Ideal Torsion Pairs for 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.