Geometric theory of algebraic space curves (Lecture notes in mathematics ; 423)

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

423 S. S. Abhyankar A. M. Sathaye

Geometric Theory of Algebraic Space Curves

Springer-Verlag Berlin.Heidelberg New York 1974

Prof. Dr. Shreeram Shankar Abhyankar Purdue University Division of Mathematical Sciences West Lafayette, IN 4?907/USA Prof. Dr. Avinash Madhav Sathaye University of Kentucky Department of Mathematics Lexington, KY 40506/USA

Library of Congress Cataloging in Publication Data

Abhy~Lkar~ Shreeram Shankar. Geometric theory of algebraic space curves. (Lecture notes in mathematics ; 423) Includes bibliographical references and indexes. i. Curves, Algebraic. 2. Algebraic varieties. I. Sathaye~ Avinash Madhav, 1948joint author. Iio Title. III. Series: Lecture notes in mathematics (Berlin) ; 423. Q~3.L28 no. 423 [QA567] 510'.8s [516'.35] 74-20717

A M S Subject Classifications (1970): 14-01, 14 H 99, 14 M 10

ISBN 3-540-06969-0 Springer-Verlag Berlin • Heidelberg • New York ISBN 0-387-06969-0 Springer-Verlag New York • Heidelberg • Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1974. Printed in Germany Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

PREFACE

The o r i g i n a l Montreal

Notes

(36.9),

namely

m a i n part of this b o o k w a s

[ 3 3that

The m a i n

"All i r r e d u c i b l e

degree

at most

ground

field are c o m p l e t e

completely

five and genus

proved

and circulated, self-contained of the proof,

it had

nonsingular

at most one over

in 1971.

published.

We

obsolete

space

intended

or rather,

the p r e p a r a t o r y

material,

finally

started

1973,

a completely

in the process,

and s o m e w h a t

in June

closed

the T h e o r e m A w a s

to give

and

clearer

of

of the proof h a v e b e e n w r i t t e n

of the Theorem,

to b e c o m e

curves

an a l g e b r a i c a l l y

treatment

October

by p r o v i n g

1973, that

however,

the size

enlarged;

sharper.

and w e

Murthy

while

the

The p r e s e n t

finally

at m o s t one o v e r an a l g e b r a i c a l l y

plete

intersections.

Conjecture" k

his p r o o f

[123-

decided

that

is a l g e b r a i c a l l y

closed

modules

closed)."

over

detailed

proofs

itself,

by u s i n g

a concrete

three

elements

given

in the M o n t r e a l

curves

the w e l l k n o w n

can sometimes

are

"Serre's

free in

be more u s e f u l

description

of a basis

nonsingular

[ 3 3 as one of the main

of

field are com-

k[X,Y,Z~

for the ideal of an i r r e d u c i b l e Notes

ground

space

H o w e v e r he also i l l u s t r a t e d

concrete

the t h e o r e m

our m a i n T h e o r e m

nonsingular

In fact he proved

that all p r o j e c t i v e

that,

rendered

"All i r r e d u c i b l e

genus

than

to the T h e o r e m

As such,

Two v e r s i o n s

to the

to be a book.

During

(when

the p r o o f

intersections."

but none

proof c o n t i n u e d version was

part was

just a s e q u e l

steps

space curve in his

proof. Another genus

and d i f f e r e n t i a l s

function developed 1973.

important

feature

of the book

of a s e p a r a b l y

field over an a r b i t r a r y by A b h y a n k a r

One v i r t u e

any a r t i f i c i a l

during

the

of the p r e s e n t

devices

is a new t r e a t m e n t

of the

generated

one-dimensional

field.

The t r e a t m e n t was

ground Purdue

Seminar

treatment

such as r e p a r t i t i o n s

is,

in S u m m e r

that it does

or d e r i v a t i o n s

of

and Fall not need of the

IV

g r o u n d field.

(See C h a p t e r III for the t r e a t m e n t and §40 for the com-

parison with other treatments.) R e t u r n i n g to the proof of the c o m p l e t e note that it has b a s i c a l l y two parts: p r o j e c t i o n of the space curve w i t h (26.12))

i n t e r s e c t i o n theorem, we

one part is to o b t a i n a "nice"

a "nice" a d j o i n t

(our T h e o r e m

and the second part is to c o n s t r u c t a basis of two elements

for the ideal of the curve

(our T h e o r e m

(36.7)).

In the present v e r s i o n of the proof,

the second part that is

needed was already d e v e l o p e d in [ 3 ], but w e include a p r o o f for the sake of completeness. namely,

In an older version,

a g e n e r a l i z a t i o n was needed;

the t r e a t m e n t of the s o - c a l l e d chains of e u c l i d e a n domains,

and

is p r e s e n t e d in §39, m a i n l y b e c a u s e it is of interest in itself. The t r e a t m e n t of the first part about p r o j e c t i o n s different

from its c o u n t e r p a r t in ~ 3 3-

is e s s e n t i a l l y

in [ 3 I, w e w r i t e down expli-

cit e q u a t i o n s of the curve and carry out the p r o o f by c o m p l i c a t e d calculations;

here a s i m i l a r m e t h o d was

too cumbersome.

W h a t we p r e s e n t h e r e is a g e o m e t r i c a r g u m e n t in w h i c h

w e never e v e n need a c o o r d i n a t e system. to c o n v i n c e anybody,

self-contained

However,

that this is geometry,

a v o i d e d the use of g e o m e t r i c terms, ours,

first tried and proved to be

it m i g h t be d i f f i c u l t

for we h a v e d e l i b e r a t e l y

so that the proof may stay rigor-

and still r e a s o n a b l y short.

Thus we h a v e taken the useful g e o m e t r i c concepts,

translated

t h e m into precise a l g e b r a i c terms and a l m o s t never gone b a c k to the g e o m e t r i c terms.

For a g e o m e t r i c m i n d e d reader, however, we h a v e pro-

v i d e d a d i c t i o n a r y in §43 so that he may be able to read the underlying g e o m e t r i c 'geometry'

argument very easily.

The name of this b o o k owes its

to this arrangement.

The p r o o f of e x i s t e n c e of the "nice p r o j e c t i o n " may be a p p r o p r i a tely d e s c r i b e d as repeated a p p l i c a t i o n s of "Bezout's Theorem." need b a s i c a l l y the special

(but m o s t w e l l known)

"Bezout's Little Theorem" in the present book,

case,

namely,

We

termed as the case of the

intersection

of a h y p e r s u r f a c e

§ (23.9).

The general

presented

in §38;

case

mainly

because

Bezout's

is, however,

study

to e m b e d

jective

terminology

Chapter

II,

is treated

in

is

p r o o f of this

To avoid clashes only

get a p r o j e c t i v e

counterpart

of the

(§26)

theorem.

and

This made

in a p r o j e c t i v e

concentrate

space,

in affine

on p r o j e c t i v e

then return

theorem

it

and

and procurves

in

about exist-

to affine

curves

in

IV.

Bezout's

theorem

multiplicity;

also needs

for p r o j e c t i v e

We d e v e l o p

the

for affine

curves

theory

in C h a p t e r

is to provide

a precise

as w e l l

for p r o j e c t i v e

The only use of C h a p t e r cerned

curve

first

of nice p r o j e c t i o n s

Chapter

accessible

a projective

completion. we

case

literature.

the g i v e n affine

its p r o j e c t i v e

This

of two h y p e r s u r f a c e s

a readily

in the

Theorem

a curve.

of i n t e r s e c t i o n

is not a v a i l a b l e

necessary

ence

case

with

III,

I §5,6;

theory of i n t e r s e c t i o n

as affine curves

curves,

in C h a p t e r

in our case. II §23,24

and put it t o g e t h e r

so far as c o m p l e t e

a p r o o f of the w e l l k n o w n

in c h a p t e r

intersections genus

and

formula

are

IV.

con-

for plane

curves. We h a v e their

taken

coordinate

coordinate sections

rings

appear

they d e s c r i b e Here

to be almost

spaces

abstract, in case

in d u p l i c a t e

results

for these

is a general

summary

of various

I gives

the theory

essentially

some general

Chapter

cepts

if they are

of a m b i e n t

at a point,

contains

sent

rings,

that v a r i e t i e s

the same

Chapter curves

the v i e w p o i n t

II gives

abstract

for p r o j e c t i v e

curves.

they

in the

are embedded.

(§5,6;

§23,24

two types

by

etc.)

Several because

of varieties.

chapters. multiplicity

or local

curves.

of two It also

terminology. a treatment

irreducible

of i n t e r s e c t i o n

and by ideals

of i n t e r s e c t i o n for affine

are r e p r e s e n t e d

of

projective

multiplicity,

"homogeneous varieties. projection,

domains"

which

It d e v e l o p s tangential

repre-

the con-

spaces

etc.

VI

C h a p t e r III gives a t r e a t m e n t of d i f f e r e n t i a l s generated

function fields.

in s e p a r a b l y

It also has various genus

a b s t r a c t and e m b e d d e d p r o j e c t i v e curves.

formulas

for

This c h a p t e r is almost

s e l f - c o n t a i n e d except for some use of C h a p t e r I and some a l t e r n a t i v e proofs u s i n g C h a p t e r II. C h a p t e r IV studies affine irreducible curves w i t h an equivalence class of affine c o o r d i n a t e systems preassigned; algebraically,

translated

such curves have c o o r d i n a t e rings w h i c h are

ered domains"

"filt-

We also study the concepts of taking a p r o j e c t i v e

c o m p l e t i o n and taking an affine piece; tion and d e h o m o g e n i z a t i o n . the main T h e o r e m

algebraically,

homogeniza-

Then we go on to finish the proof of

(36.9).

C h a p t e r V is a supplement.

It deals w i t h g e n e r a l i z a t i o n s

some concepts of the first four chapters

of

and has several statements

w h o s e p r o o f are only s k e t c h e d or referred to other sources. An e l e m e n t a r y k n o w l e d g e of general algebra is assumed to be a v a i l a b l e to the reader Lemma'

(for example,

results

, "Krull's I n t e r s e c t i o n Theorem',

There is only one

"official exercise"

like

'Nakayama's

'~ eif i = n formula'

etc.).

(in §15), but several pro-

perties stated in C h a p t e r I, II and IV may very w e l l be treated as exercises w i t h v a r y i n g degree of difficulty. The contents

are intended to give b r i e f d e s c r i p t i o n s

in

g e o m e t r i c w o r d s of w h a t is b e i n g treated in the r e l e v a n t sections. A list of i n t e r d e p e n d e n c e s of sections

follows the contents.

S h r e e r a m S. A b h y a n k a r A v i n a s h Sathaye

CONTENTS CHAPTER

I.

LOCAL

GEOMETRY

OR LENGTH

§l.

General

§2.

Principal

§3.

T o t a l q u o t i e n t r i n g and c o n d u c t o r . (3.1). L o c a l i z a t i o n of the c o n d u c t o r .

§4.

N o r m a l model. (4.1). Divisor (4.2). Divisor (4.3) . The " ~

§5.

Length affine curve. (5.1 (5.2 (5.3 (5.4)

.

(5.5 (5.6 (5.7

(5.8) (5.9

(5.10 (5.1l) (5.12).

.

terminology. ideals

and p r i m e

ideals.

of a f u n c t i o n . of zeros of a f u n c t i o n . e.f. = n " formula. ll

in a o n e - d i m e n s i o n a l n o e t h e r i a n domain, or i n t e r s e c t i o n m u l t i p l i c i t y on an i r r e d u c i b l e V a l u e s of local i n t e r s e c t i o n m u l t i p l i c i t y . V a r i o u s cases. L o c a l e x p a n s i o n of i n t e r s e c t i o n m u l t i p l i c i t y over a divisor. Global intersection multiplicity. L o c a l e x p a n s i o n of length. I n t e r s e c t i o n m u l t i p l i c i t y e q u a l s l e n g t h in the i n t e g r a l c l o s u r e (R*) . Local intersection multiplicity equals a l e n g t h (in R) for a p r i n c i p a l ideal. G l o b a l i z a t i o n of (5.6). S p e c i a l case of (5.6) - the n o r m a l case. S p e c i a l case of (5.7) - the n o r m a l case. D e f i n i t i o n . M u l t i p l i c i t y of a local d o m a i n of d i m e n s i o n one. D e f i n i t i o n and p r o p e r t i e s . C o n d u c t o r , its length; and a d j o i n t s . L e m m a on o v e r a d j o i n t s .

§6.

L e n g t h in a o n e - d i m e n s i o n a l noetherian homorphic image, or a f f i n e i n t e r s e c t i o n m u l t i p l i c i t y on an e m b e d d e d i r r e d u c i b l e curve. (6.1). V a l u e s of local i n t e r s e c t i o n m u l t i p l i c i t y . V a r i o u s cases. (6.2). L o c a l e x p a n s i o n of i n t e r s e c t i o n m u l t i p l i c i t y in the p r e i m a g e . (6.3). Global intersection multiplicity. (6.4). Case of a l g e b r a i c a l l y c l o s e d g r o u n d field (6.5). Case w h e n a c u r v e is t h o u g h t to be e m b e d d e d in itself. (6.6) to (6.9). R e s t a t e m e n t s of (5.6) to (5.9) for the case of a h o m o m o r p h i c image.

20

§7.

A c o m m u t i n g lemma for length.-(7.1). F o r two e m b e d d e d i r r e d u c i b l e curves, at a c o m m o n s i m p l e point, the i n t e r s e c t i o n m u l t i p l i c i t y of e i t h e r one w i t h the o t h e r is the same. (7.2). G l o b a l i z a t i o n of (7.1) o v e r a d i v i s o r .

27

VIII

(7.3)

C o m p l e t e g l o b a l i z a t i o n of

(7.1).

§8.

L e n g t h in a t w o - d i m e n s i o n a l regular local d o m a i n . . . . . . (8.1). I n t e r s e c t i o n m u l t i p l i c i t y of curves e m b e d d e d in a regular surface. Local case. (8.2). For two curves e m b e d d e d in a regular surface, the i n t e r s e c t i o n m u l t i p l i c i t y of either one w i t h the other is the same. Local case. (8.3). A d d i t i v i t y of i n t e r s e c t i o n m u l t i p l i c i t y of curves e m b e d d e d in a regular surface. Local case.

28

§9.

M u l t i p l i c i t y in a regular local domain (9.1). M u l t i p l i c i t y of an irreducible curve (embedded in a regular surface) at a point is the order of its d e f i n i n g equation. (9.2). T e c h n i c a l lemma for (9.1).

30

§i0. Double points of algebraic curves i0.i). Theorem. D e s c r i p t i o n of a double point of a curve. 10.2). Lemma. D e s c r i p t i o n of h i g h nodes. 10.3). Lemma. D e s c r i p t i o n of h i g h cusps. 10.4). Lemma. D e s c r i p t i o n of n o n r a t i o n a l h i g h cusps.

33

C H A P T E R II.

88

PROJECTIVE G E O M E T R Y OR H O M O G E N E O U S DOMAINS . . . . . .

§ii. F u n c t i o n fields and p r o j e c t i v e models.

86

§12. H o m o g e n e o u s h o m o m o r p h i s m .

68

§13. H o m o g e n e o u s ideals and h y p e r s u r f a c e s

68

(projective varieties)

§14. H o m o g e n e o u s subdomains, flats (linear varieties), projections, b i r a t i o n a l projections, and cones. (14.1). D i m e n s i o n and e m b e d d i n g d i m e n s i o n of a h o m o g e n e o u s subdomain. (14.2) and (14.3). D i m e n s i o n and e m b e d d i n g d i m e n s i o n of a (homogeneous) h o m o m o r p h i c image.

70

§15. Zeroset and h o m o g e n e o u s l o c a l i z a t i o n . _ (15.1), (15.2) and (15.3). E x t e n s i o n to (homogeneous) localization. (15.4) and (15.5) A l t e r n a t i v e (affinized) d e s c r i p t i o n of the (homogeneous) localization. (15.6). C o r r e s p o n d e n c e b e t w e e n h o m o g e n e o u s prime ideals and h o m o g e n e o u s localization. (15.7), (15.8) and (15.9). R e s t a t e m e n t of (15.1), (15.2) and (15.3) for e m b e d d e d varieties. (15.10). Lemma. Number of conditions imposed on a linear s y s t e m of h y p e r s u r f a c e s .

75

§16. H o m o g e n e o u s c o o r d i n a t e systems.

85

IX

§ 17.

§18.

§ 19.

§2o.

§21.

§22.

Polynomial rings as h o m o g e n e o u s domains . . . . . . (iZl) to (17.5). E q u i v a l e n t d e s c r i p t i o n s and p r o p e r t i e s of h o m o geneous domains w h i c h are p o l y n o m i a l rings over a field. O r d e r on an e m b e d d e d ( i r r e d u c i b l e ) curve and i n t e g r a l projections. (18.1) and (18.2). O r d e r of a h y p e r s u r f a c e at a v a l u a t i o n of an e m b e d d e d curve. (18.3) and (18.4). O r d e r of an ideal at a v a l u a t i o n of an e m b e d d e d curve. (18.5). Zerosets of ideals. (18.6) to (18.10). O r d e r at a v a l u a t i o n of an e m b e d d e d curve behaves like a valualtion. (18.11). P r o j e c t i o n lemma. P r o j e c t i o n of v a l u a t i o n and order, from a v e c t o r space. (18.12). Projection lemma. P r o j e c t i o n of v a l u a t i o n and order, from a flat (linear variety). (18.13). Corollary-definition. C o n d i t i o n for a ~ - i n t e g r a l p r o j e c t i o n (where ~ is a hyperplane). (18.13.1) . S p e c i a l case of ( 1 8 . 1 3 ) - p r o j e c t i o n from a c e n t e r not m e e t i n g the curve. O r d e r on an a b s t r a c t (irreducible) curve and integral p r o j e c t i o n s . (19.1) to (19.12). V e r s i o n s of (18.1) to (18.12) w h e n a curve is t h o u g h t of as e m b e d d e d in itself. (19.13) and (19.13.1). V e r s i o n s of (18.13) and (18.13.1) for an a b s t r a c t curve. (19.14). Remark. "Integral"ness of p r o j e c t i o n commutes w i t h h o m o m o r p h i c image. V a l u e d v e c t o r spaces. (20.1) to (20.13). S t r u c t u r e and p r o p e r t i e s v e c t o r space.

86

88

93

95

of a v a l u e d

O s c u l a t i n g flats and integral p r o j e c t i o n s of an e m b e d d e d (irreducible) curve. (21.1) D e f i n i t i o n and s t r u c t u r e of o s c u l a t i n g flats. (21.2), (21.3) and (21.4). A p p l i c a t i o n of §20 to the p r o p e r t i e s of o s c u l a t i n g flats. (21.5). Properties of o s c u l a t i n g flats in special cases. (21.6). C o n d i t i o n for integral p r o j e c t i o n in terms of o s c u l a t i n g flats at the center of projections. O s c u l a t i n g flats and integral p r o j e c t i o n s of an a b s t r a c t (irreducible) curve. (22.1) to (22.6). R e s t a t e m e n t s of (21.1) to (21.6) w h e n a curve is thought of as e m b e d d e d in itself (22.7). Remark. O s c u l a t i n g flats commute w i t h h o m o m o r p h i c image.

109

118

X

§23.

I n t e r s e c t i o n m u l t i p l i c i t y w i t h an e m b e d d e d (irreducible projective) curve. (23.1), (23.2) and (23.3). Properties of i n t e r s e c t i o n m u l t i p l i c i t y w i t h an embedded curve. (23.4). Case of a l g e b r a i c a l l y closed ground field. (23.5). A d d i t i v i t y of i n t e r s e c t i o n m u l t i p l i c i t y . (23.6). I n t e r s e c t i o n m u l t i p l i c i t y equals length for a p r i n c i p a l ideal. (23.7). All points of an e m b e d d e d line are simple. (23.8) and (23.9). Bezout's Little Theorem. Definition. D e g r e e of an elabedded (irreducible) curve. Remark. Affine i n t e r p r e t a t i o n of degree. (23. i0). Lemma. If there are enough rational points (23. ii). then e m b e d d i n g d i m e n s i o n of an embedded curve is less than or equal to its degree. Lemma. If the degree of an embedded curve (23. 12). is one, then its e m b e d d i n g d i m e n s i o n is one. Remark. H y p e r p l a n e s h a v e degree one. (23. 13). P r o j e c t i o n formula. P r o j e c t i o n of v a r i e t i e s (23. 14). from flats. Special p r o j e c t i o n formula. Degree of (23. 15). the projection. Remark. Case of an a l g e b r a i c a l l y closed (23. 16). ground field. Lemma. B i r a t i o n a l i t y of the p r o j e c t i o n from (23. 17). the generic point on a line. D e f i n i t i o n and p r o p e r t i e s of tangents to (23. 18). an e m b e d d e d curve. C o m m u t i n g len~as. V e r s i o n s of (23. 19) and (23.20). (7.1) and (8.2) for p r o j e c t i v e curves.

§24.

I n t e r s e c t i o n m u l t i p l i c i t y w i t h an a b s t r a c t (irreducible) curve. (24.1) to (24.17). V e r s i o n s of (23.1) to ( 2 3 . 1 7 ) for an a b s t r a c t curve. (24.18). D e f i n i t i o n and p r o p e r t i e s of tangents to an a b s t r a c t curve. (24.19). Remark. Relations b e t w e e n an e m b e d d e d curve [A,C] and an a b s t r a c t curve A/C. (24.20). Lemma on overadjoints. Projective v e r s i o n of (5.12). (24.21). Lemma on u n d e r a d j o i n t s . E x i s t e n c e of c e r t a i n type of p r o j e c t i v e u n d e r a d j o i n t s w h i c h are true adjoints in an affine piece.

§25.

T a n g e n t cones and q u a s i h y p e r p l a n e s . (25.1). Definition. L e a d i n g form of a h y p e r s u r f a c e . (25.2) and (25.3). Le~ma-definition. Definition and p r o p e r t i e s of tangent-cones. ~-quasihyperplane. (25.4) . Definition. C h a r a c t e r i z a t i o n of ~ - q u a s i h y p e r p l a n e s . (25.5) . Lemma. A h y p e r p l a n e (different from ~) (25.6) . Lemma. is a ~ - q u a s i h y p e r p l a n e . Lemma. Quadric ~ - q u a s i p l a n e s . (25.7) . Lemma. V e r s i o n of (9.1) for p r o j e c t i v e curves. (25.8) . Lemma. D e g r e e of an e m b e d d e d plane curve is (25.9). the d e g r e e of its d e f i n i n g equation.

121

138

148

XI

(25. i0) . (25.11) . (25.12) . (25.13) . (25.14) .

§26.

Lemma. Characterization o f t a n g e n t lines of plane projective curves. Definition Intersection multiplicity of two hypersurfaces. A d d i t i v i t y of the i n t e r s e c t i o n m u l t i p l i c i t y of two hypersurfaces. Lemma. V e r s i o n o f (8.2) f o r p r o j e c t i v e curves. Bezout's Theorem. Intersection of two plane projective curves.

2-equimultiple plane projections of projective space quintics. (For n o t a t i o n see b e g i n n i n g of §26.) (26.1). Lemma. Most lines through a d-fold point o f a n i r r e d u c i b l e c u r v e are d - s e c a n t s . A l s o , if d < d e g r e e of the curve, t h e n t h e r e are (d l l ) - c h o r d s t h r o u g h the p o i n t in e v e r y p l a n e t h r o u g h the p o i n t . Lemma. A n i r r e d u c i b l e c u r v e of d e g r e e m 2 26.2). h a s 2 - c h o r d s in e v e r y p l a n e . 26.3) Lemma. S u f f i c i e n t c o n d i t i o n for e x i s t a n c e of 4-chords. 26.4) Lemma. S u f f i c i e n t c o n d i t i o n for e x i s t a n c e of 2-secants. 26.5) Lemma. P r o j e c t i o n f r o m p o i n t s o n an (n-l)-secant. 26.6) Lemma. Projection from points on 2-secants o f an i r r e d u c i b l e q u a r t i c . 26.7) Lemma. P r o j e c t i o n f r o m p o i n t s on c e r t a i n 3-secants. 26.8) Cone Lemma. 26.9). Plane Lemma. (26.10) . Q u a d r i c L e m m a . (26.11). Proposition. Detailed description of proj e c t i o n s of c u r v e s of d e g r e e at m o s t 5. (26.12) . T h e o r e m . C o n d e n s e d v e r s i o n o f (26.11) for r e f e r e n c e .

CHAPTER

III.

BIRATIONAL

GEOMETRY

155

180

OR GENUS_

§27.

Different. (27.1) . D e f i n i t i o n s . (27.2) . D e d e k i n d ' s f o r m u l a . "{y = ~ dx " (27.3) . L e m m a . Condition for an u n r a m i f i e d e x t e n s i o n . (27.4) . T e c h n i c a l lemma. (27.5) . L e m m a . Another characterization of an unramified extension. Description of an unramified extension. (27.6). F i n i t e n e s s o f the s e t o f (27.7) . L e m m a . ramified primes. Separably generated (27.8) . D e f i n i t i o n . function fields. (27.9) . L e m m a . Conditions for separably generated function fields.

180

§28.

Differentials. (28.1). ~formulatiQn of ( 2 7 . 2 ) in t e r m s ~ivlsors. i n t e g r a l case.

189

of

XII

(28.2).

C o n v e r s i o n f o r m u l a for d i f f e r e n t . I n t e g r a l case. (28.3) and (28.4). E x c h a n g e lemmas. (28.5). G e n e r a l c a s e of (28.2). (28.6). G e n e r a l c a s e of (28.1). (28.7). Definition. ordv(~,x) ; (~,x) (28.8)

to

(28.11). (28.12) . (28.13) . (28.14) . (28.15) . (28.16) . (28.17) . (28.18) . (28.19). (28.20) . (28.21). (28.22) .

i n t e n d e d to be r e p l a c e d by ~dx. (28.10). Lemma. (~,x) behaves so far as ord V is c o n c e r n e d .

~dx,

T e c h n i c a l d e f i n i t i o n of genus. Theorem. A r a t i o n a l curve h a s genus zero Definition. Usual differentials and their properties. U s u a l d e f i n i t i o n of genus. Genus f o r m u l a s . Remark. A l t e r n a t i v e p r o o f to a genus formula. Remark. Definition. Uniformizing parameter and c o o r d i n a t e . Lemma. P r o p e r t i e s of u n i f o r m i z i n g coordinates. Example. Valuations with unseparable r e s i d u e fields. Remark. Lemma. (28.19) r e f o r m u l a t e d u s i n g differentials.

§29.

Genus of an a b s t r a c t curve. (29.1) and (29.2). Genus f o r m u l a s for p l a n e p r o j e c t i v e curves. R a t i o n a l i t y of a c u r v e of genus zero. (29.3). Remark. (29.4). D i r e c t p r o o f of r a t i o n a l i t y of a conic. (29.5). D i r e c t p r o o f of r a t i o n a l i t y of a line. (29.6). D i r e c t c o m p u t a t i o n of the genus of a cubic. (29.7). Theorem. A p p l i c a t i o n of (24.21) to p l a n e p r o j e c t i v e c u r v e s of genus ~ 1 and d e g r e e 4 or 5.

220

§30.

G e n u s of an e m b e d d e d curve. (30.1) and (30.2). Genus f o r m u l a s . of a c u r v e of genus zero. (30.3). Theorem. Combined version and (26.12) for r e f e r e n c e .

231

CHAPTER

IV.

AFFINE

GEOMETRY

domains.

OR FILTERED

Various

Rationality of

(29.7)

DOMAINS-

definitions.

234 234

§31.

Filtered

§32.

Homogenization or t a k i n g p r o j e c t i v e c o m p l e t i o n . (32.1). Definition. Degree (32.3) to (32.8). P r o p e r t i e s of h o m o g e n i z a t i o n .

236

§33.

Dehomogenization or t a k i n g an a f f i n e p i e c e (33.1). Definition. Dehomogenization (33.2) to (33.4). P r o p e r t i e s of d e h o m o g e n i z a t i o n

238

XIII

§34.

Relation b e t w e e n h o m o g e n i z a t i o n and d e h o m o g e n i z a t i o n . . . .

241

§35.

P r o j e c t i o n of a filtered domain. (35.1). Definition. Projection. (35.2). R e l a t i o n w i t h h o m o g e n i z a t i o n and dehomogenization. (35.3). Lemma. C o n d i t i o n s for integral projections. (35.4). Definition. Degree, genus. (35.5). Theorem. Affine v e r s i o n of (30.3) and (26.12) .

244

§36.

C o m p l e t e intersections. (36.1) . Definition. C o m p l e t e intersections, essentially hyperplanar. (36.2) . Lemma. E s s e n t i a l l y planar space curve is a c o m p l e t e intersection. (36.3) . C o r o l l a r y to (36.2). (36.4) . Lemma. A case of complete intersection. (36.5) . Theorem. A s u f f i c i e n t c o n d i t i o n for c o m p l e t e intersection. (36.6) . E l e m e n t a r y t r a n s f o r m a t i o n s . (36.7) . Theorem. Another sufficient condition for complete intersection. (36.8) . C o r o l l a r y to (36.7). (36.9) . Main theorem of complete intersection.

248

C H A P T E R V. §37.

257

APPENDIX.

Double points of a l g e b r o i d curves. t r e a t m e n t of most of §i0.

§38.

Bezout's

§39.

Chains of e u c l i d e a n curves. v e r s i o n of (36.7).,

§40. §41.

theorem.

An a l t e r n a t i v e 257 268

The general case

Treatments of d i f f e r e n t i a l s A short survey.

A generalized 274

in d i m e n s i o n one.

A g e n e r a l i z a t i o n of D e d e k i n d ' s c o n d u c t o r and d i f f e r e n t . _

280 formula about 281

§42.

The general adjoint condition.

285

§43.

G e o m e t r i c language. Geometric motivations behind the various notations.

287

§44.

Index to notations.

295

§45.

Index to topics.

298

I n t e r d e p e n d e n c e o f sections. In the following,

§b e §a I ..... §a r

d i r e c t l y referred to in §b. other p r e r e q u i s i t e s previous

sections

§l, §2, §3, §4,

for §b

Except

means

§al ..... §r

for such references,

are the notations

and d e f i n i t i o n s

and they may be located from the index.

§5,

basic.

are

the only from

XIV

§6 ~ §5.

§7 4- §6. §8, §9,

independent.

§10 ~ §s. §ii, §18 ~

§12,

§13,

§14,

§15,

§16,

§17, basic.

§15.

§19 ~ §18. §20 4- §18,§19. §21 ~ §18,§20. §22 ~ §21. §23 ~ §4,§5,§15,§17,§18. §24 4-- §5,§10,§15,§23. §25 ~ §8,§9,§15,§23. §26 ~ §18,§21,§23,§25. §27, basic. §28 ~ §27

(§3,§5,§23

and §25 optional use).

§29 ~ §4,§15,§17,§24,§25,§28. §30 ~ §25,§26,§29. §31,§32,

basic.

§33 4- §32. §34 ~ §15,§25,§32,§33. §35 ~- §19,§26,§30,§33,§34. §36 4- §10,§26,§30. §37 ~- §i0. §38 ~ §23,§25. §39 ~ §36. §40,

independent.

§41,

related to §28.

§42 ~ §29,§30 §43,§44,§45,

(also related index.

to §i0).

CHAPTER

I:

LOCAL

gEOMETRY

OR LENGTH

§i. We

shall

assume

(1.7),(1.8),(1.9) tions

and

I.

elementary

ful r e s u l t s

about

General

terminology.

the

terminology

This

includes

properties

local

introduced

general

algebraic

of m o d e l s

rings.

We

shall

in [i:

and

(1.1), (1.6),

terms,

defini-

some w e l l k n o w n

also

u s e the

and u s e -

following

nota-

tion. By card w e d e n o t e For E = a

subsets

module

positive generated

over

and A

we d e n o t e

For of

by

the

a module

n, b y

set

also put

Ix I + x 2 + . . . + X n :

E

x i ~ Ji}.

jn

[E

: A~ = sup{n:

a ring

note

that, if

dimension

of

there

exists

For

: A]

is e i t h e r

is a field, t h e n

E

over

a ring

A

n > 1

(A) = the

subgroup

a subset

J

of

of

In c a s e

is an

J

A

A

subgroup

A, b y

[E:A~

we denote

a sequence

of

ideal

the

length

of

E

E 0 c E 1 c E2c...cE n with

a nonnegative [E

: A3

integer

is s i m p l y

A.

Principal

ideals

and p r i m e

ideals.

we define:

set o f all

nonzero

principal

ideals

in

or

w.

Also

the v e c t o r - s p a c e -

w

F

is a

~ E2~.-.~En~

A

§2.

x i ~ Ji }.

i.e.,

E0 ~ E1 [E

by

the a d d i t i v e

x i e J}.

of A - s u b m o d u l e s

then

(for i n s t a n c e

integer,

additive

For

we denote

[XlX2...Xn:

over

as an A - m o d u l e ,

that

the

E

jO = A.

E

Note

A, w h e r e

we denote

[XlX2...Xn:

integer

group

is a p o s i t i v e

of a r i n g

JiJ2...Jn

set

n

set

by

generated A, w e

the

integer, the

of an a d d i t i v e

where

J i , J 2 .... 'Jn

a positive

in

a ring),

subsets

by

number.

J i , J 2 ..... Jn

Jl + J 2 + ' ' ' + J n For

cardinal

A,

1.2

~(A)

= the

~i(A)

set of all p r i m e

= [P c ~(A):

dim A/P

ideals

in

A

= i]

for any ~i(A,x)

= ~(A,x)

~(A,I)

I c p]

1

for any

= ~(A,I)

~([A, I3)

x ¢ A

,

N ~i(A)

= {p ~ ~(A):

~i(A,I)

,

N ~i(A)

I c A

,

= ~(A,I) for any

I ¢ A

or

I c A

,

~i ([A, I]) = ~i (A, I) and ~([A,I],J)

= ~(A,I)

~i([A,I~,J)

We

note

~(A)

that

= ~(A,0)

Z(A, IA) I c A rad

= ~

n 9(A,J)

([A,I~,J)

~

(A,I) = ~(A) ~ ~

I e A

or

I c A

i

and

J e A

or

J c A

any

,

I c A,

I =

for any

then

= 9([A,0],I) or

n ~i(A)

1

~

= 9([A,I],0)

and

for any

= ~([A,I])

ideals

I

and

= D(A,I) J

P I c

radA{0}

~(A,I)

=

I = A

~(A,I)

= ~(A,J)

~

9(A,I)

O 9(A,J)

= ~(A,I

n J)

~(A,I)

n ~(A,J)

= ~(A,I

+ J)

radAI

= radAJ =

A, IJ)

and

§3.

For

a nonnull

Total

quotient

ring

R

we

rlng

define

and c o n d u c t o r .

in

for any A

we have:

.

1.3

~(R)

= the

total

quotient

ring

of

R.

and we define ~(R) ~(R),and of

R

we

in

~(R)

For

= the c o n d u c t o r note ~(R), = the

we

largest

{X

~

R:

xR

=

{X

e

R

: xR

=

{x

c ~(R):

upon

in the

letting

ideal C

integral

R

to be

closure

the

of

R

integral

in

closure

c

C

in

R

which

remains

an ideal

in

R*

R} R}

xR

ideal

~([A,C])

upon

R

then have

=

a nonunit

and,

that,

of

c R}.

in a r i n g

A

we

define

= ~(A/C)

letting

f: A ~ A / C

to be

the c a n o n i c a l

epimorphism,

we

define

~([A,C~) We

take

= f-I(~(A/C)) note

of the

fact

that

the c o n d u c t o r

localizes

properly,

i.e.:

(3.1) i__nn ~(R) respect

If

R

is a d o m a i n

is a finite to some

model,

For a domain the =

For

~(R)

and

multiplicative

§4.

I(R,k)

such

R

and

set

of all

~ (V) =

local

ring.

= I (R, R)

R

we

S

Normal

in

define

and

integral

R, t h e n

closure r i n q of

~(R)S

=

of

R

R

with

(S).

model. k

of

subrings R)

the

is the q u o t i e n t

system

a subring

that

a domain

that

V V

R of

we ~(R)

define with

k c V

is a o n e - d i m e n s i o n a l

such

regular

1.4

and for any

Q e R

~(R,Q)

or

Q c R

we d e f i n e

= {V e ~(R) : O r d v Q

For a ring

A

and any

> 0}.

C e 9(A)

we define

([A,C3) = ~(A/C). F o r a ring f: A ~ A / C

A, C e ~(A),

and

to b e the c a n o n i c a l

ord[A,C~,vQ

Q e A

or

epimorphism,

= Ordvf(Q)

for any

Q c A, u p o n

letting

we d e f i n e

V ~ ~([A,C])

and w e d e f i n e

~([A,C~,Q) We observe

We note

= ~(f(A),

that,

if

field e x t e n s i o n )

(K,k)

K

c o m p l e t e m o d e l of

is a f u n c t i o n

m o d e l of K

k

over

sor o f a f u n c t i o n

i, t h e n

I__ff K

for e v e r y

is zero",

.

over k

(i.e. a f i n i t e l y t r d e g k K = i, t h e n

k

and:

I (K,k)

all w h o s e m e m b e r s

fact that "the d e g r e e

field o v e r a field

k

{ i I if if

x ~ 0 x = 0.

that b y c o n v e n t i o n

real n u m b e r

of the d i v i -

with trdegkK =

we have,

(ordVX) [V/M(V) : k~ =

positive

is the

are normal.

V e ~ (K,k)

We remark

gener-

i.e.:

is a f u n c t i o n x e K

field with

K

W e a l s o take n o t e of the w e l l - k n o w n

(4.1)

> 0~

f(Q))

o v e r a field

is a p r o j e c t i v e

unique

ord[A,C?,vQ

that t h e n

~([A,C~,Q)

ated

= {V ¢ ~ ( [ A , C ] ) :

t i m e s ~ = ~ times p o s i t i v e = ~ times ~--- c o

real n u m b e r

1.5

Further, upon

for a n y

family

a (U)us U

where

is an i n t e g e r

a(u)

or

letting

U'

= [u ¢ U:

a(u)

~ 0},

=

a(u)

< 0}.

u

= {u c U:

a(u)

= ~]

,

ww

u

(u ~ u:

by convention

we have

0

if a(u)

u~U'

a (u)

U'

=

if U' , U =

is a n o n e m p t y

finite

set and

if

U'

is a n o n e m p t y

finite

s e t and

if

U'

is a n i n f i n i t e

=

u~U

finite

We follows fact

recall from

that

(4.1)

(4.3)

about

(4.2)

If

extensions

K

U

~

is a

set.

follows

" ~ eif i = n "

set and

U

immediately

formula,

i.e.,

of Dedekind

is a f u n c t i o n

field

from

(4.2) w h i c h

in t u r n

f r o m the w e l l - k n o w n

domains.

over

a field

k

with

w

trdegkK

= I, then,

closure

of

k

upon

i__nn I<

letting

for every

(ordvX) [V/M(V) : k] VcI(K,k[xj)

(4.3) let

R

be

extension that

this

ring over

Let the K

S

be

integral o_~f

L.

assumption a field,

for every

Q c D0(S)

to be

the

(relative)

algebraic

x c K we have

EK- k ( x ) ~

i__ff x ~ k

0

if

=

,

t

domain

closure

S

Assume

of

that

R

is a u t o m a t i c a l l y if

system, we have

S

0 ~ x ¢ k

i_ff x = 0 .

a Dedekind

and also

to s o m e m u l t i p l i c a t i v e

k

with

quotient

in a f i n i t e is a f i n i t e satisfied,

algebraic S-module.

if

is the q u o t i e n t

o f an a f f i n e

"

field

S

and

field (Note

is an a f f i n e

ring,

ring over

L

with

respect

a field.)

Then

1.6

(OrdRpQ) [ R / P R p

: S/Q]

=

[K : L]

Pc ~0 (R)

§5.

Let subring

R

of

(*)

Length

be R

in a one-dimensional

a noetherian

such

domain with

P e 90(R)

we have

and

if

k

dim

domain.

R = i.

Let

k

be

a

that

for e v e r y

(Note t h a t

noetherian

(*) is s a t i s f i e d

is a s u b f i e l d

of

R

such

k N P e ~0(k)

[R/P

for

: k/(k

R = k;

that

[R/P

n p)] < ~

it is a l s o s a t i s f i e d ,

: k~

< ~,

f o r all

P e ~0(R).) Let

(**)

R

R

be

the

integral

is a f i n i t e

For any

I c R

xk(R,I,Q)

=

closure

of

R

in

~(R).

Assume

that

R-module.

or

I c R

and

any

Q £ R

~ (ordvI)[V/M(V) v e ~ (R,Q)

or

: k/(k

Q c R

we

define

n M(V))]

and X(R,I,Q)

and

for a n y

I c R

kk(R,I) (We n o t e equation ~V/M(V)

of

= IR(R,I,Q),

or

= kk(R,I,0)

that,

kk

and

if

closed

we have

kk(R,I,Q)

kk(R,I)

= X(R,I).)

k

we

define

and

k(R,I)

is a f i e l d

in a s s e r t i o n

: k(k N M(V))]

an a l g e b r a i c a l l y

I c R

= [V/M(V) field,

= k(R,I,Q),

then:

(5.3) : k].

then: and

= X(R,I,0). in the a b o v e

below, We

also

for a n y

for any

defining

we have note

that,

if

k

is

I c R or I c R or Q c R

I e R

or

I c R

we have

1.7

We

note

(5.1)

that

For

then

any

clearly

I ¢ R

we have

or

(5.1),

I c R

and

(5.2)

any

0

X(R,I,P)

=

k(Rp, I) =

(5.3):

P ¢ ~0(R)

,

a positive

and

integer, ,

we have:

if

IR

if

{o} ~ I N c P,

if

IR = {0},

~

P,

and xk(R,I,P)

= kk(Rp,I)

=

X(R,I,P)[R/P:

k/(k

~ P)]

0 =

IR~P,

,if

a positive

{o} ~ I R c p,

i n t e g e r , if , if

(Note

that,

(5.2)

For

if

k

any

is a f i e l d ,

I ~ R

or

then

I c R

JR/P:

and

any

IR = [0}.

k/(knP)] Q ~ R

= or

JR/P:

k].)

Q c R

we

have :

k(R,I,Q)

= X(R, IR, r a d R ( I R

xk(R,I,Q)

+ QR))

= kk(R, I R , r a d R ( I R

=

~ k(R,I,P) Pe ~0 (R, Q)

=

a nonnegative

+ QR))

=

and

k(R,I,Q)

= ~ = %k(R,I,Q)

= ~ ~ IR =

~ (R,I,Q)

= 0 = lk(R,I,Q)

= 0 = IR + Q R = R,

X(R,I)

k(R,I,Q)

for

any

xk(R,J,Q)

J ~ R

~ lk(R,I)

or

~ k (R,I,Q)

J c R

and

=

with

[0}

xk(R,I,Q)

JR c

lk(R,J,Q)

or

~,

~ kk(R,I,P) PC ~0 (R,Q)

= a nonnegative

=

integer

IR

and

integer

Q R ~ R,

~ QR c radRIR ,

we have

= xk(R,J,Q).

or

~,

1.8

Also

k(R,J,Q)

= k(R,I,Q)

~ lk(R,J,Q) J(Rp)

where

(Rp)

For

any

I ~ R

k(R,I)

= k (R, IR)

lk(R,I,Q)

= I(Rp)

is the integral

(5.3)

=

,

closure or

I c

of

R

for

all

P e ~0(R,Q)

Rp

in

(Rp) .

we have:

(or~I)[V/M(V):

=

R/(R

N M(V))~

Ve~(R)

l(R,p) Pc~0(R)

a nonnegative

kk(R,I)

= k k ( R , IR)

=

integer

or

~ ,

(ordvI) [V/M(V) : k/(k

Q M(V))

V ¢ ~ (R)

=

~ ~k(R,P) Pe~0(R)

= a nonnegative

and

integer

I(R,I)

= ~ ~ kk(R,I)

= ~ ~ IR =

k(R,I)

= 0 ~ kk(R,I)

= 0 = IR = R ,

for

any

J c R

or

k(R,J)

> k(R,I)

k(R,J)

= X(R,I)

J c

and

R

with

kk(R,J)

or

~ ,

{0}

JR c IR

we

have

k lk(R,I)

Also

The k (R,I)

following and

(5.4) in

R

we

= k k (R,J)

six

= k k ( R , I) ~ J R * = I R ,

lem~las p r o v i d e

alternative

definitions

of

k k ( R , I) . LEMMA.

have

Without

assuminq

condition

(**),

for

any

ideal

1.9

[Rp/IRp:

Rp][R/P:

k/(k

~ P)]

=

[R/I

: k].

P¢~0(R) (Note

that

if

PROOF.

k

We

is a f i e l d

shall

prove

then

our

[R/P

: k/(k

assertion

by

N P)]

=

[R/P

induction

on

: k].)

[R/I

: R].

Clearly

[R/I

: R~ = ~ ~ I =

{0}

[R/I

: R] = 0 ~ I = R =

=

both

sides

of

the

above

equation

are

CO,

and

So all

let

0 <

values

[R/I of

: R]

[R/I

: R~

is a n o n e m p t y

finite

ideal

RQ

J

between

in IRQ

and

(i)

moreover,

upon

set.

with

We

sides

assume

can

IRQ c J

J.

that

the

above

the

fix

Q ¢ ~0(R,I)

such

given

that lemma

there we

equation

assertion

than

By Nakayama's

c IRQ

of the

one. and is n o

then

I'

is a n

=

Now then ideal

have

;

N IRp Q~P¢~0(R) ideal

in

R

(3)

[I'/I

: R~ =

1 ,

(4)

[R/I'

: R] =

[R/I

~ J

with

: R]

,

I c

-i

I'

,

such

are

is t r u e

letting

I'

that

and

smaller

M(RQ)J

(2)

we have

< ~

both

that

0.

for

~0(R,I) take in

an RQ

i.i0

[Rp/IRp

: Rp][R/P

: k(k

n P)]

P¢~0 (R) (5) =

[R/Q

: k/(k

Q Q)]

[R/I'

+

: Rp][R/P

: k/(kflP)]

,

P~90 (R) and (6)

[R/I

In v i e w of

(7)

By

: k] =

[I'/I

(4), b y the

~ [R/I' Pe~0 (R) (i) and

(2) we

: k] + [R/I'

induction hypothesis

: Rp][R/P

QI'

[I'/I

In v i e w of

N P)) ] =

[R/I'

of

I'/I

induces

a

of

I'/I

induces

a

on it and we h a v e

: R/Q] =

[I'/I

: R]

(8), the k - m o d u l e - s t r u c t u r e

(k/(k n Q ) ) - m o d u l e - s t r u c t u r e

on it and w e h a v e

(i0)

N Q)]

[I'/I

N o w the a b o v e c i d e s w i t h the above

said

: k].

c I .

(8), the R - m o d u l e - s t r u c t u r e

(R/Q)-module-structure

(9)

: k/(k

we get

see t h a t

(8)

In v i e w o f

: k].

: k/(k

said

=

[I'/I

: k].

(k/k n Q ) ) - m o d u l e - s t r u c t u r e

(k/(k n Q ) ) - m o d u l e - s t r u c t u r e

(R/Q-module-structure,

induced

of

I'/I

on it b y its

and h e n c e b y v e c t o r

space theory

we have (ii)

[I'/I

By

(3),

(5),

: k/(k

(6),

N Q)]

(7),

=

(9),

[R/Q

: k/(k

(i0) and

10

N Q)][I'/I

(ii)

coin-

: R/Q].

it f o l l o w s

that

i.ii

E [Rp/IRp Pe~0(R) (5.5)

LEMMA.

PROOF. taking

F o r any .

k(R,I)

= [R / I R

The

k = R.

IRp

(I')

[R*/IR*

I £ R

: k].

by

: k/(k N M(V))]

: Rp][R*/P

: k / ( k n P) ]

by

: k]

o__rr I c R.

Let

(5.4).

P e 90

be s u c h

that

Then

= [Rp/IRp

S = Rp.

: Rp]

and

xk(R,I,P)

By a s s u m p t i o n

there

= [Rp/IRp

exists

: k].

x e S

such

Now

X(R,I,P)

I {

= [R*/IR*

equation we have

=

Let

we have

f r o m the s e c o n d e q u a t i o n

~ (or~I)[V/M(V) Vc5(R)

LEMMA.

xS = IS.

xk(R,I)

follows

~ . [Rp/IRp Peg0 ( R )

Let

o__[r I c R

and

=

X(R,I,P)

that

: R]

F o r the s e c o n d

is p r i n c i p a l .

PROOF.

w

: k/(k N P)] = [R/I : k].

I e R

first equation

kk(R,I) =

(5.6)

: Rp][R/P

xk(R'I'P)

= X(S,x)

and

[Rp/IRp

= k(R,I,P) [S/M(S)

: Rp] = [S/xS

: S]

,

and

: k / ( k q M(S)) ]

(2') [Rp/IRp

a n d by (3')

Upon

: k]

=

[S/xS

,

(5.4) w e h a v e

[S/xS

letting

: S][S/M(S)

S

: k/(k N M(S))]

to be the i n t e g r a l

~(S,x)

=

[S*/xS*

= [S/xS

closure

we also get

(4')

: k]

: S].

of

S

: k].

in

~(S),

by

(5.5)

1.12

In v i e w

of

(i'),

proving

that

(2'),

(i)

(3')

IS /xS

We

now p r o c e e d

and

(4'),

: S] =

to p r o v e

[S/xS:

(i).

dimensional noetherian domains, , of ( i ) are ~. So h e n c e f o r t h

our

assertion

to

S].

Since

we

is r e d u c e d

S

see

that

assume

that

xS c xS

c S

and

[xS*/xS:

S] =

[S*/xS:

and if

S

are

x = 0

x ~ 0.

one-

then both

sides

Now

xS c S c S

and h e n c e

(i)

IS /xS

Since

: S] +

0 ~ x c S, y - xy

and under S*/S

this

isomorphism

is S - i s o m o r p h i c

(2)

to

finite, Since

if w e then

x

prove

that

(i) and

is a n o n z e r o

image

IS /S:

[S/xS:

element

of

S

is

of

xS

S] +

S

[S/xS:

onto

S].

xS

; consequently

and h e n c e

S] =

(2) p r o v e

[S*/S:

an S - i s o m o r p h i s m

xS /xS

S]

S].

and

IS /S:

S~

are b o t h

(i). in the o n e - d i m e n s i o n a l

local

domain

S,

get

(3)

Now S.

[S/xS:

S

is a f i n i t e

Upon

S-module

S] < ~

and h e n c e

~(S)

is a n o n z e r o

ideal

Consequently

(4)

(5)

the

[xS /xS:

Now

we

gives

S] =

k (s,~(s)) taking

S

and

~(S)

~(s,~(s))

for

<

R

I

in

= ~s*/~(s)s*:

s]

12

and

(5.5)

we

also

get

in

1.13

Clearly W

~(s)s

and h e n c e

by

(4) and

w

= ~(s)

c s c s

(5) w e see that W

(6) By

IS /S (i),

(2),

(3) and

(5.7) principal

LEMMA~

X(R,I) PROOF.

kk(R,I)

Then

tion

= [R/IR

equality

equality

kk(R,I) gives

=

[R/IR

IRp

the

= ER/IR

Let

first by p u t t i n g

Now

Rp

follows

from

(5.6).

k = R.

: k / ( k Q P)]

: Rp~[R/P

P ~ ~0(R)

o__rr I c R

PROOF.

is

: k~.

: k / ( k Q P)]

: k~

= [Rp/IRp

= [Rp}

be such that

we have,

~ [Rp/IRp Pe~0 (R)

k(R,I,P)

: Rp~

be such

that

Rp

by

(5.6)

by

(5.4)

is regular.

we have

and

kk(R,I,P)

is a p r i n c i p a l

= [Rp/IRp

ideal d o m a i n

Alternatively

: k~.

and hence

we can note

our asser-

that now

and h e n c e =

(OrdRpI)[Rp/M(Rp)

= [Rp/IRp

and more

and

=

LEMMA.

I C R

Then

~ I(R,I,P)[R/P Pe ~0 (R)

I c R

k(R,I,P)

: R~

or

=

for any

~(R,P)

(i).

I e R

P ¢ ~0(R).

The second

For the s e c o n d

(5.8)

(6) w e get Let

for all

: S] < ~

: R / ( R n M(Rp)) ]

: Rp]

(by definition) (obviously)

generally,

13

i. 14

lk(R,I,P)

=

(OrdRpI) [Rp/M(Rp)

=

[R/IRp

: Rp][Rp/M(Rp)

= [Rp/IRp LEMMA.

(5.9) I c R

: k/(k

N M(Rp))

(by d e f i n i t i o n )

: k].

I__~f R

(obviously)

: (k/(k N M ( R p ) ) ?

(by

is normal,

then

for any

I ¢ R

(5.4) o__~r

we h a v e (R,I) = PROOF.

for all

If

[R/IR

and

: R]

k k(R,I)

R

is n o r m a l

then

P ¢ D0(R),

and h e n c e

our a s s e r t i o n

Alternatively

we can note

bijection

~0(R)

of

X(R,I) =

~

that

onto

Rp

if

~(R)

R

=

[R/IR

: k~

is a p r i n c i p a l follows

is normal

then

.

ideal d o m a i n

from

(5.7).

P - Rp

gives

a

and h e n c e

(ordvI) [V/M(V)

: R/(R Q M(V))]

(by d e f i n i t i o n )

W 9 (R) =

~ P~0(R)

= [R/IR and m o r e

[R/IRp

: Rp7

(obviously)

: R].

(by 5.4) ).

generally,

lk (R, I) =

(ordvI)[V/M(V) w~

(5.10).

=

~ [R/IRp P¢~0 (R)

=

[R/IR

n M(V))~

(by d e f i n i t i o n )

I(R) "= ~ (R,M(R))

: Rp][R/P

: k(k

and

(obv iou sly )

(by (5.4)). In case lk(R)

R

is local,

= lk(R,M(R)).

that then:

1 < card ~(R)

n P)]

: k]

DEFINITION.

We o b s e r v e

: k/(k

(R)

~ ~ (R) = a p o s i t i v e

14

integer,

we d e f i n e

i. 15

kk(R)

= k(R)[R/M(R)

and by Nakayama's

: k/(k Q M(R))]

(We note that,

if

[R/M(R)

We also note that,

field,

then

k

is regular.

is a field,

then if

[R/M(R) k

: k/(k n M(R))]

is an algebraically

=

closed

k k(R) = k (R).)

We also observe P ¢ 90(R)

integer,

Lemma we have k(R) -- 1 ~ R

: k].

= a positive

that, without

assuming

R

to he local,

for any

we now have X(R,P)

(5.11)

= k(Rp),

kk(R,P)

DEFINITION.

k~(R,Q) k(R,Q) adj(R,Q)

= kk(Rp),

For any

= k (R,~(R),Q)

Q ¢ R

and or

9(R,P) Q c R

= 9(Rp).

we define

,

= lk (R,{ (R) ,Q) = [~ ~ F

(R): ~Rp c ~(Rp)

for all

P e ~0(R,Q)],

and tradj(R,Q)

= [~ ¢ F

Rp

where in

(R): ~(Rp)

= ~(Rp)

for all

P e 90(R,Q)],

denotes the integral closure of

Rp

~ (Rp) .

We note that then: k~(R,Q)

=

~ P~0

k (R,Q) =

~ P~0

k~(R,Q) k(R,~,Q)

X~(R,P)

= a nonnegative

integer,

k(R,P) X~

= a nonnegative

integer,

(R,Q) (R, Q)

k(R,Q) = 0 ~ X~[ > X~(R,Q)

= 0 ~ Rp

and

is regular

~ k(R,~,Q)

~ adj (R,Q),

15

for all

~ lk(R,Q)

P ¢ ~0(R,Q)

for all

,

1.16

adj (R,Q) =

n adj (R,P) (R,Q)

P~0

,

and tradj (R,Q) =

N tradj(R,P) P¢~0(R,Q)

By an a d j o i n t in ~rue

adjoint

R

in

R

a__ t at

Q Q

= [~ c a d j ( R , Q ) :

k(R,#,Q)

=

Ik(R,~,Q)

[~ c adj (R,Q):

w e m e a n a m e m b e r of we mean a member of

adj (R,Q),

= X{(R,Q)] = xk(R,Q)]

and b y a

tradj(R,Q);

from these

to

R

f r o m the

where

R

two w e m a y d r o p ~' in R " w h e n the r e f e r e n c e

is c l e a r

context. W e also d e f i n e

~{(R) = k(R,~(R)) k(R)

= kk ( R , ~ ( R ) )

adj (R) = {~ e ~

(R) : ~ c ~(R)]

and *

tradj (R) = [~ e 5

*

W

(R): eR

c l o s u r e of

R

= ~(R)], in

is the

integral

~ (R)°

We note t h a t t h e n k~(R)

= X{(R,0)

=

l~k(R) = lk(R,0)

=

~ x~(R,P) pc~ 0 (R) ~

: a nonnegative

integer,

~,~k(R,P) = a n o n n e g a t i v e

integer,

P ~ 0 (R) ~(R)

= 0 ~ kk(R)

= 0 ~ R

is n o r m a l

~ Rp

is r e g u l a r

for all

P e ]30 (R), ~(R,~)

m X{(R)

and

adj (R) = adj (R,0) =

kk(R,~2)

m ~k(R)

n adj (R,P) P e ~ 0 (R)

16

,

for all

4~ ¢ adj (R) ,

1.17

tradj(R)

= tradj(R,0)

=

n P~0

tradj (R, P) (R)

= [~ c adj(R):

I(R,_~) = X~(R)]

= [~ c adj(R):

k (R) ] = X{

kk(R,~)

and w

tradj (R) @ ~ ~ ~(R) B y an a d j o i n t adjoint

in

in

R

X~ k(R,P)

that

for any

= k{(Rp)

= ~ k(Rp)

in

we mean a member of

we m e a n a m e m b e r

We o b s e r v e X{(R,P)

R

is p r i n c i p a l

of

adj (R), and b y a t r u e

tradj (R).

P c D0(R)

= a nonnegative

= X~(R,P)[R/P:

R .

we have:

integer,

k/(k Q p)] = a nonnegative

integer,

and X~(R,P)

= 0 ~ X~(R,P) Rp

is n o r m a l

Rp

is r e g u l a r

I(R,I,P) W e note that,

if

k

= OrdRpI

is a field,

Finally we observe

that,

k I{(R,Q)

for a n y

= I~(R,Q)

= 0

if

k

for e v e r y

then

[R/P

I ¢ R

: k/(k

N P)] =

is an a l g e b r a i c a l l y

Q e R

or

Q c A,

and

or

closed

I c R. [R/P

: k].

field,

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

l~k(R) = ~ ( R ) (5.12) rational

over

L E M M A ON O V E R A D J O I N T S . k.

Let

e

I

is an ideal

[R/I

in

V ¢ ~(R)

b e any n o n n e g a t i v e

I = {(R)(R

Then

Let

R

*

with

: k] = e + [R/~(R)

n M(V)

)e

17

integer,

and let

.

I c {(R)

: k]

b___e r e s i d u a l l y

and

such

that,

lk(R,I)

k(R) . = e + X~

then:

1.18

PROOF.

Now

~(R)

is a n o n z e r o

ideal

in

R , and u p o n

letting

w

Q = R

n M(V) w

we have

that

Q

(i)

prime

ideal

in

I = ~(R)Q e c ~(R)Qe-lc...C(R)Q

are

ideals

in

R

(2)

~(R)

c R, w e

(3)

that

(R)Q j

also

~(R)Q i

note

R .

So

0 = ~(R)

with

~(R)Q i ~

Since

We

is a n o n z e r o

see

i ~ 0

=

i ~ j

that,

is an ideal

for e v e r y

~(R) Q i

whenever

we

in

R

clearly

for all

i ~ 0.

have,

W

{y e R : o r d v Y

> i + ordv~(R)

and

(4)

or~

We

cleaim

whenever

d z 0

~(R) Q d+l

+ x R = { ( R ) Q d.

of

and any

x ~ {(R) Q d \ ~ ( R ) Q d + I ,

we have

(4) w e have,

ordvx

(51 )

= d + ordv~(R)

and ordwx

(5 2 )

Given

(5 3 )

any

v Jw

that,

for any

In v i e w

~ or~{(R)

> or~(R),

z e ~(R) Q d , b y

whenever

(4) w e

or%z

V ~ W

get

~ d + or%~(R)

18

~ ~(R) .

~ ~(R)]

.

1.19

and (54 )

Now

ordWZ

V

is residually

residually can find

(52 ) and

V ~ W ~ ~(R).

rational over

R

k), and hence

(because it is assumed in view of

(51 ) and

to be

(5 3 ) we

~ d + 1 + ordv~(R).

(54 ) we have,

or~(z-~x)

In view of

whenever

such that

ordV(Z-~X)

(56 )

(R)

rational over ~ e R

(55 ) By

~ ord W

~ ordw~(R),

(54 ) and

(56), by

whenever

V ~ W ~ ~(R).

(4) we get that

z - ~x ~ ~(R)Q d+l

and hence By I

z e ~(R)Q d+l + xR.

d m 0

and any ideal

(5).

J

in R

with

~(R)Q d+l c J c ~(R)Q d, we must have either

By

the proof of

(5) we see that,

for any

(6)

This completes

(i),

(2),

(3) and

(7)

J = ~(R)Q d+l

or

J = ~(R)Q d.

(6) we get

[R/I

: R~ = e + [R/~(R)

: R~.

Let

(8)

P0 = R Q M(V).

Then

P0 ~ D 0 (R)

(9) Now

and we clearly have

IRp = ~(R) Rp V

v i e w of

is assumed (7),

whenever

P0 ~ P ~ ~ 0(R)"

to be residually

(8) and

(9), by

rational over

(5.4) we conclude

~9

that

k

and hence,

in

1.20

[R/I

Since

V

: k] = e +

is r e s i d u a l l y

rational

hk ( R , I )

§6.

Length

Let

A

be Let

be

of

a subring

(*)

for

(Note

f: A ~ A

every

[A/P

such

(*)

is a s u b f i e l d

k,

is of

N P]

and

let

A/C

be

we

obviously

have,

noetherian

C c D 1 (A) the

be

canonical

homomorphic

such

image.

A/C

that

epimorphism.

is

Let

k

that

we

have

k 0 p ~ ~0

and

< ~.

satisfied A

: k]

= e + Ik ( R ) .

P e ~0([A,C])

: k/(k

that

over

in a o n e - d i m e n s i o n a l

a domain

noetherian.

[R/{(R)

such

for that

A = k; [A/P

it is a l s o

: k]

< ~

for

satisfied

if

k

every

P e Z0([A,C]).).

Assume

(**)

that

the

integral

closure

of

A/C

in

~ (A/C)

and

any

Q e. A

is a f i n i t e

(A/C)-

module.

For

any

I e A

I([A,C],I,Q)

=

or

I c A

~(f(A),f(I),f(Q)) =

and

for

I C A

~([A,C],I)

For

or

X~([A,C],Q)

Q ~ A

I c A,

or

Q c A,

we

define

;

define

and

~k([A,C],i ) =

Q c A, w e

-- k ~ ( f ( A ) , f ( Q ) )

we

~k([A,C],I,Q)

k f(k) ( f ( A ) , f ( I ) , f ( Q ) )

= X(f(A),f(I))

any

and

or

and

i

~f(k)(f(n),f(I)).

define k

20

([A,C],Q)

=

f (k) l~ (f(A),f(Q)),

1.21

w

adj ([A,C])

=

[~ ¢ F

(A) : f(~)

~ adj (f(A))},

and w

tradj ([A,C])

By an a d j o i n t true

adjoint

of

C

of

C

these

two p h r a s e s

clear

from We

(5.11)

the

note

{~ ¢ F

in in

A A

we may

(A):

f(~)

we m e a n we mean

drop

¢ tradj (f(A))}.

a member

adj ([A,C]),

and b y

of

tradj ([A,C]),

from

the

reference

to

A

(5.2),

(5.3),

(5.10)

a member

"in A '~

when

of

that

clearly

(6.1)

For

then, get

any

in v i e w (6.1),

I ~ A

of

(5.1),

(6.2),

or

= k (f(A) f ( p ) , f ( I ) )

=

(6.3),

I c A

and

(6.4)

any

, if

a positive

and

integer,

if , if

P ¢~)0([A,C])

IA~

: A/P]

P,

IA c p

and

IA ~ C,

IA c C,

and = k([A,C],I,P)[A/P

: k/(k

n p)]

= X f(k) (f(A) f ( p ) , f ( I ) ) = kk ([Ap, CAp], I) 0

=

(We o b s e r v e

[A/P

is

that,

if

IA ~ P,

if

IA c P

, if

IA c C .

,

a positive

if

k

integer,

is a field,

: k].)

21

then

[A/P:

and

(6.5):

= k ([Ap,CAp~,I)

~ VI)[V/M(V) ( [ A , C ] , P ) ( ° r d [ A ' C]'

V~

0

kk([A,C~,I,P)

a

context.

we

k ([A,C],I,P)

=

and

IA ~ C,

k/(k

Q P)] =

w e have:

1.22

For any

P e ~0([A, C3)

1 < card([A,C~,P)

kk(~A,C~,P)

w e also have:

< k([A,C~,P)

= k(f(A) f(p))

= k([A,C~,P)[A/P

: k/(k

= a positive

Q P)~ = kf(k) (f(A) f(p)) = a positive

k~(~A,C~,P)

= x([A,C3,%([A,C~,P)

= k~([A,C~,P)[A/P

integer,

= k~(f(A) f(p)) = a nonnegative

xk([A,C~,P)

integer,

: k/(k

integer

N P)~ = ~ k ( [ A , C ~ , ~ ( [ A , C ~ ) , P ) f(k) (f

= k~

(A) f (p)

= a nonnegative

) integer,

and = 0 ~ k~k([A,C~,P)

X~([A,C],P)

= 0

f(A) f(p)

is n o r m a l

f(A) f(p)

is r e g u l a r

X(~A,C3,P)

= 1

k (EA, C],I,P)

= °rdf(A) f(p) f(I)

for e v e r y (We a g a i n [A/P

observe

that,

if

k

I e A

is a field,

then

or [A/P

I c A. : k ( k N P)] =

: k] .) (6.2)

X(EA,C~,I,Q)

For any

I ~ A

or

I c A

or

= k(EA,C~,IA+C,radA(IA+QA+C)

Q c A =

w e have:

~ X([A,C~,I,P) P¢~0 ( [A,C~ ,Q)

= a nonnegative or xk([A,C~,I,Q)

= x k ( E A , C T , I A + C , radA(IA+QA+C)

~,

=

~ P~0

kk(EA,Cj,I,P) ([A,C~,Q)

= a nonnegative or

22

integer

integer

1.23

([A,C~,I,Q)

= ~ ~ Ik([A,C~,I,Q)

= ~ ~ IA c C

([A,C3,I,Q)

-- 0 ~ ~ k ( [ A , C ~ , I , Q )

= O ~ IA + QA + C = A,

~ ([A,C~,I)

= I([A,C~,Q)

~ lk([A,C3),I)

and

QA + C ~ A,

= lk([A,C~,I,Q)

QA c r a d A ( I A + C ) . Also,

for any

J ¢ A

or

J c A

I([A,C~,J,Q)

> ~([A,CT,I,Q)

I([A,C3,J,Q)

= ~ ([A,C~,I,Q)

with and

JA c IA + C

we h a v e

lk([A,C~,J,Q)

~ lk([A,C~,I,Q).

and ~ kk([A,C3,J,Q)

= lk([A,C~,I,Q)

w

f(J) (f (A) f(p) ) where

(f(A) f(p))

= f (I) (f (A) f (p) ) is the

integral

for all

closure

of

P ¢ D0([A,C~,Q) f(A)f(p)

,

in

(f(A)f(p)). For any I~([A,C~,Q)

Q e A

or

Q c A

we also have:

= I([A,C~,~([A,C~),Q)

= P~0

~ I~([A,C~,P) ([A,C~,Q)

= a nonnegative I~k([A,C~,Q)

= Ik([A,C~,~([A,C~),Q)

=

integer,

~ X~([A,C~,p)k P~D 0 ([A,C~,Q)

= a nonnegative ~([A,C~,Q)

= 0 ~ kk([A,C~,Q)

= 0 ~ f(A)f(p)

integer,

is r e g u l a r

for all

PoD 0 ([A,C~,Q), ~ ([A,CT,~,Q)

~ I~([A,C~,Q)

and

lk([A,C~,~,Q)

~ lk([A,C~,Q)

~ adj ([A,C~,Q), adj ([A,C~,Q)

=

~ adj ([A,C~,P), Pe~ 0 ([A,C~,Q)

23

for all

1.24

and

tradj ([A,C],Q)

N

=

tradj ([A,C],P)

PC% 0([~,c],P) [~5 e adj ([A,C],Q): =

(6.3)

For

I([A,C],I)

I([A,C],~,Q)

: I{([A,C],Q)}

[~ e adj ([A,C],Q) : lk([A,C],~,Q) any

I e A

= I([A,C],I,0)

or

:

I c A

= lk([A,C],Q]-

we have:

I([A,C],IA)

~ ([A,C], I,P)

= P e D 0 ([A,C])

a nonnegative

Ik([A,C],I)

= Ik([A,C],I,0)

= Ik([A,C],IA)

=

~

integer

or

xk([A,C],I,P)

pc~o ([A,c]) = a nonnegative or

I([A,C],I)

= ~ ~ xk([A,C],I)

: ~ ~ IA c C,

= 0 ~ Ik([A,C],I)

= 0 ~ IA + C = A.

integer

t

and X([A,C],I) Also,

for any

J e A

or

J c A

X([A,C],J)

m X([A,C],I)

X([A,C],J)

=

and

with

J A + IA + C

Ik([A,C],J)

we have:

a xk([A,C],J),

and X([A,C],I)

~ kk([A,C],J)

= xk([A,C],I)

w

where

f(A) We

IE([A,C])

also =

is t h e

integral

w

f(J) f(A)

=

closure

of

f(I)f(A) f(A)

, in

~(f(A)).

have: XE([A,C],0)

=

I([A,C],~([A,C]))

=

~ I{ ([A,C],P) Pe~3 0 ([A,C])

= a nonnegative

24

integer,

OO,

1.25

~,~([A,c])

k~k([A,C],0)

=

= xk([A,C],~([A,C])

=

~ kk([A,C],P) PoD 0([A,C])

= a nonnegative ~([A,C])

= 0 ~ lk([A,C])

-- 0 ~ A/C

integer

is normal

f(A) f(p)

is r e g u l a r

for all

P ¢ ~0([A,C]), ([A,C],~)

> k~([A,C])

and

k~k([A,C],~)

for all adj ([A,C])

=

m I~([A,C])

~ c adj ([A,C]),

n adj ([A,C],P), PcD 0([A,C])

and tradj([A,C])

=

n

tradj ([A, C3, P)

{4 c adj ([A,C]):

x([A,C3,~)

= I~([A,C])}

[4 ¢ adj ([A,C]) : xk([A,C],~) (6.4)

If

xk([A,C],I,Q)

k

is an a l g e b r a i c a l l y

= ~ ([A,C],I,Q)

Ik([A,C],I)

-- ~ ([A,C],I)

k([A,C~,Q)

= ~([A,C3,Q)

k = X~([A,C~)].

closed

field,

for any

I ~ A

or

I c A

and any

Q c A,

for any

I ~ A

or

I c A,

for any

Q e A

or

then:

Q c A,

and

~k([A,C]) = ~([A,C]) (6.5)

If

C = [0],

then:

\

lk([A,C],I,Q)

= kk(A,I,Q) I

for any

I ~ A

or

I cA

I ([A,C],I,Q)

= I(A,I,Q)

and any

Q ¢ A

or

Q cA

I

25

;

1.26

kk([A,C~,I)

= xk(A,I)I for any

X ([A,C~, I) =

k~([A,C~,Q)

I ~ A

or

I c A ;

k (A, I) I

= X~(A,Q) for any

adj ([A,C],Q)

Q c A

or

Q c A

;

= adj (A,Q)

tradj ([A,C~,Q)

= tradj (A,Q)

xk([A,CT)

= xk(A)~

and

k~([A,C~)

= I~(A)

;

and adj ([A,C~) In v i e w of (6.6) that

= adj (A)

(5.6)

LEMMA.

f(I) f (A) f (p)

to

Let

and

(5.9) we I C A

tradj ([A,C~) immediately

o__rr I c A

i s principal.

