Rigid Cohomology and de Rham-Witt Complexes

Rigid Cohomology and de Rham-Witt Complexes

PIERREBERTHELOT

Dedicated to Francesco Baldassarri, on his sixtieth birthday

ABSTRACT- Letkbe a perfect field of characteristicp>0,WnˆWn(k). For sepa- ratedk-schemes of finite type, we explain how rigid cohomology with compact supports can be computed as the cohomology of certain de Rham-Witt com- plexes with coefficients. This result generalizes the classical comparison theo- rem of Bloch-Illusie for proper and smooth schemes. In the proof, the key step is an extension of the Bloch-Illusie theorem to the case of cohomologies relative to Wnwith coefficients in a crystal that is only assumed to be flat overWn.

1. Introduction

Let k be a perfect field of characteristic p>0, W_n ˆW_n(k) (for all n1),WˆW(k),KˆFrac(W). IfXis aproper and smooth scheme over k, a nd ifWV_X ˆlim

n

W_nV_X denotes the de Rham-Witt complex ofX, the classical comparison theorem between crystalline and de Rham-Witt co- homologies ([8, III, Th. 2.1], [14, II, Th. 1.4]) provides canonical iso- morphisms

H_crys(X=W_n) ! H(X;W_nV_X);

(1:1:1)

H_crys(X=W) ! H(X;WV_X);

(1:1:2)

H_crys(X=W)_K ! H(X;WV_X;K);

(1:1:3)

where the subscript _K denotes tensorization with K. In particular, the

(*) Indirizzo dell'A.: IRMAR, Universite de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France.

E-mail: [email protected]

latter allows to study the slope decomposition of crystalline cohomology under the Frobenius action, thanks to the degeneracy of the spectral sequence defined by the filtration of WV_X by the subcomplexesWVⁱ_X [14, II, Cor. 3.5].

In this article, our main goal is to generalize the isomorphism (1.1.3) to the case of a separatedk-scheme of finite type. Then H_crys(X=W)_K is no longer in general a good cohomology theory, but we may use instead rigid cohomology with compact supports, which coincides with H_crys(X=W)_K whenXis proper and smooth, while retaining all the standard properties of a topological cohomology theory. The key case is the case of a properk- scheme, as the non proper case can be reduced to the proper one using a mapping cone description. We will therefore assume for the rest of the introduction thatXis proper.

WhenXis singular, the classical theory of the de Rham-Witt complex can no longer be directly applied toX. Instead, we will consider a closed immersion ofXinto asmoothk-schemeYof finite type, and we will use on Y de Rham-Witt complexes with coefficients in crystals, as introduced by EÂtesse in [11]. More precisely, we will construct on Y a WOY;K-algebra A^W_X;Y, supported in Xand closely related to the algebra of analytic func- tions on the tube of X in Y when Y admits a lifting as a smooth p-adic formal schemeYoverW. This algebra is endowed with a de Rham-Witt connection, which extends so as to define a complexA^W_X;YbWOYWV_Y (the tensor product being completed for the canonical topology of the de Rham- Witt complex). Our main result, which is part of Theorem 5.2, is then the existence of afunctorial isomorphism

H_rig(X=K) !H(Y;A^W_X;YbWOYWV_Y);

(1:1:4)

which coincides with (1.1.3) whenXis smooth andY ˆX.

A key ingredient in the proof of Theorem 5.2 is the use of Theorem 3.6 and of its Corollary 3.8, which provide generalizations of the comparison isomorphisms (1.1.1) and (1.1.2) to the case of cohomologies of a smoothk- scheme Y with coefficients in acrystal in O_Y=Wn-modules (resp. O_Y=W- modules) E. Such results have been previously obtained by EÂtesse [11, TheÂoreÁme 2.1] and Langer-Zink [18, Theorem 3.8] when E is flat over O_Y=Wn (resp. O_Y=W). However, this assumption is not verified in our sit- uation, since we work on the smooth schemeY with crystals supported in aclosed subschemeXY. We give here aproof of the comparison the- orem with coefficients inEthat only requiresE to be flat overWn, in the sense of Definition 2.3 (resp. quasi-coherent and flat relative to W). It

provides under these assumptions functorial isomorphisms H_crys(Y=Wn;E) ! H(Y;E^W_n WnOYWnV_Y);

(1:1:5)

H_crys(Y=W;E) ! H(Y;bE^Wb_WOYWV_Y);

(1:1:6)

whereE^W_n is the evaluation of the crystalEon the PD-thickening (Y;WnY) (resp.bE^Wˆlim

En ^Wn ). As a first application, we get that, when the immersion X,!Y is regular (in particular, whenXis smooth), there exists functorial isomorphisms

H_crys(X=W_n) !H(Y;P^W_X;Y;nWnOYW_nV_Y);

(1:1:7)

H_crys(X=W) !H(Y;Pb^W_X;Yb_WOYWV_Y);

(1:1:8)

whereP^W_X;Y;n is the divided power envelope of Ker(W_nO_Y !!W_nO_X) with compatibility with the canonical divided powers of VW_{n 1}O_Y, a nd Pb^W_X;Y ˆlim

Pn ^WX;Y;n.

