On a Completion of Cohomological Functors Generalising Tate Cohomology II

Abstract: Viewing group cohomology as a so-called cohomological functor, G. Mislin has generalised Tate cohomology from finite groups to all discrete groups by defining a completion for cohomological functors in [24]. For any cohomological functor $T^{\bullet}: \mathcal{C} \rightarrow \mathcal{D}$ we have constructed its Mislin completion $\widehat{T}^{\bullet}: \mathcal{C} \rightarrow \mathcal{D}$ in [15] under mild assumptions on the abelian categories $\mathcal{C}$ and $\mathcal{D}$ which generalises Tate cohomology to all $T1$ topological groups. In this paper we investigate the properties of Mislin completions. As their main feature, Mislin completions of Ext-functors detect finite projective dimension of objects in the domain category. We establish a version of dimension shifting, an Eckmann-Shapiro result as well as cohomology products such as external products, cup products and Yoneda products.