Decomposition of torsion pairs on module categories

TL;DR

This paper generalizes torsion pairs in module categories, decomposes them via projective and injective modules, and classifies all torsion pairs on specific algebra categories, advancing understanding of their structure.

Contribution

It introduces a generalized framework for torsion pairs and provides a method to classify all torsion pairs on certain algebra categories.

Findings

Decomposition of torsion pairs using projective and injective modules

Classification of torsion pairs on algebra KAn and tub categories

Identification of all torsion pairs on finite dimensional hereditary algebras

Abstract

In this article, we generalize the concept of torsion pairs and study its structure. As a trial of obtaining all torsion pairs, we decompose torsion pairs by projective modules and injective modules. Then we calculate torsion pairs on the algebra KAn and tub categories. At last we try to find all torsion pairs on the module categories of finite dimensional hereditary algebras.