文摘
Let R be an isolated Gorenstein singularity with a non-commutative resolution k to view the MathML source">A=EndR(R⊕M). In this paper, we show that the relative singularity category k 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 24654b82d45cdae6e3283" title="Click to view the MathML source">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 24654b82d45cdae6e3283" title="Click to view the MathML source">Dsg(R) determines k to view the MathML source">ΔR(Aus(R)), where k to view the MathML source">Aus(R) is the corresponding Auslander algebra. The proofs of these results use dg algebras, k to view the MathML source">A∞ Koszul duality, and the new concept of dg Auslander algebras, which may be of independent interest.