Big Singularity Category

• Big Singularity Category is a triangulated invariant that extends classical singularity categories to unbounded, coproduct-complete settings for rings and schemes.
• It employs stabilization and adjunctions to relate injective, projective, and Gorenstein modules, thereby enabling explicit computations of homological invariants.
• The framework supports classification of localizing subcategories and reveals deep connections in representation theory and tensor triangular geometry.

The big singularity category is a triangulated category that provides a robust invariant encoding non-regularity for a broad class of commutative and noncommutative rings and schemes. It generalizes the classical (or "small") singularity category by incorporating unbounded, coproduct-complete settings, and it is pivotal in the study of singularity phenomena, stable derived categories, and the relationship between projective, injective, and Gorenstein projective modules. This framework underlies modern approaches to resolutions of singularities, classification of subcategories, and functoriality properties under change of rings, and it enables explicit computation of invariants for classes of algebras via refined homological tools.

1. Definition and Core Construction

Let $R$ be a noetherian commutative ring or a separated noetherian scheme $X$. The big singularity category, often denoted as $S(R)$ or $\Dsg^{\mathrm{big}}(R)$, is defined as the full subcategory of the homotopy category $K(\Inj R)$ of complexes of injective $R$-modules whose objects are acyclic complexes: $S(R) = K_{\mathrm{ac}}(\Inj R) = \{ I^\bullet \in K(\Inj R) \mid H^n(I^\bullet)=0,\,\forall\, n \in \mathbb{Z} \}.$ Alternatively, the big singularity category may be realized as the Verdier quotient

$\Dsg^{\mathrm{big}}(R) = K(\Inj R)/K(\Proj R),$

where $K(\Proj R)$ denotes the homotopy category of complexes of projective $R$-modules. This construction produces a triangulated category admitting arbitrary (small) coproducts, and its subcategory of compact objects is naturally equivalent to the idempotent completion of the classical singularity category

$\Dsg(R) = D^b(\mathrm{mod}~R)/D^\mathrm{perf}(R),$

where $D^b(\mathrm{mod}~R)$ is the bounded derived category of finitely generated $R$-modules and $D^\mathrm{perf}(R)$ the subcategory of perfect complexes (Stevenson, 2011, Oppermann et al., 2018).

2. Triangulated and Localizing Structures

A triangulated category $\mathcal{T}$ is an additive category with an autoequivalence (the shift functor $\Sigma$) and a class of distinguished triangles satisfying Verdier’s axioms. In this context:

• Localizing subcategories are full triangulated subcategories closed under all set-indexed coproducts.
• Thick subcategories are full triangulated subcategories closed under direct summands.

The big singularity category's access to arbitrary coproducts ensures the existence of a well-behaved lattice of localizing subcategories, which is crucial for classification theorems and applications to tensor triangular geometry (Stevenson, 2011).

3. Stabilization and Equivalence to Stable Module Categories

For artinian or more general rings, the big singularity category admits an alternative and often computationally advantageous description via the stabilization of the stable module category. Let $\underline{R\textrm{-} \mathrm{Mod}}$ denote the stable module category of $R$, where morphisms factor through projective modules. The syzygy functor $\Omega_R$ induces a looped category structure, and the stabilization $$\mathcal{S}(\underline{R\text{-}\mathrm{Mod}},\Omega_R)$$ is defined by:

• Objects: pairs $(X, n)$ with $X \in \underline{R\textrm{-} \mathrm{Mod}}$, $n\in \mathbb{Z}$.
• Morphisms: $$\operatorname{colim}_{p \ge \max\{n, m\}} \, \underline{R\text{-}\mathrm{Mod}} \left(\Omega^{p-n}_R X, \Omega^{p-m}_R Y\right)$$.

The canonical functor from this stabilization to the big singularity category is a triangle equivalence

$\mathcal{S}(\underline{R\text{-}\mathrm{Mod}},\Omega_R) \xrightarrow{\sim} \Dsg^{\mathrm{big}}(R)$

for arbitrary rings $R$, recovering Buchweitz's and Keller–Vossieck's classical results in the Gorenstein case. Thus, the big singularity category can be viewed both as a Verdier quotient of derived categories and as the stabilization of the stable module category (Chen, 19 Nov 2025, Chen et al., 1 Sep 2025, Chen, 2015).

4. Functoriality, Change of Rings, and Adjoints

For a ring homomorphism $f: R \to S$ of noetherian rings meeting the finite projective dimension hypotheses on both sides, the classical module adjunction

$(- \otimes_R S) \dashv \res \dashv \Hom_R(S, -)$

induces adjunctions at the level of big singularity categories. The derived tensor functor $-\otimes_R^{\mathbb L} S$ and restriction of scalars admit explicit left and right adjoints, and under further hypotheses (e.g., $\Cone(f) \in \perf(R \otimes_k R^\mathrm{op})$), one obtains full faithfulness or equivalence on big singularity categories. Rigorous construction of right adjoints often employs the theory of 0-cocompact objects and dual Bousfield-localization; in particular, inclusions of subcategories orthogonal to a set of 0-cocompact objects always admit explicit right adjoints constructed through iterated approximations and homotopy limits (Oppermann et al., 2018).

5. Applications: Classification, Singularity Equivalences, and Leavitt Rings

The formalism of the big singularity category provides explicit classification of localizing and thick subcategories, especially for stable derived categories of hypersurface rings and locally complete intersection schemes. Tensor actions play a decisive role in this classification and underpin results relating to the telescope conjecture (Stevenson, 2011).

In representation theory, the big singularity category provides a framework for constructing singular equivalences (i.e., equivalences between the singularity categories of rings not necessarily derived equivalent). For example:

• Quadratic monomial algebras and radical-square-zero algebras can be explicitly related via chains of pre-triangle equivalences, yielding triangle equivalences at the singularity level and decompositions of the category into blocks associated with Leavitt path algebras (Chen, 2015, Chen, 19 Nov 2025).
• For artinian rings with $\operatorname{rad}^2=0$, the big singularity category is equivalent to the stable graded module category over a Leavitt path algebra associated to a modified Gabriel quiver, revealing connections with the structure of FC rings and the universality of certain Frobenius abelian categories (Chen et al., 1 Sep 2025, Chen, 19 Nov 2025).

6. Examples, Generalizations, and Computational Tools

Typical examples include:

• For commutative noetherian Gorenstein rings, the big singularity category collapses to zero exactly when the ring is regular; otherwise, it encodes the singular locus.
• In the case of finite-dimensional monomial algebras, explicit computations exploit the reducibility of the category to stable categories of semisimple modules and path algebras with radical square zero, via combinatorial constructions on quivers and relations (Chen, 2015).
• For trivial extension algebras and homological epimorphisms, functoriality and adjunctions between big singularity categories yield explicit fully faithful embeddings and stable $t$-structures determined by 0-cocompact generators (Oppermann et al., 2018).

A plausible implication is that the big singularity category serves as a technical hub integrating homological and categorical perspectives on singularities, stabilizing the interplay between projective, injective, and Gorenstein projectives, and rendering both abstract classification and explicit computation tractable in a wide range of algebraic and geometric settings.