Semistable Sheaves and Comparison Isomorphisms in the Semistable Case

Semistable Sheaves and Comparison Isomorphisms in the Semistable Case

FABRIZIOANDREATTA- ADRIANIOVITA

Dedicato a Francesco Baldassarri, con affetto

Contents

1 - Introduction. . . 133

2 - Fontaine's sheaves on Faltings' site . . . 136

2.1 - Notations. . . 136

2.1.1 - The classical period rings . . . 137

2.1.2 - Assumptions . . . 141

2.1.3 - Continuoussheaves . . . 143

2.2 - Faltings' topos . . . 144

2.2.1 - The Kummer eÂtale site ofX. . . 144

2.2.2 - The finite Kummer eÂtale sitesU^fket_L . . . 147

2.2.3 - Faltings ' s ite . . . 148

2.2.4 - ContinuousFunctors. . . 149

2.2.5 - Geometric points. . . 150

2.2.6 - The localization functors. . . 151

2.2.7 - The computation ofRⁱv^cont . . . 153

2.3 - Fontaine'ssheaves . . . 158

2.3.1 - The sheavesO_XandOb_X . . . 159

2.3.2 - The morphismU . . . 162

2.3.3 - The sheafA^r_log. . . 162

2.3.4 - The sheafA_log. . . 165

2.3.5 - PropertiesofA^r_logandA_log. . . 168

2.3.6 - The sheavesB^r_logandB_log . . . 170

2.3.7 - The sheavesB^r_log;KandB_log;K . . . 171

2.3.8 - The monodromy diagram . . . 172

2.3.9 - The fundamental exact diagram . . . 173

2.3.10 - Cohomology ofBlogandBlog . . . 174

2.4 - Semistable sheaves and their cohomology . . . 175

2.4.1 - The functorD^geo_log. . . 175

2.4.2 - Geometrically semistable sheaves . . . 176

2.4.3 - The functorD^ar_log. . . 176

2.4.4 - Semistable sheaves . . . 178

2.4.5 - The category of filtered Frobenius isocrystals . . . 178

2.4.6 - A geometric variant . . . 184

2.4.7 - Properties of semistable sheaves . . . 185

2.4.8 - Cohomology of semistable sheaves . . . 189

2.4.9 - The comparison isomorphism for semistable sheaves in the proper case . . . 191

3 - Relative Fontaine's theory . . . 200

3.1 - Notations. First properties . . . 200

3.1.1 - First properties ofR_n . . . 203

3.1.2 - The ringRb. . . 206

3.1.3 - The lift of the FrobeniustowerRe1 . . . 209

3.1.4 - The mapU . . . 211

3.1.5 - The ringA^‡_R n . . . 213

3.2 - The ringsB_dR . . . 217

3.2.1 - Explicit des criptions . . . 218

3.2.2 - Connections. . . 222

3.2.3 - Flatness and Galois invariants . . . 223

3.3 - The functorsD_dRandDe_dR. De Rham representations . . . 225

3.4 - The ringsB^cris_log and B^max_log . . . 228

3.4.1 - Explicit descriptions of B^cris_log and B^max_log . . . 229

3.4.2 - Galoisaction, filtrations, Frobenii, connections . . . 233

3.4.3 - Relation with B_dR . . . 234

3.4.4 - Descent from B^max_log . . . 236

3.4.5 - Localizations. . . 244

3.5 - The geometric cohomology of B^cris_log . . . 247

3.5.1 - Almost eÂtale des cent . . . 251

3.5.2 - De-perfectization . . . 255

3.5.3 - The cohomology ofA^geo;cris_log Re and ofA^geo;max_log Re . . . 264

3.5.4 - The cohomology of the filtration of B^cris_log(R). . . 266e 3.5.5 - The cohomology of B_log^cris(R) . . . 267e 3.5.6 - The arithmetic invariants. . . 268

3.6 - The functorsD^cris_log and D^max_log :Semistable representations . . 270

3.6.1 - Localizations. . . 276

3.6.2 - Relation with isocrystals . . . 279

3.7 - The functorsD^log;geo_cris and D^log;geo_max : Geometrically semistable repres entations . . . 280

4 - List of Symbols. . . 282

1. Introduction

LetKbe a finite extension ofQpwith ring of integersO_Kand fix for the rest of this article a uniformizing parameter p of O_K. We denote by S:ˆSpec(O_K) and byM the log structure on Sassociated to the prelog structureN ! OK sending n2Ntopⁿ2 OK. We denote by (S;M) the associated log scheme.

LetX !Sbe a morphism of schemes of finite type (or a morphism of formal schemes topologically of finite type) with semistable reduction, by which we mean that there exists a log structureNonXand a morphism of log schemes (or log formal schemes) f:(X;N) !(S;M) satisfying the assumptions of section § 2.1.2. In particularf islog smooth.

Let now W:ˆW(OK=pOK) and we denote by O:ˆW[[Z]] and by O ! O_K the natural W-algebra homomorphism sending Z to p. Write P_p(Z)2W[Z] for the monic irreducible polynomial of p over W. It is a generator of Ker O ! O_K

. We denote byeS:ˆSpf(O) and byMe the log structure oneSassociated to the prelog structureN ! Osendingn2Nto Zⁿ 2 O. Let us consider the natural diagram of log formal schemes

We assume that there exists a GLOBAL deformation~f: X;e Ne

! S;e Me of f. Such deformationsexist for example ifX isaffine or if the relative dimension ofXoverSis1, but not in general.

Our main concern in thisarticle isto:

1) Define Faltings's logarithmic sites X_K and X_K associated to f:(X;N) !(S;M) and Fontaine (ind continuous) sheaves on it associated to the deformation~f:(X;e N)e !(S;e M):e B^r_cris,B^r_log,B^r_log,B_logandB_log.

2) Define the category Sh(XK)_ss of semistable (in fact arithmetically semistable) eÂtale local systems onXKand study its properties; see § 2.4.4 and § 2.4.7.

