The Comparison Isomorphisms C cris

Transcription

1 The Comparison Isomorphisms C cris Fabrizio Andreatta July 9, 2009 Contents 1 Introduction 2 2 The crystalline conjecture with coefficients 3 3 The complex case The strategy in the p-adic setting 4 4 The p-adic case; Faltings s topology 5 5 Sheaves of periods 7 6 The computation of R i v B cris 13 7 The proof of the comparison isomorphism 16 1

2 1 Introduction Let p > 0 denote a prime integer and let k be a perfect field of characteristic p Put O K := W k and let K be its fraction field Denote by K a fixed algebraic closure of K and set G K := GalK/K We write B cris for the crystalline period ring defined by J-M Fontaine in [Fo Recall that B cris is a topological ring, endowed with a continuous action of G K, an exhaustive decreasing filtration Fil B cris and a Frobenius operator ϕ Following the paper [AI2, these notes aim at presenting a new proof of the so called crystalline conjecture formulated by J-M Fontaine in [Fo Let X be a smooth proper scheme, geometrically irreducible, of relative dimension d over O K : Conjecture 11 [Fo For every i 0 there is a canonical and functorial isomorphism commuting with all the additional structures namely, filtrations, G K actions and Frobenii H i X et K, Q p Qp B cris = H i cris X k /O K OK B cris It follows from a classical theorem for crystalline cohomology, see [BO, that H i crisx k /O K = H i dr X/O K The latter denotes the hypercohomology of the de Rham complex 0 O X Ω 1 X/O K Ω 2 X/O K Ω d X/O K 0 The first interpretation provides a Frobenius operator, the second one provides a filtration, the Hodge filtration For every n N we put Fil n H i dr X/O K = H i dr X/O K if n 0, to be 0 if n d + 1 and to be the image of the hypercohomology of the complex 0 Ω n X/O K Ω 2 X/O K Ω d X/O K 0 if 0 n d Then, in the statement of the conjecture 1 Frobenius on the left is defined by the identity operator on H i X et, Q K p and by Frobenius on B cris Frobenius on the right hand side is defined by Frobenius on H i crisx k /O K and by Frobenius on B cris ; 2 the filtration on the left hand side is defined by H i X et, Q K p Qp Fil n B cris, while on the right hand side it is given by the composite filtration Fil n H i dr X/O K OK B cris := a+b=n Fila H i dr X/O K OK Fil b B cris ; 3 the Galois action on the left hand side is induced by the action of G K on H i X et, Q K p and on B cris The Galois action on the right hand side is given by letting G K act trivially on H i crisx k /O K and through its natural action on B cris The conjecture is now a theorem, proven by G Faltings in [F2 In fact it holds without assuming that K is unramified and with non-constant coefficients But these assumptions will simplify our considerations There are various approaches to the proof of the conjecture One and the first is based on ideas of Fontaine and Messing in [FM using the syntomic cohomology on X; a full proof for constant coefficients using these methods was given by T Tsuji in [T There is also an approach for constant coefficients based on a comparison isomorphism in K- theory which is due to W Niziol [N We will follow the approach by Faltings, not in its original version but using a certain topology described in [F3 Our approach, which works also for non constant coefficients, is based on the papers [AI2 and [AB The strategy consists in defining a new cohomology theory associated to X and proving that it computes both the left hand side via the theory of almost étale extensions and the right hand side of conjecture 11 The new inputs, compared to Faltings s original approach, are: i we systematically study the underlying sheaf theory of Faltings topology Faltings uses Galois cohomology of affines and then patches his computations to global results using hypercoverings; 2

3 ii we introduce certain acyclic resolutions of sheaves of periods on Faltings topology This allows to avoid the use of Poincaré duality In Faltings original approach one needed to prove Poincaré duality in Faltings theory and to show that it is compatible both with poicaré duality on crystalline or de Rham cohomology and with Poincaré duality on étale cohomology We explain how to avoid this In order to simplify the proof, we also assume that there exists a morphism F : X X lifting Frobenius on O K and the Frobenius morphism on the special fiber X k This is a strong hypothesis For example, if X is an abelian scheme over O K with ordinary reduction, it amounts to require that X is the canonical lift of X k In fact, in [AI2 the existence of such a lift of Frobenius is assumed only Zariski locally on X and this is harmless On the other hand, this stronger assumption allows us to work with H i dr X/O K instead of H i crisx k /O K with the Frobenius operator induced by F We can then avoid the use of crystalline cohomology and its technicalities completely 2 The crystalline conjecture with coefficients We present the statement of the comparison isomorphism conjecture in the case of non constant coefficients As before X SpecO K is a proper and smooth scheme, geometrically irreducible We make no assumption on O K but we assume that X is obtained by base change from a scheme defined over W k where k is the residue field of O K this is needed in [AI2 not in Faltings original approach We consider two categories: The category Q p ShXK et of Q p-adic étale sheaves By a p adic étale sheaf L on XK et we mean an inverse system L := {L n } ShXK etn such that L n is a locally constant and locally free of finite rank sheaf of Z/p n Z modules for the étale topology of X K and L n = L n+1 /p n L n+1 for every n N Given two such objects M = {M n } n and L := {L n } n a morphism of p-adic étale sheaves f : M L is a collection of morphisms f n : M n L n of sheaves such that f n f n+1 modulo p n for every n N The category Q p ShXK et has the p-adic étale sheaves as objects and one defines the morphisms by tensoring the Z p -module of morphisms as p-adic étale sheaves with Q p The category Fil F IsocX K /K of filtered F -isocrystals on X K Let F Isoc X k /W k be the category of isocrystals E on X k /W k, endowed with a non-degenerate Frobenius ϕ: ϕ E E By isocrystal we mean a crystal of O Xk /W k-modules up to isogeny See also [B3, especially Thm 242, for a different point of view in the context of the rigid analytic spaces To any such object one can associate a coherent O XK -module E XK with integrable, quasi-nilpotent connection : E XK E XK OXK Ω 1 X K /K The category Fil F IsocX K/K consists of an F -isocrystal E on X k /W k and an exhaustive descending filtration Fil E XK on E XK by O XK - submodules satisfying Griffith s transversality with respect to the connection on E XK i e, such that Fil n E XK Fil n 1 E XK OXK Ω 1 X K /K Then, Conjecture 21 Assume that there exist a Q p -adic étale sheaf L and a filtered-f -isocrystal E which are associated Then, for every i 0 there is a canonical and functorial isomorphism commuting with all the additional structures namely, filtrations, G K actions and Frobenii H i X et K, L Q p B cris = H i cris X k /W k, E W k B cris Here, the filtration on H i crisx k /W k, E W k B cris is obtained via H i crisx k /W k, E W k B cris = H i dr XK, E XK K B cris using the Hodge filtration on H i dr XK, E XK The clarification 3

