Czechoslovak Mathematical Journal, Vol. 69, No. 3, pp. 801-810, 2019


Generalized tilting modules over ring extension

Zhen Zhang

Received November 8, 2017.   Published online February 15, 2019.

Abstract:  Let $ \Gamma$ be a ring extension of $R$. We show the left $\Gamma$-module $U=\Gammaøtimes_RC$ with the endmorphism ring End${}_{\Gamma}U=\Delta$ is a generalized tilting module when ${}_RC$ is a generalized tilting module under some conditions.
Keywords:  ring extension; generalized tilting module; faithfully balanced bimodule
Classification MSC:  13D02; 13D07; 13D05
DOI:  10.21136/CMJ.2019.0512-17


References:
[1] I. Assem, N. Marmaridis: Tilting modules over split-by-nilpotent extensions. Commun. Algebra 26 (1998), 1547-1555. DOI 10.1080/00927879808826219 | MR 1622428 | Zbl 0915.16007
[2] L. W. Christensen: Semi-dualizing complexes and their Auslander categories.. Trans. Am. Math. Soc. 353 (2001), 1839-1883. DOI 10.1090/S0002-9947-01-02627-7 | MR 1813596 | Zbl 0969.13006
[3] E. E. Enochs, O. M. G. Jenda: Relative Homological Algebra. De Gruyter Expositions in Mathematics 30, Walter de Gruyter, Berlin (2000). DOI 10.1515/9783110803662 | MR 1753146 | Zbl 0952.13001
[4] H.-B. Foxby: Gorenstein modules and related modules. Math. Scand. 31 (1972), 267-284. DOI 10.7146/math.scand.a-11434 | MR 0327752 | Zbl 0272.13009
[5] K. R. Fuller: *-modules over ring extensions. Commun. Algebra 25 (1997), 2839-2860. DOI 10.1080/00927879708826026 | MR 1458733 | Zbl 0885.16019
[6] K. R. Fuller: Ring extensions and duality. Algebra and Its Applications (D. V. Huynh et al., eds.). Contemp. Math. 259, American Mathematical Society, Providence (2000), 213-222. DOI 10.1090/conm/259 | MR 1778503 | Zbl 0965.16004
[7] R. Göbel, J. Trlifaj: Approximations and Endomorphism Algebras of Modules. De Gruyter Expositions in Mathematics 41, Walter De Gruyter, Berlin (2006). DOI 10.1090/conm/259 | MR 2251271 | Zbl 1121.16002
[8] E. S. Golod: $G$-dimension and generalized perfect ideals. Tr. Mat. Inst. Steklova 165 (1984), 62-66. (In Russian.) MR 0752933 | Zbl 0577.13008
[9] H. Holm, D. White: Foxby equivalence over associative rings. J. Math. Kyoto Univ. 47 (2007), 781-808. DOI 10.1215/kjm/1250692289 | MR 2413065 | Zbl 1154.16007
[10] Y. Miyashita: Tilting modules of finite projective dimension. Math. Z. 193 (1986), 113-146. DOI 10.1007/BF01163359 | MR 0852914 | Zbl 0578.16015
[11] S. Sather-Wagstaff, T. Sharif, D. White: Comparison of relative cohomology theories with respect to semidualizing modules. Math. Z. 264 (2010), 571-600. DOI 10.1007/s00209-009-0480-4 | MR 2591820 | Zbl 1190.13007
[12] S. Sather-Wagstaff, T. Sharif, D. White: Tate cohomology with respect to semidualizing modules. J. Algebra 324 (2010), 2336-2368. DOI 10.1016/j.jalgebra.2010.07.007 | MR 2684143 | Zbl 1207.13009
[13] S. Sather-Wagstaff, T. Sharif, D. White: AB-contexts and stability for Gorenstein flat modules with respect to semidualizing modules. Algebr. Represent. Theory 14 (2011), 403-428. DOI 10.1007/s10468-009-9195-9 | MR 2785915 | Zbl 1317.13029
[14] A. Tonolo: $n$-cotilting and $n$-tilting modules over ring extensions. Forum Math. 17 (2005), 555-567. DOI 10.1515/form.2005.17.4.555 | MR 2154419 | Zbl 1088.16011
[15] W. V. Vasconcelos: Divisor Theory in Module Categories. North-Holland Mathematics Studies 14. Notas de Matematica 53, North-Holland Publishing, Amsterdam; American Elsevier Publishing Company, New York (1974). DOI 10.1016/s0304-0208(08)x7021-5 | MR 0498530 | Zbl 0296.13005
[16] T. Wakamatsu: On modules with trivial self-extensions. J. Algebra 114 (1988), 106-114. DOI 10.1016/0021-8693(88)90215-3 | MR 0931903 | Zbl 0646.16025
[17] T. Wakamatsu: Stable equivalence for self-injective algebras and a generalization of tilting modules. J. Algebra 134 (1990), 298-325. DOI 10.1016/0021-8693(90)90055-S | MR 1074331 | Zbl 0726.16009
[18] T. Wakamatsu: Tilting modules and Auslander's Gorenstein property. J. Algebra 275 (2004), 3-39. DOI 10.1016/j.jalgebra.2003.12.008 | MR 2047438 | Zbl 1076.16006
[19] J. Wei: $n$-star modules over ring extensions. J. Algebra 310 (2007), 903-916. DOI 10.1016/j.jalgebra.2006.10.026 | MR 2308185 | Zbl 1118.16011

Affiliations:   Zhen Zhang, Department of Mathematics, Qilu Normal University, Jinan 250200, P. R. China, e-mail: 157642043@qq.com


 
PDF available at: