Some topics in geometric invariant theory over non algebraically closed fields

In this paper, we review some problems related with the study of geometric and relative orbits for the actions of algebraic groups on affine varieties defined over nonalgebraically closed fields. Mathematics Subject Classification (AMS 2000) : Primary : 14L24. Secondary: 14L30, 20G15

Some topics in geometric invariant theory over

non-algebraically closed fields Dao Phuong Bac∗ and Nguyˆe˜n Quˆo´c Thˇa´ng†

Abstract

In this paper, we review some problems related with the study ofgeometric and relative orbits for the actions of algebraic groups onaffine varieties defined over non-algebraically closed fields

Mathematics Subject Classification (AMS 2000) : Primary : 14L24.Secondary: 14L30, 20G15

Plan

I Introduction

II An overview of geometric invariant theory Observability and related tions

no-III Stability in geometric invariant theory over non-algebraically closed fields

IV Topology of relative orbits for actions of algebraic groups over completelyvalued fields

Let G be a smooth affine algebraic group acting morphically on an affine variety X, all 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

∗Department of Mathematics, VNU of Science, 334 Nguyen Trai, Hanoi, Vietnam E-mail : daophuongbac@math.harvard.edu, daophuongbac@yahoo.com.

†Institute of Mathematics, 18-Hoang Quoc Viet, Hanoi, Vietnam E-mail : nqthang@math.ac.vn Support in part by NAFOSTED and VIASM.

case, i.e., when k is an algebraically closed field However, since the very

be-ginning of modern geometric invariant theory, as presented in [Mu], [MFK],there has been a need to consider the relative case of the theory For ex-ample, Mumford has considered many aspects of the theory already oversufficiently general base schemes, with arithmetical aim (say, to constructarithmetic moduli of abelian varieties, as in Chap 3 of [Mu], [MFK]) Alsosome questions or conjectures due to Borel ([Bo1]), Tits ([Mu]) ask forextensions of results obtained (in the case of algebraically closed fields) tothe case of non-algebraically closed fields As typical examples, we cite theresults by Birkes [Bi], Kempf [Ke], Raghunathan [Ra], which gave solutions

to some of the above mentioned questions or conjectures

This article has its aim to overview some of the recent results in thisdirection, while trying to put them in a coherent form In fact, since theresults in the field are quite diverse, to give a reasonable account of (themajority of) all existing results would require a whole book Therefore, thereades will find that we are concerned mostly with some basic topics such asthe geometry and the topology of the orbits

Throughout, we consider only smooth affine (i.e linear) algebraic groups

defined over some field k, which are called also shortly as k-groups For

ba-sic theory of smooth affine (linear) algebraic groups over non-algebraically

closed field we refer to [Bo2], and for a k-group G, the notion of a rational

k-module V for G is as in [Gr1], [Gr2] However, in few places we give some

account of the recent development, which is directly related to our discussionhere

Observability and related notions

2.1 Some basic definitions and facts

To keep things simple, we describe the basic notions in their simplest form(thus not in most general form), with the hope that the readers will either

be able themselves either to extend to a more general setting later on, or findthe corresponding ones in the literature

Action and Orbit Let k be a field, ¯k an algebraic closure of k Let

G be an affine algebraic group, V an affine variety all are defined over k For

simplicity, we identify G, V with their points in ¯k Assume that there exists

a regular k-morphism ϕ : G × V → V , (g, v) 7→ g.v such that the following

holds:

1) g.(h.v) = (gh).v, ∀ g, h ∈ G, v ∈ V ;

2) e.v = v, ∀ v ∈ V , where e denotes the identity element of G.

Then one says that we are given an action of G on V defined over k, or that G acts k-regularly on V For a fixed v ∈ V , the image of G × {v} via ϕ

is called the orbit of v under G and denoted by G.v.

Consider a smooth affine algebraic group G acting morphically on an affine variety V , all are defined over a field k One of the basic subjects in our study is the orbit G.v, v ∈ V , under the action of G From the geometric point of view, the most important objects are the closed orbits and open orbits

(with respect to Zariski topology) The first natural question is the following:

Is there any closed (open) orbit in V ?