4 of what being associated means is part of the conjecture As an example, let f : E X be a relative elliptic curve or an abelian scheme Then, one can consider the Q p -adic étale sheaf L n n where L n := E K [p n is the groups scheme of p n -torsion points of E K On the other hand, one can consider the filtered F -isocrystal E defined by R 1 f cris, OE cris k /W k the first derived functor in the crystalline sense of the crystal OE cris k /W k with respect to f k The module with connection E XK on X K is the first de Rham cohomology of E K relative to X K with the Gauss-Manin connection The filtration has only one step and Fil 1 E XK E XK is f Ω 1 E K /X K Then, in this case L and E are associated and the conjecture holds See [AI2 3 The complex case The strategy in the p-adic setting Assume that X is a complex analytic projective variety Let O X denote the sheaf of holomorphic functions on X In this consider on the one hand a locally constant sheaf L of C-vector spaces and a locally free coherent sheaf E of O X -modules endowed with an integrable connection Then, L and E, are associated if L = E =0 If this is the case the de Rham complex provides a resolution for L: 0 L E E OX Ω 1 X/C This provides a canonical comparison isomorphism H i X, L = H i dr X, E Here, we strongly use the analytic topology to prove that the displayed long sequence is exact and this approach certainly will not work for schemes or even for rigid analytic p-adic spaces As a refinement of this argument, more algebraic in flavor, we consider the universal covering space v : X X Let x X be a point and let π 1 X, x be the fundamental group Then, v 1 L becomes a constant sheaf on X and defines and is defined by a representation of π 1 X, x Write v E := v 1 E v 1 O X O X e It is a O X e-module endowed with an integrable connection Then, L and E are associated if we have an isomorphism v 1 L C O X e = v 1 E v 1 O X O X e of O X e-modules, compatibly with π 1 X, x-action and connections We can consider the complex 0 v 1 L v E v E O ex Ω 1 e X/C It is π 1 X, x-equivariant and exact There is an obvious relation between the two complexes The former is locally given by the π 1 X, x-invariants sections of v of the latter This allows to recover the comparison isomorphism over X from the comparison isomorphism H i X, v 1 L = H i dr X, v E of π 1 X, x-modules Indeed, we have compatible spectral sequences and H i π 1 X, x, H j X, v 1 L = H i+j X, L H i π 1 X, x, H j dr X, v 1 E = H i+j dr X, E inducing the isomorphism H i X, L = H i dr X, E We will use the complex case as a guide for the proof in the p-adic case: Step 1: Construct the analogue of the universal covering space v : X X Step 2: Define the notion of being associated on X and construct the analogue of the exact sequence 0 v 1 L v E v E O ex Ω 1 e X/C Here, we need to find a replacement for the sheaf of holomorphic functions O e X introducing Fontaine s sheaves of periods Step 3: Deduce the comparison isomorphism for associated objects 4

5 4 The p-adic case; Faltings s topology We let X SpecO K be as before Of course we do not expect to construct a universal covering space Following Faltings we will construct the sheaves on such space After all we are interested not in the space itself but in comparing sheaves on it We give the following: Definition 41 Let E XK be the category defined as follows i the objects consist of pairs U, f : W U K such that U is an open subscheme of X and f is a finite étale morphism with U K equal to the base change of U to K We will usually denote by U, W this object to shorten notations; ii a morphism U, W U, W in E XK consists of a pair α, β, where U U is the inclusion X and β : W W is a morphism commuting with α OK Id M Let us remark that the pair X, X K is a final object in E XM Moreover, fibre products exist setting U 1, W 1 U,W U 2, W 2 := U 1 U 2, W 1 W W 2 We say that a family {U i, W i U, W } i I is a covering family if {U i U} i I is a covering of X and {W i W } i I is a covering of W i e, every K-valued point of W is in the image of a K-valued point of one of the W i s One verifies that the covering families satisfy the axioms of a pre-topology [SGAIV, Def II 13: PT0&PT1 Given a morphism U, W U, W and a covering family {U i, W i U, W } i I the base change {U i, W i U,W U, W } i I is a covering family of U, W ; PT2 Given a covering family {U i, W i U, W } i I and for every i I a covering family a covering family {U ij, W ij U i, W i } j Ji the composite family {U ij, W ij U, W } i I,j Ji is a covering family; PT3 the identity U, W U, W is a covering family We let X be the category E XK with the given pretopology It should be thought of as the open subsets of the universal cover of X It turns out that properties PT0 PT3 suffice to define a category of sheaves of abelian groups ShX on X A presheaf of abelian groups is a controvariant functor F : EX 0 K AbGr It is a sheaf if, for every object U, W and every covering family {U i, W i U, W } i I the sequence 0 F U, W i I F U i, W i F U i, W i U,W U j, W j is exact Given a presheaf F one can use the covering families to define the associated sheaf Remark 42 There are variants of the definition above which will be used in the sequel Let X be the formal scheme defined by completing X along the special fiber X k α One can consider pairs U, W where U X is Zariski open and W U K is finite étale meaning that there is a finite extension L such that W is defined by a finite and étale morphism over U L as rigid analytic spaces β One can consider pairs U, W where U X is étale and W U K is finite étale γ One can consider pairs U, W where U X is p-adically formally étale and W U K is finite étale In case α one defined the pretopology X as in the algebrai setting In cases β and γ Faltings defined in [F3, p 214 the pre-topology as before: a family {U i, W i U, W } i I 5 i,j I

