Journal of Algebra 390 (2013) 181–198

Contents lists available at SciVerse ScienceDirect

Journal of Algebra
www.elsevier.com/locate/jalgebra

On the topology of relative and geometric orbits for actions
of algebraic groups over complete fields
Dao Phuong Bac a,1 , Nguyen Quoc Thang b,∗
a
b

Department of Mathematics, VNU University of Science, 334 Nguyen Trai, Hanoi, Viet Nam
Institute of Mathematics, 18-Hoang Quoc Viet, Hanoi, Viet Nam

a r t i c l e

i n f o

Article history:
Received 31 October 2012
Available online 15 June 2013
Communicated by Gernot Stroth
MSC:
primary 14L24
secondary 14L30, 20G15

a b s t r a c t
In this paper, we investigate the problem of closedness of (relative)
orbits for the action of algebraic groups on affine varieties defined



182

D.P. Bac, N.Q. Thang / Journal of Algebra 390 (2013) 181–198

Besides, due to the need of number-theoretic applications, the local and global fields k are in the
center of such investigation. For example, let an algebraic k-group G act on a k-variety V , x ∈ V (k).
We are interested in the set G (k).x, which is called relative orbit of x (to distinguish with geometric orbit
G .x). One of the main steps in the proof of the analog of Margulis’ super-rigidity theorem in the global
function field case (see [40,19,20]) was to prove the (locally) closedness of some relative orbits G (k).x,
x ∈ V (k), for some action of an almost simple simply connected group G on a k-variety V . Moreover,
when one considers some arithmetical rings (say, the integers ring, or the adèle ring of a global field)
instead of k, this leads to Arithmetic Invariant Theory (see [5,6]) which plays an important role in the
current study of arithmetic of elliptic and related curves over global fields. In this paper we assume
that k is a field, complete with respect to a non-trivial valuation v of real rank 1 (e.g. p-adic field
or the field of real numbers R, i.e., a local field). Then for any affine k-variety X , we can endow
X (k) with the (Hausdorff) v-adic topology induced from that of k. Let x ∈ X (k) be a k-point. We are
interested in a connection between the Zariski-closedness of the orbit G .x of x in X , and Hausdorff
closedness of the relative orbit G (k).x of x in X (k). The first result of this type was obtained by Borel
and Harish-Chandra [10] and then by Birkes [7], see also Slodowy [36] in the case k = R, the real field,
and then by Bremigan (see [14]). In fact, it was shown that if G is a reductive R-group, G .x is Zariski
closed if and only if G (R).x is closed in the real topology (see [7,36]), and this was extended to p-adic
fields in [14]. Notice that some of the proofs previously obtained in [7,14], etc. do not extend to the
case of positive characteristic. The aim of this note is to see to what extent the above results still hold
for more general class of algebraic groups and complete fields. In the course of study, it turns out that
this question has a close relation with the problem of equipping a topology on cohomology groups
(or sets), which has important aspects, say in duality theory for Galois or flat cohomology of algebraic
groups in general (see [30,22,34,35]). We emphasize that, in the case char.k = p > 0, the stabilizer
of a (closed) point needs not be a smooth subgroup, and the treatment of smoothness condition


