VII - The Auslander–Reiten-quiver

Summary

In this chapter we introduce a quiver called the Auslander–Reiten quiver, or for short the AR-quiver, of any artin algebra Λ. The definition is motivated by the interpretation of the ordinary quiver of an Auslander algebra in terms of the associated algebra of finite representation type given at the end of the previous chapter.

We start by giving the construction of the AR-quiver and a criterion which can be read off directly from the AR-quiver ensuring that the composition of some irreducible morphisms in mod Λ is not zero. The AR-quiver often decomposes into a union of infinite components and the possible structures of such components and other full subquivers of the AR-quiver are studied. Here combinatorial results play a crucial role and these combinatorial results will also be applied in the next chapter dealing with hereditary artin algebras.

The Auslander–Reiten-quiver

In this section we introduce the Auslander–Reiten-quiver of an artin algebra and give some of its basic properties. We illustrate with several examples, and give the connection between the Auslander–Reiten-quiver of an algebra of finite representation type and the ordinary quiver of its Auslander algebra.

In VI Section 1 we introduced for an artin algebra Λ an equivalence relation on ind Λ. The equivalence relation is generated by M being related to N if there is an irreducible morphism from M to N or from N to M. On the other hand we have seen in VI Section 5 that the quiver of the Auslander algebra Γ of an artin algebra Λ of finite representation type has vertices in one to one correspondence with ind Λ.