Ll. AMS 78 (1972), Societe Mathematique de France Asterisque 63 (1979) p. 113-164
:imates. Ann. of rface. Ann. Sc.
~bra:i.c surfaces. SLOPE FILTRATION OF F-CRYSTALS
3.revitch theorem. by i.eudonne modules. Nicholas M. Katz 3-135. (Princeton) s Notes on crys Jgus, Mathematical
orphisms of This paper is devoted to the systematic study of the variation of the Hodge • 40 (1976), 1976). and Newton polygons of an F-crystal when that F-crystal moves in a family. As
r)ques en carac such, it constitutes a natural sequel to my report [6] on Dwork's pioneering Topologia investigations of such variation. However, I have tried to make this paper self-
orie des inter contained and accessible to non-specialists. aire de geometrie Notes in Math. Some of the results are new, and interesting, even in the "classical" case of
F-crystals over perfect fields. I have in mind particularly the "basic" and
"sharp" slope estimates ( cf l. 4, l. 5) and the "Newton-Hodge" decomposition
(cf 1.6). These "pointwise" results are in fact the key to all the "global"
results given in 2.3- 2.7 .
Special thanks are due to Arthur Ogus for suggesting the possible existence .e Paris-Sud ay of the Newton-Hodge decomposition. , bat. 425 (France)
113 N.KATZ
I.
Table of Contents For any perfecl
ring of Witt vectori
I. F-Crystals over perfect fields
1.1 basic definitions
1.2 Hodge polygons the absolute Froben, 1.3 Newton polygons of a cra-F-crystal 1.4 relations between Hodge and Newton polygons; the basic slope estimate . I' 1.5 the sharp slope estimate I generated W(k)-mod,J 1.6 the Newton-Hodge decomposition I which induces an au~' i f: (M,F) ---7- (M,F') I II. F-Crystals over F -algebras p I The category of cra~; 2.1 basic definitions cra -F-crystals by keel·I 2.2 the case of a perfect ground-ring 1 ZL -modules , over 2.3 Grothendieck's specialization theorem p :z:~ ' I 2.4 Newton-Hodge filtration over smooth rings of F-crystals whic,i 2.5 splitting slope filtrations over perfect rings The exterior ~I 2.6 isogeny theorems over curves; Newton filtrations over curves (AiM,Ai(F)) with unll 2.7 constancy theorems over k[[T]] and over its perfection i' Ai(F) defined by •! r ,I...
For i 0 , but
The iterates of 1
(M,Fn) , n = 1,2, ..• 1
(1.2)
integers defined as
M of maximal rank,
exist W(k)-bases
114 SLOPE FILTRATION OF F-CRYSTALS
I. F-Crystals over perfect fields
(1.1) Basic definitions
For any perfect field k of characteristic p > 0 , we denote by W(k) its
ring of Witt vectors, and by
o :W(k) ~ W(k)
the absolute Frobenius automorphism. For any integer a ¥ 0 , we have the notion
of a oa-F-crystal over k , namely a pair (M,F) consisting of a free finitely ~ basic slope estimate generated W(k)-module M together with a oa-linear endomorphism F:M---+ M
which induces an automorphism of Mgz m A morphism of oa-F-crystals p p f:(M,F)---+ (M,F') is a W(k)-linear map f:M---+ M' such that fF = F'f
The category of oa-F-crystalsup to isogeny is obtained from the category of
oa-F-crystals by keeping the same objects, but tensoring the Hom groups, which are
~-modules,p over ~p with m p . An isogeny between oa-F-crystals is a morphism of F-crystals which becomes an isomorphism in this new category.
The exterior powers of a oa-F-crystal (M,F) are the oa-F-crystals
3 over curves (AiM,Ai(F)) with underlying module A~(k)(M) , and with oa-linear endomorphism orfection Ai(F) defined by
For i 0 , but (M,F) ¥ 0 , we define (A 0 M,A 0 (F)) to be (W(k) ,oa)
The iterates of a oa-F-crystal (M,F) are the oan_F-crystals
(M,Fn) , n = 1,2, ••.
(1.2) Hodge polygons 2 The Hodge numbers h0, h1 , h , of a oa-F-crystal (M,F) are the
integers defined as follows (cf [9]). The image F(M) is a W(k)-submodule of
M of maximal rank, say r , so by the theory of elementary divisors, there
exist W(k)-bases {v , .•. , vr} and {w , ... , wr} of M such that 1 1
115 N.KATZ
which pr a. J. F(v.) p w. stract H J. J. 1 I with integers I of Giv I These integers are called the Hodge slopes of (M,F) . The Hodge numbers ! the Hodg!
(M,F) are defined by (~) int
Thus we have (as fol4: (r = rank(M)) {vjA .. Ji 1 :1 . i 1 M/F(M) ~ ~ (W(k)/p W(k))h Th~ i>O defined I: i' i. Notice that we have the elementary interpretation:
0 for i < A <===> 1: 1.2.1
A ,.li F-Omodp i.e. F(M) c p~ !' and the!
0 for i > B <===> !i 1.2.2
B p acr-a-linear V:M---+ M such that FV VF
According to a marvelous theorem of Mazur [9], these "abstract" Hodge numbers sometimes coincide with more traditional Hodge numbers. Thus let .X be a projec- j-' i are H (X,OX/W(k)) tive smooth W(k)-scheme, all of whose Hodge cohomology groups assumed to be free, finitely generated W(k) modules, whose ranks we denote Let X be the projective smooth k-scheme obtained from X by reduc- hi,j (X) 0 Then for each integer j ~ 0 , the crystalline cohomology groups tion modulo p Hj (X ) are free finitely generated W(k)-modules, given with a a-linear F cris o
116 SLOPE FILTRATION OF FCRYSTALS
which provides a structure of o-F-crystal. Mazur's theorem asserts that the ab-
stract Hodge numbers of these o-F-crystals are given by the formula
hi,j (X)
0 the Hodge slopes of the ith exterior power (AiM,Ai(F)) , 0 < i < r , are the (:) integers a. j Jl' ... ' i (as follows immediately from computing the matrix of F in the bases {vj /\, ...rw. } and {w. f\ .. ,tw. } 1 Ji Jl Jj_ The Hodge polygon of (M,F) is the graph of the Hodge function on [O,r] ·I defined on integers 0 < i < r by I I if i 0 1 if 1 < i < r I I and then extended linearly between successive integers. If "'e define A ord(F) greatest integer A with F =: 0 mod p least integer A with hA(M,F) ~ 0 B p then we have ract" Hodge numbers et X be a projec- ' Hj(x,~i/W(k)) are I The Hodge polygon thus looks like .nks we denote I . from X by reduc- Le cohomology groups I ;h a a-linear F I 117 N.KATZ But the direct ' I Kos zul filtrat i i Ai(F) - 0 mod pi ) a+ b i , whi ,I (h0 ,o)~ SlOPE 0 ) ( 0,0 ~~ In fact, length h length h f ~ F-crystals on 1, I I. F-crystals whidl,, 0 i 1 2 . i) at which the Hodge polygon changes slope I h + ... h '~ +2h + ... +lh ' The points ( pursue that po~i f, are called its break-points. I I tion, which we li The Hodge polygon is not at all an isogeny invariant, as simple examples show. tion. ,. The only general result I know about its isogeny-behavior is the following trivial (1. 3) Newton "specialization" property. l The Newto: I I Lemma 1.2.3. rational numbe I Suppose we have an exact seguence of aa-F-crystals .I 'I I II II I defined in anylj Then the Hodge polygon of the direct sum (M $M , F $F ) lies above the Hodge 1 2 1 2 Pick an J1~ polygon of (M,F) . aa-F-crystal ojj rank(M), we have !I ~· E~uivalently, we must show that for 1 ~ i < r li obtained from number A , wr Now we have j $ Aa(Ml)®Ab(M2) over k' def:i.Ji a+b=i t: I!I so that min (ord (Aa(F )®Ab(F )) . 1 2 1: a+b=i 118 SLOPE FILTRATION OF F-CRYSTALS But the direct sum ~ Aa(M )~Ab(M ) is exactly the associated graded of the a+b=i 1 2 Koszul filtration (by "how many m •s") of AiM . Thus any congruence 1 Ai(F) _ 0 mod FA implies the same congruence for each of the Aa(F1 )~Ab(F 2 ) a+ b i , which is to say that we have if a + b i In fact, Mazur's theorem strongly suggests the desirability of studying l F-crystals only up to "Hodge-isogeny", i.e. only regarding as equivalent two 'I F-crystals which have the same Hodge polygon and which are isogenous. We will not 50n changes Slope pursue that point of view here, except in so far as the "Newton-Hodge" decomposi- tion, which we will discuss further on, may be regarded as a step in that direc- imple examples show. I tion. e following trivial (1.3) Newton polygons The Newton slopes of a cra-F-crystal (M,F) are the sequence of r rank(M) rational numbers ) . defined in any of the following equivalent ways. above the Hodge Pick an algebraically closed overfield k' of k , and consider the cra-F-crystal over k' M), we have I (M ~ W(k') , F~cra) W(k) obtained from (M,F) by "extension of scalars". For each non-negative rational number A , written in lowest terms N/M , we denote by E(A) the cra-F-crystal over k' defined by \ E(A) )) . \ 119 N.KATZ TI endomorp According to a fundamental theorem of Dieudonne (cf [8]), the category of ( oa-F-crystals up to isogeny over an algebraically closed field k' is semisimple, and the E(A)'s give a set of representatives of the simple objects in this An equiv! category. Thus we can write the exis j "matrix" with a unique finite sequence of rational numbers N1 /M1 ~ N2/M2 ~ ... , EMi = r . The Newton slopes of (M,F) are defined to be the sequence of r rational numbers repeated M repeated times, ... ). 1 For each rational number A , we define i.e. mult(A) From the above explicit description of the Newton slopes, it is obvious that Eit~ of the i 1' L mult(A) = r (r rank(M)) AE(!l, ji 1.3.1 for each A , the product A mult(A) lies in ~ ; in l; i' particular the Newton slopes admit r! as a common denominator. 'I and that:~ For the next characterization of the Newton slopes, we choose an auxiliary integer N > l which is divisible by r! , r = rank(M) , and consider the dis- ii II I I Thei crete valuation ring interpreJJ,, W(k')[l/N]. R W(k' )[X]/(~-p) 1.3.2 X . For any We extend a to an automorphism of R by requiring that o(X) I I j~ rational number A with NA £ ~ , we may speak of 1. 