On the continuity of the finite Bloch-Kato cohomology

On the continuity of the finite Bloch-Kato cohomology

ADRIANIOVITA(*) - ADRIANOMARMORA(**)

ABSTRACT- LetK0be an unramified, complete discrete valuation field of mixed character- istics (0;p) with perfect residue field. We consider two finite, freeZp-representations of GK0,T1and T2, such thatTiZpQp, foriˆ1;2, are crystalline representations with Hodge-Tate weights between0andrp 2:LetKbe a totally ramified extension of degreeeofK0. Supposing thatp3ande(r 1)p 1, we prove that for every integer n1and iˆ1;2, the inclusion H_f¹(K;Ti)=pⁿH_f¹(K;Ti),!H¹(K;Ti=pⁿTi) of the finite Bloch-Kato cohomology into the Galois cohomology is functorial with respect to morph- isms asZ=pⁿZ[GK0]-modules fromT1=pⁿT1toT2=pⁿT2. In the appendix we give a related result forpˆ2.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 11S32 ; 14F30

KEYWORDS. Selmer groups, Galois representations, lattices in crystalline representations, Bloch-Kato finite cohomology, strongly divisible modules.

Contents

1. Introduction . . . 240

1.1 The main result . . . 242

1.2 Notation and conventions . . . 245

2. Strongly divisible modules . . . 245

2.1 The ringS . . . 245

2.2 Strongly divisible lattices overS . . . 246

2.3 Functors towards Galois representations . . . 247 (*) Indirizzo dell'A.: Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, MontreÂal, QueÂbec H3G 1M8, Canada, and Dipartimento di Matematica Pura ed Applicata, UniversitaÁ degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy.

E-mail: iovita@mathstat.concordia.ca

This author was partly funded by the grant PRIN-MIUR 2010-11, ``Arithmetic Algebraic Geometry and Number Theory'' and an NSERCDiscovery grant.

(**) Indirizzo dell'A.: IRMA, Universite de Strasbourg et CNRS, 7 rue ReneÂ-Descartes, 67084 Strasbourg, France.

E-mail: marmora@math.unistra.fr

This author was partly supported by the project ANR-09-JCJC-0048-01.

3. Extensions of crystalline representations . . . 250

3.1 The Fontaine-Lafaille theory . . . 250

3.2 Relation with strongly divisible lattices overS . . . 252

3.3 A double complex computing extensions . . . 254

4. Proof of the main result. . . 259

4.1 Construction ofH¹_r…K; †. . . 259

4.2 Some complements . . . 266

A. The characteristic two case . . . 268

1. Introduction

LetKdenote a complete discrete valuation field of mixed characteristics (0;p) with perfect residue fieldk. We choose an algebraic closureKofKand denote by G_K the Galois group ofK overK. For ap-adic representationV ofG_K, the finite cohomology groupH¹_f(K;V) is defined as

H¹_f(K;V)ˆker(H¹(K;V)!H¹(K;VQpB_cris))

whereH¹(K; ) denotes the first continuous Galois cohomology functor and B_crisis the ring of crystalline periods. IfV is a crystalline representation then the coho- mology classes inH¹_f(K;V) represent classes of extensions

0!V !E!Qp!0

which are crystalline, i.e. such thatEis a crystallineGK-representation. IfTVis aZp-lattice stable byGK, the groupH¹_f(K;T) is defined as the fiber product of the diagram

where his the map induced by the injection TV. By definition the classes in H¹_f(K;T) correspond to classes of extensions ofTbyZpwhich are crystalline (after extending scalars toQp). The canonical mapH¹_f(K;T)!H¹(K;T) is injective and identifies H¹_f(K;T) to h ¹(H¹_f(K;V)); on the other hand the map H¹_f(K;T)! H¹_f(K;V) is not injective in general (H¹_f(K;T) contains all thep-torsion ofH¹(K;T)).

The main relevance of the groups H¹_f(K;V) comes from global arithmetic.

Suppose that F is a number field and that M is a p-adic representation of GF :ˆGal(F=F). Then the Selmer group ofMoverF is the subgroup of the con- tinuous cohomology group H¹(F;M) of classes which satisfy certain local restric- tions, in particular the classes in the Selmer group restricted to the decomposition group atvofG_F, for placesvofFdividingp, lie inH¹_f(F_v;M).

If thep-adic representationMas above is thep-adic realization of a motive over a number fieldF, then the family ofp-Selmer groups over the cyclotomic tower of F (or some other Z_p-tower) encodes important arithmetic information about the motive, in particular its algebraic p-adic L-functions are defined in terms of it.

Ralph Greenberg was the first to ask the question: suppose we have two motives which are congruent modulo a power ofp. What can one say about the two families ofp-Selmer groups?

For example in [14] and [13] one studies the case of two elliptic curves defined overQ(or even modular forms of weight two) and one obtains: if the elliptic curves are both ordinary or both supersingular atpand they are congruent modulop, then under certain further hypothesis one shows that the families of non-primitive Selmer groups attached to these elliptic curves are also congruent modulo p. It follows, using results of K. Kato that if the main conjecture holds for one of the elliptic curves then it holds for the other.

Of course in order to study the behaviour of Selmer groups with respect to congruences, one has to first investigate the behaviour of the local conditions de- fining the Selmer groups with respect to the congruences, in other words to ask whetherH¹_f(K;T) variesp-adically continuously withT(where nowK is ap-adic field andTaZ_p-representation ofG_Kas at the beginning of this introduction). This is the main task of this article.

