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The Spectrum of Stable Infinity Categories with Actions

Hisato Matsukawa

TL;DR

This work defines the relative Matsui spectrum $\mathrm{Spec}(D,S)$ for a stable $\infty$-category $D$ with an action by a stable $\infty$-category $S$, unifying Balmer's tensor triangular spectrum with Matsui's approach. It proves that $\mathrm{Spc}(D,S)$ is the universal support datum and develops its fundamental properties, including descent, localization commutativity, and a morphism to the base spectrum $\mathrm{Spec}_{\otimes}(S)$, together with a finite-type version $\mathrm{Spec}^{\mathrm{fin}}(D,S)$. The framework recovers classical geometric spaces from categorical data across a wide range of settings—perfect complexes on schemes, twisted derived categories, singularity categories, and derived matrix factorization categories—thereby extending tensor triangular geometry beyond globally tensorial contexts while preserving geometric intuition. These results yield explicit descriptions in concrete examples (Koszul algebras, Severi–Brauer schemes, and DMF) and provide new tools for classifying thick submodules via prime-like subobjects in enriched categorical settings.

Abstract

We introduce the relative Matsui spectrum, a new invariant associated with a stable \(\infty\)-category equipped with an action. This construction generalizes both Balmer's tensor triangular spectra and Matsui's triangular spectra, and provides a unified framework for classifying thick submodules. We establish its fundamental properties, including universality, comparison with existing spectra, and descent, and construct a natural morphism to the Balmer spectrum of the base. Applications show that the relative Matsui spectrum recovers the underlying classical geometric spaces from categorical data in various settings: categories of perfect complexes of schemes, twisted derived categories, categories of singularities, and derived matrix factorization categories. Thus the relative Matsui spectrum extends the reach of tensor triangular geometry beyond globally tensorial settings, while preserving geometric intuition.

The Spectrum of Stable Infinity Categories with Actions

TL;DR

This work defines the relative Matsui spectrum for a stable -category with an action by a stable -category , unifying Balmer's tensor triangular spectrum with Matsui's approach. It proves that is the universal support datum and develops its fundamental properties, including descent, localization commutativity, and a morphism to the base spectrum , together with a finite-type version . The framework recovers classical geometric spaces from categorical data across a wide range of settings—perfect complexes on schemes, twisted derived categories, singularity categories, and derived matrix factorization categories—thereby extending tensor triangular geometry beyond globally tensorial contexts while preserving geometric intuition. These results yield explicit descriptions in concrete examples (Koszul algebras, Severi–Brauer schemes, and DMF) and provide new tools for classifying thick submodules via prime-like subobjects in enriched categorical settings.

Abstract

We introduce the relative Matsui spectrum, a new invariant associated with a stable -category equipped with an action. This construction generalizes both Balmer's tensor triangular spectra and Matsui's triangular spectra, and provides a unified framework for classifying thick submodules. We establish its fundamental properties, including universality, comparison with existing spectra, and descent, and construct a natural morphism to the Balmer spectrum of the base. Applications show that the relative Matsui spectrum recovers the underlying classical geometric spaces from categorical data in various settings: categories of perfect complexes of schemes, twisted derived categories, categories of singularities, and derived matrix factorization categories. Thus the relative Matsui spectrum extends the reach of tensor triangular geometry beyond globally tensorial settings, while preserving geometric intuition.
Paper Structure (13 sections, 44 theorems, 57 equations)

This paper contains 13 sections, 44 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Construction
  3. Universality
  4. Fundamental Results
  5. Commutativity of Localizations
  6. The Morphism to the Spectrum of the Base
  7. Applications
  8. Some properties for calculating relative Matsui spectra
  9. Derived categories of perfect complexes of schemes
  10. Twisted derived categories and Severi--Brauer schemes
  11. Koszul algebras
  12. The categories related to singularities
  13. Derived matrix factorization categories

Key Result

Lemma 2.3

Let $I \in S$ and $a \in T \setminus I$. Then there exists an $S$-prime $P \supset I$ such that $a \notin P$.

Theorems & Definitions (98)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Example 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 2.8
  • ...and 88 more