Dr. Katz DEq Homework Solutions 60

Dr. Katz DEq Homework Solutions 60 - 60 N.M. Katz: T he...

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60 N.M. Katz: The corresponding spectral sequence in locally free (gs.n-modules of finite rank ty-n,k+n__ nk-nIr~an Flan/~'an) Wt-'I (an) -- (~ DR ~xlil n,. .n L"i~/~ ! (4.1.1.2) i,<. .-<i, Rkf,"((a~/s(log O)) a") is of formation compatible with arbitrary change of base S' ~ S an in the category of analytic spaces, thanks to (4.0.3). The canonical morphism of spectral sequences (4.1.1.3) (4.0.1.6)--~ (4.1.1.2) induces horizontal (for the Gauss-Manin connections (4.0.2)) morphisms (4.1.1.4) E, | (gs~, ~ Er(an). d)s In fact, the morphisms (4.1.1.4) are all isomorphisms, because source and target are locally free (_gsa,-modules of finite rank of formation com- patible with arbitrary change of base, and because (by GAGA) the morphism (4.1.1.3) of spectral sequences is an isomorphism when S=Spec(C). Thus the analytic spectral sequence (4.1.1.2) gives rise, via the functor "germs of local horizontal sections" (which is an equivalence of categories between coherent sheaves on
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