On the infinite radical of a module category (Q2766378)

Over the last thirty years or so, the Auslander-Reiten quiver has become one of the most important tools used in the representation theory of finite-dimensional algebras over an algebraically closed field \(k\). However, for a \(k\)-algebra \(A\), it gives a description of the category mod-\(A\) only modulo its infinite radical \(\text{rad}_A^\omega\), but provides no information about the morphisms in the latter. The paper under review is the first contribution towards a systematic study of the ideal \(\text{rad}_A^\omega\), and it contains a number of interesting results. The first of these establishes, for each ordinal \(\omega<\alpha<\omega^2\), the existence of a finite-dimensional \(k\)-algebra \(A\) with \(\text{rad}_A^\alpha\neq 0\) and \(\text{rad}_A^{\alpha+1}=0\). Furthermore, the author obtains information concerning the infinite radical of module categories over the so-called special biserial algebras.