Note that, even if X is smooth, the possibility of computing the crystalline cohomology ofXusing the de Rham-Witt complex of a smooth embedding was not previously known. Our results imply in particular that, as objects of the derived category D^b(X;K), the complexes P^W_X;Y;nWnOY W_nV_Y, Pb^W_X;Yb_WOYWV_Y, a nd A^W_X;Yb_WOYWV_Y do not de- pend, up to canonical isomorphism, on the embeddingX,!Y. It would be interesting to have a direct proof of this fact in the form of appropriate Poincare Lemmas.

We now briefly describe the content of each section.

In section 2, we prove some Tor-independence properties of the sheaves of Witt differentialsW_nV_Y^j on asmoothk-schemeY, and, more generally, of the graded modules associated to their canonical filtration or to theirp- adic filtration. Specifically, we show in Theorems 2.6 and 2.10 the vanishing of the higher Tor's involving such sheaves and the evaluation on the PD- thickening (Y;W_nY) of acrystal onY=W_n that is flat overW_n - aresult that may seem surprising at first sight, given the intricate structure of the sheaves of Witt differentials. The flatness assumption allows to reduce to similar statements for the reduction modulopof the crystal. The key point is then the existence of a filtration of the crystal such that the corre- sponding graded pieces havep-curvature 0. Thanks to Cartier's descent, this allows to write them as Frobenius pullbacks, and it is then possible to conclude using the local freeness results proved in [14] when the graded

modules associated to WnV_Y^j are viewed asOY-modules through an ap- propriate Frobenius action.

We use these results in section 3 to prove the comparison theorem between crystalline and de Rham-Witt cohomologies with coefficients in a crystal that is flat overW_n (Theorem 3.6). As in the constant coefficient case, the proof proceeds by reduction to the associated graded complexes for the p-adic and for the canonical filtration. The Tor-independence re- sults of the previous sections allow to reduce to the fact that, in the con- stant coefficient case, one gets quasi-isomorphisms between complexes that can be viewed as strictly perfect complexes ofO_Y-modules whenO_Y acts through an appropriate power of Frobenius. The section ends with the above mentionned application to crystalline cohomology for regularly embedded subschemes.

We begin section 4 by recalling the definition of rigid cohomology for a properk-schemeX. Given aclosed immersion ofXin asmoothp-adic formal schemePoverW, letA_X;Pbe the direct image by specialization of the sheaf of analytic functions on the tube of X inP. We explain howAX;P can be identified with the inverse limit of the completed divided power envelopes (tensorized withK) of the ideals of the infinitesimal neighbourhoods ofXin P. We also give a variant of this result in which these envelopes are replaced by their quotients by the ideal ofp-torsion sections. IfXis proper overkand is embedded as aclosed subscheme in asmoothk-schemeY, we derive from this construction an isomorphism between the rigid cohomology ofXand the derived inverse limit of the crystalline cohomologies (tensorized withK) of the infinitesimal neighbourhoods ofXinY(Theorem 4.7).

In section 5, we keep these last hypotheses, and we use the crystalline nature of the previous constructions to define two inverse systems of WOY;K-algebras canonically associated toX,!Y, with isomorphic inverse limits. By definition, this common inverse limit is theWOY;K-algebraA^W_X;Y entering in the isomorphism (1.1.4). To define (1.1.4) and to prove Theorem 5.2, we combine the isomorphism of the previous section, which identifies the rigid cohomology ofXwith the limit of the crystalline cohomologies of its infinitesimal neighbourhoods inY, with an isomorphism derived from the comparison theorem 3.6, which gives an identification of this limit with the cohomology of the complexA^W_X;Y b_WOYWV_Y. The first identification uses the description ofA^W_X;Y as a limit of PD-envelopes, while, because of theW_n-flatness assumption in Theorem 3.6, the second one uses the sec- ond description given in section 4, based onp-torsion free quotients of PD- envelopes. We end the section by extending (1.1.4) to rigid cohomology with compact supports for open subschemes ofX.

To conclude this article, we explain the relation between the iso- morphism (1.1.4) and the isomorphism [5, (1.3)], which identifies the slope 51 part of rigid cohomology with Witt vector cohomology. As a con- sequence, we obtain that the part of slope1 ofH_rig(X=K) can be iden- tified with the cohomology of asubcomplex ofA^W_X;Yb_WOYWV_Y. We hope that Theorem 5.2 can also be used to provide a description of parts of higher slopes inH_rig(X=K), but this is still an open question at this point.

General conventions

1) In the whole article, we denote byka perfect field of characteristicp, byW_nˆW_n(k) (forn1) andW ˆW(k) the usual rings of Witt vectors with coefficients ink, and byK the fraction field of W. Formal schemes overW are always supposed to bep-adic formal schemes.

2) IfY is ak-scheme, we callcrystal on Y=Wn (resp.Y=W) acrystal in O_Y=Wn-modules (resp. O_Y=W-modules) on the usual crystalline site Crys(Y=W_n) (resp. Crys(Y=W)) [7, Def. 6.1].

3) From section 3 till the end of the article, we assume for simplicity that all schemes and formal schemes under consideration are quasi-com- pact and separated.

