Some remarks on the actions of the additive group schemes

J . Math. Kyoto Univ. 10-1 (1970) 189-205

Some remarks on the actions of the additive group schemes by

Masayoshi MIYANISHI (Received Dec. 14, 1969)

T h is n o te consists o f tw o parts and contains different types of results on the actions of the additive group scheme. In the first part, we consider the following problem :

k be an algebraically closed field of arbitrary characteristic and A be an integral local k-domain essentially o f finite type over k . Suppose a n iterative infinite higher derivation D= {D } a c t s o n A so that D 1(m )cm fo r a ll i > 0 , n t being the maximal ideal o f A . Then does there exist a valuation rin g r of the quotient field K o f A such that Let

1

0

.

'Y/' dominates A and that Y/' is invariant under the induced action of D in

K , i.e. Di ( V ) c 'r fo r a ll i > ? Such a 1/* exists i f w e assume moreover one of the following con-

ditions: (1)

For any a o f A ,

D1 ( a) =0 fo r all sufficiently large integer j (cf.

Theorem 1.1).

X defined over k on which the additive group o f t h e universal domain o f k acts and a fixed point x o f X such that th e lo c a l rin g (9x, x w i t h th e induced action of the (2)

There exist a projective variety

additive group is operator-isomorphic to (A , 111) w ith the action of the additive group induced from D (c f. Theorem 1.9). In the second part, w e w ill give a remark on the construction of a quotient preschem e o f a preschem e

X endowed with an action of a

group preschem e G such that t h e stabilizer group is a finite group

preschem e at each point of X (c f. Theorem 2.1).

190

M asay oshi M iy anishi

U sing this rem ark, w e prove th e next result : L e t k b e a f ie ld o f characteristic z e r o a n d V b e a variety defined

over k o n w h ich th e a d d itiv e group o r th e m u ltip lic a tiv e group of the universal domain o f k acts n o n -trivially. L e t K be the field of k-rational functions o n V a n d K b e th e fie ld o f c o n sta n ts in K b y t h e induced 0

a c tio n o f th e a d d itiv e group o r th e m u ltip lic a tiv e g ro u p . Then we have

K = K 0 (t ), t being a n indeterminate (c f. Theorem 2.2). Part 1.

1 . L e t k b e a n algebraically closed field of arbitrary characteristic a n d le t A b e a k-algebra su c h th a t th e a d d itiv e group scheme Ga acts

non-trivially o n Spec (A ) v i a a k-morphism

6 : Ga .x

Spec (A) —*Spec (A),

k

while

6

is defined by a k-homomorphism o f k-algebras, D : A -* A ® k [t],

t being a n indeterm inate. I f we write D (a )= a + D i(a )O t+

Dn(a)(31" + • • , fo r any a E A,

i s a n iterative infinite higher derivation o n A and th en D = {D} Di (a) = 0 f o r all sufficien tly la r g e integer j a n d f o r a n y a o f A , (cf. 0

[6 ]). b e a k-algebra. W e denote Homk (Spec ( R ) , G . ) and Homk (Spec (R ), Sp ec (A )) b y Ga ( R ) a n d Spec (A )(R ) respectively and 6(Spec (R )) b y 0( R ) . A n element A. o f Ga (R ) is regarded a s a n element o f R a n d a n elem ent x o f Spec (A )(R ) is given b y a n k-homomorphism L et R

x o f A to R . Then 6 ( R ) ( 2 , x ) = 2 x is g iv e n b y 'x = (x0A )D : AOk R , w h ere A is id en tified w ith t h e k-homomorphism k [t ] —*R w hich sends t to 2. I f we denote by a ( x ) th e im a g e o f a by x, th en w e have a(A x )= a(x )+ D i(a)(x ) 2+ D 2(a)(x ) ± • • • + D „(a)(x ) An + • • . .

I f A is a n integral k-domain which is finitely generated over k and i f D is a universal domain o f k , th en S p ec (A )(2) is a n affine variety

defined over k. A s above, 6 defines th e a c tio n o f th e a d d itiv e group Ga ( 2 ) o f D o n Spec (A )(S 2) such that

S om e rem ark s on the actions of the additiv e g ro u p sc h e m e s 191

a (2 x) = a (x ) D i(a )(x )2 D 2 (a ) (x ) 2 + • • • + D n (a )(x ) A" ± • — 2

for 2 E Q , a E A an d a p o in t x of Spec (A)(S2). Now l e t V b e a n affine variety d efin ed o ver k w ith th e affine coordinate rin g A a n d le t r b e a n action o f th e ad d itive group o f S2 o n V defined over k. T h e n it is e a s ily s e e n th a t for points x and t o f X and S2 respectively w hich a r e gen eric o ver k an d fo r a E A , w e have a (r (t, x ))= a ( x ) Di(a)(x) t + • • • + D n (a)(x) + • • • , 1>0 i s a n iterative infinite higher derivation o n A . If we define a k-homomorphism of k-algebras D: A — > A g k r t ] by

where D =

(a )= a+ Di(a)Ot+ • • • + Dn(a)00 + • • • ,

D defines an action

6

of the additive group scheme G a o n Spec (A ) and

r is obtained from 6* a s r = 6(2).

