An Introduction to Commutative Algebra: From the Viewpoint of Normalization

An Introduction to Commutative Algebra From t h e Viewpoint of Normalization

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From t h e Viewpoint of Normalization

I Jiaying University, China

NEW J E R S E Y * L O N D O N * S I N S A P O R E

-

EElJlNG * S H A N G H A ' * HONG K O N G * TAIPEI

CHENNAI

Published by

World Scientific Publishing Co. Re. Ltd.

5 Toh Tuck Link, Singapore 596224 USA office: Suite 202,1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library

AN INTRODUCTION TO COMMUTATIVE ALGEBRA From the Viewpoint of Normalization Copyright 0 2004 by World Scientific Publishing Co. Re.Ltd. All rights reserved. This book, or parts thereof. may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rasewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-238-951-2

Printed in Singapore by World Scientific Printers ( S ) Pte Ltd

For Pinpin and Chao

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Preface

Why normalization? Over the years I had been bothered by selecting material for teaching several one-semester courses. When I taught senior undergraduate students a first course in (algebraic) number theory, or when I taught first-year graduate students an introduction to algebraic geometry, students strongly felt the lack of some preliminaries on commutative algebra; while I taught first-year graduate students a course in commutative algebra by quoting some nontrivial examples from number theory and algebraic geometry, often times I found my students having difficulty understanding. The problem is that in a short semester there is not enough time to go through the significant background material needed in my course (if you are an instructor of mathematics, can you ask your students to find and read those material themselves and do they usually follow?). Based on my lecture notes on (algebraic) number theory, algebraic geometry, and commutative algebra used at Shaanxi Normal University and Bilkent University, I decided to fuse several things into one - the presentation of this book. As a consequence, the text consists of five chapters that are designed for a (one-semester) common course taken by senior undergraduate students or by first-year graduate students in mathematics. The goal is to introduce to the students the concrete source of commutative

vii

...

Vlll

Commutative Algebra

-

algebra through the following diagram:

DVR -+ PID I

ED

UFD 1~ D V R +

- DD

I= I

NND

where DVR = PID = UFD = ED = DD = NND -= Z[6]=

Discrete valuation ring, Principal ideal domain, Unique factorization domain, Euclidean domain, Dedekind domain, Normal Noetherian domain, The integral closure of 2[6] (or equivalently the integral closure of Z ) in the number field K = Q(d), K[C]= The normalization (or integral closure) of the coordinate ring K[C]of an algebraic curve C in its field of rational functions K(C), - and K [ V ]= The normalization (or integral closure) of the coordinate ring K [ V ]of an irreducible algebraic set V in its field of rational functions K ( V ) . Or more precisely, in terms of normal (normalized) structure the lectures demonstrate how t o get to the center -of commutative algebra by recognizing the roles that 2 [ 6 ] ,K[C]and K [ V ]play in number theory and algebraic geometry, so that, after reading this volume, interested readers may read a course in algebraic number theory a t a higher level, or start an advanced course in algebraic geometry with a better background.

I assume that the reader has taken the undergraduate courses including Linear Algebra and A First Course in Abstract Algebra (Galois theory may not be included). Moreover, very little about topological space is needed

Preface

ix

to understand the Zariski topology in Chapter 5. Other than these prerequisites, the book is self-contained. Exercises are given a t the end of each section. Though most of the exercises mainly test the understanding of the text in the usual way, the reader is involved in providing proofs and in working problems that have not been completely solved in the text; and furthermore, students are asked to extend some of the theory that is essential for the subsequent sections.

Huishi Li

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Contents

vii

Preface Chapter 1 Preliminaries 0. Conventional Review 1. Noetherian Rings 2. Factorization of Elements in a Domain 3. Field Extensions 4. Symmetric Polynomials 5. Trace and Norm 6. Free Abelian Groups of Finite Rank 7. Noetherian Modules

Chapter 2 Local Rings, DVRs, and Localization 1. SpecR, m-SpecR, and Radicals 2. Local Rings and DVRs 3. The Ring of Fractions and Localization 4. The Module of Fractions

Chapter 3 Integral Extensions and Normalization 1. Integral Extensions 2. Noether Normalization

3. Normal Domains and Normalization 4. Normal Domains and DVRs

xi

1 1 6 9 18 28 33 40 45

53 53 58 67 73

79 79 85 90 94

Contents

xii

Chapter 4 The Ring $ 2 in ~ K = Q(t9) 1. d K is Normal and Free of %Rank [K : Q] 2. and Q(u) 3. Factorization of Elements in d K 4. From d K to Dedekind Domains

a(&)

Chapter 5 Algebraic Geometry 1. Finite Field Extension and Nullstellensatz 2. Irreducible V and the Prime I(V) 3. Point P and the Local Ring Op,v 4. Nonsingular Points and DVRs 5. Normalization of Algebraic Curves 6. Parametrize a Rational Curve via Normalization 7. Rational Curves and Diophantine Equations

99 102 108 112 117

127 127 134 146 151 157 162 167

References

171

Index

173

Chapter 1

Preliminaries

This introductory chapter concentrates on basic notions and properties concerning Noetherian rings, factorization of elements in a domain, field extensions, symmetric polynomials, trace and norm, free abelian groups of finite rank, and Noetherian modules, which might not be familiar t o some readers. So we include most of necessary proofs for the reader’s convenience.

0. Conventional Review

In this book all rings are commutative associative rings with identity 1, and throughout the text, N = the set of nonnegative integers, Z= the set of integers (ring of integers), Zf= the set of positive integers, Q = the set of rational numbers (field of rational numbers), R = the set of real numbers (field of real numbers), C = the set of complex numbers (field of complex numbers). Let R be a ring and A a subring of R. Then we insist that A has identity 1~ and

And we write

R X = R - (0). If { U i } i E ~and {Vl,..., Vm} are collections of nonempty subsets of R, 1

Commutative Algebra

2

then the sum of { U i } i , = and ~ the product of {Vl,..., Vm}are defined as

vl...vm={cvl...v,

I }

ViEV,

,

where the sums involved in both C Ui and VI . . . V, are finite sums. So one understands that 0 the sum of given ideals is an ideal; the product of finitely many given subrings (ideals) is a subring (an ideal); and 0 for subrings (ideals) I , J and K , I ( J K )=IJ IK =JI KI = ( J K)I.