We will now be more precise and consider two crystallineZp-representations of GK,T1 and T2 which are isomorphic modulopⁿ (we say thatT1 andT2 are con- gruent modulo pⁿ). Consider a class j2H¹(K;T1=pⁿT1) which is the reduction modulopⁿof a class coming fromH¹_f(K;T₁). We would like to know if the image ofj via the isomorphism betweenH¹(K;T₁=pⁿT₁) andH¹(K;T₂=pⁿT₂) is the reduction of a class inH¹_f(K;T2). This could be reformulated as follows: given an isomorphism i:T1=pⁿT1!T2=pⁿT2 of Z=pⁿZ[GK]-modules, is there an isomorphism ~i of Z=pⁿZ-modules making the following diagram commutative?

The maps H¹_f(K;T_i)=pⁿH¹_f(K;T_i)!H¹(K;T_i=pⁿT_i), for iˆ1;2, in the diagram above are induced by the injections H¹_f(K;T_i),!H¹(K;T_i) and H¹(K;T_i)=

pⁿH¹(K;Ti),!H¹(K;Ti=pⁿTi). We recall that they are injective, cf. Lemma 4.1.1, and so this reformulation of the question is only apparently more general. However it makes sense to ask more generally whether H¹_f(K; )=pⁿH¹_f(K; ) defines a sub- functor ofT=pⁿT7!H¹(K;T=pⁿT).

It is known that these questions do not have positive answers in general. For example let T1ˆZp(1), T2ˆZp(p), KˆQp and p3. We have clearly T1=pT1ˆFp(1)T2=pT2 and Kummer theory implies that H¹(Qp;Fp(1))ˆ

Q_p=(Q_p)^p, which is a two dimensionalFp-vector space. Simple cohomological argu- ments show that the F_p-vector spaces H¹_f(Q_p;T₁)=pH¹_f(Q_p;T₁) and H¹_f(Q_p;T₂)=

pH¹_f(Q_p;T₂) have both codimension one inH¹(Q_p;F_p(1)). However, one can show that they are orthogonal with respect to cup-product and they correspond to sub- spaces ofQ_p=(Q_p)^pspanned respectively bypandp‡1. This example shows also that our problem cannot be solved by using only arguments based on the lengthes of theZ=pⁿZ-modulesH¹_f(K;T_i)=pⁿH¹_f(K;T_i).

Another example, due to R. Greenberg, shows that the equivariance of the morphism i:T₁=pⁿT₁!T₂=pⁿT₂ with respect to G_K is too weak in general.

Consider two ordinary elliptic curvesE1andE2overK, assume thatKcontains the coordinates of the p-torsion pointsE1[p] andE2[p] and that it has finite re- sidue field. Foriˆ1;2, letT_iˆTp(E_i), whereTpdenotes thep-adic Tate module.

A result of Coates and Greenberg (cf. [9]) states that H¹_f(K;T_i) is equal to the image ofH¹(K;T(E^_i)),!H¹(K;T_i), whereE^_iis the formal group ofE_i(which has height 1 by ordinarity). By the hypothesis onK, the reductionT_i=pT_iˆE_i[p] is a two dimensionalFp-vector space, trivial asGK-module, in particularT1andT2are congruent modulo p. So any isomorphism i:T1=pT1!T2=pT2 not sending the image of T_p(E^₁)=pT_p(E^₁) to T_p(E^₂)=pT_p(E^₂) gives a negative answer to our question.

Nevertheless several positive answers are know in the literature, some of them are listed in Remark 1.1.4-(1) : the goal of this article is to generalize them as detailed above.

1.1 ±The main result 1.1.1 -

We use the notations of the previous section and denote by K₀ the maximal absolutely unramified subextension ofK, bye:ˆ[K:K0] and byGK0the absolute Galois group ofK0.

Letn1 be an integer,W Za subset andZaZ=pⁿZ-module of finite type endowed with a continuous action ofG_K. We say thatZis crystalline with Hodge- Tate weights inWif there exists an exact sequence ofZ_p[G_K]-modules

0 !T⁰ !T !Z !0

whereTis a crystallineZp-representation with Hodge-Tate weights inW.

We denote by Rep_Qp…G_K†^W_cris (resp. Rep_Zp…G_K†^W_cris, resp. Rep_Z=pnZ…G_K†^W_cris) the category of crystalline Q_p-representations (resp. Z_p-representations, resp.

Z=pⁿZ-representations) with Hodge-Tate weights in W. For semi-stable re- presentations we use analogous notations. For any integers ab, we denote by [a;b] the set of integersi, such thataib.

Let denote by Mod(Z=pⁿZ) the category ofZ=pⁿZ-modules.

1.1.2 -

To address the problem of continuity for the Bloch-Kato finite cohomology we proceed as follows. First assume p3. For every integer 0rp 2, we con- struct explicitly a functor

H¹_r(K; ):Rep_Z=pⁿ_Z…GK0†^[0;r]_crys !Mod(Z=pⁿZ);

endowed with a morphism of functorst¹_st:H¹_r(K; )!H¹(K; ), such that for every crystallineZ_p-representationsTofG_K0with Hodge-Tate weights in [0;r], we have the canonical factorisation:

For any crystallineZp-representationT ofGK0, we denote byrmax orrmax(T) the largest Hodge-Tate weight ofT.

The following statement which is proved as Theorem 4.1.6 in section §4.1 con- stitutes the main result of this article.

THEOREM. Let p3be a prime integer and fix an integer r with0rp 2.

We denote by T a crystallineZp-representation of G_K0with Hodge-Tate weights in [0;r] and assume e(r_max 1)p 1. Then for every integer n1, H¹_f(K;T)=

pⁿH¹_f(K;T)is isomorphic to H¹_r(K;T=pⁿT)viae.

REMARK 1.1.3.

(1) We point out that ifr_max1 the condition on the ramification indexeofKis empty.

