Commutative Algebra and Algebraic Geometry

Roceedin@ of the International Mathematical Conference L.H. Y. Chen, T.B. Ng, MJ. Wicks (eds.) 0North-Hdlandhblishing C o m p y , 1982

125

COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY Masayoshi Nagata Department o f Mathematics Kyoto U n i v e r s i t y Kyoto Japan

0. INTRODUCTION.

In recent years algebraic geometry has been studied in a very abstract manner and this has brought it closer to commutative algebra. On the other hand, in order to learn commutative algebra it is often useful to look at the geometry which underlies its problems. For this purpose, the classical approach to algebraic geometry is helpful. Here we shall discuss some basic notions of algebraic geometry and their connection with commutative algebra, especially the connection with polynomial rings. A ring will always be a commutative ring with 1. We begin with the following remark:

...,

Let k, K be the rings with k a subring of K, and let XI, Xn be variables (i.e. . .,XJ commuting indeterminates) over k. A polynomial f (Xl, ,Xn) from k [X,. defines a mapping of Kn into K such that each (bl,. ,bn) of Kn is mapped to Addition in the polynomial ring k [X1 f(bl ,..., b,). Xn1 is defined by the b,) to the operations of k so that the sum of two polynomials will map (bl, sum of the images of (bl, b,) under these polynomials. There is a similar result for multiplication and products.

. ..

.. ,...,

...,

.

...,

One sees immediately from the definition of a polynomial ring that properties of k will have a lot to do with properties of k [X1, Xn] ; some representative examples of this will be discussed in the first half of 51.

...,

In the second half of 8 1 we review results which are related to Noether's normalisation theorem. Although it is important to study polynomial rings without any restriction on the coefficient ring k, it is also true that the most basic case occurs when k is a field. Here we feel that the ring k [X1, X], has rather simple structure. This does not mean that questions are easy if they are related to polynomial rings

...,

M.Naga ta

126

over a field : there are difficult problems which are due to the simplicity of the structure o f polynomial rings. Thus we shall give most attention to the case where k isa field. This corresponds geometrically to affine n-space (i.e. ndimensional affine space) over k. Again, we may feel that affine spaces have simple structure and are more easily understood than projective spaces. But the geometry of affine space is more understandable via the geometry of projective space. Some topics related to this fact will be discussed in 5 2 . As an example of the complexity of phenomena related to polynomial rings, or equivalently to rational varieties, we shall discuss in 13 the birational correspondences of ruled surfaces. The discussion is further related to the automorphism groups of polynomial rings and Cremona groups. These are considered in 14 and 1 5 . Finally, in 16, we discuss group actions on an affine ring. A list of principal references is given at the end. 1. ELEMENTARY PROPERTIES OF POLYNOMIAL RINGS

.

Consider the polynomial ring F = klXl,. ., over a ring k. Then properties of k are carried over to or reflected in F. We observe some typical examples of such properties of k.

3

The first example concerns zero-divisors. Since k may be taken as a subring of F, it is obvious that zero-divisors of k are zero-divisors of F. In the converse direction there is THEOREM 1.1. If f is a zero-divisor in F, then there is a non-zero b in k such that bf = 0 (i.e. all coefficients of f are annihilated by b). Moreover, if 0 g e F is such that fg = 0, then b may be chosen in the ideal o f k generated by the coefficients of g.

+

PROOF. We order the monomials X;l.. . Xen by the lexicographical order of the n series of exponents; thus, dl ... xnd, x;1..* x,"" > x1 if either e > dl, or there is j > 1 such that e =di for i < j and e > d i j j' Suppose f, g are as stated. If f = O , there is nothing to prove, so we suppose f # 0. We then have s,t > 0 such that s

f=

1

i=l

t

aimiy g =

1 b.n., j=1 J J

where ai, b. are non-zero elements of k and mi, n. are monomials such that 1

1

127

Commutative algebra and algebraic geometry ml

>

... >

mS

and n1 >

... > nt '

The proof i s by induction on s + t . I f s t t = 2, we have s = t = 1 and 0 = fg = a b m n 1111'

Hence albl = 0 and we may t a k e b = b l . Now suppose s + t > 2 and assume t h e r e s u l t holds f o r products with "fewer terms". The leading term of f g i s alblmlnl

and we must again have albl = 0 .

The cases

( i ) a g # 0, and ( i i ) alg = 0 a r e taken s e p a r a t e l y . 1 (i)

Since albl = 0, a g has fewer terms than g . 1

Further

f a 1g = a 1f g = 0, so t h e induction hypothesis a p p l i e s .

Hence t h e r e i s b i n k with bf = 0, b # 0,

and b is i n t h e i d e a l of k generated by t h e c o e f f i c i e n t s of alg.

The i d e a l

generated by alb *,..., albt i s contained i n t h e i d e a l generated by t h e c o e f f i c i e n t s o f g, s o b i s i n t h e l a t t e r .

-

a m we g e t ( i i ) We now have a g = 0. Taking f = f 1 1 11 0 = f g = (fl+alml)g = f l g + mlalg = f l g . The induction hypothesis now a p p l i e s t o f l , g t o give b, 0 bf

1

= 0 and b i s i n t h e i d e a l generated by bl,...,

a b . = 0, 1 1 1

5

5 t , and hence a 1b

j

= 0.

bt.

# b

e k , such t h a t

Since alg = 0, we have

I t follows t h a t

bf = b(fl+alml) = bfl = 0 .

Q.E.D.

The following i s an easy consequence. I f k i s an i n t e g r a l domain, then F = k[X

COROLLARY 1 . 2 .

lJ...JXdi s a l s o an

i n t e g r a l domain. This i s equivalent t o COROLLARY 1.3.

I f P i s a prime i d e a l i n k, then PF i s a prime i d e a l i n F, where

PF i s t h e i d e a l of F generated by P.

.,Yn]

PROOF.

Since k / P is an i n t e g r a l domain, we have (k/P) [Y1,..

domain.

I t i s easy t o see t h a t t h e l a t t e r and F/PF a r e isomorphic.

i s an i n t e g r a l Thus F/PF i s

an i n t e g r a l domain and PF is a prime i d e a l . We s t a t e without proof t h e following t h r e e r e s u l t s which a r e very familiar; they appear i n most o f t h e standard t e x t books.

THEOREM 1 . 4 . k[X]

I f k i s a unique f a c t o r i s a t i o n domain, then t h e polynomial r i n g

i s a l s o a unique f a c t o r i s a t i o n domain.

Consequently, F = k[Xl,.,.,X,,]

is a

128

M. Nagata

unique f a c t o r i s a t i o n domain THEOREM 1.5.

I f k i s a f i e l d , k[X]

is a unique f a c t o r i s a t i o n domain.

THEOREM 1.6.

I f k is a noetherian r i n g , then F i s a l s o noetherian

We r e c a l l t h a t a r i n g i s noetherian i f it s a t i s f i e s t h e ascending chain c o n d i t i o n . DEFINITION.

A normal domain is an i n t e g r a l domain which i s i n t e g r a l l y closed i n

i t s f i e l d of f r a c t i o n s .

We n o t e t h a t i n t h e c a s e k i s a f i e l d t h e f a c t t h a t k i s i n t e g r a l l y closed i s t h e same a s k being a l g e b r a i c a l l y c l o s e d . A basic r e s u l t i s

I f k i s a noetherian normal domain, t h e n P also i s a n o e t h e r i a n

THEOREM 1 . 7 .

normal domain. In o r d e r t o prove t h e theorem, besides Theorems 1.5 and 1.6, we a l s o need I f k i s a noetherian normal domain, then

THEOREM 1 . 8 .

(i)

f o r every prime i d e a l P o f height 1, t h e r i n g k i s a d i s c r e t e v a l u a t i o n r i n g ; and

P

= Ia/b

I a,b

e k

6 b 6 P 1

( i i ) k i s t h e i n t e r s e c t i o n of a l l such kp. (Here, P i s of height 1 ( i n an i n t e g r a l domain) means t h a t i f Q i s a prime i d e a l such t h a t 0 C Q

LP, then

either Q = 0 or

4

= P.)

The proof of t h i s theorem, which helps t o c h a r a c t e r i s e t h e normality of a noetherian domain, w i l l be found i n any standard t e x t book and i s omitted. For t h e proof o f Theorem 1.7 we begin with Gauss’ Lemma : LEMMA 1.9. PROOF.

If k i s a unique f a c t o r i s a t i o n domain, then k i s a normal domain.

Assume t h a t f / g ( f , g i n k and g # 0) i s i n t e g r a l over k .

a r e d > 0 and c l , . . . ,

(f/dd d- 1 Hence f d + c l f g + ,

+

..

cl(f/g)

+ cdg

a f a c t o r of f d , and then of f .

d-1

= 0.

+

... +

Cd

= 0.

I t follows t h a t any prime f a c t o r o f g i s

But we may assume t h a t f , g have no common prime

f a c t o r , so it follows t h a t g has no prime f a c t o r s , i . e . g i s a u n i t . We can proceed now t o

Then t h e r e

cd i n k such t h a t

Q.E.D.

Commutative algebra and algebraic geometry PROOF OF THEOREM 1 . 7 .

Let P be a prime i d e a l o f k o f height 1 and consider This i s a unique f a c t o r i s a t i o n domain by Theorem 1.5; hence

.,XA.

Ff’ = kp[X1,..

a normal domain by Lemma 1 . 9 .

. ., X A

F = k[X1,. F;.

Let f be i n t h e f i e l d o f f r a c t i o n s o f Since F* i s normal , f i s i n P Q.E.D.

and suppose f i s i n t e g r a l over F .

Thus f e n F ; ,

129

and by Theorem 1.8, t h e l a t t e r i s F .

Now we d i s c u s s some r e s u l t s t h a t a r e c l o s e l y r e l a t e d t o t h e famous theorem known as t h e normalisation theorem of Noether. This r e s u l t i s s t a t e d a s THEOREM 1.10.

a field k.

(i)

zl,.

Let t h e r i n g R be generated by a f i n i t e number of elements over

Then t h e r e a r e elements z l , . . . , z t

.., z t

of R such t h a t

a r e a l g e b r a i c a l l y independent over k , i . e . k[zl

n a t u r a l l y isomorphic t o k[X1,.

. . , X t ] ; and

. . ,zt] .

( i i ) R i s i n t e g r a l over k[zl,.

Moreover, i f R = k[X1

given non-zero i d e a l i n R, then we may choose z1 from P.

,.. . , z t]

,.. .,xnj

is

and P i s a

The proof r e q u i r e s some p r e l i m i n a r i e s . LEMMA 1.11.

f b k. (i)

Let f be a polynomial i n F = klX1,.

Then t h e r e a r e elements Y 1 , . . . , Y

Y1 = f ; and

( i i ) F i s i n t e g r a l over R = k[Y1,

n

. ., X A

, where

k i s a f i e l d and

i n F such t h a t

...,Yn].

