊂ L\X . Now one has a canonical isomorphism (3.7.15.2)

∼

Ind M −→ U (L\X )

⊗

M.

U (L\0 )⊗OX

Remarks. (i) The PBW filtration on U (L) defines a filtration (Ind M )i := U (L)i M on Ind M . By construction, we have a natural morphism (Sym L) ⊗ M  gr Ind M which is an isomorphism if L is OX -flat.59 (ii) Let A be any chiral algebra. The functor IndA : M(X, ALie ) → M(X, A) Ind

from 3.3.6 is the composition M(X, ALie ) −−→ M(X, U (ALie )) → M(X, A); the latter arrow is N 7→ N/JN where J ⊂ U (ALie ) is the ideal such that A = U (ALie )/J. (iii) The above constructions and statements remain valid if we consider Omodules equipped with L- and U (L)-actions (see 2.5.2, 3.3.5(ii)) instead of Dmodules. 3.7.16. Define a chiral L-module, or Lch -module, as a DX -module M equiped with a chiral pairing µ = µLM ∈ P2ch ({L, M }, M ) (called a chiral action of L on M ) which satisfies the following “truncated” version of the Lie algebra action axiom. Let j 0 : U 0 ,→ X × X × X be the complement to the diagonals x1 = x3 and x2 = x3 . Consider a chiral operation µL1 M (idL1 , µL2 M ) − µL2 M (idL2 , µL1 M ) ∈ 58 To check this, we can assume that L = Ltor and it is supported at a single point x ∈ X, i.e., L = ix∗ F for some Lie algebra F . Then L\ = jx∗ (F ⊗ OUx ), L\0 = F ⊗ OX and C is the Chevalley complex of F with coefficients in U (F ) ⊗ jx∗ OUx . Now the statement is clear. 59 The latter assertion follows from 3.7.14 together with the construction of Ind M in the proof of the proposition.

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223

∗

P3ch ({L, L, M }, M ) and its restriction ν : j∗0 j 0 (L  L  M ) → ∆(3) M . We demand that ν equals the composition ∗

(3)

j∗0 j 0 (L  L  M ) → ∆1=2 j∗ j ∗ (L  M ) → ∆∗ M. ∗

(3.7.16.1)

Here ∆1=2 : X × X ,→ X × X × X is the embedding (x, y) 7→ (x, x, y), the first arrow is [ ]L  idM , and the second one is ∆1=2 (µLM ). ∗ Example. Assume that M is supported at a point x ∈ X. Then a chiral L-action on M is the same as a ∗ action of the Lie∗ algebra jx∗ jx∗ L on M . Here jx : Ux ,→ X is the complement to x. Chiral L-modules form an abelian category which we denote by M(X, Lch ). A chiral operation ϕ ∈ PIch ({Mi }, N ), where Mi , N are chiral L-modules, is said to be compatible with the chiral L-actions (or to be a chiral Lch -operation) if P µLN (idL , ϕ) = ϕ(µLMi , idMi0 )i0 6=i . Let PLchch I ({Mi }, N ) ⊂ PIch ({Mi }, N ) be the i∈I

subspace of such an operations. They are closed under the composition, so we have defined a pseudo-tensor structure M(X, Lch )ch on M(X, Lch ). 3.7.17. If X is affine, then M(X, Lch )ch can be described as follows. The proofs are left to the reader. As in the beginning of the proof in 3.7.11, a chiral pairing µLM amounts to a morphism •µ : L\X ⊗ M → M such that its restriction to L\0 ⊗ M = Γ(X, h(L)) ⊗ M comes from a ∗ operation ∈ P2∗ ({L, M }, M ). L\X

Lemma. µLM is a chiral action if and only if •µ is an action of the Lie algebra on M . 

Let us describe chiral Lch -operations in these terms. (I) Let Mi , i ∈ I, be chiral L-modules. Then j∗ j (I)∗  Mi is naturally an L\X I module (recall that L\X I is a Lie algebra in the tensor category of left DX I -modules, see (iii) in 3.7.6). To see this, consider for i ∈ I the morphism of DX I˜ -modules ˜ (I)

˜

(I)

µLMi  (  idMi0 ) : j∗ j (I)∗ L  (Mi ) → ∆i∗ j∗ j (I)∗  Mi where ∆i : X I ,→ 0 i 6=i

˜

X I = X × X I is the diagonal x = xi . Its push-forward by the projection to X I (I) (I) is a morphism of DX I -modules ai : L\X I ⊗ (j∗ j (I)∗  Mi ) → j∗ j (I)∗  Mi . This (I)

is an L\X I -action on j∗ j (I)∗  Mi , and the actions ai for different i ∈ I mutually P (I) commute. The promised L\X I -module structure on j∗ j (I)∗  Mi is ai . i∈I

(I)

If N is another Lch -module, then ∆∗ N is an L\X I -module: the action mor(I)

∆(I) (•µ )

(I)

(I)

phism is L\X I ⊗ ∆∗ N = ∆∗ (L\X I ⊗ N ) −−∗−−−→ ∆∗ N . (I)

(I)

Lemma. A chiral operation ϕ : j∗ j (I)∗  Mi → ∆∗ N is compatible with the chiral L-actions if and only if ϕ is a morphism of L\X I -modules. 3.7.18. A chiral U (L)-module structure on a DX -module M defines in the obvious manner a chiral L-action on M . A chiral U (L)-operation between U (L)modules (see 3.3.4) is automatically an Lch -operation. So we have a pseudo-tensor functor (3.7.18.1)

M(X, U (L))ch → M(X, Lch )ch .

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Proposition. This is an equivalence of pseudo-tensor categories. Proof. Use 3.7.17 and (3.7.15.2). A chiral Lch -operation is the same as a chiral U (L)-operation by Remark (iii) in 3.3.4 (applied to P = L ⊂ U (L)).  Corollary. The functor Ind from 3.7.15 extends naturally to a pseudo-tensor functor M(X, L)ch → M(X, U (L))ch left adjoint to the obvious pseudo-tensor functor M(X, U (L))ch → M(X, L)ch . Proof. We can assume that X is affine. Let Pi , i ∈ I, be L-modules. According (I) to (3.7.15.2), the L\X I -module j∗ j (I)∗  IndPi is equal to the induced L\X I -module U (L\X I )

⊗ U (L\0 )⊗OX I

(I)

j∗ j (I)∗  Pi . We are done by the second lemma in 3.7.17.



3.7.19. Let x ∈ X be a point, jx : Ux ,→ X its complement. The topological associative algebra U (L)as x (see 3.6.2–3.6.7) can be described as follows. This algebra depends only on the restriction of U (L) to Ux , so it is convenient to change notation for the moment and assume that L is a Lie∗ algebra on Ux . Consider the ˆ Lie (L)) ˆ Lie (jx∗ L) (see 2.5.12). Denote by U (h ˆ Lie (L) := h topological Lie algebra h x x x Lie ˆ (L) defined as the completion of the plain the topological enveloping algebra of h x enveloping algebra with respect to the topology whose base is formed by the left ˆ Lie (L). ideals generated by an open subspace in h x Proposition. There is a canonical isomorphism of topological associative algebras (3.7.19.1)

∼ ˆ Lie (L)) −→ U (h U (L)as x x .

ˆ Lie (L) and U (L)as are completions of, respectively, hx (jx∗ L) and Proof. h x x hx (jx∗ U (L)) with respect to certain topologies. It follows from the definitions that the canonical morphism hx (jx∗ L) → hx (jx∗ U (L)) is continuous with respect ˆ Lie (L) → U (L)as . This to these topologies; thus we have a continuous morphism h x x is a morphism of Lie algebras (as follows from Remark (i) in 3.6.6). So it yields ˆ Lie (L)) → U (L)as . According a morphism of topological associative algebras U (h x x to 2.5.15, 3.7.18, 3.6.6, and Example in 3.7.16 this morphism yields an equivalence between the categories of discrete modules. Since our algebras are complete and separated, and their topologies are defined by systems of left ideals, this shows that our morphism of topological algebras is an isomorphism.  3.7.20. Twisted enveloping algebras. Let L[ be an (automatically central) ω-extension of a Lie∗ algebra L (see 2.5.8). Let 1[ be the corresponding horizontal section of L[` (so L` = L[` /OX 1[ ). The [-twisted chiral enveloping algebra of L is the quotient U (L)[ of U (L[ ) modulo the ideal generated by 1 − 1[ . So for any chiral algebra A a morphism U (L)[ → A amounts to a morphism of Lie∗ algebras L[ → ALie which maps 1[ to 1A . As in 3.7.13, the image of L[ in U (L)[ defines the PBW filtration on U (L)[ , and one has a canonical surjective map of coisson algebras (3.7.20.1)

Sym· L`  gr· U (L)[ .

One has the twisted version of the Poincar´e-Birkhoff-Witt theorem:

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225

Proposition. If L is OX -flat then (3.7.20.1) is an isomorphism. Proof. Notice that ε := 1 − 1[ is a central horizontal section of U (L[ )` . It defines the endomorphism ε· of the D-module U (L[ ), and U (L)[ is its cokernel. This endomorphism shifts the PBW filtration of U (L[ ) by 1, and the corresponding endomorphism of gr U (L[ )` = Sym L[` (see 3.7.14) is multiplication by 1[ ∈ L` . It is injective with the cokernel equal to Sym L` . This implies our PBW.  The coproduct on U (L[ ) (see 3.7.12) yields a coaction of the Hopf chiral algebra U (L) on the chiral algebra U (L)[ . If L is OX -flat, then U (L)[ is a U (L)-cotorsor (see 3.4.16). The functor (we use the notation of 2.5.8 and 3.4.16) (3.7.20.1)

P(L) → P(U (L)),

L[ 7→ U (L)[ ,

is naturally a morphism of Picard groupoids. Exercise. Show that (3.7.20.1) is an equivalence of Picard groupoids.60 3.7.21. The morphism L[ → U (L)[ yields a functor M(X, U (L)[ ) → M(X, L) (since any ∗ action of ωX on any module is trivial, L[ -modules are the same as Lmodules). It admits a left adjoint functor Ind[ = Ind[L : M(X, L) → M(X, U (L)[ ) (see 3.7.15). For an L-module M we have Ind[L M = IndL[ M/(1 − 1[ )IndL[ M . One defines the canonical filtration on Ind[ M as the image of the canonical filtration on IndL[ M , so we have a canonical morphism of graded Sym L` -modules (3.7.21.1)

Sym· L` ⊗ M  gr· Ind[ M.

Lemma. If L is OX -flat, then (3.7.21.1) is an isomorphism. Proof. Same as that of the proposition in 3.7.20 (see Remark (i) in 3.7.15).  3.7.22. Here is the twisted version of compatibility (3.7.19.1). Assume that we are in the situation of 3.7.19, so L is a Lie∗ algebra on Ux and let L[ be an extension of L by ωUx . Let us compute the topological associative algbera [ as U (L)[as x = (U (L) )x . ˆ Lie (L) is continuous ˆ Lie (L[ )  h The morphism of topological Lie algebras h x x [ 1 Lie ˆ ˆ and open. Its kernel is the image of k = hx (jx∗ ωUx ) −→ hx (L[ ). The twisted ˆ Lie (L))[ is the quotient of U (h ˆ Lie (L[ )) modulo topological enveloping algebra U (h x

x

the closed ideal generated by the central element 1[ (1) − 1. Proposition. There is a canonical isomorphism of topological associative algebras (3.7.22.1)

∼

ˆ Lie (L))[ −→ U (L)[as . U (h x x

ˆ Lie (L[ )) → U (L[ )as → U (L)[as Proof. The composition of the morphisms U (h x x x [ Lie ˆ (L))[ → U (L)[as . As follows from sends 1 to 1, so it yields a morphism U (h x x 3.7.19, this morphism induces an equivalence between the categories of discrete modules; hence it is an isomorphism (use the fact that our algebras are complete and separated and their topologies are defined by systems of left ideals).  60 Hint: for C ∈ P(U (L)) the corresponding ω-extension L[ is the Lie∗ subalgebra of C defined as the preimage of L × C ⊂ U (L) ⊗ C by the coaction morphism δ : C → U (L) ⊗ C.

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3.7.23. Let A be a chiral algebra and L a Lie∗ algebra that acts on A (see 3.3.3). For a chiral algebra B consider the set of pairs (i, ψ) where i: A → B is a morphism of chiral algebras and ψ: L → B a morphism of Lie∗ algebras, such that i is a morphism of L-modules (here L acts on B via ψ and the adjoint action of B). Such pairs depend on B functorially, and it is easy to see that there exists a universal pair (i, ψ); the corresponding chiral algebra B is denoted by A ∗ U (L). Indeed, consider the L-action on A as an action of L on the Lie∗ algebra ALie ,61 so we have the semi-direct product ALie o L of the Lie∗ algebras ALie and L. Our A ∗ U (L) is the quotient of U (ALie o L) modulo the obvious relations (which say that the morphism A → A ∗ U (L) is a morphism of chiral algebras, not just of Lie∗ algebras). Here is another construction of A ∗ U (L). Recall that Indch L is a pseudo-tensor ch functor (see 3.7.18), so IndL A is a Liech algebra. The image of 1A is the unit ch ch in Indch L A, so IndL A is a chiral algebra. The canonical morphism i: A → IndL A ch and the morphism ψ: L → IndL A that comes from the chiral action of L on 1A satisfy the above properties. One checks immediately the universality property, so A ∗ U (L) = IndL A. We leave it to the reader to check that the PBW filtration on IndL A is a chiral algebra filtration, and the PBW morphism (3.7.23.1)

A ⊗ Sym· L` → gr· (A ∗ U (L))

is a morphism of chiral algebras. According to Remark (i) in 3.7.15, if L is OX -flat, then (3.7.23.1) is an isomorphism. 3.7.24. The constructions of 3.7.23 and 3.7.20 easily combine. Namely, assume that we have A, L as in 3.7.23 and L[ as in 3.7.20. Look at the pairs i: A → B, ψ [ : L[ → B that satisfy the same conditions as i, ψ from 3.7.23 and assume that ψ [ (1[ ) = 1B . As in 3.7.23, there exists a universal pair (i, ψ [ ), and we denote the corresponding B by A ∗ U (L)[ . One has A ∗ U (L)[ = A ∗ U (L[ )/(1−1[ )A ∗ U (L[ ). [ [ There is a canonical identification A ∗ U (L) = IndL A. The canonical filtration on Ind[L A is a chiral algebra filtration, and the canonical morphism A ⊗ Sym· L` → gr· (A ∗ U (L)[ ) is a morphism of chiral algebras. If L is OX -flat, then this is an isomorphism (see 3.7.21). This version of the PBW theorem will be of use in 3.8. Remark. For M ∈ M(X) a structure of the A ∗ U (L)[ -module on M amounts to a chiral action on M of the semi-direct product ALie o L[ of the Lie∗ algebras ALie and L[ such that the chiral ALie -action is actually a chiral A-action and the chiral action of ω ⊂ L[ is the unit action. This follows immediately from 3.7.18 and the construction of A ∗ U (L)[ as a quotient of U (ALie o L[ ). 3.7.25. Virasoro vectors. Consider the Virasoro extension ΘcD of ΘD of central charge c ∈ k (see (c) in 2.5.10). The twisted enveloping algebra U (Θ)c := U (ΘD )c is called the chiral Virasoro algebra of central charge c. For a chiral algebra A a Virasoro vector, a.k.a. stress-energy tensor, of central charge c is a morphism of chiral algebras U (Θ)c → A which is the same as a morphism of Lie∗ algebras ιc : ΘcD → A which sends 1c : ωX ,→ ΘcD to 1A . 61 We

use notation of 3.3.2.

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227

A Virasoro vector yields two actions of the Lie algebra Θ on the sheaf A, called the adjoint and Lie action, defined, respectively, by the formulas (3.7.25.1)

Lieτ a := adτ a − a · τ.

adτ a := adι(τ ) a,

Here · is the action via the right D-module structure and ι : ΘD → A/ω1A is defined by ιc . The adjoint action is DX -linear, while the Lie action is compatible with the usual ΘX -action on DX . Example. Sugawara’s construction. Let g be a finite-dimensional Lie algebra, κ a symmetric ad-invariant bilinear form on g. So we have the Kac-Moody Lie∗ algebra gκD (see 2.5.9); let A be the twisted enveloping algebra U (gD )κ . Let γ = Σai ⊗ bi ∈ g ⊗ g be a symmetric ad-invariant tensor. It yields endomorphisms adγ : x 7→ Σ[ai , [bi , x]] and γ : x 7→ Σκ(ai , x)bi of g which commute with the adjoint action; one has Tr(γ ) = κ(γ) := Σκ(ai , bi ). We say that γ is κ-normalized if adγ + γ = idg . Remark. A κ-normalized γ is unique; it exists if and only if the bilinear form x, y → 7 κ(x, y) + 21 Tr(adx ady ) on g is non-degenerate. Consider γ as a symmetric section of gD  gD ⊂ A  A. The image of γ by the Lie∗ bracket morphism A  A → ∆∗ A equals κ(γ)φ1A where φ ∈ ∆∗ ωX is a canonical skew-symmetric section which can be written in terms of local coordinates as (dx · δ(x − y))∂x . Thus the skew-symmetric morphism of OX×X -modules (3.7.25.2)

γ

µ

OX×X (∆) − → j∗ j ∗ A  A − → ∆∗ A

kills OX×X (−2∆); i.e., it yields a skew-symmetric morphism ν → ∆∗ A where ν was defined in (c) in 2.5.10. The restriction of this morphism to ωX ⊂ ν equals κ(γ)1A . Replacing ν by the induced D-module and using (2.5.10.1), we get a morphism of DX -modules (3.7.25.3)

(−2κ(γ))

sγ : ΘD

→A

such that sγ 1−2κ(γ) = 1A ; this is the Sugawara tensor. Lemma. If γ is κ-normalized, then sγ is a Virasoro vector of central charge −2κ(γ), and the corresponding adjoint action of ΘD on A coincides with the one defined by the obvious action 62 of ΘD on gκD . Proof. Straightforward computation.



3.8. BRST, alias semi-infinite, homology We considered the BRST reduction in the classical setting in 1.4.21–1.4.26; now we turn to the quantum version. The Becchi-Rouet-Stora-Tyupin construction arose in mathematical physics. It came to representation theory in a pioneering work of B. Feigin [F] (his term “semi-infinite” reflects the view of a Tate vector space as a sum of its compact and discrete “halves”63 ) and was further developed in [FGZ]. These articles deal with a linear algebra setting; the BRST differential is given by an explicit formula written in terms of a redundant Z-grading. Its conceptual characterization (the BRST 62 See 63 Or,

the first Remark in 2.5.9. perchance, the view of our momentary existence suspended between the two eternities.

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property of 3.8.10(iii) and 3.8.21(iii) below) is due to F. Akman [A]. An important example of the BRST construction is the quantum Drinfeld-Sokolov reduction which produces from (the chiral envelope of) an affine Kac-Moody algebra the chiral W algebra (see Chapter 14 of [FBZ] and references therein). Some closely related subjects we do not touch: the BRST homology of finite quantum groups [Ar] and a general homological algebra approach to BRST-like constructions [Vo]. We discuss BRST homology first in the setting of chiral algebras (3.8.1–3.8.15) and then in the more traditional setting of Tate’s linear algebra (3.8.17–3.8.22). The two pictures are compared in 3.8.23–3.8.25. W -algebras are briefly mentioned in 3.8.16; we refer the reader to [FBZ] for a thorough treatment. We work in the DG super setting (see 1.1.16) skipping the adjective “DG super”. 3.8.1. Chiral Weyl algebras. Let T be a DX -module; we assume that it has no O-torsion. Let h iT ∈ P2∗ ({T, T }, ωX ) be a skew-symmetric64 ∗ pairing. Set T [ := T ⊕ ωX . This is a Lie∗ algebra with commutator equal to h iT (plus zero components). So T [ is an extension of a commutative Lie∗ algebra T by ωX . Let W = W(T, h iT ) be the corresponding twisted enveloping chiral algebra U (T )[ (see 3.7.20). This is the Weyl algebra of (T, h iT ). It carries the PBW filtration so that T [ = W1 , and gr· W = Sym· T (see 3.7.20). The classical, or coisson, Weyl algebra Wc is Sym T equipped with a coisson bracket that equals h iT on T = Sym1 T . It differs from gr· W as a coisson algebra: the coisson bracket on gr· W is trivial. 3.8.2. Assume our T is a vector DX -bundle and h iT is non-degenerate; i.e., the corresponding morphism T → T ◦ (see 2.2.16) is an isomorphism. Then the center of W(T ) equals ωX . Let A be any chiral algebra. The tensor product A ⊗ W(T ) (see 3.4.15) contains A and W(T ) as subalgebras, and A coincides with the centralizer of T [ ⊂ W(T ) ⊂ A ⊗ W(T ). We have a pair of adjoint functors (3.8.2.1)

M(X, A) → M(X, A ⊗ W(T )),

M(X, A ⊗ W(T )) → M(X, A),

where the left functor is M 7→ M ⊗ W(T ) and the right one sends N ∈ M(X, A ⊗ W(T )) to the centralizer of T . 3.8.3. Lemma. The functor M(X, A) → M(X, A ⊗ W(T )) is fully faithful. Its image consists of those A ⊗ W(T )-modules on which the Lie∗ algebra T [ acts in a locally nilpotent way. In particular, if T is purely odd, then the functors (3.8.2.1) are mutually inverse equivalences of categories.  3.8.4. Let us return to situation 3.8.1 (so we drop the non-degeneracy condition). Assume that we have a direct sum decomposition T = V ⊕ V 0 such that h iT vanishes on both V and V 0 . Let h i ∈ P2∗ ({V 0 , V }, ωX ) be the restriction of h iT (notice that the sign of h i depends on the order of V , V 0 ). One recovers (T, h iT ) from (V, V 0 , h i) in the obvious way, so we will write W = W(V, V 0 , h i). There are obvious embeddings of chiral algebras Sym V, Sym V 0 ,→ W(V, V 0 ). 64 As always, “skew-symmetric” refers to the “super” tensor structure: if T is purely odd, this amounts to “naive symmetric.”

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Define an extra Z-grading on T [ setting T [(−1) := V , T [(1) := V 0 , T [(0) := ωX . (−1) (1) Therefore W acquires an extra Z-grading W(·) .65 One has W1 = V , W1 = V 0 , a+n a−n and gra W(n) = Sym 2 V ⊗ Sym 2 V 0 . Remark. W carries another filtration – the one generated by (Sym V 0 , V + Sym V 0 ) (see 3.7.13). This filtration is commutative, and the associated graded algebra gr0· W equals Sym V 0 ⊗ Sym· V . So we have a canonical identification of graded algebras66 gr0 W = Sym(V 0 ⊕ V ) = Wc which is compatible with coisson brackets. So if V , V 0 are mutually dual vector DX -bundles, then W is a SymV 0 cdo (see 3.9.5 for terminology). (0)

3.8.5. The Tate extension revisited. Consider the DX -module W2 . It (0) (0) contains W0 = W0 = ω and W2 /ω = gr2 W(0) = V ⊗ V 0 . The adjoint action of (0) (±1) (0) W2 preserves W1 . Since ω ⊂ W2 acts trivially, we get ∗ actions of V ⊗ V 0 on (0) (0) V , V 0 . Since W2 = µ(V, V 0 ) + ω, we see that W2 is a Lie∗ subalgebra of W. Recall that in 1.4.2 we defined the associative∗ algebra structure on V ⊗ V 0 together with its left action on V and right action on V 0 . We leave it to the reader to check that the above action of V ⊗V 0 on V coincides with that of 1.4.2, the one on (0) V 0 is opposite to that of 1.4.2, and the Lie∗ algebra structure on V ⊗ V 0 = W2 /ω coincides with the Lie∗ algebra structure associated with the associative∗ algebra (0) structure from 1.4.2. So the quotient V ⊗ V 0 = W2 /ω is the Lie∗ algebra GLie ∗ from 2.7.2, and we have an extension of Lie algebras (3.8.5.1)

(0)

0 → ω → W2 → G → 0.

This extension canonically identifies with the extension G−[ opposite to the Tate (0) extension G[ from 2.7.2. Namely, consider the chiral product j∗ j ∗ V V 0 → ∆∗ W2 . 0 0 Its restriction to V  V equals −h i : V  V → ∆∗ ωX , and its composition with (0) the projection W2 → V ⊗ V 0 is the standard projection j∗ j ∗ V  V 0 → ∆∗ (V ⊗ V 0 ). Therefore it yields a canonical identification of extensions (see the construction of G[ in 2.7.2) (3.8.5.2)

(0) ∼

W2 −→ G−[ .

This identification is obviously compatible with the Lie∗ brackets. 3.8.6. Chiral Clifford algebras. Let L be a (DG super) DX -module; suppose that it is a finite complex of vector DX -bundles. Let L◦ be its dual; denote by ( ) ∈ P2∗ ({L◦ , L}, ωX ) the canonical non-degenerate pairing (see 2.2.16). Denote the corresponding pairing ∈ P2∗ ({L◦ [−1], L[1]}, ωX ) by h i; set Cl = Cl(L, L◦ , ( )) := W(L[1], L◦ [−1], h i). This is our Clifford chiral algebra. As in 3.8.4, Cl contains subalgebras Sym(L◦ [−1]) and Sym(L[1]). It carries an extra Z-grading Cl(·) and the PBW filtration Cl· with Cl1 = L[−1] ⊕ L[1] ⊕ ω and gr Cl = Sym(L◦ [−1] ⊕ L[1]). (0) Consider the Lie∗ algebra Cl2 . The chiral product j∗ j ∗ (L◦ L) = j∗ j ∗ (L◦ [−1] (1) (−1) (0) L[1]) = j∗ j ∗ (Cl1  Cl1 ) → Cl2 yields the ∗ pairing equal to ( ), so it defines, 65 Notice that, unless V, V 0 are purely odd, the Z-grading mod 2 does not coincide with the structure “super” Z/2-grading. 66 The grading we consider comes from the grading on W; we forget about the grading coming from the filtration.

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as in 3.8.5, an isomorphism of ω-extensions of G = L ⊗ L◦ (cf. (3.8.5.2)) (0) ∼

Cl2 −→ G[

(3.8.6.1)

where G[ is the Tate extension defined by L, L◦ , ( ). This is an isomorphism of Lie∗ (0) (±1) algebras, and it identifies the adjoint action of Cl2 /ω on Cl1 with the usual action of G on L◦ [−1], L[1]. The proof coincides with that of the similar statement for Weyl algebras plus a sign exercise. 3.8.7. Let now L be a Lie∗ algebra on X which satisfies the conditions of 3.8.6 as a mere DX -complex. So we have the Clifford algebra Cl and the Tate extension (0) gl(L)[ of the Lie∗ algebra gl(L) = L ⊗ L◦ identified with Cl(T )2 . The adjoint action of L yields a morphism of Lie∗ algebras L → gl(L). Denote by L[ the pullback to L of the extension gl(L)[ ; this is the Tate extension of L. So we have a canonical morphism of Lie∗ algebras β : L[ → gl(L)[ ⊂ Cl which sends 1[ ∈ L[` to 1 ∈ Cl` . 3.8.8. Let A be a chiral algebra and α : L[ → A a morphism of Lie∗ algebras such that α(1[ ) = −1A ; we refer to (A, α) = (A, L, α) as BRST datum. Since α and β take opposite values on 1[ ∈ L[ , we have a morphism of Lie∗ algebras `(0) := α + β : L → A ⊗ Cl(0) .

(3.8.8.1)

Consider the contractible Lie∗ algebra L† (see 1.1.16); recall that as a mere complex it equals Cone(idL ). Then `(0) extends to a morphism of mere graded Lie∗ algebras (we forget about the differential for a moment) ` : L† → A ⊗ Cl,

(3.8.8.2)

whose component `(−1) : L[1] → A ⊗ Cl(−1) is the composition L[1] ,→ Cl(−1) ,→ A ⊗ Cl. 3.8.9. To define the BRST charge, we need a simple lemma. Consider the Chevalley differential δ on the commutative! algebra Sym(L◦ [−1]) (see 1.4.10). Let us identify Sym(L◦ [−1]) with its image by the embedding Sym(L◦ [−1]) ⊂ Cl ⊂ A ⊗ Cl. The adjoint action of L† on A ⊗ Cl via ` preserves Sym(L◦ [−1]). The corresponding action of L† on the Chevalley DG algebra is compatible with the differentials: one has ad`(0) = [δ, ad`(−1) (a) ] ∈ P2∗ ({L, Sym(L◦ [−1])}, Sym(L◦ [−1])). Restricting this identity to L◦ [−1] ⊂ Sym(L◦ [−1]), we get (here [ ] is the bracket on A ⊗ Cl): Lemma. The ∗ operations [`(0) , idL◦ [−1] ] and [`(−1) , δ|L◦ [−1] ] in P2∗ ({L, L◦ [−1]}, A ⊗ Cl(1) ) coincide.  Consider a chiral operation (3.8.9.1)

χ ˜ := µ(`(0) , idL◦ [−1] ) − µ(`(−1) , δ|L◦ [−1] ) ∈ P2ch ({L, L◦ }, A ⊗ Cl(1) [1])

where µ is the chiral product on A ⊗ Cl. The corresponding ∗ operation vanishes by the lemma, so it amounts to a morphism of complexes χ : L ⊗ L◦ → A ⊗ Cl(1) [1].

(3.8.9.2) ∼

We have h(L ⊗ L◦ ) −→ End(L). Set (3.8.9.3)

d = dA,α := χ(idL ) ∈ h(A ⊗ Cl(1) [1]).

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This is the BRST charge for (A, α). Its adjoint action is an odd derivation dA,α of A ⊗ Cl called the BRST differential. It has degree 1 with respect to both the structure degree and the (·) -grading. 3.8.10. Theorem. (i) One has [d, d] = 0; hence d2A,α = 0. The DG chiral algbebra (A ⊗ Cl, dA⊗Cl + dA,α ) is denoted by CBRST (L, A). (ii) ` from (3.8.8.2) is a morphism of DG Lie∗ algebras (3.8.10.1)

` : L† → CBRST (L, A).

(iii) d is a unique element of h(A ⊗ Cl(1) [1]) which is even of structure degree 0 and such that add `(−1) = `(0) . Similarly, dA,α is a unique odd derivation of A ⊗ Cl of structure and (·) degrees 1 such that dA,α `(−1) = `(0) . Remarks. (i) The statement (ii) of the theorem says that `(−1) is a dA,α (or dA⊗Cl + dA,α ) homotopy between `(0) = α + β and 0; we refer to it as the BRST property of d or dA,α . According to (iii) of the theorem, the BRST property determines d and dA,α uniquely. (ii) The construction of the BRST element and the above theorem remain literally valid in the setting of chiral algebras on an algebraic DX -space Y (see 3.3.10). We assume that L is a Lie∗ algebra on Y which is a vector DX -bundle on Y, etc. (iii) Let ϕ : A → A0 be a morphism of chiral algebras. Then (A0 , L, ϕα) is again a BRST datum. It follows from the statement (iii) of the theorem that the morphism ϕ ⊗ idCl : A ⊗ Cl → A0 ⊗ Cl sends dA,α to dA0 ,ϕα ; hence we have a morphism of DG chiral algebras CBRST (L, A) → CBRST (L, A0 ). (iv) For the “classical” version of the theorem see 1.4.24. Proof. It is found in 3.8.11–3.8.13. 3.8.11. First we show that dA,α `(−1) = `(0) ∈ Hom(L, A ⊗ Cl); i.e., d and dA,α satisfy (iii). Consider the operation µ(χ, ˜ `(−1) ) ∈ P3ch ({L, L◦ , L}, A⊗Cl). We label the vari(0) (−1) ables by the lower indices 1, 2, 3. One has µ(χ, ˜ `(−1) ) = µ(`1 , µ(idL◦ [−1] 2 , `3 ))+ (0) (−1) (−1) (−1) (−1) (−1) µ(µ(`1 , `3 ), idL◦ [−1] 2 )−µ(`1 , µ(δ|L◦ [−1] 2 , `3 ))−µ(µ(`1 , `3 ), δ|L◦ [−1] 2 ). (0) Restricting it to j∗ (L  L◦ )  L, we get µ(`1 , h i23 ) + µ(`(−1) [ ]13 , idL◦ [−1] 2 ) + (0) (−1) (0) (−1) µ(`1 , ad◦L 32 ) = µ(`1 , h i23 ) + [µ(`1 , idL◦ [−1] 2 ), `3 ]. Both summands vanish on L  L◦  L, so we can consider them as operations in P2∗ ({L ⊗ L◦ , L}, A ⊗ Cl). Insert idL ∈ h(L ⊗ L◦ ) in the first variable; the first summand yields `(0) and the second one 0. We are done. 3.8.12. Now let us prove the uniqueness statement (iii) in the theorem 3.8.10. (a) Let F ⊂ h(A ⊗ Cl) be the centralizer of `(−1) , so F contains A ⊗ Sym(L[1]). Let us show that actually F = A ⊗ Sym(L[1]). This statement follows from its “classical” version 1.4.22. Indeed, consider the ring filtration on A ⊗ Cl generated by (A ⊗ Sym(L[1]), A ⊗ (Sym(L[1]) + L◦ [−1])) (see 3.7.13). The associated graded algebra equals A ⊗ Clc (see 1.4.21). Consider the corresponding filtration on h(A ⊗ Cl); we have gr h(A ⊗ Cl) = h(A ⊗ Clc ). Take v ∈ h(A ⊗ Cla ); let n be the smallest number such that it lies in the nth term of the filtration, so the corresponding v¯ ∈ grn h(A ⊗ Clca ) = h(A ⊗ Symn (L◦ [−1]) ⊗ Symn−a (L[1])) is non-zero (cf. the proof in 1.4.22). Then the composition adv `(−1) takes values in the previous term,

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and the corresponding morphism from L[1] to grn−1 (A ⊗ Cl(a) ) ⊂ A ⊗ Clc is the morphism ψ(¯ v ) from 1.4.22. So 1.4.22 implies that for v ∈ F one has n = 0; q.e.d. (b) Consider the maps h(A ⊗ Cl(a) ) → Dera (A ⊗ Cl) → Hom(L[1], A ⊗ Cla−1 ), where the first arrow is the adjoint action and the second one is ν 7→ ν`(−1) .67 Now both of them are injective for a ≥ 1; this obviously implies 3.8.10(iii). The proof is an obvious modification of that of its “classical” counterpart 1.4.24(iii) (replace 1.4.22 by (a) above and Clc by Cl). 3.8.13. Let us show that [d, d] = dA,α (d) vanishes. According to (a) in 3.8.12, it suffices to prove that [d, d] ∈ F , i.e., d2 `(−1) = 0. One has d`(−1) = `(0) (see 3.8.11), so we want to show that d`(0) = 0. Again by (a) in 3.8.12 it suffices to show that d`(0) takes values in F , i.e., [d`(0) , `(−1) ] = 0. One has68 [d`(0) , `(−1) ] = d[`(0) , `(−1) ] − [`(0) , d`(−1) ] = d`(−1) [ ] − [`(0) , d`(−1) ] = `(0) [ ] − [`(0) , `(0) ] = 0, and we are done. Finally 3.8.10(ii) follows from 3.8.11 and 3.8.13.  3.8.14. The DG chiral algebra CBRST (L, A) considered as an object of the homotopy category HoCA(X) (see 3.3.13) is called the BRST reduction of A with a (L, A) = 0 for a 6= 0; then the BRST respect to α. The reduction is regular if HBRST 0 reduction is the plain chiral algebra HBRST (L, A). For any graded A ⊗ Cl-module N the action dN of d on N is a dA,α -derivation of square 0 commuting with the structure differential dN . So N equipped with a differential dN + dN is a DG CBRST (L, A)-module. In particular, for an A-module M the module M ⊗ Cl (see 3.8.2) is a DG CBRST (L, A)-module. We denote it by CBRST (L, M ) and call the BRST complex of L with coefficients in M . Its · · (L, A)-module; this is the BRST, or semi(L, M ) is an HBRST cohomology HBRST infinite, homology of L with coefficients in M . As follows from Remark (iii) in 3.8.10 (applied to the morphism ϕ : U (L)−[ → A defined by α) our CBRST (L, M ) depends, as a complex of plain DX -modules, only on the chiral L−[ -module structure on M . Variant. Let x ∈ X be a point, jx : Ux ,→ X its complement. Assume that our BRST datum is given on Ux . According to 3.6.6 and 3.6.13 the category of ˆ as jx∗ A⊗Cl-modules supported at x coincides with the category of discrete Aas x ⊗Clx modules. We can consider d as an element of hx jx∗ A ⊗ Cl, so its action yields a ˆ as canonical differential on any discrete Aas x ⊗Clx -module. We will see in 3.8.25 that it can be described in terms of the Tate linear algebra. Remark. In this setting we have no canonical jx∗ Cl-module supported at x, as so to define the BRST homology of an Aas x -module M , we should choose some Clx module C and then define the BRST homology of M as the homology of M ⊗ C. If L is extended to X, i.e., if we have a Lie∗ subalgebra LX ⊂ jx∗ L, then we can take for C the module corresponding to the fiber at x of the Clifford module for LX . 3.8.15. The classical limit. Let us show that the BRST construction for coisson algebras from 1.4.21–1.4.25 can be considered as a “classical limit” of the chiral BRST construction. Consider an increasing filtration on Cl with terms Sym(L◦ [−1]) · Sym≤n (L[1]) (it differs from the PBW filtration from 3.8.6). This is a commutative filtration and the Dera are derivations of degree a with respect to the grading A ⊗ Cl(·) . first equality comes since d is a derivation, the second and the fourth ones follow since ` commutes with brackets, and the third one is 3.8.11. 67 Here 68 The

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gr· Cl = Sym(L◦ [−1]) ⊗ Sym· (L[1]); this identification is, in fact, an isomorphism of coisson algebras (see 1.4.21, 1.4.23) (3.8.15.1)

∼

gr Cl −→ Clc .

Let now A, α : L[ → A be a BRST datum. Suppose that A carries a commutative filtration A· (see 3.3.12) such that α(L[ ) ⊂ A1 . Then gr A is a coisson algebra, and αc := α mod A0 : L → gr1 A is a morphism of Lie∗ algebras, so we have the corresponding classical BRST complex CBRST (L, gr A)c (see 1.4.24). Equip A ⊗ Cl with the tensor product of our filtrations. This is a commutative filtration, and one has an isomorphism of coisson algebras gr(A⊗Cl) = gr A⊗gr Cl = (gr A) ⊗ Clc . The image of the morphism ` from (3.8.8.2) lies in the first term of the filtration, and ` mod A0 equals its classical counterpart (1.4.23.1). The BRST differential preserves our filtration, and gr d satisfies the classical BRST property, hence equals the classical BRST differential. We have proved: Lemma. There is a canonical isomorphism of DG coisson algebras (3.8.15.2)

∼

grCBRST (L, A) −→ CBRST (L, grA)c .

In particular, if the “classical” BRST reduction of gr A is regular, then the “quantum” BRST reduction of A is also regular.  3.8.16. W -algebras. An important example of the BRST construction is the quantum Drinfeld-Sokolov reduction. See Chapter 14 of [FBZ] for all details and proofs. We use the notation of (a) in 2.6.8. As in loc. cit., we have a canonical embedding α : n(F)D ,→ g(F)κD . Consider the twisted enveloping algebra U κ := U (g(F)D )κ (see 3.7.20) and the embedding αψ := iα + ψ · 1U κ : n(F)D ,→ U κ . Since the adjoint action of n is nilpotent, the extension n(F)[D of n(F)D from 3.8.7 is trivialized, so αψ extends to a BRST datum n(F)[D → U κ denoted again by αψ . Theorem. The BRST reduction of U κ with respect to αψ is regular. Thus it 0 (n(F)D , U κ ) called the W -algebra.  reduces to a plain chiral algebra W κ := HBRST Remark. The theorem for generic κ follows, by 3.8.15, from the classical counterpart (which is the first proposition in 2.6.8). 3.8.17. BRST in the Tate linear algebra setting. In the rest of this section we will show that the associative topological DG algebras CBRST (L, A)as x can be computed purely in terms of Tate’s linear algebra. We assume that the reader is acquainted with the content of 2.7.7–2.7.9 and 3.6.1. Notice that if T is a Tate vector space and R any topological vector space, then ˆ ∗ is equal to the space Hom(T, R) of continuous linear maps T → R. More R⊗T ˆ ˆ ∗⊗n generally, R⊗T equals the space of continuous polylinear maps T ×· · ·×T → R. For a topological Lie algebra L its topological enveloping algebra U (L) is a topological associative algebra equipped with a continuous morphism of Lie algebras L → U (L) which is universal in the obvious sense. Thus U (L) is a completion of the plain enveloping algebra with respect to the topology made of left ideals generated by open subspaces V ⊂ L. If the topology on L is generated by open Lie subalgebras, then U (L) satisfies the Poincar´e-Birkhoff-Witt theorem: the associated

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graded algebra gr U (L) for the PBW filtration69 equals the completed symmetric algebra Sym L. This condition holds automatically if L is a Tate vector space.70 Remark. If L[ is a central extension of L by k, then we have the corresponding twisted topological enveloping algebra U (L)[ ; this is a topological associative algebra equipped with a morphism of topological Lie algebras L[ → U (L)[ which sends 1[ ∈ k ⊂ L[ to 1U (L)[ and is a universal morphism with these properties. The PBW theorem for U (L)[ holds if the topology on L[ is generated by open Lie subalgebras (and we assume that the topology induced on k ⊂ L[ is discrete). Let T be a Tate (super) vector space, < >=< >T a skew-symmetric (continuous) bilinear form on T . Consider T [ := T ⊕ k as a Lie algebra with the non-zero component of the bracket equal to < >. This is a central extension of T (considered as a commutative Lie algebra) by k. The twisted topological enveloping algebra W = U (T )[ is called the topological Weyl algebra of (T, < >). Assume that < > is non-degenerate. Then the category W mod can be described as follows. Let V ⊂ T be a c-lattice; then its orthogonal complement V ⊥ is also a c-lattice. Assume that the restriction of < > to V is trivial; i.e., V ⊥ ⊃ V . The restriction of < > to V ⊥ comes from the finite-dimensional vector space V ⊥ /V . The Weyl algebra W (V ⊥ ) is equal to the subalgebra of W generated by V ⊥ (and coincides with the centralizer of V in W ). The projection W (V ⊥ ) → W (V ⊥ /V ) yields the functor W (V ⊥ /V )mod → W mod, M 7→ W ⊗ M . This functor is fully W (V ⊥ )

faithful. According to a variant of Kashiwara’s lemma it identifies W (V ⊥ /V )mod with the subcategory of those W -modules on which every v ∈ V ⊂ W acts in a locally nilpotent way. Every discrete W -module is a union of submodules which satisfy this property with respect to smaller and smaller V ’s. As in 3.8.6 we can replace skew-symmetric pairing by a symmetric pairing to get a topological Clifford algebra. As above, we are interested in the graded version of this construction, so we start with a Tate vector space F and consider the topological graded Clifford algebra Cl = Cl(·) generated by the graded (super) Tate vector space F ∗ [−1] ⊕ F [1] such that the commutator pairing between F ∗ [−1] and F [1] equals the canonical pairing F ∗ [−1]⊗F [1] = F ∗ ⊗F → k ⊂ Cl. Our Cl contains commutative graded subalgebras Sym(F ∗ [−1]), Sym(F [1]); here Sym denotes the completed symmetric algebra.71 It carries the PBW filtration Cl· compatible with the grading; the algebra gr Cl is equal to Sym(F ∗ [−1] ⊕ F [1]). (0) (1) (−1) We have Cl0 = k, Cl1 = F ∗ [−1] and Cl1 = F [1]. Notice that Cl2 (0) is equal to the simultaneous normalizer of F ∗ [−1], F [1] in Cl. The adjoint action (0) of this Lie subalgebra preserves both F ∗ [−1] and F [1] and identifies Cl2 /k with (0) gl(F ) (as a topological Lie algebra).72 Thus Cl2 is a central extension of gl(F ) by k. It equals the Tate extension gl(F )[ from 2.7.8: 3.8.18. Proposition. There is a canonical isomorphism of topological central extensions (0) ∼ Cl2 −→ gl(F )[ . 69 Its

terms are closures of the terms of the PBW filtration on the plain enveloping algebra. for every c-lattice V ⊂ L its normalizer in L intersected with V is an open Lie subalgebra of L contained in V . 71 I.e., the symmetric algebra in the category of topological vector spaces for the product ⊗. ˆ 72 Use Exercise in 2.7.7. 70 Proof:

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Proof. According to (ii) in 2.7.1 and 2.7.8 such an isomorphism amounts to a (0) (0) pair of continuous sections sc : glc (F ) → Cl2 , sd : gld (F ) → Cl2 which commute (0) with the adjoint action of gl(F ), sc + sd : glc (F ) ⊕ gld (F ) → Cl2 is an open map, (0) and sc − sd : glf (F ) → k ⊂ Cl2 equals tr. Recall (see Exercise in 2.7.7) that glc (F ), gld (F ) are completions of F ⊗ F ∗ with respect to appropriate topologies. We define sc , sd restricted to F ⊗ F ∗ as, respectively, compositions F ⊗ F ∗ = · · (0) (0) F ∗ [−1] ⊗ F [1] − → Cl2 , F ⊗ F ∗ = F [1] ⊗ F ∗ [−1] − → Cl2 where · is the product map. It follows immediately from Exercise in 2.7.7 that sc and sd are continuous, (0) (0) so they extend to continuous maps sc : glc (F ) → Cl2 , sd : gld (F ) → Cl2 . We leave it to the reader to check that sc , sd satisfy the properties listed above.  3.8.19. Let L be a topological Lie algebra which is a Tate vector space. The adjoint action yields a continuous morphism of Lie algebras ad : L → gl(L). The Tate extension of L is the pull-back L[ of the Tate extension gl(L)[ by ad. Remark. For every Lie subalgebra P ⊂ L which is a c-lattice the adjoint action maps P to gl(L)P , so, according to Remark in 2.7.9, we have a canonical section sP : P → L[ . If P 0 ⊂ P ⊂ L is another such subalgebra, then the restriction of sP to P 0 equals sP 0 + trP/P 0 (the trace of the adjoint action of P 0 on P/P 0 ). According to 3.8.18 we have a canonical morphism of topological Lie algebras β : L[ → gl(L)[ ⊂ Cl(0) where Cl is the Clifford algebra as in 3.8.17 (for F = L). One has β(1[ ) = 1Cl . Let A be a topological associative algebra. A BRST datum is a continuous morphism of Lie algebras α : L[ → A such that α(1[ ) = −1A . Since α and β take opposite values on 1[ , we have a morphism of topological Lie algebras (3.8.19.1)

ˆ (0) . `(0) := α + β : L → A⊗Cl

Consider the contractible topological Lie DG algebra L† (see 1.1.16). Then `(0) extends to a morphism of topological graded Lie algebras (we forget about the differential for a moment) (3.8.19.2)

ˆ ` : L† → A⊗Cl,

ˆ (−1) . where `(−1) is the composition L[1] → Cl(−1) ⊂ A⊗Cl Denote by δ the Chevalley differential on the topological commutative DG algebra Sym(L∗ [−1]) defined by the topological Lie algebra structure on L. Let us ˆ identify Sym(L∗ [−1]) with its image by the embedding Sym(L∗ [−1]) ⊂ Cl ⊂ A⊗Cl. ˆ The action of L† on A⊗Cl via ` and the adjoint action preserve Sym(L∗ [−1]). Considered as an action of L† on the Chevalley DG algebra, it is compatible with the differentials: for a ∈ L the action of ad`(0) (a) on Sym(L∗ [−1]) coincides with [δ, ad`(−1) (a) ]. ˆ 3.8.20. Lemma. The map L⊗L∗ → A⊗Cl[1], a⊗b 7→ `(0) (a)·b−`(−1) (a)·δ(b), ˆ where · is the product in A⊗Cl, extends by continuity to a morphism (3.8.20.1)

ˆ ∗ → A⊗Cl ˆ (1) [1]. χ : L⊗L

ˆ open subspaces Proof (cf. 3.8.9). We want to find for each open U ⊂ A⊗Cl P ⊂ L, Q ⊂ L∗ such that `(0) (a) · b + `(−1) (a) · δ(b) ∈ U if either a ∈ P or b ∈ Q.

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We can assume that U is a left ideal. Take P , Q such that `(0) (P ), `(−1) (P ), Q, δ(Q) ⊂ U . Then our condition is satisfied since [`(0) (a), b] − [`(−1) (a), δ(b)] = 0.  ∼

ˆ ∗ −→ End(L). Set Recall that L⊗L (3.8.20.2)

ˆ (1) [1]. d = dA,α := χ(idL ) ∈ A⊗Cl

This is the BRST charge. Its adjoint action dA,α is the BRST differential; this is a ˆ of degree 1. continuous odd derivation of A⊗Cl 3.8.21. Theorem. (i) One has [d, d] = 0; hence d2A,α = 0. ˆ The topological associative DG algbebra (A⊗Cl, dA⊗Cl + dA,α ) is denoted by ˆ CBRST (L, A). (ii) ` from (3.8.19.2) is a morphism of topological Lie DG algebras (3.8.21.1)

` : L† → CBRST (L, A).

ˆ (1) such that add `(−1) = `(0) . Simi(iii) d is a unique odd element in A⊗Cl ˆ larly, dA,α is a unique continuous odd derivation of degree 1 of A⊗Cl such that (−1) (0) dA,α ` =` . We refer to (ii) as the BRST property of d or dA,α . Proof (cf. the proof of Theorem 3.8.10). (a) One has dA,α `(−1) = `(0) . ˆ equals A⊗Sym(L[1]) ˆ ˆ (b) The centralizer F of L[1] ⊂ Cl ⊂ A⊗Cl ⊂ A⊗Cl. In particular, F has degrees ≤ 0. (c) Let us check (iii). Property [d, `(−1) ] = `(0) determines d uniquely up to an element of F 1 . So d is unique. Similarly, property d`(−1) = `(0) determines d up to a ˆ (1) derivation ∂ which kills the image of L[1]. Such a ∂ sends L∗ [−1] ⊂ Cl(1) ⊂ A⊗Cl (0) ∗ ˆ and A ⊂ A⊗Cl to F . Since ∂ has degree 1, it kills L [−1] and A according to (a). Thus ∂ = 0; i.e., d is determined uniquely. (d) Statement (i) follows from (b); the argument repeats 3.8.13. Statement (ii) follows from (i) and (a).  ˆ 3.8.22. For every discrete A⊗Cl-module N the action of d on N is an odd dA,α -derivation of degree 1 and square 0, so it defines on N a canonical structure of a DG CBRST (L, A)-module. Fix a graded discrete Cl-module C; then for a discrete A-module M the tensor product M ⊗ C becomes a DG CBRST (L, A)-module called the BRST, or semi-infinite homology, complex of M (for the Clifford module C). Remarks. (i) If L is purely even, then an irreducible Cl-module is unique up to a twist by a 1-dimensional (super) vector space. So the BRST complex with respect to an irreducible C is uniquely defined up to a twist by a 1-dimensional graded (super) vector space. (ii) d, hence BRST complexes, behave in the obvious manner with respect to morphisms of A’s. The “smallest” A for the BRST datum is U (L)−[ . So the BRST complex of M depends only on the L−[ -module structure on M (coming from α). (iii) Suppose L is discrete. Then L[ is split by means of sd , which identifies L-modules and L−[ -modules. The Chevalley homology complex of an L-module coincides with the BRST complex of the corresponding L−[ -module for the Clifford module Sym(L[1]). Similarly, if L is compact, then we use the splitting sc to identify L- and L−[ -modules. The Chevalley cohomology complex of an L-module coincides then with the BRST complex of the corresponding L−[ -module for the Clifford module Sym(L∗ [−1]). If L is finite-dimensional, then the splittings differ

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by the character tr ad : L → k, and we get the usual identification of the Chevalley homology complex of an L-module with the Chevalley cohomology complex of the L-module twisted by det L. 3.8.23. Now we can compare the chiral and the Tate linear algebra versions of BRST. Let x ∈ X be a (closed) point, jx : Ux ,→ X its complement. Let L be a vector D-bundle on Ux , L◦ the dual vector D-bundle. According ˆ x (jx∗ L◦ ) is dual to L(x) := h ˆ x (jx∗ L); i.e., to 2.7.10 the Tate vector space L◦(x) := h ◦ ∗ L(x) = (L(x) ) . Let Cl· be the graded chiral Clifford algebra generated by L◦ [−1] ⊕ L[1] (see · 3.8.6), and let Cl(x) be the topological graded Clifford algebra generated by L◦(x) [−1] ⊕L(x) [1] (see 3.8.17). According to 3.7.22, there is a canonical isomorphism of ∼ · topological graded algebras Clx·as −→ Cl(x) . Here is a simple alternative construction of the identification r[ of (2.7.14.1).73 Consider the Tate extension gl(L)[ = Cl20 (see (3.8.6.1)). The previous isomorphism 0 yields then a morphism of Lie algebras hx jx∗ gl(L)[ → Cl(x)2 . It is obviously contin-topology (see 2.7.13), so we have a morphism of topouous with respect to the Ξsp[ x [ 0 0 logical Lie algebras gl(L)(x) → Cl(x)2 . Using the identification Cl(x)2 = gl(L(x) )[ of [ [ [ 3.8.18, we can rewrite this morphism as r : gl(L)(x) → gl(L(x) ) . By construction, ∼

this r[ is a lifting of the isomorphism r : gl(L)(x) −→ gl(L(x) ) of 2.7.12 to our central k-extensions. Now we assume that our L is a Lie∗ algebra, so we have the Tate extension L[ (see 3.8.7). Then L(x) is a topological Lie algebra (see 2.5.12 and 2.5.13(ii)) and L[(x) is a central extension of L(x) by k. By definition of L[ the adjoint action of L lifts canonically to a morphism of topological extensions L[(x) → gl(L)[(x) . Thus r[ provides a canonical identification of L[(x) with the Tate linear algebra extension (L(x) )[ from 3.8.19. Assume we have a chiral BRST datum, i.e., a chiral algebra A on Ux and a morphism of Lie∗ algebras α : L[ → A such that α(1[ ) = −1A (see 3.8.8). This as yields the topological associative algebra Aas x = (jx∗ A)x (see 3.6.2–3.6.6) and the [ [ as morphism α(x) : L(x) → Ax which sends 1 to −1 which is a Tate linear algebra BRST datum (see 3.8.19). According to 3.6.8 the functor A 7→ Aas x commutes with tensor products, so as ˆ (A ⊗ Cl)as = A ⊗Cl . (x) x x 3.8.24. Lemma. The canonical morphism (3.6.6.2) of graded Lie algebras as ˆ hx jx∗ A ⊗ Cl → (A ⊗ Cl)as x = Ax ⊗Cl(x)

sends the chiral BRST charge (3.8.9.3) to its Tate linear algebra cousin (3.8.20.2). Proof. The image of chiral d satisfies the BRST property, so our statement follows from 3.8.21(iii).  3.8.25. According to 3.6.6 the categories of graded jx∗ A ⊗ Cl-modules supˆ ported at x and discrete Aas x ⊗Cl(x) -modules are canonically identified. Each of 73 Our

L is V of 2.7.14.

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these objects carries a canonical BRST differential (defined as the action of the BRST charges), and the above lemma identifies these differentials. This provides a canonical identification of the BRST homology from 3.8.14 (Variant) with those from 3.8.22. 3.9. Chiral differential operators In this section we explain what enveloping chiral algebras of Lie∗ algebroids are. In particular, we consider chiral algebras of differential operators or cdo which correspond to the tangent algebroid ΘY . The key difference with the case of Lie∗ algebras treated in 3.7 is that the construction of the enveloping algebra requires an extra structure (that of chiral extension) on the Lie∗ algebroid. In 3.9.1–3.9.3 we consider chiral RDif -algebras for a commutative DX -algebra R. The important fact is that chiral RDif -algebras are local objects with respect to Spec R, so they can live on any algebraic DX -space. An example of such algebras are R-cdo defined in 3.9.5. The notion of a chiral Lie algebroid is discussed in 3.9.6–3.9.10; in particular, we define an affine structure on the groupoid of chiral Rextensions of a given Lie∗ algebroid L in 3.9.7, construct rigidified chiral extensions in 3.9.8, and relate chiral extensions with mod t2 quantizations in 3.9.10. The enveloping chiral algebras of chiral Lie algebroids come in 3.9.11. The Poincar´eBirkhoff-Witt theorem (in a form suggested by D. Gaiitsgory), which is a main technical result of this section, is in 3.9.11–3.9.14. To make the exposition clear, we first present another proof of the PBW (actually, of a slightly more general statement) for the usual Lie algebroids,74 and then explain how to modify it to the chiral setting. The PBW theorem permits us to identify chiral extensions of ΘY with cdo (see 3.9.15). In the rest of the section we describe the groupoid of chiral R-extensions explicitly. In 3.9.16–3.9.17 we define a canonical DG chiral extension of a certain DG Lie∗ algebroid over the de Rham-Chevalley algebra of L. In particular, for L = ΘR we get a canonical DG cdo DDR over the de Rham DX -algebra DR = DRR` /X (see 3.9.18). We use it in 3.9.20 to identify chiral extensions of L with variants of Tate structures from 2.8.1; for L = Θ we get exactly Tate structures on Y. Thus concrete Tate structures from 2.8.15 and 2.8.17 provide examples of cdo. The formalism of Tate structures from 2.8 provides an obstruction to the existence of the global chiral extension of L (see 3.9.21–3.9.22). In particular, 2.8.13 yields a version of the Gorbounov-Malikov-Schechtman formula of 3.9.23 for the obstruction to the existence of a global cdo on a jet DX -scheme. In 3.9.24– 3.9.26 we identify modules over the enveloping algebra of a chiral algebroid with chiral modules over the algebroid, describe the corresponding topological associative algebras as enveloping algebras of topological Lie algebroids, and prove a version of the PBW theorem. In particular, we show that the topological associative algebras coming from a cdo are topological tdo (see 3.9.27). Much of the material was independently developed by Gorbounov-MalikovSchechtman [GMS2], see also [Sch1] and [Sch2]. They consider the setting of graded vertex algebras; in our languge, these are chiral algebras on X = A1 equivariant with respect to the action of the group Af f of affine linear transformations (see 0.15). Their cdo and chiral algebroids live on the jet scheme of a constant fibration Z = X × B over X equipped with an evident Af f -action; the “vertex OB -algebroid” from loc. cit. is the degree 1 component of (the translation invariant 74 The

proof from 2.9.2 seems not to admit a chiral version.

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part of) a chiral Lie algebroid with appropriate induced structure which permits us to recover the whole chiral algebroid. The notion of a chiral RDif -algebra is a chiral analog of that of the D-algebra from [BB] 1.1. The subject of 3.9.1–3.9.3 was also treated in [KV]. The canonical chiral DG tdo DDR from 3.9.18 first appeared (in the graded vertex algebra setting) in the work of Malikov-Schechtman-Vaintrob [MSV] under the name of chiral de Rham complex. The original construction of [MSV] was coordinate-style (the standard “linear” chiral DG algebras, attached to coordinate neighbourhoods of B, are patched together by means of certain explicit liftings of coordinate transformations); it was elucidated in [KV] and [Bre1]. The formula in 3.9.23 (in the graded vertex algebra setting)75 is due to GorbounovMalikov-Schechtman [GMS1], [GMS2] 8.7, [Sch1], see also [Bre2].76 The first, to our knowledge, “non-linear” example of cdo was found by Feigin-Frenkel [FF1], [FF2]. They constructed a cdo (as a topological associative algebra) on the jet scheme of an open Schubert cell by certain mysterious explicit formulas. In fact, as was shown in [GMS3], for X = A1 the jet scheme of a flag space admits a unique Af f -equivariant cdo; its restriction to the Schubert cell is the Feigin-Frenkel cdo. From the point of view of this article, this follows from (3.9.20.2) and the exercise in 2.8.17. Similarly, the cdo on the jet scheme of a group constructed in [AG] and [GMS3] arise from the Tate structures from 2.8.15. The family of cdo on the jet scheme of a flag space parametrized by Miura opers was suggested by E. Frenkel and D. Gaitsgory (see 3.9.20 and 2.8.17). The subject of 3.9.27 is related to the Appendix in [AG]. For an application of chiral Lie algebroids to representation theory of affine Kac-Moody Lie algebras at the critical level, see [FrG]. 3.9.1. Let R be a commutative chiral algebra; i.e., by (3.3.3.1), R` is a commutative DX -algebra. We say that a chiral R-module M is an R-differential module or simply an RDif -module if M is a union of submodules which are successive extensions of central R-modules.77 SEquivalently, this means that there is a filtration {0} = M−1 ⊂ M0 ⊂ M1 ⊂ · · · , Mi = M , such that the gri M are central.78 The subcategory M(X, R)Dif ⊂ M(X, R) of RDif -modules is closed under direct sums and subquotients. Any chiral R-module N has the maximal differential submodule N Dif . Every chiral operation preserves maximal differential submodules. We denote by ch M(X, R)ch the full pseudo-tensor subcategory of RDif -modules. Dif ⊂ M(X, R) Consider the sheaf of commutative Lie algebras h(R) on X. It acts canonically on every chiral R-module M , so M is a Sym h(R)-module. If M ∈ M(X, R) is an RDif -module, then the h(R)-action on M is locally nilpotent which means that every m ∈ M is killed by some Symi h(R), i  0. The converse is true if we know that for every i > 0 the subsheaf of M that consists of sections killed by Symi h(R) is OX -quasi-coherent. This always happens if M is either OX -flat or supported in finitely many points. Let f : R1` → R2` be a morphism of DX -algebras. If M is an R2Dif -module, then it is an R1Dif -module via f , so we have faithful pseudo-tensor functors (3.9.1.1)

ch f· : M(X, R2 )ch Dif → M(X, R1 )Dif .

75 Notice that in the translation equivariant setting ω X is trivialized, so it disappears from the formula. 76 Presumably, in the Af f -equivariant setting our construction reduces to that of [GMS2]. 77 See 3.3.7 and 3.3.10. In this definition R need not be commutative. 78 One can define M by induction taking for M the preimage of (M/M cent . · i i−1 )

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3.9.2. Consider the topological algebra with connection (Ras , ∇) defined in 3.6.14. Since R is commutative, Ras is also commutative. We have R` = Ras /IR for an open ideal IR ⊂ Ras . Let RDif be the IR -adic completion of Ras . Precisely, n we consider on Ras the topology defined by those open ideals I for which IR ⊂I Dif for n  0; our R is the corresponding completion. This is a commutative topological OX -algebra equipped with a connection ∇. Remark. The functor R 7→ RDif commutes with coproducts.79 We define discrete RDif [DX ]-modules in the same way as we defined discrete R [DX ]-modules in 3.6.15; i.e., discrete RDif [DX ]-modules are the same as discrete Ras [DX ]-modules such that the action of Ras comes from the discrete action of RDif . They form an abelian category M` (X, RDif ). The equivalence from 3.6.16 identifies RDif -modules with discrete RDif [DX ]modules. Therefore we have an equivalence as

(3.9.2.1)

∼

M(X, R)Dif −→ M` (X, RDif ).

3.9.3. Proposition. Let f : R1` → R2` be an ´etale morphism of DX -algebras. (i) The morphism of topological algebras R1Dif → R2Dif is also ´etale. (ii) The pseudo-tensor functor f· from (3.9.1.1) admits a left adjoint pseudoDif ch · tensor functor f · : M(X, R1 )ch ⊗ M. Dif → M(X, R2 )Dif . One has f M = R2 R1Dif

(iii) RDif -modules and chiral R-operations between them have the ´etale local nature. Proof. (i) Let I1 ⊂ R1Dif be an open ideal, I2 ⊂ R2Dif the closed ideal generated by f (I1 ). We want to show that I2 is open and the morphism R1Dif /I1 → R2Dif /I2 is ´etale. This follows from the fact that the morphism R1Dif → R2Dif is formally ´etale, which is an immediate corollary of 3.6.19. (ii) It follows from (i) that f · defined by the above formula sends discrete Dif R1 [DX ]-modules to discrete R2Dif [DX ]-modules. The adjunction property on the level of the usual morphisms is evident. To check it for chiral R-operations one uses the lemma from 3.4.19 (the case of J = I); the details are left to the reader. (iii) Follows from (i), (ii) and the remark in 3.9.2.  According to 3.9.3(iii) one has the notion of an ODif -module on any algebraic DX -space Y; we also call these objects O-modules on YDif . We denote the corresponding abelian pseudo-tensor category by M(Y)ch Dif . 3.9.4. For a commutative R a chiral RDif -algebra is a chiral algebra A together with a morphism of chiral algebras R → A such that A is an RDif -module. According to 3.9.3(iii) these objects have the ´etale local nature, so we have the notion of a chiral ODif -algebra on any algebraic DX -space Y; we also call them chiral algebras on YDif . The corresponding category is denoted by CA(Y)Dif . Example. Let A be a chiral algebra equipped with a commutative filtration R = A0 ⊂ A1 ⊂ · · · (see 3.3.12). Then A is a chiral RDif -algebra. Notice that in this situation gr1 A is a Lie∗ R-algebroid (see 1.4.11) in the obvious way. Let A be a chiral RDif -algebra. We say that a chiral A-module M is Rdifferential if M considered as an R-module is. This notion is ´etale local, so for 79 This

follows immediately from a similar property of the functor R 7→ Ras ; see 3.6.8, 3.6.13.

