As seen in the preceding chapters, the Auslander–Reiten quiver of an algebra is a very useful combinatorial invariant allowing us to store algebraic information about the module category. We were, for instance, able to use it to compute homomorphisms and extensions between modules, as well as to construct an algebra obtained by tilting from one that was known before. However, its usefulness is not restricted to being a device for storing information. As we shall see in this chapter, its combinatorial properties can be used to characterise classes of algebras.

We start from the results of Chapter VII on the Auslander–Reiten quiver of a representation–finite hereditary algebra A; it follows from these results that the full subquiver of Γ(mod A) consisting of the projective points is connected, acyclic, and meets each τ-orbit of Γ(mod A) exactly once and every path in Γ(mod A) having its source and target in it must entirely lie in it. These three properties characterise what is called a section in a (generally infinite) component of the Auslander–Reiten quiver.

We first generalise this remark by showing that any representation–infinite hereditary algebra has sections in two infinite components, which we call postprojective and preinjective. We then define a new class of algebras, the so-called tilted algebras, which now play a prominent rôle in the representation theory of algebras and which are obtained from hereditary algebras by tilting.