Diagonal Complexes and F-gauge Structures 2705660453, 9782705660451

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,

HERMANN, EDITEURS DES SCIENCES ET DES ARTS, PARIS

TORSTEN EKEDAHL

TRAVAUXENCOURS

(@ HERMANN.PARIS

,

Supported by the Swedish Natural Science Research Council.

ISBN 2 7056 6045 3 © 1986, Hermann, 293 rue Lecourbe, 75015 Paris

Tous droits de reproduction, meme fragmentaire sous quelque forme que ce soit, y compris photographie, microfilm, bande magnetique, disque ou autre, reserves pour tous pays.

CONTENTS

0. Preliminaries and notations I. The diagonal t-structure Hodge-Witt objects The two fundamental filtrations II. F-gauge structures III. Diagonal complexes revisited Diagonal dominoes Dominoes Mazur-Ogus objects IV. Coherent R-complexes The Hodge-Witt numbers Duality V. F-crystals VI. Examples and applications

1 17 23 28 33

55 57

68 69 79 81 88

93 99

VII. Complements

117

Bibliography

119

•

INTRODUCTION

There have .been two attempts to put extra structure on the crysta 11 i ne cohomology of a smooth and proper variety in positive characteristic and giving, in particular, a closer relation with the Hodge cohomology. The first such attempt is Mazur's theory of gauge structures, further developped by Ogus. The second is the theory of the de Rham-Witt complex of BlochDeligne-Illusie. It is the purpose of this paper to continue the study of the algebraic properties of these structures begun in [Il-Ra] and [Ma]. It turns out, in fact, that in a very precise sense the two approaches are equivalent. Hence we will study coherent R-modules, where R is the Raynaud ring, and coherent F-gauge structures, notions which capture the essential algebraic properties of the cohomology of the de Rham-Witt complex and crystalline cohomology with its gauge structure respectively. However, it is not the notions of coherent F-gauge structures and coherent R-modules which are equivalent, it is the categories of bounded coherent complexes of F-gauge structures and bounded coherent R-complexes that are and the equivalence we will in fact construct will not take coherent F-gauge structures to coherent R-modules nor vice versa. This is one of the reasons why we will consistently work with derived categories. Another reason is that a complex contains more information than its cohomology. A striking example of this is the case of the cohomology of a supersingular K3-surface with Artin invariant a . The cohomology R-modules of the de Rham-Witt complex depend only on a , whereas the whole complex determines the periods up to finite indeterminacy and so has o-1 moduli.

Let me give a more detailed description of the contents of the present paper. If we think of the canonical truncations of a complex as giving a filtration of that complex, what the first chapter is concerned with is the definition and study of nother filtration on coherent- ,_R-complexes. More -precisely, in the terminology of [B-B-D] which is eminently suited for our purposes, we will define a non-standard t-structure on the triangulated category of coherent R-complexes. This t-structure will be called the diagonal t-structure. The reason for this is that it filters a coherent R-complex M diagonally; T Mi+1)

are

•

1s0-

>> 0).

To every complex M of R-modules we may associate a complex of F-gauge structures S(M) where S(M) = s(M), the associated simple complex . and the S(M) 1 are the simple complex associated to suitable modifications a la Nygaard of M. One of the main results of this paper then says that S(-) is an equivalence of categories from bounded coherent R-complexes to 00

~

bounded complexes of F-gauge structures with coherent cohomology. It should be noted that this equivalence is used in proving the results on diagonal dominoes and Mazur-Ogus complexes mentioned above which explains why the results in the paper are given in a different order than they are presented in this introduction. The reader will no doubt notice that this paper is concerned mainly with coherent R-complexes and their diagonal and not F-gauge decomposition, most of the results are obtained using the diagonal decomposition . • One of the reasons for this is historical I discovered the diagonal decomposition first and the F~gauge structures appeared at the very end of my work when I tried to prove that every F-crystal was of the form s(M), for M a Mazur-Ogus diagonal complex. It seems that F-gauge structures are a perfectly natural generalization of Dieudonne modules, indeed they have been introduced as such by Fontaine independently of the relation with R-modules, and many results obtained here could no doubt be obtained by starting from F-gauge structures rather than R-modules. There is, however, one important case where this seems to be not true. By modification a 1~ Nygaard at different degrees we get an action of the group of functions 7l > 7l with finite support on R-modules which induces an action on bounded coherent R-complexes. This action preserves diagonal complexes but not F-gauge structures, in fact the diagonal t-structure is in some sense the t-structure closest to the F-gauge structure t-structure preserved by this action. Hence the category of diagonal complexes has a large degree of symmetry which the category of coherent F-gauge structures lacks. The filtration by type on diagonal dominoes mentioned above, for instance, is obtained by applying elements of this group to the filtrations of (I:4.5). 11

11

;

On the other hand, F-gauge structures are more concrete objects and so easier to construct. An example of this is the case of coherent F-gauge structures M with M = 0 which correspond to type O diagonal dominoes. Even for simple examples of such F-gauge structures the corresponding 00

R-complex can be difficult to understand. Consider for instance the case of such F-gauge structures which are indecomposable and killed by p . As will be seen these are extremely easy to describe and it is easy to see that the corresponding R-complexes have the property that their diagonal cohomology R-complexes have the property that their cohomology R-modules are either zero or isomorphic to a degree shift of one of the ~j:s , using the terminology of [Il-Ra], but except for some simple cases I do not know how to determine ·the j:s in question. •

•

•

In chapter IV we will introduce the Hodge-Witt numbers, h~,J , associated to a coherent R-complex M. They depend only on the slopes of the• • 1 1 F-isocrystals H (s(M)) ®w K, K := field of fractions of W , and the T 'J of [Il-Ra] which measure the sizes of the non-finitely generated, as • • W-modules, parts of HJ(M). From the definition of the h~,J it follows •

•

•

that

•

•

hw'J=b n := the rank of H (s(M)) .1C +J=n

equivalent to

1

C

1

•

•

•

(-1)Jh~,J ·

C

•

•

and a formula of Crewis(cf.[Cr])

•

(-1)Jhl,J . The main result concerning the

J J Hodge-Witt numbers is the inequality

•

•

•

•

h l ,J < hl ,J . As an immediate consew quence we get that if M fulfills the conditions of Mazur and Ogus then • • • • 1 hw l ,J = hl ,J and so the T ,J can be explicitly determined by the slopes and . . the Hodge numbers. In particular we get a criterion for when the T1 'J are zero. Another consequence is the Katz conjecture on the relation between the Newton and Hodge polygons ; by definition the Newton polygon lies above the Hodge-Witt polygon, the • Hodge polygon associated to the Hodge-Witt numbers, • • 1 and the inequality hw ,J -~1 . i i i ) VX E D the re i s a di st i ngu i shed tr i an g1e BE ~

1

( A, X, B) wi th AE ~O ,

•

We wi 11 only consider t-structures which are non-degenerate and locally of finite amplitude ,• . e. A) n ~n

= n ~n = {0}

n n B) vx E D there is an

•

n E IN

s. t.

XE ~n n ~-n .

Remark : i) Hence the standard example will be category rather than D(A) as in [B-B-0].

Db(A)

for

A an abelian

ii) It is clear that to specify a t-structure it is sufficient to 0 0 0 specify the abelian category o := -~ n~ in D and we will sometimes do just that. If D and E are triangulated categories and T: D > E a functor we say that T is triangulated if there is given a natural equivalence T(-[1]) °>'" T(-)[1] and if T takes distinguished triangles to distinguished triangles. If D and E are given t-structures we say that a · 0 0 triangulated functor T: D > E has (D ,E )-amplitude, or simply

r'\

L.

o0-amplitude mmn E t1

largest subobject

in

R and

T(-)

r\'1 EA

is an idempotent radical.

0 Definition 1.2. Let .(o,o ) be a triangulated category with a A radical filtration on

Ti + 2c

... c

0

(D ,D )

is an increasing sequence

id of sub functors of the identity on

there is a

t-structure . i i+1

... c T cT

DO s . t .

i) For every ME DO Ti (M) = 0 if i > 0 . •

•

•

•

ii) Hom5- 1 (T-J(M),id/T- 1 (N)) = 0 iii) Hom5-i + 1 ( T- j (M) , i d/T-i ( N))

=

for all 0

for all

i,j i

and

has the required properties.

i,i+1 E [m,n]. By hypothesis we may find

and distinguished triangles T>i+ x 1

>- Z"

> Y'

Y' ,Y"ED-- Z'

>- T< ix

>

and

1 Z' ,Z"E~

> Y"

>

>--. This gives us a diagram Y"[-1]

Y'

Z"[-1]

Z'

I claim that the composite

Y"[-1]

> Z'

is zero and indeed that 1 Hom(Y"'[-1] ,Z') = 0 .• By the induction assumption Y" E r:2-i + n ~O and 1 so by devi ssage we may assume that Y" = T-j (M) [-j] , j > i +1 Z' E ~in

52-

and

Z'

= id / T-k(N)[-k] , k < i

Homj-k+ 1(T-j (M), id/ T-k(N)) - 0 is a morphism

Y11 [ -1]

> ZI

so

Hom(Y"[-1],Z')

by ( 1.2 ii) as

=

j > i +1 > i > k • Hence there

making commutative the f o 11 owing di agram Y"[-1]

>

Y'

5

By [B-B-D: Prop. 1.1.11] there is a diagram commutative up to sign of dis-

tinguished triangles

~>

~>

Y11 [-1]

y1

! -1

II

!z1 !

➔

! I claim that •

YE~O

•

>-

>- T)i+1X

{

J > z

➔

z II

!

➔

J

ZED> 1 • Indeed, Y E ~i

and

>

J

>- X

>- T i and as Y1, Y11 E ~O we see that YE ~O and s i mi l a r 1y Z E-rr2. 1 . Fi na 11y , Y' ,Z' E D[m,i] and Y",Z" E D[i+ 1 ,n] so it is clear that T x -and by shiftin; we get that 1 1 1 1 - -

(o,o

0

and

)

(D,'ff°)

(o,o'°) commutes with (o,o 0 ) then clearly Ti . 1 H (X)[-i]

commute with

Hj (-)

iff 'i< (H 1 (X)[-i]) -

XE o 0)

10

r.J

Note that then TO

deg 0 •

where

•

dV- 1 = F1 d

i

deg 1

> 0 -

• •

1 • It is proved in [Ek 2] that ME 0-(R) is coherent iff H (M) 1s a successive extension of degree shifts of elementary R-modules for all i . It is also clear that a coherent R-module M with 'i' M [1] in D~(R-cut-I). 2 1 Remark : We will not particularly distinguish between D~(R-cut-I} and its essential image in Db(R) and hence we will say that a complex whose cohoc mo 1ogy i s cut at I i s cut at I • If I 1s an intervall of 7l then we denote by R-mod-I the category of R-modules of level I . The proof of the following result is then altogether similar to the proof of (5.1). •

Proposition 5.2. The f unctor

D~(R-mod-I)

> D~(R)

is full and faithful

wi t h essential image t hose complex es whose cohomoZogy is of level

I •

14

Remark : The same sort of abuse of language will be used here. Recal 1 also that if

ME D(R)

then considering

M as a double complex

we get the slope spectral sequence : •

•

•

•

•

> H1+J ( s ( M) ) •

E~ 'J = HJ ( M) 1

(5.3) 6. Let

•

ME D~(R). Then we put

hi 'j ( M) : = dim Hj ( R 1 •

•

•

®

~ M) i

•

•

•

1 J J 1 - oo J 1 T ' (M) := dim H (M) /(V ZH (M) +V)

( 6. 1)

where

K := Frac(W) = W ®~ ~ .

Then

.C

Proposition 6.2. i) bn(M) =

•

•

ml'J(M) .
M/M.

M •••

1

~

> M/M.

1 +1

~

• • .c: D (M/M.) c: D (M/M. -

l+ 1

1 -

is such that

> •.• ) c:...

'o°(-) M/M.,

0

we get

0

of subobjects of

0 - t (~1/M., )

0

0

and

for

1

t (M/M. ) = t (M/M.) 10

to

This gives us an ascending sequence

IT°(M/M . ) = 'B'°(M/M.) 10

'o°(-)

1

i

i-2_ ;

> i 0 -

M/M., = tO(M/M.) , .

..

0

~

D (M). Hence there is some

and by applying once again

but by assumption

22 •

Theorem 1 • 12. Let

ME & •

i)

1

H (M) ( i)

M is finite. tor sion iff

length fo r all · i

is a Dieudonne module of finite

M= e Hi ( M) [ - i ] • 1

and then

•

ii) M is without finite torsion iff

fo r all

i

and if

a torsion f r ee

1

is without finite torsion

H (M)(i)

00

MER-mod n & it is without finite torsion iff

V- ZMO is

W-module .

Proof : If we can prove that when

r

M is finite torsion and

Hr(M) = 0

for

0

(M)(r ) is a Dieudonne module of finite 0 0 0 length then we have proved the only if-part of i). This is because then r >r

for some

r

would be cut at

then

{r } 0

so

H

r

M = T 7l has (!)(0) < -i 1 then H (~(~i ((I))))-" 0 so HO(s(D(Ui) (-(1)-))-" 0 which by ( 1.4 i) and 1 (1.4.1 i) contradicts the assumption that D(U.)=D (U.)[-1]. Hence M is ·

.

=l

=l

a Dieudonne module. If

M is not of finite length then M has a quotient 0 which is a torsion free Dieudonne module. Then H (s(D(M)) = Homw (M,W) would 1 be non-zero again contradicting D(M)=D (M)[-1]. Conversely if M is a (!) : 7L

Dieudonne module of finite length then for every s(D(M)((!))) = RHornw(M,W) i.e. equal to

o (M)[-1] 1

so (1.6) shows that and by (1.8)

D(M)

M=D(D(M))=

>- 7l

M((!)) = M so

is of amplitude [1,1]

o (o 1

1

1

(M))=t (M). By

devissage and shift we get the if-part. Now for ii) we may assume that Hr(M) = O for

r >r

0

and as

T N

Nc

=

> M N is 1 H (,

without finite O

Ho(N) = H0 (t 1(Ho(M))) so by i) N is cut at O so by (0:5.1) 1 0 T>Ot (H (M)) is a direct factor of N so it is at the same time finite 1 torsion and without finite torsion and hence zero. Finally t (HO(M)) is a subobject of HO(M) ER-mod n /1 and by LeIT111a 1.5.1 this implies that 1 1 1 t (Ho(M)) ER-mod n Ll and so O = '>at (Ho(M)) = t (Ho(M)). Let ME &n R-mod. By Lemma 1.5 any subobject of M in Ii is in R-mod n & and if it is a Dieudonne module it is a subobject also in R-mod. By (1.5.1 ii) if NE R-mod n /l is a Dieudonne modu 1e and N > M is a monomorph ism in R-mod then it is also a monomorphism in d. Hence by i) M is without torsion iff it does not contain a sub-R-module N which also lies in ll and is finite torsion as such and those N are exactly the Dieudonne 00 modules of finite length so M is without finite torsion iff V- ZMO is torsion free it containing all sub-Dieudonne modules of M. The only ifpart of ii) is clear as extensions of objects without finite torsion are without finite torsion. Hodge-Witt objects 2. We begin by showing that the diagonal Hodge-Witt objects form a nice category. Let ¾w denote the category of diagonal complexes which also

24

are Hodge-Witt complexes, i.e. cut at

¾w

Proposition 2.1.

~.

is closed under sub and quotient objects and exten -

• si-ons .

Indeed, extensions are clear as the Hodge-Witt complexes form a triangulated category. Let ME ¾w and N a subobject in L\ • As was seen in •

•

•

•

•

•

1

the proof of Lemma 1.5.1 H1 (N)- 1 < > H1 (M)- 1 and so H1 (N)is a fini• • • • 1 1 1 1 which implies that VZH (N)= H (N)and tely generated W-module • 1 H (N)(i) is a Dieudonne module. Again the long exact sequence of Lemma • 1.5.1 shows that now H1 (M)- 1 > H1 (M/N)- 1 is surjective so the same 1 1 argument shows that H (M/N)is a Dieudonne module. 00

•

•

•

•

•

Prop os i ti on 2 . 2 . Let ME ti i) PM E ~w iff PM : = Ker p : M ii) ME for a ii

¾w

ME

iff there exists a constant

¾w . C such that

1 l gth ( H ( s (M(w))))

~ •

7l.

Indeed, only if of i) follows from (2.1). Suppose that PM E¾w and choose r such that Hr(M) = 0 if r > r 0 . Then T 0 and (2.1) shows that

pH

r0

- 0

(M)[-r ] 0

is Hodge-Witt so by shifting we may assume that

MER-modn/1. Lerrma 1.5.1 shows that

PMER-modn/1 and that

0

P(PM) = p(M

so we are reduced to showing that JM 0 ) of finite length implies that v- zMO = MO . Look however at the exact sequence 00

0

)

25

00

and apply snake lemma for multiplication by p . As V- ZMO/p is of finite length we see that M~V- ZMO is of finite l~ngth but if MO/V- ZMO is p non-zero it contains k[[V]] as a submodule. For ii) the only if is clear as M((l)) = M if ME ~W . In the other direction we may suppose again that r ro H (M) = O r > r 0 . If we can show that H (M)[-r 0] is Hodge-Witt then r M = T H +J ( s ( M((l)) ) )

is a quotient of H1{s(M{w)))

so our

r

conditions are fulfilled for H O(M)[-r 0J and by shifting we may assume that MER-mod n /1 but if M then is not Hodge-Witt it has a quotient iso1 morphic to Ui so H (s(Ui(w))) would be bounded which is clearly false. Corollary 2.2.1. An ME D~{R) is Hodge -Witt iff the length of the torsion of H*(s(M((l)))) is bounded independently of (l). This follows from Cor. 1.4.1. By (2.1) and (0:1.1) we see that for any ME& there is a largest Hodge-Witt subobject which we will denote HW(M). Then HW(-) is an idempo• tent radical. We then have •

Proposition 2.2.2. HW(H 1 (-))

•

=

H1 (HW(-)). •

•

Proof: It will be enough to show that if ME&. then H1 (HW(M)) > H1 (M) is injective and that if HW(M) = 0 then HW(H 1 (M)) = O . First 1(N) > Hi(Hi(HW(M))[-i]) Hi(HW(M))[-i] c > M with quotient N . Then Hi• H1 (HW(M)) is zero as Hi- 1(N) is generated in degree i-1 and Hi(HW(M)) is• concentrated • in degree i . By the long exact cohomology sequence 1 H (HW(M)) > H1 (M) . is injective. .