(2) The hypothesise(rmax 1)p 1 is necessary, see Proposition 4.2.3.

(3) For every r such that rmaxrp 2, we have H¹_r(K;T=pⁿT)ˆ H¹_rmax(K;T=pⁿT) canonically. It is a consequence of the theorem but actually it follows already by the construction ofH¹_r, see Remark 4.1.3.

In particular ife(r 1)p 1, H¹_r(K; ) is a subfunctor of H¹(K; ); in this case set

H¹_f(K; ):ˆH¹_r(K; ):

(1:1:3:1)

The next statement which is proved as Corollary 4.1.12 in section §4.1 states the hypothesis under which the correspondenceT=pⁿT7!H¹_f(K;T)=pⁿH¹_f(K;T) is well- defined and functorial (we have indeedH¹_f(K;T)=pⁿH¹_f(K;T)ˆH¹_f(K;T=pⁿT)).

COROLLARY. Let p3and let T1and T2be two crystallineZp-representations of G_K0with Hodge-Tate weights in[0;r][0;p 2]and assume e(r 1)p 1.

Then for every morphism (resp. isomorphism) i:T₁=pⁿT₁!T₂=pⁿT₂ of Z=pⁿZ[G_K0]-modules, there exists a morphism (resp. isomorphism)~iofZ=pⁿZ- modules making the following diagram commutative.

REMARK 1.1.4.

(1) Some cases of Corollary 4.1.12 were already known:

(a) The case whereeˆ1 and the representationsT₁andT₂have Hodge-Tate weights in [0;p 2]. In this case the result follows from [1, Lemmas 4.4 and 4.5] using Fontaine-Lafaille theory.

(b) In the case where the Hodge-Tate weights ofT1andT2are in [0;1], under the hypothesis (T₁=pⁿT₁)^GK0 ˆ(T₁=pⁿT₁)^GK and p3, the result was proved in [13].

(c) The case of representations coming from Barsotti-Tate groups have been treated by Nekovar [18, A.2.6] using flat cohomology; ifp3 this implies, by a result of Kisin [16, (2.2.6-7)], the case where representations have Hodge-Tate weights in [0;1].

(2) Under the stronger hypothesiserp 2 we can prove a more general func- toriality, see Proposition 4.2.1.

(3) The hypothesis e(r 1)p 1 is not necessary in Corollary 4.1.12. For example, assume thatkis finite and letTbe a crystallineZ_p-representation of rank one and Hodge-Tate weight r2. We have H¹_f(K;T)ˆH¹(K;T), with no restrictions on e (cf. [1, Example 3.9]); by Tate duality it follows H¹_f(K;T)=pⁿH¹_f(K;T)ˆH¹(K;T=pⁿ) and the functoriality is obvious.

(4) If the representations have Hodge Tate weights in [0;r][0;p 2], but not in [1;r], then we do not know if the hypothesise(r 1)p 1 is necessary in Corollary 4.1.12. In particular, we do not know if Corollary 4.1.12 is true even in the following particular case: the representationsT1 andT2are irreducible of rank 2 with Hodge-Tate weights exactly 0 andp 2, withp7 ande>1 (or pˆ5 ande>2).

(5) In Appendix A we deal with the case of characteristicpˆ2 which is essentially independent from the rest of the paper.

1.2 ±Notation and conventions 1.2.1 -

We denote byO_K (resp.O_K) the ring of integers ofK(resp.K).

For any ringA, we denote byW(A) the ring of Witt vectors with coefficients in A. WhenAˆk, we write simplyWˆW(k). We recall that we have denoted byK₀ the field of fractions ofWand letsdenote the arithmetic Frobenius ofW(andK₀), i.e. the isomorphism lifting the p-power map on k. We set for every n0, Wn ˆW=pⁿW.

We denote byNthe set of integersi0.

2. Strongly divisible modules

In the following, except for the appendix, we assumep>2.

2.1 ±The ring S

In this section we recall a definition of G. Faltings in [10, §2], see also [5] and [6].

2.1.1 -

Letpbe a uniformizer ofK and denote byE(u) its minimal monic polynomial overK0. It is an Eisenstein polynomial with coefficients inW of degreee.

We denote by evp:W[u]! O_K theW-algebra homomorphism sendingutop.

Its kernel is generated byE(u). We letSdenote thep-adic completion of the di- vided power hull ofW[u] with respect to the ideal ker(ev_p), on which we have di- vided powers compatible with the canonical divided powers of the idealpW[u]. The ringScan be identified to the sub-ring ofQp[[u]]of seriesP

i2Nai uⁱ

q(i)!such that the a_i's belong toW, the sequence (a_i)_i0converges to 0 forigoing to‡1andq(i) is the quotient of the Euclidean division ofibye.

2.1.2 -

The ringSis endowed with the following extra structures.

(1) A Frobenius homomorphism W:S!S, which is the unique continuous s- semi-linear morphism sendingutou^p.

(2) A decreasing filtration (FilⁱS)_i2NonS, defined as follows: for every integer i0, FilⁱSis the thep-adic closure inSof the ideal generated by^E(u)j

j! , for allji.

We have: for all 0ip 1, W(FilⁱS)pⁱS; for such an integer i, put W_iˆp ⁱWj_Filⁱ_S.

Finally, we setc1 ˆW₁(E(u)): it is a unit inS.

2.1.3 -

We denote by V¹_logˆVb¹_S=W(log) (resp. V¹ˆVb¹_S=W) the module of continuous logarithmic differential (resp. regular) 1-forms ofSover Wandd:S!V¹V¹_log the canonical differential. We haveV¹ˆSduandV¹_logˆSu ¹du.

