DSpace at VNU: On the topology of relative orbits for actions of algebraic groups over complete fields

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

over complete fields in its relation with the problem of equipping a
topology on cohomology groups (sets) and give some applications.
© 2013 Elsevier Inc. All rights reserved.

Keywords:
Algebraic groups
Relative and geometric orbits over complete
fields

Introduction
Let G be a smooth affine (i.e. linear) algebraic group acting regularly on an affine variety X , all
are defined over a field k. Many results of (geometric) invariant theory related to the orbits of the
action of G are obtained in the geometric case, i.e., when k is an algebraically closed field. However,
since the very beginning of modern geometric invariant theory, as presented in [25,26], there is a
need to consider the relative case of the theory. For example, Mumford has considered many aspects
of the theory already over sufficiently general base schemes, with arithmetical aim (say, to construct
arithmetic moduli of abelian varieties, as in Chapter 3 of [25,26]). Also some questions or conjectures
due to Borel [8], Tits [25], etc. ask for extensions of results obtained to the case of non-algebraically
closed fields. As typical examples, we just cite the results by Birkes [7], Kempf [18], Raghunathan [28],
etc. to name a few, which gave the solutions to some of the above mentioned questions or conjectures.

*
1

Corresponding author.
E-mail addresses: , (D.P. Bac), (N.Q. Thang).
Current address: Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA.

0021-8693/$ – see front matter © 2013 Elsevier Inc. All rights reserved.
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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
plays an important role here. The most satisfactory results are obtained for perfect fields, and also

for a general class of groups over local fields. We have the following general results regarding some
relations between the topology of relative orbits and that of geometric orbits.
Theorem 1. Let k be a field, complete with respect to a real valuation of rank 1, G a smooth affine k-group,
acting k-regularly on an affine k-variety V , v ∈ V (k), and G v the stabilizer of v in G.
(1) (a) The relative orbit G (k). v is Hausdorff closed in (G . v )(k). Thus if G . v is Zariski closed in V , then G (k). v
is Hausdorff closed in V (k).
(b) (See [11,12,14].) If moreover, the stabilizer G v of v is smooth over k, 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) Assume that G (k). v is Hausdorff closed in V (k). Then if either
(a) G is nilpotent, or
(b) G is reductive and the action of G is strongly separable,
then G . v is Zariski closed in V . Therefore, in these cases, G . v is Zariski closed in V if and only if G (k). v is
Hausdorff closed in V (k).
(3) Assume further that k is a perfect field, G = L ×k U , where L is a reductive and U is a unipotent subgroup
of G, L is defined over k, V , v are as above. Then G (k). v is Hausdorff closed in V (k) if and only if G . v is
Zariski closed in V .
Here the action of G is said to be strongly separable (after [29]) at v if for all x ∈ Cl(G . v ), the
stabilizer G x is smooth, or equivalently, the induced morphism G → G /G x is separable.
One of the main tools to prove the theorem is the introduction of some specific topologies on the
(Galois or flat) group cohomology and their relation with the problem of detecting the closedness of
a given relative orbit. The main ingredient is the following theorem proved in [4], where we refer to
Section 1 for the notion of special and canonical topology on the cohomology set H1flat (k, G ).
Theorem 2. (See [4].) Let G be an affine group scheme of finite type defined over a field k, complete with respect
to a valuation of real rank 1. Then


