Big Singularity Categories in Algebraic Geometry

• Big singularity categories are large triangulated categories capturing the homological and geometric complexity of singularities in algebraic contexts.
• They classify localizing and thick subcategories through tensor actions and support theory, linking subsets of the singular locus with categorical invariants.
• These categories enable practical applications such as verifying the telescope conjecture and establishing Orlov-type equivalences in derived settings.

Big singularity categories are large triangulated categories that capture the homological and geometric complexity of singularities in commutative algebraic, and derived algebraic, settings. As formalized in the paper of the homotopy categories of acyclic injectives and their tensor actions, big singularity categories generalize the classical (small) singularity category and admit fine-grained classifications of their subcategories, reflecting the geometry of the singular locus and providing a framework for stratification, support theory, and the telescope conjecture. In modern development, these categories function as robust invariants at the interface of algebraic geometry, homological algebra, and tensor-triangular geometry.

1. Structural Definition and Tensor Actions

The big singularity category of a noetherian separated scheme $X$ is constructed as the homotopy category of acyclic complexes of quasi-coherent injective sheaves:

$\mathrm{S}(X) = K_{\mathrm{ac}}(\mathrm{Inj}\,X)$

This category is compactly generated; its compact objects recover the classical singularity category $\mathrm{D}_{\mathrm{Sg}}(X)$. A central property, pivotal for the classification of subcategories, is an exact tensor action by the unbounded derived category:

$$\mathrm{D}(X) \times \mathrm{S}(X) \longrightarrow \mathrm{S}(X)$$

Here, all functors are exact and preserve coproducts. The action is constructed via K-flat resolutions to guarantee a well-defined derived tensor product and is used to define a geometry-reflecting support for objects:

$$\operatorname{supp}(A) = \{x \in X \mid \Gamma_{(x)}A \neq 0\}$$

where the localizing idempotents $\Gamma_{(x)}$ are derived from Rickard idempotents associated to the residue fields at points $x \in X$.

2. Classification of Localizing and Thick Subcategories

For affine schemes $X = \mathrm{Spec}\,R$ with $R$ a (locally Gorenstein) hypersurface or more generally a local complete intersection, the tensor action allows a complete classification of localizing subcategories of $\mathrm{S}(X)$ in terms of subsets of the singular locus:

$$\begin{aligned} \{\text{subsets of } \operatorname{Sing} R\} &\overset{\tau}{\underset{\sigma}{\rightleftarrows}} \{\text{localizing subcategories of } S(R)\} \ \{\text{specialization-closed subsets of } \operatorname{Sing} R\} &\overset{\sim}{\rightleftarrows} \{\text{thick subcategories of } \mathrm{D}_{\mathrm{Sg}}(R)\} \end{aligned}$$

Here, the support of a localizing subcategory $\mathcal{L}$ is $$\sigma(\mathcal{L}) = \operatorname{supp} \mathcal{L}$$, while $$\tau(W) = \{A \mid \operatorname{supp} A \subseteq W\}$$. The assignments are mutually inverse, and every localizing subcategory is automatically a $\mathrm{D}(R)$-submodule. The lattice of subcategories is thus determined by subsets of the singular locus.

In the case of a local complete intersection $(R, \mathfrak{m}, k)$ of codimension $c$, the stratification is more refined: the classifying space is a disjoint union of projective spaces over the residue fields at singular points,

$$\{\text{subsets of } \coprod_{p \in \operatorname{Sing} R}\mathbb{P}^{c_p-1}_{k(p)}\} \overset{\tau}{\underset{\sigma}{\rightleftarrows}} \{\text{localizing subcategories of }S(R)\}$$

where $c_p = \operatorname{cx}_{R_p}(k(p))$.