3.3 A p the image of XNA in R . Let K denote the fraction field of R . Again by Dieudonne , we know that which transforms under the oa-linear M ® K admits a K-basis vJ(k) 120 SLOPE FILTRATION OF FCRYSTALS endomorphism F®cra by the formula category of A. d k' is semisimple, l p e. l objects in this I An equivalent, and for us more useful, characterization of the Newton slopes is by I the existence of an R-basis of M ® R with respect to which the W(k) A. "matrix" of F®cra is upper-triangular, with p l along the diagonal: ! ' JM l:Mi r . 2 2.· •. , = entries) >f r rational numbers in R A ;ed M times,.··). r 2 (0 p i.e. A. . l I - p ui mod L RuJ. j is obvious that Either of these last two descriptions makes it obvious that the Newton slopes o f th e l. th ext erlor . power (AiM,Ai(F)) of (M,F) are the(:) numbers A, + ... + < r in Jl .on denominator· and that the Newton slopes of the nth iterate (M,Fn) of (M,F) are choose an auxiliary td consider the dis- , I The last description of the Newton slopes makes clear the elementary interpretations I l. 3.2 all Newtons slopes A. of (M,F) are 0 if and only if I l ;(X) X . For any I F is a cra-linear automorphism of M . l. 3.3 all Newton slopes A. of (M,F) are > 0 if and only if l F is topologically nilpotent on M , i.e. iff and only ne ' we know that if Fr (M) c pM where r = rank(M) • 121 N. KATZ Lemma 1. 3. 4. 1'. The Newton polygon of (M,F) is the graph of the Newton function on [O,r], Su ose we defined on integers 0 < i ~ r by Newto~(i) =least Newton slope of (AiM,Ai(F)) I ·r Then the Newton if i 0 Newton o on o if Proof. Extendi and then extended linearly between successive integers. In terms of the distinct I ~ semisimplici ty Newton slopes ~i of (M,F) together with their multiplicities mult(~i) , I arranged in strictly increasing order ~l < ~ 2 < ••• , the Newton polygon looks I Example. Over ~] I like The Newton i ! •I I \ I "'="''. In nj make use of the I. tend Cia to an~~ tion we will gii! ' volves no exten1: length=mult(~ 1 ) of iterates). ~I,, depends only onll Newton polygon changes slope are called its break-points. From the earlier noted a . (Of course, are all integers, it follows that the break- By Manin [: fact that the products ~. mult(~.) l l 2 points of the Newton polygon are always lattice-points in m , i.e. they have kC F a , then : p p-adic ordinals integer coordinates. I• By its very construction, the Newton-polygon is an isogeny invariant (indeed In terms of a ! over an algebraically closed field it is~ isogeny invariant). In contrast to cra -F-crystal the case of Hodge polygons, we have (M,F) need 122 SLOPE FILTRATION OF FCRYSTALS Lemma 1. 3. 4. tion on [O,r], Su:p:pos~we have an exact seguence of oa-F-crystals Then the Newton :polygon of the direct sum (M ~M ,F ~F ) coincides with the r 1 2 1 2 I Newton :polygon of (M,F) . I Proof. Extending scalars, we may suppose k algebraically closed. By Dieudonne's of the distinct I semisim:plicity theorem, our exact sequence splits in the "u:p-to-isogeny" category . mult(J.I.) , • l I {!jED :polygon looks I Example. Over JF , take M = ~ ~:ll: I p p p The Newton and Hodge :polygons are / I• I Newton Hodge I Remarks. In our characterizations of the Newton slopes of a oa-F-crystal, we make use of the integer a , (not just of the automorphism cra), in order to ex- tend cra to an algebraically closed overfield of k However, in the next sec- tion we will give an "internal" characterization (cf 1.4.4) (i.e., one that in- volves no extension of scalars) of the Newton :polygon (in terms of Hodge :polygons of iterates). Consequently, the Newton :polygon of a given oa-F-crystal over k a at which the depends only on the automorphism a of k , and not on the auxiliary choice of a ;he earlier noted a , (Of course a as automorphism of k determines a unless k is finite.) a l that the break- By Manin [8], we know that if a is the identity on k, i.e., if L. e. they have kC F a , then the Newton slopes of a oa-F-crystal (M,F) on k are :precisely the p :p-adic ordinals of the eigenvalues of "F viewed as~ endomorphism of M". invariant ~(indeed In terms of a matrix (Fij) for F , this means that the Newton polygon of the In contrast to oa-F-crystal (M,F) coincides with the Newton polygon of the "reversed" character- istic polynomial det(l- T(F .. )) of the matrix lJ However, if cra # id. on k , then the Newton :polygon of a oa-F-crystal det(l - T(F .. )) , where (M,F) need not coincide with the Newton :polygon of lJ 123 N.KATZ (Fij) is the matrix expressing the action of F on some basis of M. Here is l an example, due to B. Gross. Over W with p 3 mod consider the 2 = 4 , p universally. J o-F-crystal of rank two with matrix I I I ~ R-basi 1 1- p (p+l)i) r ( (p+l)i p - 1 I As we may c The eigenvalues of this matrix both have ordinal 1/2 (since trace = 0 , det p). I a-linearly eQuivalent, via li li) , to the matrix • But this matrix is ( I I (: :) , I we get I and hence our o-F-crystal has Newton slopes {0,1} , not {1/2,1/2} . (1.4) Newton-Hodge relations; the basic slope estimate In this section we will discuss various relations between Hodge and Newton Remarks. polygons. Theorem 1.4.1 (Mazur) For any oa-F-crystal (M,F) , the Newton polygon is above the Hodge polygon. Both polygons have the same initial point (namely (0,0)) and the same terminal point (namely (r, ord(det(F))). ~· For any oa-F-crystal of rank~· the Hodge slope and the Newton slope 0 0 l coincide. Applying this remark to (A M,A (F)) = (W(k),cra) and to (ArM,Ar(F)) , Thus I we see that the two polygons begin and end together. To show that \ r for 1 ~ i ~ r , 1.4.2 it suffices to show that for each of the exterior powers of (M,F) , we have least Newton slope > least Hodge slope , .,I . (M i.e., we must prove I Let 124 SLOPE FILTRATION OF FCRYSTALS )f M . Here is 3ider the universally. But in terms of the matrix A .. of Fecra on MeR with respect to lJ ord(F) min (ord (A .. )) . p lJ i,j As we may choose the R-base so that this matrix is ace = 0 , det p). Al entries) Latrix p • in R Newton slopes \ .::_ ... .::_ \ • A ( r 0 p we get Al min (ord (A.j)) .::_ ord(p ) p l i,j Remarks. If we compare largest rather than smallest slopes, we get '[odge and Newton '(,, I' greatest Newton slope .::_ greatest Hodge slope , simply because the two polygons are both convex, and have the same terminal point, the Hodge polygon. i.e., they end like he same terminal Newton- ~,______/ ' Hodge the Nevrton slope 1d to (J{M,I{ (F)) , Thus denoting by A and B the least and greatest Hodge slopes and by Al and ;hat Ar the least and greatest Newton slopes, we have A < A < A < B 1.4.2 - 1- r- M,F) , we have As a by-product of this method of proof, we get the (1.4.3) Basic slope estimate Let (M,F) be a cra-F-crystal of rank r , and let A > 0 be a rational 125 N. KATZ ~· For any real number x , denote by {x} the "next higher integer", i.e., {x} = - [-x]. Then (M,F) has all Newton slopes > A if and only if for all in- as required. tegers n ~ 1 we have Corollary 1. 4. 4 ord (Fn+r-l) ~ {nA} The Newton fu i.e., iterates (M,Fn) We begin with the "if" part. Let A be the smallest Newton slope of ~· 1 (M,F) . Then the smallest Newton slope of (M,Fn+r-l) is (n+r-l)A1 , while by By the previous theorem, applied valid for hypothesis its smallest Hodge slope is ~ {nA} n+r-1 ' to (M,F ) , we have Proof. Since bothi 0 < i < r , (n+r-l)A ~ {nA} > nA for all n ~ 1 , thenj 1 to prove the form 1 i A ~ A as required. X= i is whence 1 and ' eqji Conversely, we must show that, still denoting by A the smallest Newton 1 ( fliM,fli (F)) and I slope of (M,F), we have the smallest Al Nj: n+r-1 I ord (F ) ~ nA1 . I 1: Extending scalars to R ( cf 1. 3.1 ff) it suffices to show By the basic slop~; I il of M®R , we have In terms of a suitable R-basis I from which the A. 1 (F®cra)(u.) = p u. + elt. of L Ruj . I l l j (N-i+l)A1 e p M@R • I 'l The graphs of the ~ u. span M®R , we find Since i r , and the l \ \ 126 SLOPE FILTRATION OFF-CRYSTALS ~r integer", i.e. , tly if for all in- as required. Corollary l. 4. 4 The Newton function of (M,F) is obtained from the Hodge function of the iterates (M,Fn) of F by the (archimedean) limit formula ~wton slope of 1 lim -Hodge (x) n Fn ·l)A , while by n->-oo 1 : theorem, applied valid for 0 < x < r rank(M) . Proof. Since both the Newton and Hodge functions are defined first on integers 0 < i < r , then interpolated linearly between successive integers, it suffices to prove the formula for x = an integer 0 < i < r . The formula, for (M,F) and x = i , is equivalent to (indeed term by term identical with) the formula for 1allest Newton Thus it suffices to prove universally that, denoting by Al the smallest Newton slope of (M,F) , we have :: ~.. f lim n-roo By the basic slope estimate, we have ( n+r-1 )Al' ~ ord (Fn+r-1) _> {n'Al } ~ nA, , 1 from which the required limit formula is immediate. Examples. Consider the cr-F-crystal over F with M Z $ ~ and F given by p p p the matrix 1 The graphs of the functions and - Hodge are n Fn 127 N. KATZ (1. 5) 0 1 h ,h , ... , as drawn : the highest is NewtonF , the middle is that of l:. Hodge for some n Fn n ~ 2 , and the lowest is that of HodgeF If instead we take M = ~ $ ~ F given by p p Proof. The '1 the "only if'l determinants'!I I 1 then the graphs of and of all ~ Hodge coincide: Suppose I· n F 2 n given a rati4; ~(2,1) (0,0) ~ jl I' 1 !: while the graph of n+l HodgeF n+l is ., 2 2 Because the f\',. i, !i _/(2,1) ( 0, 0 ) .