=

26

•

Let now HW(M)

=

1

0 • Now HW(H (M))[-i]

c

> -r >i~1 . Let

•

1 nverse

N be the

•

•

1

1

•

in M of HW(H (M))[-i] • Then N is cut at {i } so HW(H (M))[-i] direct factor of N and so a subobject of M and hence O .

lS

a

With the results obtained so far we can give a new proof of 11 survie du coeur 11 as well as a generalization of [Il 1]. Proposition 2.3. Let

•i)

ME D~(R)

(survie du coeur ) In the spectr al sequence •

•

==>>Hl+J(~(M)) •

1

C -

Z

•

,J

•

00 •

•

•

ii) In the same spectr al sequence the tor sion of FilJHl+J(s(M)) maps onto • 1 1 the tor sion in V- ZHJ(M) /B 'J . 00

•

•

•

00

,-..J

Indeed, we have a mo. rphism -r . . 1M > M which induces an isomorphism ' 1 +Jk · 1 k i H ('i' H (M) if k+! < i+j and an injection if k+i = i+j whose 1 +J- 1 image, when

k+t = i+j

is

00

F BHk(M/' • The inclusion

~

B~'j •

1

00

F BHj(M)i •

00

•

now •

1

follows by functoriality of the spectral sequence. That Br 'J c- F BHJ(M) follows from repeated application of the following useful lemma which shows 1 1 1 1 . . M) c Ek ' (M) that Ek' ( -r . . M) = Ek ' ( M) if k+1 < i +J· and Ekr ' ( -r v· as a double complex and consider the two spectral sequences thus obtained or chase diagrams.

27

A

For the other inclusion consider the t-structure D associated to the idempotent radical HW(-) on & . Then -i M induces an isomerphism •

V- ooZH J ( -r . .

M 1 +J- 1

A

t . 1M 1 +J-

k ( H L· A

)

a

=

if 00

• )

1

00

•

--..> V- ZHJ(M)

•

1

and

k+t> i+j . •

1

•

•

1

•

As for ii), F BH (M)J/B 'J is taken care of by (1.4 i) and (1.4.1 i) by considering 'i'< . . 1M > M . For the rest we reduce using ( 1.4 i), 1 +J1 ( 1.4. 1 i) and i) to 'i'>. .M and then trivially to H +J(M) so we may 1 +J 1(M) --..> M is onto on the assume that ME/! . Theorem 1.2.2 shows that t • torsion of H1 +J(-) so we reduce to t 1(M) but it is Hodge-Witt so there is nothing to prove. 00

•

•

We are now almost prepared to give the local version of the Hodge-Witt decomposition. What remains is only to give a generalization of the notion of exotic torsion (cf. [ I 1 1]) . Def i n i ti on 2 • 4 • Let

i)

The exotic

H1(s(Hi- 1(M)))

ME D~ ( R) .

1-tor sion in degree in Hi(s(M)).

ii) The -f init e total tor sion in degree iii) The exotic a-torsion in degree Ha(s(t 1(Hi(M)))).

i

of M is the image of

i of

M is

Ha(s(ta(Hi(M)))).

i of M is the torsion of •

In this way we get a 3-step filtration of the torsion of H1 (s(M)) whose quotients are respectively the exoti c 1-torsion, the finite total torsion and the exotic a-torsion. If N= D(M ) then the formul a ~(D(-)) =RHomw(s(- ) ,W) shows that the torsion in degree i of s (M) is dual to the torsion in •degree -i+1 of s(N ) and it is clear that the filtrations just described are dual to each other . Finally, if X is a smooth

28

and proper variety then the divisorial and exotic torsion of

H~ris(X/W)

,n •

the sense of [Il 1:II 6.7] equals the total finite torsion and the exotic a-torsion respectively. •

•

Theorem 2.5. Let

ME D~(R)

and suppose that

1

d: H (M)J

zer o . Then •

•

The quotient of Hl+J(~{M)) decorrrposition ~i e H>i . -i)

by its exotic

k+r , £-r+1 ~>- E ii) The differ entials r • • ====,>► H1 +J ( s (M)) are zero when k+£ = i +j

1-torsion

admits a canonical

of the spectral sequence i -r •

Hi(M)j+

1

zero whenever

i+j = r is Hodge-Witt in degree r . We then get a Hodge-Witt decomposition for Hr(s(M)) modulo exotic 1-torsion for all M which are Hodge-Witt in degree r. Remark : This definition of Hodge-Witt in degree r differs from the one used by Illusie and Raynaud in [Il-Ra]. Their "Hodge-Witt in degree r" is equivalent to our "Hodge-Witt in degree r and r-1". I suggest the present definition because it has the desirable feature that M is Hodge-Witt in degree r iff Hr(M) is Hodge-Witt. From our point of view the only role played by the assumption "H-W in degree r-1" to insure a H-W decomposition in degree r is that it excludes the existence of exotic 1-torsion in degree r. Many other assumptions will give us that. The two fundamental filtrations

3. One important invariant of a diagonal complex is the associated simple complex. We start therefore by dividing & in classes according to the behaviour of s(-).

29

Definition 3. 1. An ME /1 is said to be i)

without · s-tor sion if

ii)

~-tor sion

-

if

iii) ~-0- torsion

H*(~(M)) is without torsion

H*(~(M)) is torsion . (s-1- torsion ) if it is

~-tor sion

and

-

1 H ( ~ (M)) = 0

( HO ( s ( M) ) = 0) •

-

iv)

s-acyclic

if

s(M)

=

0 •

The following lemma is then crucial Lemma 4.2 i) The

s-tor sion

objects are closed under sub and quotient

obj ects and ex tensions . ii)

The

-s-1- torsion -

objects ar e closed under extensions

(s-0- torsion )

and sub (quotient) objects .

iii) s-1- torsion

objects are without finite torsion . •

iv) all

has

An element •

l

s(M)

=

0

iff

1

H (M)

•

1,,S

s-acyclic

•

Proof : Proposition 1.4 shows that if 0 > M1 > M2 > M3 exact sequence in & then we get a long exact sequence ( 4. 2. 1)

for

0 0 0 0 > H ( s ( M ) ) > H ( s ( M ) ) > H ( s (M ) )

2

1

1 > H ( s ( M )) 3

3

,s an •

1 1 >- H ( ~ ( M ) ) > H ( ~ ( M2 ) )

1

> 0

1 and H (s(M)) is torsion for all ME&. From this i) and ii) follows immediately. Let M be s-1-torsion. By ii) t 1(M) i s s-1-torsion but by (1.12) t 1(M) is Hodge-Witt so HO(s(t 1(M )) = 0 implies t 1(M) = 0 . This gives iii). Finally, iv) follows from Corollary 1.4.1. The notion opposite to objects.

s-acyclic

objects is the notion of Mazur-Ogus

30

Definition 4.3. A diagonal complex M is said to be Mazur- Ogus if Hom&(N,M) = Hom&(M,N) = 0 for all s-acyclic objects N . The category of Mazur- Ogus objects will be denoted ~O . Also for Mazur-Ogus objects is there a crucial result. Lemma 4.4. Suppose without

ME ~O . Then

L

i

H ( R1 ®R M)

j

=

i + j .c O and

0 if

M is

s-tor sion . •

Su ppose H= 1 • C1ear 1y U~ .

i

L

i s s - acy c 1 i c for a 11

i . By ( 0 : 4 • 3 ) we get

j

that H (R ®RM) = 0 if i+j = 1 or -1 and combining this with (1.4 ii) 1 . L . 1 we get H ( R ® R M) J = 0 i f i +j .c O • The spectra 1 sequence ( cf . ( 0 : 6 • 2 • 1) 1

Hi ( R 1

®~

>Hi +j ( k

®~

O and the triangle

degree s(M)

M)j

s (M))

>

is concentrated in degree

shows that

s(M)

p >

s(M)

k

®~

>

s (M)

is concentrated in

L

k ®w s(M)

O and that HO(s(M))

>

shows that

is torsion free .

•

When

H.t 1 we replace

U~

We have now come to the major result on the structure of ~. Theorem 4.5. For any ME& there exists two (canonical) functorial filtra tions characterized by i) r esp . ii) 2 3 1 (4.5.1) 0 c F c F c F = M (4.5.2) such that i)

F

1

largest 1·1·)

is the largest

s-0-torsion

subobject of M and

3 1 s-1-torsion quotient object of F ; F .

G3/G 2

s- 1- torsion quotient ob ject of 2 the largest s-0-torsion subobject of G . iii) G1 = F1 n G2 ; F2 = F1+G 1 . · h e .&argest ., ~st

31

iv)

2 1 2 1 F /F = G /G is Mazur - Ogus and

v)

2 1 2 1 1 1 F /G = F /F 61 F /G . The existence of

1 1 2 2 F /G = F /G is

2 1 1 F , F , G and

2 G

s-acyclic .

follows from (0:1.1), (1.5)

(1.11 iv) and (4.2). Now, v) follows from iii) so we are left with iii) and 2 1 2 1 iv). Let us first show that F ;F and G2;G 1 are Mazur-Ogus. If F ;F is not then there exists some s-acyclic N and either a non-zero morphism 2 F2;F 1 > N or a non-zero morphism N > F ;F 1 . The image of these would, by (4.2), be

s-1-torsion -

the definition of

2 F

and

s-0-torsion -

respectively which contradicts

1 resp. F . Similarly for

Let us now show that By (4.2) Im(F 1 > G3;G 2 )

1 2 1 F /G n F is

is

2 1 G ;G .

~-acyc 1 i c

s-0-torsion

and

and that

s-1-torsion

1 2 1 G =G nF . i.e.

1 2 1 · s-acyclic. This image is isomorphic to F ;G n F which therefore is 2 1 1 1 2 s-acyclic and hence s(G nF ) > s(F ) is an isomorphism so G nF s-0-torsion. By the definition of nition of

1 G we have

1 1 F , G cF 1 and by definition

1 2 1 G nF c G

1 2 G cG

so

is

but by the defi-

1 2 1 G cG nF and con-

2 1 2 1 1 1 2 2 sequently F /F = G /G and F /G = F /G 2 1 follow from iii) and as we have already proved that F ;F is Mazur-Ogus 1 1 2 1 3 2 1 1 and that F ;G = F ;G n F = Im(F > G ;G ) is s-acyclic we are left with the equa 1 i ty F2 = F1+G 2 or what amounts to the same that G1 = G2 n F1 . As the equalities

2 1 G ;G

2 1 3 2 > F ;F is an epimorphism. Now, G ;G is s-1-torsion and by 2 1 lemma 4.4 G ;G is without s-torsion. Hence by (4.2.1) and (1.4)

0 3 1 1 0 2 H (s(G ;G )) = H (s(G ; G ))

3 1 1 and it is torsion free and H (s(G ;G )) = 1 - 3 2 = = 0 H (s(G /G )). Using (4.2.1) one more time we see that tors-H (s(M)) = H0 (s(G 1 )), H0 (s(M))/tors= H0 (s(G 2;G 1 )) and H1(s(M)) = H1(s(G 3;G 2 )). In the same way tors-H 0 (s(M)) = H0 (s(P 1 )), HO(s(M)) / tors = H0 (s(F 2;F 1 )) and 1 H (s(M)) = H1 (s(F 3;F 2 )). In particular we see that G2;G 1 > F2;F 1 induces 0 isomorphisms on H (s(-)) and on H1(s(-)) (both are zero in that case)

32

so s(G 2;G 1) F2;F 1

2 1 1 2 > s(F ;F ) is an isomorphism and the cokernel of G ;G 2

is

1

s-acyclic and hence O as F ;F is Mazur-Ogus. Let us also write down the formulas obtained in the last part of the argument. Corollary 4.5.3. 1 0 H (s(F ))

-

·

tors-H 0 (s(M)) = -

-

1 0 H (s(G ))

--

1 2 0 0 1 0 2 H (s(F ;F )) = H (s(M))/tors = H (s(G ; G )) 3 2 1 1 3 2 1 H (s(F ;F )) = H (s(M)) = H (s(G ;G )) . -

>

II

F - gauge structures

1. As we have seen, if ME&. then s(M) is almost but sometimes not completely concentrated in degree O . By changing & a little we can get the associated simple complex to be concentrated in degree O . Define a functo. . 1 1 1 r i a 1 f i 1tr at i on on &..• c M c f~ + c ... M by ~1 = O i f i < O , M = M ; f - 2 i > O and MO = G (cf. Thm. I : 4. 5). By ( O: 1• 1) and (I: 4. 2) MO is an idempotent radical. Definition 1.1. Let (~O,~O) be the t-structure associated to the radical filtration Mi and G:=~On~O, Tg i the G- truncations and Hl ( - ) : = T-g < . OT g > · ( ) [ i ] . g 1 1 Proposition 1.2. Let ME Db(R). Then M has G-amplitude [m,n] iff s(M) C . L . 1 has amplitude [m,n] and H ( R1 &R _M~ = 0 unless m-2 < i +j < n . •

Proof : We may assume exact sequence

H= 1 . Let first

MEG . By construction we have an

1 such that ~(M) has no ~-1-torsion quotients and H- (M) is s-1-torsion. 1 By (I:4.5.1) H (s(~(M)))- 0 so we see that s(M) has amplitude [0,0] . - ';';'() . Furthermore, by (I:4.2) the image of any morphism H (M) > U~ would be ~-1-torsion and hence zero by assumption. This together with (0:4.3) shows that

Hi (R

Hi (R

®~

1

®~

'R'° (M) )j = 0

M)j = O if

1 Suppose now that

if

i +j = 1

and then (I: 1.4 ii) shows that

i+j"' -2,-1,0 • Devissage now gives the only if-part.

~(M)

has amplitude

[m,n]

and

Hi(R

1

®~

M)j = 0

unless

34

m-2 -< i +j - u0 is an isomorphism. Now (I:4.2 iv) shows that the s-acyclic diagonal complexes form an abelian subcategory of /1. Then (I :1.11), (I:4.2 iv) and the fact that s-acyclic objects are a fortiori s-1-torsion imply that

35

the s-acyclic objects form an artinian category so being both artinian and • noetherian every object has finite length. We have seen that U~ are simple objects. Let M be some simple object. To show that it is isomorphic to • • • some ~~ it suffices to show that Hom&(U~,M) or Hom&(M,U~) is non-zero for some i . If this were not the case then by (0:4.3) Hi (R

1

s~ M)j = 0

for

would degenerate so

i+j = -1

R 1

&~

or

1 and by (I: 1.4 ii) the spectral sequence

M would be zero and (0:4) would give that

M= 0

contrary to the assumption that M is simple. [m,

00 [

We have thus proved that the M of the proposition has G-amplitude In exactly the same way we prove that M has G-amplitude ]- ,n] . 00

•

• Corollary 1.2.2. i ) ~(-) 1.-S G-exact ii) (-) ;LR (-) has G-amplitude [-2,0]

I

i i i ) RHom·R · ( - , - ) has G-amplitude

[0,2]

•

Indeed, i) is obvious and ii) and iii) follows from (0:3.3, 3.3.1). 2. Our purpose is now to show that cription.

(~o,~o) admits a more concrete des•

Definition 2. 1. i) An

1

structure is a graded W-module M= e M i Ellti-ve . ly t oget her with linear mappings ~ F and ~V of degree 1 an d - 1~ respe£ 00 ~FV ~VF d l . · h · M · ( . • . F>- Mi F>s.t . = = p an a a- i-near i-somorp i-sm T : : = 1lg! 00 Mi+ 1 • i i-1 ---->- ••• ) > M : = 11 m( . . . > M >- M >- • •• ) • >-

r

-

F-gauge

v

v

v

·

ii) Let I be an interval of 7l. An F-gauge stru cture is of level I if 1 for every n E 71.. be 1ow I V: Mn+ >- Mn is an i somor phism and f or every n 1 above I F: Mn> Mn is an isomorphism. If I= [0, n] nE1N we will al so say that the F-gauge structure is of level n .I f I = [ O, oo ( we will also say that the F-gauge structure is ef fective.

36

iii) The category, with the evident morphisms, of F-gauge level I) will be denoted F-g-str(-1).

structures (of

Remark: i) It may be more reasonable to require instead an isomorphism • • 1 1 -r : "li~"(M ,F) > "lig_!"(M ,V). We will only consider F-gauge structures of finite level where there is no difference . •

ii) Let

I

= [ m, n]

• On

M=

$

1

M

iEI •

1

by F : M •

':':'n - 1

F and

we may define endomorphisms -1

•

00

and

> M

1

V: M

T

> M

V

Mn >

r,.J

. 1

V > M.In this way we give M a structure of module over the W-ring n-m 0 Fa=a° F aV=Va generated by F and V and rel at ions FV =VF= p for a E W . Just as it sometimes is convenient to regard R-n1odules rather than R-modules it is sometimes convenient to consider F-gauge structures of level I where M also has been given a structure of module over the p-adic completion of this ring. If in all that is to come we replace R by R and add this extra condition to our F-gauge structures then all our results will remain true with only trivial modifications of the proofs. Examples : i) A virtual F-crystal is a triple (U,F ,N) where U is a finite dimensional K-vector space K:= Frac(W), F a o-1 inear automorphism of U and N a W-submodu 1e of U of maximal rank (i.e. KN = U). We wi 11 say that the virtual F-crystal is of finite type if N is a finitely generated W-module. Suppose now that M is a torsion free F-gauge structure such - oo that M ®wK is a finite dimensional K-module. The relations FV = p shO// r,.r,.,J

,

00

•

r,.J

0

- oo

that M ®wK = 1im(M ® K,F) = M ® w K and similarly for M ®w K • Hence > - oo - oo we get a virtual F-crystal where U: = M 6)W K ' F lS M ®w K = •

0

•

• Ml of images K and N = M C > . The K = M ® K M > M ®w M ®w ®w K w • ,n u form a decrea s ing filtration of u with the properties 00

1"

- oo

- oo

- oo

37

•

1

b) N=U• M 1

•

•

•

into and onto

c) F maps

1

u• M

•

1

Conversely, any virtual F-crystal (U,F,N) with a decreasing filtration • • 1 1 "' fulfilling a)-c) gives an F-gauge structure by putting M = M , V the "' inclusion, F multi pl i cation by p and T the isomorphism obtained by condition • 1 c). Furthermore, the virtual F-crystal is of finite type iff t; M is of finite type in which case u• p-iMi is also of finite type and t~e level is finite . Conversely, if the ievel is finite and one (and hence all) of the • M1 are of finite type then the associated virtual F-crystal is of finite type. .:

"' "'

The forgetful functor (M,F,V,T) I > (U,F,N) has a right adjoint. -1 1. 1. Namely we put for (U ,F ,N) a virtual F-crystal M = (F (p N)) n N . Then 1 1 {Mi} is decreasing, u Mi = (F- ( v piN)) n N = F- vn N --N and F( u p- i Mi ) -i

= F( F- 1Nn ~ --

i

p- i N) - F( F! 1N n V) --

=N

•- Furthermore

{Mi }

is

clearly the largest filtration fulfilling a)-c) so we get our right adjoint. We will denote this functor Hodge(-) and call the obtained F-gauge structure the Hodge F-gauge structure associated to a virtual F-crystal. 1 Note that the Hodge F-gauge structure is of level I iff p-n- -FNcN for a11 n below I and p-i +1 -FN -=> N for a 11 i above I . (This is 1eft to the reader). In particular a virtual F-crystal of finite type whose Hodge F-gauge structure is effective is just an F-crystal in the ordinary sense. ii) An . -oo 0 M =M

F-gauge

structure of level 1

00

and M = M

so we can put

1 is just an "'

R0-module. Indeed,

F> M1 = Moo T> M-oo = MO

an d

-1 "' T > M oo = M1 V> MO • I n particular we can regard Dieudonne

modules as a subcategory of F-g-str.

