Notes on Crystalline Cohomology. (MN-21) 9781400867318

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NOTES ON CRYSTALLINE COHOMOLOGY by Pierre Berthelot Arthur Ogus

Princeton University Press and University of Tokyo Press

Princeton, New Jersey 1978

Copyright (g) 1978 by Princeton University Press All Rights Reserved

Published in Japan exclusively by University of Tokyo Press in other parts of the world by Princeton University Press

Printed in the United States of America by Princeton University Press, Princeton, New Jersey

Library of Congress Cataloging in Publication Data will be found on the last printed page of this book

Contents

§0.

Preface

§1.

Introduction

§2.

Calculus and Differential Operators

§3.

Divided Powers

§•+.

Calculus with Divided Powers

§5.

The Crystalline Topos

§6.

Crystals

§ 7. §8.

The Cohomology of a Crystal Frobenius and the Hodge Filtration

Appendix A

—

The Construction of

Appendix B

—

Finiteness of

Elim

Γ.Μ

i

Preface

The first seven chapters of these notes reproduce the greatest part of a seminar held by the first author at Princeton University during the spring semester, 1974.

The

seminar was meant to provide the auditors with the basib tools used in the study of crystalline cohomology of algebraic varieties in positive characteristic, and did not

cover all

known

results

on this topic.

These notes

have the same limited purpose, and should really be considered only as an introduction to the subject. In Chapter I, we draw a rapid picture of the various cohomology theories for algebraic varieties in characteristic p, and try to explain the specific need for a p-adic cohomology, as well

as the motivations of the technical

definition of the crystalline site; this chapter is purely introductory, and contains no proof.

The second chapter

introduces some basic notions of differential calculus, and in particular various presentations of the notions of connection and stratification.

Actually, these notions, under

this particular form, are relevant to crystalline cohomology only in characteristic zero; but it seemed more convenient to give first the algebraic presentation of ordinary differential calculus, which is more or less familiar to the reader, and then to explain how it has to be modified to yield a good formalism in characteristic p. this,

we

introduce

To

do

in Chapter III the notion of a divided

ii

power ideal.

The main result in this chapter is the construc­

tion of the divided power envelope of an ideal in an arbitrary commutative ring, which is of constant use in what follows. Chapter IV reviews then the notions of Chapter II with the modifications necessary to work in characteristic p. With chapter V begins the theory of the crystalline topos. Once we have defined the crystalline site, and described the sheaves on this site, we establish the functoriality of the corresponding topos, and show in particular that if X is an S-scheme, and (I,γ) a divided power ideal in S which extends to X, the crystalline cohomology of X relatively to (S,I,Ύ) depends only upon the reduction of

X

modulo I.

The chapter

ends with a discussion of the relations between the crystal­ line and the Zariski topoi, which will be used in Chapter VII to relate crystalline and de Rham cohomologies. is devoted to the notion of crystal.

Chapter VI

First we define crystals,

and show how they can be interpreted as modules on a suitable scheme endowed with a quasi-nilpotent integrable connection. We then associate to of order

1

a complex

K"

of differential operators

on a smooth S-scheme Y a complex of crystals,

with linear differential, on the crystalline site of any Y-scheme X.

In the particular case where K" is the complex

of differential forms on Y relatively to S, we thus obtain a resolution of the structural sheaf of the crystalline topos ("Poincare' lemma" ) . In Chapter VII, we prove (in a new way) the fundamental property of crystalline cohomology:

If

X

is a closed

Ill

subscheme of a smooth S-scheme Y (on which the crystalline cohomology of

X

ρ

is nilpotent),

relatively to

morphic to the de Rham cohomology of

Y

S

relative

is iso­ to

S,

with coefficients in the divided power envelope of the ideal of

X

in

Y; more generally, if

E

is a crystal on

analogous result gives the crystalline cohomology of coefficients in

E.

X, an X

with

We derive several consequences of this

fact, the most important of which is a base changing theorem. As a particular case of this theorem, one gets

"'universal

coefficients" exact sequences, which may be used to relate torsion in crystalline cohomology and "pathologies" in characteristic p, such as varieties having too big de Rham cohomology, non reduced Picard scheme, closed 1-forms not coming from the Albanese variety, etc. crystalline cohomology relative

Finally, we define

to a p-adically complete

noetherian base A, by studying the inverse limit of crystal­ line cohomologies relatively to the A/p . Chapter VIII,due to the second author, gives an appli­ cation of these results to Katz's conjecture. briefly recall the conjecture.

Let us

If X is a proper and smooth

variety over a perfect field k, and

η < 2dim(X), one asso­

ciates to X in degree η two convex polygons.

The first one,

the Hodge polygon, is the convex polygon with sides of slope i and horizontal projection of length h3" = dim k H n - 1 (X,n^ / ). The slopes of the sides of the second one, the Newton polygon, are the slopes of the action of Frobenius on the crystalline cohomology

H (X/W(k)), and

IV

the lengths of their horizontal projections are the multiplicities of the corresponding slopes.

The conjecture then

asserts that the Newton polygon lies

above the Hodge polygon,

and leads thus to p-adic estimates on the zeroes and poles of the zeta function of

X when

k

is a finite field.

been proved by Mazur ([6], [7]) when

X

It has

is projective and

has a projective and smooth lifting on

W(Jc), assuming further

its

Following

Hodge

groups

have

no

torsion.

an

idea

of Deligne, one first proves a local theorem (which can be regarded as a p-adic version of the Cartier isomorphism), from which one can deduce Katz's conjecture for an arbitrary smooth proper X/k, as well as a stronger conjecture of Mazur [6, p. 663]. Finally, there are two appendices, the first of which gives a rapid sketch of Roby's divided power envelope of a module.

The second appendix discusses inverse limits in the

context of derived categories, using a method rather different from Houzel's [SGA 5 XV]. Among the topics not discussed in this book, but closely related, we should mention the following: a)

Poincare' duality, the construction of the cohomology class of a cycle, and the Lefschetz fixed point formula, for which the reader may refer to L21;

b)

crystalline cohomology of abelian varieties, and its relation to Dieudonne' theory, which can be found in Mazur-Messing [8];

V

c)

the slope filtration on crystalline cohomology, and its interpretation,thanks to the De Rham-Witt complex, developed by Bloch and Illusie, following a direction initiated by Mazur (cf. [3], [1], [4], [5]).

Let us finally mention that our knowledge of crystalline cohomology remains far from being as complete as it is for etale cohomology, for example; actually, little progress seems to have been made in most of the questions raised in the introduction of [2]. We hope

that our treatment will be comprehensible to

anyone with a knowledge of Grothendieck's theory of schemes, i.e. with EGA, as well as the standard facts about algebraic De Rham cohomology.

In particular, we have not assumed

familiarity with topoi or derived categories, and have tried to provide an informal development (by no means a systematic treatment) of these ideas as we need them. Apart from some minor details, these notes have been entirely written by Ogus, who tried to recover the rather informal spirit of the seminar; I wish to thank him for all the work he has done.

I also thank Princeton University for its

hospitality during the spring semester, 1974.

Finally, both au-

thors thank Neal Koblitz and Bill Messing for reading the manuscript and making innumerable suggestions, and the typist Ruthie L. Cephas for her patience and precision.

P. Berthelot

vi

References for Preface

[1]

M. Artin, B. Mazur, Formal groups arising from algebraic varieties , preprint.

[2]

P. Berthelot, Cohomologie cristalline des varietes de caracte'ristique ρ > 0, Lecture Notes in Math. 4"S7 , Springer.

[3]

, Slopes of Frobenius on crystalline cohomology, Proc. of Symp. in Pure Math. 29 (19 74), p. 315-328.

[4]

S. Bloch, Algebraic K-theory and crystalline cohomology, preprint.

[5]

L. Illusie, Complexe de de Rham-Witt et cohomologie cristalline, re'sume' d'un cours S 1'Universite" Paris-Sud, 1976.

[6]

B. Mazur, Frobenius and the Ifodge filtration I, Bull. A.M.S. 78 n* 5 (1972), p. 653-667.

[7]

, Frobenius and the Hodge filtration II, Ann. ο? Math. 9_8 (1973), p. 58-95.

[8]

B. Mazur, W. Messing, Universal extensions and onedimensional crystalline cohomology, Lecture Notes in Math. 370, Springer.

1.1

Si.

Introduction.

Let

k

be the field with

q

elements,

a smooth, projective, and geometrically connected scheme.

One

wants to know how many rational points X has, or more generally, the number

of

tension of

valued points of

X

where

is the ex-

of degree v . The values of these numbers are

conveniently summarized in the zeta function of X, given by This can also be written as the product being taken over the closed points of X, where over

k.

deg x means the degree of the residue field

It is clear from the second expression that

x

k(x) is

a power series with integral coefficients and constant term 1. The following results represent the culmination of twenty-five years of work by Weil, Dwork, Grothendieck, Deligne, and others.

Theorem I. 1.

is a rational function of

where

n = dim

Moreover 1

t

and can be written:

and is a polynomial of degree

where

is the

Betti-number of a lifting of X to characteristic zero, if

one exists.

Furthermore, one has a functional equation

where

1.2

Theorem II. ficients. bers

The polynomials

above have integral coef-

Moreover, if

, the complex num-

have absolute value The

second

theorem

was

only

recently

proved

by

Deligne [2], whereas the first is several years older.

It has

been well known for some time that Theorem I follows "formally" from the existence of a sufficiently rich cohomological machine ("Weil cohomology").

This is supposed to be (at least) a functor

from the category of smooth projective k-schemes X to the category of graded finite dimensional algebras over a field K of characteristic zero, enjoying the following properties:

A

i) ii)

iii)

There is a canonical map ("trace"), an isomorphism if X is geometrically connected. The multiplication law and trace map induce pairings which are perfect if X is geometrically connected.

B

If

(resp.

) is the canonical map, then

and multiplication induce an isomorphism:

C

Let

be the free abelian group generated by the irre-

ducible closed subsets of X of codimension natural map ("cycle map")

such that:

r.

There is a

1.3

i)

If

f:X

Y, and if

and

is the map induced from

by

Poincare duality (Aiii), then, wherever are defined on ii)

If

and

and

and

are closed subsets, then , w h e r e m e a n s via

the Kunneth map (B). iii)

If

X = Spec k, we have a commutative diagram:

The above axioms formally imply [8] a Lefschetz fixed point formula: If

f:X •* X, and if

is the diagonal and

is the graph of

f, then

where the left side means the pairing of (Aiii) applied to and for if we let is

This formula essentially proves Theorem I, be the

relative Frobenius:X

in coordinates),

smooth scheme of dimension zero with that

X, (which

is easily seen to be a points.

It follows

and elementary manipulations lead

quickly to the fundamental formula:

1.4

Note. C'

i)1

We can get away with slightly weaker axioms, viz.: The same as above, but

only defined for non-

singular cycles, ii)'

Assuming

and

are smooth and intersect

transversally. iii)'

If

is a closed immersion and X' is also

smooth,

where

is

the unit element, 1

iv)

If

is a closed point,

= deg x .

These properties A,B,C' (together with a comparison with the cohomology of a lifting, for the statement about the Betti numbers), suffice to prove Theorem I. In these notes we shall try to explain one such (weak) Weil cohomology — crystalline cohomology.

It is denoted

,

and takes its values in the (fraction field of) the ring W(k) of Witt vectors of the ground field k.

Let us first briefly

review the history of the attempts to build a Weil cohomology, and the special place of the crystalline theory among them.

1)

Serre's cohomology of coherent sheaves (FAC).

Of course

has characteristic p if X does, in-

stead of characteristic zero, but it was hoped that at least one would recover the Betti numbers of this a lifting ofwith X from Igusa showed that fails, an example of a surface X with violated the expected

dim dim Pic(X).

This Later Serre constructed

1.5

a surface X with H (X, 0„) one dimensional and H CX, βγ,,) = 0, so that

AIb(X) and therefore also the Picard variety vanish.

We

now understand that this behavior is due to singularities of the Picard scheme of X.

For a further discussion of these and

other examples, as well as precise references, we refer the reader Mumford's appendix to Chapter VII of Zariskx's book on surfaces [14]. Since we do have the inequalities β! >, β. , one might try the de Rham hypercohomology sion, we have

H1CX,ΩΛ,·,).

If

β'.' is its dimen­

β! >. β'.' , with inequality caused by the failure

of the Hodge =» de Rham spectral sequence to degenerate. ever for Serre's surface we still have

How­

β" = 1 > β, = 0 (to see

this, convince yourself that Frobenius induces an injection Η1(Χ,0χ)^Η1(Χ,Ω^/κ)).

If we set H 1 CX) = 9_ . Η ^ Χ , Ω ^ )

IT(X,Ω *.. ), we do get a (characteristic p)

or

theory satisfying

A , B , C , above, and (after some fancy footwork) a congruence formula for Z„ — but by no means a proof of Note.

Theorem I.

The proof of the congruence formula alluded to

above is due to Deligne and should appear some day in SGA.

On

the other hand, Katz has given a proof in [SGA VII, exp. XII], based on the theory of Dwork.

2)

Serre's Witt vector cohomology.

If

k

is a perfect field of characteristic

construction yields a canonical lifting of

k

p, a classical

to a discrete

valuation ring W(k) = lim W Ck). Serre generalized this to WCA) = lim W CA) for any. ring A of characteristic ρ > 0, obtained

1.6

1

a sheaf of rings Wη = W η (0λ ) , and then defined H CX 5 W) to be lim H (X,W ). This is a module over W(k) and hence yields a charac­ teristic zero theory; moreover H (X,W) is free and of finite rank.

However, since H 1 CX 5 W) = 0 if

i > dim X, it is clear

that this cannot be all of a Weil cohomology. rank of H (X5CC) is

Furthermore, the

S. 2 dim AIbX 5 and the inequality is

ordinarily strict — in fact if X is a curve, the rank is where

g

is the genus and

of X [13]. Finally, if

σ

2g-a ,

is the p-rank of the Jacobian

i >, 2, Serre showed that

even for

abelian varieties H (X,W) need not be finitely generated (too much torsion) [12]. Nevertheless, Serre [12] was able to "force" a theory for abelian varieties by defining L(X) = H 1 W 5 W ) S T (X*)8 W(k), P Sp where T (X*) is the Tate module of points of order a power of ρ

on the dual.

If φ :X •* X is a group homomorphism, one gets

an endomorphism L(V) of L(X), and Serre proved that the charac­ teristic polynomial of L(iP) agreed with the characteristic poly­ nomial in the S,-adic Tate module as studied by Weil. Thus L is a good p-adic H

for abelian varieties, and should "agree with"

crystalline H

. . Artin and Mazur (unpublished) have shed some light

on the meaning of H (X,W)Qw/,vK5 where K is the fraction field of W(k). They show that H 1 CX 5 W)SK is a quotient of H 1 ^g-(X)SK 5 and under some hypotheses on X, is the part on which Frobenius acts with slopes in [O 5 I).

(The p-adic value of an eigenvalue

of F is called the slope of the eigenvalue — it, unlike the eigenvalue itself, is well-defined.)

Recent work of Bloch [1]

using higher K-theory to generalize the W-construction should

1.7

shed more light on the rest of crystalline cohomology.* us remark that Witt vector cohomology seems

intimately related

with p-torsion phenomena, the non-smoothness of

3)

Let

Pic , etc.

Grothendieck's β-adic cohomology.

So far, this is the only theory which is sufficiently rich to prove Theorem II, and it was the first to prove all of The­ orem I.

This is not the place to even sketch Grothendieck's

magnificent idea.

Let us only say that for

I Φ ρ, one has

reasonable cohomology groups H 1 CXjZZS 11 Z); then to get the Weil cohomology we let H X (X,Q.) = lim H X ( X , Z Z Λ ) ® ¢ , where X is Xx k * 0. like

In order to integrate, one needs terms

. So, says Grothendieck, let us agree to consider,

instead of all nilpotent immersions, only those which are endowed with such "divided powers".

This gives us the so-called

"crystalline site", which seems to have just the amount of rigidity we need.

Before we explain these divided powers,

however, we shall review the formalism of differential operators, their linearization, and their relation to descent data.

1.13

Perhaps at this point it is desirable to answer the ques­ tion: "Why bother with the crystalline theory, when cohomology works so well?"

&-adic

One of the original motivations

was to have a meaningful theory of

p-torsion.

At present,

p-torsion remains almost a total mystery — although in princ­ iple crystalline cohomology does give reasonable looking p-torsion, very little is known about it.

Probably the most

important aspect of crystalline cohomology (which distinguishes it from etale cohomology) is its connection to Hodge theory. In particular, it seems to be the best way to attack Katz's conjecture about the relation of the p-adic nature of zeta functions to Hodge numbers (which first emerged in Dwork's work on hypersurfaces [3]). Perhaps even more striking is Mazur's discovery (in the course of his proof of Katz's con­ jecture) [10] that the Hodge filtration of a suitable variety in characteristic Frobenius on H

ρ > 0

is determined by the action of

. (X/W). Let me remark that for this result it

is crucial to work with cohomology with values in W-modules, not just

W 8 ^-modules.

References for §1

[1]

Bloch, S0 "K-theory and crystalline cohomology" Publ. Math. I.H.E.S.

to appear in

[2]

Deligne, P. "La conjecture de Weil I" Publ. Math. I.H.E.S. 43 (1974).

[3]

Dwork, B. "On the zeta function of a hypersurface II" Ann, of Math 80 No. 2 (1964) pp. 227-299.

[4]

Grothendieck, A. "Crystals and the de Rham cohomology of schemes" in Dix Expose's sur la Cohomologie des schemas North Holland (1968).

[5]

. "On the de Rham cohomology of algebraic varieties" Publ. Math. I.H.E.S 29 (1966) pp. 95-103.

[6]

Katz, N. "On the differential equations satisfied by period matrices" Publ. Math. I.H.E.S. 35 (1968) pp. 71-106.

[7]

Katz, N., and Oda, T. "On the differentiation of DeRham cohomology classes with respect to parameters" J. Math. Kyoto U. 8 (1968), pp. 199-213.

[8]

Kleiman, S. "Algebraic cycles and the Weil conjectures" in Dix Exposes sur la Cohomologie des Schemas, North Holland (1968) pp. 359-386.

[9]

Lubkin, S. "A p-adic proof of the Weil conjectures" Ann, of Math. 87 (1968) pp. 105-255.

[10]

Mazur, B. "Frobenius and the Hodge filtration— estimates" Ann, of Math. 98 (1973) pp. 58-95.

[11]

Monsky, P. P-adic Analysis and Zeta Functions Kinokuniya Book Store, Tokyo (1970).

[12]

Serre, J. P. "Quelques proprietes des varie'te's abeliennes en caracteristique p" Am. J. Math. 80 No. 3 (1958) pp. 715-739.

[13]

[14]

"Sur la topologie des vari£t£s algebriques en caracte'ristique p" Symp. Int. de Top. AIg. Univ. Nac. Aut. de Mexico (1958) pp. 24-53. Zariski, 0. Algebraic Surfaces Springer Verlag, (1971).

(second supplemented edition)

§2.

Calculus and Differential Operators. In this chapter we develop Grothendieck's way of geometr-

izing the notions of calculus and differential geometry, and in particular the notion of a locally (or rather infinitesimally) constant sheaf.

We begin by reviewing the formalism of dif-

ferential operators. If X -» S is a morphism of schemes, and if F and G are modules, then a differential operator from F to G, relative to S, will be an

linear map

which is "almost"

In order to make this precise, we begin by brutally linearizing

h, i.e., by forming the obvious adjoint map:

Using the

-module structure of F, we can make a natural iden-

tification:

where

(In order

to be kind to the typist, we shall often write 0 g for With this identification, a®b8x

to

maps an element of the form

ah(bx).

Notice that F has two structures of an two maps

and

respectively.

sending If we form the scheme

corresponds to one of the projections

lgebra, via the

a to a®l and l®a, then d^ In writing

we use the map d^ to construct the tensor product and the map d^ to obtain an is affine,

module structure on the result.

therefore is

If

We shall find it

convenient to refer to, and to indicate by writing, the

1.2

module structure of that from

from

as the "left" structure and

as the "right" structure.

