Dr. Katz DEq Homework Solutions 38

Dr. Katz DEq Homework Solutions 38 - sociated to...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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) Iaa) 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-
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern