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The Auslander-Reiten conjecture for Gorenstein rings


Author: Tokuji Araya
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
Published electronically: December 17, 2008
MathSciNet review: 2480274
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Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. T. Araya and Y. Yoshino, Remarks on a depth formula, a grade inequality and a conjecture of Auslander, Comm. Algebra 26 (1998), no. 11, 3793-3806. MR 1647079 (99h:13010)
  • 2. M. Auslander, Functors and morphisms determined by objects, Representation theory of algebras (Proc. Conf., Temple Univ., Philadelphia, PA, 1976), pp. 1-244. Lecture Notes in Pure Appl. Math., Vol. 37, Marcel Dekker, New York, 1978. MR 0480688 (58:844)
  • 3. M. Auslander, S. Ding and O. Solberg, Liftings and weak liftings of modules, J. Algebra 156 (1993), 273-317. MR 1216471 (94d:16007)
  • 4. M. Auslander and I. Reiten, On a generalized version of the Nakayama conjecture, Proc. Amer. Math. Soc. 52 (1975), 69-74. MR 0389977 (52:10806)
  • 5. L. L. Avramov and A. 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 (2003g:16009)
  • 6. M. Hochster, Grassmannians and their Schubert subvarieties are arithmetically Cohen-Macaulay, J. Algebra 25 (1973), 40-57. MR 0314833 (47:3383)
  • 7. C. Huneke and G. J. Leuschke, On a conjecture of Auslander and Reiten, J. Algebra 275 (2004), no. 2, 781-790. MR 2052636 (2005a:13033)
  • 8. R. Takahashi, Remarks on modules approximated by G-projective modules, J. Algebra 301 (2006), no. 2, 748-780. MR 2236766 (2007a:13010)
  • 9. Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, London Math. Soc. Lecture Note Series, vol. 146, Cambridge University Press, Cambridge, 1990. MR 1079937 (92b:13016)

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Additional Information

Tokuji Araya
Affiliation: Nara University of Education, Takabatake-cho, Nara City 630-8528, Japan
Email: araya@math.okayama-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-08-09757-8
Keywords: Auslander-Reiten conjecture, Gorenstein ring, stable cohomology module
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
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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