We can summarize our construction by saying that an linear map

induces a unique such that

linear map

where

is

the map It is easy to tell from Let

whether or not h is linear:

be the kernel of the map induced by multiplica-

tion

the ideal of the diagonal

that I is generated as with

,

Note

module by elements of the form

Indeed, if

, then

and hence: x Now i.e.

h

is

linear iff

iff

for all , i.e., iff

annihilates

This makes, I hope, the following definition of differential operator a reasonable notion of "almost" linear:

2.1

Definition.

-linear map

ferential operator of order equivalently, iff

h

annihilates

factors:

where

and In other words,

operator of order

is a "dif-

is induced by

is the universal differential

1.3

Before proceeding, let us give an explicit description of is smooth, in local coordinates. that

is generated as a left

the form

by all elements of

since any

sum

can be written as a

More generally, if E is any is

generated,

as

an

Proposition.

Suppose

are local

defining an etale map

Then

is the free

basis the image of the left and right

in place of

is smooth and

coordinates, i.e., sections of Let

module,

by

Of course, analogous statements hold with

2.2

Notice first

module with

(This holds with both structures on

symmetry allows us to

consider only the left structure.) Proof.

Since

is a locally closed immersion of smooth

S-schemes, it is a regular immersion. that

Now the hypotheses imply

are a basis of Since

we see that

and hence

is the class of

is free with basis the images of

the monomials of degree j in the

Thus the exact sequences:

and induction make the result

clear.

In order to understand composition of differential operators, let us go back to are of

Suppose

linear; how can we describe g°T: f and g?

Clearly

and in terms the problem is

2.1+

how to recover

from To do this, consider the map

given by

If we identify corresponds to the geometric

map

Clearly

homomorphism and is viewed as

linear if

and

are (simultaneously)

modules by the extreme left or right. makes sense and is

is a ring

Thus

linear (using

the left structure), hence so is

2.3

3)

Lemma.

With the above notation and the identification

The map

Proof. the form

induces maps:

To check 1), it suffices to consider elements of which go by

to

1.5

and then by have observed,

As we

2) follows; it can also be deduced from the

universal mapping property of

For

3), we must check

that the clockwise composition annihilates the ideal

I

is generated by elements of the form

Then

so

Now ' 6

Recall that

' is generated by products

and since

is a ring homomorphism

If we expand this as a sum, have at least the left.

m+1

it is clear that each term must

to the right of the

or

to

In either case, the image of the term in

is

zero. 2.4

Corollary.

If

operators of orders _< n and m tion

and

are differential

respectively, then the composi-

is a differential operator of order

and we have a commutative diagram:

,

1.6

2.3

Remark.

The nontrivial commutativity in the diagram is

the lower left triangular region; its commutativity gives another explanation of 1) and 2) in the above lemma.

It is

obtained by tensoring the square below with F .

If

E and F

are

denote

the sheaf of germs of differential operations of order from

n

which can be canonically identified with Notice that the

-module structure is com-

patible with the usual one on F ; thus if D: ferential operator and composition

a

is a dif-

is a section of

is the

regarding multiplication by a as a differen-

tial operator "a" of order 0.

We let

In fancy language, the functor is represented by the pro-object When X/S is smooth, we can locally give an explicit description of differential operators and their composition:

2.6

ProDosition.

coordinates,

and

Suppose

is smooth, x^,...xn are local

is the image of

is a multi-index, with denotes

is a basis for

and Let

in

If

then that be the so dual basis

1.7

for

and let

ferential operator.

be the corresponding dif-

Then

• is a basis for

viewed as an

module through the second

and composition is given by:

Proof.

linear and

is the

corresponding operator, then for any responding to a.

the operator cor-

is (multiplication by

forms a basis for

. This shows that

viewed as an

module as we have

described. To verify the composition formula, we must show that the map

takes is

Kronecker's

position,

where

S-function. From the definition of com-

is given by is induced by our old

where Recalling that

and is a homomorphism, we compute

If we apply left only with

we get zero unless Applying

and so we are we get zero unless

and in this case we get

2.7

Remark.

The formula implies that the D q 's all commute

(although of course they don't commute with, for instance, all

1.8

operators of order zero).

Moreover, over

we can write

and generates

Thus — a fact which fails in

characteristic p, or over We can now try to use differential operators to develop a suitable notion of locally constant sheaves."1' We begin with the familiar: 2.8

Definition.

"connection" on an

tive map: section of

module

is an addi-

such that E

and

a

if

is a section of

For example, the exterior derivative nection, called the "constant" one. set of all connections on

2.9

Proposition.

is a

is a con-

It is easy to see that the

is in one-one correspondence with

A connection on E is equivalent to

isomorphism

which, modulo the kernel

linear of

Cj,, reduces to the identity endomorphism of E .

Starting from

Since

e

above, let

is linear for the right module structure of . , so

is that

note because

e

reduces to the identity.

Now

compute: "'"For an explanation of the relationship between locally constant sheaves and connections in the classical case, we refer the reader to Deligne's Equations Differentielles... LNM 163, Springer.

2.!

(Recall that Conversely, given

then reversing

the previous calculation shows that the right

structure on

is linear for

Extension of scalars gives us

the

which clearly reduces to the

identity mod

, To see that

the involution aQb

e

is an isomorphism, consider

induced by switching the factors;

We have a

T-linear map a :

of course a is bijective and also reduces to the identity mod ft"*". The endomorphism

is

T-linear, so

is

linear, moreover it reduces to the identity modulo the square zero ideal hence (a°e)

It follows that

is an isomorphism,

is bijective, and hence so is

Let us now try to motivate Grothendieck's description of a connection in terms of a suitable site.

Recall that descent

data for a sheaf E on X relative to morphism

on

tivity conditions. E

means an isosatisfying certain transi-

If E

for some sheaf of 0g-modules K,

has natural descent data, which we shall say is "effective"

or "constant".

(A formula for the data in this case is given

on page

Now for any T lying between

2.17

descend E to a sheaf

on

and S, we can

(namely the pullback of K), and

these sheaves will be compatible in the obvious sense.

1.10

By (2.9), a connection is nothing more than first order descent data.

(We are temporarily postponing a discussion of

integrability and the cocycle condition.)

In particular, if

T lies between X and S and is such that

is a closed

immersion defined by a square zero ideal, a natural sheaf tion

we can hope to find

First suppose there were a retracwe would then be happy to take

it independent of

In fact,

were

is another retraction,

factors through the first infinitesimal

neighborhood

of the diagonal in

since g^ and g^ agree modulo a square zero ideal.

Now the data

of a connection is an isomorphism

pulling it

back via

Hence up

g

gives an isomorphism

to (so far non-canonical) isomorphism, of the retraction.

E^ is indeed independent

One deduces that if T' is another "first

order thickening" of

admitting a retraction, and if

is compatible with the retractions,there is an

isomorphism

A suitable cocycle condition would give us compatibility of these isomorphism with composition, and also would allow us to construct the sheaves natural) condition that the retractions

under the (much more : existed only

locally, (since we could then glue the local constructions). The compatibility which we shall need is the following: Suppose that T" is another thickening of is another morphism. gram should commute:

and that

Then we require that the following dia-

We shall make all this precise later. reader is convinced

For now, I hope the

that the notion of an (integrable) connec-

tion is equivalent to first order decent data, which should in turn be equivalent to the data of a sheaf on a site made up of the first order thickenings of (open subsets of) X.

This gen-

eralizes to higher orders as well: 2.10

Definition.

A "stratification" on E is a collection of

isomorphisms 1)

such that: linear.

2)

and

3)

is the identity map.

4)

are compatible, via the "restriction"

is the n^^1 infini-

The cocycle condition holds: tesimal neighborhood of

and

is projection via the coordinates i and j, then for all

Let me try to explain the cocycle condition. and Note that:

stand for the

n:

We shall let projection.

2.12

Thus, the cocycle condition says that the following diagram commutes for all

n .

In other words, it says that if we use the stratification to construct isomorphisms

and

then the composition of these is the isomorphism provided by

We leave it as an exercise for the reader to

deduce the compatibility of the Pu's described above. We can describe this algebraically: Identifying the map has three correspond to tively.

secomes the map

-module structures, and the , and

for i=l,2,3, respec-

The cocycle condition can then also be expressed:

1.13

4b)

The following diagram commutes, for all

m and n:

To motivate some of the other ways of giving a stratification on the

-module

return to the case of effective descent

data for for any

for some module

module K, then . Thus we get a

natural transformation, for any two

-modules F and G ,

We shall see that a stratification on

E

if

h

2.11

allows us to do this

is a differential operator. Proposition.

Suppose

is smooth and E is an

The following data are equivalent: 1) A stratification on E (i.e., the maps 2)

A collection of compatible, right

i

above).

Linear maps

such that the following diagram commutes (cocycle condition):

1.14

3)

An 0^-linear ring homomorphism:

3 bis) A collection of

linear maps, for any two

F and G:

modules compatible

with composition, and taking the identity of to the identity. 4)

A compatible family of sheaves thickening

of an open set in

isomorphisms

2.12

Remark.

morphisms, so topos

of

Dix txposes,

for each nilpotent with transitive

for any

Notice that in 4) the "transition maps" are isois not just any object of the infinitesimal In the terminology of Grothendieck's article in is "special", these days we might call

a crvstal in the infinitesimal topos.

Sketch of the Equivalence. has already been sketched.

The equivalence of (1) and (4)

Given the data of the

the

E'S, one gets

we leave to the

reader the fact that the cocycle condition translates as claimed. Given the as follows: If and

we get the data of Obis) (and hence of (3)) h:F •*• 6 is its

is a differential operator of order linearization, we let , which is a differen-

tial operator of order

• The fact that

is

1.15

equivalent to the fact that

, Indeed, if

is the projection, then

is given by gi

and where

is the projection.

Thus

Finally, the cocycle condition is equivalent to the fact that

V

preserves compositions.

We shall verify one direction,

and with the reader's permission, shall suppress the subscripts First of all, if

is a differential operator, I claim

that the following diagram commutes:

c.f.

Lemma (2.3) (and the pre-

ceeding discussion) for the definition of This is a straightforward consequence of the definitions of V(f) and

and the cocycle condition.

Adding the triangle on

the right below, which comes from the discussion of composition of differential operators, Lemma (2.3), we get

if

Going around the top is V(g) ° V(f), around the bottom is Given the data of 3bis) we clearly have the data of 3), which we now show gives 1). View the map

1.16

as a map where

and

map is

have the

-structure from the left.

linear, and I claim it is automatically

To see this, first observe that if then

P-linear. and

, where

by

The

is multiplication

(which we write x3) is

, with

Check this by recalling that the P-module structure of

is defined by where

of

is multiplication by the class

Thus

while because

the maps

linear.

Now since

preserve composition, Since

and since

• generates

Now apply Horn The map

to the map

is evaluation.

is locally free, so the map diagram defines e .

linear,

, it follows that

to get the map

below.

Finally, since X/S is smooth, is an isomorphism.

Thus, the

We let the reader verify t h a t h a s

the

desired properties.

Note:

i is computed using the left

structure, so that induces an isomorphism with than as shown.

nodule

(rather

1.25

2.13

Remark.

It follows automatically that any 0^-linear ring

homomorphism

takes differential

operators of order

n to differential operators of order

because an endomorphism belongs

h of E belongs to

n,

iff

for all

What are the maps connection on

, in the case of the constant

Well

is just the exterior

derivative d, which is induced

Recalling that

we see that 6(x) is the reduction of l®x, and since

is the linearization of is just the identity map.

This works for any relatively constant connection on a sheaf of the form

, where K is a sheaf on S.

Slightly more

generally, let us suppose that E is a sheaf of on X.

modules

Then we can put a constant connection on

taking the reductions

mod n

by

of the following maps:

Begin with the map

sending c®x to

There is a natural identification: which takes

to

the"inclusion" tions , we obtain a map

If we compose

T with

and make these identifica, which maps

The induced

to

then maps

The above construction "is" Grothendieck's linearization of differential operators, if we apply it to

modules

which

1.18

come from

module structures.

Indeed, recall that in this

case we can write

through which any

linear map can be linearized — as we saw at the beginning of this chapter.

The point of the above construction is that it

furnishes

with a canonical stratification (which reduces to

the usual one if If we take

then maps

cation of

of

as an

is

and the map

to

This is a stratifi-

module using the lp.ft structure, i.e.,

it is useful to notice that the map is a stratification of

using the right structure.

(Indeed, if we had done the above

construction using

instead of

Moreover, if

]

is a differential operator, and if

we use the stratification nothing other than that

to compute

we find

immediately from the definitions. Note

is a differential operator if we use the right

module structure, but is structure.

we would have obtained

linear using the left

module-

In fact it is better than that, it is compatible with

the stratifications we have just put on from the commutativity of the diagram below:

This comes

2.

The above construction (really a complicated triviality) almost provides us with a way cf going from the category of modules and differential operators to the category of stratified 0^-modules and horizontal

-linear maps.

I say "almost"

because we have not passed to infinitesmal neighborhoods of the diagonal, which requires the replacement of system

order

The reader can check that

induces maps

for all

h:F

m and n, and that if

induces a map

ing to the

by the inverse

G

for all

has m.

Pass-

(or keeping the entire system in mind),

we obtain Grothendieck's linearization of differential operators:

2.14

Construction.

If

is an

inverse limit (or system) of the

be the

modules

has a canonical stratification, and if operator, we obtain a horizontal

Then L.(F) is a differential

linear, functorial map

If X/S is smooth, we can use Construction (2.11) to obtain a sheaf on the infinitesmal site which is almost a crystal. The problem of the inverse limit is a nuisance, but it turns out that in positive characteristics we can avoid it.

There-

fore, we shall not pursue it further. Let us now recall the obstructions to extending a connection

to a stratification.

tion of on

Using the construc-

3bis) above, applied to the first order stratification

E and the exterior derivative

a differential operator of order As expected,

1.

I a subset,

x's for which

Since

prove that J 1 is an ideal.

we have only to

If

and

and since either

i or j

is >. 1 and

J is an; ideal, each term in the sum belongs to J, so and

If

and

Certain constructions can be carried out with P.D. structures without difficulty.

For example,

is a

direct system of P.D. algebras and has a unique P.D. structure Y such that each is a P.D. morphism. a restriction is required:

For tensor products,

1.5

3.7 I

Lemma. !

Suppose A is a ring, B and C are A-algebras, and

and

are augmentation ideals (i.e. there is a sec-

tion of tively.

, etc.) with P.D. structures y and 6, respecThen the ideal K = Ker

has a unique

P.D. structure e such that 1

) and

are P.D. morphisms. We say that a sub-algebra B of (A,I,y) is a "sub-P.D.

algebra" iff for each

for

_

Thus in this case there exists a (unique) P.D. structure such that

on

is a P.D. morphism.

If I is an augmentation ideal there is a useful analogue of lemma (3.6), whose proof is so similar that we omit it: 3.8

Lemma.

Let (A,I,Y) be a P.D. algebra, and assume

has a section s .

Let

let B be the subring of

and A

I be subsets, and

generated by

B is a sub P.D. algebra of (A,I,y) iff s

Then for every

and Let us recall now the basic properties of the P.D.

analogue of the symmetric algebra.

Since its construction

is fairly involved, we only sketch it in an appendix (following Roby [1,2]).

3.9

Theorem.

algebra

Let M be an A-module.

Then there exist a P.D.

and an A-linear map

with

the following universal property: If (B,J,6) is any A-P.D.

3. 6

algebra and

is a A-linear, there is a unique P.D.

morphism Moreover: 0)

is a graded algebra,

1)

If

A' is any A-algebra,

2)

If

M =

and

3) 4)

We use the notation

if

for

x

M, and

Sometimes we shall denote

the divided power structure

y

by

is gen-

erated, as an A-module, by is a basis for

.

If

M,

is a basis for If

is a basis for M , we shall also denote

r^(M) by

and call it the

P.D. polynomial

A-algebra on the indeterminates

It has the ex-

pected universal mapping property with respect to P.D. algebras: i

If

is any P.D. A-algebra and

for each

I, there is a unique P.D. homomorphism

I It is now necessary to describe some other technical features of P.D. algebras.

It would be too tedious to prove,

or even state, all of them, so we only provide a sample.

3„7

3.10

Proposition.

Let (I,y) and (J,6) be P.D. ideals in A.

Then IJ is a sub-P.D. ideal of both I and J, and on

Y and 6 agree

IJ. Proof.

IJ

is generated (as an ideal) by the set of pro-

ducts

3.11

It is therefore enough to check that

Corollary.

If I is a P.D. ideal, then

I is a sub-

P.D. ideal, for all

3.12

Proposition.

Suppose

suppose that that Y and

are P.D. ideals,

is a sub-P.D. ideal of I and J, and suppose agree on

structure on

Proof.

and

Then there is a unique P.D.

such that I and J are sub-P.D. ideals.

We have an exact sequence of A-modules: For each Then

A-modules, where

exp(A)

g:I

, let

exp(A) is a homomorphism of is the A-module of power

series of exponential type (c.f. appendix).

Similarly, for

defines an A-linear deduce that there is a unique A-linear duces

d and g.

If

It is easy to see that

. We 1

which in-

, define satisfies the first four axioms

3. 8

for a divided power structure. trick.

Let

For the last axiom we need a

be the free A-module with basis

A-module with basis Y, and write

the free

For any

with

and consider the P.D.

morphisms

and respectively.

sending

We deduce an algebra homomorphism and hence is a P.D.

algebra, and the element

maps to

axiom S holds, i.e.

3.13

iff

I

Since in

Suppose (A,I,y) is a P.D. ring and

We say that

S

iff

I

is

I is

is a "set of P.D. generators of

is the smallest sub P.D. ideal of I containing

equivalently,

r(H)

, it also holds in A.

Definition.

a subset.

z.

S

I" —

generated as an ideal by

We shall often be working with algebras over some fixed P.D. ring as "base", usually a truncated Witt ring The ideal (p)

has many P.D. structures; we shall work with

the "canonical" one, induced from the unique P.D. structure on , using (3.5) and (3.11). ings

of

We want to consider thicken-

algebras, where the P.D. strucutre

is compatible with the canonical one on this just says that

.

on

Since

is a P.D. morphism.

It is convenient, however, to have a more general notion of compatibility which does not require that the P.D. ideals be preserved.

J

•

3. 9

3.14

Definition.

algebra.

Let (A,J,Y) be a P.D. ring and B an A-

We say that " Y extends to B" iff there is a P.D.

structure

on IB such that

is a P.D.

morphism. Notes. 1)

If

2)

Y

exists it is unique.

This is easy.

extends to B iff thereis an P.D. ideal

such that

of B

is a P.D. morphism, for

if such a map exists, it is easy to see using Lemma (3.6) that IB is a sub P.D. ideal of J. 3)

In general

does not exist; for instance if

and

3.15

is not a sub P.D. ideal.

Proposition.

Suppose I is principal.

Then

Y

extends

to any B. Proof.

In this case

for some fixed

and we want to define

In fact this is well-

defined, because if is a multiple of

t, so

This shows that our definition makes sense, and it is easy to see that it is a P.D. structure. 3.16

Proposition.

Let

(A,I,Y) be a P.D. ring, B an A-algebra,

a P.D. ideal in B. (1)

Y

extends to B and

Then the following are equivalent: on

3. 10

2)

The ideal ture

has a (necessarily unique) P D

struc-

such that

are P.D. morphisms. 3)

There is an ideal

with a P.D structure

such that are P.D. morphisms.

Proof.

is just (3.12).

For (3) -*• (1) observe that and that

on

y

(2) ->• (3) is trivial.

extends to B by note 2 above

because the two maps are P, D.

morphisms.

3.17

Definition.

If the equivalent conditions of the above

proposition are fulfilled, we say that

3.18

Remark.

