Finiteness of cohomological dimension for arbitrary groups

Determine whether finiteness of the homological dimension of an arbitrary group over a commutative coefficient ring implies finiteness of its cohomological dimension.

Background

The paper places its Gorenstein results in the context of the classical homological and cohomological dimensions of groups. For countable groups, Bieri’s inequality gives a bound of one between these dimensions, and analogous cardinality bounds settle the implication for groups of cardinality less than aleph-omega. The general case remains unresolved, so the finiteness implication is an explicitly stated classical open problem rather than a question answered by the paper.

References

Thus, in the classical setting, the question whether the finiteness of $\hd_kG$ implies that of $\cd_kG$ is settled for groups of cardinality less than $\aleph_\omega$, while it remains open in general.

— $\aleph_m$-presented Gorenstein flat modules and Gorenstein dimensions of groups  (2609.24512 - Stergiopoulou, 21 Sep 2026) in Section 1, Introduction