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any Y as above and A ∈ CA(Y)Dif we know what the O-differential A-modules on Y, or simply the A-modules on YDif , are. These objects form an abelian pseudotensor category M(Y, A)ch Dif ; we also have the corresponding sheaf of pseudo-tensor categories M(Y´et , A)ch et . Dif on Y´ 3.9.5. Cdo. Assume for the moment that ΩR is a finitely generated locally projective R` [DX ]-module; e.g., R is smooth (see 2.3.15). A chiral algebra of twisted differential operators on Spec R (we abbreviate it to cdo on Spec R or R-cdo) is a chiral RDif -algebra A which admits a commutative filtration A0 ⊂ A1 ⊂ · · · (see 3.3.12) such that R = A0 and there is an isomorphism of coisson Z-graded algebras ∼ Sym· ΘR −→ gr· A identical on R.80 Notice that the filtration A· is uniquely defined: indeed, A0 = R, A1 is the normalizer of R in D, and our filtration is 1-generated (see 3.7.13). We call it the canonical filtration. The above isomorphism of the coisson graded algebras is also uniquely defined. It is clear that cdo are local objects for the ´etale topology. Thus we know what cdo on any smooth algebraic DX -space Y are. We denote the corresponding category by CDO(Y) ⊂ CA(Y)Dif . This is a groupoid. We have the corresponding sheaf of groupoids CDO(Y´et ) on Y´et . The principal difference between cdo and the usual tdo (see [BB] or [Kas1]) is that there is no canonical (“non-twisted”) cdo. We also do not know if cdo always exist locally in the ´etale topology. This is true (as follows from (3.9.20.2)) if ΘY is a locally trivial vector D-bundle (which happens when Y is a jet DX -scheme). If cdo exist locally, then they form a gerbe. We will see that this gerbe need not be trivial. Similarly to the usual tdo, one handles cdo realizing them as twisted enveloping algebras of certain Lie∗ algebroids. We explain the chiral Lie algebroid basics in 3.9.6–3.9.14 returning to tdo in 3.9.15. 3.9.6. Chiral Lie algebroids. An important class of chiral RDif -algebras (which includes cdo) is formed by chiral enveloping algebras of Lie∗ R-algebroids. Contrary to the case of mere Lie∗ algebras, to define such an enveloping algebra, one needs an extra structure of the chiral extension of our algebroid. We first describe this structure and play with it a bit; the chiral envelopes enter in 3.9.11. Let R be a commutative chiral algebra, B a chiral R-algebra, and L a Lie∗ R-algebroid. Suppose we have a D-module extension (3.9.6.1)

0 → B → L[ → L → 0

together with a Lie∗ bracket and a chiral R-module structure µRL[ on L[ such that the arrows in (3.9.12.1) are compatible with the Lie∗ algebra and chiral R-module structures. Then L[ acts as a Lie∗ algebra on itself (by adjoint action), on B (since B ⊂ L[ ), and on R (via the projection L[ → L and the L-action on R). Assume that the following properties are satisfied: (i) The chiral operations µB ∈ P2ch ({B, B}, B) and µRL[ ∈ P2ch ({R, L[ }, L[ ) are compatible with the Lie∗ actions of L[ . (ii) The structure morphism R → B is compatible with the Lie∗ L[ -actions. 80 Exercise: show that the existence of such an isomorphism amounts to the folowing conditions: the adjoint ∗ action of gr1 D on D0 = R identifies it with ΘR and gr1 D generates gr· D.

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(iii) The ∗ operation that corresponds to µRL[ is equal to −ιτLσ [ R where τL[ R is the L[ -action on R, σ is the transposition of variables, and ι is the composition of the structure morphism R → B and the embedding B ⊂ L[ . We call such an L[ a chiral B-extension of L. The particular case of chiral R-extensions (when B = R) is most interesting. We also call a pair (L, L[ ), where L[ is a chiral R-extension of L, a chiral Lie R-algebroid. The image of 1R in L[` is denoted by 1[ . Example. In the situation of the example in 3.9.4, A1 is a chiral Lie Ralgebroid, a chiral R-extension of gr1 A. More generally, suppose we have an embedding of chiral algebras B ⊂ A and a D-submodule L[ of A containing B. Assume that the structure morphism R → B is injective, L[ is a Lie∗ -subalgebra and a chiral R-submodule of A, and L[ normalizes R ⊂ A (in the ∗ sense). Then L := L[ /B is naturally a Lie∗ R-algebroid and L[ is its chiral B-extension. We will see in 3.9.11 that every L[ that satisfies a certain flatness condition comes in this way. The quadruples (R, B, L, L[ ) form a category in the obvious manner. For fixed R, B and L the chiral B-extensions of L form a groupoid Pch (L, B); if B = R, we write simply Pch (L). We denote the category of chiral Lie R-algebroids by LieAlg ch (R) = LieAlg ch (Spec R` ). By 3.9.3(iii) chiral B-extensions have the ´etale local nature (just as Lie∗ algebroids), so we can replace Spec R` by any algebraic DX -space Y. The corresponding categories are denoted by Pch (Y, L, B), etc., and the sheaves of categories on Y´et by Pch (Y´et , L, B), etc. 3.9.7 The affine structure on Pch (L). Let L be a Lie∗ R-algebroid. The important point is that Pch (L) is not a Picard groupoid: the notion of a trivial chiral extension of L makes no sense. We will see in a moment that this groupoid carries an affine structure instead. As in 2.8.2, we define a classical R-extension Lc of L as an extension of L by R in the category of Lie∗ R-algebroids. In other words, Lc is a Lie∗ R-algebroid equipped with a horizontal section 1c ∈ Lc` and an identification of Lie∗ alge∼ broids81 Lc /R1c −→ L; the morphism ι = ιLc : R → Lc , r 7→ r1c is assumed to be injective. These objects have the ´etale local nature, so, as above, we have the groupoid of classical extensions Pcl (L) = Pcl (Y, L). The Baer sum of classical extensions makes Pcl (L) a Picard groupoid. We can also form k-linear combinations of classical extensions, so Pcl (L) is actually a k-vector space category. If L[ is a chiral R-extension and Lc is a classical R-extension of L, then we have the Baer sum of extensions Lc + L[ . This is a Lie∗ algebra, a chiral R-module, and an R-extension of L in the obvious way. The compatibilities in the definition in 3.9.6 are obviously satisfied, so Lc + L[ is a chiral R-extension of L. Therefore we have defined an action of the Picard groupoid Pcl (L) on Pch (L). Lemma. If Pch (L) is non-empty, then it is naturally a Pcl (L)-torsor. Proof. The Baer difference of two chiral extensions is a classical extension.  81 Notice that R · 1c ⊂ Lc is automatically a Lie∗ ideal which acts trivially on R, so Lc /R1c is a Lie∗ R-algebroid.

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243

We will describe Pch (L) in down-to-earth terms later in 3.9.20–3.9.21. Presently we see that the whole of Pch (L) is at hand (modulo the understanding of Pcl (L)) the moment we find some chiral R-extension. Here is a way to construct one: 3.9.8. Rigidified chiral extensions. We consider the general setting of 3.9.6, so we have a chiral R-algebra B. Let L be a Lie∗ algebra acting on R and B in a compatible way; denote by τLR , τLB the L-actions. Let L be a (L, τLR )rigidified Lie∗ R-algebroid (see 1.4.13), so we have a morphism of Lie∗ algebras ∼ ψ : L → L which yields an isomorphism of R-modules R` ⊗ L −→ L. Proposition. Suppose that L is OX -flat. Then there is a chiral B-extension L[ equipped with a lifting ψ [ : L → L[ of ψ such that ψ [ is a morphism of Lie∗ algebras and the adjoint action of L on B via ψ [ equals τLB . Such (L[ , ψ [ ) is unique. We call (L[ , ψ [ ) the (L, τLB )-rigidified B-extension of L. Proof. Existence: Consider the chiral algebra B ∗ U (L) (see 3.7.23). It carries the PBW filtration and gr· B ∗ L = B ⊗ Sym· L` . Define L[ as the pull-back of the extension 0 → B → (B ∗ U (L))1 → gr1 B ∗ L → 0 by the morphism L = R` ⊗ L → ` B ⊗ L = gr1 B ∗ L. It is an (L, τLB )-rigidified B-extension of L in the evident way. Uniqueness: Suppose we have some (L[ , ψ [ ). Consider the chiral pairing η := σ by µRL[ (idR , ψ [ ) ∈ P2ch ({R, L}, L[ ). The corresponding ∗ pairing equals −ιτLR [ property (iii) in 3.9.6, and the composition of η with the projection L → L is ∼ the chiral pairing that corresponds to the rigidification R ⊗ L −→ L. Therefore η [ identifies ∆∗ L with the push-out of the extension 0 → R  L → j∗ j ∗ R  L → τLR ∆∗ R` ⊗ L → 0 by R  L −− → ∆∗ R → ∆∗ B. Therefore every two (L, τLB )-rigidified B-extensions of L are canonically identified. This identification is compatible then with the chiral R-actions (use the trick from the remark in 3.3.6, or, equivalently, consider the chiral R-module structure as the Ras -module structure; see 3.6.16) and with the Lie∗ brackets (use property (i) in 3.9.6). We are done.  3.9.9. For B = R the above proposition can be rephrased as follows. For (L, τ ) as above its R-extension is a D-module extension L[ of L by R together with a Lie∗ algebra structure on L[ such that the projection π : L[ → L is a morphism of Lie∗ algebras and the adjoint action of L[ on R ⊂ L[ coincides with τ π. Such extensions form a Picard groupoid (in fact, a k-vector space category) P(L, τ ). For an (L, τ )-rigidified Lie∗ R-algebroid (L, ψ) the pull-back by ψ of a classical or a chiral R-extension of L is an R-extension of L. So we have defined functors (3.9.9.1)

Pcl (L) → P(L, τ ),

Pch (L) → P(L, τ ).

The first of them is a morphism of Picard groupoids; the second one transforms the Pcl (L)-action on Pch (L) to the translation action of P(L, τ ) on itself. Corollary. If L is OX -flat, then these are equivalences of groupoids. Proof. Since Pch (L) is non-empty, it suffices to treat the first functor. Its inverse assigns to an R-extension L[ of L the push-forward of the R ⊗ R-extension (L[ )R of the Lie∗ R-algebroid LR by the product map R ⊗ R → R. 

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3.9.10. Quantization mod t2 . Here is a different application of the notion of a chiral extension. Suppose we are given a coisson structure on R. It yields a structure of Lie∗ algebroid on the R[DX ]-module Ω = Ω1R/X (see 1.4.17), so we have the corresponding groupoid Pch (Ω) of chiral R-extensions. Consider the groupoid Qch = Qch (R, { }) of mod t2 quantizations of the coisson structure (see 3.3.11); let Qcl = Qcl (R, { }) be the groupoid of k[t]/t2 -deformations of our coisson algebra. Just as in the conventional Poisson setting, the Baer sum operation makes Qcl a Picard groupoid (actually, a k-vector space category) and Qch a Qcl -torsor. There are canonical morphisms of Picard groupoids and their torsors82 (3.9.10.1)

Pcl (Ω) → Qcl ,

Pch (Ω) → Qch

defined as follows. For a classical extension Ωc the corresponding coisson deformation Rc , as a plain D-module, is the pull-back of Ωc by d : R → Ω. Let p : Rc → R, dc : Rc → Ωc be the projections. The coisson bracket is defined by the property that p and dc are morphisms of Lie∗ algebras. The product on Rc` is determined by the requirement that p is a morphism of algebras and dc is a derivation (Ωc is an R` -, hence Rc` -, module). The morphism of algebras k[t]/t2 → Rc` is determined by the condition that dc (t) = 2 · 1c and p(t) = 0. The chiral definitions are similar. The quantization R[ corresponding to Ω[ , as a plain D-module, equals the pull-back of Ω[ by d : R → Ω. Let p : R[ → R, d[ : R[ → Ω[ be the projections. We define { }(1) demanding that p and d[ be morphisms of Lie∗ algebras. The chiral product µR[ ∈ P2ch (R[ ) is defined by the properties pµR[ := µR (p, p), d[ µR[ := µΩ[ (p, d[ ) − µσΩ[ (d[ , p) where σ is the transposition of arguments. The k[t]/t2 -algebra structure k[t]/t2 ,→ R[` is p(1R[ ) = 1R , d[ (1R[ ) = 0 and p(t) = 0, d[ (t) = 2 · 1[ . The compatibilities are straightforward; we leave them to the reader. Proposition. Functors (3.9.10.1) are fully faithful. If R` is formally smooth, then the “classical” functor is an equivalence, and the “chiral” functor is an equivalence if Pch (Ω) is non-empty ´etale locally on Spec R` .83 Proof. The “classical” statement: we recover Ωc from Rc as ΩRc /tΩRc , where Ω = Ω1Rc /X is the Lie∗ algebroid defined by the coisson structure on Rc , and 1c ∈ Ωc` is the image of d(t)/2. For an arbitrary Rc this could be an extension of (−1) Ω by some quotient of R. It is always R itself if 84 ΩR/X = 0, as happens to be true for a formally smooth R. The “chiral” statement follows from the “classical” one.  Rc

3.9.11. Suppose we have (R, B, L, L[ ) as in 3.9.6, so B is a chiral R-algebra and L[ is a chiral B-extension of a Lie∗ R-algebroid L. For a test chiral algebra A a chiral morphism ϕ[ : L[ → A is a morphism of DX -modules L[ → A compatible with all the structures. Precisely, this means that ϕ[ is a morphism of Lie∗ algebras, φ := ϕ[ |B : B → A is a morphism of chiral algebras, and ϕ[ is a morphism of chiral φ

R-modules (where R acts on A via the morphism R → B − → A). It is easy to see “torsors” unless Pch (Ω) is empty. hope that the latter condition is redundant. 84 See 2.3.13 for the notation. 82 I.e.,

83 We

3.9. CHIRAL DIFFERENTIAL OPERATORS

245

that there is a universal ϕ[ ; we denote the corresponding A by U (L)[ = U (B, L)[ . This is the chiral envelope of L[ . It is clear that U (L)[ is a chiral RDif -algebra and the enveloping algebra functor is compatible with the ´etale localization of R. Therefore the above constructions make sense over any algebraic DX -space Y. Notation: U (L)[Y . We define the Poincar´e-Birkhoff-Witt filtration U (L)[· on U (L)[ as the filtration generated by U (L)[0 := φ(B) and U (L)[1 which is the sum of U (L)[0 and the image of µU (L)[ (ϕ[ , φ) ∈ P2ch ({L[ , B}, U (L)[ ) (see 3.7.13). Consider the chiral algebra gr· U (L)[ . This is a chiral R-algebra.85 Our ϕ[ yields a morphism ϕ : L → gr1 U (L)[ whose image is central in gr U (L)[ . By 3.4.15, ϕ and φ yield the surjective PBW morphism of graded chiral algebras B ⊗ Sym·R` L`  gr· U (L)[ .

(3.9.11.1)

R`

Theorem. If R and B are OX -flat and L is a flat R` -module, then the PBW morphism is an isomorphism. Notice that if A = R, then the PBW filtration is commutative and the PBW morphism Sym·R` L`  gr· U (L)[ is a morphism of coisson algebras. Proof. We deduce the statement from the ordinary PBW theorem for Lie∗ algebras (see 3.7.14). The reduction is of a quite general nature. A similar argument proves the PBW theorem (or its more general version parallel the above theorem) for the usual Lie algebroids. To make the exposition more transparent, we first discuss the setting of the usual Lie algebroids (see 3.9.12 and 3.9.13) and then we explain the slight modifications needed to cover the chiral situation (see 3.9.14). 3.9.12. We have already discussed the PBW theorem for the usual Lie algebroids in 2.9.2. The argument from loc. cit. does not generalize to the chiral setting though. Below we give a different proof. Here is a version of the PBW theorem parallel to the statement from 3.9.11. Let R be a commutative k-algebra, B an associative R-algebra, and L a Lie R-algebroid. Suppose we have an extension 0 → B → L[ → L → 0 together with a Lie algebra and an R-module structure on L[ such that the arrows are compatible with the Lie algebra and R-module structures. Then L[ acts on itself by adjoint action, on B (since B ⊂ L[ ), and on R (via the projection L[ → L and the L-action on R). Assume that the following properties are satisfied: (i) L[ acts on B by derivations; its action on L[ is compatible with the R-action (via the L[ -action on R), so L[ is a Lie R-algebroid. (ii) The structure morphism R → B is compatible with the L[ -actions. We call such an L[ a B-extension of L. For such a B-extension L[ we can consider morphisms L[ → A, where A is a test associative algebra, compatible in the obvious manner with the above structures. There is a universal morphism ϕ[ : L[ → U (L)[ . Then U (L)[ carries the PBW filtration U (L)[· where U (L)[0 is the image of B, U (L)[1 = ϕ[ (L[ ) · U (L)[0 , and U (L)[n = (U (L)[1 )n for n ≥ 1. One gets a canonical surjective morphism of graded algebras B ⊗ Sym·R L  gr· U (L)[ . R

85 Therefore

U (L)[ is an RDif -algebra.

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Theorem. If L is a flat R-module, then this is an isomorphism. The proof consists of three steps (a)–(c). First we consider the case when L[ admits a rigidification. Then our statement follows from the usual PBW theorem. Next, in (b), we construct for any L an associative filtered DG algebra V together with an identification H 0 V = U (L)[ such that gr V is controllable (here step (a) is used). Finally, we show that if L is R-flat, then V is quasi-isomorphic to U (L)[ as a filtered algebra, which yields the theorem. 3.9.13. (a) Suppose we have a Lie algebra L acting on R and B by derivations. Consider the rigidified Lie algebroid L := R ⊗ L. Then the trivial R-module extension L[ := B ⊕ R ⊗ L carries a unique A-extension structure in the sense of 3.9.12 such that the morphism L → L[ , l 7→ 1R ⊗ l is a morphism of Lie algebras and the adjoint action of L ⊂ L[ on A ⊂ L[ coincides with the structure action. We call L[ the rigidified B-extensions of L. Denote by U (B, L) the universal associative k-algebra equipped with an associative algebra morphism φ : B → U (B, L) and a Lie algebra morphism ϕL : L → U (B, L) which are compatible with respect to the L-action on B. Consider the morphism ϕ[ : L[ → U (B, L), rl + b 7→ φ(r)ϕL (l) + φ(b). It satisfies all the properties from 3.9.12 and is universal, so U (B, L) = U (L)[ . The algebra U (B, L) can be described explicitly as follows. The L action on B amounts to an action of the Hopf algebra U (L) on the algebra B. Denote by B ∗ U (L) the corresponding twisted tensor product of B and U (L): this is an associative algebra which equals B ⊗ U (L) as a k-module with the product (a ⊗ u) · (b ⊗ v) = Σaui (b) ⊗ u0i v where Σui ⊗ u0i is the coproduct of u. One checks in a moment that the obvious morphisms B → B ∗ U (L) and L → B ∗ U (L) satisfy the universality property, so we have a canonical isomorphism U (L)[ = ∼ U (B, L) −→ B ∗ U (L). It identifies the PBW filtration on U (L)[ with the filtration B ∗ U (L)i := B ⊗U (L)i . Therefore the PBW theorem for L[ follows from the usual PBW for L. (b) Let us return to the general situation of 3.9.12, so L is an arbitrary Lie R-algebroid and L[ is its arbitrary B-extension. Let us define the DG associative filtered algebra V that was promised in loc. cit. Denote by L our L[ considered as a mere Lie k-algebra acting on R and B. Let K be the kernel of the morphism of R-modules R ⊗ L → L, r ⊗ l[ 7→ rl. Our L acts on R ⊗ L (by the tensor product of the L-action on R and the adjoint action), so it acts on K. ˜ := SymR (K[1]) and B ˜ := B ⊗ R. ˜ So B ˜ is an associative R-algebra, ˜ Set R and R

˜ L) be the enveloping algebra (see (a)). This is L acts on these algebras. Let U (B, a Z-graded associative super algebra whose 0 component equals U (B, L). ˜ L) carries the PBW filtration; by (a), the corresponding The algebra U (B, ˜ ⊗ Symk L. We want to consider a different associated graded algebra equals B filtration, namely, the filtration which on a graded component of degree a equals the PBW filtration shifted by a; this is a ring filtration compatible with the grading. We ˜ L) equipped with this filtration by V. Shifting the PBW isomorphism denote U (B, ˜ · ⊗ Sym· L = appropriately, we get the identification of graded modules gr· V = B

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247

B ⊗ Sym·R (K[1]) ⊗ SymR (R ⊗ L) = B ⊗ Sym·R (K[1] ⊕ R ⊗ L). Therefore R

R

R

∼

gr· V −→ B ⊗ Sym·R (K[1] ⊕ R ⊗ L).

(3.9.13.1)

R

Let us define a natural derivation d on V. We have the maps α, β : K → U (B, L) where α is the composition K ,→ R⊗L → U (B, L), the second arrow is the product map, and β comes from the projection R ⊗ L → L[ , r ⊗ l[ 7→ rl[ , which sends K to B ⊂ L[ , and the morphism B → U (B, L). Lemma. There is a unique odd derivation d of degree 1 on V which kills the ˜ −1 = K[1]. images of B and L and equals α − β on R Proof of Lemma. The uniqueness of d is clear since our conditions define d on the generators of V. It remains to show that d is correctly defined. Set Λ := Sym(k[1]) = k ⊕ k[1]; this is a Z-graded commutative algebra. Consider the Z-graded associative Λ-algebra VΛ = V ⊗ Λ. We have the morphism of ˜ → VΛ and that of Lie algebras L → VΛ which are graded associative algebras B ˜ They satisfy the obvious universality property. compatible via the L-action on B. An odd derivation of degree 1 of V is the same as an automorphism of the Z-graded Λ-algebra VΛ which induces the identity on its quotient V. We construct d using the universality property of VΛ . The conditions on d mean that the corresponding automorphism a(d) of VΛ ˜ −1 = K[1] it is the morphism ν : K[1] → fixes the images of B and L, and on R VΛ = V ⊕ V[1], k 7→ k + α(k) − β(k). Now ν takes values in the centralizer ZΛ ⊂ VΛ of B; it is a morphism of R-modules, and it is compatible with the L-actions. Thus we get a(d) from the universality property of VΛ the moment we extend ν to a ˜ = SymR (K[1]) → ZΛ . Such an extension is unique, morphism of R-algebras R and its existence amounts to the fact that on the image of ν the product of VΛ is commutative. The latter property is equivalent to the commutativity of the image of the map K[1] → VΛ , k 7→ k + α(k), which is evident.  It is clear that d2 = 0 (check on the generators) and d preserves the filtration, so V is a filtered DG algebra which sits in degrees ≤ 0. The differential acts on gr1 U = K[1]⊕LR as the canonical embedding K → LR , so we can rewrite (3.9.13.1) as a canonical isomorphism of graded DG algebras gr· V = B ⊗ Sym·R Cone(K → R ⊗ L).

(3.9.13.2)

R

The projection V0 = U (B, L) → U (B, L)[ has kernel equal to the image of d, so we have an isomorphism of filtered algebras ∼

H 0 V −→ U (L)[ .

(3.9.13.3)

(c) Now suppose that L is R-flat. Then 0 → K → R ⊗ L → L → 0 is a short exact sequence of flat R-modules. So the projection Sym·R Cone(K → L)  Sym·R L is a quasi-isomorphism. By (3.9.13.2) one has H a gr· V = 0 for a 6= 0 (hence H 0 gr· V = gr· H 0 V) and H 0 gr· V = B ⊗ Sym·R L. Therefore, by (3.9.13.3), one has R

gr· U (L)[ = B ⊗ Sym·R L, and we are done. R



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3.9.14. Proof of the theorem in 3.9.11. Let us return to the chiral setting. The above arguments should be modified as follows. Again, we proceed in three steps. (a) Suppose that L is an OX -flat Lie∗ algebra acting on R and B, L[ is an (L, τLR )-rigidified Lie∗ algebroid, and L[ its (L, τLB )-rigidified B-extension (see 3.9.8). One check immediately that U (B, L)[ equals the chiral algebra B ∗ L from 3.7.23 (see the proof in 3.9.8), so the PBW theorem for L[ follows from the PBW result from 3.7.23. Steps (b) and (c) literally repeat those of 3.9.13.  3.9.15. Using the language of 3.9.4, one can rephrase the PBW theorem for chiral Lie algebroids as follows. Let Y be an OX -flat algebraic DX -space, L an OY -flat Lie∗ algebroid on Y. Then the functors L[ 7→ U (L)[Y , (A· , α) 7→ A1 are mutually inverse equivalences between the groupoid Pch (Y, L) of chiral OY -extensions of L and that of pairs (A· , α) where A· is a chiral algebra on YDif equipped with a commutative filtration ∼ A0 ⊂ A1 ⊂ · · · such that gr· A` = Sym(gr1 A` ) and α : gr1 A −→ L is an isomorphism ∗ of Lie algebroids. In particular, for smooth Y there is a canonical equivalence of groupoids (see 3.9.5) (3.9.15.1)

∼

CDOch (Y) −→ Pch (Y, ΘY ).

3.9.16. We are going to describe in “classical” terms the groupoid Pch (L) under the assumption that Y is OX -flat and L is a vector DX -bundle. This will be done in 3.9.20 below. The first step is to define certain canonical DG chiral Lie algebroid L[C related to L. The construction, inspired by [MSV], is of independent interest. At the moment we work locally, so we have an OX -flat commutative DX -algebra R` and a Lie∗ R-algebroid L which is a locally projective R` [DX ]-module of finite rank. Let C = CR (L) be the de Rham-Chevalley complex of L (see 1.4.14). This is a commutative DG DX -algebra which equals Sym(L◦ [−1]) as a mere graded module; here L◦ is the dual R` [DX ]-module. Let us define a DG Lie∗ C-algebroid LC . Our C carries an action of the Lie∗ DG algebra L† (see 1.4.14), so we have the corresponding induced DG Lie∗ Calgebroid (L† )C (see 1.4.13). Denote by K−1 ⊂ (L† )−1 C the kernel of the product · −1 ` map (L† )C = R ⊗ L − → L; let K ⊂ (L† )C be the DG C-ideal generated by K−1 . ∗ Then K is also a Lie ideal which acts trivially on C (since K−1 acts trivially on C and is normalized by L† ), and we set LC := (L† )C /K. Our LC looks as follows. By construction, it sits in degrees ≥ −1, and we ∼ have a morphism of DG Lie∗ algebras ι : L† → LC such that ι−1 : L −→ L−1 C . −1 The C-submodule LC+ generated by LC = L is not closed under the action of d. In fact, as one checks in a moment, LC+ is freely C-generated by L−1 C , and the “Kodaira-Spencer” map LC+ → LC /LC+ , ` 7→ d(`)modLC+ , is an isomorphism. So we have an exact sequence of graded C-modules (3.9.16.1)

0 → C ⊗ L[1] → LC → C ⊗ L → 0. R

R

3.9.17. The following proposition can be considered as a version of 3.9.8 for L replaced by a single super symmetry:

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249

Proposition. A DG chiral C-extension L[C of LC exists and is unique up to a unique isomorphism. Proof. Existence. Let us construct L[C . Consider the rigidified DG chiral Cextension (L† )[C that corresponds to the action of L† on C (see 3.9.8). In degree −1 it coincides with (L† )C , and we define K[ ⊂ (L† )[C as the DG C-submodule generated by K−1 . It is automatically a Lie∗ ideal, and its image in (L† )C equals K. Set L[C := (L† )[C /K[ . This is a chiral DG extension of LC by C/JC , JC := C ∩ K[ ; it remains to show that JC = 0. Since JC is closed under the action of L† , hence L[1] ⊂ L† , it suffices to check that J := JC0 = R ∩ K[ vanishes. d

Let K[+ ⊂ K[ be the C-submodule generated by K−1 . The composition K−1 − → [0 [0 −1 [0 K → K[0 /K[0 is R-linear, so K + d(K ) is an R-submodule; hence K = + + d

−1 K[0 ). Since the composition (L† )−1 → (L† )0C  (L† )0C /C1 ((L† )−1 + + d(K C − C ) is an isomorphism, we see that J = R ∩ K[0 . + Let (L† )[C+ ⊂ (L† )[C be the preimage of C ⊗ L[1] ⊂ (L† )C . This is a graded chiral C-module which is an extension of C ⊗ L[1] by C. As we saw in the proof in 3.9.8, ∆∗ (L† )[C+ is the push-forward of the extension 0 → CL[1] → j∗ j ∗ CL[1] → ∆∗ C ⊗ L[1] → 0 by the morphism C  L[1] → C defined by the ∗ action of L[1] ⊂ L† on C. The above DX×X -modules are R`  R` -modules in the obvious way and the ∗ · action is R` -bilinear, so (L† )[C+ is an R` ⊗R` -module. Set I ` := Ker(R` ⊗R` − → R` ). Then K[+ = I ` · (L† )[C+ since the R` ⊗ R` -action commutes with the chiral C-action. Since L and C are flat R` -modules, we see that C ∩ K[+ = 0, hence J = 0; q.e.d. 0 Uniqueness. Let L[C be any DG chiral C-extension of LC . The projection 0 L[C → LC is an isomorphism in degree −1, so ι lifts in a unique way to a morphism 0 0 of complexes ι[ : L† → L[C . This is automatically a morphism of Lie∗ algebras.86 0 According to 3.9.8, ι[ extends in a unique way to a morphism of DG chiral [ [ Lie C-algebroids ιC : (L† )C → L[C . It vanishes on (K[ )−1 , hence factors through a 0 morphism L[C → L[C . It remains to note that L[C has no non-trivial automorphisms since it is generated (as a DG chiral Lie C-algebroid) by the degree −1 part which coincides with that of LC . 

The next lemma (cf. 3.8.10) will not be used in the sequel. Lemma. There is a unique element d ∈ h(L[1 ), called the de Rham-Chevalley charge, whose adjoint action is the differential d = dL[ of L[C . It acts on C as dC , C and one has [d, d] = 0. Proof. Uniqueness. It suffices to show that elements of h(L[1 C ) are uniquely determined by its adjoint action on L[−1 . In fact, the adjoint action morphism C [−1 [i−1 [i h(LC ) → Hom(LC , LC ) is injective for any i ≥ 1. To see this notice that L[C , considered as a mere graded C-module, carries a canonical 3-step filtration C = L[C0 ⊂ L[C1 ⊂ L[C2 = L[C with gr L[C = C ⊕ C ⊗ L[1] ⊕ C ⊗ L (see (3.9.16.1)). The bracket with L[−1 = L[1] preserves the filtration; on C [ gr LC this is the obvious “convolution” coming from the duality. Thus for every 86 Check

0

first that ι[ is compatible with [ , ]0,−1 and then with [ , ]0,0 .

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[−1 [i−1 i i ≥ 1 the corresponding morphism gr h(L[i ) is C ) = h(gr LC ) → Hom(LC , gr LC injective, and we are done. [−1 Existence. Let us find d ∈ h(L[1 equals d. C ) whose adjoint action on LC [−1 [1 Notice that the quotient layer of the map h(LC ) ,→ Hom(LC , L[0 C ) is the usual identification of h(L◦ ⊗ L) with R` [DX ]-linear endomorphisms of L. Choose any R

0 d0 ∈ h(L[1 C ) which lifts the identity endomorphism idL . The derivation dC − τ (d ) 0 [ of C is R-linear; hence such is the derivation d := d − add0 of LC . One has [0 ∗ 0 0 0 d0 (L[−1 C ) ⊂ LC1 and the R-bilinear pairing ∈ P2 (L), `, ` 7→ [`.d (` )], is skewsymmetric (here `, `0 ∈ L = L[−1 C [−1]). Now the image of the adjoint action [−1 [0 map h(L[1 ) ,→ Hom(L , L ) consists exactly of all R` [DX ]-linear morphisms C1 C C [0 φ : L → LC1 such that the pairing `, `0 7→ [`, φ(`0 )] is skew-symmetric. Therefore, [−1 adding to d0 an element of h(L[1 as d0 , we get the promised d. C1 ) that acts on LC Let us prove that the adjoint action of d coincides with d on the whole of L[C . We check it on every L[i C by induction by i. The case i = −1 is known. Induction step: Suppose we know that d − add kills L[i C . As we have seen above, to check [i+1 that it kills LC , it suffices to verify that [`, (d − add )(f )] = 0 for every ` ∈ L[−1 C , f ∈ L[i+1 . One has [`, (d − ad )(f )] = [(d − ad )(`), f ] − (d − ad )([`, f ]) = 0, and d d d C we are done. Looking at C ⊂ L[C , we see that τ (d) acts on C as dC . Finally, [d, d] ∈ h(L[2 C) 2 equals 0 since its adjoint action on L−1 vanishes (being equal to 2d ).  C

Question. Can one find an explicit formula for the de Rham-Chevalley charge similar to the formula for the BRST charge from 3.8.9? 3.9.18. Important example. Suppose that R` is smooth and L is the tangent algebroid ΘR . Then C is equal to the relative de Rham complex DR = DRR` /X , which is a smooth DG commutative DX -algebra, and LC is its tangent algebroid. So, combining 3.9.17 with (3.9.15.1), we see that DR admits a unique DG cdo DDR . 3.9.19. Passing to the components of degree 0 in 3.9.17 and 3.9.18, we get a canonical Lie∗ R-algebroid T = T(L) := L0C and its chiral R-extension T [ = T(L)[ := L[0 C . Let us describe them explicitly. By (3.9.16.1) T is an extension of L by the ideal L◦ ⊗ L. Our T acts on L (as R

on a mere vector DX -bundle) by the adjoint action on L−1 C = L. This action is Rlinear with respect to the T-variable,87 so L is a T-module (see 1.4.12). It identifies L◦ ⊗ L ⊂ T with gl(L) in the usual way. The canonical section ι = ι0 : L → T (see R

3.9.16) is a morphism of Lie∗ algebras but not of R` -modules: for f ∈ R` , ` ∈ L the endomorphism ι0 (f `) − f ι0 (`) ∈ gl(L) is `0 7→ τL (`0 )(f )`. Remark. If ΘR , hence the Lie∗ algebroid E(L), is well defined, then the above action identifies T with the pull-back of the extension 0 → gl(L) → E(L) → ΘR → 0 by the action morphism τL : L → ΘR . Now let us consider T [ . Its restriction to gl(L) ⊂ T identifies canonically with the Tate extension gl(L)[ as defined in 2.7.6. Indeed, our extension does not depend on the Lie∗ algebra structure on L, and in case R` = OX our identification 87 Since

L−1 C acts trivially on R.

3.9. CHIRAL DIFFERENTIAL OPERATORS

251

is (3.8.6.1). The case of general R` is similar; we actually defined the identification at the end of the proof of the existence statement in 3.9.17. The canonical Lie∗ algebra splitting88 ι[ = ι[0 : L → T [ is not R-linear: precisely, the chiral operation µT [ (idR , ι[ ) − ι[ µL ∈ P2ch ({R, L}, gl(L)[ ) is equal to89 µCl (d, idL ) where µCl ∈ P2ch ({L◦ , L}, gl(L)[ ) is the canonical chiral pairing and d : R → L◦ = C1 is the differential in C.90 Summing up: as a Lie∗ algebra, T [ is the semi-direct product of L and gl(L)[ with respect to the canonical action of L on gl(L)[ coming from the adjoint action of L. The chiral action of R on T [ is the usual R` -action on gl(L)[ , and on L ⊂ T [ it is given by the above formula. ∼

3.9.20. Consider a pair (T c , ρ) where T c ∈ Pcl (T(L)) and ρ : T c |gl(L) −→ gl(L)[ is an isomorphism of extensions which identifies the adjoint action of L ⊂ T with the canonical action of L on gl(L)[ coming from the adjoint action. Such pairs form a groupoid PT (L). The Picard groupoid Pcl (L) acts naturally on PT (L): the translation by Lc ∈ cl P (L) sends (T c , ρ) to the Baer sum of T c and the pull-back of Lc by the projection T  L. This action makes PT (L) a Pcl (L)-torsor. Proposition. There is a canonical anti-equivalence of Pcl (L)-torsors ∼

Pch (L) −→ PT (L).

(3.9.20.1)

Proof. For L[ ∈ Pch (L) the corresponding (T c , ρ) is defined as follows. The pull-back of L by T  L is a chiral R-extension of T trivialized over gl(L) ⊂ T; this trivialization is invariant under the adjoint action of T. Our T c is the Baer difference of the canonical chiral extension T [ and this chiral extension. The above trivialization provides91 ρ.  Of course, the above equivalence has the local nature, so it works in the setting of an OX -flat algebraic DX -space Y and a Lie∗ algebroid L on Y which is a vector DX -bundle. For example, if Y is smooth, then T(ΘY ) = E(ΘY ) and the groupoid PT (ΘY ) coincides with the groupoid Tate(Y) of Tate structures on Y as defined in 2.8.1. Combining (3.9.20.1) with (3.9.15.1), we get a canonical anti-equivalence of Pcl (ΘY )torsors ∼

CDO(Y) −→ Tate(Y).

(3.9.20.2)

We discussed some concrete constructions of Tate structures in 2.8.14–2.8.17. Now they can be seen as examples of cdo. If Y is equipped with an action of a group DX -scheme G affine over X, then weakly G-equivariant Tate structures correspond to cdo equipped with a G-action (see 3.4.17) which extends the G-action on OY . Example. According to 2.8.17, we have a canonical family of cdo on the jet scheme of a flag space parametrized by Miura opers. 88 ι[

3.9.17.

: L† → L[C was defined in the beginning of the proof of the uniqueness statement of ∼

the fact that ι[−1 : L −→ L[−1 is R-linear and the Leibnitz formula. C the map dual to the action τL . 91 Recall that the restriction of T [ to gl(L) equals gl(L)[ ; see 3.9.19.

89 Use

90 I.e.,

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Exercise. Suppose we are in the situation of 2.8.16 and G is connected. Let A be the cdo on Y that corresponds to the Tate structure E(ΘY )[ . Show that the morphism α : L[ → E(Θ)[ from loc. cit. yields a morphism of Lie∗ algebras α : L[ → A which sends 1[ to −1A . The corresponding BRST reduction satisfies <0 0 HBRST (L, A) = 0 and HBRST (L.A) is the cdo on Z that corresponds to the Tate [ structure E(ΘZ )α . 3.9.21. Back in 2.8.3–2.8.9 we described Tate structures on a vector DX bundles in terms of connections. This material can be easily rendered to the situation when ΘR is replaced by an arbitrary L as above. One has only to replace mere connections by L-connections (see 1.4.17). Let us formulate the principal statements; for the proofs and details we refer to 2.8. Consider (Y, L) as above. Let C = COY (L) be the de Rham-Chevalley DG commutative OY [DX ]-algebra. We have the 2-term complex h(C)[1,2) := τ≤2 σ≥1 h(C) of sheaves and the Picard groupoid h(C)[1,2) -tors of h(C)[1,2) -torsors. There is a canonical equivalence of Picard torsors ∼

Pcl (L) −→ h(C)[1,2) -tors

(3.9.21.1)

which assigns to Lc the h(C1 )-torsor of L-connections on Lc and the curvature map c : C → h(C2 )closed (see 2.8.3). Now we consider the set T of pairs (T c , ∇c ) where T c = (T c , ρ) ∈ PT (L) and c ∇ is an L-connection on T c . Let C be the set of pairs (∇, c(∇)[ ) where ∇ is an L-connection on T and c(∇)[ ∈ h(C2 ⊗ gl(L)[ ) is a lifting of c(∇) ∈ h(C2 ⊗ gl(L)) such that d[∇ (c(∇)[ ) = 0. Here d[∇ is a dC -derivation of C ⊗ gl(L)[ that comes from the canonical T-action on L and ∇. There is a canonical bijection (see 2.8.5) (3.9.21.2)

∼

T −→ C,

(T c , ∇c ) 7→ (∇, c(∇c )).

The above T and C are actually groupoids (the morphisms in T are those between T c ; for C see the formulas in 2.8.6), and (3.9.21.2) lifts to an isomorphism of groupoids. They have the local nature, so one can consider the corresponding sheafified groupoids. The one for T is equivalent to PT (L), so we get an equivalence between PT (L) and the sheafification of C. As in 2.8.8 this description can be made concrete by choosing a hypercovering Y· of Y, an L-connection for T on Y0 , a lifting χ ∈ h(C2 ⊗ gl(L)[ ) of C(∇) on Y0 , and a section µ of h(C1 ⊗gl(L)[ ) on Y1 that lifts ∂0∗ ∇−∂1∗ ∇. The same formulas as in 2.8.8 ˇ produce from this datum a Cech cocycle ζ ∈ C 3 (Y· , h(C≥1 )). Its cohomology class DL 3 ≥1 ch1 (L) ∈ H (Y, h(C )) does not depend on the auxiliary choices. It vanishes if and only if PT (L) is non-empty. If this happens, then PT (L) identifies canonically with the groupoid of elements in C 2 (Y· , h(C≥1 )) whose differential equals ζ; the set 0 1 ≥1 of morphisms a → a0 is d−1 )). C (a − a ) ⊂ C (Y· , h(C Suppose now that Y is smooth, so the Lie∗ algebroids ΘY and E(L) are well defined. There is a canonical morphism T → E(L) (the action of T on L) which provides the pull-back morphism of groupoids (3.9.21.3)

Tate(L) → PT (L).

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253

Here L in the left-hand side is considered as a mere vector DX -bundle on Y. A connection for a Tate structure on L provides a connection on the corresponding extension T c , which yields a morphism of the corresponding C-groupoids. In particular, we get: 3 ≥1 3.9.22. Corollary. The obstruction chDL )) to the exis1 (L) ∈ H (Y, h(C D tence of the global chiral extension of L is the image of ch1 (L) by the canonical morphism of DG algebras 92 DRY/X → C. 

Of course, the case of L = ΘY is all in 2.8. Looking at (3.9.20.2), we see, in particular, that the obstruction to the existence of a global cdo is chD 1 (ΘY ) ∈ ≥1 4 H (Y, DRX (DRY/X )). Consider, for example, the situation when our Y is a jet space, Y = JZ for an algebraic space Z/X smooth over X (see 2.3.2). Let p : Y → Z be the canonical projection and ΘZ/X the relative tangent bundle. We will see in a moment that ΘY is locally trivial, so cdo exist locally on Y (see Remark in 2.8.1). Consider the p∗

≥2 ≥1 ≥2 map H 4 (Z, DRZ ) −→ H 4 (Y, DRY ) → H 4 (Y, DRX (DRY/X )) (see 2.8.12).

3.9.23. Corollary. The obstruction chD 1 (ΘY ) to the existence of global cdo 1/2 ≥1 is equal to the image of ch2 (ΘZ/X ωX ) in H 4 (Y, DRX (DRY/X )). Proof. The morphism of O-modules dp : p∗ Ω1Z/X → Ω1Y/X yields an isomor∼

phism of OY [DX ]-modules (p∗ Ω1Z/X )D −→ Ω1Y/X . Passing to dual modules (and using an OY [DX ]-version of (2.2.16.1)), we see that ΘY = (p∗ ΘZ/X ωX )D . Now use 2.8.13.  · · Since p has contractible fibers, the map p∗ : HDR (Z) → HDR (Y) is an isomorp∗−1

≥1 4 4 4 (Z) −−→ HDR phism. The image of chD 1 (ΘY ) by H (Y, DRX (DRY/X )) → HDR (Y) − 1/2

is the conventional de Rham Chern class ch2 (ΘZ/X ωX ) (see Remark in 2.8.12). 1/2 4 (Z) does not vanish, then Y admits no global cdo. Thus if ch2 (ΘZ/X ωX ) ∈ HDR 3.9.24. The rest of the section comprises a few comments on modules over the chiral enveloping algebras parallel to 3.7.16–3.7.19. In particular, we show that for as a chiral R-tdo A the topological algebras Aas x are topological chiral Rx -tdo. [ Let (R, B, L, L ) be as in 3.9.6. For a DX -module M a chiral (B, L[ )-module structure, or chiral (B, L[ )-action, on M is a chiral action µL[ M ∈ P2ch ({L[ , M }, M ) of the Lie∗ algebra L[ on M (see 3.7.16) such that: (i) The restriction µBM of µL[ M to B ⊂ L[ is a unital chiral B-module structure on M . (ii) The operations µL[ M (µRL[ , idM ) and µRM (idR , µL[ M ) − µL[ M (idL[ , µRM ) ch ∈ P3 ({R, L[ , M }, M ) coincide. Here µRM is the restriction of µBM to R. Denote the category of chiral (B, L[ )-modules by M(X, (B, L[ )ch ). Lemma. There is a canonical equivalence of categories (3.9.24.1)

∼

M(X, U (B, L)[ ) −→ M(X, (B, L[ )ch ).

Proof. For any U (B, L)[ -module M the restriction of the U (B, L)[ -action to L[ is an (B, L[ )-module structure on M , so we have a functor M(X, U (B, L)[ ) → 92 Which

comes from the action morphism L → ΘY .

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M(X, (B, L[ )). Its invertibility follows easily from the remark in 3.7.24 since U (B, L)[ is a quotient of B ∗ U (L)[ modulo the evident relations.  One can consider a weaker structure of the (B, L[ )-module on a DX -module M which is a pair consisting of a unital B-module structure µBM and a Lie∗ action τL[ M of L[ on M such that: (i) The Lie∗ action of B Lie on M that corresponds to µBM equals the restriction of the L[ -action on B. (ii) The restriction µRM ∈ P ch ({R, M }, M ) of µBM to R is compatible with the Lie∗ actions of L[ on R and M . Denote the category of (B, L[ )-modules by M(X, (B, L[ )). The evident forgetful functor M(X, (B, L[ )ch ) → M(X, (B, L[ )) admits a left adjoint functor (3.9.24.2)

Ind : M(X, (B, L[ )) → M(X, (B, L[ )ch ).

The proof of this fact is similar to the proof of the proposition in 3.7.15; it also provides a construction of Ind in terms of the enveloping algebra functor. As in the corollary in 3.7.18, Ind extends naturally to a pseudo-tensor functor. For a (B, L[ )-module M we define the PBW filtration on Ind M as the image by the chiral action of j∗ j ∗ U (B, L[ )·  M . If M is a central R-module, then gr· IndM is also a central R-module, and we have a surjective morphism M ⊗ SymR` L  R`

gr· IndM . Exercise. If the assumptions of the PBW theorem (see 3.9.11) are satisfied, then this is an isomorphism (cf. Remark (i) in 3.7.15). 3.9.25. Let x ∈ X be a point, jx : Ux ,→ X its complement, and suppose that on Ux we have R, a Lie∗ algebroid L, and its chiral R-extension. Let us describe (see 3.6.2–3.6.7). the topological associative algebra U (R, L)[as x [ [ [ [ as of pairs (R , L ) ∈ Ξ Consider the set ΞRL ξ x (jx∗ R)×Ξx (jx∗ L ) such that Lξ is x ξ a chiral Lie Rξ -subalgebroid of jx∗ L[ ; i.e., L[ξ is a Lie∗ subalgebra and Rξ -submodule of jx∗ L[ , and its action on jx∗ R preserves Rξ (cf. 2.5.19). The corresponding Rξ and L[ξ form topologies at x on, respectively, jx∗ R and jx∗ L[ which we denote [ [ ˆ LieR (L[ ) be the corresponding completions. . Let RxasL and h and ΞLieR by ΞasL x x x [ asL ˆ LieR (L[ ) is a topological is a commutative topological algebra and h Then R x

x

[

Lie RxasL -algebroid.93 The section 1[ provides a canonical central element 1[ ∈ ˆ LieR (L[ ) (the 1[ -image of 1 ∈ k = h ˆ x (jx∗ ω)) which acts trivially on RasL[ . h x x Let Q be any commutative topological algebra, P [ a topological Q-algebroid (see 2.5.18), and 1[ ∈ P [ a central element which acts trivially on Q. Then P := P [ /Q1[ is a topological Lie Q-algebroid. The topological enveloping algebra U (Q, P )[ of (Q, P [ ) is the quotient of the topological enveloping algebra of the topological Lie algebra P [ (see 3.8.17) modulo the following relations: ·1[

(i) The composition Q −−→ P [ → U (Q, P )[ is a morphism of unital associative algebras. (ii) The map P [ → U (Q, P )[ is a morphism of Q-modules; here Q acts on U (Q, P )[ by the left product via (i). 93 See

2.5.18 for the definition.

3.9. CHIRAL DIFFERENTIAL OPERATORS

255

The topological algebra U (Q, P )[ has a filtration U (Q, P )[· with U (Q, P )[0 := the closure of the image of Q, and U (Q, P )[i := the closure of the image of the map (Q[ )⊗i → U (Q, S P )[ , `[1 ⊗ · · · ⊗ `[i 7→ `[1 · · · `[i ; we refer to it as the PBW filtration. It is clear that U (Q, P )[i is dense in U (Q, P )[ and gr U (Q, P )[ is a topological commutative Q-algebra topologically generated by U (Q, P )[1 . [ ˆ LieR (L[ ), we get a filtered topological associative Taking Q = RxasL and P [ = h x [ asL[ ˆ LieR algebra UL := U (Rx , hx (L))[ . Lemma. There is a canonical isomorphism of filtered topological algebras ∼

UL[ −→ U (R, L)[as x .

(3.9.25.1)

Proof. The morphism UL[ → U (R, L)[as comes from the universality property x of UL[ since the morphism L[ → U (R, L)[ yields a morphism of topological Lie ˆ LieR (L[ ) → U (R, L)[as that satisfies properties (i)–(ii) above. The fact algebras h x x that it is an isomorphism of topological algebras follows from (3.9.24.1) just as in the proof of the proposition in 3.7.22. The strict compatibility with filtrations is evident.  ˆ R (L) (see 2.4.11). The evident 3.9.26. Consider the topological Rxas -module h x [ [ R as ˆ continuous morphisms Rx → UL0 , hx (L) → gr1 UL1 yield a continuous morphism ˆ R (L) → gr· U [ SymRxas h x L

(3.9.26.1)

of graded topological algebras with dense image.94 One has the following version of the PBW theorem: Proposition. Suppose that R` is a finitely generated DUx -algebra, L is a finitely generated projective R` [DUx ]-module, and L[ is a chiral R-extension of L. Then (3.9.26.1) is an isomorphism. Proof. (i) We can assume that L is a free R` [DUx ]-module. If not, choose a finitely generated projective R` [DUx ]-module T such that ˜ := L ⊕ T is free. Then L ˜ is a Lie∗ algebra with respect to a bracket whose L only non-zero component is the Lie∗ bracket on L; it is a Lie∗ algebroid in the ˜ [ := L[ ⊕ T is a chiral R-extension of L; ˜ the embedding evident way. Similarly, L [ ˜ [ are morphisms of chiral Lie R-algebroids. Suppose and the projection L  L ˜ [ ; let us show that it is also holds for L[ . Indeed, the that our PBW holds for L ˜ [ yields a morphism U [ → U [ of filtered topological algebras morphism L[ → L ˜ L L [ and hence a map gr UL[ → gr UL . Due to our assumption, we can rewrite it as ˜ [ R ˜ ˆ gr UL → SymRxas hx (L). Its image lies in the topological Rxas -subalgebra gener˜ which equals SymRas h ˆ R (L) ⊂ h ˆ R (L) ˆ R (L). We have defined a morphism ated by h x x x x ˜ it is evidently inverse to (3.9.26.1). ˆ R (L); gr U [ → SymRas h L

x

x

(ii) Let {`[i } be a finite set of sections of L[ such that the corresponding `i ∈ L freely generate L as an R` [DUx ]-module and let t be a parameter at x which we assume (by shrinking X) to be invertible on Ux . Take any Rξ ∈ Ξas x (jx∗ R) which is a finitely generated DX -algebra. For an integer n denote by L[n = L[ξn the chiral 94 Here Sym as h ˆ R (L) is the topological symmetric algebra, i.e., the topological commutative Rx x as -algebra freely generated by the topological Ras -module h ˆ R (L). Rx x x

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Rξ -submodule of jx∗ L[ generated by sections {tn `[i } and 1[ . Let Ln = Lξn be its image in jx∗ L which is a free Rξ` [DX ]-module with base {`i }. Let us show that for n sufficiently big L[n is a chiral Lie Rξ -subalgebroid of jx∗ L[ which is a chiral Rξ -extension of Ln . (a) For sufficiently big n the Lie∗ action of Ln ⊂ jx∗ L preserves Rξ ⊂ jx∗ R (since Rξ is finitely generated). ∼ [ (b) For such n, L[n is an extension of Ln by Rξ ; i.e., L˘ → Ln . To n := Ln /Rξ − [ see this, notice that L˘ n is the chiral Rξ -submodule of jx∗ L /Rξ generated by the n n [ n images t `˘i of t `i . Now (a) means that the t `˘i are Rζ -central sections (see 3.3.7). ` Thus L˘ n is actually an Rζ [DX ]-module; hence it projects isomorphically to Ln . (c) It remains to show that for n as in (a) and sufficiently big m our L[m+n is a Lie∗ subalgebra of jx∗ L[ . Equivalently, this assertion means that all the sections [tm+n `[i , tm+n `[j ] ∈ ∆∗ jx∗ L[ lie in ∆∗ L[m+n , or that their images [tm+n `[i , tm+n `[j ]˘∈ [ ∆∗ (jx∗ L[ /Rξ ) lie in ∆∗ Lm+n ˘ . Now, since L˘ n ∈ Ξx (jx∗ L /Rξ ), one can find a, b ≥ 0 ≤b [ such that [tn `[i , tn `[j ]˘∈ ∆· (t−a ΣRξ` tn `˘i ) · D≤b X×X ⊂ ∆∗ (jx∗ L /Rξ ) where DX×X is the sheaf of differential operators of degree ≤ b on X × X. It is clear that any m ≥ a + b will do. (iii) Algebras Rξ as in (ii) form a base of the topology Ξas x (jx∗ R), Lξn a base [ R . The latter fact imof the topology Ξx ξ (jx∗ L) and (Rξ , L[ξn ) a base of ΞRL x [ as plies that the U (Rξ , Lξn ) form a base of the Ξx -topology of jx∗ U (R, L)[ , so [` [ U (R, L)[as = ← lim x − U (Rξ , Lξn )x . According to 3.4.11 applied to Rξ and Lξn , we ˆ R (L). Now use (3.9.25.1) to finish the proof. as h  have gr U (R, L)[as x = SymRx x Remark. We do not know if the R` [DUx ]-projectivity assumption on L can be weakened to the R` [DUx ]-flatness assumption. 3.9.27. Let Q be a reasonable commutative topological algebra, so ΘQ is well defined (see 2.5.21), and let Q → F be a morphism of topological algebras. We say that F is a topological Q-tdo if there is a commutative ring filtration F0 ⊂ F1 ⊂ · · · by closed subspaces of F such that F0 is the image of Q, there is an isomorphism ∼ of S topological graded Poisson algebras SymQ ΘQ −→ gr F , and F is the completion of Fi with respect to the topology formed by left ideals I such that I ∩ Fi is open in Fi for each i. Corollary. Suppose that R is smooth and we have Θ[R ∈ Pch (ΘR ). Then U (R, ΘR )[as is a topological Rxas -tdo. x Proof. Our (R, Θ[R ) satisfy the conditions of 3.9.26. By the proposition in ˆ R (ΘR ) = ΘRas . We are done by (3.9.25.1) and 3.9.26. 2.5.21, one has h  x x ˆ LieR (Θ[ ). This is an Ras -extension of the topological Lie Ras Set Θ[Rxas := h x x x R as [ as algebroid ΘRxas , and U (R, ΘR )[as = U (R , Θ ) . Notice that for any classical RRx x x c cl c R c as ˆ extension ΘR ∈ P (ΘR ) the completion ΘRxas := hx (ΘR ) is also an Rx -extension of the topological Lie Rxas -algebroid ΘRxas . Consider our extensions as mere extensions of topological Rxas -modules. Those that correspond to classical R-extensions of ΘR split, so, by the lemma in 3.9.7, the class of Θ[Rxas does not depend on the choice of a particular chiral extension Θ[R ∈ Pch (ΘR ).

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257

Exercises. Suppose that R` = Sym(DnUx ), n ≥ 1, and ΘνRxas is any topological Rxas -extension of the Lie Rxas -algebroid ΘRxas . Show that if ΘνRxas splits as an extension of topological Rxas -modules, then the enveloping algebra U (Rxas , ΘRxas )ν (see 3.9.25) vanishes. In particular, the extension Θ[Rxas is non-trivial, and for any ΘcR ∈ Pcl (ΘR ) one has U (Rxas , ΘRxas )c = 0.

3.10. Lattice chiral algebras and chiral monoids The lattice chiral algebras are important examples of non-commutative chiral algebras that do not arise naturally as enveloping algebras of Lie∗ algebras. They appeared as vertex algebras in [B1] (and, to some extent, in an earlier article [FK]) and played a prominent role in [FLM]; see [K] 5.4, 5.5, for a nice presentation. From the geometric perspective, the lattice chiral algebras arise naturally from the geometry of local Picard ind-schemes, namely, from their canonical chiral monoid structure. The structure of a lattice chiral algebra at a point x ∈ X is controlled by an appropriate Heisenberg group whose commutator pairing is defined by the Contou-Carr`ere symbol [CC] of the geometric class field theory. The exposition below divides naturally into three parts: (i) The description of lattice chiral algebras in terms of θ-data. The lattice chiral algebras are defined in 3.10.1. The commutative lattice algebras are identified with the jet algebras of principal bundles for the dual torus in 3.10.2. The parallel non-commutative objects are θ-data defined in 3.10.3; the theorem identifying lattice algebras and θ-data is stated in 3.10.4. The construction of the θ-datum corresponding to a lattice algebra is given in 3.10.6. The converse construction, which will be of use in 4.9, is presented in 3.10.8. It is based on an interpretation of a θ-datum as a line bundle on the group of divisors equipped with a certain factorization structure (see 3.10.7), which is a local version of the self-duality of the Picard variety as described in [SGA 4] Exp. XVIII 1.3, 1.5. In 3.10.9 the Lie∗ Heisenberg subalgebra of a lattice algebra is considered. Subsection 3.10.10 treats a particular example (called the “boson-fermion correspondence”; see, e.g., [K] 5.2 or [FBZ] 4.3). In 3.10.11 we briefly mention another procedure of reconstruction of an arbitrary lattice chiral algebra from its θ-datum which is due to Roitman [Ro]. It is not difficult to check that in the translation equivariant setting, lattice chiral algebras are essentially the same as lattice vertex algebras from the references above.95 (ii) The Heisenberg groups. We begin with introductory material on group ind-schemes, their super extensions, and group algebras (see 3.10.12). After a brief reminder on the Contou-Carr`ere symbol (we refer to [CC], [D2] 2.9, and [BBE] 3.1–3.3 for all details), we define the Heisenberg groups and describe their representations (see 3.10.13). In 3.10.14 we assign to a lattice algebra A and a point x ∈ X a Heisenberg group whose twisted group algebra coincides with Aas x . Thus A-modules supported at x are identified with representations of the Heisenberg group, which implies the results of Dong [Don]. The Heisenberg group description of Aas x has an immediate twisted version (see 3.10.15).

95 In the vertex algebra literature the lattice vertex algebras are constructed rather than defined, and the cotorsor structure from 3.10.1 is implicit.

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(iii) The chiral monoids. These are geometric objects that underlie the construction of lattice chiral algebras from 3.10.8. We give the first definitions and explain how ind-finite chiral monoids generate chiral algebras (see 3.10.16). Let us mention that non-ind-finite chiral monoids can also be used to construct some important chiral algebras (such as the chiral Hecke algebra). Such constructions require some extra data. This subject will not be discussed in the book. 3.10.1. Let Γ be a lattice (= a free abelian group of finite rank), Γ∨ := Hom(Γ, Z) the dual lattice and h , i : Γ × Γ∨ → Z the pairing, T := Gm ⊗ Γ = Spec k[Γ∨ ] and T ∨ := Gm ⊗ Γ∨ = Spec k[Γ] the corresponding tori, t := k ⊗ Γ and t∨ = k ⊗ Γ∨ = t∗ their Lie algebras. For γ ∈ Γ we write the corresponding homomorphism Gm → T as f 7→ f ⊗γ . Let Tors(X, T ), Tors(X, T ∨ ) be the Picard groupoids of T - and T ∨ -torsors on X. ∨ ` ∨ ∨ Let TX = Spec FX := JTX be the jet DX -scheme of the group X-scheme TX := T ∨ ×X. This is a commutative group DX -scheme, so F = FX is a commutative and cocommutative counital Hopf chiral algebra (see 3.4.16). We will consider chiral ∨ algebras equipped with a TX -action (which is the same as a counital F -coaction; see 3.4.17). As in 3.4.16, we have the Picard groupoid PHeis (X, Γ) := P(X, F ) of F -cotorsors. Definition. An object of PHeis (X, Γ) is called a lattice Heisenberg, or simply lattice, chiral algebra on X (with respect to Γ). Remark. The Picard groupoid PHeis (X, Γ) carries an evident action of the group Aut (Γ). Therefore for any group acting on Γ, one can consider the corresponding equivariant lattice algebras. The lattice algebras equivariant with respect to the involution γ 7→ −γ are called symmetric lattice algebras. 3.10.2. The commutative case. Let PHeis (X, Γ)0 ⊂ PHeis (X, Γ) be the Picard subgroupoid of commutative lattice chiral algebras.96 The functor A 7→ ∨ -torsors. SpecA` identifies PHeis (X, Γ)0 with the Picard groupoid of DX -scheme TX ∨ -torsors, According to Remark (iv) in 3.4.17, the latter objects are the same as TX so we have defined a canonical equivalence of Picard groupoids (see (3.4.17.2)) (3.10.2.1)

∼

PHeis (X, Γ)0 −→ Tors(X, T ∨ ).

3.10.3. θ-data. We want to extend (3.10.2.1) to the non-commutative situa∨ -torsors are replaced by the following objects: tion. The TX Definition. A θ-datum for Γ on X is a triple θ = (κ, λ, c) where κ : Γ×Γ → Z is a symmetric bilinear form, λ is a rule that assigns to each γ ∈ Γ a super line bundle λγ on X, and c is a rule that assigns to each pair γ1 , γ2 of elements of Γ ∼ an isomorphism cγ1 ,γ2 : λγ1 ⊗ λγ2 −→ λγ1 +γ2 ⊗ ω ⊗κ(γ1 ,γ2 ) . We demand c to be associative and to satisfy a twisted commutativity property: (i) The isomorphisms cγ1 ,γ2 +γ3 (idλγ1 ⊗ cγ2 ,γ3 ) and cγ1 +γ2 ,γ3 (cγ1 ,γ2 ⊗ idλγ3 ) : ∼ γ1 λ ⊗ λγ2 ⊗ λγ3 −→ λγ1 +γ2 +γ3 ⊗ ω ⊗κ(γ1 ,γ2 )+κ(γ2 ,γ3 )+κ(γ1 ,γ3 ) coincide. ∼ γ1 ,γ2 = (−1)κ(γ1 ,γ2 ) cγ2 ,γ1 σ where σ : λγ1 ⊗ λγ2 −→ λγ2 ⊗ λγ1 is the commu(ii) c tativity constraint. 96 If

F

A1 , A2 are commutative lattice algebras, then A1 ⊗ A2 is also commutative (as a chiral subalgebra of A1 ⊗ A2 ).

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Notice that by (ii) the parity of the super line λγ equals κ(γ, γ) mod2. There ∼ is a canonical identification OX −→ λ0 which identifies 1 ∈ OX with the unique 0 0,0 section 1 ∈ λ such that c (1 ⊗ 1) = 1. Sometimes we call θ = (κ, λ, c) a κ-twisted θ-datum and write it as θκ = (λ, c). All θ-data for Γ on X form a groupoid Pθ (X, Γ) which is the disjoint union of the groupoids Pθ (X, Γ)κ of κ-twisted θ-data. Our Pθ (X, Γ) is a Picard groupoid with 0 the product (κ, λ, c) ⊗ (κ0 , λ0 , c0 ) = (κ + κ0 , λ ⊗ λ0 , c ⊗ c0 ) where (λ ⊗ λ0 )γ := λγ ⊗ λ γ , γ ,γ (c ⊗ c0 )γ1 ,γ2 := cγ1 ,γ2 ⊗ c0 1 2 . Notice that P(X, Γ) := Pθ (X, Γ)0 is a Picard subgroupoid of Pθ (X, Γ) and each θ P (X, Γ)κ is a P(X, Γ)-torsor (it is non-empty by the lemma below). Since θ-data have the ´etale local nature, we have a sheaf of Picard groupoids Pθ (Γ) on Xe´t , etc. Lemma. (i) There is a canonical equivalence of Picard groupoids (3.10.3.1)

∼

P(X, Γ) −→ Tors(X, T ∨ ).

