On profinite groups of type FP∞

Abstract

Suppose R is a profinite ring. We construct a large class of profinite groups L′HRˆF, including all soluble profinite groups and profinite groups of finite cohomological dimension over R. We show that, if G∈L′HRˆF is of type FP∞ over R, then there is some n such that HRn(G,R〚G〛)≠0, and deduce that torsion-free soluble pro-p groups of type FP∞ over Zp have finite rank, thus answering the torsion-free case of a conjecture of Kropholler.