Minimal infinite submodule-closed subcategories

TL;DR

This paper investigates subcategories of module categories over artin algebras that are closed under certain operations and contain infinitely many indecomposables, establishing the existence and properties of minimal such subcategories.

Contribution

It proves the existence of minimal infinite submodule-closed subcategories and explores their key properties, providing essential finiteness conditions for module categories.

Findings

Existence of minimal infinite submodule-closed subcategories

Characterization of properties of these minimal subcategories

Establishment of finiteness conditions for module categories

Abstract

Let $\Lambda$ be an artin algebra. We are going to consider full subcategories of $\mod\Lambda$ closed under finite direct sums and under submodules with infinitely many isomorphism classes of indecomposable modules. The main result asserts that such a subcategory contains a minimal one and we exhibit some striking properties of these minimal subcategories. These results have to be considered as essential finiteness conditions for such module categories.