Dr. Katz DEq Homework Solutions 22

Dr. Katz DEq Homework Solutions 22 - Then the conjugate...

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22 N.M. Katz: Proof The only point is that under either of the hypotheses, the canonical morphism of base change ( F* s R"f, (•bX/S (log D)) --~ R"f~, ") (a* (~bx/s (log D))) which comes from the cartesian diagram ( X <p)- ~ > X ( lf(P) lf S Fab~'S is an isomorphism. ( Remark. Under the isomorphism (, the Gauss-Manin connection deduced on F*s R"f, (~2bX/S (log D)) annihilates the image under F*s of R"f,(~x/s(log D)) (compare [24], 5.1.1). (2.3.2) Proposition. In the geometric situation 1.0, suppose that X is proper over S (and that S is a scheme of characteristic p, as it has been throughout Section 2). Suppose further that ( Each of the Hodge cohomology sheaves R"f,(~x/s(logO)) is a locally free sheaf of finite rank on S and (hence) that its formation com- mutes with arbitrary change of base S'-* S. ( The Hodge =~ De Rham spectral sequence E~ 'b = Rb f, (~x/s (log D)) ~ R a +b f, (I?~c/S (log D))
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Unformatted text preview: . Then the conjugate spectral sequence, which, thanks to ( and the hypothesis (, may be written ( r 'b = F~*s R"f, (~x/s (log D)) ~ R" +b f, (g2~/s (log D)), is degenerate at E 2 . Proof By (, it follows that the conjunction of the hypotheses ( and ( remains true after an arbitrary change of base S'~ S, and implies that the formation of the Hodge ~De Rham spectral sequence commutes with arbitrary change of base S'--, S. From ( it follows also that the formation of the conE~' b commutes with arbitrary change of base, while by general principles the formation of the entire conjugate spectral sequence commutes with any fiat base change S'-+ S. ( We may assume that S is affine, because the question is local on S. We wish to reduce to the case in which S is noetherian. So suppose S=Spec(A). Clearly there exists a subring AocA which is finitely generated over Z, a proper and smooth Ao-scheme Xo, and smooth...
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