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.