(1) The special and canonical topologies on H1flat (k, G ) coincide.
(2) Any connecting map appearing in the exact sequence of cohomology in degree 1 induced from a short
exact sequence of affine group schemes of finite type involving G is continuous with respect to canonical
(or special) topologies.
Some preliminary results on this topic are presented in Section 1, where the main result are Theorems 1.2.2, 1.2.4. In Section 2 we give some general results on the closedness of (relative) orbits,
especially over complete arbitrary fields, where the main results are Theorems 2.1, 2.2.2, and 2.2.3. In
particular, Theorem 2.2.3 complements and generalizes a result obtained earlier by Van den Dries and
Kuhlmann [39] (that the image of an additive polynomial in many variables over a local function field
has the optimal approximation property). In Section 3 we consider the converse statement (that “if
G (k). v is Hausdorff closed in V (k) then G . v is Zariski closed in V ”) in the case of arbitrary complete
fields and the action of smooth affine algebraic groups with a special class of algebraic groups, including nilpotent groups, reductive groups over any complete field, where the main result is Theorem 3.1.
In Section 4 we consider the same problem, but under the assumption that k is perfect, which gives
us finer results, where the main result is Theorem 4.5. Along the way, we give some applications to
the topology of adèlic orbits of algebraic groups which might be of interest. Some of our results have
been reported in [1–3] and [4]. In fact, the results of the present paper improve the main results
obtained there.
Notations and conventions. Q p , R, C denote the fields of p-adic numbers, real and complex numbers,
respectively. Z p denotes the ring of p-adic integers, and F p the finite field with p elements (p is
a prime). In this paper we consider strictly only affine group schemes of finite type (i.e., algebraic
affine group schemes) defined over a field k. By a smooth k-group G we always mean, by conventions,
a smooth affine k-group scheme (i.e., a linear algebraic k-group, as defined in [9]). All other terminologies related to algebraic groups we follow [9]. In particular, a reductive group means a linear algebraic
group (not necessarily connected) with trivial unipotent radical, but not linearly reductive, as usually
treated in Geometric Invariant Theory. We consider only affine k-group schemes G of finite type. For
i
(k, G ) denotes the flat cohomology of G of degree i, whenever it makes sense. We always
them, Hflat
i
denote by {1} the set consisting of the trivial cohomology class in Hflat
(k, G ). When G is smooth,

d G ,0

d G ,1

d G ,2

d G ,3

0 → G ( R ) −−→ G ( S ) −−→ G S ⊗2 −−→ G S ⊗3 −−→ G S ⊗4 → · · · ,

(1)

where G is considered as a covariant functor from the category Com. Alg R to the category Gr of
groups and the differential di := d G ,i are given by the formula (written additively in the commutative
case, for simplicity)

d G ,i = −G (e 1 ) + G (e 2 ) − · · · + (−1)i +1 G (e i +1 ).
In particular, we have d G ,0 ( f ) = f (the embedding R ⊂ S), d G ,1 ( f ) = − f 1 + f 2 , for all f ∈ G ( S ), and
for f ∈ G ( S ), f ∈ Im(G ( R ) → G ( S )) if and only if f ∈ Ker(d1 ). By convention, for x ∈ G ( S ⊗n ), we
denote

xi 1 ...it := G (e it ) ◦ G (e it −1 ) ◦ · · · ◦ G (e i 1 )(x)
whenever it makes sense.
ˇ
The cohomology group Hr ( S / R , G ) := Ker(dr +1 )/ Im(dr ) of this complex is called Cech
cohomology
ˇ
of G with respect to the covering (or layer) S / R. Then we define the Cech–Amitsur
cohomology
p


d G ,2 = G (e 1 )−1 G (e 2 )G (e 3 )−1 .

One defines

Z 1 ( S / R , G ) := g ∈ G S ⊗2

g 1−1 g 2 g 3−1 = 1 ⊂ G S ⊗2 ,

and for a, b ∈ Z 1 ( S / R , G ), a ∼ b in Z 1 ( S / R , G ) if a = c 1−1 bc 2 for some c ∈ G ( S ), and define

H1flat ( S / R , G ) = Z 1 ( S / R , G )/∼,

H1flat ( R , F ) := lim H1flat ( S / R , F ),
→S/R

where the limit is taken over all faithfully flat extensions S / R.
Now we specialize the situation to the case of fields. Let L /k be a normal field extension (resp.
¯ The Cech–Amitsur
ˇ
L = k).
cohomology is defined via the complex

1 → G (k) → G ( L ) → G ( L ⊗k L ) → · · · → G (⊗kr L ) → · · · ,


D.P. Bac, N.Q. Thang / Journal of Algebra 390 (2013) 181–198

185


with respect to a non-trivial valuation v of real rank 1. It seems that not very much is known about
how to endow canonically a topology on the set H1flat (k, G ) such that all connecting maps are continuous. First we recall a definition of a topology on H1flat (k, G ) via embedding of G into special k-groups
given in [38]. Recall that a smooth (i.e. linear) algebraic k-group H is called special (over k) (after
Grothendieck and Serre [32]), if the flat (or the same, Galois) cohomology H1flat ( L , H ) is trivial for all
extensions L /k. Given a k-embedding G → H of G into a special group H , we have the following
exact sequence of cohomology
δ

