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On Tubular Tilting Objects in the Stable Category of Vector Bundles

  • Jian Min ChenEmail author
  • Ya Nan Lin
  • Shi Quan Ruan
Article
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Abstract

The present paper focuses on the study of the stable category of vector bundles for the weighted projective lines of weight triple. We find some important triangles in this category and use them to construct tilting objects with tubular endomorphism algebras for the case of genus one via cluster tilting theory.

Keywords

Tilting sheaf weighted projective line stable category cluster category tubular algebra 

MR(2010) Subject Classification

14A22 14F05 16G70 16S99 18E30 

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References

  1. [1]
    Barot, M., Kussin, D., Lenzing, H.: The cluster category of a canonical algebra. Trans. Amer. Math. Soc., 362, 4313–4330 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Buan, A. B., Iyama, O., Reiten, I., et al.: Cluster structures for 2-Calabi–Yau categories and unipotent groups. Compos. Math., 145(4), 1035–1079 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Buan, A. B., Marsh, R., Reineke, M., et al.: Tilting theory and cluster combinatorics. Adv. Math., 204, 572–618 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Chen, J., Lin, Y., Ruan, S.: Tilting objects in the stable category of vector bundles on the weighted projective line of type (2, 2, 2, 2; λ). J. Algebra, 397, 570–588 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Chen, J., Lin, Y., Liu, P., et al.: Classifications for tilting objects on a weighted projective line of type (2, 2, 2, 2; λ). arXiv:1303.1323Google Scholar
  6. [6]
    Fomin, S., Zelevinsky, A.: Cluster algebras I. Foundations. J. Amer. Math. Soc., 15(2), 497–529 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Geigle, W., Lenzing, H.: A class of weighted projective curves arising in representation theory of finite dimensional algebras. Singularities, representations of algebras, and Vector bundles. Springer Lect. Notes Math., 1273, 265–297 (1987)CrossRefzbMATHGoogle Scholar
  8. [8]
    Happel, D.: Triangulated categories in the representation theory of finite-dimensional algebras. London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 119 (1988)Google Scholar
  9. [9]
    Iyama, O., Yoshino, Y.: Mutation in triangulated categories and rigid Cohen–Macaulay modules. Invent. Math., 172, 117–168 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Keller, B.: On triangulated orbit categories. Doc. Math., 10, 551–581 (2005)MathSciNetzbMATHGoogle Scholar
  11. [11]
    Keller, B., Reiten, I.: Acyclic Calabi–Yau categories. Compos. Math., 144(5), 1332–1348 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Kussin, D., Lenzing, H., Meltzer, H.: Triangle singularities, ADE-chains and weighted projective lines. Adv. Math., 237(1), 194–251 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Lenzing, H.: Wild canonical algebras and rings of automorphic forms. In Finite-dimensional algebras and related topics (Ottawa, ON, 1992), volume 424 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 191–212. Kluwer Acad. Publ., Dordrecht, 1994Google Scholar
  14. [14]
    Lenzing, H., Meltzer, H.: Sheaves on a weighted projective line of genus one, and representations of a tubular algebra. In Representations of algebras, Sixth International Conference, Ottawa 1992. CMS Conf. Proc., 14, 313–337 (1993)zbMATHGoogle Scholar
  15. [15]
    Lenzing, H., Ruan, S.: On vector bundles of rank two on a weighted projective line. In PreparationGoogle Scholar
  16. [16]
    Orlov, D.: Derived categories of coherent sheaves and triangulated categories of singularities. Algebra, Arithmetic, and Geometry, Progress in Mathematics, 270, 503–531 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science (CAS), Chinese Mathematical Society (CAS) and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesXiamen UniversityXiamenP. R. China

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