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Dr. Katz DEq Homework Solutions 20

# Dr. Katz DEq Homework Solutions 20 - first is the Hodge...

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20 N.M. Katz: (2.2.1.11) Tautology. Hypotheses as above, suppose that for an integer ro> 1, the formation of Ero commutes with base change, and that the spectral sequence over Z (2.2.1.12) gf 'q (Z) = R p + q f, (grF p (Kz)) =~ R" + q f, (Kz) is degenerate at Ero. Then after any arbitrary change of base ~o: Z1 -+ Z, the spectral sequence p,q __ P+q P (2.2.1.13) El (Z1)-R fz,,(gr~(Kz,)) ~RP+qfzl,(Kzx ) is degenerate at Ero. Furthermore its formation from Ero on commutes with arbitrary change of base. (2.2.2) Examples. Let's return to the geometric situation 1.0, so that our morphism f: Y~ Z in (2.2.1) becomes the morphism f: X~ S of 1.0. We take for K the complex O]/s (log D)), and for an arbitrary S-scheme S', we take Ks, = (2},/s, (log D'), the symbol ' denoting fibre product with S' over S. There are two filtrations in which we shall be interested. The
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Unformatted text preview: first is the Hodge filtration (1.4.1), (the "bestial" one in the terminology of [8]). The second is the one, noted Z<o and called "canonical" in [8], which is defined by K" if n<p (2.2.2.1) Z<p(K)= Ker(d) if n=p = tO if n>p. The spectral sequence defined by this filtration (2.2.2.2) ~<=.Ef'"=RP+qf.(gr~.K) ~RP+qf,(K) is the d~calage (cf. [8], 1.3.3) of the "second spectral sequence of hyper- cohomology" (2.2.2.3) ,,E~ 'q = RP f. (~ffq (K)) => R p +q f. (K), which by definition means that we have isomorphisms, compatible with the dr and with the standard isomorphism Er+, ~-H(E, dr), ~, l~q+2p,--p (2.2.2.4) ~ < ,E~' q - lla.~r + 1 for each integer r> 1 (cf. [8]). 2.3. The Conjugate Spectral Sequence (2.3.0) Recall that the entire "second spectral sequence" of hyper- cohomology (2.3.0.1) E~'q=RPf, (~q(f21,/s(log D))) =~ RP+qf,(t2]/s(log D))...
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