Remarks on Auslander algebras (Q1814225)

This paper is a continuation of \textit{K. Igusa, M.-I. Platzeck} and the authors [Commun. Algebra 15, 377-424 (1987; Zbl 0609.16014)]. It consists essentially of facts concerning the Auslander algebra of a standard Artin algebra, in case the former is representation finite. Sample results: Proposition. If the Auslander algebra of the standard algebra \(\Lambda\) is representation finite, then \(\Lambda\) is a zero- relations algebra. Proposition. If the Auslander algebra of the standard algebra \(\Lambda\) is representation finite, then for any indecomposable \(\Lambda\)-module \(M\), if \(\alpha_ L(M) = 1\), \(d({\mathbf P}_ 0,M) \leq 4\), and, if \(\alpha_ L(M) = 2\), \(d({\mathbf P}_ 0,M) \leq 3\). Here \(\alpha_ L(M)\) is the number of indecomposable summands in the middle of an Auslander-Reiten sequence ending up at \(M\), \({\mathbf P}_ 0\) is the set of projective indecomposables and \(d({\mathbf P}_ 0,M)\) is the distance from \({\mathbf P}_ 0\) to \(M\) in the Auslander-Reiten quiver.