2. Tor-independence properties of the de Rham-Witt complex

Let Y be asmooth k-scheme. We prove in this section Tor-in- dependence results between the sheaves of de Rham-Witt differential forms WnV_Y^j and the evaluation on the Witt thickenings (Y;WnY) of a crystal onY=W_n that is flat overW_n (see Definition 2.3).

2.1. Let E be acrystal onY=W_n for some n1 (resp. acrystal on Y=W). Recall that:

a) For each divided powers thickeningU,!T, whereUYis an open subset andTaWn-scheme (resp. aWn-scheme for somen),Edefines an O_T-moduleE_T, which we call theevaluation ofEon T.

b) For each morphism of thickenings (U⁰;T⁰)!(U;T), defined by a Wn-PD-morphismv:T⁰!T, the transition morphism

vET! ET⁰; (2:1:1)

defined by the structure of E as a sheaf ofO_Y=Wn-modules (resp. O_Y=W- modules) on the crystalline site, is an isomorphism.

We will consider more specifically two families of thickenings. The first one is the family of thickenings (U;Un), whereUY is an open subset andUn is asmooth lifting ofU over Wn (which always exist when U is affine). WhenUˆYand a smooth liftingY_nhas been given, we will simply denote byE_n theO_Yn-moduleE_Yn. From the crystal structure ofE,E_n in- herits an integrable connection r_n:E_n! E_nV¹_Yn=Wn, which is quasi- nilpotent[7, Def. 4.10]. This construction is functorial inE, and defines an equivalence between the category of crystals onY=Wnand the category of O_Yn-modules endowed with an integrable and quasi-nilpotent connection relative toW_n.

IfE is acrystal onY=W andY is lifted as a smooth formal schemeY overW, with reductionYnoverWn,Ecan be viewed as a compatible family of crystals onY=Wn for alln. Thus,Edefines for allnanO_Yn-moduleEn

endowed with an integrable connectionr_n such thatE_n ' E_n‡1=pⁿE_n‡1as amodule with connection. Then theO_Y-modulebE ˆlim

En ⁿ has a connec- tion r ˆlim

rn ⁿ, and one gets in this way an equivalence of categories between the category of crystals onY=W and the category ofp-adically separated and completeO_Y-modules endowed with an integrable and to- pologically quasi-nilpotent connection.

The second family of thickenings we are going to use is provided by the immersionsY,!WnY:ˆ(jYj;WnOY), which are divided powers thicken- ings thanks to the canonical divided powers of the idealVWn 1O_Y (defined by (Vx)^[i]ˆ(p^{i 1}=i!)V(xⁱ) for all i1). Thus, evaluating E on (Y;W_nY) defines a W_nO_Y-module, which will be denoted by E^W_n. For n⁰n, the closed immersionW_n⁰Y,!W_nYdefines amorphism of PD-thickenings of Y, hence the crystal structure ofE provides ahomomorphismE^W_n ! E^W_n0, the linear factorization of which is an isomorphism

W_n⁰O_YWnOYE^W_n E! ^W_n⁰: (2:1:2)

IfEis acrystal onY=W, these homomorphisms turn the family ofW_nO_Y- modules (E^W_n )_n1into an inverse system ofW_nO_Y-modules.

2.2. Let s:W !W be the Frobenius automorphism ofW. We now assume that we are given a smooth formal schemeYliftingYoverW, with reductionY_noverW_n. We assume in addition thatYis endowed with as- semi-linear morphism F:Y!Y lifting the absolute Frobenius endo- morphism ofY. AsOYisp-torsion free, the homomorphismF:OY! OY

defines asection sF :OY !WOY of the reduction homomorphism

WOY! OY, characterized by the fact that wi(sF(x))ˆFⁱ(x) for any x2 OY and any ghost componentwi [14, 0, 1.3]. Composing sF with the reduction homomorphisms and factorizing, we get for all n1 ahomo- morphism

t_F:O_Yn!W_nO_Y: (2:2:1)

These homomorphisms are compatible with the reduction maps when n varies, and functorial in an obvious way with respect to the couple (Y;F).

The morphism t_F is a PD-morphism, because the canonical divided powers ofVWn 1OYextend the natural divided powers of (p). Therefore, it defines amorphism of thickenings ofY, and, for any crystalEonY=Wn, we get a canonical isomorphism

W_nO_YOYn E_n E! ^W_n; (2:2:2)

where the scalar extension is taken by means of the homomorphismt_F. We will need the following condition onE:

DEFINITION2.3. LetY be asmoothk-scheme.

(i) IfEis acrystal onY=W_n, we say thatE isflat over W_n if, for any smooth liftingU_n overW_nof an open subsetUY, the evaluationE_Un of E onUn is flat overWn.

(ii) IfEis acrystal onY=W, we say thatEisflat over Wif, for alln, the induced crystal onY=Wn is flat overWn.

For any open subsetUY, two smooth liftingsUn;Un0ofU overWn

are locally isomorphic, and such local isomorphisms extend canonically to the evaluations onU_nandU_n⁰of acrystal. Therefore,Eis flat overW_nif and only if there exists an open coveringU_aofYand, for alla, asmooth lifting Ua;nofUaoverWnsuch that the evaluationEa;nofEonUa;nis flat overWn. Similarly, whenE is acrystal on Y=W,E is flat over W if and only if there exists an open coveringU_a ofYand, for alla, aliftingU_aofU_aa s a smooth formal scheme overWsuch that the corresponding Zariski sheaf bE_aˆlim

En ^a;n onU_a isp-torsion free.

