SMF

A criterion for complete reducibility and some application

A criterion for complete reducibility and some application

Marc CABANES
A criterion for complete reducibility and some application
     
                
  • Année : 1990
  • Tome : 181-182
  • Format : Électronique
  • Langue de l'ouvrage :
    Anglais
  • Pages : 93-112
  • DOI : 10.24033/ast.5

Let A be a finite dimensional algebra over a field k, let Y be a f.g. A-module, let AmodY be the full subscategory of endomorphic images of powers of Y in Amod. We show that if EndAY is self-injective then AmodY and (EndAY)mod are equivalent. The subcategory AmodY is not stable by quotients in Amod 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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