Torsion models for tensor-triangulated categories

TL;DR

This paper constructs a torsion model for certain tensor-triangulated categories, generalizing algebraic and homotopical reconstructions, and extends previous work to higher dimensions.

Contribution

It introduces a new torsion model for finite-dimensional, Noetherian tensor-triangulated categories, generalizing prior one-dimensional results.

Findings

Constructs an adelic torsion model equivalent to the original category.

Categorifies the Cousin complex in algebra and chromatic homotopy theory.

Extends previous one-dimensional models to higher-dimensional settings.

Abstract

Given a rigidly-compactly generated tensor-triangulated category whose Balmer spectrum is finite dimensional and Noetherian, we construct a torsion model for it, which is equivalent to the original tensor-triangulated category. The torsion model is determined in an adelic fashion by objects with singleton supports. This categorifies the Cousin complex from algebra, and the process of reconstructing a spectrum from its monochromatic layers in chromatic stable homotopy theory. This model is inspired by work of the second author in rational equivariant stable homotopy theory, and extends previous work of the authors from the one-dimensional setting.