IfEis flat overW_n (resp.W), then its restriction to Crys(Y=W_i) is flat overWifor anyin(resp. anyi). IfEis flat as anO_Y=Wn-module (resp. as anO_Y=W-module), thenEis flat overWn (resp. overW).

We now begin our study of the Tor-independence properties between the sheavesW_nV_Y^j and the evaluation of a crystal that is flat overW_n. We recall first from [14, I, 3.1] that, for i2Z, thei-th step of the canonical

filtration ofWnV_Y^j (resp.WV_Y^j) is the sub-WnOY-module (resp. sub-WOY- module) defined by

FilⁱW_nV_Y^j ˆ

WnV_Y^j if i0;

Ker(W_nV_Y^j !W_iV_Y^j) if 1in;

0 if i>n

8>

<

>:

resp:FilⁱWV_Y^j ˆ WV_Y^j if i0;

Ker(WV_Y^j !W_iV_Y^j) ifi1

! : (

We denote

grⁱWnV_Y^j ˆ FilⁱWnV_Y^j=Fil^i‡1WnV_Y^j; grⁱWV_Y^j ˆ FilⁱWV_Y^j=Fil^i‡1WV_Y^j: By construction, the canonical homomorphism

grⁱWV_Y^j !grⁱWnV_Y^j

is an isomorphism for i5n. Note that, under the assumptions of 2.2, the homomorphism tF defined in (2.2.1) endows the sheaves FilⁱWrV_Y^j and grⁱWrV_Y^j with anOYn-module structure for 1rnand alli,j.

The following lemma allows to relate the reduction modulopoft_Fwith the usual Frobenius endomorphism.

LEMMA2.4. Let A be a commutative ring without p-torsion, F:A!A a ring homomorphism lifting the absolute Frobenius endomrphism of A=pA, and s_F :A!W(A) the ring homomorphism such that w_i(s_F(x))ˆFⁱ(x)for all x2A and all i0 [14, 0, 1.3]. Let n1 be an integer, A_nˆA=pⁿA, t_F:A_n !W_n(A₁)the factorization of s_Fas in(2.2.1), and t_F :A₁!W_n(A₁)=pW_n(A₁)the reduction modulo p of t_F. Denote by

F:A₁'W_n(A₁)=VW_{n 1}(A₁)!W_n(A₁)=pW_n(A₁) (2:4:1)

the homomorphism induced by the action of F on W_n(A₁)=pW_n(A₁). Then F is equal to the composition

A1 !^F A1 ^t!^F Wn(A1)=pWn(A1);

(2:4:2)

where the first homomorphism F is the absolute Frobenius endomorphism of A1.

PROOF. It suffices to prove the equality of the morphisms obtained by composingFandtFFwith the surjectionA!!A1. By construction, there is a commutative diagram

in which the unlabelled arrows are the canonical surjections. Note that the commutativity of the left upper square (in which the upperFis the given homomorphism and the lower one the canonical Frobenius en- domorphism of W(A)) is aconsequence of the definition of sF [14, 0, 1.3]. The lemma follows from the equality of the exterior paths of the

diagram. p

LEMMA 2.5. Let Y be a smooth k-scheme, such that there exists a smooth formal scheme Y over W lifting Y and a semi-linear en- domorphism F of Ylifting the absolute Frobenius endomorphism of Y.

For some n1, letEbe a crystal on Y=Wn, flat over Wn. Let Yn be the reduction of Yover W_n andE_n the evaluation ofEon Y_n. Then, for the O_Yn-module structure defined on W_nO_Y-modules by t_F,

Tor^Oq^Yn(En;FilⁱWrV_Y^j)ˆ Tor^Oq^Yn(En;grⁱWrV_Y^j)ˆ0 (2:5:1)

for q1,1rn, and all i, j.

In particular, when WnO_Y is viewed as anO_Yn-algebra throught_F,En

andW_nO_Y are Tor-independent overO_Yn.

PROOF. The canonical filtration ofWrVⁱ_Yis discrete and codiscrete, so it suffices to prove the vanishing of Tor^O_q^Yn(E_n;grⁱW_rV_Y^j). Moreover, we may assume thatrˆn, since grⁱW_rV_Y^j ˆ0 fori50 orir, and the natural map grⁱW_nV_Y^j !grⁱW_rV_Y^j is an isomorphism for 0i5rn.

As grⁱWnV_Y^j is annihilated byp, itsOYn-module structure is also given by the compositionO_Yn! O_Y!^tF WnO_Y=pWnO_Y, wheret_Fis the reduction

oftFmodp. Therefore, we obtain isomorphisms E_n ^L

O_Yn grⁱW_nV_Y^j ' (E_n ^L

O_Yn O_Y)^L

OY

grⁱW_nV_Y^j: BecauseE_nandO_Ynare flat relatively toW_n, we have

En ^L

OYn OY ' En ^L

OYn (OYn ^L

Wnk) ' En ^L

Wnk E! 1; so it suffices to prove that

Tor^O_q^Y(E₁;grⁱW_nV_Y^j)ˆ0 forq1 and alli,j.

