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 ,(l|174 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(g-1))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 p-curvature of the Gauss-Manin 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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De Rham cohomology, Hodge, Homological algebra, Cech bicomplex