Image-extension-closed subcategories of module categories of hereditary algebras

TL;DR

This paper explores IE-closed subcategories in hereditary algebra module categories, linking them to torsion pairs, twin rigid modules, and mutation processes to classify and compute these subcategories.

Contribution

It introduces a classification of IE-closed subcategories via twin rigid modules and develops mutation techniques for these modules in hereditary algebras.

Findings

IE-closed subcategories relate to torsion pairs

Classification of IE-closed subcategories by twin rigid modules

Mutation of twin rigid modules enables computation in finite cases

Abstract

We study IE-closed subcategories of a module category, subcategories which are closed under taking Images and Extensions. We investigate the relation between IE-closed subcategories and torsion pairs, and characterize $\tau$-tilting finite algebras using IE-closed subcategories. For the hereditary case, we show that IE-closed subcategories can be classified by twin rigid modules, pairs of rigid modules satisfying some homological conditions. Moreover, we introduce mutation of twin rigid modules analogously to tilting modules, which gives a way to calculate all twin rigid modules for the representation-finite case.