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Unformatted text preview: 30 N.M. Katz: is a s ection o f the inclusion F~= i r
~-1, whose kernel is none other
t hen F i+lnF~o~ ~-1. T his shows that the canonical mapping F~+lc~
F~o~i - 1 ~ F~o~i-a/Ffo~iis a n isomorphism, which proves (220.127.116.11.2).
(2.3.5) In this section we wish to explain the differential equations
satisfied by certain of the higher Hasse-Witt matrices introduced in
(2.3.4). These differential equations were first noticed by Igusa  in
t he case of elliptic curves, then later explained quite generally by Manin
(18.104.22.168) We place ourselves under the hypotheses of (2.3.2), and assume
f urther that the divisor D is void, and that the geometric fibres of X /S a re
c onnected, of dimension N. Under these hypotheses, we have
(22.214.171.124.1) R2Nf,(~Q~c/s)--%RNf,(O~/s) is locally free of rank 1 and even
c anonically isomorphic to the structural sheaf Cs, via the trace morphism
(cf.  and )
tr: RaNf , (Y2]/S)~ d)s (126.96.36.199.2) w hich carries the Gauss-Manin connection on R ZNf,(f2]/s) i nto the
s tandard connection on d)s (the one given by exterior differentiation
d : (9s'-~ 12~/r).
T he cup-product pairings
(188.8.131.52.3) Rn f , ([2*x/s)@ R 2N-'f, (fd~c/s)~ tr R2N f (Q~c/s) " ' (gs a re perfect dualities of coherent locally free S modules, for which the
f iltrations F and F~o~are both self-dual in the sense that (184.108.40.206.4) ( F' R n f , (O~/s)) • = F N +1 -n R a N - nf , (f2~/s) (220.127.116.11.5) (F~'onR ' f , (O~c/s))• = F~o, ~-n RaN-n f , (12~;/s).
+ T he associated graded pairings
r iR n 12*
g F( f*(X/S)) | gF (R
f,(x/s) n-i i N+i-n ~ N 2N 9 r
gF( R f,( [2 s))) N-i a nd
, grfoo. (R2Sf, (f2~:/s))
F ~* i
,bs(Rn - - i f ,(~x/s))| /l /l
* N+i-n N-f,(f2x/si )) ~ *
. F.bsR N f,(f2x/s) "" __~(9
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- Fall '11