The Grothendieck Festschrift, Volume III: A Collection of Articles Written in Honor of the 60th Birthday of Alexander Grothendieck (Progress in Mathematics Modern Birkhäuser Classics)

0 and any sheaf F, we are done. For the rest of this section, let A be a regular local ring which is the localization of an algebra of finite type over a field K of characteristic p > 0. Let Ki(A) be the group H?t(A,T(2)) and let %\)A be the natural map from A'2(^4) t ° A'2(yl). Our eventual goal is to show that tpA 1S a n isomorphism. L e m m a 2.4. ip^ has torsion kernel and cokernel. Proof. We recall from [L2] that T(2) is defined as a two-term complex of etale sheaves C 2) i —• C 2)2 , corresponding to a map of presheaves C2,i(U) —• C2)2(f7) such that if U is equal to Specyl, Coker
- ^ -

I 0

C2>2(A)

1 P

# ( A C 2 ) 1 ) — —+

H°(A,Cit2)

that I<2(A) = Coker
S. LICHTENBAUM

42

• H2(A,T(2))

H\A,K3)

•

*I>A

K2(A)

H°(A,K2) !• K2(A) .

=

Since 6 is an isomorphism up to torsion by Lemma 2.2 and Hl{A^K^)

is

torsion by Lemma 2.3, ipA is an isomorphism up to torsion. L e m m a 2.5. Let £ be a prime different from p. Then the map induced by 4>A from K2(A)i to K2(A)t is surjective. Proof. Let F be the quotient field of A. Then we have the following commutative diagram with exact rows: 0

•

K2(A)t

•

n

K2(F)€

Vi(Hx))

I , C o d l := 1 <*F

0

• Hlt(A,in®»e)

n

> Hl(F, fit ® fit)

1-

ar,codx= 1

where the top row comes from Gersten's conjecture for A and the bottom row is the Bloch-Ogus sequence [BO]. The map ap comes from the Rummer sequence for T(2, F) : Hl(F,nt ® /i/) -* H2(F, T(2))t = K2{F)t and the commutativity is by direct computation if F contains a primitive ^-th root of unity and immediately reduces to that case, if not. This diagram then defines a unique map a : H\t(A,pi ® pi) —* K2{A)i making the diagram commute. In order to prove the surjectivity, it suffices to show that the diagram: H\t(A,in®in)

——•

K2(A)i

i Hlt(Aifn®fit)

•

K2(A)t

commutes, since the surjectivity of (3 follows from the Rummer sequence for T(2) ([L2], Corollary 8.3). Essentially the same argument as in the proof of Theorem 4.5 in [L2] now shows the commutativity, and completes the proof. Corollary 2.6. The natural map tpA from K2(A) to K2(A) is an isomorphism up to p-torsion. Proof. Theorem 9.1 of [L2] implies that K 2(A)/£K 2(A) is isomorphic to K2(A)/£K2(A). Now Lemma 4.6 of [L2] completes the proof, in view of Lemmas 2.4 and 2.5.

WEIGHT-TWO MOTIVIC COHOMOLOGY

43

Let Y be a scheme over the prime field Fp. T h e differential symbol induces a map (p from K2yy to fty/l °^ ^ a l e or Zariski sheaves on Y. Let i/2>y be the image of (p. In view of ([I] T h m . 02.4.2), if Y = S p e c F , Theorem 0.3 of the introduction implies t h a t K2F jpK2^ — V2,F in either the etale or Zariski sites. Now assume t h a t Y is regular and of finite type over a field. L e m m a 2 . 7 . There is a distinguished

triangle (up to

r(2,Y) - ^ r(2,y) - i * ( 2 ) [ - 2 ] -

nl

2-torsion)

r(2,y)[i].

P r o o f . Since the only non-vanishing homology sheaves of T(2, Y ) are = gT2I<3(Y) (up to 2-torsion) and 7i2 = A" 2 (Y) ([L2], Corollary 8.4)

it suffices to show t h a t multiplication by p is an isomorphism o n W 1 and t h a t there is a short exact sequence 0-+?{2

-^->n2

->i*(2)->0.

Let B be any strictly Hensel local ring on Y and let F be the quotient field of B. Lemma 2.1 and Theorem 0.2 show t h a t multiplication by p is an isomorphism on H1. T h e Gersten conjecture for A gives rise to the commutative diagram 0

• K2(A)/p

•

vA{2)

K2(F)/p

v i/ F (2) .

Since f3 is an isomorphism by Theorem 0.3, and since a is surjective by definition, a is an isomorphism. Since K2{F) has no p-torsion by [Su] neither does K2(A), which completes the proof. L e m m a 2.8. (i) The natural map from K2(A)p (ii) The natural map from K2(A)/p (iii) H*(A,T(2))P) = 0.

to K2(A)p is surjective. to K2(A)/p is bijective.

P r o o f . Lemma 2.7 implies K2(A)P = H2(A, T(2))p = 0, which proves (i). T h e homology sequence of the triangle in Lemma 2.7 implies also that 0-

K2(A)/p

-2U H°(A,uA(2))

-

is exact. But now the commutative diagram

H3(X,

T(2)) p -

0

S. LICHTENBAUM

44

K2(A)/p

•

!•

I 0

VA(2)

• K2(A)/p

—^-> ^ ( 2 )

tf3(X,r(2))p

•

shows that a is an isomorphism and H3(X,T(2))P

• 0

= 0.

P r o p o s i t i o n 2.9. 6 A is an isomorphism. Proof. Since / ^ ( F ) has no p-torsion, Gersten's conjecture implies K2{A)P = 0, so Corollary 2.6 implies ^ is injective. The obvious variant of Lemma 4.6 of [L2], Lemmas 2.7 and 2.8 and Corollary 2.6 imply that 6A is surjective. Still retaining the notations of [L2], if A is a regular ring, let D(A) = Coker (C2,i(^4) —• C ^ C ^ ) ) , so Corollary 1.6 of [L2] yields a natural injection of D(A) into K2{A), If all residue fields of A have at least three elements, then Lemma 1.7 of [L2] shows that the induced map on Zariski sheaves on Spec A from D to K2 is an isomorphism. On the other hand, if

then D(A) c l e a r ly maps to Coker<£>, which by the hyperco-

homology spectral sequence maps to H?t(A, T(2)). Now applying Proposition 2.9, we have T h e o r e m 2.10. Let X be a regular scheme of finite type over a field ^ 2*V Then the natural map from J^.zar to R2a*T(2,X) of Zariski sheaves is an isomorphism. P r o p o s i t i o n 2.11. Let X be a connected, regular scheme of finite type over a field. Then R1a^T(2JX) = gr^K3, which is naturally isomorphic to the constant sheaf whose stalks are all gr2K3(F(X)),

up to 2-torsion.

Proof. By looking at stalks, it suffices to show that if A is local, H°(A,giyK3) —• i7°(F,gr^A r 3 ) is bijective up to 2-torsion, where F = F(X) is the quotient field of A. But we have the commutative diagram

H°(A,tfK3)

—^

'I " p*K3{A)

H°{F,^K3)

I' —?U

&*K3(F)

.

We know that a is an isomorphism up to 2-torsion by Lemma 2.1, /? is an isomorphism by Theorem 0.1, and 6 is injective up to 2-torsion again by Lemma 2.1 and Theorem 0.1.

WEIGHT-TWO MOTIVIC COHOMOLOGY

45

P r o p o s i t i o n 2.12. R3 a+T(2,X) = 0 up to 2-torsion. Proof. This follows immediately from [L2], Corollary 9.7 and Lemma 2.8 (iii) of this paper. T h e o r e m 2.13. Up to 2-torsion, we have: (a) HUXJ<2) = Hl{X,T{2)) (b)

HUX,K2)~Hlt(X,T(2))

(c) There is a natural injection from CH2(X)

= Hl^X^K^)

to

i7| t (X,r(2)) with torsion cokernel. Proof. Consider the spectral sequence H!ar(X,R
Hlt+\X,T(2)).

=*

Ignoring 2-torsion, we have Rqa*T(2) = 0 for q = 0 and 3, and since Rxa*T(2) is constant, Hgw(X,R1 a*T(2)) = 0 for p > 0. Also, R2a*T(2) is /^2,zar- Parts (a) and (b) of the theorem then follow immediately, and we obtain an exact sequence (up to 2-torsion) 0 —• H^^X, Hl{X,V{2))

-

4

tf°ar(*,fl «*r(2))

-

HlT(X,K2).

K^) —• 4

Since i ? a i ( 2 ) is

easily seen to be torsion, part (c) follows. Remark 2.14. A consideration of the Gersten resolution of Ki shows that #£ a r (X, K2) = 0 for p > 3, hence the above map from H%[Xy T(2)) to H^r(X,R4a+T(2))

is surjective.

T h e o r e m 2.15. Let X be a smooth projective variety over C. The image of H*t(X, T(2)) contains all torsion classes in H*t(X, Tt{2)). Proof. We start with the Kummer sequence 0 - Hi(X,T(2))/en

-* H^X^f?)

-

tfft(X,r(2))„.

- 0

and apply the inverse limit functor, which is exact since the groups are finite, to obtain 0

~> # W ( 2 ) ) ~> #ft(*.Z/(2)) - TtiHfoX.TW)

- 0.

Since Ti is torsion-free, any torsion in Hft(X, Z^(2)) comes from H

tw(2)y

Let

B(X)

be the image d ^ t ( X , r ( 2 ) ) in

ff«(X,Z<(2)),

so S ( X ) ® Z/ maps onto the closure J3(X) of J5(X) in #? t (X,Z*(2)), 2

which contains all torsion classes. We claim that the map from 5 ( 1 ) 0 Z^ —> B(X)

is surjective with

2

torsion kernel. The surjectivity is evident. Let A(X)

be the image of

S. LICHTENBAUM

46

the codimension-2 cycles CH2(X) in i/| t (X,Z^(2)). Since we know by Theorem 2.13 that CH2(X) is a subgroup of tf|t(X,r(2)) with torsion quotient, it suffices to show that the map from A(X) ® J-t —• A(X) has torsion kernel. Let Zi, i— 1 , . . . , n, represent a basis for codimension-two cycles modulo numerical equivalence. Since, by a theorem of Lieberman [Li], for codimension-two cycles numerical equivalence is the same as homological equivalence (mod torsion), any cycle in A(X) has a multiple which is a linear combination of the Z{. So if /? G Ker(^(X) ® Z/ —• A(X)), we may write m/? = ^"=1 a * ® *«> a » ^ ^ - Let ^ i represent dimension-two cycles modulo numerical equivalence. By the ity of intersection product and cup product, X ^ = 1 otifayWj) det(z{,Wj) / 0, it follows that all the a,- are zero, i.e. m(3 claim is demonstrated. So any torsion class comes from a torsion element in B(X)

a basis for compatibil= 0. Since = 0, so our

® Z^, which z in turn must come from a torsion element in B(X), so be in the image of

tf£(x,r(2)). Corollary 2.16. There are projective smooth algebraic varieties over C where H%t(X}T(2)) properly contains CH2(X). Proof. There are examples of Atiyah and Hirzebruch [A-H] of such varieties with codimension-two torsion classes in cohomology which do not come from algebraic cycles. 3. The construction of Beilinson's complex Let X be a regular noetherian scheme. Let F be a sheaf on X for the big Zariski site and a* F the associated sheaf for the etale site. The adjointness of the functors a* and a* gives rise to a map from F to a* a* F. This in turn induces a map in the derived category of Zariski sheaves C t-> Rot*ot* C. If we let C = r z a r (2, X), observe that a* C is r^ t (2,X), and that since C is acyclic outside of [1,2] the map from C to Ra+a* C factors through t<2Ra*a* C. Proposition 3.1. The natural map from r z a r (2, X) to/<2 J Rof*r^ t (2,X) is an isomorphism up to 2-torsion. Proof. We must check that we have induced isomorphisms on the first and second Zariski homology sheaves W up to 2-torsion, and for this it suffices to look at stalks. Let A be a local ring on X. For Ti1 we must show that the natural map from K$(A) to H9t(A, K3) is an isomorphism up to 2-torsion. But up to 2-torsion this map is just the map 7 in the proof of Proposition 2.11, and a corollary of that proof is that 7 is an

WEIGHT-TWO MOTIVIC COHOMOLOGY

47

isomorphism up to 2-torsion. For H2 we are looking at the natural map from K^iA) to H?t(A,T(2))i which is an isomorphism by Proposition 2.9. We may conclude now (see [L2], Section 11) that r z a r ( 2 , X ) satisfies Beilinson's conjectures as described in [LI], at least up to 2- torsion, and modulo the caveat in the introduction. 4. G e r s t e n ' s conjecture and purity for motivic cohomology Along with the many other suggested properties of motivic cohomology ([Be], [Bl], [LI]), there ought to be analogues, in the etale site, of Gersten's conjecture, and, closely related to it, the theorem of purity. The general Gersten conjecture ought to be something like the following, although perhaps stronger in some sense: Conjecture 4.1. Let X be a regular scheme of finite type over a field k. Let d — dimX. Let X{ denote the set of points of X of codimension i. If x £ X, let j x : Speck(x) —• X. Then there exist objects Ui in the derived category, i— 1 , . . . , d, together with the morphisms indicated below, such that there are distinguished triangles: *(X,r)

—

® *< r + 1 JJ(i x ).Z(*(*),r) — Ui —

l(X,r)[l]

x€X0

Ui —- © UrR(jx)tZ(k(x),

(r - 1))[-1] —• U2

x£Xi

tfd-i—

0 xexd-i

tUd

and an isomorphism U

d ^ 0

t
d)[-d\.

x£Xd

The referee suggests that this conjecture could be rephrased as follows: Conjecture 4 . 1 . The complex Z(X, r) in the derived category is the image under the forgetful functor of an object (Z(X,r), F) in the filtered derived category (See [BBD]. Section 3 for basic definitions and terminology) such that the associated graded objects in the derived category Fi~1/Fi are equal to ©XGX t _^
S. L I C H T E N B A U M

48

C o n j e c t u r e 4 . 2 . Let i : Y —• X be a closed immersion of a regular scheme Y into a regular scheme X. Let c be the codimension ofY in X. Then t
'Z(X, r) ~ Z(Y, r - c ) [ - 2 c ] .

Note that Conjecture 4.2 yields a composite map Z(Y, r — c)[—2c] —+ t Z(X, r)[—2n], which could include the above ( / = z,n = —c) as a special case. This more general trace m a p could possibly follow from a more general purity conjecture with Rf] replacing Ri\ but it is not clear to the author even how to define Rf\ (One cannot expect a map / ! to exist taking sheaves to sheaves, so Rf would have to be defined directly on the level of complexes.) In this section, we will show t h a t the Gersten Conjecture (4.1) is valid for our proposed candidate T(X,2) for Z ( X , 2 ) if X is a regular scheme of finite type over a field and we work modulo 2-torsion. Under the same conditions we prove the weak purity conjecture if r = 2 and c = 1. We also show t h a t the Gersten conjecture leads to a natural construction of a cycle m a p . L e m m a 4 . 3 . Let C% and D% be complexes of objects in an abelian category which are acyclic outside of [1,2]. Let (p : C t —» D% be a map in the derived category such that

H

^

H2(D.)

Then there exists a triangle in the derived

-> W -+ 0. category

C,->D.-+W[-2]^C.[i\. P r o o f . This is essentially the same as Exercise 3.1 of [L2], i.e., a straightforward exercise in derived categories. T h e o r e m 4 . 4 . Conjecture 4.1 is true up to 2-torsion for T(X,2), i.e., there exists an object U in the derived category ofetale sheaves on X and distinguished triangles (up to 2-torsion) T(X, 2) and

0 t<3R(jx),T(k(x), xex0

2)-+U^

T(X, 2)[1]

(*)

WEIGHT-TWO MOTIVIC COHOMOLOGY

49

tf- 0 t<2R(jx),r(k(x), i)[-i] - ©< from T(X, 2) to t<sRj+r(F,2). Since j * and y* are adjoint functors, there is a natural map from C to Rj+Rj* C for any object C in the derived category of sheaves on F. Since j * is exact and j*T(X,2) = T(F,2) we get a map from r ( X , 2 ) to i?;*r(F,2), which factors through t< 2 i?j*r(F,2) since r ( X , 2 ) is acyclic in degrees > 2, so certainly through t<3Rj^T(F)2). We now compute the effect of ip on the homology sheaves TV of r ( X , 2). We first claim that ip induces an isomorphism on Ti1. It suffices to check this on stalks, so we may assume that X = Spec^l, A strictly Hensel. In this case Til(T(X,2)) = K^\A), up to 2-torsion, by Proposition 8.1 of [L2] and H\Rj+T(F,2)) = W 1 (r(F,2)) = Af°(F)> UP to 2-torsion by [L2], p. 191 and Theorem 0.1. By Lemma 2.1, A3 ^(^4) is isomorphic to K^\F), up to 2-torsion. Next we compute the effect of ift on Ti2. As above if X is strictly Hensel, X = Spec A, Ti2(T(X, 2)) = K2(A), by Proposition 2.9. Ti2(Rj*T(F, 2)) = Ti2{Y(F,2)) = K2(F) by [L2], p.191. By the Gersten conjecture for A, we have the exact sequence 0 -> K2(A) - ^ K2(F) -*+ [ J Ki(Hx)) x£Xi

-

J J K0(k(x)) xex0

- 0.

It follows that the cokernel of the map induced by ip from W 2 (T(X, 2)) to Ti2(Rj*T(F,2)) is W, where

W = Ker ( [ ] jmGm \07GA'1

[ J j*Z J . x€X0

/

Theorem 4.4 now follows from Lemma 4.3 and Hilbert Theorem 90 for T(0), T(l) and T(2) (see Proposition 2.12) with of course U = W[-2]. T h e o r e m 4.5. The weak purity conjecture (4.2) is true up to 2-torsion if r = 2 and c = 1, i.e. t<3m'T(X92)~r(Y,l)[-2]. Proof. We apply Ri to the two distinguished triangles in the conclusion of Theorem 4.4. We may again assume X is connected, and let j : Spec F(X) -+ X. We claim first that

S. LICHTENBAUM

50

Rpi(t<sRj*r(F(X),2))

=0

forp<3

or, equivalently, that t<3Ri(t<3Rj*T(F(X)12))

= 0.

But an easy argument shows that this last expression is equal to t<3R(rj+)T(F(X),2)} which is zero since rj+ is the zero functor. Hence the long exact homology sequence of (*) shows that the map from U to r(X,2)[l] induces an isomorphism oit<2RiU with t< 2 (/M ! r(X,2)[l]) = t< 3 iJrr(X,2)[l]. By definition of U and W and by Hilbert's Theorem 90, it follows from the triangle (**) that we have the exact sequence of sheaves on Y : 0 _> i'w -

0 i(jx),Gm xeXi

Z 0 i\jx)a xex2

If x £ X\ is not in Y, then i(jx)* 0 = t
- R'i'W

-

0 xeXi

B}t{jx).G„

— 0, and

= t
=

t
1

by Hilbert 90, which implies R r(jx)+Gm = 0 as well. If x E XiflY, then l ! " 0"a:)* — (^x)*, where /i r is the inclusion of x in Y. Then R1(t'(jx)*) == 0 by Hilbert 90, and R1i'(jx)*Gm = 0 by the composite-functor spectral sequence. If x G I 2 is not in Y, then r^'*)* = 0 and if x E X2 fl Y, again i\jx)* — (hx)*. This all implies that we have an exact sequence 0 - iW -

0

(fcx)*Gm - ^ 0

xeY0

{hx)JL - RliW

- 0.

x^Y1

Since Ker

y and Coker

!

hence

!

= << 0 fli W[-2] =i W[-2] = G m ,y[-2] = T(Y, l ) [ - l ] , s o t < 3 ^ ! r ( X , 2 ) = r(Y,l)[-2].

5. Cycle m a p s Let X be a regular scheme of finite type over a field K. We will give two different definitions of a map which associates an element of the group HA(X, T(2)) with any codimension-2 cycle on X, and show that they are equivalent. We will go on to show that if X is smooth, this cycle map is

WEIGHT-TWO MOTIVIC COHOMOLOGY

51

compatible with the classical ^-adic and p-adic cycle maps. Throughout this section we work modulo 2-torsion. In particular, neither £ nor p is allowed to be 2. Our first definition was given in Theorem 2.13 as a map 6 from the group H2(Xzari K2) of codimension-2 cycles modulo rational equivalence to HA{X, r(2)). Our second definition will be more refined, and will associate with any prime codimension-2 cycle Z a class tp(Z) (the fundamental class) in # f ( X , r ( 2 ) ) whose image in # 4 (X,T(2)) will be 0(cl(Z)). Let A denote any of the natural maps : ffJ(X,T(2)) -+ # 4 ( X , r ( 2 ) ) , # ! ( * z a r , r ( 2 ) ) - # 4 ( X z a r , r ( 2 ) ) , or H*(X„,K2) -> H2(XZ3TJ<2). Theorem 4.4 evidently gives us a map from t H3(Xzar,gT2yK3)

-

ff4(X2ar,r(2))

- ^ H\Xz&t,

K2)

-#4(Xzar,gr2A'3).... Since gv2(K3) is a constant sheaf (Proposition 2.11), and X is irreducible, ^ ( ^ z a n g r ^ ^ s ) is zero for i > 0, which yields the conclusion of the lemma. The proof for cohomology with supports is similar. T h e o r e m 5.2. \{^{Z))

=

0(ct(Z)).

Proof. First we observe that Theorem 4.4 remains true in the Zariski site (the proof is similar, but simpler at key points). Let

(Z). Now look at the Gersten sequence for the sheaf K2 z a r \

0 - K2 - [J (jx)J<2 - [J 0»).£i -> U 0«)'£° -* °x£X0

xexx

x£X2

52

S. LICHTENBAUM

Let z be the generic point of Z. The Gersten sequence induces a map p from (jz)*Z to /v2[2] which takes 1^ to the class of Z in H2(Xzar,K2). It is easy to see that we have a commutative diagram (jz)*Z ^ ^

T(2)zar[4]

lid

lfi

(jz)a —^— tf2[2] . S o ^ ( ^ z a r ( Z ) ) i s t h e c l a s s o f Z i n 7 / | ( X z a r , ^ 2 ) , a n d c ^ ) = A(^(^zar(Z)). Here we have written /?* for the map induced by (3 from .H"|(X zar ,r(2) to i / | ( X z a r , A' 2 ). We will also use /?* for the corresponding map without supports, and we remark that both these maps are isomorphisms by Lemma 5.1. We now go back to the definition of 0. The cycle map 0 is derived from the spectral sequence HP(XWlRqamT(2)) =» # P + 9 ( X , r(2)), which may be identified with the hypercohomology spectral sequence HP(Xzai,W(Ra,T(2)))

=>

H"+"(X,T(2)).

This evidently commutes with the spectral sequence H"(Xzar,W(t<2Ra,T(2)))

=>

m+"{X^TM2RatT{2))).

Since we have seen that tf<2ita*r(2) is r(2) z a r , we easily derive the commutative diagram H2(XZ^K2)

-^—>

,[ ' H2(Xzar,K2)

H\Xzar,T(2))

|. —L_

H*(X,T(2)).

So 6 =

But now we have A(V>(Z)) = A(y?z(V-zar(Z)) =

HfzWzKM^z)))

=
which completes the proof of Theorem 5.2. L e m m a 5.3. Our cycle class map is compatible with products. More precisely, if Z is an integral codimension 2 closed subscheme ofX which is the scheme-theoretic intersection of two divisors Y\ and Y2 on X, then the fundamental class of Z in Hg(X,T(2)) is the product of the fundamental classes of Yi and Y2 in H^XJil)) and H$2(X,T(1)). Proof. Since products in the Zariski site are compatible with etale products, it suffices to prove this in the Zariski site. Since Remark 2.6 of

WEIGHT-TWO MOTIVIC COHOMOLOGY

53

[L2] shows that the map from T(2) to A^t - 2] is compatible with products, and since we have seen that this map induces an isomorphism from # z ( ^ z a r , r ( 2 ) ) to i / | ( X z a r , K2), it suffices to prove that fundamental classes are compatible with products into H\{X7i^v,K2). Recall that the fundamental class of Y{ is obtained by taking the Gersten resolution of Gm on X

Gm - [ ] (j,),Gm - U tt*)'Z x£X0

X£XI

which give us a map from (jyt)*Z to G m [l], where yt- is the generic point of Yi. The class of 1 in / / y ( X z a r , (jy,)*Z) = Z maps to the fundamental class in # y t ( X z a r , Gm). Similarly, the fundamental class of Z is obtained by taking the Gersten resolution R2 of K2 and taking the image of 1 in # z ( ^ z a r , (J*)*Z) = Z in # ! ( X z a r , ^ 2 ) , where z is the generic point of Z. To compute products, we replace the Gersten resolutions of Gm by the complexes d defined as follows : Let U{ — X — Yi, 7,- : Ui —* X. Then Ci is the complex (ji)*Gm —• (i yi )*Z, which also represents Gm in the derived category. The fundamental class of Yi can equally well be defined by mapping (jyt)+Z to C{. But now we have an evident product map C\ ® C-2. —»• R2, which is compatible with the fundamental class maps and the product map (j y ,)*Z® (jy2)*Z —• (j*)*Z. Proposition 5.4. Let X be a smooth scheme over a field k. Let n be an odd integer prime to the characteristic of k. Let Z be a closed integral subscheme of X of codimension 2. Then the image of the cycle class ip(Z) corresponding to Z in H\{X, T(2)) via the Kummer sequence map to H^{X, u®2) agrees with the classical cycle map defined by Deligne in [SGA4\]. Proof. Let i : Z —> X. Since Deligne shows in [SGA4|] that z(x>t*$2) = H0(Z,R4rn®2) may be computed locally on Z and is determined by its restriction to any non-empty open subset of Z, we may assume that Z is a complete intersection of prime divisors Y\ and Y^Since our fundamental classes agree with Deligne's for divisors, and both are compatible with products ([SGA4|] and Lemma 5.2), as are the maps from T(l) to /i n and T(2) to //®2, the proposition follows. H

Proposition 5.5. The cycle map 0 is compatible with the "classical" p-adic cycle map into H2(X,i/2fn)- (See [M] for the precise definition). Proof. Since the p-adic cycle map factors through H2(X, K2) it suffices to show that there is a natural map a from H4(X)T(2)) such that the following diagram commutes :

to

H2(X,K2)

54

S. LICHTENBAUM

H2(Xm,K2)

•

I "

H4(X,T(2))

1-

2

H2(X,K2).

H (X,K2)

But since the natural m a p from T(2) to 7\2[—2] yields the commutative diagram H4(Xzar,T(2))

- ^ -

E\Z^K2)

I "

V H*(X,T(2))

•

H2(X,K2)

with /?* an isomorphism, and we have seen that 9 = <po(3~l, we are done.

REFERENCES [A-H] [BBD] [Be] [Bl] [BK] [BMS] [BO] [CTR] [I] [LI]

Atiyah, M. and Hirzebruch, F., Analytic cycles on complex manifolds, Topology 1 (1962), 25-45. Beilinson, A. A., Bernstein, J. and Deligne, P., Faisceaux pervers, Asterisque 100 (1982). Beilinson, A. A., Height pairings between algebraic cycles, Current Trends in Arithmetical Alg., Geometry, Cont. Math. 6 7 (1987). Bloch, S., Algebraic cycles and higher K- theory, Advances in M a t h . 6 1 (1986), 267-304. Bloch, S. and Kato, K., p-adic etale cohomology, Publ. I.H.E.S. 6 3 (1986), 107-152. Beilinson, A., MacPherson, R. and Schechtman, V., Notes on motivic cohomology, Duke Math. J. 5 4 (1987), 679-710. Bloch, S. and Ogus, A., Gersten's conjecture and the homology of schemes, Ann. Sci. Ec. Norm. Sup. I V , Ser. 7 (1974), 181-202. Colliot-Thelene, J.-L. and Raskind, W., H^,-cohomology and the second Chow group, Math. Ann. 2 7 0 (1985), 165-199. Illusie, L., Complexe de la Rham-Witt et Cohomologie crystalline, Ann. Sci. Ec. Norm. Sup. Ser. 12 (1979), 501-661. Lichtenbaum, S., Values of zeta-functions at non-negative integers, Number Theory, Noordwijkerhout 1983, Lect. Notes in M a t h . 1 0 6 8 . Springer-Verlag (1984) 127-138.

WEIGHT-TWO MOTIVIC COHOMOLOGY [L2] [Li] [Ma]

[M] [MS] [So] [Su] [SGA4|] [T]

55

Lichtenbaum, S., The construction of weight-two arithmetic cohomology, Invent. Math. 88 (1987), 183-215. Lieberman, D., Numerical and homological equivalence of algebraic cycles on Hodge manifolds, Amer. J. Math. 90 (1968), 366-374. Manin, Ju I., Correspondences, motifs and monoidal transformations, (AMS translation series), Mathematics of the USSR-Sbornik 6 (1968), 439-469. (original in Matematiceskii Sbornik 77 (119) (1968) 475-507) Milne, J., Values of zeta functions of varieties over finite fields, Amer. Jour, of Math. 108 (1986), 297-360. Mercuriev, A. S., Suslin, A. A., On the K3 of a field, preprint, LOMI (1987). Soule, C., Operations en K-theorie algebrique, Canadian J. Math. 37 (1985), 488-550. Suslin, A. A., Torsion in K2 of fields, K- theory, 1, (1987) 5-29. Deligne, P., et al, Cohomologie Etale, Lecture Notes in Math. 569. Springer-Verlag (1977). Thomason, R., Survey of algebraic vs etale topological K-theory, (preprint).

Department of Mathematics White Hall Cornell University Ithaca, New York 14853

Symmetric Spaces over a Finite Field

GEORGE LUSZTIG* Dedicated to A. Grothendieck on his 60th birthday

Introduction Let G be a (connected) reductive group defined over a finite field Fq (q odd) with a given involution 9:G —• G defined over Fq. The pair (G, 6) will be called a symmetric space (over Fq)\ we shall fix a closed subgroup K of the fixed point set Ge such that K is defined over Fq and K contains the identity component (Gd)° of Ge. Let T be a maximal torus of G defined over Fq and let A: T{Fq) —• Qt be a character (£ is a fixed prime not dividing q)\ let R^ be the virtual representation of G(Fq) attached in [2] to (T, A). This paper is concerned with the computation of the sum

9tK(Fq)

we find for it an explicit formula. (See 3.3(a).) There is a particular kind of symmetric spaces in 1 — 1 correspondence with connected reductive groups G\ defined over Fq (without involution); to G\ corresponds the symmetric space (G\ x G i , r ) where r{g\,g[) — (g[,#i). It is well known that many objects attached to reductive groups can be redefined in terms of the associated symmetric space so that they make sense for an arbitrary symmetric space. For example: (b)

the Schubert varieties in the theory of reductive groups become a special case of the closures of G^-orbits on the flag manifold of G;

(c)

the set of conjugacy class of the reductive group G\ (or G\{Fq)) becomes a special case of the set of double cosets K\G/K (or K(Fq)\G(Fq)/K(Fg));

* Supported in part by the National Science Foundation.

58

GEORGE LUSZTIG

(d)

the problem of classifying the irreducible representations of Gi(Fq) becomes a special case of the problem of classifying the irreducible representations of G(Fq) with a non-zero space of K(Fq)-invariant vectors;

(e)

the character table of G\(Fq) becomes a special case of the character table (suitably normalized) of the Hecke algebra (algebra of double cosets) of G(Fq) with respect to K(Fq).

In the same spirit, our formula for (a) is a generalization of the orthogonality formula [2,6.8] which expresses the sum

\Gi{Fq)\~l Y. Tr(^,4;)Tr(^,4j) in terms of the Weyl group. (Here T\,T{ are maximal tori of G\ defined over Fq and A^A^ are characters of Ti(Fq),T{(Fq).) The proof of the orthogonality formula given in [2] is based in part on the disjointness theorem [2,6.2]. While that disjointness theorem extends to symmetric spaces (see 3.5), it is not strong enough to imply our formula for (a) in the general case. On the other hand, the argument given in [3, p.8-10] generalizes better. In the case where A is a character in general position of T(Fq), our formula gives the dimension of the space of A"(F^)-invariant vectors in the irreducible representation dtRj, of G(Fq), hence it can be regarded as a solution of (d) as far as "generic" representations of G(Fq) are concerned; this confirms in part a conjecture of Bannai-Kawanaka-Song [1,6.7]. Acknowledgements: I wish to thank David Vogan for some useful discussions. Several ideas presented here were developed during the winter 1985-86 while I was visiting the University of Rome. 1. Recollections and notations 1.1. In this paper, G will always denote a connected reductive group defined over a finite field Fq with q elements (q odd), with a given involution 8:G —• G defined over Fq and a given closed subgroup K of the fixed point set Ge such that K is defined over Fq and K contains the identity component (G6)0. We shall denote by F:G —+ G the Frobenius map corresponding to the /^-structure. We denote S = G / K , a homogeneous variety for G.

SYMMETRIC SPACES OVER A FINITE FIELD

59

1.2. If T is a maximal torus of G, we define 6 T = { / G G\0(f~1Tf) = / T / } . Then 6 j is a union of T — K double cosets and we denote 0 y the image of 0 T under the canonical map G —• S. Then 0 ^ is a T-stable subvariety of S. The following analog of the Bruhat decomposition is well known. _1

P r o p o s i t i o n 1.3. Let T be a maximal torus of G and let B be a Borel subgroup of G containing T, with unipotent radical U. (a)

B has only finitely many orbits on S.

(b)

T has only finitely many orbits on 0 ^ .

(c) We associate to any T-orbit O on 0 ^ the unique B-orbit O1 on S such that O C O'. The resulting map {set of T-orbits on 0 T } —• {set of B-orbits on S } is a bisection. (d) IfOcO* are as in (c) then the map U x O —* O' defined by the B-action is an affine space bundle. First note that if the proposition is true for a pair ( T c B), then it follows easily that it is true for any other pair {xTx~l C xBx'1), (x G G). Using [5,7.5], we are reduced to the case where both T and B are 0-stable. In this case, a proof can be found, for example, in [4, §4] at least under the additional assumption that K = G6'; however, essentially the same proof applies in the general case. 1.4. The variety of Borel subgroups of G will be denoted by B. We shall denote by W the Weyl group of G and I: W —+ N the usual length function. To two Borel subroups B, B1 one can attach naturally an element w £ W (relative position) as in [2,1.2]; we shall then write B—-—B'. 1.5.

Let Q = {B G B\B opposed to OB) J = variety of all 0-stable maximal tori T such that T C B for some B £ Q V = variety of all pairs T C B with TeJ,BeSl. (a) If Cl is non empty, then K° acts transitively on it, as well as on J and V (by conjugation). 1.6. If / : V —• V is a morphism of a variety into itself we denote by V? its fixed point set. For a closed subgroup G\ of an algebraic group G*i we denote by Z(G\) or ZG2{G\) the centralizer of G\ in G2 and by G\ the identity component of G\.

60

GEORGE LUSZTIG 2. The homomorphism e 2.1.

We set «r(G) = ( \> * f ^ k (G) is even; v [ — 1, otherwise Let T be a maximal torus of G which is F-stable. It is well known

that (a)


where B is any Borel subgroup of G containing T and U is its unipotent radical. 2.2. Assume now that T is a maximal torus of G which is both instable and 0-stable. Let TK = TDK and let T^ be the identity component of TK . It is well known that

teTK=>t2 GT£.

(a)

We denote by Z = Z(T%) the centralizer of T£ in G; for each t G T £ we set Zt = ZC[ZQ(JL). Then Z, Z* are connected reductive groups defined over Fq. We define (6)

e=

eT:T?^±l

by e(<) =
e(t) = (-1)»W

(b)

c(<) = 1 for all i E {TQK)F

(c)

e is a group homomorphism.

Proof. Using 2.1(a) for Z and Zt instead of G, we see that for t G Tf we have ( - l ) n W = a(T)a(Z) •
SYMMETRIC SPACES OVER A FINITE FIELD

61

but not in FU. For any t G T£ let $t be the set of roots a G $ such that a(t) = 1. We have n(t) = |3>| -f |<£<|. Hence, to prove (c) it is enough to show, in view of (a), that given t,t' G T£, we have (d)

| * t | + \9f\ + \$tt>\ + | * | = O(mod 2).

From (b) and 2.2(a) we see that a(t) G {±1}, <*(*') G {±1} for all roots a. Hence we can define a partition of $ into four pieces &* ={ae

$\a(t) = i,a(t') = j},ij

G {±i}.

We have | * , | = l* 1 ^! + | * l , " l U * * ' l = l* 1,L l + I*" 1 ' 1 !*!***'! = |* 1 > l | +

I*- 1 *- 1 !,|$| = l*1'1! + |* l '" l | + |*- l ' l | +1*" 1 '" 1 !-

H e n c e t h e left h a n d

side of (d) is 4\Q1*l\ + 2\*l*-l\

+ 2\*-1'1\

+

2\9-l>-l\

and (d) follows. The proposition is proved. 3. Statement of the main results 3.1. Let T be an F-stable maximal torus of G and let X:TF —• Qt be a character. The variety QT in 1-2 is F-stable and we denote ©T,A = | / £ 6 T ; /

M(f-*Tf)* = ef~1Tf }•

Here f'1 \: (f~lTf)F -+ ~Q\ is the character defined by '""'A^') = A ( / ^ / _ 1 ) , and CJ-IT/ is as in 2.2(b). It is clear that QF x is a union of TF - KF double cosets of GF. 3.2. We recall the definition of the virtual representations R^> of [2]. Let B be a Borel subgroup of G containing T, with unipotent radical U. Let X = {geG\g-1F(g)eFU}. Then G F acts on X by
(a)

*(Y)A-

= £(-1)'dimtf«(Y) A -,

62

GEORGE LUSZTIG

In particular Hlc(X)x-i by definition [2,1.20]

is well defined; it is naturally a G F -module and,

R$ = J2(-l)iHic(X)x-l, i

is a virtual representation of GF of dimension

x(X)\-1-

T h e o r e m 3.3.

\KF\'1 £ K{9^) =

(a)

geKF

Y,^T)
sum over a set of representatives f for the double cosets TF\Qj,

F

X/K

.

T h e o r e m 3.4 (a)

£

TT(g,R)r)=\TFr £
9EKF

fee?

unipotent

T h e o r e m 3.5. Let p be an irreducible representation of GF such that the space of KF-invariants pK is non-zero. Assume that p appears in the GF-module Hlc(X)x-i where T}\,B,U,X are as in 3.1, 3.2. Then there exists f G QT and an integer n > 1 such that Fn f — f and the character \:(f-lTf)Fn -» Q*t defined by \(t') - A ^ / * ' / " 1 ) ) is trivial l on the subgroup ({f~ TfYK)Fn. (Here, N:TFn —• TF is the norm map.) For that /,n,A we have A(0(V)) = A^')" 1 , for allt' G {f~lTf)Fn. 3.6. Remark. Theorems 3.3, 3.4, 3.5 are generalizations of Theorems 6.8, 6.9, 6.2 of [2] respectively. 4. P r o o f of T h e o r e m 3.5. 4.1. (a)

Let T, 5 , U be as in 3.1 and let O be a T-orbit on 6 T . We define 6 = Go = {(*, u,<j>)eFU xU x 0\F() = xu in.}

The variety & has a TF- action (6)

t:(x,u,<j)) -*(txrl,turl,tf).

SYMMETRIC SPACES OVER A FINITE FIELD

63

Actually, the same formula defines an action of a larger group HQ on 6 , where

(c)

H0 =

{teT\rlF(t)eTF0}

and, for any T-orbit 0\ on 0 ^ , (d) To1 is the stabilizer in T of an element of 0\ (this stabilizer is independent of the choice of an element of C?i, since T is abelian). Clearly, TF C Ha. We shall need the following generalization of [2,6.7]. L e m m a 4.2. Let ip:T —• T be an endomorphism ofT which commutes with Fn for an integer n > 1, and let T\ be the identity component ofT^. Let H = {t G T\t~1F(t) e T^}. Assume that A|#o nT F is trivial Then XoN:Tpn -^Q*t is trivial on TxnTFn. (N is the norm mapT*"1 -+TF). Proof. Recall that N is the restriction to TF of the homomorphism N:T -+ T,N(t) = tF(t) • • • F " " 1 ^ ) . If t G T^, then N(t) G H; indeed, Nty^FiNit)) = t~lFn{t) and ^(t-1Fn(t)) = r 1 *"»(*) since ^ ( 0 = t n and V commutes with F . Thus, N(T*) C # , hence N(TX) C H°. It follows that X\N(Ti) CiTF is trivial, hence A o N\Ti HTFn is trivial. Proposition 4.3. Assume that for some integer i,H%c(&o)\-i (defined in terms of the TF-action, as in 3.2) is non-zero. Let / ' G 0 T be such that f'K G O, let n > 1 be such that Fnf = / ' and let f = Ff. Then A o N:TFn -> T is trivial on the subgroup f • {f~lTffK • f"1. Proof. We apply Lemma 4.2 to the automorphism ip:T -+ T,ip(t) = fOtf-Hf)r\ We have T* = / ( / " 1 T / ) G . -Z" 1 hence 7\ = ftf^Tfiy-1. l 1 We have T F O = f{f~ Tf)Kf- . Since T^,TFO have the same identity component, iif (of 4.2) and # # (of 4.1) have the same identity component. Now Ho acts on Hlc{&o) (see 4.1) and #£,, being connected, must act trivially; hence HQC\TF acts on Hlc(&o)\-i both trivially and as the restriction of A" 1 . Since Hlc{&o)\-1 ^ 0, it follows that A is trivial on HQC\TF hence on H° HTF. Hence 4.2 is applicable and the proposition follows. Proposition 4.4. Assume that LC = (G0)0 and consider the orbit space KF\X of X in 3.2 with respect to the KF-action ko'.g —• kog. ^ e map g H-+ (g~1F(g),g~1K) defines an isomorphism of KF\X with Z = {(x) eFU

x S\F = x~l4>}.

The proof is left to the reader; it uses Lang's theorem for G and K.

64

GEORGE LUSZTIG

Now Z has a T^-action t: (#,) —+ (txt~lyt<j>)\ this is compatible with the action of TF on KF\X induced by the action of TF on X. Hence we have (a)

H\{X)^

- Hi(KF\X)x-r

= ffi(Z)A-x

for all i. 4.5.

For each 5-orbit O' on S, we denote

Then the ZQI form a partition of Z into finitely many locally closed T e stable sub varieties; this partition has the property that the closure of a piece in the partition is a union of pieces in the partition. It follows that: (a) If Hlc(Z)\-i / 0 for some i, then Hl(Zof)\-i ^ 0 for some i and some Of. (6)

X(2)A-I=EO'»X(ZO')A-«

where x(Z)\-1iX(Zo,)\-1 S.

are as m

3.2(a) and O1 runs over all J9-orbits in

4.6. Let O be the unique T-orbit in QT contained in the 5-orbit O' in S (see 1.3(c)). Using 1.3(d) we see that the map &a —• ZQI, (x,u,<j)) —• (x~1F(u)~1, u), is an afflne-space bundle, compatible with the T F -actions. It follows that: (a) If Hlc(Zo')\-i T^ 0 for some i, then Hlc(&o)\-1 / 0 for some i (b)

X(^O')A-I = X ( 6 O ) A - I

4.7. P r o o f of T h e o r e m 3.5. It is sufficient to consider the case where K = (Ge)°; the general case reduces immediately to this case. By assumption, we have H^X)^^ ^ 0 for some i. Using 4.4(a), we deduce that Hlc(Z)x-i / 0 for some i. Using 4.5(a) we deduce that Hlc(Zo')\-i ^ 0 for some Of and some i and using 4.6(a) we deduce that Hl(&o)\-i ^ 0 for some O and some i. We then apply 4.3, and we obtain the first assertion of 3.5. Let / , n, A be as in that assertion. The image of the homomorphism t' —* t'9(t') of f~lTf into itself is connected and contained in G6, hence l it is contained in {f~ TffK. Hence if*' G ( / _ 1 T / ) F n , then t'0(t') G l Fn ((f~ Tf)°K) and by the first assertion we have \(t'0(t')) = 1. Hence the second assertion of 3.5 is proved.

SYMMETRIC SPACES OVER A FINITE FIELD

65

5. Computations of some Euler characteristics 5.1. In this section we shall assume that G is simply connected and that there exists a maximal torus T C G such that 6{i) — fl for all t G T. We fix such a T and a Borel subgroup B containing T. In this case we must have K = Ge = (G')°. (See [6,8.2]. Then B G ft (see 1.5) and the isotropy group of 5 for the transitive /f-action on ft (see 1.5(a)) is T* = {t G T\t2 = 1}. 5.2. For each simple coroot 6cj\Fq —• T (with respect to 5 ) , let tj = &j{— 1) G TR-. Then f i,*2» • • • Ji (^ = r a n k G) form a basis of T^ as an F2-vector space. Let XO'>TK —• ±1 be the character such that Xo(tj) = — 1 for j = 1 , . . . ,^. Let £ be the 1-dimensional, Tf-equivariant Q^-local system on ft such that the action of the isotropy group TK on the stalk of C at B is given by xo5.3. Let D be the line in B consisting of all B' G B which are contained in a fixed parabolic subgroup of semisimple rank 1. Then D = PX,D — D fl ft consists of two points and the restriction of C to D D ft(= P 1 — {0,1}) is isomorphic to the unique Q^-local system with monodromy of order exactly two. Hence (a) # j ( £ > n f t , £ ) = 0 for alii. (We denote the restriction of C to various subvarieties of B again by £.) For each w G W let Yw = {Bf G ft | B w £ ' } , a locally closed subvariety of ft. Proposition 5.4.

dim HHY n-l1'

2/ i =

^)^

Proof. When w = e, we have Yy, = { 5 } and the result is clear. Assume now that w ^ e and that the result is already known for w' of strictly smaller length than w. We can find u/, s G W with ^(s) = l,f(u/) = £(w) — l,w's = w. We consider the following varieties Y = { ( £ ' , B") € fi x 12 | B-^—B',

B'

S

°T *

B")

y' = {(B',B") e fi x n 15——s', s'——B"} Y" = { ( 5 ' , B") € n x fl I B—^—B',

B' = B")

Y'" = {(B', B") e (S - fi) x 0 | B—-—B',

B'——

B"}.

GEORGE LUSZTIG

66

We can pull back C to each of these four varieties via the second projection; this pull back is denoted again as C. We have Hi(Y, C) = Hic(Y,n, C) = 0 for all i.

(a)

Indeed, consider the first projection of Y or Y"' to B. By the Leray spectral sequence it is enough to show that each fibre of this projection has vanishing Hlc(,C). In both cases, this follows from 5.3(a). Now Y = Y' U Y" with Y' open and Y" closed in Y. Using (a) and the long exact sequence attached to this partition it follows that Hi(Y",C)^Hi+1(Y',C)

(6)

for alii.

We may identify Y' with an open subset of Yw and Y,n with its complement in Yw (via the map (JB ; , B") —• B". Using (a) and the long exact sequence attached to this partition, it follows that Hlc(Y', C)^Hlc(Yw,C)

(c)

for all i.

The map ( £ ' , B") —+ B" defines an isomorphism (d)

Y" ^ Yw, .

From (b), (c), (d) it follows that Hi(YWjC) sition follows by induction.

- ^ " H ^ s C) and the propo-

Corollary 5.5. X(YW) = ( - 1 ) ^ ) . (x denotes Euler characteristic.) The proof will be given in 5.7. It is based on the following result. Lemma 5.6[2,3.2]. Let cr:V —> V be an automorphism of finite order prime to q of an algebraic variety V over Fq. Then ^t(—l)*Tr(cr*, HJ,(V)) = X(V'). 5.7. Proof of Corollary 5.5. We can regard A" as a principal fibration over Q with finite group TK, via the map 7r: K —» Q(g —• gBg~l) and the action of TK on K by right translation. Let Yw = 7r~x(Yw). Then H1C(YW1C) (resp. Hlc(Yw)) is isomorphic to the subspace of H%C(YW) on which TK acts according to \ (resp. trivially). Hence (a)

^ ( - l y d i m f f ^ , / ; ) = \TK\-> J^ •

t£TK

^(-1),'Tr(t'Hc(Yw)k(t) i

SYMMETRIC SPACES OVER A FINITE FIELD

67

and £ ( - l ) ' d i m i y ' ( n , ) = \TK\~l J2 E ( - 1 ) , ' T r ( t ' H c ( t w ) ) .

(6)

i

t€TK

i

The right hand sides of (a),(b) can be evaluated using 5.6 (the fixed point set oft on Yw is empty for t ^ e); hence they are both equal to |Tx|~ 1 x(y i y ). Thus, the left hand sides of (a) and (b) are equal. But the left hand side of (a) is equal to ( — 1 ) ^ ) by 5.4; hence so is the left hand side of (b). The Corollary is proved. 6. Beginning of the study of © 6.1.

In this section we shall assume that

(a)

T, 5 , U, A are as in 3.1., 3.2.

(b)

K = Ge

(c)

0T=T,T%

(d)

O is the T-orbit of the base point K in 0 ^ .

Ccentre(G)

6.2. Let © = &o be the variety associated to these data as in 4.1(a). If we identify O with T/TK , we have

e = {(*,tM) EFU xu x T\F(t)~lxut e i<] the TF action on ©

modulo the free action of Tjcti' (#,u^i) —• (x}u,tti); is given by t0:(x,u,t) —• (foxt^ 1»*o«*o 1 >M). We now state the main result of this section.

(a)

P r o p o s i t i o n 6.3. F X ( 6 ) = | T I K T ^ f |-i(_i)dim(c//t/nFf/)

6.4. We make the change of variables (x,u,i) x' — t~1xt, u' — t~lut] then © becomes (a)

© = { ( * , u , t ) e F U xU x T\F(t)-ltxu

modulo the free action of T ^ , ^ : (#, u,tf) —• ( t f j action is tQ\(x,u,i) —> (x^u^ot). Let

-1

—• (x^u'^t)

where

G A'} ^ ! , ^ 1 ^ ! , ^ ) ; the TF

(6)

6x = {(a:, u,r)eFU

xU x T\rxu G A'}.

(c)

62 = ©i modulo the free T^ — action f i : ( z , u , r ) —• (x,w,fir).

68

GEORGE LUSZTIG

On ©i we have a T/r-action ti'.(x,u,r) —• (t^lxti,t^1utiyF(ti)~ltir)] the same formula defines a T^/T^-action on ©2- Indeed, when / i G T% we have ( ^ ^ l ^ r ^ i ^ i ) " 1 * ! 7 " ) = (*» u ^ i ) " 1 ^ ? - ) (by 6.1(c)) and this is in the same orbit as (X,U,T) for the Tj^-action in (c). Moreover, using Lang's theorem in T^ we see that the set of X^-orbits on ©i may be identified with the set of TK/T^-orbits on ©2 (which is an algebraic variety since TK/T^ is finite); using Lang's theorem in T, we see that the map (x,u,2) —• (z, u,F(t)~xt) defines an isomorphism TF\& = TK\&I = (TK/TK)\&2. It follows that (d)

F

X(T

\&)

=

X((TK/T°K)\&2).

6.5. Applying 1.5(a) to (T C B) G V and (T C FB) e V we see that there exists n G J\ normalizing T such that FJ5 = n~1Bn. We then also have FU = n~1Un. Let (a)

©3 = (BnB) 0 K modulo the free T^-action by left translation. We define a map

(6)

6 2 - 63

by ( x , u , r ) —v nrxu] this is well defined since for ( Z , U , T ) G ©1 we have rxu G A' hence nrxu G A' and nrxu = (urn-1) • ( n x n " 1 ) ^ ^ G TUnU = BnB] we also use the fact that T^ is central. It is clear that the map (b) is an affine-space bundle. In particular, we have (c)

x(©2) = X(©3).

6.6. On ©3 we have a free TK/T^ action by left translation. (This action is not compatible under (b) with the action of TK/T^ on ©2.) It is clear that the map g —• g~xBg defines an isomorphism (a)

(TK/T^)\&3

£ {B' G tf^opposed to 0B\

B—^—B'}

where w G W is the relative position of (Byn~lBn) = (ByFB). (We use the fact that B f] K = TK.) The variety in the right side of (a) remains unchanged when G is replaced by the simply connected covering of its derived group (with the involution corresponding to 9) hence we can apply to it Corollary 5.5; using (a), we see that (b) X((TK/T%)\&3)

6.7.

= ( - 1 ) ^ ) = (^l)dim(f//f/nn-1C/n)

=

(_^dim(U/UnFU)

When a finite group T of order prime to q acts on an algebraic

SYMMETRIC SPACES OVER A FINITE FIELD

69

variety V over Fq, we have as in 5.7: = IIT1 £

X(T\V)

£ ( - l ) « T r ( T ) H«(V0)

= |rr 1 Ex(v 7 ) 76r

where V1 is the fixed point set of 7 on V. We apply this in three cases: (a) X((TK/T°K)\&3) = )(TK/T°K)\-iX(63), since the action of

(6) (c)

F

x(T

\e)

on ©3 is free,

(TR/T^)

= \TF\-1 J2 *(et0)> to£TF

X((TK/T%)\62)

= \(TK/T%)\-*

£

X(6£).

i0eTK/T^

From (a), 6.6(b) and 6.5(c), we see that (d)

\(TK/T^)\(-l)iim(u'UnFU\

x(© 2 ) =

6.8. Assume that i\ G TK/T^ (representing t\ G Tx) has a fixed point ( # , u , r ) on ©2. Then x G Ff7nZ&(*i),ti G UDZ^ti^FfayHi G T£ hence F(*i) = i\. Thus ©21 ^s empty unless Ft\ — t\. If Ffi — t\% we may choose t\ G T£ and we see that ©2* may be identified with the variety defined exactly like © 2 but for (G,{7,T,0,...) replaced by (Zg(*i), *7 H Z g ^ ^ T , 0,...). To this variety we may apply formula 6.7(d). Hence 6.7(c) becomes: (a)

X((TK/T°K)\e2)=

£

\(l*)F\-1(-l)*™(u,l'u,lFU,l)>

t,€T£

where Uu =UC\Z°G{t{). Assume now that to G TF has a fixed point (x,u,t) on © (in the description 6.4(a)). Then x £ FU D Z%{t), u £U <~) Z%(t),t0 eTF. Hence <5'° is empty unless t0 G T£. If tf0 G T £ , we see that 6'° may be identified with the variety defined axactly as in 6 (in 6.4(a)) but for (G, U,T, 6,...) replaced by (Z%(t0), U n Z£(
(6)

X(TF\&)

= £ l^r^e*').
GEORGE LUSZTIG

70

Now 6.3(a) certainly holds when G = {e}. We may assume that 6.3(a) holds for (Z%.(t0), U n Z%(tQ),T, 0,...) instead of (G, U, T, 0,...) whenever to G Tj£ is not central in G (i.e. Zg^to) is a proper subgroup of G.) It follows that the terms corresponding to such *o in the sums in the right hand sides of (a), (b), match. Subtracting term by term the identities (a), (b) and using 6.4(d) we deduce

o=

(|rFr1x(G)-|(7i)F|-1(-i)dim^^nFi;)

£ *0eT£ncentre(G)

since &° = & for tf0 m the last sum. This implies that 6.3(a) holds. Corollary 6.9. x ( 6 ) A - i = | ( 2 j ) F r 1 E « o € T j f ( " l ) d i m ( ^ ° / l / < 0 n r o t 0 ) A ( * o ) where U** =

unz°G(t0). Proof. We have to€TF

i

1

^

x(6 to )A(
(by 5.6)

= I^T1

£

x(6«")A(fo)

(see 6.8)

= l^r

and we apply 6.3 to &° (see 6.8). Corollary 6.10. X(fA-J-|0

(

otherwise.

Proof. Using 6.9 and the definition of eT (2.2(b)) we have X(6A-o

= i(n) F r 1 (-i) d i m ( ^ n w ) £

^(^O)AW-

Now e T is a character of T£ (see 2.3(c)) hence to the last sum we may apply the orthogonality formula for the characters of T£; we also use 2.1(a). The desired result follows.

SYMMETRIC SPACES OVER A FINITE FIELD

71

7. T h e s t u d y of 6 , continued 7.1. (a)

In this section we shall assume that T, B, U, A are as in 3.1, 3.2.

(b) O is a T-orbit in 0 T such that FO = O. (Hence 0 F ^ 0.) Then F T 0 = TFO = T 0 (see 4.1). We set 0 = £ 0 = Z(7^), a connected reductive group defined over F^. Let IQ\TQ —» {±1} be defined by (C)

€0(0 = ^ ) ^ ( 0 ) .

(d)

Let O be the inverse image of O under G —> G/K.

7.2. Let eOF, and let n G G be defined by n = fO(f)'1 where / G G satisfies ^ = /A'. Then n depends only on (not on f),Fn = n and 0n = n - 1 . Let Of.G —+ G be the involution defined by 0^(

= c 0 (t) for all * G T F .

In particular, the following three conditions are equivalent: (b)

OFnefA7fc0;(9Fce?|A;A|TP=eo.

L e m m a 7.4. (a) We have \Tg\ • I T ^ I " 1 = \TF\OF\ F F F F (b) / / A ' = (G')°, then 6 ± 0 anrf \T \O \ = |T \0F/AF|. Proof, (a) Since T acts transitively on O and the stabilizer of any point (in particular a rational point) is To, the number of T F -orbits in OF is equal to the cardinal of the cokernel of F - 1: To /Tg -+ Ta/T%, hence also to the number of fixed points of F on TO/TQ. This equals 1

l^i-iiST -

(b) By Lang's theorem in K we have 0F/KF follows.

-^0F'.

The Lemma

72

GEORGE LUSZTIG

Proposition 7.5. Let & = &o be the variety associated to the data in 7.1(a), (b) as in 4.1(a). If K = Ge, then:

(a)

( \T£\ • \{T*0)F\-'o-{T)o-{G), if\\T,= lQ \ 0 otherwise. (b) IfK = Ge = (Ge)°,then: x(6)A-i = £ l(/-1r/)£|-|rJ,r1|^|-V(r)flr(z((/-1r/)i)).

X ( 6 ) A

_, =

For the proof we shall need the following known result: Lemma 7.6. Let V be a variety over Fq with an action ofTF and with an action of a torus T\ over Fq, commuting with the TF-action, with fixed point set VTl. Then x(V)\-* = x(VTl)\-^ For a proof, based on [2,3.2], see [3,1.6]. 7.7. For the proof of Proposition 7.5, we assume that K — Ge. We have FTQ = TFO = Ta and Tc C H0 (see 4.1). Since T0,H0 have the same dimension, it follows that HQ = TQ. NOW HQ acts on &o (see 4.1). Applying 7.3 to the restrictions of this action to TF and to HQ = T$, we deduce (a)

X(6)A-I=X(6T$)A-.

where &T° - {(x, u,)€ FU xU x G\F(j) = xu,

xeG,ueG},

hence (6)

&To = {(xiU,) e FU' x C / ' x 0\F<j> = xu},U' = Uf\G.

Let / , } n, 0$: G —• G be as in 7.2. We write 0' instead of 0$. Let 0 i be the T-orbit of the base point Qe in G/Ge . We have an imbedding j:G/Ge <-* G/K,j(gG0 ) — gfK; this is well defined since 1 e f~ G f C K. This imbedding is T-equivariant and it carries the T-orbit 0\ isomorphically onto the T-orbit O. From this and (b) we see that (#, u, ')) defines a T F -equivariant isomorphism ©' - ^ © Tc> , where & is defined like 6 in 6.1, in terms of (£,T,0', [ / n £ , 0 i , . . . ) . This and (a) imply that X ( ® ) A - X = X ( ® ' ) A - 1 - NOW X ( ® ' ) A ~ 1 is given by 6.10 for ((/, T, 0',...); these satisfy the hypotheses in 6.1. Thus, 7.5(a) is proved. In the case where K = Gd = (G*)°, the formula in 7.5(a) can be rewritten in the form 7.5(b) using the remarks in 7.3 and Lemma 7.4. This completes the proof.

SYMMETRIC SPACES OVER A FINITE FIELD

73

8. The study of 6 , concluded 8.1.

In this section we shall assume that

(a)

T, B, U, A are as in 3.1, 3.2

(b)

K = G9

(c)

O is a T-orbit in 0 T such that FO ^ O.

Let S = &o be the variety associated to the data above as in 4.1(a). The main result of this section is the following. Proposition 8.2. x ( 6 ) A - i = 0. The action 4.1 of HQ on & restricts to the T F -action and to an action of the torus HQ. We shall write H,H° instead of HO,HQ. Using 7.3, we have X ( © ) A - 1 = x(&H )A- 1 - Hence the proposition is a consequence of the following result. Proposition 8.3. Under the hypothesis in 8.1, the fixed point set &H is empty. Proof. We assume that (#,u,0) G &H , and we shall prove that O n GF / 0. Let Q = Z(H°) and let Q be the derived group of Q. Our assumption implies that x G Q,u G G\ since x,u are unipotent, we have x &G,u EG hence (a)

xu G G-

Our assumption also implies that t = for all t G H°, hence H° C To (see 4.1), hence H° C T£. Applying F to the equation t = we have that F(t)F() = F(0) for all t G ff°. But for t G # we have t~lF{t)F() = F(<j)) (by the definition of # , see 4.1) hence F(t)F(<j>) = tF(). Thus tF() = F(<£) for all t G # ° hence H° C T F O and H° C T £ 0 . By the definition of H, we have dimH0 = dimTp0 so that H° = Tjp^. We have FT® - T%Q hence dimT£ = dimT£ 0 = dimtf 0 . Therefore the inclusion H° C To must be an equality Thus (b)

H0 =

T%=T£o.

Our assumption also implies that F<j> = xu<j>. Hence, if / € © T is a representative for <^>, we have Ff = g'fk where g' = xu € Q (see(a)) and ib e K. Hence 6 > ( / - y " F ( / ) ) = f-lgT*F{f). In terms of

(<0

n=

f-0(f)-\

74

GEORGE LUSZTIG

the previous equality can be written (d)

F(n) =

g'ne(9r\(9'€G).

If t € H°, then t € To and t € TFO (see (b)) hence f~Hf G A' and F(f)-HF(f) € A, hence ^(/" 1
FWn^G?.

We now define 0':G -> G by 0'(#) = nO(g)n~l and 0":G -* G by #"(#) = F(n)6>(^)F(n)" 1 . Then 0',0" are involutions since 0(n) = n " 1 . It is clear that O'T = T, 0"T = T and (/)

Te' =To,T°"

=TFa.

In particular, 0' and 0" acts as the identity on H° (see (b)), hence they map Q into itself and Q into itself. Let T = T C\G, a maximal torus of Q. Since T and Q are both 0' and 0"-stable, it follows that T is both 0' and 0"-stable. From (f) it follows that f9' C TG,f9" C T F O hence, taking identity components, and using (b): f9' C H°,f9" C H°. Since f C G, it follows that f*' C H° D 0 , 7 * " C # ° fl 0. But ff°nff is finite since H° is in the centre of Q and Q is the derived group of Q. It follows that f9 , T* are finite. Hence the involutions 0',0" of T must act on f as ^ t~x. We thus have ^ ( ^ n " 1 = r 1 , ^ ) ^ ! ) ^ ) - 1 = Z"1 for all t G T. This implies n'Hn = F ( n ) - ^ F ( n ) for all * G T hence F ^ n " 1 G Z G ( f ) . Using now (e), we deduce that F(n)n-1 G ZG(f). Clearly Zg(f) = T. Hence F ^ n " 1 G T so that F(n) = txn for some ti £ T. Introducing this in (d) we have (g)

h = y'n^')"1^"1,(
We have *x = g'6'(g')-\. Since 0'£ C (/, we see that tx G £. Thus ti G £ flT, hence tfi G T. We can find t2 G T such that t\ — tx. Since 0'(/) = r 1 for all * G T, we have 0'(tf2) = / J 1 hence t2n6(t2)-ln-1 = t20'(t2)~l = t2-t2 = h. We have F(n) = ^ n = ^ n ^ ^ ) " 1 . Using Lang's theorem for T, we write ^2 = F ^ ) - 1 ^ for some ^3 G T. We have F(n) = i ^ a ) " 1 ^ ^ ) " 1 ^ ^ ) ) hence ^ f a ) " 1 G G F . Using (c), we deduce hfO^f)-1 G G F . Let / ' = * 3 /. We have f'K G 0 and hence / ' W 1 € G F , hence F ( / , 0 ( / / ) - 1 ) = f'OtfT1, / ' " ^ C H € G* = A\ Thus F ( / ' t f ) = /'A" so that / ' A is an F r rational point of O. This contradicts our assumption FO ^ O and proves the proposition.

SYMMETRIC SPACES OVER A FINITE FIELD

75

9. Proof of T h e o r e m s 3.3 and 3.4 9.1.

In this section, T, B, U, A are as in 3.1, 3.2.

9.2. We first prove Theorem 3.3 under the assumption that G is simply connected. In this case we have necessarily K = Ge = (G9)0. (See [5,8.2].) Let A be the left hand side of 3.3(a). Using 4.4(a), we see that A = x{Z)\-x (notation of 4.4); here we have used that K = (G°)° (see the hypothesis of 4.4). Using 4.5(b) and 4.6(b) we deduce that (a)

A =

J2x(6o)x-> o

where O runs over the T-orbits on 0 ^ . Using 8.2, we see that the last sum can be restricted to those O which satisfy FO = O. (Here we have used that K = Ge.) For each O such that FO = O, we can express x(&o)\-1 by 7.6. (Here we use that K = GB = (GB)°.) The resulting expression for A is exactly the right hand side of 3.3(a). 9.3. Next we prove Theorem 3.4 under the assumption that G is simply connected. We can write the right hand side of 3.3(a) as

E

\{rlTf)FK\\TF\-'\KF\
We now sum the identities 3.3(a) over all characters A of TF. The sum of the left hand sides is (a)

\TFHA^I"1

£

Tt(9,I%)

gZKF unipotent

since T r ^ i t ^ ) — T r ^ ^ ) f° r 9 unipotent and ^ A T r ( 0 , Rq>) — 0 for g non-unipotent, (see [2,4.2]). The sum of the right hand sides is (6)

where

|rFnA^rV(T) £

A(f)a(Z((f-'Tf)0K))\(f-lTf)FK\

GEORGE LUSZTIG

76

A(f) = 8{A|/ G 0 f A } = 8{A|A(/<'/- 1 ) = €f.1TJ(t'),W

G (r^/)^}

= number of characters of (/~ Tf) (/

-1

whose restriction to

T / ) £ is equal to e / _ 1 T / 1 F 1 1

= l(/- r/) |-|(/- r/)£|=

| T

F|.|

( /

-i

T / )

Fri

Hence the expression (b) is equal to


|*V £ /€©?

This must be equal to (a), and Theorem 3.4 is proved (for G simply connected). 9.4. We now prove Theorem 3.4 for general G. Let G —+ G be the simply connected covering of the derived group of G. Then 0, F,T, {/,... give rise to analogous objects 0, F,T, £/,... for G. We take K = GB. The two sides of the identity 3.4(a) for G coincide with the respective sides of the identity 3.4(a) for G. (We use the fact that the unipotent elements of K are naturally in bijection with those of K, that the Green functions of GF,GF are the same, and the fact that we have a natural bijection GFjfF-^GF/TF.) Since 3.4(a) is already known to hold for G, it follows that it also holds for G. 9.5. We now prove Theorem 3.3 for general G. We shall again make use of the character formula [2,4.2]. We have

\Kf\-1 E

(a)

Tr

(^)

F

g€K

= \i
= |A'

F

r

x£GF

l x~ sx£T

1

E seKF

£ xecF

semis

x

E Tr(«.^r*->,w)

«€K f nzO(i) unip

\Z°G(s)Fr\(x-'sx)\(xTx-y\x

x-iSX£T

E f€Z°G(,)F -f-1xTx~1J


SYMMETRIC SPACES OVER A FINITE FIELD

77

(The last step used 3.4(a) for Z%(s), 0, KnZ%(s),... instead of G, 0, K,....) We now make a change of variables (s,x,f) —• (t)x,,f),x' = f~xx,t = l x~ sx, and use the definition of 6, (see 2.2(b)); the expression (a) becomes (*) |A^|-l|TF|-l

A(tMT)0r(Z((^T^-1)?,))6r/Tr/_1(^^-1)

£ F

t£T x'£GF x'tx''1 eK e(x'Tx'~1)=x'Tx'~1

We now make a change of variables (t,x') —• (i,x'),t the expression (b) becomes (c)

\KF\-*\TF\-i

cr(T)a(Z((x'TxrlfK))

Y, x'eoF

= x'tx'

, and

x

d(x'Tx'~1)=x'Tx'~1

x

Y:

A(*'-V) W I ( <).

teix'Tx'-1)^ Since e is a homomorphism (2.3(c)), the last sum is \(x'Tx' ) £ | if ^\(x'Tx'~1)F ~ e IT i-1 anc * *s ^ otherwise; hence (c) is equal to the right hand side of 3.3(a). Thus, Theorem 3.3 is proved. Using the remarks in 7.3 we can rewrite the identity 3.3(a) in the following form. X

Corollary 9.6.

\KF\-* J2 ^g^) gZKF

= Y,\TF\dF/I
where O runs over all T-orbits in 0 y such that OF ^ 0 and \\TF = lQ (notations of 7.1). In the case where K = (G')°, we can replace \TF\6F/KF\ by \ o\ ' l T o F | _ 1 ( s e e 7-4)> w h i c h i s a P ° w e r o f 2 > s i n c e T0/To is a 2-group. T

10. Non singular characters 10.1. Let J be as in 1.5. It is known that for a 0-stable maximal torus T": (a) T' G J if and only if the involution induced by 6 on the set of coroots (or roots) with respect to T' is fixed point free.

78

GEORGE LUSZTIG

10.2. Let T be an F-stable maximal torus of G and let X:TF —• Qt be a character. Following [2,5.15], we say that A is non-singular if the following condition is satisfied: (a) for any coroot h:Fq —• T and any (or equivalently, some) integer n > 1 such that Fnh(x) = h(xqn) for all x G F* n , the restriction of A -AT to h(F*n)(C Tpn) is non-trivial. (N is the norm T F * -* T F ) . In this section we shall assume that A is non-singular. It is known that this implies that F is an actual representation of G and that "most" irreducible F representations of G are of this form. We state the main result of this section. Theorem 10.3. Recall that A is non-singular. The space of KFinvariant vectors of a(T)a(G)R^ is non-zero if and only if (T, A) is GFconjugate to a pair (T", A') such that T' G J and \'\T,F = e T /. This result confirms in part a conjecture in [1,6.7]. For the proof, we need two lemmas. Lemma 10.4. If f e 0f |A then V = f~lTf P

G J, (see 10.1).

/—i

*

Proof. Let A': T" —» Qt be given by A' =1 A (see 3.1). Assume that there exists a coroot ft: Fq —• T such that ft = 6oh. Let n > 1 be an integer such that Fnh(x) = h(xqn) for all x E Fq. Now A' is non-singular, hence there exists x G F*» such that A'(7V(ft(x))) / 1, where A^T'*" -*• T'F is the norm map. By our assumption on / , we have \'\T,F = eT,. Using 2.3(b) it follows that A'|(T£) F = 1. Let N'.T -* T be the homomorphism defined by N(t) = tF(t) • • • n l F ~ {t)- then N = 7V|T' F \ Clearly, N(Tg) C T$ (since 7 $ is F-stable), hence N((T'°)Fn) C 7 $ . We then have N((Tg)F*) = iV((r^ 0 ) F n ) C F T / F n ^ o _ ( T ^ ) . Since 0 o ft = ft, the image of ft is a connected group contained in T'9 hence in 7 $ , hence /i(F*») C ( T # ) F n . We then have 7V(ft(F;,)) C N{{T'«)Fn)

C (T*)F.

In particular, N(h(x)) G (Tj£) F . Since A' is trivial on ( T £ ) F , we have X'(N(h(x))) = 1. This contradiction proves the lemma (by the characterization 10.1(a) of J ) . L e m m a 10.5. Let V G JyFT

= T'. 77>en
79

SYMMETRIC SPACES OVER A FINITE FIELD

Proof. Choose a Borel subgroup B' containing X" such that B' is opposed to 0B'. Let U' be the unipotent radical of B' and let U — U' C\ Z(TK). Using 2.1(a) we see that it is enough to prove the congruence (a)

dim(U'/U' n FU1) = dim(u'/Tj'

n Flf)

(mod 2).

Let S be the set of all roots with respect to X"; it has a natural action of 0. Let S\ (resp. £2) be the set of roots corresponding to U' (resp. FU'). Since B1 is opposed to 6B' (hence FB' is opposed to 0(FB')), the involution 0:S —+ S,0a = 0{a)~l leaves Si and S2 stable. Let S' be the set of all roots a in S which are trivial on T£, i.e such that 0a — a~l; S' is the fixed point set of 0:S —• S. Then S'DSi (resp. S'flc^) are the roots corresponding to if (resp. Flf). NOW (a) is equivalent to \SinOS2\ = |5 / n5infl5 2 |(mod 2) and also to \(S - S')H Sx H 0S2\ = 0 (mod 2). The involution 6 leaves (S — Sf) f] Si D 0S2 stable and is fixed point free on it; hence this set has an even number of elements. The lemma is proved. 10.6. We now prove Theorem 10.3. Using 10.4 and 10.5 we can rewrite 3.3(a) in our case as follows: (a)

dim((a(T)a(G)R^)KF)

= number of double cosets

TF\0^x/KF.

This, together with 10.4, clearly imply 10.3. 10.7. Let V E J, FT' = V. Applying 10.5 to G and to Z%(t), (t e T'K ) we see that (a)

eT, (t) =
11. An e x a m p l e 11.1. In this section we shall assume that G is simply connected, i^-split and that there exists a maximal torus T of G such that 0t = t"1 for all t £T. (The maximal tori with this property will then form the set J of 1.5.) We fix T e J with FT = T. T h e o r e m 11.2. Lei X:TF -> Qt be a non-singular character. The dimension of (a(T)o-(G)R^)KF is 0 if X\Tg ^ 1 and is |T£| if \\T% = 1.