One of the first, though elementary but very basic, results is the ing well-known statement, which assures the existence of closed orbits Onshould also note that the open orbits may not exist in general

follow-2.1.1 Theorem (Cf [Bo1], [Hum]) With above assumption,

a) Each orbit G.v is a smooth locally closed subvariety of V;

b) Each orbit G.v contains an open dense subset of its closure;

c) The boundary of G.v is the union of orbits of lower dimension;

d) There exists G-orbits which are closed (in Zariski topology) in V.

Linear action Linearization Among those affine varieties which are

the most important, one can single out the class of varieties V with linear

action of G, i.e., V are vector spaces and G acts on V via a rational (i.e.

linear) representation ρ : G → GL(V ) We call such V also G-modules (or

k-G-modules if they are defined over k).

It is no doubt that the study of representation theory of G by using

(or from the point of view of) geometric invariant theory should play somedefinite role

Fortunately, the general action of affine groups on affine varieties is notfar from this linear action, as the following statement shows.

2.1.2 Theorem (Cf [Bi], [Bre], [Ke], [KSS]) Let G be an affine

alge-braic group acting regularly on an affine variety V, all defined over a field k Then there exists a closed embedding ψ : V ,→ W and a rational representa- tion ρ : G → GL(W ), all defined over k, such that ρ(g)(ψ(x)) = ψ(g.x) for all x ∈ V

One should also note that a closely related and somewhat more generalresult also holds Namely we have

2.1.3 Theorem (Cf [Su1, Theorem 1], [Su2, Theorem 2.5]) Let G be a

connected affine algebraic group acting regularly on a normal quasi-projective variety V Then there exists a projective embedding ψ : V ,→ W := P n

and a projective group representation ρ : G → PGL n = Aut(W ) such that

ρ(g)(ψ(x)) = ψ(g.x) for all x ∈ V

Therefore it is no harm to consider any affine (resp quasi-projective) variety V just as a G-stable closed k-subvariety of some k-G-module (resp some projective G-space) W The proofs of all these facts give also very ex-

G-plicit way to construct such linear (resp projective spaces), where the action

is linearized The setting in [Su2] is scheme-theoretic, thus it may be applied

to a more general situation

Quotients To study the actions of algebraic groups on algebraic varieties, it

is convenient to consider the set of orbits (the quotient sets) for such actions.However, such sets usually do not give much information and the first ques-tion comes to mind, when we are considering the action of algebraic groups

on algebraic varieties, is how to recognize if there is any other structure onsuch sets

Naturally, if an algebraic group G acts on a variety V , all defined over

a field k, it also acts on the affine algebra k[V ] To use the correspondence

Geometry ↔ Algebra, the quotient set V /G should have the algebra

of functions which, being considered as functions on V , are constant on the

G-orbits, thus the G-invariant functions In order that the quotient of V

under the action of G exist, it is necessary that the subalgebra k[V ] G of

all G-invariant functions of k[V ] be finitely generated Assuming this, we

say that a variety W , together with a morphism p : V → W is an

alge-braic (or categorical) quotient of V under the action of G, if the comorphism

p ∗ : k[W ] ' k[V ] G is a k-isomorphism of k-algebras from k[W ] onto the subalgebra of all G-invariant functions in k[G], and we denote W = V //G Further, a categorical quotient is called a geometric quotient, if all the fibers

of p are closed, in other words, all G-orbits are closed If it is the case, we denote V /G the corresponding geometric quotient One should mention also

that in order that the geometric quotient exist, it is necessary that all the

G-orbits be closed Meanwhile, the following holds

2.1.4 Theorem (Rosenlicht) (Cf [Do, Thm 6.2], [Sp, Satz 2.2]) Given

any action of an affine algebraic group G on an affine variety V, there exists

an open G-stable subvariety U ,→ V such that with the induced action of G

on U, the geometric quotient U/G exists.

One should mention the following notions An affine algebraic group G

is called linearly reductive if any linear representation ρ : G → GL(V ) is completely reducible Equivalently, G is linearly reductive, if for any such

ρ and non-zero G-invariant vector v ∈ V , there exists a linear G-invariant

form F on V such that F (v) 6= 0 G is called geometrically reductive if for any linear representation ρ : G → GL(V ) and non-zero G-invariant vector

v ∈ V , there exists a homogeneous G-invariant polynomial F on V such that

F (v) 6= 0 It is tautollogical that linearly reductive ⇒ geometrically tive The converse is also true in characteristic 0 (and this is also equivalent

reduc-to the property of being reductive (the solvable radical is a reduc-torus)) but fails

in general in characteristic p > 0 Finally, G is called reductive if its tent radical R u (G) is trivial Then we have the following criteria for linearly

unipo-reductivity due to Kemper [K].

