Faisceaux pervers intro(en)

AN INTRODUCTION TO PERVERSE SHEAVES

Antoine Chambert-Loir

Antoine Chambert-Loir Université Paris-Diderot. E-mail : [email protected]

Version of February 12, 2017, 12h28 The most up-do-date version of this text should be accessible online at address http://webusers.imj-prg.fr/~antoine.chambert-loir/enseignement/2016-17/ pervers/pervers.pdf ©2017, Antoine Chambert-Loir

CONTENTS

1. Triangulated categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Sets, universes and categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Limits, colimits, adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Additive and abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Complexes in additive categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. Triangulated categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6. Decent cohomological functors, and applications . . . . . . . . . . . . . . . . 1.7. The octahedral axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8. The homotopy category of an additive category . . . . . . . . . . . . . . . . . . 1.9. Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10. Derived categories and derived functors . . . . . . . . . . . . . . . . . . . . . . . . 1.11. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 5 7 12 18 22 28 36 41 58 67

2. Cohomology of sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. General topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Abelian categories of abelian sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Extensions by zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Cohomology with compact support . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7. Verdier duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8. The six operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9. Constructible sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 69 70 73 75 79 84 84 84 84 84

3. Truncation structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.1. Definition of truncation structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

iv

CONTENTS

3.2. The heart of a truncation structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.3. Glueing truncation structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.4. Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4. Perverse sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

CHAPTER 1 TRIANGULATED CATEGORIES

1.1. Sets, universes and categories 1.1.1. — We shall work within the classical framework of set theory, as formalized by the zfc axioms: Zermelo-Fraenkel with choice. However, the inexistence of a ‘‘set of all sets’’ makes this framework not really adequate to consider the usual categories or functors. We thus complement this theory with Grothendieck’s concept of universes. Definition (1.1.2). — An universe is a non-empty set U satisfying the following properties: a) b) c) d) e)

For every set x ∈ U and every element y ∈ x, one has y ∈ U; For every x, y ∈ U, one has {x, y} ∈ U; For every x ∈ U, one has P(x) ∈ U; For every I ∈ U and every family (x i )i∈I of elements of U, one has ⋃i∈I x i ∈ U; The set N belongs to U.

1.1.3. — In some precise sense, an universe can be seen as a model of set theory: the axioms of a universe precisely guarantee that all classical operations of sets do not leave a given universe. For example, if x, y are elements of a universe U, then the pair (x, y), defined à la Kuratowski by (x, y) = {{x}, {x, y}} belongs to U. Then the product set x × y, a subset of P(P(x ∪ y)) belongs to U, as well as all of its subsets, so that the graphs of all functions from x to y belong to U. In particular, if f is surjective, then any retraction (whose existence is asserted by the axiom of choice) belongs to U. Consequently, existence of universes does not follow from the axioms of zfc — this would indeed contradict Gödel’s second incompleteness theorem — and zfc has to be supplemented by an axiom such as the following.

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Axiom (1.1.4). — For every set x, there exists an universe U such that x ∈ U. Definition (1.1.5). — A category C is the datum of two sets ob(C ) and mor(C ), whose elements are respectively called its objects and its morphisms, of two maps o, t ∶ mor(C ) → ob(C ) (origin and target) and of a partial composition map: mor(C ) × mor(C ) → C , denoted ( f , g) ↦ g ○ f satisfying the following properties, where f , g, h ∈ mor(C ): a) the composition g ○ f is defined if and only if t( f ) = o(g); one has o(g ○ f ) = o( f ) and t(g ○ f ) = t(g); b) the composition is associative: if t( f ) = o(g) and t(g) = o(h), then h ○ (g ○ f ) = (h ○ g) ○ f ; c) for every object X ∈ ob(C ), there exists a morphism idX ∈ mor(C ) such that o(idX ) = t(idX ) = X, idX ○ f = f for every f ∈ mor(C ) such that t( f ) = X, and g ○ idX = g for every g ∈ mor(C ) such that o(g) = X. If f ∈ C , the objects o( f ) and t( f ) are called the origin and the target of f . For any two objects X, Y in a category C , one writes C (X, Y), or HomC (X, Y) to be the subset of mor(C ) consisting of all morphisms f with origin X and target Y. 1.1.6. — Let X, Y ∈ ob(C ) and f ∈ C (X, Y). One says that f is left-invertible if there exists g ∈ C (Y, X) such that g ○ f = idX ; any such element g is called a left-inverse of f . One says that g is right-invertible if there exists h ∈ C (Y, X) such that f ○ h = idY , and every such element g is called a right-inverse of g. One says that f is invertible if it is both left- and right-invertible. In this case, any left-inverse g and any right-inverse h of f coincide, since g = g ○ idY = g ○ ( f ○ h) = (g ○ f ) ○ h = idX ○h = h, and this element is simply called the inverse of f . The definitions of a left-invertible morphism and of a right-invertible are deduced one from the other by passing to the opposite category. 1.1.7. — Let X, Y ∈ ob(C ) and f ∈ C (X, Y). One says that f is a monomorphism if for every object Z ∈ ob(C ) and every h, h′ ∈ C (Z, X), the equality f ○ h = f ○ h′ implies that h = h′ .

1.1. SETS, UNIVERSES AND CATEGORIES

3

If f is left-invertible, then f is a monomorphism. Let indeed h, h′ ∈ C (Z, X) be such that f ○ h = f ○ h′ ; for every left-inverse g of f , one then has h = (g ○ f ) ○ h = g ○ ( f ○ h) = g ○ ( f ○ h′ ) = (g ○ f ) ○ h′ = h′ . One says that f is an epimorphism if for every object Z ∈ ob(C ) and every g, g ′ ∈ C (Y, Z), the equality g ○ f = g ′ ○ f implies that g = g ′ . If f is right-invertible, then f is an epimorphism. Let indeed g, g ′ ∈ C (Y, Z) be such that g ○ f = g ′ ○ f ; for every right-inverse h of f , one then has g = g ○ ( f ○ h) = (g ○ f ) ○ h = (g ′ ○ f ) ○ h = g ′ ○ ( f ○ h) = g ′ . The definitions of a monomorphism and of an epimorphism are deduced one from the other by passing to the opposite category. The reader will take care that a morphism can be both a monomorphism and an epimorphism, without being an isomorphism (exercise 1.11.1). Example (1.1.8). — All classical mathematical objects, such as sets, abelian groups, topological spaces, k-modules (where k is a ring), k-algebras, etc., give rise to categories. Since there is no set of all sets, we need to fix a universe U. The category Set U of U-sets has for objects the elements of U and for morphisms the maps between those sets, the composition being given by the usual composition of maps. More generally, the abelian groups whose underlying set belongs to U are the objects of a category Ab U , the morphisms of this category being the morphisms of abelian groups between them. One defines similarly categories Top U or Ring U whose objects are the topological spaces, or the rings, with underlying set an element of U. Or, if k is an object of Ring U , one defines categories Mod (k)U or Alg (k)U of k-modules, or of k-algebras, whose underlying set belongs to U. In practice, one can often work within an universe U which is fixed once and for all and talk about the category Set of sets, etc. Example (1.1.9). — Let C be a category. Its opposite category, denoted by C o , has the same objects and the same morphisms, but the origin/target maps are exchanged, and the order of composition is switched. When one writes down a general construction/theorem from category theory both in C and in the opposite category C o , one obtains two related statements, one being obtained from the other by ‘‘reversing the arrows’’.

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Example (1.1.10). — Let A be a set and let ⩽ be a preordering relation on A, that is, a binary relation on A which is reflexive and transitive. From (A, ⩽), one defines a category A as follows: one has ob(A) = A and mor(A) is the set of pairs (a, b) ∈ A × A such that a ⩽ b, the maps o and t being the first and second projection respectively, the composition is defined by (b, c) ○ (a, b) = (a, c) for every a, b, c ∈ A. Definition (1.1.11). — Let U be a universe. a) One says that a set X is U-small if there exists a bijection f ∶ X → X′ with an element of U. b) One says that a category C is a U-category if ob(C ) and mor(C ) belong to U. c) One says that a category C is U-small if ob(C ) and mor(C ) are U-small. d) One says that a category C is locally U-small if for every objects X, Y ∈ ob(C ), the set C (X, Y) is U-small. Since ob(C ) can be identified with the subset of mor(C ) of all identities, we observe that if mor(C ) belongs to U, then ob(C ) belongs to U as well. For example, the category Set U is locally U-small, but not U-small. However, if V is an universe such that U ∈ V, then Set U is a V-category. Definition (1.1.12). — Let C and D be two categories. A functor F from C to D is the datum of two maps ob(F) ∶ ob(C ) → ob(D) and mor(F) ∶ mor(C ) → mor(D) satisfying the following properties: a) For every f ∈ mor(C ), one has o(mor(F)( f )) = ob(F)(o( f )) and t(mor(F)( f )) = ob(F)(t( f )); b) For every pair ( f , g) in mor(C ) such that t( f ) = o(g), one has mor(F)(g ○ f ) = mor(F)(g) ○ mor(F)( f ); c) For every object X ∈ ob(C ), one has mor(F)(idX ) = idob(F)(X) . In practice, the maps mor(F) and ob(F) associated with a functor F are simply denoted by F. Definition (1.1.13). — Let C and D be two categories, and let F, G be two functors from C to D. A morphism of functors α ∶ F → G is a map α ∶ ob(C ) → mor(D) satisfying the following properties: a) For every X ∈ ob(C ), the morphism α(X) has source F(X) and target G(X);

1.2. LIMITS, COLIMITS, ADJUNCTIONS

5

b) For every X, Y ∈ ob(C ) and every f ∈ C (X, Y), one has α(Y) ○ F( f ) = G( f ) ○ α(X). Morphisms of functors are composed in the obvious way, turning the set Fun(C , D) of functors from C to D into a category. Let U be a universe. If C and D are U-categories, then Fun(C , D) is a U-category.

1.2. Limits, colimits, adjunctions 1.2.1. — A quiver Q is the datum of a set V (vertices), of a set A (arrows), and of two maps o, t ∶ V → Q (origin and target). Let Q = (V, F, o, t) be a quiver. Let C be a category. A Q-diagram in C is given by a family (Xv )v∈V of objects of C and of a family ( f a )a∈A of arrows of C such that o( f a ) = Xo(a) and t( f a ) = Xt(a) . 1.2.2. — Let C be a category and let D = ((Xv ), ( f a )) be a Q-diagram in C . A cone on the diagram D is the datum of an object X of C and of a family (pv )v∈V satisfying the following properties: a) For every v ∈ V, pv is a morphism in C with origin X and target Xv ; b) For every a ∈ A, one has f a ○ po(a) = p t(a) . One says that a cone (X, (pv )) on a diagram D = ((Xv ), ( f a )) is a limit of the diagram D if for every cone D′ = (X′ , (p′v )) on D, there exists a unique morphism φ ∈ C (X′ , X) such that pv ○ φ = p′v for every v ∈ V. Let (X, (pv )) and (X′ , (p′v )) be two limits on D. Since X is a limit and X′ is a cone on D, there exists a unique morphism φ ∈ C (X′ , X) such that pv ○ φ = p′v for every v ∈ V. Since X′ is a limit and X is a cone on D, there exists a unique morphism φ′ ∈ C (X, X′ ) such that pv = p′v ○ φ′ for every v ∈ V. Then pv = p′v ○ φ′ = pv ○ φ ○ φ′ , for every v ∈ V; since X is a limit, one has φ ○ φ′ = idX . Similarly, p′v = pv ○ φ = p′v ○ φ′ ○ φ, for every v ∈ V; since X′ is a limit, one has φ′ ○ φ = idX′ . Consequently, φ and φ′ are isomorphisms, inverse one to the other. 1.2.3. — By passing to the opposite category, one defines the notions of a cocone on a diagram D and of a colimit of D. Explicitly, a cocone on the diagram D is

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the datum of an object X of C and of a family (pv )v∈V satisfying the following properties: a) For every v ∈ V, pv is a morphism in C with origin Xv and target X; b) For every a ∈ A, one has p t(a) ○ f a = po(a) . One says that a cocone (X, (pv )) on a diagram D = ((Xv ), ( f a )) is a colimit of the diagram D if for every cocone D′ = (Y, (qv )) on D, there exists a unique morphism φ ∈ C (Y, X) such that φ ○ pv = qv for every v ∈ V. If (X′ , (p′v )) is another colimit of the diagram D, then there exists a unique morphism φ ∶ X → X′ such that φ ○ pv = p′v for every v ∈ V, and φ is an isomorphism. Example (1.2.4). — a) Let Q be the empty quiver — no vertex and no arrow. The corresponding Q-diagram D is empty — no object, no morphism. A cone on D is just an object of C ; a limit of D is an object X such that for every object X′ in C , there exists a unique morphism φ ∶ X′ → X in C . Such an object is called a terminal object of C . Passing to the opposite category, a colimit of D is called an object!initial —: this is an object X such that for every object X′ ∈ C , there exists a unique morphism φ ∶ X → X′ in C . In the case of the category of sets, the empty set is an initial object, and terminal objects are sets with one element; in the case of the category of k-modules, the initial and the terminal objects are the zero module; in the case of the category of groups, the initial and the terminal objects are the groups reduced to the unit element. In the category of rings, the ring Z is an initial object, and the zero ring is a terminal object. The category of fields has no initial object and no terminal object. b) Let Q = (V, A, o, t) be a quiver with no arrows. A Q-diagram in C is just a family (Xv )v∈V of objects, indexed by the set V of vertices of Q. A limit of D is called a product of the family (Xv ); a colimit of D is called a coproduct of the family (Xv ). In the case of the category of sets, one gets the product, resp. the disjoint union; in the case of the category of k-modules, one gets the product, resp. the direct sum; in the case of the category of groups, one gets the product, resp. the free product. In the category of rings, the product is a product, and the tensor product furnishes a coproduct. In the category of fields, products or coproducts rarely exist.

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c) Let Q be a quiver with two vertices a, b and two arrows both with origin a and target b. A Q-diagram is given by two objects A, B in C and two morphisms f , g ∶ A → B. A limit of this diagram is called an equalizer of the pair ( f , g); a colimit is called a coequalizer of the pair ( f , g). 1.3. Additive and abelian categories 1.3.1. — Let C be a category. A zero object in C is an object 0 which is both initial and terminal. Then for every pair (X, Y) of objects in C , there exists a unique morphism in C (X, Y) which factors through 0; it is denoted by 0. Definition (1.3.2). — Let C be a category. One says that C is semi-additive if the following conditions are satisfied: a) C admits finite products and finite coproducts; b) There exists an object of C , denoted by 0, which is both initial terminal; c) Let (X1 , X2 ) be a pair of objects of C , let (X1 ⊔ X2 , ( j1 , j2 )) be a coproduct, let (X1 × X2 , (p1 , p2 )) and let ε ∶ X1 ⊔ X2 → X1 × X2 be a morphism such that p a ○ ε ○ jb = idXa if a = b, and 0 otherwise. Then ε is an isomorphism. Let us detail the third condition a little bit. By definition of a coproduct, the map f ↦ ( f ○ j1 , f ○ j2 ) is a bijection from C (X1 ⊔X2 , X1 ×X2 ) to ∏2b=1 C (Xb , X1 × X2 ). Similarly, for every b ∈ {1, 2}, the definition of a product implies that the map g ↦ (p1 ○ g, p2 ○ g) is a bijection from C (Xb , X1 × X2 ) to ∏2a=1 C (Xb , Xa ). Consquently, the map f ↦ (p a ○ ε ○ jb )(a,b)∈{1,2}2 is a bijection from C (X1 ⊔ X2 , X1 × X2 ) to ∏2a,b=1 C (Xb , Xa ). Consequently, there there exists a unique morphism ε as stated, and the assertion is that ε is an isomorphism. Lemma (1.3.3). — Let C be a semi-additive category. For every pair (X1 , X2 ) of objects of C and every pair f , g ∈ C (X1 , X2 ), let f + g be the unique element of C (X1 , X2 ) such that dX1

( f ,g)

ε−1

δX2

X1 Ð→ X1 × X1 ÐÐ→ X2 × X2 Ð→ X2 ⊔ X2 Ð→ X2 , where dX1 is the unique morphism whose composition with the two canonical morphisms X1 ×X1 → X1 is idX1 , and δX2 is the unique morphism whose composition with the two canonical morphisms X2 → X2 ⊔ X2 is idX2 . Then the composition law ( f , g) ↦ f + g on C (X1 , X2 ) is commutative, associative, the zero morphism is a neutral element.

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8

Moreover, for every triple (X1 , X2 , X3 ) of objects of C , the composition map C (X1 , X2 ) × C (X2 , X3 ) → C (X1 , X3 ) given by ( f , g) ↦ g ○ f is bi-additive: for f , f ′ ∈ C (X1 , X2 ) and g, g ′ ∈ C (X2 , X3 ), one has g ○ ( f + f ′ ) = (g ○ f ) + (g ○ f ′ ) and (g + g ′ ) ○ f = (g ○ f ) + (g ′ ○ f ). Proof. — To be done. Definition (1.3.4). — One says that a semi-additive category C is additive if its semi-groups of morphisms C (X1 , X2 ) are abelian groups. Example (1.3.5). — a) Let k be a ring; the category of k-modules is an additive category. b) Let X be a topological space and let O be a sheaf of rings on X; the category of O-modules is an additive category. c) The category of complex Banach spaces, with linear continuous maps for morphisms, is an additive category. 1.3.6. — Let C be an additive category. Let (X, Y) be a pair of objects of C and let f ∈ C (X, Y) be a morphism. An equalizer of the pair ( f , 0) is called a kernel of f , and is denoted by Ker( f ); a coequalizer of the pair ( f , 0) is called a cokernel of f , and is denoted by Coker( f ). If f is a monomorphism, then 0 is a kernel of f ; if f is an epimorphism, then 0 is a cokernel of f . Let Ker( f ) be a kernel of f , and let i ∶ Ker( f ) → X be the canonical morphism. By definition of a kernel, the map C (Z, Ker( f )) → C (Z, X) given by g ↦ i ○ g is injective. In other words, j is a monomorphism. By passing to the opposite category, one deduces that the canonical morphism p ∶ Y → Coker( f ) is an epimorphism. Definition (1.3.7). — One says that an additive category C is an abelian category if the following properties hold: a) Every morphism has a kernel and a cokernel; b) Every monomorphism is a kernel; c) Every epimorphism is a cokernel. Example (1.3.8). — a) Let k be a ring. The category of k-modules is an abelian category. Epimorphisms are surjective morphisms, monomorphisms are injective morphisms; kernel and cokernels coincide with the usual notions.

1.3. ADDITIVE AND ABELIAN CATEGORIES

9

A theorem of Mitchell asserts that for every abelian category C , there exists a ring k and an exact fully faithful functor of C into a category of k-modules. In particular, kernels and cokernels are preserved by this embedding. For certain arguments, this allows to pretend objects of C are k-modules and play with their ‘‘elements’’. b) Let X be a topological space and let O be a sheaf of rings on X. The category of O-modules is an abelian category. Monomorphisms, resp. epimorphisms, are the morphisms which induce injective, resp. surjective, morphisms on all stalks. Consequently, monomorphisms are injective morphisms. However, not every epimorphism is surjective (see exercise 2.10.1). Kernels are defined naively; however, the cokernel of a morphism φ of O-modules is the sheaf associated with the presheaf U ↦ Coker(φU ). c) Let A be an abelian category. The additive category C (A) of complexes in A is an abelian category. Kernels and cokernels are computed termwise and a morphism of complexes f ∶ X → Y is a monomorphism (resp. an epimorphism, resp. an isomorphism) if and only if so is f n ∶ Xn → Yn , for every integer n ∈ Z. d) The category of Banach spaces is not an abelian category. Indeed, in this category, monomorphisms are the injective continuous morphisms, while kernels are monomorphisms with closed image. Proposition (1.3.9). — Let f ∶ X → Y be a morphism in an abelian category C . a) The morphism f is a monomorphism if and only if Ker( f ) = 0; b) The morphism f is an epimorphism if and only Coker( f ) = 0; c) The morphism f is an isomorphism if and only it is both a monomorphism and an epimorphism. Proof. — a) The conditions ‘‘ f is a monomorphism’’ and ‘‘Ker( f ) = 0’’ are both equivalent to the statement that for every object Z, the zero morphism is the only morphism h ∈ C (Z, X) such that f ○ h = 0. b) Similarly, the conditions ‘‘ f is an epimorphism’’ and ‘‘Coker( f ) = 0’’ are both equivalent to the statement that for every object Z, the zero morphism is the only morphism g ∈ C (Y, Z) such that g ○ f = 0. c) If f is an isomorphism, then it is both an epimorphism and a monomorphism. Let us assume, conversely, that f is both an epimorphism and a monomorphism. Since f is a monomorphism, it is the kernel of a morphism g ∶ Y → Z. In particular, one has g ○ f = 0 = 0 ○ f . Since f is an epimorphism, one has

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g = 0. Since f ∶ X → Y is a kernel of 0, the relation 0 ○ idY = 0 implies the existence of a unique morphism h ∶ Y → X such that idY = f ○ h. In particular, f is right-invertible. By passing to the opposite category, one proves that f is left-invertible. Consequently, f is an isomorphism, as was to be shown. Lemma (1.3.10). — Let C be an abelian category. a) Every monomorphism is a kernel of its cokernel; b) Every epimorphism is a cokernel of its kernel. Proof. — a) Let i ∶ X → Y be a monomorphism. Let p ∶ Y → Coker(i) be a cokernel of i and let j ∶ Ker(p) → Y be a kernel of p. Since q ○ i = 0, there exists a unique morphism u ∶ Ker( f ) → Ker(q) such that i = j ○ u. Let us prove that u is an isomorphism. By definition of an abelian category, there exists a morphism f ∶ Y → Z such that i is a kernel of f . Since f ○ i = 0, there exists a (unique) morphism w ∶ Coker(i) → Z such that f = w ○ q.

v

j

q

Z w

→

←

←

← → Ker(q)

→ Y →

→

→

← →

←

u

f

i

← ←

X

Coker(i)

Since f ○ j = w ○ q ○ j = 0, there exists a morphism v ∶ Ker(q) → Ker( f ) such that j = i ○ v. Since i and j are kernels, they are monomorphisms. Then the relations i = j ○ u = i ○ v ○ u and j = i ○ v = j ○ u ○ v imply that v ○ u = idKer( f ) and u ○ v = idKer(q) . In particular, u is an isomorphism. The proof of assertion b) is similar, and follows from a) by passing to the opposite category. Proposition (1.3.11). — Let C be an abelian category and let f ∶ X → Y be a morphism in C . Let i ∶ Ker( f ) → X be a kernel of f and let p ∶ Y → Coker( f ) be a cokernel of f . Let q ∶ X → Coker(i) be a cokernel of i and let j ∶ Ker(p) → Y be a kernel of p. There exists a unique morphism f¯ ∶ Coker(i) → Ker( f ) such that f = j ○ f¯ ○ q, and f¯ is an isomorphism. A kernel of p is called an image of f and is denoted by Im( f ); a cokernel of j is called a coimage of f is denoted by Coim( f ). The proposition thus says that

1.3. ADDITIVE AND ABELIAN CATEGORIES

11

any morphism f induces a canonical isomorphism from its image to its coimage. This is represented by the following diagram: f

←

q

j f¯

→ Coker( f )

←

→ ←

Coim( f )

p

→

→ Y

←

→ X

←

i

←

Ker( f )

→ Im( f )

Observes that when one passes to the opposite category, kernels and cokernels are switched, as are images and coimages. The proof of the proposition uses as an intermediate step a result which can be seen as corollary. Lemma (1.3.12). — a) There exists a unique morphism f1 ∶ X → Im( f ) such that f = j ○ f1 . b) For every factorization f = j′ ○ f1′ , where j′ ∶ T → Y is a monomorphism and f1′ ∶ X → T is a morphism, there exists a unique morphism u ∶ Im( f ) → T such that f1′ = u ○ f1 and j = j′ ○ u. c) The morphism f1 is an epimorphism. Proof. — a) By definition, Im( f ) is a kernel of p ∶ Y → Coker( f ), so that p ○ f = 0. Consequently, the assertion follows from the definition of a kernel. b) Let p′ ∶ T → Coker( j′ ) be a cokernel of j′ . Since p′ ○ f = p′ ○ j′ ○ f1′ = 0, there exists a unique morphism u ′ ∶ Coker( f ) → Coker( j′ ) such that p′ = u ′ ○ p. Then p′ ○ j = u′ ○ p○ j = 0, so that there exists a unique morphism u ∶ Ker(p) → T such that j = j′ ○ u. Then j′ ○ f1′ = f = f ○ f1 = j′ ○ u ○ f1 ; since j′ is a monomorphism, one has f1′ = u ○ f1 . c) Let s ∶ Ker(p) → S be a morphism such that s ○ f1 = 0; let us prove that s = 0. Let k ∶ Ker(s) → Ker(p) be a kernel of s; there exists a unique morphism f1′ ∶ X → Ker(s) such that f1 = k ○ f1′ . Then f = j ○ f1 = ( j ○ k) ○ f1′ . Since j ○ k is a monomorphism, part b) of lemma 1.3.12 asserts that there exists a unique morphism u ∶ Ker(p) → Ker(s) such that f1′ = u ○ f1 and j = j ○ k ○ u. Since j is a monomorphism, this implies that k ○ u = idIm( f ) . Finally, s = s ○ k ○ u = 0. Proof of proposition 1.3.11. — Since p ○ f = 0, there exists a unique morphism f1 ∶ X → Ker(p) such that f = j ○ f1 ; by lemma 1.3.12, f1 is an epimorphism. Then j ○ f1 ○ i = f ○ i = 0, hence f1 ○ i = 0, because j is a monomorphism. Consequently, there exists a unique morphism f¯ ∶ Coker(i) → Ker(p) such that f1 = f¯ ○ q.

12

CHAPTER 1. TRIANGULATED CATEGORIES

One then has f = j ○ f¯ ○ q. If f¯′ is a second morphism such that f = j ○ f¯′ ○ q, one has f¯′ ○ q = f¯ ○ q, because j is a monomorphism, hence f¯′ = f¯, because q is an epimorphism. Since f1 = f¯ ○ q is an epimorphism, f¯ is an epimorphism as well. By passing to the opposite category, we see that f¯ is a monomorphism. Consequently, f¯ is an isomorphism, as claimed.

1.4. Complexes in additive categories Let A be an additive category. 1.4.1. — A complex in A is a sequence (d n ∶ Xn → Xn+1 )n∈Z of morphisms in A such that d n+1 ○ d n = 0 for every n ∈ Z. These morphisms d n are called the differentials of this complex. One generally denotes such a complex by the letter X, remembering of the objects rather than the differentials, which may then be denoted by dXn , writing the name X of the complex as a subscript to avoid possible confusions. Let X and Y be complexes in A. A morphism of complexes f ∶ X → Y is a family f = ( f n )n∈Z where, for every n ∈ Z, f n ∈ A(Xn , Yn ), such that dYn ○ f n = f n+1 ○ dXn+1 for every n ∈ Z. Morphisms of complexes are composed in the obvious way. The complexes in C form an additive category C (A): products and coproducts are computed termwise. One also considers finite or semi-infinite complexes involving sequences indexed by an interval in Z. They amount to extending the sequence of differentials by zero morphisms to/from zero objects. 1.4.2. — Let X be a complex in an additive category A. Let m ∈ Z. The mth shift of X is the complex Σ m X defined by: (Σ m X)n = Xm+n and d Σn m X = (−1)m dXm+n for every n ∈ Z. For m = 1, one simply writes ΣX; this is the complex obtained by shifting X one step to the left. If f ∶ X → Y is a morphism of complexes in A, the morphism Σ m f ∶ Σ m X → Σ m Y is defined by (Σ m f )n = f m+n for every n ∈ Z. In this way, the assignment (X ↦ Σ m X, f ↦ Σ m f ) gives rise to a functor Σ of the category C (A) onto itself.

1.4. COMPLEXES IN ADDITIVE CATEGORIES

13

One has Σ0 = id and Σ m+p = Σ m ○ Σ p for every m, p ∈ Z. In particular, the functors Σ m are isomorphisms of categories. 1.4.3. — Let f , g ∶ X → Y be two morphisms of complexes in an additive category A. A homotopy with origin f and target g is a sequence θ = (θ n )n∈Z where, for every n ∈ Z, θ n ∈ A(Xn , Yn−1 ) such that, f n − g n = dYn−1 ○ θ n + θ n+1 ○ dXn . Let h ∶ Y → Z be a morphism of complexes; then the family h ○ θ = (h n−1 ○ θ n ) is a homotopy with origin h ○ f and target h ○ g. Let k ∶ Z → X be a morphism of complexes; then the family θ ○ k = (θ n ○ k n−1 ) is a homotopy with origin f ○ k and target g ○ k. 1.4.4. — Let X and Y be complexes in the additive category A; for every n ∈ Z, let f n ∈ A(Xn , Yn ). We define as follows the cone of f , denoted by C f : For every n ∈ Z, one sets Cnf = Yn ⊕ Xn+1 , and define a morphism dCn f ∶ Cnf → Cn+1 f by the d n f n+1

block-matrix ( 0Y −d n+1 ). X Observe that dCn+1 ○ dCn f is given by the product of matrices f (

dYn+1 f n+2 dYn f n+1 0 dYn+1 f n+1 − f n+2 dXn+1 ) ( ) = ( ), 0 −dXn+2 0 −dXn+1 0 0

so that C f is a complex if and only if f is a morphism of complexes. Let us assume that f is a morphism of complexes. The canonical morphisms n α f ∶ Yn → Cnf = Yn ⊕Xn+1 define a morphism of complexes α f ∶ Y → C f . Similarly, n n n+1 → Xn+1 define a morphism of the canonical morphisms β n+1 f ∶ Cf = Y ⊕ X complexes β f ∶ C f → ΣX. One has β f ○ α f = 0. For every n ∈ Z, let θ nf ∶ Xn → Yn−1 ⊕ Xn = Cn−1 be the canonical morphism. f One has dCn−1 f

○

θ nf

+

θ n+1 f

○ dXn

dYn−1 f n 0 =( ) dn n ) (0 idXn ) + ( 0 −dX idXn+1 X fn 0 fn = ( n) + ( n) = ( ) = α f ○ f . −dX dX 0

Consequently, the family θ f = (θ nf ) is a homotopy with origin α f ○ f and target 0. Conversely, let g ∶ Y → Z be a morphism of complexes and let η be a homotopy with origin g ○ f and target 0. Then the family (γ n ) given, for every n ∈ Z, by

14

CHAPTER 1. TRIANGULATED CATEGORIES

γ n = ( g n φ n ) is the unique morphism of complexes γ ∶ C f → Z such that γ○α f = g and γ ○ θ = η. In other words, the triple (C f , α f , θ f ) solves the universal problem of making α f ○ f the origin of a homotopy θ f with target 0. For every n ∈ Z, let φ nf ∶ Cnf → Yn = (ΣY)n−1 be the canonical morphism. One has n−1 n n n d ΣY ○ φ nf + φ n+1 0) + (idYn+1 0) ( f ○ dC f = −dY (idY

dYn f n+1 ) 0 −dXn+1

= (−dYn 0) + (dYn f n+1 ) = (0 f n+1 ) = Σ f ○ βf , so that the family φ f = (φ nf ) is a homotopy with origin Σ f ○ β f and target 0. Let f , g ∶ X → Y be two morphisms of complexes and let φ be a homotopy with origin f and target g. For every n ∈ Z, let λ n ∶ Yn ⊕ Xn+1 → Yn ⊕ Xn+1 be given n by the block-matrix ( 01 φ1 ). The family λ = (λ n ) is a morphism of complexes from C f to C g . 1.4.5. — Let X, Y be complexes in an additive category A. Two morphisms of complexes f , g ∶ X → Y are said to be homotopic if there exists a homotopy with origin f and target t. This is an equivalence relation, compatible with the group structure on C (A)(X, Y). Morphisms homotopic to 0 are called null homotopic; they form a subgroup C (A)(X, Y)0 of C (A)(X, Y). Define K (A)(X, Y) = C (A)(X, Y)/C (A)(X, Y)0 . Passing to the quotients, the composition maps in C (A) induce composition maps K (A)(X, Y) × K (A)(Y, Z) → K (A)(X, Z). These maps define a category K (C ), called the homotopy category of the additive category A. It is an additive category. The functors Σ m , for m ∈ Z, extend to K (A). 1.4.6. — Let us assume that A an abelian category. Let X be a complex in A. Let n ∈ Z. Since dXn ○ dXn−1 = 0, the canonical monomorphism Im(dXn−1 ) → Xn factors uniquely through Ker(d n ) → Xn .

