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so there is no unique "maximal affine subgroup", while the product of two copies of the multiplicative group C* is isomorphic (analytically but not algebraically) to a non-split extension of any given elliptic curve by
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Over non-perfect fields there is still a smallest normal connected linear subgroup such that the quotient is an abelian variety, but the linear subgroup need not be smooth.
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The reduced connected component of the relative Picard scheme of a proper scheme over a perfect field is an algebraic group, which is in general neither affine nor proper.
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Chevalley's original proof, and the other early proofs by
Barsotti and Rosenlicht, used the idea of mapping the algebraic group to its
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For analytic groups some of the obvious analogs of
Chevalley's theorem fail. For example, the product of the additive group
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There are several natural constructions that give connected algebraic groups that are neither affine nor complete.
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and any elliptic curve has a dense collection of closed (analytic but not algebraic) subgroups isomorphic to
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is an extension of an abelian variety by an affine algebraic group. In general this extension does not split.
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Atti della
Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali
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over a discrete valuation ring is an algebraic group, which is in general neither affine nor proper.
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A consequence of
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has a unique normal smooth connected affine algebraic subgroup such that the quotient is an
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later gave an exposition of
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Barsotti, Iacopo (1955b), "Un teorema di struttura per le varietĂ gruppali",
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and are hard to follow for anyone unfamiliar with Weil's foundations, but
43:(though he had previously announced the result in 1953), Barsotti (
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Rosenlicht, Maxwell (1956), "Some basic theorems on algebraic groups",
233:(1960), "Une démonstration d'un théorème sur les groupes algébriques",
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Barsotti, Iacopo (1955a), "Structure theorems for group-varieties",
263:"A modern proof of Chevalley's theorem on algebraic groups"
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Chevalley's structure theorem is used in the proof of the
129:The connected component of the closed fiber of a
62:. The original proofs were based on Weil's book
270:Journal of the Ramanujan Mathematical Society
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235:Journal de Mathématiques Pures et Appliquées
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64:Foundations of algebraic geometry
299:American Journal of Mathematics
160:Néron–Ogg–Shafarevich criterion
27:states that a smooth connected
357:Theorems in algebraic geometry
16:Theorem in algebraic geometry.
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25:Chevalley's structure theorem
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119:with affine kernel. So
261:Conrad, Brian (2002),
98:generalized Jacobian
39:. It was proved by
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187:10.1007/bf02413515
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306:: 401–443,
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131:Neron model
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320:0002-9327
282:0970-1249
247:0021-7824
218:: 43–50,
196:0003-4622
80:Examples
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