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The Exact Poincaré Lemma in Crystalline Cohomology of Higher Level
Journal of Algebra 240, 559᎐588 Ž2001. doi:10.1006rjabr.2001.8749, available online at http:rrwww.idealibrary.com on
The Exact Poincare ´ Lemma in Crystalline Cohomology of Higher Level Bernard Le Stum IRMAR, Uni¨ ersite´ de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France E-mail: [email protected]
and Adolfo Quiros ´1 Departamento de Matematicas, Uni¨ ersidad Autonoma de Madrid, ´ ´ Ciudad Uni¨ ersitaria de Cantoblanco, E-28049 Madrid, Spain E-mail: [email protected] Communicated by Laurent Clozel Received December 10, 1999
We prove a formal Poincare ´ Lemma in crystalline cohomology of higher level and we derive a de Rham interpretation of crystalline cohomology of higher level. We also define the Spencer complex of higher level and show that it can be used to compute the cohomology of arithmetic D-modules. 䊚 2001 Academic Press CONTENTS 1. The de Rham complex. 2. Integration of differential forms of higher le¨ el. 3. The Formal Poincare´ Lemma in higher le¨ el. 4. The Poincare´ Lemma in crystalline cohomology. 5. Complements and applications.
INTRODUCTION As long as the theory is concerned, crystalline cohomology generalizes well to higher level Žsee wB3, E-LS, Q-LS, Tx, for example.. Unfortunately, 1
Partially supported by DGICYT Project PB 96-0065. 559 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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the main tool for computing crystalline cohomology, namely, the usual de Rham complex, cannot be used in higher level. Following Lieberman wLx, Berthelot suggests in wB3x to use the so-called Jet Complex. This is what we call in this article the de Rham complex of higher level. In crystalline cohomology, the main ingredient in order to show that the de Rham complex computes de Rham cohomology is the Formal Poincare ´ Lemma. This is also the case in higher level and we show that crystalline cohomology of higher level can be computed as the cohomology of a complex whose terms are locally free, the de Rham complex of higher level. We are also able to define a Spencer complex in higher level and prove that the ˆXŽ m. cohomology of a D r S -module agrees with the cohomology of the corresponding m-crystal. Finally, using Berthelot’s Frobenius descent, we give an explicit quasi-isomorphism between the de Rham complex of level m of an m crystal and the de Rham complex of level m q 1 of its Frobenius pull-back. In a forthcoming article, we will consider the Filtered Poincare ´ Lemma and compute the cohomology of T-m-crystals. We will apply this to the study of the Hodge filtration on Dieudonne modules of level m. Con¨ entions We let p be a non-zero prime and m g ⺞. When m / 0, all schemes are ⺪ Ž p.-schemes. If I s Ž i1 , . . . , i n . g ⺞ n we will write < I < s i1 q ⭈⭈⭈ qi n and, more generally, we use the classical conventions for multi-indexing.
1. THE DE RHAM COMPLEX For the notion of m-PD-structures the fundamental reference is wB4x. 1.1. DEFINITION. We develop an idea introduced by Berthelot in wB3x, following Lieberman wLx. More precisely, we explain the construction of the de Rham complex of higher level. Let S be a scheme and X a smooth S-scheme. Everything below will be done relative to S but we will drop the reference to S in order to lighten the notations. We write PX mŽ r . for the m-PD-envelope of X in X Ž rq1.rS, d i : PX mŽ r q 1. ª PX mŽ r . for the projection that forgets the Ž i q 1.st component, and si : PX mŽ r y 1. ª PX mŽ r . for the degeneracy arrow that repeats the Ž i q 1.st component. We also write pi : PX mŽ r . ª X, for the ith projection and ⌬: X ª PX mŽ r . for the diagonal embedding. We consider the simplicial complex of schemes PX m Ž ⭈ . [ PX m Ž r . , d i , i s 0, . . . , r ; si , i s 0, . . . , r 4 ,
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
561
and the usual complex PX mŽ⭈. built out of it. Thus, PX mŽ r . is the structural sheaf of PX mŽ r .. By definition, the normalized complex associated to PX mŽ⭈. is N PX mŽ⭈. [ lKer sUi . Recall that if A is a Žnot necessarily commutative. ring and M is an A-A-bimodule, then the tensor algebra T Ž M . is defined in the obvious way. We will use this vocabulary when A is commutative and M is endowed with two Ždifferent. A-module structures, called ‘‘structure on the left’’ and ‘‘structure on the right.’’ We might sometimes call this tensor algebra the ‘‘non-commutative’’ tensor algebra. For example, as a graded algebra, PX mŽ⭈. is canonically isomorphic to the non-commutative tensor algebra T Ž PX mŽ1... If we call IX mŽ1. the m-PD-ideal of PX mŽ1., one easily checks that N PX mŽ⭈., which is a differential sub-algebra of PX mŽ⭈., is isomorphic as a graded algebra to T Ž IX mŽ1... Finally, if KX⭈ m denotes the differential ideal in N PX mŽ⭈. m generated by IX mŽ1. p q1 4, we call KXⴢ m ⍀ ⴢX m [ N PX m Ž ⭈ . rK the jet complex of order p m or the de Rham complex of le¨ el m of XrS. Note that, as a graded algebra, ⍀ ⴢX m is a quotient of T Ž ⍀ 1X m .. We will make this more precise later. We can also define the de Rham complex of a DXŽ m.-module F as follows. Recall that, for each k, DXŽ m. k denotes the sheaf of operators of level m and order at most k. By definition, this is the dual to the sheaf of principal parts PXk mŽ1. and DXŽ m. is the union of the DXŽ m. k when k varies. The action of DXŽ m. on F induces for all k a pairing DXŽ m. k = F ª F that corresponds to a morphism d: F ª F mO X PXk mŽ1.. In the particular case k s p m , we have ⍀ 1X m ; PXk mŽ1. and d takes its values in F mO X ⍀ 1X m . F ª F mO X ⍀1X m . It is Thus we get what could be called an m-connection d:F m-integrable: using the Leibniz rule, it extends to a differential on F mO X ⍀ ⴢX m . This is the de Rham complex of F and we can define the de Rham cohomology of le¨ el m of F : Hdi R m Ž X , F . [ H i Ž X , F mO X ⍀ ⴢX m . . 1.2. Description of the de Rham Complex. Unless otherwise specified, we will always see PX mŽ r ., N PX m , etc., as left OX -modules using pU1 , and tensor products will be over OX . Let us now describe the differential on PX mŽ⭈., N PX mŽ⭈., and ⍀ ⴢX m . As usual, the differential on OX is induced by the map OX ª OX mOS OX , f ¬ 1 m f y f m 1.
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It is also convenient to call ␦ : PX mŽ1. ª PX mŽ2. the ring homomorphism dU1 induced by the map OX mOS OX ª OX mOS OX mOS OX , f m g ¬ f m 1 m g . The differential on PX mŽ1., IX mŽ1. and ⍀1X m is given by d Ž ␣ . s 1 m ␣ y ␦ Ž ␣ . q ␣ m 1. For higher degree, we can use the Leibniz rule: d Ž ␣ m  . s d ␣ m  q Ž y1 .
de g␣
␣ m d .
Note also that the non-commutativity of the tensor product means that if ␣ ,  g PX mŽ1. and f g OX , then ␣ m f  s f ␣ m  q d Ž f .. ␣ m  , where the regular product is taken in the monoid PX mŽ1., IX mŽ1. or ⍀ 1X m . We assume now that we have local coordinates t 1 , . . . , t n on X and write i s d Ž t i .. Then PX mŽ1. Žresp. IX mŽ1.. is the free module on the generators J 4 Žresp. with < J < ) 0.. It follows that PX mŽ r . Žresp. IX mŽ r .. is the free module on the generators J14 m J 2 4 m ⭈⭈⭈ m J r 4 Žresp. with < J1 <, . . . , < Jr < ) 0.. Note that when < J < - 2 p m , we have J 4 s J and we will therefore drop the 4 . We will write dt i for the image of i in ⍀ 1X m . Then we see that ⍀ 1X m is the free module on the generators Ž dt . J with 0 - < J < F p m . Note that, since Ž dt . J 4 s 0 whenever < J < ) p m , such terms will never play a role in our calculations. To lighten the notations, we will therefore not make explicit the corresponding conditions of the type 0 - < J < F p m , in the hope that this will make it easier to read without baffling the reader. The local description of ⍀ rX m for r ) 1 needs some more care. Note first that ⍀ ⴢX m has a natural structure of graded algebra. We keep using the tensor notation for this structure Žusing the wedge product, as in the case m s 0, would be misleading.. Actually, ⍀ ⴢX m is the quotient of T Ž ⍀1X m . by the ideal generated by the images of the d w J 4 x with p m - < J < F 2 p m . Since T r Ž ⍀ 1X m . is the free module on the generators Ž dt . J14 m Ž dt . J 2 4 m ⭈⭈⭈ m Ž dt . J r 4 with 0 - < J1 <, . . . , < Jr < F p m and d is given by the classical formula dŽ J4. s
Ý 0-V-J
J V
¦ ;
JyV 4
m V 4 ,
we see first that is generated by the Ž dt . I m Ž dt . J with 0 - < I <, m < J < F p subject to the relations ⍀ 2X m
Ž ).
Ý 0-V-J
J V
¦ ;Ž
dt .
Jy V
V
m Ž dt . s 0
with p m - < J < F 2 p m .
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
563
For r ) 2, ⍀ rX m is generated by the Ž dt . J1 m Ž dt . J 2 m ⭈⭈⭈ m Ž dt . J r with 0 - < J1 <, . . . , < Jr < F p m subject to the same relations, tensored on the right and on the left by elements of the form Ž dt . J1 m Ž dt . J 2 m ⭈⭈⭈ m Ž dt . J i . We also have a local description of the m-connection d: F ª F mO X ⍀1X m corresponding to a DXŽ m.-module F : it is given by d Ž x . s ⌺⭸ wU xŽ x . m Ž dt .U where the ⭸ wU x are the divided partial derivatives. In general, the de Rham complex is infinite. However, we will show that it has locally free terms. First, we need a lemma. 1.3. LEMMA. Unless J s 0, . . . , 0, 2 p m , 0, . . . , 04 , in which case there is only one term, there are at least two different terms with in¨ ertible coefficients in the formula Ž).. Proof. If we write J s j1 , . . . , jn4 we may assume by symmetry that j1 G ⭈⭈⭈ G jn . If j1 ) p m we let V s Ž p m , 0, . . . , 0.. Then it follows from wB4, Lemma 1.1.3x that the coefficient ² VJ : is invertible. If not, there exits i such that j1 q ⭈⭈⭈ qji F p m and that j1 q ⭈⭈⭈ qjiq1 ) p m. In this case, we take V s Ž j1 , . . . , ji , 0, . . . , 0. and again ² VJ : is invertible. By symmetry, unless J s 2 p m , 0, . . . , 04 , in which case our relation simply says that m m Ž dt 1 . p m Ž dt 1 . p s 0, there are at least two different terms with invertible coefficients. 1.4. PROPOSITION. The de Rham complex of le¨ el m, ⍀ ⴢX m is a complex whose terms are locally free of finite rank. Proof. This is a local question. Thus, we may assume that we have local coordinates and use the notations of Subsection 1.2. For r s 0 or 1, there is nothing to do. Assume now r s 2. For different J ’s with p m - < J < F 2 p m , the relations involve different generators Ž dt .U m Ž dt .V. We know from the lemma that at least one of the coefficients in each relation is invertible. It follows that ⍀ 2X m is free. More precisely, for each I with p m - < I < F 2 p m , we can choose a couple Ž AŽ I ., B Ž I .. such that I s AŽ I . q B Ž I . and that the set U
V
S s Ž dt . m Ž dt . : Ž U, V . / Ž A Ž I . , B Ž I . . 4 is a basis for ⍀ 2X m . We consider now the case r s 3. From the case r s 2 we know that
Ž dt . U m Ž dt . V m Ž dt . W : Ž V , W . / Ž AŽ I . , B Ž I . . 4 is a set of generators for ⍀ 3X m , and this module is defined by the following relations for p m - < J < F 2 p m and < K < F p m :
Ý 0-V-J
J V
¦ ;Ž
dt .
Jy V
V
K
m Ž dt . m Ž dt . s 0
with K / B Ž I .
LE STUM AND QUIROS ´
564 and
Ý 0-V-J , V/AŽ I .
J V
¦ ;Ž
J AŽ I .
¦ ;
y
dt .
JyV
V
m Ž dt . m Ž dt .
Ý
0-V-I , V/AŽ I .
I V
¦ ;Ž
dt .
BŽ I .
Jy AŽ I .
V
m Ž dt . m Ž dt .
IyV
s 0.
Using the relations with K / B Ž I ., which all involve different generators, we can write Ž dt . AŽ J . m Ž dt . BŽ J . m Ž dt . K as a linear combination of the other terms. It follows that
Ž dt . U m Ž dt . V m Ž dt . W : if W s B Ž I . then V / AŽ I . , and if W / B Ž I . then Ž U, V . / Ž A Ž I⬘ . , B Ž I⬘ . . 4 is a set of generators for ⍀ 3X m . We can rewrite the second sum in the remaining relations in terms of these generators. Now we know from the lemma that, in these relations, at least one of the coefficients ² VJ : with V / AŽ I . is invertible and we can write the corresponding term as a linear combination of the other generators, getting a basis of ⍀ 3X m . Finally, the case r ) 3 easily follows from the above description of ⍀ rX m and the case r s 3. 1.5. Linearization. We consider now a construction analogous to that of Definition 1.1 that will be ‘‘linear’’ in OX . This is a standard technique: we look at the shifted simplicial complex PXqm Ž ⭈ . s PX m Ž r q 1 . ; d i , i s 1, . . . , r q 1; si , i s 1, . . . , r q 1 4 Ž. and we consider the associated complex of abelian sheaves Pq X m ⭈ on X. Ž . As before, we define the normalized complex N Pq ⭈ [ lKer sUi . Xm q Ž .. q Ž . Ž As a graded algebra, PX m ⭈ resp. N PX m ⭈ is isomorphic to PX mŽ1. mO X T Ž PX mŽ1.. Žresp. PX mŽ1. mO X T Ž IX mŽ1... Taking the quotient by PX mŽ1. mO X KXⴢ m , we get a differential complex PX mŽ1. mO X ⍀ ⴢX m which is a quotient of PX mŽ1. mO X T Ž ⍀1X m .. By analogy with the usual de Rham complex, we will write ⍀ Pⴢ m s PX mŽ1. mO X ⍀ ⴢX m . The differential on ⍀ Pⴢ m is given by the Leibniz rule dŽ ␣ m . s d␣ m q ␣ m d , where d: PX m Ž1. ª PX mŽ1. mO X ⍀ 1X m is the m-connection for the right structure induced by ␣ ¬ ␦ Ž ␣ . y ␣ m 1, and d:⍀ rX m ª ⍀ rq1 X m is the differential of the de Rham complex of level m defined in Subsection 1.2.
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
565
As before, we have the classical formula for moving coefficients inside products: if , g ⍀ 1P m and ␣ g PX mŽ1., then
m ␣ s ␣ m q Ž d ␣ . m . Note that, in local coordinates, d s 1 m dt, and that we have dŽ I4. s
Ý 0-UFI
I U
¦ ;
IyU 4
Ž d .
U
Žwhere Ž d .V s 0, when < V < ) p m . and d Ž Ž d .
I
. sy Ý 0-U-I
I U
¦ ;Ž . d
IyU
U
m Ž d . .
Of course, a set of generators of ⍀ rP m over PX mŽ1. is given as before by the Ž d . J1 m ⭈⭈⭈ m Ž d . J r with 0 - < J1 <, . . . , < Jr < F p m subject to the relations
Ý 0-V-J
J V
¦ ;Ž . d
Jy V
V
m Ž d . s 0
with p m - < J < F 2 p m ,
tensored on the right and on the left by elements of the form Ž d . J1 m Ž d . J 2 m ⭈⭈⭈ m Ž d . J i .
2. INTEGRATION OF DIFFERENTIAL FORMS OF HIGHER LEVEL We assume that we have local coordinates on X as in Subsection 1.2. As before, we write t s Ž t 1 , . . . , t n ., and we set ˆt s Ž t 1 , . . . , t ny1 . and ˆ s Ž 1 , . . . , ny1 .. Also, given an n-uple I s Ž i1 , . . . , i n ., we will write Iˆs Ž i1 , . . . , i ny1 .. There exists a unique linear map
2.1. PROPOSITION AND DEFINITION. h:⍀ rP m ª ⍀ ry1 P m such that J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
¡0
s
J
j
m Ž dˆ . s Ž dn . m Ž d .
J sq 1
m ⭈⭈⭈ m Ž d .
Jr
if Ž 1 . s s r and j s 0 or Ž 2 . j / 0 and Js / 0 or Ž 3 . p m ¦ i
~ Ž y1.
sy1
iqj i
y1
¦ ;
mniqj4 Ž d .
¢
J sy 1
J sq 1
J
I 4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
m ⭈⭈⭈ m Ž d .
J sy 1
Jr
if Js s 0, p m N i and either Ž 1 . 0 - j - p m or Ž 2 . j s p m and s s r .
LE STUM AND QUIROS ´
566
We will show below that this map is well defined and unique. Note that the formula makes sense because the modified binomial coefficient ² i qi j : is invertible when p m N i and 0 - j F p m wB4, Lemma 1.1.3x. We first want to look at some examples: 2.2. EXAMPLES. Ža. For r s 1 and n s 1, we get
¡ iqj Ž d . . s~¦ i ; ¢0
y1
h Ž
i4
iqj4
j
if p m N i otherwise.
This is the formula for integrating differentials of level m as we can easily verify: we have
Ž h( d . Ž i4 . s
Ý 0-uFi
¦ui ;h Ž
iyu4
u
Ž d . . .
If i s 0, this is an empty sum and we get 0. If not, we can write i s qp m q t with 0 - t F p m and we see that hŽ iyu4Ž d . u . s 0 unless p m divides i y u, which means that t s u since 0 - u F p m. Thus, we see that for i ) 0, t Ž h( d . Ž i4 . s i h Ž iyt4 Ž d . . s i
¦;
i t
y1
¦;¦;
t
t
iytqt4 s i4 .
Žb. For r s 1 and n s 2 the formula is quite similar j
h Ž 1i14 2i 2 4 Ž d 1 . 1 Ž d 2 . s
¡i ¢0
~
j2
.
q j2 i 4 i qj 4 1 1 2 2 2 i2
¦ ; 2
if j1 s 0 and p m N i 2 . otherwise
It corresponds to integration with respect to the second variable. Žc. For r s 2 and n s 1, we get a new situation; j
h Ž i4 Ž d . m Ž d .