1 → G (k) → H (k) → ( H /G )(k) −
→ H1flat (k, G ) → 0.
Here H /G is a quasi-projective scheme of finite type defined over k (cf. [15] or [33]). Let k be
equipped with a Hausdorff topology. Since δ is surjective, by using the natural (Hausdorff) topology on ( H /G )(k), induced from that of k, we may endow H1flat (k, G ) with the strongest topology such
that δ is continuous. For the moment, we call it the topology just defined the H-special topology on
H1flat (k, G ). More precisely, we have the following
1.2.2. Theorem. (See [4].) Let k be a field which is complete with respect to a non-trivial valuation of rank 1
and G an affine k-group scheme of finite type. Then the special topology on H1flat (k, G ) does not depend on the
choice of the embedding into special groups and it depends only on k-isomorphism class of G.
1.2.3.

Next we define another topology on H1flat (k, G ).

Definition. The canonical topology on H1 (k, G ) (resp. H1flat (k, G )) is the k s /k- (resp. k¯ /k-) canonical
topology.


186

D.P. Bac, N.Q. Thang / Journal of Algebra 390 (2013) 181–198

It is the same as we define the corresponding topology on H1flat (k, G ) as the limit of the topol-

(A) When is G (k). v Hausdorff closed in (G . v )(k)?
We have first the following result with its origin goes back to Borel and Harish-Chandra and which
is a motivation of our work.
2.1. Theorem.
(1) (Cf. [11,12,14,16].) Let k be a field, complete with respect to a real valuation of rank 1, G an affine k-group
scheme, acting k-regularly on an affine k-variety V , v ∈ V (k). Denote by G v the stabilizer of v in G. If the
stabilizer G v of v is a smooth k-subgroup of G, then for any w ∈ (G . v )(k), the relative orbit G (k). w is
open and closed in Hausdorff topology of (G . v )(k).
(2) Let k be a global field and A the adèle ring of k. If v, G v are as above, then for any w ∈ (G . v )(A), the
relative orbit G (A). w is open and closed in Hausdorff topology of (G . v )(A).
Proof. (1) First proof. The proof is due to Borel and Tits [11, Section 9], [12, Section 3]. Since G v
is smooth, the projection π : G → G . v = G /G v , g → g . v is separable and defined over k, thus the
differential dπ : T g G → T π ( g ) (G . v ) is surjective. It follows that for any w ∈ (G . v )(k), the projection
π : G → G . w = G . v, g → g . w is also separable and defined over k. Then it is well known that the
morphism πk of analytic varieties G (k) → (G . w )(k) also has surjective differential, thus is open by
Implicit Functions Theorem (see [31]). Therefore all G (k)-orbits G (k). w are open, and thus also closed
in Hausdorff topology of (G . v )(k).


D.P. Bac, N.Q. Thang / Journal of Algebra 390 (2013) 181–198

187

Second proof. Since G v is smooth, we know [2,4] that the special (or canonical) topology on
H1flat (k, G v ) is discrete and from the exact sequences

1 → G v → G → ( G / G v ) → 1,
δ

G (k) → (G /G v )(k) −

of finite type. Then the induced map f A : X (A) → Y (A) is a closed map.
Proof. (2) The proof is essentially the same as given in [23, Proposition 2.2.1(ii)]. The only modification we need is the following observation. Let B := A[ T ] be the ring of polynomials in the variable T
over A. We consider the norm on A by defining |x| := Max v |x v | v , where x = (x v ) ∈ A and |.| v denotes
the normalized v-adic norm on the completion k v of k at v. Next, since k has characteristic > 0,
all the valuations v are non-archimedean. Then one checks that the norm on A is non-archimedean.
Therefore the proof of the theorem on the continuity of roots given in [13, Section 3.4], still holds.
Now the proof of (2) as was given in [23] goes through and we are done. ✷
We have the following main result of this section.


