Classifying rational G-spectra for profinite G

TL;DR

This paper develops an algebraic model for rational G-spectra for any profinite group G, extending previous finite group results and linking homological dimension to the topological structure of G.

Contribution

It generalizes the algebraic modeling of rational G-spectra to profinite groups and relates homological dimension to the Cantor--Bendixson rank of subgroup spaces.

Findings

Constructed an algebraic model for rational G-spectra for profinite G.

Determined the homological dimension based on the Cantor--Bendixson rank.

Calculated the homological dimension of rational Mackey functors.

Abstract

For G an arbitrary profinite group, we construct an algebraic model for rational G-spectra in terms of G-equivariant sheaves over the space of subgroups of G. This generalises the known case of finite groups to a much wider class of topological groups, and improves upon earlier work of the first author on the case where G is the p-adic integers. As the purpose of an algebraic model is to allow one to use homological algebra to study questions of homotopy theory, we prove that the homological dimension (injective dimension) of the algebraic model is determined by the Cantor--Bendixson rank of the space of closed subgroups of the profinite group G. This also provides a calculation of the homological dimension of the category of rational Mackey functors.