We denote again W:V¹_log !V¹_log theW-semi-linear homomorphism, induced by the Frobenius onSvia the universal property of the differentials forms. It maps u ¹dutopu ¹duand we putW₁ˆp ¹W. The submoduleV¹is stable underWandW₁. Clearly we have dWˆpWd. In the literature, it is also common to introduce the continuousW-derivationN:S!Ssendinguto u.

2.2 ±Strongly divisible lattices over S (Following[5]and[17]) Letrbe an integer such that 0rp 2.

2.2.1 -

We first define the category ⁰Mod(S)^r_log. Its objects are 4-uples (M;Fil^rM;W_r;r_M), where:

(1) Mis anS-module;

(2) Fil^rMis a sub-S-module ofMcontaining (Fil^rS)M;

(3) W_ris aW-semi-linear mapW_r:Fil^rM!M, such that for allsin Fil^rSandxin Mwe haveW_r(sx)ˆc₁^rW_r(s)W_r(E(u)^rx);

(4) r_M:M!M_SV¹_logis a logarithmic connection satisfyingE(u)r_M(Fil^rM) Fil^rM_SV¹_logsuch that the following diagram commutes.

When there is no risk of confusion, we will simply denoter ˆ r_M.

The morphisms in this category are defined to be theS-linear maps preserving the submodule Fil^rand commuting withW_randr.

We remark that the integerris fixed in the definition of the category⁰Mod(S)^r_log and thus the datum of Fil^rMin the 4-uple (M;Fil^rM;W_r;r_M) refers to asinglesub- module ofM.

If no confusion is arising, we will sometimes denote an object (M;Fil^rM;W_r;r) only by its underlying moduleM.

The category ⁰Mod(S)^r_log is Zp-linear. A short sequence 0!M⁰!M! M⁰⁰!0 of objects in ⁰Mod(S)^r_log is said exact, if the induced sequences

0!M⁰!M!M⁰⁰!0 and 0!Fil^rM⁰!Fil^rM!Fil^rM⁰⁰!0 are short exact sequences ofS-modules.

2.2.2 -

We denote ⁰Mod(S)^r the full subcategory of ⁰Mod(S)^r_log of objects whose con- nection r is regular. In the literature it is common to use the derivation N_Mˆ r_M( u ^d

du) instead of the connection r: it is called the monodromy op- eratorand the regularity condition is equivalent toN(M)uN(M).

2.2.3 - Free objects

We denote Mod(S)^r (resp. Mod(S)^r_log) the full subcategory of ⁰Mod(S)^r (resp.

0Mod(S)^r_log) whose objects satisfy:

(1) Mis free of finite type as anS-module;

(2) theS-moduleM=Fil^rMhas nop-torsion;

(3) the image ofW_rgeneratesMas anS-module.

We call Mod(S)^r(resp. Mod(S)^r_log) the category ofstrongly divisible lattices(resp.

logarithmic strongly divisible lattices) overSof weightr.

2.2.4 - Torsion objects

We denote ModFI(S)^rthe full subcategory of⁰Mod(S)^rwhose objects satisfy:

(1)Mis isomorphic toL

i2IS=pⁿⁱSas anS-module, whereIis a finite set andn_i are positive integers;

(2) the image ofW_rgeneratesMoverS.

For everyn1, we denote ModFI(S=pⁿS)^rthe full subcategory of ModFI(S)^r of objects annihilated bypⁿ, asS-modules.

2.3 ±Functors towards Galois representations 2.3.1 -

We denote byRthe projective limit lim O^K=pO_K, where the transition maps are given by the absolute Frobenii x7!x^p. We denote byAcris the W(R)-algebra of crystalline periods defined by Fontaine in [11, 2.2.3].

Let pˆ(p⁽ⁿ⁾)_n2N be a compatible sequence ofpⁿ-roots ofp, i.e. everyp⁽ⁿ⁾ be- longs toK,p⁽⁰⁾ˆpandp^(n‡1)pˆp⁽ⁿ⁾. We denote by [p]2W(R)A_crisits Teich- muÈller representative, and, for anyginG_K, we pute(g)ˆg([p])=[p] inA_cris.

2.3.2 -

We recall the definition of a ring introduced by Kato in [15, § 3], and denoted afterwardsAb_st, cf. [2] and [4] for more details.

Let Acrishxibe the divided power hull of the polynomial algebraAcris[x] com- patible with the canonical divided powers of the idealpAcris[x]; forj0, we denote byg_j(x) thej-th divided power ofx. The ringAbstis thep-adic completion ofA_crishxi.

It is endowed with the following structures:

(1) A filtration: for any integeri0, FilⁱAbstˆ X^‡1

nˆ0

ang_n(x)an2A_cris; lim

n!‡1anˆ0;8ni;an2Fil^{i n}A_cris

( )

: We note that for everyn0, we have FilⁱAb_st\pⁿAb_stˆpⁿFilⁱAb_st. (2) A FrobeniusW, extending that ofA_crisby mappingxto (1‡x)^p 1.

(3) AGK-action extending the action onAcris, defined as follow: for anyginGK, g(x)ˆe(g)x‡e(g) 1.

(4) An S-algebra structure given by the monomorphism S,!Abst, u7!

[p](1‡x) ¹. It identifiesSto (Abst)^GKand it is compatible with all the others structures.

(5) Ap-adically continuous connectionr:Abst!Abst_SV¹_log, satisfyingr(A_cris)ˆ0 andr(x)ˆ (1‡x)u ¹du.

Remark that for every 0rp 2, the datum (Abst;Fil^rAbst;p ^rWj_Filr ;r) belongs to⁰Mod(S)^r_log.

