A note on algebras of finite uniserial type
Let \(K\) be a commutative Artin ring and \(R\) a \(K\)-algebra. \(R\) is an Artin (\(K\)-)algebra, if \(R\) is finitely generated as an \(R\)-module. In the paper, the following characterization of Artin algebras \(R\) of finite uniserial type is given in case \(K\) is local and its residue field is algebraically closed. Theorem: The following conditions on \(R\) are equivalent: (a) Every uniserial \(R\)-module is determined up to isomorphism by its composition series. (b) There are only finitely many isomorphism classes of uniserial \(R\)-modules. (c) For any uniserial \(R\)-module \(M\) and any simple \(R\)-module \(S\) one of the following holds: (i) Any extension of \(S\) by \(M\) has the same Loewy length as \(M\) has. (ii) \(\text{Ext}(S,M)\) is a cyclic \(\text{End }M\)-module. The theorem, to some extent, answers a question, listed as open problem in [\textit{M. Auslander}, \textit{I. Reiten} and \textit{S. O. Smalø}, Representation theory of Artin algebras (Cambridge, 1995; Zbl 0834.16001)]. It also confirms the conjecture 5.11b in [\textit{B. Zimmerman-Huisgen}, Trans. Am. Math. Soc. 348, No. 12, 4775-4812 (1996; Zbl 0862.16008)]. If \(K\) is an algebraically closed field, and \(R\) is given by a quiver and relations, conditon (c) provides a finite algorithm to decide whether \(R\) is of finite uniserial type (b) or not.
scientific article; zbMATH DE number 1013935