Skip to main content
SpringerLink
Log in

Categories of exact sequences with projective middle terms

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

Let A be a finite-dimensional algebra over an algebraically closed field k, \(\mathcal{E}\) the category of all exact sequences in A-mod, \(\mathcal{M}_P\) (respectively, \(\mathcal{M}_I\)) the full subcategory of \(\mathcal{E}\) consisting of those objects with projective (respectively, injective) middle terms. It is proved that \(\mathcal{M}_P\) (respectively, \(\mathcal{M}_I\)) is contravariantly finite (respectively, covariantly finite) in ɛ. As an application, it is shown that \(\mathcal{M}_P = \mathcal{M}_I\) is functorially finite and has Auslander-Reiten sequences provided A is selfinjective.

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. Anderson F W, Fuller K R. Rings and Categories of Modules. NewYork: Springer-Verlag, 1992

    Book  MATH  Google Scholar 

  2. Auslander M S, Reiten I. Applications of contravariantly finite subcategories. Adv Math, 1991, 86: 111–152

    Article  MATH  MathSciNet  Google Scholar 

  3. Auslander M S, Smalø O. Almost split sequences in subcategories. J Algebra, 1981, 69: 426–454

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen X W. An Auslander-type result for Gorenstein-projective modules. Adv Math, 2008, 218: 2043–2050

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen X W. Stable monomorphism category of Frobenius category. Math Res Lett, 2011, 18: 125–137

    Article  MATH  MathSciNet  Google Scholar 

  6. Ringel C M, Schmidmeier M. Submodules categories of wild representation type. J Pure Appl Algebra, 2006, 205: 412–422

    Article  MATH  MathSciNet  Google Scholar 

  7. Ringel C M, Schmidmeier M. The Auslander-Reiten translation in submodule categories. Trans Amer Math Soc, 2008, 360: 691–716

    Article  MATH  MathSciNet  Google Scholar 

  8. Ringel C M, Schmidmeier M. Invariant subspaces of nilpotent operators I. J Rein Angew Math, 2008, 614: 1–52

    Article  MATH  MathSciNet  Google Scholar 

  9. Xiong B L, Zhang P, Zhang Y H. Auslander-Reiten translations in monomorphism categories. Math Forum, doi: 10.1515/forum-2011-0003, 2012

    Google Scholar 

  10. Zhang P. Monomorphism categories, cotilting theory, and Gorenstein-projective modules. J Algebra, 2011, 339: 181–202

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China

    KeYan Song & YueHui Zhang

Authors
  1. KeYan Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. YueHui Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to YueHui Zhang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Song, K., Zhang, Y. Categories of exact sequences with projective middle terms. Sci. China Math. 57, 477–482 (2014). https://doi.org/10.1007/s11425-013-4760-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-013-4760-4

Keywords

MSC(2010)

Search

Navigation