Profinite Direct Sums with Applications to Profinite Groups of Type $Φ_R$

TL;DR

The paper introduces the profinite direct sum $\widehat{\bigoplus}$ as a robust, well-behaved analogue of the ordinary direct sum for profinite $R$-modules, connecting algebraic and categorical viewpoints via cosheaves and bundles and establishing a profinite Mackey's Formula. It proves that the projective dimension of a profinite direct sum is governed by its fibres, and develops Ext/Tor identities compatible with these constructions, enabling a powerful homological toolkit. The framework is then applied to profinite groups, defining type $\Phi_R$ and proving closure under subgroups, with finiteness results on finitistic dimension and several nontrivial examples (including groups with infinite $vcd_R$ that are nonetheless of type $\Phi_R$). These results extend cohomology and representation-theoretic methods in the profinite setting and provide practical tools for studying cohomology theories and representation categories of profinite groups.

Abstract

We show that the "profinite direct sum" is a good notion of infinite direct sums for profinite modules having properties similar to direct sums of abstract modules. For example, the profinite direct sum of projective modules is projective, and there is a Mackey's Formula for profinite modules described using these sums. As an application, we prove that the class of profinite groups of type $Φ_R$ is closed under subgroups.

Theorems & Definitions (50)

• Definition 2.1: gareth
• Proposition 2.2
• proof
• Definition 2.3: ribesgraph Section 5.1
• Proposition 2.4: ribesgraph Proposition 5.1.2
• Example 2.5
• Proposition 2.6
• proof
• Proposition 2.7
• proof