2.3.3 -

We denote by T_st:Mod(S)^r_log!Rep_Zp…G_K†^[0;r]_st , the contravariant functor de- fined as follows:

T_st(M)ˆHom⁰_Mod(S)^r_log M;Abst

; with theG_K-action induced by that onAb_st.

The functor D^r_free:Rep_Zp…G_K†^[0;r]_st !Rep_Zp…G_K†^[0;r]_st , sending a Z_p-representa- tion T to itsr-twistedZp-linear dual HomZp T;Zp

(r), is an anti-equivalence of

categories because, by definition, the objects in Rep_Zp…G_K†^[0;r]_st are free as Z_p- modules. We denoteT_st:Mod(S)^r_log!Rep_Zp…G_K†^[0;r]_st the covariant functor which is the composition ofT_st withD^r_free.

In [5, Conj. 2.2.6-(1)] Breuil conjectured that the functorT_st is an anti-equiva- lence (¹) of categories and this has been proved by T. Liu.

THEOREM 2.3.4. (Theorem 2.3.5 in [17]). For 0rp 2, the functor M7!T_st(M)establishes an anti-equivalence between the category of logarithmic strongly divisible lattices of weight r and the category of semi-stable Z_p- representations of GK with Hodge-Tate weights in [0;r].

COROLLARY2.3.5. If M is inMod(S)^r, then T_st(M)is crystalline and the re- striction T_st toMod(S)^ris an equivalence of categories withRep_Zp…G_K†^[0;r]_cris.

PROOF. By a result of Breuil [2] the monodromy operator ofD_st(T_st(M)_ZpQ_p) is the residue at zero of the connection ofM_WK₀: the corollary follows. p

EXAMPLE2.3.6. Let 0rp 2 be an integer.

(1) The datum (S;Fil^rS;W_r;d) is an object of Mod(S)^r that we denote shortly by1(r).

(2) The datum (S;S;W;d) is an object of Mod(S)^rthat we denote by1.

We haveT_st(1)ˆT_st(1(r))ˆZ_pandT_st(1(r))ˆT_st(1)ˆZ_p(r).

2.3.7 -

We denoteAb¹_st ˆAb_stW(K₀=W) and endow it with the induced structures. It is an object of ⁰Mod(S)^r_log in a straightforward way. We consider the contravariant functorT^1;_st :ModFI(S=pⁿS)^r!Rep_Z=pⁿ_Z…GK†defined by

T^1;_st (M)ˆHom⁰_Mod(S)^r_log M;Ab¹_st :

We denoteD^r_tor:Rep_Z=pⁿ_Z…GK† !Rep_Z=pⁿ_Z…GK†the Pontryagin duality twisted by r, which is the functor sendingTto HomZp T;Qp=Zp

(r). It is an anti-equivalence of categories and we have D^r_tor(T)ˆHom_Z=pⁿ_Z T;Z=pⁿZ

(r) becauseT is anni- hilated bypⁿ. We set

T_st¹(M)ˆD^r_tor(T_st^1;(M)):

(¹) The notations forT_st andTstare exchanged in [17].

By [4, 2.3.1.1±2.3.1.3] the functor T^1;_st is exact and so the functor T¹_st is also exact. In general the functorT¹_st is not full, but iferp 2 it is fully faithful, cf.

[8, TheÂoreÁme 1.0.4].

For every integern1, we have the following commutative diagram of func- tors

(2:3:7:1)

where the vertical functors are reduction modulopⁿ: indeed by [7, 1.2.2] we have 0!T_st(M)!^pnT_st(M)!T^1;_st (M=pⁿM)!0;

(2:3:7:2)

which gives the commutativity of the left square; the commutativity of the right square is obvious because the representations in Rep_Zp…G_K†^[0;r]_cris are freeZp-mod- ules of finite type.

3. Extensions of crystalline representations 3.1 ±The Fontaine-Lafaille theory

We recall some definitions and results of Fontaine and Lafaille, see [12], [21, §2]

and [1, §4].

Let 0rp 2 be an integer.

3.1.1 -

Let MF^[0;r]_W be the category whose objects (D;FilⁱD;W_i;i2Z) are the following data:

(1) aW-module of finite typeD;

(2) a decreasing filtration (FilⁱD)_i2Zby direct summands, satisfying Fil^r‡1Dˆ0 and Fil⁰DˆD;

(3) for alli, as-linear mapW_i:FilⁱD!D, such thatW_ij_Fili‡1 ˆpW_i‡1 and X^r

iˆ0

W_i(FilⁱD)ˆD:

The morphisms of MF^[0;r]_W are W-linear maps preserving the filtrations and commuting with theW_i⁰s, for alli.

3.1.2 -

We say that an objectDof MF^[0;r]_W is astrongly divisible latticeif it is free asW- module. We denote MF^[0;r]_W;free the full subcategory of MF^[0;r]_W of strongly divisible lattices. We denote MF^[0;r]_Wn the full subcategory of MF^[0;r]_W whose objects are finite lenghtWn-modules.