Furthermore, (1) f o r any given p o s i t i v e i n t e g e r q we can choose mi, q, i n such a way t h a t

mi Y. = X . + X1 1

1

,i

(2) I f k i s i n f i n i t e , we can choose c

Y. = X . + ciX1, 1

PROOF.

1

= 2,...,

m u l t i p l e s of

n.

i n k so t h a t i i = Z,..,, n.

Let q be given and t a k e t a m u l t i p l e of q such t h a t t > degf.

mi = For each monomial M = X

dl

ti-l

,

mi

Yi = X i +X1

,

i = 2,...,

.. . Xndn d e f i n e t h e weight

Now s e t

n.

of M t o be w(M) =Cdimi. Note t h a t

t h e ordering o f monomials by t h e i r weights coincides with t h e lexicographical order of t h e exponents i n t h e i r r e v e r s e o r d e r , i . e . as dn,

..., d l ’

We now l e t M be t h e monomial of g r e a t e s t weight t o occur i n f , so t h a t f = cM + monomials o f l e s s e r weights, c e k .

I f g i s t h e polynomial i n X1,

g = f(X1, Y2 then we see t h a t

Yn such t h a t

Y2,...,

~~

-xl , ..., Yn m2

X’I”

1,

M.Naga ta

130

g = ( - l ) m cXycM) + terms of lower degree i n XI where m = d

2

.. .+

t

Introducing Y1 = f , d i v i d i n g through by ( - l ) m c and

dn.

rearranging, shows t h a t X1 i s i n t e g r a l over R = k[Y1,..., Further, s i n c e X. = Y i

i

Y.I = X.1

c.X

t

+

c

11'

e k.

i

..., Yn;

it i s of t h e form

terms o f lower degree i n XI,

where a i s ( t h e value o f ) a polynomial i n c 2 , . . . , c n . may choose t h e ci s o t h a t a # 0.

Set h = height I .

The e a r l i e r argument can now be r e p e a t e d .

...,Xn]

,.. .,Yn]

;

(3) Yh+i = \+i + gi with g. e PIXl, k # 0, gi e P X],':

[ X p , ...,

t h e r e a r e Y;,

R' = k[Y;,

Let I be an i d e a l

.. ., YA

. . .,%la

,.. .,Yh;

s a t i s f y i n g corresponding c o n d i t i o n s f o r 1'. Then I '

n R'

and

In case p = t h e c h a r a c t e r i s t i c of

Take an i d e a l I ' such t h a t I ' G I , h e i g h t I' = h - 1 .

..., Y; ] .

= IiChY;R'.

Then h e i g h t ( I

n R')

1

1

Then

Set

,

= h -1

Therefore, t h e r e i s an element f e I n R ' such t h a t f # I ' n R ' .

height ( I n R I ) = h . Considering f ( 0 , .

o f F such t h a t (1)

(2) I fRli s generated by Y1

If h = 0, then I = 0, and we may s e t X. = Y . .

We employ induction on h.

5 1.

Q.E.D.

over a f i e l d k, and l e t P be t h e prime

Then t h e r e a r e elements Y 1 , . . . , Y n

F i s i n t e g r a l over R = k[Y1

PROOF.

Since k i s i n f i n i t e , we

(Normalization theorem for polynomial r i n g s )

of t h e polynomial r i n g F = k[X,,

Assume t h a t h

.

Here we s e t

Now express f as a polynomial i n X 1, Y2,

field.

ynl

mi XI , we s e e t h a t F i s i n t e g r a l over R .

-

For (2), we now assume k i s i n f i n i t e .

THEOREM 1 . 1 2 .

,

. . , 0, Y i ,.. ., Y i ) , we

may assume t h a t f e F' = k ( Y i , ,

. .,YA] .

Apply Lema 1.11 t o t h i s f and F ' , and we s e e t h e e x i s t e n c e of Yh = f , Y h + l J . . . ,

.,Yn] and Yh+i + Y; ( i 21 1. Set .., YnJ we s e e (1) and (3) immediately ( i n

Yn such t h a t F ' i s i n t e g r a l over k[YhJ.. Yi = Y;

for i < h .

p # 0, we choose m

Then, with Y1,. i

Iich YiR which i s a I i-h YiR. Q.E.D.

t o be m u l t i p l e s o f p ) .

Set zi = $Y

A s f o r (2), s i n c e I n R c o n t a i n s

prime i d e a l of height a t least h, we s e e t h a t I n R =

PROOF OF THEOREM 1.10.

polynomial r i n g k[X1

case

Let I be t h e kernel o f t h e n a t u r a l s u r j e c t i o n J, of t h e

,.. .,Xn]

. ( i =l,..., h+i

Then we have Y1

t o R. n -h).

,... ,Yn

Then R = $(k[X l J . . . , X n l )

1.

as i n Theorem 1 . 1 2 . i s i n t e g r a l over

I f g e k[ZIJ...JZn-h] (polynomial r i n g ) and J,(k[Y lJ...,Yn]) = k[z l J . . . J z n-h i f g(z l J . . . J zn-h ) = 0, then g(Yh+lJ...J Y ) e I , and g = 0 by (2) i n Tneorem 1.12.

Q.E.D.

131

Commutative algebra and algebraic geometry THEOREM 1.13.

Assume t h a t R i s an i n t e g r a l domain generated by a f i n i t e number

of elements over a f i e l d k . Po = 0, ( i i ) P

t

I f Po, P 1 , . . . ,

Pt a r e prime i d e a l s such t h a t ( i )

.

is maximal, ( i i i ) PoC P1 E.. G Pt and ( i v ) t h e r e i s no prime i d e a l

Q such t h a t P i - l c Q C P i for any i , then t must be t h e transcendence degree o f R over k .

PROOF.

We proceed by induction on t .

independent elements z l , . . . , z

.

...,

By Theorem 1.10, t h e r e a r e a l g e b r a i c a l l y

over k such t h a t R i s i n t e g r a l over

I f t = 0, then R i s f i e l d .

k[zl, zu] over F implies t h a t F is a f i e l d , and then u = 0.

Assume now t h a t t > 0.

l a s t statement of Theorem 1.10, we may assume t h a t z1 e P1. 1

which is i n t e g r a l over F/zlF

prime i d e a l s Pi/P1. COROLLARY 1.14.

By t h e

Since R i s i n t e g r a l

n F = z1F . Then we apply 1 2 k[z2, z , and t o t h e U

over F, height (P n F ) = l J 1 and t h i s implies t h a t P our induction t o R/P1,

F =

Then t h e i n t e g r a l dependence of R

Thus we have t - 1 = u - 1 and t = u .

..., 1 Q.E.D.

I f an i n t e g r a l domain R i s f i n i t e l y generated over a f i e l d k ,

then f o r any prime i d e a l P of R, i t holds t h a t height P + Krull dim R/P = Krull dim R = t r a n s . degkR. I f a r i n g R i s f i n i t e l y generated over a f i e l d k and i f M i s a

COROLLARY 1.15.

maximal i d e a l of R, then R/M i s a l g e b r a i c over k . I f a r i n g R i s f i n i t e l y generated over a f i e l d k and i f I i s an

WEOREM 1.16.

i d e a l of R, then t h e i n t e r s e c t i o n o f a l l maximal i d e a l s containing I coincides with t h e r a d i c a l

0 of

I) = { f e R l f m

I f o r some m))

PROOF.

B

I (

.

( t h e i n t e r s e c t i o n o f a l l prime i d e a l s c o n t a i n i n g ( H i l b e r t zero-point theorem.)

I t s u f f i c e s t o show t h e case where I i s prime.

i t s maximal i d e a l

MI.

Then considering R/I, we

Let f be a nonzero element o f R and consider R [ l / f ]

may assume t h a t I = 0. over k .

fi =

R[l/f]/M'

i s a l g e b r a i c over k , and R / ( M ' n R )

Thus M ' n R i s a maximal i d e a l o f R not containing f .

and

i s algebraic Q.E.D.

We add here some o t h e r important a p p l i c a t i o n s o f t h e normalization theorem.

For

t h e purpose, we f i r s t prove some f a c t s on derived normal r i n g s . LEMMA 1 . 1 7 .

Let R be a noetherian r i n g with t o t a l q u o t i e n t r i n g Q.

An

R-submodule M of Q i s f i n i t e l y generated i f and only i f t h e r e i s a nonzero-divisor a of F such t h a t aMG R .

This follows from a r e s u l t which a s s e r t s t h a t i f an i n t e g r a l domain R i s i n t e g r a l over a normal i n t e g r a l domain F, then f o r any i d e a l I o f R , it holds t h a t height I = height ( I n F ) . This r e s u l t i s a c o r o l l a r y t o t h e Going-down Theorem.

M.Nagata

132

I f p a r t : M is contained i n a-1R which i s f i n i t e R-module, and hence M

PROOF.

i s noetherian.

The only i f p a r t i s easy.

Q.E.D.

THEOREM 1.18.

Let R be a normal r i n g , f(X) a monic polynomial over R, a a r o o t of f(X) i n an extension f i e l d , and R* t h e derived normal r i n g of R[a] Let f v ( X )

.

be t h e d e r i v a t i v e of f(X), and l e t D be t h e d i s c r i m i n a n t o f f ( X ) .

Then we have

D R * C f ' ( a ) R * S R[a]. PROOF.

Let t h e r o o t s o f f(X) be u1 = a , U 2 , . . . , ~ r

( r =deg f ) , and s e t gi(X) =

li,l

Therefore (ul - u i ) . Then f'(X) = lgi(X) and f ' ( a ) = gl(a) = I t s u f f i c e s t o prove t h e i n c l u s i o n f ' ( a ) R * C R[a] assuming t h a t

f(X)/(X - u i ) . DR*E f ' ( a ) R * .

f(X) i s i r r e d u c i b l e . a i s separable.

I f a i s i n s e p a r a b l e , then f v ( a ) = 0.

Hence we assume t h a t

Let T be t h e l e a s t Galois extension o f R c o n t a i n i n g R*, l e t G be Let u1 = 1,

t h e Galois group, and l e t H be t h e subgroup corresponding t o R*.

..., u r

u2,

be elements of G such t h a t

Let b be an a r b i t r a r y element o f R*.

1'

= ui.

Then G i s t h e d i s j o i n t union o f

Write g,(X) = c r - l ~ r - l +

k i . gi(x) = gl(x)'i.

liDjbUicjuiaj =

Zi

l j ( libui

... +

c

( c ~ =- ~1; ci e R[al).

Then b f ' ( a ) = bgl(a) =

I:=,

U'

b 'g. (a) =

Since b, c . a r e H-invariants, we s e e t h a t 3 i s G-invariant and hence i s i n R. Thus b f ' ( a ) e R(a1. Q.E.D.

buicjui

COROLLARY 1.19.

cgi)aj.

Let R be a noetherian normal r i n g , K i t s f i e l d of f r a c t i o n s ,

and l e t L be a f i n i t e s e p a r a b l e a l g e b r a i c extension f i e l d of K.