6 is a covering family if {U i U} i I is a covering of X or X and {W i W } i I is a covering of W As pointed out by A Abbes this gives the wrong pre-topology for which the period sheaves behave badly The right definition, which produces a pre-topology which is coarser than Faltings, is that one takes the pre-topology generated by coverings of the following form: a {U i U} i I is a covering of X or X and W i = W U U i ; b U i = U and {W i W } i I is a covering of W We refer to [AI2 for details We next define a morphism v : ShX ShX analogous to the push forward functor associated to the morphism v : X X in the complex case Let v 1 : X Zar X, where X Zar is the Zariski topology on X, be the functor associating to an open U X the pair U, U K This induces a functor v from the category of presheaves on X to the category of presheaves on X Zar sending a presheaf F to v F U := F v 1 U One verifies that v sends covering families to covering families and sends intersection of two open subsets to the fibre products in X In particular, v sends sheaves on X to sheaves on X By construction it is left exact One can verify that v admits a left adjoint v which is exact so that v sends injective objects to injective objects and we can derive the functor v The rest of this section is devoted to the explicit computation of the functors R i v Localization functors: Fix an algebraic closure Ω of the fraction field of X K Consider a Zariski open subset U X and let G U := π 1 U K, Ω be the fundamental group of U K with base point defined by Ω Let U fet be the category of finite and étale morphisms W U K K It is endowed with a pretopology: for every object W the coverings are morphisms {W i W } such that the images of the W i s cover the whole of W By Grothendieck s formulation of Galois theory, the category sheaves of abelian groups on U fet is equivalent to the category G K U AbGr of Z-modules with discrete action of G U We have a fully faithful morphism ρ 1 U : Ufet X K sending W U, W It sends covering families to covering families so that it induces a morphism ρ U, : ShX Sh U fet K GU AbGr which is left exact, admits an exact left adjoint ρ U so that it can be derived Given a sheaf F we define ρ U, F to be the localization of F at U This construction remains valid also for the variants of the topology given in 42 One needs to take care of the choice of a base point, though We refer to [AI2 for details Example 43 Assume that U = SpecR U is affine Let R U Ω be the union of all R U OK O K - subalgebras S of Ω such that S is normal and R U OK K S [ p 1 is finite and étale Given a sheaf F on X write F R U for the direct limit lim S F U, Spec S[p 1 It is naturally endowed with a continuous action of G U and we have ρ U, F = F R U as GU -modules In particular, given a sheaf of abelian groups F on X we get a spectral sequence a map H n G U, ρ U, F R n v F U Define H n Gal,X F to be the sheaf on X associated to the presheaf U Hn U o K,fet, ρ U, F Theorem 44 The morphism H n Gal,X F Rn v F is an isomorphism of sheaves on X 6

7 To prove this it suffices to show that the given morphism induces an isomorphism at level of stalks We define a geometric point of X to be a pair x, y where a x is a geometric point of X;b y is a geometric point of O X,x OK K Write G x,y to be π 1 Spec OX,x OK K, y Let J x,y be set of pairs U, W, y where U is an open neighborhood of x and W U K is a finite and étale morphism with a point y over y One verifies that it is a directed set Given sheaf F on X define the stalk F x,y as the direct limit lim F U, W taken over all pairs U, W, y J x,y If F is a sheaf of abelian groups, then F x,y is a discrete module with continuous action of G x,y Since H n Gal,X F x = H i G x,y, F x,y to prove the theorem it suffices to verify that Lemma 45 1 A sequence of sheaves of abelian groups is exact if and only if for every geometric point x, y the induced sequence of stalks is exact 2 For every geometric point x, y and every sheaf F of abelian groups on X we have R i F x = H i G x,y, F x,y As a consequence of this theorem we can prove the relation between Faltings cohomology and étale cohomology Consider the forgetful functor u 1 : X X et sending U, W W It K induces a morphism u : ShX et ShX Then, Faltings proves in [F1 that if L is a locally K constant étale sheaf on X K we have HGal,X n u L R n v u L This and 44 imply that the natural morphism R n v u F R n v u F is an isomorphism This implies Proposition 46 We have H i X et K, L = H i X, u L 5 Sheaves of periods In this section we will introduce the sheaf theoretic analogue of the ring O K, C K and B cris of p-adic Hodge theory We start with the analogue of O K First of all notice that given a sheaf M on X we get a sheaf on X, which we denote again by M, via the formula MU, W := MU In particular, we can view O X and Ω i X/O K as sheaves on X Definition 51 We define the presheaf on X, denoted O X by O X U, W := the normalization of ΓU, O U in ΓW, O W By construction there is a natural injective ring homomorphism O X OK O K O X We have Proposition 52 The presheaf O X is a sheaf Moreover, if U = SpecR U is an open affine subset of X then O X R U = R U Before sketching the proof of the proposition we remark that this remains true also for the variants 42 For variant α we denote by Ob X the associated sheaf If one uses the original definition of Faltings the proposition is wrong in cases β and γ We refer to [AI2 for a counterexample Proof The second statement is clear For the first, we may assume that W is irreducible and we may consider only the α s such that W α Let {U α, W α U, W } α be a covering family We set U αβ := U α U U β and W αβ := W α W W β We have the following commutative diagram f 0 O X U, W α O g XU α, W α α,β O X M U αβ, W αβ 0 ΓW, O W α ΓW α, O Wα α,β ΓW αβ, O αβij 7