2.1.5 Theorem [K] Let k be a field, G a linear algebraic group defined

over k Then G is linearly reductive if and only if the following conditions hold:

a) For every affine G-scheme X, the invariant ring k[X] G is finitely generated over k,

b) For every G-module V, the invariant ring k[V ] G is a Cohen-Macaulay ring.

Quite recently, Alper (et al.) in a series of papers [Al1], [Al2], [AE] have

extended several classical results in the case of smooth affine groups over

fields to group schemes Let S be a scheme, G → S a f ppf group scheme,

BG the classifying stack for G G is called S-linearly reductive if the

mor-phism BG → S is cohomological affine, i.e., if f is quasi-compact and it induces an exact functor f ∗ : QCoh(BG) → QCoh(S) between the cate- gories of quasi-coherent modules over BG and that over S Then it was

proved in [Al1] the following characterization of the linear reductivitiy

2.1.6 Proposition ([Al1, Prop 12.6]) Let k be a field, G a separated

k-group scheme of finite type Then the following statements are equivalent i) G is k-linearly reductive (in newly introduced sense);

ii) The functor V 7→ V G from the category Rep(G) of G − representations

to the category V ec of vector spaces is exact;

iii) The functor V 7→ V G from the category Rep f in (G) of finite dimensional

G − representations to the category V ec of vector spaces is exact;

iv) Every G-representation is completely reducible.

v) Every finite dimensional G-representation is completely reducible.

vi) For every finite dimensional G-representation V and 0 6= v ∈ V G , there exists a G-invariant functional F ∈ (V ∗)G such that F (v) 6= 0.

Regarding the geometric reductivity, we have the following well-known sult, basically due to Mumford and Haboush

re-2.1.7 Theorem (Cf [Gros2, Thm A, p.3], [MFK, Thm A 1.0]) Let

G be an affine algebraic group defined over a field k The the following are equivalent.

1) G is reductive;

2) G is geometrically reductive;

3) For any affine k-algebra A with a G-action, the subalgebra of G-invariants

A G is also an affine k-algebra.

2.1.8 Corollary If G is a reductive group acting on an affine variety

V , then the categorical quotient V //G always exists.

Remarks 1) Besides the original proof of the Mumford conjecture (that

in 2.1.7, (1) ⇔ (2), given by Haboush, recently there was given a second

proof by Sastry and Sesahdri [SaSe]

2) Theorem 2.1.7 about the geometric reductivity can be stated in a more

general setting of schemes over very general bases, and we refer to [Se] as

a standard reference for this For yet in a more general stack framework,

we mention the works by Alper (et al.) [Al1], [Al2], [AE], where there weregiven a fragments of a program of extending the major results of GeometricInvariant Theory to this new setting, by employing systematically the stackapproach

3) There is a vast literature on the Hilbert’s 14th Problem We just indicate

a few sources and refer the readers to these and the reference there: [Do],[Gr1], [Kr], [MFK] Below we consider some examples related with the finite-ness of the rings of invariants Other examples, in particular those due toNagata and Roberts, are given, say, in [Gr2, Sec 8])

Examples 1) ([G-P]) Let k be any commutative ring with 1, ² the dual number (i.e., ²2 = 0, X a variable, and A := k[², X] Let G a act on A via λ.f := f + ²λ(∂f /∂X) Then one checks that the ring of invariants

AGa = k[²X, ²X2, ], which is not finitely generated over k.

2) (Weitzenb¨ock) Let Ga act regularly on a vector space V over a field of characteristic 0 Then k[V ]Ga is finitely generated

2.2 Observable and other related subgroups

Isotropy subgroups (stabilizers) For v ∈ V , we denote by G v the

isotropy (or the same stabilizer) subgroup of v in G If V is an affine (resp.

quasi-projective) variety, then by 2.1.2 (resp 2.1.3), the stabilizers can berealized also as stabilizers arising in a linear (resp projective) representa-tions

Due to the fact that there exists a natural bijection G/G v ' G.v, which

in fact, is an isomorphism of varieties in many cases, the study of the isotropy

subgroups of a given group G also plays an important role This lead to the

study of the so-called observable subgroups

Let G be an affine algebraic group defined over an algebraically closed field k Then G acts naturally on its regular function ring k[G] by right translation (r g f )(x) = f (x.g), for all x, g ∈ G, f ∈ k[G] For H a closed k-subgroup of G, we consider k[G] H := {f ∈ k[G] : r h f = f, for all h ∈ H},

the k-subalgebra of H-invariant functions of k[G].