If a ring R ' ,

such t h a t R C. R'L L, i s i n t e g r a l over R , then R ' is f i n i t e as an R-module. Let a e L be such t h a t L = K(a) and l e t c o , . .

PROOF.

coan +...+

= 0 (co f 0 ) .

c

a i s i n t e g r a l over R.

Then, t a k i n g coa i n s t e a d of a , we may assume t h a t

normal r i n g R* of R[a]

i s a f i n i t e R[a]-module, Q.E.D.

['l

...,zt D

'* *

D

hence over R , t o o .

Thus R ' is a

In Theorem 1.10, i f R i s an i n t e g r a l domain which i s s e p a r a b l e over

LEMMA 1.20.

zl,

i n R be such t h a t

Then Lemma 1.17 and Theorem 1.18 show t h a t t h e derived

noetherian R-module.

k (i.e.,

., c

t h e f i e l d of f r a c t i o n s o f R i s separably generated over k ) , then t h e can be chosen so t h a t R i s separably a l g e b r a i c (and i n t e g r a l ) over 't]

'

For t h e proof, t h e r e a d e r i s assumed t o know some b a s i c f a c t s on d e r i v a t i o n s PROOF, al,...,an

I t s u f f i c e s t o prove t h e c a s e where t h e c h a r a c t e r i s t i c p # 0 . be a s e t o f g e n e r a t o r s f o r R over k such t h a t a,,+l,...,

s e p a r a t i n g transcendence base.

Then we apply t h e proof o f Theorem 1.10.

c o n s t r u c t i o n of Yi i n Theorem 1 . 1 2 , we see t h a t ar+l P being t h e prime f i e l d .

Let

an form a

-

P zi e P(al

,

. ..,

P ah]

By our

,

Therefore, f o r every d e r i v a t i o n D o f k ( a l,..., a,,),

we

Commutative algebra and algebraic geometry

133

.

..

we have Dah+i = Dz. ( i = 1, . ,n -h; n-h = t .) This shows that if D is a derivation over k(zl, zt), then D is derivation over k(ah+l,,..,an). Since form a separating tanscendence base, it follows that D = 0, and ah+l,..., a zl, ..., zt are as required. Q.E.D.

...,

Now we come to another application of the normalization theorems. THEOREM 1.21. Let R be a finitely generated integral domain over a field k, and let L be the field of fractions of R. Then the integral closure R* of R in a finite algebraic extension field L' of L is a finite R-module, and hence R* is finitely generated over k. If L ' is separably generated over k, then we choose zl, ..., zt as in Lemma 1.20, and we prove the assertion by Corollary 1.19. In the general case, we choose z1, . , zt as in Theorem 1.10. There is a finite purely inseparable extension k' of k such that L'v K' is separable over k'. Then the integral PROOF.

..

,..

closure T* of T = k' [zl .,zt] is a finite T-module. T is a finite module over F = k[z l,...rzt , and T* is finite over F. Since F C R C_R* C T * , we see the result. Q.E.D.

3

COROLLARY 1.22. Let R be a finitely generated integral domain over a field k, and let R* be the derived normal ring of R. Let P* be a prime ideal of R*, and set P = P * n R . Then height P* = height P. PROOF. Since R* is finitely generated, we have height P* = trans.degkR* trans.degkR*/P* = trans.degkR - trans.degkR/P = height P (Corollary 1.14). Q.E.D. Although the following theorem is known for general noetherian rings, we prove this restricted form as an application of the normalization theorems. THEOREM 1.23.

If a ring R is finitely generated over a field k and if I is an

ideal of R generated by r elements ( I # R ), then for every minimal prime divisor P of I, it holds that height P 'r. PROOF. (1) The case where r = 1 : It suffices to show a contradiction assuming the existence of a chain of prime ideals P 3 P' 3 PI'. Considering R/P", we may assume that R is an integral domain. By Corollary 1.22, we may assume that R is normal. Let Q1,...,Qm be minimal prime divisors of I different from P, and let (I Q, outside of P. Considering RIg-l], we may g be an element of Q, n assume that P is the unique minimal prime divisor of I. Let zl,...,zt be as in ] Let L, M be fields of fractions of R, F, Theorem 1.10. Set F = k[zl,...,z t respectively, and consider a finite normal extension L* of M containing L, the integral closure R* of F in L*, the norm f* = N (f) of a generator f of I, and L/M

...

.

M.Nagata

134

P* i s a minimal prime d i v i s o r o f fR*.

a prime i d e a l P* o f R* l y i n g over P .

Since F i s a UFD, t h e r e i s a prime i d e a l Q o f F c o n t a i n i n g f * , contained i n P f l F By t h e going-down theorem, t h e r e i s a prime i d e a l Q* of R*

and o f height 1.

Then f * e Q*, and some conjugate o f f i s i n

lying over Q and contained i n P*. Then f e Q*'

with a a i n G(R*/F)

Q*. height Q = 1, and t h e r e f o r e Q*'n

implies t h a t Q*a

nR

.

R) = h e i g h t Q*'

We have h e i g h t (Q*'n

R i s a minimal prime d i v i s o r of fR.

=

This

= P and h e i g h t P = 1.

(2) m e general c a s e : We prove t h e a s s e r t i o n by induction on r .

be a s e t o f generators f o r I , and s e t I ' = d i v i s o r of I ' contained i n P .

li
Let f l ,

.. .,f

Let P ' be a minimal prime

By induction, height P' zr-1.

In R/P', P / P '

i s a minimal prime d i v i s o r o f t h e p r i n c i p a l i d e a l frR + P1/P' and h e i g h t P / P ' <1. Therefore, by v i r t u e o f Corollary 1.14 we s e e t h a t h e i g h t P

5 r.

Q.E.D.

2 . AFFINE SPACES AND PROJECTIVE SPACES

The s t r u c t u r e o f a f f i n e spaces seems t o be simpler than t h a t o f p r o j e c t i v e spaces. But, i n order t o develop geometry, o r t o understand geometric n a t u r e b e t t e r , p r o j e c t i v e spaces help us q u i t e a l o t . We s h a l l begin with two t y p i c a l d i f f e r e n c e s between a f f i n e and p r o j e c t i v e spaces. One example i s t h e Bezout theorem. I f we look a t two plane curves, o f degrees m and n, r e s p e c t i v e l y , and having no common component, then i n t h e p r o j e c t i v e case, we have t h e f i n e r e s u l t t h a t t h e t o t a l number o f common p o i n t s , counted with m u l t i p l i c i t i e s defined t o be one p l u s t h e degrees o f c o n t a c t s , i s equal t o mn.

This i s generalized a l s o t o t h e higher dimensional case, i . e . , t h e Bezout

theorem.

On t h e c o n t r a r y , we cannot have such an e q u a l i t y i n t h e a f f i n e c a s e .

Of course, s i n c e an a f f i n e plane i s imbedded n a t u r a l l y i n a p r o j e c t i v e plane, we have an i n e q u a l i t y i n t h e a f f i n e c a s e . Another example i s t h e completeness o f p r o j e c t i v e v a r i e t i e s .

For t h e d i s c u s s i o n ,

l e t us begin with a f f i n e v a r i e t i e s . Let k b e a f i e l d and consider an a l g e b r a i c a l l y closed f i e l d K c o n t a i n i n g k ; t h e polynomial r i n g F = k[X1,.

. ,,Xn]

; and t h e n-dimensional a f f i n e space Sn over K .

I f we f i x a coordinate system on Sn, then each element f(XI,. a function on Sn, so t h a t each p o i n t ( a l , . f(al,

...,an)

of K.

. .,an)

generated by J .

of F defines

I f a subset J o f F i s given, then we d e f i n e t h e zero-point

s e t V(J) of J t o be t h e s e t o f p o i n t s p = (pl, . . . , p n ) f o r any f e J .

. .,Xn)

o f Sn i s mapped t o t h e element e Sn such t h a t f ( p ) = 0

Obviously, V(J) c o i n c i d e s with V(J*) i f J* i s t h e i d e a l o f F Thus i n o r d e r t o observe

observe t h e c a s e where J i s an i d e a l .

s e t s of t h e form V(J), it s u f f i c e s t o

I n t h e c o n t r a r y d i r e c t i o n , i f we have a

subset T of S", then we d e f i n e I(T) f o r T t o be t h e set o f polynomials f e F

135

Commutative algebra and algebraic geometry such t h a t f ( p ) = 0 f o r any p e T.

Then I(T) i s an i d e a l o f F .

Under t h e same n o t a t i o n , we see e a s i l y t h e following t h r e e a s s e r t i o n s : (1) I f J i s an i d e a l i n F, then J E I ( V ( J ) ) . (2) I f J and J ' a r e i d e a l s i n F having t h e same r a d i c a l , then V(J) = V ( J 1 ) . ( 3 ) I f J1,

..., Jma r e .

i d e a l s i n F , then

. . . n Jm) = V(J1) u . . . l J V ( J m ) , n . .. n V(Jm).

( i ) V(J1,. .,JJ = V(J1 n ( i i ) V(J1 +. + Jm) = V(J1)

..

A l i t t l e more d i f f i c u l t r e s u l t i s t h a t , s i n c e K i s a l g e b r a i c a l l y closed:

(4) I f J i s an i d e a l of F , then I(V(J)) coincides with

fl.

O r , more p r e c i s e l y ,

i f f vanishes a t every a l g e b r a i c p o i n t p = ( p l , . . .,pn) within V(J), i . e . , p e V(J) and every pi i s a l g e b r a i c over k , then some power of f i s i n J . This (4) follows from Theorem 1.4 and i s another form o f t h e H i l b e r t zero-point theorem. A subset W o f Sn i s c a l l e d a k-closed set (or, more p r e c i s e l y , a Zariski k-closed

Such a W i s c a l l e d an a f f i n e k - v a r i e t y i f W i s not

s e t ) i f W = V(J) f o r some J .

a proper union of two closed s e t s .

I f we a r e obviously observing t h e s e sets over

k only, then k may be omitted. The r e s u l t s s t a t e d above show t h a t W i s a k - v a r i e t y i f and only i f t h e r e i s a prime i d e a l P i n F such t h a t W = V(P), o r e q u i v a l e n t l y , P = I(W) i s prime. F o r a k-closed s e t W, we s e t J = I(W). W induced from elements o f F .

k and i s denoted by k[W].

The r i n g F / J i s t h e s e t of f u n c t i o n s on

Then F / J i s c a l l e d t h e coordinate r i n g o f W over

( F o r a very general treatment o f a l g e b r a i c v a r i e t i e s ,

it i s b e t t e r t o observe F / J without assuming t h a t J = I(W), but assuming t h a t W

V(J).)

