Auslander-Reiten Sequences for Gorenstein Rings of Dimension One

Let R be a complete local Gorenstein ring of dimension one, with maximal ideal ?. 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 ? 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.