(ii) For any symmetric bilinear form κ on Γ one has Pθ (X, Γ)κ 6= ∅. Therefore P (X, Γ)κ is a P(X, Γ)-torsor and Pθ (Γ)κ is a neutral T ∨ -gerbe on Xe´t . P γ Proof. (i) Take any θ = (λ, c) ∈ P(X, Γ). Then N (θ) := λ is a Γ-graded commutative OX -algebra (with product c) and Spec N (θ) is a T ∨ -torsor over X. Our equivalence is θ 7→ Spec N (θ). (ii) Let us construct a κ-twisted θ-datum. First we choose a central super Gm -extension Γ of Γ such that the commutator pairing Γ × Γ → Gm equals γ1 , γ2 7→ (−1)κ(γ1 ,γ2 ) . This means that for every γ ∈ Γ we have a super line γ ∼ and for every γ1 , γ2 an isomorphism cγ 1 ,γ2 : γ1 ⊗ γ2 −→ γ1 +γ2 such that c is associative, i.e., cγ 1 ,γ2 +γ3 (idγ1 ⊗ cγ 2 ,γ3 ) = cγ1 +γ2 ,γ3 (cγ1 ,γ2 ⊗ idγ3 ), and one has ∼ cγ 1 ,γ2 = (−1)κ(γ1 ,γ2 ) cγ 2 ,γ1 σ where σ : γ1 ⊗ γ2 −→ γ2 ⊗ γ1 is the commutativity constraint. Such Γ always exists.97 Next, choose a line bundle of half-forms ω ⊗1/2 (this step is superfluous if κ is even). We get θκ = (λ, c) ∈ Pθ (X, Γ)κ where λγ := (ω ⊗1/2 )⊗−κ(γ,γ) ⊗ γ and c is the evident product multiplied by c .  θ

Remarks. (i) Pairs (Γ, θ) where Γ is a lattice, θ ∈ Pθ (X, Γ) form naturally a category Pθ (X). Namely, a morphism φ : (Γ, θ) → (Γ0 , θ0 ) is a pair (φΓ , φλ ) where φΓ : Γ → Γ0 is a morphism of lattices and φλ is a collection of isomorphisms ∼ φ (γ) φγλ : λγ −→ λ0 Γ , γ ∈ Γ, such that φΓ is compatible with κ and κ0 and φλ is compatible with c and c0 in the obvious way. This is a symmetric monoidal category with the operation (Γ, θ)⊕(Γ0 , θ0 ) = (Γ⊕Γ0 , θ⊕θ0 ) where θ⊕θ0 = (κ⊕κ0 , λ⊗λ0 , c⊗c0 ). (ii) The group Aut (Γ) acts on Pθ (X, Γ) by transport of structure. So, as in the remark in 3.10.1, we can consider equivariant θ-data. For example, symmetric θdata form a sheaf of Picard groupoids on Xe´t ; its κ-components are neutral µ2 ⊗Γ∨ gerbes. In particular, symmetric θ-data are rigid. 3.10.4. Here is the promised generalization of 3.10.2: Theorem. There is a canonical equivalence of Picard groupoids (3.10.4.1)

∼

PHeis (X, Γ) −→ Pθ (X, Γ).

97 To construct one, choose a morphism v : Γ → Z/2 and a bilinear pairing r : Γ × Γ → Z/2 so that κ(γ1 , γ2 ) mod2 = v(γ1 )v(γ2 )+r(γ1 , γ2 )+r(γ2 , γ1 ). Take for Γ the Gm -extension of Γ defined by the 2-cocycle (−1)r , considered as a super extension with parity of the line corresponding to γ equal to v(γ).

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The equivalence is compatible with the Aut(Γ)-actions on the Picard groupoids, so it yields the equivalences between the Picard groupoids of equivariant objects. Proof. It can be found in 3.10.5–3.10.8. We define the functor (3.10.4.1) in 3.10.6 (after the necessary preliminaries of 3.10.5), essentially repeating the construction of 3.10.2. The inverse functor is constructed in 3.10.8 after we reinterpret the notion of θ-datum in 3.10.7; for a different construction see 3.10.11. ∨ ∨ 3.10.5. Preliminaries on TX -modules. (i) The canonical projection TX  is a morphism of group X-schemes, so we have an embedding of the counital ` ∨ Hopf OX -algebras OX [Γ] ,→ FX , γ 7→ χγ . The constant connection on TX yields a ∨ ∨ horizontal section of the above projection TX ,→ TX which is a morphism of group ` schemes, so we have a morphism of the counital Hopf DX -algebras FX  OX [Γ] left inverse to the above embedding. Consider the category of OX -flat OX -modules N equipped with a counital ` ∨ FX -action (we call such N simply a TX -module). This is a tensor category: for ∨ ∨ ∨ TX -modules M , N their tensor product as TX -modules is, by definition, the TX ∨ TX

F

submodule M ⊗ N ⊂ M ⊗ N that consists of those sections n for which the images ` ⊗ M ⊗ N coincide. The of n by the morphisms δM ⊗ idN , idN ⊗ δM : M ⊗ N → FX F

` . unit object for ⊗ equals FX ∨ ∨ Every TX -module N is automatically a TX -module = the counital OX [Γ]∨ ∨ comodule (via TX ,→ TX ). Therefore it carries a canonical Γ-grading N = ⊕N γ `γ where N γ = {n ∈ N : δN (n) ∈ FX ⊗ N }. Let N0 ⊂ N be the maximal submodule ∨ ∨ ∨ . One has on which the TX -action factors through the projection TX  TX

N0γ = {n ∈ N : δN (n) = χγ ⊗ n}.

(3.10.5.1)

F

The functor N 7→ N0 commutes with the tensor products: one has (M ⊗ N )γ0 = M0γ ⊗ N0γ (as OX -submodules of M ⊗ N ). In particular, if N is invertible with F

respect to ⊗, then all N0γ are line bundles on X. (ii) It is convenient to keep in mind the dual F `∗ -module picture: ` ` `∗ defines a topo, OX ). The coalgebra structure on FX := HomOX (FX Set FX `∗ logical OX -algebra structure on FX . The topology admits a base of open ideals `∗ /Iα areSlocally free OX -modules of finite rank.98 Therefore the Iα such that FX `∗ `∗ /Iα is the inductive limit of subschemes finite := Spec FX ind-scheme Spec FX and flat over X. It is also a commutative group ind-X-scheme (the product comes ` from the algebra structure on FX ) equipped with a connection along X (defined ` ∨ ∨ `∗ by the connection on FX ). The above TX  TX yield a projection Spec FX  ΓX ∼ `∗ `∗ and its section ΓX −→ (Spec FX )red ,→ Spec FX . ∨ ` For an OX -module N , a TX -action δN : N → FX ⊗ N amounts to a discrete F

`∗ FX -module structure on N . The tensor product M ⊗ N is formed by sections of the `∗ ˆ `∗ FX ⊗FX -module M ⊗ N that are supported (scheme-theoretically) on the diagonal `∗ `∗ `∗ Spec FX ,→ Spec FX × Spec FX . The submodules N0 and N γ of N are formed by X

`∗ sections supported, respectively, on (Spec FX )red and the connected component of `∗ Spec FX labeled by γ. `∗ For a geometric interpretation of Spec FX see 3.10.8. 98 Our

` is locally free as an O -module. FX X

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261

3.10.6. From latice algebras to θ-data. Let A be any lattice chiral algebra. According to 3.10.5, A carries a Γ-grading A = ⊕Aγ . One checks in a moment that it is compatible with the chiral product: µA sends j∗ j ∗ Aγ1  Aγ2 to ∆∗ Aγ1 +γ2 . `γ By 3.10.5, for each γ ∈ Γ we have a (super) line bundle λγA := A`γ 0 ⊂ A . For γ1 , γ2 ∈ Γ let κA (γ1 , γ2 ) be the order of zero at the diagonal of the ope morphism ◦A (see 3.5.10, 3.5.11) restricted to the line bundle λγA1  λγA2 ⊂ A`  A` .99 In other words, κA (γ1 , γ2 ) is the largest integer n such that µA : j∗ j ∗ A  A → ∆∗ A vanishes on λγA1  λγA2 (n∆). The top Taylor coefficient of ◦A is then a non-zero morphism of OX -modules cγA1 ,γ2 : λγA1 ⊗ λγA2 → A` ⊗ ω ⊗κA (γ1 ,γ2 ) . γ1 ,γ2 By (3.10.5.1) (and the fact that δA is a morphism of chiral algebras), cA γ1 +γ2 ⊗κA (γ1 ,γ2 ) ` ⊗κA (γ1 ,γ2 ) takes values in λA ⊗ω ⊂A ⊗ω . F

By 3.10.5(i), our picture is compatible with the ⊗-product of lattice algebras. F

Namely, for two lattice algebras A, B the λγ line for A ⊗ B equals λγA ⊗ λγB as a submodule of A` ⊗ B ` . Due to compatibility of the ope with tensor products (see F

3.5.11), κ and c for A ⊗ B are equal to, respectively, κA + κB and cA ⊗ cB . In F

particular, for B equal to the ⊗-inverse to A, one has κA + κB = 0 and cA ⊗ cB = 1, γ1 +γ2 which implies that κA (γ1 , γ2 ) < ∞ and cγA1 ,γ2 : λγA1 ⊗ λγA2 → λA ⊗ ω ⊗κA (γ1 ,γ2 ) is always an isomorphism. Lemma. (κA , λA , cA ) is a θ-datum. Proof. For γ1 , γ2 , γ3 ∈ Γ consider the restriction of the triple ope (see 3.5.12) 1 2 3 to the line bundle A`γ  A`γ  A`γ 0 0 0 . At each diagonal xi = xj it has zero of order κA (γi , γj ). On the other hand, its top term at, say, the diagonal x1 = x2 is equal to ◦A (cA (γ1 , γ2 ), idA`γ3 ), so it has zero of order κA (γ1 + γ2 , γ3 ). Therefore 0 κA (γ1 + γ2 , γ3 ) = κA (γ1 , γ3 ) + κA (γ2 , γ3 ), so κA is bilinear. It is obviously symmetric. The associativity of cA follows from the associativity property of the ope. The twisted commutativity of cA follows from the commutativity of the ope, since ˜ ∗ A` as the transposition symmetry acts on the subquotient A` ⊗ ω κA (γ1 ,γ2 ) of ∆ κA (γ1 ,γ2 ) multiplication by (−1) .  We define the Picard functor PHeis (X, Γ) → Pθ (X, Γ) from (3.10.4.1) as A 7→ θA := (κA , λA , cA ). Its restriction to PHeis (X, Γ)0 is the composition of (3.10.2.1) and the inverse to (3.10.3.1), so our functor is fully faithful. To prove that our functor is an equivalence, it suffices to construct its right inverse. We will do this in 3.10.8 after reinterpreting the notion of θ-datum in 3.10.7. 3.10.7. Denote by Div(X) a functor which assigns to a quasi-compact scheme Z the group of Cartier divisors in X × Z/Z proper over Z. This is a sheaf with respect to the flat topology. Set Div(X, Γ) := Div(X) ⊗ Γ; we call its points Γ-divisors. Let λ be a super line bundle on Div(X, Γ). Therefore λ is a rule that assigns to any Z and a Γ-divisor D ∈ Div(X, Γ)(Z) a super line bundle λD on Z in a way compatible with the base change. Definition. A factorization structure on λ is a rule that assigns to any Z and a pair of Γ-divisors D1 , D2 ∈ Div(X, Γ)(Z) whose supports do not intersect an 99 We

will see in a moment that this morphism is non-zero, i.e., κA (γ1 , γ2 ) 6= ∞.

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isomorphism cD1 ,D2 : λD1 ⊗ λD2 −→ λD1 +D2 . We demand c to be compatible with the base change and to be associative and commutative: (i) For every D1 , D2 , D3 ∈ Div(X, Γ)(Z) whose supports are pairwise disjoint, the two morphisms cD1 +D2 ,D3 (cD1 ,D2 ⊗ idλD3 ) and cD1 ,D2 +D3 (idλD3 ⊗ cD1 ,D2 ) : ∼ λD1 ⊗ λD2 ⊗ λD3 −→ λD1 +D2 +D3 coincide. ∼ (ii) One has cD1 ,D2 = cD2 ,D1 σ, where σ : λD1 ⊗ λD2 −→ λD2 ⊗ λD1 is the commutativity constraint. Pairs (λ, c) as above form naturally a Picard groupoid (the tensor product is (λ1 , c1 ) ⊗ (λ2 , c2 ) := (λ1 ⊗ λ2 , c1 ⊗ c2 )); denote it by Picf (Div(X, Γ)). Proposition. There is a natural equivalence of Picard groupoids (3.10.7.1)

∼

Picf (Div(X, Γ)) −→ Pθ (X, Γ).

Proof. For γ ∈ Γ let iγ : X → Div(X, Γ) be the morphism x 7→ (x) ⊗ γ; here (x) is the Cartier divisor that corresponds to x. For (λ, c) ∈ Picf (Div(X, Γ)) the corresponding θ = (κ, λγ , cγ1 ,γ2 ) ∈ Pθ (X, Γ) is defined as follows. We set λγ := iγ∗ λ. For γ1 , γ2 ∈ Γ let λγ1 ,γ2 be the pull-back of λ by the morphism iγ1 ,γ2 : X × X → Div(X, Γ), (x1 , x2 ) 7→ iγ1 (x1 ) + iγ2 (x2 ). On ∼ the complement U to the diagonal ∆ our c yields an isomorphism λγ1  λγ2 |U −→ ∼ λγ1 ,γ2 |U . So we have an isomorphism λγ1  λγ2 −→ λγ1 ,γ2 (−κ(γ1 , γ2 )∆) for certain κ(γ1 , γ2 ) ∈ Z. Pulling it back to the diagonal, we get an isomorphism cγ1 ,γ2 : ∼ λγ1 ⊗ λγ2 −→ λγ1 +γ2 ⊗ ω κ(γ1 ,γ2 ) of super line bundles on X. Let us check that θ = (κ, λγ , cγ1 ,γ2 ) so defined is, indeed, a θ-datum. The commutativity property of cγ1 ,γ2 is evident. Now notice that for any I ∈ S and i) (γi ) ∈ ΓI we have a super line bundle λ(γP on X I defined as the pull-back of λ by (γi ) I the map i : X → Div(X, Γ), (xi ) 7→ iγi (xi ), and c yields an isomorphism (3.10.7.2)

∼

λγi −→ λ(γi ) (−

X

κ(γi , γi0 )∆i,i0 )

(the summation is over the set of unordered pairs i 6= i0 ∈ I and ∆i,i0 is the diagonal divisor xi = xi0 ). One checks the bilinearity of κ and the associativity of cγ1 ,γ2 looking at (3.10.7.2) on X × X × X. The functor Picf (Div(X, Γ)) → Pθ (X, Γ) we have constructed is evidently a morphism of Picard groupoids. It is faithful: this follows from (3.10.7.2) since any point of Div(X, Γ) factors flat locally through i(γi ) for some (I, (γi )). It remains to prove that our functor is essentially surjective. Consider the Picard groupoid of extensions Ext(Div(X, Γ), Gm ). Such an extension is the same as a pair (λ, c) as above except that cD1 ,D2 are defined for every D1 , D2 (the supports may intersect). So we have an evident morphism of Picard groupoids Ext(Div(X, Γ), Gm ) → Picf (Div(X, Γ)). Its composition with (3.10.7.1) takes values in P(X, Γ) = Tors(X, T ∨ ) (see (3.10.3.1)). The functor (3.10.7.3)

Ext(Div(X, Γ), Gm ) → Tors(X, T ∨ )

is an equivalence. To see this, notice that our objects are Γ-linear, so we can assume that Γ = Z and our functor is Ext(Div(X), Gm ) → Pic(X). Now the inverse functor

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263

assigns to L ∈ Pic(X) the line bundle λD := NmD/Z (p∗X L) on Div(X) with evident c; here pX : X × Z → X is the projection.100 To finish the proof, it remains to present for every symmetric bilinear form κ on Γ some (λκ , cκ ) ∈ Picf (Div(X, Γ)) which yields a κ-twisted θ-datum. Suppose for a moment that Γ = Z. For D ∈ Div(X)(Z) choose (locally on Z) some effective divisor D0 ∈ Div(X, Z) such that D ≥ −D0 . Set (3.10.7.4)

λD := det(pZ∗ OX×Z (D)/OX×Z (−D0 )) ⊗ det⊗−1 pZ∗ OD0 .

Here pZ : X × Z → Z is the projection, OD0 := OX×Z /OX×Z (−D0 ); notice that pZ∗ OX×Z (D)/OX×Z (−D0 ) and pZ∗ OD0 are vector bundles on Z. Our λD is independent of the auxiliary choice of D0 in the usual way, so we have defined a super line bundle λ on Div(X). It carries a natural factorization structure c coming from the fact that OX×Z (D1 )/OX×Z (−D10 ) ⊕ OX×Z (D2 )/OX×Z (−D20 ) = OX×Z (D1 + D2 )/OX×Z (−D10 − D20 ) and OD10 ⊕ OD20 = OD10 +D20 if our divisors do not intersect. The form κ defined by (λ, c) is the product pairing on Z. Suppose Γ is arbitrary. Any γ ∨ ∈ Γ∨ yields a bilinear form κγ ∨ : (γ1 , γ2 ) 7→ ∨ γ (γ1 )γ ∨ (γ2 ). Every integral symmetric bilinear form is an integral linear combination of the κγ ∨ ’s, so we can assume that κ = κγ ∨ . Consider the pull-back of λ from (3.10.7.4) by the projections Div(X, Γ) → Div(X) defined by γ ∨ . The bilinear form of the corresponding θ-datum equals κ, and we are done.  3.10.8. From θ-data to lattice algebras. To finish the proof of the theorem in 3.10.4, we will construct the functor inverse to (3.10.4.1). Let Z be a quasi-compact scheme and S an effective Cartier divisor in X × Z/Z proper over Z. Let Div(X)S be the functor on the category of quasi-compact Zschemes that assigns to Y /Z the subgroup Div(X)S (Y ) ⊂ Div(X)(Y ) of all Cartier divisors whose support is contained (set-theoretically) in (the pull-back of) S. Set Div(X, Γ)S = Div(X, Γ)S,Z := Div(X)S ⊗ Γ. Lemma. Div(X)S is a formally smooth ind-scheme over Z which can be represented as the inductive limit of a directed family of subschemes which are finite and flat over Z. The same is true for Div(X, Γ)S . Proof. Follows from the fact that the functor which assigns to a test scheme Y the set of all effective Cartier divisors in X × Y /Y of given degree n is representable by the scheme Symn X (see [SGA 4] Exp. XVII 6.3.8) which is smooth.  Therefore Div(X, Γ)S = − lim → Spec Rα where {Rα } is a directed system of OZ algebras connected by surjections which are locally free OZ -modules of finite rank. Remark. Notice that Div(X, Γ)S,Z depends only on the reduced scheme Sred ⊂ X ×Z, so Div(X, Γ)S,Z carries a canonical action of the universal formal groupoid on Z (whose space is the formal completion of Z × Z at the diagonal; cf. the proof of the proposition in 3.4.7). If Z is smooth, this means that the ind-Z-scheme Div(X, Γ)S,Z carries a canonical flat connection ∇. For θ ∈ Pθ (X, Γ) let (λ, c) ∈ Picf (Div(X, Γ)) be the super line bundle with factorization structure that corresponds to θ by (3.10.7.1). For any (S, Z) the pullback of λ to Div(X, Γ)S can be seen as a projective system of invertible Rα -modules 100 Recall

that for effective D the norm functor NmD/Z : Pic (X × Z) → Pic (Z) comes from × the norm homomorphism pZ∗ O× D → OZ ; one extends it to the whole Div(X) demanding NmD/Z to be multiplicative with respect to D. For details see [SGA 4] Exp. XVII 6.3, Exp. XVIII 1.3.5.

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101 λS,Z = ← lim OZ -module − λRα . Therefore we have a locally projective [ (3.10.8.1) A`θ S,Z := HomOZ (λ∗S,Z , OZ ) = λRα ⊗ HomOZ (Rα , OZ ). Rα

It is clear that A`θ S,Z depends on (S, Z) ∈ C(X) (we use the notation from 3.4.6) in a functorial way; i.e., they form a structure from (i) in 3.4.6. The factorization structure c on λ yields in the obvious way a factorization structure (see (ii) in 3.4.6) on A`θ which we denote also by c. The pair (A`θ , c) evidently satisfies properties (a) and (b) in loc. cit., so it is a factorization algebra. We denote it simply by A`θ or A`(Γ,θ) . Notice that for θ ∈ P(X, Γ) ⊂ Pθ (X, Γ) our A`θ is commutative. Proposition. A`θ is naturally a lattice chiral algebra whose θ-datum identifies canonically with θ. Proof. Notice that Pθ (X) → FA(X), (Γ, θ) 7→ A`(Γ,θ) , is naturally a tensor functor (we use the notation from Remark (i) in 3.10.3). Let 1 = 1Γ ∈ Pθ (X, Γ) be the unit θ-datum. Looking at the “diagonal” morphisms (Γ, 1) → (Γ, 1) ⊕ (Γ, 1) and (Γ, θ) → (Γ, 1) ⊕ (Γ, θ) inPθ (X), we get the morphisms of factorization algebras δ : A`1 → A`1 ⊗A`1 and δθ : A`θ → A`1 ⊗A`θ . The first arrow is a coproduct which makes A`1 a commutative and cocommutative counital Hopf factorization algebra.102 The second arrow is a counital coaction of A`1 on A`θ . The diagonal morphism (Γ, 1) → (Γ, θ) ⊕ (Γ, θ⊗−1 ) yields a morphism A`1 → A1

A`θ ⊗ A`θ⊗−1 ⊂ A`θ ⊗ A`θ⊗−1 (see 3.4.16 for the notation). One easily checks that it is an isomorphism, so Aθ is an A1 -cotorsor. To see that Aθ is a lattice chiral algebra, it remains to identify A1 and F , i.e., ∼ ` . There is a natural to define an isomorphism of the Hopf DX -algebras A`1 X −→ FX pairing of group DX -ind-schemes ∨ h i : Div(X, Γ)∆,X × TX → GmX ,

(3.10.8.2) ∨

∨

hD ⊗ γ, f ⊗γ i := NmD/Z (f )hγ,γ i . Here ∆ ⊂ X × X is the diagonal divisor, Z is a test X-scheme, D ∈ Div(X)∆ (Z), and f is a section of JGmX realized, as in 2.3.3, as an invertible function of the formal neighborhood of the pull-back of ∆ in X × Z. An immediate computation shows that the morphism of Hopf DX -algebras ` A`1 X → FX that corresponds to (3.10.8.2) is an isomorphism. It is clear from the construction that the θ-datum that corresponds to the lattice chiral algebra Aθ equals θ. We are done.  3.10.9 The Heisenberg Lie∗ subalgebra. We use the notation from 3.10.5 0 and 3.10.6. For γ ∈ Γ set α(γ) := dχγ /χγ ∈ F 0 = FX ; we extend it to α : tD → F 0 by D-linearity. Our α takes values in the submodule of Lie algebra elements with respect to the coalgebra structure, and it yields an isomorphism of DX -algebras ∼ Sym t`D −→ F `0 . Let A be any lattice chiral algebra and θ = (κ, λ, c) ∈ Pθ (X, Γ) its θ-datum. ◦A Consider the composition A`  A` −→ ∆˜∗ A`  ∆˜∗ (A` /OX 1A ). For any γ ∈ Γ γ −γ its restriction to λ  λ has zero of order −κ(γ, γ) + 1 at the diagonal. The top coefficient is a morphism λγ ⊗ λ−γ → (A0` /OX 1A ) ⊗ ω ⊗−κ(γ,γ)+1 ; since the 101 In

fact, it is locally free. counit A`1 → O comes from the (unique) morphism (Γ, 1Γ ) → (0, 10 ) in Pθ (X).

102 The

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265

left-hand side is identified, via cγ,−γ , with ω ⊗−κ(γ,γ) , we can consider it as a section αθ (γ) of A/ω1A . Consider the coaction morphism δA : A` → F ` ⊗ A` . For lγ ∈ λγ one has δA (lγ ) = χγ ⊗ lγ , so, looking at the ope, we see that (3.10.9.1)

δA (αθ (γ)) = α(γ) ⊗ 1A + 1F ⊗ αθ (γ) ∈ F ` ⊗ A/ω1F ⊗A .

Since A is an invertible F ` -comodule, this property determines the section αθ (γ) uniquely. In particular, it depends on γ in the additive way. We extend αθ to a morphism of DX -modules tD → A/ω1A . Let tθD → A be the corresponding morphism of the ω-extensions; we denote it also by αθ . Proposition. (i) The image of tD is a commutative Lie∗ subalgebra of A/ω1A , so is a central extension of the commutative Lie∗ algebra tD . The Lie∗ bracket tO × tO → h(ω) is the Heisenberg bracket (γ1 ⊗ f1 , γ2 ⊗ f2 ) 7→ κ(γ1 , γ2 )f1 df2 (see 2.5.9). (ii) The adjoint action of h(tD ) = tO on any λη ⊂ A` , η ∈ Γ, is multiplication by the character κ(η) := κ(η, ·) ∈ Γ∨ . (iii) For l ∈ λη the image of ∇(l) in A/ωλη coincides with the 0th term of the ope expansion l ◦ αθ (η).103 (iv) As an OX -module equipped with chiral U (tD )θ -action, P η A is equal to the induced U (tD )θ -module Ind θ A0 where tθD acts on A0 = λ according to (ii).104 θ θ ∼ 0 In particular, α : U (tD ) −→ A . (v) The extension tθD depends on θ in the additive way, i.e., θ 7→ tθD is a Picard functor. The automorphisms of θ act on tθD according to the homomorphism ∨ θ ∗ ⊗γ ∨ T ∨ (OX ) = O× 7→ γ ∨ ⊗ df /f . X ⊗ Γ → Aut tD = t ⊗ ω, f tθD

Proof. We check (ii) first. Let x be a local parameter on X. For γ, η ∈ Γ choose generators lη ∈ λη and l±γ ∈ λ±γ such that c(lγ , l−γ ) = 1A dx−κ(γ,γ) . ˜ (3) Consider the triple ope φ := (x1 − x2 )κ(γ,γ) lγ (x1 )l−γ (x2 )lη (x3 ) ∈ ∆ ∗ A (see 3.5.12). The orders of zero of the double ope are known by 3.10.6, so one has −x3 κ(γ,η) ) ψ where ψ ∈ Ax3 [[x1 − x2 , x2 − x3 ]]. Let ψ0 ∈ Ax3 be the constant φ = ( xx21 −x 3 term of the Taylor power series ψ. Consider φ as an element of Ax3 ((x2 −x3 ))((x1 − −x2 κ(γ,η) 2 ) ψ = (1 + κ(γ, η) xx12 −x x2 )); then φ = (1 + xx21 −x −x3 )ψ0 modulo (x1 − x2 )P 3 where P := Ax3 [[x2 − x3 ]] + (x1 − x2 )Ax3 ((x2 − x3 ))[[x1 − x2 ]]. Computing φ as 2 (x1 − x2 )κ(γ,γ) (lγ (x1 )l−γ (x2 ))lη (x3 ), we see that (1 + κ(γ, η) xx12 −x −x3 )ψ0 = lη (x3 ) + θ θ (x1 − x2 )α (γ)(x2 )lη (x3 ) modulo P . Therefore ψ0 = lη and α (γ)(x2 )lη (x3 ) = κ(γ, η)(x2 − x3 )−1 lη (x3 )+ regular terms, which is (ii). To check (i), we need to compute the polar part of the ope product of αθ (γ) and θ α (η). Consider the ope µ := (x1 −x2 )κ(γ,γ)(x3 −x4 )κ(η,η) lγ (x1 )l−γ (x2 )lη (x3 )l−η (x4 ). (x1 −x3 )(x2 −x4 ) κ(γ,η) ) ν where ν is a Taylor power series with the constant It equals ( (x 1 −x4 )(x2 −x3 ) term 1A . As an element of Ax4 ((x2 − x4 ))[[x1 − x2 , x3 − x4 ]], our µ is equal to (1 + κ(γ, η)(x1 −x2 )(x3 −x4 )(x2 −x4 )−2 )1A modulo terms which are Taylor power series or lie in the cube of the maximal ideal of k((x2 −x4 ))[[x1 −x2 , x3 −x4 ]]. On the other hand, one has µ := (x1 − x2 )κ(γ,γ) (x3 − x4 )κ(η,η) (lγ (x1 )l−γ (x2 ))(lη (x3 )l−η (x4 )) = 103 Notice that, by (ii), the ope l ◦ αθ (η) has pole of first order with the principal part in λη , so the image in A/ωλη of its 0th term is well defined. 104 For the induced modules, see 3.7.15 and 3.7.21.

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(1+α(γ)(x2 )(x1 −x2 )+· · · )(1+α(η)(x4 )(x3 −x4 )+· · · ). Thus α(γ)(x2 )α(η)(x4 ) = κ(γ, η)(x2 − x4 )−2 1A + regular terms, and we are done. We leave (iii) to the reader. To deduce (iv), we have to show that the canonical morphism Indθ (A0 ) → A is an isomorphism. This follows easily from (3.10.9.1); the details are left to the reader. The first statement in (v) is evident. Thus it suffices to check the second statement only for the trivial θ-datum, where it is immediate.  (i) Let {γi } ⊂ Γ be a subset that generates Γ as a semigroup. Then P Remarks. λγi generates A as a chiral algebra.105 (ii) Suppose that A is symmetric (see Remark in 3.10.1). The action of the involution on tκD yields then a D-module splitting tD → tθD (that identifies tD with the −1-eigenspace of the involution). Therefore tθD is identified with the Heisenberg extension tκD from 2.5.9. If A is commutative, then tθD is the twist of the trivial extension by the T ∨ torsor corresponding to A (see 3.10.2) according to the T ∨ (OX )-action described in (iv) of the above proposition. 3.10.10. The boson-fermion correspondence. Suppose that Γ = Z and κ is the product pairing. Let θ := (λ, c) be any κ-twisted θ-datum. Set λ+ := λ+1 ⊗ ω and λ− := λ−1 ⊗ ω; these are odd line bundles and c+1,−1 yields a pairing ∼ c : λ+ ⊗λ− −→ ω. Then (λ+D ⊕λ−D )[ := λ+D ⊕λ−D ⊕ω carries a Lie∗ bracket with the only non-zero component being the pairing ∈ P2∗ ({λ+D , λ−D }, ω) defined by c. Thus (λ+D ⊕ λ−D )[ is an ω-extension of the abelian Lie∗ algebra λ+D ⊕ λ−D . Let Clθ be the corresponding twisted enveloping algebra (see 3.7.20); this is a Clifford chiral algebra (see 3.8.6). Proposition. There is a natural isomorphism of chiral algebras (3.10.10.1)

∼

Clθ −→ Aθ .

Proof. The canonical embeddings λ±1 ,→ A`θ define the morphisms of Dmodules ι : λ+D ⊕ λ−D → Aθ and ι[ := ι ⊕ 1Aθ : (λ+D ⊕ λ−D )[ → Aθ . The latter is a morphism of Lie∗ algebras; it yields the promised morphism of chiral algebras Clθ → Aθ . Our morphism is injective since Clθ is a simple chiral algebra (due to irreducibility of the Clifford modules); it is surjective by Remark (i) in 3.10.9.  3.10.11. Remark. According to [Ro], an arbitrary lattice chiral algebra A can be reconstructed from its θ-datum θ = (κ, λ, c) ∈ Pθ (X, Γ) as follows. Let N = N (θ) be the direct sum of λγ , γ ∈ Γ, and let P = P (θ) be the sum of λγ1  λγ2 (κ(γ1 , γ2 )∆) ⊂ j∗ j ∗ N  N . Let A˜` be the factorization algebra freely generated by (N, P ) (see 3.4.14). So we have the universal morphisms ˜ι : N → A˜`X and P → A˜`X×X . Let ˜ι γ : λγ → A˜`X be the components of ˜ι and ⊗κ(γ ,γ ) let c˜ γ1 ,γ2 : λγ1 ⊗ λγ2 → A˜`X ⊗ ωX 1 2 be the pull-back to the diagonal of the morphism106 λγ1  λγ2 ,→ P (−κ(γ1 , γ2 )∆) → A˜`X×X (−κ(γ1 , γ2 )∆). Let A(θ) be the quotient of A˜ modulo the ideal generated by the sections ˜ι 0 (1) − 1 ˜ ∈ A˜` and A

105 Indeed,

X

P γ the chiral subalgebra generated by λ i contains all λγ (as the top coefficients of appropriate ope), hence it contains the image of αθ , and we are done by (iii) of the proposition. ⊗−n 106 Recall that ∆∗ A ∗ ˜` ˜` . X×X = AX (see (3.4.2.1)) and ∆ OX×X (n∆) = ωX

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⊗κ(γ ,γ )

c˜ γ1 ,γ2 − ˜ι γ1 +γ2 cγ1 ,γ2 ∈ A`X ⊗ ωX 1 2 ⊗ (λγ1 )⊗−1 ⊗ (λγ2 )⊗−1 . Let ιγ = ιγθ : λγ → A(θ)` be the induced morphisms; they satisfy an evident universality property. By 3.10.6 and 3.4.14 there is a unique morphism of chiral algebras A(θ) → A ` which sends ιγ to the embedding λγ = A`γ 0 ,→ A (see 3.10.6). As shown in [Ro], this is an isomorphism. 3.10.12. Group ind-scheme basics. We are going to identify modules over a lattice algebra with representations of a certain Heisenberg group. Let us recall first the relevant definitions. (a) Let G be an ind-affine group ind-scheme. To avoid unnecessary complications, we assume S it to be the inductive limit of countably many affine schemes, so G = Spf P := Spec Pα where P is a topological algebra, Pα = P/Iα , and Iα are open ideals that form a base of the topology. The product on G makes P a counital topological Hopf algebra. Example. Let x ∈ X(k) be a point on our curve and Ox ⊂ Kx the algebra of Taylor formal power series and the field of Laurent power series. Any affine group scheme T yields a group ind-scheme that we denote (by abuse of notation) by ˆ x ) (cf. 2.4.8 and 2.4.9). It contains a group subscheme T (Kx ), T (Kx )(R) := T (R⊗K ˆ x ). T (Ox ), T (Ox )(R) := T (R⊗O For a discrete vector space V a G-action on V is a rule that assigns to each test commutative algebra R an action of the group G(R) on the R-module VR := V ⊗ R which is natural with respect to morphisms of the R’s. Equivalently, this a counital ˆ := lim(P/Iα ) ⊗ V . A discrete vector space equipped with a P -coaction V → P ⊗V ←− G-action is referred to as a G-module or a representation of G; these objects form a tensor abelian category Gmod. S Consider the topological dual P ∗ := Pα∗ . This is an associative unital algebra. It carries a topology whose base is formed by all left ideals whose intersection with each profinite-dimensional vector space Pα∗ is open. The completion is a topological associative algebra (in the sense of 3.6.1) called the group algebra of G; we denote it by k[G]. A discrete k[G]-module is the same as a G-module: one has Gmod= k[G]mod. So k[G] is the topological algebra of endomorphisms of the functor which assigns to a G-module the underlying vector space. Our k[G] is naturally a counital cocommutative topological Hopf algebra. Denote by k[G]gr the functor which assigns to a test algebra R the group of the ˆ group-like invertible elements of the topological Hopf R-algebra k[G]⊗R. There is an evident natural group homomorphism G → k[G]gr , g 7→ δg . Remark. The latter map need not be an isomorphism: for example, if H is a semisimple simply connected algebraic group, then k[H(K)x ] = k. (b) The above picture has a twisted version: For a group ind-scheme G its super extension G[ is an extension of G by the Picard groupoid of super line bundles.107 Explicitly, such a G[ is a rule that assigns to any g ∈ G(R) an invertible super R-module λ[g in a functorial manner, and to ∼ any pair g1 , g2 ∈ G(R) a natural identification c[g1 ,g2 : λ[g1 ⊗ λ[g2 −→ λ[g1 g2 which is associative in the obvious sense. We have the parity homomorphism [ : G → Z/2, where [g is the parity of λ[g . The super extensions of G form a Picard groupoid in the obvious way. 107 For

details see, e.g., the appendix to section 2 in [BBE].

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Remark. In the above definition we used only the monoidal structure of the Picard groupoid (the commutativity constraint is irrelevant). Therefore a super extension amounts to a pair that consists of an extension of G by the Picard groupoid of line bundles and a parity homomorphism G → Z/2; the former structure is the same as a central Gm -extension of G. For commuting g1 , g2 ∈ G(R), we set {g1 , g2 }[ := c[g1 ,g2 /c[g2 ,g1 σ ∈ R× where σ : ∼

[

λ[g1 ⊗λ[g2 −→ λ[g2 ⊗λ[g1 is the commutativity constraint. Thus {g, g}[ = (−1)g . If G is commutative, then the commutator pairing { }[ : G×G → Gm is bimultiplicative. A G[ -module is a super vector space V together with a rule that assigns to ∼ g ∈ G(R) an isomorphism of super R-modules λ[g ⊗ V −→ VR and satisfies the usual properties; equivalently, this is an action of G[ on V , considered as a mere group ind-scheme and a mere vector space, which changes the Z/2-grading on V according to [ . Denote by G[ mod the abelian category of G[ -modules. Consider λ−[ := (λ[ )⊗−1 as a super line bundle over G. Its sections form a S −[ ∗ −[ −[ ∗ topological super P -module λ−[ P = lim ←− λPα . The topological dual (λP ) = (λPα ) is naturally an associative super algebra. It carries a natural topology formed by all ∗ left ideals whose intersection with each profinite-dimensional vector space (λ−[ Pα ) is [ open. The completion is the topological associative super algebra k[G] called the [-twisted group algebra of G. A G[ -module is the same as a discrete k[G][ -module: one has G[ mod = k[G][ mod. The diagonal morphism G[ → G×G[ yields a morphism of topological algebras [ [ ˆ δ : k[G][ → k[G]⊗k[G] which is a counital k[G]-coaction on k[G][ . (c) The Cartier duality establishes an anti-equivalence between the category of commutative affine group schemes and that of commutative group ind-finite indschemes.108 It extends readily to a self-duality on the category of commutative group ind-schemes that can be represented as an extension of a group ind-finite ind-scheme by an affine group scheme. On the level of topological Hopf algebras, the Cartier duality interchanges the topological algebras of functions and the group algebras. Let G be such a group ind-scheme and G0 its Cartier dual, so G0 = Spf k[G]. The canonical morphism G → k[G]gr (see (a)) is an isomorphism. For a topological associative (super) algebra Q a counital k[G]-coaction δQ : ˆ amounts to a G0 -action on Q (which is a rule that assigns to a test Q → k[G]⊗Q ˆ natural with respect algebra R a G(R)-action on the topological R-algebra Q⊗R to morphisms of the R’s). The category of such pairs (Q, δQ ) is naturally a tensor k[G]

k[G]

ˆ 2 is the category with respect to the tensor product ⊗ , where Q1 ⊗ Q2 ⊂ Q1 ⊗Q topological subalgebra of elements on which the G0 -actions along Q1 and along Q2 coincide. We say that (Q, δQ ) is a k[G]-cotorsor if it is invertible with respect to k[G]

⊗ . The category of k[G]-cotorsors is a naturally Picard groupoid.

Lemma. The above Picard groupoid identifies naturally with the Picard groupoid of super extensions of G. Sketch of a proof. For a super extension G[ of G the pair (k[G][ , δ [ ) is a k[G]cotorsor. Conversely, let (Q, δQ ) be a k[G]-cotorsor. For g ∈ G(R) consider the 108 See,

e.g., section 4 in Chapter II of [Dem].

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ˆ ˆ : δQ (q) = corresponding group-like element δg ∈ k[G]⊗R. Then λg := {q ∈ Q⊗R δg ⊗ q} is an invertible super R-module, and the product in Q provides a morphism c[g1 ,g2 : λ[g1 ⊗ λ[g2 → λ[g1 g2 which is an isomorphism. This datum is the super extension G[ that corresponds to Q. The details are left to the reader.  3.10.13. The Heisenberg groups. We use the notation of 3.10.12; see especially the example in loc. cit. Consider the commutative group ind-scheme Kx× := Gm (Kx ); its Lie algebra equals Kx , and the connected component of the reduced ind-scheme is the group scheme Ox× := Gm (Ox ). So an R-point of Kx× is the same as an invertible element ˆ f of Kx ⊗R. Contou-Carr`ere defines in [CC]109 a canonical skew-symmetric bimultiplicative symbol pairing (3.10.13.1)

{ }x : Kx× × Kx× → Gm .

The corresponding pairing of the groups of k-points is the tame symbol map (f, g) 7→ (−1)v(f )v(g) (f v(g) /g v(f ) )(x) (here v(f ) is the order of zero of f at x); the derivative with respect to the first variable is the pairing Kx × Kx× → Ga , (a, g) 7→ Resx ad log g; the Lie algebra pairing Kx × Kx → k is (a, b) 7→ Resx adb. If ˆ × , then f ∈ Ox× (R) = (Ox ⊗R) (3.10.13.2)

{f, g}x = Nmdivg/R f.

The symbol pairing is non-degenerate: it identifies Kx× with its own Cartier dual. This means that the corresponding morphism of the group ind-schemes Kx× → Spf k[Kx× ] is an isomorphism. Consider, as in 3.10.1, our lattice Γ and the corresponding torus T := Γ ⊗ Gm . Any symmetric bilinear form κ : Γ × Γ → Z defines a skew-symmetric pairing κ(γ ,γ ) { }κx : T (Kx ) × T (Kx ) → Gm , {f1⊗γ1 , f2⊗γ2 }κx := {f1 , f2 }x 1 2 . Definition. A Heisenberg κ-extension is a super extension of T (Kx ) whose commutator pairing equals { }−κ x . A Heisenberg extension is a super extension which is a Heisenberg κ-extension for some κ. The Baer product of Heisenberg κ- and κ0 -extensions is a Heisenberg (κ + κ0 )extension, so the Heisenberg extensions form a Picard groupoid Heis T (Kx ). It is a disjoint union of the groupoids Heisκ T (Kx ) of the κ-extensions. So Heis0 T (Kx ) is the Picard subgroupoid Ext(T (Kx ), Gm ) of commutative Gm -extensions of T (Kx ), and each Heisκ T (Kx ) is a Heis0 T (Kx )-torsor (it is non-empty: see below). Since every commutative Gm -extension of T (Kx )-splits, Ext(T (Kx ), Gm ) identifies naturally with the Picard groupoid of Hom(T (Kx ), Gm )-torsors. The symbol pairing yields an isomorphism Hom(T (Kx ), Gm ) = T ∨ (Kx ), so these objects are the same as T ∨ (Kx )-torsors. Let T (Kx )[ be a Heisenberg κ-extension. Let us describe the category of T (Kx )[ -modules. Let Z = Zκ ⊂ T (Kx ) be the kernel of the { }κx -pairing. So Z lies in the kernel of the parity homomorphism κ : T (Kx ) → Z/2, f ⊗γ 7→ v(f )κ(γ, γ)mod2, and we 109 See

also [D2] 2.9 and [BBE] 3.1–3.3.

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have the central Gm -extension Z [ ⊂ T (Kx )[ . The Cartier dual (Z [ )0 is an extension of Z by Z 0 . Let Spf R ⊂ (Z [ )0 be the preimage of 1 ∈ Z; this is a Z 0 -torsor. Any T (Kx )[ -module is automatically a discrete R-module, so the category T (Kx )[ mod is an R-category. Proposition. T (Kx )[ mod is equivalent, as an R-category, to the category Rmod of discrete R-modules. Proof. We will define a functor T (Kx )[ mod → Rmod leaving the construction of the inverse to the reader. The construction (for κ 6= 0) depends on auxiliary choices. Different choices yield (non-canonically) isomorphic functors. Choose a decomposition Γ = Γ0 × Γ1 where Γ0 is the kernel of κ; we write Ti := Gm ⊗ Γi , etc. Then Z = Z0 × Z1 where Z0 = T0 (Kx ) and Z1 is the kernel of the morphism T1 → T1∨ defined by κ which is a finite group of order | det(κ|Γ1 )|. Then Z1 ⊂ T1 (Ox ), so we have the embedding Z1[ ⊂ T1 (Ox )[ of the central Gm extensions. For each character χ : Z1[ → Gm whose restriction to Gm ⊂ Z1[ equals idGm choose its extension to a character χ ˜ of T1 (Ox )[ . [ Now let V be a T (Kx ) -module. For any χ as above let Vχ˜ ⊂ V be the subspace of all vectors on which T1 (Ox )[ ⊂ T (Kx )[ acts by the character χ. ˜ This is a Z [ [ submodule of V , hence a discrete R-module. Our functor T (Kx ) mod → Rmod is V 7→ ⊕ Vχ˜ .  χ

3.10.14. Let us return to the lattice chiral algebras. Let θ ∈ Pθ (X, Γ)κ be any θ-datum. Let Aθ be the corresponding chiral lattice algebra (see (3.10.4.1)). θ Theorem. Aas θx is the group algebra of a certain Heisenberg κ-extension T (Kx ) naturally defined by Aθ .

Remark. We see that the category of Aθ -modules supported at x identifies naturally with the category of T (Kx )θ -modules. By the above proposition, for a non-degenerate κ our category is semisimple with | det κ| isomorphism classes of irreducibles. This result (in the case when κ is even and positive definite) was first established by Dong [Don]. as Proof. (a) If θ = 1, then Aas 1x = Fx is the topological Hopf algebra of functions on the ind-group T ∨ (Kx ) (see 2.3.2 and (2.4.9.1)). It identifies canonically with ∨ hγ,γ ∨ i k[T (Kx )] via the Cartier duality T (Kx )×T ∨ (Kx ) → Gm , f ⊗γ , g ⊗γ 7→ {f, g}x . (b) For an arbitrary θ the F -coaction δθ : A`θ → F ` ⊗ A`θ yields, by transport of structure, an Fxas -coaction on Aas θx (see 3.6.8). as Lemma. Aas θx is an Fx -cotorsor.

Proof of Lemma. We want to show that the morphism F ` → A`θ ⊗ A`θ−1 yields Fxas

∼ as as ˆ as an isomorphism Fxas −→ Aas θx ⊗ Aθ −1 x ⊂ Aθ −1 x ⊗Aθ −1 x . as Let Ξθ ⊂ Ξx (Aθ ) be the subset of subalgebras Aθξ which are invariant with respect to the T ∨ -action, i.e., satisfy the property δθ (Aθξ ) ⊂ F ` ⊗A`θξ . This subset is −1 as ` cofinal in the Ξas x -topology: indeed, for any B ∈ Ξx (Aθ ) one has B ∩δθ (F ⊗B) ∈ ` Ξθ . Thus Aas θx = lim ←− Aθξx , Aθξ ∈ Ξθ . In fact, the set Ξθ does not depend on θ. Namely, the maps Ξθ1 → Ξθ2 , Aθ1 ξ 7→ Aθ2 ξ := δ −1 (Aθ1 ξ ⊗ Aθ2 /θ1 ), form a transitive system of bijections between

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F

the Ξθ ’s; here δ : Aθ1 −→ Aθ2 ⊗ Aθ2 /θ1 ,→ Aθ1 ⊗Aθ2 /θ1 is the “coproduct” morphism. So we write Ξ instead of Ξθ . For an OX -module N equipped with a counital F ` -coaction δN : N → F ` ⊗ N −1 ` set N xˆ := ← lim −(δθ Fξ ⊗ N )x ; this is a topological vector space equipped with a Ξ

counital Fxas -coaction. If N is an F ` -cotorsor, then N xˆ is an Fxas -cotorsor (since all OX -module F ` -cotorsors are equivalent Zariski locally at x, we can assume that N = F ` where the statement is evident). Moreover, if N 0 is the inverse F ` ∼

F`

cotorsor, so that we have F ` −→ N ⊗ N 0 ⊂ N ⊗ N 0 , then N 0 xˆ is the Fxas -cotorsor ˆ 0 xˆ yields an isomorphism inverse to N xˆ, and the corresponding map F ` xˆ → N xˆ⊗N Fxas

∼ ˆ 0 xˆ (by the same argument). Fxas = F ` xˆ −→ N xˆ ⊗ N 0 xˆ ⊂ N xˆ⊗N ` Now we can prove the lemma. We have seen that Aas ˆ as a topological x = A x as vector space equipped with an Fx -action. So the last isomorphism for N = A`θ , N 0 = A`θ−1 is the equality we wished to check; q.e.d. 

(c) By the lemma from (c) in 3.10.12, we have a super extension T (Kx )θ of θ [ T (Kx ) such that Aas θx = k[T (Kx )] . It remains to check that T (Kx ) is a Heisenθ κ berg κ-extension, i.e., that its commutator pairing { } equals { }x . Consider the difference < >κ := { }θ − { }κx . This pairing depends only on κ (since { }θ depends only on the isomorphism class of θ near x, which is κ) and is invariant with respect to automorphisms of Kx (for the same reason). αθ ˆ x (jx∗ j ∗ tθ ) ,→ As follows from (3.10.9.1), the Lie algebra of T (Kx )θ equals h x D κ Aas θx , so the Lie algebra pairing that comes from < > vanishes by (i) of the proposition in 3.10.9. Let us show that the pairing between T (Kx )red and the Lie algebra t(Kx ) that comes from < >κ vanishes. We know that it vanishes on T (Ox ) × t(Kx ) by the above and on T (Kx )red × t(Ox ) by (ii) in the proposition in 3.10.9. So < >κ comes from a pairing Γ × t(Kx )/t(Ox ) = T (Kx )red /T (Ox ) × t(Kx )/t(Ox ) → k. Consider the action of a group of homotheties on Kx (with respect to a parameter in Kx ). It is trivial on Γ and has trivial coinvariants on Kx /Ox . Since it preserves < >κ , we get the promised vanishing. We have shown that < >κ vanishes on T (Kx ) × T (Kx )0 where T (Kx )0 is the connected component of T (Kx ). Therefore it comes from a pairing Γ × Γ = T (Kx )/T (Kx )0 × T (Kx )/T (Kx )0 → Gm . Since < >κ depends on κ in the additive way, it suffices to consider the case when γ = Z and κ is the product pairing. Here { }θ and { }κx are controlled by the parity, so < >κ = 0, and we are done.  3.10.15. A twisted version. Suppose we have a finite group H acting on Γ, an H-equivariant lattice algebra Aθ (see Remark in 3.10.1), and an H-torsor P on Ux := Xr{x}. Consider the twisted chiral algebra Aθ (P ) (see 3.4.17); it is an F (P )cotorsor. We also have the twisted torus T (P ), which is a group scheme over Ux , and the ind-scheme T (P )(Kx ) of its Kx -points. Notice that κ, which is an H-invariant bilinear form on Γ, defines the pairing { }κx : T (P )(Kx ) × T (P )(Kx ) → Gm , so, as in 3.10.13, we have the notion of a Heisenberg κ-extension of T (P )(Kx ). Theorem. (Aθ (P ))as x is the group algebra of a certain Heisenberg κ-extension T (P )(Kx )θ of T (P )(Kx ). Proof. Same as the proof of the non-twisted version (see 3.10.14).



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Exercise. Formulate and prove the main results of [BaK]. 3.10.16. Chiral monoids. Let us highlight the geometric structure underlying the construction of 3.10.8. This section will not be used later in the book. For us, “ind-algebraic space” means a functor on the category of affine schemes representable by the inductive limit of a directed system of quasi-compact algebraic spaces connected by closed embeddings. For a quasi-compact scheme Z an indalgebraic Z-space is an ind-algebraic space G equipped with a morphism G → Z; denote by IZ the category of these objects with morphisms being closed embeddings. All the IZ ’s form a fibered category I over the category of quasi-compact schemes (with the fibered product as the pull-back functor). Suppose we have a pair (G, c) where: (i) G is a morphism from the fibered category C(X) (see 3.4.6) to I. Thus G is a rule that assigns to each S ∈ C(X)Z an ind-algebraic Z-space GS = GS,Z ; for S 0 ≤ S we have a canonical closed embedding GS 0 ,Z ,→ GS,Z ; everything is compatible with the base change. In particular, the universal divisors yield ind-algebraic Symn X-spaces GSymn X . (ii) c is a rule that assigns to every pair of mutually disjoint divisors S1 , S2 ∈ ∼ C(X)Z an identification cS1 ,S2 : GS1 ,Z × GS2 ,Z −→ GS1 +S2 ,Z . We demand that these Z

identifications be commutative and associative in the obvious manner and that they be compatible with the natural morphisms from (i). Assume that the following conditions hold: (a) For every n the closure in GSymn X of the complement to the preimage of the discriminant divisor in Symn X equals GSymn X . Equivalently, GSymn X can be represented as the inductive limit of a directed family {Gα } of algebraic Symn Xspaces and closed embeddings such that each Gα has no non-zero local functions supported over the discriminant divisor of Symn X. (b) One has GSym0 X 6= ∅. (c) The ind-algebraic spaces GSymn X are separable. Definition. Such a (G, c) is called chiral, or factorization, monoid on X. Chiral monoids form a category which we denote by CM(X). A chiral monoid is said to be commutative if c can be extended to a morphism cS1 ,S2 : GS1 × GS2 → GS1 +S2 defined for arbitrary Si ∈ C(X)Z and natural with Z

respect to morphisms in C(X) (by (a) and (c) such an extension is unique, if it exists). In many aspects chiral monoids are similar to chiral algebras; the next remarks are parallel to, respectively, Remarks (i)–(iii) in 3.4.6, subsections 3.4.7, 3.4.20, and the proposition in 3.4.6. Remarks. (i) The associativity and commutativity of c permits us to define for any finite family of mutually disjoint divisors Sα ∈ C(X)Z a canonical identification c{Sα } of the fibered product of GSα over Z and GΣSα . (ii) By (b) and (i) one has GSym0 X = Spec k. So for any (S, Z) the embedding ∅ ⊂ S yields a canonical section 1G : Z = G∅,Z → GS,Z . It is preserved by the structure embeddings G(S) ,→ G(S 0 ) and pull-backs; this is the unit section of G. (iii) As in Remark (iii) in 3.4.6, in the above definition we can replace C(X)Z by the ordered set of relative effective Cartier divisors adding the condition that all the structure embeddings GS ,→ GnS are isomorphisms.

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(iv) Suppose we have two morphisms of schemes f, g : Z 0 ⇒ Z which coincide 0 on Zred . Then the corresponding pull-back maps C(X)Z → C(X)Z 0 coincide, so ∼ one has a canonical identification f ∗ GS −→ g ∗ GS of ind-algebraic spaces. Therefore GS,Z carries a canonical action of the universal formal groupoid on Z (whose space is the formal completion of Z × Z at the diagonal). For S ⊂ S 0 the embeddings GS ,→ GS 0 are compatible with the action, as well as the pull-back identifications (in particular, 1G is fixed by the action). So we have defined a canonical integrable connection on the ind-algebraic Symn X-space GSymn X . (v) The category CM(X) admits finite projective limits. A chiral monoid G admits a monoid structure (as an object of CM(X)) if and only if it is commutative; such a structure is unique and the product G × G → G is given by the morphisms cS,S : GS × GS → GS . (vi) For a chiral monoid G and a finite set I let GX I be the pull-back of GSym|I| X by the projection X I → Sym|I| X. For π : J → I one has evident morphisms (we Q ∼ use the notation from 3.4.5) ν (π) : ∆(π)∗ GX J → GX I , c[π] : j [π]∗ GX Ji −→ j [π]∗ GX J I

satisfying the obvious versions of properties (a)–(f) from 3.4.5. Conversely, any such datum (GX I , ν (π) , c[π] ) defines a chiral monoid (one recovers GSymn X by descent). One can also consider a weaker structure taking into consideration only nonempty I’s and surjective maps between them; the resulting objects may be called chiral semigroups. In fact, a chiral monoid is the same as a chiral semigroup which admits a unit section (the definition is left to the reader); morphisms of chiral monoids are the same as morphisms of chiral semigroups that preserve the unit sections. (vii) The functor G 7→ GX on the category of chiral monoids is faithful. So a chiral monoid can be considered as an ind-algebraic X-space GX equipped with an extra structure. This structure has Xe´t -local origin. Let G be a chiral monoid and P an ind-algebraic space. For a point x ∈ X set P xˆ := P × Spf Ox ; this is an ind-algebraic X-space (which lives over the formal neighbourhood of x) equipped with an evident connection ∇P . A chiral G-action on P at x is a chiral monoid structure on the disjoint union GX t P xˆ such that the embedding GX ,→ GX t P xˆ is a morphism of chiral groupoids and the connection on P xˆ defined by the chiral monoid structure (see Remark (iv) above) equals ∇P . Definition. (i) For a chiral monoid G a line bundle on G is a rule that assigns to any S ∈ C(X)Z a line bundle λGS on GS , and to any morphism (S 0 , Z 0 ) → (S, Z) in C(X) an identification of λGS0 and the pull-back of λGS by the map GS 0 → GS in a way compatible with the composition of morphisms. (ii) For λ as above, a factorization structure on λ is a rule that assigns to any ∼ non-intersecting S1 , S2 ∈ C(X)Z an identification c : λGS1 ⊗ λGS2 −→ c∗S1 ,S2 λGS1 +S2 . These isomorphisms should be compatible with the structure morphisms from (i) and commutative and associative in the obvious sense. (iii) Suppose that G is commutative. The factorization structure is said to be commutative if the isomorphisms c from (ii) can be extended to similar isomorphisms defined for arbitrary S1 , S2 ∈ C(X)Z and compatible with the structure morphisms from (i) (by properties (a) and (c) this can be done in a unique way). Line bundles on G equipped with factorization structure are also called Gm extensions of G. Replacing mere line bundles by super line bundles, one gets the notion of super Gm -extension; these objects form a Picard groupoid denoted by

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P icf (G). Of course, one can replace Gm by any commutative group scheme (or even an algebraic Picard groupoid). Examples. (i) (The local Picard schemes). The ind-schemes Div(X, Γ)S (see 3.10.8) form a commutative chiral monoid. As in 3.10.8, the pull-back of any (λ, c) ∈ Picf (Div(X, Γ)) (see 3.10.7) is naturally a super Gm -extension of our chiral monoid. This super extension is commutative if and only if (λ, c) ∈ Ext(Div(X, Γ), Gm ) ⊂ Picf (Div(X, Γ)) (see (3.10.7.3)). It is not difficult to check that our functor identifies Picf (Div(X, Γ)) with the Picard groupoid of super Gm -extensions of our chiral groupoid. (ii) (The affine Grassmannian). Let G be any algebraic group. Then for S ∈ C(X)Z the functor which assigns to a Z-scheme Y the set of pairs (F, γ) where F is a G-torsor on X × Y and γ its trivialization over the complement to (the pull-back of) S is representable by an ind-scheme GS of ind-finite type. These ind-schemes form naturally a chiral monoid (see [BD] 5.3.12). For G = T we get the chiral monoid from (i). For any x ∈ X there is a natural chiral G-action on G(Kx ) at x. Remark. For any DX -scheme YX the corresponding multijet DX I -schemes YX I (see 3.4.21) define, according to 3.4.22, a canonical chiral semigroup structure on YX (see (vi) in the previous remarks). Notice that this chiral semigroup does not admit a unit section (unless YX = X). As was pointed out by Kapranov and Vasserot [KV] 3.2.4, for affine YX = Spec A` the ind-scheme YXmer := Spec Aas (see 3.6.18) of “horizontal meromorphic jets” is again a chiral semigroup, modulo the fact that property (a) from the definition of chiral monoid was not verified in loc. cit. It would be nice to check this property (in case of the usual multijets, this is the contents of the theorem in 3.4.22). Let G be a chiral monoid and λ its super extension. Suppose that for each S ∈ C(X)Z , Z is affine, the corresponding GS,Z is the inductive limit of its closed subschemes which are finite and flat over Z. In other words, GS,Z = Spf R where R = lim ←− Rα such that Rα are OX -algebras which are locally free OX -modules of finite rank. Set [ (3.10.16.1) A`S,Z := λ ⊗ R∗ := λ ⊗ HomOX (Rα , OX ). R

R

Here λ is λGS considered as an invertible R-module. In other words, A`S,Z = π! (λGS ⊗ π ! OZ ); here π : GS,Z → Z is the structure projection and π ! is the !-pullback functor. The factorization structures on G and λ define on A` the structure of the factorization algebra (see 3.4.6). In the situation of Example (i) above our A is the lattice chiral algebra from 3.10.8. Questions. (i) Suppose that we have (G, λ) such that GS,Z are proper over Z. Is it true that A`S,Z := Rπ! (λGS ⊗ Rπ ! OZ ) form a factorization DG algebra? If G is the affine Grassmannian for semisimple G and λ is defined by a positive integral level κ, then A should be equal to the integrable quotient of U (gD )κ . (ii) Let G be any chiral monoid and x ∈ X a point. Can one define a group space Gas x such that for an ind-algebraic space P a chiral G-action on P at x amounts to a Gas x -action on P ? Will a super extension λ of G provide a super extension of ? In the situation of Example (ii), Gas Gas x x should be equal to G(Kx ).

CHAPTER 4

Global Theory: Chiral Homology Qinovnik umiraet, i ordena ego ostats na lice zemli. Koz~ma Prutkov, “Mysli i Aforizmy”, 1860

†

4.1. The cookware This section collects some utensils and implements needed for the construction of chiral homology. A sensible reader should skip it, returning to the material when necessary. Here is the inventory together with a brief comment on the employment mode. (i) In 4.1.1–4.1.2 we consider homotopy direct limits of complexes and discuss their multiplicative properties. The material will be used in 4.2 where the de Rham complex of a D-module on Ran’s space R(X) is defined as the homotopy direct limit of de Rham complexes of the corresponding D-modules on all the X n ’s with respect to the family of all diagonal embeddings. See [BK], [Se]. (ii) In 4.1.3 and 4.1.4 we discuss the notion of the Dolbeault algebra which is an ¯ algebraic version of the ∂-resolution. Dolbeault resolutions are functorial and have nice multiplicative properties, so they are very convenient for computing the global de Rham cohomology of D-modules on R(X), in particular, the chiral homology of chiral algebras. An important example of Dolbeault algebras comes from the Thom-Sullivan construction; see [HS] §4. (iii) In 4.1.5 we recall the definitions of semi-free DG modules, semi-free commutative DG algebras, and the cotangent complex following [H] (see also [Dr1] and [KrM]). The original construction of the cotangent complex, due to Grothendieck [Gr1], [Il], was performed in the setting of simplicial algebras. We follow the DG setting of [H] (using the notation L ΩF instead of the standard LF/k ). (iv) Sections 4.1.6 and 4.1.7 deal with Batalin-Vilkovisky algebras and the corresponding homotopy categories. BV algebras are “quantum deformations” of odd Poisson (alias braid, alias Gerstenhaber) algebras. We will see in 4.3.1 that chiral chain complexes of, respectively, commutative DX -algebras, coisson algebras, and general chiral algebras are naturally commutative DG algebras, odd Poisson algebras, and BV algebras. For a more lively account, see, e.g., [Ge] and [Schw]. (v) A typical example of BV algebra is the Chevalley homology complex of a Lie algebra; from the BV viewpoint it is the BV envelope of the Lie algebra (see 4.1.8). In 4.1.9–4.1.10 we explain what BV envelopes of Lie algebroids are. To † “A civil servant dies, and regalia of his stay on the face of the earth.” Koz’ma Proutkoff, “Thoughts and Aphorisms”, 1860.