J u s t a s i n t h e c a s e o f F and Sn, we d e f i n e ( i ) V ( J I ) f o r an i d e a l J '

(T) f o r a subset T of W. V ( J 1 ) i s a k-closed subset o f W, and ( i i ) I k (T) i s an i d e a l of k[W] and I k [ w ~ ( V ( J 1 ) )is t h e r a d i c a l of J ' .

i n k[W]

Ik

IWI

IWJ

A t t h i s p o i n t , we observe a n a t u r a l correspondence between p o i n t s o f W and c e r t a i n

i d e a l s of k[W]

.

Set xi =

Xi modulo I (W)

.

A point p = (pl,

...,P,)

e S"

l i e s on W i f and only i f f ( p ) = 0 f o r every f e I(W), i . e . I ( p ) 2 I(W). not uniquely determined by I ( p ) . (I

: k[W] -K

such t h a t $xi = pi.

But p is

Indeed, p i s determined by t h e homomorphism Thus :

( 1 ) Each point o f W corresponds i n 1-1 way t o a homomorphism of klw] i n t o K .

Therefore ,

(2) lbo p o i n t s p = ( p l , . . . ,

(i.e.

pn) and q = (ql,

...,%)

correspond t o t h e same i d e a l

I ( p ) = I ( q ) ) i f and only i f t h e r e i s an isomorphism of k[pl, ...,pn]

onto k[ql,.

. .,qn]

.

which maps pi t o q . ( i = 1 , . . , n ) .

In p a r t i c u l a r ,

.

M. Nagata

136 (3)

Every a l g e b r a i c p o i n t of W corresponds t o a maximal i d e a l o f k[W]; every maximal i d e a l of k[W]

conversely,

corresponds t o t h e set o f conjugates o f an

algebraic point. If W i s an a f f i n e k - v a r i e t y , t h e n

From now on, we mainly observe k - v a r i e t i e s . I(W) i s prime and kfW] i s an i n t e g r a l domain.

The f i e l d of f r a c t i o n s o f k[W]

is

c a l l e d t h e function f i e l d o f W over k and i s denoted by k(W). Assume t h a t U and W a r e k - v a r i e t i e s i n Sm and Sn, r e s p e c t i v e l y .

I f t h e r e is given

an i n j e c t i o n a o f k(U) i n t o k(W) , then we say t h a t a r a t i o n a l mapping (or, r a t i o n a l correspondence) o f W t o U i s given; t h e r e a l meaning of t h i s i s as follows.

Write krW] = k[xl

,., .,xn]

and k[U]

= k[yl

,...,yA

.

Then consider t h e

ring

B = k[xl,...,xnJ

ayl,...,

aym3

*

We say t h a t a p o i n t p of W corresponds t o a p o i n t q i f t h e r e i s a homomorphism of B i n t o K whose r e s t r i c t i o n s on k[W], k[U]

d e f i n e p, q .

For one point p of W, t h e r e may be many q, o r none, i n U which correspond t o p . I f a l l uyi a r e r e g u l a r a t p ( i . e .

uyi = gi(x)/hi(x) with giJ hi e k(W), hi(p) #O), then we say t h a t t h e r a t i o n a l mapping i s r e g u l a r a t p, and i n t h i s c a s e q i s unique, a s i s e a s i l y seen.

Therefore any r a t i o n a l mapping i s r e a l l y a mapping on

t h e o u t s i d e o f a proper closed s u b s e t . I f k(W) = ok(U), then we say t h a t t h e correspondence i s b i r a t i o n a l , because we have r a t i o n a l mappings i n two ways.

The correspondence i s b i r e g u l a r a t a p o i n t

p o f W i f and only i f

u I g / h ( g , hek[U],

h(q) # 0

I = I g / h l g , hek[W1, h(p) # 0 I

where q i.s a p o i n t corresponding t o p . Next we observe p r o j e c t i v e v a r i e t i e s .

A homogeneous c o o r d i n a t e r i n g of

n-dimensional p r o j e c t i v e space Pn i s a polynomial r i n g H = k[XoJ.. v a r i a b l e s over t h e ground f i e l d k .

.,Xn]

in n

+

1

In t h i s case, we employ homogeneous coordi-

n a t e s and t h e r e f o r e : Elements o f H a r e not f u n c t i o n s on Pn, (2) i f f(X) i s a homogeneous form n Xn and i f p = ( p O J . . . , p ) i s a p o i n t of P , then it i s well defined n whether o r not f ( p ) = 0, and t h e r e f o r e (3) i f J i s a homogeneous i d e a l o f H, i . e .

(1)

i n Xo,

...,

J i s generated by homogeneous forms, then t h e set V(J)

of zero p o i n t s of J i n Pn i s well defined; and (4) i f f(X) and g(X) a r e homogeneous forms of t h e same

degree, then f(X)/g(X) i s a well defined function on Pn f / g i s c a l l e d a r a t i o n a l f u n c t i o n on

P".

-

The set Wi = Pn

V(g(X)); such a f u n c t i o n

-

V(Xi) =

E ( p o J . . .,pn) e

p l p i = l} i s n a t u r a l l y regarded a s an n-dimensional a f f i n e space with coordinate r i n g Ri = k[Xo/Xi

,. .., Xi-l/Xi

, Xi+l/Xi ,.. .,Xn/Xi

1 and

Pn = WoU

.. .U Wn.

Under

137

Commutative algebra and algebraic geometry

th e b i r a t i o n a l correspondence between Wi and W . (defined by t h e fact t h a t Ri and 3 1 I p = q i n Pn. Thus Pn i s t h e union of n + I a f f i n e spaces and gluing them i s made R . have t h e same f i e l d of f r a c t i o n s ) p e Wi corresponds t o q e W. i f and only i f

v i a n a t u r a l b i r a t i o n a l mappings.

If P i s a homogeneous prime i d e a l of H, then

U = V(P) i s a p r o j e c t i v e k - v ar i et y i n Pn and U i s t h e union of a t most n + I a f f i n e k - v a r i e t i e s which a r e n a t u r a l l y b i r a t i o n a l with one another.

The common

function f i e l d i s c a l l e d t h e function f i e l d o f U and t h e r i n g H/P i s c a l l e d t h e homogeneous coordinate r i n g o f U over k ; H/P w i l l be denoted by

qV].

I f W i s an a f f i n e o r a p r o j e c t i v e k - v ar i et y and i f an i n j e c t i o n of k(W) i n t o k(U) is given, then we have a r a t i o n a l mapping ( r a t i o n a l correspondence) defined

n a t u r a l l y v i a a f f i n e p i eces .

I t s r e g u l a r i t y a t a point, b i r a t i o n a l i t y , bire gula -

r i t y a t a point a r e defined s i m i l a r l y t o t h e a f f i n e c a se .

One form of t h e completeness of U i s s t a t e d a s follows : I f U and W a r e a s above, then f o r any point q of W, t h e r e i s a

THEOREM 2 . 1 .

point p i n U which corresponds t o q and t h e r e f o r e i f W i s also p r o j e c t i v e and i f Z i s a closed subset o f W then t h e s e t Z ' o f p oints o f U which corresponds t o some

p o in t s of Z i s a closed s e t . Since t h e discussion o f t h i s kind i s not our main s u b j e c t , we omit t h e d e t a i l s . What we wish t o emphasize here i s t h a t t h e importance o f t h e completeness l i e s i n t h e f a c t t h a t i n order t o i n v e s t i g a t e p r o p e r t i e s of b i r a t i o n a l l y equivalent v a r i e t i e s we o f t e n have a l o t of h el p given by t h e p r o p e r t i e s of U v i a b i r a t i o n a l correspondence. These examples, t h e Bezout theorem and completeness, show good f e a t u r e s of projective varieties.

But t h e r e a r e many examples i n which we f e e l t h a t p r o j e c t i v e

v a r i e t i e s a r e much more complicated than a f f i n e v a r i e t i e s . t h e product o f spaces.

A t y p i c a l example i s

The product o f a f f i n e spaces Sm and Sn over t h e f i e l d K

i s nothing but (mtn) -dimensional a f f i n e space Smtn; t h e p a i r of p = (pl,.

.,

...,

. .,p,)

ql,. ..,qn), But i n t h e p r o j e c t i v e .,%) e Pn, t h e similar arrangement pm+n+l which i s not uniquely determined by p and q (because, f o r i n s t an ce, q = ( t q o,...,tqJ w ith an a r b i t r a r y and q = (ql, 9,) i s t h e p o i n t ( p l , . . p,, case, i f p = (po,. ..,pm) e and q = (qo,. of t h e coordinates makes (p, pm, qo

,...,

.

,...,a)

# 0 ) . Therefore (po, . . . , p m, s o , . . . , % ) does not give t h e p a i r o f p,q. One good way t o deal with t h e p a i r i s t o as s i g n a sequence o f products of pi and q j' I t i s easy t o se e say (Poqo, P 1 q o , * ** aPmqo, Poq1,***sPmqls P o q 2 , * * * *P,%). t h a t by t h i s assignment we have a good b i r e g u l a r correspondence between Pm x Pn t

and a c e r t a i n p r o ject i v e k - v ar i et y i n P"+m+n,

M. Nagata

138

For the purpose of our later discussion, we add some discussion on birational

mappings. Assume that k is a field as before and that U is a projective k-variety in P". If a finite number of elements zo, zm of k(U) is given, then we have a k-variety W and a rational mapping of U to W as follows. Take a transcendental element t over k(U) and consider the natural homomorphism of the polynomial ring k[Yo,. ,Ym] onto k[tzo,, , ,tzm] . The kernel Q is a homogeneous prime ideal and we have W = V(Q). Thus W is the union of affine k-varieties with coordinate rings Ai = k [zo/zi,. .,zm/zi which are subrings of k(U) Hence we have a natural rational mapping of U to W; W is called the projective k-locus of (zOD...,zm). The same is applied to a sequence of elements zo, zm such that all zi/z. are in k(U).

...,

..

.

.

.

1

...,

I

Now, letting k[xo,

. . .,xn1 be

the homogeneous coordinate ring Hk [U], where U is the projective k-locus of (xO,...,xn), we consider the pair of (xo,...,xn) and (zo zm), i.e. (xozo, xnzO, xozl,..., xn z n) , Then the natural mapping of the projective k-locus T of this pair to U is regular everywhere; the inverse, i.e. the rational mapping of U to T, is called the dilatation of U by (ZO,...,~m).

,...,

...,

If W is birational to U and if W +. U is regular, then T is biregular to W. Therefore any projective variety W such that W + U is birational and regular, is biregular to a project variety obtained by a dilatation. This means that general dilatation is very difficult to deal with. But, in some special cases, dilatation behaves nicely. One sees easily If zo,, ,., zm are taken from Hk[U] so that they are homogeneous elements of the same degree, then the rational mapping of U to T is regular outside of V( IziHk[U]) ?HEOREM 2 . 2 .

.

