A proof of the strong no loop conjecture

TL;DR

This paper proves the strong no loop conjecture for finite dimensional algebras over algebraically closed fields, confirming that simple modules with finite projective dimension have no self-extensions.

Contribution

It provides a proof of the strong no loop conjecture specifically for finite dimensional algebras over algebraically closed fields, a longstanding open problem.

Findings

Confirmed the strong no loop conjecture for finite dimensional algebras over algebraically closed fields

Established that simple modules with finite projective dimension lack self-extensions

Contributed to the understanding of module theory in algebraic structures

Abstract

The strong no loop conjecture states that a simple module of finite projective dimension over an artin algebra has no non-zero self-extension. The main result of this paper establishes this well known conjecture for finite dimensional algebras over an algebraically closed field.