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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.
Barnes, David 1 ; Sugrue, Danny 1
@article{10_2140_agt_2024_24_3801,
author = {Barnes, David and Sugrue, Danny},
title = {Classifying rational {G{\textendash}spectra} for profinite {G}},
journal = {Algebraic and Geometric Topology},
pages = {3801--3825},
publisher = {mathdoc},
volume = {24},
number = {7},
year = {2024},
doi = {10.2140/agt.2024.24.3801},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2024.24.3801/}
}
TY - JOUR AU - Barnes, David AU - Sugrue, Danny TI - Classifying rational G–spectra for profinite G JO - Algebraic and Geometric Topology PY - 2024 SP - 3801 EP - 3825 VL - 24 IS - 7 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2024.24.3801/ DO - 10.2140/agt.2024.24.3801 ID - 10_2140_agt_2024_24_3801 ER -
%0 Journal Article %A Barnes, David %A Sugrue, Danny %T Classifying rational G–spectra for profinite G %J Algebraic and Geometric Topology %D 2024 %P 3801-3825 %V 24 %N 7 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2024.24.3801/ %R 10.2140/agt.2024.24.3801 %F 10_2140_agt_2024_24_3801
Barnes, David; Sugrue, Danny. Classifying rational G–spectra for profinite G. Algebraic and Geometric Topology, Tome 24 (2024) no. 7, pp. 3801-3825. doi: 10.2140/agt.2024.24.3801
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