On a Completion of Cohomological Functors Generalising Tate Cohomology II

TL;DR

This paper studies the properties of Mislin completions of cohomological functors, extending Tate cohomology to broader contexts and revealing their ability to detect finite projective dimensions and support various cohomology products.

Contribution

It advances the understanding of Mislin completions by analyzing their properties, including dimension shifting, product structures, and their role in detecting finite projective dimensions.

Findings

Mislin completions of Ext-functors detect finite projective dimension.

Established a version of dimension shifting for Mislin completions.

Proved the existence of cohomology products like cup and Yoneda products for these completions.

Abstract

Viewing group cohomology as a cohomological functor, G. Mislin has generalised Tate cohomology from finite groups to all discrete groups by defining a completion for cohomological functors in 1994. In a previous paper, we have constructed for a cohomological functor $T^{\bullet}: \mathcal{C} \rightarrow \mathcal{D}$ its Mislin completion $\widehat{T}^{\bullet}: \mathcal{C} \rightarrow \mathcal{D}$ 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.