Idempotent kG-modules with injective cohomology

Let G be a finite group andk 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 image of Benson, Carlson and Rickard is equal to the injective hull of image 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.