Unformatted text preview: . Then the conjugate spectral sequence, which, thanks to (220.127.116.11.1) and the hypothesis (18.104.22.168), may be written (22.214.171.124) r 'b = F~*s R"f, (~x/s (log D)) ~ R" +b f, (g2~/s (log D)), is degenerate at E 2 . Proof By (126.96.36.199.1), it follows that the conjunction of the hypotheses (188.8.131.52) and (184.108.40.206) 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 (220.127.116.11) 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. (18.104.22.168) 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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- Fall '11
- Characteristic, Category theory, Isomorphism, Sheaf, Hodge, spectral sequence