When is Group Cohomology Finitary?

Abstract

If $G$ is a group, then we say that the functor $H^n(G,-)$ is finitary if it commutes with all filtered colimit systems of coefficient modules. We investigate groups with cohomology almost everywhere finitary; that is, groups with $n$th cohomology functors finitary for all sufficiently large $n$. We establish sufficient conditions for a group $G$ possessing a finite dimensional model for $e.g.$ to have cohomology almost everywhere finitary. We also prove a stronger result for the subclass of groups of finite virtual cohomological dimension, and use this to answer a question of Leary and Nucinkis. Finally, we show that if $G$ is a locally (polycyclic-by-finite) group, then $G$ has cohomology almost everywhere finitary if and only if $G$ has finite virtual cohomological dimension and the normalizer of every non-trivial finite subgroup of $G$ is finitely generated.