Skip to main content
SpringerLink
Log in

Trivial extension of Koszul algebras

  • Research Article
  • Published:
Frontiers of Mathematics in China Aims and scope Submit manuscript

Abstract

Let Λ be a Koszul algebra, and let M be a graded Λ-bimodule. We prove that the trivial extension algebra of Λ by M is also a Koszul algebra whenever M is Koszul as a left Λ-module. Applications and examples are also provided.

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. Auslander M, Reiten I, Smalø S. Representation Theory of Artin Algebras. Cambridge: Cambridge Univ Press, 1997

    MATH  Google Scholar 

  2. Berger R. Koszulity of nonquadratic algebras. J Algebra, 2001, 239(2): 705–734

    Article  MathSciNet  MATH  Google Scholar 

  3. Bian N, Ye Y, Zhang P. Generalized d-Koszul modules. Math Res Lett, 2011, 18(2): 191–200

    Article  MathSciNet  MATH  Google Scholar 

  4. Green E L, Marcos E N, Martínez-Villa R, Zhang P. D-Koszul algebras. J Pure Appl Algebra, 2004, 193(1): 141–162

    Article  MathSciNet  MATH  Google Scholar 

  5. Green E L, Marcos E N, Zhang P. Koszul modules and modules with linear presentations. Comm Algebra, 2003, 31(6): 2745–2770

    Article  MathSciNet  MATH  Google Scholar 

  6. Green E L, Martínez-Villa R. Koszul and Yoneda algebras. In: Bautista R, Martínez Villa R, Peña J A D L, eds. Representation Theory of Algebras. Seventh International Conference, August 22–26, 1994, Cocoyoc, Mexico. CMS Conf Proc, Vol 18. Providence, Amer Math Soc, 1996, 247–306

    Google Scholar 

  7. Martínez-Villa R, Montano-Bermudez G. Triangular matrix and Koszul algebras, Int J Algebra, 2007, 1(10): 441–467

    Article  MathSciNet  MATH  Google Scholar 

  8. Priddy S B. Koszul resolutions. Trans Amer Math Soc, 1970, 152(1): 39–60

    Article  MathSciNet  MATH  Google Scholar 

  9. Shelton B, Tingey C. On Koszul algebras and a new construction of Artin-Schelter regular algebras. J Algebra, 2001, 241(2): 789–798

    Article  MathSciNet  MATH  Google Scholar 

  10. Ye Y. Higher Koszul algebras and higher Koszul complexes. Ph D Thesis, Hefei: University of Science and Technology of China, 2002

    Google Scholar 

  11. Ye Y, Zhang P. Higher Koszul modules. Sci China Math, 2002, 32(11): 1042–1049

    Google Scholar 

  12. Ye Y, Zhang P. Higher Koszul complexes. Sci China Math, 2003, 46(1): 118–128

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author is very grateful to the referees for helpful comments. He thanks Professor Yu Ye for numerous discussion and suggestions. This work was supported in part by the National Natural Science Foundation of China (Grant No. 11471017), Doctoral Research Foundation, and the Research Culture Foundation of Anhui Normal University (No. 2014xmpy11).

Author information

Authors and Affiliations

  1. School of Mathematics and Computer Science, Anhui Normal University, Wuhu, 241002, China

    Zhi Cheng

Authors
  1. Zhi Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhi Cheng.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cheng, Z. Trivial extension of Koszul algebras. Front. Math. China 12, 1045–1056 (2017). https://doi.org/10.1007/s11464-017-0655-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11464-017-0655-y

Keywords

MSC

Search

Navigation