Kevin Schlegel, Ideal Torsion Pairs and the Krull-Gabriel Dimension of an Artin Algebra.

Abstract:

Let A be an Artin algebra and modA the category of finitely generated (left) A-modules. The Krull-Gabriel dimension, KG(A), of A is a homological invariant which measures the complexity of modA. I.e. if A is of finite representation type, then KG(A) = 0, if A is tame and hereditary, then KG(A) = 2 and if A is wild, then KG(A) equals infinity. We introduce the notion of ideal torsion pairs in modA, generalizing torsion pairs, and a new homological dimension, the torsion dimension, TD(A), of A. We relate the two homological dimensions and show methods for computing TD(A).