GEORGE LUSZTIG

80

(Note that Tg = {te TF\t2 = 1}.) The proof is based on the following result. L e m m a 11.3. cT, = 1 for any T' G JF. Proof. The definition of cT, shows that it is enough to prove that for any g G GF such that g = g~l we have a{Z^{g)) =
= {? e ( r 1 T / ) F | < ' 2 = 1} = ri{t€TF\t2

= i}f

F

= r'T f hence the condition f A | ( / _ 1 T / ) £ = 1 is equivalent to the condition \\TF = 1. Thus 6 f A is empty if A|T£ ^ 1. Assume now that A|T£ = 1. Then 0 £ A = i? F where # = {/ G G\f~lTf G J } . It remains to show that (a)

TF\RF/KFhas

\Tg\ elements.

Since K acts transitively on J, we have R = N(T)K where AT(T) is the normalizer of T in G. Let 5 be a Borel subgroup containing T and let n G N(T). Applying 1.5(a) to (T C 5 ) G 7>, (T C nBu'1) G P , it follows that n e TK. Thus JV(T) C TK, hence R = T • K. Now T x A' acts transitively on R by (£,&):r —» trk~l and the isotropy group of 1 is T # . Since T x if is connected, it follows that TF\RF jKF is in bijection with the cokernel of 1 — F:TK —• TR-, hence has the same number of elements as T £ . The completes the proof of (a) and of the theorem.

SYMMETRIC SPACES OVER A FINITE FIELD

81

REFERENCES [1] E. Bannai, N. Kawanaka, S.Y. Song, The character table of the Hecke algebra H(GL2n(Fq),Sp2n{Fq))} preprint. [2] P. Deligne, G. Lusztig, Representations of reductive groups over finite fields, Ann. Math. 103 (1976), 103-161. [3] G. Lusztig, Representations of finite Chevalley groups, C.B.M.S. Regional Conf. Series in Math. 39, Amer. Math. Soc, 1978. [4] T.A. Springer, AIgebraic groups with involutions, in Algebraic Groups and Related Topics, Adv. Studies in Pure Math 6, Kinokuniya and North Holland, 1985. [5] R. Steinberg, Endomorphisms of linear algebraic groups, Mem. Amer Math. Soc. 80 (1968).

Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139

Le theoreme de positivite de l'irregularite pour les Px-modules

ZOGHMAN MEBKHOUT dedie a Alexandre Grothendieck a Voccasion de son soixantieme anniversaire.

Sommaire 0. 1. 2. 3. 4. 5.

Introduction Rappels de quelques resultats Le theoreme de positivite Le cycle d'irregularite Application au theoreme de semi-continuite Le theoreme des coefficients de de Rham-Grothendieck en caracteristique nulle 6. Filtrations du faisceau d'irregularite Bibliographie 0. Introduction Soient X une variete algebrique complexe non singuliere, Z une sous variete de X et M, un coefficient de la categorie D\(Vx). Dans ce travail nous definissons le complexe d'irregularite IRz(M) de M. le long de Z qui est alors un complexe de faisceaux d'espaces vectoriels complexes algebriquement cons true tible et nous montrons que si M est un T>xmodule holonome et Z est une hypersurface de X le complexe IRz(M) est un faisceau pervers sur Z. Son cycle caracteristique qui est defini de fagon purement algebrique a priori comme difference de deux cycles est alors un cycle lagrangien positif du fibre cotangent T*X. Si X est une surface de Riemann la dimension de Pespace vectoriel complexe IRz(M) pour tout point Z de X est egale au nombre classique de Fuchs attache

84

ZOGHMAN MEBKHOUT

a la singularity Z de M en vertu du theoreme de Malgrange [Mi], [M2]. Ainsi le faisceau IRz(M), objei d'une categorie derivee, apparait comme la generalisation vraiment naturelle du nombre de Fuchs en dimension superieure et rend compte de la ramification de M en tout point de Z simultanement. Le theoreme de positivite est la base de toutes les proprietes de la ramification sur le corps des nombres complexes et en particulier de la ramification moderee. La chose la plus remarquable sans doute est qu'il se substitue avantageusement au theoreme d'Hironaka [H] sur la resolution des singularites dans la demonstration du theoreme de comparaison de Grothendieck [Gi] pour la cohomologie de de Rham d'une variete algebrique complexe non singuliere et plus generalement dans la demonstration du theoreme des coefficients de de Rham-Grothendieck [G2] sur le corps des nombres complexes. D'ailleurs le faisceau IRz(Ox), & priori non nul comme nous le definissons ici, apparait tout a fait explicitement dans la demonstration de Grothendieck [Gi] comme l'obstruction au theoreme de comparaison entre cohomologies de de Rham d'une variete algebrique complexe. Cette idee de localisation d'un probleme en apparence de nature globale s'est revelee comme le point crucial dans l'etude de la cohomologie des varietes algebriques et a ete reprise a maints endroits. Cependant ceci va aussi a l'encontre de ce que pensait Grothendieck lui-meme [G3] a savoir que la resolution des singularites est vraiment au fond du probleme dans l'etude de la cohomologie des varietes algebriques. Ceci amene a penser que le theoreme de resolution des singularites n'est pas au fond du probleme pour la cohomologie de de Rham aussi bien en caracteristique pure p > 0 qu'en inegale caracteristique et invite a chercher des methodes directes de demonstration. La theorie des T>x-modules et ses succes sont avant tout une application de la theorie des categories triangulees et des categories derivees. L'harmonie interne et la portee de cette derniere theorie etaient quelque peu masquees par la difficulte du theoreme d'Hironaka. En effet le point crucial dans les demonstrations, une fois tout devissage effectue, rest ait le theoreme d'Hironaka. Ce faisant on perdait invariablement beaucoup d'informations dans 1'application systematique de ce theoreme qui a fini par denaturer certaines questions concernant les singularites. Le theoreme de positivite qui est une autre application de la theorie des categories triangulees et des categories derivees montre qu'en ce qui concerne la theorie de de Rham-Grothendieck en caracteristique nulle le theoreme de resolution des singularites n'est pas indispensable. A cette occasion particuliere le lecteur nous permettra de rendre hommage a cette theorie et a son createur. La motivation du theoreme de positivite etait le theoreme de semicontinuite de Pirregularite d'une famille d'equations differentielles [Me4]

LE THEOREME DE POSITIVITE DE L'IRREGULARITE

85

analogue au theoreme de semi-continuite de Deligne du conducteur de Swan [Li], Voici le contenu de ce travail. Dans le §1 nous rappelons quelques resultats generaux de la theorie des £>x-modules utilises dans la suite. Dans le §2 nous montrons le theoreme de positivite et ses consequences immediates. Dans le §3 nous definissons le cycle d'irregularite et nous montrons une formule de type Riemann-Roch exprimant la caracteristique d'Euler-Poincare de la cohomologie de de Rham d'un fibre a connexion integrable sur une variete ouverte non singuliere sur un corps de caracteristique nulle a l'aide de sa ramification a l'infini. Dans le §4 nous precisons le theoreme de semi continuity de Pirregularite [Me4] a partir du theoreme de positivite. Dans le §5 nous indiquons comment le theoreme de positivite remplace avantageusement le theoreme d'Hironaka dans la demonstration du theoreme des coefficients de de Rham-Grothendieck sur le corps des nombres complexes. Enfin dans le dernier § nous definissons une filtration du faisceau d'irregularite par des sous-faisceaux indexes par des nombres reels > 1 mais dont les sauts locaux sont des nombres rationnels : les pentes critiques locales du faisceau IRz(M). La situation est toute pareille a celle du theoreme de Hasse-Arf pour la filtration (numerotation superieure) du groupe de Galois d'une extention galoisienne finie d'un corps local [Sex]. Les resultats de ce travail ont ete annonces dans une Note aux Comptes rendus de l'Academie des Sciences : C. R. Acad. Sci. Paris, 303, serie I, 1986, 803-806. Terminologie: Le lecteur trouvera peut-etre paradoxal de qualifier le faisceau IRz(M) (qui n'a rien de pathologique) "d'irregulier". II serait plus correct de Pappeler faisceau de ramification (qui n'a rien de sauvage) comme dans la theorie des corps locaux. Mais sur le corps des nombres complexes la terminologie "point singulier irregulier" est profondement enracinee dans la theorie des equations differentielles depuis le siecle dernier qu'il parait difficile de changer de terminologie. Nous utilisons dans cet article la terminologie de la theorie de Grothendieck, par exemple on appelera coefficient constructible un objet de la categorie Dhc(Cx) pour mettre en valeur son caractere intrinseque. Une partie de ce travail a ete realise pendant un sejour en 1986 a l'Institut for Advanced Study de Princeton. Nous remercions les membres de cet institut de leur hospitalite. Nous remercions B. Malgrange des discussions que nous avons eues a cette occasion. II nous a fait savoir qu'il avait conjecture de son cote le theoreme de positivite. Nous remercions Y. Laurent des nombreuses discussions a propos de ses travaux. Nous remercions enfin J . J . Sansuc d'avoir relu le manuscrit.

86

ZOGHMAN M E B K H O U T 1. R a p p e l s d e q u e l q u e s r e s u l t a t s

Dans ce paragraphe nous allons rappeler quelques resultats generaux de la theorie des X>x~ m °dules utilises dans cet article. Nous renvoyons le lecteur a [Mea], par exemple, pour leurs demonstrations et leurs references. 1.1. Soit (X,Ox) une variete algebrique non singuliere sur un corps de caracteristique nulle ou une variete analytique complexe. On note T>x le faisceau des operateurs differentiels d'ordre fini a coefficients dans Ox defini et note Viffx/ki^x) dans [EGA, IV, §16] pour tout espace annele (X,Ox) en fc-algebres sur un corps k. Le faisceau T>x est un faisceau d'anneaux non commutatifs coherent a d m e t t a n t une filtration Z>j, / G N par les faisceaux des operateurs differentiels d'ordre < /. Le spectre analytique du faisceau gradue de O^-algebres commutatives gr(X>A') : = iVi+i/Vi est egal au fibre cotangent w : T*X —> X. E t a n t donne un £>x-module (a gauche) A4, une filtration croissante Aik par des 0 x - m ° d u l e s coherents telle que M. = UkMk est dite bonne si ViMk C Mi+k pour tout / G N et tout k G N avec egalite des que k est assez grand. Un D ^ - m o d u l e M est coherent si et seulement s'il admet lo~ calement une bonne filtration. Le faisceau gradue gr(A4) := ®iM\+\/Mi associe a une bonne filtration de M est un gr(X ) A') _ m 0 ( lule coherent et le cycle du fibre cotangent T*X associe au faisceau gr(A4) ne depend pas de la bonne filtration choisie. Nous noterons CCh(«M) ce cycle que Ton appelle cycle caracteristique de A4. Si done M. est un X>A'- m °dule coherent, CCh(A^) est un cycle homogene posit if globalement defini. Son support note Ch(A'f) est par definition la variete caracteristique de M. Pour tout Z>x- m °dule coherent M. non nul, on a Pinegalite de Bernstein dim(Ch(A'f)) > d i m ( X ) . En fait Tinegalite de Bernstein est aussi consequence du theoreme de l'involutivite des caracteristiques, cf. [G], mais elle est bien plus elementaire, cf. ([M4] ou [Mea]). On deduit de l'inegalite de Bernstein que la dimension homologique des fibres du faisceau T>x est egale a la dimension de X. Definition 1.1.1. On dit qu'un Z>x-rciodule M est holonome s'il est coherent et tel que dim(Ch(A^)) = d i m ( X ) ou s'il est nul. La terminologie holonome est impropre, le lecteur preferera peut-etre dire module de dimension minimale. On note Mh(Z>x) la categorie des X>x - m °dules holonomes. C'est une sous-categorie pleine de la categorie des £>x-rciodules qui est abelienne et stable par extension. Si (1.1.1)

0 — -Mi

-+M-^M2-+0

est une suite exacte de DA"-niodules holonomes on a l'egalite entre cycles caracteristiques

LE THEOREME DE POSITIVITE DE LIRREGULARITE

(1.1.2)

CCh(M)=

CCh(Mi)

+

87

CCh(M2).

On note Dbh(T>x) la sous-categorie de D(T>x), categorie derivee de la categorie des Z>x-modules a gauche, des complexes bornes a cohomologie holonome. La categorie Dbh(T>x) est une sous-categorie pleine et triangulee de D(T>x)1.2. Notons LOX le faisceau sur X des formes differentielles de degre maximum. C'est de fagon naturelle un T>x-mod\i\e a droite. Si M est un Dx- m °dule a gauche holonome on a ExtlVx(M,T>x) = 0 si i ^ dim(X) et Ext^ (M,Vx) est un £>x-module a droite holonome ou n := dim(X). Notons Mu le Vx-module a droite Ext^x{M,Vx) et M* le Vx-module a gauche homoxiwxiM"). Le foncteur de dualite M —• M* est une anti-involution de la categorie des Z>x-modules a gauche holonomes dans elle-meme. 1.3. Dans la situation de 1.1, si Z est une sous variete de X definie par un Ideal 2^, on definit deux foncteurs exacts a gauche de cohomologie locale algebrique de la categorie des 2>x- m °dules a gauche dans elle-meme en posant (1.3.1)

M(*Z)

:= \imhom0x

(lkz,M)

k

(1.3.2)

(Ox/lkz,M)

algr z (,M) := limhom0x k

pour un £>A'-module M a gauche. Ces deux foncteurs se derivent a droite pour donner naissance a des foncteurs da la categorie Dh(T>x) dans ellememe M —*• RM(*Z) et M —» Ra\gTz(M). De plus on a un triangle distingue de la categorie Db(T>x) (1.3.3)

i2al g r z (.M) -+M-+

RM(*Z)

muni d'un morphisme naturel dans le triangle de cohomologie locale topologique (1.3.4)

1

RTz(M)-^M-+RjJ-

M.

Dans le cas algebrique, si M est a cohomologie Ox quasi-coherente, le morphisme precedent entre (1.3.3) et (1.3.4) est un isomorphisme.

88

ZOGHMAN MEBKHOUT

T h e o r e m e 1.3.5. Si M. est un complexe de la categorie D\(T>x) le triangle (1.3.3) de cohomologie locale algebrique est un triangle distingue de la categorie Dbh(T>x)Le theoreme 1.3.5 resulte de l'equation fonctionnelle de Bernstein-Sato, cf. ([K3], [NMi]). On deduit du theoreme 1.3.5 que si / : X' -+ X est morphisme de varietes non singulieres l'image inverse (1.3.6)

Lf*(M)

:= Ox*

I

f~\M)

f-HOx)

d'un complexe M de la categorie Dhh{Vx) est un complexe de la categorie Dhh(Vx')- En particulier le produit tensoriel interne (1.3.7)

Mi

® M2 Ox

de deux coefficients M\ et M.o de la categorie Dhh{T>x) est encore un coefficient de la categorie Dhh{Vx)1.3.8. Si Z est une sous-variete de X et si on pose Z' := / _ 1 ( Z ) et M' '.— Lf*(M), on a un morphisme de triangles de la categorie Dbh(T>x') Lr(Ra\gTz(M))

JRalg^ z #(^f , )

• Lf*(M)

•

Lf*(RM(*Z))

>

>

RM'{*Z')

M'

qui est un isomorphisme. 1.4. Supposons que X est une variete complexe munie de la topologie transcendante. On note Dhc(Cx) la categorie triangulee des coefficients complexes algebriquement constructibles dans le cas algebrique et analytiquement constructibles dans le cas analytique. Le faisceau Ox est de fagon naturelle un 'D^-module a gauche. On peut done considerer les deux foncteurs exacts, Tun covariant, l'autre contravariant de la categorie Db(Vx) dans la categorie Db(Cx) (1.4.1)

DR(M) :=RhomVx

(1.4.2)

S(M) := RhomVx

(Ox,M) (M,Ox).

Le X>x- m °d u l e a gauche Ox est auto-dual et si M est un complexe de la categorie Db(T>x) dont la cohomologie est £>x-o)herente, on a des isomorphismes canoniques

LE T H E O R E M E DE P O S I T I V I T E D E L ' I R R E G U L A R I T E

(1.4.3)

89

DR(M)=S(M*)

(1.4.4)

S(M)*DR(M*).

On a alors le theoreme de constructibilite [K2]: T h e o r e m e 1.4.5. Si M. est un complexe DR(M) et S(M) sont constructibles.

de Dhh(Vx),

les complexes

Les foncteurs DR et S envoient done la categorie Dbh(T>x) dans la categorie Dbc(Cx)Si M est un X ^ - m o d u l e holonome, les faisceaux de cohomologie des complexes DR(M) e t S(M) sont concentres entre les degres 0 et d i m ( X ) . De plus on a la proposition suivante [ K 2 ] : P r o p o s i t i o n 1.4.6. Si M. est un Vx-module holonome, la dimension du support des faisceaux ExtlVx(Ox,M) et ExtlVx(M,Ox) est inferieure ou egale a d i m ( X ) — i pour tout i compris entre 0 et d i m ( X ) . Le lecteur trouvera dans [NM2] des demonstrations geometriques elementaires du theoreme de constructiblite 1.4.5 et de la proposition 1.4.6, independantes de tout theoreme d'analyse. 1.5. On suppose toujours que X est une variete complexe. On a alors le theoreme de dualite locale [MeJ : T h e o r e m e 1 . 5 . 1 . Pour tout complexe un morphisme canonique de dualite DR(M)->

RhomCx

M de la categorie Dh(Vx)

on a

(S(M),CX)

qui est un isomorphisme si A4 est un complexe

de la categorie

Dbh(T>x)-

Autrement dit le foncteur Ai —• M* est compatible a la dualite locale des complexes constructibles. Ceci n'est pas vrai si on enleve l'hypothese d'holonomie. Definition 1.5.2. On dit qu'un coefficient constructible T a la propriete de support si hl{T) est nul pour i £ [0,dim(A")] et si la dimension du support du faisceau hl(T) est inferieure ou egale a d i m ( X ) — i pour tout ie [0,dim(X)]. On note h%(T) le i-eme faisceau d'un complexe T. Si done M est X>x-niodule holonome les complexes DR(M) et S(M) ont la propriete de support en vertu de 1.4.6. Definition 1.5.3. On dit qu'un coefficient constructible T a la propriete de co-support si le complexe dual Tv \— Rhomcx{T,Cx) a la propriete de support.

ZOGHMAN MEBKHOUT

90

Si done M est un T>x-module holonome les complexes DR(M) et S(M) ont la propriete de co-support en vertu du theoreme de dualite locale 1.5.1. On pose alors cf. [BBD] ou cette terminologie est proposee: Definition 1.5.4. On dit qu'un coefficient constructive T est un faisceau pervers s'il a les conditions de support et de co-support. Si done M est un Vx-module holonome les complexes DR(M) etS(M) sont des faisceaux pervers. On note Perv(Cx) la categorie des faisceaux pervers. C'est une sous categorie pleine de la categorie Dbc(Cx)- Le couple Perv(Cx), Dbc(Cx) a les proprietes du couple Mh(X>x), Dhh(Vx). Par exemple on a alors le result at suivant, cf. [BBD], : T h e o r e m e 1.5.5. La categorie Perv(Cx) est une sous categorie abelienne de Dhc{Cx) e£ un triangle distingue de Dhc{Cx) donne naissance, a cote de la suite longue de cohomologie ordinaire, a une suite longue de cohomologie perverse. De plus la categorie est un champ cJest-a-dire que ses objets et ses morphismes sont de nature locale, tout comme les objets et les morphismes de la categorie Mh(X>x)Notation 1.5.6. On note ^(J7) le i-eme faisceau de cohomologie perverse d'un coefficient constructible T. Si T est un coefficient constructible sur une sous-variete Z de X on dit que c'est un faisceau pervers sur Z si le coefficient T\—codimx^] vu comme complexe de la categorie Dbc(Cx) appartient a Perv(Cx)- On note Perv(Cz) la categorie des faisceaux pervers sur Z. C'est une sous categorie-abelienne pleine de la categorie Dhc(Cz)2. Le t h e o r e m e de positivite de Pirregularite On utilise les notations du paragraphe 1. 2.1. Enonces. Soit X une variete complexe (algebrique ou analytique) non singuliere et Z une sous variete fermee de X de codimension p. Notons j :U :=X -Z

CX D Z :i

les inclusions (transcendantes) canoniques. Soit M un complexe de la categorie Dhh(T>x)> On pose (2.1.1)

IRz(M)

:=

KTz(DR(RM(*Z)))[i\

(2.1.2)

IRz(M)

:=

rlS(RM{*Z))\p).

Les morphismes canoniques de complexes de Px- m °dules

LE THEOREME DE POSITIVITE DE L'IRREGULARITE

91

RM(*Z)^Rj*j~1M Ra\gTz{M)

—M

donnent naissance aux morphismes de complexes constructibles (2.1.3)

DR(RM(*Z))

(2.1.4)

i~lS(M)

Rj,rlDB{M)

—

-

S(Ra\gYz{M))

de sorte que le complexe IRz(M), resp. IR^(M), est isomorphe au cone du morphisme (2.1.3) decale de p vers la gauche, resp. du morphisme (2.1.4). On a ainsi defini de fac,on naturelle un foncteur covariant exact IRz et un foncteur contravariant exact IR% de categories triangulees de la categorie Dbh(T>x) dans la categorie Dhc(Cz)Proposition 2.1.5. Si M. est un complexe de la categorie Dhh(Vx) les coefficients constructibles IRz(M)[—1] et IRz(M)[—p] de la categorie Dhc(Cx) sont en dualite :

IRuz(M)[-p]~RhomCx P r e u v e de 2.1.5. parce que

(IRz(M)[-l],Cx).

Cela resulte du theoreme de dualite locale 1.5.1

DR(R*\gYz(M))

^ l

(S(Ra]gTz(M))Y

RYz{DR{M))^{i- S{M)Y

.

T h e o r e m e 2.1.6. Si Z est une hypersurface, c'est-a-dire defini localement par une equation, et si M est un Vx-module holonome, les coefficients IRz(M) et IR^(M) sont des faisceaux pervers sur Z. Le preuve du theoreme 2.1.6 sera donnee en 2.2. En vertu du theoreme 2.1.6 les foncteurs IRz et IRZ envoient la categorie abelienne Mh(T>x) dans la categorie abelienne Perv(C^) si Z est une hypersurface. Corollaire 2.1.7. Si Z est une hypersurface les foncteurs IRz et IR% sont des foncteurs exacts de categories abeliennes entre Mh(£>A') et Perv(Cz). Preuve de 2.1.7. En effet si 0 —• M\ —• M —• M2 —• 0 est une suite exacte de la categorie Mh(T>x) on a par construction les deux triangles distingues de la categorie Dbc(Cz) 0 -> IRz(Mi)

— IRz(M)

— IRZ(M2)

—0

92

ZOGHMAN MEBKHOUT 0 — IRVZ{M2) — IRz(M)

— IRg(Mi)

— 0.

Prenons la suite longue de cohomologie perverse dans les triangles precedents; on trouve en vertu du theoreme 2.1.6 qu'on a, en fait, affaire a des suites exactes courtes de la categorie abelienne Perv(Cz). En particulier le faisceau IRz(M) est nul si et seulement si les faisceaux IRz{M\) et IRz{M2) sont nuls. Le lecteur pourra mesurer le chemin parcouru s'il se rappelle qu'un resultat bien plus faible que ce dernier [Me2] reposait de fa^on essentielle, avant qu'on dispose du theoreme de positivite, sur le theoreme d'Hironaka. Corollaire 2.1.8. Si M est un complexe de la categorie Dbh(T>x) et si Z est une hypersurface, on a des isomorphismes canoniques de faisceaux pervers, pour tout i : 'tiilRziMfi^IRziViM))

'h^mziMVzmsih'iM)). P r e u v e de 2.1.8. On raisonne par recurrence sur l'amplitude de M. Si M. est concentre cohomologiquement en un seul degre c'est le theoreme 2.1.6. Dans le cas general, soit hk(M) le dernier faisceau de cohomologie de M. Le complexe M s'envoie sur son dernier faisceau de cohomologie et on a un triangle distingue de la categorie Dbh(T>x) M'

^M->hk{M)

ou l'amplitude de M! est egale a celle de M diminuee d'une unite. D'ou le triangle distingue de D\{Cz) IRz(M')

- IRz(M)

-

IRz(hk(M)).

Prenons la suite longue de cohomologie dans le premier triangle et appliquons le foncteur exact IRz puis la suite longue de cohomologie perverse dans le second triangle. En comparant les deux suites longues ainsi obtenues on obtient le corollaire 2.1.8 a partir de l'hypothese de recurrence. Le deuxieme isomorphisme se demontre de la me me fagon. En particulier pour un complexe M. de la categorie D\(Vx) l e complexe IRz(M) est nul si et seulement si les faisceaux IRz(hl(M)) sont nuls pour tout i. Soient M un complexe de la categorie Dbh(T>x) et M* := homCx

(uXiRhomVx

(M,Vx))[dim(X)]

son complexe dual. Pour toute sous-variete Z les coefficients constructibles IRz(M) et IRz(M*) ne sont pas en dualite. Cependant notons x{F) ' a fonction sur X d'Euler-Poincare d'un coefficient constructible T:

LE THEOREME DE POSITIVITE DE L'IRREGULARITE

X(^)(x):=^(-l)'dimc/*

i

93

(^)x.

i

Theoreme 2.1.9. Pour toute sous-variete Z de X et tout complexe de M de la categorie Dhh(T>x) on a l'egalite entre fonctions constructibles complexes sur X x(IRz(M))

=

x(IRz(M*))-

La preuve de 2.1.9 sera donnee en 2.3. Corollaire 2.1.10. Si Z est une hypersurface et si M est un T>x~ module holonome, alors le faisceau IRz(M) est nul si et seulement si le faisceau IRz(M*) est nul. Preuve de 2.1.10. En effet un faisceau pervers est nul si et seulement si sa fonction d'Euler-Poincare est nulle. Definition 2.1.11. Si Z est une sous variete de X et si M est un complexe de la categorie D j ( D x ) , on appelle complexe d'irregularite de M le long de Z les coefficients constructibles IRz(M) et IR^(Ai). Definition 2.1.12. Si Z est une hypersurface de X et si M est un T>xmodule holonome, on appelle faisceau d'irregularite de M le long de Z les faisceaux IRZ(M) et IR%(M). Exemple 2.1.13. Si X est une surface de Riemann, Z un point de X et P := P(x, -£;) est une equation differentielle non nulle sur X qui a un point singulier en Z le nombre de Fuchs de P en Z qui est defini purement algebriquement est egal a la dimension de l'espace vectoriel IRz(M) pour M. := T>x/{P) ou (P) est l'ideal a gauche de T>x engendre par P. C'est le theoreme de comparaison de Malgrange ([Mi], [M2]). Exemple 2.1.14. Si X est une surface complexe non singuliere, M un X>x-module holonome qui est lisse en dehors d'une courbe Y sur X et Z une courbe sur X coupant Y en un seul point, alors le faisceau IRz(M) est un faisceau ponctuel place en degre un. C'est le theoreme 3.1 de [MeJ qui est done un cas particulier du theoreme de positivite. Le lecteur trouvera des exemples de faisceaux d'irregularite dans [Me^. Remarque 2.1.15. Si Z est une sous-variete de X de codimension plus grande que 1 le complexe IRz(M) n'est pas en general un faisceau pervers. Par exemple le lecteur trouvera des exemples dans [Me4] ou IRz(M) n'est pas concentre en un seul degre avec dim(X) = 2 et Z est un point de X. 2.1.16. Soit / : X —> C une fonction holomorphe non const ante sur le plan complexe C et Z l'hypersurface / _ 1 ( 0 ) . Fixons les notations a l'aide du diagramme suivant

ZOGHMAN MEBKHOUT

94

X* -^X" ::=

x--z

tx

i

DZ

I i is U * lr c* ^C* :--= c--0 c c D 0, ou C* un revetement universel de C*. Rappelons que le faisceau tff(Ox) sur Z de fonctions multiformes est defini dans [SGA 7, XIV] par

Vf(Ox):=rlj*P*p-1rlOx. C'est un T>x-mod\i\e a gauche muni d'une action de la monodromie notee T et qui commute a Taction de T>x. II contient comme sous-X>x-module a gauche muni d'une action de la monodromie le faisceau des fonctions multiformes de determination finie $/(Ox)C'est le faisceau des germes des fonctions multiformes dont Taction de T admet un polynome minimal. On a alors le theoreme suivant: T h e o r e m e 2.1.17. Pour tout polynome P G C[T] non nul Paction de P(T) sur les T>x-modules a gauche munis d'une action de la monodromie ^j(Ox) et ^ I (Ox) est surjective et pour tout complexe M. de la categorie Dhh(T>x) le morphisme canonique RhomVx

{M^J

(OX)) -> RhomVx

(M,*j

(Ox))

est un isomorphisme. P r e u v e de 2.1.17.

Voir [Me5], §4.

2.1.18. Le faisceau ^ /(Ox) contient comme sous-T>x-module a gauche muni d'une action de la monodromie le faisceau Nils/((9x) des fonctions a croissance moderee, dites de classe de Nilson. Le cone du morphisme

RkomVx (A^Nilsy (Ox)) -^ WiomVx (M,¥J

(OX))

pour un Tx-module holonome M n'est pas un faisceau pervers. Cependant il apparait comme "limite inductive" de faisceaux pervers. De fagon precise on a les isomorphismes Nils/(Ox) — lim Ker(P, Nils/(0x)) et yf(Ox) = lim Ker(PyVf(Ox)) ou la limite inductive est prise selon Tensemble filtrant des polynomes P de C[T] ordonnes par la divisibility. Mais pour chaque polynome P fixe le X>c-module Ker(P, Nilsjd (C?c)) est somme directe de £>c-modules qui sont extention successive de connexion meromorphes regulieres de rang un [SGA 7, XIV]. C'est done un VQmodule holonome. Le X>x-m°dule Ker(P, Nils/(Ox)) est image inverse

LE T H E O R E M E DE P O S I T I V I T E DE L ' I R R E G U L A R I T E

95

par la fonction / de Ker(P, Nils ld (C?c)) [SGA 7, XIV] c'est done un module holonome [K3] (voir aussi [NM2]). On peut aussi invoquer K e r ( P , N i l s / ( O x ) ) est un C?A r (*Z)-module localement libre de type [SGA 7, XIV] muni d'une action de Vx et est done holonome loc. On a alors le theoreme : T h e o r e m e 2 . 1 . 1 9 . Pour tout polynome AA le cone du morphisme canonique RhomVx

(M,

Ker(P,

Nils, (Ox)))

P et tout Vx-module

-+RhomVx

(M,

Vxque fini cit.

holonome

Ker (p,vf

(Ox)))

est un faisceau pervers sur Z. P r e u v e de 2.1.19. En effet, par construction, le cone du morphisme du theoreme (2.1.19) est isomorphe hIRz(M* ox Ker(P, N i l s , ( 0 x ) ) ) ce qui le fait apparaitre comme faisceau d'irregular ite d'un Vx -module holonome le long d'une hypersurface Z. On reduit alors au theoreme 2.1.6. 2.2. Demonstration du theoreme 2.1.6. La question est locale pour la topologie transcendante. On peut done supposer que X est une variete analytique complexe et que M est un X>x-niodule holonome analytique. 2.2.1.

Preliminaires.

L e m m e 2.2.1.1. POUT tout Vx-module holonome M il passe par tout point en dehors d'une partie analytique de X de dimension nulle une hypersurface de X lisse et non caracteristique pour Af. Preuve de 2.2.1.1. Rappelons qu'on dit qu'une sous-variete lisse X' de X est non caracteristique pour un £>x-module coherent Af si la variete caracteristique Ch(Af) de Af ne coupe le conormal T £ / X de X1 dans X que le long de la section nulle du fibre cotangent T * X . Si Af est un Vx~ module holonome, sa variete caracteristique Ch(A r ) n ' a localement qu'un nombre fini de composantes irreductibles qui en plus sont de dimension au plus egale a d i m ( X ) . Ces composantes qui sont homogenes se projettent sur X en des sous varietes de X. La partie analytique du lemme 2.2.1.1 est fournie par la reunion de ces projections qui sont de dimension nulle. En effet la fibre de C\i(Af) au dessus de tout point en dehors de cette partie n'est pas egale a toute la fibre du fibre cotangent T*X au dessus du meme point pour des raisons de dimension. Prenons un vecteur cotangent non nul au dessus de ce point qui n'est pas dans Ch(A^); alors toute hypersurface passant par ce point et conormal a ce vecteur est non caracteristique pour Af. 2.2.1.2.

Soit Z une sous variete de X definie par un Ideal lz

et

ZOGHMAN MEBKHOUT

96

°jciz-^°xlXhz k

le complete formel de X le long de Z qui est de fagon naturelle un T>xmodule a gauche. On definit le faisceau Qz par la suite exacte de T>xmodules a gauche 0 - Ox/z

- Off~z - Qz - 0

:=i~1Ox.

ou0x/z

L e m m e 2.2.1.3. Pour tout complexe J\f de Vx-modules a gauche a cohomologie bornee et coherente le triangle de la categorie Dh(Cx) RhomVx

(AT,Ox/z)

^RhomVx

(#,0^)

est isomorphe au triangle de la categorie i-1RhomVx

(Af,Ox)^RhomVx

— RhomVx Db(Cx)

(RalgVz(Af),Ox) 1

— r RhomVx Preuve de 2.2.1.3. RhomVx

(Af,Qz)

(Rtf(*Z),Ox)[+l].

II suffit de definir un isomorphisme

(Ra\grz(Af),Ox)

= RhomVx

(Af,0—) .

Mais on a un isomorphisme naturel d'adjonction RhomVx

(Ra\gTz{Af),Ox) ^RhomVx

(M,Rhom0x

{Ra\gTz

(Ox),Ox))

et un isomorphisme de complexes de T>x-modules qui permet de passer du point de vue des pro-modules au point de vue des ind-modules : O^

2 Rhom0x

( « a l g r z (Ox),

Ox).

Rappelons sa demonstration: (a) Si M et M sont des X>x-modules a gauche homox(M,Af) est un X>x-module a gauche par d<j) : m —• d<j>{m) — <j>(dm) ou d est une derivation, <j> une section locale de homox{M,N) et m est une section locale de M. Si M est un complexe de T>x-modu\es a gauche le complexe homox(M,N) est un complexe de P^-modules a gauche et si M est un complexe de D^-modules a gauche le complexe homox{M.,N) est un complexe de Px-modules a gauche. Soit Ox —> % u n e resolution

LE THEOREME DE POSITIVITE DE L'IRREGULARITE

97

X>x~injective de Ox qui est alors (9x-injective. Le complexe Rhomox (Rs\gTz{Ox),Ox) est represente par homox(&\gTz(l),l) qui est done un complexe de T>x-modules a gauche. (b) Pour tout fc on a un morphisme de bidualite qui est un isomorphisme Ox/Ikz

- homCx

(homCx

(Ox/l|,Z),I) .

D'ou un morphisme (*)

\imOx/Zz Jb

-* \unhom0x

(homCx

(0x/l|,X) ,l)

k

Sfcomox(algrz(I),I). Voyons que ce morphisme est X>x-hneaire. II suffit de voir que le diagramme suivant est commutatif pour tout k: Ox/Ikz+l

• hom0x

OxlA

• homox

(homCx

(Ox/Ikz+\lp),lp)

{homox(Oxllkzap)ap).

Notons fk une section locale de ( 9 x / j | , 9k u n e section locale de et homox(Ox/^z^z) ^* u n e section locale de homox P p (homox(Ox/Zz>3' )i2 ). Alors d(f)k+i est l'application qui a gk associe d<j>k+i(gic) ~ -^P) par 1'injection naturelle de homox(Ox/l1z,lp) +1 p dans homox(Ox /%z >Z )- Maintenant dgk est l'application qui a fk+i associe dgk(fk+i) — gk(dfk+i)- En tenant compte des ces actions des derivations on trouve que diagramme est commutatif. (c) Voyons que le morphisme (*) est un isomorphisme. Notons J \— lim Jk le complexe lim hom,ox (homox (Ox/2z^),X). Les composantes du complexe J^ sont flasques et pour un ouvert V les morphismes de transition dans le systeme projectif lim T(V]Jk) sont surjectifs. II resulte alors de ([EGA 0 III], (13.3.1), Publ. IHES 11, (1961), pp. 1-82) que les composantes de J sont acycliques pour le foncteur section globale. Pour un ouvert V de Stein les morphismes de transition entre cohomologies de degre zero de Y(V\Jk) sont surjectifs en vertu du theoreme B de Cartan et done la cohomologie commute a la limite projective loc. cit. On trouve alors pour un ouvert V de Stein assez petit que la cohomologie de r ( V; J) est nulle en degre positif et isomorphe a r(V r ;lim Ox/^z) en degre nul. Le complexe J est acyclique en degres positifs et son faisceau de cohomologie de degre nul est isomorphe a lim Ox/^z- D'ou l'isomorphisme cherche. On peut remplacer le faisceau Ox par n'importe quel fibre a

98

ZOGHMAN MEBKHOUT

connexion integrable. Mais par contre cet isomorphisme est en defaut pour un Z>x-module singulier. En particulier si M. est un 'Dx-module holonome et Z est une hypersurface de X, on a done les isomorphismes IRVZ{M) = i-lBhomVx

(M(*Z), Ox) [+1] = RhomVx

(M, Qz).

L e m m e 2.2.1.4. Soient T un coefficient constructive sur X, X = UQXa une stratification de Whitney de X telle que la restriction a chaque strate Xa de chaque faisceau de cohomologie de T est lisse et x un point de X. Alors si X' est une hypersurface lisse passant par x et transverse a la stratification UaXa au voisinage de x, le morphisme canonique rlRhomCx

{T)Cx)-^RtiomCxl

{f'^.Cx')

est un isomorphisme au voisinage de x ou f est l'inclusion canonique de X1 dans X. Preuve de 2.2.1.4. La question est locale. On peut supposer par devissage a la Grothendieck que T est un faisceau lisse sur une strate Xa et nul en dehors de cette strate. Fixons les notations a l'aide du diagramme suivant

j : xa n x' c ~xa n x' if J '-Xa

1/ C

Xa.

II suffit de voir que Ton a Tisomorphisme entre complexes d'espaces vectoriel complexes

pour tout point x de Xa — Xa. Mais si B est un voisinage assez petit de x dans X les types d'homotopie de B f] Xa H X' et de B C\ Xa sont les memes en vertu du premier theoreme d'isotopie de Thorn-Mather, cf. ([LT2], Thm. 1.2.8) et le lemme en resulte. Corollaire 2.2.1.5. Soient T un coefficient constructible, X = UaXa une stratification de Whitney de X, un point x de X et X' une hypersurface comme dans le lemme 2.2.1.4. Alors au voisinage de x on a les isomorphismes pour tout i de faisceaux pervers f-1Phi{T)^phi(f-1J:). Preuve de 2.2.1.5. Sous les conditions de 2.2.1.5, si T est un coefficient constructible qui a la condition de support, le coefficient f~lT a

LE THEOREME DE POSITIVITE DE L'IRREGULARITE

99

la condition de support. En vertu de 2.2.1.4, si T a la condition de cosupport, f~lT a la condition de co-support. Done si T est un faisceau pervers sur X, f~lJ- est un faisceau pervers sur X' et le corollaire 2.2.1.5 est vrai dans ce cas la. Dans le cas general soit [a, 6] Pamplitude perverse d'un coefficient T\ on a alors un triangle distingue de Dhc(Cx)

ou l'amplitude de T' est egale a [a, 6 — 1]. On a un triangle distingue de Dbc(Cx>)

Appliquons le foncteur i~l a la suite longue de cohomologie perverse associee au premier triangle puis prenons la suite longue de cohomologie perverse du second triangle; on deduit le corollaire 2.2.1.5 en raisonnant par recurrence sur la longueur b — a. 2.2.1.6. Soient P(x,dx) un operateur differentiel d'ordre m defini au voisinage d'un point XQ de X et X' une hypersurface lisse definie au voisinage de xo et non caracteristique pour P(x, dx)> Posons Af := Vx/^xPAlors on a, au voisinage de XQ,

ou / designe l'inclusion canonique de X' dans X. On a en vertu du theoreme de Cauchy-Kovalewski un isomorphisme canonique

r1homVx(N,Ox)^O^l et la nullite du faisceau Ext\y (Af,Ox)- En resume, on a alors un isomorphisme canonique au voisinage de XQ f-lRhomVx

(Af1Ox)=RhomVxl

{rAf,Ox>)>

On en deduit par devissage [Ki] que si A^ est un Px-module coherent defini au voisinage de xo tel que X' est non caracteristique pour Af, alors

LfAf <* f*AT, f*Af est un X>^-module coherent et on a un isomorphisme canonique au voisinage de XQ f-lRhomVx

{M,Ox)=WiomVx,

U*N>0X').

2.2.2. Preuve du t h e o r e m e 2.1.6. Nous allons raisonner par recurrence sur dim(X). Soit done un triplet (X,Z,A4) ou X est une variete analytique complexe, Z une hypersurface de X eventuellement singuliere et AA un £>x-module holonome. II s'agit de montrer que les coefficients constructibles IRz(M) et IR^(M) sont des faisceaux pervers sur Z. II

ZOGHMAN MEBKHOUT

100

suffit de montrer, en vertu du theoreme de dualite locale 1.5.1, que le coefficient IRvz{M)[-l)

= r1RhomVx

(M(*Z),Ox)

est un faisceau pervers sur X, ou i designe l'inclusion canonique de Z dans X. Remarquons qu'en vertu de (2.2.1.3) on a l'isomorphisme r1homVx

(M(*Z),Ox)

3 Ext~x

(M,QZ)

= 0.

En particulier si dim(X) = 1, le complexe IR^(M)[— 1] est un faisceau pervers sur X puisque le faisceau Ext\^ (A4(*Z), Ox) est a support de dimension nulle. D'ou le theoreme 2.1.6 en dimension un. Supposons demontre le theoreme 2.1.6 pour toutes les varietes analytiques complexes de dimension egale a dim(X) — 1 et soit UaXa une stratification de Whitney de X telle que Z est reunion de strates et que les restrictions aux strates Xa des faisceaux de cohomologie (ordinaire) de IR^(M)[— 1] sont lisses. En vertu du lemme 2.2.1.1, puisque M(*Z) est un 2>;r-module holonome, il passe par tout point de Z, en dehors d'une partie analytique de Z de dimension nulle, une hypersurface lisse non caracteristique pour M(*Z) au voisinage de ce point. II passe par tout point de Z, en dehors d'une partie analytique de Z de dimension nulle, une hypersurface lisse qui est transverse a la stratification \JaXQ. Done il passe par tout point de Z, en dehors d'une partie analytique de Z de dimension nulle, une hypersurface lisse non caracteristique pour M(*Z) et transverse a la stratification UaXa. Soient X' une telle hypersurface et / l'inclusion canonique de X' dans X. Posons Z' := Z D X' et M' := f*(M) qui est un £>x'-module holonome en vertu du 1.3.6. On a done un triplet (X^Z',M) analogue au triplet (X, Z, M) avec dim(X') = dim(X) — 1. En vertu de l'hypothese de recurrence le coefficient constructible IRz(M')[—l] est un faisceau pervers sur X'. En vertu de 1.3.8 et de 2.2.1.5, on a les isomorphismes Lf*{M{*Z))

2 Lf*{M{*Z'))

S

M\*Z').

En vertu du theoreme de Cauchy-Kovalewski on a un isomorphisme canonique f-lRhom.Vx

(M(*Z),Ox)=RhomVx,

{M'{*Z'),Ox>)

et done

riiR£{M)[-i]*m>(M')[-i]. En vertu du corollaire 2.2.1.4 et de l'hypothese de recurrence les faisceaux de cohomologie perverse de degre non nul du coefficient IRg(<M)[— 1] sont nuls en dehors d'une partie analytique de Z de dimension nulle. Le

LE THEOREME DE POSITIVITE DE L'IRREGULARITE

101

theoreme 2.1.6 est done vrai en dehors d'une partie analytique de Z de dimension nulle. Reste a exorciser ce qui se passe sur cette partie de dimension nulle. Remarquons que le coefficient IR^(M)[— 1] qui est concentre cohomologiquement en degres [l,dim(X)] a certainement la condition de support. Voyons qu'il a alors la condition de co-support ou que son complexe dual qui n'est autre que IRz(M)[—l] a la propriete de support. Notons j 1'inclusion canonique U \— X — Z dans X. Par construction, on a Tisomorphisme: IRz(M)[-l]=RhomVx

(M\Cz)[-i\

ou Cz est le D^-module a gauche defini par la suite exacte de Vx-modules a gauche 0 -> Ox(*Z) - j.j-'Ox

-> £ * - 0.

/

Done le complexe/JK^(A <)[—1] qui est a priori concentre cohomologiquement en degres [l,dim(X) -f 1] a la propriete de support en dehors d'une partie analytique de Z de dimension nulle. Ce qui Tempeche d'avoir la propriete de support est son dernier faisceau de cohomologie Ext^x (M*, Cz) ou n := dim(X). La demonstration du theoreme 2.1.6 est achevee par le lemme suivant: L e m m e 2.2.2.1. ceau Ext1j>x(Af}Cz)

Pour tout Vx-module est nul.

a gauche holonome N, le fais-

P r e u v e de 2.2.2.1. Les arguments sont caiques sur ([M2], theoreme 2.3). On a une suite exacte de faisceaux de support de dimension nulle ExtnVx (tf,Ox(*Z))

-

Extlx

{N,j*rlOx)

-

ExtnVx {M,CZ) - 0.

Nous allons voir que le premier morphisme est surjectif, ce qui demontrera le lemme 2.2.2.1. Posons Nv := Ext^x{N,Vx) qui est un Vx -module a droite coherent. Pour tout Vx -module a gauche Q on a Tisomorphisme

ExtnVx{M,Q)^Mv

® g. Vx

Soit Vqx —• Vpx — • Mv —• 0 une presentation locale de J\fu. On a done localement Tisomorphisme: Mv ® Q~Q*l<$>(Gq). Le morphisme ExiVx

(tf,Ox(*Z))

-

ExtVx

{N,j*rlOx)

ZOGHMAN MEBKHOUT

102

se represente par le morphisme naturel de faisceaux d'espaces vectoriels complexes de dimension finie

crx{*z)i4>(o'x{*z)) - (j.rlox) a {j*rlox). Soit B un voisinage de Stein d'un point o de X. Pour B assez petit on a deux suites exactes :

o— T(BA{OX(*Z))) i

— T(B,OPX(*Z)) 1

-> (ox(*z)/4>(ox(*zj))o 1

—o

o-.r(B,0(>r1^)) -^r(B1^rlox) ^(j.r1ox/*(j.r1ox))o-+o. L e m m e 2.2.2.2. Soient F un espace vectoriel complexe topologique de Frechet et E un sous-espace vectoriel de F de codimension finie image d'un espace de Frechet par une application lineaire continue; alors E est ferme dans F. P r e u v e de 2.2.2.2. C'est une consequence directe du theoreme de Banach: une application lineaire continue surjective entre espaces de Frechet est un homomorphisme, cf. [Se2], ([Gs], exercice 4, p. 57). Appliquons ceci a Y(B)j*j~1Opx) qui est un espace de Frechet. Pour B assez petit, Y(B, $(]+j-lOqx)), qui est isomorphe a <j)(Y(B, j * . ; " 1 ^ ) ) , ceci se voit en supposons que le pretention <j> de Mv est le debut d'une resolution locale finie par des modules libres de type fini et en utilisant le theoreme B de Cartan. Done Y(B,(fi(j*j~lOx)) est ferme dans TiBJJ^Ox) e n v e r t u d u lemme 2.2.2.2. L'espace T(B, j*j-lOpx)/ 1 q T(B,<j)(j*j~ O x)) muni de la topologie quotient est separe. Mais, pour B assez petit, Z est definie par une equation et l'espace T(B,OpK(*Z)) est dense dans Y(B, j*j~1Ox). Done l'image du morphisme Y{B,Opx{*Z))/ q l p l q Y{B,4>(O x(*Z))) -+ Y{B,uJ- O x)/Y{B,{uJ- O x)) est dense. C'est done un morphisme surjectif. Le morphisme (ox(*zy(ox(*z)))0^((jj-lox)/(jtj-1ox))0 est done surjectif. D'ou le lemme 2.2.2.1. Remarque 2.2.2.3. Le lemme 2.2.2.1 qui est de nature transcendante est pour Finstant le principal obstacle a une demonstration purement algebrique de theoreme de positivite, cf. §3. 2.3. Demonstration du theoreme 2.1.9. II s'agit de montrer que si M est un complexe de la categorie Dhh(T>x) les fonctions constructibles complexes, sur X, x((IRz(M)) et x((IRz(M*)) sont egales pour toute sousvariete Z de X. On peut supposer que M est un T>x-Tnodu\e holonome. La question est locale. Si Z — Z\ f) Z2, on a par construction le triangle de Mayer-Vietoris de la categorie Dhc(Cx) •

LE THEOREME DE POSITIVITE DE L'IRREGULARITE

103

IRz(M)-+IRz1(M)®IRz2(M)-+IRZluz2(M). En raisonnant par recurrence sur le nombre d'equations definissant Z on est ramene a supposer que Z est definie par une equation. D'autre part on a, par definition du faisceau d'irregularite, l'egalite: X (IRz(M))

= X {DR(R
- x (DR(RTZ(M))) •

Mais en vertu du theoreme de dualite locale 1.5.1, on a l'egalite X (DR(RTZ(M)))

=

X

(PR(RTZ(M'))).

Done, pour demontrer le theoreme 2.1.9, il suffit de montrer que Ton a l'egalite: X(DR(Ra\gTz(M)))

=

x(DR(Ra\gTz(M*)))

pour tout X>x-module holonome M et toute hypersurface Z. En prenant une equation de Z et en considerant son graphe on est reduit a supposer que Z est lisse parce que tous les foncteurs en M qui interviennent dans l'egalite precedente commutent avec une immersion fermee, cf. [Me3J. Considerant alors la V-filtration ([M3], [K4]) de T>x indexee par Z et definie par:

Vk(Vx):={PGVx,P(Jpz)clpz-k,p£z} ou Xz est l'ldeal de Z avec Xpz = Ox si p est negatif ou nul. Si t est une equation locale de Z elle est d'ordre —1 pour la V-filtration. Le gradue d'ordre 0, gr^(X>x) := Vo(Vx)/V-i(T>x), contient comme section distinguee le champ d'Euler Eu. II resulte alors de l'equation fonctionnelle de Bernstein-Sato, cf. ([K3], [NMi]), que tout Px-module holonome M. admet une unique filtration croissante de Malgrange-Kashiwara, indexee par 2T, par des Vo(£>x)-modules coherents Vk(M), telle que Taction du champ d'Euler Eu sur le gradue d'ordre k, gr^(M) := Vk(M)/Vk-i(M), admet localement un polynome minimal dont les zeros sont dans la bande — k — 1 < Rea < — k du plan complexe pour tout k £ Z, ([M3], [K4]). II en resulte alors que les T>z- modules gr^(A^) sont coherents. En fait on a le resultat suivant, cf. ([Sai], [SM]): Theoreme 2.3.1. Pour tout Vx-module holonome M les Vz-modules gr^(A^) soni holonomes pour tout k £Z. De plus les T>z-modules gr^(M) et gr]jf (.M*) sont en dualite pour tout k EZ:

gvl(M)Y=gvUM*). Pour la preuve du theoreme 2.3.1, voir [SM]. On definit avec ([M3], [K 4 ]) :

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ZOGHMAN M E B K H O U T

Definition 2.3.2. Pour tout P x - m o d u l e holonome M. et toute hypersurface lisse Z , on pose ^(M) := grwz(M) et Vg(M) := g r ^ ( A f ) . Les foncteurs Wg et 3>£ s o n ^ exacts de la categorie Mh(T>x) dans la categorie Mh(T>z). Si t est une equation locale de Z le complexe 0 —• Wrg^M) —• $2?(A4) —* 0 represente le complexe Li* M de la categorie Dhh(T>z) pour tout P x - m o d u l e holonome A4 ou i est 1'inclusion canonique de Z dans X. Mais le complexe /Jalgr^(A / f)[l] est 1'image directe par l'immersion i au sens des X^-modules du complexe Li*M [Me2J. II resulte alors du theoreme 2.3.1 et du theoreme de dualite 1.5.1 que Ton a l'egalite entre fonctions constructibles complexes: X (DR(R^\gYz(M)))

=

X

(DR(R
Ceci termine la demonstration du theoreme 2.1.9. Si / : X —» C est une fonction holomorphe non const ante et Ai un T>xmodule holonome, notons Mj l'image directe de M. par le morphisme graphe de / . Si Z := X x 0 les £> z -modules ^(Mj) et $™(A4/) sont a support dans / _ 1 ( 0 ) et done ils proviennent de £>x-modules holonomes, que nous noterons encore ^™(A4/) et ^^(Mj). Definition

233.

On pose Vp(M)

:= ^(Mj)

et <$>J(M) :=

^{Mj).

Les foncteurs W? et $7* sont exacts de la categorie des T>x-modules holonomes dans la categorie des X>x-modules holonomes a support dans /_1(0). 3. Le cycle d ' i r r e g u l a r i t e Dans le §2 nous avons defini le faisceau d'irregularite par voie transcendante. Mais comme tout faisceau pervers le faisceau d'irregularite admet un cycle caracteristique positif du fibre cotangent defini a l'aide de la theorie des cycles evanescents topologiques de SGA 7, cf. [LM]. Le cycle caracteristique de faisceau d'irregularite admet une description purement algebrique qui garde un sens sur un corps k de caracteristique nulle. Cependant le passage du faisceau d'irregularite a son cycle fait perdre beaucoup d'information sur la ramification le long de l'hypersurface, ne serait-ce que parce que tout cycle lagrangien positif n'est pas en general le cycle caracteristique d'un faisceau pervers. Aussi le theoreme de positivite suggere de construire a partir d'un X ^ - m o d u l e holonome M et d'une hypersurface Z d'une variete non singuliere X sur k un T>x-module holonome a support dans Z tel que lorsque k est le corps des nombres complexes son complexe de de Rham (transcendant ) soit isomorphe au moins virtuellement au faisceau IRz(M). Cette construction n'est pas encore au point; aussi nous nous limitons dans ce § a definir le cycle

LE THEOREME DE POSITIVITE DE L'IRREGULARITE

105

d'irregularite. Nous supposerons toujours pour simplifier que les corps de base qui interviennent sont algebriquement clos. 3.1 Cas des fibres a connexion integrable. Soient une variete algebrique (X^Ox) non singuliere sur un corps k de caracteristique nulle, Z une hypersurface de X et M un XV-module holonome lisse de rang r sur U := X — Z. Si j designe l'inclusion canonique de U dans X on a j*Ou — Ox{*Z) et on pose M(*Z)

:= j+M

m

qui est done un X>x- °dule holonome. Definition 3.1.1. On definit le cycle d'irregularite de M, le long de Z, CCh{M) en posant: CCh(IRz{M))

:= CCh(M(*Z)) - r C C h ( O x ( * Z ) ) .

En vertu du theoreme de l'involutivite des caracteristiques [G], CCh(IRz(M)) est un cycle lagrangien du fibre cotangent T*X : ses composantes irreductibles sont les adherences des fibres conormaux de la partie lisse de leur projection sur X. T h e o r e m e 3.1.2. Sous les conditions preceden tes le cycle d 'irregularite CCh(IRz(M)) est positif. P r e u v e du t h e o r e m e 3.1.2. Par le principe de Lefschetz on peut supposer que k —C. Notons alors par "an" en exposant le passage d'un objet defini sur k a Fob jet transcendant associe. En vertu de [LM] on a les egalites de cycles lagrangiens: CCh(.M(*Z)) = rCCh(Ox(*Z))=

CCh(DR(Man(*Z™)) CCh(Rj*nDR(M™)).

Done par definition le cycle CCh(IRz(M)) apparait comme egal au cycle caracteristique du faisceau IRz{M^n{^Z^n))[— 1] qui est done positif. Si Ton a une suite exacte de fibres sur U a connexion integrable

on a Pegalite CCh(IRz(M))=

CCh(IRz(Mi))+

CCh(IRz (M2)) •

D'ou un morphisme IRz : K{Ov) -

Zn{TX)

entre le groupe de Grothendieck des fibres sur U a connexion integrable et le groupe des cycles lagrangiens de T*X.

ZOGHMAN MEBKHOUT

106

Exemple 3.1.3. Si dim(X) = 1 et si Z est un point, la multiplicity de CCh(IRz(M)) est egale ail nombre de Fuchs. Ainsi le cycle CCh(IRz(M)) apparait comme la generalisation naturelle en dimension superieure de l'irregularite de Fuchs. Plus generalement si Z est une hypersurface d'une sous-variete Y de X telle que Y — Z est lisse et M est un fibre sur Y — Z de rang r a connexion integrable on definit son cycle d'irregularite le long de Z en posant CCh(IRz(M))

:= CCh(A4(*Z)) - rCCh(HvY

(Ox(*Z)))

ou M designe l'image directe au sens des I>x-modules de M par 1'immersion de Y — Z dans X et p := codimy(X). Le cycle CCh(IRz(Ai)) est encore un cycle lagrangien positif. T h e o r e m e 3.1.4. Soit un fibre vectoriel de rang r a connexion integrable M sur un ouvert U, complement aire d'une hypersurface Z d'une variete algebrique X propre et non singuliere sur un corps de caracteristique nulle. On a alors la formule de type Riemann-Roch : X(U,DR(M))

= rX{U,DR{Ov))

(-lfim^CCh(IRz(M)).T*xX.

+

P r e u v e d u t h e o r e m e 3.1.4. On a bien sur note x(U,DR(Ai)) la caracteristique d'Euler-Poincare du complexe de de Rham global iJHom-p^ (Ou,M) d'un Vu-modu\e M et CCh(IRz(M)).T%X le nombre d'intersection [F] du cycle d'irregularite et de la section nulle du fibre cotangent T*X. Si j designe l'inclusion canonique de U dans X on a: X(U,DR(M))

= X(X,RjmDR(M))

=

X(X,DR(M(*Z))).

Mais si Af est un T>x-module holonome on a la formule globale [L3]: X{X,DR(M))

(-l)dim^CCh(X).T^X.

=

Appliquons cette formule a M — M(*Z) et a M = Ox{*Z): on trouve la formule du theoreme 3.1.4. Bien sur le point important dans la formule du theoreme 3.1.4 est la positivite du cycle CCh(IRz(M)). Autrement dit a cette situation on attache canoniquement un nombre fini de couples (ma^Za) ou les ma sont des entiers naturels positifs ou nuls (les irregularites generiques) et les Za sont des sous-varietes irreductibles de Z si bien qu'on a la formule plus familiere [D]: X{U,DR(M))

= rx(U,DR(Ou))

+

(-lf,mWJ2m°TLX-TxXa

On a note comme d'habitude T% X l'adherence du conormal de la partie lisse de Za. Pour les formules analogues en theorie /-adique voir [L2], [Ka].

LE T H E O R E M E DE P O S I T I V I T E DE L ' I R R E G U L A R I T E

107

Plus generalement si M est un fibre vectoriel a connexion integrable sur un ouvert lisse {/, complement aire d'une hypersurface Z d'une sousvariete eventuellement singuliere Y d'une variete algebrique X propre et non singuliere sur un corps de caracteristique nulle, on a une formule de type Riemann-Roch analoque a celle du theoreme 3.1.4. C'est cette derniere situation qui se produit le plus souvent dans les applications. 3.2. Cas general Pour definir le cycle d ' i r r e g u l a r i s d'un X>x-module holonome le long d'une hypersurface nous avons besoin de l'isomorphisme d'Euler [Mc] entre le groupe Z(X) des cycles de X et le groupe des fonction constructibles F ( X ) a valeurs dans Z. Rappelons que si T est une sous-variete reduite de X supposee definie sur un corps de caracteristique nulle on definit de fagon purement algebrique 1'obstruction d'Euler Eux(T) en un point x de T qui est un entier, cf. [Gs], [LT]. Elle coincide avec l'obstruction d'Euler de MacPherson [Mc] definie par voie transcendante quand le corps de base est C. Ceci entraine que la fonction x —* Eux(T) est constructible et que le morphisme de groupes : Eu : Z ( X ) —

F(X)

est un isomorphisme. D'autre part Papplication qui a une sous variete irreductible de X associe l'adherence du conormal de sa partie lisse se prolonge de fagon naturelle en un isomorphisme entre Z ( X ) et le groupe Zn(T*X) des cycles lagrangiens homogenes du fibre T*X. En particulier a un Vjj-module holonome M.JJ sur le complementaire U d'une hypersurface Z de X correspond la fonction constructible Eu(CCh(A4t/)) sur U et au prolongement par zero t /iEu(CCh(Ax-module holonome on definit le cycle d'irregularite de Af, le long de Z , CCh(IRz(M)) en posant CCh(IRz(M))

:= CCh(M(*Z))

- j. (CChO'-1^)) .

Le cycle CCh(IRz(M)) est un cycle lagrangien homogene. Si on pose jij~xM := {M*(*Z))* on a en vertu du theoreme 2.1.9 l'egalite entre cycles caracteristiques CCh(.M(*Z)) = CC\i{jij~lM) et le cycle d'irregularite prend la forme plus symetrique

CCh(IRz(M))

= CCh(j
(CCh(rlM)).

ZOGHMAN M E B K H O U T

108 Theoreme 3.2.2. cycle CCh(IRz(M))

Pour tout est positif.

triplet

(X, Z, M)

comme

ci-dessus,

le

P r e u v e d u t h e o r e m e 3.2.2. On peut supposer que le corps de base est C. Dans ce cas le cycle CCh(IRz{M)) est le cycle caracteristique du faisceau IRz(M)[— 1] qui est done positif. On obtient done un morphisme de g r o u p e s :

IRz:K(Mh(Vx))->ln(T*X). Definition 3.2.3. Si M est un complexe de la categorie Dhh(Vx) definit son cycle d'irregularite en p o s a n t :

on

CCh(IRz(M)) := J2 CCh(lRz (h^M))) . l

Cette definition est imposee par le corollaire 2.1.8. C'est done un cycle positif qui est nul si et seulement si les faisceaux de cohomologie de M sont reguliers cf. §5, 5.1.5. 4. L e t h e o r e m e d e s e m i - c o n t i n u i t e d e l ' i r r e g u l a r i t e 4.1. Nous allons montrer dans ce § que le theoreme de positivite precise le theoreme de semi-continuite de l'irregularite dont il est issu [ M e ^ . Soit done / : X —• S un morphisme lisse de dimension relative egale a un entre varietes algebriques non singulieres sur un corps de caracteristique nulle, ou entre varietes analytiques complexes, et Z une hypersurface de X telle que la restriction de / a Z soit finie. Soit M. un Z>x-module holonome dont la restriction a U := X — Z est lisse de rang r. Posons Xs := f~x(s), Zs := Zf)Xs et Ms '-= j*s{M) ou j s designe l'inclusion canonique de Xs dans X, Ms est done un T>x8-module holonome lisse sur Us := Xs — Zs de rang r et on note irx(Ais) la multiplicity de T*XS dans le cycle d'irregularite IRT(MS) pour tout point x de Xs. On definit les fonctions ^ et $ sur S et a valeurs dans N en posant * ( * ) : = £ irx(.M,)

et

$(s)

:=*(*) + r#Z,,

X—+S

ou # Z 5 est egal au nombre de points de Zs. On a alors les resultats suivants [Me4]: (i) les fonctions ^ et ^ sont constructibles; (ii) la fonction est semi-continue inferieurement; (iii) si la fonction est localement constante, la variete caracteristique de M(*Z) est egale a T^X U TZX (on suppose que r est non nul). En fait on a le resultat plus precis le suivant: Theoreme 4.1.1. localement constante,

Sous les conditions precedentes, si la fonction le morphisme Z —• S est etale.

$ est

LE THEOREME DE POSITIVITE DE L'IRREGULARITE

109

P r e u v e du t h e o r e m e 4.1.1. On peut supposer que le corps de base est le corps des nombres complexes, puis que / est un morphisme entre varietes analytiques complexes. La question est locale sur S. On peut supposer par changement de base que dim(S') = 1. Dans cette situation, si la fonction 3> est localement constante, toutes les fibres Xs sont non caracteristiques pour M(*Z) [Me^. En particulier

Ainsi si SQ est un point special et st un point general voisin, le saut de la fonction ^ est egal a £

dimcIR5(M)x

X—>Si

- J2

dimcIR^M)x.

X—>SQ

Considerons le triangle distingue de la categorie Dbc(Cz0) des cycles evanescents [SGA 7, XIV]: J70lIRvz{M)

- ms

(IRZ(M))

- B$s

(IR^(M)).

Mais alors puisque IR^(M) est un faisceau per vers sur Z, le complexe R$j(IRz{M)) est un faisceau per vers sur ZQ, il est done concentre en degre zero puisque dim(Zo) = 0 et en particulier le morphisme: J701IR^M)^J7t1IRz(M) est injectif et le saut de la fonction # , egal a J2x-+s ^ m c-H^/(JKz(«A^))arj est positif ou nul. Mais le saut de la fonction $ apparait alors comme la somme de deux entiers positifs non nuls: * ( „ ) - *(,„) = *(«,) - *(«o) + r ( # Z , t - #ZSo) . Comme ce saut est nul par hypothese, il en resulte que le saut de la fonction ^ est nul et que #Z8f = #ZSQ. Le morphisme Z —»• S est etale. C'est cette propriete de positivite qui est a la base du theoreme de positivite. Remarque 4.1.2. Dans la situation precedente le saut de la fonction \I> est positif ou nul chaque fois que IR% (M(*Z)) est nul. C'est par exemple le cas si le support du faisceau IRg(M) n'est pas egal a Z tout entier, cf. §5. En fait c'est la une conjecture generale. 4.2. Dans la situation precedente supposons que dim(5) = 1 et notons T>X/s l e faisceau des operateurs differentials sur X relativement a S [EGA, IV, §16]. Alors si M. est un T>x-module holonome lisse sur U on lui associe de fagon naturelle un T>x/5-module et les fonctions \P et <£ ne dependent que de ce Vx/s-m°du\e. Si on part d'un Ox(*Z)-mod\ile coherent muni d'une structure de £>x/5-module a gauche, les fonctions ^ et 3> sont encore definies de fagon evidente et il est facile de fabriquer des exemples ou la

110

ZOGHMAN M E B K H O U T

fonction ^ n'est pas semi-continue inferieurement a cause des phenomenes bien connus de confluence. Mais la fonction <£ est toujours semi-continue inferieurement comme l'a montre Deligne dans une lettre a N. Katz datee du 1.12.1976. Conjecture 4.2.1. (Malgrange). La fonction ^ associee a un X>x-module holonome lisse sur U est semi-continue inferieurement. Autrement dit, dans le cas d'une famille d'equations differentielies provenant d'un Z>x-module holonome lisse sur U il ne se produit pas de confluences. Le theoreme 4.1.1. fournit un support substantiel a cette conjecture. Cependant les methodes de cet article ne semblent pas adaptees pour demontrer cette conjecture et qu'il faille plutot connaitre la structure des connexions meromorphes irregulieres formelles le long d'une courbe sur une surface complexe non singuliere.

5 . L e t h e o r e m e d e s coefficients d e d e R l i a m Grothendieck en caracteristique nulle Nous allons montrer dans ce § comment le theoreme de positivite remplace avantageusement le theoreme de resolution des singularites d'Hironaka [H] dans la demonstration du theoreme de comparaison entre cohomologies de de R h a m d'une variete algebrique complexe [Gi] et plus generalement dans la demonstration du theoreme des coefficients de de Rham-Grothendieck [G2]. C'est la un exemple tres important ou la resolution des singularites n'est pas au fond du probleme, contrairement a une opinion generalement bien admise jusqu'alors [G3]. Pour les demonstrations des resultats de ce § nous renvoyons le lecteur a l'article [Mes] consacre systematiquement a la situation de singularites regulieres et au theoreme d'existence de Riemann du point de vue du theoreme de positivite. Pour alleger l'expose dans ce § qui est surtout destine a informer le lecteur on se limite au cas algebrique, mais il est clair que les enonces valent aussi bien dans le cas analytique. 5.1. Soient X une variete algebrique complexe non singuliere, Z une hypersurface de X et M de T>x-module holonome. Notons Y le support de M et supposons que la trace de Z sur Y est une hypersurface. T h e o r e m e 5.1.1. Etant donne un triplet (X,Z,M) comme ci-dessus tel que Y — Z soit lisse et que M soit lisse au dessus de Y — Z, alors le faisceau IRz(M) est mil, si et seulement si la codimension dans ZC\Y de son support est au moins egale a un. P r e u v e d e 5 . 1 . 1 . Voir [Me 5 ] §3. C o r o l l a i r e 5 . 1 . 2 . (Grothendieck) [Gi].

Si X est une variete

algebrique

LE THEOREME DE POSITIVITE DE L'IRREGULARITE

111

complexe non-singuliere, les morphismes canoniques (*)

HDR(X)

-

HDR(X™)

entre cohomologie de de Rham H'DR(X) := H'(X]DR(Ox)) de X et cohomologie de de Rham H'DR(X*n) := H'(X™;DR(Ox™)) = H'(X*n;Cx™) an de la variete transcendante X associee a X sont des isomorphismes. P r e u v e de 5.1.2. La suite spectrale de de Rham-Cech nous reduit a supposer que X est affine [Gi]. Si X designe l'adherence de X dans un espace projectif Pm l'obstruction aux isomorphismes (*) est egale, en vertu du theoreme GAGA de Serre, a la cohomologie de Z := X — X a valeurs dans le faisceau IRz(H^-(Opm)) ou p =r m — dim(X) [Gi]. Mais en vertu du theoreme 5.1.1 le faisceau IRz((Opm)) est nul pour toute hypersurface Z de Pm. Ceci entraine par un argument combinatoire de Mayer-Vietoris que le complexe IRz(RTy(Opm)) est nul pour toute sousvarietes Y et Z de Pm. D'ou le corollaire 5.1.2. Corollaire 5.1.3. (Deligne) [D]. Si £ est un fibre a connexion integrable sur une variete complexe non singuliere X tel que les nombres dimcIRoo(n*£) soient nuls pour tout point a rinfini oo de toute courbe non singuliere v : C —+ X au-dessus de X, alors les morphismes canoniques

(**)

H-DR(x,e)^H-DR(xm,em)

entre cohomologie de de Rham H'DR{X,£) \— H'(X;DR(£)) de X a valeurs dans £ et cohomologie de de Rham H'DR(X^n]£^n) := H"(Xan\DR{£^n)) de la variete transcendante A' an associee a l a valeurs dans £ a n sont des isomorphismes. P r e u v e de 5.1.3. Comme dans le theoreme 5.1.2 on peut supposer que X est affine. Soit X l'adherence de X dans un espace projectif P m . Quitte a prendre la normalisation projective, bien que cela ne soit pas indispensable [Mes], on peut supposer que X est normale. Soit £ l'image directe au sens des X>x-modules d e £ par Timmersion canonique de X dans P m . C'est alors un Dp™-module holonome a support dans X. Alors en vertu du theoreme GAGA de Serre, l'obstruction aux isomorphismes (**) est egale a la cohomologie de Z := X — X a valeurs dans IRz(£). En faisant passer une courbe par un point assez general de Z on voit que ce faisceau est nul en vertu du theoreme 5.1.1. D'ou le corollaire 5.1.3. T h e o r e m e 5.1.4. Soit M un Vx-module holonome sur une variete affine complexe non singuliere X. Alors son faisceau d'irregularity le long

112

ZOGHMAN MEBKHOUT

de tout diviseur, y compris ceux de 1'infini, est nul si et settlement si son image inverse sur toute courbe non singuliere au-dessus de X n'a que des singularites regulieres, y compris a Vinfini. Preuve de 5.1.4.

Voir [Me5] §3.

Ceci amene a poser la definition: Definition 5.1.5. On dit qu'un £>x-module holonome sur une variete algebrique non singuliere sur un corps de caracteristique nulle est regulier si son image inverse sur toute courbe non singuliere au-dessus de X n'a que des singularites regulieres, resp. son cycle d'irregularite le long de tout diviseur d'un ouvert affine de X est nul. On note Mhr(T*x) la sous categorie de Mh(£>x) des T>x-modules reguliers et Dhhr(T>x) la sous categorie triangulee Dhh(T>x) des complexes dont la cohomologie est dans Mhr(T>x)Corollaire 5.1.6. Soient Ai% (i = 1, 2) des coefficients de la categorie DhriPx) pour une variete algebrique complexe non singuliere X. Alors le morphisme canonique RRomVx(X]Mi,M2)-+RKomVxAn(X*n;Mr,M2n)

(***) est un isomorphisme. Preuve de 5.1.6.

Voir [Me5] §3.

Corollaire 5.1.7. Si Y est une sous-variete d'une variete algebrique complexe non singuliere X on a les isomorphismes canoniques

2 ExiVx^

(,Yan;72algry.n ( O x „ ) , O x „ ) S H

Preuve du corollaire 5.1.7.

(Yan;CY*»).

