@article{d3e472397a7b40f5a7e859bbc00141f5, title = "Relative singularity categories I: Auslander resolutions", abstract = "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..", keywords = "math.AG, math.AC, math.CT, math.RT", author = "Martin Kalck and Dong Yang", note = "36 pages, new title, introduction restructured and clarified, new section on related work, several changes in the presentation, some typos fixed", year = "2016", month = "10", day = "1", doi = "10.1016/j.aim.2016.06.011", language = "English", volume = "301", pages = "973--1021", journal = "Advances in Mathematics", issn = "0001-8708", publisher = "Academic Press Inc.", }