On the Hereditariness of the Representations of Thread Quivers

TL;DR

This paper proves that the category of pointwise finite dimensional representations of any thread quiver is hereditary, using a new Ext criterion and structural reductions, with an alternative proof via Keller's theorem.

Contribution

It establishes hereditariness for these categories and introduces a novel Ext criterion, expanding understanding of their homological properties.

Findings

Proved hereditariness of the representation category for all thread quivers.

Developed a Yoneda Ext criterion for hereditariness.

Provided an alternative proof using Keller's theorem.

Abstract

We prove a conjecture of Paquette, Rock, and Yildirim by showing that, for every thread quiver, the abelian category of pointwise finite dimensional representations is hereditary. Since this category typically lacks enough projectives and injectives, standard homological methods do not apply directly. Our approach combines a Yoneda Ext criterion for hereditariness, established in this paper, with structural reductions to the subcategory of quasi noise free representations. We also indicate an alternative proof using a Keller's theorem on derived categories.