Dr. Katz DEq Homework Solutions 43

Dr. Katz DEq Homework Solutions 43 - Algebraic Solutions of...

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Algebraic Solutions of Differential Equations 43 (3.2.3) Corollary. Hypotheses and notations as in (3.2.2), the indecom- posable central idempotents in K [G], which all lie in (P have reductions modulo t~ in k [G] which remain indecomposable. Proof. The number of e i is equal to the number off~, both being the number of conjugacy classes in G. Hence every e i lifts an ft. Q.E.D. In terms of representations, this gives: (3.2.3 bis) Hypotheses as in (3.2.2bis), the reduction mod p of any K-irreducible representation of G in a free C-module of finite rank is irreducible (and every irreducible representation of G in a finite-dimen- sional k-space arises this way). (3.2.3.1) (3.2.2his), every irreducible repre- sentation of G in a finite-dimensional k-space is absolutely irreducible. This follows from (3.3.0bis), applied to arbitrary finite exten- sions K' of K, and arbitrary extensions of the valuation of K to K', by which we can realize arbitrary finite extensions k' of k as residue field.
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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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