Module Categories of Small Radical Nilpotency

Abstract

This paper aims to initiate a study of the representation theory of representation-finite artin algebras in terms of the nilpotency of the radical of their module category. Firstly, we shall calculate this nilpotency explicitly for hereditary algebras of type \(\mathbb {A}_n\) and for Nakayama algebras. Surprisingly, this nilpotency for a given algebra coincides with its Loewy length if and only if the algebra is a hereditary Nakayama algebra. Secondly, we shall find all artin algebras for which this nilpotency is equal to any given positive integer up to four and describe completely their module category.