Exceptional components of wild hereditary algebras

Abstract

This paper studies regular components of wild hereditary algebras. It is clear that the existence of extremely many maps in the infinite radical of the module category causes algebras to be wild. Whereas the dimension of the Horn-spaces of maps from indecomposable preprojective modules or to indecomposable preinjective modules can be calculated by linear methods via the Euler-bilinearform ( , ), these methods give only poor information if we pass to regular modules. So it is natural to study as a first step maps between regular modules which are in the same regular component V, and we see in (4.7) that this is a way to get information about the maps between arbitrary regular modules. There is another, less obvious reason for the study of single components: If A and B are connected wild hereditary algebras, then by [9], via tilting modules, there can be constructed bijections between the regular components of the Auslander-Reiten quivers T(A) and T(B). In order to find finally properties of these bijections, we have to study properties of components. Proposition 6.3 points in that direction. If G? is a regular component in T(A), with A wild hereditary, we denote by the quasi-rank rk(%‘) of V the smallest integer N such that rad(X, r”x) # 0 for all n 2 N and all XE %. We call a component V exceptional, if there is an indecomposable (quasi-simple) module XE %? such that Hom(X, YX) # 0 but Hom(X, z r + ‘1) = 0 for some r > 0. In Section 4 we show that there are only finitely many exceptional components in T(A) and that the existence of those components has strict consequences for the algebra A, that is, for its ordinary quiver 9(A). In Section 5 we present the numerical invariants of the most important exceptional components. In the last section we give applications to tilting theory. The applications are chosen under the aspect that the notion of the quasi-rank of a regular component might be a pendant to the period of a tube in the tame case. 184 0021-8693/92 $5.00