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 -