Relative singularity categories I: Auslander resolutions

Abstract

Let $R$ be an isolated Gorenstein singularity with a non-commutative resolution $A=En{d}_R(R\oplus M)$. 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.

45pages. Section 2.7 rewritten, Remark 2.11 added, one Appendix removed

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