Cycle-finite algebras
Let \(A\) be a finite dimensional algebra over an algebraically closed field, and \(\text{mod }A\) denote the category of its finitely generated right modules. Then \(A\) is said to be cycle-finite if, for every sequence \(M_0 @>f_1>> M_1 @>f_2>> M_2\to\dots @>f_n>> M_n=M_0\) of non-zero non-isomorphisms between indecomposable objects in \(\text{mod }A\), none of the \(f_i\) lies in the infinite radical \(\text{rad}^\infty(\text{mod }A)\) of \(\text{mod }A\). The main result of the paper states that if \(A\) is cycle-finite, then the Auslander-Reiten quiver \(\Gamma(\text{mod }A)\) of \(A\) has a sincere stable tube if and only if \(A\) is tame concealed or tubular. As a consequence, any cycle-finite algebra is of polynomial growth. These results generalise those obtained by the author and the reviewer for multicoil algebras [Math. Scand. 71, No. 1, 31-61 (1992; Zbl 0796.16014)]. In addition, it is shown that the following conditions are equivalent for a cycle-finite algebra \(A\): (a) \(A\) is domestic, (b) \(A\) contains no tubular algebra as a full convex subcategory, (c) \(\text{rad}^\infty(\text{mod }A)\) is nilpotent, (d) all but at most finitely many components of \(\Gamma(\text{mod }A)\) are stable tubes of rank 1.
scientific article; zbMATH DE number 810251
scientific article