1.4. COMPLEXES IN ADDITIVE CATEGORIES

15

The consideration of the opposite category furnishes a different description: the canonical morphism Xn → Im(dXn ) factors uniquely through ̃ n (X) be a kernel of the resulting epimorphism Xn → Coker(dXn−1 ); let H ψXn ∶ Coker(dXn−1 ) → Im(dXn ). The morphism f ∶ Ker(dXn ) → Coker(dXn−1 ) satisfies ψXn ○ f ○ φXn = 0; consẽ n (X). quently, it factors uniquely through a morphism f ′ ∶ Hn (X) → H → Ker(dXn )

↩

→ ψ Xn

→ Xn+1

→

↠

↠ ←

Coker(dXn−1 )

d Xn

↩

←

←

→ Xn

←

d Xn−1

←

→

← Xn−1

φ Xn

↩

↠

Im(dXn−1 ) ↩

↠ Im(dXn )

Let u ∶ Ker(dXn ) → Coker(dXn−1 ) be the composition of the two canonical morphisms indicated on the diagram. Since u ○ φXn = 0, the morphism φXn factors ∼ uniquely through Ker(u); one checks it induces an isomorphism Im(dXn−1 ) Ð → Ker(u). Since φXn ○ u = 0, the morphism φXn factors uniquely through Coker(u); ∼ one checks that it induces an isomorphism Coim(u) Ð → Im(dXn ). Any of these objects will be called the nth cohomology object of the complex X, and denoted by Hn (X). The cohomology objects are functorial: any morphism of complexes f ∶ X → Y induces morphisms of cohomology objects Hn ( f ) ∶ Hn (X) → Hn (Y) in such a way that Hn (g ○ f ) = Hn (g) ○ Hn ( f ) and Hn (idX ) = idHn (X) . These functors are additive. One says that a complex X is acyclic, or exact, if Hn (X) = 0 for every n ∈ Z. If Hn ( f ) is an isomorphism for every n ∈ Z, then one says that f is a homologism, or a quasi-isomorphism. Lemma (1.4.7). — Let A be an abelian category. a) Let f , g ∶ X → Y be morphisms of complexes in A. If f and g are homotopic, then Hn ( f ) = Hn (g) for every n ∈ Z. b) Let f ∶ X → Y be a morphism of complexes in A. If f induces an isomorphism in the homotopy category K (A), then f is a homologism.

CHAPTER 1. TRIANGULATED CATEGORIES

16

c) Let X be a complex in A. If the identity morphism idX is null homotopic, then the complex X is acyclic. Proof. — a) (Temporarily left as an exercise.) b) Let g ∶ Y → X be a morphism of complex such that f ○ g and g ○ f are homotopic to identity. Then Hn ( f ) ○ Hn (g) = id and Hn (g) ○ Hn ( f ) = id, hence Hn ( f ) is an isomorphism, for each n ∈ Z. In other words, f is a homologism. c) Assume that idX is null homotopic. Then one has idHn (X) = Hn (idX ) = Hn (0) = 0, for every n ∈ Z. Consequently, Hn (X) = 0 for every n, and X is acyclic. 1.4.8. — Let f

Z

0→XÐ →YÐ →→ 0 be an exact sequence of complexes in an abelian category A. For every n ∈ Z, the morphisms f n and g n give rise to an exact sequence n n f

n n g

0 → X Ð→ Y Ð→ Zn → 0 in A. Let C f be the cone of f , let α f ∶ Y → C f and β f ∶ C f → ΣX be the canonical morphisms, and let θ f ∶ C f → Σ−1 X be the canonical homotopy such that α f ○ f = dC f ○ θ f + θ f ○ dX . Since g ○ f = 0, the null homotopy g ○ f ≃ 0 induces a unique morphism of complexes h ∶ C f → Z such that h ○ α f = g and h ○ θ f = 0. Explicitly, α f = ( 01 ) and θ f = ( 0 1 ), so that h = ( g 0 ). Lemma (1.4.9). — of objects in A: Hn−1 (β f )

n

a) The morphisms f , α f , β f induce a long exact sequence Hn ( f )

n

Hn (α f )

n

Hn (β f )

ÐÐÐÐ→ H (X) ÐÐÐ→ H (Y) ÐÐÐ→ H (C f ) ÐÐÐ→ H

n+1

Hn+1 ( f )

(X) ÐÐÐÐ→ . . .

b) The morphism h is a homologism. c) The morphism f is a homologism if and only if C f is acyclic, if and only if Z is acyclic. Proof. — For this proof, we shall pretend that the abelian category A is a category of modules.

1.4. COMPLEXES IN ADDITIVE CATEGORIES

17

a) Since α f ○ f is null homotopic, one has Hn (α f )○Hn ( f ) = 0. Since β f ○ α f = 0, one has Hn (β f ) ○ Hn (α f ) = 0. Since Σ f ○ β f is null homotopic, one has Hn+1 ( f ) ○ Hn (β f ) = 0. Let x ∈ Xn be such that dX (x) = 0 and such that Hn ( f )([x]) = 0 in Hn (Y). Let y ∈ Yn−1 be such that f (x) = dY (y); let z = ( −y x ); one has dC f (z) = 0 and β f (z) = x. This proves exactness at Hn (X). Let y ∈ Yn be such that dY (y) = 0 and such that Hn (α f )([y]) = 0; let z ∈ Cn−1 f ′

be such that ( 0y ) = dC f (z); write z = ( xy′ ). One has d(y ′ ) + f (x ′ ) = y and d(x ′ ) = 0, hence [y] = [ f (x ′ )] = Hn ( f )([x ′ ]). This proves exactness at Hn (Y). Let z ∈ Hn (C f ) be such that dC f (z) = 0 and Hn (β f )([z]) = 0 in Hn+1 (X); write z = ( xy ). One has d(y) + f (x) = 0 and d(x) = 0; moreover, there exists x ′ ∈ Xn such that x = d(x ′ ). Consequently, d(y + f (x ′ )) = 0 and ) y+ f (x ) ) − d (( 0′ )) z = ( y+ f0(x ) ) − ( f (x Cf x 0 −x ) = ( ′

′

′

so that [z] = Hn (α f )([y + f (x ′ )]). This proves exactness at Hn (C f ). b) Let n ∈ Z. Let z ∈ Cnf be such that d(z) = 0 and Hn (h)([z]) = 0. Let z ′ ∈ Zn−1 be such that h(z) = dZ (z ′ ); write z = ( xy ). Then d(y) + f (x) = 0, d(x) = 0 and d(z ′ ) = g(y). Since g n−1 ∶ Yn−1 → Zn−1 is surjective, there exists y ′ ∈ Yn−1 such that z ′ = g(y′ ); then d(z ′ ) = dZ (g(y ′ )) = g(dY (y ′ )), so that g(y − dY (y ′ )) = 0; consequently, there exists x ∈ Xn such that y − dY (y′ ) = f (x). ′ ′ We thus have z = ( d(y )+x f (x) ) = d(( yx )), so that [z] = 0. This proves that Hn (h) is injective. Let now z ∈ Zn be such that dZn (z) = 0. Since g n is surjective, there exists y ∈ Yn such that z = g n (y), hence 0 = dZn (g n (y)) = g n (dYn (y)). Consequently, there exists x ∈ Xn−1 such that dYn (y) = f n−1 (x). One has f n (dXn−1 (x)) = dYn ( f n−1 (x)) = ′ 0, hence dX (x) = 0 since f n is injective. Let z ′ = ( −xy ) ∈ Cn−1 f . One has d(z ) = 0 and h(z ′ ) = g(y) = z, so that Hn (h)([z ′ ]) = [z]. This proves that Hn (h) is surjective. c) If Hn (C f ) = 0, the exact sequence of a) shows that Hn ( f ) is an epimorphism and Hn+1 ( f ) is a monomorphism. Consequently, if C f is acyclic, then Hn ( f ) is an isomorphism for every n, so that f is a homologism. Conversely, if Hn ( f ) is an epimorphism, then Hn (α f ) = 0, while if Hn ( f ) is a monomorphism, then Hn−1 (β f ) = 0. In particular, if f is a homologism, then Hn (α f ) = 0 and Hn (β f ) = 0 for every n. Then, 0 = Im(Hn (α f )) = Ker(Hn (β f )) = Hn (C f ) for every n, so that C f is acyclic.

18

CHAPTER 1. TRIANGULATED CATEGORIES

1.5. Triangulated categories Let C be an additive category endowed with an automorphism Σ of C (translation). Definition (1.5.1). — A triangle in C is a complex T such that dTn+3 = ΣdTn for every n ∈ Z. A morphism of triangles f ∶ T → T′ is a morphism of complexes such that f n+3 = Σ f n for every n ∈ Z. Concretely, a triangle only depends on three consecutive objects and morphisms, and is represented as follows: u

v

w

XÐ →YÐ →ZÐ → ΣX, the next differential being Σu ∶ ΣX → ΣY, etc. Conversely, a sequence (u, v, w) of three composable morphisms gives rise to a triangle if and only if v ○ u = 0, w ○ v = 0 and Σu ○ w = 0. The datum of a morphism from such a triangle to a similar triangle u′

v′

w′

X′ Ð → Y′ Ð → Z Ð→ ΣX′ is equivalent to the datum of three morphisms f ∶ X → X′ , g ∶ Y → Y′ and h ∶ Z → Z′ such that u ′ ○ f = g ○ u, v ′ ○ g = h ○ v and w ′ ○ h = Σ f ○ w. Morphisms of triangles are composed in the obvious way, and triangles in C form a category. Triangles are complexes in C , hence can be shifted; observe that a shift of a triangle is a triangle. However, to avoid confusion with the automorphism Σ of C , we shall not use the letter Σ to indicate shifts of triangles. 1.5.2. — Since triangles in C are complexes, a morphism of triangles φ ∶ T → T′ gives rise to a cone Cφ . Let us explicit the description of this cone. Let us thus consider a morphism of triangles, as represented by the diagram:

v′

←

→ ΣX

←

←

w

h

→ Z′

→

←

→ Y′

→

→ ←

→

u′

→ Z

←

←

←

v

g

f

X′

→ Y

←

u

←

X

w′

Σf

→ ΣX.

By definition, its cone is the complex ′ (u g ) 0 −v

′ h ) ( v0 −w

′ (w Σf ) 0 −Σu

X′ ⊕ Y ÐÐÐÐ→ Y′ ⊕ Z ÐÐÐÐ→ Z′ ⊕ ΣX ÐÐÐÐÐ→ ΣX′ ⊕ ΣY.

1.5. TRIANGULATED CATEGORIES

19

There is also a natural notion of homotopy between morphisms of triangles. If two morphisms of triangles F = ( f , g, h) and F′ = ( f ′ , g ′ , h ′ ) are homotopic, the choice of a homotopy η = (θ, φ, ψ) with origin ( f , g, h) and target ( f ′ , g ′ , h′ ) gives rise to a morphism of triangles λ ∶ CF → CF′ , explicitly given by the diagram

→ Y′ ⊕ Z

( 01 ψ1 )

→ Z′ ⊕ ΣX

←

′ h′ ) ( v0 −w

←

←

←

←

←

←

( 01 φ1 )

←

′ ′ (u g ) 0 −v

→ ΣX′ ⊕ ΣY →

X′ ⊕ Y

→ Z′ ⊕ ΣX →

=

( 01 θ1 )

→

→

→

λ

CF′

→ Y′ ⊕ Z

′ (w Σf ) 0 −Σu

←

X′ ⊕ Y

←

=

′ h ) ( v0 −w

←

CF

′ (u g ) 0 −v

( 01 Σθ 1 )

→ ΣX′ ⊕ ΣY

′ ′ (w Σf ) 0 −Σu

This morphism λ is an isomorphism. Finally, one says that a triangle T is contractible if idT is null homotopic. Definition (1.5.3). — A triangulated category is an additive category C endowed with an automorphism Σ and a set T of triangles such that the following properties hold: (1.5.3.1) A triangle isomorphic to a triangle in T belongs to T ; u v w (1.5.3.2) A triangle X Ð →YÐ →ZÐ → ΣX in C belongs to T if and only if the triangle Y Ð→ Z Ð→ ΣX ÐÐ→ ΣY belongs to T ; −v

−w

−Σu

idX

(1.5.3.3) For every object, the triangle X Ð→ X → 0 → ΣX belongs to T ; u →Y→ (1.5.3.4) For every morphism u ∶ X → Y in C , there exists a triangle X Ð Z → ΣX in T ; u′ v′ u v w w′ (1.5.3.5) Let X Ð →YÐ →ZÐ → ΣX and X′ Ð → Y′ Ð → Z Ð→ ΣX′ be triangles of T . For every f ∈ C (X, X′ ) and every g ∈ C (Y, Y′ ) such that u′ ○ f = g ○u, there exists a morphism h ∈ C (Z, Z′ ) such that ( f , g, h) induces a morphism of triangles:

v

′

h

→ Z′

→ ΣX ←

←

←

w

→

←

→ Y′

→

→ ←

→

u

′

→ Z

←

←

←

v

g

f

X′

→ Y

←

u

←

X

w

′

Σf

→ ΣX

whose cone belongs to T . Let us say that a triangle in C is a distinguished triangle if it belongs to T , and that a morphism of triangles is a distinguished morphism if its cone is a distinguished triangle.

20

CHAPTER 1. TRIANGULATED CATEGORIES

The axioms a triangulated category thus claim that triangles isomorphic to distinguished triangles are distinguished (1.5.3.1), as well as their shifts (1.5.3.2); they assert the existence of distinguished triangles (1.5.3.3), (1.5.3.4); they finally allow to construct distinguished morphisms between distinguished triangles with two prescribed arrows (1.5.3.5). Given two morphisms of triangles which are homotopic, if one of them is distinguished, then so is the other. If one relaxes axiom (1.5.3.5) by only requiring the existence of a morphism h, one gets the weaker notion of a pretriangulated category. 1.5.4. — Let (C , Σ, T ) be a (pre)triangulated category. Let us endow the opposite category C o with the translation functor Σ−1 . Observe that a triangle in C , when viewed as a complex in C o , is again a triangle, so that T o = T is a set of triangles in C . Let us prove that (C o , Σ−1 , T o ) is a (pre)triangulated category. Axioms (1.5.3.1) and (1.5.3.2) follow formally from their analogue in C . idΣ−1 X

Let X be an object in C . Shifting the distinguished triangle Σ−1 X ÐÐÐ→ Σ−1 X → − idX

0 → X in C , we obtain the distinguished triangle Σ−1 X → 0 → X ÐÐ→→ X. In − idXo

the opposite category, this triangle rewrites as Xo ÐÐÐ→ Xo → 0 → Σo Xo . Since idXo

it is isomorphic to the triangle Xo ÐÐ→ Xo → 0 → Σo Xo , axiom (1.5.3.3) holds in C o. A similar argument shows that axioms (1.5.3.4) and (1.5.3.5) hold as well. Example (1.5.5). — Let C be an additive category and let KC be the homotopy category of complexes in C . We have seen how the cone of a morphism of complexes f ∶ X → Y sits in a diagram f

αf

βf

XÐ → Y Ð→ C f Ð→ ΣX, where α f ○ f is homotopic to 0, β f ○ α f = 0 and Σ f ○ β f is homotopic to 0. Consequently, these diagrams give rise to triangles in the homotopy category. We shall prove below that the category KC , endowed with the triangles isomorphic to such a cone triangle, is a triangulated category. Definition (1.5.6). — Let C be a (pre)triangulated category. An additive functor H ∶ C → A to an abelian category is said to be cohomological if the complex H(T) in A is exact for every distinguished triangle T.

1.5. TRIANGULATED CATEGORIES

21

Let H ∶ C → A be a cohomological functor. For every integer m ∈ Z, set = H ○ Σ m . By definition, every distinguished triangle

Hm

u

v

w

XÐ →YÐ →ZÐ → ΣX gives rise to a long exact sequence H−1 (w)

H0 (u)

0

H0 (v)

0

H0 (w)

0

...

⋅ ⋅ ⋅ → H (Z) ÐÐÐ→ H (X) ÐÐÐ→ H (Y) ÐÐÐ→ H (Z) ÐÐÐ→ H1 (X) Ð → −1

On the other hand, a shifting argument shows that to verify that an additive functor is cohomological, it suffices to prove that for every distinguished triangle as above, the complex H(X) → H(Y) → H(Z) is exact at H(Y). Lemma (1.5.7). — Let C be a (pre)triangulated category and let A be an object of C . The functor C (A, ⋅) ∶ C → Ab is a cohomological functor. u

v

w

Proof. — Let X Ð →YÐ →ZÐ → ΣX be a distinguished triangle and let us show that the complex u∗

v∗

C (A, X) Ð→ C (A, Y) Ð → C (A, Z) is exact in the middle, where u∗ and v∗ are the group morphisms deduced from composition with u and v respectively. First of all, we recall that v∗ ○ u∗ = 0; indeed v∗ ○ u∗ maps every f ∈ C (A, X) to v ○ u ○ f = 0. Let then f ∈ C (A, Y) be such that v∗ ( f ) = v ○ f = 0. By axioms (1.5.3.3) − idΣA

and (1.5.3.2), the triangle A → 0 → ΣA ÐÐÐ→ is distinguished; by axiom (1.5.3.2), −v

−w

−Σu

A

→ 0

the triangle Y Ð→ Z Ð→ ΣX ÐÐ→ ΣY is distinguished. By axiom (1.5.3.5), there exists a morphism h1 ∈ C (ΣU, ΣX) making the following diagram

−w

←

←

←

→ ΣA

h1

→ ΣX

Σf

→ ΣY

−Σu

←

←

←

← ←

→ Z

←

−v

− idΣA

→

→

f

→ ←

→ Y

→ ΣA

commutative. In particular, Σu ○ h1 = Σ f , so that f = u ○ Σ−1 h1 belongs to Im(u∗ ).

CHAPTER 1. TRIANGULATED CATEGORIES

22

1.6. Decent cohomological functors, and applications Definition (1.6.1). — A cohomological functor H ∶ C → A is said to be decent if: a) The abelian category A satisfies (AB∗4 ): products exist, and a product of exact sequences is exact; b) The functor H respects products. For example, for every object A ∈ ob(C ), the functor C (A, ⋅) is a decent cohomological functor. Definition (1.6.2). — A triangle T in C is said to be decent if the complex H(T) is exact for every decent cohomological functor H. In particular, a distinguished triangle is decent. In this definition, we should take care about universes. In practice, we will only use decent cohomological functors of the form C (A, ⋅), where A is an object of the (pre)triangulated category C . Consequently, if U is a universe containing C (X, Y) for every X, Y ∈ ob(C ), it suffices to consider cohomological functors with values in Ab U . (Check!) Proposition (1.6.3). — Let C be a (pre)triangulated category. Let us consider a morphism u v w → Y → Z → ΣX X h

v′

→

→

→

→

u′

←

←

←

←

←

←

←

g

f

Σf

w′

X′ → Y′ → Z′ → ΣX. of decent triangles. If f and g are isomorphisms, then h is an isomorphism, so that this morphism of triangles is an isomorphism. ←

←

←

Proof. — Let A be an object of C and let us apply the functor C (A, ⋅); we obtain the diagram of abelian groups Σu

→ C (A, ΣY)

Σf

→

→

→

→

→

h

←

→ C (A, ΣX)

←

g

f

w

←

→ C (A, Z)

←

←

←

v

←

→ C (A, Y)

←

u

←

C (A, X)

Σg

C (A, X′ ) → C (A, Y′ ) → C (A, Z′ ) →′ C (A, ΣX′ ) →′ C (A, ΣY′ ) ′ ′ ←

←

←

←

u

v

w

Σu

whose two rows are exact sequences. The two left vertical morphisms are induced by f and g, hence are isomorphisms, as well as the two right vertical morphisms, which are induced by Σ f and Σg. By the five lemma, the morphism h ∶ Z → Z′

1.6. DECENT COHOMOLOGICAL FUNCTORS, AND APPLICATIONS

23

induces an isomorphism C (A, Z) → C (A, Z′ ). Since this holds for every object A, it then follows from the Yoneda lemma that h is an isomorphism. Corollary (1.6.4). — Let C be a (pre)triangulated category and let u ∶ X → Y be u u → Y → Z → ΣX and T′ ∶ X Ð → Y → Z′ → ΣX be disa morphism in C . Let T ∶ X Ð tinguished triangles. There exists a morphism of triangles of the form (idX , idY , h), and for every such morphism, the morphism h ∶ Z → Z′ is an isomorphism. This corollary furnishes a kind of ‘‘uniqueness’’ to the axiom (1.5.3.4), in the sense that any two triangles extending a given morphism are isomorphic. However, there is no canonical such isomorphism. This is in fact one of the defects of the theory of triangulated categories that it does not functorially complete morphisms into triangles. Proof. — The existence of such a morphism of triangles follows from axiom (1.5.3.5). Since a distinguished triangle is decent, proposition 1.6.3 implies that h is an isomorphism. Corollary (1.6.5). — Let C be a (pre)triangulated category. Let f ∶ X → Y be a f

morphism in C . The triangle X Ð → Y → 0 → ΣX is distinguished if and only if f is an isomorphism. Proof. — Let us consider the morphism of triangles

←

←

←

←

←

←

←

=

→ 0

→ ΣX →

→ X

f

→

f

→

←

→ X

f

→ 0

←

→ X

←

idX

X

idΣX

→ ΣX.

Two out of three vertical morphisms are isomorphisms, and the top triangle is distinguished. If the bottom triangle is distinguished, proposition 1.6.3 implies (after a shift of the diagram) that f is an isomorphism. Conversely, if f is an isomorphism, then this morphism of triangles is an isomorphism, so that the bottom triangle is a distinguished.

CHAPTER 1. TRIANGULATED CATEGORIES

24

Corollary (1.6.6). — Let us consider the following diagram of distinguished triangles (in which the dashed arrows are not supposed to exist):

← v′

←

→ ΣX

h

→ Z′

→

→ Y′

→

→ ←

→

u′

w

←

←

→ Z

←

←

←

v

g

f

X′

→ Y

←

u

←

X

w′

Σf

→ ΣX.

a) The following conditions are equivalent: 1) v ′ ○ g ○ u = 0; 2) There exists f ∶ X → X′ such that u ′ ○ f = g ○u; 3) There exists h ∶ Z → Z′ such that h ○v = v ′ ○ g; 4) There exists a morphism of triangles ( f , g, h). b) If they hold, and if C (X, Σ−1 Z′ ) = 0, then such morphisms f and h are unique. Proof. — Let us prove that 1)⇒2). Applying the decent functor C (X, ⋅) to the bottom triangle, one obtains an exact sequence u′

v′

C (X, Σ−1 Z′ ) → C (X, X′ ) Ð → C (X, Y′ ) Ð → C (X, Z′ ). Since v ′ ○(g○u) = 0, there exists a morphism f ∈ C (X, X′ ) such that g○u = u ′ ○ f . If, moreover, C (X, Σ−1 Z′ ) = 0, there exists exactly one such morphism f . Conversely, the existence of a morphism f ∶ X → X′ such that u ′ ○ f = g ○ u implies that v ′ ○ g ○ u = v ′ ○ u ′ ○ f = 0. This shows that 1) and 2) are equivalent. The proof of the equivalence 1)⇔3) follows by passing to the opposite (pre)triangulated category. The implications 4)⇒2) and 4)⇒3) are obvious. Finally, if 2) holds, the existence of a morphism of triangles as in 4) follows from axiom (1.5.3.5), so that 2)⇒4). The proof of the implication 3)⇒4) is analogous. When these conditions hold and C (X, Σ−1 Z′ ) = 0, the uniqueness of the morphisms f and h has been established during the proof of their equivalence. u

Z

w

Corollary (1.6.7). — Let X Ð →YÐ →vÐ → ΣX be a distinguished triangle. Assume −1 that C (X, Σ Z) = 0. Then: u

v

a) The morphism w is the unique morphism ∂ such that the triangle X Ð →YÐ → ∂

ZÐ → ΣX is distinguished; u

v′

w′

b) For every distinguished triangle of the form X Ð →YÐ → Z′ Ð→ ΣX, there exists a unique morphism of triangles of the form (idX , idY , h), and it is an isomorphism.

1.6. DECENT COHOMOLOGICAL FUNCTORS, AND APPLICATIONS

25

Corollary (1.6.8). — Two morphisms of distinguished triangles of the form ( f , g, h) and ( f , g, h′ ) are homotopic. Proposition (1.6.9). — Let C be a (pre)triangulated category. a) A product (resp. a coproduct) of a family of distinguished triangles is a distinguished triangle. b) A triangle which is a direct factor of a distinguished triangle is a distinguished triangle. ui

vi

wi

Proof. — a) Let (Xi Ð → Yi Ð → Zi Ð→ ΣXi )i∈I be a family of distinguished triangles. Assume that the products X = ∏i∈I Xi , Y = ∏i∈I Yi and Z = ∏i∈I Zi exist. Since Σ is an automorphism of the category C , it commutes with products, and ΣX is a product of the family ΣXi . Let u ∶ X → Y be the unique morphism u v w pYi ○ u = u i ○ pXi , for every i; define v and w similarly. Then X Ð →YÐ →ZÐ → ΣX is a triangle, and the claim is that this triangle is distinguished. We first prove that it is a decent triangle. To that aim, let A be an object of C and let us apply the cohomological functor C (A, ⋅) to this triangle. We get a complex (Σ−1 w i )

(u i )

C (A, ∏ Σ−1 Zi ) ÐÐÐ→ C (A, ∏ Xi ) ÐÐ→ i

i (v i )

(w i )

C (A, ∏ Yi ) ÐÐ→ C (A, ∏ Zi ) ÐÐ→ C (A, ∏ ΣXi ) i

i

i

which, by definition of products in a category, identifies with the product of the complexes Σ−1 w i

ui

vi

wi

C (A, Σ Zi ) ÐÐ→ C (A, Xi ) Ð → C (A, Yi ) Ð → C (A, Zi ) Ð→ C (A, ΣXi ), −1

for i ∈ I. Since each of these complexes of abelian groups is exact (proposition ??), the initial complex is exact as well. Let us now complete the morphism u ∶ X → Y into a distinguished triangle u

v ′

w′

XÐ →YÐ → Z′ Ð→ ΣX. For every i ∈ I, there exists a morphism h i ∶ Z′ → Zi which gives rise to a morphism of distinguished triangles:

vi

hi

→ Zi

→ ΣX ←

←

←

←

pYi

←

→ Yi

w′

wi

→

ui

→ Z′

←

←

←

v′

→

←

Xi

→ Y →

→

pXi

u

←

X

ΣpXi

→ ΣXi .

CHAPTER 1. TRIANGULATED CATEGORIES

26

Let h ∶ Z′ → Z = ∏i∈I Zi be the morphism (h i ). It fits in a morphism of decent triangles → Z′

w′

→ Y

v

⇐

h

→ Z

←

←

⇐

←

→ ΣX

wi

⇐

←

u

→

←

X

v′

⇐

⇐

⇐

→ Y

←

u

←

X

→ ΣXi .

Since two out of three morphisms are isomorphisms (they are identities!), so u v w →YÐ →ZÐ → ΣX is distinguished, as is h. Consequently, the initial triangle X Ð was to be proved. The case of coproducts follows by considering the opposite triangulated category. u

v

u′

w

v′

w′

→YÐ →ZÐ → ΣX and T′ ∶ X′ Ð → Y′ Ð → Z′ Ð→ ΣX′ be two triangles b) Let T ∶ X Ð whose direct sum ( u u′ )

( v v′ )

( w w′ )

X ⊕ X ÐÐÐ→ Y ⊕ Y ÐÐÐ→ Z ⊕ Z ÐÐÐÐ→ ΣX ⊕ ΣX′ ′

′

′

is a distinguished triangle. Let us prove that T is a distinguished triangle. First of all, it is decent. Indeed, for every object A ∈ C , applying the functor C (A, ⋅) to the triangle T ⊕ T′ furnishes the exact sequence −1 (Σ w

Σ−1 w ′

( u u′ )

)

C (A, Σ−1 Z)⊕C (A, Σ−1 Z′ ) ÐÐÐÐÐÐÐ→ C (A, X)⊕C (A, X′ ) ÐÐÐ→ C (A, Y)⊕C (A, Y′ ) Consequently, the complex Σ−1 w

u

v

w

C (A, Σ−1 Z) ÐÐ→ C (A, X) Ð → C (A, Y) Ð → C (A, Z) Ð → C (A, ΣX) is exact, which proves that the triangle T is decent. Let us now complete the morphism u ∶ X → Y into a distinguished triangle u v˜ w˜ ˜Ð ˜ → Z and h′ ∶ Z ˜ → Z′ be morphisms that fit in a XÐ →YÐ →Z → ΣX. Let h ∶ Z morphism of distinguished triangles:

←

→ Y ⊕ Y′ u′ )

(v

( h′ ) h

→ Z ⊕ Z′ v′ )

←

←

←

( 01 )

(w

→ ΣX →

←

(u

w˜

←

←

←

˜ → Z →

→

X ⊕ X′

v˜

←

→ Y

→

( 01 )

u

←

X

( 01 )

→ ΣX ⊕ ΣX′ . w′ )

1.6. DECENT COHOMOLOGICAL FUNCTORS, AND APPLICATIONS

27

Composing with the projections, we obtain a morphism of decent triangles: → ΣX

w

→ ΣX

←

⇐

←

⇐ ←

→ Y

→

u

w˜

v

h

→ Z

⇐

⇐

˜ → Z

⇐

⇐

←

X

v˜

←

→ Y

←

u

←

X

where two vertical arrows out of three are isomorphisms. Consequently, the remaining arrow h is an isomorphism; the triangle T is isomorphic to the top triangle, which is distinguished, hence T is a distinguished triangle.

