On a Completion of Cohomological Functors Generalising Tate Cohomology I

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.