38

F-gauge

iii) Consider an

W-module. Then

as a co

M

- co

=M = 0

may be identified with

Mn

V': Mm and

I • The

w

so

1

➔ End(M). As we shall see these

F-gauge

M

•

m M1

I = [m,n]

structures of level

together with maps

M' =

m ' V' ~> M

F ' , V'

•

s-acyc 1 i c ( diagonal comp 1exes.

Fi x some finite level

m< i

•

lS of finite length

M

such that

is zero for all but a finite number of

I

so we get a homomorphism

correspond to iv)

r~

structure

~1

Mn

r~m

1

wi th

except for F ' : Mn > Mm wh i ch i s a -linear and n 1 o - -linear and F'V' = V'F' = p except for V'F' : Mn ~> M

> Mm which are the identity. Whenever convenient we will

F'V': Mm

use this description. v)

I= [m,n] • The functor

We continue to fix

Tr (M) = Mr

is represented by a necessarily projective

~

Gr. An explicit description is given as follows

F-gauge 1

G

r

iEI CWV'sa

sElN

As

where

r

{ Tr(- )}

a r E G~

and

➔ Mr

Hom(Gr,M)

3 . We will now to an

R-module

=

Gr

for

given

structure

r, WF'sa

ssf{

r

rE I

form a

F-g-str-I •

associate a complex of

F-gauge

structures

(cf. [Ny] ) • Definition 3.1. i) Let

F-gauge

complex of •

a) S( M)

1

.• --

1

s(M((p ))

•

b) S(M)

1

•

• • •

•

Fi S(M ) i +1

M be an

• • • • •

R-module . Then

S(M)

•

1..,S

the following

structures

•

•

with (pl(n )

d i- 2 > o*M

Ii- 2

d > o* M

d

+

(p I> (p( ar).

is gjven by

is a conservative set of functors the

set of projective generators for

> Ab

Tr: F-g-str-l

= - Q•

,n

>-- o *M

i-1

Ii -1

d > o*M

•

~

•

Ml

tF d

> o*

d

>--

Mi +1

d >

• • •

d >

• • •

{P

Mi ~ M1+1

39

•

S(M)

1

•

• • • •

d

> a*M

d

i-3

i-2

> a*M

d

i -1 ~ Mi > a*M

d

>

• • •

,-..J

ip

vi S(M)i+1 c)

S(M)

00

=

•

• • • •

d

> a*M

a*s(M), S(M)

iv

i-3 d

- s(M)

and

•

d Ml >-

i-1 M

> a*M

- oo

II

d

>-

•••

is the identity.

T

ii) Let M be a complex of R-modules . Put S(M) sums) associated to the double complex S(M).

the simple complex (using

We have thus a functor from complexes of R-modules to complexes of F-gauge structures. This functor clearly preserves quasi-isomorphisms and mapping cones so it extends to a triangulated functor from D(R) to D(F-g-str). The formulas used to define F and V also show that it takes D(R-I) to D(F-g-str-I) for any interval I . ,-..J

,-..J

Definition 3.2. Let

ME D(F-g-str-I) for some finite interval I . The completion M of M is R~im{W/pn ®~ M} with its obvious structure of complex of F-gauge structures of level · I . An ME D(F-g-str-I) is said ...

to be complete if the canonical morphism ...

One shows without difficulty that ~1 = Proposition 3.3. The functor

~: D(R-I)

> M is an isomorphism .

M ... -M

>

.

D(F-g-str-I)

comrnutes with com-

pletion and hence takes complete complexes to complete complexes .

Proof : This follows from

s((-)) = s(-) -

-

...

and

~

( - ) ((p) = ( ( - ) ( (p) )

( Cf . ( 0 : 2 . 1)).

Proposition 3.4. Let ME D(R-I). Then there exists a co"rmutative up to sign diagram of distinguished triangles

40

! S(M)

>

•

F >

1

+

> t >i+ 1Hod(M)

!

iv .

,...,,

F >-

S(M)i-1

>-

1

!

1

,...,,

iv (3.4.1)

i . S(M)

,...,,

S (M) -

1

> t >iHod(M)

> o*T< iHod(M)

> o*T ( R1 ®LR.·.M) ~

>-

!

!

1

>

i

!

!

>

where the lowest and the right most triangles are those of (0:3.4). To define (3.4.1), we may assume that M is R1 - and z1 - acyclic and that p is injective on M. We will define our morphisms already on the level of double complexes so we may assume that M is just an R-module. We then define

0

\

t0 •

and

S(M)

1

/V

>

--.>

iM / VM .+dVM . 1

1

1

> •••

T- o*Mi- / P

d • • •

d • • •

>- o*M

!i-1

~ M; / V

>

0

JF •

d> z M1 / V-td V 1

> 0

•

41

To show that the triangles are distinguished we reduce by a limit argument to the case where M is bounded in all directions and then by devissage and shift to M= R in which case we compute directly (or use the argument of [N:1]).

Corollary 3.4.1. The functor

S: D(R-I)

> D(F-g-str-1), I

a finite inter-

val , is conservative on the category of complete complexes . L

•

Indeed, if S(M) = 0 then we have seen that complete this implies that M= 0 (0:4).

R1 ®RM= 0 and as

M is

4. We will now go in the other direction. Starting from an F-gauge structure we will construct a complex of R-modules. Recall that C(R-I), I a finite interval, denotes the abelian category of complexes of R-modules of level I . The functor z 0s(-): C(R-I) > F-g-str-I taking an XE C(R-I) to Ker D: S( X)O > S(X)O , where D is the total differential, commutes with inverse limits and hence has a left adjoint t . Here is an explicit description of t . We let o denote the differential of an R-complex. Definition 4.1. Let ME F-g-str . Then generated by the

-,

•

complex degree

S-l e m

t(M)

is the

R-complex

of level

S- ® m in module degree

i E I , mE M with

l

•

l

and with relations •

i)

V\ E W

m ,m E MJ 1

2

Si

® (

\ m1+m2) - \(Si

®

m1)+ S;

0 -1 Si

® (

\ m1+m2) - \

(Si

®

®

m2

m1) + Si

®

m2

if

i

>j-1

if

i

i-1 •

i,n

l

•

=

• - oo

equals

M

applied

T

J

Sr ®m.,

=

Sr ®m J. .

I leave to the reader to verify that

ml> r . s. em ,IT l

is the adjunction unit for an adjunction between Lemma 4. 2. For al l

r E I = [ m, n]

t(G ) r wher e

R( S;

®

ar)

=

R-resp .

r e sp .

W[F,F- 1]-module on a

Si® ar r esp. Sn® ar .

Proof: The relations in ii i ) and iv) show th at

t(G ) r

=

C F-g-str

Now, which is right adjoint to the inclusion. As S(-) commutes with inverse limits it also has a left adjoint T: C(F-g-str)) > C(R-1)). On the other hand we can extend t as ,usual to complexes, we take the associated simple complex of the natural double complex. For this extension we get a natural transformation id > S(t(-)) hence getting by adjunction T > t . There are many ways of seeing that this is an equivalence. One is to reduce to bounded complexes as both commute with direct limits, then using the fact that Toincl : F-g-str > C(F-g-str) > C(R-1) equals t by transitivity rf adjoints and then, starting from this argue by induction on the length of the complex X. We have O > t >iX > X > t 0 which gives exact sequences T( t >i X)

> T( X)

i

i 0

By induction

> t(t>iX)

T(t>iX)

> T( t

> t(t>iX)

> 0

> t(t 0 .

is an isomorphism so

T(t>iX)

>

T(X)

is a monomorphism and then we use five lemma. In particular we have adjunction morphisms

(4.3)

t(S(-))

> id

id

>

S(t(-)) .

We then right derive t to get Lt defined on D-(F-g-str-1) and - ,,..,,..._ then get complete to get Lt:= (Lt). As S commutes with completion we get for ME 0-(F-g-str-I) and NE 0-(R-I)

(4.4)

{f):

Lt(S(N))

>

N

...

lJJ : M

-

>- S ( Lt ( ~1) ) .

44

Theorem 4.5. For ME D-(F-g-str-1)

NE 0-(R-I)

and

-isomorphi sms .

the maps in ( 4.4) are

Proof: As ~ is conservative on complete complexes to show that ~ is an isomorphism it suffices to show that S(~) is an isomorphism. This, however, is the case of lJJ for M= S(N) so we are reduced to showing that lJJ is an isomorphism and it is enough to show that M > S(Lt(M)) is an isomorphism by (2.3). As t and S coITJTiute with direct sums we may by devissage reduce to M= Gr for some rE I . What needs to be shown is that for every j •

•

G~ ------> S(t(Gr))J , is a quasi isomorphism. Because both Gr and ~(t(Gr)) are of 1eve 1 I we may assume that j EI . Let us fix some notation. Put • t =m= n+1 where I= [m,n] . We use the description G =r.F' 1 a. e

lE1N r (j) : = VI r- j +nar r

r1f1N VI 1cxr •

F : = FI t

and put

-

j-r > 0 and

where m= 0 if •

not. Then

•

G~ = } . 'j_ ~ ~ 1

,f.fN

(4.5.1)

S

C

1ElN

£ :

-

=t

®

F s = Fk +

(J)

r

n = 0 if r-j

f_

1

Furthermore , ( 4 • 1 , i i i ) - i v) ) sh ow s that

s •

1( Sm® ar)

if

j

>r 0 and =t if

•

(n-m)k+r-jvk( a ) { P µm® ar

k Sm®.'{

= FI j-r+ma

if not and

if j

m -

(4.5.2)

, V : = VI t

if

j

r

By extension of k we may assume that it is infinite (which is only to s imp 1 i fy the argument be 1ow anyway). For each :\ E W we get, by the uni versa l property of Gr an endomorphism p:\ : Gr > Gr taking ar to :\a.r. .

.

.

By functoriality it acts on t(G) and commutes with GJ > S(t(G ))J . r r = r Using the descriptions of Gr and t(Gr) we see that they split up into sums of pieces where

pA

for all

A

acts by multiplication by

Aok

for

some k. Because the automorphisme ok, kE?l, are all distinct this sum is necessarily direct and we may restrict our attention to one piece say where

pA

acts by Ao

f

•

. The part of G~ with this property will then be

45

WF f s

if

j >r

and

f >-_0

wil 1 be generated over

and

W by

f-1 -f-1 F ( S; ® ar) ( V ( Si ® ar ) ) f

WV f cp

(V-f (S.

®

a ))

1 ( S. Ff+ r Xf

r

a )) r

if

and

f-1 -f-1 F d ( S; ® a r ) ( d V ( Si ® a r ) )

1

®

a ))

r

i < min(j ,r) ;

i < min ( j-1 , r) ;

r

( S.

i

l

if

S(t(Gr))

if

1

call the

f < O etc. The part of

Ff ( Si ®a.r) if

f

if

< i max ( j -1 , r) when f >0 ( f < 0 ) • Let ·us . . G~ Yf and the one of S(t(Gr))J Xf so of complexes and we want to show that it is a

quasi-isomorphism. We distinguish between f > 0 and f < 0. Case I : f > 0: let us first show that H1 (Xf) = O. When f > O a-I ready 6 is surjective (recall

6 is the differential of t(G) as R-complex). This is true also if r j~r so we may assume r < j • Then if we put d' the other differential of • • t(G )(cpJ)(cpJ(n) = - 6 . ) we have that d' is surjective except that r Jn d(S. ®a) is not hit (i.e. d( S -® a) generates the cokernel of d') but J- 1 r J r , it is hit by 6 so that we first find an a such that d( Sj_ 1 ® ar ) = oa mod Coker d' and then hit d ( S. ®a )- oa by d' . , J- 1 r The next step is to observe that projection of xi onto the that

Ff(Sf®a.r)-factor Yf

>

is injective on

z0xf. This first shows that

z0xf is injective as £.fE or Ff+l E maps to Ff( Sm® a.r) resp.

Ff+l (SmQII a.r)

which is non-zero, then that in order to show that

0 Yf > z xf is surjective it suffices to s how that the image of Yf by the projection onto the Ff( S ®a )-factor contains m r 0 the image z xf. However, Yf maps surjectively to this factor. Case II :

46 •

f < O. (Put >

Yf

0

z

n=-f: Arguments very similar to the preceding ones show that xf

is injective and that

1 H ( Xf) = O and that we may again project

to the Let us first assume that p(n-m)h+r-jwvh( Sm® ar)

t(G) r

where

so we have to show that the

z0xf

any element of y. l

j < r • Then (4.5.2) shows that :Yf

is divisible by

has

R-module

i , p,

/\

d' y . = oy . and we then want to show that 1 1+ 1 (n-m)h+r-j p

vh( Sm® ar)-factor

p(n-m)h+r-j . We have

degree

maps to

y.

acts on it by

1

of

m< i < n A0

f-

-

and

ym is divisible by

•

Put h

a .V ( S -® a ) l 1 r { h 1 a .V + ( f3 .® a ) 1 l r

Y· = l

Then

d I y . = cSy. for 1 l+ 1

m< i < n

i < j-1

--

aC! 1

>r

or

i

>r

is equ i va 1ent to

h+1 P a.l+ 1

a. -- Ph+1 a . 1 l+ l 1 h a a. -- P a. 1 1+ l From this we get

i

if not.

i -1

or

r

ym is divisible by

is very similar and left to the reader. ~

~

~

~

Any F-gauge structure may be regarded as a graded R := W[F ,V] / (FV-p)modu 1e wh ere deg F = 1 a nd deg V= - 1 . We a 1so get r i ngs k [F] : = R/ (V) , ~

~

~

k [ V] : = R/ ( F)

~

~~

k : = R/ ( F , V) •

Definition 4.6. i) Let

ME D(F-g-str). Then there is a conunutative , up to

sign , diagram of distinguished triangles

•

47

~

r.J

F>- M(1)

M

>

r M(-1)

(4.6.1)

!

r.J

F

>-

r.J

>- M/F(1)

rv

M/V (-1)

r.J

M

>-

M/ F

>-

l

l F> M/V

r.J

>- M/(V,FJ

>-

!

!

l

>

iv

r.J

r.J

>

!

ii) Let ME D(F-g-str). Then there is a distinguished triangle r.J

~> M/F

(4.6.2)

v>

W/p

iii) Let NE D( R) • Then ( 4. 6. 1) fo r

®~

M ~> M/V

M= S( N)

>

is isomorphic to ( 3. 4. 1) •

Proof: (4.6.1) follows from the exact sequences r.J

0

r.J

> R(-1)

V>-

,-,JR

r.J

~> k[F]

> 0

> k[V]

> 0

r.J

0 ~> R(1) 0 ~> R

F>

R

(F,-V~R(1}@ R(-1) (V,F~ R

> k ---> 0

ii) is clear and iii) follows from the proof of (3.4). Proposition 4.7. i) Lt has finite amplitude on ex te nds to all of D(g-str-1) .

D(F-g-str-1), I

...

ii)~ and Lt give inverse equivalences between complete l evel

I

and complete

F-complex es

of

finite , so

R-complexes

F-gauge - structures

of level

of I .

They take bounded complexes to bounded complexes .

Indeed, by (0:4) R 1

®~ ( - )

=

s (- )/ (F.V)

...

L

...

Lt has finite amplitude iff R1 ®R Lt has. Now as ... by (3.6) and as S is an inverse of Lt we get

48

R ® ~ Lt ( - ) = ( - ) / 1

(F,V) and (-)/(F,V) has finite amplitude. Thus i)

if we can show that a complete M D(F-g-str-1) follows and ii) will follow ,..._, ,..._, ,..._, ,..._, is bounded if M/ (F,V) is. However, the grading of M/F and M/V have,..._, amp 1i tude ]- n] resp. [m , if I= [m,n] so (3.6.1) shows that M/F ,..._, ,..._, ,..._, and M/V are bounded if M/ (F,V) are. Then (4.6.2) shows that W/p ®~ M is bounded but Nakayama s (lemma shows that a complete complex is bounded if W/p ®~ M is . 00 [

00 ,

1

5. We will now begin to impose finiteness conditions on

Definition 5.1. i) An ME D(F-g-str-1), I W/p ®~M

complete and

is a coherent

F-gauge

structures.

finite , is coher ent iff M is

k-complex

i . e . with f inite dimen -

sional cohomology spaces .

ii) An

F-gauge

complex of

structur e of level

F-gauge

I

structures of level

i s coher ent iff it is coher ent as a

I .

Proposition 5.2. i) Let ME D-(F-g-str-1). Then M is coher ent iff Vi EI j E 7l HJ (M)1 is a finitely gene rated W-module . •

ii) A complete

ME D(F-g-str-1)

M/(F,V)

is coherent iff

is a coher ent

k-complex . •

1

Proof : It is clear that if ME F-g-str-1 and M is finitely generated for all i then M is coherent. This gives one direction. For the other we may a pp 1y descend i ng i nduct ion so that we may assume that Hr (M) = 0 for • • 1 j < r and then it suffices to show that HJ (t~) is of finite type for al 1 • 1 i . The completeness of M implies that HJ(M) is a p-adically complete W-module and W/p ®~ M coherent implies that Hj(M)i/p is '.inite dimen1 sional. This together implies by Nakayama's lemma that Hr(M) is of finite ,..._, ,..._, type. As for ii) (4.6.2) shows that if M/F and M/V are coherent k-complexes then so is W/p ®~ M. However, if I= [m,n] then M/F and ,..._, M/V are zero in module degrees above n resp. below and (4.6.1) and the ,..._, ,..._, ,..._, ,..._, coherence of M/(F,V) show that modulo coherent complexes M/F resp. M/V •

49

is isomorphic to its module shift one step to the right resp. to the left which clearly implies that they are coherent. We will denote by Dc(F-g-str-I) the category of coherent complexes. Putting together (4.7 ii) and (5.2 ii) we get ...

