Ideal Torsion Pairs for Artin Algebras

TL;DR

This paper introduces ideal torsion pairs in Artin algebra module categories, characterizes their functorial finiteness, and explores their applications in generalizing preprojective modules and defining a new homological dimension linked to Krull-Gabriel dimension.

Contribution

It generalizes torsion pairs to ideals of morphisms, characterizes functorially finite cases, and introduces the torsion dimension, connecting it with existing homological measures.

Findings

Functorially finite ideal torsion pairs characterized via functors and objects.

Introduction of the torsion dimension and its relation to Krull-Gabriel dimension.

Equivalence of torsion and Krull-Gabriel dimensions for hereditary Artin algebras.

Abstract

For the module category of an Artin algebra, we generalize the notion of torsion pairs to ideal torsion pairs. Instead of full subcategories of modules, ideals of morphisms of the ambient category are considered. We characterize the functorially finite ideal torsion pairs, which are those fulfilling some nice approximation conditions, first through corresponding functors and then through the notion of ideals determined by objects introduced in this work. As an application of this theory, we generalize preprojective modules, introduce a new homological dimension, the torsion dimension, and establish its connection with the Krull-Gabriel dimension. In particular, it is shown that both dimensions coincide for hereditary Artin algebras.