On a Completion of Cohomological Functors Generalising Tate Cohomology I

TL;DR

This paper introduces a unified framework for Tate cohomology via a specific completion of cohomological functors, extending its applicability to new mathematical contexts like condensed mathematics.

Contribution

It provides a general completion method that explains various existing generalizations of Tate cohomology and applies to broader settings such as topological groups.

Findings

Unified completion explains different approaches to Tate cohomology.

Extension of Tate cohomology to condensed mathematics.

Defines Tate cohomology for any T1 topological group.

Abstract

Tate cohomology has been generalised by several authors using different constructions that have applications in group theory, ring theory and homotopical algebra. Therefore, there is a need for a uniform account that explains why their underlying approaches all lead to the same conclusions. The key notion in such a uniform theory is a specific completion of cohomological functors that is constructed under mild assumptions. This completion takes Tate cohomology to settings where it has never been introduced such as in condensed mathematics. Through the latter, one can define Tate cohomology for any $T1$ topological group.