The Torsion Theory and the Melkersson Condition

Abstract

We consider a generalization of the notion of torsion theory, which is associated with a Serre subcategory over a commutative Noetherian ring. In 2008 Aghapournahr and Melkersson investigated the question of when local cohomology modules belong to a Serre subcategory of the module category. In their study, the notion of Melkersson condition was defined as a suitable condition in local cohomology theory. One of our purposes in this paper is to show how naturally the concept of Melkersson condition appears in the context of torsion theories.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    M. Aghapournahr, L. Melkersson: Local cohomology and Serre subcategories. J. Algebra 320 (2008), 1275–1287.

    MathSciNet  Article  Google Scholar 

  2. [2]

    A. Beligiannis, I. Reiten: Homological and homotopical aspects of torsion theories. Mem. Am. Math. Soc. 883 (2007), 207 pages.

    MathSciNet  MATH  Google Scholar 

  3. [3]

    S. E. Dickson: A torsion theory for Abelian categories. Trans. Am. Math. Soc. 121 (1966), 223–235.

    MathSciNet  Article  Google Scholar 

  4. [4]

    P. Gabriel: Des catégories abéliennes. Bull. Soc. Math. Fr. 90 (1962), 323–448. (In French.)

    Article  Google Scholar 

  5. [5]

    J. Lambek: Torsion Theories, Additive Semantics, and Rings of Quotients. Lecture Notes in Mathematics 177, Springer, Berlin, 1971.

    Book  Google Scholar 

  6. [6]

    B. Stenström: Rings and Modules of Quotients. Lecture Notes in Mathematics 237, Springer, Berlin, 1971.

    Book  Google Scholar 

  7. [7]

    B. Stenström: Rings of Quotients. An Introduction to Methods of Ring Theory. Die Grundlehren der Mathematischen Wissenschaften 217, Springer, Berlin, 1975.

    MATH  Google Scholar 

  8. [8]

    T. Yoshizawa: Subcategories of extension modules by Serre subcategories. Proc. Am. Math. Soc. 140 (2012), 2293–2305.

    MathSciNet  Article  Google Scholar 

  9. [9]

    T. Yoshizawa: On the closedness of taking injective hulls of several Serre subcategories. Commun. Algebra 45 (2017), 4846–4854.

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgments

The author expresses gratitude to the referees for their kind comments and valuable suggestions.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Takeshi Yoshizawa.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yoshizawa, T. The Torsion Theory and the Melkersson Condition. Czech Math J 70, 121–145 (2020). https://doi.org/10.21136/CMJ.2019.0193-18

Download citation

Keywords

  • Melkersson condition
  • Serre subcategory
  • torsion theory

MSC 2010

  • 13C60
  • 13D30