Verdier Quotients in Triangulated Categories

• Verdier quotients are localizations of triangulated categories that collapse thick subcategories to zero by inverting morphisms whose cones lie in these subcategories.
• They are constructed via roofs or ideal quotients, preserving triangulated structures and satisfying a universal property for exact functors.
• Applications include homological algebra, singularity categories, and Calabi–Yau structures, offering tools for advanced representation theory and higher category frameworks.

A Verdier quotient is a central construction in the theory of triangulated and stable ∞-categories, formalizing the process of localization at a thick subcategory. It produces, for any triangulated category $\mathcal{T}$ and a thick subcategory $\mathcal{S}$, a new triangulated category $\mathcal{T}/\mathcal{S}$ in which all objects of $\mathcal{S}$ are identified with zero, and morphisms whose cones belong to $\mathcal{S}$ are inverted. This quotient possesses a universal property for triangulated functors: any exact functor from $\mathcal{T}$ that annihilates $\mathcal{S}$ factors uniquely through $\mathcal{T}/\mathcal{S}$ (Iyama et al., 2017, Drew, 2015, Li, 2016, Cortés-Izurdiaga, 2023, Zhou et al., 2017, Puthenpurakal, 2021, Keller et al., 2023). Verdier quotients permeate various domains: homological algebra, geometric representation theory, singularity categories, stable module categories, and the theory of Calabi–Yau structures.

1. Definitions and Universal Properties

Let $\mathcal{T}$ be a triangulated category and $\mathcal{S}\subseteq \mathcal{T}$ a thick subcategory, i.e., full triangulated and closed under direct summands. The Verdier quotient $\mathcal{T}/\mathcal{S}$ is constructed so that:

• Objects of $\mathcal{T}/\mathcal{S}$ are those of $\mathcal{T}$.
• Morphisms $f: X \to Y$ in $\mathcal{T}/\mathcal{S}$ are equivalence classes of "roofs" $X \leftarrow X' \rightarrow Y$, where the left arrow has cone in $\mathcal{S}$.
• The category inherits a triangulated structure from $\mathcal{T}$ by considering images of distinguished triangles under the quotient functor.

The defining universal property: for any triangulated category $\mathcal{U}$, a triangle functor $F: \mathcal{T}\to\mathcal{U}$ which sends all objects of $\mathcal{S}$ to zero factors uniquely (up to unique isomorphism) via $\mathcal{T}/\mathcal{S}$ (Iyama et al., 2017, Drew, 2015, Li, 2016, Keller et al., 2023).

This construction is formally a localization of categories: $\mathcal{T}/\mathcal{S} \cong \mathcal{T}[W^{-1}]$, where $W$ denotes morphisms whose cone lies in $\mathcal{S}$ (Drew, 2015).

2. Realization as Subfactor and Ideal Quotients

Under suitable conditions, a Verdier quotient can be concretely realized inside the ambient triangulated category as an ideal quotient. Specifically, consider full additive subcategories $\mathcal{Z} \supseteq \mathcal{P}$ of $\mathcal{T}$; the ideal quotient $\mathcal{Z}/[\mathcal{P}]$ has the same objects as $\mathcal{Z}$ and morphisms given by $\operatorname{Hom}_{\mathcal{T}}(X,Y)$ modulo those factoring through $\mathcal{P}$.

Iyama–Yang's theorem (Iyama et al., 2017) establishes sufficient conditions involving torsion pairs on $\mathcal{T}$ and its thick subcategory $\mathcal{S}$ for an equivalence of additive (and triangulated) categories:

$$\mathcal{Z}/[\mathcal{P}] \simeq \mathcal{T}/\mathcal{S},$$

where $\mathcal{Z}$ and $\mathcal{P}$ are defined via specific intersections of torsion pairs. This framework recovers numerous classic results as special cases, such as Buchweitz's equivalence for singularity categories, Orlov's theorem for graded singularities, and the Amiot–Guo–Keller construction for cluster categories.

Zhi–Wei Li complements this with a subfactor approach: under the existence of a so-called $\mathcal{S}$-localization triple $(\mathcal{U},\mathcal{X},\mathcal{V})$ satisfying the Verdier condition, one has a triangle equivalence $$(\mathcal{U} \cap \mathcal{V})/[\mathcal{X}] \simeq \mathcal{T}/\mathcal{S}$$ (Li, 2016). This approach unifies silting reduction, mutation–recollement results, and relative singularity categories.

