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HOMOLOGICAL FINITENESS CONDITIONS FOR MODULES OVER GROUP ALGEBRAS

Published online by Cambridge University Press:  01 August 1998

JONATHAN CORNICK
Affiliation:
Centre de Recerca Matematica, Institut d'Estudis Catalans, Apartat 50, E 08193 Bellaterra, Spain. E-mail: ICRM6@cc.uab.es
PETER H. KROPHOLLER
Affiliation:
School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS. E-mail: P.H.Kropholler@qmw.ac.uk
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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.

Type
Notes and Papers
Copyright
The London Mathematical Society 1998

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