------; I (1,2nn+l) Extending I suffices to The point of these examples is that the Newton polygon may or may not be I attained at some finite n , and that the seQuence of approximating functions need I not be monotone. The common features of the examples, namely that all approximants I At this p are convex polygons lying on or below the Newton polygon, but sharing its begin- \ ning and terminal points, are indeed common to all examples. This follows from I (M,Fn) , and the fact that l:. Newton = NewtonF . _I Theorem 1. 4.1 applied to n Fn I i.e., ( ( 128 T SLOPE FILTRATION OF FCRYSTALS f (1.5) Sharp slope estimate ' In this- section we will give a sharpening of the basic slope estimate, in which the "lag" term r- 1 is replaced by a sum of certain Hodge numbers. Sharp Slope Estimate 1.5.1 Let (M,F) be a oa-F-crystal, and A ~ 0 a rational number. Let 0 1 h ,h , ... , be the Hodge numbers of (M,F) Then all Newton slopes of (M,F) are > A if and onl;z if for all integers n > 1 we have ce for some , Fn n+ I hi i {nA} Proof. The "if" part is proved exactly as for the basic slope estimate. To prove the "only if" part, we first reduce it to a reasonable-sounding assertion about determinants, and then give an unpleasantly computational proof of that assertion. Suppose then, that Al is the least Newton slope of (M,F) and that we are given a rational A Because the function "ord" assumes only integral values, it suffices to prove that n+ I hi ord (F i nA A. A Extending scalars to a suitable R containing p as well as the p l ' it suffices to prove that r may not be n+ I hi ng functions need (F®cra) i At this point, we must observe that for ~real A > 0 , we have .ring its begin- s follows from n+ I hi ord (A i nA for n > 1 l = NewtonF . i.e.' HodgeF(n + I hi) > nA i 129 N.KATZ T To see that this is true, look at the Hodge polygon. It has a break-point at r, If we ( I hi , I i hi) , and from this point rightwards the slopes are all ~ {\} , so I F@cra , we g i<\ i<\ ( To prol that we have I the R-basisl HodgeF(n + I hi) > I ih. + n{\} ~ n\ I i<\ - i<\ l for all real n > 0 I The matrix Let us further observe that in terms of a suitable basis of .I I ' I I each of wholl l i: By the hypoj; an upper triangular matrix over R all of whose diagonal entries lie in the ideal pending iR of R. Lemma l. 5. 2 Let R be any commutative ring with 1, Ic R an ideal, and ~:R---+ R (Of course an endomorphism of R such that upper In-k , for any x E R , ~(x) E xR . Let M be a free R-module of finite rank, and let F:M ---+ M be a p-linear endomorphism of M , whose matrix relative to some R-basis {ui} of M is upper triangular, and has all of its diagonal entries in the ideal I Suppose that for some integer k ~ 0 , we have the congruences for all n ~ 1 Then we also have is left to for all n ~ 1 130 SLOPE FILTRATION OF FCRYSTALS T a ·reak-point at '( If we apply this to our R = W(k')[pl/NJ (J I M@R with ' all ~ {!.} , so F@cra , we-get the "sharp" slope estimate. To prove the lemma, let us denote by (Fij) the matrix of F relative to the R-basis {ui} of M F(u.) 0 if i > j ' F .. E: I . J._ J..,J.. The matrix of Fn is the product matrix of each of whose entries is a sum of products of the form n-1 ~ (F. . ) 1 n ' 1 n+l By the hypothesis made on ~ , each of these products is divisible by the corres- ;s lie in the ideal ponding product "without ~ " ',, 'I Fi i n' n+l , and ~:R--->- R (Of course this product vanishes unless i ~ i ~ ... ~ in+l, since (Fij) is 1 2 upper triangular.) So it suffices to show that each such n-fold product lies in In-k , for n > k Let J(n) denote the ideal generated by the n x n minors of (F. j) By J.., be a -linear hypothesis, we have J(n) c In-k for n > k , so it suffices to show that J(n-1) + •.. + In-lJ(l) +In, eal I Suppose F. . E: J(n) + I I 1 n' 1 n+l I whenever 1 ~ i ~ ... ~ in+l < r . Since the diagonal entries Fi,i lie in I , ' 1 it suffices to treat the case when i < i < ••• < in+l. This case follows I 1 2 I inductively from the following well-known determinant formula, whose verification l is left to the reader. I { 131 N.KATZ Theorem 1 I. Formula l. 5. 3 I Let I upper-triangular matrix (X. j=O if i > j) of Let (X. j) be an r x r l, l, oint of ! \) > 2 indeterminates. For each subset T c {1, ... ,r} of cardinality (M,F) T we define of two cr J 1.6.1 det(T) det xt t 0 v-1' v-1 the (v-1) x (v-1) minor indexed by :l l. 6.2 I; I! and we define il 'I In termsi: f(T) is form~~ 'I the tra11 Then we have the formula Piotod1 X det(T) (-l)#S • f(T-S) • TI s,s sE:S !I 1: ! I (1.6) Newton-Hodge decomposition In this section we give a fundamental decomposition theorem for cra-F-crystals, in terms of the interrelations between their Hodge and Newton polygons. I owe entirely to Ogus the idea that such a decomposition should exist. 132 SLOPE FILTRATION OF F-CRYSTALS T'( Theorem 1.6.1 (Newton-Hodge decomposition) ' if i > j) of Let~ (M,F) be a cra-F-crystal of rank r . Let (A,b) E ~ x Z be a break- v > 2 point of the Newton polygon of (M,F) , which also lies on the Hodge polygon of (M,F) Then there exists a unique decomposition of (M,F) as a direct sum 1. 6.1 Hodge slopes of (Ml,Fl) = {first A Hodge slopes of (M,F)} Newton slopes of (Ml,Fl) {first A Newton slopes of (M,F)} ~ (M2 ) = r- A 1.6.2 Hodge slop~s of (M2,F2) {last r -A Hodge slopes of (M,F)} 'f; Newton slopes of (M2,F2) {last r -A Newton slopes of (M,F)} In terms of polygons, this means that the Hodge (resp., Newton) polygon of (M,F) is formed by joining end-to-end the Hodge (resp. Newton) polygon of (M ,F ) with 1 1 the translate by ~,b) of the Hodge (resp. Newton) polygon of (M ,F ) . 2 2 Pictorially, we have ,s I I I t for :e and Newton I I .tion should l I ( I 133 N. KATZ T We give the proof in a series of lemmas. Lemma 1 , we mu show that L i Lemma 1. 6. 3 locally free, t Hypotheses as in Theorem 1.6.1, there exists a unigue F-stable W(k)-submodule 1 is necessarily M c M such that 1 first a Newton M is free of rank A , and M/M is free of rank r - A . 1 1 be too big). To verify if we put F = F/M , then the Newton slopes of (M ,F ) 1 1 1 1 are the first A Newton slopes of (M,F) . the PlUcker e~u 1 scalars, e.g., ,1 1: Lemma 1. 6. 4 R = W(k' Hl/NJII !I Hypotheses as in Theorem 1.6.1, and notations as in Lemma 1.6.1 above, put admits a K-basi! ,, F/M • Then we have a short exact seguence of cra-F-crystals ! 2 1: i, !, in which (M ,F ) and (M ,F ) satisfy the properties 1.6.1 and 1.6.2 of 1 1 2 2 I' the conclusion of Theorem 1.6.1. where ;,1 .s_ ... ,~ AA(M)®K , it islj Lemma 1. 6. 5 b + ... + A :1 = ;, AI The exact seguence of cra-F-crystals in Lemma 1.6.5 above admits a unigue 1 so by uni~uenesli splitting ,, e~uations. ii (M,F) It remains I polygon Proof of Lemma 1. 6. 3. We first use "PlUcker coordinates" to reduce to the case Dividing when A = 1 The hypothesis that the Newton polygon of (M,F) has a break-point at (A,b) and that the Hodge polygon of (M,F) goes through (A,b), is equi ,,,~, .. I valent to the hypothesis that the Newton polygon of (/M,l\.A(F)) has a break any such line point at (l,b), and that the Hodge polygon of (A~,AA(F)) goes through (l,b). n ~ 1 , there Admitting temporarily the truth of Lemma 1 for (A~,AA(F)) and the point automorphism. (l,b), we obtain a unigue AA(F)-stable line L in A~ , such that (L,AA(F)) all Newton has Newton slope = b . Therefore if there exists M1 c M of the sort re~uired in j. j, 134 SLOPE FILTRATION OFF-CRYSTALS Lemma 1 , we must have 1 =AA (M ), by unigueness of 1 . Conversely, if we can 1 show that 1 is of the form AA(M ) for some W(k)-submodule M c M with M/M 1 1 1 locally free, then M is uniquely determined by 1 (Plucker embedding!), M )le W(k)-submodule 1 1 is necessarily F-stable (since 1 is), and its Newton slopes are necessarily the first a Newton slopes of (M,F) (otherwise the Newton slope of (1,AA(F)) would r -A . be too big). To verify that 1 is of this form, it suffices to verify that 1 satisfies the Plucker equations, and for this we may first make any injective extension of scalars, e.g., from 1>/(k) to the fraction field K of a suitable ring I //. i: R = W(k' )[pl/N] of the sort considered in ~ But over such a K M@K , 6.1 above, put admits a K-basis e , ... ,er with respect to which the matrix of F is 1 oa-F-crystals and 1.6.2 of where Al < ,,, ~ AA < AA+l ~ ... ~ Ar are the Newton slopes of (M,F) , Inside AA(M)@K , it is now visible that there is a unique F-stable line of Newton slope But 1@K is also such a line, lmits a unigue so by uniqueness, we have 1@K = Kel''···AeA, whence 1@K satisfies the Plucker equations. It remains to treat the case (A,b) = (l,b) In this case, as the Hodge b polygon goes through (l,b), the endomorphism F of M is divisible by p luce to the case Dividing F by pb , we are reduced to the case (A,b) = (1,0); i.e., the case in has a break-point which zero occurs as a Newton slope of (M,F) with multiplicity one. We must (A,b), is equi- find an F-stable line 1 c M on which F induces an automorphism, and show that has a break- any such line is unique. For this, it suffices to show that for every integer es through (l,b). n ~ 1 , there is a unique F-stable line 1 in M/pnM on which F induces an and the point n automorphism. Because F has ~ as a Newton slope with multiplicity one, 2 all Newton slopes of A (F) are strictly positive, and hence we have he sort required in 135 N. KATZ must 2 if v ?._ rank(A M) In terms of a looks like i.e., if v ?._ n(~) . But all iterates Fv of F also have zero as a Newton slope, and hence all iterates Fv of F have zero as a Hodge slope, i.e., for v = 1,2,3, .... with !A the on mod p , but all 2 X 2 minors of For any v ?._ n(~) , we thus have Let us be This means exactly that for each v , the image of are = 0 mod pn ?