By convention, we identify the (smooth) affine algebraic groups considered

with their points in a fixed algebraically closed field For a k-subalgebra R

of k[G], we put R 0 = {g ∈ G : r g f = f, for all f ∈ R} Then for any closed

subgroup H ⊂ G we have

H ⊆ (k[G] H)0 ⊆ G.

With a motivation from representation theory, Bialynicki-Birula, Hochschildand Mostow (see [BHM, p 134]) introduced the concept of ”observable

subgroup” A closed subgroup H of G is called an observable subgroup of

G if any finite dimensional rational representation of H can be extended

to a finite dimensional rational representation on the whole group G (or, equivalently, if every finite dimensional rational H-module is a H-submodule

of a finite dimensional rational G-module) In loc cit some equivalent

conditions for a subgroup to be observable were given

Then Grosshans (see [Gr1], [Gr2, Chap 1] and reference therein) hasadded several other conditions It turned out later that for closed subgroups

the property of being observable for a subgroup H is equivalent to the ity H = (k[G] H)0 Up to now there are known several equivalent conditionsfor a subgroup to be observable, which are more or less simple to verify andthey are gathered in Theorem 2.2.1 below

equal-Epimorphic subgroups On the opposite side, a closed subgroup H ⊆ G may satisfy the equality (k[G] H)0 = G If it is so, H is called an epimorphic

subgroup of G In fact, under an equivalent condition, this notion was first

introduced and studied by Bien and Borel in [BB1] [BB2] (see also [Gr2, Sec

23, p 132] for recent treatment), which in turn, is based on similar notionfor Lie algebras, given by Bergman (unpublished) There were given severalequivalent conditions for a closed subgroup to be epimorphic (see Theorem2.2.10 below)

Grosshans subgroups In the connection with the solution of the 14thHilbert problem, the following well-known problem is of great interest As-

sume that X is an affine variety, G is a reductive group acting upon X morphically, H is a closed subgroup of G and consider the G-action on the regular function ring k[X] by left translation: (l g f )(x) = f (g −1 x) It is

natural to ask when k[X] H is a finitely generated k-algebra.

For a closed subgroup H ⊂ G, we have k[X] H = k[X] (k[G] H)0

(see [Gr1],

[Gr2, Chap 1]) On the other hand, it is well-known (loc cit) that (k[G] H)0

is the smallest observable subgroup of G containing H So the problem is duced to the case when H is an observable subgroup To solve this problem,

re-Grosshans ([Gr1], [Gr2, Chap 1, Sec 4, 5]) introduced the codimension 2condition for observable subgroups, and the subgroups satisfying this condi-

tion are called subsequently Grosshans subgroups of G (see below).

Quasi-parabolic and subparabolic subgroups Closely related to servable subgroups is the following class of subgroups Recall that ([Gr2,

ob-p 17]) a closed subgroup H of an affine algberaic group G is called parabolic if H ⊂ G ◦ and H is the isotropy subgroup of a highest weight vector for some (finite-dimensional) absolutely irreducible representation of G ◦ A

quasi-connected closed subgroup H is called subparabolic if H ⊂ Q, R u (H) ⊂

R u (Q) for some quasi-parabolic subgroup Q of G, and in general a closed subgroup H is subparabolic if H ◦ is so

In this section we are interested in some questions of rationality related

to observable, epimorphic, quasi-parabolic, subprabolic and Grosshans groups The first rationality results regarding observable (resp epimorphic)subgroups were already given in [BHM], and then in [Gr2], [W] (resp [BB1],[BB2] and [W]), where some arithmetical applications to ergodic actions werealso given We give some other new results related to rationality properties ofobservable, epimorphic and Grosshans subgroups (which were stated initiallyfor algebraically closed fields) Some arithmetic and geometric applicationswill be considered in another paper under preparation

sub-Some rationality properties for observable groups First we recall

well-known results over algebraically closed fields For an algebraic group G

we denote by G ◦ the identity connected component subgroup of G.

