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Dr. Katz DEq Homework Solutions 39

Dr. Katz DEq Homework Solutions 39 - Algebraic Solutions of...

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Algebraic Solutions of Differential Equations 39 (2.4.1) In the special case when S is a schema of characteristic p, the conjugate spectral sequence (2.4.0.1) is the second spectral sequence of hypercohomology for the (,0xcp)-linear complex F,(O}/s(log D)) on X ~p), and the functors R"J~ ~. Because f~P): X ~p)--, S is quasi-compact and separated, the various F,(f~x/s(log D)) are quasi-coherent (gx-modules, and F,(d) is Cxc~,-linear, the second spectral sequence (2.4.0.8) of the Cech bicomplex does map isomorphically to the conjugate spectral sequence (2.4.0.9), which is a spectral sequence of quasi-coherent (_9 s- modules. The simple interpretation of the conjugate spectral sequence in characteristic p as the second spectral sequence of the Cech bicomplex (2.4.0,3) makes possible effective calculations, as we shall see. On the contrary, the conjugate spectral sequence over C remains in shadow. 3. The Main Technical Result on the p-Curvature of the Gauss-Manin Connection 3.0. We return to the geometric situation 1.0, and assume as before that S, and hence T, is a scheme of characteristic p. Recall that for any (9 s-
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