Some questions from homological algebra for finitely generated modules over Noetherian rings

Homological methods are a driving force in the study of Noetherian rings. In this talk we examine two homological questions in the representation theory of artin algebras that have also found applications in commutative algebra. Along the way we recall some key concepts from homological algebra.

The first question concerns modules having only very few non-trivial self-extensions. The endomorphism rings of such modules often have revealing properties. Several questions concerning the existence of such modules still remain open. We focus on one such question, known as the Auslander-Reiten Conjecture.

The second topic involves the notion of complexity which describes the rate of growth of projective resolutions of modules. It provides a measure of how close an arbitrary module is to being projective. We use Auslander-Reiten quivers to obtain results about complexity in the setting of artin algebras.