Special precovered categories of Gorenstein categories

  • Tiwei Zhao
  • Zhaoyong Huang


Let A be an abelian category and P(A) be the subcategory of A consisting of projective objects. Let C be a full, additive and self-orthogonal subcategory of A with P(A) a generator, and let G(C) be the Gorenstein subcategory of A. Then the right 1-orthogonal category \(G{(L)^{{ \bot _1}}}\) of G(C) is both projectively resolving and injectively coresolving in A. We also get that the subcategory SPC(G(C)) of A consisting of objects admitting special G(C)-precovers is closed under extensions and C-stable direct summands (*). Furthermore, if C is a generator for \(G{(L)^{{ \bot _1}}}\), then we have that SPC(G(C)) is the minimal subcategory of A containing \(G{(L)^{{ \bot _1}}}\)G(C) with respect to the property (*), and that SPC(G(C)) is C-resolving in A with a C-proper generator C.


Gorenstein categories right 1-orthogonal categories special precovers special precovered cate- gories projectively resolving injectively coresolving 


18G25 18E10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



This work was supported by National Natural Science Foundation of China (Grant No. 11571164), a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, the University Postgraduate Research and Innovation Project of Jiangsu Province 2016 (Grant No. KYZZ16 0034), Nanjing University Innovation and Creative Program for PhD Candidate (Grant No. 2016011). The authors thank the referees for the useful suggestions.


  1. 1.
    Anderson F W, Fuller K R. Rings and Categories of Modules, 2nd ed. Graduate Texts in Mathematics, vol. 13. Berlin: Springer-Verlag, 1992Google Scholar
  2. 2.
    Asadollahi J, Dehghanpour T, Hafezi P. On the existence of Gorenstein projective precovers. Rend Semin Mat Univ Padova, 2016, 136: 257–264MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Auslander M, Bridger M. Stable Module Theory. Memoirs of the American Mathematical Society, vol. 94. Providence: Amer Math Soc, 1969Google Scholar
  4. 4.
    Bravo D, Gillespie J, Hovey M. The stable module category of a general ring. ArXiv:1405.5768, 2014Google Scholar
  5. 5.
    Christensen L W, Foxby H-B, Holm H. Beyond Totally Re exive Modules and Back: A Survey on Gorenstein Dimen-sions. New York: Springer, 2011Google Scholar
  6. 6.
    Enochs E E. Injective and at covers, envelopes and resolvents. Israel J Math, 1981, 39: 189–209MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Enochs E E, Jenda O M G. Gorenstein injective and projective modules. Math Z, 1995, 220: 611–633MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Enochs E E, Jenda O M G. Relative Homological Algebra. de Gruyter Expositions in Mathematics, vol. 30. New York: Walter de Gruyter, 2000Google Scholar
  9. 9.
    Enochs E E, Jenda O M G, López-Ramos J A. Covers and envelopes by V-Gorenstein modules. Comm Algebra, 2005, 33: 4705–4717MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Holm H. Gorenstein homological dimensions. J Pure Appl Algebra, 2004, 189: 167–193MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Huang Z Y. Proper resolutions and Gorenstein categories. J Algebra, 2013, 393: 142–167MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Huang Z Y. Homological dimensions relative to preresolving subcategories. Kyoto J Math, 2014, 54: 727–757MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Iwanaga Y. On rings with finite self-injective dimension II. Tsukuba J Math, 1980, 4: 107–113MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Sather-Wagstaff S, Sharif T, White D. Stability of Gorenstein categories. J Lond Math Soc (2), 2008, 77: 481–502MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Takahashi R. Remarks on modules approximated by G-projective modules. J Algebra, 2006, 301: 748–780MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wang J, Liang L. A characterization of Gorenstein projective modules. Comm Algebra, 2016, 44: 1420–1432MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingChina

Personalised recommendations