Example (1.6.10). — Let X and Y be objects of a (pre)triangulated category. The canonical morphisms X → X ⊕ Y and X ⊕ Y → Y fit in a triangle X → X ⊕ Y → 0 YÐ → ΣX. This triangle is distinguished. idX

Indeed, it is the direct sum of the triangles X Ð→ X → 0 → ΣX and 0 → idY

Y Ð→ Y → 0. The first one is distinguished, by axiom (1.5.3.3). The second one − idY

is isomorphic to the triangle 0 → Y ÐÐ→ Y → 0 which is a shift of the distinidY

guished triangle Y Ð→ Y → 0 → ΣY, hence is distinguished by axioms (1.5.3.1) and (1.5.3.2). Proposition (1.6.11). — Let C be a (pre)triangulated category. a) A contractible triangle is distinguished. b) A morphism between distinguished triangles which is null homotopic is distinguished. Proof. — a) Let T be a contractible triangle. Let us first prove that it is decent. Let A be an object of C and let us apply the functor C (A, ⋅). We obtain a complex C (A, T) of abelian groups. Let h ∶ T → T[−1] be a homotopy with origin idT and target 0; then C (A, h) is a homotopy with origin idC (A,T) and target 0. Consequently, the complex C (A, T) is exact. u v w Let X Ð →YÐ →ZÐ → ΣX be this triangle and let

u

→ Y

← →

θ

v

→ Z ← φ

v

→ Z

← →

→ Y ←

←

←

X

u

←

X

w

→ ΣX ← ψ

w→

ΣX

← →

28

CHAPTER 1. TRIANGULATED CATEGORIES

be a homotopy with origin idX and target 0, so that idX = θ ○ u + Σ−1 (w ○ ψ) idY = φ ○ v + u ○ θ idZ = ψ ○ w + v ○ φ. Since v ○ u = 0, this implies u = φ ○ v ○ u + u ○ θ ○ u = u ○ θ ○ u = u + u ○ Σ−1 (w ○ ψ), so that u ○ Σ−1 (w ○ ψ) = 0, hence Σu ○ (w ○ ψ) = 0. Let us complete u ∶ X → Y into a distinguished triangle u

v′

w′

XÐ →YÐ → Z′ Ð→ ΣX and let us apply the cohomological functor C (ΣX, ⋅); one gets an exact sequence Σu∗

w∗′

C (ΣX, Z′ ) Ð→ C (ΣX, ΣX) ÐÐ→ C (ΣX, ΣY). consequently, there exists ψ ′ ∈ C (ΣX, Z′ ) such that w ′ ○ ψ ′ = w ○ ψ. Let λ = ψ ′ ○ w + v ′ ○ φ ∶ Z → Z′ ; one has w ′ ○ λ = w ′ ○ ψ′ ○ w + w ′ ○ v ′ ○ φ = w ○ ψ ○ w = w − w ○ v ○ φ = w λ ○ v = ψ′ ○ w ○ v + v ′ ○ φ ○ v = v ′ − v ′ ○ u ○ θ = v ′ . Consequently, the diagram → ΣX

w′

→ ΣX

← v′

⇐

←

⇐

←

w

λ

→ Z′

←

→ Y

→

u

→ Z

⇐

⇐

v

⇐

⇐

←

X

→ Y

←

u

←

X

depicts a morphism of triangles. Since these triangles are decent and two out of three vertical morphisms are isomorphisms, it is an isomorphism of triangles. In particular, the initial triangle is distinguished. b)

1.7. The octahedral axiom Let (C , Σ, T ) be a triangulated category.

1.7. THE OCTAHEDRAL AXIOM

29

Definition (1.7.1). — One says that a commutative square f

Y

←

←

←

g

f′

g′

→

→ ←

Y′

→ Z → Z′

is homotopically cartesian if there exists a morphism h ∶ Z′ → ΣY in C so that the diagram g (−f )

( f ′ g′ )

h

Y ÐÐÐ→ Y ⊕ Z ÐÐÐ→ Z′ Ð → ΣY ′

is a distinguished triangle. Observe that the composition of the first two arrows vanished, since ( f′

) = g ′ ○ f − f ′ ○ g = 0.

a) Let f

g

f′

g′

→

→ ←

Y′

→ Z ←

←

Y

←

Lemma (1.7.2). —

g g′ ) ○ ( − f

→ Z′

be a homotopically cartesian commutative square and let

φ′

→

→ ←

Y′

→ Z ←

←

g

f

←

Y

γ′

→ P

be a commutative square in C . There exists a morphism h ∶ Z′ → P such that h ○ g ′ = γ ′ and h ○ f ′ = φ′ . b) Let Y, Y′ , Z be objects of C and let f ∶ Y → Z and g ∶ Y → Y′ be morphisms. There exists an object Z′ and morphisms g ′ ∶ Z → Z′ and f ′ ∶ Y′ → Z′ such that the diagram

←

is a homotopically cartesian square.

f′

→

→ Y′

→ Z ←

←

g

f

←

Y

g′

→ Z′

30

CHAPTER 1. TRIANGULATED CATEGORIES

c) Moreover, if Z′′ is an object of C and g ′′ ∶ Z → Z′′ , f ′′ ∶ Y′′ → Z′ are morphisms in C such that → Z ←

←

g

f

←

Y

→

→

f ′′

g ′′

Y′ → Z′′ is homotopically cartesian, there exists an isomorphism h ∶ Z′ → Z′′ such that h ○ f ′ = f ′′ and h ○ g ′ = g ′′ . ←

Proof. — a) The functor Hom(⋅, P) is a cohomological functor on the opposite triangulated category. Applying it to the distinguished triangle Y → Y′ ⊕Z → Z′ → ΣY, we obtain an exact sequence C (Z′ , P) → C (Y′ ⊕ Z, P) → C (Y, P). The image of the morphism φ′ − γ′ is φ′ ○ g − γ′ ○ f = 0. Consequently, there exists a morphism h ∈ C (Z′ , P) such that φ′ = h ○ f ′ and γ′ = h ○ g ′ . g b) It suffices to complete the morphism ( − f ) ∶ Y → Y′ ⊕ Z in a distinguished triangle. c) This follows from the uniqueness property of corollary 1.6.4. Lemma 1.7.2 provides ‘‘homotopy push-outs’’ in triangulated categories; by passing to the opposite category, one deduces the following lemma which provides ‘‘homotopy pull-backs’’. a) Let

f′

→

→ ←

Y′

→ Z ←

g

f

←

Y

←

Lemma (1.7.3). —

g′

→ Z′

be a homotopically cartesian commutative square and let

g′

f′

→

→ ←

Y′

→ Z ←

←

γ

φ

←

Q

→ Z′

be a commutative square in C . There exists a morphism h ∶ Q → Y such that f ○ h = φ and g ○ h = γ.

1.7. THE OCTAHEDRAL AXIOM

31

b) Let Y′ , Z, Z′ be objects of C and let f ′ ∶ Y′ → Z′ and g ′ ∶ Z → Z′ be morphisms. There exists an object Y and morphisms f ∶ Y → Z and g ∶ Y → Y′ such that the diagram f

Y

→ Z ←

←

←

g

→

→

f′

Y′

g′

→ Z′

←

is a homotopically cartesian square. ˜ is an object of C and g˜ ∶ Y ˜ → Y, f˜ ∶ Y ˜ → Y are morphisms c) Moreover, if Y in C such that f˜

˜ Y

←

←

←

→

→

g˜

→ Z

f′

Y′

g′

→ Z′

←

˜ → Y such that is homotopically cartesian, there exists an isomorphism h ∶ Y f ○ h = f˜ and g ○ h = g˜. Lemma (1.7.4). — Let

→

→ ΣX

w′

→ ΣX

⇐

←

→ → Y′

←

u

′

w

←

←

←

→ Z

g

⇐ ←

X

v

v

′

h

→ Z′

⇐

⇐

→ Y

←

u

←

X

be a distinguished morphism of distinguished triangles. Then the square v

←

→ Z

←

←

Y g

→

→ ←

Y′

v

′

h

→ Z′

is homotopically cartesian and the triangle g ) ( −v

(v ′ h)

Σu○w ′

Y ÐÐ→ Y′ ⊕ Z ÐÐ→ Z′ ÐÐÐ→ ΣY is distinguished. Proof. — By hypothesis, the cone C of this this morphism of triangle, ←

→ Y′ ⊕ Z

′ h ) ( v0 −w

→ Z′ ⊕ ΣX

′ (w 1 ) 0 −Σu

←

←

X⊕Y

′ (u g ) 0 −v

→ ΣX ⊕ ΣY,

CHAPTER 1. TRIANGULATED CATEGORIES

32

is a distinguished triangle. Observe that the diagram ←

←

⇐

←

→ Y′ ⊕ Z

′ −1 ( −w 1 0)

→ ΣX ⊕ Z′ −1

←

←

→ ΣX ⊕ ΣY′

←

0 0 ( v′ h )

←

g (0 0 −v )

(

→

→

⇐

X⊕Y

→ Z′ ⊕ ΣX

′ (w 1 ) 0 −Σu

←

→ Y′ ⊕ Z

→

( u1 01 )

′ h ) ( v0 −w

←

X⊕Y

′ (u g ) 0 −v

1 0) ( Σu 1

→ ΣX ⊕ ΣY′

Σu○w ′ )

is commutative. Indeed, (

1 0 w′ 1 w′ 1 −1 −w ′ −1 )( )=( ) = ( ) ( ), Σu 1 0 −Σu Σu ○ w ′ 0 Σu ○ w ′ 1 0 −w ′ 1 v ′ h 0 0 ( )( )=( ′ ) 1 0 0 −w v h u′ g 0 g 1 0 ( )=( )( ). 0 −v 0 −v u 1

Since the vertical morphisms are isomorphisms, it is an isomorphism of triangles from C to the bottom triangle, which is therefore distinguished. On the other hand, the bottom triangle is the direct sum of the triangle T of interest and the −1 triangle X → 0 → ΣX Ð→ ΣX. Consequently, the triangle T is a direct factor of a distinguished triangle, hence is distinguished. Lemma (1.7.5). — Let

→

→ ←

Y′

→ Z ←

v

←

g

←

Y

v′

h

→ Z′

be a homotopically cartesian square, and let ∂ ∶ Z′ → ΣY be such that g (−f )

( v′ h )

∂

Y ÐÐÐ→ Y′ ⊕ Z ÐÐÐ→ Z′ Ð → ΣY is a distinguished triangle. Let g

g′

g ′′

YÐ → Y′ Ð → Y′′ Ð→ ΣY be a distinguished triangle. There exists a distinguished triangle h

h′

h ′′

ZÐ → Z′ Ð → Z′′ Ð→ ΣZ

1.7. THE OCTAHEDRAL AXIOM

33

such that the diagram

→ Z′

→ ΣY ←

⇐

←

v′

← h′

→ Z′′

←

←

←

←

← h

g ′′

→

Z

→ Y′′ ⇐

→

→

v

g′

→ Y′

←

g

Y

h ′′

Σv

→ ΣZ

is commutative. Moreover, ∂ = g ′′ ○ h ′ .

Proof. — Let k ∶ Z′ → Y′′ be a morphism such that the morphism of triangles

←

⇐

→ ΣY

k

→ Y′′

← g′

←

←

←

(1 0)

→ Y′

g

∂

⇐

⇐

→ Z′ →

→

⇐ ←

Y

( v′ h )

←

→ Y′ ⊕ Z

←

Y

g ) ( −v

g ′′

→ ΣY

is a distinguished morphism (axiom (1.5.3.5)), so that its cone C is a distinguished triangle. Let us consider the diagram:

←

)

1 ( Σg 1 ) −Σv 0 1 ′ ΣY ⊕ ΣZ.

←

←

←

←

−g −1 ( 0 0) −1 0 ′′ → ΣY ⊕ 0 → ΣY ⊕ 0 ⊕ Y 1 0 ( ) ( 0 ) k Σv○g

←

←

h

→ ΣY ⊕ ΣY′ ⊕ ΣZ →

← (

→ 1

′′

→

0

→ Y′′ ⊕ ΣY

0 0 ( 1′ 0 ) v −1 ′ 0 ⊕ Y ⊕ Z′

→

Y ⊕ Y′ ⊕ Z

(g k ) 0 −∂

←

→ Y′ ⊕ Z′

→

1 ( g 1 ) −v 0 1

g 1 0 ) 0 −v ′ −h

′′ 1 ⎞ ⎛g 0 −Σg ⎝ 0 Σv ⎠

←

Y ⊕ Y′ ⊕ Z

(

′

Its first line is the triangle C; its bottom line is the direct sum of the three triangles 1

Y → 0 → ΣY Ð → ΣY 1

Y′ Ð → Y′ → 0 → ΣY′ h

k

Σv○g ′′

ZÐ → Z′ Ð → Y′′ ÐÐÐ→ ΣZ;

CHAPTER 1. TRIANGULATED CATEGORIES

34

the vertical morphisms are isomorphisms. Let us check that is is commutative:

0 0⎞ ⎛0 ⎛0 0 ⎞ ⎛ 0 ⎞⎛ 1 ⎞ ⎜ 1 0 ⎟ (g 1 0 ) = ⎜ g ⎟ ⎜ ⎟ ⎜ 1 0⎟ = ⎜ 1 ⎟ ⎜ g 1 ⎟ ⎜ ⎟ 0 −v ′ −h ⎜ ⎟ ′ ′ ⎝v −1⎠ ⎝−v ○ g 0 h⎠ ⎝ h⎠ ⎝−v 0 1⎠ ⎛−g ′′ −1⎞ ⎛ 0 0 ⎞ ⎛ 0 0 ⎞ ⎛ 0 0 ⎞ ⎛0 ⎞ ⎜ 0 0 ⎟⎜ 1 0 ⎟ = ⎜ 0 0 ⎟ = ⎜ 1 0 ⎟⎜ 0 ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ′ ′ ′ ⎝ −1 0 ⎠ ⎝v −1⎠ ⎝−g −k ⎠ ⎝v −1⎠ ⎝ k⎠ ⎛ 1 ⎞ ⎛ g ′′ 1 ⎞ ⎛1 ⎞ ⎛−g ′′ −1⎞ ⎜ Σg 1 ⎟ ⎜ 0 −Σg ⎟ = ⎜ 0 ⎟⎜ 0 0 ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝−Σv 0 1⎠ ⎝ 0 Σv ⎠ ⎝ Σv ○ g ⎠ ⎝ −1 0 ⎠

Consequently, the bottom triangle is distinguished; in particular, the triangle h

k Z′ Ð →

Y′′

Σv○g ′′

ZÐ → ÐÐÐ→ ΣZ is distinguished. Set Z′′ = Y′′ , h′ = k and h′′ = Σv ○ g ′′ ; one has h ○ v = v ′ ○ g, by hypothesis; one has ∂ = g ′′ ○ k = g ′′ ○ h′ and h′ ○ v ′ = k ○ v ′ = g ′ , by definition of k; and one has Σv ○ g ′′ = h′′ by definition of h′′ . This concludes the proof of the lemma.

Theorem (1.7.6) (Verdier’s ‘‘octahedral axiom’’). — Let u3

v3

w3

u1

v1

w1

u2

v2

w2

X1 Ð → X2 Ð → Z3 Ð→ ΣX1 X2 Ð → X3 Ð → Z1 Ð→ ΣX2 X1 Ð→ X3 Ð → Z2 Ð→ ΣX1 ,

be three distinguished triangles, where u2 = u1 ○ u3 . There exist two morphisms m1 ∶ Z3 → Z2 and m3 ∶ Z2 → Z1 such that

m1

m3

Σv 3 ○w 1

Z3 Ð→ Z2 Ð→ Z1 ÐÐÐ→ ΣZ3

1.7. THE OCTAHEDRAL AXIOM

35

is a distinguished triangle, fitting in a commutative diagram

v2

←

⇐

m2

⇐

←

Σv 3 ○w 1 Σw 3

→ Z3

←

←

←

→

Σv 3

→ X2

→ ΣX2

⇐ Z1

w1

→

Σu 3

w2

←

←

→

→

←

←

ΣX1

m1

→ Z2

v1

Z1

→ ΣX1 ⇐

→ X3

→

u2

w3

←

←

→ Z3

u1

→

⇐ X1

v3

←

←

⇐

→ X2

←

u3

←

X1

→ Σ2 X1

Proof. — Let us apply lemma 1.7.2. This furnishes a morphism m1 ∶ Z3 → Z2 giving rise to a morphism of distinguished triangles

w2

→ ΣX2

←

←

→ ΣX1

v2

m1

→ Z2

⇐

u1

←

→ X3

w3

⇐

←

⇐

→ Z3 →

u2

→

⇐ ←

X1

v3

←

→ X2

←

u3

←

X1

in which the commutative square v3

→ Z3 ←

←

←

X2

←

X3

v2

→

→

u1

m1

→ Z2

is homotopically cartesian. Lemma 1.7.4 then furnishes a morphism m2 ∶ Z2 → Z1 giving rise to a morphism of distinguished triangles X2

v3

←

←

←

→ Z3

X3

→

→

u1 v2

←

←

←

→ Z2 →

← Σv 3

→

←

→

w1

X2

m2

⇐ Z1

←

⇐

→

v1

Z1

m1

Σv 3 ○w 1

→ Z3 .

CHAPTER 1. TRIANGULATED CATEGORIES

36

This concludes the proof of the theorem. Remark (1.7.7). — The name of this axiom comes from a particular way of representing its final diagram as an octahedron. Indeed, if one identifies the vertex Xi and its shift ΣXi , for each i, as well as the two vertices of an identity morphisms, one gets a figure with eight triangles: four of them are the distinguished triangles, and the four other are the commutative triangles. The reader shall find in Hartshorne (1966), Be˘ılinson et al (1982), Neeman (1990), or in May (2001) alternative representations of the diagram, which she may find more appealing. To conclude this section, let us quote a strengthening of the octahedral axiom: Proposition (1.7.8) (‘‘3 × 3 lemma’’). — Every commutative square X

u

←

←

←

g

→

→

f

→ Y

←

X′

u′

→ Y′

can be completed to a diagram ←

←

←

← ← Σv

h′′

→ ΣZ

→ ΣX′′

(−1) Σw

→

g ′′

w ′′

Σf′

←

← ←

←

←

←

→ Z′′

←

v ′′

Σf

→ ΣX′

h′

←

←

←

← ←

←

←

←

←

←

→ Y′′

w′

→

g′

→ ΣY

→ ΣX →

Σu

w

h

→ Z′

→

←

ΣX

v′

→

u ′′

→

→

f ′′

→ Y′ →

X′′

→ Z →

u′

→

f′

v

→

X′

→ Y g

→

f

u

←

X

Σ f ′′

→ Σ2 Z

where the first three rows and first three columns are distinguished triangles, with commutative squares except for the right bottom one which is (−1)-commutative. 1.8. The homotopy category of an additive category 1.8.1. — Let A be an additive category and let C (A) be its homotopy category; recall that the objects of C (A) are complexes in A, and morphisms are homotopy classes of morphisms of complexes. It is an additive category. Let Σ be the translation functor of C (A).

1.8. THE HOMOTOPY CATEGORY OF AN ADDITIVE CATEGORY

37

For every morphism f ∶ X → Y of complexes in A, we have constructed its cone C f which is a complex, together with morphisms of complexes α f ∶ Y → C f , β f ∶ C f → ΣX, such that α f ○ f , β f ○ α f and Σ f ○ α f are null homotopic. Consequently, the cone gives rise to a triangle in K (A): f

αf

βf

XÐ → Y Ð→ C f Ð→ ΣX. We have also proved that two homotopic morphisms f , g ∶ X → Y give rise to isomorphic triangles. Definition (1.8.2). — Let T be the set of all triangles in K (A) which are isomorphic to the cone triangle of a morphism of complexes. Theorem (1.8.3). — Let A be an additive category. Together with its translation automorphism, and its set of triangles T , the homotopy category K (A) is a triangulated category. Let us say that a triangle is distinguished if it belongs to T . Proof. — 1) By definition, a triangle which is isomorphic to a distinguished triangle is distinguished. 2) Let us consider a distinguished triangle T and let us prove that the shift of T f

g

h

→ ΣX is distinguished as well. We may assume that T is the triangle X Ð →YÐ → Cf Ð −g

−h

−Σ f

(where g = α f and h = β f ), so that its shift is the triangle Y Ð→ C f Ð→ ΣX ÐÐ→. By definition, C−g = C f ⊕ ΣY = Y ⊕ ΣX ⊕ ΣY, endowed with the differential dC−g

1 ⎞ ⎛d Y Σ f dC f −g =( )=⎜ 0 ⎟ ⎜ 0 −ΣdX ⎟. 0 −dY ⎝0 0 −ΣdY ⎠

0

Let u = ( −Σ1 f ) ∶ ΣX → C−g ; let v = ( 0 1 0 ) ∶ C−g → ΣX. Observe that u is a morphism of complexes because 1 ⎞⎛ 0 ⎞ ⎛ 0 ⎛d Y Σ f ⎞ ⎜ ⎟ ⎜ ⎟ dC−g ○ u = ⎜ 0 ⎟ ⎜ 0 −ΣdX ⎟ ⎜ 1 ⎟ = ⎜ −ΣdX ⎟ , ⎝0 0 −ΣdY ⎠ ⎝−Σ f ⎠ ⎝ΣdY ○ Σ f ⎠ 0 ⎛ 0 ⎞ ⎛ ⎞ ⎟ ⎜ ⎟ u ○ d ΣX = − ⎜ ⎜ 1 ⎟ ΣdX = ⎜ −ΣdX ⎟ ⎝−Σ f ⎠ ⎝Σ f ○ ΣdX ⎠

CHAPTER 1. TRIANGULATED CATEGORIES

38

and Σ f ○ ΣdX = ΣdY ○ Σ f , since f ∶ X → Y is a morphism of complexes. Similarly, d ΣX ○ v = −ΣdX ○ (0 1 0) = (0 −ΣdX 0) , v ○ ΣdC−g

1 ⎞ ⎛d Y Σ f = (0 1 0) ⎜ 0 ⎟ ⎜ 0 −ΣdX ⎟ = (0 −ΣdX 0) . ⎝0 0 −ΣdY ⎠

One has ⎛ 0 ⎞ ⎟ v ○ u = (0 1 0) ⎜ ⎜ 1 ⎟ = 1. ⎝−Σ f ⎠ On the other hand, 0⎞ ⎛ 0 ⎞ ⎛0 0 ⎛ 1 0 0⎞ ⎟ ⎜ ⎜ ⎟ u○v = ⎜ 0⎟ ⎜ 1 ⎟ (0 1 0) = ⎜0 1 ⎟ = id − ⎜0 0 0⎟ . ⎝−Σ f ⎠ ⎝0 −Σ f 0 =⎠ ⎝0 Σ f 1 ⎠ 0

Let θ ∶ C−g → Σ−1 C−g be defined by the matrix ( 1 dC−g θ + θdC−g

0 ). 0 0

One has

1 ⎞ 1 ⎞ ⎛0 ⎞ ⎛d Y Σ f ⎞ ⎛0 ⎛d Y Σ f ⎟⎜ ⎟ ⎜ ⎜ =⎜ 0 ⎟ 0 ⎟ ⎟ ⎟ ⎜ 0 ⎟ + ⎜ 0 ⎟ ⎜ 0 −ΣdX ⎜ 0 −ΣdX ⎝0 0 −ΣdY ⎠ 0 −ΣdY ⎠ ⎝ 1 0 0⎠ ⎝ 1 0 0⎠ ⎝ 0 0 0⎞ ⎛ 0 0 0⎞ ⎛ 1 0 0⎞ ⎛ 1 ⎟ ⎜ ⎟ ⎜ =⎜ 0 0⎟ ⎟ + ⎜ 0 0 0⎟ = ⎜0 0 0⎟ ⎜ 0 ⎝−ΣdY 0 0⎠ ⎝dY Σ f 1 ⎠ ⎝0 Σ f 1 ⎠ = id −u ○ v.

Consequently, u ○ v − id is null homotopic. This proves that u is an isomorphism from C f to C−g in the homotopy category, with inverse isomorphisms v. Let us now prove that the diagram

⇐

⇐

v

⇐

←

Cf

→ ΣY

←

−g →

−Σ f

→

←

⇐

→ ΣX

⇐

⇐ ←

Y

−h

α−g→

C−g

←

→ Cf

←

−g

←

Y

→ ΣY

β−g

1.8. THE HOMOTOPY CATEGORY OF AN ADDITIVE CATEGORY

39

is commutative. Indeed, one has v ○ α−g

⎛0 0⎞ ⎟ = (0 1 0) ⎜ ⎜ 1 0⎟ = (1 0) = β f ⎝0 1 ⎠

3) Let X be a complex and let f ∶ 0 → X be the unique morphism. One 1 has C f = X, α f = id, and β f = 0. This leads to a distinguished triangle 0 → X Ð → 0 −1 0 XÐ → ΣX. This triangle is isomorphic to the triangle 0 → X Ð→ X Ð → ΣX, which is therefore distinguished. Since a shift of a distinguished triangle is distinguished, 1 this proves that the triangle X Ð → X → 0 → ΣX is distinguished. αf

f

βf

4) For every morphism f ∶ X → Y, the cone triangle X Ð → Y Ð→ C f Ð→ ΣX is distinguished, by definition. 5) Let us finally prove axiom (1.5.3.5): given two distinguished triangles and a partial morphism between them, we need to show that it can be completed into a distinguished morphism of distinguished triangles in K (A). We may assume that the two given triangles are cone triangles, and choose representatives in C (A). Let thus consider a diagram of complexes in A: ← α′

h

→ Y′ ⊕ ΣX′

β′

←

←

→ Y′

→ ΣX →

→

u′

β

←

→ Y ⊕ ΣX

←

←

←

←

α

→ ←

X′

→ Y g

→

f

u

←

X

Σf

→ ΣX′

where u, u ′ , f , g are morphisms of complexes such that g ○ u is homotopic to u′ ○ f ; let θ ∶ X → Σ−1 Y′ be a homotopy such that g ○ u − u ′ ○ f = dθ + θd. Let us show the existence of a morphism h ∶ Y ⊕ ΣX → Y′ ⊕ ΣX′ such that the triangle ′ (u g ) 0 −α

′ h ) ( α0 −β

′ (β Σf ) 0 −Σu

X′ ⊕ Y ÐÐÐÐ→ Y′ ⊕ (Y ⊕ ΣX) ÐÐÐÐ→ (Y′ ⊕ ΣX′ ) ⊕ ΣX ÐÐÐÐÐ→ ΣX′ ⊕ ΣY is distinguished. We set h = ( 0g ΣΣ fθ ). Then −1

h ○ α − α′ ○ g = (

g Σ−1 θ 1 1 ) ( ) − ( ) g = 0, 0 Σf 0 0

g Σ−1 θ Σ f ○ β − β ○ h = Σ f (0 1) − (0 1) ( ) = (0 Σ f ) − (0 Σ f ) = 0, 0 Σf ′

so that the preceding diagram is indeed a morphism of distinguished triangles. To prove that its cone is a distinguished morphism, we shall show that there

CHAPTER 1. TRIANGULATED CATEGORIES

40

exists morphisms of complexes φ and ψ, as represented by the diagram below, that give rise to isomorphisms, inverse one of the other:

→ ΣX′ ⊕ ΣY ⇐

ψ

⊕ ΣX

(0 1 f ) 0 0 −u

→ ←

⊕

ΣX′

φ

⇐

←

⇐

→

Y′

→ ←

←

→ Y′ ⊕ Y ⊕ ΣX

←

u′ g ( 0 −1 ) 0 0

1 ⎛0 ⎜0 ⎝0 0

0 1 0 0 0

′ ′ ′ → 1 0 ΣX ⊕ ΣY 0 Y ⊕ Y ⊕ ΣX ⊕ ΣX ⊕ ΣY 0 0 0 →

←

⇐

X′ ⊕ Y

⊕ Y ⊕ ΣX ⇐

⇐

→

←

⊕Y

Y′

1 g θ (0 0 f ) 0 0 −1

←

X′

u′ g ( 0 −1 ) 0 0

(0 0 0 0 1 )

0⎞ 1⎟ 0⎠ 0

Set ⎛1 ⎜0 ⎜ ⎜ φ = ⎜0 ⎜ ⎜0 ⎜ ⎝0

0 θ⎞ 0 0⎟ ⎟ ⎟ 0 −1 ⎟ ⎟ 1 f ⎟ ⎟ 0 −u ⎠

⎛ 1 0 0 0 0⎞ ⎟ and ψ = ⎜ ⎜ g 0 0 1 0⎟ . ⎝θ f −1 0 0⎠

Write A, B, C for the matrices of the first row, and A′ , B′ , C′ for those of the second row. One has ψA′ = A,

B′ φ = B, ψ ○ φ = id .

Moreover, ⎛1 ⎜0 ⎜ ⎜ φ ○ φ = ⎜0 ⎜ ⎜0 ⎜ ⎝0

g 0 0 0 0

0 0 1 0 u

0 0 0 1 0

0⎞ ⎛0 ⎜0 0⎟ ⎟ ⎜ ⎟ ⎜ = id + ⎜0 0⎟ ⎟ ⎜ ⎜0 0⎟ ⎟ ⎜ ⎝0 0⎠

g −1 0 0 0

0 0 0 0 u

0 0 0 0 0

0⎞ 0⎟ ⎟ ⎟ . 0⎟ ⎟ ⎟ 0⎟ −1⎠

Let ⎛0 ⎜0 ⎜ ⎜ H = ⎜0 ⎜ ⎜0 ⎜ ⎝0

0 0 0 0 1

0 0 0 0 0

0 0 0 0 0

0⎞ 0⎟ ⎟ ⎟ ′ ∶ Y ⊕ Y ⊕ ΣX ⊕ ΣX′ ⊕ ΣY → Σ−1 Y′ ⊕ Σ−1 Y ⊕ X ⊕ X′ ⊕ Y. 0⎟ ⎟ 0⎟ ⎟ 0⎠

1.9. LOCALIZATION

41

The differentiel of the cone Y′ ⊕ Y ⊕ ΣX ⊕ ΣX′ ⊕ ΣX is given by ⎛d ⎜0 ⎜ ⎜ δ = ⎜0 ⎜ ⎜0 ⎜ ⎝0

0 0 u′ g ⎞ d u 0 −1 ⎟ ⎟ ⎟ , 0 −d 0 0 ⎟ ⎟ 0 0 −d 0 ⎟ ⎟ 1 0 0 −d ⎠

and one checks that δH+Hδ = φ○ψ−id. Consquently, φ and ψ are isomorphisms in the homotopy category, inverse one of the other. Consequently, the first row is isomorphic to the bottom row, which is a cone, hence it is is a cone. This concludes the proof of the theorem. 1.9. Localization Definition (1.9.1). — Let C , D be triangulated categories. A functor F ∶ C → D is called a triangulated functor if it is additive, commutes with translations, and if it maps a distinguished triangle in C to a distinguished triangle in D. 1.9.2. — Let C be a triangulated category. A subcategory D of C is called a triangulated subcategory if the following properties hold: (i) Every object of C which is isomorphic to an object of D belongs to D; (ii) For every objects X, Y of D, one has D(X, Y) = C (X, Y); (iii) The subcategory D is stable under the translation functor of C and under finite coproducts; (iv) For every morphism f ∶ X → Y in D, there exists a distinguished triangle f

g

h

XÐ →YÐ →ZÐ → ΣX in C , and Z is an object of D. These axioms imply that D is a triangulated category when endowed with the restriction of the translation functor Σ and the set of triangles of C whose vertices belong to D, and that the inclusion functor is a fully faithful triangulated functor. Moreover, in a distinguished triangle X → Y → Z → ΣX, if two objects out of three belong to D, then so does the third one. This holds by hypothesis if X and Y belong to D, and the two other cases follow by considering translated triangles. The triangulated subcategory D is said to be thick if for every objects Y, Y′ of C such that Y ⊕ Y′ is an object of D, then Y, Y′ ∈ ob(D).

42

CHAPTER 1. TRIANGULATED CATEGORIES

Lemma (1.9.3). — Let F ∶ C → D be a triangulated functor. Let Ker(F) be the full subcategory of C whose objects are the objects X ∈ ob(C ) such that F(X) ≃ 0. Then Ker(F) is a thick triangulated subcategory. Proof. — a) Let X ∈ ob(Ker(F)); then F(X) ≃ 0, hence F(ΣX) = Σ(F(X)) ≃ 0, so that ΣX ∈ ob(Ker(F)). b) Let X, Y ∈ ob(C ) be such that X ≃ Y; if F(X) ≃ 0, then F(Y) ≃ F(X) ≃ 0; c) Let Y, Y′ ∈ ob(C ); if Y ⊕ Y′ ∈ Ker(F), then 0 ≃ F(Y ⊕ Y′ ) ≃ F(Y) ⊕ F(Y′ ), hence F(Y) ≃ 0 and Y ∈ Ker(F); d) Let X → Y → Z → ΣX be a distinguished triangle in C ; assume that X, Y ∈ ob(Ker(F)), so that F(X) ≃ F(Y) ≃ 0; then 0 → 0 → F(Z) → 0 is a distinguished triangle in D, so that F(Z) ≃ 0, hence Z ∈ ob(Ker(F)). The following theorem asserts that every thick triangulated subcategory appears in this way. Theorem (1.9.4) (Verdier). — Let C be a triangulated category and let N be a triangulated subcategory. There exists a triangulated category D and a triangulated functor F ∶ C → D such that: a) N ⊂ Ker(F); b) If F′ ∶ C → D ′ is a triangulated functor such that N ⊂ Ker(F′ ), there exists a unique functor G ∶ D → D ′ such that F′ = G ○ F. Moreover, Ker(F) is the smallest thick triangulated subcategory of C containing N . The proof of the theorem will occupy the rest of the section. We consider throughout a triangulated category C and a triangulated subcategory N . Definition (1.9.5). — One says that a morphism f ∶ X → Y in C is an isomorphism f

(mod N ) if there exists a distinguished triangle X Ð → Y → Z → ΣX in C such that Z ∈ ob(N ). Lemma (1.9.6). — a) If a morphism in C is an isomorphism, then it is an isomorphism (mod N ); b) Let f ∶ X → Y and g ∶ Y → Z be morphisms in C ; if two among f , g, g ○ f are isomorphisms (mod N ), then so is the third one; c) Let f ∶ X → Y be a morphism in C . Then f is an isomorphism (mod N ) if and only if Σ f is an isomorphism (mod N ).

1.9. LOCALIZATION

43

As a consequence, there exists a unique subcategory S of C whose set of morphisms is the set of isomorphisms (mod N ). Its set of objects is the set of objects of C . f

Proof. — a) Indeed, if f is an isomorphism, the triangle X Ð → Y → 0 → ΣX is distinguished, and 0 ∈ N . b) Let us consider an octahedral diagram

⇐

⇐ W

←

→ ΣU.

