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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 d0smodules with integrable Tconnection, 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 GaussManin 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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This note was uploaded on 12/21/2011 for the course MAP 4341 taught by Professor Normankatz during the Fall '11 term at UNF.
 Fall '11
 NormanKatz

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