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

183

(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,

one may consider Galois cohomology of G of degree i, denoted by Hi (k, G ). For an affine variety V ,
a point v ∈ V is always understood as a closed point. We refer to [9] for other terminologies and basic
facts of algebraic groups used here, and to [30] for basic facts concerning Galois cohomology of linear
algebraic groups over fields, and [21,22], for étale and flat cohomology of group schemes. Below, the
terminology “open” or “closed”, unless otherwise stated, always means in the sense of Zariski topology. Below, if we do not mention it explicitly, the field of definition k is assumed to be in general
non-algebraically closed, and a k-point is a closed point defined over such field.
1. Preliminaries
1.1. Galois and flat cohomology
We need in the sequel several facts concerning Galois and flat cohomology of affine algebraic
groups over a field k. We refer to [30] for most standard facts concerning Galois cohomology of linear
algebraic groups over fields, and [21,22,34,35] for étale and flat cohomology of group schemes.
Let R be a commutative ring with unity, G a flat affine R-group scheme of finite type. For any
overring S / R, we set S ⊗n := S ⊗ R · · · ⊗ R S (n-times). Let

e i : S ⊗n → S ⊗(n+1)
be the map s1 ⊗ · · · ⊗ si −1 ⊗ si ⊗ · · · ⊗ sn → s1 ⊗ · · · ⊗ si −1 ⊗ 1 ⊗ si ⊗ · · · ⊗ sn .
For any group (covariant) functor G from the category Com.Alg R of commutative R-algebras to
Groups, we denote the corresponding morphism by


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

G (e i ) : G S ⊗n → G S ⊗(n+1) .
ˇ
If G is commutative we consider the following Cech–Amitsur
complex related with faithfully flat
extension S / R (see, e.g., [17], [21, Chapter III, Section 2])
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

p

Hflat ( R , G ) := lim Hflat ( S / R , G ),
→S/R

p

0,

where the limit is taken over all faithfully flat extensions S / R.
ˇ
If G is non-commutative, then we may consider the non-abelian Cech–Amitsur
complex for a
faithfully flat extension S / R
d G ,0

d G ,1

d G ,2

1 → G ( R ) −−→ G ( S ) −−→ G S ⊗2 −−→ G S ⊗3 ,

(2)

where the differentials d G ,i are given by the formulae (written multiplicatively)

d G ,0 = id,

d G ,1 = G (e 1 )−1 G (e 2 ),

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

where the complex may go on to infinity. One defines the groups of cocycles and the group of

cochains

Z r L /k, G ( L ) := Ker(d G ,r +1 ),

B r L /k, G ( L ) := Im(d G ,r ).

ˇ
Then we define the Cech–Amitsur
cohomology

Hrflat ( L /k, G ) = Z r L /k, G ( L ) / B r L /k, G ( L ) .
ˇ
One may use this Cech
cohomology to obtain two types of cohomology for G: the Galois cohomol¯
ogy Hr (Gal(k s /k), G (k s )), by taking M = k s the separable closure of k in a fixed algebraic closure k,
¯ If G is a smooth
and the flat cohomology Hr (k¯ /k, G ) (denoted also by Hrflat (k, G )) by taking M = k.
k-group scheme, then it is known [35, Theorem 43] that

Hr Gal(k s /k), G (k s )

Hr (k¯ /k, G ).

1.2. Topology on Galois or flat cohomology sets and groups
In many problems related with cohomology, one needs to consider various topologies on the group
cohomology, such that all the connecting maps are continuous. Of course, the weakest (coarsest)
topology is not interesting since it does not give anything, thus it is excluded from consideration.
1.2.1. Special topology
Assume that G is an arbitrary affine group scheme of finite type defined over a field k, complete
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.


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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-

ogy just defined. Namely, a subset U ⊂ H1flat (k, G ) is open (closed) if and only if f L−/1k (U ) is so in
H1flat ( L /k, G ( L )) for all L. Equivalently, we may regard

H1flat (k, G ) =

f L H1flat L /k, G ( L ) ,

(∗)

L /k

and by definition, the subset U is open (closed) in H1flat (k, G ) if and only if its intersections with the
subsets Im( f L ) = f L (H1flat ( L /k, G ( L ))) are so in f L (H1flat ( L /k, G ( L ))) for all L. We call such a topology
“canonical” (since it is defined intrinsically only in term of G). It is clear that when G is commutative,
this is just the definition we gave above. The general case of affine group k-schemes can be treated
in similar fashion.
1.2.4. Theorem. (See [4].) With the same assumption as of 1.2.1:
(1) The special and canonical topologies on H1flat (k, G ) coincide.