←

←

⇐

←

←

←

←

←

←

→

→

→

→ ΣY

→ ΣX

⇐

←

→ V

←

←

→ Z W

→ ΣX ⇐

g○ f

←

→ Y g

⇐ X

→ U →

f

→

X

By definition, f , resp. g, resp. g ○ f , is an isomorphism (mod N ) if and only if U, resp. W, resp. V, is an object of N . Since N is a triangulated subgcategory of N , if, in the distinguished triangle U → V → W → ΣU, two objects out of three belong to N , then so does the third one. This implies the claim. f

g

h

→YÐ →ZÐ → c) Let us assume that f is an isomorphism (mod N ) and let X Ð ΣX be a distinguished triangle, where Z ∈ ob(N ). Translating this triangle −Σ f

−Σg

−Σh

three times, one obtains a distinguished triangle ΣX ÐÐ→ ΣY ÐÐ→ ΣZ ÐÐ→ Σ2 X, Σf

Σg

−Σh

which is isomorphic to the triangle ΣX Ð→ ΣY Ð→ ΣZ ÐÐ→ Σ2 X. Since ΣZ is an object of N , this shows that Σg is an isomorphism (mod N ). The other direction is analogous.

g

→

→ X

←

Z f

←

1.9.7. — Let X, Y be objects of C . Let D(X, Y) be the set of triples (Z, f , g) where Z ∈ ob(C ), f ∈ S (Z, X) is an isomorphism (mod N ) and g ∈ C (Z, Y); we represent such a triple by the diagram

Y

44

CHAPTER 1. TRIANGULATED CATEGORIES

Let ∼ be the relation in D(X, Y) defined as follows: (Z′ , f ′ , g ′ ) ∼ (Z′′ , f ′′ , g ′′ ) if there exists a triple (Z, f , g) ∈ D(X, Y) and isomorphisms (mod N ) u ′ ∈ S (Z, Z′ ) and u ′′ ∈ S (Z, Z′′ ) such that f = f ′ ○ u and g = g ′ ○ u. We represent this by the diagram:

← ←

u ′′

←

→

←

f ′′

→ → Y →

g

← Z

f

g′

←

→ → →

X

←u′

←

f′

→

Z′

g ′′

Z′′ a) Let

→

→ ←

Y′

→ Z ←

g

v

←

Y

←

Lemma (1.9.8). —

v′

h

→ Z′

be a homotopically cartesian square. Then v ∈ mor(S ) if and only if v ′ ∈ mor(S ); and g ∈ mor(S ) if and only if h ∈ mor(S ). b) The relation ∼ in D(X, Y) is an equivalence relation. c) Let (W1 , f1 , g1 ) ∈ D(X, Y) and (W2 , f2 , g2 ) ∈ D(Y, Z); there exists a triple (W, f , g) ∈ D(W1 , W2 ) such that the square

g1

→

→ ←

W1

→ W2 ←

←

f

g

←

W

f2

→ Y

is commutative; for such (W, f , g), the equivalence class of the triple (W, f1 ○ f , g2 ○ g) only depends on the equivalence classes of the triples (W1 , f1 , g1 ) and (W2 , f2 , g2 ). Proof. — a) With the notation of lemma 1.9.6, g is an isomorphism (mod N ) if and only if Y′′ ∈ ob(N ), if and only if h is an isomorphism (mod N ). The other assertion follows by symmetry. b) It is obvious that the given relation is reflexive and symmetric; let us establish that it is transitive. Let (Z, f , g), (Z′ , f ′ , g ′ ), (Z′′ , f ′′ , g ′′ ) be elements of D(X, Y) such that (Z, f , g) ∼ (Z′ , f ′ , g ′ ) and (Z′ , f ′ , g ′ ) ∼ (Z′′ , f ′′ , g ′′ ); by

1.9. LOCALIZATION

45

definition, there exist two diagrams

f ′′

g′

←

→ → →

→

→

Z′

←

←v

′′

←

←

←

X

W′

←

and

→ → →

→ → Y →

←v ′

←

′

f′

→

←u

g

←

← W →

f′

←

X

←u

←

f

Z′

←

Z

g′

→ → Y → g ′′

Z′′

as above, where u, u ′ , v ′ , v ′′ are isomorphisms (mod N ). By lemma 1.7.4, there exists an object W′′ and morphisms w ∈ C (W′′ , W) and w ′ ∈ C (W′′ , W′ ) that give rise to a homotopically cartesian square W′′

w

←

←

←

→ W

v′

←

W′

→

→

w′

u′

→ Z′ .

By a), the morphisms w and w ′ belong to mor(S ). Consequently, the diagram

u○w

v ′′ ○w ′

←

←

←

←

W′′

←

→

→ → → f ′′

←

X

←

←

f

→

Z g

→ → Y → g ′′

Z′′ proves that (Z, f , g) ∼ (Z′′ , f ′′ , g ′′ ), as was to be shown. It then follows from the definition of ∼ that it is the equivalence relation generated by the relation given by (Z, f , g) ∼ (Z′ , f ′ , g ′ ) if there exists a morphism u ∈ S (Z′ , Z) such that f ′ = f ○ u and g ′ = g ○ u. c) Let (W, f , g) be any element of D(W1 , W2 ); observe that f1 ○ f ∈ S (W, X), so that (W, f1 ○ f , g2 ○ g) belongs to D(W, X, Z). Let us prove the existence of such triples (W, f , g), and let us show that for every such triple, the equivalence class modulo ∼ of (W, f1 ○ f , g2 ○ g) only depends on the equivalence classes of (W1 , f1 , g1 ) and (W2 , f2 , g2 ).

46

CHAPTER 1. TRIANGULATED CATEGORIES

By lemma 1.7.3, there exists a homotopically cartesian commutative square: → W2 ←

←

f

g

←

W

g1

→

→ ←

W1

f2

→ Y.

By a), the morphism f is an isomorphism (mod N ), so that (W, f , g) ∈ D(W1 , W2 ). More generally, let (W′1 , f1′ , g1′ ) ∈ D(X, Y) and (W′2 , f2′ , g2′ ) ∈ D(Y, Z) be equivalent to (W1 , f1 , g1 ) and (W2 , f2 , g2 ) respectively. Let us choose (W′ , f ′ , g ′ ) ∈ D(W′1 , W′2 ) such that f1 ○ f ′ = g2 ○ g ′ and let us prove that the elements (W′ , f1′ ○ f ′ , g2′ ○ g ′ ) and (W, f1 ○ f , g2 ○ g) of D(X, Z) are equivalent. By the definition of the equivalence relation ∼, we may assume that there exists u1 ∈ S (W′1 , W1 ) and u2 ∈ S (W′2 , W2 ) such that f1 ○ u1 = f1′ and g1 ○ u1 = g1′ on the one side, and that f2 ○ u2 = f2′ and g2 ○ u2 = g2′ on the other side. By lemma 1.7.3, there exists a morphism h ∶ W′ → W such that f ○ h = u1 ○ f ′ and g ○ h = u2 ○ g ′ , so that the following diagram is commutative: g′

←

h

f

→ → Z

f2

→ Y

←

←

f1

→

g2

→

← ←

f 1′

→ W1

g1

→

u1

→

→ W′1

→ W2

g 2′

←

←

W

g

←

f

→

→ ′

u2

←

← ←

←

→ W′2

←

W′

X

Recall that f , f ′ , u1 are isomorphisms (mod N ); by lemma 1.9.6, u1 ○ f ′ is an isomorphism (mod N); the relation f ○ h = u1 ○ f ′ implies that h is an isomorphism (mod N ). Consequently, (W, f1 ○ f , g2 ○ g) is equivalent to (W′ , f1 ○ f ○ h, g2 ○ g ○ h) which is equal to (W′ , f1′ ○ f ′ , g2′ ○ g ′ ). Proposition (1.9.9). — a) There exists a unique category D such that ob(D) = ob(C ), D(X, Y) = D(X, Y)/∼ for every objects X, Y, and such that the composition of the classes of triples (W1 , f1 , g1 ) ∈ D(X, Y) and (W2 , f2 , g2 ) ∈ D(Y, Z) is

1.9. LOCALIZATION

47

the class of a triple (W, f1 ○ f , g2 ○ g) where (W, f , g) is any element of D(W1 , W2 ) such that g2 ○ f = f2 ○ g. b) There exists a unique functor F ∶ C → D such that for every morphism f ∶ X → Y in C , F( f ) is the equivalence class of the triple (X, idX , f ). c) For every morphism f ∈ S (X, Y), F( f ) is an isomorphism in D, and its inverse is the class of the triple (X, f , idX ). Moreover, for every triple φ = (W, f , g) ∈ D(X, Y), one has F([φ]) = F(g) ○ F( f )−1 . d) Let D ′ be a category, let F′ ∶ C → D ′ be a functor such that F′ ( f ) is invertible, for every morphism f ∈ mor(S ). Then there exists a unique functor G ∶ D → D ′ such that F′ = G ○ F.

Proof. — a) The set of objects, the set of morphisms and the composition law are prescribed; it thus remains to prove that the composition law is associative and the existence of neutral elements at each object. Let X, Y, Z, T be objects of C , let u ∈ D(X, Y), v ∈ D(Y, Z) and w ∈ D(Z, T); write [u] for the class of u in D(X, Y), etc. We build objects P, Q, R as depicted by the diagram, all of whose vertical arrows are isomorphisms (mod N ):

→ Z

← ←

→

←

→ T

←

←

← ←

→ Y

→

←

→

→ ⋅

→ ⋅

←

←

→ ⋅

→ P

←

→ Q

←

R

→ X

By construction, [v] ○ [u] is the class of the triple (P, P → X, P → Z), hence [w] ○ ([v] ○ [u]) is the class of the triple (R, R → X, R → T). Similarly, [w] ○ [v] is the class of the triple (Q, Q → Y, Q → T), hence ([w] ○ [v]) ○ [u] is the class of the triple (R, R → X, R → T). The composition law is thus associative.

48

CHAPTER 1. TRIANGULATED CATEGORIES

Let us show that the class εX of the triple (X, idX , idX ) is an identity at X. Let φ = (W, f , g) ∈ D(X, Y). By construction, the diagram ⇐ W ←

⇐

←

→ Y

→

f

→

f

g

←

W X

⇐ X

⇐

⇐

⇐

X shows that the composition [φ]○[εX ] is represented by φ, so that [φ]○[εX ] = [φ]. One proves similarly that [εY ] ○ [φ] = [φ]. b) For f ∈ C (X, Y), the origin of F( f ) is X, and the target of F( f ) is Y. Consequently, there exists at most one such functor, and the map on objects has to be the identity. By construction, F(idX ) = εX . Let f ∈ C (X, Y) and g ∈ C (Y, Z); the diagram f

←

⇐

←

→ Z

f

→

⇐

f

X

g

→ Y

←

X

→ Y

⇐

←

⇐

X and the definition of the composition law show that F(g ○ f ) = F(g) ○ F( f ). Consequently, F is a functor. c) Let f ∈ S (X, Y); let φ be the class of (X, f , idX ). The diagram

←

→ ← X

⇐

⇐

⇐

⇐

X

⇐ X

←

⇐

idY

⇐ X ⇐

f

idY

←

→ → Y

f

←

Y

f

→ → Y

1.9. LOCALIZATION

49

proves that F( f ) ○ [φ] is represented by (X, f , f ) and that (X, f , f ) is equivalent to (Y, idY , idY ). Consequently, F( f ) ○ [φ] = εY . One proves similarly that [φ] ○ F( f ) = εX . Finally, if φ = (W, f , g) ∈ D(X, Y), the diagram

←

→

f

→ Y

⇐

⇐

W

g

⇐

⇐

⇐ W

⇐

⇐

⇐ W

←

W

X shows that F(g) ○ F( f )−1 is represented by (W, f , g), hence [φ] = F(g) ○ F( f )−1 . d) Necessarily, G(X) = F′ (X) for every object X of C . Moreover, for every triple φ = (W, f , g), one has [φ] = F(g) ○ F( f )−1 so that necessarily, G([φ]) = F′ (g)−1 F′ ( f ). It remains to show that this formulae define a functor G such that G ○ F = F′ . Let φ = (W, f , g) and (W′ , f ′ , g ′ ) be equivalent triples; let us show that F′ (g) ○ F′ ( f )−1 = F′ (g ′ ) ○ F′ ( f ′ )−1 . By the definition of the equivalence relation on D(X, Y), we may assume that there exists h ∈ S (W, W′ ) such that f = f ′ ○ h and g = g ′ ○ h. Then F′ (g) = F′ (g ′ )○F′ (h) = F′ (g ′ )○F′ ( f ′ )−1 ○F′ ( f ′ )○F′ (h) = F′ (g ′ )○F′ ( f ′ )−1 ○F′ ( f ), hence F′ (g) ○ F′ ( f )−1 = F′ (g ′ ) ○ F′ ( f ′ )−1 . Consequently, G is well defined. One has F′ = G ○ F by construction. To prove that it is a functor, we need to check that it maps unit elements to unit elements, and that it is compatible with composition. Since εX = F(idX ), one has G(εX ) = F′ (idX ) = idF′ (X) . Let then φ = (W1 , f1 , g1 ) ∈ D(X, Y) and ψ = (W2 , f2 , g2 ) ∈ D(Y, Z); Let (W, f , g) ∈ D(W1 , W2 ) be such that g1 ○ f = f2 ○g. By definition, [ψ] ○ [φ] is the class of the triple (W, f1 ○ f , g2 ○ g). Consequently, G([ψ] ○ [φ]) = G([(W, f1 ○ f , g2 ○ g)]) = F′ (g2 ○ g)F′ ( f1 ○ f )−1 = F′ (g2 ) ○ F′ ( f2 )−1 ○ F′ ( f2 ) ○ F′ (g) ○ F′ ( f )−1 ○ F′ ( f1 )−1 = G([ψ]) ○ F′ ( f2 ○ g) ○ F′ ( f )−1 ○ F′ ( f1 )−1 = G([ψ]) ○ F′ (g1 ) ○ F′ ( f1 )−1 = G([ψ]) ○ G([φ]),

CHAPTER 1. TRIANGULATED CATEGORIES

50

hence G is a functor. This concludes the proof of the proposition. Remark (1.9.10). — The universal property stated in part d) of proposition 1.9.9 is preserved by passing to the opposite categories, so that the category D o is canonically isomorphic to the category obtained by this construction by starting from C o and its thick triangulated subcategory N o . Proposition (1.9.11). — a) Let f , g ∶ X → Y be morphisms in C . The following conditions are equivalent: (i) F( f ) = F(g); (ii) There exists an isomorphism h (mod N ) such that f ○ h = g ○ h; (iii) There exists an isomorphism h (mod N ) such that h ○ f = h ○ g; (iv) The morphism f − g factors through an object of N . b) Let f ∶ X → Y be a morphism in C . The following conditions are equivalent: (i) F( f ) is an isomorphism; (ii) There exists morphisms g ∶ W → X and h ∶ Y → Z such that f ○ g and h ○ f are isomorphisms (mod N ); f

→ Y → Z → ΣX in C , there exists (iii) For every distinguished triangle X Ð ′ ′ an object Z ∈ C such that Z ⊕ Z ∈ N ; f

→ Y → Z → ΣX in C and an (iv) There exists a distinguished triangle X Ð object Z′ ∈ C such that Z ⊕ Z′ ∈ N . Proof. — a) (i)⇒(ii). By hypothesis, the triples (X, idX , f ) and (X, idX , g) are equivalent; there exists a triple (W, f ′ , g ′ ) in D(X, Y) and morphisms u, v ∈ S (W, C) such that u = f ′ = v ′ , f ○ u = g ′ and g ○ v = g ′ . Consequently, f ○ f ′ = g ○ f ′ = g ′ , and f ′ is an isomorphism (mod N ). The implication (i)⇒(iii) follows from the same argument by passing to the opposite category. The implications (ii)⇒(i) and (iii)⇒(i) hold, because F(h) is an isomorphism. Let us prove that (ii)⇒(iv). Let h ∶ W → X be an isomorphism (mod N ) such that f ○ h = g ○ h; by definition of S , we may complete h into a distinguished h

k

triangle W Ð →XÐ → N → ΣW, where N ∈ N . Let us apply the cohomological functor C (⋅, Y) on C o : this gives an exact sequence k∗

h∗

C (N, Y) Ð→ C (X, Y) Ð→ C (W, Y).

1.9. LOCALIZATION

51

By assumption, h∗ ( f −g) = ( f −g)○h = 0; consequently, there exists a morphism j ∶ N → Y such that f − g = j ○ h: the morphism f − g factors through an object of N . Conversely, assume that (iv) holds and let us consider an object N of N and a factorization f − g = v ○ u, where u ∈ C (X, N ) and v ∈ C (N, Y). There u w exists a distinguished triangle X Ð → N → W Ð → ΣX, hence a distinguished Σw

u

→ N → W. Consequently, Σw ∈ S (Σ−1 W, X). Moreover, triangle Σ−1 W Ð→ X Ð ( f − g) ○ Σw = v ○ u ○ Σw = 0. This proves (ii). b) (i)⇒(ii). Let (W, s, g) ∈ D(Y, X) be a triple whose equivalence class is an inverse of F( f ). By definition of the composition, (W, s, f ○ g) is equivalent to (Y, idY , idY ). Consequently, there exists a triple (Z, h, k) and isomorphisms (mod N ), u ∶ Z → W and v ∶ Z → Y, such that s ○ u = h, f ○ g ○ u = k, h = idY ○v = v and k = idY ○v = v. Then, h = k = v and u are isomorphisms (mod N ) and ( f ○ g) ○ u = k, so that f ○ g is an isomorphism (mod N ). The other part of (ii) follows by passing to the opposite category. (ii)⇒(i). These assumptions imply that F( f ) is left-invertible and rightinvertible; consequently, F( f ) is invertible. (ii)⇒(iii). Let h ∶ Y → T be such that h ○ f is an isomorphism (mod N ). By the implication (ii)⇒(i), F( f ) is an isomorphism, as well as F(h ○ f ), so that F(h) is an isomorphism as well. The commutative diagram

← →

→ Z

←

→ T⊕Z

←

( 01 )

→ ΣX

⇐

←

h) (u

←

←

←

→ Z ⇐

T

u

←

→ Y →

→

h○ f

f

←

X

(0 1)

0

Σh○Σ f

→ ΣT,

corresponds to a morphism of distinguished triangles; since its bottom row is a contractible triangle, this morphism is distinguished(1) hence the commutative square

(1)

Pas fait!

←

← ←

T

→ Y →

→

h○ f

f

←

X

h) (u

→ T⊕Z

( 01 )

CHAPTER 1. TRIANGULATED CATEGORIES

52

is homotopically cartesian.(2) Since h ○ f is an isomorphism (mod N ), so is ( uh ). Let then p = ( 01 ) ∶ T → T ⊕ Z

and

q = ( 1 0 ) ∶ T ⊕ Z → T.

One has h○ f h○ f ( uh ) ○ f = ( u○ f ) = ( 0 ) = p ○ (h ○ f ).

Since F( f ), F(( uh )) and F(h ○ f ) are isomorphisms in D, this shows that F(p) is an isomorphism in D. One has q ○ p = idT , so that F(q) is the left-inverse of F(p). Consequently, it is also its right-inverse and the image of p ○ q = ( 01 00 ) by F coincides with that of id. By a), there exists an isomorphism (mod N ), s˜ = ( st ) ∶ W → T ⊕ Z, such that s ( 00 01 ) ○ s˜ = 0. Consequently, t = 0. Let W Ð → T → N → ΣW be a distinguished −1 triangle; its coproduct with the distinguished triangle 0 → Z Ð→ Z → 0 is the distinguished triangle ( 0s )

W ÐÐ→ T ⊕ Z → N ⊕ Z → ΣW, so that N ⊕ Z is an object of N , as was to be shown. The implication (iii)⇒(iv) is obvious since there exists a distinguished triangle f

of the form X Ð → Y → Z → ΣX. f Let us finally prove that (iv)⇒(ii). Consider a distinguished triangle X Ð →Y→ −1 Z → ΣX. Since 0 → Z′ Ð→ Z′ → 0 is a distinguished triangle, the triangle (f) 0

X ÐÐ→ Y ⊕ Z′ → Z ⊕ Z′ → ΣX is distinguished. Since Z ⊕ Z′ is an object of N , by assumption, the morphism ( 0f ) ∶ X → Y ⊕ Z′ is an isomorphism (mod N ). Let g = ( 01 ) ∶ X → Y ⊕ Z′ . One has ( 0f ) = g ○ f , which proves the first part of (ii). The second one is proved analogously. Corollary (1.9.12). — Let X be an object of C . The object F(X) is isomorphic to 0 if and only if there exists an object Y of C such that X ⊕ Y ∈ ob(N ). Proof. — Apply part b) of the proposition to the zero morphism f ∈ C (0, X). Then F(X) ≃ 0 if and only if F( f ) is an isomorphism. Given the distinguished (2)

Ça non plus...

1.9. LOCALIZATION

53

triangle 0 → X → X → 0, the corollary follows from the equivalence (i)⇔(iii). Proposition (1.9.13). — The category D is an additive category and the functor F is an additive functor. Proof. — a) Let us show that 0 is both an initial and a terminal object in D. We thus need to show that for every object X of C , all elements of D(X, 0), resp. of D(0, X), are equivalent one to the other. Let (V, f , g), (V′ , f ′ , g ′ ) ∈ D(X, 0); since 0 is a terminal object in C , one has g = g ′ = 0. Let u W → V ←

←

←

←

V′

→

→

u′

f′

f

→ X

be an homotopically cartesian square (lemma 1.7.3). Since f and f ′ are isomorphisms (mod N ), so are u and u ′ . The triple (W, f ○ u, 0) and the morphisms u, u ′ imply that (V, f , g) ∼ (V′ , f ′ , g ′ ), as was to be shown. Consequently, 0 is a terminal object in D. By considering the opposite category, 0 is an initial object in D. b) Let X, Y be objects of C , and let us show that X⊕Y is a product of X and Y in the category D. Let i ∶ X → X⊕Y, j ∶ Y → X⊕Y, p ∶ X⊕Y → X and q ∶ X⊕Y → Y be the canonical morphisms. Given an object P and two morphisms φ ∶ P → X and ψ ∶ P → Y in D, we need to show that there exists a unique morphism θ ∶ P → X ⊕ Y such that F(p) ○ θ = φ and F(q) ○ θ = ψ. The morphisms φ, ψ are represented by triples (W, f , g) ∈ D(P, X) and (W′ , f ′ , h) ∈ D(P, Y). Considering a homotopically cartesian square → W′ ←

←

←

W′′

f

→

→ ←

W

f′

→ P

we reduce to the case where W = W′ and f = f ′ . Let k = ( hg ) and let θ ∈ D(P, X ⊕ Y) be the class of the triple (W, f , k). One has F(p) ○ θ = φ and F(q) ○ θ = ψ. Let θ ′ ∈ D(P, X ⊕ Y) be a morphism such that F(p) ○ θ ′ = φ and F(q) ○ θ ′ = ψ. Up to replacing W, we may assume that θ ′ is represented by a ′ triple of the form (W, f , k ′ ), where k ′ = ( hg ′ ). Then (W, f , g) and (W, f , g ′ )

54

CHAPTER 1. TRIANGULATED CATEGORIES

are equivalent, so that, in particular, F(g) = F(g ′ ); by proposition ??, there exists an isomorphism (mod N ), u ∶ U → W, such that g ○ u = g ′ ○ u. Similarly, there exists an isomorphism (mod N ), v ∶ V → W, such that g ○ v = g ′ ○ v. Considering a homotopy pull-back of u and v, we may assume that U = V and u = v. Then k ○ u = k ′ ○ u, hence F(k) = F(k ′ ). Similarly, one proves that X ⊕ Y is a product. c) By the two preceding paragraphs, the category D is semi-additive, and the functor F is additive. It remains to prove that every morphism φ ∈ D(X, Y) has an opposite. Let (W, f , g) be a triple representing φ and let φ′ = [(W, f , −g)]. One checks that φ + φ′ = [(W, f , 0)] = 0. This concludes the proof that the category D is additive.

1.9.14. — If f is an isomorphism (mod N ), then so are Σ f and Σ−1 f ; consequently, there exists unique endofunctors of D, stil denoted by Σ and Σ−1 , such that Σ ○ F = Σ and Σ−1 ○ F = Σ−1 . One has Σ ○ Σ−1 = F = idD ○F, so that Σ ○ Σ−1 = idD ; similarly, Σ−1 ○ Σ = idD . In particular, Σ is an automorphism of the category D. Say that a triangle in D is distinguished if it is isomorphic to the image under F of a distinguished triangle of C .

Theorem (1.9.15). — The category D is a triangulated category and the functor F ∶ C → D is a triangulated functor.

Proof. — 1) By construction, a triangle which is isomorphic to a distinguished triangle is distinguished. 2) It follows from the definition and the analogous property for the triangulated category C that if T is a triangle in D, then the shift of T is distinguished if and only if T is distinguished. 1 3) For every object X, the triangle X Ð → X → 0 → ΣX in D is the image under F of the ‘‘same’’ triangle in C , hence is distinguished.

1.9. LOCALIZATION

55

4) Let φ ∶ X → Y be a morphism in D; let it be represented by a triple (W, s, f ), f

g

h

where s is an isomorphism (mod N ). Let W Ð → ΣW be a distin→YÐ →ZÐ guished triangle in C . The diagram in D F(g)

←

←

⇐

←

⇐

→ Z

←

F(g)

Σs

→ ΣX

←

→ Y

→

φ

→ ΣW

⇐

⇐ ←

X

F(h)

→ Z

←

→ Y

→

F(s)

F( f )

←

W

ΣF(s)○F(h)

shows that the bottom triangle is distinguished. 5) Let us consider a partial diagram of distinguished triangles in D: → Z

w

←

→ ΣX

←

←

←

←

v

→

→ → Y′

←

← u

′

v

′

→ Z′

←

X′

→ Y g

→

f

u

←

X

w′

Σf

→ ΣX′

and let us show that there exists a morphism h ∶ Z → Z′ in D that gives rise to a distinguished morphism of distinguished triangles. We may assume that both horizontal triangles are images by F of distinguished triangles in C . Let us show that the leftmost commutative square is isomorphic to the image of a commutative square in C . Let (U, s1 , u1 ) ∈ D(X, Y) and (V, t1 , g1 ) ∈ D(Y, Y′ ) be representatives of u and g. Let (W, t2 , u2 ) ∈ D(U, V) be a triple built from a homotopy pull-back of u1 and t1 . Similarly, let (U′ , t1′ , f1 ) ∈ D(X, X′ ) and (V′ , s1′ , u1′ ) ∈ D(X′ , Y′ ) be representatives of f and g ′ . Let (W′ , s2′ , f2 ) be a triple built from a homotopy pull-back of f1 and s1′ . Let finally (P, s3 , t3 ) ∈ D(W, W′ ) be a triple built from a homotopy pull-back of t1′ ○ s1 and s1 ○ t2 . These constructions are summarized by the diagrams:

X

→ W

W′

←

←

Moreover, t2 , s1′ , s3 and t3 are isomorphisms (mod N ).

s3

t3

←

←

←

← ←

←

←

←

→ X′

←

f1

P

s ′1

t 1′ ○s ′1

→

←

→ Y′

←

u ′1

→

U′ t 1′

→

V′

←

←

s ′2

f2

→

X

→ Y

W′

→

→

s1

t1

→

Y′

→

U

u1

→

→

t2

→ V

g1

←

W

u2

s 1 ○t 2

→ X

CHAPTER 1. TRIANGULATED CATEGORIES

56

Let us complete u2 ○ t3 ∶ P → V into a distinguished triangle P → V → Q → ΣP in C ; let us complete u1′ ∶ V′ → Y′ into a distinguished triangle V′ → Y′ → Q′ → ΣV′ in C . This furnishes the diagram of distinguished triangles in C :

←

←

←

←

←

← ←

⇐

← v′

→ Z′

→

→

→ Y′

→

→

→

→

→

⇐ ←

→ u′

Σf′

→ ΣV′

r′

←

←

←

←

→

→ Q′

←

←

→

h′

→ Y′

←

X′

→ ΣP →

g′

→ ΣX Σs

←

→ Q

V′ s′

w

←

r

→ V

f′

→ Z ←

P

t

v

←

←

s

→ Y

←

u

←

X

w′

Σs′

→ ΣX′

where s, t, s ′ are morphisms in S such that f = F(s′ ) ○ F( f ′ ) ○ F(s)−1 and g = F(g ′ ) ○ F(t)−1 . Let us choose a morphism h ′ ∶ Q → Q′ that gives rise to a distinguished morphism of distinguished triangles in C . By lemma ??, there exist morphisms r ∶ Q → Z and r ′ ∶ Q′ → Z′ in S that give rise to isomorphisms of triangles in C . The morphism h = F(r ′ )○F(h ′ )○F(r)−1 ∶ Z → Z′ in D induces a distinguished isomorphism of distinguished triangles. This shows that the category D satisfies the five axioms of a triangulated category. By construction, the functor F is triangulated. Lemma (1.9.16). — Any diagram of distinguished triangles in C

v′

h

→ Z′

→ ΣX ←

←

w

→

←

→ Y′

→

u′

→ Z

←

←

←

←

←

v

→ ←

X′

→ Y g

→

f

u

←

X

w′

Σf

→ ΣX′

where f , g are isomorphisms (mod N ) can be extended to a morphism of triangles, where h is an isomorphism (mod N ). Proof. — Let us complete the morphism u ′ ○ f ∶ X → Y′ to a distinguished u′ ○ f

v ′′

w ′′

triangle X ÐÐ→ Y′ Ð→ Z′′ Ð→ ΣX. We then decompose the given diagram as the

1.9. LOCALIZATION

57

composition of two diagrams of distinguished triangles:

→ Z′′

←

←

→ ΣX

→ Y′

v′

⇐

h′

→ Z′

→

→

⇐

←

u′

w ′′

←

←

⇐

v ′′

h ′′

w′

←

←

→ Y′

→

X′

→ ΣX

←

→

→

⇐

u′ ○ f

w

←

←

←

→ Z

g

X f

v

⇐

⇐

→ Y

←

u

←

X

Σf

→ ΣX′

←

Starting with the first two triangles, let us now apply the octahedral axiom: v

v

⇐

→ ΣX

←

←

←

←

→ Z′′ →

N

⇐ N ←

←

⇐

→ Y

Σv

→

→ ←

Σu

h ′′

→ ←

ΣX

→ ΣX

←

⇐

→ Y

w

⇐

X

→

u′ ○ f

→

⇐

g

→ Z

←

←

←

→ Z

←

→ Y

←

u

←

X

Σw

→ Σ2 X

Since g is an isomorphism (mod N ), the object N belongs to N ; consequently, h′′ is an isomorphism (mod N ) as well. We then apply the octahedral axiom in the opposite category to the last two triangles, after having shifted them to the left: −Σ−1 w ′

→

→

N′

⇐ N′ ←

←

→ ΣY ⇐

←

⇐

Σf

h′

→ Z′

→ ΣX

−Σu ′ ○Σ f

−w ′

→ ΣX′

−Σu ′

←

←

←

−w ′′

⇐

−v ′

→ Y′

→

→

⇐

←

Y′

→

Z′′

←

−v ′′

←

←

←

⇐

→

→

Y′

−u ′

←

→ X′

←

→ Σ−1 Z′

←

−Σ−1 v ′

←

Σ−1 Y′

→ ΣY′

Again, since the morphism Σ f is an isomorphism (mod N ), the object N′ belongs to N , so that h′ is an isomorphism (mod N ). Finally, we may let h = h′ ○ h′′ ; it is an isomorphism (mod N ).

CHAPTER 1. TRIANGULATED CATEGORIES

58

Proof of theorem 1.9.4. — We have constructed a triangulated category D and a triangulated functor F ∶ C → D such that N ⊂ Ker(F). Let now F′ ∶ C → D ′ be any triangulated functor to a triangulated category D ′ be such that N ⊂ Ker(F′ ). f

Let f ∶ X → Y be an isomorphism (mod N ) and let X Ð → Y → Z → ΣX be a distinguished triangle in C ; by definition, Z is an object of N . Then F′ ( f )

F′ (X) ÐÐ→ F′ (Y) → 0 → ΣF′ (X) is a distinguished triangle, because F′ is a triangulated functor and N ⊂ Ker(F′ ). Consequently, F′ ( f ) is an isomorphism. By proposition 1.9.9, there exists a unique functor G ∶ D → D ′ such that F′ = G ○ F. The functor G is additive: let indeed φ, ψ ∶ X → Y be morphisms in D; there exists an object W, an isomorphism (mod N ), s ∶ W → X, and morphisms f , g ∶ W → Y in C such that φ = F( f ) ○ F(s)−1 and ψ = F(g) ○ F(s)−1 . Then, φ + ψ = F( f + g) ○ F(s)−1 , hence G(φ + ψ) = G(F( f + g) ○ F(s)−1 ) = G(F( f + g)) ○ G(F(s))−1 = F′ ( f + g) ○ F′ (s)−1 = F′ ( f ) ○ F′ (s)−1 + F′ (g) ○ F′ (s)−1 = G(φ) + G(ψ), as was to be shown. The functor G is triangulated. Indeed, if T is a distinguished triangle in D, it is isomorphic to the image by F of a distinguished triangle T1 in C . Then G(T) = G(F(T1 )) = F′ (T1 ) is a distinguished triangle, because F′ is a triangulated functor. Finally, it follows from corollary 1.9.12 that Ker(F) is the smallest thick triangulated subcategory of C containing N . This concludes the proof of theorem 1.9.4. 1.10. Derived categories and derived functors 1.10.1. — Let A be an abelian category and let K (A) be its homotopy category. By lemma 1.4.9, the cohomology functors Hn ∶ K(A) → A are cohomological functors, for all n ∈ Z. One has Hn = H0 ○ Σ n . Lemma (1.10.2). — Let N be the full subcategory of K (A) whose objects are the acyclic complexes. a) The subcategory N is a thick triangulated subcategory of K (A).

