## Details

Original language | English |
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Publication status | E-pub ahead of print - 2023 |

## Abstract

## Keywords

- math.RT, Primary 16G10, 16E10

## Cite this

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**Selfextensions of modules over group algebras: WITH AN APPENDIX BY BERNHARD BÖHMLER AND RENÉ MARCZINZIK.**/ Böhmler, Bernhard (Contributor); Erdmann, Karin; Klász, Viktória et al.

2023.

Research output: Working paper/Preprint › Preprint

*Selfextensions of modules over group algebras: WITH AN APPENDIX BY BERNHARD BÖHMLER AND RENÉ MARCZINZIK*. Advance online publication.

}

TY - UNPB

T1 - Selfextensions of modules over group algebras

T2 - WITH AN APPENDIX BY BERNHARD BÖHMLER AND RENÉ MARCZINZIK

AU - Erdmann, Karin

AU - Klász, Viktória

AU - Marczinzik, Rene

A2 - Böhmler, Bernhard

PY - 2023

Y1 - 2023

N2 - 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.

AB - 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.

KW - math.RT

KW - Primary 16G10, 16E10

M3 - Preprint

BT - Selfextensions of modules over group algebras

ER -