SinceE₁ is the evaluation onY of acrystal, its connectionr₁ is quasi- nilpotent [7, Def. 4.10]. LetTY ˆV^1__Y be the sheaf ofk-derivations onY andc:TY ! EndOY(E1) thep-curvature ofr1[15, 5.0]. We define an in- creasing filtration ofE₁ by horizontal submodulesE^m₁,m0, by setting, for any affine open subsetUY,

G(U;E^m₁)ˆ (2:5:2)

fx2G(U;E)j 8D1;. . .;Dm2G(U;TY);c(D1) c(Dm)(x)ˆ0g [15, 5.5]. Asr1is quasi-nilpotent, this filtration is exhaustive, and it suffices to prove that, forq1,m0, and alli,j,

Tor^O_q^Y(E^m₁;grⁱW_nV_Y^j)ˆ0:

By construction,E⁰₁ˆ0 and, form1, each quotientE^m₁=E^m₁ ¹hasp-cur- vature 0. Therefore, for all m, there is an O_Y-module F^m such that E^m₁=E^m₁ ¹'FF^m. SinceF is flat, it is enough to prove that, for allq1, m0, and alli,j,

Tor^O_q^Y(F^m;grⁱWnV_Y^j)ˆ0;

grⁱW_nV_Y^j being now viewed as anO_Y-module thanks to the composition OY !^F OY ^t!^F WnOY=pWnOY:

By Lemma 2.4, this homomorphism is the factorization F:OY !WnOY=pWnOY

of the Frobenius action onW_nO_Y=pW_nO_Y. As grⁱW_nV_Y^j is alocally freeO_Y- module for this structure [14, I, 3.9], the lemmafollows. p

THEOREM2.6. Let Y be a smooth k-scheme. For some n1, letEbe a crystal on Y=Wn, flat over Wn, and letE^W_n be the evaluation ofEon the PD- thickening(Y;WnY). Then

Tor_q^WnOY(E^W_n ;FilⁱWrV_Y^j)ˆ Tor^W_q^nOY(E^W_n;grⁱWrV_Y^j)ˆ0 (2:6:1)

for q1,1rn, and all i, j.

PROOF. The statement is local onY, hence we may assume that Yis affine and has a smooth formal liftingYoverWendowed with aliftingFof the absolute Frobenius endomorphism ofY. LetYnbe the reduction ofYon W_nandE_nthe evaluation ofEonY_n. By (2.2.2),t_Fprovides an isomorphism

E_nOYn W_nO_Y E! ^W_n:

Applying (2.5.1) for iˆjˆ0 a nd rˆn, we obta in in D (W_nO_Y) a n iso- morphism

En ^L

O_Yn WnOY E! ^W_n: (2:6:2)

On the other hand, the transitivity isomorphism for the derived extension of scalars yields

E_n ^L

O_Yn FilⁱW_rV_Y^j ! (E_n ^L

O_Yn W_nO_Y) ^L

WnOY

FilⁱW_rV_Y^j E! ^W_n ^L

WnOY

FilⁱWrV_Y^j:

By (2.5.1), the left hand side is acyclic in degrees6ˆ0. The first vanishing of (2.6.1) follows, and the second one is obtained by the same argument. p

We define thecanonical filtrationofE^W_{n WnOY} W_nV_Y^j by Filⁱ(E^W_{n WnOY}W_nV_Y^j)ˆ

E^W_{n WnOY}W_nV_Y^j if i0;

Ker(E^W_{n WnOY}WnV_Y^j ! E^W_{i WiOY}W_iV_Y^j) if 1in;

0 if i>n:

8>

<

>: (2:6:3)

Using (2.1.2), the theorem implies:

COROLLARY2.7. Under the assumptions of the theorem, the natural map

E^W_n WnOY FilⁱWnV_Y^j !Filⁱ(E^W_n WnOY WnV_Y^j) (2:7:1)

is an isomorphism for all i.

2.8. Let Y be asmooth k-scheme, and assume thatn2. If E is a crystal onY=Wn, multiplication byponE^W_n WnV_Y^j vanishes on the image ofE^W_n Fil^{n 1}WnV_Y^j. Factorizing and taking the isomorphism (2.1.2) into account, one gets a canonical homomorphism

p:E^W_{n 1Wn1OY}W_{n 1}V_Y^j ! E^W_{n WnOY}W_nV_Y^j: (2:8:1)

We recall that, whenE ˆ O_Y=Wn, the morphismpis an injection p:Wn 1V_Y^j ,!WnV_Y^j

for allj[14, Prop. 3.4]. It follows that, fori1, there is an exact sequence (2:8:2) 0!Wn 1V_Y^j=p^{i 1}Wn 1V_Y^j !^p WnV_Y^j=pⁱWnV_Y^j !WnV_Y^j=pWnV_Y^j !0:

We will show later that, when E is flat over Wn, the homomorphism (2.8.1) is still injective (see Corollary 2.11). This is a consequence of another Tor-independence property, which we prove next.