¡ iqj ~ s ¦ i ; ¢0
k
.
y1
iqj4 Ž d .
k
if p m N i and j - p m , if p m ¦ i m
but our definition does not seem complete because hŽ i4Ž d . p m Ž d . j .
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
567
is not defined yet when p m N i. However, we have the following relations pm q j
Ý
jF¨ Fp m
¦
¨
;Ž . d
p m qjy¨
¨
m Ž d . s 0
that give the formulas
Ž d .
pm
j
m Ž d . s y
Ý
j-¨ Fp
m
y1
pm q j j
pm q j
¦ ;¦ ;
Ž d . p
¨
m
qjy¨
m Ž d .
¨
and therefore, if p N i, m
h Ž i4 Ž d .
pm
m Ž d .
j
. sy
Ý
j-¨ Fp m
y1
pm q j j
¨
pm q i q j y ¨ i
¦
=
pm q j
¦ ;¦ ; y1
;
p
m
qjy¨ qi 4
¨
Ž d . .
This formula may not look as natural as the one in the definition, anyway, we see in this case that h is unique and well defined. Looking at more complicated examples would not be easier than general case which we now try to explain. Let us first show that, in formula for h, and provided it is well defined, we can ‘‘move the inside.’’
but the the ni4
2.3. LEMMA. We always ha¨ e J
J sy 1
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
m Ž d .
J
j
m Ž dˆ . s Ž dn .
J sq 1
m ⭈⭈⭈ m Ž d .
J
sh ˆ I 4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
J sy 1
Jr J
m ni4 Ž dˆ . s Ž dn .
m Ž d .
J sq 1
m ⭈⭈⭈ m Ž d .
j
Jr
.
Proof. This is proved by induction on s, the case s s 1 being trivial. Using the formula for moving coefficients across the tensor products, we get J
h ˆ I 4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . J
J sy 1
J
s h ˆ I 4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
J sy 2
J
qh I 4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . J
j
m ni4 Ž dˆ . s Ž dn . m ⭈⭈⭈ m ni4 Ž dˆ .
J sy 2
J sy 1
J
j
m Ž dˆ . s Ž dn . m ⭈⭈⭈
m d Ž ni4 . Ž dˆ .
J sy 1
j
m Ž dˆ . s Ž dn . m ⭈⭈⭈ . It is therefore sufficient to show that, if Jsy1 / 0, then J
h ˆ I 4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
J sy 2
m d Ž ni4 . Ž dˆ .
J sy 1
m ⭈⭈⭈ s 0.
LE STUM AND QUIROS ´
568 Since
d Ž ni4 . s
Ý 0-uFi
¦ui ;
iyu4 n
u
Ž dn . ,
it is sufficient to show that for u / 0, we have J
J sy 2
h ˆ I 4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
J sy 1
m niyu4 Ž dˆ .
u
Ž dn . m ⭈⭈⭈ s 0.
By induction, this is J
h I 4niyu4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
J sy 2
m Ž dˆ .
J sy 1
u
Ž dn . m ⭈⭈⭈
which is 0 by definition, since u / 0 and Jsy 1 / 0. If it exists, the map h is unique. More precisely, for the
2.4. PROPOSITION. condition
Ý 0-V-J
J J J h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 V
¦;
m Ž d .
Jy V
V
m Ž d . m ⭈⭈⭈ s0
to be satisfied whene¨ er Jˆ/ 0 and jn G p m , it is necessary and sufficient that Ž writing j s jn . J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . sy
Ý
j-¨ -p m
J sy 1
y1
pm q j j
m Ž dn .
pm
ˆ
j
m Ž dˆ . J Ž dn . m ⭈⭈⭈
pm q j
¦ ;¦ ; ¨
J
= h I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . m Ž dn .
p m qjy¨
J sy 1 ¨
ˆ
m Ž dˆ . J Ž dn . m ⭈⭈⭈
when j ) 0 and that J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
J sy 1
J
m Ž dn .
s yh ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . y
Ý
0-¨ -p m J
pm
J sy 1
ˆ
m Ž dˆ . J m ⭈⭈⭈ ˆ
m Ž dˆ . J m Ž dn .
pm
m ⭈⭈⭈
pm
¦ ; ¨
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
J sy 1
m Ž dn .
p m y¨
ˆ
¨
m Ž dˆ . J Ž dn . m ⭈⭈⭈ .
Proof. First of all, note that the formulas we are going to prove do give a definition of h for all generators, although for generators of the type m
ˆ J J p ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 m Ž dn . m Ž dˆ . J m ⭈⭈⭈ ,
we only get a definition by descending induction on s.
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
569
Now, we look at the above condition
Ý 0-V-J
J J J h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 V
¦;
m Ž d .
Jy V
V
m Ž d . m ⭈⭈⭈ s0
with Jˆ/ 0 and j G p m . Writing ¨ s ¨ n , note that all the terms in this sum ˆ and that these two cases are mutually are 0 unless ¨ s j or Vˆ s J, exclusive. Thus, our sum splits into two sums. Actually, if j ) p m , we are only left with one sum
Ý
0-¨ -j
j
J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
¦; ¨
m Ž d .
jy ¨
J sy 1
¨
ˆ
m Ž dˆ . J Ž d . m ⭈⭈⭈ s0
and it gives us the required formula since all terms are 0 unless j y p m F ¨ - j. For j s p m , we get J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . q
Ý
pm
0F¨ -p m
J sy 1
ˆ
m Ž dˆ . J m Ž dn . ˆm m ⭈⭈⭈ p
J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
¦ ; ¨
J sy 1
m Ž dn .
p m y¨
¨
ˆ
m Ž dˆ . J Ž dn . m ⭈⭈⭈ s0 which gives us J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
J sy 1
m Ž dn .
J
s yh ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . y
Ý
0-¨ -p m
pm
J sy 1
ˆ
m Ž dˆ . J m ⭈⭈⭈ pm
ˆ
m Ž dˆ . J m Ž dn . ˆ m ⭈⭈⭈
J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
¦ ; ¨
pm
m Ž dn .
p m y¨
J sy 1
¨
ˆ
m Ž dˆ . J Ž dn . m ⭈⭈⭈ .
2.5. Remark. It follows from Lemma 2.3 that, unless Js s 0, p m N i, j s p m and s - r, we have J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
J sy 1
m Ž d . s Ž y1 .
sy 1 I 4
J
m Ž dˆ . s Ž dn .
J sq 1
m ⭈⭈⭈ m Ž d .
J J ˆ Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . sy 1
j
Jr
LE STUM AND QUIROS ´
570
J
m h ni4 Ž dˆ . s Ž dn .
j
J sq 1
m Ž d .
J
m ⭈⭈⭈ m Ž d . r .
Using the formulas in Proposition 2.4, we also see that we always have J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
J sy 1
m Ž d . s Ž y1 .
J
m Ž dˆ . s Ž dn .
J sq 1
m ⭈⭈⭈ m Ž d .
j
Jr
sy1 I 4
J J ˆ Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 J
j
m h ni4 Ž dˆ . s Ž dn . m Ž d .
J sq 1
m ⭈⭈⭈ m Ž d .
Jr
.
Moreover, if j / 0, then, J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . m Ž dn . s Ž y1 .
pm
sy1 I 4
m Ž d .
J sq 1
J sy 1
j J Ž dn . m ⭈⭈⭈ m Ž d . r
J J . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 Ž dˆ
m h ni4 Ž dn .
pm
m Ž d .
J sq 1
Ž dn .
j
J
m ⭈⭈⭈ m Ž d . r .
The map h is well defined.
2.6. PROPOSITION.
Proof. We need to show that our definition is stable under the relations induced by
Ý 0-V-J
J V
¦ ;Ž . d
Jy V
V
m Ž d . s 0,
for < J < ) p m . We first show that
Ý 0- V-J
J J JyV V J h I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 m Ž d . m Ž d . m ⭈⭈⭈ s 0 V
¦; ˆ
whenever < J < ) p m . As in the proof of Proposition 2.4, we write j s jn and ¨ s ¨ n . The case j s 0 is trivial and the case j G p m was done in Proposition 2.4. Thus, we are left with the case 0 - j - p m. As usual, our sum splits into two sums:
Ý
J h V
¦ ; ˆ
ˆ ˆ 0FV-J
I 4 i4 n
ˆ ˆ J J . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 m Ž dˆ . Jy V Ž dˆ ˆ
j
m Ž dˆ . V Ž dn . m ⭈⭈⭈ q
Ý
0F¨ -j
j
¦ ; ˆ ¨
h
I 4 i4 n
J J . sy 1 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ
m Ž dn .
jy ¨
¨
ˆ
m Ž dn . Ž dˆ . J m ⭈⭈⭈ .
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
571
Only the case p m N i has to be considered. The first sum reduces to J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . s Ž y1 .
s
s Ž y1
s
iqj i
q Ž y1
ˆ
I 4 d
J1
m ⭈⭈⭈ m Ž dˆ .
J sy 1
m Ž dˆ . J m niqj4 ⭈⭈⭈
I 4 d
J1
m ⭈⭈⭈ m Ž dˆ .
J sy 1
m niqj4 Ž dˆ . J m ⭈⭈⭈
y1
ˆ
y1
iqj i
s
j
m Ž dˆ . J m Ž dn . m ⭈⭈⭈
y1
¦ ; ˆ Ž ˆ. .¦ ; ˆ Ž ˆ. .¦ ; ˆ Ž ˆ. iqj i
ˆ
J sy 1
I 4 d
J1
m ⭈⭈⭈ m Ž dˆ .
ˆ
J sy 1
m dniqj4 Ž dˆ . J m ⭈⭈⭈ .
Now, we need to compute the second sum. We get
Ž y1.
sy1
Ý
iqj y¨ i
j
0F¨ -j
¦ ;¦ ¨
¨
y1 J
;ˆ
I 4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
J sy 1
ˆ
mniqjy¨ 4 Ž dn . Ž dˆ . J m ⭈⭈⭈ s Ž y1 .
sy1
q Ž y1 . s Ž y1 .
¦ ¦ ¦ ¦
sy 1
sy1
q Ž y1 .
iqj i
J1
J sy 1
Ž ˆ. ; ˆ Ž ˆ. ; Ý ¦ ; Ž ˆ. ; ˆ Ž ˆ. Ž ˆ. ; ˆ Ž ˆ. I 4 d
iqj i
iqj i
sy 1
y1
m ⭈⭈⭈ m d
y1
iqj ¨
0-¨ Fiqj
ˆ
m niqj4 Ž dˆ . J m ⭈⭈⭈ ¨
y1
J1
I 4 d
iqj i
ˆ
⭈⭈⭈ m niqjy¨ 4 Ž dn . Ž dˆ . J m ⭈⭈⭈
m ⭈⭈⭈ m d
J sy 1
ˆ
m niqj4 Ž dˆ . J m ⭈⭈⭈
y1
I 4 d
J1
m ⭈⭈⭈ m d
J sy 1
ˆ
m dniqj4 Ž dˆ . J m ⭈⭈⭈ .
Adding both formulas, we do get zero as expected. To finish the proof, we also need to make sure that
Ý 0-V-J
J h V
¦ ; ˆ
I 4 i4 n
J J p . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 m Ž dn . Ž dˆ
m Ž d .
Jy V
m
V
m Ž d . m ⭈⭈⭈ s 0
ˆ jn q p m .. Since the whenever < J < ) p m . We consider the n-uple K s Ž J, 2 differential satisfies d s 0, we always have
Ý 0-W-K
Ky W W K d Ž d . m Ž d . W
¦;
s
Ý 0-W-K
K W
¦ ;Ž . d
Ky W
m d Ž d .
W
,
LE STUM AND QUIROS ´
572 and it follows that
Ý 0-W-K
K h W
¦ ; ˆ
I 4 i4 n
J J . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 Ž dˆ
md Ž d . s
Ý 0-W-K
K h W
Ky W
¦ ; ˆ
I 4 i4 n
W
m Ž d . m ⭈⭈⭈ J J KyW . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 m Ž d . Ž dˆ
md Ž d .
W
m ⭈⭈⭈ .
In the first sum, if < W < ) p m , we obviously get 0. If not, then < K y W < ) p m and we can use the first part of the proof to conclude that the corresponding term is 0. Thus, we see that
Ý 0-W-K
K h W
¦ ; ˆ
I 4 i4 n
J J KyW . sy 1 m Ž d . Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ
md Ž d .
W
m ⭈⭈⭈ s 0.
Writing k s k n and w s wn , it is clear that all the terms are zero unless ˆ s K. ˆ Therefore, we have w s k or W ˆ ˆ Kˆ J J h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 m Ž dˆ . KyW ˆ W ˆ ˆ 0FW-K
Ý
¦;
ˆ
md Ž dˆ . W Ž dn . q
Ý 0Fw-k
¦wk ;h ˆ
I 4 i4 n
k
m ⭈⭈⭈
J J . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 Ž dˆ
m Ž dn .
ky w
w
ˆ
m d Ž dn . Ž dˆ . K m ⭈⭈⭈ s 0.
Using the first part of the proof, we see that all the terms in the first sum ˆ s 0 and k s p m . But in this case, < Kˆ y Wˆ < ) p m and we are zero unless W also get 0. Finally using Definition 2.1, we easily see that all the terms in the second sum are 0 unless k y w s p m. This gives us exactly what we want.
´ LEMMA 3. THE FORMAL POINCARE NIN HIGHER LEVEL We are going to prove in full generality a result which extends Example 2.2Ža.. First, we need a lemma.
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
573
3.1. LEMMA. With the notations of Section 2, we ha¨ e, for 0 - j - p m , J
h d Ž ni4 Ž dˆ . Ž dn .
j
J
s hd ni4 Ž dˆ . Ž dn .
. mP
j
m P.
Proof. Since h is linear and J
d Ž ni4 Ž dˆ . Ž dn .
j
. s d Žni4 . m Ž dˆ . J Ž dn . j . q ni4d
J
. Ž dn . Ž dˆ
j
.
,
it is enough to show that we can move the P outside both in hw d Žni4 . m Ž dˆ . J Ž dn . j . m P x and in hwni4d ŽŽ dˆ . J Ž dn . j . m P x. We first compute J
j
d Ž ni4 . m Ž dˆ . Ž dn . s
Ý 0-uFi
¦ui ;
iyu4 n
u
J j . Ž dn . . Ž dn . m Ž dˆ
Since j / 0 and j / p m , it follows from Remark 2.5 that we can move the P outside in hw d Žni4 . m Ž dˆ . J Ž dn . j . m P x. Now, we compute J
d Ž dˆ . Ž dn .
j
sy
Ý
0- Ž V ; ¨ .-Ž J ; j .
J V
j
¦ ;¦ ;Ž ˆ . d
¨
Jy V
Ž dn .
jy¨
V
¨
m Ž dˆ . Ž dn . .
Multiplying by ni4 on the left, tensoring with P on the right, and applying h, this expression splits into two sums as usual y
0FV-J
y
J h V
¦ ; Ž ˆ . Ý ¦; Ž .
Ý
j
0F¨ -j
¨
h
i4 n
i4 n
d
d
Jy V
m Ž dˆ . Ž dn . m P
jy ¨
m Ž dˆ . Ž dn . m P
n
V
J
j
¨
Žactually, if J s 0, then ¨ cannot be 0 in the second sum, but it does not matter.. Thanks to Remark 2.5 again, we see that we can move the P outside in hwni4d wŽ dˆ . J Ž dn . j . m P x since j - p m in the first sum and m j y ¨ - p in the second. 3.2. PROPOSITION. With the notations of Section 2, we ha¨ e h( d q d( h s Id y where is the projector that sends i and Ž d i . J to itself if i - n and to 0 otherwise. Proof. We only need to check this identity on a set of generators of ⍀ rP m , and we take the elements of the form J
j
Q m ni4 Ž dˆ . Ž dn . m P with Q [ ˆ I 4Ž dˆ . J1 m ⭈⭈⭈ m Ž dˆ . J sy 1 , P [ Ž d . J sq 1 m ⭈⭈⭈ m Ž d . J r and, ei-
LE STUM AND QUIROS ´
574
ther s s r or 0 - j - p m. We have J j . Ž dn . m P Ž hd q dh . Q m ni4 Ž dˆ J
j
J
j
J
j
s hd Q m ni4 Ž dˆ . Ž dn . m P q dh Ž Q m ni4 Ž dˆ . Ž dn . m P s h dQ m ni4 Ž dˆ . Ž dn . m P q Ž y1 .
sy 1
J
j
h Q m d Ž ni4 Ž dˆ . Ž dn .
s
J
. mP
j
q Ž y1 . h Q m ni4 Ž dˆ . Ž dn . m dP q Ž y1 .
sy 1
J
s
J
s Ž y1 . dQ m h ni4 Ž dˆ . Ž dn . J
yQ m h ni4 Ž dˆ . Ž dn .
j
J
j
.P J
P q Q m h d Ž ni4 Ž dˆ . Ž dn . sy1
dP q Ž y1 .
J
j
. mP
dQ m h ni4 Ž dˆ . Ž dn .
j
P
. m P q Q m h ni4 Ž dˆ . J Ž dn . j dP J j J j d Ž ni4 Ž dˆ . Ž dn . . m P q Q m dh Ž ni4 Ž dˆ . Ž dn . . m P ,
qQ m dh Ž ni4 Ž dˆ . Ž dn . sQmh
j
d Q m h Ž ni4 Ž dˆ . Ž dn .
j
and, using Lemma 3.1 we get J j . Ž dn . m P Ž hd q dh . Q m ni4 Ž dˆ J
s Q m Ž hd q dh . ni4 Ž dˆ . Ž dn .
j
m P.
Thus, we may assume that r F 1. In other words, we are left with showing that we always have J j J j . Ž dn . s ni4 Ž dˆ . Ž dn . Ž hd q dh . ni4 Ž dˆ
unless i s j s 0 in which case we get 0. We write [ ni4Ž dn . j and we compute Ž hd q dh.w Ž dˆ . J x. We assume first j / 0 and J / 0. The same computation as in Lemma 3.1 gives us hd Ž dˆ .
J
s
Ý 0-uFi
y
¦ui ;h
J h V
u
J j . Ž dn . Ž dn . m Ž dˆ
¦ ; Ž ˆ . Ý ¦; Ž . Ý
0FV-J
y
iyu4 n
j
0F¨ -j
¨
h
i4 n
i4 n
d
d
n
Jy V
jy ¨
V
m Ž dˆ . Ž dn . J
m Ž dˆ . Ž dn .