Y and 6

are "compatible".

If B is an augmented A-algebra with a P. D«

augmentation ideal

) and

, then

with any P.D, structure on any ideal I of A. and observe that since

is compatible To see this, let which has a

P.D. structure, as is easy to see. We are now ready to construct one of the divided power analogues of formal completion, namely, the "P.D. envelope" of an ideal.

We work systematically over a fixed P.D, algebra

(A,J,y) and consider only

3.19

Theorem.

PD structures compatible with y .

Let (A,I,y) be a P.D, algebra and let J be an

ideal in an A-algebra B.

Then there exists a B-algebra

3. 11

with a P.D. ideal

such that

such that

is compatible with Y , and with the following universal property:

For any B-algebra C containing an ideal K which

contains JC and with a P.D. structure

compatible with y, there

is a unique P.D. morphism

making the

diagram commute:

Proof.

First we do the construction in the special case:

Case 1.

.

(In this case, g

and

ijjof will also

be P.D. morphisms.) The construction in this case is as follows: the P.D. algebra

Start with

of Theorem (3.9), with

the universal map. Consider the ideal J

of

generated by elements of the following two forms:

Claim.

is a sub PD ideal of

this, note first that we get the same

. To prove

J if we replace rela-

tion (ii) by: (ii)' Let and

be the ideal generated by the elements of the form (i) the ideal generated by elements of the form (ii)', so

3. 12

that

Since

Thus, using the formula for

, we see that it suffices

to show that

belongs either to

First suppose with

say

, Since

this sum is zero.

the degree zero part of

Thus, if we write

and

, with

, we have It follows that

that

and hence

in fact. In other words, which is easily seen to be a sub PD ideal

Now suppose

By (3.6) it suffices to see that if

Obviously it belongs to

in the remaining computations,

we shall write

Compute:

This completes the proof that

is a sub P.D. ideal

and allows us to conclude that the image

in

has a P.D. structure, which we also denote by [ ]. The first set of relations in

J

imply that

and the

3. 13

second set insures that

is compatible with Y .

For the

time being we verify the universality with the additional restriction that

If this is the case, since

we get a P.D. morphism ma

P

inducing the

and since

is also a P.D. morphism,

it is easy to see that this map y

factors through V .

The General Case. let

, and

be the sub P.D. ideal generated by J

(i.e. the

ideal generated by

is a s in

the theorem, let

with its P.D. structure

case 1 there is a P.D. map

Since

the sub P.D. ideal generated by

3.20

then by is

it is contained in K.

Remarks. 1)

By the very construction of V , we see that and hence that the algebra V depends only on

Of course, the P.D. ideal

still depends on J. 2)

If the structure map A and if

Y

extends to

B factors through some on

,

, then

This is because in the set of generators of type

ii, it suffices to consider

from any generating set of I, or of 3) As a B algebra,

is generated by

since this was true already for Moreover, any set of generators of J gives us a set of P.D. generators for

y's

3. 14

4)

In general it is not true that

since

such an equality would imply that Y extends to B/J. Conversely, if y extends, we get equality, because the universal mapping property tells us that there is a map sending

to zero, which then induces an

inverse to the canonical automatically extends if

5)

or if

IB

If

is an A-module, if

M

ideal

Note that I

Y

is principal (by (3.15)),

(trivial). and if J is the

then

.

(When we write

Y=0, we mean with the trivial P.D. structure on the zero ideal of A.)

This is easy to check from the universal

mapping properties. and

In particular, if then

is the P.D. polynomial

algebra 6)

Suppose that

Y

extends to B/J and in addition that

has a section.

Then we can drop the compati-

bility conditions, i.e. let

To see this,

denote the extension of

The section allows us to

write

Then

extends to

and so by

the universal mapping property, we get a map inverse to the canonical surjective map 7)

If K is an ideal of B such that

then

This is an exercise in universal mapping properties. is an integer such that

An example arises and J has

when

m

generators.

3. 15

Then

and hence

depends only on an infinitesimal neighborhood of V(J) in Spec B. 8)

Suppose that

is a surjective P.D.

morphism, and

Then the canon-

ical map:

is an isomorphism,

It suffices to see that the image has a P.D. structure compatible with

Y

(c.f. Remark 1),

and hence it suffices to see that the kernel of meets if K

in a sub P.D. ideal.

this kernel is just I is a sub P.D. ideal and

But

and since is compatible with Y ,

this is clear from (3.16).

3.21

Proposition.

bra B and

Suppose J is an ideal in the (A,I,Y) alge-

is a B algebra.

Then there is a natural map

which is an isomorphism if flat over Proof.

B, The map comes from the map

In the flat case, flat over

is

.

is From the description of J , we see

that is isomorphic to

easily

3. 16

3.22

Corollary.

If B is a flat

-algebra, y extends

to B. Proof. ible with

In general, to give a P.D. structure on IB compaty

is equivalent to giving a section of the canoni-

cal

such that if

sub P.D. ideal of

K

is its kernel,

is a

In particular we have a map

with a P.D. kernel, and hence a map ness , this is a map

By flatand it is easy to check that

is a sub P.D. ideal.

3.23

Corollary.

The map

is an isomorphism mod

Z-torsion. Proof.

Let B1

. The map

flat, and of

course Thus the map

becomes an isomorphism when tensored

with

3.24

Definition.

Let (A,I,Y) be a P.D. ring, n > 1 an integer.

n

Then i'- -' i s the ideal generated by

3.25

Proposition.

Proof.

Compute

is a sub P.D. ideal, and

3. 17

some integer N, and hence belongs to

The

next statement is obvious.

3.26

Warning.

is not generated by

in general.

For example if

with its canonical structure, ideal

, and the

where

the sequence

Since

is not monotone increasing, this is

not just

3.27

is the Witt ring

in general.

Definition.

A

P.D. ideal I is "P.D. nilpotent" iff

for some In general, if

I

is P.D. nilpotent it is nilpotent, but

not conversely5 for example, take the ideal (2) in

3.28

Proposition.

with

Let V be a discrete valuation ring with

parameter

, Recall that Ctt) has a (unique) divided

power structure iff

e

This structure induces a nil-

potent P.D. structure on

(where

This proposition follows easily from the formula for given in (3.3). The notion of nilpotent P.D. structure gives rise to another notion of P.D. envelope which is useful for some purposes. 3.29

Definition.

If

B is an ideal and n

is an integer,

3. 18

We can sheafify the notion of a P.D. algebra, and speak of a sheaf of P.D. rings

on a space X (for now a

topological space, later a topos), meaning the obvious things. If

is a map,

on Y, and if

is a sheaf of P.D. rings

is a sheaf of P.D. rings on Y, is a sheaf of P.D. rings on X.

A "P.D. ringed

space" is a pair (X,(A,I,Y)) where X is a space and (A,I,Y) is a sheaf of P.D. rings on X.

A morphism of

is a continuous map

together with a map sheaves of

P.D. rings:

ringed spaces

(in particular,

If (A,I,y) is a P.D. algebra and see that the localization ture

it is easy to

has a canonical P.D. struc-

Y such that

is a P.D. morphism: just

set

(In fact we have already seen in

(3.22) that

Y

extends to any flat A-algebra).

sheaf of P.D. algebras on the spectrum of

Thus we get a

A, and hence

we can regard Spcc(A,I,Y) as a P.D. ringed space. Moreover we can reverse the procedure: If X = Spzc A and

is a quasi-

coherent sheaf of ideals, one sees easily by taking global sections that P.D. structures on tures on

I

correspond to P.D. struc-

Similarly, the P.D. morphisms •

SpecI

can be identified with P.D. morphisms The following result follows easily from

3.30

Proposition.

Let

S

be a scheme,

a quasi-coherent

sheaf of ideals with a P.D. structure y, and let X be an S-scheme.

3. 19

Then if

B

is a quasi-coherent

quasi-coherent ideal,

is a is a quasi-coherent

In the discussion which follows let us fix (S,I,Y) as in the proposition, and suppose S-schemes.

i:X

Y is a closed immersion of

We use the notation

defines

f

o

r

i

f

J

and because of the proposition, we can define a

scheme If

y

extends to X (i.e.

as we have seen, so that immersion that

j

factors through a closed with kernel

is a P.D. immersion).

universal:

, then

if

a P.D. ideal (we say

The P.D. immersion

j

is

is a P.D. immersion (compatible with

Y) and if the solid diagram below exists, we get a unique as shown:

3.31

Remark.

If

Y is only locally closed (and Y extends

to X) then does

still makes sense, as for some

m.

This is because

is a nil ideal, so that the underlying topological space of

( resp. D) is the same as that of X, and we can

therefore replace Y by an open neighborhood of X in which it is closed.

3. 20

We call

the

n**1 order divided power-neighborhood

of X in Y (even though it is not a subscheme of Y), and the divided power envelope or neighborhood of X in Y.

We can

compute it, locally, in the following case:

3.32

Proposition.

Suppose

S-schemes and

is an immersion of smooth

Then

is locally isomorphic to

a P.D. polynomial algebra over

Proof.

By the above remark, we may assume that

is

closed, say of codimension d, and defined by the ideal J . Since

is locally generated by where

d

sections and

and hence by Since X/S is smooth, the map

locally has a section, and we may drop the subscript

Y

by (3.20.6).

regular sequence of sections of the

Y

J

extends to X, so Let

which generate J ; using

t's and the section we get a map

of course).

If

map induces an isomorphism:

be a

is the ideal

(locally, this Thus we have:

3. 21

For many purposes it is convenient to work over a formal base, e.g., a p-adic base.

Hence we shall need to discuss some

compatibilities of the constructions of this chapter with inverse limits. Let (A,I, Y) be a Noetherian P.D. ring with P P.D.-ideal.

(The most important case is A = the Witt ring of

a perfect field and P = I = (p), with structure.)

l a sub

Recall that

Y

its unique P.D.

is a sub P.D. ideal, so that

we have a natural P.D. morphism It is easy to see that the operations Y on I induce a P.D. structure o n we have P.D. morphisms:

Now assume that

A

formal A-scheme Z, let

3.3 3

Proposition.

n

,

is P-adically complete. Z^

If

For any

denote

Suppose Y is a formal A-scheme with ideal

of definition containing PO^, and assume that Y.

so that

Y

extends to

is a sheaf of ideals, there are canonical

isomorphisms:

where the

means P-adic completion.

Moreover,

canonical P.D. structure compatible with

Y .

has a

3. 22

Proof.

The first statement is an immediate consequence

of (3.20.8), and the rest follows immediately.

3.34

Corollary.

Let

Y be a formal subscheme, and let

denote the formal completion of Y along X. ideal P contains a nonzero integer.

Assume that the

Then there is a canonical

isomorphism:

Proof.

According to the previous result, it suffices to

prove this over

instead of A.

But there we can appeal to

(3.20.7), exactly as we did in the proof of (3.32).

3.3 5

Corollary.

smooth.

Suppose that

Then

is locally isomorphic to the P-adic comple-

tion of a P.D. polynomial algebra with coefficients in a formally smooth A-algebra.

In particular, if

A

has no

Z-torsion,

has none. Proof.

Locally it is easy to find a closed

smooth over A such that

formally

. Recall from (3.20.1) that

Now the formal completion of Y along Z is locally isomorphic to the formal completion of

along Z, so

by (3.34) it suffices to consider this case - which is trivial. If A has no

-torsion, we can be quite explicit.

a formally smooth (hence flat) A-algebra, seen to be the C-subalgebra of the elements

.

If C is is easily

generated by all

Clearly any element of

3. 23

can be uniquely written as a polynomial completion of sums

, The P-adic

is then the subring of all infinite such that

tends to zero P-adically as

Evidently this ring is Z-torsion free.

3. 1

§ 4.

Calculus with Divided Powers. Suppose

is a PD scheme (with

I a quasi-coherent

ideal, as always), and suppose X is an S-scheme. be the

Cartesian product of X with itself, computed

over S, and let immersion

be the diagonal immersion.

is locally closed and has

It follows from Remark (3.20.6) that if divided power envelope of X in

4.1

Let

Definition.

Suppose

or X/S is separated.

Y

The

retractions to X. Y

extends to X, the

does not depend on Y .

extends to X and either mO^ = 0

Then we can form the divided power

envelope

The corresponding

order divided power neighborhood we shall denote by note that it makes sense even without the hypotheses or X/S separated. As a consequence of Proposition (3.32), we see that if X/S is smooth, if

and if

are local coordinates

of X, then the structure sheaf

is isomorphic

to the PD polynomial algebra

4.2

Remark.

where

The natural map

To see this, let

is an isomorphism,,

be the ideal of X, so that Since

is a square zero ideal,

(3.2.4) shows that is has a PD structure. tion and therefore is injective. by

, and in

follows from this that the map

Hence

We know that

a

has a secis generated

if is also surjective.

It For

3. 2

n

, all we can say is that the

m

a

p

i

s

an isomorphism. We shall now indicate the PD version of stratifications. Recall that we had an algebra morphism:

Since the augmentation ideal

has a PD structure

and is an augmentation ideal, (3.7) tells us that is a PD algebra. erty of V tells us that

The universal mapping prop-

induces a PD morphism , which we shall again denote by

We have the useful formula: from which we see that for all

4.3

Definition.

Let

E

be an

induces maps m and n .

A "PD stratification

onE"is a collection of isomorphisms:

such that: 1)

Each

2)

The

are compatible, in the obvious sense, and

3. 3

3)

The following diagram commutes, for all

m

and

n

(cocycle condition):

Somewhat stronger than a PD stratification is a "hyper PD stratification".

(This notion is not useful in characteristic

zero.)

4.3H

Definition.

An"HPD stratification on

is an isomorphism:

such that

2)

e

reduces to the identity mod

3)

The cocycle condition holds.

(Let the reader imagine

the diagram.) We can interpret PD stratifications in terms of an analogue of differential operators, called "PD differential operators". We have to be careful however: a

PD

differential operator

cannot be regarded as a map

4.4

Definition.

If

E and F are

tial operator E + F of order

a "PD differenlinear map

An "HPD differential operator -linear map

is an

A PDdifferential operator map

induces a

as in the diagram below.

ential operator of order necessarily determine not generate

This

n, but note that

f, if as on

is a differdoes not

This is because

E

does

-module, as it did

We are forced to define composition of PD differential operators formally: If

, then

gof

is defined to be the composite: The diagram shows that definition works, of course, for

4.5

Example.

Suppose

HPD differential operators.

is a derivation.

"is" a differential operator of order differential operator of order

The same

Then 8

1, hence also a PD the diagram)

4.5

More precisely,

where D is a unique PD differential op-

erator of order

Suppose that

Then

is again a derivation, hence a differential operator of order

On the other hand,

has

order exactly p, in general, as we shall see. but of order

Of course,

differential operator such that

Suppose now that local co-ordinates.

is smooth and Let

is a system of and recall that

is a basis as a left

for

We want to describe composition in

terms of the dual basis sheaf of P D

differential operators of order

Proposition.

Proof.

the

With the notations of the paragraph above,

We must compute

is the image of

By definition this

under the composite:

under and finally to

under

3. 6

4.7

Corollary.

If

Thus the ring of PD differential operators is generated by the first order ones — and composition is just "formal".

It

will follow that even in characteristic p > 0, a PD stratification on E is equivalent to an integrable connection on E . 4.8

Theorem.

Suppose X/S is smooth and E is an

Then the following are equivalent: i) ii)

A PD stratification A collection of

on E. maps:

which fit together to give a ring homomorphism:

ii bis)

For all

F and G , maps

space.) iii) Proof. An taking composition. (i) integrable (We => identities (ii): have skipped This connection to isthe identies the analogue same on and as E .in of compatible (2.11): (2.11.2)with Given to save

3. 7

a PD differential operator

(ii) => (i):

we set

The same method as in (2.11) will work as soon as we know that the maps

are automatically

This requires a different argument.

We

need a formula which describes in local coordinates •module structure on

Claim.

, in the

multi-index notation we have been using.

To verify the claim,

note that both sides of the equation are

maps

so it suffices to show that they agree on any (s is again a multi-integer). are

also

that

Moreover, since both sides

z, we may take is by definition

Recalling

D q (zy), we get:

This proves the claim; we use it to check that linear, still working with local coordinates.

's

are

We have to

3. 8

check that for any

Since

and any rnulti index q,

(2.11) tells us that

we at least have

for any

is z.

-linear, so that By the above claim

, where

is the

"identity" PD differential operator. is just the canonical projection.)

Thus, we

have: for any

z and m

Now by induction it is easy to prove that for any multi-index q, i.e.

as desired.

It is interesting to note that the key in (2.11)

was the fact that that

PD Vl{6

(ii) => (iii) (iii) =» (ii)

X

) generates P ; here we use the fact

generated This is the same as before. Here we have to be careful in dealing with curvature, because of this distinction between u and u b for PD differential operators.

The prob-

lem is that in the formula we gave for curvature, only

_

_

occur.

3. 9

4.9

Lemma.

Suppose

is a connection on

are PD differential operators of order ferential operator

E

and

u and v

. Then the

has order

PD dif-

(Note:

smoothness of X/S is not needed here.)

Proof.

First a warning: If we replace

arbitrary operators of order

and

by

on E, the statement is false,

even in characteristic 0, if rank E > 1 . To prove the lemma, we must show that

, the difference

between the top and bottom compositions in the diagram below, factors through

4.9.1

i.e.

Claim.

where

is generated by

To check this, recall that and Now compute:

that it annihilates

is generated by is generated by

3. 10

Recall that

is by definition the composite: Since

id

mod

and since Thus , Now we can prove the lemma. generators of

Let us follow one of our

, tensored with an

and bottom of our diagram.

m e E , along the top

On the top: Since this doesn't depend on the

order of

u and v, the same is true

along the bottom, and the

lemma is proved. The fact that (iii) gives (ii) is now clear: In local coordinates, we have from the curvature formula that, since and hence since

has order

K= 0, But

, it follows that

Thus it makes sense to define by

and to extend by

linearity.

What conditions on an integrable connection correspond to a hyper-PD stratification?

We shall answer this only for

p-torsion schemes.

4.10

Definition.

and that tion

Suppose that

that

is smooth,

is a set of local co-ordinates for on an

is said to "quasi-nilpotent" (with

respect to the co-ordinate system) iff for each open all s integers

A connec-

, there exist an open covering such that

U

X and and

3. 11

4.11 Remark.

We shall see that, if the connection is in-

tegrable, the condition for quasi-nilpotence is independent of the co-ordinate system.

Note that in any case

if

in general, and so the operators are nilpotent.

4.12

Theorem.

an

Suppose

is smooth, and

, Then the data of an

HPD

E

is

stratification on E is

equivalent to the data of an integrable connection on

E which

is quasi-nilpotent.

Proof. on E, let

Let

be an

HPD

stratification

P : HPD

be the cor-

responding map, and let Thus if

In any local co-ordinate system

we have

, with dual"basis" for

HPD

For any section

0(m) as a locally finite sum: Of course, formula is:

is just

m of E, we can write

t. 12

Since the sum is locally finite,

for almost all

q, so the connection is quasi-nilpotent (in any co-ordinate system), as claimed. Conversely, suppose

is an integrable connection, quasi-

nilpotent in some co-ordinate system. fines

for each

we can make s

e

The connection

de-

i; because of the rule for composition,

n

s

nilpotent tells us

e

T

h

e

n

quasi-

for almost all

q, so

that we can use the displayed formula above to define This map is from

linear (using the

on the tensor product); we get a

structure linear map e

by extension of scalars. The only thing we must check carefully is the cocycle condition, paying attention to the distinction between D and the following diagram:

, We need the commutativity of

4.13

4.13

Corollary.

The condition of quasi-nilpotence of an

integrable connection is independent of the co-ordinate system.

4.14-

Exercise.

tion mod ρ

is.

A connection is quasi-nilpotent iff its reduc­

5 § 5.