3) Define in §2.4.7 a Fontaine functorD^ar_logfrom the category of semi- stable eÂtale local systems onX_Kto the category of log filteredF-isocrystals onX relative to O. More precisely these are Frobenius isocrystals (con- sidering the Kummer eÂtale site on (X;N) modulop) relatively to thep-adic completion of the divided power envelope of Owith respect to the ideal generated bypandP_p(Z), with filtration on their base change viaO ! O_K defined by mappingZtop; see § 2.4.5.

4) We prove the followingcomparison isomorphism theorem, see 2.33.

Suppose thatLisap-adic Kummer eÂtale local system onXK, which when viewed asan eÂtale local system onX_K is semistable. Assume thatX isa proper and geometrically connected scheme overO_K. We have, see 2.33,

THEOREM 1.1. a) The p-adic representation Hⁱ X_K^ket;L

of G_K:ˆ

Gal(K=K)is semistable for all i0.

b) There are natural isomorphisms respecting all additional struc- tures (i.e. the filtrations, after extending the scalars to K, the Frobenii and the monodromy operators)

D_st Hⁱ(X_K^ket;L)

Hⁱ X_k=W^‡_cris

log;D^ar_log(L)^‡ :

Here, D^ar_log(L)^‡ is the Frobenius log isocrystal on X_k relative to W^‡ obtained fromD^ar_log(L) by base change via the mapO !WsendingZto 0.

Here,W^‡isWwith log structure defined byN!Wgiven by sending every n2Nto 0. In particular, Hⁱ Xk=W^‡_cris

log;D^ar_log(L)^‡

isa finite dimensional K₀-vector space, K₀ˆFracW, endowed with a Frobeniuslinear auto- morphism and a monodromy operator. Its base change toKcoincideswith the cohomology of the filtered log isocrystalD^ar_log(L)_XKgiven by base change of D^ar_log(L) via the map O ! O_K, sendingZ top. Thusthese cohomology groupsare endowed with filtrationscoming from the filtration onD^ar_log(L)_XK. For the constructions in (1)-(3) the existence of local deformations ofX toOwould suffice; namely the notion of semistable eÂtale local systems and the functorD^ar_logcan be defined locally and then glued. On the contrary, it is in (4) that we definitely need the existence of a global deformation Xe in order to guarantee the finiteness of the cohomology of Frobenius iso- crystals on the reduction of (X;N) modulo prelatively toOcris, a key in- gredient to prove the theorem. We hope to be able to remove thisas- sumption in the future.

All these constructions are generalizations to the semistable case of the analogue results in the smooth case. The comparison isomorphisms in the smooth case were recently proved in [AI2] (after having been proved be- fore in different waysand variousdegreesof generality by G. Faltings, T.

Tsuji, W. Niziol etc. see the introduction of [AI2] for an account on the history of the problem to date.)

The proof of the comparison isomorphisms in the smooth case pre- sented in [AI2] was in fact a result cumulating three sources:

i) [AI2] in which Faltings' site associated to a smooth scheme (or formal scheme) was defined (in that articleK was supposed unramified overQ_p

and so no deformation was required) and the global theory of Fontaine sheaves on the site was developed.

ii) [Bri] where the local Fontaine theory in the relative smooth case was worked out. In particular, ifRisanO_K-algebra, ``small'' (in Faltings' sense) and smooth overO_K it wasproved in [Bri] the following fundamental re- sult: the inclusionR[1=p],!B_cris(R) isfaithfully flat.

iii) [AB] where (in the notationsof ii) above) the geometric Galoisco- homology ofBcris(R) wascalculated.

The present article generalizes to the semistable case all three articles quoted above asfollows: in chapter 2 we develop the global theory, i.e., we define Faltings' logarithmic sitesX_KandX_Kand the Fontaine sheaves on it. In chapter 3 we work out the local Fontaine theory in the relative semistable case generalizing [Bri]: we define semistable representations and prove their main properties. The situation is more complicated than in the smooth case, namely letU ˆSpf(R) be a small log affine open of (X;N) andU ˆe Spf(R) a deformation of it to (e S;e M). We define relative Fontainee ringsB^cris_log(R) ande B^max_log (R), which are bothe R[1=p]-algebrasande together generalizeBcris(R) to the semistable case. More precisely:e

a) LetRemaxbe thep-adic completion of the ringRehPp(Z) p

iasa subring of R[1=p]. We prove that the inclusione Remax

p ¹

,!B^max_log(R) isclose toe being faithfully flat; see 3.31. More precisely we show:

i) If aˆ1, see the assumptions on § 3.1 (i.e. we are in the semistable reduction case) then in factRemax

p ¹

,!B^max_log (R) isfaithfully flat.e ii) Ifa>1 then the situation is more complicated, namely there exists an algebra A (denoted A^‡;log_R~o;max in the proof of theorem 3.31) such that Re_max

p ¹ ,!A

p ¹

,!B^max_log(R) and having the propertiesthat a faithfullye flatRemax

p ¹

-algebraCisa direct summand ofA p ¹

asC-module and the extension A

p ¹

,!B^max_log (R) isfaithfully flat. It followsthat if a se-e quence ofRe_max

p ¹

-modules

0 !M⁰ !M !M} !0

becomesexact after base changing it toB^max_log (R) then it wasexact to starte with and that anRemax

p ¹

-module isfinite and projective if it isso after base changing to B^max_log (R). These properties are what we call ``close toe faithful flatness'' and allow to prove thatD^ar_logof a semistable sheaf is anF- isocrystal.

b) If we denote byGR the (algebraic) fundamental group of Spm(R_K) for a geometric base point, we compute the continuous GR-cohomology ofB^cris_log(R) with results similar to those in [AB].e

c) Finally, ifG_Risthe (algebraic) fundamental group ofR[1=p] for the same choice of geometric base point as at b) above and if V isa p-adic representation ofGRthen we prove:VisB^cris_log(R)-admissible if and only ife V isB^max_log (R)-admissible if and only if the eÂtale local systeme Lattached to the representationVis semistable, in which caseVitself is called a semistable representation.