_n(~) I I• ! (in matricial terms, at least one of the columns of the is a line I; matrix of Fv is not divisible by p , and all the other columns are congruent For this, it su i L as mod pn to W(k)-multiples of this column). By the definition of the n,v from one to th~i combinations o~i images, we have goes through ~~~ F(L ) for all v . 1 n,v !i :I Since the L for v ?._ n(r ) are ~. we must have •' n,v 2 in li !I F(L ) = L L for v ?._ n(~) . I n,v n,v+l n,v Because the Ne~l Therefore if we define L to be L for any v ?._ n(r ) , Ln is an F-stable haa "" Newton n n,v 2 j: line on on which F induces an automorphism. That Ln is the such M/p~ uni~ue I• I! line results from the fact that ·!. It now follows liii, •. 1 binations of t ! ,I so that .Ln must contain any such line, and hence be e~ual to any such line. This concludes the proof of Lemma 1.6.3 QED . We now turn to the proof of Lemma 1.6.4. By construction, (M1 ,F1 ) has as Newton slopes the first A Newton slopes of (M,F) , and therefore by ~.3.4) have a common 136 SLOPE FILTRATION OF F-CRYSTALS (M2 ,F2 ) must have as its Newton slopes the last r - A Newton slopes of (M,F) In terms of a basis of M adapted to the filtration M c M , the matrix of F 1 looks like b.ence all (: :) with A the A x A matrix of F on M , and ID the (r-A) x (r-A) matrix of 1 1 F on M . inors of Fv 2 2 oe image of Let us begin by showing that this matrix has the same Hodge polygon as does ::>lumns of the re congruent For this, it suffices to show that "elementary column operations" allow us to pass b.e 1 as n,v from one to the other, i.e., to show that all the columns of IB are W(k)-linear combinations of the columns of A • By hypothesis, the Hodge polygon of (M,F) goes through (A,b), and hence all A x A minors of (Ao\ m]3) are - 0 mod pb b in particular all A x A minors of {A,JB) are - 0 mod p Because the Newton polygon of (M,F) goes through (A,b) and because (M ,F ) 1 1 an F-stable has as Newton slopes the first a Newton slopes of (M,F), we have the unique such b det (/A) p x unit. It now follows by Cramer's rule that all columns of JB are W(k)-linear com- binations of the columns of A . Hence the matrices such line. ,F ) has as 1 1 by (1.3.4) have a common Hodge polygon. 137 N.KATZ Intrinsically, this means that (M,F) and (M1~M2 , F 1~F 2 ) have a common Hodge polygon. Therefore we have a partition {Hodge slopes of (M,F)} lJ {Hodge slopes of i=l,2 To split the pr A x (r-A) matrix X So we need only verify that the Hodge slopes of (M1 ,F1 ) are the A smallest " of a W(k)-linear era among those of (M,F). But the~ of the smallest A Hodge slopes of (M,F) is morphism of b (the Hodge polygon of (M,F) goes through (A,b)). So it suffices to see that the sum of all A of the Hodge slopes of (M1 ,F1 ) is b . For this just (A,b), and hence its Hodge recall that the Newton polygon of ends at i.e., polygon ends there as well. ~ In a basis of M adopted to We now turn to the proof of Lemma 1.6.5 i i.e., M c M , the matrix of F is 1 I I i.e., [ Because either are all As we have seen in the proof of Lemma 1.6.4 , the columns of E successive iterati W(k)-linear combinations of those of ~ , so we can write this matrix Remarks. If we Dieudonne module of such a group connected group, for some integral matrix ill • Let n denote the largest Hodge slope of A , and let m denote the smallest -m Since m ~ n by and p ID are integral. Hodge slope of ID . Then p~A-l pYAn -1 is topologically nil- Lemma 1.6.4,p-~ is integral. Notice that either -a has all Newton slopes > 0) ' potent (i.e., that the cr -F-crystal or that p-nD is topologically nilpotent (i.e., (M2 ,p-nF2) has all Newton slopes > 0), or possibly both; this is immediate from the inequalities 138 SLOPE FILTRATION OF FCRYSTALS have a common -.< Ar To split the projection of (M,F) onto (M ,F ) is equivalent to finding an 2 2 e A smallest X A x (r-A) matrix X , with entries in W(k) , so that ( ) , viewed as the matrix 1 opes of (M,F) is of a W(k)-linear cross-section M ---+ M of the projection M ---+ M , is a 2 2 ffices to see l morphism of cra-F-crystals. Matricially, this means For this just I hence its Hodge 1 i.e., l a XI) ;J'\Xa + JAm adopted to I i i.e.' I i.e., I -a -a X (pnA-lXp-9D)cr - ma . I 1 ,,1 n -1 are all Because either pYA or is topologically nilpotent, the method of 1atrix successive iterations leads to a unique solution of this equation. Remarks. If we apply this Newton-Hodge decomposition to the contravariant Dieudonne module of a p-divisible group, we recover the cannonical decomposition of such a group over a perfect field into the product of an etale group, a bi- connected group, and a toroidal group. denote the smallest ;e m .::_ n by ~ologically nil- ~n slopes > 0) , s all Newton slopes etale biconnected toroidal 139 N. KATZ on A up to i 0 the same F -algebras II. F-crystals over p with (2.1) Basic definitions which becomes In this section we recall the basic notions concerning crystals on arbitrary isogeny if and affine schemes in characteristic p > 0 (compare [2]). By an absolute test object, uv = pn = vu). we mean a triple (B,I,y) consisting of a p-adically complete and separated Suppose w ~ -algebra B , a closed ideal I c B with p E I , and a divided power structure B/I , i.e., together with a homomorphism of F -algebras A - algebra in p 0 A map of A -test objects f : (B,I,y;s)---+ (B' ,I' ,y' ,s') is I s : A ---+ B/I Similarly, gi 0 0 an algebra homomorphism f : B---+ C which maps I to I' , "commutes" with the u(¢) : M(¢) For any given divided power structures y,y' , and induces an A0 -homomorphism I B/I----+ B'/I' (for the given structures s,s'). endomorphism is rule which assigns to every A -test object A crystal M on A 0 I 0 \ (B,I,y;s) a p-adically complete and separated B-module, noted M(B,I,y;s) , and on A 0 which assigns to every map f: (B,I,y;s)----+ (B' ,I',y' ;s') of A0 -test objects on A 0 a B'-isomorphism such that M(f) the notion M(B,I,y;s) g B' ~ M(B' ,I' ,y' ;s') B in a way compatible with composition of maps of test objects. A crystal M is ¢:A ----+ 0 said to be locally free of rank r if for all A -test objects (B,I,y;s) , the a 0 a ¢ = ¢ B-module M(B,I,y;s) is a locally free B-module of rank r . A morphism of (2.2) u : M---+ N , is a rule which assigns to each A -test object crystals on A 0 0 I (B,I,y;s) a B-module map phism, the u(B,I,y;s) M(B,I,y;s) ---+ N(B,I,y;s) category of structure y in a way compatible with the isomorphisms M(f),N(f) . The category of crystals SLOPE FILTRATION OF FCRYSTALS on A up to isogeny is obtained from the category of crystals on A by keeping 0 0 the sam~objects, but tensoring the Hom groups, which are Z -modules, over ~ p p with a), • An isogeny between crystals on A is a morphism of crystals on A p 0 0 which becomes an isomorphism in this new category (explicitly, u N ----->- M is an ;als on arbitrary isogeny if and only if for some integer n > 0 , there exists v M ----->- N vri th :olute test object, uv=pn=vu). td separated Suppose we are given two JF -algebras, A and B , and a homomorphism p 0 0 ~d power structure If (B,I,y;s) is a B -test object, then (B,I,y;s·¢) is an 0 in 2: p A -test object. Given a crystal M on A , the "inverse image" crystal M(¢) 0 0 1ple (B,I,y;s) on B is defined by the formula 0 Jtructure s of Llgebras M(B,I,y;s¢) . (B',I',y',s') is Similarly, given a morphism u : M----->- N of crystals on A , its "inverse image" 0 ~ornmutes" with the u(B,I,y;s¢) . ephism For any JFP-algebra A , we denote by o : A ---+ A the absolute Frobenius 0 0 0 j endomorphism o(x) = xp , and by_ oa a > 1 , its ath iterate. By a 3t object oa-F-crystal (M,F) on A , we mean a locally free (of some rank r ) crystal M 0 11(B,I,y;s) , and I on A together with an isogeny F:M(oa) ----+ M . A morphism of oa-F-crystals 0 f A -test objects 0 on A f: (M,F) ----+ (M' ,F') , is a morphism f:M ----+ M' of crystals on A 0 0 I such that The category of oa-F-crystals up to isogeny, and f the notion of an isogeny between oa-F-crystals, are defined in the expected way. Given a oa-F-crystal (M,F) on A , and any homomorphism of JFP-algebras 0 crystal M is ¢:A ----+ B , the inverse image (M(¢) ,F(¢)) is a oa-F-crystal on B (because 0 0 0 B,I,y;s) , the a a 0 ¢ = ¢ • 0 for any homomorphism of JF -algebras). p morphism of h A -test object (2.2) Perfect rings. 