The Crystalline Tooos. We are ready to assemble the constructions of the first

four chapters into the notion of the "crystalline site", which will then give rise to the "crystalline topos".

In this

and the next section, all schemes will be killed by a power of a prime p, unless otherwise specified.

This assumption

will allow us to postpone the technical difficulties of in­ verse limits. Let

S = (S5I,Ύ) be a PD-scheme, which will play the role

of the "base".

For any S-scheme X to which γ extends (in the

sense of (3.14)), we want to define the "crystalline site X relative to

S", which we denote by Cris(X/S).

of

It is the site

whose objects are pairs (υ*—»Τ,δ), where U is a Zariski open sub set of X, U=->Τ is a closed S-immersion defined by an ideal J , and

S

is a PD structure on

in the sense of (3.17).

J

which is compatible with

γ ,

Note that since 0 T is killed by a

power of p, J is a nil ideal, so that U-* T is a homeomorphism. We shall often abuse notation by writing (U5T,6) for (U=-*T,6), or even by just writing T for the whole thing. T = ( U ^ T 1 J ) an "S-PD thickening of U". Y

We shall call

The assumption that

extends to X insures us that for each Zariski open U of X, id„

(U

s-υ,Ο) is an object of Cris(X/S), because then

γ

is com­

patible with the trivial PD-structure 0 on the zero ideal of U. (The converse is also true — in fact, the reader can easily check that if U has any S-PD thickening at all, Ύ

extends to U.

We must also specify the morphisms of the site Cris(X/S) and the covering families.

A morphism Τ-ϋ*Τ' in Cris(X/S) is

5.2

just a commutative square:

Uc

,.τ

U'-+ 0 T defines a sheaf of

rings on Cris(X/S), which we call the "structure sheaf" and

denote by

This is the most im-

portant object of our study. 2)

The cofunctor T i+ (J^ defines another sheaf of rings on Cris(X/S), which we denote by 0 o r

(in a nota-

tion to be developed later) 3)

The cofunctor

defines a sheaf of

PD ideals in t>x/s' which we denote by J^/galso extremely important.

is

We have an exact sequence:

5.5

5.3 Remark. A useful consequence of the Zariski interpreta­ tion (5.1) of (X/S) . is the fact that it has enough points, cris " r For us, this means that we can tell if a map of sheaves: v:F -»• G

is an isomorphism by looking at stalks: It is enough

to check that for each Zariski

neighborhood

of

χ e X and each S-PD thickening T of a x,

the map of stalks: ( F T )

"* (Sm)

is an isomorphism. In order to exploit Grothendieck's philosophy we recall the natural embedding of a (suitable) site into its associated topos.

If T is an object

in any category

X, T = How[ , T]

is

an object in the category X of presheaves on X, and for any F e X , there is a canonical identification a la Yoneda: Ham [T,F] a F(T).

For most sites, and certainly for Cris(X/S)

(as the reader will easily verify), T is in fact a sheaf, so that one has a fully faithful functor from associated topos.

the site into its

Incidentally, we shall often find it useful

to abuse notation a bit and write F(G) for Ηοιηχίβ,Έΐ

if

F and G are presheaves, even if G is not representable by an object of the site. The first major advantage of the crystalline topos over the crystalline site is its functoriality.

If

g : X' •*• X

is

an S-morphism, there is no way to pull back S-PD thickenings in X to S-PD thickenings in X', in general.

However, we will

be able to pull back the sheaves they represent, and hence obtain a morphism of topoi ( X V S ) to recall what this means:

.

•* (X/S)

. .

It is wise

5.6

5.M- Definition.

A morphism of topoi

is a functor:

which has a left adjoint with finite inverse limits."'" 5.5

Remark.

Moreover

Of course,

and

which commutes

determine each other uniquely.

commutes with arbitrary inverse limits and

arbitrary direct limits just from the adjointness. tion behind the extra condition on

with

The intui-

f* is that the "stalks" of

f*F are supposed to be the same as the stalks of F, and finite inverse limits can be computed stalk by stalk.

It is remark-

able that this condition is sufficient to give

f

meaning.

geometric

If there is danger of confusion with module pull-back,

we sometimes write Suppose that in the commutative diagram below, S' ->• S a PD morphism.

Then we want to obtain a morphism of topoi g We begin by specifying

is .:

if

(5.6.1)

5.6

Definition.

Suppose

Then

is the sheaf on Cris(X'/S') defined by

This means the following: , and if h: (T',- b c

are two maps, then there

such that

wou = w°v .

The properties above follow from three divided

power constructions, the first of which goes back to Grothendieck's letter to Tate.

5.12

5.11

Lemma.

with U

1

Suppose (U'jT^fi1) is an object of

affine and

g(U')

contained in an open affine U of X.

Then there exists an S-PD thickening U ^ T T' ->• T extending the map

Proof.

g:U' ->• U.

respectively, with U = Spec C.

Since

and an S'-PD morphism

We may assume S and S1 are affine, given by (A,I,Y)

and and

Cris(XVS')

U' = Spec C', T' = Spec B',

Let B = C x B' in the following diagram: C'

it' is surjective, so is it •, its

kernel

is

the

ideal

where J' is the kernel of it ' . Clearly we can set on J and making

h

compatible with Y .

to get a PD structure a PD morphism.

We must check that

First, note that

Y' to B', by hypothesis, and since A also extends to B'.

y

extends to

c 6 IC,

with

y , so

Thus we can extend

y

Since 6' is compatible with 6

is

and

A' is a PD morphism, y to B by setting if

and

C

6

b' G IBr

y', it is compatible

is also compatible with y

The above lemma shows that the category

.

^ Ix

5 1 16

is not

empty; the following lemma shows that it satisfies conditions (ii) and (iii).

5.13

5.12

Lemma.

Suppose we are given T' 6 Cris(X'/S'), T1 and

T 2 e Cris(X/S), an S-scheme Y, and finally the solid arrows in the diagram below.

Then there exists a T e Cris(X/S) and dotted

arrows making the diagram commute.

Proof.

Let

U =

n U 2 in X, which maps to the fiber

product T^ * 1 2 by a locally closed immersion.

We take T to be

the divided power envelope D n ? , (T, x T„) of U in I, x T, . ' 1 2 ^ The subscript 6 jfi

means

two PD structures ^

and S 2 on ^

to the construction of Dy 2.1.3:. 5.13

that we impose compatibility with the

.

and T 2 , in a manner analogous

For details, see [Berthelot III,

•

Lemma.

Suppose

Y and Z

are schemes, and we are given the solid arrows in the diagram. Assume that

u^oh = u ^ h

and that

u-Jy

= U

2^U

*

Then

"there

exists a T e Cris(X/S) and dotted arrows shown, such that U ^ u = UjOU .

5.14

Proof. u1 and

Let

u2

J, be the PD ideal . Ker(0 iT i l

agree on

0 U U Let

sub PD ideal of J1 generated by subscheme of T1 it defines.

).

Since

1

J

be the

and let T be the

•

These lemmas complete the proof of Proposition (5.10)

—

(in order to get condition (iv) of Proposition (5.10), one takes Z = S and Y some object of Cris(X/S) in Lemma (5.13)). The general statement of the lemma is needed when one has a morphism g': X" g

X' over

cris ° g cris =

S"

(gogt)

S', and one wants an identification

cris'

The t r i c k is to show

that

for

every

T

of

that

Cris(X/S).

shall leave this as an exercise for the assiduous reader.

5.14 Remark.

We •

The construction of Lemma 5.12 tells us, in

essence, that finite products are representable in

Cris(X/S).

If we combine this with the Lemma 5.13 (and the universal mapping property of the construction not stated there) we conclude that inverse limits over finite nonempty index sets are representable in

Cris(X/S).

For details, cf. [Berthelot, III, 2.1].

The structure sheaf

"x/5 is

a

sheaf of rings in

and we shall want to consider ( x / s cr -L s J "x/S^ systematically as a ringed topos. g

cris :

i.e.

(X,/,S

'^cris

If

g: X'

(X/S)

cris

X is as in (5.6.1), then is a mor

Phism

there is a natural map

an object of

Cris(X/S), we need

of

ringed topoi, if

a map

^

xs

5.15

But if is an object of T

'

T and

CrdsCX'/S') and

is

a

T

map

hence provides us with a map of rings:

We are now in a good position to use general nonsense about topoi to define cohomology.

First, the notion of global

sections. 5.15

Definition.

an object of X.

Let X be a site with topos T and let T be Then "r(T, )" is the functor T ->• ((Sets))

given by F h f(T).

More generally, if T is any object of T ,

T(T, ) is the functor F h- HomT(T,F).

If

of T , we write

r(e,F).

r(T,F) or

T(f) for

e

is the final

The final object e of a topos T is just the sheafification of the constant presheaf whose value at any U is the set {0} consisting of a single element. In the case of the category of sheaves on an ordinary topological space X, this sheaf is represented by the open set X of the site, but in the case of the crystalline topos, it is not representible.

In general, a section

s € r(T,F) = Hom(e,F) is just a compatible collection of sections s T e F(T) for every object T of X, i.e. an element of iiffi. F(T). TfeX If

A

is any sheaf of rings on a site X, the category of

sheaves of A-modules has enough injectives, and for any T € T we can define

H 1 (T, ) to be the

in this category. L

i t h derived functor of r(T, ),

As usual, the abelian group structure of

H" (T,F) does not depend on A, i.e., can be computed in the

5.16

category of abelian sheaves on X.

Moreover, one has a Leray

spectral sequence for any morphism of topoi.

5.16

Proposition.

S' ->• S. g

Then

g

Suppose

As a consequence:

g:X' -> X covers a PD morphism

induces a morphism of topoi:

1

. : (X'/S ) . oris crxs

(X/S) . . crxs

If

E'

is an abelian sheaf

in (X'/S') cr £ g , there is a Leray spectral sequence:

Proof.

Let us sketch the construction of the Leray spec-

tral sequence of a morphism of topoi the fact that

f: T' -> T . The key is

f* preserves finite inverse limits.

ular, it takes the final object of

T

In partic-

to the final object of

T' , because the final object is the inverse limit over the empty category.

Then if E1 is a sheaf in T', because We can therefore

apply the spectral sequence of a composite functor, if injectives to injectives. exact,

this is automatic.

But since its left adjoint

takes f*

is

•

We can already prove the first important rigidity property of crystalline cohomology, namely its invariance under certain PD thickenings:

5.17

Theorem.

Suppose we have a Cartesian square as shown,

with Sq'-^—defined by a sub PD ideal K of I . natural isomorphism:

Then there is a

5.17

Proof. ''"cris'

(X

We have a morphism of topoi

0 / S ) cris

(X/S)

spectral sequence.

c ris

' and

hence

a

corresponding Leray

It is therefore clear that the last two of

the following statements will imply the theorem:

5.17.1

If

T e Cris(X/S).

5.17.2

The functor

is representable in CrisCX'/S').

i . cris*

is exact.

5.17.3 Indeed, let U be the open set of X defined by the PD ideal (J,6) of 0 T , and let U Q = U n X Q .

Then

»U is defined by the

ideal KOy of Oy, and hence U0-T is defined by the ideal KO^ + J. By definition of the crystalline site, y extends to a PD structure Y on I0T.

It is easy to see that the fact that K is a sub PD

ideal of I implies that K0T becomes a sub PD ideal of I0T. Since Y is compatible with 6, K0T + J has a PD structure compatible with Y > by (3.16).

Thus (UqjT) becomes an object of Cris(XQ/S), and

it is clear that the morphism (UQ,T) that (UQ,T) represents

i*(U,T) = Hom(

(U,T) is universal, i.e. ,(U,T)).

It is now easy i . (F) for any sheaf F on J to compute * cris4 Cris(Xn/S).J By definition, i . 0 ' oriSj, In terms of the associated Zariski sheaves, we have therefore:

Now

since a sequence of sheaves is exact iff the associated sequence

5.18

of Zariski sheaves for every PD thickening is, it is clear that i

is exact

cris 4

Remark. whenever

i

-

Also,

We shall see later (6.2) that

is exact

is a closed S-immersion.

We shall now describe some constructions which will help us to localize certain calculations on the crystalline site. The first of these is a projection from the crystalline topos x

to the Zariski topos

zar

•

This projection will be a fancy

way of fitting together the crystalline cohomology of the various Zariski open subsets of X.

5.18 Proposition.

There is a natural morphism of topoi:

uv/0: X/S

X , given by: J zar ' &

(X/S) cris

(1)

For

F e (X/S)

(2)

For

E e x

zar

Notice that

. and cris

Jj

:U = — a n

open immersion, *

and

is the set of global sections of

u

F over ( / S ) C r i s > (which should be thought of as "horizontal" sections

over U).

It is quite easy to see that

are adjoint to each other.

and

Moreover,

for any (U,T,6), and from this it is

5.19

clear that

commutes with arbitrary inverse limits.

we really do have a morphism of topoi.

Thus

It is not however, a

morphism of ringed topoi, because there is,in general, no map The Leray spectral sequence of u ^ g

never-

theless exists. (In terms of de Rham cohomology, it is, in fact, just the so-called conjugate spectral sequence: is smooth.

This will

become apparent later.)

5.19

To justify calling

u

x /s

a

projection we provide

it with a section

.

The functor

itself has a left adjoint, which deserves to be called given by

U

x/S! '

the Zariski sheaf given by F on

the object

of

Cris(X/S).

Clearly Since

u

x/s;

clearly

commutes with inverse limits, we get our morphism of topoi by setting

and Unlike

u

• a mor

x/s ' ^X/S

^"X/S

Obviously, Phi s,n

ringed topoi, because there is a map Since of

is obviously exact, the Leray spectral sequence

i^/S

^ e S ene:pa ' te > and we have:

5.2 0

for any Zariski sheaf E on X .

5.21

There is another, more general, notion of localization we

shall need, which makes sense in any topos.

In fact, the

5.20

definition works in any category, we shall give only a sketch, for more details, compatibilities, and points of view, the reader can look in [SGA 4 IV 8 and SGA 3 I].

5.22

Proposition.

If T is an object in a category C, let C| T

denote the category of arrows in C with target T, and let s:C|T

C denote the functor which takes an arrow

f: S

T

into its source S .

5.22.1

If X x T exists in C for all X, then

adjoint

r T : C -»• C|T , given by

5.22.2

If

s

has a right

means the corresponding presheaf category,

there is a natural equivalence of categories:

5.22.3

If z e e , then the functor

adjoint

jz .

Proof.

has a right

The first statement is obvious.

For the next one,

let us content ourselves with a description of a

n

d

t

h

e

reader who so desires can easily check • ^i

that they are quasi-inverses. then

n(F)

If

is an object of

is the cofunctor

given by the

following: take induced by F.

, and we just 1

where Going back, if G

is

an

is the function

object

of

5.21 Let

S Q : C + ((Sets)) by

is the obvious map.

then

Note that a morphism

object

G: S

T induces an

and an object

and it is immediate that

We shall make these

identifications without further reference. Before explaining (5.22.3), let us remark that products, hence r z exists (and is compatible with and

rT

C

has

r T , if

Z =T

exists). A

Using this construction one gets internal Horn A. If

in

C:

is the presheaf of cross-sections of

F.

Z' and Z are objects of C,

5.22.4

Now we can describe the functor j„ : u 5.2 2.5 If F: Z 1 ->- Z is an object of j^(F)

C| z ,

Let us verify that for any Hom[G,j2(F)].

An arrow

F

a

commutative triangle:

This means a compatible collection of triangles, for each T G Ob(C),

5.22

These triangles are clearly equivalent to the data: for each and each

g e G(T) a cross-section of F^: Z'(T)

Z(T)...

T i.e.

a morphism of cofunctors: Now if C has a topology, one can perform the same constructions in the category C of sheaves on C, and if the functor C •* C factors through C, all the above constructions go through without change.

One obtains:

5.23

Proposition.

Then

T|z

in which

Let

T

be a topos, Z

an object of

T .

is also a topos, and there is a morphism of topoi:

j

Moreover,

is given as in (5.22."+), and

j^*

has a left adjoint

jz,

(5.22.1).

given by the functor

s in (5.22.1).

Proof.

Let's just observe that since

j *

adjoint, it preserves arbitrary inverse limits.

5.2t

Proposition.

If E is an abelian sheaf in

has a left •

T there is a

canonical isomorphism

Proof.

The final object of

is just

Z.

Thus, if

T|z

is

id : Z -> Z , and

E e T, we have

This proves the statement for

i= 0; the

5.23

general case will follow from the facts that j* takes injectives to injectives.

is exact and

The exactness of j* (true for

any morphism of topoi) follows in this case from the existence of its right adjoint j_

and its left adjoint

(which is not

the same for abelian sheaves as it is for sheaves of sets). The statement about injectives will be a consequence of the fact that j z , is also exact, as is clear from its description:

If F G

is an abelian sheaf, then j z ,(F) is the sheaf associated to the presheaf

We can make the localization construction somewhat more concrete in the case of the crystalline topos by giving a description in terms of compatible families of associated Zariski sheaves, in analogy with (5.1).

5.25

Proposition.

then

We omit the proof.

If Z = (U,Z,e) is an object of Cris(X/S),

may be described as follows:

It is (equivalent

to) the category of compatible collections of pairs (u,Fu), where

u: Y ->• Z is a morphism in Cris(X/S) and F u is a Zariski

sheaf on Y.

j,, : (X/S) . , (X/S) is the ens I ^ ens canonical morphism, and if we also use the description (5.1) of (X/S)

cris'

5.25.1

Moreover, if

then

and

^z*

are

eiven

If

E e (X/S) . and crxs then is the Zariski sheaf

on Y .

5.24

5.25.2

If

then and

F = { (u,F )} £ (X/S)G 2?.1S, | rand T = (U,T,6)e Cris(X/S), ^ is the Zariski sheaf

on T, where

p^ are the PD-morphisms:

(Here

D

is the double PD-envelope used in (5.12).)

Proof.

If f: G

Z

is an object of

here is

how to obtain the corresponding collection u: Y

pT

Z is a morphism in Cris(X/S), and if

If ~

is a Zariski

open subset, then F11(Y»)on = Horn-(Y',G). We we lethave the reader verify that this gives a sheaf Y zar Li , and that an equivalence of categories. Now it is easy to verify (5.25.1), using the description (5.22.1) of

jz* .

Horn2

Since

is,by definition,

and since

jugt

is

. this is

Localizing on the Zariski topology of

Y gives (5.25.1). The final statement is only a trifle more complicated, is a Zariski open subset,

But the "sheaf"

is

represented by the "object" , and hence by the description of F u , Hom^Cpr^j F) is

Now formation of

D(T'xZ) is compatible

with localization in the Zariski topology of T, so (5.25.2) follows. •

5.25

We can relate our two notions of localization by an important diagram.

This diagram will be our main computational

device in the next two chapters.

5.26

Proposition.

Cris(X/S).

Suppose

Z = (V,Z,e) is an object of

Then there is a commutative diagram of topoi:

Moreover: 5.26.1

If E is a sheaf in (X/S) . , cris '

5.2 6.2 5.26.3

is exact. If E is an abelian sheaf in (X/S) . , cris '

Proof.

The morphism

is defined as follows:

If

is an element of (X/S)cris|2 . . , is just the Zariski sheaf Zzar

.

If E is a sheaf on The reader can easily

check that these form an adjoint pair, and that

preserves

finite inverse limits.

To see that the diagram commutes, it

suffices to check that

for every Zariski

sheaf E on X.

But

5.26

is just Ey , if

But also.

Now We

suppose

recall

from

E

is

a

(5.25.1)

sheaf

in

that , hence

proving (5.26.1).

A morphism F + G in (X/S) . , cris|z

epimorphism iff each morphism of Zariski sheaves F y hence

is exact.

5.2 7

Corollary.

G

is,

u

It follows that the Leray spectral

sequence for the morphism abelian sheaf in

is an

degenerates, so that if E is an 5

we have from (5.24):

Suppose that in the localization diagram

(5.26) above, V = X, i.e., Z is an S-PD thickening of X.

