Logarithmic Hodge Witt Forms and Hyodo Kato Cohomology

Transcription

1 Journal of Algebra 249, (2002) doi:0.006/jabr , available online at on Logarithmic Hodge Witt Forms and Hyodo Kato Cohomology Pierre Lorenzon Mathematische Institut, Einstein Strasse 62, 4849 Münster, Germany Communicated by Laurent Clozel Received October 6, 998 INTRODUCTION The aim of this paper is to extend some results of Illusie and Raynaud [3] to the Hyodo Kato cohomology [ ]. Let S be the spectrum of a perfect field k of characteristic p>0endowed with a fine log structure and let X be a proper, log smooth, and of Cartiertype fine log scheme over S. Recall that the Hyodo Kato cohomology groups H m X/W S of X are defined as the limit over n of the crystalline cohomology groups H m X/W n S of X over the Teichmüller lifting W n S. These are finitely generated W -modules, on which the Frobenius endomorphism of X induces a σ-linear isogeny φ, where W = W k and σ is the Frobenius automorphism of W. We study the slopes of the corresponding crystals, and especially their integral slopes, using the alternative description of the Hyodo Kato cohomology groups as H m X W X/S, where W X/S is the de Rham Witt complex of X/S [9]. First of all, generalizing the Illusie Raynaud finiteness theorem [3, (II, 2.2)] we prove that RƔ X W X/S, as an objectof D R, where R is the Raynaud ring W F V +W F V d, has bounded cohomology, consisting of coherent complexes of R-modules (Theorem 3.). This implies the degeneration modulo torsion of the slope spectral sequence as well as the Mazur Ogus-type results [2, Sect. 8; 22, Sect. 7] concerning the Newton polygon of H m X/W S and the Hodge polygon given by the H m q X q X/S. Next, we study the integral slopes. To do this we Current address: Dept. Math., Bât. 425, Univ. Paris Sud, 9405 Orsay Cedex, France /02 $ Elsevier Science (USA) All rights reserved.

2 248 pierre lorenzon consider the logarithmic Hodge Witt sheaves W n q X/S log, consisting of sections of W n q X/S which are étale locally sums of sections of the form d log m d log m q where m m q are sections of M gp X (cf. [4]). Using a basic resultof Tsuji [23] (see also [8]), we prove thatthese sheaves are in a suitable sense the fixed points of the operator F on the Hodge Witt sheaves W n q X/S (2.3), as in the classical case where the log structures are trivial [4, 0]. We then extend the structure theorem of Illusie and Raynaud [3] for the cohomology groups H m q X et W n q X/S log and their limits. In particular, we show that when k is algebraically closed, the latter give a basis over K of the part of slope q of H m X/W S K, where K = Frac W (3.4.6). We also discuss the notion of ordinariness ((4.), (4.3)), generalizing criteria of Bloch and Kato [3] and Illusie and Raynaud [3], thus completing a program sketched in [4, (2.2) and (2.3)].. REVIEW OF THE DE RHAM WITT COMPLEX ON LOG SCHEMES AND THE HYODO KATO COHOMOLOGY In the following we will denote by M X the log structure of a fine log scheme X. Let S be a perfect scheme of characteristic p>0, equipped with a fine log structure, and f X S be a smooth of Cartier-type [5, (4.8)] fine log scheme over S. For any integer n, we will denote by S n the log scheme W n S endowed with the log structure defined in [9, (3.l)]. The log scheme S n is equipped with an automorphism σ given by raising the coefficients of the Witt vectors to the pth power. The ideal pw n S is given its usual PD-structure. We refer the reader to [9, (2.5)] for the definitions of the crystalline topos X/S n Cris and of the sheaf X/Sn on X/S n Cris. We denote by X et the étale topos of X (where X is the underlying scheme of X) and by u X/Sn X/S n Cris X et the canonical morphism of topoi. Since the theory of de Rham Witt complex developed in [9 4 ] over a perfect base field easily generalizes to the case of a perfect base, we shall freely use this generalization and give the following definition: For each n and q 0, the qth component of the de Rham Witt complex of index n is given by W n q X/S = σn Rq u X/Sn X/Sn (.) For any embedding system (cf. [9, (2.8)]) X Z, C X/Sn denotes the crystalline complex (cf. loc. cit., (2.9)) associated to the crystal X/Sn.Ifθ