The simplest case is therefore the case where IziHk[U] = I(p) with a k-rational point p = (po,...,pn), i.e. pi/p. e k for all possible pairs (i,j). Then U -+ T 1 is regular except at p. This dilitation is called the quadratic dilatation with center p. If one checks carefully, one sees easily that the set E of points on T which correspond to p is a subset of T which is biregular to the base variety of the tangent cone of U at p; if p is a simple point then E is biregular to Pr (r =dim 0 - 1 ) and every point of E is a simple point of T. Quadratic dilatations play a very important role in the theory of surfaces. One reason is the validity of the following.

'MEOREM 2.3. If U and W are birationally equivalent non-singular projective surfaces and if the birational mapping W -+ U is regular, then W is biregular to a surface which is obtained by certain successive quadratic dilatations.

Commutative algebra and algebraic geometry

139

In the higher dimensional case, the situation is much more difficult. For instance, it is usual that the set of points at which a given dilatation is not biregular has some positive dimension. Therefore we are obliged to observe dilatation other than quadratic dilatation. An immediate generalization of a quadratic dilatation is a dilatation defined by (zo,.. .,zm) such that cziHk[U] is a prime ideal or (a prime ideal)n(a primary ideal belonging to the ideal generated by all positive degree elements) (under the notation of Theorem 2.2). Such a dilatation is called the monoidal dilatation with center C = V(lziHk[U]). In this case, if C is a nonsingular k-variety and if every point of C is a simple point of U, then the dilatation appears to be understandable. But, in the general case, phenomena along singular subsets of C are very complicated. On the other hand, if IziHk[U] is the intersection of certain prime ideals, then some complication happens at the intersections of the components of V(lziHk[U]) as one sees easily by the following example : Consider P3 with homogeneous coordinate system (Xo, XI, X2, X3). Take lines Li = V(X.H + XmH) ( H=k[X 0,...,X3]; {i,j,ml = {1,2,3 1 ) . Then I(L1UL2UL3) = 3 (X1X2, X2X3, X3X1). Then we have the dilatation P3 -+ W defined by W has four exceptional surfaces, one of which corresponds (X1X2, X2X3, X3X1). to the origin ( 1, 0, 0, 0). Furthermore, this dilatation cannot be factored into a product of successive monoidal dilatations. In connection with this fact, the writer notes that the following is an important question on birational mappings : QUESTION 2 . 4 . Assume that U and W are birationally equivalent nonsingular projective varieties. Does there exist a projective variety T such that both of the birational correspondences U + T and W + T are factored into products of successive monoidal dilatations with nonsingular centers ? In closing this section, we make a few remarks on the correspondence of subvarieties under a birational mapping. First of all, we should not restrict our observation only to set-theoretical aspects. For instance, a divisor is a linear combination of divisorial subvarieties (i.e. subvarieties of one dimension less) under certain restrictions (no restriction in the nonsingular case), and by virtue of this notion we can discuss the Riemann-Roch theorem, for instance. Therefore, let us observe ideals instead of subvarieties. In the projective case, it suffices to observe homogeneous ideals of the homogeneous coordinate rings; in the affine case, just ideals of the affine coordinate ring. If we want to observe more general cases (abstract varieties or schemes) we observe sheaves of ideals. If an ideal is locally principal everywhere, it defines a positive Cartier divisor. But, in the general case, the geometric feature of an ideal

M.Nagata

140

i s s t i l l not very c l e a r and must be one o b j e c t t o be s t u d i e d i n t h e f u t u r e .

Anyway, i f W and U a r e v a r i e t i e s and i f a r e g u l a r r a t i o n a l mapping o f W and U i s given, then an i d e a l J on U generates an i d e a l J ' on W ( i n t h e p r o j e c t i v e c a s e , i r r e l e v a n t primary components may be d i s r e g a r d e d ) .

One disadvantage i s t h a t two

mutually d i s t i n c t i d e a l s on U may generate t h e same i d e a l on W .

But t h i s does

not happen i n c e r t a i n good c a s e s : i n c l u d i n g t h e case of i d e a l s f o r d i v i s o r s on U, i f U i s p r o j e c t i v e nonsingular and i f W i s p r o j e c t i v e ,

Anyhow, we have a

mapping of t h e s e t o f i d e a l s on U t o t h e one on W a s s o c i a t e d with t h e r e g u l a r r a t i o n a l mapping W

+

U.

By t h i s mapping, we can d e f i n e t h e t o t a l transform.

For i n s t a n c e , i f U and W a r e p r o j e c t i v e and i f U i s nonsingular, then a d i v i s o r D on U i s expressed a s D1 - D 2 with p o s i t i v e d i v i s o r s Di.

l o c a l l y p r i n c i p a l i d e a l s J1, J

W by t h e mapping given above. Another

2

on U and they d e f i n e C a r t e r d i v i s o r s Di, D; on The t o t a l transform o f D i s defined t o be D;

I t i s defined a s follows i n t h e b i r a t i o n a l

If U and W a r e b i r a t i o n a l , then t h e s e t F o f p o i n t s on U a t which t h e

r a t i o n a l map U

-+

W i s not b i r e g u l a r forms a proper closed subset of U .

subvariety C of U i s n o t contained i n F, then C t o a subset C ' o f W. + U

-

If a

( C n F ) corresponds b i r e g u l a r l y

The c l o s u r e C* o f C ' i s a s u b v a r i e t y o f W; C* i s c a l l e d

t h e proper transform of C on W .

W

- D;.

important subvariety i n W corresponding t o a given s u b v a r i e t y i n U i s

t h e one c a l l e d t h e proper transform. case.

The Di correspond t o

This n o t i o n may be g e n e r a l i z e d t o t h e case where

is not b i r a t i o n a l , but we omit t h e case.

F o r d e t a i l s of t h e t o p i c s i n t h i s s e c t i o n , s e e Z a r i s k i [25]

.

3 . RULED SURFACES AND RATIONAL SURFACES A s we learned from Theorem 2 . 3 , q u a d r a t i c d i l a t a t i o n s p l a y an important r o l e i n

t h e theory of s u r f a c e s .

As a matter o f f a c t , t h e r e i s a n i c e r e s u l t which

On a non-singular p r o j e c t i v e s u r f a c e F , we d e f i n e f i r s t t h e i n t e r s e c t i o n number (D, D') o f two d i v i s o r s D, D ' . If D characterizes quadratic d i l a t a t i o n s .

and D ' have no common component, then (D, 0 ' ) i s t h e number of i n t e r s e c t i o n s counted with properly defined m u l t i p l i c i t i e s ; i n t h e general c a s e , (D, 0 ' ) i s defined t o be (D, D") with D" l i n e a r l y equivalent t o D ' and having no common component with D.

Then one s e e s t h a t i f W i s a nonsingular p r o j e c t i v e s u r f a c e ,

W + F i s r e g u l a r b i r a t i o n a l and i f D*, D** a r e t h e t o t a l transforms o f D, D' on W, then it holds t h a t (D, 0 ' ) = (D*, D**). Now t h e c h a r a c t e r i z a t i o n we wanted can be s t a t e d a s follows : THEOREM 3.1.

Let F ' be a nonsingular p r o j e c t i v e v a r i e t y and l e t C be an i r r e -

d u c i b l e curve on F ' .

Then t h e r e i s a r e g u l a r b i r a t i o n a l mapping o f F ' t o a

nonsingular p r o j e c t i v e s u r f a c e F such t h a t (i) C corresponds t o a p o i n t , say p,

141

Commutative algebra and algebraic geometry of F, and ( i i ) t h e r a t i o n a l mapping F + F ' i s t h e q u a d r a t i c d i l a t a t i o n with

c e n t e r p, i f and only i f ( i i i ) C i s a nonsigular r a t i o n a l curve and ( i v ) (C, C ) = - 1 . In computing i n t e r s e c t i o n numbers, t h e following i s u s e f u l : Let F be a nonsingular p r o j e c t i v e s u r f a c e and l e t D be a curve on

THEOREM 3.2. F.

I f p e F i s an m-ple point of D, then l e t t i n g D ' ,

E be t h e proper transform

of D and t h e curve corresponding t o p, we have t h e following e q u a l i t i e s : (DI, E) = m, PROOFS.

(D, D) = ( D ' ,

0') + m

2

.

The proof of t h e i f p a r t of Theorem 3 . 1 i s d i f f i c u l t and we r e f e r t h e 1

E is b i r e g u l a r t o P by our statement a f t e r Theorem 2 . 2 . I f m = 0 then t h e b i r a t i o n a l mapping i s b i r e g u l a r a t every p o i n t of D and we have

reader t o Zariski [ 2 7 ] . (D, D) = ( D ' ,

D'),

(DI, E) = 0 .

Assume now t h a t m > 0.

That m i s t h e m u l t i p l i -

c i t y of p on D means t h a t t h e l o c a l equation f o r D a t p starts with t h e degree i m-i m p a r t : l o c a l equation = 1 a . x y + (higher degree terms). This shows t h a t t h e i n t e r s e c t i o n o f D ' with E co:es from la.xiym-i = 0 viewing E as P1 and a l s o t h a t D t + mE i s t h e t o t a l transform of D.

1

The f i r s t h a l f implies t h a t

(Ill,

E) = m.

D i s l i n e a r l y equivalent t o a d i v i s o r D" not going through p . Therefore (D' + mE, E) = 0 and (D', E) = -m(E, E). Thus (E, E) = -1. Now, (D, D) = 2 2 (D' + mE, D ' + mE) = ( D ' , 0 ' ) + 2 m ( D ' , E) + m (E, E) = (D', D ' ) + m Q.E.D.

.

These f a c t s enable us t o c l a r i f y t h e notion of i n f i n i t e l y near p o i n t s e f f e c t i v e l y used by Castelnuovo.

Indeed, l e t F be a nonsingular p r o j e c t i v e s u r f a c e .

In t h e

c l a s s of b i r a t i o n a l l y equivalent p r o j e c t i v e s u r f a c e s , we f i x b i r a t i o n a l correspondences and

we i d e n t i f y two p o i n t s i f they correspond b i r e g u l a r l y under t h e

fixed b i r a t i o n a l correspondences.

Then

An i n f i n i t e l y near point t o a p o i n t p o f F of o r d e r one i s an ordinary (i) point of E under t h e n o t a t i o n a s above. ( i i ) An i n f i n i t e l y near point t o p of o r d e r m > 1 i s an i n f i n i t e l y near p o i n t t o a point q o f o r d e r one which i s an i n f i n i t e l y near p o i n t t o p o f o r d e r m - 1. ( i i i ) Let D be a d i v i s o r on F and l e t pi ( i = 1, i n f i n i t e l y near p o i n t s t o some p o i n t s o f F . c o e f f i c i e n t s mi.

-

li2mipi;

be ordinary p o i n t s o r

-

I m p i with i n t e g r a l

L lmipi t o F' obtained by t h e i s defined t o be ( t o t a l transform o f D)

Then t h e t o t a l transform of D

quadratic d i l a t a t i o n with c e n t e r p = p

mlE

...,s)

We consider D

1 t h i s can happen only i f p1 i s an o r d i n a r y point o f F .

