Selfextensions of modules for Nakayama and Brauer tree algebras

TL;DR

This paper investigates the self-extension properties of modules over Nakayama and Brauer tree algebras, revealing conditions under which Ext groups are non-zero and deriving implications for module projective dimensions and algebra bounds.

Contribution

It provides new insights into the relationship between self-extensions and projective dimensions for modules over Nakayama and Brauer tree algebras, including a new proof of a classical Loewy length bound.

Findings

Non-zero Ext^1(M,M) implies infinite projective dimension for Nakayama algebra modules.

In Brauer tree algebras, non-zero Ext^1(M,M) implies non-zero Ext^i(M,M) for all i>0.

A new proof of bounds on Loewy length for Nakayama algebras with finite global dimension.

Abstract

For Nakayama algebras $A$, we prove that in case $Ext_A^1(M,M) \neq 0$ for an indecomposable $A$-module $M$, we have that the projective dimension of $M$ is infinite. As an application we give a new proof of a classical result from \cite{Gus} on bounds of the Loewy length for Nakayama algebras with finite global dimension. For Brauer tree algebras $A$ with an indecomposable module $M$, we prove that $Ext_A^1(M,M) \neq 0$ implies $Ext_A^i(M,M) \neq 0$ for all $i>0$.