Duality and local group cohomology

Abstract

Let G G be a group, let k k be a field, and let L {\mathcal {L}} be a local system — an upwardly directed collection of subgroups whose union is G G . In this paper we give a short, elementary proof of the following result: If either A A is a k k - k G kG -bimodule, or else k k is finite dimensional over its center, then Ext G ∗ ⁡ ( A , B ∨ ) = lim ← L ∈ L ⁡ Ext L ∗ ⁡ ( A , B ∨ ) \operatorname {Ext}^{*}_{G}(A,B^{\vee }) =\varprojlim _{L\in \mathcal {L}} \operatorname {Ext}^{*}_{L}(A,B^{\vee }) . From this we deduce as easy corollaries some recent results of Meierfrankenfeld and Wehrfritz on the cohomology of a finitary module.