Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

GORENSTEIN QUASI-RESOLVING SUBCATEGORIES

  • Cao, Weiqing (School of Mathematics and Statistics Jiangsu Normal University) ;
  • Wei, Jiaqun (School of Mathematical Sciences Nanjing Normal University)
  • Received : 2021.09.16
  • Accepted : 2022.02.23
  • Published : 2022.07.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we introduce the notion of Gorenstein quasiresolving subcategories (denoted by 𝒢𝒬𝓡𝒳 (𝓐)) in term of quasi-resolving subcategory 𝒳. We define a resolution dimension relative to the Gorenstein quasi-resolving categories 𝒢𝒬𝓡𝒳 (𝓐). In addition, we study the stability of 𝒢𝒬𝓡𝒳 (𝓐) and apply the obtained properties to special subcategories and in particular to modules categories. Finally, we use the restricted flat dimension of right B-module M to characterize the finitistic dimension of the endomorphism algebra B of a 𝒢𝒬𝒳-projective A-module M.

Keywords

Acknowledgement

Weiqing Cao was supported by the Science Foundation of Jiangsu Normal University (No. 21XFRS024). Jiaqun Wei was supported by the National Natural Science Foundation of China (Grant No. 11771212) and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

References

  1. M. Auslander and M. Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, RI, 1969.
  2. M. Behboodi, A. Ghorbani, A. Moradzadeh-Dehkordi, and S. H. Shojaee, C-pure projective modules, Comm. Algebra 41 (2013), no. 12, 4559-4575. https://doi.org/10.1080/00927872.2012.705934
  3. L. W. Christensen, H.-B. Foxby, and A. Frankild, Restricted homological dimensions and Cohen-Macaulayness, J. Algebra 251 (2002), no. 1, 479-502. https://doi.org/10.1006/jabr.2001.9115
  4. N. Ding, Y. Li, and L. Mao, Strongly Gorenstein flat modules, J. Aust. Math. Soc. 86 (2009), no. 3, 323-338. https://doi.org/10.1017/S1446788708000761
  5. S. Eilenberg and J. C. Moore, Foundations of relative homological algebra, Mem. Amer. Math. Soc. 55 (1965), 39 pp.
  6. E. E. Enochs and O. M. G. Jenda, Gorenstein injective and projective modules, Math. Z. 220 (1995), no. 4, 611-633. https://doi.org/10.1007/BF02572634
  7. E. E. Enochs and O. M. G. Jenda, Relative homological algebra, De Gruyter Expositions in Mathematics, 30, Walter de Gruyter & Co., Berlin, 2000. https://doi.org/10.1515/9783110803662
  8. E. E. Enochs, O. M. G. Jenda, and J. A. Lopez-Ramos, Covers and envelopes by V - Gorenstein modules, Comm. Algebra 33 (2005), no. 12, 4705-4717. https://doi.org/10.1080/00927870500328766
  9. S. M. Fakhruddin, Pure-injective modules, Glasgow Math. J. 14 (1973), 120-122. https://doi.org/10.1017/S0017089500001853
  10. H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004), no. 1-3, 167-193. https://doi.org/10.1016/j.jpaa.2003.11.007
  11. L. Mao and N. Ding, Gorenstein FP-injective and Gorenstein flat modules, J. Algebra Appl. 7 (2008), no. 4, 491-506. https://doi.org/10.1142/S0219498808002953
  12. S. Sather-Wagstaff, T. Sharif, and D. White, Stability of Gorenstein categories, J. Lond. Math. Soc. (2) 77 (2008), no. 2, 481-502. https://doi.org/10.1112/jlms/jdm124
  13. J. Simmons, Cyclic-purity: a generalization of purity for modules, Houston J. Math. 13 (1987), no. 1, 135-150.
  14. J. Wei, Finitistic dimension and restricted flat dimension, J. Algebra 320 (2008), no. 1, 116-127. https://doi.org/10.1016/j.jalgebra.2008.03.017
  15. A. Zhang, Finitistic dimension and endomorphism algebras of Gorenstein projective modules, arXiv:1802.00669.
  16. X. Zhu, The homological theory of quasi-resolving subcategories, J. Algebra 414 (2014), 6-40. https://doi.org/10.1016/j.jalgebra.2014.05.018