SpringerLink
Log in

Notes on piecewise-Koszul algebras

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

Abstract

The relationships between piecewise-Koszul algebras and other “Koszul-type” algebras are discussed. The Yoneda-Ext algebra and the dual algebra of a piecewise-Koszul algebra are studied, and a sufficient condition for the dual algebra A ! to be piecewise-Koszul is given. Finally, by studying the trivial extension algebras of the path algebras of Dynkin quivers in bipartite orientation, we give explicit constructions for piecewise-Koszul algebras with arbitrary “period” and piecewise-Koszul algebras with arbitrary “jump-degree”.

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

Access this article

Log in via an institution

Price includes VAT (Germany)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chen, Z., Ye, Y.: One-point extensions of t-Koszul algebras. Acta Mathematica Sinica, English Series, 23(6), 965–972 (2007)

    Article  MathSciNet  Google Scholar 

  2. Green, E. L., Marcos, E. N., Martinez-Villa, R., et al.: D-Koszul algebras. J. Pure Appl. Alg., 193, 141–162 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Lü, J. F.: On modules with d-Koszul-type submodules. Acta Mathematica Sinica, English Series, 25(6), 1015–1030 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Lü, J. F., He, J. W., Lu, D. M.: Piecewise-Koszul algebras. Sci. China, Ser. A, 50, 1785–1794 (2007)

    Google Scholar 

  7. Brenner, S., Butler, M. C. R., King, A. D.: Periodic algebras which are almost Koszul. Alg. Represent. Theory, 5, 331–367 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cassidy, T., Shelton, B.: Generalizing the notion of a Koszul algebra. Math. Z., 260, 93–114 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Green, E. L., Snashall, N.: Finite generation of Ext for a generalization of D-Koszul algebras. J. Algebra, 295, 458–472 (2006)

    MATH  MathSciNet  Google Scholar 

  10. Green, E. L., Marcos, E. N.: δ-Koszul algebras. Comm. Alg., 33(6), 1753–1764 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Polishchuk, A., Positselski, L.: Quadratic algebras, University Lectures Series, Vol. 37, American Mathematics Sosiety, Providence, 2005

    MATH  Google Scholar 

  12. Cibils, C., Zhang, P.: Calabi-Yau objects in triangulated categories. Trans. Amer. Math. Soc., 361, 6501–6519 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Lü, J. F.: On an example of δ-Koszul algebras. Proc. Amer. Math. Soc., 138, 3777–3781 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Lü, J. F.: Notes on δ-Koszul algebras. Applied Cate. Struc., DOI: 10.1007/s10485-010-9224-1, Online First

  15. Lü, J. F., Yu, X. L.: A note on the bound of the generation degree for Ext-algebras (in Chinese). Sci. Sin. Math., 40(3), 223–226 (2010)

    Google Scholar 

  16. Auslander, M., Reiten, I., Smalø, S.: Representation Theory of Artin Algebras, Cambridge studies in Advanced Mathethematics, Vol. 36, Cambridge, UK, Cambridge University Press, 1995

    Book  MATH  Google Scholar 

  17. Aquino, R. M., Green, E. L.: On modules with linear presentations over Koszul algebras. Comm. Alg., 33, 19–36 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Artin, M., Schelter, W. F.: Graded algebras of global dimension 3. Adv. Math., 66, 171–216 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  19. Beilinson, A., Ginszburg, V., Soergel, W.: Koszul duality patterns in representation theory. J. Amer. Math. Soc., 9, 473–525 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  20. Si, J. R.: Higher order Koszul modules. Acta Mathematica Sinica, Chinese Series, 52(5), 101–110 (2009)

    MathSciNet  Google Scholar 

  21. Guo, J. Y., Wan, Q. H., Wu, Q. X.: On the Koszul modules of exterior algebras. Acta Mathematica Sinica, English Series, 23(11), 1967–1984 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. Guo, J. Y., Li, A. H., Wu, Q. X.: Selfinjective Koszul algebras of finite complexity. Acta Mathematica Sinica, English Series, 25(12), 2179–2198 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Weibel, C. A.: An Introduction to Homological Algebra, Cambridge Studies in Avanced Mathematics, Vol. 38, Cambridge Univ. Press, Cambridge, 1995

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Zhejiang Normal University, **hua, 321004, P. R. China

    Jia Feng Lü

  2. Department of Mathematics, Zhejiang University, Hangzhou, 310027, P. R. China

    **ao Lan Yu

Authors
  1. Jia Feng Lü
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. **ao Lan Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jia Feng Lü.

Additional information

Supported by National Natural Science Foundation of China (Grant No. 11001245), Zhejiang Province Department of Education Fund (Grant No. Y201016432) and Zhejiang Innovation Project (Grant No. T200905)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lü, J.F., Yu, X.L. Notes on piecewise-Koszul algebras. Acta. Math. Sin.-English Ser. 27, 943–952 (2011). https://doi.org/10.1007/s10114-011-8238-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-011-8238-4

Keywords

MR(2000) Subject Classification

Search

Navigation