2.2.1 Theorem ([BHM], [Gr2, Theorems 2.1 and 1.12]) Let G be a

lin-ear algebraic group defined over an algebraically closed field k and let H be

a closed k-subgroup of G Then the following conditions are equivalent a) H = (k[G] H)0

b) There exists a finite dimensional rational representation ρ : G → GL(V ) and a vector v ∈ V , all defined over k such that

H = G v = {g ∈ G : ρ(g).v = v}.

c) There are finitely many functions f ∈ k[G/H] which separate the points

in G/H.

(as algebraic varieties).

g) The quotient field of the ring of G ◦ ∩ H-invariants in k[G ◦ ] is equal to the

field of G ◦ ∩ H-invariants in k(G ◦ ).

h) If 1-dimensional rational H-module M is a H-submodule of a finite mensional rational G-module then the H-dual module M ∗ of M is also a H-submodule of a finite dimensional rational G-module.

di-Examples 1) If H is a normal closed subgroup of G, then H is able in G.

observ-2) Let χ be a character of G and let H χ := Ker(χ) Then H is observable

in G In particular, if H has no character, then H is observable in G 3) If H1, H2 are observable subgroups of G, then so is H1∩ H2

4) If H is observable in K and K is observable in G then so is H in G 5) Let B be a Borel subgroup of SL2 Then the quotient SL2/B is isomorphic

to the projective line P1, so B is not observable in SL2

Now let k be any field If a closed k-subgroup H of a linear algebraic

k-group G satisfies the condition b) (resp e)) in Theorem 1 where v ∈ V (k)

and the corresponding representation ρ is defined over k, then we say that

H is an isotropy k-subgroup of G (resp has extension property over k).

First we recall the following rationality results proved in [BHM, rems 5, 8]

Theo-2.2.2 Theorem ([BHM, Theorem 5]) Let G be a linear algebraic k-group,

H a closed k-subgroup of G, K an algebraic field extension of k Then H has extension property over k if and only if it has one over K.

2.2.3 Theorem ([BHM, Theorem 8]) If H is a closed k-subgroup of a

lin-ear algebraic k-group G with extension property over k, then H is an isotropy

k-subgroup of G Conversely, if k is algebraically closed and H is a isotropy k-subgroup then it has extension property over k.

From Theorem 2.2.2 and Theorem 2.2.3, we derive the following

2.2.4 Proposition ([TB]) Let k be an arbitrary field and let H be a

closed k-subgroup of a k-group G The following two conditions are lent.

equiva-a) H is an isotropy subgroup of G over k.

b) H is an isotropy subgroup of G over k, i.e., there exists a finite sional k-rational representation ρ : G → GL(V ) and a vector v ∈ V (k) such that H = G v

dimen-Remark In [W], another proof of Proposition 2.2.4 was given, which isbased on some ideas of Grosshans [Gr1], under the condition (which is not

essential) that k = Q and H is connected.

We consider

k[G] H(k) = {f ∈ k[G] : r h f = f, ∀h ∈ H(k)},

and

(k[G] H(k))0 = {g ∈ G : r g f = f, ∀f ∈ k[G] H(k) }.

Then k[G] H(k) and k[G] H := {f ∈ k[G] : r h f = f, ∀h ∈ H} are

k-subalgebras of k[G] In general we have the following diagram

If, moreover, H(k) is Zariski dense in H then we have

k[G] H(k) = k[G] H = k[G] H ∩ k[G].

We say that H is relatively observable over k if H = (k[G] H(k))0 , and H is

k-observable, if (k[G] H)0 = H It is clear that if k is algebraically closed, then

these notions coincide with the observability We have the following obviousimplication

H is k-observable ⇒ H is observable.

2.2.5 Proposition ([TB]) Let k be a field, and let H be a closed k-subgroup

of a k-group G Then

a) H 0 = k[G] H = k ⊗ k k[G] H;

b) H is observable if and only if H is k-observable;

c) Assume that H(k) is Zariski dense in H Then H is observable ⇔ H is k-observable ⇔ H is relatively observable over k.

2.2.6 Proposition ([TB]) Let H be a k-subgroup of a k-group G The

following are equivalent:

a) There exist finitely many functions in k[G/H] which separate the points

in G/H.

b) There exist finitely many functions in k[G/H] which separate the points

in G/H.

