A filtration of the modular representation functor

Let F and K be algebraically closed fields of characteristics p>0 and 0, respectively. For any finite group G we denote by KRF(G)=K⊗ZG0(FG) the modular representation algebra of G over K where G0(FG) is the Grothendieck group of finitely generated FG-modules with respect to exact sequences. The usual operations induction, inflation, restriction, and transport of structure with a group isomorphism between the finitely generated modules of group algebras over F induce maps between modular representation algebras making KRF an inflation functor. We show that the composition factors of KRF are precisely the simple inflation functors where C ranges over all nonisomorphic cyclic p′-groups and V ranges over all nonisomorphic simple KOut(C)-modules. Moreover each composition factor has multiplicity 1. We also give a filtration of KRF.