3. Explicit Constructions and Localizations

Regarded as a Gabriel–Zisman localization, the Verdier quotient is computed by formally inverting all morphisms whose cone lies in $\mathcal{S}$, typically represented by roofs and equivalence relations driven by common refinements and commutative squares (Drew, 2015, Keller et al., 2023). In practical terms:

• The morphisms in $\mathcal{T}/\mathcal{S}$ are described by colimits over "fractions" with cones in $\mathcal{S}$.
• Distinguished triangles, shifts, and other triangulated operations descend directly to the quotient, as the localization respects these structures.

For stable quasi-categories (i.e., stable $\infty$-categories), the Verdier quotient is defined as the cofiber (pushout) in the $\infty$-category of stable quasi-categories. The localization theorem asserts an equivalence between this cofiber and the quasi-category obtained by inverting morphisms with cofibers in a stable subcategory. This perspective admits symmetric monoidal enhancements and compatibility with tensor products (Drew, 2015).

4. Applications in Homological and Representation Theory

• Singularity Categories: For a Noetherian ring $A$ and its projectives, the singularity category $D_{sg}(A) = D^b(A\text{-mod})/K^b(A\text{-proj})$ is a Verdier quotient; it plays a foundational role in the study of stable Cohen–Macaulay categories and singularities (Zhou et al., 2017, Iyama et al., 2017, Li, 2016).
• Gorenstein-projective Precovers: In the homotopy category of projective complexes, Verdier quotients by the thick subcategory of totally acyclic complexes control the existence of Gorenstein-projective precovers and totally acyclic precovers, contingent on the "smallness" of Hom-sets ensured by enough injectives and the Baer lemma (Cortés-Izurdiaga, 2023).
• Support Varieties: For complete intersection rings, Verdier quotients of the stable category by thick subcategories enable definitions of punctured support varieties, and cohomological phenomena such as symmetry and rigidity, generalizing the entire landscape of support theory in modular representations (Puthenpurakal, 2021).

These structures facilitate localization, window, and reduction phenomena—illustrated across derived categories, cluster categories, and homotopy categories.

5. Calabi–Yau Structures and Derived Enhancements

Amiot's construction demonstrates that if a thick subcategory $N$ of a Hom-finite, $k$-linear triangulated category $T$ carries a $d$-Calabi–Yau structure, then the Verdier quotient $T/N$ acquires a canonical $(d-1)$-Calabi–Yau structure via explicit bilinear forms on morphism spaces (encoded by roof representatives and their compatibility with cyclic symmetry) (Keller et al., 2023). This extends naturally to dg categories via Drinfeld quotients, with the relevant forms arising from negative cyclic homology.

Such enhancements provide the technical core for Amiot's generalized cluster categories and underpin results on 2–Calabi–Yau triangulated categories with cluster-tilting objects, connecting cyclic homology with the computation of potentials in Jacobian algebras and their categorical incarnations (Keller et al., 2023, Iyama et al., 2017).

6. Localisation Sequences, Recollements, and Category Splittings

The formalism of Verdier quotients induces recollement diagrams and stable $t$-structures, explaining, for large families of homotopy and derived categories, why many quotients split as direct products or admit stable decompositions. Zhou–Zimmermann's analysis of filtrations by boundedness properties and their interrelations via Hasse diagrams demonstrates that a considerable portion of the derived category landscape can be understood in terms of Verdier quotients and their associated localization sequences (Zhou et al., 2017).

This organization yields both classic and new infinite singularity categories and systematically accounts for the structure of almost all nontrivial Verdier quotients in finite and infinite settings.

7. Generalizations and Future Directions

Research has extended the Verdier quotient beyond classical triangulated categories:

• Stable $\infty$-categories and symmetric monoidal enhancements enable applications in higher algebra and derived algebraic geometry (Drew, 2015).
• The subfactor realization approach unifies silting, mutation, and relative singularity reductions, suggesting further generalizations to cotorsion pairs, twin cotorsion pairs, and analytic settings such as derivators (Li, 2016, Iyama et al., 2017).
• Calabi–Yau structures and their propagation via quotients highlight the interplay between categorical, homological, and cyclic structures in modern representation theory and topology (Keller et al., 2023).

The Verdier quotient's ability to render thick subcategories as zero, while controlling the morphism space structure and triangulated properties, continues to be central to advancements in homological algebra, representation theory, and higher category theory.