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define it, an extra structure – that of the BV extension – is needed. The situation is parallel to that of chiral envelopes from 3.9; we will see in 4.8 that the chiral homology functor transforms chiral envelopes to BV envelopes. The homotopy properties of this construction are discussed in 4.1.11 and 4.1.12; in 4.1.13 we prove a technical statement (to be used in 4.1.18) asserting that while considering BV Lie R-algebroids from the homotopy point of view, one can change R at will by any cofibrant representative in its homotopy class. If R is a plain commutative algebra (i.e., a DG algebra sitting in degree 0) and L a plain Lie R-algebroid, then BV extensions of L are the same as right L-module structures on R. Then the BV envelope is the de Rham-Chevalley complex for this L-module (see 2.9.1); as a mere graded algebra it equals Sym(L[1]). They were considered in [Kos] (the case of the tangent algebroid), [Hue], [X], and (under the name of Calabi-Yau or vertex 0-algebroid structures) in §11 of [GMS2]. (vi) In 4.1.14 and 4.1.15 we consider homotopy unital commutative algebras and BV algebras and show that the corresponding homotopy categories are the same as the homotopy categories of the corresponding strictly unital objects. We need this material since the chiral chain complexes of unital chiral algebras are naturally homotopy unital BV algebras (not the strict ones); see 4.3.4. (vii) In 4.1.16–4.1.18 we discuss perfect complexes, perfect commutative DG algebras, and perfect BV algebras. Perfect commutative algebras are immediate counterparts in the homotopy DG world of the usual smooth algebras, and perfect BV algebras correspond to de Rham complexes for a flat connection on the line bundle ω −1 . The cohomology of a perfect BV algebra is finite-dimensional. We show in 4.6.9 that the chiral homology of a (very) smooth commutative DX -algebra is a perfect commutative algebra, and that of a cdo is a perfect BV algebra (see 4.8.5). We refer to [CFK] for a geometric discussion of commutative DG algebras supported in non-positive degrees. The perfectness property for commutative DG algebras should be compared with a much stronger (and deeper) condition on a DG algebra F – perfectness of F as an F ⊗ F -bimodule – introduced by M. Kontsevich (the latter property makes sense for arbitrary associative DG algebras). 4.1.1. Homotopy direct limits. Below A is a category which we tacitly assume to be closed under direct sums of sufficiently high cardinality. (i) Let B be a simplicial set. So for every n ≥ 0 we have a set Bn and for every monotonous map of intervals α : [0, . . . , m] → [0, . . . , n] we have the corresponding map ∂α : Bn → Bm compatible with the composition of the α’s. Denote by C(B, A) the category of homology type coefficient systems on B with coefficients in A. Thus F ∈ C(B, A) is a rule that assigns to every simplex b ∈ Bn an object Fb ∈ A and to every α as above a morphism ∂α = ∂αF : Fb → F∂α (b) compatible with the composition of the α’s. Denote by Cs (A) the category of simplicial objects in A. We have a functor Cs : C(B, A) → Cs (A)

(4.1.1.1)

where Cs (F )n := ⊕ Fb and the structure maps ∂α : Cs (F )n → Cs (F )m are direct b∈Bn

sums of the corresponding morphisms ∂αF . (ii) Let P be a small category. It yields a simplicial set BP (the classifying simplicial set; see [Se] or [Q3]); an n-simplex p˜ ∈ BPn is a diagram (p0 → · · · → pn )

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in P. Consider the category Fun(P, A) of functors F : P → A, p 7→ Fp . There is a fully faithful embedding Fun(P, A) ,→ C(BP, A)

(4.1.1.2)

which identifies a functor F with the system F(p0 →···→pn ) = Fp0 . (iii) Suppose that A is a k-category. Let C(A) be the category of complexes in A. By Dold-Puppe one has the fully faithful embedding Norm : Cs (A) ,→ C(A)

(4.1.1.3)

which identifies Cs (A) with the full subcategory of complexes having degrees ≤ 0. (iv) Suppose that A is a DG (super) k-category.1 We have the functor tot : C(A) → A which sends a complex C = (C · , d) to an object tot C ∈ A such that tot C = ⊕C n [−n] as a plain object without differential; the structure differential is the sum of d and the structure differentials of C n [−n]. Set Norm := tot Norm : Cs (A) → A

(4.1.1.4)

For F ∈ C(B, A) as in (i) set C(B, F ) := Norm Cs (B, F ). For a functor F : P → A as in (ii) set C(P, F ) := C(BP, F ); this is the homotopy direct P-limit of F . Notice that the DG structure on A yields DG structures on all the above categories, and (4.1.1.1)–(4.1.1.4) are DG functors. Remark. Suppose that A = C(B) for an abelian category B. Let lim −→ F ∈ C(B) be the plain direct P-limit of F . There is an obvious canonical morphism of complexes C(P, F ) → lim −→ F . If F takes values in B ⊂ A, then the complex C(P, F∼) has degrees ≤ 0 and the above morphism yields an isomorphism H 0 C(P, F ) −→ lim −→ F . Exercise. Let S ∈ P be an object such that the group G := Aut P acts freely on every set Hom(S, T ), T ∈ P. Let A ∈ A be an object equipped with an Aut S-action, and let F =Ind A be the corresponding induced functor, F (T ) := A[Hom(S, T )]G . Then the complex C(P, F ) computes the homology of G with coefficients in A. In particular, if G is finite and we are dealing with k-categories where k is a field of characteristic 0, then the map C(P, F ) → lim −→ F = AG is a quasi-isomorphism. 4.1.2. Operations. From now on A is a pseudo-tensor category; we assume that direct sums in A are compatible with operations.2 (i) Consider the category of k-modules. This is a tensor category, so both categories of simplicial k-modules and k-complexes are tensor categories. Norm is not a tensor functor. However for any finite set of simplicial k-modules Ni there is a canonical functorial morphism of complexes c = c{Ni } : ⊗Norm(Ni ) → Norm(⊗Ni )

(4.1.2.1)

1 We always assume A to be pretriangulated (the cones of morphisms are well defined, see [BoKa] or [Dr1]) and closed under Q appropriate direct limits. 2 I.e., P ({ ⊕ F }, G) = PI ({Fαi }, G). αi I α∈Ai

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which equals id⊗Ni if the Ni are constant simplicial k-modules;3 such a c is unique. According to 1.1.6(ii), this makes Norm a pseudo-tensor functor.4 Now if C is any pseudo-tensor simplicial k-category, we define its normalization as a pseudo-tensor DG k-category Norm C having the same objects as C and operations P Norm C := Norm P C . The composition of operation is defined using the above c. Remarks. (a) The plain k-categories corresponding to C and Norm C coincide. (b) Morphisms (4.1.2.1) are homotopy equivalences of complexes. (ii) Cs (A) is a simplicial pseudo-tensor catregory. Namely, for Fi , G ∈ Cs (A) the simplicial set PI ({Fi }, G)s is defined as follows. Consider Pba := PI ({Fi a }, Gb ); they form a cosimplicial-simplicial set P . Our PI ({Fi }, G)s is the corresponding total simplicial set Tot P (for the definition of Tot see, e.g., [BK]). Remark. If A is a tensor category, then Cs (A) has an obvious structure of a tensor simplicial category. The above pseudo-tensor simplicial structure comes from this tensor simplicial structure. (iii) If A is a pseudo-tensor k-category, then C(A) is a pseudo-tensor DG kcategory. The normalization functor extends in the obvious way to a pseudo-tensor DG functor (4.1.2.2)

Norm : NormCs (A) → C(A).

(iv) Assume that A is a pseudo-tensor DG (super) k-category. Then the categories in (4.1.2.2) are bi-DG categories and Norm is a pseudo-tensor bi-DG functor. Let us consider them as DG categories (with respect to the total grading and differential). The functor tot: C(A) → A is a pseudo-tensor DG functor in the obvious way. We get a pseudo-tensor DG functor (4.1.2.3)

Norm := tot Norm : Cs (A) → A.

4.1.3. Dolbeault algebras. Below “scheme” means “k-scheme” where k is our base field of characteristic 0. Definition. Let X be a scheme. A Dolbeault OX -algebra is a commutative unital DG OX -algebra Q, quasi-coherent as an OX -module, such that: α (a) The structure morphism OX − → Q is a quasi-isomorphism. (b) Q is homotopically OX -flat (see 2.1.1). (c) Spec Q0 is an affine scheme. ∼ By (c) for each OX -quasi-coherent Q-module N one has Γ(X, N ) −→ RΓ(X, N ). A Dolbeault DX -algebra is a DG DX -algebra which is a Dolbeault OX -algebra. Lemma. If X is separated and quasi-compact, then it admits a Dolbeault OX algebra. If, in addition, X is smooth, then it admits a Dolbeault DX -algebra. Proof. We present two constructions; for the second one X has to be quasiprojective. In both situations the Dolbeault algebras we define satisfy Q<0 = 0. (i) Let us begin with the Thom-Sullivan construction (see [HS] §4). amounts to the fact that the Norm(Ni ) are complexes supported in degree 0. associativity diagram from 1.1.6(ii) for τ = Norm, ν = c is commutative due to the uniqueness property of c. 3 Which

4 The

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For a finite set I we denote by ∆I the subscheme of AI defined by the equation Σti = 1. Let ΩI := Γ(∆I , DR∆I ) be the de Rham algebra of ∆I ; this is a resolution of k. For any I 0 ⊂ I we have an evident embedding ∆I 0 ⊂ ∆I , hence a projection ΩI → ΩI 0 . Choose a finite affine covering {Us },Ts ∈ S, of X. For a non-empty subset I ⊂ S we have an affine embedding jI : UI := Ui ,→ X. Our Q is a subalgebra of the DG i∈I Q algebra jI∗ OUI ⊗ ΩI which consists of those collections (fI ) that for every I 0 ⊂ I I

the images of fI 0 , fI under the maps jI 0 ∗ OUI 0 ⊗ ΩI 0 → jI∗ OUI ⊗ ΩI 0 ← jI∗ OUI ⊗ ΩI coincide. It follows easily from [HS] 4.1 that Q is a Dolbeault OX -algebra. If X is smooth, then it is a DX -algebra in the evident way. Exercise. If X is a projective curve, then the OX -algebra Q0 is not finitely generated. (ii) Here is another construction. We suppose that X is quasi-projective. The Dolbeault algebras we will construct have the property that each Qi is a locally free OX -module. Let π : Y → X be a Jouanolou map; i.e., Y is a torsor over X with respect to an action of some vector bundle such that Y is an affine scheme.5 Consider now the relative de Rham complex DR(Y /X). It is clear that Q := π∗ DRY /X is a Dolbeault OX -algebra. Suppose X is smooth. Then the jet algebra JQ = DR(JQ0 /OX )6 (see 2.3.2) is a Dolbeault DX -algebra. Indeed, it obviously satisfies conditions (a) and (b), and condition (c) follows since the morphism Spec JQ0 = JY → Y is affine.  In the next lemma (parallel to the lemma from 2.2.10) we consider the category of all Dolbeault algebras in either OX - or DX -setting. Since the tensor product of two Dolbeault algebras is obviously a Dolbeault algebra, this is a tensor category. Lemma. Every functor from the category of Dolbeault algebras to a groupoid is isomorphic to a trivial functor. Proof. Denote our functor by ¯. (i) Let ζ, η : Q1 → Q2 be any two morphisms. Let us show that ζ¯ = η¯. Consider ⊗2 ⊗2 the morphisms χ, χ0 : Q1 → Q⊗2 1 , µ : Q1 → Q1 , ξ : Q1 → Q2 where χ(a) = a ⊗ 1, 0 χ (a) = 1 ⊗ a, µ(a ⊗ b) = ab, ξ(a ⊗ b) = ζ(a)η(b). Since µχ = µχ0 , one has χ ¯=χ ¯0 . 0 Since ξχ = ζ, ξχ = η, we have ζ¯ = η¯. (ii) For any finite collection {Qα } of Dolbeault algebras one can find a Dolbeault algebra Q such that for every α there is a morphism Qα → Q. Indeed, one can take Q = ⊗Qα and the standard morphisms. (iii) To finish the proof, let us show that for a pair Q0 , Qn and a chain of χ1 ψ1 χ2 χn ψn ¯−1 · · · ψ¯1 χ ¯−1 : morphisms Q0 ← Q0 → Q1 ← · · · ← Q0 → Qn the composition ψ¯n χ n

1

n

1

5 Recall that one constructs Y as follows. Choose an open embedding X ⊂ X ¯ such that X ¯ is ¯ r X up if necessary, we can assume that X ¯ rX ⊂ X ¯ is locally a projective variety. Blowing X ¯ ,→ Pn . Let V ⊂ Pn∗ × Pn be the complement defined by one equation. Choose an embedding X to the incidence divisor. Set Y := X × V . Pn

fact that functors DR and J commute follows from their universal properties. Precisely, for an OX -algebra B both DR(JB) and JDR(B) represent the same functor F 7→ HomOX (B, F 0 ) on the category of commutative DG DX -algebras. 6 The

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¯0 → Q ¯ n does not depend on the chain. To see this, choose φi : Qi → Q as in (ii). Q ¯ ¯ Since φi−1 χ ¯i = φ¯i ψ¯i by (i), our composition equals φ¯−1  n φ0 ; q.e.d. 4.1.4. Dolbeault resolutions. Let Q be a Dolbeault DX -algebra. For a DX -complex the morphism αM : M → M ⊗ Q is called a Dolbeault resolution of M . It is a quasi-isomorphism by (a), (b) from the definition in 4.1.3, and by (c) the morphisms Γ(X, M ⊗ Q) → RΓ(X, M ⊗ Q), ΓDR (X, M ⊗ Q) → RΓDR (X, M ⊗ Q) are quasi-isomorphisms. Thus we have canonical quasi-isomorphisms (4.1.4.1)

∼

Γ(X, M ⊗ Q) −→ RΓ(X, M ),

∼

ΓDR (X, M ⊗ Q) −→ RΓDR (X, M ).

Remarks. (i) Another way to compute RΓDR (X, M ) is to use a smaller complex ΓDR (Y, π ∗ M ) where π : Y → X is a Jouanolou map. We will use Dolbeault D-resolutions for the computations of the de Rham homology of D-modules on R(X) (see 4.2); it is not clear if one can do this using the above type of complexes. (ii) Let i : X ,→ Z be a closed embedding of smooth varieties. The for Q, M ∼ as above one has ΓDR (Z, i∗ (M ⊗ Q)) −→ RΓDR (Z, i∗ M ).7 (iii) According to Remark (i) in 1.4.6 the functor CM(X) → CM(X, Q), M 7→ M ⊗Q, is a compound tensor DG functor (here C denotes the category of complexes or DG modules). The “forgetting of the Q-action” functor CM(X, Q) → CM(X) is a ∗ pseudo-tensor functor. So αM : M → M ⊗ Q is a morphism of ∗ pseudotensor functors. Thus for ϕ ∈ PI∗ ({Mi }, N ) its action on RΓDR coincides with the (I) morphism of complexes  ΓDR (X, Mi ⊗ Q) → ΓDR (X I , ∆∗ N ⊗ Q). I

Lemma. Suppose X is a curve and N ∈ M(X) is such that H 1 (X, N ) = 0. Then H ≥1 (X, h(N )) = 0. In particular, if Q is a Dolbeault DX -algebra, then for ∼ any M ∈ CM(X) one has Γ(X, h(M ⊗Q)) −→ RΓ(X, h(M ⊗Q)). So, if M is homotopically quasi-induced (see 2.1.11), then Γ(X, h(M ⊗ Q)) computes RΓDR (X, M ). Proof. Set K := Ker(N  h(N )). Then H i (X, h(N )) = H i+1 (X, K) for i ≥ 1, and the latter groups vanish since dim X = 1.  Sometimes it is convenient to deal with non-quasi-coherent versions of Dolbeault algebras. Precisely, suppose we have a commutative unital DG OX -algebra Q (which we do not assume to be OX -quasi-coherent). We say that Q is a Dolbeaultstyle OX -algebra if for every quasi-coherent OX -module M the obvious morphisms Γ(X, M ⊗Q) → RΓ(X, M ⊗Q), RΓ(X, M ) → RΓ(X, M ⊗Q) are quasi-isomorphisms. For a smooth X, a Dolbeault-style DX -algebra is a Dolbeault-style OX -algebra equipped with a flat connection. The protoexample is the classical Dolbeault algebra: assuming that k = C, X is smooth and proper, we take for Q(U ), where U/X ∂¯ ∂¯ ¯ is ´etale, the sections of the ∂-resolution Ω0,0 → Ω0,1 → · · · of the algebra of holoU − U − morphic functions. Quasi-isomorphisms (4.1.4.1) are valid for any Dolbeault-style DX -algebra Q. 4.1.5. A reminder on semi-free objects and the cotangent complex. We follow the super conventions of 1.1.16, as always dropping the adjective “super”; the base field k is of characteristic 0. Commutative and associative DG k-algebras are called simply commutative algebras; the corresponding category is 7 To see this, notice that the terms of the complex DR(i (M ⊗ Q)) admit an increasing ∗ bounded below filtration whose successive quotients are Q0 -modules; hence their higher cohomology vanishes.

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denoted by Com. The subcategory of unital algebras is denoted by Comu. These are tensor categories. For all details for this section we refer to, e.g., [H] or [Dr1]. Suppose we have F ∈ Comu. A (DG) F -module P is said to S be semi-free if it admits a filtration P0 ⊂ P1 ⊂ · · · by DG submodules such that Pi = P and each gri P is a free F -module.8 Equivalently, P is semi-free if and only if it admits a system of semi-free generators, i.e., a base {ei } of P considered as a mere graded F module, with index set I equipped with a projection I → Z≥0 , i 7→ n(i), such that for every i ∈ I the element d(ei ) ∈ P is a linear combination of ej ’s with n(j) < n(i) (so d(ei ) = 0 if n(i) = 0). Homotopically projective (= K-projective in the sense of [Sp]) F -modules are the same as direct summands of semi-free F -modules. Every F -module admits a semi-free left resolution. An algebra F ∈ Comu is said to be semi-free if it admits a system of semi-free generators, i.e., a system of elements {fi } ⊂ F indexed by a set I equipped with a projection I → Z≥0 , i 7→ n(i), such that for every i ∈ I the element d(fi ) belongs to the subalgebra generated by fj , n(j) < n(i), and the {fi } freely generate F as a mere graded commutative algebra. Any commutative algebra admits a semi-free resolution. The category Comu of commutative unital DG algebras is naturally a closed model category (with quasi-isomorphisms as weak equivalences, surjective morphisms as fibrations). Its cofibrant objects are the same as retracts of semi-free algebras. For F ∈ Comu its cotangent complex (or the cotangent F -module) L ΩF is an object of the derived category D(F ) defined as follows. The cotangent module is equipped with a canonical morphism L ΩF → ΩF . If F is semi-free, then this morphism is an isomorphism in D(F ). Otherwise one chooses a semi-free resolution of F and defines L ΩF as the image of ΩF˜ under the canonical equivalence of categories ∼ D(F˜ ) −→ D(F ); one checks that this definition does not depend on the choice of F˜ ; i.e., for different F˜ the corresponding objects are canonically identified. 4.1.6. Batalin-Vilkovisky algebras. The Batalin-Vilkovisky operad BV is a DG k-operad which coincides with the 1-Poisson (alias braid, alias Gerstenhaber) operad as a mere Z-graded super operad. So it is generated by binary operations · of degree 0 (the product) and { } of degree 1 (the 1-Poisson bracket) that satisfy the usual relations. The differential is determined by the property d(·) = { }. Therefore for a complex C a BV algebra structure on C is a 1-Poisson structure on C considered as a mere graded vector space (see 1.4.18) such that d(·C ) := dC ·C − ·C dC⊗C = { }C . Remark. So { }C measures the extent to which dC is not a derivation of ·C . In fact, dC is a second order differential operator with respect to the commutative algebra structure; its symbol equals { }. Notice that the action of the Lie algebra L := C[−1] on C extends canonically to an action of the contractible Lie algebra L† (see 1.1.16). Namely, the component L[1] = C ⊂ L† acts on C as ·. For a BV algebra C a structure of a BV C-module on a complex M amounts to a BV algebra structure on C ⊕ M such that C  C ⊕ M are morphisms of BV algebras and ·, hence { }, vanishes on M ⊂ C ⊕ M . For m ∈ M its centralizer 8 I.e.,

is isomorphic to a direct sum of modules isomorphic to F [n], n ∈ Z.

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Cent(m) consists of all c ∈ C such that {c, m} = 0; m is C-central if Cent(m) = C. Notice that C-central elements form a subcomplex Mc ⊂ M , and the product map defines a morphism of complexes · : C ⊗ Mc → M . The quotient operad of BV modulo the relation { } = 0 is the unit operad Com. Therefore a commutative (DG) algebra is the same as a commutative BV algebra, i.e., a BV algebra C with { }C = 0. Suppose we have a k[t]-flat family of BV algebras Ct such that C0 is commutative. Then C0 acquires a 1-Poisson bracket { }, {a0 , b0 } := (t−1 {at , bt }Ct ) mod t. We refer to Ct as BV quantization of the 1-Poisson algebra (C0 , { }). Example. Let C be any BV algebra. Consider Ct which coincides with C[t] as a mere commutative graded algebra and has differential dCt := tdC (so { }Ct = t{ }C ). Then Ct is a BV quantization of C considered as a 1-Poisson algebra with zero differential. Notice that the more non-degenerate { } on H · C0 is, the fewer cohomology classes of C0 survive the quantization. The BV operad is acyclic: one has H · (BVn ) = 0 for n > 0. So the homotopy category of BV algebras coincides with that D(k). An interesting homotopy BV theory arises in a filtered setting. Namely, BV is naturally a DG filtered operad: the (increasing) filtration is the stupid one BV ≥−n . Notice that gr BV equals the 1-Poisson operad (the differential is trivial). A filtered BV algebra is a complex C equipped with a BV algebra structure and an increasing filtration which is compatible with the BV algebra structure (i.e., the products BVn ⊗ C ⊗n → C are compatible with the filtrations). This amounts to the property that the filtration on C is compatible with the product · and with the differential, and the induced product on gr· C is compatible with the differential. So gr· C is a 1-Poisson DG algebra (and Ct := ⊕Ci is its BV quantization). S We denote be¯ BV the category of filtered BV algebras C such that C−1 = 0, Cn = C. Let BV ⊂ BV be the full subcategory of those C for which C0 = 0. Notice that for any C ∈ BV the odd Poisson bracket on C0 vanishes, C0 is a commutative DG algebra, and C1 [−1] is a Lie DG algebra with respect to { }. ¯ ,→ BV admits a right adjoint BV → BV ¯ which Remark. The embedding BV assigns to C· ∈ BV the same C with a new filtration which is the old Ci for i > 0, and the new C0 equals 0. The BV operad is augmented in the obvious way. So we have the notion of a unital BV algebra. Explicitly, a BV algebra C is unital if it has a unit 1 ∈ C 0 with respect to · such that d(1) = 0 (then 1 lies in the { }-center of C). In the filtered setting we assume that 1 ∈ C0 . The subcategory of unital filtered algebras in BV is denoted by BVu . The embedding BVu → BV admits an obvious left adjoint (adding the unit) BV → BVu . The DG operad BV has a canonical coproduct δ : BV→BV⊗BV, δ(·) = · ⊗ ·, so we know what the tensor product of the BV algebras is (this is the usual tensor product of 1-Poisson algebras with the obvious differential). The tensor product of BV compatible filtrations is BV compatible, so we know what the tensor product of filtered BV algebras is. The tensor product of unital algebras is obviously unital. ¯ and BVu are closed model categories with weak 4.1.7. Proposition. BV, BV, equivalences being filtered quasi-isomorphisms and fibrations those morphisms f for which gr f is surjective.

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Sketch of a proof. The reference [H], strictly speaking, does not cover our filtered setting, but the arguments easily adopt to it. Here are the needed changes; we consider the case of BV. The general Theorem 2.2.1 of [H] remains valid if we replace the category ofScomplexes C(k) by the DG category of filtered complexes C· such that C−1 = 0, Ci = C. In Axiom (H1) from [H] 2.2 we assume that M is contractible as a filtered complex, and in the definition of the CMC structure on C one takes for weak equivalences filtered quasi-isomorphisms, for fibrations those f for which gr(f \ ) is surjective. The only modifications in the proof are that in the definition of standard cofibration [H] 2.2.3(i) one takes for M any filtered complex with gr(d) = 0, and in that of the standard acyclic cofibration [H] 2.2.3(ii) a contractible filtered complex. The proof that BV fits into this framework coincides with that of Theorem 4.1.1 of [H].  Below, the homotopy category of a closed model category C is denoted by HoC. ¯ ,→ BV identifies HoBV ¯ with the full subcategory of HoBV The embedding BV that consists of BV algebras C such that C0 is acyclic.9 The forgetting and adding of the unit functors HoBVu  HoBV remain adjoint on the level of homotopy categories. The notion of a filtered BV algebra makes sense in any abelian tensor k-category A; if A has a unit object, then we can consider unital filtered BV algebras. The ¯ corresponding categories are denoted by BV(A), BV(A), and BVu (A). Remark. In fact, the notion of a BV algebra makes sense in any DG pseudotensor category and that of a filtered BV algebra in a filtered DG pseudo-tensor category. Unital algebras make sense in the augmented setting (see 1.2.8). 4.1.8. BV enveloping algebras. Below we write down several constructions of BV algebras. Let us begin with the BV envelopes of Lie algebras which are the same as (homological) Chevalley complexes. ¯ → Lie, (a) Let Lie be the category of Lie DG algebras. The obvious functor BV ¯ ¯ C 7→ C1 [−1], admits left adjoint C : Lie → BV. Similarly, we have a pair of adjoint functors BVu → Lie, C : Lie → BVu where C is the composition of C¯ and the adding of the unit. For L ∈ Lie the corresponding C(L) is the Chevalley complex of L, and ¯ C(L) is the reduced Chevalley complex. As a plain graded commutative algebra, C(L) equals Sym(L[1]), the filtration C(L)i is Sym≤i (L[1]), the differential and the 1-Poisson bracket are determined by the condition that the embedding L = Sym1 (L[1])[−1] ⊂ C[−1] is a morphism of Lie DG algebras. Similarly, as a ¯ plain graded commutative algebra, C(L) equals Sym>0 (L[1]), etc. Our functors preserve quasi-isomorphisms so they descend to homotopy categories; we get pairs of adjoint functors HoLie  HoBV, HoLie  HoBVu . Remark. The above definitions make sense in any abelian tensor k-category A (for the unital setting we have to assume that A has a unit). (b) S More generally, suppose we have a filtration L0 ⊂ L1 ⊂ · · · on L such that Li = L, [Li , Lj ] ⊂ Li+j−1 (we call such an L· a commutative filtration on L). Then the filtration C(L)· on C(L) generated by L· (as on a commutative algebra generated by L· [1]) is compatible with the BV structure. Denote by Lie· the category of Lie DG algebras equipped with a commutative filtration. Then 9 The

inverse functor comes from Remark above.

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the functor Lie· → BVu which assigns to L· its Chevalley complex filtered in the above manner is left adjoint to the functor BVu → Lie which assigns to a filtered BV algebra C ∈ BVu the filtered Lie DG algebra C[−1]. The same is true for a non-unital version. The filtration on C(L) we considered in (a) corresponds to L0 = 0, L1 = L. (c) Suppose we have a central extension L[ of L by k[−1]. Consider the Chevalley complex C(L[ ) filtered according to the filtration L[0 := k[−1], L[1 = L[ . Then C(L[ )0 = Sym(k) = k[t], so C(L[ ) is a filtered BV k[t]-algebra. Set C(L)[ := C(L[ )t=1 ∈ BVu ; this is the [-twisted Chevalley complex of L. The filtration on C(L)[ is called the standard filtration. 4.1.9. BV extensions. To define the BV envelope of a Lie algebroid, one needs an extra structure of its BV extension. The situation is parallel to that of chiral envelopes of Lie∗ algebroids considered in see 3.9. Suppose we have R ∈ Comu and a DG Lie R-algebroid L (see 2.9.1). A BV ι π extension of L is an extension of complexes 0 → R[−1] − → L[ − → L → 0 together with a Lie R-algebroid structure on L[ as on a mere graded module (i.e., with differentials forgotten). The following properties should hold: (i) π is a morphism of graded Lie R-algebroids, ι a morphism of R-modules, and ι(k[−1]) belongs to the center of L[ . The Lie bracket on L[ is compatible with the differential. d(·) (ii) The morphism10 L[ ⊗ R = R ⊗ L[ −−→ L[ [1] equals −ι composed with the [ structure action of L on R. We call such (L, L[ ) a BV Lie R-algebroid, and abbreviate it to L[ . The pairs (R, L) and (R, L[ ) form the categories LieAlg and LieAlg BV . So a morphism φ : (R1 , L1 ) → (R1 , L2 ) is a pair (φR , φL ) of a morphism φR : R1 → R2 in Comu and a morphism φL : L1 → L2 in Lie which are compatible in the obvious sense, a morphism in LieAlg BV is a triple φ[ = (φR , φL , φL[ ), etc. For a fixed R the BV categories of (BV) Lie R-algebroids are denoted by LieAlgR , LieAlgR . Example. Let L be a Lie algebra acting on R, L[ a central extension of L by k[−1]. It yields the L-rigidified Lie R-algebroid LR := R ⊗ L and L[R ∈ Pc (LR ) ∈ PBV (L) which equals L[R as a (see 2.9.1). We also have a BV extension L[BV R mere graded Lie R-algebroid and whose differential is r ⊗ `[ 7→ dL[ (r ⊗ `[ ) − ι(`(r)). R We refer to L[BV as the L[ -rigidified BV extension of LR . If L[ is the trivialized R extension of L, then L[BV is called the L-rigidified BV extension. R Denote by Pc (L) the groupoid of Lie R-algebroid extensions Lc of L by R[−1] (we assume that k[−1] ⊂ R[−1] belongs to the center of Lc ) and by PBV (L) the groupoid of BV extensions. The Baer sum defines a Picard groupoid (in fact, a k-vector space groupoid) structure on Pc (L) and makes PBV (L), if non-empty, a Pc (L)-torsor. [ Let PBV s (L) be the set of (isomorphism classes of) pairs (L , s) where s : L → [ L is an R-linear section of π compatible with the grading, but not necessary with the bracket and the differential. Define Pcs (L) in a similar way. Then Pcs (L) is a c vector space, and PBV s (L), if non-empty, is a Ps (L)-torsor. Let us describe them explicitly. 10 Here

· is the R-action on L[ .

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BV For (L[ , s) ∈ PBV , µBV ) where ω BV : L × L → R[−1], s (L) consider a pair (ω BV 0 µ : L → R are maps ιω (`, ` ) := [s(`), s(`0 )] − s([`, `0 ]), ιµBV := [d, s]. For [ c c (L , s) ∈ PBV s (L) we have a similarly defined pair (ω , µ ). BV

Lemma. Both ω BV , ω c ∈ HomR (Λ2 L, R[−1]) are 2-cocycles of L considered as a mere graded R-algebroid. The map µc : L → R is R-linear, and for r ∈ R, ` ∈ L one has µBV (r`) = rµBV (`) − `(r). Both in BV and in the c setting, one has [d, µ] = 0 and [d, ω](`, `0 ) = `(µ(`0 )) − `0 (µ(`)) − µ([`, `0 ]). The maps (L[ , s) 7→ (ω BV , µBV ), (Lc , s) 7→ (ω c , µc ) are bijections between BV Ps (L), Pcs (L) and the sets of pairs (ω, µ) that satisfy the above conditions.  Remarks. (i) Suppose that s : L → L[ commutes with the bracket; i.e., it is a morphism of graded Lie R-algebroids. This amounts to the vanishing of ω BV , and the equation on µBV just means that the formula r · ` := −`(r) + rµBV (`) is a right L-module structure on the R-module R. Therefore one gets a bijection between the set of pairs (L[ , s) as above and the set of right L-module structures on R. (ii) Suppose that R, L have degree 0 (i.e., R is a plain commutative algebra, L a plain Lie algebroid). Then any BV extension L[ admits a unique splitting s ∼ compatible with the grading s : L −→ L[0 ⊂ L[ , which is automatically a morphism of graded Lie R-algebroids. We see that BV extensions of L are the same as CalabiYau, or vertex 0-algebroid, structures on L from §11 of [GMS2]. (iii) For arbitrary L consider the de Rham-Chevalley DG algebra CR (L) (see 2.9.1). Recall that as a mere graded algebra, CR (L) equals HomR (SymR (L[1]), R). Now the conditions on (ω c , µc ) from the lemma just mean that it is an even 0-cocycle in CR (L)[1], i.e., an odd 1-cocycle in CR (L). 4.1.10. The BV envelopes of BV algebroids. If C is a unital filtered BV algebra, then RC := C0 = gr0 C is a commutative unital DG algebra, L := gr1 C[−1] is a Lie DG RC -algebroid, and C1 [−1] is a BV extension of L. We have defined a functor BVu → LieAlg BV . It admits a left adjoint LieAlg BV → BVu , (R, L[ ) 7→ CBV (R, L)[ (the BV envelope of L[ ). As a plain commutative graded algebra, CBV (R, L)[ equals Sym[R (L[1]) := the quotient of SymR (L[ [1]) modulo the relation 1[ = 1 where 1[ := ι(1) ∈ L[ [1]. The filtration subspaces CBV (R, L)[a are images of Sym≤a (L[ ), and the 1-Poisson bracket and differential are uniquely determined by the condition that L[ → CBV (R, L)[1 [−1] is a morphism of Lie (DG) algebras. The morphism of Lie algebras L → gr1 CBV (R, L)[ [−1] defines a morphism of 1Poisson algebras SymR (L[1]) → gr CBV (R, L)[ which is a quasi-isomorphism if L is homotopically R-flat. Example. Suppose that R is a plain smooth algebra, X := Spec R. Then a BV extension of ΘR is the same as a right D-module structure on OX (see Remark (ii) in 4.1.9), which is the same as a left D-module structure (= the flat connection) −1 on ωX , and CBV (R, ΘR )[ is the corresponding de Rham complex. In particular, it has finite-dimensional cohomology. Consider the de Rham-Chevalley DG algebra CR (L) as in Remark (iii) in 4.1.9. Notice that CBV (R, L)[ , considered as a mere graded module, is naturally a CR (L)module.11 One checks in a moment that this action is compatible with the differentials. Therefore CBV (R, L)[ is a DG CR (L)-module. 11 An

element ϕ ∈ C1R (L) = HomR (L[1], R) acts on CBV (R, L)[ as a derivation whose ϕ

restriction to CBV (R, L)[1 is the composition L[ → L −→ R.

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Remarks. (i) Take any element of Pcs (L); let γ = (ω c , µc ) be the corresponding odd 1-cocycle in CR (L) (see Remark (iii) in 4.1.9). Let L[γ the translation of L[ by the corresponding “classical” extension of L. Then CBV (R, L)[γ identifies naturally with CBV (RL)[ as a mere graded CR (L)-module, so that its differential dγ becomes d + γ where d is the differential of CBV (RL)[ . (ii) The DG algebra CR (L) carries a natural filtration CR (L)≥i such that gri CR (L) = HomR (Symi (L[1]), R). The cocycles γ as above lie in CR (L)≥1 . In fact, any odd 1-cocycle γ of CR (L)≥1 (not necessary coming from Pcs (L)) defines a twisted differential dγ := d + γ on CBV (R, L)[ . If f is an even element of degree 0 in CR (L)≥1 , then dγ+df is equal to12 (1 + f )dγ (1 + f )−1 , so the cohomology of CBV (R, L)[ with respect to dγ depends only on the class of γ in H 1 CR (L)≥1 . 4.1.11. Let us discuss the homotopy aspects of the above construction. Proposition. (i) The categories LieAlg, LieAlg BV have natural closed model category structures with weak equivalences being those morphisms φ, resp. φ[ , for which both φR , φL are quasi-isomorphisms, and fibrations those morphisms for which both φR , φL are surjective. BV are naturally (ii) For a fixed R ∈ Comu the categories LieAlgR , LieAlgR closed model categories with quasi-isomorphisms as weak equivalences and surjective morphisms as fibrations. Sketch of a proof. Our situation does not fit into the setting of [H] 2.2 directly, but the arguments of loc. cit. can be easily modified to do the job. (i) Let us replace C(k) by C(k) × C(k) in the general setting of [H] 2.2. With conditions (H0), (H1) in [H] 2.2 modified in the obvious way, Theorem 2.2.1 of loc. cit. (together with its proof) remains valid in the present situation. Consider a functor LieAlg → C(k) × C(k) which assigns to (R, L) the same pair considered as mere complexes; there is a similar functor LieAlg BV → C(k) × C(k), (R, L[ ) 7→ (R, L). These functors admit left adjoints F : C(k) × C(k) → LieAlg, F BV : C(k) × C(k) → LieAlg BV . One has F (P, Q) = (R, L) where R = Sym(P ⊗ F r(Q)), where F r(Q) is the free Lie algebra generated by Q, and L = F r(Q)R = R ⊗ F r(Q); similarly, F BV (P, Q) = (R, F r(Q)[R ) where F r(Q)[R is the F r(Q)-rigidified BV extension (see 4.1.9). Our functors satisfy conditions (H0), (H1), and we are done. (ii) Replace C(k) in [H] 2.2 by the category C(k)ΘR of pairs (Q, τ ) where Q ∈ C(k), τ : Q → ΘR := Der(R, R) is a morphism of complexes. Theorem 2.2.1 from loc. cit. (with conditions (H0), (H1) modified in the evident way) remains valid. BV The functors LieAlgR → C(k)ΘR , LieAlgR → C(k)ΘR sending L or L[ to BV (L, τL ) admit left adjoints F and F . Namely, F (Q, τ ) = F r(Q)R , where the free Lie algebra F r(Q) acts on R according to τ , and F BV (Q, τ ) is its F r(Q)-rigidified BV-extension. Our functors satisfy conditions (H0), (H1), and we are done.  The usual constructions, such as adding a variable to kill a cycle (see [H] 2.2.2), work for the above closed model categories. For example, consider the case of LieAlg. Let L be a Lie R-algebroid, and suppose we have a datum (Q, τQ , ) where Q is a k-complex with zero differential, τQ : Q → ΘR a morphism of graded 12 Since

f ∈ CR (L)≥1 , the multiplication by 1 + f is an automorphism of CBV (R, L)[ .

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vector spaces,  : Q[−1] → L a morphism of complexes such that τL  = dτQ . Then we have (Cone(), τ ) ∈ C(k)ΘR where τ := (τL , τQ ). Define a new Lie R-algebroid L(Q, τQ , ) such that for any L0 ∈ LieAlgR a morphism L(Q, τQ , ) → L0 is the same as a morphism ν : (Cone(), τ ) → L0 in C(k)ΘR such that ν|L : L → L0 is a morphism of Lie R-algebroids. There is an evident morphism L → L(Q, τQ , ) of Lie R-algebroids; any morphism isomorphic to such an arrow for some datum as above is called an elementary cofibration. A standard cofibration is aSmorphism L → L0 such that L0 admits a filtration L00 ⊂ L01 ⊂ · · · such that L0i = L0 , ∼ L −→ L00 , and each L0i → L0i+1 is an elementary cofibration. Each cofibration in LieAlgR is a retract of a standard one. We say that a Lie R-algebroid L is semi-free if the morphism 0R → L is a standard cofibration. Every Lie R-algebroid admits a left semi-free resolution. Remark. A morphism of graded vector spaces χ : Q → L yields an identifica∼ tion L(Q, τQ , ) −→ L(Q, τQ + τL χ,  + dχ) coming from the standard isomorphism ∼ Cone() −→ Cone( + dχ) defined by χ. 4.1.12. The evident functors LieAlg BV → LieAlg → Comu, LieAlgR → BV → LieAlg BV preserve (co)fibrations and weak equivalences; LieAlg, and LieAlgR we have the corresponding functors between the homotopy categories. There is a fully faithful embedding Comu ,→ LieAlg left adjoint to the projection LieAlg → Comu, which assigns to R the trivial R-algebroid 0R and its lifting Comu → LieAlg BV , R 7→ 0[R . They also preserve (co)fibrations and weak equivalences The functor LieAlg BV → BV, (R, L[ ) 7→ CBV (R, L)[ does not preserve weak equivalences. But its restriction to the subcategory of those (R, L[ ) for which BV ) preL is homotopically R-flat (which includes cofibrant objects of LieAlgR serves them, so we have a well-defined functor between the homotopy categories L (R, L)[ . HoLieAlg BV → HoBV, (R, L[ ) 7→ CBV [ L To compute CBV (R, L) , one should consider a left resolution LL → L which is homotopically R-flat as an R-module (for example, one can always find LL which is a semi-free R-module). The BV extension L[ defines, by pull-back, a BV extension L L of LL . One has CBV (R, L)[ = CBV (R, LL )[ . Notice that gr CBV (R, L)[ = SymL R L. 4.1.13. The next technical proposition assures that while doing homotopy computations with Lie R-algebroids, or BV Lie R-algebroids, one can replace R by any cofibrant algebra of the same homotopy class. For R ∈ Comu denote by [R] ∈ HoComu our R considered as an object of the homotopy category; the same notation is used for morphisms in Comu. Let HoLieAlg[R] be the fiber of HoLieAlg over [R]; i.e., it is the category of ∼ pairs (LP , [φ]) where LP = (P, LP ) ∈ HoLieAlg and [φ] : [R] −→ [P ] an isomorBV phism in HoComu. One has a similar category HoLieAlg[R] . There are evident functors (4.1.13.1)

HoLieAlgR → HoLieAlg[R] ,

BV BV HoLieAlgR → HoLieAlg[R] .

Proposition. For a cofibrant R these functors are essentially surjective. Proof. We will consider the case of Lie algebroids; the BV setting is treated similarly. Below we denote Lie R-algebroids as LR , etc.

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Surjectivity on objects: (i) Suppose we have (LP , [ϕ]) ∈ HoLieAlg[R] . We want to define weak equivalences LR ← LF → LP in LieAlg such that the composition R ← F → P is of homotopy class [ϕ]. We can assume that LP is a semi-free Lie P -algebroid. We move step-by-step by elementary layers of LP . The steps are constructed as follows. (ii) Suppose we have a weak equivalence ν : LS → LT in LieAlg such that S ∈ Comu is cofibrant and an elementary cofibration iT : LT → L0T in LieAlgT . We will construct an elementary cofibration iS : LS → L0S in LieAlgS homotopy µ π equivalent to iT . More precisely, we construct morphisms L0S ← LU − → L0T and j : LS → LU in LieAlg such that πj = iS , µj = iT ν, π is a trivial fibration, µ is a weak equivalence. T T Write L0T = LT (Q, τQ ,  ) (see the end of 4.1.11). We can find a morphism of complexes S : Q[−1] → LS such that νS is homotopic to T (since ν is a quasiisomorphism). By Remark at the end of 4.1.11, replacing our datum by an equivalent one, we can assume that νS = F . Let νS : S → T be the morphism of algebras corresponding to ν. Consider the morphisms of complexes ΘS →DerνS (S, T ) ← ΘT which send derivations θS ∈ ΘS , θT ∈ ΘT to νS θS , θT νS ∈DerνS (S, T ). Since S is cofibrant, ΩS is a homotopically projective S-module; hence the morphism ΘS →DerνS (S, T ) is a quasi-isomorphism. Therefore one can find morphisms of S S = τLS S : Q → ΘS and κT : Q[1] →DerνS (S, T ) such that dτQ graded modules τQ S T . νS − νS τQ and dκT = τQ S S We define iS as the elementary cofibration LS → L0S := LS (Q, τQ ,  ). Define LU as the Lie algebroid whose morphisms to any LF are the same as the triples (φ, χ, κ), where φ : LS → LF is a morphism of Lie algebroids and χ : Q → LF , κ : Q[1] →DerφS (S, F ) are morphisms of graded modules, such that dχ = φS S and dκ = τLF (χ)φS − φS τQ . So one has an evident morphism j : LS → LU and π

µ

morphisms L0S ← LU − → L0T corresponding to the triples (iS , χS , 0) and (ν, χT , κT ) where χS , χT are the structure embeddings of Q for our elementary cofibrations. The promised relations between these morphisms are evident, so it remains only to check that π and µ are weak equivalences. We leave it as an exercise to the reader. (iii) Let us return to (i). Our LPSis a semi-free Lie P -algebroid, so we have a filtration 0P = LP 0 ⊂ LP 1 ⊂ · · · , LP n = LP , such that each iP n : LP n ,→ LP n+1 is an elementary cofibration in LieAlgP . We will define by induction the µn πn morphisms LRn ← LFn −→ LT n , iRn : LRn ,→ LR n+1 , and jn : LFn → LFn+1 such that the obvious diagram is commutative (i.e., πn+1 jn = iRn πn , µn+1 jn = iP n µn ), iRn are elementary cofibrations, πn are trivial fibrations, µn are weak equivalences. Passing to the inductive limit by iRn , jn , iP n , we get the promised LR ← LF → LP . Step 0: Our R is cofibrant, so we can realize the homotopy class [ϕ] by an actual morphism ϕ : R → P . Set LR0 := 0R , LF0 := 0R , π0 := idR , µ0 = ϕ. µn πn Induction step: Suppose we have already defined LRn ← LFn −→ LP n . Let us apply the construction of (ii) to LS = LFn , LT = LP n , ν = µn , and iT = iP n . The LU , j, and µ we get are our LFn+1 , jn , and µn+1 . Notice that the construction S of (ii) depends on a choice of S , τQ , and κT from loc. cit. subject to certain conditions, and we have to choose them properly in order to define πn+1 . Since πFn : Fn → R is a trivial fibration and R is cofibrant, the subcomplex ΘFn ,R ⊂ ΘFn of vector fields preserving πn is quasi-isomorphic to the whole of ΘFn . This S S implies that for given S we can choose τQ and κT so that τQ ∈ ΘFn ,R , and we

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S R do it that way. Then τQ yields τQ : Q → ΘR . Now let iRn be the elementary R cofibration LRn ,→ LR n+1 := LRn (Q, τQ , πn S ) and let πn+1 be the composition S S of the trivial fibration π : LU → L0S = LFn (Q, τQ ,  ) from (ii) and the projection S S R S LFn (Q, τQ ,  ) → LRn (Q, τQ , πn  ) which equals πn on LFn and idQ on Q. We leave it to the reader to check that this is a trivial fibration, and we are done.

Surjectivity on morphisms: Suppose we have LR , L0R ∈ LieAlgR and a morphism ν : LR → L0R in HoLieAlg which lifts id[R] . We want to show that it comes from a morphism LR → L0R in HoLieAlgR . One uses a lifting of homotopy argument. Let π : LT → LR be a trivial fibration13 in LieAlg such that LT is cofibrant. One can find a morphism µ : LT → L0R in LieAlg such that µ is homotopic to νπ. The morphism πT : T → R is a trivial fibration, so it identifies R with T /I where I ⊂ T is a contractible ideal. So LT /ILT is a Lie R-algebroid. The morphism µT : T → R is homotopic to πT . We will show that µ is homotopic to a morphism κ : LT → L0R such that κT : T → R equals πT . Then π, κ yield morphisms of Lie R-algebroids LR ← LT /ILT → L0R . The left one is a trivial fibration, the composition is homotopic to ν, and we are done. To define κ, we first choose a homotopy between πT and µT , i.e., a cofibrant S, a trivial fibration ψ : S → T in Comu, its sections γπ , γµ : T → S, and a morphism ρ : S → R such that ργπ = πT , ργµ = µT . γµ Let LQ be the colimit of the diagram LT ← 0T −→ 0S in LieAlg, and let iµ



S LT −→ LQ ← 0S be the structure morphisms. Notice that iµ is a trivial cofibration since γµ is. There is a natural projection p : LQ → LT such that pS equals

ψ

0S − → 0T → LT and piπ = idLT . Our p is a trivial fibration, so the morphism γπ 0T −→ 0S → LQ extends to a morphism iπ : LT → LQ such that piπ = idLT . Finally, one has a morphism β : LQ → L0R such that βiµ = µ and βS is the ρ  composition 0S − → 0R → L0R . Our κ is βiπ : LT → L0R . 4.1.14. Homotopy unital commutative algebras and modules. Let Com be the category of possibly non-unital commutative DG algebras. We say that E ∈ Com is a homotopy unit algebra if the corresponding graded cohomology algebra H · E is the unit algebra. Thus H 6=0 E = 0, H 0 E = k. A morphism of homotopy unit algebras is their morphism as commutative algebras which commutes with the identification H 0 = k (or, equivalently, is a quasi-isomorphism). For every homotopy unit E one has canonical morphisms of homotopy unit algebras E ← τ≤0 E → H 0 E = k. The tensor product of homotopy unit algebras is a homotopy unit algebra. For R ∈ Com a homotopy unit in R is a morphism E → C such that E is a homotopy unit algebra and k = H 0 E → H · C is a unit in the (graded) algebra H · R. A homotopy unital commutative algebra is a commutative algebra R equipped with a homotopy unit iR : ER → R; we often denote it simply as R. A morphism of homotopy unital algebras R → R0 is a pair (f, fE ) where f : R → R0 is a morphism of commutative algebras, fE : ER → ER0 a morphism of the homotopy unit algebras, such that f iR = iR0 fE ; we often abbreviate (f, fE ) to f . The category Comhu of homotopy unital commutative algebras is a tensor category. One has an obvious fully faithful embedding of tensor categories Comu ,→ Comhu . 13 Which

means that both T → R and LT → LR are surjective quasi-isomorphisms.

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Proposition. (i) Comhu is a closed model category with weak equivalences being quasi-isomorphisms and fibrations those morphisms (f, fE ) for which both f , fE are surjective. Denote by HoComhu the corresponding homotopy category. (ii) The functor HoComu → HoComhu is an equivalence of categories. Sketch of a proof. (i) Arguments of [H] work with obvious small modifications. One should only replace the definition of a standard cofibration from [H] 2.2.3(i) by the following: a morphism f : R → R0 in Comhu is a standard cofibration if fE : ER → ER0 and the morphism R ∗ ER0 → R0 are standard cofibrations in ER

Com. Here R ∗ ER0 is the coproduct of R and ER0 over ER in Com. ER

(ii) We consider a sequence of fully faithful embeddings Comu ⊂ Com00hu ⊂ Com0hu ⊂ Comhu each of which becomes an equivalence on the level of homotopy categories: >0 Let Com0hu ⊂ Comhu be the subcategory of R with ER = 0. The left adjoint functor to this embedding is (R, ER ) 7→ (R, τ≤0 ER ). It is clear that the corresponding homotopy categories are equivalent. Let Com00hu ⊂ Com0hu be the subcategory of R with ER = k. The left adjoint functor to this embedding is R 7→ R ∗ k (notice that for R ∈ Com0hu there is a ER

unique morphism ER → k). If R is cofibrant, then the morphism R → R ∗ k is a ER

weak equivalence, so the homotopy categories of Com00hu and Com0hu are equivalent. Finally, consider the embedding Comu ⊂ Com00hu . For R ∈ BV00hu the element iR (1) ∈ R is idempotent. It is clear that Ru := iR (1) · R ∈ Comu, the functor Com00hu → Comu, R 7→ Ru , is both left and right adjoint to the embedding, and the obvious arrows Ru  R are quasi-isomorphisms. It is clear that we get mutually inverse equivalences of the homotopy categories.  Suppose we have a homotopy unital commutative algebra R. An R-module M is said to be homotopy unital if H · M is a unital H · R-module. This amounts to the fact that the multiplication by 1 ∈ H 0 R endomorphism of M , considered as an object of the derived category of R-modules, is equal to idM .14 Denote the DG category of homotopy unital R-modules by C(R)hu and the corresponding derived category by D(R)hu . A morphism f : R → R0 of homotopy unital algebras yields an obvious exact DG functor C(R0 )hu → C(R)hu . We leave it to the reader to show that if f is a quasi-isomorphism, then the corresponding ∼ functor D(R0 )hu −→ D(R)hu is an equivalence and also to check that for R ∈ Comu the category D(R)hu is canonically equivalent to the derived category of unital Rmodules. Therefore, by the above proposition, from the homotopy point of view homotopy unital modules over homotopy unital algebras are the same as unital modules over unital algebras. The above category C(R)hu is a tensor category in the obvious manner. For every M ∈ C(R)hu the morphism R ⊗ M → M , r ⊗ m 7→ m, is a quasi-isomorphism whose homotopy inverse is a morphism M → R ⊗ M , m 7→ ˜1 ⊗ m, where ˜1 ∈ R0 is any lifting of 1 ∈ H 0 R. Similarly, the morphism M → Hom(R, M ) which assigns to m ∈ M the morphism R → M , r 7→ rm, is a quasi-isomorphism whose homotopy inverse is f 7→ f (˜ 1). Thus one can compute morphisms and the tensor product in D(R)hu using, say, semi-free resolutions just as we do in the case of unital algebras. 14 Indeed, if M is homotopy unital, then this is an idempotent automorphism in the derived category; hence it is the identity.

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4.1.15. Homotopy unital BV algebras. Here is a BV version of the above definitions. For a C ∈ BV a homotopy unit in C is a morphism of commutative algebras E → C0 such that E is a homotopy unit commutative algebra, the image of E is central in C,15 and k = H 0 E → H · gr C is a unit in H · gr C. A homotopy unital BV algebra is C ∈ BV equipped with a homotopy unit iC : EC → C. The category of homotopy unital BV algebras (cf. 4.1.14) is denoted by BVhu ; it is a tensor category which contains BVu as a full tensor subcategory. Proposition. (i) BVhu is a closed model category with weak equivalences being filtered quasi-isomorphisms and fibrations those morphisms (f, fE ) for which both gr fC , gr fE are surjective. ∼ (ii) One has an equivalence of the homotopy categories HoBVu −→ HoBVhu . Proof. An immediate modification of the proof in 4.1.14.



The construction from 4.1.9 generalizes to the homotopy unital setting as follows. For R ∈ Comhu a homotopy unital Lie R-algebroid L is a Lie R-algebroid16 such that L is a homotopy unital R-module and the image of ER → R is annihilated by the action of L. For such an L its BV extension is defined exactly as in 4.1.9 with an obvious modification (we demand that the ι-image of ER [−1] lies in the center of L[ ). One calls (L, L[ ) a homotopy unital BV Lie R-algebroid; we BV abbreviate it often to L[ . The pairs (R, L[ ) form a category LieAlghu . There BV is an obvious functor BVhu → LieAlghu , C 7→ (C0 , C1 [−1]) (we tacitly assume that EC0 := EC and the Lie algebroid L is gr1 C). It admits a left adjoint BV LieAlghu → BVhu , (R, L[ ) 7→ CBV (R, L)[ . We have an obvious morphism of 1-Poisson algebras SymR L → gr CBV (R, L)[ which is a quasi-isomorphism if L is L (R, L)[ ∈ HoBVhu = HoBVu as in 4.1.12. homotopically R-flat. One defines CBV 4.1.16. Perfect complexes. Let D be an additive category which admits arbitrary direct sums. An object P ∈ D is said to be compact if the functor Hom(P, ·) commutes with direct sums; i.e., for any family Mα of objects of D the obvious map ⊕Hom(P, Mα ) → Hom(P, ⊕Mα ) is an isomorphism. Exercise. Suppose D is the category of S-modules where S is a plain associative algebra. Then compact objects of D are the same as finitely generated S-modules. If D is a triangulated category, then its compact objects are also called perfect objects; they form a thick subcategory Dperf ⊂ D. Let F be a commutative algebra (:= a commutative unital DG super algebra) having degrees ≤ 0. Consider the derived category D(F ) of DG F -modules. Notice that D(F ) is a t-category, so we have the subcategories D(F )>a , D(F )a ) = 0 = Hom(P, D(F )
C is an E-algebra. axioms of a Lie R-algebroid for a non-unital R are the same as in the unital setting. 17 This follows from the following observation: suppose we have M ∈ D(F )>ai such that i ∼ ai → ∞; then ⊕Mi −→ ΠMi . The same is true for the b situation. 16 The

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Notice that for every f ∈ F 0 whose image in H 0 F is invertible, the functor D(Ff ) → D(F ) is an equivalence of categories (here Ff is the f -localization of F ). Take any f¯ ∈ H 0 F ; we see that the category D(Ff ) for f ∈ F lifting f¯ does not depend on the choice of f ; we denote it by D(Ff¯). Thus D(F ) can be localized with respect to the Zariski topology of Spec(H 0 F ), so we can speak of properties of objects of D(F ) that hold Zariski locally on Spec(H 0 F ). Notice that semi-free DG F -modules (see 4.1.5) with generators in degrees bounded from above are the same as DG F -modules which are, as mere graded F -modules, free with generators in degrees bounded from above. Lemma. For P ∈ D(F ) and an interval [b, a] the next conditions are equivalent: (i) P is perfect of span in [b, a]; (ii) P is a retract of an object of D(F ) represented by a semi-free F -module P˜ with finitely many generators whose degrees are in [b, a]; (iii) locally on Spec(H 0 F ) our P can be represented by a semi-free module P f with finitely many generators of degrees in [b, a]. Proof. (ii)⇒(i): Clear. (i)⇒(ii): (a) First we construct a semi-free resolution φ : T → P with finitely many generators in each degree ≤ a. The construction goes as follows. Suppose we have already defined T ; set Ti := F T ≥a−i ⊂ T . This is a semi-free F -module with finitely many generators in degrees [a − i, a], and H j Ti → H j P is an isomorphism for j > a − i and is surjective for j = a − i. We will define Ti and φi := φ|Ti by induction by i. The first step: We know that H >a P = 0. Also H a P is a finitely generated 0 H F -module (by Exercise above), so one can find (φ0 , T0 ). Induction step: Suppose we have (Ti , φi ). Then Cone(φi ) is perfect and its cohomology vanishes in degrees ≥ a − i. Then Ti+1 is obtained from Ti by adding finitely many free generators eα in degree a − i − 1, where d(eα ) ∈ Tia−i and φi+1 (eα ) ∈ P a−i−1 are chosen so that the cycles (−d(eα ), φi+1 (eα )) generate H a−i−1 Cone(φi ). (b) So we have defined our T . Set P˜ = Ta−b , ϕ := φa−b ; denote by ψ the composition P˜ → P → τ≥a P . Since Cone(ψ) ∈ D(F ) b. Such a P˜ was constructed in step (i)⇒(ii) above. Then B is a perfect complex of span [b, b]. L ¯ Since B ⊗ H 0 F is a perfect H 0 F -complex of span [b, b], it equals Q[−b] where F

¯ is a finitely generated projective H 0 F -module. Notice that Q ¯ = H b B. Passing Q 0 ¯ is a free H F -module. Let Q be to a Zariski localization, we can assume that Q ¯ = H b B. This a free F -module with generators in degree b such that H b Q = Q identification amounts to a morphism Q → B in D(F ) which is automatically a quasi-isomorphism (because a morphism of perfect F -modules which becomes an isomorphism after tensoring by H 0 F is a quasi-isomorphism). The morphisms P˜  B in D(F ) yield homotopy classes of morphisms of DG F -modules P˜  Q. Choose some true morphisms from the homotopy classes; after possible further Zariski localization, we can assume that the composition Q → P˜ → Q is an isomorphism, so Q is a retract of P˜ . The kernel P f of the

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retraction represents P . The F 0 -module of degree b generators (P f /F <0 P f )b is projective and hence locally free, and we are done. (iii)⇒(i): Perfectness is a local property: Indeed, suppose P is perfect Zariski locally on Spec(H 0 F ). Then one can find a finite covering Spec Ff0α of Spec F 0 such ˇ that each Pfα ∈ D(Ffα ) is perfect. For any DG F -module M consider its Cech complex C(M ) with respect to this covering; this is a resolution of M . If g is a finite product of fα ’s, then the functor M 7→ HomD(F ) (P, Mg ) = HomD(Fg ) (Pg , Mg ) commutes with direct sums. Hence the functor M 7→ HomD(F ) (P, C(M )) = HomD(F ) (P, M ) commutes with direct sums, and we are done. The fact that the span of a perfect P is determined locally is clear from the following observations. The upper bound of the span is the largest a such that H a P 6= 0. The lower bound is the largest b such that for every plain H 0 F -module M the complex HomF (T, M ) is acyclic in degrees > −b; here T is a resolution of P from the step (i)⇒(ii).  Corollary. For any perfect P ∈ D(F ) the complex P ∗ := RHom(P, R) ∈ D(F ) is perfect and the canonical morphism P → (P ∗ )∗ is an isomorphism.  4.1.17. Perfect DG algebras. As above, let F be a commutative algebra having degrees ≤ 0. We say that F is perfect if, as an object of the homotopy category of commutative DG algebras, it is a retract of a DG algebra F˜ which, as a mere graded commutative super algebra, is free with finitely many generators of degrees ≤ 0. Such F has span ≤ n if one can find F˜ with generators in degrees ∈ [−n, 0]. The next proposition is a non-linear version of the lemma in 4.1.16. Let us make first two remarks: (a) Notice that if f ∈ F 0 is such that its image in H 0 F is invertible, then the morphism F → Ff is a quasi-isomorphism. Thus for any f¯ ∈ H 0 F the localizations Ff for f ∈ F lifting f¯ define the same object of the homotopy category of DG algebras; denote it by Ff¯. So we can localize F , as an object of the homotopy category, with respect to the Zariski topology of Spec(H 0 F ), and one can speak of (homotopy) properties of F that hold Zariski locally on Spec(H 0 F ). (b) F is semi-free if and only if it is free as a mere graded algebra. Proposition. For an algebra F having degrees ≤ 0 and an integer n ≥ 1 the following conditions are equivalent: (i) F is perfect of span ≤ n; (ii) Zariski locally on Spec(H 0 F ), our F is homotopy equivalent to a semi-free DG algebra with finitely many generators of degrees −n, . . . , 0; (iii) H 0 F is a finitely generated algebra and the cotangent complex L ΩF ∈ D(F ) (see 4.1.5) is perfect of span in [−n, 0]. Proof. It is clear that (i)⇒(iii)⇐(ii). Suppose F satisfies (iii); let us check (i) and (ii). (a) Let us show that F admits a resolution φ : G → F such that G is semi-free with finitely many S generators in each (non-positive) degree. We write G = Gi where Gi is the subalgebra of G generated by G≥−i , and we construct Gi by induction getting on the ith step a morphism of DG algebras ∼ φi = φ|Gi : Gi → F such that H a (Gi ) −→ H a F for a > −i, H −i (Gi ) → H −i F is surjective, and Gi is semi-free with finitely many generators in degrees −i, . . . , 0. Then Gi+1 is obtained from Gi by adding finitely many free generators in degree

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−i − 1 and extending φ to them. To be able to do this, we need to know that H −i−1 F is a finitely generated H 0 F -module.18 Replacing F by an appropriate resolution, we can assume that F is obtained from Gi by adding (possibly infinitely many) semi-free generators in degrees < −i. The exact sequence of DG F -modules 0 → ΩGi ⊗ F → ΩF → ΩF/Gi → 0 shows Gi

that ΩF/Gi is perfect. It has generators in degrees ≤ −i − 1, so H −i−1 ΩF/Gi is a finitely generated H 0 F -module. If i ≥ 1, then the universal derivation yields ∼ F −i−1 /(d(F −i−2 ) + G−i−1 ) −→ H −i−1 ΩF/Gi ; hence H −i−1 F is a finitely generated i H 0 F -module, and we are done. Consider the case i = 0; set I := d(F −1 ) ⊂ F 0 = G0 (which is a finitely generated polynomial algebra), so H 0 F = F 0 /I. ∼ Then F −1 /(d(F −2 ) + IF −1 ) −→ H −1 ΩF/G0 , so F −1 /(d(F −2 ) + IF −1 ) is a finitely d

generated H 0 F -module. The exact sequence 0 → H −1 F → F −1 /d(F −2 ) − →I→0 shows then that H −1 F is a finitely generated H 0 F -module; q.e.d. From now on we assume, as is possible by (a), that F is a polynomial algebra with finitely many generators in each (non-positive) degree. (b) Let B be any commutative DG algebra, I ⊂ B a DG ideal. Recall that n ≥ 1 is an integer such that the span of L ΩF lies in [−n, 0]. Lemma. If H i I = 0 for every i > −n, then any morphism ρ¯ : F → B/I can be lifted to ρ : F → B. If, in addition, H −n I = 0, then such ρ is unique up to a homotopy. m Proof. One has B = ← lim − B/I , so one can construct a lifting, and a homotopy between two liftings, passing successively from B/I m to B/I m+1 . Therefore we can assume that I 2 = 0. Replacing B by the pull-back of B → B/I by ρ, we can assume that ρ¯ = idF . So B is an extension of F by a DG F -module I, and we want to construct a section. Since F is semi-free, we can find a section F → B which is a morphism of graded algebras but may not commute with the differential d. Commuting it with d, we get a derivation F → I[1], i.e., a morphism of DG F -modules c : ΩF → I[1]. One can modify our section to make it commute with d if and only if c is homotopic to zero, i.e., since ΩF is semi-free, if c vanishes as an element of HomD(F ) (ΩF , I[1]). If it happens, then the homotopy classes of ρ form a HomD(F ) (ΩF , I)-torsor. We are done by the condition on n. 

(c) Let Fn ⊂ F be the (DG) subalgebra generated by F ≥−n and let I ⊂ F be the DG ideal in F generated by F <−n . By the lemma, the projection ρ¯ : F → F/I = Fn /Fn ∩ I can be lifted to a morphism of DG algebras ρ : F → Fn . ρ Again by the lemma (applied to B = F , I = I) the composition F − → Fn → F is homotopic to idF , so F is a retract of Fn in the homotopy category, and we have proved (i). To prove (ii) consider step (i)⇒(iii) of the proof of the lemma in 4.1.16 for P := ΩF , P˜ := ΩFn ⊗ F , b := −n. We can choose, after a Zariski localization of Fn

Spec(H 0 F ), a direct summand P f of P˜ as in loc. cit. Then the universal derivation identifies the F 0 -module (Fn /(Fn<0 )2 )−n of degree −n generators of Fn with 18 We assume that i ≥ 0; one constructs G in the evident way using the fact that H 0 F is a 0 finitely generated algebra.