Theorem 5.3. S and Lt give inverse equivalences between b b Dc(F-g-str-I) and Dc(R-I) and Dc(F-g-str-I).

Dc(R-I)

and

,,,.

Of course S and Lt do not preserve the standard t-structures. Let 0 0 0 0 (~ -1,~ -1) denote the t-structure induced on D~(R-1) by (~ .~ ). Proposition 5.4. ~

takes

0 0 (~ -r,~ -I)

It clearly suffices to show that if then ME G or by ( 1. 2) that s ( ~1) E W-mo d [-2,0] . However ~(M) = S(M) so clearly L R1 ®RM has amplitude [-2,0] . 00

to the standard

t-structure .

ME Db(R-I) and S(M) E F-g-str-I C L and that R ®RM has amplitude 1 s(~1) E W-mod and (3.4) shows that -

It is clear that Li(M) is not a very convenient description of the R-complex associated to M. Our purpose is now to give a more explicit description in some cases. Here comes a point where we will prefer to work with the slight variant of F-gauge structures of level I mentioned before i.e. where we require an extension of the action of W [F,V] to its a-.... p-adic completion and work with R instead of R. Let us call such objects .... .... F-gauge-structures. We will denote by t'(-) the functor from F-gauge .... structures to complexes of R-modules corresponding to t(-). As coherent F-gauge structures are p-adically complete they may be regarded as .... F-gauge structures. More generally we can take the p-adic completion of any F-gauge structure of finite level. ...

In particular the completions of the Gr we will denote Gr. It is ... clear that they form a set of projective generators for F-g-str-1 .

50

LeITTTia 5.5. The

....

Gr ar e noether ian

....

F-gauge

structur es of level

Indeed, they are finite type modules over the W [F,V] which is well known to be noetherian. a --

p-adic

1 .

completion of

Theorem 5.6. Let M be a coher ent F-gauge structur e of level .... .... that Lt ( M) E il- I ( : = L\ n D( R- I ) ) . Then Lt ( M) = t ' (M) .

such

1

Proof: As M is coherent we can by lemma 5.5 find a resolution H· > M • 1 · of M such that H is a finite sum of Gr:s . Then Lt(M) is the completion of t'(H·). I claim that t'(H·) already is complete. For this it suffices to show that t'(Gr) is complete. The analogue of (4.2) gives that t'(Gr) =

~

1

R( 8i® a r) + W{ F,F- } ( 8n®ar)

where

1

W{ F,F- }

is the

p-adic

m Hj d ~ Hj+ 1 d ~ ... ) . Note first that by the • definition of t'(-) t'(HJ) 1 resp. t'(M) 1 are concentrated in degrees -i •

Ei'

1

•

•

and -i+1 . As the two spectral sequences are spectral sequences of R-modules this shows that d = 0 if r >3 . Furthermore, t' (-) is a left adjoint so right exact which ~hows that E~•i(H") phism for each

i

and as

dr = O if

r

~3

> E~'i(M)

E~ ' i (H·)

is an isomor-

> E~ ' i (M)

is still

•

an isomorphism. Now the spectral sequences converge to H1(t'(M)) so we get a commutative diagram •

1

H

(Lt(M))

resp .

51

"> EO'i(M) 00

•

i~

1

~>- H (t'(M))

•

...

This shows that Lt(M) > t'(M) induces a . split surjection on cohomology. . 1 ... l Furthermore, the kernel of H (Lt(M)) > H (t'(M)) being an extension of subquotients of the E~'k(H") for k < i is concentrated in module degrees < -i (so far we have not used that Lt(M) E ~. I claim that this implies that t' (M) E & , that Lt(M) > t' (M) is surjective in fl and that its kernel is Hodge-Witt. Indeed, construct a distinguished triangle ... --.> K > Lt(M) > t'(M) > and consider the long exact cohomology 1 1 1 sequence ... > H (K) > H (Lt(M) > H (t'(M)) > ..• The fact that • • 1 1 1 H. (Lt ( M) ) >-. H ( t' ( M) ) is a s p1 it su r j ec ti on imp 1 i es that. H (Lt ( M) ) = . 1 1 1 1 H (t'(M))@H (K). This gives first that H (t'(M)) and H (K) are coherent -being direct factors of something coherent, hence that t'(M) and K are • coherent and secondly that H1 (t'(M)) and H1 (K) are generated by its degree -i part being images of something that is, which by definition means that K,t'(M) E &. • Finally H1 (K) is concentrated in degrees< -i and is generated by its degree -i part and therefore it is a shifted Dieudonne module so K is Hodge-Witt. As K,t'(M) E& the distinguished triangle >- K > Lt(M) >- t' (M) > is in fact a short exact sequence in ~ . •

•

•

•

•

•

We have to any NE Li-In G-I associated two objects a(N) := t' (S(N)) and b(N) in G-I n Li-I such that b(N) is Hodge-Witt and we have obtained a functorial short exact sequence 0

> b(N)

> N

> a(N)

> 0 .

We want to show that b(N) is O • One case is clear namely when tains no Hodge-Witt subobjects. Suppose that we also know it when Hodge-Witt. Consider the exact sequence in &

N conN 1S •

52

0

>- HW ( N)

>- N

> N/HW ( N)

>- 0 .

By definition of G an NE & is in G iff it has no non-trivial s-1torsion quotient. ~ence any quotient in & of N is also in G . On the hand, any Hodge-Witt diagonal complex is also in G so both HW(N) and N/HW(N) are in G . Consider the diagram 0

! HW(N) 0

> b(N)

>--

r,,J

>

a(HW(N))

t

!N

>

J N/HW ( N)

r,,J>-

a(N)

> 0

J a(N/HW(N))

t 0

The fact that N/HW(N) "> a(N/HW(N)) implies that b(N) c HW(N). If we can show that a(HW(N)) >-- a(N) is a monomorphism we immediately get b(N) = 0. Let L be the kernel of a(HW(N)) >- a(N). As a ( HW ( N) ) = HW( N) i s Hodge • Witt L is Hodge-Witt and by (I:1.5.1) H1 (L) __._.>- H1 (a(HW(N))) is a mono• morphism. However, we have seen that , H1 ( N) >- H1 (a(N)) is split and the • 1 1 splitting is clearly functorial so H (a(HW(N))) --.> H (a(N)) is a direct factor of H1• (HW(N)) > H1 (N) which again by (I:2.2.2) is injective . 1 The ref ore H ( L) = 0 and so L = 0 . •

•

•

•

•

We are hence left with the case when NE~w and we want to show that N= t' ( S( N)) or proper 1 y speaking N = t' ( HO ( S( N))) . However, by ( 0: 5. 1) we may . a s s ume that N= •$ H1 ( N) [ - i ] and then S( N) i s an F- gauge st r uctu re and •

1

not just quasi-isomorphic to one. Hence we have a morphism of complexes Lt(S(N)) = t'(H·) >- t'(S(N)) where H· >- S(N) is a finite type projec-

53

tive resolution of S(N)

as before. The composite

Lt(S(N))

1

t (S(N))

>

adj>- N is an isomorphism. However, we have already seen that ...

Lt(S(N)) ----> t (S(N)) isomorphism. 1

Corollary 5.6.1. Let t (M) = ~(Lt(M)).

is a split epimorphism on cohomology so it is an

M be a coheren t

F-gauge

structure . Then

I

... 1

Proof : We saw in the proof of the theorem that Lt(M) > t (M) induced a 1 split surjection on cohomology and that by construction t (M) is concentrated i n degree - i and - i +1 . As Lt ( M) E Gc ~O Hi ( L-E ( M). ) i s generated 1 by its part of degree less than or equal to -i and so is H (t (M)) being a direct factor which also shows that H1 (t (M)) is coherent. As it is also concentrated in degrees -i and -i+1 it is generated by its degree -i pa rt so t' ( M) is a diagonal complex. Consider the diagram in D(R) •

1

1

•

1

...

LtM

__..> Ho(Lt(M))

ijJt /// t'(M)~

/

tp

>- t 1 (H (S(~(Lt(M))) .

By the theorem tp is an isomorphism. As t'(M) is in ~ ~ factors througi ~ ( Lt ( M) ) and as tp i s an i so morph i s m ~ ( Lt ( M) ) > t M) i s a s p1i t monomorphism as is a fortiori the map induced on cohomology. However, the map on cohomology induced by ~ is a split surjection so ~(Lt(M)) >t'(M) is a quasi-isomorphism. 1

(

III

Diagonal colllplexes revisited

1. We can now obtain more information about di ag onal complexes by using the results of II. Proposition 1.1. The functor gory of

s-acyclic

stru,ctures

M with

S((-)[1])

gives an equivalence from the cate -

diagonal complexes to the category of coherent

F-gauge

00

M =0 •

Proof: An s-acyclic diagonal complex N is ~-1-torsion so by definioo tion N[1] E G and S(N[1]) E F-g-str . Furthermore, S(N[1]) = s(N[1]) = 0 . Conversely, if M is a coherent F-gauge structure with M = 0 then • ... • ,n s(Lt(M))=O so by (I :4.2 iv) H1(Lt(M)) lS s-acyclic for all and .1 particular s-1-torsion. This implies that H ( Lt (r~ ) ) = o if i 7 -1 so ... Lt(M)[-1] 1s an s-acycli c diagonal complex . 00

•

•

Definition 1.2. Let I= [m,n] be a finite interval and J a subset of [ m, n-1 ] . Let M( I , J) be the F-gauge s true ture with M( I , J) 1 = k if i E I , 0 if not , F : M( I , J) i ~> M( I , J) i +1 is the identity if i E J and O if 1 not and V: M(I ,J) i ~> M(I ,J ) i+ is the identity if i EI J and O if not . We will denote by M(I,J) also the associated s-acyclic diagonal •

complex.

It is convenient to represent M(I,J ) by a directed graph r as follows. The vertex set of r is I and there is an arrow from i to i+1 if F: M(I,J)i > M(I,J)i+ 1 is the identity and an arrow from i +1 to 1 V:M(I,J)i+ 1 > M(I,J)i is the identity. Thus M( {0 ,1,2 },{1}) will have 0 1 2 ..c. >• as graph. •

56

Proposition 1.3. The

M(I,J)

are precisely the indecomposable

diagonal complexes killed by

p

if

s-acyclic

H= 1 •

Indeed, let N be the F-gauge structure associated to an indecomposable s-acyclic diagonal complex killed by p and let I= [m,n] be its level. By induction we may assume the result true for level I := [m+1,n] . Let N denote the F-gauge structure with N11 = N1 if i .cm and NI m = 0 and the same F : s and V : s as N except between N m and NI m+ 1 • By induction N1 is a sum of M(I ,J): s with I c I and in fact necessarily have m+1 E: I because a factor M(I ,J) in N with m+1 (/. I wo u1d g i ve a non - tr i vi a 1 factor of N . 1

•

•

1

,-...J

,-...J

1

11

11

11

I

11

1

II

Let us admit for the moment that for any two pairs (I ,J ) and 1 1 (I 2 ,J 2 ) with r 1 ,I 2 cI m-1E:I 1 ,r , J cr n[m-1,n-1] and 2 1 1 J 2 c I 2 r1 [m+1,n-1J there is either a morphism M( I ,J ) > M( I ,J ) or one 1 1 1 2 M(I 2 ,J 2 ) > M(I 1 ,J 1) inducing the identity in degree m+1 . 1

Suppose that F: Nm >- Nm+ 1 is non-zero, pick some kcNm such that F k.cO and fix some decomposition ~N. of N into indecomposables. U~ng l the admitted resulted it is clear we can find an automorphism ~ of N which is a product of "elementary mat r i ces of the form i d-Kp , tp : N. ----..> N., l J i .c j , such that the image of k by F lies in one of t~e factors Ni of

,-...J

1

1

II

,-...J

the decomposition

0

,-...J

of

N1

•

As

F : k ~>- N~+1

is an isomorphism we

10

can find a complement T of

k

•

1n

such that the composite

,-...J

T' >- Nm _F~>- Nm+1

>- N~+1

is zero and hence

10

.. . )

,-...J

F F 0 -< >- k >- N~+1 < > ,-...J

is a non-zero direct factor of

NII .• -_ ( N

...

and so equals

,-...J

10

N

•

Clearly

i s of the form M( Y, J ) . Si mi 1a r 1y , i f V i s non - zero . I f both V and F a re zero then N= Nm~ N so either N= Nm in which case N = k ( -m) or N= N where the result has been assumed by induction. ,-...J

N

V

II

I

1

It remains to show the result we have so far admitted as the M(I,J) are clearly indecomposable. Let r be the largest integer

) < •

-11-

-11-

s

>-

• • •

< •

• • •

• >-

• • •

In the first case M(I ,J ) c > M(I ,J ) and in the second 1 1 2 2 M(I 2 ,J ) >> M(I ,J 1) and both morphisms induce the identity in degree 2 1 m+1 . In the general case the graphs of M(I ,J ) and M(I 2 ,J 2) are the 1 1 same up to degree r+1 and then one continues with f+ 1 r+~ and the other ~+ 1 r+ 2•. ln the • > -case there is an epimorphism to M( [m+1,r+1] , J n [ m+1 , r ] ) = M( [m+1 , r +1] , J n [ m+1 , r ] ) an d i n the < •-ca s e we have 1 2 M( [m+1,r+1] ,J n [m+1,r]) as subobject in both cases inducing the identity 1 in degree m+1 . Composing these two morphisms we get what we want. Diagonal dominoes

Definition 2.1. A diagonal domino is an

s-tors ion

diagonal complex without

fini te tor sion .

It is clear that a subobject of a diagonal domino and an extension of diagonal dominoes is again a diagonal domino. Also any s-1-torsion object contains no finite torsion as its finite torsion would be at the same time ~-1-torsion and Hodge-Witt and so zero by "survie du coeur In particular ~-acyclic objects are diagonal dominoes. Finally, by (I:2.2.2), any domino is a diagonal domino. 11

•

58

Lemma 2 . 2. i) ( H= 1) Let

ME Li be

s-torsion . Then

M has no non- zero

-

s-0-tok sion ( s-1-torsiok ) subobject (res p . quo tient ob j ec t ) iff Hom&( U ,~1) = 0 · ( Hom&(~1, U0 ) = 0) for all k . 0 ii) I f

~

M is a diagonal domino then so is O ( M) 0 s-0-tor sion iff (M) is s-1 - tor sion .

~

and

rvQ

M = 0 ( 0 ( M)). Also

o

M is

M be an

iii) Let

s-0-torsion

( s-1-torsion) is an

diagonal domino and let (re sp . s-1-torsion)

s-0-torsion

diagonal domino . Suppose

M s -torsion

image of any sely, if

with no non-zero

>- M is

U~

Homli(U~,M)

0

=

s-0-torsion

by ( I :4.2)

Hom

subobject. As the

(~~ ,M) 6

= 0 • Conver-

then this is also true. for any subobject so it will

0 H (s(M))

suffice to prove that

s-0-torsion

=

O • As

-1

HO(s(M))

L

is torsion by assumption

-

it will suffice to prove that H (k ®w s(M)) = 0 • By (0 :4.3) Hi(R ®~ M)j = 0 if i+j = -1 so the spectral sequence 1 Hi(R 1 H- (k

gives

Let

®~

s(M))

1

®~

► Hi+j(k

M)j

formula

O(M)

s ( D( M) )

=

=

s(M))

0 • The second part of i) is proved in a similar way.

=

M be a diagonal domino. Then

(I: 1.11) and

®~

is without finite torsion by 0 (M) is s-torsion by the by (I : 1.8). Hence

o0 (M)

RHomW ( s ( M) , W)

s(D(M)) = RHomw(s(M) ,W)

'B'°(M)

and

shows that

M=

oD(M)

o

'o° (o'° (M) ) is

by (I : 1. 11) • Final l y

s-0-torsion

iff

M is

s-1-torsion. To prove iii) we may by ii), the fact that finite torsion and

M(~) (lJJ ) = M(~+lJJ )

1

that

s(M(~)) = S(M) • However by definition s(M(~)) is concentrated in degree 1. Let us introduce• the notation •

all

n . Hence

is clearly without

0((-)(~)) = 0(-)(-~), assume that we are in the resp.

case and by the formula . that

M(~)

mi ( u5)

=

U~ i

and

~(n) = o.,n for some i M[ 1] E G so by ( II :4.4)

so

for (-)(~) with (n) = -i for 1 mi takes s- 0-torsion (s-1-torsion) m. ( - )

59

diagonal dominoes to s-0-torsion (s-1-torsion) diagonal dominoes if ( i > 0) . Note a 1so that m. ( m. ( - ) ) = m. . ( - ) and m = id . J

1

i

Ws => Ws+1 / 2

M => ••• => W

=> ••• =>

0

2s E 7l

such that a) if

s E 7l

b) if s = r-1 /2 , r E 71. , then Vk E 7l Hom&( U~ ,mr (Ws /Ws+ 112 )) = Hom&(mr (Ws /Ws+ 1I 2 ), U~ ) = O • 1

ii) The filtration is determined by a) and b), Ws and m ( Ws ( - ) ) = Ws - r ( m ( - ) ) • r

is an idempotent radical

r

i ii ) 0o(ws ( _) ) = o'° (_);w- s +1; 200 ( _) . This filtration will be referred to as the type filtration or the filtration by type. 1 2 1 2 Proof: Let M be a diagonal domino and let F , F , G and G have the 2 1 2 1 2 1 meanings of (I:4.5). As F ;F = G ;G is Mazur-Ogus H (s(F ;F 1) = 0 and - 0

2

1

(I:4.5.3) and the fact that M is s-torsion show that H (s(F /F )) so 2 1 F ;F is s-acyclic but by the definition of the Mazur-Ogus-property this 2 2 1 2 1 gives F /Fi= G /G = 0 so F = F and G2 = G1 . Hence 0 c G1c F1 c M , F1 is the largest s-0-torsion subobject, M/ G1 the largest s-1-torsion quotient of M, by- (I:4.5) F1; G1 is s-acyclic and by (I:4~5.3) M/F 1 is 1 ~-1-torsion so a diagonal domino. Thus G and F1 are admissible subobjects and they are also clearly idempotent radicals. I claim that if r < 0 1 1 1(m M) < m G1 . Indeed, consider then mr F cG (m M) and if r > 0 then F r r - r the first inclusion. By lemma 2.2

1 mr F

mrF 1

1 1 > F (m M). ·Now consider mrF r

is

s-0-torsion

so

· ·

1(m M). The left hand side ► F1(m-M) / G r r

60

is

s-1-torsion so by (I:4.2) the . 1 image, N , is s-acyclic so if it is non-zero it has some u0 as image and 11 i i . 1 1 1 so would. mrF • Hence F would have m_rUO = Ur as image but F (F ) = F and

