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Gheorghiu, Max

University of British Columbia

Viewing group cohomology as a so-called cohomological functor, G. Mislin has generalized Tate cohomology from finite groups to all discrete groups by defining a completion for cohomological functors in his paper "Tate Cohomology for Arbitrary Groups via Satellites". We construct a Mislin completion for any cohomological functor whose domain category is an abelian category with enough projectives and whose codomain category is an abelian category in which all countable direct limits exist and are exact. This 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 topological group all whose points are closed. More specifically, we generalize four constructions of Mislin completions from the literature, prove that they yield isomorphic cohomological functors and provide explicit formulae for their connecting homomorphisms. For any morphism in the domain category we develop formulae for the induced morphism in each degree of the Mislin completion in terms of each construction. As their main feature, Mislin completions of Ext-functors detect finite projective dimension of objects in the domain category. Moreover, we establish a version of dimension shifting, an Eckmann-Shapiro result as well as Yoneda and external products.

Text

2024-04-17

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/87860

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/