Triangulated Tensor Ideals

• Triangulated tensor ideals are thick subcategories in a tensor triangulated category that respect both the triangulated and monoidal structures, providing a framework for support theory.
• They enable the classification of objects via the Balmer spectrum, linking algebra, geometry, and topology through spectral and frame-theoretic approaches.
• Their study informs scheme reconstruction, the absence of strong generators, and insights into infinite Rouquier dimensions in both commutative and noncommutative settings.

A triangulated tensor ideal is a fundamental structure in tensor triangular geometry, providing a categorical and topological framework that enables the classification, analysis, and reconstruction of various objects in algebra, geometry, and topology. The concept organizes thick subcategories respecting both triangulated and monoidal structures and serves as the arena for universal support theory, spectrality, and the study of dimension in triangulated categories.

1. Definition and Foundational Structures

Let $\mathcal{T}$ be an essentially small triangulated category with a symmetric monoidal structure $(\otimes, \mathbb{1})$ that is exact in each variable. A tensor triangulated category (tt-category) is such a triple $(\mathcal{T}, \otimes, \mathbb{1})$, where all objects admit shifts $\Sigma$ and cones, and the tensor product is exact on triangles (Balmer, 2019).

A thick tensor ideal $\mathcal{I} \subseteq \mathcal{T}$ is a full triangulated subcategory that is closed under direct summands (i.e., additive, idempotent-complete), and satisfies: for every $X \in \mathcal{I}$ and $T \in \mathcal{T}$, $X \otimes T \in \mathcal{I}$. In noncommutative or non-symmetric settings, a thick tensor ideal is closed under both left and right tensoring.

A thick tensor ideal is called radical if $X^{\otimes n} \in \mathcal{I}$ for some $n \geq 1$ implies $X \in \mathcal{I}$. For rigid (strongly self-dual) tt-categories, all thick tensor ideals are automatically radical (Balmer, 2019).

2. Prime Ideals and the Balmer Spectrum

Prime tensor ideals are the building blocks for the topological structure associated with a tt-category.

• A prime thick tensor ideal $P \subsetneqq \mathcal{T}$ is a proper thick ideal such that, for all $X, Y \in \mathcal{T}$, $X \otimes Y \in P$ implies $X \in P$ or $Y \in P$.
• The set of all prime thick tensor ideals forms the Balmer spectrum $\mathrm{Spc}(\mathcal{T})$, topologized via basic open sets:

$$U(x) = \{\,P \in \mathrm{Spc}(\mathcal{T}) : x \notin P\,\}.$$

For symmetric monoidal categories, this is the original construction by Balmer; in noncommutative monoidal settings, the construction generalizes with prime and semiprime ideals, and sometimes requires further notions such as "completely prime" ideals (Deyn et al., 27 Oct 2025, Mallick et al., 2022, Rowe, 2023).

• The support of an object $x \in \mathcal{T}$ is:

$$\mathrm{Supp}(x) = \{\,P \in \mathrm{Spc}(\mathcal{T}) : x \notin P\,\}$$

and the closed sets in the Zariski topology are generated by such supports.

• In the presence of rigidity and suitable duality, the spectrum is a spectral space in Hochster's sense—quasi-compact, $T_0$, with a basis of quasi-compact opens stable under finite intersections, and every irreducible closed set has a unique generic point (Balmer, 2019, Rowe, 2023).

3. Classification of Thick Tensor Ideals

The classification of thick tensor ideals in rigid tt-categories is governed by the geometry of the Balmer spectrum.

Balmer's Classification Theorem (Balmer, 2019, Boe et al., 2014): There is an order-preserving bijection between radical thick tensor ideals of $\mathcal{T}$ and Thomason subsets of $\mathrm{Spc}(\mathcal{T})$—unions of closed sets with quasi-compact complements:

$$\begin{align*} & \{\text{radical thick tt-ideals in }\mathcal{T}\} \longleftrightarrow \{\text{Thomason subsets }V \subseteq \mathrm{Spc}(\mathcal{T})\} \ & \mathcal{J} \mapsto \mathrm{Supp}(\mathcal{J}) := \bigcup_{x \in \mathcal{J}} \mathrm{Supp}(x), \qquad V \mapsto \mathcal{T}_V := \{ x \in \mathcal{T} : \mathrm{Supp}(x) \subseteq V \}. \end{align*}$$

This result holds in numerous contexts:

• Perfect complexes on schemes (Balmer, 2019), bounded derived categories over finite EI categories (Xu, 2013), and representations of classical Lie superalgebras (Boe et al., 2014).
• Stable module categories of finite group schemes correspond $\mathrm{Spc}$ to $\operatorname{Proj} H^*(G,k)$, and thick tensor ideals correspond to support varieties (Bloom, 2024).

In noncommutative settings, classification remains valid (with semiprime ideals replacing radical ones) whenever the spectrum is noetherian (Deyn et al., 27 Oct 2025, Rowe, 2023).

Thick subcategories which are compactly generated correspond bijectively to specialization-closed subsets of (commutative) spectra; in derived categories of commutative noetherian rings, all compactly generated thick tensor ideals are of the form

$\mathrm{thick}^\otimes\{ R/p : p \in W \}$

for specialization-closed $W \subseteq \operatorname{Spec} R$ (Matsui et al., 2016).

