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

Contents lists available atSciVerse ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra On the topology of relative and geometric orbits for actions of algebraic groups ove

Contents lists available atSciVerse 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 Baca,1, Nguyen Quoc Thangb, ∗

aDepartment of Mathematics, VNU University of Science, 334 Nguyen Trai, Hanoi, Viet Nam

bInstitute of Mathematics, 18-Hoang Quoc Viet, Hanoi, Viet Nam

a r t i c l e i n f o a b s t r a c t

Article history:

Received 31 October 2012

Available online 15 June 2013

Communicated by Gernot Stroth

MSC:

primary 14L24

secondary 14L30, 20G15

Keywords:

Algebraic groups

Relative and geometric orbits over complete

fields

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

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

* Corresponding author.

E-mail addresses:bacdp@math.harvard.edu , daophuongbac@yahoo.com (D.P Bac), nqthang@math.ac.vn (N.Q Thang).

1 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.

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∈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 top-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 Section1for 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

(1) The special and canonical topologies on H1flat(k,G)coincide.

(2) Any connecting map appearing in the exact sequence of cohomology in degree1 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 Section1, where the main result are The-orems 1.2.2, 1.2.4 In Section 2we give some general results on the closedness of (relative) orbits, especially over complete arbitrary fields, where the main results areTheorems 2.1, 2.2.2, and 2.2.3 In particular,Theorem 2.2.3complements 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 Section3 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, includ-ing nilpotent groups, reductive groups over any complete field, where the main result isTheorem 3.1

In Section4we consider the same problem, but under the assumption that k is perfect, which gives

us finer results, where the main result isTheorem 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 Zp 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 terminolo-gies 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

them, Hi

flat(k,G)denotes the flat cohomology of G of degree i, whenever it makes sense We always

denote by {1} the set consisting of the trivial cohomology class in Hi flat(k,G) When G is smooth, one may consider Galois cohomology of G of degree i, denoted by H i(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

topol-ogy 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⊗ · · · ⊗s i−1⊗s i⊗ · · · ⊗s n→s1⊗ · · · ⊗s i−1⊗1⊗s i⊗ · · · ⊗s n

For any group (covariant) functor G from the category Com.Alg R of commutative R-algebras to Groups, we denote the corresponding morphism by

G(e i): G

S⊗n

S ⊗( n+1)

.

If G is commutative we consider the following ˇCech–Amitsur complex related with faithfully flat extension S/R (see, e.g.,[17],[21, Chapter III, Section 2])

0→G(R) −−→d G ,0 G(S) −−→d G ,1 G

S⊗2 d G ,2

S⊗3d G ,3

S⊗4

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

d G , i= −G(e1) +G(e2) − · · · + (−1)i+1G(e i+1).

In particular, we have d G ,0(f) = f (the embedding R⊂S), d G ,1(f) = −f1+f2, 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

x i1 i t:=G(e i t) ◦G(e i t− 1) ◦ · · · ◦G(e i1)(x)

whenever it makes sense

The cohomology group Hr(S R,G) :=Ker(d r+1)/Im(d r)of this complex is called ˇCech cohomology

of G with respect to the covering (or layer) S/R Then we define the ˇCech–Amitsur cohomology

Hflat p (R,G) := lim

→S RHp flat(S/ ,G), p0,

where the limit is taken over all faithfully flat extensions S/R.

If G is non-commutative, then we may consider the non-abelian ˇCech–Amitsur complex for a faithfully flat extension S/R

1→G(R) −−→d G ,0 G(S) −−→d G ,1 G

S⊗2d G ,2

S⊗3

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

d G ,0=id, d G ,1=G(e1)−1G(e2), d G ,2=G(e1)−1G(e2)G(e3)−1.

One defines

Z1(S/ ,G) := g∈G

S⊗2 g−1

1 g2g−1

3 =1

⊂G

S⊗2

,

and for a,b∈Z1(S R,G), a∼b in Z1(S R,G)if a=c−1

1 bc2 for some c∈G(S), and define

H1flat(S/ ,G) =Z1(S/ ,G)/ ∼, H1flat(R,F) := lim

→S RH1flat(S/ ,F),

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.

L= ¯k) The ˇCech–Amitsur cohomology is defined via the complex

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

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

Z r

L/ ,G(L) 

:=Ker(d G ,+1), B r

L/ ,G(L) 

:=Im(d G ,).

Then we define the ˇCech–Amitsur cohomology

Hr flat(L/ ,G) =Z r

L/ ,G(L) 

/ r

L/ ,G(L) 

.

One may use this ˇCech cohomology to obtain two types of cohomology for G: the Galois

cohomol-ogy Hr(Gal(k s/ ),G(k s)), by taking M=k s the separable closure of k in a fixed algebraic closure k,¯

and the flat cohomology Hr(¯k k,G)(denoted also by Hr flat(k,G)) by taking M= ¯k If G is a smooth k-group scheme, then it is known[35, Theorem 43]that

Hr

Gal(k s/ ),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 H1

flat(k,G)such that all connecting maps are contin-uous First we recall a definition of a topology on H1

flat(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

1→G(k) →H(k) → (H/G)(k) − →δ 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) topol-ogy on(H/G)(k), induced from that of k, we may endow H1

flat(k,G)with the strongest topology such that δis 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 H1

flat(k,G)

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

topology

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−1

L / (U) is so in

H1flat(L k,G(L))for all L Equivalently, we may regard

H1flat(k,G) = 

L

f L

H1flat

L/ ,G(L) 

and by definition, the subset U is open (closed) in H1

flat(k,G)if and only if its intersections with the subsets Im(f L) = f L(H1

flat(L k,G(L)))are so in f L(H1

flat(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 of1.2.1:

(1) The special and canonical topologies on H1

flat(k,G)coincide.

