Technical Summary: The Spectrum of Stable Infinity Categories with Actions

Problem Statement

The study of spectra associated with triangulated or stable $\infty$-categories serves as a bridge between abstract categorical structures and underlying geometry. Two primary frameworks exist:

1. Balmer's Tensor Triangular Spectrum: Introduced by Balmer [Bal05], this associates a locally ringed space $\text{Spec}^\otimes(\mathcal{T})$ to a tensor triangulated category $\mathcal{T}$, classifying thick tensor ideals. It successfully recovers geometric spaces (e.g., schemes) from categories of perfect complexes. However, it requires a global tensor product structure.
2. Matsui's Triangular Spectrum: Proposed by Matsui [Mat21] and further developed by Hirano, Ouchi, and others [HO22, HO24, HKO24, Ito23, IM24], this generalizes the concept to triangulated categories lacking a tensor structure, classifying thick subcategories.

A gap remains in unifying these approaches and extending them to settings where a category $\mathcal{D}$ possesses an action by another category $\mathcal{S}$ (an $\mathcal{S}$-module) but lacks a global tensor product on $\mathcal{D}$ itself. The paper addresses the need for a unified invariant that classifies thick $\mathcal{S}$-submodules of a stable $\infty$-category $\mathcal{D}$ equipped with an action of a stable $\infty$-category $\mathcal{S}$, while recovering classical geometric intuition in various non-tensorial or relative settings.

Methodology

The author constructs a new invariant called the relative Matsui spectrum, denoted $\text{Spec}(\mathcal{D}, \mathcal{S}, \mathcal{B})$, for a datum $(\mathcal{D}, \mathcal{S}, \mathcal{B})$ consisting of algebra objects $\mathcal{B}, \mathcal{S} \in \text{Cat}^{\text{perf}}_\infty$, a homomorphism $\phi: \mathcal{B} \to \mathcal{S}$, and a left $\mathcal{S}$-module $\mathcal{D} \in \text{Cat}^{\text{perf}}_\infty$.

Construction:

1. Primes and Topology: The underlying topological space is defined via $\mathcal{S}$-primes. A thick $\mathcal{S}$-submodule $P \subset \mathcal{D}$ is an $\mathcal{S}$-prime if the intersection of any family of $\mathcal{S}$-submodules strictly containing $P$ is also strictly containing $P$. The space $\text{Spc}(\mathcal{D}, \mathcal{S})$ is the set of such primes equipped with a topology generated by supports $\text{Supp}(W) = \{P \mid W \not\subset P\}$.
2. Structure Sheaf: The structure sheaf $\mathcal{O}$ is constructed using natural transformations of identity functors on localized categories, specifically $\mathcal{O}(U) = \text{Nat}^0_{\mathcal{B}}(\text{Id}_{\mathcal{D}_U}, \text{Id}_{\mathcal{D}_U})$, where $\mathcal{D}_U$ is the localization of $\mathcal{D}$ at the open set $U$.
3. Universality: The spectrum is characterized as the universal support datum. A support datum is a topological $T_0$-space equipped with a map from objects of $\mathcal{D}$ to closed subsets satisfying shift invariance, action compatibility, direct sum additivity, exactness, and distinguishability. The relative Matsui spectrum is shown to be the initial object in the category of such support data.

Key Technical Tools:

• Commutativity of Localizations: The paper utilizes conditions under which localizations of stable $\infty$-categories commute (Lemma 5.1, Lemma 5.4), which is crucial for establishing descent properties and fiberwise decompositions.
• Admissibility and Smashing: The author introduces conditions (admissible, smashing, rigid) on the datum $(\mathcal{D}, \mathcal{S})$ to ensure the existence of a natural morphism to the spectrum of the base $\text{Spec}(\mathcal{S}, \mathcal{S})$ and to guarantee that the spectrum behaves well under base change and localization.