Moreover ifVis a semistable representation thenD^cris_log(V) andD^max_log (V) determine one another andD^cris_log(V) providesD^ar_log(L).

Using all these results in the second part of chapter 2 we prove the semistable comparison isomorphism (theorem 1.1 stated above).

We'd like to point out that T. Tsuji has a preprint [T2] where the theory of semistable eÂtale sheaves on a semistable proper scheme over O_K is developed. On the one hand hiswork ismore general than oursashe has less restrictive assumptions on the logarithmic structures allowed and on the existence of a global deformation over (eS;M). On the other hand nei-e ther does the author prove in that article any faithful flatness result nor does he derive comparison isomorphisms for the cohomology of the semistable eÂtale local systems defined there.

Finally, recent work of P. Scholze [Sc] might lead in the future to results in thedirectionofprovingthatdeRhameÂtalesheavesarepotentiallysemistable.

2. Fontaine's sheaves on Faltings' site 2.1 ±Notations

Letp>0 denote a prime integer andK a complete discrete valuation field of characteristic 0 and perfect residue fieldkof characteristicp. Let K₀be the field of fractionsofW(k). LetO_Kbe the ring of integersofKand choose a uniformizerp2 OK. Fix an algebraic closureKofKand writeGK

for the Galoisgroup ofKK. InK choose:

(a) a compatible systems ofn!-rootsp^n!1 ofp;

(b) a compatible systems of primitiven-rootsenof 1 for varyingn2N.

DefineK_n⁰ :ˆK[p^n!1] and K⁰₁:ˆ [_nK_n⁰. Since T^m p is an Eisenstein polynomial overOK, thenOK⁰_n:ˆ OK[p^n!1]ˆ OK[T]= T^n! p

isa complete dvr with fraction field preciselyK_n⁰.

LetM be the log structure onS:ˆSpec(OK) associated to the prelog structure c:N! OK given by 17!p. Letc_K:OK[N]! OK be the asso- ciated map of O_K-algebras. For every n2N we write Sn;Mn

for the compatible system of log schemes given byS_n:ˆSpec O_K=pⁿO_K

and log structureM_nassociated to the prelog structureN! O_K=pⁿO_K, 17!p. We refer to [K2] for generalitieson logarithmic geometry.

WriteO:ˆW(k)[[Z]] for the power series ring in the variableZand let NO be the log structure associated to the prelog structure c_O:N! O defined by 17!Z. We define Frobeniuson O to be the homomorphism given by the usual Frobenius on W(k) and by Z7!Z^p. It extendsto a morphism of log schemes inducing multiplication byponN. LetP_p(Z) be the minimal polynomial ofpoverW(k). It isan Eisenstein polynomial and uO:O ! O_K, defined byZ7!p, induces an isomorphism, compatibly with the log structures,O= P_p(Z)

! O_K.

2.1.1 - The classical period rings

Write A_cris for the classical ring of periods constructed by Fontaine [Fo, & 2.3] andA_logthe classical ring of periods constructed by Kato [K1,

§ 3]. More precisely, let Ee^‡_O

K :ˆlimOb_K where the transition maps are given by raising to thep-th power. Consider the elementsp:ˆ p;p^1p;. . .

, p:ˆ p;p^1p;

and e:ˆ 1;ep;

. The s et Ee^‡_O

K hasa natural ring structure [Fo, § 1.2.2] in whichp0 and a log structure associated to the morphism of monoidsN!Ee^‡_O

Kgiven by 17!p. WriteA_inf O_K

, or s imply A_inf, for the Witt ringW Ee^‡_O

K

. It is endowed with the log structure as- sociated to the morphism of monoids N!W Ee^‡_O

K

given by 17!

p . There isa natural ring homomorphism u:W Ee^‡_O

K

! Ob_K [Fo, § 1.2.2]

such thatu p

ˆp. In particular, it is surjective and strict considering onOb_Kthe log structure associated toN!Ob_Kgiven by 17!p. Its kernel isprincipal and generated by Pp

p

or by the element j:ˆ

p p.

WriteI for the ideal ofW Ee^‡_O

K

generated by [e]^pn1 1 forn2Nand by the TeichmuÈller lifts[x] for x2Ee^‡_O

K such that x⁽⁰⁾ liesin the maximal ideal of Ob_K.

We recall that A_cris isthe p-adic completion of the DP envelope of W Ee^‡_O

K

with respect to the ideal generated by p and the kernel of u. Similarly, A_log isthe p-adic completion of the log DP envelope of the morphism W Ee^‡_O

K

_W(k)O with respect to the morphism

uuO:W Ee^‡_O

K

_W(k)O !Ob_K. In particular, A_logA_crisfhu 1ig;

by which we mean that there exists an isomorphism of A_cris-algebras from the p-adic completion A_crisfhVig of the DP polynomial ring over A_cris in the variable V and A_log sending V to u 1 with u:ˆ

p Z; cf.

