Dr. Katz DEq Homework Solutions 27

Dr. Katz DEq Homework Solutions 27 - 144tt...

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Algebraic Solutions of Differential Equations 27 and the degeneration of the Hodge =~ De Rham spectral sequence gives us a surjection ( R"f, (f2~/s (log D)) --~ R"f, (d~x) = E ~ Putting these together, we obtain the diagram which defines the Hasse- Witt operations: r "~ ~ R"f, (f2;/s (log D)) --~ E ~ (2, /l ]l Fg~ R"f, (ex) . .................. n_:__w_ ................ :,~ R"f, (6x) The matrix of this operation in a local base of R"f, ((_gx) is called the Hasse- Witt matrix of X/S in dimension n. The composite p-linear mapping ( R,f,(Cx) - F*bs ,F~R,f,(Cgx ) n-w ,Rnf,(Cx) is the one induced by the p-th power endomorphism of d~ x. From the definition (, it follows that we have: ( Proposition. Hypotheses as in (2.3.2),
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Unformatted text preview: 144tt operation ( H-W: F* s R"f, (d~x) ~ R"f, ((gx) is none other than the intersection (F~"on ~ F 1) (R"f, (f2~/s(log D))). ( Corollary (Assumptions as in (2.3.2)). In order to have a direct sum decomposition ( R"f,(f2~/s(log D)), ~ F ~ GFc"on it is necessary and sufficient that the Hasse-Witt operation ( H-W: F*s R"f,((~x)~ R"f, ff)x) be an isomorphism. Proof. Since the statements whose equivalence is asserted are both of the form "a certain homomorphism of locally free S-modules of finite rank is an isomorphism", and because the formation of these modules commutes with all changes of base S' --* S, we are immediately reduced to the case in which S is the spectrum of a field K. The isomorphisms ~g ( co~E~ 'b ~,~ F~*~E~ 'a...
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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.

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