Lang's theorem: Difference between revisions

Revision as of 17:33, 11 March 2014

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{{mergefrom|Lang–Steinberg theorem|date=March 2014}}

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, writing <math>\sigma: G \to G, \, x \mapsto x^q</math> for the Frobenius, the [[morphism of varieties]]

:<math>G \to G, \, x \mapsto x^{-1} \sigma(x)</math>&nbsp;

is surjective. ThisThe theorem implies that <math>H^1(\mathbf{F}_q, G) = H_{\text{ét}}^1(\operatorname{Spec}\mathbf{F}_q, G)</math> &nbsp;

vanishes, and, consequently, any [[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]].