Vanishing of self-extensions over symmetric algebras

Abstract

We study self-extensions of modules over symmetric artin algebras. We show that non-projective modules with eventually vanishing self-extensions must lie in AR components of stable type $\mathbb{Z}A_{\infty}$. Moreover, the degree of the highest non-vanishing self-extension of these modules is determined by their quasilength. This has implications for the Auslander-Reiten Conjecture and the Extension Conjecture.