Selfextensions of modules over group algebras

with an appendix by Bernhard Böhmler and René Marczinzik

verfasst von

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.

Organisationseinheit(en)

Institut für Algebra, Zahlentheorie und Diskrete Mathematik

Externe Organisation(en)

University of Oxford
Rheinische Friedrich-Wilhelms-Universität Bonn

Typ

Artikel

Journal

Journal of Algebra

Band

649

Seiten

319-346

Anzahl der Seiten

28

ISSN

0021-8693

Publikationsdatum

01.07.2024

Publikationsstatus

Veröffentlicht

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Algebra und Zahlentheorie

Elektronische Version(en)

https://doi.org/10.1016/j.jalgebra.2024.03.014 (Zugang: Offen)
https://doi.org/10.48550/arXiv.2310.12748 (Zugang: Offen)