= tradj (A).

get

and

(6.6)

to

(6.9):

P ~ ~0([A,C~)

be

such

Then

C)Ap Ap~

~([A,cT,I,P)

= [A/(IA

+

kk([A,CT,I,P)

= [A/(IA

+ C)Ap:

and

(6.7)

LEMMA.

is p r i n c i p a l

for all

X([A,C],I) (6.8) reqular.

LEMMA. Then,

Let

I ~ A

o__rr I c A

P ¢ ~0([A,C~). = [A/(IA+C) : A~ Let

P CD0([A,C~)

for any

I ~ A

k].

be such

f(I)f(A)f(p)

Then and be

xk([A,C~,D

= [A~IA+C):

such that

f(A)f(p)

o__[r I c A, we have,

k([A,C~,I,P)

= [Ap/(IA+C)Ap:

Ap~

~k([A,C~,I,p)

= [Ap/(IA+C)Ap:

kl.

and

26

that

k~. i__ss

1.27

(6.9)

I_~f A/C

LEMMA.

is normal,

then,

for any

I e A

o__rr

I c A, we have k([A,C],I)

-- [A/(IA+C):

§7. Let

A

A~

A commuting

be a domain.

Let

n o e t h e r i a n and the integral (A/C)-module. the integral Let

(*)

k

Let

be a subring of

for every

[

and

k

P ¢ D0([A,C])

I([A,C],P)

~ (A/D)

in

~ (A/C)

A/D

is

is a finite

is noetherian

is a finite

and

(A/D)-module.

n ~0([A,D~)

we have

k N P ¢ ~0(k)

(*)

is satisfied A

for

A--k.

such that

It is also satisfied

[A/P : k~ < ~ , for every

N ~0 [A,D3).)

LEMMA.

I__ff P ~ D0([A,C ~) n

= 1 = ~ [A,D~,P),

([A,C~,D,P) PROOF.

A/C

A/C

such that

P ~ D0([A,C~)

is a subfield of

(7.1)

k].

[A/P : k/(k n P)~ < ~.

(Note that if

be such that

be such that in

= [A/(IA+C):

lemma for length.

closure of

A/D A

lk([A,C~,I)

C ¢ 91 (A)

D ~ 91 (A)

closure of

I

and

= I [A,D~,C,P)

It suffices

~0(EA,D])

is such that

then and

lk([A,C~,D,P)

to note that by

= ~k([A,D~,C,P).

(6.1) and

by([A,C~,D,P) P ¢ 90([A,C ~) with I([A,C~,P)

= 1

27

~-- [ A / ( C + D ) A p

:Ap~

-- [Ap/(C+D)Ap

: k~

and xk(EA,C~,D,P)

and

(6.8) we have that:

1.28

k ([A,D3,C,P) p ¢ D0([A,D~)

with

I([A,D~,P)

=

LEMMA.

1 = k([A,D~),P) ([A,C~,D,Q)

for all

Follows

LEMMA.

P ~ ~0([A,C~)

PROOF.

§8. Let

In

and

I_~f k ( [ A , C ~ , P )

(7.2)

take

N D0([A,D~)

: kl

k ([A,C~,P)

n D0([A,Q]),

=

then

= ~k([A,D~,C,Q) •

(7.1).

= 1 = X([A,D~,P)

for all

then and l k ( [ A , C ~ , D )

= lk([A,D~,C).

Q = 0.

in a t w o - d i m e n s i o n a l

b e a domain,

[A/(C+D)Ap

=

is such that

kk([A,C~,D,Q)

(6.2) and

= I([A,D~,C)

Length A

from

N ~0([A,D~),

k([A,C~,D)

o__rr Q c A

P c ~0([A,C3)

= ~ ([A,D~,C,Q)

PROOF.

(7.3)

I__ff Q c A

: Ap~

and

= 1 =

~k([A,D~,C,P)

(7.2)

[Ap/(C+D)Ap

let

regular

P c D0(A)

be

local d o m a i n .

such t h a t

Ap

is a t w o W

dimensional

regular

l o c a l domain,

and

let

~ ¢ F

and

(A)

~

c F

(A).

We d e f i n e

W e note that then:

= a nonnegative ([A,~,~,P)

= 0 ~ ~ +

~

i n t e g e r or

P ,

and 0 ~ I([A,~,~,P)

~ ~ ~

(MAp

28

is p r i m a r y

for

M(Ap).

o0

1.29

The

next

I ([A,#],~,P)

LEMMA.

such

finite

that

Assume

the

X([A,~,~],P)

PROOF.

domain

that

(A/45)-module, the

module.

([A,~],~,P)

A/~

assume

time when

~ ([A,~],~,P)

and

coincide.

closure

from

(6.2)

Assume

that

the

and

integral

A/~

integral Also

they

that

Also

of the

is a o n e - d i m e n s i o n a l

noetherian

of

is a

that

A/~

in

either

~ (A/~)

~ c p

or

P c ~0 (A).

= X([A,~],~,P).

LEMMA.

such

most

integral

Follows

(8.2)

that

defined

(A/~)-module.

Then

that

says

are b o t h

(8.1) domain

lemma

(6.6).

A/~

closure

is a o n e - d i m e n s i o n a l

noetherian

of

is a finite

A/~

is a o n e - d i m e n s i o n a l

closure

assume

and

of

that

A/~

i__nn ~ (A/R) noetherian

i__n ~ (A/~)

either

~ + ~

c P

domain

is a f i n i t e

such

(A/~)-

o__rr p ¢ ~30 (A).

Then

-- ~ ( [ A , ~ , ~ , P ) .

PROOF.

Follows

from

(8.1). w

(8.3)

LEMMA.

Let

also

@'

~([A,ee, ~ , p ) PROOF. zero.

So n o w

canonical that

= Ap

,then b o t h

suppose

that

~Ap

for a n y

b e qiven.

~ Ap

B

+ ~([A,~', ~ , P )

sides

of the

Let

f: A p

.

Then

above

equation

- B = A~Ap

is a o n e - d i m e n s i o n a l

local

ring

are be

the

such

z c B, w e have,

[B/zB

can t a k e

Then

(A)

= ~([A,e,~],P)

~Ap

epimorphism.

(i)

We

If

e F

: B~ = ~ = z

elements

x

is a z e r o d i v i s o r

and

x'

in

B

such

that

clearly l([A,gg,~],p)

=

k ([A,~', ~ ] , P ) I ([A,~X~', ~ ] , p )

[B/xB =

: B~

[B/x'B =

[B/xx'B

29

,

: B]

in

,

: B],

B.

xB = f(~'Ap).

Now

i. 30

and h e n c e

(*)

If

w e are

[B/xx'B

x

or

sides

of

x'

reduced

: B]

in

(i) w e k n o w

[B/xB

=

is a zero

(*) are

divisors

to p r o v i n g

B.

~.

all

divisor

xx' the

in

B,

xx'B

: B].

then by

that

is a l s o

three

[B/x'B

+

So n o w a s s u m e

Then

that

: B]

that

x

(i) w e

and

x'

nonzerodivisor

terms

in

(*) are

see t h a t b o t h are n o n z e r o -

in

B,

and

noninfinite.

so b y Now

C xB c B

and h e n c e

~)

[B/xx'B

Since B

x

: B] =

is a n o n z e r o d i v i s o r

ontO

xB

and u n d e r

Consequently,

B/x'B

(2)

this

(i) and

(2) w e g e t

§9. Let

A

in

: B] + [ x B / x x ' B

B, y - x y

isomorphism

is B - i s o m o r p h i c

[B/x'B

NOW by

[B/xB

: B] =

to

image

xB/xx'B

[xB/xx'B

a B-isomorphism of

x'B

is

local

domain,

and

let

: B].

in a r e g u l a r

let

P e D(A)

~ e F

l' ([A,~, ],P) We

note

that

X' ([A,~],P)

be

local such

domain. that

Ap

(A).

is a r e g u -

We d e f i n e

= OrdAp~

then:

= k' ([Ap,~Ap],

M(Ap))

= a nonnegative

~' ([A,¢],P)

= o ~ ¢ ~

1 ~ Ap/~Ap

is a r e g u l a r

integer,

P,

and

k' ([A,~],P)

=

xx'B.

and h e n c e

w

lar

of

(*).

Multiplicity

b e a domain,

the

gives

: B].

30

local

domain.

1.31

The

next

~ ([A,~],P)

lemma

says

are b o t h

(9.1)

LEMMA.

defined

Assume

dimensional

noetherian

i__nn ~(A/~)

is a f i n i t e

PROOF. then upon follows

If

(9.2) and

let

domain Then

be

and

let

First then we

nomial k,

upon

be

integer t(X,Y),

such

[b e k:

ord h(~)

A/~

is a o n e -

t.he i n t e q r a ! c l o s u r e Then

~ ¢ •

that;

upon

X' ([A,~],P)

of

A/~

= X([A,~3,P).

= 0 = l([A,~],P).

with

If

~ c p

~ A = ~, the a s s e r t i o n

lettinq

that,

closure

t(l,b)

that: = e.

we

shall

of

shall

reduce is

a basis and

R R

regular

local

domain

f: S - R = S/~S

to b e

is a o n e - d i m e n s i o n a l

local

i~n

~(R)

is a f i n i t e

E

the a s s e r t i o n

the g e n e r a l

infinite. of

there

M(S),

exists e,

~ - t(x,y)

¢ M(S)]. h:

prove

Then

Let and

when

case k

let

a unique

the

R-module.

¢ M(S) e+l. kI

e =

Let

to be

is a o n e - d i m e n s i o n a l

to t h a t

ords~.

nonzero

is finite,

S/M(S)

is in-

special

be a coefficient

in i n d e t e r m i n a t e s

S - E = S/(y-ax)S

Now

e

homogeneous

X,Y,

with

: R] =

and

the

for a n y

poly-

canonical local

a ¢ k\k I , epimorphism,

domain

and

: E] = e ,

31

is a

kI =

regular

[S/(~Q,y-ax)S

for

coefficients

clearly

[R/f(y-ax)R

case.

set

Consequently [E/h(~0)E

and

and

be a two-dimensional

of degree

that

letting

we have

that

I'([A,~],P)

we have

.S/M(S)

(x,y)

positive

and

= ords~-

Case when

in

such

and th e i n t e q r a l

finite

P e 90(A)

such

and

X' ([A,~],P)

coincide.

(A/~)-module.

Le___tt S

ep!morphism,

PROOF.

S,

domain

S = Ap

LEMMA.

X(R)

that

o f the t i m e w h e n

(9.2):

~ ~ S

canonical

they

~ ~ P, t h e n

taking

from

that most

: S] =

[E/h(~))E

: E]

1.32

By

(5.6) w e a l s o h a v e k(R,f(y-ax))

= [R/f(y-ax)R

B y the a b o v e t h r e e d i s p l a y e d

: R].

equations we get that

k (R,f(y-ax))

= e.

Thus

(i)

X (R,f(y-ax))

Now

for a n y

such t h a t

= ords~

V e ~ (R)

or~f(y-bx)

there

for all

a ¢ k \ k I.

is at m o s t o n e e l e m e n t

~ ordvM(R).

Since

~(R)

b

in

is finite,

k

upon

letting

k 2 = [b ~ k: o r d v f ( y - b x ) we conclude k(R,M(R))

that

k(R)

Since

kI

S/M(S)

and

k2

By

V ~ ~(R)},

a e k \ k 2, w e c l e a r l y h a v e

By definition

~,(R) = X ( R , M ( R ) ) ,

are

for all

finite,

then

by

and h e n c e

a e k \ k 2.

(i) and

(2) it f o l l o w s

that,

k (R) = ords%0.

(9.2) a n d the a r g u m e n t s

seen that a c t u a l l y ,

without

assuming

used

in its proof,

S/M(S)

to b e

it c a n

infinite,

we

M(S)S[X~

is a

k I = k 2. ) General

prime

ideal

that

S'

ordSZ,

case in

Let

S[X~

X

and,

b e an i n d e t e r m i n a t e . upon

is a t w o - d i m e n s i o n a l

for a l l

z e~ S.

(3)

letting regular

Now

S' = S [ X ~ M ( S ) S [ X ~, w e h a v e local d o m a i n a n d

ords,Z =

In p a r t i c u l a r

ords,~ = ords~. Let

R[Y];

For any

-- l ( R , f ( y - a x ) )

is infinite,

(REMARK.

have

for some

that

(2)

if

is finite.

= k(R,f(y-ax)).

we conclude

be

k2

~ OrdvM(R)

Y

b e an i n d e t e r m i n a t e .

and upon

one-dimensional

letting

Now

M(R) R[Y]

is a p r i m e

R' = R [ Y ] M ( R ) R [ y ~ , we h a v e

local d o m a i n

and

M(R) R' = M ( R ' ) .

32

that

R'

ideal is a

Also we have a

in

1.33

unique

epimorphism

s ¢ S

and Let

S'

f' (X) = Y. R

be

the q u o t i e n t

R

the

r i n g of

R [ Y T \ M ( R ) R[Y]. Since

f':

The

~ R'

such

Clearly

Ker

that

f' (s) = f(s)

f' = ~S'.

i n t e g r a l c l o s u r e of , R [Y] with respect

R

R'*

closure

is a f i n i t e

is the

in

(R).

Let

R'

be

to the m u l t i p l i c a t i v e

integral

R-module,

for all

we also have

of

that

R' R'

in

set ~(R').

is a f i n i t e

R'-module. Let V~l

be

V I , V 2 ..... V p

the q u o t i e n t

M(Vi)Vi[Y ] . of

9(R').

b e all

ring of

the d i s t i n c t

Vi[Y]

V I', V 2',...V'P

Then

with

are

members

respect

exactly

For

1 ~ i ~ p, we have:

[V[/M(V[)

: R'/(R'nM(V[))]

=

Ordv!Z

for all

all

at

~(R).

to the p r i m e the d i s t i n c t

[ V i / M ( V i)

: R/(R

Let ideal members

n M(Vi))]

and = Ordv

1

Since

M(R)R'

= M(R'),

(4)

z ~ Vi .

we conclude

that

k (R') = I (R)

Clearly proved

S'/M(S')

above

is i n f i n i t e

k(R') (3),

(4) and

(5) w e get

§i0. Let

R

closure

R

subring

of

In this (i0.I)

and h e n c e

b y the

special

case

we have

(5) By

z 1

be

that

I (R) = o r d s ~ .

points

a one-dimensional

of R

Double

= OrdS,m

R

in

~(R)

over which section

THEOREM.

we

Assume

R

of a l g e b r a i c

curves.

local

such

is a f i n i t e

prove

that

k(R) the

the = 2

finite

following and

following.

33

that

R-module.

is r e s i d u a l l y

shall

Then we have

domain

let

Let

the k

integral be any

algebraic. theorem .

d =

[R/~(R)

: R~.

i. 34

(i0.i.i) Case

Exactly ~(R)

(i).

one =

of

{V,W],

V/M(V)

(2).

=

followinq

for

= W/M(W)

ord W(M(R)) Case

the

some

three

V ~ W.

= R/M(R)

cases Then

and

occurs.

necessarily

or~(M(R))

=

I.

~(R)

-- IV],

for

Then

necessarily

~(R)

=

Then

necessarily

some

V

such

OrdvM(R)

that

V/M(V)

= R/M(R).

V/M(V)

~

= 2.

W

Case

(3).

IV],

o r d v M (R) = (10.1.2)

There

for

some

V

such

[V/M(V)

that

: R/M(R) ] =

2

R/M(R).

and

i.

exists

x e M(R)

such

that

l(R,x)

= 2.

Further

we have: in c a s e

(i),

X (R,x)

=

2 ~ ordVX

in c a s e

(2),

X (R,x)

= 2 ~ ord

X (R,x)

=

V

= ordWX

x =

=

1 ;

2 ;

w

in case

(3),

2 ~ ordVX

=

1.

w

(10.1.3)

There

exists

z e R

(10.1.4)

There

exists

y

w

such

e M(R)

that

such

in c a s e

(i),

ordVY

= d < orgy

in c a s e

(2*),

orgy

=

in case

(3),

OrdvY

= d

and

M(V)

does

not belonq

(10.1.5)

I__ff x

and

y

2d +

are

R

= R[z].

that:

;

1

as

residue

in

of too

(10.1.2)

y/x d

modulo

R/M(R). and

(10.1.4)

respectively,

then M(R) (10.1.6) Jl' =

I_~f x ~(R)

Further,

in c a s e

i_~f J ~

(i),

[x,y}R.

i__~s a__{si__nn ( 1 0 . 1 . 2 ) ,

+ xlR

0 ~ i < d.

=

for

i =

~(R)

In m o r e J = Ji

for

upon

lettin q

0,i, ... ,d, w e h a v e

i__{sa n y

detail

then,

ideal

in

R,

that

then

[ R / J i = R] = J = Ji

for

i.

some

we have: some

0 ~ i ~ q.

34

In m o r e

detail

we have

i. 35

in case

(2),

in case

(3),

J = Ji' w h e r e

i = 1/2 o r ~ J

J = Ji" w h e r e

i = ordvJ.

;

W

(10.1.7)

W e have: ~(R)

= M(v)dM(w) d

(2),

~(R)

: M(V) 2d

(3),

~(R) : M ( v ) d

i__nn case

(i*) ,

in c a s e in c a s e

w

(10.1.8)

;

;

W__eehave:

in case

(i),

d = min{s:

ordVY

= s < orgy

i__nn case

(2),

d = minis:

orgy

= 2s + 1

i_~n case

(3),

d = min{s

: for

for for

some

some

y ~ R]

y e R]

;

;

W

and

the

does The followinq (10.1.9)

(10.1.10)

can b e d e d u c e d

emdim

some

y,~

c R, o r g y

residue

of

y/£

not b e l o n g

to

R/M(R)].

from

k~(R)

M(V)

t h e above.

= 2d,

kk(R)

= 2d[R/M(R)

: k] : 2[R/~(R)

: k].

w

[a /~(R)

(10.1.12)

For

i = 0, i ..... d,

Ji D

~(R)

Further

: R] = 2d.

i__nn R,

there

such

exist

that

unique

[R/J i

w e have:

particular ~

~(R)

~R * =

Let

A

domain Let

X(R, Ji) ~ lk(R,e)

@(R)~

X(R,~)

ideals

: R] = i.

k k ( R , J i) = 2 [ R / J i : k] = 2 i [ R / M ( R )

(10.1.14)

modulo

R = 2.

(10.1.11)

(10.1.13)

= Ordv@

and

z 6 R

let

= X

(R), ~ 0(2),

dimensional f: A - R

such

i__nn

~ X~(R),

~ ~ (R,~)

b e a two

and

= 2i.

k k (R, ~)

< I~(R)

: k],

that

be R

35

for e v e r y

reqular

local

an ~ i m o r p h i s m .

= R[z]

and

~ e R.

=

s,

1.36

let

~,~

f(~)R

e A

such

= {(R) .

that

Note

that,

in case

(i),

in case in case

= f(~)

and

Then

Kerf

PROOF.

f(~)z

c

for

({~,~]A) 2

I c R

or

i e R

k(R,I)

= ordvI

+ or~I

(2),

k(R,I)

= ordvI

;

(3),

k(R,I)

= 2 ordvI.

w e have: ;

w

In v i e w (I0.i.i0), have

(10.1.8)

by

(10.1.5)

(10.1.2)

Proof

unique card

and

(5.10))

Now

of

~(R)

letting the some

V

Also

~(R)

R.

X(R)

= 2,

V

and

W

and

W

are and

I x0 =

c a n be

R ~ 2 R

and

member

(i) h o l d s

to be

2 = 1

deduced

that

from

(10.1.9)

(10.1.3) (2)

~(R)

then:

rational for some

emdim

is

card

members over

over

since

exist

k(R)

= 2

R = 2.

= 1

and

R, or

the

(3)

is n o t r e s i d u a l l y

~(R)

= 2,

of

~(R),

and,

x 2 ~ R ; now u p o n

if

or~x

1 = 1

x2

if

ordwx I ~ 1 = ordvx 2

x I + x2

if

ordg~x I ~ 1 ~ o r d v x 2

upon

we have

R, o r d V X 1 = 1

xI

36

x,y

is as follows:

c a r d ~(R)

rational of

the said

regular

Thus

the d i s t i n c t

residually or~x

(since

is not

is r e s i d u a l l y

if

(since w e m u s t

easily

claim

to d e d u c e

(10.1.5).

~)(R) ~ i, or

the u n i q u e

(10.1.6)

R > i.

(10.1.2)

(I) c a r d

over

xI e R

emdim

it is e a s y

Now we

and

(10.1.4)).

and

Since

(i0.1.4)

emdim

= 1

from

(10.1.13)

we have

of

(5.5)

(10.1.12)

and h e n c e

either

and

(I0.i.i0).

(i0.i.i),

member

rational

and

and

(10.1.2)

From

(using

(5.4)

= Jd ) ; and a l s o

(10.1.7),

by

and

(i0.I.ii)

~(R)

implied

of this,

for letting

1.37

w e get

x0 ~ R

clearly

have

with

ordvx 0 = 1 = ordwx 0

ordVX

finally,

clearly

there

z, b y N a k a y a m a ' s

-- 1 = o r d w x

exists

lemma,

z ~

we h a v e

; also

~ l(R,x)

(R

R

for a n y

x ~ R

we

= 2 ;

~ M(V))\M(W)

and

for any

such

= R[z3. W

Since have

l(R)

that

R n

~(R)

= 2, =

IV],

(M(V)\M(V) 2) =

clearly

if

(2) h o l d s ,

V

then:

is r e s i d u a l l y

~, R N

upon

rational

(M(V) 2\M(V) 3) ~

clearly

there

lemma,

Since

and

l(R)

that

9(R)

for a n y

exists

= 2, =

R

if

2

z ~ M(V)\M(V)

we have

(3) h o l d s

then:

[V/M(V)

: R/M(R)]

we

clearly

have

there

exists

g: V - V/M(V)

by Nakayama's Now, deduction, and

as w e l l shall

for the

z ¢ V

is the

lemma,

(10.1.14)

(10.1.8)

(10.4)

for a n y

x ¢ R

we

and

such

upon

such

z, b y

V = R

we

(M(V)kM(V2))

= ~,

= 2 ;

V/M(V)

epimorphism;

-- g ( R ) ( g ( z ) ) for any

such

z,

R* = R[z3.

can b e d e d u c e d as p r o o f s

of

be p r e s e n t e d (I),

letting

= 2, R N

that

canonical

we have

cases

for a n y

= R[z~.

{V],

x ~ R

clearly

where

R,

-- 2 ;

x ~ M ( V ) \ M ( V ) 2 ~ l(R,x)

also,

V = R , we

have

Nakayama's

have

over

~, and

x e M ( V ) 2 \ M ( V ) 3 ~ k (R,x)

also,

letting

(2)

from

(10.1.5)

(10.1.4), in the and

(10.1.9).

(10.1.5),

Lemmas

(3)

and

This

(10.1.6),

(10.2),

(10.3)

(10.1.7) and

respectively. w

REMARK. R the

is like

Geometrically

the

singularity

local is

ring what

speaking:

In c a s e

of a s i n g u l a r i t y may be called

(i),

R

represents

of an a l g e b r a i c

a "high

node";

curve

the h i g h

(or,

when

node

is

w

an " o r d i n a r y called

node"

a "high

if

cusp";

d =

I.

the h i g h

In case cusp 37

(2),

R

represents

is an " o r d i n a r y

cusp"

what if

may be

d = i.

1.38

In case

(3),

R

represents

what

may be

called

a "nonrational

high

cusp" °

(10.2) V

and

are

W

LEMMA be

the d i s t i n c t

residually

such

that

n_ote the

ON H I G H

rational

ordVX

NODES.

Assume

members

of

over

R.

= 1 = ordwx,

set o f a l l p a i r s

that

~(R).

Assume

that

a_nd fix a n y

(p,q)

card~(R)

= 2, and

Assume

that

there

exists

such

x ¢ R.

of nQnneqative

V

let

and

w

x ¢ R

Let

inteqers,

G

d__ee-

let

W

G

and

--- { (p,q)

for e v e r y n o n n e g a t i v e

G

For

everv

I c R

inteqer

=

n

~ G: p = q]

n

[ (p,q)

le__~t

~ G: p a n ~ q}.

le__~t

G(I)

=

{ (ordvr,

or~r):

0 ~ r e I}

; w

(Note t h a t G(I)

then

G(I)

c G,

is a s u b s e m i g r o u p

every

(p,q)

e G(I)

and

of

and

if

G,

I

is an ideal

i.e.,

p',q')

(p + p ' , q

in

+ q')

R

or

e G(I)

R

, then when-

e G(I).).

Let w

P = R

(Note t h a t

then:

ideals

R

in

.

P

and

Further

R

For

every

w

N M(V)

Q

are

and

Q = R

exactly

N M(W).

all the

distinct

maximal

we h a v e

Q

P =

R

n

Q

=

R

Q

(PQ)

-- M(R).

(re,n) ¢ G, w e h a v e w

S(V) m N M(W) n = pm Q Qn = pmQn =

{r ¢ R : o r ~ r

~ m

and o r d w R

and G ( p m Q n) =

[ (p,q)

38

~ G: p ~ m

and

q ~ n}

;

a n~

1.39

and,

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

for

every

nonnegative

M(V) n n M(W) n = pn Q Q n = p n Q n

integer

= {r c R * :

n

we

(ordVr, ordwr)

have

e G n U {~,~]]

and G(pnQn)

= Gn .)

Let d =

Then

we

have

the

(10.2.1)

: R].

following:

For

a = min(ordVY,

[R/~(R)

any

y ~ R

orgy),

we

with

have

ordVY

paQa

c

~ orgy,

(x,y)R

upon

; whence

letting

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

G a c G((x,y)R). (10.2.2)

We

have

ordv~(R ) = or~(R) ~(R)

= pdQd

G(R)

= G

and

= d = a positive k~(R)

integer

= 2d

and W

For

every

unique

integer

ideal

moreover

we

Ji

have

every

with

i__nn R

and

emdim

0 ~ i ~ d

with

~(R)

R = 2.

we have

c Ji

such

that that

there [R/J i

exists

a

: R] = i ;

have

Ji = ~(R)

For

i

U Gd

+ xiR

y c R

(x,y)R

and

with

= M(R).

k ( R , J i)

orgy For

c ~(R) ~R

any

= 2i

~ orgy

= ~(R)

we

> 2d

~ k(R,e)

39

0 < i ~ d

and

~ ~ R

~ k(R,~)

for

min(ordVY,or~y)

have

,

= 2d

.

,

= d, w__ee

1.40

.a_nd

l(R,~)

(10.2.3) dimensional

Let

f: A - R

regular such

< 2d = I (R,~)

local

anv

elements

that

Then

Ker

f c

((~,~)A) 2

PROOF

OF

(10.2.1).

be

ring.

R

an epimorphism Let

= R[z],

Let

=- 0 ( 2 ) .

any

z ~ R

f(~)R

, ~ c A

=

y £ R

where

~(R),

A

is a t w o -

and

and

~ e A

f(~)z

be

given.

First

there

exists

r e

we

b__ee

=

f(8).

claim

that:

if

a = ordVY

such

that

Namely, 8 e R\M(R)

< ordWY

or~r

since such

= a

W

that

= b,

then

and

ordwr

(x,y) R

> b.

is r e s i d u a l l y

rational

ordW(Y

- 8x b)

> b;

(I) w e

get:

over

R,

it s u f f i c e s

there

exists

to t a k e

r = y - 8x b. By

I

induction

if

on

a = orgy

m,

from

< ordWY

and

m

is a n y

nonnegative

integer,

then

(2) there

Next

(3) I

if

<

exists

we

r ¢

claim

Namely,

exists

by

and

ordwr

and

s = xP-ar Now we

< ordWY s e

(2) w e

> q - p + a + xq

claim

such

that

ordvr

= a

and

ordwr

> m.

that:

a = ordVY

there

(x,y)R

and

(x,y)R

can

(p,q) such

find

r e

; it s u f f i c e s

in c a s e

p < q.

that:

40

e G

that

with ordVS

(x,y)R to t a k e

such

a ~ p = p

that

s = xp

~ q,

and

then

ordws

ordvr in c a s e

= q.

= a p = q,

1.41

if

a = ordVY

such

that

Namely, e R\M(R)

y,

= y - 6,x a.

such

(3)

I if

and

ordvY

then

a = orgy'

since

6'

By

< orgy,

V

there

ordV(Y

(4) w e

get

~ orgy,

y'

¢

(x,y)R

< ordvY°

is r e s i d u a l l y

that

exists

rational

over

R,

there

- 6'x a) > a ; it s u f f i c e s

exists

to take

that:

then

G a c G((x,y)R)

,

(5) where

a = min(ordvY

Now we

claim

if o r d V Y (6)

that:

~ orgy

min(or~y,or~y), (x,y) R Namely,

elements

I, and

(since

residually or~(D

now

there

in

R

that

or~

,

62

e R\M(R) ~'

(since

that

61

to take W

such

R

we

and n o w

~'

over

find

R) w e and

such

V

can

find

- 62t2)

such

W

~'

= ~ - 60t 0 rational

ordv( ~ - 61t I) > e

and

in case

R) w e now

is

that

is r e s i d u a l l y

over

> e,

find

(since

to take

that

rational

41

~ - ~'

= e = ordvD,

can

then

p e R

(since

= ~ - 62t 2.

(5) w e

ordv~

= ~ - 61t I ; finally,

or~(~

that

O r d v t 0 = e = o r d v t 0, o r d v t I =

it s u f f i c e s

~ R\M(R)

such

by

2 ; in c a s e

can

is r e s i d u a l l y that

e R

m a =

> min(ordV~,or~).

rational

= e < or~,

find

~'

ordv( ~ - 60t 0) > e

over

ordv~

can

exists

min(ordv~,Or~)

e = min(or~,or~), such

- 60t 0 - pt I) > e,

R) w e

with

min(or~',ordwD')

rational

it s u f f i c e s

take

then

is r e s i d u a l l y

such

pt I ; in case over

0 ~ ~ c R

ordvt 2 > e = or~t

v

60 c R\M(R)

and

letting

t0,tlt 2

e < or~t first

and

upon

, orgy).

can

ordv~

, and > e =

find

it s u f f i c e s

to

1.42

By induction if

l

(7)

on

i, from

ordVY ~ ordwY ., i

any element

in

NOW

~' c R*

, ordw~')

l

ordVY ~ ordwY

paQa c

integer,

min(ordv~,Ordw~) such that

and

~

is

~ min(ordVY,ordwY)

~ - 4' ¢ (x,y)R

and

> i.

(7) can clearly be reformulated

if

(8)

is any nonnegative

R , with

then there exists

min(ordv~'

(6) we get:

and

(x,y)R + piQi

thus:

a = min(ordVY,or%y),

for every nonnegative

then

integer

i.

Next we claim that:

f there exists

a positive

integer

u

such that

(9) pUjQuj Namely, positive get

c M(R) j since

integer

~(R) u

puQU c M(R);

integer

for every positive is a nonzero

such that

it follows

integer

ideal

puQU c ~(R):

that

pUjQuj

in

j. R , we can find a

since

c M(R) j

~(R) c R, we then for every positive

j.

Finally we claim that:

(lO)

I~ f

ordVY / o r % y

aQa c

Namely, positive

and

a = min(ordVY,OrdwY)

then

(x,y)R. by

integer

(8) and

(9) we get

paQa c

j, and by the Krull [(x,y)R + M(R) j] =

(x,y)R + M(R) j

intersection

for every

theorem we have

(x,y)R.

j=l This completes PROOF OF w

such that

the proof of

(10.2.2). for some

Let ~ ~ R

Q

(10.2.1). be the set of all nonnegative

we have

OrdW~ = w.

42

ordv~ ~ ordw~

and

integers

min(ordv~,

1.43

w

Since

~(R)

Q ~ ~.

c R

Upon

and

is a n o n z e r o

~R)

a = min{w:

that

in

R , we

see t h a t

letting

(1) we get

ideal

a

(2)

w

is a p o s i t i v e

ordVY

h e n c e f o r t ~ ........fix . any

~ orgy

such

e Q}

integer,

and

and

for s o m e

rain (Ordvy,

y 8 R

ordWY)

= a

we have

;

y ¢ R.

By a s s u m p t i o n

x e R

(3) and

so

G*\{0,0]

we have

with

ordvx

--- G ( [ x , x 2,x 3,...])

G a c G((x,y)R);

(4)

G~(R))

In v i e w

of

(10.2.1)

and

paQa

clearly

c G((x,y)R)

consequently,

= G((x,y) R) =

(5)

and hence

= 1 = ordwx

c

in p a r t i c u l a r

; by

in v i e w of

(G \[ (0,0)]

U Ga

(10.2.1)

(i) we g e t

and

G(R)

and that

= G

U G a-

(2) w e h a v e

(x,y)R

paQa

c

~(R)

; also,

for any

r e R*\paQ a

have min(ordvr,ordwr)

~- a

w

and we

can

find

s ~ rR

such

that

m i n ( o r d V r , o r d w r) = m i n ( o r d V s , o r d W S )

and then

in v i e w

thus we have

proved

(6)

By

(7)

of

(2)

(4) w e

< max(ordVs,ordw

see t h a t

s ~ R

that

~(R)

= paQa

.

(6), w e get ordv~(R ) = ordw~R) 43

= a

and h e n c e

s)

r ~

~(R)

;

we

i. 44

and,

since

V

and

W

are

(8)

residually

k~(R)

Since (6) w e

V

and

W

are

rational

over

R, w e

also

get

rational

over

R, b y

(4) and

= 2a.

residually

see t h a t

(9)

for any

~ e R

w e have:

~ e

~(R)

(i0)

for a n y

~ ¢ R

w e have:

c~R

for any

~ e R

w e have:

I(R,~)

~ k (R,~)

a 2a

w

=

~(R)

~ ~ (R,~)

= 2a

and

(ii)

We

claim

(R A

rational then

{ ¢ R

given

over

in v i e w

of

reverse

can

we

inclusion

claim

[(R N

Namely,

e R n find

= ordw@

have

= i < a

we have

(piQl).

(piQi),

p ~ R

(since

such

ordw(g

v

that

-p~)

is r e s i d u a l l y

Ordv(~

> i

-Og)

> i

and h e n c e

; thus

(pi+iQi+l))

+ ~R D

is o f c o u r s e

induction

(R n ( p a Q a ) )

Next

(14)

~

(4) w e m u s t

By decreasing

(13)

ordv{

+ {R = R N

(pi+iQi+l)

(R n

the

any

R) w e

- pg e R n

and

with

(pi+iQi+l))

Namely,

~ 0(2).

that

for any (12) I

~ 2a = k(R,~)

on

(piAi. U ) ,

obvious.

i, in v i e w o f

+ xlR = R N

.pi_i, ~ ~ )

(3) and

for

(12) w e get:

0 ~ i ~ a.

that:

(piQi))/(R

N

(pi+IQi+l))

for e v e r y

44

: R3 =

1

for

0 ~ i < a.

and

1.45

~

by

(4) w e m u s t h a v e

(R n

o r d v ~ -- i = o r d w ~

(R n this shows

(pi+iQi+l))+

(pi+iQi+l))

and h e n c e b y

~R = R N

(piQl)

(12) w e h a v e

;

that [ (R N

(i4 i) by

(piQi))\(R n

(piQi))/(R n

(pi+iQi+l))

: R~ s 1 ;

(3) xi e

(R n

(piQ1))\(R N

(pi+iQi+l))

and h e n c e (142 )

R N

now by

(141 ) and

(piQi)~

R Q

view of

(pi+iQi+l)

; hence by applying

: R~ = 1

(13) w i t h

i = I, in

(x,y)R = M(R)

(15) we h a v e

e m d i m R < 2; n o w

R

is not r e g u l a r b e c a u s e

we m u s t h a v e

(16)

emdim R = 2 . Since

R N

(17)

(6),

(3),

(pOQ0) = R, u p o n

J. = 1

(13) and

(18) By

;

(5) w e get

therefore

by

(PIQI))/(R Q

(PQ) = M(R)

(15)

By

(pi+iQi+l)

(142 ) we get [ (R n

Now

R N

setting

~(R)

,

(14) w e get

[R/J i : R~ = i (6) and

+ xiR

(17) we h a v e 45

for

0 < i s a .

~(R) ~ i;

1.46

(19)

X ( R , J i) =

We

claim

given

2i

for

0 ~ i ~ a.

that:

any

ideal

J

in

R

with

~(R)

c J,

upon

letting

i = ordvJ, (20) we have

Namely, 0 < i < a,

that

i

in v i e w

is a n

(4)

and

J c R n

i = a

that

then

since

~(R)

J = J a ; so h e n c e f o r t h

ordv~

=

i

and

then

=

(203 )

c J)

(2)

and

in v i e w

6x I + ~ y

(6) w e h a v e

y

(204 )

e

xi c

since

(R) c J, b y that

is

an

J = Ji"

integer

with

;

by

(3) , (6), (17)

that

(6),

~(R)

(13),

hence

of

i < a; w e

(2),

with

~(R),

and can

(201 ) w e

see

take

(3)

8 e R\M(R)

and hence

+ ~R

(17),

and

by

(15) w e

and

(pOQ0)

= R, b y

by

the

definition

of

(6),

(203 ) w e

(201),

(13)

and

we have

a = d.

46

get

;

: R] = a

d

get

~ e R

(202 ) a n d

(204 ) w e

1

R N

[R/~(R)

(21)

i

that

and

J = J..

Since

and

0 ~ i ~ a,

~ e J

with

clude

see

(piQi)

assume

(202 )

by

(6) w e

with

and

(201 )

if

integer

(14) w e

get

con-

1.47

Now

in v i e w

(Ii),

and

PROOF

of

(21),

(15) to

OF

the p r o o f

of

(10.2.2)

R

= R[z],

is c o m p l e t e

by

(4),

(20).

(10.2.3).

Since

and

so,

upon

relabelling

z ~

P.

Now,

since

V

V

and

W

we must

suitably,

is r e s i d u a l l y

(I)

rational

have

we may over

z ~

that

R, w e c a n

find

that

(2)

z + f(8)

¢ P.

w

NOW

R

= R[z

+ f(8)],

and h e n c e

(3)

z + f(8)

Since

f(~)R*

=

~(R),

(4)

by

.

we get

that

-- d = o r d w f ( ~ )

~ =

clearly

~ +

66

have

(6)

By a s s u m p t i o n

(~,9)A =

f(~)z

=

consequently,

(7)

By a s s u m p t i o n

by

(2),

=

and h e n c e

f(~)(z

(3) a n d

ordwf(~)

OrdVx

(~,~)A

f(8),

f(N)

(8)

have

letting

(5)

we

~ Q

(10.2.2)

ordvf(~)

Upon

we must

+ f(6))

(5) w e h a v e

;

(4) w e g e t t h a t

= d < Ordvf(~)

= 1 = OrdwX,

{ e A

by

with

and h e n c e

f(~)

47

-- x

.

upon

P n Q,

suppose

8 ~ A\M(A)

such

(6) to

fixing

any

i. 48

we h a v e

(9)

Ordvf(~)

NOW,

in v i e w

of

(7) and

(I0)

By

= 1 = ordwf(<)

(8), b y

(f({),f(9))R

(10.2.2)

we h a v e

emdim

(10.2.2)

;

we g e t

that

= M(R)

R = 2, and h e n c e

in v i e w

of

(I0) w e c o n c l u d e

that

(ii)

and

(~,~)A = M(A).

Since

d > 0,

in v i e w of

(7) w e

also h a v e

(4) w e g e t

~ ~ ~A

~ e M(A)

; consequently,

; in v i e w

in v i e w

of

of

(4)

(ii) we

can

write

(12)

and

~ =

b

6'~ b + D~

is a p o s i t i v e

where

integer;

(13)

By

now,

b = d

(12)

and

(13) w e

see t h a t

6' c A\M(A)

and

by

(9) a n d

(4),

(7),

p e A

(12) w e g e t

.

({d,n)A =

(e,9)A,

and h e n c e

by

(6) w e

get (~d,~)A =

(14)

Let then NOW,

y ¢

any

(({d,9)A)2,

in v i e w of

(15) such

y e A

with and

(11), w e

f(y)

(~,~)A.

= 0

in v i e w

b e given.

of

(14)

We

this will

shall

show that

complete

the p r o o f .

can w r i t e

y = r + s~ + t~ 2

with

r,s,t

in

A

that I

either

r = 0

(16) or

r = 60 ~a

with

60 e A\M(A)

48

and n o n n e g a t i v e

integer

a

1.49

l

either

(17) [ o r

By

(9),

s = 0

s = 81 gb

(16) and

(18)

f(~)

(19) N o w by

= or~f(r)

= 0, in v i e w o f

ordvf(r) (15),

(16),

and

integer

b.

that

and

ordvf(S)

(7),

= oral(r) (17)

and n o n n e g a t i v e

8 1 e A\M(A)

(17) w e d e d u c e

ordvf(r)

Since

with

(15) and

a 2d (19)

and

= or~f(s)

(18) w e c o n c l u d e

ordvf(S)

it f o l l o w s

that

that

= or~f(s)

z d .

y c ((~d,9)A)2.

w

(10.3) {V] R N

LEMMA ON HIGH CUSPS.

and that

V

is r e s i d u a l l y

for e v e r y

n c G

I c V

[2m

Assume R.

that

(M(V)2\M(V)3)-

Let

let

: m ~ G]

{2m+l

: m ~ G

,

with

m ~ n]

let G(I)

= {ordvr

: 0 ~ r e I]

Let d = [R/~ (R) Then we have

: R]

the f o l l o w i n g ,

(10.3.1).

d

is a p o s i t i v e

ordv~(R)

integer,

= 2d = X~(R)

and

49

and

that

Assume

let

Gn For every

over

x c R N

integers,

G

V = R .

rational

(M(V)\M(V) 2) = ~, and fix any

b the s e t of all n o n n e g a t i v e

and

Let

~(R)

= M(V)

2d

~(R)

G

=

1.50

G(R)

For

every

unique

inteqer

ideal

moreover

i

Ji

= G

with

i__nn R

U Gd

and

0 ~ i ~ d

with

~(R)

emdim

we have

~ Ji

R =

2.

that

such

that

2i

for

there

exists

a

[ R / J i : R~ =

i ;

we have

Ji = ~(R)

_FOr e v e r y F__oor an%<

y

¢ R

~ e R

+ xiR

with

and

ordVY

k ( R , J i) =

= 2d + 1

we have

0 ~ i ~ d.

(x,y)R = M(R).

we have

e ~(R) ~R

= X(R,~)

= {(R)

z 2d

~ X(R,~)

=

2d

and X(R,~)

(10.3.2)

Let

~ 2d = I ( R , ~ )

f: A - R

be

~ 0(2).

an epimorphism

where

A

is a t w o -

w

dimensional

reqular

local

r i n q.

Let

z c R , ~ c A,

w

any

elements

Then

Ker

such

that

R

and

8 ¢ A

be

w

= R[z~,

f(~)R

= ~(R),

and

f(~)z

=

f(~).

f c((~,~)A) 2

PROOF

OF

(10.3.1).

Given

any

0 ~

~ e V

we

can write

~ =

~'/~

W

with

0 ~

ordv~

must

~'

¢ R

; since

contain

and ordv~

some odd

0 ~ =

l

upon

positive

and

some

then

we have

~ ¢ V, w e

ordv~

conclude

= or%~'

that

G(R)

i.e.,

: 2w + 1 ~ G(R)}

=

0-

letting a = min{w

(since by

~ R,

for

integer,

[w ~ G

Now

~

assumption integer

and

R n for

e G

: 2w + 1 ¢ G(R)}

( M ( V ) \ M ( V ) 2) = some

y

~ R

50

0)

we have

we

get

that

a

is a

-

1.51

(2)

orgy

henceforth

fix any such

=

2a + 1 ;

y e R.

By a s s u m p t i o n

(3)

x ~ R

and h e n c e b y

(4)

(i) a n d

G(M(R)) We c l a i m

with

ordvX = 2

(2) w e see t h a t

= G((x,y)R)

=

(G*\[0})U G A

and

G(R)

= G

U Ga

that

l (5)

Igiven

any

0 ~ 9 e V

such t h a t

Namely, such t h a t

~ - ~'

upon

with ordv~

¢

(x,y)R

letting

ordvt = e

and

e = or~9,

and t h e n

~ 2a,

ordv~'

by

(since

V

there exists

e V

> ordvg.

(4) w e can find

t ¢

is r e s i d u a l l y

rational

w

R) w e can find

~'

(x,y)R over

w

6

c R\M(R)

such that

or~(9-6

t) > e ; n o w it

W

suffices

to take

0' = 9 - 6 t.

By i n d u c t i o n on (6)

M(V) 2a c

i, from

(5) we get:

(x,y)R + M(V) i

for e v e r y n o n n e g a t i v e

integer

i.

N e x t w e c l a i m that:

(7) I

there

exists

a positive

M(V) uj c M(R) j Namely, positive t h e n get positive By

since

integer

(6) and

is a n o n z e r o

such t h a t

M(V) u c M(R) integer

u

for e v e r y p o s i t i v e

~(R) u

integer

j. (7) w e get

51

integer

ideal

M(V) u ~

; it f o l l o w s

such t h a t

that

~(R)

in

j V, w e can find a

; since

~(R) c R, we

M(V) uj c M(R) j

for e v e r y

1.52

M(V) 2a c and by

the

(x,y)R + M(R) j

Krull

for e v e r y

intersection

[(x,y)R

theorem

+ M(R)J~

=

positive

integer

j

we have

(x,y)R

;

therefore

j=l (8)

M(V) 2a c

and h e n c e we

can

s ~

in p a r t i c u l a r

find

R

s ¢rR

and h e n c e

r ~ ~(R)

; thus

(i0)

ordv~(R)

(i) w e

~(R)

M(V) 2a c ~(R) ordVS

~(R)

Since

.

with

(9)

and

(x,y)R

=

=

2a - 1

we h a v e

= M(v)2a

see that,

and

X~(R)

= 2a

(12)

~ c {(R)

(13)

~R

and

any

then by

proved

*

r ~ R \M(V)

2a

(4) w e h a v e

that

. V

for any

(ll)

given

and hence

= 2a

IV]

; also

is r e s i d u a l l y

rational

~ ¢ R, we h a v e

the

over

R, b y

(4)

following.

.

~ l(R,~)

~ 2a

w

= {(R)

~ k(R,~)

-- 2a

.

and (14)

I (R,~)

We

claim

{

2a

=

l(R,~)

~

0(2)

.

that

for a n y

{ e R

with

ordw{

we have

(R N M(V) 2i+2)

= 2i

where

i

is an

integer

< a ,

(15) + ~R = R N M(V) 2i

w

Namely, rational

given

over

any

{

R) we c a n

e R N M(V) 2i, find

p e R

(since

such

that

V

is r e s i d u a l l y

o r d v ( { -p~)

> 2i

w

then *

in v i e w of

(4) w e m u s t

- p{ ¢ R N M(V)

2i+2

have

the

reverse

By d e c r e a s i n g

~ 2i + 2, and h e n c e

; thus

(R N M(V) 2i+2) and

o r d v ( { -p{)

inclusion

+ {R D R Q M(V) 2i

is of c o u r s e

induction

on

i,

52

,

obvious.

in v i e w

of

(3) and

(15) w e get

and

1.53

(16)

(R N M(V) 2a)

+ x i R = R O M(V) 2i

for

0 ~ i < a .

for

0 < i < a .

N e x t we c l a i m that (17)

[(R N M(v) 2 i ) / ( R

Namely,

(R n M ( v ) 2 i ) \ ( R

(4) we m u s t h a v e

o r d v ~ = 2i

(R N M(V) 2i+2)

this

shows

N M(V) 2i+2)

and h e n c e b y

+ ~R

=

(15) we h a v e

R Q M(V) 2i

that [(R N M(v) 2 i ) / ( R

(171 ) by

: R]

for e v e r y ~

by

N M(V) 2i+2)

N M(V) 2i+2)

: R~ ~ 1 ;

(3) we h a v e x

i

C

(R N M(V)

2i\

(R N M

(V) 2i+2)

and h e n c e (17 2 )

(R Q M(v) 2 i ) \ ( R

now by

(171 ) and

~ ~ ;

(172 ) we get [(R N M ( v ) 2 i ) / ( R

Now by assumption M(R)

N M(V) 2i+2)

; consequently,

R P

N M(V) 2i+2)

: R~ = 1 .

(M(V)\M(V) 2) = ~, and h e n c e

by applying

(16) w i t h

R e M(V) 2 =

i = I, in v i e w of

get (18) By

(x,y)R = M(R).

(18 we get

R N (19)

emdim

R < 2 ; now

(M(V)\M(V) 2) = ~ ; t h e r e f o r e emdim

R

is not r e g u l a r

we m u s t h a v e

R = 2

53

because

(8) we

i. 54

Since

R n M(V) 0 = R,

J

(20)

by

(9),

(16)

and

(21)

By

(3),

(9) a n d

(22)

claim

given

f

i =

(23) I

J

we have

that

i

in v i e w and

0 < i < a .

in

R

with

~R)

c J,

upon

letting

is an

of

(4)

integer

and

with

(9) w e

0 ~ i ~ a,

see

that

(2),

(3)

i

and

J = Ji

is an

integer

clearly ;

~cJ

and

=

2i

and

~ =

6x I + ~y

then

(9) w e h a v e

(234 )

clude

.

take

(233 )

since

for

J c R N M(V) 2i

ordv~

(2)

0 < i ~ a

for

(i/2)or~J,

(232 )

by

i

2i

ideal

(231 )

with

that

that:

0 ~ i ~ a,

can

,

(20) w e h a v e

any

Namely, with

letting

+ xlR

see

: R~ =

(R,J i) = We

we

--- ~(R)

l

(17) w e

[R/J i

upon

xi c

~(R) that

c J, b y

(9),

in v i e w

y ~

~(R)

of

with

8 e R\M(R)

~(R),

and

+ ~R

(16),

hence

and

and

by

(18)

~ ~ R

(233 ) w e

we

get

;

get

;

(20),

J = J.

l

54

(231),

(232 ) a n d

(234 ) w e

con-

1.5~

Since

R N M(V) 0 = R, b y

(9),

[R/~(R)

and h e n c e

b y the d e f i n i t i o n

(24)

Now

in v i e w

(14),

and

(18)

PROOF

OF

rational 8 e A

of

(24), to

such

(17) we get

d

we h a v e

of

(10.3.1)

.

the p r o o f

is c o m p l e t e

R, and

By assumption R n

R

= R[z],

V

(M(V)\M(V) 2 = ~, t h e r e f o r e

that

(I)

o r d V (z+f (8)) =

1

and

(2)

Since

f(6)R

either

8 e A\M(A)

or

=

by

w e get

~(R),

(3)

~ =

clearly

that

.

~ + 66

have

(5)

(~,~)A =

By a s s u m p t i o n

f(e)z

=

f(~)

consequently (6)

= 2d

6 = 0 .

letting

(4)

we

(i0.3.1)

ordvf(C~)

Upon

by

by

(4),

(9) to

(23).

(10.3.2).

over

and

: R] = a

of

a = d

(16)

(i) and

f(~),

=

(~,8)A

and h e n c e

f(~)(z

+ f(6))

(3) w e g e t

ordvf(D)

by

that

--- 2d + 1

55

(4) we h a v e

;

is r e s i d u a l l y we

can

find

1.56

By a s s u m p t i o n

ordvx

(7)

=

2

{ e A

and hence,

upon

with

= x

f({)

fixing

any

,

we have

(s)

ordvf(~)

NOW,

in v i e w

of

(6) and

(9)

(7), b y

(f(~),

By

(10.3.1)

clude

we h a v e

(10.3.1)

we g e t

that

f ( ~ ) ) R = M(R).

emdim

R =

2, and h e n c e

in v i e w

of

(9) w e

con-

that

(i0)

(~,~)A = M(A)

Since and

= 2

d > 0,

in v i e w

(6) w e a l s o h a v e

~ ~

of

.

(3) w e g e t

~A

~ ~ M(A)

; consequently,

; in v i e w

in v i e w

of

of

(3)

(i0) w e

can

write

=

(II)

and

p

6' {P + p~

is a p o s i t i v e

integer;

(12)

By

where

nOw,

p = d

(ii)

and

(12)we

6' e A \ M ( A )

and

by

(8) and

(3),

(6),

p e A

(ii) w e g e t

°

see that

({d,~)A =

(~,~)A,

and h e n c e

by

(5)

w e get

(13)

(@d,~)A =

Let a n y then proof.

(14)

y ~

y ¢ A

((~d,z)A)2,

Now,

in v i e w

with

f(y)

and of

(~,$)A.

= 0

in v i e w

(i0),

y = r + sD + t~ 2

we

of

be given. (13)

We

this will

can w r i t e

with

56

r,s,t

in

A

shall

show that

complete

the

i. 57

such

that

[

either

r = 0

or

60 ~a

(15) r =

with

6 0 e A\M(A)

and

nonnegative

integer

a

with

81 ¢ A\M(A)

and

nonnegative

integer

b

and

( I either

s = 0

(16) or

Since

s = 61 ~b

f(y)

= 0, b y

(6),

(8),

r ~ {2dA

(17)

Now by

(14)

and

(17)

(14), and

it f o l l o w s

(15) s

and

¢ {dA

that

y ~

(16) w e d e d u c e

that

.

((@d,9)A)2 @

(10.4) that

~(R)

=

LEMMA

ON N O N R A T I O N A L

IV}.

Assume

R n M(V)\M(V) 2 ~ g: V ~ V/M(V)

~

and

b e the

that

T h e n we h a v e

the

(10.4.1)

CUSPS.

[V/M(V)

fix an v

canonical

d =

HIGH

x

Let

: R/M(R)]

¢ R n

= 2.

(M(V)\M(V) 2) .

epimorphism.

[R/~(R):

V = R .

Assume

Assume

that

Let

Let

R]

followinq

We h a v e

or%{(R)

= d = a positive

~(R) = M ( v ) d

and

integer

k ~(R) = 2d

and emdim For

everv

unique

inteqer

ideal

J

in 1

moreover

i

with R

R = 2 .

0 ~ i ~ d

with

~(R)

we h a v e c J

such

that that

- -

there

exists

[R/J.

: R~ 1

we have Ji ~

~(R)

+ x±R

and

k(R, Ji)

57

= 2i

for

0 ~ i ~ d

o

a =

i

;

1.58

For any

0 ~ ~ ¢ R

we have:

d ~ ordv@

For every

y ¢ R

(x,y)R = M(R).

I

there exists

[

ordv@'

~ith

= ordv@

ordVY = d

For an~

@' ¢ R

such that

and

and

g(9'/@)

~ g(R).

g(y/x d) ~ g(R), we have

~ 6 R

we have

-- {(R)

~ X(R,~)

= ~(R)

~ X (R,~) -- 2d

w

aR

~ 2d

w

~R and

X(R,~) (10.4.2). dimensional

Let

~ 2d = I (R,~) -= 0(2).

f: A - R

reqular l o c a l

be a n eDimorphis m where

rinq.

Let

z ¢ R

W

an y elements Then

such that

Ker f c

((~,~)A) 2

PROOF OF

(10.4.1).

R

, ~ C R

A and

is a two~ ~ R

be

*

= R[z],

f(~)R

= {(R),

and

f(~)z = f(~).

We claim that:

there exists a positive

integer

u

such that

(I) I M ( V ) uj c M(R) j Namely, positive get

since

integer

M(V) u c M(R);

integer

for every positive

{(R) u

is a nonzero

such that

integer

ideal in

M(V) u c ~(R);

it follows that

j.

V, we can find a

since

M(V) uj c M(R) j

~(R)

c R, we then

for every p o s i t i v e

j.

Recall that b y assumption (2)

x c R Let

some

Q

~ ~ R

g(V) ~ g(R),

with

ordVX = I.

be the set of all nonnegative with

or%~

and hence

= w

we have

in v i e w of

integers

w

such that for

g (~/x w) ~ g (R).

By assumption

(i) we see that 58

~ ~ ~.

1.59

Now upon

letting

(3)

a = min[w:

we get that

a

w ¢ Q}

is a p o s i t i v e

i n t e g e r and t h e r e

exists

y ¢ R

such

that

(4)

ordvY = a

henceforth

fix a n y

such

We c l a i m that

(5)

a

OrdV8

I OrdVS'

if

ordve = ordvxb cOnsequently if and

e

and

g(@'/e)

hence by

and e i t h e r

;

such that

and

g(e'/8)

then by

to t a k e in

then either

(3) w e m u s t h a v e

we h a v e

0' e R

g(yxb-a/8)

are e l e m e n t s

~ g(R),

@ c R

= OrdV8

ordv8 = b m a

it s u f f i c e s 8'

0 ~

exists

~ Namely,

g ( y / x a) ~ g(R)

y ~ R.

for any

I there

and

(4) w e see t h a t ~ g(R)

6' = y x b - a R

~ g(R).

or or

such that g(~'/xh)~g(R)

ordVYX

b-a

g(xb/8) ~ g(R) 8' = x b.

or~8' or

= ;

Conversely,

= ordv@ = b ~ g ( 8 / x b) ~

g(R)

b ~ a.

Next we claim that

given any

0 ~ ~ e V

such that

q - ~' e

Namely, elements

p

since

and n o w it s u f f i c e s By induction

M(V) a c

(7) By

(i) and

(x,y)R

[V/M(V) p*

and

w i t h ord V = e ~ a, t h e r e

on

in

R

to t a k e i, from

and

: R/M(R)]

OrdV~'

> ordv~

= 2, in v i e w of

such t h a t 4' = ~ - px

exists

~' ¢ V

.

(4) we c a n

find

o r d v (qx -e - p - p*yx -a ) > 0 e

* e-a - p yx

(6) w e get:

(x,y)R + M(V) i

for e v e r y n o n n e g a t i v e

(7) w e g e t

59

integer

i.

;

i. 60

M(V) a c

and by

the

Krull

(x,y)R + M(R) j

intersection

for

theorem

every

positive

integer

j

we have

¢0

n j=l

[x,y)R

+ M(R) j] -- ( x , y ) R

;

therefore

(8)

and

M(V) a c

hence

ordvr

= p < a,

s ~ V that

in p a r t i c u l a r

with either

upon

OrdvS r ~

R

M(V) a c ~(R)

letting = p

and

or

s ~

; also,

s = r y / x a, b y g(s/r) R

~(R)

(9)

(x,y)R

~

; thus

given

any

(2) a n d

g(R),

and

we have

(4) w e

then

proved

r ¢ V

by

get

with that

(5) w e

see

that

= M(V) a

and hence

(io)

ordv~(R)

~(I{) =

Since see

{V}

and

[V/M(V)

= a

.

: R/M(R)]

=

2,

in v i e w

¢

~(R)

of

(i0) w e

that

(n)

X~(R)

= 2a

(12)

for any

~ ¢ R

we have:

(13)

for

any

~ ¢ R

we have:

~R

for any

~ ¢ R

we have:

X(R,~)

~ k (R,~)

m 2a

@

=

~(R)

~ k(R,~)

= 2a

and

(14) We

claim

that

60

&

2a = k ( R , ~ )

=- 0(2)

1.61

/ for any

~ ¢ R

with

o r d v ~ = i < a, we h a v e

(15) (R N M(V) i+l)

Namely,

given

+ ~R = R N M(V) i

any

~ * ~ R N M(V) i

we must h a v e

*

then o b v i o u s l y ,

and

if

ordv~

(if

ordv~*

>

= i, then b y

(5))

g(~ /~)

e g(R)

W

h e n c e we can find

p e R

such that

(R N M(V) i+l)

and the r e v e r s e

inclusion

By d e c r e a s i n g (16)

~

- p~ ¢ R N M ( v ) i + I

; thus

+ ~R ~ R D M(V) i ,

is of course

induction

on

obvious.

i, in v i e w of

(R q M(V) a) + x i R = R Q M(V) i

for

(2) and

(15) we get

0 ~ i < a .

Next we claim that (17)

[ (R D M(v) i/(R D M(V) i+l)

Namely,

by

for

0 ~ i < a .

(R N M(v) i ) \ ( R N M(V) i+l)

(15) we h a v e (R N M(V) i+l)

which

+ {R = R D M(V) i

shows that

(171 ) by

: R~ = 1

for e v e r y ~

[ (R n M ( v ) i ) / ( R

N M(V) i+l)

: R] ~ 1 ;

(2) we h a v e xi ~

(R n M(v) i ) \ ( R D M(V) I+I)

and h e n c e

(172 )

i,

W

(R N M(v) i ) \ ( R Q M(V) i+l) @ @ ;

61

and

1.62

now by

(171 ) a n d

(172 ) w e

get

[ (R n M ( v ) i ) / ( R

Now view

of

R N M(V) (8) w e

= M(R)

and hence

by

; R] = 1 .

applying

i =

(16) w i t h

I,

in

get

(18)

BY

D M ( V i+l)

(x,y)R = M(R)

(18) w e

g(V)

get

~ g(R)

emdim

R < 2 ; now

; therefore

we must

(19)

emdim

Since

R N M(V)

0

= R,

J

(20)

R

is n o t

regular

because

have

R =

upon

2 .

letting

=

~(R)

see

that

+ xlR

,

1

by

(9),

(16)

and

(17) w e

(21)

for

[ R / J i : R] = i

Since

[V/M(V)

: R/M(R)]

(22)

=

2 , by

k ( R , J i) =

We

claim

given

2i

0 < i < a

(2),

for

(9) a n d

0 < i < a

.

(20) w e h a v e

.

that:

any

ideal

i = or~

J

we have

that

J

in

R

with

~(R)

c J,

upon

letting

,

(23)

Namely, 0 < i < a,

in v i e w and

i

is a n

of

(9) w e

i = a

see

with

that

i

0 ~ i ~ a,

is a n

and

integer

J = Ji

"

with

clearly

J c R N M(V) i ;

(231 )

if

integer

then

(since

~(R)

c J) b y

62

(2),

(9),

(20)

and

(231 ) w e

see

i. 63

that

J = J a ; so h e n c e f o r t h

(232)

with

ordv~

= i

and

then

~ =

(4) and

(9) w e h a v e

~(R)

c J, b y

that

J = J

Since

(9),

i

(16),

by

in v i e w

0

have of

and

(18)

PROOF

OF

g(z) ~ (10.4.1),

(1)

(24), to

(9),

by

and

(233)

~ e R

;

w e get

(231),

(232 ) and

(234 )

we

con-

(16)

and

(17) w e g e t

of

d

we h a v e

of

(10.4.1)

.

is c o m p l e t e

by

(5),

(9) to

(23).

Since

g(R).

Since

we get

that

ordvf(~)

(2)

6 ¢ R\M(R)

(18) we get

: R] = a

the p r o o f

(10.4.2).

By assumption

V = R[z]

f(~)R

--- d = o r d v f ( ~ )

Ordvx

e A

= 1

with

=

and

~(R)

and hence,

f(~)

or~f(~)

= x

= 1 .

63

and

g(V)

and

f(~)z

g(f(~)/f(~))

we h a v e

(3)

take

+ ~R ;

(20),

= R, b y

the d e f i n i t i o n

of

(4) and

and h e n c e

¢ ~(R)

a = d

(14),

can

l

R n M(V)

(24)

Now

i < a ; we

(2),

with

[R/~(R)

and h e n c e

of

y ~ ~(R),

x

since

in v i e w

6x I + ~y

(234 )

clude

that

~ e J

(233 )

by

assume

upon

,

~ g(R), =

f(~),

~ g(R).

fixing

any

we m u s t in v i e w

i. 64

In v i e w

(i), t h e r e

g(f(9)/f([d))

that A

of

such

is a p e r m u t a t i o n

~ g(R).

Note

that

now

of

(9,{) ~

(~,~)

such

are e l e m e n t s

and

in

that

(4)

o r d v f (~3) = d

(5)

g(f(~])/f(~d))

(6)

ordvf(~)

(7)

g(f({)/f(~))

(~ g(R)

= d

~ g(R)

and

(8)

(~,~])A =

In v i e w

of

(2),

(4) and

(9)

By

(5), b y

(f(~),

(10.4.1)

(c~,8)A

.

(10.4.1)

we get

f ( ~ ) ) R = M(R)

we also have

emdim

R = 2, and h e n c e

by

(9) we

conclude

that

(i0)

(~,~)A = M(A)

Since (6) and

d > 0,

in v i e w

(7) w e a l s o h a v e

of

(6) w e get

~ ~

~(A)

~ ~ M(A)

; consequently,

; in v i e w in v i e w

of

of

(4),

(i0) w e

can w r i t e

(Ii)

and

~ =

p

6'~ p + p~

is a p o s i t i v e

where

integer;

6'

now by

e A\M(A)

(3),

(4),

and

p e A

(6),

(7) and

(ii) w e

get

(12) By

(ii)

p = d and

(12) w e

see t h a t

. (~d,~)A =

get

64

(~,~)A,

and h e n c e

by

(8) w e

1.65

(13)

(~d,~)A = Let any

then

y e A

with

y ¢ ((~d,~)A)2,

Now, in view of (14)

(~,~)A .

f(y) = 0

be given.

and in view of

We shall show that

(13) this will complete the proof.

(i0), we can write

y = r + s~ + t~ 2

with

r, s, t

in

A

such that

(15) I either i or

r = 0

r = 60 ~a

with

60 ¢ A\M(A)

and nonnegative

integer

a

with

61 ¢ A\M(A)

and nonnegative

integer

b .

and I either

s = 0

(16) I

or

s = 81gb

Since

f(y) = 0, by

(4),

r e ~2dA

(17) NOW by

(3),

(14) and

(5), and

(14),

(15) and

(16) we deduce that

s e ~dA .

(17) it follows that

65

y c ((~d,~)A)2

CHAPTER

IIo

PROJECTIVE

By a h o m o g e n e o u s family

[Hn(A)]0mn<~

underlying

additive

GEOMETRY

OR HOMOGENEOU S DOMAINS

domain we mean a domain of a d d i t i v e

g r o u p of

subgroups

A

space

of

A

(Note t h a t t h e n

for all

We may refer

H0(A)

and

n ; HI(A) n = Hn(A)

for all

n ; A

that

B0-vector-subspace

B0

of

B0 N

(Bl)n = [0]

[0]

for all d i s t i n c t

a homogeneous

and

for all

such that

for all p o s i t i v e postive

and an

: H0(A)]

= Hm+n(A)

; and

of domain

B = B0[BI],

H0(B)

B

= B0

for all

: H0(A)] A

< ~

is n o e t h e r i a m

and

B1

is a

0 < [B 1 : B 0] < ~ ,

n, and and

<

A.

H0(A)

m

is a

H0(A)-vector-

0 < [HI(A)

integers

the

n ; H0(A)

n > 0 ; 0 < [Hn(A)

integers

domain upon taking

m

Hm(A)Hn(A)

is a s u b f i e l d

B

s u c h that:

a

s u m of the f a m i l y

field of

have:

together with A

becomes

is an a f f i n e d o m a i n o v e r

Now assume

all

Hn(A)

as the g r o u n d

Note that we automatically m

for all

n.); A = H 0 ( A ) [ H I ( A ) ] ; to

of

is the d i r e c t

[Hn(A)]0~n< ~ ; Hm(A)Hn(A ) c Hm+n(A ) subfield

A

n. and

(Bl)m n Then

B

Hn(B ) =

(Bl)n = becomes (BI)n

for

n > 0. In the r e s t o f C h a p t e r

§ll.

Function

II,

let

A

be a homogeneous

fields and projective

domain.

models.

We define H(A)

=

U

H n (A) .

0gn<~ F o r any

0 ~ x ~ ~(A)

D e g A x = the u n i q u e

we define

nonnegative

and w e a l s o d e f i n e

DegA0

= _

We define

66

integer

n

such that

x c Hn(A)

2.2

R(A) Note

=

U {x/y 0 ~n<~

that:

0 ~ y ¢ H n(A)

R (A)

, and

H0(A)-vector-space, function z

we have

field over

~(A) R(A)

and

0 ~ y ¢ Hn(A)

is a s u b f i e l d o f I ~ H n(A)

is t r a n s c e n d e n t a l

have

: x ¢ H n(A)

~ (A) ; for e v e r y

such that

I

generates

R(A) = H 0 ( A ) ( [ x / y

H 0 (A)

; for e v e r y

over

R (A)

: x ¢ I~)

z ¢ H (A) \H 0 (A)

; and for e v e r y

z ¢

n > 0 , H n(A)

as an

: R(A)

is a

we have that H (A)\H 0(A)

we

= R(A) (z). m a y be c a l l e d

the

function

f i e l d of

A

We d e f i n e Dim A = trdegHO(A)R(A) and Emdim A =

[H I(A)

: H0(A)]

- 1

a n d w e n o t e that then: Emdim A ~

Dim A =

[trdegH0(A)~(A)3

- 1

and Dim A =

[dim of the u n d e r l y i n g

(nonhomogeneous)

ring of A~ - i.

