A proof of extension conjecture

TL;DR

This paper proves the extension conjecture for finite-dimensional elementary algebras over a field using triangulated categories and homological algebra, and introduces a bimodule approach to the strong no loop conjecture, offering new proofs.

Contribution

It establishes the extension conjecture for a broad class of algebras and presents a novel bimodule method for the strong no loop conjecture, enhancing existing proof techniques.

Findings

Extension conjecture holds for finite-dimensional elementary algebras.

New bimodule approach provides alternative proofs of Igusa-Liu-Paquette theorem.

Methodology combines triangulated categories and differential graded homological algebra.

Abstract

Extension conjecture states that if a simple module over an artin algebra has nonzero first self-extension group then it has nonzero i-th self-extension group for infinitely many positive integers i. It is shown by recollement of triangulated categories and differential graded homological algebra approach that extension conjecture is true for finite-dimensional elementary algebras over a field, particularly, for finite-dimensional algebras over an algebraically closed field. Moreover, bimodule approach is introduced to strong no loop conjecture, which provides two new proofs of Igusa-Liu-Paquette theorem.