Selfextensions of modules over group algebras

with an appendix by Bernhard Böhmler and René Marczinzik

authored by

Bernhard Böhmler, Karin Erdmann, Viktória Klász, Rene Marczinzik

Abstract

Let \(KG\) be a group algebra with \(G\) a finite group and \(K\) a field and \(M\) an indecomposable \(KG\)-module. We pose the question, whether \(Ext_{KG}^1(M,M) \neq 0\) implies that \(Ext_{KG}^i(M,M) \neq 0\) for all \(i \geq 1\). We give a positive answer in several important special cases such as for periodic groups and give a positive answer also for all Nakayama algebras, which allows us to improve a classical result of Gustafson. We then specialise the question to the case where the module \(M\) is simple, where we obtain a positive answer also for all tame blocks of group algebras. For simple modules \(M\), the appendix provides a Magma program that gives strong evidence for a positive answer to this question for groups of small order.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

External Organisation(s)

University of Oxford
University of Bonn

Type

Article

Journal

Journal of Algebra

Volume

649

Pages

319-346

ISSN

0021-8693

Publication date

01.07.2024

Publication status

Published

Peer reviewed

Yes

Electronic version(s)

https://doi.org/10.1016/j.jalgebra.2024.03.014 (Access: Open)
https://doi.org/10.48550/arXiv.2310.12748 (Access: Open)