Torsion pairs and rigid objects in tubes

TL;DR

This paper classifies torsion pairs in tube categories and establishes a geometric interpretation linking them to maximal rigid objects, extending existing models to include Pruefer and adic modules, with applications to type A Dynkin quivers.

Contribution

It provides a complete classification of torsion pairs in tube categories and extends geometric models to include larger categories with new objects.

Findings

Torsion pairs correspond to maximal rigid objects in extended categories.

Geometric models are extended to include Pruefer and adic modules.

The classification applies to linear orientations of Dynkin quivers of type A.

Abstract

We classify the torsion pairs in a tube category and show that they are in bijection with maximal rigid objects in the extension of the tube category containing the Pruefer and adic modules. We show that the annulus geometric model for the tube category can be extended to the larger category and interpret torsion pairs, maximal rigid objects and the bijection between them geometrically. We also give a similar geometric description in the case of the linear orientation of a Dynkin quiver of type A.