LEMMA2.9. Let Y be a smooth k-scheme. For n1and all j, the sheaf WnV_Y^j=pWnV_Y^j is a locally freeOY-module of finite rank for the structure defined by the homomorphism F:OY !WnOY=pWnOY (2.4.1).

PROOF. We proceed by induction onn, the claim being clear fornˆ1.

We set W_nV_Y^j ˆ0 for nˆ0. For n1, the commutative diagram with exact rows

yields an exact sequence

in which the morphisms areOY-linear for the module structure defined by F. AspWn 1V_Y^j ˆpWnV_Y^j for alln1, the induction hypothesis reduces to

proving that grⁿWn‡1V_Y^j=pgr^{n 1}WnV_Y^j is a locally free finitely generated OY-module for the structure defined byF. By [14, I, (3.10.4)], there is a commutative diagram

in which the exterior columns are defined by the inverse Cartier operator C ¹ and we setB0V_Y^j ˆ0,Z0V_Y^j ˆV_Y^j. All maps in this diagram become OY-linear maps if we endow the terms of the middle column with the structure defined by F, the exterior terms of the upper row with the structure defined byFⁿ, and the exterior terms of the lower row with the structure defined by F^n‡1 (see [14, I, Cor. 3.9]). Therefore, the corre- sponding cokernel sequence, which can be written as

0!BV_Y^j‡1!grⁿWn‡1V_Y^j=pgr^{n 1}WnV_Y^j !BV_Y^j !0;

is an O_Y-linear exact sequence when BV_Y^j andBV_Y^j‡1 are viewed as O_Y- modules thanks toF^n‡1. By [14, 0, Prop. 2.2.8], they are then locally free of

finite rank overOY, which ends the proof. p

THEOREM2.10. Let Y be a smooth k-scheme, and letEbe a crystal on Y=W_n, flat over W_n. LetE^W_n be the evaluation ofEon(Y;W_nY). Then

Tor^W_q^nOY(E^W_n ;W_rV_Y^j=pⁱW_rV_Y^j)ˆ Tor^W_q^nOY(E^W_n;pⁱW_rV_Y^j)ˆ0 (2:10:1)

for q1,1rn, i0, and all j.

PROOF. Thanks to Theorem 2.6, it suffices to prove the vanishing of the left hand side.

The statement is trivial foriˆ0. Let us prove it first foriˆ1. We may assume that Y is affine, and we can choose a smooth formal scheme Y lifting Y overW, together with aliftingF of the absolute Frobenius en- domorphism ofY. Then the isomorphism (2.6.2) and the transitivity of the derived extension of scalars show as in the proof of Theorem 2.6 that it is equivalent to prove the relation

Tor^O_q^Yn(E_n;W_rV_Y^j=pW_rV_Y^j)ˆ0;

(2:10:2)

where WrV_Y^j=pWrV_Y^j is viewed as an OYn-module thanks totF. As this

module is annihilated byp, the flatness ofEnrelative toWn implies as in the proof of Lemma 2.5 that it is equivalent to prove that

Tor^O_q^Y(E1;WrV_Y^j=pWrV_Y^j)ˆ0;

(2:10:3)

whereE1ˆ En=pEnandWrV_Y^j=pWrV_Y^j is viewed as anOY-module thanks to the reductiontF :OY!WrOY=pWrOYoftF. Repeating again the proof of Lemma2.5, we can use thep-curvature filtration ofE₁to reduce (2.10.3) to the relation

Tor^O_q^Y(F;W_rV_Y^j=pW_rV_Y^j)ˆ0;

(2:10:4)

whereF is anO_Y-module andW_rV_Y^j=pW_rV_Y^j is viewed as anO_Y-module through the compositiont_FF. As the latter is equal toFby Lemma2.4, relation (2.10.4) is then aconsequence of Lemma2.9.

This proves that, for iˆ1,Tor^W_q^nOY(E^W_n;WrV_Y^j=pⁱWrV_Y^j) vanishes for q1, 1rn, and allj. Moreover, by Theorem 2.6, it also vanishes when rˆ1 forq1,i0, and allj. Using the exact sequences (2.8.2) and the previous result foriˆ1, one can then argue by induction onrto prove that the same vanishing holds for 1rnand alli0. p COROLLARY2.11. Let Y be a smooth k-scheme, and letEbe a crystal on Y=W_nfor some n2. IfEis flat over W_n, the homomorphism(2.8.1)is an injection

p:E^W_{n 1Wn1OY}W_{n 1}V_Y^j ,! E^W_{n WnOY}W_nV_Y^j:

PROOF. Since p:W_{n 1}V_Y^j !W_nV_Y^j is injective, it follows from the theorem that

Idp:E^W_{n WnOY} W_{n 1}V_Y^j ! E^W_{n WnOY} W_nV_Y^j is injective too. Using (2.1.2), we get an isomorphism

E^W_{n WnOY}W_{n 1}V_Y^j E! ^W_{n 1Wn1OY}W_{n 1}V_Y^j;

which identifies Idp to the homomorphism p defined by (2.8.1) and

completes the proof. p

2.12. We now assume thatEis acrystal onY=W, flat overW, and we consider the inverse system ofWnOY-modulesE^W_n defined in 2.1 by taking the evaluation of E at all thickenings (Y;WnY) when n varies. We also assume thatEis aquasi-coherent crystal, i.e., that, for any PD-thickening

(U;T) in Crys(Y=W), the evaluationETofEonTis aquasi-coherentOT- module. It is equivalent to ask that, for anyn1, this condition be verified for thickenings of the form (Ua;Ua;n), whereUavaries in an open covering ofYandU_a;nis asmooth lifting ofU_aonW_n. Thanks to [14, I, Prop. 1.13.1], the W_nO_Y-module E^W_{n WnOY}W_nV_Y^j is then quasi-coherent for all n1 and allj.

