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Dr. Katz DEq Homework Solutions 21

# Dr. Katz DEq Homework Solutions 21 - Algebraic Solutions of...

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Algebraic Solutions of Differential Equations 21 is endowed with the Gauss-Manin connection (cf. [24], 3.5). For reasons which will soon become apparent (cf. (2.3.3)below), we call this the conjugate spectral sequence, and the corresponding filtration of R"f,(f2~/s(log D)) the conjugate filtration, noted i F~'on. This filtration is horizontal, and is every bit as interesting as the Hodge filtration. (2.3.1) Let us explicate the Ez term of the conjugate spectral sequence with the aid of Cartier's isomorphism (2.1.1). (2.3.1.1) Lemma. The Cartier isomorphism ~ defines an isomorphism of (gs-modules for each pair a, b of non-negative integers (2.3.1.1.1) Ea,~_ ~ b co, 2 - R f, (Jr~ (f2x/s(log D))) , R"f~,P)(tr*(Qbxls(log D))). Proof. From the commutativity of the diagram X- r ~ X~p) S and the fact that F is a homeomorphism, we have isomorphisms (2.3.1.1.2) ~o,E3 'b = Raf, (~'r ~b (t2~,./s (log D))) ~-
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