Duality and local group cohomology

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