We d e f i n e

~(A)

= ~ ( H 0 ( A ) , H I(A))

for a n y d o m a i n Note that In c a s e

K, S(R) ~(A)

=

U ~(H0(A) E{y 0 ~ y c H 1 (A)

denotes

{Rp : p ¢ ~(R)}

is a p r o j e c t i v e

D i m A = i, we d e f i n e

(A) = ~(R(A), In case

model of

H 0(A)).

D i m A = 0, we d e f i n e

Deg A =

JR(A)

67

: H 0(A)~

: x ¢ H I ( A ) } ~ ~ where, .

R (A)

over

H 0 (A).

2.3

and w e

note

that

We observe

then

Deg A

is a p o s i t i v e

integer.

that

Emdim A = 0 =

R(A)

= H 0 = Dim A = 0 ,

and Dim

A = 0

and

§12. Let (ring)

A'

be also

H 0(A)

algebraically

Homogeneous

f

c H n(A' )

for a l l

f ( H n(A))

c H n(A' )

for

geneous,

then:

is s u r j e c t i v e

f ( H n(A))

f

= H n(A')

§13. An

ideal

for a l l

in

A

all

~ x ~ Q 0 ~n<~ n sufficiently

(2)

Q =

(Q n H ( A ) ) A

(3)

Q =

IA

Note and

that

that:

Also

(of d e g r e e

f

note

= f(Hn(A))

f: A - A' zero)

a

if

is h o m o g e n e o u s

that,

if

f

= Hn(A')

is h o m o -

for

n = 0,

1

n.

ideals

conditions

x

be

and hypersurfaces.

¢ H

n

n

are

(A)

if t h e

following

satisfied:

for a l l for a l l

n

(where x

n

= 0

for

n ;

;

for s o m e HI(A)A

or

Note

n) = x n c Q

for a n y h o m o g e n e o u s

radAQ = A

let

is s a i d to b e h o m o q e n e o u s

with

large

and

homoqeneous

n -- 0,i.

three mutually equivalent

(i)

n.

Homogeneous

Q

domain

is c a l l e d

f(H n(A))

= E m d i m A = 0.

homomorphimsms.

a homogeneous

homomorphism,

closed

I c H(A). is the o n l y h o m o g e n e o u s ideal

HI(A)A

For any nonmaximal

Q

in

A

c

Q

for

Hn(A)

c

Q

for a l l

68

ideal

in

A,

we have:

~ Hn(A)

homogeneous

maximal

prime

some

ideal

n

sufficiently P

in

A, w e

large regard

n.

2.4

A\P

as a h o m o g e n e o u s

n, w h e r e f

f: A - A / P

becomes

morphism maximal

d o m a i n by s e t t i n g is the c a n o n i c a l

homogeneous.

of

A

Conversely,

onto a homogeneous

homogeneous

prime

F o r any n o n m a x i m a l

R([A, C3) and,

in c a s e

ideal

homogeneous

= R(A/C)

and

= f(Hn(A))

epimorphism.

if

f

domain

Note

for all

that,

is a h o m o g e n e o u s A',

then

Ker

f

then

epiis a non-

A. prime

ideal

~([A,C])

C

in

A

we define

= ~(A/C)

D i m A / C = i, w e a l s o d e f i n e

~([A,C~)

F o r any h o m o g e n e o u s nonnegative

in

Hn(A\P)

integer)

= ~(A/C)

ideals

QI'Q2 .... 'Qs

in

A

(where

s

is a

we define

A ( A ' Q I ' Q 2 ..... Qs ) =

(QI N Q 2 N . . . N Q s

Q HI(A))A

and Emdim[A, Q i , Q 2 ..... Qs 3 = (Emdim A)

- [QI N Q 2 Q . . . A H I ( A )

: H0(A)~

and w e note t h a t then: ~(A)

= HI(A)A

and

Emdim[A~

= EmdimEA,~(A)~

= - 1 ;

&(A, Q I , Q 2 ..... Qs ) = A(A, Q I , Q 2 ..... Qs,~(A)) = a nonunit homogeneous E m d i m A m E m d i m [ A , Q i , Q 2 ..... Qs ~ = E m d i m [ A , Q i , Q 2 ..... Qs,A(A) 3 = E m d i m [ A , ~ ( A , QI,Q2 ..... Qs)~ = E m d i m [ A , A ( A , Q I , Q 2 ..... Q s , A ( A ) ) ~ ~-i; and

69

ideal

in

A ;

2.5

E m d i m [ A , Q i , Q 2 ..... Qs~ = - 1 A ( A , Q I , Q 2 ..... Qs ) = for

1 < i ~ s

(depending

We observe in

A

we have

on

that

or

A(A)

,

for any n o n m a x i m a l

homogeneous

prime

ideal

we have

We

= Emdim

For

set o f all n o n z e r o

.

any

• ~ H

(A)

homogeneous

Note

that,

n H(A),

then

principal

ideals

in

A.

we define

Deg A# = min{DegA~

~

A/P

define

(A) = the

~

Qi = A

i).

Emdim[A,P~

H

~ (A)

Deg A#

: 0 ~ M ¢ ~ D H(A)}

is a n o n n e g a t i v e

integer

and

for any

we have

Deg A~=

DegA~

~

~=

~A

.

We define Hn(A)

=

{• ~ H

(A)

: DegAS=

n]

W

and w e

note

§14.

that

H0(A)

Homogeneous

jections,

and

such

that

B D Hn(A)

B

that,

is a h o m o g e n e o u s ideal

in

subdomains,

subdomain

is a s u b r i n g

for all

We note

[A}.

flats,

projections,

birational

pro-

cones.

By a h o m o g e n e o u s B

=

of

of

A

we mean

A, H0(B)

a homogeneous

= H0(A),

and

domain

Hn(B)

=

n > 0. if

ideal

B in

is a h o m o g e n e o u s A,

then

B.

70

B n Q

subdomain is c l e a r l y

of

A

and

if

a homogeneous

Q

2.6

For a h o m o g e n e o u s a subfield

of

~(A),

subdomain

and

B

R(B)

of

A, w e m a y r e g a r d

to b e a s u b f i e l d

of

~ (B)

R(A).

to b e

We note

that clearly

Dim B ~ Dim A D i m B = D i m A ~ JR(A)

, : R(B)~

< ~ ,

and (14.1)

Emdim B ~ 1 = Dim A

and

H 0 (A)

algebraically

closed

= Dim B = 1 .

We observe C

that,

is a h o m o g e n e o u s

upon

letting

canonical

if

prime

prime

ideal

geneous

subdomain of

and so

~(f(B))

homogeneous

is a h o m o g e n e o u s

ideal

D = B Q C, and,

epimorphisms,

homogeneous

B

in

such t h a t

f: A - A / C

and

the f o l l o w i n g :

in

f(B)

B.

may be regarded g(B)

and

D

Hn(f(B))

to b e the

is a n o n m a x i m a l to b e a h o m o -

= f(Hn(B))

to b e a s u b f i e l d f(B)

and if

H 1 (B) ~ C, then,

may be regarded

by taking

A

g: B ~ B / D

we have

f(A)

domains,

A

s u b d o m a i n of

of

are n a t u r a l l y

for all

~([A,C3).

As

isomorphic.

There

W

exists

a unique

isomorphism

h

*

: 5(g(B))

~ 5(f(B))

such that

h (g(~))

w

= f(~)

for all

get a u n i q u e h

(~)

~ ¢ B ; we have

isomorphism

for all

h:

~ e R([B,D])

ical i s o m o r p h i s m s .

Via

h

(R([B,D3))

R([B,D3)

= R(f(B))

~ R(f(B))

; we designate

h, R ( [ A , C 3 )

becomes

h

such that and

a

and so we

h

h(~)

=

to b e c a n o n -

R([B,D3)-vector-

s p a c e and w e h a v e

Dim B/D I

~ Dim A/C

,

D i m B / D = D i m A / C ~ [R([A,C3)

: R([B,D~)3

< ~ ,

and (14.2)

Emdim[B,D3

~ 1 = Dim A/C

and

H0(A)

algebraically

= D i m B / D = I. F i n a l l y w e n o t e that, w i t h o u t

n,

assuming

71

HI(B)

~ C, w e h a v e

closed

2.7

1 + Emdim[B,D]

=

[HI(B)

: H0(A)]

- [C n HI(B)

= Emdim~A,C,HI(B)A

: H0(A)]

~ - Emdim[A, H I ( B ) A ~

and ( 14.3) Emdim[B,D~ in We

= the

any

set of all

= the

spaces

in

observe

that

that

L -- A L

nonzero

subdomains

of

Given

A,

any

we

a homogeneous

gives

(inclusion

members and

jL,A

the

to

of

and w e

jL,A note

A

the

= A L N Hn(A),

is then a of

~(A)

inverse

{0} ~ L c ~(A)

reference

HI(A)

(Emdim A - i ) - d i m e n s i o n a l

Hn(AL)

jL,A

call

of

vector

define

and

AL

s e t of all

We

prime

~(A) .

{0] ~ L e ~(A)

note

H0(A)-vector-subspaces

set of all

A L = H0(A)~L ~

and w e

homogeneous

define

~i(A)

and w e

is a n o n m a x i m a l

B.

~(A)

For

~ 0 = D

from

may simply

preserving)

onto

any

n ,

of

A.

We

bijection

of the

the s e t of all h o m o g e n e o u s is g i v e n

J c A

by

B -- HI(B).

we define

= AL N J

projection

is c l e a r

subdomain

bijection

and

for all

of

J

from

the c o n t e x t ,

call

L

in

A.

When

we may write

it the p r o j e c t i o n

of

jL

J

the instead

from

L.

that

A L = A L ' A = the p r o j e c t i o n

By a flat (N D H I ( A ) ) A . A.

in

A

we mean

We

note

that,

ideal

in

B y an

that

Emdim~A,N~

= i.

i-flat We

an i d e a l N in

is t h e n A

define

72

of

N

A

in

clearly

we mean

from

A

L

such

(in A)

that

a nonunit

a flat

N

in

N =

homogeneous A

such

2.8

(A) = the set of all

flats

in

A

and

We

observe

~(A)

that

onto

~

~i(A)

= the

set of all

L - LA

given

a (inclusion

(A),

N - N N HI(A).

and

the

We note

for any h o m o g e n e o u s

nonnegative

integer)

=

A.

preserving)bijection

bijection

{A(A)]

ideals

is g i v e n

of

by

,

QI'Q2 ..... Qs

in

A

(where

s

is a

w e have,

A(A, Q I , Q 2 ..... Qs ) e ~ e ( A ) ,

where

e = E m d i m [ A , Q i , Q 2 ..... Qs ] ,

~(A, Q I , Q 2 ..... Qs ) c Qi

' for

1 ~ j ~ s ,

and:

N c Qj

, for

N e ~

in

that

~_I(A) and

inverse

i-flats

(A)

with

i ~ j ~ s

N c & ( A , Q I , Q 2 ..... Qs ) In case

s > 0, w e m a y

flat w h e r e

We

also

£(A, Q I , Q 2 ..... Qs )

e = E m d i m [ A , Q i , Q 2 .... ,Qs])

QI,Q2,...,Qs reference

call

to

note

; from A

these

is c l e a r

phrases from

i__n A

" in A

the

flat

spanned

(or the e-

by

" m a y be d r o p p e d

when

the

the context.

that H I(A)

= ~r-l(A)

F o r any

where

[0] fi N ~ ~

r = Emdim (A), w e

A.

define

A N = A N N H I (A)

Again tion

we

observe

that

of the s e t of all

homogeneous

subdomains

N - AN

gives

nonzero

members

of

A,

and

a

(inclusion

preserving)

of

onto

~

the

inverse

{0}

~ N e ~

(A)

bijection

the

set of all

is g i v e n

B - HI(B)A. In the r e s t

of §14,

let any

73

(A)

biject-

be qiven.

by

2.9

For

any

J c A

we d e f i n e jN,A = A N N J

and w e

call

jN,A

words,

projection

corresponding to

A

from

member

is c l e a r

and w e

the ~ r o j e c t i o n

simply

from

call

of

N

is the

N

n HI(A)

the

context,

J

from

same

thing

of

~(A).

N

of

J

A N = A N'A = the p r o j e c t i o n

A.

In o t h e r

as p r o j e c t i o n Again,

we m a y w r i t e

it the p r o j e c t i o n

in

jN

from

of

when

the

instead N.

A

from the

We

from

reference

of

note

N

jN,A that,

(in A)

,

and Emdim A N = Emdim We o b s e r v e

inverse

particular c N] ~

j ~ jN

* [J ¢ ~i(A)

tion of and the

that

: J c N]

bijection

~ , N

onto

gives

HI(AN),

and

A - Emdim[A,N]

gives

a

(inclusion

onto

~ i*- e - i (A N )

preserving) where

is g i v e n b y

K ~ KA.

a

preserving)

(inclusion

the

inverse

- 1 .

We

bijection

bijec-

e = Emdim[A,N~,

note

that

then,

[~ c HI(A)

is a g a i n

in

:

given by

@A.

We

define H

(A,N)

=

{~ ¢ H

(A)

: ~ =

and w

*

Hn(A,N) and we n o t e Hn(A,N) We

also

that

onto note

H

= H

w

(A,N)

~ , ~N (AN),

and

*

to m e a n ideal

in

gives

(inclusion

inverse

W

(A,A (A)) = H

shall

say t h a t

that

~ ( A N) = ~(A).

C

the

a

preserving)

bijection

bijection

is g i v e n

by

of

~ ~ ~A.

that

H We

n Hn(A)

in

A, we

W

(A)

and

the p r o j e c t i o n

shall

Given

from

N

= H n(A) in

any n o n m a x i m a l

say t h a t

74

*

H n ( A , A (A))

A

is b i r a t i o n a l

homogeneous

the p r o j e c t i o n

of

C

prime from

N

in

2.10

A

is b i r a t i o n a l

to m e a n t h a t

N ~ C

and

Again,

f r o m t h e s e two p h r a s e s w e m a y d r o p

to

is c l e a r

A

LR([A,C~):R([AN,cN)~ " in A " w h e n

= i.

the r e f e r e n c e

f r o m the c o n t e x t .

§15.

Zeroset

and h o m o g e n e o u s

localization.

We define ~(A)

= the s e t of all n o n m a x i m a l

homogeneous

prime

ideals

in

A.

W e also d e f i n e

~I(A)

= {P ¢ n(A)

: E m d i m A / P = D i m A/P]

hi(A)

= {P ~ n(A)

: D i m A / P = i}

I(A)

and w e n o t e

= ~I(A)

N n i (A)

that

~ i (A) c ~ i + l (A) We observe A ~ A/P, With

every

1

hi(A)

and

t h a t for any

*

c ~i(A)

P c ~(A),

[R([A, P3)

i.

v i a the c a n o n i c a l

H 0 ( A / P ) - v e c t o r - s p a c e becomes a

this u n d e r s t a n d i n g

for all

epimorphism

H0(A)-vector space.

we have

: H0(A)~

= [R([A,P])

: H 0 (A/P)

= {P ~ •(A)

: [R([A,P~)

: H 0(A)]

and 9 0(A)

< ~}

We define neg~A, P3 = [R([A, P3)

: H 0(A) ] for all

and w e n o t e t h a t t h e n

Deg[A, P3 = D e g ( A / P )

75

for all

P c Do(A)

P e ~0(A)

2.11

We also note that 1 n0(A ) = [p ¢ n0(A)

: Deg[A,P]

= i]

and in particular: H0(A)

algebraically

closed = n0(A) = ~ ( A )

We define n(A,x)

= [P ¢ n(A)

~l(A,x)

: x ¢ P]

= ~(A,x)

n ZI(A)

~i (A,x) = •(A,x)

Q ~i(A)

for any

x ¢ H(A)

~}(A,x) : ~(A,x) n ~(A) 1 n(A,I)

: [P c ~(A)

: I n H(A) c P]

n I(A,I) = [%(A,I) e ~I(A) for any

I cA

n i (A, I) = n (A, I) N n i (A) nI(A,I)

= n(A,I)

n nl(A~l

n(A,\I)

= ~(A)\n(A,I)

nl(A,\I)

= n(A,\I)

N nl(n)

ni(A,\I)

= ~3(A,\I)

N ~i(A)

~(A,\I)

= ~(a,\i) n e~(a)

n([A,I])

= •(A,I)

nl([A,I])

= •I(A,I)

ni([A,I])

= ni(A,I)

1 ni([A'I])

I(A,I ) = [%i

I

76

for any

I c H(A)

or

I cA

for any

I ¢ H(A)

or

I cA

2.12

n([A,I~,J)

= n(A,I)

N ~(A.,J)

nl([A,I~,J)

= n([A,I~,J)

n nl(A)

for a n y

I ¢ H(A)

or

ni([A,I~,J)

= n([A,IT,J)

N hi(A)

and a n y

J ~ H(A)

or J c A

nl([A,I],J) i

= ~([A, IT,J)

n nl(A) i

n([A,I~,\J)

= ~3(A,I)\~3(A,J)

~I(A,I],\J)

= ~I([A,I~,\J)

I c A

and

Q ~I(A)

~i([A,I~,\J)

= ni([A,I~,\J)

n ~i(A)

~ i1( [ A , I ] , \ J )

= n Ii ( [ A , J ~ , \ J )

N ~I(A)

We note that then ~(A) •(A,xA)

= •([A,0],x)

= ~(A,0)

= ~([A,x],O)

= n([A,0],I)

n(A,I)

= ~ ~ HI(A)

~(A,I)

is a finite

set ~ n(A,I)

~ n0(A,I) (radAI)

=

J

= e ~ ~0(A,I)

c n(A,J)

~ n0(A,~)

and

n(A,I)

(radAI) n(A,I)

I

= n(A)

~ n0(A,I)

and a n y

J ~ H(A)

or J c A

= Z(A,x) ,

= ~([A,I])

= n(A,I)

I c A ,

n(A,i)

= n(A,J)

or

,

= n([A,I],0)

ideals

I e H(A)

l

x e H(A)

for a n y and for a n y h o m o g e n e o u s

for a n y

= Z([A,x])

for a n y n(A, (I N H ( A ) ) A )

I

D0(A)

in ~

A Z =

w e have: [0]

c radAI

,

,

= ~0(A,I)

= n0(A,J) P

(HI(A)A)=

(radAJ)

Q

(HI(A)A)

•

(radAJ)

Q

(HI(A)A)

,

c n0(A,J ) n

(HI(A)A)

77

~

I c A

2.13

n(A,I)

U n(A,J)

= ~(A,I

N J) = n(A, IJ)

n(A,I)

n n(A,J)

= n(A,I

+ J)

,

and

We observe sing)

injection

~(A). A/C

that,

in p a r t i c u l a r of

~(A)

Finally we note

bijection

G i v e n any

t h a t for any

~([A,C~)

P e ~(A)

= {x/z

~(A,I,P)

=

; ~(A,P)

~(A,P)/M(~(A,P))

for e v e r y

n

=

I c A

subsets

letting

gives

a

of

f: A

(inclusion

~(A/C).

for any

x ~ Hn(A)

for any

I c A ,

,

(A)

is a local d o m a i n w i t h q u o t i e n t

= dim A - Dim A/P

isomorphic with

R([A,P~)

we have

~ ~ ~(A,x,P)

(x/z)~(A,P)

we have

for e v e r y

~(A,I,P)

= ~(A, (I N H(A))A,

M(~(A,P))

field

H 0(A) ; d i m ~ ( A , P )

= ~(A,P,P)

and

78

;

P)

;

;

;

c ~(A,P)

z ¢ Hn(A)\P

c ~(A,P)

{ ~(A,I,P)~(A,P)

(15.3)

~(A,P)

x c H

~(A,x,P)~(A,P)

(inclusion rever-

= ~(A,A, P) ,

is n a t u r a l l y

I1511 (15.2)

onto

: z c Hn(A)\P ~

is a s p o t o v e r

for e v e r y

upon

p - f(P)

U ~(A,x,P) XcINH(A)

a n d w e note t h a t then:

an

we define

(A,P)

(A)

C e ~(A),

epimorphism,

of

~(A,x,P)

gives

into the set of all n o n e m p t y

to be the c a n o n i c a l

preserving)

P - ~(A,P)

and

and

;

2.14

given

any

I' =

z ¢ HI(A)\P

U {xz - n 0~n<~

we have

, upon

: x c I n H

letting (A)}

for

I cA

every

,

n

that

(15.4) A'

= H0(A) [HI(A)z-I],

P n Hn(A)

We

also

I'A'

= P'

observe

that

given

I

any

upon

(15.5)

=

P =

rn =

of

0

that (15.4)

note

onto

= the and

that

observe

becomes

a

any

,

and

(I n H ( A ) ) A

R'

that

center

sufficiently

(15.5)

a

= P

.

¢ ~(H0(A) [HI(A)z-I])

we

(inclusion

the

9%(A,P)

n]

domain

A

of

on

i.e.,

~B(A),

,

H 0(A)-vector-space

, and

by

of

with such

dim A

= 1

that

definition

n ~(A,P). We

projective canonical

n

,

reversing)bijection

P ¢ [3(A)

= M(V)

all

that:

a unique

V

for

= R'

exists

P ¢ f~0 (A).

= the

¢ M(R')

large

any homogeneous

M(9% (A,P))

via

rnz-n

see

there

clearly

and

and

given

¢ ~(A),

c V

with

and

~(A) .

V

I c A

and

= A'p,

e M(9%(A,P))},

z ~ P ¢ f~(A)

f~(A)

(A,V) We

all

gives

~(A,P)

We

for

P ~ ~(A,P)

any

~(A,P)

every

0 ~ z c HI(A)

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

for

: xz -n

~ r : r ¢ Hn(A) 0
We have

(15.6)

{x e H n ( A )

~(A'),9%(A,P)

letting {

In v i e w

P'e

define

center

of

epimorphism

as w e l l

we have

79

as

V

in

A = P.

V ~ V/M(V),

V/M(V)

R([A,P])-vector-space

and

2.15

[V/M(V)

: H0(A)~

< ~

[V/M(V)

: H0(A)]

=

and

[V/M(V)

:R([A,P])]

<

and

Finally

H 0(A)

we

note

[V/M(V)

closed

= [V/M(V) =

any h o m o g e n e o u s

Q c H(A)

or

.

that:

algebraically

For

: R([A,P])TDeg[A,P]

1:

domain

Q c A

we d e f i n e

~(A,Q)

=

: H 0(A) ]

[v/M(v)

: ~([A,P])3

.

with

Dim A = 1

and

A

any

w

IV e ~(A)

: ~

(A,V)

¢ •(A,Q)]

and ~(A,\Q)

we

note

that

for a n y o t h e r

~(A,Q')

We o b s e r v e

that,

(A,Q)

= ~(A)\~(A,Q)

Q'

= ~(A,Q)

;

or

c H(A)

~ n(A,Q')

Q'

c A

we

then have

~ n(A,Q)

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

= ~(A)

~ ~(A,Q)

~(A,Q) n(A,Q) [0]

is an i n f i n i t e

set

is an

set

infinite

= ~(A)

c n(A,Q)

~
Q N H(A)

c

[0]

in case

Q e H(A)

in case

Q c A

,

and 0 ~ Q ~ H 0(A) ~(A,Q)

-- ~ ~ n(A,Q)

in case

Q c H(A)

in case

Q c A

= ~

I

HI(A)

c radA(Q

80

n H(A))A

;

2.16

we

also

observe

that

for

any

V

c ~(A)

we have

w

(A,V)

Given

= the

any

f: A - A / C

unique

C c ~(A)

to b e

the

P ¢ ~0(A)

and

canonical

epimorphism,

=

[f(x)/f(z)

~([A,C],I,P)

=

0 ~([A,C],x,P) xc I n H (A)

and we

note

that

field

R([A,C])

R([A,C],P)

then

V ~ ~(A,P)

upon

letting

define

for

any

for any

x

I c A

e Hn(A) ,

= ~([A,C],P,A)

; R([A,C],P)

for every

we

: z e Hn(A)\P]

~([A,C],P)

= Dim A/C

that

P ¢ ~]([A,C]),

~([A,C],x,P)

~([A,C],P)

dim

any

such

is a l o c a l

is a s p o t

- Dim

A/P.

x ¢ Hn(A)

domain

over

Also

we have

with

H0(A/C)(~

quotient

H0(A))

;

we have:

~ @ ~([A,C],x,P)

c ~([A,C],P)

and (15.7) (f(x)/f(z))~([A,C],P)

~ ([A, C ] , x , P) ~ ([A, C], P) = for every

z ¢ Hn(A)\P

(

for every

(15.8)

I c A

;

we have

~{([A,C],I,P)~([A,C],P)

~([A,C],I,P)

M(~([A,C~,P)

and

= ~{([A,C], (I N H ( A ) ) A , P ) : ~([A,C],

(15.9)

c ~([A,C],P)

(I n H ( A ) ) A

+ C,P)

;

= ~([A,C],P,P)

Further, w

f

induces

a unique

epimorphism

f : ~(A,P)

- ~([A,C~,P)

such

that

w

f

(X/Z)

=

f(x)/f(z),

whenever

x ¢ Hn(A)

and

z ¢ Hn(A)\P

; we

w

have f

Ker

f

(~(A,I,P))

= ~(A,c,P)

; for any

= ~([A,C],I,P)

• ([A, C], P ) / M (~ ([A, C~, P) )

I ¢ H(A)

; and and

f

induces

R([A,P]) o

81

or an

I c A

we have

isomorphism

between

2.17

We

again

observe

P ~ ~([A, C3,P) onto

gives

~([A,C~)

a

a unique

note

that

any

C ¢ ~(A),

(inclusion

given

any

in view

reversing)

C ~ Ill(A)

P e ~([A,C~)

~([A,C],P)

We

for

of

bijection

(15.6), of

[I([A,C~)

.

In particular, exists

that

clearly

such

= the

and

V

c ~([A,C~),

there

that

center

of

P e ~0([A,C~).

We

V

on

~([A,C~).

define

w

([A,C~,V)

From

the

drop

" in A

We

note

phrase "

that,

"the when

f: A -

projective

projective the

center

center

reference

to

of A

of

V

V

on

o__nn C

C

is c l e a r

in

from

i__nn A = P.

A the

"

we

= ~ * (A/C,V)

A/C

is

the

canonical

an

H0(A)-vector-space

context.

the

~ * (~A,C~,V)

and

canonical

epimorphisms

epimorphism.

A ~ A/C

as w e l l

[V/M(V)

: H0(A) ~ < ~

[V/M(V)

: H0(A) 3 =

= f - i (~ * ( A / C , V ) )

as

and

We

observe

V - V/M(V),

V/M(V)

R([A,P~)-vector-space

and

[V/M(V)

that

we

note

H0(A)

For

any

EV/M(V)

: R ( E A , P3) ~ <

: R ( [ A , P3) ~ D e g E A , P~

C e ~I(A)

and

closed

any

=

[V/M(V)

: H0(A)~

=

[v/M(v)

: R([A,p~)3

Q e H(A)

or

Q c A

= 1

we

w

~([A,C~,Q)

= {V e ~([A,C~)

: ~

([A,C~,V)

e n(~A,C~,Q)}

and ~([A,C~,\Q)

= ~(EA, C~)\~([A,C~,Q)

82

via

and we have

that: algebraically

,

becomes

and

Finally

may

then

f(~ * (~A,C~,V))

where

= the

define

2.18

we

note

that,

then,

~([A,C],Q')

We observe

that,

for any o t h e r

= ~([A,C],Q)

Q'

~ H(A)

or

~ ~([A,C~,Q')

Q' c A, w e h a v e

= n([A,C],Q)

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

we have

= ~([A,C])

~ ~([A,C],Q)

is an i n f i n i t e

set

~([A,C],Q)

is an i n f i n i t e

set

~([A,C],Q)

n([A,C],Q)

= ~([A,C])

C e n (A,Q) ! I~

c c Q N H(A)

c C

in c a s e

Q ~ H(A)

in case

Q c A,

and ~([A,C],Q)

= ~ ~ ~(EA,C],Q)

= in case

/0 ~ Q e H0(A) HI(A ) c radA((Q

We

also

observe

~*([A,C],V)

We end

this

section

LEMMA

domain,

i__nn ~(R,P)

for any

= the u n i q u e

(15.10) geneous

that,

be

and

V e ~([A,C]),

P e ~o([A,C])

in case

+ C)

Q c A.

we have

such

that

V c ~([A,C],P).

with:

ON IMPOSED for e a c h

given.

n H(A))A

Q ~ H(A)

For

LINEAR

CONDITIONS.

P ¢ D0(R)

every

let any

nonnegative

Let

R

ideals

integer

m,

be

a homo-

I(P)

c I' (P)

let

E m = [x ~ Hm(R)

: ~(R,x,P)

c I(P)

for all

P ~ ~0(R)]

E'm = {x c Hm(R)

: ~ ( R , x , P ) c I'(P)

for all

P ~ no(R) } .

and

Then we have

that

E m c E'm

are

Ho(R)-vector-subspaces

83

of

Hm(R)

and

2.19

[E m : H0(R)]

[~(R,P)/I(P)

+

: H0(R)]

Pe~0 (R)

[~(R,P)/I'(P)

[E m : Ho(R)] +

: H0(R)]

P¢~0 (R) [H m(R)

: H 0(R)]

(We o b s e r v e

.

that,

if

[~(R,P)/I(P)

: ~(R,P)]

< ~ , then

it

P ~ n 0 (R) can e a s i l y be are a c t u a l l y

seen that, equalities;

this o b s e r v a t i o n PROOF.

that,

and the

we shall o n l y p r o v e Now, g(P)

for each

: Hm(R)

the

not p r o v e

to use

it.)

p ¢ D0(R),

from the

first.

then, Hence

fix

y(P)

¢ Hm(R)\P

and d e f i n e

Clearly

g(P)

is an

and we h a v e

[~(R,P)/I(p)

of

c g(p)-l(I' (P))

H m (R)

and

: H0(R) ] = [Hm(R)/g(p)-l(!(p)):

H0(R)].

similarly,

(3)

[~(R,P)/I' (P) First

note that,

: H0(R) ] = [Hm(R)/g(p)-I(I'

then there

the i n e q u a l i t y are o n l y

(p))

: H0(R)]

if

E [~(R,P)/I(P) P¢~]0 (R)

(4)

inequalities

no o c c a s i o n

for all

follows

g(P) (x) = X/y(p)

H 0 (R)-vector-subspaces

(2)

we shall

two

inequality.

g(p)-l(I~P))

are

the above

as we shall h a v e

inequality

first

by

m

I' (P) = ~(R,P)

P ¢ ~]0(R)'

(i)

and

if

second

~ ~(R,P)

H 0(R)-isOmOrphism

however,

in the p r e s e n t book,

Note

E'm = Hm(R)

for all large

is t r i v i a l l y

finitely

many

= H0(R) ] = - ,

true.

P's,

say

H e n c e we m a y assume PI,...,Pr

P ¢ ~ 0 ( R ) \ { P 1 ..... Pr] = I(P) = ~(R,P)

84

that

, such that or e q u i v a l e n t l y

2.20

g(p)-I(I(P)) Now

put

D i = g(pi)-l(I(Pi

i = 1,2 ..... r.

(5)

We

Qi

The

and

given.

using

Let

For

[

and h e n c e

i =

r Q Qi i=l

(2),

deduced

(3),

r E' = n Q~ . m i=l I

and

(5)

in t h e and

the

following result

of

easy

exercise

(7) b y

taking

Q

be

a finite-dimensional

1,2 ..... r

vector

; let k - v e c t o r - s p a c e s

space I

~i

c Qi

over

a

c ~

be

(Hint.

r

Both

known

r

: k] + i=l~[Q~ : k] ~ [i=l~iN' : k] + i=l~[~i : k]

: k]

results

fact

+

r ~ [Q/Qi i=l

are

easily

: k]

r ~ [ N n' : k] i=l ±

deduced

by

r ~ [p/f~' : k] i=l ±

+

induction

on

r

and

.

the

that,

[A 1 n A 2 for

any

§16. By

r

(or s i m i l a r l y ) r [ n Pi i=l

(7)

of

is n o w

for

g ( P i )-I (I' (Pi) )

(i),

r fl Q i i=l

=

m

from

~i

Then

(6)

well

'

and

(R).

m

k = H0(R).

EXERCISE. k.

))

have

E

result

spaces,

= Hm(R)

field

c Q~ l

required

in v e c t o r

clearly

= H

a homo@eneous

: k] two

+

[i I +

subspaces

Homogeneous coordinate

A2

: k] =

iI , A2

[A 1

: k]

of

~. )

coordinate

systems.

system

A

in

we

+

mean

a

[A2:

k]

H 0(A)-basis

H 1 (A). Let

where

(X0,X I, .... X r)

r = Emdim

A.

Let

be

a homogeneous

H 0(A) (r+l)

a =

be

(a0,a I ..... ar) with a 0 , a I ..... a r (r+l) \ * a c Ho(A) (0,0 ..... 0), let a be

85

coordinate the

set

of

in

H0(A).

the

ideal

in

system

in

A,

all

(r+l)-tuples

For

any

A

generated

by

2..21

the

set

gives any

{aqXp

a surjective a

and

exists any

- apXq

a

b

m a p of

in

(r+l) \

¢ H0(A)

by

the

Similarly, members

of

~

(r+l) \

such

(0,0 ..... 0), [aqXp

obvious

matrix

ap

~bp

we h a v e

- apXq

that

then:

onto *

~ 0 ( A ) ; for

we have

a

for

0,i,.

that,

p if

a - a

= b

*

~ there ,r

; for

aq ~ 0, t h e n

a

*

: p = 0,i, .... r].

representations

can b e g i v e n

for o t h e r

(A).

Polynomial

A polynomial over

ring

A

rings

as h o m o g e n e o u s

= k [ Y 0 , Y 1 ..... Yr]

a field

k

obviously

domains.

in i n d e t e r m i n a t e s

becomes

a homogeneous

domain

taking

Hn(A

) =

[0] U

(the set of

all n o n z e r o

of d e g r e e

and

(0,0 ..... 0)

that

set

§17.

Y 0 ' Y I .... 'Yr

We note

H0(A) ( r + l ) \ ( 0 , 0 ..... 0)

H0(A)

0 ~ ~ e H0(A)

is g e n e r a t e d

by

: p , q = 0,i ..... r}.

homogeneous

polynomials

n)

t h e n we h a v e

Emdim

Moreover,

we

clearly

A

have

-- D i m A

(17.1)

to

= r.

(17.5):

Emdim A = Dim A = r (17.1)

A

is i s o m o r p h i c ,

ring

in

r + 1

as a h o m o g e n e o u s indeterminates

86

over

domain,

to a p o l y n o m i a l

a field.

2.22

Emdim A = Dim A some h o m o g e n e o u s algebraically

coordinate

independent

every h o m o g e n e o u s (17.2)

algebraically

~0(A)

system in

consists of

elements over

coordinate

independent

A

system in

H 0 (A) A

elements over

consists of H 0 (A)

c ~(A)

~i* (A) = ~i1 (A)

for all

Im

i

+ Emdim A 1

[Hm(A) :H0(A) ] =

m .

for all Emdim A

Emdim A = Dim A = r = nr_l(A)

= H

and for any

(A} A n(A)

• ¢ Hn(A)

N ~(A)

we have

[r)r

(17.3)

for all

m < n

[~m (A/e) :H 0 (A)/¢~ =

for all

(17.4)

Emdim A = Dim A = the local ring

~ (A,P)

is regular

m a n .

for all

P ~ e(A). Emdim A = the smallest nonnegative

(17.5)

there exists a h o m o g e n e o u s Dim A' = r

domain

integer A'

with

and a h o m o g e n e o u s h o m o m o r p h i s m

87

r

such that Emdim A' = A' ~ A.

2.23

§18.

Order

Let Given

V

We

C e ¢

on

an

%(A)

embedded

and

~([A,C]),

let

let

curve

f: A

P =

~

and

integral

- A/C

be

the

projections.

canonical

epimorphism.

([A,C],V).

define

ord([A,C],x,V)

= ordv~([A,C~,x,P.)

for

all

x

e H(A)

ord([A,C],I,V)

= ordv~([A,C],I,P)

for

all

I c A.

and

We have

observe

that,

(18.1),

(18.2)

(18.1)

For

in v i e w

and

any

of

(15.7),

(15.8)

and

(15.9),

we

(18.3):

x ~ H(A)

ord([A,C~,x,V)

we

have

= ord([A,C],xA,V) =

a nonnegative

integer

or

~

,

and

(18.2)

ord([A,C~,x,V)

= ~

~ x

~ C,

ord([A,C~,x,V)

> 0 = x

e P.

For

ord([A,C~,x,V)

(18.3)

For

ord([A,C~,I,V)

any

x

e Hn(A)

we

have

= ordvf(x)/f(z)

any

I c A

we

for

all

z

have

= ord([A,C~,

(I D H ( A ) ) A , V )

--- o r d ( [ A , C ~ ,

(I n H ( A ) ) A

= min[ord([A,C~,x,V): =

£ Hn(A) \ P

a nonnegative

+ C,V) x

integer

~ I n H(A)} or

and ord([A,C~,I,V)

= ~ ~

I n H(A)

c C

,

ord([A,C~,I,V)

> 0 ~

I Q H(A)

c P

.

88

~

,

.

then

2.24

In v i e w o f

(18.1)

and

(18.3) we get

(18.4)

ord([A,C],J,V)

(18.5)

For any

@([A,C],Q)

= IV'

In v i e w o f (18.6),

(18.7),

m ord([A,C],I,V)

Q e H(A)

(18.8)

(18.2) and

ord([A,C],a,V)

(18.7)

For any

ord ([A,C],x+y,V)

where

H n (A)

If

of

0]

.

or~

w e a l s o get

for all

and

0 # a e H0(A).

0 ~ a e H0(A)

we have

= ord([A,C],x,V)

and

y

in

Hn(A)

we h a v e

m min(ord([A,C],x,V),ord([A,C],y,V))

equality holds

(18.9)

J c I c A.

we have

and p r o p e r t i e s

x ~ H(A)

x

(18.5):

for all

: ord([A,C],Q,V')>

= 0

ord ([A,C],ax,V)

For a n y

Q c A

and

(18.9):

(18.6)

(18.8)

or

~ ,~([A,C])

(18.1),

(18.4)

in case

[V/M(V)

ord([A,C],x,V)

: H0(A) ] =I,

,

/ ord([A,C],y,V)

t h e n g i v e n any

x

and

,

y

in

s u c h that ord ([A,C],x,V)

there exists

a unique

a ¢ H 0(A)

ord([A,C],x-

moreover:

m ord ([A,C],y,V)

ay,V)

a = 0 ~ ord([A,C],x,V)

# ~ ,

such that > ord ([A,C],y,V)

>ord

;

([A, C],y,V)

W e s h a l l n o w prove:

(18. I0) L E M M A . ord([A,C],xy,V)

For any

x e Hm(A)\C

= ord([A,C],x,V)

89

and

y ~ Hn(A)\C

+ ord([A,C],y,V)

we have

2.25

and ord ([A,C],xn,v)

PROOF.

- ord([A,C],ym,v)

We can take

= Ordvf(xn)/f(ym)

z ~ H 1 (A)\P

and t h e n we h a v e

ord ([A, C],xy,V) = o r d v f (xy)/f (zre+n) = [ordvf(x)/f(zm ) + -- o r d ( [ A , C ] , x , V )

[or~f(y)/f(z

by

(18.2)

by

(18.2),

by

(18.2)

n)

+ ord([A,C],y,V)

and we also h a v e ([A,C],xn,v)

- ord([A,C],ym,v)

= [Ordvf(xn)/f(zmn)]

- [Ordvf(ym)/f(zmn)]

= Ordvf(xn)/f(ym ) N o w w e shall prove: (18.11) and

D = C N.

PROJECTION Assume

(Note that,

Let

h:

that

if

D ~ %(B)

LEMMA.

Given

[0] ~ N e ~(A),

let

B = AN

D e ~3I(B).

H 0(A)

is a l g e b r a i c a l l y

closed,

then

~ IN : H0(A) ] - [C A N : H0(A) ] ~ 2.)

R([B,D])

~ R(f(B))

b e the c a n o n i c a l

isomorphism,

and let

W = h - l ( v D R(f(B))). (Note that now: h(W)

= v ~ R(f(B)), Then

for all

ord([A,e],x,V) and

for all

ord([A,C],N,V) and

x ¢ N\C

= N, W ~ ~([B,D]),

is a p o s i t i v e

integer.)

we h a v e

- ord ([A,C],N,V)

J c 7](A)

ord([A,e],J,V)

ordvM(h(W))

~ ~, HI(B)

= [ordvM(h(W)) ]lord ([B,D],x,W) ] ,

wit____hh J c N

- ord([A,C],N,V)

and

J ~ C

we h a v e

-- [ordvM(h(W)) ] [ o r d ( [ B , D ] , J , W )

90

2.26

PROOF. Q = ~

Let

g: B ~ B / D

([B,D],W).

b e the c a n o n i c a l

epimorphism.

Let

We can fix

z ¢ N\Q

and then,

in v i e w of

ordWg(x)/g(z)

Upon multiplying

and

(18.2),

= ord([B,D~,x,W)

the a b o v e

I for all

(i)

(18.1)

ordvf(x)/f(z)

=

ordvM(h(W))

(18.10)

(2)

in v i e w of

(3)

ord([A,C~,x,V)

z ¢ N, b y

(5)

for all

(18.1)

(18.3)

and

(2) a n d

and

ord([A,C],x,V)

[

=

(18.3)

and

we have

J ~ C

and

ord ([A,C],J,V)

that

for all

x c N.

(3) w e get

= o r d ([A,C~,z,V) that

- ord([A,C],N,V)

[ordvM(h(W))][ord([B,D],x,W)

In v i e w of

,

(i), w e c o n c l u d e

(4) it f o l l o w s

/

x ~ N\C

~ ord([A,C~,z,V)

ord ([A,C~,N,V) (i),

integer.

- ord([A,C~,z,V)

and hence,

Now by

w e see t h a t

[ordvM(h(W)) ] [ o r d ( [ B , D ] , x , W )

= ordvf(X)/f(z)

(4)

integer.

we h a v e

ord([A,CT,x,V)

Since

we h a v e

we have:

= a nonnegative By

x ¢ N\D

= a nonnegative

equation by

x ¢ N\D

for all

(5) w e see t h a t

]

for all

for all

x ¢ N\C

J c ~(A)

- ord([A,C],N,V) =

[ordvM(h(W)) ][ord([B,D],J,W) ] .

91

.

with

JcN

2.27

In v i e w of

(18.12) and

D = C N.

(18.3) b y

PROJECTION Assume

(Note that,

h:

LEMMA.

that

if

D e ~(B)

Let

(18.11) w e get:

Given

is a l g e b r a i c a l l y

~ Emdim[A,e~N]

-

*

(A), let

B

=

AN

D e ~I(B)-

H0(A)

R([B,D])

{0] ~ N e ~

R(f(B))

closed,

- Emdim[A,N]

then:

~ 2.)

b e the c a n o n i c a l

isomorphism,

and let

W = h - l ( v N R(f(B))).

(Note t h a t now: V n R(f(B)), Then

and

and

for all

ord([A,C],N,V)

ordvM(h(W) ) x e

is a p o s i t i v e

(N D H I ( A ) ) \ C ,

ord([A,C],x,V)

- ord([A,C],N,V)

for all

(A)

J ¢ ~

~ ~, W e ~ ( [ B , D ] , h(W))

with

=

integer.)

w_e have,

-- [ o r d v M ( h ( W ) ) ] [ o r d ( [ B , D ] , x , W ) ) ] ,

J c N

and

J ~ C, u p o n

lettinq

K = jR, we h a v e

ord

([A,C],J,V)

-

ord([A,C],N,V)

= [OrdvM(h(W)) ][ord([B,D],K,W) ] . (18.13) be given with t h a t the

COROLLARY-DEFINITION. ~ c N.

following

D ¢ ~(B)

I

Let

B = AN

two c o n d i t i o n s

and,

upon

to b e the c a n o n i c a l

(*)

h - l ( v ' n 9(f(B)))

D ¢ ~I(B)

Let and

~ e HI(A) D = C N.

and

By

N e ~

(A)

(18.12) we

see

are e q u i v a l e n t .

letting

h:

9([B,D])

isomorphism,

w e have:

e ~([B,D],17 N)

for all

- R(f(B))

V'

~ ,~([A,C],~).

and:

(**) ord([A,C],n,V')

> ord([A,C],N,V')

92

for all

V'

e ~([A,C],~)

2.28

We

shall

integral

to m e a n

satisfied. to

A

tion

say that

that one

From

is c l e a r

the p r o j e c t i o n

this

(and h e n c e

C

both)

phrase we may drop

f r o m the c o n t e x t .

(**), w h e r e a s

of

in C h a p t e r

N

o f the

" in A

In t h i s

IV w e

from

is n -

two conditions

" when

Chapter

shall use

i__nn A

we

the

is

reference

shall use

condition

(*).

condi-

We

note

that obviously the p r o j e c t i o n (18.13.1)

D ¢ nl(B)

and n0([A,C3,N)

of C from N

= ~ = is m - i n t e g r a l .

§19.

(19.1)

Order

to

R

P = ~

be

celarly

(19.i) by

assertion

and

integral

get

by

projections.

{0];

in addition B

identity

map

(18.i)

and

f

and

by

as

to

by

G i v e n V c ~(R),

R = i.

~B,D~

(19.10)

where, A

or

for

map

R - R.

(18.11)

and

(18.12)

replacements by

S;

D

we by

I c A.

1 ~ i ~ 10,

as w e l l

identity

from

above

I ¢ H(A)

replacing:

the

(19.12)

to the

for any

(19.1)

from

as w e l l R(S)

Dim

We define

assertions

(19.11)

replace:

domain with

= ord([R,{0}~,I,V)

is o b t a i n e d

R; C

where

a homogeneous

(R,V).

ord(R,I,V)

We

curve

(19.12).

Let let

on an abstract

as We

[A, C3 also get

respectively,

let

S = RN

and

{0};

and

by

h

the

- R(S).

w

(19.13) be

given with

COROLLARY-DEFINITION. ~ c N.

ing two conditions

(*) (**)

Dim

then

Let

Let By

S = R N.

*

~ c HI(R) (19.12)

we

and

N e ~

see t h a t

the

(R) follow-

are e q u i v a l e n t .

S = 1

and:

V'

N R(S)

Dim S = 1

and:

ord(R,~,V')

93

e ~ ( S , ~ N)

for all V'

> ord(R,N,V)

c O(R,~).

for a l l V'

¢ ~(R,n).

2.29

W e s h a l l say t h a t the p r o j e c t i o n to m e a n that o n e satisfied. to

R

tion

(and h e n c e both)

from

from the context.

(**), w h e r e a s

in C h a p t e r

in

R

of the c o n d i t i o n s

F r o m this p h r a s e we m a y o m i t

is c l e a r

N

i__ss~ - i n t e q r a l (*) and

(**)

is

,r in R '~ w h e n the r e f e r e n c e

In this C h a p t e r w e s h a l l use c o n d i -

IV w e s h a l l u s e c o n d i t i o n

(*).

We note that obviously

I

the p r o j e c t i o n

(19.13.1)

D i m S = 1 and

~0(R,N)

= ~

from

N

in R

is n - i n t e g r a l

and the p r o j e c t i o n (19.13.2)

from

N

in

R

is ~ - i n t e g r a l

I the p r o j e c t i o n

(19.14) canonical

REMARK.

of

G i v e n any

{0] from N in R

C ~ ~I(A),

let

is n - i n t e g r a l .

f: A - A / C

b e the

epimorphism. w

We n o t e that,

then,

for any

V

c ~([A,C~),

ord([A,C~,x,V

) = o r d (f (A) , f (x) ,V )

ord([A,C~,I,V

) -- o r d ( f ( A ) , f ( ( I

we c l e a r l y h a v e

for a n y

x c H(A)

and N H(A))A),V

)

*

We also n o t e that, n c N

and

for any

for any W

~ c HI(A)

and

N ~ ~

(A), w i t h

~ ~ C, w e c l e a r l y h a v e that:

the projection

of

the p r o j e c t i o n

C from

from

N

f(N)

in in

g4

A f(A)

is f ( ~ ) - i n t e g r a l is

f(~)-integral.

I c A.

2.30

§20.

By a v a l u e d a field,

A

Valued

vector

space we m e a n

is a k - v e c t o r - s p a c e

v: is a m a p p i n g , and

Z

where

is the

vector

Q(A)

a triple

with

[A : k~

A U fi(A)

is the

spaces.

(k,A,v)

where:

k

is

< ~ , and

~ Z

set of all k - v e c t o r - s u b s p a c e s

set o f all n o n n e g a t i v e

integers

together

of

with

A

, such

that: (1)

v(0)

=

(2)

v(x+y)

> m i n (v (x) ,v (y) )

(3)

v(x+y)

= min(v(x),

v(x)

~

;

~ v(y)

v(y))

v(zx)

= v(x)

for a l l

(5)

given

any

and

there

exists

such (6)

that v(L)

(*) [V/M(V)

(**)

(H 0(R),

(18.1),

: H0(A)]

H I(R),

in

> v(y)

(18.3),

and x

y

and

in y

]% ;

in

]% w i t h

0 ~ a ¢ k v(x)

;

> v(y)

~ ~ ,

; and for e v e r y

(18.7),

and

= i, we h a v e

and

I% w i t h

: x ¢ L}

any

that

(18.8)

V

L ¢ Q(A) and

(18.9)

¢ ~([A,C~)

(H0(A) , HI(A),

we

see that:

with ord([A,C3,.,V)

space.

(19.1),

(19.3),

F_or any h o m o q e n e o u s with

for all

x ¢ A

y

C ¢ ~I(A)

vector

In v i e w o f

V ¢ ~(R)

v(x-ay)

For any

is a v a l u e d

x

= min{v(x)

In v i e w o f

x

;

(4)

a ¢ k

for all

[V/M(V)

(19.7), domain

(19.8) R

with

: H 0(A) ] = i, we h a v e

ord(R,.,V))

is a v a l u e d

95

vector

and

(19.9)

Dim R = 1 that space.

we

see that:

and

any

2.31

In

§21

and

groundwork lemmas like

about

to

topic of

for

this

shall

defining

define

osculating

osculating vector

in m i n d

flats

flats,

space.

situations

osculating

To

(*)

much

shall

fix

and

more

we

flats.

the

(**).

thoroughly

To

now

idea, We

prove the

shall

than

prepare

several

reader

deal

needed

in t h e

defined For

every

=

[]i(L) Z(L)

rest

of

§20,

=

let

in

(k,A,v)

be

a valued

vector

let:

~ ( A ) : I c L}

[Ie

Q(L):

[I

=

[v(x): =

Y' (L) = =

Ix

x e L}

~ L:

[I~

,

: k]

Iv(I) : I ~ 0 (L)]

Y(L,j)

Y(L)

L ¢ Q (A)

[Ie =

Z' (L)

=

i}

,

,

v(x)

~ j}

,

q (L) : I = Y ( L , j )

[Ie

Q(L):

,

for

I = Y(L,v(I))]

some

j e 7}

,

,

w

(L)

=

{X e L:

v(x)

> v(L)}

(L)

=

[x e L:

V(X)

= ~]

, and

w

A

*

We

note

that,

in v i e w

w

(i),

w

(2)

and

: k]

.

(4),

we

then

have

w

A

(L)

e O (L).

Let w

p(n)

we

also

may

with

above.

Q(L)

T

the

the

book.

So, as

we

a valued

keep

of

§23

observe

that

=

[A

in v i e w C N L

A

in t h e

of

(18.1) in

case

and of

(19.1) (*)

(L)= [0]

Now,

(L)

rest

of

§20,

in c a s e

let

of

(**)

L ~ n d (A)

p = p (L) .

96

be

given

and

let

space

the rest

2.32

We

shall

first

state Lemmas

(20.1)

to

(20.13)

and t h e n p r o v e

t h e m o n e b y one. (20.1)

LEMMA.

Z(L) = Z' (L)

(20.2)

LEMMA.

c a r d Z'(L)

(20.3)

LEMMA.

Y(L)

(20.4)

LEMMA.

For any

over

j e Z(L),

then

{ d - p + i.

= Y' (L).

Y(L,j)

j ~ Z

we h a v e

~ Y(L)

(20.5)

LEMMA.

c a r d Y(L)

(20.6)

LEMMA.

I ~ Y(L)

(20.7)

LEMMA.

I__~f d > p

and

Y(L,j)

v(Y(L,j))

¢ ~(L);

if m o r e -

= j.

= c a r d Z(L). = Y(I)

c Y(L)

then

~

and

(L) c T

~

(I) = A (L).

(L) ~

Y(L)

N Qd_I(L)

W

and

v(T

that

(L)) > v(L);

v(I)

> v(L),

(20.8) Ld = L

then

LEMMA.

such t h a t

Moreover

we have

There

have

v(L and

i)

> v(

~i+I )

L'~I = Li

for

for

, i) > v ( L

whenever the said

characterized

sequenc e

and

sequence

v(L

Alternatively,

unique

a unique

sequence

v ( L i) > v ( L i + I) p ~ i ~ d, L i = T

%,

~

0d_l(L)

such

Lp c L p + i C . . . c for

p { i < d.

(Li+ I)

with

L ,e c L , e + i c . . . c L ,i+ I)

for

p ~ e ~ d,

,d = L

fo___~r ~ ~ i < d.

such that

Moreover

we

p { ~ { ~ ~ i ~ d. sequence

by saying

the

c Lp,+ic...c_L~ = L

for

of

= ~.

w e c a n say that g i v e n a n y

a unique and

is a n y m e m b e r

(L).

(L), an___~d V(Lp)

L , i = L~, i

completely

i_~f I

exists

L i e Y(L)

More generally

L , i ~ Qi(L)

I = T

Li ~ Qi(L)

p { i < d, Lp = ~

there exists

moreover,

%

c Lp+iC...eL

following: such that

P' ~ i ~ d, v( % ', , = -.

p ~ i ~ d.

97

-- L

can be a

there exists L:l ~ Qi (L)

Moreover

we h a v e

a and..... p' = p

2.33

(20.9)

We have

card

Y(L)

= card

Y' (L) = c a r d

card

Y(L)

n Qi(L)

Z' (L) = c a r d

Z(L)

= d - p + 1

and

Moreover,

with

~ = V(Lp)

is t h e

the

= card

Y' (L)

notation

N Qi(L)

of

(20.8)

=

1

for

we have

p

< i < d.

that:

> V(Lp+l)>...>V(Ld)

unique

descending

Y(L)

= Y' (L) =

Y(L)

N Qi(L)

= Y' (L)

LEMMA.

Let

labelling

{ L p , L p + 1 ..... Ld]

of

Z(L)

,

,

and

(20. i0 ) Given the Ji

J c ~ b (L)

unique c ~i(J)

and

In v i e w

of

and

L D J, w e

r(b)

~ d

of

Lp c Lp+l

n --- p (J)

let

sequence

(obtained

v(Ji) the

Q Ni(L)

fo r

set-theoretic

integers

get such

p

~ i < d.

be

L

(20.8)

as

in

(20.8)

to

J)

such

Ib e

that

n < i < b.

inclusions

a unique

for

J n c J n + l c" " °c Jb -- J

let

applying

> v ( J i + I)

clearly

[Li}

c. " . c L d =

and

by

=

Lp c Lp+iC...

sequence

p = r(~)

Ld = L

< r(~+l)<...<

that

J n L r (b) = J n L s , f o r

r (b)

< s ~ d

,

and J N L r(i) r(i)

I

we

..... r ( b ) }

Lp c Lp+l c...c

J n L p = Jn"

for

~ < i < b

and

< s < r(i+l).

Alternatively,

[r(~),r(~+l) that

= J N L s / J Q L r(i+l)

it

can

instead

in

[ p , p + l ..... d}

L d = L,

follows

that:

Ls

define

c ~ s (L)

upon

98

the as

for

letting

complement follows. p

~ s ~ d,

of From

the

J ¢ ~

facts

(L),

and

2.34

Gi =

we

we

have

have

Is

~

[p,p+l ..... d}:

Gi ~

~

for

1

~ i ~ d

gi

=

min{s:

s

e Gi}

J +

- b

Ls

- p

for

1

¢ ~b+p_n+i(L)}

+

n

_< i

and

then

~ d - b

upon

- p

+

letting

n

that

P

< gl

< g2<'''
~ d

and

[p,p+l ..... d}\{r(v),r(~+l)

whence, of

d

in - p

particular, > b

-

u =

n,

we

upon

the

must

J +

have

v =

-- [ g l g 2 . . . . . g d _ b _ p + n

n.

We

also

note

}

that

;

in

case

letting

unique

Lu_ 1 c

..... r(b)}

integer

Lp

and

~

[p+l,p+2

with

Lu ~

p

J +

<

u

~ d

such

that

Lp

and u

we

clearly

=min[s

follows

J

n Ls_ 1 =

J

N L s}

,

+

n - p

~ b

for

n

have

u= It

..... d}:

that

if

r(i)

=

r(i)

=

u

=

d

- p

-- b

p

-

n +

i

p

-

n +

i +

gl -

" n

and

+

then:

1

Lp_n+ i c

n

J

+

<

u

Lp

+

1

,

~ i < u+n-p,

and

We

claim

as

defined

we

have

that,

above,

we

with have

the

1

for

sequence

~ =

n

and,

99

u

+ n

p

=

u_Rpn

- p

r(~)

~ i

<

lettin~

~ b.]

r(v+l)<...
=

d

< d +

I,

2.35

J. = J N L 1 s

for

r(i)

< s < r(i+l) for

v(Ji)

= V(Lr(i))

In particular, in

the

above

n < i < b

if

bracketed

Ji

= J n L

Ji

= J Q Lp-n+i+l

remark,

and

p-n+i

then,

d - p = b - n + 1

with

u

as d e f i n e d

we have

for

v ( J i) = V ( L p _ n + i)

n < i < u+n

- p

and and

(we n o t e

that

in case

J.

= L.

for

0 < i < u.)

1

1

(20.11) d

of

LEMMA.

of

integers

with

given

sequence

is

v(Ji) A

Given

= V(Lp_n+i+l)

(L)

= [0],

we

any

sequence

then

have

p = r(n)

0 < n < p,

there

exists

sequence

derived

from

the

for u + n - p < i < b

J the

p = 0 = n

;

and

< r(n+l)<...
e ~b(L)

such

that

pair

and

L

J

the

as

in

(20.10) . (We o b s e r v e J - Z(J) empty

gives

subsets

(20.12) given and

and

(Note D e ~I(B) Let ~(f(B)) (Note W

in view

a surjective

of

map

In the

q = p(N); Assume

that

of

(20.9) of

and

~(L)

(20.10),

onto

the

this

set

implies

of

all

that

non-

Z(L) .)

LEMMA.

let

D + C N.

that,

if

situation

(i.e.,

that

q =

(*),

[C N N

let

{0]

: H0(A)]).

~ N ~ ~e(L) Let

b__ee

B = An

D ¢ ~I(B)-

H0(A )

is

algebraically

closed

then:

epimorphism.

Let

~ e - q ~ 2.) f: A - A / C be

the

that

c ~([B,D]),

the

canonical

now:

h(W)

be

canonical isomorphism,

ord([A,C],N,V)

= V n ~(f(B)),

and

integer. )

100

and

let

~ ~, H 0 ( B ) ordvM(h(W))

h:

W = h-l(v = HI(A),HI(B)

~([B,D]) N ~(f(B))). = N,

is a p o s i t i v e

2.36

Given any = J

b e the u n i q u e

tha-t

Ji ¢ Qi (J)

n ~ i < b. Also

J ¢ ~b(N)'

let

sequence and

Kn

(20.8) b y r e p l a c i n g

Jn c Jn+l c . . . C (20.8)

to

J)

o r d ( [ A , C ] , J i ,V) > o r d ( [ A , C ] , J i + 1 ,V)

K n + l ~. .. C

C

and let

(obtained b y a p p l y i n g

(Note t h a t n o w

let

n = p(J)

[J: H 0(B) ] = b

~

J

=

A,C,V,L

K i ¢ Qi(L)

and

n ~ i < b.

Then we have

be the u n i q u e

by

B,D,W,J

o r d ( [ B , D ] , K i ,W)

K

= J 1

and

for

s e q u e n c e... (obtained

respectively)

n ~ i < b

such

[D n J: H 0(B) ] = n.)

> o r d ( [ B , D ] , K i + 1 ,W)

for

Jb

from

such t h a t for

,

1

and

ord ([B,D]K i ,W) for =

[ord([A,C],Ji,V) (Observe that

(20.10)

gives

recipes

for o b t a i n i n g

(Ji)n
and

(ord([A,C],Ji,V))n~i~e

(Li)p~i~ d

and

( o r d ( [ A , C ] , L i , V ) p < i ~ d ; this

ficant when

L = A, i.e.,

(20.13) Given.

Let

LEMMA.

if

from the

.

the

sequences

sequences

is p a r t i c u l a r l y

signi-

L = HI(A).)

In the s i t u a t i o n

S = R n.

(Note that,

n < i < b

- ord([A,C],N,V) ]/ordvM(h(W))

Assume H 0(R)

that

(**), l e t

[0} ~ N ~ ~e(L)

b__e_e

D i m S = i.

is a l g e b r a i c a l l y

closed,

then:

D i m S = 1 = e ~ 2.) Let

W = V N R(S).

(Note that now: W c ~(S),

and o r d v M ( W )

Given se~uenc9 and

any

J e ~b(N)'

ord(R,Ji,V)

R,V,L

let

ord(S,Ki,W)

S,W,J

= H0(R),

(20.8)

to

J)

(obtained

respectively)

such t h a t

for

0 ~ i < b.

101

= N ,

b e the u n i q u e

such t h a t

fo___~r 0 ~ i < b.

sequence

> ord(S,Ki+l,W)

HI(S)

integer.)

J0 c Jl c . . . c Jb = J

> ord(R, Ji+l,V)

be the u n i q u e by

~ ~ , H0(S)

is a p o s i t i v e

(obtained b y a p p l y i n g

c . . . c Kb = J ing

ord(R,N,V)

from

Also

Ji ¢ ~i (J) let

K0 c Ki

(20.8) b y r e p l a c -

K i ¢ Qi(J) Then we have

and

2.37

K. = J. 1

ord(S,Ki,W)

and

and

(20.10)

gives

L = A, i.e., P R O O F OF

P R O O F OF

(20.1).

By

for

for o b t a i n i n g

from the

0 < i ~ b

.

the s e q u e n c e s

sequences

is p a r t i c u l a r l y

(6)

and b y

(20.2).

V(Xl),

given

any e l e m e n t s

we have

(i),

Z' (L) c Z(L).

that

nonzero,

,

(Li)0~i~ d

significant when

L = HI(R).)

x k ¢ Q(L),

consequently

recipes

(ord(R,Ji,V)~i~b

(ord(R, L i , V ) ) 0 ~ i ~ d ; t h i s

we have

0 ~ i ~ b

= [ o r d ( R , J i , V ) - ord(R,N,V)]/orcIvM(W)

(Observe t h a t (Ji)0~i~b

for

1

(4)

Z(L) and

Therefore

(6)

in

w e get

v(xk)

X l , X 2 ..... x m

are all d i s t i n c t

al,a2, .... a m

For any

x ¢ L,

--- v(x)

Z(L) = Z' (L).

Given any elements

v ( x 2) ...... v ( x m)

c Z' (L).

k

in

L

and n o n i n f i n i t e ,

at l e a s t one o f w h i c h

such and is

we have for some

v ( a l x I + a 2 x 2 + . . . + a m X m) = v ( x i)

i

by

(3)

and

(4)

and h e n c e alx I + a 2 x 2 + . . . + a m X m It f o l l o w s

A (L)

that c a r d Z' (L)\[~} =

[L/A

(L)

= k]

and hence c a r d Z' (L) ~ [L/A*(L)

Clearly

[L/A*(L)

: k] + 1 .

: k~ = d - p, and h e n c e

P R O O F OF

(20.3).

Obvious

P R O O F OF

(20.4).

By

(I*),

in v i e w o f

(2*) a n d

102

c a r d Z' (L) ~ d - p + 1 .

(6*)

(4*) w e

see t h a t

;

2,38

W

Y(L,j)

~ O(L),

and n o w b y

Y(L,j)

e Y(L)

and

PROOF Y(L)

OF

into

Therefore

and by

OF

(20.6).

in v i e w o f

In v i e w

(i)

and

we

(20.3),

(6)

if

j ¢ Z(L),

then

gives

see t h a t

an

the

injective

said m a p

m a p of

is s u r j e c t i v e .

Z(L).

Obviously:

Therefore, of

I - v(I)

(20.4)

= card

see that,

= j.

Clearly

card Y(L)

PROOF

we

v(Y(L,j))

(20.5).

Z(L),

(6)

I ¢ Y' (L) = Y' (1) c y' (L)

w e get

we also

I ¢ Y(L)

see t h a t

= Y(I)

I ~ Y(L)

.

c Y(L)

= A

.

(I) = A

(L)

W

PROOF

OF

(20.7)°

Assume

that

d > p.

Then

in v i e w of

(6)

we have (i)

v (L) <

Consequently,

in v i e w o f

W

W

(i),

(2),

A * (L) c T * (L) ~ Y(L) and

there

(2)

W

W

(4)

and

and

(6)

we

v ( T * (L))

see t h a t

> v(L)

exists

y ¢ L

such

that

v(y)

(2) and

(5)

-- v(L)

= min[v(x)

: x e L}

*

In v i e w o f

(i),

W

we

see t h a t

y ~ L\T

(L)

and

the k -

W

vector-space

L/T

(L)

is

generated

by the

image of

y ; consequently

W

[L/T

(L)

: k] = 1

(3) Let

and h e n c e

T

(L) ¢ ~ d _ l ( L )

any

(4) be given

I ¢ ~d_l(L) such t h a t

(5)

then by

v(I)

(5) and

(6*) w e g e t

> v(L)

T*(L)

103

;

c I, and h e n c e

by

(4) and

(3) w e

2.39

get

I = T

(L)

PROOF

OF

paragraphs of

(i)

(20.8).

follow

and

(6)

The

assertions

from the assertio~ we

clearly

have

in the in the

L ~ Y(L)

second

and the third

first paragraph. N Qd(L)

In v i e w

and:

w

v(L)

Therefore

the assertions

d - p = 0.

Now,

first paragraph

PROOF

OF

= m ~ L =

in t h e

in v i e w

of

(20.9).

L e t the

(L)

.

first paragraph

(20.6)

in the g e n e r a l

A

and

case

(20.7),

the

follow by

notation

are o b v i o u s

assertions

induction

b e as in

in c a s e

(20.8).

on

in t h e

d - p

By

.

(20.8)

we have

{ L p , L p + 1 ..... Ld} c Y(L)

consequently

by

(20.1),

:

~, (~) :

Y(L)

(i)

(20.2),

(20.3)

£~,L+I

.....

and

(20.5)

we

conclude

that

T,d}

(20.8)

= c a r d Y' (L) = c a r d

Z' (L) = c a r d

Z(L)

= d - p + 1 .

we have

L i ¢ Y(L) and

c a r d [ L p , L p + 1 ..... Ld} = d - p + 1 ;

and

c a r d Y(L)

By

and

N Qi(L)

for

p ~ i ~ d

clearly

c a r d Y(L)

n ~i(L)

= c a r d Y(L)

p < i gd Consequently,

by

c a r d Y(L)

(i) w e

conclude

N ~i(L)

that

= card

Y' (L) N Q i ( L )

= 1

for

p

~ i ~ d

and Y(L)

N Qi(L)

= y' (L) N ~ i ( L )

104

=

{Li}

for

p ~ i ~ d

.

2.40

In v i e w o f

(i), b y

I

~ = V(Lp)

(20.8) we also

> V ( L p + l ) > . . . > v ( L d)

l is the u n i q u e PROOF

OF

see that

descending

(20.10).

labelling

By a p p l y i n g

(20.9)

of to

Z(L) L

and

J

we get that:

(I) {~ = V(Lp)> L(Lp+l)>...>v(L d) is the u n i q u e

descending

labelling

of

Z(L)

,

of

Z(J)

,

= v ( J n) > V ( J n + l ) > . . . > v ( J b) (2) is the u n i q u e

(3)

descending

labelling

Y(L)

Q ~i(L)

= {Li}

for

p ~ i ~ d ,

Y(J)

D Qi(J)

= [Ji}

for

n ~ i ~ b .

and

(4)

J c Q (L), we c l e a r l y h a v e

Since and

(2), t h e r e

d

(5) Since

(6) Since

of i n t e g e r s

Upon

a unique

J c L, b y

(3),

Ji = J n Lt(i) Ji c Qi (J) by

~ Z(L)

sequence

; consequently,

p = t(n)

by

(I)

< t(n+l)<...
such that

v ( J i) = v(Lt(i))

consequently

(7)

exists

Z(J)

for

for

(4) and

for

n ~ i ~ b

.

(5) we get that

n ~ i < b

.

n < i ~ b, we h a v e

Ji ~ Ji+l

(6) w e get

J n Lt(i)

~ J N Lt(i+l)

for

letting

105

n ~ i < b

.

for

n ~ i ~ b;

2.41

I

Fb =

Ix e L

: v(Jb)

> v(x)}

{x

: v(Ji ) > v(x)

and Fi =

in v i e w

of

(i),

e L

(3)

and

L s c L t (b)

(s)

we

n

< i
,

t(i)

~ s < t(i+l)

get

for

Fb

for

t(b)

~ s ~ d

,

and c L t (i)

Ls

now

U

(5)

> v ( J i + l )}

by

(20.1)

we

for

U F1

have

n

Z' (J)

=

~ i < b

Z(J),

and

and

hence

in v i e w

of

(2)

;

we

also

get

(9)

By

J N

(7),

(8)

and

J N

(io) I

and

(9)

F. = 1

it

~

for

follows

n

< i ~ b

.

that

L t(b)

=

J N Ls

for

J N Lt(i)

=

J N Ls ~

and

~ s < t(i+l)

t (b)

~ s < d

,

and

hence

we

Consequently,

t(i)

must

have

upon

~ =

letting

J D

n

for

Lt(i+l)

n

~ i < b

,

and

t (i)

r(b+l)

= d

= r (i)

+ i, b y

for

n

(5),

(6)

~ i g b and

(i0)

. we

get

Ji =

J D Ls

for

r(i)

< s < r(i+l)1 for

v ( J i ) = v ( L r(i) )

The follows

special from

PROOF Then,

OF

in v i e w

the

claim

bracketed

(20.11). of

for

(20.9),

n

~ i gb

.