On a note bien sur Q\—~ le complete X/Y

formel du complexe de de Rham de X le long de Y. Ce corollaire resulte du fait que RYy{Ox) appartient a Dhhr{T>x) par le theoreme 5.1.1 et du theoreme 5.1.6. Ce resultat a ete obtenu par plusieurs auteurs (Deligne (1969), Herrera-Libermann,...) a Taide du theoreme d'Hironaka. Remarquons cependant que le corollaire 5.1.7 apparait dans la theorie des Vx~ modules comme equivalent au theoreme de comparaison de Grothendieck [ d ] par le theoreme de dualite locale pour les T>x-modules holonomes [MeJ. On obtient done, puisque la cohomologie du complete formel du complexe de de Rham de X le long de Y est isomorphe a la cohomologie cristalline de Y, une demonstration, sans faire appel au theoreme

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113

de la resolution des singulariies, de Tisomorphisme entre la cohomologie cristalline d'une variete algebrique complexe et la cohomologie de l'espace localement compact sous-ajacent. Si / est un morphisme entre varietes algebriques non singulieres sur un corps de caracteristique nulle, notons / * et / c * les foncteurs image inverse et image directe cristalline (cf. par exemple [Me 3 ]) entre categories derivees de Vx-modules. Posons pour tout complexe M de Vx-modules f\M) := (f*(My)* et fc,(M) := ( / C - ( . M ) T ou ( )* designe le foncteur de dualite de X>x-modules. II resulte du theoreme de dualite pour les complexes de Vx- modules a cohomologie bornee et coherente que le foncteur / c , est isomorphe au foncteur fc* si / est propre [Me 3 ]. Les categories Dhh(Vx) sont stables par les operations / * , / ' , / c *, fc,, x-module a droite. Mais on a un formalisme et des compatibilites entre les operations tout a fait similaires, cf. [Mea]. Le theoreme 5.1.8 contient comme cas particuliers plusieurs resultats dont les demonstrations reposaient de fagon essentielle sur le theoreme d'Hironaka. Ainsi, par exemple, le fait que les categories Dhhr(Vx) sont stables par image directe / c * contient comme cas particulier le theoreme de regularite de la connexion de Gauss-Manin [Kat]. Ou encore le fait que le foncteur DRr commute avec le foncteur ^ ^ a pour consequence que le theoreme de monodromie locale d'un germe de fonction holomorphe (cf. [Le]) est rigoureusement equivalent au theoreme de rationalite des zeros du polynome de Bernstein-Sato associe a ce germe [M3] realisant ainsi un peu mieux l'idee originale de Malgrange. On obtient done une demonstration du theoreme de rationalite des zeros du polynome de Bernstein-Sato d'un germe de fonction holomorphe sans faire appel au theoreme d'Hironaka comme souhaite par Le [Le]. II n'est pas fortuit de constater que e'est le me me homme qui est a Forigine aussi bien du theoreme de comparaison [Gi] que du theoreme de monodromie ([SGA 7, I] ou l'Appendice

114

ZOGHMAN M E B K H O U T

de Particle de J. P. Serre, J. Tate, Good reduction of abelian varieties, Ann. of Math. 88 (1968), 492-517). Le theoreme 5.1.8 a visiblement ete predit par Grothendieck [G3] page 312, ligne 7 du bas. Nous regardons son enonce et sa demonstration comme un des succes les plus eclatants de la theorie de Grothendieck, Tune des plus admirables s'il en est, par son unite, sa generality et son originalite. Une fois aquis le theoreme de comparaison on a alors une meilleure approche, beaucoup plus precise, du theoreme d'existence de Riemann, proche de celle de G r a u e r t - R a m m e r t pour le theoreme analogue pour les revetements finis, et sans faire appel au theoreme d'Hironaka [Mea] §5. II devient alors interessant de chercher une demonstration du theoreme des coefficients de Hodge-Deligne sans faire appel au theoreme d'Hironaka. A partir du theoreme 5.1.8 on deduit [Me2] le theoreme analogue pour les coefficients holonomes d'ordre infini, voir ([Mea], Chap. II, t h m , 9.5.7) pour l'enonce complet. Nous rappelons cela parce que ce sont les coefficients holonomes complexes d'ordre infini qui doivent servir de modele aux coefficients p-adiques en caracteristique p > 0. En effet il y a une analogic naturelle entre une algebre de Dwork-Monsky-Washnitzer, completee faible d'une algebre localisee le long d'une hypersurface, dans le cas padique et une algebre de fonctions holomorphes ayant des singularites essentielles le long d'une hypersurface dans le cas complexe, cf. [NM2]. Pour cette raison la theorie des modules d'ordre infini a commencee a etre developpee par P. Berthelot, cf. [B] et, independament, par L. Narvaez et l'auteur, cf. ([NM2], [NM3]). Bien que Ton n ' a pas encore demontre tout ce que suggere la theorie complexe on commence a avoir de nombreuses indications qui montrent que c'est la bonne theorie de de Rham-Grothendieck en caracteristique p > 0. Cette theorie doit fournir les coefficients padiques analogues aux coefficients /-adiques (/ ^ p) et aux coefficients de de Rham-Grothendieck en caracteristique nulle. Soient W := W(k) l'anneau des vecteurs de W i t t d'un corps parfait k de caracteristique p > 0, X —• W un schema formel lisse topologiquement de type fini sur W , resp. un schema formal faible lisse topologiquement de type fini sur W, alors le faisceau T^X/w ^ e s ° P ^ r a t e u r s differentiels p-adiques d'ordre infini est defini [B], resp. ([NM2], [NM3]), comme le complete de DworkMonsky-Washnitzer du faisceau des operateurs differentiels T>if[x/w(Ox) de ([EGA IV], §16), resp. du faisceau defini dans ([NM 2 ], 4.4.3). Le faisceau V*x,w est l'analogue p-adique du faisceau V^ des operateurs differentiels d'ordre infini dans le cas complexe. Mais comme toujours la situation est bien meilleure en geometrie algebrique qu'en geometrie analytique. En effet P. Berthelot [B] a demontre que le faisceau VXl\v e s ^ limite inductive de ses sous-faisceaux des operateurs qu'il apelle d'echelon j(j G N) qui sont noetheriens et coherents. II en deduit que le fais-

LE T H E O R E M E DE P O S I T I V I T E DE L ' I R R E G U L A R I T E ceau Vx,w

115

(g)Q est un faisceau coherent d'anneaux [B]. A partir de la on

peut definir, cf. [NM3], la categorie des T>x>w ® Q-modules holonomes qui doit etre la categorie de base pour line theorie des coefficients padiques realisant ainsi l'idee de Grothendieck ([G3], 7.3, p.355) du topos de Monsky-Washnitzer. Cependant pour l'instant on n ' a pas encore demontre le point clef de cette theorie a savoir qu'une sous-categorie convenable de T>*x/W 0 Q-modules holonomes est stable par image directe par une immersion ouverte d'ou doit resulter que ces categories des coefficients holonomes p-adiques sont stables par les six operations de Grothendieck, en particulier le theoreme de finitude de la cohomologie de Dwork-MonskyWashnitzer et de la cohomologie rigide de Berthelot. L'analogue complexe pour les coefficients holonomes d'ordre infini [Me 2 ] repose dans Pet at actuel de nos connaissances sur le theoreme d'existence de Riemann. Mais la encore on peut penser que la situation dans la cas p-adique est plus favorable et doit etre intermediaire entre le cas algebrique de caracteristique nulle et le cas analytique complexe bien que la theorie du polynome de Bernstein-Sato pour les algebres de Dwork-Monsky-Washnitzer est insuffisante [NM2]. Mais il y a lieu d'etre optimiste d ' a u t a n t plus que l'hypotheque du theoreme de resolution des singularites est levee en caracteristique nulle. Le lecteur trouvera de plus amples renseignements sur cette theorie dans ([B], [NM 3 ]). 6. F i l t r a t i o n d u f a i s c e a u d'irregularity Soient X une variete algebrique complexe non singuliere, Z une hypersurface de X et M un P x - m o d u l e holonome. Nous allons definir des filtrations des faisceaux IRz(M) et IR^(M) par des sous-faisceaux indexes par les nombres reels r compris entre 1 et 00, mais dont les sauts locaux sont des nombres rationnels. La situation est toute analogue a celle du theoreme de Hasse-Arf pour la filtration (numerotation superieure) du groupe de Galois d'une extension galoisienne finie d'un corps local [Sex]. De plus ces filtrations sont purement algebriques au niveau du cycle caracteristique et gardent un sens sur tout corps de caracteristique nulle. La encore comme dans le §4 la situation doit etre considered comme provisoire. II nous faut pour cela generaliser le theoreme de positivite. On utilise la theorie de Y. Laurent ([Lai], [ L ^ ] , [Las], [La«4]). Les ingredients sont le theoreme de constructibilite et le theoreme de dualite locale. Pour les resultats sur les espaces vectoriels topologiques et en particulier sur les espaces nucleaires nous renvoyons le lecteur aux articles de Grothendieck ([G4], [G5], [Ge], [G7], [Gs])- II y trouvera toutes les demonstrations qui peuvent lui manquer. 6.1.

Soient X une variete analytique complexe et Z une sous-variete

ZOGHMAN MEBKHOUT

116

lisse de X de codimension p. On a alors les deux suites exactes de T>x~ modules a gauche 0 - Ox/z 0-

- 0^~z

-

Q z -> 0

algtfP (Ox) - Hpz (Ox) - £ * - 0.

Si (x,z) = (a?i,... ,xp,zi,... ,2 n _ p ) est un systeme de coordonnees locales tel que Z soit definie par X\ — ...xp = 0, un germe au point x = z = 0 du faisceau O-rr-, resp. du faisceau H^(Ox), est represente par une serie ^ a € A r p a a z a , r e s P- YlpeNp bpx~P~l, ou a a , resp. bp, est une suite de fonctions holomorphes definies sur le meme ouvert U de Z, resp. telle que pour tout compact K de £/, pour tout e positif, il existe une constante positive ou nulle K€ telle que sup^ \bp\ < Kt{j3\)~lv^. On a note fi+1 pour /? + ( l , . . . ,1) appartenant kNp, /3\ pour /?i!.. ./? p ! et |/?| pour /?i + . . . /3r. Pour tout reel r superieur ou egal a 1, on definit le sousfaisceau des germes de fonctions, resp. des ultradistributions, de classe de Gevrey d'ordre r, Ox/z{r], r e s P #z/x{ r }> du faisceau O-rrz, resp. du faisceau H^(Ox)^ comme le faisceau des germes des series YlaeN* a<*x<*> resp. Y^p£NpbPx~P~1^ t e l l e x-modules a gauche de O^r-, resp. xIz de J7£(0x), cf. ([L a i ], [La2], [La3], [La4]). Pour tout r, 1 < r < oo, on a les isomorphismes de faisceaux d'espaces vectoriels complexes

Ox/z{r}-+Ox/z{l} Bz/x{l}^Bz/x{r} qui a la serie ^2a£NP ao<xa, resp. YlpzNp bpx~P~l, associe la serie convergente £ a € A r P aa(or!)1~ra?°f, resp. la serie YlpeNp hp{p\)l~rx'13'1. Pour tout polycylindre K, resp. tout ouvert de Stein V', assez petit de Z l'espace r(/v; C?x/z{l}) r e s P- ^V^S^/A^il})* e s t u n espace vectoriel topologique de type DFN (dual de Frechet-nucleaire), resp. FN (Frechetnucleaire) cf. ([G4], [G5], [Ge], [G7], [Gg]). Choississons des equations locales de Z on a alors les isomorphsimes d'espaces vectoriels topologiques qui commutent a Taction de T(K\Vx/z)i resp. T(V]T>x/z)i r(K;Oz)®OCP/0{l}

-*T

(K;Ox,z{l})

T(V;Oz)®B0,c,{l}

- r (V;Bz/x{l})

•

LE THEOREME DE POSITIVITE DE L'IRREGULARITE

117

Le produit tensoriel topologique complete est defini sans ambiguite puisque on a affaire a des espaces nucleaires ([G4], [G5]). On en deduit, pour tout r, 1 < r < 00, un diagramme commutatif d'isomorphismes 011 les isomorphismes verticaux ne sont pas compatibles a Taction de T(K;Vx/z)> resp. T(V]Vx/z), rnais ou les isomorphismes horizontaux sont compatibles a Taction de T{K\Vx/z)i resp. de T(V;Vx/z), T(K;Oz)®OCP/0{r} • T (K;Ox/z{r))

i

I

T(K;Oz)®OCP/0{l}

•

T{K;0X/Z{1})

T(V;Oz)®B0/Cr{l}

•

T(V;BZ/X{1})

resp.

1

I

T(V;Oz)®B0/Cp{r}

• T (V;BZ/X{r})

.

D'ou une topologie DFN, resp. FN, sur T(K;Ox/z{r})i resp. T(y\Bz/x{r}), P o u r tout r,\
Bo/cAr} [G6].

Considerons le complexe de Dolbeaut V0' (Ocp/o{r}), resp. £ (Bo/Cp{r}), des courants, resp. des formes differentielles indefiniment differentiates, sur Z a valeurs dans Ocp/o{r], resp. dans #o/Cp{ r } ([Sci], I). Pour tout compact K, resp. tout ouvert V, de Z on a les isomorphismes d'espaces vectoriels topologiques ([Sci], I) compatibles a Taction des operateurs differentiels holomorphes 0,

r (A';2>'0' •) ®OCP/0{r}

- T (K;V'0''

(Gc,/o{r}))

resp. T (V;£°--)®B0/CP{r}

^T

(V;£0'- (B0/Cp{r}))

.

Rappelons que si F' est un complexe acyclique d'espaces de type F dont les difFerentielles sont continues (done des homomorphismes) et F est un espace de type FN alors le complexe F'®F est acy clique. Ceci est un cas particulier de la formule de Kunneth topologique [SC2], voir aussi [Do]. On en deduit que si E' est un complexe acyclique d'espaces de type DF reflexifs dont les differentielles sont continues et E un espace de type DFN reflexif le complexe E'®E est acyclique. En effet le complexe dual fort E'f est un complexe d'espaces de type F dont les differentielles

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ZOGHMAN M E B K H O U T

sont done des homomorphismes [Gs]- Par dualite ce complexe est acyclique [Se2]. Done le complexe E'f®E' est acyclique et son complexe dual qui est E'®E ([G4], [G 5 ]) est acyclique. Appliquons ceci, toutes les hypotheses precedentes sont requises, on trouve que si A', resp. V, est un polycylindre, resp. un ouvert de Stein, assez petit de Z le complexe T{K;D,0>'(0CP/o{r})), resp. T(V\E0> '(B0/CP{r})), est une resolution de r res en l'espace T(K;Ox/z{ })i P F(V;Bz/x{r}) vertu du theoreme B de C a r t a n . Par consequent au dessus d'un ouvert assez petit de Z le complexe £>/0' '(Ocp/o{r})i r^sp. £ 0 ' (Bo/Cp{r}), est une resolution Vx/z~ res lineaire du faisceau Ox/z{r), P Bz/x{r}, pour un choix d'equations locales de Z. Nous avons besoin de la generalisation suivante du theoreme de dualite de Serre [Se2] : T h e o r e m e 6.1.1. (i) Pour tout r, I < r < 00, et tout ouvert de Stein es V assez petit de X l'espace Hlc(V]Ox/z{r}) t nul S i * est different de l dim(Z); Vespace H (V; B^/X{r}) est nul si i est different de 0. (ii) II existe un unique (a isomorphisme pres) couple de toplogieFN—DFN es surT{V]B^,x{r}) —H^~p(V',Ox/z{r}) ^ un accouplement compatible a Faction de T(V;Vx) qui est une dualite parfait e entres ces espaces e'est a dire qu'il identifie un espace au dual fort de l'autre. P r e u v e de (i). On a pose B^,x{r} := u>x ®ox Bz/xir}Les com0, n r son posantes du complexe £ ( # o / c { } ) ) t acycliques pour le foncteur sections globales. En choisissant des equations locales de Z on trouve en vertu de ce qui precede et du theoreme B de Cartan que Hl(V]B^,x{r}) est nul si i est non nul et l'isomorphisme de r(V;Z>x)-niodules a droite r(VnZ;u>z)®Bp0/CP{r}-+r(vnZ;Bnz/x{r}). son Les composantes du complexes V0, (Ocp/o{r}) ^ acycliques pour le foncteur sections a support compact. Mais l'espace Ocp/o{r} etant precisement de type DF on les isomorphismes ([Sci], C h a p . I §3 p. 62)

r c (vn Z;V°-•) ®oc,/0{r} - r c (v nz-v'°<' (oCP/0{r})). On trouve en vertu de ce qui precede et du theoreme de dualite de es Serre [Se2] que l'espace Hlc(V; Ox/z{r}) ^ n u l s i 2 e s ^ different de n — p et l'isomorphisme de r ( V ; D ; r ) - m o d u l e s a gauche H?~p

(V n Z; Oz) ®0CP/o{r}

-

# c n _ P (V;Ox/z{r})

.

P r e u v e d e (ii). Pour un choix d'equations locales de Z le couple est T(V]B^,x{r}) — H"~p(V]Ox/z{r}) isomorphe au couple d'espaces P FN-DFN, T(V D Z\ L>Z)®B 0/CP {r} - H?-*(V D Z; Oz)%OCP,0{r}. Le

LE T H E O R E M E DE P O S I T I V I T E DE L ' I R R E G U L A R I T E

119

couple d'espaces FN-DFN T(V n Z;uz) - H?~P(V D Z; Oz) est muni d'un accouplement qui est une dualite parfaite [Se2J. Le couple d'espaces FN—DFN Bp,CP{r}—Ocp/o{r} est muni d'un accouplement residu qui est compatible avec l'accouplement residu du couple isomorphe BP,CP{1} — OCP/Q{Y). Supposons que p > 2 ( le cas p — 1 se traite de la meme fagon) alors le couple BP,CP{1} — OCP/O{1} est isomorphe au couple Hp~l{Cp - 0 ; C J C P ) - Hlc{Cp - 0;Oz). II en resulte en vertu du theoreme de dualite de Serre que l'accouplement residu qui est compatible a Taction des operateurs differentiels est une dualite parfaite entre le couple FN — DFN,Bp0/CP{r} — Ocp/o{r}. En vertu de ([G4], [G 5 ]) le couple d'espaces

FN-DFN

J(VnZ;uz)0Bpo/CP{r}-

H^iVnZ-.Oz^OcP/oir}

est

en dualite parfaite. II en resulte des topologies FN — DFN sur le couple Y(V\Bnzjx{r) - H?-P(V;Ox/z{r}) et une dualite parfaite compatible a Taction de T(V;T>x). La seule chose qui reste a voir est que cet accouplement ne depend pas des equations locales de Z. Mais Ton a un morphisme naturel

Ox/z{r}®Bz/x{r}^Bz/x{r} Ox

d'ou un morphisme par cup-produit Hnc~P (V;Ox/z{r})

® T (V;BZ/X{r})

-

Hnc~v {V;Bz,x{r])

.

Ce morphisme suivi du morphisme residu H^~p{V\Br^jX{r}) —• H™(V',<JJX) —* C coincide avec l'accouplement precedent une fois restreint au couple T(V D Z^z) ® B*/C,{r] - H?~P{V n Z\ Oz) Oc,/o{r}. II suffit done de demontrer que ce morphisme est continu. Mais on a un diagramme commutatif pour V assez petit Ov,z{r}

® B"z/X{r}

1 T>"-{Oc,l«{r}) ® / " - ' ' ' (KK,M)

•

B"z/X{r}

I ' D"-'--(BS / c ,W)

011 le deuxieme morphisme horizontal est le produit croise ([ScJ, C h a p . II §2) du morphisme X>'°>' ®ov, £ n ~ p ' ' -+ V'n-p>' au dessus d'un petit voisinage V de Torigine dans Cn~p et de Tapplication bilineaire continue Ocp/o{r} x Bp0/CP{r} -> Bp/CP{r}. En vertu de ([Go], C h a p . II, t h m . 6.6.1) le cup produit

H?-p

(V;Ox/z{r})®T(V;Bz/x{r})

-+ ff""" ( ^ z / x M )

ZOGHMAN MEBKHOUT

120

est represents par le morphisme de complexe

rc (v n Z;V0' • (ocr/0{r}))

® r (v n z-tn-?> • (B£/C,{r}))

-r e (i/nz;^'(B; c ,{r})). En vertu de ([Sci], C h a p . II, §2, cor. de la prop. 3 p . 38) le morphisme entre ces deux complexes est forme d'applications bilineaires continues. P a r ailleurs le morphisme, pour un petit voisinage de l'origine V" dans C p , (^o/cp{r}) ~~^ HP{V"\UJCP) induit un morphisme continu m r r c ( F n Z ; I ) - ^ X ^ o / C p { } ) ^ r C ( ^ n Z ^ m " P ' ( ^ c P ( V r , , ; a ; c p ) ) . Comme HP(V"',UCP) est un espace DFN la cohomologie du dernier complexe est isomorphe a H^(V'\un'p)®HP(V';uCp) = H?(V;u>x)D'ou un morphisme continu

HrP(V;Ox/z{r}®r(y;Bnz/x{r}))-«H?(V;u,x), qui suivi du morphisme residu qui est continu est l'accouplement obtenu par cup produit entre le couple H"~p(V',Ox/z{r}) ~ r ( V ; # £ / A r { r } ) qui est done continu. Remarque 6.1.2. Hc~p(V]Ox/z{r})

Pour tout r, 1 < r < oo l'accouplement entre le couple - r(V;Z?5/;r{r})

est

compatible avec l'accouplement

du couple qui lui est isomorphe H?-P(V;OX/Z{1})

~ T(V\ ^ z / x i 1 ) ) -

Remarque 6.1.3. En prenant un recouvrement de X par des ouverts de Stein assez petit on deduit de 6.1.1 le theoreme analogue global. 6.2. Supposons que Z est une sous-variete lisse de codimension p d'une variete analytique complexe X. Le theoreme suivant est du a Ramis [R] en dimension un et a Laurent ([Lai], [La2]) en dimension superieure. T h e o r e m e 6 . 2 . 1 . ( [ L a J , [La 2 ]). Pour tout Vx-module holonome M., le complexe Rhomi>x {M, Bz/x {r}) est constructible pour tout r. De plus pour tout e positif assez petit le morphisme canonique RhomVx

(M,BZ/x{r

+ e}) -^RhomVx

(M,BZ/x{r})

est localement un isomorphisme pour tout r, sauf pour un nombre fini d'entre eux qui sont les pentes critiques locales de M, le long de Z et qui sont des nombres rationnels. T h e o r e m e 6.2.2. Pour tout complexe M de la categorie Dbh(T>x), et tout nombre reel r, 1 < r < oo, on a un morphisme canonique de la categorie Db(Cx) RhomVx(M*,Bz/x{r})

-+Rhomcx(RhomVx(M,Ox/z{r}),Cx)

[+p]

LE THEOREME DE POSITIVITE DE L'IRREGULARITE

121

qui est un isomorphisme. Si r = l,oo, le theoreme 6.2.2 se reduit au theoreme de dualite locale 1.5.1 qui correspond au cas p = 0. Pour 1 < r < 00 nour allons demontrer le theoreme 6.2.2 en reprenant les arguments de la demonstration du theoreme de dualite locale [Mei]. L e m m e 6.2.3. (i) Pour tout 1 < r < 00 lefaisceau Extlc(Ox/z{r}^x) est nul si i ^ dim(X) + p. (ii) On a un morphisme canonique de T>x-modules a droite

- Extnc+P (Ox/z{r},Cx)

ux ® Bz/X{r}

ou

n := dim(X).

Ox

P r e u v e de 6.2.3 (i). Pour tout r, 1 < r < 00, le faisceau Extlc r (@x/z{ }>Cx) est associe au prefaisceau V —> Ext^(V'; Ox/z{r},Cx)• H suffit done de montrer que pour tout ouvert de Stein V assez petit l'espace es vectoriel complexe Ext^>(l/; Ox/z{l},Cx) t nul si i est different de n+p. Mais en vertu du theoreme de dualite de Poincare pour les coefficients generaux sur les espaces localement compacts [V] on a les isomorphismes de dualite Home (HlcX,Ox/z{r},C)

« Ext**-*'

(V;Ox/z{r},Cx).

Mais en vertu de 6.1.1 (i) l'espace vectoriel Hlc{V,Ov/z{r}) est different de n — p. D'ou 6.2.3.(i).

est nu

l si *

P r e u v e de 6.2.3 (ii). II resulte de 6.1.1 (ii) que si V est de Stein assez petit l'espace H^~P(V — Z,Oy/Z{r}) est de type DFN dont le dual topologique est l'espace de type FN T(V, B^,x{r}). D'ou un morphisme canonique T (y,Bz,x{r})

- Ext£ + P

(V;Ox/z{r},Cx)

qui est r(V, 2V)-lineaire. En passant aux faisceaux associes on trouve un morphisme de T>x-modules a droite « x ® Bzix{r}

- Extnc+P

(Ox/z{r},Cx).

Ox

D'ou 6.2.3.(ii). P r e u v e du t h e o r e m e 6.2.2. Pour tout complexe M de T>x-modules a cohomologie bornee et coherente on a un isomorphisme canonique

Rhom,Vx

(M\BZ/X{r})

« M ® (LJX ® Bz/X{r}) Vx \

Ox

[-n]. J

ZOGHMAN MEBKHOUT

122

D'oii en vertu du lemme 6.2.3 un morphisme

RhomVx

{M*,BZ;x{r})

- M I RhomCx

(Ox/z{r},Cx)\p]-

Mais en vertu du lemme du "way out" foncteur on a un isomorphisme canonique pour tout complexe M de Z>x-modules a cohomologie bornee et coherente:

M I RhomCx (Ox/z{r},Cx) \p] KRhomCx

(RhomVx

{M,0Xfz{r})

,Cx) [p]-

D'ou le morphisme du theoreme 6.2.1. Pour montrer que c'est un isomorphisme si M est un complexe de la categorie Dbh(Vx), on peut supposer que M est un Vx-module holonome puis qu'il admet une resolution libre, puisque la question est locale sur X, par des 1>x-modules libres de type fini 0

-> VxN ~*

-* Vx

— °-

D'ou un representant local du morphisme du theoreme 6.2.2 ux®BrzNix{r]

0—

—

"X®Brz°/x{r}

—

Ox

Ox

I n

r

^ 0

'

1

0^Ext c+r(o »/z{r},Cx)

-

^

Ext^*

(O%/Z{T},CX)

-

0.

Pour montrer que c'est un isomorphisme il suffit de montrer que l'analogue global BrzN/x{r}J —

0— r(v,»x®

—

r(v,ux

1 0_ExtJ

+p

(v,O^z{r},Cx) —

® tf?/x{r}J ~ " ° 1

n

^Kxt c+*

(v,Or£/z{r},Cx)

— 0.

pour V de Stein assez petit est un isomorphisme. Mais en vertu du theoreme de constructibilite, pour V assez petit, la cohomologie du premier complexe d'espaces de type FN est de dimension finie et done ses differentielles sont des homomorphismes [Se2J. En vertu du theoreme 6.1.1 c'est le dual fort topologique du complexe

LE THEOREME DE POSITIVITE DE L'IRREGULARITE

0 - H?-" (v,Orx°/z{r))

-

Hnc~p (V,0%z{r})

123

- 0.

Ce dernier complexe d'espaces de typeDFNest a differentielles d'images fermees puisque dual d'un complexe d'espaces de type F dont les differentielles sont des homomorphismes [Gs]. II en resulte que ce complexe est a cohomologie de dimension finie et separee. En vertu du theoreme de dualite de Poincare pour les coefficients generaux sur les espaces localement compacts [V] le dual algebrique de ce dernier complexe est precisement le deuxieme complexe du diagramme precedent. Les morphismes verticaux induisent des isomorphismes en cohomologie puisque le dual topologique et le dual algebrique d'un espace topologique separe de dimension finie coincident. D'ou le theoreme 6.2.2. Corollaire 6.2.4. Pour toute sous-variete lisse Z de X, tout complexe M. de la categorie D\(Vx) e£ tout reel r, 1 < r < oo, le complexe es RhomT>x(M,Ox/z{r}) t constructible. P r e u v e du corollaire 6.2.4. Cela resulte du theoreme de constructibilite 6.2.1, du theoreme de dualite locale 6.2.2 et du theoreme de bidualite locale pour les coefficients constructibles qui implique qu'un complexe borne de faisceaux d'espaces vectoriels complexes est constructible si et seulement si son complexe dual est constructible. 6.3. Supposons que Z est une sous-variete lisse de codimension p d'une variete analytique complexe X. Pour tous reels r', r, 1 < r' < r < oo, definissons les deux Z>x-modules Qz{r,r'} et Cz{r,r'} par les suites exactes de T>x -modules 0 - Ox/z{r'}

- Ox/z{r)

- Qz{r,r'}

- 0

0 - Bz/X{r)

-* Bz/X{r'}

- Cz{r, r'} - 0.

Definition 6.3.1. Pour tout complexe M. de la categorie Dbh(Vx), tous reels r', r, 1 < r' < r < oo, et toute sous-variete lisse Z de codimension p posons m{ry}(M)

:=RhomVx

IRz{r,r'}{M)

:= Rhom,Vx

(M,Qz{r,r'}) (M\Cz{r,r'}).

On a ainsi defini deux foncteurs exacts IRvz{r,r'},IRz{r,r'} de la categorie triangulee Dbh(T>x) dans la categorie triangulee Dbc{Cz)- Par definition done

ZOGHMAN MEBKHOUT

124

IRz{oo,l}(M)=IRz(M)

et

IRS{oo,l}(M)=IRZ(M).

T h e o r e m e 6.3.2. Pour tout complexe M de la categorie Dbh(Vx), tous reels r',r, 1 < r' < r < oo, et toute sous variete lisse Z de codimension p les complexes I/£|{r, r'}(A4) et IRz{r,r'}(M) sont en dualite: IRZ{r,r'}{M)[-p]nKhomCx P r e u v e de 6.3.2.

(IRz{ry}[-l),Cx)

•

Cela resulte du theoreme de dualite locale 6.2.3.

T h e o r e m e 6.3.3. Pour tout Vx-module holonome M., tous reels r', r, / < r' < r < oo, et toute hypersurface lisse Z, les complexesIR%{r, r'}(M) et IRz{r,r'}(M) sont des faisceaux pervers sur Z. Remarquons que IRz{r,r'}(RalgTz(M)) puisqu'on a les isomorphismes algff£(0jr) S f l a l g r z (Bz/x{r'})

est nul pour tous les r et r' S flalgrz ( £ z / x { r } ) .

On a done les isomorphismes IRz{r, rf}(A4) = IRz{r, r'}(A4(*Z)). On peut done supposer que M est isomorphe a son localise M(*Z). Nous allons proceder en deux etapes pour demontrer le theoreme 6.3.3. On peut supposer en vertu du theoreme 6.2.1 que r et r' sont des nombres rationnels. P r e u v e d u t h e o r e m e 6.3.3. Cas (i); r = oo. Si r = oo et r' = 1 le theoreme 6.3.3 se reduit au theoreme de positivite 2.1.6. Si r = oo et rf ^ 1 il suffit de demontrer que le complexe Rhom-px(M.,OxJzir})[l] es est un faisceau pervers sur Z puisque que Rhomx>x(MyOx/z{°°}) ^ nul si M. est isomorphe a son localise A4(*Z). Considerons le faisceau T*x/z{r} des operateurs micro-differentiels de type r qui opere sur le faisceau Ox/z{r} [Las]. Rappelons [La3] que le faisceau T>x/z{r] es ^ l a restriction a Z du faisceau ^ { r } sur A := T%X lui meme restriction du faisceau des operateurs 2-microdifferentiels, de type r, £\{r,r} sur T*A. Le faisceau T>x/z{r} es ^ u n faisceau coherent d'anneaux muni d'une filtration croissante indexee par Z dont la dimension homologique est egale an — dim(X). En coordonnees locales (x,t) ou t est une equation locale de Z une section locale P d'ordre m(m G Z) de T>x/z{r] s'ecrit

p(x,t,dx,dt)=

£

aatPn(x)tadfd2

ou aa}pn(x) est une suite de fonctions holomorphes, a, /? G N, 7 G Nn~l tels qu'il existe une constant C positive avec les deux conditions

LE T H E O R E M E DE P O S I T I V I T E DE L ' I R R E G U L A R I T E

125

p(/3 - a ) + q(a + | 7 | ) < m et \aa,Pn{x)\ < Ca+f>+M((a - / ? ) ! ) r / ( « + \l\Vsur tout compact o u r = p/q. Le faisceau T>xjz\X\ s e reduit simplement a la restriction a Z d u faises ceau Vx- L'extension T*x/z{l} "-+ ^x/z{r} t plate mais non fidelement plate [Las]. II nous suffit done de demontrer que le complexe Rhomx>x/z{r} est un {^xiz{r}®vx-M, Ox/z{r})[l] faisceau pervers sur Z. Le T>X/z{r}module Vx/z{r}®vx M est holonome [La^. Mais on a de fagon evidente 0(*Z) ®ox Vx/z{r} - Vxiz{r) ®ox 0(*Z) et la localisation le long de Z commute au changement de base T>x/z{^} ~* ^x/z{r}H suffit de es demontrer que le complexe Rh.orn,Vx ^(M,Ox/z{r})[^] *' u n faisceau pervers sur Z pour tout T > x / z { r } - m ° d u l e holonome M isomorphe a son localise le long de Z'. T h e o r e m e 6.3.4. Pour tout r, 1 < r < oo, et pour tout triplet (X,Z,M) comme ci-dessus ou M. est un Vx/z{r}-m°dule holonome M. isomorphe a son localise le long de Z le complexe RhomVx/z{r} es (M, Ox/z{r})[l] t un faisceau pervers sur Z. P r e u v e d u t h e o r e m e 6.3.4. Les arguments de la demonstration du theoreme 2.1.6 se transposent puisque on a les memes ingredients. On raisonne par recurrence sur dim(A r ). Si d i m ( X ) = 1 le faisceau homVx/z{r} es {M,Ox/z{r}) ^ nul parce que si t est equation locale de Z l'operateur t qui est dans T>x/z{r} e s ^ inversible dans M.. D'ou le theoreme dans ce cas la. En dimension superieure on a (1) le theoreme de constructibilite de Laurent 6.2.1; (2) le theoreme de Cauchy-Kowalewski de type {r} a valeur dans le faisceau Ox/z{r} ([1^3], 3.2.2): si X1 est une hypersurface non caracteristique de type {r} pour M le systeme induit Ai' := Ox' 0 O x M. est un £ > x ' / Z ' { r } _ m o d u l e holonome isomorphe a son localise le long de Z' \— Z fl X' et on a l'isomorphisme RhomT>x/z{r}(Ai, Ox/z{r})\X' — Mom.vxi/zi{r](M',Oxl/z,{r}y, (3) en dehors d'un ensemble de dimension nulle de Z il passe une hypersurface X1 transverse a Z non micro-caracteristique de type { r } , C h ( A f ) { r } fl T^,A ou A' := T%,Z puisque la variete caracteristique Ch(A^){r} [Las] de type {r} d'un module holonome est lagrangienne [La 4 ]; (4) le theoreme de dualite locale pour les Z>x/ z {r}-modules holonomes qui s'enonce et se demontre exactement comme le theoreme 6.2.2 a partir du 6.1.1 parce que les operateurs de T>x/z{r} operent continument sur les espace T(V, Bz/x{r}) et par transposition sur les espaces H"~P(V, Ox/z{r}) pour V assez petit; (5) et enfin l'hypothese de recurrence sur dim(A r ) entraine que le faisceau Ext1^ r }(M,Cz{00, r}) est a support de dimension nulle puis qu'il est

ZOGHMAN MEBKHOUT

126

nul par les memes arguments qui demontrent le lemme 2.2.2.1. P r e u v e du t h e o r e m e 6.3.3 Cas (ii); r ^ oo. En vertu du cas (i) les complexes Wiomx>x{M, ^ x / z { r } ) [ l ] s o n ^ des faisceaux pervers sur Z si M est isomorphe a son localise le long de Z et done par dualite les son complexes Rhomx>x(M* ,BZjx{r}) ^ des faisceaux pervers sur Z. Du triangle distingue pour r' < r RhomVx

(M\BZ/X{r})

- RhomVx

(M\BZ/X{r'}) -+RhomVx(M*,Cz{ry})

on deduit que le complexe Rhom.x>x{M* ,Cz{ryrf}) a la condition de support. Reste a voir qu'il a la condition de co-support. Par dualite il suffit de montrer que pour toute variete lisse S de Z le complexe KTs(Rhomx>x(M*,Cz{r,r,}))[cod\mz(S)] est concentre en degres positifs. Mais RTs(Bz/x{r})[codimz(S)} = Hqs(BzjX{r}) pour tout r ou q — codim^(5). De plus si r1 < r le morphisme Hg(BZ/x{r}) —• Hl(Bz/x{rf} est injectif. D'ou la suite exacte de X^-modules 0 - H% (Bz/X{r})

-* Hi (Bz/X{r'})

-+ fff (£ Z {r,r'}) - 0

qui montre bien que le complexe RTs(RIiom'Dx(M* ,Cz{r, r'})) [codimz(S')] qui est isomorphe au complexe Rhomx>x(M* )Hqs(Cz{r) r'})) est concentre en degres positifs. Par consequent les complexes IR%{r, r'}(M) et IRz{r, r'}(A4) sont des faisceaux pervers sur Z pour tous r et r'. D'ou le theoreme 6.3.3. Corollaire 6.3.5. Pour toute hypersurface lisse Z de X, ZR^{r, r 7 }, IRz{r, r'} sont des foncteurs exacts de categories abeliennes entre la categorie Mh(T>x) et la categorie Perv(Cz). P r e u v e du corollaire 6.3.5. Cela resulte du theoreme 6.3.3 et de la suite longue de cohomologie perverse. Corollaire 6.3.6. Pour toute hypersurface lisse Z de X, pour r, r', r" tels que 1 < r" < r' < r < oo le faisceau IRz{r, r'}{M) est un sousfaisceau du faisceau IRz{r, r"}(A4) et le faisceau IRg{rf, r"}(M) est une sous-faisceau du faisceau IRz{r,r"}(M). P r e u v e du corollaire 6.3.6. On a alors deux suites exactes de T>xmodules 0 - QZ{r',r")

-

Qz{r,r"}

0^Cz{r,r'}

-+Cz{r,r"}

-

Qz{r,r'}

- Cz{r\

- 0

r") - 0.

LE THEOREME DE POSITIVITE DE L'IRREGULARITE

127

Et le corollaire 6.3.6 resulte du theoreme 6.3.3 et de la suite longue de cohomologie perverse. Definition 6.3.7. Toujours pour une hypersurface lisse Z et M un X>x_m°dule holonome, posons pour tout reel r, 1 < r < oo IRz{r}(M)-IRvz{r,l}{M)

et

IRz{r}(M)

:=

IRz{oo,r}(M).

On obtient une filtration decroissante avec r du faisceau IRz(M) := IRz{l}(M) D IRz{r}(M) D ... D IRz{oo}(M) = 0 et une filtration croissante avec r du faisceau Illjf(-M) := JR£{oo}(Al) D 7iJ|{r}(A<) D . . . D ZR£{1}(A4) = 0. Ces titrations, qui se determinent Tune l'autre par dualite, sont localement finies, et leurs sauts locaux sont des nombres rationnels, qui sont les pentes topologiques critiques de M le long de Z. En particulier si IRZ(M) est nul ou si IRz{r}(M) est nul, les faisceaux IRz{r}(M) et IRz{r}(M) sont nuls pour tout r. 6.4. Si Z est une hypersurface de X qui est singuliere nous allons encore definir des filtrations des faisceaux IRz(Ai) et IRZ(M) pour un T>x-module holonome M par des sous-faisceaux indexes par les nombres reels r, 1 < r < oo, localement finie dont les sauts locaux sont encore des nombres rationnels qui seront les pentes topologiques critiques de M le long de Z. Si / est une equation locale reduite de Z au dessus d'un ouvert U de X, notons 8f le morphisme graphe U —• U' := U x C, Z' Thypersurface non singuliere U x 0 de U' et Mj le V\j> -module holonome image directe au sens des Z>x-modules ou C designe le plan complexe. On a des isomorphismes de faisceaux pervers sur Z' IRz>(Mf)=IRz(M)[-l]

IRvz,{Mj)=IRvz{M)[-\].

et

Les filtrations IRz>{r}(Mj) et IRz,{r}(M/) se transposent au-dessus de U' aux faisceaux IRZ(M) et IRZ(M). II faut voir maintenant que ces filtrations ne dependent pas de l'equation / de Z. Ceci resulte de la proposition suivante: Proposition 6.4.1. Si S est une immersion d'un couple lisse (X,Z) dans un autre couple lisse (X1 ,Z'\ on a un morphisme canonique de projection

Ms 0

lux'

T>x' \

0 Bz*/x'{r}) Ox,

'

-+£* [N ® \ux J

\

Vx\

® Ox

BZ/x{r] '

K

128

ZOGHMAN MEBKHOUT

qui est un isomorphisme pour tout Vx-module tout reel r, 1 < r < oo.

a droite coherent Af et

P r e u v e de 6.4.1. Le morphisme est precisement le morphisme de projection. Pour verifier que c'est un isomorphisme, le lemme du "way out" foncteur nous ramene a supposer que Af = Vx, auquel cas c'est immediat. Pour un triplet (X,Z,M) les faisceaux IRz(M) et IR^M) se trouvent localement munis de filtrations IRz{r}(M) et IR^{r}(M) par des faisceaux (pervers). Mais comme la nature fait bien les choses, ces filtrations locales se recollent pour donner naissance a des filtrations globales cf. [BBD]. Ainsi si Z est un hypersurface on a defini des foncteurs exacts IRz{r} et IRz{r] entre categories abeliennes Mh(T>x) et Perv(Cx) qui ont toutes les proprietes des foncteurs analogues quand Z est lisse. 6.5. Pour tout r les cycles caracteristiques CCh(IRz{r}(M)), CCh(IRz{r}(Ai)) sont des cycles lagrangiens positifs du fibre cotangent T*X pour tout triplet (X,Z,M) ou Z est une hypersurface et M un Vx -module holonome. Ces cycles sont differences de deux cycles lagrangiens qui sont dermis de fagon purement algebrique. En effet si Z est lisse le complexe Rhomx>x(A4)Bz/x{r}) apparait dans la theorie de Y. Laurent ([Lai], [La2]) comme la restriction a Z vue comme section nulle du fibre normal T%X d'un complexe constructible sur T%X dont le cycle caracteristique est defini par voie algebrique. Par Tintermediaire de l'isomorphisme d'Euler le cycle caracteristique du complexe Rhom-px es (M,Bz/x{r}) t defini par voie purement algebrique. La encore la situation doit etre consideree comme provisoire. On doit pouvoir definir de fac,on purement algebrique une filtration par des modules holonomes du module "d'irregularite", qui doit contenir toute l'information precedente. 6.6. Pour un triplet (X, Z, M) comme ci-dessus ou Z est une hypersurface lisse le faisceau IRz(Ai) est done muni d'une filtration par des sousfaisceaux dont les sauts sont localement finis. Ce sont des nombres rationnels que Ton appellera les penies topologiques de M. le long de Z. Ces pentes varient de fagon constructibles sur Z. Elles sont contenues en vertu du theoreme de Laurent 6.2.1 dans les pentes algebriques definies pour tout module coherent et toute sous-variete lisse [La4]. Les pentes algebriques sont aussi localement finies et varient de fa^on constructible sur Z ([La 4 ], [Sa2]). Nous demontrerons dans un travail en colaboration avec Y. Laurent que pour un X>x-module holonome les pentes topologiques coincident avec les pentes algebriques le long d'une hypersurface. Nous definirons l'analogue du polygone de Newton pour les X^-modules holonomes le long d'une hypersurface et nous preciserons les resultats de ce numero en donnant un analogue complexe du theoreme de Hasse-Arf [Sei] en dimension superieure.

LE THEOREME DE POSITIVITE DE L'IRREGULARITE

129

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[BDK]

[D] [Do] [F] [Ga] [Gi] [Go] [Gs] [Gi] [G2]

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[GQ] [G7] [Gs]

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132 [SC2]

[Sei] [Se2] [R] [V]

ZOGHMAN M E B K H O U T L. Schwartz, Seminaire sur les produits tensoriels topologiques (1953-54); Operations algebriques sur les distributions a valeurs vectouelles; le theoreme de Runneth, expose 24, Secretariat de Mathematiques, Paris. J.-P. Serre, Corps locaux, Hermann, Paris, 1962. J.-P. Serre, Un theoreme de dualite,, Comm-Math. Helv. 2 9 (1955), 9-26. J.-P. Ramis, Theoremes d'indices Gevrey pour les equations differentielles ordinnaires, Mem. A. M. S. 2 9 6 (1984). J.-L. Verdier, Dualite dans les espaces localements compacts, Sem. Bourbaki, expose n° 300 (1965-66). SIGLES

[EGA IV]

[SGA 7]

Elements de Geometrie Algebnque, par A. Grothendieck and J. Dieudonne, C h a p . IV, Etude locale de schemas et de morphismes de schemas (quatrieme partie), Publ. I. H. E. S. 3 2 (1966). Seminaire de Geometrie algebrique du Bois Marie (1967-69), Groupes de monodromie en geometrie algebrique, par A. Grothendieck, P. Deligne, et N. Katz, Lecture Notes in Math. 2 8 8 et 3 4 0 , SpringerVerlag, 1972 et 1973. UA 212, Mathematiques Universite de Paris VII 2 Place Jussieu 75251 Paris Cedex 05, France

The Convergent Topos in Characteristic p

ARTHUR OGUS

l

Dedicated to A. Grothendieck on his 60th birthday

The purpose of this note is to investigate some of the foundational questions concerning convergent cohomology as introduced in [?] and [?], using the language and techniques of Grothendieck topologies. In particular, if X is a scheme of finite type over a perfect field k of characteristic p and with Witt ring W, we define the "convergent topos (X/W)conv" and we study the cohomology of its structure sheaf Ox/w a n d of K-x/w : ~ Q ® @x/wSince the topos (X/W)conv is not noetherian, formation of cohomology does not commute with tensor products, and these are potentially quite different. In fact we find the following, when k is algebraically closed: T h e o r e m 0.0.1. If X/k phisms:

is smooth and proper, there are natural isomor-

Hionv(X/W,Ox/w) Hionv(X/W,ICx/w)

3 H\Xiu

Zp) 0 W

= Q® Hiris(X/W,Ox/w)

S

HUg{X).

Actually many of the results we prove are more general, dealing with cohomology with coefficients in a crystal and with schemes over ramified DVR's and which are not necessarily smooth. When X/k is smooth, k is perfect, and V is the Witt ring of Ar, we also verify that the category of convergent isocrystals on X/V is a full subcategory of the category of crystals (in the usual divided power sense) up to isogeny. I decided to look carefully at these questions in response to Grothendieck 's remark that most mathematicians don't have the patience to think carefully about foundational matters. In addition to motivation, nearly all the fundamental ideas and techniques—indeed the very concept of geometry—in this paper are due to Grothendieck, with influences from Dwork Martially supported by NSF Grant No. DMS-8502783

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and Berthelot as well. It is an honor to be able to dedicate this work to him. 1. Widenings and Enlargements We begin by recalling the definition of enlargements made in [7], incorporating some slight modifications (one of which was introduced by Faltings in [5]). We will also find it convenient to define a more general object which we call a "widening," after Coleman's notion of a "wide open set" in rigid geometry. Let V be a complete DVR of mixed characteristic p and let n be a nonzero element of the maximal ideal of V; k is the residue field of V and K is its fraction field. Let V\ denote V/nV and let X be a V-scheme of finite type. By a "formal V'-scheme" we mean a morphism of formal schemes S —* SpfV, which we tacitly assume is essentially of finite type. (We do not assume that S has the p-adic topology.) For any such S we denote by Si the closed formal subscheme defined by nOs, and by So the largest reduced formal subscheme of Si. Definition 0.1.1. A "widening of(X/V)" is a triple (S, Z, z), where S is a flat formal V-scheme, Z C S is a subscheme of definition, and z: Z —+ X is a V-morphism. We say a widening (S,Z,z) is an "enlargement of(X/V)" l ff So C Z. If f\X —• X' is a morphism of V-schemes and {S'^Z'^z') is a widening of (X'/V), then a "morphism of widenings g:(S,Z,z) —* (S',Z',z') over f" is a morphism gs'S —• S' inducing a commutative diagram: X <4- Z

[s

[**

x' S- z'

9s

S'

In particular, we let Wide(X/V) denote the category whose objects are the widenings of X/V and whose morphisms are the morphisms of widenings over idx] Enl(X/V) stands for the analogous category of enlargements. We say a widening {S,Z,z) is "affine" iff the morphism z is an affine morphism, and we say (S,Z,z) is "absolutely affine" iff Z is an affine scheme. Notice that the category Wide(X/V) has products and fiber products. In fact, if V := (S\Z',z') and T" := (S , ; , Z\ z"), we let S'xyS" be the formal completion of S' x S" along Z' Xx Z", and it is clear that (S'xyS", Z' Xx Z",prx) is the product of T' and T" in the category Wide(X/V). Similarly, if T := (S,Z,z) is another widening and g':V -> T and g":T" -+ T are morphisms, then (S'xsS",Z' xz Z"\prx) is the fiber product V xTT" in Wide(X/V). UV and T" are enlargements

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of X/V it need not be the case that T' x T" is an enlargement, because {Z'xxZ")red ± (S'xvS")o in general. On the other hand, if T, T', and T " are all enlargements of X/V then so is the fiber product X" Xj T". Thus, Enl(X/V) has fiber products, but not products, in general. We endow the category Enl(X/V) with the Zariski topology and denote the topos of sheaves on Enl(X/V) by (X/V)conv. For example, define Ox/v(T) := Os(S), OxiT) := Oz(Z), IX/V(T) := IZ(S), and Kx,v := K ® Os{S) for any enlargement T = (5, Z, z) of X/V. One sees easily that these are all sheaves on Enl(X/V). Remark 0.1.2. The definition we have given above works even if X is purely in characteristic zero, and gives something quite close to the infinitesimal site of X. In fact, if we took ir = 0, we would have So = Sred, and then a widening is an enlargement iff S is an infinitesimal thickening of Z. In particular, if X is a K-scheme and 7T = 0, then the category Enl(X/V) is just the category of infinitesimal thickenings of schemes over X, and (X/V)conv is the "big" infinitesimal topos of X/K as defined by Grothendieck. However, due to lack of time and space, we shall in this paper always assume that 7r annihilates X) i.e. that X is in fact a V\-scheme. Then it follows that Z is also a Vi-scheme, and hence Z C. S\ automatically and the condition that Z contain So just says that S has the 7r-adic topology. We define crystals following Grothendieck's model [3]. Definition 0.1.3. A "'crystal of Ox/v"m°dules" is a sheaf E of Ox/v modules in (X/V)conv such that each morphism g\S' —> S in Enl(X/V) gives rise to an isomorphism pg:g^(Es) —+ Es>Similarly a crystal of £x/v- m °dules is a crystal of Ox/v~m°du\es such that each Es is a sheaf of /Cs-modules. We have to refer to [7] for the basic facts about sheaves of /Cs-modules on formal schemes. It should be clear that a crystal of JCx/v-modules is the same notion as that of a "convergent isocrystal" as discussed there. For any enlargement T of X/V, widening T' of X'/V, and morphism f\X -» X', we let /*(T / )(T) denote the set of morphisms T -> T over / . If / = idx we denote this just by hT'(T). One sees immediately that f*(V) is a sheaf on Enl(X/V), i.e. an object of (X/V)conv. We will see later that if T' is a widening of X/V, then the sheaf HT* determines T' uniquely and will simply write T' in place of /ijv. By abuse of language, we will say that an object of (X/V)COnv is a "widening" (resp., an "affine widening") if it is isomorphic to hx for some (affine) widening T of X/V. In particular, the product of two (affine) widenings in (X/V)COnv is again

136

ARTHUR OGUS

an (affine) widening. Similarly if i:X —• X' is a closed immersion and (S'yZ'^z') is an (affine) enlargement of X') then i*(S',Z',z') is an (affine) f 1 widening of X. Indeed, Z \— X Xx Z is a closed subscheme of Z' and hence of S', and we let S be the formal completion of S' along Z and z := prx- Z —• X; note that z is affine if z' is. Then (5, Z, z) is a widening oiXjV such that i*(S\Z',z') £ fys'.z',*')If # is an object of (X/V) c o n v and T — (5, 2T, z) is a an object of Enl(X/V), we can construct a sheaf ET on the Zariski topology of S (which is the same as the Zariski topology of Z) in the obvious way: for any open set U C S, ET(U) := E(U,Z C\ U,z\Znuy For any morphism g:T' -+ T in Enl(X/V) there is an evident transition map pg:g^Ex —• £ T ; > and it is clear that £* is determined by the collection of Zariski sheaves ET and transition maps pg. Conversely, one can easily check that any such collection will define an an object of (X/V)conv if the transition maps satisfy the obvious compatibility conditions. For a complete discussion in the crystalline case, c.f [2]. The verification of the following lemma is completely straightforward and involves nothing special about our definitions. L e m m a 0.1.4. Suppose T is an enlargement of X/V and E. is a finite inverse system of sheaves in (X/V)conv. Then there is a natural isomorphism (lim£.) T £ l i m ( £ T . ) . A sequence E of abelian objects of (X/V)conv

is exact iff each E T is.

The functoriality of our construction of the convergent topos is an immediate consequence of its definition. Proposition 0.1.5. A morphism of V\-schemes f:X canonical morphism of topoi:

—• X' induces a

conv

Proof: We define a functor: :Enl(X/V) — Enl(X'/V) by (5, Z, z) ^ (5, Z,foz). Then according to [4] we have by abstract nonsense a constellation of functors on presheaves: u*:Enl(X/Vy-+ Enl(X'/VJ *:Enl(X'/Vj-+

Enl(X/Vj

Here 0* is right adjoint to <j>* and <j>\ is left adjoint to <j>* and extends . The functors * and <j>* make up a morphism of topoi provided that is

CONVERGENT TOPOS IN CHARACTERISTIC p

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"cocontinuous." This means that for any object (S, Z, z) of Enl(X/V), and any covering sieve R of 0(5, Z, z), the set of arrows g: (S) Z,z) —• (5, Z, z) such that <j>{g) factors through R generates a covering sieve of (5, Z, z)—a triviality in this case, because S and its Zariski topology do not change when we apply <j>. It implies (in fact is equivalent to) the fact that the functor 0* above takes sheaves to sheaves. We should also note that if (S,Z,z) is an object of Enl(X/V) and (T',Z',z') is an object of Enl(X'/V) it follows trivially from our definitions that Mor(cj>(S, Z, z), (S', Z', z\)) S Mor((S, Z, z),f(S',

Z', z'))

is the set of morphisms of enlargements (5, Z, z) —• (Sf\Z',z') over / . Furthermore, one can check easily that * takes sheaves to sheaves, so that we have /* ^ <j)\ Q.E.D. 2. Universal Enlargements A crucial tool for the study of the convergent topos is the construction of "universal enlargements." These are the analogs of Berthelot's "tubes" in rigid geometry. We have to modify slightly the construction given in [7]. Proposition 0.2.1. Suppose T — (5, Z, z) is a widening of X/V. Then the sheaf hx in (X/V)conv is canonically a direct limit of enlargements: hT = \\mTx,n(S,Z,z). Moreover, each Tx,n(S,Z,z) := (Tn(S),Zn,zn) is (absolutely) affine if(S,Z,z) is (absolutely) affine. Proof: Note that -K £ Iz- Denote by Bz(S) the formal blow up of S by the ideal Iz, and let Tz{S) denote the affine piece of Bz(S) defined by 7r, with natural map A: Tz(S) —• S. Then A - 1 Z C Tz(S) is defined by the ideal 7r, and A: Tz(S) —• S is universal with this property. If n > 0, let Xn:Tz}n(S) —• S denote this construction with Z replaced by the intersection Yn of its (n — l)st infinitesimal neighborhood in S with S\, i.e. the subscheme define by 1% + nOs- Let us note that An maps Tz>n(S)i to Yn; we denote X~1(Z) by Zn. We find a commutative diagram: Zn

—

TZ>n(S)l

—

Z

—>

Yn

—+

I x

TZtn(S)

S

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138

This diagram defines a map zn:Zn Tx,n{S,Z,z)

—> X, and then :=

{TZin{S))Zn,zn)

is an enlargement of X endowed with an element of hT(Tx,n(5, Z, z)). T h u s we have morphisms of functors {Tx}n{S, Z, z) —• h(S,z,z) : n £ N } which patch together to give a morphism: \im{Txtn(S, Z, z)} —• /ITTo see that this morphism is surjective, let ( 5 , Z,z) be an enlargement of X/V and g: ( 5 , Z , z) —• ( 5 , Z, z) a morphism. Then ^ | ^ factors through Z and so g*IzO^ C / ^ . Since 5o C Z, Ig C 7rC?s for some n and hence 9*!z°s c * 0 5- ! t f ° U ° w s ^ a t g*(I% -f T T 0 5 ) = ^ s , and by the universal property of A n : T ^ n ( 5 ) —+ S, g factors through A n . One verifies immediately t h a t the restriction of g to Z factors through Zn, so that we have a morphism of enlargements of X/V: (S,Z,z) —• Tx,n(S, Z,z). T h e injectivity of our morphism will follow from the following stronger result, which we will need again later. L e m m a 0 . 2 . 2 . Suppose T = ( 5 , Z, z) is a widening of X/V, with T = l i m T n as above. Then the morphisms: hxn —• ^T n + i and JiTn —• hx are monomorphisms. Proof: It suffices to prove t h a t each hxn —• hr is a monomorphism, i.e. t h a t for every enlargement X" of X/V, the map fiTn(T') —• /ix(T') is injective. We may suppose without loss of generality t h a t T' = (S*, Zf, z'), with 5 ' = SpfC, and t h a t 5 = SpfB, with Z defined by an ideal 7; then T n is affine, say SpfBn. We claim that two homomorphisms Bn —* C with the same composition B —» Bn —*• C must be equal. Since C is flat over V, it suffices to check this after tensoring with the fraction field K of V. Then our claim follows from the fact that C is separated in the 7r-adic topology and t h a t B 0 K has dense image in Bn 0 K. Let us record this fact and its proof explicitly. L e m m a 0 . 2 . 3 . The image of K 0 B —> K 0 Bn topology.

is dense in the

w-adic

Proof: Let Cn denote the ring of the affine piece of the blow up of In + wB cut out by 7r, so that Bn is the 7r-adic completion of Cn. Evidently A' 0 B —* K ® Cn is an isomorphism. Suppose x G K 0 Bn and m > 0. T h e n there is a positive integer j such t h a t n* x G B n . Now we can write nJ x = y + wm+i z, with y G C n and z £ Bn. Then x = 7r" J 'j/ + 7r m z, where now 7T-Jy G A' ® C n = A 0 B. Q.E.D. We have already remarked t h a t if i: X —• X 7 is a closed immersion and ( 5 ' , Z ' , z') is an (affine) enlargement of X\ then i*(S', Z ' , z') is a widening

CONVERGENT TOPOS IN CHARACTERISTIC p

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(resp, an affine widening) of X. In particular, it is canonically a direct limit of enlargements \imTxtn(S,Z,z). Similarly we see that the product of two widenings is canonically a direct limit of enlargements. The following lemma follows immediately from the universal property of the construction of Tz(S). Lemma 0.2.4. Suppose g: (S', Z', z') —*• (£, Z, z) is a morphism of widenings of X/V, and Z' = g~1(Z). Then for any n > 0, there is a natural isomorphism: Tz>}n{S') —» Tz>n(S) Xs Sf. The next lemma gives us a bound on the influence of the choice of Z on the construction of the direct system {Tx,n(S, Z, z)}. Lemma 0.2.5. Let (S,Z,z) and (S,Z\zf) be widenings of X/V with Z C Z' and z'\z = z, and suppose there exists a sheaf of ideals J in Os such that J m + 1 C wOs and Iz C Iz> + J . Then there are maps hn'-Tz',n(S,Z',z')

—• Tz}n+m(S,

Z, z)

which when composed with the natural maps Tz,n+m{S,

Z, z) —• T Z / ) n + m ( 5 , Z', Z1)

in either direction become the obvious transition morphisms. In particular, the natural maps: Tz,n(S,Z,z)->Tz.,n(S,Z',z') induce an isomorphism of Artin-Rees ind-objects. Proof: It follows from our hypothesis that 7^ + m C Jg, + J m + 1 C Ify + wOs for any n > 0. Then 7£ + m + TTOS C V%, + TTOS for any n > 0, and it follows from the construction that we get an induced map Tz'tn(S) —+ Tz)n+m(S). It is apparent that this map has the desired properties. Q.E.D. Since the map Tz'tn(S) —» Tz,n+m(S) constructed above does not map Z'n to Zn+m in general, we cannot assert the existence of a morphism of enlargements Tx,n(S,Z',z') —» Tx^+miS, Z,z). However, if we let Zn denote the inverse image of Z in Tz'yn{S,Z'', z') we do get a commutative diagram of morphisms of enlargements: fn:=(Tz.n(S),Zn,zn)

J-n+m

,n+m

(S)

±+

(Tz>n(S),Z'n,z'n):=rn

ARTHUR OGUS

140

Now if E is a sheaf of Ox/v-modules, we have a natural map

and since the morphism of formal schemes underlying hn is the identity map, we can identify h*nET>n with £7^. If in addition E is a crystal, p~h is an isomorphism, and we can therefore identify ET1 with Ef . Finally, we have an isomorphism ph : hn ET . —• i?^ • Putting these all together, we get the following result: Proposition 0.2.6. Let (5, Z, z) —+ (5, Z1, z') be as in the previous lemma, and let E be a crystal of Ox/v-modules in (X/V)conv. Then there are natural isomorphisms: h E

n (Tz,n+m(S),Zn

+ m,zn+rn)

-+

E

(Tz,tn(S),Z>nz
For example, if (S',Z', zf) and (5', Z", z") are enlargements of X'', with Z' C Z" and z " ^ ' = z1, and if f: X —* X ' is a closed immersion, the above lemma applies to the morphism of widenings i*(Sf, Z\ z') —• i*(S', Z'\ z"). Indeed, let Y := X xx> Z and Y' := X xx> Z" = Y H Z". Then IY> = Jy + /z'S and since (S",Z",z") is an enlargement, J™,*1 C TT for some integer m. 3. Sheaves on Widenings We shall have to deal with widenings as systematically as enlargements. To this end we generalize the construction of the Zariski sheaf ET associated to a sheaf E in (X/V)conv and an enlargement T to allow T to be a widening. If U C S is a Zariski open subset of 5, then T\u := {U%Z O U,z\zc\u) is again a widening of X/V, and so if Z? is a sheaf in (X/V)conV) E(T\u) =: Mor(T|c/, £") is defined. Define a presheaf ET on the topological space S = Z by ET(U) := E(T\u)\ one checks immediately that in fact ET is a sheaf. Now by Proposition 0.2.1, we have an isomorphism: fiT — lim{ftyn}, where each Tn is an enlargement, and hence E(T) ^ Mor(hT,E) S lim{Mor(/i Tn ,£*)} ^ h m £ ( T n ) . Restricting to open subsets, we find that ET — \\m{\n* ETU}We will find it convenient to construct a sort of direct limit topos T as follows. We form the category whose objects are the open subsets of some Tn and where, if Un is open in Tn and Um is open in T m with n < m, Mor(Un,Um) is the set of maps Un —• J7m covering the given transition

CONVERGENT TOPOS IN CHARACTERISTIC p

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morphism <j>m,n'-Tn —• T m . (Note that this set has at most one element, so we can write U CU'iiU and U' are objects and there exists a morphism U —• U'.) The coverings of Un are just the ordinary coverings of Un C T n in the Zariski topology. One verifies immediately that we have thus constructed a pretopology and that to give a sheaf on this pretopology is equivalent to giving a family of sheaves Fn on each Tn and transition morphisms ipn: Fn+\ —• <j)n(Fn), where <j)n := <£n+i,n- We denote the topos consisting of this category of sheaves by T. Similarly, we can construct a topos S by setting Sn := S for every n with the identity maps as transition maps. This topos is equivalent to the category of inverse system of sheaves on 5, and we have an obvious morphism: 7 : 5 —» S for which 7* is just the inverse limit functor. If E is a sheaf in (X/V)COnv, then for each n we have a sheaf E?n on Tn and a transition map ipn: En+i —•
—•

f

It follows from the definitions and the remarks at the beginning of the previous paragraph that jr*Ef — ET for any sheaf E in (X/V)COnv> We shall call a sheaf E of (9^-modules "coherent" iff each En is coherent as an Orn-module. We refer to [7] for the precise definition of a coherent sheaf of Ar-modules on a V-adic formal scheme T; a coherent sheaf of /C^-modules is a similarly compatible collection of coherent sheaves of /CT H modules and transition maps. We say that a sheaf E of Of is "crystalline" iff the morphisms xj)n: En+\ ®oT Orn —* En are isomorphisms; we say that E is "admissible" iff these morphisms are surjections. For example, the sheaves /CXfVf, Ox,vf) a,ndOxf are crystalline, and lx,vf is admissible. We are now ready for the following key result. P r o p o s i t i o n 0.3.1. Let T := (S,Z,z) Ox/v(T)^Os(S)

be a widening of X/V. and

Then

Ox/V)T^Os.

Before giving the proof, let us note the following immediate consequence. Corollary 0.3.2. The functor h: Wide(X/V) ful

- • (X/V)conv

is fully faith-

142

ARTHUR OGUS

Proof of Proposition 0.3.1: It suffices to show that T(f, Of) = T(S,Os), or that jr*Of — Os- In fact, we will establish a more general result. Proposition 0.3.3. For any coherent sheaf of Os-modules F on S, the natural map F —• JT*1T^ 25 an isomorphism. For any coherent crystalline sheaf of Of-modules E in T, the natural map J^JT^E —• E is an isomorphism. In fact, JT induces an equivalence between the categories of coherent sheaves of Os-modules on S and of coherent crystalline Of-modules in T. Proof: We may assume without loss of generality that T is absolutely affine; say S = SpfB, with Z the subscheme of definition defined by the ideal L For each positive integer n, let Jn be the ideal In -f KB, and let Cn be the affine piece of the blowup of Jn in B defined by 7r. (That is, Cn is the affine ring of D+(TT) C ProjGn, where Gn is the graded ring (B 0 Jn 0 J% 0 • • •).) Finally, we let Bn denote the 7r-adic completion of C n , so that Tn = SpfBn. Let //„: B —> Cn and An: B —• Bn be the natural maps. Lemma 0.3.4. For any two positive integers n and m, the image of the map Bn+mn —• Bn is contained in Im(\n) -f TrmBn. Proof: It suffices to prove this before passing to completions. If x £ Jm+mn we denote by [x] the element y regarded as an element of degree 1 in Gm~*~mn. Then C m + n m is the degree zero part of the localization of G m + m n by [7r], which as a B-algebra is generated by elements y of the form {[^]/[7r] : x e 7 n + m n }. But the map C n + m n -> Cn takes [J"+™] to Imn[In], and any such element y will map to an element of ImnCn C 7r m C n . It follows immediately that 7m(C n + m n —• Cn) C Im(fin) -f irmCn. Q.E.D. Lemma 0.3.5. For any positive integers m and n, let J m , n := Ker(B -

Bn/irmBn).

The Furth ermore, there exists an integer N such that Jm n C m I for all n > N. In particular, for n > N, the topology on B induced by the n-adic topology on Bn is the I-adic topology, and \n.B —+ Bn is injectwe. Proof: It suffices to prove this statement with Cn instead of Bn. The inclusion Inm C J™ holds for any n and m, and hence /i n (7 n m ) C /in( J™) C irmBn. For the reverse inclusion, note that by the Artin-Rees lemma there exists an integer N such that Iv+N C\ nB C Iuw for all v > 0. I claim that for n > N, fi-1(7rmBn) C Im. Suppose that 6 £ / i " 1 ^ ™ ^ ) . If Hn(b) — 7Tmg with g £ Bn, we can write g = x/[7r]d, where x £ GJ, say.

CONVERGENT TOPOS IN CHARACTERISTIC p

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Since B is 7r-torsion-free, so is G n , and this means that [V]d6 = wmx in Gnd — Jd. We can express this in the original ring B by simply saying that there is an element x of Jd such that 7rdb = 7rm#. Since n G / , we conclude that irdb G I m J * . If d = 0 we see that 6 G 7 m , as claimed. If d > 0, note that J* = (J n + TT)** = Ind + J"^- 1 )*- -f • • • 7rd = Ind + ^ n " 1 * and so ?rd6 G 7 m + n d + I™^'1. Write 7rd6 = y + TTZ, where 2/ G 7 m + n d m d l and z e I J ~ . Then since d > 0 we have y G / m + n d n TT£, and since n > N, y G 7 r / m + n ( d - 1 ) . We can now write y — ny' and wd~lb — y' -f 2: G jm+n(d-i) + imjd-i _ jm jd-i Continuing by induction, we find that in fact belm as desired. Q.E.D. Lemma 0.3.4 implies that the image of the map Bn+mn just B/Jm,n. Thus we have maps: JO —• C7n4-mn —+ BjJmn

—• Bn/7rmBn

is

—• Cn/7r C n .

The composition of either pair of maps is just the natural transition map. Tensoring with any finitely generated 2?-module F, passing to the limit in n and rn, and applying Lemma 0.3.5, we find that the natural map F —• l i m F 0 Bn is an isomorphism. This proves half of Proposition 0.3.3. For the other half, suppose E = {En} is a compatible collection of finitely generated i? n -modules, with En+\ ®£ n + 1 Bn = En. By Lemmas 0.3.4 and 0.3.5, we have a natural map Bn+mn —• B/Jmyn —* J 5 / / m , and we set Fm := En+mn ®JBn+mn B/Im for any n > N. Then there are natural isomorphisms: jFm+i®B//m+i B/Im —• F m , and hence F := limF m is a finitely generated B-module such that F ®# B/Im = F m for each m. Moreover, F®BBn?iF®B

\unBn/irmBn

l i m F ® B ® # / / m n ® B / / — Bn/7rmBn S l i m ^ w ® B Bn/7rmBn

S limF ® B Bn/7rmBn ~ l i m F m n ® B / / m*

~ Bn/wmBn

£ limFn/7rmF £ F n . Q.E.D.

R e m a r k 0.3.6. If T = (5,Z,z) is a widening of X / 7 , /CX/v(T) is the set of rigid analytic functions on the open tubular neighborhood ]5[z of radius one of Z in 5 as defined by Berthelot [1]. We shall also have to consider the higher derived functors of the functor 7T*.

144

A R T H U R OGUS

P r o p o s i t i o n 0.3.7. Let T be a widening of X/V. sheaf of coherent Of or JCf-modules, then Rqyx*Ef absolutely affine, Hq{f, Ef) = 0 for q > 0.

If Ef is an admissible — 0 for q > 0. IfT is

Proof: Clearly the second statement implies the first, so we may as well suppose t h a t T — (S, Z, z) with S = SpfB and Z defined by an ideal J. Let N denote the topos consisting of inverse systems of sheaves indexed by the natural numbers N . We have an obvious morphism y: T —• N for which 7* takes a sheaf F of T to the inverse system of abelian groups {T(Tn, F n ) } ; we are going to use the Leray spectral sequence for this morphism. Evidently (Rqy+F)n = Hq(Tn,Fn). Since Tn is an affine formal scheme and En is coherent, each Hq(Tn,En) = 0 for q > 0. (Cf [7] for the case of JCTnmodules.) Thus, Rqy*Ef — 0 for q > 0, and it suffices to show t h a t Hp(N,y*Ef) = 0 for p > 0. Now each T(Tni En) is a finitely generated Bn (or A r n )-module, which we denote again by En. Since Hp(N,y+E) = Rp\im{En : n G N } , our proposition will follow from the following result in commutative algebra. P r o p o s i t i o n 0 . 3 . 8 . Let {Enjipn : n G N } be a collection of finitely generated Bn or K 0 Bn modules and transitions maps inducing surjections: ipn:En+l Then Rp \im{En}

®Bn+! Bn

-+En.

= 0 for p > 0.

Proof: For p > 1 this is trivial, and for p — 1 it amounts to the following statement: any sequence e. G HEn is a coboundary, i.e. there exists a sequence / G UEn such t h a t VVi+i(/n+i) _ fn — e n for all n. C l a i m 0.3.9. For any positive integers m. and n, let FmEn

:=Im(En+mn^En).

Then for any m' > m and n' > n, FmEn

C Im(Fm'Enl

-> En) +

7rmEn.

Proof: It suffices to show t h a t for j > n + run, the image of the map Ej —• En/irmEn is independent of j , i.e., t h a t it contains the image of En+mn. To see this, note t h a t since the m a p : Ej <S>B3 Bn+mn —• En+mn is surjective, the image of En+mn in Ej-KrnEn is contained in the s u b - i ? n + m n module generated by the image of Ej. But Lemma 0.3.4 above implies t h a t Bn+mn and Bj have the same image in Bn/-KmBn, and hence this image already is a j B n + m n - s u b m o d u l e . Q.E.D.

CONVERGENT TOPOS IN CHARACTERISTIC p

145

Now let fn

:= e n -}-e n + i H

e

:

n

~

Cn2+n+1 H

en2+n C(n+1)2+n + 1 .

