Dr. Katz DEq Homework Solutions 77

Dr. Katz DEq Homework Solutions 77 - Algebraic Solutions of...

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Algebraic Solutions of Differential Equations 77 Then after the change of base Spec(C)--~ T (cf. the diagram ( each of the coherent sheaves with integrable connection on Sc (P(x) R"fc.(t2~Cc/sc(log Oc)) V), x~A becomes trivial on a finite &ale covering q9: S' ~ Sc, i, e., becomes isomorphic to (((gs,) b"(z), d), where b,(z) is the rank of P(X) R'f,(f~/s(log D)) over the generic point of S. 5.4. The Birational Point of View (5.4.0) Let Sc be a connected smooth C-scheme, and (Me, 17c) a locally free sheaf of finite rank with integrable connection on Sc. (5.4.1) Remark. In order that there exist a finite 6tale covering of Sc on which (Me, Vc) becomes trivial (as locally free sheaf with integrable connection), it is necessary and sufficient that there exist a finite exten- sion K of the function field L of Sc such that (Me, Vc)| K is trivial (i. e., so that Mc| is spanned over K by horizontal sections). (5.4.2)
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