3.1.3 -

The Fontaine-Lafaille theory gives a commutative diagram of functors

where the vertical functors are reduction modulopⁿ; the functorsT_cris andT^1;_cris are (exact) equivalences of categories defined by

T_cris ˆHom_W;Fil;W ;A_cris

andT_cris^1;ˆHom_W;Fil;W ;A_cris=pⁿA_cris

; and the quasi-inverses ofT_crisandT^1;_cris are given respectively by

D_crisˆHom_Zp[GK0] ;A_cris

andD^1;_cris ˆHom_Z=pⁿ_Z[GK0] ;A_cris=pⁿA_cris : We will need also covariant versions of these equivalences, defined by composing withD^r_freeorD^r_torrespectively:

Tcris:ˆD^r_freeT_cris:D7!Fil^r(AcrisWD)^WrˆId;

T_cris¹ :ˆD^r_torT^1;_cris:D7!Fil^r(Acris=pⁿAcrisWnD)^WrˆId; Dcris:ˆD_crisD^r_free:T7!(AcrisWT( r))^GK0;

D¹_cris:ˆD^1;_crisD^r_tor:T7!(Acris=pⁿAcrisWnT( r))^GK0:

These covariant equivalences are compatible with reduction modulo pⁿ as in the diagram above for contravariant ones. In particular we will need the following: letT be in Rep_Zp…GK0†^[0;r]_cris. To the short exact sequence 0!T!^pnT!T=pⁿT!0 we associate the sequence of filtered modules

0!D_cris(T)!^pnD_cris(T)!D¹_cris(T=pⁿT)!0 (3:1:3:1)

which is exact in the sense that, for everyi2Z, the induced sequence 0!FilⁱD_cris(T)!^pnFilⁱD_cris(T)!FilⁱD¹_cris(T=pⁿT)!0 is a short exact sequence ofW-modules, cf. [21, 2.2.3.1].

REMARK 3.1.4. The covariant functorsD_crisandD¹_crisdepend on the choice ofr, unlike their controvariant variantsD_crisandD^1;_cris. When we need to be precise we shall denote them byD^[0;r]_cris :ˆD_crisD^r_free andD^1;[0;r]_cris :ˆD_crisD^r_tor. For any in- tegers 0r1r2p 2, and any representation T2Rep_Zp…GK0†^[0;r_crys^1], the strongly divisible lattices D^[0;r_cris^1](T) and D^[0;r_cris^2](T) have the same underlying W- module but have filtrations (and Frobenii) shifted as follow: for alli2Z,

Filⁱ(D^[0;r_cris^2](T))ˆFil^{i‡(r1 r2)}(D^[0;r_cris^1](T)) and W_i^[0;r2]ˆW_i‡(r^[0;r1₁^]:_r2)

In particular the breaks of the filtration ofD^[0;r_cris^2](T) are betweenr₂ r₁andr₂. The same formula is true forD^1;[0;r]_cris .

3.2 ±Relation with strongly divisible lattices over S 3.2.1 -

Let (D;FilⁱD;W_i;i2Z) be an object of MF^[0;r]_W . We define an object M(D)ˆ(D_WS;Fil^rM(D);W_r;r)

of⁰Mod(S)^ras follow:

Fil^rM(D)ˆX^r

iˆ0

FilⁱDWFil^{r i}SDWS;

(3:2:1:1) W_rˆP_r

iˆ0W_iWW_{r i}andr ˆId_Dd, i.e.r(m_Wa)ˆm_Wd(a). By using the fact thatSbelongs to⁰Mod(S)^r, it is a straightforward exercise to verify thatM(D) belongs to⁰Mod(S)^r; it is also clear thatD7!M(D) is functorial.

By the definitions of the categories MF^[0;r]_W , MF^[0;r]_Wn , Mod(S)^rand ModFI(S=pⁿS)^r, we can prove that:

- ifDis free overW, thenM(D) belongs to Mod(S)^r;

- ifDbelongs to MF^[0;r]_Wn, thenM(D) belongs to ModFI(S=pⁿS)^r; - we haveM(D)=pⁿM(D)ˆM(D=pⁿD).

The only non trivial property to check is thatW_r(Fil^rM(D)) generatesM(D) as anS- module. Using (3.2.1.1), it is enough to check that every element ofM(D)ˆD_WSof the formv_W1 is in the image ofP_r

iˆ0W_iWW_{r i}. By hypothesisvˆP_r

iˆ0W_i(f_i), for

somef_i2FilⁱD. Putm_iˆf_iWE(u)^{r i}2FilⁱD_WFil^{r i}S. SinceW_{r i}(E(u)^{r i})ˆ c^{r i}₁ inSwe have

X^r

iˆ0

W_r(m_i)c₁^{(r i)}ˆX^r

iˆ0

W_i(f_i)_WW_{r i}(E(u)^{r i})c₁^{(r i)} ˆX^r

iˆ0

W_i(f_i)_W1ˆv_W1:

PROPOSITION3.2.2. (1)Let D be inMF^[0;r]_W;free. We have a natural isomorphism of crystallineZp-representations of G_K

T_st(M(D))(T_cris(D))_jGK:

(2) Let D be inMF^[0;r]_Wn . We have a natural isomorphism of crystallineZ=pⁿZ- representations of G_K

T¹_st(M(D))(T_cris¹ (D))_jGK:

PROOF. The proofs of (1) and (2) are analogue: let us prove (2) and leave (1) to the reader. By composing with the dual functor D^r_tor ˆHomZp ;Z=pⁿZ

(r) we are reduced to prove the analogous statements on contravariant functors. Since A_cris=pⁿA_crisˆ(Ab_st=pⁿAb_st)^rˆ0, cf. [3, Lemme 3.1.2.3], we have

T^1;_cris(D)ˆHomW;Fil;W D;Acris=pⁿAcris

ˆHom_W;Fil;W D;(Ab_st=pⁿAb_st)^rˆ0

ˆHom_W;Fil;W;r D;Ab_st=pⁿAb_st

ˆHom_S;Fil;W;r DWS;Abst=pⁿAbst Hom0Mod(S)^r_log D_WS;Ab_st=pⁿAb_st

ˆT_st(M(D)):