(2) If α ∈ Z 1 (k¯ /k, G ) is a 1-cocycle with values in G, α G is the twist of G by means of α , then there exists a
homeomorphism H1flat (k, G ) H1flat (k, α G ) with respect to special topology on these cohomology sets.
(3) All connecting maps arising from a short exact sequence of algebraic groups are continuous with respect
to special and canonical topology.
2. Relative orbits for actions of algebraic groups over arbitrary complete fields are closed
First we consider the following situation. Let k be a field, complete with respect to a real valuation
of rank 1, G an affine k-group scheme of finite type, acting k-regularly on an affine k-variety V ,
v ∈ V (k). Denote by G v the stabilizer of v in G. Recall that the stabilizer G v of v is always defined
over k. Then G (k). v is naturally a subspace of (G . v )(k) ⊂ V (k) with induced Hausdorff topology. The
first question is
(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) −
→ H1flat (k, G v )

we derive that G (k). v = δ −1 (1) so G (k). v is open and closed in (G . v )(k). Since δ is continuous [2,4],
any other G (k)-orbit is the preimage of an element from H1flat (k, G v ), thus is also closed and open.
(2) The two proofs remain the same as above, by making use the corresponding results (namely
[27, Chapter I, Section 3.6], for the first proof, and our results regarding adèlic special topology for
the second proof). ✷
2.1.1. Remarks. (1) This theorem corresponds to the part (1)(b) of Theorem 1 of the Introduction.
(2) The statement and the idea of the first proof above has its origin in Borel and Harish-Chandra
[10] for the case of real field R, and the general case and its arguments of the proof are in [11,
Section 9, Proof of Lemma 9.2] and [12, Section 3], which make use of the general Implicit Functions
Theorem. Later on such arguments appear also in [14, Section 5] and [16]. Then, the converse was
proved for reductive groups over the reals by Birkes [7] (see also [36]), and for reductive groups over
any local fields of characteristic 0 case, by Bremigan [14]. Here we may also treat the fields which
are complete with respect to a non-trivial valuation of real rank 1, for which the general Implicit
Functions Theorem holds.
2.2. Next we treat the cases where the stabilizer group G v need not be smooth. We will show that
in this case the closedness of relative orbits still holds, while the openness may fail. For the converse
(see below), the best result one can achieve is the case where k is a perfect field.
First we recall the following new versions of the Open Mappings Theorem in any characteristic,
due to L. Moret-Bailly, which is very useful in the study of topology of orbits. For a scheme X of finite
type over a field k equipped with a real valuation of rank 1 denote by X top the set X (k) of k-rational
points of X endowed with the Hausdorff topology induced from that of k.
2.2.1. Proposition.
(1) (See [23, Proposition 2.2.1(ii)], [24, Theorem 1.4].) Let k be a field equipped with a real valuation v of
rank 1, which is algebraically closed in its completion k v (e.g. if k is complete or algebraically closed) and
let f : X → Y be a finite k-morphism of k-schemes of finite type. Then the induced map

f top : X top → Y top
is a closed map.
(2) Let k be a global function field, A the adèle ring of k and let f : X → Y be a finite k-morphism of k-schemes
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.


