대한수학회지 (Journal of the Korean Mathematical Society)

  • 제61권3호
  • /
  • Pages.603-620
  • /
  • 2024
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

(𝒱, 𝒲, 𝑦, 𝒳)-GORENSTEIN COMPLEXES

  • Yanjie Li (College of Mathematics and Statistics Northwest Normal University) ;
  • Renyu Zhao (College of Mathematics and Statistics Northwest Normal University, Gansu Provincial Research Center for Basic Disciplines of Mathematics and Statistics)
  • 투고 : 2023.08.31
  • 심사 : 2023.12.29
  • 발행 : 2024.05.01
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

Let 𝒱, 𝒲, 𝑦, 𝒳 be four classes of left R-modules. The notion of (𝒱, 𝒲, 𝑦, 𝒳)-Gorenstein R-complexes is introduced, and it is shown that under certain mild technical assumptions on 𝒱, 𝒲, 𝑦, 𝒳, an R-complex 𝑴 is (𝒱, 𝒲, 𝑦, 𝒳)-Gorenstein if and only if the module in each degree of 𝑴 is (𝒱, 𝒲, 𝑦, 𝒳)-Gorenstein and the total Hom complexs HomR(𝒀, 𝑴), HomR(𝑴, 𝑿) are exact for any ${\mathbf{Y}}\,{\in}\,{\tilde{\mathcal{Y}}}$ and any ${\mathbf{X}}\,{\in}\,{\tilde{\mathcal{X}}}$. Many known results are recovered, and some new cases are also naturally generated.

키워드

과제정보

The authors are grateful to the referee for the careful checking of this article and some helpful comments.