188

D.P. Bac, N.Q. Thang / Journal of Algebra 390 (2013) 181–198

2.2.2. Theorem.
(1) Let k be a field, complete with respect to a non-trivial real valuation of rank 1, G an affine k-group scheme
of finite type acting k-regularly on an affine k-variety V and assume that v ∈ V (k). Then the relative orbit
G (k). v is Hausdorff closed in (G . v )(k). Thus, it is closed in V (k), if G . v is Zariski closed in V .
(2) Let k be a global field, A the adèle ring of k and G, H , v, V be as above. Then the relative orbit G (A). v is
Hausdorff closed in (G . v )(A), hence it is so in V (A), if G . v is Zariski closed in V .
Before proving the theorem, we need some auxiliary results. For X a scheme of finite type over an
n
affine base scheme S = Spec( A ) of characteristic p, let us denote by F the Frobenius map, X ( p ) the
n
n
pn
n
S-scheme obtained from X by base change F : A → A . Denote by the same symbol F : X → X ( p )
the Frobenius mapping and if X = G is a group scheme, let F G := Ker( F n ). It is well known that if
S = Spec(k), then for n sufficiently large, the quotient group scheme H := G / F G is a smooth affine

( p ) is
α c α T . Then X
n

defined by f ( p ) ( T 1 , . . . , T m ) = 0, thus it is also defined over A p . By induction we see that X ( p ) is
n
defined over A p .
(2) The assertion follows from above.
(3) Let A = k be a field. By induction, one may reduce everything to the case n = 1 and that X is
affine and given by a single equation

cα T α ,

f (T 1 , . . . , T m ) =
α

where α = (α1 , . . . , αm ), c α ∈ k. For, if a point y = ( y 1 , . . . , ym ) ∈ Im( X (k) → X ( p ) (k)) is the image
p
of x = (x1 , . . . , xm ) ∈ X (k), then by the proof of (1) we see that y i = xi , i = 1, . . . , m. Then it clearly
(
p)
implies that if x, x ∈ X (k) have the same image in X (k), then x = x and we are done.
If A = A, then the same argument also works through. The lemma is proved. ✷
2.2.2.2. Lemma. Let X be a scheme of finite type over a complete valued field k of characteristic p > 0 with
n
respect to a valuation v of rank 1. Then with induced Hausdorff topology, X (k p ) is closed in X (k).
Proof. We may assume that X is affine. Then, by using induction on n, we are reduced to showing
p
that k p is closed in k, or the same, k p is complete. If {xn } is a Cauchy sequence of k p , then as a
sequence in k, it has a limit x ∈ k. Let y = x1/ p ∈ k1/ p . Denote by |.| the v-adic norm on k1/ p . Then


↓

↓

F

1 →
1 →

K

↓

K
↓ Fn
n
K (p )

FG

→
→

F

↓

G
↓ Fn

1

↓
G
F (k)
↓
α
1 →
K (k)
−
→
G (k)
↓ Fn
↓ Fn
β
(
pn ) pn
(
pn ) pn
1 → K
(k ) −
→ G
(k )
↓i
n
G ( p ) (k)

↓
γ
−

= i δ F n G (k)

= i δ G (p

n

)

kp

n

.
n

We know that δ is an open map by Implicit Functions Theorem (since K ( p ) is smooth), so
n
n
n
n
δ(G ( p ) (k p )) is an open, thus also closed subgroup of H ( p ) (k p ). Since the latter is a closed subn
(
pn )
group of H
(k) by 2.2.2.2, it follows that ζ (γ (G (k))) is a closed subgroup of H ( p ) (k). Since ζ is
continuous, it follows that γ (G (k)) = ζ −1 (ζ (γ (G (k)))) is also closed in H (k) as desired. The statement
(1) of Theorem 2.2.3 is proved.
(2) The proof is the same as above. Namely let us consider the exact sequence

1 → K → G → H → 1,

Let (k, v ) be a valued field with a valuation v (written additively), S a non-empty subset of k. We
say after Van den Dries and Kuhlmann [39] that S has optimal approximation property (OA) in (k, v ) if
for any x ∈ k, there exists s ∈ S such that v (x − s) = min{ v (x − z) | z ∈ S }. The following implications
hold [39].
2.3.1. Proposition. (See [39, Section 1].)
(1) With the above notation, we have

S is compact

⇒

S has OA

⇒

S is closed.

