HOMOLOGICAL FINITENESS CONDITIONS FOR MODULES OVER GROUP ALGEBRAS

Abstract

We develop a theory of modules of type FP_∞ over group algebras of hierarchically decomposable groups. This class of groups is denoted H[Fscr ] and contains many different kinds of discrete groups including all countable polylinear groups. Amongst various results, we show that if G is an h[Fscr ]-group and Mis a ℤG-module of type FP_∞ then M has finite projective dimension over ℤH for all torsion-free subgroups H of G. We also show that if G is an h[Fscr ]-group of type FP_∞ and M is a ℤG-module which is ℤF-projective for all finite subgroups F of G, then M has finite projective dimension over ℤG. Both of these results have as a special case the striking fact that if G is an h[Fscr ]-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[Fscr ]-group of type FP_∞ has finite virtual cohomological dimension.