Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The homological spectrum via definable subcategories

Isaac Bird, Jordan Williamson

TL;DR

The paper develops a purity- and definability-based framework to study the homological spectrum of a rigidly-compactly generated tensor-triangulated category. It forges a precise link between the homological spectrum and quotients of the Ziegler spectrum by introducing tensor-closed definable subcategories and the corresponding topologies, culminating in a homeomorphism between $ extsf{Spc}^{ extsf{h}}( extsf{T}^{ extsf{c}})$ and the $GZ$-topologised quotient ${{ extsf{KZg}}_{ ext{Cl}}^{igotimes}}( extsf{T})^{ ext{GZ}}$. The work characterises injective objects in homological residue fields via definable subcategories, relates the homological and Balmer spectra through a functorial and separation-theoretic lens, and provides a concrete example with $ extsf{D}( extbf{Z})$ in which the homological spectrum recovers $ ext{Spec}( extbf{Z})$. This offers a new, purity-driven viewpoint on tensor-triangular geometry and its invariants. The results suggest that the homological spectrum can be effectively studied through Ziegler-theoretic methods, with implications for functoriality and structural comparisons with the Balmer spectrum.

Abstract

We develop an alternative approach to the homological spectrum of a tensor-triangulated category through the lens of definable subcategories. This culminates in a proof that the homological spectrum is homeomorphic to a quotient of the Ziegler spectrum. Along the way, we characterise injective objects in homological residue fields in terms of the definable subcategory corresponding to a given homological prime. We use these results to give a purity perspective on the relationship between the homological and Balmer spectrum.

The homological spectrum via definable subcategories

TL;DR

The paper develops a purity- and definability-based framework to study the homological spectrum of a rigidly-compactly generated tensor-triangulated category. It forges a precise link between the homological spectrum and quotients of the Ziegler spectrum by introducing tensor-closed definable subcategories and the corresponding topologies, culminating in a homeomorphism between and the -topologised quotient . The work characterises injective objects in homological residue fields via definable subcategories, relates the homological and Balmer spectra through a functorial and separation-theoretic lens, and provides a concrete example with in which the homological spectrum recovers . This offers a new, purity-driven viewpoint on tensor-triangular geometry and its invariants. The results suggest that the homological spectrum can be effectively studied through Ziegler-theoretic methods, with implications for functoriality and structural comparisons with the Balmer spectrum.

Abstract

We develop an alternative approach to the homological spectrum of a tensor-triangulated category through the lens of definable subcategories. This culminates in a proof that the homological spectrum is homeomorphic to a quotient of the Ziegler spectrum. Along the way, we characterise injective objects in homological residue fields in terms of the definable subcategory corresponding to a given homological prime. We use these results to give a purity perspective on the relationship between the homological and Balmer spectrum.
Paper Structure (15 sections, 27 theorems, 50 equations, 2 figures)

This paper contains 15 sections, 27 theorems, 50 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Purity and definability
  3. Modules
  4. Purity and definability
  5. The fundamental correspondence
  6. The Ziegler spectrum
  7. Definability along the Yoneda embedding
  8. Localisations and definable subcategories
  9. An atlas of spectra
  10. Tensors and modules
  11. Definable tensor closure
  12. Homological residue fields
  13. Comparing the Ziegler and homological spectrum
  14. Separation properties
  15. An example: the spectrum of $\mathsf{D}(\mathbb{Z})$

Key Result

Theorem 1

Let $\mathsf{T}$ be a rigidly-compactly generated tensor-triangulated category. Then there is a homeomorphism

Figures (2)

  • Figure 1: A schematic of the underlying sets of the topological spaces under consideration.
  • Figure 2: Solid maps are continuous, whereas dashed maps are only functions. A bullet $\bullet$ on the map indicates that the map is closed (and hence open since all the maps are bijections).

Theorems & Definitions (54)

  • Theorem : \ref{['hspecisgz']}
  • Theorem : \ref{['determineinjs']}
  • Theorem : \ref{['simple']}
  • Definition 2.1
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Proposition 2.8: krsmash
  • Lemma 2.9
  • proof
  • ...and 44 more