(2) (See [39, Section 2].) If k is a local field, then

S is closed

⇒

S has OA.

Therefore, Proposition 2.3.1 combined with Theorem 2.2.3 gives us the following


D.P. Bac, N.Q. Thang / Journal of Algebra 390 (2013) 181–198


3.1. Theorem.
(1) Let k be a field, which is complete with respect to a non-trivial valuation of real rank 1, G a smooth affine
group scheme of finite type over k acting k-regularly on an affine k-variety V . Let v ∈ V (k) be a k-point.
Assume that G (k). v is closed in Hausdorff topology induced from V (k). Then G . v is closed (in Zariski
topology) in V in either of the following cases:
(a) G is nilpotent;
(b) G is reductive and the action of G is strongly separable at v in the sense of [29].
(2) Let k be a global field, A the adèle ring of k, G (A). v be Hausdorff closed in V (A). If one of the assumptions
(a), (b) above holds for G, then G . v is Zariski closed in V .


192

D.P. Bac, N.Q. Thang / Journal of Algebra 390 (2013) 181–198

Before going to the proof, we need the following result due to Birkes. Let f : Gm → V be a morphism of algebraic varieties. If f can be extended to a morphism ˜f : Ga → V , with ˜f (0) = v, then we
write f (t ) → v while t → 0, or limt →0 f (t ) = v.
3.2. Theorem. (See [7, Proposition 9.10].) Let k be an arbitrary field, G a smooth affine nilpotent algebraic
k-group acting linearly on a finite dimensional vector space V via a representation ρ : G → G L ( V ), all defined
over k. If v ∈ V (k) is a point, Y is a non-empty G-stable closed subset of Cl(G . v ) \ G v, then there exist an
element y ∈ Y ∩ V (k), a one-parameter subgroup λ : Gm → G defined over k, such that λ(t ). v → y while
t → 0.
(The property stated in 3.2 is the so-called Property A figured in [7,28].)
From Theorems 2.2.2 and 3.1 we derive immediately the following
3.2.1. Corollary. Let k be as above, G a smooth affine group scheme of finite type or over k acting k-regularly
on an affine k-variety V , and let v ∈ V (k). If G is either nilpotent or reductive and the action of G is strongly
separable at v, then the relative orbit G (k). v is closed in Hausdorff topology in V (k) if and only if the geometric
orbit G . v is Zariski closed.
Proof of Theorem 3.1(1)(a)(b). The assertion follows from the following
3.2.2. Proposition. Let k be a field, which is complete with respect to a non-trivial valuation of real rank 1,


D.P. Bac, N.Q. Thang / Journal of Algebra 390 (2013) 181–198

193

4. Zariski closed orbits for actions of algebraic groups over perfect complete fields
In this section we continue the investigation of question (B) under the assumption that the base
field is perfect. Namely, we state and prove a closedness result for geometric orbits under action of a
large class of algebraic groups over perfect fields under the assumption of closedness of relative orbits.
Before going to main results, we need some auxiliary results, some of which have their independent
interest.
4.1. Lemma. Let G be an algebraic group acting regularly on a variety V , v ∈ V and let G ◦ be the connected
component of G. Then G . v is Zariski closed (resp. Zariski open) in V if and only if G ◦ . v is so.
gi G ◦ ,
Proof. First assume that G . v is Zariski closed and we show that so is G ◦ . v. For, let G =
◦
◦
◦
the decomposition of G into left cosets modulo G . Thus G . v = i ∈ I g i G . v = j ∈ J g j G . v, where
the latter is the disjoint union with respect to J ⊆ I , thus it is the decomposition of G . v into its
irreducible components. Since G ◦ . v is the image of the Zariski open (in G) subset G ◦ via projection
(orbit map), it is also Zariski open in G . v, hence so is each g j G ◦ . v, being homeomorphic to G ◦ . v.
Therefore being the complement of the union of Zariski open subsets, G ◦ . v is also Zariski closed. The
converse is obvious.
Next assume that G . v is Zariski open in V . Then as above, G ◦ . v is Zariski open in G . v, thus also
is Zariski open in V . The converse is trivial; in fact if H is any (not necessarily closed) subgroup of G,
such that H . v is Zariski open in V , then so is G . v, since we can consider the decomposition into left
cosets of G modulo H , to get G . v = i ∈ I g i H . v. ✷
4.2. Proposition. With notation and assumptions as in Lemma 4.1, assume that H is a closed subgroup of G
and v ∈ V .

