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, 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. Note that the [[kernel (algebra)|kernel]] of this map (i.e., <math>G = G(\overline{\mathbf{F}_q}) \to G(\overline{\mathbf{F}_q})</math>) is precisely <math>G(\mathbf{F}_q)</math>.

The theorem implies that <math>H^1(\mathbf{F}_q, G) = H_{\mathrm{\acute{e}t}}^1(\operatorname{Spec}\mathbf{F}_q, G)</math> &nbsp;