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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
k-group. First we need the following

2.2.2.1. Lemma. Let X be a scheme of finite type over an affine base scheme S = Spec A of characteristic p.
Then
n

n

n

(1) X ( p ) is defined over A p , i.e. Spec( A p );
n
n
(2) F n ( X ( A )) = X ( p ) ( A p );
(3) If A is a field k, or a subring of a direct product of fields (e.g. the adèle ring A of a global field), then the
n
natural map X (k) → X ( p ) (k) is injective.
Proof. (1) By induction, one is reduced to the case n = 1. Also, we may assume that X is also affine
and that X is defined in the affine space Am
S by a single affine equation

cα T α ,

f (T 1 , . . . , T m ) =
α

where

α = (α1 , . . . , αm ), cα ∈ A, T α := T 1α1 · · · T mαm . Let f ( p) ( T 1 , . . . , T m ) :=

p α
( 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
p

xn − x = (xn − y ) p → 0, so |xn − y | → 0, i.e., xn → y in k1/ p . Hence {xn } is a Cauchy sequence. Since k
is complete, it follows that y ∈ k. Lemma 2.2.2.2 is proved. ✷


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

189

2.2.3. Theorem. Let k be a complete valued field with respect to a valuation of rank 1.
(1) Let G, H be affine algebraic k-group schemes and let γ : G → H be a k-morphism of algebraic groups.
Then the image γk (G (k)) is a closed subgroup of H (k).
(2) Let G (resp. H = G / K ) be an affine algebraic k-group scheme (resp. k-homogeneous space under G, where
K is a k-subgroup of G) and let γ : G → H be the canonical k-morphism. Then the image γk (G (k)) is a
closed subset of H (k).
Proof. (1) Without loss of generality, we may assume that γ (G ) = H . Let K = Ker(γ ). If K is smooth,
then γ is separable, so γk : G (k) → H (k) is an open map with respect to Hausdorff topology. Therefore
γk (G (k)) is an open hence also closed subgroup of H (k).
Next we assume that K is not a smooth group. Then it is known [33, Exp. XVII], that for n sufn
n
ficiently large the Frobenius iteration K ( p ) is a smooth k-group. Moreover, it is defined over k p by
2.2.2.1. We consider the following commutative diagram with exact rows and columns

1

1

1

↓

↓

↓

F

1 →
1 →

K

↓

K
↓ Fn
n
K (p )

FG

→
→

F

↓

G
↓ Fn
n

G (p )

γ

−
→
δ
−
→

H

↓

→ 1

H

↓ Fn

n
H (p )

→ 1

and by functoriality, we derive from this the following commutative diagram

1

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
−
→

H
F (k)
↓
γ
−
→
H (k)
↓ Fn
δ
(
pn ) pn
−
→ H
(k )
↓i
n
θ
−
→
H ( p ) (k)

Here all the rows are exact and i means a closed embedding by 2.2.2.2. Since F H (k) = 1, it foln
lows that ζ := i ◦ F n : H (k) → H ( p ) (k) is injective. Hence on the one hand we have γ (G (k)) =
−
1
ζ (ζ (γ (G (k)))). On the other hand, we have

ζ γ G (k)

= 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,

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where by assumption, K is a k-subgroup scheme of G, and H = G / K is a quasi-projective k-scheme.
By 2.1, we see that the image of G (k) is open and closed in H (k). Then the whole argument of the
proof of Theorem 2.2.3(1) extends to our case, by making use of Lemmas 2.2.2.1 and 2.2.2.2. ✷
Proof of Theorem 2.2.2. (1) We may assume that G v is a non-smooth affine k-group scheme, thus
char . = p > 0. We need to show that the image of G (k) in (G /G v )(k) is a closed set there, which
follows from Theorem 2.2.3.
(2) follows from the same arguments, by using Lemma 2.2.2.1. ✷
2.2.3.1. Remarks. (1) Theorem 1(1)(a) of the Introduction follows from Theorem 2.2.3.
(2) The proof of 2.2.2 and the fact that the canonical topology is not changed via the twisting (see
1.2.3 or [4]) imply the following
2.2.3.2. Corollary. Let k be either a complete field (resp. a global field with adèle ring A), G an affine k-group
scheme of finite type. Then the special (or canonical) (resp. adèlic) topology on H1flat (k, G ) (resp. H1flat (A, G ))
is T 1 .
Now we consider the case of commutative affine group schemes over local fields. As a consequence
of results proved above, we give more details of the proof of [22, Chapter III, Section 6, Lemma 6.5(a)]
regarding the topological structure of the flat cohomology groups Hrflat ( L /k, G ). We have the following
2.2.3.3. Corollary. (Cf. [22, Chapter III, Section 6].) Let k be a local field, G an affine commutative group scheme
of finite type over k. Equipped with the canonical topology, the group Hrflat ( L /k, G ) is T 1 , σ -compact and locally
compact.
Proof. By shifting the dimension, we may assume that r = 1. That the space is T 1 follows from Theorem 2.2.3. The only new ingredient needed is the fact (Theorem 2.2.3) that over a completely valued
field with real valuation of rank 1, if f : G → H is a k-morphism of affine k-group schemes of finite
type, then the image f (G (k)) is closed in H (k) in Hausdorff topology. The other arguments, related
with the (σ -)compactness follow from this result combined with the original arguments in [22]. ✷
2.3. Applications
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