For allj, we define

bE^WbWOYWV_Y^j :ˆlim

n

(E^W_n WnOYWnV_Y^j):

By construction, we have projections

bE^WbWOYWV_Y^j ! E^W_i WiOY WiV_Y^j

for each i1, and, since the inverse system has surjective transition maps and quasi-coherent terms, the Mittag-Leffler criterion implies that these projections are surjective. We define the canonical filtration of bE^WbWOYWV_Y^j by

Filⁱ(bE^Wb_WOYWV_Y^j)ˆ bE bWOYWV_Y^j if i0;

Ker(bE^Wb_WOYWV_Y^j ! E^W_{i WiOY}W_iV_Y^j) if i1:

(

Note that, fori1,

Filⁱ(bE^WbWOYWV_Y^j) !lim

n

Ker(E^W_n WnOY WnV_Y^j ! E^W_i WiOYW_iV_Y^j);

hence the Mittage-Leffler criterion implies that, for any affine open subset UY,

H^q(U;Filⁱ(bE^Wb_WOYWV_Y^j))ˆ0 (2:12:1)

forq1 and alli,j.

For each affine open subset UY, the canonical filtration endows G(U;bE^WbWOYWV_Y^j) with atopology, which will be called the canonical topology. From (2.12.1), we deduce the isomorphism

(2:12:2) G(U;bE^WbWO_YWV_Y^j)=FilⁱG(U;bE^WbWO_YWV_Y^j) !G(U;E^W_i W_iO_YWiV_Y^j) fori1. It follows thatG(U;bE^WbWO_YWV_Y^j) is separated and complete for the canonical topology.

PROPOSITION2.13. LetY be a smoothk-scheme, and letEbe a crystal onY =W. Assume thatEis flat overW and quasi-coherent.

(i) For all j, multiplication by p is injective on bE^WbWOYWV_Y^j. (ii) If UY is an affine open subset, then, for all i0and all j, pⁱG(U;bE^Wb_WOYWV_Y^j)is closed inG(U;bE^Wb_WOYWV_Y^j)for the canonical topology, and G(U;bE^WbWOYWV_Y^j) is separated and complete for the p- adic topology.

PROOF. As multiplication byponbE^Wb_WOYWV_Y^j is the inverse limit of the mapspdefined in (2.8.1), assertion (i) results from Corollary 2.11.

LetUbe an affine open subset ofY, and leti2N. Using assertion (i) and Corollary 2.11, we can write a commutative diagram with exact columns

in which the surjectivity in the right hand side column results from the surjectivity of the transition maps in the inverse system (G(U;E^W_{n iWn iOY} W_{n i}V_Y^j))_n>i. It follows that the bottom horizontal arrow is an isomorphism. The quasi-coherence assumption onEimplies that

pⁱG(U;E^W_{n iWn iOY}W_{n 1}V_Y^j))ˆpⁱG(U;E^W_{n WnOY}W_nV_Y^j) and that the map (2.12.2)

G(U;Eb^Wb_WOYWV_Y^j)!G(U;E^W_{n WnOY}W_nV_Y^j)

is surjective. Therefore the quotient in the right hand side column can be rewritten as

lim

n>iƒ

G(U;bE^Wb_WOYWV_Y^j)

pⁱG(U;bE^Wb_WOYWV_Y^j)‡FilⁿG(U;bE^Wb_WOYWV_Y^j):

This proves the first part of assertion (ii). As G(U;bE^WbWOYWV_Y^j) is separated and complete for the canonical topology, which is coarser than the p-adic topology, the second part of assertion (ii) follows by

[9, Ch. III, § 3, në 5, Cor. 2 to Prop. 10]. p

3. De Rham-Witt cohomology with coefficients

In this section, we extend the classical comparison theorem between the crystalline and de Rham-Witt cohomologies of a smoothk-schemeYto the case of cohomologies with coefficients in a crystal onY=Wnthat is flat overWn.

As indicated in our general conventions, we now assume until the end of the article that all schemes are quasi-compact and separated. We first recall the construction of the comparison morphism in the case of constant coef- ficients [14, II, 1], starting at the level of complexes of Zariski sheaves onY.

