A criterion for complete reducibility and some application
A criterion for complete reducibility and some application
Astérisque | 1990

Anglais
Let A be a finite dimensional algebra over a field k, let Y be a f.g. A-module, let A−modY be the full subscategory of endomorphic images of powers of Y in A−mod. We show that if EndAY is self-injective then A−modY and (EndAY)∘−mod are equivalent. The subcategory A−modY is not stable by quotients in A−mod but when it contains the simple modules this leads to a criterion of complete reducibility. This applies to A=kG, Y=IndGUk when G is a finite BN-pair of characteristic p=char(k) and U is one of its Sylow p-subgroups ; EndkGY is a modular Hecke algebra. The equivalence of categories and the criterion of complete redfucibility are applied to certain questions on kG-modules stemming from the study of finite geometries and extensions of simple modules.
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