I

the

case

when

d - p = b

- n +

1

now

remark.

Let we

Lp c know

Lp+iC...~L that

106

d

=

L

be

as

in

(20.8).

2.42

(1)

v (Lp) = ~

and

V(Lp)

> V ( L p + l ) > . . . > ( L d)

(2) is the u n i q u e

Since

0 < n < p, w e can take

(2) w e can take

(3) By

descending

x.

c L

l

J

(2) and

(3)we

(4)

¢ nn(~

of

Z (L)

In view of

(L)).

(20.1)

and

with

v(x i) = V(Lr(i)) (i),

labelling

for

n < i < b

get

> V ( X n + l ) > . . . > v ( x b)

Let W

J = J Then in v i e w of *

J

+ Xn+l k + X n + 2 k + . . . + X b k

(4), b y

(l),

(3)

and

(4)

. we see that

J c ~b(L),

W

= A (J), n = p(J),

and

!

(5) Let ing

Z (J) = [ ~ , V ( X n + l ) , V ( X n + 2) ..... V(Xb)] Jn c Jn+l ~'" .c J n (20.8)

n < i < b,

to

J, such

= J

be the u n i q u e

that

V(Jn)

I

and

(obtained

v(J i) ~ v(Ji+l)

and

(6) By a p p l y i n g

Ji e ~i(J)

sequence

(20.1)

and

(20.9)

= to

J

we know

that

v(J n) > V ( J n + l ) > . . . > v ( J b) is the u n i q u e

and hence,

in view of

descending (4) and

labelling

of

(5), we m u s t h a v e

107

Z' (J)

by applyfor

2.43

(7) By

v(J i) = v(xi) (I),

(3),

(6) and

for

(7) we get

(8)

v(J i) = V(Lr(i))

In v i e w of p = r(n) the pair

(2) and

(8), by

< r(n+l)<...< J

and

{

PROOF OF

n < i ~ b

L

(20.12).

By

n < i ~ b .

(20.10) we conclude that the given sequence

r(b)

as in

for

~ d

must be the sequence derived

from

(20.10). (18.11) we have

ord ([B,D], Ji,W)

(I)

= [ord([A,C],Ji,V) n < i ~b

- ord([A,C],N,V) ]/ordvM(h(W))

for

.

Since ord([A,C],Ji,V) by

> ord([A,C],Ji+l,V)

for

n ~ i < b ,

> ord([B,D~,Ji+l,W)

for

n ~ i < b

(I) we get that

ord([B,D],Ji,W)

and hence by the uniqueness part of

(20.8),

applied

to

B,D,W,J,

we

conclude that K.1 = J 1 therefore,

in view of

for

n ~ i < b ;

(i), it now follows that

ord ([B,D],Ki,W)

I for

= [ord(~A,C],Ji,V) PROOF OF

(20.13).

- ord([A,C],N,V) ]/ordvM(h(W)) By

(19.11)we

108

have

n < i ~ b.

2.44

(1)

ord (S,Ji,W) for

= [0rd(R,Ji,V)

0 < i ~b

- ord (R,N,V) ]/ordvM(W)

.

Since ord(R, Ji,V) by

> ord(R, Ji+l,V)

for

0 ~ i < b

> ord (S,Ji+I,W)

for

0 ~ i < b

(20.8),

applied

,

(i) w e g e t t h a t

ord(S,Ji,W)

and h e n c e b y the u n i q u e n e s s conclude

therefore,

in v i e w o f

for

C ~

(21.1)

1

for

= [ord(R, Ji,V)

S,W,J,

we

;

that

- ord(R,N,V)]/ordvM(W)

.

flats and i n t e g r ~

projections

of an e m b e d d e d

curve.

DI(A) LEMMA-DEFINITION.

: H0(A) ] = i.

Emdim[A,C,L],

0 ~ i ~ b

(I), it n o w f o l l o w s

0 < i ~b

Osculating

Let

= J

1

ord(S,Ki,W)

[V/M(V)

to

that K

§21.

p a r t of

F o r any

Let

V ~ O([A,C])

L e @d(A),

it is e a s y to get the

upon

be such that

letting

following by applying

p = (18.3)

and

(20.8). The______ree x i s t s L i c ~i(A) Moreover

an d

sequence

L = Ld D Ld+I~...~L p

ord([A,C],Li_l,V ) < ord([A,C],Li,V)

for

such t h a t d < i ~ p.

we have

L i ~ HI(A) for

a unique

= {x ~ Li_ 1 ~ HI(A)

: ord([A,C],x,V)

d < i ~ p ,

109

> ord([A,C],Li_l,V~

2.45

L p = A (A,C,L) , and

ord ([A, C] ,Lp, V) = ~

M o r e q e n e r a l l y w e can say that q i v e n there exists

a unique

L , i e ~i(A) d < i ~ ~.

and

Moreover we have

unique

sequence

,i_l,V)

L , i = LS, i

by sayinq

d ~ ~ ~ p , such that for

d ~ i ~ ~ ~ 8 ~ p.

L = L d D Ld+ID...DLp

L = L'd ~ Ld+l'D...~L'p,

,V) = ~.

whenever

the f o l l o w i n q :

ord([A,C],L~_I,V ) < ord([A,C3,LI,V) ord([A,C],L~,

with

< ord([A,C~,L~,i,V)

the s a i d s e q u e n c e

characterized

~

L = L ,d ~ L ~ , d + I D . . . ~ L ~ , ~

ord([A,C],L

Alternatively, completely

sequence

any

There exists

such that

for

L!z e ~i(A)

d < i ~ p'

Moreover we have

p' = p

c a n be a and

, and

and

L~z = L.l

for

d ~ i ~ b. We d e f i n e

Ti([A, C3,L,V) = Li

Ti([A,C],L,V) and w e n o t e t h a t then: negative

integer

for

1

for

d ~ i ~ p

= ord([A,C],Li,V ) Tp([A,C],L,V)

= ~, ~ i ( [ A , C ] , L , V )

d ~ i < p, and in v i e w of

is a non-

(18.3) w e h a v e

w

~d([A,C],L,V)

> 0 ~ L c ~

([A,C],V)

We also define

Ti([A,C],V ) = Ti([A,C],A(A),V) for

-i < i < E m d i m [ A , C ]

Ti([A,C3,V ) = Ti([A,C~,A(A),V )

W e note t h a t in v i e w of

(18.3) w e h a v e w

T_I([A,C'],V)

= 0

and

110

To([A,C],V)

= ~ ([A,C'],V)

2.46

Ti([A, C3,L,V) at

V

relative

f l a t of when

in

=.

L.

the r e f e r e n c e

to

A

By

(20.1)

(18.3) and

Let

A

and

(21.4)

Z(L)

m a y be c a l l e d

to

(20.12) w e

A

the o s c u l a t i n g

i-

" in A

immqediately g e t L e m m a s

L c ~d(A),

= [ord([A,C~,x,V):

Y'(L)

= [Ie

~

be s u c h t h a t

th@ set of all n o n n e g a t i v e

= {ord([A,C~,I,V):

upon

[V/M(V)

integers

" ,

(21.2),

: H0(A)~

together with

letting

I e ~

(A)

with

I c L}

x e L n HI(A)}

(A): I n HI(A)

= [x ~ L N Hl(A): j

for some = {I e M

C

f r o m the c o n t e x t .

V ¢ 3([A, C3)

Z'(L)

Y(L)

in

From these phrases we may drop is c l e a r

Let

denote

T h e n for any

i - f l a t of

as s t a t e d b e l o w .

LEMMA. Z

the o s c u l a t i n g

Ti([A, C3,V) V.

(21.2) = I.

to

at

(21.3)

C

m a y be c a l l e d

(A) : I N HI(A)

= [x e L n HI(A):

ord([A,C~,x,V)

~ j ]

e Z~ ord([A, C3,x,V)

> ord([A,e~,I,V)}} p = Emdim[A, C, L~

we have

the f o l l o w i n g :

(i)

Y(L)

(2)

c a r d Y(L)

(3)

Y(L)

Q 91i(A) = Y' (L) n ~

= c a r d Y' (L) n ~ i ( A )

= Y' (L) = [ T i ( [ A , C ~ , L , V ) :

Z(L)

c a r d Y(L)

for d ~ i ~ p.

I

is the u n i q u e

= 1

for

d ~ i < p.

d ~ i ~ p}.

= Z' (L).

= c a r d Y'(L)

Td([A, C3,L,V) (6)

(A) = { T i ( [ A , C ~ , L , V ) } w

Q ~i(A)

(4) (5)

;

= c a r d Z' (L) = c a r d Z(L)

< Td+I([A, C 3 , L , V ) < . . . < T p ( [ A , C ~ , L , V )

ascending

labelling

111

of

Z(L).

= p - d + I. = =

2.47

J

J - [ord([A,C],I,V):

I ~ ~

w

(A)

with

I c J]

qives a surjective map of (7) [J e

(A) : J c L]

onto the set of all nonempty

subsets of

Z(L) (21.3) =

LEMMA.

Let

V e ~ ([A,C])

be such that

[V/M(V):

H0(A)]

i.

Given any Emdim[A,C,L]

L ~ ~d(A) and

and

J e ~b(A)

with

J c L, let

p =

n = Emdim[A,C,J].

In view of the relations L = T d ([A,C],L,V)

(1')

Ts([A,C],L,V)

we clearly qet a unique (2')

e~(A)

for

L D J eg~b(A)

,

d ~ s ~ p

and

sequence

d < r(b) < r(b+l)<...
of inteqers I

D Td+I([A,C],L,V)D...~3Tp([A,C],L,V)

= p

such that

(A,J,Td([A,C],L,V))

= ~(A,J,Ts([A,C],L,V))

for

d ~ s mr(b)

,

an___dd

[~

(A,J,T(r_I) ([A,C],L,V))

/

A(A,J,Ts([A,C],L,V))

A (A,J,T r(i) ([A,C],L,V)) for

Equivalently, terized

recursively

b < i ~ ~

in v i e w of

and

r(i-l)

< s < r(i)

(i'), the sequence

(2') can be charac-

thus:

r(b) = maxis

e [d,d+l ..... p]: Ts([A,C],L,V)

112

~ J]

,

2.48

r(i)

= maxis

¢ Jr(i-l)

+ l,r(i-l)

+ 2 ..... p} for

T s([A,C~,L,V)

b < i < ~ ,

D ~(A,J,T r(i_I)+I([A,C~,L,V))]

and r (~) = p

.

IAlternatively, [r(b),r(b+l) relations

we

..... r(~}

(i')

and

the

can

instead

in

[ d , d + l ..... p}

follows

Gi =

we have

Is

that:

upon

@,

for

thus.

N HI(A)

In v i e w

= Tn([A,C~,J,V)

J + Ts([A,C~,L,V

of of

the

~ HI(A)

,

) ~b+p_n_i(A)}

1 ~ i ~ b + p - n - d,

gi = maxis:

we have

complement

letting

e [ d , d + l ..... p]:

Gi ~

the

equation

J N Tp([A,C~,L,V)

it

define

s E Gi}

for

and

then

upon

letting

1 ~ i ~ b + p - n - d

that

P > gl

> g2>'''>gb+p-n-d

~ d

and { d , d + l ..... p } \ { r ( b ) , r ( b + l ) ..... r ( v ) ~ Whence, of

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

p - d > n - b,

the

H

----

unique

upon

we must

v = n.

with

d ~ u < p

) c J + Tp([A,C~,L,V)

and Tu([A,CI,L,V)

We

also

note

letting

integer

Tu+I([A,C~,L,V

have

= [gl,g 2 ..... g b + p _ n _ d ~

~Z J + T p ( [ A , C ~ , L , V )

113

such

that

that

in c a s e

2.49

and u

= maxis

¢ [d,d+l ..... p-l}: =

we

clearly

u= It

u

A(A,J,Ts+I([A,C],L,V))

~ ,

have

= gl

follows

r(i)

~(A,J,Ts([A,C],L,V))

that,

" if

p - d = n - b + i, then:

= p - n + i - 1

for

b

b - 1 ~ u + n - p < n

~ i ~ u + n - p

,

and r(i)

We v = n

= p - n + i

claim and,

that,

upon

r(i-1)

=

in t h e

=

particular, above

T i([A,C],J,V)

for

u + n - p < i ~ n.]

with

the

r(b-l)

(2')

J + Tp([A,C],L,V)

as d e f i n e d

=

we have

| for

b

~ i ~ n

.

Tr(i) ( [ A , C ] , L , V ) if

p - d = n - b + 1 remark,

the_____nn, w i t h

u

as d e f i n e d

we h a v e

A(A,J, T p _ n + i _ I ( [ A , C ] , L , V ) )

] for

T i ([A,C],J,V)

above,

= d - I, w e h a v e

A(A,J,Ts([A,C],L,V))

bracketed

=

sequence

c

< s ~ r(i)

~i([A,C],J,V) ID

Tp_n+i([A,C],L,V)

Letting

Ti([A,C],J,V) for

and

Tp_n+i_ I([A,C],L,V)

b

< i ~ u + n - p

I i

and

Ti([A'C]'J'V)

=

~(A'J'Tp-n+i([A'C]'L'V))

} fQr u + n - p < i

T i([A,C],J,V)

=

~p-n+i([A'C]'L'V)

114

~n

.

,

2.50

(We n o t e

that,

in case

Ti([A,C~,J,V)

= Ti([A,C],L,V),

Conversely, d ~ r(b)

A(A,C,L)

given

any

=

[0}, w e h a v e

for

u < i g n.)

L e '~d(A)

< r(b+l)<...
= p

of

p = Emdim

and give n any

integers

with

A : n

and

sequence

Emdim[A,C,L~

= p

W

n ~ E m d i m A, = n

suqh

pai r

J

there

that the n____dd a L

exists

qiven

J e ~b(A)

sequence

with

is the

J c L

sequenqe

and

Emdim[A,C,J]

derived

f r o m the

as above. W

(21.4) with

LEMMA.

J c N,

let

B = A N , D = C N, (Note t h a t Assume (Note D e ~31(B)

h:

now:

K ¢ ~-e-I

an Z

J g ~b(A)

n = Emdim[A,C,J~.

Let

(B)

and

Emdim[B,D,K]

= n - e - I.)

if

H 0(A)

is a l g e b r a i c a l l y

closed,

then:

e ~ 2.)

that

(Note t h a t V N R(f(B)),

and

and

D ¢ ~(B).

be

~ R(f(B))

such

*

{0} ~ N ¢ ~e(A)

K = jN.

f: A - A / C

R([B,D])

be qiven

and

that,

Let

any

q = Emdim[A,C,N]

that

~ q-

Given

the c a n o n i c a l

be the

canonical

epimorphism isomorphism.

[V/M(V) : H0(A) ~ -- i.

now:

ord([A,C~,N,V)

and o r d v M ( h ( W ) )

and

let

Let

Le__~t W = h - l ( v

~ =, W ~ ~ ( [ B , D ~ ) ,

is a p o s i t i v e

V ~ ~([A,C~) Q R(f(B))). h(W)

=

integer.)

Then we have

Ti([B,D],K,W)

= Te+I+i([A,C],J,v)N

T i([B,D~,K,W)

=

fo r

b - e - 1 < i ~ n - e - i,

an__Ad [Te+l+ i([A,C~,J,V)

- ord ([A,C~,N,V) ~ / o r d v M ( h ( W ) ) for

b - e - 1 < i < n - e - i. w

(In c o n n e c t i o n with

J c L,

(21.3)

with

gives

(21.4),

recipes

observe

that

given

for obtaining

115

any

L ¢ ~ d (A)

the sequences

2.51

Ti([A,C],J,V)b
n

and

Ti(EA,C],J,V)b~i<

T i ([A,C],L,V)d~i~ p

and

Ti([A'C]'L'V)d~i~p"

This

is e s p e c i a l l y

draw

the

the

reader's

case

of

attention

(21.4)

(21.5). Lemmas

significant

when

and

to t h e

L = case

from

the

where

~ (A). of

sequences

p = Emdim[A,C,L].

We would

(21.3)

when

also L =

like

to

A (A)

and

drawn

from

J = N.)

REMARK.

(21.3)

when

n

Several

(21.4).

concrete

As

corollaries

a sample,

we write

can be

down

the

following

information. Assume Given Note

that

H0(A)

is a l g e b r a i c a l l y

[0] ~ N e ~ * (A), e that

now

Emdim[B,D]

Q =

and

recall

f: A - A / C ~f(B))

be

V N = h-l(v

We

be

= q - e - i.

[I e ~ e + l ( A )

the

J

gives

p = Emdim[A,C].

B = A N , and

canonical

canonical

Let

i.e.,

of

~

onto

equivalently:

epimorphism

isomorphism.

and For

~0(B)

q - e > 2.

let

h:

every

Let

~ ([B,D])

V ~

~[A,C])

let

n f(B)).

=

that

[s

(21.3)

Te+I([A,C],N,V)

for

every

V ~

e [ 0 , i ..... p]:

and

=

Te+ I([A,C],N,V) N =

(21.4)

Ts([A,C],V)

(with

A(A,N,Tm(v)

,q([A,C]),

J = N

([A,C],V))

and

upon

letting

J N]

,

L =

A(A)),

¢

~ * ( [ B , D ] , V N)

and w

m(V)

D = C N.

: I c N]

a bijection

D ~ ZI(B),

the

observe

of

~

that

m(V)

in v i e w

q = Emdim[A,C,N],

Let

that

J

Assume

let

closed.

= 0 ~ Tm(V) ([A.C],V)

= .~ ( [ A , C ] , V )

116

= V (~ ~ ( [ A , C ] , N )

we

get:

2.52

We

also

F(J)

observe

=

that

for any

[V ¢ ~ ( [ A , C ] )

J ¢ Q,

upon

letting

: Te+I([A,C],N,V)

= J}

,

we have F(J)

=

IV e ~ ( [ A , C ] )

=

{V ¢ ~ ( [ A , C ] , J )

= a finite

and,

in v i e w

of

Finally p = Emdim

we

:

we

: ord([A,C],J,V)

> ord([A,C],N,V)]

get

V ¢ F(J) } = ~ ( [ B , D ]

observe

A = q =

~ ord([A,C],N,V}

set

(21.4),

{V N

: ord([A,C],J,V)

that,

if

, jN)

A(A,C)

( E m d i m B)

+ 1

and

integer

with

-i

for

=

,

[0}

any

and V

e

e = 0,

then

.~[A,C]),

upon

letting

u(v)

=/

the

unique

Tu(V)+I

in v i e w

u(V)

of

(21.3)

= maxis

([A,C],V)

and

c N

(21.4),

¢ [ - I , 0 ..... p - l ] :

and

with

=

1 ~ N = ~

T i ( [ B , D ] , V N)

< p

such

Tu(V) ([A,C],V)

J = N

and

L =

that

~ N

,

A(A),

A(A,N,Ts+I([A,C],V))}

get:

,

([A,C~,V)

= Ti+ I([A,C],V) N for -i

T i ( [ B , D ~ , V N)

we

A(A,N,Ts([A,C],V)) =

u(V)

< u(v)

=

Ti+I([A,C~,V)

and

117

- ~0([A,C],V)

~ i ~ q-i

,

2.53

u(v) # -1 - N ~ S ([A,C],V) T i([B,D],v N) = A(A,N,T i([A,C],v))N for

-i ~ i < u(V)

for

u(V)

Ti([B,D],V N) = Ti([A,C],V) =

and T i ( [ B , D ] , V N) = T i + I ( [ A , C ] , v ) N ~ i < q - 1 .

T i([B,D],V N) = Ti+ I([A,C],V)

(21.6) Let

REMARK.

~ ~ H 1 (A)

and

q = Emdim[A,C,N].

I

Assume that N ~ ~ e (A)

is a l g e b r a i c a l l y

be given with

~ c N.

closed.

Let

We note that then clearly:

the p r o j e c t i o n

(21.6.1)~

H 0 (A)

of

C

from

N

in

A

is w-integral

q-e ~ 2

and

~ c Te+I([A,C],N,V)

for all

V e ~([A,C],~)

q-e a 2

and

~ c Te+I([A,C],N,V)

for all

V ¢ ~([A,C],N).

!

~

We also note that, the p r o j e c t i o n

of

C from N in A

is n-integral (21.6.2)

if e = 0,then:

{ q ~ 2

and

~ ~ TI([A,C~,V)

for all

V c ~([A,C],N).

§22.

Osculating Let

R

(22.1)

flats and integral projections

be a homogeneous

domain with

LEMMA-DEF INITION.

Let

in an abstract

curve.

Dim R = i.

p = Emdim R.

Let

V e ~(R)

be

w

such that

[V/M(V)

: H0(R) ] = i.

For any

(20.8) we get the following:

118

L ~ ~d(R),

by

(19.3)

and

2.54

There

exists

L i ¢ ~i(A) Moreover

and

a unique

ord(R, Li_l,V)

=

Ix ¢ L i _ for

clearly

< ord(R,Li,V)

for

{0}

M__ore q e n e r a l l y

we

exists

1 N H I(A) : ord (R,x,V)

d < i Kp

Lp =

there

L = L d ~ Ld+I~...oL

such

P

that

d < i ~ p.

we have

L i n H I(R)

and

sequence

a unique

and can

> o r d (R,Li_I,V)

,

o r d ( R , Lp,V) say

that

sequence

= =.

qiven

L = L

any

,d ~ L

ff

with

d ~ ~

,d+l~...~Lff,ff

~ p

such

, that

w

L

,i ¢ ~i(R)

Moreover

and

ord(R,L

we have

L

, i = LS, i

alternatively, completely

,i_l,V)

the

characterized

said by

< ord(R,L

whenever sequence

sayinq

, i , V ) for

d ~ i ~ ~

d < i ~ ~

~ 8 ~ P-

L =L d D Ld+I~...DLp

the

followinq:

.

There

can be exists

a

w

unique

sequence

ord(R,L[_l,V) Moreover We

L = L'd D Ld+l~..' .DL'p < ord(R,L~,V)

we have

T i(R,L,V)

integer

= p

and

also

d < i g p'

L!

1

= L

for

1

and

LIt ¢ ~ i ( R ) ord(R,L'p,,V)

and = =.

d ~ i ~ p.

-- L i for

d ~ i < p

= ~

, mi(R,L,V)

= ord(R,Li,V)

note

that,

then

for

d ~ i < p,

Tp(R,L,V) and

in v i e w

Td(R,L,V) We

that

define T i(R,L,V)

We

p'

for

such

of

(19.3)

> 0 ~ L c

is a n o n n e g a t i v e

we have

~*(R,V)

define T i(R,v)

= T i (R, ~ (R) ,V) for

T i(R,v)

=

T i ( R , A (R),V)

119

-i

~ i ~ p

2.55

and we n o t e that,

in v i e w o f

T_I(R,V) Ti(R,L,V) relative a__tt V. R

to

T0(R,V)

Ti(R,V)

= ,O*(R,V)

the o s c u l a t i n q

may be called

From these phrases we may drop

(22.2)

to

to

(19.3),

[0]; B

identity map

(22.7)

(20.1)

(22.6) w h i c h

(21.6) b y l e t t i n g

R

the o s c u l a t i n q

a__tt V

i-flat

in

R

" in R " w h e n the r e f e r e n c e

S = RN

to

(20.11),

and

are r e s p e c t i v e l y

and r e p l a c i n g :

A

(20.13),

to

obtained as w e l l

from as

(21.2)

[A,C]

as

[B,D]

by

by

[0];

f

R - R; and

h

b y the

identity map

R(S)

~ 9(S).

REMARK.

Given

any

C c ~I(A),

b e the c a n o n i c a l

[V/M(V)

We n o t e that,

: H0(A)]

S; D

we g e t a s s e r -

as w e l l

f: A ~ A / C

such that

in

(22.6).

(22.2)

R; C b y

i-flat

from the context.

In v i e w of

let

and

may be called

L.

is c l e a r

tions

= 0

(19.3), we h a v e ,

let

epimorphism.

by

b y the

r = Emdim[A,C] Let

to

and

V e ~([A,C])

be

= i.

t h e n clearly:

Emdim

f(T i([A,C],V)) = T i(F(A),V)

f(A) = r

and

,

T i([A,C],V)

= f-l(T i ( f ( A ) , V ) )

Ti([A'C]'V)

= Ti (f (A) 'V) '

for

-i < i < r .

' I #

More generally, p = Emdim[A,C,L],

we n o t e that,

w e c l e a r l y have:

for any f(L)

L e ~ d (A), u p o n

e 9~d+r_p(f(A))

f(T i([A,C],L,v)) = T i + r _ p (f (A) , f (L) ,v)

= L n f-i (Ti+r_ p (f (A) , f (L) ,V) ) ,

T i([A,C],L,v)

= T i + r - p (f (A) ' f (L) 'V)

120

and

,

Ti([A,C],L,V)

,

letting

for

d ~ i ~ p .

2.56

§23.

Intersection

Given

C ¢ ~(A),

For any

multiplicity let

I ¢ H(A)

or

with

f: A ~ A / C I c A

an e m b e d d e d

curve.

b e the c a n o n i c a l

and any

Q ¢ H(A)

epimorphism.

or

Q c A

we define: H ([A,C], I,Q) w

o r d ( [ A , C ] , I , V ) [V/M(V) : R ( [ A , ~ V~

([A,C],V) ]) ] ,

([A,C]) ,Q)

([A,C], I,\Q) =

([A, c],v) ]) ]

~ ord ([A, el, I,V) [V/M (V) : ~ ([A,~ Ve,~ ([A, C] ,\Q) ([A,C],I,Q)

=

~ ord([A,C],I,V)[V/M(V): V ¢ ~ ([A,C],Q)

H0(A)]

•

and o r d ( [ A , C ] , I , V ) [V/M(V) : H0(A) ]

([A.C], I . \ Q ) =

vc,q ([A,c],hQ) For any

I e H(A)

or

I c A

we define: w

u([A,C],I) For any

= u([A,C],I,0) Q ~ H(A)

~([A,C],Q)

=

U~([A,C],\Q)

U ~* ([A'C]'Q)

Adj ([A,C],Q)

Q c A

U

([A,C],I)

----

= U

,

~ ~.~(~([A,C],P)) Pe~]0 ([A,C],kQ)

.

f(k) (~ ([A,C], P) ~ ~"(Z P6~]0 ([A,C3,Q)

where

~ I~f(k) (~([A,C],P) PCZ]0 ([A,C],\Q) = [~ e H

(A): ~ ( [ A , C ] , # , P )

for all Adj ([A,C], \Q) = {~ e H

(A): 9 ( [ A , C ] , ~ , P )

k = H0(A)

, c adj(~([A,C],P))

p ¢ n 0([A,C~,\Q)]

121

where

k = H 0 (A) ,

C adj(9([A,C],P))

P ¢ ~0([A,C],Q)]

for all

([A,C],I,0)

we define:

~ I{(~([A,C],P)) P e n 0 ([A,C],Q) =

U * ([A.C~.\Q)=

or

and

,

2.57

Tradj([A,C3,Q)

--- [~ e H

(A): ~ ( [ A , C ] , ~ , P )

for a l l

¢ Tradj (~([A,C3,P))

P e n0([A,C~,Q)},

and

w

Tradj([A,C~,\Q)

= {~ ¢ H

(A): ~ ( [ A , C 3 , P )

for all By

an a d j o i n t

¢ tradj(~([A,C~,P)

P ¢ n 0([A,C3,\Q) ] ,

of

C

in

A

a__tt Q, we m e a n a m e m b e r of

B y an a d j o i n t o f

C

in

A

outside

Adj ([A,C3,\Q).

Q, we m e a n a m e m b e r o f

B y a _true a d j o i n t o f

m e m b e r of

Tradj ([A,C],Q).

Q, w e m e a n

a m e m b e r of

c

in

A

a_~t Q, w e m e a n a

B y a true a d j o i n t of

Tradj ([A,C],\Q).

d r o p " in A ", w h e n the r e f e r e n c e

Adj ([A,C~,Q).

to

A

C

in

A

outside

From these phrases is c l e a r

we may

from the context.

Finally we define and

Ad9 ([~,C~) = Adj ([A,C~,0) B y an a d j o i n t o f

C

a true adj0int of

in C

and

U~([A,C],0)

in

and

T r a d j ([A,C~) = T r a d j ([A,C],0).

A, we m e a n a m e m b e r of

Adj ([A,C~),

A, w e m e a n a m e m b e r o f

from these phrases we may drop clear

~([A,C])--

and b y

Tradj ([A,C~).

" in A ", w h e n t h e r e f e r e n c e

to

Again, A

is

from the c o n t e x t . In v i e w of

(5.1), (5.6), (5,8), (5.10), (5.11), (17.4), (18.1), (18.2),

(18.3), (18.4), (18.5), (23.1)

Let

and

(18.10), w e c l e a r l y g e t

k = H0(A).

any

P e ~0([A,CT),

I' =

(I N H ( A ) ) A

upon in c a s e

([A,cl,I,P)

Then

letting

for any I' = IA

(23.1)to

I ¢ H(A) in case

or

(23.7): I c A

I ¢ H(A)

and

and

I c A, w e have:

= ~ (~([A,c3,p),

~([A,C3,T,P))

= ~ (~(A,P),~(A,C,P) ~,~(~,T,p))

positive

integer,

if , if

U

([A,C~,I,P)

I' c P I' c C;

= u([A,c3,I,P)Deg[A'P~ = if(k) ( ~ ( [ A , C ~ , P ) , ~ ( [ A , C ~ , I , P ) )

= ~k ([~ (A, p) ,~ (A,C,p) 7,~ (A, I,P) )

122

and and

I' ~ C ,

2.58

0

I

a positive

For

any

P ¢ ~0([A,C]),

1 < card

we

~([A,C],P)

also

integer,

if

I' 9 ~ P

if

I' c p

if

I' c C

([A,C],P)

U~([A,C],P)

< u([A,C],P)

.

=

I(~([A,C],P))

=

~,([N(A,P),~(A,C,P)

= u([A,C],P)Deg[A,P]

=

=

xk([R(A,P),~(A,C,P)

],~(A,P,P))

=

X~(~([A,C],P))

integer

,

= a positive

= X~([~(A,P),~(A,C,P)

= ~([A,C],P)Deg[A,P]

],~(A,P,P))

I f(k) ( ~ ( [ A , C ] , P ) )

= a nonnegative U ~*( [ A , C ] , P )

I' ~ c ,

and

have:

= a positive U

,

f(k) },~

=

integer,

I)

integer,

(~ ( [ A , C ] , P ) )

k([~(A,P),~(A,C,P) = a nonnegative

])

integer.

and U~([A,C],P)

= 0

U~([A,C],P)

= 0

~ ([A,C],P)

is n o r m a l

~ ([A,C],P)

is r e g u l a r

u([A,C],I,P)

= ord~([A,C~,p)~([A,C],I,P) for

(23.2) Q c A,

upon

For

any

every

I ¢ H(A)

I c H(A) or

I c A

letting

IA + C

in c a s e

I ~ H(A)

if = (I n H ( A ) ) A

+ C,in

I cA

case

123

,

or

I cA

and

any

.

Q

c H(A)

or

2.59

f

QA + C

and

, in c a s e

Q e H(A)

,

Q' = 1 (Q N H ( A ) ) A

+ C,

Q cA

in c a s e

,

t we have :

u([A,C],I,Q)

= ~([A,C],I',

=

U

(radA(I'+Q'))

~

Pen0([A,C~,Q)

integer

-- U ( [ A , C ] , I ' ,

(radA(I'+Q')

=

~ u Pe~0 ([A,C],Q)

= a nonnegative

u([A,C],I,Q)

= ~ ~ U and

U ([A,C],I,Q)

N

(HI(A)A))

or

([A,C],I,P) integer

([A,C],I,Q) n([A,Cg,Q)

= 0 ~ ~

(HI(A)A))

u ([~,c],i,P)

= a nonnegative

([A,C],!,Q)

~

~

([A,C],I,Q)

or

~

,

= ~ ~ ~([A,C],I)

= ~([A,C])

~ ~ I' -- C

HI(A)

and

= 0 ~ ZI~,[A,C],I)

N

~ radAQ'

n([A,C~,Q)

HI(A ) c radA(I'+Q') *

[i ([A,C],I)

= U ([A,C],I,Q)

W

~ ~

([A,C],I)

= ~

~3([A,C],I) Q' N

U ([A,C],!,\Q)

c ~([A,C],Q)

(HI(A)A)

= U ([A,C],I',\(radAQ')

N

([A,C],I,Q)

c radAI'

(H I ( A ) A ) )

L~ ([A,C], I,P)

P~0 ([A,C],\Q) a nonnegative

U

([A,C], I,\Q)

= U

integer

or

([A, C T , I ' , \ ( r a d A Q ' ) N

~

,

(HI(A)A))

W

=

~

~

([A,c],i,p)

P~ n0 ([A,C], \Q) = a nonnegative

integer

124

or

~

,

,

,

=

e

2.60

([A,C~, I,\Q) = ~ ~ ~

([A,C~,\Q)

= ~ ~ ~([A.C~,I) ~I'

and

= n([A,C~)

=C~Q'

n ( [ A , C ~ , \Q) ~

,

and

U([A,C],I\Q) = 0 ~U

([A,C~,I,\Q)

= 0 ~ ~([A,C~,I) 9'

Moreover, c I'

for any

J ~ H(A)

with

N

J ~ I'

n n([A.C],\Q)

(HI(A)A)

or

J c A

=

c radAI' with

J n H(A)

w e have:

([A,C],J,Q)

U ([A,C~,J,Q)

~ ~ ([A,C~,I,Q) w ([A,C~,J,Q)

= ~I([A,C~,I,Q)

and . ~ U

([A,C~,I,Q)

~ U*([A,C~,J,Q)

= U

,

([A,C],I,Q)

p ([A.c3.J. p)~ ([A.c3.p)*

,

I = ~([a,c~,I,p)~([a,c],P~, for all u([A,C],J,\Q)

~ u([A,C3,I,\Q)

P e •0([A,C],Q)

and

([A,C~,J,XQ)

~ U

([A,C~,I.\Q)

,

and

u ([A,c],J,XQ) = u ([A,CT,I,\Q)

U

[

([A,C~,J,\Q)

= U

([A,C3,I,\Q)

~ ([A,C~, J, P)~ ([A,C~, P) * *

I = ~ ([A,C~, I, P)~ ([A, C~, P)

for all

where

~([A,C],P)

is the i n t e g r a l

P e ~0([A,C3,P)

closure

of

~([A.C~,P)

, in

([A,c~). For any

Q ¢ H(A)

or

Q c A

also h a v e

125

(in v i e w o f

(15.4)

and

(15.5) we

2.61

U~([A,C],Q)

~

W

([A,C],Q)

~([A,C],Q)

=

~ Q)~([A,C],P) Pcn0(~A,C~,

=

~ Pen 0 ( [A, C~, Q ) ~

W

= a nonnegative

integer

,

[A,C], P) = a n o n n e g a t i v e

integer

,

(

= 0 ~ ~([A,C~,Q)

= 0 ~ ~([A, C3,P)

is r e g u l a r

p ~ n0([A,C],Q) U ([A,C],~,Q)

~ U~([A,C],Q)

and

U

(~A,C],~,Q)

for all Adj ([A,C],Q)

Tradj([A,C],Q)

=

B

~ ~

(~A,C],Q)

~ c Adj([A,C],Q)

~ Adj ([A,C],p) Pen 0 ( [A, C], Q)

=

for all

,

,

Tradj(LA, C~,P) Pen0(~A,C~,Q)

= {~ e Adj([A,C],Q)

: ~([A,C],~,Q)}

= [~ c Adj([A, Cj,Q)

: ~

~(EA,

C],\Q)

=

A,~C],\Q)~([A,C], P6n 0 ( [

~([A,

C3,\Q)

=

~ ~([A,C],P) P~e O([A,c3,\Q)

~([A,C],\Q)

= 0 ~ U~([A,C],\Q)

([A,C~,~,Q)

> U~([A,C~,\Q)

integer

,

= a nonnegative

integer

,

= 0 ~ ~([A,C],P)

and

U

for all

([A,C],~,\Q)

is r e g u l a r

for all

, ~ ~([A,Cq~,\Q)

• c Adj (EA, C],\Q)

Adj([A,C~,\Q)

=

N Adj ([A,C],P) PeSO ( [A, C], \Q)

Tradj ([A,C],\Q)

=

N Tradj ([A,C],P) Pc• 0 ([A, C], \Q)

and

126

([A,C],Q)}

P) = a n o n n e g a t i v e

P ~ n0([A,C],\Q) ([A,C],~,\Q)

= ~

, ,

2.62

=

{~ ~ Adj([A,C3,\Q):

u([A,C],~,\Q)

=

[~ ¢ Adj([A,C3,\Q):

U

(23.3)

For

any

([A,C~,~,\Q)

I ¢ H(A)

IA + C

= ~([A,C~,\Q))

or

I c A,

, in case

(I n H ( A ) ) A

+ C,

= N~([A,C3,\Q)}

upon

letting

I ¢ H(A)

I cA

in case

•

,

,

we have:

u ([A,c~,i) = u ([A,C~,I')

ord([A,C],I,V)[V/M(V):

=

R([A,~

([A,C~,V)])]

V¢~([A,C3)

=

~

u ([A,c3,I,P)

p~n 0 ([A,C~, I,P) = a nonnegative

u

([A,c~,z)

= u

([A,c],z')

integer

or

o r d ([A, C ~, I,V) [ V / M (V) : H 0 (A) ~

= vc,q ([A,c))

w

=

~ u P e n 0 ([A,c3)

= a nonnegative

([A,c3, I,P) integer

or

~ ,

w

([A,C~,I)

u([A,C~,I) and

for any

=

~ ~ U

([A,C3,I)

= = ~ •([A,C3,I)

= ~([A,C3)

= 0 ~ ~

([A,C~,I)

= 0 ~ n([A,C~,I)

=

J ~ H(A),

with

J ~ I'

or

J c A

# ~ HI(A) with

~ u([A,C~,I)

and

U

([A,C~,J)

and

127

> U

([A,C~,I)

,

c radAI'

J N H(A)

we have u([A,C~,J)

= I' = C

c I'

2.63

~([A,C],J)

= ~([A,C],I)

~ Z

([A,C],J)

= Z

([A,C],I)

~ ([A,C], J, P) ~ ([A, C], P)

f where

~([A,C],P)

=

~ ([A,C], I, P) ~ ([A, C], P) * for all

P ¢ [30([A,C])

is the i n t e g r a l

closure

of

"

~([A,C],P)

in

([A,c]). Finally ~([A,C])

(in v i e w of

=

~

(15.4)

~

and

(15.5) we also have:

([A,C~,P)

= a nonnegative

integer

U~([A,C],P)

= a nonnegative

integer

P ~ 0 ([A,c]) ~ u~ ([A'c])

=

~

pe~0 ([A,c])

,

W

U~([A,C])

= 0 ~ U~([A,C])

= 0 ~ .~([A,C],P) p c n0([A,c])

is r e g u l a r ,

w

L~([A,C],~)

~ ~([A,C])

and

U

w

([A,C],~)

~ U~([A,C])

¢ Adj ([A,C]) Adj ([A,C]

=

N Pen 0

for all

Adj ([A,C],P)

for all

, ,

([A,C])

and Tradj ([A,C~

=

N Tradj ([A,C], P) P e n 0 ([A,C])

= {~ ¢ A d j ( [ A , c ~ ) :

U([A,C],~)

= {~ ¢ A d j ( [ A , C ] ) :

U*([A,C],~)

= U~([A,c])] w

(23.4). ([A,C],I,Q)

is a l g e b r a i c a l l y

If

H 0(A)

= ~

[A,C~,I,Q)

closed,

-- U ~ ( [ A , C ] ) ] then:

for any

I ¢ H(A)

or

I cA

and any

Q ¢ H(A)

or

Q cA

W

([A,C],I,\Q) ([A,C],I)

= ~ ([A,C],I,\Q) = ~([A,C],I)

for any

128

I ¢ H(A)

or

I cA

,

;

2.64

= ~([A,C],Q) Q

for a n y ~([A,C],\Q)

= ~([A,C],\Q)

or

H(A)

Q cA

,

f

,

and W

k~{([A,C]) = k ~ ( [ A , C ] ) (23.5).

We have

U ([A,C],xy,Q)

= U ([A,C],x,Q)

+ U ([A,C],y,Q)

for any

,

U{([A,C],xy,Q)

= [I*([A,C],x,Q)

+ U

([A,C],y,Q)

,

in

U ([A,C],xy,\Q)

= U ([A,C],x,\Q)

+ u ([A,C],y,\Q)

,

and any

W

U

W

([A,C],xy,\Q)=

U

x

and

Y

H (A)\C

W

([A,C],x,\Q)+

Q 6 H(A)

t] ([A,C],y,\Q)

or Q c A

and %

U ([A,Cq,xy)

= U ([A,C],x)

.

u

+ u ([a,C],y)

w

([A,C],xy)=

(23.6) Either

U

,

1 )

([A,C],y)

,]

w

([A,C~,x)+

Let

U

x

and

y

(

P ¢ ~0([A,C]),

assume that

for any

and let

in

H(A)\C

I e H(A)

~([A,C],I,P)~([A,C],P)

or

.

I c A.

is p r i n c i p a l

(note t h a t

w

this is c e r t a i n l y U ([A,C],P)

= i.

U ([A,C],I,P)

so if

I ~ H(A)

or

I z H

(A)); o r a s s u m e

Then,

= [~([A,C],P)/~([A,C],I,P)~([A,Cj~,P) =

that:

[~(A,P)/(~{(A,I,P)~{(A,P)

+ ~(A,C,P))

: ~([A,C],P)] : ~{(A,P)]

and ([A,C],I,P)

(23.7).

= [[~([A,C],P)/~([A,C],I,P)~([A,Cq,P) = [~(A,P)/(~(A,I,P)~(A,P)

+ ~(A,C,P))

If

U ([A,C],P)

Emdim[A,C]

= i, t h e n

P ~ n0([A,C].

129

: H0(A) ] : H0(A)] = 1

for all

2.65

Next we claim that: (23.8)~

LEMMA.

For any

x ¢ Hm(A)\C

and

y ¢ Hn(A)\C

w__ee

have n[z PROOF. n[~

([A,C],x) ]

=

m[~

([A,C],y) ] .

Namely

([A,C],x) ] - m[u

([A,C],y) ~_

=

~ Ve3([A,C])

[n(ord([A,C~,x,V))

=

~ [or~f(xn)/f(ym)~[V/M(V) Ve~([A,C])

= 0

by

(23.9)

- m(ord([A,C~,y,V))~[V/M(V)

(4.1),

since

DEFINITION-BEZOUT'S

there exists a unique positive

: H0(A) ~

: H0(A)]

by (18.10)

0 ~ f(xn)/f(y m) ~ R([A,C~)

LITTLE THEOREM. integer,

In v i e w of

t 9 be denoted by

(23.8),

Deg[A,C~,

such that U

([A,C~,~)

=

(23.10)

REMARK°

characterize Q ¢ H(A)\C

(Deg[A,C]) (DegAS)

Q c A

with

Q N H(A) ~ C

([A,C~,~,\Q)/degA~:

= max{u

([A,C~,~,\Q)/n:

Deg[A,C~

if

H0(A)

= max{~

way.

with

~ ~ C .

we can clearly Let any

be given.

Then

W

= max{~

is algebraically

([A,C~,~,\Q)/n: for all

~ c H

~ ¢ Hn(A)

for all large enough Moreover,

(A)

formula,

also in the following

*

Deg[A,C~

~ ~ H

In view of the above

Deg[A,C] or

for all

130

with

with

A ~ ~ ~ C]

~ ~ C}

n. closed,

~ c Hn(A)

n > 0 .

(A)

then

with

~) ~ C]

2.66

(23.11)

LEMMA.

,Assume t h a t

card{P

e ~30([A,C]):

(Note that this is a l g e b r a i c a l l y and,if

H0(A)

assumption

closed;

Deg[A,P]

= i] m 1 + D e g [ A , C ]

is a u t o m a t i c a l l y

namely, ~ 0 ( [ A , C ] )

is a l g e b r a i c a l l y

closed,

satisfied

is a l w a y s

then

.

if

H0(A)

an i n f i n i t e

Deg[A,P]

= 1

set,

for all

P c n0([A,c]).) Then

Emdim[A,C]

PROOF.

Let

d = Deg[A,C],

(i)

s =

By assumption D~ (A)

~ Deg[A,C]

there

[H I(A)

and

let

: H 0 (A) ]

exist pairwise

distinct members

L0,LI,...,L d

of

such t h a t

(2)

[L i : H 0 ( A ) ]

= s - 1 , for

0 < i < d ,

and (3)

LiA C ~ 0 ( [ A , C ] ) ,

Suppose,

if p o s s i b l e ,

~ H I(A) LoA,

such that

LIA ..... LdA

Z~0([A,~ ])

that

for

L 0 D LID...NL d ~ C ; then there exists

• ~ C

and

are p a i r w i s e

and hence,

0 ~ i ~ d .

~ c LiA

distinct members

in v i e w of

(23.1)

•

U

for

and

0 ~ i < d. of

Now

£0([A,C])

N

(23.3), w e see that w

([A,C],~)

> d + i.

However,

since

~ £ HI(A)

and

~ ~ C, b y

w

w e get

U

([A,C],~)) = d, w h i c h

L 0 D L I N . . . N L d c C, and h e n c e (4)

By

(5)

is a c o n t r a d i c t i o n . in v i e w of

(3) w e h a v e

L 0 n L I N . . . N L d = C N H I(A) (i),

(2) and

(4) w e see t h a t

[C N HI(A)

: H0(A ) ] ~ s - d - i

131

Therefore

(23.9)

2.67

By

(i) a n d

(5) w e g e t

(23.12)

Emdim[A,C]

LEMMA.

PROOF.

First

Deg[A,C3

suppose

g d = Deg[A,C]

.

= 1 ~ Emdim[A,C]

that

Emdim[A,C]

= 1 .

= I.

Then there exist w

P ¢ D0([A,C]) c p

and

(23.3),

with

~ ~ C.

(23.1)

Deg[A,P] Clearly

and

= i, a n d t h e r e e x i s t s ~ + C = P

and h e n c e ,

~ ¢ HI(A) in v i e w o f

with (23.9),

(23.6) w e get

w

Deg[A,C]

= U

([A,C],~)--~

Conversely

suppose

([A,C~,~,P)=

that

[~(A,P)/~(A,~C,P)

: ~(A,P)]

=

[~(A,P)/M(~(A,P)):

~(A,P)]

=

1

Deg[A,C]

•

= i.

Let

e = 1 + Emdim[A,C]. w

Then

e ~ 2

that

@i i C

and clearly for

1 ~ i ~ e

(i)

~(A,C)

Since for

there exist

Deg[A,C~

n0([A,C],@i)

=

in

HI(A)

such

and

+ ~i + ~2 + ' ' ' + %

= I, in v i e w o f

1 < i ~ e, w e h a v e

~ I , ~ 2 ..... %

(23.1),

= A(A)

(23.3)

and

c a r d ~ 0 ( [ A , C ] , ~ i) = 1

{Pi], w e h a v e

Deg[A,Pi]

= I.

. (23.9), w e see that,

and, By

upon

letting

(i) w e g e t

n 0([A,c3,~ 1)n...n~ 0([A,c3,~ e) = and hence

card[P

e n0([A,C]):

Therefore by (23.13) Deg[A,C]

Deg[A,P]

= i] ~ c a r d { P i , P 2 ..... Pe]

(23.11) w e c o n c l u d e REM~ARKo

= i, t h e n

Emdim[A,C~

As a consequence

u([A,C],P)

As a r e f o r m u l a t i o n

that

of

= 1

of

for all

= 1 .

(23.12), w e

nl(A):

132

see that,

P ¢ D0([A,C~)

(23.12) w e have:

£)I(A) = [ E ¢

z 2 .

Deg[A,E]

= i]

;

if

2.68

this motivates

As a consequence

if

~i1 (A)

the n o t a t i o n of

(23.12) w e a l s o

E m d i m A = D i m A, t h e n

The a b o v e o b s e r v a t i o n s

(23.14) be given

PROJECTION

such t h a t

~I(A)

= {E ¢ nl(A):

may henceforth

FORMULA.

J c N

see that:

and

(Note that,

if

H0(A)

C N ¢ hi(AN)

~ Emdim[A,C,N]

= i]

be used tacitly.

Let a n y

J c C.

Deg[A,E]

N ¢ ~

Assume

is a l g e b r a i c a l l y - Emdim[A,N]

(A)

that

and

J ¢ ~

(A)

cN ¢ ~i (AN) "

closed,

then:

~ 2.)

T h e n we h a v e w

.

u ([A,c],J) - ~ ([A,C],N) = [~([A,c]): ~([AN, cN])] ~*([A~,cN~,J N) PROOF°

R (f (B))

Let

B = A N , D = C N, and

b e the c a n o n i c a l

K = jN.

isomorphism.

Let

For every

h: R([B,D])

-

W ¢ ~([B,D])

let G(W) = {V ¢ .~([A,C~): V n R(f(B))

For e v e r y

W c ~([B,D])

h: R([B,D])

~ R(f(B))

V ~ V/M(V),

V/M(V)

g(V) =

and

V ¢ G(W),

and the c a n o n i c a l

becomes

[V/M(V):

a

= h(W)]

v i a the c a n o n i c a l epimorphisms

(W/M(W))-vector-space

W/M(W)]

and

p(W)

=

isomorphism

W ~ W/M(W) and,

[W/M(W):

upon

(1)

[V/M(V):

H0(A) ] = g(V)p(W)

Now

([A,c],J) =

~

- ~

([A,c],~)

[ord([A,C],J,V)

- ord([A,C],N,V)][V/M(V):

VC~([A,C])

133

letting

H0(B) ] ,

we have

H0(A) ]

and

2.69

=

~ We ~([B,D])

=

~ [ord([A,C],J,V) V~G(W)

~

(i)

(18o 1 2 )

V e G (W)

=

[R([A,C]): R([B,D])]

=

[R([A,C]) : R([B,D]) ] ~ ([B,D],K)

=

[~ ([A,C])

~ [ord([B,D],K,W)]p(W) w~|[B,O])

by

(4.3)

~([Am,cN]) ]~* tt~AN, cN~,~). J

:

(23.15) b e given.

by

[ordvM(h(W)) ] [ o r d ( [ B , D ] , K , W ] g ( V ) p ( W )

~

W e ~ ([B,D])

by

- ord([A,C],N,V)]g(v)p(W)

SPECIAL

Assume

(Note that,

PROJECTION

that if

FORMULA.

{0] J N ~ ~

Let any

(A)

C N e ~I(AN).

H0(A)

is a l g e b r a i c a l l y

C N ~ ~ I ( A N) ~ E m d i m [ A , C , N ]

closed,

- Emdim[A,N]

then:

~ 2.)

T h e n we h a v e (23.15.1)

Deg[A,C]

- ~/ ([A,C],N) = [R ([A,C])

: R ([AN, cN]) ]Deg[AN, c N]

and Deg[A,C]

- ~

([A,C],N)

. ~ ([~,CN]) ]

= [~([A,C])

Deg[AN, c N] = 1 (23.15.2)

PROOF. taking

= 1

Emdim[A,C,N]

- Emdim[A,N]

In v i e w of

J -- xA

(23.15.1),

Emdim[AN,cN]

with

(23.12)

and

(23.16)

REMARK.

*

~*

(23.9),

x c

(23.15.1)

(N N H I(A))\C.

= 2 . follows Now

from

(23.15.2)

= {N ~

follows

by from

(14.3). Assume

that

H 0(A)

is a l g e b r a i c a l l y

Let

N

(23.14)

(A): E m d i m [ A , C , N ]

- Emdim[A,N]

134

= 2]

closed.

2~70

i.e., N

= [0 / N ~ ~

~'

=

(A) : E m d i m [ A N , cN] = i}

.

Let

Then clearly Deg[A,C~

{N

N' ~ ~

~

N

n0([A,c~)

:

and b y

n n0([A,Nl)

(23.15) w e

see t h a t

= m a x { [ R([A,C~) : R ( [ A N , c N ~ ) ~ : = [ R([A,C])

: R([AN, cN])]

= ~}

N e N*]

for all

N ¢ N'

W

(23.17)

LEMMA.

(Note that,

if

Let

[0} ~ N eg~ I(A).

H0(A)

Assume

is a l g e b r a i c a l l y

that

closed,

C N e ~ I ( A N) ~ E m d i m [ A , C , N ~

C N e nl(A).

then:

a 3.)

Let = {P e ~ 0 ( A ) :

N c P

and

[R([A, C3) : R ( [ A P , c P ~ ) ~ ~ i}

and the set o f a l l s u b f i e l d s o f

R(f(AN))

T h e n we h a v e

and w h i c h

R([A,C])

are d i f f e r e n t

which

from

contain

R([A,C])

the f o l l o w i n q :

(23.17.1)

card ~ ~ c a r d ~'

o

(23.17.2)

If

([A,C~,N)

(Although we

Deg[A,C]

- ~

s h a l l not u s e this r e m a r k

~ 3, ~he_D

card ~ ~ 1 .

in t h i s b o o k ,

we note that

w

by a well-known overfield

fact

of a field

from algebra K',

then:

(namely, K

if

K

is an a l g e b r a i c

has a primitive

element over W

K' ~

there

are o n l y a f i n i t e n u m b e r of s u b f i e l d s

K') we k n o w t h a t over

~(f(AN)).

~'

is f i n i t e ~

Hence by

(23.17.1) 135

R([A,C~)

of

K

containing

has a p r i m i t i v e

w e see t h a t

if

R([A,C3)

element has

a

2.71

primitive cular,

element

if

over

H0(A)

R([A,C~)

has

R(f(AN))

(i')

~

is finite.

zero c h a r a c t e r i s t i c ,

is s e p a r a b l e

PROOF°

then

over

R(f(AN)),

or, more

then

~

So,

in p a r t i -

generally

if

is finite.)

Clearly

R(f(AN))

c R(f(AP))

c R([A,C])

for all

W

P e ~0(A) and h e n c e

to prove

(23.17.1)

with

N c p

it s u f f i c e s

to show

that

w

(*)

card{P

In v i e w of

(i'),

e ~0(A):

(*)

R(f(AP))

c K} ~ 1

is e q u i v a l e n t

for e v e r y

K e Q'

to:

R(f(AN)) ( R ( f ( A P ) ) , R ( f ( A Q ) ) )

= R([A,C~)

,

(**) for e v e r y

P ~ Q

in

~(A)

with

N c P

and

N c Q .

w

TO prove given.

(**)

let any

P ~ Q

in

~0(A)

Then w e can take e l e m e n t s N + X A = p,

with

X,Y,Z

N + Y A = Q,

in and

N c P HI(A) Z £ N\C

and

N c Q

such that .

Now clearly R(f(AP))

= R(f(AN)) (f(X)/f(Z)) R(f(AQ))

and

= R(f(AN))(f(Y)/f(Z))

;

also c l e a r l y (N n H I(A))

+ X H 0(A)

+ Y H 0(A)

= H I(A)

and h e n c e R([A,C3)

= R(f(AN))(f(X)/f(Z),

f(Y)/f(Z))

consequently R(f(AN)) (R(f(AP)),

R(f(AQ))) 136

=

R ( [ A , C ~) ~

;

;

be

2.72

and this proves By

(**).

(23.15)

(R([A,C]):

w e have, R(f(AN))~

~ Deg[A,C~

- ~*

([A, C3,N)

and h e n c e Deg[A,C3

- ~*([A,C3,N)

[R(LA, C~q): R ( f ( A N ) ) 3

~ 3 = 1 or 2 or 3

= card ~' < 1 ; therefore

(23.17.2)

(23.18) (*)

follows

DEFINITION°

[V/M(V):

([A,C~,P)

(23.17.1).

Let

H0(A) 3 = 1

(Note that c o n d i t i o n U

from

= l, or: H 0 ( A )

P ~ Do([A, C3). for all

that

V c ~([A,C3,P)

(*) is a u t o m a t i c a l l y is a l g e b r a i c a l l y

*

Assume

satisfied

closed.

if, either:

Also

note

that:

W

(*) = P ¢ ~o(A),

Deg[A, P3 = l, and

U

([A,C3,P)

= ~([A,C],P).)

We define: W

*

*

TI([A, c I,P) = {L 6 ~l(A) : L c P We note

that,

and

~

*

(EA,C],L,P)

g ~

([A,C3,P)}.

then clearly, W

TI(EA,C~,P)

= {L ¢ ~ l ( A ) :

*

([A,C],L,P)

> ~

([A,C],P)]

(23.18.1) W

for some

= {L ¢ ~I(A) : L = TI([A,C~,V) v c ~([A, C3,P)} and h e n c e (23.18.2)

1 < card TI([A, C3,P)

and so in p a r t i c u l a r

137

< c a r d 0([A, C3,P)

<

2.73

(23.18.3)

u

([A,C~,P)

By a tanqent

1-flat

TI([A,C],P):

From

reference

A

to

Finally

of

= 1 = card

C

in

this p h r a s e

is c l e a r

we observe

A

TI([A,C3,P)

at

P

we may drop

f r o m the

=

1 .

we mean

a member

" in A ,r, w h e n

of

the

context.

that by

(23.2)

LEMMA.

Let

and

(23.6)

we get

(23.19)

and

(23.20) :

(23.19) Q c A

be

COMMUTING

such t h a t

P ¢ n0([A,C~)

n n0([A,D3)

U ([A,C~,D,Q)

(23.20) D ¢ ~II(A)

U ([A,C3,P)

and anv

([A,C],D,Q) in v i e w o f the

n ~0([A,Q3).

LEMMA.

Q ¢ H(A)

and

If

that,

U

then,

or

for all

([A,C~,D,Q)

Emdim[A,C]

and

Q ¢ H(A)

Then

o__rr Q c A

= ~ ([A,D],C,Q)

fact

and

= 1 = U ([A,D3,P)

= N ([A,C3,C,Q)

COMMUTING

D ¢ 91 (A)

=

= N

2, t h e n

([A,D),C,Q)

for a n y

we have

N

([A,C~,D,Q)

~([A,C3,D,P)

= N

is p r i n c i p a l

([A,D],C,Q) for all

P ¢ n0([A,C]).

§24. (24.1)

to

Let

Intersection

multiplicity

with

an a b s t r a c t

with

D i m R = i.

and any

Q ¢ H(R)

curve.

(24.17). R

be a homogeneous

For a n y

I ¢ H(R)

or

domain I c R

or

Q c R, w e

define

~(R,I,Q)

N(R,I,\Q)

=

~ ord(R,I,V)[V/M(V) V e ~ (R, Q) =

~ ord(R,I,V)[V/M(V) V ¢ ~ (R, \Q)

: R([R,~(R,V)])]

: R([R,~

(R,V)])]

-k

(R,I,Q)

=

~

o r d ( R , I , V ) [V/M(V)

V ¢ ~ (R,Q)

138

: H0(R)]

,

,

,

,

2.74

and

ord (R, I,V) [V/M (V) : H 0 (R) ]

(R, I, \ Q ) = V ¢ ~ (R, \Q) For

any

I ¢ H(R)

or

I c R, w e define: W

U (R,I)

For

= U (R,I,0)

any

and

Q ¢ H(R)

U

or

W

(R,I)

=

~ X~(~(R,P)) PCZ0 (R,Q)

~{(R,\Q)

=

~ X~(~(R,P)) P e n 0 (R, \Q)

=

~ k P e n 0 (R,Q)

,

,

(~(R,P)),

*

(R,I,0)

Q c R, w e define:

~{(R,\Q)

u~(R,Q)

= ~

where

k(9~(R,p))

k = H 0 (A)

where

k = H 0(A)

Pc~30 (R, \Q) Adj (R,Q)

=

[~ c H

(R): ~(R,~_,P)

¢ adj (~(R,P))

P C •0(R,Q)] Adj(R,\Q)

=

{~ ~ H

(R): ~(R,~,P)

,

C adj (~(R,P))

P ¢ ~0(R,kQ)] Tradj(R,Q)

=

[~ ¢ H * ( R ) :

for all

~(R,~,P)

for all

•

¢ tradj(~(R,P))

P c Z~0(R,Q)]

for a l l

,

and Tradj (R,\Q)

=

[~ ¢ H

(R): 9~(R,~,P)

¢ tradj (~(R,P))

for all

P e n0(R,\Q) ] B y an a d j o i n t

in

R

at

adjoint

in

R

outside

adjoint

in

R

at

adjoint

in

R

outside

these from

phrases the

Q, w e

Q, w e m e a n

Q, we m e a n

we may

mean

of

w e define:

139

of of

Adj (R,Q).

the

of

By an

Adj (R,\Q).

Tradj(R,Q),

a member

" in R ", w h e n

context.

Finally

a member

a member

Q, we m e a n

drop

a member

By

By a t r u e

Tradj (R,\Q).

reference

a true

to

R

From is c l e a r

2.75

~(R)

= H~(R, 0)

,

~(R)

= u~(R, 0)

,

Tradj(R)

;

and Adj(R)

= Adj(R, 0)

B y an a d j o i n t i n

R, w e m e a n a m e m b e r

i__nn R, w e m e a n a m e m b e r of

[A,C]

as Note

by

that

(24.14)

by

R N,

(24.18)

by

{0];

now g i v e s to

in a d d i t i o n

[AN,c N]

[V/M(V)

(R,P) = I, or:

and

H0(R)

to

and by true a d j o i n t

(24.13) w h e r e ,

A

b y the i d e n t i t y m a p of

(23.14)

Deg

to

R.

We

(23.17)

replacements, by

for

(23.i) b y r e p l a c i n g : f

from

~AP, c p]

DEFINITION.

Adj(R),

the d e f i n i t i o n

(24.17)

we

as w e l l R - R.

also g e t

respectively,

also r e p l a c e

R P.

Let

P e £0(R).

: H0(R)]

(Note that c o n d i t i o n H

from

to the a b o v e

and

(*)

(24.1)

is o b t a i n e d

R; C

(24.9)

assertions where,

(24.i)

of

Tradj(R).

We clearly get assertions 1 ~ i ~ 13,

= Tradj(R,0)

= 1

Assume

for all

V e ~(R,P)

(*) is a u t o m a t i c a l l y is a l g e b r a i c a l l y

e

that

satisfied

closed.

if, e i t h e r :

A l s o n o t e that:

w

(*) = P e ~ 0 ( R ) '

Deg[R,P]

= i, and

U

(R,P) = U (R,P).)

We define

TI(R,P)

= {L e ~I(R) : L c P

and

U

(R,L,P)

~ U

(R,P)}

W e note t h a t t h e n c l e a r l y *

TI(R,P)

*

w

= [L 6 ~l(R) : U

w

(R,L,P)

> U

(R,P)]

w

= [L e 9]I(R): L -- TI(R,V)

for some

and h e n c e (24.18.2)

w

1 ~ c a r d T I(R,P)

and so in p a r t i c u l a r

140

~ card ~(R,p)

< ~

V ¢ 0(R,P)]

2.76

(24.18.3)

By

a %angent

this

phrase

from

the

U

1-flat

drop

REMARK.

note

then,

g(x)

=

~ H(A))A)

the

=

1 .

a member

of

reference

TI(R,P).

to

R

From

is c l e a r

any

upon

letting

for

any

C ~ ~I(A)

any

be

f: A ~ A / C

x c H(A),

J c A, w e

given.

clearly

and

to b e

the

canonical

g (J) =

have:

= ~ (f (A) ,g (I) ,g (Q) ) ,

([A,C],I,\Q)

= ~ (f(A),g(I),\g(Q))

,

for

any

(f (A) ,g (I) ,g (Q) ) ,

and

any

or

Q cA

I ¢ H(A)

or

I c A

w

([A,C],I,Q)=

U

*

Q ~ H(A)

w

([A,C],I,\Q)=

([A,C],I)

U

(f(A),g(I),\g(Q)),

1

= ~ (f (A) ,g (I) ) ,

for

w

U

TI(R,P)

we mean

" in R " w h e n

f(x)

for

w

U

1 = card

a__tt P

Let

that

([A,C],I,Q)

U

R

=

context.

epimorphism, f((J

in

we may

(24.19) We

(R,P)

w

([A,C],I)=

U{([A,C],Q)

U

(f(A),g(I))

= ~/~(f(A),g(Q))

U~([A,C],kQ) U~([A,C],Q)

= u~(f(A),kg(Q)) = u~(f(A),g(Q))

U[([A,C],\Q)

,

= u~(f(A)

,

~ ¢ Adj ( [ A , C ] , \ Q ) ]

[f(~):

~ e Tradj ([A,C],Q)]

,

= Adj ( f ( A ) , \ g ( Q ) )

,

Q cA ,

= T r a d j (f (A) ,g (Q) ) ,

= ~(f(A)),

w

w

U~([A,C])

= u~(f(A))

•

[f(~):

• ¢ Adj([A,C])]

[f(~_):

~ c Tradj([A,C])]=

I c A

= Adj(f(A))

,

Tradj(f(A)) 14'I

any

Q ~ H(A)

[ f (4) : • e T r a d j ([A, C], \Q) ] = T r a d j (f (A) , \g (Q)) ,

~([A,C])

or

for

= Adj (f(A),g(Q))

[f(~):

I c H(A)

•

,\g (Q))

[f(~) : ~ ~ Adj ( [ A , C ] , Q ) ]

any

:

or

;

2.77

and for any {f(L) : L ¢ T I ( [ A , C ] , P ) ] = Tl(f(A),f(p))

We

also

note

~([A,C],I,Q)

1

[V/M(V) : H 0 ( A ) ]

for all

•

if

that,

C = [0],

= ~(A,I,Q)

~(EA, C],I,\Q)

such

that

= 1

V ¢ ~([A,C],P).

then clearly:

,

= ~(A,I,\Q)

*

P ¢ 20(CA,C])

f o r any

I e H(A)

or

IcA

and

Q c H(A)

or

QCA;

w

U

([A,C],I,Q)=

•

H

([A,C],I,\Q)=

(A,I,Q)

,

any

W

~(EA,C],I)

~

(A,I,\Q)

= u(A,I)

, for any

([A,C],I)

= U

U~([A,C],Q) U~([A'C]'hQ)

(A,I)

,

= u@(A,Q)

,

= ~(A,\Q)

U~([A,C],Q) W

Adj ([A,C],Q)

= Adj (A,Q)

Adj([A,C~,\Q)

= T r a d j (A,Q)

Tradj ([A,C],\Q)

,

= T r a d j (A,\Q)

[I~([A,C])

= ~(A)

,

U~([A,C])

= u~(A)

•

= Adj (A)

T r a d j ([A,C])

,

= Adj(A,\Q

T r a d j (EA,C],Q)

A d j ([A,C])

f or any

.

= u~(A,\Q),

or

I c A

;

,

-- u ~ ( A , Q ) ,

U~([A,C~,\Q)

I e H(A)

,

= T r a d j (A)

and

142

Q e H(A)

or

QcA;

2.78

%

*

*

TI([A'C]'P)

|

= TI(A'P)I

for any

for all (24.20) rational over e ~ e'

V ¢ ~(A,P)

LEMMA ON OVERADJOINTS. H 0(R).

such that

P ¢ Z]0(A)

[V/M(V): H0(A)] = 1

Let

V

Given any nonnegative

~(R)

¢

be residually

integers

m,e,e'

with

let

E(m,e) = {x , Hm(R): ord(R,x,V)

~ e + or~(~(R,O*(R,V))), ord(R,x,W)

and

~ or%~(~(R,~

whenever

(W)))

,

V F6 W e ~(R)}

and similarly E(m,e') = Ix ¢ Hm(R): ord(R,x,V)

~ e' + ordv~(~(R, ~ (R,V))), ord(R,x,W)

and

z or%~(~(a,~

whenever

(R,W)))

V @ W ¢ ~(R)}

Then:

(i)

E(M,e) c E(m,e')

are

[E(m,e): H0(R) ] + e + (2)

H0(R)-vector ~

P~o (R)

> [E(m,e'): H0(R) ] + e' +

subspaces of

U

(R,x) ~ e + u~(R),

~ [~(R,P)/~(~(R,P)): PCno (R)

for all

x ¢ E(m,e)

i

and

(4)

(R,x) m e' + u~(R), PROOF.

for all

Let P0 = ~ (R,V)

and

143

,

[~(R,P)/~(9%(R,P)): H0(R) ]

[Hre(R) : H 0(R) ] , (3)

Hm(R)

x ¢ E(m,e')

H0(R)]

2.79

• (R,P 0) Then

in v i e w of

b y taking,

= the

(5.12),

for all

integral

of

~ ( R , P 0)

the f i r s t two a s s e r t i o n s

D M(V)) e

I

E(~(R,P))

and

]

I' (P) =

in

R(R)

follow from

(15.10)

P c ~0 (R) ,

E(~(R, P0)) (~(R,P 0)

I(P) =

closure

E(~(R(R, P0 )) (~(R,P0 )*

N M(V)

if

P = P0

if

P @ P0

)e'

[ E(~(R, P0))

if

P = P0

if

P ~ P0

"

W

The

last two a s s e r t i o n s

(24.21)

(*)

Let

~ 2

n = Deg R, a n d let for e v e r y

(**)

(***)

from the d e f i n i t i o n

L E M M A ON U N D E R A D J O I N T S .

U (R,P)

s(P),

follow

for all

~

[Hm(R):

~

for a l l

(I/2)~E(R)

-

ideal

-

Tradj (R,\n)

Moreover,

N H;(R)

/ ¢

i_~f m ~ 2

and

n / 0(2)

144

.

Let

such that.

P e ~0(R,~)

s(P)Deg[R,P] E pe~ 0 (R,~)

E 2s(P)Deg[R,P] P e ~ 0 (R, ~)

Then

R.

inteqers

W

m n ~ u~(R)

in

that

and (****)

u

~0(R)

be n o n n e g a t i v e

(I/2)uE(R,P)

H0(R) ] >

e

be a homogeneous

P ¢ ~0(R,~),

s(P)

P

Assume

of

.

m

and

2.80

then there

exists ¢ Tradj (R,\~)

such that

~

is i r r e d u c i b l e

® --- ~i~2 PROOF.

with

®i

~ Hm(R) in the

and

sense

~2

that:

i__nn H

(R) = ~i = R

o__rr ~2 --- R.