3 logarithmic hogde witt forms 249 denotes the morphism of topoi X et X et (cf. loc. cit. (2.8)), we have the canonical isomorphism (cf. loc.cit., (2.20)), Ru X/Sn X/Sn Rθ C X/Sn (.2) Since C X/S is flatover S n (cf. [9, (2.22)]), we have the exact sequence We have the operators 0 C X/Sn p m C X/Sn+m C X/Sm 0 (.3) d W n q X/S W n q+ X/S π n W n+ q X/S W n q X/S p W n q X/S V W n q X/S F W n+ q X/S W n+ q X/S W n+ q X/S W n q X/S (.4) (cf. [9, (4.)]).. The differential map d corresponds to the connecting map in the exactsequence (.3) for m = n. 2. The map π n is defined as in The map p is the unique map whose composition with the projection π n W n+ q X/S W n q X/S is the multiplication by p on W n+ q X/S. 4. The map V corresponds to the map p C X/Sn C X/Sn+ in the exact sequence (.3) for n =. 5. The map F corresponds to the projection in the exact sequence (.3) for m =. W n X/S d is the de Rham Witt complex of index n that we denote W n X or W n if no confusion is possible. One can show Proposition.5. W n d π n n is a projective system of graded differential algebras over f Sn. Moreover, for each n the operators (.4) satisfy the following properties:. (cf. [9, (4..)]) FV = VF = p df = pfd Vd = pdv FdV = d 2. F π n = π n F V π n = π n V. 3. F x y =F x F y x V y =V F x y

4 250 pierre lorenzon Furthermore, Proposition.6 (cf. [9 4 6 ] with q = 0). There exists a canonical isomorphism of sn -algebras W n X/S 0 = W n X compatible with F, V, and π n. Especially, W n is a W n X -algebra. The projective system W n X/S π n denoted by W X/S (.7) is called the pro-complex of de Rham Witt of X over S and W X/S is called the de Rham Witt complex of X over S. We will set = lim W n X/S (.8) ZW n q = Ker d W n q W n q+ BW n q = Im d W n q W n q H q W n = ZW n q /BW n q The canonical filtration is defined as in [0, (I 3.A)].. Fil n W r q = W r q,ifn 0orr Fil n W r q = Ker π r n W r q W n q,if n<r. 3. Fil n W r q = 0, if n r. We recall that there exists a Cartier isomorphism (.9) C σ q X/S = H q X/S =W q X/S (.0) which is X -linear and compatible with the differential map (cf , and 2 2 ). We will identify q X/S and W q X/S by (.0). Moreover, the diagram W q F W q /dw q = = (.) q C q /d q (where the upper horizontal map F is deduced from F W 2 q W q is commutative. We define étale subsheaves of q X/S, 0 = B 0 q X/S B q X/S B 2 q X/S Z 2 q X/S Z q X/S Z 0 q X/S = q X/S

5 logarithmic hogde witt forms 25 by the formulas B 0 q X/S = 0 Z 0 q X/S = q X/S B q X/S = d q X/S ( ) Z q X/S = Ker d q X/S q+ X/S (.2) (cf. [0, (0.(2.2.2)); 9, (4.3)]). B n q C X/S B n+ q X/S /B q X/S Z n q C X/S Z n+ q X/S /B q X/S Remark 3 Itis easily seen, by induction on n, that B n q X/S and Z n q X/S are locally free of finite type (over /S (isomorphic to X)) Xpn and of formation compatible with any base change T S with T perfect and endowed with the inverse image log structure. We will omitthe index X/S if no confusion is possible. We put gr n W r = Fil n W r /Fil n+ W r For r n +, we note that the natural map gr n W r gr n W n+ is an isomorphism. We define gr n W q by gr n W q = gr n W n+ q Lemma.4. We have the exact sequences 0 B q n Bq n+ C n B q 0 (.4.) q and n Proof. sequence, 0 Z q n+ Zq n dc n B q 0 (.4.2) We have B q n+ Zq n and Bn q B q n+. Moreover, we have an exact 0 B q n Zq n C n q 0 and finally C n B q n+ Bq is surjective. This proves (.4.).