¨
j
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
s
Ý 0-uFi
¦ui ;h
iyu4 n
u
j J . Ž dn . m Ž dn . Ž dˆ
J
q Ž dˆ . h ni4 Ž dn . y
Ý
j
0-¨ -j
¦; h
¨
y h ni4 Ž dn .
j
575
i4 n
j
Ž dn .
m Ž dˆ .
jy ¨
m Ž dn .
¨
. Ž dˆ
J
J
J
J
J
s hd Ž . m Ž dˆ . y h Ž . Ž dˆ . q Ž dˆ . hŽ w . J
J
s hd Ž . m Ž dˆ . y dh Ž w . Ž dˆ . . Adding to dh Ž dˆ .
J
s d h Ž . Ž dˆ .
J
J
s dh Ž . m Ž dˆ . q h Ž . d Ž dˆ .
J
,
we see that J J . s Ž hd q dh . Ž . m Ž dˆ . . Ž hd q dh . Ž dˆ
Obviously, this formula still holds when J s 0. What happens if j s 0 and J / 0? We have hd ni4 Ž dˆ .
J
s
Ý 0-uFi
y
¦ui ;h
Ý 0-V-J
J h V
iyu4 n
¦;
i4 n
u
J . Ž dn . m Ž dˆ
. Ž dˆ
Jy V
m Ž dˆ .
V
J
s hd Ž ni4 . m Ž dˆ . .
On the other hand, we have dhwni4Ž dˆ . J x s 0 s dhŽni4 .Ž dˆ . J since, by definition, h s 0 when r F 0. Thus, we see that J J . s Ž hd q dh . Ž ni4 . Ž dˆ . . Ž hd q dh . ni4 Ž dˆ
In other words, we may assume that n s 1 Žand still r F 1.. Thus, we want to show that we always have j j Ž hd q dh . i4 Ž d . s i4 Ž d .
unless i s j s 0, in which case we get 0. For j s 0, this is done in Example 2.2Ža. so we assume now j / 0. If i s 0, then we have
Ž hd q dh . Ž d .
j
s hd Ž d . sy
Ý
0-¨ -j
j
q dh Ž d . j
h Ž d .
¦; ¨
j
jy ¨
m Ž d .
¨
q d Ž j4 .
LE STUM AND QUIROS ´
576
j
jy¨ 4
Ž d .
j
jy¨ 4
j Ž d . s Ž d . .
¦ ; Ý ¦ ;
sy
Ý
¨
0-¨ -j
q
¨
0-¨ Fj
¨
¨
Thus, we are left with the case i / 0 and j / 0. We have hd i4 Ž d .
j
s h d i4 m Ž d . s
Ý 0-uFi
y
¦ui ;h
Ý
j
0-¨ -j
j
iyu4
¦; h
¨
j
q h i4d Ž Ž d .
.
u
j Ž d . m Ž d .
i4
jy ¨
Ž d .
m Ž d .
¨
.
Assume first that p m ¦ i, then the second sum is 0. Moreover, in the first sum, if we write i s qp m q t with 0 - t - p m , the only u for which we do not get a 0 term is u s t. Thus, we see that hd i4 Ž d .
j
s
i t
y1
i t
j
j
i4 Ž d . s i4 Ž d . .
¦;¦;
Since in this case dhw i4Ž d . j x s 0, we are done. Finally, we are left with the case p m N i and i / 0. We have dh i4 Ž d .
j
y1
iqj i
¦ ; ¦ ;
sd
iqj4
y1
iqj s i
Ý
0-uFp m
iqj u
¦ ;
iqjyu4
Ž d .
u
and hd i4 Ž d .
j
s
Ý
0-uFp m
y
Ý
¦ui ;h
0-¨ -j
s
j
iyu4
¦; ¨
h
i4
u
j Ž d . m Ž d .
Ž d .
jy ¨
m Ž d .
¨
m i j pm h iyp 4 Ž d . m Ž d . pm
¦ ; y
Ý
0-¨ -j
j
iqjy¨ i
¦ ;¦ ¨
y1
;
¨
iqjy¨ 4 Ž d . .
Now, we use the formula of Proposition 2.3, or more precisely in this case,
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
577
the formula of Example 2.2Žc.: m
h iyp 4 Ž d . sy
pm
Ý
j-¨ Fp
sy
Ý
j-¨ Fp
m Ž d .
m
m
j y1
pm q j j
pm q j
¦ ;¦ ; ¦ ; ¦ ;¦ ¨
y1
pm q j j
p m qjy¨
m
h iyp 4 Ž d . iqjy¨ i y pm
pm q j ¨
m Ž d .
¨
y1 ¨
;
iqjy¨ 4 Ž d . .
Now, we have i pm
pm q j j
¦ ;¦
y1
pm q j
iqjy¨ i y pm
; ¦ ;¦ ¨
y1
iqj s i
y1
iqj
; ¦ ;¦ ; ¨
and it follows that m i j pm h iyp 4 Ž d . m Ž d . pm
¦ ;
iqj sy i
y1
¦ ;
Ý
j-¨ Fp
iqj
m
¦ ; ¨
iqjy¨ 4
¨
Ž d . .
Since we also have j
iqjy¨ i
¦ ;¦ ¨
y1
iqj s i
y1
iqj
; ¦ ;¦ ; ¨
we get that hd i4 Ž d .
j
iqj i
y1
¦ ;
sy
Ý
0-¨ Fp
m
but ¨ /j
iqj
¦ ; ¨
iqjy¨ 4
¨
Ž d . .
Adding both formulas, we get j j Ž hd q dh . i4 Ž d . s i4 Ž d .
and the proposition is proved. As a consequence, we get the Formal Poincare ´ Lemma in higher level: 3.3. THEOREM. The de Rham complex ⍀ Pⴢ m is a resolution of OX . Proof. We proceed by induction on the dimension on X. The assertion being local in nature, it is sufficient to show that if f : X ª X ⬘ is a smooth morphism of relative dimension one of smooth schemes over S, then the map of complexes on X, f *: f *⍀ Pⴢ ⬘m ª ⍀ Pⴢ m is a quasi-isomorphism. More precisely, locally, there are coordinates t 1 , . . . , t n on X such that t 1 , . . . , t ny1
578
LE STUM AND QUIROS ´
are coordinates on X ⬘ and f * is a homotopy equivalence. Actually, since this map, which is just the inclusion i ¬ i and Ž d i . J ¬ Ž d i . J for i s 1, . . . , n y 1, has an obvious retraction given by n ¬ 0 and Ž dn . J ¬ 0, it is sufficient to show that the projector :n ¬ 0 and Ž dn . J ¬ 0 on ⍀ Pⴢ m is homotopic to the identity. In other words, we need to find a homotopy h on ⍀ Pⴢ m such that h( d q d( h s Id y . But this is just the content of Proposition 3.2.
´ LEMMA IN 4. THE POINCARE CRYSTALLINE COHOMOLOGY We show that, as in the case m s 0, one can derive a de Rham interpretation of crystalline cohomology from the Formal Poincare ´ Lemma. What follows is a straightforward generalization of the classical proof Žsee wB1, B-Ox.. For the basic notions concerning the crystalline site and topos of level m, the reader can look at wLS-Q, 4.1x. We let Ž S, ᑾ, ᑿ . be an m-PD-scheme with p locally nilpotent such that p g ᑾ and let X be a smooth S-scheme to which the m-PD-structure of S extends. 4.1. Crystals and the de Rham Complex. If E is an abelian sheaf on ˇ Cris Ž m. Ž XrS ., the Cech᎐Alexander complex EP Ž⭈. of E is the complex of abelian sheaves on X obtained by applying E to the simplicial complex ˇ PX mŽ⭈.. The normalized Cech᎐Alexander complex is the complex NEP Ž⭈. s lKer sUi . Assume E is an m-crystal. Then for each r, the first projection p1: PXŽ m. Ž r . ª X gives us an isomorphism EP Ž r . ( E X mO X PX mŽ r . and we will sometimes write E X mO X PX mŽ⭈. instead of EP Ž⭈. . However, the reader should keep in mind that, at this point, the differential is not linear in E X . We also have NEP Ž r . s E X mO X N PX mŽ r . and we will likewise sometimes write NEP Ž⭈. s E X mO X N PX mŽ⭈.. Recall that KX⭈ m denotes the differential m ideal in N PX mŽ⭈. generated by IX mŽ1. p q1 4. Then E X mO X KXⴢ m is stable under the differential of E X mO X N PX mŽ⭈. and we get another description of the de Rham complex of the DX m -module E X : E X m ⍀ ⴢX m s Ž E X m N PX m Ž ⭈ . . r Ž E X m KXⴢ m . . We can also consider the linearized construction: we build the complex of abelian sheaves Eq P Ž⭈. on X obtained by applying the sheaf E to the simplicial complex PXqmŽ⭈.. The map of simplicial complexes d 0 : PXqmŽ⭈. ª PX mŽ⭈. induces a morphism of complexes dU0 : EP Ž⭈. ª Eq P Ž⭈. . If we consider PXqmŽ⭈. as a simplicial complex over X by using the first projection p1 , if E is an m-crystal and if we identify EP Ž r . with E X mO X
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Ž . PX mŽ r . as before, then Eq P Ž⭈. s E X mO X PX m ⭈q 1 with a differential that q Ž . is linear in E X . In other words, we see that Eq P Ž⭈. s E X mO X PX m ⭈ . With U q these notations, the map d 0 : E X mO X PX mŽ⭈. ª E X mO X PX mŽ⭈. is given for each r by the composition of the scalar extension E X m PX m Ž r . ª PX m Ž 1 . m E X m PX m Ž r . and the map
m Id: PX m Ž 1 . m E X m PX m Ž r . ª E X m PX m Ž 1 . m PX m Ž r . s E X m PX m Ž r q 1 . , where : PX m Ž1. m E X ª E X m PX mŽ1. is the Taylor isomorphism. 4.2. Crystalline Interpretation of Linearization. We consider the crystalline site of level m, Cris Ž m. Ž XrS . and the associated topos Ž XrS .Žcrm.i s . We have a canonical morphism of topoi Ž m.
u s uŽXm. r S : Ž XrS . cr i s ª X Z ar and a localization morphism Ž m.
Ž m.
j X : Ž XrS . cr i sNX ª Ž XrS . cr i s . Note that j X # is exact. If we compose u with j X we get a morphism of topoi Ž m.
u N X s Ž XrS . cr i sNX ª X Z ar . It is actually is a morphism of ringed topoi. To see this, note that the structural sheaf of X Z ar is OX and the structural sheaf of Ž XrS .Žcrm.i s N X is the restriction of OXŽ m. r S . Thus, we need to have a morphism of rings Ž m. uy1 O ª O . In other words, given an open subset U of X, an NX X XrSN X m-PD-thickening U ¨ Y, and a morphism p:Y ª X extending the inclusion of U in X, we need a morphism OU ª O Y and we just take py1 . One can check that u N X# is exact and that the adjunction map u N X# uUN X F ª F is an isomorphism. If F is a sheaf of OX -modules, we let LŽ m. Ž F . [ j X #uUN X F and LŽXm. Ž F . [ LŽ m. Ž F . X . 4.3. PROPOSITION. Ž1. If F is a sheaf of OX -modules, then LŽ m. Ž F . is an m-crystal and we ha¨ e LŽXm. Ž F . s PX mŽ1. mO X F. Ž m. Ž . Ž m. Ž . Ž2. We ha¨ e uŽXm. F s F and R i uŽXm. F s 0 for i ) 0. r S# L r S# L Ž3. If E is an m-crystal, there is a canonical isomorphism E m LŽ m. Ž F . ( LŽ m. Ž E X m F .. Proof. Note first of all that, by definition, given any open subset U of X and any m-PD-thickening U ¨ Y, we have LŽ m. Ž F . Y s p1# pU2 F where p1: PU mŽ Y =S X . ª Y and p 2 : PX mŽ Y =S X . ª X denote the projections. In particular, LŽXm. Ž F . s PX mŽ1. mO X F. Now, in order to prove that
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LŽ m. Ž F . is an m-crystal, we need to check that, given a morphism of Žwhere pX1 and m-PD-thickenings f :Y ⬘ ª Y, we have f *p1# pU2 s pX1# pXU 2 X p 2 are defined in the same way as p1 and p 2 .. We may clearly assume that U s X. Moreover, the question is local. Thus, since X is smooth, we may assume that X ¨ Y has a retraction. It is therefore sufficient to consider the case Y s X. Writing Y instead of Y ⬘, g for the canonical map PX mŽ Y =S X . ª PX m Ž1., p1: PX mŽ1. ª X and q1: PX mŽ Y =S X . ª Y for the first projection, we are led to show that f *p1# s q1# g*. But this follows from the fact that, PX mŽ1. being flat over X, the canonical map PX mŽ Y =S X . ª Y =X PX mŽ1. is an isomorphism. We now prove the second assertion. Since the adjunction map u N X #uUN X F ª F is an isomorphism, we have u# LŽ m. Ž F . s u N X #uUN X F s F. Now, since u N X # and j X # are exact, we obtain Ru# LŽ m. Ž F . s Ru# j X #uUN X F s R Ž u# j X # . uUN X F s R Ž u N X # . uUN X F s u N X #uUN X F s u# LŽ m. Ž F . . For the last assertion, we use the same notations as in the beginning of the proof. We need a compatible family of isomorphisms E Y m ; p1# pU2 F ª p1# pU2 Ž E X m F ., that is, ;
E Y m PX m Ž Y = X . mO X F ª PX m Ž Y = X . m E X m F . But, E being a crystal, we have E Y mO Y PX m Ž Y = X . s EP ŽY=U . s PX m Ž Y = X . mO X E X . 4.4. Remark. Before going any further, we would like to describe the Taylor isomorphism of LŽXm. Ž F . [ PX mŽ1. m F. By definition, it is obtained by composing the isomorphism ;
PX m Ž 1 . m LŽXm. Ž F . s PX m Ž 1 . m Ž PX m Ž 1 . m F . ª PX m Ž 2 . m F
Ž f m g . m Ž h m k . m x ¬ f m gh m k m x with the inverse of ;
LŽXm. Ž F . m PX m Ž 1 . s Ž PX m Ž 1 . m F . m⬘ PX m Ž 1 . ª PX m Ž 2 . m F
Ž f m g . m x m ⬘ Ž h m k . ¬ fh m k m g m x, where m⬘ means that the tensor product uses both left structures. Thus, the Taylor isomorphism is given by ;
: PX m Ž 1 . m LŽXm. Ž F . ª LŽXm. Ž F . m PX m Ž 1 .
Ž f m g . m Ž h m k . m x ¬ Ž f m k . m x m⬘ Ž 1 m gh . . 4.5. PROPOSITION. Ž1. There exists a unique OXŽ m. r S -linear differential on L ⍀ ⴢX m . such that LŽXm. Ž ⍀ ⴢX m . s ⍀ Pⴢ m . Ž m. Ž
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
Ž2.
581
If E is an m-crystal, we ha¨ e Ž m. RuŽXm. Ž ⍀ ⴢX m . . s E X m ⍀ ⴢX m . r S# Ž E m L
Proof. We saw in Proposition 4.3 that LŽXm. Ž ⍀ rX m . is an m-crystal and that, for all r, we have LŽXm. Ž ⍀ rX m . s ⍀ rP m . To prove the first assertion it is sufficient to check that the differential of ⍀ Pⴢ m is compatible with the Taylor isomorphisms. In order to do so, we describe the Taylor isomorphism of LŽXm. Ž F . in the particular case F s PX mŽ k .. We have canonical isomorphisms PX mŽ1. m LŽ m. Ž F . s PX mŽ k q 2. and LŽXm. Ž F . m PX mŽ1. s PX mŽ k q 2.. From Remark 4.4, we see that the Taylor isomorphism is induced by the automorphism f m g m h1 m ⭈⭈⭈ m h k ¬ f m h1 m ⭈⭈⭈ m h k m g of PX mŽ k q 2.. In particular, we see that the maps d i : LŽ m. Ž PX m Ž k . . s PX m Ž k q 1 . ª LŽ m. Ž PX m Ž k q 1 . . s PX m Ž k q 2 . are compatible with the Taylor isomorphisms, for i ) 0. It follows that the Ž m. Ž . same is true for the differential of the complex PXq ⭈ and also for the ⴢ Ž m. Ž . differential of the complex ⍀ P m which is a subquotient of PXq ⭈. The second assertion will follow from Proposition 4.2 once we know that u# sends the differential of E X m ⍀ Pⴢ m to the differential of E X m ⍀ ⴢX m . Note that, in general, the map dU0 F s u# LŽ m. Ž F . ª LŽXm. Ž F . s PX m Ž 1 . m F is just the adjunction map for u N X . In particular, it is functorial in LŽ m. Ž F . and injective. In order to show that u# sends the differential of E X m ⍀ rP m to the differential of E X m ⍀ rX m , it is therefore sufficient to check that dU0 preserves differentials. But as we mentioned in Subsection 4.1, dU0 comes from a morphism of simplicial complexes. 4.6. Remark. It is not difficult to describe the DXŽ m.-structure on Let us first consider the m-connection
LŽXm. Ž F ..
;
ⵜ : LŽXm. Ž F . ª LŽXm. Ž F . m PX m Ž 1 . induced by the Taylor isomorphism Žwe write ⵜ and not d, so as to avoid confusion with the differential of the de Rham complex.. We have ⵜ Ž f m g . m x s Ž 1 m g . m x m ⬘ Ž 1 m f . y Ž f m g . m x m ⬘ Ž 1 m 1. s Ž 1 m g . m x m ⬘ Ž 1 m f . y Ž 1 m g . m x m ⬘ Ž f m 1. s Ž 1 m g . m x m ⬘df . If we have local coordinates on X, this last formula gives ⵜŽ m x . s ⵜ Ž 1 m t . m x y ⵜ Ž t m 1. m x s y Ž 1 m 1. m x m
LE STUM AND QUIROS ´
582 from which we derive ⵜŽ J 4 m x . s
J V
< < Ý Ž y1. V
¦ ;Ž
0-VFJ
JyV 4
m x . m V 4 .
At this point, it is convenient to use partial derivatives. Thus, we see that ⵜ² I : Ž f m 1 . m x s ⭸ ² I : Ž f . m x and ⵜ² I : Ž J 4 m x . s Ž y1 .
