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Dr. Katz DEq Homework Solutions 16

Dr. Katz DEq Homework Solutions 16 - 16 N.M Katz Then gr~(M...

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Unformatted text preview: 16 N.M. Katz: Then gr~ (M) = F p (M)/F p +a(M) ~ gr~ R p +qf, (f2]/s (log D)) ( l[ Rq f, (~2P/s(log O)). B ecause M is horizontal, it follows from ( t hat we have a commutative d iagram a f, (C ,,s(log D))- J , | (O ,/s(log O)) F V(M) v ~/r| grV(M) v __~f2~/r | grP- 1(M) ( 1] R ' f , ( ~ / s ( l o g D)) P , ~/r| Rq+~f, (~xV})~(logD)) We deduce from it that the Hodge filtration Fi(M) of M is horizontal (i. e., each Fi(M) is horizontal) if and only if the restriction to Op gr~(M) o f the mapping "cup-product with the Kodaira-Spencer class" ( p: @ gr~(M)-, @ f2s~/r| p -I(M) p vanishes. ( R emark. I n practice, the M in ( will be either all of R "f,(f2~/s(log D)), or the p rimitive part o f R"f,(f2]/s) in case O is void a nd X / S is projective and smooth, or the part of R"f, (f2~c/s(logD)) which t ransforms according to a prechosen irreducible representation of a finite group (of order prime to all residue characteristics of S) which acts a s a group of S-automorphisms of X and preserves D. This last case will a rise when we discuss Schwartz's list (cf. 6.0). 2. The Cartier Operation and the Conjugate Spectral Sequence 2.0. Throughout this section, we will consider the situation of 1.0, with t he additional assumption that S is a scheme of characteristic p (a prime n umber), i.e. that p. ls = 0 in ~Ys. (2.0.1) Recall that for any S-scheme n: Y---, S, the S-scheme y~v) is by d efinition the fibre product of 7~: Y---,S and the a bsolute F robenius ...
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