4. Frames, Spectral Spaces, and Support Theory

The set of radical (or, in general, semiprime) thick tensor ideals, equipped with inclusion, forms a frame—a complete distributive lattice in which finite meets distribute over arbitrary joins (Mallick et al., 2022, Balmer et al., 2017). This underlies the point-free approach to tensor triangular geometry:

• Radical thick tensor ideals: complete frame $\mathcal{F}$.
• Frame points: frame maps $p: \mathcal{F} \to \{0,1\}$ correspond to prime tensor ideals.
• The Hochster dual of the spectral space $\mathrm{Spc}(\mathcal{T})$ is homeomorphic to the space of frame points, and the frame of opens in the dual topology is isomorphic to the frame of radical thick tensor ideals.

This frame-theoretic approach yields:

• Universal initial support datum: $s(x) = \sqrt{x}$ for each $x \in \mathcal{T}$.
• Classification theorems via Stone duality and coherent frame theory, including homeomorphisms between spectra and spectrum spaces of radical ideals (Mallick et al., 2022, Deyn et al., 27 Oct 2025).

In the general monoidal (possibly noncommutative) triangulated category, every semiprime thick tensor ideal corresponds to an open of the pseudo-Hochster dual of $\mathrm{Spc}(\mathcal{T})$, and the lattice of semiprime tensor ideals is always a spatial frame (Deyn et al., 27 Oct 2025).

5. Generation, Dimension, and Strong Generators

A key property is the failure of nontrivial thick tensor ideals to admit strong generators in connected spectra. Let $G \in \mathcal{T}$ and define the sequence:

$$\langle G \rangle_1 = \mathrm{add}\{\Sigma^n G : n \in \mathbb{Z}\}, \qquad \langle G \rangle_{k+1} = \{ Z : \exists \text{ triangle } A \to Z \oplus Z' \to B \to \Sigma A,\, A \in \langle G \rangle_k,\, B \in \langle G \rangle_1 \}$$

$G$ is a strong generator of a thick subcategory $\mathcal{I}$ if $\langle G \rangle_N = \mathcal{I}$ for some $N$. In an essentially small rigid tt-category with connected Balmer spectrum, no proper nonzero thick tensor ideal admits a strong generator (Steen et al., 2014). For instance, this applies to the category of perfect complexes $\mathrm{Perf}(R)$ over a commutative ring $R$ with connected $\operatorname{Spec} R$.

The absence of strong generators implies infinite Rouquier dimension for proper ideals, and is reflected in numerous contexts such as:

• Stable module categories over finite groups;
• The Spanier–Whitehead category in stable homotopy theory (Steen et al., 2014).

6. Examples and Structural Variants

Classical commutative cases

Category	Spectrum	Ideals classified by	References
$\mathrm{Perf}(R)$, $R$ comm.	$\operatorname{Spec} R$	Thomason subsets	(Balmer, 2019)
$D^b(kC)$, $C$ finite EI	$$\bigsqcup \operatorname{Spec}^h H^*(\operatorname{Aut}_C(x),k)$$	Union over iso-classes	(Xu, 2013)
$SH^c$ (finite spectra)	Chromatic tower	Thick subcat. theorem	(Balmer, 2019)

Noncommutative and generalized cases

• Noncommutative tensor tt-categories: radical thick tensor ideals correspond to points of a coherent frame with associated spectral space, and the spectrum is homeomorphic to this dual space (Mallick et al., 2022, Deyn et al., 27 Oct 2025).
• Quantum group representations and Lie superalgebras: thick tensor ideals correspond to $G$-stable closed subsets of appropriate projective varieties (e.g., nilpotent cone, detecting subalgebra) (Boe et al., 2017, Boe et al., 2014).
• Filtered module categories: spectrum computed as a union of two copies of $\operatorname{Spec} R$, with thick ideal classification via pairs of Thomason subsets (Gallauer, 2017).

Additional perspectives

• For right-bounded derived categories $D^-(R)$ over a commutative noetherian ring $R$, various bijections are established between different classes of tensor ideals and specialization-closed subsets of $\operatorname{Spec} R$ or Thomason subsets of $\mathrm{Spc} D^-(R)$, with counterexamples to injectivity conjectures and infinite-dimensional phenomena in the DVR case (Matsui et al., 2016).
• For Tate motives with integral coefficients, the tt-spectrum organizes as a "cofinite" set corresponding to primes and the rational point, with thick tensor ideals classified by finite sets of primes (Gallauer, 2017).

7. Structural and Topological Implications

The triangulated tensor ideal formalism underpins the following major structural results:

• Universal Support Theory: The universal property of spectra implies any rule $\sigma$ satisfying support axioms induces a homeomorphism of its support space with the Balmer spectrum when the classification theorem holds (Balmer, 2019, Mallick et al., 2022).
• Scheme Reconstruction: The spectrum, equipped with a suitable structure sheaf, can reconstruct the original scheme from its category of perfect complexes (Balmer, 2019).
• Extensions to Big Categories: In well-generated triangulated categories, the lattice of localizing tensor ideals with closure under arbitrary coproducts forms a frame, whose spectrum refines that of the compacts (Krause et al., 2022). Dualizable localizing ideals correspond to convex subsets of the spectrum in cohomologically stratified environments (Zou, 14 Jun 2025).
• Frame-Theoretic and Stone Duality Viewpoint: The lattice-theoretic approach provides a categorical analog of the classical Nullstellensatz, relating radical tensor ideals to opens of the Hochster dual spectrum (Mallick et al., 2022, Deyn et al., 27 Oct 2025).

The classification of triangulated tensor ideals thus provides a bridge between the algebraic, categorical, and geometric aspects of modern algebra, facilitating explicit calculations and deep conceptual understanding across multiple mathematical domains.