8 Note that for every α the map {W α W U U α } i is surjective because W U U α is irreducible and W α Since the {U α U} α is a covering of X it follows that {W α W } α is a covering in XK et In particular, the bottom row of the above diagram is exact Moreover the vertical maps are all inclusions therefore f is injective, ie O X is a separated presheaf Let x Kerg Then x ΓW, O W O XM U α, W α We are left to prove that x is integral α over ΓU, O U Without loss of generality we may assume that U α = SpecA α is affine for every α Let us denote by x α the image of x in Γ W U U α, O W U U α Because Wα W U U α is surjective, the image x α of x α in ΓW α, O Wα is in fact in O X U α, W α, hence integral over A α, it follows that x α is integral over A α Let P α X A α [X be the monic characteristic polynomial of x α over A α Then P α X Uαβ = P β X Uαβ for all α, β, therefore there is a monic polynomial P X ΓU, O U such that P X Uα = P α X As P x Uα = P α x α = 0 for every α it follows that P x = 0, ie that x is integral over ΓU, O U We come next to the sheaf theoretic analogue of C K This coincides with ÔK [p 1 ; here ÔK is the p-adic completion of O K and it is a topological ring and we can view ÔK [p 1 as the limit of the inductive system ÔK with transition maps given by multiplication by p It is crucial that our definition captures the topology For example, we would like that the localization of our continuous sheaf at an affine R U gives the G U -module R U endowed not with the discrete topology but with the p-adic topology To take this into account we use the category of inverse systems Categories of inverse systems We review some of the results of [J Let A be an abelian category Denote by A N the category of inverse systems indexed by the set of natural numbers Objects are inverse systems {A n } n := A n+1 A n A 2 A 1, where the A i s are objects of A and the arrows denote morphisms in A The morphisms in A N are commutative diagrams A n+1 A n A 2 A 1 B n+1 B n B 2 B 1, where the vertical arrows are morphisms in A Then, A N is an abelian category with kernels and cokernels taken componentwise and if A has enough injectives, then A N also has enough injectives Let h: A B be a left exact functor of abelian categories It induces a left exact functor h N : A N B N which, by abuse of notation and if no confusion is possible, we denote again by h If A has enough injectives, then also A N does One can derive the functor h N and R i h N = R i h N If inverse limits over N exist in B, define the left exact functor lim h: A N B as the composite of h N and the inverse limit functor lim : B N B Assume that A and B have enough injectives For every A = {A n } n A N one then has a spectral sequence lim p R q ha n = R p+q lim ha, where lim p is the p th derived functor of lim in B If in B infinite products exist and are exact functors, then lim p = 0 for p 2 and the above spectral sequence reduces to the simpler exact sequence 0 lim 1 R i 1 ha n R i lim ha lim R i ha n 0 8

9 Generalities on inductive systems Let A be an abelian category We denote by Ind A, called the category of inductive systems of objects of A, the following category The objects are Ai, γ i with A i Z i object of A and γ i : A i A i+1 morphism in A for every i Z Given an integer N Z a morphism f : A := A i, γ i B := B i Z j, δ j of degree N is a system j Z of morphisms f i : A i B i+n for i Z such that δ i+n f i = f i+1 γ i Since A is an additive category, the set of morphisms of degree N form an abelian group with the zero map, the sum of two functions and the inverse of a function defined componentwise Given a morphism f = f i i Z : A B of degree N we get a morphism of degree N +1 given by δ i+n f i i Z This defines a group homomorphism from the morphisms Hom N A, B of degree N to the morphisms Hom N+1 A, B of degree N +1 We define the group of morphisms f : A i, γ i B i Z j, δ j j Z in Ind A to be the inductive limit lim N Z Hom N A, B with respect to the transition maps just defined One verifies that the category Ind A is an abelian category Let B be an abelian category in which direct limits of inductive systems indexed by Z exist Consider the induced functor lim : Ind B B Suppose we are given δ-functors T n : B A with n N Define lim T n : Ind A B as the composite of the functor Ind A Ind B, given by A i inz T n A i, and of the i Z functor lim Then, if lim is left exact in B, the functors limt n, for varying n N, define a δ-functor as well Given F = F n ShX N define H 0 X, F := lim n H 0 X, F n Define H i X, F as the i-th derived functor of this If G = G m m N Ind ShX N with G m ShX N define H i X, G = lim m Hi X, G m This bit of abstract non-sense allows us to define inductive limits of topological sheaves Ind ShX N on X where Fontaine s sheaves of periods naturally live For example, the analogue of Ô K is the inverse system ÔX := O X /p n O X where the transition maps are the natural n reduction maps Similarly, the analogue of C K is the inductive limit ÔX[ p 1 := ÔX n where the transition maps are given by multiplication by p We next consider the analogue of B cris Since we d like to use the technology above we will first show how to construct B cris as a direct limit of an inverse system of rings We will then generalize this construction to sheaves Review of a construction of B cris Put W n := W n O K /po K the Witt vectors of length n with values in O K /po K and write ϕ for Frobenius on W n We have a ring homomorphism θ n : W n : O K /p n O K given by s 0,, s n 1 n 1 pn i i=0 pi s i where s i O K /p n O K is a lift of s i for every i Choose a compatible sequence of roots p 1/pn 1 n 1 in O K i e, p 1/pn p = p 1/pn 1 Denote by p n := [ p 1/pn Wn the Teichmüller lift of p 1/pn O K /po K Write ξ n := p n p W n ; it is a generator of Kerθ n Let u n+1 : W n+1 W n be the composite of the natural projection composed with Frobenius ϕ Then, θ n [ u n+1 u n modulo p n and u n+1 ξ n+1 = ξ n Put A cris as the inverse limit of the rings W ξ m n n with respect to the transition morphisms 9 m N

