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 nth 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.