Lang's theorem (original) (raw)
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. The proof (given below) actually goes through for any that induces a nilpotent operator on the Lie algebra of G.
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| dbo: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. (en) |
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| rdfs: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. The proof (given below) actually goes through for any that induces a nilpotent operator on the Lie algebra of G. (en) |
| rdfs:label | Lang's theorem (en) |
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