First we shall prove Theorem 1 .1 . L e t k b e a n algebraically closed field o f arbitrary

characteristic a n d le t A b e a n in teg ral lo c al k-domain w hich is essentially of f inite ty pe ov er k.

Suppose th at an iterativ e inf inite higher k-

deriv ation D = {D i }1>0 acts o n A so th at f o r any a E A , w e hav e D i (a) = 0 f o r all sufficiently large integer j and th at D (m) g at f o r all integer i

i > 0, nt being the m ax im al ideal o f A . L e t K be the quotient f ield of

A . T hen there ex ists a v aluation rin g ^V of K w hich dom inates A and w hich is inv ariant under th e induced action of D on K , i.e . D i ( V )g Y / ' f o r all in te g e r i > 0 . P r o o f . 1) F irst w e sh all assume th at A is n o rm a l. There exists a non-zero elem ent a o f n t such that D ( a ) = O fo r a ll i > O. In fact, ta k e a n y non-zero elem ent b o f pt. T h e n th e r e e x is ts a n in teg er n such that D ( b ) \ 0 and D ( b ) = O fo r a ll j > n . Then D ( b ) belongs t o m a n d h a s t h e d esired p ro p erty. L e t I - = a A a n d p b e a prim e divisor o f a. T h en w e have the following result.

Masayoshi Miyanishi

192

L em m a 1 .2 . I an d

p are in v arian t u n d e r D , i.e. D 1 (I) ..g I and

Di (p)g).) f o r an y i > O. P r o o f . Since D i (ax )=

E D (a)D

f+k=i

i

k

(x )=aD i ( x ) , it is obvious that

Di(I) c I. A s for p, y) for some element y o f A , since p is a prime divisor o f I. L e t x be a n element of p. W e shall prove D i (x )

E ( I : y i I ).

If i = 0 , this is trivial. S u p p o se we have Di (x) E (I: yi ' 1 ) fo r j < i . Then since D i (x y )= E Di (x )D k (y ) E I a n d since Di (x) y i E I , w e have D 1 (x) E ( I : y i

). On the other hand, since A is normal, any prime divisor of a principal ideal has height 1. Therefore ( I : y i +1 ) = ( I : y ) . For otherwise, ( I : y 1 ') is contained in a p rim e divisor p' o f I which has height > 1 . Hence Di(x ) E ( I : y)=1). q.e.d. L e t I an d p b e a s above. Since A is norm al and p is a prime ideal of height 1 , A , is a discrete valuation r i n g . We shall prove our theorem by induction on dim ( A ) . If dim (A) = 1, w e have only to take A a s 1/* since A is a discrete valuation rin g . S u p p o se our theorem were proved if dim ( A ) < n . Take a , I and

=Q(A/p)

is the quotient field of

A/p and

p as

above. T h en A p /pA,

dim (A/p) =dim (A ) —1. B y

our assumption, there exists therefore a valuation r in g 1/- o f Q(A/p) which dominates derivation

D

A/p and

is invariant under the iterative infinite higher

induced from D on A / p . L et V be the composite of A p

w ith i7 , i.e . '1/. =

E A p ; a (m od pA,)EY/'} .

T h en 17 . i s a valuation

rin g o f K such that l r p Ap = 4 , and 1./Vp./1 - = 'r , hence 1/' dominates A. O n th e other h a n d , D(a) E Y/' f o r any a E 1/ s in c e D ( a) (mod = D (a (mod pA )). Thus Y/' is a required valuation ring. 0

-

p

I f A is not necessarily norm al, A has such a form a s A = B „ where B is a k-algebra o f finite type and p is a prim e ideal o f B . Let x1 be elements o f B which generate B over k . By our assumption o n th e actio n o f D , a k-algebra C w hich is generated by D i (x i ) over k for a ll i > 0 an d j , 1 < j < 1 , is also a k -algeb ra o f finite type an d is invariant under D . Let q=miTh C , n i being th e m axim al ideal o f A . Then q is invariant under D a n d A = C , because A If we define an action 6 o f th e ad d itive group scheme Ga o n Spec (C) (2)

S om e rem ark s on the actions of the additiv e g ro u p sc h e m e s 193 from th e actio n o f D o n C, 6 also defines th e actio n r = 0 ( 2 ) of the

additive group o f a universal domain 2 o f k on Spec (C )(2 ). T h is z can be extended to the action o f 2 o n Spec (C )(2 ), where C is the normalization o f C , (c f. [2 1). 't is g iv e n b y an actio n a o f Ga o n -

Spec (C ) which is defined by a n a c tio n o f D o n C a n d which makes

th e following diagram commutative, Ga x Spec (C)

Spec (C)

Ga x Spec (C)

+ Spec (C)

r

where 7r denotes th e canonical morphism. N ow le t S= C — q . Then (C) s i s th e normalization o f A = C , and is therefore a finite A-module since A is a k-algebra which is essen-

Let A'= (C ) 5 a n d 131 , • , 43, be the prime divisors of litA i n A '. Then since A ' is a finite A-module, w e have r > 1 , T 1 r1A = in for a ll i , 1< i < r , an d th ere is no inclusion among . M o re o v e r s in c e inA i s in varian t under D , V i is a ls o invariant under D fo r e v e ry i, 1< i < r , b y th e same a rg u m e n t a s in Lemma tially of finite type over k. t

t

1.2.

