Dr. Katz DEq Homework Solutions 59

# Dr. Katz DEq Homework Solutions 59 - A l g e b r a i c S o...

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Algebraic Solutions of Differential Equations 59 Because the d,o are horizontal, all the cohomology groups of this complex are coherent d0s-modules with integrable T-connection, hence are locally free, hence their formation commutes with arbitrary change of base. To prove degeneration at E2, it suffices, by the above, to prove it after an arbitrary change of base S'--. S where S' is the spectrum of a field (because S is reduced). But when S is the spectrum of a field of characteristic zero, the degeneration is proved by Deligne in [8]. Q.E.D. 4.1. Topological Interpretation Suppose, in addition to the hypotheses of (4.0.3), that T=Spec(C). By the regularity theorem (cf. [7, 11, 24]), the Gauss-Manin connectioo on the terms of the spectral sequence (4.0.1.6) has "regular singular points". By the fundamental comparison theorem of [7], the functor "germs of analytic local horizontal sections" from the category of coherent sheaves on S with integrable connections relative to C, with regular singular points, to the category of local coefficient systems of
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