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 Aoscheme Xo, and smooth...
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 Fall '11
 NormanKatz
 Characteristic, Category theory, Isomorphism, Sheaf, Hodge, spectral sequence

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