6 252 pierre lorenzon Moreover, in the following commutative diagram, the bottom row and the left and central columns are exact: B q n 0 Z q B q n n+ Zq n C n C n dc n B q Id 0 Z q q d B q 0 We deduce from this the exactness of the middle row and the proof of (.4.2). Q.E.D. Proposition 5 (cf. [0, (I 3.8)]). exact sequences: We have the following morphisms of 0 Bn q q V n W n+ q 0 B q n+ q V n W n+ q /dv n q (.5.) 0 Z q n+ q dv n W n+ q 0 Z q n q dv n W n+ q /V n q (.5.2) Proof. We have the exact sequences (cf. [9, (4.4)]) 0 R q n q q V n dv n Kerπ n 0 0 R q n B q n+ Zq n C n dc n B q 0 (.5.3) The proposition follows from them and from the exact sequences (.4.) and (.4.2). Q.E.D. In the same way we get Proposition.6 (cf. [0, (I 3.9)]).. For all r n and for every q we have Fil n W r q X = V n W r n q X + dv n W r n q X

7 logarithmic hogde witt forms The associated graded is the center term of the cross, of which the row and column are exact, 0 F n+ q /Bn q 0 F n+ q /Z q n+ dv n V n gr n W q β F n+ q /Zn q 0 β F n+ q /B q n+ 0 where gr n W q is considered as an X -module via F X = W n+ X / VW n X W n+ X /pw n+ X and the maps β and β are defined by and for x (resp. y) a local section of q X/S β V n x +dv n y = class of y β V n x +dv n y = class of x (resp. q x/s. Proposition (.6) has the following consequence. Corollary.7. For any q, W n q X/S is a quasi-coherent W n X -module and is coherent when S is Noetherian. Corollary.8. For all n the map p W n W n+ (.4) is injective. Proof. The proof is the same as that in [0, (I 3.0)] and is a consequence of (.6). Q.E.D. We now recall the following properties (Corollaries.9 and.20) of the operators defined in (.4). Corollary.9. W q X/S are injective. Corollary.20. For all q, the endomorphisms p F V of the pro-object The image of F n is characterized as d p n W q+ =F n W q 20

8 254 pierre lorenzon Proof. that It follows directly from the definitions of the operators F and V Im F n W 2n q W n q =Ker d W n q W n q Since we have the quasi-isomorphism W /p n W = Wn 20 3 (cf. [0, (I )]), which follows from the quasi-isomorphisms W m /p n W m = W n m n (cf. [9, (4.5)]) and the fact that W n q is a quasi-coherent W n X -module (.7), Eq. (.20.2) implies (.20.). Q.E.D. Proposition.2. For all i 0, Ker p i W n+ q W n+ q = Fil n+ i W n q Proof. We obtain this result by induction on i (cf. [0, (I 3.4.)]). Q.E.D. Proposition.22 (cf. [9, (4.9)]). the derived category D X f Sn, We have a canonical isomorphism in W n X/S = Ru X/Sn X/Sn compatible with Frobenius and with the transition morphisms. The Frobenius is given by p q F in degree q on W n X/S and induced on the right-hand side by the absolute Frobenius of X and the endomorphism σ of S n. Corollary.23. The isomorphism of (.22) induces an isomorphism Rf X W n = Rf X/Sn If X/S is proper, Rf W X/S is a perfect complex and the natural map Rf W X/S Rlim Rf X/Sn is an isomorphism. Proof. This follows from Proposition.22 as in [0, Sect. II 2.B]. To see the perfectness of K n = Rf W n one is reduced to n =, in which case we have the facts that each q X/S is locally free of finite type and f is proper and flat (being log smooth of Cartier type). The rest follows from the fact that for n m the natural map K n L X/S K m m is an isomorphism using the standard argument (cf. [8, XV, Sect. 3, Lemme ]). Q.E.D.

