{
  "id": "1405.1888",
  "version": "v2",
  "published": "2014-05-08T11:40:01.000Z",
  "updated": "2015-12-03T20:51:06.000Z",
  "title": "One positive and two negative results for derived categories of algebraic stacks",
  "authors": [
    "Jack Hall",
    "Amnon Neeman",
    "David Rydh"
  ],
  "comment": "main results strengthened in the noetherian situation; rewrite of Introduction; additional details on well-generation",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $X$ be a quasi-compact and quasi-separated scheme. There are two fundamental and pervasive facts about the unbounded derived category of $X$: (1) $\\mathsf{D}_{\\mathrm{qc}}(X)$ is compactly generated by perfect complexes and (2) if $X$ is noetherian or has affine diagonal, then the functor $\\Psi_X \\colon \\mathsf{D}(\\mathsf{QCoh}(X)) \\to \\mathsf{D}_{\\mathrm{qc}}(X)$ is an equivalence. Our main results are that for algebraic stacks in positive characteristic, the assertions (1) and (2) are typically false.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-08T11:40:01.000Z",
      "abstract": "Let $X$ be a quasi-compact algebraic stack with affine diagonal and let $\\Psi_X \\colon \\mathsf{D}(\\mathsf{QCoh}(X)) \\to \\mathsf{D}_{\\mathrm{qc}}(X)$ be the natural functor. We prove that $\\Psi_X$ is an equivalence of categories if $\\mathsf{D}_{\\mathrm{qc}}(X)$ is compactly generated. We also define a large class of algebraic stacks, the poorly stabilized stacks, and prove that if $X$ is poorly stabilized with affine diagonal, then $\\Psi_X$ is not an equivalence. For poorly stabilized stacks $X$, the derived category $\\mathsf{D}_{\\mathrm{qc}}(X)$ is not compactly generated. None of these subtleties arise for schemes or algebraic spaces.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-12-03T20:51:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14F05",
      "13D09",
      "14A20",
      "18G10"
    ],
    "keywords": [
      "derived category",
      "negative results",
      "poorly stabilized stacks",
      "affine diagonal",
      "quasi-compact algebraic stack"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.1888H"
    }
  }
}