+

+

+

+

Let S be a nonempty subset of R and A a subring of R. We set

Z[S]= the subring of R generated by S ={

~

s ...s:; z

I

si, E S, m E

z+,aj EN}

A[S]= the subring of R generated by S over A = ~ a ( , , ~ ...s:; ) s ~ a(,,j) E A , si, E S, m E

1

{

z+,aj E N}

(S) = the ideal of R generated by S

=

I

{ c r i s i ri E R, si E

s},

where the sums involved in Z[S], A [ S ]and ( S ) are finite sums. If S = {sl,..., s,} is finite, we write Z[S]= Z[sl, ..., sn],A [ S ]= A [ S I..., , s,], ( S ) = (s1, ..., sn), and call them a finitely generated subring, a finitely generated subring over A , and a finitely generated ideal of R, respectively. Clearly we n can also write ( ~ 1 ,...,Sn) = C,=lRsi. Let R L R ' be a ring homomorphism. Then we insist that cp is not the zero-homomorphism and cp(1~) =l ~ , ,

and we write Kercp, Imcp for the kernel and image of cp, respectively. A very useful consequence of the first isomorphism theorem on ring homomorphism states that

Preliminaries

3

*

if R-%A and R-B are ring homomorphisms, cp is surjective, and Kercp C Ker$, then there is a ring homomorphism A-%B defined by p ( a ) = $ ( r ) ,where cp(r)= a , such that the following diagram commutes:

*I

A

Kercp L) R

JP

pocp=$

B if furthermore

1c, is surjective, then p is surjective as well.

Let R be a ring with identity 1 = 1 ~If. R has no divisors of zero, i.e., a, b E R and ab = 0 implies a = 0 or b = 0, then R is called an integral domain, or simply a domain. If R is a domain, then so is the polynomial ring R[zl, ..., zn]in variables z1 , ...,zn over R. If a , b E R and ab = 1, then a (hence b) is called a unit of R. If every nonzero a E R is a unit, then R is called a field. 0.1. Proposition Every finite domain is a field.

Proof Exercise.

17

Thus, if p E Zis a prime number, then the ring Z / ( p ) of integers modulo p , usually denoted Z,,is a field. Let K be a field. Consider the set of integers

I

o ( ~=) { m E Z+ m~ = o for some

1

x E K~ .

If o ( K ) = 0, then the characteristic of K , denoted charK, is defined to be zero, i.e., charK = 0; if o ( K ) # 0, then charK is defined to be the smallest integer p E o ( K ) . In the second case p is a prime number (exercise 3). Every field has a smallest subfield P (with respect to the inclusion relation on subfields), the prime subfield, which is either isomorphic to

Q, if charK = 0, or t o

Z p , if charK = p p

> 0.

Clearly, every finite field F has charF > 0. If a field K has charK = ( u b ) P = ap bP for all a , b E K (exercise 4).

> 0, then

+

+

Commutative Algebra

4

If R is a domain, then the field of fractions of R is constructed via the equivalence relation on R x R X:

( a ,b)

-

(c,d ) if and only if bc = ad.

Write f for the equivalence class represented by ( a ,b), and write Q(R)for the quotient set R x R X /-. Then

I

Q ( R ) = { % a,bER, b f o } where the addition and multiplication are defined the same as that for rational numbers. Thus, in Q(R), = 0 is the zero of the additive group ( Q ( R )+), , = ~ Q ( Ris) the identity of the multiplicative group (Q(R),.), and if a = f # 0 then a-1 = $. The ring homomorphism

4

XR : R

--+Q ( R )

r w

r

-

1

is injective. In the case where R is a field, XR is an isomorphism. So we may view R as a subring of Q(R)and write R Q(R).Consequently, if Q(R)[x] is the polynomial ring in variable x over Q(R),then R [ z ] Q(R)[x]. If R' is another domain and 9:R R' is an injective ring homomorphism, then cp induces an injective ring homomorphism Cp: Q ( R )-+ Q(R'), where (f) = Hence Q(R) may be viewed as a subfield of Q(R'), and consequently Q(R)[z] may be viewed as a subring of Q(R')[x].It turns out that if cp is an isomorphism then so is q. In particular, if Q ( R )is the field of fractions of the domain R and R c B c Q(R),where B is a subring of Q ( R ) ,then Q ( B )= Q(R). --f

3.

We assume that the reader is familiar with the basic structural properties of a polynomial ring R [ q ,...,x,] in variables x1, ..., x, over the ring R , for instance, R is a subring of R [ q ,...,z,] consisting of constant polynomials, every f E R [ Q ,..., z,] has a unique expression into the linear combination of monomials: f = r,xy' . . .xEn, where cy = (a1,...,a,) E N" = { (ai,...,a,) I a1,...,a, E N}, and the degree of f is defined as degf

= max

{ a1 + . . + a,

r,xy'

1

. . . xzn # 0 is a term of f .

If f = 0 then conventionally degf is defined as -m. Thus, for f = C Tax:' . . .xC,an, g = Crp$ one knows how to determine the

...xk,

Preliminaries

5

+

degree of f g and f . g according to the addition and multiplication of polynomials. In particular, we recall the following important properties of a polynomial ring. I f f E R [ z ]is a nonconstant monic polynomial, i.e., degf 2 1 and f is of the form

f = rcn

+ an-lrcn-l + . . . +ao,

ai

E

R,

then a division algorithm on g E R [ z ]by f exists: g =qf

+

T,

q,r E R [ z ] ,degr

< degf.

Let R be a ring. If Z[sl, ..., s,] is the subring of R generated by .,., s,, then there is an onto ring homomorphism from the polynomial ring Z [ z l ,..., rc,] to Z[sl, ..., s,]: s1,

Z[Zl, ...,%I

-

f(z1, .-,4 H

q s 1 , ..., S n ] f(Sl,...,S,)

Let A be a subring of R. If A [ s l ,..., sn] is the subring of R generated by sl, ..., s, over A , then there is an onto ring homomorphism from the polynomial ring A [ z l ,...,rc,] t o Ajsl, ..., s,]:

A [ ~ I. -,. , ~ 2 ]

-

A [ s l ,'''7

sn]

Exercises 1. Let A be a subring of the ring R and S C R a nonempty subset of R. Show that Z[S]is the smallest subring of R containing S , that A[S] is the smallest subring of R containing A and S , and that (S) is the smallest ideal of R containing S. (Here the ordering on subrings and ideals is the usual inclusion ordering on subsets.) 2. Complete the proof of Proposition 0.1. (Hint: If R = { a l , ..., a,} is a finite domain and 0 # ai E R, then a i R = R.) 3. Let K be a field. Show that if charK = p # 0, then p is a prime number. 4. Let K be a field of charK = p # 0. Show that ( a b ) P = ap bP for all a , b E K . (Hint: Check the binomial coefficients of the expansion of ( a b)p.)

+

+

+

Commutative Algebra

6

1. Noetherian Rings Since Noetherian ring plays a leading role in commutative algebra, we start with this notion. Let R be a ring. R is said to satisfy the maximal condition if every nonempty set of ideals contains a maximal member with respect to the inclusion relation on ideals. R is said to satisfy the ascending chain condition if for every ascending chain of ideals

I1 there is some k such that Ik

c I2 5 . . * c In = Ij

...

for all j 2 Ic.

1.1. Theorem Let R be a ring. The following are equivalent. (i) R satisfies the maximal condition. (ii) Every ideal of R is finitely generated. (iii) R satisfies the ascending chain condition.