5.2 7.1

The functor

5.27.2

If E is an abelian sheaf in

Then:

is exact. is

acyclic for

Proof.

Look at the description (5.25.2) of the functor

is a thickening of P T : D -> T is a homeomorphism, so follows.

p^

U.

But then the map is exact, and (5.27.1)

5.27

Now if E is an abelian sheaf in

we have a

spectral sequence: which degenerates to an isomorphism: But

and

are exact, so

5.2 8

d

Let us conclude this section with a discussion of cover-

ings of the final object

e

of ( X / S ) c r i s • Although

e

is

not representable in

Cris(X/S), it can often be covered by a

representable sheaf.

In fact, I claim that this is the case if

there is a closed immersion

with Y/S smooth.

it is easy to see that envelope

Then

is represented by the S-PD

of X in Y, and I claim that the map Y — e

is an epimorphism.

This just means that the associated map of

Zariski sheaves ? T

e T is an epimorphism, i.e., that for suf-

ficiently small open subsets T ! of T, Y(T') is not empty. This in turn

just means that the map T' n X

Y can be lifted to T',

which follows from the fact that Y/S is smooth and is a nilimmersion. (N.B.

It is a standard fact from SGA1 that a morphism

with a smooth target Y locally extends over a nilpotent immersion; here we need it for nilimmersions as well. that this is also true, suppose T

1

To see

= Spec B, with X n T'

defined by the nil ideal J.

To see that the map Y(B) -* Y(B/J)

is surjective, write

, with

ideal.

Then

a finitely generated

, and since Y/S is locally of finite

5 o 28

presentation,

[EGA IV, vol. 3].

But

each J^ is a nil ideal and finitely generated, hence a nilpotent ideal, hence each

is surjective, and what

we need is a consequence of exactness of direct limits.) According to SGA 4 V

(4.4 and 5.2), the above means that

Y -* e is a covering in

•

For us

j this will mean that

one can verify exactness by pulling back to Y. to see at least in the crystalline case: is a O-sequence of abelian sheaves on

If

This is easy I = E^

•* E^

, and

is exact, then I claim that for each T S Cris(X/S) (and hence also £ ) is exact.

Since

the claim is local on T, we may assume that there is a morphism u: T -»• Y, and by assumption the sequence

is exact.

But according to (5.25.1), this is just

5.29

The above remarks furnish the basis of the Cech-

Alexander technique of computing cohomology in a topos. Although we shall not need this technique for the main comparison theorems, let us sketch it. final object in a topos T

If Y + e

is a covering of the

(i.e., an epimorphism) consider the

semi-simplicial object in T ;

with the usual

projection maps:

If E is an abelian sheaf in T, we can form a complex of sheaves in T, with

and with coboundary

maps induced by the alternating sum of the projection maps.

5.29

If T is an object of T , we have by (5.23) that The value of this construction is in the fact that there is a natural resolution:

The map is clear, and we may verify that it is a quasiisomorphism after restricting to

.

In other words, it

suffices to check that the complex:

is acyclic if T maps to Y.

But in this case it is easy to conv+1

struct a chain homotopy, using the maps induced from the given indices.

T •+• Y.

TxY

T xY

The reader can work out the

6.1

§ 6.

Crystals. A "crystal", says Grothendieck, is characterized by two

properties: it is rigid, and it grows.

Any sheaf F on Cris(X/S)

"grows" over PD thickenings (U,T,6) of open subsets of X construction.

by

In order for F to be a crystal, we impose a

rigidity which we shall only make precise for sheaves of flx/s-modules.

From now on, we shall write

of a sheaves of sets, and

u - 1 for pull-back

u* for module pull-back (i.e. u-"'"

followed by tensor product with the structure sheaf) — unless there is no danger of confusion.

6.1

Definition.

A "crystal" of (?^g-modules is a sheaf F of

Ox/s-modules such that for any morphism in Cris(X/S), the map

is an

isomorphism. A trivial example of a crystal is the sheaf ^ / S itself. An extremely nontrivial and useful example is furnished by the following:

6.2

Proposition.

Suppose

is a closed immersion of

(S,I,Y)-schemes

(to which

functor

is exact, and

i

cris& O^^g-algebras. Proof.

sheaf

Y

extends, as always).

Then the

i . (£>v/c) is a crystal of c n s ^ i/o

The key to this result is a close look at the

i - 1 (T) = i*(T)

(5.6) ,

if T = (U,T,S) e Cris(X/S).

First of all, I claim that it is representable.

Indeed,

6.2

U Q = U n Y is a closed subscheme of T, and it is natural to expect that

i*(T) is represented by a suitable PD envelope of

U Q in T, compatible with of the base.

6

as well as with the PD structure Y

This is easily constructed.

Suppose TJc—>T is de-

fined by the PD ideal (J, 6); then by compatibility of 6 extends to a PD structure

6.2.1

Lemma.

S

on

y and 6,

J 1 = I0T + J, and we have:

With the notations of the previous paragraph,

i*(T) is represented by If F is a sheaf in (*/S) cri , then the Zariski sheaf associated to

where

on T is given by

X: D ->• T is the natural map.

Proof of

6.2.1.

First we should check that D really is

a PD thickening of U Q . need to know that J-, -

5

Recall from (3.20.4) that for this we

extends to

+ J in 0.. is just u 0

0.. . Since the image of J 0 , to which Y is assumed to

70 0

extend, this is clear, as is the compatibility of [ Thus D really is an object of Cris(Y/S).

]

with y .

We obtain the map X

from the universal mapping property of D, and it is easy to see that

it makes D represent

we can now easily compute

i*(T).

If F is a sheaf on Cris(Y/S),

icrxs ^(F), as follows: By definition, In terms of associated

R Zariski sheaves, this gives the statement of the lemma.

•

5.3

Proof of 6.2.

The exactness of

follows from the

lemma, because as far as underlying Zariski spaces go, X: D is the closed immersion U^ -»• U, so

T

is exact.

It is clear from the above construction that to prove that i

. (0 v/ _) is a crystal, we have to establish a compatibility cris£ i/o

of the divided power envelope construction with base change. Namely, if

u: (U',T',6') + (U,T,fi) is a morphism in Cris(X/S),

we have to show that the map i.e.

is an isomorphism. This fact is one of the key technicalities in the theory of

crystals, and to prove it we shall have to look again at the construction of PD envelopes.

We may assume that U = U' and

that all our schemes are affine, with T = Spec B, T* = Spec B 1 , and

u: (B,J,6) ->- (B',J',6') a PD morphism.

map B/J -»• B'/J'

is an isomorphism.

the closed subscheme U Q = U n Y.

Since U = U', the

Let L C B and L' C B' define

Then we have a commutative

diagram as shown, and the theorem amounts to establishing the claim below:

Claim.

The arrow

p

above is an isomorphism.

Recall that since

we construct

by setting L^ = L + J ^ , forming (P,L15[ ]), and taking L to be the sub-PD ideal of L^ generated by L. B/J a B'/J' L' = LB' + J'.

Now in our case

and L' and LB' have the same image in B'/J', so It follows that

n(L') is contained in the image

5.14

of L8B1

J'

L^B'eDSJ^ .

in D0B', and hence also in the image L^ of I claim that L^ has a PD structure compatible with

6* . The universal mapping property of V will then provide us with an inverse to the arrow

P.

To get the desired PD structure on L" we have to go back to the construction of is a

certain

where

ideal of

(3.19).

K

Now

, by (A2), which in degree zero is just B', and in particular contains the PD ideal (J|,6'). the PD ideal

Because

is an augmentation ideal,

its PD structure [ ] is compatible with 6' , by (3.20.6). That is, the ideal L* = J-jTgd.) + ^ ( L ^ B B 1 has a PD structure 6*, compatible with [

] and

. The image of L*

in

1

P D -(L, )9B is our ideal hi' , so to endow the latter with the B, $ l J. compatible PD structure, we have to prove:

&

Claim.

!

is a sub PD ideal of L^ , where K

the kernel of

, or

is

equivalently, the

1

image of K8B' in T8B , and where In the ensuing calculation, we shall drop the subscript "1", and write L for L^ , etc.Furthermore, if denote

where

Recall from (3.19)that generated by

x £ L, x'""-' will

is the canonical map. where K^ is the ideal e L} and K 2 is the ideal generated by Recall also that K + = K n Tg(L) is a

sub-PD ideal of

Tg(L)

(this was the key to the D-construction).

It follows that K , + = K1 n(rt(L)8B') is a sub-PD ideal of D rg(L)0B' , since K , + equals the image of K+0B' .

6.5

Because

, we have

Now if If we can prove the same for

we are finished,

using the formula for Suppose

, with If we write

with , we get a similar decom-

position:

_ with

lying in the sub-PD ideal

Thus it suffices to consider the term i.e.

we may assume that , B1 is generated as a B-module by 1

Since and j' .

It follows that

x

can be rewritten as a sum Notice that

each term

belongs to

x also belongs to

so does

, and since Since the terms

belong to the sub PD ideal

they

cause no difficulty. It remains to check that if so do its divided powers.

Since

and

we see that in fact

and since

is an isomorphism, it follows that together with the fact that allows

This, is

a PD

m0 rphism,

us to make the following computation (in which we drop

the useless subscript).

6.6

In the last line, the terms

belong to the ideal K,

and the second term is zero.

It follows that

, as desired.

6.3

Corollary.

The canonical maps:

are isomorphisms.

Proof.

We have diagrams:

The morphisms

are morphisms in Cris(X/S), and the fact that

is a crystal tells us that the maps:

are isomorphisms.

Recalling that

6.7

we are reduced to seeing that This follows easily from the universal mapping properties.

6.4

Exercise.

Deduce from the Corollary a natural isomorphism Show that

HPD stratification on

e

is an

If

the corresponding integrable connection, show that , for any section y of the ideal of Y in X. The complex

can be thought of as the deRham com-

plex of

, and we shall sometimes denote it by

6.5

Suppose

Exercise.

PD morphism

is a morphism covering the and E is a crystal on X/S. Then

is a crystal on from an object in

Cris

If

h:

to an object in

is a morphism Cris(X/S),

Let us now change our point of view and notation.

We want

to study crystals over an S-scheme X which is not smooth by embedding X in some smooth S-scheme Y. 6.6

Theorem.

If

is a closed immersion of S-schemes,

with Y/S smooth, the following categories are naturally equivalent: (i)

The category of crystals of

on

Cris(X/S).

6.8

(ii)

The category of cation (as an

-modules with an HPD stratifi-module) which is compatible with the

canonical HPD stratification (iii)

The category of

-modules with an integrable,

quasi-nilpotent connection (as an

module) which is

compatible with the canonical connection on Before giving the proof, let me remark that in (iii), compatibility of a connection nection

-module E with the con-

means that if

a

is a section of V and

a section of

e

, under the canonical

identification

Thus if we (abusively) view as the exterior derivative, we can

view

as a connection on the

Proof of 6.6.

module E in the usual sense.

It suffices to check the equivalence of (i)

and (ii), and to refer to (4.12) for the equivalence of (ii) and (iii). Suppose E is a crystal on Cris(X/S). morphisms in Cris(X/S)

There are two and since E is

a crystal, we get isomorphisms Combining these, we get an isomorphism: (The tensor product is taken over

Since there are isomorphisms (6.4): we can interpret

the above as an isomorphism:

6.9

, an HPD stratification on stratification on

in other words, as

Notice that the HPD is built into the construction of e ;

this insures the compatibility we claimed. Conversely, given a

module F with compatible

•module stratification, we construct a crystal E on Cris(X/S) as follows:

It suffices to specify E^, for sufficiently small

Cris(X/S), e.g.

if there exists an S-morphism

extending sheaf on

, we define Erp to be

does not depend on

Viewing F as a The fact that

h, up to canonical isomorphism, comes from

the HPD stratification on F, viewed as an isomorphism This part of the argument

is the same as it was without divided powers (2.10ff),

so we do not repeat it.

6.7

Corollary.

Suppose we have a Cartesian square, where

is locally of finite type and

X/S

is a PD im-

mersion defined by a sub-PD ideal of I . Then the natural map ((Crystals on X/S)) -> ((Crystals on

is an equivalence of

categories.

Proof.

Since sheaves can be determined and even constructed

locally, the question is of a local nature on X.

Hence we may

6.10

assume that X can be embedded in a scheme Y which is smooth over S and to which

y

also extends.

In (5.17) we showed that

, and in (6.2) we showed that is our embedding. Applying (6.2) with

we see that Now using description (6.6) (ii) or (iii)

of crystals, we get the corollary immediately.

This proof also

proves the next corollary.

6.8

Corollary.

is smooth.

Assume also in the situation of (6.7) that X/S

Then the category of crystals on Cris(XQ/S) is

equivalent to the category of

modules with integrable, quasi-

nilpotent connection relative to S.

We shall now indicate the divided power analogue of the linearization of differential operators we sketched in (2.14), a construction we shall use in the next section to calculate the cohomology of a crystal.

If Y is an S-scheme (to which Y ,

as always, extends), and if E is a sheaf of C^-modules, we define from the left.

, with the O^-module structure An HPD differential operator

induce an fly-linear map u

is an Oy-linear map:

to be the composition:

This makes

into a functor.

will asfollows:

, and

is defined

6.11

Just as before

has a canonical HPD stratification: induced from the map given

in (2.14)

Here is another description: The left and right

•module structures on

and

, respectively), each have a canonical HPD stratification, the second of which corresponds to is an

module,

is computed using the right

structure, and it inherits an HPD stratification (and an module structure) from the left-structure. an HPD differential operator,

If

u:

is

is the map

induced from the stratification on

via (4.8) (it is

an HPD differential operator using the right structure of but is

linear and even horizontal using the left structure.

In summary: 6.9

Construction.

There is a functor

from the category of

Oy-modules and HPD differential operators to the category of HPD stratified

-modules and

Y/S is smooth, and hence The

linear horizontal maps.

If

is locally free by the remark in (4.1),

is exact. construction furnishes us with a large collection

of HPD stratifications, and hence with a large collection of crystals, via (6.6). immersion and

More generally, if

is a closed

is smooth, we shall use the following notation:

If E is a sheaf of (^-modules,

will be the sheaf of

modules with HPD stratification indicated above.

will

be the crystal we get on Y/S by construction (6.6), and (or just L(E)) will be

, a crystal on X/S by (6.5).

6.12

We can give another description of

in terms of our locali-

zation diagram

(represented,

the reader will recall,

6.10

Proposition.

If

We have a diagram:

E

is a sheaf of

modules and

is the natural map, there is a natural isomorphism:

Proof.

An arrow

is equivalent to an

arrow

Using

(5.25.1) we see

that the source of this hoped for arrow assigns to any the sheaf to

u

the sheaf

, and by (5.26) its target assigns Now

is by definition

, and by (6.5) we see that this is Thus we have only to give a compatible collection of maps for every certainly do.

But

u, and a map

E

will

and we can just take the

map induced by To show that our arrow recall from (5.25.2) that if

is an isomorphism,

6.13

are the natural maps.

Thus, we have

where natural projection.

is the

On the other hand, if there is a map

(which we may assume, by

is just

It is clear that we are thus reduced to proving that the map is an isomorphism. show that the ideal of X in

The problem is to

has a PD structure com-

patible with Y . This ideal is is the ideal of an augmentation ideal.

, where

and Since

is the ideal of

Y —

is flat, the PD structure

J extends to a PD structure 6' on

, and since K

is an augmentation ideal, its PD structure is compatible with This gives us the desired extension, establishes the isomorphism

and proves the proposition.

6.10.1

Remark.

E

P-modules instead of a sheaf of

of

Actually, we could have started with a sheaf -modules.

Then if we

it can be shown that L(E) is a crystal. If

where

maps from

to

T

and to

Let us apply the functor

are the

respectively.

to the complex

-modules and HPD differential operators. a complex

and

of

To see that we get

, the reader can show that the composition

6.14

is zero as an HPD differential operator. We shall take a more explicit approach, attempting to demystify the

construction by working it out in detail using local

coordinates.

(All we are really doing is Exercise 6.4

in a

special case.)

6.11

Lemma.

Suppose

are local coordinates on Y, is a section of

a

is a section of

, and

. Then the map behaves as follows:

Proof. Since

d

Recall that

is a differential operator of order one, we have a

diagram:

In this diagram,

is the composition of the two horizontal

arrows,

is the natural projection, and is the

'linearization of

d.

Now since

6.15

T.

If

n = l this is clear;

the complex is just

with It is simplest to proceed by induction on

n . Assuming that

is a quasi-isomorphism,

so is because But

this

consists arrow

can

be

of

locally

identified

free

with

the

-modules. arrow:

and since the arrow a quasi-isomorphism, we are done. A refinement of the above result will provide us with a beautiful crystalline interpretation of the Hodge filtration.

is

6.17

Let us keep the notation of the previous theorem. We begin by restricting the exact sequence (5.2.3) , obtaining:

Since

is exact (5.27.1), since

is just

again, and since

(6.10), we ob-

tain an exact sequence: (6.13.1) It is clear that

is a P.D. ideal in

terms of local coordinates, if ideal generated by Let

.

In

^ is the

and the ideal

denote the PD filtration (3.24); it follows from the

local formula (6.11) that L(d) maps

into , so that

is a subcomplex of map

sends

Noting that the canonical into K , we see that we have an

augmentation:

6.13

Theorem.

Proof.

The above map is a quasi-isomorphism:

First let us describe the ideal

in terms of

local coordinates, where

I claim that it consists

of the set of PD polynomials

such that

6.18 for all

k .

It is easy to see that this set is a sub PD ideal

of

. Hence it suffices to check that it contains a set of

PD generators of

, for instance This is clear.

In particular, if

is the set of PD polynomials

such that

, The inclusion of

complexes:

. There

is a section

(the

to zero for chain homotopy

linear mapping sending

which sends

s:

, (the under which

, and a linear map sending _ is invariant. Thus, when

the result is clear. To proceed by induction on

n

we have to show that the

map isomorphism. Recall that then

can be identified with:

All that remains is to observe that the projection the homotopy

id®s

and

are compatible with the filtration F m .

Since this is immediate, the proof follows. Using the same technique we can also obtain resolutions of other crystals of

6.19

6.14

Theorem.

6.14.1

If

M

With the notations of (6.13):

is any sheaf of

on Cris(X/S), Then there is a

canonical quasi-isomorphism:

6.14.2

If

E

is a crystal of

the corresponding sheaf of

and if

is

with integrable

connection {6.6), then there is a natural quasi-isomorphism:

Proof.

We shall leave (6.14.1) to the reader, who needs

only transcribe the proof of (6.13), replacing by

M.

everywhere

To deduce (6.14.2), we need to find a nice

quasi-isomorphism:

In fact, there is even an isomorphism of complexes of which, since it is linear, preserves the ideals

hence the filtration

To find this isomorphism, we work with the associated HPDstratified

-modules, in the following proposition.

6.20

6.15

Proposition.

and let

Let

be any

E

be an

-module.

isomorphism:

6.15.1

-module with HPD-stratification,

Then there is a canonical Moreover:

is compatible with the HPD-stratifications on and

6.15.2

If

u:

is an HPD differential operator

and if

is the induced operator into

Proof.

Recall that

so that

supposed to be an isomorphism: we can use cation on

£

is

Evidently is the given HPD stratifi-

E .

To prove that

B

is horizontal (i.e., that it preserves

the HPD stratifications) we have to chase a tedious diagram. The essential point

of the calculation is the cocycle condi-

tion for First we must give a more explicit formula for the HPDstratification of the any

module,

/.-construction.

Recall that if

and is deduced from the map:

F

is

6.21

To write this, we need the map:

Notice that this map is

linear for the left

module struc-

tures; to enforce this we shall sometimes use a subscript The reader can now easily see that

II .

is given by the

following diagram:

To see that the isomorphism

6 is compatible with the HPD-

stratifications, consider the diagram below:

6.22

The top triangle is exactly the cocycle condition for The slanted arrows come from the morphisms: The commutativity of the three parallelograms is trivial.