J I
¦ ;
JyI 4
m x.
Finally, we get ⵜ² I :
< < Ž f J 4 m x . s Ý Ž y1. U
0FUFI
I U
J U
½ 5¦ ;⭸
² IyU :
Ž f . JyU 4 m x.
Using this formula, it is not hard to check directly that
Ž ⵜ² I : ( d .
Ž f J 4 m Ž d . J
s Ž d(ⵜ² I : .
1
m ⭈⭈⭈ m Ž d .
Ž f J 4 m Ž d . J
1
Jr
.
m ⭈⭈⭈ m Ž d .
Jr
..
This gives another proof that the differential of the complex LŽ m. Ž ⍀ ⴢX m . is DXŽ m.-linear. We prove now the Poincare ´ Lemma for crystals of level m and show that de Rham cohomology of level m can be used to compute crystalline cohomology of level m. 4.7. THEOREM. If E is an m-crystal on XrS then the complex E m LŽ m. Ž ⍀ ⴢX m . is a resolution of E, we ha¨ e an isomorphism in the deri¨ ed category ;
ⴢ RuŽXm. r S# E ª E X m ⍀ X m ,
and, for all i, an isomorphism ;
Hcri i s, m Ž XrS, E . ª Hdi R , m Ž X , E X . . Proof. The first assertion is not completely formal since the functor from the category of locally quasi-nilpotent DXŽ m.-modules to the category of sheaves on Cris Ž m. Ž XrS . is not exact in general wB1, IV, 1.7.8x. We need to show that the obvious map E ª E m LŽ m. Ž ⍀ ⴢX m . is a quasi-isomorphism. It is sufficient to check that if U ¨ T is a sufficiently small m-PD-thickening, the map ET ª ET m LŽ m. Ž ⍀ ⴢX m .T is a quasi-isomorphism. We may assume that U s X and that X ¨ T has a restriction. Moreover, as we saw in Proposition 1.4, ⍀ ⴢX m is a complex of locally free sheaves, and we may therefore assume that T s X and E s OXŽ m. r S . Then, our assertion follows from the Formal Poincare ´ Lemma ŽTheorem 3.3..
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Applying RuŽXm. r S# gives the second assertion thanks to Proposition 4.5. Finally, we take cohomology on both sides. 4.8. Generalization to Singular Schemes. Even if we no longer assume that X is smooth over S, we can still consider the crystalline topos Ž XrS .Žcrm.i s of level m and the canonical morphism of topoi Ž m.
u s uŽXm. r S : Ž XrS . cr i s ª X Z ar . Let E be an m-crystal on XrS. If i: X ¨ Y is an embedding into a smooth S-scheme, we know that i# E is a crystal on YrS and that R q i# E s 0 for q ) 0 Žsee wB-O, 7.1.2x in the case m s 0.. We write E Y [ Ž i# E . Y . We can form the de Rham complex E Y m ⍀ Yⴢ m Žwhich is actually supported on X . and it follows from Theorem 4.7 that Ž m. Ž m. R⌫ Ž X , RuŽXm. r S# E . s R⌫ Ž Y , Ri# Ru X r S# E . s R⌫ Ž Y , Ru Y r S# Ri# E . ⴢ s R⌫ Ž Y , RuŽYm. r S# i# E . s R⌫ Ž Y , Ž i# E . Y m ⍀ Y m . .
Therefore, we have ;
Hcri i s, m Ž XrS, E . ª Hdi R , m Ž Y , E Y . . 5. COMPLEMENTS AND APPLICATIONS All formal schemes are p-adic formal schemes over a fixed complete discrete valuation ring V of characteristic 0 with residue field of characteristic p. 5.1. The de Rham Complex of Finite Order. Let S be a scheme or a formal scheme and X be a smooth Žformal. S-scheme. In the formal scheme case, we cannot use divided power envelopes of infinite order. However, for each integer k, we can consider the ˇ Cech᎐Alexander complex PXk mŽ⭈. of order k wB4, Sect. 2.1x, and just as we did in Subsection 1.1, the normalized complex N PXk mŽ⭈. associated to it, ⴢ k Ž . and the quotient ⍀ k, X m of N PX m ⭈ by the differential ideal generated by k Ž . k Ž . p mq1 4 IX m 1 . Note that PX m ⭈ is not in general the tensor algebra on ⴢ k Ž . m PXk mŽ1. and that ⍀ k, X m s N PX m ⭈ when k F p . Moreover, since as in Subsection 1.2 all the generators and all the elements occurring in the ⴢ relations are of order at most p m , ⍀ k, X m does not depend on k when m ⴢ ⴢ k G 2 p . It is important to remark that for schemes we have ⍀ k, Xm s ⍀Xm m when k G 2 p and, taking into account the independence, it will do no ⴢ m harm to write in general ⍀ ⴢX m s ⍀ k, X m for k G 2 p . This complex has locally free terms. We can define the de Rham complex of a DXŽ m.-module F as before: the action of DXŽ m. on F induces, for all k, a morphism d: F ª F mO X PXk m that gives rise for k G p m , and in particular for k G 2 p m , to an m-connec-
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584
F ª F mO X ⍀ 1X m . Using the Leibniz rule, it extends to a differential tion d:F on F mO X ⍀ ⴢX m . This is the de Rham complex of F and we can define the de Rham cohomology of le¨ el m of F : Hdi R , m Ž X , F . [ H i Ž X , F mO X ⍀ ⴢX m . . We now want to describe a construction analogous to that of Subsection 1.5 that works over formal schemes and has also some intrinsic interest for schemes. An integer k being fixed as before, we consider the shifted simplicial complex k PXkq m Ž ⭈ . s PX m Ž r q 1 . ; d i , i s 1, . . . , r q 1; s i , i s 1, . . . , r q 1 4 ,
Ž. the associated complex of abelian sheaves PXkq m ⭈ , and the normalized U kq Ž . complex N PXkq ⭈ [ lKer s . We let K be the differential ideal of m i Xm Ž . N PXkq ⭈ generated by m dy1 IXkm Ž 1 . 0
ž
p m q1 4
/ ; NP
kq Xm
Ž 2.
ⴢ and we write ⍀ k, P m for the quotient complex, called the linearized de Rham complex of order k. Only the case k G 2 p m is of interest to us. By construction, the Ž. morphism of complexes dU0 : PXk m ª PXkq m ⭈ induces a morphism of comⴢ k, ⴢ r plexes ⍀ X m ª ⍀ P m . For each r, the map ⍀ rX m ª ⍀ k, P m extends by linear; k Ž . r k, ⴢ Ž ity to an isomorphism PX m 1 mO X ⍀ X m ª ⍀ P m as one easily checks by a local computation.. Thus, there should not be any confusion in writing PXk mŽ1. mO X ⍀ ⴢX m for the linearized de Rham complex of order k G 2 p m . As in Section 2, we can define a homotopy on this complex and deduce that it is a resolution of OX . By duality, the linearized de Rham complex PXk mŽ1. mO X ⍀ ⴢX m of order . which is a right k G 2 p m gives rise to a complex H om O XŽ ⍀ ⴢX m , DXŽ m. k resolution of OX . Since the de Rham complex has locally free terms, going to the limit on k gives rise to a complex H om O XŽ ⍀ ⴢX m , DXŽ m. . of DXŽ m.-modules called the Spencer complex of le¨ el m.
5.2. PROPOSITION. Let S be a scheme or a formal scheme and X be a smooth Ž formal . S-scheme. Then, Ž1. The Spencer complex of le¨ el m o¨ er XrS is a right resolution of OX by locally free DXŽ m.-modules. Ž2. For any DXŽ m.-module F , there is an isomorphism R H om DXŽ m . Ž OX , F . s F mO X ⍀ ⴢX m . Ž3.
For any DXŽ m.-module F , we ha¨ e RHom DXŽ m . Ž OX , F . s Hdi R m Ž X , F . .
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
585
Proof. Ž1. Since, by construction, the Spencer complex of level m is a right resolution of OX , there only remains to check that the differential in the Spencer complex is DXŽ m.-linear. Fix two integers k and l with k G 2 p m. It follows from the universal property of the divided power envelope that, for each r, the map X Ž rq2.rS ª X Ž rq1.rS that forgets the second component induces a map Ž kql .q PXl m Ž 1 . =X PXkq Ž r.. m Ž r . ª PX m
By functoriality, we actually get a morphism of simplicial complexes Ž kql .q PXl m Ž 1 . =X PXkq Ž ⭈. . m Ž ⭈ . ª PX m
From this, we deduce a morphism of complexes .q PXŽ kql Ž ⭈ . ª PXl m Ž 1 . mO X PXkqm Ž ⭈ . m
from which we derive a morphism of complexes l ⴢ l k ⴢ PXkq m Ž 1 . mO X ⍀ X m ª PX m Ž 1 . mO X PX m Ž 1 . mO X ⍀ X .
Now we apply duality in order to get a pairing ⴢ Ž m. ⴢ Ž m. DXŽ m. l = H om O X Ž ⍀ X m , DX k . ª H om O X Ž ⍀ X m , DX Ž kql . . .
Taking the limit on k and l gives a pairing DXŽ m. = H om O X Ž ⍀ ⴢX m , DXŽ m. . ª H om O X Ž ⍀ ⴢX m , DXŽ m. . which, by construction, is nothing but the canonical action of DXŽ m. on the Spencer complex. Ž2. It follows from the first part that R H om DXŽ m . Ž OX , F . s H om DXŽ m . Ž H om O X Ž ⍀ ⴢX m , DXŽ m. . , F . s F mO X ⍀ ⴢX m where the left hand side is the de Rham complex of F. Ž3. Take cohomology. 5.3. Crystalline Cohomology o¨ er a Formal Base. Let S be a formal scheme. If X is a smooth formal S-scheme, completing the Spencer ˆXŽ m. . complex H om O XŽ ⍀ ⴢX m , DXŽ m. . gives a right resolution H om O XŽ ⍀ ⴢX m , D of OX because the de Rham complex is made of locally free terms. Thus, if ˆXŽ m.-module, we have as before F is a D
ˆXŽ m. . , F s F mO X ⍀ ⴢX m R H om DˆXŽ m . Ž OX , F . s H om DˆXŽ m . H om O X Ž ⍀ ⴢX m , D
ž
/
and RHom DˆXŽ m . Ž OX , F . s Hdi R , m Ž X , F . . Assume now that the ramification index of V is F Ž p y 1. p m , that S is endowed with an m-PD-structure Ž ᑾ, ᑿ . compatible with p, and let X be
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6
an S-scheme to which the m-PD-structure of S extends. Let E be a quasi-coherent crystal on XrS. Assume i: X ¨ Y is an embedding into a smooth formal S-scheme and, for all n 4 0, let i n : X ¨ Yn be the map induced mod ᑾ nq 1. We know that E Y [ Ž i# E . Y n is a DYŽ m. -module and it follows that E Y :s lim E Y n n ˆXŽ m.-module. We can therefore form the de has a natural structure of D Rham complex E Y m ⍀ Yⴢ m . 5.4. PROPOSITION.
We ha¨ e
Hcri i s, m
;
Ž XrS, E . ª Hdi R , m Ž Y , E Y . .
Proof. Using Subsection 4.8, one proceeds as in wB-O, Theorem 7.23x. To finish this article, we want to explain how Berthelot’s Frobenius descent wB5x applies in our context. So, we let Ž S, ᑾ, ᑿ . be a formal m-PD-scheme compatible with p and X 0 a smooth formal S0-scheme to which the m-PD-structure of S extends. It follows from wB5, Proposition 2.2.3 and Sect. 4.1x that if F0 : X 0 ª X 0X is the sth iterate of the relative Frobenius of X 0rS0 and E⬘ is a quasi-coherent m-crystal over X 0X rS, then E [ F0U E⬘ is in a natural way an Ž m q s .-crystal over X 0rS. 5.5. PROPOSITION. F0U :
Ž1.
Hcri i s, m
The morphism F0 induces an isomorphism ;
Ž X 0X rS, E⬘ . ª Hcri i s, mqs Ž X 0rS, E . .
Ž2. Assume S is scheme killed by p Nq 1. If m G N and if F: X ª X ⬘ is s a lifting of F0 gi¨ en in local coordinates by F*Ž t i . s t ip , then F induces a quasi-isomorphism F* : EXX ⬘ mO X ⬘ ⍀ ⴢX ⬘m ª E X mO X ⍀ ⴢX Ž mqs. . Proof. Ž1. This is a local question and we may assume that F0 lifts to F : X ª X ⬘. Then, the result follows from the previous results and wB5, Proposition 2.3.8 and Sect. 4.1x. More precisely, the isomorphism ;
F* : RHom DˆXŽ m . Ž OX ⬘ , E X ⬘ . ª RHom DˆXŽ mq s. Ž OX , E X . of wB5x, combined with the identifications of Proposition 5.3 tells us that Frobenius is an isomorphism ;
F* : Hdi R , m Ž X , E X ⬘ . ª Hdi R , mq1 Ž X , E X . and Proposition 5.4 gives the announced isomorphism. Ž2. It follows from wB5, Proposition 2.2.2x that F induces a morphism of simplicial complexs F : PX Ž mqs.Ž⭈. ª PX ⬘mŽ⭈.. Thus, we get morphisms of complexs F* : EXX ⬘ mO X PX ⬘m Ž ⭈ . ª E X mO X PX Ž mqs. Ž ⭈ . and F* : E X ⬘ mO X N PX ⬘m Ž ⭈ . ª E X mO X N PX Ž mqs. Ž ⭈ . .
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587
Since S is killed by p Nq 1, it follows from Lemma 5.6 below that, for m G N, F* sends the differential ideal KXⴢ ⬘m into KXⴢ Ž mqs. . Thus, we get a morphism of complexs F* : E X ⬘ mO X ⍀ ⴢX ⬘m ª E X mO X ⍀ ⴢX Ž mqs. . Thanks to Proposition 5.2, it corresponds to the canonical map F* : H om DXŽ m⬘ . Ž H om O X ⬘Ž ⍀ ⴢX ⬘m , DXŽ m. ⬘ . , EX ⬘ . ª H om DXŽ mq s. H om O X Ž ⍀ ⴢX Ž mqs. , DXŽ mqs. . , E X
ž
/
which, in the derived category, is the canonical map F* : R H om DXŽ m⬘ . Ž OX ⬘ , E X ⬘ . ª R H om DXŽ mq s. Ž OX , E X . . We know from wB5, Proposition 3.3.8x that this is an isomorphism. There remains to prove the following lemma that generalizes wB5, 2.2.4x. m 5.6. LEMMA. Ž1. For u / 0, the integer Ž pu . has p-adic order G m y u q 1. m mq s Ž2. For m G N, we ha¨ e F*Ž i p . s i p mod p Nq 1. Ž3. Let N be such that X is killed by p Nq 1. Then, for m G N, the image m mq s of IX mŽ1. p q1 4 by the Frobenius map is inside IX Ž mqs.Ž1. p q1 4.
Proof. Ž1. We can write u s qp q r with r - p. In case q s 0, we have ord
pm s m G m y u q 1. u
ž /
If q ) 0, we proceed by induction on m, the case m s 1 being trivial. So let m ) 1. Then, we have ord
pm p my1 pm G ord s ord , qp q u
ž /
ž /
ž /
and by induction, this is G m y q G m y u q 1. s s Ž2. We have F*Ž i . s Ž t i q i . p y t ip q p for some g PX mŽ1. and it follows that F* Ž i p
m
. s Ž ip
s
q p .
pm
s
Ý 0Fu-p
s
Ý
0Fu-p n
n
pm u
ž / Ž
p m u u p sŽ p m yu. p i . u
ž /
ps i
.
p myu
Ž p .
u
588
LE STUM AND QUIROS ´
m If u ) 0, then Ž pu . p u s 0 mod p Nq1 thanks to the first part, and we get m mq s p p F*Ž i . s i mod p Nq 1. m Ž3. It is sufficient to show that iq p qk 4 g PX mŽ1. is sent into mq s q p qk 4 IX Ž mqs.Ž1. . We already know by wB5, Proposition 2.2.2x that, for all m l, the image of IX mŽ1.l4 under F is inside IX Ž mqs.Ž1.l4. Since F*Ž iq p qk 4 . m s F*Ž i p .w q x F*Ž ik ., our assertion follows from the second part of this lemma.
ACKNOWLEDGMENTS We heartily thank P. Berthelot for providing access to some of his unpublished material. The main ideas in this article were inspired by his work. The collaboration on this article was made possible by the financial support of the European Union through the TMR Network Arithmetic Algebraic Geometry ŽERB FMRX 960006..
REFERENCES wB1x
P. Berthelot, ‘‘Cohomologie cristalline des schemas de caracteristique p ) 0,’’ ´ ´ Lecture Notes in Math., Vol. 407, Springer-Verlag, New YorkrBerlin, 1974. wB2x P. Berthelot, Cohomologie rigide et theorie des D-modules, in ‘‘Proc. Conf. p-adic ´ Analysis, Trento 1989,’’ Lecture Notes in Math., Vol. 1454, Springer-Verlag, New YorkrBerlin, 1990. wB3x P. Berthelot, Letter to Illusie, 1990. wB4x P. Berthelot, D⺡† -modules coherent. I. Operateurs differentiels de niveau fini, Ann. ´ ´ Sci. Ecole Norm. Sup. 29 Ž1996., 185᎐272. wB5x P. Berthelot, D⺡† -modules coherent. II. Descente par Frobenius, prepublication ´ 98-32 de l’IRMAR, Rennes, 1998. wB-Ox P. Berthelot and A. Ogus, ‘‘Notes on Crystalline Cohomology,’’ Princeton Univ. Press, Princeton, NJ, 1978. wE-LSx J.-Y. Etesse and B. Le Stum, Fonctions L associees ´ aux F-isocristaux surconvergents. II. Zeros ´ et poles ˆ unites, ´ In¨ ent. Math. 127 Ž1997., 1᎐31. wLS-Qx B. Le Stum and A. Quiros, ´ Transversal crystals of higher level, Ann. Inst. Fourier Ž Grenoble. 47 Ž1997., 69᎐100. wLx D. Lieberman, Generalizations of the de Rham complex with applications to duality theory and the cohomology of singular varieties, in ‘‘Proc. Conf. in Complex Analysis, Rice Univ. Studies, 1972.’’ wTx F. Trihan, Fonction L unite ´ d’un groupe de Barsotti᎐Tate, Manuscripta Math. 96 Ž1998., 397᎐419.