191

2.3.2. Proposition. Let k be a local field, G an affine k-group scheme with a k-subgroup K , H = G / K ,
π : G → H the corresponding projection. Then the image of G (k) in H (k) via π is a closed subset, thus also has
OA property.
This result implies the following main result of [39, Theorem 1].
2.3.3. Theorem. (See [39, Section 1, Theorem 1].) If f = f ( T 1 , . . . , T n ) is an additive polynomial in n variables with coefficients in Fq ((t )), then the image of Fq ((t )) under f (i.e., the set { f (a1 , . . . , an ) | a1 , . . . , an ∈
Fq ((t ))}) has OA in (Fq ((t )), v t ), where v t denotes the discrete valuation corresponding to t.
Proof. Indeed, we just apply 2.3.2 to the case, where k is a local function field (Fq ((t )), v t ), G = Gna ,
H = Ga , and π is given by an additive polynomial f in n variables T 1 , . . . , T n . ✷
2.3.4. Remarks. (1) In certain sense, Theorem 2.2.3 is a complement to the results obtained in [39].
(Of course, if k is a local function field and G = Gna , H = Ga , γ is an additive polynomial in n variables, then our result 2.2.3(1) is a consequence of [39, Theorem 1].) (See also the remark right after
Theorem 1 of [39], where it was expressed the desire to extend the results to more general class of
fields.) However, it would be of interest to investigate the OA property of the image of the map γ of
Theorem 2.2.3.
(2) As mentioned in the Introduction, our result should be useful in finding a shorter way to the
approach by Venkataramana [40].
3. Zariski closed orbits for actions of algebraic groups over arbitrary complete fields
In this section we are interested in the following second question.
(B) With notation as above, what is the relation between the properties that G (k). v is Hausdorff
closed and G . v is Zariski closed?
By Theorem 2.2.2, we know that if G . v is Zariski closed in V , then G (k). v is Hausdorff closed in
V (k). So we are interested mainly in the following converse statement: “If G (k). v is Hausdorff closed
in V (k), then G . v is Zariski closed in V ”.
First we recall the notion of strongly separable action of algebraic groups after [29]. Let G be a
smooth affine algebraic group acting regularly on an affine variety V , all are defined over a field k.
Let v ∈ V (k) be a k-point, G v the corresponding stabilizer, and Cl(G . v ) the Zariski closure of G . v in V .
Definition. (After [29].) The action of G is said to be strongly separable at v if for all x ∈ Cl(G . v ), the
stabilizer G x is smooth, or equivalently, the induced morphism G → G /G x is separable.
The following is the main result of this section.
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 .


