Modules with Injective Cohomology, and Local Duality for a Finite Group

Let G be a finite group and k a field of characteristic p dividing |G|. Then Greenlees has developed a spectral sequence whose E2 term is the local cohomology of H∗(G, k) with respect to the maximal ideal, and which converges to H∗(G, k). Greenlees and Lyubeznik have used Grothendieck’s dual localization to provide a localized form of this spectral sequence with respect to a homogeneous prime ideal p in H∗(G, k), and converging to the injective hull Ip of H∗(G, k)/p. The purpose of this paper is give a representation theoretic interpretation of these local cohomology spectral sequences. We construct a double complex based on Rickard’s idempotent kG-modules, and agreeing with the Greenlees spectral sequence from the E2 page onwards. We do the same for the Greenlees-Lyubeznik spectral sequence, except that we can only prove that the E2 pages are isomorphic, not that the spectral sequences are. Ours converges to the Tate cohomology of the certain modules κp introduced in a paper of Benson, Carlson and Rickard. This leads us to conjecture that Ĥ∗(G, κp) ∼= Ip , after a suitable shift in degree. We draw some consequences of this conjecture, including the statement that κp is a pure injective module. We are able to prove the conjecture in some cases, including the case where H∗(G, k)p is Cohen–Macaulay.