Proof (i)

+ (ii) Let I

be a nonzero ideal of

R.Set

S = {all finitely generated ideals contained in I } . Then S # 0, and by (i) there is a maximal member in S , say J = C;=, Rai with ai E I . If J # I , then there is some x E I, x 6 J. Thus, J is properly contained in J' = J Rx and J' E S , contradicting the choice of J . Therefore I = J , a finitely generated ideal. (ii) + (iii) Let

+

be an ascending chain of ideals in R. Set I = UIi. Then I is an ideal of R and hence finitely generated, say I = CE, Raj with aj E I . Suppose aj E Ii3 with il < i2 < ... < i,. Then aj E Ii,, j = 1,...,m, and consequently I = Ii,. Let k = i,. Then Ik = Ij for all j 2 k. (iii) + (i) Let S = { I i } be a nonempty set of ideals in R. If S did not have a maximal member, there would be a strictly ascending chain of ideals 0 out of S , which does not satisfy the chain condition.

1.2. Definition A ring R satisfying one of the equivalent conditions of Theorem 1.1 is called a Noetherian ring.

7

Preliminaries

Let R be a ring. If every ideal I of R is a principal ideal, i.e., I = ( a ) = Ra for some a E I , then R is called a principal ideal ring. Principal ideal rings are special Noetherian rings. If a principal ideal ring R is also a domain, then we simply call R a PID. It is a result of the division algorithm in Z and the division algorithm in the polynomial ring K [ z ] where , K is a field, that both Z and k[z]are PIDs (exercise 1). Concerning polynomial rings in finitely many variables over a Noetherian ring, we have the following celebrated result. 1.3. Theorem (Hilbert basis theorem) If R is a Noetherian ring then so is the polynomial ring R[z]in variable z over R. Hence, the polynomial ring R[zl,..., z,], in any finitely n variables z1, ...,z,, is Noetherian.

Proof We show that if R[x]is not Noetherian then neither is R, by adopting a well-known argumentation (as one may easily find a t the site ' ) . Suppose that I is an ideal of R[z]which is not finitely generated. Then a sequence of polynomials from I can be chosen as follows.

fi

E I with least degree nl,

f2

E

I - Rfl with least degree n2, E I - ( R f i Rf2) with least degree n3,

f3

+

k

fk+1

E

I - I R f i with least degree nk+l, i=l

where n1 5 722 I 123 5 ' . . I nk+l 5 . . . . Claim Let ai be the leading coefficient of

Then

fi.

n

Ral

c Ral + Ra2 c . . c C R a i c . . . *

i=l

is a strictly ascending chain of ideals in R. If the claim was not true, then k Rai = Cfz; Rai for some k, and this would yield ak+l = k riai, ri E R. Note that for i = 1,..., k, we http://planetmath.org/encyclopedia/Proofo~ilbertBasisTheorem.html

Commutative Algebra

8

have

fi

= aizna

+ strictly lower degree terms, + strictly lower degree terms.

~ i f i P k + l -= ~ ira.xnk+1 2 2

It follows that

while /

\

k

k

by the choice of f k + l . But clearly degg(s) < degfk+l, contradicting the 0 choice of f k + l . Therefore the claim holds, i.e., R is not Noetherian. The polynomial ring K [ z l ,..., z, . .I in infinitely many variables over a field K is non-Noetherian, due t o the existence of a strictly ascending chain of ideals: (51)

c

(21,52)

c .. c *

”’,5,) c . . .

(51,

*

In Chapter 4 we will see that if A is the set of all algebraic integers, then A forms a i.e., the set of complex zeros of monic polynomials in Z[x], ring and it is not Noetherian; while for a finite dimensional field extension Q C K with K a subfield of C, A n K is always Noetherian. Noetherian rings stemming from algebraic geometry are given in Chapter 5. Exercises 1. Show that Z and K [ z ]are PIDs, where K[z] is the polynomial ring in x over a field K . 2. Let R + R‘ be an onto ring homomorphism. Show that if R is Noetherian then so is R’. 3. Let A be a Noetherian subring of the ring R, and let { T I , ...,r s } be a finite subset of R. Show that the subring A[rl,...,T,] of R is Noetherian. 4. Let K be a field which, as a subring, is contained in the ring R. Assume that R is finite dimensional over K . Show that R is Noetherian.

Preliminaries

9

5. Let R be a Noetherian ring. The ring of formal power series over R is ] of the formal series the associative ring R [ [ z ]consisting

i=O

where f(z)= 0 if and only if ri = 0 for all i = 0,1, ..., and the addition and multiplication are defined as for the power series with real coeffcients in calculus. Show that R [ [ s ]is] Noetherian. (Hint: Define the degree of a series as the lowest power of x.) 6. By Theorem 1.3, Z[x]is Noetherian. Show that the ideal I = (2, x) is not a principal ideal. 7. Let Zz[z,y]be the polynomial ring over the field Z2. Show that in &[x, y]/(z2 z y3 1) the ideal (TIg ) is not a principal ideal.

+ + +

2. Factorization of Elements in a Domain Let R be a domain. It is easy to see that the set of units in R , denoted

I

1

U ( R ) = {u E R u is a unit in R , forms a group with respect to the multiplication of R. U ( R )is called the group of units in R. 2.1. Definition (i) For r E R, u E U ( R ) ,the element y = ur = ru is called an associate of r. (ii) Let r , s E R. r is said to be divisible by s, denoted s [ r ,if r = sz for some z E R, where s (hence t)is called a divisor (or a factor) of r. For u E U ( R )and r E R, u and ur are called the trivial divisors of r (note that r = (ur)u-l = ( u - l r ) u ) . (iii) For r E R, if r has only trivial divisors in R , then we say that r is irreducible in R; otherwise, r is reducible in R. (So zero is reducible in any domain.) (iv) For r E R, if r is reducible, then r = sz with nontrivial divisors s, t. In this case we say that r has a proper factorization.

Example (i) Let R = Z.Then U ( R )= {fl}. (ii) Let R

= Z [ i ]where

i = fl. Then U ( R )= {fl, f i } (see Chapter 4

Commutative Algebra

10

section 3). (iii) Let R = K [ z ]be the polynomial ring in z over a field K . Then

U ( R )= K X . Thus, one easily finds elements in each R that have proper factorization. 2.2. Proposition Let R be a domain, T , s E R. The following hold: (i) r E R is a unit if and only if 7-11. (ii) Any two units are associates t o each other, and any associate of a unit is a unit. (iii) r , s are associates to each other if and only if rIs and SIT. (iv) T is irreducible if and only if every divisor of T is either an associate of r or a unit. (v) Any associate of an irreducible element is irreducible. Proof Exercise. In terms of ideal structure, we may characterize units, divisibility, associates and irreducibility, as follows. 2.3. Proposition Let R be a domain and let r , s be nonzero elements of

R. (i) r E U ( R ) if and only if ( r )= R. (ii) T I S if and only if ( r ) 2 ( s ) . (iii) r , s are associates to each other if and only if ( r ) = ( s ) . (iv) T is irreducible if and only if ( r ) is maximal among the principal ideals of R (with respect to the inclusion ordering on ideals). Proof Exercise.