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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,
G a smooth affine algebraic group acting regularly on an affine variety V , all of which are defined over k. Let

v ∈ V (k). Assume further that the relative orbit G (k) v is closed in Hausdorff topology induced from V (k). If
G is either nilpotent or reductive with strongly separable action at v, then the orbit G . v is closed (in Zariski
topology) in V .
Proof. (a) Assume that a nilpotent k-group G acts k-linearly on a k-vector space V , v ∈ V (k), and
that G (k). v is closed in Hausdorff topology. We assume the contrary that G . v is not closed in V .
Then we set Y := Cl(G . v ) \ G . v = ∅. Clearly Y is a closed subset of Cl(G . v ) which is also G-stable.
By Birkes result, there exist y ∈ Y ∩ V (k), a one-parameter subgroup λ : Gm → G defined over k, such
that λ(t ). v → y, t → 0. Denote by Cl the closure in Hausdorff topology. Thus by this choice, y ∈
Cl (λ(k∗ ). v ) ⊂ Cl (G (k). v ) = G (k). v ⊂ G . v, since G (k) v is closed, which is a contradiction. This shows
that G . v is closed as required.
(b) Assume that G . v is not closed, i.e., v is a k-point of instability for the action of G. Since the
action of G is strongly separable, by Theorem 2.3 of [29], there exists a one-parameter subgroup
λ : Gm → G, defined over k, such that limt →0 λ(t ). v = v 0 ∈ Cl(G . v ) \ G . v. Let X be the closure in
/ G (k). v, but
Hausdorff topology of the set {λ(t ). v | t ∈ k∗ } in V (k). It is clear that v 0 ∈ V (k), and v 0 ∈
from above, it is clear that v 0 ∈ X and the latter is contained in G (k). v, since G (k). v is closed in
Hausdorff topology by assumption. This contradiction shows that G . v is Zariski closed. ✷
Proof of Theorem 3.1(2). It is easy to see that if G (A). v is Hausdorff closed in V (A) then for any
completion L = k w of k, G ( L ). v is Hausdorff closed in V ( L ). Then by (1) G . v is Zariski closed in V . ✷
3.2.3. Remarks. (1) Theorem 1(2) of the Introduction follows from Theorem 3.1.
(2) If k is perfect, then this theorem is contained in the main result of Section 4 below. Thus it is
especially interesting in the case of non-perfect fields, e.g. local function fields.
(3) In non-perfect field case, the stabilizer G v in general needs not be smooth. In [3] there were
given some counter-examples to the effect that in any characteristic, if one of the conditions on
G in Theorem 3.1 (i.e., the nilpotency, or the strong separability of the action), is removed, then
the assertion (1) does not hold. Also, the statement of Theorem 3.1(1)(b) does not hold already for
G = SL2 , if one removes some hypotheses on strong separability. Thus the strong separability condition
in Theorem 3.1 is essential.

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 .
(1) If G . v is closed in V then there is a conjugate H of H in G such that H . v is closed in V . In particular,

