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

TL;DR

This paper develops a torsion model framework for tensor-triangulated categories using Tate squares, enabling decomposition of objects and establishing Quillen equivalences, with applications to rational circle-equivariant spectra.

Contribution

It introduces a one-step torsion model approach for tensor-triangulated categories and applies it to rational circle-equivariant spectra, highlighting features for future generalizations.

Findings

Decomposition of objects via Tate squares in tensor-triangulated categories

Establishment of Quillen equivalences with local torsion models

Application to rational circle-equivariant spectra

Abstract

Given a suitable stable monoidal model category $\mathscr{C}$ 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 $\mathscr{C}$ 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 from [18] 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.