0 When A is a perfect JF -algebra, i.e., when o:A ---+A is an automor- o p 0 0 phism, the ring W(A ) of Witt vectors of A provides an initial object in the 0 0 category of all A -test objects, namely (W(A ), (p), y ; s) . The divided power 0 0 gory of crystals structure y on pW(A ) is uniquely determined by the requirement 0 141 N. KATZ is the inverse of the isomorphism T the homomorphism s:A ---+ W(A )/(p) [ Theorem 0 0 W(A )/(p) ~ A obrained by sending a Witt vector to its first component. I Let 0 0 • ,1 Let /.. > Evaluation at this initial object provides an equivalence of categories between I (res Ne A and the category of p-adically complete and the category of crystals on 0 Spec(A ) 0 separated W(A )-modules. Given a homomorphism ~:A ---+ B0 of perfect 0 0 F -algebras, the construction M ~ M(~) on crystals corresponds to the con Corollar p struction on modules Let linear be M 1--r M @ W(B ) dfn the W(B )-module M(~) W(A ) o o 0 Hodge (rd 11 in which W(B) is viewed as a W(A )-algebra by means of W(~):W(A)---+ W(B) locally ori1 0 0 0 0 \, W(cr) :W(A ) ~ W(A ) , then the If we denote by (J the automorphism 0 0 Proof. category of cra-F-crystals on our perfect A is equivalent to the category of 0 powers pairs (M,F) consisting of a locally free (of some rank r ) W(A0 )-module M I. Hodge slo:fi together with a cra-linear map F:M---+ M which induces an automorphism of Because 11 M®z m . p p for cra-Fj' In particular, when A is a perfect field k , we recover the more mundane 0 to the ba ! notion of cra-F-crystal with which we were concerned in Chapter 1. il :jI (2.3) Grothendieck's specialization theorem. !I,, We now turn to the consideration of a cra-F-crystal (M,F) over an arbitrary ii ~:A ---+ k with k a perfect field, d JF -algebra A For any homomorphism 0 ii p 0 Therefore li (M,F) (~) is a cra-F-crystal over k . Its Newton and Hodge polygons depend only ,, !i on the underlying point ker(~) E Spec(A ), and not on the particular choice of a 0 perfect overfield of the residue field at this point. This allows us to speak of then we the Newton and Hodge polygons and slopes of (M,F) at the various points of Spec(A ) . The following theorem and corollary are a slight strengthening of F 0 Grothendieck's specialization theorem. So we are II As on Aperf 0 142 SLOPE FILTRATION OF F CRYSTALS phism Theorem 2.3.1 (Grothendieck) component. over an arbitrary F -algebra A Let (M,F) be a oa-F-crystal of rank r p 0 Let A > 0 be a real number. The set of points in Spec(A ) at which all Hodge >ries between 0 (resp. Newton) slopes of (M,F) are > A is Zariski closed, and locally on Jmplete and Spec(A ) it is the zero-set of a finitely generated ideal· )erfect 0 is to the con- Corollary 2.3.2 Let P be the graph of any continuous ffi-valued function on [O,r] which is The set of points in Spec(A ) at which the linear between successive integers. 0 Hodge (resp. Newton) polygon of (M,F) lies above P is Zariski closed, and is r(A)->- W(B) locally on Spec(A ) the zero-set of a finitely generated ideal. 0 0 0 . then the Proof. The Corollary follows by applying the theorem to the various exterior 1e category of powers of (M,F) . The theorem for Newton slopes follows from the theorem for \ )-module M 0 Hodge slopes, applied to a suitable iterate (M,Fn) of (M,F) , as follows. norphism of Because Hodge slopes are always integers, and Newton slopes are always in ~~r. According for oa-F-crystals of rank r , we may assume that lies in the more mundane to the basic slope estimate, we have F has all Newton slopes > A has Hodge slopes > nA ! nA >ver an arbitrary F has all Newton slopes .::_ n+:t;"-l Jerfect field, Therefore, if we choose n so large that ;;ons depend only A- _L < n • A < ,ular choice of a r! n+r-1 A ' ws us to speak of then we have us points of engthening of F has all Newton slopes > A ~ Fn+r-l has all Hodge slopes > nA So we are reduced to proving the theorem for Hodge slopes. Aperf image As replacing A by its perfection and (M,F) by its inverse 0 0 (M,F)(.p) of on Aperf alters neither Spec(A ) nor the perfect-field-fibres 0 0 143 N. KATZ are the identity' A is perfect. As the at the points of Spec(A ) , we may assume that 0 (M,F) ~ 0 solute Frobeniusl Spec(A ) , we may further assume that (M,F) is a ~ theorem is ~ on 0 Because Hodge slopes ring homomorphis W(A )-module of rank r with a cra-linear endomorphism F 0 The algebr are integers, we may also assume that A is an integer. (F .. ) s is the inver F is now given by an r x r matrix l,J In terms of a ~ of M , initial" with entries in W(A ) • For any homomorphism ~:A0 ---+ k with k a perfect 0 need not (W(~) (F .. ) ) obtained by applying field, (M,F)(~) is given by the r x r matrix l,J Witt vectors. Now valence of cate i to the individually thought of as ~ component-wise F i,j ' lie in pairs (M, V ) and only if all the W(~)(F .. ) (M,F) (~) has all Hodge slopes > A if l,J I' 2 gether with an of each of the r Witt- ~~ and only if the first A components lw(k) i.e., if i! ' If we fix 1 vectors W(~)(F; .) all vanish, i.e., if and only if ~ annihilates the ideal in l,J 2 lence of catego~· A generated by the first A Witt-vector components of each of the r matrix 0 of triples (M I: ' I~ coefficients F. j e: W(A ) l' 0 M together wit ' I A guestion. Is there a natural structure of closed subscheme on these Zariski morphism FE : j subsets of Spec(A ) defined by "slopes .::_A"? Given a cra-F-crystal over tensoring M(E) j 0 1 JF [£]/(e:2 ) , does it make sense to ask if its Newton or Hodge slopes are "every- Let us den; p . II lterations allo;l where" > A ? ,, i(E): A-lioo I (2.4) Newton-Hodge filtration. and which sits A is an JF -algebra of In this section we will consider the case in which 0 p one of the following two kinds: A of A A is smooth over a perfect subring 00 0 0 A is a formal power series ring in finitely many variables 0 A of A over a perfect subring 00 0 Z -algebra A In both cases, there exists a p-adically complete and separated p 00 A /pA ~ A , such that point" ~ , together with an isomorphism 00 00 0 which is flat over p ""/pn+ 1"' • The al- A dfn A I n+lA is formally smoo th over ~ ~ for each n > 1 , n oop oo sition maps W(A )-algebra; it is unigue up to automorphisms which A is naturally a 00 gebra 00 144 SLOPE FILTRATION OF FCRYSTALS As the are the identity on W(A ) and "~>Thich reduce mod p to the identity. The ab- 'ect. 00 is a free solute Frobenius map a:A -->- A may be lifted, non-uniquely in general, to a 0 0 1se Hodge slopes ring homomorphism L: : A -->- A which is necessarily a-linear over lti(A ) 00 00 00 The algebra A provides an A -test object, namely (A , (p), y ;s) , in "I>Thich 00 0 00 brix (F .. ) s is the inverse of the given A /pA ~ A . This A -test object is "pseudo- l,J 00 00 0 0 k a perfect initial" in the sense that~ A -test object receives a map from it, 1Jut this map 0 ained by applying need not be unique. Evaluation at this "pseudo-initial" object provides an equi- ectors. Now valence of categories between the category of crystals on A and the category of 0 F .. ) lie in pairs (M, V ) consisting of a p-adically complete and separated A -module M to- l,J 00 2 the r vlitt- gether with an integrable, nilpotent W(A )-connection. 00 .tes the ideal in If we fix a lifting i: : A -->- A of a , we similarly obtain an equiva- 00 00 2 .he r matrix lence of categories between the category of aa-F-crystals on A and the category 0 of triples consisting of a locally free (of some rank r ) A -module 00 M together with an integrable, nilpotent W(A )-connection V and a horizontal ;hese Zariski 00 morphism Fi: : (M(~) ,V(l:a)) -->- (M, V) which induces an isomorphism after j!' 3tal over (l:)a tensoring M and M over ~ with Jes are "every- p Let us denote by the perfection of A The method of successive 0 iterations allows us to construct for each choice of L: , a unique homomorphism which reduces mod p to the inclusion A ~ Aperf 0 0 ' n JF -algebra of and which sits in a commutative diagram p i(i:) A , W(Aperf) 00 " 0 'o ll: ~lw(a) i(i:) A c , W(Aperf) my variables 00 0 z -algebra A This homomorphism i(l:) should be thought of as the universal "i:-Teichmuller p 00 4 A , such that point" of A In fact, i(i:) provides a construction of as the 0 00 /pn+lZ£ • The al- p-adic completion of the "i:-perfection" lim A (in which the successive tran- ---->- 00 tomorphisms which sition maps A ---+ A are all L: ) Notice that 00 00 145 N.KATZ hence~. in W(Aperf) fJA = i(L) (A) n p 0 (2.4.1) 00 lies in pbA 00 i the matrix coe ! simply because tending scalar l Given a aa-F-crystal on A , thought of as (M,V,FL) , its inverse image on 0 ideal in Aper I 0 < obtained from (M,FL) on A by the extension is the pair (M,FL)(i(L)) 00 Fij mod of scalars i(L) : A ----+ W(Aperf) . 00 0 ideal in generate the Theorem 2.4.2 (Newton-Hodge filtration) be a aa-F-crystal over an F -algebra A of the type dis- at each point , ~ (M,V,F) p 0 I; the same Witt-il .::C.::U.::;S.::;S.::;e;;:;d_;a,_b:::;o~v::.:e::....:i=-'n;!....!2::.!.~4:.:•c...... !:S:.eu"'p~p~o~s.o:e_:::t!!h::::at::.. (A, b) € Z x ZL is a break point of the Newton ,, !· line 1 c M a~! polygon of (M,V,FL) at every point of Spec(A ) , and that (A,b) lies on the 0 Then there exists a as we did in Hodge polygon of (M,V,FL) at every point of Spec(A0 ) t~;;. of F makes oi! unigue FL-stable horizontal A -submodule M c M , ~ M locally free of 00 1 1 \; horizontal). ~ a , and M dfn M/M locally free of rank r - A , such that 2 1 are at every point of Spec(A ) , the Hodge (resp. Newton) 0 slopes of (M , V\M , FL\M ) are the A smallest of 1 1 1 the Hodge (resp. Newton) slopes of (M,V,FL) , locally at every point of Spec(A ) the Hodge (resp. Newton) 0 slopes of (M , V\M , FL\M ) are the r- A greatest 2 2 2 Hodge (resp. Newton) slopes of (M,V,FL) Furthermore, when A is itself perfect, the exact sequence of aa-F-crystals 0 to do so first admits a unique splitting. a perfect M is a free A -module of Localizing on Spec(A ) , we may suppose that 00 !2:22!.