-

(Of course, i f we consider a q u a d r a t i c d i l a t a t i o n with c e n t e r not equal t o any o f pi,

then our i d e n t i f i c a t i o n of p o i n t s implies t h a t D

- lmipi

i s unchanged under t h e

d i l a t a t ion. ) (iv)

We then d e f i n e t h a t D

-

[mipi

i s p o s i t i v e i f a f t e r some successive q u a d r a t i c

M. Nagata

142

dilatation, the total transform becomes a positive divisor ; in such a case we say that D goes through cmipi. One should note that even if a divisor D goes through both Imipi and 1n.q. and 1 1 if p. # q . for any i, j, it may not hold that D goes through lmipi + lnjqj. 1

3

In this way, base conditions for linear systems are rigorously interpreted and we see that Castelnuovo's computation is right. Furthermore, any divisor on a birationally equivalent nonsingular projective surface is expressed in the form of D - lmipi on F. Let us apply some easy part of these facts to the case of ruled surfaces. By a ruled surface F, we understand a nonsingular projective surface which has a regular rational mapping J, to a nonsingular projective curve B so that for any is biregular to P1 . Each $- 1 (p) is called a generator of F. point p of B, $-'(p) If B is rational, then F is called rational; if B is irrational, then F is called irrational. Two generators G , G ' are algebraically equivalent to each other and we see that ( G , G) = 0. Take p e G and consider the quadratic dilatation with center

p : F + F'. Let G ' and E be the proper transforms of G and p. Theorem 3 . 2 shows that ( G I , GI) = -1 and therefore Theorem 3 . 1 shows that there is a nonsingular projective surface F" such that F' -+ F" is obtained by the contraction of G' to a point. Then F" is obviously a ruled surface. Thus we obtain a birational

mapping of a ruled surface to another ruled surface, which we call the elementary transformation with center p; we denote it by elm Note that if G' is contracted P' to q, then elm is the inverse of elm q P' Since, on an irrational ruled surface, there are no rational curves except for generators, we see the following result easily. 'IHEOREM 3 . 3 .

If F and F* are birationally equivalent ruled surfaces, then the birational transformation F -+ F* is the product of successive elementary transformations. In the case of rational ruled surfaces, similar results hold. The only difference is that 'P xP1 is a rational ruled surface and the projection to each factor gives a structure as a ruled surface. This means that if F is a rational ruled surface biregular to p1 xp', then F has one more structure as a ruled surface. Hence the process of elementary transformations may be followed after changing the ruled surface structure. The proof of this fact is rather difficult, and we refer the reader to Nagata [13].

Commutative algebra and algebraic geometry

143

We f e e l t h a t t h e f a c t t h a t t h e r e a r e only a small number of b i r a t i o n a l t r a n s f o r mations between r u l e d s u r f a c e s shows t h a t t h e r e a r e many b i r e g u l a r c l a s s e s of i r r a t i o n a l ruled surfaces. r u l e d s u r f a c e s i n not easy.

Actually, t h e b i r e g u l a r c l a s s i f i c a t i o n of i r r a t i o n a l In t h e r a t i o n a l case, t h e c l a s s i f i c a t i o n is easy

and we can s t a t e t h e r e s u l t a s follows : THEOREM 3.4. Consider Fo = P1 x P 1 a s a r u l e d s u r f a c e with g e n e r a t o r s p X P1 (p e P 1) Take mutually d i s t i n c t n p o i n t s q l , . ., q on one P1 x r ( r e P 1 ) n Let Fn be t h e r u l e d s u r f a c e obtained by t h e successive elementary transformations

.

.

.

with c e n t e r s ql, . . . , q n .

Then ( i ) every r a t i o n a l r u l e d s u r f a c e i s b i r e g u l a r t o

some Fi; and ( i i ) i f i # j , then Fi cannot be b i r e g u l a r t o F b i r e g u l a r mappings a r e not t h e previously f i x e d ones.) For t h e proof, see f o r i n s t a n c e Nagata (13

( O f course, t h e s e

j'

1,

By a s e c t i o n C on a ruled s u r f a c e F, we understand an i r r e d u c i b l e curve C such t h a t (C, G) = 1 f o r a generator G .

Then t h e number n f o r Fn above can be charac-

t e r i z e d a l s o by t h e following f a c t . THEOREM 3.5.

Let F be a r a t i o n a l r u l e d s u r f a c e .

(1) I f t h e r e i s a s e c t i o n C

on F such t h a t (C, C) = -n < 0, then F i s b i r e g u l a r t o Fn.

(2) If t h e r e i s no

such s e c t i o n , then F i s b i r e g u l a r t o Fo. PROOF.

Let B be t h e proper transform of r x P1 on F under t h e n o t a t i o n o f n Then (B, B) = -n. I t s u f f i c e s t o show t h a t i f n > 0, then B i s t h e

Theorem 3.4.

If C i s a s e c t i o n , C i s l i n e a r l y equivalent t o B + mg with a generator g and a nonnegative i n t e g e r m.

unique s e c t i o n having negative s e l f i n t e r s e c t i o n number.

(C, C) = -n + 2m

'> -n

+ m = (B, C ) .

Therefore, if (C,C) < 0, then (B, C) < 0 and B, C must have a common component. Q. E . D. 2

These r e s u l t s have a l o t t o do with Cremona transformations o f P

, because

of t h e

following f a c t . (1) I f p e P 2 , then d i l P 2 i s b i r e g u l a r t o F1 ; and (2) i f F i s a P nonsingular r a t i o n a l p r o j e c t i v e surface, then e i t h e r F i s b i r e g u l a r t o P 2 o r t h e r e 'MEOREM 3 . 6 .

is one Fn and a r e g u l a r r a t i o n a l mapping o f F t o Fn.

Here, F1, Fn a r e those

given i n Theorem 3.4. For t h e proof, s e e f o r i n s t a n c e Nagata [13

1.

As f o r Cremona transformations of P2, we s h a l l have some d i s c u s s i o n o f them i n 85.

M. Naga ta

144

This theorem o f t e n helps our understanding of r a t i o n a l s u r f a c e s , because nons i n g u l a r r a t i o n a l p r o j e c t i v e s u r f a c e s a r e e i t h e r P2 o r Fn o r a s u r f a c e blown up from some Fn.

It

One simple example i n t h i s d i r e c t i o n may be cubic s u r f a c e s .

i s well known t h a t a cubic nonsingular s u r f a c e i n P 3 h a s 27 l i n e s on i t .

But

it i s a l s o i n t e r e s t i n g t o c h a r a c t e r i z e t h e s e s u r f a c e s i n t h e following way : THEOREM 3.7.

Let p l , .

. . ,p6 be

ordinary p o i n t s on p2 such t h a t ( i ) no t h r e e o f

them a r e c o l l i n e a r ; and ( i i ) t h e r e i s no conic going through a l l o f them.

Then

t h e r a t i o n a l s u r f a c e obtained by q u a d r a t i c d i l a t a t i o n s with c e n t e r s a t a l l o f pi i s a nonsingular cubic s u r f a c e i n P3.

Conversely, any nonsingular c u b i c s u r f a c e

i n P3 i s b i r e g u l a r t o some one obtained a s above. (See, f o r

We omit t h e proof, because we need some knowledge of cubic s u r f a c e s . and van d e r Waerden [22]

i n s t a n c e , Nagata [14]

.)

By t h e way, we n o t e t h a t t h e 27 l i n e s a r e ( i ) t o t a l transforms of pl, . . . , p 6 ; ( i i ) proper transforms o f conics going through f i v e o f pl, . . . , p 6 ;

and ( i i i ) proper

transforms of l i n e s going through two o f p1,...,p6.

4 . AUTOMORPHISM GROUPS OF POLYNOMIAL RINGS

. ., Xn

Let k be a r i n g , X1,.

indeterminates and s e t F = k[XID., ,,Xn]

.

Then t h e

o b j e c t we s h a l l observe i n t h i s s e c t i o n i s t h e automorphism group AutkF, i . e . t h e group of automorphisms of F which f i x every element o f k. I f n = 1, then t h e s t r u c t u r e of AutkF i s e a s i l y seen, and we omit t h e case.

If

n = 2 and i f k i s a f i e l d , then we have a good r e s u l t which w i l l be s t a t e d l a t e r (Theorem 4.1).

Even when n = 2 , i f k i s not a f i e l d , we do not have any s a t i s -

factory r e s u l t yet.

If n

3, then even i f k i s a f i e l d , we do not have any good

result yet. In t h i s s e c t i o n , we review some f a c t s on AutkF. THEOREM 4.1.

F i r s t we s t a t e :

Assume t h a t k i s a f i e l d and n = 2 .

We write x,y i n s t e a d o f X1,

Then Aut F i s generated as an amalgamated product by two subgroups A and J :

X2. k A i s t h e s e t of

(I

i n AutkF such t h a t

ox = ax + by + c

(a, b, c e k)

oy = ex + f y + g

(e, f , g e k)

af

-

be # 0.

J i s t h e s e t of T i n AutkF such t h a t TX

= ax + f ( y )

TY

= by + c

# a e k ; f ( y ) e kry]) (b, c e k, b # 0 ) . (0

145

Commutative algebra and algebraic geometry

There a r e several known proofs of t h i s r e s u l t ; one proof i s found i n Nagata [19]. One immediate adaption of t h i s r e s u l t t o t h e case n = 3 would be t h e following question : Assume t h a t k i s a f i e l d and n = 3.

QUESTION 4.2.

Let A be t h e subgroup o f

AutkF whose elements a r e a f f i n e ( l i n e a r ) transformations.

Let J be t h e group con-

s i s t i n g of elements u o f AutkF such t h a t

i s AutkF generated by A and J ?

Our c o n j e c t u r e on t h i s question i s negative, because t h e following element

'I

of

AutkF seems not t o be i n t h e group generated by A and J .

In order t o s e e t h a t t h i s d e f i n e s an element o f AutkF, we n o t e f i r s t t h a t T(X3X1

t

2

x2) =

x3x1

t

x2.

x1

-+

x1

Then we f i n d t h a t t h e following gives t h e i n v e r s e of

T : +

2X2(X3X1

2

X2)

+

-

X3(X3X1

+

2 X2),

Several people, including David Wright, have asked t h e following questions :

...,

F for i = 1, n and l e t A be k[X1,. .,Xi] t h e subgroup o f AutkF c o n s i s t i n g o f a f f i n e transformations. Do A, H1, Hn-l

In Aut F, s e t Hi = Aut

QUESTION 4.3.

.

k

...,

generate AutkF ? QUESTION 4.4.

I u

e AutkF

I uVi

QUESTION 4 . 5 . s e t Fn = k[X1,

Set Vi = k 4 Vi 1

.

t

kX1

Do K1,.

t

... + kXi

. ., Kn

for i = l,...,n.

