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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 F q {\displaystyle \mathbf {F} _{q}} , then, writing σ : G → G , x ↦ x q {\displaystyle \sigma :G\to G,\,x\mapsto x^{q}} for the Frobenius, the morphism of varieties
is surjective. Note that the kernel of this map → G {\displaystyle G=G\to G} ] is precisely G {\displaystyle G}.
The theorem implies that H 1 = H e ´ t 1 {\displaystyle H^{1}=H_{\mathrm {{\acute {e}}t} }^{1}} vanishes, and, consequently, any G-bundle on Spec F q {\displaystyle \operatorname {Spec} \mathbf {F} _{q}} 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 In fact, this application was Lang's initial motivation. If G is affine, the Frobenius σ {\displaystyle \sigma } may be replaced by any surjective map with finitely many fixed points