Gorenstein Properties of Simple Gluing Algebras

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Abstract

For given bound quiver algebras A and B, we obtain a new algebra Λ, called the simple gluing algebra, by identifying two vertices. We investigate the Gorenstein homological property, the singularity category, the Gorenstein defect category and the Cohen-Macaulay Auslander algebra of Λ in terms of that of A and B. Finally, we give applications to cluster-tilted algebras.

Keywords

Gorenstein projective module Singularity category Gorenstein defect category Simple gluing algebra 

Mathematics Subject Classification (2010)

18E30 18E35 

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Notes

Acknowledgments

The work was done during the stay of the author at the Department of Mathematics, University of Bielefeld. He is deeply indebted to Professor Henning Krause for his kind hospitality, inspiration and continuous encouragement. The author thanks Professor Liangang Peng very much for his guidance and constant support. The author was supported by the National Natural Science Foundation of China (No. 11401401 and No. 11601441).

References

  1. 1.
    Assem, I., Simson, D., Skowroński, A.: Elements of the representation theory of associative algebras. Vol. 1. Techniques of representation theory London Mathematical Society Student Texts, vol. 65. Cambridge University Press, Cambridge (2006)CrossRefMATHGoogle Scholar
  2. 2.
    Auslander, M., Reiten, I.: Application of contravariantly finite subcategories. Adv. Math. 86(1), 111–152 (1991)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Auslander, M., Reiten, I.: Cohen-Macaulay and Gorenstein Artin algebras. In: Progress in Math. 95, pp. 221–245. Birkhäuser, Verlag Basel (1991)Google Scholar
  4. 4.
    Auslander, M., Reiten, I., Smalø, S.O.: Representation Theory of Artin Algebras Cambridge Studies in Advanced Mathematics, vol. 36. Cambridge University Press, Cambridge (1995)CrossRefMATHGoogle Scholar
  5. 5.
    Auslander, M., Smalø, S.O.: Almost split sequences in subcategories. J. Algebra 69, 426–454 (1981)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Avramov, L.L., Martsinkovsky, A.: Absolute, relative and Tate cohomology of modules of finite Gorenstein dimensions. Proc. Lond. Math. Soc. 85(3), 393–440 (2002)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Beligiannis, A.: Cohen-Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras. J. Algebra 288, 137–211 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bergh, P.A., Jørgensen, D.A., Oppermann, S.: The Gorenstein defect category. Preprint, available at arXiv:1202.2876 [math.CT]
  9. 9.
    Brüstle, T.: Kit algebras. J. Algebra 240(1), 1–24 (2001)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Buan, A.B., Vatne, D.F.: Derived equivalence classification for cluster-tilted algebras of type a n. J. Algebra 319(7), 2723–2738 (2008)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Buchweitz, R.: Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein Rings. Unpublished Manuscript. Availble at: http://hdl.handle.net/1807/16682 (1987)
  12. 12.
    Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters (a n case). Trans. AMS 358, 1347–1364 (2006)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Chen, X., Geng, S., Lu, M.: The singularity categories of the Cluster-tilted algebras of Dynkin type. Algebr. Represent. Theory 18(2), 531–554 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Chen, X., Lu, M.: Singularity categories of skewed-gentle algebras. Colloq. Math. 141(2), 183–198 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Chen, X.-W.: Singularity categories, Schur functors and triangular matrix rings. Algebr. Represent. Theor. 12, 181–191 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Chen, X.-W.: A recollement of vector bundles. Bull. London Math. Soc. 44, 271–284 (2012)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Chen, X.-W.: The singularity category of a quadratic monomial algebra. Preprint, available at arXiv:1502.02094 [math.RT]
  18. 18.
    Chen, X.-W., Shen, D., Zhou, G.: The Gorentein-projective modules over a monomial algebra. To appear In: Proceedings of the Royal Society of Edinburgh Section A: MathematicsGoogle Scholar
  19. 19.
    Cline, E., Parshall, B., Scott, L.: Finite dimensional algebras and highest weight categories. J. Reine Angew. Math. 391, 85–99 (1988)MathSciNetMATHGoogle Scholar
  20. 20.
    Enochs, E.E., Jenda, O.M.G.: Relative homological algebra. De Gruyter Exp. Math. 30 Walter De Gruyter Co (2000)Google Scholar
  21. 21.
    Happel, D.: On Gorenstein Algebras. In: Progress in Math, vol. 95, pp. 389–404. Basel, Birkhäuser Verlag (1991)Google Scholar
  22. 22.
    Kalck, M.: Singularity categories of gentle algebras. Bull. London Math. Soc. 47 (1), 65–74 (2015)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Keller, B., Vossieck, D.: Sous les catgories drives, (French) [Beneath the derived categories]. C. R. Acad. Sci. Paris sér. I Math. 305(6), 225–228 (1987)MathSciNetMATHGoogle Scholar
  24. 24.
    Kong, F., Zhang, P.: From CM-finite to CM-free. J. Pure Appl. Algebra 220 (2), 782–801 (2016)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Liu, P., Lu, M.: Recollements of singularity categories and monomorphism categories. Comm. Algebra 43, 2443–2456 (2015)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Liu, S.: Auslander-reiten theory in a Krull-Schmidt category. São Paulo J. Math. Sci. 4(3), 425–472 (2010)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Lu, M.: Gorenstein defect categories of triangular matrix algebras. J. Algebra 480, 346–367 (2017)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Orlov, D.: Triangulated categories of singularities and D-branes in Landau-Ginzburg models. Proc. Steklv. Inst. Math. 246(3), 227–248 (2004)MathSciNetMATHGoogle Scholar
  29. 29.
    Orlov, D.: Triangulated categories of singularities and equivalences between Landau-Ginzburg models. Mat. Sb. 197, 1827–1840 (2006). See also arXiv:0503630 [math.AG]MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Orlov, D.: Derived categories of coherent sheaves and triangulated categories of singularities. Algebra, Arithmetic, and Geometry, vol. II. Progr. Math., vol. 270, pp. 503–531. Birkhäuser Boston, Inc., Boston (2009)MATHGoogle Scholar
  31. 31.
    Ringel, C.M.: The Gorenstein projective modules for the Nakayama algebras. I. J. Algebra 385, 241–261 (2013)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Ringel, C.M., Zhang, P.: Representations of quivers over the algebra of dual numbers. J. Algebra 475, 327–360 (2017)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Vatne, D.F.: The mutation class of d n quivers. Comm. Algebra 38(3), 1137–1146 (2010)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Vatne, D.F.: Endomorphism rings of maximal rigid objects in cluster tubes. Colloq. Math. 123, 63–93 (2011)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Xiong, B.L., Zhang, P.: Gorenstein-projective modules over triangular matrix Artin algebras. J. Algebra Appl. 11(4), 1250066 (2012)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Yang, D.: Endomorphism algebras of maximal rigid objects in cluster tubes. Comm. Algebra 40(12), 4347–4371 (2012)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Zhang, P.: Gorenstein-projective modules and symmetric recollements. J. Algebra 388, 65–80 (2013)MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsSichuan UniversityChengduPeople’s Republic of China

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