Lang's theorem: Difference between revisions

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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'').

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>.

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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 | url=http://www.jstor.org/stable/=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}}