Relative singularity categories I: Auslander resolutions

2012/05/31 by Martin Kalck, Dong Yang · 2 citations
Mathematics · #math.AC #math.AG #math.CT #math.RT #msc:14B05 #msc:14E15 #msc:18E30

paper · pdf · doi:10.1016/j.aim.2016.06.011

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

arxiv created 2016/07/29 · arxiv updated 2016/08/01

Abstract

Let R be an isolated Gorenstein singularity with a non-commutative resolution A=End_R(R⊕ M). In this paper, we show that the relative singularity category Δ_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 Δ_R(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.

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