The projective dimension of profinite modules for pro-p groups

TL;DR

This paper develops homological criteria to determine projective dimensions of modules over pro-p groups, leading to new characterizations of free pro-p groups and structural insights into groups with infinitely many ends.

Contribution

It introduces a homology-based criterion for finite projective dimension, providing new characterizations of free pro-p groups and structural theorems for groups with infinitely many ends.

Findings

A homology criterion for finite projective dimension of modules.

A new characterization of free pro-p groups.

A structure theorem for pro-p groups with infinitely many ends.

Abstract

The homology groups introduced by A. Brumer can be used to establish a criterion ensuring that a profinite $\mathbb{F}_p[[G]]$-module of a pro-$p$ group $G$ has projective dimension $d<\infty$ (cf. Thm. A). This criterion yields a new characterization of free pro-$p$ groups (cf. Cor. B). Applied to a semi-direct factor $G\to\mathbb{Z}_p\to G$ isomorphic to $\mathbb{Z}_p$ which defines a non-trivial end in the sense of A.A. Korenev one concludes that the closure of the normal closure of the image of $\sigma$ is a free pro-$p$ subgroup (cf. Thm. C). From this result we will deduce a structure theorem (cf. Thm. D) for finitely generated pro-$p$ groups with infinitely many ends.