The first Brauer-Thrall conjecture for extriangulated length categories
Abstract
Let (A,) be a length category. We introduce the notation of Gabriel-Roiter measure with respect to and extend Gabriel's main property to this setting. Using this measure, when (A,) satisfies some technical conditions, we prove that 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.
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