Hereditary Uniserial Categories with Serre Duality

An abelian Krull-Schmidt category is said to be uniserial if the isomorphism classes of subobjects of a given indecomposable object form a linearly ordered poset. In this paper, we classify the hereditary uniserial categories with Serre duality. They fall into two types: the first type is given by the representations of the quiver An with linear orientation (and infinite variants thereof), the second type by tubes (and an infinite variant). These last categories give a new class of hereditary categories with Serre duality, called big tubes.