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

Scott Balchin, J. P. C. Greenlees, Luca Pol, Jordan Williamson

TL;DR

This work develops a comprehensive framework to reconstruct tensor-triangulated categories from local torsion data via an adelic torsion model. Central to the approach is a one-step Tate-type diagram that expresses the unit as a homotopy pullback of localized and completed pieces, yielding a Quillen equivalence between the original model category and a cellularization of a diagram of module categories built from $L ext{1}$, $oldsymbol{ extLambda} ext{1}$, and their smash products. The paper introduces the pre-torsion model and its extension to the full torsion model, showing how adelic data enforces algebraicity and enabling abelian/differential models in special cases such as rational $ ext{T}$-spectra. Along the way, it develops a robust theory of localization, completion, and support in finite-dimensional Noetherian Balmer spectra, with concrete applications to derived algebra, chromatic homotopy, and rational equivariant spectra. The results culminate in a Quillen equivalence between rational $ ext{T}$-spectra and a differential model, establishing a concrete, algebraic foothold for the torsion-based reconstruction program in tensor-triangulated settings.

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.

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

TL;DR

This work develops a comprehensive framework to reconstruct tensor-triangulated categories from local torsion data via an adelic torsion model. Central to the approach is a one-step Tate-type diagram that expresses the unit as a homotopy pullback of localized and completed pieces, yielding a Quillen equivalence between the original model category and a cellularization of a diagram of module categories built from , , and their smash products. The paper introduces the pre-torsion model and its extension to the full torsion model, showing how adelic data enforces algebraicity and enabling abelian/differential models in special cases such as rational -spectra. Along the way, it develops a robust theory of localization, completion, and support in finite-dimensional Noetherian Balmer spectra, with concrete applications to derived algebra, chromatic homotopy, and rational equivariant spectra. The results culminate in a Quillen equivalence between rational -spectra and a differential model, establishing a concrete, algebraic foothold for the torsion-based reconstruction program in tensor-triangulated settings.

Abstract

Given a suitable stable monoidal model category and a specialization closed subset of its Balmer spectrum one can produce a Tate square for decomposing objects into the part supported over and the part supported over spliced with the Tate object. Using this one can show that 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.

Paper Structure

This paper contains 37 sections, 35 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. The dimension filtration
  4. The one-step context
  5. The torsion model
  6. Rational $\mathbb{T}$-spectra
  7. Diagrams of module categories
  8. Module categories
  9. Diagrams of model categories
  10. Tensor-triangulated categories and their enhancements
  11. Balmer spectrum
  12. Sufficiently nice model categories
  13. Localization, completion and support
  14. Localization, completion and support
  15. Properties of the functors
  16. ...and 22 more sections

Key Result

Theorem 1

There is a Quillen equivalence The effect of the cellularization is that an object of the tensor-triangulated category $h\mathscr{C}$ is equivalent to one specified by:

Theorems & Definitions (89)

  • Theorem
  • Theorem
  • Example 2.1
  • Example 2.2
  • Lemma 2.3: changeofgroups
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Theorem 2.7: GreenleesShipley14b
  • Definition 3.1
  • ...and 79 more