Science China Mathematics

, Volume 62, Issue 12, pp 2487–2496 | Cite as

Gorenstein homological invariant properties under Frobenius extensions

  • Zhibing ZhaoEmail author


We prove that for a Frobenius extension, a module over the extension ring is Gorenstein projective if and only if its underlying module over the base ring is Gorenstein projective. For a separable Frobenius extension between Artin algebras, we obtain that the extension algebra is CM (Cohen-Macaulay)-finite (resp. CM-free) if and only if so is the base algebra. Furthermore, we prove that the reprensentation dimension of Artin algebras is invariant under separable Frobenius extensions.


Frobenius extensions separable extensions Gorenstein projective modules reprensenation dimension 


13B02 16E10 16G10 16G50 


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This work was supported by National Natural Science Foundation of China (Grant No. 11571329) and the Natural Science Foundation of Anhui Province (Grant No. 1708085MA01). The research was completed during the author’s visit at the University of Washington. He thanks Professor James Zhang for his hospitality. The author thanks Dr. Jie Li for pointing out an error in the manuscript. The author thanks the referees for the helpful comments and valuable suggestions.


  1. 1.
    Auslander M. Representation Dimension of Artin Algebras. Queen Mary College Mathematics Notes. London: Queen Mary College, 1971zbMATHGoogle Scholar
  2. 2.
    Auslander M, Reiten I. Applications of contravariantly finite subcategories. Adv Math, 1991, 86: 111–152MathSciNetCrossRefGoogle Scholar
  3. 3.
    Beligiannis A. Cohen-Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras. J Algebra, 2005, 288: 137–211MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bennis D, Mahdou N. Gorenstein global dimension. Proc Amer Math Soc, 2010, 138: 461–465MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen X. Totally re exive extensions and modules. J Algebra, 2013, 379: 322–332MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen X. An Auslander-type result for Gorenstein-projective modules. Adv Math, 2008, 218: 2043–2050MathSciNetCrossRefGoogle Scholar
  7. 7.
    Enochs E E, Jenda O M G. Gorenstein injective and projective modules. Math Z, 1995, 220: 611–633MathSciNetCrossRefGoogle Scholar
  8. 8.
    Enochs E E, Jenda O M G. Relative Homological Algebra. de Gruyter Expositions in Mathematics, vol. 30. Berlin-New York: Walter de Gruyter, 2000CrossRefGoogle Scholar
  9. 9.
    Ermann K, Holm T, Iyama O, et al. Radical embeddings and representation dimension. Adv Math, 2004, 185: 159–177MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fischman D, Montgomery S, Schneider H-J. Frobenius extensions of subalgebras of Hopf algebras. Trans Amer Math Soc, 1997, 349: 4857–4895MathSciNetCrossRefGoogle Scholar
  11. 11.
    Guo X. Representation dimension: An invariant under stable equivalence. Trans Amer Math Soc, 2005, 357: 3255–3263MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hirata K, Sugano K. On semisimple extensions and separable extensions over noncommutative rings. J Math Soc Japan, 1966, 18: 360–373MathSciNetCrossRefGoogle Scholar
  13. 13.
    Holm H. Gorenstein homological dimensions. J Pure Appl Algebra, 2004, 189: 167–193MathSciNetCrossRefGoogle Scholar
  14. 14.
    Huang Z, Sun J. Invariant properties of represenations under excellent extensions. J Algebra, 2012, 358: 87–101MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kadison L. The Jones polynomial and certain separable Frobenius extensions. J Algebra, 1996, 186: 461–475MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kadison L. New Examples of Frobenius Extension. University Lecture Series, vol. 14. Provedence: Amer Math Soc, 1999CrossRefGoogle Scholar
  17. 17.
    Kasch F. Grundlagen einer theorie der Frobenius-Erweiterungen. Math Ann, 1954, 127: 453–474MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kasch F. Projektive Frobenius-Erweiterungen, Sizungsber. Heidelberger Akad Wiss Math-Natur Kl, 1961, 1961: 89–109zbMATHGoogle Scholar
  19. 19.
    Li Z, Zhang P. A construction of Gorenstein-projective modules. J Algebra, 2010, 323: 1802–1812MathSciNetCrossRefGoogle Scholar
  20. 20.
    Morita K. Adojint pairs of functors and Frobenius extensions. Sci Rep Toyko Kyoiku Daigaku Sect A, 1965, 9: 40–71zbMATHGoogle Scholar
  21. 21.
    Müller B. Quasi-Frobenius Erweiterungen I. Math Z, 1964, 85: 345–368MathSciNetCrossRefGoogle Scholar
  22. 22.
    Nakamaya T, Tsuzuku T. On Frobenius extension I. Nagoya Math J, 1960, 17: 89–110MathSciNetCrossRefGoogle Scholar
  23. 23.
    Pierce R S. Associative Algebras. New York: Springer, 1982CrossRefGoogle Scholar
  24. 24.
    Ringel C M. The Gorenstein-projective modules for the Nakayama algebras I. J Algebra, 2013, 385: 241–261MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ren W. Gorenstein projective modules and Frobenius extensions. Sci China Math, 2018, 61: 1175–1186MathSciNetCrossRefGoogle Scholar
  26. 26.
    Schneider H-J. Normal basis and transitivity of crossed products for Hopf algebras. J Algebra, 1992, 151: 289–312MathSciNetCrossRefGoogle Scholar
  27. 27.
    Sugano K. Separable extensions and Frobenius extensions. Osaka J Math, 1970, 7: 291–299MathSciNetzbMATHGoogle Scholar

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© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesAnhui UniversityHefeiChina

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