Lang's theorem: Difference between revisions

Revision as of 15:08, 11 March 2014

Revision as of 17:02, 11 March 2014

Line 1:

{{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>H^1(G \mathbf{F}_q,to G), =x H_{\textmapsto x^{ét}}^-1(\operatorname{Spec} \mathbf{F}_q, Gsigma(x)</math> &nbsp;

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

vanishes., Consequentlyand, 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]].

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.

== Proof ==

Define

:<math>f_a: G \to G, \quad f_a(x) = x^{-1}a\sigma(x)</math>.

Then we have: <math>(d f_a)_e = d(h \circ (x \mapsto (x^{-1}, a, \sigma(x))e = dh_{(e, e, e)} \circ (-1, 0, d\sigma_e) = -1 + d \sigma_e</math> where <math>h(x, y, z) = xyz</math>. It follows <math>(d f_a)_e</math> is bijective. Since <math>f_a(bx) = f_{f_a(b)}(x)</math>, we also see that <math>(df_a)_b</math> is bijective for any ''b''. Let ''X'' be the closure of the image of <math>f_1</math>. Then smooth points of ''X'' form a open dense subset; thus, there is some ''b'' in ''G'' such that <math>f_1(b)</math> is a smooth point. Since the tangent space to ''X'' at <math>f_1(b)</math> and the tangent space to ''G'' at ''b'' have the same dimension, it follows that ''X'' and ''G'' have the same dimension, since ''G'' is connected. Thus, the image of <math>f_1</math> contains an open dense subset ''U'' of ''G''. Now, given an element ''a'' in ''G'', for the same reasoning, the image of <math>f_a</math> contains an open dense subset ''V'' of ''G''. The intersection <math>U \cap V</math> is then nonempty but then this implies ''a'' is in the image of <math>f_1</math>.

== See also ==