Hereditary uniserial categories with Serre duality

TL;DR

This paper classifies hereditary uniserial categories with Serre duality, identifying two main types: quiver representations of A_n and tubes, including a new class called big tubes.

Contribution

It provides a complete classification of hereditary uniserial categories with Serre duality, introducing the concept of big tubes as a new class.

Findings

Hereditary uniserial categories with Serre duality are either A_n quiver representations or tubes.

Introduction of big tubes as a new class of hereditary categories with Serre duality.

Classification results extend understanding of Serre duality in abelian categories.

Abstract

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 A_n 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.