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]]

{{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. TheNote theoremthat impliesthe that[[kernel (algebra)|kernel]] of this map (i.e., <math>H^1G = G(\overline{\mathbf{F}_q, G}) =\to H_G(\overline{\textmathbf{étF}_q}^1)</math>) is precisely <math>G(\operatorname{Spec}\mathbf{F}_q, G)</math> &nbsp;.

vanishes,<ref>This is "unwinding definition". Here, <math>H^1(\mathbf{F}_q, G) = H^1(\operatorname{Gal}(\overline{\mathbf{F}_q}/\mathbf{F}_q), G(\mathbf{F}_q))</math> is [[Galois cohomology]]; cf. Milne, Class field theory.</ref> 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]].

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;

It is not necessary that ''G'' is affine. Thus, the theorem also applies to [[abelian variety|abelian varieties]] (e.g., [[elliptic curve]]s.) In fact, this application was Lang's initial motivation.

vanishes,<ref>This is "unwinding definition". Here, <math>H^1(\mathbf{F}_q, G) = H^1(\operatorname{Gal}(\overline{\mathbf{F}_q}/\mathbf{F}_q), G(\overline{\mathbf{F}_q}))</math> is [[Galois cohomology]]; cf. Milne, Class field theory.</ref> 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]].

It is not necessary that ''G'' is affine. Thus, the theorem also applies to [[abelian variety|abelian varieties]] (e.g., [[elliptic curve]]s.) In fact, this application was Lang's initial motivation. If ''G'' is affine, the Frobenius <math>\sigma</math> may be replaced by any surjective map with finitely many fixed points (see below for the precise statement.)

The proof (given below) actually goes through for any <math>\sigma</math> that induces a [[nilpotent operator]] on the Lie algebra of ''G''.<ref>{{harvnb|Springer|1998|loc=Exercise 4.4.18.}}</ref>

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

== Proof ==

{{harvs|txt|last=Steinberg|year=1968|authorlink=Robert Steinberg}} gave a useful improvement to the theorem.

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, a, e)} \circ (-1, 0, d\sigma_e) = -1 + d \sigma_e</math>&nbsp;

where <math>h(x, y, z) = xyz</math>. It follows <math>(d f_a)_e</math> is bijective since the differential of the Frobenius <math>\sigma</math> vanishes. 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>. The [[smooth point]]s of ''X'' form an open dense subset; thus, there is some ''b'' in ''G'' such that <math>f_1(b)</math> is a smooth point of ''X''. 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 smooth. Since ''G'' is connected, the image of <math>f_1</math> then contains an open dense subset ''U'' of ''G''. Now, given an arbitrary element ''a'' in ''G'', by 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>.

Suppose that ''F'' is an endomorphism of an algebraic group ''G''. The '''Lang map''' is the map from ''G'' to ''G'' taking ''g'' to ''g''⁻¹''F''(''g'').

== See also ==

*[[Lang–Steinberg theorem]]

The '''Lang–Steinberg theorem''' states<ref>{{harvnb|Steinberg|1968|loc=Theorem 10.1}}</ref> that if ''F'' is surjective and has a finite number of fixed points, and ''G'' is a connected affine algebraic group over an [[algebraically closed field]], then the Lang map is surjective.

== Proof of Lang's theorem ==

Define:

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

Then, by identifying the [[tangent space]] at ''a'' with the tangent space at the [[identity element]], we have:

:<math>(d f_a)_e = d(h \circ (x \mapsto (x^{-1}, a, \sigma(x))))_e = dh_{(e, a, e)} \circ (-1, 0, d\sigma_e) = -1 + d \sigma_e</math>&nbsp;

where <math>h(x, y, z) = xyz</math>. It follows <math>(d f_a)_e</math> is bijective since the differential of the Frobenius <math>\sigma</math> vanishes. 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''.<ref>This implies that <math>f_a</math> is [[étale morphism|étale]].</ref> Let ''X'' be the closure of the image of <math>f_1</math>. The [[smooth point]]s of ''X'' form an open dense subset; thus, there is some ''b'' in ''G'' such that <math>f_1(b)</math> is a smooth point of ''X''. 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 smooth. Since ''G'' is connected, the image of <math>f_1</math> then contains an open dense subset ''U'' of ''G''. Now, given an arbitrary element ''a'' in ''G'', by 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>.

== Notes ==

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== References ==

* {{cite book | last=Springer | first=T. A. Springer,| "title=Linear algebraic groups", 2nd| ed.publisher=Birkhäuser | year=1998. | isbn=0-8176-4021-5 | oclc=38179868 |edition=2nd}}

*{{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Algebraic groups over finite fields | jstor=2372673 |mr=0086367 | year=1956 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=78 | pages=555–563 | doi=10.2307/2372673}}

*{{Citation | last1=Steinberg | first1=Robert | title=Endomorphisms of linear algebraic groups | url=https://books.google.com/books?id=54HO1wDNM_YC | publisher=[[American Mathematical Society]] | ___location=Providence, R.I. | series=Memoirs of the American Mathematical Society, No. 80 |mr=0230728 | year=1968}}

[[Category:Algebraic groups]]

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[[Category:Theorems in algebraic geometry]]