Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

FOXBY EQUIVALENCE RELATIVE TO C-WEAK INJECTIVE AND C-WEAK FLAT MODULES

  • Gao, Zenghui (College of Applied Mathematics Chengdu University of Information Technology) ;
  • Zhao, Tiwei (Department of Mathematics Nanjing University)
  • Received : 2016.08.10
  • Accepted : 2017.06.01
  • Published : 2017.09.01
Download PDF
⟨ Previous Next ⟩

Abstract

Let S and R be rings and $_SC_R$ a (faithfully) semidualizing bimodule. We introduce and study C-weak flat and C-weak injective modules as a generalization of C-flat and C-injective modules ([21]) respectively, and use them to provide additional information concerning the important Foxby equivalence between the subclasses of the Auslander class ${\mathcal{A}}_C$ (R) and that of the Bass class ${\mathcal{B}}_C$ (S). Then we study the stability of Auslander and Bass classes, which enables us to give some alternative characterizations of the modules in ${\mathcal{A}}_C$ (R) and ${\mathcal{B}}_C$ (S). Finally we consider an open question which is closely relative to the main results ([11]), and discuss the relationship between the Bass class ${\mathcal{B}}_C$(S) and the class of Gorenstein injective modules.

Keywords

References

  1. T. Araya, R. Takahashi, and Y. Yoshino, Homological invariants associated to semi-dualizing bimodules, J. Math. Kyoto Univ. 45 (2005), no. 2, 287-306. https://doi.org/10.1215/kjm/1250281991
  2. D. Bennis, J. R. Garcia Rozas, and L. Oyonarte, Relative Gorenstein dimensions, Mediterr. J. Math. 13 (2016), no. 1, 65-91. https://doi.org/10.1007/s00009-014-0489-8
  3. D. Bennis, J. R. Garcia Rozas, and L. Oyonarte, Relative projective and injective dimensions, Comm. Algebra 44 (2016), no. 8, 3383-3396. https://doi.org/10.1080/00927872.2015.1065852
  4. D. Bennis, J. R. Garcia Rozas, and L. Oyonarte, When do Foxby classes coincide with modules of finite Gorenstein dimension?, Kyoto J. Math. 56 (2016), no. 4, 785-802. https://doi.org/10.1215/21562261-3664914
  5. D. Bennis and N. Mahdou, Strongly Gorenstein projective, injective, and flat modules, J. Pure Appl. Algebra, 210 (2007), no. 2, 437-445. https://doi.org/10.1016/j.jpaa.2006.10.010
  6. D. Bravo, J. Gillespie, and M. Hovey, The stable modules category of a general ring, arXiv:1405.5768v1, 2014.
  7. K. S. Brown, Cohomology of Groups, Springer-Verlag, New York, 1982.
  8. L. W. Christensen, Semi-dualizing complexes and their Auslander categories, Trans. Amer. Math. Soc. 353 (2001), no. 5, 1839-1883. https://doi.org/10.1090/S0002-9947-01-02627-7
  9. E. E. Enochs and H. Holm, Cotorsion pairs asociated with Auslander categories, Israel J. Math. 174 (2009), 253-268. https://doi.org/10.1007/s11856-009-0113-y
  10. E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, de Gruyter Expositions in Math. 30, Walter de Gruyter GmbH & Co. KG, Berlin, 2000.
  11. E. E. Enochs, O. M. G. Jenda, and J. A. Lopez-Ramos, Dualizing modules and n-perfect rings, Proc. Edinb. Math. Soc. 48 (2005), no. 1, 75-90. https://doi.org/10.1017/S0013091503001056
  12. E. E. Enochs and J. A. Lopez-Ramos, Kaplansky classes, Rend. Sem. Mat. Univ. Padova 107 (2002), 67-79.
  13. H.-B. Foxby, Gorenstein modules and related modules, Math. Scand. 31 (1973), 267-284.
  14. Z. Gao and Z. Huang, Weak injective covers and dimension of modules, Acta Math. Hungar. 147 (2015), no. 1, 135-157. https://doi.org/10.1007/s10474-015-0540-7
  15. Z. Gao and F. Wang, Coherent rings and Gorenstein FP-injective modules, Comm. Algebra 40 (2012), no. 5, 1669-1679. https://doi.org/10.1080/00927872.2011.554473
  16. Z. Gao and F. Wang, Weak injective and weak flat modules, Comm. Algebra 43 (2015), no. 9, 3857-3868. https://doi.org/10.1080/00927872.2014.924128
  17. E. S. Golod, G-dimension and generalized perfect ideals, Trudy Mat. Inst. Steklov. 165 (1984), 62-66.
  18. R. Gobel and J. Trlifaj, Approximations and Endomorphism Algebras of Modules, de Gruyter Expositions in Math. 41, 2nd revised and extended edition, Walter de Gruyter GmbH & Co. KG, Berlin-Boston 2012.
  19. H. Holm and P. Jorgensen, Semi-dualizing modules and related Gorenstein homological dimensions, J. Pure Appl. Algebra 205 (2006), no. 2, 423-445. https://doi.org/10.1016/j.jpaa.2005.07.010
  20. H. Holm and P. Jorgensen, Covers, precovers and purity, Illinois J. Math. 52 (2008), no. 2, 691-703.
  21. H. Holm and D. White, Foxby equivalence over associative rings, J. Math. Kyoto Univ. 47 (2007), no. 4, 781-808. https://doi.org/10.1215/kjm/1250692289
  22. Z. Huang, Proper resolutions and Gorenstein categories, J. Algebra 393 (2013), 142-169. https://doi.org/10.1016/j.jalgebra.2013.07.008
  23. L. Hummel and T. Marley, The Auslander-Bridger formula and the Gorenstein property for coherent rings, J. Commut. Algebra 1 (2009), no. 1, 283-314. https://doi.org/10.1216/JCA-2009-1-2-283
  24. J. J. Rotman, An Introduction to Homological Algebra, Academic Press, New York, 1979.
  25. S. Sather-Wagstaff, Semidualizing Modules, https://www.ndsu.edu/pubweb/ssatherw/DOCS/survey.pdf.
  26. S. Sather-Wagstaff, T. Sharif, and D. White, Stability of Gorenstein categories, J. London Math. Soc. 77 (2008), no. 2, 481-502. https://doi.org/10.1112/jlms/jdm124
  27. R. Takahashi and D.White, Homological aspects of semidualizing modules, Math. Scand. 106 (2010), no. 1, 5-22. https://doi.org/10.7146/math.scand.a-15121
  28. W. V. Vasconcelos, Divisor Theory in Module Categories, North-Holland Mathematics Studies, Vol. 14, North-Holland Publ. Co., Amsterdam, 1974.