1.10. DERIVED CATEGORIES AND DERIVED FUNCTORS

59

b) Let f ∶ X → Y be a morphism of complexes in A. Then f is an isomorphism (mod N ) if and only if its cone C f is acyclic, if and only if f is a homologism. Proof. — a) First of all, N is an additive subcategory of K (A), invariant under the translation automorphism. Let then X, Y be acyclic complexes and u ∶ X → Y be a morphism in K (A); let us extend it to a distinguished triangle u v w →YÐ →ZÐ → ΣX in K (A). We need to show that Z is acyclic as well. By XÐ definition of the distinguished triangles of K (A), we may assume that Z is the cone of a morphism of complexes representing u. Since X and Y are acyclic, u is a homologism. By lemma 1.4.9, Z is acyclic. b) This follows from lemma 1.4.9. Definition (1.10.3). — Let A be an abelian category. The derived category D(A) is the triangulated category quotient of the homotopy category K (A) by its thick triangulated subcategory of acyclic complexes. There are canonical functors k

h

A → C (A) Ð → K (A) Ð → D(A). The functor A → C (A) considers an object X of A as the unique complex such that Xn = 0 for n ≠ 0 and X0 = X,a all differentials being 0. It is fully faithful. Lemma (1.10.4). — Let B be an abelian category. A cohomological functor H ∶ K (A) → B factors through h if and only if it maps homologisms to isomorphisms. The functor C (A) → D(A) is a ∂-functor. 1.10.5. — Let A be an abelian category and let C be a triangulated category. A ∂-functor F ∶ A → C is an additive functor endowed, for every exact sequence S = (0 → X → Y → Z → 0) in A, with a morphism ∂(S) ∶ F(Z) → ΣF(X), satisfying the following properties: u

v

a) For every exact sequence S = (0 → X Ð →YÐ → Z → 0), the diagram F(u)

F(v)

∂(S)

F(X) ÐÐ→ F(Y) ÐÐ→ F(Z) ÐÐ→ ΣF(X) is a distinguished triangle;

CHAPTER 1. TRIANGULATED CATEGORIES

60

b) For every morphism of exact sequences

→ 0,

←

←

←

←

→ X′

→ Y

→ Z

←

←

←

←

→ 0

h

→

←

0

g

→

S′

→ Z

→

→

f

→ Y

←

→ X

←

0

←

S

the diagram

∂(S)

←

F(Z′ )

←

←

→ ΣF(X) →

→

F(h)

∂(S)

←

F(Z)

ΣF( f )

→ ΣF(X′ )

is commutative. 1.10.6. — Let X be a complex in an abelian category A. The truncations of X are the following complexes: τ⩽n (X) = ⋅ ⋅ ⋅ → Xn−1 → Ker(dXn ) → 0 → 0 → . . . ′ τ⩽n (X) = ⋅ ⋅ ⋅ → Xn−1 → Xn → Im(dXn ) → 0 → . . .

τ⩾n (X) = ⋅ ⋅ ⋅ → 0 → 0 → Coker(dXn−1 ) → Xn+1 → . . . ′ τ⩾n (X) = ⋅ ⋅ ⋅ → 0 → Im(dXn−1 ) → Xn → Xn+1 → . . .

There are obvious morphisms ′ ′ τ⩾n (X) → τ⩾n (X) → X → τ⩽n (X) → τ⩽n (X). ′ (X) → τ (X) and τ (X) → τ ′ (X) are homologisms. The The morphisms τ⩾n ⩾n ⩽n ⩽n morphism τ⩾n (X) → X induces an isomorphism Hi (τ⩾n (X)) → Hi (X) for every integer i such that i ⩾ n, and Hi (τ⩾n (X)) = 0 for i < n. The morphism X → τ⩽n (X)) induces an isomorphism Hi (X) → Hi (τ⩽n (X)) for every integer i such that i ⩽ n, and Hi (τ⩽n (X)) = 0 for i > n. Moreover, the diagrams ′ 0 → τ⩾n (X) → X → τ⩽n (X) → 0 ′ 0 → τ⩾n (X) → X → τ⩽n (X) → 0

are exact sequence of complexes. Every morphism of complexes f ∶ X → Y induces in an obvious way a morphism of complexes τ⩽n ( f ) ∶ τ⩽n (X) → τ⩽n (Y), and τ⩽n defines a functor from the category C (A) to itself. If f is null homotopic (resp. a homologism), then

1.10. DERIVED CATEGORIES AND DERIVED FUNCTORS

61

so is τ⩽n ( f ). Consequently, the functor τ⩽n extends to an endofunctor of the homotopy category K (A) (resp. of the derived category D(A)). Similar properties hold for the other truncations. Remark (1.10.7). — Let X ∈ K ⩽0 (A) and Y ∈ K ⩾0 (A). The cohomological functor H0 induces an isomorphism D(A)(X, Y) → A(H0 (X), H0 (Y)). In particular, the canonical functor A → D(A) is fully faithful. Definition (1.10.8). — Let X be a complex in an abelian category A. One says that X is homotopically injective if K (A)(N, X) = 0 for every acyclic complex N. By definition, this notion only depends on the isomorphism class of X in the homotopy category K (A). Lemma (1.10.9). — Homotopically injective complexes form a thick triangulated subcategory of K (A). Proof. — Let X → Y → Z → ΣX be an exact triangle. Assume that X and Z are homotopically injective. Let N be an acyclic complex. Since the functor K (A)(N, ⋅) is a cohomological functor, it induces an exact sequence K (A)(N, X) → K (A)(N, Y) → K (A)(N, Z) of abelian groups. Consequently, K (A)(N, Y) = 0 and Y is homotopically injective. This implies that the full subcategory of homotopically injective objects is stable under forming distinguished triangles. In particular, it is stable under forming finite coproducts, hence is an additive subcategory. It is therefore a triangulated subcategory. Finally, let X and Y be complexes such that X ⊕ Y is homotopically injective. For every acyclic complex N, K (A)(N, X) ⊕ K (A)(N, Y) ≃ K (A)(N, X ⊕ Y) = 0, hence K (A)(N, X) = 0. Proposition (1.10.10). — Let Y be a complex in an abelian category A. The following assertions are equivalent: (i) The complex Y is homotopically injective;

62

CHAPTER 1. TRIANGULATED CATEGORIES

s

←

W

f

→

→ ←

X

←

(ii) For every diagram

g

→ Y

of morphisms of complexes, where s is a homologism, there exists a unique morphism g ∶ X → Y in K (A) such that g ○ s is homotopic to f ; (iii) For every complex X, the functor h induces an isomorphism K (A)(X, Y) → D(A)(X, Y); (iv) For every complex Z and every homologism f ∶ Y → Z, there exists a morphism g ∶ Z → Y such that g ○ f is homotopic to idY .

s

←

W

f

→

→ ←

X

←

Proof. — (i)⇒(ii). Let

g

→ Y s

be a diagram of morphisms of complexes, where s is a homologism. Let W Ð → X → Z → ΣW be a distinguished triangle in K (A); since s is a homologism, the complex Z is acyclic. Applying the (contravariant) cohomological functor K (A)(⋅, Y), we obtain an exact sequence s

K (A)(Z, Y) → K (A)(X, Y) Ð → K (A)(W, Y) → K (A)(Σ−1 Z, Y). Since Y is homotopically injective and Z is acyclic, it follows that K (A)(Z, Y) = K (A)(Σ−1 Z, Y) = 0. ∼

Consequently, the morphism s induces an isomorphism K (A)(X, Y) Ð → K (A)(W, Y). In particular, there exists a unique morphism g ∶ X → Y in K (A) such that g ○ s = f . (ii)⇒(iii). By definition of morphisms in D(A), assertion (ii) implies that the canonical morphism K (A)(X, Y) → D(A)(X, Y) is surjective. On the other hand, let g ∈ K (A)(X, Y) be such that h(g) = 0. By proposition 1.9.11, there exists a homologism s ∶ W → X such that g ○ s = 0 in K (A). Assertion (i) applied with f = 0 then implies that g = 0. (iii)⇒(iv). Let f ∶ Y → Z be a homologism. Since it induces an isomorphism in D(A), assertion (iii) implies that there exists a morphism g ∈ K (A)(Z, Y)

1.10. DERIVED CATEGORIES AND DERIVED FUNCTORS

63

such that h(g) = h( f )−1 . One has h(g ○ f ) = idY . By (iii) again, g ○ f = idY in K (A)(Y, Y), as was to be shown. (iv)⇒(i). Let X be an acyclic complex and let f ∶ X → Y be a morphism of f

g

complexes. Let X Ð →YÐ → Z → ΣX be a distinguished triangle. The triangle g

Σf

YÐ → Z → ΣX Ð→ ΣY is distinguished as well, so that g is a homologism. By (iv), there exists a morphism h ∶ Z → Y such that h ○ g = idY in K (A). Consequently, f = idY ○ f = h ○ g ○ f = 0 in K (A). This proves that K (A)(X, Y) = 0, as claimed. Theorem (1.10.11). — If A is a Grothendieck abelian category(3) , then every complex is homologous to a homotopically injective complex. This theorem is due to Spaltenstein (1988) in the particular case where A is the category of abelian sheaves on a topological space. His methods have then be extended to reach the result in this form by Alonso Tarrío et al (2000) and Serpé (2003). More precisely, these authors show that the functor h ∶ K (A) → D(A) admits a right adjoint. Corollary (1.10.12). — Let A be an abelian category and let I be the thick triangulated category of K (A) consisting of all homotopically injective complexes. The triangulated functor I → D(A) is fully faithful; if A is a Grothendieck abelian category, it is an equivalence of categories. Corollary (1.10.13). — Let A be a Grothendieck abelian category. If U is an universe such that A is a locally U-small, then D(A) is locally U-small. Example (1.10.14). — Let U be a universe, let k be a ring whose underlying set belongs to U and let Mod (k)U be the category of k-modules whose underlying set belongs to U. For every objects X, Y of Mod (k)U , one has Hom(X, Y) ∈ U, so that the category Mod (k)U is locally U-small. Consequently, the derived category D(Mod (k)U ) is locally U-small. This means that the abelian category A satisfies the axiom AB∗5 (existence and are exactness of colimits) and admits a generator (an object P such that that A(P, X) = 0 if and only if P = 0). The category of k-modules, the category of O-modules over a topological space are Grothendieck categories. (3)

64

CHAPTER 1. TRIANGULATED CATEGORIES

1.10.15. — Let A, B be abelian categories and let F ∶ A → B be an additive functor. The functor F induces an additive functor on complexes, F ∶ C (A) → C (B), and F ∶ K (A) → K (B), because it preserves homotopies. However, if X is an acyclic complex in A, the complex F(X) may not be acyclic (unless F is exact) so that this functor F does not pass to the derived categories. The theory of derived functors aims at associating with F a functor RF between the derived categories which reflects the properties of F. We assume that A is a Grothendieck category and let I be the thick triangulated subcategory of K (A) of homotopically injective complexes. Then the functor I → D(A) is an equivalence of triangulated categories. The derived functor RF is defined as the composition of a (chosen) quasi-inverse of this equivalence, the functor F, and the canonical functor to D(B). This is depicted by the diagram → K (B)

← ←

←

F

h

h

RF

←

D(A)

→

→

→

∼

←

→ K (A)

←

I

→ D(B)

in which, we insist, the square is usually not commutative! Explicitly, given a complex X ∈ K (A), the ‘‘recipe’’ to compute RF(X) consists in taking the chosen homotopically injective resolution of X, namely a homologism ε ∶ X → I to a homotopically injective complex I, and defining RF(X) = F(I). Let f ∶ X → X′ be a morphism in K (A), let ε′ ∶ X′ → I′ be the chosen homotopically injective resolution of X′ . Since I′ is homotopically injective, there exists a unique morphism f ′ ∶ I → I′ in K (A) such that f ′ ○ ε = f ○ ε′ (proposition 1.10.10); let us define RF( f ) = F( f ′ ) ∶ F(I) → F(I′ ). These maps define a functor K (A) → D(B). If f ∶ X → X′ is a homologism, then the morphism f ′ ∶ I → I′ constructed above is an isomorphism in K (A) (proposition 1.10.10), so that RF( f ) is an isomorphism. Consequently, the functor we have just defined extends to a functor RF ∶ D(A) → D(B), called the right derived functor of F. For every object X of K (A), the morphism F(ε) ∶ F(X) → F(I) = RF(X) in K (B) satisfies an universal property: for every object Y of K (B), the canonical

1.10. DERIVED CATEGORIES AND DERIVED FUNCTORS

65

morphism F(ε) ∶ h(F(X)) → RF(X) induces an isomorphism ∼

colim K (A)(Y, F(X′ )) Ð → D(B)(h(Y), RF(X). s ′ XÐ →X (In fact, the system (F(X′ ))X→sX′ is ‘‘eventually constant’’.) Proposition (1.10.16). — The functor RF ∶ D(A) → D(B) is triangulated. Proof. — First of all, if ε ∶ X → I is a homotopically injective resolution of X, then Σε ∶ ΣX → ΣI is a homotopically injective resolution of ΣX. This furnishes an isomorphism RF(ΣX) ≃ ΣRF(X). Let indeed X → Y → Z → ΣX be a distinguished triangle in D(A); let us show that the triangle RF(X) → RF(Y) → RF(Z) → ΣRF(X) in D(B) is distinguished. We may assume that X → Y → Z → ΣX is the image by h of a distinguished triangle in K (A), which we denote in the same way. Consider homotopically injective resolutions ε ∶ X → I and η ∶ Y → J. By proposition 1.10.10, these data can be completed to a morphism of disinguished triangles ←

←

←

←

←

→

I

→ J

→ K

→ ΣX →

→

ζ

←

←

η

→ Z

→

ε

→ Y

←

X

Σε

→ ΣI

←

←

in K (A). Observe that the complex K is homotopically injective. Apply the functor h to this diagram. One obtains a morphism of distinguished triangles in D(A), two morphisms out of three are isomorphisms; consequently, the morphism ζ ∶ Z → K is a homologism. Since F is a triangulated functor, the triangle F(I) → F(J) → F(K) → ΣF(I) is distinguished in K (B). it is thus distinguished in D(B) as well. This shows that the functor RF is triangulated. Proposition (1.10.17). — Let Y be a complex in an abelian category. a) Assume that Yn = 0 for n ≠ 0. Then Y is homotopically injective if and only if Y0 is an injective object of A. b) Assume that Yn = 0 for n < 0 and that Yn is an injective object of A for n ⩾ 0. Then Y is homotopically injective. Proof. — a) Let us assume that Y is homotopically injective and let us show that Y0 is injective. Let j ∶ X′ → X be a monomorphism in A, let f ′ ∶ X′ → Y0 be a morphism; we need to show that there exists f ∶ X → Y0 such that f ○ j = f ′ .

CHAPTER 1. TRIANGULATED CATEGORIES

66

j

k

→ X′′ → 0) Let k ∶ X → X′′ be a cokernel of j, so that N = (0 → X′ Ð → X Ð is an acyclic complex in A (we put the term X′ in degree 0). Morphisms of complexes u ∶ N → Y correspond to morphisms u 0 ∶ X′ → Y0 ; a morphism is null homotopic if and only if extends to X. Since K (A)(N, Y) = 0, the morphism f ′ extends to X, as claimed. The converse assertion follows from b). b) Let us now assume that Yn = 0 for n < 0 and that Yn is injective for n ⩾ 0, and let us prove that Y is homotopically injective. Let f ∶ N → X be a morphism of complexes. Let N be an acyclic complex. We will construct morphisms θ n ∶ Nn+1 → Yn such that f n = θ n ○ dXn + dYn−1 ○ θ n−1 , by induction on n. If n < 0, it suffices to define θ n = 0. We then assume that θ m is constructed for m < n and proceed to defining θ n . Consider the diagram

θ n−1

→ Yn

← →

d n−1

→ Nn+1 ← f n+1 ←

←

dn

θn dn

→

→ Yn−1

→ Nn ←f n

← →

←

←

←

←

d n−1

→

← →

Yn−2

θ n−2

→

→

f n−2

→ Nn−1 ← f n−1

←

Nn−2

→ Yn+1

One has f n ○ dNn−1 = dYn−1 ○ f n−1 = dYn−1 (θ n−1 ○ dNn−1 + dYn−2 ○ θ n−2 ) = dYn−1 ○ θ n−1 ○ dNn−1 , hence f n − dYn−1 θ n−1 ∶ Nn → Yn factors through Nn / Im(dNn−1 ). Since N is acyclic, Im(dNn−1 ) = Ker(dNn ). The morphism dNn induces a monomorphism Nn / Ker(dNn ) → Nn+1 . Since Yn is injective, there exists a morphism θ n ∶ Nn+1 → Yn such that f n − dYn−1 θ n−1 = θ n ○ dNn . This provides the required morphism. This shows that the morphism f is null homotopic and concludes the proof of the proposition. Example (1.10.18). — Let A be an abelian category; assume that every object of A admits a monomorphism to an injective object (for example, a Grothendieck abelian category). Let us show that every complex X ∈ C ⩾0 (A) is homologous to an object I ∈ C ⩾0 (A) whose terms are injective. By proposition 1.10.17, this is a partial result in the direction of theorem 1.10.11. In particular, if I is the additive full subcategory of injective objects of A, it implies that the canonical functor K + (I ) → D + (A) is an equivalence of categories. (To be done.)

1.11. EXERCISES

67

Remark (1.10.19). — Assume that F is left exact. For every integer n, let R n F ∶ A → B be the composition of the inclusion functor A → D(A), of RF, and of Hn . Then for every object X ∈ A, the canonical morphism F(X) → RF(X) induces an isomorphism F(X) ≃ R0 F(X). Moreover, every exact sequence 0 → X → Y → Z → 0 in A gives rise to a long exact sequence 0 → F(X) → F(Y) → F(Z) → R1 F(X) → . . . 1.11. Exercises Exercise (1.11.1). — In the category of rings, let f ∶ Z → Q be the inclusion morphism. Show that f is both a monomorphism and an epimorphism but is not an isomorphism. u

v

w

Exercise (1.11.2). — Let X Ð →YÐ →ZÐ → ΣX be a distinguished triangle in a (pre)triangulated category. Establish the equivalence of the following conditions: (i) u is an isomorphism; (ii) v = 0; (iii) w = 0. If they hold, prove that Z = 0. Exercise (1.11.3). — Let C be an abelian category and let X be a complex in C . One says that X is contractible if idX is null homotopic. a) Prove that X is contractible if and only if it is acyclic and if the exact sequence 0 → Ker(dXn ) → Xn → Im(dXn ) → 0 is split, for every integer n ∈ Z. b) Let k be a ring and let C be the category of k-modules. Assume that X is an acyclic complex such that Xn is a free k-module, for every n ∈ Z. Prove that X is contractible if, moreover, k is a field or a principal ideal domain. c) Prove that X is contractible if, moreover one has Xn = 0 for every n < 0. d) Let k = Z/4Z and Xn = Z/4Z for every n ∈ Z, let dXn be given by the multiplication by 2. Prove that X is acyclic but not contractible.

CHAPTER 2 COHOMOLOGY OF SHEAVES

2.1. General topology Definition (2.1.1). — A continuous map f ∶ X → Y is separated if for every pair (x, x ′ ) of points of X such that f (x) = f (x ′ ) and x ≠ x ′ , there exist disjoint open subsets U and U′ of X such that x ∈ U and x ′ ∈ U′ . If X is separated, then every continuous map with origin X is separated. Definition (2.1.2). — A continuous map f ∶ X → Y is closed if f (A) is a closed subset of Y, for every closed subset A of X. It is proper if it is universally closed, that is, if the map f × idZ ∶ X × Z → Y × Z is closed, for every topological space Z. Proposition (2.1.3). — Let f ∶ X → Y be a continuous map; let us assume that X is separated. a) The map f is proper if and only if f −1 (y) is compact, for every y ∈ Y. b) If X and Y are locally compact (in particular, separated), the map f is proper if and only if f −1 (A) is a compact subset of X, for every compact subset A of Y. Proposition (2.1.4). — Let f ∶ X → Y be a continuous map. a) Assume that f is proper (resp. separated). Then for every subspace A of Y, the map fA ∶ f −1 (A) → A induced by f by restriction is proper (resp. separated). b) Assume that f is proper (resp. separated). Then for every closed subspace Z of X, the map f ∣Z ∶ Z → Y is proper (resp. separated). c) Assume that there exists an open covering V of V such that for every V ∈ V , the map fV ∶ f −1 (V) → V is proper (resp. separated). Then f is proper (resp. separated).

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Definition (2.1.5). — A topological space X is paracompact if for every open covering U of X, there exists an open covering V of X satisfying the following properties: a) For every V ∈ V , there exists U ∈ U such that V ⊂ U (the covering V refines the covering U ); b) Every point of X has an open neighborhood A such that the set of V ∈ V such that A ∩ V ≠ ∅ is finite. A compact topological space is paracompact; a metrizable topological space is paracompact; every subspace of a cellular space is paracompact. Lemma (2.1.6). — Let X be a locally compact topological space. For every compact subset A of X and every open neighborhood U of A, there exists an open neighborhood V of A such that V ⊂ U. Proof. — For every a ∈ A, let Va be a compact neighborhood of a which is ˚ a form an open covering of A; since contained in U. When a ∈ A, the interiors V ˚ a . The latter A is compact, there exists a finite subset S of A such that A ⊂ ⋃a∈S V set is an open neighborhood V of A; since S is finite, V ⊂ ⋃a∈S Va ⊂ U.

2.2. Abelian categories of abelian sheaves 2.2.1. — Let X be a topological space. The set Op(X) of open subsets of X is ordered by inclusion; we consider the associated category Op(X). Let C be a category (for instance, the category of sets, or the category of abelian groups). A C -presheaf on X is a contravariant functor F from Op(X) to the category C . A morphism of presheaves is a morphism of functors. It the category C admits limits (resp. colimits), then the category of C presheaves on X admits limits (resp. colimits), which are computed pointwise. 2.2.2. — Let U be an open subset of X and let U be an open covering of U. To these data, we attach a quiver whose vertices are the pairs {V, V′ } of elements of U , this vertex being the target of two arrows of respective origins {V} and {V′ }. Every C -presheaf F defines a diagram, with value F (V ∩ V′ ) at the vertex {V, V′ }, and with morphisms induced by the inclusions V ∩ V′ ⊂ V (resp. V ∩ V′ ⊂ V′ ), for V, V′ ∈ U .

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One says that a C -presheaf is a sheaf if for every open subset U of X, and every open covering U of U, the cone (F (U) → F (V ∩ V′ ))V,V′ ∈U is a limit. 2.2.3. — Assume that the category C admits limits. Then the natural functor from the category of sheaves on X to the category of presheaves admit a left adjoint F → F † , inducing, for every presheaf F and every sheaf G , a bijection Hom(F † , G ) Ð → Hom(F , G )). ∼

Let us consider a diagram of sheaves. Viewed as a diagram of presheaves, its limit is a sheaf, and is its limit in the category of sheaves. However, the presheaf-colimit of this diagram is usually not a sheaf; the associated sheaf furnishes a colimit in the category of sheaves. 2.2.4. — We denote by Ab(X) the category of sheaves of abelian groups on X. If OX is sheaf of rings on X, we denote by Mod (OX ) the category of OX -modules on X. Let k be an abelian group (resp. a ring); we write kX for the constant sheaf on X associated with k. One has Mod (ZX ) = Ab(X). These categories admit limits and colimits. Limits (for example products or equalizers), are computed as presheaves. Colimits, for example coproducts or coequalizers, require to consider the sheaf associated with the colimit-presheaf. These are abelian categories. 2.2.5. — Assume that OX is a sheaf of commutative rings. Then the category of OX -modules admits an internal Hom bifunctor Hom(⋅, ⋅), and a tensor product. These are additive functors. 2.2.6. — Let x ∈ X. The fiber of a an abelian sheaf F at a point x is the colimit Fx = colim F (U). U∋x

The functor Ab(X) → Ab given by F ↦ Fx is exact. 2.2.7. — Let f ∶ X → Y be a continuous map. For every open subset V of Y, f −1 (V) is an open subset of X. Consequently, f induces a functor Op(Y) → Op(X). By composition, every presheaf F on X induces a presheaf f∗ F on Y. If F is a sheaf, then f∗ F is a sheaf as well. The mapping F ↦ f∗ F gives rise to a functor f∗ ∶ Ab(X) → Ab(Y).

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This functor has a left adjoint, denoted f ∗ , which can be defined as fol∗ G (U) = lows. For every sheaf G on Y and every open subset U of X, let fpre ∗ G with the structure of a colimV⊃ f (U) G (V). The universal maps endow fpre presheaf on Y. Let f ∗ G be the associated sheaf. For every open subset V of Y, one has f ( f −1 (V)) ⊂ V. The canonical mor∗ G ( f −1 (V)). furnishes a morphism G (V) → colimW⊃ f ( f −1 (W)) G (W) = fpre phism of sheaves εG ∶ G → f∗ f ∗ G . This morphism is functorial in G . On the other hand, for every open subset U of X and every open subset V of Y ∗ ( f F )(U) = containing f (U), one has U ⊂ f −1 (V); then, the canonical map fpre ∗ −1 colimV⊃ f (U) f∗ F (V) = colimV⊃ f (U) F ( f (V)) → F (U) defines a morphism ∗ ( f F ) → F , hence a morphism of sheaves ε ∶ f ∗ ( f F ) → of presheaves fpre ∗ F ∗ F . This morphism is functorial in F . The morphisms ε⋅ and η⋅ are the unit and the counit of an adjunction f ∗ ⊣ f∗ , and furnish functorial bijections: HomX ( f ∗ G , F ) ≃ HomY (G , f∗ F ) for every sheaf F on X and every sheaf G on Y. 2.2.8. — When C is the category of sets, sheaves have an alternative, topological, definition in terms of étale spaces over X, that is, a topological space E endowed with a continuous map p ∶ E → X which is a local homeomorphism (every point of E has an open neighborhood U such that p∣U is a homeomorphism from U to an open subset of X). To every étale space p ∶ E → X is associated its sheaf F of continuous sections. The fiber Fx is identified with p−1 (x). Conversely, given a sheaf F , one endowes the set EF = ∐x∈X Fx with a topology so that the projection EF → X is étale and its sheaf of continuous sections identifies with F . In the setting of étale spaces, the functor f ∗ corresponds to the fiber product of topological spaces. 2.2.9. — Let F be an abelian sheaf on X. Let U be an open subset of X and let s ∈ F (U). By the sheaf property of F , if V is a family of open subsets of U with union V, such that s∣W = 0 for every W ∈ V , then s∣V = 0. The support of s, supp(s), is the intersection of all closed subsets Z of U such that s∣U Z = 0. It is a closed subset of U, and the restriction of s to its complement is 0; consequently, it is the smallest such closed subset.

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Theorem (2.2.10). — Let F be sheaf on X, let A be a subspace of X and let j ∶ A → X be the canonical inclusion. Let us make one of the following hypotheses: (i) The subspace A admits a basis of paracompact neighborhoods; (ii) The space X is paracompact and A is closed; (iii) The space X is metrizable; (iv) The space X is Hausdorff and A is compact. Then the canonical morphism of presheaves j∗pre F → j∗ F induces a bijection colim F (U) → j∗ F (A). U⊃A

For the proof, I refer to (Godement, 1973, théorème 3.3.1, p. 150) or (Bourbaki, 2016, I, p. 37, théorème 2). 2.3. Extensions by zero 2.3.1. — Let j ∶ W → X be the inclusion of a locally closed subset of X. Let F be a sheaf on W. For every open subset U of X, let j! F (U) be the subset of j∗ F (U) = F (W∩U) consisting of all sections s whose support (which is closed in W ∩ U) is closed in U. This is a sub-presheaf of j∗ F , actually a subsheaf. Moreover, the construction j! gives rise to a functor Ab(W) → Ab(X). Lemma (2.3.2). — a) For every x ∈ X W, one has j! (F )x = 0. b) For every x ∈ W, the canonical morphism ( j∗ F )x → Fx induces an isomorphism ( j! F )x → Fx . Proof. — a) Let x ∈ X W. Let U be an open subset of X such that x ∈ U and let s ∈ F (W∩U) be an element of j! F (U). By hypothesis, one has supp(s) ⊂ W, so that x ∈/ supp(s), hence there exists an open neighborhood V of x such that V ⊂ U and s∣V = 0. This implies that the germ s x of s at x is 0. We thus have ( j! F )x = 0. b) Let x ∈ W. Since W is locally closed, there exists an open neighborhood V of x in X such that W∩V is closed in V. For every an open subset U of V and every s ∈ j∗ F (U) = F (W∩U), the support supp(s) of x is closed in W∩U, hence in U, because W∩U is closed in U. Consequently, j! F (U) = j∗ F (U) for every such U. In particular, the inclusion j! F → j∗ F induces an isomorphism ( j! F )∣U → ( j∗ F )∣U . In particular, the canonical morphisms ( j! F )x → ( j∗ F )x → Fx are isomorphisms.

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Proposition (2.3.3). — The functor f! is exact and fully faithful. It induces an equivalence of categories from Ab(W) to the full subcategory of Ab(X) consisting of abelian sheaves G such that Gx = 0 for every x ∈ X W. On that subcategory, the functor f ∗ induces a quasi-inverse.

Proof. — At the level of fibers, the functor f! induces the identity functor, or the zero functor; it is in particular exact. Let G be an abelian sheaf on X such that Gx = 0 for every x ∈ X W. Let us show that the canonical morphism ηG ∶ G → f∗ f ∗ G factors through f! f ∗ G . Let indeed U be an open subset of X. Then f∗ f ∗ G (U) = f ∗ G (W ∩ U) and the morphism ηG (U) ∶ G (U) → f ∗ G (W ∩ U) factors through the morphism s ↦ (s∣V )V from ∗ G (W ∩ U) = colim G (U) to fpre V⊃W∩U G (V) ≃ colimU⊃V⊃W∩U G (V). Moreover, for every s ∈ G (U), the support of s is closed in U, hence the support of s∣V is closed in V, for every open subset V of X such that U ⊃ V ⊃ W ∩ U. This implies that the image of s belongs to f! f ∗ G (U). The resulting morphism of sheaves, G → f! f ∗ G , induces an isomorphism on fibers: this is tautological for x ∈ W, and follows from the fact that Gx = 0 otherwise. This morphism is thus an isomorphism. Let then F be an abelian sheaf on W. The canonical morphism εF ∶ f ∗ f∗ F → F induces a morphism f ∗ f! F → F . Let O be an open subset of X containing W such that W is closed in O. Let V be an open subset of W. The canonical morphisms ∗ fpre f! F (V) = colim f! F (U) ≃ colim f! F (U) U⊃V

O⊃U⊃V

→ colim f∗ F (U) = colim F (W ∩ U) → F (V) O⊃U⊃V

O⊃U⊃V

∗ fF are isomorphisms. They induce an isomorphism from the presheaf fpre ! ∗ to the sheaf F , so that the corresponding morphism from f f! F to F is an isomorphism as well. Let F and G be abelian sheaves on W and let v ∶ f! F → f! G be a morphism of abelian sheaves. There exists a a unique morphism of sheaves u ∶ F → G the

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diagram → f ∗ f! G ←

← ←

F

→

→

εF

f ∗v

←

f ∗ f! F

u

εG

→ G.

Since f! u induces the morphism v x on the fibers at x, one then has f! u = v. This concludes the proof of the proposition. 2.3.4. — Let G be a sheaf on X. For every open subset U of X, let ΓW (G )(U) be the set of all sections s ∈ G (U) such that supp(s) ⊂ W. This is a subsheaf of G ; moreover, the construction G ↦ ΓW (G ) is functorial. Let x ∈ X W, let U be an open neighborhood of x and let s ∈ ΓW (G ). By assumption, x ∈/ supp(s); by definition of the support, there exists an open neighborhood V of x which is contained in U such that s∣V = 0. This shows that ΓW (G )x = 0. Moreover, if Gx = 0 for every x ∈ X W, then ΓW (G ) = G . Let us define j! G = j∗ ΓW (G ). This construction defines a functor Ab(X) → Ab(W). The morphisms F → j∗ j ! F = j! j ! F and j! j! G → ΓW (G ) → G are the unit and the counit of an adjunction ( j! , j! ): they provide functorial isomorphisms HomX ( j! F , G ) ≃ HomW (F , j! G ) for every abelian sheaf F on W and every abelian sheaf G on X. 2.3.5. — If W is closed in X, then j! = j∗ . If W is open in X, then ΓW (G )∣W = G , for every abelian sheaf G on X, so that ! j = j∗ . 2.4. Cohomology Let X be a topological space. Definition (2.4.1). — One says that a sheaf F on X is flasque if for every open subset U of X, the restriction map F (X) → F (U) is surjective.

