Groups with many finitary cohomology functors

TL;DR

This paper investigates the behavior of cohomology functors in groups, revealing a dichotomy in the set of degrees where these functors commute with filtered colimits, and explores examples and recent literature on the topic.

Contribution

It establishes a dichotomy for hierarchically decomposable groups regarding the set of degrees of cohomology functors that commute with filtered colimits, and demonstrates that any finite or cofinite set can occur.

Findings

For hierarchically decomposable groups, the set of degrees where cohomology functors commute is either finite or cofinite.

Any finite or cofinite set of natural numbers can be realized as such a set in suitable examples.

Includes a survey of recent literature, especially Martin Hamilton's work.

Abstract

For a group G we consider the set of natural numbers n for which the nth cohomology functor of G commutes with filtered colimit systems of coefficient modules. We find that for the large class of hierarchically decomposable groups there is a dichotomy: this set of natural numbers is either finite or cofinite. Moreover any finite or cofinite set is shown to arise in suitable examples. A survey of recent literature on this topic, especially the work of Martin Hamilton, is included.