{
  "id": "1710.06003",
  "version": "v1",
  "published": "2017-10-16T21:39:50.000Z",
  "updated": "2017-10-16T21:39:50.000Z",
  "title": "On the category of finitely presented mod $p$ representations of $GL_2(F)$, $F/\\mathbb{Q}_p$ finite",
  "authors": [
    "Jack Shotton"
  ],
  "comment": "7 pages. Comments and corrections welcomed",
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "Let $F$ be a finite extension of $\\mathbb{Q}_p$, and let $\\mathbb{F}$ be a finite field of characteristic $p$. A smooth representation of $GL_2(F)$ over $\\mathbb{F}$ is finitely presented if it can be written as the cokernel of a map between representations induced from compact-mod-centre open subgroups of $GL_2(F)$. We prove that the category of finitely presented smooth representations is an abelian subcategory of all smooth representations. This amounts to showing that the kernel of a map between finitely presented smooth representations is finitely presented. The proof uses amalgamated products of completed group rings.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-16T21:39:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E50",
      "11F70"
    ],
    "keywords": [
      "smooth representation",
      "compact-mod-centre open subgroups",
      "finite field",
      "finite extension",
      "abelian subcategory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}