The Exact Poincare ´ Lemma in Crystalline Cohomology of Higher Level Bernard Le Stum IRMAR, Uni¨ ersite´ de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France E-mail: [email protected]
and Adolfo Quiros ´1 Departamento de Matematicas, Uni¨ ersidad Autonoma de Madrid, ´ ´ Ciudad Uni¨ ersitaria de Cantoblanco, E-28049 Madrid, Spain E-mail: [email protected] Communicated by Laurent Clozel Received December 10, 1999
We prove a formal Poincare ´ Lemma in crystalline cohomology of higher level and we derive a de Rham interpretation of crystalline cohomology of higher level. We also define the Spencer complex of higher level and show that it can be used to compute the cohomology of arithmetic D-modules. 䊚 2001 Academic Press CONTENTS 1. The de Rham complex. 2. Integration of differential forms of higher le¨ el. 3. The Formal Poincare´ Lemma in higher le¨ el. 4. The Poincare´ Lemma in crystalline cohomology. 5. Complements and applications.
INTRODUCTION As long as the theory is concerned, crystalline cohomology generalizes well to higher level Žsee wB3, E-LS, Q-LS, Tx, for example.. Unfortunately, 1
Partially supported by DGICYT Project PB 96-0065. 559 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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the main tool for computing crystalline cohomology, namely, the usual de Rham complex, cannot be used in higher level. Following Lieberman wLx, Berthelot suggests in wB3x to use the so-called Jet Complex. This is what we call in this article the de Rham complex of higher level. In crystalline cohomology, the main ingredient in order to show that the de Rham complex computes de Rham cohomology is the Formal Poincare ´ Lemma. This is also the case in higher level and we show that crystalline cohomology of higher level can be computed as the cohomology of a complex whose terms are locally free, the de Rham complex of higher level. We are also able to define a Spencer complex in higher level and prove that the ˆXŽ m. cohomology of a D r S -module agrees with the cohomology of the corresponding m-crystal. Finally, using Berthelot’s Frobenius descent, we give an explicit quasi-isomorphism between the de Rham complex of level m of an m crystal and the de Rham complex of level m q 1 of its Frobenius pull-back. In a forthcoming article, we will consider the Filtered Poincare ´ Lemma and compute the cohomology of T-m-crystals. We will apply this to the study of the Hodge filtration on Dieudonne modules of level m. Con¨ entions We let p be a non-zero prime and m g ⺞. When m / 0, all schemes are ⺪ Ž p.-schemes. If I s Ž i1 , . . . , i n . g ⺞ n we will write < I < s i1 q ⭈⭈⭈ qi n and, more generally, we use the classical conventions for multi-indexing.
1. THE DE RHAM COMPLEX For the notion of m-PD-structures the fundamental reference is wB4x. 1.1. DEFINITION. We develop an idea introduced by Berthelot in wB3x, following Lieberman wLx. More precisely, we explain the construction of the de Rham complex of higher level. Let S be a scheme and X a smooth S-scheme. Everything below will be done relative to S but we will drop the reference to S in order to lighten the notations. We write PX mŽ r . for the m-PD-envelope of X in X Ž rq1.rS, d i : PX mŽ r q 1. ª PX mŽ r . for the projection that forgets the Ž i q 1.st component, and si : PX mŽ r y 1. ª PX mŽ r . for the degeneracy arrow that repeats the Ž i q 1.st component. We also write pi : PX mŽ r . ª X, for the ith projection and ⌬: X ª PX mŽ r . for the diagonal embedding. We consider the simplicial complex of schemes PX m Ž ⭈ . [ PX m Ž r . , d i , i s 0, . . . , r ; si , i s 0, . . . , r 4 ,
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
561
and the usual complex PX mŽ⭈. built out of it. Thus, PX mŽ r . is the structural sheaf of PX mŽ r .. By definition, the normalized complex associated to PX mŽ⭈. is N PX mŽ⭈. [ lKer sUi . Recall that if A is a Žnot necessarily commutative. ring and M is an A-A-bimodule, then the tensor algebra T Ž M . is defined in the obvious way. We will use this vocabulary when A is commutative and M is endowed with two Ždifferent. A-module structures, called ‘‘structure on the left’’ and ‘‘structure on the right.’’ We might sometimes call this tensor algebra the ‘‘non-commutative’’ tensor algebra. For example, as a graded algebra, PX mŽ⭈. is canonically isomorphic to the non-commutative tensor algebra T Ž PX mŽ1... If we call IX mŽ1. the m-PD-ideal of PX mŽ1., one easily checks that N PX mŽ⭈., which is a differential sub-algebra of PX mŽ⭈., is isomorphic as a graded algebra to T Ž IX mŽ1... Finally, if KX⭈ m denotes the differential ideal in N PX mŽ⭈. m generated by IX mŽ1. p q1 4, we call KXⴢ m ⍀ ⴢX m [ N PX m Ž ⭈ . rK the jet complex of order p m or the de Rham complex of le¨ el m of XrS. Note that, as a graded algebra, ⍀ ⴢX m is a quotient of T Ž ⍀ 1X m .. We will make this more precise later. We can also define the de Rham complex of a DXŽ m.-module F as follows. Recall that, for each k, DXŽ m. k denotes the sheaf of operators of level m and order at most k. By definition, this is the dual to the sheaf of principal parts PXk mŽ1. and DXŽ m. is the union of the DXŽ m. k when k varies. The action of DXŽ m. on F induces for all k a pairing DXŽ m. k = F ª F that corresponds to a morphism d: F ª F mO X PXk mŽ1.. In the particular case k s p m , we have ⍀ 1X m ; PXk mŽ1. and d takes its values in F mO X ⍀ 1X m . F ª F mO X ⍀1X m . It is Thus we get what could be called an m-connection d:F m-integrable: using the Leibniz rule, it extends to a differential on F mO X ⍀ ⴢX m . This is the de Rham complex of F and we can define the de Rham cohomology of le¨ el m of F : Hdi R m Ž X , F . [ H i Ž X , F mO X ⍀ ⴢX m . . 1.2. Description of the de Rham Complex. Unless otherwise specified, we will always see PX mŽ r ., N PX m , etc., as left OX -modules using pU1 , and tensor products will be over OX . Let us now describe the differential on PX mŽ⭈., N PX mŽ⭈., and ⍀ ⴢX m . As usual, the differential on OX is induced by the map OX ª OX mOS OX , f ¬ 1 m f y f m 1.
LE STUM AND QUIROS ´
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It is also convenient to call ␦ : PX mŽ1. ª PX mŽ2. the ring homomorphism dU1 induced by the map OX mOS OX ª OX mOS OX mOS OX , f m g ¬ f m 1 m g . The differential on PX mŽ1., IX mŽ1. and ⍀1X m is given by d Ž ␣ . s 1 m ␣ y ␦ Ž ␣ . q ␣ m 1. For higher degree, we can use the Leibniz rule: d Ž ␣ m  . s d ␣ m  q Ž y1 .
de g␣
␣ m d .
Note also that the non-commutativity of the tensor product means that if ␣ ,  g PX mŽ1. and f g OX , then ␣ m f  s f ␣ m  q d Ž f .. ␣ m  , where the regular product is taken in the monoid PX mŽ1., IX mŽ1. or ⍀ 1X m . We assume now that we have local coordinates t 1 , . . . , t n on X and write i s d Ž t i .. Then PX mŽ1. Žresp. IX mŽ1.. is the free module on the generators J 4 Žresp. with < J < ) 0.. It follows that PX mŽ r . Žresp. IX mŽ r .. is the free module on the generators J14 m J 2 4 m ⭈⭈⭈ m J r 4 Žresp. with < J1 <, . . . , < Jr < ) 0.. Note that when < J < - 2 p m , we have J 4 s J and we will therefore drop the 4 . We will write dt i for the image of i in ⍀ 1X m . Then we see that ⍀ 1X m is the free module on the generators Ž dt . J with 0 - < J < F p m . Note that, since Ž dt . J 4 s 0 whenever < J < ) p m , such terms will never play a role in our calculations. To lighten the notations, we will therefore not make explicit the corresponding conditions of the type 0 - < J < F p m , in the hope that this will make it easier to read without baffling the reader. The local description of ⍀ rX m for r ) 1 needs some more care. Note first that ⍀ ⴢX m has a natural structure of graded algebra. We keep using the tensor notation for this structure Žusing the wedge product, as in the case m s 0, would be misleading.. Actually, ⍀ ⴢX m is the quotient of T Ž ⍀1X m . by the ideal generated by the images of the d w J 4 x with p m - < J < F 2 p m . Since T r Ž ⍀ 1X m . is the free module on the generators Ž dt . J14 m Ž dt . J 2 4 m ⭈⭈⭈ m Ž dt . J r 4 with 0 - < J1 <, . . . , < Jr < F p m and d is given by the classical formula dŽ J4. s
Ý 0-V-J
J V
¦ ;
JyV 4
m V 4 ,
we see first that is generated by the Ž dt . I m Ž dt . J with 0 - < I <, m < J < F p subject to the relations ⍀ 2X m
Ž ).
Ý 0-V-J
J V
¦ ;Ž
dt .
Jy V
V
m Ž dt . s 0
with p m - < J < F 2 p m .
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
563
For r ) 2, ⍀ rX m is generated by the Ž dt . J1 m Ž dt . J 2 m ⭈⭈⭈ m Ž dt . J r with 0 - < J1 <, . . . , < Jr < F p m subject to the same relations, tensored on the right and on the left by elements of the form Ž dt . J1 m Ž dt . J 2 m ⭈⭈⭈ m Ž dt . J i . We also have a local description of the m-connection d: F ª F mO X ⍀1X m corresponding to a DXŽ m.-module F : it is given by d Ž x . s ⌺⭸ wU xŽ x . m Ž dt .U where the ⭸ wU x are the divided partial derivatives. In general, the de Rham complex is infinite. However, we will show that it has locally free terms. First, we need a lemma. 1.3. LEMMA. Unless J s 0, . . . , 0, 2 p m , 0, . . . , 04 , in which case there is only one term, there are at least two different terms with in¨ ertible coefficients in the formula Ž).. Proof. If we write J s j1 , . . . , jn4 we may assume by symmetry that j1 G ⭈⭈⭈ G jn . If j1 ) p m we let V s Ž p m , 0, . . . , 0.. Then it follows from wB4, Lemma 1.1.3x that the coefficient ² VJ : is invertible. If not, there exits i such that j1 q ⭈⭈⭈ qji F p m and that j1 q ⭈⭈⭈ qjiq1 ) p m. In this case, we take V s Ž j1 , . . . , ji , 0, . . . , 0. and again ² VJ : is invertible. By symmetry, unless J s 2 p m , 0, . . . , 04 , in which case our relation simply says that m m Ž dt 1 . p m Ž dt 1 . p s 0, there are at least two different terms with invertible coefficients. 1.4. PROPOSITION. The de Rham complex of le¨ el m, ⍀ ⴢX m is a complex whose terms are locally free of finite rank. Proof. This is a local question. Thus, we may assume that we have local coordinates and use the notations of Subsection 1.2. For r s 0 or 1, there is nothing to do. Assume now r s 2. For different J ’s with p m - < J < F 2 p m , the relations involve different generators Ž dt .U m Ž dt .V. We know from the lemma that at least one of the coefficients in each relation is invertible. It follows that ⍀ 2X m is free. More precisely, for each I with p m - < I < F 2 p m , we can choose a couple Ž AŽ I ., B Ž I .. such that I s AŽ I . q B Ž I . and that the set U
V
S s Ž dt . m Ž dt . : Ž U, V . / Ž A Ž I . , B Ž I . . 4 is a basis for ⍀ 2X m . We consider now the case r s 3. From the case r s 2 we know that
Ž dt . U m Ž dt . V m Ž dt . W : Ž V , W . / Ž AŽ I . , B Ž I . . 4 is a set of generators for ⍀ 3X m , and this module is defined by the following relations for p m - < J < F 2 p m and < K < F p m :
Ý 0-V-J
J V
¦ ;Ž
dt .
Jy V
V
K
m Ž dt . m Ž dt . s 0
with K / B Ž I .
LE STUM AND QUIROS ´
564 and
Ý 0-V-J , V/AŽ I .
J V
¦ ;Ž
J AŽ I .
¦ ;
y
dt .
JyV
V
m Ž dt . m Ž dt .
Ý
0-V-I , V/AŽ I .
I V
¦ ;Ž
dt .
BŽ I .
Jy AŽ I .
V
m Ž dt . m Ž dt .
IyV
s 0.
Using the relations with K / B Ž I ., which all involve different generators, we can write Ž dt . AŽ J . m Ž dt . BŽ J . m Ž dt . K as a linear combination of the other terms. It follows that
Ž dt . U m Ž dt . V m Ž dt . W : if W s B Ž I . then V / AŽ I . , and if W / B Ž I . then Ž U, V . / Ž A Ž I⬘ . , B Ž I⬘ . . 4 is a set of generators for ⍀ 3X m . We can rewrite the second sum in the remaining relations in terms of these generators. Now we know from the lemma that, in these relations, at least one of the coefficients ² VJ : with V / AŽ I . is invertible and we can write the corresponding term as a linear combination of the other generators, getting a basis of ⍀ 3X m . Finally, the case r ) 3 easily follows from the above description of ⍀ rX m and the case r s 3. 1.5. Linearization. We consider now a construction analogous to that of Definition 1.1 that will be ‘‘linear’’ in OX . This is a standard technique: we look at the shifted simplicial complex PXqm Ž ⭈ . s PX m Ž r q 1 . ; d i , i s 1, . . . , r q 1; si , i s 1, . . . , r q 1 4 Ž. and we consider the associated complex of abelian sheaves Pq X m ⭈ on X. Ž . As before, we define the normalized complex N Pq ⭈ [ lKer sUi . Xm q Ž .. q Ž . Ž As a graded algebra, PX m ⭈ resp. N PX m ⭈ is isomorphic to PX mŽ1. mO X T Ž PX mŽ1.. Žresp. PX mŽ1. mO X T Ž IX mŽ1... Taking the quotient by PX mŽ1. mO X KXⴢ m , we get a differential complex PX mŽ1. mO X ⍀ ⴢX m which is a quotient of PX mŽ1. mO X T Ž ⍀1X m .. By analogy with the usual de Rham complex, we will write ⍀ Pⴢ m s PX mŽ1. mO X ⍀ ⴢX m . The differential on ⍀ Pⴢ m is given by the Leibniz rule dŽ ␣ m . s d␣ m q ␣ m d , where d: PX m Ž1. ª PX mŽ1. mO X ⍀ 1X m is the m-connection for the right structure induced by ␣ ¬ ␦ Ž ␣ . y ␣ m 1, and d:⍀ rX m ª ⍀ rq1 X m is the differential of the de Rham complex of level m defined in Subsection 1.2.
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
565
As before, we have the classical formula for moving coefficients inside products: if , g ⍀ 1P m and ␣ g PX mŽ1., then
m ␣ s ␣ m q Ž d ␣ . m . Note that, in local coordinates, d s 1 m dt, and that we have dŽ I4. s
Ý 0-UFI
I U
¦ ;
IyU 4
Ž d .
U
Žwhere Ž d .V s 0, when < V < ) p m . and d Ž Ž d .
I
. sy Ý 0-U-I
I U
¦ ;Ž . d
IyU
U
m Ž d . .
Of course, a set of generators of ⍀ rP m over PX mŽ1. is given as before by the Ž d . J1 m ⭈⭈⭈ m Ž d . J r with 0 - < J1 <, . . . , < Jr < F p m subject to the relations
Ý 0-V-J
J V
¦ ;Ž . d
Jy V
V
m Ž d . s 0
with p m - < J < F 2 p m ,
tensored on the right and on the left by elements of the form Ž d . J1 m Ž d . J 2 m ⭈⭈⭈ m Ž d . J i .
2. INTEGRATION OF DIFFERENTIAL FORMS OF HIGHER LEVEL We assume that we have local coordinates on X as in Subsection 1.2. As before, we write t s Ž t 1 , . . . , t n ., and we set ˆt s Ž t 1 , . . . , t ny1 . and ˆ s Ž 1 , . . . , ny1 .. Also, given an n-uple I s Ž i1 , . . . , i n ., we will write Iˆs Ž i1 , . . . , i ny1 .. There exists a unique linear map
2.1. PROPOSITION AND DEFINITION. h:⍀ rP m ª ⍀ ry1 P m such that J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
¡0
s
J
j
m Ž dˆ . s Ž dn . m Ž d .
J sq 1
m ⭈⭈⭈ m Ž d .
Jr
if Ž 1 . s s r and j s 0 or Ž 2 . j / 0 and Js / 0 or Ž 3 . p m ¦ i
~ Ž y1.
sy1
iqj i
y1
¦ ;
mniqj4 Ž d .
¢
J sy 1
J sq 1
J
I 4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
m ⭈⭈⭈ m Ž d .
J sy 1
Jr
if Js s 0, p m N i and either Ž 1 . 0 - j - p m or Ž 2 . j s p m and s s r .
LE STUM AND QUIROS ´
566
We will show below that this map is well defined and unique. Note that the formula makes sense because the modified binomial coefficient ² i qi j : is invertible when p m N i and 0 - j F p m wB4, Lemma 1.1.3x. We first want to look at some examples: 2.2. EXAMPLES. Ža. For r s 1 and n s 1, we get
¡ iqj Ž d . . s~¦ i ; ¢0
y1
h Ž
i4
iqj4
j
if p m N i otherwise.
This is the formula for integrating differentials of level m as we can easily verify: we have
Ž h( d . Ž i4 . s
Ý 0-uFi
¦ui ;h Ž
iyu4
u
Ž d . . .
If i s 0, this is an empty sum and we get 0. If not, we can write i s qp m q t with 0 - t F p m and we see that hŽ iyu4Ž d . u . s 0 unless p m divides i y u, which means that t s u since 0 - u F p m. Thus, we see that for i ) 0, t Ž h( d . Ž i4 . s i h Ž iyt4 Ž d . . s i
¦;
i t
y1
¦;¦;
t
t
iytqt4 s i4 .
Žb. For r s 1 and n s 2 the formula is quite similar j
h Ž 1i14 2i 2 4 Ž d 1 . 1 Ž d 2 . s
¡i ¢0
~
j2
.
q j2 i 4 i qj 4 1 1 2 2 2 i2
¦ ; 2
if j1 s 0 and p m N i 2 . otherwise
It corresponds to integration with respect to the second variable. Žc. For r s 2 and n s 1, we get a new situation; j
h Ž i4 Ž d . m Ž d .