9 logarithmic hogde witt forms FIXED POINTS OF F ON HODGE WITT SHEAVES The assumptions on X S are the same as in the previous section. Since we assume that f X S is of Cartier type, the absolute Frobenius F X of X factors through X, where X is the pullback of X (in the category of fine log schemes) by the absolute Frobenius F S of S. The factorization F X/S X X is exact(cf. [5, (4.0)]). We denote by w the canonical projection X X, hence we have F X = w F X/S. Since X is the pullback of X by F S in the category of fine log schemes, we have the isomorphism We denote by X /S = w X/S (2.) C X/S q X /S = H q X/S the Cartier isomorphism (cf. [5, (4.2)]) and by C X/S Z q X/S X /S the Cartier operator (cf. [0, (0.(2..9))]). We have the exact sequence 0 B X/S Z X/S C X/S X /S 0 (2.2) Letus now denote by X n the pullback of X by the nth power of the Frobenius F S and by F n X/S X X n the natural map. We observe that the pullback of X n by the Frobenius F S is canonically isomorphic to X n+. Since f is of Cartier type, f is integral [5, (4.9)]; then the pullback by F S in the category of fine log schemes is also the pullback in the category of log schemes. The underlying scheme of the pullback is the pullback of the underlying scheme. It follows from this that all X n for n have the same underlying topological space. Proposition 2.3. The kernel of the map d log M gp X X/S is pmgp X + f M gp S Proof. Let x be a geometric pointand letus consider the same morphism as in [5, (3.5)], k x k x M gp X x / X x + f xm gp S (2.3.) d log a a Suppose that a M gp X and that d log a = 0. Then a = 0 in k x M gp X x / X x + f xm gp S, hence a = uvbp with u X, v f M gp S, and b M gp X. The factthatd log a = 0 implies that d log u = 0 which, in turn, implies du = 0, hence u X p by the Cartier isomorphism. Q.E.D.

10 256 pierre lorenzon We have an exactsequence 0 M gp X d log Mgp X (2.4) (cf. [20, (6..2)]). We will denote M n gp = M X n. Since all X n have the same underlying topological space, we can consider all M n gp as abelian sheaves on X. We have the inclusions M n+ gp M n gp M gp = M gp X Then (by definition of a log fiber product) M n gp = p n M gp + f M gp S (2.5) Since f X S is smooth, for every n we may construct (locally on X) closed immersions X Z n over S n compatible with the immersions S n S n+, with Z n log smooth over S n.foru an open setsuch thatthe morphism U Z n exists, we can define a map d log M gp X U W n U/S following [9, (4.9)]. For all V U and m Ɣ V M gp X we define d log m = d log m, where m is a lifting of m in M gp D n, D n being the PD-envelope of U in Z n. One can show (loc. cit.) that this map is well defined and compatible with the immersions S n S n+. Itis easy to show Proposition 2.6. The map d log defined above satisfies the following properties:. π n d log = d log. 2. F d log = d log. 3. d d log = d log x = x d x for x X, where x = x 0 0 W n X. 5. d log is compatible with the identification = W. Proposition 2.7. We have the exact sequence n gp 0 MX M gp d log X Wn Proof. Itis clear thatd log M n gp = 0. Suppose that a is a local section of M gp such that d log a = 0. Letus assume thata is a local section m gp of M with m<n. Then we have a = p m b + c with b (resp. c) a local section of M gp X (resp., f M gp X ). The factthatd log a = 0 implies that p m d log b + d log c = 0 and consequently p m d log b = 0. Then we have d log b Fil n+ m. Itfollows thatd log b Fil, so its image in vanishes. Hence b M gp. Q.E.D.

11 logarithmic hogde witt forms 257 For every q and every n we denote by W n q X/S log the abelian étale subsheaf of W n q X/S generated by the sections of the type d log m dlog m q, for m m q local sections of M gp X. We will simply write W n q log if no confusion is possible. The sheaves W n q log form a projective subsystem W q log of W q. Hence the pro-object W q log is p-torsion free (cf..9). The isomorphism (2.) gives us a X -linear map ψ X/S w X /S Proposition 2.8 (cf. [0, (0.(2.4.2))]). For any q 0 the following sequence is exact for the étale topology: 0 q X/S log Z q X/S ψ C X/S q X /S 0 Proof. See [23, (6..)]. Q.E.D. Proposition 2.8 and the morphisms of the exact sequences (.5.) and (.5.2) imply thatwe have an exactsequence for the étale topology on X, 0 q log q /B q n C q /B q n+ 0 (2.9) (cf. [0, (I 5.7.3)]). Using the same morphisms and the equality Im F n W n+ q q =Z n q we also see that for every n and every m 0 we have Ker F Fil m W n+ q W n q p m W n+ q log + Filn W n+ q (2.0) (cf. [0, (I 5.7.4)]). The inclusion (2.0) and the fact that W q log is p-torsion free (.9) imply that, for any n, the map W q log /pn W n q log (2.) is an isomorphism of pro-objects. As a consequence of (2.0), the following sequence is exact for all m n q of 0 W m q log p n W m+n q log W n q log 0 (2.2) Proposition 2.3 (cf. [4, lemma 2]). The following sequence is exact: 0 W n q log W n q F W n q /dv n q 0 Proof. This result is a consequence of (2.0) and the compatibility between F and C, (.) (cf. [0, (I 3.26)]). Q.E.D.

