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Singularity categories and singular equivalences for resolving subcategories

Abstract

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.

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Acknowledgments

The authors are grateful to Kazuhiko Kurano for his comment on the proof of Theorem 6.5, to Henning Krause for his question on Remark 5.9 and to Osamu Iyama for his suggestions on all parts of the paper. The authors also thank the referee for his/her kind and helpful advice.

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Correspondence to Ryo Takahashi.

Additional information

Dedicated to Professor Yuji Yoshino on the occasion of his sixtieth birthday.

RT was partly supported by JSPS Grant-in-Aid for Scientific Research (C) 25400038.

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Matsui, H., Takahashi, R. Singularity categories and singular equivalences for resolving subcategories. Math. Z. 285, 251–286 (2017). https://doi.org/10.1007/s00209-016-1706-x

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Keywords

  • Complete intersection
  • Finitely presented functor
  • Functor category
  • Gorenstein ring
  • Resolving subcategory
  • Simple hypersurface singularity
  • Singular equivalence
  • Singularity category
  • Stable category

Mathematics Subject Classification

  • 13C60
  • 13D09
  • 16G60
  • 16G70
  • 18A25
  • 18E30