¡ iqj ~ s ¦ i ; ¢0
k
.
y1
iqj4 Ž d .
k
if p m N i and j - p m , if p m ¦ i m
but our definition does not seem complete because hŽ i4Ž d . p m Ž d . j .
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
567
is not defined yet when p m N i. However, we have the following relations pm q j
Ý
jF¨ Fp m
¦
¨
;Ž . d
p m qjy¨
¨
m Ž d . s 0
that give the formulas
Ž d .
pm
j
m Ž d . s y
Ý
j-¨ Fp
m
y1
pm q j j
pm q j
¦ ;¦ ;
Ž d . p
¨
m
qjy¨
m Ž d .
¨
and therefore, if p N i, m
h Ž i4 Ž d .
pm
m Ž d .
j
. sy
Ý
j-¨ Fp m
y1
pm q j j
¨
pm q i q j y ¨ i
¦
=
pm q j
¦ ;¦ ; y1
;
p
m
qjy¨ qi 4
¨
Ž d . .
This formula may not look as natural as the one in the definition, anyway, we see in this case that h is unique and well defined. Looking at more complicated examples would not be easier than general case which we now try to explain. Let us first show that, in formula for h, and provided it is well defined, we can ‘‘move the inside.’’
but the the ni4
2.3. LEMMA. We always ha¨ e J
J sy 1
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
m Ž d .
J
j
m Ž dˆ . s Ž dn .
J sq 1
m ⭈⭈⭈ m Ž d .
J
sh ˆ I 4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
J sy 1
Jr J
m ni4 Ž dˆ . s Ž dn .
m Ž d .
J sq 1
m ⭈⭈⭈ m Ž d .
j
Jr
.
Proof. This is proved by induction on s, the case s s 1 being trivial. Using the formula for moving coefficients across the tensor products, we get J
h ˆ I 4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . J
J sy 1
J
s h ˆ I 4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
J sy 2
J
qh I 4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . J
j
m ni4 Ž dˆ . s Ž dn . m ⭈⭈⭈ m ni4 Ž dˆ .
J sy 2
J sy 1
J
j
m Ž dˆ . s Ž dn . m ⭈⭈⭈
m d Ž ni4 . Ž dˆ .
J sy 1
j
m Ž dˆ . s Ž dn . m ⭈⭈⭈ . It is therefore sufficient to show that, if Jsy1 / 0, then J
h ˆ I 4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
J sy 2
m d Ž ni4 . Ž dˆ .
J sy 1
m ⭈⭈⭈ s 0.
LE STUM AND QUIROS ´
568 Since
d Ž ni4 . s
Ý 0-uFi
¦ui ;
iyu4 n
u
Ž dn . ,
it is sufficient to show that for u / 0, we have J
J sy 2
h ˆ I 4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
J sy 1
m niyu4 Ž dˆ .
u
Ž dn . m ⭈⭈⭈ s 0.
By induction, this is J
h I 4niyu4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
J sy 2
m Ž dˆ .
J sy 1
u
Ž dn . m ⭈⭈⭈
which is 0 by definition, since u / 0 and Jsy 1 / 0. If it exists, the map h is unique. More precisely, for the
2.4. PROPOSITION. condition
Ý 0-V-J
J J J h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 V
¦;
m Ž d .
Jy V
V
m Ž d . m ⭈⭈⭈ s0
to be satisfied whene¨ er Jˆ/ 0 and jn G p m , it is necessary and sufficient that Ž writing j s jn . J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . sy
Ý
j-¨ -p m
J sy 1
y1
pm q j j
m Ž dn .
pm
ˆ
j
m Ž dˆ . J Ž dn . m ⭈⭈⭈
pm q j
¦ ;¦ ; ¨
J
= h I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . m Ž dn .
p m qjy¨
J sy 1 ¨
ˆ
m Ž dˆ . J Ž dn . m ⭈⭈⭈
when j ) 0 and that J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
J sy 1
J
m Ž dn .
s yh ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . y
Ý
0-¨ -p m J
pm
J sy 1
ˆ
m Ž dˆ . J m ⭈⭈⭈ ˆ
m Ž dˆ . J m Ž dn .
pm
m ⭈⭈⭈
pm
¦ ; ¨
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
J sy 1
m Ž dn .
p m y¨
ˆ
¨
m Ž dˆ . J Ž dn . m ⭈⭈⭈ .
Proof. First of all, note that the formulas we are going to prove do give a definition of h for all generators, although for generators of the type m
ˆ J J p ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 m Ž dn . m Ž dˆ . J m ⭈⭈⭈ ,
we only get a definition by descending induction on s.
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
569
Now, we look at the above condition
Ý 0-V-J
J J J h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 V
¦;
m Ž d .
Jy V
V
m Ž d . m ⭈⭈⭈ s0
with Jˆ/ 0 and j G p m . Writing ¨ s ¨ n , note that all the terms in this sum ˆ and that these two cases are mutually are 0 unless ¨ s j or Vˆ s J, exclusive. Thus, our sum splits into two sums. Actually, if j ) p m , we are only left with one sum
Ý
0-¨ -j
j
J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
¦; ¨
m Ž d .
jy ¨
J sy 1
¨
ˆ
m Ž dˆ . J Ž d . m ⭈⭈⭈ s0
and it gives us the required formula since all terms are 0 unless j y p m F ¨ - j. For j s p m , we get J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . q
Ý
pm
0F¨ -p m
J sy 1
ˆ
m Ž dˆ . J m Ž dn . ˆm m ⭈⭈⭈ p
J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
¦ ; ¨
J sy 1
m Ž dn .
p m y¨
¨
ˆ
m Ž dˆ . J Ž dn . m ⭈⭈⭈ s0 which gives us J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
J sy 1
m Ž dn .
J
s yh ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . y
Ý
0-¨ -p m
pm
J sy 1
ˆ
m Ž dˆ . J m ⭈⭈⭈ pm
ˆ
m Ž dˆ . J m Ž dn . ˆ m ⭈⭈⭈
J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
¦ ; ¨
pm
m Ž dn .
p m y¨
J sy 1
¨
ˆ
m Ž dˆ . J Ž dn . m ⭈⭈⭈ .
2.5. Remark. It follows from Lemma 2.3 that, unless Js s 0, p m N i, j s p m and s - r, we have J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
J sy 1
m Ž d . s Ž y1 .
sy 1 I 4
J
m Ž dˆ . s Ž dn .
J sq 1
m ⭈⭈⭈ m Ž d .
J J ˆ Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . sy 1
j
Jr
LE STUM AND QUIROS ´
570
J
m h ni4 Ž dˆ . s Ž dn .
j
J sq 1
m Ž d .
J
m ⭈⭈⭈ m Ž d . r .
Using the formulas in Proposition 2.4, we also see that we always have J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
J sy 1
m Ž d . s Ž y1 .
J
m Ž dˆ . s Ž dn .
J sq 1
m ⭈⭈⭈ m Ž d .
j
Jr
sy1 I 4
J J ˆ Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 J
j
m h ni4 Ž dˆ . s Ž dn . m Ž d .
J sq 1
m ⭈⭈⭈ m Ž d .
Jr
.
Moreover, if j / 0, then, J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . m Ž dn . s Ž y1 .
pm
sy1 I 4
m Ž d .
J sq 1
J sy 1
j J Ž dn . m ⭈⭈⭈ m Ž d . r
J J . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 Ž dˆ
m h ni4 Ž dn .
pm
m Ž d .
J sq 1
Ž dn .
j
J
m ⭈⭈⭈ m Ž d . r .
The map h is well defined.
2.6. PROPOSITION.
Proof. We need to show that our definition is stable under the relations induced by
Ý 0-V-J
J V
¦ ;Ž . d
Jy V
V
m Ž d . s 0,
for < J < ) p m . We first show that
Ý 0- V-J
J J JyV V J h I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 m Ž d . m Ž d . m ⭈⭈⭈ s 0 V
¦; ˆ
whenever < J < ) p m . As in the proof of Proposition 2.4, we write j s jn and ¨ s ¨ n . The case j s 0 is trivial and the case j G p m was done in Proposition 2.4. Thus, we are left with the case 0 - j - p m. As usual, our sum splits into two sums:
Ý
J h V
¦ ; ˆ
ˆ ˆ 0FV-J
I 4 i4 n
ˆ ˆ J J . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 m Ž dˆ . Jy V Ž dˆ ˆ
j
m Ž dˆ . V Ž dn . m ⭈⭈⭈ q
Ý
0F¨ -j
j
¦ ; ˆ ¨
h
I 4 i4 n
J J . sy 1 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ
m Ž dn .
jy ¨
¨
ˆ
m Ž dn . Ž dˆ . J m ⭈⭈⭈ .
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
571
Only the case p m N i has to be considered. The first sum reduces to J
h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . s Ž y1 .
s
s Ž y1
s
iqj i
q Ž y1
ˆ
I 4 d
J1
m ⭈⭈⭈ m Ž dˆ .
J sy 1
m Ž dˆ . J m niqj4 ⭈⭈⭈
I 4 d
J1
m ⭈⭈⭈ m Ž dˆ .
J sy 1
m niqj4 Ž dˆ . J m ⭈⭈⭈
y1
ˆ
y1
iqj i
s
j
m Ž dˆ . J m Ž dn . m ⭈⭈⭈
y1
¦ ; ˆ Ž ˆ. .¦ ; ˆ Ž ˆ. .¦ ; ˆ Ž ˆ. iqj i
ˆ
J sy 1
I 4 d
J1
m ⭈⭈⭈ m Ž dˆ .
ˆ
J sy 1
m dniqj4 Ž dˆ . J m ⭈⭈⭈ .
Now, we need to compute the second sum. We get
Ž y1.
sy1
Ý
iqj y¨ i
j
0F¨ -j
¦ ;¦ ¨
¨
y1 J
;ˆ
I 4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ .
J sy 1
ˆ
mniqjy¨ 4 Ž dn . Ž dˆ . J m ⭈⭈⭈ s Ž y1 .
sy1
q Ž y1 . s Ž y1 .
¦ ¦ ¦ ¦
sy 1
sy1
q Ž y1 .
iqj i
J1
J sy 1
Ž ˆ. ; ˆ Ž ˆ. ; Ý ¦ ; Ž ˆ. ; ˆ Ž ˆ. Ž ˆ. ; ˆ Ž ˆ. I 4 d
iqj i
iqj i
sy 1
y1
m ⭈⭈⭈ m d
y1
iqj ¨
0-¨ Fiqj
ˆ
m niqj4 Ž dˆ . J m ⭈⭈⭈ ¨
y1
J1
I 4 d
iqj i
ˆ
⭈⭈⭈ m niqjy¨ 4 Ž dn . Ž dˆ . J m ⭈⭈⭈
m ⭈⭈⭈ m d
J sy 1
ˆ
m niqj4 Ž dˆ . J m ⭈⭈⭈
y1
I 4 d
J1
m ⭈⭈⭈ m d
J sy 1
ˆ
m dniqj4 Ž dˆ . J m ⭈⭈⭈ .
Adding both formulas, we do get zero as expected. To finish the proof, we also need to make sure that
Ý 0-V-J
J h V
¦ ; ˆ
I 4 i4 n
J J p . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 m Ž dn . Ž dˆ
m Ž d .
Jy V
m
V
m Ž d . m ⭈⭈⭈ s 0
ˆ jn q p m .. Since the whenever < J < ) p m . We consider the n-uple K s Ž J, 2 differential satisfies d s 0, we always have
Ý 0-W-K
Ky W W K d Ž d . m Ž d . W
¦;
s
Ý 0-W-K
K W
¦ ;Ž . d
Ky W
m d Ž d .
W
,
LE STUM AND QUIROS ´
572 and it follows that
Ý 0-W-K
K h W
¦ ; ˆ
I 4 i4 n
J J . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 Ž dˆ
md Ž d . s
Ý 0-W-K
K h W
Ky W
¦ ; ˆ
I 4 i4 n
W
m Ž d . m ⭈⭈⭈ J J KyW . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 m Ž d . Ž dˆ
md Ž d .
W
m ⭈⭈⭈ .
In the first sum, if < W < ) p m , we obviously get 0. If not, then < K y W < ) p m and we can use the first part of the proof to conclude that the corresponding term is 0. Thus, we see that
Ý 0-W-K
K h W
¦ ; ˆ
I 4 i4 n
J J KyW . sy 1 m Ž d . Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ
md Ž d .
W
m ⭈⭈⭈ s 0.
Writing k s k n and w s wn , it is clear that all the terms are zero unless ˆ s K. ˆ Therefore, we have w s k or W ˆ ˆ Kˆ J J h ˆ I 4ni4 Ž dˆ . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 m Ž dˆ . KyW ˆ W ˆ ˆ 0FW-K
Ý
¦;
ˆ
md Ž dˆ . W Ž dn . q
Ý 0Fw-k
¦wk ;h ˆ
I 4 i4 n
k
m ⭈⭈⭈
J J . 1 m ⭈⭈⭈ m Ž dˆ . sy 1 Ž dˆ
m Ž dn .
ky w
w
ˆ
m d Ž dn . Ž dˆ . K m ⭈⭈⭈ s 0.
Using the first part of the proof, we see that all the terms in the first sum ˆ s 0 and k s p m . But in this case, < Kˆ y Wˆ < ) p m and we are zero unless W also get 0. Finally using Definition 2.1, we easily see that all the terms in the second sum are 0 unless k y w s p m. This gives us exactly what we want.
´ LEMMA 3. THE FORMAL POINCARE NIN HIGHER LEVEL We are going to prove in full generality a result which extends Example 2.2Ža.. First, we need a lemma.
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
573
3.1. LEMMA. With the notations of Section 2, we ha¨ e, for 0 - j - p m , J
h d Ž ni4 Ž dˆ . Ž dn .
j
J
s hd ni4 Ž dˆ . Ž dn .
. mP
j
m P.
Proof. Since h is linear and J
d Ž ni4 Ž dˆ . Ž dn .
j
. s d Žni4 . m Ž dˆ . J Ž dn . j . q ni4d
J
. Ž dn . Ž dˆ
j
.
,
it is enough to show that we can move the P outside both in hw d Žni4 . m Ž dˆ . J Ž dn . j . m P x and in hwni4d ŽŽ dˆ . J Ž dn . j . m P x. We first compute J
j
d Ž ni4 . m Ž dˆ . Ž dn . s
Ý 0-uFi
¦ui ;
iyu4 n
u
J j . Ž dn . . Ž dn . m Ž dˆ
Since j / 0 and j / p m , it follows from Remark 2.5 that we can move the P outside in hw d Žni4 . m Ž dˆ . J Ž dn . j . m P x. Now, we compute J
d Ž dˆ . Ž dn .
j
sy
Ý
0- Ž V ; ¨ .-Ž J ; j .
J V
j
¦ ;¦ ;Ž ˆ . d
¨
Jy V
Ž dn .
jy¨
V
¨
m Ž dˆ . Ž dn . .
Multiplying by ni4 on the left, tensoring with P on the right, and applying h, this expression splits into two sums as usual y
0FV-J
y
J h V
¦ ; Ž ˆ . Ý ¦; Ž .
Ý
j
0F¨ -j
¨
h
i4 n
i4 n
d
d
Jy V
m Ž dˆ . Ž dn . m P
jy ¨
m Ž dˆ . Ž dn . m P
n
V
J
j
¨
Žactually, if J s 0, then ¨ cannot be 0 in the second sum, but it does not matter.. Thanks to Remark 2.5 again, we see that we can move the P outside in hwni4d wŽ dˆ . J Ž dn . j . m P x since j - p m in the first sum and m j y ¨ - p in the second. 3.2. PROPOSITION. With the notations of Section 2, we ha¨ e h( d q d( h s Id y where is the projector that sends i and Ž d i . J to itself if i - n and to 0 otherwise. Proof. We only need to check this identity on a set of generators of ⍀ rP m , and we take the elements of the form J
j
Q m ni4 Ž dˆ . Ž dn . m P with Q [ ˆ I 4Ž dˆ . J1 m ⭈⭈⭈ m Ž dˆ . J sy 1 , P [ Ž d . J sq 1 m ⭈⭈⭈ m Ž d . J r and, ei-
LE STUM AND QUIROS ´
574
ther s s r or 0 - j - p m. We have J j . Ž dn . m P Ž hd q dh . Q m ni4 Ž dˆ J
j
J
j
J
j
s hd Q m ni4 Ž dˆ . Ž dn . m P q dh Ž Q m ni4 Ž dˆ . Ž dn . m P s h dQ m ni4 Ž dˆ . Ž dn . m P q Ž y1 .
sy 1
J
j
h Q m d Ž ni4 Ž dˆ . Ž dn .
s
J
. mP
j
q Ž y1 . h Q m ni4 Ž dˆ . Ž dn . m dP q Ž y1 .
sy 1
J
s
J
s Ž y1 . dQ m h ni4 Ž dˆ . Ž dn . J
yQ m h ni4 Ž dˆ . Ž dn .
j
J
j
.P J
P q Q m h d Ž ni4 Ž dˆ . Ž dn . sy1
dP q Ž y1 .
J
j
. mP
dQ m h ni4 Ž dˆ . Ž dn .
j
P
. m P q Q m h ni4 Ž dˆ . J Ž dn . j dP J j J j d Ž ni4 Ž dˆ . Ž dn . . m P q Q m dh Ž ni4 Ž dˆ . Ž dn . . m P ,
qQ m dh Ž ni4 Ž dˆ . Ž dn . sQmh
j
d Q m h Ž ni4 Ž dˆ . Ž dn .
j
and, using Lemma 3.1 we get J j . Ž dn . m P Ž hd q dh . Q m ni4 Ž dˆ J
s Q m Ž hd q dh . ni4 Ž dˆ . Ž dn .
j
m P.
Thus, we may assume that r F 1. In other words, we are left with showing that we always have J j J j . Ž dn . s ni4 Ž dˆ . Ž dn . Ž hd q dh . ni4 Ž dˆ
unless i s j s 0 in which case we get 0. We write [ ni4Ž dn . j and we compute Ž hd q dh.w Ž dˆ . J x. We assume first j / 0 and J / 0. The same computation as in Lemma 3.1 gives us hd Ž dˆ .
J
s
Ý 0-uFi
y
¦ui ;h
J h V
u
J j . Ž dn . Ž dn . m Ž dˆ
¦ ; Ž ˆ . Ý ¦; Ž . Ý
0FV-J
y
iyu4 n
j
0F¨ -j
¨
h
i4 n
i4 n
d
d
n
Jy V
jy ¨
V
m Ž dˆ . Ž dn . J
m Ž dˆ . Ž dn .
