Advertisement

Gorenstein Properties of Simple Gluing Algebras

Article
  • 32 Downloads

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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)CrossRefzbMATHGoogle Scholar
  2. 2.
    Auslander, M., Reiten, I.: Application of contravariantly finite subcategories. Adv. Math. 86(1), 111–152 (1991)MathSciNetCrossRefzbMATHGoogle 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)CrossRefzbMATHGoogle Scholar
  5. 5.
    Auslander, M., Smalø, S.O.: Almost split sequences in subcategories. J. Algebra 69, 426–454 (1981)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Beligiannis, A.: Cohen-Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras. J. Algebra 288, 137–211 (2005)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chen, X., Lu, M.: Singularity categories of skewed-gentle algebras. Colloq. Math. 141(2), 183–198 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chen, X.-W.: Singularity categories, Schur functors and triangular matrix rings. Algebr. Represent. Theor. 12, 181–191 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chen, X.-W.: A recollement of vector bundles. Bull. London Math. Soc. 44, 271–284 (2012)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Kong, F., Zhang, P.: From CM-finite to CM-free. J. Pure Appl. Algebra 220 (2), 782–801 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Liu, P., Lu, M.: Recollements of singularity categories and monomorphism categories. Comm. Algebra 43, 2443–2456 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Liu, S.: Auslander-reiten theory in a Krull-Schmidt category. São Paulo J. Math. Sci. 4(3), 425–472 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lu, M.: Gorenstein defect categories of triangular matrix algebras. J. Algebra 480, 346–367 (2017)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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]MathSciNetCrossRefzbMATHGoogle 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)zbMATHGoogle Scholar
  31. 31.
    Ringel, C.M.: The Gorenstein projective modules for the Nakayama algebras. I. J. Algebra 385, 241–261 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Ringel, C.M., Zhang, P.: Representations of quivers over the algebra of dual numbers. J. Algebra 475, 327–360 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Vatne, D.F.: The mutation class of d n quivers. Comm. Algebra 38(3), 1137–1146 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Vatne, D.F.: Endomorphism rings of maximal rigid objects in cluster tubes. Colloq. Math. 123, 63–93 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Xiong, B.L., Zhang, P.: Gorenstein-projective modules over triangular matrix Artin algebras. J. Algebra Appl. 11(4), 1250066 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Yang, D.: Endomorphism algebras of maximal rigid objects in cluster tubes. Comm. Algebra 40(12), 4347–4371 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Zhang, P.: Gorenstein-projective modules and symmetric recollements. J. Algebra 388, 65–80 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

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

Personalised recommendations