Enhanced Auslander-Reiten duality and tilting theory for singularity categories

Abstract

We establish a general method 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. We prove that such an equivalence exists as soon as there is a triangle equivalence between the graded singularity category of a Gorenstein ring and the derived category of a finite dimensional algebra. This is based on two key results on dg enhancements of cluster categories and of singularity categories, which are of independent interest. First we give a characterization of dg orbit categories among Z-graded dg categories in terms of a generating condition. 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 Calabi-Yau property of 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. ContentsPart 2. Tilting theory for singularity categories and realizations as cluster categories 27 6. Rings of dimension 0 and 1 27 7. Quotient singularities 33 8. Geigle-Lenzing complete intersections and Grassmannian cluster categories 38 Appendix A. Group graded rings 43 Appendix B. Covering functors and tilting subcategories 46 References 47