~-0-torsion

and the right hand sid€ is

u;= O

1 so functoriality of F gives a contradiction. Thus 1 1 1 1 >- r (mrM)/G (mrM) is O so mrF > G (mrM). The second inclusion

F1

mrFl

1

is proved in a similar fashion. We now put, for s E -U.. , WsM := m (F (m M)) s-1/2 1 -s s and W : = m ( G ( m M)). As m preserves diagonal dominoes these a re -s s -s admissible subobjects and we have just sho\-1n that it is a descending filtra1

tion. Property a) is clear as we have shown that

2

F (msM)/G (msM)

is

~-acyclic. As for b) we need to show that for any diagonal domino N , (k 1 1 1 , 1 k Homll U ,m G (m_ N)/F N) = Homll(m G (m_ 1N)/F N,U_ 1 ) = 0 for all k . Put 0 1 1 1 1 H :=m G (m_ N)/F 1N. Then by the definition of 1 1

non-trivial

s-0-torsion

subobject and

F

1

and

1 G , H contains no

m_ 1H has no non-zero

~-1-torsion

quotient object. We then conclude by lemma 2.2. Let us now show that a) and b) characterize {Ws } . By applying m. it 1 0 112 is clear that it suffices to show that w and w are characterized and 1 1 we want in fact to show that they equal F resp. G . By a) and lerm1a 2.2 Uk) = 0 we see that Hom&(Ws /Ws+ 112 ,U~) = O if s > 1/2 . Hence Hom fl. (W 112 '=0 · 1/2 1 k 0 and by lemma 2. 2 W c G . In the same way one shows that Hom&(U 0 ,M/W ) =0 0;w 112 is s-0-torsion so by lemma 2.2 M/WO is s-1-torsion and as w -

-

M;w 112

is similarly.

s-torsion

-

G1 cW 112 . The equality

and

w0 = F1 is shown

N,ow mr(W,-))=Ws-r(myi-)) fol lows from the definition and the Ws are idempotent 1 1 1 1 radicals as G and F are. By (2.2 ii) 'ff°(F (-)) ='ff°(-)/G 'ff°(-) and o'°(G 1 (-)) =n'°(-)/F 1o0 (-)

s.uch that

Ws = wso

if

so iii) follows. By (1:1.11 iv) there is an

s >s -

0

and

Ws

then has the property that

s

mrws

0

is

~--0-torsionfor all large r . From (2.2 iii) it then follows that Ws(cp) is ~-a-torsion for all -U.. and then (I :2.2) implies that wso is ·finite torsion and wso is zero as it is without finite torsion. This gives separation and finiteness in one direction. Exhaustiveness and finiteness in

61

the other direction follows by duality from this and iii). Definition 2.4. The type of a diagonal domino a : 1/2. 7l > ™ll defined by

(c f . (0:6)). We will say that •

f unction whenever

t

~

M is of type

M is the function

s if a(t) equals the zer o

s .

Thus the diagonal dominoes of type O are exactly the s-acyclic objects and with the aid of mr the study of diagonal dominoes of type is reduced to the study of those of type O . Similarly type r-1 / 2 is reduced to type -1 / 2 . Propos it i o·n 2. 5. A diagonal domino of type obj ect in G is a diagonal domino iff

-1 / 2

lies in

r

G . A torsion

Indeed, a diagonal domino M of type -1 / 2 has no non-trivial - ~-1torsion quotients so is in G and by definition HomG(U~ ,M) = Homl\(U~ ,M) = 0 • Conversely, let MEG with HomG ( ~~ ,M) = 0 for a11 k • We have an exact sequence in G

so

· k r-J-1 HomG (U , H ( M) [ 1] ) 1

=

0

for a11

k •

Lemma 2.5.1. Let NE& and suppose that for some Hom0 ( R) ( ~ ~ , N{ 1] )

=

0 • Then

i

H (R

1

tt~

M)j =

k

N= 0 •

Proof : Ap.p lying 1

and all

m; ( - ) we may assume i = 0 . By ( O: 4 . 3 ) we get 0 if i +j = 0 so the sp-e ctra 1 sequence ==>> Hi+j(k • L s (N))

w=

62

shows that

HO(k

®~

1 is torsion~ H ( s ( N) ) = 0

1 H ( s ( N))

together with (I : 1 • 4 i i ) and

L R ®R N = 0 1

and, as

s(N))=O and Nakayama's lemma gives

so by ( I : 1 •4)

•

N

lS

s-acyclic

N) j = 0

when

i+j = 0 gives

gives that

ME ll.

and now

Hi ( R 1

®~

but this

N= 0

so

by ( 0: 4) . 1 The lemma applied to H- (M)

HomA( U~ ,M) = 0

k

by definition and . Homf1(M,U ) = O by lemma 2.2 and the first equality gives 0 1

t (M)

=

0 •

Corollary 2.5.1. The category of diagonal complex es of type valent to the categor y of torsion coher ent

-1/2

str uctur e s

F-gauge

is equiM f ul-

filling the following pr oper ty

For any { of

x in

x E Mi + 1 , y E Mi - 1 and

i , if 00

M

•

equals the &mage of

T

in

y

applied to the image - oo

M

then

x

=y = 0

Proof: To prove the corollary it will suffice to show that for any structure

x

in

M , Hom(S(U~) ,M) = { (x,y) EPMk+ 1xpMk-

00

M

equals the image of

S(U~) i = k

if

identity if

i

>k

i"' k i

>k

and

O

y

if

in

M-

00 }

V:

and zero if not, that

and the identity if not and that

:

T

F:

· applied to the image of

S(U~) i

S(U~) i

is the

T

F-gauge

However, it is clear that

•

i = k , that

1

•

> S(U~) i +

> S(U~) i-l

p'th

1

is the

is zero if

power map. This

gives the desired description. We will finish our study of type

-1/2

by constructing a few examples

and show that we have obtained at least all weakly simple (c f. ( I, 1.5)) objects when H= 1 . . Let ME & be. s-acycl ic. Put I : = { i E 7L : Hom&(M ,U~) ~ O} and I'= {i E 7L: Hom&(U~,M) ~ O} and suppose that Ic J

a nd J n I ' = 0 and put

if

nE J . I clai~ that

If

i (/_ J

1

U

then

> M bu t

tori a 1 i ty of

1 1

W112 ( - ) • If

wh e re

N_ is of type

U~(- .•. or ... I- 0 > M2 > M3 • 0 • >- Hl(M1) lS zero except in degree -i +1

> M1

Hi-1(M)

definition of

3 (

•

where

the n

1

•

•

Hl(M1)

lS killed by a power of

Hi - 1 (M ) - i+ 1/ Vn for some 3 Thus

n and so the image of •

0

factors t .hrough

V so

•

•

1

> H (M )

1

is of finite 1ength.

1

1

> H (M )

> H (M ) 3

2

> 0 •

•

1 1 T- ' (-).

is exact modulo finite length which gives additivity for Now ii) is obvious as Ti ' -i (-) only depends on th-e R0-complex structure and so is independent of d. Finally, iii) follows f -rom i), devissage and explicit •

calculation for

U~ •

It is not difficult to convince oneself that M(I,J) is characterized . by the facts that dimk f~(I ,J) 1 = 1 if i EI , 0 if not and that for all iE I it has M([i,i+1], {i } nJ) as a subquotient. Us i ng th i s we see that i n order to prove that

p·h i c to

N(-- Nm,n

with as

M'

is of type or

N;w 112 N by -1/2

Nm • Put

is a quotient of

U~

it is without finite torsion.

and so there is some admissible

-

N' := N/N

r~•

and

M • However, MI

-

:= M/N • Now

is not of ty,pe

Ncw

1I2

N

1/2 -

W

-1 / 2

M' = M'

as

M

is

weakly simple and so, for some k, there is a non-zero U~ > M'. As 1 2 12 w~ N' = Wl N;N. is of type -1 /2 Hom&( U~, N' ) = 0 and hence the composite ~

1

> M'

> U~

is non-zero. By (0:4.3), and [Il-Ra:I.3.7] and modification

by m1 , Hom11(u~ ,U~) = 0 if k ~ i is an isomor~hism whence

M' = N'

&

":'hence k = i U~ • Let

N"

and by (0:4.2) the composite be the inver~e image in

M of the U~-factor. Thus N" is an extension of N by 1 1 I2 W N =N and being a subobject of M , w N = 0 so N" 11

11

11

On the other hand, N' and so of type

-1/2

is a quotient of by assumption. As

M so

w1I2 N

N 1 = M/N"

1

and

U~

whence

is of type

= N', so

N" ~ 0

-1/2. 1I2 N' = w N/N

M = N"

as it

67

is weakly simple. We have therefore obtained a short exact sequence

- ----->0 ----->- N

(2.7.1)

•

M

l

--.:a..>- 0 • >- U =1

We now have two cases i) N= M . Suppose m,n •

N=

i E [m,n] . By repeated use of (2.6) we get

1

and so (2.7 .1) shows that M(-c.p{i}) is an extenand so . is . s-acyclic sion of something s-acyclic by something s-acyclic and is indecomposable and killed by p as M is. However, Tl ' - l ( M( - O and as t (Fj)i are complete in the standard topology by the proof of (II:4.6) it suffices to show that the • 1 1 image of ~ is closed. In module degree -i this is clear as t (FJ) in 1 degree -i are of finite type over RO resp. W{F,F- } which are noetherian. In degree -i+1 we have t ()i: (R 1)k > (R 1),Q, and furthermore as 1 ~ is the completion of a morphism between G :s t•(~) is a matrix with r elements in RO . Let Vj be the highest power of V which occurs in this matrix. Then t O

5

r>O

s > 0 and the C-s i gn means series convergent in the p-ad i c topology. Note that R1 = TT WdVr ~ / , WFrd with the C-sign having the r >O r)Q

F-sd = dV

,

72

same meaning as before and the product resp. p-adic series-topology on the • first resp. second factor. Now t'(~) 1 restricted to M+ has image in ~-j and M+ resp. ~-j are of finite type over W{F } which is noetherian. Hence the image of M+ is closed in N~-j and so in. (R 1)£ as ~-j is. Therefore the image of t'(~)i is closed iff the image of (R 1)k/M+ > (R 1)£/ Im(M+) is closed. Now (R 1)k/M+ is profinite so this image is closed. Let now M=• (V,F,N) be a virtual F-crystal. We will say that M is • 1 1 divisible by p , i E 7l, if p- F takes N into itself. It is clear that for every i there• is a unique largest sub-virtual F-crystal of M which is divisible by p1 which we will denote d.M. Clearly the d.M form a 1 1 descending filtration. Let now NEG and consider V= ~(N)- "wK with its associated structure of virtual F-crystal. We have a spectral sequence 00

~> HO ( s ( N)) = S-

00 (

N) .

Let F.N be the image in V of the FiliHO(s(N)) coming from this 1 spectral sequence. From the definitions it is clear that (V,F,F.N) is . 1 1 divisible by p so F.Ncd.M where M= (V,F S(N)- / tors). 00

1

Theorem 4. 5.

Let

NEG

-

1

-

be Mazur- Ogus . Then

-

F .N = d .N 1

1

for aZZ

•

1 •

Indeed, one direction has already been noted, so we need only show that the composite d.M > N/F.N is zero. If we put T := S(N), then we may 1 1 assume that N=t'(T) and we want to show that if mEd .M then 8-~m is 1 J 0 i n t T) for a 11 j < i . As d . M i s a de c re as i ng f i 1tr at i on i t wi 11 suffice to do it for j = i-1 . It is clear that d.McMicMi- 1 so that 1 m= -r n where n E Mi - 1 and -r : Mi > MC1ear 1y a 1so n E nF-£p£ i Mi - 1 . 1

(

00

•

i-

•

1

As Si ® m= p ( Si ® n) it suffices to show that 8- ®n=O . Suppose that 1 . is of amplitude [r,s] with t := s-r . We have Fn = p1 n' for some •

and as F1

is injective we get

N

73

~- i n :

vi n

•

(Here we let

F Ms >

F:M • ,n relations

00

> M

t '. ( T)

f3. e n E n Vi t' ( T) i ' - i 1

r.J

F resp . V also denote the composite

r.J

s-1

r.J

I

r.J

£

- oo

> M

we get

Mr and similarly for v.). Using the Iterating we get f3 l. s n= V( f3 1. s n >

1

).

which is zero by 1emma 4. 4.

Theorem 4.6. The category of diagonal Mazur- Ogus complexes is equivalent to the category of tuples (M ,M ,N ,N ,~ ,~, P) where M and M are torsion 1 2 1 2 1 2 f ree diagonal Hodge - Witt complexes , N and N are diagonal dominoes with 1 2 WON = 0 and w112N = N , 1jJ is a monomorphism M > M , with finite torsion 1 2 2 1 2 coker nel , ~ is a monomor phism N > N with finite torsion cokernel and 1

p

is an isomorphism

Coker ~

2

>Coker~.

Proof: Let MEL\io and put M1 := HW(M) (cf. (1:2)), M2 := D(HW(D(M))), so M2 is the largest torsion free Hodge-Witt quotient of M , N := Ker(M >M ) 2 1 and N2 : = M/M 1 . Let ~ be the composite M1 c > M > M and ~ the com2 po s i te N1 c > M > N2 . Then Coker ~ = M/ N1/ I m M1 = M/ N1+M 1 = M/M 1 . I m N1 = Coker~. Let p be this identification. Let us now verify that (M 1 ,M ,N 1 ,N ,~ ,~, P) fulfills the conditions of 2 2 the theorem. It is clear that N and N are torsion objects without 1 2 finite torsion subobjects and hence diagonal dominoes. Now N n M is a 1 1 diagonal domino as a subobject of N1 and Hodge-Witt as a subobject of M 1 and so zero. However, N1 n M1 = Ker ~ = Ker ~ and so ~ and ~ are monomorphi sms. Coker~= Coker ~ is torsion being a quotient of N2 and Hodge-Witt being a quotient of M and so finite torsion. Finally, w0N cWOM= O and 2 1 112 0 = M/W M >> N /W 112 N . Converse 1y, 1et ( M ,M ,N , N , iµ ,c.p , p ) have the 2 2 1 2 1 2 properties of the theorem. Let N' be a cokerne 1 of th e rnorph i sn1 Coker ~ 4 and M the pullback of the diagram

Coker~

74

•

with morphisms the composites N ~Coker~ ">- N' resp. M >> Coker w 2 2 ":> N'. I claim that M is Mazur-Ogus. Indeed, for this it will suffice to show that Hamil (A ,M) = 0 and Hom& (M ,A) = 0 for any · s·-acyc 1 i c A . By assumption we get a short exact sequence O > M > M > N > 0 and 1 2 Hom&(M 1 ,A)= Hom~(N 2 ,A)= 0 the first as HW(M ) = M and HW(A) = 0 and the 1

112

1

112

second as W N2 = N2 and W A = O . Again by assumption we get an exact sequence O > N1 > M > M2 > 0 and Hom&(A,N ) = Homl\(A,M ) = 0 the 1 2 first as

0 w A =A

and

0 w N1 = 0 , the second because

M 2

is Mazur-Ogus.

It is clear that the two constructions just described give functors which are inverses to each other.

'\io .

ME

Then

D(M)

=o'°(M)

i\io

and if M= (M ,M ,N , 1 2 1 N , 1/! , D(M) (resp. D(M) > N) with N s-acyclic then, by duality, there is a non-zero M > D(N) (resp. D(N) > M) but D(N) is s-acyclic in & and M is Mazur-Ogus so we get a contradiction. That

o'°(M ) • > o'°(M) is the maximal Hodge-Witt 2 subobject is clear by duality and similarly o'°(M) > 'B'°(M ) is the maxi1

mal Hodge-Witt-quotient-object. Thi s clearly gives the second part. As for the fina 1 part 1~e have

Hom(s(M 1) ,W). Now

~

vp ( disc(s(M )))

D (M ) = M and 1 2

1

=

lgth s(-rr°(M ) / M )

1 1 M / M = N ;N = N / ~ D (N ) 2 1 2 1 2 2

as

s(-rr°(M )) =

and so

1

75

0 lgth s(OO(M ) ; M ) = lgth s(N ) - lgth s('o°(N )) = lgth H (s(N )) + lgth 2 2 2 1 1

1

~ -

0

H (s(D (N ))) = 2.lgth H (s(N )), the last by duality. 2 2 Inspired by (4.6) we introduce the following notation · Definition 4. 7. Let M-HW : = M/p-M

L\,o .

ME

It is clear that the

As

ME

L\1o

(M ,M ,N ,N ) 1 2 1 2

Hi (Rn

-1, 0 ®~

and

M)j = 0

f or all

j .

i\io .

M,K/ W)

®~

no non-zero morphisms

Suppose we know that

M

>

Rn

=

O when

E1'i

is concen-

for

i+j = -1 ® LR

or

1 •

M)-i = 0

for all

i

iff •

Homil( U~n+ 1 ,M) = Hom&( M, U~_ 1)=O

Hi+l(Rn

R (j)[-j] NE Ii

®~

M)-i = 0

for all

Un-l

c

for all

i

,

iff there are

j . By shifting we are then V

Hom&(N,Rn) = 0

> ~n

a

iff

HomLl.(N,Un_ 1) = 0.

has · o( i)=O

if

so one implication is clear. If

Hom&(N,Un_ )=0 then as Ui c > Un_ 1 for all i < n=1 1 so by devissage Hom~(N,T) = O for any domino T with

on the other hand

a( i )

O and

is a domino whose type

o (n-1) = 1 • Then

Hom (N,Ui) = 0 11

(HW(M),M-HW,p-M,M-p).

= RHomR(M,HomW ( Rn,K / W)) = RHomR(M,~n)(1)

n reduced to showing that for any

and

M,

•

~n := Homw(Rn,K/ W)(-1), so that

>n-1

are the

Then ( 4. 8) n degenerates iff

Indeed, HomW(Rn

i

subobject of

1, by (I:1.4), it is clear that (4.8)n

1 Hi+ ( R n

Lemma 4. 9. i) Let ME Li • Then . Hom&(M, U~_- ) = 0 for all j . 1 ME

p-torsion

is concentrated in degree

trated in total degrees

ii) Let

the

and consider the spectra 1 sequences

W/pn ®~ s(M)

degenerates iff

p-M

M-p : = M/HW (M).

and

Let again

Put

i

> n- 1

•

76

That

Rn

to show that

is a domino is easily checked and by (111:3.2) it only remains dVn-l : ~~ > ~~ is injective and that dimkKer dVn- 2 = 1 or

by duality that

dVn-l : R~

> R~

is surjective and

dimk Coker dVn- 2 = 1

which is easily verified by inspection. We have thus proved i). Now ii) follows from i), the discussion preceding the lemma and duality. 1-n Proposition 4.10. Let ME Llrv,o . Then (4.8)n degenerates iff W (p-M) and Wn- 112 (M-p) = M-p .