Let k = H 0 (R)

In v i e w of (i)

(*), b y

(i0.i.i0)

[~(R,P)/~(~(R,P))

and b y

(10.1.13)

I

: k3 =

we h a v e

w e H

we h a v e

(I/2)u~(R,P)

for all

P ¢ n0(R)

w < u~(R,P)

for all

P ¢ n0(R)

that, w

(R)

with

U

that

(R,~,P)

(2) w

= U

(R,~)

-= 0(2)

Let (3)

~2 = [P e •0(R,rr): k~(R,P) ~ 2}

Then b y by

(*) we h a v e

= 0

for all

P e £0(R,I~)\Q,

and h e n c e

(**) we get

(4)

s(P) = 0 For each

ideal (5)

#~(R,P)

.

J(P)

P e. Q, in

in v i e w of

~(R,P)

[~(R,P)/J(P)

for all

with

P e ~0(R,~)\Q

(**), b y

~(~(R,P))

: k~ = [ ~ ( R , P ) / ~ ( ~ ( R , P ) )

(10.1.12) c J(P)

there

exists

an

such that

: k~ - s ( P ) D e g [ R , P ~

and (6)

xk(9~(R,P),J(P))

For each

P e D0(R)

= k~k(~ (R,P))

- 2s(P)Deg[R,P~

we get an ideal

t45

I(P)

in

~(R,P)

by setting

2.81

(7)

J(P)

if

P e

~(~(R,P))

if

P ~ ~ o

I(P)

By

(i),

(3),

(4),

(8)

(5

and

(7)we

then have

: k] =

[~(R,P)/I(P) Pen 0 (R)

( 1 / 2 ) U ~ (R)

s(P)Deg[R,P]

-

.

Pcn0(R,~) Upon

(9)

letting

E = {x ¢ Hm(R):

by

(15.10)

(io) By

we see that

[E:k] +

(***) ,

9~(R,x,P) E

(8) and

for all

P ¢ f]0(R)]

is a k - v e c t o r - s u b s p a c e

~ [9(R,P)/I(P) p c ~ 0 (R)

: k]

of

z [Hm(R)

Hm(R)

: k]

with

.

(I0) we get

(ii)

[E:k] Upon

¢ I(P)

> 0

letting *

(12) by

A = [@ e Hm(R):

(9) we o b v i o u s l y U

(R,~,P)

and h e n c e

in v i e w of

(i3)

x ¢ E]

(3),

@ ¢ A

(4),

for all (6) and

e ~ A

(7) we

and

P ¢ ~]0(R)

see that

we have:

~ ~(R,P)

- 2s(P)Deg[R,P]

for all

P ¢ 90(R,~)

for all

P ~ ~]0(R'~)

and u

By

(R,G,P)

for some

get

m xk(~(R,P),I(p))

for e v e r y U

@ = xR

(R,~,P)

>_ ki{(m,P)

(13) we get

146

2.82

*

(14)

w

U

(R,®)

~ u~(R)

-

2s(P)Deg[R,p]

for all

® c A .

P¢~0 (R, ~) By

(3),

(7),

(9) and

(15) By

(12) w e

A c Adj (R,\~)

(9),

(Ii) and

A ~ ~ •

In v i e w of

(12), b y B e z o u t ' s

(17

u

(****),

(13),

U

( 18

(R,e,P)

Little

(R,®) = m n

(14) and

for e v e r y

(17)

® g A

Theorem

for all

(24.9) w e a l s o h a v e

~ g A .

it f o l l o w s

that

w e have:

= ~(R,P)

- 2s(P)Deg[R,p]

for all

P g ~0(R, TT)

for all

P c ~0(R,\n).

and U

By

A Hm(R)

(12) w e h a v e

(16)

By

also get

(R,®,P)

(15) and

(18) w e see t h a t

(19)

A c Tradj (R,\n)

NOW 2.

= u~(R,P)

let

In v i e w of

® ¢ A

such that

(2) and

= ®i~2

~ = ~i®2

with

®i ~ H

(R) for i = i,

(18) w e see that, U

and hence by Bezout's

N Hm(R)

(R,® i) =- 0(2)

Little

with

Theorem

® g A

and

,

(24.9) w e g e t t h a t

®i e H

(R)

for

i = 1,2

(20) = n ( D e g R ® i) =- 0(2) Now,

(21)

if

m ~ 2

and

D e g R ® i = 0(2),

and

for

i = 1,2

n # 0(2),

.

then by

(20)

DegR~ 1 + DegR® 2 ~ 2

147

it f o l l o w s

that

2.83

It

follows

®2 = R.

This

that

together

§25. Assume

we

with

Emdim

and

cones

A = Dim

DEFINITION.

for

(16)

Tangent

that

(25.1

DegR@ i = 0

For

some (19)

and

and hence

finishes

the

@I = R

or

proof.

quasihyperplanes.

A = r

any

i

.

• ~ H

(A)

and

any

P ~ ~[A,~])

define

([A,~],P)

We

note

that

=

k' ([~ ( A , P ) , ~ (A,~,P) ] , M ( ~ (A,P)))

then:

~' ( [ A , ~ ] , P )

= a nonnegative

U' ( [ A , ~ ] , P )

-- 0 ~ ~ ~ P

U' ( [ A , ~ , P ] )

and

so,

if

• ¢ e(A),

=

integer

,

,

1 ~ ~(A,P)/~(A,~,P)

is a r e g u l a r

local

domain

,

in p a r t i c u l a r ~

then:

U' ( [ A , ~ ] , P )

=

1 ~ • c P

Let

4} C H

and

~([A,~],P)

is

regular.

(25.2)

LEMMA-DEFINITION.

given.

We

can

take

P + XrA

= }{I (A)A.

Now

d

be

the

we

unique

integer

~i = 0

We the

claim

choice

of

Namely,

that ~ let

can

n n-i ~ ~iXr i---0

= Let

~ c H (A)

for

<0dA a nd~

any

with

(A)

~ A = ~,

uniquely

and

<0i e H i ( A P )

with

0 ~ d ~ n

(whence

P c ~ 0 (A)

X r e H 1 (A)

be with

write

with

0 ~ i < d,

and

and

such

~d ~ 0

that

.

in p a r t i c u l a r

d)

is

with

and

any

independent

x r~'

~ H(A)

148

<0'A = ~

X r c HI(A)

of

2.84

with

P + XrA = H I(A)A

b e given,

and

let

n ~0' = be

the

x'n-i r

)~ Mi' i=0

corresponding

with

expression.

<0' = a
with

' e H 4 ( A P) ~i

Then

0 @ a e H 0(A)

and w

X'r = b X r

Substituting

*

+ X

with

in the a b o v e

0 ~ b

e H 0(A)

expression

of

and

M'

X

H 1 (A P)

we get

n ~' = aM =

~

a ~ i ( b X r + x*)n-i

i=0 and h e n c e

it f o l l o w s

' = 0

which

proves

that

for

our

0 < i < d,


~ 0, and

M ~ A = ~0dA ,

claim.

We define T([A,~],P)

and w e note

that

then

= %0dA

T([A,~],P)

£ Hd(A,P) w

(25.3)

LEMMA.

Let

• e Hn(A),

Then we have

d = D e g A T ([A,~],P) .

(25.3.1)

U' ([A,~],P)

(25.3.2)

For ~

= d

P e~0(A),

the

and

followinq

.

w

T([A,~],P)

(25.3.3)

L 69~r_l(A)

c L ~ ~ (A,~,P)

~ 6

H* (A,P)

with

c ~ (A,L,P)

L c P

+ M(~ (A,P)) d + l

~ d = n ~ • = T([A,~],P)

~P ~ ~(A P) ~ P ¢ (25.3.4)

If

~ ~ H

we have :

(A,P)

and

149

{o}

~ ¢ ~(A),

then

[~([A,~])

: ~([AP,~P])]

=

n

-

d

.

2.85

PROOF. over in

k. A

Let

and let

W e can t a k e a h o m o g e n e o u s

such t h a t

= ~A

where

k -- H 0(A)

P =

coordinate

(X0,X 1 ..... X r _ I ) A

with

indeterminates

(X0,X 1 ..... X r)

and t h e n we h a v e

~ k [ Y 0 , Y 1 ..... Y r _ l ]

polynomial

be

system

n ~ ~ i ( X 0 ' X l ..... X r i=d

~ =

~0i(Y0,Y 1 ..... Yr_l)

nonzero homogeneous

Y0'YI '''''Yr-I

of d e g r e e

i

1 )xn-1

is e i t h e r for

zero or a

d ~ i ~ n ; we also

have

~0d(Y0,Y 1 ..... Y r _ l ) ~ 0 (25.3.3) ~ H

(A,P)

• P = [0}

and

is n o w o b v i o u s . and

~ ¢ ~(A).

To p r o v e Then

; so in p a r t i c u l a r

ical e p i m o r p h i s m .

T([A,~,P)

= ~ d ( X 0 , X l ..... X r _ I ) A

(25.3.4),

assume

that

r > 0, A p = k [ X 0 , X 1 ..... Xr_l~,

X 0 (~ ~.

Let

f: A ~ A / ~

and

Now

= R ( f ( A m )) ( f ( X r ) / f ( X 0))

f ( X r ) / f ( X 0)

and

be the c a n o n -

R (f(A m )) = f(k) ( f ( X l ) / f ( X 0),f (X2)/f(X0) ..... f ( X r _ l ) / f ( X 0)) R([A,~)

.

satisfies

the

,

,

irreducible

equation

n [f([0i(X0,Xl ..... X r _ l ) ) / f ( x 0 ) i 3 [ f ( x r ) / f ( x 0 ) ~n-i = 0 . i=d of degree

n - d

[R([A,~)

which

proves

R ( f ( A P)).

: R([AP,~P~)~

= n - d ,

(25.3.1)

(25.3.4))

and

that

(25.3.2),

~ ~ H

(A,P)

A' = k [ X 0 / X r , X l / X r ..... X r _ I / X r ~ N o w the e l e m e n t s ent o v e r

! k, Ap,

coefficient

Therefore

(25.3.4)

To p r o v e proving

over

and

w e d r o p the a s s u m p t i o n and P' =

Xo/Xr,Xl/X r ..... X r . i / X r is an r - d i m e n s i o n a l

set for

i Ap,

, M(A~,)

=

150

• G ~(A).

(made for

~et

(Xo/Xr,Xl/X r ..... X r _ I / X r ) A ' . are a l g e b r a i c a l l y

regular

l o c a l ring,

k

independis a

( X 0 / X r , X l / X r , . . . ,Xr_±~/Xr)A!,~

,

2.86

and

(in v i e w o f

(15.1),

(15.2)

*

and

*

(A,~,P) --- ~0 ~ (A,P) w h e r e

(15.4) w e h a v e n

~

~(A,P)

! = Ap,

and

n

-- ~0/Xr =

~0i (X0/X r •X l / X r ..... X r _ i/Xr) i=d

Therefore let a n y

U' ([A,~],P) L c ~ r - I (A)

= d

which proves

with

(a0X0+alXl+...+ar_iXr_l)A at l e a s t o n e o f w h i c h

L c P

(25.3.1).

b e given.

where

Now

To p r o v e

(25.3.2),

L =

a 0 , a I .... ,ar_ 1

are e l e m e n t s

In v i e w o f

and

is not zero.

(15.1)

in

(15.2) w e

have

(A,L,P) = and hence

it f o l l o w s t h a t

T([A,~),P)

which

(a 0 ( x 0 / x r) + a l ( X l / x r ) + . . . + a r _ I ( x r _ I / X r ) ) ~ ( A , P )

proves

(25.4)

c L ~ ~(A,~,P)

c ~(A,L,P)

+ M(~(A,P)) d°~l

(25.3.2).

DEFINITION.

Let

~ ¢ HI(A)

and

~ 6 Hn(A).

We c a l l

w

~

a ~-~uasihvperplane

in

A,

if,

for

some

P ¢ ~o(A)

with

~ c P,

we have

U' ([A,~],P)

= n-I

We n o t e that then,

and

T([A,~],P)

in v i e w o f

is a ~ - q u a s i h y p e r p l a n e

in

= n-i

(25.3.1),

w e have:

A

t

i

for some = n-

and

P ~ [10(A,~)

Deg[A,P~

-- 1 w e h a v e u' ([A,~],P)

1 n(A,T([A,#],P))

F r o m the p h r a s e case

with

= ~(A,~)

"~-quasihyperplane

in A

" we may drop

"hyper"

in

r = 3.

(25.5)

LEMMAo

~-quasihyperplane ordinate

syste m

in

Le__tt ~ ¢ HI(A) A ~ n > 1

X0,Xl,.o~,X r

and

and t h e r e i__nn A

151

~ 6 Hn(A). exists

such that

Then

~

a homogeneous ~ = X0A

and

is a co-

k

2.87

=

n-i X r + <0n)A w i t h (X 0

~n

¢ Hn(AP)

PROOF.

Obvious

in v i e w

of

(25.6)

LEMMA.

I__~f ~ ¢ HI(A)

where

(X0,X 1 .... ,Xr_l) A

P =

(25.3.1).

and

~

e HI(A)

are

such

that

w

~,

~¢

then

PROOF. ordinate

H2(A)

Clearly

system

= XrA.

Now

~

~

~=

(X0Xr)A in

LEMMA.

is a ~ - q u a s i h y p e r p l a n e

H2(A),

X0,Xl, .... X r

a ~-quasihyperplane

(25.7)

and

and w e

in

A

can t a k e

such

and h e n c e

by

A.

a homogeneous

that (25.5)

in

~ = X0A we

co-

and

see t h a t

~

is

A.

Suppose

r = 3

and

let

~ e HI(A)

and

~ ~ H2(A)

w

be

such

that

L ¢ l~(A,rr)

is a ~ - q u a s i p l a n e with

N ~(A,~5)

in

A,

for

o__rr • ¢ H

some

L ¢ ~I(A).

(A,P)

for

some

Then either P ¢ 9~0(A )

L c P.

PROOF. HI(A)

Since

such

that

L ¢ ~ (A,n), ~ = X0A

and

we

can t a k e

L =

elements

(X0,Xl)A.

Since

X0

and

X1

L ¢ ~ (A,~),

in we

get

=

(X0X 3 + X I X 2 ) A

with

X2

and

X3

in

H I(A)

If

then,

upon

taking

~-quasiplane

in

A.

[ (X0,XI,X2,X3)H0(A)

: H 0(A) ] > 3

P =

(25.5)

(X0,Xl,X2)A,

by

we

see

that

~

is a

If

[ ( X 0 , X l , X 2 , X 3 ) H 0(A)

: H 0(A) ] { 3

w

t h e n we hence

can t a k e L c P

(25.8) P ¢ ~]0(A)

P eD~0(A)

and

LEMMAo w e have:

such

that

Xi ¢ P

for

0 { i m 3, and

~ ¢ H* (A,P).

Suppose

r : 2.

~' ([A,C],P)

Then

for any

= ~ ([A,C],P).

152

C ¢ QI(A)

and

any

2.88

PROOF.

Follows

from

(25.9)

LEMMA.

SuPPose

Deg[A,C]

(23.1) and

r = 2.

(23.2).

Then for anv

we have

C e £1(A)

= DegAC.

PROOF. DegAC = 1 (23.12).

Take any and

If

[R([A,C])

and hence by (25.10)

T([A,C],P)

Moreover, (*)

(9.1),

if

P ¢ ~0(A).

Emdim[A,C] C ~ H

then by

:R([AP,cP]) ] =

(A,P)

(25.3) and

(degAC)

(23.15) we get

Deg[A,C]

= l by

(25.8) we have

r = 2

= degAc. and l e t an Z

C ~ ~ I (A)

Then for an•

L e 9~1 (A)

with

([A,C3,L,P)

~U([A,C],P)

.

P e •0([A,C])

then clearly

- ~/ ([A,C],P)

Deg[A,C]

Suppose

= L ~U

[V/M(V)

C ~ H

= i, and hence also

(A,P)

LEMMA.

If

L C P

and any_ we have

an__~d

: H0(A) ] = 1

for all

V 6 ~ ([A,C],P)

,

the____nn W

= • lI(A,T([A,C],p) ) -- T I ( [ A , C ] , P ) .

£ i (A'T([A'C]'P)) (Note that condition U

(*) is a u t o m a t i c a l l y

([A,C],P) = i, or PROOF.

In view of

first assertion. with

L c P

H0(A)

is a l g e b r a i c a l l y

(23.18),

if either

closed.)

the second assertion

To prove the first assertion,

be given.

follows

let any

from the

L e~l(A)

Now

*

*

([A,C],L, P ) =

(25.3)

and

by

(~FA,L],C, P) = or%

and by

satisfied,

([A,L],p)~ ([A,L],C,P)

(25.8) we have

153

by

(23.20)

(23.1) and

(23.7)

2.89

T([A,C],P)

c L ~ ord~([A,L],p)~([A,L],C,P)

> W ([A,C],P)

Therefore w

T([A,C],P) (25o11)

c L ~ ~

DEFINITION°

P ¢ ~ r - 2 (A)

([A,C~,L,P) For any

~

> ~([A,C],P) and

~

in

H

(A)

and any

we define

([A,~,~],P)

= X([~(A,P),~(A,~,P),~(A,~,P)

], M ( ~ ( A , P ) ) )

and w e note t h a t then: U ([A,~,~],P)

= u([A,~,~],P)

u([A,#,~],P)

=

= a nonnegative

0 = P In(A,~)

integer or

~ n(A,~)

= ,

,

and U ([A,~,~],P)

= = ~ P G n(A,P') P' e n r _ l ( A , ~ )

(25.12) P c Z r - 2 (A)

LEMMAo

F o r any

PROOF°

Follows

(25.13) D ¢ H

n ~r_l(A,~)

~, ~'

and

~

i__nn H

(A)

a__nnda n t

we have

u ( [ A , ~ ' ,@],P)

w

for some

PROOF°

Suppose

from

BEZOUT'S

+ u ([A,~',~],P)

(8.3). r = 2.

P ¢ ~0(A),

Follows

(25.14)

from

LEMMA.

(A), a n d a n y

= u ([A,~,~],p)

The n for any

we have

u([A,C,D],P)

C ¢ ~I(A),

= u([A,C],D,P)

(23.6).

THEOREM

FOR PLANE CURVES.

Suppose

(A)

N ~I(A,D)

r = 2.

w

Then

for any

C

and

D

any

i__nn H

with

~I(A,C)

we have

~ ([A,C,D],P)Deg[A,P] P c ~ 0 (A)

154

=

(degAC) (degAD)

=

2.90

PROOF.

§26.

Follows

from

2-equimultiple Assume

closed.

that

Let

(23.9),

plane projections

TT e HI(A).

Let

(26.9).

(26.10).

(of

C

in

A

of

projection

C

from

is ~-integral, for some

C

in

n = Deg[A,C].

(26.1)

to

some suggestive

terms.

they are to be in force

center

mean a member

N

of

< 2

~ 2

center

~0([A,n],\C )

C

in

Q c

A

and

relative to U ([A,C],N)

the p r o j e c t i o n

for all

of

L

TT) we

= I, the of

C

from

Q e f~0 (AN),H ([AN, cN],Q0 )

Deg[AN, c N] = n - i.

(for

C

in

A

such that:

the p r o j e c t i o n

for all

we mean a m e m b e r

such that:

Q0 ~ ~0 ([AN'CN]'TTN)"

is birational,

U ([AN, cN],Q)

(for

is birational,

U ([AN, cN],Q)

TT)

L

= d.

~0([A,TT],C) N

~) we mean a member

lying on

~([A,C],L)

By a better projeqtin~

N

i.e.,

lying on

(of

N

of

A

~ d.

mean a member

from

first introduce

U ([A,C],L)

~II(A,TT) such that

= 2

and let

we shall not even use them in the statement of

By a c[oQd projectinq

N

is a l g e b r a i c a l l y

and its detailed version

are meant to be local,

such that

By a d-secant of

(26.11)

space quintics.

(26.11).

By a d-chord I(A,TT) 91

H 0 (A)

(25.13).

These will be preceded by Lemmas

only in this section; Theorem

and

C e ~I(A,\~)

To avoid repetition we

These definitions

(25.12), and

of p r o j e c t i v e

Emdim A = Dim A = 3

We want to prove Theorem Proposition

(25.9),

of

C

relative to

the p r o j e c t i o n from

N

~) we of

C

is TT-integral,

~0(AN), u([AN, cN],Q 0) = 2

for some

Q0 ~ £~0 ([AN'CN]'TTN)" and Deg[AN, c N] = n. By a best projecting mean a member from

N

N

of

(for

[~o([A,~],\C )

is birational,

U ([AN, cN],Q) = 1

center

C

such that:

the p r o j e c t i o n of

for all

in

C

A

relative

the p r o j e c t i o n from

Q ~ ~0([AN, cN],\~N),, and

155

to

N

~) we of

C

is TT-integral, Deg[AN, c N] = n.

2.91

(26. i)

LEMMA.

Given

any

d = U ( [ A , C ] , P 0)

PO

¢ ~30 (A, ~) , l e t

•

A =

[L ¢ n l ( A , ~ ) :

T =

[L ¢ A

: L

L c Po} is

, and

a d-secant}

Then card(A\T) Moreover, then

if

there

is

exists

PROOFo card

b

By

Now

=

card

{L

any

card

F

inteqer

a b-chord

< ~.

L

: L =

Now

6 A

that

b

~ d + 1

and

b

< n,

•

Theorem

(23.9)

we

know

that

let

&(A,P0,P)

[~0([A,C~,~)

.

such

Bezout' s Little

~0([A,C~,TT)

F

< ~ =

< ~

for

some

implies

P c ~0(FA,C~,TT)\[P0}}

that

, card

Now

obviously,

(or

say

F

<

in v i e w

of

F A~F

if

Hence

take

b

*

U

( T I ( [ A , C ~ , P O)

card(A\T)

Now

let

~

d + 1

L

d =

we

have

0

-*

If

(23.18))

any

to b e

< =

integer

then any

we

and

b

would

member

of

N n

1

since

be

card

given

have

b

F.

If

156

(A,~))

such < d

, if

A =

~

that and

b = d +

d ~

we

b

0

must

< d +

so

it w o u l d

1

and

have

1

card

and suffice

b

F

< no to

2.92

~ {p0 }

n 0([A,c],~) it s u f f i c e s

to t a k e

n 0([A,C],~) we

then there exists L = ~(A,P0,P). then:

c {p0}

see t h a t

n0([A,C],n)

there

= {Po }

exists

ord([A,C],n,V) and h e n c e

Finally,

if

b = d + 1

Little

and

and

Theorem

(23.9)

and

= n > d = U ([A,C],P 0)

V ¢ ~ ([A,C],P 0)

such t h a t

> ord([A,C],P0,V )

upon letting

L = TI([A,C],P0,V) in v i e w of

now obviously

L

and

(or say in v i e w of

U ([A,C],L) and h e n c e

,

(21.1), w e h a v e

L ¢ nl(A,~)

P0 e ~ 0 (A'L)

> U ([A,C],P 0) = d = b - 1 ,

is a b - c h o r d .

LEMMA.

If

PROOF.

Clearly

(or say b y B e z o u t ' s

P0 e ~ 0 (A'TT)

follows by

;

(23.18))

(26.2)

exists

P ¢ ~o([A,C],~)\{Po}

in v i e w of B e z o u t ' s

U ( [ A , C ] , n , P O) = U ([A,C],~) Consequently

some

n ~ 2, t h e n t h e r e

with

exists

Little

a 2-chord

Theorem

L.

(23.9))

there

U ([A,C],P 0) >_ i, and h e n c e o u r a s s e r t i o n

(26. i) .

(26.3)

Assume

(*)

that

n ~ 4

~

and

[u ([A,C],P)

- 1] ~ 2

p~n 0 ([A,C],~) Then

there exists

PROOF. our assertion

If

a 4-chord

L.

U ([A,C],P 0) > 3

follows

from

for some

(26.1).

157

If

P0 e n 0 ( [ A , C ] , v ) ,

U ([A,C],P)

< 3

then

for all

2.93

P c £o(EA, C],~) bers

Pl

then,

and

P2

in v i e w

of

now

it c l e a r l y

(26.4).

(*) Then

that

u([A,C],~,P)

there

exists

PROOF. 3-secant,

By

then

Then,

Let

PI,P2,P3

(*),

by Bezout's

~o(A,L')

be

= 2.

So n o w

If

also

must have dicated

there

Little

to take

Let

Theorem

N2

and Let

L

is not

assume

that

N 3 e ~o(A,L*).

b y the

exists

the d i s t i n c t

following

N3 L

that

is not a 3 - s e c a n t .

a 2-chord L = L'

members

of

(23.9)

we

be

get

i = 1,2

158

L'

get

figure

that

a

L'

In v i e w

of

card ~0([A,C],n)\ members

Then

Then

of

clearly

it s u f f i c e s

let

is n o t

c a r d ~ o ( [ A , C ] , L ') = 3.

the d i s t i n c t

then

If

~o([A,C~,L').

is a 3 - s e c a n t .

suggestive

L'

So n o w a s s u m e

= A(A, PI,N2).

For

mem-

P e ~o(A,~)

(*), w e

a 3-secant L

two d i s t i n c t

and

which

of

find

L = &(A, PI,P2).

for all

L

in v i e w

~o(EA,C],~)\£0(A,L'). 2-chord.

~ 1

it s u f f i c e s

is a 3 - s e c a n t .

n = 5

can

i = 1,2,

to take

a 2-chord

(26.2)

such

a 2 , for

suffices

Assume

(*), w e

~o([A,C],~)

U ([A,C],Pi)

and

of

L

is a

to take

in v i e w

Li ~ £ iI(A,~)

of

L = L

.

(*) we be

as in-

2.94

| l

L2 %% L*I ~

N

.....

n I

P1

fl ~'/

$

% % %

i.e.,

let

2-secant

L i = A(A,Ni,Pi). for

i = 1,2.

REMARK.

Note

Namely,

Q0(A,~)

and we are

looking

So

Then

in v i e w

it s u f f i c e s

that this

lemma

of

to

(*) w e

take

is j u s t

see t h a t

L = L1

or

an e l e m e n t a r y

Li

is

L = L2 .

combinatorial

fact:

n,

contain

exactly

distinct,

lines

the

five points

and

since

the ten

ten

to choose then

(n-l)-chord.

There

from.

it a p p e a r

is n o t d i v i s i b l e does

LEMMA. Then

not

Assume L

five d i s t i n c t

for a l i n e w h i c h

three points.

lines which

(26.5)

is a set o f

is a n

joins

are

2

If a l i n e exactly

contain

that

contains

Emdim[A,C7

(n-l)-secant,

159

times

there must

exactly

in t h e p l a n e

two o f t h e m a n d d o e s = i0,

three

by three,

points

three

not

not

necessarily

exactly in t h e exist

three of

ten

lines;

a line

in

points.

= 3, and

let

L

be

any

2.95

card n 0 ( [ A , L ] , C ) and every

N ¢ ~0([A,L],\C)

PROOF. that

is a best projecting

The assertions

L ~ ~iI(A,~)

and

(i)

about card

center.

follow from our assumption

C ~ ~I(A,\~).

Now let any

N ¢ n0([A,L],\C)

be given.

Then clearly

C N ~ ~I(AN),

that the p r o j e c t i o n of Emdim[A,C]

Since

N

for any ~

~ n

(18.13.1)

is w-integral.

U ([A,C],~)

we also know

Since

(23.9) we get

e HI(A)

we see that,

for any

J e ZI(A)

A(A,L,J)

e HI(A)

U ([A,C],L) because

from

Little Theorem

N ~ ~0(A,C),

(3)

C

and by

= 3, by Bezout's

(2)

with

N e ~0(A,J)

and

J ~ L, we have

and

+ U ([A,C],J) ~ u([A,C],a(A,L,J))

clearly

L c ~](A,A(A,L,J)),J By

< ~ = card Z 0 ( [ A , L ] , \ C )

(2) and

e ~(A,h(A,L,J)),

and

n0(A,L)

N n0(A,J)

= IN].

(3) we get that

I for every

J ¢ ~iI(A)

with

N ¢ 90(A,J)

and

J ~ L, we have

(4) U ([A,C],L)

+ U ([A,C],J)

~ n .

By assumption (5)

U ([A,C],L)

Clearly there exists J' = ~(A,P,N) U ([A,C],J')

we get

~ i.

~ n - i .

P e ~0([A,C],\L) J' ~ ~31 I(A)

C o n s e q u e n t l y by

160

and then upon letting

with (4) and

N ~ ZI0(A,J'),

J' ~ L, and

(5) we see that

2.96

(6)

L1 ( [ A , C 3 , L )

i.e.,

L

is an

(n-l)-secant.

jection Formulas C

from

N

(23.14)

= n - 1

In v i e w of

and

(4) a n d

(23.15) w e d e d u c e

(6), b y the P r o -

t h a t the p r o j e c t i o n

is b i r a t i o n a l , Deg[AN, cN~ = D e g [ A , C ~

--- n ,

and U ([AN,cN~,Q)

Also

= 1

for all

Q e [30([AN,cN~,\LN)

clearly n 0 ( [ A N , c N ~ , \ N) c n 0 ( [ A N , C N ~ ,\LN)

Therefore

N

(26.6)

is a b e s t p r o j e c t i n g

LEMMA.

Assume

that

.

center.

n = 4.

Let

L

be a 2 - s e c a n t

and

le____t

(*)

N 6 ~0([A,L~,\C)

be such t h a t (**)

the project.ion of

Then

N

is a b e t t e r

PROOF.

Now

Deg[A,C3

In v i e w of

(i),

from

projectinq

N

of

= 4, and (*) and

C

from

N

L ¢ £iI(A,~)

and h e n c e b y

Deg[AN,cN3

= 4

and

161

(18.13.1)

is rT-integral. with

we k n o w

By a s s u m p t i o n

U ([A,C~ L) -- 2.

(**), b y the P r o j e c t i o n

see t h a t (2)

is b i r a t i o n a l .

center.

N e £0([A,~3,\C)

that the p r o j e c t i o n

(i)

C

Formula

(23.14),

we

of

2.97

U ([AN, cN], LN) = 2

(3)

Given

any

and

L N e ~ 0 ( [ A N , c N ] , ~ N)

Q e 9 0 ( A N , \ L N),

upon

letting

D = &(A N ,L N,Q)

we c l e a r l y

have

(4)

D ~ H I ( A N)

and (5)

U ([AN,cN],Q)

In v i e w of

(2) and

(4), b y B e z o u t ' s

(6)

Little

Theorem

(23.9)

(3),

(5) and

(7)

(6), we get U ([AN,cN],Q)

Therefore (26.7)

N

is a b e t t e r

LEMMAo

N e ~30(A,L)

Assume

be

~ 2 .

projecting

that

n = 5.

center. Let

L

be a 3-secant,

such that

(*)

U ([A,C],N)

(**)

U ([A,C],~,N)

> 1 > 1

and (***) Then

the p r o j e c t i o n N

is a qood

PROOF. (i)

we get

U ([AN, cN], D) = 4 ;

now b y

let

+ U ([AN,cN~, LN)~ U ([AN, cN], D)

By

of

C

from

projectinq

N

center.

(***) we h a v e EmdimEA,C,N]

~ 2 .

Since

162

is b i r a t i o n a l .

and

2.98

(2)

N ~ •0(A,L)

and (3)

L e ~i1 (A,Tt) ,

we get

(4)

N ~ f]0(A,n)

By(*),

(**)

and

(4) we get that

(5)

u ([A,c],~)

=

i

and (6)

card

G([A,C],N)

= i

and

rf c T I ([A,C],V)

where In v i e w of C

from

N

(i),

(4) and

(6), by

is ~-integral.

(7)

(23.14)

and

(***), (23.15)

(8)

(2),

(3),

(21.6.2)

~([A,C],N) we

see that the p r o j e c t i o n

Now by a s s u m p t i o n

U ([A,C],L)

In v i e w of

IV] =

= 3 (4) and

(7), by the P r o j e c t i o n

we see that Deg[AN, c N] = 4

and (9)

N ([AN,cN],L N) = 2 Given

any

and

L N ~ I]0([AN, cN],TT N)

Q e fl0 (AN,\L N) , upon D = A ( A N,L N,Q)

we c l e a r l y (i0)

have n £ H I (AN)

and

163

letting

Formulas

of

2,99

(ii)

~ ([AN,cN],Q)

In v i e w o f

(7) and

(12)

+ U ([AN, cN], LN)

(9), b y B e z o u t ' s

~ ~ ([AN,cN], D)

Little

Theorem

(23.9) w e g e t

U ([AN,cN], D) - 4 ;

now, b y

(8),

(I0) and

(13)

(ii), w e get,

U ([AN,cN],Q)

Therefore

(26.8)

N

is a g o o d p r o j e c t i n g

C O N E LEMMA.

(1)

~ 2 .

Let

Emdim[A,C]

L

center.

b e a 2-chord.

Assume

that

n ~ 5,

= 3 ,

and

(*)

~ ([A,C],P)

Also assume (**)

4 = 2 times

that

The 2

C

from

only composite

and

~ = cN'A

we clearly have

(3)

is not b i r a t i o n a l .

positive

integer

factorization

letting

(2)

such t h a t

Projection

-< 5

of 4. Formula

and

D e g [ A N' ,C N' ~ = 2 Upon

N'

.

i S a b e s t project_iin__g c e n t e r .

(*), b y the S p e c i a l

e ~ ( A N')

P ~ ~([A,C])

~ ~(A,L)

is the o n l y p r o p e r

(i) a n d

C N'

of

N'

N e %([A,L],\C)\[N']

PROOF.

view of

for all

that there exists

the projection

Then every

= 1

~ ¢ n(A)

,

164

u ([A,C],N')

~ 1 .

is 4, and

Therefore, (23.15), w e

in see

2.100

(4)

~ ¢ H

(5)

C ¢ nl(A,#)

and,

in v i e w o f

(i), b y

(25.9)

(6)

¢

NOW by

and h e n c e

~: n 0 ( A , L )

by

the

and

Projection

T,~'

(7)

(2) and

,

also

see t h a t

H 2 (A)

U ([A,C~,L)

Formula

a 2 > 1 a U ([A,C3,N')

(23.14)

c ~0 ( A N ' ' c

N l

we

see t h a t

)

(7) w e g e t

(8) By

we

,

assumption

~,

By

(A,N')

L ¢ ~(A,~)

(3) and

(6) w e

see that

(9)

Q(A,~)

Now

= ~ .

let any

(i0)

b e given.

N H I(A)

N ~ n0([A,L~,\c)\[N'] Let us k e e p

in m i n d

the

165

following

suggestive

figure:

2. i01

S,T

: Points of

C

necessarily

on

L, not

distinct

may contain

N'

and

S f

(') and

projection

C

(4) and

of

/

.

N

NOW in view of

T

(18.3.1)

from

N

we know that

is n-integral.

J

C N e ~ ( A N)

Since

and the

N' ¢ ~ ( A , L ) ,

by

(I0) we see that for any

J 6 ~(A)

with

N ~ ~(A,J)

and

J @ L, we have

(ii) w

By

A(A,N',J)

e HI(A)

A(A,N',J)

E n(A,@)

(9) and

and if

J ¢ ~(A,~),

then

(ii) we get that

for any

J e ~(A)

with

N e ~(A,J)

and J / L, we have

(12) I J ~ In view of

[~(A,~) (6) and

(12), by Bezout's

166

Little Theorem

(23.9) we conclude

2,102

that I for any (13)

J ¢ £]II(A) with

and

J ~ L, we have

/ I U ([A,J],~)

By

N 6 ~]0(A,J)

(8) and

~ 2

I

(i0) we know that

N ¢ Z]0(A,#)

and hence by

(13) we get

that I for any

J e 2 I(A)

with

N e D 0 (A, J)

and

J ~ L, we have

and hence by

(5) and

N e ~]0(A,J)

and

(14) ([A,J],~,\~)

By

(i0) we know that

~ 1 .

N ~ ~0(A,C),

(14) we get

that for any

J e 211 (A)

with

J ~ L, we have

(15) u ([A,J],C)

(because

~ 1

U ([A, J],C) = ~ ([A,J],C,\N)

by

(10)

u ([A,J],~,\N)

by

(5)

1

by

(14).)

In view of (*), by the Commuting Lemma ([A,J],C) = U ([A,C],J) and hence by I

for any

(23.19) we know that 1 (A) J c ~31

(15) we get that

for any

J e ~(A)

with

N e [30 (A,J)

and

J ~ L, we have

(16) ([A,c],J)

~ 1

Now in view of (16), by the Projection Formula that the projection of

C

from

N

167

(23.14) we conclude

is birational,

2.103

D e g [ A N , C N] = D e g [ A , C ]

= n

and (17)

u([AN,cN],Q)

Since

L e ~(A,~),

-- 1

by

U ([AN,cN],Q)

Therefore

N

Z ~ ( A , J I)

that, and

Further

O e 90 ([AN,cN],\~ N)

Assume

that

are d i s t i n c t

fo___~r A ( A , L , J I , J 2) = 9, say, we h a v e

that,

~(A,L)

then,

there

Z 0 ( A , J 2) assume

for all

L, Ji,J2

such that,

Note

= 1

LEMMA.

(A)

assume

(17) we get

center.

PLANE

Further

Q e ~ 0 ( [ A N , c N ] , \ L N)

is a b e s t p r o j e c t i n g

(26.9) o__~f ~

for all

N ~ ( A , J I) O ~ ( A , J 2 ) is a u n i q u e

; and then

member,

members

~ e HI(A).

= ~.

say

N, c.ommon to

N~(A,L).

that

(*)

U ([A,C],N)

{ 1

an___dd 2

(**)

t +

t = ~ U

~ U ([A,C],J i) ~ n + 2, w h e r e i--i

([A,C],L,P),

the s u m m a t i o n

P e ~(A,C)\(~0(A,JI)

U n0(A,J2)).

beinq

extended

over

all

T h e n we h a v e C ¢ ~(A,%) PROOF.

In v i e w of

(*), by

U ([A,C],Ji,N) and h e n c e

(i)

upon r e l a b e l l i n g

Jl

and h e n c e

Emdim[A,C]

(23.18.3)

we see that

for

~ 1 and

U ([A,C~,JI,N)

168

J2

i = 1 or 2

suitably

~ i.

< 2 .

we may

suppose

that

2. 104

By the d e f i n i t i o n

(2) By

of

~([A,C~,JI,N) (i) and

N

we clearly have

+ u([A,C],JI,\J2 ) = u(~A,C],JI)

(2) w e g e t

(3)

1 + u ( ~ A , C ] , J I , \ J 2) ~ u ( ~ A , C ] , J I)

By the d e f i n i t i o n

(4)

of

~

L ~ ~(A,@)

we have

and

Ji ¢ ~(A,~)

for

i = 1,2

.

Now clearly

U (~A,C],~)

~ t + ~ ( [ A , C ] , J I , \ J 2) + ~ ( ~ A , C ] , J 2) n + 1

and h e n c e b y B e z o u t ' s

(26.10) n

=

QUADRIC

by by

Little

Theorem

LEMMA.

Let

(23.9) w e g e t

L

(4)

(**) and

(3)

C ¢ ~I(A,~).

be a 2 - s e c a n t .

Assume

that

5p

(*)

~(~A,C],P)

= 1

for all

P e ~0([A,C],\L)

and t h e r e does

not e x i s t any

~ c H 2 (A)

such that

C c ~(A,~)

.

(**) is a n - q u a s i p l a n e

and

Let = {N ~ ~0(LA, L],\C) :

N

is a b e t t e r

projection

center}.

Then card ~0(~A,L],\C)\~) PROOF°

(i)

In v i e w of

~ 2

(**), by

and

card n0(A,L)\~

(25.6) w e

Emdim[A,C]

169

= 3.

see that

< ~ = c a r d ~.

2.105

Given jection number and

any

of

C

and

N ~ ~0([A,L],\C), from

N

we

(18.13.1)

is ~ - i n t e g r a l ,

Emdim[A,C]

(23.15)

by

and

we k n o w that

(because

5 is a p r i m e

= 3) in v i e w of £he P r o j e c t i o n

see that the p r o j e c t i o n

C

of

the pro-

from

Formula

N

(23.14)

is b i r a t i o n a l ,

Deg[AN, c N] = 5 ,

U ([AN,cN], JN) = H ([A,C],J)

and h e n c e

for e v e r y

j e ~II(A)

with

N ~ 90(A,J)

in p a r t i c u l a r

U ([AN, cN], LN) = U ([A,C],L)

= 2

LN~ n O ([AN,cN],~ N)

and

w

Thus

it o n l y r e m a i n s

to p r o v e

that

card

w

= {N ~ n 0 ( [ A , L ] , \ C ) : J ¢ ~(A)

u([A,C],J)

with

f)

£

2

where

for some

~ 3

N C ~0(A,J)] w

Suppose, (2)

if possible,

three distinct

that

members

> 3.

card NI,N2,N 3

of

T h e n we

can take

20 ([A, L], \C)

for w h i c h w e can find 1 Ji ~ nl(A)

(3)

with

N i ~ ~0(A,Ji)

for

i = 1,2,3

such that

(4) Since (5)

u ( [ A , C ] , J i) ~ 3 is a 2-secant,

L

n0(A,L)

Now we can choose suggestive

in v l e w of

N n 0 ( A , J i) = [Ni] Pij

~ g0(A)

for

i = 1,2,3

(2),

(3) and

and

Ni~

as i n d i c a t e d

figure

170

.

(4) we get

~0(A,C) in the

for

i = 1,2,3.

following

2 . lOG

I

I

I I

I l

/i

S

J2

J3

N2

N3

~P21

11

P22

13

IP32

P23

!

t P23

!

I l

I S,T

: Points

of

I

C

on

L, not n e c e s s a r i l y

distinct.

n0(~A, L3,C) = { S , T ]

i.e.,

w e choose

nine d i s t i n c t

members

P

. , i,j = 1,2,3, 13

of

0 (A)

such that

(6)

Pij ~ ~ 0 ( A ' J i )

for

i,j = 1,2,3

.

N o w we have,

[H 2 (A) : H 0(A) ] = number

of d i s t i n c t

monomials

of d e g r e e

2 in 4 i n d e t e r m i n a t e s

= i0 , (i.e.,

geometrically

projective

(7)

3-space)

speaking,

there

are

and h e n c e we can find

¢ ¢ H 2 (A)

such t h a t

171

~gquadrie

surfaces

in

2.107

(8) By

Pij 6 ~0(A,~) (6),

(7) and

for

(8) we have

~([A,Ji3,{)

~ 3 > 2 = (Deg[A, Ji3) (DegAS),

and hence by Bezout's (9)

Little Theorem

Ji 6 ~ ( A , ~ )

Consequently

by

Ni (2),

6

(7) and

(23.9) we conclude

for

i = 1,2,3.

~0(A,~)

for

i = 1,2,3.

that

~ 3 > 2 = (Deg[A, L3) (DegAS)

and hence by Bezout's

Little

(II)

L c £~(A,~). L

i = 1,2,3,

(I0) we have

~([A,L~,~)

Since

for

(3) we get

(I0) Now by

i,j = 1,2,3.

is a 2-secant,

(12)

Theorem we conclude

that

we have ([A,C],L)

= 2.

Now let s = ~ u([A,C~,~,P),

(131 )

over all In view of

(5),

Emdim[A,C3

NOW

(15)

(ii),

if possible

i,j c {I,2,3}.

(132), we see that

(26.9)

= 2 .

~0(A, Ji) Q ~0(A, Jj) ~ ~

In view of

for some dis-

(*), (2),(3), (4), (5), (12), (13 I)

can be applied

~ i, a contradiction

~0(A,Ji)

is extended

3 U £0(A, Ji) i=l (12) and (131) we have

s ~ u([A, C3,L)

Now suppose,

(14)

the summation

P e ~0(A,C)\

(132 )

tinct

where

to

(i).

to

L,J i

and

i,j e {1,2,3].

(9) gives, u([A,C~,~,J i) > ~([A,C~,Ji),

NOW

172

Jj; and we g e t

Thus we have proved

N £0(A, Jj) = ~, for distinct

for

i = 1,2,3.

and

2,108

([A,c],~) > s +

3 ~ U ([A,C],~,J i) i=l

by

(131 ) and

(14)

by

(15)

3

> s +

~ U ([A,C],J i) i=l

> ii

by

> 5 times

=

(4)

and

2 by

(Deg[A,C](DegA~)

and h e n c e b y B e z o u t ' s

Little

(16)

Theorem

(7)

(23.9) we m u s t h a v e

C ~ nl(A,~)

In v i e w of

(**),

(7),

(ii)

and

(16), b y

(25.7)

we c o n c l u d e

w

(17)

~ ¢ H

In v i e w of

(A,P)

for some

(14), we can take

(18)

permutation

P e ~0(A,L)

a

(e(1),

e(2),

e(3))

of

(1,2,3)

such that

(19) Upon

P ~ E]0(A,Je(i))

(14),(18)

(21)

for

4 i = A ( A , p , j e(i) ) and

4i

for

i = 1,2,

i = 1,2

with

(19) we get

~ H I(A)

for

and (22)

i = 1,2.

letting

(20) by

(132 )

Je (3) (~ ~ (A'4142)

173

41#42

that

2. 109

By

(7),

(9),

(17),

(18),

(20)

and

(21) w e

see

that

(231 By

(9)

and

(18) w e

also h a v e

(24)

Je(3)

Now

(22),

card Q

(23)

and

¢ ~ (A, ~)

(24) y i e l d

a contradiction;

therefore

we m u s t h a v e

< 2 .

(26. ii)

PROPOSITION. W

(26.11.1) such

that

If

Emdim[A,C3

C £ ~(A,~),

and

~ 3

there

then:

exists

a ~-quasiplane

and

(26.11.2)

If

n ~ 2

(26.11.3)

If

n = 3 = Emdim[A,C},

every

2-chord

every

then

Emdim~A,C]

and

N ¢ ~o(EA, L},\C)

any

there

I_ff n = 4, EmdimEA, C~

3-secant

exists L

a 3-chord;

we have

2-secant

exists

L

i__~s

Assume

that

~([A,C~,P)

= l

projecting

center.

PO e ~ 0 ( [ A , C ] , ~ )

3-chord

is a 3 - s e c a n t ;

projecting

Emdim[A,C~

for all

174

that

= 3, and

is a b e s t

n = 4,

a 2-chord;

we have

that

N e ~ o ( E A , L3,\C)

(26.11.5)

~

£0([A,L~,\C)

for some

every

there

card ~ 0 ( [ A , L ) , C ) < ~ = c a r d £ o ( [ A , L ] , \ C )

and e v e r y

that

~ 3

then:

for any

is a b e s t

# ( E A , C 3 , P O) ~ 1

then:

such

~ 6 H I(A)

C e ~(A,~).

is a 2 - s e c a n t ;

(26.11.4)

exists

~ ~ H 2 (A)

card no([A,L~,C ) < ~ = card and

there

= 3

center.

and

P 6 £o([A,C},~)

and

for

2.110

Then there

exists

fQ l l o w ! n g .

If

a 2-chord,

L

and for any 2-chord

is not a 2 - s e c a n t

card n 0 ( [ A , L ~ , C )

L

N ~ ~ 0 ( [ A , L3,\C)

a 2-secant

and

-- 1

,

is a b e s t p r o j e c t i n g

for all

we h a v e the

is a 3-secan______~t,

< ~ = card n 0 ( [ A , L ~ , \ C )

and every

~ ([A,C~,P)

then

L

center.

P e ~0([A,C~,\77)

I__ff L

i_ss

,

then upon letting

= {N ~ n 0 ( [ ~ , L ~ , \ c ) :

is a b e t t e r p r o j e c t i n g

N

center}

we have

card n0([A,L~,\C)\ ~ (26.11.6)

If

~ 1

and

n = 5, E m d i m [ A , C ~ [u([A,C~,P)

then:

there

exists

any 4-secant

L

a 4-chord;

Assume

(*)

2 >

(**)

u([A,C],P)

that:

assuming

- i] ~ 2 ,

every 4-chord

and

is a 4 - s e c a n t ;

is a best projecting

and

for

= 1

for all

C e ~I(A,~).

that there

exists

center.

n = 5;

~ [U ([A,C],P) P ~ n 0 ([A, C~, ~)

and t h e r e d o e s not e x i s t a n y ~uasiplane

-- 3, and

< ~ = card D 0 ( [ A , L ] , \ C )

N ¢ ~0([A,L~,\C)

(26.11.7)

< = = card Q .

we have that

card D 0 ( [ A , L ] , C )

and e v e r v

card D 0 ( A , L ) \ D

- i] ~ 0 ;

P e D0([A,C],\~)

e H 2 (A) [Note t h a t

such that (*) +

PO ~ D o ( [ A ' C ~ ' W )

175

(**)

is a Wis e q u i v a l e n t

such that:

to

2.111

U ([A,C],P 0) = 2, and

Then

there

exists

~ ([A,C],P)

a 2-secant

~ ([A,C],P)

and

for a n y

=

such

= 1

2-secant

= 1

L

such

for all

L

upon

IN e ~ 0 ( [ A , L ] , \ C ) :

N

~0([A,L],\C)\Q

and

for all

P e %([A,C],\P

0)

that

P ¢ ~([A,C],\L)

,

lettinq

is a b e t t e r

projecting

center]

we h a v e

card

(26.11.8)

Assume

~ 2

that:

card

e0(A,L)hQ

< = = card Q

n = 5;

(*)

~ [u([A,C],P) me n0 ([A, C],~)

- i]

(**)

U ([A,C],P)

P e n0([A,C],\~)

= 1

u([A,C],~,N')

and

there

does

quasiplane the

not

and

exist

such

ugon

any

some

=

N'

;

e ~0(A,~)

~ e H2(A) [Note

0

such

that

;

;

that

(*) +

~

(**)

is a ~is e q u i v a l e n t

to

that

~ ([A,C],P)

any

for

C e ~I(A,~).

assumption

Then

for all

> 1

.

= 1

there

exists

2-chord

L

for all

P ~ Z0([A,C]).]

a 2-chord

we have

the

L

such

that

following.

If

N' L

e ~0(A,L),

and

is a 2 - s e c a n t

for then

letting

=

{~ ~ ~ 0 ( [ A , T ~ , \ C ) :

N

is a b e t t e r

projecting

center}

we have

card

I_~f L

~0([A,L],\C)\Q

is a 3 - s e c a n t

and

~ 2

and

the p r o j e c t i o n

176

card~(A,L)\n

of

C

from

< = = card f]

N'

o

is b i r a t i o n a l

2.112

then

N'

is a g o o d

projection

of

C

from

card

and

every

neither

N

is a b e s t

is a 3 - s e c a n t

and

the

then

projectinq

then.

L

center.

is a b e s t

n =

I_~f

L

i__ss

is a 4 - s e c a n t e

< ~ = card n0([A,L~,\C)

that:

~

L

< ~ = card([A,L],\C)

a 3-secant,

Assume

If

is n o t b i r a t i o n a l

~ ~0([A,L],\C)

(26.11.9)

(*)

nor

n0([A,L],C)

N

center.

~0([A,L~,C)

a 2-secant

every

N'

c ~30([A,L],\C)

card

and

projecting

,

projectinq

center.

5;

13

[~ ( [ A , C ] , P )

-

-- 0

;

1

for all

P ¢ n0([A,C],\~)

for

P ¢ ~0(A,n)

Pen 0 ([A,CL~)

(**)

U ([A,C],P)

=

(***)

u([A,CT,n,P)

< 1

all

;

;

e

and

there

does

quasiplane

and

(*) +

(**)

not

exist

and

for

is e q u i v a l e n t

then

upon

Q =

there

any

~ ~ H 2 (A)

C ¢ Ill(A,#).

u[A,C],P)

Then

any_

=

such

L

to t h e

1

exists

[Note

we have

the

that

assumption

for all

a 2-chord

such

that

(***)

=

~ (*)

is a ~; also

note

that

P e n0([A,Cl).3

L

such

following.

that

L

is n o t

I__ff L

a 3-secant,

is a 2 - s e c a n t

letting

{N c ~ 0 ( [ A , L ~ , \ C ) :

N

is a b e t t e r

projecting

center~

we have

card n0([A,L~,\C)\O

I_~f

L

is n o t

a 2-secant

< 2

then

and

L

that

card n0(A,L)\Q.

is a 4 - s e c a n t ~

177

< ~ = card

n

.

2. 113

card ~ o ( [ A , L ] , C ) < ~ = card no([A,L],\C) and every

N 6 £0([A,L],\C )

PROOF. of

(25.6),

(26.11.2) (26.5), from and

the second assertion

(26.11.4)

(26.2),

To prove

there exists

such that

(26.11.9)

follows

(26.12)

L

(26.8).

(26.7),

Assume

~([A,C],P)

= 1

and

There exists

3 ~ n { 5

p r o j e c t i o n of

and

follows

follows

from

U ([A,C],P)

(*) there exists

follows (26.8)

= 1

from

and

(26.1)

for all (26.10).

(26.5).

(26.5).

that

n ~ 5

and

~ e H2(A )

(26.3)

~0 (A'L)" and finally note

and

for all

(26.11.8)

Finally,

P 6 ~]0([A,C],\n)

three situations such that:

C

from

and there exists N

~

min(4,n)

prevails. is a ~ - q u a s i p l a n e

the projection of

for all

C

from

and there exists N

C

Q 6 £ 0 ( [ A N , c N ] , \ ~ N)

') = 2

for some

N ~ ~o(A,TT)

is birational,

i__ssTT-integral, u([AN,cN],Q)

~([AN,cN],Q0

= 1

such that:

the from ,

~ Deg[AN,c N] < n .

4 < n ~ 5

p r o j e c t i o n of

N c ~o(A,~)

is birationa!,

i__ssTT-inteqral, ~([AN, cN],Q)

(3)

N

(26.11.6)

and

C ¢ £1(A'~) . (2)

N

in view

(26.2)

(26.11.5)

(26.10)

Then at least one of the following (i)

(26.5).

PO~

(26.11.7)

(26.4),

THEOREM.

from

first note that by

such that

(26.10),

from

and,

U ([A,C],P O) = 2, then note that by

the rest of

(26.1),

follows

and

(**) we must now have

P c ~o([A,C],\L); from

and

(26.11.7),

a 2-secant

(*) and

(26.1)

is obvious

from the first assertion.

(26.11.3)

from

(26.6)

in (26.11.1)

follows

(23.11).

follows

PO 6 ~o([A,C],~)

follows

from

(26.5),

(26.5).

that by

is a best projecting center.

The first assertion

follows

,

~ 2

the projection of

for all

C

the from

Q ¢ ~]o(AN),

Q0 6 ~o([AN,cN],\~N),

178

such that:

z([AN,cN],QI)

= 2

2.114

for some

Q1 ~ ~

PROOF.

([AN'CN~'~N)'

Follows

from

and min(4,n)

(26.11).

179

~ Deg[AN, cN~

< n .

CHAPTER

III:

BIRATIONAL

In this ~(Y)

in

chapter

Y , with

derivative

of

X,Y,Z

~(Y);

for a n y

X,Y,Z,

corresponding

subscript;

X

and

Y, w i t h

partial

derivative

denotes

the p a r t i a l

shall

(27.1) field

L.

field

now

thus

for i n s t a n c e ,

~(X,Y)

derivative

of

(S,K)

the

K

of

of

and

we

R c ~

note

that

(S,K).

We

S

L.

~

that

(27.2)

we

O(S,K)

~ {0};

shall

closure

indicated

in-

b y the

X,

with

~(X,Y)

denotes

and

the

~y(X,Y)

respect

to

Y.

o f the d i f f e r e n t .

domain with

of

S

quotient

in a f i n i t e

algebraic

define

module

TraceK/L(~)

(S,K)

is

an

of

~ S

S

i__nn K

for a l l

R-submodule

8 ~ R]

of

K

with

define

= {a ~ K:

note

t h e Y-

of the

~x(X,Y)

to

be a n o r m a l

We

~3(S,I<) = the d i f f e r e n t

and we

theory

= the c o m p l e m e n t a r y

then

are

ring,

~(X,¥)

integral

= {~ e K:

denotes

Different

Let

be

~' (Y)

for a polynomial

respect

review Dedekind's

R

extension

in s o m e

with

for a n y p o l y n o m i a l

in t w o o r m o r e

derivatives

coefficients of

ring,

polynomial

the p a r t i a l

DEFINITION. Let

indeterminates; in s o m e

§27. We

OR GENUS

are

coefficients

determinates

in

GEOMETRY

then see

after

as

of

e R

S

for all

~)(S,K) c R

that,

if

proving

K

in

and

K ~ ~ ~

O(S,K)

is s e p a r a b l e

(27.2)

we

(S,K)}

is an i d e a l over

shall use

this

L,

in

R.

In

then

observation

tacitly. (27.2) be

a normal

DEDEKINDS domain with

FORMULA

FOR DIFFERENT

quotient

field

180

L.

AND CONDUCTOR. Let

R

be

the

Let

S

integral

3.2

closure L.

of

Let

S

y

in a f i n i t e

~ R

be

such

alqebraic

that

seaprable

K = L(y),

(y) = yn + { n _ i Y n - l + . . "+{0

be

the m i n i m a l

is n o r m a l

monic

we have

polynomial

of

and

{0 ..... {n-i

over

~i e S

for all

i.)

{(S[y])

= {' (y)~

(S,K)

extension

K

o_~f

let

with

y

field

L.

(Note

i__nn L

that

since

S

Then we have

Also we have

{' (y)

(Whence i__ff S

~ ~(S[y])

N u(S,K)

in p a r t i c u l a r

and

~(S,K)

is a D e d e k i n d

domain,

~

~(S[y])~(S,K)

[0], b e c a u s e

such

that,

We

can

y = Yl

take

elements

any

((Y)

{' (y) ~ 0.)

.

Moreover,

=

~' (y)R

Yl ..... Yn

.

in an o v e r f i e l d

of

K

and

(Y) = For

~' (y)R

then we have

~(S[y])~](S,K)

PROOF.

c

~ L[Y]

(Y - yl ) (Y - y2)... (Y-Yn).

of d e g r e e

< n, b y

Lagrange

interpolation,

we

have

(i)

(Y)

=

n

E

i=l For proof, and

their

note values

Y I ' Y 2 ..... Y n the d i v i s i o n

(2)

of

that both coincide Y

.

algorithm

~ (Y) =

{ (Yi)

- -

<(Y)

~' (Yi)

Y-Yi

sides

are p o l y n o m i a l s

for the

Since

~(Y)

n

distinct

~ S[Y]

n-i ~

~j (Z)Y j + ~ (Z)

j=0

with

181

Y

of d e g r e e

< n

values

is m o n i c

we have

(Y-Z)

in

of degree

n, b y

3.3

(3)

~j (Z) ¢ S[Z],

for

0 < j ~ n-i ,

and (4) By

~n-i (Z) = i. (2) we have (y) Y-Yi

and substituting

n-i D nj (Yi) Yj j=0

,

for 1 < i < n ,

these in (i) and collecting the coefficients

of

YJ

we get (5)

n-i {~ (y) 9j (y)h { (Y) -- j=0 ~ TraceK/L <'' {' (y) ] Yj

for every

(Y) e L[Y]

of degree

Given any

z e K

we can write

z = { (y)

of degree < n, and then substituting

(6)

z =

n-i {Z~]j (y)) ~ Trace j=0 < ~' (y)

Given any

yj

y

for

for all

< n .

for some Y

{ (Y) ~ L[Y]

in (5) we get that

z ~ K

z ¢ S[y], we can write

z = [ (y), where

[ (Y) =

n-i ~ {jYJ j=0

with

~j ¢ S ,

and then we get ~n-i = TraceK/L

(zt

by

~ g(y

(4) and

(5).

Thus we have (7)

TraceK/L < ~ I To prove that 8 e ~(S[y])

C S,

for all

~ S [ y ] c {' (y)~ and

z ¢ S[y]

(S,K)

~ ¢ R ~ TraceK/L

182

.

it suffices to show that g, (y)

3.4

and

this

in t u r n

To p r o v e suffices

follows

the

to s h o w

by t a k i n g

reverse

z = 88

inclusion

in

~(S[y])

~

(7). ~' (y)~

(S,K)

, it

that,

W

e ~

So

let any

by

(3) w e

(S,K)

~ £ ~

and

any

(S,K)

and

any

y~j (y) ~ R

~ ~ ~

(S,K),

(9)

by

for

letting This

(8) w e

z = ~' (y)~y,

completes

(i0) From

it i m m e d i a t e l y

multiplying

both

in c a s e

S

then

R-submodule

R

follows

I

of

b ~ I], w e h a v e (27.3) K

be a f i n i t e

X, u p o n

.Let

alqebraic

(9) w e g e t

z c S[y~.

S

n ~(S,K) by

O(S,K),

we

get

, in the g e n e r a l

domain,

we

further

letting

have,

~

domain

and h e n c e

J = {~ e K: a b e (S,K)~(S,K)

extension

183

of

for R

any

finite

for all

= R.

b e a d°main with_quotient

field

case,

= {'(y) R ;

a Dedekind

IJ = R ; h e n c e

LEMaMA.

(i0)

c {' (y)

is a D e d e k i n d

of

0 ~ j < n - i.

that

s ~(S[y])

sides

is a l s o

for

of the e q u a t i o n

~(S[y])~(S,K) because

y ~ R ,

= ~'(y)~*(S,K).

{(S[y])D(S,K) and,

¢ S

(6) and

the p r o o f

~'(y) Upon

Since

get

by

~(SEy]) (I0)

be given.

0 s j ~ n - I.

T r a e e K / L ( ~ y ~ j (y))

Upon

y ¢ R

~ S[y].

get

(8)

Since

y ~ R ~ ~' (y)~y

L,

let

.fie! d y e K

L, let be s u c h

3.5

that

K = L(y),

over

L.

Then:

and

Assume

that

{' (y) ~ Q ~

separable

let

over

{(Y) {(Y)

be

e S[Y].

(Q N S ) S [ y ] Q

quasilocal

Let

Q

= M(S[y]Q)

monic

polvnomial

be

a prime

and

S[y]Q

of

ideal

y

in

SLY].

is r e s i d u a l l y

S. w

PROOF.

the m i n i m a l

Let

w

S

domain

= SSQQ;

and

P

w

R = S [y]

is a m a x i m a l

and

P = QR.

ideal

in

R

Then with

S

is a

P Q S

=

w

M(S and

).

Let

k =

f: R -- R/P

f(S

algebraic

).

Then

field

polynomial

of

be k

over

canonical

is a f i e l d

extension z

the

of

k.

k. Then

and

Let our

epimorphism. f(R)

~(Y)

be

assertion

Let

= k(z)

= a

finite

the m i n i m a l is c l e a r l y

z = f(y)

monic

equivalent

to : w

(*)

~' (y) ~{ P ~ M(S

)Rp = M(Rp)

and

9' (z) ~ 0 .

w

Let then

u:

clearly

epimorphism then

f

(i) Then

- R

Ker u = such

clearly

applying

S [Y]

that

be

{(Y)S*[Y]. v(Y)

Ker v = M(S to the

the u n i q u e

= Y

Let

I = v(u-I(M(S*)R))

v:

and

)S [Y].

coefficients

S -epimorphism

v(a)

Let of

and

S*[Y] =

{(Y),

for all

~ k[Y]

and

be

u(Y)

= y

;

the u n i q u e a E S

be obtained

; by

let

J = v(u-l(P))

clearly

(2)

(Y)k[Y]

(3)

= I c J = ~ (Y)k[Y]

= a maximal

~' (Y) ~ P = ~' (y) ~

J

and

(4)

~ k[Y]

f(a)

((Y)

with

~' (z) ~ 0 = ~' (Y) ~ ~ ( Y ) k [ Y ]

In v i e w o f

(2) we h a v e

(Y) = ~(Y) is(Y)

184

.

ideal

in

k[Y].

3.6

where D(Y);

i

is a p o s i t i v e

differentiating

integer

both

and

~(Y)

e k[Y]

is c o p r i m e w i t h

sides w e get ,

~' (Y) = i~(y) i-i ~. (y)~(y)

+ ~(Y) i

{' (Y) ~ 9 ( Y ) k [ Y ]

and

. (y)

and h e n c e

(5) By

(2) and

~ i = 1

q' (Y) ~ ~ ( Y ) k [ Y ] .

(5) w e see that

(6)

~' (Y) ~ J ~ Ik[Y]j = M ( k [ Y ] j )

We observe

that,

h

is an e p i m o r p h i s m

of a domain

and

W

are

such t h a t K e r h c V

E

and

c W

W

is prime,

M(~(W)).

In v i e w of

~' (Y) ~ q ( Y ) k [ Y ] .

if

a domain and

V

and

ideals

t h e n clearly:

in

D

D

V D W = M ( D W) ~ h ( V ) ~ ( W )

(i), u p o n a p p l y i n g

this o b s e r v a t i o n

onto

=

t w i c e we

see t h a t

(7)

M(S*)Rp

NOw

(*) f o l l o w s REMARK.

= M(Rp)

from

(3),

~ Ik[Y]j -- M ( k [ Y ] j )

(4),

(6) a n d

(7)

N o t e t h a t w e have:

M(s

* )s * [Y]

U' ~ p z =

f(y)

= q (Y)

{ (Y) S [Y]

s [Y]

u

~

s [y] = a

~ 0

)

k[z]

>

51v ' n (Y)k[Y]

C

,

U

> k[Y] 0

where

the c o l u m n s

restrictions

of

ally by diagram

= R/P

0

a n d last two r o w s a r e e x a c t u

0

and

chasing

v

respectively.

u'

and

W e are p r o v i n g

in the a b o v e d i a g r a m .

185

and

v'

are

(*) e s s e n t i -

3.7

(27.4)

LEMMA.

R

be

P

be a prime

an o v e r d o m a i n ideal

Rp ~ Rp/M(Rp) all

prime

be

ideals

Then we have

(I) y ¢ P' S[y~Q

the

for all = Rp

y ¢ R

PROOF.

Let

with

quotient

is a f i n i t e

field

that

R

in

R

such

that

(S A P ) R p = M ( R p ) .

canonical

P'

in

R

epimorphism.

such

that

P'

Let

L.

S-module.

Q

Let

Let

be

n s = P N s

the

Let

g: set of

and

P' ~ P.

followinq:

Rp that

a domain

such

the

is such

P'

such

be

S

¢ ~,

and h e n c e

If

S

of

I_~f y ~ R

(2)

p'

Let

that

then

y ~ P, g(Rp)

upon

lettin~

in particular

is residually y ~

L(y)

separable

P, g(Rp)

= g ( S s D P) (g(y)),

Q = S[y~

N P

we have

= the

quotient ' field

over

S,

= g ( S s D P) (g(y)),

then

an___dd

there

and

y e P'

r i n g of

R

of

R.

exists for all

~ ~. E = SSN P , D = the

quotient

with

respect

w

to the m u l t i p l i c a t i v e

set

maximal

different

ring,

ideals

E c

in

D, D

D I = Rp , and

D

S\P,

is a f i n i t e

P'

- P'D

I = PD, from

I.

E-module,

gives

and

~

Then

I

a bijection

of

let a n y

y e R\P

be given

g ( S s N P ) (g(y))

and

y ¢ P'

for all

P'

is the o n l y m a x i m a l finite by

Ely,-module,

Nakayama's

we clearly

have

(2),

find

D

which

E[y~j

~ D\I

set and h e n c e

z ¢ D

such

that

g(z)

can

find

t ~ S\P

field

Rp

such

upon

and

I

is a

; therefore Q ~- S[y~

= Rp

N P

, and h e n c e

R. separable

remainder

z ~ I

that,

186

S[y~Q

=

D

g(D I) = g(E) (g(x)).

Chinese

and

of

J ; also

letting

D,

letting

E[y~,

JD I = M(DI)

Upon

g(Rp)

upon

in

is r e s i d u a l l y

that

b y the

such

), and

; consequently

that

= g(x)

ideal

in

Q

that

Then,

contains

= D I.

= the q u o t i e n t

is a f i n i t e

then we

in

ideal

onto

such

e ~'.

is a m a x i m a l

= E[y~j

assume x

J

g ( D I) = g ( E [ y ~

S[y~Q L(y)

To p r o v e can

ideal

lemma w e get

in p a r t i c u l a r

T h e n we

that,

is a q u a s i l o c a l

~'

(i),

N I, w e h a v e

D

set of all

is a m a x i m a l

To p r o v e

J = E[y~

= the

for all letting

theorem I

e Q

y = tz,

over Now we

S. Q

can

find

, and we have

3.8

y e R.

Clearly

y ~ P, g(Rp) = g(SsD P) (g(y)), and

y ¢ P'

for all

P' e ~'. (27.5) L, let

R

and let

LEMMA.

Le___~t S

me a n o r m a l

be an o v e r d o m a i n of P

S

b ~ a prime ideal in

domain w i t h q u o t i e n t

such that R.

R

field

is a finite S-module,

T h e n the f o l l o w i n g two c o n d i t i o n s

are equivalent:

(*)

(S N P)Rp = M(Rp)

(**) and

T h e r e exists

and

y ¢ R

Rp

is r e s i d u a l l y separable over

such that, upon lettinq

{(Y) = the m i n i m a l m o n i c p o l y n o m i a l of

S[y]Q = Rp

and

{' (y) ~ P ; (note that,

y

over

since

S

Q = S[y]

S.

Q P

L, we h a v e

is normal we h a v e

{(Y) ¢ slY]).

let

PROOF.

Follows

from

(27.6)

Le__~t S

be a normal domain w i t h q u o t i e n t

R

(27.3) and

be the inteqral c l o s u r e o f

extension

K

o__ff L.

(27.4).

S

A s s u m e that

i n a finite a l q e b r a i c

R

is a finite S-module.

this c o n d i t i o n is c e r t a i n l y satisfied, respect to some m u l t i p l i c a t i v e Let

P

be a Prime ideal i n

over

By

S.

Then:

N P

we have

S[y]Q = Rp

we h a v e

~ (Y) e~ S[Y]).

and

£(S,K) ~ P.

quently

K

such that D

is a q u o t i e n t ring, w i t h

(S n P)Rp = M(Rp) f~(S,K) ~ P ; K

D: K -- K

(27.5) there exists

Q = S[y]

ing

(Note that

with

an d

Rp

is s e p a r a b l e

D(SsQp)

c SSQ P ,

D(Rp) c Rp .

PROOF.

we get

S

such that

L ; and for every d e r i v a t i o n

we have

if

field

set, of an affine ring o v e r a field.)

R

is r e s i d u a l l y separable over

field . L, and

y ~ R

such that, upon letting

{(Y) = the m i n i m a l monic p o l y n o m i a l of and

{' (y) ~ By

Since

(27.2) we have {' (y) ~

is separable o v e r D (SsN P) c Ssn p , let

to the c o e f f i c i e n t s

P ; (note that since

in

L.

