{
  "id": "2310.12748",
  "version": "v1",
  "published": "2023-10-19T13:52:21.000Z",
  "updated": "2023-10-19T13:52:21.000Z",
  "title": "Selfextensions of modules over group algebras",
  "authors": [
    "Bernhard Böhmler",
    "Karin Erdmann",
    "Viktoria Klasz",
    "Rene Marczinzik"
  ],
  "categories": [
    "math.RT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-19T13:52:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G10",
      "16E10"
    ],
    "keywords": [
      "group algebra",
      "positive answer",
      "selfextensions",
      "important special cases",
      "small order"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}