The radical and almost split sequences

Abstract

As in Chapter I, we let k be an arbitrary (commutative) field. Our algebras are finite dimensional k-algebras, associative and with an identity. The main working tool in this book is the notion of almost split sequences. It arose from an attempt to understand the morphisms lying in the radical of a module category. From this attempt, Auslander and Reiten extracted the notions of irreducible morphisms and almost split sequences, which allow all irreducible morphisms to be arranged in a neat way. We start our discussion in Section II.1 with a short description of the radical of a module category. We define and study irreducible morphisms and almost split sequences in Section II.2. We prove in Section II.3 the existence theorem for almost split sequences and we proceed to apply these sequences to the study of the radical in Section II.4.