2.4. Definition Let R be a domain. We say that factorization into irreducible elements is feasible in R if every nonzero nonunit element may be expressed as a product of finitely many irreducible elements. 2.5. Proposition Factorization into irreducible elements is feasible in a

Noetherian domain R.

Proof Let R be a Noetherian domain. Suppose that the assertion was not true. Then the set R of nonzero nonunit elements which cannot be factorized into finite products of irreducible elements would be nonempty.

Preliminaries

11

Since R is Noetherian, let (y) be a maximal member in

S={(x)

IXFQ}.

Then y is reducible because y E R, and y = rs for r , s # U ( R ) . Thus, (y) is properly contained in ( r ) n (s) (otherwise r or s would be a unit by Proposition 2.3). By the choice of (y) we have

where pi’s are irreducible elements. But then y = P I * *.p,p,+l .-.p,, a product of finitely many irreducible elements. This is a contradiction and hence R = 0.

2.6. Definition Let R be a domain in which factorization into irreducible elements is feasible. For a nonzero nonunit x E R, if any two factorizations

satisfy n = m and (up to the arrangement of divisors) pi = Uiqi, i = 1, ..., n, where ui E U ( R ) ,then x is said to have a unique factorization in R. If every nonzero nonunit element of R has a unique factorization in R, we say that R is a UFD (abbreviation of the phrase “unique factorization domain”).

Remark At this stage, it is better to be aware of two facts. (i) There are Noetherian domains which are not UFDs (see exercise 4 of this section and Chapter 4 section 3). (ii) There are UFDs which are not Noetherian (exercise 5). In order t o discuss the uniqueness of factorization into irreducible elements, we introduce the notion of a prime in a domain. 2.7. Definition Let R be a domain, 0 # x E R, and x to be a prime if xlab implies xla or xlb for any a, b E R.

# U ( R ) . x is said

2.8. Proposition Let p be a prime in a domain R. The following hold: (i) Any associate of p is a prime in R. (ii) p is irreducible in R.

Proof Exercise.

Commutative Algebra

12

2.9. Theorem If factorization into irreducible elements is feasible in a domain R, then R is a UFD if and only if every irreducible element is a prime.

Proof Since factorization into irreducible elements is feasible in R, by Proposition 2.2(v), every nonzero nonunit x E R has a factorization 2

= P I f ..pel

where pi may be an associate of some irreducible element. First suppose that factorization in R is unique. Let p be an irreducible element and plab where a # 0, b # 0. Then ab = pc for some 0 # c E R. Consider the unique factorizations: a = p l . . pn, b = 4 1 . . . qm, c = T I . . .T ~ Then +

PC = ~ ( ~ * *1. i rs) 1

= (PI. . *pn)(ql.. ' qm) = ab.

By the uniqueness, p divides some pi or some q j . Hence pja or p ) b , and this shows that p is a prime. Conversely, suppose every irreducible element is a prime. Consider the factorization into primes

Then pl1q1 (q2 . . . qm). Without loss of generality we may assume plIq1. Then, 41 = ulpl for some u1 E U ( R ) because q1 has only trivial divisors. Thus, z = ~ 1 . e . p=~ ( u i p 1 ) ( ~ 2 * * * q mand ) p 2 * * * p n= (ulq2)(q3*'.qm)* After repeating this process n times, up t o the arrangements of divisors we derive qi = uipi with ui E U ( R ) ,and m 5 n. Similarly, n 5 m. So n = m. This shows that factorization is unique in R. CI 2.10. Theorem Every PID is a UFD.

Proof Let R be a PID. Then factorization into irreducible elements is feasible in R because R is Noetherian. Let p be an irreducible element in R. Then by Proposition 2.3(iv), ( p ) is maximal among all ideals. Suppose plab but p /y a. Then ( p ) is properly contained in the ideal ( p ,a ) . By the maximality of ( p ) we have ( p ,u ) = R. It follows that 1 = ph aq and b = bph abq. This yields pjb, showing that p is a prime. By Theorem 2.9, R is a UFD. 0

+

+

Remark Recall that before learning a systematic theory on UFDs, in the

,

Preliminaries

13

arithmetic theory on R = Z (or in R = K [ x ]where K is a field) a prime p is defined as the element which has only the divisors fl (A E K " ) , f p (Ap). If a, b E R, plab but p /y a, then the Euclidean algorithm output the greatest common divisor gcd(p, a ) = 1 in the form af+pg=L

f,gER,

that yields plb immediately as in the above proof. That is why we know, without arguing that R is a PID, that R is a UFD. Indeed, there is a class of UFDs that hold a version of Euclidean algorithm, as described below.

Definition A Euclidean domain is a domain R with a function (called a Euclidean function):

2.11.

4 : RX-N that satisfies (i) if a , b E R X and a [ bthen $ ( a ) 5 4(b); and (ii) if a, b E R X then there exist q, r E R such that a = qb

+

T,

where either r = 0 or $ ( r ) < 4(b).

Example (iv) Z is a Euclidean domain with the Euclidean function given by the absolute value function. K [ x ]is a Euclidean domain with the Euclidean function given by the degree function. (A consequence of applying the division algorithm to both Z and K [ x ] . ) 2.12. Theorem Every Euclidean domain R is a PID.

Proof Let I be a nonzero ideal of R. If function on R, let us set

4(x*)=min { 4 ( x ) E N

4 is the associated Euclidean O#x E I .

}

+

For any 0 # y E I , y = qx* r with T = 0 or 4(r) < 4(x*). But T = y - qx' E I . By the choice of xi,T = 0. Thus, y = qx'. This shows

that I = (x*).

0

2.13. Corollary Every Euclidean domain is a UFD. Proof This follows from Theorems 2.10-2.12.

0

14

Commutative Algebra

Except for Zand K [ x ]other , Euclidean domains will be given in Chapter 4 section 3.

Remark Let K = Q ( a ) . By Theorem 3.4 of Chapter 4, the ring dK of algebraic integers in K is not a Euclidean domain. However, AK is a PID. The reader is referred to http://www.mathreference.com/id,npid.htmlfor a beautiful proof on this fact. We now proceed to show that the polynomial ring R[x]in variable x over a UFD R is a UFD.

2.14. Lemma (Gauss) Let R be a domain. Then any prime of R is a prime in R[x]. Proof Let p be a prime in R and R = R/(p). Then a direct verification shows that R is a domain, and so is the polynomial ring R[x].If f , g E R[x] and p l f g , then f g E ( p ) . For T E R, write F for the image of T in R. Consider the ring homomorphism

R[x] 5 R[x]

Then p ( f g ) = 7 . 3 = 0. Since R[x]is a domain, it follows that 9 = 0, i.e., p l f or pjg, as desired.

= 0 or

17

Let R be a UFD. Then for any T I , ..., T, E R, not all zero, the greatest common divisor gcd(a1 , ..., a,) exists in R (exercise 6).