· 0 Consider first the case (A,b) = (l,b) . Then the least Hodge slope is rank r F .. word the same b at every point. This means that each matrix coefficient l,J i(L)(F. j) e W(Aperf) with its first b Witt-vector components nilpotent, and l, 0 146 T SLOPE FILTRATION OF F-CRYSTALS hence zero, in Aperf . Therefore i(E)(Fij) lies in p b W ( Aperf) , and so 0 0 . . - bA b I' 1 ~es ~n p oo So dividing F by p , we may assume b = 0 . This means that l the matrix coefficients Fij generate the unit ideal in A (because after ex 00 I tending scalars to W(Aperf) , their first Witt-vector components generate the unit inverse image on 0 I Aperf ideal in ; as these first components are just the mod p 0 Fij in A the by the extension I ' 0 ' Fij mod p generate the unit ideal in A and hence the generate the unit I 0 Fij ideal in A ) For every iterate Fv of F its matrix coefficients still I 00 ' generate the unit ideal. But for v ~ n(~) , all Hodge slopes of > n ,f the type dis- at each point of Spec(A ) , so that all 2 x 2 minors of Fv lie in pnA (by I 0 00 1int of the Newton ' the same Witt-vector argument in W(Aperf)) So we can construct the required 0 n lies on the line 1 c M as the "limit" of the images mod p of v ~ n(~) , just there exists a as we did in the case of a perfect field. This construction via images of iterates .ocally free of of F makes obvious that 1 is F-stable and horizontal (since F itself is horizontal). The slope assertions about F on 1 and on M/1 are pointwise, so are already proven. We do the general case (A,b) by constructing the required line 1 in AA(M) . It remains only to see that this line is of the form AA(M ) for some 1 locally free M c M of rank A with M/M locally free. [The F-stability and 1 1 ~ewton) horizontality of M then are consequences of the F-stability and horizontality reatest 1 cf the line; the slope assertions about M and M/M are pointwise, so are 1 1 already proven.] To see that 1 satisfies the PlUcker equations, it suffices oa-F-crystals to do so after an arbitrary injective extension of scalars. For this purpose we first embed A in W(Aperf) . Then, because is reduced we can embed 00 0 W(Aperf) in the product, indexed by all homomorphisms $ : Aperf ---+ k with k 0 0 a perfect field, of the W(k)'s. This reduces us to the case A =a perfect 0 A -module of free 00 field, in which case we have already proven it. east Hodge slope is As for the splitting in the case of a perfect A , the proof is word-for 0 in A has . ,j 00 word the same as in the case of a perfect field . :s nilpotent, and 147 N.KATZ slopes ~ Al , whi Remarks of the type dis- M adopted to the be a cra -F-;rystal over an JF -algebra A Let (M,'V,F) p 0 Spec(A ) , the Newton and I Suppose that at every point of cussed above in 2.4. 0 I ! Hodge polygons coincide with each other, and that they are constant, i.e.,~ h 0 ,hl , ... ,the Hodge numbers. Then the pendent of the point. Let us denote by A splitting of th. associated graded pieces of the Newton-Hodge filtration are cra-F-crystals a "unit-root" cross-section of I F. with of rank hi , such that l splitting is an (all Newton slopes = 0) cra-F-crystal. But a unit-root cra-F-crystal of rank hi is equivalent (cf [7]) to a continuous representation of the fundamental group of It would be interesting to understand the "meaning" GL(hi ,W(F ) ) in a p cra-F-crystals in question of these p-adic representations, especially when the i where X is r I' arise as crystalline cohomology groups of families of varieties. I 1· satisfies the ma1l I i (2.5) Splitting Theorems. i, I' I I In this section, we give a splitting theorem up to isogeny for slope filtra . i.e.' i tion of cra-F-crystals over perfect rings. I i.e., Theorem 2.5.1 \ Let A be a perfect ring, and let I 0 I We must l entries in Suppose that for some rational cra-F-crystals over A be an exact sequence of 0 I are all at every point of Spec(A ) the Newton slopes of 0 I at every point of Spec(A ) are all > A. V of Mat 0 ~A , while the Newton slopes of (M2 ,F2 ) Then in the category of cra-F-crystals up to isogeny, this exact seguence splits uniquely. Suppose we can are free W(A )-modules 0 Localizing on A , we may assume that M, M1 , M2 equation ~· 0 of ranks r, r , r respectively. Because the Newton slopes of (M,F) at any 2 1 ;!1 Z , we may in fact choose rational Spec(A ) lie in the discrete set point of 0 numbers Al < A such that at all points of Spec(A0 ) , (M1 ,F1 ) has all Newton 2 148 SLOPE FILTRATION OFF CRYSTALS slopes ~ Al , while (M ,F ) has all Newton slopes ~ A . In terms of a basis of 2 2 2 e type dis- M adopted to the filtration, the matrix of F has the shape Newton and i.e., inde- (: ~) ·s. Then the A splitting of the exact is a morphism (M ,F ) ---+ (M,F) which is a ·ystals se~uence 2 2 ~ "unit-root" cross-section of the projection. In terms of the given bases, the matrix of a d of rank splitting is an matrix of the form ontal group of 1 the "meaning" in question where X is 1 denotes the identity matrix, and where X satisfies the matrix e~uation a (J slope filtra- AX + II: XID i.e., i.e., We must show that this matrix e~uation has a uni~ue solution matrix X with entries in W(A ) e ~ Let us denote by Mat the space of all r x r 0 p 1 2 · some rational matrices with entries in W(A ) e ~ , with the linear topology defined by the 0 p !(A ) are all entry-by-entry congruence modulo p~(A ) . Consider the a-a-linear endomorphism 0 0 ~ ) are all > A. V of Mat defined by 0 :oguence splits -a X t--->- V(X) dfn (,11.-lXID)cr Suppose we can prove that v is topologically nilpotent. Then our matrix e W(A )-modules 0 e~uation (M,F) at any t choose rational X - V(X) has all Newton 149 N. KATZ 1' (2.6) obviously has the uni~ue solution l -a In X L Vn(-(/A-lrl:)cr ) ' ! n>O curves. Thus 1 ' and we are done. We will deduce the topological nilpotence of V from the basic slope estimate, applied' to fA -1 and to ID • Because (M ,F ) has all slopes _:::_ ;\ , we have 2 2 2 Theorem 2.6.1 Because JM ,F ) has all Newton slopes ~ ;\l , its determinant has its single 1 1 I. Let (M,'V •1i Newton=Hodge j; sort. Suppose 1 endomorphism defines a o-a-F-crystal over A , all of whose slopes are > N - ;\l . So by the Speo(A0 ) , a: I 0 a ' to a cr -F-e sl basic slope estimate, we have i: { (n+l-r ) (N-;\ )} t, 1 1 _ 0 mod p 1: (where [x] del! and hence we have {(n+l-r)(;\ -;\ )-N(r-1)} 2 1 Proof. In the jl 0 mod p ·I gives (r den1: -1 In terms of the matrices lA , ID , these estimates may be rewritten i {(n+l-r );\ } 2 2 0 mod p which in {(n+l-r)(;\ -;\ )-N(r-1)} 2 1 0 mod p which together give the estimate for the endomorphism V Therefore we {(n+l-r)(;\ -;\ )-N(r-1)} 2 1 Vn - 0 mod p As ;\ > ;\l , this estimate establishes the re~uired topological nilpotence of V . by 2 SLOPE FILTRATION OF F-CRYSTALS I (2.6) IsogenY theorems I In this section, we will give an isogeny theorem for oa-F-crystals over Thus let A be an F -algebra which is either l ~· 0 p I an integral domain, smooth of dimension slope estimate, I < l over a perfect field k , we have I a formal power series ring in one vari able k[[T]] over a perfect field k . Theorem 2.6.1 its single oa-F-crystal over an F -algebra A of the above (2.6) (M,V,F) p 0 a-a -linear 1 sort. Suppose that A is a positive real number, such that at every point of .ce (M ,l(F )- ) 1 1 Then (M,V,F) is isogenous Spec(A ) , all Newton slopes of (M,V,F) ~ > A Al . So by the 0 to a oa-F-crystal (M' ,V' ,F') which is divisible by A in the sense that for all n ~ l , (F1 )n _ 0 mod p[nA] (where [x] denotes the integral part of the real number x ). Proof. In the case when A is itself a perfect field, the basic slope estimate 0 gives (r denoting the rank of M ) n Fn+r-l - 0 mod p{nA} for all n > l which in turn implies that if we put v = {(r-l)A}, we have for all n > 0 Therefore we can define a W(k)-module M' with M c M' c p-VM nilpotence of V . by M' L image M--+p-v M ) . n>O 151 N. KATZ T see that an The basic inequality [x+y] .::_ [x] + [y] allows one to check that for all n .::_ 0 , the operator Fn/p[nA] maps M' to itself. The fact that M' is "caught" be- -\) C(K ) . tween M and p M guarentees that M' is a free W(k)-module of the correct 0 between M rank. The inclusion of (M,F) into (M' ,F) is the reg_uired isogeny. In the general case, the basic slope estimate applied pointwise together with the "Witt-vector-component"-argument already used shows that over A00 we still have for all n > 0 We define So it is natural to define an -module M' with A00 M c M' c p-v M by I This descripl: M' I n>O the operator!. To eee Clearly M' is horizontal(it's defined in terms of the horizontal maps FL:n), and li F n/p[nA] now viewed as t:an_linear endomorphisms is stable by all the operators L: ' ji of M' . The only problem is that I cannot prove (or disprove!) that M' is a i, locally free A -module. (Even the fact that M' is finitely generated depends on (this becausl; 00 A is noetherian. Can one give an the fact that, in the case envisioned, 0 effective bound on the number of terms needed in the apparently infinite sum of Because A images which defines M' ?) 0 remains To circumvent this difficulty, we will define a larger A00-module M" Let us M c M' c M" c p -vM c M~ ®Ql p p Let us denote by K the fraction which will have all the reg_uired properties. 