Details
| Original language | English |
|---|---|
| Pages (from-to) | 319-346 |
| Number of pages | 28 |
| Journal | Journal of Algebra |
| Volume | 649 |
| Early online date | 26 Mar 2024 |
| Publication status | Published - 1 Jul 2024 |
Abstract
Keywords
- math.RT, Primary 16G10, 16E10, Global dimension, Nakayama algebra, Group algebras, Extension groups
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
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In: Journal of Algebra, Vol. 649, 01.07.2024, p. 319-346.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
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 - 2024/7/1
Y1 - 2024/7/1
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
KW - Global dimension
KW - Nakayama algebra
KW - Group algebras
KW - Extension groups
UR - http://www.scopus.com/inward/record.url?scp=85189696509&partnerID=8YFLogxK
U2 - 10.1016/j.jalgebra.2024.03.014
DO - 10.1016/j.jalgebra.2024.03.014
M3 - Article
VL - 649
SP - 319
EP - 346
JO - Journal of Algebra
JF - Journal of Algebra
SN - 0021-8693
ER -