{
  "id": "1405.1887",
  "version": "v2",
  "published": "2014-05-08T11:35:27.000Z",
  "updated": "2015-12-03T20:37:58.000Z",
  "title": "Perfect complexes on algebraic stacks",
  "authors": [
    "Jack Hall",
    "David Rydh"
  ],
  "comment": "reordering of the Introduction and sections 3 and 4; additional material on perfect complexes; Appendix A removed to be incorporated into another paper",
  "categories": [
    "math.AG"
  ],
  "abstract": "We develop a theory of unbounded derived categories of quasi-coherent sheaves on algebraic stacks. In particular, we show that these categories are compactly generated by perfect complexes for stacks that either have finite stabilizers or are local quotient stacks. We also extend To\\\"en and Antieau--Gepner's results on derived Azumaya algebras and compact generation of sheaves on linear categories from derived schemes to derived Deligne--Mumford stacks. These are all consequences of our main theorem: compact generation of a presheaf of triangulated categories on an algebraic stack is local for the quasi-finite flat topology.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-08T11:35:27.000Z",
      "abstract": "Let $X$ be a quasi-compact algebraic stack with quasi-compact and separated diagonal. We show that the unbounded derived category $\\mathsf{D}_{\\mathrm{qc}}(X)$ is compactly generated if (i) $X$ has quasi-finite and separated diagonal or (ii) $X$ is a $\\mathbb{Q}$-stack of s-global type. These are both simple consequences of our main result: compact generation of $\\mathsf{D}_{\\mathrm{qc}}(X)$ is quasi-finite flat local on $X$.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-12-03T20:37:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14F05",
      "13D09",
      "14A20",
      "18G10"
    ],
    "keywords": [
      "perfect complexes",
      "quasi-compact algebraic stack",
      "separated diagonal",
      "quasi-finite flat local",
      "s-global type"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.1887H"
    }
  }
}