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Singularity categories via higher McKay quivers with potential

Junyang Liu

TL;DR

The paper generalizes the Kalck–Yang framework to arbitrary dimensions by introducing higher McKay quivers with potential and connecting the resulting $n$-dimensional Ginzburg dg algebras to singularity and cosingularity categories. It establishes a GLn structure theorem that identifies the singularity category of invariant rings with a cosingularity category of an explicit dg algebra via a monoidal transfer from skew group algebras, and a SLn refinement that expresses the singularity category as a small cluster category of a higher Ginzburg algebra, with maximal Cohen–Macaulay modules corresponding to a Higgs category in isolated cases. The approach yields explicit quasi-isomorphisms and minimal models, and extends the non-Gorenstein GLn case, providing a uniform dg-model for quotient singularities across dimensions. Overall, the results deepen the links between singularity theory, higher cluster categories, and Higgs categories, offering concrete tools for analyzing quotient singularities beyond dimension three.

Abstract

In 2018, Kalck and Yang showed that the singularity categories associated with $3$-dimensional Gorenstein quotient singularities are triangle equivalent (up to direct summands) to small cluster categories associated with McKay quivers with potential. We introduce higher McKay quivers with potential and generalize Kalck and Yang's theorem to arbitrary dimensions. The singularity categories we consider occur as the stable categories of categories of maximal Cohen-Macaulay modules. We refine our description of the singularity categories by showing that these categories of maximal Cohen-Macaulay modules are equivalent to Higgs categories in the sense of Wu. Moreover, we describe the singularity categories in the non-Gorenstein case.

Singularity categories via higher McKay quivers with potential

TL;DR

The paper generalizes the Kalck–Yang framework to arbitrary dimensions by introducing higher McKay quivers with potential and connecting the resulting -dimensional Ginzburg dg algebras to singularity and cosingularity categories. It establishes a GLn structure theorem that identifies the singularity category of invariant rings with a cosingularity category of an explicit dg algebra via a monoidal transfer from skew group algebras, and a SLn refinement that expresses the singularity category as a small cluster category of a higher Ginzburg algebra, with maximal Cohen–Macaulay modules corresponding to a Higgs category in isolated cases. The approach yields explicit quasi-isomorphisms and minimal models, and extends the non-Gorenstein GLn case, providing a uniform dg-model for quotient singularities across dimensions. Overall, the results deepen the links between singularity theory, higher cluster categories, and Higgs categories, offering concrete tools for analyzing quotient singularities beyond dimension three.

Abstract

In 2018, Kalck and Yang showed that the singularity categories associated with -dimensional Gorenstein quotient singularities are triangle equivalent (up to direct summands) to small cluster categories associated with McKay quivers with potential. We introduce higher McKay quivers with potential and generalize Kalck and Yang's theorem to arbitrary dimensions. The singularity categories we consider occur as the stable categories of categories of maximal Cohen-Macaulay modules. We refine our description of the singularity categories by showing that these categories of maximal Cohen-Macaulay modules are equivalent to Higgs categories in the sense of Wu. Moreover, we describe the singularity categories in the non-Gorenstein case.
Paper Structure (10 sections, 13 theorems, 55 equations)

This paper contains 10 sections, 13 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Quivers with potential
  5. Singularity categories and cluster categories
  6. The structure theorems
  7. Quivers with potential for polynomial algebras
  8. A monoidal functor
  9. The structure theorem in the $\mathrm{GL}_n$ case
  10. The structure theorem in the $\mathrm{SL}_n$ case

Key Result

Theorem A

Let $G\subset \mathrm{GL}_n$ be a finite subgroup. Then the singularity category associated with $k[x_1, \ldots, x_n]^G$ is triangle equivalent up to direct summands to the small cosingularity category associated with an explicit dg tensor algebra $A'$. If $k[x_1, \ldots, x_n]^G$ has isolated singul

Theorems & Definitions (28)

  • Theorem A: see Theorem \ref{['thm:GLn']} for details
  • Theorem B: see Theorem \ref{['thm:SLn']} for details
  • Proposition 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof
  • ...and 18 more