D.P. Bac, N.Q. Thang / Journal of Algebra 390 (2013) 181–198

4.3.1. Remark. In fact, in the reductive case, original theorem of Kempf gives more information about
the nature of instable orbits and we recall here only one of its consequences.
4.3.2. Theorem. (See Kempf [18].) Let k be a perfect field, L a reductive k-group. Let L act k-regularly on an
affine k-variety V , and let v be a point of instability of V (k), i.e., the orbit L . v is not closed. Let Y be any closed
L-invariant subset of Cl( L . v ) \ L . v. Then there exist a one-parameter subgroup λ : Gm → L, defined over k,
a point y ∈ Y ∩ V (k), such that when t → 0, λ(t ). v → y.
Proof of Theorem 4.3. The proof uses the ideas of [7, Proposition 9.10], and in fact we give more
details here for the sake of completeness. First, we fix a closed non-empty G-invariant subset Y
of Cl(G . v ) \ G . v, and consider the assertion of the theorem as regarding the tuple (G , Y , v ). Then,
since G . v is not closed, from Proposition 4.2 and the fact that G = L × U , it follows easily that
the L-orbit L . v is not closed. It is clear that L acts on G . v and all orbits Lg . v, g ∈ G, have the
same dimension, since L is a direct factor of G. Therefore they are all closed in G . v. In particular,
L . v is closed in G . v. Now we use induction on the dimension of G . v. There is nothing to prove
when dim(G . v ) = 0. By Kempf’s Theorem, applied to the closed L-stable set Y 0 := Cl( L . v ) \ L . v = ∅,
there exist a one-parameter subgroup λ : Gm → L defined over k, a point y 0 ∈ Y 0 ∩ V (k), such that
λ(t ). v → y 0 , while t → 0. Since L . v is closed in G . v and L . v is open in Cl( L . v ), it follows that
Cl( L . v ) ∩ G . v = L . v. Hence we have y 0 ∈ [Cl(λ(Gm ). v ) \ L . v ] ⊂ [Cl(G . v ) \ G . v ]. Let Z be the Zariski
closure Cl(G . v ). Then L acts on Z and we may consider the algebraic (categorical) quotient Z // L. Let
A := { z ∈ Z | Cl( L . z) ∩ Cl(G . y 0 ) = ∅}. By considering the projection π : Z → Z // L, one can check that
the set A is closed in V . Moreover, A is G-stable; in fact, if a ∈ A, then Cl( L .a) ∩ Cl(G . y 0 ) = ∅ if and
only if for any g ∈ G, g Cl( L .a) ∩ g Cl(G . y 0 ) = ∅, i.e., Cl( Lg .a) ∩ Cl(G . y 0 ) = ∅, since g Cl( L . z) = Cl( L . g . z),
g Cl(G . y 0 ) = Cl(G . y 0 ) for all g ∈ G. Also, v ∈ A, since λ(t ). v → y 0 , while t → 0. Thus A = Cl(G . v ).
Let y ∈ Y (⊂ A ). Then we have Cl( L . y ) ∩ Cl(G . y 0 ) = ∅. Since Y is closed and is L-stable, we have
Cl( L . y ) ⊂ Y , thus Y 1 := Y ∩ Cl(G . y 0 ) = ∅. It is clear that Y 1 is a closed G-stable subset of Z . Also
Y 1 ∩ G . y 0 = ∅, since Y ∩ G . y 0 = ∅. Hence Y 1 ⊂ Cl(G . y 0 ) \ G . y 0 . Since y 0 ∈ Cl(G . v ) \ G . v, it follows
that dim(G . y 0 ) < dim(G . v ). By applying the induction hypothesis to the tuple (G , Y 1 , y 0 ), there exists
a one-parameter subgroup μ : Gm → G (in fact μ : Gm → L, since L is the unique maximal reductive
subgroup of G) defined over k, such that μ(t ). y 0 → y 1 ∈ Y 1 , while t → 0. Since λ(t ). v → y 0 , while