(2) Ifα ∈Z1(¯k k,G)is a 1-cocycle with values in G, α G is the twist of G by means ofα, then there exists a homeomorphism H1

flat(k,G) H1

flat(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→Tπ ( 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)

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,

G(k) → (G/G v)(k) − →δ 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) Sinceδis continuous[2,4],

any other G(k)-orbit is the preimage of an element from H1

flat(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

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 : Xtop→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 fA: 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

modifica-tion we need is the following observamodifica-tion Let B:=A[T] be the ring of polynomials in the variable T

over A We consider the norm on A by defining|x| :=Maxv|x v|v , where x= (x v) ∈A and|.|vdenotes

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 2

We have the following main result of this section

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 affine base scheme S=Spec(A)of characteristic p, let us denote by F the Frobenius map, X ( p n )the

S-scheme obtained from X by base change F n : A→A p n Denote by the same symbol F n : X→X ( p n )

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

(1) X ( p n ) is defined over A p n

, i.e Spec(A p n

);

(2) F n(X(A)) =X ( p n )(A p n);

(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

natural map X(k) →X ( p n )(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 A m S by a single affine equation

f(T1, ,T m) = 

α

c α T α,

where α = ( α1, , αm), cα∈ A, T α:=T α1

1 · · ·T α m

m Let f ( p )(T1, ,T m) := α c

p

α T α Then X ( p ) is

defined by f ( p )(T1, ,T m) =0, thus it is also defined over A p By induction we see that X ( p n ) is

defined over A p n

(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

f(T1, ,T m) = 

α

c α T α,

where α = ( α1, , αm), cα∈k For, if a point y= (y1, ,y m) ∈Im(X(k) →X ( p )(k)) is the image

of x= (x1, ,x m) ∈X(k), then by the proof of (1) we see that y i=x i p , i=1, ,m Then it clearly implies that if x,x∈X(k)have the same image in X ( p )(k), then x=xand we are done.

If A=A, then the same argument also works through The lemma is proved. 2

2.2.2.2 Lemma Let X be a scheme of finite type over a complete valued field k of characteristic p>0 with respect to a valuation v of rank 1 Then with induced Hausdorff topology, X(k p n)is closed in X(k).

Proof We may assume that X is affine Then, by using induction on n, we are reduced to showing

that k p is closed in k, or the same, k p is complete If {x n p} is a Cauchy sequence of k p, then as a

sequence in k, it has a limit x∈k Let y=x1 p∈k1p Denote by|.|the v-adic norm on k1 p Then

x p−x= (x n−y)p→0, so|x n−y| →0, i.e., x n→y in k1 p Hence{x n}is a Cauchy sequence Since k

is complete, it follows that y∈k.Lemma 2.2.2.2is proved 2

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 letγ: G→H be a k-morphism of algebraic groups Then the imageγk(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 letγ: G→H be the canonical k-morphism Then the imageγk(G(k))is a closed subset of H(k).

Proof (1) Without loss of generality, we may assume thatγ (G) =H Let K=Ker( γ ) If K is smooth,

thenγ is separable, soγk : 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 suf-ficiently large the Frobenius iteration K ( p n ) is a smooth k-group Moreover, it is defined over k p n by 2.2.2.1 We consider the following commutative diagram with exact rows and columns

1 → K ( p n ) → G ( p n ) − →δ H ( p n ) → 1 and by functoriality, we derive from this the following commutative diagram

F G(k) −→γ F H(k)

1 → K ( p n )(k p n) − →β G ( p n )(k p n) − →δ H ( p n )(k p n)

G ( p n )(k) − →θ H ( p n )(k)

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

ζ−1(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 )

k p n

.

We know that δ is an open map by Implicit Functions Theorem (since K ( p n ) is smooth), so

δ(G ( p n )(k p n)) is an open, thus also closed subgroup of H ( p n )(k p n) Since the latter is a closed

sub-group of H ( p n )(k)by 2.2.2.2, it follows that ζ ( γ (G(k))) is a closed subgroup of H ( p n )(k) Since ζ is continuous, it follows thatγ (G(k)) = ζ−1(ζ ( γ (G(k))))is also closed in H(k)as desired The statement (1) ofTheorem 2.2.3is proved

(2) The proof is the same as above Namely let us consider the exact sequence

where by assumption, K is a k-subgroup scheme of G, and H=G/K is a quasi-projective k-scheme.

By2.1, we see that the image of G(k)is open and closed in H(k) Then the whole argument of the proof ofTheorem 2.2.3(1) extends to our case, by making use ofLemmas 2.2.2.1 and 2.2.2.2 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 fromTheorem 2.2.3

(2) follows from the same arguments, by usingLemma 2.2.2.1 2

2.2.3.1 Remarks (1) Theorem 1(1)(a) of the Introduction follows fromTheorem 2.2.3

(2) The proof of2.2.2and the fact that the canonical topology is not changed via the twisting (see 1.2.3or[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 T1

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 Hr flat(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 H r flat(L k,G)is T1,σ-compact and locally compact.

Proof By shifting the dimension, we may assume that r=1 That the space is T1follows from Theo-rem 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

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 2.3.1combined withTheorem 2.2.3gives us the following