Indecomposables live in all smaller lengths

Abstract

We show that there are no gaps in the lengths of the indecomposable objects in an abelian k k -linear category over a field k k provided all simples are absolutely simple. To derive this natural result we prove that any distributive minimal representation-infinite k k -category is isomorphic to the linearization of the associated ray category which is shown to have an interval-finite universal cover with a free fundamental group so that the well-known theory of representation-finite algebras applies.