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Singularity Categories of Stable Resolving Subcategories and Applications to Gorenstein Rings

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Extended Abstracts Spring 2015

Part of the book series: Trends in Mathematics ((RPCRMB,volume 5))

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Abstract

We study the singularity category of the stable category of a resolving subcategory of an abelian category. We consider when the stable categories of two resolving subcategories have triangle equivalent singularity categories. As an application, we obtain a characterization of simple hypersurface singularities of type \((\mathsf {A}_1)\).

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References

  1. L. Angeleri Hügel, D. Pospíšil, J. Š\({\check{\rm t}}\)ovíček, J. Trlifaj, Tilting, cotilting, and spectra of commutative Noetherian rings. Trans. Am. Math. Soc. 366(7), 3487–3517 (2014)

    Google Scholar 

  2. M. Auslander, Coherent functors, in Proceedings Conference Categorical Algebra (La Jolla, Calif., 1965), (Springer, New York, 1966), pp. 189–231

    Google Scholar 

  3. M. Auslander, M. Bridger, Stable module theory. Mem. Am. Math. Soc. 94, American Mathematical Society, (Providence, R.I, 1969)

    Google Scholar 

  4. M. Auslander, I. Reiten, Stable equivalence of Artin algebras, in Proceedings of the Conference on Orders, Group Rings and Related Topics (Ohio State Univ., Columbus, Ohio, 1972), Lecture Notes in Mathematics 353, (Springer, Berlin, 1973) pp. 8–71

    Google Scholar 

  5. M. Auslander, I. Reiten, Applications of contravariantly finite subcategories. Adv. Math. 86(1), 111–152 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Auslander, I. Reiten, \(D\)Tr-periodic modules and functors. Representation theory of algebras (Cocoyoc, 1994), CMS Conference Proceedings, 18, American Mathematical Society, (Providence, RI, 1996), pp. 39–50

    Google Scholar 

  7. A. Beligiannis, The homological theory of contravariantly finite subcategories: Auslander-Buchweitz contexts, Gorenstein categories and (co-)stabilization. Commun. Algebra 28(10), 4547–4596 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. R.O. Buchweitz, Maximal Cohen–Macaulay modules and Tate cohomology over Gorenstein rings. Preprint, http://hdl.handle.net/1807/16682

  9. X.W. Chen, The singularity category of an algebra with radical square zero. Doc. Math. 16, 921–936 (2011)

    MathSciNet  MATH  Google Scholar 

  10. H. Dao, R. Takahashi, The radius of a subcategory of modules. Algebra Number Theory 8(1), 141–172 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. H. Dao, R. Takahashi, Classification of resolving subcategories and grade consistent functions. Int. Math. Res. Not. IMRN 1, 119–149 (2015)

    MathSciNet  MATH  Google Scholar 

  12. H. Krause, The stable derived category of a Noetherian scheme. Compos. Math. 141(5), 1128–1162 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. H. Matsui, R. Takahashi, Singular equivalences of stable resolving subcategories. Preprint, arXiv:1412.8061

  14. D.O. Orlov, Triangulated categories of singularities and \(D\)-branes in Landau-Ginzburg models. Proc. Steklov Inst. Math. 246(3), 227–248 (2004)

    MathSciNet  MATH  Google Scholar 

  15. G. Stevenson, Subcategories of singularity categories via tensor actions. Compos. Math. 150(2), 229–272 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. R. Takahashi, Classifying thick subcategories of the stable category of Cohen-Macaulay modules. Adv. Math. 225(4), 2076–2116 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. R. Takahashi, Contravariantly finite resolving subcategories over commutative rings. Am. J. Math. 133(2), 417–436 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. R. Takahashi, Reconstruction from Koszul homology and applications to module and derived categories. Pacific J. Math. 268(1), 231–248 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Y. Yoshino, Cohen–Macaulay modules over Cohen–Macaulay rings. London Mathematical Society Lecture Note Series 146, (Cambridge University Press, Cambridge, 1990)

    Google Scholar 

  20. Y. Yoshino, A functorial approach to modules of \(G\)-dimension zero. Illinois J. Math. 49(2), 345–367 (2014)

    MathSciNet  MATH  Google Scholar 

  21. G. Zhou, A. Zimmermann, On singular equivalences of Morita type. J. Algebra 385, 64–79 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Graduate School of Mathematics, Nagoya University Furocho, Chikusa-ku, Nagoya, Aichi, 464-8602, Japan

    Ryo Takahashi

Authors
  1. Ryo Takahashi
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Correspondence to Ryo Takahashi .

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Editors and Affiliations

  1. Universitat Autònoma de Barcelona, Departament de Matemàtiques Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain

    Dolors Herbera

  2. Universitat Autònoma de Barcelona, Departament de Matemàtiques Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain

    Wolfgang Pitsch

  3. Universitat de Barcelona, Departament de MatemĂ tiques i InformĂ tica, Barcelona, Barcelona, Spain

    Santiago Zarzuela

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Takahashi, R. (2016). Singularity Categories of Stable Resolving Subcategories and Applications to Gorenstein Rings. In: Herbera, D., Pitsch, W., Zarzuela, S. (eds) Extended Abstracts Spring 2015. Trends in Mathematics(), vol 5. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-45441-2_28

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