Since the outer circuit of

the diagram is just the following, we have proved that

6

is

horizontal:

To prove (6.15.2), let

v = p(u):

The assertion is the commutativity of the first diagram below, which expands into the second diagram.

Note that again, the

essential point is the cocycle condition.

6.23

6.1

§ 7 . The Cohomology of- a Crystal. We are now ready to establish the fundamental property of crystalline cohomology, namely, its relation to de Rham cohomology. Our first goal is the following result, from which all the finiteness and base changing properties to come will be deduced:

7.1

Theorem.

Suppose

with Y/S smooth. cation, let

E

Let

i: E

is a closed immersion of S-schemes,

be a

be the crystal on X obtained from

and let

is the complex E

E

by (6.6),

be the complex of sheaves on

tained from the connection on E .

on

module with HPD stratifi-

as

ob-

(Recall that the complex , obtained from the connection

-module.) Then there is a canonical isomorphism:

If

is the canonical projection (5.18),

there is a natural isomorphism in the derived category of sheaves of abelian groups on

Proof.

Of course, (7.1.2) is

a

fancy

local

form of

(7.1.1), which in fact it implies, because Let us begin by proving (7.1,2) in the special case in which E arises from an

module

F

with HPD stratification, i.e.

(with the HPD stratification of the tensor

6.2

product).

In this case,-the Poincare lemma (6.1H) gives us a

quasi-isomorphism in

We now use the

localization description of L (6.10):

Recall that

is represented by

and that we have a com-

mutative diagram:

Now (6.10) tells us that note that each of these is acyclic for

by (5.27).

It follows that there

is an isomorphism in the derived category:

The latter is just the complex of Zariski sheaves the theorem is proved in our special case.

, so

For the general case,

we need a lemma:

7.1.2 of

Lemma.

and the latter is a crystal

modules.

Proof.

Since (6.2) tells us that Suppose

u:T'

is exact, T is a morphism in

Cris(Y/S), we must show that an isomorphism.

Recall from (6.2) that

is where

6.3

D is a certain PD envelope of

T in T, whose formation is

compatible with change of T . Thus, , since

E

is a crystal on X/S.

In particular, and

is the crystal on Y/S defined by E viewed as an

HPD stratified

module.

Applying the above special case to

on Y/S, we see that there is a canonical isomorphism:

Since the theorem follows. Using the filtered Poincare' lemma and (6.10.1), we can prove a more precise result:

7.2

Theorem.

and let

complex of

which in degree

is the ideal of

X in

Proof.

D).

q

is

be the sub(where

Then:

The filtered Poincare' lemma (6.14) tells us that

we have a quasi-isomorphism:

J

6.4

Recall that for

is acyclic by (5.27.2), and hence applying

get

The theorem

follows.

7.3

Corollary.

There is a natural isomorphism:

An immediate consequence is the fact that crystalline cohomology captures the de Rham cohomology of a lifting, if it exists.

7.4

Corollary.

sub PD ideal of

Suppose X/S is smooth, I, and

is defined by a

Then there is a natural

quasi-isomorphism:

Proof.

7.5

Just recall that we showed in (5.17) that

Remark.

Because the isomorphism of (7.3) is natural, we

can use it to compute the crystalline cohomology of maps, as well as spaces.

If

is a map over a PD morphism and X can be embedded in schemes

which are smooth over S' and S, and if X/S is separated, then in fact we can find such embeddings which induces

f.

and a map

We obtain a commutative diagram:

7.5

It follows that

depends only on

f, and not on

A crystal E of

is said to be

coherent" iff each C>T-modules on T.

is a quasi-coherent sheaf of Our next task is the establishment of finiteness

and base changing properties of the cohomology of quasi-coherent crystals.

7.6

Theorem.

Suppose S is quasi-compact,

is

quasi-compact and quasi-separated, and

is

the composite

Then for any quasi-coherent crystal E

in

is quasi-coherent on S.

exists an

Proof.

r

such that

Moreover there and all

E .

First suppose X can be embedded in a scheme Y which

is smooth, quasi-compact,

and quasi-separated over S.

Then we

get from (7.1) a natural isomorphism in the derived category:

is

a

bounded

complex

of

quasi-coherent

-modules, the

theorem for our X follows from standard considerations. We can reduce to the above case by using Cech cohomology. Let each

be a finite affine open covering of X, and for If F is a sheaf of

6.6

modules on Cris(X/S), let where

is the natural map.

Then we can form a

complex C "(F) in a natural way, which is easily seen to be a resolution of F.

Now let

tive sheaves of

be a resolution of E by injec-

modules.

One sees easily that the total

complex C'(E') is a resolution of E by sheaves which are acyclic for

Thus the cohomology of the double complex is the cohomology of

The spectral

sequence of the double complex verges to

therefore con-

so we need only prove that each E ^ is

quasi-coherent and that already clear for

p.

for

Now

p and q

large — as is

is the cohomology of the complex and since the complex

is a resolution of

by sheaves which are acyclic for Thus, we have

reduced the problem to the spaces coherent crystal on

since

is a quasi-

Clearly it suffices to prove the re-

sult for the sets Since X/S is quasi-separated, sets of the above form are not necessarily affine, but are quasi-compact. obviously quasi-affine, and hence separated.

They are also In other words,

we are reduced to proving the theorem for X quasi-compact and separated.

Repeat the above Cech procedure with an affine

covering.

This time one gets an affine covering with affine

intersections, so the result follows from the affine case. Of course it would be tedious to attempt to write down a specific r which worked in the above theorem. However, it is

7.7

important to notice that the process of embedding pieces of X in smooth schemes is compatible with base change, so that (7.6) really gives an any

r

which works not just for X/S, but also for

X'/S' obtained from base change via a PD morphism

(S',Ι»,Ύ·) ->· (S,I,Y). In order to study the base changing properties of the cohomology of a crystal, we first need a fancy form of the ad­ junction formula in the derived category.

This holds for any

morphism of ringed topoi, as explained in [SGA 4 XVII], where it is called "trivial duality".

Let us give the reader a rough

sketch: f: (T1,A') •+• (T,A) is a morphism of

Recall first that if

ringed topoi, and if F* is a complex of A'-modules which is bounded below, then it is always possible to find

a

quasi-

isomorphism F* ->· I* , where I* is a complex of injective T1

A'-modules

in

definition

the image of

Α-modules in

and the hyperderived functor IRf4F* f*I*

is, by

in the derived category of

T .

We need a similar construction

for

f* .

Since there do

not exist, in general, enough projectives in the category of Α-modules in a topos, the construction is not so standard, so we will indicate a few more details. is a functor L

which associates to

First observe that there

any Α-module E a flat

Α-module L (E) and an epimorphism L (E) -* E.

Namely

the sheafification

takes

the

free

Α-module

insures us that L that

there

is

of with

the

presheaf

basis

which

E(U) - {0} . L*(E)

any

(Leaving

takes zero maps to zero maps.)

a functorial complex

L (E) is U

to

out 0

It follows

of flat A-modules

7.8

3-

0

with H (L^(E)) = O if to E.

i Φ O, and H (L'(E)) naturally isomorphic

Now if E" is a complex of A~modules, L"(E") is a double

complex of flat Α-modules, and the associated simple complex L*(E*) is easily seen to be naturally quasi-isomorphic to E E° is bounded above. natural map

if

If E* and F" are two such complexes, the

L'(E') ® L"(F') -* L*(Ε*βΓ")

is hence a quasi-

isomorphism, and since L" is also compatible with shifts, one see that if plexes

C ( u ) is the mapping cone of a morphism of com­

u: E* -* F' , then there is a natural quasi-isomorphism

C(L'(u)) -• L'(C'(u)), now define

(That is, L" preserves triangles.)

lf*(E*) to be the class of

category of Α-modules.

We

f*L"(E') in the derived

Since the modules comprising L*(E *) are

flat, I f * = f*L* takes acyclic complexes to acyclic complexes, hence quasi-isomorphism to quasi-isomorphisms, (and triangles to triangles).

Note that If*E"

is only defined on complexes

which are bounded above. Finally, if

M* is bounded above and N" is bounded below,

K Hom^CM'jN"] is defined to be the class of the complex Hom^CM*,1"], where I" is an injective resolution of

N'.

(Recall that this is the complex which in degree

is

"TTHomCM , I i

x

3, with the usual boundary maps.)

k

Note that since M'

is bounded above and I" is bounded below, the product "Jf is really i only a finite one.

7.7

Proposition.

(Adjunction formula).

If

E* is a complex

of Α-modules bounded above, if F' is a complex of A*-modules bounded below, and if

f: (T1,A') -»• (T,A) is a morphism, there

6.7

is a canonical, transitive, functorial isomorphism:

in the derived category of abelian groups.

Proof. and

Let

be a flat resolution, and let be injective resolutions.

Then is

Using the adjointness of

f* and

, we obtain a

natural map:

(The fact that the arrow in the second line is a quasiisomorphism depends on the injectivity of the modules This gives us the desired arrow.

We shall not verify, or

even list, all the compatibilities it satisfies.

However we

shall use them freely, even in our sketch of the proof of the fact that it is an isomorphism.

Clearly the problem is in the

arrow We begin by factoring

f

into two morphisms: (the reader can, I hope, easily

imagine the meaning of the middle term). sitivity of functor

By appealing to tran-

Ad^ , we reduce the problem to is exact, so its adjoint

injectives, so the problem is trivial. problem to maps of the form

f:

g and h . Now the

takes injectives to We have thus reduced the

6.10

To prove that the arrow

is

a quasi-isomorphism, we have to show that the complex comprises modules which are acyclic for

In other

words, we have to prove that if L is A-flat and

I

is B-injective,

then Choose an injective A-module J containing in a natural way, and since

; then

I

is injective,

it is in fact a direct summand, and it is enough to prove the statement with this Horn Let

in place of

P* be a resolution of

flatness of the

P's

that

I . by flat A-modules.

and the adjointness of

and

The

Horn

show

is a complex of injective A-modules, in fact

an injective resolution of

Thus we have only to

calculate the cohomology of the complex But this is just

and since

L

is flat and

injective, it's just a resolution of

J

is

and in

particular is acyclic. Perhaps we should remark that an A-module E in ringed topos (T,A) is

defined to be flat iff

is exact.

to prove that a sheaf E of to be flat iff each

It is not hard turns out

is a flat

module [Berthelot III 3.5.2],

but we may as well take this as our definition. We are now ready to study the base changing properties of the cohomology of a flat crystal.

Suppose S is quasi-compact, are quasi-compact and quasi-

separated, and g:

covers a PD morphism u:

6.11

Let E be a quasi-coherent, flat crystal on X. Then (7.6) tells us that the complex is defined. and

is actually bounded, so is a crystal on X 1 ,

On the other hand,

is a bounded complex on S', which is reasonable to

compare with In fact it is easy to obtain at-map (in the derived category) i.e.

an element of

of the complex

This can be done from the adjunction formula in several ways, here is one: The adjunction formula for

g

gives an isomorphism: and since and we obtain (from the identity map

morphism in the derived category: functor

E

is flat,

idj,,) a Applying the

, we obtain an element of

By the adjunction formula for

u, this in turn is isomorphic to which is where we wanted to be.

Some additional hypotheses will allow us to assert that the base changing map above is actually an isomorphism. suppose that the morphisms

f and f'

We

fit into the diagram:

6.12

7.8

Theorem. Suppose that in the above diagram, S is quasi-compact, is smooth,quasi-compact, and quasi-separated, u is a PD morphism are defined by sub

PD ideals of

I and I', respectively. and that

E

Suppose further that

is a flat, quasicoherent crystal on

X.

Then the base changing arrow is an isomorphism:

Proof.

We begin with a very special case, assuming that X

lifts to a smooth affine scheme Y over S.

Then

lifts

X' , and we can write everything down explicitly: Since S^ C S is defined by a sub PD ideal,

and the crystal E

corresponds to an 0^-module

with integrable quasi-

nilpotent

connection.

Since E is flat, each

Orp so

is a flat

module.

£

We know from (7.1.2) that and since

this is just the complex crystal where

is flat over

f

is affine,

On the ether

corresponds to the

hand, the

module

is the natural map, and we have , we have a natural

isomorphism of complexes of i

-modules: —

this is just the base chang-

ing theorem for affine morphines in the Zariski topology, together with the fact that formation of base change.

is compatible with

Of course, this arrow is the same as our fancy

looking base changing arrow, and the theorem is proved in the special case.

6.13

To deduce the general case it is necessary, unfortunately, to resort to the technique of cohomological descent.

This is

because it is not clear a priori that the adjunction map is compatible with Cech resolution.

In order to overcome this

difficulty, we have to provide the Cech resolution with a geometric msaning — i.e. topoi.

we have to express it in terms of

We shall give here only a rough sketch and refer to for details.

The idea of cohomological descent is to replace the topos by a topos

which stands for a finite covering

together with all the gluing data.

Thus if

is a multi-index with

let

and consider the simplicial scheme:

We construct a topos (X'/S) . cris tion of sheaves between the

on each

whose'feheaves" are collec-

together with compatible maps

covering the inclusions

There is

an obvious morphism of topoi: deduced from the inclusions

The key fact of cohomological

descent, which we shall use without proof, is that for any abelian sheaf E, the natural map,

is an isomorphism.

Using this, we can reduce our proof to the topos Construct a topos covering a commutative diagram:

in a similar manner, using the etc., we get

;

7.14

Thus, it suffices to prove that the top arrow is an isomorphism. We would like to reduce to the individual opens IK in the open covering. To do this, we have to construct a stupid topos S" whose sheaves are families of sheaves on as before.

The point is that

map

indexed by the same indices factors through an obvious , and to prove the base chang-

ing theorem for

fjc/g

Since it is easy for

it suffices to do it for we are reduced to the morphism

and the amounts to checking it for each map

D^ + S, as is easy

to see. We can now easily complete the proof of the theorem.

So

far, we know it if either: Ci) X

lifts to a smooth affine scheme Y over S, or

(ii) X has a finite open covering such that the theorem holds for any intersection Suppose first that X satisfies (i) and U C X is open and quasicompact.

Then it is clear that U can be covered by finitely

many special affine sets IK which satisfy (i), and it is also clear that the intersections Thus the theorem is true for U.

also satisfy (i). Finally, suppose

is smooth,

7.15

separated, and quasi-compact.

If x 6 X, it is easy to see, by

choosing local coordinates at

x, that x has an open neighborhood

U x satisfying (i) — the local coordinates mean that we have to lift an

étale

possible because

which is

is a nil immersion — c.f. [EGA IV 18.1.2].

Since X is quasi-compact, finitely many of these neighborhoods will do, and since X is quasi-separated, any intersection U of them is quasi-compact. Hence the covering we get satisfies (ii), so the theorem is true for X.

7.9

•

Corollary. Let S be quasi-compact, let f: X -»• SQ be smooth,

quasi-compact, and quasi-separated, with a sub PD ideal of I. E = E^ to SQ)„

with

defined by

Let E be a flat crystal on X/S, and let

its associated integrable connection (relative

Then there is an isomorphism:

Proof.

The base changing theorem tells us that

is just

Remark. pOg = 0), if

and we know that the latter is

Note that in the equicharacteristic case (i.e. f

is proper and flat, it is still not known

whether or not each individually —

i.e.

commutes with base change whether or not each is locally free.

7.16

7.11

Corollary.

Suppose

is smooth, quasi-compact, and

quasi-separated, and Y is quasi-compact. coherent crystal on

Then if E is a flat quasi-

is a crystal in the derived cate-

gory of

Proof.

Suppose

is an object of Cris(Y/S).

Let

It is easy to see that there is a natural equivalence of categories

, and using this, one

can identify v:

Now if is a morphism in , and since U

T and

are defined by

PD ideals, (7.8) tells us that the map

is an isomorphism, as desired. This result is a crystalline version of the Gauss-Manin connection.

We can make this explicit when Y/S is smooth.

case the maps

are flat, so

In this , and we

see from (7.11) that there is an isomorphism:

Since

is smooth

and

we have exactly an integrable, quasi-nilpotent connection on the Gauss-Manin connection.

7.17

Here is a strong form of the base changing theorem for a smooth map:

7-12

Corollary.

Suppose, in the diagram below, that

quasi-separated and smooth, that X' - XXyY'.

Y

f

is

is quasi-compact, and that

Then if E is a flat quasi-coherent crystal on E,

the base changing map:

is an isomorphism.

and

Proof.

Suppose

v: T'

T

is a PD morphism.

Thanks to (7.11) and Exercise

(6=5), we can describe But now if we identify

and

with

we see that the

arrow

is an isomorphism,

as desired.

•

The reader may have already noticed an important consequence of the

assertion (7.8) that the arrow:

is a quasi-isomorphism: the target is a bounded complex by (7.6), hence

for almost all

i . This will enable

us to prove the following finiteness results for

7.18

7.13

Corollary.

Let

f: X—>(S,I,Y) satisfy the hypotheses of

(7.8), and let E be a flat quasi-coherent crystal on Cris(X/S). Then

has finite tor-dimension, i.e.

is isomorphic in

the derived category to a bounded complex of flat Proof.

I claim first of all that there exists an in-

teger n such that for all

Og-modules

is

acyclic except in degrees within (0,n). we can assume S is affine.

Since S is quasi-compact,

First suppose that M is quasi-coherent,

and consider the generated by

The ideal I'

M and I in 0s', has a PD structure

is a PD morphism

Y' , and there

Now the base changing

theorem tells us that

a complex

which we can bound (independent of s') by (7.6). But a direct summand o bounded.

f

a

n

d

is

hence it too is uniformly

The above argument works only for quasi-coherent M, but if M is arbitrary, it implies that the stalks of

are uni-

formly bounded, hence so is Because L'

of flat

For each

q, let:

is bounded above, we can find a complex representing it, still bounded above.

7.19

Then the obvious maps

and

isomorphisms.

are flat, we have

Since the

are quasi-

for any M

Thus, A

and any

complex of flat the theorem.

0

is flat, and hence

Og-modules quasi-isomorphic to

A' i s L'.

a

bounded

This proves

•

We can now prove the fundamental finiteness property of the cohomology of a crystal. perfect complex.

The key concept is that of a

This notion is defined and studied in [SGA 6 I]

for nonnoetherian ringed topoi.

We will content ourselves with

a much more prosaic situation:

7.14 Definition.

Let A be a noetherian ring.

A complex K° of

A-modules is called "strictly perfect" iff it is bounded and if each

7.15

is finitely generated and projective.

Lemma.

A complex K" of A-modules is quasi-isomorphic to

a strictly perfect complex iff it has finite tor-dimension and finitely generated cohomology.

Proof.

One constructs a complex P" of finitely generated

projectives and a quasi-isomorphism P* ->K', inductively, in the usual manner. if P*

Then as in the proof of the previous result,

K" is acyclic in degrees < m, one has a quasi-isomorphism A' , with

if projective.

is flat and finitely generated, hence p

7.20

7.16

Theorem.

Suppose

f:X—>Sp is a smooth proper map,

is defined by a sub PD ideal of I, and S is noetherian.

If E

is a crystal of locally free, finite rank is a perfect complex of Os-modules —

i.e.

is, locally on

S,

quasi-isomorphic to a strictly perfect complex. Proof. is affine.

Since the assertion is local on S, we may assume Moreover, by the quasi-coherence of crystalline

cohomology (7.6), we may work over

instead of

Then (7.15) applies, so we need only check that coherent on

S. is

S.

Since p is nilpotent on S and the ideal K of S^ in S i s a PD ideal, it is a nil ideal, and since S is noetherian, for some

m . The proof is by induction on

m.