=

0

Indeed, HW(M) and M-HW are torsion free Hodge-Witt so for them • (4.8) n degenerates for any n . Hence by lemma 4.9 Hom~(UJ ,M) = u =-n+ 1 • • • Hom&(u:n+ 1,p-M) and Hom~(M,U~_ 1) = Homli(M-p,U~_ 1) for all j and it follows easily from the characterization of the type filtration that for a diagonal domino. N

1 w -n(N)

=0

(resp. Hom11(N,U~_ 1) = 0)

for all

Corollary 4.10.1. Let

ME ~O .

i)

2 (Wn-l/ (N) j .

= N)

iff

Hom&(U~n+l 'N)

=0

We then conclude by (4.9 ii).

If (4.8)n degenerates then (4. 8) m degenerates if

m M-p and as dimk HomLl(~n-+1 ,Un_ 1) = 2n-1 we see that dimk Homli(U_n+ 1,M-p) ~ 2n-1 . By (4.9) HomL\\.(U_n+ 1 ,M) = 0 and so Hom~(U_n+l'M-p)' >- Ext~(U_n+l'HW(M))

1(

d imk Ext •

1

mq. x ( m 1

U-n+ ,HW ( M)) 1

•

' -

1

( M)

)•

=

1 m '-l ( HW ( M))

=

and as 1 m '-l ( M)

HW(M) and so

is Hodge-Witt 2n-1

Hi+j(s(M))

equals the filtration by maximal virtual sive powers of

p.

iv) ( H = 1) Let

ME: D~ ( R)

r.J

n

1 F1('if1 + ( M) ) = 0 •

and

equals the reduction modulo

H (s(M))/p

Hn(s(M))

'ifl- 1( M)

is Mazur-Ogus in degree

F-crystals

be Mazur- Ogus in degree

divisible by succes-

n • Then

r.J

M!:=:! T n- M . In particu Zar if 1

M is Mazur- Ogus then

MC:! E& t' (Hodge(Hn(s(M) )~K)) [-n] . n -

v)

Suppose

degree Then

ME Dbc(R)

m. Suppose also that m+i,-i 2n- 1 < max m . . • -

1

m but not Hodge- Witt in . L . 1 lgth H ( Rn ®R M)J for some n.

is Mazur- Ogus in degree nbm(M) =

.C l+J=m

•

•

Proof: We have that

bn(M) = b (Hn(M))

0

•

•

1

h ,-, ('i{l(M))

for a 11

i • As .

i) gives

1

0

.

1

•

1

•

h 'J ('if1 (M) ) = 0

l

•

•

1

h,,n-,(M)

>

•

,-,

('i{l(M))

the condition of

l

and so

Hn(M)

is Mazur-Ogus by

•

h '- ('i{l(M))=h 1 'n-,(M)

(III:4.1). Also 1

'

~c h

b ('i{l(M)) 0

b (Hn(M)) =Ch ,-, ('H'1(M)) •

•

and by (I:1.4.1 ii)

and as

'if1(M)

is

MO

i +j ~ 0 . The s • s •

if

( 1. 2 • 1)

Hi ( R

1

®

~

> Hi +j ( R ®~ M) 1

Hj ( M) )

and (I:1.4 ii) then gives hi,j(H'1- 1 (M)) = o· if i+j = 1 and 1 h i ' j (Hn + ( M) ) = 0 i f i +j = - 1 • Th i s i s equ i v a1e n t to G2 ( w1- 1( M) ) = Tf- 1 ( M) 1 1 and F ( H11 + ( M) ) = 0 ( cf • ( I I I : Lemma 2 • 2 ) ) • Converse 1y , i f H'1 ( M) i s Ma z u r-0 gus , G2 ( ~- 1 ( M) ) = H11- 1 ( M) and 1 1 F ( 'if1 + ( M) ) = 0 then h i 'j (Hn - 1 ( M) ) = 0 i f i +j = 1 a n d h i 'j ('R"1 + 1 ( M) ) = 0 •

if

i+j=-1

bn(M)

=

1

=c

hi,-i(H'1(M))

=.C

l

l

Hn(s(M))

(i.e. Hn-l(M)EG) TO

iff the whole positive

y-axis

belongs to

Hodge(a) •

•

and a 11 the other

Example : In ( 2 . 1 . 1 ) a = ( ••• ,a

Definition 2.2. A sequence

a

1

=

ment

0

1

-2

,a

-1

0

1

2

,a ,a ,a, ..• )

b =O •

with almost all

•

possibly the first non- zero ele is normaliz ed if all a -> 0 except . . . . 1 aJ and that aJ = -aJ + if aJ < 0 and j < 0 • If a is any sequence 1

•

1

with almost all a = 0 then the unique normalized b with Hodge (a) = Hodge ( b) is cal led the associated normalized sequence . •

•

1

1

Definition 2.3. If a= (a )iE?l is a sequence of real numbers with a = 0 1 for i ~ 2a . If

part of

a

and

b

>Hodge ( b))

zero if

b are sequences with finite support s . t .

( I 2a ) i

Hodge(a)

is contained in the slope

b

2 (I 2a) i = ( I b) i

=

( I 2b ) i

Hodge(a)

L,

and conversely if iii) If

and

1

'l

(in the pointwise ordering) then

sely if

i+1

1

Hb . •

Proposition 2.4. i) If

2 2 I b >I a

a

•

and

a

>Hodge(b)

and conver-

is normalized then the slope i+1

par t of Hodge(b)

is also normalized . for all

i

>> 0

then

Hodge(a)

and

Hodge(b)

have

the same endpoints where the endpoint of a Hodge polygon is the vertex lying on the rightmost vertical side and conversely if

a and

b

are normalized .

84

To prove this let us first note that

i a,i+ 1 meets the

y-axis

,n •

(0,-(E 2a)i) and similarly for 1b,i+ 1 • This is easy linear algebra. Hence 2 2 2 2 ( E b)i > ( E a)i iff Ha,i+ 1 cHb,i+ 1 and 1a,i+ 1 = 1b,i+ 1 iff ( E a)i = ( E b{ If therefore

2 2 L b>L a

then

vi : Ha,1. c- Hb ,,.

so

Hodge(a) c- in Ha,,. c- n i Hb ,,.

and, by definition, Hodge (a)~ Hodge ( b) . As for the converse, if . n n Hb . ;t 0 and so i . c Hb . which t a . n H·odge (a) ;t 0 th-e·n £ a,1+ 1 . ,J a,1+ 1 ,1+ 1 '1 + 1 2 1. J 2 1. ( L b) > ( L a) . However as a > 0 it is clear gives Ha,1+ . 1cHb . and ,l+ 1 that t a . n Hodge(a) ;t 0 for all i . (Note that a >> 0 is equivalent to ,1

(0,0) E Hod,ge(a)

and that

In ii) as the slope

a

is normalized).

i+1

part of

Hodge(a)

is

i a,,+ . 1 n Hodge(a)

and

by assumption Hodge(a) > Hodge(b) and £ a,1+ . = £b . ,1+ 1 we g.et either 1 £ . n Hodge (a) = 0 or i b . n Hodge ( b) ;t 0 and then it is the s 1ope i +1 a,1+ 1 · ,l+ 1 part of Hodge(b) and £ . n Hodge(a)c £b . n Hodge(b) and in either a,1+ 1 ,l+ 1 case we get what we want. The converse is also clear. As for iii), the endpoint is equal to the slope

•

1

part for all

i >> 0

and we conclude by ii). We will need two more concepts. If P is a finite sided convex polygon P beginning with { (0,y) : a < y } for some a > 0 and ending with { (b,y) c < y} for some b ,c > 0 then there is a uni qu-e norma 1 i zed sequence a s. t. Pen H . and Hodge(a) > Hodge(b) for all b with Pc() Hb .• Simply let - i a,, -, ,, i a,i

be the line of slope

i

which touches

P but does not cut it. Note

that £a , 1. will touch P where P changes slopes from < i to will say that Hodge(a) is the highest Hodge polygon below P.

> i • We

If \ ~ \ ~ \ •• -~ As is an increasing sequence of real numbers we 1 2 3 define its Newton polygon as follows. We start with the positive y-axis, ·then the line segment from (0,0) to (1, \ 1), then the line segment from (1, :\ ) to (2, :\ +:\ ) etc . and we f i n i sh by { ( s , y ) : Y ~ :\ +:\ + .. · +A5 } • 1 2 1 1 2

85

A .s_A 2 i . 3. Let us now def i-ne the Hodge-Witt numbers. Definition 3. 1. Let

ME D~(R). Define fo r each

m.,n-.(M), h.,n-.(M)

h~,n-.(M)

and

h.,n-.

:=

w

•

•

T • ,n- • (M),

(Hodge -Witt nwnber s) by

m.,n-.+~2(T.,n-.).

•

h1,n-1 = (h.,n-.)1 w w •

Put

Theorem 3.2. Let

i)

sequences

n E "1I..

b (M) n

=

C .

ME D~(R). •

1

•

h ,J (M)

w , +J=n

i i ) ( Crew ' s f orrrru la)

for all •

C (-1 ) J h

•

1

n • •

•

'J (M) =

•

•

C (-j ) Jh ~ 'J ( M)

J

fo r all

•

1

•

J

Proof: As -Hence for k >> 0

.C l

+J=n

•

•

.C 1 +J=-n

-m 1 ,J , the last equality as

m•,-n-.

and

r•~n-.

·have finite suppo-r t

86

but

C

.

•

•

1 m 'J = bn by (0:6.2). In ii) the right hand side is additive on

1+J=n distinguished triangles and one verifies easily that so is the left hand side (cf. (III:2.6)). We reduce then to elementary R-modules and compute. Remark:Inhi s th es i s,Cr ew ,[Cr],proved a formula which up to a rearrangement of the terms is 3.2 ii). His proof is the same as the one given here. Another rearrangement has been proved, independently, by Milne (to appear) by a s lightly different method. •

Theorem 3.3. Let

•

•

•

ME D~(R). Then

•

fo r all

1

•

,J .

By shifting we may assume i +j = 0 . We then have hi ,j (M) > hi ,j (TI° (M)) by (I:Cor. 1.4.1) and evidently h: ,j (M) = h: ,j (TI° (M)), so we may assume •

1

ME ti . By ( I : 1 • 4)

•

•

•

if I i+j I> 1 and clearly h:'J(M) = 0 if • 1 1 i +j ~ 0 ( a s th i s i s true of m 'J and T 'J ) . Hence by ( 3 . 2 i i ) hi 'j = hi 'j - (hi 'j - 1+hi 'j +1) < hi 'j . h 'J ( M) = 0

•

w

•

•

-

Corollary 3.3.1. Suppose that h: ,j (M)

= hi ,j ( M)

for

i +j

is Ma zur-Ogus in degr ee

ME D~(R)

= n and

so

T • 'n- ·

= L 2 ( h • 'n- • -m • ,n- ·) •

Proof: By (3.2 i), (3.3) and assumption we have b whence h1 ,J = h1 ,J i +J· = n . •

n '

•

w

Definition 3.4. Let i)

•

n. Then

b = n

•

C .

1+J=n

•

< C w - 1+J=n . •

•

h1 ,J

•

h1 ,J=

'

ME D~(R).

The Newton polygon in degree

gon of the slopes of the virtual

n, Newtonn(M) of M is the Newton polyF-crystal Hn ( s (M)) ® K •

ii) The Newton - Hodge polygon in degree n, Newton-Hodgen(M), of M is the highest Hodge polygon below Newtonn(M). iii) The abstract Hodge polygon in degree

n, Hodgeabs(M), of M is the

Hodge polygon of the Hodge numbers of the virtual

n

F-cr ystal

n H ( s (M)

where these Hodge numbers · are defined with the aid of the ele,nentary

®

K),

87

divisors of the Frobenius (cf . [Ka]).

iv) The Hodge - Witt polygon in degree

n, Hodge-Wittn(M) , of M is

Hodge(h · ,n- ·). w

v)

The Hodge polygon in degree

Theorem 3.5. Let

n , Hodge (M), of M is n

b

ME Dc (R). Then , for all

Hodge(h.,n-.).

n ,

Newtonn (M) .?_ Newton-Hodgen (M) .?_ Hodge~bs ( M) .?_ Hodge-Wi ttn (M) .?_ Hodgen (M)

and all but

Hodge (M)

have the same endpoints .

n

n = 0 . Newton 0 (~1) ~ Newton-Hodge 0(M) and so, a fortiori Hodge-Witt 0 (M) ~

Indeed, by shifting we may assume •

and by (3.3) we have

•

•

•

h:,-, ~ h1 ,-,

Hodge (M). By (2.5) and (0:6.2) Newton-Hodge (M) = Hodge(m· ,o-.) and 0 0 L 2 ( h~ ' O- • ) = L 2 ( m• ' O- • ) + T • ' O- • .?_ L 2 ( m• ' O- • ) and so , by ( 2 • 4 ) , Newton - Hodge 0 > 1 2 Hodge-Witt • Consider now N = F 'H°(M) / F 'H°(M). By (I :4.5.3) and (I :Cor. 0 1 2 0 1.4.1) the virtual F-crystals HO(s(M)) ® K and H (s(F 'H°(M)/F 'H°(M))) -

~

K

are equal and so have the same abstract Hodge polygons. On the other hand is Mazur-Ogus so by (1.2 iii) Hodge-Witto(N) as

=

=

Hodge~bs(N)

Hodgeo(N). Finally, by (III:2.6)

m• ' O- • ( N) = m• ' O- • ( M)

which gives

Hodge (N) 0

N

and by (3.3)

T.,o-.(N)

~T

0

we get Hod ge-W i t t 0 ( N) .?_ Hod ge-W i t t 0 ( M)

,o-.(M)

and

by ( 2 • 4 )

Hodge~bs(M) = Hodge~bs(N) .?_ Hodge-Witt (M). 0

For the endpoints it is now sufficient to prove that Newton-Hodge n (M) and Hodge-Wittn(M) have the same endpoints. This follows from ( 2 .4 iii) as . n-. . T' (M) has finite support. Proposition 3.6. Let ME Db(R). Then Newton (M) touches the slope i part • ';";11 C n . _. . _. of Hodge M -iff H (M) is cut at { i-1 } and hJ ,n J (M) = hJ ,n J (M) for n w j < i-1 .

88

Indeed, as

. , n-. m

NewtonnM

touches

that

(E2h•,n-.)i-1 . But

M ;M for some i . As u [i+1](-i) is sim,p le 0 2 1 0 as object of G (e.g. as its associated F-gauge structure visibly is) this morphism is a monomorphisme so obj,ect with •

that

•

h1 ' J ( M/ M ) = O unl es s

1

see that

hand

M ;M would contain a non-zero sub2 1 s(-)=0 contrary to definition of M . The same argument shows 1

D(M/M ) 1

s(M )=0 1

D(M ) 1

has

and

G-ampl itude •

or

- 1 . Us i ng ( 0 : 3 . 3 ) and ( I I : 1 . 2 ) w,e

[0, 1]

so

D~(M) = D~(M 1)

on the other

•

h1 'J(M )=0 1

has amplitude

I claim that

i +j = 0

[2 ,2]

so

unless

i+j=0,-1

D~(M) = D~(M/M ) 1

or

and

D(M ;M ) = D~ (M ;M )[ -1] • In fact as for 2 1 2 1

-2

so by (0:3.3)

D~(M) = D~(M/M 1 ). M/M 1

we see that

D~(M ;M ) = 0 • Now as hi ,j (M ;M ) = 0 unless i+j. ~ or -1 and s(M 2;M 1) 2 1 2 1 1 is torsion as M ;M is we see by (0:3.3) that h 'J(D(M 2;M 1 )) = 0 unless 2 1 i+j = 0 or 1 and s(D(M ;M )) is concentrated in degree 1 . Hence 2 1 2 2 ~ ( D~ (M2/M1)) = Ho ( s ( D(M2/M 1)) . 0 and hi '-i - ( D~ (M2/M 1)) = hi '-i- ( D(M2/M1)) =0. •

However, s(D~(M ;M ))= 0 so D~(M ;M )[-1] 2 1 2 1

is an

s-acyclic

diagonal complex

D~(M ;M } if non-zero contains some u [i+1](-i) as subobject which 0 2 1 contradicts h i+ 1 ,-i- 3 (o~(M ;M )) = 0 by (0:4.3). In the same way as 2 1 2 1 1 2 hi ,-i- (o~(M /M;➔ l = hi,-i- (o(M)) = 0 we see that o!(M ;M ) contains no

and so

2

2 1 no,n -zero object with s ( - ) = O and as it is torsion as M /M is by what we 2 2 1 1 2 1 1 2 0 have just shown o (o (M ;M )) = o (o (M ;M )) = O . As for o (M 1) = D(M 1) we g g g g g 0 2 see that it has s(-) = O and so, again by what has been shown, og (o g (~1 1)) =

D~(o;(M )) = 0 1

M/M

2 tient

and

and as

D~(M ) = D~(M) 1

the same is true for

is without torsion so it embeds in N

that has

oO(M')

s(-)=0. As

is again in

L\,o

M' is in and so in

M • Further

M' = t'(Hodge(s(M/M )) 2 '\,o• D(M') ='o°(M') G • Thus

with a quo-

by (III:4.11)

D(M') = D~(M'). The long

90

exact sequence

O

> D~(N)

2

= D~(M/ M2)

> D~(M/ M ) 2

> D~(M/M ) > 0 2

> ...

1 2 1 2 DO(N)=D (N)=O=D (M/ M )=D (M')=D (M') g g g 2 g g

and the fact t hat D~(M')

> D~(M')

and

D~(M/M 2 )

= D~(N).

As

D~(M')

is in

i'IMO

give D~(DO(M'))

=

0

og(o (M')) = O . To summarize we have obtained og0 (M) = og0 (M'), an exact sequence

O

> D~(N)

> D~(M)

> D~(M ;M ) 2 1 1 0 2

> 0

D~(M ). 2 1 2 0 0 0 2 1 2 1 Hence, og (o (M)) = o (o (M)) = o (o (M)) = o (o (M)) = o (o (M)) = 0 , o (o (M)) =

g g g g g g g Dg2 ( Dg1( N) ) = D( D( N) ) , DOg( DOg( M) ) = DOg ( DOg( M = D( D( M 1

)

D( D( M2/ M )) 1

and

we are finished.