.+

2.15. Definition Let R be a UFD. If a polynomial T , Z ~ + T ~ - I Z ~ - ' + . . = f(x) E R[x]has the property that gcd(Tn,rn-l, ..., T O ) = d E U ( R ) , then f ( x ) is called a primitive polynomial. TO

2.16. Proposition Let R be a UFD. If f,g E R[x]are primitive then so is the product f g. Proof This follows immediately from Gauss lemma. 2.17. Theorem Let R be a UFD with the field of fractions K = Q ( R ) . (i) If f E R[x]and f = gh for some g, h E K [ x ] ,then there is a unit

Preliminaries

15

a E K [ z ]such that gal a-lh E R[x]. (ii) Let f , g E R [ x ]where , g is primitive. If 91f in K [ x ]then g l f in R[z]. Proof (i) Let f = g h be as assumed. Let TO E R be the common denominator of the coefficients of 9. Then rog E R[x].Let d be the greatest common divisor of all coefficients of rag. Then g1 = ag is primitive in R[x],where a = 3 E K , Similarly, there exists /3 E K such that hl = ph is primitive = f , where a and b have only common divisors which come in R[x].Set from U ( R ) . Then

a

,f

= aPgh = g l h l

and a f

= bglhl.

Now, if a E U ( R ) ,then since b a P = a , we have ag = 91, a-lh = a - l b p h = a - l b h l have coefficients in R, as desired. So it remains to show that a E U ( R ) . If not;, there would be some prime p dividing a. Hence p l b g l h l but p /y b by the choice of a and b, and p /y 91, p 1hl because both g1 and hl are primitive. This contradicts Gauss lemma. Therefore, a must be a unit. 0 (ii) This follows from part (i). Let R be a UFD and f(x) E R[x]with degf (x)2 1. If d is the greatest common divisor of all coefficients of f(z), then f(z)= d f l ( z ) where f l ( z ) is a primitive polynomial. Bearing this fact in mind, Theorem 2.17 enables us to derive immediately the following. 2.18. Proposition Let R be a UFD with the field of fractions K = Q ( R ) , p ( z ) E R[z]with degp(z) 2 1. Then p ( z ) is irreducible in R[x]if and only if p(x) is irreducible in K [ z ] .

17 2.19. Theorem If R is a UFD then so is R[z] Proof Since R is a UFD and R c R [ s ]by , Gauss lemma we need only t o consider polynomials of degree 2 1. Let K = Q ( R )be the field of fractions of R. Then K [ x ]is a UFD. Thus every f (x)E R [ x ]with degf (x)2 1 is factorized into a product of finitely many irreducible elements in K [ z ] .By Theorem 2.17 and Proposition 2.18, factorization of polynomials of degree 2 1 into irreducible polynomials is feasible in R [ z ]and , irreducible polynomials in R [ x ]are primes. Hence R[z] is a UFD. 17

Commutative Algebra

16

2.20. Corollary For any field K , the polynomial ring finitely many variables 2 1 , ...,z, over K is a UFD.

K[zl,

..., z],

in 0

We finish this section by Eisenstein’s criterion concerning the irreducibility of polynomials in R[z],where R is a domain. 2.21. Theorem Let R be a domain and

f(.)

= a,zn

+ . . . + a15 + a0

a polynomial in R[z].Suppose there is a prime p E R such that (a) P X an, (b) plai, i = 0, ...,71 - 1, (c) P2 1.0. Then f ( z ) is irreducible in R[z].

Proof Suppose f(z)= g ( z ) h ( z )for g(z), h ( z ) E R[z]where g(.)

= c,zT

+ . . . + c1z + co

+

with ci,dj E R and T , s > 1, T s = n. Then by (b) and (c), p)ao = cod0 and hence p divides co or do but not both. Suppose ~ 1 % . By (a), we may let c,be the first coefficient of g(z) not divisible by p . But note that

This implies p a,, contradicting (b) because m < n. Hence g ( 5 ) or h ( z ) must be a unit of R. 2.22. Corollary If p is a prime number, then the polynomial f ( z ) = zp-l

+zp-2 + . + 5 + 1 * *

is irreducible in Z[z]and hence irreducible in Q[z].

Preliminaries

Proof Note that f(z)=

f(x

=

+

9. If we use the translation z = X + 1, then

(X + 1)” - 1 + ( X + 1) - 1

1 (xp+ (;) X

=x p - l +

17

(;)

xp-l+

xp-2

+

(;)

(;)

xp-2

XP-3

+ . .. +

(P - 1 ) x +

1 - 1)

+ .. . + p .

Now, using p as the prime needed in Theorem 2.21, we conclude that f(z) is irreducible in Z[z] and hence irreducible in Q [ x ]by Proposition 2.18.

Exercises Complete the proof of Proposition 2.2. Complete the proof of Proposition 2.3. Complete the proof of Proposition 2.8. Let R = K [ t 2t3] , be the subring generated by t2 and t3 in the polynomial ring K [ t ]over a field K . Show that both t2 and t3 are irreducible in R but none is a prime. However t6 = t2t2t2= t3t3. (See also Chapter 3 (section 2, exercise 2) and Chapter 3 (section 3, Example (iii)).) 5. Show that the polynomial ring R = K[z1,z2,..., 2, ,...] in infinitely many variables over a field K is a UFD. (Hint: Any polynomial in R belongs to a polynomial ring in finitely many variables over K . ) 6. Let R be a domain, a , b E R not all zero. Up to a unit multiple, define the greatest common divisor of a and b, denoted gcd(a, b), and the least common multiple of a and b (in case a # 0, 6 # 0), denoted lcm[a, b], as in Z (or as in K [ z ]with K a field). (In a similar way, for a l , ..., a, E R, gcd(a1, ..., a,) and lcm(a1, ...,a,) may be defined.) Show that the following statements are equivalent for a domain R in which factorization into irreducible elements is feasible. (a) R is a UFD. (b) Every irreducible element of R is a prime. (c) For every a, b E R, not all zero, gcd(a, b) (or lcd[a, b] in case a # 0, b # 0) exists. (d) The intersection of two principal ideals of R is another principal ideal. 7. Let R be a UFD, f , g E R [ s ] .Use Theorem 2.17 to show that if f,g

1. 2. 3. 4.

Commutative Algebra

18

do not have common irreducible divisors in R[x]then f,g do not have common irreducible divisors in K [ x ]either, where K = Q ( R )is the field of fractions of R. 8. Let p be a prime number. Show that xn - p is irreducible in Z[x]and hence in Q[x]. 9. Prove that f = l l y x 8 3y7x5 9x5 - 7y7 - 21 is irreducible in Z[x,y ] . (Hint: Consider f in Z[y][x].)

+

+

3. Field Extensions The study of field extensions stems from the study of zeros of polynomials and the study of irreducibility of polynomials. Let K be a field and f E K [ x l ,...,x,] a polynomial of degree 2 2. Then the property that f has or does not have a zero in K , and the property that f is reducible or irreducible over K , all depends on the ground field K , for instance, first consider the zeros of x2 - 1, x2 - 2 in Q and the zeros of x2 - 3, x2 1 in R, and then consider the zeros of the given polynomials by extending Q to EX,R to C.A full demonstration of this aspect is given in Chapter 4 and Chapter 5. In this section we focus on several fundamental topics concerning field extensions.