(In the above sums, the elements e n +j are first mapped into En and the addition is performed there.) Then we find that 1pn(fn + l) ~ I n - e'n ~ ^n

Hence it suffices to prove that t! is a coboundary. Note that e'n G FnEn for all n. We now proceed to choose sequences g. and e." with gn G FnEn and e^ G 7rnEn such that

We do this inductively, starting with #1 = 0, using Claim 0.3.9 above. Then e/ (and hence also e.) will be a coboundary if e." is. But now as m tends to infinity, the image of e^ + m in En tends to zero, and hence the infinite sum hn := e^ + e^ + 1 H (where again all the elements e ^ + m are mapped to En) converges. Since hn — ifin(hn+i) = ~~en^ o u r result is proved. The case of K 0 O^-modules was treated by Kiehl [6] with slightly different hypotheses; we include it here for the sake of completeness. Suppose En is a finitely generated A" 0 Bn-module for each n and we have maps il>n: En+i —+ En inducing surjections \j)n as above. If Ln is a lattice in En and Z/n+i is a lattice in En+\, the intersection of L n +i with the inverse image of Ln is a lattice in En+\. Thus if we replace Ln+\ by this intersection we may assume that the transition maps ifin are compatible with the lattice structure. Moreover, it follows from Lemma 0.2.3 that the image of tpn is dense in En. Now if we are given a sequence e. G HEn, we can inductively find elements fn G En and e'n G 7rnLn such that en -h fn — ipn(fn+i) + e'n. Then the sequence e. is cohomologous to e'., and since the latter converges to zero, we can construct a coboundary for it as in the last step above. Q.E.D. 4. Cohomology of Widenings We are now ready to begin the study of the cohomology of sheaves in the convergent topos. By and large we can follow the techniques used in [2] to study crystalline cohomology. Of course, these techniques are variations on themes of Grothendieck and developments of Berthelot. First of all we can construct a morphism of topoi: ux/v: (X/V)COnv —* Xzar just as in the crystalline case: if j:U —> X is a Zariski open immersion and E is a

ARTHUR OGUS

146

sheaf in (X/V)conv, uXjv*{E){U) := i*conv(E){Vr/Vconv), and if F is a sheaf on Xzar and (S,Z,z) is an enlargement of X/V, U*X,V(F)T := z*F. The reader can easily check that these are adjoint, and trie exactness of u*x,v follows from the exactness of z* and Lemma 0.1.4. If T is an object of (X/V)COnv we let H%onv(T, ) denote the qth derived functor of the functor T(T, ). To compare this with the cohomology of the entire topos (X/V)conv, it is useful to introduce the restricted topos (X/V)conv\T, ie. the category of morphisms g in (X/V)COnv with target T. There is amorphism jr' {X/V)COnv\T —y (X/V)conv; recall that jj>{G) = prx'- G x T —> T, that JT*(9- F —» T) is the sheaf of cross-sections of g, and that 2T n a s a n exact ^ft adjoint J'T« which takes a morphism g: F —• T to its source i7*. All this is standard; c.f. [2] or [4]. It follows easily that, for any abelian object E of (X/V)conv, there is a natural isomorphism: Hlonv{T,E)

-

H*((X/V)conv\TJ*TE).

Let UT:(X/V)conv\T —• Xzar denote the composite u ^ / v ° JT- If T1 is an enlargement (5, Z, z) of Ar/V we can construct a commutative diagram of topoi which will allow us to compare various topologies, just as in the crystalline case [2]. Although the Zariski topologies of S and Z are the same, as ringed topoi they are different, and we emphasize this by the choice of notation in the diagram below. {X/V)conv\T

•

Vs

n| \X/V)conv

Szar

•

/

Xzar

To construct the morphism T5 let F be a sheaf on Zzar. It suffices to specify <j>^{F) on objects g.T' —• T of (X/V)conv\T whose source is an enlargement T" := {S',Z',z') of X/V. Then g induces a morphism Qz'-Z'zar ~~* Zzar, and we define <j)^(F)(g) to be ^ ( F ) ( Z ' ) . If G is a sheaf on (X/V)conv \T and U C Z is a Zariski open subset, we define T*(G)(U) to be G(gu), where gu is the inclusion (5|c/, U, z\u) —+ (5, Z, -ar). The necessary adjointness and exactness properties of these constructions, as well as the commutativity of the above, are immediate. Note that <J>T*JTF is just the Zariski sheaf FT associated to G and T — (5, Z, z). If T is a widening of X/V we have to modify these techniques a little. The techniques above define a morphism of topoi T''{X/V)conv\T —* S, and a functor <j)f^{XlV)conv\T —* T, but it does not seem to be possible to construct an adjoint <j>\, in any natural way. We still have a commutative diagram:

CONVERGENT TOPOS IN CHARACTERISTIC p

(X/V)COnv\T

+

T

{X/V)conv

•

Xzar

+

<

147

Szar

Zzar

It is useful to remark that we can define <j>*~ on the category of "crystalline" sheaves. For example, if F is a crystalline sheaf of O^-modules and g:T' -+T is an object of (X/V)conv|T, with V = (S", Z', z') an enlargement of X/V, we have seen that for n large enough # factors uniquely through a map gn\Tf —• T n . The crystalline nature of F implies that the sheaf #*F n is independent of the choice of n, so we can define %(F) to be g^(Fn). It is clear that this construction is compatible with morphisms in (X/V)conv\T, and thus %{F) is a crystal of Ox/v-modules m (X/V)conv\TIn particular, if F is a sheaf of 0s-modules, then y^(F) is a crystalline sheaf of (^-modules and < ^ 7 j ( F ) = (j>j>{F) is a crystal of ( 9 x / v m ° d u l e s m (X/V)conv\T- Conversely, it is apparent from the definition that if E is any crystal of (9x/j/-modules in {X/V)COYIV\T, <j)f^(F) is a crystalline sheaf of (9^-modules, and the natural map <j&f+(E) —• E is an isomorphism. If g: T' —• T is an object of {X/V)Conv |T with T' a widening of A"/V and if E is any object of (X/V)conv |T, we can define in the obvious way a sheaf E§ in the topos T'; this sheaf will be a sheaf of O^-modules if E is a sheaf of 0 x / y - m ° d u l e s . We will say that E is "admissible" iff each E$ is coherent and admissible, with a similar definition for sheaves of JCx/v-modules. Because of the unfortunate fact that <j>f^ is not part of a morphism of topoi it is not a priori clear that it will be compatible with Leray spectral sequences. We shall have to verify that <j>f^ takes injectives to acyclics "by hand." We shall say that a sheaf F in T is "flasque" iff whenever U C [/', the transition map F(Uf) —• F(U) is surjective. It is quite standard to verify that an abelian flasque sheaf is acyclic for the global section functor T(T, ) and for the functor JTL e m m a 0.4.1. IfT is a widening ofX/V, the functor <j>f^\ (X/V)conv\T T takes injectives sheaves to flasque sheaves. Proof: We have a morphism of topoi j n : (X/V)conv\Tn —• (X/V)conv\Tn+iThe functor j * has an exact adjoint j n ». Its abelian incarnation (for which we abusively use the same notation) is also exact. Specifically, if G is an abelian sheaf in (X/V)COnv\Tn, then j n \ is the sheaf associated to the presheaf P: P(h: S - Tn+1) = ®{G(g) :<j>nog = h}.

—•

ARTHUR OGUS

148

Since the morphism Tn+\ is a monomorphism by Lemma 0.2.2, we see t h a t P(h) is zero unless h factors through <j>n, and if it does factor, the factorization <j>n og = h is unique and P(h) = G(g). Thus, j n i deserves to be called "extension by zero," as one would hope. If H is an abelian sheaf in ( X / V ) c o n t , | T n + 1 , j„(H)(g) = H(n o —• Zt// is injective, the map I{U') —+ /({/) is surjective. Q.E.D. P r o p o s i t i o n 0 . 4 . 2 . Let T := (S,Z,z) be a widening of X/V. Then the functor rf^:(X/V)conv\T —• T is exact. If E is an abelian sheaf in (X/V)conv\T> there is a natural isomorphism: Hq(T,Ef) = q H {{X/V)COTIV\T5 E). IfT is absolutely affine and E is an admissible sheaf °f @x/v or )Cx/v-modules in (X/V)COTIV\T then these groups vanish for q>0. Proof: T h e first statement follows immediately from the definitions and the analogue of Lemma 0.1.4 for the localized topos (X/V)COnv\T> We have seen that T((X/V)conv\T, ) c a n be written as a composite of the two functors T(T, ) and f^- Lemma 0.4.1 above tells us t h a t the spectral sequence for the composite of these functors exists, and the exactness of f+E = Erf, the second statement is proved. Finally, Proposition 0.3.7 implies t h a t Hq(T,Ef) — 0 if E is a coherent sheaf of Ox/v °r Kx/v-modules in (X/V)conv\T such t h a t Ef is admissible. Q.E.D. P r o p o s i t i o n 0 . 4 . 3 . If T = (S,Z,z) is an affine widening of X/V E is an admissible sheaf of Ox/v or Kx/v-modules in (X/V)conv\T RqUT*E and RqJT*E vanish for q > 0, while HT*E = ET-

and then

Proof: We may assume without loss of generality t h a t X is affine. Then so is Z, and the previous proposition then tells us t h a t Hq((X/V)conv\T,E) = 0 for q > 0. Since Rqur*E is the sheaf associated to the presheaf U i-* Hq((U/V)conv\T, E), in fact RquT«E = 0 for q > 0. To see t h a t RqjT*(E) = 0, it is enought to show that RqJT*(E)T' = 0 for each enlargement T'. We have a diagram:

S'is

"tU

T'xT

*£*-*" (X/V)conv\T,xT

\ prs> \prf S>

^

r TlT " tlL

(X/V)conv\T,

(X/V)conv\T lJr

1Z*

(X/V)conv

C O N V E R G E N T T O P O S IN C H A R A C T E R I S T I C p

149

One verifies immediately t h a t j j , o jT+ = JT\T,* ° 3 T ' \ T ' Moreover, because the left adjoint jV" of j?, is exact, t h e functor JT1 takes injectives to injectives. This argument applies as well to the functor JT'\TUsing these facts, Lemma 0.4.1, and the exactness of j ^ , , , <j>T'*, <}>T'XT* a n ( ^ JT' > w e see t h a t {Rq3T*E)T,

S

T>*3T'Rqh+E

= =

T'*RqJT\T,*JT'\TE R prT<*T\TE q

All this holds for any abelian sheaf E on (X/V)conv\TAssume t h a t V = (S',Z',z') is an absolutely affine enlargement of X/V. Then T' x T is also absolutely affine, and if E is admissible, Proposition 0.4.2 implies the vanishing of Hq{T'xT, 0. Sheafifying on S we find t h a t RqprT'*E = 0 for q > 0, and the proposition is proved. Q.E.D. T h e following corollary is an immediate consequence of the two previous propositions. C o r o l l a r y 0 . 4 . 4 . IfT is an affine widening of X/V sible sheaf of QTJV or ^T/V modules on {X/V)conv forux/v*.

and if E is an admis| T , JT*(E) is acyclic

5. C r y s t a l s a n d C o n n e c t i o n s We begin with a description of the behaviour of crystals under pushing forward via closed immersions. Although this result will not be used explicitly, it does provide a prototype for dealing with some of t h e technicalities that arise later. P r o p o s i t i o n 0 . 5 . 1 . Let E be a crystal of Ox/vm°dules (X/V)conv and let i:X —• X' be a closed immersion. Then i*E Artin-Rees pro-crystal in (Xf/V)COnv in the following sense: there is ily {En : n G N } of'O]x>/v-modules in (X'/V)conv such that i+E = and such that for every morphism g:Tf —+ T in Enl(Xf /V), the maps P9En:9*EniT

induce an isomorphism,

of Artin-Rees

—•

EntT'

pro-objects.

in is an a famlimjE^ natural

ARTHUR OGUS

150

Proof: Let T := (S, Z, z) be an enlargement of X'/V. Then i*E(T) = E(i*T) and i*T = lim{T„ : n £ N } . Hence t*£(T) ^ lim#(T„). If we define i? n to be An*(i?Tn), where A n :T n —• S is the natural map, it is clear that i*E = lim{£'n : n £ N } . Suppose that T' —+ T is a morphism of enlargements of X'/V, where T := (S\Z',z') and T := (S,Z,*)« Let Z" := ^ / ( Z ) C 5 ' , let z" := Z " -> Z — X'', and let T" := ( S ' , Z " , z " ) . Then we can factor the morphism g as g = g" o gf, where g'lT' —• T " and 0":T"-+r. We observe first that the maps gn*Enj —* Enji> are isomorphisms. Indeed, we have by Lemma 0.2.4 a cartesian diagram: Tz/,„(5')

S'

- ^

T Z ,„(5)

12+

S.

Since the maps X'^ and A^ are affine, our diagram will commute with pushfoward and pullback. We find: g*S'En,T

-

05'^*£rx,n(S,Z,*)

-

Kn9*ETx,n(S,Z,z)

-

*"n^Tjr,n(5' f Z",«")

= En,T" Next we consider the map g'.T' —• T". Here we are in the situation of lemmas 0.2.5 and 0.2.6, and we obtain maps as follows:

^(T^ n + T O (s'),z; + m f < + m ) A

«+m*%,n+m(SV;+m,<+m)

ft

»*£(Tzi/fn(s'),z;',*'n')

-> —

A

n*%>)n(5'),^'n')

En+m,T'

—> EUyT"

En+m,T'

—•

g'*En,T"-

It should be clear that these maps define the requisite inverse of the associated Artin-Rees pro-object. Q.E.D. Theorem 0.5.2. There is a natural equivalence of categories between the categories of coherent convergent and coherent infinitesimal crystals ofOx/v modules. Proof: This is an immediate consequence of Proposition 0.3.3 above. Q.E.D.

CONVERGENT TOPOS IN CHARACTERISTIC p

151

Note that if {S,Z,z) is an enlargement of X/V, then any nilpotent immersion i: S —> 5 ' defines a new enlargement (Sf, Z, z) of X/V if S' is flat over V. In particular, if E is a crystal of Ox/v-modules on (X/V)Conv, the value E(s\z,z) °f ^ o n (Sf ,Z, z) provides a canonical extension of E(stz,z) to S'. This construction almost defines a crystal of C?5-modules on the nilpotent site of S/V—the only trouble is the flatness requirement on Sf. If E is a crystal of /C^/^-modules this flatness becomes irrelevant, and we obtain for each (5, Z, z) a crystal of K ® O^-modules as explained in [7]. We need to give a similar construction for the Artin-Rees objects associated to widenings. Let T := (S, Z, z) be a widening of X/V, let E be a crystal of C?x/v-modules in (X/V)conv, and let Ej> := lim{i? n } be the associated Zariski sheaf on S. We shall see that ET has an integrable connection as an (9s-module, in a suitable sense. Let S* denote the rath infinitesimal neighborhood of the diagonal of S x £, with its two projections prt: Sf —• S. Note that Sf will be flat over V if, for example, S/V is smooth, as we shall assume. Let Z[ := prf (Z), let z\\Z\ —• X := z o pri and let Z —+ Sf via the diagonal. Then we have morphisms of widenings:

(S',Z,z)

—

{S',Z\,z[)

\

[* (S,Z,z)

Let J be the ideal of the diagonal in S', so that J m + 1 = 0. Since Iz — Iz' -h «/, we can construct as in Proposition 0.2.6 maps E

TZtn+Tn(S',Z,z)

->

ETz,n(S',Z[,z't)-

It follows from the base change lemma 0.2.4 that Tn(S', Z'it z'i) S Tn(S, Z, z) x s , p r , S', so that there are canonical isomorphisms: pr* En = £V n (5' } z',*')• We deduce the existence of maps:

pr*2En+m ^ £T n+m (s',z^;) -* £rn+m(s',z,*) -+ £r z , n (5',z>;)

=pr{En.

These maps fit together to form a stratification on the Artin-Rees proobject associated to {En}. Let us summarize this as follows: Proposition 0.5.3. Let T := (S,Z,z) be a widening of X/V, with S/V formally smooth, let E be a crystal of Ox-modules in (X/V)conv, and let {En} be the associated inverse system of sheaves of Os-modules. Then for each n there is a map VES- £n + l —• fts/V ® En

152

A R T H U R OGUS

inducing an integrable connection ciated to jT*Ef

on the Artin-Rees

pro-Os-module

asso-

We now must develop the Poincare lemma in the context of convergent cohomology. This can be done just as in the crystalline case. We choose a method of exposition t h a t avoids explicit mention of the linearization of differential operators, but in fact t h a t is what is happening. We suppose again that X can be embedded as a closed subscheme of a formal scheme Y which is formally smooth over V. Then T := (Y,X,idx) is a widening of X/V, and for any coherent sheaf F of Oy-modules, (j>^F is a crystal of Ox/v^^odu\es in (X/V)conv\TFor any crystal E of Ox/v~ modules in (X/V)COTlVl we set QlT(E) := JT*{<1>T^Y/V ® 3T^)^ a s n e a f in (X/V)COnv We shall construct maps of sheaves:

dl:Q^(E)-^Q^1(E) which form a complex

QT(E).

P r o p o s i t i o n 0 . 5 . 4 . Let E be a coherent crystal of Kxfv-modules in (X/V)conv and let T — (Y,X, idx), with Y/V formally smooth. Then there exist a natural complex £lT(E) and a quasi-isomorphism E —• QT(E) of complexes of sheaves in (X/V)COnv Proof: Let V = (S',Z',z') be an enlargement of X/V. To simplify the notation, we assume that T and T' are absolutely afflne; the general case follows by gluing. Now Q%(E) = 3TJTE, so Q^(E)(T') = E(T' x T). Note t h a t this group has the structure of an Qy{Y)-module, we shall see t h a t in fact it has an integrable connection as Oy{Y)-module; we then obtain the complex QT(E) as the De R h a m complex of this connection. T h e construction will be essentially the same as the construction in Proposition 0.5.3. Let E' := fr(E). Then (jT>*Ef)(T') S l i m j ^ } , where E'n := E'T . (T'xT)' Each of these modules E'n can be viewed as an Oy(Y)modules, and we will see that the corresponding Artin-Rees pro-object has a stratification. Let Yf denote the m t h infinitesimal neighborhood of the diagonal of Y x y , let X[ := pr~x(X), and let z[\ X[ —• X be the obvious projection. We can also embed X in Y' via the diagonal. Then (Y\Xy idx) and ( Y ' , X / , z { ) are widenings oiX/V, with V x (Y'.XMx) = (S'xY',Z',*') and V x {J*,X[,z\) = ( S ' x Y ' , ^ ' , z<), where Z[ := Z xx X[. There is a commutative diagram:

(S'xY>,Z',z>)

—

\

(S'iY',Z'itzi)

j(S'iY,Z',z')

CONVERGENT TOPOS IN CHARACTERISTIC p

153

Let J be the ideal of the diagonal in Y1 and let J' be its pullback to S'xY'. Then the ideal of Z' in S'xY' is the sum of J' and Iz*. so again we can construct maps ETx,n+m(S'xY',Z',z')

~>

^Tx,n{S'xY\Z't,z[)'

As Z[ = pr~ (Zf), the base change lemma basechange tells us that

TXtn(S'iY,Z',z')

Y1 * Tx,„(S'xY'XzO-

xs,iYpri

We conclude as before the existence of maps: pr^E^^ —• pr\E'n, which fit together to form a stratification as desired. The previous paragraph applies to any crystal E' of Kx/v-modules on (X/V)conv \T- If E' = fyE, we get a natural map E —• jT^j^E. We shall see that this map determines a quasi-isomorphism E —* Q'T(E). This verification is a local question; i.e. it suffices to check that for each absolutely affine enlargement T' of X/V,

the map E^> —• QT(E)T'

is a quasi-isomorphism.

Now (JT^T^Y/V

®3TE)T'

=PrT'*(T^Y/V

®JTE)T'XT,

and T'xT 9* (S'xY, Z, z). It is clear from 0.3.6 that the value of <^(fi'y/v)<8) j^E) on T' x T can be identified with the global sections of the sheaf &S'XY/Y ® # T on the rigid analytic tube (S' x Y)]z[> a n d w e c a n argue as does Berthelot in his proof of the rigid analytic Poincare lemma [1] to conclude that our map is a quasi-isomorphism. Q.E.D. 6. Cohomology of the Convergent Topos Let us first note that convergent cohomology is invariant under infinitesimal thickenings. Proposition 0.6.1. If i: X —+ X' is a nilpotent immersion, the functor conv

is exact. Moreover, i*Ox/v — Ox'/v have natural isomorphisms:

and i*ICx/v — ICx'/v,

H
-

H*onv(X/V,Ox/v)

Hlnv{X'/V,ICx'iv)

-

Hlnv(X/V,Kx/v).

so that we

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154

Proof: If (S',Z',z') is an enlargement of X'/V, Z := X xx> Z' -> Z' is a nilpotent immersion, and hence Z is a subscheme of definition of S''. Then z := z'\z defines the structure of an enlargement of X; and (S',Z,z) represents the functor i*(S',Z', z'). It follows that i*(F)(s\z',z') — F(S',ztz), and this makes everything clear. Q.E.D. Now suppose that i: X —> Y is a, closed immersion from X to a formal scheme Y which is formally smooth over V. It is immediate that the object i*Y of (X/V) conv covers the final object, which we denote by X/V. In particular, a sequence E of abelian sheaves in (X/V)conv is exact iff j*0(Y^T, is. We can use this remark to construct resolutions in the usual way. Let Y(n) := i*(YxYx . . . Y) (where we have taken n-f 1 copies of Y in the product). If E is an abelian sheaf on (X/V)conv let Cy(E) := JY(n)*Jy(n)(E). Then the obvious maps C$(E) -+ Cy+l(E) make this collection into a complex, and it follows that this complex is a resolution of E. If the sheaf E is admissible as Ox/v or /C^/v-module, our resolution is even acyclic for uxfv*, by 0.4.4. Moreover, ux/v*Cy(E) = Ey(n)\ w e denote the corresponding complex of Zariski sheaves on X by CAY(E). Theorem 0.6.2. There is a canonical isomorphism in the derived category: Rux/v*E^CAy(E). For example, if E is a coherent sheaf of Ox-modules, define a sheaf iX/v*(E) on (X/V)conv by ix/v*(E)(S, Z, z) := z*(E)(Z) for any enlargement (5, Z, z) of X/V. This obviously defines an admissible sheaf of Ox/v~ modules, so the sheaves Cy(E) are ux/v*-acyclic. It follows from Proposition 0.3.3 that Ey = E for all n. Since the boundary maps are zero in even degrees and the identity in odd degrees, we can conclude: Corollary 0.6.3. If E is a coherent sheaf of Ox-modules, there are canonical isomorphisms: Hlnv(X/V,

ix/v.(E))

S HUAX, E).

Theorem 0.6.4. Let E be a coherent crystal ofOxiv-m°dules in (X/V)coru m and let E be the corresponding crystal ofOx/v- °dules in (X/V)inj. Then there is a canonical isomorphism: H?nf(X/V,E) = H%onv(X/V,E) for all qProof: We prove this under the assumption that X can be embedded in a smooth V-scheme Y, leaving the to the reader the task of applying cohomological descent to deduce the general case. Our language expresses

CONVERGENT TOPOS IN CHARACTERISTIC p

155

Grothendieck's calculation of the infinitesimal cohomology of E as the existence of a canonical isomorphism: RuX/v*E — CA'(E)Y, where EY is constructed in the infinitesimal topos just as EY was in the convergent topos. But Proposition 0.3.3 tells us that there is a canonical isomorphism CAn(E)y — Ey for each n. Since these isomorphisms are evidently compatible with the boundary maps, the theorem follows. Q.E.D. Combining this with the main result of [8], we find: Corollary 0.6.5. If X/V canonical isomorphisms:

H<mv(X/V,Ox/v)

is proper and k is algebraically closed, there are

S H
SJ Hjt(X,Zp)

® V.

T h e o r e m 0.6.6. / / E is a coherent sheaf of JCx/v~modules in (X/V)conv and X —» Y is an embedding of X in a formally smooth formal V-scheme Y, then there is a natural isomorphism in the derived category: RuX/v*E

= uTm(Q'Y ® Ey),

where (£l'Y/y 0 Ey) is the De Rham complex of the integrable connection on Ey constructed in Proposition 0.5.3. Proof: We have just seen that there is a quasi-isomorphism: E —• CtT(E) in (X/V)conv. Moreover, it follows from Proposition 0.4.4 that each sheaf of the complex QT(E") is acyclic for ux/v*- Finally, we have only to note that uX/v*ttT(E) S (£l'Y/v ® Ey). Q.E.D. T h e o r e m 0.6.7. If X/k is proper, there is a canonical isomorphism Hlnv(X/V,ICx/v)~Hiig(X/K)

Proof: In fact, this follows directly from the above result and Berthelot's definition of rigid cohomology. Q.E.D. Corollary 0.6.8. IfY/V is a smooth formal lifting of X/k, a canonical isomorphism: Hionv(X/V,JCx/v)

~K®

HhR(Y/V).

then there is

A R T H U R OGUS

156

Proof: This is an immediate consequence of 0.6.6.

Q.E.D.

7. C r y s t a l s a n d i s o c r y s t a l s For most of this section we suppose t h a t V is the W i t t ring of k, w = p, and X is a smooth k scheme of finite type. Our goal is to justify our habit of referring to coherent crystals of JCx/v-modules in (X/V)COnv as "convergent isocrystals." Let us recall the definition of the category of isocrystals. D e f i n i t i o n 0 . 7 . 1 . The category of isocrystals in CrisX/W whose objects are the crystals E of finite type Ox/v-modules phisms given by Mori80(E,

F) := K 0 Morcris(E,

is the category and with mor-

F)

for any two crystals E and F. Thus, we have an essentially surjective functor from the category of crystals of finite type Ox/v -modules in CrisX/W to the category of isocrystals. It is natural to denote this functor by E »—• K 0 E. We are now ready to state the main result of this section. T h e o r e m 0.7.2. Suppose that V = W and X/k is smooth. Then the category of coherent crystals of Kx/v-modules in (X/V)COnv is equivalent to a full subcategory of the category of isocrystals in CrisX/W. Before beginning the proof, we have to investigate the category of isocrystals in some detail. If L is a crystal of finite type Ox/v -modules in CrisX/W and T is an object of CrisX/W we can set (K 0 L)(T) := K 0 L ( T ) . This defines a functor from the category of isocrystals to the category of abelian sheaves in CrisX/W. Now if X C Y is a closed immersion of A' into a formally smooth p-adic formal V-scheme Y, let D = Dx(Y) be the (p-adically complete) divided power envelope of X in Y , and let D(l) = DX(Y x Y) and D(2) = DX(Y xY x Y ) , with the usual maps PiiPij- We obtain a sheaf ED of finite type K 0 (9jr> x (y)-modules together with an element 6G HomDW(p*2ED,p\ED) satisfying the cocycle coditions: 1. A*(e) = ide

EndD(ED)

2- P12OO 0P23M = Pia( f ) £

HomD{2)(vlED,p\ED)

C O N V E R G E N T T O P O S IN C H A R A C T E R I S T I C p

157

We shall refer to such an e as an "HPD iso-stratification" on ED- Evidently we have a functor which associates to any isocrystal E a sheaf of finite type A' 0 OD-modules ED with H P D iso-stratification. P r o p o s i t i o n 0 . 7 . 3 . The category of isocrystals in CrisX/W to the category of finite type K 0 (Do-modules with HPD

is equivalent iso-stratification.

Proof: It is easy to see t h a t our functor is fully faithful. T h e main point is to show t h a t it is essentially surjective. We begin by discussing the lift able case. The next lemma will serve as our technical key; evidently it proves the proposition in the liftable case. L e m m a 0 . 7 . 4 . Suppose Y/V is a formally smooth p-adic formal scheme and L is a coherent sheaf of Oy-modules with HPD iso-stratification c on K 0 L. Let U C Y be an open subset of Y, possibly empty, and let V be a coherent sheaf of Oy modules on U with an isomorphism a:K 0 V —• A' 0 L\u such that V is stable under the HPD iso-stratification c\u- Then there exist an e-stable coherent extension L" of V toY and an isomorphism a ' : A 0 L" —• K 0 L extending a and compatible with e. Proof: Replacing a by pna for a suitable n, we may assume t h a t a comes from an actual morphism L' —• L\JJ. Replacing V by its image in A|(/, we may assume t h a t L' C L\u> Choose m so t h a t pm annihilates L\ujV\ and let Lm be the restriction of L to the subscheme of definition Ym defined by pm and L'm C Lm the image of V in Lm. Choose a coherent extension Mm C Lm of L ^ to Ym and let M C L be the inverse image of Mm. T h e n K 0 M = A' 0 L and M\u = V. Thus, without loss of generality, we may assume in our proof of the lemma t h a t L\u = V. Let E be the sheaf A 0 L on y . T h e rest of the argument is based on the proof of rigid analytic faithfully flat descent due to O. Gabber c.f (1.9) of [7]. Let Dy(l) be the p-adically complete divided power envelope of the diagonal of Y x Y and let Dy(2) be the divided power envelope of the diagonal of Y x Y x Y. We have maps Pi:DY(l)^Y,

Pl}:DY(2)^DY(l).

Let 0: E —+ p\E be the m a p given by x h-» e(p^x). Recall from (3.32) of [2] t h a t the two projections maps pi'.Dy(l) —> Y are flat; it follows t h a t p\L C p\E. Let <j> := 6\L and L" := 6~lp\L; we shall show t h a t L" is coherent and compatible with 6. Because L\u = V is already stable under 0, it is clear from the construction t h a t L"\u = L\u. Note t h a t A' 0 L" = E: because tensoring with A' is exact, K 0 9~lp\L = 0-!p*A0L^£.

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First let us check that L" C L. In fact, the first cocyle condition implies that if we let AE:p*iE-+

E

by

I®7HXA*(7)

for x G ED)y

G OJD(I),

then A# o 9 = id. Since L is an Oy-submodule of E, A^ maps p\L to L. If x is a local section of L", 6{x) G p\L by definition, and hence x = AE(0(X)) G L also. Since Y is noetherian, we can also conclude that L" is coherent. The flatness of pi implies that p\L" C p\L\ our goal is to prove that the map ' \— 6\L>>'-L" —* p\L factors through p\L". This can be checked locally, so we may assume that Y is affine, say Y — Spec A. Let A' := A A and A!' := A 0 A 0 A Then £)y(l) and Dy(2) are also affine, say DY(1) = Sp/J3',and Dy(2) = S p / £ " , and write £ for £ ( 5 ) , e/c. As usual we denote by B' <S>B B1 the tensor product of B' with itself over J3, using P2 for the Bf which appears on the left and p* for the B' which appears on the right. Note that here we really do mean the ordinary tensor product—not its 7r-adic completion, to which we add a "~". Recall that there is a natural isomorphism: A' <S>A A! —± A!'

given by

a\ 0 a 2 0 (1$ 0 a$ 1—• a\ 0 02^3 0 04.

This isomorphism identifies the ideal V®A' + A'®V with the ideal I" of the multidiagonal of A". By (3.7) of [2], the ideal VB'(&BB' + B'&BB'V has divided powers, and so this isomorphism induces a map /?*: Bn —• B'&BB' . We denote by 6* the map £?' —> B'&BB' in the bottom row of the following commutative diagram: A'

?h

A1'

—+

A'®A A'

i . 1 _ 1 B,

£1^

B„

l_^

B'®BB'

Then the cocycle condition (2) above can be expressed as the commutativity of the following diagram: E

-^ 9

E®BBf

E 0 * B' id<8>6*

^

E®BB'®BB'

We can now prove that L" is compatible with 0. Consider the following

CONVERGENT TOPOS IN CHARACTERISTIC p

159

diagram: *^d

L"®BB'

1 L"

1

-*-

L®BB'

-^

£®*fi'

1 E

L®BB'®BB'

t®I?

I

E®BB'®B®B'

II ^

E®BB'®BB'

By definition, the square on the bottom left is Cartesian. Since p\ is flat, it remains Cartesian when tensored on the right by JB', and so the large rectangle on the right is also Cartesian. Thus it suffices to prove that the composite arrow: L"

_> E-^E

0 5 B'$^E

® B B' ® B B'pr-^>dE/L ® B 5 ' ® B # '

is the zero map. Since E/L is 7r-torsion, the natural map E/L ® B # ' ® B £ ' -+ £ / £ ®B B'®BB' is an isomorphism, so it suffices to show that our map becomes zero after we follow it with this isomorphism. This follows easily from the cocycle condition: if ar E L", 6(x) E LBB' and hence (6®id)0(x) = id®6*(0(x)) G L®BB'®BB'. Q.E.D. It is now clear that Proposition 0.7.3 will follow from the following lemma. L e m m a 0.7.5. The category of isocrystals on X/k is of a local nature. Proof: This means that if we are given an open cover {X{} of X, a finite type crystal E{ on each X{, and compatible isomorphisms of isocrystals ctij: K <S> E{ = K<S>Ej on each X% HXj, then there exists a finite type crystal ^ o n l and isomorphisms K E\xt — E{ giving back the isomorphisms otij. To prove this we may as well assume that each X( has a lifting YJ, and then we argue by induction on the number n of elements in the covering (which we may assume is finite because X is quasicompact). Let X[ := U{Xi : i < n}; then by the induction assumption there exists a finite type crystal E[ on X[ with the desired properties, and hence inducing an isomorphism K ® E'\x'r\X' — K ® Enlx'nx' • Thus, we may assume without loss of generality that n = 2 and that X
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160

V with HPD-stratification 62. Moreover, we are given an isomorphism a: K 0 L —• K 0 V compatible with the stratifications. Lemma 0.7.4 tells us that we may replace L by an isogenous L" which is still stable under its stratification and which prolongs V. This V defines a crystal on X2, and since the category of crystals is of a local nature, the proof of the lemma is complete. Q.E.D. Before beginning the proof of Theorem 0.7.2 it will be helpful to discuss briefly the following way of passing between the crystalline and convergent topoi. Remark 0.7.6. Note that if X C Y is a closed immersion of X into any smooth formal V-scheme Y, the divided power envelope Dx{Y) of X in Y (p-adically completed) is flat over V. Since the ideal J of X in Dx(Y) has divided powers, the pth power of any of its elements is divisible by p. If Y is sufficiently small and affine, the ideal I of X in Y has r := codim(X, Y) generators, and Irp becomes divisible by p in Dx(Y). It follows that the natural map Dx(Y) —* Y factors through Tx, r p(Y). Proof of Theorem 0.7.2: By Lemma 0.7.5 we can prove this locally, and hence may assume X lifts to a smooth S/V. Suppose that E is a coherent crystal of fCx/v-modules in (X/V)conv, choose n large and let S' and S" be the formal schemes underlying Tx,n(S x S) and Txin(S x S x S). For n large enough we will have maps D(l) := DX(S xS)^

5',and D{2) := DX(S

x S x 5) -+ S".

By Lemma (1.5) of [7], we can find a coherent sheaf Ls of Os-modules such that K 0 L = Es- We obtain a sheaf Ls of finite type 5-modules together with an element e GK 0 HomD(i)(plLs,plLs) satisfying the cocycle coditions. Thus any crystal of finite type /Cx/v-modules in (X/V)conv gives rise to a module with HPD iso-stratification {E,c) and in fact this gives us a functor from the category of isocrystals in (X/V)COnv to the category of finite type (9s-modules with HPD iso-stratification. By Lemma 0.7.4 (E,e) "is" an isocrystal. Also, theorem (2.15) of [7] allows us to identify FConv(X/V, E) with the set of horizontal sections of Ey with respect to its connection Vy, which is the same as A 0 Ycris(<X/\^) Z/). This shows us that our functor is fully faithful and completes the proof. Q.E.D. Let us recall the following result of Berthelot:

CONVERGENT TOPOS IN CHARACTERISTIC p

161

Theorem 0.7.7. Suppose X/k is smooth and proper, V is the Witt ring 2n of k, and L is a coherent crystal of Ox/v-m°du^es CrisX/W which is convergent, i.e. which corresponds via Theorem 0.7.2 to a coherent crystal °f fcx/v-modules E in (X/V)conv. Then there is a canonical isomorphism: K ® HlrU{X/V, L) S H*conv(X/V, E) Proof: Let us sketch the idea of the proof. Embed X in a smooth Y/V as usual, and note that for any n > rp we have a map Dx(Y) —• T ^ n ( y ) , where r — codim(X,F). Passing to the limit as n —• oo, we obtain a map of complexes of abelian sheaves on X: QY ® EY

-+ &D ®

E

D

We can check that it is a quasi-isomorphism locally on X, and hence we may assume that there is a lifting Y' of X to V. These complexes both map quasi-isomorphically to their analogues with Y' in place of Y, so the theorem is reduced to the case in which Y is itself a smooth lifting of X. In that case, Dx(Y) = Y and Txyn(Y) = Y for every n, and our morphism is just the identity map. Q.E.D. Corollary 0.7.8. If X/k is proper, there is a canonical isomorphism Hionv(X/V,

Kxlv)

~K®

Hlris(X/W).

We can deduce from this comparison theorem and 0.7.2 a new proof of the following finiteness result, first proved by Berthelot by rather different methods. Theorem 0.7.9. If E is a convergent isocrystal on a smooth proper X/k, then the cohomology groups H%onv(X/V, E) are finite dimensional K-vector spaces. Proof: If V — W, this is immediate. If there is a convergent isocrystal E' on X/W such that E = V 0 E, it is a consequence of the base change formula: Hlnv{X/V, V ® E) S V 0 Hlonv(X/V,E). In general, we may find a finite extension V of V which is Galois over W, and it will suffice to prove that V 0 E is a direct summmand of some E" which descends to W. This is clear: take for E" the sum of the conjugates of E under Aut(V"/W). Q.E.D.

162

ARTHUR OGUS BIBLIOGRAPHY

[1] P. Berthelot, Geometrie Rigide et Cohomologie des Varietes Algebriques de Caracteristique p, Colloque "Analyse p-adique et ses applications," CIRM -Lummy (1982). [2] P. Berthelot and A. Ogus, Notes on Crystalline Cohomology, Mathematical Notes, (1978) Princeton University Press. [3] A. Grothendieck, Crystals and the De Rham Cohomology of Schemes, Dix Exposes sur la Cohomologie des Schemas," North Holland (1968). [4] A. Grothendieck et. ai, Seminaire de Geometrie Algegbrique 4, Lecture Notes in Mathematics 269(1972), Springer Verlag. [5] G. Faltings, F-isocrystals on Open Varieties—Results and Conjectures, preprint (1988). [6] R. Kiehl, Theorem A und Theorem B in der nichtarchimedischen Funktionentheorie, Inventiones Math. 2(1967), 256-273. [7] A. Ogus, F-isocrystals and De Rham Cohomology II—Convergent Isocrystals Duke Journal of Mathematics 51(1984), 765-850. [8] A. Ogus, Cohomology of the Infinitesimal Site, Annales Scientifiques de PEcole Normale Superieure 8 (1975), 296-308.

Department of Mathematics University of California Berkeley, California 94720

Finiteness Theorems and Hyperbolic Manifolds

A. N. PARSHIN Dedicated to A. Grothendieck

Introduction Let / : X —» S be a proper smooth holomorphic family of projective algebraic varieties. If we fix a base point SQ £ S, then we have monodromy action of the fundamental group />:*i(S,so)— Aut

Hp{X0,l)

where Xo is a fiber over so- The following results were proved using a hyperbolic metric which was introduced by S. Kobayashi [10], [11]. Borel's Theorem. Let S = A* be a punctured disk and T be an image of a generator of 7Ti(5, 0) under p. Then there exist non-negative integers M and N such that (TM - ld)N = 0. Deligne's Theorem. Let S be an algebraic variety. Fix a Hodge type of the fibers of f. Then the set of characters of representations p over R arising from all families with given type is finite. For a more complete description of the applications of hyperbolic analysis to monodromy problems and related problems (Griffiths' theory), we refer to [5]. A proof of Deligne's theorem can be found in [8] (and an outline in [24]). In this paper we apply these kind of ideas to a study of sections of the families f : X —> S. Motivated by diophantine geometry over functional fields we are interested here in the finiteness theorems for the set of sections. The paper is organized as follows. Section 1 contains a short survey of hyperbolic manifolds. In Section 2, we give a new proof of MordelPs

A. N. PARSHIN

164

conjecture for algebraic curves over functional fields using a hyperbolic argument. At last in Section 3, these ideas are applied to a study of rational points on the projective subvarieties of abelian varieties (a new proof of Raynaud's theorem [17]) and of integer points on the affine open subsets of abelian varieties. In the last case we prove Lang's finiteness conjecture for integer points [12] (under a mild restriction on the hyperplane section). These results can be considered as some arguments in favour of the general Lang conjecture t h a t the Mordell property is valid for the hyperbolic varieties (see [13]). Some hints about the possible application of these ideas to the number case are collected in an appendix. This paper is indebted very much to the works of Borel, Griffiths, and especially Deligne, mentioned earlier. Recently I was familiarized with Grothendieck's letter [7] in which he has outlined new conjectures about the relation between the maps of algebraic varieties defined over number fields and the corresponding maps of their fundamental groups. Going back probably to his older paper [6] these conjectures should be valid for the "anabelschen" manifolds (in his terminology) which have something in common with hyperbolic manifolds (and coincide with them in dimension 1). The problems which Grothendieck considers in [7] can be also formulated for the holomorphic maps of algebraic varieties. Under the strong conditions of the "negativeness" type a positive solution of the problems is contained in the theorems of Borel-Narasimhan [1] and Siu [20] (the last result is the rigidity theorem for Kahler manifolds - a holomorphic variant of Mostow's rigidity theorem for locally symmetric spaces). All these results and conjectures show the deep link between the hyperbolicity and the arithmetical properties of the variety. This work was finished during my stay at IHES in September-October 1986 and I would like to express my gratitude to Professors M. Berger and N. Kuiper for their warm hospitality. 1. H y p e r b o l i c M a n i f o l d s We consider here complex manifolds X not necessarily compact. General references about the hyperbolic manifolds are [10], [11], and [14]. Let i , t / G l . Choose the following structure a : the points xo, x\,..., xn with xo = x, xn — y, holomorphic maps /,• : A —»• X (where A = {Z G C : \Z\ < 1}) and points at,b{ G A, i = 0 , l , . . . , n — 1 such t h a t fi(at)

= Xi,fi(bi)

= a?,- +1 .

Definition 1. Let dx(x,y) = inf a ^ p(a,i,bt) where p is the Poincare metric in A . On every complex manifold, dx satisfies symmetry and triangle conditions but it can happen that dx{x,y) — 0 for different points x,y.

FINITENESS THEOREMS

165

Definition 2. A complex manifold X is hyperbolic iff dx(x,y) > 0 for all points x / y. In this case, dx is a hyperbolic metric on X. It has the following properties: 1. For any holomorphic map / : X —• Y and x, y G X

dx(x,y)>dY(f(x)J(y)) 2.

C?A

= p-

3. rfjxy(xi, 2/i), (a?2, !fe)) > max (d* (xi, ^2), d*(yi, 2/2)) 4. If 7r : X —• Y is an unramified covering x,y £ X, then c/y(7r(x),7r(y)) =

inf

dx(x,y')

It means that the product of hyperbolic manifolds is hyperbolic, that is, if / : X —• Y is a covering then X is hyperbolic iff Y is. By definition dc = 0 and we get that a compact Riemann surface is hyperbolic iff its genus g > 1. Also, if we remove from some compact Riemann surface (of arbitrary genus) a finite number of disjoint closed disks (with sufficiently smooth boundaries), then we get a hyperbolic surface with a complete metric. 5. Let / : X —» S be a proper holomorphic map. Then the set of s (E S' such that fiber Xs is hyperbolic is open in the classical topology [2]. 6. Let y be a relatively compact complex subspace of a complex manifold X. Denote by Y its closure in X and assume that there is no non-constant holomorphic maps C —• Y. Then X is hyperbolic [4], [13]. 7. Let W be a relatively compact open subset of a complex manifold X, D is a closed complex subspace in W and W,D are their closures in X. __ Assume that there are no holomorphic maps g : C —• D and g : C —• W — D. Then W — D is hyperbolic and there exists an Hermitian metric p on X such that d\v-D > p\w-D> This result is actually proved by Green [4, proof of Theorem 3] but not stated as above. His proof can be applied to this more general situation without any change (see [13, p. 175]). To the examples of hyperbolic manifolds already mentioned, we can add bounded domains in C ^ (because they are contained in A ^ ) and also 8. If A is a complex torus and D C A is a closed complex subspace, then D is hyperbolic iff D does not contain a translation of some complex subtorus [4]. This result is quite obvious in one direction because d^N — 0, and consequently d& = 0 for any complex torus.

A. N. PARSHIN

166

2. Algebraic Curves The main purpose here is to prove the following: Theorem 1. Let X be an algebraic curve defined over the field K = C(B) of algebraic functions of one variable. Assume that X is projective, smooth and of genus g > 1, and cannot be defined over C even after some finite extension of K. Then, the set X{K) of rational points is finite. Here B is a projective algebraic curve (= compact Riemann surface) over C. This result is a geometrical analogy of Mordell's conjecture and was proved by Manin. The other proofs were given later by Grauert and the author (see [12]).* The theorem can be reformulated in purely geometrical terms. Theorem 2. Let V be a projective smooth surface, and B be a projective smooth curve over C. Consider a fibration f : V —• B having as fibers projective smooth curves of genus g > 1, possibly with a finite number of exceptions. Assume that for any finite covering B1 —* B the fibration V x B' —• B' is not trivial. B

Then the set of sections of the map f is finite. First of all we introduce the following notations. Let S be a finite subset in B, corresponding to non-smooth fibers of / . Denote by U a disjoint union of small closed disks around the points from S. Let

D = rnu) W = V-D b0£B-U = B0 wo € / - 1 ( M

Fix a point wo and consider sections C of the map / which are going through the point WQ of the fiber F. *A very nice effective proof (our proof here is not effective) was found by H. Esnault and E. Viehweg. It gives the best known bounds for the height of rational points.

FINITENESS THEOREMS

167

The fibration / : W —• BQ is smooth and that gives an exact sequence of fundamental groups

(1)

1 - 7n(F, Wo) - *l(W, W0) ^

7Ti(So,60) - 1

All the manifolds are topologically K(TT, l)-spaces. If the section C goes through w0 it defines a splitting i c of a sequence (1). If not, take the point w\ of intersection C and F and connect u>i and WQ by some path 7 on F . As above, C defines a splitting of a sequence like (1) but with the point w\ instead of wo- Then the path j gives an isomorphism of the new exact sequence with (1). It means that we can associate with C a conjugacy class of splittings ic. Theorem 2 will follow from the two statements. Proposition 1. A class of homomorphisms ic defines the corresponding section C up to a finite number of possibilities. Proposition 2. The set {ic} of conjugacy classes of the splittings coming from the sections C is finite. Proof of Proposition 1. We have a commutative diagram with exact rows (2)

H2(V,D)+-H2V)+-H2(D)

T 7 = Image of

T

T

NS(V) *- NS(K) <- { F j

where NS(l^) is the Neron-Severi group of the surface V and {F;} is a subgroup generated by the components of non-smooth fibers. A curve C gives us elements (denoted also by C) in NS(V) and by (2) in H2(V) and H2(V,D). The conjugacy class ic defines C as an element of H2(V, D) because all the spaces involved here are K(TT, l)-spaces and their maps are conjugacy classes of the homomorphisms of fundamental groups. If we know C in H2(V, D) then we know it in the group H2(V) by Lemma 1. Let C,C be the sections and C = C + T
A. N. PARSHIN

168

i

lp 3

So we can assume that C is equivalent to C I claim t h a t C — C. If not, remember t h a t on a non-isotrivial fibration we have C.C < 0 and C.C < 0 (Arakelov's Theorem). By assumption C.C > 0 and we see that

C.C = C.C' + C'.C = C'.C' +

Xnl(Fi.C)<0 Zni(Fi.C,)>0.

T h e definition of equivalency implies t h a t E n , ( F t . C ) = T,nt(Fi.Cf) we get a contradiction.

and

It remains to show that the knowledge of C in H2(V) defines the curve C up to a finite number of possibilities. If we imbed our surface V to Pjv, then the degree of C in this imbedding is defined by its intersection with a hyperplane section as with the classes in HiiV). The above mentioned inequality C.C < 0 means t h a t the sections are impossible to deform as the curves on V. This gives the finiteness statement. P r o p o s i t i o n 3 . The surface W is

hyperbolic.

This follows immediately from Proposition 6, Section 1 because b o t h Bo and the fibers F are hyperbolic. Remark. It is possible to give two other proofs of this fact. T h e first way is to remove some smooth fibers also and then apply Griffiths' Theorem [5] t h a t the surface W can be uniformized by a bounded domain in C 2 . T h e other way is to construct over W a smooth non-trivial family of algebraic curves of some genus (using an author's construction of ramified coverings, see [24]). This gives a finite map to the product of moduli varieties ( = Teichmiiller space) by the base curve BQ which is hyperbolic. P r o p o s i t i o n 4. Any section C considered as a submanifold of the surface W with an hyperbolic metric dx is a totally geodesic submanifold. It means t h a t for any points x,y G C

169

FINITENESS T H E O R E M S

(3)

dc(x,y)

=

dw(x,y).

Proof. Applying Proposition 1, Section 1 to the maps C <—* W —+ B - ^ C we get dc(x,y)

> dw(x,y)

> dB(f(x)J(y))

>

dc(x,y)

and t h a t implies (3). P r o o f o f P r o p o s i t i o n 2. Choose the loops 7 1 , . . . , j n on (Bo, 60) which are represented by the generators of wi(Bo,bo). Also choose a system of p a t h s connecting all pairs WQ^W\ of points on F. This gives some representative i : 7r 1 ( J 0,6o) - * KI(W,WQ) in any conjugacy class ic such t h a t the images 2(71), • • • ,i(jn) have the following expression in iti(W,wo) (4)

f(7Jb) = 7 7 * T " 1 ,

*=l,...,n.

Here 7* are the loops jk themselves b u t considered on the section C —• B and 7 is a p a t h connecting WQ with w\ = C f) F. By Proposition 4 the length of jk in the hyperbolic metric of W is bounded and it is true for Z

(TJO °y t n e compactness of F.

We can enlarge slightly the set U t o Uf without touching the point 60 and the loops 7 1 , . . . , 7 n - Let W = f~l(U ), U be a closure of [/'. Then 7Ti(W',wo) = 7Ti(W, wo) and all loops 2 ( 7 ) , . . . ,i(yn) belong to WW is a compact manifold with a boundary and we have L e m m a 2 . The set of elements of bounded length is finite.

of 7Ti(py/, iu 0 ) represented

by the loops

T h e proof can be done along different lines. Take, for example, a universal covering W —• W. Its Galois group T is 7Ti(W) and the loops of bounded length on W correspond to the paths on W contained in some ball. T h e intersection of any (compact) ball with an orbit of T is finite and we are done. To get the statement of Proposition 2, we need to mention only t h a t our representative i is completely defined by its values (4) which constitute a finite set by the lemma and by the bound for the length in the hyperbolic metric. Remark. If the hyperbolic metric would be a Riemannian metric of non-negative curvature, then the classical argument on the uniqueness of geodesical loops in each homotopy class states t h a t the class ic defines the corresponding section C in a unique way.

170

A. N. PARSHIN 3. Subvarieties of Abelian Varieties

To get finiteness theorems on the abelian varieties, it is necessary to avoid some trivial exceptions. Let A be an abelian variety defined over a field K = C(B) of algebraic functions on some projective curve B. Definition 1. Let AQ be a maximal abelian subvariety of A which is defined over C(/\/C-trace, it exists and is unique [12]). Then we say that a set M C A(K) is a finite modulo trace iff the M intersects with only finitely many cosets of A0(C). Theorem 1. [17]. Let X C A be a projective subvariety of an abelian variety, defined over a functional field K — C(B). If X does not contain a translation of an abelian subvariety, then the set X(K) of rational points is finite modulo trace. Denote by SA the subset of B where A has a bad reduction. Then there exists a group scheme f : A —* B (Neron's minimal model) with a general fiber A and which is smooth and projective over the base outside SA- Rational points from ^4(A') define then the sections C e—• A of the map / . Definition 2. Fix some finite subset S of B — SA and some projective imbedding of A. Let H be a hyperplane section of A and let D be its closure in A. A rational point P G A(K) is an S-integer point iff

f{CnD)csAvs for the corresponding section C. Theorem 2. Let A be an abelian variety over a functional field K = C(B) and let H be a hyperplane section in some projective imbedding of A. Assume that H does not contain a translation of an abelian subvariety. Then any S set of S-integer points on A is finite modulo trace. Without the assumption (*) it is Lang's conjecture [12]. I do not know whether the assumption (*) is really necessary or not. There are examples of abelian varieties with hyperplane sections containing an abelian subvariety [13, p. 174]. Corollary 1. If A is a simple abelian variety, then the set of S-integer points is always finite modulo trace. Corollary 2. Let H be a general hyperplane section on an abelian variety A. Then the set of S-integer points is finite modulo trace.

FINITENESS THEOREMS

171

Proof of the theorems will be done simultaneously along the same lines as in Section 2. We remove from B a finite number of disjoint closed disks and let Bo be a remaining part of B0 and W = f~1(B0), D is a closure in W of the subvariety X (Theorem 1) or a closure of the hyperplane section H (Theorem 2). We assume that at least SA <£ Bo but the exact description of removed disks will be done later. There is an exact sequence of fundamental groups (we fix the base points as in Section 2): (1)

1 - HMK)

- * i ( W > o ) - i * *n(Bo,M - 1.

Any rational point P G A(K) defines a section C of the fibration / : W —» Bo and a splitting ic (or conjugacy class of splittings) of the sequence (1). Proposition 1. The set of rational points P G A(K) class ic is finite modulo trace.

with a given

Proof. Let G = 7Ti(J5o,6o). The manifold W is an abelian group scheme over BQ. It has a zero section which defines some splitting i 0 of (1). The difference ic — %o is a map from G to H\(Ab0) and actually is a 1-cocycle on G. It gives a homomorphism a : A(K) —• /f 1 (G, T), T = H1(Abo,l). On the other side, if S D SA are the centers of the removed disks then G = Ga\(K8/K) where Ks is a maximal Galois extension of A' unramified outside S. An exact sequence 0 _> Aln - A(KS) -£+ A(KS) - 0 induces the following commutative diagram A(K)/lnA(K)

^

T A{K)

H\G,Aln)

i -2U H\G, V) J-+

H'iCAin)

where the map /? is coming from the natural maps T —> i/^^Bo) Z / l n Z ) = Ain. Now an equality a ( P ) = 0 implies that P is infinitely divisible in A(K). Because the group A{K) is finitely generated modulo trace (Mordell-Weil theorem) the point P is contained in ^4o(C). The proposition is proved.

172

A. N. PARSHIN

Proposition 2. There exists a finite number of disjoint closed disks U C B such that the subvarieties D and W — D lying over Bo = B — U are hyperbolic. There exists an Hermitian metric p on W such that d\y-D > p\w-DProof. Consider the family Db,b G B. By assumption (from both theorems) a general fiber of the family does not contain a translation of an abelian subvariety. From the general theorems about Hilbert schemes of subvarieties, it is easy to deduce that there exists a countable subset R C B such that outside of R any fiber Df, does not contain a translation of an abelian variety. By Proposition 8, Section 1, it is hyperbolic. From Proposition 5, Section 1, we know that the set {6 G B : D}, is hyperbolic} is open in complex topology. Consequently, the set {6 G B : Db is not hyperbolic} is closed and no more than countable. Lemma. Let R be a closed countable subset of a compact Riemann surface B. Then R is contained in a finite number of disjoint closed disks. Proof. For any point x G R there exists a closed disk Ux such that x G Ux and dUx 0 R = 0. Taking these disks sufficiently small, we can assume that they do not intersect each other. By compactness, there are only finitely many of them. Now we apply the construction of the manifolds W and D introduced above to the set of disks which are slightly more than the disks from the lemma and which contain the points S C B m the case of Theorem 2. If<7 : C —> D or g : C —> W — D are holomorphic maps then / o g are trivial maps by the hyperbolicity of Bo. But the fibers Df, and (W — D)&, are hyperbolic by Propositions 8 and 7. So our maps g are trivial. Applying again Proposition 7 (for Theorem 2) and Proposition 6 (for Theorem 1) we get that both D and W — D are hyperbolic, as well as an existence of metric p. The theorems follow now from Proposition 3. The set of classes of splittings {ic} arising from the sections belonging to D (Theorem 1), or from the sections corresponding to S-integer points in W, are finite. Proof. First, we consider the case of Theorem 1. Then we proceed exactly as in the proof of Proposition 2, Section 2, using Proposition 2 instead of Proposition 3, Section 2. This gives a finite number of (classes) splittings of the exact sequence *(3) 1 -+ TTI(£)6O,U;O) - • 7Ti(D,w0) —• ffiCBo,M -+ 1. There is a map of the sequence (3) to (1) and thus our splittings completely define the maps ic from (1). An application of Proposition 1 gives us the theorem.

FINITENESS T H E O R E M S

173

In the case of Theorem 2, we are working with the fibrations / : W — D —• BQ which are not proper. Using the exact sequence for TTI(W — D), wo) we get the bounds for the lengths of i'c(jT) in the hyperbolic metric d ^ - D - Here i'c : 7Ti(Bo,bo) —* TTI(W — D,w0) is a splitting corresponding to the section C which goes through wo. We do the same for any section C with C 0 (W — D)b0 = w\. The bounds for the images i'c(yx) in 7Ti(W — D,w\) will be the same. Using the second part of Proposition 2 we get the upper bounds in some metric p on W} and because Wi,0 is compact, the distances p(wo,Wi) are bounded for all w\. Returning as in Section 2 to conjugacy classes in -K\{W — D, WQ) and applying the natural m a p from it\{W — D, w$) to n\{W, WO), we get the bound for the images of ic in the sequence (1). Now we can use Lemma 2 for the metric p on W, and then apply Proposition 1. Remark 1. These results are related to the following general problem. Let / : X —* S be a proper algebraic family of hyperbolic manifolds over a Riemannian surface S (S can either be an algebraic curve, or a compact Riemann surface without a finite number of disks). Is the set of algebraic sections considered modulo deformations finite? See the discussions in [13] and [18]. A positive answer is known for families with strong additional restrictions (triviality of / ; the fibers should have a negative tangent bundle in Grauert's sense. See a review in [18]). The most general result along these lines is Noguchi's finiteness theorem [15] for the families of hyperbolic manifolds. He uses the following condition: if / : V —• B is the family and U C V is the set of all smooth fibers then U should be hyperbolically imbedded into V in some sense. T h e general theorem is applied in [15, Section 5] to prove the Mordell conjecture in the functional case by using hyperbolic metric. It is done there by checking t h a t U is hyperbolically imbedded for some model V of the general fiber. M. G. Seidenberg has developed a different approach. He could prove t h a t the desired property can be satisfied after some base change (see [19]). Remark 2. Theorem 1, Section 2 is a particular case of Theorem 1. If we imbed a curve X in its Jacobian A, then the intersection X C\ Jo(C) is finite for non-isotrivial X. Also Proposition 1, Section 2 can be replaced by the argument used in the proof of Proposition 1 above.

A. N. PARSHIN

174

Appendix The loops, primes and Frobenius Let K be a number field, A# its integers, and X an algebraic curve of genus > 1 over K. Is it possible to apply the approach developed above to this situation? (Same question for problems from Section 3). First of all, we can consider Neron's minimal model over Spec A# instead of the fibration f : V —> B. A definition of fundamental groups was done by Grothendieck (see SGA 1 and [9]). We have an analogue of the exact sequence (1) and the finiteness problem can be also divided in two parts. It can be shown that Proposition 1 remains valid in the new situation, and the main difficulty is to find some substitute for the loops and their lengths from the proof of Proposition 2. Not pretending to answer this question, we give here a remarkable analogy between loops on complex manifolds and closed points (= primes) on algebraic varieties defined over a finite field Fq.

1. a complex manifold X over C with a base point XQ

an algebraic variety X over Fq

2. a loop 7 on X

a closed point vonX

3. length fx- (7) of the loop 7 in hyperbolic metric

norm Nv of the point v

4. The set of homotopy classes of loops 7

the set of closed points v with Nv < C is finite

with£x(l)
FINITENESS THEOREMS 5.

175

/ : Y — X is a holomorphic map preserving base points, or an algebraic morphism defined over Fq If 7 is a loop on Y, then a loop fx(l) is defined on X and

If v is a closed point on Y, F*(v) is a closed point on X and

M7)>'x(/.7)

Nv>N(f,v)

7r : y —• X is a covering Let y G 7T_1(^o) and a = (j( T , 2/) G Gal(Y/X) is such that 7 is covered with a path

Let y G 7r_1(i;) and F = FTV be the Frobenius automorphism of k(y) over k(v)



y /

=jy

1

Ff =

CT = 7- CT7

y~1Fj

f:Y-+X gives a map /•OKT.J/))

=
/ . : xi(y) -

jn(x)

and a map f:Y-+X of universal covering

7T : Y - • X is a finite covering

/.(Fr?) _

FJ(y)

A. N. PARSHIN

176 Fix y on Y V a E Gal(Y/X) 3 ja =
9. a loop of minimal length in the class a

10. Ruelle zeta function (for the conjugacy classes of geodesic loops)

Chebotarev's theorem: every a G Gd\(Y/X) is represented by a Frobenius automorphism

Frobenius automorphism of smallest n o r m in the class a

Weil zeta function (for the rational points over Fq)

This type of analogue was used already at least twice: (1) If we consider a compact Riemann surface of genus > 2 and the distribution of lengths of closed geodesies on it, then one has an asymptotical law which is exactly the same as for primes (from A ^ , or for closed points on curves defined over F^). This is also true for Riemannian manifolds with negative sectional curvature (see [16], [21], [22]). (2) Faltings has proved an analogy of Deligne's theorem from the introduction for the number case [3]. His use of the Riemann-Weil hypothesis to bound the norms of Frobenius automorphisms has great similarity with Deligne's bounds for the trace of monodromy operators. Remark 1. Still, there is a great difference between the objects which we have compared above. We can associate with the loops the elements of the fundamental group. In the number case, the Frobenius automorphisms are defined only up to conjugacy. They look much more like the free loops (without a base point). It is exactly the difficulty which appears in the straightforward a t t e m p t s to construct non-abelian class field theory. Remark 2. In [23] P. Vojta has introduced a new kind of analogy between number and functional cases (see, in particular, the dictionary on p.34). It would be interesting to find a relation of his analogy with the previous one.

FINITENESS THEOREMS

177

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8]

[9] [10] [11] [12] [13] [14] [15]

[16] [17]

[18]

A. Borel and R. Narasimhan, Uniqueness conditions for certain holomorphic mappings, Inv. Math. 2 (1967), 247-255. R. Brody, Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc. 235 (1978), 213-219. G. Faltings, Endlichkeitssatze fur abelsche Varietaten uber Zahlkorpern, Inv. Math. 73 (1983), 349-366. M. Green, Holomorphic maps to complex tori, Amer. J. Math. 100 (1978), 615-620. Ph. Griffiths, Periods of integrals on algebraic manifolds, Bull. Amer. Math. Soc. 76 (1970), 228-296. A. Grothendieck, Un Theoreme sur les Homomorphismes de Schemas Abehens, Inv. Math. 2 (1966), 59-78. A. Grothendieck, Letter to G. Faltings, 27 June 1983. P. Deligne, Un Theoreme de Finitude pour la Monodromie,, inbook "Discrete Groups in Geometry and Analysis", Birkhauser, BostonBasel-Stuttgart, 1987, 1-19. N. Katz and S. Lang, Finiteness theorems in geometric class field theory, l'Enseignement Mathematique 27 (1981), 285-320. S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings, New York, 1970. S. Kobayashi, Intrinsic distances, measures and geometric function theory, Bull. Amer. Math. Soc. 82 (1976), 357-416. S. Lang, Fundamentals of Diophantine Geometry, Springer, New York-Berlin-Heidelberg, 1983. S. Lang, Hyperbolic and diophantine analysis, Bull. Amer. Math. Soc. 14 (1986), 159-205. S. Lang, Introduction to Complex Hyperbolic Spaces, Springer, New York-Berlin-Heidelberg, 1987. J. Noguchi, Hyperbolic fibre spaces and Mordell's conjecture over function fields, Pub. Res. Inst. Math. Sci. Kyoto Univ. 21 (1985), 27-46. W. Parry and M. Policott, An analogue of the prime number theorem for closed orbits of Axiom A flows, Ann. Math. 118 (1983), 573-591. M. Raynaud, Around the Mordell conjecture for function fields and a conjecture of Serge Lang, Lecture Notes in Mathematics 1016 (1983), 1-19. M. G. Seidenberg and V. Ya. Lin, Finiteness theorems for the holomorphic maps, in "Encyclopedia of Mathematical Sciences," 9, Several Complex Variables III, Springer, Berlin-Heidelberg-New York, 1987.

178 [19]

[20]

[21] [22]

[23] [24]

A. N. PARSHIN M. G. Seidenberg, Functional analogue of the Mordell conjecture: A non-compact version, Izvestija AN SSSR, ser. matem. 53 (1989). (In Russian). Y.-T. Siu, Strong rigidity for Kahler manifolds and the construction of bounded holomorphic functions, in "Discrete Groups in Geometry and Analysis," Birkhauser, Boston-Basel-Stuttgart, 1987, pp. 124-151. T. Sunada, L-functions in geometry and some applications, Lecture Notes in Mathematics 1201 (1986), 266-284. A. B. Venkov, Spectral theory of automorphic functions, Selberg zeta-function and some problems of analytic number theory and mathematical physics, Uspekhi Mathem. Nauk. 34 (1979), 69-135. P. Vojta, Diophantine Approximations and Value Distribution Theory, Lecture Notes in Mathematics 1239 (1987). Yu. G. Zarhin and A. N. Parshin, Finiteness theorems in diophantine geometry, an appendix to Russian edition of [12], Moscow, 1986, 369-438. (English translation by the AMS to appear).

Steklov Mathematical Institute Academy of Sciences of USSR Ul. Vavilova 42 117966 Moscow GSP-1, USSR

p-groupes et reduction semi-stable des courbes MICHEL RAYNAUD**) dedie a A.

Grothendieck

Soit R un anneau de valuation discrete complet, de corps des fractions K, de corps residuel k de caracteristique p > 0. Considerons une i^-courbe propre et lisse X , a fibres geometriques connexes et un revetement fini etale galoisien de groupe G : YK —*• XK de la fibre generique XK de X. Lorsque l'ordre de G est premier a p, il resulte de la theorie du groupe fondamental de Grothendieck [9], qu'apres extension moderee eventuelle de R, YK s'etend en un revetement fini etale Y —+ X. Nous nous proposons d'examiner ici le cas ou G est un p-groupe. T h e o r e m e 1. Supposons (i) La jacobienne

J{YK)

que G soit un p-groupe.

de YK a potentiellement

bonne

Alors: reduction.

(ii) Apres extension finie eventuelle de R, la courbe YK admet une reduction semi-stable Y sur R et les composantes connexes du graphe de la fibre speciale Y^ de Y sont des arbres. De plus, les composantes irreductibles eclatees de Yk (i.e., celles qui se projettent sur des points de Xk), ont une cohomologie cristalline dont les pentes se situent strictement entre 0 e i l . Le theoreme 1 se deduit d'une version locale (theoreme 2) et repose sur des proprietes elementaires des actions de groupes nilpotents sur les schemas reguliers (proposition 1). Nous examinons ensuite quelques exemples de revetements cycliques d'ordre p. 1. A c t i o n d'un g r o u p e n i l p o t e n t Rappelons qu'un schema X est geometriquement unibranche (confer [8] Ojy 23.2.1 ou [13] p. 100) si, pour tout point x de X, Thenselise strict de Ox,x a un spectre irreductible. Si X est integre, il revient au meme de W Unite associee au CNRS N° 752.

180

MICHEL RAYNAUD

dire que X —• X est un homeomorphisme, ou X est le normalise de X. Si X est geometriquement unibranche, il en est de meme de tout schema X' etale sur X. Proposition 1. Soient S un schema regulier localement noetherien integre, M son corps des fractions, N une M-algebre finie Hale galoisienne de growpe de galois G nilpotent, T le normalise de S dans N, f : T —* S le morphisme canonique. Soil S le ferme de S, purement de codimension 1, au-dessus duquel f est ramifie. On suppose que S est geometriquement unibranche. Alors: (i) T = f~1(S)

est geometriquement

unibranche.

(ii) Soient T,- une composante irreductible de T, Si son image dans S. Alors le sous-groupe d'inertie I de G est constant en tous les points de T{. Soit T' le quotient de T par I et T{ Vimage de T{ dans T'. Alors T1 est etale sur S au voisinage de Tt-; T,- —* Ti est un homeomorphisme et Ti —» Si est un revetement etale galoisien de groupe de Galois D/I, oil D est le sous-groupe de decomposition de Ti. Comme S est geometriquement unibranche et localement noetherien, S est localement irreductible. Quite a restreindre S a un ouvert, on peut done supposer que S = Si est irreductible. Soit s le point generique de S et soit t au-dessus de s le point generique de T,-. Le groupe de decomposition D est done le stabilisateur de t. Notons / le sous-groupe d'inertie de G en t. Supposons d'abord D ^ G. Comme G est nilpotent, on peut trouver un sous-groupe invariant H de G, contenant D et distinct de G ([4] Section 6.3 Proposition 8). Notons T" = T/H le quotient de T par H. T" est un revetement galoisien de 5, de groupe G" — G/H. Notons t" Pimage de t dans T". T" est evidemment etale sur S au-dessus de S — S. D'autre part, vu le choix de # , T" est aussi etale sur S au-dessus de s ([13] p. 103), done est etale sur S en codimension < 1. D'apres le theoreme de purete de Zariski, ([9] Chapter X 3.1), T" est alors etale sur S. Comme H contient D, T" est decompose au-dessus de s. S etant geometriquement unibranche, T" est alors decompose au-dessus de S. Quitte alors a remplacer S par T / ; , qui est encore regulier, on est ramene au cas ou G = H. De proche en proche, on peut done supposer que G = D. Maintenant T' = T/I est un revetement galoisien de 5, de groupe D/I, et T = Ti est irreductible. Comme plus haut, on voit que T' est etale sur S. L'image reciproque T de S dans T1 est irreductible et est un revetement etale de S de groupe de Galois D/I, en particulier T est

P-GROUPES ET REDUCTION SEMI-STABLE DES COURBES

181

geometriquement unibranche. Finalement T —• T est un morphisme entier, radiciel au point generique, done est un homeomorphisme puisque T est geometriquement unibranche. Comme T" est etale sur S en tout point de T" le groupe d'inertie I en t est aussi le groupe d'inertie en tout point de T, d'ou la proposition. Desormais, on considere un anneau de valuation discrete R complet, de corps des fractions K, de corps residuel k\ on note 7r une uniformisante de R. Corollaire 1. Soit X un R-schema reguher et plat, YK —• XK un revetement fini etale galoisien, de groupe G nilpotent et soit Y le normalise de X dans YK . Supposons que Xk soit geometriquement unibranche, alors il en est de mime de Y^. Exemple. On prend pour X une i?-courbe lisse. Alors Xk est geometriquement unibranche et le corollaire 1 s'applique; il equivaut au fait que les seules singularites (Yfc)red sont des points de rebroussement. L'exemple precedent peut etre generalise en acceptant un diviseur relatif a croisements normaux, le long duquel le revetement est moderement ramifie. De fagon precise, on a le result at suivant : Corollaire 2. Soient X un R-schema lisse, D un diviseur relatif a croisements normaux de X, U Vouvert X — D, VK —* UK un revetement fini etale galoisien de groupe de Galois G nilpotent, moderement ramifie le long de DK- Soit Y le normalise de X dans Vk. Alors Yk est geometriquement unibranche. D e m o n s t r a t i o n . Plagons nous en un point x de Xk. Apres localisation stricte de I en x, on peut supposer Y local et on doit montrer que Yk est irreductible. Le diviseur D a pour equation t{- • tr = 0, ou les t{ font partie d'un systeme de coordonnees relatives de X/R en x. De plus, il existe des entiers n,- > 1, premiers a la caracteristique de K, tels que l'indice de ramification de VK le long de la composante de D d'equation ti — 0, soit rii. Notons X1 le schema fini sur X obtenu en extrayant pour tout i une racine n t -eme de t{. Alors X1 est l'henselise strict d'un schema lisse sur R et done X\ est integre. Notons Y1 le normalise de Y x x X1. Commes YK est moderement ramifie le long de Dk, il resulte du lemme d'Abhyankar ([9], Chapter XIII Proposition 5.2) que Y^ —> Xj^ est etale. On peut appliquer la proposition 1 et conclure que Y£ est geometriquement unibranche. Soit alors un composant local de Y1. II domine par un morphisme fini surjectif le schema local normal V, done le schema irreductible

182

MICHEL RAYNAUD

Yj.1 domine Yk et celui-ci est aussi irreductible. 2. D e m o n s t r a t i o n d u t h e o r e m e 1 On suppose desormais que R est de caracteristique residuelle p > 0. Soit ( X , x) un germe de courbe relative lisse sur R. Done X = Spec ( 0 ) , ou O est un anneau local regulier de dimension 2, localise d'une R courbe lisse. Soit D un diviseur a croisements normaux relatifs sur X (e'est a dire un sous-schema ferme de X, etale sur R) et U = X — D. Soit G un p-groupe fini et considerons VK —* UK un revetement etale galoisien de groupe G, etale en dehors de DK, et moderement ramifie le long de DK (lorsque K est egalement de caracteristique p e t G ^ {1}, D est necessairement vide). Soit Y le normalise de X dans VKOn suppose que Y admet un i?-modele semi-stable, e'est a dire qu'il existe un eclatement Y —> Y de Y tel que : (i) La fibre fermee Yj. est geometriquement reduite. (ii) Les seules singularites de Yjt sont des points doubles ordinaires. (iii) Toute droite projective de Yk rencontre les autres composantes en au moins trois points geometriques distincts. II resulte du theoreme de reduction semi-stable pour les courbes ([2], [6], [12]) que, quitte a faire une extension finie de R, Y admet un Rmodele semi-stable. De plus, celui-ci est essentiellement unique et s'obtient a partir d'une desingularisation minimale en contractant les chaines de droites projectives de self-intersection (—2). En particulier Taction de G sur Y s'etend en une action de G sur Y. Notons que toute composante irreductible de Yk est en correspondance birationnelle avec sa transformee stricte dans Y; en particulier Yk est reduit. D'autre part, il resulte du corollaire 2 a la proposition 1 que Yk est geometriquement unibranche. T h e o r e m e 2. Les composantes connexes de Yk out des graphes qui sont des arbres. De plus, les composantes propres de Yk ont une cohomologies cristalline dont les pentes appariiennent a ]0,1[. On peut supposer k algebriquement clos. Commengons par etablir la premiere assertion. Puisque Yk est geometriquement unibranche et est reduit, ses seules singularites sont des points de rebroussement (cusps). En particulier, Yk est somme disjointe de ses composantes irreductibles. Quitte a remplacer G par le groupe de decomposition D d'une composante irreductible de Yfc,

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et X par un localise du quotient Y/D, on est done ramene au cas ou Yk est irreductible. Notons j/,-, i G / , la famille des points fermes du schema semi-local Y. Soit Y° la composante irreductible de Y^, transformed stricte de Y^. Comme Yk a pour seules singularites des points doubles ordinaires et que Yk est geometriquement unibranche, Y° n'est autre que la normalised de Yk et Y° —• Yk est un homeomorphisme. On note yi Vunique point ferme de Y° au-dessus du point ferme yi de Yk. Prenons Y° comme composante origine dans le graphe de Yk et soit g : Y —• Y la projection. Les fibres fermees de g sont les g~1(yi) et g~l{yi) est liee a Y° par le seul point yt-. Pour voir que le graphe de Yk est un arbre T, il suffit de montrer que le graphe de g~l{yi) est un arbre I\- : On passe en effet des T,- a T en branchant les I\- sur la composante origine Y ° , via les ponts yt-.