1

)

)

)

and

,

D~(M)

g g g g Dg1( Dg1(M) ) = D91( Dg1( M / M1))= 2

D~ ( D~ ( M) ) = D~ ( D~ ( M )) = D( D( M )) • As we have 1 1

Corollary 4.1.1. The coher ent

F-gauge

=

D( D( - ) ) = i d

N with the proper ties

structur es

•

that they are torsion and that for no non-zer o element of in

00

N

and

eleme nt of

- oo

N

G.

1

N

1 zer o ar e ex actly those of the for m ~(D (M)) 9

ar e the images for a tor s ion

Indeed, it is clear from (4.1) and its proof that the elements of G of the form

D~(M)

for

M torsion are exactly the torsion elements of

G

containing no non-zero subobject with s(-) = 0 • It is also clear that an . ME G contains no non-zero subobject with s ( - ) = 0 i ff HomG ( ~~ [ 1] ,M) = 0 •

for all

•

i • However, HomG(U~[1],M) = HomF-g-str(S(U~[1]),S(M))

•

S

is an

•

•

e qu i va 1en ce of ca te go r i es and as

as

S( U~ [ 1] ) J =

k if

i

=

j

and

O

i f not, we

•

1 1 see that HomF -g-s t r (S(u [ 1]) ,N) = {x E N : Fx = Vx = O} . It is clear however that = 0 the condition that there be no such non-zero x for all i is equivalent to the second condition of the corollary.

Remark : i) The coherent F-gauge structures fulfilling the condition of the corollary are precisely those F-gauge structures cons idered by Fontaine and Me ssing (to appear). That they have the fo rm of the co rollary sugge sts that in their theory the dual point of view could be more useful .

91

ii) It is no doubt possible to define directly on F-gauge structures a tensor product and internal Hom-functor which would correspond to (-)*~(-) resp. RHom~(-,-) through S(-) but I have not tried to do that.

V

F-Crystals

1. If we apply (IV:3.6) to t'(Hodge(M)) where M is a virtual we get the Newton-Hodge decomposition of [Ka].

F-crystal

Theorem 1. 1. Let M be a vir tual F-crystal . Then Newton(M) touches • • • 1 par t and its M slope< Hodge(M) at slope 1 iff -is the swn of its slope

>1 •

par t .

Proof By (0:5.1) it cut at {i} • • •

•

lS

clear that M

.

lS

such a sum iff

t' (Hodge(M))

•

lS

'

2. Let us put S the Zariski topos of k, Sperf the topos of sheaves on perfect k-schemes with the etale topology (cf.[Be 2]) and sf£ the topos of sheaves on k-schemes with the fpqc-topo 1ogy. We ring S by vJ and sf'l and Sperf by the ring scheme W . Then we have natural morphisms of ringed topoi n : s f,Q, > s , n : sperf >- S and p : Sf,Q, >- Sperf where p* is the restriction. We then have np= n . Note that

(~ 4

Ln*(W/pn) =

n

~)E D(Sft_~) (resp. D(Sperf_~)) and so has cohomology ~n and Fn~ in degree O and -1 (resp. ~n in degree 0) and the rest O • In particular we see that if M is a finitely generated W-module Ln*M is an affi~e (resp. an affine perfect) group scheme. I claim that for any W-comp 1ex M , Rp*L n~M = Ln*M . Indeed we reduce to• showing that for any perfeet affine k-scheme U and any index set I H;£ ( U,€t W) = 0 for i >0 . I -

As U is quasi-compact we may assume that I contains just one element and then it is clear as U is affine. One also easily checks that for any fini~ type affine k-group scheme G Rp*G= p*G which is simply the perfection of

94

G. Clearly every on-linear map, for n >0 , between W-comp 1exes induces naturally a morphism between Ln* applied to these complexes. (In the case of sperf we can even allow n to be an arbitrary integer) . •

1

If M=~M is a complex of F-gauge structures then we define a complex of graded sheaves MF E D(SF£ ) and a complex of graded inverse systems of sheaves M~ E D(Sperff) as follows. We let (MF):= Cone(Ln*Mi £-r> Ln*M-

00 )

(M~) i := 21: / p"®i Cone(Ln*Mi

and

R.-r➔_

21:/p"

®i

where

)

is induced

R.

1

00

00

> tvf' and T : M> M N is the complex of inverse systems (... > Z1:/pn+ 1 Z1: 11~ N

from the composite of the natura 1 mapping and

Ln*M-

00

M

> ..• ) • By what has been observed before

21: / pn Z1: 11~ N

21:/p •

11~

►

Rp*MF = M~

and if M is coherent then the cohomology of MF (resp. M~) is an affine group scheme (resp. an inverse system of affine perfect group schemes). Remark : The reason why we use inverse systems in the case of Sperf and not in the case of sf£ is that an exact sequence of affine group schemes gi~es an exact sequence of sheaves in sf £ whereas only exact sequences of finite type affine perfect group schemes give exact sequences in Sperf_ The reduction modulo successive powers of p will ensure that in the cases of interest all our perfect group schemes will be of finite type. By introducing a topology slightly finer than the etale we could have avoided this reduction. Recall that for M a complex of R-modules one introduces M~ a . perf F L ( F-1 graded inverse system of sheaves ,n S by M. = 7l/p· ®?L Cone Ln*M > Ln*M). We now have Proposition 2.1. Let R-complex

of level

l = [m,n] be a finite interval and M a complete I . Then (M~) i = ( (~( ;.1) )F.) i [-i] .

F . L F-1 Indeed, M. = 7l/p ®?L s(L n*M >- Ln*M) s(R.11~Ln*M

£-r> ~;p·

1

F- ► ®~

R.11~L n*M). On the other hand

S(Ln*M)m)

which by

[Ek 2:I :8.1]

( (S(M) )~) i = s(il'. / p"

11~

equals S(Ln*M)i

where we in the obvious fashion have extended the •

95

notion of F-gauge structures to F-gauge structures on S is the obvious functor from complexes of R5perf-modules to complexes of F-gauge structures on Sperf_ We then see by (the analogue for sperf of) (II:2.3) that we may replace Ln*M by its completion so that we want to prove that for a complete R5perf-complex N of level I (?l/p"

®~

s(N

F-1 ► N)) i

= "ll / p"

®i ~(S(N) i

i -r> S(N)m)[-i]. Now exac t ly as

before one proves that S is an equivalence of categories from complete perf R5perf-complexes of level I and complete :F-gauge structures on S of level I. Now, (cf.[Ek2:III.1.5.4D s(N

1

F-l> N) [i]

=

RHomR

(W(i)[-il ,N)

5perf and as S is an equivalence of categories and ~(i) and N are complete ... this equals RHomF-g-str(S(~(i)[-i]),S(N)). As the Gr are loca~ly proj~c. . F,1-m_V,n-1+1 tive if we can show that there is an exact sequence O > Gm ... Gi > ~(~( i) [-i]) > 0 then we immediately get RHomF-g-str(~(W( i )),S(N))= s(S(N)i

£-r> S(N)m)

which is what we want. This exact sequence is easily

established however. Let now

M be a virtual

F-crystal

and consider

1 H (Hodge(M)F) =:MF.

Being a quotient of a connected and pro-smooth group scheme it is connected and pro-smooth. -However, by ( 2. 1) and the fact that Hodge (M) E Ll we see that the perfection of MF is killed by some power of p and hence so is MF itself. It is therefore a quotient of a finite sum of n*M :s divided by some power of p and so a smooth connected unipotent (graded) group scheme. Definition 2.2. Let M and N be vir tual

F-cr ystals . A morphism

M

> N

is said t o be a str ong i sogeny if it is an isogeny in the usual sense and

MF

> NF is an i sogeny . We say that M and N ar e s t r ongly isogenous if t her e is a vir t ual F-cr ys t al L and str ong isogenies L > M and L >N. Proposition 2.3. i)

A

morphism M > N between virtual

strong isogeny i ff a cone of

t'(Hodge(M))

>

F-cr ystals

t'(Hodge(N))

is a

has cohomology

96

of finite length as

W-moduZe .

ii) Strong isogenies are stable by pullbacks and compositions . iii) The relation of being strongly isogenous is an equivalence relation. iv) The Newton and Hodge polygons are the same for strongly isogenous

F-crystals . •

v)

1

T

•

•

'-

1

(t'(Hodge(M))=dim(MF)

1

•

Proof: By (2.1) to prove i) it is sufficient to show that if

KE D~(R),

s(K) has finite length cohomology and the cohomology of K~ is a finite perfect group scheme then the cohomology of K has finite length. Now FO . s(-) and ((-).) have ~-amplitude [0,1] and so we reduce immediately to KE&. The first condition says that K is s-torsion so we need only show that there is no diagonal domino part. I now claim that for any LE~ 1 H (L~)O is a connected unipotent perfect group scheme and that 1 L • > H (L~)O is exact up to isogeny. The first part is clear by devissage 1 as H ((-)~)0 is right exact on & • The second part is then clear as devissage again shows that HO(L~)O always is etale. Using this result, dev i ssage and (II I : 2. 6. 1), we see that Ti ,-i ( L) = dim H-i + 1 ( L~) i for any L . 1 1 This shows that T ,- (K) = 0 for all i and hence gives i) and gives also, through (2.1), v).Now ,i) irrrnediately gives that strong isogenies are stable under compositions as the class of R-complexes with cohomology of finite length as W-module is stable under cones. Let L ► N and M > N be two morphisms between F-crystals, suppose that L > N is a strong isogery and let K be the pullback of L > N and M > N . As Hodge(-) is 00 right adjoint to (-) it preserves pullbacks and so Hodge(K) is the pullback of Hodge(M) > Hodge(N) and Hodge(L) > Hodge(N). Th i s i mp l i es that cones of t'(Hodge(L)) > t'(Hodge(N)) and t'(Hodge(K)) > t'(Hodge(M)) are isomorphic and so by i) K > M is a strong isogeny. This finishes the proof of ii) and iii) is a formal consequence of ii). • 1 -1 Finally the statement concerning Newton polygons is clear and the m' •

•

'

•

97

.• •

1

are the same. By v) the T • • h,,-, are the same but w complexes.

•

'-

1

are the same and so by definition the as we are dealing with Mazur-Ogus •

If M 1s an F-crystal then (MF) 1 , being a unipotent algebraic group, is isogenous to a sum of ~n:s and the ordered partition n p . Locally, in the Zari ski topology, the long exact sequence of cohomology obtained from (1.1) shows tha_t any X is of this form. In general it is easy to see that there is a filtration • • 1 0 1 p-1 1 M • MJ c M +J when OX= M c M c ... M = n*°x of sub-Ox -modules such that i+j < p and Mi / Mi- 1 =L-i 1~ i ~ p-1. L

•

•

r-.J

ii) X is an integral local comple t e i ntersection scheme and 14( = TI* ( wx ® Lp- 1) • ,-...,J

Indeed, to see that X is a local complete intersection it suffices, X being smooth, to show that X - ➔ X is a local con1plete intersection mo~ phism. This is local in the flat topology on X so we may trivialize X so • it is enough to prove that aL > X is an l .c .i morphism. However, aL 1S a flat group scheme. In parti cular depth X= 2 so to prove that X is reduced it suffices to prove that it i s generically reduced. Consider Xred ~> 'x >X. If X is not generically reduced then as X > X generically is of the form Spec(k( X)[t] / (tp-f)) >- Spec k(X ) we see that fE k(X)p and Xred > X is birationally an isomorphism. It is finite, as it is the composite of a closed immersion and a finite morphism and X is normal. Therefore Xre d > X is an isomorphism by Zari ski's main theorem. This gives a section X ">- Xred > X contrary to the non-triviality of X > X. l",J

l",J

r-.J

r-.J

r-.J

l",J

r-.J

l",J

Fina 11 y, 14( = TI* wx ® 14(; x

r-.J

r-.J

so it suffices to show that

14(;x = TI* ( Lp- l) • By an

easily proven result on group scheme torsors this equals

TI*(s*waL/X )

where

s: X > aL is the zero section and the adjun ct ion formula applied to the 1 immersion aL > L gives s*w = LP- . a,L/X iii) Suppose A. Then

2 H ( X,OX) r-.J

1 1 H (X ,OX) - > H (X,°>f)

2

2 is an i s omorphism , H (X , OX)

2 i s a monomorphism and dimkH ( X •°x) / H ( X ,O X) r-.J

P-1

=~

2

: dimkH ( X,L 1

. -,)

> .

101

To see this it will suffice to show that 1

H ( X, TI*OX/OX)

and that

0 H (X,TI*OX/Ox) =0 =

p- 1 . 2 -1 d~ mkH ( X, TI*OX/OX) = ~ _ : d1 mkH ( X, L ) : Now

.

2

1 1 has a filtration by the M /OX with successive quotients 1 -< i - that the X TI> X

H2( C) , is ki 11 ed by p . As p = VF it wi 11 be sufficient to show 2 2 maps H (X,W°x) into H (X,WOX). It is clear that the Frobenius X trivializes X so that there exists a mapping X > X such 1T composite X > X > X equals F. It is easy to see that > X is also F: X > X . Now F: H2(X,w°x) > H2(X,wox) is ,-..J

,-..J

,-..,,J

induced by functorial ity from F: X ~> X so it factors through 2 2 TI: H (X,WOX) > H (X,wo.;r) which is what we want. Hence H2(c) is killed

102

by p with V injective and so contains no submodule of finite type over W. Therefore TH 2(X ,W°x) is contained in H2 (X,WOX) and thus equals 2 TH (X ,WOX). v)

2

Assume A. Then

dimkH (X,L

2

~

2 ,. . ,

2

dimk(H (",W°x)/ H (X,WOx)) / V(H (X,W X) / I1 (X,W X)) -

).

This follows from iii) and iv) • •

1

Let us consider H (X,ffim). Put N:= n*Gim/(6m. If we pull back X by TT p-1 1. then it becomes trivial dans n*X= Spec(@ L- ) with multiplication the . . . . i=O p-1 . -, -J 1 -J -, obvious one; L ~L > L if i+j < p and zero if not. Also n*N = 1+ @L p-1 . i=1 as multiplicative group is through exp and log isomorphic to @ L-, • i=1

1

1

2 is an isomorphi sm and H (X,16m)

rv

vi) Assume A. H (X,(6m) > H (X,ffim) H2('x,16m) is i njective with cokernel killed by

>

p.

0 1 H (X,N) = H (X,N) = 0

Indeed, it is clear that we need to prove that 2 and that H (X,N) is killed by p. As n is faithfully flat and TT*N , as we have just seen, is killed by p the last statement follows. Again as V TT is faithfully flat we get a Cech spectral sequence Hi (X( j) ,N) ==>>Hi+j- 1(X,N) where

x{j) := Xxx

Xxx· .. xxX

{j

times)

so it will be sufficient to show that 1 rv Q rv{1} H ( X, n*N) = H ( X, n*N) = H ( X , n N) = 0 Q

rv

2

p-1 -i rv(2) > X . Now n*N!:::! _@ L , X = a.LxXX with rv

n2 the com-

,= 1

posite of the projection q2 onto the second factor and TT . Hence we need Q -i 1 -i Q -i to prove that H (X,n*L ) = H (X, TT*L ) = H (a.LxX X, q2TT*L ) = 0 for rv

-i Jii

are isomorphisms . •

•

Proof: Recall that H1 (Y,7lp{1)) := lim H1 (Y,µpn). The exact sequences ---.> ffi

0

4 m

.

n ffi

m

> 0

- "' X be a mi ni ma l res o1uti on of the s i ngu1a r i ti es of

p :-X'

"'

X•

ix) Assume A and c • a ) wx I = p *TT* (u.lx ® Lp- 1 ) • •

•

1 "'

1

b) H (X,W~)

>- H (X' ,WOX,)

1 c) H (X ,LZP ( 1))

2

H (X, LZP ( 1))

1 > H ( X' ,a'.p ( 1)) is an isomorphism and

2

> H ( X' ,LZP ( 1)) is injective with cokerne l a _free

module of rank equal to

p*(Wx)

I claim first that

wX '

=

R p*6Jpn

spectrum of a local ring and of X are rational

and that

2

T: H (X,L2p(1))

o1 which

X has in fact only ratio-

0 . To prove this we a re reduced to pro-

=

is injective on Pie x

pn

5

where

S is the

> "' X a flat morphism. As the s ingularities

S

1 H (xs,ox,)=0

so this follows from

s

This immediately gives that

=

so we get a). To prove c) we consider the

1

ving that multiplication by

Rp*OX'

"'

Rp*WOX , = WO'x which in turn gives b). As

nal double points spectral sequence

12'.p -

2 ( p-1 ) ( ( L. L) p + ( wX. L) p + c ( X) ) .

2 Indeed, as the singularities of "'X are rational

gives

i .

is an isomorphism for all

1 H (X,L2P(1))

1 > H (X' ,L2p(1))

2

> H (X' ,L2p(1))

1 H (X' ,OJ p ) i s a n i so morph i s m a nd is surjective on the kernel of p that Coker T is torsion free. I

[Li: Thm 12. 1].

is injective. As

is an isomorphism 1

H (X,11-1P)

>

2 H (X,OJ p ) >- H2( X' , 0-1 p ) a n i nj e ct i on , T and injective on the cokernel which shows 2 2 claim that T H (X,(6 ) > T H (X' ,Oim) i s P

an isogeny. Indeed by vii) we can replace diagram -

X ---.>-

-X

"'

p

X by X and then we have a

>-X

\,...,,, F \ "' '{ X

m

> X ~>- X

105

2 where X > X and X > X are blowing ups.As T H (~,~) is a biraP m tional invariant on smooth, proper surfaces we conclude. Hence the rank of Coker t equals the rank of Pie X' minus the rank of Pie X and this difference is equal to the number of irreducible components of the exceptional fibers. As we have an Ap_ -singularity we get p-1 such for each ...-...._

_......._

rv

1

singular point so we conclude by viii). We can now compare invariants of x) As ~ume A and C.

a)

c ( x' ) 1

= p ( c~ (

X and X'.

x) + 2 ( p - 1 ) ( (.i)x•L ) + ( r - 1) 2 ( L • L ) ) •

2 1 1 b) H (X,WOX) > H (X',WOX,) is an isomorphism and H (X,WOX) > H2(X' ,WOX,) is an injection with a V-torsion free cokernel killed by p . The dimension of the cokernel as Dieudonne module is p(p-l~(p- 2 )(L.L) + p( ~- l ) (wX ® L- l . L ) c) H1 (X,7l (1)) p

+ ( p-1) X •

2

> H1(X' ,ll (1)) is an isomorphism and p

H (X,7l p (1))

H2 (X' ,ll ( 1)) is an injection with torsion free cokernel o.f rank p 2 ( p - 1) ( ( L • L ) p + p ( wX • L ) + c ( X) ) • 2

This is just putting together ix), v), vii), R-R and A to compute •

X(L- 1 ).

