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 I_p 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 κ_p introduced in a paper of Benson, Carlson and Rickard. This leads us to conjecture that \hatH*(G,κ_p) ≅ I_p, after a suitable shift in degree. We draw some consequences of this conjecture, including the statement that κ_p 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.

The author is partly supported by a grant from the NSF