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Lemma (2.4.2). — Let F be a flasque sheaf on X. a) The sheaf F ∣U is flasque, for every open subset U of X; b) Let f ∶ X → Y be a continuous map. The sheaf f∗ F is flasque. Example (2.4.3). — If X is discrete, every section of an étale space is continuous, so that a sheaf F on X is flasque if and only if its fibers Fx are non-empty, for all x ∈ X. Indeed, In particulier, every abelian sheaf on X is flasque in this case. Let Xδ be the set X endowed with the discrete topology. The identity p ∶ Xδ → X is continuous. For every abelian sheaf F on X, one lets G(F ) = p∗ p∗ F . This is a flasque sheaf on X, and the unit ηF ∶ F → p∗ p∗ F is a monomorphism. Explicitly, one has G(F )(U) = ∏x∈U Fx , for every open subset U of X, the restriction morphisms are the morphisms ∏x∈U Fx → ∏x∈V Fx , for V ⊂ U which are not only surjective, but have a section. Example (2.4.4). — An injective sheaf F on X is flasque. Let indeed U be an open subset of X and let s ∈ F (U). Let j ∶ U → X be the canonical inclusion; let f ∶ j! ZU → F be the unique morphism corresponding to the morphism ZU → j∗ F which maps 1 to s. The canonical morphism u ∶ j! ZU → ZX is injective; since F is injective, there exists a unique morphism g ∶ ZX → F such that g ○ u = f ; it corresponds to a section t ∈ F (X) such that t∣U = s. Consequently, the restriction morphism F (X) → F (U) is surjective, as as to be shown. Proposition (2.4.5). — Let F be a sheaf on X. Assume that for every open subset U of X and every s ∈ F (U), there exists an open covering V of X such that for every V ∈ V , there exists tV ∈ F (V) such that tV ∣U∩V = s∣U∩V . Then F is flasque. Proof. — Let U be an open subset of X and let s ∈ F (U). Let us show that there exists t ∈ F (X) such that t∣U = s. Let R be the set of pairs (V, t), where V is an open subset of X and t ∈ F (V). The relation ⪯ defined by (V, t) ⪯ (V′ , t ′ ) if and only if V ⊂ V′ and t ′ ∣V = t is an ordering relation on R; moreover, the ordered set R is inductive. By Zorn’s lemma, we may consider a maximal element (W, t) of R such that (U, s) ⪯ (W, t); let us show that W = X. By hypothesis, there exists an open covering V of X and, for every V ∈ V , an element tV ∈ F (V) such that tV ∣U∩W = t∣W∩V . If W ≠ X, there exists an open subset V ∈ V such that V ⊂/ W; then, there exists a unique section t ′ ∈ (W ∪ V) which restricts to t on W and to tV on V. In particular, (W ∪ V, t ′ ) is an element

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of R such that (W, t) ⪯ (W ∪ V, t ′ ), contradicting the hypothesis that (W, t) were maximal. Corollary (2.4.6). — Let F be a sheaf on X. Assume that there exists an open covering V of X such that F ∣V is flasque, for every V ∈ F . Then F is flasque. Proof. — Indeed, the condition of the proposition is satisfied: since F ∣V is flasque, there exists t ∈ F (V) which restricts to s∣U∩V . Proposition (2.4.7). — Let 0 → F ′ → F → F ′′ → 0 be an exact sequence of abelian sheaves. Assume that F ′ is flasque. a) For every open subset U of X, the sequence 0 → F ′ (U) → F (U) → F ′′ (U) → 0 is exact. b) If F is flasque, then F ′′ is flasque as well. Proof. — (1) It is a general fact that the sequence 0 → F ′ (X) → F (X) → F ′′ (X) → 0 is exact, except possibly at F ′′ (X). Let s ∈ F ′′ (X). Let us show that there exists t ∈ F (X) with image s. Let R be the set of pairs (V, t), where V is an open subset of X and t ∈ F (V) maps to s∣V in F ′′ (V). The relation ⪯ defined by (V, t) ⪯ (V′ , t ′ ) if and only if V ⊂ V′ and t ′ ∣V = t is an ordering relation on R; moreover, the ordered set R is inductive. By Zorn’s lemma, we may consider a maximal element (W, t) of R; let us show that W = X. By hypothesis, there exists an open covering V of X and, for every V ∈ V , an element tV ∈ F (V) which maps to s∣V in F ′′ (V). If W ≠ X, there exists an open subset V ∈ V such that V ⊂/ W. Then the elements t∣V∩W and tV ∣V∩W of F (V ∩ W) both map to s∣V∩W in F ′′ (V ∩ W); consequently, their difference belongs to F ′ (V ∩ W). Since F ′ is flasque, there exists u ∈ F ′ (X) such that t∣V∩W − tV ∣V∩W = u∣V∩W . In other words, the elements t ∈ F (W) and tV + u∣V ∈ F (V) agree on V ∩ W; consequently, there exists a unique section t ′ ∈ (W ∪ V) which restricts to t on W and to tV + u∣V on V; it maps to s∣W∪V in F ′ (W ∪ V). In particular, (W ∪ V, t ′ ) is an element of R such that (W, t) ⪯ (W ∪ V, t ′ ), contradicting the hypothesis that (W, t) were maximal. Let U be an open subset of X. By restriction to U, the initial exact sequence of sheaves on X furnishes an exact sequence of sheaves on U, and F ′ ∣U is flasque. By the case already treated, the diagram 0 → F ′ (U) → F (U) → F ′′ (U) → 0 is exact. (1)

This seems to be another instance of the same reasoning. Find a unifying statement?

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Assume now that F is flasque as well; let us show that F ′′ is flasque. Let U be an open subset of X and let s ∈ F ′′ (U). By what precedes, there exists t ∈ F (U) which maps to s. Since F is flasque, there exists t ′ ∈ F (X) such that t ′ ∣U = t. Then the image s′ of t ′ in F ′′ (X) satisfies s′ ∣U = s. Consequently, F ′′ is flasque. Corollary (2.4.8). — Let f ∶ X → Y be a continuous map of topological spaces. The full subcategory of Ab(X) consisting of flasque sheaves is adapted to the functor f∗ . In particular, it is adapted to the functor Γ(X, ⋅) of global sections. In particular, if F ● is a complex in D+ (Ab(X)) such that F j is flasque, for every j, then R f∗ F ● = f∗ F ● . Proof. — We need to check two properties: a) Every abelian sheaf on X embeds in a flasque sheaf. b) For every exact sequence 0 → F ′ → F → F ′′ → 0 of flasque abelian sheaves on X, the diagram 0 → f∗ F ′ → f∗ F → f∗ F ′′ → 0 of abelian sheaves on Y is exact. The first property follows from example ??. Let us then check the second one. Let V be an open subset of Y. Taking sections on V, the given diagram induces a diagram 0 → f∗ F ′ (V) → f∗ F (V) → f∗ F ′′ (V) → 0 of abelian groups, which identifies with the diagram 0 → F ′ ( f −1 (V)) → F ( f −1 (V)) → F ′′ ( f −1 (V)) → 0. By proposition ??, the latter diagram is exact. In particular, the initial diagram is exact, as was to be shown. Corollary (2.4.9). — Let f ∶ X → Y and g ∶ Y → Z be continuous maps of topological spaces. The canonical morphism of functors Rg∗ ○ Rf∗ → R(g ○ f )∗ from D+ (Ab(X)) to D+ (Ab(Z)) is an isomorphism. Proof. — Consider an element of D+ (Ab(X)), represented by a complex F ● in K+ (Ab(X)) with flasque terms. Then the canonical morphisms (g ○ f )∗ F ● → R(g ○ f )∗ F ● and f∗ F ● → R f∗ F ● is are homologisms. Since F j is flasque for every j, so is f∗ F j ; consequently, the canonical morphism g∗ ( f∗ F ● ) → Rg∗ ( f∗ F ● ) is an isomorphism. The corollary follows.

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2.5. Cohomology with compact support 2.5.1. — Let f ∶ X → Y be a continuous map. Let F be an abelian sheaf on X. For every open subset V of Y, let f! F (V) be the set of all sections s ∈ F ( f −1 (V)) whose support is proper and separated over V, that is, such that the map f induces by restriction a proper map f ∣supp(s) ∶ supp(s) → f −1 (V). This is a subset of f∗ F (V). Proposition (2.5.2). — Let f ∶ X → Y be a continuous map. For every abelian sheaf F on X, f! F is an abelian subsheaf of f∗ F . Proof. — Let V be an open subset of Y. The support of the zero section of f∗ F is empty, hence is proper over V. Let then s, s′ ∈ F ( f −1 (V)); consider them as elements the support of the element s + s′ is contained in the union supp(s) ∪ supp(s′ ); it is proper over V; consequently, s + s′ ∈ f! F (V). Similarly, the support of −s is equal to the support of s, so that −s ∈ f! F (V) if s ∈ f! F (V). This shows that f! F (V) is a subgroup of f∗ F (V). Let U, V be open subsets of Y such that V ⊂ U. Let s ∈ f∗ F (U) = F ( f −1 (U)). One has supp(s∣ f −1 (V) ) = supp(s) ∩ f −1 (V). Consequently, supp(s) is proper over U, then supp(s∣ f −1 (V) ) is proper over V. Consequently, the restriction morphism f∗ F (U) → f∗ F (V) maps f! F (U) into f! F (V). In other words, f! F is a sub-presheaf of f∗ F . Let finally U be an open subset of Y, let V be an open covering of U and let (sV )V∈V be a family, where sV ∈ f! F (V), such that sV ∣V∩W = sW ∣V∩W for every V, W ∈ V . Viewing the section sV as an element of f∗ F (V), for every V ∈ V , we see that there exists a unique element s ∈ f∗ F (U) such that s∣V = sV for every V ∈ V ; indeed, f∗ F is a sheaf. Then, for every V ∈ V , the intersection supp(s) ∩ f −1 (V) is proper over V; consequently, supp(s) is proper over U, hence s ∈ f! F (U). This proves that f! F is a subsheaf of f∗ F . 2.5.3. — Let f ∶ X → Y be a continuous map of topological spaces. Let u ∶ F → G be a morphism of abelian sheaves on X. For every open subset V of Y, the morphism f∗ (u) ∶ f∗ F (V) → f∗ G (V) maps f! F (V) to f! G (V). Indeed, the image f∗ (u)(s) of a section s ∈ f! F (V) is the section u(s) of f∗ G ( f −1 (V)); its support is a closed subset of the support of s, hence is proper over V.

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Consequently, the maps F ↦ f! F and u ↦ f! (u) define a functor from the category Ab(X) to the category Ab(Y). Lemma (2.5.4). — The functor f! is a left-exact additive functor. (2) Proof. — It follows from its definition that the functor f! is additive. Let 0 → F ′ → F → F ′′ be an exact sequence of abelian sheaves on X and let us show that the diagram 0 → f! F ′ → f! F → f! F ′′ of abelian sheaves on Y is an exact sequence. Let V be an open subset of Y and let s ∈ f! F ′ (V) map to 0 in f! F (V). Then s, viewed as an element of F ′ ( f −1 (V)), maps to 0 in F ( f −1 (V)). Since the morphism from F ′ to F is a monomorphism, one has s = 0. This shows that the morphism f! F ′ → f! F is a monomorphism. Let then s ∈ f! F (V) map to 0 in f! F ′′ (V). Again, the section s, when viewed as an element of F ( f −1 (V)) maps to 0 in F ′′ ( f −1 (V)). By definition of the exactness of the sequence 0 → F ′ → F → F ′′ , there exists a unique section s ′ ∈ F ′ ( f −1 (V)) which maps to s. Since the morphism F ′ → F is a monomorphism, the support of s′ is equal to the support of s, hence is proper over f −1 (V). This proves that the section s of f! F (V) is the image of a section of fF′ (V), as was to be shown. 2.5.5. — By the general theory of derived functors, the functor f! , for a continuous map f ∶ X → Y, gives rise to a functor R f! ∶ D(Ab(X)) → D(Ab(Y)). When f is the canonical map from X to a point, the functor f! identifies with the functor Γc of sections with compact support. To be able to compute it conveniently, it is important to find a suitable category of abelian sheaves which is adapted to these functors and is as large as possible. On locally compact topological spaces, a convenient category is that of soft sheaves. (In the book of (Godement, 1973), which considers general support conditions, this notion is called c-soft.) Definition (2.5.6). — Let X be a locally compact topological space. An abelian sheaf F is soft if for every compact subspace A of X, the canonical morphism F (X) → F ∣A (A) is surjective. (2)

Prove a general commutation with limits?

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Example (2.5.7). — Let X be a locally compact topological space? The sheaf CX of continuous functions on X is soft. Let indeed A be a compact subset of X and let f ∈ CX ∣A (A). The sheaf CX ∣A is the sheaf of germs of continuous functions in a neighborhood of A, but the extension theorem 2.2.10 implies that f is induced by a continuous function f ′ defined in an open neighborhood W of A. We may assume that W is compact. By Urysohn’s theorem (which is valid on compact spaces), there exists a continuous function h ∶ W → [0; 1] such that h ≡ 1 in a neighborhood of A and h ≡ 0 on ∂W. Let g ∶ X → R be defined by g(x) = h(x) f ′ (x) for x ∈ W and g(x) = 0 for x ∈ X W. Its restriction to W is continuous; since X W and W is a covering of X by closed subsets, the function g is continuous. By construction, it coincides with f ′ on neighborhood of A, so that the section g∣A of CX ∣A (A) is equal to f . By the same argument, every sheaf of CX -modules on the locally compact topological space X is soft. More generally, if O is a soft sheaf of rings on a topological space X, then any sheaf of O-modules on X is soft. Lemma (2.5.8). — Let F be a soft abelian sheaf on a topological space X. For every subspace W of X, the sheaf F ∣W is soft Proof. — Let A be a compact subset of W and let s ∈ (F ∣W ∣A )(A) = F ∣A (A). Then A is a compact subset of X, hence there exists t ∈ F (X) such that t∣A = s. Then t∣W is a section of F ∣W (W) which restricts to s on A. Proposition (2.5.9). — Let X be a locally compact topological space and let F be a soft abelian sheaf on X. Let A be a compact subset of X and let U be an open neighborhood of A. The canonical morphism Γc (U, F ) → Γ(A, F ∣A ) is surjective. Proof. — Let V be an open neighborhood of A such that V is compact and contained in U. Let s ∈ Γ(A, F ∣A ). Let B = A ∪ ∂V; since A ∩ ∂V = ∅, there exists a unique section s′ of F ∣B (B) whose restriction to A is equal to s and whose restriction to ∂V is zero. Since F is soft, there exists a section t ′ ∈ Γ(X, F ) such that t ′ ∣A = s and t ′ ∣∂V = 0. There exists an open neighborhood W′ of ∂V such that t ′ ∣W′ = 0; let W = W′ ∪ (X V). Then there exists a unique section t ∈ F (X) such that t∣V = t ′ and t∣W = 0; indeed, W ∩ V = W′ ∩ V and t ′ ∣W′ ∩V = 0. By construction, the support of t is contained in X W ⊂ V. Since V is compact and contained in U, so is the support of t.

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Proposition (2.5.10). — Let X be a locally compact topological space. Let 0 → F ′ → F → F ′′ → 0 be an exact sequence of abelian sheaves. Assume that F ′ is soft. a) For every open subset U of X, the sequence 0 → Γc (U, F ′ ) → Γc (U, F ) → Γc (U, F ′′ ) → 0 is exact. b) If F is soft, then F ′′ is flasque as well. Proof. — Let us first prove that if X is compact, then the sequence 0 → Γ(X, F ′ ) → Γ(X, F ) → Γ(X, F ′′ ) → 0. By left-exactness of the functor Γ(X, ⋅), it is exact except possibly at Γ(X, F ′′ ), so that we just need to prove the surjectivity of the morphism Γ(X, F ) → Γ(X, F ′′ ). Let s ′′ ∈ F ′′ (X). There exists a finite covering (Ui )1⩽i⩽n of X and a family (s i ), where s i ∈ F (Ui ) for every i, such that s i lifts s ′′ ∣Ui . Let (Vi )1⩽i⩽n be an open covering of X such that Vi ⊂ Ui for every i; for m ∈ {1, . . . , n}, let Wm = V1 ∪ ⋅ ⋅ ⋅ ∪ Vm . Let us show by induction on m that s′′ ∣Wm lifts to a section of F . For m = 1, the section s∣W1 lifts s ′′ ∣W1 . Let s ∈ F (Wm ) be a section that lifts s ′′ ∣Wm ; the restrictions of s and s m+1 to Wm ∩ Vm+1 both lift s′′ ∣Wm ∩Vm+1 , so that their differences belong to F ′ (Wm ∩ Vm+1 ). Since F ′ is soft, there exists a section s ′ ∈ F ′ (X) such that s∣Wm ∩Vm+1 − s m+1 ∣Wm ∩Vm+1 = s ′ ∣Wm ∩Vm+1 . Then s and s m+1 +s′ ∣Vm+1 coincide on Wm ∩Vm+1 , hence can be glued to a section of F (Wm+1 ) that lifts s′′ ∣Wm+1 . This proves the desired surjectivity when X is compact. Let now U be an open subset of X and let us prove that the sequence 0 → Γc (U, F ′ ) → Γc (U, F ) → Γc (U, F ′′ ) → 0 is compact. By left exactness of the functor Γc (U, ⋅), it suffices to prove the surjectivity of the morphism Γc (U, F ) → Γc (U, F ′′ ). Let s′′ ∈ Γc (U, F ′′ ) and let V be an open neighborhood of s′′ such that V is compact. Since F ′ ∣V is soft and V is compact, the section s′′ lifts to a section s ∈ F (V). The restriction s∣∂V maps to 0 in F ′′ (∂V), hence belongs to F ′ (∂V). Since F ′ is soft, there exists a section t ∈ F ′ (V) such that t∣∂V = s∣∂V . The section s − t ∈ F (V) lifts s′′ ∣V and restricts to 0 on ∂V. Consequently, its extension by 0 is an element of F (U) which lifts s′′ ; moreover, its support is contained in V, hence is compact. Let us finally assume moreover that F is soft and let us prove that F ′′ is soft as well. Let A be a compact subset of X and let s′′ ∈ Γ(A, F ′′ ∣A ). Let V be an open neighborhood of A such that V is compact and s′′ is induced by a section of F ′′ on V; we still denote it by s′′ . Since F ′ ∣V is soft, there exists s ∈ F (V) that

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lifts s′′ . Since F is soft, there exists t ∈ Γc (X, F ) that extends s. The image t ′′ of t in Γc (X, F ′′ ) extends s′′ . That concludes the proof of the proposition. Corollary (2.5.11). — Let f ∶ X → Y be a continuous map of topological spaces. Assume that X is locally compact. Then the functor f! ∶ Ab(X) → Ab(Y) is left exact and the full subcategory of Ab(X) consisting of soft abelian sheaves is adapted to f! . In particular, it is adapted to the functor Γc (X, ⋅) of global sections with compact support. Proof. — Every abelian sheaf can be embedded into a flasque sheaf, hence into a soft sheaf. Let 0 → F ′ → F → F ′′ → 0 be an exact sequence of abelian sheaves on X, where F ′ and F are soft; by the proposition, the sheaf F ′′ is soft as well, so that we just need to prove that the sequence 0 → f! F ′ → f! F → f! F ′′ → 0 of abelian sheaves on Y is exact. By left exactness of the functor f! , it suffices to prove the surjectivity of the morphism f! F → f! F ′′ . By proposition ??, its fiber at a point y ∈ Y identifies with the morphism Γc (X y , F ∣X y ) → Γc (X y , FX′′y ). The sequence 0 → F ′ ∣X y → F ∣X y → F ∣′′X y → 0 is exact, and FX′ y is flasque. Consequently, the sequence obtained by applying the functor Γc (X y , ⋅) is exact, as was to be shown. Corollary (2.5.12). — Let X be a locally compact topological space. Let F be an abelian sheaf on X. The following properties are equivalent: (i) The abelian sheaf F is soft; j (ii) One has Hc (U, F ) = 0 for every open subset U of X and every integer j ⩾ 1; (iii) One has H1c (U, F ) = 0 for every open subset U of X. Proof. — The implication (i)⇒(ii) follows from the preceding corollary, and the implication (ii)⇒(iii) is obvious. Let us thus assume that H1c (U, F ) = 0 for every open subset U of X and let us prove that F is soft. (Unfinished). Theorem (2.5.13). — Let f ∶ X → Y be a separated proper map of topological spaces. Let y ∈ Y, let X y = f −1 (y). a) For every sheaf F on X, the canonical map ( f∗ F ) y → Γ(X y , F ∣X y ) is an isomorphism; b) This isomorphism extends uniquely to an isomorphism of cohomological functors (R j f∗ ⋅) y ≃ R j Γ(X y , ⋅∣X y ) on D+ (Ab(X)).

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2.6. Tensor products 2.7. Verdier duality 2.8. The six operations 2.9. Constructible sheaves 2.10. Exercises Exercise (2.10.1). — Let X be a complex manifold, for example X = C, and let OX be the sheaf of holomorphic functions on X. By associating with a holomorphic function f on an open subset U of X the (non-vanishing) function exp( f ), one defines a morphism of sheaves ε ∶ OX → OX× . a) Prove that if U is simply connected (say, contractible), then the induced morphism εU ∶ OX (U) → OX× (U) is surjective. b) Prove that the morphism ε is surjective. c) Let X = C and U = C× . Prove that the function z ∈ OX× (C× ) does not belong to the image of εU .

CHAPTER 3 TRUNCATION STRUCTURES

3.1. Definition of truncation structures Definition (3.1.1). — Let D be a triangulated category. A truncation structure, in short t-structure, on D is the datum of two full subcategories D ⩽0 and D ⩾1 of D satisfying the following conditions: a) Every object isomorphic to an object of D ⩽0 (resp. of D ⩽0 ) belongs to D ⩽0 (resp. of D ⩽0 ); b) One has D(X, Y) = 0 for every X ∈ ob(D ⩽0 ) and every Y ∈ ob(D ⩾1 ); c) One has ΣD ⩽0 ⊂ D ⩽0 and D ⩾1 ⊂ Σ−1 D ⩾1 ; d) For every object X ∈ ob(D), there exists a distinguished triangle A → X → B → ΣA in D, where A ∈ ob(D ⩽0 ), B ∈ ob(D ⩾1 ). Example (3.1.2). — Let (D ⩽0 , D ⩾1 ) be a truncation structure on the triangulated category D. Let n ∈ Z. Let us write D ⩽n = Σ−n D ⩽0 and D ⩾n+1 = Σ−n D ⩾1 , for every integer n. The second condition of the definition can then be written D ⩽0 ⊂ D ⩽1 and D ⩾1 ⊂ D ⩾0 . By translation, the pair (D ⩽n , D ⩾n+1 ) is a translation structure on D. Example (3.1.3). — Let (D ⩽0 , D ⩾1 ) be a truncation structure on the triangulated category D. Then (D ⩾1,o , D ⩽0,o ) is a truncation structure on the opposite triangulated category D o . Example (3.1.4). — Let A be an abelian category and let D(A) be its derived category. Let n ∈ Z. Recall that D⩽n (A) is the full subcategory of D(A) consisting of complexes X such that H j (X) = 0 for j > 0, while D⩾n+1 (A) is the full subcategory of D(A) consisting of complexes Y such that H j (Y) = 0 for j ⩽ 0. Given a complex X ∈ D(A), recall that τ⩽0 X ∈ D⩽0 (A), τ⩾1 X ∈ D⩾1 (A), and that

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there is a distinguished triangle τ⩽0 X → X → τ⩾1 X → Στ⩽0 X. This shows that the pair (D⩽0 (A), D⩾1 (A)) is a truncation structure on D(A). Moreover, for every integer n, one has D⩽n (A) = Σ−n D⩽0 (A) and D⩾n+1 (A) = Σ−n D⩾1 (A). Example (3.1.5). — Let D be a triangulated category. Then the pairs (D , 0) and (0, D) are ‘‘degenerate’’ truncation structures on D. Proposition (3.1.6). — Let (D ⩽0 , D ⩾1 ) be a truncation structure on the triangulated category D. a) The inclusion D ⩽0 → D admits a right adjoint τ⩽0 ; b) The inclusion D ⩾1 → D admits a left adjoint τ⩾1 ; c) For every object X of D, there exists a unique morphism ∂ ∶ τ⩾1 X → Στ⩽0 X ∂

such that the triangle τ⩽0 X → X → τ⩾1 X Ð → Στ⩽0 X is distinguished. ⩽0 ⩾1 d) Let A ∈ ob(D ), B ∈ ob(D ) and let A → X → B → ΣA be a distinguished triangle. There exists a unique morphism of distinguished triangles

←

⇐

← → Στ⩽0 .

←

←

←

←

→ τ⩾1

→ ΣA

←

←

→ X

→

⇐

→ τ⩽0

→ B

→

→ X

←

A

Proof. — a) We need to find, for every object Y ∈ ob(D), an object τ⩽0 of D ⩽0 , a morphism εY ∶ τ⩽0 Y → Y in D inducing bi-functorial isomorphisms D(X, Y) ≃ D(X, τ⩽0 Y), for X ∈ ob(D ⩽0 ). Let A → Y → B → ΣA be a distinguished triangle, where A ∈ D ⩽0 and B ∈ D ⩾1 . Applying the cohomological functor D(X, ⋅) to the translated distinguished triangle Σ−1 B → A → Y → B, we obtain an exact sequence D(X, Σ−1 B) → D(X, A) → D(X, Y) → D(X, B). Since Σ−1 B ∈ D ⩾1 and B ∈ D ⩾1 , the two extreme groups vanish, so that the morphism A → Y induces an isomorphism D(X, A) → D(X, Y) for every X ∈ ob(D ⩽0 ). u →Y→B→ For every object Y in D, let us choose a distinguished triangle A Ð ΣA and let us set τ⩽0 Y = A and εY = u. Let f ∶ Y → Z be a morphism in D. By what precedes, the morphism εZ induces an isomorphism D(τ⩽0 Y, τ⩽0 Z) → D(τ⩽0 Y, Z). Let then τ⩽0 ( f ) ∶ τ⩽0 Y → τ⩽0 Z be the unique morphism such that εZ ○ τ⩽0 ( f ) = f ○ εY .

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87

One checks readily that τ⩽0 is a functor and that the morphisms εY , for Y ∈ ob(D), are the counits of an adjunction, making τ⩽0 a right adjoint of the inclusion of D ⩽0 in D. b) This is proved analogously to a), or can be deduced from a) by passing to the opposite category and shifting. In fact, one may choose for every object Y a distinguished triangle A → Y → B → ΣA as above and set τ⩾1 Y = B. c) Let Y be an object of D. By construction, the counit εY of the adjunction (⋅, τ⩽0 ) and the unit ηY of the adjunction (τ⩾1 , ⋅) stand in a distinguished triangle ηY

εY

∂

τ⩽0 Y Ð→ Y Ð→ τ⩾1 Ð → Στ⩽0 Y. Since D(τ⩽0 Y, τ⩾1 Y) = 0, the uniqueness of the differential ∂ follows from corollary 1.6.6. u v w d) Let Y be an object of D, let A Ð →YÐ →BÐ → ΣA be a distinguished triangle, where A ∈ D ⩽0 and B ∈ D ⩾1 . Let us show that there exist a unique morphism of distinguished triangles of the form ( f , idY , h):

ηY

h

→ τ⩾1 Y

→ ΣA ←

← ←

←

w

→

→ Y

→ B

←

←

⇐

v

→

εY

⇐

←

→

f

τ⩽0 Y

→ Y

←

u

←

A

∂

Σf

→ Στ⩽0 Y.

Since A ∈ ob(D ⩽0 ) and τ⩾1 Y ∈ ob(D ⩾1 ), one has ηY ○ idY ○u ∈ D(A, τ⩾1 Y) = 0, by definition of a truncation structure. Consequently, the assertion follows from corollary 1.6.6. Corollary (3.1.7). — a) Let X ∈ ob(D). The following properties are equivalent: 1) One has τ⩽0 X = 0; 2) One has D(A, X) = 0 for every object A ∈ ob(D ⩽0 ); 3) The morphism ηX ∶ X → τ⩾1 X is an isomorphism; 4) One has X ∈ ob(D ⩾1 ). b) The category D ⩾1 is a thick additive subcategory of D; it is stable under products and under extensions (if X → Y → Z → ΣX is a distinguished triangle and X, Z ∈ ob(D ⩾1 ), then Y ∈ ob(D ⩾1 )). Proof. — a) Assume that τ⩽0 X = 0; by adjunction of (⋅, τ⩽0 ), one then has D(A, X) ≃ D(A, τ⩽0 X) = 0. Assume conversely that D(A, τ⩽0 X) = 0. By adjunction of (⋅, τ⩽0 ), one has 0 = D(τ⩽0 X, X) = D(τ⩽0 X, τ⩽0 X), so that idτ⩽0 X = 0 and τ⩽0 X = 0. This proves that 1)⇔2). ηX 1)⇒3). Assume that τ⩽0 X = 0. The canonical distinguished triangle 0 → X Ð→ τ⩾1 X → 0 then implies that ηX is an isomorphism.

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3)⇒4). If ηX is an isomorphism, it follows from the definition of a truncation structure that X ∈ ob(D ⩾1 ). 4)⇒1). Finally, let us assume that X ∈ ob(D ⩾1 ). One thus has 0 = D(τ⩽0 X, X) = D(τ⩽0 X, τ⩽0 X), hence τ⩽0 X = 0. b) The characterization 2)⇔4) implies that it is stable under products. More precisely, for every family (Xi )i∈I with product X and for every object A ∈ D ⩽0 , the isomorphism D(A, X) ≃ ∏i D(A, Xi ) implies that X ∈ D ⩾1 if and only if Xi ∈ D ⩾1 for every i. In particular, D ⩾1 is a thick additive subcategory of D. Let X → Y → Z → ΣX be a distinguished triangle, where X and Z belong to D ⩾1 . Let A ∈ D ⩽0 ; let us apply the cohomological functor D(A, ⋅) to this triangle. This furnishes an exact sequence D(A, X) → D(A, Y) → D(A, Z). Since D(A, X) = D(A, Z) = 0, we thus have D(A, Y) = 0, hence Y ∈ ob(D ⩾1 ) by a). Recall that if Y ≃ X ⊕ Z, then there exists a distinguished triangle X → Y → Z → ΣX; in particular Y belongs to D ⩾1 if both X and Z do. Either by a similar reasoning, or by passing to the opposite category, one has the following corollary. Corollary (3.1.8). — a) Let X ∈ ob(D). The following properties are equivalent: 1) One has τ⩾1 X = 0; 2) One has D(X, B) = 0 for every object A ∈ ob(D ⩾1 ); 3) The morphism ηX ∶ τ⩽0 X → X is an isomorphism; 4) One has X ∈ ob(D ⩽0 ). b) The category D ⩽0 is a thick additive subcategory of D; it is stable under coproducts and under extensions (if X → Y → Z → ΣX is a distinguished triangle and X, Z ∈ ob(D ⩽0 ), then Y ∈ ob(D ⩽0 )). 3.1.9. — In category theory, an adjoint is only unique up to a canonical isomorphism, and the construction of the functors τ⩽0 and τ⩾1 involved the choice of a distinguished triangle A → X → B → ΣA, for every object X ∈ D, where A ∈ D ⩽ and B ∈ D ⩾1 . idX If X ∈ D ⩽0 , we may assume that the chosen distinguished triangle is X Ð→ X → 0 → ΣX, where 0 is a zero object, chosen to be X if X ≃ 0. In this case, one has τ⩽0 X = X for every object X ∈ D ⩽0 , and εX = idX . Similarly, if X ∈ D ⩾1 , we assume that the chosen distinguished triangle is idX

0 → X Ð→ X → Σ0, where 0 is a zero object chosen to be X if X ≃ 0, so that τ⩾1 X = X and ηX = idX .

