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

Suppose $R$ is a profinite ring. We construct a large class of profinite groups $\widehat{{\mathbf{L}}^{\mathrm{\prime }}{\mathbf{H}}_R}\mathfrak{F}$, including all soluble profinite groups and profinite groups of finite cohomological dimension over $R$. We show that, if $G \in \widehat{{\scriptstyle\bf L}'{\scriptstyle\bf H}R}\mathfrak{F}$isoftype$\operatorname{FP}\infty$over$R$,thenthereissome$n$suchthat$H_R^n(G,R [[ G ]]) \neq 0$,anddeducethattorsion-freesolublepro-$p$groupsoftype$\operatorname{FP}_\infty$over$$\mathbb{Z}$_p$ have finite rank, thus answering the torsion-free case of a conjecture of Kropholler.