Czechoslovak Mathematical Journal, Vol. 72, No. 4, pp. 989-1002, 2022

On the invariance of certain types of generalized Cohen-Macaulay modules under Foxby equivalence

Kosar Abolfath Beigi, Kamran Divaani-Aazar, Massoud Tousi

Received June 22, 2021.   Published online March 21, 2022.

Abstract:  Let $R$ be a local ring and $C$ a semidualizing module of $R$. We investigate the behavior of certain classes of generalized Cohen-Macaulay $R$-modules under the Foxby equivalence between the Auslander and Bass classes with respect to $C$. In particular, we show that generalized Cohen-Macaulay $R$-modules are invariant under this equivalence and if $M$ is a finitely generated $R$-module in the Auslander class with respect to $C$ such that $C\otimes_RM$ is surjective Buchsbaum, then $M$ is also surjective Buchsbaum.
Keywords:  Auslander class; Bass class; Buchsbaum module; dualizing module; generalized Cohen-Macaulay module; local cohomology; semidualizing module; surjective Buchsbaum module
Classification MSC:  13C14, 13D05, 13D45
DOI:  10.21136/CMJ.2022.0227-21

[1] N. Bourbaki: Elements of Mathematics. Commutative Algebra. Chapters 1-7. Springer, Berlin (1998). MR 1727221 | Zbl 0902.13001
[2] W. Bruns, J. Herzog: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics 39. Cambridge University Press, Cambridge (1993). DOI 10.1017/cbo9780511608681.004 | MR 1251956 | Zbl 0788.13005
[3] 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
[4] L. W. Christensen, A. Frankild, H. Holm: On Gorenstein projective, injective and flat dimensions: A functorial description with applications. J. Algebra 302 (2006), 231-279. DOI 10.1016/j.jalgebra.2005.12.007 | MR 2236602 | Zbl 1104.13008
[5] H.-B. Foxby: Hyperhomological algebra & commutative rings, notes in preparation.
[6] J. Herzog: Komplex Auflösungen und Dualität in der lokalen Algebra: Habilitationsschrift. Universität Regensburg, Regensburg (1974). (In German.)
[7] T. Kawasaki: Surjective-Buchsbaum modules over Cohen-Macaulay local rings. Math. Z. 218 (1995), 191-205. DOI 10.1007/BF02571897 | MR 1318153 | Zbl 0814.13017
[8] C. Miyazaki: Graded Buchsbaum algebras and Segre products. Tokyo J. Math. 12 (1989), 1-20. DOI 10.3836/tjm/1270133544 | MR 1001728 | Zbl 0696.13016
[9] J. J. Rotman: An Introduction to Homological Algebra. Universitext. Springer, New York (2009). DOI 10.1007/b98977 | MR 2455920 | Zbl 1157.18001
[10] S. Sather-Wagstaff: Semidualizing modules. Available at (2000), 109 pages.
[11] P. Schenzel, N. V. Trung, N. T. Cuong: Verallgemeinerte Cohen-Macaulay-Moduln. Math. Nachr. 85 (1978), 57-73. (In German.) DOI 10.1002/mana.19780850106 | MR 0517641 | Zbl 0398.13014
[12] R. Y. Sharp: Finitely generated modules of finite injective dimension over certain Cohen-Macaulay rings. Proc. Lond. Math. Soc., III. Ser. 25 (1972), 303-328. DOI 10.1112/plms/s3-25.2.303 | MR 0306188 | Zbl 0244.13015
[13] J. Stückrad, W. Vogel: Buchsbaum Rings and Applications: An Interaction Between Algebra, Geometry and Topology. Springer, Berlin (1986). DOI 10.1007/978-3-662-02500-0 | MR 0881220 | Zbl 0606.13018
[14] 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
[15] N. V. Trung: Absolutely superficial sequences. Math. Proc. Camb. Philos. Soc. 93 (1983), 35-47. DOI 10.1017/S0305004100060308 | MR 0684272 | Zbl 0509.13024
[16] K. Yamagishi: Bass number characterization of surjective Buchsbaum modules. Math. Proc. Camb. Philos. Soc. 110 (1991), 261-279. DOI 10.1017/S0305004100070341 | MR 1113425 | Zbl 0760.13010

Affiliations:   Kosar Abolfath Beigi, Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, North Sheikh Bahaee St., 1993891176 Tehran, Iran, e-mail:; Kamran Divaani-Aazar, Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran, e-mail:; Massoud Tousi, Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, P.O. Box 19395-5746, Tehran, Iran, e-mail:

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