Remark : c) was essentially proven by W. Lang under the stronger assumption B (cf. [La]). Xi) Assume

A

and B • Then

Ho ( X.n~) = Ho (X .~)

Indeed, we have an exact sequence on 0 ----.>-

TI*

.

rv

X

L-p

0

-p

O

-1

1

-p (TI* L ) =

It is c1ear that we a re finished if H ( TI* L ) = H ( TI* L ) = H 0 and 0 0 H (n!) = H (TI*n!), By projecting down to X we see that the first part is implied by A and as TI* TI*n! = TI*°'>< ®o n! the second is imp l i ed by B if X

106

•

we use the filtration

{M

1

} •

0 xii) Assume C. Then ever y closed form in H (n~,) is in the image of HO(~). Proof : The statement is Zari ski 1oca 1 on X so we ma_y assume that 1 is . of the form Spec(OX[t]/(tP-f)) for fE OX. Then we have a cartesian diagram ,...,, X

i where

F is the Frobenius. We have exact sequences 0

1 > p*~1

1

> p*nX'

i

sf 0

>

1 n/A 1

1 > p*nx '//A 1

>

~

i 1

>-

---.>- 0

~/fA1

and it is sufficient to show that the image in

1

p*nX'/~1

of a closed form

cones from ~;~1 • However, [Sz:Exp. V Le11me 2) shows that it actually comes from n~/~1 xiii) Asswne

as the global assumptions are never used there.

A , B and C .

Then the closed

the injective mapping

1-forms

HO(X, n~)

of

X' are exactly those in the image of > HO(X' ,Q~ 1 ).

As it is obvious that all forms in the image are closed this follows from xi) and xii·). 2 We wil 1 now further specialize to X=IP and L = O(n) n >0 . Then A and B are fulfilled and a simple dimension count shows that we can find sE O(pn) such that 0(-pn) > fl ~2 has only isolated simple singularities.

107

1 H (X' ,WOx,)

From x) we then get that 2 H (WOX,)

is killed by

p

= H0 (wn!,) = H0 (wn~,) = o ,

that

1 and is without finite torsion,that H (\JniJ is tor-

sion free of slope O and that it equals Pie X' ®71. W. Now Pie X' contains the inverse image of a hyperplane and all the exceptional divisors. The square of the inverse image of a hyperplane is p and as the singularities are of type Ap- 1 each connected component of the exceptional divisors gives an intersection matrix of determinant (-1)P- 1p . Hence the p-adic ordinal of the discriminant of the group of cycles generated by the cycles 2 2 that has ju st been des c r i bed i s 1 + p n - 3p n + 3 . By i x ) this group has finite index in Pie X' so Pie X' has discriminant of p-adic

1 1 H ( x , wn ) • If

a


O • h O( n! , )

by

and so

r-'. o (i)(-i+1) =

JE7l

( x ) , b)

we get :

=/ . a ( - i )( i + 1 ) ~ } . a ( i )( - i + 1) > lEW m

p(p-1~(p-2)n2 _ p(~-1 )(n+ 3 ) + p-l _ ~(p2n2+ 3pn- 4 ) _ p[(p-1 )~p-2)-3p]n2-

p(p+5)n - 3p2-7p

4

4

1 +

•

Also by xi) and xii)

2 duality d : H (OX ,) o( i) = 0 if

d:

Ho (n x1,)

2 1 > H (S"2X , )

>-

Ho (n ~,)

is injective and so by

is surjective . This shows by (III: 3. 3) that

p-1) then we can get more crystalline torsion than classical torsion for two reasons. First the example (cf. [Ra]) of Raynaud gives a non-flat N and so the order of N at the generic point is strictly smaller than its order at the 2 special point and so the length of H (X' .~P) is strictly smaller than the 2 length of the finite torsion of H (X/W). Secondly we will see in a moment that we may also get exotic torsion. Remark : Raynaud's example is constructed starting from a finite IFP-vector space V, a non-degenerate alternating pairing ~ on V and the associated Heisenberg group N . If one lets dim V~ 6 and considers the split extension G= N~ Sp(~) then we may replace N by G in Raynaud' s argument and

109

then the general fiber of X will have a perfect fundamental group so 2 H (X' ,?l t ) will be torsion free for all primes t (including p) but H2 (X/W) will have torsion. As our first example of how to produce exotic torsion let us again consider a line bundle L on a smooth and proper surface fulfilling condition A of 1 and n: X' ~> X the resolution of an a.L-torsor for which -p ---a.> n1 has only simple isolated zeroes. L X

I claim first that

takes HO(X,Znn!)

n*

Indeed, this only uses the fact that dC: znn 1

kernel of n

>- n 2

and

n

n*: HD(x,n~)

into HO(X',Zn+l n~J.

is inseparable ; Zn+ 1n > Ho( X',n~,)

is inseparable. I further claim that n*: HO(X,z 00n!)

1

is the

is zero as >- HO(X',Z00n!J is

By [Il 1:0.2.5.3.3-5] it will suffice to show that n* 1 induces an isomorphism on global sections of ffi m/pffi m and Bnn for all n. The first part follows from (1;x). For the second part we get from [11 1: I.3.11.4] that it will be sufficient to show that n induces an isomorphism on the kernel of F on H1(-,WnO). However, from (1 ;x) it follows that

an isomorphism.

n*

induces an isomorphism already on

of xi) that

A implies that

We therefore see that

1 H (-,Wn0). It is clear from the proof

n*: HD(x,n!)

> HO( X',n!J

is injective.

dimkHO( X',Zn+l n~J / HO(X',Z 00n!J 2. dimkHO(X,zn n~) /

0 1 H ( X, Z00nX) •

For

n= 0

II.6.16], X'

we get that if

then , by [ I 1 1 :

has exotic torsion .

We can take as of

X has non - closed 1-forms

X one of the coverings of IP

2

1 , from the formula given there we see that if

always find such an

X

with non- closed

constructed at the end p >7

then we may

1-forms .

As another type of example let us consider a smooth and proper surface X with a line bundle L s. t H1 ( x·, L- 1) = O and s. t there is a Lef chetz penc i 1

11 0

1

be this pencil ( where y is a blowing 1 is a locally free sheaf on IP hence isomorphic for some integers r i . I claim that r. < 0 for all i . ~> IP ( V)

= IP

l -

Indeed, let CcX be a curve in IV . Then the strict transform of C on X' is isomorphic to C as on 1y smooth points of C are blown up and we have a commutative diagram

1 H (x,OX)

By assumption

1

1 :,. H (c,OC)

1 :,. H (C,Oc). Let

H (Y ,Oy)

( canon i ca 11 y) isomorphic to

is injective and hence so is

x :=(l)(C). Then 1

1 HO (IP , R (l)*O y)

is injective. This clearly implies that f : IP

1

>- IP

1

1 H (Y ,Oy) 1 >- ( R (l)*0 y) x

1 :,. H (c,Oc)

i _s

which therefore

< 0 for al 1 i . Let now 1-:--

r.

be a non-constant morphism. We then get a diagram Y"

where the square is cartesian and T a (minimal) resolution of singulari1 ti es. As R c.p*O y ~@ 0 ( r. ) , r. i

1

1 HO(JP ,f*R (l)*0Y)

ri

l

is an isomorphism and irrespective of the values of the

1 1 1 H (IP , R ,~n* 0 y )

1 1 f*R (l)*0 y = R p*O y 1 H ( Y , 0 y)

l -

1(IP 1 f * R1 0 ) ----.. H >, c.p* y

. a monomorph ism. . By base change 1s

and as is we 11 known

RT*O Y" = Oy, • Hence we get that

1

:,. H ( Y11

,

0 y 11 )

is an isomoY'phism and H2 ( Y , 0 y)

is a monomorphism .

:,.

111

As in the proof of ( 1, iv) this implies that is an isomorphism for all

n

and so as above that

►

1 H ( Y , WnO)

1 H ( Y" ,WnO)

>

HO(Y, B001"2 ~)

is an isomorphism. I claim that there is a bound for the which depends only on

not on

f .

Indeed, we have taken care of B n1 so as above it suffices to bound 00

3

the order of

PPic X( ). It is clear that this equals

1 1 Pf(k(IP ) ,Jae( Y" /lP ))

whose order clearly is bounded by the order of the kernel of p on the Jacobian of a geometric generic fiber of ~ so in particular bounded by something depending only on ~. •

1

Let now

f

be

Fi

and let

x(P )

be the resolution of the pullback •

1

of

Y

by

Fi • Then as we have seen the dimensions of the

H0 (x(P )

,z n1) 00

are bounded. •

If we can show : ( * ) F or every

n the dimensions of

HO(X(p l) ,znn1)

~>- oo

we get a situation similar to the one discussed , on o. 111 .

By• the same argument as before to show(*) it suffices to show that the Ho(X(pl) ,n1) ----> oo .but this is proved in [Sz:Exp. 5]. •

( p1)

Remark : By choosing X and V suitably we may make all the X liftable. Indeed if we start with X and V lifted the Lefschetz pencil 1 1 ~: X' >IP lifts and as Fi on IP lifts the pullback by Fi of ~ lifts. The singularities of this pullback are rational so we may simultaneously resolve the singularities in the fibers of the lifting (cf. [Ar]) so • the X( p

1

)

lift. In the last step we may be forced to ramify the base of the lifting so even if we start with an unramified lifting, which wecertaimy may, we have no guarantee that we will end up with one.

112

3. Let now X be a smooth and proper variety over a perfect field k of • characteristic p> 0 • We will say that X fulfills RH if there lS a smooth and proper family T > S with S of finite type over IFP , a morphism ~=Speck > S such that Xk is isomorphic to the pullback along ~ of T and such that for every closed point x of S the fiber Tx has the property that the eigenvalues of the Frobenius morphism Fx with respect •

to

k(x)

on

H~ris(Tx/W(k(x))) are algebraic with all its absolute values 2 equal to k(x)li/ for all i . It is no doubt true that X always fulfills RH but at the moment this is known only when X is dominated by a smooth and projective variety (possibly over an extension of k) (cf. [KaMe]) and so in particular when dim X 0 as it equals respectively h1 ' 1 (lifting), w 1 1 1 1 m' or h ' . Ho we ver , as observed by Szpi ro ( cf . [ Sz ] ) i f X >- C lS a non-isotrivial smooth fibration with C a smooth curve then for the sequence X(n) := Xxcc, with Fn: C >- C the second projection,of smooth and proper surfaces, the Betti numbers are constant and C~ > oo so 1 1 hw, > - oo and we have examples of surfaces which can be deformed to neither liftable nor Hodge-Witt nor Mazur-Ogus surfaces. •

Corollary 3.3 enables us to give a proof of a result of Nygaard which very closely resembles the proof given by Serre in characteristic 0 .

11 5

Proposition 3.4.[Ny 1] Let X be a smooth and pr oper 3- f old and suppos e t hat ther e exists a dominant rat i onal separable mapping

0 hi 'O = O as hi,O > O and by (3.3) hO,i=O · for i > O. w w w =C (-1)ih0,i = 1 Then Crew's formula gives x (O) X w · 1

Remark: Only a very small part of the general theory is really needed in the proof. 4. We will end by giving a generalization of a part of Nygaard s proof of the Rudakov-Shafarevich theorem. 1

Theorem 4.1. Let

X be a smooth and pr oper sur face over k, perfect of char acter istic p > 0 and suppose that the rank of the Ner on-Severi group of 2 b (X). If h~• (X) is odd then the dimension of 2 2 is str ictly smaller than h~• . X equals

Imd=:HO(X,n~)

Remark. i) As will be seen in the proof this dimension is always less than or equal to ho, 2 so we exclude the possibility of an equality. w ii) In Nygaard s case h0 ' 2 = 1 so his conclusion is that all the 1-forms 1

w

are closed. Proof: By duality we need to prove that the codimension of H2(X,OX)

is less than

h~•

2

• The image of

00

2

V- ZH (X,WOX)

Ker d in in

lies in the kernel so this codimension equals the codimension of 2

00

2

2

(H (X,WOX)/V- ZH (X,WOX) )/VH (X,WOX) in

2

00

2

2 H (X,OX)

Ker d:

2 2 2 > H (X,WQ~)/dVH (X,WOX) + VH (X,WQ~)

2

(H (X,WOX)/V- ZH (X,WOX)) / VH (X,WOX)' as

[ 11-Ra ]) ,. which has dimension

2 h~ •

as

R 111~RI'(WQ)() = Rf ( Q)() (cf. 1 2 0 m • = O by the condition on the

Neron-Severi group. Hence if the theorem is fa 1se is the type of

0 2 dom (H (X,WOX)

2 > H (x,ws~ ))

Ker d = O and so if o

we get from (111:3.3) that

116

a(i

) =0

if

i E 7l

However as

+ ord disc NS/tors= 2 } . i a ( i) p 16N'

tors

and as . NS

&

74_

NS

8

1 W= H ( X,

and by

2 = H ( Xk ,74_ ( 1))

1

wn

a ( i) = 0

we get by

NS/tors_ is unimodular. By the R-R theorem 2 and so by ([Se:V:Thm 2]) K = -r (NS/tors) (8)

)

we get from (II I : 3. 3) that 'v'i E 71..+

i -adi c KF = F where

2

we get

pf di SC NS/

Poi ncare dua 1 i ty that (2)

-r (-)

for every

FE NS

is the index but

-r (NS/tors) = -b +2 • Noether' s formula gives 2 2 1 12x = c 2+K = -b +2+c = 4+4h~ • ( 8) and together with Crew's formu 1a (cf. 2 2 0 1 0 2 0 1 0 2 (iV:3.2)) this gives 1-hw' +h w' = 1+hw' (2) i . e . hw' = O (2) contrary

by Hodge ' s index theorem

to assumption.

VII

Complements

I will quickly sketch some possible extensions of the theory. L

b

L

1. One could, as in [Ny], study Rn QtR M , ME Dc(R), instead of R1 ®RM . If we put h~'j(M) := lgth Hj(Rn ~~ M)i then we get, with the same proof as for

n= 1 ,

C

•

•

•

(-1)Jh~,J = n

C

•

•

•

(-1)Jh~,J

•

•

•

•

and so nh:,J ~ h~,J . From this

1

follows that the

n-Hodge •

polygon, Hodge(h~,m-•)

lies below the Hodge-Witt.

•

nb m(M) = .C hn1 ,J we get the expected result, obtained by Nygaard 1 +J=m in [Ny] in the geometric case. This could be proved by redoing our argument for arbitrary n or by paraphrasing Nygaard's argument. (The last thing could of course al so have been done for n = 1 . ) Of course even if we paraphrased Nygaard's argument we would have to do it for 'ifTI(M) and not for M. In case

2. We have not treated the conjugate spectral sequences. They can in fact be . b defined for any ME Dc(R). For n = 1 we have the sequence of k-complexes ... > T< i Hodge(M) > T .•. and the spectral sequence associated to such a "filtration" is the conjugate spectral sequence of level 1 of M . The ge·neral ization to arbitrary levels is immediate and then we pass to the limit as in [Il-Ra] and, again as in loc. cit., we define F' 2 and v•. We can th.en prove the coherence of the E -term by d~vissage and explicit computation or paraphrasing loc. cit. One can then with the methods used in this paper prove analogues to the results obtained on the slope spectra.1 sequence.

11 8

3. If one wants to continue the study of the spectra 1 sequence for •

one should note that it is really the degree of F-gauge structures •

( 3. 1)

•

•

E~ 'J = S( HJ ( M)

b ME Dc ( R)

•

oo

part of a spectral sequence

•

1

)

Note that,from the E2-term on,it is a spectral sequence of coherent F-gauge structures and is the spectral sequence associated to the cohomological functor HO(~(-)) on D~(R) with its standard t-structure. Furthermore, we can consider this spectra 1 sequence as a spectra 1 sequence in G and so of R-cornplexes. As such it is the spectral sequence associated to the cohomological fut1ctor HO(-) on Db(R) with the standard t-structure. Note g C b , finally that even though the standard t-structure on Dc(R) commutes with the diagonal t-structure and the diagonal t-structure commutes with the F-gauge t-structure, (~o,~o), the standard t-structure does not commute with the F-gauge t-structure, because if it did, by the description just given, (3.1) would degenerate at E which it does not always do (cf. 2 [Ek 1:V.2]). Hence commutation among t-structures is not a transitive relation.

BIBLIOGRAPHY

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[B:O] : P. BERTHELOT, A. OGUS.- Notes on crystalline cohomology. Princeton University Press. [Cr] : Thesis. Princeton University 1981 [De] : P. DELIGNE.- La conjecture de Weil. II Publ. IHES 52 (1980), p. 137-252. [Ek 1] : T. EKEDAHL.- On the multiplicative properties of the de Rham-Witt complex. I Ark. for mat. Vol. 22:2 (1984), p. 185-239. [Ek 2] : T. EKEDAHL.- On the multiplicative properties of the de Rham-Witt camp l ex . II Ark . for mat . Vol . 23 : 1 ( 1985 ) . [Il 1] : L. ILLUSIE.- Complexe de de Rham-Witt et cohomologie cristall ine. Ann. scient. de l'ENS, 4e serie, 12 (1979), p. 501-661. [Il 2] : L. ILLUSIE.- Report on crystalline cohomology. Proc. of symp. in pure math. Vol. 29 AMS. [11-Ra] : L. ILLUSIE, M. RAYNAUD.- Les suites spectrales associees au complexe de de Rham-Witt. Publ. IHES 57 (1983), p. 73-212.

120

[Ka] : N. KATZ.- Slope filtrations of F-crystals. Asterisque 63 (1979), p. 113-164. [Ka 1] : N. KATZ.- On the ubiquity of pathology in products. Progress in Mathematics vol. 35, Birkhauser, p. 139-154. [Ka-Me] : N. KATZ, W.MESSING.- Some consequences of the Riemann hypothesis for varieties over finite fields. Inv. Math. 23 (1974), p. 73-77. [La] : W. LANG.- Remarks on Comp. Math.).

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3-fold.

I. SLN 341, p. 85-147.

[Ra] : M. RAYNAUD.- "p-torsion" du schema de Picard. Asterisque 64 (1979), p. 87-148. [Se] : J.-P. SERRE.- A course in arithmetic. Springer Verlag. [Sz] : L. SZPIRO.- Seminaire sur les pinceaux de courbes de genre au moins deux. Asterisque 86 (1981).

•

....

71I011PANS

•