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When X ∈ D ⩽0 ∩ D ⩾1 , it is a zero object and the two chosen distinguished triangles coincide. 3.1.10. — Let n be an integer. The functor τ⩽n = Σ−n τ⩽0 Σ n is a right adjoint of the inclusion functor D ⩽n → D. The functor τ⩾n+1 = Σ−n τ⩾1 Σ n is a left adjoint of the inclusion functor D ⩾n+1 → D. These truncation functors are additive. To simplify the notation, we also let τn = τ⩾n+1 , for every integer n. In particular, an object X of D belongs to D ⩽n if and only if τ>n X = 0; it belongs to D ⩾n if and only if τb τ⩾a X = τ>b X = 0, hence τ⩾a X ∈ D ⩽b . Similarly, if X ∈ D ⩾a , then τ⩽b X ∈ D ⩾a as well. Proposition (3.1.12). — Let a and b be integers such that a ⩽ b. For every object X ∈ D, there exists a unique morphism fX ∶ τ⩾a ○ τ⩽b X → τ⩽b ○ τ⩾a X such that the following diagram is commutative

← ←

τ⩾a τ⩽b X

→ τ⩾a X →

←

→

⩽b X

η Xa

←

η τa

→ X

←

τ⩽b X

ε Xb

fX

ε bτ⩾a X

→ τ⩽b τ⩾a X

Moreover, the morphisms fX give rise to an isomorphism of functors f ∶ τ⩾a ○ τ⩽b → τ⩽b ○ τ⩾a . Proof. — Since τ⩽b X ∈ D ⩽b , the morphism ηXa ○ εXb factors uniquely through τ⩽b τ⩾a X: there exists a unique morphism fX′ ∶ τ⩽b X → τ⩽b τ⩾a X such that ηbτ⩾a X ○ fX′ = ηXa ○ εXb . As we have seen, one has τ⩽b τ⩾a X ∈ D ⩾a ; consequently, the morphism fX′ factors uniquely through τ⩾a τ⩽b X: there exists a unique morphism fX ∶ τ⩾a τ⩽b X → τ⩽b τ⩾a X such that fX ○ η τa⩽b X = fX′ . The morphism fX satisfies εbτ⩾a X ○ fX ○ η τa⩽b X = εbτ⩾a X ○ fX′ = ηXa ○ εXb .

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The uniqueness of such a morphism is established by the same argument, reversed: the relation εbτ⩾a X ○ ( fX ○ η τa⩽b X ) = ηXa ○ εXb implies that fX ○ η τa⩽b X = fX′ , and this in turns characterizes fX . Let us show that fX is an isomorphism. We build an octahedron

η Xa

⇐

⇐ τ>b X

←

→ Στ⩾a τ⩽b X

→ Σ2 τ
←

←

← ←

←

→

→

→ Στ⩽b X

→

→

←

Στ
⇐ ←

→ Στ
←

→ τ⩾a X

η Xb+1

τ>b X

⇐

ε Xb

→ X

→ Στ
←

←

→ τ⩾a τ⩽b X

←

←

⇐

⩽b X

→

ε Xa−1

→

⇐ τ
←

→ τ⩽b X

←

τ
η τa

where both horizontal triangles are the canonical truncation triangles, as well as the left vertical triangle. This furnishes a distinguished triangle τ⩾a τ⩽b X → τ⩾a X → τ>b X → Στ⩾a τ⩽b X. One has τ⩾a τ⩽b X ∈ D ⩽b and τ>b X ∈ D >b , so that there exists a unique isomorphism of distinguished triangles → τ>b X

←

→ τ>b τ⩾a X

←

⇐

← ←

←

←

→ Στ⩾a τ⩽b X →

→ τ⩾a X

→

ε bτ⩾a X

⇐

←

→ τ⩽b τ⩾a X

←

→ τ⩾a X

←

τ⩾a τ⩽b X

→ Στ⩽b τ⩾a X.

The left vertical morphism is equal to fX . Indeed, the morphism εbτ⩾a X ○ fX ∶ τ⩾a τ⩽b X → τ⩾a X is the only one that makes the middle upper square of the octahedron commute, and u = fX is the only morphism such that εbτ⩾a X ○ u = εbτ⩾a X ○ fX . Consequently, fX is an isomorphism. Finally, one deduces from the characterization of the morphism fX that it induces an isomorphism of functors. Definition (3.1.13). — Let D and D ′ be triangulated categories endowed with truncation structures and let F ∶ D → D ′ be a triangulated functor. One says that

3.2. THE HEART OF A TRUNCATION STRUCTURE

91

F is right t-exact if F(D ⩽0 ) ⊂ D ′⩽0 , and that it is left t-exact if F(D ⩾1 ) ⊂ D ′⩾1 . One says that F is t-exact if it is both left t-exact and right t-exact. 3.1.14. — Assume that F is left t-exact. By translation, one observes that F(D ⩾n+1 ) ⊂ D ′⩾n+1 for every integer n. Let moreover X ∈ D, let τ⩽0 X → X → τ>0 X → Στ⩽0 X be the canonical triangle. Applying F, we obtain a distinguished triangle in D ′ : F(τ⩽0 X) → F(X) → F(τ>0 X) → ΣF(τ⩽0 X). By assumption, F(τ>0 X) ∈ D ′>0 ; consequently, the morphism F(X) → F(τ>0 X) factors through a unique morphism τ>0 F(X) → F(τ>0 X). 3.1.15. — Assume that F is right t-exact. By translation, one observes similarly that F(D ⩽n ) ⊂ D ′⩽n for every integer n. Moreover, for every object, the morphism F(τ⩽0 X) → F(X) factors through a unique morphism F(τ⩽0 X) → τ⩽0 F(X).

3.2. The heart of a truncation structure Definition (3.2.1). — Let D be a triangulated category. The heart of a truncation structure (D ⩽0 , D ⩾1 ) on D is the full subcategory D ⩽0 ∩ D ⩾0 . Theorem (3.2.2). — Let D be a triangulated category and let C be the heart of a truncation structure (D ⩽0 , D ⩾1 ) on D. a) The category C is an abelian category; as it subcategory of D, it is thick and stable under finite products and extensions. u v b) A complex 0 → X Ð →YÐ → Z → 0 in C is an exact sequence if and only if there u v w →YÐ →ZÐ → ΣX is a distinguished exists a morphism w ∶ Z → ΣX such that X Ð triangle in D. c) The functor H0 = τ⩾0 τ⩽0 ∶ D → C is a cohomological functor. Proof. — a) First of all, the category C is a thick additive subcategory of D, because both D ⩽0 and D ⩾0 are themselves thick additive subcategories of D. Let us show that any morphism in C admits a kernel and a cokernel. Let thus u ∶ X → Y be a morphism in C and let us choose a distinguished triangle u v w XÐ →YÐ →ZÐ → ΣX in D. The vertices Y and ΣX of the translated triangle Y → Z → ΣX → ΣY belong to D ⩽0 and to D ⩾−1 , hence Z ∈ D ⩽0 ∩ D ⩾−1 .

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Let us then prove that the composition v ′ = ηZ ○ v ∶ Y → τ⩾0 Z is a cokernel of u. One has v ′ ○ u = ηZ ○ v ○ u = 0. Let moreover f ∶ Y → W be a morphism in C such that f ○ u = 0; applying the (contravariant) cohomological functor D(⋅, W) to the previous distinguished triangle, we obtain an exact sequence w∗

v∗

u∗

D(ΣX, W) Ð→ D(Z, W) Ð→ D(Y, W) Ð→ D(X, W). Since ΣX ∈ D ⩽−1 and W ∈ D ⩾0 , one has D(ΣX, W) = 0. Since u ∗ ( f ) = f ○ u = 0, there exists a unique morphism g ′ ∈ D(Z, W) such that f = v ∗ (g ′ ) = g ′ ○ v. Since W ∈ D ⩾0 , there exists a unique morphism g ∈ D(τ⩾0 Z, W) such that g ′ = g ○ ηZ . The morphism g satisfies f = g ′ ○ v = g ○ ηZ ○ v = g ○ v ′ , and it is the unique such morphism. In a similar manner, we show that the composition w ′ = Σ−1w○ε Σ−1 Z ∶ τ⩽0 Σ−1 Z → X is a kernel of f . Retaining the notation, let us moreover assume that u is a monomorphism. Then its kernel vanishes, one has τ⩽0 Σ−1 Z = 0, hence Σ−1 Z ∈ D ⩾1 , hence Z ∈ D ⩾0 . Since we had Z ∈ D ⩽0 , this shows that Z ∈ C . The preceding construction then shows that u ∶ X → Y is a kernel of v. Similarly, if u is an epimorphism, its cokernel vanishes, hence τ⩾0 Z = 0. This implies that Z ∈ D ⩽−1 ∩ D ⩾−1 , hence Σ−1 Z ∈ C . The preceding construction shows that u is the cokernel of the morphism Σ−1w ∶ Σ−1 Z → X. We have shown that C is an additive category in which every morphism admits a kernel and a cokernel, such that every monomorphism is a kernel, and every epimorphism is a cokernel. Consequently, C is an abelian category. u v b) By definition, a complex 0 → X Ð →YÐ → Z → 0 in C is an exact sequence if and only if u is a monomorphism and v ∶ Y → Z is its cokernel. The description of the cokernel of v shows that there exists a morphism w ∶ Z → ΣX such that u v w →YÐ →ZÐ → ΣX is a distinguished triangle. Conversely, given the diagram X Ð such a distinguished triangle with vertices in C , the construction of the kernel u v and cokernel of u proves that 0 → X Ð →YÐ → Z → 0 is an exact sequence. u v w c) Let X Ð →YÐ →ZÐ → ΣX be a distinguished triangle. Let us show that the H0 (u)

H0 (v)

induced complex H0 (X) ÐÐÐ→ H0 (Y) ÐÐÐ→ H0 (Z) is exact at H0 (Y).

H0 (u)

1) We first assume that X, Y, Z belong to D ⩽0 and prove that H0 (X) ÐÐÐ→ H0 (Y)

H0 (v)

ÐÐÐ→ H0 (Z) → 0 is exact.

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Let T ∈ C . Applying the (contravariant) cohomological functor D(⋅, T) to the given triangle, we obtain an exact sequence of abelian groups: w∗

v∗

u∗

D(ΣX, T) Ð→ D(Z, T) Ð→ D(Y, T) Ð→ D(X, T). Since ΣX ∈ D ⩽−1 and T ∈ D ⩾0 , one has D(ΣX, T) = 0. On the other hand, since ∼ T ∈ D ⩾0 , the morphism η0Z ∶ Z → τ⩾0 Z induces an isomorphism D(τ⩾0 Z, T) Ð → D(Z, T). Since T ∈ D ⩽0 , the morphism ε τ⩾0 Z ∶ τ⩽0 τ⩾0 Z → τ⩾0 Z induces an ∼ isomorphism D(τ⩾0 Z, T) Ð → D(H0 (Z), T). In this way, the previous exact sequence rewrites as the exact sequence H0 (v)∗

H0 (u)∗

0 → C (H0 (Z), T) ÐÐÐ→ C (H0 (Y), T) ÐÐÐ→ C (H0 (X), T), since H0 (X), H0 (Y), H0 (Z) belong to C . Since this holds for every object T of C , the initial diagram is exact. Let us now just assume that X ∈ D ⩽0 and let us prove that the diagram H0 (u)

H0 (v)

H0 (X) ÐÐÐ→ H0 (Y) ÐÐÐ→ H0 (Z) → 0 is still exact. Let us first prove that the morphism τ⩾1v ∈ D(τ⩾1 Y, τ⩾1 Z) is an isomorphism. Let T ∈ D ⩾1 ; applying the contravariant cohomological functor D(⋅, T) to the initial distinguished triangle furnishes an exact sequence w∗

v∗

u∗

D(ΣX, T) Ð→ D(Z, T) Ð→ D(Y, T) Ð→ D(X, T). Since X ∈ D ⩽0 , one has ΣX ∈ D ⩽0 as well, and D(ΣX, T) = D(X, T) = 0, because T ∈ D ⩾1 . Consequently, the morphism v ∗ ∶ D(Z, T) → D(Y, T) is an isomorphism. Making use of the adjunction (τ⩾1 , ⋅), we obtain that τ⩾1 (v)∗ ∶ D(τ⩾1 Z, T) → D(τ⩾1 Y, T) is an isomorphism. Since this holds for every object of D ⩾1 , we finally deduce that τ⩾1v is an isomorphism, as claimed. Let us now build an octahedron

← ← →

→ Στ⩽0 Y

⇐

←

⇐ ←

←

←

→ ΣU

←

⇐

⇐ τ⩾1 Y

→

←

ΣX

→ ΣX

→

→

ηY

τ⩾1 Y

→ Z

←

←

←

v

←

→ Y

←

εY

⇐

u

→ ΣX

→

→

⇐ X

→ U

←

→ τ⩽0 X

←

X

→ Σ2 X

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94

and let us consider the right vertical distinguished triangle U → Z → τ⩾1 Y → ΣU. The morphism Z → τ⩾1 Y is precisely the composition of the unit Z → τ⩾1 Z and of the inverse of the isomorphism τ⩾1v. Consequently, the object U is isomorphic to τ⩽0 Z, and this distinguished triangle is isomorphic to the canonical truncation triangle of Z. The top horizontal distinguished triangle then identifies with the triangle τ⩽0 v

u′

XÐ → τ⩽0 Y ÐÐ→ τ⩽0 Z → ΣX, where u ′ is the unique morphism such that εY ○ u ′ = u. The three objects X, τ⩽0 Y, τ⩽0 Z belong to D ⩽0 ; applying the case already established, we obtain the desired exact sequence. 2) One proves similarly (or by passing to the opposite category), that if Z ∈ D ⩾0 , H0 (u)

H0 (v)

then the diagram 0 → H0 (X) ÐÐÐ→ H0 (Y) ÐÐÐ→ H0 (Z) is an exact sequence. 3) Let us finally establish the general case. We begin with an octahedron

→ Y

←

→ Στ⩾1 X

←

← ←

←

v

w

→ ΣX

→

→ ←

Στ⩽0 X

⇐

⇐

⇐ Z

→ Στ⩽0 X

←

⇐

←

Z

←

←

→ U →

u′

→

τ⩽0 X

→ Στ⩽0 X ⇐

⇐

u

←

←

←

←

→

→ X

←

→ τ⩾1 X

→

τ⩽0 X

→ Σ2 τ⩽0 X.

Since τ⩽0 X ∈ D ⩽0 , the second horizontal distinguished triangle furnishes an exact sequence 0

H0 (u)

H (X) ÐÐÐ→ H0 (Y) → H0 (U) → 0. Since τ⩾1 X ∈ D ⩾1 , one has Στ⩾1 X ∈ D ⩾0 and the the second vertical distinguished triangle (shifted once) furnishes an exact sequence 0 → H0 (U) → H0 (Z) → H0 (Στ⩾1 X). The composition of the epimorphism H0 (Y) → H0 (U) and of the monomorphism H0 (U) → H0 (Z) is the morphism H0 (v); we thus have established the exactness of H0 (X) → H0 (Y) → H0 (Z) at the middle object. This concludes the proof of c), hence of the theorem.

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95

3.2.3. — If X ∈ D <0 , then H0 (X) = τ⩾0 τ⩽0 X = τ⩾0 X = 0. Similarly, if X ∈ D >0 , then τ⩽0 X = 0 hence H0 (X) = 0. For every integer n and every object X of D, we set Hn (X) = H0 (Σ n X). With this notation, any distinguished triangle X → Y → Z → ΣX in D gives rise to a long exact sequence ⋅ ⋅ ⋅ → Hn−1 (Z) → Hn (X) → Hn (Y) → Hn (Z) → Hn+1 (X) → . . . in C . Observe that Hn (X) = Σ n τ⩾n τ⩽n X. If X ∈ D n , then Hn (X) = 0. Definition (3.2.4). — A truncation structure (D ⩽0 , D ⩾1 ) on D is said to be nondegenerate if ⋂n D⩾n and ⋂n D⩽n are reduced to zero objects. The canonical truncation structure on the derived category D(A) of an abelian category (example 3.1.4) is nondegenerate. However, the ‘‘degenerate’’ truncation structures of example 3.1.5 are not nondegenerate. Proposition (3.2.5). — Let (D ⩽0 , D ⩾1 ) be a nondegenerate truncation structure on D. Then the following properties hold: a) An object X ∈ D is zero if and only if H j (X) = 0 for every integer j; b) An object X ∈ D belongs to D ⩽n if and only if H j (X) = 0 for every integer j such that j > n; c) An object X ∈ D belongs to D ⩾n if and only if H j (X) = 0 for every integer j such that j < n; d) A morphism u ∶ X → Y in D is an isomorphism if and only if H j (u) ∶ H j (X) → H j (Y) is an isomorphism for every integer j. Proof. — a) If X = 0, then H j (X) = 0 for every j, because H j is an additive functor. Conversely let X be an object of D such that H j (X) = 0 for every integer j. First assume that there exists an integer n such that X ∈ D ⩽n . Then 0 = Hn (X) = τ⩾n (X), so that X ∈ D ⩽n−1 . By induction, one has X ∈ ⋂m D ⩽m X = {0}. Similarly, if there exists an integer n such that X ∈ D ⩾n , then one has 0 = Hn (X) = τ⩽n (X), hence X ∈ D ⩾n+1 and, by induction, X ∈ ⋂m D ⩾m = {0}. In the general case, let us consider the canonical triangle τ⩽0 X → X → τ⩾1 X → Στ⩽0 X. Applying the functor H0 , it induces a long exact sequence . . . Hn−1 (X) → Hn−1 (τ⩾1 X) → Hn (τ⩽0 X) → Hn (X) → . . .

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so that Hn−1 (τ⩾1 X) ≃ Hn (τ⩽0 X) for every integer n. Let n ∈ Z; if n > 0, we have Hn (τ⩽0 X) = 0; otherwise, one has n ⩽ 0, then Hn (τ⩽0 X) ≃ Hn−1 (τ⩾1 X) = 0 since n − 1 ⩽ 0. Since τ⩽0 X ∈ D ⩽0 , the particular case already treated shows that τ⩽ X = 0. One proves similarly that τ⩾1 X = 0. Then, the exact triangle 0 → X → 0 → 0 proves that X = 0. b) We already know that if X ∈ D ⩽n , then H j (X) = 0 for every integer j > n. Conversely, let X be an object of D such that H j (X) = 0 for every integer j > n. The canonical triangle τ⩽n X → X → τ⩾n+1 X → Στ⩽n X induces a long exact sequence . . . H j−1 (X) → H j−1 (τ⩾n+1 X) → H j (τ⩽n X) → H j (X) → . . . If j − 1 > n, then H j−1 (X) = H j (X) = 0, hence H j−1 (τ⩾n+1 X) ≃ H j (τ⩽n ) = 0 since j > n. Otherwise, j −1 ⩽ n, and H j−1 (τ⩾n+1 X) = 0. Consequently, H j (τ⩾n+1 X) = 0 for every integer j. By assertion a), one has τ⩾n+1 X = 0, hence X ∈ D ⩽n . c) This is analogous: one proves that H j (τ⩽n−1 X) = 0 for every integer j, hence τ⩽n−1 X = 0, hence X ∈ D ⩾n . d) If u is an isomorphism, then so is Hn (u) for every integer n. Let us assume, conversely, that Hn (u) is an isomorphism for every integer n. Let us complete u u v w into a distinguished triangle X Ð →YÐ →ZÐ → ΣX. Applying the functor H0 , we obtain a long exact sequence: Hn−1 (w)

Hn (u)

Hn (v)

Hn (w)

⋅ ⋅ ⋅ → Hn−1 (Z) ÐÐÐÐ→ Hn (X) ÐÐÐ→ Hn (Y) ÐÐÐ→ Hn (Z) ÐÐÐ→ Hn+1 (X) → . . . in C . Using that Hn (u) is an isomorphism for every n, one deduces Hn (Z) = 0 for every n. By a), this implies that Z = 0. Consquently, u is an isomorphism. 3.2.6. — Let F ∶ D → D ′ be a triangulated functor between triangulated categories endowed with truncation structures. Let C and C ′ be their hearts, and let F˜ = H0 ○ F ∶ C → C ′ ; it is an additive functor. Proposition (3.2.7). — a) If F is left t-exact, then F˜ is left exact; moreover, for ˜ 0 (X)) ≃ H0 (F(X)). every object X ∈ D ⩾0 , the morphism H0 F(εX ) ∶ F(H b) If F is right t-exact, then F˜ is right exact; moreover, for every object X ∈ D ⩽0 , ˜ 0 (X)) is an isomorphism. the morphism H0 F(ηX ) ∶ H0 (F(X)) → F(H u

v

Proof. — a) Let us assume that F is left t-exact. Let 0 → X Ð →YÐ →Z→0 be an exact sequence in C . By theorem 3.2.2, there exists a morphism w ∶ Z → u v w ΣX such that X Ð →YÐ →ZÐ → ΣX is a distinguished triangle. Applying the

3.2. THE HEART OF A TRUNCATION STRUCTURE

97 F(u)

F(v)

triangulated functor F, we obtain a distinguished triangle F(X) ÐÐ→ F(Y) ÐÐ→ F(w)

F(Z) ÐÐ→ ΣF(X) in D ′ . Since F is left t-exact and X, Y, Z belong to D ⩾0 , their images F(X), F(Y), F(Z) belong to D ′⩾0 ; in particular, H−1 (F(Z)) = 0. The long exact sequence associated with the previous triangle and the cohomological functor H0 furnishes an exact sequence H0 F(u)

0

0

H0 F(v)

0 → H (F(X)) ÐÐÐ→ H (F(Y)) ÐÐÐ→ H0 (F(Z)). This proves that the functor H0 ○ F is left exact. Let then X ∈ D ⩾0 , so that τ⩽0 X = H0 (X). As above, the canonical triangle εX τ⩽0 X Ð→ X → τ>0 X → Στ⩽0 X leads to an exact sequence 0

H0 F(ε X )

0 → H F(τ⩽0 X) ÐÐÐÐ→ H0 F(X) → H0 F(τ>0 X). Since τ>0 X ∈ D ⩾1 , the left t-exactness of F implies that F(τ>0 X) ∈ D ′⩾1 , hence ˜ 0 (X)) = H0 (F(τ>0 X)) = 0. Consequently, the canonical morphism H0 F(εX ) ∶ F(H H0 F(H0 (X)) → H0 F(X) is an isomorphism, as claimed. b) The case of a right t-exact functor is treated similarly. Corollary (3.2.8). — Let D , D ′ , D ′′ be triangulated categories endowed with truncation structures. Let F ∶ D → D ′ and G ∶ D ′ → D ′′ be triangulated functors. a) If F and G are left t-exact, then G ○ F is left t-exact and the morphism of ̃ ○̃ ̃ functors H0 G(ε) ∶ G F→G ○ F is an isomorphism, where ε ∶ τ⩽0 → id is the ̃ ̃ ○̃ counit of the adjunction (⋅, τ⩽0 ) in D. and G ○F=G F. b) If F and G are right t-exact, then G ○ F is right t-exact and the morphism ̃ ̃ ○̃ of functors H0 G(η) ∶ G ○F→G F is an isomorphism, where η ∶ id → τ⩾0 is the unit of the adjunction (τ⩾0 , ⋅) in D. Proof. — a) Assume that F and G are left t-exact. Then G ○ F(D ⩾1 ) ⊂ G(D ′⩾1 ) ⊂ D ′′⩾1 , so that G ○ F is left t-exact as well. ̃ Let then X ∈ C . By definition, G ○ F(X) = H0 (G(F(X))). Let Y = F(X); since ˜ F is left t-exact, Y ∈ D ′⩾0 , and F(X) = H0 (Y). Consequently, the morphism ̃ ˜ F(X)) ˜ H0 G(εY ) ∶ G( → H0 G(Y) = G ○ F(X) is an isomorphism. b) The case of right t-exact functors is analogous. Proposition (3.2.9). — Let D and D ′ be triangulated categories endowed with truncation structures. Let G ∶ D → D ′ and F ∶ D ′ → D be triangulated functors. Assume that F is right adjoint to G.

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Then, F is left t-exact if and only if G is right t-exact. If these properties hold, ˜ then F˜ is right adjoint to G. Proof. — Let us assume that F is left t-exact and let us prove that G is right t-exact. Let X be an object of D ⩽0 and let us prove that G(X) ∈ D ′⩽0 . We use the criterion of corollary 3.1.8; let Y ∈ D ′⩾1 ; since F is left t-exact, one has F(Y) ∈ D ⩾1 ; consequently, D ′ (G(X), Y) ≃ D(X, F(Y)) = 0; this proves that G(X) ∈ D ⩽0 . Conversely, these arguments prove that if G is right t-exact, then F is left t-exact. Let X be an object of C ′ and Y be an object of C . Since F(Y) ∈ D ⩾0 , one has ˜ ˜ F(Y) = H0 (F(Y)) = τ⩽0 F(Y); similarly, G(X) = H0 (G(X)) = τ⩾0 G(X). The adjunction (G, F) furnishes a bifunctorial isomorphism D(X, F(Y)) ≃ D ′ (G(X), Y). Since X ∈ D ⩽0 , the adjunction (⋅, τ⩽0 ) furnishes a bifunctorial ˜ isomorphism C (X, F(Y)) ≃ D(X, F(Y)). Similarly, the adjunction (τ⩾0 , ⋅) ˜ furnishes a bifunctorial isomorphism C ′ (G(X), Y) ≃ D ′ (G(X), Y). The composition of these isomorphisms is a bifunctorial isomorphism ˜ ˜ ˜ C (X, F(Y)) ≃ C ′ (G(X), Y). In particular, F˜ is right adjoint to G. 3.3. Glueing truncation structures Proposition (3.3.1). — Let D be a triangulated category, let M , N be two full triangulated subcategories such that every object isomorphic to an object of M (resp. N ) belongs to M (resp. N ). We make the following hypotheses: (i) For every A ∈ M and every B ∈ N , one has D(A, B) = 0; (ii) For every object X ∈ D, there exists a distinguished triangle A → X → B → ΣA, where A ∈ M and B ∈ N . Let QM ∶ D → D/M and QN ∶ D → D/N be the localization functor from D to its quotients by the subcategories M and N respectively. a) The functor QM ∣N N → D/M is an equivalence of categories. It admits a quasi-inverse whose composition with QM is a left adjoint of the inclusion N → D. b) The functor QN ∣M M → D/N is an equivalence of categories. It admits a quasi-inverse whose composition with QN is a left adjoint of the inclusion M → D. We shall sum up the conclusion of the proposition by saying that the diagrams τN

0 → M → D Ð→ N → 0

and

τM

0 → N → D Ð→ M → 0

3.3. GLUEING TRUNCATION STRUCTURES

99

are exact sequences of triangulated categories. Proof. — The two statements are obtained one from another by passing to the opposite category, so that we only prove the first one. Let us observe that (M , N ) is a truncation structure on D. By corollaries 3.1.7 and 3.1.8, an object X of D belongs to N if and only if D(A, X) = 0 for every object A of M , and an object X of D belongs to M if and only if D(X, B) = 0 for every object B of N . Moreover, the subcategories M and N are thick. Let X, Y be objects of N . By generalizing the argument done in proposition 1.10.10, the localization functor Q ∶ D → D/M induces an isomorphism ∼

N (X, Y) = D(X, Y) Ð → D/M )(Q(X), Q(Y)), so that the functor QM is fully faithful. Let τM ∶ D → M and τN ∶ D → N be the truncation functors associated with this truncation structure. Let X be an object of D; in the canonical distinguished triangle τM X → X → τN X → ΣτM X, the morphism X → τN X induces an isomorphism in D/N , by construction of the localized category, because τM ∈ → → Q(τN X). This proves that the functor QN is M . Consequently, Q(X) Ð essentially surjective. We then observe that the functor τN ∶ D → M is a right adjoint of the inclusion functor M ↪ D. By what precedes, it induces a quasi-inverse of the functor QN . 3.3.2. The general context of glueing. — We fix some notation which will remain in force in the following sections; the motivation for this situation will be explained in example 3.3.3. We assume given three triangulated categories D , DU , DF and six triangulated functors:

→ ← →

i!

j!

← → D ←

→ ← →

DF

i∗ i∗

j∗

j∗

← → DU . ←

We also make the following hypotheses: (i) The two pairs (i ∗ , i∗ ) and (i∗ , i ! ) are adjoint. (ii) The two pairs ( j! , j∗ ) and ( j∗ , j∗ ) are adjoint. In each case, the corresponding units will be denoted by a letter η, the corresponding counits by a letter ε.

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(iii) The functors i∗ , j! and j∗ are fully faithful. Equivalently, the counits ∗ ! ∗ → id and j j∗ → id are isomorphisms, as well as the units id → i i∗ and id → j∗ j! . (iv) One has j∗ i∗ = 0. As a consequence, its left adjoint i ∗ j! = 0, and its right adjoint i ! j∗ = 0. Moreover, for X ∈ DF and Y ∈ DU , one has D( j! Y, i∗ X) = DU (Y, j∗ i∗ X) = 0 and D(i∗ X, j∗ Y) = DU ( j∗ i∗ X, Y) = 0. (v) For every object X ∈ D, there exists distinguished triangles i∗ i

ε

η

ε

η

j ! j∗ X Ð →XÐ → i ∗ i ∗ X → Σ j ! j∗ X and i∗ i ! X Ð →XÐ → j∗ j∗ X → Σi∗ i ! X. By corollary 1.6.7, the unlabeled arrows of these triangles are uniquely determined and these triangles are functorial in X. Observe the symmetry: passing to the opposite categories interchanges i ! and i ∗ on the one hand, and j! and j∗ on the other hand. Example (3.3.3). — The important example of such a glueing context, and the motivation for the notation, comes from topology. Then, D, DU and DF are the derived categories D(Ab(X)), D(Ab(U)) and D(Ab(F)) of the categories of abelian sheaves on a topological space X, an open subset U, and the closed complement subset F = X U. Let i ∶ F → X and j ∶ U → X be the inclusions. (i) The extension by zero functor i! = i∗ ∶ Ab(F) → Ab(X) is exact, and induces a functor, still denoted i∗ , from DF to D. The functor of restriction to F, i ∗ ∶ Ab(X) → Ab(F), is also exact, and induces a triangulated functor i ∗ D → DF . The functor i∗ does not admit a right adjoint at the level of categories of sheaves, but Verdier duality provides a right adjoint i ! ∶ D → DF at the level of derived categories. (ii) Similarly, the extension by zero functor j! ∶ Ab(U) → Ab(X) is exact and fully faithful and induces a functor j! ∶ DU → D. The functor of restriction to U, j∗ ∶ Ab(X) → Ab(U), is exact as well, and induces a functor j∗ D → DU . On the other hand, the functor j∗ ∶ Ab(U) → Ab(X) is only left exact and we denote by j∗ ∶ DU → D its right derived functor. (iii) The full faithfulness of i∗ , j! and j∗ holds at the level of categories of sheaves, it remains true at the level of derived categories.

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(iv) The relation j∗ i∗ = 0 holds at the level of categories sheaves, and remains true at the level of derived categories. (v) Every abelian sheaf F on X gives rise to an exact sequence 0 → j! j∗ F → F → i∗ i ∗ F → 0. Applied to every term of complex C of abelian sheaves on X, these exact sequences furnish an exact sequences of complexes of abelian sheaves, hence, passing to the homotopy category K (Ab(X)), a distinguished triangle j! j∗ C → C → i∗ i ∗ C → Σ j! j∗ C. The second required distinguished triangle is deduced from this one by applying the duality functor. Proposition (3.3.4). — With the hypotheses of §3.3.2, one has the following three exact sequences of triangulated categories: j!

i∗

0 → DF ← ÐD ← Ð DF → 0 i∗

j∗

i!

j∗

0 → DF Ð →D Ð → DU → 0 0 → DF ← ÐD ← Ð DU → 0. Proof. — Since the triangulated functors i∗ ∶ DF → D and j! ∶ DU → D are fully faithful, they induce equivalences of triangulated categories from DF and DU to their images. The hypothesis (iv) and the first distinguished triangle (v) of §3.3.2 allow us to apply proposition 3.3.1 to the triangulated category D and to its pair (i∗ DF , j! DU ) of triangulated subcategories. It furnishes two exact sequences of triangulated categories i∗

j∗

0 → DF Ð →D Ð → DU → 0

and

j!

i∗

0 → DU Ð →D Ð → DF → 0.

Indeed, with the notation of that proposition, the two functors τDU and τDF are induced by the distinguished triangle (v), hence are given by τDU X = j! j∗ X and τDF X = i∗ i ∗ X; finally, we have identified DF and DU as a subcategory of D via the functors j! and i∗ respectively. The same argument applied to the second distinguished triangle of §3.3.2 (v) furnishes two exact sequences of triangulated categories: i∗

j∗

0 → DF Ð →D Ð → DU → 0

and

j∗

i!

