Enhanced Auslander-Reiten duality and Morita theorem for singularity categories

TL;DR

This paper establishes a Morita theorem linking singularity categories of Gorenstein rings with cluster categories of finite dimensional algebras, using dg enhancements and tilting objects to create equivalences.

Contribution

It introduces new Morita-type theorems for dg categories and demonstrates bimodule Calabi-Yau properties of singularity categories, advancing the understanding of their structure and equivalences.

Findings

Constructs triangle equivalences between singularity and cluster categories.

Proves dg enhancements of singularity categories have bimodule Calabi-Yau property.

Provides applications to Gorenstein rings, quotient singularities, and complete intersections.

Abstract

We establish a Morita theorem to construct triangle equivalences between the singularity categories of (commutative and non-commutative) Gorenstein rings and the cluster categories of finite dimensional algebras over fields, and more strongly, quasi-equivalences between their canonical dg enhancements. More precisely, we prove that such an equivalence exists as soon as we find a quasi-equivalence between the graded dg singularity category of a Gorenstein ring and the derived category of a finite dimensional algebra which can be done by finding a single tilting object. Our result is based on two key theorems on dg enhancements of cluster categories and of singularity categories, which are of independent interest. First we give a Morita-type theorem which realizes certain $\mathbb{Z}$-graded dg categories as dg orbit categories. Secondly, we show that the canonical dg enhancements of the singularity categories of symmetric orders have the bimodule Calabi-Yau property, which lifts the classical Auslander-Reiten duality on singularity categories. We apply our results to such classes of rings as Gorenstein rings of dimension at most $1$, quotient singularities, and Geigle-Lenzing complete intersections, including finite or infinite Grassmannian cluster categories, to realize their singularity categories as cluster categories of finite dimensional algebras.