{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $S$ be a [[scheme|5A35TYZD02]], and let $\\mathcal{C}$ be the category of schemes over $S$ with the fppf topology.\nAn *algebraic stack* over $S$ is a category $\\mathcal{X}$ with a functor $p \\colon \\mathcal{X} \\to \\mathcal{C}$ such that:\n\n1. $\\mathcal{X}$ is a stack in groupoids over $\\mathcal{C}$: every fibre category is a groupoid, and objects and morphisms glue along fppf coverings.\n2. The diagonal $\\mathcal{X} \\to \\mathcal{X} \\times_{\\mathcal{C}} \\mathcal{X}$ is representable by [[algebraic spaces|MTV9TFKF3X]].\n3. There are an $S$-scheme $U$ and a 1-morphism from $U$ to $\\mathcal{X}$ that is surjective and [[smooth|EXKGG9X6J0]].\n\nIn clause 3, a scheme $U$ is seen as the stack of $S$-schemes over $U$.\nClause 2 makes this 1-morphism representable by algebraic spaces, so surjective and smooth make sense.\n\n## Remarks\n\nLet $k$ be a field and $G$ a finite group, seen as a constant group scheme over $k$.\nThe classifying stack $BG$, whose objects over $T$ are $G$-torsors over $T$, is an algebraic stack.\nThe map $\\operatorname{Spec} k \\to BG$ given by the trivial torsor is surjective and étale, hence smooth.\nEvery algebraic space is an algebraic stack whose fibre categories have only identity morphisms.\nA functor that fails the sheaf condition, such as the constant presheaf with value $\\set{0, 1}$, fails clause 1.\nIf in clause 3 the map can be chosen étale, the stack is called a Deligne–Mumford stack.\nThis is the definition of the Stacks Project; some authors also require the diagonal to be quasi-compact and separated.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "VNX79CRH90",
    "kind": "definition",
    "msc": [
      "14A20",
      "18F20"
    ],
    "name": "Algebraic stack",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Algebraic_stack"
      }
    ],
    "tag": "VP1N9S4J4F",
    "uses": [
      "5A35TYZD02",
      "MTV9TFKF3X",
      "EXKGG9X6J0",
      "1DZMT4S9FB"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/VP1N9S4J4F/",
  "links": [
    {
      "field": "defines",
      "target": "VNX79CRH90"
    },
    {
      "field": "uses",
      "target": "5A35TYZD02"
    },
    {
      "field": "uses",
      "target": "MTV9TFKF3X"
    },
    {
      "field": "uses",
      "target": "EXKGG9X6J0"
    },
    {
      "field": "uses",
      "target": "1DZMT4S9FB"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/VP1N9S4J4F.md",
  "msc": [
    "14A20",
    "18F20"
  ],
  "name": "Algebraic stack",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VP1/VP1N9S4J4F.md",
  "status": null,
  "tag": "VP1N9S4J4F",
  "uri": "m2://VP1N9S4J4F",
  "url": "https://tamnd.github.io/math2/wiki/algebraic-stack-definition/",
  "writing": {
    "tag": "VP1N9S4J4F",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}