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(P˜ /F <0 P˜ )−n . Define K ⊂ (Fn /(Fn<0 )2 )−n as the submodule that corresponds to (P f /F <0 P f )−n ⊂ (P˜ /F <0 P˜ )−n . Let F f be the subalgebra of Fn generated by Fn>−n and the preimage of K in Fn−n . Since K is a free F 0 -module, our F f is semifree with finitely many generators in degrees −n, . . . , 0. Finally, the morphism ∼ F f → F is a quasi-isomorphism. To see this, notice that F fi −→ F i for i > −n and ∼ −n −n H ΩF f −→ H ΩF , and apply then the lemma from (b).  4.1.18. Perfect BV algebras. Let C be a 1-Poisson DG algebra C having degrees ≤ 0. We say that C is perfect if it is perfect as a commutative algebra and ∼ the pairing ΩR ⊗ ΩR → R[1] defined by { } yields a quasi-isomorphism L ΩR −→ RHomR (L ΩR , R)[1]. Notice that such a C has span ≤ 1. A filtered unital BV algebra C having degrees ≤ 0 is said to be perfect if the 1-Poisson algebra gr C is. L Suppose that C = CBV (R, L)[ for some BV Lie R-algebroid L[ . Then C is perfect if and only if R is a perfect commutative algebra and the composition L → ΘR = Hom(ΩR , R) → RHom(L ΩR , R) is a quasi-isomorphism. Here the first arrow comes from the L-action on R. Question. Is it true that the cohomology of any perfect BV algebra is finitedimensional? L (R, L)[ as above: This happens for C = CBV

Proposition. For such a C one has dim H · C < ∞. Proof. By 4.1.13, replacing R by a quasi-isomorphic algebra, we can assume that R is a semi-free algebra sitting in degrees ≤ 0. Our objects have the Zariski local nature with respect to Spec R0 , so it suffices to show that our cohomology is finite-dimensional locally on Spec R0 . By 4.1.17 and 4.1.13 we are reduced to the situation when R is a semi-free algebra with finitely many generators in degrees 0 and −1. We can also assume that L is semi-free. Let us show that CBV (R, L)[ is quasi-isomorphic to the de Rham complex for a certain twisted right D-module structure on R (cf. Example in 4.1.10). We know that ΩR is semi-free and ΩR = L ΩR . So the morphism L → ΘR is a quasi-isomorphism. Therefore the corresponding morphism of the de Rham-Chevalley DG algebras DR(R) = CR (R, ΘR ) → CR (L) is a filtered quasiisomorphism. The conditions on R imply that R admits a right D-module structure,19 so ΘR admits a BV extension Θ[R (see Remark (i) in 4.1.9). Its pull-back to L differs from L[ by some “classical” extension of L given by an appropriate cocycle of CR (L)≥1 . By Remark (ii) in 4.1.10 we can replace this cocycle by a homologous one without changing CBV (R, L)[ . So we can assume that our cocycle actually comes from an odd 1-cocycle γ in DR(R)≥1 . Let P be CBV (R, ΘR )[ equipped with the twisted differential dγ . The evident morphism of graded modules CBV (R, L)[ → P commutes with the differential, and it is a filtered quasi-isomorphism. It remains to show that dim H · P < ∞. Our P is a twisted de Rham complex, hence a DG DR(R)-module. Forgetting about the differentials, one has DR(R) = ≥n lim ←− SymR (ΩR [−1])/SymR (ΩR [−1]), P = SymR (ΘR [1]), and the DR(R)-action on 19 Indeed, the dualizing module of R is isomorphic to a shift of a copy of R (as a DG Rmodule).

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P is the usual convolution product. Notice that both P and DR(R) are supported in finitely many degrees. Consider P as a DG DR(R0 )-module. We have a finite decreasing filtration p,q i of the corresponding spectral F := Ω≥i R0 P . It suffices to prove that the terms E2 sequence are finite-dimensional. Set P¯ := gr0 P ; this is a complex of free R0 -modules, and gri P = ΩiR0 ⊗ P¯ . R0

Thus E1p,q := H p+q grp P = ΩpR0 ⊗ H q P¯ and the differential d1 is the de Rham differential for a DR0 -module structure on H q P¯ . So it suffices to show that all H q P¯ are holonomic DR0 -modules. This amounts to the existence of a stratification Zα of Spec R0 such that each complex P¯ |Zα has coherent cohomology. Now the proposition follows from the next two facts whose proof is left to the reader:20 - the complex of H 0 R-modules P¯ ⊗R0 H 0 R is quasi-isomorphic to (H 0 R)[dimR]; - the restriction of P¯ to Spec R0 r SpecH 0 R is acyclic.  4.2. The construction and first properties Chiral homology is a “quantum” version of (the algebra of functions on) the space of global horizontal sections of an affine DX -scheme (i.e., the space of global solutions of a system of non-linear differential equations). To define it, recall that chiral algebras amount to factorization algebras which are D-modules on Ran’s space R(X) (see 3.4.1 and 3.4.9). Now the chiral homology of X with coefficients in a chiral algebra is the de Rham homology of R(X) with coefficients in the corresponding D-module. Below we make the above informal definition rigorous. The technical problem to deal with is the fact that R(X) is not an ind-scheme in the strict sense21 (recall that the R(X)i are not algebraic varieties for i ≥ 3; see 3.4.2). So we are outside of the documented grounds of algebraic geometry, and one has to explain what the D-modules on R(X) are and how to compute the de Rham homology. We know that R(X) is an ind-scheme in a broad sense: it is an inductive limit of the X n ’s with respect to the non-directed family of all diagonal embeddings (see 3.4.1(ii)). Luckily this inductive system satisfies a number of specific properties (listed in 4.2.1) which permit us to handle sheaves on R(X) almost as easily as if it were an ind-scheme in the strict sense. We begin with the definition of !-sheaves on X S and consider some special types of complexes in 4.2.1. For a general formalism of sheaves on diagrams of spaces see [SGA 4] Exp. Vbis ; we consider the dual setting of !-sheaves. The cohomology of !-sheaves is defined in 4.2.2. In 4.2.3 we describe a usual spectral sequence which computes the homology beginning with the homology of the configuration spaces strata of R(X). In 4.2.4 the !-sheaves on X S are compared, in the case of the classical topology, with plain sheaves on R(X). The convolution tensor product ⊗∗ of !-sheaves is defined in 4.2.5. The D-module setting and the de Rham homology are considered in 4.2.6. We explain that the de Rham homology can be computed using Dolbeault resolutions in 4.2.7, prove the compatibility with the ⊗∗ product of coefficients in 4.2.8, and prove a stabilization property in 4.2.9–4.2.10. The chiral homology of a chiral algebra is defined in 4.2.11 using the Chevalley-Coisin complex from 3.4.11. In 4.2.12 we show that the chiral homology can be computed by P¯ carries a natural filtration with gra P¯ isomorphic to Syma (ΘR/R0 [1])[dim R0 ]. R(X) is not a union of a directed system of closed subschemes.

20 Hint: 21 I.e.,

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ch certain functorial chiral chain complexes C˜ ch and (more economic) CPQ . Another ch chiral chain complex Clog is defined in 4.2.14 after a digression on forms with logarithmic singularities (this complex will not be used in subsequent sections, so the reader can skip 4.2.13 and 4.2.14). Section 4.2.15 contains a remark on non-quasi-coherent Dolbeault resolutions. In 4.2.16 we consider the zero homology H0ch . In the commutative case it coincides with the algebra of functions on the space of global solutions from 2.4.1–2.4.5. Compatibility with filtrations and chiral homology with coefficients are discussed, respectively, in 4.2.18 and 4.2.19.

4.2.1 Sheaves on X S . We begin with general definitions. Below, k is our base field of characteristic 0. Let I be a category equivalent to a small one, and let YI , I 7→ YI , be an Idiagram of topological spaces or of schemes. We assume that every YI has a finite cohomological dimension (to be able to consider painlessly infinite complexes of sheaves) and that for each ϕ : I → J the map ϕY : YI → YJ is a closed embedding. A !-sheaf on YI is a rule P that assigns to each I ∈ I a sheaf of k-vector spaces PYI on YI and to each ϕ : I → J a morphism θ(ϕ) : ϕY · PYI → PYJ . We demand that the θ’s are compatible with the composition of the ϕ’s and that θ(id) is the identity map. In the scheme-theoretic setting we consider sheaves with respect to the ´etale topology. The !-sheaves form an abelian k-category Sh! (YI ). We denote by CSh! (YI ) the corresponding DG category of complexes and by D(YI ) the derived category. We say that a complex P of !-sheaves on YI is admissible if for each morphism ∼ ϕ : I → J the morphism θ(ϕ) yields a quasi-isomorphism PYI −→ Rϕ!Y PYJ . Admis! sible complexes form a full DG subcategory CSh (YI )adm ⊂ CSh! (YI ) closed under quasi-isomorphisms, and a full triangulated subcategory D(YI )adm ⊂ D(YI ). We apply the above considerations to I = S◦ and the diagram X S , I 7→ X I ; the structure embeddings are diagonal ones ∆(I/J) : X J ,→ X I . Here X is either a separated topological space of finite cohomological dimension or a separated kscheme of finite type. In the case of schemes we consider sheaves with respect to the ´etale topology. The diagram X S satisfies the following specific properties. Fix any I ∈ S. Then - The category S◦/I of all arrows I  T is equivalent to a finite ordered set Q(I). - The diagonals X T , T ∈ Q(I), form a stratification of X I . - Let j (I) : U (I) ,→ X I be the complement to all strata X T , T ∈ Q(I), T 6= I. Then the action of Aut I on U (I) is free. The arguments below work for any diagram that satisfies these properties. Every !-sheaf P on X S carries a canonical Cousin filtration P1 ⊂ P2 ⊂ · · · where Pa ⊂ P is the smallest subsheaf such that PaX J = PX J for |J| ≤ a. We say that P (I/T ) is nice if for every I and n the evident morphism ⊕ ∆· gra PX T → gra PX I T ∈Q(I,a)

is an isomorphism. Lemma. Suppose that P is nice and admissible. Then for any I the identification gr|I| PU (I) = PU (I) yields a quasi-isomorphism (4.2.1.1)

∼

(I)

gr|I| PX I −→ Rj∗ PU (I) .

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Proof. We use induction by n := |I|. Assume that our assertion is true for all I 0 of order < n. Consider a filtration X1I ⊂ · · · ⊂ XnI = X I on X I where I XaI is the union of the diagonals of dimension ≤ a, so XaI r Xa−1 is the dis(T ) joint union of U , T ∈ Q(I, a). It yields a filtration on PX I (as on an object of the derived category of sheaves on X I ) with successive quotients gr0a PX I = (I/T ) (T ) ⊕ R(∆(I/T ) j (T ) )· R(∆(I/T ) j (T ) )! PX I = ⊕ ∆· Rj· j (T )·R∆(I/T )! PX I . T ∈Q(I,a)

T ∈Q(I,a)

Since PaX I are supported on XaI , the identity morphism of PX I lifts canonically to a morphism φ in the filtered derived category from PX I equipped with the Cousin filtration to PX I equipped with the above filtration. Since P is nice, one (I/T ) (T ) has gra φ = ⊕ ∆· (φT ) where φT : gra PX T → Rj· j (T )·R∆(I/T )! PX I is T ∈Q(I,a)

the evident morphism. Since P is admisible, we can rewrite φT as the canonical (T ) morphism PX T → Rj· j (T )· PX T . By the induction assumption, φT is a quasiisomorphism for |T | < n, so gra φ is a quasi-isomorphism for a < n. Our filtration has length n and φ lifts the identity morphism, so grn φ, which is (4.2.1.1), is a quasi-isomorphism as well, q.e.d.  Every P ∈ CSh! (X S ) admits a canonical nice resolution P˜ → P. Namely, (I/T ) P˜X I is the homotopy direct limit C(Q(I), ∆∗ PX T ) (see 4.1.1(iv)) where the (I/J) ˜ PX J ,→ P˜X I coming from structure morphisms are the embeddings θ(I/J) : ∆∗ ˜ the embedding Q(J) ⊂ Q(I). It is clear that P is nice, the DG functor P 7→ P˜ is exact, and the obvious projection P˜ → P is a quasi-isomorphism. Notice that any quasi-isomorphism between nice complexes is automatically a filtered quasi-isomorphism with respect to the Cousin filtrations. 4.2.2. Cohomology with coefficients in a !-sheaf. A !-sheaf P on X S yields an S◦ -diagram of vector spaces, I 7→ Γ(X I , P ) := Γ(X I , PX I ). Denote by Γ(X S , P ) its inductive limit. The Cousin filtration on P defines a filtration on Γ(X S , P ). We say that P is handsome 22 if it is nice and the following extra conditions hold: (a) The morphisms Γ(X n , PX n ) → RΓ(X n , PX n ) are quasi-isomorphisms. (b) The projections PX n → grn PX n admit splittings (that need not commute with the differential). ∼

Lemma. If P is handsome, then grn Γ(X S , P ) −→ Γ(X n , grn P )Σn and the morphisms Γ(X n , grn P ) → RΓ(X n , grn P ) are quasi-isomorphisms. Proof. By (b) the Cousin filtration on P splits (the splitting need not commute with the differential), so grn Γ(X S , P ) = grn Γ(X S , grn P ), and the first claim follows since grn P is nice. The second claim is checked by induction by n.  Remarks. (i) For any P ∈ CSh! (X) the nice complex P˜ automatically satisfies (b); if P satisfies (a), then P˜ is handsome. (ii) Every P ∈ CSh! (X) admits a flabby resolution, i.e., a quasi-isomorphism 0 P → P 0 such that P 0 is flabby (which means that each PX n is a flabby complex of n 23 sheaves on X ). 22 Cf.

“Teddy Bear” from “When we were very young” by A. A. Milne. that a complex of sheaves F on Y is flabby if for every closed Z ⊂ Y the morphism ΓZ (Y, F ) → RΓZ (Y, F ) is a quasi-isomorphism. 23 Recall

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Let CSh! (X)h ⊂ CSh! (X S ) be the full DG subcategory of handsome complexes. The above remarks imply that its localization by quasi-isomorphisms coincides with D(X S ). On the other hand, the lemma shows that the functor Γ(X S , ·) sends quasi-isomorphisms in CSh! (X)h to filtered quasi-isomorphisms. Therefore one has an exact functor RΓ(X S , ·) : D(X S ) → DF (k).24 We write H · (X S , P ) := H · RΓ(X S , P ). By Remark (i), for any P ∈ CSh! (X) one has RΓ(X S , P ) = Γ(X S , P˜0 ) where P → P 0 is a resolution such that P 0 satisfies (a); e.g., P 0 is flabby. 4.2.3. Cohomology of configuration spaces. Set R(X)◦n := U (n) /Σn ;25 this is the space of configurations of n-points on X. The projection U (n) → R(X)◦n is an ´etale Σn -covering. For P ∈ Sh! (X) the sheaf PU (n) is Σn -equivariant, so by descent it defines a sheaf PR(X)◦n on R(X)◦n . We write Γ(R(X)◦n , P ) := Γ(R(X)◦n , PR(X)◦n ), etc. Lemma. For admissible P there is a canonical quasi-isomorphism (4.2.3.1)

∼

grn RΓ(X S , P ) −→ RΓ(R(X)◦n , P ). ∼

Proof. By Lemmas from 4.2.1, 4.2.2 one has grn RΓ(X S , P ) −→ RΓ(U (n) , P )Σn . ∼ Now the trace map yields RΓ(U (n) , P )Σn −→ RΓ(R(X)◦n , P ), and we are done.  Consider the spectral sequence Erp,q for the Cousin filtration converging to H (X S , P ) (we call it the Cousin spectral sequence). If P is admissible then, by the lemma, one has ·

(4.2.3.2)

E1p,q = H p+q (R(X)o−p , P ).

4.2.4. Comparison with sheaves on R(X). This section plays a purely motivational role. Suppose that our X is a locally compact separated topological space of finite cohomological dimension. Consider the topological space R(X) defined in 3.4.1; the obvious projections rI : X I → R(X) identify R(X) with the inductive limit of the diagram X S . We deal only with those sheaves of k-vector spaces on R(X) which are inductive limits of subsheaves supported on the R(X)i ’s. Such sheaves form an abelian k-category Sh(R(X)); one has the corresponding DG category of complexes CSh(R(X)) and the derived category D(R(X)). For G ∈ Sh(R(X)) the space of those sections of G that are supported on some R(X)i is denoted (by abuse of notation) by Γ(R(X), G). It is naturally filtered by the subspaces Γ(R(X), G)n of sections supported on R(X)n (the Cousin filtration). The functor Γ admits a right derived functor RΓ(R(X), ·) : D(R(X)) → DF (k) which can be computed by means of flabby resolutions.26 We write H · (R(X), G) := H · RΓ(R(X), G). The maps rI : X I → R(X) are finite, so the push-forward functor rI∗ : Sh(X I ) → Sh(R(X)) is exact. It admits a right adjoint functor rI! : Sh(R(X)) → Sh(X I ). We have a right exact functor r∗ : Sh! (X S ) → Sh(R(X)) which sends P 24 For us, the filtered derived category is formed by complexes F equipped with an increasing S filtration F· such that Fa = 0 for a  0 and Fa = F ; the morphisms are morphisms of complexes preserving filtrations localized by filtered quasi-isomorphisms (:= the morphisms that induce quasi-isomorphisms between the successive quotients). 25 Here U (n) := U ({1,... ,n}) ⊂ X n , etc. 26 G ∈ CSh(R(X)) is said to be flabby if for every n the complex of sheaves i! G on R(X) n n is flabby; here in : R(X)n ,→ R(X).

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to the inductive S◦ -limit of sheaves rI∗ PX I . It admits a right adjoint which is a left exact functor r! : Sh(R(X)) → Sh! (X S ); one has r! FX I := rI! F . The functor r∗ admits a left derived functor Lr∗ : D(X S ) → D(R(X)); for nice ∼ ∼ P one has Lr∗ P −→ r∗ P . Thus for any P one has r∗ P˜ −→ Lr∗ P . Similarly, r! ! S admits a right derived functor Rr : D(R(X)) → D(X ). For a flabby complex ∼ of sheaves F on R(X) one has r! F −→ Rr! F ; moreover, the complex r! F is nice ∼ ! and admissible, and r∗ r F −→ F . If P is admissible, then P → Rr! Lr∗ P is a quasi-isomorphism. So the functors Lr∗ and Rr! are adjoint, and they yield mutually inverse equivalences Lr!

D(X S )adm  DR(X).

(4.2.4.1)

Rr !

For any P ∈ CSh (X ) an obvious projection Γ(X S , P ) → Γ(R(X), r∗ P ) is compatible with the Cousin filtrations. If P is handsome, then this projection is ∼ an isomorphism of mere complexes;27 moreover Γ(R(X), r∗ P ) −→ RΓ(R(X), r∗ P ). If, in addition, P is admissible, then this is a filtered quasi-isomorphism. We get !

S

Lemma. For P ∈ D(X S ) there is a natural morphism RΓ(X S , P ) → RΓ(R(X), Lr∗ P )

(4.2.4.2)

in DF (k) which is an isomorphism in D(k). If P ∈ D(X S )adm , then this is a filtered quasi-isomorphism.  Remark. The spectral sequence (4.2.3.2) becomes the usual spectral sequence for the Cousin filtration R(X)· converging to H · (R(X), Lr∗ P ). Notation. Suppose we are in the scheme-theoretic setting. In view of (4.2.4.1) we will write D(R(X)) := D(X S )adm ; admissible complexes will be also called complexes of sheaves on R(X). Thus D(R(X)) is a full subcategory of D(X S ). The restriction of the functor RΓ(X S , ·) to D(R(X)) is denoted by RΓ(R(X), ·); the terms of its Cousin filtration by RΓ(R(X), ·)n = RΓ(R(X)n , ·). One has ∼ grn RΓ(R(X), P ) −→ RΓ(R(X)◦n , P ) (see (4.2.3.1)), etc. 4.2.5. The convolution product. The category Sh! (X S ) carries a natural tensor structure. Namely, for a finite non-empty family Pα , α ∈ A, of !-sheaves on X S we define their convolution tensor product ⊗∗ Pα as (cf. (3.4.10.1)) (⊗∗ Pα )X I := ⊕ (Pα X Iα ),

(4.2.5.1)

IA A

where the structure arrows θ(π) are the obvious ones. Our tensor category is denoted by Sh! (X S )∗ . The obvious natural morphisms ν{Pα } : ⊗Γ(X S , Pα ) → Γ(X S , ⊗∗ Pα ) are compatible with the Cousin filtrations; they make Γ(X S , ·) a pseudo-tensor functor (see 1.1.6(ii)) Γ(X S , ·) : Sh! (X S )∗ → CF (k)⊗ .

(4.2.5.2)

Notice that ⊗∗ Pα is nice if the Pα are nice. The same is true for “handsome”, so RΓ(X S , ·) is also a pseudo-tensor functor. 27 Not

filtered ones!

4.2. THE CONSTRUCTION AND FIRST PROPERTIES

301

Lemma. If X is quasi-compact, then Γ(X S , ·) and RΓ(X S , ·) are tensor functors. Proof. Our condition assures that Γ(X I , ⊗∗ Pα ) = ⊕ ⊗ Γ(X Iα , Pα ), so ν{Pα } IA A

are filtered isomorphisms. The fact about RΓ(X S , ·) follows as above.



Notice that P 7→ P˜ is naturally a pseudo-tensor functor, as follows from ∗ P (see 1.1.6(ii)) is ^ (4.1.2.3). Namely, the compatibility morphism ⊗∗ P˜α → ⊗ α (I /T ) (Iα /Tα ) formed by the maps C(Q(Iα ), ∆∗ PαX Tα ) → C(ΠQ(Iα ), ∆∗ α α PαX Tα ) ∗ P ) I where the arrow comes from (4.1.2.1). The projection P ˜ → P is a ^ ⊂ (⊗ α X morphism of pseudo-tensor functors. Let B be a DG k-operad. We see that if P is a B algebra in CSh! (X)∗ , then so ˜ is P . By (4.2.5.2) the complexes of sections Γ(X S , P˜ ) and Γ(X S , P ) are B algebras, and the morphism Γ(X S , P˜ ) → Γ(X S , P ) is a morphism of B algebras. Remark. The tensor product ⊗∗ comes very naturally in the R(X) setting of 4.2.4. Then the commutative semigroup structure on R(X) (see (iii) in 3.4.1) yields the convolution tensor structure on Sh(R(X)): for F, G ∈ Sh(R(X)) one has F ⊗∗ G := ◦∗ (F G) where ◦ : R(X)×R(X) → R(X) is the “disjoint sum” operation. Notice that ⊗∗ is an exact functor (since ◦ is finite). If X is quasi-compact, then there is an obvious identification Γ(R(X), F ⊗∗ G) = Γ(R(X), F )⊗Γ(R(X), G), and the same for the RΓ’s. So Γ(R(X), ·) and RΓ(R(X), ·) are tensor functors. Now r∗ : Sh! (X S )∗ → Sh(R(X))∗ , Lr∗ : D(X S )∗ → D(R(X))∗ are tensor functors in the obvious way. This explains the name “convolution tensor product” for ⊗∗ . 4.2.6. D-complexes on X S . (i) Suppose we have any diagram I 7→ YI as in 4.2.1 where the YI are smooth algebraic varieties and morphisms are closed embeddings. In such a situation a right D-module on Y is a rule that assigns to each I a right D-module MI = MYI on YI and to every ϕ : I → J a morphism θ(ϕ) : ϕY ∗ MI → MJ ; we demand that θ(ϕ) is compatible with the composition of the ϕ’s and θ(idI ) = idMI . Right D-modules on Y form an abelian k-category M(Y ). A complex M of right D-modules on Y (a.k.a. a right D-complex on Y ) is admissible if for each ϕ the morphism MI → Rϕ!Y MJ defined by θ(ϕ) is a quasi-isomorphism. Every complex quasi-isomorphic to an admissible complex is admissible. Our prime diagram is X S where X is our curve; here the above notion of right D-module coincides with that from 3.4.10. Admissible right D-complexes on X S are also called right D-complexes on R(X). The corresponding DG category is denoted by CM(R(X)) and the derived category is denoted by DM(R(X)) (cf. 4.2.4). We have fully faithful embeddings CM(R(X)) ,→ CM(X S ), DM(R(X)) ,→ DM(X S ). Below we skip the word “right” when it is clear that we are dealing with right D-modules. An increasing filtration on a D-complex M is admissible if gri M are admissible complexes. The corresponding filtered derived category is denoted by DF M(R(X)). (ii) Recall that the tensor category M` (R(X)) acts on M(X S ) (see (3.4.10.6)), so the tensor DG category of complexes CM` (R(X)) acts on CM(X S ). The action of the tensor DG subcategory of homotopically flat complexes (see 3.4.3) preserves admissible complexes and quasi-isomorphisms, so it yields an action on DM(R(X)). (iii) When needed, resolutions of D-complexes on X S can be built by induction. Here is the induction step: Suppose we have M ∈ CM(X S ) and a quasi-isomorphic

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embedding of Σn -equivariant D-complexes αn : MX n ,→ FX n which is a quasiisomorphism. Then there is a quasi-isomorphic embedding M ,→ F of D-complexes on X S which is an isomorphism on X
(4.2.6.1)

One also has RΓDR (X S , RjV ∗ jV∗ M ) = RΓDR (V, jV∗ M ). (v) A D-complex M on X S also yields a complex h(M ) of !-sheaves on X S , g hence the filtered complex Γ(X S , h(M )). The canonical projections DR(M ) → 28 Our

MX I ⊕ (

conditions define F on X Sn . ⊕

T ∈Q(I,n)

(I/T ) ∆∗ MX T

(I/T )

∆∗

,→ MX I ⊕

Suppose |I| > n.

Define FX I as the quotient of (I/T )

FT ) modulo the sum of the images of the maps (θ(I/T ) , −∆∗

(I/T ) ∆∗ FX T

.

(αn )) :

4.2. THE CONSTRUCTION AND FIRST PROPERTIES

303

DR(M ) → h(M ) yield morphisms of filtered complexes (4.2.6.2)

S S S ΓDR g (X , M ) → ΓDR (X , M ) → Γ(X , h(M )).

4.2.7. Dolbeault resolutions. One can represent RΓDR (X S , ·) by appropriate functorial complexes. Namely, let Q = QX be a Dolbeault DX -algebra (see 4.1.3). According to 3.4.9 and 3.4.20, it gives rise to a commutative factorization DG algebra which we denote also by Q by abuse of notation. Thus for every n ≥ 1 we have a commutative DG DX n -algebras QX n . We will call such a factorization algebra a Dolbeault DR(X) algebra. Lemma. Every QX n is a Dolbeault DX n -algebra. Proof. Condition (a) of 4.1.3 holds since it obviously holds on the level of the Cousin complexes. Condition (b) follows from Remark (i) in 3.4.4. By construction, n QX n is a Qn X -algebra, so one has an affine morphism Spec QX n → (Spec QX ) . The latter is an affine scheme which implies condition (c).  Let M be a right D-complex M on X S . The above lemma together with (4.1.4.1) show that DR(M ⊗ Q) satisfies condition (a) of 4.2.2. Thus, by Remark g (i) in loc. cit., the complex DR(M ⊗ Q) is handsome, so we have a canonical filtered quasi-isomorphism (4.2.7.1)

∼

S → RΓDR (X S , M ). ΓDR g (X , M ⊗ Q) −

4.2.8. ⊗∗ compatibilities. According to 3.4.10 the category M(X S ) carries two canonical tensor structures ⊗∗ and ⊗ch . The de Rham functor is a pseudo-tensor functor in an evident way: (4.2.8.1)

DR : CM(X S )∗ → CSh! (X S )∗ .

Lemma. The functor (4.2.8.2)

ΓDR (X S , ·) : CM(X S )∗ → CF (k)⊗

is actually a tensor functor. Proof. ΓDR (X S , ·) is a pseudo-tensor functor by (4.2.8.1) and (4.2.5.2). We want to show that the compatibility arrows (see 1.1.6(ii)) are isomorphisms; this follows from the fact that Γ(X tIα , DR(Mα X Iα )) = ⊗Γ(X Iα , DR(Mα X Iα )).  Composing DR and ΓDR (X S , ·) with P 7→ P˜ (see 4.2.1 and 4.2.5), we get pseudo-tensor functors (4.2.8.3)

g : CM(X S )∗ → CSh! (X S )∗ , DR

(4.2.8.4)

S S ∗ ⊗ ΓDR g (X , ·) : CM(X ) → CF (k) .

Let Q ∈ CM` (R(X)) be any commutative factorization algebra. Then M 7→ M ⊗ Q (see 4.2.6(ii)) is a pseudo-tensor endofunctor of M(X S )∗ . Namely, the compatibility morphisms ⊗∗ (Mα ⊗ Q) → (⊗∗ Mα ) ⊗ Q (see 1.1.6(ii)) are formed by the maps (MX Iα ⊗ QX Iα ) = (MX Iα ) ⊗ (QX Iα ) → (MX Iα ) ⊗ QX tIα where the arrow is the tensor product of the identity map for MX Iα and the factorization product for Q.

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Suppose that Q is a Dolbeault algebra. Composing (4.2.8.4) with · ⊗ Q, we get a pseudo-tensor functor S S ∗ ⊗ ΓDR g (X , · ⊗ Q) : CM(X ) → CF (k) .

(4.2.8.5)

Since the compatibility morphisms for · ⊗ Q are quasi-isomorphisms, it descends, by (4.2.7.1), to a tensor functor RΓDR (X S , ·) : DM(X S )∗ → CF (k)⊗ .

(4.2.8.6)

Notice that h : M(X S )∗ → Sh! (X S )∗ is evidently a pseudo-tensor functor, so composing it with Γ(X S , ·), we get a pseudo-tensor functor Γ(X S , h(·)) : CM(X S )∗ → CF (k)⊗ .

(4.2.8.7)

Morphisms (4.2.6.2) are actually morphisms of pseudo-tensor functors. 4.2.9. Cousin D-complexes. Repeating the construction from 4.2.3, we see that a D-complex M on X S defines a complex MR(X)on of D-modules on every R(X)on . One has DR(MR(X)on ) = DR(M )R(X)on , so, by (4.2.3.1), for M ∈ DM(R(X)) we get a canonical quasi-isomorphism ∼

grn RΓDR (R(X), M ) −→ RΓDR (R(X)on , M )

(4.2.9.1)

· (R(X), M ) with the first and the Cousin spectral sequence Erp,q converging to HDR term p+q E1p,q = HDR (R(X)o−p , M ).

(4.2.9.2)

There is an exact fully faithful functor jR∗ : M(R(X)on ) ,→ M(X S ) (n)

(4.2.9.3) (n)

defined by (jR∗ N )X I :=

⊕ T ∈Q(I,n)

(I/T ) (I) j∗ NU (T ) ,

∆∗

N ∈ M(R(X)on ). Here NU (T )

is the pull-back of N by the ´etale projection U (T ) → R(X)on . One has N = (n) (jR∗ N )R(X)on . (n)

Notice that for N as above and a left D-module L on R(X) one has L⊗jR∗ N = (n) (n) jR∗ (LR(X)on ⊗ N ), so the image of jR∗ is preserved by the action of the tensor ` category M (R(X)). A Cousin D-complex on R(X) is a complex M of right D-modules on X S such (n) that M −n ∈ jR∗ (M(R(X)on )) (in particular, M a = 0 for a ≥ 0). Such M is automatically admissible. Cousin D-complexes form an abelian category Cous(R(X)). Remark. Clearly, the functor Cous(R(X)) → DM(R(X)) is a fully faithful embedding. In fact, Cous(R(X)) is the core of a certain canonical t-structure on DM(R(X)). Let M be a Cousin D-complex. According to (4.2.9.1) we have a canonical quasi-isomorphism (4.2.9.4)

∼

grn RΓDR (R(X), M ) −→ RΓDR (R(X)on , M −n ).

i Since dim R(X)on = n, the cohomology groups HDR (R(X), M ) vanish for i > 0.

4.2. THE CONSTRUCTION AND FIRST PROPERTIES

305

a a 4.2.10. Lemma. The morphisms HDR (R(X)n , M ) → HDR (R(X), M ) are surjective for a > −n and are isomorphisms for a > −n + 1. If X is affine, then this is true, respectively, for a > −n − 1 and a > −n.

Proof. We use (4.2.9.4). If X is affine, then so are the varieties R(X)om . So the groups H a grm RΓDR (R(X), M ) vanish for a > −m, which implies our statement. If X is compact, then we consider a coordinate projection U (m) → X. It is affine and a a X is a curve, so HDR (U (m) , M −m ) = 0 for a > 1. Since HDR (R(X)om , M −m ) ⊂ a (m) −m a HDR (U , M ), we see that H grm RΓDR (R(X), M ) = 0 for a > −m + 1, and we are done.  4.2.11. Chiral homology: definition. For the rest of this chapter we assume (if not explicitly stated otherwise) that X is proper and connected. We work in the DG setting skipping the letters “DG” whenever possible, so “chiral algebra” means “DG chiral super algebra”; a chiral algebra which sits in degree 0 is called a “plain chiral algebra”. Let A be a not necessary unital chiral algebra on X. Consider the corresponding Chevalley-Cousin complex C(A) ∈ CM(X S ) as defined in 3.4.11. This complex is obviously admissible, so C(A) is a D-complex on R(X). If A is a plain chiral algebra, then C(A) is a Cousin complex. We define the chiral homology of X with coefficients in A or, simply, the chiral homology of A as the de Rham cohomology of C(A): (4.2.11.1)

C ch (X, A) := RΓDR (R(X), C(A)),

Hach (X, A) := H −a C ch (X, A).

Since C ch preserves quasi-isomorphisms, it can be considered as a functor on the homotopy category HoCA(X) (see 3.3.13). One has C(A)R(X)on = (Symn (A[1]))R(X)on (here Symn is the exterior symmetric power), so the Cousin spectral sequence (4.2.9.2) converging to Hnch (X, A) looks as (4.2.11.2)

−p−q 1 Ep,q = HDR (R(X)op , Symp (A[1])).

Remark. The Cousin filtration on C ch (X, A) seems not to be a part of any fundamental structure and plays mere technical role. In most cases the spectral sequence is highly non-degenerate (of course, it degenerates when µA = 0) and of no help for computations. Following the notation from 2.1.12, for a plain chiral algebra A we can rewrite (4.2.11.2) as (4.2.11.3)

1 DR Ep,q = Hp+q (R(X)op , Λpext A).

Here the D-module Λpext A on R(X)op is Symp (A[1])R(X)op [−p]. Notice that the vector spaces Hach (X, A) vanish for a < 0. 4.2.12. Chiral chain complexes. For a chiral algebra A we have defined C ch (X, A) as an object of the derived category. Often it is important to represent it by means of some actual functorial complexes; we refer to any such construction as a chiral chain complex. One defines a chiral chain complex replacing DR(C(A)) by a quasi-isomorphic handsome complex (see 4.2.2). Two nuisances had to be dealt with: the global one (each complex DR(C(A)X n ) needs to be resolved in order to compute RΓDR (X, C(A)X n )) and the local one (the complexes DR(C(A)) are not nice). The global problem is treated by means of Dolbeault resolutions (or their

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non-quasi-coherent version; see 4.2.15). To make DR(C(A)) nice, one can either g or use a modified de Rham functor replace DR by its canonical nice resolution DR from 2.2.10. One gets chiral chain complexes denoted, respectively, by C˜ ch (X, A)Q and C ch (X, A)PQ . Another possibility is to use forms with logarithmic singularities ch along the diagonals; it leads to the complex Clog (X, A)Q to be discussed in 4.2.14. Let us define the first chiral chain complex C˜ ch . Choose a Dolbeault DX S algebra Q and set AQ := A ⊗ Q. Our complex is C˜ ch (X, A)Q := ΓDR g (X , C(AQ )). Since C(AQ ) = C(A) ⊗ Q, we have a canonical filtered quasi-isomorphism (see (4.2.7.1)) (4.2.12.1)

∼ C˜ ch (X, A)Q −→ C ch (X, A).

Unfortunately, the complex C˜ ch (X, A)Q is unpleasantly huge. Indeed, its sub(n) quotient grn C˜ ch (X, A)Q comprises, apart from relevant sections of An , Q over U a pile of contractible debris from lower dimensional strata. The chiral chain complexes C ch (X, A)PQ we are going to consider next have the advantage of being reasonably small. To define it, we need to choose, apart from Q, a (non-unital) commutative DX algebra resolution P : P → OX such that P>0 = 0 and each Pa is DX -flat (see 2.2.10). For example, one can take P from Example in 2.2.10. Consider the non-unital chiral algebra APQ := A ⊗ P ⊗ Q. The promised chiral chain complex is C ch (X, A)PQ := Γ(X S , h(C(APQ ))). Proposition. There is a canonical filtered quasi-isomorphism (4.2.12.2)

∼

C ch (X, A)PQ −→ C ch (X, A). ∼

Proof. The quasi-isomorphisms A → AQ ← APQ yield one C ch (X, A) −→ C (X, APQ ). Consider a canonical morphism p : DR(C(APQ )) → h(C(APQ )) in CSh! (X S ). We will show that (i) p is a quasi-isomorphism (thus h(C(APQ )) is ∼ admissible), and (ii) h(C(APQ )) is handsome. Now (i) implies that C ch (X, APQ ) −→ ∼ RΓ(X S , h(C(APQ ))), and (ii) implies that C ch (X, A)PQ −→ RΓ(X S , h(C(APQ ))) (see 4.2.2). Our (4.2.12.2) is the composition. (i) Consider the Cousin filtration on C(APQ )X I . It splits (in a way that does not respect the differential), so it suffices to show that DR(gr C(APQ )X I ) → h(gr C(APQ )X I ) is a quasi-isomorphism. This follows since PT is DX T -flat, for (I/T ) (T ) gr C(APQ )X I is a direct sum of complexes ∆∗ ((j∗ j (T )∗ (AQ [1]T )) ⊗ PT ). (ii) h(C(APQ )) is evidently nice and satisfies condition (b) of 4.2.2. To check condition (a) of loc. cit., it suffices, by the above argument, to verify that each term (I) of the complex h(j∗ j (I)∗ (AQ [1]I ) ⊗ PI ) has no higher cohomology. In fact, for every QI [DX I ]-module N and a DX I -flat DX I -module R the sheaf h(N ⊗ R) has no higher cohomology. Indeed, the complex DR(N ⊗ R) is a left resolution of h(N ⊗ R), and each term of this resolution has no higher cohomology by condition (c) of 4.1.3.  ch

Lemma. One has (4.2.12.3)

∼

grn C ch (X, A)PQ −→ Γ(U (n) , h((APQ [1])n ))Σn .

4.2. THE CONSTRUCTION AND FIRST PROPERTIES

307

Proof. It suffices to show that (4.2.12.4)

∼

(n)

(n)

h(j∗ j (n)∗ (APQ )n ) −→ j∗ j (n)∗ h((APQ )n ).

We will check this replacing the complex (APQ )n by each of its terms. As in the end of the proof of the proposition, these are direct sums of modules of type N ⊗ R where R is DX I -flat; hence DR(N ⊗ R) is a left resolution of h(N ⊗ R) and (n) (n) DR(j∗ j (n)∗ N ⊗ R) is a left resolution of h(j∗ j (n)∗ N ⊗ R). Since the latter com(n) (n) (n) plex equals j∗ j (n)∗ DR(N ⊗R) = Rj∗ j (n)∗ DR(N ⊗R), we see that h(j∗ j (n)∗ N ⊗ ∼ (n) (n)∗ (n) (n)∗ R) −→ Rj∗ j h(N ⊗ R). Thus, being a mere sheaf, h(j∗ j N ⊗ R) equals (n) j∗ j (n)∗ h(N ⊗ R), and we are done.  Therefore we have an identification of mere graded modules (4.2.12.5)

C ch (X, A)PQ = ⊕ Γ(U (n) , h((APQ [1])n ))Σn , n≥1

the Cousin filtration is the filtration by n. The canonical morphism APQ → AQ provides filtered quasi-isomorphisms (4.2.12.6)

C ch (X, A)PQ ← C˜ ch (X, AP )Q → C˜ ch (X, A)Q

which compare the two types of chiral chain complexes. Remark. One can define the chiral homology functor directly by formulas (4.2.12.1) or (4.2.12.2). One has to show then that our complexes as objects of the filtered derived category do not depend on the auxiliary choice of Q or P, Q. This follows from the lemma in 2.2.10 (or rather the remarks after it) and the second lemma in 4.1.3. 4.2.13. A digression on forms with logarithmic singularities. The material of 4.2.13 and 4.2.14 will not be used in the subsequent sections and can be skipped. ch We will construct another chiral chain complex Clog (X, A)Q using forms with logarithmic singularities along the diagonal divisor. This section collects some basic facts about the logarithmic de Rham complex. For I ∈ S consider the de Rham DG algebra DRX I which is contained in (I) the larger DG algebra j∗ j (I)∗ DRX I of forms with possible singularities along the (I) (I)∗ log diagonal divisor. Let DRX DRX I be the DG subalgebra generated by I ⊂ j∗ j OX I and 1-forms df /f where f = 0 is an equation of a component of the diagonal divisor. It carries an increasing filtration W0 ⊂ W1 ⊂ · · · where W0 = DRX I and Wa is the DRX I -submodule generated by the products of ≤ a forms df /f as above. (I)

log (I)∗ Proposition. (i) The embedding ι : DRX DRX I is a quasiI ,→ j∗ j isomorphism. (ii) There is a canonical identification (cf. (3.1.10.1))

(4.2.13.1)

∼

log grW → a DRX I −

⊕ T ∈Q(I,|I|−a)

(I/T )

∆·

DRX T ⊗ Lie∗I/T ⊗ (λI /λT )[−a]. (I)

(I)

I Proof. Recall (see 3.1.7 and (3.1.10.1)) that j∗ j (I)∗ ωX I = j∗ j (I)∗ ωX ⊗ λI ∈ (I) (I/T ) (I)∗ I M(X I ) has a canonical filtration W· with grW j j ω = ⊕ ∆ ωX T ⊗ ∗ X −` ∗ T ∈Q(I,`)

Lie∗I/T ⊗ (λI /λT ). For the corresponding filtration on the de Rham complex one

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(I)

(I)∗ has grW ωX I ) = ` DR(j∗ j (I/T )

ΦT := ∆·

⊕ T ∈Q(I,−`)

ωX T ) ⊗ Lie∗I/T ⊗ (λI /λT ). Set

DR(ωX T ) ⊗ Lie∗I/T ⊗ (λI /λT ). The usual quasi-isomorphic embed-

(I/T )

dings ∆· Φ`X I :=

(I/T )

DR(∆∗

(I/T )

DR(ωX T ) ,→ DR(∆∗ ωX T ) define a quasi-isomorphic embedding (I) (I)∗ W ⊕ ΦT ,→ gr` DR(j∗ j ωX I ).

T ∈Q(I,−`)

(I)

(I)

log One has DR(j∗ j (I)∗ ωX I ) = j∗ j (I)∗ DRX I [I]. Set DRlog (ωX I ) := DRX I [I], (I)

so we have an embedding ι : DRlog (ωX I ) ,→ DR(j∗ j (I)∗ ωX I ). Define the W log filtration on DRlog (ωX I ) by W` DRlog (ωX I ) := (W`+|I| DRX I )[I]. We will show that ι is compatible with the W filtrations and (4.2.13.2)

∼

(I)

W log (I)∗ grW (ωX I ) −→ Φ`X I ⊂ grW ωX I ). ` ι : gr` DR ` DR(j∗ j

This implies the proposition. Let us check that ι sends W` to W` and that the image of grW ` ι lies in Φ`X I . By induction we can assume that the statement is known for every `0 < `. A section of W` DRlog (ωX I ) is a linear combination of forms of type µ = ν ∧ df1 /f1 ∧ · · · ∧ dfa /fa where a = |I| + `, the fi are local equations of some irreducible components of the diagonal divisor, and ν is a regular form. We can assume that the divisors fi = 0 have normal crossings (otherwise our form lies in W`−1 ); let X T be their (I) intersection. If µ has top degree, then it evidently lies in W` j∗ j (I)∗ ωX I . Its image (I) (I)∗ µ ¯ in grW ωX I is killed by multiplication by each fi ; hence µ ¯ ∈ ΦT . If µ is ` j∗ j not of top degree, then it can be written as a convolution of a similar form of top degree with a polyvector field along the fibers of the projection (fi ) : X I → Aa ; (I) (I)∗ (I) ωX I lies in ΦT . hence µ ∈ W` DR(j∗ j (I)∗ ωX I ) and its image in grW ` j∗ j W W log The surjectivity of gr` ι : gr` DR (ωX I ) → Φ`X I follows from the above argument together with the fact that the Aut(I/T )-module Lie∗I/T ⊗ (λI /λT ) is irreducible. It remains to show that grW ` ι is injective. By surjectivity it suffices to check that (I) (I) (I)∗ log ¯ ∈ grW ωX I ) lies for µ ∈ DR (ωX I ) ∩ W` DR(j∗ j (I)∗ ωX I ) its image µ ` DR(j∗ j in Φ`X I . We will prove this by induction by |I|. The case ` ≤ −|I| is evident, so we can assume that ` > −|I|. (I) (I/T ) (T ) (T )∗ Consider the residue map r : j∗ j (I)∗ ωX I → ⊕ ∆∗ j∗ j ωX T .

T ∈Q(I,|I|−1) (I) (I)∗ It sends the subcomplex DR (ωX I ) ⊂ DR(j∗ j ωX I ) to the sum of subcom(I/T ) (I/T ) (T ) (T )∗ plexes ∆· DRlog (ωX T ) ⊂ DR(∆∗ j∗ j ωX T ). (I) The kernel of r equals ωX I = W−|I| j∗ j (I)∗ ωX I and r is strictly compatible with W filtrations (see 3.1.6 and 3.1.7). Therefore grW ` r is injective. Thus Φ`X I = (I/T ) (I/T ) −1 r (⊕∆· Φ`X T ). So we need to check that r(¯ µ) ∈ ⊕∆· Φ`X T . Since r(µ) ∈ (I/T ) ⊕∆· DRlog (ωX T ), this follows from the induction assumption.  log

Remark. Another way to prove the above proposition is to notice that we can assume that X = A1 and use then the Orlik-Solomon theorem [OS]. ch 4.2.14. Now we can define the promised chiral chain complex Clog (X, A)Q . ` We assume that A is unital. Let AX S be the corresponding factorization algebra (see 3.4.9). This is a left D-module on X S , and one has a canonical isomorphism ∼ C(ω) ⊗ A`X S −→ C(A) of right D-modules on X S (see (3.4.13.1)).

4.2. THE CONSTRUCTION AND FIRST PROPERTIES

309

Consider the Cousin complex C(ω)X I , I ∈ S. As a mere graded module, it is the (I/T ) (I/T ) (I/T )∗ (I/T ) (I/T ) (I/T )∗ sum of components ∆∗ j∗ j (ωX [1])T = ∆∗ j∗ j ωX T [|T |], (I/T ) (I/T ) (I/T ) (I/T )∗ log T ∈ Q(I). Then ∆· DR (ωX T )[|T |] ⊂ DR(∆∗ j∗ j ωX T )[|T |] form log a DG DRX I -submodule of DR(C(ω)X I ) which we denote by DRCX I . When I log varies, we get a !-subcomplex DRC ⊂ DR(C(ω)). Now consider DG DRX I -modules DRlog (AX I ) := DRlog (ωX I ) ⊗ A`X I and log ` ` DRC log (A)X I := DRCX I ⊗ AX I . Since AX I is flat along the diagonals, one has ∼

(I/T )

(I/T )

∆· DRlog (ωX T )[T ] ⊗ A`X I −→ ∆· identification of mere graded modules

DRlog (AX T )[T ], so there is a canonical

∼

DRC log (A)X I −→

(4.2.14.1)

⊕ T ∈Q(I)

(I/T )

∆·

DRlog (AX T )[T ].

(I)

The obvious embedding DRlog (ωX I ) ,→ DR(j∗ j (I)∗ ωX I ) yields a morphism (I) (I) log DR (AX I ) → DR(j∗ j (I)∗ AX I ) = DR(j∗ j (I)∗ (AX [1])I )[−|I|] which is an injective quasi-isomorphism by (4.2.13.2) since A`X I is flat along the diagonals. Similarly, we have a quasi-isomorphic embedding DRC log (A)X I ,→ DR(C(A))X I . When I varies, we get a quasi-isomorphic embedding of !-complexes on X S DRC log (A) ,→ DR(C(A)).

(4.2.14.2)

The !-complex DRC log (A) is admissible by (4.2.14.1) and (i) in the proposition in 4.2.13, and nice according to (4.2.14.1).29 Choose a Dolbeault DX -algebra Q ch and set AQ := A ⊗ Q. Then DRC log (AQ ) is handsome.30 Set Clog (X, A)Q := S log Γ(X , DRC (AQ )). According to 4.2.2, the arrow (4.2.14.2) defines a filtered quasi-isomorphism ∼

ch Clog (X, A)Q −→ C ch (X, A).

(4.2.14.3)

ch The chiral chain complexes Clog (X, A)Q and C˜ ch (X, A)Q (see 4.2.12) are connected by natural quasi-isomorphisms. Consider the morphisms DRC log (AQ ) ← g ^ log (AQ ) → DR(C(A DRC Q )) where the left arrow is the canonical nice resolution (see 4.2.1) and the right one comes from (4.2.14.2). Applying Γ(X S , ·), we get the promised quasi-isomorphisms ch ^ log (AQ )) → C˜ ch (X, A)Q . Clog (X, A)Q ← Γ(X S , DRC

(4.2.14.4)

4.2.15. Variant. Sometimes it is convenient to use instead of Q some nonquasi-coherent Dolbeault-style algebras; see 4.1.4. Namely (cf. 3.4.2), suppose that for each I ∈ S we are given a Dolbeault-style DX I -algebra QX I and for each π : J  I a horizontal morphism of unital DG OX I -algebras ν (π) = ν (J/I) : ∆(J/I)∗ QX J → QX I ; one assumes that the ν (π) are compatible with the composition of the π’s, ν (idI ) = idQX I . Let us call such datum a Dolbeault-style DR(X) algebra. For example, for k = C and X compact the classical Dolbeault algebras on X I (see 4.1.4) form a Dolbeault-style DR(X) -algebra. Of course, any Dolbeault DR(X) -algebra (see 4.2.7) is automatically a Dolbeault-style DR(X) -algebra. Now such Q defines a resolution C(A)Q of C(A). As a mere graded (nonquasi-coherent) DX I -module, C(A)QX I is equal to the direct sum of components 29 Notice

that DR(C(A)) is not nice. (a) and (b) from 4.2.2 are evident.

30 Properties

310

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GLOBAL THEORY: CHIRAL HOMOLOGY

(T )

∆∗ (j∗ j (T )∗ (A[1])T ⊗ QX T ); the definition of the differential is left to the reader. If Q is a Dolbeault DR(X) -algebra, then C(A)Q = C(A) ⊗ Q = C(AQ ). One can use C(A)Q in the same way that we have used the Dolbeault resolutions in the previous sections. Therefore we have the corresponding chiral chain comch S g plexes C˜ ch (X, A)Q := Γ(X S , DR(C(A) Q )), C (X, A)PQ := Γ(X , h(C(AP )Q )), ch S log ` C (X, A)Q := Γ(X , DRC ⊗ Q ⊗ AX S ), etc. The details are left to the reader. 4.2.16. The 0th chiral homology. For a plain chiral algebra A set hAi = hAi(X) := H0ch (X, A). By construction and 3.4.12 one has (4.2.16.1)

DR I hAi = lim −→ H0 (X , AX I )

(the inductive limit of the S◦ -system of vector spaces). It follows from 4.2.10 that (4.2.16.2)

1 1 hAi = Coker(HDR (U, j ∗ A2 ) → HDR (X, A))

where the arrow comes from the chiral product µ : j∗ j ∗ A  A → ∆∗ A. Suppose A is commutative. Then, by (4.2.16.2) and 2.4.5, hAi coincides with the same noted vector space from 2.4.1. Therefore hAi is a commutative unital 

algebra. Its product · is31 the quotient map of the composition H0DR (X, A)⊗2 −→ H0DR (X ×X, A2 ) → H0DR (X ×X, AX 2 ) → hAi; here the last arrow is the canonical morphism and the middle one comes since A2 ⊂ AX 2 . ∼

Example. For the unit chiral algebra ω the identification H0ch (X, ω) = hωi −→ ∼ k comes from the trace isomorphisms H0DR (X I , ωX I ) −→ k. For any unital chiral algebra A we denote by 1ch = 1ch A ∈ hAi the image of 1 ∈ hωi = k by 1A . 4.2.17. The construction of 4.2.16 can be rendered to the DG setting as follows. For a DG super chiral algebra A let A♥ be a copy of A considered as a plain super chiral algebra equipped with an extra Z-grading and an odd derivation δ of degree 1 and square 0. Set hAi := hA♥ i; the Z-grading and δ make it a super complex. Similarly, C ch (X, A♥ ) is naturally a complex in the abelian category CVects of super complexes. Let H a C ch (X, A♥ ) ∈ CVects be its cohomology and C τ≤0 C ch (X, A♥ ) the corresponding truncation. Since H 0 C ch (X, A♥ ) = hAi, one has C a projection τ≤0 C ch (X, A♥ ) → hAi. A complex in CVects is the same as a super bicomplex, and we can pass to the total super complex. Then C ch (X, A♥ ) becomes C ch (X, A); denote by C C τ≤0 C ch (X, A) the total complex of τ≤0 C ch (X, A♥ ). Since H >0 C ch (X, A♥ ) = 0, C the map τ≤0 C ch (X, A) → C ch (X, A) is a quasi-isomorphism. So the above projection yields a canonical morphism in the derived category φA : C ch (X, A) → hAi.

(4.2.17.1)

Question. Is it true that C ch is equal to the left derived functor of the functor h i? In other words, can one find for every A a morphism of chiral algebras A0 → A which is a quasi-isomorphism and such that (4.2.17.1) for A0 is a quasi-isomorphism? If A is commutative, then hAi is a commutative DG algebra in a natural way, and the canonical morphism of DX -modules A` → hAi ⊗ OX is a morphism of DG 31 See,

e.g., the proof of 2.4.5.

4.2. THE CONSTRUCTION AND FIRST PROPERTIES

311

commutative DX -algebras which identifies the right-hand side with the maximal constant DX -algebra quotient of A`X (see 2.4.1–2.4.5). We will see in 4.6.1 that the above question has a positive answer if we restrict ourselves to commutative chiral algebras. 4.2.18. Compatibility with filtrations. A filtration A0 ⊂ A1 ⊂ · · · on a (not necessary unital) chiral algebra A (see 3.3.12) yields an admissible filtration P C(A)0 ⊂ C(A)1 ⊂ · · · on C(A). Namely, for I ∈ S set (A[1])I := ⊗ (A li [1]) ⊂ n i∈I P (A[1])I , the summation is over the set of all collections (li ) ∈ ZI≥0 such that li ≤ i∈I

n. Now one has C(A)nX I :=

⊕ T ∈Q(I)

(I/T ) (T ) (T )∗ j∗ j (A[1])T n

∆∗

(see (3.4.11.1)).

Our filtration on C(A) yields a filtration on C ch (X, A). Since gr C(A) = C(gr A), one has (4.2.18.1)

gr C ch (X, A) = C ch (X, gr A).

So one has a spectral sequence converging to H·ch (X, A) with (4.2.18.2)

1 ch Ep,q = Hp+q (X, gr A)p .

Here the upper index p is the grading on H ch (X, gr A) that comes from the grading gr· A. The compatibility with filtrations can be seen on the level of concrete chiral chain complexes C˜ ch (X, A)Q and C ch (X, A)PQ from 4.2.12. Here it is convenient to choose the Dolbeault algebra Q so that each component Qi is OX -flat. Then An ⊗ Q, An ⊗ P ⊗ Q form filtrations on AQ and APQ . The corresponding filtrations on the Chevalley-Cousin complexes satisfy gr C(AQ ) = C(gr AQ ) and gr C(APQ ) = C(gr APQ ). They yield filtrations on C˜ ch (X, A)Q , C ch (X, A)PQ , and (4.2.18.3) gr· C˜ ch (X, A)Q = C˜ ch (X, gr· A)Q ,

gr· C ch (X, A)PQ = C ch (X, gr· A)PQ

(the second equality needs an argument similar to the one used in the proof of the proposition in 4.2.12; the details are left to the reader). Example. Every A carries a (non-unital) filtration A0 = 0, A1 = A.32 The corresponding filtration on C ch (X, A)PQ is the Cousin filtration. 4.2.19. Chiral homology with coefficients. (i) Let A be a (not necessarily unital) chiral algebra and {Ms }, s ∈ S, a finite family of (possibly non-unital) chiral A-modules. Consider the chiral algebra A{Ms } := A ⊕ (⊕Ms [−1]) (see 3.3.5(i)). The homotheties of Ms define a GSm -action on A{Ms } . Therefore C(A{Ms } ) is a ZS -graded complex. Denote by C(A, {Ms }) its component of degree 1S . Set (4.2.19.1)

C ch (X, A, {Ms }) := RΓDR (R(X), C(A, {Ms }))

and Hach (X, A, {Ms }) = Hach (A, {Ms }) := H −a C ch (X, A, {Ms }); this is a chiral homology of A with coefficients in {Ms }. In other words, C ch (X, A, {Ms }) is the component of degree 1S of C ch (X, A{Ms } ). If S = ∅, we get the chiral homology of A. Our complex is equipped with the Cousin filtration which is the translation by |S| of the filtration induced by the Cousin filtration of C ch (X, A{Ms } ). 32 Notice

that this filtration is commutative (see 3.3.12).

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If all the Ms are equal to M , we write C ch (X, A, MS ) := C ch (X, A, {Ms }); if |S| = 1, we write simply C ch (X, A, M ). Let us describe C ch (X, A, {Ms }) a bit more explicitly. Denote by SS the category whose objects are non-empty finite sets equipped with an embedding S ,→ I; morphisms are surjections identical on S. We have an S◦S -diagram of closed embeddings X SS , I 7→ X I which carries a D-complex C 0 (A, {Ms }) defined as follows. The graded D-module C 0 (A, {Ms })·X I has an additional grading by the subset Q(I, S) ⊂ Q(I) that consists of all I  T in SS . For such T the correspond(I/T ) (T ) (T )∗ ing component is ∆∗ j∗ j ((Ms )  (A[1])T rS )[|T r S|]. The differential is the sum of two components: the first one comes from µA and µMs and the second one comes from the differentials on A and Ms . We have an obvious canonical identification RΓDR (X SS , C 0 (A, {Ms })) = C ch (X, A, {Ms }). We can also use the chiral chain complexes from 4.2.12. Namely, let us define C˜ ch (X, A, {Ms })Q , C ch (X, A, {Ms })PQ as the components of degree 1S of the complexes C˜ ch (X, A{Ms } )Q , C ch (X, A{Ms } )PQ . Then (4.2.12.1) and (4.2.12.2) yield canonical quasi-isomorphisms (4.2.19.2)

∼ ∼ C ch (X, A, {Ms })PQ −→ C ch (X, A, {Ms }) ← C˜ ch (X, A, {Ms })Q .

As a mere graded module, our C ch (X, A, {Ms })PQ is a direct sum of components (S+n) (S+n)∗ Cnch (X, A, {Ms })PQ := Γ(X S × X n , h(j∗ j ((MsPQ )  (APQ [1])n )))Σn , n ≥ 0, where MsPQ := Ms ⊗ P ⊗ Q. The chiral chain complexes are functorial with respect to chiral operations in the following sense. For S  T and a T -family of A-modules Nt each operation ϕ ∈ PAchS/T ({Ms }, {Nt }) := ⊗PAchSt ({Ms }, Nt ) (see 3.3.4) yields a morphism C ch (ϕ) : C ch (X, A, {Ms })PQ → C ch (X, A, {Nt })PQ ; one has C ch (ϕψ) = C ch (ϕ)C ch (ψ). The same is true for C˜ ch complexes. (ii) Consider the case when S 6= ∅ and each Ms is supported at a single closed point xs ∈ X. We can assume that these points are pairwise different (otherwise C ch (X, A, {Ms }) = 0). Let jS : US := X r {xs } ,→ X be the complement. Notice that in the definition of the chiral chain complex there occur now only affine varieties, so there is no need for using the Dolbeault DX -algebra Q. We also do not need to use P to compute the de Rham cohomology of Ms . Therefore we see that C ch (X, A, {Ms }) can be represented by a smaller complex C ch (X, A, {Ms })P with components (4.2.19.3)

(n)

Cnch (X, A, {Ms })P := (⊗h(Ms )) ⊗ Γ(US , h((AP [1])n ))Σn

(n)

where US is the complement to the diagonal divisor on (US )n , n ≥ 0. The differential comes from the chiral product on AP and the AP -module structure on (n) Ms 33 in the usual manner, using the fact that sections of h((AP [1])n ) over US (n) (n)∗ are the same as sections of h(j∗ j (jS∗ jS∗ AP [1])n ) over the whole of X n (see (4.2.12.4)). We can consider Ms as jS∗ jS∗ A-modules (see 3.6.3) and one has (4.2.19.4)

C ch (X, A, {Ms })P = C ch (X, jS∗ jS∗ A, {Ms })P .

Sometimes it is convenient to identify Ms with the corresponding Aas s -module h(Ms ) (see 3.6.7) and to write C ch (X, A, {h(Ms )}) := C ch (X, A, {Ms }), etc. 33 Defined

by the projection idA ⊗ P : AP → A.

4.3. THE BV STRUCTURE AND PRODUCTS

313

The complexes C ch (X, A, {Ms })PQ and C ch (X, A, {Ms })P are connected by evident natural quasi-isomorphisms (4.2.19.5)

C ch (X, A, {Ms })PQ → C ch (X, AQ , {MsQ })P ← C ch (X, A, {Ms })P .

Suppose in addition that A is a plain chiral algebra and the Ms are plain Amodules. The spectral sequence converging to Hnch (X, A, {Ms }) for the Cousin filtration is (4.2.19.6)

(p)

1 DR Ep,q = (⊗h(Ms )) ⊗ Hp+q (US , A)sgn Σp

where the right indices mean skew-coinvariants of the action of the symmetric (p) 1 group. Since US is affine, Ep,q vanishes unless p ≥ q ≥ 0. In particular, one has ch H<0 (X, A, {Ms }) = 0 and H0ch (X, A, {Ms }) is the space of the coinvariants of the Lie algebra Γ(US , h(A)) acting on ⊗h(Ms ). (iii) The above constructions make sense for families of A-modules. Namely, suppose Ms is a Ys -family of A-modules where Ys = Spec Rs ; i.e., Ms is an Rs ⊗ Amodule. Then C ch (X, A, {Ms })PQ is naturally an ⊗Rs -module, i.e., an O-module Q on Ys . This construction is compatible with the flat base change, so one can take for Ys any scheme (or an algebraic stack). If the Ms are DYs -modules (inQa way compatible with the A-action), then C ch (X, A, {Ms })PQ is a D-module on Ys . For example, for any A-module M the DX×X -module ∆∗ M can be considered as an X-family of A-modules where A acts along the second variable. Therefore the A-modules {Ms }, s ∈ S, yield a complex of DX S -modules (4.2.19.7)

Cch (X, A, {Ms })PQ := C ch (X, A, {∆∗ Ms })PQ . (I)

One has Cch (X, A, {Ms })PQ = j∗ j (I)∗ Cch (X, A, {Ms })PQ and C ch (X,A,{Ms })PQ = Γ(X S , hCch (X, A, {Ms })PQ ) = RΓDR (X S , Cch (X, A, {Ms })PQ ). If all Ms are equal to M , we write Cch (X, A, MS )PQ := Cch (X, A, {Ms })PQ . As in the end of (i), our complexes are functorial with respect to chiral operations: every ϕ as in loc. cit. yields a morphism of complexes of DX S -modules (4.2.19.8)

(S/T ) ch

Cch (ϕ) : Cch (X, A, {Ms })PQ X S → ∆∗

C (X, A, {Nt })PQ X T

and Cch (ϕψ) = Cch (ϕ)Cch (ψ) in the obvious sense. Remark. As in (ii), in the definition of the DX S -complexes Cch (X, A, {Ms }) there is no need to use Q (for S 6= ∅). 4.3. The BV structure and products The principal result of this section is theorem 4.3.6 which says that the chiral homology functor commutes with the tensor product. In the case of commutative algebras and 0th chiral homology this becomes an obvious statement (see 4.2.16 and (ii) in the lemma in 2.4.1): the functor which assigns to a DX -scheme the space of its horizontal sections commutes with the direct products. The key tool is a canonical Batalin-Vilkovisky algebra structure on the chiral complex C ch defined in 4.3.1. Its “classical” counterpart is an 1-Poisson algebra structure on C ch (R) for a coisson algebra R. If R is any commutative chiral algebra and A a chiral R-algebra, then C ch (X, R) is a homotopy commutative algebra and C ch (X, A) is a homotopy C ch (X, R)-module (see 4.3.2 and 4.3.4). In 4.3.3 we show that the higher chiral homology of the unit chiral algebra ω is trivial by an adaptation of

314

4.

