Skip to main content
SpringerLink
Log in

Weak dimensions and Gorenstein weak dimensions of group rings

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

Abstract

Let K be a field, and let G be a group. In the present paper, we investigate when the group ring K[G] has finite weak dimension and finite Gorenstein weak dimension. We give some analogous versions of Serre’s theorem for the weak dimension and the Gorenstein weak dimension.

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. M. Auslander: On regular group rings. Proc. Am. Math. Soc. 8 (1957), 658–664.

    Article  MathSciNet  Google Scholar 

  2. M. Auslander, M. Bridger: Stable Module Theory. Memoirs of the American Mathematical Society 94. American Mathematical Society, Providence, 1969.

    MATH  Google Scholar 

  3. D. Bennis, N. Mahdou: Global Gorenstein dimensions. Proc. Am. Math. Soc. 138 (2010), 461–465.

    Article  MathSciNet  Google Scholar 

  4. D. J. Benson, K. R. Goodearl: Periodic flat modules, and flat modules for finite groups. Pac. J. Math. 196 (2000), 45–67.

    Article  MathSciNet  Google Scholar 

  5. R. R. Colby: Rings which have flat injective modules. J. Algebra 35 (1975), 239–252.

    Article  MathSciNet  Google Scholar 

  6. I. G. Connell: On the group ring. Can. J. Math. 15 (1963), 650–685.

    Article  Google Scholar 

  7. I. Emmanouil: On the finiteness of Gorenstein homological dimensions. J. Algebra 372 (2012), 376–396.

    Article  MathSciNet  Google Scholar 

  8. E. E. Enochs, O. M. G. Jenda: Gorenstein injective and projective modules. Math. Z. 220 (1995), 611–633.

    Article  MathSciNet  Google Scholar 

  9. S. Glaz: Commutative Coherent Rings. Lecture Notes in Mathematics 1371. Springer, Berlin, 1989.

    Book  Google Scholar 

  10. V. E. Govorov: On flat modules. Sib. Mat. Zh. 6 (1965), 300–304. (In Russian.)

    MathSciNet  MATH  Google Scholar 

  11. H. Holm: Gorenstein Homological Algebra: Ph.D. Thesis. University of Copenhagen, Copenhagen, 2004.

    MATH  Google Scholar 

  12. H. Holm: Gorenstein homological dimensions. J. Pure Appl. Algebra 189 (2004), 167–193.

    Article  MathSciNet  Google Scholar 

  13. T. Y. Lam: A First Course in Noncommutative Rings. Graduate Texts in Mathematics 131. Springer, New York, 2001.

    Book  Google Scholar 

  14. L. Li: Representations of modular skew group algebras. Trans. Am. Math. Soc. 367 (2015), 6293–6314.

    Article  MathSciNet  Google Scholar 

  15. D. S. Passman: The Algebraic Structure of Group Rings. John Wiley, New York, 1977.

    MATH  Google Scholar 

  16. J. J. Rotman: An Introduction to Homological Algebra. Pure and Applied Mathematics 85. Academic Press, New York, 1979.

    MATH  Google Scholar 

  17. U. Stammbach: On the weak homological dimension of the group algebra of solvable groups. J. Lond. Math. Soc., II. Ser. 2 (1970), 567–570.

    Article  MathSciNet  Google Scholar 

  18. R. G. Swan: Groups of cohomological dimension one. J. Algebra 12 (1969), 585–601.

    Article  MathSciNet  Google Scholar 

  19. Y. Xiang: Homological dimensions of skew group rings. Algebra Colloq. 27 (2020), 319–330.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Computational Science, Huaihua University, Huaihua, 418000, China

    Yueming Xiang

Authors
  1. Yueming Xiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yueming Xiang.

Additional information

The work is supported by the Scientific Research Foundation of Hunan Provincial Education Department (18C0997).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xiang, Y. Weak dimensions and Gorenstein weak dimensions of group rings. Czech Math J 71, 803–816 (2021). https://doi.org/10.21136/CMJ.2021.0102-20

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.21136/CMJ.2021.0102-20

Keywords

MSC 2020

Search

Navigation