10 [ defined by the u n s One can prove that A cris /p n A cris ξ m = Wn n This admits a filtration [ m N where Fil r A cris /p n A cris ξ m = Wn n Furthermore, A cris is endowed with 1 an action of G K, m r via its action on O K, which is continuous for the p-adic topology; 2 a filtration defined by Fil r A cris := lim Fil r A cris /p n A cris ; 3 a Frobenius ϕ defined by Frobenius on the W n s Let {ζ n } n N be a compatible system of primitive p n th roots of unity so that ζ p n+1 = ζ n It defines an element [ζ n+1 W n Note that u n+1 [ζn+2 = [ζ n+1 and [ζ n+1 1 Kerθ n Let ε = lim[ζ n+1 A + cris be the induced element and write t := log ε = n/n ε 1 n=1 Is an element of Fil 1 A cris and B cris = A cris [t 1 It inherits a G K -action, a filtration and a Frobenius from those defined on A cris Note that t [p := t p /p! A cris so that p!t [p = t p and p is invertible in B cris More precisely, for every m N write A cris m = A cris t m Since G K acts on t via the cyclotomic character i e, for g G K we have gt = χgt, then A cris m is G K -stable and as a Galois module it is a Tate twist of A cris Put Fil r A cris m := Fil r+m A cris t m Eventually, since ϕt = pt then Frobenius on B cris sends A cris m A cris pm We note that B cris = lim A cris m compatibly with G K -action, a filtration and a Frobenius The sheaf B cris We switch to Faltings site Consider the sheaves W n := W n OX /po X We have a Frobenius ϕ on W n and on W n,x As in the classical case we have a homomorphism of sheaves θ n : W n O X /p n O X We further have sheaf homomorphisms u n+1 : W n+1 W n, defined be the composite of the natural projection composed with Frobenius Fix an object U, W of X Write S = O X U, W Lemma 53 The element ξ n generates the kernel of θ n : W n S/pS S/p n S Proof Let us first remark that θ n ξ n = p 1/pn pn p = 0, therefore ξ n Kerθ n one needs to show that if x Kerθ n then x ξ n W n S/pS One proceeds by induction on n For n = 1, then θ 1 is Frobenius on S/pS whose kernel is generated by p 1/p since S is normal For the inductive step we refer to [AI2 In particular, we conclude that the kernel of the map of sheaves θ n is generated by ξ n One defines A cris as the inverse system A cris /p n A cris for varying n N where A cris/p n A cris is the sheaf W n Wn Acris /p n A cris The transition morphisms are defined by un Then, A cris is endowed with a Frobenius morphism, a filtration defined by Fil r A cris /p n A cris := W n Wn Fil r A cris /p n A cris and a GK -action One defines similarly the inverse system A crism as the inverse system A crism/p n A crism := W n Wn Acris m/p n A cris m for m N with induced filtration and G K -action and we let B cris Ind ShX N be the induced inductive system of inverse systems of sheaves The sheaf B cris Consider W n,x := W n O X /po X OK O X Using our assumption that X admits a global lift of Frobenius, we get a Frobenius on W n,x Extending θ n and u n+1 to O X -linearly morphisms we get compatible morphisms θ n,x : W n,x O X /p n O X and u n+1,x : W n+1,x W n,x We analyze the kernel of θ n,x Let U, W X with U = SpecR U affine and put S := O X U, W Assume that U admits an étale morphism to Spec O K [T 1 ±1,, T ±1 d in this case we say that U is small Assume furthermore that S contains p n+1 -th roots T 1/pn+1 i for all 1 i d of the variable T i Denote by T i,n := [T 1/pn+1 i W n S/pS and Xi,n :== 1 T i T i,n 1 10

11 W n S/pS OK R U Since the kernel of the ring homomorphism R U /p n R U R U /p n R U R U /p n R U defined by x y xy is the ideal I = T T 1,, T d 1 1 T d, we conclude that the kernel of the map θ n,s : W n S/pS OK R U S/p n S is the ideal generated by ξ n, X 1,n,, X d,n The derivation d: R U Ω 1 R U /O K = d i=1 R U dt i extends to a W n S/pS A cris Wn p n A cris linear connection S : [ [ ξ m W n S/pS OK R n U, Xm 1,n,, Xm d,n ξ m W n S/pS OK R n U, Xm 1,n,, Xm d,n RU Ω 1 R U /O K sending Xm 1 1,n m 1! X m d d,n m d! d X m 1 1,n i=1 m i 1 1,n m 1 X X! m i 1! [ ξ m W n S/pS OK R n U, Xm 1,n,, Xm d,n It follows from this that de Rham complex m d d,n m d! 0 W n S/pS Wn A cris p n A cris W n S/pS OK R U dt i One can prove that in fact [ X = Wn S/pS Wn Acris /p n 1,n m A cris,, Xm d,n [ ξ m n, Xm 1,n,, Xm d,n [ is exact Note that W n S/pS Wn Acris /p n X1,n A m cris,, Xm d,n X m 1 1,n RU Ω R U /O K 0 admits a filtration where Fil r is generated by the elements ξm 0 n m 0! m d with m! 0 + m 1 + m d r Such filtration satisfies Griffith s tranversality with respect to the connection S We also notice that the action of the m 1! X m d d,n [ Galois group G U on S extends to a an action on W n S/pS OK R ξ m n U, Xm 1,n,, Xm d,n let c i : G U Z p be defined by σ T 1/pn+1 c i = ζ i σ n+1 T 1/pn+1 i Then, σ Xi m / m X m s i 1 [ζ n+1 ciσ s := T s m s! s! i,n 1 s=0 Indeed, Such action preserves the filtration and commutes with the connection S Since the pairs U, W with the properties above define a basis for the given pre-topology on X, one can sheafify this construction to get a sheaf A cris /p n A cris of O X A cris /p n A cris -modules, endowed with an integrable connection, a decreasing filtration Fil A cris /p n A cris which satisfies Griffiths transversality, a Frobenius ϕ Moreover, {A cris /p n A cris } n define an inverse system A cris ShX N of O X A cris -modules, endowed with a connection : A cris A cris OX Ω 1 X/O K, filtration Fil A cris := { Fil A cris /p n A cris and a Frobenius ϕ Moreover, }n Proposition 54 Consider the complex A 1 cris A cris OX Ω 1 2 X/O K A cris OX Ω 2 X/O d A cris OX Ω d X/O K 0 i it is exact; ii the natural inclusion A cris A cris identifies Ker 1 with A cris; iii Griffith s transversality we have Fil r A cris Fil r 1 A cris OX Ω 1 X/O K for every r; iv for every r N the sequence 0 Fil r A cris Fil r A cris 1 Fil r 1 A cris OX Ω 1 X/O K 2 Fil r 2 A cris OX Ω 2 X/O K 3, with the convention that Fil s A cris,m = A cris for s < 0, is exact; 11