~3(Y) e S S Q p [ Y ]

187

L,

is normal

{' (y) ~ 0 ; conse-

G i v e n any d e r i v a t i o n

Then

over

{' (y) ¢ ~)(S,K), and h e n c e

P, we have

{ (Y).

S

y

D: K ~ K

be o b t a i n e d b y apply-

3.9

0 = D{(y)

therefore, S[y]Q

since

, we conclude

(27.7) degree

over

Ordv~(L

PROOF. V c

Then

By

~(K,k)\~.

such

that

E(K,k)

N V,K)

(27.6)

K = L(y)

such

Now,

a function

field

k,

and

let

a finite

Let

~

since

Rp =

.

K

b e the

~ M(W)

be

set of

o_~r W

is

all

not

set ., a n d

of t r a n s c e n d e n c e a lqebraic W

separable

c K(K,k)

residually

for e v e r y

such

separable

V e z(K,k)\A

= 0.

it f o l l o w s that

and

that

&

then,

= the m i n i m a l

~' (y) ~ 0

Rp.

be

is a finite

To p r o v e

~(Y)

we have

&

Dye

L

(L Q M ( W ) ) W

L D W.

we have

L.

, we get

;

c Rp

Let

of

+ {' (y)Dy

D(Rp)

a field

extension either

c R\P

that

LEMMA.

one over

field that

~' (y)

= 9(y)

is a f i n i t e upon

we

Q V,K)

set,

we

of

y

= 0

for all

can t a k e

y e K

letting

monic

and h e n c e

ordvO(L

polynomial

can

find

a finite

over

subset

L ,

~

of

that

~(Y)

¢

(L N V ) [ Y ]

and

~' (y) e V\M(V)

for all

w

V

Now by

(27.5)

(27.8) degree

w e get

DEFINITION.

one over

we mean

~ c ~

an e l e m e n t

k.

x

in

that

over

k(x).

Note

there

exists

a separating theorem,

erated

this

k;

K

also

be

K

is a f i n i t e

a function

such K

that

k

K

of

is p e r f e c t

follows

from

188

the

field

set.

of t r a n s c e n d e n c e

t r a n s c e n d e n t a ! of is s e p a r a b l e

is s e p a r a b l y

transcendental if

~

By a s e p a r a t i n ~

then:

F. K. S c h m i d t ' s over

, and hence

Let

a field

c i(K,k)\~

qenerated K/k.

then

Also K

following

K/k

algebraic over note

k that by

is s e p a r a b l y lemma:

gen-

3.10

(27.9)

LEMMA.

one o v e r a field (i)

For

able over

k,

some

the .[ollowing t h r e e V e Z(K,k)

(2)

K

i___s s e p a r a b l y

(3)

For i n f i n i t e l y m a n y separable

PROOF. separable

over

k

then,

x

each of w h i c h then c l e a r l y

of

K/k;

k(x),

by

(27.7) which

if

rational

is r e s i d u a l l y

w e have

S = k[x]

that

k

are

separable

we deduce

that

separ-

V

is

is r e s i d u a l l y x e M(V)\M(V) 2,

over

k(x);

then c l e a r l y

set of m e m b e r s k,

and,

if

is r e s i d u a l l y

infinitely

over

k;

there

are

thus

(2) and take a s e p a r a t i n g

is infinite,

over

separable

V

with

algebraic

Z(k(x),k)

are r e s i d u a l l y

~(k(x),k)

k

is finite,

separable

many members

therefore,

k.

of

over

Z(k(x),k)

upon t a k i n g

infinitely

over

of

L =

many members

The

of

implication

is of course obvious.

K

be a s e p a r a b l y

degree one over

d~(x) dx

are re e_qqivalent:

such that

is an infinite

§28.

there

V

degree

k.

(3), a s s u m e

now,

case there

are r e s i d u a l l y

(i)

(2) =

e v e r y m e m b e r of

which

Note

that

V ~Z(K,k)

is s e p a r a b l e

is r e s i d u a l l y

in e i t h e r

Let

conditions

qve[

upon t a k i n g

K

[ k [ X ] ( x _ a ) k [ x ] : a e k]

(3) =

o[ t r a n s c e n d e n c e

k.

To show that

transcendental

Z(K,k)

K

V ~ ~(K,k)

If there exists

(2).

k; thus

we have

generated

ove~

(27.6) we get that

(i) =

field

k.

residually

by

For a function

that

We o b s e r v e transcendentals

generated

function

field of t r a n s c e n d e n c e

a field. if

is a u n i q u e = ~' (x)

Differentials.

x

is a s e p a r a t i n g

derivation

for all

~(Y)

We also note

K/k

then

of

K ~ K, t o be d e n o t e d b y

K/k,

then

ddx _ ' such that

e k[Y~.

the chain rule: of

transcendental

If

for any

the c r i t e r i o n

x

and z ¢ K

y

are any we h a v e

for a s e p a r a t i n g

189

separating dz dx

dz dv dy dx

tran__@scendental:

" If

3.11

x

is a n y

have

separating

that:

z

transcendental

is a s e p a r a t i n g

As a c o n s e q u e n c e and

z

are a n y e l e m e n t s

cendenal then

of the

of

K/k

x + z

in

z

xz

K

then of

criterion

such

that

x

transcendental

made

above

K/k

z c K

that:

transcendental

of

K/k,

three

of

we

~ d ~ ~ 0.

is a s e p a r a t i n g

transcendental

in the

for a n y

we o b s e r v e

is not a s e p a r a t i n g

is a s e p a r a t i n g

observati0ns

K/k,

transcendental

above

is a s e p a r a t i n g

z ~ 0, t h e n The

and

of

and

if

x

transof

K/k,

if a l s o

K/k.

paragraph s may be used

tacitly. For (27.2)

the p r e s e n t

of

function

fields,

we

can

reformulate

thus: (28.1)

SPECIAL

DIFFERENT. separating

Let

and

~(X,Y)

.V ~ I(K,k)

be

that

(*)

x

CASE

transcendental

irreducible

Assume

situation

y

such is

Upon

(1)

y

be of

that

any

be

x ¢ V

integral

FORMULA

elements

K/k,

¢ k[X,Y]

ordv{(k[x,y]Q)

PROOF.

OF D E D E K I N D ' S

and

k(x,y)

such

that

and

y

over

k(x)

+ ordv~(k(x)

in

FOR CONDUCTOR

AND

K

x

= K.

~(x,y)

e V.

Let

n V°

such t h a t Let = 0.

nonconstant Let

Q = k[x,y]

~ M(V).

Then

q V,K)

= ordv~y(X,y )

letting

s = k(x)

N V

and ~(Y) we

clearly

= the m i n i m a l

monic

polynomial

have {(Y)/¢(x,Y)

~ s\M(v)

and h e n c e

(2) Upon

{' (y)V = letting

190

{y(X,y)V

.

of

y

is a

over

k(x)

3.12

= the i n t e g r a l by

closure

of

S

in

K,

(27.2) we h a v e {(S[y])~(S,K)

= {' (y)R .

and h e n c e

(3)

ordv{(S[y])

Upon

+ or~D(S,K)

in v i e w of

n M(v)

{(S[y])V

,

= {(S[y]p)V

;

clearly

(5)

S[y]p = k[x,y]Q (*) follows

(28.2)

V c I(K,k) k(x,y)

over

(**)

(i) to

CASE OF C O N V E R S I O N

PROOF.

Assume

that

y

k(y)

Q V.

Then

~(x,y)

(i)

ordv~(k[x,y]Q) (28.1)

= 0.

x e V

with

o_~f K/k

and

over

Assume

k(x)

y

n M(V). n V,K)

O r d v ~ ( k [ x , y ] Q) + ordv~O (k (y) N V,K)

191

that

n v, and

~ (X,Y) Then b y

i__{s

~ k[X,Y] (28.1)

we also get

= ordv~x~,y)

x

n V,K)

= ordv~y(X,y)

interchanged,

Let

and any

y e V.

irreducible

Q = k[x,y]

and

FOR DIFFERENT.

-- ord V ~ x + °rdv~(k(x)

+ Ordv~(k(x) x

y

is i n t e q r a l

Q V,K)

Let

FORMULA

and

We can take n o n c o n s t a n t

such that

By a p p l y i n g

x

such that

ordv~(k(y)

.

(5).

transcendentals

be q i v e n

= K.

inteqral

from

SPECIAL

any s e p a r a t i n q

(2)

S[y]

(3.1) we h a v e

(4)

Now

(y)

letting P =

also

= or~{'

we get

3.13

B y the c h a i n rule we h a v e 0 =

d ~ (x,v)

= ~ (x,y)

dx

d_z

~y(X,y)

+

dx

and h e n c e

(3) Now

ordv(x(X,y) (**)

follows

We want assumptions

from

(2) and

(28.3)

(28.1)

integralness

For this purpose we

such t h a t

k(x,y)

= K. {(x,y)

x

in t h e i r

Let

x

irreducible

= 0.

(K,k)

irreducible

integral over

k[z]

"

last b u t o n e

be

and

y

~(X,Y)

~ k[Z,Y]

; k[x,y]

= k[z,y]

sentences.

be any elements K/k,

x ~ V

z

of

such that: ; ordv ~

of

¢ k[X,Y]

such that

transcendental

~(Z,Y)

the

lemmas

transcendental

Let n o n c o n s t a n t V e

~dx

remain valid without

two e x c h a n g e

LEMMA.

Then there exists ' a separating constant

(28.2)

i~ a s e p a r a t i n q

Let

+ or%

(3).

and

made

first prove

FIRST EXCHANGE

i__nn K

that

(i),

to s h o w t h a t about

= ordv{y(X,y)

be and

K/k

~(z,y)

= 0 ; and

and such y ¢

v.

and a n o n = 0; y

i_~s

ordv~y(Z,y)

= ordv~y(X,y). PROOF.

We can find

y (Y) e k [ Y ] \ k

t h e n we can find a p o s i t i v e n m nO

integer

nO

such t h a t

y(y)

such t h a t

~ M(V),

for e v e r y

and

integer

w e have:

(i)

(2)

((Z + y(y)n,y)

is a m o n i c

polynomial

in

Y

over

k[Z],

d_z

o r d v [ n Y (y)n-Iy, (y) dx ] > 0

and

(3)

o r d v [ n Y (y)n-Iy, ( y ) ~ X ( x , y ) ~ > o r d v ~ y ( x , y )

(note t h a t

(3) c a n b e a r r a n g e d

because,

192

since

x

is a s e p a r a t i n g

3.14

transcendental Fix any

of

K/k,

integer

we have

(y(x,y)

n z n O , and

~ 0).

let

Z = X - y(y)n

Now clearly

k[z,y]

= k[x,y]

.

Also dz _ 1 - ny (y)n-iy, (y) dv dx dx and h e n c e

by

(2) w e h a v e dz

° r d v ~xx = 0 ; consequently,

dz d~x ~ 0, a n d h e n c e

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

transcendental

of

K/k.

obviously

Clearly tegral

~(Z,Y)

9(z,y) over

by

~(Z + ~ ( y ) n , y )

is a n o n c o n s t a n t

B y the

qy(Z,y)

and h e n c e

=

= 0, and h e n c e

k[x].

(3) w e

chain

x

and

y

o_~f K / k

and

y c V.

K/k

with

k(x)

N V

over

SECOND

Then z e V

; x

k(y)

N V

(y){X(x,y)

and

V e I(K,k)

Let

any be

; and

that:

y

over is

is in-

+ ~y(X,y)

separatinq qiven

t h e r e .exists a s e p a r a t i n q

is i n t e q r a l

y

= o r d v ~ Y(x,y)

LEMMA.

such

see t h a t

that

EXCHANGE anv

(i) w e

polynomial.

rule we g e t

= ~y(y)n-i,

conclude

e k[Z,Y]

irreducible

in v i e w o f

o r d v q Y(z,y) (28.4)

is a s e p a r a t i n g

Let

~(Z,Y)

Then

z

= K

; z

k(z)

q V

; k(y,z)

over

k(z)

193

s u c h that

transcendental

k(x,z)

inteqral

transcendentals

is i n t e q r a l = K

N V.

; z

x ~ V z

o__ff

over i s inteqral

3.15

PROOF.

Let

A(u)

= [W ~ Z(K,k)

: u ~

W]

for e v e r y

u C K

and

r(u,w)

= {w'

~ ~(~,k)

: W'

o k(u)

= W n k(u)]

for e v e r y

u e K

and e v e r y r

(u,w)

= F(u,w)\{w]

w

~ ~ (K,k)

Let

(i)

~ = A(x)

Then,

since

x .c V

and

U F * (x,V)

y ~ V, w e h a v e

(2)

V ~ Q

Now

~ O {V]

(27.7),

(3)

is a f i n i t e

we can

I

subset

and

J

of

Z(k,k)

in

~ (K,k)

and hence, such

(F(x,I)

U F(x,J))

(4)

(k(x)

N M(I))I

= M(I)

(5)

(k(y)

N M(J))J

= M(J)

such

n F(y,J)

.

= # =

and

F(x,I)

find

U F* (y,V)

U 5(y)

that,

upon fI

to be

the c a n o n i c a l

(6)

fi(I)

in v i e w

that

N

(Q U

{V])

letting for

: I - I/M(I) epimorphisms,

is a f i n i t e fI(k(x)

fj

we have

separable

: J ~ J/M(J)

that

algebraic

field

extension

of

algebraic

field

extension

of

n I)

and

(7)

fj(J)

is a finite fj(k(y)

separable Q J)

194

of

3.16

Clearly

(8)

F(x,I)

and h e n c e

we

can

find

~ M(W) and

then upon

U F(y,J)

~ ~ k[x]\k

for all

W

~

is a finite

such

e F(X,I)

set

that

U T(y,J)

O Q U

Iv]

,

letting

D = the we have

U [V]J

that

D

integral

closure

is a D e d e k i n d

domain

of

k[i/~]

with

in K

quotient

field

K

such

that

(9)

D c W In v i e w

of

for all

(9) w e h a v e

W

e F(x,I)

D c I

O F(y,J)

and h e n c e

O ~ U [V]

in v i e w

of

(6) w e

can

find

(io)

a ~ D\M(I)

In v i e w

of

such

that

(9) w e h a v e

fi(I)

D c J

= fi(k(x)

and h e n c e

N I) (fi(a))

in v i e w

of

(7) w e

can

find

(11)

b e D\M(J)

In v i e w theorem we

of

can

(2),

such

(3),

(8) and

fj(J)

= fj(k(x)

N J) (fj(b)).

(9), b y the C h i n e s e

remainder

find

(12)

such

that

teD

that

(13)

t ~ M(V)

(14)

t - a e M(I)

and

t (~ M(W)

for all

W e Q

, w

and

t e M(W)

and

195

for all

W

e F

(x,I)

,

3.17

(15)

t - b e M(J)

N o w by

(i) and

and

(9) to

t e M(W)

for all

W ~ F

(y,J)

(15) w e g e t that:

(16)

t 6 M(V)

and

t 6 W\M(W)

for all

W ~ ~ (x,V) U A(X)

(17)

t 6 M(V)

and

t e W\M(W)

for all

W 6 £

(18)

t c M(W)

for all

W

and

W c V (x,I),

fi(I)

= fi(k(x)

(y,V) U A(y)

t e I\M(I), N I) (fi(t))

and w

(19)

t c M(W)

for all and

In v i e w of

x

= fj(k(x)

find

for all

find a

6nx 6 M(W) since

fj(J)

(8) w e can

c M(W) and then w e c a n

W ~ r (y,J),

for all

is a s e p a r a t i n g

N J) (fj(t)).

~ e k(x)\k

W e ['(x,I)

9ositive

t c J\M(J),

such that

U F(x,J)

integer

n

W c F(x,I)

transcendental

U ~ U IV]

such

that

U F(x,J)

U ~ U [V]

of

upon

K/k,

letting

2

I ~n

, if

K

is s e p a r a b l e

over

~nx , if

K

is not s e p a r a b l e

k(~)

Y

we clearly

(20)

over

for all

W e F(x,I)

U V(x,J)

U ~ U {V}

and

y

(21) Upon

k(~)

get that

y 6 M(W)

is a s e p a r a t i n g

transcendental

letting

196

of

K/k.

;

3.18

z=It

,if

t+y,if by

(i) a n d

(16) to

t

is a s e p a r a t i n g

t

is not

transcendental

a separating

of

K/k

transcendental

of

K/k

(21) w e g e t that:

(22)

z

(23)

z e M(V)

and

z ~ W\M(W)

for a l l

W e F

(x,V)

O A(x)

(24)

z e M(V)

and

z ~ W\M(W)

for all

W ~ F

(y,V)

U A(y)

(25)

z ~ M(W)

for a l l

is a s e p a r a t i n g

and

transcendental

W ~ F

fi(I)

(x,I),

= fi(k(x)

of

K/k

z ~ I\M(I)

,

Q I)(fi(z))

and

(26)

z ~ M(W)

for all and

In v i e w of Since x

y

k(z)

transcendentals such t h a t

over

and and

ordvO(k(y) PROOF.

k(x)

x

x e V

(**)

K/k D V

By

with ; x

k(y)

k(z)

CONVERSION

z

(5) and

N V

z ¢ V

FORMULA

y

N V,K) there

(26) w e get t h a t z

K/k

= K.

k(x)

N V, and

k(y,z)

is i n t e g r a l o v e r

= K.

k(y)

N V, and

Let any separatinq

add any

V e I(K,k)

be qiven

Then

= o r d v ~ x + °rdv~D(k(x) exists

such that:

y

is i n t e g r a l o v e r

FOR DIFFERENT.

of

is i n t e g r a l o v e r ; and

k(x,z)

N V.

y ¢ V.

(28.4)

n J) (fj(z))

(25) w e get t h a t

y ~ V, l~y (24) we see t h a t

(28.5)

z ~ J\M(J)

n v.

(27.4), b y

is i n t e g r a l o v e r

of

(4) and

(23) w e see t h a t

is i n t e g r a l o v e r In v i e w o f

(y,J),

fj(J) = fj(k(y)

(27.4), b y

x e V, b y

Since

W ~ F

a separating

k(x,z) k(z)

Q V

is i n t e g r a l o v e r

197

N V,K)

.

transcendental

z

= K ; z

is i n t e g r a l o v e r

; k(y,z)

= K ; z

k(z)

N V.

is i n t e g r a l

By a p p l y i n g

3.19

(28.2) to the pair

(x,z)

(i)

n V,K) = ord V ~dz x + or~©(k(x)

or~Q(k(z)

By applying

(2)

we get

(28.2) to the pair

or~©(k(y)

q V,K)

(z,y) we get

N V,K) = ord V dd--~z+ ordvO(k(z)

N V,K)

By the chain rule we have

d r = dvd.__~z dx

dz dx

and h e n c e

(3)

or~

NOW

(**) follows (28.6)

and

y

that

(2) and

K

k(x,y)

be such that Let

(3).

FORMULA FOR CONDUCTOR AND DIFFERENT. in

K/k, and

e k[X,Y]

Q = k[x,y]

such that = K.

By

isa

~ (x,y) = 0. N M(V).

Let

irreducible

= 0, y is integral over

(i)

k[x,y]

Let

x

separatinq transirreducible

v e ~(K,k)

be such

Then

N V,K) = ordv~y(x,y)

(28.3) there exists a separating

and a nonconstant

~(z,y)

x

Let nonconstant

ordv{(k[x,y] Q) + ordvQ(k(x) PROOF.

K/k

(i),

~elements

x,y e V.

(*)

from

DEDEKIND'S

cendental of ~(X,Y)

dd-~x= o r d V ~ z + o r d V 3dz gx"

~(Z,Y)

transcendental

¢ k[X,Y]

z

of

such that

k[z],

= k[z,y]

,

and

(2)

dz

°rdv ~ x = 0

N o w in p a r t i c u l a r k(z)

and

k(z,y)

OrdV~y(Z,y)

= ordv~y(x,y)

= K, z ~ V, and

N V ; consequently b y applying

198

y

is integral over

(28.1) to the pair

(z,y),

in view

3.20

of

(I) w e g e t

(3)

ordv~(k~x,y~Q)+

By a p p l y i n g (4) Now

(28.5)

ordVD(k(z) (*) f o l l o w s (28.7)

(~,x)

ordvO(ki(z)

to the p a i r ~ V,K)

from

(2),

~

ordv~

(x,z)

= ordv~y(Z,y)

(3) and

x

Q V,K)

(4).

F o r any

and

V e ~(K,k)

in

and any

x

if

is a s e p a r a t i n g

=

and

x

is a s e p a r a t i n g

if

of

~ ~ 0

transcendental

and w e n o t e

of

is not a s e p a r a t i n g

K/k

~ = 0

transcendental

of

K/k

that then:

ordv(~,x) ordv(~,x) of

x

K/k x ~ V

, if e i t h e r or

pair

~ ~ 0, x ¢ V ,

transcendental

ordv~. + ordv~](k(1) N V,K) + o r d v x 2 ordv(~,x)

(ordered)

K, w e d e f i n e

+ ordvig (k (x) Q V, K) and

.

w e a l s o get

dz + o r d ~ ( k ( x ) = ord V ~xx

DEFINITION.

of elements

n V,K)

= an i n t e g e r or

~ ~ ~ ~ ~ 0

and

x

~ ,

is a s e p a r a t i n g

transcendental

I
and ordva + ordv(~,x) (When w e w a n t ordV,k(~,x)

to m a k e an e x p l i c i t instead

REMARK.

let

~

!(K,k).

of

" ~dx

LEMMA.

an_dd y

" after

Let

for all

reference

a e K . to

k

we may write

o r d v ( ~ , x ) .)

The r e a d e r m a y k e e p

be r e p l a c e d by

(28.8)

= ordv(a~,x)

x

in m i n d

that

its d e f i n i t i o n

in

(~,x)

is i n t e n d e d

(28.13).

be any s e p a r a t i n g ....... transcendental

be a n y e l e m e n t s

in

Then

199

K, and

let

to

V

of

be any member

K/k, of

3.21

(1)

°rdv($'Y)

= °rdv ~Id Ix '

and

(2)

= °rdv I~ ddxy )

°rdv(~'Y) PROOF.

dental are

of

If either K/k

~

(28.5)

x e V

(28.5);

y ~ V

y ~ v

then that

and obviously

(28.9)

If

if

dx

(28.10)

~ ~ 0 and

and

(2) follows

If

for some

and

y c V

and noting

y

that

dy dx

from

i) follows

(i) follows by apply-

dy _ x 2 dy ; if dx-i dx (28.5) to the pair

by applying

finally,

if

to the pair

This completes

"

(2)

is a separating

then

then

(28.5)

transcen-

as both sides of

y c V

(i) follows by applying

and

Fellows

x ~ V (x-l,y-1)

the proof of

(i)

(i).

x,y,~,8

are any elements

(and hence every)

= ordv(~,x)

separating

If

~

~

and

for all

by twice applying

LEMMA.

in

K

such that

transcendental

z

x

and

y

V e I(K,k)

formula

(i) of

(28.8).

are any separating

are any nonzero

elements

of

transcendentals

K, then:

O~c~

(28.10.1) (28.10.2)

is not a separating

dy-i dy ; and, dx = - y -2 dx

dy -I = y-2x2 dx-i

0 ~ ~ c K.

then

PROOF.

K/k

x c V

x ~ V

for all

(i) as well

(i) follows

that

Ordv(8,y)

of

y

that

(x-l,y)

then

LEMMA.

= ~ ~ K/k

K/k.

and noting

and noting

of

of

suppose

to the pair

and

(x,y -I) and

from

or

then both sides of

transcendental

ing

~ = 0

So henceforth

directly

+ ordv(~,x)

°rdv(~'Y)

= °rdv (~-6xl d + ordv(~,x)

200

for all

v ~ ~ (K,k)

3.22

and [ o r d v ( 8 , y ) ][V/M(V)

: k]

V ¢ I (K,k) =

~ [or~(~,x) V ¢ I(K,k)

][V/M(V)

: k]

(28.10.3) = an i n t e q e r .

PROOF. of

(28.10.1)

(28.8).

In v i e w

of

is o b v i o u s . (28.10.1)

(28.10.2)

and

follows

(28.10.2),

from

28.10.3)

formula follows

(2) from

(4.1).

(28.11) number,

DEFINITION.

t__oob e d e n o t e d

By

(28.10.3)

b y g e n u s (K,k),

[ordv(~,x)][V/M(V) Ve I(K,k) for e v e r y element

separating ~

in

(28.12)

there

exists

a unique

rational

such that

: k] = 2 g e n u s ( K , k )

transcendental

x

of

K/k

- 2

and

every

nonzero

K.

THEOREM.

If

K = k(x)

for some

x e K

then

g e n u s (K,k) = 0.

PROOF. for a l l in

Now obviously

V e

I(K,k),

(or,

and h e n c e

say b y our

(27.6))

assertion

ordv~(k(x)

follows

by

N V,K)

taking

= 0 ~ = 1

(28.11).

(28.13) o f the

form

DEFINITION. ~dx

with

a

By and

a differential x

in

of

K, w h e r e

K/k we mean these

an o b j e c t

objects

are

to

satisfy:

~dy = ~ d x

~ B

dx = a ~z

for

some

separating

(and h e n c e

every)

transcendetal

z

of

K/k. In o t h e r w o r d s ,

in t h e

set o f a l l p a i r s

201

(~,x)

of elements

in

3.23

K

we

introduce

(8,Y)

~

the

equivalence

(~,x)

~

relation

dx = ~ d~z

~

for

some

ting

and

then

we

let

~dx

stand

for

the

(and h e n c e

every)

transcendental

equivalence

z

class

separa-

of

K/k

containing

(~,x). (When w e w a n t could

x

write By

dx

We

let

in

K

something

we

mean

idx.

0

stand

for

then

have:

of

K/k.

space

convert

by

all

K/k. 8,~

*

+ ~ dx

~,~

Note ,Y,Y

dy

that

,u

observe

by

dx

=

~z

,z

these any

also

reference

instead

and

we

0 ~ ~ ~

of

to

K

and

k,

we

for

any

~

and

~dx.)

observe

that

0

x

and

differentials

other

~*

du = the

and

that

note

transcendental

dx

*

K

dz

of

with

and

z

is a s e p a r a t i n g

K/k

into

are

elements

d~z +

y(~dx)

a K-vector-

K

such

is g e n e r a t e d

d----~ dz

of

the

for

any

that

for

z

of

any K/k

by

form of

~dx

i.e.,

that

and

any

of

nonzero

with

if

8dy = ~dx of

(yS)dy

the

of

=

said

member

0 ~ G

¢ K

K

have

K/k,

then

(y~)dx

K-vectorof

and

it, x

i.e., a

K/k.

x

and

x

) = xdx

x

transcendental

transcendental

K-vector-space-dimension

it

(y~)dx

"well-defined",

in

~

=

a separating

is a s e p a r a t i n g

transcendental

note

+ ~

definitions

d(xx

We

~

all

in

u

differential

separating We

~dx

, and

that

is one,

any

*

are

= ~ dx

space

~dK/kX

0d0,

set of

,y,x,x

+ We

explicit

defining

~dx

for

the

an

like

we

transcendental we

to m a k e

¢ K, we

in

+ x dx

any

have

202

we

.

~(Y)

¢ k[Y~,

and

any

separating

3.24

d D(x)

<~ ' (x) dd--z ~ I d z = ddx q(x)

= ~' (x)dx =

For any differential

~dx

of

K/k

and any

dz

.

V e I(K,k)

we

define

ordv~dX = Ordv(~,x) and we n o t e that, i.e.,

:

in v i e w of

(28.9),

8dy = ~ d x = o r d v ( ~ , y )

this d e f i n i t i o n

= O r d v ( ~ , x ).

is " w e l l - d e f i n e d '

We o b s e r v e

o r d v ~ d X = an i n t e g e r or

that:

~ ,

o r d v ~ d X / ~ ~ ~dx ~ 0 , and o r d v Y (~dx) = O r d v Y we also observe

that

if

+ Ordv~dX

~ dx

for all

7 ¢ K ;

is a n y o t h e r d i f f e r e n t i a l

of

K/k

then: W

ord v ( ~ d x

*

W

+ ~ dx ) ~ min(ordv~.dx

W

, o r d v ~ dx

)

w

with equality

if

(28.14) (i.e., w i t h

Ordvctdx @ o r d v ~ dx

THEOREM. ~ ~ 0

F o r an V

and

x

~

and

a separating

x

i__nn K

with

transcendental

~dx ~ 0 of

K/k)

w__ee

FOR CONDUCTOR AND DIFFERENTIAL°

Let

have (or~dx)[V/M(V)

: k] = 2 g e n u s ( K , k )

- 2

V ¢~ (K,k) PROOF.

This

(28.15)

DEDEKIND'S

x

and

x

is a s e p a r a t i n g

ble

y

~ (X,Y)

degree of

is s i m p l y a r e s t a t e m e n t FORMULA

b e any e l e m e n t s

K

such t h a t

transcendental

e k[X,Y~ ~ (X,Y).

in

be

of

of

such that

Let

203

K/ko

(28.11).

k(x,y)

= K.

Assume

Let n o n c o n s t a n t

~ (x,y) = 0.

Let

that

irreduci-

n b e the

(total)

3.25

W = {V c I (K,k): x e V

and

y e V],

W' = [V ~ I(K,k) : x ~ V

and

yx -I c V], and

W

and

xy -I e M(V)]

= {v e I(K,k):

(Note that then clearly: W' N W

= ¢.)

s (v)

For any

=

y ~ V

W = 9(k[x,y]), V ~ I(K,k)

W' U W

, and

let

k[x,y]

if

k[x-l,yx -I]

i_~f V c W'

k[y-l,xy -I]

f

V e W

V c W*

R(v) = s(V)s(v)

= I(K,k)\W

and

n M(V)"

Also let R = k[x,y]

and

R

= the integral

closure ' of

k[x,y]

i__nn K.

Then we have

(28.15.1)

ordv~(R(V))

{

+ or~dx

- ordv~y(X,y )

~n_3)ordVX_ 1

for all for all

V c W V ¢ W'

(n-3) ordVy-i

for all

V ~ W*

=

2 +

(28.15.2) I

~ [ordvdX Vc~ (~,k) \~ (R)

= 2 genus(K,k)

- or~;y(X,y)][V/M(V)

+

5~ [ord~(R(V))][V/M(V) vc!9 (R) + [R*/~(R) : k]

= 2 genus(K,k)

: k~

: k]

and (28.15.3)

2 genus(K,k)

= (n-l) (n-2) -

(Note that since

~ [ordv~(R(V))][V/M(V) VcI(K,k)

K = k(x,y)

204

is separably

generated

: k] .

over

k, by

3.26

Maclane's of

K/k;

theorem

consequently,

assumption

of

PROOF. (i)

either

x By

x

or

y

(28.15.3)

being

is a s e p a r a t i n g

would

a separating

remain

transcendental

valid w i t h o u t

transcendental

of

the

K/k.)

(28.6) we get

ordv~(R(V))

+ ordVdX

- ordv~y(X,y)

= 0

for all

v e W.

Now ordv~y(X,y)-idx

= ordvdX

- ordv~y(X,y )

for all and hence

by taking

2 genus(K,k)

{y(X,y) -I = 2 +

V c I(K,k) for

~ [ordvdX VoW

~

in

(28.14)

- ordv{y(X,y)

we get ][V/M(V)

: k]

(2) + In view of (3) Now

(3.1)

and

[R /~(R) (28.15.2) Given

from

V ~ W'

we clearly

have

= 0, and h e n c e

(x-l,yx -I)

we get

ordv{(R(V))

: k].

[ordv~(R(V))][V/M(V)

(2) and

: k].

(3).

letting

= xn{ (X-I,yx -I)

that n o n c o n s t a n t

{(x-l,yx -I)

(41 )

~ V~W

(I),

, upon

(X,Y)

- ordv~y(X,y)][V/M(V)

(5.5) we have

: k] =

follows

any

~ ,[ordvdX VeW'UW

irreducible

by a p p l y i n g

+ ordvdX-i

now dx -I = _x-2dx

205

(28.6)

~(X,Y)

¢ k[X,Y]

to the pair

= ordv{y(x-l,yx-i

) ;

with

3.27

and hence o r % d x -I = o r % x -2 + o r % d x

(42 )

;

by the chain rule we have (43 )

{y(x-l,yx -!) = xl-n
and hence (44 )

ordv~y(x-l,yx-l) By (41),

(4)

--- ordVX-2 + ordvx3-n + ordv{y(X,y)

(42 ) and (44 ) we conclude that

ordv~(R(V))

+ ordvdx - ordv
= (n-3)ordVX-i

for all

V ~ W' Now let

V e W

be given.

(X,Y)

Upon letting

yn{ (Xy-i,y-l)

=

we clearly have that nonconstant 9(xy-l,y -I) = 0.

Now (by Maclane's theorem)

ing transcendental K/k.

If

xy -I

of

K/k

or

y-i

~(X,Y) ~ k[X,Y] either

ordv~(R(V))

(xy-l,y -I)

xy -I

of

we get

since =

d(xv -I) dx = ly-i - xy -2 dd~xx~ dx dx

we get (*2)

ordvd (xy -I) = ordv
-

xy

+ ordvdX ;

by the chain rule we have (*3)

~y(Xy-!,y -I) = y - n [ - x y & (x ,y) - y2{y(X,y) ]

206

is a separaof

K/k, then by apply-

+ ordVd(xy -I) = ordvgy(xy-l,y-I ) ;

d(xy -I)

and

is a separating transcendental

is a separating transcendental

ing (28.6) to the pair (*i)

irreducible

3.28 and d~ (x,v) 0 = dx = ~X (x'y) + ~y(X,y) dv dx ;

(*4)

by (*3) and (*4) we get (*5)

9y(xy-l,y -I) = - y3-nly-i - xy -2 dd~x~y(X,y)

and hence (.6)

ordv~]y(Xy l,y-l) = ordv y3-n + ordv

- xy

+ ordv{y(X,y)

;

now by (*i), (*2) and (*6) we get (*)

If

ordvG(R(V)) y-i

is a separating transcendental of

(28.6) to the pair (**i)

+ ordvdX - ordv(y(X,y) = (n-3)ordvy-I

(y-I xy-l)

ordv{(R(V))

K/k

then: by applying

we get

+ ordvdy-i = ordv~x(xy-l,y-I ) ;

since dy-i =

d.y.-I dv dx = - y-2 ~x dx dy dx

we get (**2)

°rdvdy-i = °rdvy-2 + °rdv ~x + o r ~ d x

;

by the chain rule we have ~x(xy-l,y -I) = y l-n.6X~"x ,Y )

(**3) and (**4)

0 = d{(x,v)

dx

= Cx(X,y)

+ Cy(X,y)

by (**3) and (**4) we get

207

dv

dx

;

3.29

(**5)

~x(xy-l,y -I) = -

y

1-n. "x ) dX £yt ,Y dx

and hence (**6)

ordv~x(XY

now by

,Y

) -- ordVy3-n

(*'1) , (**4) and

(**)

ordv~(R(V)) Thus

(5)

+ Ordv~y(X,y)

- Ordv~y(X,y)

in the above paragraph

we have

(i),

=

(4) and

(5)we

get

(n-3)ordVy-i

shown that

+ ordvdX - ordv~y(X,y) for all

By

;

(**6) we get

+ ordvdX

ordv~(R(V))

or( 1

+

-- ( n - 3 ) o r ~ y

-I

V c W

(28.15.1).

By

(i),

(2),

(4) and

(5)

we get I 2 genus(K,k) (6)

= 2 +

(n-3) [ ~ (ordvx-l)[v/M(V) V~W'

-

Let

=

~ [ordv~(R(V)) ~[V/M(V) V~ I(K,k)

A = k[X,Y,Z~

in the obvious

manner.

such that

~ (X,Y,I)

C e ~ I(A)'

and by

consequently

A

We have a unique

nonzero

Deg[A,C~ ~ = ZA, we have by Bezout's ~ (15.4)

.

to be a homogeneous

Upon letting

~ (X,Y,Z)

C -- ~ (X,Y,Z)A

domain

~ Hn(A) we have

(25.9) we get

(8) In view of

: k~

where we regard

= ~ (X,Y).

(7) Upon letting

~ ,(ordvy-l) [V/M(V) V~W

: k~ +

and

= n. ~ ¢ H

(A)

little theorem ([A,C~,~)

with

(23.9) we get

= Deg[A,C~

(15.5) we easily

208

DegA~ = 1

o

see that

and

~ ~ C ;

3.30

([A,C],~) (9) =

~

(or~x-l)[v/M(V)

: k] +

~ ,(ordvy'l)[v/M(V)

V~W'

Now

(28.15.3)

follows

(28.16)

REMARK.

(28.15.3)

: k]

.

V~W

is thus.

from

Firstly,

(28.15.1)

(7),

(8) and

A slight variation

graph we can "identify" check that

(6),

with

A,9

R([A,C])

2s161{

of the above proof of

and

with

is equivalent

(9).

C

as in the last para-

K, and then we can either

to

Ordvdx = ord ([A,C],~y(X,Y,Z),V) - or~([A,C],Z2,V) for all

or, alternatively, as a proof of

9y(X,y,l)

- ordv~(R(V))

V e I(K,k)

,

we can also read off the above proof of

(28.16.1) ; namely:

-- {y(X,y)

because by definition

\9 (x,y),l) X29y(l,yx-l,x -I) = -

Iby chain rule,

formula

Xgx(X,Y,Z) and hence,

= ¢ (x,y)

{y(x-l,yx -I) dx-z--- because clearly dx -I <)9(l,yx-l,x -I) = ~(x-l,yx -I)

= x3-n~y (x, y) by Euler's

(28.15.1)

+ Y%y(X,Y,Z)

assuming

+ Zgz(X,y,z)

y ~ 0,

209

= n@(X,Y,Z)

see

(44 ) ;

3.31

y2~y(xy-l, l,y -I) = -XY~x(xy-l,y-i ) _ y~y(xy-l,y -I)

because

clearly

~(xy-l,l,y -I) = ~(xy-l,y -1)

I~y(xy-l,y-l)~_l

• y --I is

, if

ay

a separating

transcendental

of

K/k

because (xy-l,y -I) = 0

|

,

~x~XY

-i

eY

-l,dx

-i

if xy is a separating " transcendental of K/k

}I~ y

3-n

{y(X,y)

by chain rule,

see

(*5) and

(**5);

etc. Secondly,

by Bezout's

Little

theorem

(23.9) we have

D ord([A,C],~y(X,Y,Z),V)[V/M(V) VCZ (K,k)

: k] = (n-l)n

and ord([A,C],Z

2,V)[V/M(V)

: k~

= 2n

.

W~(K,k)

and by

(28.14) we have [ordvdX][V/M(V)

: k] = 2 genus(K,k)

- 2 ;

Ve~(K,k) consequently,

upon multiplying

and then summing over all

2 genus(K,k)

both sides of

V e I(K,k)

- 2 = (n-l)n-2n

(28.16.1)

by [V/M(V):k]

we get

-

[ordv~(R(V))][V/M(V)

: k]

VeI(K,k) and hence 2 genus(K,k) (28.17) V e ~(R), R

by

= (n-l) (n-2) REMARK° (28.15.1)

Let the notation be as in (28.15). we see that

and not on its particular

be shown directly

~ [ordv~(R(V))][V/M(V ) : k3 Ve~ (K,k)

or~y(X,y)-idx

generators

x

and

depends

only on

y; this can actually

(see §2 of [4 ]); also one can easily 210

For any

see that:

3.32

o r d v ~ y ( X , y ) - i d x = 0 ~ R(V) can guess at D e d e k i n d ' s

is regular

formula

ing in

(27.2),

x In

and

(28.19) and

to those m e m b e r s of

ial from

I(K,k)

(28.18),

(28.4) to

p a r a m e t e r of

and not on its p a r t i c u l a r gener(see §2 of [4]).

w h i c h are r e s i d u a l l y s e p a r a b l e over

k.

(28.20) are i n d e p e n d e n t of the mater-

(28.5). Given

V e

we m e a n an element of

inq coordinate of

Let us

(28.17) except that in an a l t e r n a t i v e p r o o f of

DEFINITION. V

(28.6).

and

(28.20) we shall give special a t t e n t i o n

(28.19) and

(28.19.5) we shall use (28.18)

R

y; this too can be shown d i r e c t l y

(28.18),

These items

(28.1), and

one

(28.15.2) we see that the first s u m m a t i o n o c c u r r -

(28.15.2) depends only on

ators

= 0; thus,

o r d v ~ y ( X , y ) - i d x = ordv~(R(V))

h e n c e also guess at his formulas also remark that b y

~ Ordv~(R(V))

V/k

I(K,k), by a u n i f o r m i z i n q M(V)\M(V) 2, and by a u n i f o r m -

we m e a n an element

x ¢ V

such that

(k[x] n M ( V ) ) V = M(V); we note that every u n i f o r m i z i n g p a r a m e t e r of V

is c l e a r l y a u n i f o r m i z i n g c o o r d i n a t e of

(28.19) ble over

LEMMA.

k, w e h a v e t h e

(28.19.1) t

For any

I_~f t

V ¢ i(K,k)

x ¢ V

(28.19.2)

i s gny u n i f o r m i z i n q c o o r d i n a t e 0f

for every_

we h a v e

I_~f t

0 ~ x ~ K

w h i c h is r e s i d u a l l y separ-

f011owinq.

is a s e p a r a t i n g t r a n s c e n d e n t a l of

for ever V

V.

K/k, o r % ~ ( k ( t )

V/k,

then

N V,K) = 0, an__~d

dx ~ e V.

is any u n i f o r m i z i n g c o o r d i n a t e of

V/k,

then

we have:

o r d v d__xx= dt

(ordVX) - 1

if

ordvx ~ 0

dx ord v ~ >

(ordVX) - 1

i_~f ordvx -= 0

(characteristic of

k)

and

(28.19.3)

I_~f t

(characteristic of

is any u n i f o r m i z i n ~ c o o r d i n a t e of

211

V/k

k). and

u

3.33

is any element of

U

K

then:

is a u n i f o r m i z i n q c o o r d i n a t e of and

(28.19.4) of

If

t

or~

and

v/k, then for e v e r y

u

~du=

V/k ~ u ~ V

0.

are any two u n i f o r m i z i n q c o o r d i n a t e s

x ~ K

we h a v e

o r d v dx

--

7ju

(28.19.5)

If

t

Ordv~k(x)

If by

t

(28.19.1) and

u

n V)

follows by t a k i n g

if

x c V

if

x 6E V.

S = k[t]

d__u_u dt ~ V

and

dt ~u e V

and b y the chain rule we h a v e

du dt d-Tdu = i ; c o n s e q u e n t l y we must have

moreover,

du dt ord v ~ = 0 = ord v d~u ; for any

x e K, by the chain rule we have dx dx du dt - du dt

and h e n c e by

(12 ) w h i c h proves

in

(! I) we get o r d v dx ~U

=

dx ° r d v d-~

(28.19.4).

212

an d

(27.6).

are any two u n i f o r m i z i n g c o o r d i n a t e s of

(28.19.1) we have

(i i )

V/k

K/k, then

ordv~O(k (x) n V) + ordvx2

PROOF.

dx

~-

iS any u n i f o r m i z i n g c o o r d i n a t e of

is any separatinq t r a n s c e n d e n t a l of

o r d v dx dt

or~ =

V/k

then

3.34

In the

if

above

t

paragraph

and

u

are

we have

any two

shown

that:

uniformizing

coordinates

of

V/k

then

(2) du = 0 = or% ord v ~-~

TO p r o v e and

any

(28.19.2)

0 ~ x ~ K

ing R a r a m e t e r

u

be of

"

let a n y u n i f o r m i z i n g given;

V;

let

then

coordinate

n = ordvX

x = 6u n

8 c V \ M (V)

(31 )

~dt u

and

where

t

of

V/k

fix any u n i f o r m i z -

n

is an i n t e g e r

and

;

now ~,t32~

by

(28.19.1)

d__86 due

and b y

u n ~u

+ 8nun-i

1

dtd--uu;

we have

(33)

V

(2) w e h a v e du ord v ~ = 0 ;

(34) by

d8

dx _ dx du dt d u dt

(3 l) tO

(34 ) we

see that

dx ordv ~ = n - 1

if

(characteristic

n ~ 0

of

k)

and dx Ordv ~

this

completes Since

an e l e m e n t

the p r o o f

every of

V,

To p r o v e

the

of

and any

(41)

V/k

> n-i

if

of

uniformizing the

reverse

n m 0

of

k)

;

(28.19.2). coordinate

implication implication,

u ~ V

(characteristic

" = " let

with

du ord v ~-~ = 0

213

of of

V/k

is, b y d e f i n i t i o n ,

(28.19.3)

any u n i f o r m i z i n g

follows

from

coordinate

(2). t

3.35

b e given;

then

transcendental x ¢ k[u]

~

of

letting

K/k

; upon

@ 0

and h e n c e

u

W = k(u)

is a s e p a r a t i n g

Q V, we can take

with

(42 )

ordWX = 1 ;

now clearly

W e ~(k(u),k)

is a u n i f o r m i z i n g applying

is r e s i d u a l l y

coordinate

(28.19.2)

to

W

of

W/k,

separable

and h e n c e ,

over

k

in v i e w of

and

u

(42 ), b y

we get dx

(43) by

du

in p a r t i c u l a r

ord W ~

= 0 ;

(42 ) w e g e t

(44)

ordVX

and b y

> 0

(43 ) we g e t

(45 )

or~

dx = ~-ju

0

;

b y the c h a i n r u l e we h a v e

dx

dx du du dt

dt

and h e n c e b y

(41) and

dx ord v ~ = 0 ;

(46) now,

in v i e w of

we deduce

(44 ) and

(45), this time b y a p p l y i n g

(28.19.2)

to

V

that

(47 ) since

(45 ) we get

ordVX = 1 ; x ¢ k[u],

ordinate

of

by

V/k.

(47 ) we c o n c l u d e This c o m p l e t e s

For any u n i f o r m i z i n g transcendental

x

of

K/k

that

u

is a u n i f o r m i z i n g

the p r o o f of

parameter

t

with x ~ V

214

of

V/k

we h a v e

co-

(28.19.3). and any s e p a r a t i n g that

x-lis a separating

3.36

transcendental

of

K/k, x -I c V

and

-Idt °rd v ~dx = ordv x2 + ord v dx consequently,

to prove

(28.19.5)

I for every u n i f o r m i n g (*)

separating

it suffices to show that:

coordinate

transcendental

x

t of

of

for

(x,y).

(28.19.1),

Now,

(*) follows

So let any u n i f o r m i z i n g x

of

integral closure of the canonical in over

R

K/k k(x)

different

from

M(V)

(28.5) by taking from

(28.4) to

t

in

K.

of

x ~ V.

Let

and let

Q

N R.

Since

V

~ ¢ P

for all

P e Q

be the unique monic p o l y n o m i a l that upon applying

f of

(5)

• M(V)

{(~)

f (~)

(28.17),

Let

and

in

Y

over and

R

f: R - R/(M(V)

be the N R)

be

be the set of all maximal V

is residually

w i t h coefficients of

f (k).

{ (Y)

separable

Let

in

k

{(Y) such

we get the mini-

Now we have

{' (~) ~ M(V)

Clearly (6)

{(Y+Z) = {(Y) + Z{' (Y) + z 2

By the Chinese remainder (7)

y ~

times an element

in

k[Y,Z]

theorem we can find

(R N M(V))\M(V) 2

with

Upon letting

215

y e P

ideals

~ e R\M(V)

f(R) = f(k) (f(~)).

to the coefficients

mal monic p o l y n o m i a l

(t,x)

and any separating

k, by the Chinese remainder theorem we can find

such that

we have

(*).

be given with

epimorphism,

x ~ V

N V,K).

from

coordinate

n V

and every

with

without using any material

we shall give another proof of

transcendental

V/k

K/k

dx ord v ~ = Ordv~(k(x)

In v i e w of

;

for all

P ~ Q

.

3.37

e+y by

(5),

(6) and

(8)

y

(7) w e

if

{(~) ~

if

{(~) e M(V) 2

see

¢ R\M(V),

M(V) 2

that

f(R)

= f(k)(f(y)),

{(y)

c

y £ P

for all

P e

and

(9) In v i e w

of

(8) and

(9),

by

(27.4)

(i0)

By

we

k(x,y)

(9) w e

see

(ii)

by

(12)

is a u n i f o r m i z i n g

(28.19.1)

y

We can

we

get

find

a nonconstant Q = k[x,y]

= 0.

Let

In v i e w

of

(8) and

(i0),

by

o r d v { ( k [ x , Y ~ Q)

(in v i e w

of

(8) and

or~{(k[x,y~Q) In v i e w changed)

we

nonconstant

= K

.

(9),

by

= 0; h o w e v e r , of can

(12), find

by

coordinate

transcendental

irreducible

x

V/k ,

of

K/k

{ (X,Y)

.

e k[x,Y]

such

that

N M(V)

(28.1)

we

get

+ ordv~(k(x)

(3.1) we

and

shall

applying

a separating

irreducible

of

that

is a s e p a r a t i n g

(x,y)

(13)

get

that

y

and h e n c e

(14)

(R n M ( V ) ) \ M ( V ) 2

~(X,Z)

N V,K)

(27.4)

= or~{y(~,y)

it can be

not use

this

(28.3) (with transcendental

c K[X,Z]

is i n t e g r a l

216

over

with

k[z]

x

;

seen

that

observation). and z

y of

9(x,z)

interK/k

= 0

and such

a that

3.38

(15)

k[x,z]

(16)

o r d v ay dz = 0

= k[x,y]

and (17) By

ordvq x(x,z)

(15) we h a v e

(18)

z 6 V In V i e w of

in

= ordv{ X(x,y) .

(14),

(15)

.

and

(18),by

taking

(z,x)

for

(x,y)

(28.1) we get

(19)

o r d v ~ ( k [ x , Y ] Q) + ordv~(k(z)

By the c h a i n

rule w e h a v e 0 = d[ (x,y) dy

=

{x(X,y)

N V,K)

= ordvgx(X,Z).

dx ~yy + {y(X,y)

and h e n c e dx o r d v { y ( X , y ) = ord V ~ + o r d v { x ( X , y ),

(20) By

(13),

(21)

In view of uniformizing

(17),

(19)

ordv~(k(x)

(ii),

(16)

and

(20) we get that

N V,K)

and

coordinate

(18),

of

V/k

= ord V ~dx + ordv~(k(z)

by

(28.19.3)

and h e n c e

that

(22)

ordv~(k(z)

Q V,K)

217

= 0.

by

w e see

N V,K) .

that

(28.19.1)

z

is a

we c o n c l u d e

3.39

Now

t

is a u n i f o r m i z i n g

consequently

by

coordinate

(28.19.4)

dx ordv~y by

(21),

(22) and

and this c o m p l e t e s

the p r o o f of

(28.20)

The

of

V

being

residually

even when we restrict which

are s e p a r a t i n g Take

fix any

k

determinate.

Let

o r d v ( x P + a ) = I.

separable

we

over

clearly have

as a s e p a r a t i n g

of

K = k(x)

=

of

integer

!(K,k)

q, u p o n

is a u n i f o r m i z i n g

transcendental

(xP+a)p q

of

(28.19)

parameters

where

= x(xP+a) pq + x P ( x P + a )

t(q)

in

of

K/k,

and h e n c e

p # 0 x

and

is an in-

such that

letting ,

parameter

of

V

as w e l l

and w e h a v e

dx dt(q)

(xP+a)-p q

consequently dx ° r d v dt(q) although x ¢ V

clearly

x

- -pq < 0

is a s e p a r a t i n g

transcendental

and ordv~(k(x)

F o r any p o s i t i v e

integers

q

and

N V,K)

r

218

V

K/k.

be the u n i q u e m e m b e r

that

that the c o n d i t i o n

is e s s e n t i a l

to u n i f o r m i z i n g

Take

F o r any p o s i t i v e

dt(q) dx

k

shows

field of c h a r a c t e r i s t i c

a I/p ~ k.

t(q)

dx

= or~

following example

transcendentals

V

y ;

(*).

our a t t e n t i o n

with

(ii) so is

dx = ordv~--C .

N V,K)

to be a n o n p e r f e c t

a ~ k

and by

(23) w e h a v e

ordv~(k(x)

EXAMPLE.

V/k

we get

(23) Thus,

of

= 0 .

we get

of

K/k

with

3.40

dt (q) = dt (q) dx dt (r) dx dt (r) -

(xP+a) (pq _ pr)

and h e n c e

dt (q) dt (r)

°rdv although both

t (q)

= pq_ pri and

t (r)

> 0

if

q > r

< 0

if

q < r

are u n i f o r m i z i n g p a r a m e t e r of

w e l l as s e p a r a t i n g t r a n s c e n d e n t a l s of

(28.21)

REMARK.

In v i e w of

K/k

we could h a v e d e f i n e d

k

ordv~dX

as

K/k.

(28.19.4),

w h i c h is r e s i d u a l l y separable over

V

for any

V ¢ Z(K,k)

and any d i f f e r e n t i a l

~dx

of

by the formula

dx o r d v ~ d X = Ordv~ + OrdV ~ where

t

is a u n i f o r m i z i n g coordinate of

K/k.

Thus,

if

k

p e r f e c t we could h a v e p r o c e e d e d w i t h o u t the i n t e r v e n t i o n of (28.12).

However,

(28.20)

followed for imperfect

were (28.4) to

shows that this p r o c e d u r e could not be

k.

So, m o t i v a t e d b y

(28.19.5), we p r o c e e d e d

as we did.

(28.22) k.

LEMMA.

Then for any

t (whence,

Let

t 6 K

V 6 I(K,k) we have:

is_____aau n i f o < m S z i n q c o o r d i n a t e of in particular,

coordinate

t

and

K)

in

o__[f V/k

K/k).

Moreover,

and any d i f f e r e n t i a l

and

Ordvdt = 0; V/k

is a

for any u n i f o r m i z i n q ~dx

of

K/k

(with

we h a v e

= OrdV~ +

(ordVX) -i OrdvX ~ 0

Ordvedx = ordve

V/K ~ t ~ V

every u n i f o r m i z i n g c o o r d i n a t e of

s e p a r a t i n g t r a n s c e n d e n t a l of

x

be r e s i d u a l l y separable over

dxI

> ord v& +

(Ordvx) -i ordVX ~ 0

219

i__~f & / 0 ~ x

and

(.characteristic of i__ff ~ ~ 0 ~ x

k)

and

(.characteristic of

k).

3.41

PROOF.

This

is simply a r e f o r m u l a t i o n

of

(28.19)

in the

language of differentials.

§29. Let that

R

Genus of an abstract curve.

be a homogeneous

R(R)

is separably generated

Note that clearly H0(A)

is a l g e b r a i c a l l y

H0(A )

= some member of

Assume

H0(R).

(or, say in view of

(27.6)) we have:

closed

~(R) ~(R)

is residually is residually

is separably generated

R(R)

over

Dim R = i.

is perfect

= every member of

=

domain such that

over

separable separable

over

H0(A)

over

H0(A)

rational over

H0(A)

H0(A)

and H0(A)

is algebraically

= every member of = some member of

=

I R(R)

~(R) ~(R)

closed is residually is residually

is separably generated

[

H0(A)

is (relatively)

rational

over

H0(A),

algebraically

over

H0(A)

and

cloed in

R(R).

We define genus R = genus(R(R),H0(R)) Now let n = Deg R and g = genus In view of (29.1)

(15.4),

(15.5)

GENUS FORMULA.

If

and

R .

(25.9), by

(28.15.3) we get:

Emdim R ~ 2, then

220

3.42

g = Now we

(1/2) (n-l) (n-2)

-

(I/2)~(R)

s h a l l prove:

(29.2)

THEOREM.

(*)

some

Also assume

Assume

V e ~(R)

that

Emdim

R ~ 2

and

is r e s i d u a l l y

rational

upon

~(R,P)

over

H0(A)

that

for e v e r y (**) 1 i n t e g r a l

P e D 0(R), closure

t [ e ( R , P ) "/ ~ ( e ( R , P ) ) (Note t h a t by

of

letting

~ (R, P)

: e

(i0.I.ii):

w

to be the

i_nn ~ (R) , w_~e hav__ee:

R,P)~

= 2[~(R,P)/~(~(R,P))

u(R,P)

~ 2

for all

: e(R,P)]

P e D0(R)

~

.

(**)).

Then we have (1/2)~(R)

=

[~(R, P)/~ (9(R, P) ) : H0(A) ] Pen 0 (R)

and

I

=

(i/2) (n-!) (n-2)

- (i/2)~

w

(R)

(29.2.1) a nonnegative Moreover

(as a c o n v e r s e

i_~f g = 0 (29.2.2)

PROOF.

and

and

By

integer

y

inn

(17.4), m

.

(28.12)) w e h a v e

m

m m max(l,n-2), x

negative

of

integer

is aDY i n t e g e r

then t h e r e e x i s t Hm(R )

(17.5)

that:

such

and

we h a v e

221

that

(25.9) w e

such that nonzero elements R(R)

= H0(R) (x/y).

see t h a t

for e v e r y

non-

3.43

m+

2) if

m < n

if

m m n

[~m(R) :H o (R) ] =

i

2

m - n + 21 2

i.e., {

(i)

[Hm(R) :H 0(R) ] =

mn+ In view of

(**), by

(2)

(1/2)u~(R)

if

m < n

if

m m n

I (1/2) (re+l) (m+2) 1 - (1/2) (n-l) (n-2)

(5.4) and =

(5.5) we have [~(R,P)/~(~(R,P))

: H0(A) ]

and hence in particular (3)

(I/2)u~(R) = an integer For all nonnegative

integers

m

and

E(m,e) = Ix ¢ Hm(R): ord(R,x,V)

~ ordw~(~(R,~*(R,W)))

whenever

(24.20) we see that: Hm(R)

integers

E(m,e)

is an

m

let

~ e + ordv~(~(R, 3 (R,V))), and

ord(R,x,W)

Then for any nonnegative

e

.

V ~ W e ~(R)]. and

e, in view of

H0(R)-vector-subspace

(2), by of

•

(5)

[E(m,e)

(6)

~

: H0(R) ] + e + (I/2)u~(R)

(R,x) ~ e + ~ ( R )

for all

and

222

a [Hm(R)

x e E(m,e)

,

: H0(R)]

,

3.44

if

e > 0

I

(7)

[E(m,e)

For

then

: H0(R)

every

e(m)

(3) w e

e(m)

~ 0

m m n

see t h a t for all

such

that

c E(m,e-l)

m [E(m,e-l)

nonnegative

(8) by

E(m,e)

=

e(m) large e(m)

: H0(R)]

integer

[Hm(R)

and

m

- 1 .

upon

letting

: H0(R) ] - 1 -

is an integer, m.

So

(I/2)u~(R)

and by

for a m o m e n t

is a n o n n e g a t i v e

we

integer

(i) w e can

see t h a t

fix any

and

then by

integer (5) and

(8) w e h a v e [E(m,e(m)) and h e n c e

we

can t a k e

~ H m (R)

now by

(i),

with

(6) and

~ mn+

Little

(I/2)u~(R)

In v i e w

of

some

~0 z E ( m , e ( m ) )

;

(8) w e g e t

by Bezout's

(9)

for

• = DR

(R,~)

and h e n c e

: H0(A) ] > 0

(3) and

-

(I/2)u~(R)

Theorem

-

(24.9)

(1/2) (n-l) (n-2)

we m u s t

(1/2) (n-l) (n-2)

(9), b y

(29.1)

~ 0

we get

have

;

that

/

/g (i0)

= (1/2)(n-l)(n-2) = a nonnegative

Henceforth such

(Ii)

-

(1/2)~(R)

integer.

assume t h a t

g = 0

that

m m max(l,

n-2)

223

and

let

any

integer

m

be

given

3.45

Then by

(i0) w e h a v e w

(12)

u{(R)

and h e n c e b y

(i),

=

(8) and

(13)

(n-l) (n-2)

(ii) we get t h a t

e (m) m 1

and w

(14)

e(m)

If

[E(m,e(m)

~£

Hm(R)

and t h e n b y

+ I)

with

(6) and

+ N~(R)

: H0(R)]

~ = CR

=mn

.

> 0, t h e n we c o u l d t a k e

for some

~ ~ E(m,e(m+l))

(14) we w o u l d h a v e w

(R,~) in c o n t r a d i c t i o n Therefore

[E(m,

(5) and

(15) a n d

(17) Again,

+ i)

in v i e w of

and h e n c e

(24.9).

: H0(R)]

= 0 .

: H0(R)]

~ 1.

(16) w e g e t

[E(m,

(18)

(19)

e(m)

[E(m,e(m))

(7),

Theorem

(8) w e h a v e

(16) By

Little

we m u s t h a v e

(15) By

to B e z o u t ' s

~ 1 + mn

e (m))

(13),by

[E(m, in v i e w o f

E(m,

e(m)

: H 0(R) ] = 1 .

(5) and

- i)

(8) w e h a v e

: H0(R) ] ~ 2

(7) w e g e t

e(m)

-I)\E(m,

224

e(m))

/

3.46

In v i e w

of

(17) w e

can

(20)

and

0 ~ x e E(m,e(m))

in v i e w o f

(19) w e

(21)

can

take

0 ~ y ¢ E(m,e(m)

In v i e w we

take

of

(4),

conclude

(14),

(20)

ord(R,x,V)

l

(21),

ord(R,x,W)

= e(m)

+ ordv~(~(R, ~

W

= or~(~(R,O

(23)

(R,W)))

whenever

In v i e w

and,

by Bezout's

of

(22)

in v i e w

and

of

(23),

(22)

and

by

by

we

(19.10)

= I, and

consequently

(29.3) above,

conic. But

by

REMARK. and

so also

latter

that would

be

can

genesis a good

(24.9)

Also

Let

assume

some

we

see

now

x/y

e R(R)

that

~ 0

V @ W

e 3(R)

;

of

(29.2.2)

ancient

as w e l l

pad

for m u c h

as of

its

of b i r a t i o n a l

i d e a of p a r a m e t r i z a t i o n

be d e d u c e d So h e r e

proof

as a c o r o l l a r y

of a

of

is the e l e m e n t a r y

(29.2.2). version,

§25.

OF A C O N I C .

k = H0(R) that

that

R (R) = H 0 ( R ) ( x / y ) .

facetious! after

= 1 + ord(R,y,V)

~ ~(R).

or~(x/y)

launching

and

PARAMETRIZATION

0 ~ z e HI(R).

have

of c o u r s e

a bit

any m a t e r i a l

(29.4)

n = 2.

The

is the e l e m e n t a r y

The

not using

(4.2) w e m u s t

(R,V)))

see

whenever

geometry,

Theorem

< ord(R,y,W)

V ~ W

(19.10)

(23),

or~(x/y)

given

Little

that

(22)

and

and

- l) \E (m, e (m) )

and

R' = k

V e ~(R,~)

225

Let

~ = zR

with

[HI(R)z-lJ. is r e s i d u a l l y

any

Assume

that

rational

over

3.47

k,

and

let

euclidean

P = ~

(R,V).

domain.

Then

Moreover,

i__nn H 1 (R)

with

xk

+ yk

(*)

R'

= k[x/z]

R'

= k[x/z,z/x]

Emdim

there

R ~

exist

+ zk = HI(R)

and

zy =

x

2

2,

g =

0,

nonzero and

case

z2

in

R'

elements

(x,z)R

in

and

=

P

~o(R,~)

is x

such

=

[P]

an

~nd

y

that:

,

and

(**) NOTE. figures P

is

conic

xy

writing

down

the

two

cases

where,

at

~

Before

for the

and

the

point

birationally

vertical

(i.e.,

along

onto

parallel

the

the

proof

case

let

us

in b o t h

cases,

y-axis,

and

(and h e n c e to

=

draw the

so w e

parametrizing

y-axis)

~0(R,n)

[P]

suggestive

projecting are

by)

center

projecting

the

x-axis

the along

directions.

P

2P

i/

i

I1

i//

~

y

I

/

/ /

/

\

!

/

z = 0/ ./__

/

\

z=O// /

X

/ ---X

/

Y

i

z = 0

Hyperbola:

xy

= z

no(R,~ ) =

[P,Q].

Projection

from

outside the

z = 0

: line

2

at

=.

Parabola: ~o(R,~)

P

is

except

zy =

= x

Projection

from

above

everywhere

outside

226

.

[P}.

integral

origin.

2

P

is z =

integral 0.

3.48

PROOF.

By

(24.12)

Projection

Formula

birational

and

U

=

get

(24.15)

Emdim

(R,P)<

we

we

Emdim see

R P = i.

R a 2, and

that

the p r o j e c t i o n

So w e m u s t h a v e

~

(R, T I ( R , V ) , P )

~

(R, TI (R,V)) by B e z o u t ' s

2

then by

Little

the S p e c i a l

from

P

Emdim

R = 2.

Theorem

(24.9)

is Now

and h e n c e

(I)

~

First Theorem

(24.9)

(24.18), zero

suppose

£0(R,~)

we have

(25.2),

elements

~

(25.3),

x

(x,z)R = P, and Next

that

and

(25.8),

y

in

that

= i.

= {P).

(R,~,P)

zy = x 2

suppose

(R,P)

= 2

Then

a n d hence,

(25.9)

HI(R)

and

such

It f o l l o w s ~0(R,~)

by B e z o u t ' s

that

~ {P}.

in v i e w

(25.10)

that

Little

we

of

can

(I), by

find n o n -

xk + yk + zk = HI(R),

R' = k [ x / z ~

Since

by Bezout's

Little

w

Theorem Q ~ P

~

(R,~)

= 2, w e m u s t

then have

= [P,Q}

where

and (R,~,P)

Consequently, and

~0(R,~)

y'

in

in v i e w HI(R)

of

(24.18),

such

xk

= 1 = U

we

(R,~,Q)

can

find

nonzero

elements

x

that

+ y'k

+ zk = HI(R) w

qx, z ) R = P

and

TI(R,P)

= {xR}

and (y',z)R N o w by z

2

and

(25.2),

for some xy = z 2.

(25.3),

= Q

and

(25.8),

0 ~ a e k. It f o l l o w s

Let that

Tz(R,Q)

(25.9) y = ay'

and

= {y'R} (25.10)we Then

R' = k [ x / z , z / x ~ . 227

see

that

axy'

x k + y k + zk = HI(R)

=

3.49

Obviously (28.12)

we

Let tively

also

us

by

we

that

Let

V 1

and

we

euclidean g = 0

1

and

OF

= k[x/z];

whence

us

round

the

e ~(R) and

OF

is

Emdim

Emdim

A

LINE.

the

cases.

By

cases. by

(28.12)

Assume

that

taking

any

k = H0(R)

in p a r t i c u l a r

above

A CUBIC.

residually R = 2,

upon

lettinq

R'

GENUS

the

(24.12),

Moreover,

upon

off

of

in b o t h

(or a l t e r n a -

have:

g = 0. and

domain

in both

in view

PARAMETRIZATION

(29.6) some

an

that,

o_ff H I ( R )

have

g =

see

(29.2.1))

R =

(x,z)

is

record

(29.5) Emdim

R'

o__[r:

by

is

i.

free

and

R'

discussion

n =

R' an

Then

H0(R)-basis

= k[Hl(R) z-l]; euclidean

calculating:

Assume

that

n = 3.

Also

assume

rational

over

H0(R).

Then

either:

g = 0

and

domain.

Emdim

R = 2,

or:

that

g = 0

R = 3. w

PROOF.

Let

P = ~

(R,V),

e = Emdim

R,

and

~

= Te_I(R,P).

Then 2 g e < e -

by

(24.12)

by

(22. i)

1 + u(R,P) w

< e - 1 + u < ~

(R,P)

(R,~, P) w

U =

and

hence

we

(R,~) by

3

must

be

in

one

of

the

Bezout's

Little

following

Theorem

three

(24.9)

(mutually

exclusive)

cases:

Case

(i).

e =

2 = ~

Case

(2).

e = 2,~/ (R,P) and

Case

(3)

(R, P)

= U (R, P)

= ~/(R,P) T I(R,P)

= =

I,%(R,P)

=

IV]

,

[~].

W

e = 3, ~

(R,P)

= ~(R,P)

228

=

I,~(R,P)

=

[V],

and u(R,~,p)

= 3.

3.50

In C a s e

(I), b y

that

the p r o j e c t i o n

now

Deg

from

P

R p = 1 and h e n c e

In C a s e Theorem

the S p e c i a l

(2):

(24.9),

Projection

is b i r a t i o n a l genus

First

Rp = 0

note

that

we

clearly

and h e n c e

~

Formula

(24.15)

and h e n c e

by

{29.5);

(R,~)

we

see

g = genus therefore

= 3

by B e z o u t ' s

for all

Q e ~o(R,~)

RP ; g = 0.

Little

get

w

(*)

~

Next,

in v i e w

letting and

of

(R,Q)

(25.2),

k = H0(R),

a nonconstant

z R = ~,

= u(R,Q)

we

(25.3),

can

find

irreducible

( x , z ) R = P, ~ ( x , y , z )

~(X,Y,Z)

= 1 (25.8),

(25.9)

a free k - b a s i s ~(X,Y,Z)

= O,

and

(25.10),

(x,y,z)

¢ k[X,Y,Z]

of

such

upon HI(R)

that:

and

= ZY 2 + ~2(X,Z)y

+ ~3(X,Z)

where nonzero

and w h e r e

homogeneous

either

nonzero

Upon

0 = ~02(X,Z)

homogeneous

clearly

(Y) we

e k[X,Z]

~ k[X,Z]

~2(X,Z)

is of d e g r e e

3

is of d e g r e e

2

or

e k[X,Z]

.

letting

= x/z

we

~3(X,Z)

, ~ = y/z

~(Y)

= ~

have

that

is i r r e d u c i b l e

immediately

see

+ ~2(~,I)Y ~ in

(15.4)

and

= k[~

, R'

+ ~03(~,i)

is t r a n s c e n d e n t a l k(~) [Y3,

and

= S'[~]

¢ k[~3[Y3 over

, and

,

k,

R' = k E H I ( R ) z - 1 3 ,

% (4) = 0 ; n o w in v i e w of

that

I(E)

and b y

, S'

(15.5)we S(R')

< 2

for all

E ~ ~(R')

,

have = {~(R,Q)

229

: Q e ~0(R,\~)}

;

(4.2)

3.51

therefore

(**)

Thus

U(R,Q)

by

(*) and

~ 2

by

In C a s e projection we

get

Q e ~0(R,\~)

for all

Q ¢ ~]u(R)

(**) w e h a v e

u(R,Q)

and h e n c e

for all

(29.2.1)

< 2

we

get

g = 0

(3), b y the S p e c i a l

from

P

or

i.

