{
  "id": "1508.04353",
  "version": "v1",
  "published": "2015-08-18T15:30:15.000Z",
  "updated": "2015-08-18T15:30:15.000Z",
  "title": "Irreducible morphisms and locally finite dimensional representations",
  "authors": [
    "Charles Paquette"
  ],
  "comment": "16 pages, 1 figure",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\mathcal{A}$ be a Hom-finite additive Krull-Schmidt $k$-category where $k$ is an algebraically closed field. Let ${\\rm mod} \\mathcal{A}$ denote the category of locally finite dimensional $\\mathcal{A}$-modules, that is, the category of covariant functors $\\mathcal{A} \\to {\\rm mod} k$. We prove that an irreducible monomorphism in ${\\rm mod} \\mathcal{A}$ has a finitely generated cokernel, and that an irreducible epimorphism in ${\\rm mod} \\mathcal{A}$ has a finitely co-generated kernel. Using this, we get that an almost split sequence in ${\\rm mod} \\mathcal{A}$ has to start with a finitely co-presented module and end with a finitely presented one. Finally, we apply our results in the study of ${\\rm rep}(Q)$, the category of locally finite dimensional representations of a strongly locally finite quiver. We describe all possible shapes of the Auslander-Reiten quiver of ${\\rm rep}(Q)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-18T15:30:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G20",
      "16G70",
      "16D90"
    ],
    "keywords": [
      "locally finite dimensional representations",
      "irreducible morphisms",
      "hom-finite additive krull-schmidt",
      "auslander-reiten quiver",
      "strongly locally finite quiver"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}