12 v the connection : A cris,m A cris,m OX Ω 1 X/O K is quasi nilpotent; vi Frobenius ϕ on A cris is horizontal with respect to i e, ϕ = ϕ df [ ξ m Example 55 We consider the case that n = 1 Take the ring S/pS OK R 1 U, Xm 1,1,, Xm d,1 We claim that it is isomorphic to S/pS[δ 0, δ 1, X i,0, X i,1, 1 i d /δ p m, X p i,m 1 i d,m 0 where δ m = γ m+1 ξ 1 and X i,j = γ j+1 X i and γ : z zp In particular, it is a free S/pS-module p This allows to compute the localization of A cris One can also define A cris m = A cris t m for every m N with its filtration and connection Taking inductive limits of the A cris m as in the definition of B cris one constructs the object B cris Ind ShX N, with a filtration, connection and Frobenius so that the analogue of 54 holds One can also prove that an analogue of the fundamental exact sequence holds: Lemma 56 We have the following exact sequence in Ind Sh X N 0 Q p Fil 0 B cris ϕ 1 B cris 0 Proof We prove the weaker statement that the sequence of inverse systems in Sh X N One reduces to prove that is exact and from this that 0 Z p A cris 0 Z/pZ A cris/pa cris ϕ 1 A cris 0 ϕ 1 A cris/pa cris 0 0 Z/pZ Ob X /pob X ϕ 1 Ob X /pob X 0 This can be proven on stalks and, hence, it suffices to show that for U X affine the map 0 Z/pZ R U /pr U ϕ 1 R U /pr U 0 is exact By Artin-Schreier theory, the kernel of ϕ 1 is H 0 SpecR U /pr U, Z/pZ which is Z/pZ since R U /pr U is connected here we use that R U is p-adically complete and separated so that its finite normal extensions are connected if and only if they are connected modulo p The cokernel of ϕ 1 coincides with H 1 SpecR U /pr U, Z/pZ which is alos zero since every Z/pZ-torsor over R U /pr U can be lifted to R U and, thus, is trivial by definition of R U here again we use that R U is p-adically complete and separated so that every finite extension is henselian with respect to the ideal p 12

13 6 The computation of R i v B cris Consider the functor v : Ind ShX N ShX given by { F n,m n Then, }m { v Fn,m Theorem 61 We have R i v B cris = 0 for i 1 and it is equal to O X OK B cris if i = 0 Similarly, R i v Fil r B cris = 0 for i 1 and it is equal to O X OK Fil r B cris if i = 0 Here, O X OK B cris stands for the inductive limit O X OK A cris [t 1 where O X OK A cris stands for the inverse system of sheaves O X OK A cris /p n A cris Similarly for O X OK Fil r B cris In this section we will only sketch the proof that R i v B cris = 0 for i 1 For the other statements we refer to [AI2 also considering the variants in 42 Recall that an open affine subset U = SpecR U of X is called small if it admits an étale morphism to Spec [ O K [T 1 ±1,, T ±1 d In this case we write R U, := n R U OK O K T 1/pn 1,, T 1/pn d We will freely use the following result proven in [F1, Thm I24ii as a consequence of his theory of almost étale extensions Given normal extensions R U, S T contained in Ω such that S[p 1 T [p 1 is finite, étale and Galois with group G T/S we have that H i G T/S, T is annihilated by any element of the maximal ideal of O K for every i 1 As an application we prove the following: Lemma 62 The presheaf O X /po X is separated i e, if U, W U, W is a covering, the natural map O X U, W/pO X U, W O X U, W /p n O X U, W is injective In particular, if U = SpecR U is a small affine open subscheme of X, the map R U /pr U = O X R U /po X R U O X /po X RU is injective Furthermore, its cokernel is annihilated by the maximal ideal of O K Proof We first prove the first statement We may assume that U and U are affine We may write O X U, W = i S i resp O X U, W = j S j as the union of normal and finite R U algebras resp R U algebras of Ω, étale after inverting p such that for every i there exists j i so that S i is contained in S j i and the map SpecS j i SpecS i is surjective on prime ideals containing p Let x S i p n S j i Let P S i be a prime ideal over p and let P S j i be a height one prime ideal over it Then, x S i,p p n S j i,p Hence, x pn S i,p Thus, x lies in the intersection of all height one prime ideals of S i so that x S i We conclude that the map S i /p n S i S j i /p n S j i is injective The claimed injectivity follows We now pass to the second statement It follows from the first statement that the value of the sheaf O X /po X at U, W is given by the direct limit, over all coverings U, W of U, W with U affine, of the elements b in O X U, W /po X U, W such that the image of b in the ring O X U, W /po X U, W is 0 where U, W is the fiber product of U, W with itself over U, W Hence, OX /po X RU = lim Ker S,T,n S,T where the notation is as follows The direct limit is taken over all normal R U, subalgebras S of R U, finite and étale after inverting p over R U, [1/p, all affine covers U U and all normal extensions R U, RU S T, finite, étale and Galois after inverting p Eventually, we put U := SpecR U to be the fiber product of U with itself over U i e, R U := R U RU R U We let Ker S,T,n := Ker T/pT T S /p T S, n } m 13

