Krull-Gabriel dimension of domestic string algebras

TL;DR

This paper computes the Krull-Gabriel dimension and the Cantor-Bendixson rank of the Ziegler spectrum for domestic string algebras, confirming a conjecture and describing the topology of the spectrum.

Contribution

It provides the first complete calculation of these invariants for domestic string algebras, confirming Schröer's conjecture and detailing the Ziegler spectrum topology.

Findings

Krull-Gabriel dimension of domestic string algebras calculated

Cantor-Bendixson rank of Ziegler spectrum points determined

Topology of the Ziegler spectrum explicitly described

Abstract

We calculate, confirming a conjecture of Schr\"{o}er, the Krull-Gabriel dimension of the category of modules over any domestic string algebra, as well as the Cantor-Bendixson rank of each point of its Ziegler spectrum. We also determine the topology on this space.