L et 13=

T h en a n o rm a l rin g .24

dominates A an d satisfies

th e same condition a s f o r A in th e statement of our theorem. Hence b y (1), w e h a v e a valuation rin g r o f K which dominates 7 4 , hence A and which is invariant under D. q.e.d. .

I f th e characteristic o f th e ground field k is z e ro , th e i-th part D o f a n iterative infinite higher derivation D =1./311 1 > 0 is e q u a l to (i !) (D i ) . Therefore an action of Ga o n Spec (A ) is equivalent to an a c tio n o f a derivation D o n A . In this case, w e h a v e th e following i

- i

i

result. Proposition 1 . 3 . S u p p o se th e characteristic o f k i s z ero . L et

(A , nt) be a noetherian integral local k-domain an d let D be a k-derivation in A such that f or any elem ent a o f A , D i (a )= 0 f o r all sufficiently

large integer j.

T h e n w e hav e D(m) g n i an d th e induced action of D

o n th e residue f ield k' A/111 is trivial.

194

Masayoshi Miyanishi P r o o f . Suppose the co n trary. T ake an element x of ut such that

D ( x ) in , i.e . D (x ) is in vertib le. T h en the following Lemma 1.4 shows t h a t A A o E x], the polynomial ring of one variable over A o , A o being the ring of D-constants. However since A is a local ring, this is con-

tradictory. T hus D (m ) g in. N e x t , le t x b e an element o f A —ru. I f D ( x ) ni, w e have a contradiction by L em m a 1.4. Hence D (x) E ni, i.e . the induced action o f D on k ' is trivial.

q.e.d.

Lemma 1 . 4 . (cf . [1 0 1 ). S uppose the characteristic o f k i s zero.

L e t A b e a noetherian, in teg ral k-domain an d D be a k-derivation A such that f o r any elem ent a o f A , D j(a)= 0 f o r all sufficiently large Suppose that there ex ists an elem ent x o f A such that D (x )

integer j.

is invertible. T h e n A

A

o

[ x ] , the poly nom ial ring of one variable over

th e rin g of D-constants Ao. P r o o f . Replacing D b y D '= (1 / D ( x ))D , w e c a n suppose D (x )

= 1 . Define a k-algebra homomorphism L : A

A

L(a) = a — x D (a) + (2 !) - 1 - x 2 D2 (a) —• • • = E i=c1

-

by xi D (a). i

This m ap is w ell d efin ed b y virtue o f th e hypothesis on the action of

D . W e have L - 1 (0) x A and /m (L )= A o . In fact sin ce L(x) = 0, it

is o b v io u s th a t L (x A )= O. C o n v e r s e ly if L (a)= 0 , th e n a= xD (a ) _ ( 2 ! ) -1 x 2D 2, a ‘) + • • • E x A . Therefore x A = L ( 0 ) . On the other hand, sin ce the restriction of L o n A o i s th e id en tity map, Irn(L )jj A o i s -1

o b vio us. H o w ever sin ce D (L (a ))= 0 , L ( A ) A 0 . H en ce Im (L) = A 0 . M oreover since L = L , w e have a — L(a) E x A fo r a n y a E A . Write 2

a— L (a )= x a i . T h en a= L(a)+ xa i = L(a)H- x L(ai) + x 2 a2 , w h ere xa2 = a — L(ai). Thus we can write a -= bo + bix + • • • + bn x"(mod x 1 A) for an y in teger n > 0 , w h e re bo , • , b„ E A 0 . T h e n it is e a sy to se e th a t (i !) bi = L(D` ( a ) ) f o r 0 < i < 71. H o w ever th ere exists a n in teg er N su ch th at D i (a )= 0 fo r a ll j> N . L e t aN = E (i !) - 1 L(D i (a)) x . Then l

i

i=o

a —aN E r\xn A and r\x" A is zero b y Krull's intersection theorem. Thus n >N

a= a . N

n >N

W e sh all p ro v e th at x is algebraically independent over Ao.

S om e rem ark s on the actions of the additiv e g ro u p sc h e m e s 195

Indeed, le t bo bi x ± • • -i - b i x i = 0 f o r bo , • • •, bi E A o . Then operating L , w e have bo = L ( b o b1 x + • • • +1)1 x ') - = 0 . Therefore we h ave 6 1 + • = O. T h e n b y induction on 1 , w e c o n c lu d e th a t b i =•••=b i bi x = 0 . T h e re fo re w e h ave sh o w n th at A = A o r x ] and th a t A i s the q . e . d . polynomial ring of one variable over A o . 1-

1

2 . H ere w e suppose again k is an algebraically closed field and let X b e a complete variety defined over k on which the additive group o f k acts. In the following, we denote the additive group G a (k ) o f k b y Ga, i f th e r e is n o fe a r of confusion. W e sh all recall so m e m ore or less well-known results on the Ga -actions on a complete variety. Lem m a 1 . 5 . L e t X be a com plete variety on w hich G a acts. T hen

X h as a fix ed point. P r o o f . Take any closed point x o f X an d let Y be the closure of the orbit o f x.