If

m = 0,

SQ = S, X/S is smooth, and we know from (7.1) that Since EAv is coherent, so is each and hence so is Now let

be the subscheme defined by

so that

we have an exact sequence:

(*) Tensoring with the complex

in the derived category,

and using the base change isomorphisms

for

v = n

and

n-1, we get a triangle:

S

7.21

*f

Kf

E \

/ \

X/Ss E n-1

lf


0, so (7.21.2) is certainly satisfied, and

a Because of their functoriality, the constructions of and of (X/S.)crxs .

in the Zariski topology of

exactly as in (5.18).

are compatible with localization r

X, and we can define morphisms

There is even a morphism

and a commutative diagram:

7.28

7.22

Proposition.

Let

in the above diagram.

7.22.1

If

F

is an abelian sheaf in (X/S.) . , ens '

7.22.2

If

E

is a quasi-coherent sheaf in

then

and

Proof.

We had best begin by remarking that we have abused

language in the second statement, since we have written in a derived categroy. it meaning, for

However, the diagram and (7.22.1) gives will be a well-defined object in the

derived category of abelian sheaves on JJ, and

7.29

We first observe that the functor

crxs

has an exact left adjoint, given (for abelian sheaves, of course) by

The same is

true of

and

to injectives.

and it follows that they take injectives

Since they are also exact, they satisfy We conclude that

The rest of the proposition is proved similarly.

•

We are especially interested in the cohomology of a crystal of

If

X

can be embedded in a smooth

S-scheme Y, we can obtain an "explicit" complex representing this cohomology.

7.23 E

Theorem.

Suppose

Y/S is smooth,

is closed, and

is a quasi-coherent crystal of and let

Let

be its P-adic completion.

.Then there

is a D-module with integrable connection (E,V) inducing and for each

m, there is a natural isomorphism:

Since

we have a canonical iso-

morphism:

But (7.19)

tells us that that Cris(X/S.). in place of

E,

and and that

one

j*E

sees

easily

is a crystal on

Hence it suffices to prove the theorem with (X/S.)

7.30

For each

n, the crystal E n on

ponds to a

with integrable connection

(En , V),

w

i

t

h

(3.20.8), II x X

Cris(X/Sn) corres-

S

i

n

also.

corresponding to

{Dn}

c

e

Let V.

denote the sheaf on

let (E. ,V) the sheaf of

P.-modules with connection corresponding to

Then with integrable

connection, inducing

E.

Even though we have compatible isomorphisms for each

n , this doesn't quite

give us the theorem, because one cannot work locally in the derived category. Therefore we must copy over the proof of (7.2). Let D.

denote the sheaf in (X/S.) . defined by cris , and form the diagram below (as in (6.10)).

I claim that:

7.23.1

7.23.2

and

If

F

are exact.

is any abelian sheaf in is acyclic for

,

7.31

Indeed, it suffices to check (7.23.1) after restricting to (X/Sn) for all n

n, so that it follows from (5.26) and ('5.27).

Moreover, (7.23.1) implies (7.23.2) (c.f. (5.27.2)). There is a natural morphism of complexes: where the latter in degree q is It is a quasi-isomorphism because it is after restricting to each , by (6.13), and hence we obtain an isomorphism:

Since (7.21) tells us that the sheaves of the complex on follows.

are acyclic for

the theorem

•

We now come to the main statement of P-adic crystalline cohomology.

7.24 Theorem.

Suppose

quasi-separated, and crystal of

7.24.1

E

X/SQ is smooth, quasi-compact, and is a locally free, finitely generated in

There is an object

. Then:

D

in the derived category of

inverse systems of A-modules such that each and

7.24.2 For each isomorphism:

n, there is a natural base-changing

7.32

Moreover,

7.24.3 fect.

has finite tor-dimension.

If

is proper, the complex

is per-

Moreover, the inverse systems

satisfy

ML, and there is an isomorphism:

Proof.

The first statement is just (7.22.2) (and in fact

is true more generally).

Now the base changing theorem (7.8)

implies that the arrows isomorphisms for all

are (quasi-)

n, so that

D

is what Appendix B calls

a "quasi-consistent complex of A.-modules" B6) and (7.13) imply that

D

and

(c.f. B4).

Then

have finite tor-

dimension, and (B5) implies that the arrows are quasi-isomorphisms. proper, we know that implies that

This proves (7.24.2).

If

^q^O

is perfect, and so (Bll) is also perfect.

Let us remark that

(Bll) shows that in the proper case only,

is func-

torially determined by the inverse system

7.25 A

Corollary.

Suppose that in the notation of (7.17) above,

is a complete discrete valuation ring (necessarily of mixed

characteristic p and absolute ramification index

with

7.33

P = I = the maximal ideal of

A.

If

E and X are as in (7.25),

there are exact sequences:

Proof.

Let

K"

representing of A. Hence:

be a strictly perfect complex A-modules Let

be the uniformizing parameter

There is an exact sequence:

Since

and

result is clear.

the

•

We have at last succeeded in giving at least the definition of a reasonable p-adic Weil cohomology.

Here is what we

have proved: 7.2 6 Summary. Suppose A is as in (7. 25 ). Then there is a functor from the category of smooth proper AQ-schemes to the category of finitely generated graded A-modules such that 7.26.1 7.2 6.2

There are natural exact sequences:

7.34

7.2 6.3

If Y/S is a smooth lifting of isomorphism:

Proof.

X, there is a natural

It remains only to explain the last statement.

But by (7.4), which is turn is the same as fundamental theorem for a proper morphism.

by the •

In particular, we obtain the desired relationship between the crystalline and DeRham Betti numbers. is the rank of the free part of

Explicitly, if and if

number of its torsion factors, then the dimension of is

r)^ is the

References for Chapter 7

[ADRC]

Hartshorne, R. "On the De Rham Cohomology of Algebraic Varieties" Pub. Math. I.H.E.S. No. 45 (1976) 5-99.

[CC]

Berthelot, P. "Cohomologie cristalline des schemas de caractéristique p > 0" Lecture Notes in Mathematics No. 407, Springer Verlag (19 76).

[EGA III] Grothendieck, A and Dieudonne J. "Elements de Géométrie Algébrique" Publ. Math. I.H.E.S. No. 11 (1961). [H III]

[LCD]

Deligne, P. Théorie de Hodge III No. 44 (1975) 5-77.

Publ. Math. I.H.E.S.

Ogus, A. "Local Cohomological Dimension of Algebraic Varieties" Ann, of Math. 98 (1973) 327-365.

CSGA 4; 6] Grothendieck, A et. al. "Se'minaire de Geometrie Algebrique" Lecture Notes in Mathematics No.'s 269, 270, 305; 225, Springer Verlag.

8-1

§8.

Frobenius and the Hodge Filtration. Suppose

W

characteristic

is the Witt ring of a perfect field p, and X is a smooth k-scheme.

k

of

The Frobenius auto­

morphism of W is a PD morphism, covered by the absolute Frobenius endomorphism F„ of X, and it follows that F„ acts on the crystalline cohomology of X relative to W.

In this chapter we shall study

this action, in particular, its relationship to the Hodge filtra­ tion on crystalline cohomology (as determined from the ideal

;„,„).

The main global applications are Mazur's theorem (8.26), which says that (with suitable hypotheses on X) the action of Frobenius de­ termines the Hodge filtration on

H* (X/k) , and Katz's conjecture

(8.39), which says how the Hodge filtration limits the possible "slopes" of Frobenius. The above results generalize somewhat the work

pf Mazur [4,5].

Our technique of proof is, however, rather different,since we follow a suggestion of Deligne, proving a local result of which the above global statements are formal consequences. We approach the local problem in two parts.

In the first

part we study the DeRham cohomology of a lifting to obtain a state­ ment (18. 3) and its generalization (8.8) only valid in the liftable case.

In the second part we interpret the calculations in terms

of crystalline cohomology and obtain (8.20), which does have global meaning.

(8. 1)

Let us fix some notation.

Let (A,I,γ) be a P-adic

base (7.17), and use the notations of (7.17) except write S= Sp£ A instead of

S = Spec. A.

We assume as additional hypotheses:

8-2

P = (p), and

A

is a p-torsion free.

selves an endomorphism endomorphism

(8.2)

F_ b 0

of

Fg of

In addition, we give our-

S, lifting the absolute Frobenius

Sn. U

The local calculations take place in what we shall call

a "lifted situation over (S,Fg)". formal scheme

This means a formally smooth

Y/S, together with an Fg-morphism F y : Y Y

such that

the absolute Frobenius endomorphism of We shall also need "relative Frobenius", whose construction we recall. Fs: S

Form the fiber pro4uct

S . Let

pr: Y

S, pr': Y'

using the map S, and

be

the natural projections, and let that

be the

S-map such

Thus we have a diagram:

(8.2.1)

Notice that

are homeomorphisms.

Indeed, since we

are working with p-adic formal schemes, it is enough to check this mod p, and we have

and Notice also that the above diagram makes

sense for any

Y/S, not necessarily smooth.

Let me now explain a special case of the main local result, which gives a precise description of the image of Begin by noting that so that the image of

is contained in

kills

8-3

Since

is compatible with exterior

products, it follows that the image of

is contained in

We can say slightly more than this:

Since

is a morphism of complexes, it follows that the image of is in fact contained in Our main result says, in a sense, that this is exactly the image of

Precisely:

8.3 Theorem.

In a lifted situation (Y/S,FY), the map induces a quasi-isomorphism into the

largest subcomplex for all

k

N"

of

such that

(described explicitly above).

Both for the proof and applications, we shall in fact need to make a more general statement, ( 8.8 ).

Let us first explain

the main tools of the proof, which are the Cartier isomorphism and its relationship to Frobenius.

For the reader's convenience,

we recall the following description of the Cartier isomorphism, proved in [2, 7.2].

8.4- Theorem

(Cartier).

Suppose

X/SQ is smooth.

Then there

is a unique morphism of

such that: and the image in

for any section

8-4

C-"1" is an isomorphism,

Furthermore,

•

Here is the relationship between the action of Frobenius and C ^ in a lifted situation.

8.5

Theorem.

Suppose (Y/S,Fy) is a lifted situation (8.2).

Then for each

where

there is a commutative diagram

is "reduction mod p" and

Proof.

of

is the image in

We have already observed that

lies in

it follows that makes sense and that its image in

is a cocycle.

that the diagram commutes, first observe that both

Moreover, if

diagram commutes.

If

an

j=0

both map

Now Thus ,

and

are

1 to 1, so the is generated as

elements of the form We can even take

tion of 0 Y .

To see

Then

is the class of

•> some is the class of

with with

a sec-

8-5

which is the same as the class of _1

(8.4), this is

(a?~ d a . ) Λ . . .Λ (α?~ da.).

-1

0 ( ω Ί ) Λ . . . Λ C (u>.).

By

D

In order to state the generalization of Theorem (8.8) which we will need, we must first introduce some notation which permits us to describe the p-adic divisibility of a morphism of complexes. For the moment, the following ad hoc definition will do, later we shall use a more systematic notion (8.15 ff).

8.6

Definition.

Let

ρ

an abelian category. a complex in

If

be a fixed prime number, and let ε:

Z •* M

A

is a function and if

A , then "K'" denotes the subcomplex of

K

-

be

K'

is

given

by:

Thus,

K^

1

is the largest subcomplex of

1

K C P^ V

K"

such that

for all i .

Notice that if

K"

is a complex of

p-torsion free sheaves

of abelian groups on some site, then K^(U) = {x e ρ U. and

ε(ι)

1

Κ ( υ ) : dx e ρ

ε(i+1)

K

i+1

( U ) } , for each object

(In the presence of torsion, this may only define a presheaf, K

is the associated sheaf.)

It is clear that a morphism f: A* •*• B' induces a morphism f : A* -*• B*

for all

equally clear that if

e , and that this defines a functor. ε 1 . Any tame

e

Notice that if p = 2, = 1 is a gauge (8.7).

8-18

8.18.2

Suppose

is any function, and set Then it is not hard to see

that

is the maximal tame gauge less than or equal to e .

Moreover, we h a v e I n d e e d , need only check that

since

, we

, i.e., that each

This is why we can only consider tame gauges.

8.18.3

For any

i > 0, set

tame, and

is

This gives us a "gauge theoretic"

interpretation of the

8.18.4-

Then

Suppose

e

P.D. filtration

is tame, and

Y/S is smooth and

Then

X = YQ.

. This tells us

that the p-adic interpolation of the Hodge filtration Y defined by the tame gauge

e

depends only on X, not Y.

O

The above remarks provide us with a crystalline interpretation of the source of the arrow

of (8.8).

The target turns

out to be quite easy to handle, for abstract nonsense will tell us that "formation of if

n

is increasing.

homotopic maps Indeed, if

passes over to the derived category, Notice first that if

A" -»• B" , then

and

f and g

are

are homotopic.

is the homotopy, it follows immediately

from the definitions and the fact that

is increasing that

8-19

maps

and hence induces a homotopy

Of course, formation of

is not compatible with translation,

hence doesn't preserve triangles, nor is it exact. it has

Nevertheless,

a left derived functor, in a slightly extended sense,

because of the following simple result:

8.19

Proposition.

Let

f: A" -»• B"

sheaves of abelian groups. free and that

ri:

be a quasi-isomorphism of

Assume that

A* and B*

is increasing.

are p-torsion

Then

is

also a quasi-isomorphism.

Proof.

We just check the stalks, so we work with groups

instead of sheaves. (i)

Assuming that

is injective:

is an isomorphism:

Any class in

represented by some is torsion free.

with

is repreda=0, since

If for some

Then

(this makes sense because increasing), and since

is

is injective,

for some

(ii)

is surjective: sented by some

Any class in , with

is repreSince

is surjective, we can choose an da = 0 and a Then

with

such that Certainly

8-20

To construct the left derived functor Xn: let

D(X) -* D(X),

F" be a complex of abelian sheaves on X, and let be

a

flat resolution.

Since

Z

has finite

projective dimension, this makes sense even if we can t

a

k

e

b

1.ti(F" ) to be

e

F* is unbounded, and

bounded below if F'is. Then define This makes sense in the derived cate-

gory, thanks to the previous result. We now can state and prove the crystalline version of the main local theorem:

8.20 e

Theorem.

Let

be smooth, with

S

as in (8.1).

is any tame gauge and if

for all i ,

there is a commutative diagram as shown, in which

is an

isomorphism.

This diagram is functorial in

X and

, and agrees with the

diagram (8.8) in a lifted situation.

Proof.

If

The meaning of the horizontal arrows is clear.

The main remaining point is the existence of the arrow

8-21

It would be nice to have an intrinsic proof; we have to resort to an unpleasant local calculation.

8.21

Lemma.

Suppose

is a lifted situation, and

is a closed So-scheme. Then

Let

induces a natural morphism of complexes: Moreover:

8.21.1

For each

maps

where

8.21.2

For any gauge

m a p s i n t o

where

Proof.

Look at the relative Frobenius diagram (8.2.1).

Since

and

inducing Fx: X

X.

Moreover,

X to X', inducing

and

get induced maps

and

Now and Since the ideal hence also that

maps It follows that we on the

PD

envelopes, and

is in characteristic

p > 0,

is its absolute Frobenius morphism. is a PD ideal, Since and since

, this implies is a

8-22

PD-morphism, it follows that for all

i .

Now as we observed in the lifted situation (8.2),

maps

so the same is true for is by definition

Since

it is clearly mapped

by

Since

is a

morphism of complexes, (8.21.1) follows, and (8.21.2) is an immediate consequences. Now if

•

is tame,

for all

i, hence

Because the complex is

p-torsion free, we see that

induces the desired

arrow It is important to note that if

X—*1

is another closed

immersion into a lifted situation, and if we can find a morphism

compatible with the inclusions of X,

then the morphisms of complexes we have constructed are compatible, in the evident sense.

That is, there is a commutative

diagram:

Moreover, if (Y,FY) and (ZjF^) are any two lifted situations in which X embeds, then we can also embed X in (as a locally closed subscheme, but no matter) —

and this maps

8-23

both to (Y,Fy) and to (Z.,FZ).

It follows that the arrow in the derived category

that we have defined is independent of the choice of embedding. Let us observe that the theorem is now proved for quasiprojective X/S. to

so

X

Indeed, the absolute Frobenius of

lifts

can be embedded in a lifted situation.

over, once the arrow

More-

is defined, it must be a quasi-

isomorphism, because this is a local question, and we may therefore assume that ( X , F l i f t s .

Because the arrow

is

independent of the choice of embedding, we can use the lifting to calculate it, so that (8.8) implies that it is a quasiisomorphism. It remains only to use cohomological descent to define the arrow in the general case, as Deligne suggests. find an open covering a lifted situation:

such that

We can easily with

Then is locally closed, and

is a lifted situation.

Because each

is locally an open immersion, for any abelian

We now have natural

isomorphisms (using the notation of (7.8)):

8-13

In order to draw the cohomological consequences of the main local theorem (8.20), it is convenient to know some of the properties of the functors

It is also convenient to make a

slight additional restriction on n .

8.22 Definition.

A function

is a "cogauge" iff for all

8.22.1 n

Remarks.

is a cogauge.

i .

Suppose that K* is a p-torsion free complex and Then multiplication by

morphism:

induces an iso-

This translates into a statement in the

derived category:

There is a commutative diagram (not a triangle!):

We call this the "shifting diagram".

8.22.2

If

ri and ?

are cogauges, so are

(but

need not be).

and If

K" is torsion

free, it is easy to verify that and

(The first of these requires the cogauge

condition.) These statements translate in various ways into the derived category, which the reader can imagine.

8.23.3

If

there is a natural exact sequence: In the derived category, this trans-

lates into a triangle:

8-25

where ILn/n'

is the mapping cone of

A diagram of inclusions:

induces a morphism of triangles, which we prefer to notate as short exact sequences:

These diagrams are functorial, compatible with shifts, etc. A convenient way to express the content of the previous remark is to observe: 8.2 3.4

then the canonical map is an isomorphism.

It suffices to check this for

torsion free complexes, with the triangles replaced by short exact sequences.

Then in fact

as the reader can easily verify.

is an isomorphism

8-26

8.2 3.5

Suppose that

allowed.

We call

r

the "turn-on" value of

n . If

K'

is

p-torsion free, there is an exact sequence:

where

T.

is the "canonical" filtration, given by:

This statement passes over to the derived category. formation of

Tr

Indeed,

preserves homotopies and quasi-isomorphisms,

hence passes over to the derived category, and Thus, we have a canonical triangle:

8.2 3.6

Suppose that n' is a simple augmentation of

n at j .

Then a duplication of the calculation (8.10.1) shows that there is a natural triangle:

Formation of these triangles is natural, compatible with shifts, etc.1 • If K" is a bounded complex of sheaves on X, the assignment defines a rather intricate structure — essentially the "gauge structure" of Mazur

[5].

I do not know much about

its meaning — perhaps it is correct to think of it as a p-adic elaboration of the "conjugate filtration" of hypercohomology. ^The reader who so desires can now skip to Katz's conjecture, p.8.42.

8-27

Here we shall develop only those properties of this structure we need for the applications, and refer the reader to Mazur's papers for more details.

8.24

Proposition.

Here is an important special case:

Suppose that K" is a bounded complex of

abelian sheaves on X such that: (a) (b)

is p-torsion free. The spectral sequence: is degenerate at E^-

Then for any two cogauges 8.24.1

If

8.24.2

If

n and

the map

is injective.

(etc.) is the image of

we have

Proof.

Replace

by a flat complex

bounded), so that

for any n . The

hypotheses (a) and (b) hold with we may as well assume that

(which is still

L" in place of

K' . Thus,

K' itself is torsion free.

We prove (8.24.1) by studying a succession of cases. Case 1.

Let

Then the map

is injective.

To prove this, observe that of

at

r , so we have

is a simple augmentation

commutative diagrams:

8-28

First of all, because K' is bounded, the maps

and

are isomorphisms for tion (a) implies that all

i . Therefore

induction on

and assumpis surjective for

is surjective for

r >> 0.

Descending

r , using the above diagram, implies that

surjective for all

r . Assumption (b) implies that

jective, and the diagram implies that

is is sur-

is surjective.