14 where T S is the normalization of the base change to R U of T RU, RU, S T Study of Ker S,T,n For every S and T as above, write G S,T for the Galois group of T OK K over S RU, R U, OK K Then, T S is simply the product g G S,T T RU R U where we view R U as R U algebra choosing the left action Hence, we have Ker S,T,n = Ker T/pT T RU R U, pt RU g G S,T R U where the two maps in the display are a a,, a and a ga g G S,T Study of CokerS/pS Ker S,T,n For the rest of this proof we make the following notations: if B is a normal R U, -algebra we denote by B := B RU, R U, = B RU R U, also B := B RU, R U, = B RU R U Note that B and B are normal We then get a commutative diagram 0 S/pS S /ps S /ps α β 0 Ker S,T,n T/pT T S /p T S = g G S,T T /pt The top row is exact and the bottom row is exact by construction Since S T and S T are finite extensions of normal rings, the maps α and β are injective Define Z as Z := Coker S /ps T/pT G S,T Cokerα and Y as CokerS/pS KerS,T,n Since Ker S,T,n is G S,T -invariant, the image of Y in Cokerα is contained in Z Since α and β are injective, the map Y Z is injective Consider the exact sequence 0 S /ps = T G S,T /pt G S,T T/pT G S,T H 1 G S,T, T Then, Y Z H 1 G S,T, T Since R U, T is almost étale, the group H 1 G S,T, T is annihilated by any element of the maximal ideal of O K thanks to Faltings results recalled above This implies the last claim From this one can deduce the following: Corollary 63 Let U = SpecR U be a small open affine subset of X For every n N we have an injective map [ X m W n RU /pr U Wn Acris /p n 1,n A cris,, Xm d,n A cris /p n A cris RU with cokernel annihilated by the elements [ζ m 1 W n for every m N We refer to [AI2 for details Since we now know how to almost compute the localizations of A cris /p n A cris we can proceed to the computation of R i v A cris /p n A cris using 44 Our main theorem, stating that R i v B cris = 0 for i 1 will then amount to prove, see [AB, that [ X m H i G U, W n RU /pr U Wn Acris /p n 1,n A cris,, Xm d,n is annihilated by a fixed power of t, independent of n and U, for every i 1 These computations, and similar ones for the H 0 and for the cohomology of the filtration, are the content of [AB We simply sketch some of the ideas involved 14

15 Consider the extension R U OK K R U, [p 1 It is Galois with group Γ U isomorphic to Z d p Due to Faltings almost étale theory, it suffices to prove that [ X m H i Γ U, W n RU, /pr U, Wn Acris /p n 1,n A cris,, Xm d,n is annihilated by a fixed power of t for every i 1 Note that there exists a unique homomorphism of O K -algebras R U W n RU, /pr U, sending Ti T i,n Such map is not G U -equivariant! Write R U,n for its image One can prove that W n RU, /pr U, Wn Acris /p n A cris = RU,n Wnk Acris /p n A cris [ T 1/p m 1,n,, 1/pm T d,n m N Set A n = R U,n Wn k Acris /p n A cris and let Bn be the A n -submodule generated by the elements T 1/pm i,n for some m 1 Then, [ X m 1,n A n,, Xm d,n = RU /p n R U Wnk [ X m Acris /p n 1,n A cris,, Xm d,n [ X m so that, in particular, it is stable under the action of Γ U and similarly B 1,n n,, Xm d,n is stable under the action of Γ U The problem is reduced to prove: [ Lemma 64 1 The group H i X m Γ U, B 1,n n,, Xm d,n is annihilated by a fixed power of t for every i 1 2 The group H Γ i U, R U /p n R U Wn k fixed power of t for every i 1 [ Acris /p n X1,n A m cris,, Xm d,n is annihilated by a Proof We limit ourself to the case that d = 1 A Koszul complex argument reduces the general case to this case and we refer to [AB for details We remark that in this case Γ U is the free Z p -module generated by the element γ which acts on T 1/pn via multiplication by ζ n Then, H 1 is simply the cokernel of the operator γ 1 We write T for T 1, T for T 1,n and X for X 1,n Since and hence γx 1 = T ε T = 1 εt + εx γ 1 X [m = 1 εt + εx [m X [m = ε m 1X [m + where y [m := y m /, µ m = εm 1 [ε 1 1 Take b = N m=0 m 1 ε [j T j [ε m j X [m j j=1 = 1 ε µ m X [m + T ε m 1 X [m 1 + m β j T j ε m j X [m j j=2 and β j = εj 1 ε 1 which is an element in Fil1 A cris for j 2 b m X [m with b m B n for every m Suppose N > 0 Since the cokernel of γ 1 on X n is annihilated by ε 1/p 1, there exists a N X such that γ 1a N = 1 [ε 1 p bn Then, γ 1 a N X [N = γa N γ 1 X [N + γ 1a N X [N 15

16 and hence 1 1 ε p b γ 1 an X [N = γa N γ 1 X [N + N 1 1 ε 1 p b m X [m 1 ε 1 p N 1 m=0 B n X [m Proceeding by descending induction on N we conclude that [ 1 ε 1 X m p 1,n Xn,, Xm d,n is contained in the image of 1 γ this proves 1 2 The matrix of γ 1 on the module N i=0 R U/p n R U Wnk Acris /p n A cris X [i with respect to 1, X,, X [N is 1 εg n,n with G N n = m=0 0 T i T 2 β 2 T 3 β 3 T N 1 β N 1 T N β N 0 µ 1 T ε T 2 β 2 ε T N 2 β N 2 ε T N 1 β N 1 ε µ2 T ε 2 T N 2 β N 2 ε 2 T 2 β 2 ε N 3 T 3 β 3 ε N 3 µ N 2 T ε N 2 T 2 β 2 ε N 2 µ N 1 T ε N µ N Let G n,n be the matrix obtained erasing the first column and the last row Then, Gn,N = Ũ n,n + Ñn,N with T 0 0 µ 1 T ε 0 µ Ũ n,n = 2 T ε 2 µ N 2 T ε N µ N 1 T ε N 1 is invertible and Ñn,N has nilpotent coefficients and, hence, it is nilpotent Hence, also Ũ n,nñn,n 1 is nilpotent and G n,n = Ũn,N IN + Ũ n,nñn,n 1 is invertible This implies that the cokernel of [ γ 1 on R U /p n R U Wnk Acris /p n X1,n A m cris,, Xm d,n is annihilated by ε 1 7 The proof of the comparison isomorphism Definition 71 We say that a Q p -adic étale sheaf L = L n and a filtered-f -isocrystal E on X K are associated if we have an isomorphism E OX B cris = L Zp B cris in IndbigShX N compatibly with all extra structures ie, Frobenius, Filtrations, connections and A cris -module structures are associ- It follows from [Bri that E = Hom E, O X, d and L = Hom L n, Z/p n Z n ated since E and L are 16