there exists a maximal torus (resp. Cartan subgroup) and for each standard parabolic subgroup P θ of type
θ of G, there is a parabolic subgroup P ⊂ G, a conjugate of P θ such that P . v is closed.
(2) With the above assumption and notation, assume that G = L × U is a direct product, where L is a reductive
subgroup of G, and U a unipotent subgroup of G. Then G . v is closed if and only if L . v is closed.
Proof. (1) The algebraic group H acts regularly on G . v, thus has a closed orbit H . y, y ∈ G . v. Let y =
g . v, then H . y = H . g . v = g ( g −1 H g ). v. Since there is a homeomorphism g ( g −1 H g ). v
( g −1 H g ). v,
H . v is closed in V , where H = g −1 H g. The rest follows from above, and the fact that the conjugate
of maximal tori (resp. parabolic, Borel, Cartan subgroup) is again such a group.
(2) Assume that G . v is closed. Then by (1), there is a conjugate L = g Lg −1 of L such that L . v
is closed, but for g = lu, l ∈ L , u ∈ U , L = luL (lu )−1 = L, thus L . v is closed. For the converse, the
proof uses similar arguments as in [7, Lemma 9.7], but for the sake of completeness, we present
it here. Assume that L . v is closed. Let Y be the closure of G . v in V , Y / L the geometric quotient,
which exists since L is reductive. Let f : Y → Y / L be quotient morphism, which is surjective. Since
G = L × U , G also acts on Y / L by the formula g . f ( y ) = f ( g . y ), thus U also, and f is a G-equivariant
map. The image of G . v in Y / L is exactly the U -orbit of f ( v ) (i.e., also the G-orbit), thus is closed
since U is unipotent. Hence the preimage Z := f −1 (G . f ( v )) is a closed subset of Y containing G .x,
thus Z is equal to Y . On the other hand, L .x is closed by assumption, so f −1 ( f ( v )) = L . v, and Z =
f −1 (G . f ( v )) = G . f −1 ( f ( v )) = G . L . v = G . v, thus G . v is closed. ✷
Next we need an extension of a theorem of Kempf to a case of non-reductive groups.
4.3. Theorem (An extension of a theorem of Kempf). Let k be a perfect field, G = L × U , where L is reductive
and U is smooth unipotent k-groups. Let G act k-regularly on an affine k-variety V , and let v be a point of
instability of V (k), i.e., the orbit G . v is not closed. Let Y be any closed G-invariant subset of Cl(G . v ) \ G . v.
Then there exist a one-parameter subgroup λ : Gm → G, defined over k and a point y ∈ Y ∩ V (k), such that
when t → 0, λ(t ). v → y.


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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
t → 0, it follows that μ(t )(λ(t ). v ) = (μ(t )λ(t )). v → y 1 ∈ Y while t → 0 as required. ✷

The following corollary shows that if the stabilizer of a k-point is sufficiently big in certain sense,
then the corresponding orbit is Zariski closed. This result complements a result obtained earlier by
Steinberg (see [37, Corollary 1, p. 70]).
4.4. Corollary. Let the notation be as above, z ∈ V (k) a point such that its stabilizer G z contains all maximal
k-split tori of G. Then the orbit G . z is closed in V. In particular, if G contains no k-split subtori, i.e., if G is
k-anisotropic, then G . z is closed in V for any z ∈ V (k).
Proof. First we claim that the assertion holds for L instead of G. Assume the contrary that L . z is
not closed. Thus Y := Cl( L . z) \ L . z = ∅ and is L-stable and closed subvariety of X . Then by Kempf’s
original theorem [18], there exist a one-parameter subgroup λ : Gm → L, defined over k, y ∈ Y (k), such
that λ(t ). z → y, while t → 0. In other words, z = y ∈ Cl( L . z) \ L . z, a contradiction. Now consider the
general case. If G . z were not closed, then neither were L . z by 3.2. Since L is the unique maximal
reductive subgroup of G, all maximal k-split tori are contained in L z , where L z is the stabilizer of z
in L, and we are reduced to the case considered above. ✷
With these preparations we have the following result regarding the topology of the orbits.
4.5. Theorem. Let k be a perfect field, complete with respect to a non-trivial valuation of real rank 1. Let G be
a smooth affine algebraic k-group acting k-regularly on an affine k-variety V , v ∈ V (k), a k-point of V . Then:


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

195

(1) Assume that G = L × U , where L is reductive and U is unipotent, all defined over k. Then G . v is Zariski
closed if G (k). v is closed in the Hausdorff topology on V (k).
(2) With assumption as in (1), G (k). v is Hausdorff closed in V (k) if and only if G ◦ (k). v is so in V (k).
(3) Let k be a number field, G a smooth affine k-group. Assume that G = L × U , where L is reductive and U is
unipotent, all defined over k. Then G .x is Zariski closed in V if G (A).x is closed in the Hausdorff topology
on V (A).
(4) With assumption as in (3), G (A).x is Hausdorff closed in V (A) if and only if G ◦ (A).x is so in V (A).
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:
(a) G · v is closed in Zariski topology;