Let us prove that the inclusion in the last line is an equality. Let f:DWS! Abst=pⁿAbst be in T_st(M(D)). A priori we have f(FilⁱM(D))Filⁱ(Abst=pⁿAbst) and fW_iˆW_if only for the last step of the filtration:iˆr; let us prove this is true for any i. Let d be in FilⁱD, for i2[0;r]. For every a2Fil^{r i}Sand d2D, we have af(d1)ˆf(da)2Fil^r(Abst=pⁿAbst)ˆFil^rAbst=pⁿFil^rAbst. In particular for aˆE(u)^{r i}, we get

E([p](1‡x) ¹)^{r i}f(d1)2Fil^rAb_st=pⁿFil^rAb_st:

We can re-write it as E([p])^{r i}f(d1)‡xb, withb2Abst. Sincef(d1) belongs to A_cris=pⁿA_cris, we getE([p])^{r i}f(d1)2Fil^r(A_cris=pⁿA_cris). Considering thatE([p]) is a generatorofker(u:W(R)! O_Cp),weconcludethatf(d1) belongstoFilⁱ(A_cris=pⁿA_cris).

Let us check the commutation off withW_i, fori2[0;r]. For everya2Fil^{r i}S andd2FilⁱD, we have

W_{r i}(a)f(W_i(d)1)ˆf(W_i(d)W_{r i}(a))ˆf(W_r(da))ˆW_r(f(da))

ˆW_r(af(d1))ˆW(a)W_r(f(d1))ˆW_{r i}(a)W_i(f(d1)):

By takingaˆE(u)^{r i}, we getW_{r i}(a)ˆc^{r i}₁ , which is invertible; thus we have, for all i2[0;r] andd2FilⁱD,f(W_i(d1))ˆW_i(f(d1)). This finishes the proof. p

3.3 ±A double complex computing extensions

The goal of this subsection is to compute the groups of extensions of strongly divisible modules overSby using explicit complexes.

3.3.1 -

For (M;Fil^rM;W_r;r) in ⁰Mod(S)^r and 0rp 2 we set Fil^{r 1}Mˆ fm2MjE(u)m2Fil^rMg and define W_{r 1}:Fil^{r 1}M!M by putting, for m2Fil^{r 1}M,

W_{r 1}(m)ˆc₁¹W_r(E(u)m);

where c1ˆW₁(E(u)), cf. (2.1.2). Analogously, we define W:M!M by W(m)ˆ c₁^rW_r(E(u)^rm).

We will need later the following.

LEMMA 3.3.2. (1) Let f 2S and j1 be an integer. If E(u)f 2Fil^jS then f 2Fil^{j 1}S.

(2) Let f 2S=pⁿS and1jp 1be an integer. If E(u)f 2Fil^j(S=pⁿS), then f 2Fil^{j 1}(S=pⁿS).

(3) Let D be either inMF^[0;r]_W;free or inMF^[0;r]_Wn. Set MˆM(D). If rˆ0we have Fil ¹MˆM andW ₁ˆpW; if r1we have

Fil^{r 1}MˆX^{r 1}

iˆ0

FilⁱDWFil^{r 1 i}SDWS andW_{r 1}ˆP_{r 1}

iˆ0W_i^DW_{r 1 i}:Fil^{r 1}M!M.

(4) Let0!D1!D2!D3!0be a short exact sequence inMF^[0;r]_W . Then the sequence0!Fil^{r 1}M(D1)!Fil^{r 1}M(D2)!Fil^{r 1}M(D3)!0induced by functoriality is exact.

PROOF. Let us prove (1) and (2). Every element f 2S(resp.S=pⁿ) can be written as f ˆP

i0a_i(u)E(u)^[i], where E(u)^[i] denotes the divided power of E(u) (i.e.

i!E(u)^[i]ˆE(u)ⁱ) anda_i(u) are polynomials inW[u] of degree smaller thane, con- verging to zero (forigoing to infinity). We haveE(u)f ˆP

i0a_i(u)(i‡1)E(u)^[i‡1]. IfE(u)fbelongs to Fil^jS(resp. Fil^j(S=pⁿS)) thena_i(u)(i‡1)ˆ0 forij 1, which

impliesa_i(u)ˆ0 (whenf 2S=pⁿS,i‡1jp 1 by hypothesis, so thati‡1 is invertible).

Let us prove (3). The statement forrˆ0 is clear so let us assumer1. Let assume firstDin MF^[0;r]_W;free. We recall thatDis filtered free, and let (e_i)_1idbe a base adapted to the filtration;i.e.for everyi2[0;r], (ej)_{1jrk Fil}ⁱ_Dis a base of FilⁱD (cf. [21, 2.2.2]). Put

Fil^{r 1}MˆX^{r 1}

iˆ0

FilⁱDWFil^{r 1 i}SDWS:

The inclusion Fil^{r 1}MFil^{r 1}M is obvious so let us prove the converse. Let x2Fil^{r 1}M, we may assumexˆe_sf, for some f 2Sand 1sd. By hypoth- esis we have E(u)xˆe_sE(u)f 2Fil^r(M)ˆP_r

iˆ0FilⁱD_WFil^{r i}SD_WS.

We can write

E(u)xˆe_sWE(u)f ˆX^r

iˆ0

m_iWg_i (m_i2FilⁱD;g_i2Fil^{r i}S)

ˆX^r

iˆ0

X

rk FilⁱD jˆ1

(e_jWh_j;i) (h_j;i2Fil^{r i}S)

ˆX^d

jˆ1

e_{j W} X

rk FiliDj0ir

h_j;i:

Therefore E(u)f ˆP

ih_s;i, where the sum is taken over all i2[0;r] such that rkFilⁱDs. If we denote bymthe maximum of the integersisuch that rkFilⁱDs (or equivalently e_s2FilⁱD), then E(u)f belongs to Fil^{r m}S. By (1) we get f 2Fil^{r m 1}Sandxˆesf 2Fil^mDFil^{r m 1}SFil^{r 1}M.

