Dr. Katz DEq Homework Solutions 38

Dr. Katz DEq Homework Solutions 38 - sociated to the

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38 N.M. Katz: is by definition the "second spectral sequence of hypercohomology" (cf. (2.2.2.2)). In general, we do not know whether or not either the E2 terms or the E~ terms of this spectral sequence are quasi-coherent d~s-modules (though of course the R"f, (127,75(log D)) are quasicoherent). (2.4.0.2) Consider a coveting { V/} of X by affine open sets. From it we construct a Cech bicomplex of quasicoherent (gs-modules c"= c"({ v,}, ; /s(log 09; (2.4.0.3) C a'l* = C a ({ V/}, ~X/S (log D)) = [I(flaa) , ((~x/s (log D) I aa) ffa where aa runs over the a-simplices V/o n. .. n V/o of the nerve of the cover- ing. The homology sheaves of the associated simple complex are the R" f, (g2~;/s(log D)), just because f is quasi-compact and separated, the various f2~x/s (log D) are quasi-coherent (gx-modules, and the d in f2~/s(lOg D) is f- 1 (Cs).linea r.(cf. [ 12], III). (2.4.0.4) The "first" spectral sequence of this bicomplex, the one as-
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Unformatted text preview: sociated to the "Hodge" filtration (2.4.0.5) F j C ~176 = ~ C'({V~}, Ob/s(1Og D)), b>=j gives the usual Hodge ~ De Rham spectral sequence: (2.4.0.6) E~ "b = Rbf, (f~x/s(log D)). The second spectral sequence of this bicomplex, the one associated to the "conjugate" filtration (2.4.0.7) F k C"= Z Ca({V~} , Q~/s(l~ D)) a>k gives rise to a spectral sequence (2.4.0.8) E~'b=/~({V~}, ~sh~,(f2]/s(1Og D)))~ Ra+bf,(f2]/s(logD)) of quasi-coherent d~s-modules (~ denoting the quasi-coherent sheaf associated to a F(S, (.0s)-module), which maps canonically to the conjugate spectral sequence (2.4.0.1), (2.4.0.9) E~2'b=Raf,(~t~ ) ~ Ra+bf,(Q~/s(logD)). Of course, for each affine open coveting of X, the spectral sequence (2.4.0.8) is a spectral sequence of quasi-coherent (gs-modules, but the canonical mapping from (2.4.0.9) need not be an isomorphism, even if we replace (2.4,0.8) by its direct limit over all affine open coverings of X....
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