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On the Auslander-Reiten conjecture for Cohen-Macaulay local rings


Authors: Shiro Goto and Ryo Takahashi
Journal: Proc. Amer. Math. Soc. 145 (2017), 3289-3296
MSC (2010): Primary 13D22, 13H10
DOI: https://doi.org/10.1090/proc/13487
Published electronically: February 20, 2017
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Abstract: This paper studies vanishing of Ext modules over Cohen-Macaulay local rings. The main result of this paper implies that the Auslander-Reiten conjecture holds for maximal Cohen-Macaulay modules of rank one over
Cohen-Macaulay normal local rings. It also recovers a theorem of Avramov-Buchweitz-Şega and Hanes-Huneke, which shows that the Tachikawa conjecture holds for Cohen-Macaulay generically Gorenstein local rings.


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Shiro Goto
Affiliation: Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan
Email: goto@math.meiji.ac.jp, shirogoto@gmail.com

Ryo Takahashi
Affiliation: Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, Aichi 464-8602, Japan
Email: takahashi@math.nagoya-u.ac.jp

DOI: https://doi.org/10.1090/proc/13487
Keywords: Auslander--Reiten conjecture, Tachikawa conjecture, Gorenstein ring, Cohen--Macaulay ring, semidualizing module
Received by editor(s): March 1, 2016
Received by editor(s) in revised form: September 9, 2016
Published electronically: February 20, 2017
Additional Notes: The first author was partly supported by JSPS Grant-in-Aid for Scientific Research (C) 25400051. The second author was partly supported by JSPS Grant-in-Aid for Scientific Research (C) 25400038.
Communicated by: Jerzy Weyman
Article copyright: © Copyright 2017 American Mathematical Society