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  "body": "\n## Definition\n\nLet $k$ be a field, and let $G$ be an [[algebraic group|GH81SYAPPN]] over $k$ with multiplication $m$ and unit $e$.\nLet $X$ be a [[variety|HV6YBS0YDH]] over $k$.\nAn *action* of $G$ on $X$ is a [[morphism|CDS1VWB7A3]] $\\sigma \\colon G \\times_k X \\to X$ such that:\n\n1. $\\sigma \\circ (e \\times \\id_X)$ is the identity of $X$, where $\\operatorname{Spec} k \\times_k X$ is identified with $X$;\n2. $\\sigma \\circ (m \\times \\id_X) = \\sigma \\circ (\\id_G \\times \\sigma)$ as morphisms $G \\times_k G \\times_k X \\to X$.\n\n## Remarks\n\nOn points, with $g \\cdot x = \\sigma(g, x)$, the clauses say $e \\cdot x = x$ and $(gh) \\cdot x = g \\cdot (h \\cdot x)$.\nSo for each $k$-algebra $R$ one gets a [[group action|0WCY42EV9V]] of $G(R)$ on $X(R)$.\n\nExample: $\\mathrm{GL}_n$ acts on $\\mathbb{A}^n$ by $\\sigma(g, v) = g v$.\nThe coordinates of $g v$ are polynomials in the entries of $g$ and $v$, the identity matrix fixes every $v$, and $(gh) v = g (h v)$.\n\nNon-example: the morphism $\\sigma \\colon \\mathbb{G}_m \\times \\mathbb{A}^1 \\to \\mathbb{A}^1$, $\\sigma(t, x) = x + t - 1$, satisfies clause 1.\nIt fails clause 2, since $\\sigma(st, x) = x + st - 1$ while $\\sigma(s, \\sigma(t, x)) = x + s + t - 2$.\n",
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