The Auslander-Reiten conjecture for Gorenstein rings. (English) Zbl 1163.13015
Summary: The Nakayama conjecture is one of the most important conjectures in ring theory. The Auslander-Reiten conjecture is closely related to it. The purpose of this paper is to show that if the Auslander-Reiten conjecture holds in codimension one for a commutative Gorenstein ring \( R\), then it holds for \( R\).
MSC:
| 13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |
| 13D02 | Syzygies, resolutions, complexes and commutative rings |
| 13D07 | Homological functors on modules of commutative rings (Tor, Ext, etc.) |
References:
| [1] | Tokuji Araya and Yuji Yoshino, Remarks on a depth formula, a grade inequality and a conjecture of Auslander, Comm. Algebra 26 (1998), no. 11, 3793 – 3806. · Zbl 0906.13002 · doi:10.1080/00927879808826375 |
| [2] | Maurice Auslander, Functors and morphisms determined by objects, Representation theory of algebras (Proc. Conf., Temple Univ., Philadelphia, Pa., 1976) Dekker, New York, 1978, pp. 1 – 244. Lecture Notes in Pure Appl. Math., Vol. 37. |
| [3] | Maurice Auslander, Songqing Ding, and Øyvind Solberg, Liftings and weak liftings of modules, J. Algebra 156 (1993), no. 2, 273 – 317. · Zbl 0778.13007 · doi:10.1006/jabr.1993.1076 |
| [4] | Maurice Auslander and Idun Reiten, On a generalized version of the Nakayama conjecture, Proc. Amer. Math. Soc. 52 (1975), 69 – 74. · Zbl 0337.16004 |
| [5] | Luchezar L. Avramov and Alex Martsinkovsky, Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. London Math. Soc. (3) 85 (2002), no. 2, 393 – 440. · Zbl 1047.16002 · doi:10.1112/S0024611502013527 |
| [6] | M. Hochster, Grassmannians and their Schubert subvarieties are arithmetically Cohen-Macaulay, J. Algebra 25 (1973), 40 – 57. · Zbl 0256.14024 · doi:10.1016/0021-8693(73)90074-4 |
| [7] | Craig Huneke and Graham J. Leuschke, On a conjecture of Auslander and Reiten, J. Algebra 275 (2004), no. 2, 781 – 790. · Zbl 1096.13011 · doi:10.1016/j.jalgebra.2003.07.018 |
| [8] | Ryo Takahashi, Remarks on modules approximated by G-projective modules, J. Algebra 301 (2006), no. 2, 748 – 780. · Zbl 1109.13012 · doi:10.1016/j.jalgebra.2005.09.033 |
| [9] | Yuji Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series, vol. 146, Cambridge University Press, Cambridge, 1990. · Zbl 0745.13003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
