A characterization of finite Auslander—Reiten quivers

Abstract

(a) We give necessary and sufficient conditions for a finite k-modulated translation quiver to be an Auslander-Reiten quiver in terms of certain homology groups associated to the quiver (Section 3). (b) We show that whether a finite k-modulated translation quiver is an Auslander-Reiten quiver depends only on the underlying valued translation quiver and the characteristic of k and give necessary and sufficient conditions only in terms of the valued quiver and char k. We also show that any valued quiver with valuations (1, n) or (n, l), n Q 3, admits a modulation over any prime field (Sections 5,6, 7). (c) To a valued translation quiver r we associate an unvalued translation quiver r (i.e., all valuations are 1) and show that a finite k-modulated translation quiver is an Auslander-Reiten quiver if and only if the associated unvalued translation quiver with the trivial K-modulation (K = prime field of k) is an Auslander-Reiten quiver (Section 8).