I f x is a fixed point, we are d o n e . If n o t, Y is a closed

subvariety o f X which is isomorphic to the projective lin e P 1 . H e n c e q.e.d. Y— (the orbit o f x ) = {a point} . This point is a fixed point. I f G a a c t s o n X , th en Ga a c t s on the k-rational function field K an d x E X . H ere we denotes a set of sub-local rings of K . If we denote can consider that X b y x A the im age of A b y the action of A for A E k and a local ring A o f K , w e have x 6x,x = O x ,x x - T h e lo c a l r in g o f a fixed point x o f X

o f X b y V ( x ) = f ((— 2 ) x ) , w here 2E k , f E K

is characterized as follow s; (1) ''ex,x -=.0x ,a, ( 2 ) in c n i m x b ein g the m axim al ideal o f Ox, a a n d (3 ) the induced action of Ga o n th e residue field k ( x ) is trivial. T herefore all the fixed points of X form a closed ,

set w hich w e denote by F and c a ll the fixed point locus. Lem m a 1 . 6 . L e t X be a com plete v ariety defined ov er k on w hich

Ga acts. T h e n the fix ed point locus F is connected. For the proof, w e refer to [21 o r [41. Lem m a 1 . 7 . L e t X b e an ab e lian v ariety def ined ov er k . T hen

th e re is no non-triv ial action of a linear group G o n X. P r o o f . L e t x b e an y clo sed p o in t o f X and consider a regular

196

M asayoshi M iyanishi

morphism 0, from G to X defined by A E G---> A x E X . T h en b y the

well known theorem o f a rational mapping from a group variety to an abelian v a rie ty , 0 x = 0 + x , w h ere 0 i s a homomorphism of G t o X. Since G is a linear variety, 0 is trivial. Therefore 0 x = x. q.e.d. T h e first part of the follow ing result w as given by T. Kambayashi [5 ] b u t h ere w e sh all give a simpler proof. C orollary 1 .8 . L e t G b e a connected linear group an d le t X be

a complete norm al v ariety def ined ov er k o n w hich G acts.

L e t D be a

d iv iso r o f X . T h en an y tran sf o rm o f D by an elem ent of G is linearly equiv alent to D. M oreov er i f G=G a a n d i f X is projective, then there ex ists a div isor D o w h ic h is stable under G , i.e . x D 0 =D 0 f o r any A of k an d is linearly equiv alent to D . P r o o f . Th e divisor classes which are algebraically equivalent to D

form a principal homogeneous space of the Picard variety P= Pic (X/ k) o f X . On the other hand, G a c ts o n th e s e t o f all divisors on X by (resp. D'), t h e n 'D (A , D)----->'D and if D — D ' (resp. D - " D ') . Hence G a c ts on the principal homogeneous space o f P in w hich th e d iviso r class o f D is c o n ta in e d . B y v irtu e o f Lemma 1.7, th is action is trivial. Therefore fo r an y d iv iso r D o f X , w e have 'D D . If G =G a an d if X is projective, consider a complete linear system o f divisors which are linearly equivalent to D . T hen since X D--, - D fo r a l l A Ek , G a a c t s o n D . L e t Y b e th e C h o w variety o f D . Then Ga a c t s o n Y . B y v ir tu e o f Lemma 1.5, Y h as a fixed point y. L e t Do b e th e divisor corresponding to y. Then "D o =D o f o r any q.e.d. E k and Do --- D. N ow w e can state the second theorem of Part 1. T h e o rm 1 .9 . L e t k b e a n algebraically closed f ield o f arbitrary

characteristic, (A , In) be an integral, local k -dom ain w hich is essentially of finite type ov er k an d le t K =Q (A ) be the quotient field of A . Suppose th at an iterativ e inf inite higher deriv ation D ac ts o n A s o th at N u t) g in f o r a l l i > 0 an d th at the f ollow ing condition is satisf ied;

S om e rem ark s on the actions of the additiv e g ro u p sc h e m e s 197 T here ex ist a projectiv e variety X defined ov er k o n w hich Ga acts

(*)

an d a fix ed point x o f X such that the local ring

2 X

,x

w ith th e induced

ac tio n o f Ga is o p erato r-iso m o rp h ic to (A , m ) w ith th e ac tio n o f Ga induced f rom D . T hen there ex ists v aluation rin g Y/' o f K w hich is inv ariant under D an d dom inating A. P r o o f . Passing to th e normalization S o f X , w e can suppose X .

is normal, hence (A , In ) is n orm al. L et D be a k-rational divisor o f X which passes through x and consider a complete linear system ID I and its Chow variety Y. L e t y be th e poin t of Y which corresponds to the divisor D. Then y is a closed point of Y . A s in th e proof o f Lemma 1.5, let z be the fixed point which is contained in th e closure of the orbit o f y . Then the divisor D o corresponding to z is stable under G. and passes through x . Since the components o f D o which pass through

x a re also stable under Ga , we m ay assume D o is irred u cib le. L et

p

be the prime ideal o f A which corresponds to the generic point of D o . Then

p

is o f height 1 and invariant under the action of Ga .

The local ring

A/p

also satisfies the condition (*). Therefore using

an induction argument as in the proof o f Theorem

1.1,

we can find a

valuation rin g of K w h ich is invariant under th e higher derivation D and dominating A.

q.e.d.

Part 2. 1.

