Brauer-Thrall type theorems for submodule categories

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.

Keywords:

• Brauer-Thrall conjectures
• functor categories
• submodule categories

2020 Mathematics Subject Classification:

• 18E10
• 16G60
• 16G10
• 16G50

Additional information

Funding

The research of the second author was in part supported by a grant from IPM.