Proof. (1) Assume the contrary that G . v is not Zariski closed, so Y := Cl(G . v ) \ G . v = ∅. Then as
a corollary of (an extension of) Kempf’s Theorem 4.3 which holds for perfect fields, there exist a
one-parameter subgroup λ : Gm → G defined over k, a point y ∈ Y ∩ X (k) such that λ(t ). v → y, while
t → 0. Since λ and v are defined over k, it follows that y belongs to the v-adic closure of λ(k∗ ). v ⊂
G (k). v in V (k), thus also to the closure Z of G (k). v in V (k). Meanwhile, y ∈
/ G . v, thus y ∈ Z \ G (k). v,
i.e., G (k). v is not closed in the Hausdorff topology, a contradiction.
(2) Let [G : G ◦ ] = r < ∞. Then

G (k) : G (k) ∩ G ◦

G : G ◦ = r < ∞,

so G ◦ (k) is of finite index in G (k), G (k) = s g s G ◦ (k), g s ∈ G (k), s = 1, . . . , n. If G ◦ (k). v is closed in
V (k), then so is G (k). v = s g s G ◦ (k). v. Conversely, let G (k). v be closed in V (k). From (1) above, we
see that G . v is Zariski closed in V , hence so is G ◦ . v by 4.1. Hence by (1), G ◦ (k). v is closed.
(3) and (4) follow in the same way. ✷
4.5.1. Corollary. Let k be a perfect field, completely valued with a non-trivial valuation of rank 1, G a smooth
affine k-group acting k-regularly on an affine k-variety V . If v ∈ V (k) is such that G v contains all maximal
k-split tori of G then G (k). v is closed in V (k). In particular, if G is k-anisotropic, then any relative orbit G (k). v
is Hausdorff closed in V (k).
Proof. By Corollary 4.4, G . v is Zariski closed in V , and by Theorem 2.2.3 or 4.5(1), G (k). v is closed
in (G . v )(k), thus G (k). v is closed in V (k). ✷
From above we derive immediately the following result, which generalizes several results due to
Borel, Harish-Chandra, Birkes, and Bremigan (see the Introduction).
4.5.2. Corollary. Let k, G, V be as in 4.5.
(1) If G = L × U is as in 4.5(1) then G . v is Zariski closed if and only if G (k). v is Hausdorff closed. In particular,
if G is reductive or nilpotent, then G . v is Zariski closed if and only if G (k). v is Hausdorff closed.
(2) Let G be a smooth nilpotent k-group, T the unique maximal k-torus of G. Then the following statements
are equivalent:

G · v is closed in Zariski topology;
for any above T , T · v is closed in Zariski topology;
G (k) · v is closed in Hausdorff topology;
for any above T , T (k) · v is closed in Hausdorff topology.

Then we have the following logical scheme (b) ⇔ (d), (a) ⇒ (c), (a)
(c) (a).

(b), (b)

(a), (c)

(d), (d)

(c),

Proof. First we assume that k is a p-adic field. Then the assertions (a) ⇒ (c), (b) ⇒ (d) (d) ⇒ (b)
hold: see Theorem 2.2.2, Theorem 4.5, or [7,14].
If k is the real field then the assertions (a) ⇒ (c), (b) ⇒ (d) hold: see [10, Proposition 2.3, p. 495],
or [7,14,36], and also (d) ⇒ (b) holds: see [7, Corollary 5.3, p. 465], or [36].
(b) and (c) (d), we denote K = k¯ an algebraic closure of k and choose
To see that (a)

x
0
0

G=

0

So we have T . v = {(x, y , x−1 )t | x, y ∈ K ∗ }. Hence, T . v is not Zariski closed in K × K × K . Thus
(a) (b).
On the other hand, this example shows that, for k = R or p-adic field, we have G (k) · v = {(x, y + z,
x−1 ) | x, y ∈ k∗ , z ∈ k} is closed in Hausdorff topology. Nevertheless, T (k) · v = {x, y , x−1 | x, y ∈ k∗ } is
(d).
not closed in Hausdorff topology. Hence, (c)
(b) (a) and (c) (d). We choose