Projection

is b i r a t i o n a l

and

Deg

Formula

we

see

that

R P = 2, and h e n c e

by

the (29.4)

g = 0.

(29.7) rational

THEOREM.

over

Assume Also

H0(R) .

that

some

member

assume

that

Emdim

of

~(R)

is r e s i d u a l l y

R ~ 2, 4 ~ n ~ 5 ,

g ~ 1 ,

~(R,P)

< 2

for

all

P e D0(R)

and (R,P 0) = 2 Let

m = n - 3.

Then

there

for s o m e

exists

® e Tradj (R,\P0) such

that

@

= ~i®2

is i r r e d u c i b l e

with

®i

PO • Z~O I(R)

N Hm(R)

in the

and

®2

sense

i_nn H

that:

(R) =

either o[r

PROOF. our

assertion

s ( P 0) = 1 - g

In v i e w follows

of

(17.4),

from

(17.5),

(24.21)

.

230

by

(25.9),

taking

®i = R

®2 = R . (i0.i.ii)

~ = P0

and

and

(29.2.1),

s ( p 0)

and

3.52

§30. Let

A

R([A,C~)

Genus of an embedded

be a h o m o g e n e o u s

curve.

domain and let

is separably generated

over

be such that

C e ~I(A)

H0(A/C).

We define genus[A,C]

= genus A/C

.

Now let n = Deg[A,C~ By

GENUS FORMULA. g =

.

i_~f Emdim[A,C~

~ 2, the___~n

(1/2) (n-l) (n-2) - ~ ( [ A , C ] )

(29.2) we get:

(30.2) (*)

g = genus[A,C~

(29.1) we get:

(30.1)

By

and

some

THEOREM.

Assume

V ~ ~([A,C~)

that

Emdim[A,C~

~ 2

and

is re sidua!ly rational o v e r

H0(A/C)

Also assume that for e v e r ~

P ~ ~0([A,C~, u _ ~ n

.integral c l o s u r e of

(**)I-

[~([A,C~,P)

=

~([A,C~,P)

/~(~([A,C~,P))

(!0.i. Ii):

~([A,C~,P)

i__nn ~([A,C~),

to be the

we have:

: ~([A,C~,P)~

2[~([A,cl,P)/~(~([A,c3,P))

(Note that by

lettinq

:

~([A,C],P)

~([A,cJ,P)7

~ 2 for all

.

P ~ ~0([A,C])

Then we have (I/2)~([A,C~)

=

~ [~([A,C~,P)/~(~([A,C~,P)) P~0([A,C])

231

: H0(A/C)~

= (**).)

3.53

and

I

(30.2.1)

g =

(1/2) (n-l) (n-2) -

= a nonnegative

Moreover

(as a c o n v e r s e

i_ff g = 0

of

(I/2)~{([A,C])

integer.

(28.12))

and

m

m m max(l,n-2),

we have

that:

is any i n t e g e r

such that

t h e n there e x i s t e l e m e n t s

x

and

y

(30.2.2) i__nn H m ( A ) \ C where Finally, (26.12)

QUNITICS. closed,

f: A - C

in v i e w of

yields

(30.3)

s u c h that

R([A, C3 = H 0 ( A \ C ) (f(x)/f(y))

is the c a n o n i c a l

(25.9),

its a u g m e n t e d

(29.4),

version

that

(Note t h a t b y

(29.4),

and

(29.7),

PROJECTIONS

let

= 1

~ = zA

for all

(29.5)

and

Theorem

OF P R O J E C T I V E

E m d i m A = D i m A = 3, H 0 (A)

n < 5, g ~ i, and

u(~A,C~,P)

(29.5)

as s t a t e d below:

THEOREM ON 2-EQUIMULTIPLE Assume

epimorphism.

with

SPACE

is a l g e b r a i c a l l y

z e HI(A)\C.

Assume

that

P ~ n0([A,C~,\~).

(29.6),

g ~ 1

is a u t o m a t i c

if

n < 3.) Assume

t h a t t h e r e does

is a ~ - q u a s i p l a n e e x i s t any

projection N = 2 Q1

is

of

C

~-!ntegral, for some

of

C

and t h e r e

from

N

from and

exists

~ 2

and

assume

min(4,n)

232

such

min(4,n)

that:

t h a t there does of

C

from

i__{s~ - i n t e q r a l ,

N

i_~s

~ n.

s u c h that:

the p r o j e c t i o n

not

~ ([AN,cN~,Q)

~ Deg[AN,cN~

N ~ ~0(A,~)

for all

QO ¢ ~ 0 ( [ A N ' C N ~ ' \ ~ N ) '

e ~0([AN,cN~,~N),

N

is b i r a t i o n a l ,

~([AN,cN~,Q)

• e H 2 (A)

the p r o j e c t i o n

Q e ~0([AN,cN3,\~N),

4 < n < 5

Also

s u c h that:

the p r o j e c t i o n

for a l l Then

C ¢ ~I(A,~).

N e ~0(A,~)

birational, = 1

and

not e x i s t a n y

of

C

the from

Q £ n 0 ( A N ) , u ( [ A N , c N ~ , Q 0)

u([AN'cN3"QI)

= 2

~ D e g [ A N , c N] ~ n.

for some Moreover,

for

3.54

any such

N, upon lettinq

m = n - 3, there exists

0 ~ 8 ¢ Hm(AN)

such that ¢ Tradj([AN,cN],\~ N) R H~(A N) n ~I(AN,\~ N)

Finally,

for any such

N

and

8, upon

B' = H0(A)[HI(AN)z -I] we have

that

B'/®'

exist elements (x,y,z)A = P

x,y,t

i__nn HI(A )

such that, ~: B' - B'/@'

upon

@' = (8/zm) B '

domain;

with

in qrea,ter detail,

(x,y,t,z)A

= ~(A)

lettinq

to b e the canonical

epimorDhism

and k

we have

that

t

=

~(H0(A))

and

t

=

~(x/z)

,

is transcendental

over the field

B'/®'

k[t,t -I]

= kit]

or

233

® = 8A N.

letting

and

is an euclidean

where

k

and

and

there

CHAPTER

IV.

AFFINE

DEFINITION. sequence

Let

(Gn)0~n< ~

(i)

A =

(ii)

a filtered

domain,

by

that,

we

F0(A)

unless

F m ( A ) F n(A)

that:

of

A,

will

be said

A

Example.

Let

Sk)

say

to b e that

In the r e s t said

a filtered A

domain.

is a f i l t e r e d

of the c h a p t e r ,

otherwise.

Note

that,

A

= F l(A) n and 0 <

integers

Also

m,n.

note

If

domain

will

and

denote

by definition

= F m + n ( A ) , F n(A)

to be

is a f i l t e r e d

generated of

is s a i d

that

it

IF n(A) :F 0(A) A

is c l e a r l y

domain.

say that

filtered

such

is a

0 ~ m,n <

simply

Fn(A).

for a l l n o n n e g a t i v e

a noetherian

space

A

A

....

(A, (Gn) 0 ~ n < ~)

Gn

S

of

on

< ~ , and

we denote

set

A filtration

subgroups

, for a l l

is no c o n f u s i o n ,

finitely

domains

A = G0[GI].

The pair

also

a domain.

be

is a s u b f i e l d

EGI:G0]

(vi)

< ~

A

DOMAINS

U G , n= 0 n

GO

(v)

follows

Filtered

of additive

GmG n c Gm+n

(iv)

OR FILTERED

§31.

GO c G1 c G2

(iii)

there

GEOMETRY

R

domain

field of

over

an a f f i n e

domain

domain

extension

of

k).

the

finite

over

k

be

fixed with over

(or in f a c t k c Sk.

the g r o u n d

Then

field

]

k

, if

n = 0

Snk

, if

0 < n <

G(Sk, n)

234

over

and w e w i l l field

a field k

Let

any

finite

dimensional

F0(A).

(i.e.

a

generating k vector

(R, ( G ( S k , n ) ) 0 ~ n < ~)

k, w h e r e

/

A

the g r o u n d

be

R

domain

the g r o u n d

is a

4.2

(G(Sk, n ) o ~ n < ~ S

(or

Sk).

It is e a s y

is o b t a i n e d For

shall

domain

emdim note

that

In c a s e

filtered

homomorphism,

duced

a filtered

f

filtered

on

if

to be by

that,

the

If

B

inclusion

that

map

of

some

subfield

1

of

Sk)

taking

FI(A)

k

the

a filtered

natural

for any

get

we

by

f

and

if

B

unless

domain

with

define

is a f i l t e r e d

subdomain

of

A

subdomain

of

F0(A ) the

B and

A

some

finite

the n a t u r a l

domain,

provided

homomorphism.

with

is a

= emdim(A/p)

is a f i l t e r e d

of

let

A/p and,

further

in-

if w e

that

is

homo-

filtration

P ~ ~(A), we

filtration,

(or in fact

B = k[Sk~

f(A)

to be a f i l t e r e d

into

a

Fn(f(A) ) = f(Fn(A)),

assumed

A,

to be

f(A)

becomes

~.

be

B

filtered

is s a i d

induced

emdim[A,P~

A

f

filtration

of this

of

is a f i l t e r e d

If

homomorphism,

is a f i l t e r e d

selecting taining

called

A

for all

f: A ~ f(A)

B

every

(or

will

and

that

S

natural

domain,

c Fn(B )

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

A/p

Note

and

domain

[~(A):F0(A) ~

where

taking

m a y be

is a s u b d o m a i n

say

upon

and

In v i e w

= Fn(A/p)

we denote

a filtered

the c a n o n i c a l

mentioned,

F n([A,P3)

filtered

define

f(Fn(A))

domain

filtration.

then w e

is a l s o

f(A).

domain

otherwise

we

be a h o m o m o r p h i s m

(f(Fn(A))0~n<~

f: A - A / p

this

B

it is c l e a r

by

every

by

A ~ dim A = trdegFO(A)~(A)

f: A - B

In c a s e

morphism.

in fact

induced

then

domain.

becomes

A

d i m A = 0, by d e g A

let

a domain,

that

filtration

A = IF I(A) :F 0(A) 3 - 1 .

emdim

Now

to see

the n a t u r a l

this way.

a filtered

We

be c a l l e d

that

is o b t a i n e d

finite

subset

dimensional filtration

the

by S

con-

vector

space

induced

by

Sk).

Finally

for a n y

filtered

domain 235

A, w e

shall write

~

instead

4.3

F (A) k 0

of

as

.

k

in various

definitions

i n §5

and

Thus

§6.

for

example

F 0 (A)

(A,I,J)

= k

(A,I,J)

etc.

§32. DEGREE.

(32.1) we

Let

Homogenization. A

be

a filtered

For

domain.

any

x e A

define

degAx

= I

- =

'

if

min{n:

If

~ e F

(A), w e

g,h

e A,

negative Fn(A)

that

integer

for

an element

For

all degA~

0 ~ ~ e ~}

we

have,

for

any

g,h

¢ F

degA~

(A) is

or a non-

= n}

Let

an overdomain any

.

+ degAh. Also note that e ~ e F (A). We d e f i n e

DEFINITION. in

x ~ 0

,

a consequence

(A):

, if

= min[degA~:

~ degAg

: [~ e F

A.

define

degAgh

(32.2)

over

as

and

x ~ Fn(A)]

degA~

Note

x = 0,

A

of

x ¢ A, w e

be A

a filtered

such

that

domain z

is

and

let

transcendental

define

s

, if

x ~ F0(A),

xz n

, if

x e Fn(A)\Fn_I(A),

and

H ( A , z ) (x) =

For

any

I c A

H ( A , z , n ) I)

and

= The

0 ~ n < =

additive

A[z] {xzn:

generated

, we

subgroup by

x e I Q Fn(A) ] ,

and

236

for

define of

z

some

n > O.

be

4.4

H ( A , z ) (I) = The a d d i t i v e

s u b g r o u p of

generated

A[z]

e0

U H (A,z,n) (I) n---0

by We may drop Hz(X), clear

" A "

H ( z , n ) (I)

and

Hz(I)

respectively,

and

simply write

if the r e f e r e n c e

to

A

is

f r o m the c o n t e x t .

(32.3)

Observe

0 g n < ~, Hz(A) homogeneous N o t e that, then

if

z

a homogeneous

Hz(A) and

and

upon defining

z'

that

H z (A)

d i m A = Dim H

of

elements

Hz, (A)

A.

over

i.e.,

A,

there

which makes

for e v e r y

from the d e f i n i t i o n

for

W e say t h a t the

isomorphic,

and

to e a c h o t h e r

= H(z,n) (A)

z-homogenization

are c a n o n i c a l l y

It is c l e a r

it f o l l o w s

domain.

are two t r a n s c e n d e n t a l

Hz, (A)

to c o r r e s p o n d

(32.4)

Hn(Hz(A))

is the n a t u r a l

isomorphism between

az 'n

Thus

that,

becomes

domain

Hz(A)

a unique and

f r o m the a b o v e n o t a t i o n

az

is n

a g Fn(A).

that

~(Hz(A))

= ~(A).

(A). Z

(32.5)

Also

clearly

F 0 (A) = H 0 (H z (A))

vector

(32.6)

For

FI(A)

is i s o m o r p h i c

space and h e n c e

in p a r t i c u l a r It a l s o

~ Hz(~) (32.7) I, we h a v e {Hz(X): x £ I

For that

x e S}. and

degAx

as an

x e A, c l e a r l y w e h a v e

x ~ Hz(X)

follows

gives

HI(Hz(A))

e m d i m A = Emdim H z (A)

D e g H z (A)Hz (x) = d e g A x and

to

that

gives

for a n y

an i n j e c t i v e m a p I c A Hz(1) Further

an ideal

an i n j e c t i v e m a p H z(~)

Hz:

~ Hn(Hz(A)).

in

it is c l e a r

H z(x)

Fn(A) A

and

237

S

ideal

that

It f o l l o w s

e H z(I)

Hz: A - H ( H z ( A ) ) .

• e Fn(A),

is a h o m o g e n e o u s

~ n < ~].

,

~ x e I.

a generating in

Hz(I)

that

¢ H n ( H z(A))

Hz(A)

set for

generated

n H(Hz(A))

for a n y

and

by

= [xz n :

x ~ A, we h a v e

4.5

In p a r t i c u l a r w e have,

for any two ideals

Hz(I) It also f o l l o w s I ~ A

then

in

A,

~ I = I'

I = A, ~ 1 e I ~ z ¢ Hz(I),

so t h a t if

z ~ H z(I).

(32.8). homomorphism. a unique

that,

= Hz(I')

I,I'

Now

let

If

z'

extension

f' (a) = f(a) isomorphism

P e ~(A)

let

f: A ~ A / p

is any t r a n s c e n d e n t a l

f': A[z3

for e v e r y

-

(A/p)[z'3

a ¢ A.

of the h o m o g e n e o u s

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

and

w e m a y take

A/p,

s u c h that

It is t h e n c l e a r domains

Hz(P)

then

f

has

f' (z) = z'

and

that

Hz(A)/Hz(P)

~ = z modulo

= H_(A/P). z In p a r t i c u l a r it follows

over

be the c a n o n i c a l

f'

and

induces Hz, (A/P).

and t h e n w e h a v e

H z ( A ) / H z(P)

~(A)

- n(Hz(A)).

~i(Hz(A))

In v i e w of

and by

an i n j e c t i v e

(32.7),

p - Hz(P)

(32.4),

it follows

gives

it follows that

Hz:

a map

that ~i(A)

Hz:

Hz(~i(A))

c

- ~i(Hz(A))

is

map. §33.

(33.1)

DEFINITION.

0 ~ z ¢ HI(B).

lh

I c B

B

be a h o m o g e n e o u s

domain.

Let

h ¢ H(B), w e d e f i n e

, if

hz -n, for

Dehomogenization. Let

F o r any

F(B,z) (h) =

Also

that

and

if

h c H0(B),

and

h e Hn(B)

for

0 < n <

0 ~ n < ~ , we define

(B,z,n) (I) = The a d d i t i v e generated by

subgroup

of

R(B)

[hz-n: h e I n Hn(B) } ;

and F ( B , z ) (I)

= The a d d i t i v e generated

J u s t as b e f o r e w e m a y d r o p simply write ference

to

Fz(h), B

F(z,n) (I)

is c l e a r

by

subgroup~ of

R(B)

U F ( B , z , n ) (I) . n=0

" B " f r o m the a b o v e n o t a t i o n and

Fz(I)

f r o m the c o n t e x t .

238

respectively,

and

if the re-

an

4.6

Observe 0 ~ n < = the

that,

, F

z

(natural)

Fn(Fz(B)) Fz, (B)

upon

(B)

defining

becomes

Fn(Fz(B))

a filtered

z-dehomogenization

= Fn(F z, (B)), are naturally

we may write

F

(B)

d ehomogenization

of

(33.2)

of

for all

B.

Fz(B)

h,h'

some

Further,

the

above

with

z

zB = z'B

thus

it the

call

is said

to be =

Fz(B)

Thus,

and

if

(natural)

zB = ~,

~-

T h e n w e have,

* h = zrh '

it is c l e a r

h'

= z~

,

that,

~ Fz(h)=

g, H(B)

consider

it f o l l o w s that

and

< DegBh

x e Fz(B),

such

or

0 % r <

(B)Fz(h)

observations

F z ( h 0) = x,

, and

and

h'

any

that

domains.

d e g F z (B)Fz (h) < D e g B h

Now given

Note

(B)

z

filtered

£ H(B).

F z (h) = F z(h')

deg F

F

for

B.

Let

for

domain.

0 ~ n < =

isomorphic for

= F ( z , n ) (B),

h0

that

F

with

(h')

z

for

DegBh'

< DegBh

[h e H(B) : Fz(h) there

satisfies

exists

some

= x}.

a unique

o n e o f the

.

three

From

h 0 e H(B) equivalent

conditions (i)

degFz(B)X

(2)

h 0 e H(B)\zB

(3)

F z(h)

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

H(B)\zB

given by

-- Fz(B)

Note h = F

= x ~ h = h 0 zr

it f o l l o w s

that

is a b i j e c t i v e

for

some

H z (x) = h 0 map with

0 ~ r < ~

.

and h e n c e

the

inverse

map being

z

follows

that

map w i t h t h e that

(h) z D e g B h z

.

H

It also

bijeetive

= DegBh 0

Fz

inverse

for e v e r y

h

e H z(F z(B))

: Hn(B)\(zB)H

being

¢ H(B),

given by

Hz

w e have,

and h e n c e

239

(B) - Fn(B)

H z ( F z(B))

= B.

is a l s o

a

4.7

(33.3) that

If

h i ¢ H(B)

for

, for

i =

n m DegBh i

this

homogeneous

observation,

ideal,

it

(I)

Fz(I)

h'

c H(B),

By we

(i)

(33.2)

let

is

, we

and

for

to

F

see

n

such

I c B

is

a

ideal

I

and

have, t ~ F (h i ) i=l z

(h)= z

easy

any

that

if

=

{Fz(h) : h

¢ I N Hn(B) }

=

{Fz(h):

¢ I ~ H(B)}

and

h

(33.3),

for

any

and .

homogeneous

have

F z(h')

Now

1,2 ..... t

then

then

F(z,n)

(33.4)

1 , 2 ..... t,

t n-DegBh i ~ h.z e Hn(B) i=l i

h =

From

all

i =

us

e F z(I)

call

~ zrh ' e I

a homogeneous

for

some

0

< r <

ideal

I

prime

to

and

h

e B = h

z

,

if

satisfies

(*)

zrh

Note

that

(2)

¢ I

we

with

clearly

I e ~(B,\z)

(3)

Now

we

If

J

shall

is

ideal Let We

Hence degF

(B) (x)

=

I

ideal

z

is p r i m e

to

z

.

that:

prime

zrh ' ¢ H

must

zrh ,

B

~ I.

have,

=

prove

any

in

0 ~ r < ~

(J),

in

F z(B),

in

z.

for

some

then

0

H z(J)

is

a homogeneous

< r <

have XZ n

h'

where

= xz DegBh'

~ DegBh'

and

0

~ n ~

r + DegBh'

Since hence

h'

z

x

e Fz(B),

e H

(J). z

240

and

it

x

e J.

follows

that

I

4.8

Now

for a h o m o g e n e o u s

ideal

I

prime

to

z,

in v i e w o f

(i), w e

get:

(4)

F

We get the

(5)

If

(h')

z

e F

following

I

and

z

If

I

I'

for any

of

= Fz(I')

~ I =

h'

e H(B).

(4).

are homogeneous

is a h o m o g e n e o u s

(4) and

e I,

consequences

Fz(I)

(6)

(I) ~ h'

ideals

prime

to

z, t h e n

I'

ideal prime

to

z,

then by

(3) w e get,

H z (Fz (I)) = I .

(7)

If

P ~ ~(B,\z),

can

see t h a t

Further Fz:

~(B,\z)

given by

H z.

Relation

A

that

Fz(P)

(5),

be

the

exists

(6) and

inverse

between

of

(2), b y

(4) w e

~ ~(Fz(B))

also have

a filtered

there

in v i e w

(32.8)

is a b i j e c t i v e

In fact w e

map with

§34.

Assume

of

~ ~(Fz(B))

a bijective

Let

in v i e w

then,

map,

that

given by

homogenization

domain

and

z e HI(B)

we get

let

the

Fz:

that

inverse

~i(B,\z)

map being - ~i(Fz(B))

is

H z.

and dehomogenization.

B

be

a homogeneous

s u c h t h a t o n e o f the

domain.

following

holds:

(*)

B = H

(**)

A = F

Note hand

t h a t as

indicated

from the definition

that both

(*) and

in

z z

(A) (B)

(32.3.1),

it is c l e a r

(**) h o l d .

241

that

(**)

=

(*) =

(*). (**).

On the o t h e r Thus we

assume

4.9

the

We have

the

following

(34.1)

Hz:

A - H(B)\(zB)H*(B)

inverse Both

verse

and

being

are

given by

Hz

being

given

(34.3)

Let

: ~i(A) by

whenever p

=

and

A. map with

. (33.2).

(This

is a b i j e c t i o n is a r e s t a t e m e n t

and

z = z modulo

= B / H z (P)

(34.2),

R(B)

B

H

map with from

(P)

z

the

in-

(33.4),

and

(7).)

Q e n(B,\zB)

Then we have

H_ (A/P) z of

between

is a b i j e c t i o n

~ ~i(B,\z)

F z.

Q.

z

from

P e ~(A)

t = z modulo

(34.4)

F

restatements

(34.2)

In v i e w

relations

this

= ~(A)

Q e ~(B,\zB),

and

F t (B/Q)

follows

and

from

R(~B, Q3)

P e ~(A)

and

= A / F z (Q) .

(32.8).

= ~(A/P)

Q = Hz(P)

= ~([A,P~)

,

or e q u i v a l e n t l y

F z (Q) •

This

follows

from

Also we

clearly

(34.5)

emdim

Emdim[B,Q3,

Emdim(B/Q)

P e ~(A)

for any (15.1)

and

I e H(B) and

and

(15.4)). or

I c B

the d e f i n i t i o n

of

emdim(A/P) P e ~(A)

follows

follows

P = Fz(Q)

in

B

This

Q e ~(B,\z).

observed

(34.3).

Q ~ ~(B,\zB),

= Emdim[B,Q]

Let

and

H 0 (B) = F 0 (A) .

A = Emdim

P = Fz(Q).

(34.6)

already

have

whenever

equivalently

(32.4)

from

= Emdim(B/Q) and

(32.5)

from definitions,

or e q u i v a l e n t l y Then

clearly

Further ; this

we have

follows

~(B,I,Q).

242

Q = Hz(P) together

by

or with

as r e m a r k e d

Q = Hz(P),

we have

=

~(B,Q)

~(B,I,Q)Ap (33.3)

(34.3).

in §13.

for some = Ap

(as

= F z ( I ) A P,

in v i e w

of

4.10

Further, checked

from definitions

that we have

and the a b o v e r e s u l t

the f o l l o w i n g

d i m A = D i m B = I, P ¢ ~0(A) w

(34.6.1)

and

(B,I,Q)

U

(B,Hz(J),Q)

Q e ~0(B).

= k

(A, F z ( I ) , P ) = k

, and

(A,J,P).

= X~(A, P), and

u~(B,kz)

= X~(A).

F z ( A d j ( B , Q ) ) = adj(A,P),

(34.6.3)

where we assume

.

U

~{(B,Q)

(34.6.2)

relations,

it is e a s i l y

F z ( T r a d j (B,Q)) Fz(Adj(B,\z))

Hz(adj(A,P))

= Adj(B,Q)

;

= tradj ( A , P ) , H z(tradj(A,p)) = Tradj (B,Q) ; = adj(A),

Hz(adj(A))

= Adj(B,\z)

;

and F z(Tradj (B,\z))

(34.7) v i e w of

LEMMA.

(34.4)

Let

and

zB = ~

Assume (34.5))

and

A = F 0 ( A ) [ y I ..... y r ]. (the one

where

Observe

equivalently

geneous

polynomial

Ho(B).

C l e a r l y w e have, and

F z(~)

~ ~-quasihyperplane.

of degree

= yr A

and

in w h i c h

X i e HI(B), n

in

Let

(in

X0 = z

X 0 ..... X r _ 1

~ = Fz(~).

~ = yr A

is a n o t h e r deg ~

(25.5) w e c a n w r i t e

upon taking

Yr = Xr/X0

such that

there

{i Yl ..... Yr ]

t h a t by

and h e n c e

E m d i m B = D i m B = r.

Yl .... 'Yr ~ A

B = H 0 ( B ) [ X 0 ..... Xr],

1 ~ i ~ r-i

emdim A = dim A = r

In o t h e r w o r d s ,

induced by

PROOF.

that

@ £ Hn(B)

Then there exist elements

A

= tradj (A), H z(tradj (A)) = Tradj (B,\z).

@ =

and

filtration becomes

i.

{•x n0- I x r + ~n)B,

and

on

with

coefficients

Yi = Fz(Xi)

on

is a h o m o -

= Xi/X0

in

for

+ Fz(~n)'

A = F 0(A) [ X I / X 0 ..... X r / X 0 ] = F 0 ( A ) [Yl ..... Y r 3"

REMARK. in s o m e sense, motivation

The a b o v e

L e m m a is s a y i n g

"essentially

behind

a hyperplane

that a ~-quasihyperplane outside

the t e r m ~ - q u a s i h y p e r p l a n e . 243

~ ".

This

is,

is the

4.11

§35. (35.1) subdomain F0(A').

Projection

DEFINITION. A'

of

A

Let

Fo(A).

A

be a filtered domain.

is said to be a projection

Note that then

containing

of a filtered domain.

FI(A' )

We define

jection and we say that

A'

is an

F0(A)

FI(A')

of

A filtered

A, if

subspace

F0(A9

of

=

FI(A)

to be the center of the pro-

is the projection

of

A

from

FI(A').

We define ~(A)

= [L D F0(A ) : L

~(A) 1

= {L e S~(A)

Note that then

L - F0(A)EL~

Clearly

of

A, where

{M e ~i(Hz(A))

and

L ~ H(z,l) (L)

: z ¢ M},

of

FI(A) }

~(A)

the filtration of

and

onto

F0(A)[L3

Fn(F0(A)[L~) = L n , for

the inverse b i j e c t i o n

Also note that

subspace

gives a b i j e c t i o n of

F 0 ( F o ( A ) [ L ]) = F0(A)

0 < n < ~.

F0(A)

: [L : F0(A) 3 = emdim A - i}

the set of all projections is given by

is an

is given by

gives a b i j e c t i o n

A' - FI(A')of

Si

onto

the inverse m a p p i n g being given by

M - F(z,l ) (M) . Now let

A'

be the projection

that the p r o j e c t i o n ~(A)

from

L

of

is b i r a t i o n a l

= ~(A')

(respectivel ~

A/A'

For any

I c A, the set

I N A'

if reference

to

of

L

I

from

A

is clear;

(in A).

A

from

L e ~(A).

(resDectively

We say

integral),

is integral). is denoted by

and we say that

IL

I L'A

Or by

IL ,

is the p r o j e c t i o n

Note that then, A L'A = A L = A'

If from

C ¢ ~(A),

L = L modulo C.

birational from

then clearly

L

is b i r a t i o n a l

(35.2)

AL/c L

is the projection

We say that the projection of

(respectively

integral)

whenever

(respectively

C

of

A/C

from

L

the projection

of

integral).

RELATION W I T H H O M O G E N I Z A T I O N AND DEHOMOGENIZATION.

244

if

A/C

is

4.12

Let

z

be transcendental

are o b v i o u s

in v i e w o f

over

(33.4),

A.

(34.4)

T h e n the

following

relations

and the d e f i n i t i o n s .

H (L) (35.2.1)

Hz(A ) z

H z ( A L)

(35.2.2)

(F z ( Q ) L = F z ( Q z

for

L e

(A).

H (L) H z (A)

prime

to

(35.2.3) only

L e

z; a n d h e n c e

The p r o j e c t i o n

if the p r o j e c t i o n

(35.3) (A).

followinq

), for e v e r y h o m o g e n e o u s i d e a l Q H z (L) in p a r t i c u l a r A L = F z (H z (A) ).

LEMMA. Let

A

from

Let = H

conditions

from Hz(L)

A

(in A)

in

= H

Z

is b i r a t i o n a l

H z(A)

be a filtered

(A), C

Z

L

if and

is b i r a t i o n a l .

domain,

(C), a n d

in

L

C e ~I(A),

= H

Z

(L).

and

T h e n the

are e q u i v a l e n t :

(*)

The p r o j e c t i o n

of

C

(**)

The projection

of

C

from

L

is integral. w

from

L

i__{s~-inteqral, w h e r e

= ZHz(A).

(***)

C L ¢ ~ I ( A L)

and

o r d v ( F 1(A/C)) PROOF.

for all

< 0 = o r d V(L)

First note that

are e q u i v a l e n t ,

where

< 0 , where

for a n y f i l t e r e d

we h a v e

~ = L modulo

domain

B,

(i') to

C (5')

W e ~ (B,F 0(B)).

(i')

ordwB

(2')

ordWFl(B)

< 0

(3')

F o r some

u e FI(B),

(4')

For some

x e HI(Hz(B)),

< 0

ord(H z(B),x,W)

(5')

V e I(A/C,F0(A/C))

or%u

< 0. ordW(X/Z)

< 0 ; i.e.,

< ord (Hz(B),z,W)

W e ~ (Hz(B),zH z(B)). It is e a s y to see that, w e also g e t

is e q u i v a l e n t

to

(*). 245

that e a c h of

(I),

(2),

(3)

4.13

(i)

A/C

(2)

C L e ~ I ( A L)

is i n t e g r a l

ordv(A/C)

over

and

AL/c L

for e v e r y

V e I(A/C,F0(A/C))

< 0 = ord

w e have,

(AL/c L) < 0 . V N ~ ( A L / c L)

C L e ~ I ( A L)

(3)

and for e v e r y

V ¢ %(H_(A/C),zH_(A/C)) Z

where

of

(*) ~

(i)

(2) ~

(35.4) D e g Hz(A/C)

(19.13) (*) and

of

(I') and

(19.14),

(i') ~

define

deg A = degEA,{0]].

N o w let

is w e l l

F0(A)

an a l t e r n a t i v e

defined.

z

f r o m the

of

Then we define

over

from

closed.

deg[A, C3

degLA,C]

=

A/C ; and n o t e t h a t by

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

be a l g e b r a i c a l l y

description

follows

(***) c a n be d e d u c e d

C e ~I(A).

for any i n d e t e r m i n a t e

deg[A,C]

(*) ~

(**)

(2').

Let

(32.3)

(5').

(*) ~

(3); w h i l e

DEFINITION.

Z

C .

and

and

, (AL/cL)),

Z

(we u s e the e q u i v a l e n c e In v i e w of

we have

= V n ~ ( A L / c L) ~ ~ ( H _ ( A L / c L ) , z H

Z

z = z modulo

equivalence

V ¢ ~' (A/C,F0(A/C))

when

d i m A = i, w e

U s i n g §34, w e

(when

C e ~I(A))

can g e t as in

(24.16): deg[A,C]

We

= max{[A/C

: A L / c L]

: L e ~(A),

[L : F 0 ( A ) ]

= max{[A/C

: F0(A) (x) ] : x ¢ F I ( A / C ) \ F 0 ( A / C ) ]

= 2]

further define

genus[A,C]

= g e n u s [ H z ( A ) , H z(C) ] ,

and , genus A = genus

(35.5)

THEOREM.

Let

Hz(A),

A

whenever

be a f i l t e r e d

dim A = 1 .

domain with

emdim A =

w

d i m A = 3.

Let

A

= Hz(A)

and note

246

that then

E m d i m A* = 3.

Assume

4.14

that C

H0(A

) = F0(A)

= Hz(C).

Let

n = D e g [ A * ,C * ] Applying

= ~.

Clearly

applying

(30.3)

ON 2 - E Q U I M U L T I P L E and

let

There

exists ~

3 ~ n ~ 5 of

and

(26.12)

C e ~I(A)

,\~).

and

Let

and

t_~o A

C , and

quantities

involved we 9et

OF A F F I N E

SPACE QUNITICS.

PROJECTIONS

k([A,C~,P)

= 1

C

for all

following

M e C Q F2(A)

is a ~ - q u a s i p l a n e ,

there exist (2)

¢ ~I(A

" F z " to v a r i o u s

T h e n at l e a s t one of the

and

C

Let

theorem.

THEOREM

(1)

closed.

g = g e n u s [ A * ,C * ] = g e n u s [ A , C ] .

the T h e o r e m

the f o l l o w i n g

n ~ 5

zA and

systematically

Let

is a l g e b r a i c a l l y

Yl,Y2,Y3

such that

in

and t h e r e e x i s t s from

X([AL,cL3,P)

L

situations

and h e n c e

= M

(in A)

P c ~0([A,C~).

A

~ = H(z,2) (M)A

e H2(A

(by (34.7)) w e h a v e

that

s u c h that

L e g0(A)

A = F0(A)[Yl,~,y3~

and inteqral;

p ~ ~0([AL,cL~)

: and

further,

min(4,n)

d e g [ A L , c L] ~ n.

o~

(3)

4 < n < 5 letting (i)

and there e x i s t s B = H z ( A L) .

L e S0(A)

D = H z ( C L)

the p r o j e c t i o n o f

C

from

and L

such that upon A = zB

(in

A)

we have is b i r a t i o n a l

and i n t e g r a l . (ii)

max{!([B,D],Q)

: Q e ~0([B,D~,\A)}

= max{~(~B,D],Q) in p a r t i c u l a r ,

(iii)

min(4,n)

F__urther a s s u m e

: Q c n0([B,D]3,A)}

: P ~ ~0([AL,cL])}

< deg[AL, cL~

that

= 2 ;

we have

m a x { k ( [ A n , cL~,P)

g < i.

= 2, and

< n .

Then we have

247

)

such t h a t the p r o j e c t i o n

~s birational

= i, for all

prevails.

in c a s e

(3) above:

4.15

with

L

as above,

e Fm(AL)\F0(AL) m = 1

o r 2.)

Specifically YF0(A) image

with

Further we have

+ F0(A)

upon

= L,

of x modulo

we have

such

that

P

is a c o m p l e t e

Now

let

~

if

planar

that

"hyper"

if

Then w

if, P

Then

Let

Let C

C

exist

C N F0(A)[u,v]

(36.4) situation PROOF.

(in A)

and

xF0(A)

+

where

modulo

if

P ¢ ~(A).

P

t =

@.

We

say

can be g e n e r a t e d

be a f i l t r a t i o n

say

We say

that

P

that

filtration

on

A

and

is h y p e r - p l a n a r

P ~

relative

dim A = 3

be

u,v,w

we

planar

by

is e s s e n t i a l l y

with to

let

relative hyper-

dim A = emdim(A,~),

~.

We drop

the w o r d

see

that

then

Let

(2) p r e v a i l s .

C

A,C Then

emdim

be

essenti-

C =

(Fn(A~0~n~

A = F0(A) Eu,v,w]

n F 0 ( A ) [ u , v ~.

taking

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

(i) p r e v a i l s ,

C ~ ~I(A)

to a f i l t r a t i o n

that

A/C ~ F0(A)~u,v~/C

COROLLARY.

We have

such

and

let

intersection.

relative

¢ FI(A),

easily

LEMMA.

and

is a c o m p l e t e

c ~l(F0(A)[u,v]

C Q F0(A)[u,v ~

situation

We

~ {0~.

T h e n w e have,

(36.3)

with

= F0(AL)

that

r = 3.

planar.

~ C.

F0(A)

(Note domain.

o__[r k [ t , t - l ~

a domain

say,

is h y p e r p l a n a r

LEMMA.

there

e FI(A),

= kit]

be

intersectio ~

for some

(36.2)

PROOF.

n ~I(AL).

is an e u c l i d e a n

x,y

of

exists

intersections.

A

= r, say.

P N FI(A)

we have

ally

Complete Let

AL/~

exist

AL/®

( F n ( A ) ) 0 ~ n ~ ~ = ~,

(in A),

there

generators.

dim A = emdim(A,~) to

there

an___dd k = image

DEFINITION.

dim A - dim[A,P]

that

that

§36. (36.1)

m = n - 3,

@A L = 8 ¢ t r a d j ( [ A L , c L ] )

that,

®

putting

~

Hence

if in T h e o r e m

is a c o m p l e t e

Hence of

the proof. (35.5),

th___~e

intersection.

be

as in T h e o r e m

(35.5)

C

is a c o m p l e t e

intersection.

A L = d i m A L = 2, and

248

and

to be any g e n e r a t o r

(¥,w)A.

.

and

assume

C L e ~I(AL).

that

Let

4. 16

y

~

AL

such that

AL/c L

y A L = C L.

is normal,

A L / c L.

~(A/C)

It f o l l o w s

L + wF0(A)

that

= FI(A).

C =

A L / c L.

=

w - h ~ C

A SUFFICIENT

following

a restatement

generalization (36.5) A

For our

of e u c l i d e a n

THEOREM. such t h a t

A/C =

It is n o w o b v i o u s

t h a t the p r o j e c t i o n c e ~I(A)

P e ~0(A,C).

Further

yB e ~I(B).

(p,~)B = B.

(1')

p ~ yB.

(2')

for some

~y, w e h a v e n-i

e

exist

of

assume

By

and

terminology

Let

is r e l a t e d

domain

the old

to the

out

and

such t h a t

FI(B)

in §39.

B

a projeg-

and in-

= i, for all for some

(2') ~ D ~

n > I, u p o n p u t t i n g

A = B[Z~.

is b i r a t i o n a l

such that

(I'),

for the

e m d i m A = d i m A = 3, an d

k ([A,C],P)

e B

and a

(36.7) w a s n e e d e d

as w o u l d b e c a r r i e d

C N B = yB

that

Y e B

~Z - 8 ¢ C

and

and

(3') are s a t i s f i e d .

~' = ~p

and

~' = ~p +

({~',~'}B) n. for a n y

Q e ~0(B,{~',~'})-

a0,al,...,a n e B

{ ~ ' Z - ~ ' , a 0 zN + a l Z n - l + . . . + a n } A , PROOF.

(36.9)

from

that

respectively,

(36.9), h o w e v e r ,

Z ¢ FI(A)

C

are,

We include both proofs

~ ~0(A).

~,~,p,~

~ Q({~',B}B) n

Then there

(36.7))

be a filtered

Let

assume

integer

7pn-i

domains

such t h a t

Let

Further

INTERSECTION.

This generalization

FI(B)

Assume

Let

and

final Theorem

Le__~t A

e m d i m B = d i m B = 2.

yp

h ¢ A L.

((36.5)

3' in [31.

hence

(3')

such that

In v i e w of

FOR COMPLETE

for an o l d e r v e r s i o n of the p r o o f of

hence

w ¢ FI(A)

2' in [31 in o u r p r e s e n t

3' in [3] is enough.

teqral.

is i n t e g r a l o v e r

A = AL[w].

CONDITION

of T h e o r e m

s a k e of c o m p l e t e n e s s .

tion of

A/C

Let

for some

two T h e o r e m s

of T h e o r e m

generalization

Theorem

A/C

and

w e c l e a r l y h a v e that,

(y,w-h)A.

(36.5) The

= ~ ( A L / c L)

T h e n we h a v e

A L / c L, w e see that that

Now by hypothesis

a n d thus

(2') w e c a n w r i t e

249

such that C

c =

is a complete

i n t e r s ectiono

4.17

(i)

yp

n-1

= a0(8')n

+ al(8')n-l~'+...÷an(~')

n, w h e r e

B.

a 0 .... a n

Put (2)

f = a0zn Now we

(3)

+ alZn-l+...+a n

claim

Q e ~0(A) %'Pn-i

n-i

let

that

Hence

of

In v i e w of (5)

if

i

and

where all

particular, We

First

Assuming, using

Now

f,g

e P e ~I(A)

of

are u n i q u e

1

and

primary

~ Pi N B,

and

claim

f,g e C.

Now

for all Pi =

Q B = yB

for o t h e r w i s e

= 1

of

for all

components);

and ~'

in v i e w

in

with i.

(3). Note

(i),

of

•

that,

since

Y e Pin

p e Pi Q B

Y ~ Pi"

(i)

in the

and h e n c e

B. and h e n c e ,

a contradiction°

It is e n o u g h

¢ C N B = yB 250

Because

to p r o v e

Pi N B, w e get b y conclude

i.

(Pi NB)A

for all

C ¢ [PI' .... Pr }"

g e C,

dim[A,Pi]

in c o n t r a d i c t i o n

(Q,~)B = B, w e that

that•

decomposition

isolated

Pi N B e D 1 (B), we o n l y h a v e

g e Pi

w e get

Q1 = P1 = C.

Pi Q B ¢ 9 0 ( B ) ,

r ~

theorem

= P ~ [PI ..... Pr }"

Pi Q B e ~I(B)

1

ehB.

is I.

(being

that

P.

(i) we h a v e

proved,

; then

r =

~hat

exist

a0,a I ..... an,~',~'

unmixedness

Pi

will

and h e n c e

Then by

f,g

for

there

(3').

we h a v e

that

we h a v e

claim

with

is an i r r e d u n d e n t

f,g

claim

e Q,

e hA.

Thus

divisor

Qi

case

consequently

are p r i m a r y

if p o s s i b l e ,

~' ~ C,

contrary

1 ehB.

and

and

Now we that

and

Pi

case,

N o w we yB

e B

r N Qi i=l

Qi

we

in the

(4), b y M a c a u l a y ' s

a 0 ' a l ..... an'~''8'

both

that

h

shall p r o v e

contrary

such

common

(f,g)A =

(f,g)A,

that,

in c o n t r a d i c t i o n

(3) we get

the g r e a t e s t

- ~'

= B.

a0,a I ..... an,~',B'

h e A

~ hA.

In v i e w (4)

observe

e Q([~' ,9' ]Bn), Now

yp

such

g = ~'Z

that

[a0,a I ..... a n , e ' , 8 ' ] B For proof•

and

(6')nf

and h e n c e

to e

show (ypn-l,

~'

(by g)A

(5)) c

C.

e C N B = yB,

4.18

and

(i),

follows

(I')

and

that

n-i

= u

(6)

fact

that

n > 1

yield

a contradiction.

It

f ¢ C.

Applying yp

the

Proposition

and

yB = Q, w e

1 = o r d Q ( y p n-l) = ~

AI[3, P.97],

lAp/PAp

extended

with

get

= ordQ(Z-resultant

: BQ/QBQ][A~(f,g)A

over

B = R, A = S, Z = t,

all

of

f,g)

p : Ap~,

P c ~I(A)

the s u m m a t i o n

with

f,g e P

: BQ/QBQ] [APi/QiAp i

: APi] .

and

being

P n B = Q

r = i=l ~ [APi/PiAPi It f o l l o w s

from

(6)

that,

(36.6)

ELEMENTARY

Let

be an c o m m u t a t i v e

over

R

R

is said

distinct

i,j

(r I ..... r n)

and

s

i.

n-step

t e R

mentary R.

denote

n-steps

Whenever

the w o r d s

in

the

I

=

clear. easy

R

elementary

The

by

by

n

the d e t e r m i n a n t

and

where

if

~

of

if t h e r e for

each

M

exist n-tuple

M

to be

for

from

about

is

+i

of ele-

n-transformation we

may

j u s t use

to

R

transformations

(in the

n-transformation elements

in

respectively.

if r e f e r e n c e

of e l e m e n t a r y known

and h e n c e

251

product

transformation

elementary

are w e l l

.

the c o n t e x t

~R(i,j;t),

is an e l e m e n t a r y M

~ = j

an e l e m e n t a r y

on the n u m b e r and

j

A finite

elementary

results

induction

If

R

(s 1 ..... Sn),

is c l e a r

~(i,j;t)

transformation,

(36.6.1)

in

n × n matrix

that,

~R(i,j;t).

is said

step

following

to p r o v e

elementary

M

number

Also we may write

An

such

rj + tr.i , if

shall

as c l a i m e d .

w e have.

(r I ..... r n ) M = r r

We

Q1 = P1 = C,

ring with

an e l e m e n t a r y

c [i ..... n] R

and

TRANSFORMATIONS.

to be

over

r = 1

steps

is are

in an

substance). in

R, t h e n

in any row or c o l u m n

4.19

of

M

generate

(36.6.2) mentary

the

unit

If

ideal

(r I ..... r n ) M

transformation,

where

All

=

elementary

~(i,j,t) - I = ~

(36.6.4)

Let

R.

=

(s I ..... Sn),

where

M

is an e l e -

then

[r I ..... r n } R (36.6.3)

in

Is I ..... S n ] R

.

n-transformations

in

R

form

a group

(i,j,-t).

%: R - S

be

an

epimorphism.

be

the

Assume

that

S =

W

SI~...~S m ponent. ~(r)

and For

the

an n - t u p l e

there

r =

in

an

constructing

N

A (i) , w h e r e Usi

taking

N

(i)

R

S.

we

Let

the

i th

shall

com-

denote

N 1 ..... N m

by

be

S I , . . . , S m.

for

n-transformation

every

clearly

have

(i)

~j chosen

the

first

let

~j(i) = (i)

= ~i

~2

"'"

n-tuple

r

N over

required

N

be written

1

~S i ( p ( i , j ) , q ( i , j )

(i) ~s I

......

~m)-

in R

R

such

and

where

we

such

%(p(i,j),q(i,j)

; t

that

Let

(s I ..... Sn)

(depending

on

1

=

...

~(m)-" sm

'

i,j)

e R

Then N

will

take w

(i,j)),

where

t

is

!

~S(~(t

(36.6.5)

N.

; t(i,j)).

w

=

as

""

property,

*

M

onto

1,2,...,m.

@~i) 0~i) ...

s =

over

over

elementary

= Vi(~(r)N i

projection

(r I ..... r n)

(~(r I) ..... ~(rn))

exists

~i(%(rN))

For

upon

: S ~ Si

n-transformations

Then

i =

~i

n-tuple

elementary

that

let

S

over s)

sM =

I

0

, if

s ~ 1

t(i,j)

, if

S =

(i,j))=

be

an e u c l i d e a n

domain.

S,

there

an

such

exists

i .

Then

elementary

for

any

n-tuple

n-transformation

that

(s I, " .. ' s*) n

with

252

s*--

0

for

2 ~ i < n

.

4.20

Hence

by

(36.6.2)

we have siS =

For

constructing

M

we

n-steps

M

which

elementary algorithm

on

Let

r =

M

~(r)M

=

there

Ker and

in

is t h e follows

Further that the

first

such

that

single

with

n-steps

stages

S 1

and

exists

= 0

that

an

for

S

as

of the

euclidean

= Ker

~ +

assume

that

elementary

2 ~ i ~ n

over

R

for an

n-

.

is a n e u c l i d e a n

(m I ..... m n)

of

the

domain.

such

that

(mlrl+...+mnrn)R

that

n

must

some

column

that

i~ a true

~ e B

in

R

as

have r i e Ker

for

N

of

~,

for

2 ~ i ~ n

mi e R

where

N.

The

rest

are

as

y

in

.

(m I ..... m n) of the

proof

(36.6.2).

Assume ~

n-transformation

where

first

and

the

we

(r I ..... r n)

exis~

there

assume

take

Then

THEOREM.

such

A,B,C,r adjoint

that

of

~ Z - $ e C.

in T h e o r e m B.

Assume

(36.5).

It f o l l o w s that

~

has

property,

(*) such

obvious

epimorphism

an n-tuple

(36.6.1)

assume

there

R

r I = mlrl+...+mnr

(36.7)

the

= R.

we

transpose

an

over

we may

(36.6.4).

from

be

~ + [r I ..... r n ~ R

rN =

Then

take

perform

(Sl, ..., Sn)

exists

proof,

obtained

S

case

{m I ..... m n ] R For

R - S

in

a special

Then

~:

(r I ..... r n)

transformation

As

of

simply

(s I ..... Sn).

(36.6.6) n-tuple

[s I ..... S n } S

for that Further

some

{p',~'}B assume

~

with

= B

and

~ Z - ~ ~ C, {~'

that,

253

there

+ ~'Y,~]B

exist

= [~,y,~}B

p' .

, ~'

~ B

4.21

w

(**)

IX

([B,~],P)

T h e n there Theorem

(36.5)

as above,

exist

:

p,~

P

e

c

B

are satisfied,

a true a d j o i n t w i t h

Further,

~

has

90([B,y])}

= {1,2]

such that when

all the c o n d i t i o n s

a suitable

property

the p r o p e r t y

(*),

is c h o s e n

of and

~

i__ss,

(*). i_~f B/~B

is an e u c l i d e a n

domain. PROOF.

Choose

(i)

~z

-

~

c

(2)

{p',~']B

(3)

{Sp'

c

8,p',~'

e B, such

that,

,

= B, and

+ ~'y,~}B

[~,y,&]B

=

.

Let (4)

Q = {P e ~0(B) We claim

(5)

For proof,

(51 )

y

+ ~'y

e P]

that,

P c ~ ~ ~ ~

(*)

: p' ,Sp'

P.

assume

if possible,

that

~ ¢ P.

By

we have P

¢

Since

and

~

~ ~[SP'

is a true

+ ~'~,~}Bp

adjoint,

in view of

(**),

(4),

(51)

and

(I0.i. Ii), w e get (52 )

8 6 P({~]Bp) Let

that and (53 )

+ ~Bp

~: A - A/C

~(B) [~(z) ]

and c o n s e q u e n t l y

be the c a n o n i c a l

is the i n t e g r a l

~ e ~Bp

homomorphism.

closure

of

~(B).

. Then w e h a v e In v i e w of

(i)

(52 ) w e get ~(Z)

~ ~(B)~0(p)

Hence we must have

and in p a r t i c u l a r k~([B,y],P)

254

= 0

~p(B) (p) (by the

is normal.

last r e s u l t

in

(6.1))

4. 22

and h e n c e concludes

~

being

a true adjoint

the p r o o f of

From definition

(6)

P e ~ - p',y

~ P ; a contradiction.

This

(5).

of

Q, b y

e P

and

(I) a n d

(5) it f o l l o w s

~([B,u],P)

that

= !, and h e n c e b y

(9.1),

Y ~ p2.

Clearly (7)

~

is a f i n i t e

p' + c ~ ' y

e p\p2,

set and w e can c h o o s e

for all

c ~ B

such t h a t

p ~ ~.

Now put (8)

p =

p'

+

c~'y

,

It is c l e a r t h a t (9)

{p,~]B = B In v i e w o f

(i0)

=

~'

,

~'

=

~p

+

r%, =

~p'

+

~'U

+

c~'U

.

(*) i m p l i e s

and

[~',~}B = [~,y,~}B

(6) a n d

(**) w e g e t that

there exists P' ~ P0

~

P0 e

~0([B,u])

; in p a r t i c u l a r

with

p' ~ uB

.

I ( [ B , u ] , P 0) = 2 and h e n c e

and h e n c e

p ~ uB.

N o w let

Q1 = [P e ~ 0 ( B )

: :,~'

e P],

0 2 --" {P e ~ 0 ( B )

: p,8'

e P]

and

w

T h e n w e have,

(Ii)

~2 = ~ Let

by

(ii)

(12)

(5)

' ~ i N Q2 = ¢

P ¢ ~i"

Then by

D ~ P, 6'

P e Q 1 = UP N o w let

(13)

using

and

~

= ~i U Q2

(**) we m u s t h a v e

e tradj ([Bp,U],P),

e

((~',~')Bp) 2

P ¢ O 2.

Then

from

and (ii),

" I ([B,y],P)

and b y a p p l y i n g

(I0.i.ii)

UP ~ P B p ( { ~ ' , ~ ' } B p ) 2 o (6) and

(7) we get,

P ~ ~2 ~ P'Y ¢ P ; ~,~T ~ P ; [ 0 ' , ~ ' } B p = [ p , y } B p yp e

([~.',8'~Bp) 2 , yp ~ P B p ( { ~ ' , ~ ' } B p ) 2

255

= 2.

and

Further we get,

4.23

(14)

P ~ Q ~ {~',8'}Bp = Bp

(by (ii) and hence)

¥p 6 <{~',~']Bp] 2

From (I'),

(i0),

(2') and

(12~,

(13) and

(14), it follows that the c o n d i t i o n s

(3') for T h e o r e m

(36.5) are satisfied.

The last s t a t e m e n t readily follows

from

(36.8)

if in T h e o r e m

situation

COROLLARY.

(3) prevails,

(36.7) we get that

(36.9)

C

THEOREM.

In particular,

C

¢

~I(A)

F u r t h e r assume that C

is a complete PROOF. (36.3),

the

(36.5) and

is a c o m p l e t e intersection.

Let

A

be a filtered d o m a i n o v e r an a l g e b r a i -

with

dim A = e m d i m A = 3.

deg[A,C~

X([A,C~,P)

= 1

~ 5

an__~d genus

for all

[A,C] ~ i.

P 6 Z0([A,C ]) .

Then

intersection.

In view of T h e o r e m

(36.4)

(35.5),

then as a c o n s e q u e n c e of Theorem

cally closed g r o u n d field and w i t h Let

(36.6.6).

and

(35.5),

(36.8).

256

the proof is finished by

CHAPTER

V.

APPENDIX §37.

In §i0 w e R

Double

proved

of d i m e n s i o n

one

points

certain

of a l g e b r o i d

properties

and m u l t i p l i c i t y

curves.

o f an a b s t r a c t

two

such

that

local

the

domain

integral

w

closure

R

proof.

of

One

and

in

can also

completion", advantage

R

i.e.,

~(R)

give

with

standard-looking

a lot of b a c k g r o u n d

this

section

proof

(i0.i)°

same

shall

of

The

as in §i0.

pletion

~

R

method")

(37.1)

of

material

proofs

of

(10.1.9) as

better

still

e(O)

~(O)

If

(10.1.2) (i0.I.i0)

have

also

completion

of

a much

"completion

the c o m p l e t i o n

by =

and to

(10.1.3)

more

a direct "going R.

to

The

concrete

is t h a t w e

proof".

proof

shall

(10.1.14)

shall

alternatively

R

as in T h e o r e m

In

can be

be t h e s a m e be

also

as

the

(not b y a "com-

X(R)

= e(R),

where

e(R)

by u s i n g

formula

300

(8"),

(i0.i),

w e have,

R = 2.

of Z a r i s k i - S a m u e l , page

is the m u l t i p l i c i t y

Corollary

L 18 ~.

Thus

i, p a g e e(R)

of

299,

= k(R)

or

=

2.

R/M(R)

is an i n f i n i t e

field,

then we have

Abhyankar's

result

[ 2 3,

since emdim

R

is not

R ~ dim

regular

(R)

+ e(R)

(otherwise

- 1 ; X(R)

= 1

by

(5.10))

we

must

R = 2 = e m d i m O.

In the g e n e r a l by

of

The d i s a d v a n t a g e

can be proved

With

emdim and

gave

method

c a n get

O.

how

the

We

follows:

W e have,

in the s e n s e

(1)

O,

for the

illustrate

(10.1.1),

LEMMA.

PROOF.

=

so-called

is t h a t w e

emdim

R

by

R-module.

out.

The in

b y the

description

need

carried

a proof

by replacing

of w o r k i n g

we

is a f i n i t e

case w e

R' = R[YJM(R) R[y] a one d i m e n s i o n a l

where local

proceed Y

as in §9;

namely,

we

is an i n d e t e r m i n a t e .

domain 257

with

I(R')

= k(R),

replace

Then

R'

R is

M(R) R' = M(R')

5.2

and

an i n f i n i t e

emdim

residue

R' = 2 = e m d i m

(37.2) is e n o u g h

R

R'/M(R')

~

R/M(R) (Y).

Hence

as r e q u i r e d .

REDUCTION

to p r o v e

field

TO THE

(10.1.4)

COMPLETE

to

CASE.

(10.1.8)

by

We will

now show

replacing

R

by

that

it

its c o m w

pletion

O.

Let

{V 1 ..... Vr}

= ~(R)

and

let

Pi = M(Vi)

n R .

Let

integral

closure

w

O

be

0

in

the c o m p l e t i o n ~(O).

dimension

Using

i, w e

of

R

and

that

R

can e a s i l y

let

O

be

is a finite deduce

the

the

R-module

following

and t h a t

R

description

of

of has ~

and

e

using [ 181 18,

any

standard

Chapter

Theorem

VIII.

24,

the d i r e c t is a f i n i t e

27 a n d

with

O-module.

in

R

verse

bijections

for n o n z e r o

Further,

between

corresponding

a subring for

M(R),

nonzero

:

w e have,

:

R

which

of

V i , say

O

is a one

of

3(0)-

I - IO

in t u r n

Oi

~]

In v i e w

dimensional

and

J - J N R

R

I, J

in

R,O

: R3 =

[O/J

every

and

~.

local

ring

nonzero are

in-

In p a r t i c u l a r

respectively

we have

: 0].

and h e n c e

[R/~(R)

: R] --

I ¢ R

or

I c R

we have

[R*/IR*

: R~ =

and hence = X(O,I)

of t h e s e

= k(~,I~)

remarks,

(by

it is c l e a r

(5.5)).

that

to p r o v e *

we may

completion,

coincides

In p a r t i c u l a r

Since

of

~(R) O = ~(~)

for any

k(R,I)

(10.1.8)

Ii, T h e o r e m

~].

Similarly [0 / I O

of

ideals

ideals

[R/I

[~/~(o)

to T h e o r e m

R.

is p r i m a r y

Further

refer

say Z a r i s k i - S a m u e l

corollaries.)

a completion

is c a n o n i c a l l y

ideal

we

their

s u m of c o m p l e t i o n s

and e m d i m ~ = e m d i m ~(R)

of c o m p l e t i o n s ,

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

Theorem

coincides with

treatment

say

replace ~/,

and

everywhere, W

by

R

by

O,

its c o m p l e t i o n ,

258

R

(10.1.4)

to

e

by say

0 ~f.

, V

by

its

5.3

This

gives

(I).

O

Case

(2*)

Case

the

following

is not

a domain

dimensional

or%M(O)

= ord~(M(~) ) = 1

rational

over

is a d o m a i n

O

local

to u s e

the

and ~ / a n d

and ~ a r e

domains.

Further

~

are

residually

a complete

one

dimensional

over

O.

= ~/,

a complete

and

O

: O / M ( O ) ) ] = 2.

completeness

local

rational

[9//M(~

domain.

WITH

is y e t

regular

cases:

where ~

Further,

local

description

~

three

domain.

is a d o m a i n

DESCRIPTION

=~/~

O * =~/,

and

regular

(37.3)

for the

O.

is r e s i d u a l l y

above

O

one

O

(3)

and

complete

regular

Case

description

Further,

AN EXTRA

no s i m p l e r of

O

ord~M(O))

one

ord~(M(O))

ASSUMPTION.

than

the one

effectively

= 2 and

we

dimensional = 1

Note

that

in §i0.

shall

and

the

To be

make

able

an e x t r a

assumption:

(*)

O

and

Note

that

algebraic tained

O/M(O)

curves

in b o t h

Assuming There of

0

if

morphism O

defined Ker

is such

Cohen's

by

where over

~ = y'B.

= x

In v i e w

since

field

B K

and of

for

onto

for the

case of points

the g r o u n d

and

~

O,

(9.1)

i.e.

field

of

is con-

field

then

The map

~

if

ring

identity x,y

: K[[X,Y]]

= y.

Also

Ker ~

we g e t

from

k (O) = 2

259

canonical

exists

series

to

is a s u b f i e l d

the

there

power

restricts

9 [8]):

there

under

case w e g e t that,

o(Y)

(Theorem

O/M(O)

is a f o r m a l

O = K[[x,y]].

~(X)

good

Structure

a coefficient

in o u r p a r t i c u l a r then

field,

isomorphically

variables

M(O),

(*) h o l d s

a coefficient

maps

characteristic.

O/M(~).

(*) w e get b y

K

same

a ground

and

~: B - O

Thus, of

O

the

assumption

over

exists

which

further

emdim

the

have

on

map;

an epiin K.

is a n y b a s i s

= B - O

may be

is p r i n c i p a l , that

say

5.4

O r d M ( B )7'

= 2

Further, is e a s y

if

to s e e

x ¢ O

that we

7' = rlY2

Thus we have If

x,y

= 2, t h e n

there

shall

exists

now

following

NOTATION. If w e

ri ¢ B

Theorem

say

then we write

of

M(O),~(X)

= x,~(Y)

+ q

such *

that

, where

the

(8.2)

it

is a u n i t .

= Y,

~

,

,q

,

eases.

is a u n i t .

and

K e r {p = y B

p

three

rI

and

statement:

p*y

be a n y

¢ Kilt]I,

h = h(t).

h(X)

k(O,x)

and

¢ XK[[X]]. First we

int~Qduce

formal

power

ring over

K.

we mean

Also

if

¢ K[[t]]. X

is a n y o t h e r

= h 0 + h l X + . . . + h n xn+.

in

K[[t]]

Since

0

by

variable,

. ¢ K[[X]].

we mean

y

as u s u a l

is r e d u c i b l e

Then we may write:

~ =

(Y-p) (Y-q)

(3)

e = ordx(p-q). prove

ordth

is n o t a d o m a i n ,

(2)

shall

series

to m e a n

K[[X]][X].

We

of

we may write:

following

redescribe

h

h(x)

(i).

and

the

y ¢ B

K[[t]]

and we may write

Case

in v i e w

* * p ,q ¢ XK[EX]]

h = h 0 + hlt+...+hntn+...

Further

then

notation:

Let

simply

• where

, where

proved

Y = y2 + We

*

is a b a s i s

(I)

the

+ q

= 2,

can write

Preparation

p*y

+

k(O,x)

+ r 2 X Y + r3 X 2

By Weierstrass , = y2 y ~

with

later

, p,q

¢ K[[X]],

that

e = d =

260

and

[O/~(O)

: O].

min{i:

in

h i ~ 0}.

5.5

(4)

O

= KE[T]]

~2/= K ~ T ] ] ,

~

KILO]3,

~/'= K [ [ ~ ] ] ,

o r d ~,(h,h ' ) = o r d

(5)

~:

B - O

(6)

C~

=

the

=

are

ord~2/(h,h')

independent

= ord~(h(~))

variables, and

by

(h(T,p(~)),

definition

{ (h,h')

~,T

h' (~) .

is d e f i n e d

~(h(X,Y))

From

where

6 O

of

h(c,q(o))).

~,

it

is e a s y

: ordx(h(X)

to check

- h' (X))

> e}

that

.

w

Case

(2).

In this and we

(7)

(~

(8)

O = K[[x,y]]

we

is

(9)

(3). (}

where

, y = y(~)

In t h i s

case

= ~/ = K ' [ [ T ] ] of

0 = K[[x,y]] x = x(~) of

x,y

If

any

e K[[T]].

x

K.

T

x(~)

y

basis

of

Further

by

M(O) our

and

choice

of

= 2.

is

where

irreducible K'

where

, y = y(T)

x,y

and we

= ~//M(~

In particular

is

we

any

e K'[[T]].

may write:

is a q u a d r a t i c

have

basis Further

of

M(~)

by our

and choice

we have ord

(ii)

is

have

extension

(I0)

K[ Ix] ] [Y]

in

ordm/= o r d T

and

ord

Case

irreducible

may write:

= 9/= K [ [ ~ ] ]

x = x(~) x

case

g:~"

then we h = h(T)

~9//M(~2,~ have

g(h)

= K' = h0

T

x(T)

= 1 .

denotes

the

, for any

= h 0 + hl~+...+hn~n+...

261

canonical

residue

map,

5.6

We

shall

completion)

now

in

(37.4) in

(37.3)

each

CASE

i.e.,

First (i)

we

E(O)

=

of

OF

claim

HIGH

in

three

(i0.i.4) cases

NODES.

(i)

e n M~

to

to

above

Assume

(6)

in

(i0.i.8) with

the

(after

the

going

assumption

description

of

to (*).

case

(i)

(37.3).

e

e O

let

(6)

the

of

that

[ (h,h')

proof

From

proofs

assume

= M(~

For

give

~

: ord be

(37.3)

T

h(~)

> e

and

the

ideal

on

is

clear

that

it

the

ord

right ~

is

h' (~)

~ e~

side. an

ideal

in

O

as w e l l

w

as

~

.

Hence

On

the

(h(T),h' from

hand

(~)) (i,0)

and

in

E(O) Now

to

see

(37.3) c ~.

let

that

we

x = we

(3)

a0 = b0 =

In we

(5)

can

the

of

then

h' (~)) (0,i) have

ord

we

must

belong

b(r)

~ e

to

have ~,

and

that

and

ord

hence

h' (~)

> e,

claim.

¢ O

0 <

0

such

1 < e -

and we

that

k (O,x)

= 2.

Then

it

is

easy

(6)

0

that

and

in

i.

aI = bI ~

see

= K[[a~

view

K~[~

= K~b~

(37.3)

again

(0,f)

where

we

see

that

given

any

(h,h')

¢

write

(h,h') such

are

ord

f(~)

> e,

and

~ ¢ K[[a~

= h(T) .

= ~(x)

x e ¢ E(O)

O/E(O)

i = 0,

+

~ (a)

(h,h')

Since of

= ~(x)

that

Thus (6)

clearly

¢ E(O)

have

for

In particular

(h,h')

(h(T),

(a,b)

must

a. = b. l l

K[[T]]

if

Hence

(2)

(4)

~ c ~ (O).

other

(6)

i.e.,

clearly

by those

(modulo (i),

we

E(O)). clearly

generated

1,2 ..... e - l .

262

by

see x

i

that modulo

the

only

E(~)

ideals

for

5.7

In particular, (7)

[O/~(O) In v i e w

the

proof

of

of

ord

r

Then

=

(6)

e in

(10.1.4),

It o n l y with

: O~

remains

a' (~)

and

(37.3)

have

e = d.

and

(10.1.6), to prove

= d < ord

we

hence

(i),

(10.1.7)

0

for

i =

0 , i .... ,d-l,

(9)

b: = x

0

for

i =

0, i ..... d.

T

a

let

y* = y

- a~a i dxd

(T)

and

have:

we

a

=

0

for

i =

0 , i ..... d,

(11)

b i =

0

for

i =

0 , 1 ..... d - l ,

Now the

[x,y]O

is

it

is

(I0.i.8)

Thus,

clearly

a d' ~

0

(a * ,b * )

=

(10)

1

(7)

let

clear

is n o w

y =

that complete.

(a',b')

¢ O

b' (~).

a' = l

d < ord

and and

(10.1.5).

(8)

Also

(6)

and

primary

and

say.

bd ~

for

Then

ord

b . (~)

=

0.

M(O)

and

hence

we

have

that

number

(12)

m = max{min{ordTh(r), is a We

(13)

claim

m = For

(131 )

finite

if p o s s i b l e , ¢ O\[x,y}

that

m

m d,

(h,h')

-

(h m

min{ord (in v i e w If

(h,h')

¢ O\[x,y}O}

integer.

that

let,

Note =

nonnegative

h' (~) } :

0.

(h,h')

(f,f')

ord

m

of

~ d,

the then

m

> 0

and

with

min[ord

since

otherwise

aim)xm

f(T), fact

we

ord

clearly

we

h(~),ord

hm

=

can

263

h' (~)}

= m

> 0

.

for

clearly

f' (~)}

that

T

have

> m h 'm

choose

and

a I = b I) ~i,~2,q3

e K

such

that

5.8

m

(132)

h'

=

~2 a~ al-d

m

m

it is easy to see that

(131 ) we have,

~3Y*xm-d),

(133 )

and

* m-d 93bd b l

+

~ibl

Then and

+

hm-- ~lal

in v i e w of (f,f') =

upon letting

(3),

(h,h')

(8), -

(9),

(i0),

(91 x m + ~2 y x

(ll)

m-d

that:

min{ord

T

f(T)

'

ord

O

f' (~)] > m

and

(f,f') c O\[x,y]O Since

this

is a c o n t r a d i c t i o n

to the m a x i m a l i t y

of

m,

(13)

is

proved. From proving

(12)

and

(13),

it f o l l o w s

that

M(O)

= [x,y]~,

thus

(10.1.5). .

(37.5) in

(37.3),

i.e.,

Since h(T) (i)

CASE OF H I G H CUSPS. assume

(7) and

Assume (8) in

the d e s c r i p t i o n

of case

(2)

(37.3).

ord x(<) = 2, it is c l e a r t h a t e v e r y e l e m e n t T

e K[[T]] h(~)

can u n i q u e l y w r i t t e n

as

= h (I) (x) + T h (2) (x), w h e r e

h (I) (x), h (2) (x) ¢ K[[x]].

Let (2)

e = m i n [ o r d x h ( 2 ) (x)

: h(1) (x) + T h(2) (x) ¢ O

for some Since (3)

c O

e = min{ordxh(X) If

every

K[[x]]

~(K[[x]]) 0 ~ 8 ¢ 3(O)

O ~ K[[x]]. positive Also

Also

h(1) (x)].

we clearly have : ~ h(x)

¢ O].

= 3(0)

; we w o u l d

get t h a t

; and

since this

is a c o n t r a d i c t i o n

clearly

~ ~

K[[x]]

or~(8)

and h e n c e w e

~ 0

(2)

for

we h a v e

see t h a t e is a

integer. note t h a t

in

(3), we h a v e

hence

264

ord

T

Th(x) = 1 + 2 ord h(x) x

and

5.9

(4)

e = min{s

:ord

h(T)

=

2s + 1

of

(i)

for

some

h

¢ O}.

T

Note

(5)

that

O =

in v i e w

and

[h (I) (x) + ~ h (2) (x)

(3), w e h a v e

: ord

h (2) (x)

~ e}.

