Idempotent kG-modules with injective cohomology

Let G be a finite group and k a field of characteristic p. This paper gives a short proof of a recent theorem of Benson and Greenlees, stating that the cohomology of the kappa module κp of Benson, Carlson and Rickard is equal to the injective hull of H∗(G,k)/p as a graded H∗(G,k)-module. This shorter proof does not naturally extend to compact Lie groups, whereas the version of Benson and Greenlees is proved in this broader context.