Graphe dual

Soit / le sous-groupe d'inertie de G commun a tout les points de Yk (Proposition 1). Alors I est invariant dans G et Z = Y/I est etale sur X , done essentiellement lisse et Yk —+ Zk est un homeomorphisme. Le groupe / agit sur Y en respectant chacune des fibres g~l{yi). Fixons Tun des points yi et soit Z{ son image dans Z. Quitte a remplacer X par Z localise en Z{ et Y par Y localise en yi, on est ramene au cas ou G = / et Y possede un seul point ferme note y. Soit y son unique relevement dans Y 0 . Nous allons proceder par recurrence sur la longueur h du graphe de Yk. Si h = 0, Y = Y et T est bien un arbre. Supposons h > 0. Alors y est un point double de Yk et il existe une unique composante Y1 de Yk qui passe par y et est distincte de Y ° . Elle est necessairement stable par G.

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Considerons le quotient X de Y par G. C'est un eclatement de X en x, dont la fibre speciale est reduite avec pour seules singularites des points doubles ordinaires (confer appendice). On note / : X —•» X le morphisme canonique. Comme X est regulier, la fibre speciale de X au-dessus de x est un arbre de droites projectives. Le transforme strict de Xk par / est une composante X°. II rencontre f~x(x) en un point double x. Notons X 1 la droite projective de Xk qui passe par x. Le morphisme fini Y —• X envoie respectivement Y°,Yl,y, sur X°,Xl,x. Introduisons le schema Y' deduit de Y en contractant les composantes irreductibles de Yk autres que Y° et Y 1 ([1] ou [3] chapitre 6.6). Le morphisme g : Y —• Y se factorise en les morphismes propres : Y —• Y1 —• Y. De plus, g' est un isomorphisme au voisinage de y. Notons y' = g'(y). Y£ a deux composantes irreductibles, images respectives par g' de Y° et Y 1 et notees Y / 0 et Y ' 1 . Ces deux composantes se coupent au point double y1. Soit de meme X' le schema deduit de X en contractant les composantes autres que X° et X 1 . On obtient une factorisation de / :

X

• X'

!

-

> X.

Alors X'k a deux composantes Xf0 et Xn qui se coupent transversalement en x'. Comme la fibre X'k est reduite et que X est lisse sur R, X'1 est necessairement une droite projective et done X1 est lisse sur R aux points de Xn autres que x'. Finalement on obtient un diagramme commutatif dont les fleches vert i c a l s sont des morphismes finis de passage au quotient sous Taction de G :

L

>

-

>

Y

X

f

> Y

Y'

X'

f

-

•

X

Considerons le morphisme fini Y' —» X'. II induit un homeomorphisme Y / 0 —• X / 0 ; en particulier y' est Tunique point au-dessus de x'. Soit D' le transforme strict dans X' du diviseur relatif D de Ar. Aux points de X'1 autres que x'', on a vu que X' etait lisse sur R et en ces points D' est un diviseur relatif a croisements normaux. II resulte alors du corollaire 2 a la proposition 1 que, en dehors de y1', Yn presente pour seules singularites des points de rebroussement.

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Bien sur g' : Y —» Y' est une reduction semi-stable de Y'. Si alors on choisit un point ferme xfl de A'' 1 , autre que x' et si on localise le morphisme Y' —*• X' au-dessus de xn, on obtient une situation analogue a la situation de depart, avec une reduction semi-stable de la singularity eventuelle au dessus de x'1 dont le graphe est clairement de longueur < ft. Par hypothese de recurrence, les composantes connexes de ce graphe sont des arbres. Chacun de ces arbres se branche en un point de Y 1 , d'ou le fait que le graphe de Yk est un arbre, ce qui prouve la premiere assertion du theoreme 2. Avant d'etablir la derniere assertion, revenons sur la demonstration precedente et sur le morphisme Y' —» X'. II y a deux cas possibles : (a) Le groupe G opere trivialement sur Y 1 , done aussi sur Y ' 1 . Y 1 est alors radiciel sur A 1 , done est une droite projective. Vu le caractere minimal du modele semi-stable Y,Yl rencontre au moins trois autres composantes de 1*, done au moins deux composantes autres que Y ° . C'est dire que Y ' 1 presente au moins deux points de rebroussement. (b) Le groupe G n'opere pas trivialement sur Y 1 . Soit alors H le sousgroupe d'inertie au point generique de Y 1 . H est invariant dans G et distinct de G. Considerons le morphisme Y ' —• X' que Ton factorise a travers Z' — Y'/H. Z' est alors etale au-dessus du point generique de Xn. D'apres le theoreme de purete de Zariski, Z' —> X' est done etale au-dessus des points de Xn — { # ' } , c'est a dire au-dessus des points d'une droite affine sur k. Par suite, l'inertie en chacun des points de Y ' 1 , autres que y' est egale a H. De plus, — ou bien Y 1 est une droite projective, auquel cas Y ' 1 presente au moins deux points de rebroussement et G/H est un p-sous-groupe du groupe additif de k done est ismorphe a (I/pZ)d. — ou bien Y 1 est de genre > 0.

186

MICHEL RAYNAUD D'ou le complement suivant au theoreme 2 :

Proposition 2. Sous les conditions du theoreme 2, supposons de plus k algebriquement clos et Yk connexe (done irreductible d'apres le corollaire 1 a la proposition I). Alors : (i) le graphe T de Yk est un arbre d'origine Y° le transforme strict de Yk. (ii) Soit Z une composante irreductible propre de Yk, e'est a dire distincte de Y°. Notons y le point de Z qui lie Z aux autres composantes en direction de Y°, z^, j G J les points de Z qui lient Z a d'autres composantes Z3 quand on s'eloigne de Y°. Soient D et I les groupes de decomposition et d'inertie au point generique de Z. Alors : Le quotient de Z par H = D/I est une droite projective P sur k. Si H ^ {1}, le revetement galoisien Z —• P est ramifie au seul point y; il est mime totalement ramifie en ce point. Le groupe de decomposition D3 d'une composante ZJ est egal a I; le groupe H opere librement sur les points z-7 et sur les composantes ZK Les groupes de decomposition et groupes d'inertie des diverses composantes de Yk vont en decroissant quand on se deplace sur T en s'eloignant de Y°. (iii) Les bouts de T, autres que Y°, correspondent a des courbes de genre >0. La derniere assertion traduit le caractere minimal de la reduction semistable Y: un bout de T qui est une droite projective peut etre contracte sans introduire de singularity. L'assertion (ii) a pratiquement ete demontree. En effet, par la recurrence deja utilisee, il suffit de traiter le cas ou Z est la composante contigue a Y° et notee Y 1 . Avec les notations ci-dessus, l'assertion resulte alors du fait que Y' jL —• X' est un revetement etale au-dessus de la droite affine Xn — {x1} et est totalement ramifie au-dessus de x'. Le stabilisateur de tout point de Y' 1 , autre que y' est done egal a 7, d'ou le fait que les groupes de decomposition JDJ soient egaux a I. Exemples : (1) G — Z/pZ. Les sommets de T autres que les bouts correspondent a des droites projectives sur lesquelles G opere trivialement. Les bouts de T distincts de Y° sont des courbes de genre > 0, revetements d'Artin-Schreier de la droite projective, ramifies en un seul point.

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o : Droite projective. * : Courbe de genre > 0. (2) G = Z/9Z.

Z/9Z est le stabilisateur de Y°. Z/3Z est le stabilisateur de Y 1 . Le quotient Z/3Z de Z/9Z agit transitivement sur {^1,^2, A3}. Prouvons maintenant l'assertion du theoreme 2 qui concerne les pentes. Si C est une courbe propre et lisse sur k algebriquement clos de caracteristique p > 0, on note pc = dimQp

Hl{CMP),

le p-rang de C. Soit alors Z une composante propre de Y&. Dire que les pentes de la cohomologie cristalline de Z sont dans ]0,1[ equivaut k pz = 0. D'apres la proposition 2 (ii), Z est un revetement galoisien, de groupe H un p-groupe, de la droite projective P , ramifie en un seul point 00. D'apres la formule de Crew ([5] corollaire 1.8), on a : 1-

Pz

= (Card H)(l - pP) - (eoo - 1)

ou eoo est l'indice de ramification en 00, done est egal ici a Card(i7). Comme pp = 0, on trouve bien que pz — 0. Ceci acheve la demonstration du theoreme 2. QED Demontrons maintenant le theoreme 1, ou plutot la variante suivante ou Ton accepte une ramification moderee horizontale.

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T h e o r e m e V'. Soient X une R-courbe propre et lisse, D un diviseur relaiif etale sur R, U — X — D, YK —• XK un revetement fini galoisien de groupe G nilpotent, etale au-dessus de UK ^ moderement ramifie le long de DK- Alors : Apres extension finie de R, le normalise Y de X dans YK admet un eclatement Y qui est une reduction semi-stable de YK- Les composantes connexes du graphe de Yk sont des arbres. Les composantes contractees en des points dans le morphisme Y —• Y ont une cohomolgie cristalline dont les pentes sont dans ]0,1[. La jacobienne de YK a potentiellement bonne reduction sur R. La premiere assertion resulte aisement du theeoreme de reduction semi-stable pour YK] notons toutefois que lorsque Yk contient des composantes rationnelles, les transformers strictes de ces composantes dans Y sont des droites rationnelles qui ne rencontrent pas necessairement les autres composantes en au moins trois points. Pour etablir les autres assertions, on se ramene au cas oii YK est geometriquement connexe; il en est alors de meme de Yk qui est done geometriquement irreductible d'apres le corollaire 1 a la proposition 1. Soit p la caracteristique residuelle de R. Le groupe nilpotent G est produit de ses / sous-groupes de Sylow pour tous les nombres premiers / ([4] Section 7 theoreme 4). Si G est d'ordre premier a p , il resulte de la theorie du groupe fondamental modere([9] Chapitre XIII)) que Y est lisse sur R. On peut done se limiter au cas ou G est un p-groupe. Soit alors x un point de X au-dessus duquel Y n'est pas lisse sur R et appliquons le theoreme 2 apres localisation en x. On conclut que le graphe de Yk est un arbre d'origine la transformed stricte de Yk dans Y, avec un branchement pour chaque point y ou Y n'est pas lisse sur R. Enfin il est classique (confer. [3] Chapitre 9) que la jacobienne de YK a potentiellement bonne reduction si et seulement si la reduction semistable de YK a un graphe dont les composantes connexes sont simplement connexes. Donnons maintenant un enonce analogue au theoreme 1', mais dans lequel on ne considere plus la reduction semi-stable de YKCorollaire 1/'. Sous les hypotheses du theoreme V', considerons la normalisee Y de X dans YK et une desingularisation Y de Y. Alors Pic(Yk/k) ne contient pas de sous-tore non nul ou, de fagon equivalente, les composantes irreductibles reduites de Yk sont homeomorphes a leur normalisee et les composantes connexes du graphe de Yk sont des arbres. Soit JK la jacobienne de YK et J son modele de Neron sur R. D'apres

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le theoreme 1', Jk a potentiellement bonne reduction. On en deduit facilement que Jk ne contient pas de sous-tore non nul. Lorsque le pgcd des multiplicites geometriques des composantes irreductibles de Yj. est egal a 1, Jk est la composante neutre de Pic Yk/k ([14] Theoreme 8.2.1). D'ou le resultat dans ce cas. Dans le cas general, on a une suite exacte fppf de faisceaux : 0 -+ E -+ Pic°(Ylb/Jfc) -* J° -+ 0, ou E est l'adherence schematique de la section unite, dans Pic°(Y/ J R) ([14] Section 4). De plus, la fibre fermee E xR k de E est un fc-groupe algebrique dont la composante neutre ne contient pas de sous-tore non nul (loc. cit. 6.3.8), d'ou la conclusion. Enfin il est classique (confer [3] Chapitre 9) que Pic (Yk/k) ne contient pas de sous-tore non nul si et seulement si les composantes irreductibles de Yk sont geometriquement unibranches et si le graphe de Yj. est sans circuit. Remarque. Si dans les theoremes 1 ou V on ne suppose plus X lisse mais simplement que XK est lisse et que Xk est semi-stable, les conclusions ne sont plus valables, meme lorsque le graphe de Xk est un arbre. Par exemple, prenons pour X une courbe de genre 2, a fibre generique lisse degenerant en deux courbes elliptiques ordinaires ayant un point commun. Apres extension eventuelle de R} et pour tout / premier, on peut trouver un revetement etale Y —• X, galoisien de groupe Z//Z qui induit sur chacune des courbes elliptiques un revetement connexe. Alors le graphe de Yk n'est plus un arbre.

graphe de Yk

pour £ = 3.

graphe de Xk3. Exemples On suppose desormais que la caracteristique residuelle de R est p > 0. Partons d'une iZ-courbe propre et lisse X, a fibres geometriquement connexes de genre g et soit J la jacobienne de X.

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190

Pour tout R schema en groupes commutatif fini et plat G, de dual de Cartier G', on a une bijection canonique : 771(X,G)-^Hom(G,,J), compatible avec les changements de base sur R et ou la cohomologie est calculee pour la topologie fppf ([14] 6.2.3). Prenons pour G' un sous-schema en groupes de J, annule par une puissance de p a fibre generique de type multiplicatif. A rimmersion G' —• J correspond done un G-torseur Y —• Xy avec GK etale. Apres extension eventuelle de K qui deploie GK , on obtient ainsi tous le revetements etales YK —• XK, geometriquement connexes, galoisiens de groupe un p-groupe commutatif. On suppose desormais que G est d'ordre p. Comme G'k est un sousgroupe non trivial de Jk,Yk —» Xk est un torseur non trivial sous Gk] en particulier Yk est irreductible, geometriquement reduit, ayant pour seules singularites des points de rebroussement; Y est normal. II y a essentiellement trois cas suivant le type de G : — G' est de type multiplicatif. Alors G est etale, done Y —• X est etale. — G' est etale. Comme G'K est suppose de type multiplicatif, ce cas ne peut se produire que lorque K est de caracteristique 0. Apres extension etale de R on se ramene au cas ou G' = Z/pZ. Le revetement Y —*• X est alors un torseur de groupe /i p decrit par un faisceau inversible L sur X d'ordre exactement p sur XK et sur Xk. Si h : Y —• X est le morphisme structural, /i*(Oy) = ®,-L*, i = 0 , 1 , . . . ,p— 1 et la multiplication est donnee par la trivialisation de Lp. — G'fc = Gjk = a p . Alors G degenere d'un groupe etale sur K en un groupe ap sur k. Lorque Gk est radiciel, le genre topologique de Yk est egal a celui g de X/R, tandis que le genre arithmetique est celui de la fibre generique YK : p(g-l)

+ l-g

=

(p-l)(g-l).

Mis a part le cas g = 1, la courbe Y^ presente done effectivement des points de rebroussement. Supposons k algebriquement clos et etudions localement les singularites de Yk lorsque Gk = p>P ou ap. Soient x un point ferme de Xk, t un parametre local de Xk centre en #, y le point de Yk au-dessus de x. Au voisinage de x le torseur Yk —•> X& admet une equation de la forme i>p = a, ou a est une section de Oxtx- Le torseur n'est pas generiquement trivial, done a n'est pas une puissance p-eme. Localement pour la topologie etale,

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l'equation du torseur se ramene k vp — tn, pour un certain entier n avec (n,p) = 1. Yk est singulier en y si et seulement si n > 1. Calculons la contribution au genre arithmetique de Y^ de la singularity local en y, c'est a dire la longueur de OjO ou O est l'anneau local de Y en y et O l'anneau local de la normalised. O admet comme parametre centre une racine p-eme r de t. Modulo une unite de (9, on peut ecrire : v = rn , dv =

rn-ldr.

Le module dualisant Q de la courbe d'equation vp — / n , admet pour base dv/tn~l, celui de la normalisee Q est engendre par dr. On a c ( r = (t/r)n~1 dv/tn~l. Done fi/fi est de longeur (n — l)(p — 1). II resulte alors de ([16] Chapitre IV Section 3 N° 11) que la contribution de la singularite au genre arithmetique est (n — l)(p — l)/2. Considerons maintenant un revetement d'Artin Schreier non constant Z de la droite affine de parametre t. II admet une equation de la forme Up — U = g(i)> ou g{t) est un polynome de degre m > 0, que Ton peut supposer sans monome de degre multiple de p. Etudions le genre de la courbe normale completee Z de Z. Apres localisation etale a Pinflni et changement de t en l/t, on se ramene a l'equation :

Up

-U=l/tm.

Soit oo 1'unique point a l'infini de Z et soit r un parametre de Z centre en oo. La fonction U admet en oo un pole d'ordre m. D'ou, a des unites pres, les relations : dr/rm+1 = du = dt/tm+1. dt =

(t/r)m+1dr.

La valuation de la differente au point oo est done (p — l)(m + 1 ) . Si alors h est le genre de Z, on a : 2 f t - 2 = p(-2) + ( p - l ) ( m + l ) , d'ou h = ( r a - l ) ( p - l)/2. Revenons a notre i?-torseur Y sous G. Proposition 3. Supposons que Yk ait au point y une singularite minimale done, localement pour la topologie etale, admette une equation de la forme vp = t2 si p > 2 et v2 = t3 pour p = 2. Alors, la contribution de la singularite en y, a la reduction semi-stable Y de YK est une seule

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composante qui est une courbe de genre (p— l)/2 pour p > 2 et une courbe de genre 1 pour p = 2. En effet, on sait (proposition 2) que les bouts de l'arbre relatif a cette singularity sont des revetements d'Artin-Schreier de la droite projective, ramifies en un seul point et de genre > 0. La somme de ces genres est egale a la contribution de la singularity au genre arithmetique a savoir (p — l)/2 si p > 2 et 1 si p = 2. La formule precedente donnant le genre d'un revetement d'Artin-Schreier ne laisse alors qu'une possiblite : il y a un seul bout, dont le genre est (p — l)/2 pour p > 2 et 1 pour p = 2. Lorsque la singularity de Yj., n'est plus minimale, il est possible que la reduction semi-stable de YK depende non seulement de la singularity de Yfc, mais aussi de l'equation du torseur Y —• X sur R. Exemple : torseur sous fip sur une courbe generique Partons d'une courbe X propre et lisse de genre g definie sur un corps k de caracteristique p > 0 et d'un torseur non trivial Y sous /i p , associe a un faisceau inversible L sur X, d'ordre p. Si L est defini par le 1-cocycle gij, avec gp- — fif~l, le torseur Y a pour equation locale vp = ft. Au faisceau L on associe la forme differentielle globale v = dfijft qui est une forme fixe sous l'operation de Cartier. Lorsque k est parfait, 1'application L »—• UJ etablit une bijection entre faisceaux inversibles d'ordre divisant p et formes fixes sous l'operation de Cartier ([17] Proposition 10). Lorsque k n'est pas parfait, a une forme sur X fixe par l'operation de Cartier correspond un faisceau inversible L d'ordre p qui est defini seulement sur la courbe X1, image reciproque de X par le Frobenius du corps k. Les singularites de Y correspondent aux zeros de la forme u. Plus precisement, supposons k algebriquement clos. En un point ferme # de X, choisissons une coordonnee centree /. Quitte a modifier le 1-cocycle qui definit L, on peut choisir /; en x, de la forme /,- = 1 + /", mod i n + 1 , avec (n,p) = 1. Y est singulier au-dessus de x si et seulement si n > 1 et w presente alors en x un zero d'ordre n — 1. Exemples. Si p — 2, en un point singulier de Y, on a n > 3. Si n = 3, uj presente en x un zero double; la courbe Y admet, localement pour la topologie etale, une equation de la forme v2 = t3; sa normalised a pour equation w2 = t avec w = v/t. La contribution de la singularity au genre arithmetique de Y est 1. Si p > 2, en un point singulier on a n > 3. Si n = 2, u> presente un zero simple, Y a pour equation localement pour la topologie etale, vp = t2.

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Posons p = 2s — 1, w = v$ /t. Alors la normalisee de Y a pour equation wp = t. Dans l'anneau local O de la normalisee de Y,w a pour valuation 1, v pour valuation 2, ( 5 / 0 est de longueur s = (p — l ) / 2 , engendre par les images de w, wv,..., wvs~l. La contribution de la singularity au genre arithmetique est (p — l ) / 2 , ce qui est en accord avec la formule generale donnee plus h a u t . P r o p o s i t i o n 4. Soit X une courbe algebrique generique de genre g > 1 sur un corps k algebriquement clos de caracteristique p > 0. Alors : X est ordinaire et possede pg — 1 formes differentielles u, non nulles, fixes par l'operation de Cartier. Pour p = 2, chacune de ces formes possede g — 1 zeros doubles. Le torseur sous pLp associe a u possede g — 1 points de rebroussement; chacun d}eux contribue au genre arithmetique pour 1. Pour p > 2, chacune des formes u> a 2g — 2 zeros simples; le torseur sous fip associe possede 2g — 2 points de rebroussement; chacun d'eux contribue au genre arithmetique pour (p — l ) / 2 . Montrons d'abord qu'il existe une forme fixe par Cartier, non nulle, dont les zeros ont les proprietes mentionnees ci-dessus. Pour cela, considerons une courbe elliptique ordinaire E sur k et une forme u sur E, non nulle, fixe par Cartier. Si p > 3, on construit une courbe X , de genre g comme revetement double de E en extrayant une racine carree d'un faisceau inversible sur E, de degre g — \ assez general. L'image reciproque de u sur X a les proprietes requises. Si p = 2, on procede de meme a partir d'un revetement cyclique triple X de E obtenu en extrayant une racine troisieme d'un faisceau inversible de degre g — 1 assez general. Soit alors rj le point generique en caracteristique p, de la variete modulaire Mg des courbes de genre g et soit X^ une courbe generique de genre g au-dessus de 77 {X^ est unique des que g est > 3). Les considerations qui precedent et un argument de specialisation montrent que, sur une cloture algebrique de k(rj), il existe une forme invariante par l'operation de Cartier sur Xrj, dont les zeros ont les multiplicity souhaitees. Par ailleurs, on sait que Xv est ordinaire [11]. Soit Jv la jacobienne de Xq. Le noyau pj^ de la multiplication par p dans J^ est extension d'une partie etale pJet de rang g par une partie de type multiplicatif. D'apres Ekedahl [7], la representation de monodromie a pour image Glg(Fp). En particulier, elle agit transitivement sur les points d'ordre p, a valeurs dans une cloture algebrique de k(rj). L'action de Galois est done aussi transitive sur les formes differentielles non nulles invariantes par l'operation de Cartier et toutes ces formes ont des zeros de la multiplicite souhaitee.

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MICHEL RAYNAUD

Corollaire 4. Soit R un anneau de valuation discrete complet d'megale caracteristique, de caracteristique residuelle p > 0. Soit X une courbe propre et lisse sur R, a fibres geometriques connexes de genre g > 1. On suppose que Xk est generique. Soit Y —• X un torseur sous \iv, non trivial en reduction sur Xk- Alors le graphe de la fibre speciale de la reduction semi-stable de YK est un arbre qui admet : 2g — 2 bouts, chacun d'eux etant une courbe de genre (p — l ) / 2 , pour P>2, g — 1 bouts, chacun d'eux etant une courbe de genre 1 si p — 2. Appendice : quotient d'une courbe semi-stable. Faute de reference nous allons etablir le resultat suivant utilise dans la demonstration du theoreme V : Proposition 5. Soient R un anneau de valuation discrete complet, X une R-courbe de type fini, normale, mtegre de corps des fractions L, M une extension finie de K, Y la normalisee de X dans M. On suppose que Yk est geometriquement reduite et a pour seules singularites des points doubles ordinaires. Alors il en est de mime de XkCorollaire. Soit Y une R-courbe plate, normale, separee, sur laquelle opere un groupe fini G et soit X = Y/G. On suppose que Yk est geometriquement reduite avec pour seules singularites des points doubles ordinaires. Alors il en est de meme de Xk(Noter que R etant complet, Y est quasi-projective et le schema quotient Y/G existe). Demonstration de la proposition. Soit 77 un point generique d'une composante de Xk et r un point de Yk au-dessus de 77. L'anneau local Oyj est un anneau de valuation discrete et il en est done de meme de Ox,n- P a r ailleurs, Yk etant reduit, une uniformisante w de R est aussi une uniformisante de OyiT, done aussi de 0 j i f | . II en resulte que Xk est reduit en rj. Par ailleurs X est normal done S2; par suite Xk est 5i, done reduit. Ces proprietes sont conservees apres extension plate de l'anneau de valuation discrete R. On en deduit que Xk est geometriquement reduit. Pour etablir la proposition, on peut done gonfler R et supposer k algebriquement clos. Soit x un point ferme de Xk et y un point ferme de Yk au-dessus de x. Pour etablir la proposition, on peut faire une localisation etale de X en x et se ramener au cas ou y est 1'unique point de Y au-dessus de x.

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Supposons tout d'abord que Y soit lisse en y et montrons que X est lisse sur Ren x. Soit t un parametre de Y* centre en y et soit t un relevement de T dans Oy,y Considerons s = Norme y/x(i)Notons que Xk est integre et soit Xk son normalise et x le point de Xk au-dessous de y. La reduction S de s mod w est une section de Ox,x qui induit une uniformisante de l'anneau de valuation discrete 0% x- On en deduit immediatement que l'anneau integre O^ x est un anneau de valuation discrete, puis que X est lisse sur R en x. Supposons que Yk ait un point double en y. Apres henselisation, on peut supposer que OyiV est l'henselise de R[t,t']/tt' — ire. Distinguons deux cas : Premier cas. Xk possede deux branches en x. Le morphisme de schemas integres : Spec(Oy,y)-,Spec(C^>x) est fini et plat en dehors de x, d'un certain rang n. Notons s et s' les normes respectives de t et tf. On a ssf = 7r ne . D'ou un morphisme : R[s, sf]/ss' — 7rne —• Ox,x, dont on verifie qu'il est formellement etale (c'est a dire induit un isomorphisme sur les completes). En particulier, x est un point double ordinaire de XkDeuxieme cas. X possede une seule branche en x. On doit montrer que X est alors lisse en x. Notons que Y reste normale apres extension finie, normale et generiquement etale de l'anneau de valuation discrete R\ par suite la formation de X commute a un telle extension. Quitte a faire une telle extension de R, on peut supposer que Pepaisseur e de la singularity de Y en y est paire. Soit Y —+ Y la desingularisation minimale de Y en y. La fibre speciale de Y est encore geometriquement reduite, a croisements normaux et consiste en une chaine de e—1 droites projectives joignant les transformees strictes des deux composantes de Yk en y.

Considerons Y Xx Y et ses deux projections sur Y. Soit ZK la fibre generique de Y XxY et Z son adherence schematique dans Y XxY. Alors,

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les projections de Z sur Y sont finies et plates en codimension < 1. Comme Y est regulier, le sature S2 de Z, soit Z est alors fini et plat sur Y . De plus Z —• Y definit une relation d'equivalence en codimension < 1, done Z —• Y definit un groupoi'de fini et plat sur Y. Soit X le quotient de Y par ce groupoi'de ([10] Expose V). Par hypothese le groupoi'de fini Y —• X echange les deux composantes de Yk, done le groupoi'de Y —+ Z echange les deux bouts du graphe de Yk. Par suite, ce groupoi'de echange les composantes symetriques des extremites du graphe de Yk et en particulier fixe la composante mediane. D'apres les cas deja traites, on voit alors que Xk est reduit avec pour seules singularites des points doubles ordinaires. De plus Xk est une chaine de e/2 droites projectives liee a un bout lisse : le transforme strict de XkOn obtient X par contraction de droites projectives et done X est lisse sur R. Remarque. Sous les hypotheses de la proposition 5, supposons de plus que X soit propre. Alors on peut voir que Xk a pour seules singularites des points doubles ordinaires par voie globale, de la fa^on suivante : Soit Jy/R la composante neutre de Picy/R. C'est un schema en groupes dont la fibre special est de rang unipotent nul (confer. [3] chapitre 9). II en est done de meme de la fibre generique et en fait YK etant normale est lisse. Soit de meme la composante neutre JX/R d e ^1CX/R Qui e s t un schema en groupes lisse {loc. cit.). L'application de norme relative au morhpisme fini Y —> X , fournit un morphisme u : Jy/R —*• Jx/R Q m e s ^ surjectif sur la fibre generique. Alors u se factorise a travers v : G —• Jx/R, a v e c G semi-abelien et VK un isomorphisme. L'etude des points de / n -torsion, avec / premier inversible, montre alors que Jx/R e s t semi-abelien. La fibre speciale Xk etant geometriquement reduite, on en deduit que localement pour la topologie etale, les singularites de Xk sont "comme des axes de coordonnees" (loc. cit.). Mais Xk est domine par Y&, done les singularites de Xk sont au plus des points doubles ordinaires.

BIBLIOGRAPHIE [1] M. Artin, Some numerical criteria for contractiblity of curves on algebraic surfaces, Amer. J. of Math 8 4 (1962), 485-496. [2] S. Bosch et W . Liitkebohmert, Stable Reduction and Uniformization of Abelian Varieties 1; Math. Annalen 2 7 0 (1985), 349-379.

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[3] S. Bosch, W. Liitkebohmert et M. Raynaud, Neron Models (a paraitre dans Ergebnisse). [4] N. Bourbaki, Algebre chapitre 1, Hermann. [5] R. Crew, Etale p-covers in characteristic p, Compositio Math. 52 (1984), 31-45. [6] P. Deligne et D. Mumford, The irreducibility of the space of curves of given genus, Pub. IHES 36 (1969), 75-109. [7] T. Ekedahl, The Action of Monodromy on torsion points of a Jacobian (a paraitre). [8] A. Grothendieck, Elements de Geometrie algebrique, Chapitre IV, seconde partie; Pub. IHES 24 (1965). [9] A. Grothendieck, SGA 1 : Revetements Etales et Groupe Fondamental; Lecture Notes 224(1971), Springer Verlag . 10] A. Grothendieck, SGA 3 : Schemas en Groupes I, Lecture Notes 151 (1970). 11] I. Koblitz, P-adic variation of the Zeta-function over families of varieties defined over finite fields. Compositio Math. 31 (1975), 120-216. 12] M. Var der Put; Stable Reduction of Algebraic Curves; Report Rijksuniversiteit Gronigen, (1981). 13] M. Rayanud, Anneaux henseliens; Lecture Notes 169, Springer-Verlag, (1970). 14] M. Rayanud, Specialisation du foncteur de Picard., Pub. IHES 38, (1970), 27-76. 15] I. Safarevic, On minimal models and birational transformations; Pub. Tata Institute of Fund. Research, Bombay (1966). 16] J.-P. Serre, Groupes algebriques et corps de classes; Hermann (1959). 17] J.-P. Serre, Sur la topologie des vanetes algebriques en caracteristique p\ Symp. Int. Top. Alg. Mexico (1958), 24-53.

Universite de Paris-Sud Centre d'Orsay Mathematique - Bat. 425 91405 Orsay Cedex, France

Drawing Curves Over Number Fields G.B. SHABAT and V.A. VOEVODSKY "Lucky we know the Forest so well, or we might get lost"— said Rabbit, and he gave the careless laugh you give when you know the Forest so well that you can't get lost. A.A. Milne, The world of Winnie-the-Pooh.

Introduction 0.0. This paper develops some of the ideas outlined by Alexander Grothendieck in his unpublished Esquisse d'un programme [0] in 1984. We draw our curves by means of what Grothendieck called "dessins d'enfant" on the topological Riemann surfaces. In the sequel we shall call them simply "dessins." By definition, a desssin D o n a compact oriented connected surface X is a pair D=

(K(D),[t]),

where K(D) is a connected 1-complex ; [t] is an isotopical class of inclusions i : K(D) <-» X. We denote by Ko(D) the set of vertices of K{D). It is supposed that (a) the complement of

L(K(D))

in X is a disjoint union of open cells ;

(b) the complement of KQ(D) in K{D) is a disjoint union of open segments. The main construction we work with is based on the theorem of Gennady Belyi [1]. To a pair D = (A', [A]) it assigns a smooth algebraic curve together with some non-constant rational function on it over some number

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field. Throughout the paper we denote this curve by XD and this function by (3D. We called them the Belyi pair associated to the dessin D. According to [0], the realization of the possibility of such an assignment was one of the most striking events in Grothendieck's mathematical life. The only one he could compare it with was the following: " . . . vers Page de douze ans, j'etais interne au camp de concentration de Riecros (pres de Mende). C'est la que j'ai appris, par une detenue, Maria, qui me donnait des lemons particulieres bene voles, la definition du cercle. Celle-ci m'avait impressionne par sa simplicity et son evidence, alors que la propriete de "rotondite parfaite" du cercle m'apparaissait auparavant comme une realite mysterieuse au dela des mots". The correspondence between the curves over number fields and dessins indeed seems to be very fundamental; in the end of this introduction we outline this construction in both directions. We are interested in the constructive aspects of this correspondence. In the spirit of D. Mumford's monograph [2] we consider 5 ways of defining complex algebraic curves : (1) Writing an equation. (2) Defining generators of the uniformizing fuchsian groups. (3) (4) (5) Our

Specifying a point in the moduli space. Introducing a metric. Defining jacobian. general approach is: given D, can we say anything about XD and /?#? Overview Of The Main Results

0.1. We use the following terminology. Let D be a dessin on a surface X. When K(D) has no loops and each edge of K{D) lies in the closure of exactly 2 components of X\K(D), we call (a) a valency of a vertex V G Ko(D) the number of edges from K(D)\ Ko(D), whose closures contain V. (b) a valency of a component W of X\K(D) the number of edges from K(D)\K0(D) that lie in the closure of W. For the general dessins, see 1.4 below. For a dessin D on a surface X we call its 0-valency the least common multiple of the valencies of all the vertices from Ko(D) and its 2-valency the least common multiple of the valencies of all the components of X\K(D). Sometimes we denote them Vo(D) and v2{D)) respectively. Call the dessin D trigonal if the valencies of all the components of X\K(D) are 3. The trigonal dessins with some regularity assumptions

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define the triangulations of X. We call the dessin D on X balanced if all the valencies of the vertices of KQ(D) are equal (to vo(D)) and all the valencies of all the components of X\K(D) are equal (to v2(D)). Similar objects were explored in the classical topology from the combinatorial point of view (see e.g., [3], [15]). We do not pretend any terminological compatibility with this line of research because the technique we use is completely different. (Threlfall, e.g., called our balanced dessins the "regelmassig Zellsystem"). 0 . 1 . 1 . Equations. Here we have no general theory and only give a number of examples. We also consider the opposite question: can we actually draw a given curve? T h e answer is yes for some famous ones: Fermat curves, Klein quartic, and some modular curves are among them. T h e completeness of our results decrease rapidly with growing genus; we are able to give some complete lists (of non-trivial experimental material) for genus 0, but for genera exceeding 3 we are able to give only some general remarks. 0.1.2. Uniformizaiion. Let p = vo(D), q — v^{D). Consider the subgroup TPA of PSL2(R) consisting of those transformations of the Poincare upper half-plane 7i = {z G C|Im(z) > 0} t h a t respect the tesselation obtained by the reflections of the regular p-gon with the angles 2w/q (see Coxeter [3]). We show t h a t there exists a subgroup T of finite index in TP}q such t h a t the quotient H/T is isomorphic to Xp. We describe this T explicitly by the combinatorics of D. For the balanced dessins D this construction describes the universal covering of Xp. 0 . 1 . 3 . Moduli Here we consider only the trigonal dessins D and use the results from R.C. Penner's preprint [4], where one finds several constructions equivalent to ours (without reference to the Grothendieck program). Penner introduced the extended Teichmuller spaces T^}„, consisting of marked Riemann surfaces together with the horocycles about each puncture. Using dessins, Penner coordinates T ^ n ' s , essentially by considering the metric on X\KQ(D) of constant curvature —1 and by assigning to it some functions of the lengths of parts of edges of K{D) lying between the horocycles. By this construction T9tn turns out to be homeomorphic to U ">o~ - We deduce from Penner's results, t h a t under this coordinatisation XD corresponds to ( \ / 2 , . . . , \ / 2 ) .

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SHABAT AND VOEVODSKY Penner builds the cell decomposition

dessins D on curves of genus g with n vertices

where the trigonal dessins D correspond to the open cells. For n = 1 it gives the cell decomposition of the space T9)\ itself. We deduce from Penner that in some sense Xp G C&.

0.1.4. Metric. Here we consider only triangulations of XD • Instead of riemannian metrics we work with "piecewise-euclidean" ones; they also allow us to define a complex structure on Ar#. We show that XD corresponds to the equilateral metrics.

0.1.5. Jacobians. Here we also work only with such dessins D that triangulate X. We define "approximate" jacobians J p ( A ) , about which we think, that their limit, when D becomes finer, is the usual jacobian J(X). 0.2. In the rest of the introduction we explain the essence of the assignment of Belyi pairs to dessins. In both directions we use : T h e o r e m [1] (Belyi). A complex non-singular complete complex curve can be defined over some number field if and only if there exists a meromorphic function on it with only three critical values.

0.2.1. From dessins to curves. We are supposed to be given a dessin D o n a surface X. Choose a point in each 2-cell, connect the points of the neighbouring 2-cells by an edge of a different type, and connect all these points with the vertices of the original graph inside the 2-cells when it is possible.

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Now we have three types of vertices and three types of edges. Since we are going to map the whole picture onto P 1 (C), we mark them : o over "0", • over "oo", * over " 1 " ; we find the edges over (0,1), over (l,oo), over (oo,0). Note that the whole surface has become divided into triangles, each with vertices of all three types. The orientability of the surface results in the possibility of painting it in two colours in such a way that if we move around the black and the white triangles counter-clockwise, the order of vertices is "0"-
If all the butterflies put their wings together to become SMike and are identified, we get the desired map 0D-XD ramified only over {0, l,co}.

^(C),

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After we restore the unique algebraic structure of XQ (by the Riemann existence theorem) in which /?# is rational, we use the easier part of Belyi ("if" in the above formulation) theorem to conclude that Xp and /?# are defined over Q. The Belyi pair ( X D , / ? D ) thus obtained depends only on the original dessin D. The Belyi function f3p obtained through this construction has all the ramifications over "1" or order 2. We shall call such Belyi functions clean. The image of K(D) in X can be reconstructed as the /?p-preimage of the segment [l,oo]. 0.2.2. From curves to dessins. In this direction we outline the proof of the more striking half of the Belyi theorem ("only if" in the above formulation), closely following [1]. Suppose we are given a curve X over some number field L ; take a non-constant element / of the function field L{X) and consider it as a ramified covering

/:*(€)

^(C);

its ramification points belong to P X (Q). We are going to transform / to the desired covering by a sequence of replacements of / by P o / , where P is a polynomial with rational coefficients, considered as a map P*(C) •P 1 (C). After such replacements, infinity will always go to infinity; denote for F : X(C) ^ ( C ) by WF the set of finite critical values of F. By the first step of the construction, we reduce the situation to the case Wp C Q. We proceed by induction on # ( W F \ Q ) . At each inductive step we take for P a generator of the annihilator in Q[T] of WF- It is clear that #Wp0F = if^Wp — 1. So we suppose WF C Q and proceed to the second step. Replacing / by Af -f B with suitable A, B we can assume {0,1, oo} C WF- Proceed by the induction in #(VFp\{0,1, oo}). Assuming that there exists r G W F \ { 0 , 1, oo} with 0 < r < 1 (which can again be achieved by a change of / to Af + B, denote r = -^h^ with natural n, ra. Now use P(z)={ v }

+

] m n

m n

zm(l-z)n v '

and check that P(0) = P ( l ) = 0, P(r) = 1 and that if Pl(z) = 0, then z G {0, l , r } ; so replacing / by Po f reduces #(WV\{0, l,oo}) by 1, which completes the inductive argument. We have obtained the Belyi function / ; to get a clean one put p = 4/(1 - / ) .

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The desired dessin D is defined by setting K{D) = /? 1([l,cx>]) with K0{D) = p-\Q). 0.3. The style of our exposition is far from the standards of modern mathematics; we hope that this flaw is partially compensated for by the explicitness of the results. Besides, we do not prove some of our assertions. This is not only the result of space and time constraints, but rather of the feeling that the proper language for the mathematics of the Grothendieck Program has not yet been found. The main reasons to publish our results in the present state is our eagerness to invite our colleagues into the world of the divine beauty and simplicity we have been living in since we have been guided by the Esquisse [0]. We emphasize that in the present text we use only a very small part of the ideas one can find in the epoch-making paper. We are indebted to Yu. I. Manin for the useful discussions and to A.A. Migdal and his colleagues for their interest. We are grateful to I. Gabitov, without whose assistance the preparation of the present text would have been impossible. Part 1. Generalities Let D b e a dessin on a surface X. We choose a representative of the isotopical class of the inclusions K(D) <—• X and in what follows consider K(D) as a subset of X; all our constructions are independent of this choice. 1.1.1 The flag set F(D) is, by one of the definitions, a set of triples (U, E, V) such that U is a component of X\K(D)y E a component of K(D)\K0(D), V a vertice from K0(D) and (i) E lies in the closure of U\ (ii) V lies in the closure of E. This definition is suitable only for the fine enough dessins; later, we will give a universal definition. 1.1.2.

Consider a "cartographica?' group C2 = (
It acts on the flags in the following way :

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SHABAT AND VOEVODSKY (the operator
Because of the connectedness of X, the flag space F(D) is C2-homogeneous. In analogy with the linear case, we call a (non-oriented) Borel subgroup of the flag F G F(D) the stationary group of this action: BD>F

= {ceC2\cF

= F}

1.1.3. The onentability of X (in the spirit of the Introduction) results in the possibility of defining the map o:F(D)

>{±1}

satisfying o(aq • F) = -o(F) for all F E V{D),q G {0,1,2}. We consider the set of positively oriented flags F+(D) = o - 1 ( l ) . The oriented cartographical group C^ is the one that respects all the sets F+(D) C F(D). It is the subgroup of index 2 of C2 generated by the words in cr0,<7i, a2 of even length. We take the generators PO

=0~2(Ti

Pi

=
p2

—O-ICTQ

satisfying P2P1P0 = 1

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207

Since all the Borel subgroups corresponding to one dessin are conjugate, in cases where we need only the conjugacy class of BD,F, we shall omit the reference to F. The dessin is called a Galois one if the group BD is normal in C% • We call an automorphism of the dessin a compatible triple consisting of a permutation KQ(D) >Ko(D) and isotopical classes of homeomorphisms K(D)—>K(D), X >X. It is clear, that in the case of a Galois dessin D with Borel subgroup B the factor C%/B acts on D, transitively on the vertices, on the edges and on the components of X\K(D). Therefore, the Galois dessins are balanced. The converse is wrong: the dessin

with identified opposite sides gives an example of a balanced non-Galois dessin on the torus. All the automorphisms of the dessin D o n a surface X are realizable as conformal automorphisms of X. 1.1.4. The positively oriented flags are in canonical one-to-one correspondence with the oriented edges of K{D). We use the following convention:

Under this convention the oriented cartographical group acts on the oriented edges in the following way:

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SHABAT AND VOEVODSKY

In this way the definition of the action of the oriented cartographical group on the oriented flags is naturally extended to all the dessins, with no regularity assumptions. In the degenerate cases this action may look, for instance, like this:

p2

The orders of F E F + ( D ) with respect to po,/>2 a r e related with the general valencies (see 0.1) as follows : #{(Po)}^ =i>o(0-component of F) #{(p2)}F =U2(2-component of F). In particular, if VQ(D) = p, V2{D) = q, then for any F G F + (£)) /4) C BD)F 1.1.5. For a dessin D o n a surface X the dual dessin D* is defined as follows. The set of vertices K$(D*) is in canonical one-to-one correspondence with the components of X\K{D), and the set of edges of D—with that of D. The pair of dual dessins looks like

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209

1.1.6. It is very important [0] t h a t the correspondence between the dessins and the Belyi pairs defines the action of the Galois group Gal ( Q / Q ) o n the dessins. In particular, the FIELD O F DEFINITION O F A DESSIN makes sense. P a r t 2. R e s u l t s 2 . 1 . Equations. We start with some of the most simple dessins and try to determine which Belyi pairs over what number fields they define. Grothendieck [0] doubts t h a t the problem can be solved by a uniform method. We can confirm from our experience t h a t if such a method exists, it is quite involved. We adopt the following drawing conventions: (a) for genus 0 we put the dessins in the euclidian plane; if we have a vertex at infinity, we put arrow marks on the edges going there; (b) for genus 1 we draw everything in the period parallelogram; (c) for genus 2 and higher we realize the surfaces as polygons with identified boundary edges and draw our dessins on these polygons; the pairs of identified edges are directed controversially (which means t h a t one edge goes clockwise while another one goes counterclockwise). 2 . 1 . 0 . Genus 0. Since the curves of genus 0 have no moduli, the only thing to calculate is the Belyi function, which in this case is just a rational m a p P X (C) >-P1(C) with three critical values. It is defined up to composition with PST2(C)-transformations from the left and from the right. T h e proper choice of the representatives seems to be connected with delicate arithmetical questions which we do not discuss here; in the examples below we try to choose them in the shortest form. As a result, the critical values of the Belyi functions vary.

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210

We consider first the simplest case : suppose that X\K(D) is connected. Then K{D) is just a tree inside an oriented 2-sphere 5 2 , and we shall draw it in the plane (where the orientation is essential!). Using our right to choose the PSZ/2(C)-representatives as we like, we normalize the Belyi maps in such a way that the "centre" of the only 2-cell lies at oo and goes to oo and the intersections of K{D) and K(D*) will go to 0. Then the Belyi map is represented by a polynomial, all the zeroes of which are double; therefore it is a square of a polynomial, which we denote hp or simply h. We are going to present a number of h for some simple dessins D. We present three infinite series of tree-like dessins for which h can be specified: 1. h(z) = zn

2.

h(z) is the n-th Chebyshev polynomial: cos(n/) = h(cos i)

h(z) = zm(l - z)n (up to multiplication by constant the Belyi function from the introduction)

To cover all the trees with the number of edges not exceeding 6, we should add eight individual dessins (three of them equivalent under the Gal (Q/Q)action).

1.

X

h(z) = (z + l) 3 (3z 2 - 9z + 8)

DRAWING CURVES OVER NUMBER FIELDS h(z) = z3(9z2 - 15z + 40)

<

h(z) = z4(36z2 + 36z + 25)

h(z) = z3(6z2 + 96z + 25)

h(z) = z3(z3 + 1)

211

SHABAT AND VOEVODSKY

212

Note that all these dessins, as well as the infinite series above, are defined over the rationals Q. The remaining three dessins are defined over three cubic fields, permutable by the Gal (Q/Q). They lie in the field of decomposition of the polynomial 2523 - I2t2 - 24/ - 16 = 25(* - a)(t - a+)(t - a_) (we agree that a G R, Im (a+) > 0, Im (a~) < 0). The dessin Da:

<

is defined over Q(a),

Da+:

over Q(a + ), Da_

• •

• i

• •

over Q(a_).

For each 6 G {a, a±] we have hDc(z)

= z3{z+l)2(z

+ b).

Now we turn our attention to the Galois dessins. They constitute two families: corresponding to the plane polygons and to the Platonic solids. As for the plane n-gons, in one of the normalizations they are described by the rational function zn + (l/z)n To discuss the platonic solids, introduce the notation XD =

XD\K0(D)

and denote by Yn the affine modular curve H/T(n), where T(n) is a principal congruence subgroup of PS'L2(Z) (the kernel of the natural homomorphism PSL2(l •P£L2(Z/nZ)). We claim that the platonic solids

DRAWING CURVES OVER NUMBER FIELDS

213

with the triangle 2-cells correspond to the modular curves in the following way: -for tetrahedral D

XD=Y3

-for octahedral D

XD=Y4

-for icosahedral D —

XD = Y5.

For all of them, the Belyi map corresponds to the canonical projection Yn = n/T(n)

^(C) ~

?{/PSL2(l).

The curve Y2 did not enter this list because the corresponding

D

is not a platonic solid. But this dessin is a remarkable one: the corresponding rational function in one of the normalizations is 27(z3 - z + 1) Az2(z-1) ' which is exactly the expression for the J-invariant of the elliptic curve y2 = x(x-

l)(z-2).

214

SHABAT AND VOEVODSKY

Left composition of this function with any canonically normalized Belyi function corresponds to the barycentric subdivision of the corresponding dessin. 2.1.1. Genus 1. On the curves of genus 1 there exist balanced dessins of valencies VQ = v^ — 4, v0 = 6, v^ — 3 and v0 — 3, v^ — 6. The corresponding curves are isogenic to the elliptic curve 9

3

y —x —x in the first case and to the curve y =x - 1 in the second and in the third cases. The Galois dessins represent exactly these curves. (See [3] for the proofs of the classification results.) In the period parallelogram these dessins look like

{vo,v 2 } = {4,4}

{vo,t;2} = {6,3}

The problem of determining the J-invariant of the curves drawn by a dessin on the torus seems to be rather hard. For instance, consider the dessin on the torus which in the fundamental parallelogram looks like

The corresponding curve is defined over Q(v/7). The (V7 conjugated dessin looks like

-^7)-

DRAWING CURVES OVER NUMBER FIELDS

215

The J-invariants are - ^ ( 8 T 3 V / 7) 2 (2 ± >/7)6(10 ± 3>/7)3. Thiis calculation answers the question of A. A. Migdal, 1986. Sometimes the curve can be determined by the additional symmetries of the dessin. For instance, for D= XD is determined by the equation x —x with Belyi function -27x4/(x2

- 4 ) 3 because of the symmetry of 4th order.

2.1.2. Genus 2. We are able to give some results only for Galois dessins. They are listed in [3]; there are 10 of them, but we choose only one from each pair of the dual ones. DESSIN 1.

2.

BELYI PAIR ; 2 ==xx 5 — x y~ .,4

y2=x5

--x6-l /3=x6

SHABAT AND VOEVODSKY

216

y2 = x 6 - 1

4.

P=(l +

x6)2/4x6

y2 = x5 — x /?=31 2

! 6.

fYl ~7

Ax2(l+x4) 4\4 [27(l + z 4 ) 2 - 8 : r 4 ] 3

y

—x 4

P = 4x /(l

—X

+ x4)2

2.1.3. Genus 3. Here we discuss only three curves. The Klein quartic is defined in the homogeneous coordinates by the equation XQx\ 4- ^1*2 + *2#0 = 0

Its full automorphism group has 168 elements; the quotient by this group is isomorphic to P 1 (C), and the natural projection defines a clean Belyi map. We do not attempt to draw the corresponding dessin and refer the reader to Klein's paper [5], where one finds a very beautiful triangular tesselation of the fundamental domain of the Klein quartic on the universal covering (this was the figure from which the uniformization started). After suitable identifications, we get some triangular dessin D o n a curve of genus 3. Thus the Klein quartic itself has been drawn; indeed, the automorphism group of D and consequently of the curve X has 168 elements, and Klein quartic is the only curve of genus 3 with this (highest possible) number of automorphisms [6]. Here is the dessin for the Fermat quartic x4 + y4 = 1 :

DRAWING CURVES OVER NUMBER FIELDS

217

with the Belyi function /? = (x4 + I)2/(Ax4). Here is the dessin for the Picard curve [7] y3 = x4 — 1

with the Belyi function of the same form /? = (x4 +

l)2/(4x4).

2.1.4. Higher Genera. We propose two infinite families of curves for which something can be done. One is the family of the generalized Fermat curves x

-f y — 1

It is easy to check that the functions xm and yn are the Belyi ones. Taking n = 2, we get a drawable curve of any given genus. The other series is the family of modular curves Xo(n) (the factors ?i/ro(n), where To(n) is the preimage of the upper triangular matrices under the canonical projection PSL2{1.) >PSL2(Z/n2)). The dessins on these curves are the projections of the P5i>2(Z)-orbits of the arc from exp (2iri/3) to exp (7r i/3) on the boundary of the modular figure. For the discussion of an explicit description of modular curves see [8], [9]. 2.2 Uniformization Let D be an arbitrary dessin on a surface X. Let p = VQ(D), q = V2(D). We canonically associate t o D a conjugacy class of discrete subgroups TD C

PSL2(R)

SHABAT AND VOEVODSKY

218 and then show that

XD ~

H/YD

2.2.0. Construction. The reference for the material below is [3]. Consider in the Poincare upper half-plane 7i the regular
the rotation of n p q about the centre of angle 27r/q. the rotation of H about one of the vertices of Upq of angle 27r/p.

It is geometrically obvious that these elements satisfy the relations /?< =T P = 1, (/?7)2 = 1 (the transformation (3j being a symmetry around the centre of the side of These relations allow us to define the homomorphism Kp,q •
>rp,g C >7

PSL2(R)

•/?

Fix some F € F(D). The desired group is ^D,F = ^iP)q(BD,F) Its conjugacy class is independent of all the choices involved. Denote it by

2.2.1.

Theorem.

n/vD - xD Sketch of the proof. If we conect the centre of 11^^ by geodesies with all the vertices and all the centres of the sides and then paint (as Klein did in [5]) these triangles in black and white this way

DRAWING CURVES OVER NUMBER FIELDS

219

we realize that for every subgroup T C Tp>q of finite index these triangles are in one-to-one correspondence with the flags of the dessin on H/T obtained by the projection of the above infinite dessin on H. The orientation of the flags corresponds to the colour of the triangles. It remains to realize that this dessin is isomorphic to the original one and that the natural projection

•«/rM~p1(C)

n/rD

is the same as the one defined by this dessin by our construction (see Introduction). 2.2.2. Theorem. If D is balanced, then the natural map

n

m/TD

is isomorphic to the universal covering of X. Indeed, for general D the ramification index of the map 7i >7i/Y equals p/vo(V) over V £ Ko(D) and q/v2(S) over the "centre" of the components ofX\K(D). Therefore, this map is unramified for the balanced D. In this way we can effectively describe the universal coverings of all the curves that were drawn by the balanced dessins in the previous section. For the Klein quartic it was done by Klein [5]. 2.2.3. Now we turn to the universal coverings of the curves XQ with trigonal D. Proposition. The group C* with the additional relation p% = 1 is isomorphic to PSX2(Z).

SHABAT AND VOEVODSKY

220

Using [3], define this isomorphism by the assignment P2\

-(-!o)

PI I Pol

•(Ji)

T h e o r e m . For trigonal dessins D the curve XQ is isomorphic to Ti/T, where V C PSL2(l) corresponds to the Borel subgroup BQ under the above isomorphism. 2.3 M o d u l i 2.3.0. The results of this part are based on Penner's paper [4]. Now we work with the above curve X p . Introduce the notation [D] = #(Ko(D)). The curves Xry after a suitable marking turn into the points of the Teichmuller space Tg>n (see, e.g., [16]). We always assume 2g — 2 -f [D] > 0 and interpret the points of Tg}U as metrics with constant curvature —1 on XQ. We start with a review of Penner's approach to the Teichmuller spaces Tg)n with n > 0. Penner introduces the augmented Teichmuller spaces Tg>n, whose points correspond to the points of the usual Teichmuller space T together with the horocycles about each puncture (a closed curve, orthogonal to all the geodesies, going to the cusp). Denote for any dessin D — (A", [t]) by Ki(D) the set of (nonoriented!) edges of AT, i.e., the set of the connected components of K(D)\Ko(D). Also denote by I<2{D) the set of connected components of X\K(D). In this section we consider only trigonal dessins D. In what follows, n = [D].

L e m m a . dim R f 5>n = #A'i(£>) = 6g - 6 + 3[D]. Proof. For the Euler characteristic we have #A'o(D) - #*!(£>) + #A' 2 (J9) = 2 - 2 $ , and the number of "1,2-flags" in the pair (X, K{D)) equals 2## 2 (2?) = 3#ffi(£>).

DRAWING CURVES OVER NUMBER FIELDS

221

From these two equalities we have #K,{D)

= #K0{D) + #A'2(£>) + 2g - 2 = #K0{D) + | # A ' i ( D ) +

2g-2,

and # / f i ( D ) = 3([D] + 20 - 2) = [£>] + 2(3n So it is natural to try to coordinatize Tg>n by the functions on the set of edges K(D). On the part of T^ n , on which the horocycles are so small that they do not intersect, there exists a natural function: think of every edge from K(D) as an (infinite) line from puncture to puncture in X(D), deform it to the geodesic and take the lengths of the part between the horocycles. It turns out that the analytic continuation of this construction gives the global coordinatization of T9in\ To establish it, Penner uses the Minkowski space M 3 with the coordinates x = (#o,#i) #2) and with the metric ds2 = — dx% -f dx\ -f dx\ induced from the scalar product ( , ) of signature (—h + ) . It follows from the local isomorphity of SL (R) and SO (1,2) and some easy Lie group considerations, that the Poincare upper half-plane is isometric to a connected component of the hyperboloid

(x,x) = - 1 , and the space of horocycles on it to the future light cone (x,x) = 0,x > 0. Denote by £ the Poincare length of the above part of geodesic; let the intersections of this geodesic with the horocycles be represented by the Minkowski space points w, v. Then (see Lemma 2.1 from [4]) — (u,v) — 2exp(^), and this formula allows the length coordinatization to be continued to the whole fgtn. Denote the length map thus described by

Pen:f,, n

>R6>V

222

SHABAT AND VOEVODSKY

Theorem 3.1 from [4] says, that this map is a real-analytic homeomorphism. Now we can state our results. 2.3.1. Theorem. There exists a set of horocycles on Xp, endowed with the metric of the constant curvature, such that the above map Pen sends the corresponding point ofTgn to (>/2,...,V2)€R>V6+3n. The proof follows from Penner's proposition 6.5 from [4], which states that the point Pen-1^,...,^) is uniformizable by a subgroup of PSL^iJ^), and from our Theorem 2.3 from the previous section. 2.3.2. Next, we describe Penner's universal cell decomposition of the space Tg^n. Its points are interpreted as the conjugacy classes of the fuchsian groups T C 50" l "(2,1) together with the T-invariant set B on the light cone (the only point of B on each light ray corresponds to the choice of the horcycle). To such a pair ^T, B) the convex hull of the discrete set r • B is associated. The cells of Tgnys correspond to the pairs (T, B) with the fixed combinatorics of the boundary of this convex hull. Projecting the edges of this boundary to the hyperboloid (x, x) — — 1, we get a T-invariant tesselation of it. Dividing then by T, we get the dessins (with fixed number n = [D] of vertices) that parametrize the cells of the decomposition we are describing. From now on, we suppose [D] = 1; then the above construction defines a decomposition of the Teichmuller space Tg^\ itself. Since this decomposition is invariant under the Teichmuller modular group, it induces a finite cell decomposition of the moduli space M9}\. For a dessin D, denote by CD the cell corresponding to it. The cells CD of maximal dimension correspond to the trigonal dessins D. 2.3.3. Theorem. For [D] = 1 the point of Mg\ corresponding to XQ lies inside CDThe proof follows from Penner's reasoning on the non-emptiness of CD (Corollary 6.3 of [4]), combined with Proposition 6.5. 2.3.4. Without proof we state one more result concerning the position of the curves XD with the Galois D's in the moduli spaces. Since they

DRAWING CURVES O V E R N U M B E R FIELDS

223

have many automorphisms, they correspond to some strong singularities of the moduli spaces. T h e o r e m . The curve X over Q can be realized as Xp with a Galois dessin D if and only if it is a projection of an isolated fixed point of some finite subgroup of the Teichmuller modular group. 2.4 M e t r i c s In this section, we consider only piecewise-euclidean metrics—the ones t h a t are formed by putting together compatible flat polygons; the resulting metric is flat away from the isolated points where the vertices meet and where the discrete curvature occurs (as a difference of 27T and the sum of the flat angles). T h e piecewise-euclidean metrics define complex structures as well as the riemannian ones; any complex structure on a Riemann surface can be obtained in this way. A nice proof of this fact can be obtained using Strebel differentials; for a modern exposition of this theory see Douady-Hubbard [11] (though they do not formulate explicitly the result we need). 2.4.0. We are going to work only with the piecewise-euclidean metrics, in which all the polygons are the equilateral triangles; we call t h e m the equilateral metrics. T h e equilateral metrics have no continuous moduli and depend only on the combinatorics of the triangles; so the trigonal dessins D appear. Denote the corresponding curves Y D . They define the countable set of points in the moduli spaces. 2 . 4 . 1 . T h e o r e m . For any trigonal dessin D the curve Yr> is phic to Xj).

isomor-

2 . 4 . 2 . T h e o r e m . The set of curves YD for all the dessins D on the surfaces of genus g "is" exactly the set M^(Q) of curves over all the number fields. For the proof, see our paper [10]. 2.4.3. This result can be interpreted in terms of string physics (see, for instance, [12]; in fact, we were influenced by the authors of this p a p e r ) . It shows t h a t integration over all the metrics on Riemann surfaces using the lattice-like method of approximation of Riemann metrics uncovers the arithmetical n a t u r e of the subject.

SHABAT AND VOEVODSKY

224

In the spirit of fashionable ideas of modern theoretical physics, it is natural to suggest that the non-archimedian components would also be taken into account in this approach. For the discussion of these ideas see Manin [13]. We suppose, that the dessins on the Riemann surfaces may turn to be quite fundamental in quantum physics, being the natural analogues of Feynman graphs in the pre-string theories. 2.5 Jacobians In this section we work with the trigonal dessins D on the Riemann surfaces of positive genus g. 2.5.0. The dessin D defines on X the structure of a cell complex, which will be used in the realisation of the cohomology classes. Denote by O the structure sheaf, Q the sheaf of germs of holomorphic differentials on the curve Xp. Using the diagram 0 Hl(XD,l)

=

H\XD,1)

n

0

>H°(XD,Q)

>H\XD,C)

I

>H\X,0)

>0

J(XD)

I 0 where the horizontal exact sequence comes from the exact sheaf sequence 0

>C

>0

•0

we realize the jacobian J(X) as the double coset space J(XD) 2.5.1.

=

I(HD(X,Q))\H1(XD,C)/H\XD,1).

The steps of our construction are:

•O,

DRAWING CURVES OVER NUMBER FIELDS

225

(i) Construct a piecewise-linear analogue of the abelian differentials on X and as a result obtain a ^-dimensional space LDCH^XDX),

where the RHS is interpreted as the space of the cell cohomologies of XD. (ii) Iterate the refinements

and use the canonical isomorphisms of XD and XaD, which results in the canonical isomorphism H1(XD,C)

=

H1(XQDX)-

Under this isomorphism the spaces LanD form a sequence of the g-dimensional subspaces of H1(XD,C), whose limit is (we hope)* the /-image of the space of abelian differentials in H1(XD,C). 2.5.2. The space Lp is constructed in the following way: Denote by Cl(D) the space of the cell cochains of XD with complex coefficients. To a 1-cochain on D we associate the 1-cochain on D as it is shown below:

Thus we get an operator *D:C\D)

>C\D')

which, we suppose, is the proper analogue of the harmonic Hodge operator (see also [14]). It enjoys the following properties: (a)

*D*D* = — 1.

* (Added in proof). It is really so. The demonstration will be published in a forthcoming paper.

226

SHABAT AND VOEVODSKY l{?{l(D)

(b)

Z1(D)n(*-D1(Z\D<)))

=

(Zl denoting the cocycles), then the projection H\D)

>H\XDjC)

is an isomorphism. Thus we have obtained the diagram of isomorphisms ^(D)

,

*D

Hl(D*)

*D*

H\XD,C)

=

which defines the operators on ±D and ±D*) they also satisfy

H\XD.,C)

H1(XD,C>)

for which we use the notations

i o i o * = — 1Denote by A the cup-product on For x,y from HX(XD,C), define

H1(XD,C).

(x,y) =•/>(«) Ay,

Proposition. ( , ) is symmetric and positively defined. Denote by {£a\a — 1,...,2
(a) Va,£a
DRAWING CURVES OVER NUMBER FIELDS

227

REFERENCES Grothendieck A., Esquisse d'un programme, Preprint 1984. Belyi G.V., On Galois extensions of the maximal cyclotomic field (in Russian). Izvestiya An SSSR, ser. matem., 43:2 (1979), 269-276. Mumford D., Curves and their Jacobians, Univ. of Michigan Press, 1975. Coxeter H.S.M. and Moser W.O.J., Generators and relations for discrete groups, Springer-Verlag, 1972. Penner R.C., The Teichmuller Space of a Punctured Surface, preprint. Klein F., Ueber die Transformation siebenter Ordnung der elhptischen Funktionen, Math. Ann. 14 B.3 (1878), 428-471. Hartshorne, R.S., Algebraic Geometry, Springer-Verlag, 1977. Holzapfel, R.-P., Around Euler Partial Differential Equations, Berlin: Deutscher Verlag der Wissenschaften, 1986. Yui, N., Explicit form of modular equations, J. Reine und Angew Math, 1978, Bd. 299-300. Vladutz, S.G., Modular curves and the codes of polynomial complexity (in Russian). Preprint, 1978. Voevodsky V.A., and Shabat G.B., Equilateral tnangulations of Riemann surfaces and curves over algebraic number fields (in Russian). Doklady ANSSSR (1989), 204:2, 265-268. Douady A. and Hubbard J., On the density of Strebel differentials. Invent. Math. (1975), 30 , N2, 175-179. Boulatov D.V., Kazakov V.A., Kostov I.K., Migdal A.A., Analytical and numerical study of dynamically triangulated surfaces. Nucl. Phys. B275 [FS17] (1986), 641-686. Manin Yu. I., Reflections on arithmetical physics, Talk at the PoianaBrashov School on Strings and Conformal Field Theory, Sept. 1987, 1-14. Itzyczon, C , Random Geometry, Lattices and Fields. "New perspectives in Quantum Field Theory", Jaca (Spain), 1985. Threlfall, W., Gruppenbilder, Abh. Sachs. Akad. Wiss. Math.-Phys. K1.41, S. 1-59. Abikoff, W., The Real-Analytic Theory of Teichmuller Space, Lecture Notes in Mathematics, Springer-Verlag, 820, 1980. G.B. Shabat Profsoyuznaya ul., 115-1-253, Moscow 117321, USSR

V.A. Voevodsky Moscow State University, Mech.-Math faculty, Moscow, USSR

Sur les proprieties numeriques du dualisant relatif d'une surface arithmethique L. SZPIRO

pour A. Grothendieck

Une courbe lisse, geometriquement connexe et de genre non nul sur un corps de n ombres, possede un modele regulier (Abhyankar), minimal (Shafarevich), unique. C'est cet objet qu'on appelle surface arithmetique. Arakelov, ayant introduit en 1972 une theorie des intersections pour les diviseurs "compactifies" sur une telle surface, il est tres tentant d'essayer de developper une "geometrie italienne" dans le cadre diophantien. J'ai indique ailleurs [Sz 4] les difficultes d'une telle tache. On a cependant un corpus de base du a Arakelov [A] et Faltings [F 1]. Nous appliquons ici ces fondements pour essayer de "mesurer" le dualisant relatif. Ce dernier faisceau inversible est central dans bien des questions. Nous avons, par exemple, dans [Sz 1], exploite au maximum la situation, dans le cas geometrique, pour obtenir un "Mordell effectif" en toute caracteristique par la construction de "petits points". En genre un, nous montrons plus bas, qu'une borne superieure, polynomiale en le conducteur, pour le "degre d'Arakelov" du dualisant relatif d'une courbe elliptique donnerait Fermat par la construction de Frey. II est remarquable que pour ce genre de questions, une borne superieure pour la self-intersection du dualisant relatif soit suffisante. Cependant l'experience montre qu'on n'obtient celle-ci qu'apres avoir obtenu des resultats de positivite (i.e., une borne inferieure pour les invariants numeriques de ce faisceau). Si les bornes superieures en genre un, comme en genre au moins deux, sont toujours conjecturales, (1-4 et 2-3) nous apportons ici un modeste eclair age sur les bornes inferieures. Le plan de cet article est le suivant :

230

L. SZPIRO 1. C o u r b e s e l l i p t i q u e s 1.1.

Isogenics

et fonctions

1.2.

Points

1.3.

Points d'ordre deux et courbes de Frey

1.4.

Problemes

de torsion

et

de Green

et degre d'Arakelov

du dualisant

relatif

conjectures.

2. C o u r b e s d e g e n r e ail m o i n s d e u x

2.1.

La conjecture

2.2.

La question de la nullite dualisant relatif

2.3.

Problemes

et

de

Bogomolov eventuelle

de la self-intersection

du

conjectures.

On se reportera a ^introduction de chacun des chapitres pour une description plus detaillee de leur contenu. Indiquons que dans les deux cas (genre un et genre au moins deux) nous reussissons a donner une interpretation diophantienne d'invariants obtenus de maniere transcendante par la theorie d'Arakelov (Theoreme 2, Proposition 1 et son corollaire; Theoreme 3, Section 2.3). La lecture de ce travail requiert une certaine familiarite avec la theorie d'Arakelov. On peut acquerir celle-ci en consultant [A], [F 1], [Sz 4] et surtout [Sz 2]. L'objet mathematique le plus utilise dans ce travail est le "dualisant relatif". C'est A. Grothendieck ([G]) qui a degage le premier cette notion. D ' a u t r e part il est clair que la theorie des schemas a "magnifie" le traitement identique des corps de nombres et des corps de fonctions apres 1'intuition qu'en avaient Dedekind, Chevalley et Weil. On voit ainsi que, non seulement ce lignes sont respectueusement dediees a A. Grothendieck pour son soixantieme anniversaire, mais que comme bien d'autres, je n'aurais pas pu "penser" a ces problemes sans les travaux du maitre de la geometrie algebrique de cette deuxieme moitie du siecle. 1. C o u r b e s e l l i p t i q u e s Dans ce chapitre nous etudions les isogenies de courbes elliptiques sur un corps de nombres du point de vue de la theorie d'Arakelov. Le premier resultat notable est le theoreme 2 qui dit que le degre d'Arakelov

LES PROPRIETES NUMERIQUES DU DUALISANT RELATIF 231 d'une courbe elliptique semi-stable est un douzieme du logarithme de la norme du discriminant minimal. On voit ainsi que ce degre n'est jamais negatif et est different de la "hauteur de Faltings", resultat qui semble avoir echappe a certains auteurs. Dans la section 3 nous calculons explicitement les valeurs des fonctions de Green d'Arakelov aux points de deux torsions des courbes de Frey. Nous rappelons enfin a la section 4 la conjecture du discriminant et ses variantes. Toute reproduction et demonstration fausse de cette conjecture, peut etre librement traduite ou adaptee meme sans indication d'origine. 1.1. Isogenics et fonctions de Green. Arakelov a deja note le manque de "commutation raisonnable" de sa theorie des intersections pour les morphismes finis, meme etales, entre courbes de genre au moins deux. Le lemme 1 qui suit exprime que pour line isogenie de courbes elliptiques un modeste miracle se produit. L e m m e 1. Soient 7r : E —> E1 une isogenie de courbes elliptiques sur un corps de nombres} L\ et L2 deux faisceaux inversibles sur E' mums de metriques permises. Alors pour chaque i = 1,2, 7r*Lt- est muni par image reciproque d'une metrique permise et on a : (7r*L1.^L2) = (deg^r)(Li-L2).

C o r o l l a i r e 1. Soit n : E —• E' une isogenie de courbes elliptiques sur un corps de nombres K. Supposons que son noyau H soit fini et deploye sur OK- Notons H = J2i=i E% ou &i es^ une section de E sur OK et h le degre de w. Alors on a : Y,(Ei

• E:) = h(d - d')

oil d et 61 sont les degres d'Arakelov

de LOE ct UJEI

respectivement.