12 258 pierre lorenzon Corollary 2.4. Corollary 2.5. The following sequence of pro-objects is exact: 0 W q log W q F W q 0 The following sequence is exact: 0 q log W q /p F W q /p 0 Proof. This resultfollows from (2.4), (.8), and (2.). Q.E.D. We assume from now on that S = Spec k, where k is a perfectfield of characteristic p > 0. Letus recall thatthe Raynaud ring R is the graded W k -algebra generated by generators F and V (of degree 0) and d (of degree ) with the relations Fa = a σ F av = Va σ FV = VF = p ad = da d 2 = 0 FdV = d for a W k (cf. [3, (I.)]). As in [0], W (.8) is a sheaf of graded R-modules on X, and the projections W n+r W n induce an epimorphism W q W n q with kernel V n W q X + dv n W q X (cf. [0, (I 3.3)]). (One can show this by taking the projective limit with respect to n of the exactsequences 0 W m n q /F n W m q dv n W m q /V n W m n q W n q 0 Hence we have an isomorphism Proposition 2.7 (cf. [3, (II.2)]). isomorphism in D b X W n d, Proof. W X R R n = Wn (2.6) R n L R W = Wn The isomorphism (2.6) induces an This proposition is equivalent to the exactness of the sequence 0 W q F n F n d W q W q dv n +V n W q W n q 0 (cf. [3, (I 3.3)]). The exactness at W q follows from (2.6). The exactness at W q is a consequence of the fact that F is injective (.9). The proof of the exactness at W q W q is the same as that given in [3, (II.2.2)]. (We use (.20.) and (.9).) Q.E.D.

13 logarithmic hogde witt forms INTEGRAL SLOPES ON HYODO KATO COHOMOLOGY In this section, S is the spectrum of a perfect field k of characteristic p>0, endowed with a fine log structure, and X denotes a proper log smooth fine log scheme of Cartier type over S. Theorem 3.. The object RƔ X W X of Db R belongs to D b c R, the full subcategory of D b R consisting of complexes of R-modules with coherent cohomology (cf. [, (2.4.6); 5, p. 9]). Proof. Since we have the isomorphism (2.7) the proof is analogous to that given in [, (3..)]. Q.E.D. As in (loc. cit.), we deduce from this results similar to those of Illusie and Raynaud in [3] and Ekedahl in [5 7] concerning the first spectral sequence of de Rham Witt (the slopes spectral sequence), E qr = H r X W q H q+r X W = H q+r X/W S 3 In particular (cf. [6, III (.)]), (3..) degenerates at E modulo torsion, and H r W q K p q F is canonically isomorphic to the part of slopes in q q + of H r+q X/W S K φ (where K is the fraction field of W k ). Furthermore, using Ekedahl s theory, [7] (cf. also [20]), we obtain the following Mazur Ogus form of Katz s conjecture (a particular case of [22, (8.3.3)]). Corollary 3.2. Let us denote by Nwt m X the Newton polygon of H m X/W S /torsion φ, where H m X/W S is a Hyodo Kato cohomology (.23) of X over W, and φ is its Frobenius endomorphism. Define the Hodge polygon of X in degree m, denoted by Hdg m X, asthe increasing polygonal line starting at (0 0), having slope i with multiplicity the ith Hodge number h i m i = dim k H m i X X/S i, then. Nwt m X lies on or above Hdg m X. 2. Assume that H X/W S is torsion free and that the Hodge to de Rham spectral sequence E qr = H r X q H q+r DR X/S degenerates at E ; then h i m i is the multiplicity of the elementary divisor p i of the W -linear map φ σ H m X/W H m X/W induced by φ. Moreover, Nwt m X and Hdg m X have the same endpoint b m c m, where b m = rk H m X/W and c m = ih m i = lg H m X/W /Im φ i