X

Now

we

for

(6)

claim h

¢ O

that , h

¢

~(O)

~ ord

h 'I'( % (x)

> e.

X

Namely, observe (61 )

¢ 6(O)

and T

(5) a n d

(I), =

is

obvious;

while

for

2

= Th¢

--- a(x)

.

+ ~ b(x)

from

thus

, a(x),b(x)

(61 ) a n d

Ordx(h(1) and

O

if w e w r i t e

then (6 3 )

of

that

h

(6 2 )

in v i e w

(5) w e

h(1)(x)

> e

,

get

(x) + b ( x ) h (2) (x) ord

¢, K [ [ x ] ~

~ e

, ordxh(2)

(x)

~ e

.

.

X

From

(5) a n d

(6),

(O) = M ~ / ~ 2 e

(7)

it

follows

that

written

as

, and W

(8)

every

h

¢ 0

can

be

*W

h

(9)

~(0)

and

at most

(e-l).

From

it

(8)

generated 0,

by

is n o w

+ h

in

x

where

(x)

the

is

that

images

a polynomial

the of

only x

i

ideals

modulo

of

~(O),

of

degree

0/6(0)

are

for

1 ..... e - l . we have

[ O / ~ ( O ) ] = e, Clearly

h

follows

In p a r t i c u l a r (i0)

WW

(x)

W

¢

i =

h = h

the

complete

It r e m a i n s

and hence

proof

of

in v i e w to p r o v e

e = d.

(10.1.4),

of

(i0) b y

(10.1.5).

265

(i0.i.6), (5), So

(9), let

(10.i.7) (7)

y

¢ O

and

and (4)

with

(10.1.8)

respectively.

5.10

y(T)

(ii)

-- y(1) (x) + Ty (2) (x)

Hence

(12)

we must

ordxy(1) Upon

(13)

(14)

=

from

have > d

taking

[x,y]O But

(x)

y

*

, ordxy(2)(x)

(~) =

Ty

(2)

(x)

= d

.

we

have

(5) a n d

(I0)

we

see

+ ~ h (2) (x)

Thus clearly

O = K[[x]]

(10.1.5)

¢ O =

T h (2) (x) = y * h * ( x )

(x)

where

c KE[x]].

+ y K[[x]]

and

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

is p r o v e d .

(37.6)

CASE

(3 ) in

(37.3),

Choose of

,

[x,y]o.

Thus

case

¢ xO)

that

h

=

y(l)(x)

(since

{x,y*]O.

h = h (I) (x)

M(o)

= 2d + i.

ord Y(T

with

K'\K

any

OF N O N R A T I O N A L

field

would

do.)

note

that

First

i.e.,

CUSPS.

assume

(9),

generator

K'[[T]]

Assume

c

the

(i0)

and

K'

over

of

= K' [[X]],

description

(ii)

since

K.

ord

in

of

(37.3).

(Any e l e m e n t

x(T)

=

i.

NOW

we

T

have: (i)

every

~h (2) (x),

h

6 K'[[x]]

where

can be

h (I) (x),

uniquely

h (2) (x)

written

as

h = h (I) (x)

+

¢ KE[x]].

Let (2)

e = min[ord

h (2) (x)

: h (I) (x)

+ ~h (2) (x)

¢ O

for

some

h (i) (x) ].

X

Since (3)

K[[x]~

e : min[ordxh(X) Since Also Note

(4)

c O

O =

¢ K[[x]]

and hence

in v i e w

{h (1) (x)

clearly

C~ ~ I<' [[x~],

# e ~ (O) that

we

of

(i)

+ ~h (2) (x)

have

: #h(x) c ~ O e < ~.

and

(3),

¢ O].

and

Thus

e

we have:

: ordxh(2)(x)

266

e ~ 0.

~ e].

is a p o s i t i v e

integer.

5.11

Now

(5)

we

for

claim h

observe h

in v i e w

¢ ~(O)

¢ ~(O) of

~

ord

(5) a n d

~

= ~h ~ O

if w e

2

h (I) (x)

~ e

.

(i), =

is o b v i o u s ,

while

for

=

from

thus

, a,b

¢ K

(51 ) a n d

ordx(bh(2) and

,

let

= a + ~b

Then (5 3 )

, h

that

and (5 2 )

¢ O

X

Namely

(51 )

that

t

(4) w e

get

(x) + h (I) (x))

~ e

ord

h ( 1 ) (x)

, o r d x h (2) (x)

z

e

,

~ e.

X

From

(4) a n d

(6)

~(0)

(7)

every

(5)

h

h -- h

(8)

c O

(x) + h

x

of

From

(7)

it

by

[O/~(O)

Ordx8

images

: O] = e,

of

from

(8)

=

s

and

g(y/8) Thus

~

for

d = min{s

(3),

(10.1.8), if

the

of

only

h

ideals

x i modulo

and hence of

proving

and

(x)

is a p o l y n o m i a l

~(O),

of

C)/~(O)

for

are

gen-

i = 0 , i .... , e - l .

we have

the proof

For

¢ ~(O) (e-l).

that

Clearly (9),

h

at most

follows

the

as

WW

where

degree

In p a r t i c u l a r

view

that

can be written WW

erated

(9)

follows

= a(~2z9e , a n d

W

in

it

h(x)

(10.1.8) : OrdxY

(10.1.4), and note

~

K

that

if

with or

it is e n o u g h =

(10.1.6)

~

K, 267

y,8

(10.1.7)

e O

ordxh(X)

follows,

in

=

with s,

ordxY

=

then

g ( 8 / h ( x ) ) ~ K. to prove

s = ordxh(X)

g(y/h(x))

and

(6).

c K[[x]]

K ~ g(y/h(x)

e = d.

for

that

and some

y

¢ 0

and h(x)

¢ K[[x]]].

5.12

This

readily

It r e m a i n s (lO)

y(x)

follows

from

to p r o v e

(9) and

(10.1.5).

= y(1) (x) + ~y(2) (x)

(3).

So

let

and

y e O

ordxY

= d

such and

that

g ( y / x d) ~

K.

Hence we must have

(ii)

o r d x y ( 1 ) (x) > d , Ordxy(2) (x) = d Upon

taking

y

*

(2)

(x) = ~y

(x)

we

. clearly

have

(since

d > 0)

w

(12)

{x,y]O But

(13)

= {x,y

from

](9 .

(4) and

(9) w e

see

that

h = h (1) (x) + ~h (2) (x) ¢ O ~ ~h (2) (x) = y h

(x)

where

W

h Thus

clearly

(x) c K [ [ X ] ] .

O = K[[x]] M(o)

Thus

(10.1.5)

§38. Let We which

A

(38.1)

Now

in p a r t i c u l a r

.

domain.

and w h i c h

Assume

N e ~(A)

radA(P+N)

it follows

for two h y p e r s u r f a c e s .

(essentially)

resultant

and

(12.1.5)])

theorem

DEFINITION.

P ¢ ~r_2(A) Q = ~.

Bezout's

now give

the

{x,y]O

and

is p r o v e d .

be a h o m o g e n e o u s

shall

uses

=

+ y K•[x]]

does

that

be s u c h

= HI(A)A that

Bezout's not

assume

Emdim that

of

(25.14)

r = 2.

A = D i m A = r ~ 2.

P ~ N.

and h e n c e

[R(~A, P3)

own proof

Let

B = AN

(for i n s t a n c e

: R([B,Q])~

< ~.

see

Let and

[i :

Consequently

w

Q e H

(B) N ~(B),

[R([A,P])

(i.e.,

: R([B,Q3)]

are p o s i t i v e

~(A,P,N) and w e note (38.2)

that LEMMAo

Q c ~r_l(B)),

= [R([A,P])

g(A, P,N) Assume

so

integers.

DegBQ

and

We define

: R([B,Q])]DegBQ

is then that

and

a positive

Emdim

268

integer.

A = Dim A = r = 2

or

3. Let

5.13

P e ~r-2(A)

and

N e 30(A)

be such that

P ~ N.

Then

g(A,P,N)

=

D e g [A, P~.

PROOF.

If

r = 2

t h e n the a s s e r t i o n We shall given

follows

now d e d u c e

in [3:

(38.3)

t h e n the a s s e r t i o n from

(23.15)

BEZOUT'S

and

the f o l l o w i n g

P r o p o s i t i o n A.I on p a g e

LEMMAo

Assume

is o b v i o u s .

If

r = 3

(25.9).

lemma

f r o m its a f f i n t e

version

67].

that

E m d i m A = D i m A = r m 2.

w

Let

~ e Hm(A), ~

e Hn(A),

nr_l(A,#)

N ~r_l(A,~)

= ~

~r-2(A'~)

N ~r-2(A'~) .

and

N e 30(A)

be such that

and

N ~ n(A,~)

U n(A,~).

Let

~ =

Then ~ ( E A , ~ , ~ , P) g(A,P,N)

=mn

.

Pc~

PROOF.

If

mn = 0

show.

So s u p p o s e

H0(A)

and

let

coordinate

(X0,X 1 ..... X r _ l ) A .

~ = #

system

with

polynomial

We can

(X0,X 1 ..... Xr)

in

A

~ i ( Y o ..... Yr_l)

homogeneous 9(Y0 ..... Yr )

polynomial

U ~(A,~),

k =

such that

w e now h a v e

M ( Y 0 ..... ¥r ) = ~r

is e i t h e r

z e r o or a n o n z e r o

i, and s i m i l a r l y

with +

Let

~ Mi(Y0' " ' ' ' Y r - i )Ym-ir l
of d e g r e e

= ~ (X 0 ..... X r ) A

n > 0. k.

M i ( Y 0 ..... Y r _ l ) ~ K [ Y 0 ..... Y r _ l ]

homogeneous

and

over

N ~ ~(A,~)

+

where

m > 0

be i n d e t e r m i n a t e s

Since

= ~0(X0 ..... X r ) A

where

and w e h a v e n o t h i n g m o r e to

m n ~ 0, i.e.,

Y 0 , Y l .... 'Yr

take a h o m o g e n e o u s N =

that

then

~(¥0 ..... Yr ) = ~r

~ ~ i ( Y 0 ..... Yr_l) ~r -i l
e K [ Y 0 ..... Yr_l] of d e g r e e

i.

h a v e no n o n c o n s t a n t

By a s s u m p t i o n

common

letting 269

is e i t h e r

factor,

zero or a n o n z e r o %0(Y0 ..... Yr )

and h e n c e u p o n

and

5.14

(i)

8 (Y0 ..... Y r _ l ) = R e S y

(~0(Y0 ..... Yr ) ' 9 (Y0 ..... Yr ) ) r

where

Res

(2)

denotes

Yr

resultant with

respect

0 ~ 8(Y 0 ..... Y r _ l ) e k [ Y 0 ..... Y r _ l ]

to

Y

r '

we have

is h o m o g e n e o u s

that

of

d e g r e e mn. Let

B

AN

=

T h e n in v i e w of (3)

, ~ = ~r_2(B)

Q(Q)

® = @(X 0 ..... X r _ I ) A

.

(2) w e c l e a r l y h a v e

~ ~' ( [ B , ® ] , Q ) D e g B Q QeZ For every

, and

Q £ ~

= mn.

let

= {P e Or-2(A)

: pN = Q} .

Then clearly

~ ([A,~,~], P) g(A,P,N) P¢~ (4)

We shall

now p r o v e

t h a t for e v e r y

(5)

~ ~ ([A,~,~], P)[R([A,P]) P~(Q)

and,

in v i e w of

(3) and

So let any

Q ~ Z

suitably,

we may suppose

X l / X 0, .... X r / X 0

(4), this w i l l c o m p l e t e be given. that

are a l g e b r a i c a l l y

B'[t]

270

the proof.

Upon relabelling

X 0 ~ Q.

t = Xr/X 0 =

we have

: R([B,Q]) ] = ~' ([B,®],Q)

B' = k [ X ! / X 0 ..... X r _ I / X 0 ]

A'

Q e ~

X 0 , X 1 ..... Xr_ 1

we n o t e t h a t the e l e m e n t s

independent

over

k, and w e

let:

5.15

~0' (t) = ~ ( X 0 ..... Xr)/X~0 = t m

+

~ ~i (i' X I / X 0 ..... X r - i / X 0 ) t m - i l
~' (t) = ~ (X 0 ..... X r ) / X 0 = t n

+

~ l
~i (i' X l / X 0 ..... X r - I / X 0 ) t n - i

~, = 8(X 0 ..... Xr_l)/X~0 n = 8 ( l , X l / X 0 ..... X r _ i / X 0)

Q' =

U 0~i~

~' -- {P' f]

[~/x 0 : ~ c Q N H i ( B ) }

~ ~r_2(A')

= [P ¢ Q (Q)

: P' N B' = Q']

: X 0 ~ P]

and p, =

U [(/X~ 0~i~=

: ( e P N H i(A) ] for e v e r y

Now,

in v i e w of

Q'

e F

..~ pI

(B')

and

(15.5), w e h a v e t h a t

n ~(B')

,

is a o n e d i m e n s i o n a l

B' Q' = • ( B , Q ) p

(15.4)

P e

gives

a bijection

of

regular ~

onto

local r i n g , Q' ,

u(EA,~,~3,P) = x ( E A , ~ (t)A ,~'(t)A'],P')

(6)

I

[~([A,P])

and

(7) Since (8)

for all

: e(EB,Q])]

U' (EB,B],Q) N ~ O(A,~)

=

[A~,/P'A~,

: B~,/Q'B~,]

= X' ([B' ,8'B'],Q') U O(A,~

U ([A,45,~],P) = 0

we c l e a r l y h a v e

for all

P ~ [~(Q)\D*

271

P

e

f~

5.16

By

(6) and

(8) w e g e t ~ ([A,~,~;],P)[R([A,P])

P~

: R([B,Q])]

(Q)

(9) =

B6/QE

I ~ k ([A' ,~' (t)A' , %'(t)A' ], P' ) [Ap,/P'A~, P' C~Q'

In v i e w o f

6, 3

(i) w e h a v e

8' = Res t(~' (t),~' (t)) and hence by

[3: P r o p o s i t i o n A . I o n p a g e

67] w e g e t

~([A',~'(t)A',~ (t)A'],P')[A~,/P'A~,

I

: B 6 /QBQ,

P' C~'

(lO)

= ~' ([B',e'B' ],Q' ] Now

(5) f o l l o w s

(38.4)

from

(7),

THEOREM

BEZOUT'S

THREE SPACE.

Assume

that

*

e H

(9) and

(i0).

FOR PLANE CURVES AND FOR SURFACES

E m d i m A -- D i m A = r = 2 or 3.

IN

Let

W

(A)

and

~/ = n r _ 2 (A,~)

~

¢ H

(A)

be

such t h a t

~r_l(A,~)

N ~r_l(A,~

= ~.Let

Then

n n r _ 2(A,~).

U ([A,~,~3,P)Deg[A,P]

=

(DegAS) (DegAS;)

P~9/

PROOF. (38.2) If

H 0(A)

thus. and

and

If

is i n f i n i t e

(38.3) b y t a k i n g

then our assertion

N ¢ 9~0(A)

is f i n i t e t h e n such an

Let

Y

for all

generated by

N

b e an i n d e t e r m i n a t e n

let

H n(A') Then

Let

~' = ~A'

are m e m b e r s ~r_l(A',~')

of

H

(A)

A'

m a y not e x i s t A.

and

~'

Let

= ~A'.

272

and w e p r o c e e d

A' =

(H 0(A) (Y)) [A],

Also

of

A'

domain with

Then clearly

= DegAS, P ~ PA'

from

U f~(A,~).

((H 0(A) ( Y ) ) - s u b m o d u l e

DegA,~' = @.

follows

N ~ ~(A,~)

is a h o m o g e n e o u s

with

n ~r_l(A',~')

with

over

b e the

Hn(A).

= D i m A' = r.

and

H 0(A)

DegA, ~' gives

E m d i m A' ~'

and

= DegA~

,

a bijection

5.17

of

~r-2(A'~)

easily

n ~r-2 (A,~)

seen that

onto

for e v e r y

nr_ 2 (A' ,~')

P ¢ ~r-2 (A,~)

N ~r-2 (A',~'),

n ~r_2(A,~)

and it is

we h a v e

u ([~',~ ,~' 3,PA') = u ([A,~,~3,P) and Thus,

since

previous

H 0(A' ) = H 0(A) (Y)

DEFINITION.

~ radAQ,

g(A,Q)

as in [i:

by means

g(A,[0]).

The

(38.5.1) domain A' =

obtained

is infinite,

we are reduced

by

For any h o m o g e n e o u s (12.13)3,

of the H i l b e r t

following

three

g(A',QA')

=

by taking

(H0(A) (Y))[A3

generated

to the

and

we d e f i n e

polynomial; facts

g(A,Q)

where

= the

Q

in

the p o s i t i v e

A

A'

with

integer

we also d e f i n e

are e a s i l y

an i n d e t e r m i n a t e Hn(A')

ideal

@(A) =

seen:

is the h o m o g e n e o u s Y

over

A

and p u t t i n g

(H0(A) (Y))-submodule

of

A'

Hn(A).

(38.5.2) if

.

case.

(38.5) HI(A)

Deg[A',PA' 3 = Deg[A, P3

If

Dim A = 0

Q ~ D0(A),

then

(38.5.3)

If

g(A,Q)

then

= Deg[A,Q~.

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

g(A) = Deg A. E m d i m A = Dim A

and

Q ~ H

(A), then

(A, Q)

DegAQ • In [i:

(12.3.4) (1)I one

(38.5.4) with

PROJECTION

Q ~ N, then g(A,Q)

(38.5.5) notation

~i).

FORMULA°

QN c ~i (AN)

= [R([A,Q3)

In v i e w of

finds:

(38.5.1) nl(A)

If

Q ~ ~i(A)

and

N ¢ ~ 0 (A)

and

: R ( [ A N , Q N ~ ) ~ g ( A N , Q N)

and

(38.5.4)

= [Q ¢ ~(A):

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

we e a s i l y

g(A,Q) = i];

get (this e x p l a i n s

g(A) ---- 1 ~ E m d i m A = Dim A.

273

our

=

5.18

In v i e w

of

(38.5.6) . and

(38.5.2)

If

It is e a s i l y

(38.5.7) QA'

and

get

then

g(A,P,N)

=

for any

P c ~r_2(A)

g(A,P).

that:

Deg[A'

is as

we

we have

Q e ~I(A)

and

Q'

(38.5.4)

A = D i m A = r > 2,

P ~ N

seen

If

e ~I(A')

= 1

Emdim

and

N e ~ 0 (A)

and

in

and

,QA'~

A'

is as

in

= Deg~A,Q~.

(38.5.1),

then

(38.5.1),

then

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

D i m A'

= 1

and

if

D e g A'

Dim A =

D e g A. Finally, Projection

in v i e w

Formulas

(38.5.8) if

If

D i m A = i, (38.6)

SPACE.

of

(23.15)

and

Q ¢ ~I(A),

then

g(A)

BEZOUT'S

Assume

(38.5.1),

that

(38.5.6)

(38.5.4)

then

and

we

g(A,Q)

(38.5.7),

by

the

get: = DegEA,Q~.

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

= D e g A.

THEOREM Emdim

F O R TWO H Y P E R S U R F A C E S

A = D i m A = r a 2.

IN A P R O J E C T I V E

Let

~ ¢ H

(A)

and

w

¢ H

(A)

be

such

= ~ r - 2 (A,~)

that

~r_l(A,~)

n ~ r - 2 (A,~) .

n nr_l(A, ~)

= ~.

Let

Then

~ (L A , ~ , ~ , P )

0(A,P)

=

(DegAS) (DegAS) .

P¢~ PROOF.

If

H0(A)

is i n f i n i t e ,

then

our

assertion

follows

from

w

(38.5.4) If

and

H0(A)

the p r o o f

(38.3)

by taking

is finite, of

turn w a s

proof

by w h a t sents

Chains

of T h e o r e m

satisfied

an e u c l i d e a n

in v i e w

N ~ ~(A,~)

(38.5.1),

domain

when

of e u c l i d e a n

(36.7)

the

((36.6).)

m a y be g e o m e t r i c a l l y

either

of

with

we

U ~(A,~).

can p r o c e e d

as in

(38.4).

§39. The

then,

N c ~0(A)

true

was

curves.

based

adjoint

In a c t u a l described

a line or a n o n s i n g u l a r

274

on c o n d i t i o n ~

was

such

application

as

conic

finding in the

an

(*), w h i c h

that we ~

affine

B/~B

in was

arrange

this

which

repre-

plane.

(6

5.19

was

actually

proof

of Theorem

represented the

constructed

curve

(36.7),

which

we

also had

by

y. here

DEFINITION.

Let

the p r o j e c t i o n

Let

onto

~i:

k i " Ei

and

i,

the

in

Ei+l

~i: k i " E i + l

then

the r i n g

with

the c o n n e c t i n g D

is s a i d

" k.1

in

~i:

D

D

relation

Let ideal of

in

~I(A).

euclidean chain)

A

be

A,

such We

if

an affine that,

over

of euclidean

Note domains.

that,

if

Further,

domain

I

say

curves over k

~i-i

denote

are

fields

ki ,

that:

' and

(A,{~i},{~i},{ri},{~i})

D c E

is d e f i n e d

to

by

= ~i o ~ i + l ( d ) at t h e

a field

k

if

i

th

component

if

over

that

I

i_ff A / I

a field

is a c h a i n is i s o m o r p h i c

defined

is empty,

in t h a t case,

over

then D

275

k•

= k

for a l l

l

A = {i,2 ..... n-l}.

is an i n t e r s e c t i o n

domains A

E - Ei

i e A, s u c h

is c o n n e c t e d

to b e c o n n e c t e d

shall

of

(~i,~i) .

to b e d e f i n e d

is s a i d

of Theorem

and monomorphisms

with

: ~ i o ~i(D) that

there

- -

i e A.

outside

ki

together

domains,

D = {d ~ E say

point

a sequence

let

that

Ker ~i ~ Ker

D

of euclidean

i e AJ w e

common

be

for e v e r y

~i o Ti = ~i o oi = i d e n t i t y

I_~f

the

the c a s e w h e n

a generalization

and

and a s s u m e

(2)

a chain

of

i th c o m p o n e n t .

if

be

their

version

(39.3).

(i)

We define

needed

with

EI,E2,...,E n

E 1• ~ k.l , ~i:

i-i e A~

to w o r k

E = EI~...~E n

A c {1,2 ..... n-l]

epimorphisms

In a n o l d e r

lines w i t h This

present

domains.

Let

we

(24.21)).

a p a i r of d i s t i n c t

(39.1)

~i:

(36.9)

represented

euclidean

in

k, of

and

I c A

finitely

(connected

be

many members

chain)

to a c h a i n

an

o__ff

(connected

k.

D = E

is a c h a i n

is c o n n e c t e d

of e u c l i d e a n

if a n d o n l y

if

n = i.

5.20

EXAMPLES. nominal are we

linear

see

and

and

over

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

a field

expressions

easily

A/fiB k

ring

(i)

that

A

I

B/glB

+ g2 B % k.

the u n i q u e

restricted

B - k to

Let

k

B/glg2B

Let

LEMMA. such be

and

the

dean

domains.

9,Y

c B • such

is

(~,y)N

=

there

~B

(8'

PROOF. {i,2 ..... n} either

of the

,y')

restricted

component

the

We u s e

same

{ s , s + l ..... s+t}

{s}

with

curves

over

in

ideal

X,Y, in

and

such

B

with

to be

~i: b / g 2 B

" k

be

to

Upon k,

T1 = hl

it is e a s y

chain

by

taking

to see

of e u c l i d e a n

(~i,~i) . ring

of e u c l i d e a n

in

X,Y

curves

over

canonical

homomorphism

where

is a c h a i n

notation

D as in

(39.1)

over k.

of e u c l i -

assume

that

~i o n i ( u ( y ) )

~

(0,0).

2-transformation

N

i__nn B

with

c &B.

union

with

E. = i

i 6 A , we have

the n o t a t i o n

as a d i s j o i n t form

- D

an e l e m e n t a r y y'

where

B ~ B/g2B

a polynomial

of the

B/~B

for e v e r y

where

" k

is a c h a i n

isomorphism

exists

h2:

gl B + g2 B.

B = k[X,Y],

exactly

expressions

is a c o n n e c t e d

first

i ~ j,

~i o h I = 91 o h 2 = the c a n o n i c a l

(~0i o ~ i ( u ( 8 ) ) , Then

B/glB

that

B/g2B

that

that,

linear

~i:

whenever

B\k

n = i.

and

let

fig

of e u c l i d e a n

B - B/glB

the c o m p o s i t i o n

With

if

a poly-

where

domains

is a m a x i m a l

kernel

Let

is a c h a i n

be

~i = h2

at the

B - D

B - B/~B

whose

(flf2...fn)B

gl B + g2 B

such

c B/glB ~

~ c B

u:

and

and

connected

(39.2) k.

hl:

epimorphisms

epimorphism

domains

Take

B = k[X,Y],

fi B + fjB = B

only

gl,g 2

taking

of e u c l i d e a n

I

if and

Then

epimorphisms

with

Thus

Let

gl B + g2 B ~ B.

I =

is a c h a i n

is empty.

that

and

X,Y

B/I

(2)

that

in

is c o n n e c t e d

canonical

k

upon

in

(39.1)

~iO...U~m

s,s-i

{s ..... s+t-l}

276

for

D.

such

First write that

~ A , or of the c A , s+t ~

A.

each

form

~i

is

5.21

Let

euclidean

E (i) =

~ Ei icQ i

and

domain

contained

A (i) =

in

E (i)

A N ~i"

with

It is c l e a r t h a t e a c h (~ J" ~J) j e A (i) " m euclidean domains and D = ~ D (i) i=l In v i e w mentary

of

where

i

assume,

~

in

the p r o j e c t i o n

replace

D

by

connecting

D (i)

such

for

of

D

the

relations

that

there

chain

exist

of

ele-

that

some

8 i ~ D (i)

onto

D (i)

by

~i o u

D (i) , u

be

is a c o n n e c t e d

to p r o v e

D (i)

(8i,0)

D (i)

and h e n c e

that,

D

is a c o n n e c t e d

We are Let

then

for any

chain

reduced

f = u(8)

(~0i(ni(f)),

~

i c A. T h e n w e N

of e u c l i d e a n

to p r o v i n g

g = u(y).

~i(~i(g)))

2-transformation

the

and

such

f,g

¢ D

(~i(~i+l(f)),

w a n t to p r o v e D

following:

T h e n we h a v e

(0,0)

in

domains.

the

that

that

~i(~i+l(g)))

existance

(f,g)N =

such

~

(0,0)

of an e l e m e n t a r y

(h,0)

for

some

¢ D.

known

We will

prove

If

l, t h e n

n =

the

lenuna b y D

(or as e x p l a i n e d Now assume

D

Ni

(u(y))N i =

denotes

Thus we may

h

it is e n o u g h

2-transformations

(~i(u(8)),

(i)

(36.6.4)

Let

is an e u c l i d e a n in

n > 1.

= [ (dI ..... dn_ I)

e E

induction

n.

domain

and the p r o o f

is w e l l

(36.6.5)). Let

:

on

D

there

c

E

-- E l ( 9 . . . ~ E n _

exists

1

(d I ..... d n)

be

defined

c D

for

by

some

.

d n ¢ En].

Clearly

connecting

relations

D

is a c o n n e c t e d (~J'~J)l~j
chain Let

of e u c l i d e a n ~ = D ~ D

be

domains

with

the

ep i m o r p h i s m . ~ ( ( d I ..... d n)) Let

~ (f) =

f*

and

= (d ! ..... dn_ I)-

D (g) = g

277

.

By

induction

hypothesis,

there

5.22

exists

an e l e m e n t a r y

2-transformation w

*

(f ,g

By

(36.6.4)

applied

trans formation

where

(2)

~ (g')

M

in

(f',g')

may

thus

assume

g =

(gl ..... g n ), w e let

euclidean

D

w

in

D

such

that

w

)M

=

(d ,0).

~: D -- D

such

that

(f,g)M

=

, there

exists

an e l e m e n t a r y

2-

(f',g')

= 0.

Clearly

NOW

to

M

satisfy

the

loss,

that,

without have

domain

En

conditions

upon

such

be

8{ = ~E

(f,g)

f =

and w e

(fl' .... fn )

and

= 0.

an elementary

that

as

writing

gl = g 2 = ' ' ' = g n - i

= 8 { @ ~ . . °8's

N

same

2-transformation

(p(i),q(i)

; t'(i)).

in t h e Define

n

8 i = ~m(P(i),q(i)

;t(i)),

where

rj o 9j o T j + I o nj(t(i))

t(i)

e m

is d e f i n e d

~j+l . . . . . Tn_ 1 o 9 n _ l ( t ' ( i ) )

= t'(i)

Let (f,g)N

=

Then

N = 8182...8n. (f",g"),

(3)

~j(f")

1 < j < n,

and

by

it is e a s y

,if

1 ~ j ~ n-i

,if

j = n

to c h e c k

that

,

.

if

then

= ~jfj

, ~j(g")

(~n(f"),

~n(g"))

= ~.f. 3 3

for

some

~j,mj

e ~j(k)

, if

w

(4) In p a r t i c u l a r

which

certainly

if w e

exists

as

take

=

N

such

explained

(4) ,

(5)

(fn, g n ) N

~ n ( g '') = 0.

278

in

that

(36.6.5),

(fn,gn)N

we h a v e ,

=

(d,0),

in

view

of

5.23

(6)

N o w assume,

Clearly, assume

that

(7)

~. ~ 3

if p o s s i b l e ,

if

~

= 0

3 for some

0 ; and

T h e n w e have,

g~' ~ 0.

for

j = 1,2 ..... n-l.

1 ~ j ~ n-i

we have

j = n-I

v. = 0 i

for

,

if

j < n-i

if

j = n-l°

since

or

Hence

i = j+l ..... n-l.

= ~n_l(~n(g") ) = 0

Thus w e h a v e

~j (vj)~j (fj) = 0.

~j o Tj = i d e n t i t y

(8)

g" = 0.

g" c D,

~j ( v j + i f j + I) = 0 ~0j (vjfj)

Then

on

[Qj(fj) = 0

But clearly

Since

0 ~ vj c Tj (k)

and

k, w e m u s t h a v e

and h e n c e

(or b y

~j(~j(f"))

(36.6.2))

= ~j(~j(g"))

= 0.

we have

[~j (fl,~j (gl]Ej = {~j (f"l,~j (g")}Ej and h e n c e

(9)

~j (~j(f))

Thus

= ~j (~j (g)) = 0.

(9) is a c o n t r a d i c t i o n

contrary

to the a s s u m p t i o n

(f,g)N =

(f",0)

(39.3)

euclidean

curves

described

in

B

in

assuming and

(39.1).

Le t

~

D = B/~B Let

s u c h t h a t for e a c h

K e r ~i =

(6) w e m u s t h a v e

on

f,g

and h e n c e

g" = 0, and h e n c e

be as in the s t a t e m e n t

the p r o p e r t y

(*).

M.

Let

~

there exists

(~i(u(Mj))

the i m a g e of o__rr K e r

Mj

some

i__n D,

i c A

then e i t h e r

~i = ~ i + l ( U ( M j ) ) "

279

domains

be the set of m a x i m a l

3

denotes

of T h e o r e m

be a c h a i n of

be a c h a i n of e u c l i d e a n

[M I, .... Mn}

--

i___f u(Mj)

to the h y p o t h e s i s

as r e q u i r e d .

COROLLARY.

(36.7), w i t h o u t

in

(8) i m p l i e s

as

ideals

s u c h that:

5.24

I_~f (8,y)B ~ Mj perty of

for all

j = 1,2 ..... r ; t h e n

~

has the pro-

(*)

REMARK.

In particular,

represents

(39.1), ~ = glg 2

a pair of parallel or i n t e r s e c t i n g lines)

gl B = g2 B = B

or

c o m m o n point, the property

if, as in Examples

y ~ gl B + g2 B

(i.e.

y

and if either

does not pass through the

if any, of the lines r e p r e s e n t e d by

~), then

~

has

(*). §40.

Other treatments of d i f f e r e n t i a l s .

The study of d i f f e r e n t i a l s

in §28

was devoted to the p r o b l e m of

d e f i n i n g the d i v i s o r of a d i f f e r e n t i a l of a s e p a r a b l y g e n e r a t e d tion field ordv(~dx)

(i.e.

~/k

of t r a n s c e n d e n c e degree

for every

V c I(K,k)

one;

namely,

func-

to d e f i n i n g

and every d i f f e r e n t i a l

~dx of

K/k.

The d e f i n i t i o n should of course be such that ordV~dX = or~

+ ordvdx

,

~dx = ~dy ~ o r d v ~ d X = ordv~dY In case

V

is r e s i d u a l l y separable over

.

k, the p r o b l e m is easy

to solve and the answer is erdV~dX = o r ~ V/k.

dx , w h e r e

t

is any u n i f o r m i z i n g coordinate of

(28.10). In particular, w h e n the ground field

k

is perfect,

the p r o b l e m

is c o m p l e t e l y solved. In the general case one uses d i f f e r e n t

"tricks" to get at the

"correct" answer. The c o r r e c t answer is of course the one given in (28.13) ordVdX = o r d ~ ( k ( x )

N V,K)

and in general if

, if

x e V ,

x ~ V

ordvdX = ordVX2 + ordvd(i/x). C h e v a l l e y starts w i t h an ad hoc d e f i n i t i o n of d i f f e r e n t i a l s c e r t a i n linear maps on repartitions,

280

-

and then finally shows that they

5.25

behave

like

Other

the

point

coincides relatively

formal

that

genus

algebraically genus

cally

closed

in

K.

some

V ~ I(K,k)

Another

formula formula

sections section

by u s i n g we

shall

for B e r g e r ' s are

(*)

the n o t a t i o n

is a d i s c r e e t

K

is a finite

T

is an i n t e g r a l

R

=

separable ring

use

k

genus(K,k) (when

k

2 p.106

[9].)

is r e l a t i v e

algebrai-

of genus,

is g i v e n

is

when

in §42.

and d i f f e r e n t .

c a n be g e n e r a l i z e d Berger

and

repeat

to c o m p l e t e

Roquette.

is r e p e a t e d

In this

is o b t a i n e d .

any p r o o f s

Except

or definitions

because

inter-

which

it is s h o r t

and

the

following

algebraic

extension of

ideal

~(S)

in

= L.

extension

of

T

notation.

S

in

of

with

L.

~(T)

= K.

K.

T.

T M-

B = TM = {a/b For

any

: a ~ T , b ~ T\M} T D A ~ S

we

= the

integral

closure

define

w

A

but make

Dedekind's

ring with

closure

nonzero

in C h e v a l l e y

(See T h e o r e m

rational,

proof

[13],

clearification.

valuation

integral

is a p r i m e

shall

as

the g e n e r a l i z a t i o n

do n o t

Berger's

We

S

M

we

how

Nastold

and of

of n o n n e g a t i v i t y

of Kunz,

indicate

in it n e e d s

NOTATION.

is the

(28.1)

[i0],

ordvdX

least w h e n

for c o n d u c t o r

results

not e s s e n t i a l .

at

Generalized

in

result,

of

K).

proof

is r e s i d u a l l y

§41.

The

~ 0,

Kunz

[9]).

k.

K/k

in

and VI,

of d i f f e r e n t i a l s ,

field

of

closed

(K,k)

III

[17],

our definition

the u s u a l

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

II,

definition

of the g r o u n d

out

with

(Chapter

by Zarisk-Falb

"usual"

of d e r i v a t i o n s We

ones

treatments

start with use

the u s u a l

= {x e K

O(A/S)

: TraceK/LXY

= OD(A/S)

= {x ~ K

e S : xy

for all e A

281

y e A},

for all

and

y e A*}.

of

R

in

K.

5.26

Further w

R

we

define,

w

=

R-T

,

~3(R/S) = ~3D ( R / S ) w _w B = B-T , and (B/S)

=

-- ~3D (B/S)

R-~3 D ( T / S )

,

--- B'~3 D (T/S) w

REMARK. A =

R

or

Note

B)

A =

A

~ A

as

is

a

finite

D

stands

opposed

to t h e K ~ h l e r d i f f e r e n t , _e , T = E (S,K) and O(T/S)

(R/S)

is =

easy

{x

c K

suffix

A

in

It

The

for

ideal

that

A.

that

to

check

: xy

¢ R

defined

above

A-module for

which

the we

and

may

later.

old

have

is

different

use

in o u r

we

~D(A/S)

Dedekind

shall

= ~(S,K)

(where

an as

Also

notation

note

in § 2 7 .

that, for

all

y

~ R*],

and

w

~(B/S)

=

{x

e K

(41.1)

: xy

LEMMA.

¢ B

for

all

(Berger).

y

With

B-D(R/S)

c

c B

the

].

notation

{(R)f}(B/S)

as

above

we

have

then

we

;

w

and

further,

equality

if

in t h e

R

is

above

invertible,

i.e.

(i)

We

B

B Now

that (3)

-~(R/S)

clearly

have

have

w

B

c

R

and

hence

c R. is a B - m o d u l e ,

"fD(R/S) clearly

every B

B

is

B'~D(R/S)

it

follows

from

(1)

that

then

it

c ~(R). B

is

fractionary

both

a Dedekind ideal

invertible,

Multiplying (4)

R,

w

Since (2)

=

relation. .

PROOF.

f3(R/S)-R

of

i.e.,

sides

of

domain, B

B

and

c [(R)'~3(B/S).

282

by

is w e l l

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

-~(B/S)

(2)

and

= Bo

O(B/S),

we

have

by

(3),

known

5.27

Now we claim 9:

(5)

that

_9:

~(T)T

c T 9:

F o r proof,

we

simply

xz

¢ T

z ~ ~, w e h a v e

have

to n o t e

and hence

that

if

x c ~(T),

TraceK/L(xY)z

y ~ T

and

= TraceK/L(xz)y

¢ S

9:

by definition From (6)

of

(5),

~(R)R

T

.

it f o l l o w s

c B

that

;

since

it is e a s y

to c h e c k

that

~(R)

= R-~(T).

w

Now

if w e

both

sides

(7)

~(R)

of

further (6) b y

assume O(R/S)

9: c B D(R/S)

~(R)~(B/S)

§2,

Lemma

The

9.5,

ambiguous.

minates - T

Then

T

that w R

Ker

See

(41.3)

LEMMA

is a c o m p l e t e

well

jacobian

known

that

in his

(Roquette).

over ~

(4) and

(8).

r e p e a t e d from B e r g e r 9: , of R and B are

definitions

chosen

as d e f i n i t i o n s , Lemma

Assume

over

S,

S, w e h a v e

is generated

is i n v e r t i b l e ,

PROOF.

o f the

simply

by

is e s s e n t i a l l y

that h i s

intersection

X 1 ..... X n

such

proof

earlier

LEMMA

is a c o m p l e t e

we get

is c o m p l e t e

except

proved

(41.2)

then by multiplying

(3) w e g e t

above

We h a v e

as he h a s

= R,

c ~(R/S).

The p r o o f

REMARK.

R -~3(R/S)

,

and a g a i n b y

(s)

that

their

in t h e a b o v e for

some

notation, n ~ 0

an S - e D i m o r D h i s m elements 9: ~3(R/S)-R = R.

i.e.

descriptions,

9.

i.e.,

by

[7],

n

that

and

T

indeter-

% : S [ X 1 ..... Xn~ gl'''''gn"

[ 16~. (Kunz).

Usinq

intersection

the a b o v e

over

S,

let

J

of

gl,...,g n

with

the

ideal

is i n d e p e n d e n t

JT

283

respect

notation

to

denote

and

assuminq

the

that

imaqe b y

X 1 .... ,X n. o f the , c h o i c e

Then of

it is %

an__d_d

5.28

gl .... 'gn"

Then we have

JB = B'~(R/S)

PROOF. showing

= the K ~ h l e r

"reduction =

of the

DIFFERENT. x e K

In

proof"

K

be

(41.2)

the p r o o f

the d i s c u s s i o n

of

R

over

establishes

that

be

Assume by

S;

is r e d u c e d

after

Satz

and the

the K ~ h l e r

be

that

of the

~

to

i, proves

following different

n

jacobian

of

k [ x , y I ..... yn]

with

is the

Q = k [ x , Y 1 ..... Y n ] N M(V).

(*)

of

Let

k [ x , Y 1 ..... y n ] Q above

S = k(x) = TTQM(V)

notation

BBQM(V).

applies

It follows

or~(~(k(x)

k[x].

over

k[x]

k.

9(x)

= x

gl,...,g n with c V. and

Let

and

~: k [ x ~ [ Y l ..... Yn

with

gl ..... gn

over

yl,...,y n e K

Let

generators

AND

and and

~ ~

9(Yi)

=

J = the

respect

to Yi .... 'Yn"

Further

assume

that

let

Then we have

o r d v ~ ( k [ x , Y 1 ..... Yn]Q)

PROOF.

Let

k [ x , y I ..... yn]

integral

FOR CONDUCTOR

of one v a r i a b l e

epimorPhism

has

V e I(K,k)

field

over

the u n i q u e Ker

FORMULA

transcendental.

indeterminates

9

Let

DEDEKIND'S a function

be a s e p a r a t i n g

k [ x , y I ..... y n ]

the

and

[iI],

different

COROLLARY. Let

Y I ..... Yn

image

(41.1)

JR.

(41.4)

Yi"

of

JR = ~(R/S).

(R/S)

R/s

In v i e w

= ~(R)D(B/S) .

+ ordv~(k(x)

Q V

and

= R, say. and

N V,K)

= ordvJ

T = S l y I ..... y n ] . Not

it is e a s y

f u r t h e r w e have,

-

Then

to see

B c V

and

clearly

that

all

V =

that

n V),K)

= ordvD(T/S ) . = ordvB

e(T/S)

= ordv~(B/S ) . The

proof

now

follows

from

(41.3),

284

by

taking

ord V

of b o t h

sides.

5.29

§42. The

assertion

is v a l i d w i t h o u t Theorem,

which

stringent

list

and survey, see

(42.1) ring

such

divisor, finite

without

see

THEOREM

the

For

integral

assertion

we have

the

(i0.i.i0)

following

of M a x N o e t h e r ' s

and w h o s e

abound

proofs

in the

Famous

(under m o r e

literature.

in the

CONDITIONS.

Then of

ER*/~(R) (42.1)

REMARK.

In

is the e l e m e n t a r y

a conic,

and

in

parabola

and

a hyperbola.

(29.4)

illustrated

R

spirit

we

o_~f

R

R

i__n ~(R)

of N o e t h e r ' s

local

a nonzerois a

: R~.

w e have:

(29.2)

and

(30.2)

remain

true

(**).

(29.3) and gave

we

remarked

ancient

idea

a version

(of radius

Y

with

another

x

=

2

+ y2 = I.

cos

y = sin

@

of

by a

familiar

i) :

/ x

285

of

of p a r a m e t r i z a t i o n

circle:

x

the g e n e s i s

of it i l l u s t r a t e d

L e t us c o n c l u d e

by a c i r c l e

that

or

For a

be any

contains

: R~ = 2ER/~(R)

Theorems

condition

Let

R ~ 2, M(R)

closure

(29.2.2)

version

the

Indeed

a new proof

ON ADJOINT

COROLLARY.

assuming

(42.3)

[15~),

really

[83.

also

as a v e r s i o n

d i m R = !, e m d i m

a corollary

(42.2)

and

= 2.

condition.

[5~.

R-module.

As

regarded [14~

adjoint

(and h e n c e

k(R)

hypothesis)

that: and

assuming

(see

less

general

(i0.i. Ii)

m a y be

Af + B~ Theorem

Theorem,

The

5.30

To get rid of trigometric functions, make the substitution which enables us to compute all trigometric integral by converting them into integrals of rational functions. putting

t = tan@/2 I

x = (l-t2)/(l+t 2)

we get y = 2t/(l+t 2).

286

To wit:

5.31

§43.

GEOMETRIC

LANGUAGE.

Geometrically in an a m b i e n t varieties.

speaking,

space,

varieties

and s u b v a r i e t i e s

For a l g e b r a i c

varieties,

c a l l y defined".

If one starts

defined"

means,

exactly

ties n a t u r a l l y

present

they can of course times

in a rather

ters.

Here w e

relavent is,

intate

rings,

variety,

(43.1)

dimensional space

if

by an a b s t r a c t A

abstract

plane

Also

A

the algebraic

of various principal

[embedding

varieties

as

or h y p e r s u r f a c e s .

A

A

projective

variety

is said to be r-

is said to be a p r o j e c t i v e

curve,

are r e s p e c t i v e l y

be an a b s t r a c t

we m e a n

In particular, a homogeneous

projective

a projective

sense,

irreducible

any h o m o g e n e o u s

a subvariety

ideals w h i c h

are their own radicals.

however,

two ideas a g r e e

the

an i r r e d u c i b l e A

reducible

an a b s t r a c t

representing

of

chap-

line and an 1-space

and a

2-space.

N o w let

C

some-

of as t h e i r coord-

if, D i m A = E m d i m A.

projective

D i m A = !.

definitions

By an a b s t r a c t

The v a r i e t y

and

in the previous

of as pairs

treating

of varie-

(although

The g e n e r a l

The variety

(Dim A)-space)

projective

projective

A.

"algebraically rings"

are t h o u g h t

of points

VARIETIES.

irreducible

with

algebraic

are t h o u g h t

try to avoid

D i m A = r.

(projective

domain

varieties

domain

what

"algebrai-

to the problem;

as i l l u s t r a t e d

varieties

of such

should be

"coordinate

that we h a v e used.

abstract

PROJECTIVE

subsets

of all the t e r m i n o l o g y

they are a b u n c h

a homogeneous

and

as a s o l u t i o n

manner)

concepts

and we

unless

rigorous,

the r e a d e r w i t h

embedded

ideal~

varieties

we mean

present

irreducible

to make

themselves

awkward

are c e r t a i n

of as sets of points

all such sets

"ideals"

take care

geometric

that;

the

are t h o u g h t

subvariety

m a y be d e n o t e d

of

of by

A.

in

Geometrically

and w e A.

ideal

projective

For

287

In

may be t h o u g h t one should

irreducible,

say that any m e m b e r

Formally

[A,C~,

A

variety.

restrict

subvarieties, of

an i r r e d u o i b l e

and this is h o w

to be

~(A)

is

subvariety

it appears

in

to

5.32

various

formulas.

homogeneous variety

Since

ideal

of

A,

C e ~(A)

A/C

is an a b s t r a c t

and w e may say that

the a b s t r a c t

ing d i m e n s i o n embedded

A/C

variety

In the above

sense,

of

in

~

is a p r o j e c t i v e

3-space

variety.

~

ring of

whose

denominators

although

C

o__nn A,

i.e.,

notations

are also e x t e n d e d

projective function c

=

field

R.

in

(or

of

H

(A)\{A}.

In the special plane,

defines

can be

The deqree cases w h e n

a hypersurface

or a p r o j e c t i v e

curve. sense,

of

if,

A, by

C.

I c A,

to c o v e r p

~(A,C)

Note and we

it a subA

~(A,I,C)

w e denote

functions denotes

in the ring the case w h e n

A

on

is

subvariety

as the case may be,)

A

The above

itself

the above n o t a t i o n

the

the ideal

~(A,C).

is an i r r e d u c i b l e

~([R,P~),

A

~ c ~(A).

the ring of r a t i o n a l to

of

is res-

to be h y p e r p l a n e s .

and only

In particular,

R(A)

: A

formally w e do not call

C

ER, P], w h e r e

variety

C

meaning,"embedd-

in the a l g e b r a i c

if,

do not b e l o n g

by h y p e r s u r f a c e s

is

or that

Emdim A = min[r

1 are said

variety

generated

i.e.

surface

of degree

For an i r r e d u c i b l e

embedded

DegAS.

is i r r e d u c i b l e

local

namely,

is a s u b v a r i e t y

Hypersurfaces

A

an a p p r o p r i a t e

or a p r o j e c t i v e

it as such,

hypersurface

gets

A",

to

a projective

that a h y p e r s u r f a c e

in

projective

A.

is any m e m b e r

is equal

called

irreducible

maximal

r-space.]. A

a hypersurface

to treat

in

is not the u n i q u e

is e m b e d d e d

Emdim A

in a p r o j e c t i v e

pectively

A/C

(the variety)

A hypersurface

have

any

of a

gives

the

by taking

{0}.

Ideals

generated

by a set of h y p e r p l a n e s

algebraic

sense)

linear

varieties

of

in case

A,

subvarieties A

of

A

in

A

which

is a p r o j e c t i v e

define

(in the

are i r r e d u c i b l e

space.

sub-

In the general

w

case w e

call such

vector-space (especially by

~(A).

ideals

"flats"

or members

of all the h y p e r p l a n e s for s t u d y i n g

Members

of

affine

~(A)

~

(A).

in a flat is also

pieces)

The

H0(A)-

a useful

and their c o l l e c t i o n

may be called 288

of

linearities

in

A.

concept

is d e n o t e d If

A

5.33

W

is a p r o j e c t i v e Dim(A/N).

space,

Hence

then

for all

Emdim[A,N~

N c ~

is t h o u g h t

(A), w e have,

of as the

Emdim[A,N~

formal

=

dimension

w

of

N

in the g e n e r a l

flats

in

A.

If

~i(A)

TO e x p r e s s joining

w e use

or

Now concepts

~*(A)

which the

"the

plane

of

let us p o i n t

out

linear

irreducible

,

linear

= member

of

~I(A)

,

= member

as p a r t i c u l a r

cases,

of

= member

of

D0(A)

,

curve

= member

of

~I(A)

,

line

= member

of

~(A)

,

plane

= member

of

~(A)

, etc.

let

C

c a n be e a s i l y irreducible

be

an i r r e d u c i b l e

generalized

subvarieties

to the of

n([A,C]) A whole

array

ies of s e v e r a l o f some

a projective

space

c a n be

Thus A.

let

spanned

b//" (like

intersecting

spanned

i

of

~(A)

lines

the etc.)

ideals

by

counterparts

Q1 ...... Qs

of some

more

A:

;

subvariety

embedded

are m e m b e r s

in §15.

Q1 ..... Qs

Then we have

289

be the

A.

Above

[A,C3.

concepts

F o r example,

of

P]

to d e s c r i b e

located

of

variety

= [P e ~(A) : C c

of n o t a t i o n s

types

o f them.

C

and of d e g r e e

we have

point

Now

two

flat

subvarieties ~i(A)

N

for h o m o g e n e o u s

algebraic

of

and hence,

variety

the

= member

i-dimensional

vertex

containing

denotes the

i-dimensional

Hi(A,N).

A ( A , - , - .... ); thus

" ~ ( A ' Q I ..... Qs )

set of all

similarly.

i-dimensional

linear

= the

a cone with

of

two p o i n t s ,

about

~i(A)

is a m e m b e r

idea

the n o t a t i o n

Q1 ..... Qs

then

is d e f i n e d

N e ~(A)

is a h y p e r s u r f a c e

line

case;

(algebraically) We

illustrate

irreducible following:

subvarietthe use

subvarieties

of

5.34

Q1

lies on

Q2 ~ Q2 ~Q2

passes

through

CQl

Q1 e ~(A'Q2) Q1

lies o u t s i d e

Q1

'

Q2 ~ Q2 ~ Q1 Q1 c n ( A , \ Q 2)

QI,Q2

are s k e w ~ QI, Q2

,

do n o t m e e t

~ ( A , Q I) N ~ ( A , Q 2) = # , and in case QI' .... Qs

Dim A ~ 1 , are c o l i i n e a r

~ some

line

Q

passes

through

Q1 ..... Qs Q c O 1 n ~2 N . . . N Q s , for 1 some Q e ~I(A) , E m d i m [ A , Q 1 ..... Qs] ~ 1 , etc. Also, w h e n lating N

A

is a p r o j e c t i v e

linear varieties

where

~

~,~ at of

T([A,~_],N)

is a h y p e r s u r f a c e

Finally we come There

and

space,

passing

to the c o n c e p t

are two types of s i t u a t i o n s

osculating

= the t a n g e n t c o n e of through

which we

have

the ! n t e r e s e c t i o n

(along)

A

a

is d e f i n e d If

variety

C

to be

A, t h e n the f u n c t i o n

the set of p o i n t s

of

branches

at v a r i o u s

(places)

irreducible

C

curve

C

m a y be

at

N.

multiplicity.

space

A

multiplicity

(irreducible)

such that of

~

subvariety

and P

u([A,~,~3,P).

is an i r r e d u c i b l e

t h a t in this,

in a p r o j e c t i v e

(Dim A - 2 ) - d i m e n s i o n a l

~

study.

~,~

factor,

a point

of i n t e r s e c t i o n

F o r two h y p e r s u r f a c e s no c o m m o n

flats are o s c u -

curve

field of

(in A)

{0}

A, w h i c h

in an i r r e d u c i b l e

is

points

C

is

D0([A,C~) of

C

is

projective

R([A,C~)

= R(A/C)

and the set of all ~([A,C~)

and w e m a y be t a l k i n g

= ~(A/C).

Note

a b o u t the a b s t r a c t

m a y a l s o be t h o u g h t of as the c u r v e

290

,

5.35

[A, {03 ]

e m b e d d e d in

A.

For each b r a n c h in

A

at w h i c h

A n y set

I

there is a unique point

is c e n t e r e d and we denote

of h o m o g e n e o u s

a subvariety IA.

V

V e ~([A,C])

(divisor)

P

elements represents,

of

C

in

A

by

V

of

C

to be

u([A,C],I,V).

than

H0(A)

Points of

C

theorem; city

~

ity of

C

IA

i__nn A

To get the i n t e r s e c t i o n C

w e extend the defi-

w h i c h have a b i g g e r residue field (each p o i n t as

in the "proper" c o u n t i n g for Bezout's

and h e n c e we introduce the "corrected" ([A,C],I,-).

([A,C],V)

and

should be t h o u g h t of as several points

m a n y times as its residue degree)

C

w h i c h depends only on the ideal

m u l t i p l i c i t y at points or various pointsets of nition by linearity.

of

in the algebraic sense,

We define the i n t e r s e c t i o n m u l t i p l i c i t y of

alonq a branch

~

P

intersection multipli-

Of special i n t e r e s t is the measure of singular-

C

(or the i n t e r s e c t i o n m u l t i p l i c i t y of local c o n d u c t o r ideals) , w h i c h w e denote by the symbol ~ (uncorrected) or by ~ (corrected). Hypersurfaces, C

w h o s e i n t e r s e c t i o n m u l t i p l i c i t y a l o n g each b r a n c h of

is b i g g e r than or equal to the measure of s i n g u l a r i t y are called

adjoints, w h i l e those for w h i c h e q u a l i t y holds are true adjoints.

The

concepts of adjoints or true adjoints are d e f i n e d for individual points, Dointsets or all points of

C.

The genus of the curve function field

R([A,C])

studied in C h a p t e r III included a proof,

some and

t).

(43.2) ground

(§30 and for

field

as a b i r a t i o n a l i n v a r i a n t of the

~

(R([A,C]) (29.2.2)

genus(R([A,C]))

C = [0]

(genus(R([A,C])

(See (28.12) and

(30.2.2)

~A,C]

is d e f i n e d as

that

d u a l l y r a t i o n a l place)

(See §23,24 for details,)

case,

and is

in §29).

We have

is zero and there is a resi-

is of the form for the a b s t r a c t

H0(A/C) (t)

for

(C = 0) case,

for the e m b e d d e d case.) AFFINE V A R I E T I E S . k

An affine i r r e d u c i b l e v a r i e t y o v e r a

is b a s i c a l l y any v a r i e t y w h o s e c o o r d i n a t e ring is an

affine domain over

k; but we c o n s i d e r those varieties w h i c h also

possess an affine c o o r d i n a t e system,

291

or more intrinsically,

an

5,36

equivalence affine

class

transformations).

over a q r o u n d An affine space),

field

if

Now

sense,

of

subvariety

C

A

embedded

in

dimension embedded

of

jective

defines gets A,"

reason

are able

of

their natural

A

completion

the natural

is

A/C

meaning,

"embedding

A

in

A.

can be

(including projective

those

that the h y p e r p l a n e

(filtration)

"inside"

projective of

A,

of an affine

projective

completion

i.e.,

r-space

is u s u a l l y

termed

"

of

of

taking

Hz(A). is a A,

sense)

as s u b v a r i e t i e s i • e. of

a pro-

completion

completion

in the a l g e b r a i c

ZHz(A)

may

an ab-

variety

structure

as a natural

Note

Conversely,

in

it

A/C

also say that

naturally

of h o m o g e n i z a t i o n

for

A,

is h o w

is n a t u r a l l y

as a h o m o g e n i z a t i o n

projective Under

A

given by the same process

at in f i n i t y

of

an i r r e d u c i b l e

e m d i m A = min{r:

to put

is w h a t w e d e f i n e d

all s u b v a r i e t i e s

A/C

the affine

variety w h i c h may be termed

r-space.

any

are their own

and this

an a p p r o p r i a t e

for a t t a c h i n g

is that then w e

projective

are

that an i r r e d u c i b l e

the a b s t r a c t

namely,

plane

a subvariety

[A, C3,

and w e may

(dim A)-

As before,

Formally

that

k.

r-space}.

that any natural

acquire

C

emdim A

in an affine

o__ff A, w h i c h Note

or that

by

field

(affine

to ideals w h i c h

~(A).

We h a v e

variety

(the variety)

The b a s i c A,

formulas.

sense,

of

variet~

2-space.

variety.

Again we have

may be d e n o t e d

irreducible A

In this

to

A

in various affine

set-up.

the ground

space

and an affine

restrict

irreducible

line and an affine

of as r e p r e s e n t i n g

is any m e m b e r

of

affine

is an affine

irreducible

w h i l e we

(under n o n s i n g u l a r

domain over

An affine

be an affine

in a g e o m e t r i c

subvariety

stract

A

1-space

may be thought

the a l g e b r a i c

appears

an affine

A

A

radica l s

variety

systems

an a b s t r a c t

is a filtered

d i m A = e m d i m A.

let

in

coordinate

Thus,

k

irreducible

are r e s p e c t i v e l y

ideal

of affine

also Hz(A)

Hz

of".

as the h y p e r p l a n e

A. starting with

fix a h y p e r p l a n e

~

an a b s t r a c t

and replace

292

projective

all s u b v a r i e t i e s

variety

B, w e

(including

those

5.37

in the algebraic sense) by their "parts outside process of d e h o m o g e n i z a t i o n ,

d e n o t e d by

Fz(-)

W e i n t r o d u c e the a p p r o p r i a t e notations C h a p t e r IV, and study various properties. course,

that part of

v a r i e t y to of

Fz(B)

B

outside

and

B

~

~ " or apply the where

for

~ = zB .

Hz(-)

and

Fz(- )

in

The b a s i c p h i l o s o p h y is, of

is "isomorphic" as an affine

is in turn a natural p r o j e c t i v e c o m p l e t i o n

F z (B).

Again,

let

A

be an abstract irreducible affine variety.

A

w

hypersurface surface

~

in in

affine plane,

A A

is a m e m b e r of is

degAS.

If

F A

(A)\{A}, and degree of a hyperis an affine 3-space or an

then a h y p e r s u r f a c e is called an affine surface and an

affine curve respectively.

H y p e r s u r f a c e s of degree i are called hyper-

planes.

A

A hypersurface

belongs to

in

is irreducible if, and only if, it

~(A).

F0(A)-vector-spaces as linearities

of h y p e r p l a n e s

of

A

serve the same function

in an a b s t r a c t i r r e d u c i b l e p r o j e c t i v e variety.

p a r t i c u l a r l y i n t e r e s t e d in those v e c t o r - s p a c e s w h i c h c o n t a i n

We are F0(A)

or e q u i v a l e n t l y w h o s e natural p r o j e c t i v e c o m p l e t i o n under a natural p r o j e c t i v e c o m p l e t i o n of finity.

A

These v e c t o r - s p a c e s

is a linearity in the h y p e r p l a n e at inform ~ ( A ) . ~ ( A )

spaces w h i c h have d i m e n s i o n equal to and

~i(A)

except that

(emdim A - i).

A,

linear etc.

irreducible subvar-

run e x a c t l y parallel to those of p r o j e c t i v e varieties, ~

is replaced by

~ throughout.

(See §2.)

F u n c t i o n field of an affine irreducible variety tient field

In general ~(A)

can be s i m i l a r l y defined, but we had no use for them.

D e s c r i p t i o n s of i-dimensional, ieties of

consists of those vector-

~(A)

[R,P], we take

and in the e m b e d d e d case w h e n

~([R,P~)

is its quo-

is replaced by

= ~(R/P).

The genus of an affine irreducible curve e m b e d d e d case)

A

A

A

(or

[R,P~

in the

is defined to be the genus of its natural p r o j e c t i v e

completion, w h i c h in fact is simply

293

g e n u s ( ~ (A), F0(A))

5.38

(genus (5 (R/P) ,F 0 (R/P)) ) ° In particular,

let us note

curve w i t h

only

rational),

it follows

A =

one place

has genus

A =k[t']

for

also runs

point all

A

is the

usual

subvariety).

localization,

localization

, where

What the

affine

is r e s i d u a l l y

affine

ring:

3(-,-)).

the

letter

place

at

multiplicity. case;

and p r o j e c t i v e are d e f i n e d

ring of the p o i n t

is d i f f e r e n t

k = F0(A)

valuation.

of i n t e r s e c t i o n

in fact incases

(or of the irre-

in the two cases

case,

we

is the pro-

take the

in place

of

~,

take the

"homogeneous"

To k e e p the d i s t i n c t i o n l

at a

by e x t e n d i n g

in the affine case, w e

in the p r o j e c t i v e

w e use

where

the u n i q u e

subvariety)

local

local

t

to the p r o j e c t i v e

in both

(denoted by

the two cases,

t'

parallel

to the

of c o n s t r u c t i n g

then,

for some

(i/t')-adic

(or along an i r r e d u c i b l e

ducible cess

entirely

ideals

is an i r r e d u c i b l e

(which,

= k(t)

to the c o n c e p t

multiplicities

relavent

A

that

zero ~ ~(A)

F i n a l l y we come

tersection

at infinity

for a s u i t a b l e

infinity

This

that if

between

throughout.

*

F0(A)

takes

The various k

appear

the place

of

definitions

in §5,6.

the d e f i n i t i o n s

H0(A)

related

Note

and to the

that the

of i n t e r s e c t i o n

k

Also W

takes

local

filtration

multiplicity

the place

intersection on

A

of multiplicity

is not

except

involved

in

in the d e f i n i t i o n

w

of

k . The concepts

similar §5.

to those

Again

the

that w e have which

of conductor, in p r o j e c t i v e

filtration

studied

do not depend

of them are studied

curves;

is not

in C h a p t e r on the

adjoint,

adjoint

the d e f i n i t i o n s

involved

IV.

294

etc.

are also

are given

in the d e f i n i t i o n s .

I, those concepts

filtration

in C h a p t e r

true

(affine

of affine

structure);

in

Note

curves

and the rest

5.39

§44.

NOTE.

Index to notations.

The following list of notations

appearance.

Whenever a

(

)

appears

is prepared in the order of in the following list,

in

the actual n o t a t i o n it may be filled w i t h any tuple

(as defined)

or

sometimes dropped altogether.

list " ~()

()()

"

we are r e f e r i n g to any of

For example, w h e n w e

" n I (A,I)

" ' " ~iI([A,I~)

", " ~(A,x,P)",

... etc.

Notation

Page number I.i

Jl + J 2 + ' ' ' J n ' J i J 2 " ' ' J n ' J n ' J °

:( )]

E(

I.i

F ()

i.i 1.2

5(

1.3 1.4

ord oo

)(

1.4

conventions of)

x(

adj

)

1.4

)

1.6, 1.20,i.28

)

1.15, 1.16,1.20

), tradj(

)

1.15,1.16,1.21

k'(

1.30

H n ( ) , H( ) Deg( )( )

2.1,2.2,2.5,

~(),~()

2.10,2.64,4.3 4.13 2.2,2.4

2.1

Dim ( )

2.2

Emdim ( )

2.2,2.4

3()

2.2,2.4,2.15 2.17

6()

2.4

295

0

LO

(D

g

0

0

V

,

.

~

0"I

.

0

o O

•r.l

0 Ch

~

.,,.-I

d

~

,.--I

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5.41

Page Number

Notation

~()() ~( )

4.19 5.13, 5.18

297

5.42

§45.

(45.1)

Index

to topics.

INDEX TO TERMINOLOGY.

The

following

index provides

and c e r t a i n

theorems.

in lo c a t i n g

the d i f f e r e n t

listings

for various

terminology

We p o i n t out that §43 may be further u s e f u l meanings

of a w o r d w i t h

several

listings.

Page N u m b e r adjoint,

true adjoint,

1.16, 1.17,1.21 1.28, 1.30

,over-,an u n d e f i n e d ,under-,an

descriptive,

undefined

descriptive,

1.17,2.78 2.79

Adjoint,

true Adjoint,

2.57,2.74,2.75

Bezou t ' s

theorem,-"little",

2.65

- for plane

curves,

- for h y p e r s u r f a c e s , d-chord,

2.89 5.12 2.90

complementary complete

module,

3.1

intersection,

4.15

conductor, conversion

1.3 formula

for d i f f e r e n t

3.12,3.18

c u s p types Dedekind's

1.37 formula,

- for c o n d u c t o r

and d i f f e r e n t

3.1,3.11,3.19 5.28

- for c o n d u c t o r

and differential,

3.24

degree

(filtered -

case),

of e l e m e n t s

- of p r i n c i p a l - of filtered Degree

(homogeneous -

4.3 ideals

4.3

domains,

4.13

case),

of elements,

- of p r i n c i p a l

2.1 ideals,

2.5

- of a h o m o g e n e o u s d o m a i n w i t h d i m e n s i o n zero, d i m e n s i o n one,

2.65 2.75

dehomogenization, natural

4.5 4.6

-

298

5.43

3.1,5.26,5.28

different,

3.22,5.24

differential, - order double

3.24

of,

points

(geometric words

for),

1.37

elementary

step

4.18

elementary

transformation,

4.18

essentially

4.15

hyperplanar,-planar,

euclidean, 5.19 5.19

,chain of - domains ,chain of - curves, filterd,

4.1 4.2 4.2

-domain, -homomorphism, -subdomain filtration, -natural

4.1 4.2

flat,

2.7

function

2.2

field,

3.22,3.41,3.52; 4.13 3.25,3.42,3.52,

genus, - formulas,

3.53 ground

field, 2.1 4.1

- of a f i l t e r e d d o m a i n , of a homogeneous domain -

homogeneous, 2.1 2.2 2.2,2.4 2.3 2.3 2.3 2.13,2.16 2.20

- domain - dimension, - embedding dimension, ideal, - homomorphism, subdomain, localization, coordinate system, -

-

-

-

4.3 4.4

homogenization, natural, -

length, of a module, - as a f f i n e i n t e r s e c t i o n m u l t i p l i c i t y , - as p r o j e c t i v e i n t e r s e c t i o n m u l t i c i t y ,

-

I.i 1.6,1.20,4.2 2.53,2.73, 2.89,5.12

model, normal, - of a homogeneous

1.3 2.2

-

node

domain,

1.37

types,

299

5.44

2.46,2.55

Osculating

flat,

projection

(filtered

4.11

case).

4.11 4.11

- birational - integral, projection

(homogeneous

2.7,2 .9

case).

2.9 2.28, 2.29 2.68,2.69,5.17 2.25,2.27

- birational, - ~-integral, formulas, nemmas, projecting

center

- good, projective

better,

center

~-quasihyperplane,

2.90

best,

2.14,2.17

of a v a l u a t i o n

2.86

~-quasiplane

2.90

d-secant, separably separating

3.9

generated,

3.9

transcendental,

tangent cone to a h y p e r s u r f a c e (algebraic definition),

2 . 8 4

tangent

2.73,2.76

flat,

uniformizing, 3.32 3.32

- coordinate, parameter,

-

valued

vector

2.30

space,

300

5.45

(45.2)

Eli

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: Local rings of h i g h e m b e d d i n g Amer. J. Math., 8__99,No.4, 1073-1077

L3]

: A l g e b r a i c space curves, de Z'Universite de m o n t r e a l (1971).

[4]

and Mob Tzuong-tsieng: Embeddings of the line in the plane, submitted to J. Reine Angew. Math.

[5]

: appear in Math.

Les

Presses

The adjoint c o n d i t i o n revisited, Student.

to

[6]

Bass,

[7]

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[8]

Cohen,

[9]

Chevalley, Claude: I n t r o d u c t i o n to the theory of algebraic functions of one variable, A m e r i c a n M a t h e m a t i c a l Society, M a t h e m a t i c a l Surveys VI (1951).

[lO] [ii]

[12]

Kunz,

Hyman: On the u b i q u i t y of G o r e n s t e i n rings, 8__22, 8-28 (1963).

Math.

Z.,

I.S.: On the structure and ideal theory of complete rings, Trans. Amer. Math. Soc., 5__99,No.l, 54-106 (1946).

Ernst: Differentialformen inseparabler algebraischer Funktionenk~per, Math. Z., 7_~6, 56-74 (1961). : V o l s t M n d i g e D u r c h s c h n i t t e und Differente, Arch. Math. (Basel), XIX , 47-58 (1968).

Murthy,

M.P.

and Towber J.:

~3 are trivial, (1974).

A l g e b r a i c v e c t o r bundles over

Invent. Math.,

2_~4, Fasc 3, 173-189

[13]

Nastold, Hans-Joachim: Zum D u a l i t M t s s a t z i n s e p a r b l e n F u n k t i o n e n k ~ r p e r der D i m e n s i o n i, Math. Z., 7_~6, 75-84 (1961).

[14]

N~ther, Max: Zur Theorie des e i n d e u t i g e n E n t s p r e c h e n s a l g e b r a i s c h e r Gebilde von b e l i e b i g v i ~ l e n Dimension, Math. Ann., [, 293-316 (1870).

[15]

: U b e r einen Satz aus der Theorie der algebraischen Functionen, Math. Ann., ~, 351-359 (1873).

[16]

Roquette, P.: U b e r den S i n g u l a r i t ~ t s g r a d e n d i m e n s i o n a l e r Ringe II, J. Reine Angew. Math., 209, 12-16 (1962).

301

5.46

[17-I

Zariski, O s c a r and Falb, Peter: On d i f f e r e n t i a l s function fields, Amer. J. Math., 88, No.3, 542-566 (1961).

[18]

Zariski, O s c a r and Samuel Pierre: (vol. I,II), Van Nonstrand,

302

in

C o m m u t a t i v e algebra 1958.