3.1. LetPbe asmooth formal scheme overW, with reductionPnover Wnand special fibrePˆP1. LetXPbe aclosed subscheme. We denote by J_n (resp. J) the ideal of X in P_n (resp. P) and by P_n (resp. P) theb divided power envelope ofJ_n (resp. thep-adic completion of the divided power envelope of J). In these constructions, we impose that all divided powers be compatible with the natural divided powers ofp, which implies thatPn ' Pn‡1=pⁿPn‡1for allnandP ˆb lim

Pn ⁿ. Divided power envelopes have a natural connection, which allows to define the de Rham complexes P_nV_Pn and P b V_P; these complexes are supported in X. Let u_X=Wn (resp.u_X=W) be the projection from the crystalline topos ofX relative to Wn (resp.W) to the Zariski topos ofX. In its local form, the comparison theorem between crystalline and de Rham cohomologies [7, (7.1.2)] pro- vides functorial isomorphisms

Ru_X=WnO_X=Wn' P_nV_Pn=Wn; (3:1:1)

Ru_X=WO_X=W'P b V_P=W: (3:1:2)

inD^b(X;Wn) a ndD^b(X;W) (note that the tensor product does not need to be completed, sinceV_P=W is a locally free finitely generatedOP-module).

Taking sections on X, one gets in D^b(W_n) a nd D^b(W) the global com- parison isomorphisms

RG_crys(X=W_n;O_X=Wn)'RG(X;P_nV_Pn=Wn);

(3:1:3)

RGcrys(X=W;O_X=W)'RG(X;P b V_P=W):

(3:1:4)

These isomorphisms can be generalized to the case where the datum of the embeddingX,!P is replaced by the data of an affine open covering Uˆ(Ua) of X and of closed embeddings Ua,!Pa into smooth formal schemes [14, 0, 3.2.6]. For each multi-index aˆ(a₀;. . .;a_i), let U_aˆU_a0\ \U_ai, P_aˆP_{a0W W}P_ai, let J_a O_Pa be the ideal defining the diagonal embeddings U_a,!P_a, Pb_a its completed divided power envelope,Pa;n,Ja;n,Pa;nthe reductions modpⁿofPa,Ja,Pa, and let ja:Ua,!X be the inclusion. One can define CÏech double complexes j (P_;nV_P;n) (resp.j (PbV_P)) with general term

Y

aˆ(a0;...;ai)

j_a(P_a;nV_P^j_a;n) (resp: Y

aˆ(a0;...;ai)

j_a(Pb_aV_P^j_a))

in bidegree (j;i). If one uses the subscript ``t'' to denote the total complex associated to a multicomplex, one can then generalize the isomorphisms (3.1.1) and (3.1.2) as

Ru_X=WnO_X=Wn'(j (P;nV_P;n))_t; (3:1:5)

Ru_X=WO_X=W'(j (PbV_P))_t; (3:1:6)

and one gets similar generalizations of (3.1.3) and (3.1.4).

DEFINITION3.2. LetPbe asmooth formal scheme overW, endowed with a s-semilinear morphism F:P!P lifting the absolute Frobenius endomorphism of its special fibreP, and letXPbe aclosed subscheme.

For anyn1, lettF :OPn!WnOPbe the homomorphism (2.2.1) defined byF. Then the composition

O_Pn ^t!^F W_nO_P!W_nO_X (3:2:1)

mapsJn toVWn 1OX WnOX. Using the natural divided powers of the idealVWn 1OX (which are compatible with the divided powers ofp), the universal property of divided power envelopes provides a unique factor- ization of this composition through a homomorphism denoted

h_F :P_n!W_nO_X; (3:2:2)

which commutes with the divided powers.

3.3. Let Y be asmooth k-scheme, embedded through aclosed im- mersionY,!Pinto the special fibre of a smooth formal schemePoverW.

Assume thatPis endowed with aFrobenius liftingFas in 3.2, and keep for Ythe notation introduced forXin 3.1-3.2. Then the homomorphism (3.2.1)

extends as a morphism of complexes

V_Pn!WnV_P!WnV_Y: (3:3:1)

Thanks to the structure of gradedP_n-algebra defined byh_FonW_nV_Y, this morphism defines by extension of scalars fromOPntoPnanhF-semi-linear morphism

PnV_Pn !WnV_Y; (3:3:2)

which is still amorphism of complexes. Whennvaries, these morphisms are compatible, and their inverse limit gives a morphism of complexes

P b V_P!WV_Y: (3:3:3)

The morphisms (3.3.2) and (3.3.3) are functorial with respect to the triple (Y;P;F) in the following sense. Let P⁰ be asecond smooth formalW- scheme, with special fibreP⁰, letY⁰P⁰be asmooth closed subscheme, and let F⁰:P⁰!P⁰ be alifting of the Frobenius endomorphism of P⁰. If u:P⁰!Pis aW-morphism commuting withFandF⁰, and inducing ak- morphismf :Y⁰!Y, then we get commutative diagrams

(3:3:4)

(3:3:5)

where the vertical maps are the functoriality morphisms.

The morphisms (3.3.2) and (3.3.3) are quasi-isomorphisms, and, via (3.1.1) and (3.1.2), they define the comparison isomorphism between crystalline and de Rham-Witt cohomologies [14, II, 1.4]. In the derived categoryD^b(Y;Wn) (resp.D^b(Y;W)) of sheaves ofW_n-modules (resp. W-modules) on Y, this comparison isomorphism does not depend on the choice of (P;F) (by the standard argument comparing two embeddingsY,!(P;F) a ndY,!(P⁰;F⁰) to the diagonal embedding into (PWP⁰;FF⁰) viathe projection maps).