44
N.M. Katz:
(3.3)(=3.2bis)
Theorem.
Under the hypotheses of
(3.2.4),
the following
subdiagram
of(3.2.0)
is commutative.
(p)
*
1
a, b
p(z~v,)(~o,,E~b)_
o
, (I
))
P( Z
'p))
F~ (R~f, (fdbx/s)) ~
l)b+'
Fa*b'( P)~' (1 
P(x~P')) (F~,~ (f2~/r)) 
F*~(R ~ +'f, (s
~
(log D)))
bs
bs
ab~
P(x) R~f,(~x/s(log D))
~_l)b+,p ,(l174
Proof
(Assuming 3.2). Thu upper square is deduced from the upper
square of (3.2.0), which is a diagram of k [G]modules, by applying the
projector P(xC~ The lower square is deduced from the lower square of
(3.2.0) by applying P(Z) to the lower horizontal line, and noting that
( deg (Z)
gt
F*so(P(z))=F*s \
~
~,
trace(z(g~))
/ oF~s
(3.3.1)
= ( deg (Z) '~P
\
#G
I Z(tracez(g1))Vg~
Q.E.D.
(3.3.2)
Corollary.
Hypotheses as in
(3.2.4),
suppose that the absolute
Frobenius endomorphism of C s is injective (which is the case if T is reduced,
for example). If for a fixed integer n, the pcurvature of the GaussManin
connection on the "part of R"f,(s
D)) which transforms by Z~P.
)'',
i.e., on the submodule P(ztP~)(R"f,
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 Fall '11
 NormanKatz
 Vector Space, De Rham cohomology, Hodge, Homological algebra, Cech bicomplex

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