Kevin Schlegel, University of Stuttgart

Ideal Torsion Pairs and the Krull-Gabriel Dimension of an Artin algebra

Abstract: For the module category of an Artin algebra, we generalize the notion of torsion pairs to ideal torsion pairs. Instead of full subcategories, 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. As an application of this theory, we 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.