We shall begin with some remarks on the construction of quo-

tient preschemes. L e t S b e a prescheme, G a S-group prescheme and

X be a S-prescheme on which an action of G is given by a S-morphism 6 : GX X—> X

. We shall consider the (f.p.p.0-topology in the category

of S-preschemes (Sch/S), where (f.p.p.0 means "faithfully flat and of finite presentation", (cf.

[11,

Exp. I V ) . Define a presheaf Q on (Sch/S)

by

TE (Sch/S)

Q (T )= X (T )/G (T ).

Here since the group G (T ) acts on th e set X ( T ) b y 6 ( T ), we define

M asayoshi M iyanishi

198

on X ( T ) by

an equivalence relation

x•-•--x g x = x ' fo r some g E G (T ). f

Then X (T )/G (T ) is the factor set of X ( T ) by this equivalence relation. L e t P be the sheafication of Q with respect to the (f.p.p.0-topology on (S c h / S ). P is called the (f.p.p.f)-quotient sheaf o f X b y G . I f P is represented by a S-prescheme Y, w e sa y Y a quotient prescheme of X b y G , (cf. D C ).

f : F— > H be a morphism of (f.p.p.0-sheaves on (Sch/S). We say f a (f.p.p.f)-surjection if for any T E (Sch/S) and for any element Let

b o f H ( T ) , there exist an open covering IT — > T} „ o f T in the a

A

sense of (f.p.p.f)-topology and a family o f elements (a.)aeA of such that bl

y

,

F a a)

=a a f o r any a E A , where b 7• is the restriction of b

o n Ta . L e t 7r : X —> P b e th e canonical projection. Then P and i t satisfy the following conditions :

(i)

7t.6 -=n•pr 2 , where pr i s the projection of G x X onto X.

(ii)

7r : X—> P and O p = (6 , p r2 )p : GX X—> X X X are (f.p.p.0-surjec-

2

tions. Conversely, if a (f.p .p .0 -s h e a f P ' on (S ch / S ) an d a m orp h ism of sheaves

: 1 3 ' satisfy th e same co n d itio n s a s (i) a n d (ii) for P

and i t , th en P ' and 7I a r e isomorphic to P and i t ,, i.e. there exists an '

isomorphism o f sheaves r: P — >P ' su ch th at Tc' =y •n . .

=

(o, pr 2) : G x X — > X x X is decomposed to G x

Moreover 0

X X X X

X

X

and j is a monomorphism of sheaves. We shall prove the following result. T h e o r e m 2 .1 . L e t S b e a noetherian preschem e, G be a S -group

preschem e w h ich is f lat an d o f f inite presentation ov er S an d le t X be a n integral S -preschem e o f f inite presentation o n w hich G ac ts v ia a S-morphism

o: G x

X — > Y . Suppose th at 0 =(o , p r 2 ): Gx X — > X x X is

quasi-finite an d th at if s q is th e im ag e o f th e generic p o in t e o f X in S , the characteristic of th e residue f ield k (71) is z ero. T hen there ex ists

S om e rem ark s o n th e actions of the additiv e g ro u p sc h e m e s 199 a dense open s e t W o f X s u c h th at W is G -stable and th at W h as a quotient preschem e by th e induced action of G. P r o o f . The proof consists o f three steps.

B y Gabriel ([1 1 , Exp. V , Théorèm e 8.1), we know that there exists a dense open subset V o f X satisfying the following conditions: (a)

(i)

V is G-stable, i.e . 6 (G X V) g V.

(ii)

T here exist a S-preschem e U o f finite presentation over S and

a S-m orphism Tr : V

> U such th at it i s a surjective open m orphism of

-

finite presentation and th a t n' .6 =lt.pr2. (iii)

T h e S-m orphism Ou= (6, prz)u: Gx V — * V x V is surjective.

(iv)

I f 0 : Gx V — > V x V is a monomorphism,

0 u

i s an isomorphism

and i t is fa ith fu lly flat.

Since X is in te g ra l, V and U a re in teg ral. T h erefo re b y the generic flatness theorem (EGA, I V (6 .9 .1 )), th ere exists a non-empty open set I f o f U su c h th a t 7r -1(E p ) : 7r - 1 ( U)— > U' is a flat m o r p h is m . Hence

w e c a n assume t h a t i t i s a fla t m orphism , rep lacin g V b y 7r ( U'). -1

N ext w e shall prove that i f w e replace U by its appropriate open s e t, th e n Ou : Gx V — > V x V i s a faith fu lly fla t m o rp h ism o f finite p resen tatio n . F irstly, sin ce 7r: V U is fa ith fu lly fla t a n d o f finite presentation and sin c e w e have supposed that the characteristic of the residue field k CO a t the image o f t h e generic point e o f X in S is zero, the generic fibre of it is geom etrically reduced (E G A , IV (4.6.2)).

Moreover the fibre G, is also geometrically reduced by Cartier's theorem ([1 1, E xp. V /B ). Let E= {yE U; 7r ( y ) = V y and G a re geometrically reduced} . Then E is a locally constructible set of U (EGA, IV (9.7.7)) and contains the generic point of U. ,

- 1

y

S in c e U i s a n o e th e ria n sp ace, E i s a constructible set (EG A , III (9 .1 .1 2 )), h en ce E contains a non-empty o p e n s e t U " o f U . Then replacing U b y U " , w e c a n assume th a t V , and Gy a r e geometrically reduced fo r a ll y E U. W e claim th at O u is a faithfully fla t m orphism o f finite presentation under these assumptions.