+

Let K , L be fields. If K is a subfield of L (including the case where K Z L is a ring monomorphism), then we call L an extension field of K , and from now on K C L is referred to a field extension. Let K C L be a field extension and S intersection

cL

a subset of L. Consider the

of all subfields in L containing S. Then it is an easy exercise to verify that

(a) K ( S ) is the smallest subfield of L containing S , and (b) K ( S ) = Q ( K [ S ] )the , field of fractions of K [ S ](hence K ( S ) is also the smallest subfield of L containing K [ S ] ) . In view of the above (a)-(b), we call K ( S )the subfield of L generated by S over K . If S = {sl, ..., s,} is finite, then we write K ( S ) = K(s1, ..., s,) and call it a finitely generated extension field of K . If S consists of a single element s, then K ( s ) is called a simple extension field of K .

Preliminaries

19

Splitting field 3.1. Definition Let K be a field, and let f(z)be a polynomial in K [ z ] . If K 2 L is a field extension such that f(x) factors completely into linear factors over L , i.e., f(z)= a n ( . - ai)in L [ z ] ,and f(x) does not factor completely into linear factors over any proper subfield of L containing K , then L is called a splitting field of f(z).

Let K be a field. To see the existence of a splitting field for an arbitrary f(z)E K [ z ] we , start with an irreducible polynomial p ( z ) . Note that the quotient ring

+

then p(x)h(z) where z is the image of z in L , is a field, for, if p ( z ) ,Y$(z) $(z)g(z) = 1 for some h ( z ) , g ( z )E K [ z ] ,and hence $(z) is invertible in L. Note that via the natural ring homomorphism K [ z ]-+ L we may write K c L = K [ T ] .Thus,

K [ z ]c L [ z ]and consequently p ( l ) = 0. It follows from the division algorithm that p ( z ) is factorized in L [ x ]as P ( Z ) = .( - T)Pl(z),P l ( X ) E L M .

Now, since K [ x ]is a UFD, an induction on the degree of polynomials, or a procedure of adding the zeros of each irreducible factor of f(z)successively to the predecessor extension field, yields the following fact. 3.2. Theorem Let K be a field. Every f(z)E K [ z ]with degf(z) = n has a splitting field.

Example (i) The field and z3 x2 3x 3.

+ + +

>0

Q(G) serves as a splitting field for both x2 + 3

Remark Indeed, any splitting field of f(z)is isomorphic to the one constructed before Theorem 3.2. The reader can refer to any textbook specifying field theory for a detailed proof.

20

Commutative Algebra

Repeated zeros and separability Let K be a field and let f(z) E K [ z ] .In view of Theorem 3.2 we may always talk about the zeros of f(z) in some extension field of K . Furthermore, we explore the following Q u e s t i o n When does f(x) have no repeated zeros? 3.3. Proposition f(z) E K [ z ]has no repeated zeros if and only if f(z) and f'(z) = are coprime, i.e., they do not have nonconstant common divisor.

P r o o f Over a splitting field E of f(z),we have

f ( z ) = (z - C

y

p

. . . (z - a,)n"

where the ails are distinct. Then it is clear that f(z)and f'(z) have no nonconstant common divisor over E if and only if ni = 1 for i = 1,..., rn.

0 3.4. Proposition Let E be a splitting field of zn - 1 = f(z) E K [ x ]where , n 2 1. Suppose that charK does not divide n. Then the following hold: (i) f(z)has exactly n distinct zeros (the nth roots of unity over K ) in E. (ii) Let

un = { a E E Then

I

f ( a )= o}.

U, is a cyclic multiplicative subgroup of EX.

P r o o f (i) By the assumption, this follows from Proposition 3.3. (ii) That U, forms a subgroup of E X is clear. We show that U, contains an element of order n. To this end, let n = Plel

.. . p z s

2

be the factorization of n into primes, and let qz = for i = 1,...,s. Then, since the polynomial xQ*- 1 has exactly qz zeros in U,,for each i, there is

at E U, such that a: # 1. Set pt

e,-1

Then p,". # 1 but 'p : = 1. It follows that each pzhas order p:%. Since p ; l , ...,pss are pairwise coprime, . . pS is the desired generator for U,. ,O = =

The last proposition makes the multiplicative structure of a finite field clear.

Preliminaries

21

3.5. Theorem Let K be a finite field. Then the multiplicative group K X of K is cyclic. Proof If charK = p > 0, then [K : Z,] = m for some m and hence K X has n = p" - 1 elements which are all zeros of f(z)= zn - 1 E Z,[zc]. Since p /y n, Proposition 3.4 can be applied to this case. I3 Since K[z] is a UFD, the general discussion may be further reduced to irreducible elements. 3.6. Theorem Let K be a field and let q(z) E K[z] be irreducible. (i) If charK = 0, then q ( z )does not have repeated zeros. (ii) If charK = p > 0, then q(z) has repeated zeros if and only if q ( z ) = g(zP) for some g(z) E K[z].

Proof We apply Proposition 3.3 to both cases. (i) If charK = 0, then since q(z)is irreducible, we have q'(z)# 0 (otherwise p ( z ) would be a constant), degq'(z) < degq(z), and hence q(z) and q'(z) are coprime. (ii) Suppose charK = p > 0. Let q ( z )= a,zn an-1zn-l+ . . . alz a0 with a, # 0. Then q'(z) = na,zn-l (n - l ) a , - l ~ ~ - ~.. a1 with degq'(z) = n - 1 < degq(z) = n. Thus,

+

+

+

+ +

+

q(z) and q'(z) have a nonconstant common divisor H ra, = 0

* plr, say r = s,p.

Consequently, q(z) has repeated zeros if and only if q(z)= atpxtP

+ ... +

azpz2P+apxP+ao.Therefore, q(z)= g(zP) where g(y) = ao+a,y+a2py2+ 0 . . . + a t p y t , as claimed. 3.7. Corollary Let K be a finite field and let q(z)E K[z] be irreducible. Then q ( z ) has no repeated zeros.

Proof Since K is finite, we know that charK = p > 0 for some prime number p . Then Z,is the prime field of K and K is a finite dimensional Z,-vector space, say dimz,K = n. Hence K has pn elements. Thus, the multiplicative group of K , which is K X , has order pn - 1 and XP" = X for all X E K . (We assumed that the reader is familiar with elementary group theory.) It follows that ifg(zp) E K[z],sayg(zP) = a o + a l z P + ~ ~ ~ + a , z R P ,

22

then, after setting a:"-'

Commutative Algebra

= bi,

i = 0,1, ..., n,

which can never be irreducible. This shows that the irreducible g(x) cannot have repeated zeros by Theorem 3.6. cl 3.8. Definition If a polynomial f(x) E K [ x ]has no repeated zeros, then f(x) is called a separable polynomial over K , and otherwise an inseparable polynomial over K . (See also Definition 3.11 below.)