D e m o n s t r a t i o n d u corollaire. Si 0 ; est la section nulle de E' on a h(OE'(0') - 0E'(V)) = -hd! = {XEi . EE{) = -hd + E w ( ^ " ' £ j ) c.q.f.d. D e m o n s t r a t i o n d u l e m m e 1. Rappelons que si dfi designe la metrique plate, invariante par translation, de volume total egal a 1, sur une courbe elliptique E sur C, une metrique | |, sur un faisceau inversible L sur E est dite permise si T e q u a t i o n de courbure" suivante est satisfaite :

- — d d l o g I s I = (deg L)d^i - SD

232

L. SZPIRO

ou s est une section meromorphe de L, D le diviseur lui correspondant et SJJ la fonction de Dirac de support D. Dans la situation du lemme 1, munissons n* L de la metrique image reciproque de celle donnee sur L. On a pour tout point P de E \(**8){P)\=\8(*{P))\ donc

~^ddlo&

\**s\ = v*(~^ddlo&

Is I) = **((de& LWE' - 6D)

d S e g ^ L ) ^ g ( d e g *) " *'*D = (deg **L)dn 6,.D. Nous avons utilise dans les deux dernieres egalites les trois faits suivants (faciles a etablir):

=

(d

deg 7r* L = (deg 7r)deg L 7r*d/i£/ = (deg 7r)dfiE

Nous avons donc etabli la premiere partie du lemme 1. Pour la deuxieme partie, considerant que L\ et L2 sont associes a des diviseurs D\ et D2, en exprimant I n t e r s e c t i o n d'Arakelov comme somme de termes locaux on voit : (a) que la partie a distance finie ne pose pas de probleme, (b) q u ' a Pinfini, il suffit de verifier Ylw(pt)=p 1 7r *( 5 )(^») I — I S(P) \h est clair.

ce

Qui

C o r o l l a i r e 2. Soil E une courbe elhpiique sur un corps de nombre K et n un entier au moins egal a 2. Supposons que tous les points de n torsion de E soient rationnels sur K et notons E{, i — 1 , 2 , . . . , n2, les sections du modele minimal geometrique de E sur OK correspondantes. Alors on a la relation suivante :

Le corollaire 2 se deduit du corollaire 1 ci-dessus en prenant w = [n] et Ef = E (donc d = d' et h = n 2 ) . Note. Le corollaire 2 donne un exemple agreable d'un des obstacles au developpement de la theorie d'Arakelov : deux diviseurs effectifs, irreductibles, horizontaux d'une surface arithmetique peuvent tres bien avoir une intersection negative. (On verra plus bas dans Pexemple des courbes de Frey que les (Ei • Ej) ne sont pas tous nuls). Dans le cas geometrique si n est premier a la caracteristique chacun des (Ei • Ej) est

LES PROPRIETIES NUMERIQUES DU DUALISANT RELATIF 233 mil. Par contre quand n — p est la caracteristique du corps de base tous les (E{ • Ej) sont positifs ou nuls et certains sont non nuls. T h e o r e m e 1 (Energie du noyau d'une isogenie). Soil ir : E —> E' une isogenie de courbes elliptiques semi-stables definies sur un corps de nombre K. Soit H le noyau de cette isogenie et h son degre. Pour chaque place a de K notons P?, i = 1 , . . . , h, les points de H sur C. Alors on a

£ £ , ( / ? , ; ? ) = ^MogA +A £ I o g

A«)(ImQ6| A ^ X I m r„)« '

Remarquons de suite qu'on peut se demander si une telle formule (ou Ton retire les J^^; les cr, et K : Q) est valable pour une isogenie de courbes elliptiques sur C. C'est en tout cas un fait quand K — Q et que l'hypothese sur H est satisfaite !! Notons dans ce sens le manuscrit de N. Elkies [E]. D e m o n s t r a t i o n . Par [F 2] ou [S i] on a log | A(r)(Im r ) 6 |)

hF = A(log A - £ a

h'F = A(log A' - £ > g | A(r')(Im r ' ) 6 | ) . a

On a d'autre part [F2] (formule du changement de la hauteur modulaire par isogenie) [K : Q](hF - h'F) = log 6H,0K

- ^y^log h

ou 6H/OK e st 1'image directe du dualisant relatif du schema en groupes H sur OK (suppose fini et plat) (cf. [R]). On a par adjonction [Sz 4]

(wB H) + (HoH) = log 6HIOK - 2 Y,9{P?, Pf)a

i^j

Si d est le degre d'Arakelov de uE, on a done :

hd-hd+ Y,(Ei -E3) = h\og 6H/oK - E

E9(P?,Pj)-

Par le corollaire 1 du lemme 1 on obtient done

h(d-d') = h log lH,oK -J2 529(P?,P?)(7

i^j

L. SZPIRO

234

Par le theoreme 2 et la formule du changement de hauteur modulaire par isogenie [F2] on obtient le resultat voulu. 1.2. Points de torsion et degre d'Arakelov du dualisant relatif. Nous commenQons par etablir une formule, surprenante a premiere vue, pour le degre d'Arakelov du dualisant relatif d'une courbe elliptique semi-stable. T h e o r e m e 2. Soit E une courbe elliptique semi-stable sur un corps de nombre K. Soit A m i n son discriminant minimal et | A m i n | 5a norme. Alors le degre d'Arakelov de Vimage directe du dualisant relatif de E sur OK satisfait : 12 degAvujE = log| A m i n |. Remarque!.Certains auteurs ont fortement conjecture que deg Ar u;£ = [K : Q]hp(E). Le theoreme 1 leur donne tort. Le lecteur curieux de ces choses s'amusera a trouver dans la bibliographie de cet article, au moins un auteur repute qui s'est laisse prendre. Remarque 2. La formule donnee par le theoreme 2 peut se traduire par :

I [ I A Ur,<7 = 1 a

ou A est considere comme une section de u^' et | A \AT}O e s t s a norme d'Arakelov a la place a. II est naturel de se demander si un tel enonce est valable pour une courbe elliptique sur C (i.e., a chaque place a l'infini). Avant de passer a la demonstration du theoreme 1 nous inserons le calcul des diviseurs verticaux 0 correspondants a deux sections en une place ou la reduction est un "cycle". L e m m e 2. Soit V un anneau de valuation discrete f : E —> V une courbe elliptique dont la fibre generique est lisse et la fibre speciale est reduite et est un cycle de P 1 de self-intersection —2. Soient Fo • • • F n _i les composantes de cette fibre speciale. Si E\ et E2 sont deux sections de f telles que : (Ei-F0) / 0 et (E2'Fk) / 0, le diviseur a coefficients rationnels vertical 0 defini a un multiple de la fibre speciale pres, par Vequation ((Ei — £2 + 0) • F{) = 0 pour tout i satisfait la formule suivante _02

fc =

("-fr)log n

JV(t,).

Notons que si V est d'inegale caracteristique a corps residuel fini (situation arithmetique) N(v) — cardinal du corps residuel, et si V est d'egale caracteristique on prendra log N(v) = 1.

LES PROPRIETIES N U M E R I Q U E S DU DUALISANT R E L A T I F

235

D e m o n s t r a t i o n d u l e m m e 2. Ecrivons 0 = Hx{Fi ou les X{ sont dans Q. On doit avoir ((YtX%Fi-\-E\ — E2) -Fj) = 0 pour tout j . Notons d'abord que (Ei - E2 + 0 • 0) = 0, done - 0 2 = (an -a?jb)log

Nv,

un calcul facile permet alors de conclure. Passons maintenant a la p r e u v e d u t h e o r e m e 2 : E etant semistable la formule a montrer est invariante par changement de base. On peut done supposer que les points de torsion sont rationnels sur K. Soient E\,..., En2 les sections du modele minimal geometrique de E sur OK correspondantes. Pour tout i ^ j on definit comme plus haut 0,-j diviseur vertical a coefficients rationnels par (E{ - Ej + 9ij -V) = 0 pour tout diviseur vertical V. On a done : ({Ei-Ej+9iJ).hj)=0 et

( ^ - £ > + 0 t - i i ) 2 = O. Cette derniere formule etant due au fait que E{ — Ej est de torsion et a la formule de comparaison entre l'intersection d'Arakelov et la hauteur de Neron-Tate ([F 1], [H], [SPA] p. 83). Ecrivant X^j^»(-^» ~ ^3 + 0»,i) 2 = 0 et utilisant la premiere formule ci-dessus on obtient :

(n 4 - n2)degAruE

= ~ ^ h '

En effet —E? = deg A r u>£ pour tout i. Comme on sait que les points de n-torsion sur une fibre singuliere qui est un cycle sont distribues en n paquets de n points equidistribues, le lemme 2 donne une formule pour — Ylijtj ®!j- Appliquant ceci a n = 2, on obtient 12 degAru>£? = J2v\N nv\°g N(v) ou A^ est le conducteur de E, et nv le nombre de composantes de la fibre de v. Maintenant E etant semi-stable on sait que v ( A m i n ) = nv pour tout v\N. D'ou le resultat. Au cours de la demonstration precedente nous avons etabli le lemme suivant qui nous sera utile :

L. SZPIRO

236

Lemme 3. Soil E une courbe elliptique sur un corps de nombre Soient n un point positif et P un point rationnel de E sur K d'ordre actement n. Soient 0 la section nulle du modele geometrique minimal E sur OK et Ep la section de ce modele correspondant a P. Soit $p diviseur vertical a coefficients rationnels tel que (0 — Ep -f 0p • V) = 0 tout diviseur vertical V. Alors on a (0-Ep)

K. exde un pour

-degAru;E-
=

1.3. Points d'ordre deux et courbes de Frey. Les formules etablies precedemment permettent de "pousser" les calculs assez loin pour les courbes elliptiques definies sur Q dont les points d'ordre deux sont rationnels. (Ces courbes meritent desormais le nom de "courbes de Frey" [Fr 1], [Fr 2], [Fr 3]. Nous suivons la presentation de G. Frey. Soient a, 6, c des entiers sans facteur commun tels que a + b = c. Considerons la courbe elliptique Ea^fC definie par y1 = x{x-\-a)(x — b) (*). Frey a etabli les faits suivants : (1) Notant a celui de ces trois nombres qui est pair, si 2 4 divise a et si on a choisi b tel que 6 = —1(4), alors Ea,b,c e s t semi-stable, son equation minimale est donnee par : 2

Q

y +xy = x +

a — b — 1 2 ab x x --jg"

(2) Son discriminant minimal vaut alors : Amin = 2 - 8 a W . Les trois points d'ordre 2 significatifs correspondent, dans l'equation initiate (*) respectivement a x = 0, x = —a, x = b. Nous noterons ces points Po, Pa, Pb, les sections du modele minimal geometrique J^o, Ea, Ef,, le discriminant minimal des courbes elliptiques quotientes de E par, respectivement, chacun de ces points, AQ, A a , Aj. Proposition 1. Avec les notations precedentes on a : g(0,P0) = l l o g \c2/2~4ab\;

|A 0 | = |2- 4 a6c 4 |

9(0, Pa) = ^ l o g | 2 - 8 a 2 / 6 c | ; |A a | = |2- 1 6 a 4 6c| g(0,Pb) = ^ l o g \28b2/ac\

; |A»| = |2" 4 a6 4 C |.

LES PROPRIETES NUMERIQUES DU DUALISANT RELATIF

237

Par le lemme 3 et lemme 1 appliques aux isogenics a noyau cyclique d'ordre 2 engendres par respectivement chaque Pt} i = 0,a,6, on a = -d-H±

(0-Ei)

0?

=

d-di

ou di = ^ l o g |At-|. (i) Calcul des —02. Considerons la contribution en un nombre premier p divisant A est nulle si E{ ne passe pas par le point singulier de la fibre en p, dans le modele minimal arithmetique. Ceci a lieu : pour Eo si p | c pour Ea si p | 2" 4 a pour E\i si p | 6. Par contre, dans le cas contraire, par le lemme 2, la contribution est de —Vp4 log p. Ceci a lieu : pour EQ si p | 2~4a6 pour Ea si p | be pour Eh si p | 2~ 4 ac. On obtient done -0g |log 2" 4 ac.

=

±log 2" 4 a6, - 0 ^

=

|log 6c, -02

=

(ii) Calcul des (0 • Ei)j (contribution locale a distance finie de Pintersection). Pour ceci notons que dans 1'equation minimale homogeneisee : .2* zl y

-f Xyt = x° +

a —6 — 1

4

tx

£

o

-

abxt2

16

les quatres points de 2 torsions sont donnes par : E0:t Eq:t Eb : t 0 :*

= = = =

l , ar = 0 , y = o l 'x 4 >»=f 8 , x = 26 ,y = -b 0 , x = 0 , y = i.

On voit done que (O-E^f est le seul des (O-Ei)j qui est non nul. Le nombre reel (0 • £&)/ est done un multiple entier a de log 2. Nous calculerons a plus bas. On obtient ainsi (i) do = ^ l o g | A 0 | = ^ l o g | 2 - 4 a 6 c 4 |

L. SZPIRO

238

(ii) 4 = ^ l o g | A a | = ^ l o g | 2 - 1 6 a 4 6 c | (iii) dh = i l o g | A 6 | = ^ l o g | 2 - 4 a 6 4 c | . II nous reste a calculer les valeurs des fonctions de Green. Par le theoreme 1 on trouve

J2 g(0,P,) = \og2 i=0,a,6

(on a utilise g(Pt,Pj) = g(0,Pt - Pj)). Par le (ii) ci-dessus — #(0, Po) = 7^'°&

rV&V 4 4

2" a6c 2-8,2 a2
log

soit flf(0,Po) =

de meme et 2~4ab J_log b2 a d'ou a — 12. On en deduit la proposition. 2 12 2~ 4 ac 12


=

Corollaire. Si E est une courbe de Frey correspondant a a + b = c avec a > 0, 2 4 |a, et b = —1(4), a/ors /e maximum de la fonction de Green sur E(C) est 1 24c2 -log —jsi 6 es/ positif a6 12 2862 si b est negatif. l 0 g

T2

^

Par ce que nous venons de montrer ces valeurs sont bien le maximum des g(0,Pi), i — 0, a, b. II est facile de voir que les points de deux torsions sont des extremes pour g(0,P). Pour conclure nous avons besoin du resultat plus delicat suivant : T h e o r e m e (Showu-Zang). Soit E une courbe elliptique sur C, dont le r dans le domaine fondamental du demi-plan de Poincare pour faction de 5L(2,Z) est purement imaginaire. Alors le maximum des g(0,z) est obtenu pour z — —-—. La demonstration sera publiee prochainement dans la these de Showu-Zang [Z]. 1.4. Problem.es et conjectures. II est facile par la methode de la classe de Kodaira-Spencer introduite dans [Sz 1] expose 3, de montrer que le degre de discriminant minimal d'une courbe lisse de genre g, sur un corps k est majore par pe6(2g — 2 -h s) ou, e est le degre d'inseparabilite

LES PROPRIETIES NUMERIQUES DU DUALISANT RELATIF

239

de l'application j : C —> P 1 correspondant a cette courbe elliptique, et s le nombre de points geometriques de C dont la fibre n'est pas lisse. II est done naturel de se poser la question suivante : C D . Conjecture du discriminant. Soit K un corps de nombres, il existe une constante k ne dependant que de K telle que Norme (A^;) < N& pour toute courbe elliptique E sur K ou A# est le discriminant minimal de E sur K et NE son conducteur. Si on veut se rapprocher de 1'enonce geometrique sign ale plus haut on peut faire une conjecture plus audacieuse : C.D.F. Conjecture du discriminant (forme forte). Soit A' un corps de nombres, alors pour tout e > 0, il existe une constante C(K,e) telle que Norme (AE) < C(K,e)N%+c pour toute courbe elliptique E sur K. Notons d'abord qu'on ne peut prendre e = 0, ni meme N^(\og NE)01 pour tout a (ce dernier point a ete montre par Masser [M]). Appliquant ceci aux courbes de Frey du paragraphe precedent on obtient 1'enonce conjectural suivant : II existe une constante k telle que : si a, 6,c sont des entiers non nuls sans facteurs communs tels que a + b = c, alors

\abc\<

I 11 p) \p|abc

)

(sous la forme forte : pour tout e > 0 il existe une constante c(c) telle que /

\3+e

I a6c I < c(e) [Ylp\abcP) • ^ n s a ^ l'etonnant tour de passe-passe que de tels enonces autorisent avec une solution non triviale de l'equation de Fermat xp + yp = zp. Signalons Texpose de Oesterle [O], le livre [Ba] et le seminaire a paraitre [Sz 7] ou d'autres implications et des raffinements de ces conjectures sont etudies (cf. aussi Vojta [V]). On notera, dans un genre totalement different, que Ribet a reduit recemment la conjecture de Fermat a montrer que les courbes de Frey sont modulaires. Par Weil il suffit done de montrer que la fonction L d'une telle courbe a une equation fonctionnelle raisonnable. Ce travail de Ribet donne la premiere evidence serieuse de la conjecture de Fermat.

240

L. SZPIRO 2. C o u r b e s d e g e n r e a u m o i i i s d e u x

G. Faltings a montre dans [F 1] que la self-intersection du dualisant relatif d'une courbe de genre au moins deux n'est jamais strictement negative. Nous montrons ici (au moins pour les courbes ayant bonne reduction partout) que la non nullite de cette self-intersetion est essentiellement equivalente a une conjecture de Bogomolov qui dit que les points arithmetiques d'une courbe plongee dans sa jacobienne sont discrets pour la "topologie de Neron-Tate". Nous etablissons cette conjecture pour les points dont l'image canonique dans la jacobienne n'est pas de torsion (Theoreme 3). L'ensemble fini de points restants (M. Raynaud), donne du fil a retordre quand j u s t e m e n t la self-intersection du dualisant relatif est nulle. Nous ne savons pas si une telle situation existe. Nous Panalysons cependant au Section 2. On notera qu'un theoreme d'evanouissement de la partie fixe (conjectural) rendrait bien des services. Au Section 3 nous indiquons une nouvelle "conjecture des petits points". Cette fois c'est la hauteur de Neron-Tate qui doit etre "petite". Nous deduisons de cette conjecture quelque corollaire mediatique. 2.1. La conjecture de Bogomolov. Soit X une courbe de genre g > 2 sur un corps de nombres K. Nous etablissons dans ce chapitre une generalisation du theoreme de Raynaud (conjecture de Lang) sur la finitude de 1'ensemble des points de X(K) qui sont de torsion apres plongement dans la jacobienne. Cette generalisation remplace "hauteur de Neron-Tate nulle" par "hauteur de Neron-Tate" inferieure a c > 0. (Un resultat conjecture par Bogomolov). Malheureusement, par manque d'un theoreme de disparition asymptotique des parties fixes, nous n'etablissons pas le resultat, pour les points P de X(K) qui sont proportionnels a ux dans Pic(A r )(J\) (nous appelons ces points : points qui divisent UJ) lorsque (UX/QK -UJXJQK) = 0. Notons que nous ne connaissons pas d'exemple ou cette egalite est satisfaite. Rappelons la formule etablie dans [Sz 3] qui relie la hauteur de NeronTate (xp)2y de l'image xp (dite ici canonique) d'un point P de X(K) dans sa jacobienne par 1'application P -+ classe de fi^/K ® Ox ((2 -

2g)P),

a l'intersection d'Arakelov notee (. , .). (xPl)2-(xp2)2 i ; • xP2) = 2 ^ .

(xPl

V

2/

+

Y T

1 A 1W Q ( - - l)(u>*,oK

-(2g - 2)\EPl

• EP2) - ^ h p k

-"*loK) +

(0Pi

.

0P2)(**).

LES PROPRIETIES N U M E R I Q U E S DU DUALISANT R E L A T I F

241

Comme a l'habitude dans cette for mule u>x/oK e s ^ le dualisant relatif du modele minimal regulier de X sur OK, Ept la section de X sur OK correspondant a P,- et 0p t un diviseur vertical a coefficients rationnels defini par (ux/oK ® Ox((2 - 2g)EPt) Ox(HPt) • Ox(V)) = 0 pour tout diviseur vertical V. En particulier on a : ([SPA] expose XI) 2g(2g - 2)(-EPt)

-

U%/0K

= [K : P](xPf

- 9%.

(***)

T h e o r e m e 3 . (Conjecture de Bogomolov pour les courbes lisses). Soit X une courbe lisse, geometriquement irreductible, de genre g > 2, sur un corps de nombre K. Soit P un point rationnel sur la cloture algebrique K de K, dont Vintage canonique xp dans la jacobienne est de hauteur de NeronTate (xp)2 non nulle. Supposons que X ait bonne reduction partout. Alors il existe un nombre reel ep > 0, calculable, tels que : I'ensemble des points Q de X(K), tels que la hauteur de Neron-Tate de P — Q soit inferieure a ep, sont en nombre fini. De plus s'il existe une suite infinie de points distincts dans X(K) tels (xp)2 tend vers 0, alors wx/o2, — 0Puisque nous avons suppose que X avait une bonne reduction, les termes "0" dans les formules (**) et (***) disparaissent. Par un theoreme classique, deja utilise par G. Faltings dans [F 1] il existe, pour tout e > 0 un entier N tel que si les P i , . . . , P n , n > N sont n points distincts de X(K) et L un corps de nombres ou les P{ sont rationnels et si a est une place a Tinfini d'un tel corps, on a : X ^ j 9(P?iP?) < cn2 ( o u 9 e s t ^ a fonction de Green d'Arakelov). Pour un point P tel que (Xp)2 ^ 0 choisissons ep de telle fa^on que si ( P — Q)2 < cp on ait : (xp-xQ)

= ({xp)2(xQ)i)1"co8

0P,Q

avec (cos9pyQ — 1) assez petit (done | # P , Q | p e t i t ) . S'il existait un ensemble infini de points proches de P pour la hauteur de Neron-Tate, leurs corps de rationalite aurait un degre tendant vers Tinfini. On aurait done pour tout e > 0 un point P i , possedant TV conjugues distincts sur K, P i , . . . , P/v, soit en calculant sur un corps L de rationalite de ces N points, et en addition ant les egalites (**) :

i?j

L. SZPIRO

242

ou $ij est Tangle pour raccouplement de Neron-Tate entre xpt et xp . Comme g > 2, on s'est assure, que

cos 6{j < — — ($ij petit). On ob' 3 tient done, puisque U2X/0L = U2X/CK[L : A], U2X/0K(1 - - ) < -(xp)2 • - + 9 & N2 (2g — 2)2e—r — pour tout c > 0 et N assez grand, ce qui force w2x,n , < 9

0, contrairement au theoreme de Faltings [F 2] affirmant u;^/^, > 0. Ceci termine la demonstration de la premiere partie du theoreme. Pour la deuxieme partie, reprenant l'argument de le demonstration precedente et choisissant e > 0 et N points distincts dans cette suite tels que (xpt)2 < 6, TV grand, on obtient :

ZTQ<"2 " " ^ W 1 - j) ^ <\ + D + ^nSL •Q) G ^ d'ou ^ /

0

, < 3e pour tout e > 0.

Remarque. Le resultat s'etend par la meme demonstration a une courbe (qu'on peut supposer semi-stable) n'ayant pas necessairement bonne reduction partout si on remplace (xp)2 > 0 dans l'enonce par : il existe une constante a(X) ne dependant que du nombre de composantes des mauvaises fibres de X sur OK telles que, si (xp)2 > a(X) alors... En effet les "(0i :0i)" ne dependent que du nombre de composantes des fibres. K :Q 2.2. La question de la nullite eventuelle de ^x/a , • Pour une courbe X ayant bonne reduction partout sur A", si u2x,c , > 0 l'argument precedent donne aussi la conjecture de Bogomolov pour les points tels que (xp)2 = 0. On prendra alors cp = u2x,0 /\Q par exemple. La quest ion de savoir si w2x,0 r est strictement positif reste ouverte meme pour une courbe lisse. La remarque qui suit renforce la liaison de cette question a la conjecture de Bogomolov. Supposons que X ait bonne reduction partout et supposons que U\IQ , = 0. Nous allons essayer, sous ces hypotheses d'etablir la reciproque de theoreme precedent. Pour tout LJ > 0, UXJQK 0 Ox(tF) ou F est une fibre a l'infini, est de self-intersection positive. Par le theoreme d'existence de Faltings ([F 2]), il existe N(e) tel que pour n > N(u>) (ux/oK ® Ox{cF))®n ait une section positive au sens d'Arakelov. Ecrivons le diviseur positif D apres changement de base, a un corps L D K assez grand : D — ]Tj g~"' E{ + V

LES PROPRIETES NUMERIQUES DU DUALISANT RELATIF 243 ou les Ei sont des sections de X sur Oi et V est vertical et positif. On a : n(2j-2)

(ux/oL-Ox(D))=

J^

(-Et)2 + (wx/OL-Ox(V))

=

ne(2g-2)[L:K].

1

Comme (u>X/oL * Ox(V))

E2 > 0, il existe un i tel que — '

=

2

(XA

2g(2g-2)

< e. On pourrait croire qu'on a la reciproque cherchee. Mai-

heureusement on ne sait pas qu'un tel point correspondant a Ei satisfait a — E2 ^ 0. Par M. Raynaud il n'y a qu'un nombre fini de tels points sur K. II manque done, pour conclure, de savoir si un faisceau du type eut UX/OK ® Ox{tF) P avoir une partie fixe "horizontale" asymptotiquement ! Exemple. Prenons une courbe X de genre deux sur un corps de nombre K qui ait une bonne reduction partout. Soient E\,..., EQ les sections de X sur OK correspondantes aux points de Weierstrass supposes rationnels sur K. Alors pour chaque i — 1 , . . . , 6 on a : Wx/oK numeriquement equivalent a Ox(2Ei

+ <*iF) (F fibre a l'infini)

d'ou Ton tire : a{ = 3E? et W\JOK

= -AEf.

Comme la courbe a bonne reduction sur OK pour tout f / j , on a (Ei — Ej)2 = 0, d'ou AE2 — A(Ei -Ej) = —^x/oK* ^ n v o ^ a n i s * c l u ' o u ^ien U)2X,0 r — 0 ou bien Intersection d'Arakelov (Ei • Ej), i ^ j de deux points de Weierstrass est negative. On trouvera dans [M-B,M,B] des exemples ou "x/oK > 02.3. Problemes et conjectures. Aux paragraphes precedents nous nous sommes heurtes au probleme de bonier inferieurement ^\i0 , • La question d'une borne superieure, si elle a une reponse heureuse, donne, on le sait par la construction de Kodai'ra-Parshin, un "Mordell effectif" (cf.

[Szl]). Nous avons a ce propos propose ailleurs [Sz 6] une conjecture dite des peiits points. En fait par le theoreme de l'index de Hodge en theorie d'Arakelov ([F 1], [H]) on a ([Sz 1]) : 2g(2g-2)(-E2) > U2X/0K done toute borne superieure pour un point donne une borne superieure pour w2x,0 r. Cette conjecture s'enonce en termes d'intersections d'Arakelov, nous en proposons une plus bas en termes uniquement de hauteur de Neron-Tate.

L. SZPIRO

244

Conjecture des deux petits points pour Neron-Tate (C.P.P.N.T.). Soit g un entier et K un corps de nombres, alors il existe deux constantes a(g) et b(g) ne dependant que de g, telles que : pour toute courbe X de genre g sur K, il existe deux points distincts P\ et P^ dans X(K) dont la hauteur de Neron-Tate de l'image canonique dans la jacobienne (xpt)2 satisfait : (xp,) 2 < a(g)A(X)\og

\ DK/(i

\ + b(g)B(X)

ou A(X) et B(X) ne dependent que de la geometrie de la mauvaise reduction de X, et DK/Q est le discriminant de K. Nous allons montrer que (C.P.P.N.T.) implique u

xQ/oK

< a'(g,X)log

DK/Q

+ b'{g,X)

pour XQ la fibre de la famille de Koda'ira-Parshin correspondant au point Q rationnel sur K d'une courbe fixe X. Les constantes a'(g,X) et b'(g,X) ne dependent que du genre et de la geometrie de la mauvaise reduction de la courbe fixe X. On sait, par Parshin [P], qu'une telle conjecture implique la conjecture du discriminant A < Nk du Chapitre 1, Section 3. Est-il utile de rappeler que la construction de G. Frey lui a permis de remarquer que cette derniere conjecture donne des renseignements etonnants sur les equations diophantiennes ? Passons a la demonstration de l'implication. Soit L un corps ou les Pi sont rationnels (rappelons qu'on peut supposer que X et les XQ sont semi-stables si necessaire), on a : 2 (*p,) 2 + (*p,) 2i |

<**•**> <"ru::

i

/

A

+ — K Q 2/ o . ( " i ) i + (2g - 2)zc(X) + a(X)

ou c(X) est une borne superieure pour les fonctions de Green sur toute les XQ, Q G X(K) (la famille de Koda'ira-Parshin est a base compacte!!), et a(x) est une borne pour les termes en "0" dans (**) qui done ne depend que de g et de la matrice d'intersection des mauvaises fibres de XQ et done, que de la geometrie des mauvaises fibres de X (On peut lire les details sur la construction de Koda'ira-Parshin dans [D]). L'implication s'en deduit facile me nt.

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BIBLIOGRAPHIE [A] Arakelov, S.J., Intersection theory of divisors on an arithmetic surface, Math. USSR Izvestia 8, 1974. [Ba] Bataille, G., VAbbe C, Editions de Minuit, 1950. B,M,M-B] Bost, J.M., Mestre, J.F., Moret-Bailly, L., Seminaire sur les pinceaux de courbes elliptiques, a paraitre dans SPE, [D] Deschamps, M., La construction de Kodaira-Parshin in SPA, Seminaire sur les pinceaux arithmetiques, Asterisque 126, 1985, Societe Mathematique de France. [E] Elkies, N., A problem of minimum energy, manuscrit Columbia University, 1983. [Fl] Faltings, G., Calculus on arithmetic surfaces, Annals of Math. 119, 1984. [F2] Faltings, G., Endlichkeitssatze fur abelesche varietdten uber ZhalenKorpen, Inv. Math. 73 (1983), 349-366. [Frl] Frey, G., Rationale Punkte auf Fermatkurven und getwistete Modulkurven, J. Reine Anyew Math. 331, 1982. [Fr2] Frey, G., Links between stable elliptic curves and certain diophantine equations, Annales universitatis Saraviensis, series mathematicae I, 1986. [Fr3] Frey, G., Links between elliptic curves and solutions of A — B = C, preprint 1987. [G] Grothendieck, A., Fondements de la geometrie algebrique, Secretariat mathematique 11 rue Pierre et Marie Curie Paris 5e, 1962. [H] Hriljac, P., Heights and Arakelov intersection theory, American Journal of Math., 1983. [M] Masser, P.W., On Szpiro's conjecture, preprint Univ. of Michigan at Ann Arbor, 1982. [O] Oesterle, J., Nouvelles approches du "theoreme" de Fermat, Seminaire Bourbaki n° 694, 1988. [P] Parshin, A.N., The Bogomolov, Miyaoka, Yau inequality for arithmetical surfaces and its applications, Seminaire de theorie des nombres de Paris, 1986 (a paraitre). [Rl] Raynaud, M., Courbes sur une variete abehenne et points de torsion, Inventiones Matematicae 71, 1983. [R2] Raynaud, M., Hauteurs et isogenics in SPA, Seminaire sur les pinceaux arithmetiques, Asterisque 127, (1985), Societe mathematique de France. [S] Silverman, J.H., Heights and elliptic curves in Arithmetic Geometry, Graduate text in Mathematics, Springer-Verlag, New York, 1986.

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[Szl] Szpiro, L., Proprietes numeriques du faisceau dualisant relatif, in S P C , Seminaire sur les pinceaux de courbes de genre au moins deux, Asterisque 8 6 , Societe Mathematique de France, 1981. [Sz2] Szpiro, L., Arakelov intersection theory on arithmetic surfaces (en japonais), in arithmetic algebraic geometry, Tokyo, Janvier 1987. [Sz3] Szpiro, L., Un peu d'effectivite in SPA, Seminaire sur les pinceaux arithmetiques, Asterisque 127, Societe mathematique de France, 1985. [Sz4] Szpiro, L., Presentation de la theorie d'Arakelov, Contemporary Mathematics, 6 7 , 1987. [Sz5] Szpiro, L., Small points and torsion points pour le centieme anniversaire de S. Lefchetz, Mexico, 1985, in Contemporary Mathematics, 5 8 , part 1, 1986. [Sz6] Szpiro, L., La conjecture de Mordell d'apres G. Faltings, Seminaire Bourbaki, n° 619. [Sz7] Szpiro, L., Seminaire sur les pinceaux de courbes elliptiques, S P E , a paraitre. [V] Vojta, P., Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, 1 2 3 9 , Springer-Verlag, 1987. [Z] Zang, Showu, These a paraitre, Columbia University, New York.

C.N.R.S. Institut Henri Poincare 11 rue Pierre et Marie Curie 75005 Paris, France

Higher Algebraic K-Theory of Schemes and of Derived Categories

R. W. THOMASON* and THOMAS TROBAUGH to Alexander Grothendieck on his 60th birthday

In this paper we prove a localization theorem for the A-theory of commutative rings and of schemes, Theorem 7.4, relating the A'-groups of a scheme, of an open subscheme, and of the category of those perfect complexes on the scheme which are acyclic on the open subscheme. The localization theorem of Quillen [Ql] for A'- or G-theory is the main support of his many results on the G-theory of noetherian schemes. The previous lack of an adequate localization theorem for A'-theory has obstructed development of this theory for the fifteen years since 1973. Hence our theorem unleashes a pack of new basic results hitherto known only under very restrictive hypotheses like regularity. These new results include the "Bass fundamental theorem" 6.6, the Zariski (Nisnevich) cohomological descent spectral sequence that reduces problems to the case of local (hensel local) rings 10.3 and 19.8, the Mayer-Vietoris theorem for open covers 8.1, invariance mod £ under polynomial extensions 9.5, Vorst-van der Kallen theory for NK 9.12, Goodwillie and Ogle-Weibel theorems relating A-theory to cyclic cohomology 9.10, mod £ Mayer-Vietoris for closed covers 9.8, and mod £ comparison between algebraic and topological A'-theory 11.5 and 11.9. Indeed most known results in A'-theory can be improved by the methods of this paper, by removing now unnecessary regularity, affineness, and other hypotheses. We also develop the higher A'-theory of derived categories, which is an essential tool in the above results. Our techniques here rest on the brilliant work of Waldhausen [W], who has extended and deepened the foundation of A'-theory beyond that laid down by Quillen, allowing it to bear a heavier load. ^partially supported by NSF and the Sloan Foundation.

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The key ideas t h a t make all our results possible go back to the theory of A'o of the derived category, which was conceived by Grothendieck, and was developed by him with Illusie and Berthelot in [SGA 6]. We especially need his concept of a perfect complex, a sheaf of chain complexes t h a t is locally quasi-isomorphic to a bounded complex of algebraic vector bundles. These ideas have remained dormant for some time, especially because Quillen [Ql] discovered the higher A'-theory of exact categories in a form which did not immediately extend to define a higher A'-theory of derived categories. Thus one worked with the exact category of algebraic vector bundles, and not with the derived category of perfect complexes. Waldhausen's work [W] first made it clear how to define such a 7\-theory of a derived category, or more precisely, of a category of chain complexes provided with a notion of "weak equivalence" like quasi-isomorphism. Several people, among t h e m Brinkmann [Bri], Gabber, Gillet [Gi2], [Gi4], Hinich and Shekhtman [HS], Landsburg, Waldhausen, and ourselves then became aware of this possibility of returning to the ideas of [SGA 6]. T h e intrinsic appeal of those ideas did not instantly overcome public inertia, and they did not at once appear strictly necessary to further progress. However, they t u r n out to be essential to the very statement of our localization theorem, if not to all its consequences. Moreover, the key geometric fact behind the theorem is the fact t h a t the only obstruction to extending up to quasi-isomorphism a perfect complex on the open subscheme to the full scheme is its class in A'o- The naive analogous statement for algebraic vector bundles is false, as shown long ago by Serre ([Se], Section 5, a)). Furthermore, the proof of this extension fact depends essentially on the very Grothendieckian idea that perfect complexes are finitely presented objects in the derived category. Of course, Grothendieck's ideas completely pervade modern m a t h e matics, and it would be a hopeless task to isolate and acknowledge all intellectual debts to him. But we hope our case illustrates t h a t despite their widespread influence, and nearly two decades after Grothendieck's withdrawal from public mathematical life, many of Grothendieck's ideas are still full of unexhausted potential, and will amply repay further development. Remarkably, his by now classic works can still surprise and instruct the serious reader. We dedicate this paper to him with profound admiration. T h e reader may find a brief sketch of the contents of this long paper useful. Section 1 recalls for the convenience of the reader the results of Waldhausen [W]. An expert might skip this section, but should glance at biWaldhausen categories 1.2.4 and 1.2.11 to allow dualization of arguments, the inductive construction of chain complexes 1.9.5, the fact t h a t A-theory is invariant under functors inducing equivalences of derived categories 1.9.8, and the cofinality theorem 1.10.1. Section 2 recalls the

HIGHER ALGEBRAIC K-THEORY OF SCHEMES

249

theory of perfect complexes on schemes from [SGA 6]. The expert might skip this, but should look at the characterization of perfect complexes as finitely presented objects 2.4, the fact that a complex with quasi-coherent cohomology on a nice scheme is a direct colimit of perfect complexes 2.3, and the basis for the excision theorem laid down in 2.6. Section 3 contains the definitions and basic functorialities of A'-theory. Section 4 contains the projective space bundle theorem. Section 5 proves the key extension result for perfect complexes, and contains the first crude form of the localization theorem. Section 6 proves the Bass fundamental theorem, and defines A'-groups also in negative degrees. Section 7 extends the previous results into negative degrees, putting them in their final form. In particular, Section 7 contains the good form of the localization theorem. This section gives the best quick summary of the fundamental results. The other basic results occur as consequences in Sections 8 - 1 1 . The appendices contain various results needed in the text, but which are not limited to Ar-theory. In particular, Appendix B merely summarizes from all points of EGA and SGA the relations between the categories of Ox~ modules and of quasi-coherent Ox-modules, as a help to the conscientious reader when he becomes as confused about this as we were. The paper should be comprehensible to anyone with a good first year graduate knowledge in algebraic geometry, and with a bit of algebraic topology. We must formulate our results in the language of spectra in the sense of topology but this may be picked up easily by skimming through [A] III Sections 1-6 (ignore any pointless examples involving baroque curiosities like "MU," "MSO," "MSpin," or the "Steenrod algebra"), and a glance at [Thl] Section 5 and A. This formulation in terms of spectra is much more powerful than the naive formulation in terms of disembodied abelian groups, and is not subject to certain unstable pathologies like the formulation in terms of spaces as in [Ql] and [W]. Indeed, the spectral formulation works just like Grothendieck's formulation of homological algebra in terms of the derived category, as explained in [Thl] Sections 5. To see the proofs of the results quoted in Section 1, the reader must see [W], although the sufficiently trusting need not. [Ql] is still good reading, although not necessary for this paper, except for the homotopy theory of categories of [Ql] Section 1. The first author must state that his coauthor and close friend, Tom Trobaugh, quite intelligent, singularly original, and inordinately generous, killed himself consequent to endogenous depression. Ninety-four days later, in my dream, Tom's simulacrum remarked, "The direct limit characterization of perfect complexes shows that they extend, just as one extends a coherent sheaf." Awaking with a start, I knew this idea had to be wrong, since some perfect complexes have a non-vanishing KQ obstruction to extension. I had worked on this problem for 3 years, and

250

THOMASON AND TROBAUGH

saw this approach to be hopeless. But Tom's simulacrum had been so insistent, I knew he wouldn't let me sleep undisturbed until I had worked out the argument and could point to the gap. This work quickly led to the key results of this paper. To Tom, I could have explained why he must be listed as a coauthor. During his lifetime, Tom also pointed out the interesting comparison of the careers of Grothendieck and Newton. For more mundance assistance and useful conversations, I would like to thank Gillet, Grayson, Karoubi, Kassel, Levine, Loday, Nisnevich, Ogle, Soule, Waldhausen, Weibel, and D. Yao. 1. Waldhausen /f-theory and iT-theory of derived categories 1.0. In this section we review some definitions and results of Waldhausen's framework for A'-theory, [W]. Our only claims to some originality in Section 1 are the general cofinality theorem 1.10.1 which is slightly different from previous results, and the results 1.9.5 and 1.9.8 which make it easier to apply Waldhausen's approximation theorem. 1.1.1. Let A be an abelian category. Consider chain complexes C in *4. We use the algebraic geometer's indexing, so differentials increase degree: d : Cn - + C n + 1 . Recall the standard notation ZkC — ker d : Ck —• C* + 1 , and k B C = im d:Ck~l — Ck. A complex C is (strictly) bounded above if there is an integer N such that Cn = 0 for all n > N. The category of bounded above complexes is denoted C~(A). A complex C is cohomologically bounded above if there is an N such that Hn(C) — 0 for all n > N. Dually for bounded below, C+(A), and cohomologically bounded below. A complex is bounded if it is bounded both above and below. The category of strict bounded complexes is Cb(A). A chain map / : C —* D' is a chain homotopy equivalence if there is a chain map g : D' —• C and chain homotopies fg~lr>, gf ~ lc- More generally, a chain map / : C —+ D' is a quasi-isomorphism if it induces an isomorphism on all cohomology groups H*(f) : H*(C) = H*{D'). For any integer m, a chain map / is an m-quasi-isomorphism if Hk(f) is an isomorphism for k > m and an epimorphism for k = m. The derived category D(A) (cf. [H], [V]) is formed from the category of all chain complexes in A by localizing this category of complexes so that precisely its quasi-isomorphisms become isomorphisms in D(A). The variant subcategories D~(A), D+{A), Db(A) are formed similarly from the categories of cohomologically bounded above, cohomologically bounded below, and cohomologically bounded complexes, respectively. D(A) ad-

HIGHER ALGEBRAIC K-THEORY OF SCHEMES

251

mits a 2-sided calculus of fractions as a localization of the chain homotopy category t h a t results from the category of complexes by identifying chain homotopic maps, as in 1.9.6 below, or in [V] I Section 2, [H] I Section 3. T h e additional structure of D(A) as a triangulated category results from the construction of homotopy pushouts and pullbacks, which we review next. 1.1.2. Let / : A' —• F' and g : A' —• G' be chain maps of complexes. T h e canonical homotopy pushout h

F'

UG' A'

is the complex given by

(1.1.2.1)

( V U G'J

= Fn 0 An+l

0 Gn

with differential, (1.1.2.2)

d(x,a,y)

= (dFx + fa,

-dAa>

dGy -

go).

(We describe d as if objects of A had "elements," by the standard abuse.) Chain maps from this canonical homotopy pushout to a complex C correspond bijectively to d a t a (h,k,H) where h : F' —• C and k : G' —• C are chain maps and H is a chain homotopy hf ~ kg : A' —• C*. T h u s /f consists of maps A n —» C n _ 1 for all n such t h a t dH+Hd = hf — kg. To (h,k,H) corresponds the map from the homotopy pushout to C sending (x, a, y) to hx + i / a -f A:y. Given another / ' : A'' —• iP*' y' : A #/ —• G*;, suppose there are maps a : A' —± A'', b : F ' —• F ' , c : G' —• G' ; and chain homotopies fa ~ 6/, g'a ~ eg. T h e maps a, 6, c, and choice of chain homotopies then determine a m a p of canonical homotopy pushouts F U G —• F ' U G', as one sees by the universal mapping property of the preceding paragraph. This map will be a quasi-isomorphism if each of the maps a, 6, c is a quasi-isomorphism. This last fact follows from the 5-lemma and the long exact sequence of cohomology groups (1.1.2.3) . . . A Hn(A)

-+ Hn(F')

0

tfn(G')

— Hn(F'

U G') - i i / n + 1 ( ^ - ) — . . . A*

which results from the short exact sequence of complexes (1.1.2.4)

O - ^ F e G ^ F U G ^ A'

J 4 ' [ 1 ] -> 0.

THOMASON AND TROBAUGH

252

Here A'[k] is the complex A' shifted in degree, so ^4*[&]n = Ak+n, and H (A[k]) = Hn+k(A'). Several special cases of the homotopy pushout construction are particularly important. When f : A' = F' is the identity map, the canonical homotopy pushout is the mapping cylinder of g : A' —• G\ considered in 1.3.4. When / : A' —• F' — 0 is the map to 0, the canonical homotopy pushout is the mapping cone of g : A' —• G\ There is a canonical map from the homotopy pushout to the (strict) pushout, induced by (x,a,y) i—• (x,y) mod A'. n

F U G ^ F U A'

G\

A'

This map is a quasi-isomorphism whenever An —• F n 0 G n is a monomorphism for all n, as is seen by the 5-lemma applied to the map of the long exact sequence 1.1.2.3 to its analog resulting from the short exact sequence of complexes 0 —• A' —• F' 0 G' —• F' U G' —> 0. A'

Dually, given f : F' —> A' and g : G' —» A' one has a canonical homotopy pullback

(1.1.2.5)

\

A J d(x,a,y) = (dFX, -dAa + fx-gy,

dGy).

This indeed corresponds to the homotopy pushout in the dual category of complexes in *4op, and so has all the dual properties. As special cases, dual to the mapping cylinder is the mapping path space, and dual to the mapping cone is the homotopy fibre. 1.1.3. Several standard truncation functors are useful. Let C be a complex. There is brutal truncation akC

= a^kC

>0-+0^Ck

=

-> Ck+1 — Ck+2 - • . . . .

This is a sub complex of C. The quotient C /akC truncation, denoted a-k~lC\ There is also the good truncation C'(-k) = rkC

= r^kC

=

is another brutal

• 0 -^ im dCk~l -H. Ck -* Ck+1

-•....

There is a quotient map C -» rkC which induces an isomorphism on cohomology Hn for all n > k. For n < k- 1, Hn(rC) = 0. The kernel of

HIGHER ALGEBRAIC K-THEORY OF SCHEMES C -+ rkC is denoted T^k~lC. For n < jfc-1, while Hn{r^k-lC) = 0 for n > fc.

ff^r**-1^)

253

= #n(C"),

1.2.1. Definition ([W]). A category with cofibrations A is a category with a zero object 0, together with a chosen subcategory co{A) satisfying the three axioms: 1.2.1.1. Any isomorphism in A is a morphism in co{A). 1.2.1.2. For every object A in A, the unique map 0 —• A is in co{A). 1.2.1.3. If A —• B is a m a p in co(^4), and yl —• C is a m a p in A, then the pushout B U C exists in J4, and the canonical map C —+ B U C is in CO(J4). In particular, A has finite coproducts. 1.2.2. One calls the morphisms in co(A) cofibrations. To distinguish them in diagrams, one usually denotes them by a feathered arrow ">—•." Given A >—• B, let the quotient B/A be the pushout B U 0 along A —• 0. One says t h a t yi >—• 5 —> C is a cofibration sequence if 5 —• C is the canonical m a p B —»• 5 / y l up to an isomorphism C =. B/A. One then says B - * C is a quotient map and denotes it by a double- headed arrow. Cofibration sequences are also called exact sequences. In general, the set of quotient maps need not be closed under composition. Suppose however for a given category with cofibrations A t h a t the set of quotient maps do form a subcategory quot(j4), and moreover t h a t the opposite category Aop is a category with cofibration co(Aop) = quot(j4) o p . Suppose further t h a t the canonical m a p AU B —+ A x B is always an isomorphism, where Ax B is the product in A, and the coproduct in Aop. Suppose also t h a t A —• B —» C is a cofibration sequence in A iff the dual sequence A <— B <— C is a cofibration sequence in Aop. Under these conditions, one says A is a category with bifibrations. This concept is self dual, so Aop is then a category with bifibrations. Note t h a t in a category with bifibrations, given a cofibration sequence A >-+ B -» C, t h a t A is the quotient of B by C in Aop, so dually A must be the kernel of B - * C in A 1.2.3. Definition. A Waldhausen category (in [W], "a category with cofibrations and weak equivalences") is a category with cofibrations A, co(A), together with a subcategory w(A) of A satisfying the two axioms: 1.2.3.1. Any isomorphism in A is a morphism in w(A). 1.2.3.2. ("gluing lemma") Given a commutative diagram in A

THOMASON AND TROBAUGH

254

B*-< A^

u

C

u u

B' +-C with the two maps A >-+ B) A! >-• 5 ' being cofibrations, and with the three maps A -^ A', B -^ B', and C -^ C being in «^>l), then the induced map 5 U C —• 5 ' U C" is also in w(A). A

A'

The Waldhausen category consists of the triple data A, co(A), w(A), but one usually abbreviates it as A, or as wA when the choice of w(A) is particularly important. One says the maps in w(A) are "weak equivalences," and denotes them by arrows with tildes "- : V. 1.2.4. Definition. A biWaldhausen category is a category with bifibrations A, CO(J4), quot(A), together with a subcategory w(A) such that both (A, co(A), w(A)) and the dual (A0? quot(^)°P, w(A)°v) are Waldhausen categories. That is, 1.2.3.1, 1.2.3.2, and the dual of 1.2.3.2 concerning pullbacks with A <«- B and A' <<- B' being in quot(^4) all hold in the category with bifibrations A. This concept is self-dual, in that if A is a biWaldhausen category, so is 1.2.5. Definition. A saturated Waldhausen or biWaldhausen category is one where w(A) satisfies the saturation axiom: Given A —• B —• C composible morphisms in A, if any two of a, 6, 6a are in w(A), then so is the third. 1.2.6. Definition. An extensional Waldhausen or biWaldhausen category is one that satisfies the extension axiom: Given a commutative diagram whose rows are cofibration sequences

A^ B-» C {a [b [c A1 y^B' — C if both a and c are in w(A), then so is 6. 1.2.7. Definition. A functor F : A —> B between two Waldhausen categories is exact if F(co(A)) C co(U), if F(w(A)) C 10(B), and if F preserves pushouts along a cofibration. The last condition means that the canonical map FC U FB —• F(C U B) is an isomorphism whenever A y-• B is in co(A).

HIGHER ALGEBRAIC K-THEORY OF SCHEMES

255

A functor F : A —• B between two biWaldhausen categories is exact if both F : (A co(A), W(A)) — (fl, co(B),w(B)) and the dual F ° P : (A op , quot(A)°v ,w(A)op) -» (B°P, quot(B)°P,w(B)0P) are exact functors between Waldhausen categories. This is equivalent to saying that F : (A, co(A),ti;(A)) —• (B, co(B),w(B)) is exact and F preserves pullbacks where one of the maps is a quotient map. 1.2.8. Any category with cofibrations A becomes a Waldhausen category by taking w(A) to have as morphisms all the isomorphisms in A. Henceforth, we identify all categories with cofibrations to Waldhausen categories in this way. 1.2.9. Example. Let A be an abelian category (or more generally an exact category in the sense of Quillen [Ql]). Let co(A) consist of all monomorphisms in A (in the exact category case, let co(A) consist of all admissible monomorphisms [Ql]). Let w(A) consist of all isomorphisms in A. Then A is a Waldhausen category, and in fact a biWaldhausen category. 1.2.10. Example (optional). Let A be the category of simplicial sets. Let co(A) consist of all monomorphisms. Let w(A) consist of all "weak equivalences," i.e., all maps that induce homotopy equivalences between geometric realizations. Then A is a Waldhausen category. 1.2.11. Definition. A complicial biWaldhausen category is a saturated extensional biWaldhausen category A formed from a category of chain complexes as follows: One takes an abelian category A, whose choice is part of the structure. A is to be a full additive subcategory of the category C(A) of chain complexes in A. co(A) is to contain at least all maps of complexes in A that are degree-wise split monomorphisms such that the quotient chain complex lies in A. That is, co(A) contains all maps C —• D' in A such that for all integers n the map Cn —• Dn is a split monomorphism in A and such that moreover the quotient chain complex D'/C in C{A) is also isomorphic to a complex in A. co(A) may possibly contain other maps. One does require that if C —> D' is in co(A), then for all n, Cn —* Dn is a monomorphism in A, but not necessarily split. Of course co(A) and the corresponding quot(A) must satisfy axiom 1.2.1.3 and its corresponding dual. We also demand that the pushouts and pullbacks required in A by these axioms are also the pushouts and pullbacks in the category of C{A)\ i.e., that A is a subcategory closed under the required pushouts and pullbacks. w(A) is to contain all maps in A which are quasi-isomorphisms in the full category of complexes in A. w(A) may contain other morphisms. Of course w(A) must satisfy the usual axioms 1.2.3.2 and its dual 1.2.4, and also the saturation and extension axioms 1.2.5 and 1.2.6.

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In specifying a complicial biWaldhausen category, we make the default convention that unless explicitly specified otherwise, w(A) is to consist of exactly the quasi-isomorphisms, and that CO(J4) is to consist of exactly those degree-wise split monomorphisms whose cokernel is in A. 1.2.12. Example. Let £ be an exact category [Ql]. Then there is an abelian category A and a fully-faithful Gabriel-Quillen embedding £ —• A, reflecting exactness and such that £ is closed under extensions in A, (cf Appendix A). Let A be the full subcategory of complexes C" in A with each Ck in £, and with Ck = 0 unless k = 0. Take the cofibrations to be admissible monomorphisms, and the weak equivalences to be the quasi-isomorphisms. As H°(C) = C° for C in A = £, these weak equivalences are just the isomorphisms in £. This A is a complicial biWaldhausen category, which in fact is the biWaldhausen category of 1.2.9. A more interesting example would be take A the category of complexes in A which are degree-wise in £. See 1.11.6 below. 1.2.13. Example. Let A be an abelian category, and let A be the category of all chain complexes in A. Then with the default conventions for co(>l) and w(A), A is a complicial biWaldhausen category. There are variants where A consists of the bounded complexes, the bounded above complexes, the cohomologically bounded complexes, etc.. 1.2.14. Example. Let A be an abelian category, and let A be the category of all bounded below complexes C in A such that each Ck is an injective object of A. Then A is complicial biWaldhausen. 1.2.15. Example. Let A be an abelian category with a thick abelian subcategory B. Let A be the category of complexes C in A such that C is cohomologically bounded and with all cohomology groups Hk(C) in B. Then A is complicial biWaldhausen. 1.2.16. Definition. Let A, B be complicial biWaldhausen categories. A complicial exact functor F : A —• B is an exact functor of biWaldhausen categories in the sense of 1.2.7, with the additional property that the functor F of complexes is induced by degree-wise application of some additive functor / : A —• B between the abelian categories chosen as part of the complicial structure. Hence F(C) = • f(Ck) — f(Ck+1) - . . . . 1.3.1. Definition ([W] 1.6). Let A be a Waldhausen category. A cylinder functor T on A is a functor T : Cat(l, J4) —• A to A from the category of morphisms in A, together with three natural transformations p, j j , jf2,

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satisfying the conditions below. Thus to each morphism / : A —* B in A, T assigns an object Tf of A. To each commutative square (1.3.1.1) in A

B

I'

(1.3.1.1) f

A'

B'

T assigns functorially a morphism T{a, b) : Tf —• Tf. The natural transformations are maps j \ : A —• T / , J2 : B -+ Tf} and p : T / —• B such that pji = / : A —• B, pj'2 = 1 : B —• B, and such that (1.3.1.2) commutes. ;'iU; 2

AUB (1.3.1.2)

p

——-+ T/

T(a 6)

• B

l ' J6

aUS

A'UB'

T/'

B'

J1UJ2

We also require conditions 1.3.1.3 - 1.3.1.6 to hold. 1.3.1.3. ii U j 2 : , 4 U £ > - > T / i s in co(A). 1.3.1.4. If a and 6 are in w(A), then T(a,6) is in w(A). 1.3.1.5. If a and b are in co(^4), then not only is T(a,6) in co(i4), but also the map Tf U A'UB' - • Tf induced by the left square of (1.3.1.2) is in co(A). 1.3.1.6. T(0 —• ^4) = A, with p and 22 the identity map. To define a "cylinder functor satisfying the cylinder axiom," one imposes the extra cylinder axiom: 1.3.1.7. For all / , p :Tf -+ B is in w(A). (Note our 1.3.1.3 - 1.3.1.5 are equivalent to [W] 1.6 "Cyl 1"). 1.3.2. Definition. If A is a biWaldhausen category, a cocylinder functor or mapping path space functor is a functor M : Cat(l,>l) —» A and natural transformations

M(f : A - B) (1.3.2.1)

h A

/

\ * i

B

\<7 A

THOMASON AND TROBAUGH

258

such t h a t M is a cylinder functor on the dual (A o p , quot(j4) o p , w(A)op). M satisfies the cocylinder axiom if the dual of 1.3.1.7 holds, i.e., if q : A —• M(A —• B) is always a weak equivalence. 1.3.3. Example (optional). T h e usual mapping cylinder of algebraic topology is a cylinder functor satisfying the cylinder axiom in the Waldhausen category of simplicial sets 1.2.10. 1.3.4. Example. Let A be an abelian category, and A the biWaldhausen category of all chain complexes in A. Then A has well-known cylinder and cocylinder functors satisfying the cylinder and cocylinder axioms respectively. For given / : A —• 5 , let Tf be the canonical homoh

topy pushout A [J B of I : A —> A and / : A —• B constructed as in 1.1.2. A

h

T h e maps j \ : A —> AUB A

ji(a)

h

and j2 : B —• AUB A

are the canonical inclusions

= (a, 0,0), J2(b) = (0, 0, b). T h e map p : J 4 U B —• B is the m o r p h i s m

induced by / : A —• £ , 1 : 5 — » 5 , and the trivial homotopy / l ~ 1 / so t h a t p ( a , a ' , 6 ) = fa + b. Dually, the canonical homotopy pullback A x JB provides a cocylinder. 1.3.5. Example. Let i4 be a complicial biWaldhausen category with associated chosen abelian category A. Suppose that those canonical homotopy pullbacks and canonical homotopy pushouts, formed in the category of complexes in A starting from diagrams in A, are in fact objects of A Then we claim that the mapping cylinder and cocylinder functors of 1.3.4 induce mapping cylinder and cocylinder functors on the subcategory A, provided only that the cofibration axiom 1.3.1.5 and its dual hold. Moreover, the cylinder axiom 1.3.1.7 and its dual cocylinder axiom hold automatically. Note 1.3.1.3 holds automatically, as J1UJ2 is a degree-wise split monomorphism whose cokernel is the homotopy pushout of A —> 0 along A —• 0, and hence in A. Thus j \ U ji i s m CO(J4). A S p is a quasi-isomorphism, it is in w(A) and so 1.3.1.7 holds. Axiom 1.3.1.4 now follows from saturation 1.2.5, and axiom 1.3.1.6 is trivial. This leaves only 1.3.1.5 in doubt, proving the claim. To verify 1.3.1.5 it suffices to show Tf U A' U B' -+ Tf is in c o M ) , v AUB ' for the canonical m a p of Tf into Tf U A! U B' is a cofibration by 1.2.1.3. AUB

Hence if the first map is a cofibration so is the composite Tf —• Tf. T h e map Tf U A' U B' —* Tf is given degree-wise as a sum of An+l —• A'n+1 with identity maps

(1.3.5.1)

A'n e A n + 1 e B'n -> A'n e A'n+1 e B'n.

HIGHER ALGEBRAIC K-THEORY OF SCHEMES

259

If co(A) consists exactly of the degree-wise split monomorphisms whose quotients lie in A, or else consists of all monomorphisms whose quotients lie in A then 1.3.1.5 and its dual also hold automatically. This also works if co(A) is defined to be all maps that are degree-wise admissible monomorphisms whose quotients lie in A for some exact subcategory £ C A that contains An for all A' in A and all degrees n. Thus in all the usual examples, and in fact in all cases arising in Sections 2-11, axiom 1.3.1.5 also holds automatically. 1.3.6. Example. Let A be a complicial biWaldhausen category with associated abelian category A. Suppose co(A) is one of the cases listed in 1.3.5 that make 1.3.1.5 automatic, e.g., all degree-wise admissible monomorphisms with quotients lying in A. Suppose A is closed under finite degree shifts so that if A' is in A, so is A'[k], Suppose A is closed under extensions, i.e., that if 0 —• A' —• B' —*• C —• 0 is an exact sequence of complexes in A with A' and C in A then B' is isomorphic to a complex in A. Then A has a mapping cylinder and cocylinder satisfying the cylinder and cocylinder axioms. This will follow from 1.3.5 once we see A is closed under formation of canonical homotopy pushouts and canonical homotopy pullbacks. But this follows from the hypotheses that A is closed under extensions and degree shifts and the exact sequence (1.1.2.4) and is dual. 1.4. We henceforth consider only small Waldhausen categories, those with a set, as opposed to a class of morphisms. Hence, when we speak of a Waldhausen category of all chain complexes of abelian groups or of (9x~modules on a scheme, it is implicit that we are looking at the category of such complexes in a Grothendieck universe so that it is small with respect to a larger universe [SGA 4] I Appendice. (As in [SGA 4], the A'-theory spectrum of the biWaldhausen categories of Section 2 and Section 3 will be independent of the choice of universe at least up to homotopy, as these biWaldhausen categories in the various universes will have equivalent derived categories, so 1.9.8 applies. See Appendix F). 1.5.1. Definition [W]. For A a Waldhausen category with weak equivalences w = w(A) define wS.A to be the following simplicial category. The objects of the category in degree n, wSnA are the functors A, meeting the conditions below, to J4 from the partially ordered set of pairs of integers (i,j), with 0 < i < j < n. The partial order is defined by (hj) < ( 2 'ii') ^ both i < i' and j < j ' . The functors A must meet the conditions that for all j , A(j,j) = 0, and that for all (ij.k) with 1 < J< &> the maps A(i,j) >—• A(i, k) -* A(j, k) form a cofibration sequence. The morphisms of wSn(A) are the natural transformations A —• A!

260

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such that for all (i>j), A(i,j) —• A ; (i, j) is in w(A). By the gluing Lemma 1.2.3.2 and the cofibration sequence condition on A for 0 < i < j , it suffices that all ,4(0, k) —• ,4'(0, k) are in w(A). Given

- • . . . > - • An — .4(0,1) >-• .4(0,2) >-• •••>-> ,4(0,n). Indeed A(z,j) is A(0,j)/A(0,i) == Aj/A{ up to canonical natural isomorphism, so specifying the A(i,j) for i ^ 0 just adds choices of objects determined up to isomorphism. However, it is necessary to specify these choices to make the simplicial identities hold strictly in wS.A, instead of just up to natural isomorphism. 1.5.2. For A a small Waldhausen category, taking the nerve in each degree of the simplicial category wS.A yields a bisimplicial set NwS,A. Waldhausen defines the A'-theory space of A to be the loops on the geometric realization of this bisimplicial set, K{A) — Sl\NwSmA\. One also denotes this K(wA) when it is important to distinguish among several possible choices of weak equivalences. The A'-groups of A are the homotopy groups of K(A). This space K(A) is in fact an "infinite loop space" by [W] 1.3.3 and 1.5.3, as Waldhausen shows that it is the zero-th space of a spectrum, i.e., of a sequence of spaces each of which is homotopy equivalent by a given map to the loops on the next space in the sequence. It is in fact better to work with this spectrum than with the space. The proofs of [W] (and of [Ql], [Grl], etc.) immediately generalize to give "infinite loop space" versions of its results which are then valid for K(A) as a spectrum. 1.5.3. Definition. For A a small Waldhausen category with weak equivalences w, define K(A) = K(wA) to be the spectrum constructed from A by the process of [W] 1.3.3 and remark, 1.5.3, and whose 0th space is Sl\NwSA\. Define the Waldhausen A-groups I\n(wA) to be the homotopy groups of the spectrum, 7TnK(wA). Note these groups are 0 if n < — 1, and are isomorphic to the homotopy groups of the space Q\NwSA\ for n > 0. 1.5.4. An exact functor F : A —• B induces a simplicial functor wSA —> wS.B, and a map of spectra KF : K(A) —• K(B). This makes K a functor. If 7] : F —• G is a natural transformation of exact functors A —• B, and if for all objects A in A, r]A : FA -^ GA is in w(B), then rj induces a homotopy wS.F ~ wS.G, and in fact a homotopy of maps of spectra

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261

KF ~ KG. See [W] for details, and [Th3] A for homotopies of maps of spectra. 1.5.5. If A is a biWaldhausen category, we define K(A) using the underlying Waldhausen category (Ay co(A)} w(A)). But since A —• B —» C is a cofibration sequence in A iff C —* B —• A is a cofibration sequence in Aop, there is a canonical isomorphism of simplicial categories (wSA)op = wS,(Aop). From the canonical isomorphism between the classifying space \N,( )| of a category and of its dual ([Ql] Section 1(3)), we deduce a canonical duality isomorphism of spectra K(A) = K(Aop). This allows us to dualize all theorems of [W] when applied to biWaldhausen categories. 1.5.6. It is easy to derive the following formula for Ko(wA) from the edge-path group presentation of KQ(WA) = 7TQQ\NWS.A\ = ni\NwSmA\. KQ(WA) is the free group (or the free abelian groups) on generators [A] as A runs over the objects of A, modulo the two relations 1.5.6.1. [A] = [B] if there is a map A -=• B in w(A). 1.5.6.2. [B] = [A][B/A] for all cofibration sequences A >-> B -» B/A. Note that relation 1.5.6.2 applied to A y-> A U B -» B and B >—• AUB -* A forces [A][B] = [AUB] = [B][A]. Thus K0(wA) is abelian and we usually write the relation additively: [B] = [A] + [B/A]. (Also 1.5.6.2 forces [0] = 0.) 1.5.7. If A has a mapping cylinder satisfying the cylinder axiom, let TiA be the cone of A —• 0. Thus A >-• T(A —> 0) ->• XM is a cofibration sequence. As T(A —»- 0) ~ 0, it follows that -[A] = [EA] in I—• C -» B, which have A in A and B in B. The morphism of E(A,C,B) are commutative diagrams in C A>^ C — B A'y->C'

-»B'

THOMASON AND TROBAUGH

262

with A —» A' a morphism in A and B —* B' a morphism in B . Such a morphism in E(A,C,B) is a cofibration if A —• A', J5 —• 5 ' , and i ' U C - ^ C are cofibrations in A, B , and C respectively. Note then t h a t A

C —• C is a cofibration in C, as it is the composite AUC >~• A ' U C >-> C". A morphism in E(A,C,B) is a weak equivalence if A —• A', B -+ B', and C —* C1 are weak equivalences in J4, 5 and C respectively. This gives E(AJC,B) the structure of a Waldhausen category. T h e r e are exact functors s, t, q from E(A,C,B) to .A, C, and 5 respectively, sending A >-* C -» 5 to J4, C, and 5 respectively. There is also an exact functor U:AxB^ E(A,C,B) sending (A,B) to A>-+AUB-» B. This functor splits (s, q) : E{A,C,B) -+AxB. 1.7.2. A d d i t i v i t y T h e o r e m ( [ W ] 1.3.2, 1.4.2). Taice the tion and make the hypotheses of 1.7.1. Then the exact functors induce a homotopy equivalence of K- theory spectra K(s,q) A natural homotopy E{A,C,B). Proof.

: K(E(A,C,B))

^

K(A)

x

nota(s,q)

K(B).

inverse to this map is K(U) induced by U : AxB

—*

See [W] 1.3.2, Section 1.4.

1.7.3. C o r o l l a r y ([W] 1.3.2(4)). Let A and B be small Waldhausen categories, and let F, F', F" : A —+ B be three exact functors. Suppose there are natural transformations F' —• F and F —• F" such that the following two conditions hold: 1.7.3.1. For all A in A} F'A >-+ FA -» F"A is a cofibration 1.7.3.2. For any cofibration A' F'A U FA' -+ FA is a cofibration.

>—• A

in A,

sequence.

the induced

map

F'A'

Then there is a homotopy K{A)-+K{B)-

of maps of spectra

KF

~ KF'

-f KF"

:

Proof. The natural cofibration sequence F' >-+ F -» F" induces an exact functor A —• E(B,B,B). The additivity theorem 1.7.2 implies a homotopy Kt ~ Ks + Kq of maps K(E(B,B,B)) -+ K(B), since these maps become equal after composing with the homotopy equivalence K(U). Composing the homotopy Kt ~ Ks -f Kq with the map K(A) -> K(E(B,B,B)) yields a homotopy KF ~ KF' + KF". 1.7.4. When B is a complicial biWaldhausen category, hypothesis 1.7.3.2 is superfluous as it follows automatically from 1.7.3.1 and exactness of F ' , F, and F". For one notes t h a t F'A

U FA'/F'A^O

F'A1

U FA'* F'A'

FA'/F'A'* '

F"A'.

HIGHER ALGEBRAIC K-THEORY OF SCHEMES

263

Thus the diagram below has cofibration sequences as rows F'A

>-+ F'A U FA'

I F'A

-»

F"A'

-»

F"A

I

i >^

FA

F"(A/A') As B is biWaldhausen, the composite map FA -* F"(A/A') is in quot(#), and its kernel is a cofibration into FA. But as B is complicial, this kernel is the same as the kernel taken in the category of chain complexes in the associated abelian category B. Applying the snake lemma to the above diagram in the category of chain complexes, we see that the kernel is F'A U FA' —• FA, and so this map is a cofibration as required F'A1

by 1.7.3.2. 1.8.1. Let A be a small category with cofibrations. Suppose A has two subcategories v(A) and w(A)) each of which is the category of weak equivalences for a Waldhausen category structure on A, vA and wA. Suppose v(A) C u?(A), and that wA satisfies the extension and saturation axioms. Let Aw be the full subcategory of A whose objects are the A such that 0 —• A is in w(A), i.e., which are t^acyclic. This Aw becomes a Waldhausen category vAw with c o ( ^ ) = CO(J4) nAw and v(Aw) = v(A) H Aw. If vA and wA are biWaldhausen, so is vAw. If A has a functor T which is a cylinder functor both for vA and for wA, T induces a cylinder functor onAw. 1.8.2. Localization T h e o r e m ([W] 1.6.4 Fibration Theorem). With the notation and hypotheses of 1.8.1, suppose also that A has a functor T which is a cylinder functor both for vA and forwA, and that T satisfies the cylinder axiom 1.3.1.7 for wA. Then the exact inclusion functors vAw —> vA, vA —+ wA, induce a homotopy fibre sequence of spectra K(vAw) — K(vA) -+ K(wA). (The requisite chosen nullhomotopy ofK(vAw) —• K(wA) is induced by the natural weak equivalence 0 -^ A in wA for A in vAw.) Proof.

[W] 1.6.4.

1.9.1. A p p r o x i m a t i o n T h e o r e m ([W] 1.6.7). Let A and B be small saturated Waldhausen categories. Suppose A has a cylinder functor satisfying the cylinder axiom 1.3.1.7. Let F : A—> B be an exact functor satisfying the two conditions: 1.9.1.1. A morphism f of A is in w(A) if and only if Ff is in w(B).

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T H O M A S O N AND T R O B A U G H

1.9.1.2. Given any A in A and any x : FA —> B in B, there is an A' in A, a map a : A —+ A! in A, and a weak equivalence x' : FA' - ^ B in w(B) such that x = x' o Fa. Then under these conditions, F induces a homotopy equivalence KF : K(A) K(B). P r o o f . This results from Waldhausen's version [W] 1.6.7. Condition 1.9.1.2 appears to be weaker than the corresponding condition "App 2" of [W] in t h a t 1.9.1.2 does not require the map a : A —» A' to be a cofibration. But given x = x' o Fa as in 1.9.1.2, one applies the cylinder functor to a : A —+ A' to factor a = a" o a', with a! the cofibration A >-+ A" = T ( a ) , and a" the weak equivalence A!1 = T ( a ) -=+ A 7 . T h e n z " — x' o Fa" : FA" -^ J5 is a weak equivalence, a' : A >-> A " is a cofibration, and x = x" o F a ' . Hence 1.9.1.2 implies "App 2" of [W] in the presence of the other hypotheses. 1.9.2.Theorem. Let A be a small complicial biWaldhausen category. Let A* be the new complicial biWaldhausen structure on A where w(Af) = w(A), but where co(Af) consists exactly of those degree-wise split monomorphisms whose quotient lies in A. Suppose that A' has a cylinder functor satisfying the cylinder axiom, as often occurs (cf. 1.3.5, 1.3.6). Then the exact inclusion functor A' —• A induces a homotopy equivalence K[A!) ^ K(A). P r o o f . Apply 1.9.1 to the inclusion A —» A. Condition 1.9.1.1 is obvious, and 1.9.1.2 holds trivially with a = x, x' — 1. Something much like 1.9.2 was proved in [HS] by Hinich and Shekhtman before the unveiling of Waldhausen's approximation theorem. 1.9.2.1 Remark. Theorem 1.9.2 shows it is usually harmless in Ktheory to impose the condition t h a t co(A) consists of precisely the degreewise split monomorphisms with quotients lying in A, at least in the presence of cylinders. This is convenient, since then any complicial functor F : A—> B induced by an additive / : A —* B automatically will preserve cofibrations and pushouts along cofibrations. 1.9.3. In applications of the approximation theorem, most of the effort involved is expended in verifying condition 1.9.1.2. T h e following lemmas 1.9.4 and 1.9.5 are useful tools for this, and also for some other purposes. Lemma 1.9.5 serves to build bounded above complexes quasi-isomorphic t o a given complex, and lemma 1.9.4 serves to truncate them to strict bounded complexes.

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265

1.9.4. L e m m a . Let A be an abelian category, and let B be a full additive subcategory of A. Suppose that B is closed under extensions in Aj and is closed under kernels of epimorphisms. (More precisely, B is to be closed under taking kernels of maps in B that are epimorphisms in A.) Let C be a strictly bounded above complex in B C A. Then: 1.9.4(a). If for an integer n, one has Hk(C') = 0 for k ^ n and Ck = 0 for k n, then n Z C is an object of B, and the complex C is quasi-isomorphic to the subcomplex r-nC, which is the complex in B > Cn~2

-> Cn~l

-+ ZnC

-> 0 -+ 0 -> • • • .

1.9.4(c). If for an integer n, one has Hk(C) H (C) has a resolution by objects of B

= 0 for k ^ n, then

n

> (jn-2 _^ Cn-l IfC

_> znC

_^ Hn{C)

-+ 0.

is also strictly bounded below, this resolution has Unite length.

1.9.4(d). IfC is an acyclic complex, so Hk(C) = 0 for all k, then all Z C = BkC are objects of B, and C has a natural filtration by acyclic complexes in B, FnC = r-nC, so that Fn+\C'/FnC is isomorphic to the acyclic complex k

> 0 -+ 0 -» 5 n + 1 C £ Bn+1C

-+ 0 -* 0 - • . . . .

Proof. As C is strict bounded above, there is an integer N such that Cp = 0 for p > N. If N < n, ZnC = Cn and 1.9.4(b) is obvious. One proceeds to prove 1.9.4(b) by induction on N — n. If it is already known to be true for smaller values o£ N — n, and N — n > 1, consider C = .. — C ^ " 1 — C * -> 0 — •••. As TV > n, HN(C) = 0 and c^-i - cN is an epimorphism in A of objects in B. AS B is closed under taking kernels of such epimorphisms, the kernel ZN~lC is in B. Then C is quasi-isomorphic to the shorter subcomplex C = r^N-lC =

• CN~2 -* ZN~1C

-> 0 - 0 -+ • • •

which is also a complex in B. Clearly ZkC = ZkC for all k < N — 1, and as C' is shorter we get ZkC' is in B for all k > n by the induction hypotheses. Thus ZkC is in B for n < k < N — 1. We have that ZNC = CN is in B, and clearly ZkC = 0 is in B for k > TV. Thus ZkC is in B for all k > n, completing the induction step, and hence the proof of 1.9.4(b). Now 1.9.4(a) and 1.9.4(c) are immediate corollaries and 1.9.4(d) is a porism (i.e., follows from the proof of 1.9.4(b)).

