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

摘要

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 convergesto H*(G,k). Greenlees and Lyubeznik have used Grothendieck's dual localizationto 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 hullIp of H*(G,k)/p.The purpose of this paper is give a representation theoretic interpretation ofthese local cohomology spectral sequences. We construct a double complex based onRickard's idempotent kG-modules, and agreeing with the Greenlees spectral sequencefrom 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, notthat the spectral sequences are. Ours converges to the Tate cohomology of thecertain modules κp introduced in a paper of Benson, Carlsonand Rickard. This leads us to conjecture that hatH*(G,κp)≅ Ip, after a suitable shift in degree. We draw some consequencesof 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.