To finish the proof of (3) forD2MF^[0;r]_W;free, let us compute the Frobenius:

W_{r 1}(x)ˆc₁¹W_r(E(u)x)ˆc₁¹W_r(esE(u)f)ˆ

c₁¹W_m(el)W_{r m}(E(u)f)ˆW_m(es)W_{r 1 m}(f):

The proof of (3) forD2MF^[0;r]_Wn is similar becauseDadmits a liftingD~ 2MF^[0;r]_W;free: so MˆM(D)ˆD~WS=pⁿSand we use (2) instead of (1). Finally (4) follows from (3) and the fact that for alli,

0!FilⁱD₁!FilⁱD₂!FilⁱD₃!0

is exact (cf. [21, 2.2.3.1]). p

DEFINITION3.3.3. For (M;Fil^rM;W_r;r) in⁰Mod(S)^r, we denote Fil^rC^;(M) the double complex ofZp-modules

Fil^rC^;(M):

where the module Fil^rMis the (0;0)-entry of the complex, and the square is com- mutative by (2.2.1)-(4) and the definition of W_{r 1}, cf. (3.3.1). The simple complex attached to Fil^rC^;(M) is

Fil^rC(M): 0!Fil^rM!^g0 M(Fil^{r 1}M_SV¹)!^g1M_SV¹!0;

whereg⁰(m)ˆ(W_r(m) m;r_M(m)) and fora2M,v2Fil^{r 1}M_SV¹, g¹(a;v)ˆ r_M(a)‡v W_{r 1}W₁(v):

By constructionHⁱ(Fil^rC(M))ˆ0 fori6ˆ0;1;2 and H⁰(Fil^rC(M))Hom⁰_Mod(S)^r 1;M

:

PROPOSITION3.3.4. We have the following functorial isomorphisms:

(1) if M is inModFI(S=pⁿS)^r, then

H¹(Fil^rC(M))Ext¹_ModFI(S=pnS)^r(1=pⁿ1;M); (2) if M is inMod(S)^r, then H¹(Fil^rC(M))Ext¹_Mod(S)^r(1;M);

whereExt¹denotes the group of Yoneda extensions.

PROOF. The proofs of (1) and (2) are analogous. We will prove them by com- puting explicitly the group of Yoneda extensions and showing that it coincides with H¹(Fil^rC(M)). An alternative approach would have been to work in the bigger category⁰Mod(S)^rand prove thatH¹(Fil^rC( )) is aneffacËablefunctor. The problem is that the category⁰Mod(S)^ris not abelian in general and we do not know if there are enough injectives.

The proof takes the rest of this section. Let us start by fixing a convention in order to treat cases (1) and (2) at the same time: forn2N[ f1g, we set

1_nˆ 1=pⁿ1 ifn2N;

1 if nˆ 1; S_n ˆ S=pⁿS if n2N; S if nˆ 1;

( (

Mod_nˆ ModFI(S=pⁿS)^r if n2N;

Mod(S)^r if nˆ 1:

(

LetM be in Modn, for somen2N[ f1g, we denote respectively byW_r^M and r_Mthe Frobenius and the connection ofM.

Let [X] be a class in Ext¹_Modn(1_n;M), represented by a couple of extensions

These extensions clearly split as extensions of Sn-modules (indeed if n2N, pⁿXˆ0 by hypothesis); therefore we can write

Xˆ(MSn;Fil^rMSn;W_r^X;rX):

Let us describe more precisely the FrobeniusW_r^X:Fil^rM_nS_n !MS_nand the regular connectionrX:MSn!(MSn)SnSndu.

We haveW_r^X(0;1)ˆ(a;1), for a uniquea2M. For all (m;s)2Fil^rMSn, we get W^X_r(m;s)ˆ(W^M_r (m);0)‡W(s)W^X_r(0;1)ˆ(W^M_r (m);0)‡W(s)(a;1)

ˆ(W^M_r (m)‡W(s)a;W(s)):

(3:3:4:1)

We haverX(0;1)ˆ(g;0)du, for a unique g2M. For alls2Sn, we get rX(0;s)ˆsrX(0;1)‡(0;1)dsˆs(g;0)du‡(0;_du^ds)du

ˆ(sg;_du^ds)du; thus, for all (m;s)2MSn, we have

r_X(m;s)ˆ(r_M(_du^d)(m)‡sg;_du^ds)du:

The elements a;g satisfy some compatibility conditions corresponding to those satisfied byW_r^X andr_X (cf. 2.2.1, 2.2.3 and 2.2.4): let us write them explicitly.

Condition 2.2.1-(3): for all a in Fil^rS and x in X we have W^X_r(ax)ˆ c₁^rW_r(a)W^X_r(E(u)^rx). Forxˆ(m;s), by using (3.3.4.1), we get

c₁^rW_r(a)W_r^X(E(u)^rx)ˆ

c₁^rW_r(a)(W^M_r (E(u)^rm)‡W(E(u)^rs)a;W(E(u)^rs))ˆ

(c₁^rW_r(a)W_r^M(E(u)^rm)‡c₁^rW_r(a)W(E(u)^rs)a;c₁^rW_r(a)W(E(u)^rs)):

By usingW(E(u)^r)ˆW(E(u))^rˆ(pW₁(E(u)))^rˆp^rc^r₁, this is equal to (W_r^M(am)‡c₁^rW_r(a)p^rc^r₁W(s)a;W(as))ˆ (W_r^M(am)‡W_r(a)p^rW(s)a;W(as))ˆ (W_r^M(am)‡W(as)a;W(as))ˆW_r^X(ax);

which gives no-condition on (a;g).