Relative singularity categories I: Auslander resolutions

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• 作者：Martin Kalck^a ; ¹ ; ^{m.kalck@ed.ac.uk" class="auth_mail" title="E-mail the corresponding author} ; Dong Yang^b ; ² ; ^{yangdong@nju.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 14B05 ; 14E15 ; secondary ; 18E30
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：1 October 2016
• 年：2016
• 卷：301
• 期：Complete
• 页码：973-1021
• 全文大小：931 K

文摘

Let R   be an isolated Gorenstein singularity with a non-commutative resolution A=End_R(R⊕M)$A={\mathsf{End}}_R(R\oplus M)$. In this paper, we show that the relative singularity category 18e350e65b3" title="Click to view the MathML source">Δ_R(A)${\mathrm{\Delta }}_R(A)$ of A   has a number of pleasant properties, such as being Hom-finite. Moreover, it determines the classical singularity category D_sg(R)${\mathcal{D}}_{sg}(R)$ 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)${\mathcal{D}}_{sg}(R)$ determines Δ_R(Aus(R))${\mathrm{\Delta }}_R(\mathsf{Aus}(R))$, where Aus(R)$\mathsf{Aus}(R)$ is the corresponding Auslander algebra. The proofs of these results use dg algebras, A_∞$A_{\infty }$ Koszul duality, and the new concept of dg Auslander algebras, which may be of independent interest.