Relative singularity categories I: Auslander resolutions

By Martin Kalck, Dong Ping Yang. 2012

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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 s g ( 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 s g ( 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b70dc920b4d821418beefb13278333f82f3ae74b

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