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  "body": "\n## Statement\n\nLet $k$ be a perfect field.\nLet $G$ be a smooth connected [[algebraic group|GH81SYAPPN]] over $k$.\nThen there is a unique smooth connected [[normal subgroup|0CB0DX5BVV]] $H$ of $G$ such that:\n\n1. $H$ is a [[linear algebraic group|GA81VVQHEP]];\n2. the [[quotient|0KYX2N69J0]] $G / H$ is an [[abelian variety|R7W03TXRS9]].\n\n## Remarks\n\nUniqueness is the easy part.\nIf $H$ and $H'$ both satisfy the conclusion, the image of $H$ in $G / H'$ is a connected linear algebraic group and also a closed subvariety of an abelian variety.\nSo it is both affine and complete, hence a point, and $H \\subseteq H'$.\nBy symmetry $H = H'$.\nExistence is the deep part of the theorem.\n\nExample: let $E$ be an elliptic curve over $k$ and $G = E \\times \\mathbb{G}_m$.\nThen $H = \\set{0} \\times \\mathbb{G}_m$ is linear, and $G / H \\cong E$ is an abelian variety.\n\nOver imperfect fields the statement can fail, so the hypothesis that $k$ is perfect cannot simply be dropped.\n\n## Explanations\n\n### L1 human\n\nEvery connected algebraic group is assembled from two kinds of groups that are opposite in nature.\nOne kind is a group of matrices, which is affine.\nThe other kind is an abelian variety, which is complete.\nThe theorem says the matrix part sits inside as a normal subgroup, with an abelian variety as quotient.\n\n### L2 human\n\nThe theorem reduces many questions about algebraic groups to the two separate theories of linear algebraic groups and abelian varieties.\nFor example, a connected algebraic group that is affine has trivial abelian quotient, and one that is complete has trivial linear part.\nA standard family of examples are the generalized Jacobians of singular curves.\nSuch a group is an extension of the Jacobian of the normalized curve by a linear group.\n",
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  "msc": [
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  "name": "Chevalley's structure theorem",
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