[K1, Prop. 3.3] and [Bre, § 2] where the ring isdenoted Ab_st. We endow A_cris and A_log with the p-adic topology and the divided power filtration. We write Bcris:ˆAcris

t ¹

and Blog:ˆAlog t ¹

, where t:ˆlog [e]

, with the inductive limit topology and the filtration FilⁿB_cris:ˆ P

m2NFil^n‡mA_crist ^m and FilⁿB_log:ˆ P

m2NFil^n‡mA_logt ^m. LetB^‡_dRbe the classical ring of Fontaine defined as the completion of W Ee^‡_O

K

[p ¹] with respect to the ideal generated bykeruwith the filtra- tion defined by thisideal. Similarly, we constructB^‡_dR(O) asfollows. Define A_inf(O) asthe completion of W Ee^‡_O

K

_W(k)O with respect to the ideal uuO ₁

pOb_K

and simply denote uuO:Ainf(O)!Ob_K the map ex- tendinguu_O. Then, we setB^‡_dR(O) to be the completion ofA_inf(O)

p ¹ with respect to the ideal generated by keruu_O, with the filtration de- fined by thisideal. DefineB_dR:ˆB^‡_dR

t ¹

andB_dR(O):ˆB^‡_dR(O) t ¹

. We extend the filtrationsto B_dR and B_dR(O) asbefore. Note that B^‡_dR(O)B^‡_dR[[u 1]]B^‡_dR[[Z p]] where the filtration isthe composite of the filtration on B^‡_dR and the (u 1)-adic or (Z p)-adic filtration;

cf. 3.15 (4). We have an inclusionB_logB_dR(O), strict with respect to the filtrations. We also have the classical subrings B_cris;K:ˆB_crisK0K and B_st;K:ˆB_stK0K of B_dR introduced by Fontaine; see [Fo, § 3.1.6] and [Fo, Thm. 4.2.4]. Define Blog to be the image of the composite map fp:Blog!BdR(O)!BdR defined in [Bre, § 7], and given by Z7!p. We consider the image filtration which is the filtration inherited byB_dR. For later use we remark

LEMMA2.1. We have natural morphisms Bcris;KBst;KB_logB_dR, which are G_K-equivariant, are strictly compatible with the filtrations and induce isomorphisms on the associated graded rings.

PROOF. The mapf_pisclearly compatible withG_K-action and the filtra- tions. It sendsP_p(Z) to 0. In particular,A_log= P_p(Z)

A_logisanA_crisW(k)O_K algebra and containsthe divided powersof the element

p

=p 1. In par-

ticular,Alog= Pp(Z)

containsthe element log [p]=p

which generatesBstas Bcris-algebra by [Bre, lemma 7.1]. See also [Fo, § 3.1.6]. This provides the claimed inclusions. As the mapB_cris!B_dRinducesan isomorphism on the

associated graded rings, the claim follows. p

The ringsA_crisandA_log, and henceB_crisandB_log, are endowed with a Frobeniushaving the property that W(u)ˆu^p and W(t)ˆpt and a con- tinuousaction of the GaloisgroupGK. Moreover, there isa derivation

d:A_log !A_logdZ Z which isA_crislinear and satisfiesd (u 1)^[n]

ˆ(u 1)^{[n 1]}udZ

Z; see [K1, Prop. 3.3] and [Bre, Lemma 7.1]. Itskernel isAcris and the inclusion AcrisA_log is split injective where the left inverse is defined by setting (u 1)^[n]7!0 for every n2N. We let N be the A_cris-linear operator on A_logsuch thatd(f)ˆN(f)dZ

Z . In particular,dandNextend toB_log. It is proven in [K1, Thm. 3.7] that Fontaine'speriod ringBst, see [Fo, § 3.1.6], is isomorphic to the subring of B_log whereNactsnilpotently.

B_log-admissible representations. According to [Bre, Def. 3.2] aQ_p-adic representationV ofGK iscalledBlog-admissible if

(1)D_log(V):ˆ B_logQpV_GK

isa freeB^G_log^K-module;

(2) the morphism Blog_BGK

log Dlog(V) !BlogQpV is an isomorphism, strictly compatible with the filtrations.

In thiscaseDlog(V) isan object in the categoryMF_BGK

log(W;N) of finite and free B^G_log^K-modules M, endowed with (i) a monodromy operator NM

compatible via Leibniz rule with the one on B^G_log^K, (ii) a decreasing ex- haustive filtration FilⁿM which satisfies Griffiths' transversality with re- spect toN_M and such that the multiplication mapB^G_log^KM!M iscom- patible with the filtrations, (iii) a semilinear Frobenius morphism W_M:M!M such that N_MW_MˆpW_MN_M and detW_M isinvertible in B^G_log^K. See [Bre, § 6.1].

Comparison with semistable representations. Consider the category MF_K(W;N) of finite dimensionalK₀-vector spaces D endowed with (i) a monodromy operator N_D, (ii) a descending and exhaustive filtration FilⁿDK onDK:ˆDK0K, (iii) a Frobenius W_D such that detW_D6ˆ0 and

NDW_DˆpW_DND; see [CF]. Such a module is called Bst-admissible if there exists aQp-representationVofGKsuch thatDst(V):ˆ VQpBst_GK isisomorphic toDcompatibly with monodromy operator, Frobeniusand filtration after extending scalars toK. Consider the functor

T:MF_K(W;N) ! MF_BGK

log(W;N)

sendingD7!T(D):ˆDK0B^G_log^Kwith monodromy operatorND1‡1N, FrobeniusW_DWand filtration defined on [Bre, p. 201] using the filtration on DK and the monodromy operator. More precisely, the map fp:B_log!B_dR defined in 2.1 by sendingZtopinducesa mapB^G_log^K !B^G_dR^KˆK. This provi- desa morphismr:T(D)!D_K. Then, FilⁿT(D) isdefined inductively onnby setting FilⁿT(D):ˆ

x2T(D)jr(x)2FilⁿDK;N(x)2Fil^{n 1}T(D) .

PROPOSITION2.2 [Bre]. (1) The functorT is an equivalence of cate- gories.

(2) The notions of B_log-admissible representations of G_K and of B_st- admissible representations are equivalent. For any such, we have T Dst(V)

D_log(V).