0 field of A , and by C(K ) the completion of the local ring of A00 at the 0 0 (The notation C(K ) is to remind us that this is a Cohen prime ideal 0 From it, we ring for the field K , i.e., a mixed characteristic, complete, discrete, abso- 0 that M" i lutely unramified valuation ring with residue field K • ) By its construction we 0 A at all 00 152 SLOPE FILTRATION OF F CRYSTALS see that any derivation of the A into itself, as well as any endomorphism r-all n.::_O, 00 E=Aoo-----+ A: lifting the absolute Frobenius, extends by continuity to this "caught" be- C(K ) • Because C(K ) is flat over A , we can tensor the chain of inclusions the correct I 0 0 00 between M and M' to obtain y. I together with I -\) M c M' c p M c M0~ Cll p p , we still I 00 n n n n I I I We define M" as the intersection (inside (M0C(K ))0~ (\\ I 0 p p • I M" dfn (M' 0 C(K )) n (M0,., Cll ) ' Aoo 0 ""p p I I This description of M" shows that it is both horizontal, and stable under all I the operators (FE)n/P[nA] I To see that M" c p -vM , simply notice that I -\) ar endomorphisms I p M ,at M' is a C(K ) (A @CQ, ) A ) . Thus we have (this because M is locally free, and n 00 ·ated depends on 0 00 p give an MC M" c p-\IM , :inite sum of Because A is noetherian, this shows that M" is finitely generated. So it 0 A remains only to show that M" is flat over 00 iule M" Let us admit for the moment the following assertion about M" for any f E A with f i pA M" 00 00 the fraction has no f-torsion, and M"/fM" has no p-torsion. A at the 00 s is a Cohen From it, we easily deduce the flatness of M" , as follows. It suffices to show lis crete, abso- that M" is flat after we extend scalars from A00 to the complete local rings of 1 construction we But such a complete local ring A is of the A at all closed points of A 00 00 0 153 N.KATZ isogeny the form W(k')[[T]] , where k' is the residue field at the closed point, and T may I I is a uniformizing over-a-perf I A whose reduction mod p in A l I be chosen as any element in 00 0 is in fact Because A is flat over A we deduce from the I parameter at the closed point. 00 00 I admitted assertion that I Corollar Let M"@W(k')[[T]] has noT-torsion and I I over a erfi M"0W(k' )[ [TJ]/(T) has no p-torsion. I (M,V,F) Since M"0W(k')[[T]] is finitely generated, its flatness over W(k')[[T]] follows I Expose IV, Thm 5.6). isogenous "til from the local criterion of flatness (SGA l I• A is a domain (being Zi. -flat, I sits in a J! To prove the assertion, notice first that 00 p , ! Since I p-adically separated, and having A00/pA00 a domain) , and f # 0 -\) there is no I M" c p-vM , and p M is locally free and hence flat over in which C(K ) , and so by the , I Because f i pA it becomes a unit in 0 I: f-torsion in M" 00 j i, definition of M" we have 1: has no f-torsion ll !l Therefore the inclusion M" c p-\IM gives an inclusion !j :i ,,li :I1: d So to have M"/fM" without p-torsion, it suffices if M/fM has no p-torsion. As ~· Byjj :\ 1'1 p For ' M is flat over A , being locally free, it suffices if A00/fA00 has no p-torsion. 00 A is flat over divisible I! f f_ pA , while 00 This follows from the fact that 00 ji of Spec( J I p-adically separated, and A /pA is a domain. I 00 00 ' ,I plicity I ,I i ~ ; If we allow A to be a domain which is smooth of arbitrary dimension (cf 2.4) ~· 0 n over a perfect field k , exactly the same argument shows that M" will be flat sequence over the complete local rings A of A at all points of codimension one in J: 00 00 .! Spec(A ) (there A will be of the form C(k' )[[T]] with k' the no-longer- li o 00 perfect residue field and C(k') a Cohen ring of k' ) . In other words, the which fori 154 SLOPE FILTRATION OF F CRYSTALS point, and T may l isogeny theorem is true outside a closed set of codimension ~ 2 on ~ smooth- is a uniformizing over-a-perfect field A It would be interesting to know if the isogeny theorem .1 0 e deduce from the I is in fact true, without exceptional sets, in this more general case. I I Corollary 2.6.2 (Newton filtration) Let A be an F -algebra which is either a smooth domain of dimension one I 0 -- p - I over a perfect field k , or is k[[T]] Suppose we are given a oa-F-crystal (M,V,F) over A of rank r which at every point of Spec(A ) has the same I 0 0 ;(k 1 ) [ [T]] follows I first Newton slope A , with the same multiplicity A . Then (M,V,F) is A isogenous to a (M 1 ,V' ,F') which is divisible by p , and which •I being ZL -flat, I sits in a short exact sequence of oa-F-crystals over A p , 0 Since I 0->- (M~,V' ,F') ->- (M' ,V' ,F') ->- (M' V' F') ->- 0 there is no I 2' ' >) , and so by the I in which 1 ( MI ,V I ,F I) has rank is divisible by 1 A ' pA and at each point of Spec(A ) all its 0 Newton slopes are A (M 1 V' F') has rank r- A, is divisible 2' / ~ p and at each point of Spec(A ) all 0 its Newton slopes are > A . no p-torsion. As Proof. By the isogeny theorem, we may suppose (M,V,F) itself to be divisible by A has no p-torsion. p For any integer n > 1 such that nA 8 ZL, the nth iterate (M,V,Fn) is nA or divisible by p , i.e., all Hodge slopes of are > nA , at each point of Spec(A ) Since the first Newton slope of (M,V,Fn) is nA , with multi- 0 plicity A , at each point of Spec(A ) , we can apply the Newton-Hodge theorem brary dimension 0 (cf 2.4) to (M,V,Fn) and the point (A,AnA) . This produces a short exact b M" will be flat sequence of oan_F-crystals nension one in e no-longer- er words, the which for n = 1 (i.e., the case A E ZL) completes the proof. In general, we 155 N.KATZ need only observe that in terms of a lifting R of a:A ---+ A , the submodule . 0 0 M c M is simply the intersection 1 F nN image of ( p~NA nN>l and M is therefore F-stable. Then we can take (M ,V,F) c (M,V,F) as the 1 1 solution to our problem. It remains to see why the short exact seQuence we have constructed splits of A We have proven that, over uniQuely over the perfection 0 it splits uniQuely in the "up-to-isogeny" category i.e., by an F-compatible map M ---+ M with coefficients in We must show this map has coef- 2 ficients in W(Aperf) . But this same map also provides an "up-to-isogeny" 0 splitting of the exact seQuence of nth iterates But for any n with nA £ ~ , this is just the Newton-Hodge filtration of (M,Fn) By the induction (A,AnA) , which over Aperf has a uniQue splitting attached to the point 0 The underlying map M ---+ M of this splitting, which 2 has coefficients in \f(Aperf) , must, by uniqueness, coincide with the underlying whose source 0 '': map of our "up-to-isogeny" splitting. back" by this an extension Corollary 2.6.3 Hypotheses as in the previous Corollary 2.6.2, suppose in addition that the entire Newton polygon of (M,V,F) is constant, i.e., independent of the point in "i I be the distinct Newton slopes, and let A1 , ... ,As be exact seQuence, their multiplicities. Then (M,V,F) is isogenous to a aa-F-crystal (M' ,V' ,F') provides which is divisible by and which admits a filtration The (M 1 ,V' ,F') 0 c (M~,V' ,F') c (M2,V' ,F') c ... c (M:,v• ,F') perfection of in which :I . I 156 I I SLOPE FILTRATION OF FCRYSTALS 1 1 1.e submodule . (M~,V ,F ) has rank A + ... + Ai , and has 1 Newton slopes (A repeated A times, ... , 1 1 Ai repeated Ai times) at each point of Spec(A ). 0 1 1 1 the quotient (M /M: , V ,F ) has rank l Ai+l Ai+l + ... +As , and it is divisible by p as the 1 1 the associated graded (M:/M: ,'1 ,F ) has rank A.l l-1 1 Ai , is divisible by P , and has all Newton slopes A. ed splits l Aperf 0 ' This filtration splits unigue~y when we pass to the perfection of A 1patible map 0 , has coef- Proof. We proceed by induction on the number s of distinct Newton slopes. For :ogeny" s = 1 , the previous Corollary applies. In general, -,re construct 1 1 1 1 (M '1 1 F 1 ) c (M '1 F ) as in the previous Corollary. Then we have 1' ' ' ' 0----+ (M~,V 1 ,F 1 )----+ (M 1 ,V 1 ,F 1 )----+ (M 1 /M~,V 1 ,F 1 )----+ 0. 1 1 1 By the induction hypothesis applied to (M /M~,V ,F ) , we get an isogeny Ltting I II II II 1 1 1 1 1 ((M /M ) ,V ,F)·--->- (M /M ,V ,F ) Ging, which 1 1 he underlying whose source satisfies all the conclusions of the Corollary. Taking the "pull- 1 1 1 1 1 back" by this map of the above extension of (M /M~,V ,F ) by (M~,V ,F ) , we get an extension ion that the 0----+ (M~,V 1 ,F 1 )----+?----+ ((M 1 /M~)",v",F")-----+ 0. ' I the point in I The middle term, ? , together with the filtration of it defined first by this A , ... ,As be 1 l 11 11 II ( (M 1 /M ) , V ,F ) , I 1 1 1 exact se~uence, then by the inductively given filtration on ,1 (M ,'1 ,F ) 1 provides a solution to the problem. .I The existence of a uni~ue splitting of the filtration when we pass to the c I ) perfection of A follows, by induction, from the previous Corollary. 0 157 N. KATZ Corollary 2.6.4 cra-1 Hypotheses as in the previous Corollary, any (M,V,F) with constant Newton I morphism polygon is isogenous to an (M 1 ,V' ,F') with the property that for any integer ( k . n > 1 which is a common denominator for the Newton slopes, the Hodge and Newton j I I polygons of the cran_F-crystal at each point of Spec(A ) coincide. Remarks. 