¨
j
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
s
Ý 0-uFi
¦ui ;h
iyu4 n
u
j J . Ž dn . m Ž dn . Ž dˆ
J
q Ž dˆ . h ni4 Ž dn . y
Ý
j
0-¨ -j
¦; h
¨
y h ni4 Ž dn .
j
575
i4 n
j
Ž dn .
m Ž dˆ .
jy ¨
m Ž dn .
¨
. Ž dˆ
J
J
J
J
J
s hd Ž . m Ž dˆ . y h Ž . Ž dˆ . q Ž dˆ . hŽ w . J
J
s hd Ž . m Ž dˆ . y dh Ž w . Ž dˆ . . Adding to dh Ž dˆ .
J
s d h Ž . Ž dˆ .
J
J
s dh Ž . m Ž dˆ . q h Ž . d Ž dˆ .
J
,
we see that J J . s Ž hd q dh . Ž . m Ž dˆ . . Ž hd q dh . Ž dˆ
Obviously, this formula still holds when J s 0. What happens if j s 0 and J / 0? We have hd ni4 Ž dˆ .
J
s
Ý 0-uFi
y
¦ui ;h
Ý 0-V-J
J h V
iyu4 n
¦;
i4 n
u
J . Ž dn . m Ž dˆ
. Ž dˆ
Jy V
m Ž dˆ .
V
J
s hd Ž ni4 . m Ž dˆ . .
On the other hand, we have dhwni4Ž dˆ . J x s 0 s dhŽni4 .Ž dˆ . J since, by definition, h s 0 when r F 0. Thus, we see that J J . s Ž hd q dh . Ž ni4 . Ž dˆ . . Ž hd q dh . ni4 Ž dˆ
In other words, we may assume that n s 1 Žand still r F 1.. Thus, we want to show that we always have j j Ž hd q dh . i4 Ž d . s i4 Ž d .
unless i s j s 0, in which case we get 0. For j s 0, this is done in Example 2.2Ža. so we assume now j / 0. If i s 0, then we have
Ž hd q dh . Ž d .
j
s hd Ž d . sy
Ý
0-¨ -j
j
q dh Ž d . j
h Ž d .
¦; ¨
j
jy ¨
m Ž d .
¨
q d Ž j4 .
LE STUM AND QUIROS ´
576
j
jy¨ 4
Ž d .
j
jy¨ 4
j Ž d . s Ž d . .
¦ ; Ý ¦ ;
sy
Ý
¨
0-¨ -j
q
¨
0-¨ Fj
¨
¨
Thus, we are left with the case i / 0 and j / 0. We have hd i4 Ž d .
j
s h d i4 m Ž d . s
Ý 0-uFi
y
¦ui ;h
Ý
j
0-¨ -j
j
iyu4
¦; h
¨
j
q h i4d Ž Ž d .
.
u
j Ž d . m Ž d .
i4
jy ¨
Ž d .
m Ž d .
¨
.
Assume first that p m ¦ i, then the second sum is 0. Moreover, in the first sum, if we write i s qp m q t with 0 - t - p m , the only u for which we do not get a 0 term is u s t. Thus, we see that hd i4 Ž d .
j
s
i t
y1
i t
j
j
i4 Ž d . s i4 Ž d . .
¦;¦;
Since in this case dhw i4Ž d . j x s 0, we are done. Finally, we are left with the case p m N i and i / 0. We have dh i4 Ž d .
j
y1
iqj i
¦ ; ¦ ;
sd
iqj4
y1
iqj s i
Ý
0-uFp m
iqj u
¦ ;
iqjyu4
Ž d .
u
and hd i4 Ž d .
j
s
Ý
0-uFp m
y
Ý
¦ui ;h
0-¨ -j
s
j
iyu4
¦; ¨
h
i4
u
j Ž d . m Ž d .
Ž d .
jy ¨
m Ž d .
¨
m i j pm h iyp 4 Ž d . m Ž d . pm
¦ ; y
Ý
0-¨ -j
j
iqjy¨ i
¦ ;¦ ¨
y1
;
¨
iqjy¨ 4 Ž d . .
Now, we use the formula of Proposition 2.3, or more precisely in this case,
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
577
the formula of Example 2.2Žc.: m
h iyp 4 Ž d . sy
pm
Ý
j-¨ Fp
sy
Ý
j-¨ Fp
m Ž d .
m
m
j y1
pm q j j
pm q j
¦ ;¦ ; ¦ ; ¦ ;¦ ¨
y1
pm q j j
p m qjy¨
m
h iyp 4 Ž d . iqjy¨ i y pm
pm q j ¨
m Ž d .
¨
y1 ¨
;
iqjy¨ 4 Ž d . .
Now, we have i pm
pm q j j
¦ ;¦
y1
pm q j
iqjy¨ i y pm
; ¦ ;¦ ¨
y1
iqj s i
y1
iqj
; ¦ ;¦ ; ¨
and it follows that m i j pm h iyp 4 Ž d . m Ž d . pm
¦ ;
iqj sy i
y1
¦ ;
Ý
j-¨ Fp
iqj
m
¦ ; ¨
iqjy¨ 4
¨
Ž d . .
Since we also have j
iqjy¨ i
¦ ;¦ ¨
y1
iqj s i
y1
iqj
; ¦ ;¦ ; ¨
we get that hd i4 Ž d .
j
iqj i
y1
¦ ;
sy
Ý
0-¨ Fp
m
but ¨ /j
iqj
¦ ; ¨
iqjy¨ 4
¨
Ž d . .
Adding both formulas, we get j j Ž hd q dh . i4 Ž d . s i4 Ž d .
and the proposition is proved. As a consequence, we get the Formal Poincare ´ Lemma in higher level: 3.3. THEOREM. The de Rham complex ⍀ Pⴢ m is a resolution of OX . Proof. We proceed by induction on the dimension on X. The assertion being local in nature, it is sufficient to show that if f : X ª X ⬘ is a smooth morphism of relative dimension one of smooth schemes over S, then the map of complexes on X, f *: f *⍀ Pⴢ ⬘m ª ⍀ Pⴢ m is a quasi-isomorphism. More precisely, locally, there are coordinates t 1 , . . . , t n on X such that t 1 , . . . , t ny1
578
LE STUM AND QUIROS ´
are coordinates on X ⬘ and f * is a homotopy equivalence. Actually, since this map, which is just the inclusion i ¬ i and Ž d i . J ¬ Ž d i . J for i s 1, . . . , n y 1, has an obvious retraction given by n ¬ 0 and Ž dn . J ¬ 0, it is sufficient to show that the projector :n ¬ 0 and Ž dn . J ¬ 0 on ⍀ Pⴢ m is homotopic to the identity. In other words, we need to find a homotopy h on ⍀ Pⴢ m such that h( d q d( h s Id y . But this is just the content of Proposition 3.2.
´ LEMMA IN 4. THE POINCARE CRYSTALLINE COHOMOLOGY We show that, as in the case m s 0, one can derive a de Rham interpretation of crystalline cohomology from the Formal Poincare ´ Lemma. What follows is a straightforward generalization of the classical proof Žsee wB1, B-Ox.. For the basic notions concerning the crystalline site and topos of level m, the reader can look at wLS-Q, 4.1x. We let Ž S, ᑾ, ᑿ . be an m-PD-scheme with p locally nilpotent such that p g ᑾ and let X be a smooth S-scheme to which the m-PD-structure of S extends. 4.1. Crystals and the de Rham Complex. If E is an abelian sheaf on ˇ Cris Ž m. Ž XrS ., the Cech᎐Alexander complex EP Ž⭈. of E is the complex of abelian sheaves on X obtained by applying E to the simplicial complex ˇ PX mŽ⭈.. The normalized Cech᎐Alexander complex is the complex NEP Ž⭈. s lKer sUi . Assume E is an m-crystal. Then for each r, the first projection p1: PXŽ m. Ž r . ª X gives us an isomorphism EP Ž r . ( E X mO X PX mŽ r . and we will sometimes write E X mO X PX mŽ⭈. instead of EP Ž⭈. . However, the reader should keep in mind that, at this point, the differential is not linear in E X . We also have NEP Ž r . s E X mO X N PX mŽ r . and we will likewise sometimes write NEP Ž⭈. s E X mO X N PX mŽ⭈.. Recall that KX⭈ m denotes the differential m ideal in N PX mŽ⭈. generated by IX mŽ1. p q1 4. Then E X mO X KXⴢ m is stable under the differential of E X mO X N PX mŽ⭈. and we get another description of the de Rham complex of the DX m -module E X : E X m ⍀ ⴢX m s Ž E X m N PX m Ž ⭈ . . r Ž E X m KXⴢ m . . We can also consider the linearized construction: we build the complex of abelian sheaves Eq P Ž⭈. on X obtained by applying the sheaf E to the simplicial complex PXqmŽ⭈.. The map of simplicial complexes d 0 : PXqmŽ⭈. ª PX mŽ⭈. induces a morphism of complexes dU0 : EP Ž⭈. ª Eq P Ž⭈. . If we consider PXqmŽ⭈. as a simplicial complex over X by using the first projection p1 , if E is an m-crystal and if we identify EP Ž r . with E X mO X
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
579
Ž . PX mŽ r . as before, then Eq P Ž⭈. s E X mO X PX m ⭈q 1 with a differential that q Ž . is linear in E X . In other words, we see that Eq P Ž⭈. s E X mO X PX m ⭈ . With U q these notations, the map d 0 : E X mO X PX mŽ⭈. ª E X mO X PX mŽ⭈. is given for each r by the composition of the scalar extension E X m PX m Ž r . ª PX m Ž 1 . m E X m PX m Ž r . and the map
m Id: PX m Ž 1 . m E X m PX m Ž r . ª E X m PX m Ž 1 . m PX m Ž r . s E X m PX m Ž r q 1 . , where : PX m Ž1. m E X ª E X m PX mŽ1. is the Taylor isomorphism. 4.2. Crystalline Interpretation of Linearization. We consider the crystalline site of level m, Cris Ž m. Ž XrS . and the associated topos Ž XrS .Žcrm.i s . We have a canonical morphism of topoi Ž m.
u s uŽXm. r S : Ž XrS . cr i s ª X Z ar and a localization morphism Ž m.
Ž m.
j X : Ž XrS . cr i sNX ª Ž XrS . cr i s . Note that j X # is exact. If we compose u with j X we get a morphism of topoi Ž m.
u N X s Ž XrS . cr i sNX ª X Z ar . It is actually is a morphism of ringed topoi. To see this, note that the structural sheaf of X Z ar is OX and the structural sheaf of Ž XrS .Žcrm.i s N X is the restriction of OXŽ m. r S . Thus, we need to have a morphism of rings Ž m. uy1 O ª O . In other words, given an open subset U of X, an NX X XrSN X m-PD-thickening U ¨ Y, and a morphism p:Y ª X extending the inclusion of U in X, we need a morphism OU ª O Y and we just take py1 . One can check that u N X# is exact and that the adjunction map u N X# uUN X F ª F is an isomorphism. If F is a sheaf of OX -modules, we let LŽ m. Ž F . [ j X #uUN X F and LŽXm. Ž F . [ LŽ m. Ž F . X . 4.3. PROPOSITION. Ž1. If F is a sheaf of OX -modules, then LŽ m. Ž F . is an m-crystal and we ha¨ e LŽXm. Ž F . s PX mŽ1. mO X F. Ž m. Ž . Ž m. Ž . Ž2. We ha¨ e uŽXm. F s F and R i uŽXm. F s 0 for i ) 0. r S# L r S# L Ž3. If E is an m-crystal, there is a canonical isomorphism E m LŽ m. Ž F . ( LŽ m. Ž E X m F .. Proof. Note first of all that, by definition, given any open subset U of X and any m-PD-thickening U ¨ Y, we have LŽ m. Ž F . Y s p1# pU2 F where p1: PU mŽ Y =S X . ª Y and p 2 : PX mŽ Y =S X . ª X denote the projections. In particular, LŽXm. Ž F . s PX mŽ1. mO X F. Now, in order to prove that
LE STUM AND QUIROS ´
580
LŽ m. Ž F . is an m-crystal, we need to check that, given a morphism of Žwhere pX1 and m-PD-thickenings f :Y ⬘ ª Y, we have f *p1# pU2 s pX1# pXU 2 X p 2 are defined in the same way as p1 and p 2 .. We may clearly assume that U s X. Moreover, the question is local. Thus, since X is smooth, we may assume that X ¨ Y has a retraction. It is therefore sufficient to consider the case Y s X. Writing Y instead of Y ⬘, g for the canonical map PX mŽ Y =S X . ª PX m Ž1., p1: PX mŽ1. ª X and q1: PX mŽ Y =S X . ª Y for the first projection, we are led to show that f *p1# s q1# g*. But this follows from the fact that, PX mŽ1. being flat over X, the canonical map PX mŽ Y =S X . ª Y =X PX mŽ1. is an isomorphism. We now prove the second assertion. Since the adjunction map u N X #uUN X F ª F is an isomorphism, we have u# LŽ m. Ž F . s u N X #uUN X F s F. Now, since u N X # and j X # are exact, we obtain Ru# LŽ m. Ž F . s Ru# j X #uUN X F s R Ž u# j X # . uUN X F s R Ž u N X # . uUN X F s u N X #uUN X F s u# LŽ m. Ž F . . For the last assertion, we use the same notations as in the beginning of the proof. We need a compatible family of isomorphisms E Y m ; p1# pU2 F ª p1# pU2 Ž E X m F ., that is, ;
E Y m PX m Ž Y = X . mO X F ª PX m Ž Y = X . m E X m F . But, E being a crystal, we have E Y mO Y PX m Ž Y = X . s EP ŽY=U . s PX m Ž Y = X . mO X E X . 4.4. Remark. Before going any further, we would like to describe the Taylor isomorphism of LŽXm. Ž F . [ PX mŽ1. m F. By definition, it is obtained by composing the isomorphism ;
PX m Ž 1 . m LŽXm. Ž F . s PX m Ž 1 . m Ž PX m Ž 1 . m F . ª PX m Ž 2 . m F
Ž f m g . m Ž h m k . m x ¬ f m gh m k m x with the inverse of ;
LŽXm. Ž F . m PX m Ž 1 . s Ž PX m Ž 1 . m F . m⬘ PX m Ž 1 . ª PX m Ž 2 . m F
Ž f m g . m x m ⬘ Ž h m k . ¬ fh m k m g m x, where m⬘ means that the tensor product uses both left structures. Thus, the Taylor isomorphism is given by ;
: PX m Ž 1 . m LŽXm. Ž F . ª LŽXm. Ž F . m PX m Ž 1 .
Ž f m g . m Ž h m k . m x ¬ Ž f m k . m x m⬘ Ž 1 m gh . . 4.5. PROPOSITION. Ž1. There exists a unique OXŽ m. r S -linear differential on L ⍀ ⴢX m . such that LŽXm. Ž ⍀ ⴢX m . s ⍀ Pⴢ m . Ž m. Ž
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
Ž2.
581
If E is an m-crystal, we ha¨ e Ž m. RuŽXm. Ž ⍀ ⴢX m . . s E X m ⍀ ⴢX m . r S# Ž E m L
Proof. We saw in Proposition 4.3 that LŽXm. Ž ⍀ rX m . is an m-crystal and that, for all r, we have LŽXm. Ž ⍀ rX m . s ⍀ rP m . To prove the first assertion it is sufficient to check that the differential of ⍀ Pⴢ m is compatible with the Taylor isomorphisms. In order to do so, we describe the Taylor isomorphism of LŽXm. Ž F . in the particular case F s PX mŽ k .. We have canonical isomorphisms PX mŽ1. m LŽ m. Ž F . s PX mŽ k q 2. and LŽXm. Ž F . m PX mŽ1. s PX mŽ k q 2.. From Remark 4.4, we see that the Taylor isomorphism is induced by the automorphism f m g m h1 m ⭈⭈⭈ m h k ¬ f m h1 m ⭈⭈⭈ m h k m g of PX mŽ k q 2.. In particular, we see that the maps d i : LŽ m. Ž PX m Ž k . . s PX m Ž k q 1 . ª LŽ m. Ž PX m Ž k q 1 . . s PX m Ž k q 2 . are compatible with the Taylor isomorphisms, for i ) 0. It follows that the Ž m. Ž . same is true for the differential of the complex PXq ⭈ and also for the ⴢ Ž m. Ž . differential of the complex ⍀ P m which is a subquotient of PXq ⭈. The second assertion will follow from Proposition 4.2 once we know that u# sends the differential of E X m ⍀ Pⴢ m to the differential of E X m ⍀ ⴢX m . Note that, in general, the map dU0 F s u# LŽ m. Ž F . ª LŽXm. Ž F . s PX m Ž 1 . m F is just the adjunction map for u N X . In particular, it is functorial in LŽ m. Ž F . and injective. In order to show that u# sends the differential of E X m ⍀ rP m to the differential of E X m ⍀ rX m , it is therefore sufficient to check that dU0 preserves differentials. But as we mentioned in Subsection 4.1, dU0 comes from a morphism of simplicial complexes. 4.6. Remark. It is not difficult to describe the DXŽ m.-structure on Let us first consider the m-connection
LŽXm. Ž F ..
;
ⵜ : LŽXm. Ž F . ª LŽXm. Ž F . m PX m Ž 1 . induced by the Taylor isomorphism Žwe write ⵜ and not d, so as to avoid confusion with the differential of the de Rham complex.. We have ⵜ Ž f m g . m x s Ž 1 m g . m x m ⬘ Ž 1 m f . y Ž f m g . m x m ⬘ Ž 1 m 1. s Ž 1 m g . m x m ⬘ Ž 1 m f . y Ž 1 m g . m x m ⬘ Ž f m 1. s Ž 1 m g . m x m ⬘df . If we have local coordinates on X, this last formula gives ⵜŽ m x . s ⵜ Ž 1 m t . m x y ⵜ Ž t m 1. m x s y Ž 1 m 1. m x m
LE STUM AND QUIROS ´
582 from which we derive ⵜŽ J 4 m x . s
J V
< < Ý Ž y1. V
¦ ;Ž
0-VFJ
JyV 4
m x . m V 4 .
At this point, it is convenient to use partial derivatives. Thus, we see that ⵜ² I : Ž f m 1 . m x s ⭸ ² I : Ž f . m x and ⵜ² I : Ž J 4 m x . s Ž y1 .
J I
¦ ;
JyI 4
m x.