2.2.7 Proposition ([TB]) Let G be a k-group, H a closed k-subgroup

of G Assume that, there exists finite dimensional k-rational representation

ρ : G → GL(V ), and v ∈ V (k) such that H = G v Then there is a finite dimensional k-rational representation ρ 0 : G → GL(W ) and w ∈ W (k) such

2.2.9 Theorem ([TB]) Let G be a linear algebraic group defined over

a field k and let H be a closed k-subgroup of G Then the following are alent.

d 0 ) G/H is a quasiaffine variety defined over k.

e 0 ) Any k-rational representation ρ : H −→ GL(V ), can be extended to a

k-rational representation ρ 0 : G −→ GL(V 0 ).

f 0 ) There is a k-rational representation ρ : G −→ GL(V ) and a vector

v ∈ V (k) such that H = G v and

G/H ∼=k G.v = {ρ(g)v : g ∈ G}.

g 0 ) The quotient field of the ring of G ◦ ∩ H-invariants in k[G ◦ ] is equal to

the field of G ◦ ∩ H-invariants in k(G ◦ ).

If, moreover, H(k) is Zariski dense in H, then the above conditions are alent to the relative observability of H over k.

equiv-Some further extensions Recently, in a series of papers, J Alper et

al (see e.g [AE]) have generalized some of the results considered above forobservable subgroup schemes basing on a very general framework of stacks.The following passage is based on [Al1, Al2, AE]

Let S be any scheme, G → S a flat finitely presented quasi-affine group scheme Denote by O X the structure sheaf of the scheme X, BX the clas- sifying stack of X Then a flat, finitely presented quasi-affine subgroup scheme H ⊆ G is called observable if every quasi-coherent O S [H]-module

is a quotient of a quasi-coherent O S [G]-module In the case S = Spec(k),

k is a field, this definition is equivalent to ours, namely every finite

dimen-sional H-representation is a subH-representation of a finite dimendimen-sional

G-representation We have the following

2.2.9 Theorem (bis) [AE, Thm 1.3] Let S be any scheme, G → S

a flat finitely presented quasi-affine group scheme Consider a flat, finitely presented quasi-affine subgroup scheme H ⊆ G Then the following are equiv- alent.

d) The quotient G/H → S is quasi-affine.

If, moreover, S is a noetherian scheme, then all the above are equivalent to the following

e) Every coherent O S [H]-module F is a quotient of a coherent O S [G]-modules;

f ) For every coherent O S [H]-module F, the induced map Ind G

H F → F is a surjection of O S [H]-modules.

Some rationality properties for epimorphic subgroups We recall

(after Grosshans [Gr2, p 132]) that (¯k-)epimorphic subgroups H ⊆ G are those closed subgroups of G satisfying the condition (¯k[G] H)0 = G We have

the following characterizations of epimorphic subgroups over an algebraicallyclosed fields

2.2.10 Theorem ([BB1, Th´eor`eme 1], [Gr2, Lemma 23.7]) Let H be a

closed subgroup of G, all defined over an algebraically closed field k Then the following are equivalent.

a) H is epimorphic, i e., (k[G] H)0 = G.

b) k[G/H] = k.

c) k[G/H] is finite dimensional over k.

d) If V is any rational G-module then the spaces of fixed points of G and H

Remark The initial definition of epimorphic subgroups was given in [BB1],

by only requiring that the condition f ) above hold.

Examples 1) Let H be a closed subgroup of an affine algebraic group

G Then H is epimorphic in (k[G] H)0 (k is algebraically closed).

2) If H is a closed subgroup epimoprhic in G then so is H ∩ G ◦ in G ◦

Let notation be as above and let k be an arbitrary field Then for a subgroup H of a k-group G we say that H is relatively epimorphic over k if (k[G] H(k))0 = G, and that H is k−epimorphic if (k[G] H)0 = G Recall that

k-we have the following inclusions

2.2.11 Proposition ([TB]) With above notation, if H is either (a)

rela-tively epimorphic over k or (b) k-epimorphic, then it is also epimorphic.