0 w(JF aN) Proof. Indeed, the (M',V',F') given by the previous Corollary has the required p FN/pNA property. of (2.7) Constancy theorems Let k be an algebraically closed field of characteristic p , and let A 0 theorem, lJ be a k-algebra. We say that a cra-F-crystal on A is constant if it is (iso- 0 Oort (cf! morphic to) the inverse image of a cra-F-crystal on k , by the given algebra map I considers!; k ---r A I 0 this prod Theorem 2.7.1 k , we ge 1 I Let (M,V,F) be a cra-F-crystal of rank r on k[[T]] , with k algebra homogeneo1i ically closed. Suppose that at the two points of Spec(k[[T]]) , the Newton poly- I, IIl! gons coincide, and that this common Newton polygon has only a single slope, say I ji' A , repeated r times. Then (M,V,F) is isogenous to a constant cra-F-crystal. lj\ li A Proof. By the isogeny theorem, we may assume that (M,V,F) is divisible by p If we divji in the sense of 2.5. We will prove it constant. Let N be the denominator of we get a NA A . Then is divisible by p , and all of its Newton slopes, at each point of Spec(k[(T]]) are NA . Therefore (M,V,FN/pNA) is a "unit-root" craN_F-crystal,so equivalent to a representation of n (Spec(k[[T]]) in 1 n (Spec(k[[T]]) ~ n (Spec(k)) is trivial, because k is GL(r,WOF aN)) But 1 1 p algebraically closed. Therefore (M,V,FN/pNA) is trivial as a unit-root craN_F-crystal. In particular, (M,V) is trivial as a crystal, i.e., the W(k) module MV of all horizontal sections of (M,V) over A00 = W(k)[[T]J is free of rank r , and Because F is horizontal, it induces a 158 SLOPE FILTRATION OFF-CRYSTALS oa-linear endomorphism of MV , such that FN/pNA induces a oaN_linear auto- mstant Newton morphism of MV . Then (M,V,F) on k[[T]] is the inverse image of on any integer k. Hodge and Newton Remarks. 1) The (trivial) representation of ~ (Spec(k[[T]]) on a free Jec (A ) coincide. 1 - 0 v WOF aN)-module of rank r is provided by the set (M )fix of fixed points of p :>s the required v FN/pNA acting a aNl.- lnear 1 y on Mv . In fact, ((M )fix'F) provides a descent of (M,V,F) to lF a p 2) If we omit the words "isogenous to" from the statement of the , and let A 0 theorem, it can become false. -The simplest geometric counterexample is due to f it is (iso- \ He begins with a supersingular elliptic curve E over k , and Oort ( cf [10]). 0 ven algebra map l considers the product E (p"') x E (p"') of its p-divisible group with itself. In I 0 0 1 this product, the kernel of F is a x a Over the projective line F over I p p •t• "TDl k , we get a family of a 's sitting in a x a over a poln ln -"' with I p p p h k algebra- homogeneous coordinates ()J,V) sits the image of the closed immersion the Newton poly- a '------+ a x a p p p ;le slope, say [ x--r()Jx,vx) oa-F-crystal. A If we divide the constant group by this variable a .visible by p p 1 we get a non-constant p-divisible group over IP Restricting to the complete ienominator of 1 local ring at any closed point of IP , we get a non-constant p-divisible group 3, at each point over k[[T]] , whose Dieudonne module provides the required counterexample. ~oot" \ Concretely, this means we begin with the constant o-F-crystal (M,V,F) on ]) in I, k[ [T]] given by h• because k is M: free W(k)[[T]]-module with basis e ,e ,e ,e4 nit-root I 1 2 3 d V: the trivial connection with V(dT)(ei) = 0 for i = 1,2,3,4 .e., the W(k)- 'J in terms of the endomorphism L of W(k)[[T]] which is [[T]] is free of i F: a-linear and maps T---+ Tp , the L-linear map FL: M--+ M induces a \ is given by I 159 N.KATZ Theorem 2.7.2 Let (M,V,F The W(k)[[T]]-submodule M' c M spanned by are all (r-1)/r is constant. + TPe e pe e el 3 ' 2 ' 3 ' 4 Proof. The Newt 17 is stable under 17 and F 1: , and we have pM c M' c M But (M' ) the free W(k)-module spanned by and so that 1 17 Therefore (M' ,V,F) is not constant. Alternately, one (M ) @W(k)[[TJ] ?j M' 2 could observe that the Hodge polygon of (M' ,v,F ) is not constant (its least Hodge slope is 1 at the closed point 0 at the generic point) , and hence (M' ,V,F) cannot be constant. at both points ': i .. i Another very recent counterexample is due to Lubin. He constructs a 11 Hodge slopes are I cr-F-crystal (M,V,F) over k[[T]] of rank 5 , whose Newton slopes are all 2/5 , estimate (with 0 and whose Hodge numbers are h = 3 at both points of Spec(k[[T]]). In Lubin's example, the Hodge polygon of (M,V,F5 ) is not constant; at the closed 5 point, the least Hodge slope of F is 2 , but at the generic point it is 1 . In particular, Therefore (M,V,F5) , and a fortiori (M,V,F) , cannot be constant. root over Here is the actual example. The module M is free on the inverse W(k)[[T]]. For the endomorphism 1: of W(k)[[T]] which is a-linear and sends F is the linear map M(E) ---+ M with matrix 0 1 0 0 0 brai 0 0 1 0 0 are all 0 0 0 1 0 0 0 0 0 p Proof. p 0 0 T 0 The connection 17 on M is the unique one for which F is horizontal. In the positive direction, we do have the following two constancy theorems the 160 SLOPE FILTRATION OF F CRYSTALS Theorem 2.7.2 Let (M,9,F) be a oa-F-crystal of rank r > 1 on k[[T]], with k alge- braically closed. Suppose that at both points of Spec(k[[T]]), the Newton slopes are all (r-1)/r and the Hodge numbers hi vanish for i > 1 . Then (M,9,F) is constant. ,. I Proof. The Newton and Hodge polygons must be Mvn M' is .,I I (r,r-1) rnately, one Newton ~ ~ (its least Hodge -d(l,O) nd hence at both points of Spec(k[[T]]) (because they start and end together, and the 0 ts a Hodge slopes are 0 and 1 ). Therefore h = 1 . Applying the sharp slope are all 2/5 , estimate (with A r-l/r ) , we get leC (k[ [T]]) · {r-l·n} - 0 mod p r at the closed ; it is 1 . r-1 In particular, is divisible by p , and hence is a unit- root F-crystal. Just as in the proof of 2.7.1, this implies that (M,9,F) is over the inverse image of on k • ~r and sends Theorem 2.7.3 Let (M,9,F) be a oa-F-crystal of rank r > 1 on k[[T]], with k alge braically closed. Suppose that at both points of Spec(k[[T]]), the Newton slopes are all 1/r . Then (M,V,F) is constant. Proof. This time the basic slope estimate shows that (M,9,Fr/p) is a unit-root F-crystal, and we conclude as in 2.7.2. ,tal. The theorem 2.7.1 of constancy up to isogeny becomes false as soon as we allow lCY theorems the common Newton polygon to have more than one distinct slope, for there can be highly non-trivial extensions of constant F-crystals over k[[T]] The simplest 161 N. KATZ and most important example of such an extension is given by first taking an or- the aa-F-crys dinary elliptic curve E ~over k , then constructing its equicharacteristic aa -F-crystal ) 0 universal formal deformation E over k[[T]] , and finally taking the p-divisible attached to it! i 00 group E(p ) of E . This p-divisible group over k[[T]] sits an exact sequence ; : I 1:: t~ I 00 i The Dieudonne crystal (M,V,F) of E(p ) , or equivalently the first crystalline cohomology of E/k[[T]] , is a a-F-crystal of rank two, which is an extension of 1. P. Berthei to two constant a-F-crystals of rank one. Even the underlying crystal (M,V) is 2. P. Berthef highly non-trivial; indeed, MV is free of rank one over W(k) . (For a suitable voli .I ' choice of parameter T of W(k)[[T]] , the Serre-Tate or the Dwork theory tells us 3. s. Bloch, j: pre~' that (M,V) , viewed as a module with connection on W(k)[[T]] , admits a basis 4. B.H. e ,e in terms of which the connection is given by Gros~j,. 0 1 I 5. A. Grothd. 6. N. Katz, 1 Lee i I ""'l! [ 7. ~ j PP·JI Because the series log(l+T) has unbounded coefficients, the module MV of Lee .I I'I horizontal sections consists only of the W(k) multiples of e .) 8. Ju. I. 0 Theorem 2.7.4 9. Let (M,V,F) be a aa-F-crystal on k[[T]] , with k algebraically closed. 10. Suppose that at the two points of Spec(k[[T]]), the Newton polygons coincide. Then (M,V,F) is isogenous to a aa-F-crystal (M' ,V' ,F') whose inverse image on (k[[T]])perf is constant. Proof. This follows by combining Corollary 2.6.2 and Theorem 2.7.1. Remarks. 1) This gives an alternate proof of Berthelot's Theorem 4.7.1 in [1]. 2) B. Gross ( cf [4]) attaches to any a a -F-crystal over k( (T) )perf a ,:i representation of Gal(k((T))alg.cl./k((T))perf.) which is trivial if and only if ';il SLOPE FILTRATION OF FCRYSTALS the oa-F-crystal is isogenous to a constant one. Therefore if we begin with a ing an or- oa-F-crystal over k[[T]] with constant Newton polygon, Gross's representation, teristic attached to its inverse image on k((T))perf. , is trivial. Is the converse true? .e p-divisible ,xact sequence -:-:-:-:- References ; crystalline ,i extension of 1. P. Berthelot, Theorie de Dieudonne sur un anneau de valuation parfait, I• to appear. (M,Il) is I 2. P. Berthelot and ltJ. Messing, Theorie de Dieudonne Cristalline I, this ~or a suitable volume. theory tells us 3. S. Bloch, Dieudonne crystals associated top-divisible formal groups, preprint, 1974. its a basis 4. B.H. Gross, Ramification in p-adic Lie extensions, these proceedings. 5. A. Grothendieck, Groupes de Barsotti-Tate et cristaux de Dieudonne, Sem. Math. Sup 45, Presses de l'Universite de Montreal (1970). 6. N. Katz, Travaux de Dwork, Expose 409, Seminaire Bourbaki 1971/72, Springer Lecture Notes in Math. 317, 1973, 167-200. 7. ------, P-adic properties of modular schemes and modular forms (esp. pp. 142-146), in Modular Functions of One Variable III, Springer Lecture Notes in Math. 350, ( 1973) , 69-190. 8. Ju.I. Manin, The theory of commutative formal groups over fields of finite characteristic, Russian Math. Surveys 18, (1963), 1-83. 9. B. Mazur, Frobenius and the Hodge filtration, B.A.M.S. 78, No.5 (1972), 653-667. lcally closed. 10. F. Oort, Subvarieties of moduli spaces, Inv. Math. 24 (1974), 95-119. 3 coincide. 1verse image on -:-:-:-:- Nicholas M. Katz Department of Mathematics Princeton University Princeton, NJ 08540 U.S.A. 4.7.1 in [1]. k( (T) )perf a if and only if 163