Singularity categories and singular equivalences for resolving subcategories

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• 作者：Hiroki Matsui ; Ryo Takahashi
• 关键词：Complete intersection ; Finitely presented functor ; Functor category ; Gorenstein ring ; Resolving subcategory ; Simple hypersurface singularity ; Singular equivalence ; Singularity category ; Stable category
• 刊名：Mathematische Zeitschrift
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：285
• 期：1-2
• 页码：251-286
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-1823
• 卷排序：285

文摘

Let \(\mathcal {X}\) be a resolving subcategory of an abelian category. In this paper we investigate the singularity category \(\mathsf {D_{sg}}(\underline{\mathcal {X}})=\mathsf {D^b}({\mathsf {mod}}\,\underline{\mathcal {X}})/\mathsf {K^b}({\mathsf {proj}}({\mathsf {mod}}\,\underline{\mathcal {X}}))\) of the stable category \(\underline{\mathcal {X}}\) of \(\mathcal {X}\). We consider when the singularity category is triangle equivalent to the stable category of Gorenstein projective objects, and when the stable categories of two resolving subcategories have triangle equivalent singularity categories. Applying this to the module category of a Gorenstein ring, we prove that the complete intersections over which the stable categories of resolving subcategories have trivial singularity categories are the simple hypersurface singularities of type \((\mathsf {A}_1)\). We also generalize several results of Yoshino on totally reflexive modules.