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 (2.3.1.2.2) F* s R"f, (•bX/S (log D)) --~ R"f~, ") (a* (~bx/s (log D))) which comes from the cartesian diagram (2.0.1.1) X <p)- ~ > X (2.3.1.2.3) lf(P) lf S Fab~'S is an isomorphism. (2.3.1.3) Remark. Under the isomorphism (2.3.1.2.1), 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 (2.3.2.1) 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. (2.3.2.2) 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 (2.3.1.2.1) and the hypothesis (2.3.2.1), may be written (2.3.2.3) r 'b = F~*s R"f, (~x/s (log D)) ~ R" +b f, (g2~/s (log D)), is degenerate at E 2 . Proof By (2.3.1.4.1), it follows that the conjunction of the hypotheses (2.3.2.1) and (2.3.2.2) 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 (2.2.1.2) 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. (2.3.2.4) 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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## This note was uploaded on 12/21/2011 for the course MAP 4341 taught by Professor Normankatz during the Fall '11 term at UNF.

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