Algebraic extension and primitive elements We now start with a field extension K C_ L and consider CY E L. If there is some f(x) E K [ x ]such that ~ ( c Y = ) 0, then we say that cy is an algebraic element over K ; otherwise, we say that a is a transcendental element over K . If every element of L is algebraic over K , then L is called an algebraic extension field of K , and we refer K 2 L to an algebraic field extension. If L contains a transcendental element over K , then K c L is referred to a transcendental field extension. Let K C L be a field extension. Then L is naturally viewed as a K vector space. In the literature, the dimension dimKL is also called the degree of L over K , denoted [L : K ] . Clearly, if a field extension K C L has finite [ L : K ] ,then L is algebraic over K . For instance, [C : R] = 2. If L contains a transcendental element over K , then [L : K ] = co. It is known that e and 7r are transcendental over Q. So [R : Q] = 00. Another familiar transcendental extension is K c K ( z ) ,where K ( x ) is the field of fractions of the polynomial ring K [ x ] . Also, not every algebraic field extension is finite dimensional (exercise 4).

To understand the structure of a field extension K C L , simple extension plays a key role. Let CY E L. Consider the subring K[cy]C L and the ring

Preliminaries

homomorphism p : K[z]

-

23

K[a]

If a is a transcendental element over K, then Kerp = (0) and K[z]

K[a].

If a is algebraic over K , then kerp # (0) and hence Kerp = ( p ( z ) )for some nonconstant p ( z ) E K[x] because K [ z ]is a PID. We may assume that p ( z ) is monic. It is a consequence of the division algorithm in K[z] that p ( z ) has the smallest positive degree among all polynomials in Kerp. This leads to the following 3.9. Definition For an algebraic element a over K, the monic polynomial p ( z ) , which is the generator of kerp, is called the minimal polynomial of a over K. 3.10. Theorem Let K C L be a field extension and a E L. If a is algebraic over K and p ( z ) is its minimal polynomial, the following hold: (i) p ( z ) is irreducible and unique in K[z]. (ii) K[rc]/(p(z)) K [ a ]is a field containing K . Thus, k [ a ]= K ( a ) . (iii) If degp(z) = n,then every element p E t ( a )has a unique expression

+

= Xn-1an-’ X n - 1 ~ n - 2

+ . . + Xla + Xo,

X i E K.

Thus, {an-’,...,a, 1) forms a K-basis for K ( a ) , [K(a) : K] = n. Consequently, K(a) is a simple algebraic extension field of K .

Proof Using division algorithm by p ( z ) in K[z], all conclusions are easy exercises. 0 Later in exercise 2 the reader will be asked to show that if a1, ..., a, E L are finitely many algebraic elements over K, then K C K(a1, ...,a,) is an algebraic field extension and [ F : K ] < 00. When K plays the role as in the case of Theorem 3.6(i) and Corollary 3.7, our next goal is to show that the finitely generated algebraic field extension K(a1, ..., a,) is actually a simple extension. But first, we need the notion of a separable extension. 3.11. Definition (i) Let K

CL

be a field extension and let a E L be an

Commutative Algebra

24

algebraic element over K . If the minimal polynomial p ( z ) of a over K is separable in the sense of Definition 3.8, then a is said to be separable over K ; otherwise a is inseparable over K . (ii) Let K L be an algebraic field extension. If every element of L is separable over K , then L is said to be separable over K ; otherwise L is inseparable over K . By Theorem 3.6 and Corollary 3.7, inseparable field extension is, indeed, quite rare. 3.12. Theorem Let K C F = K(a1,a2, ...,a,) be a finitely generated algebraic field extension. Suppose that 0 2 , ...,am are separable over K . Then F = K ( 6 ) for some 6 E F .

Proof If K is finite then so is F (by Exercise 2), and the conclusion follows from Theorem 3.5. Suppose that K is infinite. We consider only the case where F = K ( a ,p) with p separable over K since the general conclusion may be obtained by an induction. Let L be a field over which the minimal polynomial p ( z ) of a and the minimal polynomial q ( z ) of P are factorized as n

m

i= 1

j=1

where al,a2 ,..., an,,&,...,Pm E L , and a1 = a, p1 = 0. (The existence of L is guaranteed by Theorem 3.2.) By the assumption, PI, ...,Dm are distinct. Thus, the equations

ai - a1 = &(PI have only finitely many solutions that

ai - a1

+

Xik

# c(p1 - Pk),

-Pk),

k # 1,

E K . Hence, there exists c E K such

1 5 2 5 n, 2

5 k 5 m.

Set 6 = a cP. Then clearly K ( 6 ) C F . Below we show that ,B E K(1Y) and then it follows that F = K ( 6 ) . Note that a = 6 - cP. We have p ( 6 - cp) = p ( a ) = 0. Consider the polynomial r ( z ) = p ( 6 - a ) E K ( d ) [ z ] .Then, by the choice of c, ,f?is the only common zero of q(z) and ~ ( x in ) F . This shows that the

Preliminaries

25

minimal polynomial of p in K(t9)[x]is of the form t - p for some p E K(t9). Therefore, p = p E K(t9) as expected. 3.13. Definition The element 6 that appears in Theorem 3.12 is called a primitive element of F .

Let F be a field. If every nonconstant polynomial f(z)E F [ x ]splits in F , i.e., f(z)= A(z - Ai), A, X i E F , then F is said to be algebraically closed. Clearly, if F is algebraically closed, then there is no proper algebraic extension of F . For instance, the field C of complex numbers is algebraically closed (this is also known as the content of the fundamental theorem of algebra). Without proof we mention the following theorem (the reader is referred to any textbook specializing field theory for the classical proof given by Emil Artin).

ny=l

Theorem Let K be a field. Then there is an extension field L of K that is algebraically closed. Liiroth's theorem Within the context of Theorem 3.6(i), Corollary 3.7 and Theorem 3.12, it is easy t o see that if K C L = K(t9) is a simple field extension, then F any intermediate field extension F of K with K L is a simple extension. The final part of this section deals with a similar situation on simple transcendental field extension. Let K be a field and x a transcendental element over K . Given coprime polynomials u ( x ) , v ( x )E K [ x ] ,consider h = - E K ( z ) and the simple extension K c K ( h ) . Set

where K ( h ) [ t ]is the polynomial ring in t over K ( h ) . 3.14. Lemma With notation as above, the following hold: (i) h is transcendental over K . (ii) q ( t ) is irreducible in K ( h ) [ t ] . (iii) [ K ( z ): K ( h ) ]= degq(t) = max{degu(rc), degv(x)}.

Commutative Algebra

26

Proof (i) Exercise. (ii) Note that q ( t ) is linear with respect to h in the polynomial ring K [ h ,t ] which is a UFD. Hence q ( t ) is irreducible in K [ h ,t ] ,for u(z) and ~ ( zare ) coprime by the assumption. It follows from Proposition 2.18 that q ( t ) is irreducible in K ( h ) [ t ] . (iii) By the construction of q ( t ) , q(z) = 0. It follows from part (ii) that q ( t ) (assuming monic) is the minimal polynomial of z over K ( h ) . Thus, I3 [ K ( z ): K ( h ) ]= degq(t) = max{degu(s), degv(z)}, as desired. 3.15. Corollary (i) Let E be any intermediate extension field of K with K E C K ( z ) . Then [ K ( z ): El < co. (ii) Every automorphism of the ring K ( z ) which is K-linear is given by

Proof Exercise.

