Algebr Represent Theor
Auslander-Reiten Sequences for Gorenstein Rings
of Dimension One
Received: 14 September 2017 / Accepted: 22 May 2018
© Springer Science+Business Media B.V., part of Springer Nature 2018
Abstract Let R be a complete local Gorenstein ring of dimension one, with maximal ideal
m. We show that if M is a Cohen-Macaulay R-module which begins an AR-sequence, then
this sequence is produced by a particular endomorphism of m corresponding to a minimal
prime ideal of R. We apply this result to determining the shape of some components of
stable Auslander-Reiten quivers, which in the considered examples are shown to be tubes.
Keywords Commutative algebra · Auslander-reiten theory · Gorenstein
Mathematics Subject Classification (2010) 13H10 · 16G70
The theory of Auslander-Reiten (AR) quivers is central in the study of artin algebras.
Regarding AR theory for maximal Cohen-Macaulay modules over a complete Cohen-
Macaulay local ring, the cases of finite AR quivers have been studied thoroughly (see ),
but in the more common case of infinite type, shapes of AR quivers seem to be largely
The paper  agrees with this assessment (cf. its introduction), and begins to bridge this
gap. It establishes a variety of lemmas in the context of symmetric orders over a DVR,
and applies these lemmas to proving the shape (namely, a tube) of some components of the
Presented by: Jon F. Carlson
This work will appear as part of the author’s dissertation at Syracuse University. The author gratefully
acknowledges support from U.S. National Science Foundation grant DMS 1502107.
215 Carnegie Building, Syracuse, NY 13244, USA