Idempotent Modules in the Stable Category

Let G be a finite group and k be an algebraically closed field of prime characteristic. Corresponding to each closed homogeneous subvariety W of the maximal ideal spectrum of H*(G, k) we construct (usually infinite-dimensional) kG-modules E(W) and F(W) which are idempotent in the sense that E(W) and F(W) are isomorphic (up to projective summands) to E(W) ⊗ E(W) and F(W) ⊗ F(W) respectively. We study the properties of these modules, and as an application we use them to describe natural direct sum decompositions of modules in quotient categories.