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the topological argument of the proposition in 3.4.1. The compatibility with tensor products is proven in 4.3.6 by a closely related argument. After the homotopy algebra preliminaries of 4.3.7 and 4.3.8 we show in 4.3.9 that the chiral homology of chiral R-algebras is compatible with the base change of R. This implies, in particular, that chiral homology is compatible with relative tensor products and direct products (see 4.3.10 and 4.3.11). It would be very interesting to understand if the chiral homology of a chiral R-algebra has local origin with respect to Spec hRi (see 4.3.13 for a more general question); a weaker result is established in 4.3.12. In 4.3.3–4.3.13 all chiral algebras are assumed to be unital. 4.3.1. The BV structure. For a review of basic facts on homotopy BatalinVilkovisky algebras see 4.1.6–4.1.15. Proposition. (i) For a (not necessary unital) chiral algebra A the chiral chain complex C ch (X, A)PQ from 4.2.12 is naturally a BV algebra. (ii) Suppose that A is commutative. Then the above BV structure is commutative; i.e., C ch (X, A)PQ is a DG commutative algebra. Therefore C ch (X, A) has a canonical structure of the homotopy commutative algebra. (iii) A coisson bracket on A yields a 1-Poisson bracket (see 1.4.18) on the chiral chain complex. If At is a quantization of the coisson structure (see 3.3.11), then the BV algebras C ch (X, At )PQ form a BV quantization of the 1-Poisson algebra C ch (X, A)PQ . (iv) More generally, an n-coisson bracket on A (see 1.4.18) yields an n + 1Poisson bracket on the chiral chain complex. Proof. (i) Recall that the Chevalley-Cousin complex C(A) is the Chevalley (S) complex of the Lie algebra ∆∗ A in the tensor DG category CM(X S )ch (see 3.4.11). Thus (see 4.1.6) C(A) carries a canonical structure of the BV algebra with respect to ⊗ch .34 The same is true for C(APQ ). By (3.4.10.4) these are automatically BV algebras with respect to ⊗∗ . We are done by (4.2.8.7). (ii) is clear. The 1-Poisson structure on C ch (X, A)PQ for coisson A comes from a natural 1-Poisson structure on the commutative algebra C(A) in the tensor category CM(X S )∗ ; the definition is left to the reader. The n-coisson brackets are treated similarly.  ch Remarks. (i) Consider the embedding Γ(X, h(ALie PQ )) = C1 (X,A)PQ [−1] ,→ C (X, A)PQ [−1]. This is a morphism of Lie algebras, so it extends naturally to a ch ¯ morphism C(Γ(X, h(ALie PQ ))) → C (X, A)PQ of BV algebras (see 4.1.8(a)). We get a canonical morphism of filtered complexes35 ch

(4.3.1.1)

Lie ¯ C(RΓ )) → C ch (X, A). DR (X, A

Explicitly, on the nth component of the Chevalley complex it is the composition ∼ Symn (Γ(X, h(APQ ))[1]) → Γ(X n , h((APQ [1])n ))Σn → Γ(U (n) , h((APQ [1])n ))Σn . (ii) The above proposition (and its proof with (4.2.8.7) replaced by (4.2.8.4)) remains true for the chiral chain complex C˜ ch (X, A)Q , and quasi-isomorphisms (4.2.12.6) are compatible with the BV structure. ch (iii) The chiral chain complex Clog (X, A)Q from 4.2.14 is not a BV algebra for general A. However it is a commutative BV algebra if A is commutative. Indeed, 34 See 35 See

Remark in 3.4.11 for an explicit description of the BV operations. 4.5.1 for the discussion of the homotopy Lie algebra structure on RΓDR (X, ALie ).

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315

by (4.2.8.1) the complex DR(C(AQ )) is always a BV algebra in CSh! (X S )∗ . If A is commutative then DRC log (AQ ) is a subalgebra of DR(C(AQ )) (which is a commutative BV algebra). Now apply (4.2.5.2). Quasi-isomorphisms (4.2.14.4) are morphisms of commutative BV algebras. 4.3.2. Further remarks. (i) For a plain commutative unital chiral algebra R ∼ the isomorphism H0ch (X, R) −→ hRi (see 4.2.16) identifies the product on H0ch (R) with the product on hRi from 2.4.1. More generally, for any commutative R the morphism φR : C ch (X, R) → hRi from (4.2.17.1) is naturally a morphism in the homotopy category of commutative algebras. (ii) Let R → A be a central morphism of chiral algebras, so A is a chiral Ralgebra. Then, in the same way as above, C ch (X, A)PQ is a C ch (X, R)PQ -module. In fact, it is a BV C ch (X, R)PQ -algebra. Thus C ch (X, A) carries a canonical structure of the homotopy C ch (X, R)-module; i.e., it lifts canonically to an object of the homotopy category of C ch (X, R)-modules. (iii) If A is a chiral algebra and {Ms } a finite family of A-modules, then the complex C ch (X, A, {Ms })PQ is a BV C ch (X, A)PQ -module.36 Here are a couple of ways to use this structure: (a) Suppose Ns ⊂ Ms are DX -submodules such that Cent(Ns ) = A (see 3.3.7). The image of Γ(U (S) , h(NsPQ )) → Γ(U (S) , h(MsPQ )) ⊂ C ch (X, A, {Ms })PQ consists of C ch (X, A)PQ -central elements, so we have a morphism of complexes · : C ch (X, A)PQ ⊗ Γ(U (S) , h(NsPQ )) → C ch (X, A, {Ms })PQ , hence a canonical morphism C ch (X, A) ⊗ RΓDR (U (S) , Ns ) → C ch (X, A, {Ms }). (b) Suppose we are in situation (ii) and each Ms is a central R-module (see 3.3.7). Then C ch (X, A, {Ms })PQ is a C ch (X, R)PQ -module; hence C ch (X, A, {Ms }) is naturally a homotopy C ch (X, R)-module. (iv) Suppose A is any (not necessary unital) chiral algebra equipped with a commutative filtration A0 ⊂ A1 ⊂ · · · (see 3.3.12). Then the corresponding filtration on C ch (X, A)PQ (see 4.2.18) is compatible with the BV structure, so C ch (X, A) is canonically an object of the homotopy category of filtered BV algebras HoBV (see 4.1.6) which we denote by C ch (X, A· ) by abuse of notation. (v) Let R be a commutative chiral algebra, L a Lie∗ R-algebroid. Then the ch C (X, R)PQ -module C ch (X, R, L)PQ is naturally a Lie C ch (X, R)PQ -algebroid.37 Indeed, SymR L is a coisson algebra (see 1.4.18, Example (ii)); its bracket shifts the Z-grading by −1. So, by (iii) in the proposition in 4.3.1, C ch (X, SymR L)PQ is a 1Poisson algebra. Now C ch (X, R, L)PQ is the component of C ch (X, SymR L)PQ [−1] of degree 1.38 The Lie algebroid structure is the restriction of the 1-Poisson structure. ch 4.3.3. Proposition. (i) For the unit chiral algebra ω one has H6= 0 (X, ω) = 0, ∼ ch so there is a canonical identification C (X, ω) −→ hωi = k. (ii) For any unital chiral algebra A the multiplication by the generator 1ch ∈ ch H0 (X, ω) is the identity endomorphism of C ch (X, A).

Remark. Statement (i) follows from the contractibility of R(X) in the classical topology (see the proposition in 3.4.1) combined with 4.2.4 and the usual 36 This follows from (i) in the proposition in 4.3.1 and the definition of C ch (X, A, {M }) s PQ in 4.2.19. 37 In the non-unital setting. 38 See 4.6.4 for a description of the whole C ch (X, Sym L) R PQ .

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comparison between the de Rham and topological homology. The argument below is an algebraic version of the topological proof of the proposition in 3.4.1. For the chiral homology of ω with arbitrary coefficients see (4.4.7.2). Proof. Statement (ii) requires an extra construction to be introduced in the beginning of the proof of theorem 4.3.6; it will be proven in part (iii) of loc. cit. We deal now with statement (i). Let tr be the projection C ch (X, ω) → H0ch (X, ω) = k and tr1 , tr2 : C ch (X, ω)⊗2 → C ch (X, ω) the maps tr1 (a ⊗ b) := tr(b)a, tr2 (a ⊗ b) := tr(a)b. We will define a morphism δ : C ch (X, ω) → C ch (X, ω)⊗2 such that ·δ, tr1 δ, and tr2 δ are the identity morphisms of C ch (X, ω). This yields the vanishing. Indeed, suppose Hnch (X, ω) 6= 0 for some n > 0. Take the smallest such n; then one has H −n C ch (X, ω)⊗2 = H0ch (X, ω) ⊗ Hnch (X, ω) ⊕ Hnch (X, ω) ⊗ H0ch (X, ω). For h 6= 0 ∈ Hnch (X, ω) write δ(h) = h1 ⊗ 1ch + 1ch ⊗ h2 . Then hi = tri δ(h) = h; hence h = ·δ(h) = 2(1ch · h). Since (1ch )2 = 1ch , we come to a contradiction. To define δ, it is convenient to represent C ch (X, ω) not by complexes from 4.2.12, but by means of the Cousin resolution, i.e., as the homotopy direct limit of the S◦ -diagram of complexes I 7→ CI := Γ(X I , DR(CX I )) where CX I is the (whole) Cousin resolution of ωX I [|I|] = (ω[1])I . So CI has degrees in the interval [−2|I|, 0] and we have a canonical trace map trC : CI → k. Now C ch (X, ω)⊗2 can be represented as the homotopy direct limit of the S◦ × S◦ -diagram I, J 7→ CI ⊗ CJ . Consider another such diagram I, J 7→ CI,J := Γ(X I × X J , DR(CX I ×X J )). There is an obvious quasi-isomorphic embedding of the diagrams CI ⊗ CJ ,→ CI,J . The map trC1 := idCI ⊗ trC : CI ⊗ CJ → CI extends in the usual way to the morphism of diagrams trC1 : CI,J → CI and the same for trC2 . Similarly, the exterior tensor ∼ product map ◦C : CI ⊗ CJ → CItJ extends to ◦C : CI,J −→ CItJ . Let δC : CI → CI,I be the morphism of Cousin complexes defined by the diagonal embedding X I ,→ X I ×X I . Let us represent C ch (X, ω)⊗2 as the homotopy direct limit of the diagram CI,J . Our δ is the morphism defined by δC . It remains to check that the compositions tri δ and ·δ are identity morphisms. Notice that · and tri come from the morphisms of diagrams ◦C , trCi . Since trCi δC is the identity map, the morphism tri δ is the identity. The morphism of diagrams ◦C δC is not the identity, but it becomes canonically homotopic to the identity after passing to the homotopy limit (by the definition of the homotopy direct limit).  4.3.4. The unital setting. From now until the end of 4.3 all chiral algebras are assumed to be unital. Suppose that R is a commutative chiral algebra. Then, by 4.3.3, the morphism 1R : C ch (X, ω)PQ → C ch (X, R)PQ is a homotopy unit in C ch (X, R)PQ (see 4.1.14). Therefore, by the proposition in 4.1.14, C ch (X, R) lifts canonicaly to an object of the homotopy category of unital commutative algebras, which we denote again by C ch (X, R) by abuse of notation. If A is a chiral R-algebra, then C ch (X, A)PQ is a homotopy unital C ch (X, R)module (see 4.1.14). Thus C ch (X, A) lives naturally in the derived category of the unital C ch (X, R)-modules. If {Ms } is a finite family of unital A-modules such that each Ms is a central R-module, then C ch (X, A, {Ms }) is a unital C ch (X, R)-module. ch ch In particular, if 1ch R ∈ H0 (X, R) vanishes, then C (X, A) = 0. An example:

4.3. THE BV STRUCTURE AND PRODUCTS

317

Lemma. Let jU : U ,→ X be an open subset of X, U 6= X. Then for any chiral algebra A one has C ch (X, jU ∗ jU∗ A) = 0. Proof. Notice that jU ∗ jU∗ A is a chiral jU ∗ OU -algebra, and H0ch (X, jU ∗ OU ) = hjU ∗ OU i = 0.  Remarks. (i) For non-commutative unital A the vanishing of 1ch ∈ H0ch (X, A) need not imply that C ch (X, A) = 0. (ii) In the above lemma the condition that A is unital is essential: for example, ·+1 if µA = 0, then H·ch (X, jU ∗ jU∗ A) ⊃ HDR (U, A). If a chiral algebra A is equipped with a unital commutative filtration, then C ch (X, A)PQ is a filtered BV algebra (see 4.3.2), and the map 1A : C ch (X, ω)PQ → C ch (X, A)PQ is a homotopy unit. Thus C ch (X, A) lifts canonically to an object of HoBVu (see 4.1.6 and the proposition in 4.1.15) which we denote by C ch (X, A· ) by abuse of notation. 4.3.5. Suppose we have a finite family of chiral algebras {Ai }i∈I ; set A := ⊗Ai . For every i ∈ I the canonical morphism νi : Ai → A (see 3.4.15) yields a morphism of complexes C ch (X, Ai ) → C ch (X, A). One has a natural morphism of complexes ◦I : ⊗ C ch (X, Ai ) → C ch (X, A), ⊗ai 7→ ·I (⊗νi (ai )), where ·I ∈ BVI is the I-fold product.39 It is a morphism of BV algebras. It is clear that ◦I define an extension of C ch to a DG pseudo-tensor functor C ch (X, ·) : CA(X)⊗ → BV⊗ . To get interesting objects, we should infuse P ⊗ Q as was done in 4.2.12. Replacing νi by νiPQ := ⊗idP⊗Q : AiPQ → APQ , we get ◦I : ⊗ C ch (X, Ai )PQ → C ch (X, A)PQ .

(4.3.5.1)

These morphisms define a pseudo-tensor extension of our functor C ch (X, ·)PQ : CA(X)⊗ → BV⊗ .

(4.3.5.2)

Passing to homotopy categories, we get canonical morphisms ◦I : ⊗C ch (X, Ai ) → C ch (X, ⊗Ai )

(4.3.5.3)

which define a pseudo-tensor extension C ch : HoCA(X)⊗ → D(k)⊗ .

(4.3.5.4)

If we play with the tensor homotopy category of chiral algebras equipped with commutative (unital) filtrations, then (4.3.5.2) defines a pseudo-tensor extension of the functor (A, A· ) 7→ C ch (X, A· ) with values in HoBV⊗ u (see 4.3.4). 4.3.6. Theorem. If the Ai are pairwise homotopically OX -Tor-independent,40 then the canonical morphism (4.3.5.3) is a quasi-isomorphism. Together with 4.3.3, this shows that C ch is a unital tensor functor on the tensor category of homotopically OX -flat chiral algebras. 39 ◦

I

commutes with the differential since {νi (ai ), νj (aj )} = 0 for every i 6= j.

40 I.e.,

L

∼

Ai ⊗ Aj −→ Ai ⊗ Aj for every i 6= j ∈ I.

318

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Remark. Suppose that the Ai carry filtrations as in 4.2.18, and let us equip ⊗Ai with the tensor product of this filtrations. Then (4.3.5.1) is a morphism of filtered complexes which is a filtered quasi-isomorphism if the gr Ai are pairwise homotopically OX -Tor-independent.41 Proof. It suffices to consider the case of two algebras A, B. Set C ch (A) := C ch (X, A)PQ ; define C ch (B), C ch (A ⊗ B) similarly. We want to show that the morphism ◦ = ◦A,B : C ch (A) ⊗ C ch (B) → C ch (A ⊗ B) of (4.3.5.1) is a quasi-isomorphism. To do this, we will construct a certain diagram (4.3.6.1)

i

˜ ◦

C ch (A) ⊗ C ch (B) → C ch (A, B)  C ch (A ⊗ B) δ

such that i is a quasi-isomorphism, ◦ = ˜ ◦i, ˜◦δ is the identity map for C ch (A ⊗ B), and δ◦ is a quasi-isomorphism. This will clearly do the job. (i) For I, J ∈ S let C(A, B)X I ×X J be the Cousin complex of C(A)X I  C(B)X J with respect to the diagonal stratification of X ItJ . It looks as follows (cf. 3.4.11). As a mere graded D-module, C(A, B)X I ×X J is a direct sum of components labeled (ItJ/T ) (T ) (T )∗ by T ∈ Q(I t J). The T -component is equal to ∆∗ j∗ j  Ft where t∈T

Ft ∈ CM(X) is A[1] if t ∈ / πT (J), B[1] if t ∈ / πT (I), or (A ⊗ B)[1] otherwise. The differential comes from the chiral products of A, B, A ⊗ B, the chiral pairing ∈ P2ch ({A, B}, A ⊗ B), and differentials of A, B, A ⊗ B in the usual way (see 3.4.11). We have the obvious morphisms of D-complexes C(A)X I  C(B)X J → C(A, B)X I ×X J → C(A ⊗ B)X ItJ . The left arrow is a quasi-isomorphism since A, B are homotopically OX -Tor-independent. Our C(A, B)X I ×X J form a D-complex on the S◦ × S◦ -diagram X S × X S in the obvious way (see 4.2.1 and 4.2.6), and the above arrows are morphisms of such complexes. To compute the cohomology, we modify C(A, B)X I ×X J replacing it by a quasiisomorphic complex C(A, B, PQ)X I ×X J which is again a direct sum of Q(I t J) (ItJ/T ) (T ) (T )∗ components where the T -component is ∆∗ j∗ j  (Ft ⊗ P ⊗ Q), and t∈T

the differential is defined in the obvious way. We have the similar morphisms C(APQ )X I  C(BPQ )X J → C(A, B, PQ)X I ×X J → C((A ⊗ B)PQ )X ItJ where the left arrow is a quasi-isomorphism. (ii) We define C ch (A, B) as the naive direct limit of the S◦ × S◦ -diagram of complexes Γ(X I × X J , h(C(A, B, PQ)X I ×X J )). The above arrows yield morphisms of complexes i = iA,B : C ch (A)⊗C ch (B) → C ch (A, B) and ˜◦ = ˜◦A,B : C ch (A, B) → C ch (A ⊗ B) of (4.3.6.1). It is clear that ˜ ◦i = ◦. As a mere graded vector space our C ch (A, B) decomposes into a direct sum ch of subspaces Cm,n (A, B) := Γ(X m × X n , h(C(A, B, PQ)X m ×X n ))Σm ×Σn . The differential is compatible with the corresponding bifiltration. As in 4.2.12, one shows ∼ · that HDR (X I ×X J , C(A, B, PQ)X I ×X J ) −→ H · Γ(X I ×X J , h(C(A, B, PQ)X I ×X J )). Therefore i is a (bifiltered) quasi-isomorphism. The obvious embeddings ∆∗ C((A ⊗ B)PQ )X I ,→ C(A, B, PQ)X I ×X I , where ∆ : X I → X I × X I is the diagonal, define a morphism δ = δA,B : C ch (A ⊗ B) → C ch (A, B) of (4.3.6.1). It is clear that ˜ ◦ is left inverse to δ. To finish the proof, it remains to check that δ◦ is a quasi-isomorphism. 41 This

follows from 4.3.6 and (4.2.18.1) since gr· (⊗Ai ) = ⊗gr· Ai .

4.3. THE BV STRUCTURE AND PRODUCTS

319

(iii) Let us prove first the statement (ii) in the proposition in 4.3.3. Consider the above picture for B = ω. By (i) in the proposition in 4.3.3, we have a canonical ∼ identification C ch (X, A) −→ C ch (X, A) ⊗ C ch (X, ω). Its composition with ◦ is the ch multiplication by 1 map. Since δ is a right inverse to ◦, the multiplication by 1ch admits a right inverse. Since (1ch )2 = 1ch , the multiplication by 1ch is an idempotent. Thus it is the identity map; q.e.d. Therefore we know that ◦A,ω and δA,ω are mutually inverse quasi-isomorphisms, as well as ◦ω,B and δω,B . (iv) Let us return to the general situation. Consider a natural morphism of complexes κ : C ch (A, ω) ⊗ C ch (ω, B) → C ch (A, B) defined by the maps C(A, ω, PQ)X I ×X J  C(ω, B, PQ)X I 0 ×X J 0 → C(A, B, PQ)X ItI 0 ×X JtJ 0 which are the composition of the exterior product maps with the morphisms 1A : ω → A along the I 0 -variables and 1B : ω → B along the J-variables. Our κ is a quasi-isomorphism. To see this, it suffices to check that the composition κ(iA,ω ⊗ iω,B ) : C ch (A) ⊗ C ch (ω) ⊗ C ch (ω) ⊗ C ch (B) → C ch (A, B) is a quasi-isomorphism. The latter map is equal to the composition iA,B (◦A,ω ⊗ ◦ω,B )σ, where σ is the transposition of the middle multiples C ch (ω), and we are done. One checks immediately that the composition δA,B ◦A,B : C ch (A) ⊗ C ch (B) → ch C (A, B) is equal to κ(δA,ω ⊗ δω,B ). Thus it is a quasi-isomorphism; q.e.d.  4.3.7. Resolutions of commutative DX -algebras. In this section we deal with commutative unital DG DX -algebras and call them simply DX -algebras. If X is affine, then DX -algebras form naturally a closed model category (with quasiisomorphisms as weak equivalences and surjective morphisms as fibrations; arguments of [H] work in this situation). When X is proper, this is no longer true literally. We will not use seriously the formalism of closed model categories, but just some constructions that we are going to recall now. Let ϕ : R → F be a morphism of DX -algebras. We say that ϕ (or F ) is homotopically R-flat if F is homotopically R-flat as a DG R-module.42 It is elementary if one can find a Z-graded DX -submodule V ⊂ F such that V is a locally projective ∼ DX -module, R ⊗ SymV −→ F , and dF (V ) ⊂ R. Finally, S F is R-semi-free if one can find a sequence of R[DX ]-subalgebras F0 ⊂ F1 ⊂ · · · , Fi = F , such that R → F0 and all Fi → Fi+1 are elementary morphisms. If F is R-semi-free, then it is homotopically R-flat. For any ϕ its resolution is a morphism of R[DX ]-algebras G → F which is a quasi-isomorphism. A resolution is homotopically R-flat or R-semi-free if G is. Resolutions of ϕ form a category in the obvious way. Lemma. (i) Any ϕ admits an R-semi-free resolution. (ii) The groupoid obtained from the category of homotopically R-flat resolutions by localization is contractible. Proof. (i) One constructs an R-semi-free resolution ψ : G → F as follows. Notation: let G0 ⊂ G1 ⊂ · · · be a sequence of subalgebras of G as above, Vi ⊂ Gi the corresponding Z-graded DX -submodules, ψi := ψ|Vi , and di := dG |Vi : Vi· → G·+1 i−1 . We will define (Vi , di , ψi ) by induction by i. Then Gi equals Gi−1 ⊗ Sym(Vi ) as a Z-graded R[DX ]-algebra, its differential dGi is determined by di and dGi−1 , and ψi together with ψ|Gi−1 determines ψ|Gi (for i = 0 it is ψ0 and ϕ). 42 I.e.,

for every acyclic DG R-module N the complex N ⊗ F is acyclic; see [Sp]. R

320

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First we choose a locally projective Z-graded DX -module V0 = V0· and a morphism of Z-graded DX -modules ψ0 : V0 → F such that dF ψ0 = 0 and the corresponding morphism ψ¯0 : V0· → H · (F ) is surjective. Set G0 := R ⊗ Sym(V0 ) (we consider V0 as a complex with zero differential). Now suppose we have defined (Gi , ψ|Gi ), i ≥ 0. Then ψ|Gi is surjective on the cohomology (since ψ0 is). Choose a locally projective Z-graded DX -module Vi+1 · · · · and morphisms of DX -modules d·i+1 : Vi+1 → G·+1 i , ψi+1 : Vi+1 → F such that · a · ¯ dGi di+1 = 0, dF ψi+1 = ψ|Gi di+1 , and the maps di+1 : Vi+1 → Ker(H ·+1 Gi → H ·+1 F ) are surjective. Remarks. (a) A variant of the induction procedure in the proof of the first part of the lemma establishes the following fact: Suppose we have R-algebras M , N such that for every a the maps H a R → H a M, H a N are surjective and have equal kernels. Then M , N admit a simultaneous R-semi-free resolution; i.e., there exists an R-semi-free algebra L together with morphisms of R-algebras L → M, N which are quasi-isomorphisms. (b) As is clear from the proof, an R-semi-free resolution G of ϕ can be chosen so that the Vi are isomorphic to a direct sum of DX -modules of type LD = L ⊗ DX where L is a line bundle on X of degree bounded from above by any constant.43 In particular, we can choose it so that Γ(X, h(Vi )) = 0. We can also assume that d(V0 ) = 0. (ii) Notice that for every finite family of homotopically R-flat resolutions {Gα } of F one can find another homotopically R-flat resolution K together with morphisms Gα → K. Namely, consider F as a ⊗ Gα -algebra, and take for K a homoR

topically ⊗ Gα -flat resolution of F . R

To finish the proof, it suffices to prove that for every morphisms ζ, ζ 0 : M → N of homotopically R-flat resolutions the corresponding arrows in the groupoid coincide.44 To do this, it suffices to find another homotopically R-flat resolution L and morphisms η : L → M , ξ, ξ 0 : M → L, χ : L → N such that ηξ = ηξ 0 = idM , χξ = ζ, χξ 0 = ζ 0 . Consider M , N as M ⊗ M -algebras via the morphisms m ⊗ m0 7→ R

mm0 and m ⊗ m0 7→ ζ(m)ζ 0 (m0 ). Our L is the corresponding simultaneous M ⊗ M R

semi-free resolution (see Remark (a) above), η, χ are the corresponding quasiisomorphisms, and ξ, ξ 0 come from the maps M → L ⊗ M , m 7→ m ⊗ 1, 1 ⊗ m. R

 Exercise. Show that statement (ii) of the lemma remains valid if we replace “homotopically R-flat” by “R-semi-free.” 4.3.8. Base change. Let ϕ : R → F be a morphism of commutative chiral algebras. It yields the base change functor ϕ∗ : CA(X, R) → CA(X, F ), A 7→ ϕ∗ A = A ⊗ F . If F is homotopically R-flat, then our functor preserves quasi-isomorphisms R

and hence defines a functor between the homotopy categories (4.3.8.1)

ϕ∗ : HoCA(X, R) → HoCA(X, F ).

suffices to take L equal to tensor powers of a given negative line bundle. show that this fact implies the lemma, one repeats the argument of part (iii) of the proof of the second lemma in 4.1.3. 43 It

44 To

4.3. THE BV STRUCTURE AND PRODUCTS

321

To treat a non-flat ϕ, we have to change our homotopy categories. Anyway, the homotopy category HoCA(X, R) is not a right object for it may change if we replace R by a quasi-isomorphic algebra. To dispatch this nuisance, one considers the category CA(X, R)˜of pairs (A, R0 ) = (A, R0 , θ) where θ : R0 → R is a morphism of commutative DX -algebras which is a quasi-isomorphism, A a chiral R0 -algebra. Its localization with respect to quasi-isomorphisms is denoted by HoCA(X, R)˜. There is an obvious functor HoCA(X, R) → HoCA(X, R)˜. Now any morphism of commutative DX -algebras ϕ : R → F yields a functor Lϕ∗ : HoCA(X, R)˜→ HoCA(X, F )˜.

(4.3.8.2)

Namely, Lϕ∗ sends (A, R0 ) to (A ⊗ G, G) where R0 → G → F is any homotopically R0

R0 -flat resolution of F . According to the lemma in 4.3.7, this is a well-defined object of HoCA(X, F )˜. The functors Lϕ∗ are compatible with the composition of the ϕ’s. For any (A, R0 ) as above, C ch (X, A) is a unital homotopy C ch (X, R0 )-module (see 4.3.2 and 4.3.4). Since θ : C ch (X, R0 ) → C ch (X, R) is a quasi-isomorphism, it identifies the corresponding derived categories of C ch (X, R)- and C ch (X, R0 )modules, and we can consider C ch (X, A) as a homotopy C ch (X, R)-module. Its C ch ϕ base change is a C ch (X, F )-module L(C ch ϕ)∗ C ch (X, A) := C ch (X, A)

(4.3.8.3)

L

⊗

C ch (X, F )

C ch (X,R)

(see e.g. [H] for details). There is a canonical base change morphism βϕ : L(C ch ϕ)∗ C ch (X, A) → C ch (X, Lϕ∗ A)

(4.3.8.4)

in the derived category of homotopy C ch (X, F )-modules. To define it, we can assume that F is homotopically R-flat. The morphism of chiral R-algebras A → A ⊗ F R

yields a morphism of homotopy C ch (X, R)-modules C ch (X, A) → C ch (X, A ⊗ F ), R

hence, by adjunction, a morphism L(C ch ϕ)∗ C ch (X, A) → C ch (X, A ⊗ F ) of homoR

topy C ch (X, F )-modules, which is our βϕ . 4.3.9. Theorem. The base change map (4.3.8.4) is a quasi-isomorphism. Proof. We can assume that R0 = R and, by (i) in the lemma in 4.3.7, that F is R-semi-free. Since chiral homology commutes with inductive limits, it suffices to consider the case of an elementary morphism ϕ.45 Let V ⊂ F be as in 4.3.7. We have a filtration R ⊗ Sym≤a V on F , so gr F equals R ⊗ Sym V as a DG DX -algebra (we consider V as a complex with zero differential). It defines filtrations on C ch (X, F ), hence on Lϕ∗ C ch (X, A) := C ch (X, A)

L

⊗ C ch (X,R)

C ch (X, F ) and on C ch (X, A ⊗ F ) (see 4.2.18). The base change R

morphism is compatible with filtrations,46 so it suffices to check that gr βϕ is a quasi-isomorphism. But, by (4.2.18.1), gr βϕ is the base change morphism for R → gr F = R ⊗ Sym V , so we have reduced our problem to the situation when V ⊂ F 45 For if we have two composable morphisms ϕ and our statement holds for each of them, then it holds for the composition. 46 Recall that this means that we have a canonical morphism in the filtered derived category.

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is killed by the differential. Then A ⊗ F = A ⊗ Sym V , so 4.3.6 provides canoniR

cal isomorphisms C ch (X, F ) = C ch (X, R) ⊗ C ch (X, Sym V ) and C ch (X, A ⊗ F ) = R

C ch (X, A) ⊗ C ch (X, Sym V ). They identify βϕ with the identity map for C ch (X, A) ⊗C ch (X, Sym V ), and we are done.  Here are some corollaries. Let R be a commutative chiral algebra, {Ai }i∈I a finite collection of chiral R-algebras. Suppose that R is homotopically OX -flat and that the Ai are pairwise Tor-OX -independent. Then ⊗Ai is a chiral R⊗I -algebra; L

set ⊗ Ai := Lδ ∗ (⊗Ai ) where δ : R⊗I → R is the product map. One has the R

following relative form of 4.3.6: 4.3.10. Corollary. There is a canonical isomorphism of the homotopy C ch (X, R)-modules (4.3.10.1)

L

⊗ C ch (X,R)

∼

L

C ch (X, Ai ) −→ C ch (X, ⊗ Ai ). R

Proof. Use 4.3.9 for δ and ⊗Ai together with 4.3.6.



4.3.11. Corollary. The chiral homology commutes with direct products: for any finite collection of chiral algebras Ai the projection morphisms yield an isomorphism (4.3.11.1)

∼

C ch (X, ΠAi ) −→ ΠC ch (X, Ai ).

The inverse map comes from the obvious (non-unital) morphisms Aj → ΠAi . Remark. The assumption that our chiral algebras are unital is essential here (consider the case of µAi = 0). Proof. The latter map is right inverse to (4.3.11.1), so it is enough to check that (4.3.11.1) is an isomorphism. It suffices to consider the case when all Ai equal OX : Indeed, for arbitrary Ai we can consider ΠAi as a chiral OIX -algebra. Then (4.3.11.1) follows from 4.3.9 for A = ΠAi , R = OIX , F = OX . Case Ai = OX : We know that H0ch (X, OIX ) = hOIX i = k I (see the end of 4.2.16). Applying 4.3.9 to R = OIX , A = F = OX , and R → A, F a projection map, we see that the higher chiral homology of OIX vanishes. Indeed, by 4.3.9 and 4.3.3, L

ch the first non-trivial Hach (X, R), a > 0, yields non-trivial Ha+1 (X, A ⊗ F ), which R

L

contradicts the vanishing of the higher chiral homology of A ⊗ F = OX . We are R

done.



4.3.12. Here is a more general statement. Let ϕ : R → F be an ´etale morphism of plain commutative DX -algebras, A any chiral R-algebra. Set AF := A ⊗ F . The chiral homologies Hach (X, A) are hRiR

modules, and the Hach (X, AF ) are hF i-modules (see 4.2.16), so we have a canonical morphism (4.3.12.1)

H·ch (X, A) ⊗ hF i → H·ch (X, AF ). hRi

4.4. CORRELATORS AND COINVARIANTS

323

In particular, for A = R we get a morphism H·ch (X, R) ⊗ hF i → H·ch (X, F ).

(4.3.12.2)

hRi

Proposition. The morphism hϕi : hRi → hF i is ´etale, and (4.3.12.1) and (4.3.12.2) are isomorphisms. Proof. (i) hϕi is ´etale: Set S := SpechRi, T := SpechF i, SX := S × X, etc. We have closed embeddings SX ,→ Spec R, TX ,→ Spec F . Consider ϕ|SX : Spec F |SX → SX ; then TX is the maximal constant closed DX -subscheme of Spec F |SX . Since ϕ|SX is ´etale, it is also an open subscheme, and we are done. (ii) By 4.3.9 it suffices to consider the case A = R, i.e., (4.3.12.2). (iii) Set C := C ch (X, R), D := C ch (X, F ). These are commutative unital ¯ := H0 D. We have a homotopy algebras having degrees ≤ 0. Set C¯ := H0 C, D morphism C → D. We want to show that the corresponding morphism H˜· C := ¯ → H· D is an isomorphism. (H· C) ⊗ D ¯ C

L

Since F is R-flat, 4.3.10 implies that D ⊗ D = C ch (X, F ⊗ F ). Since F/R is C

R

´etale, one has a DX -algebra decomposition F ⊗ F = F × Q where the projection R

¯ := H0 E. By 4.3.11 one F ⊗ F → F is the product map. Set E := C ch (X, Q), E R

L

L

C

C

has D ⊗ D = D × E where the projection D ⊗ D → D is the product map. L

One has a spectral sequence converging to H· (D ⊗ D) with the second term C

equal to T orpH· C (H· D, H· D)q . The above decomposition then gives a spectral se2 quence converging to H· D with Ep,q = T orpH˜· C (H· D, H· D)q . Notice that the map 2 E0,· → H· D is just the product map H · D ⊗ H· D → H· D and hence it is surjecH˜· C

∞ tive, i.e., E>0,q = 0. Suppose that H˜a C 6= Ha D for some a ≥ 1; consider the first such a. We have 2 2 = 0 for p ≥ 1 and q ≤ 2a − 1. This implies E0,a = Ha D. On the other hand, Ep,q 2 E0,a = (H· D ⊗ H· D)a which is the cokernel of the diagonal map H˜a C → Ha D ⊕ H˜· C

2 Ha D. Thus H˜a C → Ha D is surjective. We have E1,2a = Ker(H˜a C → Ha D). 2 ∞ Since E1,2a = E1,2a = 0, we arrive at a contradiction. 

4.3.13. Questions. Let R be a plain commutative DX -algebra. Is it true that for any chiral R-algebra A its chiral homology has a local origin with respect to the Zariski or ´etale topology of SpechRi? More generally, is this true if A is a chiral RDif -algebra (see 3.9.4)? Can one define the chiral homology for chiral algebras on any algebraic DX -space Y or on YDif (see loc. cit.)? 4.4. Correlators and coinvariants We begin with the definition of correlator functions for a plain chiral algebra A; these functions (and the differential equations they satisfy) are of primary interest for mathematical physicists (see, e.g., [BPZ]). In general, the “correlator-style” approach to chiral homology stems from the following observation: for any finite subset {xs } ⊂ X the complement to the subspace R(X)(xs ) ⊂ R(X) whose points are finite subsets containing {xs }, is acyclic (see 4.4.2). It permits us to identify

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the “absolute” chiral homology of A with the chiral homology of A with coefficients in fibers A`xs (see 4.4.3). In particular, hAi identifies with coinvariants of the action of the Lie algebra h(US , A) on ⊗A`xs where US := X r {xs } (see 4.4.4), so hAi is dual to the “space of conformal blocks” (see [FBZ] 8.2). To compute the coinvariants, it suffices to consider instead of the whole of h(US , A) the space h(US , P ) where P ⊂ A is any sub-D-module generating A (see 4.4.5).47 A relative version of this statement (when our points vary) is discussed in 4.4.6; we briefly mention the example of the Knizhnik-Zamolodchikov equation (see [KZ], [EFK], and Chapter 12 of [FBZ]). In 4.4.7 the chiral homology of the unit chiral algebra ω with arbitrary coefficients is computed. In 4.4.8 we show that the chiral homology functor commutes with adding of unit. In 4.4.9 a spectral sequence (similar to the Hochschild-Serre spectral sequence) for computing the chiral homology of a chiral algebra with a given subalgebra is constructed. Since the early days of conformal field theory, the geometry of R(X) was used in order to write down explicit integral formulas for some correlators (the Feigin-Fuchs integrals), see [DoF]. It is used in [BFS] to construct geometrically the category of representations of a quantum group. We do not touch these subjects. 4.4.1 Correlators. Let us return to 4.2.16, so A is a plain chiral algebra. We see that for every S ∈ S one has a canonical morphism of DX S -modules (4.4.1.1)

h iI : A`X S → hAi ⊗ OX S .

These morphisms are compatible with pull-backs to the diagonals, so they form a morphism h i : A`R(X) → hAi ⊗ OR(X) of the left D-modules on R(X) (see 3.4.2 for the terminology). So for a finite subset {xs } ⊂ X, s ∈ S, and a ∈ A`(xs ) = ⊗ A`xs s∈S

we have hai ∈ hAi. The compatibility with respect to the restriction to the diagonals implies that for every {xt } ⊂ X r {xs } one has ha ⊗ (⊗ 1xt )i = hai

(4.4.1.2)

where 1 is the unit section of A` . In particular, 1ch = h⊗1t i (see 4.2.16). Restricting h i to the complement R(X)on of the diagonal divisor on Symn X, we get the n-point correlator morphisms (4.4.1.3)

h in : (Symn A` )R(X)on → hAi ⊗ OR(X)on .

Exercise. Show that the following diagram commutes:

(4.4.1.4)

j∗ j ∗ A  A −−−−→ hAi ⊗ j∗ j ∗ ω  ω     y y ∆∗ A

−−−−→

hAi ⊗ ∆∗ ω

Here the horizontal arrows are the correlator morphisms and the vertical ones are the chiral products for A and ω. 47 A particular case of this statement when P is a Lie∗ subalgebra of A was considered in [FBZ] 8.3.

4.4. CORRELATORS AND COINVARIANTS

325

Remark. Suppose that the 2-point correlator pairing is non-degenerate; i.e., for every non-zero (local) section a(x) ∈ A` there exists another section b(y) such that ha(x)b(y)i = 6 0. Then the chiral algebra structure on the OX -module A is uniquely determined by the vector space hAi and the 2- and 3-point correlator maps. Indeed, we know that the chiral algebra structure is determined by the sub-DX×X ∼ module A`X×X ⊂ j∗ j ∗ AX  AX together with the identification ∆∗ AX×X −→ A`X . Now A`X×X consists of all sections a(x, y) ∈ j∗ j ∗ AX  AX such that for every section b(z) ∈ A` the correlator ha(x, y)b(z)i is regular at the divisor x = y. If a is such a section, then a(x, x) ∈ A` is determined by the condition ha(x, x)b(z)i = ha(x, y)b(z)ix=y for any b(z) ∈ A`X . 4.4.2. Let A be a unital chiral algebra and {xs } ⊂ X, s ∈ S, a finite nonempty subset. Consider the subspace R(X)(xs ) ,→ R(X) of those points tfor which the corresponding finite subsets of X contain {xs }. We will consider the complex C ch (X, A)(xs ) defined as the de Rham cohomology of R(X) with support in R(X)(xs ) and coefficients C(A). The construction of the cohomology with support was explained in (iii) and S (iv) in 4.2.6. Precisely, let X(x ,→ X S be the r-preimage of R(X)(xs ) ; this s) is a diagram of closed subvarieties of X S . By (iii) in 4.2.6 we have an admissible D-complex C(A)(xs ) := C(A)X S ∈ DM(X S )X S ⊂ DM(X S ) equipped (xs )

(xs )

with a canonical morphism C(A)(xs ) → C(A). Now one defines C ch (X, A)(xs ) as RΓDR (X S , C(A)(xs ) ) = RΓDR (X S , C(A))X S . (xs )

Proposition. The canonical morphism C(A)(xs ) → C(A) yields a quasi-isomorphism (4.4.2.1)

∼

C ch (X, A)(xs ) −→ C ch (X, A).

S Proof. Let jV(xs ) : V(xs ) ,→ X S be the complement of X(x , so we have the s) ∗ exact triangle C(A)(xs ) → C(A) → RjV(xs ) ∗ jV(x ) C(A). Let us show that the de s Rham cohomology of R(X) r R(X)(xs ) with coefficients in C(A) vanishes; i.e., the complex RΓDR (X S , RjV(xs ) ∗ jV∗(x ) C(A)) is acyclic. s Notice that R(X) r R(X)(xs ) = ∪ R(Us ) where Us ⊂ X is the complement to s∈S

xs . Any intersection of these subsets is the complement UT to a non-empty subset ˇ {xt }t∈T of X. We can write RjV(xs ) ∗ jV∗(x ) C(A) as the Cech resolution Cˇ which is the s total complex of a bicomplex whose nth column is isomorphic to ⊕ C(jT ∗ jT∗ A) |T |=n+1

where jT : UT ,→ X. By the lemma in 4.3.4 one has RΓDR (R(X), C(jT ∗ jT∗ A)) = C ch (X, jT ∗ jT∗ A) = 0; q.e.d.  4.4.3. Let us describe the complex C(A)(xs ) . Consider the embeddings is : {xs } ,→ X and js : Us := X r {xs } ,→ X. For any s ∈ S we have an A-module A˜s := Cone(A → js∗ js∗ A) acyclic off xs . The morphism of A-modules A˜s → Coker(A → js∗ js∗ A) = is∗ A`xs is a quasi-isomorphism when A is homotopically OX -flat at xs . Consider the complex C(A, {A˜s }) ∈ CM(X S ) (see 4.2.19). Proposition. There is a canonical quasi-isomorphism in DM(X S ) (4.4.3.1)

∼

C(A, {A˜s }) −→ C(A)(xs ) .

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Therefore, passing to de Rham cohomology and applying (4.4.2.1), we get a canonical quasi-isomorphism ∼ C ch (X, A, {A˜s }) −→ C ch (X, A).

(4.4.3.2)

If A is homotopically OX -flat at {xs }, it can be rewritten as ∼

C ch (X, A, {A`xs }) −→ C ch (X, A).

(4.4.3.3)

S ˜ Proof. We will define a complex C(A, {A˜s }) ∈ CM(X S ) acyclic off X(x and s) morphisms α

β

˜ C(A, {A˜s }) ← C(A, {A˜s }) − → C(A)

(4.4.3.4)

˜ such that α is a quasi-isomorphism and the morphism βS : C(A, {A˜s }) → C(A)(xs ) defined by β is also a quasi-isomorphism. This defines (4.4.3.1). Set P := ⊕ Cone(OX → js∗ OUs )[−1] ∈ CM` (X). The commutative DX s∈S

algebra Sym P is naturally ZS -graded, so the chiral algebra A⊗P is also ZS -graded. The projection P → OSX → OX , where the right arrow is (fs ) 7→ Σfs , yields a morphism of DX -algebras Sym P → OX , hence a morphism of chiral algebras β˜ : A ⊗ Sym P → A. We also have a morphism of chiral algebras α ˜ : A ⊗ Sym P → ˜ A ⊗ (Sym P/Sym≥2 P ) = A{As } compatible with the ZS -gradings (see 4.2.19 for notation). ˜ ˜ Our C(A, {A˜s }) is the component of degree 1S of C(A⊗Sym P ), α : C(A, {A˜s }) 48 ˜ ˜ ˜ → C(A, {As }) is the morphism defined by α ˜ , and β : C(A, {As }) → C(A) is the ˜ morphism defined by β. The projection Sym P → Sym P/Sym≥2 P induces a quasi-isomorphism be˜ tween the components of degree ≤ 1S . Therefore α ˜ : C(A ⊗ Sym P ) → C(A{As } ) S is a quasi-isomorphism on the components of degree ≤ 1 ; hence α is a quasiS ˜ isomorphism. Since P is acyclic off S, our C(A, {A˜s }) is acyclic off X(x . s) ˜ ˜ It remains to check that βS : C(A, {As }) → C(A)(x ) is a quasi-isomorphism. s

We are playing with admissible complexes, so it suffices to check this on U (I) ⊂ X I ˜ for any I ∈ S. There the complex C(A, {A˜s })U (I) is naturally ZS×I -graded. The S (I) intersection of X(x with U is a disjoint union of components ((xs )×X IrS )∩U (I) s) with respect to all embeddings S ,→ I. Our problem is local, so it suffices to check that βS is a quasi-isomorphism on the complement in U (I) to all the above ˜ components but one. Here each of the ZS×I -components of C(A, {A˜s })U (I) is acyclic IrS ˜ except (As )  (A[1]) . On the other hand, C(A)U (I) = (A[1])I , and the U (I) restriction of β to the above component is the morphism (A˜s )  (A[1])IrS → (A[1])I equal to the tensor product of the projections A˜s → A[1] and the identity ˜ map for (A[1])IrS . It evidently identifies C(A, {A˜s }) with C(A)S , and we are done.  Remarks. (i) The above subject generalizes in an evident manner to the case of the chiral homology with coefficients (see 4.2.19). We leave the exact formulation of the general statement to the reader. Let us consider a particular situation when 48 Recall

˜

˜s }) is the degree 1S component of C(A{As } ). (see 4.2.19) that C(A, {A

4.4. CORRELATORS AND COINVARIANTS

327

A is homotopically OX -flat at {xs }, and in addition we have {xt } ⊂ X r {xs } and A-modules Mt supported at xt . Then there is a canonical quasi-isomorphism (4.4.3.5)

∼

C ch (X, A, {A`xs , h(Mt )}) −→ C ch (X, A, {h(Mt )}). ∼

(ii) Suppose that A is a plain chiral algebra. Then the morphism C ch (X, A) −→ ∼ ch C (X, A, {A˜s }) → C ch (X, A, {A`xs }) always yields an isomorphism H0ch (X, A) −→ H0ch (X, A, {A`xs }) (regardless of whether A is OX -flat at {xs } or not). 4.4.4. H0ch as coinvariants. Let A be a not necessary unital plain chiral algebra. Suppose we have a finite non-empty subset {xs } ⊂ X and for each s ∈ S a plain A-module Ms supported at xs ; let jS : US ,→ X be the complement to ch {xs }. The spectral sequence (4.2.19.6) shows then that H<0 (X, A, {Ms }) = 0 and ch 1 0 H0 (X, A, {Ms }) = Coker(d1,0 : (⊗h(Ms )) ⊗ HDR (US , A) → ⊗h(Ms )). The latter differential can be described as follows. By 3.6.3 Ms is a jS∗ jS∗ A-module, so h(Ms ) is a module over the Lie algebra h(US , A) := Γ(US , h(ALie )). By the lemma in 2.1.7 0 (US , A) → h(US , A) is surjective, and d11,0 is the composition the projection HDR of this projection and the action of h(US , A) on the tensor product ⊗h(Ms ). Thus H0ch equals the coinvariants of this action: (4.4.4.1)

H0ch (X, A, {Ms }) = (⊗h(Ms ))h(US ,A) .

Remark. Suppose A is unital. Then the correlator map h i : ⊗A`xs → hAi from 4.4.1 coincides, via the identifications hAi = H0ch (X, {A`xs }) = (⊗A`xs )h(US ,A) (see Remark (ii) in 4.4.3) and (4.4.4.1), with the obvious projection ⊗A`xs → (⊗A`xs )h(US ,A) . 4.4.5. One usually computes H0ch (X, A, {Ms }) in a more practical way: Proposition. Let PUS ⊂ AUS be any D-submodule which generates AUS as a (non-unital) chiral algebra. Then the coinvariants for the actions of h(US , P ) := Γ(US , h(PUS )) and h(US , A) on ⊗h(Ms ) coincide. So, by (4.4.4.1), one has (4.4.5.1)

H0ch (X, A, {Ms }) = (⊗h(Ms ))h(US ,P ) .

In the unital setting it suffices to assume that P generates A as a unital chiral algebra. Proof. For a D-submodule QUS ⊂ AUS denote by I(Q) the image of the com· position h(US , Q) ⊗ (⊗h(Ms )) → h(US , A) ⊗ (⊗h(Ms )) − → ⊗h(Ms ) where · is the action map. We want to show that I(P ) = I(A). Adding a unit to A, we can assume to be in the unital setting (see 3.3.3, 3.3.4). Since h(US , ω · 1A ) acts trivially on ⊗h(Ms ), we can assume that 1A ∈ PU` S . (i) Suppose |S| = 1. Set QUS := µA (j∗ j ∗ PUS  PUS ). It suffices to show that I(Q) ⊂ I(P ). The maps Γ(US × US , j∗ j ∗ PUS  PUS ) → Γ(US × US , ∆∗ QUS ) → h(US , Q) are surjective (the first one because US is affine and the second one since, by the lemma in 2.1.7, such is its restriction to QUS ⊂ ∆∗ QUS ). So I(Q) is the image · of the composition ξ of Γ(US × US , j∗ j ∗ PUS  PUS ) ⊗ h(Ms ) → h(US , Q) ⊗ h(Ms ) − → h(Ms ) where the first arrow is µA ⊗ idh(Ms ) . By the Jacobi identity, one has ξ = ξ 0 −ξ 00 where ξ 0 , ξ 00 are the compositions Γ(US ×US , j∗ j ∗ PUS PUS )⊗h(Ms ) → · Γ(US , PUS ) ⊗ h(Ms ) − → h(Ms ) and the first arrow is the chiral jS∗ jS∗ A-action on

328

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Ms along the second, resp. first, variable. The images of both ξ 0 , ξ 00 are contained in I(P ), and we are done. (ii) It suffices to consider the case h(Ms ) = A`xs . Indeed, both functors of the coinvariants commute with the inductive limits of the Ms , so we can assume that the Ms are finitely generated. The statement depends only on the restriction of A to US , so, modifying our A at S if necessary, we can assume that each Ms is a quotient of a sum of several copies of is∗ A`xs (see 3.6.6). If proven for some modules, our statement is automatically true for each their quotients, and also it is compatible with direct sums. So we can assume that h(Ms ) = A`xs . Let us extend PUS to a D-submodule P ⊂ A. Replacing A by its chiral subalgebra generated by P , we can assume that P generates A. ∼ (iii) Suppose that |S| > 1. Let us show that (⊗A`xs )h(US ,P ) −→ (⊗A`xs )h(US ,A) . We use induction by |S|. Pick s0 ∈ S and set T := S r {s0 }; we denote the elements of T by t. Consider the map ⊗A`xt → ⊗A`xs , ⊗at 7→ 1s0 ⊗ (⊗at ). We know that H0ch (X, A) = H0ch (X, A, {A`xt }) = H0ch (X, A, {A`xs }) (see Remark (ii) ∼ in 4.4.3). So, by (4.4.4.1),49 our map induces an isomorphism (⊗A`xt )h(UT ,A) −→ ∼ (⊗A`xs )h(US ,A) . One has (⊗A`xt )h(UT ,P ) −→ (⊗A`xt )h(UT ,A) by the induction assumption. Thus it suffices to prove that (⊗A`xt )h(UT ,P ) → (⊗A`xs )h(US ,P ) is surjective. This follows immediately from the fact that the image of h(US , P ) → Aas xs0 generates 50 as  Axs0 as a topological associative algebra. 4.4.6. The material of 4.4.2–4.4.5 immediately generalizes to the situation where points xs vary, i.e., to the relative situation over U (S) . For example, the relative version of 4.4.5 looks as follows. Suppose A is a plain chiral algebra and P ⊂ A a DX -submodule that generates A as a chiral algebra. Let {Ms } be an S-family of A-modules, S ∈ S. We have \ \ the left DU (S) -modules PU\ (S) := j (S)∗ PX The latter S and AU (S) (see (3.7.6.1)). is a Lie algebra in the tensor category of left DU (S) -modules which acts naturally on j (S)∗  Ms . Consider the DU (S) -modules of coinvariants (j (S)∗  Ms )A\ and U (S)

(j (S)∗  Ms )P \

U (S)

:= Coker(PU\ (S) ⊗ j (S)∗  Ms → j (S)∗  Ms ). The relative version

of 4.4.4 and 4.4.5 says that (see (4.2.19.7) for the notation) (4.4.6.1)

(S)

H 0 Cch (X, A, {Ms }) = j∗ (j (S)∗  Ms )A\

U (S)

(S)

= j∗ (j (S)∗  Ms )P \

.

U (S)

Example. Let A be the twisted enveloping algebra U (gD )κ of the Kac-Moody extension gκD ; see 2.5.9. Then one can take P = gκD , and the DU (S) -module from (4.4.6.1) is called the Knizhnik-Zamolodchikov (KZ) equation. 4.4.7. Let A be a chiral algebra. Then for any DX -module M the tensor product M ⊗ A is naturally a chiral A-module (see Remark (i) in 3.3.4). Suppose we have a finite non-empty collection {Ms }, s ∈ S, of DX -modules such that the Ms are Tor-independent from A (i.e., the supports of the OX -torsion of Ms are disjoint from that of A). 49 See 50 To

A`xs

0n

also Remark in 4.4.4 and (4.4.1.2). see this, consider a filtration A`xs n := h(US , P )n · 1s0 on A`xs . Then the images of 0

⊗ (⊗A`xt ) form a constant filtration in (⊗A`xs )h(US ,P ) .

0

4.4. CORRELATORS AND COINVARIANTS

329

Proposition. There is a canonical quasi-isomorphism (4.4.7.1)

∼

(S)

C ch (X, A) ⊗ j∗ j (S)∗  Ms −→ Cch (X, A, {Ms ⊗ A}).

Proof. Notice that the image of the morphism M → M ⊗ A, m 7→ m ⊗ 1A , consists of A-central sections (see 3.3.7). Therefore, by (a) in 4.3.2(iii), we have the (S) product morphism · : C ch (X, A) ⊗ j∗ j (S)∗  Ms → Cch (X, A, {Ms ⊗ A}). To check that it is a quasi-isomorphism, we use the relative version of 4.4.3. When (xs ) ∈ U (S) varies, the complexes C ch (X, A, {A˜s }) form a complex C of left D-modules on U (S) , and the quasi-isomorphism (4.4.3.1) (coming from (4.4.3.4)) identifies it with the constant D-module C ch (X, A) ⊗ OU (S) . Tensoring our D∼ (S) (S) modules by (Ms )|U (S) , we get α : j∗ C ⊗ (Ms ) −→ C ch (X, A) ⊗ j∗ j (S)∗  Ms . ∼ ∼ (S) On the other hand, the projections A˜s −→ is∗ A`xs yield β : j∗ C ⊗ (Ms ) −→ ch −1 C (X, A, {Ms ⊗ A}). One checks immediately that · = βα , and we are done.  For A = ω the functor M(X) → M(X, ω), M 7→ M ⊗ ω, is an equivalence of categories (see Example in 3.3.4). Combining the proposition with 4.3.3(i), we get Corollary. For any finite collection {Ms }, s ∈ S, of DX -modules one has (4.4.7.2)

(S)

∼

j∗ j (S)∗  Ms −→ Cch (X, ω, {Ms }).

4.4.8. Proposition. Let A be a non-unital chiral algebra, A+ := A ⊕ ω the corresponding unital algebra (see 3.3.3). Then the embeddings A, ω ,→ A+ yield a quasi-isomorphism (4.4.8.1)

∼

C ch (X, A) ⊕ k −→ C ch (X, A+ ).

ch Proof. Notice that C ch (X, A+ )PQ = C ch (X, A)+ PQ ⊕ C (X, ω)PQ where the + ch I subcomplex C (X, A)PQ is generated by all chains f  ai ∈ Γ(U (I) , (A+ ), PQ ) + f ∈ OU (I) , ai ∈ APQ such that at least one of the ai ’s belongs to A ⊂ A+ . By 4.3.3(i) it suffices to show that C ch (X, A)PQ ,→ C ch (X, A)+ PQ is a quasi-isomorphism. Both complexes are filtered: the first one by the Cousin filtration and the second one by the number of ai ’s from A as above. The embedding is compatible with filtrations. It is a filtered quasi-isomorphism: indeed, grn C ch (X, A)PQ = ∼ Γ(U (n) , h((APQ [1])n ))Σn −→ RΓDR (U (n) , An )Σn , and also grn C ch (X, A)+ PQ = C ch (X, ω, {A}n copies )Σn , so we are done by (4.4.7.2). 

Remark. Suppose that A is equipped with a commutative filtration; it extends in the obvious manner to A+ . We have C ch (X, A· ) ∈ HoBV and C ch (X, A+ · ) ∈ HoBVu (see 4.3.2(iv) and 4.3.4). The above lemma (together with 4.1.7 and 4.1.15) ch shows that C ch (X, A+ · ) comes from C (X, A· ) by adding the unit. The same is true in the setting of commutative chiral algebras. 4.4.9. We return to the unital setting. For a chiral subalgebra B ⊂ A one defines the relative Chevalley-Cousin complex C(B, A)PQ ∈ CM(R(X)) as follows. Consider A as a B-module. For each I ∈ S we have a complex of DX I -modules Cch (X, B, A[1]I )PQ (see 4.2.19). As a mere graded DX I -module, our complex C(B, A)PQ X I equals (4.4.9.1)

C(B, A)·PQ X I :=

⊕ T ∈Q(I)

(I/T ) ch

∆∗

C (X, B, A[1]T )PQ .

330

4.

GLOBAL THEORY: CHIRAL HOMOLOGY (I/T )

Its differential is the sum of the differentials of ∆∗ Cch (X, B, A[1]T )PQ and the (I/T ) ch (I/S) ch morphisms ∆∗ C (X, B, A[1]T )PQ [−1] → ∆∗ C (X, B, A[1]S )PQ for S ∈ Q(T, |T | − 1) coming from the binary chiral operation µA (cf. 3.4.11). For any (J/I) J  I one has an evident embedding ∆∗ C(B, A)PQ X I ,→ C(B, A)PQ X J . We have defined the right D-complex C(B, A)PQ on X S ; it is clearly admissible. Set C ch (X, B,P A)PQ := Γ(X S , h(C(B, A)PQ )). As a mere graded module, our complex equals Γ(U (m+n) , h((BPQ [1])m (APQ [1])n )). As in the proof of m≥0,n>0 ∼

the proposition in 4.2.12 we see that C ch (X, B, A)PQ −→ RΓDR (R(X), C(B, A)PQ ). We have the Cousin spectral sequence (see 4.2.3) (4.4.9.2)

1 ch Ep,q = Hp+q (X, B, A[1][1,p] )Σp

converging to H −p−q C ch (X, B, A)PQ . Consider an evident embedding of complexes C ch (X, A)PQ ,→ C ch (X, B.A)PQ . It has a left inverse C ch (X, B, A)PQ  C ch (X, A)PQ formed by the morphisms Γ(U (m+n) , h((BPQ [1])m  (APQ [1])n )) → Γ(U (m+n) , h((APQ [1])m+n )). Proposition. The embedding C ch (X, A)PQ ,→ C ch (X, B.A)PQ is a quasich isomorphism. Therefore the spectral sequence (4.4.9.2) converges to Hp+q (X, A). Proof. The complex C = C ch (X, B, A)PQ carries an increasing filtration Cm that corresponds to the first grading (by m), and C0 = C ch (X, A)PQ . It suffices to check that grm C is acyclic for m > 0. Consider the embedding ˜j : U (1+m) ,→ X × U (m) . Set A(m) := ˜j∗ ˜j ∗ A  OU (m) ; this is a U (m) -family of chiral algebras on X. Let C(m) be the relative version of the chiral cochain complex for A(m) ; this is a complex of OU (m) -modules. It is acyclic by the relative version of the lemma from 4.3.4. Now C(m) is a left DU (m) -module (since A(m) is), and grm C = RΓDR (U (m) , B m ⊗ C(m) ) = 0; q.e.d.  4.5. Rigidity and flat projective connections Suppose one has a Z-family of curves X = {Xz } equipped with (not necessary unital) chiral algebras Az . The fiberwise chiral homologies H·ch (Xz , Az ) form quasicoherent sheaves on Z. In this section we discuss an X-local structure on A which provides a flat projective connection ∇ on the chiral homology sheaves. An example of such a structure is an extension of the DX/Z -action on A to a DX -action compatible with the chiral product: then the C(Az ) form complexes of D-modules on the fibration of Ran’s spaces R(Xz ), and ∇ is the corresponding Gauss-Manin connection. Such a simple picture occurs quite rarely though. Usually the action of “horizontal” vector fields on A is well defined only up to the adjoint action of ALie ; i.e., more precisely, A carries an action of an extension of the Lie algebra of horizontal vector fields by the (relative) de Rham complex of ALie (which is naturally a homotopy Lie algebra). This suffices for the definition of ∇ due to a key rigidity property of chiral homology: the action of the homotopy Lie algebra ch RΓDR (Xz , ALie z ) on C (Xz , Az ) is canonically homotopically trivialized. A weaker Lie structure, when A is replaced by ALie /ωX/Z 1A , leads to a projective connection. In practice, one always considers instead of the whole of ALie its smaller Lie∗ subalgebra determined by the geometry of the situation. For example, suppose that our family of chiral algebras comes from a universal setting (i.e., from a vertex algebra). Then a flat projective on the chiral homology is produced by a Virasoro

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331

vector (see 3.7.25). For a family of Kac-Moody algebras, the Sugawara tensor, and the 0th chiral homology, we get the Knizhnik-Zamolodchikov connection. Another example: suppose we have a chiral algebra A on a curve X equipped with an action of the group of G-valued functions, G is an algebraic group. We get a family of twisted chiral algebras parametrized by the moduli space BunG of G-bundles on X (see 3.4.17). Then any Kac-Moody tensor in A provides a flat projective connection on the corresponding sheaves of chiral homology on BunG . Connections of this type on the 0th chiral homology are treated in Chapters 16–17 of [FBZ]. In 4.5.1 we explain why the de Rham complex of a Lie∗ algebra is a homotopy Lie algebra. In 4.5.2 the above-mentioned rigidity property of chiral homology is established; the key tool here is the BV algebra structure on the chiral chain complex. Some variants of the rigidity property, in the format needed for the construction of the connection, are discussed in 4.5.3. The input package for the construction of the connection on chiral homology is defined in 4.5.4; the corresponding connection is constructed in 4.5.5. The twisted seting, leading to a projective connection, is discussed in 4.5.6. Section 4.5.7 contains a streamlined construction of the O-extension of the Lie algebra of vector fields on Z acting on chiral homology, and also compatibility with tensor products. Section 4.5.8 considers the case of chiral homology with coefficients, 4.5.9 compares the two settings, 4.5.10 gives a different construction of the connection in the case when the coefficient sheaves are supported at points (for a convenient explicit formula, see 4.5.12), and 4.5.11 compares the two constructions. In 4.5.13 we discuss the above-mentioned examples in more detail. As always, we deal with differential graded super objects, so “Lie algebra” means “DG super Lie algebra”, etc. 4.5.1. Let L be a Lie∗ algebra on X. Then the de Rham complex DR(L) is naturally a homotopy Lie algebra. This means that there is a canonical object in the homotopy category of sheaves of Lie algebras identified with DR(L) in the derived category of sheaves. Similarly, RΓDR (X, L) is naturally a homotopy Lie algebra; i.e., there is a canonical object in the homotopy category of Lie algebras identified with RΓDR (X, L) as a mere object of D(k). We denote these homotopy Lie algebras by DR(L), RΓDR (X, L) by abuse of notation. One constructs them as follows. Take P, Q as in 4.2.12 and write LP := L ⊗ P, LPQ := L ⊗ P ⊗ Q, etc. We have the quasi-isomorphisms of Lie∗ algebras L ← LP → LPQ which yield quasiisomorphisms of Lie algebras h(LP ) → h(LPQ ). They are canonically identified with DR(L) in DSh(X) (see 2.2.10), so we have defined the homotopy Lie algebra structure on DR(L). The Lie algebra Γ(X, h(LPQ )) is identified with RΓDR (X, L) in D(k);51 it provides the homotopy Lie algebra structure on RΓDR (X, L). The independence of the auxiliary choice of P, Q follows from the lemma in 2.2.10 (or rather Remark after it) and the second lemma in 4.1.3. Suppose that L acts on a not necessary unital chiral algebra A (see 3.3.3). Then LPQ acts on APQ , so the Lie algebra Γ(X, h(LPQ )) acts on APQ by derivations. Therefore Γ(X, h(LPQ )) acts on C ch (X, A)PQ by transport of structure. In particular, this action is compatible with the BV structure (see 4.3.1). We see that C ch (X, A) is naturally an RΓDR (X, L)-module.52 51 This 52 The

above.

follows by an argument from the proof of the proposition in 4.2.12. independence of this construction from the auxiliary choice of P, Q can be seen as

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4.5.2. Rigidity. Suppose that the L-action on A comes from a morphism of Lie∗ algebras ι : L → ALie and the adjoint action of A. Lemma. The homotopy action of RΓDR (X, L) on C ch (X, A) is canonically homotopically trivialized. Proof. Let us show that the action of Γ(X, h(LPQ )) extends naturally to an action on C ch (X, A)PQ of the contractible Lie algebra Γ(X, h(LPQ ))† (see 1.1.16). This action comes from the BV structure on C ch (X, A)PQ (see 4.3.1). Indeed, we ι ch have morphisms Γ(X, h(LPQ )) − → Γ(X, h(ALie PQ )) ,→ C (X, A)PQ [−1] of Lie algebras, so Γ(X, h(LPQ ))† acts via the canonical action of C ch (X, A)PQ [−1]† defined by the BV structure (see 4.1.6).  4.5.3. Variants. The material of this section will not be used until 4.5.5; the reader can skip it at the moment. Below, our L is a Lie∗ algebra which is homotopically quasi-induced as a DX complex (see 2.1.11), so the natural projections DR(LQ ) → h(LQ ), ΓDR (X, LQ ) → Γ(X, h(LQ )) are quasi-isomorphisms (see the lemma in 4.1.4). The corresponding projection Γ(X, h(LPQ )) → Γ(X, h(LQ )) is a quasi-isomorphism of Lie algebras. If L acts on a (not necessary unital) chiral algebra A, then LQ acts on AQ and APQ and so Γ(X, h(LQ )) acts on C ch (X, A)PQ . (i) Suppose that we are in the situation of 4.5.2. Let us show that the action of Γ(X, h(LQ )) on C ch (X, A)PQ is canonically homotopically trivialized. One proceeds by providing a homotopy between this action and the one considered in the proof in 4.5.253 and then applying 4.5.2; for technical reasons we have to pass to a larger chiral chain complex C ch (X, A)PQ ˜ . Here is a precise construction. Let P+ be the unital DX -algebra corresponding to P, so P = P ⊕ OX as an + DX -module, + → OX the morphism of unital algebras defined by P . Set P : P + + + + I := KerP , I := Ker+ P . Then P, I, I are ideals in P ; one has I · P ⊂ I, and I 54 is contractible and DX -flat. ˜ := Cone(I → P), P ˜ + := Cone(I+ → P+ ). Then P ˜ + is a unital DX Set P ˜ + is an embedding of algebras, and P ˜ is an ideal in P ˜ +. algebra, so that P+ ,→ P + + ˜ Our P extends to a quasi-isomorphism of OX -algebras P → OX . Its restriction ˜ → OX is also a quasi-isomorphism, so (P, ˜  ˜ ) satisfies the same conditions P˜ : P P + + ˜ := Cone(P ˜→P ˜ ); this is a contractible DX -algebra. as (P, P ). Set P ‡ ˜ ⊗ Q. The Lie∗ algebra L ˜ + := Consider the chiral algebra APQ := A ⊗ P ˜ P Q ˜ + ⊗ Q acts on it. The action of the normal Lie∗ subalgebra L ˜ ⊂ L ˜ + L⊗P PQ P Q Lie coincides with the adjoint action via ιPQ ˜ : LPQ ˜ → APQ ˜ + Q acts via ˜ , and LQ ⊂ LP ˜ ιQ by the adjoint action of AQ and the trivial action on P. Now the contractible Lie algebra Γ(X, h(LP˜ + Q ))‡ := Γ(X, h(LP˜ + Q )) acts nat‡

urally on C ch (X, A)PQ ˜ . Namely, the subalgebra Γ(X, h(LP ˜ + Q )) ⊂ Γ(X, h(LP ˜ + Q ))‡ , and the ideal Γ(X, h(L )) ⊂ Γ(X, h(L acts via the above action on APQ ˜ ˜ ˜ + Q ))‡ † PQ P ˜ acts as in the proof in 4.5.2 (with P replaced by P). This action provides the homotopical trivialization of the Γ(X, h(LQ ))-action we promised. 53 Note that the L -action cannot be realized directly as a part of the L Q PQ -action used in 4.5.2: since P is non-unital, there is no embedding LQ → LPQ , and the projection LPQ → LQ is not compatible with the actions. 54 The Tor-dimension of D X equals 1 since dim X = 1.