G=

a
b
0 a −1

a ∈ K ∗, b ∈ K ,

T=

a
0
0 a −1

v = (1, 1)t . Then we have

G·v=
which is not Zariski closed in K × K .

a+b
a −1


pp. 524–530.
[2] D.P. Bac, N.Q. Thang, On the topology of relative orbits for actions of algebraic groups over complete fields, Proc. Japan
Acad. Ser. A Math. Sci. 86 (2010) 133–138.
[3] D.P. Bac, N.Q. Thang, On the topology of relative orbits for actions of algebraic tori over local fields, J. Lie Theory 22 (2012)
1025–1038.
[4] D.P. Bac, N.Q. Thang, On the topology on group cohomology of algebraic groups over complete valued fields, preprint, 2012.
[5] M. Bhargava, B. Gross, Arithmetic Invariant Theory, I, see http://www.math.harvard.edu/~gross.
[6] M. Bhargava, B. Gross, X. Wang, Arithmetic invariant theory, II, see http://www.math.harvard.edu/~gross.
[7] D. Birkes, Orbits of algebraic groups, Ann. of Math. 93 (1971) 459–475.
[8] A. Borel, Introduction aux groupes arithmétiques, Hermann, Paris, 1968.
[9] A. Borel, Linear Algebraic Groups (second enlarged version), Grad. Texts in Math., vol. 126, Springer-Verlag, 1991.
[10] A. Borel, Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. 75 (1962) 485–535.
[11] A. Borel, J. Tits, Groupes réductifs, Publ. Math. Inst. Hautes Etudes Sci. 27 (1967) 50–151.
[12] A. Borel, J. Tits, Homomorphismes « abstraits » de groupes algébriques simples, Ann. of Math. 97 (1973) 499–571.
[13] S. Bosch, U. Günter, R. Remmert, Non-Archimedean Analysis, Grundlehren Math. Wiss., vol. 261, Springer-Verlag, 1984.
[14] R. Bremigan, Quotients for algebraic group actions over non-algebraically closed fields, J. Reine Angew. Math. 453 (1994)
21–47.
[15] M. Demazure, P. Gabriel, Groupes algébriques, Tome I, Masson, Paris, 1970.
[16] P. Gille, L. Moret-Bailly, Actions algébriques des groupes arithmétiques, in: V. Batyrev, A. Skorobogatov (Eds.), Proceedings
of the Conference “Torsors, Theory and Applications”, Edinburgh, 2011, Proc. Lond. Math. Soc. (2013), in press.
[17] A. Grothendieck, Technique de descente et théorèmes d’existence en géometrie algébrique. I. Généralités. Descente par
morphismes fidèlement plats, Sémin. Bourbaki 190 (1959).
[18] G. Kempf, Instability in invariant theory, Ann. of Math. 108 (1978) 299–316.
[19] L. Lifschitz, Superrigidity theorems in positive characteristic, J. Algebra 229 (2000) 375–404.
[20] G. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, 1991.
[21] J. Milne, Étale cohomology, Princeton Math. Ser., vol. 33, Princeton Univ. Press, Princeton, NJ, 1980.
[22] J. Milne, Arithmetic Duality Theorems, second ed., BookSurge, LLC, Charleston, SC, 2006.
[23] L. Moret-Bailly, Un théorème d’application ouverte sur les corps valués algébriquement clos, Math. Scand. 111 (2012)
161–168.
[24] L. Moret-Bailly, An extension of Greenberg’s theorem to general valuation rings, Manuscripta Math. 139 (2012) 153–166.