文摘
Let R be an isolated Gorenstein singularity with a non-commutative resolution A=EndR(R⊕M). In this paper, we show that the relative singularity category 18e350e65b3" title="Click to view the MathML source">ΔR(A) of A has a number of pleasant properties, such as being Hom-finite. Moreover, it determines the classical singularity category 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) determines e05ca1810c391aca6e7e95f1bc41ad6" title="Click to view the MathML source">ΔR(Aus(R)), where Aus(R) is the corresponding Auslander algebra. The proofs of these results use dg algebras, A∞ Koszul duality, and the new concept of dg Auslander algebras, which may be of independent interest.