Lang's theorem: Difference between revisions

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In algebraic geometry, '''Lang's theorem''', introduced by [[Serge Lang]], states: if ''G'' is a connected smooth algebraic group over a finite field <math>\mathbf{F}_q</math>, then

:<math>H^1(\mathbf{F}_q, G) = H_{\text{ét}}^1(\operatorname{Spec}\mathbf{F}_q, G)</math> &nbsp;

vanishes. Consequently, aany [[torsor (algebraic geometry)|''G''-bundle]] on <math>\operatorname{Spec} \mathbf{F}_q</math> 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 variety|abelian varieties]]. In fact, this application was Lang's initial motivation.