Lang's theorem

In algebraic geometry, Lang's theorem, introduced by Serge Lang, states: if G is a connected smooth algebraic group over a finite field $\mathbf {F} _{q}$, then

$H^{1}(\mathbf {F} _{q},G)=H_{\text{ét}}^{1}(\operatorname {Spec} \mathbf {F} _{q},G)$  

vanishes. Consequently, a G-bundle on $\operatorname {Spec} \mathbf {F} _{q}$ 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.

See also

• Lang–Steinberg theorem

References

• T.A. Springer, "Linear algebraic groups", 2nd ed. 1998.