Module categories with infinite radical cube zero
Let \(A\) be an Artin algebra and let \(\text{mod }A\) be the category of finitely generated \(A\)-modules. Denote by \(\text{rad}(\text{mod }A)\) the radical of \(\text{mod }A\) and by \(\text{rad}^\infty(\text{mod }A)\) the intersection of all powers \(\text{rad}^i(\text{mod }A)\). It is well known that \(\text{rad}^\infty(\text{mod }A)=0\) if and only if \(A\) is representation finite, and it was shown by the authors [Commun. Algebra 22, No. 11, 4511-4517 (1994; Zbl 0812.16019)] that \((\text{rad}^\infty(\text{mod }A))^2=0\) characterizes finite type. In the paper under review, representation finite Artin algebras with \((\text{rad}^\infty(\text{mod }A))^3=0\) are studied and a rather complete description of their module categories is given. (1) If \(\Gamma(A)\) denotes the Auslander-Reiten quiver of a representation infinite Artin algebra \(A\) with \((\text{rad}^\infty(\text{mod }A))^3=0\), then all regular components of \(\Gamma(A)\) are pairwise orthogonal, they are generalized stable standard tubes and almost all of them have rank 1. The nonregular components may be quite complicated. (2) An Artin algebra \(A\) is tame concealed, if and only if \((\text{rad}^\infty(\text{mod }A))^3=0\) and there is a faithful regular component. Consequently an Artin algebra with \((\text{rad}^\infty(\text{mod }A))^3=0\) has tame concealed factor-algebras \(B=A/I\). The embedding \(\text{mod }B\to\text{mod }A\) is studied in section 4. It is shown that almost all tubes in \(\Gamma(B)\), the Auslander-Reiten quiver of \(B\), are full components in \(\Gamma(A)\). (3) In the final part it is shown that an Artin algebra \(A\) with \((\text{rad}^\infty(\text{mod }A))^3=0\) has only finitely many ideals \(I_1,\dots,I_m\) such that \(A/I_j\) is tame concealed. Consequently \(A\) is a domestic algebra.
categories of finitely generated modules
scientific article; zbMATH DE number 923669