Vanishing of self-extensions over symmetric algebras

TL;DR

This paper investigates the behavior of self-extensions of modules over symmetric artin algebras, linking vanishing patterns to AR components and module properties, with implications for longstanding conjectures.

Contribution

It establishes a connection between vanishing self-extensions and AR components of stable type ZA_infinity, and relates the highest non-vanishing extension degree to quasilength.

Findings

Non-projective modules with eventually vanishing self-extensions lie in ZA_infinity components.

The degree of the highest non-vanishing self-extension is determined by quasilength.

Results have implications for the Auslander-Reiten and Extension Conjectures.

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.