{
  "id": "2210.12419",
  "version": "v1",
  "published": "2022-10-22T11:26:17.000Z",
  "updated": "2022-10-22T11:26:17.000Z",
  "title": "The central sheaf of a Grothendieck category",
  "authors": [
    "Konstantin Ardakov",
    "Peter Schneider"
  ],
  "categories": [
    "math.RT",
    "math.CT"
  ],
  "abstract": "The center $Z(\\mathcal{A})$ of an abelian category $\\mathcal{A}$ is the endomorphism ring of the identity functor on that category. A localizing subcategory of a Grothendieck category $\\mathcal{C}$ is said to be stable if it is stable under essential extensions. The set $\\mathbf{L}^{st}(\\mathcal{C})$ of stable localizing subcategories of $\\mathcal{C}$ is partially ordered under reverse inclusion. We show $\\mathcal{L} \\mapsto Z(\\mathcal{C}/\\mathcal{L})$ defines a sheaf of commutative rings on $\\mathbf{L}^{st}(\\mathcal{C})$ with respect to finite coverings. When $\\mathcal{C}$ is assumed to be locally noetherian, we also show that the sheaf condition holds for arbitrary coverings.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-22T11:26:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "grothendieck category",
      "central sheaf",
      "sheaf condition holds",
      "localizing subcategory",
      "abelian category"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}