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333

(ii) Suppose that A is unital and the L-action on A comes from a morphism of Lie∗ algebras ¯ι : L → ALie /ω1A and the adjoint action of ALie /ω1A on A. Let us show that the Γ(X, h(LQ ))-action on C ch (X, A)PQ is homotopically equivalent to ˜ ˜ etc., are as in (i). the multiplication by a character. Below P, Denote by L[ an ω-extension of L defined as the pull-back of the ω-extension Lie A of ALie /ω1A by ¯ι. So we have a morphism of Lie∗ algebras ι[ : L[ → ALie [ ˜ with i[ 1[ = 1A . Set L[PQ ˜ := L ⊗ P ⊗ Q, etc. ∗ As in (i), we have the Lie algebra LP˜ + Q which acts naturally on APQ ˜ , and a [ contractible Lie∗ algebra LP˜ + Q . Set L♦ := Cone(L → L ); this is a Lie∗ + ˜ + ˜ ˜ Q P Q PQ P ‡

‡

algebra55 which is a central ωPQ ˜ [1]-extension of LP ˜ +Q. ‡

The Lie algebra Γ(X, h(L♦ ˜ + Q )) is a central extension of the contractible Lie P ‡

algebra Γ(X, h(LP˜ + Q ))‡ := Γ(X, h(LP˜ + Q )) from (i) by Γ(X, h(ωPQ ˜ ))[1]. Denote by ‡

ch Γ(X, h(LP˜ + Q ))[‡ its push-out by the embedding Γ(X, h(ωPQ ˜ ))[1] ,→ C (X, ω)PQ ˜ . [ ch Now the Lie algebra Γ(X, h(LP˜ + Q ))‡ acts naturally on C (X, A)PQ in a way ˜ ch [ that the central subalgebra C (X, ω)PQ ⊂ Γ(X, h(LP˜ + Q ))‡ acts by homotheties ˜ ch according to the C ch (X, ω)PQ ˜ -module structure on C (X, A)PQ ˜ . This action is determined by the property that Γ(X, h(LP˜ + Q )) ⊂ Γ(X, h(LP˜ + Q ))[‡ acts according ♦ [ to the LP˜ + Q -action on APQ ˜ , and the image of Γ(X, h(LPQ ˜ ))† in Γ(X, h(LP ˜ + Q )) ⊂ ‡

˜ Γ(X, h(LP˜ + Q ))[‡ acts as in the proof in 4.5.2 (with L, ι, P replaced by L[ , ι[ , P). ch [ Our Γ(X, h(LP˜ + Q ))‡ is homotopically equivalent to k acting on C (X, A)PQ ˜ by homotheties (see 4.3.3). Since the Γ(X, h(LQ ))-action on C ch (X, A)PQ is a part ˜ of the Γ(X, h(LP˜ + Q ))[‡ -action, it is homotopically multiplication by a character, as was promised. 0 (X, L) acts on Remark. As follows from the above, the Lie algebra HDR tr 0 1 according to the character HDR (X, L) → HDR (X, ω) −→ k where the first arrow is minus the boundary map for the extension L[ .

H·ch (X, A)

4.5.4. Concocting a connection. Let Z be an affine k-scheme. Let π : X → Z be a smooth proper Z-family of curves;56 for a point z ∈ Z the corresponding curve is denoted by Xz . The notions we dealt with have an obvious relative version: we consider DX/Z -modules, and it is clear what chiral algebras on X/Z (= Zfamilies of chiral algebras) are, Lie∗ algebras on X/Z, etc. A Lie∗ algebra L on X/Z yields a sheaf h/Z (L) := L/(LΘX/Z ) of Lie π −1 OZ -algebras. So let A be a (not necessary unital) chiral algebra on X/Z which we assume to be OZ -flat. It defines a complex of quasi-coherent OZ -modules π·ch (X/Z, A) which is a relative version of the complex Γch (X, A) from 4.2.11. Replacing A by A ⊗ Q, where Q is a Dolbeault DX/Z -algebra, we get a chiral chain complex Rπ·ch (X/Z, A), which is an object of the derived category D(Z, OZ ) of quasi-coherent OZ -modules (= Γ(Z, OZ )-modules). Also choosing P as in 4.2.12, we can represent Rπ·ch (X/Z, A) by a complex C ch (X/Z, A)PQ (see (4.2.12.2)). We are going to describe a certain structure of X-local origin which yields an integrable connection on Rπ·ch (X/Z, A). In fact, we consider a slightly more general 55 Since 56 We

[ LP and the arrow is compatible with the LP ˜ + Q acts on LPQ ˜ + Q -actions. ˜ are sorry for the abuse of notation: before X meant an individual curve.

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situation starting with a given Lie algebroid L on Z (see 2.9.1); the output is a homotopy L-action on Rπ·ch (X/Z, A). The case of the connection corresponds to L = ΘZ (for a smooth Z). So let L be a Lie algebroid on Z. It yields a Lie π −1 OX -algebroid π ] L acting on OX which is an extension of π −1 L by ΘX/Z (see 2.9.5).57 Suppose we have the following package: (a) a Lie∗ algebra L on X/Z, (b) a Lie π −1 OZ -algebroid extension K of π ] L by h/Z (L) and a section s : ΘX/Z → K, (c) an action of K on L, (d) an action of K on A and a morphism of Lie∗ algebras ι : L → ALie . The following properties should hold: (i) The K-action on L and A is compatible with the DX/Z -module structure on them,58 the Lie∗ bracket on L, and the chiral product on A. It is π −1 OZ -linear with respect to the K-variable.59 The morphism ι : L → A commutes with the K-actions. (ii) s(ΘX/Z ) ⊂ K is a normal Lie π −1 OZ -subalgebra. Therefore K is an extension of π −1 L by K0 := h/Z (L) × ΘX/Z . (iii) The Lie subalgebra ΘX/Z ⊂ K0 acts on A and L via the DX/Z -module structures, and h/Z (L) ⊂ K0 acts on L via the adjoint action and on A via ι and the adjoint action. The adjoint action of K on h/Z (L) ⊂ K0 coincides with the K-action on h/Z (L) coming from the K-action on L. We call such a package an L-action on A governed by ι : L → ALie . 4.5.5. Proposition. Suppose that as a mere DX/Z -complex our L is homotopically quasi-induced. Then our package yields a homotopy left L-module structure on the complex Rπ·ch (X/Z, A). In particular, if L is a plain Lie OZ -algebroid, then the Ri π·ch (X/Z, A) are left L-modules. Proof. We get the homotopy L-action from the obvious Rπ· K-action trivializing homotopically the action of Rπ· K0 by means of (i) in 4.5.3. Here is the precise construction. (i) We use C ch chiral chain complexes, so one has to make some auxiliary choices of resolutions: (a) Notice that our datum is contravariantly functorial with respect to morphisms of L. Replacing L by its appropriate left resolution, we can assume that L is homotopically flat (or even semi-free) as a complex of OZ -modules. (b) Choose a Dolbeault OX -algebra Q equipped with a left π † L-action. To construct Q one can essentially repeat the construction from the proof of the first lemma in 4.1.3. Namely, we pick a Jouanolou map p : Y → X which yields a Dolbeault OX -algebra P := p· ΩY /X . Let Q be a DG OX -algebra equipped with a left π † L-action (see 2.9.5) and a morphism of OX -algebras P → Q which is universal with respect to this structure. Our Q is a Dolbeault OX -algebra.60 57 We apologize for the discrepancy of notation: the smooth map π : Y → X from 2.9.5 is now π : X → Z. 58 Our K acts on O † −1 O ⊂ O ; hence it acts on D X (via K → π L) preserving π Z X X/Z . 59 I.e., A and L are left K-modules with respect to the Lie π −1 O -algebroid structure on K. Z 60 Let us check that Q is a homotopically flat resolution of O . Since π † L is O -flat, the X X enveloping algebra U (π † L) is also homotopically OX -flat (see 2.9.2). Locally on X our P is

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(c) Choose a non-unital OX -algebra resolution P : P → OX equipped with a left π † L-action such that P>0 = 0 and each Pa is DX/Z -flat. Here we consider P as a DX/Z -module via the standard embedding ΘX/Z ,→ π † L. One can construct P by an obvious modification of Example in 4.2.12. Set P0 := Sym>0 U (π † L), where U (π † L) is considered as a left π † L-module. Let P : P0 → OX be the morphism of algebras with π † L-action defined by the morphism of left π † L-modules U (π † L) → OX , 1U (π† L) 7→ 1OX . Finally set P−1 := Ker P , and Pa = 0 for a 6= 0, −1. (ii) Since ΘX/Z ⊂ π † LZ , our P, Q are automatically DX/Z -algebras. The Lie −1 π OZ -algebroid K acts on them via K → π † L. Set KQ0 := h/Z (LQ ) × (Q ⊗ ΘX/Z ); let KQ be the push-out of K by the obvious morphism of Lie π −1 OZ -algebras K0 → KQ0 . Since K acts on the target of this ¯ Q := morphism, our KQ is a Lie π −1 OZ -algebroid extension of π −1 L by KQ0 . Set K KQ /Q ⊗ ΘX/Z ; this is a Lie π −1 OZ -algebroid which is an extension of π −1 L by h/Z (LQ ). ˜ (which satisfies the same (iii) As in (i) in 4.5.3, our P yields modified algebras P + + ˜ ˜ conditions as (P, P ), see (i)(c) above), P , and P‡ . Each of them is naturally π † Lequivariant. We have the chiral algebra APQ together with the action of the Lie∗ ˜ algebra LP˜ + Q on it (see (i) in 4.5.3). The Lie π −1 OZ -algebroid KQ acts naturally on APQ ˜ . The restriction of this action to h/Z (LQ ) ⊂ KQ0 coincides with the action via the embedding LQ ,→ LP˜ + Q . Our algebroid acts also on LP˜ + Q , and the above constructions are compatible with this action. (iv) Consider the complex of OZ -modules C ch (X, A)PQ which represents the ˜ object of the derived category Rπ·ch (X/Z, A). It carries an action of the OZ -algebra π· h/Z (LP˜ + Q )‡ := π· h/Z (LP˜ + Q ), defined in (i) in 4.5.3, and the Lie OZ -algebroid ‡

π· KQ . Notice that π· h/Z (LP˜ + Q )‡ is contractible, as follows from the relative version of the lemma in 4.1.4 applied to LQ .61 The action of π· KQ vanishes on π· (Q⊗ΘX/Z ), ¯ Q which is an extension of L by π· h/Z (LQ ). so it factors through the quotient π· K ˜ ¯ Let L be the push-out of π· KQ by the morphism π· h/Z (LQ ) ,→ π· h/Z (LP˜ + Q )‡ . This is a Lie OZ -algebroid62 which is an extension of L by a contractible ideal ˜ π· h/Z (LP˜ + Q )‡ . The above actions form an L-action on C ch (X, A)PQ ˜ . ˜ The homotopy category of left L-modules is naturally equivalent to that of Lmodules, so C ch (X, A)PQ ˜ defines an object of the latter category. Its independence from the auxiliary choices from (i) above is left to the reader. This is the promised homotopy left L-module structure on Rπ·ch (X/Z, A). 

˜ used only data (a)–(c) of Remark. The construction of the Lie algebroid L 4.5.4. 4.5.6. A twisted version. In practice, one usually finds a weaker variant of the package from 4.5.4 which leads to a twisted L-action (i.e., an action of a central extension of L) on chiral homology. If L = ΘZ , we get a flat projective connection. isomorphic to SymV where V is an acyclic complex of free OX -modules, hence Q is isomorphic to Sym(U (π † L) ⊗ V ). 61 The relative version of the lemma in 4.1.4, together with its proof, remains true because the functor Rπ· has homological dimension 1 (since π is proper of relative dimension 1). 62 Since π K · Q0 → π· h/Z (LP ˜ + Q )‡ is a π· KQ -equivariant morphism of Lie OZ -algebras.

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Below we assume that our chiral algebra A is unital. So suppose we have L and data (a)–(c) as in 4.5.4, and a version of (d) that looks as follows: ¯ an action of K on A and a morphism of Lie∗ algebras ¯ι : L → ALie /ω1A . (d) We demand properties (i)–(iii) of 4.5.4 with ι replaced with ¯ι. Our ¯ι defines, as in (ii) in 4.5.3, an ω-extension L[ of L and a morphism of Lie∗ algebras ι[ : L[ → ALie which lifts ¯ι and such that ι[ 1[ = 1A . The K-action on L lifts to L[ so that ι[ commutes with the K-actions. The ω-extension L[ is important for the reasons explained in the remark after the proposition below.63 Our package is called a twisted L-action on A governed by ι[ : L[ → ALie . Proposition. Suppose that L is a homotopically quasi-induced DX/Z -complex. Then the above package yields a homotopy OZ -extension L[ of L and a left unital homotopy L[ -action on the chiral complex Rπ·ch (X/Z, A). Remark. The extension L[ depends only on data (a)–(c), L[ , and the K-action on L[ . It does not depend on A and ι[ . Proof. Let us repeat steps (i)–(iii) of the proof in 4.5.5. So we have the complex C ch (X/Z, A)PQ which represents Rπ·ch (X/Z, A). It is a module over the ˜ commutative non-unital algebra C ch (X/Z, ω)PQ (where ω := ωX/Z ) which is a ˜ homotopy unit OZ -algebra (see 4.3.4). We will define a C ch (X/Z, ω)PQ ˜ -extension ˜ [ of the Lie OZ -algebroid L ˜ from step (iv) of the proof in 4.5.5 and an action L ˜ [ on C ch (X/Z, A) ˜ such that C ch (X/Z, ω) ˜ ⊂ L ˜ [ acts according to the of L PQ

PQ

C ch (X/Z, ω)PQ ˜ -module structure. Let π· h/Z (LP˜ + Q )[‡ be the relative version of the Lie algebra Γ(X, h(LP˜ + Q ))[‡ defined in (ii) in 4.5.3. This is a Lie OZ -algebra which is a central extension of the contractible Lie algebra π· h/Z (LP˜ + Q )‡ by C ch (X/Z, ω)PQ ˜ . [ Now C ch (X/Z, A)PQ carries a natural action of π h ˜ ˜ + Q )‡ (see (ii) in 4.5.3) · /Z (LP ¯ Q (see steps (ii) and (iv) in 4.5.5). The latter and of the Lie OZ -algebroid π· K ˜ [ be the push-out of π· K ¯Q algebroid is an extension of L by π· h/Z (LQ ). Let L [ by the morphism π· h/Z (LQ ) ,→ π· h/Z (LP˜ + Q )‡ (see (II) in 4.5.3). This is a Lie ˜ OZ -algebroid64 which is a C ch (X/Z, ω)PQ ˜ -extension of L. Our actions define the [ ch ˜ promised action of L on C (X/Z, A)PQ  ˜ . ˜ [ . Consider the Lie∗ alRemark. One can rephrase slightly the definition of L ♦  −1 ¯ gebra LP˜ + Q from (ii) in 4.5.3. Let K‡ be an extension of π L by h/Z (L♦ ˜ + Q ) defined P ‡

‡

¯ Q by the morphism h/Z (LQ ) → h/Z (L♦+ ). This is naturally a as the push-out of K ˜ Q P ‡

¯ Q is an extension of L ˜ by π· h/Z (ω ˜ )[1]. It is clear Lie π −1 OZ -algebroid. Then π· K PQ ˜ [ is the push-out of this extension by π· h/Z (ω ˜ )[1] ,→ C ch (X/Z, A) ˜ . that L PQ PQ 4.5.7. Remarks. (i) Suppose we are in the situation of 4.5.6 and L is a ˜ [ = 0, and the definition of L[ = H 0 L ˜ [ can be plain Lie algebroid. Then H 6=0 L streamlined as follows. ¯ := K/ΘX/Z ; this is a Lie π −1 OZ -algebroid which is an extension of Set K π −1 L by h/Z (L). Since L is a quasi-induced DX/Z -module, h/Z (L[ ) is a central 63 The

package can be easily reformulated so that L[ and ι[ become entry data. [ ¯ π· h/Z (LQ ) ,→ π· h/Z (LP ˜ + Q )‡ is a π· KQ -equivariant morphism of Lie OZ -algebras.

64 Since

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337

¯ this is a complex extension of h/Z (L) by h/Z (ω). Set N := Cone(h/Z (L[ ) → K); 0 −1 with cohomology H N = L, H N = h/Z (ω). The K-action on h/Z (L[ ) factors ¯ so N is naturally a Lie π −1 OZ -algebroid. Thus R0 π· N is a Lie OZ through K, algebroid which is an extension of L by R1 π· h/Z (ω) = OZ . Now there is a canonical isomorphism of the Lie algebroid OZ -extensions (4.5.7.1)

∼

L[ −→ H 0 Rπ· N.

˜ [ from the remark at the end of 4.5.6. To see this, we use the description of L ch Notice that as a morphism in the derived category π· h/Z (ωPQ ˜ )[1] → C (X/Z, A)PQ ˜ ∼ amounts to the trace morphism π· h/Z (ωPQ )[1] → τ (π h (ω )[1]) − → O ˜ ˜ ≥0 · /Z Z. PQ [ 0 ¯ Therefore L = H π· KQ . ˜ + → OX , OX → Q yield morphisms of Lie Now the standard morphisms P −1  ¯ ¯ Q ). The second π OZ -algebroids K‡ → NQ ← N where NQ := Cone(h/Z (L[Q ) → K arrow is a quasi-isomorphism; the first one induces an isomorphism of cohomology ∼ ¯ Q −→ in degrees ≥ −1. Therefore we get H 0 π· K H 0 Rπ· N which is (4.5.7.1). (ii) Suppose that for a given L we have a finite family of chiral algebras Aα together with twisted L-actions on each Aα . Then one has a natural twisted Laction on ⊗Aα such that the corresponding OX -extension of L is (homotopically equivalent to) the Baer sum of the corresponding extensions L[α . Namely, we take L = ΠLα , K the fibered product of Kα over π \ L, and ι = Σια . The product map (4.3.5.1) is compatible with the (twisted) L-actions. 4.5.8. Suppose we are in the situation of 4.5.6; let us show that the construction of loc. cit. generalizes immediately to the case of chiral homology with coefficients. Let {Ms }s∈S be a finite non-empty family of OZ -flat chiral A-modules. By 4.2.19, we have the corresponding chiral homology complex Rπ·ch (X/Z, A, {Ms }) of OZ -modules. Now suppose that the Lie π −1 OS -algebroid K acts on each Ms . We assume that this action is compatible with the DX/Z -module and the chiral A-module structure on Ms , and is π −1 OS -linear with respect to K, that the Lie subalgebra h/Z (L) ⊂ K acts via ¯ι and the h/Z (ALie /1A ω)-action on Ms , and that ΘX/Z ⊂ K acts according to the DX/Z -module structure on Ms . Such a package yields then a left unital homotopy action of L[ on the complex ch Rπ· (X/Z, A, {Ms }). Namely, it defines in the obvious manner a twisted L-action ˜[ governed by ι[ on the chiral algebra A{Ms } (see 4.2.19), so the Lie algebroid L ch from the proof in 4.5.6 acts on C (X/Z, A, {Ms })PQ as on the direct summand ˜ ch of C ch (X/Z, A{Ms } )PQ ˜ ˜ . This action is homotopy unital, i.e., C (X/Z, A, {Ms })PQ ch ˜[ is naturally a homotopy unital C (X/Z, ω) ˜ -module, and C ch (X/Z, ω) ˜ ⊂ L PQ

acts on C ch (X/Z, A, {Ms })PQ ˜ according to this module structure.

PQ

4.5.9. Suppose we have a finite non-empty set S and for each s ∈ S a section xs : Z → X whose images for different s do not intersect. As in 4.4.3 we get the A-modules A˜s . The K-action on A yields one on A˜s which evidently satisfies the above compatibilities, so L[ acts on Rπ·ch (X/Z, A, {A˜s }). ∼

Lemma. The identification Rπ·ch (X/Z, A, {A˜s }) −→ Rπ·ch (X/Z, A) of (4.4.3.2) is compatible with the L[ -actions.

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Proof. We want to show that the above canonical isomorphism in the derived category of OZ -modules lifts naturally to an isomorphism between the objects in the ˜ [ -actions. derived category of OZ -modules equipped with homotopically unital left L Recall (see the proof in 4.4.3) that our isomorphism comes from diagram α ˜ ˜ (4.4.3.4) which, in turn, arises from the morphisms of chiral algebras A{As } ← β˜

A ⊗ Sym P − → A. The twisted L-action governed by ι[ on A defines one on each of the above chiral algebras; the morphisms α ˜ , β˜ are compatible with this action, ˜s } S {A and it preserves the Z -gradings on A and A ⊗ Sym P . Passing to the corα ch 65 responding CPQ -complexes, we get quasi-isomorphisms C ch (X/Z, A, {A˜s })PQ ˜ ← ˜ β ˜ [ -modules, C˜ ch (X/Z, A, {A˜s })PQ → C ch (X/Z, A)PQ of left homotopically unital L ˜ ˜ which form the promised lifting.  4.5.10. Suppose we are in the situation of 4.5.8 and each Ms is supported at the image of a section xs : Z → X; assume that the images of xs for different s do not intersect. Then one can compute Rπ·ch (X/Z, A, {Ms }) by means of an economic complex C ch (X/Z, A, {Ms })P˜ that does not use Q; see (4.2.19.3). Let us describe ˆ [ of the extension L ˜ [ that acts naturally on C ch (X/Z, A, {Ms }) ˜ . This a variant L P construction will be compared with the generalSone from 4.5.8 in 4.5.11. ˜ etc., are as in (iii) in Below, jS : US ,→ X is the complement to xs (Z); P, P, 4.5.5 and in 4.5.2. ¯ := K/s(ΘX/Z ) which is an extension (i) Consider the Lie π −1 OZ -algebroid K [ of π −1 L by h/Z (L), and the Lie∗ algebra L♦ ˜ + ) (cf. (ii) in ˜ → LP ˜ + := Cone(LP P ‡

¯ by, respectively, h/Z (L) → h/Z (jS∗ j ∗ L ˜ + ) ˜ be the push-outs of K 4.5.3). Let Φ, Φ S P ˜ are naturally Lie π −1 OZ -algebroids, so and h/Z (L) → h/Z (jS∗ jS∗ LP˜ + ). Thus Φ ⊂ Φ ‡

˜ are Lie OZ -algebroids. We have an embedding 1[ : π· h/Z (jS∗ j ∗ ω ˜ )[1] ,→ π· Φ ⊂ π· Φ S P ˆ be the push-out of ˜ whose cokernel maps quasi-isomorphically onto L.66 Let L π· Φ ˜ by π· h/Z (jS∗ j ∗ ω ˜ )[1] ,→ C ch (X/Z, jS∗ j ∗ ω) ˜ . The latter complex (which is the π· Φ S P S P ˆ → L is a quasi-isomorphism relative version of Γ(X S , h(jS∗ jS∗ ωP˜ ))) is acyclic, so L of Lie algebroids. ˆ [ of L ˆ that acts on C ch (X/Z, A, {Ms }) ˜ . We will define an OZ -extension L P (ii) For s ∈ S set Ss := S r {s} and consider a Lie∗ algebra jS∗ jS∗ L[ /1[ jSs ∗ jS∗s ω which is a central xs∗ OZ -extension of jS∗ jS∗ L. Applying h/Z , we get a central xs· OZ -extension h/Z (jS∗ jS∗ L)[s of h/Z (jS∗ jS∗ L). Pulling this extension back to h/Z (jS∗ jS∗ LP˜ + ), we get a central xs· OZ -extension h/Z (jS∗ jS∗ LP˜ + )[s of the latter Lie algebra. Notice that the morphisms h/Z (jS∗ jS∗ L[ ) → h/Z (jS∗ jS∗ L) ← h/Z (L) lift naturally to h/Z (jS∗ jS∗ L)[s . Denote by h/Z (jS∗ jS∗ LP˜ + )[s the cone of the morphism ‡

h/Z (jS∗ jS∗ L[ ) → h/Z (jS∗ jS∗ LP˜ + )[s ; this is a Lie algebra which is a central s· OZ ¯ by, respectively, ˜ [s be the push-outs of K extension of h/Z (jS∗ jS∗ LP˜ + ). Let Φ[s ⊂ Φ ‡

the morphisms h/Z (L) → h/Z (jS∗ jS∗ LP˜ + )[s , h/Z (L) → h/Z (jS∗ jS∗ LP˜ + )[s . These ‡

˜ are Lie π −1 OZ -algebroid s· OZ -extensions of Φ and Φ. ˜ ch (X/Z, A, {A ˜s }) ˜ is the component of C ch (X/Z, A ⊗ SymP ) ˜ of degree 1S . C PQ PQ use here, as in the proof of 4.5.5, a relative version of the lemma from 4.1.4.

65 Here 66 We

4.5. RIGIDITY AND FLAT PROJECTIVE CONNECTIONS

339

˜ [s of the Lie OZ -algebroid (iii) So for each s ∈ S we have an OZ -extension π· Φ [ ˜ ˜ ˜ [ in a π· Φ; denote by π· Φ the Baer sum of these extensions. Define π· Φ[ ⊂ π· Φ [ ∗ [ ˜ similar way. By construction, π· Φ contains π· h/Z (jS∗ jS LP˜ )[1] as a submodule. By the sum of residues formula π· h/Z (jS∗ jS∗ ωP˜ )[1] ⊂ π· h/Z (jS∗ jS∗ L[P˜ )[1] is a subcomplex ˆ [ be the push-out of π· Φ ˜ [ . Let L ˜ [ by the morphism π· h/Z (jS∗ j ∗ ω ˜ )[1] ,→ in π· Φ S P ˆ [ is naturally a Lie OZ -algebroid which is an OZ C ch (X/Z, jS∗ jS∗ ω)P˜ . Our L ˆ extension of L. ˆ [ -action on C ch (X/Z, A, {Ms }) ˜ . Our L ˆ [ is the sum of (iv) Let us define the L P the ideals C ch (X/Z, jS∗ jS∗ ω)P˜ , π· h/Z (jS∗ jS∗ L[P˜ )[1], and the subalgebroid π· Φ[ . We define the action on these submodules separately, leaving it to the reader to check the compatibilities. Take any chain c = (⊗ms ) ⊗ cA ∈ Cnch (X/Z, A, {Ms })P˜ := (n) (⊗h/Z (Ms )) ⊗ Γ(US , h/Z ((AP˜ [1])n ))Σn . For any α ∈ C ch (X/Z, jS∗ jS∗ ω)P˜ and β ∈ π· h/Z (jS∗ jS∗ L[P˜ )[1] one has α(c) := c  1A (α), β(c) := c  ι[ (β). A section φ[ ∈ (π· Φ)[ can be represented by a collection (φ, {`[s }) where φ ∈ π· Φ and `[s ∈ ¯ is π· h/Z (jS∗ jS∗ L[ ) are such that for any s ∈ S the image of φ − `s in Γ(US , K) ¯ regular at xs (Z); i.e., it belongs to Γ(USs , K). Now X (4.5.10.1) φ[ (c) := (⊗ms ) ⊗ φ(cA ) + ( ⊗ ms0 ) ⊗ ((φ − `s )ms + `[s ms ) ⊗ cA . s∈S

s0 6=s

4.5.11. Let us show that the construction from 4.5.10 is naturally homotopically equivalent to the general construction from 4.5.8. Consider the quasiisomorphisms (4.5.11.1)

ch ch C ch (X, A, {Ms })PQ ˜ → C (X, AQ , {MsQ })P ˜ ← C (X, A, {Ms })P ˜,

comparing the chiral chain complexes we consider (see (4.2.19.5)). The left complex ˜ [ from 4.5.8 and 4.5.6; the right one carries the carries the action of the extension L [ ˆ ˆ 0[ action of L from 4.5.10. The middle complex carries an action of an extension L ˆ [ with L, L[ replaced by LQ := L⊗Q, L[ := L[ ⊗Q and defined in the same way as L Q ¯ of π −1 L replaced by its push-out K ¯ Q by h/Z (L) → h/Z (LQ ). the h/Z (L)-extension K ˆ 0 [ is an extension of L ˆ 0 by ⊗Qs ; the details of the construction are left to Thus L the reader. There are evident quasi-isomorphisms (4.5.11.2)

ˆ[ ˜[ → L ˆ 0[ ← L L

of Lie algebroid extensions of L, and (4.5.11.1) is equivariant with respect to these morphisms. This establishes the promised homotopy equivalence. 4.5.12. Suppose we are in the situation of 4.5.10 and L is a plain Lie algebroid, ∼ so L[ −→ H 0 L[ . Let us extract from 4.5.10 a description of the action of L[ on C ch (X/Z, A, {Ms })P˜ as on a mere object of the derived category of OZ -modules. First, as in the end of 4.5.10, for γ ∈ L its lifting γ [ ∈ L[ is given by a collection ¯ over US that lifts γ and the `[ ∈ h/Z (jS∗ j ∗ L[ ) (φ, {`[s }) where φ is a section of K s S ¯ lie in Γ(US , K), ¯ i.e., they are regular are such that the sections φ − `s ∈ Γ(US , K) s at xs (Z). Now γ [ acts on the chiral chain complex as the endomorphism given by formula (4.5.10.1). In particular, suppose that A is a plain chiral algebra and the Ms are plain Amodules. Then H0ch (X/Z, A, {Ms }) = hAi/Z = (⊗h/Z (Ms ))h/Z (US ,A) (see (4.4.4.1)),

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and L[ acts on it as (4.5.12.1)

γ [ (⊗ms ) =

X s∈S

( ⊗ ms0 ) ⊗ ((φ − `s )ms + `[s ms ). s0 6=s

4.5.13. Examples. Below we assume for simplicity that Z is smooth. (i) Let X be a Z-family of curves as in 4.5.4. Consider the following canonical package (a)–(c) from 4.5.4: Take L = ΘZ , so π † L = ΘX and π ] L is the algebra of vector fields on X preserving π. Let L = ΘX/ZD = ΘX/Z ⊗ DX/Z be the Lie∗ algebra corresponding to the Lie algebra of vertical vector fields (see Example (i) in 2.5.6(b)), so h/Z (L) = ΘX/Z . Then π ] L acts on L by transport of structure. Let K be the push-out of π ] L by the diagonal embedding morphism ΘX/Z ,→ ΘX/Z × h/Z (L), and let s : ΘX/Z ,→ K be the embedding of the first multiple. Our K is a Lie π −1 OZ algebroid in the obvious manner. It acts naturally on L so that π ] L ⊂ K acts s by transport of structure, h/Z (L) by the adjoint action, and ΘX/Z ,→ K via the DX/Z -module structure on L. Our datum satisfies the properties (i)–(iii) in 4.5.4. Now let A be a (not necessary unital) chiral algebra on X equipped with an action of π ] L. For example, such is any universal chiral algebra in the sense of s 3.3.14. The π ] L-action extends then to a K-action so that ΘX/Z ,→ K acts via the DX/Z -module structure on A. To complete datum (d) of 4.5.4, we need a Virasoro vector, i.e., a morphism of Lie∗ algebras ι : L → ALie . The conditions from 4.5.4 mean that ι commutes with the L] -actions and ΘX/Z ⊂ L] acts on A according to the Lie action defined by ι (see (3.7.25.1)); i.e., for any vertical field τ ∈ ΘX/Z its L] -action on A is a 7→ adι(τ ) a − a · τ where the first term is the adjoint action and the second term is the right DX/Z -module action. According to 4.5.5, such datum provides a flat connection on the chiral homology sheaves. Usually one finds a twisted version of the above situation, when A is unital and ι is replaced by ¯ι : L → ALie /ω1A . It leads, by 4.5.6, to a flat projective connection. The corresponding ι[ : L[ → A is then a Virasoro vector of a certain central charge c (see 3.7.25). (ii) Suppose we are in the situation of 3.4.17, so X is a single curve and we have a group DX -scheme G acting on a chiral algebra A. Assume that G is smooth; let L = Lie(G) be its Lie∗ algebra (see (iv) in 2.5.7) which is a vector DX -bundle. Suppose we have a G-equivariant morphism of Lie∗ algebras ι : L → ALie ; here G acts on L by the adjoint action. Let P = PZ be a Z-family of DX -scheme G-torsors on X, which is the same as a G-torsor on X × Z 67 equipped with a relative connection68 compatible with the G-action. By 3.4.17, our P yields then a Z-family of twisted chiral algebras A(P ) on X × Z. Let us show that our datum defines then a package (a)–(d) from 4.5.4 for A(P ) and the Lie OZ -algebroid L = ΘZ . The Lie∗ algebra from (a) is L(P ) := the P -twist of L with respect to the adjoint action. Then h/Z (L(P )) identifies in the usual way with the Lie algebra of vertical vector fields on P which commute with the G-action and are compatible with the connection. 67 I.e., 68 I.e.,

a torsor with respect to the pull-back of G by the projection X × Z → X. a connection along X-directions.

4.6. THE CASE OF COMMUTATIVE DX -ALGEBRAS

341

Our family of curves is constant, so L] is a semi-direct product of π −1 L (acting ¯ be the sheaf of pairs (τ, τ P ) by vector fields along X-directions) and ΘX/Z . Let K −1 P where τ ∈ π L ⊂ ΘX×Z and τ is its lifting to a vector field on P which commutes with the G-action and is compatible with the connection. This is naturally a Lie π −1 OZ -algebroid which is an extension of π −1 L by h/Z (L(P )).69 Set K := ¯ n ΘX/Z . Then K acts in an evident manner on L(P ), so we have defined data (b) K and (c) from 4.5.4. The morphism from (d) is ι(P ) : L(P ) → A(P )Lie , the P -twist of ι. Properties (i)–(iii) of 4.5.4 are immediate, so 4.5.5 defines a flat connection on the chiral cohomology of the family A(P ). One finds more often a twisted version of the above situation, when instead of ι one has a G-equivariant morphism of Lie∗ algebras ¯ι : L → ALie . Then we get a twisted package from 4.5.6, hence a flat projective connection on the chiral homology. The most interesting particular case of the above situation occurs when G is the jet scheme of an algebraic group H (see Remark (iv) in 3.4.17) and L[ corresponding to ¯ι is a Kac-Moody extension (see 2.5.9). Then DX -scheme G-torsors are the same as G-torsors on X (see (3.4.17.2)), so the chiral homology of A(P ) forms a twisted D-module on the moduli space BunH of H-bundles on X. 4.6. The case of commutative DX -algebras For a commutative chiral algebra its chiral homology is naturally a homotopy commutative algebra (see 4.3.1 and 4.3.4). So we have a functor from the homotopy category of commutative DX -algebras to that of commutative DG algebras. The theorem in 4.6.1 says that it is equal to the left derived functor of the functor R` 7→ hRi from 2.4.1. The key point is the computation of the chiral homology of a polynomial DX -algebra done in 4.6.2. Its relative version is presented in 4.6.4 after some needed preliminaries on semi-free modules (see 4.6.3). The linearized version of 4.6.1 (that deals with the chiral homology of R` [DX ]-modules) is considered in 4.6.5. It implies that the chiral homology commutes with the cotangent complex functor (see 4.6.6). We show also that chiral homology functor preserves perfect complexes (see 4.6.7) and commutes with their duality (see 4.6.8). By the proposition in 4.1.17, these facts imply that for a (very) smooth DX -algebra its chiral homology algebra is essentially a complete intersection (see 4.6.9). As in 4.3.7, “DX -algebra” means “commutative unital DG super DX -algebra” and “plain DX -algebra” means “DX -algebra supported in degree 0”. 4.6.1. The chiral homology as a left derived functor. Let F ` be a DX -algebra. We say that F is semi-free if it is OX -semi-free in the sense of 4.3.7 and is convenient if it is semi-free and one can find F0 ⊂ F1 ⊂ · · · ⊂ F as in loc. cit. such that the corresponding Vi have the property Γ(X, h(Vi )) = 0. We call F· a convenient filtration on F and {Vi } ⊂ F convenient generators. By Remark (b) in (the proof of) 4.3.7 every DX -algebra admits a convenient left resolution. Consider the morphism φF : C ch (X, F ) → hF i from (4.2.17.1); it is naturally a morphism in the homotopy category of commutative algebras (see 4.3.2(i)). 69 The

¯ → π −1 L is surjective since G is smooth as a DX -scheme. projection K

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Theorem. If F is convenient, then φF is a quasi-isomorphism: (4.6.1.1)

∼

φF : C ch (X, F ) −→ hF i.

Therefore on the category of commutative unital chiral algebras, C ch coincides with the left derived functor of h i. Proof. (a) Since both parts of (4.6.1.1) are compatible with inductive limits, our problem reduces to the following statement: Let R → F be an elementary morphism (see 4.3.7) such that the corresponding V ⊂ F has the property Γ(X, h(V )) = 0. Suppose that φR : C ch (X, R) → hRi is a quasi-isomorphism. Then φF : C ch (X, F ) → hF i is also a quasi-isomorphism. (b) Let us show that the above statement follows from its particular case when R` = OX and the differential of F vanishes. Recall that as a mere Z-graded DX -algebra, F equals R ⊗ Sym V , and the ring filtration Fa := R ⊗ Sym≤a V is compatible with the differential. The functor h i commutes with tensor products, so as a mere Z-graded algebra, hF i equals 0 (X, V [1]) since, by hRi ⊗ hSym V i. Notice that hSym V i = Sym K where K = HDR ` 2.1.12, the maximal constant quotient of V equals K⊗OX . The filtration Fa defines a filtration on C ch (X, F ) (see 4.2.18); we also have a filtration hRi⊗Sym≤a K on hF i compatible with the differential. Our morphism is compatible with filtrations, so, by (4.2.18.1), it suffices to check that gr φF is a quasi-isomorphism. Now gr φF = φgrF , and gr F equals R ⊗ Sym V as a DG algebra (the differential kills V ). By 4.3.6 we are reduced to the situation of R` = OX ; i.e., F = Sym V . 0 (X, V [1]). We (c) Now F = Sym V , dF = 0, so hF i = Sym K where K := HDR ch >0 >0 want to show that φF : C (X, Sym V ) → Sym K is a quasi-isomorphism. 4.6.2. We will deduce this fact from the next proposition which is valid under less restrictive assumptions on V . Take any V ∈ CM(X); set F := Sym V . Let us represent C ch (X, F ) by the commutative algebra C ch (X, F )PQ (see 4.3.2) and RΓDR (X, V ) by the complex Γ(X, h(VPQ )). The embedding Γ(X, h(VPQ ))[1] ⊂ C1ch (X, F ) ,→ C ch (X, F ) (see 4.2.12) yields a morphism of commutative algebras (4.6.2.1)

Sym(Γ(X, h(VPQ ))[1]) → C ch (X, F ).

Proposition. If V is homotopically OX -flat, then this is a quasi-isomorphism. End of the proof of the theorem. Consider V as in part (c) of the proof in 4.6.1. ∼ ∼ 0 We know that Γ(X, h(VPQ ))[1] −→ RΓDR (X, V [1]) −→ HDR (X, V [1]) = K, and it is clear that (4.6.2.1) is right inverse to φF . So the proposition implies that φF is a quasi-isomorphism, and we are done. Proof of Proposition. (i) Set F >0 := Sym>0 V . By 4.4.8 it suffices to show that (4.6.2.2)

Sym>0 (Γ(X, h(VPQ ))[1]) → C ch (X, F >0 )

is a quasi-isomorphism. (ii) Let us show first that (4.6.2.2) comes from a morphism of certain Dcomplexes on X S . Recall that the Chevalley-Cousin complex C(F >0 ) is a com(S) mutative algebra in CM(X S )∗ . Set P := Sym>0 ∗ (∆∗ V [1]) (see 3.4.10); this is (S) (S) again a commutative algebra in CM(X S )∗ . The embeddings ∆∗ V ,→ ∆∗ F >0 ,→

4.6. THE CASE OF COMMUTATIVE DX -ALGEBRAS

343

C(F >0 ) define a morphism of commutative algebras P → C(F >0 ). Passing to the de Rham cohomology, we get (4.6.2.3)

RΓDR (X S , P ) → RΓDR (X S , C(F >0 )) = C ch (X, F >0 ).

Now there is a canonical quasi-isomorphism (4.6.2.4)

∼

Sym>0 RΓDR (X, V [1]) −→ RΓDR (X S , P ). ∼

Namely, we know that RΓDR (X, V [1]) −→ RΓDR (X S , ∆∗ V [1]), and (4.6.2.3) is the symmetric power of this quasi-isomorphism via (4.2.8.6). It is immediate that (4.6.2.2) is the composition of (4.6.2.3) and (4.6.2.4). So to prove the proposition, one needs to check that (4.6.2.3) is a quasi-isomorphism. (iii) Consider a filtration W· on C(F >0 ) defined as follows. On each C(F >0 )X I our filtration is compatible with the Q(I)-grading (see (3.4.11.1)), and for T ∈ Q(I) (I/T ) (T ) (T )∗ (I/T ) (T ) one has Wn ∆∗ j∗ j (F >0 [1])T := ∆∗ ((F >0 [1])T · Wn j∗ OU (T ) ); see 3.1.6 and 3.1.7. Our C(F >0 ) also carries a Z>0 -grading defined by the action of homotheties on V ; denote the components by C(F )(n) . Subcomplex P and filtration W· are compatible with the grading. One has W−1 C(F >0 ) = C(F >0 ), and W−n−1 C(F )(n) = 0, W−n C(F >0 )(n) = P (n) . To prove the proposition, it suffices to show that (4.6.2.5)

(S)

∼

>0 RΓDR (X S , P (n) ) −→ RΓDR (X S , grW )). −n C(F

>0 )X I = One has grW −n C(F

⊕

⊕

S∈Q(I) T ∈Q(S,n)

(I/T )

∆∗

(T (F >0 [1])⊗St ) ⊗ Lie∗S/T ;

see (3.1.10.1) (we forget about the differential). Therefore (4.6.2.6)

>0 grW )X I = −1 C(F

⊕ S∈Q(I)

(I)

∆∗ (F >0 [1])⊗S ⊗ Lie∗S ,

and there is an obvious identification (4.6.2.7)

>0 >0 grW ) = Symn∗ grW ). −n C(F −1 C(F

Here Symn∗ is nth symmetric power with respect to the ⊗∗ tensor structure (see 3.4.10). Now (4.6.2.7) is compatible with the differentials; i.e., it is an isomorphism of complexes. Since P (n) = Symn∗ P (1) , (4.6.2.5) for arbitrary n follows from the case n = 1 by (4.2.8.6). >0 (iv) The complex grW ) is supported on the diagonal X ⊂ X S , so we can −1 C(F ◦ consider it as an S -diagram Φ of complexes of D-modules on X. The subcomplex >0 P (1) identifies with a constant subdiagram V ⊂ Φ. Now RΓDR (X S , grW )) −1 C(F is the de Rham cohomology of X with coefficients in the homotopy direct limit C(S◦ , Φ) (see 4.1.1(iv)). Similarly, RΓDR (X S , P (1) ) = RΓDR (X, C(S◦ , V )), and ∼ C(S◦ , V ) −→ V . We will show that V → C(S◦ , Φ) is a quasi-isomorphism. This implies (4.6.2.5) for n = 1, hence finishes the proof. Define a grading Φ = ⊕Φm so that the subdiagram Φm collects all summands in (4.6.2.6) with |S| = m. Then d(Φm ) ⊂ Φm−1 , and every Φm is the induced S◦ -diagram that corresponds to a representation of the symmetric group Σm on (F >0 [1])⊗n ⊗ Lie∗n (see Exercise in 4.1.1(iv)). Therefore, by loc. cit., the projection C(S◦ , Φ) → lim lim −→ Φ is a quasi-isomorphism, and − → Φ is a complex with terms ` ((F >0 [1])⊗m ⊗ Lie∗m )Σm . In other words, lim Φ , as a mere graded D-module, coin−→ cides with the cofree Lie coalgebra generated by the left D-module (Sym>0 V ` )[1].

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The differential equals the canonical differential defined by the commutative algebra structure on F >0` . Therefore V → lim −→ Φ is a quasi-isomorphism (see theorem 7.5 in [Q1]),70 and we are done.  4.6.3. Semi-free modules. We need to fix some terminology. Let R` be a DX -algebra. The DG category of R` [DX ]-modules (which are the same as central chiral R-modules) is denoted by M(X, R` ) and its derived category by DM(X, R` ). An R` [DX ]-module S M is said to be semi-free if it admits a filtration M−1 = 0 ⊂ M0 ⊂ M1 ⊂ · · · , Mi = M , such that for each i there exists a Z-graded DX submodule Ni ⊂ Mi which is a locally projective DX -module, d(Ni ) ⊂ Mi−1 , and the morphism R` ⊗ Ni → gri M is an isomorphism. We refer to the Ni as semi-free generators of M . We say that the generators Ni are convenient if Γ(X, h(Ni )) = 0; in this situation M is said to be convenient. A semi-free module is automatically R` -flat. A linearized version of the proof of part (i) of the lemma in 4.3.7 (which is an immediate modification of the usual construction of [Sp]) shows that every R` [DX ]module admits a semi-free resolution. Moreover, one can choose it so that Ni is isomorphic to a direct sum of (shifts of) DX -modules LD where L is a line bundle on X. If wanted, we can assume that the L are of sufficiently negative degree (cf. Remark (b) in 4.3.7), so the resolution is convenient. A morphism of DX -algebras f : R → F yields an evident exact DG functor f· : M(X, F ` ) → M(X, R` ) and its left adjoint f ∗ : M(X, R` ) → M(X, F ` ), f ∗ M := F` ⊗ M. R`

Lemma. The left derived functor Lf ∗ : DM(X, R` ) → DM(X, F ` ) is well defined and is left adjoint to f· : DM(X, F ` ) → DM(X, R` ). If f is a quasiisomorphism, then Lf ∗

(4.6.3.1)

DM(X, R` )  DM(X, F ` ) f·

are mutually inverse equivalences. Proof. Use semi-free resolutions of R` [DX ]-modules.



4.6.4. The proposition in 4.6.2 admits a version with parameters. Let R` be a DX -algebra, V an R` [DX ]-module. Consider the symmetric R` -algebra F ` := SymR` V ` . Fix auxiliary P, Q and set C ch (·) := C ch (X, ·)PQ (see 4.2.12). We have a commutative homotopy unital C ch (R)-algebra C ch (F ) and a homotopy unital C ch (R)ch module C ch (R, V ) := C ch (X, R, {V })PQ (see 4.2.19). Let SymL C ch (R) C (R, V ) be the symmetric algebra of a homotopically C ch (R)-flat resolution of C ch (R, V ). The obvious morphism of C ch (R)-modules C ch (R, V ) → C ch (F ) yields a morphism of the homotopy unital C ch (R)-algebras (4.6.4.1)

ch ch SymL C ch (R) C (R, V ) → C (F ).

70 Strictly speaking, [Q1] considers the setting of complexes over a field of characteristic 0 subject to some boundedness condition. From the modern point of view, the statement is a consequence of the Koszul duality of the operads Com and Lie (in characteristic 0); see [GK].

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Theorem. If V is homotopically R-flat, then (4.6.4.1) is a quasi-isomorphism. Proof. Choose a semi-free resolution W → V . Since V is homotopically R-flat, the morphism SymR W → SymV is a quasi-isomorphism. Replacing V by W , we can assume that V is semi-free (we use the fact that SymL C ch (R) preserves quasiisomorphisms). The corresponding filtration on V makes (4.6.4.1) a morphism of filtered C ch (R)-algebras. Using (4.2.18.3), one can replace V by gr V ; hence we are reduced to the case when V = R` ⊗ N where N is a locally projective Z-graded DX -module (considered as a complex with zero differential). So F = R ⊗ Sym N ` . ∼ By (4.4.7.1), one has C ch (X, R) ⊗ RΓDR (X, N ) −→ C ch (X, R, V ); by 4.3.6, ∼ C ch (X, R) ⊗ C ch (X, Sym N ) −→ C ch (X, F ). This identifies (4.6.4.1) with (4.6.2.1) ch (for V = P ) tensored by C (X, R), so we are done by the proposition in 4.6.2.  4.6.5. Here is a version of 4.6.1 for the chiral homology with coefficients. Let R` be a DX -algebra. Recall that the constant DX -algebra hRi ⊗ OX is a quotient of R` . Let M be a (DG) R` [DX ]-module (= the central chiral R-module). Consider MhRi := (hRi ⊗ OX ) ⊗ M ; this is the maximal quotient of M which is a R`

hRi ⊗ DX -module. L The functor M → MhRi is right exact. Let M 7→ MhRi be its left derived ` functor DM(X, R ) → D(X, hRi ⊗ OX ). So if M is homotopically R` -flat, then ∼ L MhRi −→ MhRi . Passing to the de Rham cohomology, we get a triangulated functor DM(X, R` ) L → D(hRi), M 7→ RΓDR (X, MhRi ); here D(hRi) is the derived category of unital hRi-modules. ` Now let hM iR be the maximal constant quotient of MhRi .71 This is naturally an hRi-module. The functor M 7→ hM iR is right exact. As is clear from the proof of the next lemma, it admits the left derived functor M 7→ hM iL R. Lemma. One has a natural isomorphism (4.6.5.1)

∼

L RΓDR (X, MhRi ) −→ hM iL R [−1].

L Proof. One has a natural morphism RΓDR (X, MhRi ) → hM iR [−1] coming L from the canonical morphisms MhRi → MhRi → hM iR ⊗ ω and the trace map tr

RΓDR (X, hM iR ⊗ ω) = hM iR ⊗ RΓDR (X, ω) −→ hM iR [−1]. For convenient M it is a quasi-isomorphism (see 4.6.3), and we are done.  Suppose now that R is convenient (see 3.6.1), so hRi = C ch (X, R). By 4.3.4, we have a triangulated functor DM(X, R` ) → D(hRi), M 7→ C ch (X, R, M ). Proposition. One has a natural isomorphism (4.6.5.2)

∼

C ch (X, R, M ) −→ hM iL R [−1].

Proof. Let us define a natural morphism (4.6.5.3)

C ch (X, R, M ) → hM iR [−1].

71 So hM i R is obtained from the complex MhRi by term-by-term application of the functor H0DR (X, ·) (see 2.1.12).

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We have the Z-graded DX -algebra R{M }` with components R{M }0 = R, R{M }1 = M [−1], and C ch (X, R, M ) is the degree 1 component of C ch (X, R{M } ) (see 4.2.19). The Z-graded algebra hR{M } i has components hR{M } i0 = hRi, hR{M } i1 = hM iR [−1]. The morphism φR{M } : C ch (X, R{M } ) → hR{M } i of commutative algebras (see 4.2.17 and (i) in 4.3.2) then yields (4.6.5.3). Let us check that (4.6.5.3) is a quasi-isomorphism for convenient M (see 4.6.3). This establishes (4.6.5.2). The filtration Mi induces on hM iR a filtration such that gri hM iR = hgri M i. The same is true for C ch (X, R, M ), so it suffices to check our statement for M replaced by gr M . Thus we can assume that M = R` ⊗ N where N is a complex of locally projective DX -modules with zero differential such that Γ(X, h(N )) = 0. Consider S ` := SymR` (M ` [−1]) = R` ⊗ Sym(N ` [−1]). One has an evident projection S → R{M } compatible with the Z-gradings on both algebras. It yields an isomorphism between the degree 1 components of the chiral homology, so one has C ch (X, R, M ) = C ch (X, S)1 . We also have hSi = SymhRi (hM iR ), so our projection ∼ yields an isomorphism of the degree 1 components hSi1 −→ hR{M } i1 = hM iR [−1]. ch Since S is convenient, we have C (X, S) = hSi (see 4.6.1). Therefore (4.6.5.3) is a quasi-isomorphism; q.e.d.  Corollary. The functor C ch (X, R, ·) : DM(X, R` ) → D(hRi) is left adjoint to the functor D(hRi) → DM(X, R` ), P 7→ P ⊗ ωX [1].  Remark. Here is a version of the above statements for several R` [DX ]-modules Ms (we will not use it). One has an hRi ⊗ DX S -module  MshRi which yields hRi

RΓDR (U (S) ,  MshRi ) ∈ D(hRi). Now for R convenient and Ms homotopically hRi

R-flat there is a canonical isomorphism (4.6.5.4)

∼

C ch (X, R, {Ms }) −→ RΓDR (U (S) ,  MshRi ). hRi

4.6.6. The cotangent complex. The definition of the cotangent complex (we recalled it in 4.1.5) renders itself immediately into the setting of DX -algebras. Namely, for a commutative DX -algebra R` we have its R` [DX ]-module of differentials ΩR := ΩR` /X . Now L ΩR is an object of DM(X, R` ) equipped with a morphism L ΩR → ΩR defined as follows. If R` is semi-free, then L ΩR = ΩR . One checks that if φ : R` → F ` is a quasi-isomorphism of semi-free algebras, then dφ : Lφ∗ ΩR → ΩF is a quasi-isomorphism. If R is arbitrary, then one chooses a ˜ → R and defines L ΩR as the image of Ω ˜ by the equivalence semi-free resolution R R ∼ ` ˜ follows from the ˜ ) −→ DM(X, R` ); its independence of the choice of R DM(X, R exercise in 4.3.7. Notice that the image of L ΩR in the derived category of DG R` -modules (forgetting the DX -action) equals the usual cotangent complex of R` relative to OX . Proposition. There is a canonical quasi-isomorphism (4.6.6.1)

L

∼

ΩC ch (X,R) −→ C ch (X, R, L ΩR )[1].

Proof. Let ΩC ch (X,R)PQ be the module of C ch (X, R)PQ -differentials relative to C ch (X, ω)PQ . The canonical odd derivation of the algebra R{ΩR } (see 4.2.19 for

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347

notation) yields a derivation C ch (X, R)PQ → C ch (X, R, ΩR )PQ [1] by transport of structure. So we have a morphism of C ch (X, R)PQ -modules (4.6.6.2)

ΩC ch (X,R)PQ → C ch (X, R, ΩR )PQ [1].

Its composition with the canonical morphism L ΩC ch (X,R) → ΩC ch (X,R)PQ is a morphism L ΩC ch (X,R) → C ch (X, R, ΩR )[1] in the derived category of unital C ch (X, R)modules. Replacing R by a semi-free resolution, we get an arrow L ΩC ch (X,R) → C ch (X, R, L ΩR )[1]. It remains to show that this is a quasi-isomorphism. We need a lemma. For arbitrary R consider the hRi-module hΩR iR (see 4.6.5). The universal derivation R` → ΩR yields a derivation hRi → hΩR iR between the constant quotients, hence a morphism of DG hRi-modules (4.6.6.3)

ΩhRi → hΩR iR .

Lemma. Suppose that, as a mere graded DX -algebra, R is freely generated by some DX -module. Then (4.6.6.3) is an isomorphism. Proof of Lemma. Our statement has nothing to do with the differential on R, so we can forget about it. Suppose that R` = Sym V ` . Denote by hV i the maximal constant quotient of V ` . Then hRi = SymhV i; hence ΩhRi = hRi ⊗ hV i (we identify hV i with its image in ΩhRi by the universal derivation). Similarly, ΩR = R` ⊗ V , so hΩR iR = hRi ⊗ hV i. Now (4.6.6.3) identifies the generators of the free hRi-module, and we are done.  Let us finish the proof of the proposition. We can assume that R is conve∼ nient. Then, by the theorem in 4.6.1, one has C ch (X, R) −→ hRi. The arrow ch C (X, R, ΩR )[1] → hΩR iR from (4.6.5.3) is also a quasi-isomorphism. Indeed, ΩR is convenient (since the images of convenient generators of R by the universal derivation are convenient generators of ΩR ), and our assertion was checked in the proof of the proposition in 4.6.5. The above isomorphisms identify our morphism L ΩC ch (X,R) → C ch (X, R, L ΩR )[1] with (4.6.6.3). We are done by the lemma.  4.6.7. Perfect R` [DX ]-complexes. Let us check that the chiral homology preserves perfect complexes. Let R` be any commutative DX -algebra. We have the derived category of ` R [DX ]-modules DM(X, R` ) and its subcategory DM(X, R` )perf of perfect complexes (see 4.1.16). Proposition. (i) Perfectness is a local property for the Zariski topology of X. (ii) The functor C ch (X, R, ·) : DM(X, R` ) → D(C ch (X, R)) preserves perfect complexes. Proof. (i) Suppose we have P ∈ DM(X, R` ) and a finite covering {Uα } of X such that the PUα are perfect. For an R` [DX ]-module M let M → C(M ) be the ˇ Cech resolution of M with respect to this covering. If j : U ,→ X is an intersection of some Uα ’s, then P |U is perfect; hence the functor M 7→ Hom(P, j∗ j ∗ M ) = Hom(P |U , MU ) commutes with direct sums. Thus M 7→ Hom(P, M ) = Hom(P, C(M )) also commutes with direct sums, so P is perfect; q.e.d. (ii) Follows from the fact that the right adjoint functor to C ch (X, R, ·), as described in the corollary in 4.6.5, commutes with direct sums. 

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Suppose that R has non-positive degrees, so DM(X, R` ) is a t-category. We define the span of its objects as in 4.1.16; every perfect complex has finite span. Lemma. (i) If an object M ∈ DM(X, R` ) has span in [b, a] locally on X, then the span of M lies in [b − 1, a]. (ii) If M ∈ DM(X, R` ) has span [b, a], then C ch (X, R, M ) ∈ D(C ch (X, R)) has span in [b + 1, a + 1]. Proof. (i) Clear since the cohomological dimension of X equals 1. (ii) Use the corollary from 4.6.5.



Corollary. If R` is a plain DX -algebra and M a finitely generated locally projective R` [DX ]-module, then C ch (X, R, M ) is a perfect C ch (X, R)-module of span [0, 1].  4.6.8. Compatibility with duality. Suppose that R has non-positive degrees. Proposition. (i) An object P ∈ DM(X, R` ) is perfect of span [b, a] if and only if it can be represented as a retract in DM(X, R` ) of a semi-free R` [DX ]-module with finitely many generators whose degrees are in [b, a]. (ii) For a perfect P its dual is perfect, and the functor C ch (X, R, ·) commutes with duality.  4.6.9. Theorem. For a plain very smooth DX -algebra R` (see 2.3.15) the algebra C ch (X, R) is perfect of span ≤ 1 (see 4.1.17). In particular, if the higher chiral homology of R vanishes, then hRi is a complete intersection. Remark. Probably, the theorem remains true if X is a projective variety of arbitrary dimension n, and one defines C ch (X, R) as the left derived functor of R 7→ hRi (see 3.4); the estimate for the span is ≤ n. Proof of Theorem. We know that hRi = H 0 C ch (X, R) is finitely generated (see (ii) in the proposition in 2.4.2), so, by 4.1.17, it suffices to check that the cotangent complex L ΩC ch (X,R) is a perfect C ch (X, R)-module of span in [−1, 0]. ∼ Since R is very smooth, one has L ΩR −→ ΩR . Since ΩR is a locally projective DX -module, C ch (X, R, ΩR ) is a perfect C ch (X, R)-module of span in [0, 1] (see the corollary in 4.6.7). By 4.6.6, it equals L ΩC ch (X,R) [−1], and we are done.  4.7. Chiral homology of the de Rham-Chevalley algebras In this section we consider the chiral homology of (the DG version of) the orbit space of a Lie∗ R-algebroid. In particular, we interpret the homotopy 1Poisson structure on the chiral homology of a coisson algebra R. Namely, under appropriate regularity condition it defines a Lagrangian embedding of SpechRi into a formal symplectic tube. If R is the Gelfand-Dikii coisson algebra (see 2.6.8), then this is the formal neighbourhood of the space of global opers on X in the symplectic space of all local systems. 4.7.1. The orbit space of a Lie∗ algebroid. Let R` be a plain commutative DX -algebra and L a Lie∗ R-algebroid which is a vector DX -bundle on SpecR` . Let us show that the DG version of the orbit space functor commutes with the chiral homology.

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349

The dual DX -bundle L◦ is a Lie R-coalgebroid, so we have the corresponding de Rham-Chevalley DG DX -algebra CR (L◦ ) =: CR (L) (see (ii) and (iii) in 1.4.14). It carries a decreasing filtration by DG ideals F n := CR (L)≥n with grF CR (L) = n SymR (L◦ [−1]),72 and we consider CR (L) = ← lim − CRn (L)/F as a filtered topological DG DX -algebra. Thus CR (L)PQ := lim ←−(CR (L)/F )PQ is also a filtered topological algebra, grF CR (L)PQ = SymR (L◦ [−1])PQ . We get a filtered topological homotopy ch n unital commutative DG algebra C ch (X, CR (L))PQ := ← lim − C (X, CR (L)/F )PQ ; ch ch ◦ one has grF C (X, CR (L))PQ = C (X, SymR (L [−1]))PQ . As an object of the corresponding homotopy category HoComuT,73 our algebra does not depend on the auxiliary choice of P, Q, so we write simply C ch (X, CR (L)). Consider now a homotopy unital commutative algebra C ch (X, R)PQ and a homotopy unital Lie C ch (X, R)PQ -algebroid C ch (X, R, L)PQ (see (v) in 4.3.2).74 Choose any left resolution (which is a quasi-isomorphism of homotopy unital Lie C ch (X, R)-algebroids) P → C ch (X, R, L)PQ which is homotopically C ch (X, R)PQ projective. Consider the de Rham-Chevalley complex C(P ) := CC ch (X,R)PQ (P ) which is a filtered homotopy unital topological DG algebra. In the unital setting it was defined in 2.9.1, and in the present homotopy unital setting the definition is similar. Namely, as a mere topological graded module our C(P ) is equal to C ch (X, R)PQ ⊕ HomC ch (X,R)PQ ( ⊕ SymiC ch (X,R)PQ (P [1]), C ch (X, R)PQ ), the ali>0

gebra structure on C(P ) comes since Sym>0 (P [1]) is naturally a C ch (X, R)PQ coalgebra (the coproduct comes from the diagonal map P [1] → P [1] × P [1]). As an object of HoComuT, C(P ) does not depend on the auxiliary choice of P , P, Q, so we denote it simply by CC ch (X,R) (C ch (X, R, L)). Proposition. There is a canonical isomorphism in HoComuT (4.7.1.1)

∼

C ch (X, CR (L)) −→ CC ch (X,R) (C ch (X, R, L)).

