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Relative syzygies and grade of modules

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Abstract

Recently Takahashi established a new approximation theory for finitely generated modules over commutative Noetherian rings, which unifies the spherical approximation theorem due to Auslander and Bridger and the Cohen-Macaulay approximation theorem due to Auslander and Buchweitz. In this paper we generalize these results to much more general case over non-commutative rings. As an application, we establish a relation between the injective dimension of a generalized tilting module ω and the finitistic dimension with respect to ω.

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References

  1. Auslander, M., Bridger, M.: Stable Module Theory. Memoirs Amer. Math. Soc., 94, Amer. Math. Soc., Providence, RI, 1969

    Google Scholar 

  2. Auslander, M., Buchweitz, R. O.: The homological theory of Cohen-Macaulay approximations. Soc. Math. France, 38, 5–37 (1989)

    MathSciNet  MATH  Google Scholar 

  3. Takahashi, R.: A new approximation theory which unifies spherical and Cohen-Macaulay approximations. J. Pure Appl. Algebra, 208, 617–634 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Wakamatsu, T.: On modules with trivial self-extensions. J. Algebra, 114, 106–114 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Wakamatsu, T.: Tilting modules and Auslander’s Gorenstein property. J. Algebra, 275, 3–39 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Huang, Z. Y.: ω-k-torsionfree modules and ω-left approximation dimension. Sci. China Ser. A, 44, 184–192 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Huang, Z. Y.: Extension closure of relative syzygy modules. Sci. China Ser. A, 46, 611–620 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Anderson, F. W., Fuller, K. R.: Rings and Categories of Modules, 2nd edition. In: Grad. Texts in Math., Vol. 13, Springer-Verlag, Berlin, 1992

    Google Scholar 

  9. Auslander, M., Reiten, I.: Applications of contravariantly finite subcategories. Adv. Math., 86, 111–152 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  10. Auslander, M., Reiten, I.: Cohen-Macaulay and Gorenstein artin algebras. In: G.O. Michler, C.M. Ringel, eds. Representation Theory of Finite Groups and Finite Dimensional Algebras, Bielefeld, 1991, Prog.Math. 95, Birkhäuser, Basel, 1991, 221–245

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, P. R. China

    Zeng Feng Liu & Zhao Yong Huang

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  1. Zeng Feng Liu
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  2. Zhao Yong Huang
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Correspondence to Zhao Yong Huang.

Additional information

Supported by the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20100091110034), National Natural Science Foundation of China (Grant No. 11171142), Natural Science Foundation of Jiangsu Province of China (Grant Nos. BK2010047, BK2010007) and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions

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Liu, Z.F., Huang, Z.Y. Relative syzygies and grade of modules. Acta. Math. Sin.-English Ser. 29, 489–504 (2013). https://doi.org/10.1007/s10114-012-1383-6

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  • DOI: https://doi.org/10.1007/s10114-012-1383-6

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