0 → DU Ð →D Ð → DF → 0.

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102

The second exact sequence is the one that was missing. Definition (3.3.5). — Let (DU⩽0 , DU⩾1 ) and (DF⩽0 , DF⩾1 ) be truncation structures on DU and DF respectively. Let D ⩽0 and D ⩾1 be the full subcategories of D whose objects are given by (3.3.5.1)

ob(D ⩽0 ) = {X ∈ ob(D) ; j∗ X ∈ DU⩽0 and i ∗ X ∈ DF⩽0 }

(3.3.5.2)

ob(D ⩾1 ) = {X ∈ ob(D) ; j∗ X ∈ DU⩾1 and i ! X ∈ DF⩾1 }.

Theorem (3.3.6). — The pair (D ⩽0 , D ⩾1 ) given by definition 3.3.5 is a truncation structure on D. We say that this truncation structure of D is obtained by glueing the given truncation structures on DU and DF . Proof. — We check the axioms of a truncation structure. a) By construction of the categories D ⩽0 and D ⩾0 , they contain any object of D which is isomorphic to one of their objects. b) Let X ∈ ob(D ⩽0 ) and let Y ∈ ob(D ⩾1 ). Applying the contravariant cohomological functor D(⋅, Y) to the distinguished triangle j! j∗ X → X → i∗ i ∗ X → Σ j! j∗ X, we obtain an exact sequence D(i∗ i ∗ X, Y) → D(X, Y) → D( j! j∗ X, Y). By adjunction of the pair (i∗ , i ! ), one has D(i∗ i ∗ X, Y) ≃ DF (i ∗ X, i ! Y) = 0, since i ∗ X ∈ DF⩽0 and i ! Y ∈ DF⩾1 . Similarly, the pair ( j! , j∗ ) is adjoint, hence D( j! j∗ X, Y) ≃ DU ( j∗ X, j∗ Y) ≃ 0, since j∗ X ∈ DU⩽0 and j∗ Y ∈ DU⩾1 . Consequently, D(X, Y) = 0. c) The inclusions ΣD ⩽0 ⊂ D ⩽0 and Σ−1 D ⩾1 ⊂ D ⩾1 follow from the fact that the functors i ∗ , j∗ and i ! are triangulated. d) Let X be an object of D; let us construct a distinguished triangle A → X → B → ΣA, where A ∈ D ⩽0 and B ∈ D ⩾1 . Let g ∶ X → j∗ τ>0 j∗ X be the unique morphism whose image under the adjunction isomorphism D(X, j∗ τ>0 j∗ X) ≃ DU ( j∗ X, τ>0 j∗ X) is the canonical morphism ε j∗ X ∶ j∗ X → τ>0 j∗ X. We complete g into a distinguished triangle f

g

YÐ →XÐ → j∗ τ>0 j∗ X → ΣY. Similarly, let v ∶ Y → i∗ τ>0 i ∗ Y be the unique morphism whose image under the adjunction isomorphism D(Y, i∗ τ>0 i ∗ Y) ≃ DF (i ∗ Y, τ>0 i ∗ Y) is the canonical

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103

morphism ε i ∗ Y ∶ i ∗ Y → τ>0 i ∗ Y. Let us complete v to a distinguished triangle u v AÐ →YÐ → i∗ τ>0 i ∗ Y → ΣA. Let us then build an octahedron: ← w

←

g

→

→

→ Σi∗ τ>0 i ∗ Y

← v

←

⇐ j∗ τ>0 i ∗ X →

→ u

k

⇐

j∗ τ>0 i ∗ X → ΣY

←

←

ΣA

→ ΣA

←

→ B

←

←

→ X

h

⇐

→

→

f ○u

→ ΣA ⇐

←

←

←

⇐

→ i∗ τ>0 i ∗ Y

f

⇐ A

v

←

→ Y

←

u

←

A

→ Σ2 A.

It suffices to prove that A ∈ D ⩽0 and B ∈ D ⩾0 . Applying the functor j∗ to the second vertical triangle, we obtain a distinguished triangle j∗ (h)

j∗ (k)

j∗ i∗ τ>0 i ∗ Y ÐÐ→ j∗ B ÐÐ→ j∗ j∗ τ>0 j∗ X → Σ j∗ i∗ τ>0 i ∗ Y. Since j∗ i∗ = 0, the morphism j∗ (k) is an isomorphism. Composed with the ∼ counit j∗ j∗ Ð → id (which is an isomorphism, because j∗ is fully faithful), we τ obtain an isomorphism ε ○ j∗ (k) ∶ j∗ B Ð →>0 j∗ X. In particular, j∗ B ∈ DU⩾1 . Similarly, applying the functor i ! to this second vertical triangle, we obtain the distinguished triangle !

i ! (h)

!

i ! (k)

i i∗ τ>0 i Y ÐÐ→ i B ÐÐ→ i ! j∗ τ>0 i ! X → Σi ! i∗ τ>0 i ∗ Y. ∗

Since i ! j∗ = 0, the morphism i ! (h) is an isomorphism. Composed with the unit ∼ id Ð → i ! i∗ (which is an isomorphism, because i∗ is fully faithful), this shows that τ>0 i ∗ Y ≃ i ! B, hence i ! B ∈ DF⩾1 . Let us now apply j∗ to the second horizontal triangle; we obtain a distinguished triangle j∗ ( f ○u)

j∗ (w)

j A ÐÐÐ→ j X ÐÐ→ j∗ B → Σ j∗ A. ∗

∗

Observe that ε ○ j∗ (k) ○ j∗ (w) = ε ○ j∗ (k ○ w) = ε ○ j∗ (g) = ε j∗ X . Consequently, we have a distinguished triangle j∗ ( f ○u)

ε j∗ X

j A ÐÐÐ→ j∗ X ÐÐ→ τ>0 j∗ X → Σ j∗ A. ∗

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104

Equivalently, the morphism j∗ ( f ○ u) factors uniquely through an isomorphism ∼ j∗ A Ð → τ⩽0 j∗ X. In particular, j∗ A ∈ DU⩽0 . Let us apply i ∗ to the first horizontal triangle; this furnishes a distinguished triangle i ∗ (u)

i ∗ (v)

i A ÐÐ→ i Y ÐÐ→ i ∗ i∗ τ>0 i ∗ Y → Σi ∗ A, ∗

∗

∼

hence, making use of the counit i ∗ i∗ Ð → id (which, again, is an isomorphism because i∗ is fully faithful), a distinguished triangle i ∗ (u)

εi∗ Y

i A ÐÐ→ i ∗ Y ÐÐ→ τ>0 i ∗ Y → Σi ∗ A. ∗

Consequently, the morphism i ∗ (u) factors uniquely through an isomorphism ∼ i ∗A Ð → τ⩽0 i ∗ Y. In particular, i ∗ ∈ DF⩽0 . We thus have proved that A ∈ D ⩽0 and B ∈ D ⩾0 , as was to be shown. Consequently, (D ⩽0 , D ⩾1 ) is a truncation structure on D. Proposition (3.3.7). — a) The functors j! and i ∗ are right t-exact; b) The functors j∗ and i∗ are t-exact; c) The functors j∗ and i ! are left t-exact. Proof. — If X ∈ D ⩽0 , then j∗ X ∈ DU⩽0 and i ∗ X ∈ DF⩽0 ; consequently, j∗ and i ∗ are right t-exact. If X ∈ D ⩾1 , then j∗ X ∈ DU⩾1 and i ! X ∈ DF⩾1 ; consequently, j∗ and i ! are left t-exact. By proposition 3.2.9, the functor j! is right t-exact and the functor j∗ is left t-exact, because j! is a left adjoint of j∗ , and j∗ is a right adjoint of j∗ . Since it is a right adjoint of i ∗ and a left adjoint of i ! , proposition 3.2.9 implies that the functor i∗ is t-exact. Corollary (3.3.8). — Let CU and CF denote the hearts of the given truncation structures on DU and DF , and let C be the heart of the truncation structure on D which is obtained by glueing. For each of the six functors F ∈ {i∗ , i ! , i ∗ , j∗ , j! , j∗ }, we let F˜ = H0 ○ F the corresponding functors between the hearts:

˜ı !

˜ȷ !

← → C ←

→ ← →

→ ← →

CF

˜ı ∗ ˜ı∗

˜ȷ∗ ˜ȷ∗

← → CU . ←

3.3. GLUEING TRUNCATION STRUCTURES

105

a) The adjoint pairs ( j! , j∗ ), ( j∗ , j∗ ), (i ∗ , i∗ ) and (i∗ , i ! ) give rise to adjoint pairs ( ˜ȷ! , ˜ȷ∗ ), ( ˜ȷ∗ , ˜ȷ∗ ), (˜ı ∗ , ˜ı∗ ) and (˜ı∗ , ˜ı ! ). Moreover, the functors ˜ı∗ , ˜ȷ! and ˜ı ! are fully faithful. b) One has ˜ȷ∗ ○ ˜ı∗ = 0, ˜ı ∗ ○ ˜ȷ! = 0 and ˜ı ! ○ ˜ȷ∗ = 0. For every object X ∈ CF and every object Y ∈ CU , one has C (˜ı∗ X, ˜ȷ∗ Y) = C ( ˜ȷ! Y, ˜ı∗ X) = 0. c) For every object X ∈ C , there are exact sequences ˜ȷ! ˜ȷ∗ X → X → ˜ı∗ ˜ı ∗ X → 0

and

0 → ˜ı∗ ˜ı ! X → X → ˜ȷ∗ ˜ȷ∗ X.

Proof. — a) The first part follows from proposition 3.2.9. Let X, Y ∈ CU . One has bifunctorial isomorphisms C ( ˜ȷ! X, ˜ȷ! Y) ≃ D(τ⩾0 j! X, τ⩾0 j! Y)

( j! is right t-exact)

≃ D( j! X, τ⩾0 j! Y)

(by adjunction of (τ⩾0 , ⋅))

≃ DU (X, j∗ τ⩾0 j! Y)

(by adjunction of ( j! , j∗ ))

≃ DU (X, τ⩾0 j∗ j! Y)

(by t-exactness of j∗ )

≃ DU (X, τ⩾0 Y)

(because id ≃ j∗ j! )

≃ D(X, Y) ≃ C (X, Y), under which a morphism f ∈ C (X, Y) corresponds with ˜ȷ! f ∈ C ( ˜ȷ! X, ˜ȷ! Y). This proves that the functor ˜ȷ! is fully faithful. One proves similarly that the functor ˜ȷ∗ is fully faithful. Let then X, Y ∈ CF . Since the functor i∗ is t-exact, its restriction to CF coincides with the functor ˜ı∗ . Since i∗ is fully faithful, the functor ˜ı∗ is fully faithful as well. b) The three equalities follow from corollary 3.2.8 and the fact that j∗ and i∗ are both left (or right) t-exact, that i ∗ and j! are both right t-exact, and that j∗ and i ! are both left t-exact. c) Let X ∈ C and let us apply the functor H0 to the canonical distinguished triangle j! j∗ X → X → i∗ i ∗ X → Σ j! j∗ X. One obtains an exact sequence H0 ( j! j∗ X) → H0 (X) → H0 (i∗ i ∗ X) → H0 (Σ j! j∗ X). Since X ∈ C , one has H0 (X) = X. Since j∗ and j! are both right t-exact, one has H0 ( j! j∗ X) = ˜ȷ! ˜ȷ∗ X. Since i∗ and i ∗ are both right t-exact, one has H0 (i∗ i ∗ X) = ˜ı∗ ˜ı ∗ X. Finally, ΣX ∈ D ⩽−1 ; since j! and j∗ are both right t-exact, this implies

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CHAPTER 3. TRUNCATION STRUCTURES

j! j∗ X ∈ D ⩽0 , hence Σ j! j∗ X ∈ D ⩽−1 and H0 (Σ j! j∗ X) = 0. This furnishes the desired exact sequence ˜ȷ! ˜ȷ∗ X → X → ˜ı∗ ˜ı ∗ X → 0. The second exact sequence is established similarly, by applying the functor H0 to the distinguished triangle i∗ i ! X → X → j∗ j∗ X → Σi∗ i ! X. Proposition (3.3.9). — The truncation structure on D is nondegenerate if and only if the given truncation structures on DU and DF are nondegenerate. Proof. — Let us assume that the given truncation structures on DU and DF are nondegenerate, and let us prove that the truncation structure (D ⩽0 , D ⩾1 ) is nondegenerate as well. Let X ∈ ⋂ D ⩽n . Consequently, j∗ X ∈ DU⩽n and i ∗ X ∈ DF⩽n for every integer n, so that j∗ X = 0 and i ∗ X = 0. The distinguished triangle j! j∗ X → X → i∗ i ∗ X → Σ j! j∗ X then proves that X = 0. Similarly, let X ∈ ⋂n D ⩾n . This implies that j∗ X ∈ ⋂n DU⩾n and i ! X ∈ ⋂n DF⩾n , so that j∗ X = 0 and i ! X = 0. The distinguished triangle i∗ i ! X → X → j∗ j∗ X → Σi∗ i ! X then proves that X = 0. Conversely, let us assume that the truncation structure on D is nondegenerate. It follows from the fact that the functor i∗ is t-exact and fully faithful that the truncation structure on DF is nondegenerate. Let indeed X ∈ ⋂ DF⩽n . Since i∗ is t-exact, one has i∗ X ∈ DF⩽n , hence i∗ X = 0. Since X ≃ i ∗ i∗ X, one has X = 0. Let then X ∈ ⋂ DF⩾n . Since i∗ is t-exact, one has i∗ X ∈ DF⩾n , hence i∗ X = 0. Since X ≃ i ∗ i∗ X, one has X = 0. This proves that the truncation structure on DF is nondegenerate, as claimed. Using that the functor j! is right t-exact and fully faithful, one proves that ⋂ DU⩽n = 0. Using that the functor j∗ is left t-exact and fully faithful, one proves that ⋂ DU⩾n = 0. This shows that the truncation structure on DF is nondegenerate. Example (3.3.10). — Let us start with a given truncation structure (DF , DF⩾1 ) on DF but with the degenerate truncation structure (DU , 0) on DU and let us consider the resulting truncation structure on D. F ∶ D → D ⩽0 be the corresponding truncation functor, right adjoint to Let τ⩽0 the inclusion of the subcategory D ⩽0 whose objects X are characterized by the condition i ∗ X ∈ DF⩽0 .

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107

Let us go back to the proof of theorem 3.3.6, especially part d). With the notation of that proof, we have τ > 0 j∗ X = 0 (because of the choice of the F X. Also, h truncation structure on DU ), hence Y = X, f = id; moreover, A = τ⩽0 is an isomorphism (two out of three arrows defining the morphism of horizontal triangles are isomorphisms), so that B ≃ i∗ τ>0 i ∗ Y. Then, the second horizontal distinguished triangle of the octahedron furnishes the canonical truncation triangle of X: F F τ⩽0 X → X → i∗ τ>0 i ∗ X → Στ⩽0 X. Finally, for every X ∈ D, one has F F H0 (X) = τ⩽0 τ⩾0 X = τ⩽0 i∗ τ⩾0 i ∗ X = i∗ H0 i ∗ X,

since i∗ is t-exact. Example (3.3.11). — Still starting with the truncation structure (DF⩽0 , DF⩾1 ) on DF , let us consider the degenerate truncation structure (0, DU ) on DU . In that case, an object X of D belongs to D ⩾0 if and only if i ! X ∈ DF⩾0 . Let us F the left adjoint of the inclusion functor D ⩾1 → D. For every obdenote by τ⩾0 ject X, the canonical truncation triangle of X relative to this truncation structure writes F i∗ τ⩽0 i ! X → X → τ⩾1 X → Σi∗ τ⩽0 i ! X, and the cohomological functor is computed as H0 X = i∗ H0 i ! X. Example (3.3.12). — Let us now start with the degenerate truncation structure (DF , 0) on DF and with the given truncation structure (DU⩽0 , DU⩾0 ) on DU . An U object X of D belongs to D ⩽0 if and only if j∗ X ∈ DU⩽0 . We denote by τ⩽0 be ⩽0 the associated truncation functor, right adjoint of the inclusion D → D. The canonical truncation triangle associated with an object X writes U U τ⩽0 X → X → j∗ τ>0 j∗ X → Στ⩽0 X,

and the cohomological functor is given by H0 X = j∗ H0 j∗ X. Example (3.3.13). — Finally, we start with the degenerate truncation structure (0, DF ) on DF and with the given truncation structure (DU⩽0 , DU⩾0 ) on DU . An U object X of D belongs to D ⩾1 if and only if j∗ X ∈ DU⩾1 . Let us denote by τ⩾1 the associated truncation functor, left adjoint of the inclusion D ⩾1 → D. For every object X of D, the canonical truncation functor writes U j! τ⩽0 j∗ X → X → τ⩾1 X → Σ j! τ⩽0 j∗ X,

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CHAPTER 3. TRUNCATION STRUCTURES

and the cohomological functor is given by H0 X = j! H0 j∗ X. Remark (3.3.14). — The four truncation structures on D described in examples 3.3.10, 3.3.11, 3.3.12 and 3.3.13 can be used to describe the truncation structure given by theorem 3.3.6. Indeed, one has the formulas F U τ⩽0 = τ⩽0 τ⩽0

F U τ⩾0 = τ⩾0 τ⩾0 .

and

To prove the first formula, let us go back to the octahedron drawn in the course of the proof of theorem 3.3.6. In that diagram, the first vertical distinguished triangle identifies with the canonical truncation triangle associated with the U F X= truncation structure of example 3.3.12, so that Y = τ⩽0 X. Then τ⩽0 X = A = τ⩽0 F τ U X. τ⩽0 ⩽0 The other formula is deduced from that one by passing to the opposite categories (and exchanging i ! and i ∗ on the one hand, and j∗ and j! on the other hand).

3.4. Extensions We retain the notation of the previous section, as given in §3.3.2. Definition (3.4.1). — Let Y ∈ DU . An extension of Y is an object X of D endowed ∼ with an isomorphism u ∶ j∗ X Ð → Y. Let (X, u) be an extension of Y. By the adjunction ( j∗ , j∗ ), the datum of the morphism u is equivalent with that of a morphism u ♯ ∈ D(X, j∗ Y). On the other hand, the morphism u −1 ∶ Y → j∗ X corresponds with a morphism u ♭ ∈ D( j! Y, X) under the adjunction ( j! , j∗ ). This furnishes a diagram u♭

u♯

j! Y Ð →XÐ → j∗ Y whose composition is the element of D( j! Y, j∗ Y) ≃ DU (Y, j∗ j∗ Y) ≃ DU (Y, Y) corresponding to idY . After translation and identifying j∗ X with Y, the canonical distinguished triangle i∗ i ! X → X → j∗ j∗ X → Σi∗ i ! X furnishes a distinguished triangle: (3.4.1.1)

u♯

XÐ → j∗ Y → i∗ i ! ΣX → ΣX.

3.4. EXTENSIONS

109

Applying the functor i ∗ to this triangle and using the isomorphism of functors ∼ i ∗ i∗ Ð → id, we obtain another distinguished triangle: (3.4.1.2)

i ∗ (u ♯ )

i ∗ X ÐÐÐ→ i ∗ j∗ Y → i ! ΣX → Σi ∗ X.

←

←

3.4.2. — If (X, u) and (X′ , u′ ) are two extensions of Y, a morphism of extensions from X to X′ is an element f ∈ D(X, X′ ) that makes the following diagram commutative j∗ X u

→

j∗ ( f )

Y,

→

→

←

u′

j ∗ X′

in other words, such that u ′ ○ j∗ ( f ) = u. This furnishes the commutative diagram ←

X

→

← j! Y

←

u♭

u♯

→ j∗ Y

f

X′

←

←

→

→

u′ ♭

→ u′ ♯

Example (3.4.3). — Let Y be an object of DU . The object j! Y, endowed with the ∼ → j∗ j! Y, is an extension of Y. The object j∗ Y, endowed with isomorphism ε ∶ Y Ð ∼ the isomorphism η ∶ j∗ j∗ Y Ð → Y, is an extension of Y. Moreover, the canonical morphism j! Y → j∗ Y is a morphism of extensions. Example (3.4.4). — Let Y be an object of DU , let p be an integer and let X = F j Y. τ⩾p ! Let us apply the functor j∗ to the canonical distinguished triangle i∗ τ

F j Y → 0, so that j∗ (v) is an isomorphism. Letting u = j∗ (v) ○ η j∗ j! Y ÐÐ→ j∗ τ⩾p ! ∼ → j∗ j! Y, this furnishes an extension be its composition with the unit η ∶ Y Ð (X, u) of Y. More precisely, let (X′ , u ′ ) be another extension of Y such that X is isomorphic to X′ , as objects of D. Let us show that there exists at most one isomorphism f ∶ X → X′ such that u = u ′ ○ j∗ ( f ), that is an isomorphism of extensions. Let us

CHAPTER 3. TRUNCATION STRUCTURES

110

complete the morphism u ′ ♭ ∶ j! Y → X to a distinguished triangle Z → u ′ ♭ ∶ j! Y → X → ΣZ and consider a partial morphism of distinguished triangles:

→ X′

←

← ←

←

⇐

←

←

f

u′ ♭

→ Σi∗ τ

←

→ j! Y

→ X →

→ ←

Z

u♭

⇐

g

→ j! Y

←

i∗ τ

Σg

→ ΣZ.

Relative to the truncation structure on D of example 3.3.13, the object i∗ τ

p . τ

p ! F F Proof. — Let us prove that the following properties are equivalent: (i) i ∗ X ∈ DF and i ! X ∈ DF ; (ii) i ! X = τ>p (i ∗ j∗ Y); (iii) i ∗ X = τ

p j! Y;
>p

where, in (ii) and (iii), the equalities are assumed to be induced by the morphisms of the distinguished triangle (3.4.1.2). Let u ∶ j∗ X → Y be the canonical morphism; let u ♭ ∶ j! Y → X and u ♯ ∶ X → j∗ Y. be the morphisms deduced by adjunction. If condition (i) holds, then the distinguished triangle (3.4.1.2) writes i ∗ j∗ Y as

p i ∗ j∗ Y. This shows the implications (i)⇒(ii) and (i)⇒(iii).

3.4. EXTENSIONS

111

(ii) implies that i ! X ∈ DF and that τ>p i ∗ X = 0, hence (i). Similarly, (iii)

p

Proposition (3.4.7). — The functor ˜ȷ∗ ∶ C → CU identifies the abelian category CU with the quotient of the abelian category C by the essential image C F of the functor ˜ı∗ . For every object X of C , ˜ı∗ ˜ı ∗ X is the largest quotient of X that belongs to C F , and ˜ı∗ ˜ı ! X is the largest subobject of X that belongs to C F .

3.4.8. — Let Y ∈ CU . Since the functor j! is right t-exact, one has j! Y ∈ D ⩽0 , and ˜ȷ! Y = τ⩾0 j! Y. Since the functor j∗ is left t-exact, one has ˜ȷ∗ Y = τ⩽0 j∗ Y. ˜ is the Moreover, there exists a unique morphism u˜ ∶ ˜ȷ! Y → ˜ȷ∗ Y such that ˜ȷ∗ (u) composition of the counit ˜ȷ∗ ˜ȷ∗ → id and the unit id → ˜ȷ∗ ˜ȷ! associated with the adjoint pairs ( ˜ȷ∗ , ˜ȷ∗ ) and ( ˜ȷ! , ˜ȷ∗ ). This leads to a canonical diagram ←

→ ˜ȷ! Y

u˜

→ ˜ȷ∗ Y

←

← ←

j! Y

→ j∗ Y, →

u

where u ∶ j! Y → j∗ Y is the canonical morphism. Definition (3.4.9). — Let Y ∈ CU . One defines ˜ȷ!∗ = Im( ˜ȷ! Y → ˜ȷ∗ Y). It is called the middle extension of Y.

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112

The middle extension fits naturally in a diagram → ˜ȷ!∗ Y

←

←

→ ˜ȷ! Y

→ ˜ȷ∗ Y →

←

← ←

j! Y

→ j∗ Y.

u

When one applies the functor j∗ to this diagram, the first term j∗ j! Y and the last term j∗ j∗ Y are isomorphic to Y via the unit of the adjunction ( j! , j∗ ) and the counit of the adjunction ( j∗ , j∗ ) respectively, and all morphisms are isomorphisms. In this way, ˜ȷ!∗ is naturally an extension of Y. Proposition (3.4.10). — Let Y ∈ CU . One has the following relations: F j Y = τ F j Y is the only extension of Y such that i ∗ X ∈ D <−1 a) X = ˜ȷ! Y = τ>−1 ! <−1 ∗ F >−1 ! and i X ∈ DF ; F j Y = τ F j Y; is the only extension of Y such that i ∗ X ∈ D <0 b) X = ˜ȷ!∗ Y = τ>0 ! <0 ∗ F >0 ! and i X ∈ DF ; F j Y = τ F j Y is the only extension of Y such that i ∗ X ∈ D <1 c) X = ˜ȷ∗ Y = τ>1 ! <1 ∗ F >1 ! and i X ∈ DF .

Proof. — All these assertions follow from theorem 2.2.10, except for the identification of ˜ȷ! Y, ˜ȷ!∗ Y and ˜ȷ∗ Y with the indicated extensions. F τ U j Y = τ F j Y. One has j∗ j! Y ≃ Y ∈ CU ; consequently, ˜ȷ! Y = τ⩾0 j! Y = τ⩾0 ⩾0 ! ⩾0 ! U F ∗ a) c) One has j j∗ Y ≃ Y ∈ CU ; consequently, ˜ȷ∗ Y = τ⩽0 j∗ Y = τ⩽0 τ⩽0 j∗ Y = F j Y. τ⩽0 ∗ F j Y = τ F j Y; let us first show that X ∈ C . Since i ∗ X ∈ D <0 and b) Let X = τ>0 ! <0 ∗ F ∗ j X ≃ Y ∈ CU , one has X ∈ D ⩽0 ; since i ! X ∈ DF>0 and j∗ X ≃ Y ∈ CU , one has X ∈ D ⩾0 ; consequently, X ∈ C , as claimed. The cohomology functor associated with the truncation structure of examF j Y → τF j Y ple 3.3.10 is i∗ H0 i ∗ . Consequently, the canonical morphism τ<−1 ∗ <0 ∗ can be completed in a distinguished triangle F F F τ<−1 j∗ Y → τ<0 j∗ Y → i∗ H−1 i ∗ j∗ Y → Στ<−1 j∗ Y,

or, after rotation: i∗ H−1 i ∗ j∗ Y → ˜ȷ! Y → X → Σi∗ H−1 i ∗ j∗ Y. Since the three vertices of this triangle belong to C , the diagram 0 → i∗ H−1 i ∗ j∗ Y → ˜ȷ! Y → X → 0

3.4. EXTENSIONS

113

is an exact sequence. F j Y → τ F j Y furnishes an exact seBy duality, the canonical morphism τ>−1 ! >1 ! quence 0 → X → ˜ȷ∗ Y → i∗ H1 i ! j! Y → 0. This proves that X is the image of the canonical morphism ˜ȷ! Y → ˜ȷ∗ Y, hence X = ˜ȷ!∗ Y, as was to be shown. Corollary (3.4.11). — Let Y ∈ CU . Then X = ˜ȷ!∗ Y is the unique extension of Y which as no non-trivial subobject and no non-trivial quotient in C F . Corollary (3.4.12). — The simple objects of the category C are the objects ˜ȷ!∗ S, for S ∈ CU simple, and the objects ˜ı∗ T, for T ∈ CF simple.

CHAPTER 4 PERVERSE SHEAVES

BIBLIOGRAPHY

L. Alonso Tarrío, A. Jeremías López & M. J. Souto Salorio (2000), “Localization in categories of complexes and unbounded resolutions”. Canad. J. Math., 52 (2), pp. 225–247. A. A. Be˘ılinson, J. Bernstein & P. Deligne (1982), “Faisceaux pervers”. Analyse et topologie sur les espaces singuliers, I (Luminy, 1981), Astérisque 100, pp. 5–171, Soc. Math. France, Paris. A. Borel (editor) (1984), Intersection cohomology, Progress in Mathematics 50, Birkhäuser Boston, Inc., Boston, MA. Notes on the seminar held at the University of Bern, Bern, 1983, Swiss Seminars. N. Bourbaki (2016), Éléments de mathématique. Topologie algébrique. Chapitres 1 à 4., Springer-Verlag. J.-L. Brylinski (1982), “(Co)-homologie d’intersection et faisceaux pervers”. Bourbaki Seminar, Vol. 1981/1982, Astérisque 92, pp. 129–157, Soc. Math. France, Paris. S. I. Gelfand & Y. I. Manin (2003), Methods of homological algebra, Springer Monographs in Mathematics, Springer-Verlag, Berlin, second edition. R. Godement (1973), Topologie algébrique et théorie des faisceaux, Hermann, Paris. Troisième édition revue et corrigée, Publications de l’Institut de Mathématique de l’Université de Strasbourg, XIII, Actualités Scientifiques et Industrielles, No. 1252. M. Goresky & R. MacPherson (1980), “Intersection homology theory”. Topology, 19 (2), pp. 135–162. M. Goresky & R. MacPherson (1983), “Intersection homology. II”. Invent. Math., 72 (1), pp. 77–129. M. Goresky & R. MacPherson (1988), Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 14, Springer-Verlag, Berlin. A. Grothendieck (1957), “Sur quelques points d’algèbre homologique”. Tôhoku Math. J. (2), 9, pp. 119–221.

118

BIBLIOGRAPHY

R. Hartshorne (1966), Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20, Springer-Verlag, BerlinNew York. R. Hotta, K. Takeuchi & T. Tanisaki (2008), D-modules, perverse sheaves, and representation theory, Progress in Mathematics 236, Birkhäuser Boston, Inc., Boston, MA. Translated from the 1995 Japanese edition by Takeuchi. M. Kashiwara & P. Schapira (1994), Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 292, Springer-Verlag, Berlin. With a chapter in French by Christian Houzel, Corrected reprint of the 1990 original. M. Kashiwara & P. Schapira (2006), Categories and sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 332, Springer-Verlag, Berlin. J. P. May (2001), “The additivity of traces in triangulated categories”. Adv. Math., 163 (1), pp. 34–73. A. Neeman (1990), “The derived category of an exact category”. J. Algebra, 135 (2), pp. 388–394. C. A. M. Peters & J. H. M. Steenbrink (2008), Mixed Hodge structures, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 52, Springer-Verlag, Berlin. C. Serpé (2003), “Resolution of unbounded complexes in Grothendieck categories”. J. Pure Appl. Algebra, 177 (1), pp. 103–112. N. Spaltenstein (1988), “Resolutions of unbounded complexes”. Compositio Math., 65 (2), pp. 121–154. J.-L. Verdier (1995), “Dualité dans la cohomologie des espaces localement compacts”. Séminaire Bourbaki, Vol. 9, pp. Exp. No. 300, 337–349, Soc. Math. France, Paris. J.-L. Verdier (1996), “Des catégories dérivées des catégories abéliennes”. Astérisque, (239), pp. xii+253 pp. (1997). With a preface by Luc Illusie, Edited and with a note by Georges Maltsiniotis.

INDEX

middle —, 111

C category, 2 abelian —, 8 additive —, 8 opposite —, 3 semi-additive —, 7 cocone on a diagram, 5 coequalizer, 7 coimage of a morphism, 10 cokernel of a morphism, 8 colimit of a diagram, 6 complex homotopically injective, 61 complex in an additive category, 12 cone of a morphism of complexes, 13 cone on a diagram, 5 coproduct in a category, 6

F functor, 4 exact —, 64 right derived —, 64 triangulated —, 41 G glueing truncation structures, 102 H heart, 91 homotopy, 13 homotopy category of an additive category, 14, 36

D derived category of an abelian category, 59 diagram in a category, 5 differential of a complex, 12

I image of a morphism, 10 inverse, 2 K kernel of a morphism, 8

E epimorphism, 3 equalizer, 7 extension, 108

L limit of a diagram, 5

119

120

M map proper continuous map —, 69 separated continuous —, 69 monomorphism, 2 morphism in a category, 2 invertible, 2 morphism of complexes, 12 homotopic —, 14 morphism of complexes null homotopic, 14

INDEX

R resolution homotopically injective —, 64 S sheaf soft —, 80 shift of a complex, 12 subcategory thick —, 41 triangulated —, 41

O object terminal —, 6 zero —, 7 object in a category, 2 origin of a morphism, 2

T target of a morphism, 2 truncation functor, 89 truncation of a complex, 60 truncation structure, 85

P product in a category, 6

U universe, 1

Q quiver, 5

Z zero object, 7