The Auslander-Reiten conjecture for Gorenstein rings
HTML articles powered by AMS MathViewer
- by Tokuji Araya PDF
- Proc. Amer. Math. Soc. 137 (2009), 1941-1944 Request permission
Abstract:
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 $R$.References
- 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. MR 1647079, DOI 10.1080/00927879808826375
- Maurice Auslander, Functors and morphisms determined by objects, Representation theory of algebras (Proc. Conf., Temple Univ., Philadelphia, Pa., 1976) Lecture Notes in Pure Appl. Math., Vol. 37, Dekker, New York, 1978, pp. 1–244. MR 0480688
- Maurice Auslander, Songqing Ding, and Øyvind Solberg, Liftings and weak liftings of modules, J. Algebra 156 (1993), no. 2, 273–317. MR 1216471, DOI 10.1006/jabr.1993.1076
- Maurice Auslander and Idun Reiten, On a generalized version of the Nakayama conjecture, Proc. Amer. Math. Soc. 52 (1975), 69–74. MR 389977, DOI 10.1090/S0002-9939-1975-0389977-6
- 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. MR 1912056, DOI 10.1112/S0024611502013527
- M. Hochster, Grassmannians and their Schubert subvarieties are arithmetically Cohen-Macaulay, J. Algebra 25 (1973), 40–57. MR 314833, DOI 10.1016/0021-8693(73)90074-4
- Craig Huneke and Graham J. Leuschke, On a conjecture of Auslander and Reiten, J. Algebra 275 (2004), no. 2, 781–790. MR 2052636, DOI 10.1016/j.jalgebra.2003.07.018
- Ryo Takahashi, Remarks on modules approximated by G-projective modules, J. Algebra 301 (2006), no. 2, 748–780. MR 2236766, DOI 10.1016/j.jalgebra.2005.09.033
- Yuji Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series, vol. 146, Cambridge University Press, Cambridge, 1990. MR 1079937, DOI 10.1017/CBO9780511600685
Additional Information
- Tokuji Araya
- Affiliation: Nara University of Education, Takabatake-cho, Nara City 630-8528, Japan
- MR Author ID: 639398
- Email: araya@math.okayama-u.ac.jp
- Received by editor(s): April 28, 2008
- Received by editor(s) in revised form: August 25, 2008
- Published electronically: December 17, 2008
- Communicated by: Bernd Ulrich
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 1941-1944
- MSC (2000): Primary 13H10, 13D02, 13D07
- DOI: https://doi.org/10.1090/S0002-9939-08-09757-8
- MathSciNet review: 2480274