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Classifying rational $G$–spectra for profinite $G$

David Barnes and Danny Sugrue

Algebraic & Geometric Topology 24 (2024) 3801–3825
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. It 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.

Keywords
equivariant spectra, algebraic models, rational equivariant stable homotopy theory, equivariant sheaves
Mathematical Subject Classification
Primary: 55P91
Secondary: 54B40, 55P42, 55Q91
References
Publication
Received: 10 September 2022
Revised: 4 October 2023
Accepted: 18 November 2023
Published: 9 December 2024
Authors
David Barnes
Mathematical Sciences Research Centre
Queen’s University Belfast
Belfast
Northern Ireland
United Kingdom
Danny Sugrue
Mathematical Sciences Research Centre
Queen’s University Belfast
Belfast
Northern Ireland
United Kingdom

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