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Keywords:
Auslander class; Bass class; Buchsbaum module; dualizing module; generalized Cohen-Macaulay module; local cohomology; semidualizing module; surjective Buchsbaum module
Auslander class; Bass class; Buchsbaum module; dualizing module; generalized Cohen-Macaulay module; local cohomology; semidualizing module; surjective Buchsbaum module
Summary:
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 \hbox {Buchsbaum}.\looseness +1
References:
[1] Bourbaki, N.: Elements of Mathematics. Commutative Algebra. Chapters 1-7. Springer, Berlin (1998). MR 1727221 | Zbl 0902.13001
[2] Bruns, W., Herzog, J.: 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] 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] Christensen, L. W., Frankild, A., Holm, H.: 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] Foxby, H.-B.: Hyperhomological algebra & commutative rings, notes in preparation.
[6] Herzog, J.: Komplex Auflösungen und Dualität in der lokalen Algebra: Habilitationsschrift. Universität Regensburg, Regensburg (1974), German.
[7] Kawasaki, T.: Surjective-Buchsbaum modules over Cohen-Macaulay local rings. Math. Z. 218 (1995), 191-205. DOI 10.1007/BF02571897 | MR 1318153 | Zbl 0814.13017
[8] Miyazaki, C.: Graded Buchsbaum algebras and Segre products. Tokyo J. Math. 12 (1989), 1-20. DOI 10.3836/tjm/1270133544 | MR 1001728 | Zbl 0696.13016
[9] Rotman, J. J.: An Introduction to Homological Algebra. Universitext. Springer, New York (2009). DOI 10.1007/b98977 | MR 2455920 | Zbl 1157.18001
[10] Sather-Wagstaff, S.: Semidualizing modules. Available at https://www.ndsu.edu/pubweb/ {ssatherw/DOCS/sdm.pdf} (2000), 109 pages.
[11] Schenzel, P., Trung, N. V., Cuong, N. T.: Verallgemeinerte Cohen-Macaulay-Moduln. Math. Nachr. 85 (1978), 57-73 German. DOI 10.1002/mana.19780850106 | MR 0517641 | Zbl 0398.13014
[12] Sharp, R. Y.: 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] Stückrad, J., Vogel, W.: 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] 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
[15] Trung, N. V.: Absolutely superficial sequences. Math. Proc. Camb. Philos. Soc. 93 (1983), 35-47. DOI 10.1017/S0305004100060308 | MR 0684272 | Zbl 0509.13024
[16] Yamagishi, K.: 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
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