200

Masayoshi Miyanishi

( b ) L e t P b e the (f.p.p.0-quotient sheaf o f V b y G and Tr b e the '

canonical projection from V to P . A s shown above, P and rc' are then characterized by the conditions, .6 = • p r 2 .

(1)

n ' and ø p = ( û , p r 2 ) p : G x V --÷V x V are (f.p.p.f)-surjections.

(2)

is faithfully flat and of finite presentation and since 7r .6 = lr.pr , 77 is decom posed to V = - - P = 4 U , w here 7r i s a (f.p.p.0-surjection.

Since

7r

2

"

O n t h e o th er hand, 0 : G x V -> V x V is d eco m p o sed to G x V x V , w h ere Ou-=- i•Op and w h ere i and j are

V---

monomorphisms of (f.p.p.0-sheaves. L e t x b e a p o in t o f V and r : Spec (k (x))— > V b e th e canonical

is id en tified w ith a n elem ent ço D efine a (f.p .p .0 -sh ea f V „ o n (S c h / S ) b y V,

injection. T hen e•r : Spec(k(x)) P o f P(Spec (k (x ))).

= Vx (Spec (k (x)), ço). P

Precisely saying, V . is a functor such that for

a l l 7' E (Sch/k (x )), V ,( T )= 7 c (T ) (çoT), w here gOT is the image of ço b y the morphism P(r): P(Spec (k (x )) > P ( T ), w here r is the structure morphism o f T . T h en 0,-= O px (Spec (k (x )), r ) : Gs Ok (x)—> V . i s a k(s) V (f.p.p.0-surjection, where s is the im age of x in S. - i

- 1

—

L et F be the stabilizer group o f x , i.e . the fibre product o f 0 : G XV

—

>. V x V b y the m orphism (r, r): Spec (k (x))—> V x V . T h en it is

e a s y t o s e e t h a t V , i s a (f.p.p.0-quotient sh eaf o f Gs O k ( x ) b y F. k(s)

Here Gs O k ( x ) and F a r e group preschemes o f fin ite typ e over k (x ) k(s)

and F is a sub-group prescheme o f f iG s O k ( x ) . Hence V . is representable n ik(s) a n d 0 , is fa ith fu lly fla t a n d o f te p re se n ta tio n (D a V IA , (3.2)).

M eanwhile i x = i x (Spec (k (x )), r ) : V„—> V 0 k ( x ) is a surjective monoy

k(y)

morphism of (f.p.p.f)-sheaves because 0 : G s O k (x ) k(s)

-

Vy 0 k (X ) k(y)

is a surjection (cf. the condition (iii)) and because i is a monomorphism of (f.p.p.f)-sheaves.

W e claim that i x i s in fact an isomorphism. Firstly w e can assume th a t V . and Vy 0 k ( x ) are algebraic varieties defined over k ( x ) . In fact, k(y)

V . is geom etrically reduced fo r Gs O k (x ) is geometrically reduced and k(s)

S om e rem ark s on the actions of the additiv e g ro u p sc h e m e s 201

0 , is faith fu lly flat and of finite presentation. V y 0 k ( x ) is geometrically k(y)

reduced by the assum p tio n . On the other hand, in order to see that i x i s an isomorphism, w e can exten d ap p ro p riately the base field k(x). Moreover since the actions of Gs O k ( x ) o n V , and V y 0 k ( x ) are transikcy) k(s) tiv e, V , and V y 0 k ( x ) are norm al. Therefore the irreducible components k(y)

o f V , an d V y g k ( x ) a re th e connected components. Those connected k (y)

components are the trasitiv e transformation spaces under th e connected component (G sO k (x ))o of G s O k ( x ) a t the origin. Hence we can assume k(s)

k(s)

th at the irreducible components o f V , and V y 0 k ( x ) a re geometrically k (y)

irreducib le. T h us w e can assume t h a t V . and V y 0 k ( x ) are algebraic k( y)

varieties defined over k-(x). L et K , and K r b e th e function fields of V , and V y 0 k ( x ) . Then k(y)

K , i s a purely inseparable extension of K r b ecause i i s a dominating monomorphism (c f. E G A , I (3 .5 .8 )). W e sh a ll show th at K ,= K . In fact, assum ing th e contrary, le t L b e a subfield o f K , containing K r su ch th at K „ i s a simple extension L (a) o f L , w here a is an element r

-

r

o f K , such that aP E L, p b ein g th e characteristic. T hen there exists a L-trivial derivation D in K , . L et / K ,- =K ,[t ]/ ( t 2 ) and / L = L i t 1 / ( t 2 ).

b y a ( 2 ) = A. D (2 )1 for 2 E Define a IL-automorphism a of where is the residue class of t . T h e n the following diagram is commutative, Spec ( I K ) - - > Spec (L)

Spec (K r )

