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A note on Frobenius functors

Funds:  Supported by Natural Science Foundation of China (11571329), the Natural Science Foundation of Anhui Province (1708085MA01), Project of University Natural Science Research of Anhui Province (KJ2015A101).
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https://doi.org/10.3969/j.issn.0253-2778.2018.08.003
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  • Author Bio:

    ZHAO Zhibing, male, born in 1979, PhD/lecturer. Research field: Homological algebra, representation theory of algebras. E-mail: zbzhao@ahu.edu.cn

  • Received Date: 24 April 2018
  • Accepted Date: 08 May 2018
  • Rev Recd Date: 08 May 2018
  • Publish Date: 31 August 2018
    Abstract

  • Abstract

    Some new characterizations of Frobenius bimodules in terms of Frobenius functors were given. It was proved that a bimodule is Frobenius if and only if it is finitely generated projective on both sides, and that the restriction of the corresponding tensor functor to the categories of finitely generated projective modules is a Frobenius functor. The characterizations allow us to give a new proof of the endomorphism ring theorem by a functorial method.

    Abstract

    Some new characterizations of Frobenius bimodules in terms of Frobenius functors were given. It was proved that a bimodule is Frobenius if and only if it is finitely generated projective on both sides, and that the restriction of the corresponding tensor functor to the categories of finitely generated projective modules is a Frobenius functor. The characterizations allow us to give a new proof of the endomorphism ring theorem by a functorial method.
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    • References(8)
  • [1]
    KASCH F. Grundlagen einer theorie der Frobenius-Erweiterungen[J]. Math Ann, 1954, 127: 453-474.
    [2]
    MORITA K. Adjoint pairs of functors and Frobenius extensions[J]. Sci Rep Toyko Kyoiku Daigaku (Sect A), 1965, 9: 40-71.
    [3]
    CAENEPEEL S, MILITARU G, ZHU S. Frobenius and Separable Functors for Generalized Module Categories and Nonlinear Equations[M]. Berlin: Springer-Verlag, 2002.
    [4]
    CASTAEO IGLESIAS F, GMEZ TORRECILLAS J, NSTSESCU C. Frobenius functors: Applications[J]. Comm Algebra, 1999, 27(10): 4879-4900.
    [5]
    KADISON L. New Example of Frobenius Extensions[M]. Providence, RI: AMS, 1999.
    [6]
    ANDERSON F W, FULLER K R. Ring and Categories of Modules[M] . Berlin: Springer, 1974.
    [7]
    WISBAUER R. Foundations of Module and Ring Theory[M]. Reading/ Paris: Gordon and Breach, 1991.
    [8]
    KASCH F. Projektive Frobenius-Erweiterungen, Sizungsber[J]. Heidelberger Akad Wiss Math Natur Kl, 1960, 10: 89-109.)
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    [1]
    KASCH F. Grundlagen einer theorie der Frobenius-Erweiterungen[J]. Math Ann, 1954, 127: 453-474.
    [2]
    MORITA K. Adjoint pairs of functors and Frobenius extensions[J]. Sci Rep Toyko Kyoiku Daigaku (Sect A), 1965, 9: 40-71.
    [3]
    CAENEPEEL S, MILITARU G, ZHU S. Frobenius and Separable Functors for Generalized Module Categories and Nonlinear Equations[M]. Berlin: Springer-Verlag, 2002.
    [4]
    CASTAEO IGLESIAS F, GMEZ TORRECILLAS J, NSTSESCU C. Frobenius functors: Applications[J]. Comm Algebra, 1999, 27(10): 4879-4900.
    [5]
    KADISON L. New Example of Frobenius Extensions[M]. Providence, RI: AMS, 1999.
    [6]
    ANDERSON F W, FULLER K R. Ring and Categories of Modules[M] . Berlin: Springer, 1974.
    [7]
    WISBAUER R. Foundations of Module and Ring Theory[M]. Reading/ Paris: Gordon and Breach, 1991.
    [8]
    KASCH F. Projektive Frobenius-Erweiterungen, Sizungsber[J]. Heidelberger Akad Wiss Math Natur Kl, 1960, 10: 89-109.)
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