PROOF. (1) isproven in [Bre, Thm. 6.1.1]. (2) isproven in [Bre,

Thm, 33]. p

An admissibility criterion.We prove a criterion of admissibility very similar to the ones in [CF]. LetMbe an object ofMF_BGK

log(W;N). The map B_log!B_dRsendingZtophasimageB_logby 2.1. Define

V_log⁰ (M):ˆ Blog_BGK

log M

_Nˆ0;Wˆ1

and

V_log¹ (M):ˆ Blog_BGK

log M

=Fil⁰ Blog_BGK

log M

: Let d(M):V_log⁰ (M) !V_log¹ (M)

be the map given by the composite of the inclusionV_log⁰ (M)B_logBGK

log M

and the projectionB_logBGK

log M!B_logBGK

log M. We simply writeV_log(M) for the kernel ofd(M). Then,

PROPOSITION2.3. (1)A filtered (W;N)-module M over B^G_log^K is admis- sible if and only if(a)Vlog(M)is a finite dimensionalQp-vector space and (b)d(M)is surjective.

Moreover, if V ˆVlog(M)is finite dimensional asQp-vector space then it is a semistable representation of GK and Dlog(V)M. The latter is an equality if and only if M is admissible.

(2)The functors V_log⁰ and V_log¹ on the category MF_BGK

log(W;N)are exact and the morphismd(M)is not an isomorphism if M6ˆ0.

PROOF. (1) Let D;W;N;FilD_K

be a filtered (W;N)-module overK, cf.

2.1. Asin [CF, § 5.1 & 5.2] we define V_st⁰(D):ˆ…BstK0D†^Nˆ0;Wˆ1 and V_st¹(D):ˆBdRKDK=Fil⁰ BdRKDK

. We letd(D):V_st⁰(D) !V_st¹(D) be the natural map.

First of all we claim that the proposition holds replacing the category MF_BGK

log(W;N) with the category of filtered (W;N)-modulesover K and V_logⁱ (M),iˆnothing;0;1 withV_stⁱ(D). Indeed, it isproven in [CF, Prop. 4.5]

that theQ_p-vector spaceV_st(D) isfinite dimensional if and only if for every subobject D⁰D we have tH(D⁰)tN(D⁰) (these are the Hodge and Newton numbersattached toD⁰, respectively). Moreover, it is also shown in loc. cit. that in thiscase Vst(D) is a semistable representation of G_K whose associated filtered (W;N)-module iscontained inD. It coincideswith Dif and only if dim_QpV_st(D)ˆdim_K0D. It followsfrom the proof of [CF, Prop. 5.7] that, ifV_st(D) isfinite dimensional, then dim_QpV_st(D)ˆdim_K0D if and only ifd(D) issurjective. The claim followsfor filtered (W;N)-mod- ulesoverK.

Since B^Nˆ0_log ˆB^Nˆ0_st ˆB_cris, it followsthat V_st⁰(D)V_log⁰ T(D) . Since V_log¹ T(D)

 B_logKD_K

=Fil⁰ B_logKD_K

and GrB_logˆGrB_dR by 2.1, we deduce that the complexes d(D):V_st⁰(D) !V_st¹(D) and d T(D)

:V_log¹ T(D)

!V_log¹ T(D)

are identified. Thus, via the equiva- lence of categoriesTof 2.2, claim (1) followsfrom itsanalogue for filtered (W;N)-modulesoverKdiscussed above. This concludes the proof of (1).

To prove (2) it suffices to show the exactness ofV_st⁰ andV_st¹ and the fact thatdisnot an isomorphism for non zero objectson the category of filtered (W;N)-modulesoverK. Thisisproven in [CF, Prop. 5.1 & Prop. 5.2]. p 2.1.2 - Assumptions

Fix a positive integera. We assume that we are in one of the following two situations:

(ALG) X;N

isa log scheme andf: X;N

! S;M

isa morphism of log schemes of finite type admitting a covering by eÂtale open subschemes Spec(R)X, by which we mean that Spec(R)!Xisan eÂtale morphism,

of the form:

where (i)P:ˆPaPbwithPa:ˆN^aandPb:ˆN^b, (ii) the left vertical map isthe morphism of OK-algebrasdefined by the map on monoids N!PˆP_aP_bgiven byn7! (n;. . .;n);(0;. . .;0)

, (iii)c_aisthe map of O_K-algebraswithN317!p^a.

We require that the morphism O_K[P]_OK[N]O_K!R on associated spectra is eÂtale, in the classical sense, and that the log structure on Spec(R) induced by (X;N) isthe pullback of the fibred product log structure on Spec O_K[P]_OK[N]O_K

. We further assume that for every subset J_a f1;. . .;ag and every subset J_b f1;. . .;bg the ideal in R generated by c_R N^JaN^Jb

defines an irreducible closed subscheme of Spec(R), that the ideal of R generated by c_R Pa

isnot the unit ideal and that the image of the monoid O_Kc_R Pb

issaturated in ROKO_K.

(FORM) for every n2N we have a log scheme X_n;N_n and a morphism of log schemes of finite typef_n: X_n;N_n

! S_n;M_n

such that Xn;Nn

is isomorphic as log scheme over (Sn;Mn) to the fibred product of Xn‡1;Nn‡1

and Sn;Mn

over Sn‡1;Mn‡1

. WriteXformfor the formal scheme associated to the Xn's. We require that eÂtale locally on X1 the formal schemeX_form!Spf(O_K) isof the form

where the left vertical map andc_aare defined asin the algebraic case and c_R;n inducesa morphismO_K[P]_OK[N]O_K=pⁿO_K !R=pⁿRwhich istale and the log structure on Spec(R=pⁿR) induced from (X_n;N_n) isthe pull- back of the fibred product log structure on Spec OK[P]OK[N]OK=pⁿOK

. As in the algebraic case we require that for every subsetJa f1;. . .;ag and every subset J_b f1;. . .;bg the ideal of R=pR generated by c_R;1 N^JaN^Jb

defines an irreducible closed subscheme of Spec(R=pR), that the ideal ofRgenerated byc_R P_a

isnot the unit ideal and that the image of the monoidO_Kc_R Pb

issaturated inROKO_K.

We deduce from 3.1 that

(i) in the algebraic, respectively in the formal setting, X;N (re- spectively X_n;N_n

) is a fine and saturated log scheme;

(ii) f (resp.fn) is a log smooth morphism.

