Abstract

Let Λ be an artin algebra and $\mathcal{X}$ be a quasi-resolving subcategory of $\text{mod}-\Lambda$ which is of finite type. Let ${\mathcal{S}}_{\mathcal{X}}(\Lambda )$ be the full subcategory of the morphism category $\mathcal{H}(\Lambda )$ consisting of all monomorphisms $f:A\to B$ in $\mathcal{X}$ such that $\text{Coker}f$ also lies in $\mathcal{X}$. In this paper, we state and prove Brauer-Thrall type theorems for ${\mathcal{S}}_{\mathcal{X}}(\Lambda )$. As applications, we provide necessary and sufficient conditions for the submodule category $\mathcal{S}(\Lambda )$ to be of finite type, whenever Λ is of finite representation type, as well as, for the lower 2 × 2 triangular matrix algebra $T_2(\Lambda )$ to be of finite CM-type, whenever Λ is of finite CM-type.

中文翻译：

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子模块类别的 Brauer-Thrall 类型定理

摘要

令Λ成为艺术代数并且 $\mathcal{X}$ 是一个准解析子类别 $\text{模组}——\Lambda$这是有限类型的。让${\mathcal{秒}}_{\mathcal{X}}(\Lambda )$ 是态射范畴的完整子范畴 $\mathcal{H}(\Lambda )$ 由所有单态组成 $F：\mathrm{一种}\to 乙$ 在 $\mathcal{X}$ 以至于 $\text{焦化器}F$ 还在于 $\mathcal{X}$. 在本文中，我们陈述并证明了 Brauer-Thrall 类型定理${\mathcal{秒}}_{\mathcal{X}}(\Lambda )$. 作为应用程序，我们为子模块类别提供充要条件$\mathcal{秒}(\Lambda )$ 是有限类型，只要 Λ 是有限表示类型，以及对于下 2 × 2 三角矩阵代数 ${吨}_2(\Lambda )$ 为有限 CM 类型，只要 Λ 为有限 CM 类型。