{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $G$ be a group.\nA *$G$-torsor* is a nonempty set $X$ with a [[group action|0WCY42EV9V]] of $G$ such that for all $x, y \\in X$ there is exactly one $g \\in G$ with $g \\cdot x = y$.\n\nLet $k$ be a field and let $G$ be an [[algebraic group|GH81SYAPPN]] over $k$.\nA *$G$-torsor* over $k$ is a nonempty variety $X$ over $k$ with an [[action|RW85C34GQW]] $\\sigma$ of $G$ such that the morphism\n$$G \\times_k X \\to X \\times_k X, \\quad (g, x) \\mapsto (\\sigma(g, x), x)$$\nis an isomorphism.\n\n## Remarks\n\nIn the set case, the condition says that the action is free and transitive.\nIn the variety case, $X$ becomes isomorphic to $G$, with $G$ acting by left translation, after a finite extension of $k$.\nThe torsor is *trivial* if $X \\cong G$ over $k$, and this holds if and only if $X$ has a $k$-point.\n\nExample: let $V$ be a vector space and let $A$ be an affine space with translation space $V$.\nFor any two points $a, b \\in A$ there is exactly one vector $v$ with $a + v = b$, so $A$ is a $V$-torsor.\n\nExample over a field: let $G = \\mu_2 = \\set{\\pm 1}$ over $\\R$ act on $X = \\set{x^2 = -1}$ by $x \\mapsto \\pm x$.\nOver $\\C$, $X$ has the two points $\\pm i$, and $G$ swaps them freely.\nSo $X$ is a $\\mu_2$-torsor, and it is not trivial, since $X(\\R)$ is empty.\n\nNon-example: $\\mathbb{G}_m$ acting on $\\mathbb{A}^1$ by multiplication is not a torsor.\nThe point $0$ is fixed by every $t$, so the action is neither free nor transitive.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "5CYEQY4SWR",
    "kind": "definition",
    "msc": [
      "14L15",
      "14L30",
      "20G10"
    ],
    "name": "Torsor",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Principal_homogeneous_space"
      }
    ],
    "tag": "CPA1C45GQR",
    "uses": [
      "0WCY42EV9V",
      "RW85C34GQW",
      "GH81SYAPPN"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/CPA1C45GQR/",
  "links": [
    {
      "field": "defines",
      "target": "5CYEQY4SWR"
    },
    {
      "field": "uses",
      "target": "0WCY42EV9V"
    },
    {
      "field": "uses",
      "target": "RW85C34GQW"
    },
    {
      "field": "uses",
      "target": "GH81SYAPPN"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/CPA1C45GQR.md",
  "msc": [
    "14L15",
    "14L30",
    "20G10"
  ],
  "name": "Torsor",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CPA/CPA1C45GQR.md",
  "status": null,
  "tag": "CPA1C45GQR",
  "uri": "m2://CPA1C45GQR",
  "url": "https://tamnd.github.io/math2/wiki/torsor-definition/",
  "writing": {
    "tag": "CPA1C45GQR",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}