On Profinite Groups of Type ${\mathrm{FP}}_{\infty }$

Abstract

Suppose $R$ is a profinite ring. We construct a large class of profinite groups $\widehat{{\mathbf{L}}^{\prime }{\mathbf{H}}_R}\mathfrak{F}$, including all soluble profinite groups and profinite groups of finite cohomological dimension over $R$. We show that, if $G\in \widehat{{\mathbf{L}}^{\prime }{\mathbf{H}}_R}\mathfrak{F}$ is of type ${\mathrm{FP}}_{\infty }$ over $R$, then there is some $n$ such that $H_R^n(G,R[[G]])\ne 0$, and deduce that torsion-free soluble pro-$p$ groups of type ${\mathrm{FP}}_{\infty }$ over ${\mathbb{Z}}_p$ have finite rank, thus answering the torsion-free case of a conjecture of Kropholler.