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T H O M A S O N AND T R O B A U G H

1.9.5. L e m m a (cf. [SGA 6] I 1.4). Let A be an abelian category, let V be an additive category, and let F : V —> A be an additive functor. Let C be a full subcategory of the category C(A) of chain complexes in A, such that any complex quasi-isomorphic to a complex in C is also in C. Suppose that every complex in C is cohomologically bounded above. Suppose that if D' is any strict bounded complex in V, the F(D') is in C, and that C contains the mapping cone of any map of complexes F(D') -»> C with C in C and D' strict bounded in V. Suppose the key condition 1.9.5.1 holds, so "D has enough objects to resolve": 1.9.5.1. For any integer n, any C in C such that Hl{C') — 0 for i > n, and any epimorphism in A, A -» Hn~l{C), then there exists a D in V and a map FD —• A such that the composite FD -» Hn~l(C) is an epimorphism in A. Suppose finally that the functor F : V —+ A satisfies the following two conditions (which trivially hold in the usual case where F is a fully faithful inclusion): 1.9.5.2. Given a morphism f : FD2 —• FD\ in A, there is a map d : D3 —> D2 in V such that Fd : FD% —• FD2 is an epimorphism in A and such that f o Fd = F f for some map f'.Dz—* D\ in V. 1.9.5.3. Given h : D2 —+ D\ in V with Fh — 0, there is a map d : D3 —+ D2 in V with Fd : FD3 —• FD2 an epimorphism in A and hd = 0 inV. Then: For any D' in C~(V) with F(D') in C, any C in C, and any map x : FD' - • C', there exists a D' in C~(V) with F{D') in C, a degree-wise split monomorphism a : D' —+ D'', and a quasiisomorphism x' : FD'' -=• C such that x = x' o Fa. Moreover, if x : FD' —» C is an n-quasi-isomorphism for some integer n (i.e., Hl(x) is iso for i > n and epi for i = n), then one may choose D1' above so that a : Dk —• D'k is an isomorphism for k > n. Proof. We construct D1' by induction, constructing what will become the brutal truncation n and an epimorphism for i = n. This is possible because FD' and C are cohomologically bounded above, so Hl(FD') = 0 = Hl(C) for i » 0. n f n For this large n, we set
HIGHER ALGEBRAIC K-THEORY OF SCHEMES

267

1.9.5.4.2.
AT =

• Cn~3 -> Cn~2 -> FZ3 /n 0 C n _ 1 -+FD/n+10Cn-+....

Then M' is in C. The long exact cohomology sequence for the mapping cone and the fact that crnx' is an n-quasi-isomorphism yield that Hl(M') — 0 for i > n. By the key hypothesis 1.9.5.1, there is a D ^ i n V and a map FD~-+ Zn~lM' -* Hn~\M') so that FD~-» Hn~l{M') n_1 ,n n_1 is an epimorphism. As M = FD 0 C , and (1.1.2.2) shows that the differential is given by d(d,c) = (-dd, dc - (anx')(d)) in Mn = n l = ker 5 is the fibre prodFD>n+i 0 Cn^ o n e e a s i l y c h e c k s t h a t z ~ M' n l n n uct of d : C ~ - • Z C and o^a?' : Z FanD'' -+ ZnC (1.9.5.7)

Zn~lM'

= ZnFanDf'

x

C""1 c r o

m

®Cn'1.

To simplify notation we write ZnFDf' for ZnFanD'\ as if the rest of D ' already existed. Consider now the composite map induced by the canonical projection and inclusion maps 1

FD~-+ Zn~lM'

£ ZnFD''

x

Cn'1

-> Z n F£>'' ->

FD,n.

By 1.9.5.2, there is a map L>~ -+ D^with FD~ -» FD~-» Hn~l{M') fn an epimorphism and with the map FD~ —*• FD^-+ FD being F of a map D ~ —» D / n . Replacing the old D ^ b y £)~, we may assume that FD~-^ FD'n is F of a map D~-> D,n. As FD~-+ FD'n factors through ZnFD'\ the composite F D ^ - * FL>'n -+ FL> ,n+1 is 0. By 1.9.5.3 there is a D~ - • D ^ i n £> so that FD~ -* FD~-» Hn~l{M') is still an epimorphism, and such that the composite Z)~ —• D~-+ D'n —• JT)/n+1 is 0. Replacing the old D^with this J9~, we may assume that D^—> D'n factors through Zn£>'* (= Zn^-^ £>/n on the summand D* As D " - ^ D'n factors through ZnD'\ and as dad - ad2 = 0 on D n _ 1 , we see that the composite d2 : D,n~l -+ L>/n ->

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jj/n+i - s Q JSJOW j e t an-ijj/> k e the chain complex formed from anD'' by replacing the old 0 in degree n — 1 by D'n~l, and with 3 : D,n~l —• Dfn as the new boundary operator here. Let the m a p an~1(a) agree with cr n (a) in degrees above n — 1, and to be the inclusion of the s u m m a n d £ ) * - i _ Dn~l 0 D ^ D ' " " 1 in degree n - 1. Let the m a p c r " - 1 ^ ' ) agree with c n (a:') in degrees above n — 1, and to be given in degree n - 1 on FD,n-1 S F D ^ 1 e F D ^ a s x " " 1 : F D " ' 1 -+ Cn~l on the n l s u m m a n d FD ~ and on the summand F £ ) ~ a s the composite of the m a p FD~-+ Zn~lM> and the projection Zn"lM' -> C n _ 1 determined n 1 by the isomorphism 1.9.5.7 of Z ~ M' with a fibre product. It is easy to verify t h a t an~1(xf) and an~1(a) are chain maps and t h a t 1.9.5.4.1 and 1.9.5.4.2 hold with n — 1 in place of n. To verify condition 1.9.5.4.3 that an~l(x') is an (n—l)-quasi-isomorphism, consider its mapping cone M*. This is the mapping cone M' of crn(x') with an additional term FDn~l 0 FD~= F{Dln~l) added: (1.9^5.8) M' =

> C n ~ 3 -> F D " " 1 0 FD~®

Cn~2

-+ FD'n

0 Cn~l

— •••

By construction the boundary map from the summand FD^m Mn~2 n l n 1 n l m a p s onto H ~ {M') = ^ - ^ M • ) / B - ( A f ) . Hence FD~®B - (M') n 2 n 2 1 and a / o r t u m FD~ ® M " = FD^ ® C ~ and ^ D " " 0 F £ > ~ 0 C n - 2 limp onto Z n - 1 A f = ^ n ~ 1 M ' . Then Hn-l(M') = 0, as well as Hl(M') = Hl(M') — 0 for i > n. Now the long exact cohomology sequence for the mapping cone M' of cr n "" 1 (x / ) shows t h a t an~l(x') is an (n — l)-quasi-isomorphism as required. This completes the induction step. Now given the inductively constructed anD'' for all n, we set D'm — \imo-nD'' as n —• —oo. As anDf' and an~1Df' agree in degrees above n — 1, we have D'k — (anD')k for any n < k. We define a, x' similarly. It is then clear from 1.9.5.4 that Df\ a, and x' meet the requirements, completing the proof of Lemma 1.9.5. 1.9.5.9. Porism. If 1.9.5.1 holds only for those n > N + 1 for some fixed N, the proof still constructs a aN D1' in Cb(V), and A r -quasi-isomorphism aN(x') : F(aND1') —*• C", and a degree-wise split monomorphism (TN(a) : (T^D' — ^ ( Z T ) such that * * ( * ) = 2 so t h a t the inclusion V\-* A satisfies 1.9.5.1 for n > N + 1 and V2 —» A satisfies 1.9.5.1 for all n, then the proof constructs a quasi-isomorphism x' : D1' —• C with Dlk in X>i for k > N and D,k in T>2 for all k. Moreover, if x : D' —> C is given with D ' G C~(T>2) and with D* in P i for k > N, then we may construct the quasi-isomorphism x' : D'' ^ C so t h a t there is a degree-wise split monomorphism a : D' —* Df' with a:' = a: o a.

HIGHER ALGEBRAIC K-THEORY OF SCHEMES

269

These variants of Lemma 1.9.5 will come to seem less bizarre in Section 2.2 below. 1.9.6. Let A be a complicial biWaldhausen category with associated abelian category A. So A is a full subcategory of the category of chain complexes C(A). Suppose A is closed under the formation of canonical homotopy pushouts and canonical homotopy pullbacks in C(A) 1.1.2, as is often the case 1.3.6. Then the "derived" or homotopy category of A will be a "triangulated category," and there will be a "calculus of fractions." We recall parts of this theory of Grothendieck and Verdier in our context (cf. [V] or [H] I)Let A/ ~ be the quotient category where two maps of A are identified to each other if they are chain homotopic as maps of complexes. The category Aj ~ is a full subcategory of the chain homotopy category K{A) = C(A)/~. If one defines the distinguished triangles in Aj ~ to be those chain homotopy equivalent to those coming in the usual way from mapping cone sequences in A, then Aj ~ satisfies the axioms for a triangulated category ([V] I Section 1, nos. 1-2, or [H] I Section 1), and is a subtriangulated category of K,{A) with its usual mapping cone sequence triangulated structure ([V] I Section 1 no. 2, or [H] I Section 2). Let w denote the image of w(A) \nA/~. As the complicial A is saturated and extensional, it is easy to see that w is a saturated multiplicative system in AJ ~ in the sense of ([V] I Section 2, nos. 1-2, or [H] I Section 3). The corresponding thick triangulated subcategory of Aj ~ is Aw/ ~, the full subcategory of objects A such that the unique map 0 —• A is in w. Let Ro(A) = w~lA be the "derived" or homotopy category formed from A by localizing the category A to make the maps in w(A) isomorphisms in w~1A. As chain homotopy equivalences are quasi-isomorphisms, hence are in w(A), they become isomorphisms in w~lA. In particular, in w~1A a complex A becomes isomorphic to the complex "A x 7" = A U A that A

parameterizes chain homotopies of maps out of A. Thus w~lA is also the localization w~lA/ ~ of Aj ~ at w. The work in ([V] I Section 2 or [H] I Sections 3-4) shows that u; is exactly the set of all morphisms in A/~ (or in A) that become isomorphisms in w~lA — HO(J4). Also w~lA has an induced triangulated structure. (Not only mapping cone sequences, but also general cofibration sequences turn out to yield distinguished triangles in w~lA, as in ([V] II Section 1, nos. 1-5, or [H] I 6.1).) Furthermore, the passage from Aj ~ to w~xA — w~1A/ ~ admits a "calculus of fractions" ([H] I Section 3 or [V] I Section 2 no. 3). In particular, the morphisms in w~lA from A to A1 correspond to the equivalence classes of data (1.9.6.1) in A/~,

270

T H O M A S O N AND T R O B A U G H

(1.9.6.1)

A-> A" <=- A!

where A —• A" is a morphism in A/ ~ , and A! - ^ A " is a m a p in w in T 1 / ~ . Two such d a t a A —»• A'/ £- A' and A —• J4 2 ' ^ A' are equivalent if there exists a commutative diagram (1.9.6.2) in A/~

A';

(1.9.6.2)

A,

T h e calculus of fractions insures t h a t this is an equivalence relation, and yields the composition of morphisms represented as data by constructing the C" and the b o t t o m arrows in (1.9.6.3). Here C" is the canonical homotopy pushout in A of a choice of maps in A to represent the chain homotopy classes of maps B -^ A!' and B —» B" i n y l / ~ , and the b o t t o m arrows are the canonical maps into the homotopy pushout. (It is easy to check that B" -^ C" is a weak equivalence, cf. the construction of 1.9.8.4 below.)

C

B

B

(1.9.6.3)

C Dually, morphisms in w~lA from A to A' may be represented by equivalence classes of data A £- A" —» A') with equivalence relation

(1.9.6.4)

A

A1

and with composition coming from homotopy pullbacks.

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271

When there is danger of confusion as to whether a map A —• A' is to be a map in A or in Ro(A) = w~1A, we refer to morphisms in A as strict maps. 1.9.7. Let F : A —*• 5 be a complicial exact functor between biWaldhausen categories (1.2.16). As F is induced by degree-wise application to complexes of an additive functor / : A —• B of associated abelian categories, the functor F preserves canonical homotopy pushouts and canonical homotopy pullbacks. In particular, F preserves mapping cones and mapping cylinders. Thus if A and B are closed under the formation of homotopy pushouts and homotopy pullbacks, F : A)'~—• B/~ and w~1F :w~1A—+w~1B are triangulated functors. From the calculus of fractions we see that w~lF : w~1A —• w~1B is an equivalence of homotopy categories if F : A —• B satisfies the following four conditions: 1.9.7.0. For any map a in A, a is in w(A) if and only if Fa is in w(B). 1.9.7.1. For any B in B) there is an A in A and a map FA -^ I? in w(fl). 1.9.7.2. For any map b : FA' —• FA" in B, there are maps a! \ A^ A! in ti/(.A) and a" : A —+ A" in >t, such that there is a chain homotopy 6 o Fa' ~ Fa" in B. 1.9.7.3. For any map a' : A1 —• A" in A such that Fa' ~ 0 is chain nullhomotopic in B, there is a map a : A -^ A' in u>(A) such that a' • a ~ 0 is chain nullhomotopic in A. The last two conditions (1.9.7.2 and 1.9.7.3) trivially hold whenever F : A —• B is fully faithful. Note then that F is also full and faithful for chain homotopies between maps, as a chain homotopy between maps h

C -+ D corresponds to a chain map " C x i " = C U C - ^ D. 1.9.8. Theorem. Let A and B be two complicial biWaldhausen categories, each of which is closed under the formation of canonical homotopy pushouts and canonical homotopy pullbacks (1.9.6, 1.2.11). Let F : A —• B be a complicial exact functor (1.2.16). Suppose that F induces an equivalence of the derived homotopy categories w~1F : w~lA —+ w~1B. Then F induces a homotopy equivalence of K-theory spectra K(F) : K(A) -+ K{B). Proof. By 1.9.2, we reduce to the case where cofibrations in A and B are precisely the degree-wise split monomorphisms whose quotients lie in A and B respectively. Then A and B have cylinder and cocylinder functors satisfying the cylinder and cocylinder axioms, thanks to 1.3.5. Let C be the category whose objects are data {A) FA -^ B) where A is an object of A, B is an object of B, and FA -^ B is a map in w(B).

272

THOMASON AND TROBAUGH

A map in C from (A, FA -^ B) to (A', FA! -^ B') consists of a map A —* A' in J4 and a map B —+ Bf in B such that Fi4 — ^

FA ;

5

• B'

commutes. Call a map in C a cofibration (or respectively, weak equivalence) if both maps A —• A' in A and B —> B' in 2? are cofibrations (resp. weak equivalences). This makes C a biWaldhausen category. (Indeed, it is even a complicial biWaldhausen category, with associated abelian category of data (Ay fA —• B) with Am A, B in #, and / A —* B a, map in /?.) As in 1.9.7, F : A-^ B preserves the mapping cylinders and cocylinders. Thus C has a cylinder functor induced by the cylinder functors of A and B. Dually C has a cocylinder functor. These satisfy the cylinder and cocylinder axiom 1.3.1.7. The functor F : A —* B factors as the composite of exact functors A -+ C and C -+ B. Here A - • C sends A to (A, FA = FA), and C -+ J? sends (A, FA -^ 5 ) to 5 . We will show both these functors induce homotopy equivalences on A'-theory spectra. The functor A —• C is split by an exact functor C —• J4 sending (A, FA ^> £ ) to A. So A — C -> A is the identity functor. The composite C —• A —• C is naturally weak equivalent to the identity by the natural transformation (A, FA = FA) —» (A, FA -^ J3) induced by A = A and FA ^ B. Thus K(A) — A(C) and A(C) -+ A(ji) are inverse homotopy equivalences by 1.5.4. Consider now the exact functor C —• B of biWaldhausen categories. We claim it induces a homotopy equivalence K(C) -^ K(B) by the dual to the approximation theorem 1.9.1. First we note that the dual hypothesis to 1.9.1.1 holds. For suppose (a, 6) : (A, FA -^ B) —• (A', FA' -^ £ ' ) is a map in C, whose image 6 : B -^ B' in C is a weak equivalence. Then by saturation, Fa : FA A FA' is also a weak equivalence. Hence Fa becomes an isomorphism in w~lB. As by hypothesis, w~lF :w~xA—^ w~lB is an equivalence of categories, the map a becomes an isomorphism in w~1A. Hence as in 1.9.6, the map a is in w(A). Thus both maps a and b are weak equivalences, so (a, 6) is a weak equivalence in C, as required by 1.9.1.1. To verify the dual hypothesis to 1.9.1.2, we must show that given a diagram (1.9.8.1) corresponding to a map x : B' —• (B of (FA —• £?)), this can be completed to a commutative diagram (1.9.8.2) corresponding to the factorization required by 1.9.1.2

HIGHER ALGEBRAIC K-THEORY OF SCHEMES

FA

273

B

(1.9.8.1) B'

(1.9.8.2)

As a first approximation, we construct a diagram like (1.9.8.2) inB/~, i.e., a version that chain homotopy commutes. We begin by noting that as w-1F is an equivalence of homotopy categories, B' is isomorphic in w~lB to some FAi. By calculus of fractions, this isomorphism corresponds to a datum in B : FA\ -^ B\ £- B'. Composing this isomorphism in w~lB with the map b : B' —• B and with the inverse isomorphism to FA -^ B yields a map in w~lB from FA\ to FA. As w~lF is an equivalence of categories, this map is w~lF of some map from A\ to A in w~lA. This map from A\ to A is represented by a datum A\ —• A2 £- A in A. Applying the formulae of the calculus of fractions for composition and equivalence of data representing maps in w~lB, as given in 1.9.6, we deduce that there exists in B/~ a commutative diagram, which after removal of intermediate constructions becomes:

(1.9.8.3)

We choose representatives of these maps in A and 2?, so that (1.9.8.3) becomes a chain homotopy commutative diagram in B) where the indicated maps are F of maps in A.

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THOMASON AND TROBAUGH

Now let A3 be the canonical homotopy pullback of A\ —• A? and A ^ A2, and let B3 be the canonical homotopy pullback of B\ —+ B2 and B -^ B2, as in 1.1.2. The FA3 is the canonical homotopy pullback of FAX - • FA2 and FA -> FA 2 , as in 1.9.7. By 1.1.2, the chain homotopy commutative diagram (1.9.8.3), after a choice of chain homotopies, induces a map of homotopy pullbacks FA3 —• i?3, which is an edge of a homotopy commutative cube with vertices FA3, FA2, FAi, FA, £3, B2, Bu and 5 . The map FA3 -> 5 3 is a weak equivalence. To see this, we first note that the projection map B3 —• B from the homotopy pullback is the pullback along B —> B2 of the canonical map h

ki : B2 x B\ = cocylinder (B\ —• .82) -* 5 2 . B2

As &i is a degree-wise split epimorphism, and even a map in quot(B), it follows that the pullback B3 -» B is in quot(2?). Similarly FA3 -» FA is in quot(#). Now as FA3 —• £3 is induced by the horizontal arrows in (1.9.8.3), each of which is a weak equivalence, it follows from the dual of the gluing lemma axiom 1.2.3.2 that FA3 -^ B3 is a weak equivalence. Also, as B —• i?2 is a weak equivalence, the extension axiom 1.2.6 shows that the induced map h

B3 -^ B2 x B\ = cocylinder (£1 —• #1) is a weak equivalence. As the projection of the cocylinder onto B\ is even a quasi-isomorphism, it follows that the canonical map B3 -^ B\ is a weak equivalence. The homotopy commutative right half of (1.9.8.3), together with a choice of homotopy, determines a map B' —• £3 by the universal mapping property of homotopy pullbacks, dual to the mapping property of homotopy pushouts explained in 1.1.2. As B3 —• B\ and 6' : B' —* B\ are weak equivalences, the saturation axiom implies that B' -^ B3 is a weak equivalence. Thus we have constructed a homotopy commutative diagram (1.9.8.4), the desired first approximation to (1.9.8.2).

(1.9.8.4)

FA

— ^

B

I FA3

—z—*

I B3

V

To finish, it remains to replace (1.9.8.4) by a strictly commutative diagram in B, as opposed to Bj ~. Let B" be the homotopy pullback

HIGHER ALGEBRAIC K-THEORY OF SCHEMES

275

of B —• B and B3 —» B. Then the projection B" -^ B$ is a quasiisomorphism, hence a weak equivalence. Choices of homotopies in the homotopy commutative diagram (1.9.8.4) determine maps FA3 —* £ " and B1 -» 5 " . As £ " -^ £3 and i*M3 -^ B3 are weak equivalences, the saturation axiom shows that FA3 -^ B" is a weak equivalence. Similarly, B' -^ 5 " is a weak equivalence. Now consider the map Bn —*• 5 which is the canonical projection of the homotopy fibre product onto B. By construction, the composite FA3 —• B" —* B is ^ ^ 3 —• FA —• 5 , and Bf —> B" —> B is b' : Bf —* B. Thus we have a strictly commutative (1.9.8.2) on taking i M " = F A 3 and B" = B". This completes the verification of the dual of hypothesis 1.9.1.2. Now 1.9.1 applies to complete the proof of the theorem. 1.9.9. The theorem 1.9.8 is very useful in providing AT-theoretic equivalences directly from off-the-shelf data, as found in [SGA 6] for example. Morally, it says that K(A) essentially depends only on the derived category w~lA, and thus that Waldhausen AT-theory gives essentially a Ktheory of the derived category. However, it is true that to so define a AT-theory of a derived category, one must find some underlying model A which is complicial biWaldhausen. Also, we know independence of the choice of model only w7hen the models are related by some additive functor exact in the sense of 1.2.16. These caveats are annoying, but do not seem to cause serious problems in practice. We also note an equivalence w~~1A —• w~lB often induces equivalences of homotopy categories tu - 1 A —> w~lB' for various naturally defined complicial BiWaldhausen subcategories yl', B', of A, B. Then 1.9.8 shows that K{A!) —• K(Bf) will also be a homotopy equivalence of A'-theory spectra. Thus the equivalences of 1.9.8 have a nice tendency towards inheritance by natural subcategories. 1.10.1. Cofinality T h e o r e m . Let vA be a Waldhausen category with a cylinder functor satisfying the cylinder axiom. Let G be an abelian group, and w : KQ(VA) —+ G an epimorphism. Let Aw be the full subcategory of those A in A for which the class [A] in Ko(vA) has ir[A] = 0 in G. Make Aw a Waldhausen category with v(Aw) = Aw Dv(A)f co(Aw) = Aw D co(A). Let "G" denote G considered as a EilenbergMacLane spectrum whose only non-zero homotopy group is a G in dimension 0. Then there is a homotopy fibre sequence (1.10.1.1) In particular,

K{vAw) -> K(vA) — "G"

276

T H O M A S O N AND T R O B A U G H

(110 12)

Ki(nA")

= Ki(nA)

for i > 0

P r o o f . Define w(A) to be t h e set of maps in A whose mapping cones have their KQ class in t h e kernel of 7r : Ko(vA) —+ G. It is easy t o check using 1.5.6 t h a t wA is a Waldhausen category and v(A) C w(A). Clearly wA satisfies t h e extension and saturation axioms. Appealing then t o t h e homotopy fibre sequence given by the localization theorem 1.8.2, it suffices to show t h a t K(wA) is homotopy equivalent to "G". n

For each non-negative integer n, consider IIG as a category whose obn

jects are n-tuples of elements of G, (<7i, <72> • • •,
only identity morphisms. There is a functor IT : wSnA —• I I G induced in terms of the description 1.5.1.2 by sending t h e object A\ >—• A^ >—• •••>-• An ofwSnA to the n-tuple (TT[J4I], 7r[A2] —TT[AI], 7r[v43] — 7 ^ 2 ] , . . . ,

v[An]-ir[An-i\). (In t h e more precise description 1.5.1, this functor sends A( , ) t o ( T T [ A ( 1 , 0 ) ] , 7 r [ ^ ( 2 , 0 ) ] - 7 r [ ^ ( l , 0 ) ] , . . . , ic[A(n, 0)] - 7r[A(n - 1, 0)]) = (TT[A(1,0)],

TT[A(2,1)], TT[A(3,2)],

...,

We claim t h a t for each n, 7r : wSnA

7r[A(n,n-l)]).) n

—• IIG induces a homotopy equiv-

n

alence of nerves of categories. As IIG is a discrete category, i.e., has only identity morphisms, it suffices t o show for all (<7i,<72> • • • ,9n) t h a t t h e category w~ * ( # 1 , . . . gn) has contractible nerve. T h e fibre n~l(0, 0 , . . . , 0) has initial object 0 >—• 0 >—••.•>—• 0, and so is contractible. We plead t h a t all fibres 7r - 1 (<7i,...,#„) are homotopy equivalent to 7 T _ 1 ( 0 , 0 , . . . , 0 ) , and hence are contractible. First note by 1.5.7 and the hypothesis t h a t 7r : KQ (VA) —• G is onto every element g in G is 7r[G] for some C in A. Given ( # 1 , . . . , gn) then choose G; in A so 7r[Ci) = gi. Consider the objects G. = Gi >-+ Gi U G 2 >-+ Gi U G 2 U G 3 •-+ • • • ^ C\ U G 2 U • • • U G„, and EG. = E G i > - E G i U EG 2 ^ • >-• EG X U EG 2 U • • • U E G n in wSnA. Clearly ir(C.) = (01,02, • • • ,0n). Also TT(EG.) = ( - £ i , - # 2 , • • •, - 0 n ) as [EG] = — [G] by 1.5.7. Then t h e functor UG. : wSnA —• wSnA sending ( ^ >_ . . . ^ ^ n ) t o ( i 4 i U C i >-• A 2 U G i U G 2 >-> •••>-• ^ n U G i U - - - U G n ) restricts t o a functor UG. : 7 r _ 1 ( 0 , 0 , . . . ,0) —• T r " 1 ^ , #2, • • • ,9n)> Similarly U EG. gives a functor n~l{g\,g2,• • ,9n) —* 7 r _ 1 ( 0 , 0 , . . . 0 ) . T h e maps 0 -+ CiUEC,-, G U E G , -+ 0 are in wA as [GUEG] = [C]-[C] = 0, and they induce natural transformations between the identity functors on 7 r _ 1 ( 0 , 0 , . . . ,0) and n~l(gi,g2, • • • ,9n) a n d t h e composites of the functor U G. a n d U E G . Thus these functors induce homotopy equivalences between 7 r _ 1 ( 0 , 0 , . . . ,0) and 7r_1(<7i, <72) • • • ,9n), as was to be shown. This

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277

n

completes the proof of the claim t h a t w : wSnA equivalence.

—• IIG is a homotopy

n

T h e m a p n : wSnA —• UG for various n are compatible with the simplicial operators, and so induce a functor between simplicial categories which is a homotopy equivalence of classifying spaces in each degree. Here the simplicial operators on wSnA are as in 1.5.1 - 1.5.2, and on IIG are denned so t h a t d 0 ( # i , . . . , # n ) = (#2, • •. ,0n), dt(giy... ,gn) = (9i,92^ . . , 0i_i,0,-0,-+i,0,-+2, --.,9n) for 1 < i < n, dn(glj... ,gn) = ( # i , . . . , # n _ i ) , and S i ( 0 i , . . . , 0 n ) = (flfi,...,flfi,0,^ + i,...,flf n ). W i t h this structure, the simplicial category IIG is actually the simplicial set BG = 7VG, the bar construction on G. T h e nerve of this degree-wise discrete simplicial category thus collapses to NG. T h e degree-wise homotopy equivalence 7r induces a homotopy equivalence of spaces

\NwS.(A)\-+\NG\~BG. In fact this homotopy equivalence is a map of infinite loop spaces. For one checks easily that the iterated S. construction on wA t h a t defines the Waldhausen spectrum structure ([W] 1.3.5 Remark) corresponds under n t o the iterated bar construction on G t h a t defines the Eilenberg- MacLane spectrum "G". (Or one notes that 7r is a simplicial symmetric monoidal functor, and feeds it to an infinite loop space machine [Th3] A). T h u s w induces a homotopy equivalence of spectra K(wA) - ^ "G", as required. (We found this proof in 1985; it has since become folklore.) 1.10.2. Exercise (optional). Theorem 1.10.1 is all the cofinality t h a t we need for Sections 2 - 1 1 . Other well-known cofinality results often have a different flavor, (see [Gr2] 6.1 and [Sta] 2.1 for some latest versions). In particular, the Waldhausen strict cofinality theorem [W] 1.5.9 at first seems quite different in purpose from our 1.10.1. Combining Waldhausen strict cofinality, our 1.10.1 and Grayson's cofinality trick [Gr3] Section 1, prove the following cofinality result: Let A and B be Waldhausen categories. Suppose A is a full subcategory of B closed under extensions, t h a t w(A) = j 4 n w ( J 5 ) , and t h a t a m a p in A is a cofibration in A iff it is a cofibration in B with quotient isomorphic to an object of A. Suppose that B has mapping cylinders satisfying the cylinder axiom, and that A is closed under them. Suppose finally t h a t A is cofinal in B in t h a t for all B in B there is a B' in B such t h a t B U B' is isomorphic to an object of A. Then K(A) -+ K(B) - * «K0(B)/K0(A)" is a homotopy fibre sequence. 1.11.1. To close Section 1 we compare the Quillen K-theory [Ql] of an exact category £ to the Waldhausen /^-theory of £, and to the Waldhausen A'-theory of a category of bounded complexes in S. T h e latter

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category has cylinders and cocylinders and is complicial biWaldhausen. This allows one to apply the results 1.10.1, 1.10.2, 1.9.8, 1.9.1, 1.8.2 to Quillen /^-theory of exact categories. See [Gr2] and [Sta] for rederivations of all Quillen's basic results on K-theory of exact categories in the Waldhausen framework. We pause to mention two open problems. First, find a general result for Waldhausen categories that specializes to Quillen's devissage theorem when applied to the category of bounded complexes in an abelian category. Second, show under the conditions 1.2.15, that K(A) ~ K{B). This would make Quillen's localization theorem for abelian categories an immediate consequence of 1.8.2. Logically, one should now read Appendix A, and then return to 1.11.2. 1.11.2. Theorem (Waldhausen). Let £ be an exact category in the sense of Quillen [Ql]. Consider £ as a biWaldhausen category as in 1.2.9. Then the Quillen and the Waldhausen K-theory spectra of £ are naturally homotopy equivalent. Proof. [W] 1.9, or [Gi2] 9.3. 1.11.3. Let A be an abelian category, and let i : £ —• A be an exact functor which is full and faithful. Assume that £ is closed under extensions in .4, and that if a sequence in £ is exact in A, then it must be exact in £. We also make the following stronger assumption (which will be harmless by 1.11.10): 1.11.3.1. If / is a map in £ such that i(f) is an epimorphism in A, then / is an admissible epimorphism in £. 1.11.4. Example. Let X be a scheme, and let £ be the exact category of algebraic vector bundles on X. Let A be either the abelian category of all Ox-modules, or else the abelian subcategory of all quasi-coherent Ox~ modules. Let i : £ —• A be the canonical inclusion. Then this inclusion satisfies all the conditions of 1.11.3. 1.11.5. Example. Let £ be an exact category satisfying the condition that all weakly split epimorphisms of £ are admissible epimorphisms. That is, suppose that for any r : E —+ E" in £ such that there is an s : E" —• E with rs = 1, then r : E -» E" is an admissible epimorphism in £. Let i : £ —• A be the Gabriel-Quillen embedding (cf. A 7.1), i.e., let A be the abelian category of left exact additive functors £op —• Z-modules, with i the Yoneda embedding i(E) = horri£( ,£"). Then i : £ —• A satisfies the hypotheses of 1.11.3, including 1.11.3.1. For a proof, see Appendix A.7.1 and A.7.16. 1.11.6. Given i : £ —• A as in 1.11.3, consider the category E~ of bounded chain complexes in £ as a full subcategory of the category of chain complexes C(A). Define co(E~) to be the degree-wise admissible

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279

monomorphisms. Define w(E^) to be those maps in E~ which are quasiisomorphisms in A. (This appears to depend on the choice of A, b u t in fact does not given 1.11.3.1, see 1.11.8.) Then E~ is a complicial biWaldhausen category. There is a canonical complicial exact functor £ —» E~ sending E in £ to the complex which is E in degree 0 and 0 in other degrees. Let E be J 5 ~ , but now with co(E) being the degree-wise split monomorphisms whose quotients lie in E. Then E is a complicial biWaldhausen category, and the inclusion functor E —• E~ is complicial exact. By 1.3.6, b o t h E and E~ have cylinder and cocylinder functors, satisfying the cylinder and cocylinder axioms 1.3.1.7. 1.11.7. T h e o r e m (Gillet-Waldhausen). Under the hypotheses of 1.11.3 and with the notation of 1.11.6. the canonical exact inclusions induce homotopy equivalences of K-theory spectra K{£) ^+K{E~)

*-K(E)

P r o o f . T h e m a p K{E) —» K{E~) is a homotopy equivalence by 1.9.2. T h e homotopy equivalence K{£) - ^ K{E ) is due to Gillet, who patterned his argument [Gi2] 6.2 on a proof for special cases due to Waldhausen. Gillet's statement [Gi2] 6.2 does not make the extra hypothesis 1.11.3.1 on the embedding £ —• A, but the proof given there needs 1.11.3.1 in order to work. (Gillet a t t e m p t s to evade 1.11.3.1 by appealing to Quillen's resolution theorem, but in fact 1.11.3.1 is needed to verify one of the hypotheses of the resolution theorem, [Ql] Section 4 T h m 3i).) We will give the complete proof t h a t K(£) -—> K(E ) is a homotopy equivalence. For integers a < 6, let E~b be the full subcategory of those complexes E' i n i £ ~ such t h a t El = 0 for i < a — 1 and for i > 6 + 1 . Hence £ = E^°, and E~ is the direct colimit of the E~b as b goes to + c o and a goes to —oo. Let w(E~) be the quasi-isomorphisms of complexes as in 1.11.6, and let i(E~) be isomorphisms of complexes. Set w(E^h) = w(E~) f\E~b i(E~h) = b b b i(E~) C\E~\ and co(E~ ) = c o ( # ~ ) PiE~ . T h e n wE~ and iE~b bw are Waldhausen categories. Let E~ be the full subcategory of E~b of those complexes quasi-isomorphic to 0, with co(E~bw) = E~hwC\ co(E~). Then iE~bw is a Waldhausen category. Consider the exact functor

(i.n.7.1)

«a~6-+

fc-o+l n £

sending a complex E' to (Ea, Ea+1,..., Eb). We claim t h a t this functor induces a homotopy equivalence on Waldhausen A'-theory. For b = a, this is clear, as the exact functor is then an isomorphism. T h e proof of the

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claim now proceeds by induction on b — a, and consists of showing that a functor (1.11.7.2)

iE^^US&xS

induces a homotopy equivalence on A'-theory. This functor sends a complex (Ea —+ • • • —• Eb) to the pair consisting of the subcomplex (0 —• Ea+1 -+ • • • -+ Eh) and the quotient Ea = (Ea -+ 0 -+ • • • - • 0). This functor does induce a homotopy equivalence on A'-theory by the Additivity Theorem 1.7.2 with A = iE~Jv B = £ = iE~a and C = iE~b, as the canonical filtration defining our functor induces an equivalence of categories iE~b ~ E(A,C,B). This proves of claim. On the other hand, we also claim that similarly, K(iE~bw) is homotopy b-a

equivalent to II K{£). If 6 = a, this holds trivially as E~aw = £w is equivalent to the 0 category. It also holds if a — b — 1, as the category iE^J™ is the category of complexes d : Eb~l —• Eb with d an isomorphism, and this category is equivalent to £. The proof now proceeds by induction on b — a and consists of producing a homotopy equivalence (1.11.7.4)

A (iE~bw) ^ K(iE~(b~l>)

x A (iE?J? = £)

This homotopy equivalence results by applying the additivity theorem 1.7.2 to an equivalence of categories (cf. 1.9.4(d)) (1.11.7.5)

iE~bw * E(iE?(b-Vw,

iE~bw,

iE?b?)

To see the equivalence, we must associate an extension to a complex E' in iE~bw. As E' is acyclic and ZbE = Eb as Eb+1 = 0, the map Eb~l -» Eb is an epimorphism in A. Hence Eb~1 -»• Eb is an admissible epimorphism in £ by 1.11.3.1. Thus its kernel Zb~lE' is in £ and Zb~lE >-• Eb~l -» Eb is an exact sequence in £. The complex r-b~lE' — (Ea —• i ? a + 1 —•...—» ^ 6 _ 2 ^ Z J - I J i s t h u g i n £E~(*-i)". T h e complex r*£* = ( £ 6 E Eb) is in i^b-\~ > a n d £" fits into a canonical cofibration sequence r-b~lE' >—• £• — r 6 £ ' which defines an object of jE7(flC (6 ~ 1)u \ i £ ~ 6 u \ ^ ^ D

HIGHER ALGEBRAIC K-THEORY OF SCHEMES

i

i

0

0.

I Ea

281

I Ea

_____

i i

I

(1.11.7.6)

1 I 1

1

Eb-2

h 1

Z-

0

1

b 2

_

E~

• 0

1 >

1

h l

• E~

»

B

1 y

Eb

0

=

E

0

I The inverse equivalence of the categories takes the total complex E' and forgets the extensions. It is easy to check that both the equivalence of categories and its inverse are exact functors. This completes the proof b a of our second claim that K(iE~bw) is homotopy equivalent to ~ n K(£). In fact our proof shows that for E' in E~bw, the ZkE' = BkE' are objects off, that E' \-+ Bk(E') is an exact functor, and that our claimed

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b a homotopy equivalence is induced by the exact functor iE~bw —• ~ II £ given by sending E' to (Ba+lE', Ba+2E\ . . . , BbE). Now consider the exact inclusion iE~bw —• iE~b. This induces a map on K-theory spectra, which by the two claims above is homotopy equivalent

b-a

6-a+l

to a map II K{£) -+

II

K(£)

K(iE~b") (1.11.7.7)

K(iE~b)

•

[~

[~

b-a

6-a+l

n i<(£)

•

For E' in iE~bw, the corresponding term in

n

i<{£)

6-a+l

II £ is (Ea, Ea+l,

. . . , Eb)

b-a

while the corresponding term in II £ is (i? a + 1 , . . . , Bb). From the exact sequence Zk >-• Ek -» Bk+1 for E' and the fact that Zk — Bk by acyclicity, the Additivity Theorem 1.7.2 shows the map on A'-theory spectra induced by sending E' to Ek is homotopic to the sum of the maps induced on K-theory induced by sending E' to Bk and to Bk+1. Considering also b-a

6-a+l

that Ba - imEa~1 - 0, we see that our map II K(£) -> U K(£) in (1.11.7.7) is that induced by the exact functor: (1.11.7.8) (Ba+\ . . . , Bh)*—+(Ba+\ 5 a + 1 0 5 a + 2 , ..., Bb~1®Bh). 6-a+l

The homotopy cofibre of this map is K(£), with II K(£) —• K{£) induced by (x a , . . . , Xh) \-+ E(—1)^^^• Taking the direct colimit as a goes to — oo and b goes to -foo, we get a homotopy cofibre sequence (1.11.7.9)

K (iE ~w) — K(iE~) -+ K(£)

where K(iE~) —»- K(£) sends E' to its Euler characteristic £( — \)kEk. But by the localization theorem 1.8.2, the homotopy cofibre spectrum of K(iE~w) -+ K(iE~) is K(wE~) = K(E~). Thus there is a homotopy equivalence A'(£) -^ K(E~), which in fact is the map induced by the exact functor £ —>E~. This proves the theorem. 1.11.8. Porism. Under the hypotheses of 1.11.3 and with the notation of 1.11.6, a complex E' in £ is acyclic in C(A) if and only if all cycle and boundary objects ZkE' and BkE' are in £ C A, BkE' - ZkE' in £, and the sequences Zk y-• Ek -»• J B /r+1 are exact in £. In particular, this is independent of the choice of A, provided only it satisfies 1.11.3 and especially 1.11.3.1.

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Also, the set w(E) of quasi-isomorphisms of complexes in £ is independent of the choice of A. For the first paragraph is a porism of the proof of 1.11.7. To verify the second paragraph, we note t h a t a map in E is a quasi-isomorphism (with respect to A) iff its mapping cone is acyclic. T h e formation of mapping cones uses only the additive structure of £, and acyclicity is independent of the choice of A by the first paragraph. 1.11.9. Remark. An exact functor / : £ —• £' induces a compatible additive functor of the associated Gabriel-Quillen abelian categories / * : A —• A', by A.8.2. Although / * : A —» A! need not preserve exact sequences in A, it does preserve exact sequences in £, A.8.5. Hence it preserves quasi-isomorphisms of complexes in £ by 1.11.8, and induces complicial exact functors / * : E —> E', E~ —> E'~. T h u s 1.11.7 becomes natural in £ with £ —> A chosen as the Gabriel- Quillen embedding 1.11.5. 1.11.10. Remark. If £ is an exact category not satisfying the hypothesis of 1.11.5 t h a t weakly split epimorphisms are admissible epimorphisms, we let £' be its Karoubianization A.9.1. Then 1.11.7 applies t o the Gabriel-Quillen embedding of £', as £' satisfies the hypothesis of 1.11.5. As K{£) is a cover of K(£') so #,-(£) = #,•(£') for i > 0, by classical cofinality A.9.1, this shows the extra hypothesis of 1.11.3.1 is essentially harmless. Indeed by 1.11.1 we get in general t h a t K(£) is homotopy equivalent to the K-theory of the category of those bounded chain complexes in £' whose Euler characteristic lies in Ko(£) C Ko(£f). 2. P e r f e c t c o m p l e x e s o n s c h e m e s 2.0. In this section, we review and extend the theory of perfect complexes on a scheme. This theory was discovered and developed by Grothendieck and his school (especially Illusie) in [SGA 6] as a more flexible replacement for the naive theory of algebraic vector bundles on a scheme in K-theory. The somewhat new results of this section are the characterizations of pseudo-coherence and perfection in 2.4, and some of the functoriality statements in 2.6 and 2.7. Aside from a few minor improvements, the rest of this material is already in [SGA 6]. 2.1.1. Definition. A scheme with an ample family of line bundles is a scheme X , which is quasi-compact and quasi-separated, and which has a family of line bundles {Ca} which satisfy any one of the following equivalent conditions ([SGA 6] II 2.2.3 and proof thereof).

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(a) Let n run over all positive integers n > 1 and let Ca run over the family of line bundles. Let / G T(X,C®n) run over all the global sections of all the £® n . Then the resulting family of open set Xf = {x G X\f(x) £ 0} is a basis for the Zariski topology of X. (b) One may choose a set of positive integers n > 1, a set of line bundles Ca in the family, and a set of global sections / G ^(X, £® n ) such that the set {Xj} is a basis for the Zariski topology of X and all these Xf are affine schemes. (c) One may choose a set of positive integers n > 1, line bundles Ca in the family, and global sections / G T(X, £® n ) such that {Xf} is a cover of X by affine schemes. (d) For any quasi-coherent Ox-module T, the evaluation map a,n>l

is an epimorphism. 2.1.2. Examples, (a) Any scheme with an ample line bundle in the sense of [EGA] II 4.5.3 and IV 1.7 has an ample family (consisting of one line bundle) in the present sense. As special cases we have (b) and (c): (b) Any affine scheme has an ample family of line bundles. (c) Any scheme quasi-projective over an affine scheme has an ample family of line bundles. In particular, any quasi-affine scheme has an ample family of line bundles. (d) Any separated regular noetherian scheme has an ample family of line bundles ([SGA 6] II 2.2.7.1). (e) If Y has an ample family of line bundles, and U C Y is a quasicompact open, then U has an ample family of line bundles, given as the restriction of the family on Y. This is clear by 2.1.1(a). (f) Let Y be a scheme with an ample family of line bundles { £ a } . Let / : X —• Y be a quasi-compact and quasi-separated map of schemes. Suppose X has an /-ample family of line bundles {/C/?}. That is, suppose Y is covered by (affine) opens U CY such that {1Cp\f~l{U)} is an ample family on each f~l{U). Then {f*C®k ® ICfn\k > 1, n > 1} is an ample family on X) as clearly follows from criterion 2.1.1(a) and [EGA] I 6.8.1. As special cases we have (g) and (h): (g) If Y has an ample family of line bundles and / : X —» Y is an affine map of schemes, then X has an ample family of line bundles. (h) If Y has an ample family of line bundles and / : X —• Y is a quasiprojective map of schemes, then X has an ample family of line bundles. 2.1.3.

Lemma.

Let X have an ample family of linebundies.

Then

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285

(a) For any quasi-coherent Ox-module T', there is a locally free Oxmodule £ and an epimorphism £ -» T. (b) For any quasi-coherent Ox-module T of finite type, there is an algebraic vector bundle (i.e., a locally free Ox-module of finite type) £ on X, and an epimorphism £ -+• T. (c) For any epimorphism Q —• T of quasi-coherent Ox-modules with T of finite type, there is an algebraic vector bundle £ and a map £ —• Q such that the composite map £ —+ Q —• T is an epimorphism onto T. In all cases (a), (b), (c), £ may be taken to be a direct sum of tensor powers of line bundles of the family. Proof. First note by quasi-compactness of X and 2.1.1(c) that X has an ample finite subfamily of line bundles, £ a i , . . . , £<*n. As each CQt is locally free, X is covered by open sets where all the CQt are simultaneously free. Thus any direct sum of C~®n is locally free on X, even if it is an infinite sum. To prove (a) we appeal to 2.1.1(d), and let £ be the sum with one factor £~® n for each global section in T(X, T® £® n ). Case (b) follows from case (c) on setting Q = T. To prove case (c), we consider the epimorphism 0 £ ~ ® n -* Q constructed in the proof of case (a). The composite of this epimorphism with Q -* T is also an epimorphism. As T has finite type and X is quasicompact, some finite subsum of the factors C~fn maps epimorphically to T. We let £ be this finite subsum, and take £ C ®£~® n -» Q the induced map. 2.2. Logically, the reader should now examine Appendix B before returning to 2.2.1. We note the convention that the word "C?x-module" means "a sheaf on the scheme X which is a sheaf of modules over the sheaf of rings Ox? and does not apply to a general presheaf of modules over the sheaf of rings Ox- That is, an " 0 x - m ° d u l e " is a Ox-modu\e in the Zariski topos of X. 2.1.1. Definition ([SGA 6] I 2.1). For any integer m, a chain complex E' of Ox-modules on a scheme X is said to be strictly m-pseudo-coherent if E% is an algebraic vector bundle on X for all i > m and E% — 0 for all i sufficiently large. A complex E' is strictly pseudo-coherent if it is strictly m-pseudo-coherent for all m, i.e., if it is a bounded above complex of algebraic vector bundles. 2.2.2. Definition ([SGA 6] I 2.1). A complex E' of Ox-modules is strictly perfect if it is strictly pseudo-coherent and strictly bounded below. That is, a strict perfect complex is a strict bounded complex of algebraic vector bundles.

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2 . 2 . 3 . L e m m a ([SGA 6] I 2.10b). Let A' be a complex of Oxmodules with H((A') = 0 for i > m + 1. Then Hm(A) is an Ox-module of finite type iff for each x £ X there is an open neighborhood U of x and an isomorphism in the derived category D(Ou~Mod) between the restriction A'\U and a strictly m-pseudo-coherent complex on U. Proof. Suppose Hm(A') is of finite type, and let x £ X. T h e n for some positive integer k there is an epimorphism of stalks (BkOx,x ~* Hm(A)x. As (&kOx,x is a free module over Ox,x, this m a p lifts to the group of cycles Zm(A')x -» Hm(A')x. The lifted map extends over some open nbd U of x to a map ®kOu -+ Zm(A')\U C Am\U. Shrinking U, we k m may assume t h a t (& Ou —• H (A')\U is epimorphic. Then the complex consisting only of $dkOu in degree m maps to the complex A'\U by a m a p inducing an isomorphism on the cohomology Hp = 0 for p > m , and an epimorphism ®kOu - * Hm(A')\U on Hm. We now apply Lemma 1.9.5 with A — V = category of (9t/-modules and C — C~(A). This inductive construction lemma produces a complex D' on U and a quasi-isomorphism D' A ^'If/. Moreover, D' satisfies Dm = §kOu, and D? = 0 for p > m . T h u s D" is strict m-pseudo-coherent on {/, as required. Conversely, suppose A' is locally quasi-isomorphic to a strict m-pseudocoherent complex. As Hm(A') is a quasi-isomorphism invariant, and as being of finite type is a local question, passing to U C X we may assume t h a t A' is strict m-pseudo-coherent. Recall t h a t Hl(A') = 0 for i > m-f 1. Applying 1.9.4(a), with ^4 = (9^-Mod and S = the category of algebraic vector bundles on U, to the truncated complex of vector bundles amA', m m m we see t h a t Z (A') — Z (a A') is an algebraic vector bundle on U, and hence is of finite type. As Hm(A') is a quotient of ZmA', it is also of finite type, as required. 2 . 2 . 4 . L e m m a ( [ S G A 6]). Let U be a scheme, and x a point of U. Consider the solid arrow diagram of complexes of 0\j-modules, F', G', E':

d (2.2.4.1)

r

S

\c

^ U

a

E' Then under any of the following sets of conditions, there exists a smaller nbd V of x, a complex E1' of Oy-modules on V, and maps c : E'm —>

H I G H E R A L G E B R A I C K-THEORY O F S C H E M E S G'\V, d : F'\V —• E1' such that cd — a\V, and which satisfy conclusions attached to the corresponding set of conditions:

287 the

extra

(a) If E' and F' are strict m-pseudo-coherent, and the truncation rm(b) : rmG' - ^ rmE' is a quasi-isomorphism, then we may take E1' to he strict m-pseudo-coherent and c to be a quasi-isomorphism. (b) If E' is strict m-pseudo-coherent, and F' is strict perfect, with rm(b) : rmG' - ^ rmE' a quasi-isomorphism, then we may take E1' to be strict perfect and c to be an m-quasi-isomorphism. (c) IfE' and F' are strict perfect and b : G' - ^ E' is a quasi-isomorphism, then we may take E1' to be strict perfect and c to be a quasiisomorphism. Proof. In all cases E' and F' are strictly bounded above. Hence G' is cohomologically bounded above, and replacing G' by the sub complex r-kG' for some sufficiently large k, we may assume t h a t G' is strictly bounded above. Consider the germs of complexes Ex, Fx, Gx at the point x £ U. We apply to Fx —• Gx the inductive construction Lemma 1.9.5 with A the category of modules over Oxyx> *D ^ n e category of free modules of finite type over Ox,x, and C the category of complexes in A with a map 6 to a strict m-pseudo-coherent complex such that rm(b) is a quasi- isomorphism. T h e quasi-isomorphism r m ( 6 ) and Lemma 2.2.3 show t h a t hypothesis 1.9.5.1 is met for n— 1 > m. For if C' is in C with Hl(C') = 0 for i > n > m + 1, Hn~l(C') is isomorphic to an Ox,x module of finite type via r m ( 6 ) and 2.2.3. So there is an epimorphism ®kOx,x —• Hn~l{C'). As ®kOx,x is a free module over the local ring Ox,x, this epimorphism lifts along any epimorphism of modules over Ox,x, Ax -*• Hn~l{C'). This verifies 1.9.5.1 as long as n — 1 > m. Now the variant 1.9.5.9 of 1.9.5 provides a strict perfect complex amE'x , a map crmd : o-mFx —± amEx, and an m-quasi-isomorphism crmE'x —• m cr Gx —»• Gx, which is the germ at x of a truncated version of (2.2.4.1). As crmEx , amF', and amE' are all bounded complexes of free modules of finite type, there is a small open nbd V of x in U over which <jmEx extends to a strict perfect complex amE'', over which the maps d and c extend, and over which the map be : E1' —+ E' is an m-quasi-isomorphism (i.e., its mapping cone is acyclic in degrees > m ) . Then as r m ( 6 ) is a quasi-isomorphism, it follows that am(c) : amE'' —»• G' is an m-quasiisomorphism on V. Now we apply 1.9.5 again, now with A = CV-Mod, V = CV-Mod to construct the rest of E1' and d, c, leaving amEf' unchanged. Then E' is strict m-pseudo-coherent, as arnE1' is strict perfect. This proves 2.2.4(a). To prove (b), let E'1 be the pushout of amF' —• F' and the map m a F' —> amE'f constructed above. Let c : E'1 —> G' be the m a p induced by a : F' - * G' and am(c) : amE'' -+ G'. Then E'f -+ G' is

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an m-quasi-isomorphism as both crm(c) : amE'' —» G' and amE'' —+ £"' are. As the pushout of strict perfect complexes along a degree-wise split monomorphism, E1' is strict perfect. To prove (c), we take m < 0 so that E{ = 0 = F* for i < m + 1. We apply the proof of (b), which gives a strict perfect £"' such that crmE'' = E1' (as amF' = F ' ) , and a map 6c : i?'" —• E' which is an m-quasiisomorphism on V. The mapping cone M' of be is strict perfect, is 0 below degree ra — 1 and is acyclic except for Hm~l(M') — ker (Hm(E'') -» m m l m l H (E')). By 1.9.4(a) H - {M') = Z ~ M' is an algebraic vector bundle. As E'm-1 = 0 = Em, we note that Zm-lM' = kev(ZmEf' -> m m , Z E') = Z E \ As the map 6 : G' —• £" is a quasi-isomorphism, Hm~\M') goes to 0 in # m ( G ' ) S Hm{E'). Then Z m £ £ -> Z m G ; lifts along <9G™-1 —• ZmGx as a map of germs of Ox-modules over the local ring Ox,x. As ZmE'' = Zm~lM' is locally free of finite type, this lift extends to a map of Oy-modules ZmE'' -+ G m _ 1 lifting ZmE'' -> ZmG' on some smaller open nbd V of x. We now extend amE1' to a new E'' by £/m-i = zmE'' with boundary 3 : J^™"1 -+ £ / m given by the inclusion ZmE1' C £" m . As Z m £ ' ' = Z " 1 " ^ " is a vector bundle, the new E1' is still strict perfect. The map amc : cr m £"' —• G' is extended to the new £"' by using ZmE'' -> G m _ 1 in degree m - 1. Now clearly c : E'f - • G* is a quasi- isomorphism, and the other conditions of (c) are met. 2.2.5. L e m m a ([SGA 6] I 2.2). On a scheme X, the following conditions are equivalent for any complex E' of Ox-modules: 2.2.5.1. For every point x G X, there is a nbd U ofx, a strict n-pseudocoherent complex F', and a quasi-isomorphism F' -^ E'\U. 2.2.5.2. For every point x G X, there is a nbd U of x, a strict perfect complex F't and an n-quasi-isomorphism F' —» E'\U. 2.2.5.3. For every point x G X, there is a nbd U ofx, a strict n-pseudocoherent complex F', and an isomorphism between F' and E'\U in the derived category D(Ou~Mod). 2.2.5.4. For every point x G X, there is a nbd U of x, a strict perfect complex F', and an n-quasi-isomorphism F' —• E'\U in the derived category D{Ou-Mod) (that is, there is a map in the derived category inducing an epimorphism on Hn and an isomorphism on Hk for k > n + I). Proof. We see that (1) => (2) by replacing f in (1) by the strict perfect (4) and (1) => (3). It suffices then to show that (3) => (1) and (4) => (1). To see that (3) => (1), we consider an isomorphism F' —• E'\U in D(Ou-Mod). By the calculus of fractions, this is represented by a datum of strict maps which are quasi-isomorphisms F'^-G'^ E'\U. By 2.2.4(a), after shrinking the nbd U, there is a strict n-pseudo-coherent

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F1' and a quasi-isomorphism F1' -^ G'. Then the composite Ff' -^ G' -^ E'\U is the strict quasi-isomorphism required by (1). To see that (4) => (1), we represent the n-quasi-isomorphism in D(Ou~Mod) by a datum of strict maps F' £- G' -+ E'\U where G' —• E'\U is an ?i-quasi- isomorphism. After shrinking U, by 2.2.4(c) there is a strict perfect F1' and a quasi-isomorphism F'' —» G'. Then Ff' —> £'|£/ is an n-quasi-isomorphism. Applying the Inductive Construction Lemma 1.9.5 with A ~ V = 0c/-Mod and C = C~(A)} we obtain a complex F" and a quasi-isomorphism F" -^ £"|£/, such that anF" — an F'. Thus anF" is strict perfect, so F" is strict n-pseudo-coherent as required. 2.2.6. Definition ([SGA 6] I 2.3). A complex E' of (9x-modules on a scheme X is said to be n-pseudo-coherent if any of the equivalent conditions 2.2.5.1 - 2.2.5.4 hold. The complex E' is said to be pseudo-coherent if it is n- pseudo-coherent for all integers n. 2.2.7. Clearly pseudo-coherence of E' depends only on the quasi-isomorphism class of E\ and is a local property on X. The cohomology sheaves H*(E') of a pseudo-coherent complex are all quasi-coherent Ox~ modules. If X is quasi-compact, and E' is pseudo-coherent, there is an N such that Hk(E') = 0 for all k > TV, as this is true locally on X and since any open cover of X has a finite subcover. A strict pseudo-coherent complex is pseudo-coherent. It will follow from 2.3.1(b) below that any pseudo-coherent complex of quasi-coherent Ox~ modules is locally quasi-isomorphic to a strict pseudo-coherent complex. In fact, it will be quasi-isomorphic to a strict pseudo-coherent complex on any affine open subscheme. For a pseudo-coherent complex of general (9A"~ modules, there will locally be n-quasi-isomorphisms with a strict pseudocoherent complex, but the local n&ds where the n-quasi-isomorphisms are defined may shrink as n goes to —oo, and so may fail to exist in the limit. So there may not be a local quasi-isomorphism with a strict pseudocoherent complex. This phenomenon renders the auxiliary concept of n-pseudo-coherent necessary to our work. 2.2.8. Example ([SGA 6] I Section 3). A complex E' of (9*-modules on a noetherian scheme X is pseudo-coherent iff E' is cohomologically bounded above and all the Hk(E') are coherent (9x-naodules. Proof. If E is pseudo-coherent, then E' is locally (k — 1 ^quasi-isomorphic to a complex of coherent locally free sheaves. Computing Hk(E') as Hk of the latter complex, we see that Hk(E') is coherent. Conversely, suppose E' is cohomologically bounded above with coherent cohomology. Let m be an integer and x a point of X. We apply the Inductive Construction Lemma 1.9.5 with A — C?x,a;-niodules, V — free Ox,x-

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modules of finite type, and C = cohomologically bounded above complexes with coherent cohomology to produce a strict pseudo-coherent complex of Ox,r~niodules Fx and a quasi-isomorphism Fx ^ Ex. Then cmFx is strict perfect, and crmFx —• Ex is an m-quasi-isomorphic. The strict perfect complex (TmFx and the map cmFx —• Ex extend to a strict perfect complex (2). To show (2) => (1), we represent the isomorphism in the derived category via calculus of fractions as a datum of strict quasi-isomorphisms F'^-G'^ E'\U. Then we apply 2.2.4 (c) to G' -^ F' to produce after shrinking U a strict perfect F'\ and a strict quasi-isomorphism Fh -^ G' -^ E'\U. 2.2.10. Definition ([SGA 6] I 4.2). A complex E' of Ox-modules on a scheme is perfect if it is locally quasi-isomorphic to a strict perfect complex, i.e., if 2.2.9.1 or 2.2.9.2 hold for E\ 2.2.11. Definition ([SGA 6] I 5.1). A complex E' of Ox-modules has Tor-amplitude contained in [a, 6] for integers a < b if for all Ox-modules T, Hk(E' (&Q T) — 0 unless a < k < b. If such an a and 6 exist, one says E' has (globally) finite Tor-amplitude. If X is covered by opens U such that a and b exist on each U for E'\U, one says that E' has locally finite Tor-amplitude. 2.2.12. Proposition ([SGA 6] I 5.8.1). A complex E' of Oxmodules is perfect iff E' is pseudo-coherent and has locally finite Toramplitude. Proof. A strict perfect complex E' is flat and strictly bounded, so that E(' 0 L T — E1' 0 T is cohomologically bounded. So E1' is pseudocoherent and of finite Tor-amplitude. As a perfect complex E' is locally quasi-isomorphic to a strict perfect E'\ it is pseudo-coherent and of locally finite Tor-amplitude.

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Conversely, suppose E' is pseudo-coherent of locally finite Tor-amplitude. Take any point x E X, and pick a nbd U of x on which E' has Tor-amplitude in [a, 6]. By 2.2.5.3, after shrinking U, we may replace E' up to quasi-isomorphism and assume t h a t E' is strict (a — 2)-pseudocoherent. T h e n as aaE' is a strict perfect complex strictly bounded below by a, it has Tor-amplitude bounded below by a. Applying 2.2.11 with T — Oxi we see t h a t E' is cohomologically bounded between a and b. T h u s aaE' —> E' induces an isomorphism on cohomology Hk for k ^ a, and on Ha induces an epimorphism Ha(aaE') = Za£** -» Ha(E'). Let 5 be the kernel of this epimorphism, and B[a] the complex consisting of B in degree a. Then J3[a] —* aaE' —• £" is a homotopy fibre sequence, i.e., 2 sides of a distinguished triangle in the derived category. Considering the induced long exact sequence for H*(!F ®L ( )), we get t h a t T o r f c ( ^ , 5 ) = Ha-k(T (g)L B[a]) is 0 for k > 1. T h u s 5 is a flat (9t/-module. 5 is also the only non-vanishing cohomology group Ha~1 of the mapping cone M' of aaE' —• £ " . As cr a £" is strictly perfect and £" is strict (a — 2)-pseudo-coherent, the mapping cone M' is strict (a — 2)-pseudo-coherent. By 1.9.4(b), Za~1(M') is a vector bundle, as is M f l " 2 . T h e exact sequence Ma~2 -* Za~l(M') -> Ha~l(M') a 1 —+ 0 shows t h a t 5 = H " (M') is a finitely presented Ojy-module. But as B is also flat, it is then a locally free 0[/-module of finite type, i.e., a vector bundle (e.g., [SGA 6] I 5.8.3, or Bourbaki). Now consider the truncated complex raE'. This differs from the strict perfect aaE only in degree a - 1, where raE' has Ba(E') = ker(ZaE' -» Ha(E')) = B. As a B is a vector bundle, r E' is strict perfect. But as E' is cohomologically bounded below by a, E" - ^ r a £ " is a quasi-isomorphism on [/. T h u s £" is perfect locally, and hence perfect, as required. 2.2.13.

P r o p o s i t i o n ([SGA 6] I ) .

Let x be a scheme.

(a) Suppose A' -+ B' —> C is a homotopy fibre sequence in D(Ox-Mod), i.e., forms two sides of a distinguished triangle. Then: If A' is (n + 1)-pseudo-coherent and B' is n-pseudo-coherent, then C is n-pseudo-coherent. If A' and C are n-pseudo-coherent, then B' is n-pseudo-coherent. If B' is n-pseudo-coherent, and C is (n — l)-pseudo-coherent, then A' is n-pseudo-coherent. (b) If A', B', C are the three vertices of a distinguished triangle in D(Ox-Mod), and 2 of these 3 vertices are pseudo-coherent (resp. perfect), then the third vertex is also pseudo-coherent (resp. perfect). (c) The complex F' 0 G' is n-pseudo-coherent (resp. pseudo-coherent, resp. perfect) iff both summands F' and G' are n-pseudo-coherent (resp. pseudo-coherent, resp. perfect). Proof.

To prove (a), we first note by "rotating the triangle" ([V]

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TR2 I Section 1 no. 1-1, or [H] I Section 1, TR2) that A' -* B' - • C is a homotopy fibre sequence iff C [ - l ] Z+ X -+ B' and B'[-l] -> C'[— 1] —+ A' are. Also, it is clear that a shifted complex F'[k] is (n -f k)pseudo-coherent iff F' is n-pseudo-coherent. So it suffices to prove the first statement of (a). So assume that A' is (n + l)-pseudo-coherent and that B' is n-pseudo-coherent. We need to show that C is n-pseudo-coherent. This question is local, so it suffices to show it in an arbitrarily small open nbd U of each point x £ X. By definition, for U small, we may choose representatives of the quasi-isomorphism classes of X and B' so that X is strict (n-hi)—pseudo-coherent and B' is strict n-pseudo-coherent. The map X —* B' in the derived category D(CV-Mod) is represented by a datum of strict maps X £- G' —* B'. Applying 2.2.4(a) and shrinking U, there is a strict (n-f l)-pseudo-coherent X\ and a quasi- isomorphism X'-^G-^X. Replacing A' by X', we may assume we have a strict map X —> B'. Now C is quasi-isomorphic to the mapping cone of X —* B'. In degree k this cone is ^4fc+1 0 Bk, and so is a vector bundle for fc > n. So the cone is strict n-pseudo-coherent. So C is n-pseudo-coherent as required, proving (a). Clearly (a) implies (b) for pseudo-coherence. To prove (b) for perfection, we reduce by rotating the triangle to show that if X and B' are perfect, then the mapping cone C is perfect. We now argue as in the proof of (a), locally taking strict perfect representatives for X and B\ using 2.2.4(c) to reduce to the case where there is a strict map of complexes A' —• B'. Then the mapping cone is strict perfect, and is quasi-isomorphic to C", which is hence perfect. This proves (b). The non-trivial part of (c) is to show that if F'0G* is n-pseudo-coherent (resp. pseudo-coherent, resp. perfect) then both factors F' and G' have the same property. Suppose F ' S G " is n-pseudo-coherent. We must show that for a small nbd U of x £ X that F'\U is n-pseudo-coherent. As the question is local, we may assume that F' 0 G' is quasi-isomorphic to a strict npseudo-coherent complex. Then there is an integer N >• 0 such that Hk(F') 0 Hk(G) = Hk(F 0 G) = 0 for k > N. Then F' and G' are trivially AT-pseudo-coherent, as 0 —• F is an A^-quasi-isomorphism. By descending induction on p for n < p < N we show that F' and G' are p-pseudo-coherent. To do the induction step, suppose we already know that F' and G' are (p-f l)-pseudo-coherent. Shrinking U, we may assume that F' and G' are strict (p + l)-pseudo-coherent, on replacing their representatives up to quasi-isomorphism. Consider the canonical homotopy fibre sequence, noting that a-p~1A = A/apA

(2.2.13.1)

ap^F'

0 ap+1G' -> F' 0 G' -*
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As F' and G' are strict (p-f l)-pseudo-coherent, cr p + 1 F"0<j p + 1 G' is strict perfect. As p > n, F' © G' is p-pseudo-coherent, by 2.2.13(a) proved above, F' any strict map, then there exists a strict pseudo-coherent complex F1' on X, a map a : E' —• F'', and a quasi-isomorphism x' : F'' -^ F' such that x' • a = x. (c) If E' is any strict n-pseudo-coherent complex, F' any n-pseudocoherent complex of quasi-coherent Ox-modules, and x : E' —• F' is any map, then there is a strict n-pseudo-coherent complex F'' on X, a map a : E' —» F'', and a quasi-isomorphism x' : F1' -^ F' such that x' • a = x. (d) If F' is any perfect complex of Ox-modules (perhaps not quasicoherent), then there is a strict perfect complex E' and an isomorphism in the derived category D(Ox-Mod) between E' and F'. (e) Let F' be any pseudo-coherent complex not quasi-coherent). Suppose either that F' 6 logically bounded, or else that X is noetherian Then there is a strict pseudo-coherent complex phism in D{Ox~Mod) between E' and F'.

of Ox-modules (perhaps Db(Ox~Mod) is cohomoof finite Krull dimension. E' on X and an isomor-

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Proof. To prove (b), we apply the Inductive Construction Lemma 1.9.5 with A = Q c o h ( X ) , V = the category of algebraic vector bundles, and C = the category of pseudo-coherent complexes in Q c o h ( X ) . T h e hypothesis 1.9.5.1 holds by 2.2.3 and the ample family 2.1.3(c). T h e proof of (c) is similar, using 1.9.5.9. To prove (a), we note that (b) yields E -+ F"' and F"' -=• F' with F" strict pseudo-coherent. F"' is perfect as F' is. We take an integer n < 0 so t h a t Ek = Fk = 0 = Hk(F"') for all Jfe < n. As in the proof of 2.2.12, Bn = k e r Z n ( F " ) - * Hn(F") is perfect, and hence a module of locally finite Tor-dimension, as Bn[n] is quasi-isomorphic to the homotopy fibre of the map of perfect complexes anF"' —• F"'. As X is quasi-compact, Bn _ ^ n ^ / / ) h a s globally finite Tor-dimension, say N. As # * ( F " ) = 0 for k < n, ZkF" = 5 * F " for Jb < n, and 0 -> £ * F " -+ F " * -> 5 * + * ^ " _> 0 is exact for k < n. Considering the induced long exact sequence for Tor£, ( , JF) and the fact that F"k is a vector bundle and hence is flat, we see t h a t Bn~1(F//') has Tor-dimension W - 1, etc. T h u s Bn-N{F"') ,,n N 2 1 n N is flat. T h e exact sequence F - ~ -* F ^ - ^ " -+ B ~ {F"') -> 0 shows t h a t Bn~N(F//') is also finitely presented. Thus Bn-N{F"') is a vector bundle ([SGA 6] I 5.6.3, or Bourbaki). Thus the good truncation -ls s t r i c t perfect. By choice of n, F"' —• F ' factors into quasiTn-N^pt/-j isomorphisms F"* -=• r ^ ^ F " " ) ^ F \ Setting F'* = r ^ ^ F " " ) then proves (a). To prove (d), we note t h a t the perfect F' is cohomologically bounded. T h e coherator B.16 yields an isomorphism in D(G x~Mod) between F' and a perfect complex of quasi-coherent modules RQ(F'). (Note X satisfies the semiseparation hypothesis of B.16 because of the ample family of line bundles, cf. B.7.) For n « 0}RQ(F') is quasi-isomorphic to rnRQ(F'), n which is still quasi-coherent. Now we apply (a) to 0 —• r RQ(F') to produce a strict perfect complex E' quasi-isomorphic to RQ(F') and F'. T h e proof of (e) is similar to t h a t of (d), using (b) instead of (a) at the last step. 2.3.2. P r o p o s i t i o n . Let X be a scheme with an ample family of line bundles. Let E' be a complex of quasi-coherent Ox-modules. Then there is a direct system of strict perfect complexes {Fa}, and a quasiisomorphism

(2.3.2.1) Proof.

(2.3.2.2)

limF0-=•£". Let E'(n)

E\n)

be the subcomplex r-n(E')

= . . . - * En~2

— En~l

-+ ZnE'

of E':

-> 0 -> 0 — . . . .

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Then E' is the direct colimit E' = lim is" (n) as n goes to oo. By increasing induction on n, starting with n = 0, we construct a complex E'{n}, which in each degree is an infinite direct sum of line bundles, and which is 0 in degrees above n. We construct this so that there is a quasi-isomorphism E'°{n} -^ E'(n), and a degree-wise split monomorphism E1'{n — 1} >-> E''{n) such that (2.3.2.3) commutes. E'{n-l)

(2.3.2.3)

—^—

J E'{n}

E'(n-1)

J —=—•

E'(n)

This construction is possible by the Inductive Construction Lemma 1.9.5 with A = Qcoh(X) and V — the category of sums of line bundles. Hypothesis 1.9.5.1 holds because of the ample family 2.1.3(a). Now we consider the directed system whose objects consist of an integer n > 0 and a strict bounded subcomplex Fa of E1'{n) such that in each degree Fa is a finite subsum of the given direct sum of line bundles which is E''{n} in that degree. The morphisms in the direct system are the obvious increases in n with inclusions of subcomplexes of lim is"'{n}. Given any finite subsum in E'{n}, d of it is of finite type, so is contained in a finite subsum with all degrees shifted one higher. Continuing in this way until we hit the bounding degree n, we see that any finite subsum in E1'{n} is contained in a complex Fa in the directed system. Thus for the subsystem with n fixed, limF a = Ef'{n}. Thus over the full directed system lim Fa — lim is 7 '{n}, which is quasi-isomorphic to lim£"(n) = E\ as required. 2.3.2.4. Porism. If E' in 2.3.2 also has Hk(E') = 0 for k > N, we can choose the Fa to be strictly 0 in degrees k > N. Indeed, then ^'{TV} ^ F'(n) ~ E' are quasi-isomorphisms, and we take the subsystem of Fa in E''{N}. 2.3.2.5. Remark. A general scheme is locally afflne, and hence locally has an ample family of line bundles. Thus 2.3.2 holds locally on a general scheme. In Deligne's terms ([SGA 4] V 8.2) a quasi-coherent complex is a local inductive limit of strict perfect complexes. 2.3.3. Corollary. Let X have an ample family of line bundles. Let E' be a complex of Ox-modules with quasi-coherent cohomology. Suppose either E' G D+(Ox-Mod) is cohomologically bounded below, or else that X is noetherian of finite Krull dimension. Then there is a direct system of strict perfect complexes Fa in C(Qcoh(X)), and an isomorphism in D](Ox-Mod) between l i m F a and

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E'. If E' is also cohomologically bounded above (by N), then all the F'a may be chosen to be strictly bounded above (by N). Proof. Recall that X is semi-separated because of the ample family, B.7. Then by B.16, B.17, the coherator gives a quasi-isomorphism of E' to a complex of quasi-coherent modules RQ(E'). We conclude by applying 2.3.2 to this RQ(E'). 2.4.1. Theorem. Let X be a scheme, and E' a perfect complex of Ox-modules. Then: (a) The derived functor RHom(E\ ) : D(Ox-Mod) -» D(Ox-Mod) is defined on all D(Ox-Mod), not just on D+(Ox-Mod). (b) RHom(E\ ) is locally of finite cohomological dimension. That is, for each point x £ X, there is an open nbd U of x and integers a < b such that if F' is any complex of Ox-modules on U with Hk(F') = 0 unless c < k < d (resp. unless c < k; resp. unless k < d), then Hk(RKom(E',F')) — 0 unless c — b < k < d — a (resp. unless c — b < k, resp. unless k < d — a). (c) IfF' is a complex with quasi-coherent cohomology, the RHom (E', F') has quasi-coherent cohomology (d) For any U open in X, for any direct system FQ of complexes of Ou-modules, and for any integer k, the canonical map (2.4.1.1) is an isomorphism of sheaves of Ou-modules: (2.4.1.1) \imHk (REom(E

\U,Fa)) -=•+ Hk (#Hom ( £ ' | { / , h m F a ) ) .

a

(e) If X has noetherian underlying space of finite Krull dimension, and Fa is any direct system of complexes of Ox-modules, then the canonical map (2.4.1.2) is an isomorphism (2.4.1.2) lim Mor^o^.Mod) (E\Fa) a

-=-» Mor z >( 0 x _ M o d ) I \

E\ljmFa a

This says roughly that MOTD(E\ ) preserves direct colimits, with the qualification that the direct system and its colimit are taken in the strict category of complexes C(Ox—Mod), not in D(Ox—Mod). (f) IfX is quasi-compact and quasi-separated, and Fa is any direct system of complexes of Ox-modules with quasi-coherent cohomology, then (2.4.1.2) is an isomorphism. (g) If X is quasi-compact and quasi-separated, and F'a is any direct system of complexes of Ox-modules which is uniformly cohomologically

HIGHER ALGEBRAIC K-THEORY OF SCHEMES bounded below (i.e., BnVaVfc < n Hk(Fa) isomorphism.