14 260 pierre lorenzon 3.3. As in [3, IV (3.2)], denote by S parf the topos of sheaves over the perfect site of S, consisting of fine log schemes strict over S whose underlying scheme is perfect, endowed with the étale topology. Denote by X/S parf the topos of sheaves over the relative perfect site of X over S, consisting of pairs T Y where T is in the perfect site of S and Y is a fine log scheme strict and étale over X T = X S T, endowed with the étale topology. The maps Y T define a map f X/S parf S parf. For every n, W n q X extends canonically to a sheaf on X/S parf (also written W n q ) whose restriction to Y is W n q Y, and W n q X log extends to an abelian subsheaf of W n q whose restriction to Y (with Y as above) is equal to W n q Y log. The sheaves Z W n q X and H q W n also extend to X/S parf. We denote by G p n the full subcategory of the category of abelian sheaves on S parf consisting of quasi-algebraic commutative affine groups annihilated by p n (cf. [, (2.5)]). We define G p = n G p n. Proposition 3.3. (cf. [3, (IV 3.2.2)]). For every n and for every q r 0, the sheaf R r f W n q (resp. R r f W n q log of S parf belongs to G p n. Proof. The proof is the same as in (loc. cit.) since, as in the classical case, R r f gr m W n q is representable by a finite-dimensional k-vector space. Use Remark.3. This implies the first statement. By (.6), for R r f W n q log, the general case follows from the case n = using (2.). Then we may conclude by (2.8). Q.E.D. Corollary The exact sequence (2.4) induces an exact sequence of pro-algebraic groups H r X W q log Hr X W q F H r X W q Theorem 3.4. (cf. [3, (IV 3.3)]). (a) The endomorphism F of H r X W q is surjective. Hence we have, by (3.3.2), an isomorphism H r X W q log = H r X W q F 3 4 (b) The group H r X W q log has a canonical dévissage 0 H r X W q log 0 H r X W q log Dqr where (i) H r X W q log 0 is a quasi-algebraic unipotent group. (ii) D qr is a commutative profinite étale group. Proof. The proof is similar to that of (loc. cit.). It uses the fact that RƔ X W D b c R, (3.) (cf. [3, (IV 3.2)]). Q.E.D.

15 logarithmic hogde witt forms 26 For the rest of this section, we assume that k is algebraically closed. The following exactsequences are consequences of (3.4): 0 H r X W q log Hr W q F H r X W q 0 (3.4.3) 0 H r X W q log 0 H r X W q log Dqr k 0 (3.4.4) Furthermore, H r X W q log p is finite dimensional, where H r X W q log =lim Hr X W n q log and we have a canonical isomorphism H r X W q log p p = H r X W q W K F where K is the fraction field of W k. Since H r X W q K is the part with slopes in q q + [ (cf. the remarks following (3.)), we deduce as in [3, (IV 3.6)] canonical isomorphisms H r X W q log p K = H r+q X/W S W K q (where q denotes the slope-q part ). 4. ORDINARY LOG SCHEMES In this section we assume that the underlying scheme of X is proper over k. We denote by k an algebraic closure of k S = Spec k endowed with the log structure induced by that of S. We write X = X S S in the category of fine log schemes. Following Bloch and Kato [3], we say that X is ordinary if for every q 0 and for every r we have (cf. [3, (7.2)]). Theorem 4... X/S is ordinary. H r X B q =0 The following conditions are equivalent: 2. For every r and every q, the canonical map of k-vector spaces H r( ) X q X/ S log p k H r( ) X q X/ S induced from the inclusion q X/ S is an isomorphism. log q X/ S

16 262 pierre lorenzon 3. For every q r, and n, the canonical map H r( ) X W n q X/ S log /p n W n k H r( ) X W n q X/ S is an isomorphism. 4. For every r and every q, the canonical map H r( ) X W q X/ S log p W ( k ) H r( ) X W q X/ S is an isomorphism. 5. For every r and every q, the map is bijective. 6. For every n, q, and r, F H r( X W q X/S) H r X W q X/S H r X BW n q X/S =0 where BW n q X/S is the sheaf of boundaries (.9). Proof. The proof follows the lines of [3, (7.3)]. Letus firstnote thatx/s is ordinary if and only if X/ S is ordinary. Hence we may assume that k is algebraically closed. The equivalence of Statements and 2 follows from the exact sequences 0 q X/S log q X/S C X/S q X/S /B q X/S 0 (which is the case n = 0 of the sequence (2.9)) and 0 B q X/S q X/S q X/S /B q X/S 0 We use an argument similar to that in [3, (7., 7.3)]. The equivalence between Statements 2 and 3 is a consequence of (.6), (.2), and (2.2). The factthat3 implies 4 is clear. Using the isomorphism (3.4.), we see that Statement 4 implies 5. Let us now prove that Statement 5 implies 2. If F is bijective, the inverse limitof cohomology groups of the pro-sheaf W /F (.7) vanishes. We have the exact sequences of pro-sheaves (cf..9) (cf. [0, (I 3.5)]), 0 W q /F V W q /p W q /V 0 0 W q /F dv W q /V q 0