3. Local-to-Global Principle and Support Theory

The classification uses a local-to-global principle leveraging the minimality of the localizing subcategories supported at a single prime: for each $p \in \operatorname{Sing} R$, the localizing subcategory $\Gamma_{(p)}S(R)$ is minimal, meaning it has no nontrivial proper localizing D(R)-submodules. This minimality is fundamental to the proof that the support assignment $\mathcal{L} \mapsto \operatorname{supp}\mathcal{L}$ sets up the bijection.

The geometric support theory is underpinned by the structure of tensor triangular geometry: the spectrum $\operatorname{Spc}(\mathrm{D}(X)^\mathrm{c})$, identifying with $X$ in geometric cases, and the idempotents $\Gamma_{(x)}$ encode local cohomological information on singularities.

4. Applications to Complete Intersections and Orlov-Type Equivalences

When $X$ is a local complete intersection embedded into a regular scheme, Orlov's theorem ensures an equivalence between the classical singularity category of $X$ and of a certain hypersurface $Y$, typically realized as a Koszul hypersurface in a projective bundle. This result globalizes: the equivalence extends to the big singularity categories $S(X) \simeq S(Y)$. The explicit stratification by projective spaces (arising from the complexities of residue fields at singularities) translates between $X$ and the model hypersurface $Y$.

Consequently, all aforementioned classification statements about localizing and thick subcategories, as well as geometric support, apply uniformly across both hypersurface and local complete intersection cases.

5. Telescope Conjecture in Big Singularity Categories

The relative telescope conjecture in this context asserts that every smashing $\mathrm{D}(X)$-submodule of $S(X)$ is generated by its compact objects:

$$\forall \mathcal{L} \subseteq S(X) \text{ smashing and a } \mathrm{D}(X)\text{-submodule} \implies \mathcal{L} = \langle \mathcal{L} \cap S(X)^c\rangle_\mathrm{loc}$$

The proof leverages the established support theory: smashing submodules correspond to specialization-closed subsets of the singular locus, and their compacts determine the entire submodule. Consequently, the big singularity categories for schemes with (locally) hypersurface or complete intersection singularities satisfy the telescope conjecture, mirroring the behavior in stable homotopy theory and enabling a well-controlled theory of localizations and recollements.

6. Representative Formulas and Classification Schemes

Key mathematical expressions integral to the structure of big singularity categories include:

• Definition:

$S(X) = K_{\mathrm{ac}}(\mathrm{Inj}\,X)$

• Support via idempotents:

$$\operatorname{supp} A = \{x \in X \mid \Gamma_{(x)}A \neq 0\}$$

$$\Gamma_{(x)}\mathcal{O}_X = \Gamma_{\mathcal{V}(x)}\mathcal{O}_X \otimes^{L}_{\mathcal{O}_X} L_{\mathcal{Z}(x)}\mathcal{O}_X$$

• Classification bijections:

$$\{\text{subsets of } \operatorname{Sing} R\} \longleftrightarrow \{\text{localizing subcategories of } S(R)\}$$

$$\{\text{specialization closed subsets of } \operatorname{Sing} R\} \longleftrightarrow \{\text{thick subcategories of }\mathrm{D}_{\mathrm{Sg}}(R)\}$$

• Complete intersection case:

$$\{\text{subsets of } \coprod_{p \in \operatorname{Sing} R}\mathbb{P}^{c_p-1}_{k(p)}\} \longleftrightarrow \{\text{localizing subcategories of }S(R)\}$$

7. Implications for Tensor Triangular Geometry and Homological Algebra

These results on big singularity categories advance tensor triangular geometry by providing a classification of subcategories entirely grounded in the geometry of the singular locus. This deep structural understanding has consequences for the behavior of supports, stratifications, and localizations in triangulated categories arising in algebraic geometry. It establishes the big singularity category as a setting where algebraic, geometric, and categorical methods coalesce, and offers a paradigm in which geometric singularities and categorical support correspond directly, significantly influencing strategies in homological and tensor-triangular approaches to algebraic and non-commutative geometry.