Case 1

follows. Because

is bounded above, the following is true for all

s >> 0: L(s):

The map

is injective if for all

Because K' is bounded below, it suffices to prove that L(s) is true for all Case 2.

s, and we can use descending induction on

s .

Let

Then if L(r) holds, the m a p i where is the simple augmentation of

s at

injective, r .

8-29

We have commutative squares:

The assumption implies that Case 3.

L(r) implies that c If

is injective.

b

is injective, and Case 1

Therefore

a

ri is a simple augmentation of

and if L(r) holds, then the map Proof.

is injective.

is injective.

First of all, shifting Case 2 implies that the map is injective, where

r

1

> r, the claim is trivial, and if

Now the map

a is surjective, hence

r

1

= r

Now if we have:

is also surjective, and

Case 3 follows. To finish the proof of (8.24.1), we show that L(r) implies Suppose that

and

8-30

We may as well assume that

for

i >> 0.

But then it

is easy to find a chain of simple augmentations (for instance, one can use Lemma 8.12 by considering the gauges

Of

course, all these are simple augmentations at some by case 3, the maps

so

are all injective.

It is now easy to derive (8.24.2).

We have an exact sequence

of complexes:

(8.24.1) tells us that this gives us short exact sequences:

The following result describes the cohomology of

in

a special case. 8.2 5 Proposition.

Suppose that

A

is as in (8.1), that

is smooth and proper, and that all its Hodge groups are p-torsion free. Then if ing

is any

decreas-

function, there is a canonical isomorphism: where

F

denotes the Hodge

filtration. Proof.

Thanks to Grothendieck's fundamental theorem, we can work

with the scheme Y rather than its p-adic completion.

There is a

8-31

natural morphism of filtered- complexes: and hence a morphism of spectral sequences:

Let

and observe that

morphism.

is an iso-

Since taking cohomology commutes with flat base change,

it follows that Now the map

is an isomorphism for all is just the obvious one: and since

s,t,r .

is p-torsion free,

this map is an isomorphism onto

In particular,

is also p-torsion free for all

s,t . Furthermore,

Hodge theory tells us that all

s,t,r.

for

By induction on

r we see that

and

are torsion free and that for all

r .

Thus, the map

induces an

isomorphism:

It follows that

is injective, and (by induction on

s),

induces an isomorphism

Taking If

s = i, one has the proposition. e

D

is tame, (8.18.4) shows that

and hence depends only on X = Y^. relative Frobenius

Moreover, (8.20) tells us that

induces a map (in the derived category):

8-32

Since this implies that on cohomology, maps

provided, of

course, that

e

is tame.

really is essential.

Let me remark that tameness of e

Indeed, it is easy to see that if the above

"divisibility" held for all ε , then (equivalently, in fact), and this is false, in general. It is nonetheless apparent that there should be some relation between the Hodge filtration and

F x / S , and from (8.23.5),

we can also expect the conjugate filtration

F

o n ^associated

the spectral sequence play a role.

to

to

Amazingly, Mazur's theorem asserts that

determines the (mod p) Hodge and conjugate spectral sequences, (with suitable hypotheses on X).

8.26

Theorem.

Suppose

is smooth and proper, where (S,I,Y)

is a (torsion free) p-adic base (8.1).

Let

be the natural map (reduction mod p), and let relative Frobenius

be the map induced by (8.2.1).

the Hodge spectral sequence of and that

is a flat

Assume that for each is degenerate at Then:

E^ ,

8-33

onto

maps

(8.26.2)

(8.26.3)

The diagram below commutes:

(The diagram makes sense because of Lemma 8.27, which tells us that the Hodge and conjugate spectral sequences degenerate suitably.) Proof.

Actually, various sets of hypotheses and conclu-

sions are possible.

For instance, (8.26.1) and (8.26.2) hold

assuming only that conjugate spectral sequence of

is p-torsion free and that the is degenerate at Ej.

in the lemmas which follow, we assume only that

Thus,

X/SQ is proper

and smooth, and state the additional hypotheses as we need them. If

X

does satisfy all the hypotheses of the theorem, the

base changing theorem for crystalline cohomology and the flatness assumption show that

and in

particular, the latter is locally free.

Therefore, X/SQ satis-

fies the hypotheses of the following lemma: 8.27

Lemma.

Suppose

Hodge spectral sequence of each Then:

is a flat

and that the

is degenerate at E^, for

8-34

8.27.1

The Hodge spectral sequence of

E^, consists of flat

is degenerate at

and commutes with arbitrary

base change

8.27.2

The conjugate spectral sequence of

at EJJ consists of flat

is degenerate

and commutes with arbi-

trary base change.

Proof.

Fix an integer

and consider the following

three statements: The map

is an

isomorphism. is flat. is flat. For

is trivial and

by induction on

k.

is given.

First observe that

Indeed,

is flat for all

Let us proceed implies

i, and hence its forma-

tion commutes with arbitrary base change

Consider the

diagram:

The base changing we just established implies that

a

is sur-

jective; the degeneracy of the Hodge spectral sequence of implies that

is surjective.

This implies that

8-35

Y

is surjective.

that

The theorem of exchange [EGA] then implies is locally free and commutes with arbitrary

base change, so that we have (c^). and

Observe that the above implies that

surjection to £

Let us next deduce

and so by Nakayama's lemma,

is surjective.

We obtain a short exact sequence:

follows immediately.

Assumption , and

imply all

e induces a

says that and the above sequence

The statements (a), (b), and (c) are thus valid for k.

Taken together, they imply (8.27.1).

Now we use the Cartier isomorphism:

Since

(via the absolute Frobenius of

k(s)) we see that Thus, has the same dimension as Counting dimensions of each

shows

that the conjugate spectral sequence

X(s)/k(s) degenerates at E^.

Now this spectral sequence

is, after renumbering, the spectral sequence of the canonical filtration [1,1.4] T.

on

Moreover, an easy argument using

the Cartier isomorphism shows that the complexes

T.

consist of

coherent and flat -modules. Therefore the same argument as before can be applied to T. , and (8.27.2) follows. •

8-36

Theorem (8.26) will follow from the local theorem (8.20) applied to some simple gauges, and from the calculus of cogauges (8.23) ff.

8.28

Let us introduce the following notation:

er(i) = {1

if

i < r, 0

n r (i) = ier(i) +i ς Γ (ϊ) = {0 M

if

if

i > r}

i > 0, 0

if

if

i < 0}

i < r, i-r if i > r}

= Ln K U X / S A 0 X / S

We have a picture:

ζ + r

(8.28.1)

/

"r \_

Notice that

e

is a simple augmentation of

This is (essentially) the only case of

ε

at

r .

a simple augmentation of

gauges in which we can find a global analogue of the diagram (8.10), expressing the behaviour of tions.

Ψ

under simple augmenta­

I find it convenient to write triangles as short exact

sequences.

8.29

Lemma.

There is a natural isomorphism of triangles:

8-37

Proof.

I claim that there is a canonical isomorphism:

where

is the natural map.

recall that if

Indeed,

is represented by

Now Since for any abelian F, this proves the claim. is exact.

Clearly

Thus, we have canonically:

Plugging this into the proper place in the triangle induced by applying

to the exact sequence:

you get the top triangle of the lemma. from (8.23.6).

The bottom one follows

To check that the diagram commutes, we are re-

duced to the local calculation (8.10).

•

8-38

8.30

Lemma.

8.30.1

Suppose that

is p-torsion free.

The map

is

surjective. 8.30.2

The image

is

the inverse image of 8.30.3

If, additionally, the Hodge spectral sequence of generates at is injective.

Proof.

For

de-

the map

r = 0, the first statement is a consequence of

the base changing formula,

and the other statements are trivial.

There is an exact ladder:

This diagram proves (8.30.1), by induction on ately implies (8.30.2) because

r . This immedi-

hence

If the Hodge spectral sequence of degenerates a t E ^ , one sees from the diagram that the maps are all surjective, hence (8.30.3) follows by induction.

•

Thanks to the local theorem (8.20), we have an enormous diagram:

8-39

(8.31)

Consider the hypotheses: (a)

is p-torsion free.

(b) The conjugate spectral sequence of

(c) The Hodge spectral sequence of

8.32

Lemma.

is degenerate

is degenerate at E^.

Hypotheses (a) and (b) imply (8.26.1) and (8.26.2)

of the theorem.

Proof.

Proposition (8.24) tells us that if

gauge, the map image

and that

n

is any co-

is an isomorphism onto its is compatible with the lattice

operations. To prove (8.26.1), let

denote the image of

by Lemma (8.30) is the inverse image of

which We

8-40

have to prove that the

and

M r have the same image in

i.e., that diagram (8.31).

Since

Contemplate and

are isomorphisms and

injective,this is equivalent to:

y is

But

To prove (8.26.2), first note that and So it suffices to prove that

maps

(Recall that the image of

onto

in

is

But (8.2 3.5) gives us an exact sequence:

Since Proposition (8.24) implies that the the

are surj ective.

8.33 Remark. of

are injective,

•

Assuming only that the conjugate spectral sequence

is degenerate at E^, it is still possible to prove part

of (8.26.1) and (8.26.2):

8.33.1

it,

8.33.2

If some

then y

for

such that

Since this remark seems useless, we do not include its proof.

8-41

The following lemma completes the proof of Theorem (8.26).

8.34

Lemma.

Proof.

Moreover,

Hypotheses (a), (b), and (c) imply (8.26.3).

We have a commutative diagram of triangles:

is contained in the image of the injective map

so that

factors through

(8.32) that

Recalling from

we obtain a commutative:

It is clear that Theorem (8.26) implies that the relative Frobenius morphism the (mod p) Hodge filtration of of

determines and conjugate filtration

assuming the stated degeneracy and torsion hypotheses.

Even without these hypotheses we can find a relationship, in

8

the form of inequalities, between the p-adic divisibility prop­ erties of

Φ

and the Hodge numbers of

X.

For simplicity, we restrict attention to the case in which S

is the formal spectrum of the p-Witt ring W of a perfect

field k, and X is smooth and proper over S n = Spec k . Then the absolute Frobenius endomorphism F„ of X induces a

σ-linear

endomorphism T of r morphism of

H* . (X/S), where σ is the Frobenius autocrxs Moreover, it is easy to see that

W,

H* . (X'/S) = H* . (X/S) 8 W , and crxs crxs g zation of

T.

duality that

Φ

is the obvious lineari-

It is a well known consequence of Poincare Φ

(and hence

F)

is injective modulo torsion —

and we shall indicate another proof of this fact below. Killing torsion, we thus ket what Mazur calls and

F-crystal,

and its associated span.

8.35

Definition.

An "F-crystal over W" is an injective

σ-linear endomorphism

T

of a free finite rank W-module

M.

A "span over W" is an injective W-linear map of free finite rank W-modules of the same rank.

If

T: M •*• M

its associated F-span is its linearization

is an F-crystal,

M 8 W •+• W.

The span of an F-crystal measures its p-adic divisibility. Precisely, if

,c

M —*M

is any span, there is a direct sum

co

decomposition ranks

e

of M

M = S M i=0

CO 1

,

such that

1

Im(M') = β P M i=0

1

. The

then determine the span up to isomorphism.

They are called the "Hodge numbers" of the span. In order to state the inequalities between the Hodge num­ bers of the crystalline span:

8-4 3

and the Hodge numbers of X/k, it is convenient to introduce the so-called "Hodge polygon" defined by a sequence (a0, a1,...) of nonnegative integers.

This is the continuous graph consist-

ing of the straight line segment of slQpe

0 over the interval

[0,ag], of slope 1 over the interval over

and slope

The "Hodge polygon of

i

(in degree n)

means the Hodge polygon determined by the numbers the "Hodge polygon of means the Hodge polygon determined by the Hodge numbers

of

its associated span. 8.36 Theorem. of

If

X/k

is smooth and proper, the Hodge polygon

lies on or above the Hodge polygon of

(In particular, the endpoint

X/k.

of the first lies to the left of

the endpoint of the second.)

Proof.

To prove the theorem, and perhaps to give some

insight into its meaning, it is helpful to baldly list the inequalities which it asserts.

8.37

Lemma.

If (ag,a^...) and (bg,b^...) are sequences of

nonnegative integers defining respective Hodge polygons A and B, then

B

lies on or above

A

iff for all

I(m): Proof.

Suppose the inequalities

Observe first that the domain of

B

I(m) hold for all

m.

is contained in the domain

8-44

of A,

Indeed, this is trivial

unless there exists a in that case

k

such that

and

I(m) implies that whenever .

the limit as

m

Taking

approaches infinity, we see that for any

n, as required.

Now to prove that B lies over A, it suffices to check that each of the endpoints of the line segments comprising B lie above A, since the region above A is convex.

Suppose (i,B(i))

is such an endpoint,

for some

B(i)

To calculate

previous paragraph that there exists an

n, and

A(i), note from the m

with

Thus the desired inequality

i reduces to , which is an immediate

consequence of

I(m).

The proof of the converse is similar —

investigate the consequences of the fact that the breakpoints of

A

8.37.1

lie under

Remark.

B.

It is perhaps more natural to work with the

partial sums

, for all

m .

Then if

, one sees from the above that for all

8.37.2

m .

is a span with Hodge numbers

it is trivial to observe that

if

e'^e"'".

8-45

length It is now clear that the following lemma proves Theorem (8.36).

3.38

Note that it also implies that

Lemma.

Let

(torsion), and let

image in for any

really is injective.

has finite length.

be the Moreover,

m,

Proof. We leave the first statement as an exercise for the reader - use (8.28) and the method of the following proof of the second statement. Use the notation (8.28), and notice that We have a commutative diagram:

Theorem (8.20) implies that the image of

y in M

is exactly

and the diagram implies that the image of is contained in the image of

q

in

, M1

Thus, we get an induced map from to

, necessarily surjective.

Therefore it suffices to prove that

has

length so the statement is trivial. proceed by induction on

m, using the exact sequence:

We

8-46

by (8.23.5).

Since the length of this is less than or equal to (by the Cartier isomorphism), the desired

inequality follows immediately from the induction assumption. The reader may wish to note that we have in fact proved a slightly stronger inequality than claimed, which he can work out according to his needs. Katz's conjecture is an immediate consequence of the above result.

Let me remark that this conjecture (inspired by Dwork's

fundamental work) is what led Mazur to his investigations.

To

express it, recall that the Newton polygon of an F-crystal T: M

M

over

W

is defined as follows:

Choose a basis for

M, and express

T

as a matrix, as if it were linear.

The

eigenvalues of the matrix depend on the choice of basis, but their p-adic ordinals do not.

For each

the number of eigenvalues with ordinal

be r , and form the con-

tinuous graph which begins at (0,0), with slope interval of length

r

over an

, arranged in increasing order of

r .

This (convex) graph is called the Newton polygon of the F-crystal T: M •+• M.

It is worth remarking that it depends, in fact, only

on

where

is the fraction field of

In fact a theorem of Dieudonn£ the Newton polygon of

and Manin [

3 ] asserts that

classifies it up to isomorphism.

Katz conjectured the following:

8-47

8.39 M

r

Theorem.

If

X'/k

is smooth and proper, and

H* ris (X/W)/(torsion), then the Newton polygon of (M,F*)

lies on or above the Hodge polygon of

Proof.

X/k .

Thanks to the previous result, we can follow Mazur

and reduce to an elementary calculation in linear algebra:

8.40

Lemma.

The Newton polygon of an F-crystal lies on or above

its Hodge polygon, and both have the same endpoint. Proof.

There is another way to calculate the Newton polygon

of an F-crystal which is often more convenient: representative of

T

Choose a matrix n

as above, and let p(X) = X + a ^ " +•••+an

be the characteristic polynomial of the matrix. (i, ord (a.)); then the Newton polygon of of this graph.

1

e ^ ,...

.

is the convex hull

(We leave this verification for the reader.) M 1 c—*M

Now suppose that the span 0

T

Plot the points

of

It is clear that the span of

T

has Hodge numbers 1

A T

is A 1 M* -+A 1 M,

and it is easy to calculate the Hodge numbers of this span.

In

particular, the index of its first nonzero Hodge number is clearly „ 22+· · - + (J-De^" 1 + j[i-(e°+-««+e:i"1)] e1, 2e + "if

1

value

(2)

is obvious.

To prove that

(2) =» (1), we are reduced to checking that the map is flat complex representing

an

isomorphism.

M, this is obvious.

(1) => (3), we are reduced to:

Taking a

To prove that

B. 9

"that

if

-module

t

for every

In particular,

From the long exact sequence:

and induction, we see that

for

so that

This proves (B6.1); the proofs of (B6.2) and the first part of (B6.3) are similar. Let us verify that does.

has finite tor-dimension if

Choose a flat and flasque

that

as in (B2.1), so

D; since it is still flat, for an arbitrary A-module

that there exists an i < m

and all

erated.

m

M.

such that

We must show for all

M, and we can assume that

M

is finitely gen-

Then (B2.2) tells us that But (B6.3) says that

D.

has finite tor-dimension, so there exists an

that all

for all n.

sequence:

and hence

Since the complex

m

such

i < m

and

is flasque, there is an exact

B. 10

We now come to the main finiteness result: B.7

Proposition.

Suppose

and

is quasi-consistent

Then:

B.7.1

The inverse systems

B.7.2

The natural maps

satisfy

ML . are isomorphisms.

B. 7. 3

Proof.

By

the

previous

lemma

It

is quite easy to prove the proposition if uniformizing parameter

.

cohomology modules

is a DVR with

Then (1) is automatic, since the are Artinian.

one concludes easily that and complete.

A

is

(2) follows, and then -adically separated

Since we have hence is finitely generated, hence

so is The general proof is a bit more complicated. plying (B.5), which allows us to replace D by Since

M

lies in

, where

by (B.6.1) and (B.1.6), we

can choose a flat complex L" representing M. complex S(L') represents

Begin by ap-

Then the flasque

, hence

represents

In particular, the natural map is a quasi-isomorphism. Moreover, since finitely generated projective is a noetherian A^-algebra that

we can choose a complex

of

-modules representingD n , and if represents

is finitely generated, for all

so i .

Since

L'

B. 11

is flat,

, so that, in particular,

is a finitely generated

, for all

i.

This proves

that L" satisfies the hypotheses of the following Lemma, which therefore will prove the Proposition.

B.8

Lemma. a)

Suppose L* is a complex of flat A-modules such that

The natural map

b)

is a quasi-isomorphism. is a finitely generated

Then: B.8.1

The inverse systems

B.8.2

The natural maps

B.8.3

Each

Proof.

satisfy

M.L. are isomorphisms.

is a finitely generated A-module.

This follows, I believe, from

However, I prefer to give a slightly different proof.

Let us

begin by recalling the following easy result, of which the above is "the derived category version". Step 0:

Suppose F" is a filtration on the A-module M which is

compatible with the J-adic filtration, i.e. all

k and v .

(a) (b)

Then:

has a natural If

grjA-module structure.

is generated as a

then (c)

If M

for

for all

module in degrees k

gr^M is finitely generated as a is F-adically separated, then

erated as an A-module.

M

and module and if is finitely gen-

B .12

Fix

n

and consider the filtration F" on the complex

given by

We

have a spectral sequence

The terms

with abuttment

all are

and the dif-

ferentials are Step 1; is finitely generated as a

is (as a group) a directsummand, consisting of those terms of degree > n.

Hence

is a submodule of the finitely generated

grjA-module

This proves that E^ is finitely generated, hence so is

, hence

Proof.

For each

, and let Then

is finitely generated.

be the image of be the preimage of forms an ascending chain of

hence is eventually stationary. for

Let

, we have Step 2.

This filtration

is compatible with the J-adic filtration, and finitely generated

Proof.

-submodules,

This implies that

Since the same is true for

Step 3:

if

For

module.

m j> n, we have an exact sequence: hence a diagram:

is a

B.l 3

It follows from this that

i.e. Hence

a finitely generated

Step

module.

The inverse systems

The statement follows.

satisfy

ML .

(This proves

(B8.1).) Proof.

Suppose

is generated in degrees