297

= 0), then (2.4.1.2) is an

Proof. Recall the construction of the mapping complex Hom(£", F') = fl Horn (Ep,Fq). (See [H] II Section 3 for the usual details). The statement (a) on extension from D + to D follows from the finite cohomological dimension given by (b) in the usual way, (cf. [H] I 5.3 7). To prove (b), (c), and (d), we note these are local questions. Given a point x G X} we take V to be a small nbd so that E'\V is quasiisomorphic to a strict perfect complex, and so reduce to the case where E' is a bounded complex of vector bundles (2.4.1.3)

K = . . . - • 0 -+ Ea -+ Ea+1 - > . . . _ • £ * _ > ( ) - + . . . .

Shrinking V further, we may assume all the E% are free of finite ranks Jfcj. Then Hom(2? , ',F') is ® fct F', and Hom(£",F') is the total complex of the bicomplex (2.4.1.4) consisting of finitely many finite sums of shifted copies of F': (2.4.1.4) Eom(E\F')\V

= 0 * - F ' [ - a ] <- e * - + 1 f [-a - 1] +- . . . <- e* f c F'[-6].

Clearly this Hom(£",F*)|V is exact in F\ and so represents REom(E',F')\V. Now it is clear that (b) holds with a and 6 the given strict bounds on E\ Also (c) is clear, and (d) is clear as (2.4.1.4) commutes with direct colimits. To prove (g) we recall ([H] I 4.7, [V] II Section 1 no. 2-3 4, or our 1.9.5 dualized) that any F' in D+(C?x~Mod) is quasi-isomorphic to a complex of injective Ox-modules F. Then Hom(J5 , ',7) is a complex of flasque sheaves ([SGA 4] V 4.10), and thus is deployed for computing i^r(A', ). Hence RT(X, REom(E\F')) is represented by T(X, Eom(E'J')) = Romx(E,r), which also represents RRomx(E'1 F'). The cohomology in degree 0, H°, of the complex Romx(E,F) is the group of chain homotopy classes of maps E' —* / ' , and as F is injective this is exactly Mor

Z?(O x -Mod)(^" 5

F').

Consider the Grothendieck spectral sequence (2.4.1.5) E™ = Hp (X;HgREom(E\Fa))

=> Hp+qREomx

(E\FQ)

.

By (d), HqREom(E\ ) preserves direct colimits, and by B.6, HP(X; ) preserves direct colimits, so the Ep'q term of the spectral sequence for l i m F a is the direct colimit of the E\A terms for the Fa. By the hypothesis of (g) that 3nVaVAr < n Hk(Fa) = 0, the convergence of the spectral sequences is uniform in a, and it follows that H*REomx(E\, )

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preserves direct colimits of such systems FQ (cf. [Thl] 5.50, 1.40). For H°(RRomx(E', )), this yields the desired conclusion of (g). It remains to prove (e) and (f). First we do this under the extra hypothesis that E' is globally quasi-isomorphic to a strict perfect complex. Then on replacing E' by a strict perfect complex, we get that KRom(E', ) is represented by Hom(£", ), which is a well-defined functor into the category of complexes, not just into D(0x-Mod). We consider the Grothendieck spectral sequence. As above, the E^q term, Hp(X]HqRRom(E\ ) preserves direct colimits. Also, there is an integer TV such that Hp(X;HqRKom(E',Fa) = 0 for p > N. In case (e), this is because X of finite Krull dimension has finite Zariski cohomological dimension by [Gro] 3.6.5. In case (f), this is because all HqREom(E\) Fa) are quasi-coherent Ox-modules by (c), and because X has finite cohomological dimension for quasi-coherent modules by B.ll applied to X —• Spec(Z). This N gives uniform convergence of the Grothendieck spectral sequence to Hp+q(RT(X; RRom(E', Fa)). From this uniform convergence and the fact that the E\A terms preserve direct colimits, it follows that the H*(RT(X; RRom(E\ )) preserve the appropriate direct colimits (cf. [Thl] 1.40). It remains in cases (e) and (f) to identify RT(X, RH.om(E\ F')) to RRomx(E', F ' ) , or more precisely to show that the canonical augmentation map (2.4.1.6) is an isomorphism (2.4.1.6)

MoTD(0x.Mod)(E\F')

-*

H°(X,REom(E\F')).

As we are considering the case where E' is strict perfect, the obvious devissage shows it suffices to prove (2.4.1.6) is an isomorphism when E' is a single vector bundle El. But as H°(X] ) is clearly isomorphic to Mor£>(ox-Mod)(^A' ), this reduces to the obvious adjointness of the functors ®E% and Rom(E\ ). This proves (e) and (f) when E' satisfies the extra hypothesis that it is globally quasi-isomorphic to a strict perfect complex. This extra hypothesis is always met if X has an ample family of line bundles 2.3.1(d). We prove (e) and (f) that the map (2.4.1.2) is an isomorphism without the extra hypothesis by induction on the number of open quasi-affine schemes needed to cover X. The induction starts because a quasi-affine scheme has an ample line bundle Ox- The induction step follows immediately by comparing the exact sequence of 2.4.1.7 below for l i m F a and the direct colimit of these exact sequences for the Fa. Thus the proof of Lemma 2.4.1.7 below will complete the proof of Theorem 2.4.1. 2.4.1.7. Lemma. Let U U V be a scheme, covered by open subschemes U and V. Denote the open immersions by j : U —> U U V, k :

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V —> U U V, and I : U D V -* U U V. Then for any complexes E' and F' of Ouuv-modules, there is a long exact sequence Mayer-Vietoris of groups of morphisms in the derived categories D(X) = D(Ox-modules) for X = U U V, U, V, and U n V;

(2.4.1.8) MorD(if)0"^W,i'^)

T®

MoTD{V)(k'E[l},k*F)

MoTD(unv)(CE[l],t'F) 9

I

Mo^D(yu^')(^'>•f , ')

Mor 0 ( t ,)(j*F,j*F)

© MoTD{Unv)(t'E 9

MotDiv)(k'E-,k'F) ,l'F)

I

MOTD{UUV)(E[~1],F)

i

Proof. Recall that j is flat, so j * = Lj* is exact. Let j \ : Ojj-Mod —»• Ot/uv-Mod extension by 0, the functor left adjoint to j * , ([SGA 4] IV 11.3.1). Then j \ is also exact. As both functors are exact and adjoint, they induce adjoint functors on the derived category. So there is a canonical isomorphism for all complexes G' in D(U) and H' in D(U U V) (2.4.1.9)

MovD{UuV)(j\G-,H)^

MorD(u)

(G',j*H)

There is a canonical exact sequence of complexes on U U V (2.4.1.10)

0 -+ t\t*E' -+ j\j*E'

0 k\k*E ~> E' - • 0

The maps in (2.4.1.10) are induced by the adjunction maps. Locally, and in fact after restriction to U and to V) this sequence is split exact, hence exact globally (cf. 3.20.4 below). This exact sequence induces a long exact sequence oiExt*D,UuVJ , F') = MOYD^UUV)(( )[—*]> ^") m t n e usual way ([V] I Section 1 1-2, or [H] I 6.1). Interpreting this long exact sequence by means of the isomorphisms (2.4.1.9) for j , k, and £ yields the long exact sequence (2.4.1.8).

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300

2.4.2. T h e o r e m . Let X have an ample family of line bundles. Let E' be a complex of Ox-modules, with quasi-coherent cohomology. Then the following are equivalent: (a) E' is pseudo-coherent. (b) For all integers n and k, and all direct systems {F^} of complexes of Ox-modules, the canonical map (2.4.2.1) is an isomorphism of Oxmodules (2.4.2.1) \imHk (REom(E',rnFa))

-=•> Hk (REom

IE\\imrnFa\

J.

(c) For all n, k, and all direct systems of strict perfect complexes F'a, the map (2.4.2.1) is an isomorphism. (d) E' is in D~(Ox-Mod), and for all n, k and all direct systems of strict perfect complexes Fa which are uniformly bounded above ( 3 m V a Vi > m F# — 0), the map (2.4.2.1) is an isomorphism. (e) For all integers n, and all direct systems {Fa} of complexes of Oxmodules, the canonical map (2.4.2.2) is an isomorphism (2.4.2.2) lim MorD(0x_Mod)(E',TnFa) a

^>

IE',\imrnFQ\

MoTD(Cx_Mod) \

a

/

That is, roughly speaking, Morr>(E', ) preserves direct colimits of uniformly cohomologically bounded below systems. (f) For all integers n, and all direct systems {FQ} of strict perfect complexes, the map (2.4.2.2) is an isomorphism. (g) E' in D~ (Ox-Mod), and for all n and all direct systems of strict perfect complexes Fa which are uniformly bounded above, the map (2.4.2.2) is an isomorphism. Proof. We fix an integer n, and note that the good truncation rn preserves direct colimits, so rn(\imFa) = l i m r n F a . Also, any map / : E' —• rnF' factors uniquely through E' —• rnE' as the composite with rnf : rnE' -> rnrnF' = rnF'. Similarly, REom(E', rnF') is quasin k l n isomorphic to RUon\(r ~ ~ E',r F') in degrees less than or equal to k. So all the conclusions of statements (b), (c), . . . , (g) depend only on rmE' for some m, where m possibly depends on n and k. Now as E' has quasi-coherent cohomology, rm~lE' is quasi-isomorphic to a complex of quasi-coherent modules by the coherator B.16. We show (a) => (b). As E' is pseudo-coherent, rm~lE' is (m — 1)pseudo-coherent. By 2.3.1(c) applied to a complex of quasi-coherent modules quasi-isomorphic to rm~1E', rm~1E' is quasi-isomorphic to some

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301

strict (m — l)-pseudo-coherent complex £"'. Then am~lEf' is strict perfect, and rmam~1Ef' is quasi-isomorphic to rmE\ So to prove (b) for given n and k, we take an appropriate m, and replace E' with a strict perfect complex which has truncation r m quasi-isomorphic to rm E'. Then it suffices to prove (b) for E' perfect, which follows from 2.4.1(d). Clearly (b) => (c). To show (c) => (d), the problem is to show t h a t E' is cohomologically bounded above. It suffices to show r°E' is so. As r°E' is in D + ( O x - M o d ) with quasi-coherent cohomology, it is quasi-isomorphic to the direct colimit of a direct system of strict perfect complexes Fa by 2.3.3. Consider the induced quasi- isomorphism in D + ( 0 x - M o d ) , T°E —• r°(\imFa) = \imr°Fa.

This gives a section of the sheaf i 7 0 ( / f f l o m ( r ° £ ^ r 0 ( H m F a ) ) ) ,

and hence via the m a p induced by E' —• r°E\ it gives a section of H°(REom(E', ^ ( l i m F ^ ) ) ) . By hypothesis (c), at every point x G X , the germ of this section comes from some section of a H°(RUom(E' ,T°(F^)))X. Replacing r°Fa by a quasi-isomorphic complex of injectives / ' to compute 7?,Hom, we get a germ in H°(Rom(E\ I'))x. This means there is a nbd U of x, and a chain m a p E' —• / ' on £/, representing the class of a m a p E -> r°Fa in D+(Ov-Mod), and such t h a t E' - • r°Fa -+ r ° ( l i m F a ) ~ r°(JE?') is the canonical map E' -> ^ E " in D + ( O t / - M o d ) . T h u s T°E' splits off r ° F a in D+(Ou- Mod). As F a is strict perfect, it is bounded above. Hence on U, r°Fa and so r°E' are cohomologically bounded above. T h u s E' is locally cohomologically bounded above. As X is quasi-compact (2.1.1), it follows that E' is globally cohomologically bounded above, as required. To see t h a t (b) => (e), (c) =» (f) and (d) => (g), we note t h a t the hypotheses imply t h a t E' is in D~ ( O x - M o d ) as above. Also, the TnF'a are uniformly cohomologically bounded below. Combining these facts, we see that REom(E,rnFa) is uniformly cohomologically bounded below. This shows t h a t the Grothendieck spectral sequence (2.4.1.5) computing H0(RRomx(E',rnF')) = MOTD(X)(E',TUF') converges uniformly in a . Now (b) => (e), (c) => (f), and (d) => (g) follow as in the proof of 2.4.1(g). Clearly, (e) => (f) and (f) => (g) as (f) implies t h a t E' is cohomologically bounded above similarly to the proof of (d) above. Finally (g) => (a). For it suffices to show for all n t h a t rnE' is n-pseudocoherent, since then E' is (n — l)-quasi-isomorphic to the n-pseudocoherent rnE\ and thus locally n-quasi-isomorphic to a strict perfect complex, 2.2.5.4. As rnE' is cohomologically bounded, 2.3.3 shows t h a t rnE' is quasi-isomorphic with a direct colimit of a direct system of strict perfect complexes Fa which are uniformly bounded above. Then there are quasi-isomorphisms rnE' ~ linijP a c^ l i m r n F ^ . By hypothesis (g), the m a p E' —> rnE'a

~ lim rnFa

factors E' —• rnFa

—»• l i m r n F a for some a.

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T H O M A S O N AND T R O B A U G H

Then rnE' is a retract of rnFa. As Fa is strict pseudo-coherent, rnFa is ra-pseudo-coherent, and hence rnE' is n-pseudo-coherent by 2.2.13(c). This completes the proof. 2.4.2.3. As pseudo-coherence is a local property, and any scheme is locally affine, hence locally has an ample family of line bundles, 2.4.2 is a useful criterion to apply locally on a general scheme. 2.4.3. Theorem. Let X be a scheme with an ample family of line bundles. Let E' be a complex of Ox-modules. Suppose that E' has quasi-coherent cohomology. Then the following are equivalent: (a) E' is

perfect.

(b) E' is quasi-isomorphic

with a strict perfect

complex.

(c) E' is cohomologically bounded below, and for any direct system of complexes Fa with quasi-coherent cohomology, Morf)(x)(E , ) preserves the direct colimit in that the map (2.4.1.2) is an isomorphism. (d) E' is cohomologically bounded, and the map (2.4.1.2) is an isomorphism for any direct system Fa of strict perfect complexes which are uniformly cohomologically bounded above. Proof, (a) => (b) by 2.3.1(d). We have (b) => (c) by 2.4.1(f). To show (c) => (d), the main point is to show t h a t E' is also cohomologically bounded above. But this follows from the proof of 2.4.2(c) => (d). Now to show (d) => (a), we note by 2.3.3 t h a t E' is quasi-isomorphic to the direct colimit of a direct system of strict perfect complexes which are uniformly bounded above. By (d), the quasi-isomorphism in D(Ox-Mod), E' —» l i m F a , factors through some F'a. T h u s in D((9;c-Mod), E' is a s u m m a n d of the perfect Fa. Hence by 2.2.13(c), E' is perfect as required. 2.4.3.1. As perfection is a local property, the theorem may be applied to the open affines on a general scheme to test perfection there. 2.4.4. To summarize, 2.4.3 roughly characterizes perfect complexes on schemes with ample families of line bundles as the finitely presented objects (in the sense of Grothendieck [EGA] IV 8.14 that Mor out of t h e m preserves direct colimits) in the derived category D(Ox-Mod)qc of complexes with quasi-coherent cohomology. On a general scheme, the prefect complexes are the locally finitely presented objects in the "homotopystack" of derived categories. (We must say "roughly characterizes" as we always take our direct systems in the category C ( O x - M o d ) of chain complexes, and have not examined the question of lifting a direct system if D(Ox-Mod) to C(Ox-Mod) up to cofinality.) This characterization, with Lemma 2.3.3, will be the basis for the key extension Lemma 5.5.1. A more immediate application will be to the

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functoriality statement 2.6.3 below. 2.5.1. ([SGA 6] 1.2). Let / : X - • Y be a map of schemes. For E' a strict perfect complex on Y, f*E' is clearly a strict perfect complex on X. This complex represents Lf*E\ as the vector bundles E% are flat over Oy and hence deployed for Lf*. In general, Lf* : D~(Oy-Mod) —» D~(Ox- Mod) sends perfect complexes to perfect complexes. For the question is local on Y, hence reduces to the case where Y is affine and where any perfect complex is quasiisomorphic to a strict perfect one. Similarly, Lf* = f* preserves strict pseudo-coherent complexes, and it follows t h a t Lf* : D~(Oy-Mod) —• D~ {OxMod) preserves pseudocoherence. If the m a p / also has finite Tor-dimension, and so induces a functor Lf* : Db(Oy-Mod) —• D 6 ( ( 9 x - M o d ) , Lf* preserves cohomologically bounded pseudo-coherent complexes. (Recall [SGA 6] III 3.1 t h a t / is said to have finite Tor-dimension if Ox is of finite Tor-dimension as a sheaf of modules over the sheaf of rings f~l(Oy) on X.) If E' and F' are strict pseudo-coherent, E' (&ox F ~ F' ®ox F ls also strict pseudo-coherent. If E' and F' are strict n-pseudo-coherent and strict m-pseudo-coherent respectively, E' <S>Q F' m a v be taken to be isomorphic to the strict (m + n)-pseudo-coherent E' ox F' in degrees greater than or equal to m+n. (We apply 1.9.5 with V = flat C?x- m °dules to replace E' by a quasi-isomorphic flat complex without changing E' in degree > m, etc.) It follows t h a t if E' and F' are pseudo-coherent, then E' ®Q F' is pseudo-coherent. If E' is perfect, hence of finite Toramplitude, and F' is cohomologically bounded and pseudo-coherent, then E' <S>o F' is also cohomologically bounded and pseudo-coherent. 2.5.2. Definition ([SGA 6] III). Let / : X -> Y be a m a p locally of finite type between schemes. T h e map / is n-pseudo-coherent at x E X if there is a nbd U of x and an open V C Y with / : U —• V factoring as / = gi, where i : U —+ Z is a closed immersion with i*Ofj n-pseudo-coherent as a complex on Z , and where g : Z —• V is smooth. (The property t h a t i*Ojj is n-pseudocoherent is independent of the choice of Z meeting the other conditions by [SGA 6] III 1.1.4. Hence this property depends only on / . ) T h e map / is n-pseudo-coherent if / is n-pseudo-coherent at x for all points x E X. T h e m a p / is pseudo-coherent if it is n-pseudo-coherent for all integers n. The m a p / is perfect if / is pseudo-coherent and of locally finite Tordimension. 2.5.3.

Examples

([SGA 6]).

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T H O M A S O N AND T R O B A U G H

(a) For Y noetherian, any / : X —• Y locally of finite type is pseudocoherent. (b) Any smooth map / : X —* Y is perfect. (c) Any regular closed immersion ([SGA 6] VII 1.4) / perfect ([SGA 6] III 1.1.2).

: X —*• Y is

(d) Any locally complete intersection morphism ([SGA 6] VIII 1.1) is perfect. (e) For Y not noetherian, a closed immersion need not be pseudocoherent, for Ox need not be even finitely presented over Oy. 2.5.4. T h e o r e m ([SGA6] III 2.5, 4 . 8 . 1 ) . Let f : X - • Y be a proper map of schemes. Suppose either that f is projective, or that Y is locally noetherian. Suppose that f is a pseudo-coherent (respectively, a perfect) map. Then if E' is a pseudo-coherent (resp. perfect) complex on X, Rf+(E') is pseudo-coherent (resp. perfect) on Y. Proof. T h e case / projective will follow from the slightly more general results 2.7 below on taking Z there to be a projective space bundle over Y locally. T h e case Y locally neotherian follows from the Grothendieck Finiteness Theorem ([EGA] III 3.2) t h a t i ? p / * preserves coherence, the finite cohomological dimension of Rf* ( B . l l ) , the strongly converging spectral sequence Rpf*(Hq(E')) =» # ? + ? ( # / * ( £ ' ) ) and criterion 2.2.8 t h a t a complex on a noetherian scheme is pseudo-coherent iff it is cohomologically bounded above with coherent cohomology. This shows Rf*(E') is pseudo-coherent when E' is. When / and E' are also perfect, we show t h a t Rf*(E) has locally finite-Tor-amplitude using base-change 2.5.5 locally. Then criterion 2.2.12 shows t h a t Rf*(E') is perfect. 2 . 5 . 5 . T h e o r e m ( [ S G A 6] III 3.7). Let Y be a quasi-compact scheme, and let f : X —• Y be a quasi-compact and quasi-separated map. Let E' be a cohomologically bounded complex of Ox-modules with quasicoherent cohomology, and let F' be a cohomologically bounded complex of Oy-modules with quasi-coherent cohomology. Assume either that F' has finite Tor-amplitude over Oy, or else that E' has finite Tor-amplitude over Ox • Then the canonical map is a quasi-isomorphism in D(Oy — Mod) (2.5.5.1)

RUE')

® £ y F' ^

Rf* (E' ®%x Lf*F')

.

Proof. See [SGA 6] III 3.7. We sketch the argument for F' perfect, the main case used in this paper. T h e question is local, so we may assume Y is affine. Then F' is quasi-isomorphic to a strict perfect complex 2.3.1(d). For F' strict perfect, filtering F' by crnF and comparing the long exact sequences of cohomology of the two sides of (2.5.5.1) induced

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305

by 0 —•
Let (2.5.6.1) be a pull-back

g'

X *——

(2.5.6.1)

/J

X'

•

J/'

y <

v 9

Suppose Y is quasi-compact, and that f is a quasi-compact and quasiseparated map. Suppose that f and g are Tor-independent over Y so that given x G X, y' £ y with f(x) = y = g(yf), then for all integers p > 1 we have (2.5.6.2)

ToTpOYy(Ox,x,

OY>,y>) = 0.

Let E' be a cohomologically bounded complex on X', with quasi-coherent cohomology. Suppose either that E' has finite Tor-amplitude over the sheaf of rings f~l{Oy) on the space X, or else that the map g has finite Tor-dimension. Then there is a canonical base-change quasi-isomorphism (2.5.6.3)

Lg*Rf*E' -=•

RflLg'"E'.

Proof. [SGA 4] XVII 4.2.12 defines the map, and [SGA 6] IV 3.1 shows it is an isomorphism in the derived category (cf. 3.18. below). 2.5.6.4. An examination of the proof shows that the Tor-independence hypothesis (2.5.6.2) may be weakened to be required only for those x in a closed subspace Z C X such that E' is acyclic on X — Z. 2.5.7. Proposition (Trivial Duality). Let f : X —• Y be a map of schemes. Then for E' in £ > - ( 0 y - M o d ) and G' in D+(Ox-Mod) there is a canonical isomorphism derived from adjointness between f* and /* (2.5.7.1) MoTDi0x-Mod){Lf*E\

G) 2

Mor D ( c y _ M od) ( # ' , Rf*G') .

Proof. This is [SGA 4] XVII 2.3.7. See also [SGA 6] IV 3, [H] II 5.1, [V] II, Section 3 no. 3. Also see [SGA 4 | ] , Erratum to [SGA 4], to correct an argument preceding all our cited base-change results.

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2.6.1. Lemma. Let X be a quasi-compact scheme, \Y\ a closed subspace, and U = X — \Y\ the complementary open subscheme. Then (a) If there is a finitely presented closed immersion i : Y —• X with underlying space \Y\, then U is quasi-compact and j : U —• X is a quasicompact map. (b) IfY = Spec(Ox/J) and Y' = Spec(0 x /X) are two finitely presented closed subschemes ofX, both with underlying space \Y\, then there exists an integer n > 1 such that Jn C X and Xn C J. (c) If X is also quasi-separated} and if U is quasi-compact, then there exists a finitely presented closed immersion i : Y —• X with underlying space \Y\. Proof, (a) It suffices to show that j is a quasi-compact map. Let Spec(^4) C X be an affine open, and let Y D Spec(yt) = Spec(A/J). As Y is finitely presented, J C A is a finitely generated ideal, say J = ( c r i , . . . , a n ) . Then U fl Spec(^4) is quasi-compact as required, for it is covered by the finitely many affine opens Spec(A[l/ai]) for i = 1 , . . . , n. (b) As the ideals J and 2 are of finite type, and as the support of both O/J and O/l are |Y|, there is a positive integer n so that Jn{0/1) = 0 = ln(0/J) by [EGA] I 6.8.4. Then Jn C 1 and Jn C J. (c) \Y\ is the underlying space of a reduced closed subscheme Yred = Spec(Ox/fC). The ideal K, is the direct colimit of its finitely generated subideals Ja, so K - lim Jat by [EGA] I 6.9.9. Each Ya = Spec{0/Ja) is a finitely presented closed subscheme of X. As OjK — WmO/Ja, Yred = limY a , and U = X — Yred is the direct colimit of the open subschemes X — Ya of X, U — lim(X — Ya). As U is quasi-compact, there is an a such that U — X — Ya. Then Ya has underlying space \Y\ as required. 2.6.2.1. Definition. Let / : X' —+ X be a map of schemes, with X quasi-compact. Let \Y\ C X be a closed subspace, which is the underlying space of some finitely presented closed immersion i : Y —> X (see 2.6.1(a,c)). We say / is an isomorphism infinitely near Y if the following two conditions hold: (a) / is flat over the points of Y; that is, for all x' 6 X' with y — f(x') in \Y\ C X, 0X',x' is flat over Ox,y (b) / induces an isomorphism of schemes Y' — Y x X' —• Y. 2.6.2.2. Lemma-Definition. In the presence of hypothesis 2.6.2.1(a), the condition 2.6.2.1(b) does not depend on the choice of finitely presented closed subscheme Y with underlying space \Y\. In particular, if (b) holds for Y — Spec(Ox/J), it also holds for any of the

H I G H E R A L G E B R A I C K-THEORY O F S C H E M E S infinitesimal thickenings Y(n) = Spec(Ox/Jn)an isomorphism infinitely near the subspace \Y\.

307

We may then say f is

P r o o f . For Y = S p e c ( O x / J ) , Y' is Spec(0X'/J') C Xf with J' the l ideal generated by f~ (J). As / is flat over the points of |Y|, and these are the only points x where Jx ^ Ox,x, we see t h a t f*J—> f*Ox = Ox1, is a monomorphism. So, in fact, J' = f*J. Similarly f*Jn —*• O x ' is a monomorphism, so J m = / * Jn. It follows that J,k/J,k+n = / * ( i 7 A 7 t 7 * + n ) , ^ / * preserves cokernels. Consider t h e exact sequence (2.6.2.2.1) of sheaves supported on the space \Y\: 0 — Jk+i/jk+"

(2.6.2.2.1)

— jk/jk+n _ jk/jk+i

_^ 0

As / is flat over the points of |Y|, / * of this sequence is also exact, and is in fact the exact sequence on | Y ' | where Jm is replaced by J,m everywhere. T h e m a p / induces a map between these two exact sequences. As / : Y' —*• Y is an isomorphism of schemes,the induced m a p O/ J —• 013', and even Jk/Jk+1 = Jk ® Of J — J* ® W = J'k/J'k+1 are isomorphisms of sheaves on the space | Y | = | Y ' | . By induction on n , using the 5-lemma on the m a p between the two exact sequences, we get t h a t / induces an isomorphism for all k and n > 1 (2.6.2.2.2)

jkjjk + n s

J>k/J>k+n

In particular, forfc= 0 we get Ox/Jn = 0X'/J'n, so t h a t Y ' ^ -+ n Y^ ) is an isomorphism of schemes for all n > 1. Now for Z = Spec(C?A~/^-) a finitely presented closed subscheme of X with underlying space |Y|, there exists an n such t h a t , 7 n C X by 2.6.1(b). T h e n the closed immersion Z -> X factors as Z - • Y -* X. T h u s Z' = Z x X ' - » £ is the pullback of the isomorphism Y'( n ) = Y, x x and so Z' -^ Z is an isomorphism as required. 2.6.3. T h e o r e m . Let f : Ar/ —*• X be a quasi-separated map of quasi-compact schemes. Let i : Y —» X be a finitely presented closed immersion. Suppose that f is an isomorphism infinitely near Y (2.6.2.1). Set Y' = f-x(Y) = YxX'. Then (a) For E' in D~ (Ox— Mod) with quasi-coherent cohomology and which is acyclic on X — Y, the canonical map E' —> Rf*Lf*E' is an isomorphism

in D ~ ( ( 9 A ' — M o d ) . f

(b) For E ' in D~(Ox'— Mod) with quasi-coherent cohomology and which is acyclic on X' — Y', the canonical map Lf*Rf*E'' —• El% is an isomorphism in D~(Ox'— Mod).

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(c) IfEf' is pseudo-coherent on X' and acyclic on X' — Y', then Rf+E'' is pseudo-coherent on X. (d) IfE'' is perfect on X' and acyclic on X' — Yf, then Rf*E'' is perfect on X. Proof. First we note that / is a quasi-compact map ([EGA] I 6.1.10 iii). Thus Rf* has bounded cohomological dimension on complexes with quasi-coherent cohomology (B.6, B.ll), and so Rf* is defined on D(0X'-Mod) and D-(Ox>-Mod), as well as the usual £>+(£>*/-Mod). Using the functorial Godement resolution T to compute sheaf cohomology as in Deligne's treatment in [SGA 4] XVII 4.2, we realize Rf* as an exact functor defined on the level of complexes, as Rf* = /* o T (cf. [Thl] Section 1). As Rp f* preserves direct colimits of modules by B.6, the usual uniformly converging spectral sequence argument a la 2.4.1 shows that Rf* preserves up to quasi-isomorphism the colimits of directed systems of complexes with quasi-coherent cohomology. Let j : U —• X be the open complement to Y. Then j is a quasi-compact map by 2.6.1(a), and is also quasi-separated. Hence the discussion of the preceding paragraph applies also to Rj*. We define local cohomology RTy as the canonical homotopy fibre of the map of complexes 1 —• Rj*j*, as justified by [SGA 4] V 6.5. (More precisely we use the map 1 —• j+j* —* j+Tj* = Rj*j* induced by the augmentation 1 —• T into the Godement resolution.) Then RTy preserves up to quasi-isomorphism the colimits of directed systems of complexes with quasi-coherent cohomology, as this is true of 1 and Rj+j*. Similarly, RTy has finite cohomological dimension on complexes with quasi-coherent cohomology, and preserves quasi-coherence of cohomology. Now to prove (a). Using the finite cohomological dimension of i2/*, for any k, Hk(Rf*Lf*E) equals Hk{Rf+Lf*TnE) for all n sufficiently small. Thus we reduce to the case where E' is cohomologically bounded below as well as above. Now the usual devissage reduces us to the case where E' is a single quasi-coherent module considered as a complex concentrated in one degree. For we induct on n using the homotopy fibre sequence Hn(E') —> rnE' —» r n + 1 jE" and the 5-lemma to reduce to proving the theorem for the Hn(E') as complexes. Now as Rf* and Lf* preserve direct colimits, writing the module E as the direct colimit of its submodules of finite type ([EGA] I 6.9.9), we reduce to the case where E is a quasi-coherent module of finite type, and which vanishes off Y. As the defining ideal J of Y is of finite type, there is a positive integer n such that JnE = 0 ([EGA] I 6.8.4). By devissage, it suffices to prove that the canonical map 1 —• Rf+Lf* is a quasi-isomorphism for JkE/Jk+1E with k = 0 , 1 , . . . ,n - 1. Thus we reduce to the case where E is an Ox/J = Oy module, E = i*E = Ri+E. Then using

HIGHER ALGEBRAIC K-THEORY OF SCHEMES

309

2.5.6, which applies as / is flat over the points of Y (recall 2.5.6.4), we have Rf*Lf*E ~ Rf*Lf*uE~ ~ RfJ'*Lf'*E~ ~ uRflLf'*E~. But as / ' : Y' —• Y is an isomorphism, we continue the chain of quasiisomorphisms: i*Rf+Lf'*E~ ~ i*E~ ~ E. This proves (a). The proof of (b) requires only a change of notation in this argument (e.g., Lf*Rf* in place of i£/*L/*, X' in place of X, etc.). To prepare to prove (c) and (d) we first note that if E' is acyclic on X — Y, and if F' is any complex on X with quasi-coherent cohomology, then RTyF' —• F' induces an isomorphism: (2.6.3.1)

MorD{0x_Mod)(E\

RTyF')^

MoTD(0x.Mod)

(E\

F') .

For the homotopy fibre sequence RTyF' —» F' —> Rj*j*F induces a long exact sequence:

1 Mor D (E[l],Rj.j*F) MorD (2.6.3.2)

4 (E , R r

2

Mor D ( u ) (j*E[l],j*F)

=

0

S

Mor D ( u ) (j*E,j*F)

=

0

YF ) | Mor D (E , F )

I

Mor D (E , R y * F )

I The horizontal isomorphisms result from trivial duality 2.5.7. for j : U —+ X, and the fact that j*E' ^ 0 as E' is acyclic on U. The exactness of (2.6.3.2) yields (2.6.3.1), as required. In particular, (2.6.3.1) holds for E' -RUE'' in (c) and (d). As final preparation, we note that RYy is also represented on the level of complexes as Ty of the Godement resolution on X. This yields RTy as a complex of modules over the localization of Ox along Y. As / is flat over the points of Y, f*RTy would then represent Lf*RTy on the level of complexes. We now switch over to this representative of RTy. As a statement in the derived category, (2.6.3.1) remains true. Now we prove (c) that RUE'' is pseudo-coherent. As the question is local on X, we may assume X is affine and hence has an ample family of line bundles. We appeal to criterion 2.4.2(g). Let {Fa} be a direct system of strict perfect complexes on X. Then we have a sequence of isomorphisms:

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THOMASON AND TROBAUGH

(2.6.3.3) lira M o t D ( 0 x . M o d ) {Rf.E'\

rnFa)

a

Slim

RVYrnFa)

MoTDi0x_Mod)(Rf,E'\

a

Slim

MOT D(0x-Mod)

(Rf*E'\

Rf.Lf*

RTYrn

Fa)

a

Slim

Lf*RTYTnFa)

M0TD{Ox,_Mad)(LrRf,E'-,

a

Slim

Lf*RrYTnFa)

MorD(0x,_Mod){E'-,

a

=

M o r p ( 0 x , _ M o d ) (E'\ \

\imLf*RrYTnFQ <* LrRTYrn\imFa)

=

M0TDiOxl_Mod)(E'\

=

MoTDi0x,-Mod)

=

MorD{0x-Mod)

=

MorD{0x_Mod)(Rf*E'\

=

MOTD{0x-Mod)

LrRTYrnlimFQ)

(Lf*Rf*E", (Rf*E'\

(Rf*E'\

Rf*Lf*RrYTnlunF^ RTYTn\imFa) TnKmF^).

Here the isomorphisms are successively justified by (2.6.3.1), (a), trivial duality 2.5.7, (b), pseudo-coherence of E1' with the finite Tor-dimension of Lf*RYY = f*RTY and 2.4.2(e), the fact t h a t Lf* and RTY preserve direct colimits of complexes with quasi-coherent cohomology, (b), trivial duality 2.5.7, (a), and (2.6.3.1). By 2.4.2(g), the isomorphism between the first and last terms of (2.6.3.3) shows that Rf*E'' is pseudo-coherent, proving (c). T h e proof of (d) t h a t Rf*E'' is perfect requires only removal of the truncation rn in the proof of (c), and the use of 2.4.3(d) on X and 2.4.1(f) on X' in place of 2.4.2(g) and 2.4.2(e). 2.7. P r o p o s i t i o n ( [ S G A 6]). . Let f : X —• Y be a proper map of schemes. Suppose that locally on Y, the map factors as f = hoi, with h : Z —*• Y a flat and finitely presented map, and i : X —» Z a closed immersion. Then (a) If uOx is perfect Rf*E' is perfect on Y.

on Z, and E' is a perfect

complex

on X,

then

H I G H E R A L G E B R A I C K-THEORY O F S C H E M E S (b) Ifi*Ox is pseudo-coherent on Z, and if h : Z —• Y has an family of line bundles (2.1.2(f)), and if E' is a pseudo-coherent on X, then Rf*E' is pseudo-coherent on Y.

311 h-ample complex

P r o o f . T h e question is local on Y, so we may assume t h a t Y = Spec(A) is affine, and t h a t the factorization / = h o i exists globally over Y . In case (b), Z will then have an ample family of line bundles 2.1.2(f). We write A — lim Aa as a direct colimit of its noetherian subrings of finite type over Z. On passing to a cofinal system of a, there will be flat and finitely presented maps ha : Za —* Ya — S p e c ( A a ) and closed immersions ia : Xa —• Zai such t h a t fa = ha o ia is proper, Zp = Za x (Ap), Z = limZp, etc. This all follows by [EGA] IV 8.9.1, 8.10.5, 11.2.6. Let ga : Y —• Y a , ga : Z —• Z a be the canonical maps. We use this noetherian approximation to do case (a). T h e complex i+E' is pseudo-coherent on Z by [SGA 6] III 1.1.1. As £ " has finite Toramplitude over X and i*Ox has finite Tor-dimension over Z, the complex n £ ' has finite Tor-amplitude over Z ([SGA 6] III 3.7.2). T h u s i*E' is perfect on Z by 2.2.12. By 3.20 below, whose proof does not depend on this 2.7 (or by [SGA 6] IV 3.2), there is an a in the direct system and a perfect complex E'Q on Za such that i*E' is quasi-isomorphic to Lgf*Ea. As i*E' is acyclic on Z — X, by taking a larger we may assume t h a t Ea is acyclic on Za — Xa, again by 3.20. (Danger: We do not know t h a t the maps ia and g'a are Tor-independent, and so cannot appeal to 2.5.6. Thus we do not claim t h a t we can take Ea to be i a * of a perfect complex on Xa.) As Za is noetherian, Ea has coherent cohomology sheaves by 2.2.8, which are modules over Oxa(n) f ° r some infinitesimal thickening Xa(n) of Xa in Za ([EGA] I 6.8.4). T h e maps fQ(n) = ha\ : Xa(n) -> Ya are proper as / a r e d = /<*(™)red is proper ([EGA] II 5.4.6, I 5.3.1). T h u s RhQ*(E'a) has coherent cohomology, and in fact is pseudo-coherent by the noetherian case of 2.5.4 proved above. As ha : Za —» Ya is flat and E'a has finite Tor-amplitude over Oza, it follows t h a t Ea has finite Tor-amplitude over / ~ 1 ( O y a ) . Also, Rha* has finite cohomological dimension by B . l l . Then the projection formula 2.5.5 applies to show t h a t Rha*(Ea) has finite Tor-amplitude. T h u s Rha*(EQ) is perfect on Ya by 2.2.12. Then LgaRha*(Ea) is perfect on Y. But as ha is flat, the Base-change Theorem 2.5.6 yields a quasi-isomorphism LgaRha*(Ea) ^ Rh*(LgaEa). But this complex is quasi-isomorphic to Rh*(i*E) 2^ Rf*(E'). Thus Rf*(E) is perfect, proving (a). To prove (b), it suffices to show for all n t h a t the truncation rnRf*E' = rnRh*(i*E) is n-pseudo-coherent. By B . l l , Rh+ has finite cohomological dimension, say, k. Then rnRh*(i*E) is quasi-isomorphic to TnRh*(rn~k

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T H O M A S O N AND T R O B A U G H

i*E'). By the hypothesis of pseudo-coherence of i*Ox on Z and the devissage of [SGA 6] III 1.1.1, the complex i*E is pseudo-coherent on Z'. T h e complex rn~ki*E is then (n — k — l)-pseudo-coherent by 2.2.6 and 2.2.5, as i*E —• rn~ki*E is an (n —fc— l)-quasi-isomorphism. By use of the coherator B.16, we see t h a t the cohomologically bounded rn~kuE' is quasiisomorphic to a complex of quasi-coherent O^-modules. (Note Z meets the semi-separated hypothesis of B.16, as Z has an ample family of line bundles.) Then using 2.3.1(c), we see t h a t rn~ki*E' is quasi-isomorphic to a strict (n — k — l)-pseudo-coherent complex £"'*. Then y with y — h(z), for all z in Z — U. Suppose for some locally finitely presented closed subscheme X - • Z with Z -X -V (2.6.1) t h a t /i|X - • Z is proper. Then for any perfect complex E' on Z which is acyclic on U, is perfect on Y.

Rh*(E')

If there is an /i-ample family of line bundles on Z and if E' is a pseudocoherent complex on Z which is acyclic on U, then Rh*E is pseudocoherent on y . Proof. This is the main point of the argument proving 2.7, with attention now paid to the fact t h a t the appeals to 2.5.6 and 2.5.5 really need only the flatness hypothesis in 2.7.1. 2.7.2. Remark. If / = hoi with h i : X —» Z a closed immersion, then i*Ox is a pseudo-coherent morphism by ([SGA t h a t i*C?x is perfect on Z iff / is a perfect 3.6, 4.1, or III 4.4).

: Z —• Y a smooth m a p and is pseudo-coherent on Z iff / 6] III 1.1.4, 1.2, 1.1). We have morphism by ([SGA 6] III 1.1,

3. K - t h e o r y o f s c h e m e s : definition, m o d e l s , functorialities, excision, limits 3.1. Definition. For X a scheme K(X) is the K-theory spectrum of the complicial biWaldhausen category (1.2.11) of perfect complexes of globally finite Tor-amplitude (2.2.11), in the abelian category of all Oxmodules. (By the default conventions of 1.2.11, the cofibrations in this biWaldhausen category are the degree-wise split monomorphisms, and the weak equivalences are the quasi-isomorphisms.)

HIGHER ALGEBRAIC K-THEORY OF SCHEMES

313

For Y a closed subspace of X, K(X on Y) is the A'-theory spectrum of the complicial biWaldhausen subcategory of those perfect complexes on X which are acyclic on X — Y. 3.1.1. It is clear from the description of A'0 m 1.5.6. that K0(X) indeed the Grothendieck group "K (X)n of [SGA 6] IV 2.2.

is

3.1.2. For X quasi-compact, any complex of locally finite Tor-amplitude, and in particular any perfect complex, automatically has globally finite Tor-amplitude. 3.2. Definition. For X a scheme, A' naive (X) is the AT-theory spectrum of the complicial biWaldhausen category of strict perfect complexes in the category of Ox -modules. Similarly for A n a l v e (X on Y). 3.2.1. Indeed, K™W€(X) is the naive Grothendieck group " A * ( * W of [SGA 6] IV 2.2. Soon we will see that Knaiye(X) is the Quillen Atheory of X in general, and that A' naive (X) is homotopy equivalent to K(X) when X has an ample family of line bundles. Thus locally A' naive and K agree, but it will be K that has good local-to-global properties (cf. Sections 8, 10, esp. 8.5, 8.6). 3.3. Definition. For X a scheme, G(X) is the A'-theory spectrum of the complicial biWaldhausen category of all pseudo-coherent complexes with globally bounded cohomology in the abelian category of all Oxmodules. For Y a closed subspace of X, G(X on Y) is the A'-theory spectrum of the subcategory of those complexes acyclic on X — Y. 3.3.1. Indeed G0(X) is the Grothendieck group "A (X)" of [SGA 6] IV 2.2. Quillen [Ql] defined higher K1- or G-theory only for noetherian schemes, and for these his K'{X) is homotopy equivalent to G(X) as we will see. 3.4. The construction of the A'-theory spectrum proceeds not from the derived category, but from the underlying complicial biWaldhausen category, and is known to be functorial only in the biWaldhausen category. By 1.9.8, the choice of underlying biWaldhausen category is not critical in that all related choices give the same A'-theory. But to exhibit all the required functorialities in A'-theory, many different underlying model categories must be employed. Hence we proceed to compile lists of the most useful models. 3.5. L e m m a . Let X be a quasi-compact scheme. Consider the following list of complicial biWaldhausen categories (cf. 1.2.11) in the abelian category of all Ox-modules (or in cases 6, 7, 8 in the abelian

314

T H O M A S O N AND T R O B A U G H

category of diagrams A —• B of 3.5.1. perfect

Ox-modules):

complexes,

3.5.2. perfect strict bounded

complexes,

3.5.3. perfect

bounded

above complexes

of flat

3.5.4. perfect

bounded

below complexes

of injective

3.5.5. perfect

bounded

below complexes

of Basque

3.5.6. diagrams E' -^ F' consisting perfect complexes,

Ox-modules, Ox-modules, Ox-modules,

of a quasi-isomorphism

3.5.7. diagrams as in 3.5.6, but with E' degree-wise Bat bounded and F' degree-wise inject ive bounded below, perfect complexes,

between above

3.5.8. diagrams as in 3.5.6, but with E' degree-wise Bat bounded above and F' degree-wise Basque bounded below perfect complexes. Then the obvious inclusion functors induce homotopy equivalences of all their K-theory spectra, so all are homotopy equivalent to K(X). Similarly there are homotopy equivalences to K(X on Y) from the K-theory spectra of the various subcategories of complexes which are also acyclic on X — Y. Moreover, the results will hold for non-quasi-compact X if we add everywhere the extra condition that the perfect complexes have globally Bnite Tor-amplitude. Proof. More precisely, we need to show that the inclusion functors of categories 3.5.2 through 3.5.5 into 3.5.1 induce homotopy equivalences on K-theory spectra. We also show t h a t the inclusions of 3.5.8 and 3.5.7 into 3.5.6 induce homotopy equivalences. The two functors from 3.5.6 to 3.5.1 sending E' -^ F' to E' and to F' respectively both induce homotopic homotopy equivalences, inverse to the homotopy equivalence induced by the functor from 3.5.1 to 3.5.6 sending E' to the diagram 1 : E' = E''. T h e last statement follows immediately from 1.5.4. T h e inclusion functors will induce homotopy equivalences in A'-theory because they will induce equivalences on the derived categories of the complicial biWaldhausen categories, allowing appeal to 1.9.8. (One could also appeal directly to Waldhausen approximation 1.9.1 given 1.9.5 and the facts below.) By 1.9.7 (or its dual), to prove the equivalence of derived categories it suffices to show for all B in the target category t h a t there is an A in the source category and a quasi-isomorphism A - ^ B (resp., B - ^ A). As the category of O^-modules has enough flat and enough injective objects (e.g., [H] II Section 1), and as all injectives are flasque, 1.9.5 yields the well-known fact that for any cohomologically bounded complex B\ one has quasi-isomorphisms A' A. B' -^ C where A' is bounded above and degree-wise flat, and C is bounded below and degree-wise injective, a fortiori degree-wise flasque. This immediately proves the equivalence of

HIGHER ALGEBRAIC K-THEORY OF SCHEMES

315

derived categories for the inclusions of 3.5.3, 3.5.4, and 3.5.5 into 3.5.1. If one considers an intermediate biWaldhausen category whose objects are diagrams of quasi-isomorphisms E' A F' of perfect complexes with E' degree-wise flat and bounded above, one sees similarly that the inclusion of this category into 3.5.6 and the inclusions of 3.5.7 and 3.5.8 into this intermediate category induce equivalences of derived categories. It remains only to show that the inclusion of 3.5.2 into 3.5.1 induces an equivalence of derived categories. But a perfect complex of globally finite Tor-amplitude is globally cohomologically bounded. Thus such a perfect E' is quasi-isomorphic to the truncations for suitable n < 0 and m > 0 : E' -=• rnE' <=- r^m(rnE'). As rnE' is strict bounded below and m n r- (r E') is strict bounded, these quasi-isomorphisms show that the inclusion of 3.5.2 into the intermediate biWaldhausen category of perfect strict bounded below complexes and the further inclusion of this intermediate category into 3.5.1 induce equivalences of derived categories. Hence all the inclusions induce equivalences of derived categories, and 1.9.8 yields the result. Clearly the proof works if we impose everywhere the extra condition that the complexes are to be acyclic on X — Y, yielding the result for K(X on Y). 3.6. L e m m a . For X either quasi-compact and semi-separated (B.7), or else noetherian, Lemma 3.5 remains true if the following categories are added to the list in 3.5. So all have K-theory spectra homotopy equivalent to K(X); 3.6.1. perfect complexes of quasi-coherent Ox-modules, 3.6.2. perfect complexes of injective objects in Qcoh(Ar). Proof. The inclusion of 3.6.2 into 3.6.1 induces an equivalence of derived categories as Qcoh(X) has enough injectives (B.3). The inclusion of 3.6.1 into 3.5.1 has an inverse equivalence on the derived category, given by the coherator (B.16). Now we apply 1.9.8. 3.7. L e m m a . For X noetherian, Lemmas 3.5 and 3.6 remain true if the following categories are added to the lists. Hence all have K-theory spectra homotopy equivalent to K(X): 3.7.1. perfect complexes of coherent Ox-modules, 3.7.2. perfect strict bounded complexes of coherent Ox-modules. Similarly for K(X on Y) if we add the extra condition that the complexes are acyclic on X — Y. Proof. The inclusion of 3.7.1 into 3.6.1 induces an equivalence of derived categories, as we prove by the Inductive Construction Lemma 1.9.5, whose hypothesis 1.9.5.1 is met by 2.2.8 and [EGA] I 6.9.9. We conclude by the usual 1.9.7 and 1.9.8.

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THOMASON AND TROBAUGH

The inclusion of 3.7.2 into 3.7.1 induces an equivalence of derived categories, as we see using truncations just as for the inclusion of 3.5.2 into 3.5.1. 3.8. L e m m a . For X with an ample family of line bundles (2.1.1), Lemma 3.5 remains true if the following categories are added to the list in 3.5 and 3.6. In particular, all their K-theory spectra are homotopy equivalent to K(X): 3.8.1. perfect complexes of quasi-coherent Ox-modules, 3.8.2. perfect bounded above complexes of Bat quasi-coherent Oxmodules, 3.8.3. strict perfect complexes. Similarly for K(X onY) if we add everywhere the extra condition that the complexes are acyclic on X — Y. Proof. We note by 2.1.1 and B.7 that X is quasi-compact and semiseparated, so that 3.6 applies. We note 3.8.1 = 3.6.1. Finally the inclusion of 3.8.3 into 3.8.2 and into 3.8.1 induces an equivalence of derived categories by 2.3.1(d). 3.9. Corollary. For X a scheme with an ample family of line bundles, there is a natural homotopy equivalence Knaive(X) -^ K(X). Proof.

Lemma 3.8.3 and Definition 3.2. (cf. [SGA 6] IV 2.9).

3.10. Proposition. For X any scheme, A' naive (X) is naturally homotopy equivalent to Quillen's K-theory spectrum of X ([Ql]). For X with an ample family of line bundles, K(X) is naturally homotopy equivalent to the Quillen K-theory spectrum of X. Proof. The first statement follows from 1.11.7, as strict perfect complexes are precisely bounded complexes in the exact category of algebraic vector bundles on X. The second statement follows by 3.9. 3.11. Lemma. The obvious inclusions of the complicial biWaldhausen categories in the following lists induce homotopy equivalences on K-theory under the conditions preceding each list. In particular, the K-theory spectra are all homotopy equivalent to G(X). For X a scheme: 3.11.1. cohomologically bounded pseudo-coherent complexes of Oxmodules, 3.11.2. pseudo-coherent strict bounded complexes of Ox-modules, 3.11.3. cohomologically bounded pseudo-coherent complexes of flat Ox-modules,

HIGHER ALGEBRAIC K-THEORY OF SCHEMES

317

3.11.4. cohomologically bounded pseudo-coherent complexes of injective Ox-modules, 3.11.5. cohomologically bounded pseudo-coherent complexes of Basque Ox-modules, 3.11.6. diagrams of quasi-isomorphisms E-^F' of cohomologically bounded pseudo-coherent complexes, with E' degree-wise flat and F' degree-wise Basque, 3.11.7. as in 3.11.6., except F' is degree-wise injective instead of merely Basque. For X either a quasi-compact and semi-separated scheme, or else noetherian, one may add to the list: 3.11.8. cohomologically bounded pseudo-coherent complexes of quasicoherent Ox-modules, 3.11.9. cohomologically bounded pseudo-coherent complexes of infectives in Qcoh(X), For X with an ample family of line bundles, one may add to the list: 3.11.10. cohomologically bounded strict pseudo-coherent

complexes.

Proof. Note "cohomologically bounded" means "globally cohomologically bounded," and recall the default conventions of 1.2.11 for cofibrations and weak equivalences. The proof of 3.11 exactly parallels that of 3.5-3.8, and hence we leave it to the reader. 3.12. following lences of 3.12.1. 3.12.2.

Lemma. For X a noetherian scheme, the inclusion of the biWaldhausen categories into 3.11.8 induce homotopy equivatheir K-theory spectra to G(X). cohomologically bounded complexes of coherent Ox-modules, strict bounded complexes of coherent Ox-modules.

Proof. One argues as in 3.7, recalling also that coherent complexes are pseudo-coherent by 2.2.8. 3.13. Corollary. For X a noetherian scheme, G(X) is naturally homotopy equivalent to the Quillen G- or K'-spectrum of X defined in

[Qi]. Proof. This follows from 3.12 and 1.11.7. 3.14. K(X) and Knmve(X) are contravariant functors in the scheme X, as a map of schemes / : X —• X' induces a complicial exact (1.2.16) functor Lf* = /* between the biWaldhausen categories of perfect bounded

THOMASON AND TROBAUGH

318

above complexes of flat modules (3.5.3), or those of strict perfect complexes. Similarly / induces a m a p / * : K(X' on Y') -* K(X on f~l{Y')). 3.14.1. If / : X —• X' is a map of globally finite Tor-dimension, then Lf*E' is cohomologically bounded if E' is. It follows then /* is a complicial exact functor between categories of cohomologically bounded pseudo-coherent complexes of flat modules (3.11.3), and so induces a map /* : G(X') —• G(X). This makes G( ) a contravariant functor for maps of finite Tor-dimension (cf. [SGA 6] IV 2.12 for G 0 , [Ql] Section 7 2.5). 3.15.

The functor (E\F')

»—• E' ® F' preserves degree-wise split L

monomorphisms in either variable. This functor represents E' ® F' and Ox

preserves quasi-isomorphisms if either E' or F' runs over a category of bounded above complexes of flat Ox-modules. Thus it is biexact, i.e., exact in each variable, on a pair of biWaldhausen categories if either category consists of flat complexes. It is clear that E' ® F' is strict Ox

perfect of both E' and F' are strict perfect. Applied locally on X, this L shows E' 0 F' is perfect if both E' and F' are. It is easy to see that Ox L

if E' has finite Tor-amplitude, then E' 0 F' has finite Tor-amplitude Ox

(respectively, is cohomologically bounded) if F' has finite Tor-amplitude (resp. is cohomologically bounded). (See [SGA 6] I 5.6, 5.3 if you get L

stuck.) We have already seen that E% 0 F' is pseudo-coherent if both E' Ox

is perfect (even pseudo-coherent) and F' is pseudo-coherent, back in 2.5.1. Thus 0 induces biexact functors between various biWaldhausen categories 3.5.3, 3.8.3, 3.11.3. As biexact functors induce pairings between A'-theory spectra ([W] just after 1.5.3), we get various pairings (cf. [SGA 6] IV 2.7, 2.10): (3.15.1)

K(X) A K(X) — K(X) nBiye

(3.15.2)

K

(3.15.3)

(X)

A A n a i v e (X) — A n a i v e (X)

K{X) A G(X) -+ G(X)

and even: (3.15.4) K(X

on Y)AK(X

on Z)-+K(X

on Y n Z)

(3.15.5) K{X on Y)AG(X

on Z)-> G(X on Y n Z ) .

HIGHER ALGEBRAIC K-THEORY OF SCHEMES

319

As the tensor product ® is associative and commutative up to "coherent natural isomorphism," K(X) and 7\ najve (X) are in fact "homotopyeverything" ring spectra, and the canonical map Kn3,lwe(X) —• K(X) and also /* : K{X') — K(X) and /* : A' naive (X') -+ A' naive (X) are maps of such ring spectra. The spectrum G(X) is a module spectrum over K(X), and when it exists, /* : G(X') —• G(X) is a map of module spectra over K(X') (cf. e.g., [Ma2]). Moreover K(X on Y) has a commutative and associative multiplication up to "coherent homotopy," but fails to have a unit when X ^ Y. There are also external pairings induced by (E\F') —• E' ® F' for X flat over S and Z over S. (3.15.6)

K(X) A K[Z) -> K(X x Z)

(3.15.7)

K(X) A G{Z) — G(X x Z)

See [SGA 6] IV 3.3 for how to go further. 3.16. Let / : X —» Y be a map of schemes. Then /* is a complicial exact functor in the sense of 1.2.16 between the complicial biWaldhausen categories of all bounded below complexes of flasque 0-modules on X and on Y. For flasque modules are deployed for /*, so /* represents i?/* and preserves quasi-isomorphisms on such complexes; and /* also preserves flasqueness ([SGA 4] V 4.9, and [H] I 5.3 /? or [V] II Section 2 no. 2). To conclude that /* induces maps /* : K(X) —• K(Y) or /* : G{X) —* G(Y), we need only find conditions that make Rf* to preserve the required perfection, pseudo-coherence, and the global bounds on cohomological dimension or Tor-amplitude in the definitions of the biWaldhausen categories 3.5.5 and 3.11.5. Considering B.ll, 2.5.4 (= [SGA 6] III 2.5, 4.8.1) and 2.7, we get variously (cf. [SGA 6] IV 2.11, [Th4] 1.13, [Ql] Section 7 2.7): 3.16.1. G( ) is a covariant functor on the category of noetherian schemes and proper maps. (Note this improves upon [Ql] 7.2.7, which only made G( ) a functor up to homotopy and for finite or projective maps. Our method avoids the fuss of Gillet's Chow envelope method [Gil] of proving 3.16.1. Also after the usual rectification to make f*g* = (/#)* on O-modules strictly, instead of up to natural isomorphism, our method yields a strictly functorial G( ), instead of a functor up to homotopy.) 3.16.2. G{ ) is a covariant functor on the category of quasi-compact schemes and flat proper maps with a relatively ample family of line bundles.

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3.16.3. G( ) is a covariant functor on the category of quasi-compact schemes and pseudo-coherent projective morphisms. 3.16.4. K( ) is a covariant functor on the category of noetherian schemes, with proper maps of finite Tor-dimension (i.e., perfect proper maps). 3.16.5. K( ) is a covariant functor on the category of quasi-compact schemes and perfect projective morphisms. 3.16.6. K( ) is a covariant functor on the category of quasi-compact schemes and fiat proper morphisms. 3.16.7. There are analogs of 3.16.2 - 3.16.6 for G(X on Y) and K(X on Y ) , using 2.6.3 and 2.7.1. We also note that if Z C Y C X with Z and Y closed subspaces in X, the exact functor forgetting part of the acyclicity requirement yields a canonical m a p K(X on Z) —• K(X o n 7 ) . 3.17. P r o p o s i t i o n . Projection Formula (cf. [SGA 6] IV 2.12, [Ql] Section 7 2.10). Let f : X —• Y be a quasi-compact and quasi-separated map of schemes with Y quasi-compact. Suppose that f is such that Rf* preserves pseudo-coherence (respectively, preserves perfection), hence induces a map /* : G(X) — G(Y), (resp. /„ : K(X) -> K(Y)). For example, let f be as in 3.16.1 - 3.16.3 (resp. 3.16.4 - 3.16.6J. Then /* is a m a p of module spectra over the ring spectra K(Y). That is, the diagram f3.17.lj (resp. the similar diagram where all G( )'s are replaced by K( )'s) commutes up to canonically chosen homotopy:

K(X)AG(X)

—2—•

G(X)

/* A l (3.17.1)

K(Y)AG(X)

/*

1A/* K(Y)AG(Y)

•

G(Y)

Proof. It is convenient to represent Rf+ on the chain level as /* o T where T is the total complex (totaled via sums, and not products) of the functorial flasque Godement resolution indexed by all the points of X, as in [SGA 4] XVII 4.2. This Rf* = /* o T is exact on all complexes. Now consider F' a bounded above perfect complex of flat CV-modules, and E' a pseudo-coherent (resp. perfect) complex of Ox -modules. By L

deployment, we have F' 0 ( ) — F' 0 ( ), etc. Then 2.5.5 shows t h a t the canonical map (3.17.2) is a natural quasi-isomorphism:

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321

(3.17.2)

F- ® RUE' = r ® f,TE- - /» (f*F- ® TE') - f.T (fF' Oy

Oy

\

OY

J

\

® £' Ox

(rr&r)As in 1.5.4, this quasi-isomorphism of two functors biexact in (E\F') yields the desired homotopy between two maps K(Y) A G(X) —+ G(Y) (resp., X ( r ) A # ( * ) -> A'(Y)). 3.18. P r o p o s i t i o n (cf. [SGA 6] IV 3.1.1, [Ql] Section 7 2.11). Let (3.18.1) be a pullback diagram of quasi-coherent schemes, with f a quasiseparated map.

X <-i

(3.18.1)

/J Y

• <

X'

J/' r

Suppose f and g are Tor-independent over Y (2.5.6.2). Suppose that g has finite Tor-dimension (resp., that f has finite Tor-dimension) and that f and f are such that Rf* and Rfl preserve pseudo-coherence (resp., that Rf* and Rfl preserve perfection), and so define maps /* : G(X) —+ G(Y) and fi : G{X') — G(Yf), (resp., /„ : K(X) — K(Y) and /J : K(X') — K{Y')). For example, we could suppose that f and f are as in 3.16.1 3.16.3 (resp., 3.16.4- 3.16.6;. Then there is a canonical homotopy between g*/* ~ fig'* * G(X) —• G(Y') (resp., g*f. ~ fig'* : K(X) -+ tf(y');. Proof. The idea is to use the base change Theorem 2.5.6. We follow Deligne's proof of 2.5.6 in [SGA 4] XVII 4.2, and build yet another model of G(X). Consider E' in the category of bounded above pseudo-coherent complexes of flat Ox-modules. Let T(E') be the total complex, totaled by sums, of the Godement resolution as in [SGA 4] XVII Section 4. Then each stalk T{EX) is chain homotopic to Ex so g'*T(E')x ~ g'*Ex = Lg'*Ex, and T(E') is deployed for Lg'\ As T(E') is flasque, it is deployed for Rf+ (using B.ll to allow T'(E) to be unbounded below). Using this functor T, the augmentation quasiisomorphism E' ~ T(E'), 3.11, and the methods of 3.5, we see the

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complicial biWaldhausen category of cohomologically bounded pseudocoherent complexes deployed for both Lg'* and Rf*, that of cohomologically bounded above pseudo-coherent complexes of flat Ox-modules, and that of cohomologically bounded pseudo-coherent complexes, all have equivalent derived categories. Indeed, after identifying the last two derived categories by their known equivalence, the inverse equivalences to the bideployed derived category are induced by T and the inclusion (cf. [SGA 4] XVII 4.2.10). So by 1.9.8, the bideployed biWaldhausen category has /f-theory spectrum homotopy equivalent to G{X). The techniques of proof of 3.11 and 3.5 show that the derived category of the bideployed biWaldhausen category is equivalent to the derived category of the complicial biWaldhausen category whose objects are data consisting of a cohomologically bounded pseudo-coherent E' on X which is deployed for both /* and g'*, a bounded above degree-wise flat F' on Y and a quasi-isomorphism F' -^ /*£", and a bounded below degree-wise flasque G' on X' and a quasi-isomorphism gf* E' -^ G' on X'. The associated abelian category is that of all diagrams (A —• / * # , gf* B —* C) with A an C?y-module, B an Ox-module , and C an Ox'-module. (Compare [SGA 4] XVII 4.2.12.) The K-theory spectrum of this biWaldhausen category is thus also homotopy equivalent to G(X). In this model of G(X)) g*/* is represented by the exact functor sending (F' ^ /*£", g'*E' ^ G') to g*F' (recall, F' is flat). The map fig'* is represented by the exact functor sending the object (F' -^ f*E\ g'*E' Jr» G) to flG'. There is a natural transformation

g*F- - g'f.E- - g'f.gW'E- = g'g.fWE'

- fj*E'

-

f'.G.

This is the canonical base change map of Deligne ([SGA 4] XVII 4.2.12), and is a quasi-isomorphism by 2.5.6 — [SGA 6] IV 3.1. This natural quasiisomorphism then induces the desired homotopy g*/* ~ fig'* of maps on G( ) by 1.5.4. The proof for K{ ) is essentially the same. Clearly, there are analogs for K(X on Z), etc. 3.19. Proposition (Excision). Let f : X' —> X be a map of quasicompact and quasi-separated schemes. Let Y C X be a closed subspace such that X — Y is quasi-compact. Set Y' — f~x(Y) C X. Suppose that f is an isomorphism infinitely near Y in the sense of 2.6.2.2 , 2.6.1. Then f* induces homotopy equivalences C\ 1Q 1^ K }

f* : K(X

on Y) •=• K(Xf

on Y')

r : G(X on Y) ^ G{X' on Y').

Proof. We note that the map / is quasi-separated ([EGA] I 6.1.10). By definition 2.6.2.1, we may choose a scheme structure on Y so that

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323

i : Y —• X is a finitely presented closed immersion with / inducing an isomorphism f"1(Y) — Y xXf —• Y. The proposition then results from 2.6.3 which shows Rf+ and Lf* are inverse up to natural quasi-isomorphism, once we realize i?/* and Lf* as exact functors on appropriate model biWaldhausen categories. We represent Rf* by the exact /* o T, for T the Godement resolution, and note that Rf+ preserves perfection and pseudo-coherence of complexes acyclic on X' — Y1 by 2.6.3, and preserves cohomological boundness by the finite cohomological dimension of #/*, B.ll. The most appropriate models for K(X on Y) and K(X' on Yf) (resp., G(X on 7 ) . . . ) are the complicial biWaldhausen category of perfect (resp., cohomologically bounded pseudo-coherent) complexes of Ox~ modules which are strictly 0 on X — Y. The inclusion functor of this new model into the category of perfect complexes of Ox -modules which are acyclic on X — Y is exact, and Ty o T provides an exact functor which is inverse to the inclusion up to natural quasi-isomorphism. Thus the new model category indeed has A'-theory spectrum homotopy equivalent to K(X onY), (resp. . . . ) . As / is flat over the points of Y (2.6.2.1), /* is exact on the category of complexes strictly 0 on X — Y. Thus /* and /* o T induce exact functors on the new models, which are inverse up to natural quasi-isomorphism by 2.6.3. By 1.5.4, this yields the result. 3.19.2. Examples. Let A be a noetherian ring, and / C i a n ideal. Then the map to the completion A —• AJ induces a homotopy equivalence: X(Spec(A) on Spec(i4/J)) -> K(Spec(^?)

on Spec(.4?/L4?))

For A}/IAj = A JI, and for A noetherian A —+ A} is flat over the points of Spec(u4/7), as A^ is a Zariski ring at (/). For a general commutative ring A, and / C A a finitely generated ideal, let A!\ be the henselization of A along Spec(^4/7). Then the map A —• A1} induces a homotopy equivalence A'((^4) on (A/1)) -^ K((Aj) on

{A)/IA))). If j : V —•>• X is an open immersion with Y C V, and V is quasicompact, then K(X on Y) -^ K(V on Y) is a homotopy equivalence. For this example, it is in fact trivial to prove 2.6.3 directly. 3.20. P r o p o s i t i o n (cf. [SGA 6] IV 3.2, [Ql] Section 7 2.2). Let X = limX a be the limit of an inverse system of schemes, where the "bonding" maps fap : Xa —• Xp in the system are a/fine. Suppose that all the Xa are quasi-compact and quasi-separated. Let Ya C Xa be a system of closed subspaces with fap(Yp) = Ya, and all Xa — Ya quasicompact. Then:

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3.20.1. The derived category of strict perfect (resp., perfect) complexes on X is the direct colimit of the derived categories of strict perfect (resp., perfect) complexes on the Xa, where the maps in the direct system are the Lf^p • Similarly, there is an equivalence of derived categories where one imposes the conditions of acyclicity on X — Y and Xa — Ya on the complexes in the derived categories. 3.20.2. The canonical maps induced by the Lf* are homotopy equivalences: \jmK(Xa)^K(X) a

\imK{Xa

on Ya) •=• K(X

on Y).

a

Proof. The construction of K( ), see 1.5.2 -1.5.3, clearly preserves direct colimits of biWaldhausen categories, and also converts complicial exact functors inducing an equivalence of derived categories into homotopy equivalences of A'-theory spectra, by 1.9.8. Hence 3.20.2 follows from 3.20.1. To prove 3.20.1, we first consider the biWaldhausen category of strict perfect complexes (possibly imposing the condition of acyclicity on Xa — Ya, X — Y). As a strict perfect complex is a finite complex of finitely presented modules, it follows from [EGA] IV Section 8 as quoted in C.4 that the biWaldhausen category on X is equivalent to the direct colimit of the biWaldhausen categories on the Xa, and a fortiori that the derived categories are equivalent. In more detail, we see that [EGA] IV 8.5.2(ii) applied to the modules in each degree and [EGA] IV 8.5.2(i) applied to the differentials, each applied finitely any times, shows that each strict perfect E' on X is isomorphic to f^Ea for a strict perfect E'a on some Xa. Given a morphism e : E' —+ E1' between strict perfect complexes on X, we apply [EGA] IV 8.5.2(i), first to get maps E{a -+ E'j defined on some Xa, and then to make them satisfy the identities del = et+1d, and so to obtain a chain map ea : Ea —• E'a on Xa for a sufficiently large, such that e = / ^ ( e a ) . Now if Va is an affine open in Xa, and E' is acyclic on V = Va x X, then as a complex of projective modules it is chain homotopic to 0 on the affine V. As above, [EGA] IV 8.5.2 shows that this chain nulhomotopy is defined on E'p\Vp for all sufficiently large /?, so Ep\Vp is acyclic. Applied to affine covers, this shows that if the strict perfect complex E' is acyclic on X, or on X — Y, then for all sufficiently large /? the complex E'p is acyclic on Xp or Xp — Yp. Finally e is a quasi-isomorphism iff its mapping cone cone(e) = /*cone(e a ) is acyclic on X. By the above, this occurs iff for some /3 sufficiently large cone (ep)

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325

is acyclic on Xp, i.e., iff for /? sufficiently large ep is a quasi-isomorphism on Xp. This completes the proof of 3.20.1 for categories of strict perfect complexes. To prove 3.20.1 for perfect complexes, we work in the biWaldhausen category of perfect strict bounded above complexes of flat modules. This has the correct derived category by 3.5, and all the / * and / * are exact on this category. By replacing the system of Xa by the cofinal system of all a > c*0) we may assume t h a t there is a terminal XQ. Then X and all Xa are affine over XQ. If Xo, and hence Xa and X, has an ample family of line bundles, the derived category of perfect complexes is equivalent to the derived category of strict perfect complexes as in 3.8, and 3.20.1 reduces to the strict case proved above. We now prove in 3.20.3 - 3.20.6 the result 3.20.1 by induction on the number n of affine open subschemes needed to cover XQ. If n — 1, XQ is affine, hence has an ample family of line bundles, and 3.20.1 is known. To do the induction step, let n > 1, and suppose the result is known for schemes covered by fewer affines. T h e n we write Xo = Uo U Vo with Uo open affine, and VQ open and covered by n — 1 open affines. Let Ua = Xa x Uo, etc. T h e n 3.20.1 is known for U = \imUaj V = limV^, and UC\V = limU a C\V a . We note t h a t UoHVQ is quasi-affine, and so has an ample family of line bundles, so the above indeed yields 3.20.1 for it. 3.20.3. If E'a is a bounded above flat perfect complex on Xa such t h a t faE'a is acyclic on X (or on X — Y ) , then by induction hypothesis for all /? sufficiently large fpQEa — Ep has Ep\Up acyclic on Up (or on Up—Yp), and Ep\Vp acyclic on Vp (or on Vp — Yp). As acyclicity is a local question, Eg is then acyclic on Xp (resp., on Xp —Yp) for all sufficiently large f3. We apply this to the mapping cone of a m a p ea : Ea —> E'a to show t h a t if f*ea is a quasi-isomorphism on X , then ep = fpaea is a quasi-isomorphism on Xp for all sufficiently large /?. The next step is to show t h a t any bounded above flat perfect complex on X is quasi-isomorphic to / * of a bounded above flat perfect complex on X a , for some a. For this, we make use of the patching construction of 3.20.4. 3.20.4. Let U and V form an open cover of a scheme X. various open immersions as in:

U (3.20.4.1)

*' T

—^—> t/

X T*

unv

vv 3'

Denote the

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Suppose (Ejj, Fy, G'UnV, ^>, ip) is a d a t u m consisting of flat perfect complexes E'v, Fy, G'UuV on U, V, UUV respectively, and of quasi-isomorphisms