대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제57권6호
  • /
  • Pages.1567-1579
  • /
  • 2020
  • /
  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

GORENSTEIN MODULES UNDER FROBENIUS EXTENSIONS

  • Kong, Fangdi (Department of Applied Mathematics Lanzhou University of Technology) ;
  • Wu, Dejun (Department of Applied Mathematics Lanzhou University of Technology)
  • 투고 : 2020.01.18
  • 심사 : 2020.07.09
  • 발행 : 2020.11.30
PDF 다운로드
⟨ 이전 논문 13", [BULL. KOREAN MATH. SOC. 54 (2017), NO. 3, 953-965] " href="/article/JAKO202034352378603.page?lang=ko"> 다음 논문 ⟩

초록

Let R ⊂ S be a Frobenius extension of rings and M a left S-module and let 𝓧 be a class of left R-modules and 𝒚 a class of left S-modules. Under some conditions it is proven that M is a 𝒚-Gorenstein left S-module if and only if M is an 𝓧-Gorenstein left R-module if and only if S ⊗R M and HomR(S, M) are 𝒚-Gorenstein left S-modules. This statement extends a known corresponding result. In addition, the situations of Ding modules, Gorenstein AC modules and projectively coresolved Gorenstein flat modules are considered under Frobenius extensions.

키워드

과제정보

We thank the anonymous referee for numerous pertinent comments that helped improve the exposition.

참고문헌

  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. Auslander and D. A. Buchsbaum, Homological dimension in Noetherian rings, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 36-38. https://doi.org/10.1073/pnas.42.1.36
  3. D. Bravo, J. Gillespie, and M. Hovey, The stable module category of a general ring, preprint arXiv:1405.5768, 2014.
  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. 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
  6. 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
  7. 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.
  8. 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
  9. Y. Geng and N. Ding, W-Gorenstein modules, J. Algebra 325 (2011), 132-146. https://doi.org/10.1016/j.jalgebra.2010.09.040
  10. J. Gillespie, Model structures on modules over Ding-Chen rings, Homology Homotopy Appl. 12 (2010), no. 1, 61-73. http://projecteuclid.org/euclid.hha/1296223822 https://doi.org/10.4310/HHA.2010.v12.n1.a6
  11. 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
  12. Z. Huang, Proper resolutions and Gorenstein categories, J. Algebra 393 (2013), 142-169. https://doi.org/10.1016/j.jalgebra.2013.07.008
  13. Z. Huang and J. Sun, Invariant properties of representations under excellent extensions, J. Algebra 358 (2012), 87-101. https://doi.org/10.1016/j.jalgebra.2012.03.004
  14. L. Kadison, New examples of Frobenius extensions, University Lecture Series, 14, American Mathematical Society, Providence, RI, 1999. https://doi.org/10.1090/ulect/014
  15. F. Kasch, Grundlagen einer Theorie der Frobeniuserweiterungen, Math. Ann. 127 (1954), 453-474. https://doi.org/10.1007/BF01361137
  16. 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
  17. K. Morita, Adjoint pairs of functors and Frobenius extensions, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 9 (1965), 40-71 (1965).
  18. T. Nakayama and T. Tsuzuku, On Frobenius extensions. I, Nagoya Math. J. 17 (1960), 89-110. http://projecteuclid.org/euclid.nmj/1118800455 https://doi.org/10.1017/S0027763000002075
  19. T. Nakayama and T. Tsuzuku, On Frobenius extensions. II, Nagoya Math. J. 19 (1961), 127-148. http://projecteuclid.org/euclid.nmj/1118800865 https://doi.org/10.1017/S0027763000002427
  20. W. Ren, Gorenstein projective and injective dimensions over Frobenius extensions, Comm. Algebra 46 (2018), no. 12, 5348-5354. https://doi.org/10.1080/00927872.2018.1464173
  21. W. Ren, Gorenstein projective modules and Frobenius extensions, Sci. China Math. 61 (2018), no. 7, 1175-1186. https://doi.org/10.1007/s11425-017-9138-y
  22. J. Saroch and J. Stovicek, Singular compactness and definability for Σ-cotorsion and Gorenstein modules, Selecta Math. (N.S.) 26 (2020), no. 2, Paper No. 23, 40 pp. https://doi.org/10.1007/s00029-020-0543-2
  23. 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
  24. G. Yang, Z. Liu, and L. Liang, Ding projective and Ding injective modules, Algebra Colloq. 20 (2013), no. 4, 601-612. https://doi.org/10.1142/S1005386713000576
  25. Z. Zhao, Gorenstein homological invariant properties under Frobenius extensions, Sci. China Math. 62 (2019), no. 12, 2487-2496. https://doi.org/10.1007/s11425-018-9432-2