Relative singularity categories I: Auslander resolutions

Let R   be an isolated Gorenstein singularity with a non-commutative resolution A=EndR(R⊕M)A=EndR(R⊕M). In this paper, we show that the relative singularity category ΔR(A)ΔR(A) of A   has a number of pleasant properties, such as being Hom-finite. Moreover, it determines the classical singularity category Dsg(R)Dsg(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 Dsg(R)Dsg(R) determines ΔR(Aus(R))ΔR(Aus(R)), where Aus(R)Aus(R) is the corresponding Auslander algebra. The proofs of these results use dg algebras, A∞A∞ Koszul duality, and the new concept of dg Auslander algebras, which may be of independent interest.