In the algebraic case, by abuse of notation we write Xfor (X;M). An object UˆSpec(R)2X^et with induced log structure satisfying the re- quirementsabove will be calledsmall.

In the formal case we writeX_rigfor the rigid analytic fibre ofX_form. The inverse limit of the log structures N_n defines a morphism of sheaves of monoidsfrom the inverse limitNformˆ lim

1 nNntoOXform. It coincideswith the inverse image ofN₁via the canonical mapO_Xform! O_X1. We call it the formal log structureonX_form. We also writeXor (X;N) for the inductive system X_n;N_n

n2N. It followsfrom our assumptionsthat X_form isa noetherian and p-adic formal scheme. An eÂtale open Spf(R)!Xform sat- isfying the requirements above is calledsmall. By assumption we have a covering ofX_formby small objects. For any such small affine Spf(R) ofX_form we also havep-adic formally eÂtale morphisms

Spf O_K[P]b_OK[N]O_K c^_R

Spf(R) !X_form;

whereb stands for thep-adic completion of the tensor product, with the property that the formal log structure N_formon Spf(R) isinduced by the formal log structure on the fibred product Spf O_K[P]b_OK[N]O_K

. We call any such diagram aformal chartof X_form;N_form

.

Example:Assume thatX is a regular scheme with a normal crossing divisorDX. Then eÂtale locally onXwe can choose local chartsc_Rsat- isfying the conditions above. For example, for every closed point ofXone can take P and c_R eÂtale locally to be defined by a regular sequence of elementsgenerating the maximal ideal atxso thatDisdefined by part of such a sequence. In this case also the ideal generated byc_R(P_b) inRisnot the unit ideal and the conditions above are satisfied.

2.1.3 - Continuoussheaves

Given an abelian categoryA admitting enough injectveswe consider the categoryA^Nof inverse systems of objects ofAindexed byN. It isalso abelian with enough injectives. Given a left exact functorFfromAto an abelian category B we have a left exact functor F^N:A^N ! B^N sending (C_n)_n2N7! F(C_n)

n2N and its i-th derived functor Rⁱ F^N

iscanonically

RⁱF_N

. If projective limitsexist in B, one can derive the functor F^cont:A^N! B sending (Cn)_n2N7! lim

1 nF(Cn). We refer [AI1, § 5.1] for details.

We also consider the category Ind A

of inductive systems of objects in A indexed by Z, i.e. A_h;g_h

h2Z with g_h:A_h!A_h‡1. Consider a non decreasing function a:Z!Z. Given objects A:ˆ A_i;g_i

i2Z and B:ˆ Bj;dj

j2Zwe define a morphismf:A!Bof typeato be a collection of morphisms f_i:A_i!B_a(i) such that f_i‡1g_iˆ Q

a(i)j5a(i‡1)d_jf_i. We denote by Hom^a A;B

the group of homomorphisms of type a. We s ay that two morphisms f and g of type a (resp. b) are equivalent if there exists N2N such that f_i composed with B_a(i)!Bmax (a(i);b(i))‡N and g_i composed with B_b(i) !Bmax (a(i);b(i))‡N coincide. One checksthat thisde- finesan equivalence relation. We define a morphismA!Bin Ind A

to be a class of morphisms with respect to this equivalence relation.

One can prove that Ind A

isan abelian category. If B admits inductive limitsand F:A ! B isa left exact functor, we define RⁱF^cont:Ind A^N

! Bby RⁱF^cont (A_h;g_h)_h

:ˆ lim

h!1RⁱF^cont(A_h). Then the familyfRⁿF^contg_n definesa cohomologicald-functor on Ind A

.

2.2 ±Faltings' topos

2.2.1 - The Kummer eÂtale site of X

The notationsare asin the previoussection. Both in the algebraic and in the formal case we writeX^ketfor the Kummer eÂtale site of (X;N).

In the algebraic case the category is the full subcategory of log schemes endowed with a Kummer eÂtale morphism (Y;N_Y)!(X;N) in the sense of [Il, § 2.1], i.e. morphisms which are log eÂtale and Kummer or equivalently log eÂtale and exact. The coveringsare collectionsof Kummer eÂtale morphism (Yi;Ni)!(X;N) such thatXisset theoretically the union of the imagesof theY_i's. One verifies that this defines a site; see loc. cit.

In the formal case the objects are Kummer eÂtale morphisms g_n: Y_n;N_Yn

!(X_n;N_n)

n2Nsuch thatg_nisthe base change ofg_n‡1via (X_n;N_n)!(X_n‡1;N_n‡1) for every n2N. We simply write g:(Y;N_Y)! (X;NX) for such inductive system of morphisms. The morphisms from an object (Y;NY)!(X;N) to an object (Z;NZ):ˆ hn: Zn;NZn

!

(Xn;Nn)g_n2Nare collectionsof morphisms tn: Yn;N_Yn

! Zn;N_Zn

aslog schemesover (X_n;N_n) such that t_n isthe base change oft_n‡1n2Nvia (X_n;N_n)!(X_n‡1;N_n‡1) for every n2N. We simply write t:(Y;N_Y)!

(Z;NZ) for such an inductive system of morphisms. The coverings are collectionsof Kummer eÂtale morphisms (Yf i;Ni)!(X;N)g_isuch thatX1

isthe set theoretic union of the imagesof theY_i;1's. This defines a site. Due to the characterization of log eÂtale morphisms in [K2, prop 3.14] the natural forgetful morphism of sites X^ket !X₁^ket, sending g:(Y;N_Y)!(X;N) to g₁: Y₁;N_Y1

!(X₁;N₁), isan equivalence of categories.

LEMMA2.4. Let(Y;H)2X^ket. Then,

(1)Y (resp. Spec(R)if Y_formˆSpf(R) in the formal case) are Cohen- Macaulay and normal schemes;

(2) (Y;H)(resp. Spec(R);H_form

if Y_formˆSpf(R)in the formal case) are log regular in the sense of[K3, Def. 2.1].

