 

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


Published: 
October 17, 2001

Keywords: 
Modular representation, idempotent module, Tate cohomology, local cohomology, injective module, Grothendieck duality, spectral sequence, CohenMacaulay ring, Matlis duality, depth, Koszul complex 
Subject: 
Primary 20C20, 20J06; Secondary 13D45, 13H10 


Abstract
Let G be a finite group and k a field of characteristic p dividing G.
Then Greenlees has developed a spectral sequence whose E_{2} 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 kGmodules, and agreeing with the Greenlees spectral sequence
from the E_{2} page onwards. We do the same for the GreenleesLyubeznik spectral
sequence, except that we can only prove that the E_{2} 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 CohenMacaulay.


Acknowledgements
The author is partly supported by a grant from the NSF


Author information
Department of Mathematics, University of Georgia, Athens GA 30602, USA
djb@byrd.math.uga.edu
http://www.math.uga.edu/~djb/