(b) T · v is closed in Zariski topology;
(c) G (k) · v is closed in Hausdorff topology;
(d) T (k) · v is closed in Hausdorff topology.
4.5.3. Remarks. (1) Theorem 4.5 implies Theorem 1(3).
(2) Recall that by a well-known theorem of Mostow, any smooth affine connected algebraic group
G over a field k of characteristic 0 has a decomposition G = L .U into semi-direct product, where
U is the largest connected normal unipotent k-subgroup of G, L a maximal connected reductive
k-subgroup. The groups which are direct product of a reductive group and a unipotent group are
perhaps the best possible for (1) to hold, i.e., in order that the orbit G . v be Zariski closed. Namely in
[3] we gave a minimum example among solvable non-nilpotent algebraic groups, for which the orbit
G . v is not Zariski closed. (In this case, the Levi decomposition is just semi-direct product, and is not


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a direct one.) It also shows that in general the conditions that G . v is Zariski closed and G (k). v is
Hausdorff closed are not equivalent.
(3) Also, in the case of solvable groups, in contrast with the nilpotent case (see 4.5.2) some of the
relations between the closedness of orbits of closed subgroups and that of the ambient groups may
not hold (4.5.2), as the following examples show.
4.5.4. Proposition. Let G be a smooth affine solvable algebraic group defined over a local field k of characteristic 0, T an arbitrary maximal k-torus of G, and let G act k-regularly on an affine k-variety V . Let v ∈ V (k)
be a k-point. We consider the following statements:
(a)
(b)
(c)
(d)

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
y

0

0
z

x, y ∈ K ∗ , z ∈ K ,

x−1

v = (1, 1, 1)t . Then G · v = {(x, y + z, x−1 )t | x, y ∈ K ∗ , z ∈ K }. Hence, G · v = {(x, y , z)t | xz = 1} is
Zariski closed in K × K × K . We choose

T=

x
0
0

0
y
0

0
0

x−1

x, y ∈ K ∗ .

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

a ∈ K ∗, b ∈ K ,

a ∈ K∗ ,


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

197

However, T · v = {(a, a−1 ) | a ∈ K ∗ } = {(x, y ) | xy = 1} is Zariski closed. Hence, (b)
(a). In this
example, for k = R or p-adic field, we have G (k) · v = {(a + b, a−1 ) | a ∈ k∗ , b ∈ k} is not closed in
Hausdorff topology, but T (k) · v = {(a, a−1 ) | a ∈ k∗ } is closed in Hausdorff topology. Hence, (d)
(c).
(a) follows from Example 5.2 of [3]. ✷
Finally, the assertion (c)
4.5.5. Remarks. (1) Also, the fact that G (k). v is closed (while G . v is not) shows that beyond the
perfect field case, one cannot hope for an equivalence “G (k). v is closed ⇔ G . v is closed” to hold.
(2) L. Moret-Bailly suggested that the completeness of k may be relaxed in many cases to the
property of being henselian. It does not seem to be easy to us to handle all the cases at the moment
and it will be our future project.
Acknowledgments
We warmly thank M. Van der Put for several discussions, especially for his suggestion to one of
the authors to consider the formula 2.2.2.1(2), which plays an important role in the proof of Theorem 2.2.3 and the short proof of Lemma 2.2.2.2. We thank R. Bremigan for making his papers available
to us, J. Milne and S. Shatz for e-mail correspondences related with Section 1 and their encouragement. We thank L. Moret-Bailly, J.-P. Serre and the referee for the criticism on earlier versions of the
manuscript, which leads to the better presentation of the paper. The first author thanks the Vietnam Education Foundation (VEF), Department of Mathematics of Harvard University and especially
B. Gross for financial support and hospitality. This research is funded in part by VIASM and Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number
101.01-2011.40.
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