{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $k$ be a field.\nAn *algebraic group* over $k$ is a [[group scheme|TJ04VQP4NB]] $G$ over $k$ whose underlying scheme is [[of finite type|AHTE5DVX61]] over $k$.\nSo $G$ comes with morphisms $m \\colon G \\times_k G \\to G$, $i \\colon G \\to G$ and $e \\colon \\operatorname{Spec} k \\to G$.\nThey satisfy associativity, the unit law and the inverse law as identities of morphisms.\n\n## Remarks\n\nMany authors also require $G$ to be smooth, or to be a variety; such groups are called group varieties.\nOver a field of characteristic $0$, every algebraic group is smooth.\n\nExample: the general linear group $\\mathrm{GL}_n = \\operatorname{Spec} k[x_{ij}, 1/\\det]$, with matrix multiplication and the inverse matrix.\nThe entries of a product and of an inverse are polynomials in the $x_{ij}$ and $1/\\det$, so these are morphisms, and the group axioms hold.\n\nNon-example: the constant group scheme $\\Z_k = \\coprod_{n \\in \\Z} \\operatorname{Spec} k$ is a group scheme over $k$.\nIt has infinitely many connected components, so it is not of finite type over $k$.\n\nThe group of points $G(k)$ is an abstract group, and $T \\mapsto G(T)$ is a functor from $k$-schemes to groups.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "GH81SYAPPN",
    "kind": "definition",
    "msc": [
      "14L10",
      "14L15",
      "20G15"
    ],
    "name": "Algebraic group",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Algebraic_group"
      }
    ],
    "tag": "HVMYKZ2PB5",
    "uses": [
      "TJ04VQP4NB",
      "AHTE5DVX61"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/HVMYKZ2PB5/",
  "links": [
    {
      "field": "defines",
      "target": "GH81SYAPPN"
    },
    {
      "field": "uses",
      "target": "TJ04VQP4NB"
    },
    {
      "field": "uses",
      "target": "AHTE5DVX61"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/HVMYKZ2PB5.md",
  "msc": [
    "14L10",
    "14L15",
    "20G15"
  ],
  "name": "Algebraic group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HVM/HVMYKZ2PB5.md",
  "status": null,
  "tag": "HVMYKZ2PB5",
  "uri": "m2://HVMYKZ2PB5",
  "url": "https://tamnd.github.io/math2/wiki/algebraic-group-definition/",
  "writing": {
    "tag": "HVMYKZ2PB5",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}