Relative singularity categories I: Auslander resolutions

TL;DR

This paper explores the properties of relative singularity categories associated with Gorenstein singularities and non-commutative resolutions, revealing their connections to classical singularity categories and introducing dg Auslander algebras.

Contribution

It establishes that the relative singularity category is Hom-finite and determines the classical singularity category, and shows that for finite CM type, the classical category determines the dg Auslander algebra, using advanced algebraic techniques.

Findings

Relative singularity category is Hom-finite.

It determines the classical singularity category $D_{sg}(R)$.

For finite CM type, $D_{sg}(R)$ determines the dg Auslander algebra.

Abstract

Let $R$ be an isolated Gorenstein singularity with a non-commutative resolution $A=End_R(R\oplus M)$. In this paper, we show that the relative singularity category $\Delta_R(A)$ of $A$ has a number of pleasant properties, such as being Hom-finite. Moreover, it determines the classical singularity category $D_{sg}(R)$ of Buchweitz and Orlov as a certain canonical quotient category. If $R$ has finite CM type, which includes for example Kleinian singularities, then we show the much more surprising result that $D_{sg}(R)$ determines $\Delta_R(Aus(R))$, where $Aus(R)$ is the corresponding Auslander algebra. The proofs of these results use dg algebras, $A_\infty$ Koszul duality, and the new concept of dg Auslander algebras, which may be of independent interest.