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Dr. Katz DEq Homework Solutions 19

# Dr. Katz DEq Homework Solutions 19 - with the cohomology...

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Algebraic Solutions of Differential Equations 19 We are to be given a morphism K(q~) of filtered complexes of f~l(t~z)- modules on Yz~ (2.2.1.4) q~l(Kz~, F) ~ ~ fz~X((-0z~) ~(~) -~ (Kz~, F) ~" fz~ (r which satisfies the natural transitivity condition for a composition of morphisms of Z-schemes. Consider now the spectral sequence of d~z~-modules (2.2.1.5) E~' "(Z~)=RP+qfz,.(gr~(Kz,) ) ~ RP+"fz,.(Kz), on the Z-scheme Z1, whose E~ 'q term we denote E~'q(ZO. From the given functoriality of (Kz~, F) in ZI, we deduce, for every morphism q~: Z2--, Z1 of Z-schemes, morphisms of (gz~-modules called "change of base mor- phisms ", (2.2.1.6) ~0" (E~' ~ (ZI)) ---~) ~ Er p'" (Z2) which render commutative all diagrams (2.2.1.7) * E p, q ( ,, (zo) q,*~d~' q)l K(~o) Ep.q~Z r ~, 2t I da, q K(o).~ Ep + r, q + 1-, (Z2). These morphisms are compatible with the usual isomorphism of E,+t
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Unformatted text preview: with the cohomology of (E, d,), in the sense that the diagram ~o* (Ker d~ 'q in EP'q(ZO) -K3~') ! (2.2.1.8) ~,*~ ..... icalprolection) 4, commutes. , Ker d~' q in E~' q (Z2) canonical projection Further, the induced mapping on Eoo is the associated graded of the change of base morphism deduced from (2.2.1.4): (2.2.1.9) (P* RP fzl ,(Kz,) r~) , Rp fz2,(Kz2). For each integer ro~ 1, we say that the formation of E,o commutes with base change if for every Z-scheme q~: Zt~ Z, and all pairs (p, q) of integers, the morphism (2.2.1.6) (2.2.1.10) ~o* (EP, o q (Z1)) -r~,)~ EPg q (Zz) is an isomorphism. We say that the formation of the spectral sequence from E,o on commutes with base change if for all r > to, the formation of E, commutes with base change. 2*...
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