참고문헌

  1. M. Auslander and M. Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94, Amer. Math. Soc., Providence, RI, 1969. 
  2. D. Bennis and K. Ouarghi, X?????-Gorenstein projective modules, Int. Math. Forum 5 (2010), no. 9-12, 487-491. 
  3. D. Bravo and J. Gillespie, Absolutely clean, level, and Gorenstein AC-injective complexes, Comm. Algebra 44 (2016), no. 5, 2213-2233. https://doi.org/10.1080/00927872.2015.1044100 
  4. D. Bravo, J. Gillespie, and M. Hovey, The stable module category of a general ring, arXiv:1405.5768, 2014. 
  5. 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 
  6. E. E. Enochs and J. R. Garc'ia Rozas, Gorenstein injective and projective complexes, Comm. Algebra 26 (1998), no. 5, 1657-1674. https://doi.org/10.1080/00927879808826229 
  7. 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 
  8. E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, De Gruyter Expositions in Mathematics, 30, Walter de Gruyter & Co., Berlin, 2000. 
  9. E. E. Enochs, O. M. G. Jenda, and B. Torrecillas, Gorenstein flat modules, Nanjing Daxue Xuebao Shuxue Bannian Kan 10 (1993), no. 1, 1-9. 
  10. Z. Gao, X. Ma, and T. Zhao, Gorenstein weak injective modules with respect to a semidualizing bimodule, J. Korean Math. Soc. 55 (2018), no. 6, 1389-1421. https://doi.org/10.4134/JKMS.j170718 
  11. 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 
  12. Z. Gao and T. Zhao, Foxby equivalence relative to C-weak injective and C-weak flat modules, J. Korean Math. Soc. 54 (2017), no. 5, 1457-1482. https://doi.org/10.4134/JKMS.j160528 
  13. J. R. Garcia Rozas, Covers and envelopes in the category of complexes of modules, Chapman & Hall/CRC Research Notes in Mathematics, 407, Chapman & Hall/CRC, Boca Raton, FL, 1999. 
  14. Y. Geng and N. Ding, W-Gorenstein modules, J. Algebra 325 (2011), 132-146. https://doi.org/10.1016/j.jalgebra.2010.09.040 
  15. J. Gillespie, The flat model structure on Ch(R), Trans. Amer. Math. Soc. 356 (2004), no. 8, 3369-3390. https://doi.org/10.1090/S0002-9947-04-03416-6 
  16. R. Gobel and J. Trlifaj, Approximations and endomorphism algebras of modules, De Gruyter Expositions in Mathematics, 41, Walter de Gruyter, Berlin, 2006. https://doi.org/10.1515/9783110199727 
  17. 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 
  18. 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 
  19. 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 
  20. J. Hu and A. Xu, On stability of F-Gorenstein flat categories, Algebra Colloq. 23 (2016), no. 2, 251-262. https://doi.org/10.1142/S1005386716000286 
  21. J. Hu and D. Zhang, Weak AB-context for FP-injective modules with respect to semidualizing modules, J. Algebra Appl. 12 (2013), no. 7, 1350039, 17 pp. https://doi.org/10.1142/S0219498813500394 
  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. F. Kong and P. Zhang, From CM-finite to CM-free, J. Pure Appl. Algebra 220 (2016), no. 2, 782-801. https://doi.org/10.1016/j.jpaa.2015.07.016 
  24. L. Liang, N. Ding, and G. Yang, Some remarks on projective generators and injective cogenerators, Acta Math. Sin. (Engl. Ser.) 30 (2014), no. 12, 2063-2078. https://doi.org/10.1007/s10114-014-3227-z 
  25. Z. Liu, Z. Huang, and A. Xu, Gorenstein projective dimension relative to a semidualizing bimodule, Comm. Algebra 41 (2013), no. 1, 1-18. https://doi.org/10.1080/00927872.2011.602782 
  26. B. H. Maddox, Absolutely pure modules, Proc. Amer. Math. Soc. 18 (1967), 155-158. https://doi.org/10.2307/2035245 
  27. L. Mao and N. Ding, Relative FP-projective modules, Comm. Algebra 33 (2005), no. 5, 1587-1602. https://doi.org/10.1081/AGB-200061047 
  28. 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 
  29. F. Meng and Q. X. Pan, X?????-Gorenstein projective and Y?????-Gorenstein injective modules, Hacet. J. Math. Stat. 40 (2011), no. 4, 537-554. 
  30. 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 
  31. B. Stenstrom, Coherent rings and F P-injective modules, J. London Math. Soc. (2) 2 (1970), 323-329. https://doi.org/10.1112/jlms/s2-2.2.323 
  32. D. White, Gorenstein projective dimension with respect to a semidualizing module, J. Commut. Algebra 2 (2010), no. 1, 111-137. https://doi.org/10.1216/JCA-2010-2-1-111 
  33. D. Xin, J. Chen, and X. Zhang, Completely W?????-resolved complexes, Comm. Algebra 41 (2013), no. 3, 1094-1106. https://doi.org/10.1080/00927872.2011.630707 
  34. G. Yang, Gorenstein projective, injective and flat complexes, Acta Math. Sinica (Chinese Ser.) 54 (2011), no. 3, 451-460. 
  35. X. Yang, Gorenstein categories 𝒢(𝒳, 𝑦, Z?????) and dimensions, Rocky Mountain J. Math. 45 (2015), no. 6, 2043-2064. https://doi.org/10.1216/RMJ-2015-45-6-2043 
  36. X. Yang and W. Chen, Relative homological dimensions and Tate cohomology of complexes with respect to cotorsion pairs, Comm. Algebra 45 (2017), no. 7, 2875-2888. https://doi.org/10.1080/00927872.2016.1233226 
  37. X. Yang and N. Ding, On a question of Gillespie, Forum Math. 27 (2015), no. 6, 3205- 3231. https://doi.org/10.1515/forum-2013-6014 
  38. C. Yang and L. Liang, Gorenstein injective and projective complexes with respect to a semidualizing module, Comm. Algebra 40 (2012), no. 9, 3352-3364. https://doi.org/10.1080/00927872.2011.568030 
  39. G. Yang and Z. Liu, Cotorsion pairs and model structures on Ch(R), Proc. Edinb. Math. Soc. (2) 54 (2011), no. 3, 783-797. https://doi.org/10.1017/S0013091510000489 
  40. X. Yang and Z. Liu, Gorenstein projective, injective, and flat complexes, Comm. Algebra 39 (2011), no. 5, 1705-1721. https://doi.org/10.1080/00927871003741497 
  41. G. Yang, Z. Liu, and L. Liang, Model structures on categories of complexes over Ding-Chen rings, Comm. Algebra 41 (2013), no. 1, 50-69. https://doi.org/10.1080/00927872.2011.622326 
  42. D. Zhang and B. Ouyang, Semidualizing modules and related modules, J. Algebra Appl. 10 (2011), no. 6, 1261-1282. https://doi.org/10.1142/S0219498811005695 
  43. C. Zhang, L. Wang, and Z. Liu, Ding projective modules with respect to a semidualizing module, Bull. Korean Math. Soc. 51 (2014), no. 2, 339-356. https://doi.org/10.4134/BKMS.2014.51.2.339 
  44. C. Zhang, L. Wang, and Z. Liu, Ding projective modules with respect to a semidualizing bimodule, Rocky Mountain J. Math. 45 (2015), no. 4, 1389-1411. https://doi.org/10.1216/RMJ-2015-45-4-1389 
  45. R. Zhao and N. Ding, (W?????, Y?????, X?????)-Gorenstein complexes, Comm. Algebra 45 (2017), no. 7, 3075-3090. https://doi.org/10.1080/00927872.2016.1235173 
  46. R. Zhao and Y. Li, A unified approach to various Gorenstein modules, Rocky Mountain J. Math. 52 (2022), no. 6, 2229-2246. https://doi.org/10.1216/rmj.2022.52.2229 
  47. R. Zhao and P. Ma, Gorenstein projective complexes with respect to cotorsion pairs, Czechoslovak Math. J. 69(144) (2019), no. 1, 117-129. https://doi.org/10.21136/CMJ.2018.0194-17 
  48. R. Zhao and W. Ren, 𝒱𝒲-Gorenstein complexes, Turkish J. Math. 41 (2017), no. 3, 537-547. https://doi.org/10.3906/mat-1603-88 
  49. G. Zhao and J. Sun, 𝒱𝒲-Gorenstein categories, Turkish J. Math. 40 (2016), no. 2, 365-375. https://doi.org/10.3906/mat-1502-37