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Keywords:
semidualizing module; $G_{C}$-flat module; $G _{C}$-injective module; cover; envelope
semidualizing module; $G_{C}$-flat module; $G _{C}$-injective module; cover; envelope
Summary:
Let $R$ be a commutative Noetherian ring and let $C$ be a semidualizing $R$-module. We prove a result about the covering properties of the class of relative Gorenstein injective modules with respect to $C$ which is a generalization of Theorem 1 by Enochs and Iacob (2015). Specifically, we prove that if for every $G_{C}$-injective module $G$, the character module $G^{+}$ is $G_{C}$-flat, then the class $\mathcal {GI}_{C}(R)\cap \mathcal {A}_{C}(R)$ is closed under direct sums and direct limits. Also, it is proved that under the above hypotheses the class $\mathcal {GI}_{C}(R)\cap \mathcal {A}_{C}(R)$ is covering.
References:
[1] Auslander, M., Bridger, M.: Stable Module Theory. Memoirs of the American Mathematical Society 94 American Mathematical Society, Providence (1969). DOI 10.1090/memo/0094 | MR 0269685 | Zbl 0204.36402
[2] Avramov, L. L., Foxby, H. B.: Ring homomorphisms and finite Gorenstein dimension. Proc. Lond. Math. Soc., III. Ser. 75 (1997), 241-270. DOI 10.1112/S0024611597000348 | MR 1455856 | Zbl 0901.13011
[3] Christensen, L. W.: 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
[4] Enochs, E. E., Holm, H.: Cotorsion pairs associated with Auslander categories. Isr. J. Math. 174 253-268 (2009). DOI 10.1007/s11856-009-0113-y | MR 2581218 | Zbl 1184.13029
[5] Enochs, E. E., Iacob, A.: Gorenstein injective covers and envelopes over Noetherian rings. Proc. Am. Math. Soc. 143 (2015), 5-12. DOI 10.1090/S0002-9939-2014-12232-5 | MR 3272726 | Zbl 1307.18013
[6] Enochs, E. E., Jenda, O. M. G.: Relative Homological Algebra. De Gruyter Expositions in Mathematics 30 Walter de Gruyter, Berlin (2000). DOI 10.1515/9783110215212 | MR 2857612 | Zbl 1238.13001
[7] Enochs, E. E., Jenda, O. M. G., López-Ramos, J. A.: The existence of Gorenstein flat covers. Math. Scand. 94 (2004), 46-62. DOI 10.7146/math.scand.a-14429 | MR 2032335 | Zbl 1061.16003
[8] Enochs, E. E., López-Ramos, J. A.: Kaplansky classes. Rend. Semin. Math. Univ. Padova 107 (2002), 67-79. MR 1926201 | Zbl 1099.13019
[9] Foxby, H. B.: Gorenstein modules and related modules. Math. Scand. 31 (1972), 267-284. DOI 10.7146/math.scand.a-11434 | MR 0327752 | Zbl 0272.13009
[10] Golod, E. S.: G-dimension and generalized perfect ideals. Tr. Mat. Inst. Steklova 165 (1984), Russian 62-66. MR 0752933 | Zbl 0577.13008
[11] Hashimoto, M.: Auslander-Buchweitz Approximations of Equivariant Modules. London Mathematical Society Lecture Note Series 282 Cambridge University Press, Cambridge (2000). DOI 10.1017/CBO9780511565762 | MR 1797672 | Zbl 0993.13007
[12] Holm, H.: Gorenstein homological dimensions. J. Pure Appl. Algebra 189 (2004), 167-193. DOI 10.1016/j.jpaa.2003.11.007 | MR 2038564 | Zbl 1050.16003
[13] Holm, H., Jørgensen, P.: Semi-dualizing modules and related Gorenstein homological dimension. J. Pure Appl. Algebra 205 (2006), 423-445. DOI 10.1016/j.jpaa.2005.07.010 | MR 2203625 | Zbl 1094.13021
[14] Holm, H., Jørgensen, P.: Cotorsion pairs induced by duality pairs. J. Commut. Algebra 1 (2009), 621-633. DOI 10.1216/JCA-2009-1-4-621 | MR 2575834 | Zbl 1184.13042
[15] Krause, H.: The stable derived category of a noetherian scheme. Compos. Math. 141 (2005), 1128-1162. DOI 10.1112/S0010437X05001375 | MR 2157133 | Zbl 1090.18006
[16] Nagata, M.: Local Rings. Interscience Tracts in Pure and Applied Mathematics 13 Interscience Publisher a division of John Wiley and Sons, New York (1962). MR 0155856 | Zbl 0123.03402
[17] Reiten, I.: The converse of a theorem of Sharp on Gorenstein modules. Proc. Am. Math. Soc. 32 (1972), 417-420. DOI 10.1090/S0002-9939-1972-0296067-7 | MR 0296067 | Zbl 0235.13016
[18] Salimi, M., Tavasoli, E., Yassemi, S.: Gorenstein homological dimension with respect to a semidualizing module and a generalization of a theorem of Bass. Commun. Algebra 42 (2014), 2213-2221. DOI 10.1080/00927872.2012.717654 | MR 3169700 | Zbl 1291.13026
[19] Sather-Wagstaff, S.: Semidualizing Modules. https://ssather.people.clemson.edu/DOCS/sdm.pdf Zbl 1282.13021
[20] Sather-Wagstaff, S., Sharif, T., White, D.: Stability of Gorenstein categories. J. Lond. Math. Soc., II. Ser. 77 (2008), 481-502. DOI 10.1112/jlms/jdm124 | MR 2400403 | Zbl 1140.18010
[21] Sather-Wagstaff, S., Sharif, T., White, D.: 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
[22] Takahashi, R., White, D.: Homological aspects of semidualizing modules. Math. Scand. 106 (2010), 5-22. DOI 10.7146/math.scand.a-15121 | MR 2603458 | Zbl 1193.13012
[23] Vasconcelos, W. V.: Divisor Theory in Module Categories. North-Holland Mathematics Studies 14. Notas de Matematica 5 North-Holland Publishing, Amsterdam (1974). MR 0498530 | Zbl 0296.13005
[24] White, D.: 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
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