PROOF. We provide a proof in the algebraic case. Since f:(Y;H)! (X;N) isKummer eÂtale, in particular it islog eÂtale. Since f: X;N

! S;M

is log smooth the composite (Y;H)! S;M

islog smooth. Recall that S;M

isSpec(OK) with the log structure defined by its maximal ideal.

In particular it islog regular. Arguing asin [T1, Lemma 1.5.1] we deduce from [K3, Thm. 8.2] that also (Y;H) islog regular. Due to [K3, Thm. 4.1] the schemeYisthen Cohen-Macaulay and normal. Thisprovesthe claimsin the algebraic case.

For the proof in the formal case we make some preliminary remarks in the algebraic case. Lety2Yand setxto be itsimage inX. WriteHyand N_xfor the stalk of the sheaves of monoidsHandNand putH_y:ˆH_y=O_Y;y andN_x:ˆH_y=O_X;x. Since the log structures are fine, H_y andN_x are fi- nitely generated and we have inclusions HyH^gp_y and NxN^gp_x . The morphism (Y;H)!(X;N) being Kummer eÂtale, the induced map i:Nx!Hyisinjective and there existsan integerninvertible inO_Y;ysuch that nH_yN_x. Since N^gp_x isa finite and free Z-module we can find a splitting of the group homomorphismN_x^gp!N^gp_x which composed with the inclusionNxN^gp_x providesa chartP!Nin a neighborhoodUxofxcf.

[K2, Lemma 2.10]. Proceeding similarly withH^gp_y we can find a splitting of H^gp_y !H^gp_y . Since the local ringO_Y;y istaken with respect to the eÂtale topology andnisinvertible inO_Y;y, the groupO_Y;yisn-divisible and we can take the splitting compatible with the first splitting ofN_x^gp!N^gp_x . Com- posing with the inclusionH_yH^gp_y we get a chartQ!Hin a neighbor- hoodVyofycompatible withP!N via the map of sheavesf ¹(N)!H.

To check that R is Cohen-Macaulay in the formal case it suffices to prove that the complete local ring ofRat every maximal idealyisCohen- Macaulay at the image xof y inX. To prove that it isnormal it further

suffices to show thatRis regular in codimension 1. Due to the assumptions and the proof in the algebraic case, (1) and (2) hold if Spf(R) isa formal chart of (X;N), i.e.f is the identity map. In the general case, using the considerations above, we have

Ob_Y;yOb_X;xb_Z[P]Z[Q]

whereP!Qis a morphism of monoids as above. By the construction of the chart P, we have that Pˆ f1g. We conclude from [K3, Thm. 3.2] that Ob_X;xR[[P]][[T1;. . .;Tr]]=(u) forRˆW k(x)

andupmodulo the ideal Pnf1g;T₁;. . .;T_r

. Then,Ob_Y;yR[[Q]][[T₁;. . .;T_r]]=(u). Since Qissatu- rated andQˆ f1gby construction, alsoOb_Y;yisof the same form. The proof of [K3, Thm. 4.1] appliesto deduce thatOb_Y;yisCohen-Macaulay and regular in codimension 1. This concludes the proof of (1) and (2) in the formal case as

well. p

In the algebraic case consider the presheavesO_Xket andN_Xket respec- tively defined by

X^ket3(U;N_U) !G U;O_U

; X^ket3(U;N_U) !G U;N_U

: Similarly, in the formal case for everyh2Ndefine the presheavesO_Xket

andN_X^ket h h

X^ket3 Un;NUn

n !G Uh;OUh

; X^ket3 Un;NUn

n !G Uh;NUh

:

We write O_X^ket

form andN_X^ket

form for the presheaves defined as lim

1 nO_X^ket

form and

1 nlim N_Xket

formrespectively.

PROPOSITION2.5. (1)In the algebraic case the presheavesO_Xket,O_Xket

and N_X^ket are sheaves and N_X^ket ! O_X^ket is a morphism of sheaves of multiplicative monoids such that the inverse image ofO_Xket is identified withO_Xket.

(2)In the formal case the presheaves O_Xket h ,O_Xket

h and N_Xket

h for every h2Nand the presheavesO_X^ket

form,O_Xket

form and N_X^ket

form are sheaves. Moreover, N_X^ket

h ! O_X^ket

h and N_X^ket

form! O_X^ket

form is a morphism of sheaves of multi- plicative monoids such that the inverse image of O_Xket

h (resp. O_Xket form) is identified withO_Xket

h (resp.O_Xket form).

PROOF. An unpublished result of K. Kato implies that the Kummer eÂtale topology iscoarser than the canonical topology. Thisimpliesthe

claims that the given presheaves are sheaves, see [Il, § 2.7(a)&(b)]. The

other propertiesare clear. p

2.2.2 - The finite Kummer eÂtale sites U_L^{f k e t}

LetU2X^ketand letKLK. In the algebraic case we letU_L^fket be the site offinite Kummer eÂtale coversofUL endowed with the log struc- ture defined by N; see [Il, Def. 3.1]. As remarked in [Il, Rmk. 3.11] a Kummer eÂtale mapY!U_L, inducing a finite and surjective morphism at the level of underlying schemes, is a Kummer eÂtale cover. Viceversa [Il, Cor. 3.10 & Prop 3.12] impliesthat any Kummer eÂtale cover Y!U_L is Kummer eÂtale and induces a finite and surjective morphism on the un- derlying schemes.

In the formal case we proceed differently. IfKLisa finite extension, letU_Lbe the rigid analytic space associated toU_formb_OKO_Land letU_L^fketbe the site whose objects consist of finite surjective morphismsW!U_LofL- rigid analytic spaces such that

(1)Wissmooth overL;

(2) for every formal chart

Spf O_K[P]b_OK[N]O_K c^R Spf(R) !U_form; the induced morphism WULSpm ROKL

!Spm OK[P]bOK[N]L definesa finite and eÂtale morphism of rigid analytic spaces over the open subspace of Spm LfPg

given by Spm LfP^gpg .