°at Spec (1 e ) -

I f C : Spec (IK ) — *V is a morphism defined by th e canonical K„-homo-

morphism

w

e

have C * C •act and ix • C = ix • C - u a . A s i r i s a

monomorphism o f (f.p.p.f)-sheaves, w e have C = C • a a . T h is i s a contradiction. H ence K ,= K r . S in ce V „ and V y ( g k ( x ) are transitive under k (y)

G s 0 k (X ) k(s)

and since i r commutes w ith the actions of G ,O k (x ), ix i s an

isomorphism.

k (s)

202 (c)

M asayoshi M iyanishi N ow w e shall prove that Ou: Gx V — > V x V is faithfully flat and

o f finite presentation. I f w e regard G x V a n d V x V a s V-preschemes b y th e se c o n d p ro je c tio n s, G x V a n d V x V a r e o f finite presentation o v e r V . H ence O u i s a morphism o f finite presentation (EGA, IV

(1 .6 .2 )). Since Ou i s su rje c tiv e (c f. th e c o n d itio n (iii)), w e h a v e only to prove that Ou is flat. F o r this, it is sufficient to prove t h e following results by virtue o f (E G A, I V (11.3.11)) ;

( * ) p r 2 : G xs V —> V is fiat. ( * * ) F o r e v e ry x E V,

(x)—> V .,,O k ( x ) is f la t, where Y k(y) a n d s a r e th e im a g e s o f x in U an d S respectively. However ( * ) is (Dr : G ,O k k (s)

obvious f o r G is flat over S.

A s f o r ( * * ) , since O x is decomposed to

V y 0 k (x) a n d since 0 , is f la t a n d i r i s an

Ox : Gs Ok k (s)

k (y )

isomorphism a s shown i n ( b ) , 0 , is flat.

Therefore Ou is a faithfully

fla t morphism o f finite presentation. Hence U is a q u o tie n t prescheme o f V b y G , b y t h e remark preceding th e statement o f th e theorem. q.e.d. R e m a r k s . 1 ) I n t h e statement o f our theorem , t h e assumption that th e characteristic o f th e residue field k ( .) at th e im a g e p o in t 7? of

t h e generic point

e

o f X is only to ensure that th e locally constructible

s e t E in t h e proof is not em pty. I f (D is a monomorphism, this condition is unnecessary by th e c o n d itio n (iv ) in th e proof.

2)

T h e morphism = (6, pr 2 ): Gx X — > X x X is quasi-finite if and only

i f f o r an y p o in t x o f X t h e stabilizer group S t ( x ) is a finite group sch em e. H ere S t ( x ) is defin ed a s t h e f i b r e product o f 0 b y the

morphism (r, r) : Spec (k (x )) —>X x X ; r being th e canonical injection. 3)

It is not necessary to a ssu m e that 0 is quasi-finite.

F o r our con-

clusion, it is sufficient only to a ssu m e that there exists a p o in t whose stabilizer group is finite. I n d e e d , l e t IF1=-

E G x X ; dimz 0 - 1 ((D(z))

= 0 )-. T h e n W 1 is a n o p e n s e t o f G x X (EGA, IV (13.1.3)). I f we put

W =pr 2 (

W is a n o p e n s e t o f X such that a p o in t x o f X belongs

t o IV if a n d on ly i f S t (x ) is finite. T h e r e fo r e IV is G-stable a n d TV1

S om e rem ark s on the actions of the additiv e g ro u p sc h e m e s 203 =G X W . I t is n o w e a s y t o s e e t h a t O l w is quasi-finite (E G A , II (6.2.2)). 2 . L et k b e a f ie ld o f characteristic zero a n d le t V b e a variety de-

fined over k on w hich the additive group or the m ultiplicative group of th e universal dom ain 2 o f k acts n o n -triv ially. T h en th ere ex ist a geom etrically irreducible, reduced algebraic k-schem e X and an action of the additive group scheme Ga or the m ultiplicative group scheme Gm defined by a k-m orphism 6 : Ga x X (or Gni x X)— > X s u c h th a t X(S2) =V and that 6 ( 9 ) gives the action of the additive group o r th e multiplicative group of S2 o n V . T h e actio n 6 i s non-trivial, hence by the above remark 3 ) , w e c a n assum e th a t 0 = (6, pr o) : Ga x X (o r Gm x X ) —› X X X is quasi-finite since G , and Gm a r e 1-dimensional over k. Therefore Ga (or G m ), X and Spec (k ) satisfy th e assumptions for G , X and S in Theorem 2.1. T h u s th ere ex ists a d en se o p en su b se t W o f X su ch th at W is Ga ( o r Gm )-stable and th a t W h as a quotient prescheme U o f W b y th e induced actio n o f Ga ( o r Gm ). L e t 7r: U b e the can o n ical projection. T h e n w e h a v e rt.0-=7.pr 2 a n d 7r a n d Ou : Ga x W (or G x )--> Wx W are faithfully flat and of finite presentation. L e t K o b e th e function field of U , i.e . the quotient field of the local m

r in g Ou,„ a t th e gen eric point u o f U . T h e n it is e a sily se e n th a t 7 K : W K , — >Spec(Ko) is a Ga ( o r Gm )-hom ogeneous space and th at the function field of W K , i s th e function fie ld o f W . In fact, W K , h a s a 0

K -rational p o in t, b y v irtu e o f th e well-known theorem o f Rosenlicht ( [ 9 ] ) . Hence th e function field K o f W K , is isom orphic to K o (t ), t being a n indeterminate. 0

T hus w e have shown T h eo rem 2 . 2 . L e t k be a f ield of characteristic z ero and let V be

a variety defined over k on w hich the additiv e group or the m ultiplicativ e group o f th e univ ersal dom ain acts non-triv ially . L e t K be the f ield of k-rational f unctions o n V an d le t K o b e the f ield of constants in K by

t h e induced ac tio n o f th e ad d itiv e g ro u p or the m ultiplicativ e group. T hen w e hav e K = K 0 ( t ) , t being an indeterminate.

