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Torsion models for tensor-triangulated categories: the one-step case

Scott Balchin, John Greenlees, Luca Pol and Jordan Williamson

Algebraic & Geometric Topology 22 (2022) 2805–2856
DOI: 10.2140/agt.2022.22.2805
Abstract

Given a suitable stable monoidal model category 𝒞 and a specialization closed subset V of its Balmer spectrum, one can produce a Tate square for decomposing objects into the part supported over V and the part supported over V c spliced with the Tate object. Using this one can show that 𝒞 is Quillen equivalent to a model built from the data of local torsion objects, and the splicing data lies in a rather rich category. As an application, we promote the torsion model for the homotopy category of rational circle-equivariant spectra of Greenlees (1999) to a Quillen equivalence. In addition, a close analysis of the one-step case highlights important features needed for general torsion models, which we will return to in future work.

Keywords
algebraic model, tensor-triangulated category, Balmer spectrum, rational equivariant spectra
Mathematical Subject Classification
Primary: 55P60
Secondary: 13D09, 18N40, 55P91
References
Publication
Received: 8 December 2020
Revised: 11 June 2021
Accepted: 17 July 2021
Published: 13 December 2022
Authors
Scott Balchin
Max Planck Institute for Mathematics
Bonn
Germany
John Greenlees
Mathematics Institute
University of Warwick
Coventry
United Kingdom
Luca Pol
Fakultät für Mathematik
Universität Regensburg
Regensburg
Germany
Jordan Williamson
Department of Algebra
Charles University in Prague
Praha
Czech Republic