Homological Finiteness Conditions for Modules Over Group Algebras

We develop a theory of modules of type FP ∞ over group algebras of hierarchically decomposable groups. This class of groups is denoted h F and contains many different kinds of discrete groups including all countable polylinear groups. Amongst various results, we show that if G is an h F‐group and M a Z G ‐module of type FP ∞ then M has finite projective dimension over Z H for all torsion‐free subgroups H of G . We also show that if G is an h F‐group of type FP ∞ and M is a Z G ‐module which is Z F ‐projective for all finite subgroups F of G , then M has finite projective dimension over Z G . Both of these results have as a special case the striking fact that if G is an h F‐group of type FP ∞ then the torsion‐free subgroups of G have finite cohomological dimension. A further result in this spirit states that every residually finite h F‐group of type FP ∞ has finite virtual cohomological dimension.