TY - JOUR
T1 - Relative singularity categories I: Auslander resolutions
AU - Kalck, Martin
AU - Yang, Dong
N1 - 36 pages, new title, introduction restructured and clarified, new section on related work, several changes in the presentation, some typos fixed
PY - 2016/10/1
Y1 - 2016/10/1
N2 - 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..
AB - 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..
KW - math.AG
KW - math.AC
KW - math.CT
KW - math.RT
U2 - 10.1016/j.aim.2016.06.011
DO - 10.1016/j.aim.2016.06.011
M3 - Article
VL - 301
SP - 973
EP - 1021
JO - Advances in Mathematics
JF - Advances in Mathematics
SN - 0001-8708
ER -