Czechoslovak Mathematical Journal, Vol. 69, No. 3, pp. 781-800, 2019


Relative tilting modules with respect to a semidualizing module

Maryam Salimi

Received November 7, 2017.   Published online February 6, 2019.

Abstract:  Let $R$ be a commutative Noetherian ring, and let $C$ be a semidualizing $R$-module. The notion of $C$-tilting $R$-modules is introduced as the relative setting of the notion of tilting $R$-modules with respect to $C$. Some properties of tilting and $C$-tilting modules and the relations between them are mentioned. It is shown that every finitely generated $C$-tilting $R$-module is $C$-projective. Finally, we investigate some kernel subcategories related to $C$-tilting modules.
Keywords:  tilting module; semidualizing module; $C$-projective
Classification MSC:  13D05, 13D45
DOI:  10.21136/CMJ.2019.0510-17


References:
[1] L. L. Avramov, H.-B. Foxby: Ring homomorphisms and finite Gorenstein dimension. Proc. Lond. Math. Soc. III. 75 (1997), 241-270. DOI 10.1112/S0024611597000348 | MR 1455856 | Zbl 0901.13011
[2] S. Bazzoni, F. Mantese, A. Tonolo: Derived equivalence induced by infinitely generated $n$-tilting modules. Proc. Am. Math. Soc. 139 (2011), 4225-4234. DOI 10.1090/S0002-9939-2011-10900-6 | MR 2823068 | Zbl 1232.16004
[3] K. Bongratz: Tilted algebras. Representations of Algebras. Proc. 3rd Int. Conf., Puebla, 1980 Lect. Notes Math. 903, Springer, Berlin (1981), 26-38. DOI 10.1007/bfb0092982 | MR 0654701 | Zbl 0478.16025
[4] 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
[5] E. E. Enochs, O. M. G. Jenda: Relative Homological Algebra. Gruyter Expositions in Mathematics 30, Walter de Gruyter, Berlin (2011). DOI 10.1515/9783110215212 | MR 2857612 | Zbl 1238.13001
[6] H.-B. Foxby: Gorenstein modules and related modules. Math. Scand. 31 (1973), 267-284. DOI 10.7146/math.scand.a-11434 | MR 0327752 | Zbl 0272.13009
[7] E. S. Golod: $G$-dimension and generalized perfect ideals. Tr. Mat. Inst. Steklova 165 (1984), 62-66. (In Russian.) MR 0752933 | Zbl 0577.13008
[8] D. Happel, C. M. Ringel: Tilted algebras. Trans. Am. Math. Soc. 274 (1982), 399-443. DOI 10.2307/1999116 | MR 0675063 | Zbl 0503.16024
[9] H. Holm, P. Jørgensen: Semi-dualizing modules and related Gorenstein homological dimensions. J. Pure Appl. Algebra 205 (2006), 423-445. DOI 10.1016/j.jpaa.2005.07.010 | MR 2203625 | Zbl 1094.13021
[10] 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
[11] Y. Miyashita: Tilting modules of finite projective dimension. Math. Z. 193 (1986), 113-146. DOI 10.1007/BF01163359 | MR 0852914 | Zbl 0578.16015
[12] M. Salimi: On relative Gorenstein homological dimensions with respect to a dualizing module. Mat. Vesnik 69 (2017), 118-125. MR 3621408
[13] M. Salimi, S. Sather-Wagstaff, E. Tavasoli, S. Yassemi: Relative Tor functors with respect to a semidualizing module. Algebr. Represent. Theory 17 (2014), 103-120. DOI 10.1007/s10468-012-9389-4 | MR 3160715 | Zbl 1295.13023
[14] M. Salimi, E. Tavasoli, S. Yassemi: Top local cohomology modules and Gorenstein injectivity with respect to a semidualizing module. Arch. Math. 98 (2012), 299-305. DOI 10.1007/s00013-012-0371-5 | MR 2914346 | Zbl 1246.13021
[15] S. Sather-Wagstaff: Semidualizing Modules. Available at https://ssather.people.clemson.edu/DOCS/sdm.pdf.
[16] 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
[17] R. Takahashi, D. White: Homological aspects of semidualizing modules. Math. Scand. 106 (2010), 5-22. DOI 10.7146/math.scand.a-15121 | MR 2603458 | Zbl 1193.13012
[18] X. Tang: New characterizations of dualizing modules. Commun. Algebra 40 (2012), 845-861. DOI 10.1080/00927872.2010.540285 | MR 2899912 | Zbl 1246.13022
[19] W. V. Vasconcelos: Divisor Theory in Module Categories. North-Holland Mathematics Studies 14, Elsevier, Amsterdam (1974). DOI 10.1016/s0304-0208(08)x7021-5 | MR 0498530 | Zbl 0296.13005
[20] D. White: Gorenstein projective dimension with respect to a semidualizing module. J. Commut. Algebra 2 (2010), 111-137. DOI 10.1216/JCA-2010-2-1-111 | MR 2607104 | Zbl 1237.13029

Affiliations:   Maryam Salimi, Department of Mathematics, East Tehran Branch, Islamic Azad University, Tehran, Iran, e-mail: maryamsalimi@ipm.ir


 
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