Constructing almost split sequences

Abstract

The previous chapter was mainly of a theoretical nature: we defined irreducible morphisms and almost split sequences and started to explore their use for the understanding of the radical of a module category. However, we did not say much about the explicit construction of almost split sequences, even though we pointed out that the proof of Theorem II.3.10 suggests the idea of a construction. Carrying out this construction in practice is quite difficult, and our objective in the present chapter is to explain how it can be done, at least in the easiest cases. In the first section, we prove that the indecomposable end terms of an almost split sequence are related by functors, which are called the Auslander–Reiten translations. In Section III.2, we derive the so-called Auslander–Reiten formulae, which lead us to a second existence proof for almost split sequences. Next, in Section III.3, we show how to apply these results to construct examples of almost split sequences. In the final Section III.4, we relate the Auslander–Reiten translates of a given module over an algebra to that over a quotient algebra.