Consider i n f i n i t e l y many indeterminates X1,

.. ., Xn]

f o r n = 0,1,2,

Let Ki be

generate AutkF ?

.. . .

X2,

..., Xn, ... and

For each n > 1, l e t Jn be t h e subgroup

of Aut F c o n s i s t i n g of elements u such t h a t uXi = aiXi t f i ( 0 # ai e k; f i eFi-l) k n f o r i = n , n-1, , 1. Let An be Kn above ( t h e same as A) and l e t Gn be t h e subgroup generated by An and Jn. For m > n we d e f i n e t h a t uXm = Xm i f u eAutkFn

. .,

M.Nagata

146

and we make AutkFn a subgroup o f AutkFm. Now we ask i f every AutkFn i s a subgroup o f t h e group generated by a l l t h e G i . There i s a completely d i f f e r e n t problem, on which one may f e e l t h a t t h e r e may be something t o do with t h e s e groups :

QUESTION 4 . 6 . (Jacobian problem).

. ..

Assume t h a t k i s a f i e l d o f c h a r a c t e r i s t i c

,..

zero. With polynomials f l , , f i n F = k[Xl , ,Xn] , consider t h e polynomial n Assume t h a t t h e jacobian matrix (afi/aX.) mapping (X1,...,Xn) -+ ( f l , . . . , f n ) . J has a nonzero constant determinant. Does it follow t h a t t h e mapping i s b i r e g u l a r ? One s e e s e a s i l y t h a t i f b i r a t i o n a l i t y o f t h e mapping i s known, then t h e biregul a r i t y follows.

There a r e s e v e r a l people who t r i e d t h e problem, b u t , a t t h e

moment, even i f n = 2 ( t h e c a s e n = 1 i s t r i v i a l l y e a s y ) , t h e r e a r e known only some p a r t i a l r e s u l t s , most o f which a r e due t o S .

Abhyankar and T. T . Moh.

5. CREMONA GROUPS The Cremona group of Pn i s t h e group o f b i r a t i o n a l mappings o f Pn t o i t s e l f , o r , equivalently, t h e group of K-automorphisms o f t h e r a t i o n a l f u n c t i o n f i e l d K(x l , . , . , ~ n ) of Pn.

For n = 1, every b i r a t i o n a l mapping o f P1 i s a l i n e a r ( p r o j e c t i v e ) transformation, hence, i n t h e form of AutKK(x), it i s obtained a s {x

+.

(ax + B ) / ( c x + d )

I a,b,c,d

e K , ad

For n = 2 , t h e r e is a good r e s u l t , f i r s t claimed by M .

by Castelnuovo [3 Jung [ 7

1 , etc.) .

THEOREM 5.1.

]

.

# bc 1 . Noether and then proved

Several o t h e r proofs were given l a t e r (Alexander [l]

,

One Eormulation of t h e r e s u l t can be s t a t e d as follows :

The Cremona group of P2 is generated by l i n e a r p r o j e c t i v e t r a n s -

formations and an a r b i t r a r y q u a d r a t i c transformation. Here, a q u a d r a t i c transformation i s defined a s follows. p o i n t s which a r e not c o l l i n e a r .

Let p , q, r be t h r e e

Take l i n e a r forms f , g, h, i n t h e homogeneous

coordinate r i n g H = K[x, y, z] which d e f i n e t h e t h r e e l i n e s C, C ' , C" going through two of p , q, r .

Then t h e correspondence (x, y, z) -+ (gh, h f , f g ) =

( f - l , g-l, h - l ) defines a b i r a t i o n a l mapping of P2 t o i t s e l f such t h a t ( i ) it i s b i r e g u l a r a t any point o u t s i d e o f C U C ' U C" ; ( i i ) t h e proper transform o f each o f C , C ' , C" i s a point ; and ( i i i ) each of p, q, r corresponds t o a l i n e . p r e c i s e l y , t h i s can be s t a t e d as follows : c e n t e r s p, q, r .

More

Consider q u a d r a t i c d i l a t a t i o n s with

Then t h e proper transforms o f C, C ' ,

C t t a r e nonsingular r a t i o n a l

curves with s e l f - i n t e r s e c t i o n number -1, hence they can be c o n t r a c t e d t o p o i n t s

141

Commutative algebra and algebraic geometry

(noting that they do not intersect each other),

Then the new surface is a P

2

.

We should note here that in the statement of Theorem 5.1, since all linear trans-

formations are allowed, to allow one special quadratic transformation amounts the same as to allow all quadratic transformations. Therefore, in terms of AutkK(x,y), Theorem 5.1 can be stated as : THEOREM 5 . 2 . If x,y are algebraically independent elements over the algebraically closed field k, then Aut K(x,y) is generated by a quadratic transformation k defined by (x,y) -+ (x-’, y-l) and the group of projective linear transformations

{

(x,t)

-[

,

m]

I a,b,c,d,e,f,g,h,i

eK,

i:,”fI1. d e f # 0

These formulations are made from the coordinate-wise viewpoint. In order to analyze the geometric feature of a Cremona transformation, it is worthwhile to observe it in the birational class of rational surfaces with identification of points as in 13. At this point, Theorem 5.1 is equivalent to : THEOREM 5.3. An arbitrary birational mapping of a P 2 to another P2 is the product of quadratic transformations. (This means that there is a sequence of P2 and quadratic transformations ii :

so that the given birational transformation is the product of these T ~ . )

But, as a matter of fact, Theorem 5 . 3 is too inefficient in the sense that, in order to practice the transformation, blowing up of biregularly corresponding points is often required. In order to explain such a situation and to have a more efficient formulation, we review two kinds of Cremona transformations : generalized quadratic transformations and Jonquieres transformations. The term “quadratic” transformation derived from the fact that the transformation is defined by forms of degree two (gh, hf, fg in the previous notation). For this fact, the three points need not be ordinary points. So let us observe linear systems of conics of P‘ having three base points p, q, r where p is assumed to be an ordinary point. If q, r are also ordinary points, then (i) p, q, r are not collinear, then we have the usual quadratic transformation; (ii) if they are collinear, then the system has a fixed component, and this case is omitted. So we assume that q is an infinitely near point to p of order one. CASE 1.

Assume that r is an ordinary point. Let L and L ’ be the lines going through p + q and p + r respectively. If L = L’, then the linear system has L as a fixed component, and we omit the case. So we assume that L # L’. Then the

148

M.Nagata

l i n e a r system d e f i n e s a Cremona transformation which t u r n s t o t h e following (modulo l i n e a r transformation) : (x,y,z)

+

(yz, z2, xy).

This i s obtained by

successive q u a d r a t i c d i l i t a t i o n s with c e n t e r s p, q, r and c o n t r a c t i o n s o f proper transforms of L, L ' .

Then t h e proper transform o f t h e curve on t h e f i r s t

d i l a t a t i o n came from p. CASE 2 .

Assume t h a t r i s i n f i n i t e l y near t o p .

If r i s not i n f i n i t e l y near

t o q, then t o go through p + q + r implies t h a t p must be a double p o i n t and every member of t h e l i n e a r system i s t h e sum o f two l i n e s going through p . a system does not d e f i n e a b i r a t i o n a l mapping. t o q.

Such

Therefore r i s i n f i n i t e l y n e a r Then r cannot be on t h e proper

Let L be t h e l i n e going through p + q.

transform of L a f t e r t h e d i l a t a t i o n s with c e n t e r s p, q (otherwise L becomes a fixed component o f t h e l i n e a r system).

If C i s an i r r e d u c i b l e member o f t h e

l i n e a r system, p = (0, 0, 1) and i f L = V(x), then t h e equation f o r C is of t h e form xz + ax2 + bxy + cy2 ( c # 0 ) .

On t h e o t h e r hand, L + (any l i n e going

through p) i s i n t h e l i n e a r system, a s we can compute by our r u l e i n 13.

We s e e

t h a t t h e transformation i s defined (modulo l i n e a r transformations) by :

where c depends on t h e p o s i t i o n of r and hence may be made 1 by a l i n e a r t r a n s f o r mation.

In terms of d i l a t a t i o n , we s e e t h a t t h e transformation i s made by :

First apply q u a d r a t i c d i l a t a t i o n s with c e n t e r s p, q, r ( r can be assumed t o be

of o r d e r two), then c o n t r a c t t h e proper transform of L, then t h e proper transform of t h e r a t i o n a l curve t h a t came from q , and then t h e one t h a t came from p . These two types of Cremona transformations a r e generalized q u a d r a t i c transformat i o n s ; as we have seen, they and t h e usual q u a d r a t i c transformations a r e t h e only Cremona transformations defined properly by l i n e a r systems o f c o n i c s . Let u s observe h e r e Theorem 5.3 i n t h e s e c a s e s .

As f o r Case 1, what we do is t o

t a k e a p o i n t s which i s not on e i t h e r o f t h e l i n e s L, L ' . q u a d r a t i c transformation with c e n t e r s p, r , s .

p o i n t s from t h e l i n e s going through p + s , r + s . q u a d r a t i c transformation with c e n t e r s q , q u a d r a t i c transformation.

rl,

Then f i r s t apply

Let r ' , p ' be t h e c o n t r a c t e d

p'.

Then t h e second s t e p i s t h e Thus we o b t a i n t h e generalized

In Case 1, we need t o blow up one e x t r a p o i n t .

Case 2

can be handled s i m i l a r l y , but we have t o blow up two e x t r a p o i n t s . Let us go t o t h e notion of a Jonquieres transformation.

I t i s defined t o be a

Cremona plane transformation defined by an i r r e d u c i b l e l i n e a r system L o f curves o f degree d having a (d-1)-ple p o i n t , d

2.

We have seen above t h a t t h e c a s e

d = 2 i s nothing but a usual o r a generalized q u a d r a t i c transformation, and we s h a l l mainly observe t h e c a s e d > 2 .

By t h e genus formula, we s e e t h a t an irre-

d u c i b l e curve o f degree d having a (d-1)-ple is a r a t i o n a l curve.

Computing t h e

149

Commutative algebra and algebraic geometry

dimension of L, o r variable intersections of members of L, we see that L must be a linear system having the (d-1)-ple point and 2d - 2 other base points. These base points may be infinitely near to some points, If one observes carefully, one sees that Jonquieres transformations are interpreted in terms of elementary transformations as follows : Take a point p1 of P2 and apply quadratic dilatation. Then we obtain a ruled surface F1 (under the notation of Theorem 3 . 4 ) . Let E be the total transform of pl. Apply successive elementary transformations with centers p2, ...,pd all on either E or its proper transforms and we obtain Fd. Then apply successive elementary transformations with centers P ~ + ~ , . , .p2d-l , all outside of proper transforms of E and we obtain another F1. Then F1 is contracted to P 2 . p1 is the (d-1)-ple point of L and p2, ..., are the other base points. (Only in P2d-l case d = 2 , we may take p2 outside of E and we obtain a usual quadratic transformation .) It terms of the linear system L, it is hard to describe the positions of p2, ..., p2d-l (because there is some restriction, as we saw even in the case d =2). But, in terms of elementary transformations, we can state the restriction to be that the elementary transformations can be done successively as described above. Furthermore, that p1 is a (d-1)-ple point of members of L implies that each line going through p1 meets each member of L at only one point (unless the member contains the line). This means that when we blow up pl, then on F1, every member of L becomes a section ( o r , a section t some generators). Therefore Jonquieres transformations are rather understandable Cremona transformations. The following version of Theorem 5.1 would be better than Theorem 5.3, though Theorem 5.3 looks more beautiful : Every birational mapping of a P2 to another P2 is factored into the product of Jonquieres transformation. THEOREM 5.4.

