Skip to main content
SpringerLink
Log in

Hochschild (Co)homology of a class of Nakayama algebras

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

Let Λ = kQ/I be a finite-dimensional Nakayama algebra, where Q is an Euclidean diagram

for some n with cyclic orientation, and I is an admissible ideal generated by a single monomial relation. In this note we determine explicitly all the Hochschild homology and cohomology groups of Λ based on a detailed description of the Bardzell complex. Moreover, the cyclic homology of Λ can be calculated in the case that the underlying field is of characteristic zero.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Mac Lane, S.: Homology, Grundlehren Math. Wiss. 114, Springer, Berlin, 1975

    Google Scholar 

  2. Happel, D.: Hochschild cohomology of finite-dimensional algebras. Lecture Notes in Mathematics, 1404, 108–126 (1989)

    Article  MathSciNet  Google Scholar 

  3. Skowroński, A.: Simply connected algebras and Hochschild cohomology. Proc. ICRA IV (Ottawa, 1992), Can. Math. Soc. Proc., 14, 431–447 (1993)

    Google Scholar 

  4. Assem, I., de la Peña, J. A.: The foundamental groups of a triangular algebra. Comm. Algebra, 24, 187–208 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  5. Gerstenhaber, M.: On the deformation of rings and algebras. Ann. Math., 79, 59–103 (1964)

    Article  MathSciNet  Google Scholar 

  6. Igusa, K.: Notes on the no loop conjecture. J. Pure Appl. Algebra, 69, 161–176 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  7. Liu, S. X., Zhang, P.: Hochschild homology of truncated algebras. Bull. London Math. Soc., 26, 427–430(1994)

    Article  MATH  MathSciNet  Google Scholar 

  8. Avramov, L. L., Vigueé-Poirrier, M.: Hochschild homology criteria for smoothness. Internat. Math. Research Notices, 1, 17–25 (1992)

    Article  MathSciNet  Google Scholar 

  9. Han, Y.: Hochschild (co)homology dimension. J. London Math. Soc., 73(2), 657–668 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Keller, B.: Invariance and localization for cyclic homology of DG algebras. J. Pure Appl. Algebra, 123, 223–273 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  11. Erdmann, K., Holm, T.: Twisted bimodules and Hochschild cohomology for self-injective algebras of class An. Forum Math., 11, 177–201 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bardzell, M. J., Locateli, A. C., Marcos, E. N.: On the Hochschild cohomology of truncated cycle algebras. Comm Alg., 28(3), 1615–1639 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras, London Math. Soc. Student Texts 65, Cambridge Univ. Press, Cambridge, 2005

    Google Scholar 

  14. Snashall, N., Solberg, Ø.: Support varieties and Hochschild cohomology rings. Proc. London Math. Soc, 88(3), 705–732 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  15. Bardzell, M. J.: The alternating syzygy behavior of monomial algebras. J. Alg., 188, 69–89 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  16. Holm, T.: Hochschild cohomology rings k[X]/(f). Beitrage Algebra Geom., 41, 291–301 (2000)

    MATH  MathSciNet  Google Scholar 

  17. Loday, J. L.: Cyclic homology, Grundlehren 301, Springer, Berlin, 1992

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics & Computer Science, Hubei University, Wuhan, 430062, P. R. China

    Yun Ge Xu & Dan Wang

Authors
  1. Yun Ge Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yun Ge Xu.

Additional information

Supported by National Natural Science Foundation of China (Grant Nos. 10426014 and 10501010)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xu, Y.G., Wang, D. Hochschild (Co)homology of a class of Nakayama algebras. Acta. Math. Sin.-English Ser. 24, 1097–1106 (2008). https://doi.org/10.1007/s10114-007-6072-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-007-6072-5

Keywords

MR(2000) Subject Classification

Search

Navigation