On the vanishing of self extensions over Cohen-Macaulay local rings

Araya, Tokuji; Celikbas, Olgur; Sadeghi, Arash; Takahashi, Ryo

doi:10.1090/proc/13944

On the vanishing of self extensions over Cohen-Macaulay local rings

Authors: Tokuji Araya, Olgur Celikbas, Arash Sadeghi and Ryo Takahashi
Journal: Proc. Amer. Math. Soc. 146 (2018), 4563-4570
MSC (2010): Primary 13D07, 13H10
DOI: https://doi.org/10.1090/proc/13944
Published electronically: August 7, 2018
MathSciNet review: 3856128
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Abstract: The celebrated Auslander-Reiten Conjecture, on the vanishing of self extensions of a module, is one of the longstanding conjectures in ring theory. Although it is still open, there are several results in the literature that establish the conjecture over Gorenstein rings under certain conditions. The purpose of this article is to obtain extensions of such results over Cohen-Macaulay local rings that admit canonical modules. In particular, our main result recovers theorems of Araya, and Ono and Yoshino simultaneously.

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Additional Information

Tokuji Araya
Affiliation: Department of Applied Science, Faculty of Science, Okayama University of Science, Ridaicho, Kitaku, Okayama 700-0005, Japan
Email: araya@das.ous.ac.jp

Olgur Celikbas
Affiliation: Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506-6310
Email: olgur.celikbas@math.wvu.edu

Arash Sadeghi
Affiliation: School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran
Email: sadeghiarash61@gmail.com

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

DOI: https://doi.org/10.1090/proc/13944
Keywords: Auslander-Reiten Conjecture, vanishing of Ext and Tor, canonical modules
Received by editor(s): March 17, 2017
Received by editor(s) in revised form: June 8, 2017
Published electronically: August 7, 2018
Additional Notes: The first and fourth authors were partly supported by JSPS Grants-in-Aid for Scientific Research 26400056 and 16K05098, respectively. The third author’s research was supported by a grant from IPM
Communicated by: Jerzy Weyman
Article copyright: © Copyright 2018 American Mathematical Society