For the detail of these topics, see for instance, Nagata [13]

.

we pointed out in 12, our knowledge on higher dimensional cases is very little, and how to control the cases would be an important subject to be clarified in the future. As

6 . GROUP ACTIONS ON AFFINE RINGS

When we say that a group G acts on a ring R, we understand that there is given a homomorphism $ of G into Aut R ( =the automorphism group of R) and the action of G on R is defined by oa = ($o)a for u e G, a e R .

M.Nagata

150

We d e f i n e a r a t i o n a l a c t i o n a l i t t l e d i f f e r e n t l y .

Namely, i f G i s an a l g e b r a i c

group over a f i e l d k, R a f i n i t e l y generated r i n g over k and i f K i s an a l g e b r a i c a l l y closed f i e l d i n which we consider coordinates o f p o i n t s (k C-K), then by saying t h a t a r a t i o n a l a c t i o n o f G on R i s defined over k , we understand t h a t ( i ) G a c t s on RO k K i n t h e sense s t a t e d above and ( i i ) f o r each a e R and f o r bn of R = R f 1 each connected component Gi of G , t h e r e a r e elements b l , and r a t i o n a l functions f l ,

..., f n e k(Gi)

oa = l f i ( o ) b i f o r a l l o e G i .

...,

on t h e connected component Gi,

so t h a t

Note t h a t ( i i ) above i s n e a r l y equivalent t o

assuming t h e same f o r only one component, but n o t e x a c t l y . In our observations i n t h i s s e c t i o n , t h e r o l e o f k i s not important, and we consider t h e c a s e k = K f o r s i m p l i c i t y . The condition on t h e e x i s t e n c e o f f i and bi looks t o be s l i g h t , but i s r e a l l y very strong, because o f t h e following f a c t : THEOREM 6 . 1 .

I f an a l g e b r a i c group G a c t s r a t i o n a l l y on an a f f i n e r i n g R , then

f o r each a e R , t h e module M a PROOF.

loeG oaK i s a f i n i t e

With bi a s above, we have

1oeCO oaK E. 1b.K;

K-module. s i n c e t h e right-hand s i d e

i s a f i n i t e K-module, t h e left-hand s i d e i s a l s o a f i n i t e K-module.

applied t o any Ta( T B G), o f t h e index of Go.

The same i s

and t h e f i n i t e n e s s o f Ma follows from t h e f i n i t e n e s s

Q.E.D.

This r e s u l t means t h a t i f we t a k e a s e t o f g e n e r a t o r s a l , consider t h e sum M of t h e modules Mai,

...,

a f o r R and i f we m then M i s a r e p r e s e n t a t i o n module f o r G

and we o b t a i n a r a t i o n a l r e p r e s e n t a t i o n of G, i . e . a r e p r e s e n t a t i o n p of G i n CL(n,K) ( f o r some n) such t h a t every e n t r y p . . ( o ) of pa i s a r a t i o n a l f u n c t i o n on 11

Go;

i f N i s t h e kernel of t h e r e p r e s e n t a t i o n , then every element o f N a c t s

t r i v i a l l y on R. THEOREM 6 . 2 .

Thus : I f an a l g e b r a i c group G a c t s r a t i o n a l l y on a f i n i t e l y generated

r i n g R , then t h e a c t i o n i s given v i a an a f f i n e a l g e b r a i c group.

Namely, t h e r e is

a r a t i o n a l r e p r e s e n t a t i o n p of G i n GL(n; K) (with n a n a t u r a l number) and a s e t o f elements a l ,

..., an o f

R which generate R and such t h a t laiK i s a r e p r e s e n t a -

t i o n module f o r G and t h e a c t i o n o f each element

(I

o f G is given by

Note t h a t t h i s i s a very d i f f e r e n t phenomenon from t h e p r o j e c t i v e c a s e We add h e r e two remarks whose proofs a r e easy.

?he f i r s t i s

151

Commutative algebra and algebraic geometry THEOREM 6 . 3 .

Assume t h a t a r i n g R i s generated by al,...,an over t h e f i e l d K ;

that al,

a r e l i n e a r l y independent over K and t h a t a subgroup G of GL(n,K)

...,a

i s given so t h a t f o r each p B G

(a,,

. ..,an)

(aly

. . . ,an)v

Let G* be t h e c l o s u r e o f G i n GL(n,K) (under t h e

defines a K-automorphism o f R . Zariski topology).

-

Then G* is an a l g e b r a i c group and t h e a c t i o n o f G i s n a t u r a l l y

extended t o t h a t o f G*. The o t h e r is r e l a t e d t o t h e f i e l d s o f r e f e r e n c e s and t h e r e f o r e we must consider t h e case k # K . Assume t h a t a r a t i o n a l a c t i o n o f an a l g e b r a i c group G on a f i n i t e l y

THEOREM 6 . 4 .

generated r i n g R = k[a,,

. . .,an)

i s defined over t h e f i e l d k .

If k ' is a field

k ' c K , then t h e r a t i o n a l a c t i o n is n a t u r a l l y extended t o t h a t o f

such t h a t k

G on R O k k ' , and i n t h i s c a s e i f u e G i s a k ' - r a t i o n a l p o i n t , then u ' d e f i n e s

a k'-automorphism of R e k ' . k

This i s s i m i l a r l y a p p l i e d t o an extension o f K .

A s f o r group a c t i o n s , t h e r i n g s of i n v a r i a n t s play an important r o l e on s e v e r a l

occasions. In general, i f a group G a c t s on a r i n g R, i . e . t h e r e is given a homomorphism J1 of G i n t o Aut R and t h e a c t i o n i s defined so t h a t ua = $ua f o r any u e G, then t h e G s e t of G-invariants i n R forms a subring of R . The subring i s denoted by R

.

Under t h e given circumstances, i f N i s a normal subgroup of G, then N G a c t s n a t u r a l l y on R and t h e a c t i o n i s p r a c t i c a l l y an a c t i o n o f G / N . LEMMA 6 . 5 .

I f R i s a f i n i t e l y generated r i n g over a noetherian r i n g k and i f R

LEMMA 6.6.

i s i n t e g r a l over i t s subring S containing k , then S i s f i n i t e l y generated over k.

In p a r t i c u l a r , i f a f i n i t e group G a c t s on a f i n i t e l y generated r i n g R over a G noetherian r i n g k so t h a t t h e a c t i o n of G on k i s t r i v i a l , then R i s f i n i t e l y generated over k . Proofs a r e easy, and we omit them.

Our main i n t e r e s t l i e s i n t h e case where an a l g e b r a i c group G a c t s r a t i o n a l l y on a f i n i t e l y generated r i n g R over a ground f i e l d k .

As we can s e e e a s i l y by our

observation above, t h e c a s e i s p r a c t i c a l l y equivalent t o t h e following case : (1)

G i s a subgroup of GL(n,K) f o r some n ;

(2)

al, an generate t h e r i n g R over K (with t h e same n a s above), and a r e l i n e a r l y independent over K ;

(3)

For any u

...,

(al,.

6

G, t h e automorphism o f R defined by o maps ( a l ,

. .,aJu .

...,an)

to

M.Nagata

152

Here K denotes as before an algebraically closed field in which we consider coordinates of points. Under the circumstances, the closure G* of G in GL(n,K) under the Zariski topology is an algebraic group; and G* acts rationally on R. Furthermore, RG = RG* . We recalled this formulation for two reasons. One is that we also have to discuss the case where k # K. The other is to explain one classical proof of the following fact. Here, we define as follows : linear algebraic group G is said to be linearly reductive if every rational representation of G is completely reducible.

A

A linear algebraic group G is said to be geometrically reductive (or simply

reductive) if the radical of G is a t o r u s group. It is known that a linear algebraic group G is linearly reductive if and only if either (i) the ground field is of characteristic zero and the radical of G is a torus group; or (ii) the ground field is of non-zero characteristic, say p, and the connected component Go of 1 of G is a torus group and furthermore [G : Go] is not a multiple of p. As for geometrically reductive groups, the validity of the Mumford conjecture below, proved by Haboush [4 ] , is important.

Mumford conjecture. Let G be a linear algebraic group, G E GL(n,K), such that its radical is a torus group and assume that G acts rationally on an affine ring R over K. If M is a G-stable finite K-module and if f is an element of R which is G-invariant modulo M, i.e. f - af e M for any u e G, then there is a G-invariant g of the form fm + clfm-’ + , , + c with coefficients ci in the K-module Mi = m the K-module generated by { nl...n. I n e M 1 , 1 j

.

THEOREM 6.7. If G is a linearly reductive algebraic group and if G acts rationally on a finitely generated ring R over the field K, then RG is finitely

generated over K. A proof of Theorem 6 . 7 in the classical case (see Weyl [24]

, cf.

1

Nagata [12 ) proceeds along the following lines : On one hand, we prove the theorem in the case that G is a compact Lie group (instead of an algebraic group) where the case can be proved using inte grations; another basis for the proof is that any linearly reductive algebraic group G with K = (the complex number field) contains a compact Lie subgroup which is dense in G under the Zariski topology.

Commutative algebra and algebraic geometry

153

Although this itself cannot be applied to the positive characteristic case, the idea can be adapted in many cases. A proof o f Theorem 6.7 good for an arbitrary characteristic case was given by Nagata [18] by proving the following : THEOREM 6.8. Under the circumstances of Theorem 6.7, (i) if I is an ideal of RG, then I R n R G = I ; and (ii) if $ is a G-admissible K-homomorphism of the ring R onto another ring R', then it holds that $(RG) = RIG, generalization of these results to the geometric reductive case was given by Nagata [17] ; Theorem 6.7 with the same conclusion, namely, Theorem 6.10 below; and Theorem 6.8 with modified conclusion, namely Theorem 6.9 below. A

THEOREM 6.9. Assume that a geometrically reductive group G acts rationally on a finitely generated ring R over the field K. (i) If hl,...,hr are elements of (ii) If $ is a G-admissible RG , then ( lhiR)nRG is nilpotent modulo lhiRG K-homorphism of the ring R onto another ring R' and if h e RIG, then there is a G power hS of h so that hS e $(RG). Consequently, R" is integral over @(R ) .

.

THEOREM 6.10. K.

Under the circumstances as above, RG is finitely generated over

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