{
  "id": "1710.07239",
  "version": "v1",
  "published": "2017-10-19T16:43:39.000Z",
  "updated": "2017-10-19T16:43:39.000Z",
  "title": "Generators versus projective generators in abelian categories",
  "authors": [
    "Charles Paquette"
  ],
  "comment": "10 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\mathcal{A}$ be an essentially small abelian category. We prove that if $\\mathcal{A}$ admits a generator $M$ with ${\\rm End}_{\\mathcal{A}}(M)$ right artinian, then $\\mathcal{A}$ admits a projective generator. If $\\mathcal{A}$ is further assumed to be Grothendieck, then this implies that $\\mathcal{A}$ is equivalent to a module category. When $\\mathcal{A}$ is Hom-finite over a field $k$, the existence of a generator is the same as the existence of a projective generator, and in case there is such a generator, $\\mathcal{A}$ has to be equivalent to the category of finite dimensional right modules over a finite dimensional $k$-algebra. We also show that when $\\mathcal{A}$ is a length category, then there is a one-to-one correspondence between exact abelian extension closed subcategories of $\\mathcal{A}$ and collections of Hom-orthogonal Schur objects in $\\mathcal{A}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-19T16:43:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G20",
      "18E15",
      "18E10"
    ],
    "keywords": [
      "projective generator",
      "finite dimensional right modules",
      "exact abelian extension closed subcategories",
      "essentially small abelian category",
      "hom-orthogonal schur objects"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}