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  "body": "\n## Definition\n\nLet $S$ be a [[scheme|5A35TYZD02]], and let $\\pi \\colon G \\to S$ be a scheme over $S$.\nProducts below are [[fiber products|1DZMT4S9FB]] over $S$, and all maps are [[morphisms|JXRKH6P1Z4]] over $S$.\nA *group scheme* over $S$ is such a $G$ with morphisms $m \\colon G \\times_S G \\to G$, $e \\colon S \\to G$ and $i \\colon G \\to G$ such that:\n\n1. $m \\circ (m \\times \\id_G) = m \\circ (\\id_G \\times m)$ as morphisms $G \\times_S G \\times_S G \\to G$.\n2. $m \\circ (e \\circ \\pi, \\id_G) = \\id_G = m \\circ (\\id_G, e \\circ \\pi)$.\n3. $m \\circ (i, \\id_G) = e \\circ \\pi = m \\circ (\\id_G, i)$.\n\n## Remarks\n\nEquivalently, for every $S$-scheme $T$, the set $G(T)$ of $S$-morphisms $T \\to G$ carries a [[group|G5S1M9]] structure, and the maps $G(T') \\to G(T)$ induced by $S$-morphisms $T \\to T'$ are group homomorphisms.\nThis is the [[functor of points|SWWW9X4A3N]] form of the definition.\nFor example, the multiplicative group $\\mathbb{G}_{m,S} = \\operatorname{Spec} \\Z[t, t^{-1}] \\times_{\\operatorname{Spec} \\Z} S$ is a group scheme over $S$.\nFor an affine $S$-scheme $T = \\operatorname{Spec} R$, its points are $\\mathbb{G}_m(T) = R^\\times$, the units of $R$ under multiplication.\nThe affine line $\\mathbb{A}^1_S$ with multiplication $(a, b) \\mapsto ab$ is not a group scheme.\nClause 3 fails, since the point $0 \\in R$ has no multiplicative inverse in a nonzero ring $R$.\n",
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