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- Lang's_theorem abstract "In algebraic geometry, Lang's theorem, introduced by Serge Lang, states: if G is a connected smooth algebraic group over a finite field , then, writing for the Frobenius, the morphism of varieties is surjective. Note that the kernel of this map (i.e., ) is precisely .The theorem implies that vanishes, and, consequently, any G-bundle on is isomorphic to the trivial one. Also, the theorem plays a basic role in the theory of finite groups of Lie type.It is not necessary that G is affine. Thus, the theorem also applies to abelian varieties (e.g., elliptic curves.) In fact, this application was Lang's initial motivation. If G is affine, the Frobenius may be replaced by any surjective map with finitely many fixed points (see below for the precise statement.)The proof (given below) actually goes through for any that induces a nilpotent operator on the Lie algebra of G.".
- Lang's_theorem wikiPageExternalLink books?id=54HO1wDNM_YC.
- Lang's_theorem wikiPageExternalLink 2372673.
- Lang's_theorem wikiPageID "42146937".
- Lang's_theorem wikiPageRevisionID "605154457".
- Lang's_theorem subject Category:Algebraic_geometry.
- Lang's_theorem subject Category:Algebraic_groups.
- Lang's_theorem comment "In algebraic geometry, Lang's theorem, introduced by Serge Lang, states: if G is a connected smooth algebraic group over a finite field , then, writing for the Frobenius, the morphism of varieties is surjective. Note that the kernel of this map (i.e., ) is precisely .The theorem implies that vanishes, and, consequently, any G-bundle on is isomorphic to the trivial one. Also, the theorem plays a basic role in the theory of finite groups of Lie type.It is not necessary that G is affine.".
- Lang's_theorem label "Lang's theorem".
- Lang's_theorem sameAs m.0_yf4yr.
- Lang's_theorem sameAs Q6487014.
- Lang's_theorem sameAs Q6487014.
- Lang's_theorem wasDerivedFrom Lang's_theorem?oldid=605154457.
- Lang's_theorem isPrimaryTopicOf Lang's_theorem.