Proof. We will define a natural morphism of homotopy unital filtered topological DG algebras (4.7.1.2)

C ch (X, CR (L))PQ → C(P )

which induces a quasi-isomorphism of the associated graded algebras. The action of L on R extends naturally to an action of the contractible Lie∗ algebra L† on CR (L) (see 1.4.14). Thus C ch (X,R, L)† , hence P† , acts on C ch (X, CR (L)). If we forget about the differential, then the filtrations F · naturally split, so both terms of (4.7.1.2) acquire an extra Z≥0 -grading, not merely a filtration. We define (4.7.1.2) as a unique C ch (X, R)PQ -linear morphism which is compatible with the grading, commutes with the action of the P [1]-part of P† , and is the identity map on the degree 0 component C ch (X, R)PQ . We leave it to the reader to check that it is actually a morphism of DG algebras. It is a quasi-isomorphism on grF due to 4.6.8 and 4.6.4.  72 This is not a filtration in the sense of 3.3.12 and 4.2.18, so the corresponding spectral sequence need not converge. 73 It is the same as the homotopy category of commutative unital DG algebras A equipped with a filtration by DG ideals A = F 0 ⊃ F 1 ⊃ · · · with homotopy equivalences being morphisms f such that grF f is a quasi-isomorphism. 74 Here “homotopy unital Lie algebroid” means that we have a Lie algebroid which is a homotopy unital C ch (X, R)PQ -module and whose action kills the homotopy unit in C ch (X, R)PQ .

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4.7.2. A description of H0ch . We are in the situation of 4.7.1. Suppose, in 1 addition, that the top cohomology H 1 C ch (X, R, L◦ ) = HDR (X, L◦hRi ) vanishes (see 2.4.7 and 4.6.5 for notation). By 2.2.17 (or 4.6.8 and 4.6.7), this amounts to the 0 1 fact that HDR (X, LhRi ) = 0 and HDR (X, LhRi ) is a projective hRi-module. Then 0 ch ◦ 0 ◦ 1 H C (X, R, L ) = HDR (X, LhRi ) and HDR (X, LhRi ) are mutually dual projective hRi-modules of finite rank. By 4.7.1, the positive cohomology groups H >0 C ch (X, CR (L)/F n ) vanish for each n, so · · · → H 0 C ch (X, CR (L)/F n+1 ) → H 0 C ch (X, CR (L)/F n ) → · · · are surjective maps . The projective limit H0ch (X, CR (L)) is a complete topological algebra, and hRi = H0ch (X, CR (L))/I where I is an open ideal. As follows from 4.7.1, the topology on H0ch (X, CR (L)) coincides with the I-adic topology, and one 0 has an exact sequence H1ch (X, R) → HDR (X, L◦hRi ) → I/I 2 → 0. We see that the formal scheme Spf H0ch (X, CR (L)) is a “formal tube” around hYi where Y := Spec R` . Let us describe it in geometric terms using the formal DX -scheme groupoid G on Y defined by L (see 1.4.15). We have the sheaf of ind-affine ind-schemes hYiX on Xe´t and a formal groupoid hGiX on it (see 2.4.1),75 hence for a test commutative algebra F the sheaf of F -points hYiX (F ) and the sheaf of groupoids hGiX (F ) on it. Denote by hY/Gi(F ) the stack of sections of the quotient stack hYiX (F )/hGiX (F ). Its objects can be seen explicitly as pairs (U· , f· ) where U· is an ´etale hypercovering of X and f· : U· × Spec F → G· is a morphism of simplicial DX -spaces; here G· , Gi = G × · · · × G (i copies), is Y

Y

the classifying simplicial formal DX -scheme.76 We leave the description of the morphisms in hY/Gi(F ) to the reader. The stack hY/Gi(F ) depends on F in a functorial way. Proposition. (i) Automorphisms of objects of hY/Gi(F ) are all trivial, so hY/Gi is a set-valued functor on the category of commutative algebras. (ii) There is a natural isomorphism (4.7.2.1)

∼

hY/Gi −→ Spf H0ch (X, CR (L)).

Proof. (i) Take any object φ ∈ hY/Gi(F ) and its automorphism ν. Since G is a formal groupoid, there are nilpotent ideals I and J of F such that ν modI is the identity and φ modJ comes from some ψ ∈ hYi(F/J). It suffices to show that ν¯ := ν modI(I + J) also equals the identity. Consider ψ as a morphism of DX -algebras R` → F/J ⊗ OX . One can view ν¯ as a section of h(L ⊗ (I/(I + J) ⊗ OX )). This is R`

a trivial vector space by our conditions on L, and we are done. (ii) Take any φ ∈ hY/Gi(F ). We will show that φ defines naturally a homotopy morphism φC : CR (L)` → OX ⊗ F . Then (4.7.2.1) is the map φ 7→ φ˜ := H0ch (φC ). Recall first the construction of G· from part (a) of the proof of the proposition in 1.4.15. Fix n ≥ 0 and set Cn := CR (L)` /F n+1 . Choose a Cn -semi-free resolution (a) Ψn → R` (see 4.3.7); let Tn be the Cn -tensor product of a + 1 copies of Ψn . extend functor h i to ind-affine DX -schemes in the obvious way. course, it suffices to consider U· associated to a covering {Uα }; then f· amounts to a collection (fα , gαβ ) where fα : Uα ×Spec F → YUα and gαβ : Uαβ ×Spec F → GUαβ are morphisms of DUα - and DUαβ -schemes (here Uαβ := Uα × Uβ ) such that gαβ connects the restriction of fα , 75 We 76 Of

X

fβ to Uαβ , and gαβ gβγ = gαγ on Uαβγ .

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(·)

Then Tn is naturally a cosimplicial DG DX -algebra. One has H >0 Tn = 0, so (·) (·) (·) (·) (·) τ≤0 Tn ,→ Tn is a quasi-isomorphism. Set En := H 0 Tn ; then Spec En is the ` nth infinitesimal neighborhood of Spec R ⊂ G· . Write φ as a pair (U· , f· ) (see above). Choose n sufficiently large so that f· (·) takes values in Spec En ⊂ G· . Consider the evident morphisms of cosimplicial DG DX -algebras77 (4.7.2.2)

f·

CR (L)`  Cn → Tn(·) ←- τ≤0 Tn(·) → En(·) −→ OU· ⊗ F ←- OX ⊗ F.

Choose a homotopy inverse functor which assigns to a cosimplicial DG DX -algebra a DG DX -algebra (e.g., the Thom-Sullivan construction from [HS] will do). Apply it to (4.7.2.2); we get a sequence morphisms of DG DX -algebras where every arrow directed to the left is a homotopy equivalence. Our φC is the composition. As a morphism in the homotopy category, it does not depend on the auxiliary choices involved. ˜ (iii) It remains to check that the morphism hY/Gi → Spf H0ch (X, CR (L)), φ 7→ φ, ch is an isomorphism. Both spaces are formal neighborhoods of Spec H0 (X, R), so it suffices to show that for any φ as above we have an isomorphism between the first ˜ Fix an extension F 0 of F by an ideal J of infinitesimal neighborhoods of φ and φ. ˜ of φ, φ˜ to F 0 . We want to show that square 0, and consider the sets of liftings E, E 0 0 ˜ φ 7→ φ˜ , is bijective. the map E → E, L

Set K := L ΩCR (L) ⊗ (OX ⊗ F ) ∈ DM(X, OX ⊗ F ) where L ΩCR (L) is the cotanφC

gent complex (see 4.6.6). It follows from (4.6.6.1), (4.6.5.1), (4.6.5.2) that the pull-back of the cotangent complex L ΩC ch (X,CR (L)) by the composition of morφC

phisms C ch (X, CR (L)) → H0ch (X, CR (L)) −−→ F equals RΓDR (X, K)[1] ∈ D(F ). ˜ is controlled by this complex in the usual way. Namely, there is a class Then E c˜ ∈ Ext1F (RΓDR (X, K)[1], J) = Ext1 (K, ωX ⊗ J) which vanishes if and only if ˜ 6= ∅; if c = 0, then E ˜ is a HomF (RΓDR (X, K)[1], J) = Hom(K, ωX ⊗ J)-torsor.78 E Locally on X our φC factors through a morphism f : R` → OX ⊗ F . Therefore L

the canonical exact triangle L ΩCR (L) ⊗ R → L ΩR → L◦ in DM(X, R` ) shows CR (L)

that H >1 K = 0 and c vanishes locally on X if and only if f lifts (locally) to f 0 : R` → OX ⊗ F 0 . We can assume that the latter condition holds (otherwise both ˜ are empty). Then c˜ ∈ Ext1 (τ≥0 K, ωX ⊗ J) ⊂ Ext1 (K, ωX ⊗ J);79 if c˜ = 0, E and E ˜ then E is a Hom(τ≥0 K, ωX ⊗ J)-torsor. Let us describe τ≥0 K. For any a ≥ 1 and i ∈ [0, a] the sheaf of relative 1forms with respect to the ith boundary projection Ga → Ga−1 identifies canonically with p∗i L◦ where πi : Ga → Y is the ith structure projection (see the proof of the proposition in 1.4.15), so we have a canonical morphism χa : ΩGa → ⊕ πi∗ L◦ . i∈[o,a]

Let χ0 : ΩY → L◦ be the coaction morphism. Then the χ· form a morphism of simplicial DX -modules on G· ; set Ξ· := Cone(χ· )[−1]. This complex has the property that for every simplicial structure map ∂ : Ga → Gb the corresponding morphism ∂ ∗ Ξb → Ξa is a quasi-isomorphism. OU· is the cosimplicial OX -algebra such that Spec OU· = U· . use the global duality for the de Rham cohomology. 79 More precisely, c ˜ ∈ Ker(Ext1 (τ≥0 K, ωX ⊗ J) → Ext1 (H 0 K, ωX ⊗ J)). 77 Here

78 We

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Thus we get a complex of cosimplicial (OU· ⊗ F )[DX ]-modules f·∗ Ξ. The total complex tot f·∗ Ξ is a complex of (OX ⊗ F )[DX ]-modules. It follows from the construction of φC that it identifies canonically with τ≥0 K. Now E is controlled by the Hom complex Hom(tot f·∗ Ξ, tot ωU· ⊗ J) (we assume that U0 is affine). Since L◦ is locally projective, one has H 1 Hom(tot f·∗ Ξ, tot ωU· ⊗ ∼ J) ⊂ Ext1 (τ≥0 K, ωX ⊗ J), H 0 Hom(tot f·∗ Ξ, tot ωU· ⊗ J) −→ Hom(τ≥0 K, ωX ⊗ J). One checks that the embedding transforms the obstraction c to non-emptiness of ˜ is a morphism of torsors with respect to E to c˜, and if c = 0, then our map E → E the second isomorphism map, and we are done.  0 Remark. Suppose in addition that R` is a smooth DX -algebra, HDR (X, ΩhRi ) 1 = 0, and HDR (X, ΩhRi ) is a projective hRi-module, so, by (ii) in the proposition in 2.4.7, hYi is smooth. Then hY/Gi is a smooth formal scheme, and the normal 1 bundle to hSpec R` i ,→ hY/Gi is equal to HDR (X, LhRi ).

4.7.3. The case of a coisson algebra. Suppose that R is a plain very smooth coisson algebra, so Ω = ΩR is a Lie∗ R-algebroid. As in the end of 1.4.18, consider the filtered topological (−1)-coisson algebra Ccois (R). According to (iv) in the proposition in 4.3.1, C ch (X, Ccois (R))PQ = C ch (X, CR (Ω))PQ is a homotopy unital topological filtered Poisson algebra (we follow the notation of 4.7.1). As an object of the corresponding homotopy category,80 it does not depend on the auxiliary choices of P, Q, so we use the notation C ch (X, Ccois (R)). On the other hand, by (iii) in the proposition in 4.3.1, C ch (X, R)PQ is a homotopy unital 1-Poisson algebra. Choose its semi-free resolution ψ : Φ → C ch (X, R)PQ as a 1-Poisson algebra. Then ΩΦ [−1] is a Lie Φ-algebroid. It is semi-free as a Φmodule and is perfect by 4.6.9. Consider the topological filtered Poisson algebra Cpois (Φ); as a mere topological filtered algebra it equals the de Rham-Chevalley algebra of the Lie algebroid ΩΦ [−1] (see 2.9.1).81 As an object of the homotopy category of topological filtered Poisson algebras, Cpois (Φ) does not depend on the auxiliary choices of P, Q, Φ, so we denote it simply by Cpois (C ch (X, R)). Proposition. There is a canonical homotopy equivalence of topological filtered Poisson algebras (4.7.3.1)

∼

C ch (X, Ccois (R)) −→ Cpois (C ch (X, R)).

Proof. We can assume that the quasi-isomorphism ψ : Φ → C ch (X, R)PQ of 1-Poisson algebras is surjective. Then ψ yields a morphism (Φ, ΩΦ [−1]) → (C ch (X, R)PQ , ΩC ch (X,R)PQ [−1]) in LieAlg. The arrow (coming from (4.6.6.2)) ΩC ch (X,R)PQ [−1] → C ch (X, R, Ω)PQ is a morphism of Lie C ch (X, R)PQ -algebroids; the composition ΩΦ [−1] → C ch (X, R, Ω)PQ is a quasi-isomorphism by (4.6.6.1). Set P := C ch (X, R)PQ ⊗ ΩΦ [−1]; the morphism P → C ch (X, R, Ω)PQ is a quasiΦ

isomorphism of Lie C ch (X, R)PQ -algebroids since ΩΦ is a semi-free Φ-module. We 80 Which coincides with the homotopy category of (unital) filtered Poisson algebras, i.e., Poisson algebras A equipped with a filtration A = A0 ⊃ A1 ⊃ · · · by DG ideals such that {Ai , Aj } ⊂ Ai+j−1 , and homotopy equivalences are morphisms that induce quasi-isomorphisms i between gr’s. One can assume that filtrations are complete, i.e., A = ← lim − A/A . 81 In 2.9.1 we considered the unital setting; in the present homotopy unital setting the definition of the de Rham-Chevalley complex should be modified as in 4.7.1, so Cpois (Φ) equals Φ ⊕ HomΦ (Sym>0 ΩΦ , Φ) as a mere topological graded module.

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353

get the filtered quasi-isomorphisms of DG algebras (4.7.3.2)

C ch (X, Ccois (R))PQ → CC ch (X,R)PQ (P )  Cpois (Φ)

where the left arrow is (4.7.1.2). Let T ,→ C ch (X, Ccois (R))PQ × Cpois (Φ) be the fibered product of these arrows. Our T carries the induced filtration, and both projections (4.7.3.3)

C ch (X, Ccois (R))PQ ← T → Cpois (Φ)

are filtered quasi-isomorphisms. One checks that T is a Poisson subalgebra of C ch (X, Ccois (R))PQ × Cpois (Φ), and we are done.  4.7.4. The formal neighbourhood of global opers. We are in the situ0 1 ation of 4.7.3. Suppose in addition that HDR (X, ΩhRi ) = 0 and HDR (X, ΩhRi ) is ∼ ch a projective hRi-module, so C (X, R) −→ hRi is a smooth algebra. Its homotopy 1-Poisson algebra structure yields a filtered topological Poisson algebra Cpois (hRi) which is concentrated in degree 0. The corresponding reduced algebra equals hRi, and SpechRi ,→ Spf Cpois (hRi) is a Lagrangian embedding. According to 4.7.3 and (4.7.2.1), one has a canonical identification (4.7.4.1)

∼

Spf Cpois (hRi) −→ hSpec R` /Gi

where G is the formal DX -scheme groupoid on Spec R` defined by the Lie∗ algebroid ΩR . Here is an important example of this situation. Let R be the Gelfand-Dikii coisson algebra Wcκ for a non-degenerate κ, so Spec R` = Opg (see 2.6.8). If our curve X has genus > 1, then the above conditions are satisfied, so SpechRi = Opg (X) = the space of global g-opers on X has a canonical Lagrangian embedding into a symplectic formal tube Spf Cpois (hRi). It is known (see [BD]) that the forgetting-of-B-structure map (FB , ∇) 7→ (FG , ∇) is a closed embedding of Opg (X) into the moduli space LocSysG of G-bundles with connection on X. It follows then from (4.7.4.1) and the second proposition in 2.6.8 that Spf Cpois (hRi) coincides with the formal neighbourhood of Opg in LocSysG . Remark. One can show that the latter identification is compatible with symplectic structures where LocSysG is equipped with the usual symplectic structure defined by κ. 4.8. Chiral homology of chiral envelopes We show that the chiral homology of the chiral enveloping algebra of a Lie∗ algebra L is equal to the homology of the homotopy Lie algebra RΓDR (X, L). Similar facts hold for the chiral homology with coefficients, in the twisted setting, and in the chiral Lie algebroid setting. As an application, we show that the chiral homology of a cdo on a very smooth DX -scheme Spec R` can be interpreted as a certain twisted de Rham homology of Spec C ch (X, R` ). In particular, it is finitedimensional. 4.8.1. Enveloping algebras of Lie∗ algebras. Let L be a Lie∗ DG algebra on X. We have a homotopy Lie algebra RΓDR (X, L) (see 4.5.1) which yields the Chevalley homology complex C(RΓDR (X, L)) canonically defined as an object of

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¯ ¯ HoBVu ; similarly, we have the reduced complex C(RΓ DR (X, L)) ∈ HoBV (see (a) in 4.1.8). Consider the chiral envelope U (L). The standard commutative filtration defines, by 4.3.4, a homotopy filtered unital BV algebra structure on C ch (X, U (L)); i.e., we have C ch (X, U (L)) ∈ HoBVu . Theorem. There is a canonical morphism in HoBVu (4.8.1.1)

C(RΓDR (X, L)) → C ch (X, U (L))

which is an isomorphism if L is OX -flat. Therefore H·ch (X, U (L)) is the homology of the homotopy Lie algebra RΓDR (X, L). Proof. Denote by U (L)>0 the kernel of the augmentation morphism U (L) → ω (which sends L to 0). This is a non-unital chiral algebra, and U (L) = (U (L)>0 )+ . Thus, by Remark in 4.4.8, C ch (X, U (L)) ∈ HoBVu is obtained from the non-unital ¯ by adding the unit. We will define a canonical algebra C ch (X, U (L)>0 ) ∈ HoBV ¯ morphism in HoBV (4.8.1.2)

ch >0 ¯ C(RΓ ) DR (X, L)) → C (X, U (L)

which is an isomorphism when L is OX -flat. Adding the unit, one gets (4.8.1.1). Let us represent C ch (X, U (L)>0 ) by a filtered BV algebra C ch (X, U (L)>0 )PQ (see (iv) in 4.3.2 and 4.2.18) and RΓDR (X, L) by a Lie algebra Γ(X, h(LPQ )) (see 4.5.1). The embeddings of the Lie algebras Γ(X, h(LPQ )) ,→ Γ(X, h(U (L)>0 PQ )) ,→ C ch (X, U (L)>0 )PQ [−1] identify Γ(X, h(LPQ )) with the first term of the filtration on C ch (X, U (L)>0 )PQ [−1]. By adjunction (see (a) in 4.1.8), one has a morphism ¯ α : C(Γ(X, h(LPQ ))) → C ch (X, U (L)>0 )PQ of filtered BV algebras which is our (4.8.1.2). gr α Consider the morphism Sym>0 (Γ(X, h(LPQ ))[1]) −−→ gr· C ch (X, U (L)>0 )PQ = C ch (X, gr U (L)>0 )PQ (see (4.2.18.3)). It equals the composition of the map (4.6.2.2) for V = L and the map C ch (X, Sym>0 L) → C ch (X, gr U (L)>0 ) coming from the canonical morphism Sym>0 L → grU (L)>0 . If L is OX -flat, then the latter map is a quasi-isomorphism by the Poincar´e-Birkhoff-Witt (see 3.7.14), and the former one is a quasi-isomorphism by the proposition in 4.6.2. We are done.  Here are some variants of the above theorem: 4.8.2. Coefficients. Let T ⊂ X be a finite non-empty subset and jUT : UT := X r T ,→ X its complement. Suppose we have a Lie∗ algebra L on UT and for every t ∈ T a chiral L-module Mt supported at t (see 3.7.16–3.7.19). We have the enveloping chiral algebra U (L) and the Mt are U (L)-modules, so one has the corresponding chiral homology complex C ch (X, U (L), {Ms }); the standard filtration on U (L) makes it a filtered complex. On the other hand, we have a homotopy Lie algebra RΓDR (UT , L) and each h(Mt ) is an RΓDR (UT , L)-module, so we have the corresponding Lie algebra homology complex C(RΓDR (UT, L), ⊗h(Mt )) filtered in the obvious way. Proposition. There is a canonical morphism of filtered complexes (4.8.2.1)

C(RΓDR (UT , L), ⊗h(Mt )) → C ch (X, U (L), {Mt }),

which is a filtered quasi-isomorphism if L is OX -flat.

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355

Proof. Let us represent RΓDR (UT , L) by a Lie algebra Γ(UT , h(LP )), so the h(Mt ) are Γ(UT , h(LP ))-modules, and we have the corresponding filtered Chevalley complex C(Γ(UT, h(LP )), ⊗h(Mt )). Similarly, C ch (X, U (L), {Mt }) comes as the filtered complex C ch (UT , U (L), {Mt })P (see (4.2.19.3)). The morphism of Lie∗ algebras LP → U (L)P yields then in the obvious way a morphism of filtered complexes C(Γ(UT , h(LP )), ⊗h(Mt )) → C ch (UT , U (L), {Mt })P which is our (4.8.2.1). Let us show that (4.8.2.1) is a filtered quasi-isomorphism if L is OX -flat. By (4.2.18.3) and the PBW theorem in 3.7.14, one can pass to gr reducing our statement to the case when L is commutative and its action on Mt is trivial. Then one has C(Γ(UT , h(LP )), ⊗h(Mt )) = (⊗h(Mt )) ⊗ (k ⊕ Sym>0 (Γ(X, h(jT ∗ jT∗ LP ))[1])) and C ch (UT , U (L), {Mt })P = (⊗h(Mt )) ⊗ (k ⊕ C ch (X, jT ∗ jT∗ (Sym>0 L)P )). Our morphism is the tensor product of the identity map for ⊗h(Mt ) and the direct sum of idk and the arrow (4.6.2.2) for V = jT ∗ jT∗ L . We are done by 4.6.2.  4.8.3. Twisted enveloping algebras. Let L be a Lie∗ DG algebra on X, L[ its ω-extension, and U (L)[ the corresponding twisted chiral envelope (see 3.7.20). The standard filtration U (L)[· defines a homotopy unital BV structure on the chiral complex, so we have C ch (X, U (L)[· ) ∈ HoBVu (see 4.3.4). Explicitly, it is represented by the homotopy unital filtered BV algebra C ch (X, U (L)[ )PQ . As in 4.5.1, we have a Lie algebra Γ(X, h(L[PQ )) which is a central extension of Γ(X, h(LPQ )) by Γ(X, h(ωPQ )). The latter complex computes the de Rham homology of X (shifted by 1), so we have a trace map tr : Γ(X, h(ωPQ )) → k[−1] which is canonical up to a homotopy. Pushing out our central extension by tr, we get a central k[−1]-extension Γ(X, h(LPQ ))[ of Γ(X, h(LPQ )) by k[−1]. It yields the twisted Chevalley complex C(Γ(X, h(LPQ )))[ which is a filtered unital BV algebra (see (c) in 4.1.8). As an object of the homotopy category HoBVu , it does not depend on the auxiliary choices; we denote it by C(RΓDR (X, L))[ ∈ HoBVu . Proposition. For an OX -flat L there is a canonical isomorphism in HoBVu (4.8.3.1)

∼

C ch (X, U (L)[· ) −→ C(RΓDR (X, L))[ .

Sketch of a proof. One can either repeat the arguments in the non-twisted case or reduce the twisted situation to the non-twisted one using the fact that U (L)[ is a specialization of U (L[ ) and applying 4.3.9. The details are left to the reader.  The picture of 4.8.2 admits the following twisted versions (the proofs are similar to the proof of the proposition in 4.8.2): (i) Let T and UT be as in 4.8.2, let L be an O-flat Lie∗ algebra on UT and L[ its ω-extension. We have the Lie algebra Γ(UT , h(LP )) and its central Γ(UT , h(ωP ))extension Γ(UT , h(L[P )). Let {Mt }, t ∈ T , be chiral U (L)[ -modules supported at t ∈ T . Then ⊗h(Mt ) is naturally a Γ(UT , h(LP ))-module (indeed, it is a Γ(UT , h(L[P ))-module, and the action of Γ(UT , h(ωP )) is trivial). We have an evident morphism (4.8.3.2)

C(Γ(UT , h(LP )), ⊗h(Mt )) → C ch (X, U (L)[ , {Mt })P

which is a filtered quasi-isomorphism. (ii) Let L be an O-flat Lie∗ algebra on X and L[ its ω-extension. For T ∈ S consider the corresponding Lie algebra L\PX T in the tensor category of left DX T \ modules (see 3.7.6) and its central extension L[\ by ωPX T. PX T

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Let {Mt } be a T -family of chiral U (L)[ -modules. Then j (T )∗ Mt is an L\PU (T ) module (as in (i), j (T )∗  Mt carries a natural L[\ -action which factors through PU (T ) \ LPU (T ) ). We have the Lie algebra homology complex C(L\PU (T ) , j (T )∗  Mt ) and an evident morphism of filtered complexes of DX T -modules (T )

j∗ C(L\PU (T ) , j (T )∗  Mt ) → Cch (X, U (L)[ , {Mt })P

(4.8.3.3)

(T )

which is a filtered quasi-isomorphism. Notice also that j∗ C(L\PU (T ) , j (T )∗ Mt ) = (T )

C(L\PX T , j∗ j (T )∗  Mt ).

4.8.4. Chiral differential operators. Let R` be a commutative unital DX algebra, L a Lie∗ R-algebroid, L[ its chiral R-extension (see 3.9.6). Below we fix P, Q and write C ch (·) for C ch (X, ·)PQ . Consider the chiral envelope U (L)[ (see 3.9.11) equipped with the PBW filtration. It yields a filtered homotopy unital BV algebra C ch (U (R, L)[ ) (see 4.3.2 and 4.3.4). On the other hand, we have a homotopy unital commutative algebra C ch (R) and a homotopy unital C ch (R)-module C ch (R, L) := C ch (X, R, L)PQ (see 4.3.2 and 4.3.4). The latter is naturally a Lie C ch (R)-algebroid: the Lie bracket comes in the obvious manner from the Lie∗ bracket on LPQ and the LPQ -action on RPQ , and the action on C ch (R) comes from the LPQ -action on RPQ . The complex C ch (R, L[ ) is an extension of C ch (R, L) by C ch (R, R). Let ch C (R, L)[ be its push-out by the obvious map C ch (R, R) → C ch (R)[−1]. Our C ch (R, L)[ is naturally a homotopy unital BV extension of C ch (R, L) (see 4.1.9 and 4.1.15). Namely, the Lie bracket on C ch (R, L)[ comes from the Lie∗ bracket on L[PQ and the LPQ -action on RPQ , and the C ch (R)-module structure is the obvious “exterior product” map C ch (R) ⊗ C ch (R, L)[ → C ch (R, L)[ . The C ch (R)-action does not commute with the differential due to the fact that L[ is not a central R-module; the discrepancy is given by axiom (ii) of BV extensions (see 4.1.9)82 following from the definition of chiral extension (see 3.9.6). There is an obvious map C ch (R, L)[ → C ch (U (R, L)[ )[−1] compatible with the embeddings of C ch (R)[−1]. It is also compatible with the Lie bracket and the C ch (R)-action, and its image lies in the first term of the filtration. By universality we get a morphism CBV (C ch (R), C ch (R, L))[ → C ch (U (R, L)[ ) of the homotopy unital filtered BV algebras, hence a morphism in HoBVu (see 4.1.9 and 4.1.15) (4.8.4.1)

L CBV (C ch (R), C ch (R, L))[ → C ch (U (R, L)[ ).

Theorem. If R is homotopically OX -flat and L is a homotopically flat R` module, then this is a filtered quasi-isomorphism. ∼

Proof. By PBW, SymR L −→ gr U (R, L)[ . Now use (4.2.18.3) and 4.6.4.



Exercise. Suppose that R, L have degree 0. By (3.9.21.1), the category of chiral extensions of L is a torsor over the 2-term complex τ≤2 RΓ(X, h(F 1 CR (L))) = τ≤2 RΓDR (X, F 1 CR (L)) where F is the stupid filtration. The category of homotopy BV extensions of the homotopy Lie C ch (R)-algebroid C ch (R, L) is a torsor over the 2-term complex τ≤2 τ≥1 F 1 CC ch (R) (C ch (R, L)) = τ≤2 τ≥1 C ch (F 1 CR (L)) 82 We sincerely apologize for a hideous incongruency of notation: R and L[ from 4.1.9 are now C ch (R) and C ch (R, L)[ .

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(see (4.6.10.1)). The canonical morphism RΓDR (X, F 1 CR (L)) → C ch (X, F 1 CR (L)) maps the first truncated complex to the second one. Show that the functor L[ 7→ C ch (R, L)[ from the category of chiral extensions of L to the one of homotopy BV extensions of C ch (R, L) is affine with respect to the map e : τ≤2 RΓDR (X, F 1 CR (L)) → τ≤2 τ≥1 C ch (F 1 CR (L)). Example. Let Y be a smooth affine variety, X = P1 , JYX = Spec R` the jet ch scheme for Y × X/X, L = ΘR . Then SpechRi = Y and H>0 (X, R) = 0. So for ch a cdo A on Y the complex C (X, A) is the de Rham complex of Y with coefficients in OY with respect to certain right DY -module structure on OY , which is the same as a flat connection ∇A on ωY−1 . The isomorphism classes of cdo form an 2 HDR (X, DR(JYX /X)≥1 ) = H 2 (X, hDR(JYX /X)≥1 )-torsor. The “constant jet” embedding of DX -schemes Y × X ,→ JYX yields a morphism of DX -modules DR(JYX /X) → DR(Y ) ⊗ ωX . Passing to de Rham cohomology and applying ∼ 2 tr : RΓDR (X, ωX )[1] −→ k, we get a morphism e : HDR (X, DR(JYX /X)≥1 ) → 0 ≥1 H DR(Y ) = the closed 1-forms on Y . By the exercise above, the map A 7→ ∇A is e-affine. Notice that the canonical projection JYX → Y × X yields a morphism of complexes DR(Y ) ⊗ OX → DR(JYX /X)` , hence the one DR(Y )≥1 ⊗ ωX → DR(JYX /X)≥1 → hDR(JYX /X)≥1 . Applying the functor H 2 (X, ·), we get a right inverse to e. Thus e is a surjective map, so ∇A can have arbitrary irregular singularities at infinity. Question. According to §12 of [GMS2], any right DY -module structure on OY , i.e., a flat connection ∇ on ωY−1 , yields a Virasoro vector in any cdo on the “universal” jet scheme of Y in the setting of graded vertex algebras. The latter produces an action of the the group ind-scheme Autk[[t]] on the cdo, which can be planted then on the jet scheme of Y over any curve X. Take for A in the above example such a cdo. Is it true that ∇A = ∇? 4.8.5. Corollary. Suppose R` is a very smooth plain DX -algebra and A is a chiral R` -cdo. Then dim H·ch (X, A) < ∞. Proof. By the above theorem, 4.6.9, 4.6.8, and 4.6.6, C ch (X, A) is a perfect BV algebra (see 4.1.18), and we are done by the proposition in 4.1.18.  Question. Is it true that the chiral homology of a (formal) quantization of any symplectic coisson algebra is finite-dimensional? Presumably, the finiteness can be seen from the first order of deformation, so the picture of 3.9.10 may provide the clue. 4.9. Chiral homology of lattice chiral algebras The fact that conformal blocks of a lattice Heisenberg algebra with positive κ are appropriate θ-functions is standard in mathematical physics; for a mathematical proof see [Ga] 6.2.2 and also [FS]. The proof of the theorem in 4.9.3 presented below uses the Fourier-Mukai transform which helps to reduce it to the particular case of the simplest commutative lattice chiral algebra. The descent construction in 4.9.1 is a particular case of [BD] 4.3.12, 4.3.13.

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4.9.1. We follow the notation of 4.10, so we have a lattice Γ and the corresponding torus T = Gm ⊗ Γ. Let Tors(X, T ) be the algebraic Picard stack of T -torsors on X; this is an algebraic stack. For a test affine scheme Z set T (X)rat (Z) := − lim → T (U ) where the limit is over the set of all open U ⊂ X × Z such that the fiber of U over any point of Z is non-empty. In other words, an element of T (X)rat (Z) is a family of rational maps X → T parametrized by Z. One easily checks that T (X)rat is a sheaf for the fppf topology. We also have a sheaf of Γ-divisors Div(X, Γ) := Div(X) ⊗ Γ (see 3.10.7), a morphism of sheaves div: T (X)rat → Div(X, Γ), and a canonical identification of the Picard stacks (4.9.1.1)

∼

π : Cone(div) −→ Tors(X, T )

where the projection Div(X, Γ) → Tors(X, T ) is D ⊗ γ 7→ O(D)⊗γ . Proposition. This projection yields an equivalence of the Picard groupoids of line bundles (4.9.1.2)

∼

Pic(Tors(X, T )) −→ Pic(Div(X, Γ)).

Proof. It suffices to check that for any test scheme Z - every regular function ϕ on T (X)rat × Z comes from Z; - every line bundle M on T (X)rat × Z comes from Z. It suffices to prove our statements for T = Gm . Choose an ample line bundle L on X and set Vn := H 0 (X, L⊗n ), Vn0 := Vn r {0}. Define pn : Vn0 × Vn0 → Gm (X)rat by (f, g) 7→ f /g. Our ϕ defines a regular function p∗n ϕ on Vn0 × Vn0 × Z which is invariant with respect to the obvious action of Gm on Vn0 × Vn0 . Suppose that n is big enough, so dim Vn > 1. Then p∗n ϕ extends to a Gm -invariant regular function on Vn × Vn × Z, which necessarily comes from Z. Similarly, p∗n M extends to a Gm -equivariant line bundle on Vn × Vn × Z whose restriction to the diagonally embedded Vn × Z comes from Z; such an object comes from a uniquely defined line bundle on Z.  4.9.2. By 3.10.7 and the above proposition we have canonical morphisms of the Picard groupoids (4.9.2.1)

∼

Pθ (X, Γ) −→ Picf (Div(X, Γ)) → Pic(Tors(X, T ))

where Pic is the Picard groupoid of super line bundles. For λ ∈ Picf (Div(X, Γ)) we denote the corresponding super line bundle on Tors(X, T ) also by λ. Example. Suppose Γ = Z and λ ∈ Picf (Div(X)) is defined by formula (3.10.7.4). The corresponding super line bundle on Tors(X, Gm ) = Pic(X) is λL = detRΓ(X, L) ⊗ det⊗−1 RΓ(X, OX ). The group of connected F components of Tors(X, T ) equals Γ (the degree of the T -torsor), Tors(X, T ) = Tors(X, T )γ . Each line bundle λ on Tors(X, T ) yields a map δλ : Γ → Γ∨ so that for F ∈ Tors(X, T )γ the group T = Aut(F) acts on the fiber λF by the character δλ (γ) : T → Gm . The map λ 7→ δλ is Z-linear: one has δλ⊗λ0 = δλ + δλ0 . For θ ∈ Pθ (X, Γ) we write δθ := δλ where λ corresponds to θ by (4.9.2.1).

4.9. CHIRAL HOMOLOGY OF LATTICE CHIRAL ALGEBRAS

359

Lemma. (i) For θ ∈ Pθ (X, Γ)κ (see 3.10.3) the map δθ : Γ → Γ∨ is affine with the linear part equal to κ, hκ(γ), γ 0 i = κ(γ, γ 0 ). If θ is symmetric,83 then δθ = κ. (ii) For θ ∈ Pθ (X, Γ)0 = P(X, Γ) = T ors(X, T ∨ ) (see (3.10.3.1) the map δθ is constant with image equal to the degree of the T ∨ -torsor. Proof. Let us check (ii) first. For θ ∈ P(X, Γ) the corresponding λ is an extension of Tors(X, T ) by Gm (see (3.10.7.3)). So δθ is constant and its value is clear from [SGA 4] Exp. XVIII 1.3. Let us prove (i). By linearity and the argument from the end of the proof of the proposition in 3.10.7, it suffices to check the first statement for Γ = Z and κ the product pairing, which follows from the example above. The second statement follows from the first one.  Remark. If δλ (γ) = 0, then the restriction of λ to Tors(X, T )γ is the pullback of a uniquely defined line bundle on Tors (X, T )γ (the coarse moduli space of classes of isomorphisms of T -torsors of degree γ) which we denote also by λ. 4.9.3. Let A be a lattice chiral algebra. Let θ ∈ Pθ (X, Γ)κ be its θ-datum (see 3.10.4), λ the super line bundle on Tors(X, T ) defined by θ, and λ∗ = λ⊗−1 the dual bundle. Theorem. There is a canonical quasi-isomorphism (4.9.3.1)

∼

C ch (X, A) −→ RΓ(Tors(X, T ), λ∗ )∗ = ⊕ RΓ(Tors(X, T )γ , λ∗ )∗ γ∈Γ

compatible with the Γ-gradings. Here the Γ-grading of C ch (X, A) comes from the Γ-grading of A. Remarks. (i) The γ-components of the right-hand side of (4.9.3.1) for which δλ (γ) 6= 0 vanish. If δλ (γ) = 0, then RΓ(Tors(X, T )γ , λ∗ ) = RΓ(Tors(X, T )γ , λ∗ ). If κ is non-degenerate, then there is only one such γ; in particular, the chiral homology is finite-dimensional. (ii) More precisely, the numerical class of λ∗ on T ors(X, T )γ (which is a J(X)⊗ Γ-torsor) equals ξ ⊗ κ where ξ is the canonical polarization of the Jacobian J(X). Therefore for κ non-degenerate, the only non-zero chiral homology group occurs in the degree equal to the product of g and the number of negative squares in κ; its dimension is equal to |det κ|g . (iii) The description of the chiral homology of A with coefficients84 is left to the inquisitive reader. 4.9.4. Proof of the theorem. First let us check that the chiral homology satisfies the same vanishing property as the right-hand side of (4.9.3.1): Lemma. If δλ (γ) 6= 0, then C ch (X, A)γ = 0. Proof. Consider the Lie∗ subalgebra αθ : tθD ,→ A0 (see 3.10.9). Therefore t = Γ(X, h(tD )) acts on A by the adjoint action. By (ii) and (iv) in the proposition in 3.10.9, t acts on each Aγ by the character κ(γ) ∈ Γ∨ , so it acts on C(X, A)γ in the same manner. On the other hand, according to Remark in 4.5.3 and Remark (ii) in 3.10.9, t acts on C(X, A) by the character − deg F∨ ∈ Γ∨ where θ = θsym F, θsym ∈ Pθ (X, Γ)κ is symmetric and F ∈ Tors(X, T ∨ ) = P(X, Γ). So if C ch (X, A)γ 6= 0, then κ(γ) + deg F∨ = 0, and we are done by the lemma from 4.9.2.  83 I.e., 84 The

invariant with respect to the involution γ 7→ −γ. category of A-modules was described in 3.10.13 and 3.10.14.

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4.9.5. We use the notation from 3.10.8. For each I ∈ S we have an ind-X I -scheme Div(X, Γ)X I := Div(X, Γ)S,X I (where S is the standard divisor ∪xi ). These ind-schemes form naturally an S◦ diagram. Interpreting Γ-divisors as T -torsors equipped with meromorphic trivializations, we get a canonical projection ϕ = ϕI : Div(X, Γ)X I → Tors(X, T ) which is formally smooth. So our diagram lives over Tors(X, T ). Notice that the canonical connection ∇ from the remark in 3.10.8 acts along the fibers of ϕ. Our ϕ yields the projection φ = φI : Div(X, Γ)X I → Tors(X, T ). Let us fix some γ ∈ Γ such that δλ (γ) = 0, and consider the γ-component of our picture. Set P := Tors(X, T )γ , P := Tors(X, T )γ , and GX I := Div(X, Γ)γX I (the Γ-divisors of degree γ). The GX I ’s form an S◦ -diagram GX S of ind-schemes. ϕ → P → P ; the composition of We have the projections π : GX S → X S and GX S − the latter arrows is φ. As was mentioned in the remark in 4.9.2, our λ is a super line bundle on P . 4.9.6. Let us recall some terminology from [BD] 7.11.4. Let Y be an indscheme of ind-finite type. An O! -module M on Y is a rule that assigns to any closed subscheme Z ⊂ Y a quasi-coherent OZ -module MZ , and to any Z 0 ⊂ Z an embedding MZ 0 ⊂ MZ which identifies MZ 0 with the submodule of sections of MZ killed by the ideal IZ 0 of Z 0 ; the embeddings should be transitive. If L is a line bundle on Y , then L ⊗ M , (L ⊗ M )Z := LZ ⊗ MZ , is an O! -module on Y . S For a morphism p : Y → B where B is a scheme, we set p! M := p|Z∗ MZ ; for B = Spec k we write p! M =: Γc (Y, M ). All O! -modules on Y form an abelian category. For a scheme Q of finite type we denote by DQ the dualizing complex of Q realized as the Cousin complex (see [Ha]). For Y as above the complexes DZ form an O! -module DY on Y . If p : Y → B is ind-proper and B is of finite type, then we have the canonical trace map trp : p! DY → DB . 4.9.7. We will consider O! -modules on the ind-schemes GX I . Recall (see 3.10.8) that GX I = Spf R where R = RX I is a topological OX I -algebra which is the projective limit of its quotients Rα = R/Iα which are finite and flat over OX I . For any O! -module M on GX I its image π! M is naturally a discrete R-module; this establishes an identification of the category of O! -modules on GX I and that of discrete R-modules (which are quasi-coherent as OX I -modules). The functor π! admits a right adjoint π ! which S assigns to an OX I -module N the discrete R-module π ! N = HomOX I (R, N ) := HomOX I (Rα , N ); both π! and π ! are exact functors. A line bundle on GX I is the same as an invertible topological R-module. One has DGX I = π ! DX I . Since DX I is a right DX I -complex and ΘX I acts on R via ∇, our π! DGX I is a right DX I -complex. Since ∇ acts along the fibers of φ, the line bundle φ∗ λ on GX I is equipped with a left ΘX I -action. Therefore π! (φ∗ λ ⊗ DGX I ) = φ∗ λ ⊗ π! DGX I is a right DX I -complex. R

The above objects are compatible with the embeddings coming from arrows in S, so the DGX I form a !-complex on GX S , etc. The right D-complex π! (φ∗ λ⊗DGX S ) on X S is evidently admissible. Proposition. There is a canonical quasi-isomorphic embedding in CM(X S ) (4.9.7.1)

C(A)γX S ,→ π! (φ∗ λ ⊗ DGX S ).

4.9. CHIRAL HOMOLOGY OF LATTICE CHIRAL ALGEBRAS

361

= π! (φ∗ λ ⊗ π ! OX I ). Therefore Proof. According to (3.10.8.1), one has A`γ XI `γ γ ∗ ! C(A) := AX I ⊗ C(ω)X I = π! (φ λ ⊗ π C(ω)X I ). Since C(ω)X I is the Cousin resolution of ωX I [|I|] with respect to the diagonal stratification and DX I is its whole Cousin resolution, we have a canonical quasi-isomorphic embedding C(ω) ,→ DX I . Our (4.9.7.1) is the embedding π! (φ∗ λ ⊗ π ! C(ω)X I ) ,→ π! (φ∗ λ ⊗ π ! DX I ).  4.9.8. Passing to the de Rham cohomology, the proposition yields an identification (see 4.2.6(iv) for the notation) (4.9.8.1)

∼

S ∗ C ch (X, A)γ −→ ΓDR g (X , π! (φ λ ⊗ DGX S )).

Let DR∇ (DGX S ) and DR∇ (φ∗ λ ⊗ DGX S ) be the de Rham complexes along g ∇ the corresponding canonical nice resolutions (see the ∇-foliation; denote by DR S ∗ 4.2.6(iv)). These are complexes of !-sheaves on GX S , and ΓDR g (X , π! (φ λ⊗DGX S )) ∗ g ∇ (φ λ ⊗ DG )). = Γc (GX S , DR XS

The differential of the DR∇ complexes is φ−1 OP -linear. Therefore we have S◦ systems of complexes φ! DR∇ (DGX S) and φ! DR∇ (φ∗ λ⊗DGX S ) = λ⊗φ! DR∇ (DGX S). g ∇ -complexes. One has Γc (GX S , DR g ∇ (DG ⊗ φ∗ λ)) = The same is true for the DR XS

◦ g Γ(P, λ ⊗ lim −→ φ! DR∇ (DGX S )) where lim −→ is the inductive S -limit. Combining the identifications, we get

(4.9.8.2)

∼ g C ch (X, A)γ −→ Γ(P, λ ⊗ lim −→ φ! DR∇ (DGX S )).

4.9.9. We have the trace maps trφ : φ! DGX I → DP which extend canonically to the morphisms φ! DR∇ (DGX I ) → DP (killing all the other components of the DR∇ -complex). Passing to the inductive limit, we get lim −→ φ! DR∇ (DGX S ) → DP . g Composing it with the projection DR∇ → DR∇ , we get a morphism of complexes of OP -modules (4.9.9.1)

g trφ∇ : − lim → φ! DR∇ (DGX S ) → DP .

We define the γ-component of (4.9.3.1) as the composition of (4.9.8.2) and the ∼ ∗ ∗ g morphisms Γ(P, λ ⊗ lim −→ φ! DR∇ (DGX S )) → Γ(P, λ ⊗ DP ) −→ RΓ(P, λ ) where the first arrow comes from idλ ⊗ trφ∇ and the second arrow is the Serre duality. To finish the proof of the theorem, we need to prove that the constructed arrow is a quasi-isomorphism. This follows from the next proposition: Proposition. trφ∇ is a quasi-isomorphism of complexes of OP -modules. Proof. A morphism f of complexes of OP -modules is a quasi-isomorphism if (and only if) its Fourier-Mukai transform Φ(f ) is. We will check that it happens with trφ∇ . The Fourier-Mukai transform85 Φ takes values in the derived category of complexes of O-modules on the (coarse) moduli space of line bundles on P which are algebraically equivalent to 0. The latter identifies in the usual way with the moduli space Q := Tors(X, T ∨ )0 of T ∨ -torsors of degree 0. Namely, for q ∈ Q the corresponding line bundle λq on P is constructed as follows. Let Fq be the T ∨ -torsor of degree 0. It defines a line bundle on Tors(X, T ) according to, say, (3.10.7.3) and (4.9.1.2); since Fq has degree 0, our line bundle comes from a (uniquely defined) 85 See

the recent book [Po] on the subject.

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line bundle on Tors(X, T ). Our Lq is its restriction to P := Tors(X, T )γ . Therefore λq coincides with the line bundle λ that corresponds to the commutative lattice chiral algebra Aq such that Spec A`q = JFq (the jet scheme of F; see 3.10.2). Recall that Φ is defined by means of a “kernel” which is a line bundle on Q × P whose restriction to the fiber over each q equals λq . Looking back at (4.9.8.2), we see that Φ(trφ∇ ) is equal to the γ-component of the morphism (4.9.3.1) for the family of lattice chiral algebras AQ parametrized by Q.86 So, what we want to do is to prove our theorem for our special family of commutative lattice algebras. Notice that φ(DP ) = δ0 := the skyscraper O-module at 0 ∈ Q. Summing up with respect to all γ ∈ Γ, we get a morphism of OQ -complexes τ : C ch (X, AQ ) → k[Γ] ⊗ δ0 . We want to check that this is a quasi-isomorphism. ∼ First notice that H 0 τ : hAQ i = H0ch (X, AQ ) −→ k[Γ]⊗δ0 . Indeed, the Q-scheme Spec hAQ i is the space of horizontal sections of Spec A`Q , so it is a copy of T ∨ which lives over 0 ∈ Q. One checks in a moment that H 0 τ does not vanish on each of the γ-components, so it is an isomorphism. Since H·ch (X, AQ ) is a unital hAQ i-module, we see that it vanishes outside 0 ∈ Q. It remains to check that the morphism Li∗0 τ , where i0 : {0} ,→ Q, is a quasi-isomorphism. We have Li∗0 C ch (X, AQ ) = C ch (X, A0 ) and Li∗0 k[Γ] ⊗ δ0 ' k[Γ] ⊗ Tor· (δ0 , δ0 ) = k[Γ] ⊗ Sym(V [1]) where V is the cotangent space to Q at 0. It follows from the construction that our map H·ch (X, A0 ) → k[Γ] ⊗ Sym(V [1]) is a morphism of commutative algebras. It is isomorphism in degree 0 and surjective in degree −1 (since H 0 τ is an isomorphism). To finish the proof, it suffices to check that the commutative algebra H·ch (X, A0 ) is generated by H0ch and H1ch , and dim H1ch is equal to dim Q = rk(Γ)g where g is the genus of X. Let us embed our torus T ∨ into a vector space K of the same dimension. We have the open embedding of the jet schemes JT ∨ = Spec A`0 ,→ JK =: Spec R` . Therefore H·ch (X, A0 ) = H·ch (X, R) ⊗ hA0 i (see (4.3.12.2)), and hRi

H·ch (X, R) = Sym(K ∗ ⊗ RΓ(X, ω)[1]) by the proposition in 4.6.2. We are done.  4.9.10. Questions. What would be an analog of the above theorem for a chiral algebra coming from an arbitrary ind-finite chiral mononoid (see 3.10.16)? Suppose A is the integrable quotient of the Kac-Moody algebra U (gD )κ defined by a semi-simple algebra g and a positive integral level κ. Is it true that all the higher chiral homologies of A vanish?

86 The above considerations generalize to the situation of families of lattice algebras in a straightforward way.

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Index and Notation action of a Lie∗ algebra on a commutative! algebra 1.4.9 action of a Lie∗ algebra on a chiral algebra 3.3.3 action of a Lie algebroid on a chiral algebra 4.5.4, twisted 4.5.6 action of a pseudo-tensor category on a category 1.2.11 action of a tensor category on a pseudo-tensor category 1.1.6(v) admissible complex of sheaves on X S 4.2.1 admissible D-complex on X S 4.2.6 algebraic DX -space 2.3.1 annihilator 3.3.7 augmentation functor, non-degenerate 1.2.5, reliable 1.4.7 augmentation functor in compound setting 1.3.10, in D-module setting 2.2.7 augmented compound tensor category, functor 1.3.16 augmented operad 1.2.4 augmented pseudo-tensor category 1.2.4 augmented pseudo-tensor functor, unit 1.2.8 Batalin-Vilkovisky (BV) algebras 4.1.6 BV extension of a Lie algebroid 4.1.9 BV quantization of an odd Poisson algebra 4.1.6 BV structure on the chiral chain complex 4.3.1 BRST reduction, charge, differential: classical 1.4.23, quantum 3.8.9, 3.8.20 BRST property 1.4.24, 3.8.10, 3.8.21 calculus of variations 2.3.20 cdo 3.9.5 centralizer 3.3.7 center of a chiral algebra 3.3.7 central chiral modules 3.3.7 Chern classes chD n 2.8.10 Chevalley-Cousin complex of a chiral algebra 3.4.11, relative version 4.4.9 Chevalley complex of a Lie∗ algebra, inner 1.4.5, 1.4.10 chiral action of a Lie∗ algebra 3.7.16, of a chiral Lie algebroid 3.9.24 chiral action of a chiral monoid 3.10.17 chiral algebras 3.3.3, commutative 3.3.3, non-unital 3.3.2, universal 3.3.14 chiral algebra freely generated by (N, P ) 3.4.14, chiral RDif -algebras 3.9.4 chiral enveloping algebra of a Lie∗ algebra 3.7.1, of a chiral Lie algebroid 3.9.11 chiral extension of a Lie∗ algebroid 3.9.6, rigidified 3.9.8 chiral homology 4.2.11 chiral lattice algebras 3.10.1 369

370

INDEX AND NOTATION

chiral Lie algebroids 3.9.6 chiral modules 3.3.4 chiral L-modules 3.7.16 chiral RDif -modules 3.9.1 chiral monoid 3.10.17 chiral operations 3.1.1, for (g, K)-modules 3.1.16 chiral A-operations 3.3.4 chiral Lch -operations 3.7.16 chiral product 3.3.2 chiral pseudo-tensor structure 3.1.2 Clifford algebra, coisson 1.4.21, chiral 3.8.6, linear algebra version 3.8.17 coisson algebras 1.4.18, modules 1.4.20, D-module setting 2.6.1, elliptic 2.6.6 commutative! algebras, modules 1.4.6 commutative DX -algebras 2.3.1 complementary quotients 1.3.1 compound operad 1.3.18 compound pseudo-tensor category 1.3.7, augmented 1.3.10 compound pseudo-tensor functor 1.3.9 compound tensor category, functor 1.3.14 compound tensor product maps 1.3.12, binary 1.3.13 connections for Lie∗ algebroids 1.4.17 connections on chiral homology 4.5 Contou-Carr`ere symbol 3.10.13 convenient DX -algebra 4.6.1 convenient R` -modules 4.6.3 coordinate system on a DX -scheme 2.3.17 c operations 1.4.27, in D-module setting 3.2.5 correlators 4.4.1 cotangent complex 2.3.15, 4.1.5, 4.6.6 cotorsor 3.4.16, 3.10.12 Cousin D-complex 4.2.9 Cousin filtration 4.2.1, 4.2.19 Cousin spectral sequence 4.2.3, 4.2.11, 4.2.19 c-stack, c-rank 2.9.10 DX -algebras 2.3.1 D-algebra point 2.3.1 D-modules: left and right 2.1.1, functoriality 2.1.2, induced 2.1.8 D-modules: quasi-induced 2.1.11, maximal constant quotient 2.1.12 D-modules: topology at a point x 2.1.13, universal 2.9.9 D-modules on R(X), left 3.4.2 D-modules on X S , right 3.4.10 de Rham-Chevalley complex of a Lie∗ algebroid, inner 1.4.14 de Rham complex of a D-module 2.1.7 de Rham homology 2.1.12, cohomology 2.1.7 DX -scheme 2.3.1, formally smooth, smooth, very smooth 2.3.15 DX -scheme: the global space of horizontal sections 2.4.1 DX -scheme: the local space of horizontal sections 2.4.8 Dolbeault algebras 4.1.3

INDEX AND NOTATION

371

Dolbeault DR(X) -algebra 4.2.7 Dolbeault resolutions 4.1.4 Dolbeault-style algebra 4.1.4 Dolbeault-style DR(X) -algebra 4.2.16 duality 1.3.11, 2.2.16 duality for de Rham cohomology, global 2.2.17, local 2.7.10 elliptic morphism, Lie∗ algebroid 2.5.22, coisson algebra 2.6.6 enveloping algebra of an operadic algebra 1.2.16, of a Lie algebroid 2.9.2 enveloping BV algebra of a BV algebroid 4.1.8. 4.1.9 enveloping chiral algebra of a chiral Lie∗ algebroid 3.9.11 Euler-Lagrange equations 2.3.20 factorization algebras 3.4.1, D-module setting 3.4.4, truncated 3.4.13 factorization algebras: canonical D-module structure 3.4.7, commutative 3.4.20 factorization algebra freely generated by (N, P ) 3.4.14 factorization B-modules 3.4.18 factorization structure 3.4.4 filtration on a chiral algebra, commutative, unital 3.3.12 flabby complex of sheaves 4.2.2 formal groupoid 1.4.15 formally smooth/´etale morphism of DX -schemes 2.3.16 Fourier-Mukai trnasform 4.9.9 (g, K)-modules 2.9.7, chiral structure 3.1.16 (g, K)-structure 2.9.8 Gelfand-Dikii coisson algebra 2.6.8 Gelfand-Kazhdan structure 2.9.9 group action on a chiral algebra 3.4.17 hamiltonian reduction 1.4.19 handsome complexes of !-sheaves on X S 4.2.2 Harish-Chandra pair, module 2.9.7 Heisenberg Lie∗ algebra 2.5.9 Heisenberg group 3.10.13 homotopically OX - and DX -flat complexes 2.1.1 homotopy unit commutative algebra 4.1.14 homotopy unital commutative algebra 4.1.14, BV algebra 4.1.15 Hopf chiral algebras 3.4.16 ind-scheme 2.4.1 induced modules 3.7.15, 3.9.24 inner Hom, inner P objects 1.2.1, in augmented sense 1.2.7 jet scheme 2.3.2–2.3.3 Kac-Moody extension 2.5.9 Kashiwara’s lemma 2.1.3 Knizhnik-Zamolodchikov (KZ) equations 4.4.6 lattices, c- and d- 2.7.7 Lie∗ algebras and modules 1.4.4, D-module setting 2.5.3–2.5.4 Lie algebroid 2.9.1

372

INDEX AND NOTATION

Lie∗ algebroid 1.4.11, D-module setting 2.5.16, elliptic 2.6.6 Lie coalgebroid 1.4.14 matrix ∗ algebra 1.4.2 middle de Rham cohomology sheaf h 2.1.6 Miura torsor 2.8.17 module operads 1.2.11 modules over operadic algebras 1.2.13 morphisms of DG DX -algebras: semi-free, elementary 4.3.7 multijet scheme 3.4.21 mutually commuting morphisms of chiral algebras 3.4.15 n-coisson algebra 1.4.18 n-Poisson algebra 1.4.18 nice complexes of !-sheaves on X S 4.2.1 non-degenerate pairs (F, ∇h ) 2.5.23 normally ordered tensor product 3.6.1 O-modules on R(X) 3.4.2 odd coisson algebra 1.4.18 odd Poisson algebra 1.4.18 ope algebra, associative, commutative 3.5.9 oper 2.6.8 operad 1.1.4, of Jacobi type 3.7.1 operadic algebra 1.1.6(iii), augmented 1.2.8 operator product expansion 3.5.8 perfect BV algebra 4.1.18 perfect commutative DG algebra 4.1.17 perfect complexes 4.1.16 Poincar´e-Birkhoff-Witt theorem for Lie∗ algebras 3.7.14, twisted case 3.7.20 PBW theorem for usual Lie algebroids 2.9.2, 3.9.12 PBW theorem for chiral Lie algebroids 3.9.11 polydifferential ∗ operations 1.4.8, D-module setting 2.3.12 pre-factorization algebra 3.4.14 projective connections 2.5.10 pseudo-tensor category 1.1.1 pseudo-tensor category: A-, k-, additive, abelian 1.1.7 pseudo-tensor functors, adjointness 1.1.5 pseudo-tensor k-category 1.1.7 pseudo-tensor product 1.1.3 pseudo-tensor structure 1.1.2 pseudo-tensor subcategory, full 1.1.6(i) quantization of a coisson algebra 3.3.11, mod t2 3.9.10 quasi-factorization algebra 3.4.14 Ran’s space 3.4.1 reasonable topological algebra 2.4.8 representable pseudo-tensor structure 1.1.3 resolutions of commutative DX -algebras 4.3.7 rigidification of a Lie∗ algebroid 1.4.13, of a Lie algebroid 2.9.1

INDEX AND NOTATION

rigidified B-extension 3.9.8 A-structure 1.2.11 ∗ algebras 1.4.1, D-module setting 2.5.1 ∗ pairing, non-degenerate 1.4.2 ∗ operations for D-modules 2.2.3, induced case 2.2.4(i) Schouten-Nijenhuis bracket 1.4.18 semi-free DX -algebras 4.3.7, 4.6.1 semi-free modules 4.1.5, 4.6.3 smooth DX -algebras 2.3.15 special DX I -module 3.1.6 stress-energy tensor 3.7.25 Sugawara’s construction 3.7.25 super complex 1.1.16 super conventions 1.1.16 I-topology 2.1.17 Ξ-topologies: Ξx - 2.1.13, ΞLie 2.5.12, ΞRL 2.5.18, Ξsp x x x 2.7.11 as cois Ξ-topologies: Ξx 2.6.3, Ξx 3.6.4 tangent Lie∗ algebroid 1.4.16, for a D-scheme: Remark (ii) in 2.3.12, 2.3.15 Tate extension: D-module setting 2.7.2, 2.7.3, on a DX -scheme 2.7.6 Tate extension: linear algebra setting 2.7.8, chiral approach 3.8.5 Tate extension of a Lie∗ algebra 2.8.15, 3.8.7 Tate structure on a vector D-bundle 2.8.1 Tate vector space, compact, discrete 2.7.7 tensor product of chiral algebras 3.4.15 tensor product of pseudo-tensor categories 1.1.9 topological associative algebra 3.6.1 topological commutative algebra 2.4.1 topological Lie algebroid 2.5.18 transversal quotients 1.3.1 twists of chiral algebras 3.4.17 unit object in a compound tensor category 1.3.16, strong 1.3.17 vector D-bundles 2.1.5, on a DX -scheme 2.3.10 vector DX -scheme 2.3.19 vertex operator 3.5.14 very smooth DX -algebras 2.3.15 Virasoro extension 2.5.10 Virasoro vector 3.7.25 W -algebras 3.8.16 Weyl algebra, chiral and coisson 3.8.1, linear algebra version 3.8.17 Wick algebra 3.6.11, global 3.6.20 θ-datum 3.10.3 Notation hAi 2.4.1, 4.2.16 Aas x 2.4.8, 3.6.2, 3.6.4, 3.6.13

373

374

INDEX AND NOTATION

Aas 3.6.14 ALie 3.3.3 A(P ) 3.4.17 Aw x 3.6.11 Aw (X) 3.6.20 B(M ) 3.4.18 ¯ 4.1.6 BV, BVu , BV C(A) 3.4.11 C(B, A) 4.4.9 C ch (X, A) 4.2.11 C˜ ch (X, A)Q , C ch (X, A)PQ 4.2.12 C ch (X, A, {Ms }), C ch (X, A, {Ms })PQ , Cch (X, A, {Ms })PQ 4.2.19 Cch (X, A, M )PQ , Cch (X, A, MI )PQ 4.2.19 C ch (X, B, A)PQ 4.4.9 CA(X) 3.3.3 CA(X)F 3.4.16 C`R (L, ·), CrR (L, ·), CR (L) 2.9.1 Cpois (R) 2.9.1 CM(R(X)) 4.2.6 Ccois (R) 1.4.18 Comu 1.4.6, 4.1.6 ComuD (X) 2.3.1 (S) ∆∗ 3.4.10 DDR 3.9.18 DM(R(X)) 4.2.6 End∗ (V ) 2.2.15, 2.5.6(a) Exp(X) 3.4.1 gl(V ) 2.5.6(a) gl(V )[ 2.7.4, 2.7.8 H·ch (X, A) 4.2.11 · (X, M ) 2.1.7 HDR H·DR (X, M ) 2.1.12 HoC 4.1.7 JZ 2.3.2 L† 1.1.16 L♦ 1.4.16, 2.9.1 M ` , N r 2.1.1 Mconst 2.1.12 MX 2.8.17 Mr (X), M` (X), M(X) 2.1.1 M(X)∗! 2.2.6 M(X)ch 3.1.2, M(X)cl 3.2.5 M(X, R` ) 2.3.5 M(X, A), M(X, A)ch 3.3.4 M(X S ) 3.4.10 M` (R(X)) 3.4.2 OI 3.5.8 Pch (A, L) 3.9.6

INDEX AND NOTATION

Pch (L) 3.9.6 Pcl (L) 2.8.2 P(F ) 3.4.16 Q(I) 1.3.1 Q(I, m) 1.4.27 R(X) 3.4.1 S 1.1.1 ˆ 1.2.4 S SchD (X) 2.3.1 Spec hR, Li 2.5.18 Spf Q 2.4.1 Tate(V ), Tate(Y) 2.8.1 U (I) 3.1.1 U [J/I] 3.4.4 U (L) 3.7.1 V(a) 3.6.14 Z(A) 3.3.7 λI 2.2.2, 3.1.4 ⊗∗ , ⊗ch 3.4.10 ~ 3.6.1 ⊗ Ξas x 2.4.8, 3.6.4 Ξx 2.1.13 2.5.12 ΞLie x ΞR x 2.4.11 ΞRL 2.5.19 x 1ch = 1ch A 4.2.16.

375