Finally, we get ⵜ² I :
< < Ž f J 4 m x . s Ý Ž y1. U
0FUFI
I U
J U
½ 5¦ ;⭸
² IyU :
Ž f . JyU 4 m x.
Using this formula, it is not hard to check directly that
Ž ⵜ² I : ( d .
Ž f J 4 m Ž d . J
s Ž d(ⵜ² I : .
1
m ⭈⭈⭈ m Ž d .
Ž f J 4 m Ž d . J
1
Jr
.
m ⭈⭈⭈ m Ž d .
Jr
..
This gives another proof that the differential of the complex LŽ m. Ž ⍀ ⴢX m . is DXŽ m.-linear. We prove now the Poincare ´ Lemma for crystals of level m and show that de Rham cohomology of level m can be used to compute crystalline cohomology of level m. 4.7. THEOREM. If E is an m-crystal on XrS then the complex E m LŽ m. Ž ⍀ ⴢX m . is a resolution of E, we ha¨ e an isomorphism in the deri¨ ed category ;
ⴢ RuŽXm. r S# E ª E X m ⍀ X m ,
and, for all i, an isomorphism ;
Hcri i s, m Ž XrS, E . ª Hdi R , m Ž X , E X . . Proof. The first assertion is not completely formal since the functor from the category of locally quasi-nilpotent DXŽ m.-modules to the category of sheaves on Cris Ž m. Ž XrS . is not exact in general wB1, IV, 1.7.8x. We need to show that the obvious map E ª E m LŽ m. Ž ⍀ ⴢX m . is a quasi-isomorphism. It is sufficient to check that if U ¨ T is a sufficiently small m-PD-thickening, the map ET ª ET m LŽ m. Ž ⍀ ⴢX m .T is a quasi-isomorphism. We may assume that U s X and that X ¨ T has a restriction. Moreover, as we saw in Proposition 1.4, ⍀ ⴢX m is a complex of locally free sheaves, and we may therefore assume that T s X and E s OXŽ m. r S . Then, our assertion follows from the Formal Poincare ´ Lemma ŽTheorem 3.3..
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
583
Applying RuŽXm. r S# gives the second assertion thanks to Proposition 4.5. Finally, we take cohomology on both sides. 4.8. Generalization to Singular Schemes. Even if we no longer assume that X is smooth over S, we can still consider the crystalline topos Ž XrS .Žcrm.i s of level m and the canonical morphism of topoi Ž m.
u s uŽXm. r S : Ž XrS . cr i s ª X Z ar . Let E be an m-crystal on XrS. If i: X ¨ Y is an embedding into a smooth S-scheme, we know that i# E is a crystal on YrS and that R q i# E s 0 for q ) 0 Žsee wB-O, 7.1.2x in the case m s 0.. We write E Y [ Ž i# E . Y . We can form the de Rham complex E Y m ⍀ Yⴢ m Žwhich is actually supported on X . and it follows from Theorem 4.7 that Ž m. Ž m. R⌫ Ž X , RuŽXm. r S# E . s R⌫ Ž Y , Ri# Ru X r S# E . s R⌫ Ž Y , Ru Y r S# Ri# E . ⴢ s R⌫ Ž Y , RuŽYm. r S# i# E . s R⌫ Ž Y , Ž i# E . Y m ⍀ Y m . .
Therefore, we have ;
Hcri i s, m Ž XrS, E . ª Hdi R , m Ž Y , E Y . . 5. COMPLEMENTS AND APPLICATIONS All formal schemes are p-adic formal schemes over a fixed complete discrete valuation ring V of characteristic 0 with residue field of characteristic p. 5.1. The de Rham Complex of Finite Order. Let S be a scheme or a formal scheme and X be a smooth Žformal. S-scheme. In the formal scheme case, we cannot use divided power envelopes of infinite order. However, for each integer k, we can consider the ˇ Cech᎐Alexander complex PXk mŽ⭈. of order k wB4, Sect. 2.1x, and just as we did in Subsection 1.1, the normalized complex N PXk mŽ⭈. associated to it, ⴢ k Ž . and the quotient ⍀ k, X m of N PX m ⭈ by the differential ideal generated by k Ž . k Ž . p mq1 4 IX m 1 . Note that PX m ⭈ is not in general the tensor algebra on ⴢ k Ž . m PXk mŽ1. and that ⍀ k, X m s N PX m ⭈ when k F p . Moreover, since as in Subsection 1.2 all the generators and all the elements occurring in the ⴢ relations are of order at most p m , ⍀ k, X m does not depend on k when m ⴢ ⴢ k G 2 p . It is important to remark that for schemes we have ⍀ k, Xm s ⍀Xm m when k G 2 p and, taking into account the independence, it will do no ⴢ m harm to write in general ⍀ ⴢX m s ⍀ k, X m for k G 2 p . This complex has locally free terms. We can define the de Rham complex of a DXŽ m.-module F as before: the action of DXŽ m. on F induces, for all k, a morphism d: F ª F mO X PXk m that gives rise for k G p m , and in particular for k G 2 p m , to an m-connec-
LE STUM AND QUIROS ´
584
F ª F mO X ⍀ 1X m . Using the Leibniz rule, it extends to a differential tion d:F on F mO X ⍀ ⴢX m . This is the de Rham complex of F and we can define the de Rham cohomology of le¨ el m of F : Hdi R , m Ž X , F . [ H i Ž X , F mO X ⍀ ⴢX m . . We now want to describe a construction analogous to that of Subsection 1.5 that works over formal schemes and has also some intrinsic interest for schemes. An integer k being fixed as before, we consider the shifted simplicial complex k PXkq m Ž ⭈ . s PX m Ž r q 1 . ; d i , i s 1, . . . , r q 1; s i , i s 1, . . . , r q 1 4 ,
Ž. the associated complex of abelian sheaves PXkq m ⭈ , and the normalized U kq Ž . complex N PXkq ⭈ [ lKer s . We let K be the differential ideal of m i Xm Ž . N PXkq ⭈ generated by m dy1 IXkm Ž 1 . 0
ž
p m q1 4
/ ; NP
kq Xm
Ž 2.
ⴢ and we write ⍀ k, P m for the quotient complex, called the linearized de Rham complex of order k. Only the case k G 2 p m is of interest to us. By construction, the Ž. morphism of complexes dU0 : PXk m ª PXkq m ⭈ induces a morphism of comⴢ k, ⴢ r plexes ⍀ X m ª ⍀ P m . For each r, the map ⍀ rX m ª ⍀ k, P m extends by linear; k Ž . r k, ⴢ Ž ity to an isomorphism PX m 1 mO X ⍀ X m ª ⍀ P m as one easily checks by a local computation.. Thus, there should not be any confusion in writing PXk mŽ1. mO X ⍀ ⴢX m for the linearized de Rham complex of order k G 2 p m . As in Section 2, we can define a homotopy on this complex and deduce that it is a resolution of OX . By duality, the linearized de Rham complex PXk mŽ1. mO X ⍀ ⴢX m of order . which is a right k G 2 p m gives rise to a complex H om O XŽ ⍀ ⴢX m , DXŽ m. k resolution of OX . Since the de Rham complex has locally free terms, going to the limit on k gives rise to a complex H om O XŽ ⍀ ⴢX m , DXŽ m. . of DXŽ m.-modules called the Spencer complex of le¨ el m.
5.2. PROPOSITION. Let S be a scheme or a formal scheme and X be a smooth Ž formal . S-scheme. Then, Ž1. The Spencer complex of le¨ el m o¨ er XrS is a right resolution of OX by locally free DXŽ m.-modules. Ž2. For any DXŽ m.-module F , there is an isomorphism R H om DXŽ m . Ž OX , F . s F mO X ⍀ ⴢX m . Ž3.
For any DXŽ m.-module F , we ha¨ e RHom DXŽ m . Ž OX , F . s Hdi R m Ž X , F . .
POINCARE ´ LEMMA IN CRYSTALLINE COHOMOLOGY
585
Proof. Ž1. Since, by construction, the Spencer complex of level m is a right resolution of OX , there only remains to check that the differential in the Spencer complex is DXŽ m.-linear. Fix two integers k and l with k G 2 p m. It follows from the universal property of the divided power envelope that, for each r, the map X Ž rq2.rS ª X Ž rq1.rS that forgets the second component induces a map Ž kql .q PXl m Ž 1 . =X PXkq Ž r.. m Ž r . ª PX m
By functoriality, we actually get a morphism of simplicial complexes Ž kql .q PXl m Ž 1 . =X PXkq Ž ⭈. . m Ž ⭈ . ª PX m
From this, we deduce a morphism of complexes .q PXŽ kql Ž ⭈ . ª PXl m Ž 1 . mO X PXkqm Ž ⭈ . m
from which we derive a morphism of complexes l ⴢ l k ⴢ PXkq m Ž 1 . mO X ⍀ X m ª PX m Ž 1 . mO X PX m Ž 1 . mO X ⍀ X .
Now we apply duality in order to get a pairing ⴢ Ž m. ⴢ Ž m. DXŽ m. l = H om O X Ž ⍀ X m , DX k . ª H om O X Ž ⍀ X m , DX Ž kql . . .
Taking the limit on k and l gives a pairing DXŽ m. = H om O X Ž ⍀ ⴢX m , DXŽ m. . ª H om O X Ž ⍀ ⴢX m , DXŽ m. . which, by construction, is nothing but the canonical action of DXŽ m. on the Spencer complex. Ž2. It follows from the first part that R H om DXŽ m . Ž OX , F . s H om DXŽ m . Ž H om O X Ž ⍀ ⴢX m , DXŽ m. . , F . s F mO X ⍀ ⴢX m where the left hand side is the de Rham complex of F. Ž3. Take cohomology. 5.3. Crystalline Cohomology o¨ er a Formal Base. Let S be a formal scheme. If X is a smooth formal S-scheme, completing the Spencer ˆXŽ m. . complex H om O XŽ ⍀ ⴢX m , DXŽ m. . gives a right resolution H om O XŽ ⍀ ⴢX m , D of OX because the de Rham complex is made of locally free terms. Thus, if ˆXŽ m.-module, we have as before F is a D
ˆXŽ m. . , F s F mO X ⍀ ⴢX m R H om DˆXŽ m . Ž OX , F . s H om DˆXŽ m . H om O X Ž ⍀ ⴢX m , D
ž
/
and RHom DˆXŽ m . Ž OX , F . s Hdi R , m Ž X , F . . Assume now that the ramification index of V is F Ž p y 1. p m , that S is endowed with an m-PD-structure Ž ᑾ, ᑿ . compatible with p, and let X be
LE STUM AND QUIROS ´
586
6
an S-scheme to which the m-PD-structure of S extends. Let E be a quasi-coherent crystal on XrS. Assume i: X ¨ Y is an embedding into a smooth formal S-scheme and, for all n 4 0, let i n : X ¨ Yn be the map induced mod ᑾ nq 1. We know that E Y [ Ž i# E . Y n is a DYŽ m. -module and it follows that E Y :s lim E Y n n ˆXŽ m.-module. We can therefore form the de has a natural structure of D Rham complex E Y m ⍀ Yⴢ m . 5.4. PROPOSITION.
We ha¨ e
Hcri i s, m
;
Ž XrS, E . ª Hdi R , m Ž Y , E Y . .
Proof. Using Subsection 4.8, one proceeds as in wB-O, Theorem 7.23x. To finish this article, we want to explain how Berthelot’s Frobenius descent wB5x applies in our context. So, we let Ž S, ᑾ, ᑿ . be a formal m-PD-scheme compatible with p and X 0 a smooth formal S0-scheme to which the m-PD-structure of S extends. It follows from wB5, Proposition 2.2.3 and Sect. 4.1x that if F0 : X 0 ª X 0X is the sth iterate of the relative Frobenius of X 0rS0 and E⬘ is a quasi-coherent m-crystal over X 0X rS, then E [ F0U E⬘ is in a natural way an Ž m q s .-crystal over X 0rS. 5.5. PROPOSITION. F0U :
Ž1.
Hcri i s, m
The morphism F0 induces an isomorphism ;
Ž X 0X rS, E⬘ . ª Hcri i s, mqs Ž X 0rS, E . .
Ž2. Assume S is scheme killed by p Nq 1. If m G N and if F: X ª X ⬘ is s a lifting of F0 gi¨ en in local coordinates by F*Ž t i . s t ip , then F induces a quasi-isomorphism F* : EXX ⬘ mO X ⬘ ⍀ ⴢX ⬘m ª E X mO X ⍀ ⴢX Ž mqs. . Proof. Ž1. This is a local question and we may assume that F0 lifts to F : X ª X ⬘. Then, the result follows from the previous results and wB5, Proposition 2.3.8 and Sect. 4.1x. More precisely, the isomorphism ;
F* : RHom DˆXŽ m . Ž OX ⬘ , E X ⬘ . ª RHom DˆXŽ mq s. Ž OX , E X . of wB5x, combined with the identifications of Proposition 5.3 tells us that Frobenius is an isomorphism ;
F* : Hdi R , m Ž X , E X ⬘ . ª Hdi R , mq1 Ž X , E X . and Proposition 5.4 gives the announced isomorphism. Ž2. It follows from wB5, Proposition 2.2.2x that F induces a morphism of simplicial complexs F : PX Ž mqs.Ž⭈. ª PX ⬘mŽ⭈.. Thus, we get morphisms of complexs F* : EXX ⬘ mO X PX ⬘m Ž ⭈ . ª E X mO X PX Ž mqs. Ž ⭈ . and F* : E X ⬘ mO X N PX ⬘m Ž ⭈ . ª E X mO X N PX Ž mqs. Ž ⭈ . .
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Since S is killed by p Nq 1, it follows from Lemma 5.6 below that, for m G N, F* sends the differential ideal KXⴢ ⬘m into KXⴢ Ž mqs. . Thus, we get a morphism of complexs F* : E X ⬘ mO X ⍀ ⴢX ⬘m ª E X mO X ⍀ ⴢX Ž mqs. . Thanks to Proposition 5.2, it corresponds to the canonical map F* : H om DXŽ m⬘ . Ž H om O X ⬘Ž ⍀ ⴢX ⬘m , DXŽ m. ⬘ . , EX ⬘ . ª H om DXŽ mq s. H om O X Ž ⍀ ⴢX Ž mqs. , DXŽ mqs. . , E X
ž
/
which, in the derived category, is the canonical map F* : R H om DXŽ m⬘ . Ž OX ⬘ , E X ⬘ . ª R H om DXŽ mq s. Ž OX , E X . . We know from wB5, Proposition 3.3.8x that this is an isomorphism. There remains to prove the following lemma that generalizes wB5, 2.2.4x. m 5.6. LEMMA. Ž1. For u / 0, the integer Ž pu . has p-adic order G m y u q 1. m mq s Ž2. For m G N, we ha¨ e F*Ž i p . s i p mod p Nq 1. Ž3. Let N be such that X is killed by p Nq 1. Then, for m G N, the image m mq s of IX mŽ1. p q1 4 by the Frobenius map is inside IX Ž mqs.Ž1. p q1 4.
Proof. Ž1. We can write u s qp q r with r - p. In case q s 0, we have ord
pm s m G m y u q 1. u
ž /
If q ) 0, we proceed by induction on m, the case m s 1 being trivial. So let m ) 1. Then, we have ord
pm p my1 pm G ord s ord , qp q u
ž /
ž /
ž /
and by induction, this is G m y q G m y u q 1. s s Ž2. We have F*Ž i . s Ž t i q i . p y t ip q p for some g PX mŽ1. and it follows that F* Ž i p
m
. s Ž ip
s
q p .
pm
s
Ý 0Fu-p
s
Ý
0Fu-p n
n
pm u
ž / Ž
p m u u p sŽ p m yu. p i . u
ž /
ps i
.
p myu
Ž p .
u
588
LE STUM AND QUIROS ´
m If u ) 0, then Ž pu . p u s 0 mod p Nq1 thanks to the first part, and we get m mq s p p F*Ž i . s i mod p Nq 1. m Ž3. It is sufficient to show that iq p qk 4 g PX mŽ1. is sent into mq s q p qk 4 IX Ž mqs.Ž1. . We already know by wB5, Proposition 2.2.2x that, for all m l, the image of IX mŽ1.l4 under F is inside IX Ž mqs.Ž1.l4. Since F*Ž iq p qk 4 . m s F*Ž i p .w q x F*Ž ik ., our assertion follows from the second part of this lemma.
ACKNOWLEDGMENTS We heartily thank P. Berthelot for providing access to some of his unpublished material. The main ideas in this article were inspired by his work. The collaboration on this article was made possible by the financial support of the European Union through the TMR Network Arithmetic Algebraic Geometry ŽERB FMRX 960006..
REFERENCES wB1x
P. Berthelot, ‘‘Cohomologie cristalline des schemas de caracteristique p ) 0,’’ ´ ´ Lecture Notes in Math., Vol. 407, Springer-Verlag, New YorkrBerlin, 1974. wB2x P. Berthelot, Cohomologie rigide et theorie des D-modules, in ‘‘Proc. Conf. p-adic ´ Analysis, Trento 1989,’’ Lecture Notes in Math., Vol. 1454, Springer-Verlag, New YorkrBerlin, 1990. wB3x P. Berthelot, Letter to Illusie, 1990. wB4x P. Berthelot, D⺡† -modules coherent. I. Operateurs differentiels de niveau fini, Ann. ´ ´ Sci. Ecole Norm. Sup. 29 Ž1996., 185᎐272. wB5x P. Berthelot, D⺡† -modules coherent. II. Descente par Frobenius, prepublication ´ 98-32 de l’IRMAR, Rennes, 1998. wB-Ox P. Berthelot and A. Ogus, ‘‘Notes on Crystalline Cohomology,’’ Princeton Univ. Press, Princeton, NJ, 1978. wE-LSx J.-Y. Etesse and B. Le Stum, Fonctions L associees ´ aux F-isocristaux surconvergents. II. Zeros ´ et poles ˆ unites, ´ In¨ ent. Math. 127 Ž1997., 1᎐31. wLS-Qx B. Le Stum and A. Quiros, ´ Transversal crystals of higher level, Ann. Inst. Fourier Ž Grenoble. 47 Ž1997., 69᎐100. wLx D. Lieberman, Generalizations of the de Rham complex with applications to duality theory and the cohomology of singular varieties, in ‘‘Proc. Conf. in Complex Analysis, Rice Univ. Studies, 1972.’’ wTx F. Trihan, Fonction L unite ´ d’un groupe de Barsotti᎐Tate, Manuscripta Math. 96 Ž1998., 397᎐419.