The first Brauer-Thrall conjecture for extriangulated length categories

TL;DR

This paper proves the first Brauer-Thrall conjecture for extriangulated length categories by extending Gabriel-Roiter measure and establishing a key equivalence between infinite indecomposables and arbitrarily large length objects.

Contribution

It introduces a Gabriel-Roiter measure for extriangulated categories and proves the conjecture under certain technical conditions, extending classical representation theory results.

Findings

Proves the first Brauer-Thrall conjecture for extriangulated length categories.

Extends Gabriel-Roiter measure to this setting.

Establishes equivalence between infinite indecomposables and arbitrarily large length objects.

Abstract

Let $(\mathcal{A},\Theta)$ be a length category. We introduce the notation of Gabriel-Roiter measure with respect to $\Theta$ and extend Gabriel's main property to this setting. Using this measure, when $(\mathcal{A},\Theta)$ satisfies some technical conditions, we prove that $\mathcal{A}$ has an infinite number of pairwise nonisomorphic indecomposable objects if and only if it has indecomposable objects of arbitrarily large length. That is, the first Brauer-Thrall conjecture holds.