17 Due to 54 we have an exact sequence In particular, 0 L Zp B cris E OX B cris OX Ω X/O 0 H i X, L Zp B cris = H i X, E OX B cris OX Ω X/O The isomorphism is compatible with filtration, G K -action and Frobenius Due to 61 the latter coincides with H i X, E OX Ω X/O K OK B cris i e, H i dr X, E OK B cris The isomorphism is compatible with Frobenius and G K -action and it can be proven to be compatible with filtrations as well we refer to [AI2 for the non-trivial proof of this fact The same isomorphisms hold if we work with X instead of X In particular, H i X, L Zp B cris = H i X, L Zp B cris Put Vi := H i X, L = H 1 X et K, L and D i := H i X, H i X, E OX Ω X/O K One can prove that V i = H i X, L; this is a GAGA type theorem and passes via the variants β and γ of 42; see [AI1 It then follows from 56 that we have long exact sequences α V i i Fil 0 1 ϕ ɛ D i K B cris D i K B i cris Vi+1 β i γ i β i+1 ϕ=1 ωi 1 ϕ D i K B cris Di K B cris D i K B cris ϕ=1 D i+1 K B cris We recall a criterion from [CF for a filtered-f -module D over K to be admissibile i e, to be associated to a crystalline representation of G K Let δd: D K B cris ϕ=1 be the natural map Put V cris D := Kerδ D D K B cris Fil 0 D K B cris Proposition 72 [CF The filtered ϕ-module D over K is admissible if and only if V cris D is a finite dimensional Q p -vector space and b δd is surjective Moreover, if D is admissible then V := V cris D is a crystalline representation of G M D cris V = D a and We apply this to the above exact sequence Consider the part of the above diagram in degrees 0 and 2d ɛ2d 1 α V 2d 2d Fil 0 D 2d M0 B cris 1 ϕ D 2d M0 B cris 0 β 2d γ 2d ϕ=1 ω2d 1 ϕ D 2d M0 B cris D2d M0 B cris D 2d M0 B cris 0 Note that δd 2d is the composite of D2d M0 B cris ϕ=1 ω2d D2d M0 B cris Cokerγ 2d and also that KerδD 2d = Ker 1 ϕ: Fil 0 D 2d M0 B cris D 2d M0 B cris It follows that α 2d induces a surjective Q p -linear map V 2d KerδD 2d and that δd 2d is surjective We deduce from 72 that D 2d is admissible and that we have a Q p -linear, surjective homomorphism V 2d V cris D 2d which is G K -equivariant Let Di be H i X, H i X, E OX Ω X/O K and V i be H i X, L = H 1 X et, K L Then, D0 = D 2d as filtered F -modules up to shifting the filtration and twisting Frobenius accordingly to Poincaré 17

18 duality for filtered F -modules In particular, D 0 is admissible Similarly V 0 is the dual of V 0 up to Tate twist according to Poincaré duality in étale cohomology We also have the exact sequence α 0 0 V0 Fil 0 D0 M0 B cris D0 ɛ M0 B 0 cris β0 γ0 0 D0 ϕ=1 ω0 M0 B cris D0 1 ϕ M0 B cris D0 M0 B cris It follows that V 0 = KerδD 0 = V cris D 0 Hence, dim Qp V 2d = dim Qp V 0 = dim K D 0 = dim K D 2d = dim Qp V cris D 2d and therefore V 2d = Vcris D 2d This proves our statement for i = 0 and i = 2d Let us remark at the same time that as α 2d is injective, ɛ 2d 1 = 0 An easy diagram chase shows that ɛ 0 = 0 and therefore we can continue with i = 1 along exactly the same lines as for i = 0 By induction the Theorem follows References [AB F Andreatta, O Brinon: Acyclicité géométrique de B cris, Submitted Available at brinon [AI1 F Andreatta, A Iovita: Global applications of relative ϕ, Γ-modules, Astérisque [AI2 F Andreatta, A Iovita: Comparison isomorphisms for formal schemes, Work in progress [B3 P Berthelot: Cohomologie rigide et cohomologie rigide à supports propres Premiére partie, Preprint IRMAR 96-03, 89 pages 1996 Available at [Bri O Brinon: Représentations p-adiques cristallines et de de Rham dans le cas relatif, Mémoires de la SMF [BO P Berthelot, A Ogus: Notes on crystalline cohomology, Princeton University Press, 1978 [CF P Colmez, J-M Fontaine: Construction des représentations p-adiques semi-stables, Invent Math 140, 1 43, 2000 [F1 G Faltings: p-adic Hodge Theory, Journal AMS 1, , 1988 [F2 G Faltings: Crystalline cohomology and p-adic Galois representations, In Algebraic Analysis, Geomtery and Number Theory JIIgusa ed, John Hopkins University Press, Baltimore, 25 80, 1998 [F3 G Faltings: Almost étale extensions, In Cohomologies p-adiques et applications arithmétiques, vol II P Berthelot, J-M Fontaine, L Illusie, K Kato, M Rapoport eds Astérisque , [Fo J-M Fontaine, Sur certains types de représentations p-adiques du group de Galois d un corps local; construction d un anneau de Barsotti-Tate, Ann of Math 115, , 1982 [FM J-M Fontaine, W Messing: p adic periods and p adic étale cohomology, In Current trends in arithmetical algebraic geometry Arcata, Calif, 1985, Contemp Math 67, , ϕ

19 [Fo J-M Fontaine: Le corps des périodes p-adiques With an appendix by Pierre Colmez In Périodes p-adiques Bures-sur-Yvette, 1988 Astérisque 223, , 1994 [SGAIV Théorie des topos et cohomologie étale des schémas Tome 1: Théorie des topos Séminaire de Géométrie Algébrique du Bois-Marie SGA 4 Dirigé par M Artin, A Grothendieck, et J L Verdier Avec la collaboration de N Bourbaki, P Deligne et B Saint-Donat LNM [J U Jannsen: Continuous Étale cohomology, Math Ann 280, , 1988 [N W Niziol: Crystalline conjecture via K-theory, Ann Sci 1998 École Norm Sup 31, , [T T Tsuji, p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, Invent Math 137, ,