Key Contributions and Results

1. Unification and Generalization

The relative Matsui spectrum generalizes both Balmer's and Matsui's constructions:

• If $\mathcal{D} = \mathcal{S} = \mathcal{T}$ (a rigid tensor triangulated category), $\text{Spec}(\mathcal{D}, \mathcal{S})$ recovers the Balmer spectrum $\text{Spec}^\otimes(\mathcal{T})$.
• If $\mathcal{S} = \text{Perf}(Z)$ or $\text{Sp}^\omega$ (finite spectra), it recovers the Matsui spectrum $\text{Spec}_\triangle(\mathcal{D})$.
• It provides a unified framework for classifying thick $\mathcal{S}$-submodules, extending tensor triangular geometry to settings without a global tensor product.

2. Fundamental Properties

• Classification of Submodules: There is a bijection between thick $\mathcal{S}$-submodules of $\mathcal{D}$ and specialization-closed subsets of $\text{Spc}(\mathcal{D}, \mathcal{S})$ (Theorem 4.3). Every $\mathcal{S}$-submodule is an intersection of prime $\mathcal{S}$-submodules.
• Morphism to the Base: For admissible data, there is a canonical morphism $\text{Spec}(\mathcal{D}, \mathcal{S}) \to \text{Spec}(\mathcal{S}, \mathcal{S})$. This allows for a fiberwise analysis of the spectrum.
• Descent and Base Change: The spectrum satisfies descent properties (Lemma 7.2) and behaves well under base change (Theorem 6.7), including embeddings when the base is locally generated by the image of a sub-algebra.

3. Geometric Applications

The paper demonstrates that the relative Matsui spectrum recovers classical geometric spaces from categorical data in diverse settings:

• Schemes: For a flat morphism $X \to Y$ with $Y$ regular, the spectrum of $\text{Perf}(X)$ as a $\text{Perf}(Y)$-module is a locally ringed space over $Y$, fiberwise given by the Matsui spectrum of $\text{Perf}(X_y)$. Relatively derived equivalent schemes appear as open subschemes.
• Twisted Derived Categories: For twisted derived categories $\text{Perf}(X, \mu)$, the spectrum is canonically isomorphic to the base scheme $X$.
• Severi–Brauer Schemes: For a Severi–Brauer scheme $Y \to X$, the spectrum encodes $Y$ along with additional copies of the base $X$, reflecting semi-orthogonal decompositions. In relative dimension one, the spectrum is explicitly described as $Y \sqcup (\coprod_{\mathbb{Z}} X)$.
• Singularities:
  • For the category of singularities $\text{Sing}(X)$, if $X$ is locally a hypersurface, the spectrum recovers the singular locus of $X$.
  • For $\text{Coh}(X)$, the fiber over a point $x$ relates to the embedding codimension of the local ring $\mathcal{O}_{X,x}$. The spectrum classifies submodules of $\text{Coh}(X)$ via specialization-closed subsets of a space related to the singular locus.
• Matrix Factorizations: For derived matrix factorization categories $\text{DMF}(X, L, W)$, the spectrum recovers the relative singular locus $\text{Sing}(X_0/X)$, extending results by Hirano [Hir19].
• Koszul Algebras: The spectrum classifies thick subcategories over Koszul algebras of regular rings, generalizing results by Liu and Pollitz [LP25].

Significance

The paper claims that the relative Matsui spectrum extends the reach of tensor triangular geometry to settings where a global tensor product is absent or where the category is defined relative to a base. By preserving geometric intuition through the classification of subcategories, it allows for the recovery of underlying classical geometric spaces (such as schemes, singular loci, and twisted structures) from purely categorical data.

The construction provides a unified framework that subsumes previous spectral theories (Balmer and Matsui) and offers a robust tool for analyzing the geometry of stable $\infty$-categories equipped with actions. The results establish that even in complex relative settings (e.g., twisted categories, matrix factorizations, and categories of singularities), the categorical structure retains enough information to reconstruct the relevant geometric loci and classify their submodules.