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
, then, writing
σ
:
G
→
G
,
x
↦
x
q
for the Frobenius, the morphism of varieties
G
→
G
,
x
↦
x
−
1
σ
(
x
)
is surjective. Note that the kernel of this map (i.e.,
G
=
G
(
F
q
¯
)
→
G
(
F
q
¯
)
) is precisely
G
(
F
q
)
.
The theorem implies that
H
1
(
F
q
,
G
)
=
H
e
´
t
1
(
Spec
F
q
,
G
)
vanishes, and, consequently, any G-bundle on
Spec
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 (e.g., elliptic curves.) In fact, this application was Lang's initial motivation. If G is affine, the Frobenius
σ
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
σ
that induces a nilpotent operator on the Lie algebra of G.
Steinberg (1968) gave a useful improvement to the theorem.
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−1F(g).
The Lang–Steinberg theorem states 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.
Define:
f
a
:
G
→
G
,
f
a
(
x
)
=
x
−
1
a
σ
(
x
)
.
Then (identifying the tangent space at a with the tangent space at the identity element) we have:
(
d
f
a
)
e
=
d
(
h
∘
(
x
↦
(
x
−
1
,
a
,
σ
(
x
)
)
)
)
e
=
d
h
(
e
,
a
,
e
)
∘
(
−
1
,
0
,
d
σ
e
)
=
−
1
+
d
σ
e
where
h
(
x
,
y
,
z
)
=
x
y
z
. It follows
(
d
f
a
)
e
is bijective since the differential of the Frobenius
σ
vanishes. Since
f
a
(
b
x
)
=
f
f
a
(
b
)
(
x
)
, we also see that
(
d
f
a
)
b
is bijective for any b. Let X be the closure of the image of
f
1
. The smooth points of X form an open dense subset; thus, there is some b in G such that
f
1
(
b
)
is a smooth point of X. Since the tangent space to X at
f
1
(
b
)
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
f
1
then contains an open dense subset U of G. Now, given an arbitrary element a in G, by the same reasoning, the image of
f
a
contains an open dense subset V of G. The intersection
U
∩
V
is then nonempty but then this implies a is in the image of
f
1
.