The morphisms are morphisms as rigid analytic spaces over U_L. The coveringsare collectionsof morphismsWi!W, fori2I, whose images coverWset theoretically.

REMARK 2.6. (i) If W!U_L isa finite morphism of rigid analytic spaces, then for every formal chart ofUthe map

r:W_ULSpm R_OKL

!Spm R_OKL

isfinite by [FdP, Th. III.6.2] so that it isof the form Spm(B)! Spm R_OKL

for a R_OKL-algebra B which isfinite asa R_OKL- module. Then, r isfinite and eÂtale over Spm LfP^gpg

if and only if R_OKL

P^gp !B

P^gp is a finite and separable extension of algebras.

Since thiscondition can be checked on K-points, this holds if and only if ROKL

P^gp

!B P^gp

isfinite and eÂtale in the usual sense.

(ii) LetW !U_Lbe a finite morphism ofL-rigid analytic spaces withW

smooth overL. Then, condition (2) holdsif and only if there exist formal chartsofUformwhich coverUformand for which condition (2) holds.

(iii) We remark that the definition in the algebraic case coincides with the one provided by the analoguesof requirements(1) and (2). Indeed, givenU2X^ketandW!U_La Kummer eÂtale cover,W !U_LisKummer eÂtale. Thus, W !Spec(L) is log smooth, and in fact smooth as the log structure onListrivial. The analogue of condition (2) holdsthanksto [K2, Prop. 3.8]. Viceversa assume thatW !ULis a finite surjective morphism satisfying conditions (1) and (2). Leti:U_L^o,!U_Lbe the locusof triviality of the log structure and letj:W^o,!Wbe itsinverse image inW. AsU_Lislog regular, see 2.4, the log structure onU_Lisdefined byO_UL\i(O_U^o_L) O_UL thanksto [K3, 11.6]. AsWissmoothOW\j(OW^o) OWdefinesa fine and saturated log structure onW, cf. [Il, § 1.7]. Using this log structure we get a map of log schemesW !U_Land, asW !U_Lisfinite and surjective, it is exact and log eÂtale, i.e., Kummer eÂtale.

Given a finite extensionKLL⁰Kthe base change fromLtoL⁰ provides a morphism of sites U^fket_L !U^fket_L0 . For arbitrary extensions KLK, we then get a fibred siteU^fketover the category of finite ex- tensions of K contained in Lin the sense of [SGAIV, § VI.7.2.1]. We let U_L^fketbe the site defined by the projective limit of the fibred siteU^fket; s ee [SGAIV, Def. VI.8.2.5].

REMARK 2.7. For example, one hasthe following explicit descrip- tion of U_K^fket. The objects inU^fket_K consist of pairs (W;L) where L isa finite, extension of K contained in K andW 2U_L^fket. Given (W;L) and (W⁰;L⁰) define Hom_U^fket

K (W⁰;L⁰);(W;L)

to be the direct limit lim! HomL⁰⁰ W⁰L⁰L⁰⁰;W LL⁰⁰

over all finite extensions L⁰⁰ of K, contained in K and containing both L and L⁰, of the morphisms W⁰L⁰L⁰⁰! W LL⁰⁰ asrigid analytic spacesoverUL⁰⁰.

2.2.3 - Faltings' site

LetKLK be any extension. LetEXL be the category defined as follows

i) the objects consist of pairs U;W

such that U2X^ket and W2U_L^fket;

ii) a morphism (U⁰;W⁰) !(U;W) inE_XLconsists of a pair (a;b), where a:U⁰ !Uisa morphism inX^ketandb:W⁰ !WUKU_K⁰ isa morphism in U_L^0;fket.

The pair (X;XL) isa final object in EXL. Moreover, finite projective limitsare representable in EXL and, in particular, fibred productsexist:

the fibred product of the objects(U⁰;W⁰) and (U⁰⁰;W⁰⁰) over (U;W) is U⁰_UU⁰⁰;W⁰_WW⁰⁰

whereW⁰_WW⁰⁰isthe fibred product of the base- changesof W⁰ and W⁰⁰ to U⁰_UU⁰⁰_fket

L over the base-change of W to U⁰_UU⁰⁰_fket

L . See [Err, Prop. 2.6].

We say that a family f(U_i;W_i) !(U;W)g_i2I isa covering family if either

a) fU_i !Ug_i2Iisa covering inX^fketandW_iW_UKU_i;Kfor every i2I.

or

b) U_iUfor alli2IandfW_i !Wg_i2Iisa covering inU^fket_L . We endowE_XL with the topology T_XL generated by the covering fam- iliesdescribed above and denote by X_L the associated site. We call T_XL Faltings' topology and XL Faltings' site associated to X. Asin [Err, Lemma 2.8] one proves that the so called strict coverings of (U;W) (see definition 2.8 below) are cofinal in the collection of all covering familiesof (U;W).

DEFINITION2.8. A familyf(Uij;Wij) !(U;W)g_i2I;j2Jof morphisms in EXLiscalled astrict covering familyif

a) For eachi2Iand for everyj2Jwe have an objectU_i2X^ketand isomorphismsUiUijinX^fket.

b)fUi !Ug_i2Iisa covering inX^fket.

c) For every i2Ithe familyfW_ij !W_UKU_i;Kg_j2J isa covering in U_L;i^fket.

This is not Faltings' original definition of the site given in [F3]. We refer to [AI2] for a discussion of the differences between the two ap- proachesand motivationsfor our definition.

2.2.4 - ContinuousFunctors ForKLKwe let

vX;L:X^ket !XL; zX;L:X^et !XL

be given by vX;L(U):ˆ U;UL

in the algebraic case and by vX;L(U):ˆ

U;U_K

, viewingU_Kasan object ofU_K^fket, in the formal case and similarly forz_X;L. We simply writev_Landz_L.