204

Masayoshi Miyanishi Theorem 2 . 3 . T heorem 2 .1 is als o v alid if the condition on the

characteristic of k (v ) is replaced by the f ollow ing condition:

G , is geom etrically reduced an d geom etrically irreducible. P r o o f . In fact, w hen w e assume th at the characterisic o f lc(77) is zero, it is o n ly to e n su re th a t the fibre G , and the generic fib re o f n .

a re g eo m etrically red u ced . H en ce w e sh all see th at th is h o ld s tru e

under a new assumption. L e t L and K b e th e fie ld s o f rational functions o n V a n d U

resp ectively. T h en the generic fib re o f 7r,

/

K = ' V O K , i s an integral

alg eb raic sch em e o v er K a n d O K : GX V K —> V K X V K i s surjective. le 0 )

Therefore GI( =-.G ,O K acts tran sitiv ely o n V K . H ence, VK is normal. 1“ fl ,

L e t K , b e an algebraic separable closure of K . T h en V K is geome-

trically reduced over K if a n d o n ly i f V K O K , is so o v e r K ,, (EGA, I V (4 .6 .1 0 )). O n th e other h an d, th e set of 1( -rational points of G, 3

G (K ,), is dense in G O K ,, since G O K , is geom etrically reduced and

geo m etrically irred ucib le. T h erefo re w e can assume th a t th e set o f K-rational points, G (K ), is dense in GK. L et K 1 b e the ring of constants in L under the induced action of

G ( K ) . Then K 1 is a lg e b ra ic o v e r K becase G K a c ts tra n sitiv e ly on V K . S in ce

VK

is normal and sin c e U i s th e quotient space o f V by

G in th e catego ry o f rin ged sp aces (cf. Gabriel, ibid., this property

h o ld s u n d e r a n y fla t b a se ch a n ge U' U .) , K c o in c id e s w ith K. Moreover K is algebraically closed in L because GK is connected. 1

N ow w e shall show th at L is separably generated over K . L et K' b e a purely inseparable algebraic extension of K such that the composite field L(K ) is separably generated over K ' . T ake an y sub-extension K " o f degree p in K ', p being the ch aracteristic. I f L ® K " is not reduced, 1

L (K " )= L . Then K " is contained in L . This is a contradiction because K is algeb raically clo sed in L . Therefore L O K " is red u ced and iso-

K morphic to L ( K " ) . On the other hand, K " is the ring of constants in L O K " under th e action of G ( K " ) . N ow proceeding by induction on

S om e rem ark s on the actions o f the additiv e g ro u p sch e m es 205

th e degree [K ': K ], w e k n o w th a t L O K '= L (K '), i.e . L is separably closed. Therefore VK is geometrically reduced over K (E A G , IV (4.6.1)). q.e.d.

U sing this remark * , w e can drop the assumption on the characteristic o f k from Theorem 2.2. Theorem 2 .2 is , in tu rn , generalized as u su al for the case w h ere a k-solvable algebraic group G acts o n V so (

)

th at the stabilizer group S t ( x ) is fin ite at least on one point x o f V. Bibliography

[1]

[2]

M. Démazure et 64, Schémas en

A . Grothendieck, Séminaire de géométrie algébrique, 1963/ gro u p es, Exp. IV , V , VI , VI B , In st. H autes Études Sci., mimeographed notes. J. F o garty, A lgeb raic fam ilies o n algeb raic surfaces, A m er. J . M ath ., 90 A

(1968), 510-521.

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A. Grothendieck et J . Dieudonné, Éléments de géométrie algébrique (EGA), Publ. M ath . In st. Hautes Études Sci., No. 4, 8, 11, 20, 24, 28. G. Horrocks, F ix ed point schemes o f ad d itiv e gro up actions, Topology, 8

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T . Kambayashi, Projective rep resen tatio n o f algeb raic lin ear gro u p s o f

(1969), 233-242.

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transformations, Amer. J. M ath., 88 (1966), 199-205. M . Miyanishi, G e actions on the affine plane, to appear in Nagoya M ath. J. M . Nagata, Local rings, Interscience Publisher, New York. M . Raynaud, P a ssa g e a u q u o tie n t p a r u n e re la tio n d'équivalence plate, Proc. o f a conferance on local fields a t Driebergen, Springer Verlag, 1967. M . Rosenlicht, Som e basic theorem s on algebraic groups, A m er. J. Math., 78 (1956), 401-443.

[10] J . Lipm an, Free derivation modules on algebraic varieties, A m er. J. M ath., 87 (1965), 874-898.

(* )

T he author is indebted to M . Maruyama for valuable suggestions.