Generic Idempotent Modules for a Finite Group

Let G be a nite group and k an algebraically closed eld of characteristic p. Let F U be the Rickard idempotent kG-module corresponding to the set U of subvarieties of the cohomology variety VG which are not irreducible components. We show that F U is a nite sum of generic modules corresponding to the irreducible components of VG. In this context, a generic module is an indecomposable module of innnite length over kG but nite length as a module over its endomorphism ring. 1. Introduction The purpose of this paper is to draw attention to a particular innnite dimensional module for a nite group algebra, which plays the role of a generic module in a certain precise sense which will be described. The module in question is a particular case of Rickard's construction 14] of idempotent modules in the stable category. These idempotent modules have played a pivotal role in some recent developments in modular representation theory, see for example 1, 3, 4, 5].