17 logarithmic hogde witt forms 263 (cf. [0, (I 3.9)]), which (as in the classical case) are deduced from (.6) and (.20.3). By Statement 5 we have H X W q X/S /F =0, hence H r X W q /p = H r X q 4 Moreover H X W q /p is of finite dimension over k by (4..) and F is bijective on H X W q /p. Therefore, the long exact sequence of cohomology associated to (2.5) gives an isomorphism k H r X q log = H r X W q /p From the isomorphism (4..), we obtain Statement 2. To prove that Statement 5 implies 6, we use the exact sequence 0 H r X W q / F n H r X W q +V n H r X W q d H r X BW n q V n F n H r+ X W q / F n H r+ X W q 0 which we obtain just as in [3, (IV 4.9, 4.0)], using (.20.2), (2.6), and (.20.). The factthatf is bijective on H r W q and the exact sequence above imply Statement 6. And finally, Statement 6 clearly implies Statement. Q.E.D. Remark 4 2 As in [3, (IV 4.3)], one can prove that Conditions 6 of (4.) are equivalent to the following one: for all q and r, H r X BW q =0 4 2 where H r X BW q =lim H r X BW n q Moreover, when Conditions 6 are satisfied the degeneration of the slope spectral sequence at E follows by the same argument as in [3, (4.3)] and we have a canonical isomorphism (Hodge Witt decomposition), H m X/W S = q H m q X W q Indeed, by (.20.2) and Corollary.9, F is bijective on the pro-object ZW q and hence F is bijective on H r X ZW q =lim H r X ZW n q. n On the other hand, by Corollary.6 and an analogue of (b and c ))], we see that W n q is a coherent W n X -module and BW n q and ZW n q are coherentsub-w n X -modules of F n W n q.

18 264 pierre lorenzon Hence H r X W n q H r X ZW n q, and H r X BW n q are finitely generated W n -modules, and we have a long exactsequence H r X ZW q H r X W q H r X BW q From these two facts, we obtain that (4.2.) Condition 5 of Theorem 4.. If Condition 6 of Theorem 4. is satisfied, the morphism of spectral sequences from to E q r = H r X ZW n q H q+r( X q ) ZW n q q E q r = H r X W n q H q+r X W n induced by the morphism of complexes q ZW n q q W n is an isomorphism. Hence the spectral sequence (3..) degenerates and we have the isomorphism (4.2.2). Theorem 4.3 (cf. [3, (7.3)]). The following conditions are equivalent:. X/S is ordinary and H m X/W S (.23) is torsion free for every m. 2. The Newton polygon Nwt m X (3.2) coincides with the Hodge polygon Hdg m X (3.2) for each m. Proof. The proof is the same as in (loc. cit.). Q.E.D. ACKNOWLEDGMENTS I thank Luc Illusie for having suggested the problems I considered in this paper and for his useful comments. I especially thank the referee for his very careful reading of this paper and the extremely detailed suggestions and comments he made about this work. I also thank Y. Nakkajima for his remarks about the first version of this paper. I am very grateful to Carol Hamer, Emmanuel Kowalski, and Stephan Engelhard for their help in the preparation of this text. REFERENCES. P. Berthelot, Le théorème de dualité plate pour les surfaces (d après J. Milne), in Surfaces Algébriques, Séminaire de Géométrie Algébrique d Orsay, , Lecture Notes in Mathematics 868, Springer-Verlag, New York/Berlin, P. Berthelot and A. Ogus, Notes on Crystalline Cohomology, Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, S. Bloch and K. Kato, p-adic étale cohomology, Publ. Math. Inst. Hautes Etudes Sci. 63 (986),

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