We have the following analog of Theorem 2.2.10 over an arbitrary field

2.2.12 Theorem Let k be any field and let H be a closed k-subgroup

of a k-group G Then the following are equivalent.

a 0 ) H is k-epimorphic, i.e., (k[G] H)0 = G.

b 0 ) k[G/H] = k.

c 0 ) k[G/H] is finite dimensional over k.

d 0 ) For any rational G-module V defined over k, the spaces of fixed points of

G and H in V coincide.

e 0 ) For any rational G-module V defined over k, if V = X ⊕ Y , where X, Y

are H-invariant, then X, Y are also G-invariant.

f 0 ) Morphisms defined over k of algebraic k-groups from G to another one

are defined by their values on H.

Remark It was mentioned in [W, p 195], that Bien and Borel

(unpub-lished) have also proved that if G is connected, then d) ⇔ d 0) (Here d) fromTheorem 2.2.10 and d’) from 2.2.12.)

Some rationality properties for Grosshans subgroups One of themain results related with the finite generation problem (hence also with theHilbert’s 14-th problem) mentioned in the Introduction is the following re-sult of Grosshans (Theorem 2.2.14) First we recall the following very usefulresult which reduces to the case of connected groups

2.2.13 Theorem ([Gr2, Theorem 4.1]) Let k be an algebraically closed

field For any closed subgroup H of G, if one of the following k-algebras k[G] H , k[G] H ◦

, k[G ◦]H∩G ◦

, k[G ◦]H ◦

is a finitely generated k-algebra, then the same holds for the other.

2.2.14 Theorem ([Gr2, Theorem 4.3]) For an observable subgroup H

of a linear algebraic group G, all defined over an algebraically closed field k, the following are equivalent.

a) There is a finite dimensional rational representation ϕ : G → GL(V ),

an element v ∈ V , such that H = G v and each irreducible component of G.v − G.v has codimension ≥ 2 in G.v.

b) The k-algebra k[G] H is a finitely generated k-algebra.

If b) holds, let X be an affine variety with k[X] = k[G] H , and with G-action via left translations of G on G/H There is a point x ∈ X such that G.x is open in X and G.x ' G/H via gH 7→ g.x and each irreducible component of

X \ G.x has codimension ≥ 2 in X.

The observable subgroups which satisfy one of the equivalent conditions in

Theorem 2.2.14 are called Grosshans subgroups (see [Gr2, Chap 1, Sec 4]).

There are some nice geometrical characterizations and examples of Grosshanssubgroups presented in there and the reference therein

For a field k, a k-group G and an observable k-subgroup H ⊂ G, we say that H satisfies the codimension 2 condition over k if H satisfies condition

a) above where V, ϕ are all defined over k and v ∈ V (k).

We call H a Grosshans subgroup relatively over k (resp k-Grosshans

sub-group) of G if k[G] H(k) (resp k[G] H ) is a finitely generated k-algebra.

Examples 1) Let H be a closed subgroup of G Then the following are equivalent: a) H is a Grosshans subgroup of G; b) H ∩ G ◦ is a Grosshans

subgroup of G ◦ ; c) H ◦ is a Grosshans subgroup of G ◦

2) If K < H are observable subgroups of G and K is a Grosshans subgroup

of G, then so is K in H.

3) If H is a reductive subgroup of G, then H is a Grosshans subgroup in G.

Now let k be any field We have a result similar to Theorem 2.2.14 for

k-Grosshans subgroups.

2.2.15 Theorem ([TB]) Let k be an infinite perfect field, G a connected

k-group Assume that H is a observable k-subgroup of G Consider the lowing conditions.

fol-a 0 ) H satisfies codimension 2 condition over k.

b 0 ) One of the k-algebras k[G] H , k[G] H ◦

, k[G ◦]H∩G ◦

, k[G ◦]H ◦

is a finitely erated k-algebra.

gen-c 0 ) H is a Grosshans subgroup relatively over k of G (i.e k[G] H(k) is finitely generated k-algebra).

Then, together with conditions in Theorem 15, we have the following cations

of char.k > 0 (Various extensions of classical results in (geometric) ant theory to the case of characteristic p > 0 were discussed at length in [MFK, Appendices].) It will be more interesting to have examples with G, H

invari-connected groups

A relation with the subalgebra of invariants of a Grosshans subgroup of

a reductive group acting rationally upon a finitely generated commutativealgebra is given in the following

2.2.16 Theorem [Gr2, Theorem 9.3].) Let k be an algebraically closed

field For any closed subgroup H of a reductive group G, all defined over k,