0

3.16. Theorem (Luroth) Let K be a field and z a transcendental element over K . Let E be an intermediate extension field of K with K $ E C K ( z ) . Then E = K(y) for some y E K ( z ) (hence E E K ( z ) )and [ K ( z ): El < 00.

Proof By Corollary 3.15, [ K ( z ): E ] < co. Let p ( t ) E E[t]be the minimal polynomial of z over El say

If we multiply p ( t ) by the least common multiple, say s, of the denominators of ri’s, the obtained polynomial

is primitive in K [ z ] [ twith ] respect to t (check it!). Write degtf(z,t) for the degree of f ( z ,t ) in t. Then

n = deg,f(z,t) Note that s, = s and all

= degp(t) =

2 E E. As

#

E E - K . Set h = at least one and u(z)are coprime in K [ z ] .Then

[ K ( z ): El.

is transcendental over K , there is =

$ for convenience, where u ( z )

q ( t ) = hv(t)- u ( t )

Preliminaries

27

is irreducible in K ( h ) [ t ]and

[ K ( z ): K ( h ) ]= degq(t) = max{degu(z), degv(z)}

(2)

by Lemma 3.14. Since K having the equality

2 K ( h ) & E C K ( z ) ,we complete the proof by

[ K ( z ): El

=

[ K ( z ): K ( h ) ] .

To this end, note that q(z) = 0 and q ( t ) E E[t]. Hence q ( t ) = p ( t ) p ~ ( t ) with p l ( t ) E E [ t ] for , p ( t ) is the minimal polynomial of z over E . Thus, by formula (I),

But f ( z ,t ) is primitive in K [ z ] [ twith ] respect to t. It follows from Theorem 2.17( ii) that

u ( z ) v ( t )- v ( ~ ) u ( = t )df(z,t), d E K [ z ] [ t ] .

(3)

Suppose deg5f(z,t) = m. Then max{degu(z), degv(z)} 5 m by formula (1). So the above formula ( 3 ) implies that deg,(u(z)v(t) - v ( z ) u ( t ) = ) m

(4)

and d is a constant. Note that u ( z ) v ( t )- w(z)u(t)is antisymmetric in z and t. Therefore, (2) (4) yields

+

[ K ( z ): K ( h ) ]= max{degu(z), degv(z)} = m

=n =

as desired.

[ K ( z ): El,

Commutative Algebra

28

More results concerning field extensions are given in section 5 and (Chapter 3 Theorems 1.8 and 2.4).

Exercises L be a field extension and a1, ...,a, E L. Show that if a1, ..., a, are algebraic over K , then K 2 F = K(cr1,...,am) is an algebraic field extension and [F : K ] is finite. (Hint: Note that F = K ( a l ) ( a 2 ) . (a,) and use Theorem 3.10(iii).) Let K L E be a tower of algebraic field extension, i.e., L is algebraic over K and E is algebraic over L. Use exercise 1 to show that E is also algebraic over K . Moreover, show that if [ L : K ] < co and [E : L] < 00, then [ E : K ] = [ L : K ] [ E: L ] . (Hint: If a E E and Anan + .. . + Xla + XO = 0 for X i E L , then consider K C: K(X,, ..., XO) C K(X,, ...,X o ) ( a ) . ) Use Theorem 3.10(iii) to show that if K C: L is a field extension, then all elements of L which are algebraic over K form a subfield of L containing K . 2 is called the algebraic closure of K in L. (Hint: For a l p E L , algebraic over K , consider K & K [ a ] k [ a ] [ p ] . ) Let F be the subfield of C consisting of all algebraic elements over Q. Use (section 2, exercise 8) to show that [F : Q]= co. Show that if K is an algebraically closed field, then K is infinite (or

1. Let K

+

2.

3.

4.

5.

equivalently, that a finite field cannot be algebraically closed). (Hint: If F = {ul, ..., u n } is a finite field, consider the polynomial p ( x ) = ny="=,x- ui) 1 in F [ x ] . ) 6. Let d E Z be square-free. Then every element a E is of the form a = r s d , where r, s E Q. Show that a has the minimal polynomial

+

a(&)

+

pa(.)

= x2 - 2 r x

+ (r2- s2d).

Q(a,

7. Let F = 8). Find a primitive element for F . 8. Prove Lemma 3.14(i). 9. Complete the proof of Corollary 3.15.

4. Symmetric Polynomials Let R be a ring and R [ x l ,...,x,] the polynomial ring in variables 51, ..., x , over R. P u t

N" = { a = (a1,..., a,)

I

>

ai E N .

Preliminaries

Then every element f(z1,...,2,) E R [ q ,..., z],

29

has a unique expression

Let S, denote the permutation group of {1,2, ..., n}. A polynomial f ( s l..., , 2,) E R[zl,..., z], is said to be symmetric if

+ +

+ + +

For example, 5: 5; xi, ( 2 1 2 2 5 3 ~ ) ( ~ 1 1 ~ 2 ~ 3 a ) ~ . Important symmetric polynomials are those elementary symmetric polynomials:

Let R[sl,..., s,] be the subring of R [ x l ,..., xn] generated by R and ,...,s,}. Then i t is clear that every g ( s 1 , s 2 ,..., s,) E R[sl,..., ]s, is a symmetric polynomial. The next theorem, due to Newton, shows that the converse is also true. {s1,s2

Commutative Algebra

30

4.1. Theorem If f = f(z1,..., 2 , ) is a symmetric polynomial in R [ z l ,...,z,], then f(21,...,2n)E R[si,..., sn].

c,

Proof To reduce f = f(sl,...,2,) = c,z~' . . . zEn into a polynomial in elementary symmetric polynomials, in view of previous (*) we order the set of monomials

I

{2yiZ;z..*2,:-

(a1,...,a,)

ENn

by the lexicographic ordering:

. . .2:n

2 ~ 1

+lez

271

. . . 25

if and only if a1 = P1

,

a2 =

Pz, ..., a,-1

= a3-1 while

a, < p, for some s 5 n.

Thus, the terms of f are ordered lexicographically (note that +lez is a total ordering), and we may assume that the leading monomial of f is xy'x;' . . . x;n. Since f is symmetric, zztl)zzi2,. . . x;;,, occurs in f for every 7r E S,. It follows that the leading monomial of f has the property that 011 2 a2 2 2 a,. For example, the leading monomial of ki k 2 . . $1 s 2

.sk

-

(21

+ .. . + 2 , ) k l

. . . ( 2 1 . ..2 , ) k n

is ++"'

By choosing kl = a1 -aZ, ..., k,-1 = a,-1 -a,, k, = a,, we can make this the same as the leading monomial o f f . Suppose that the leading coefficient of f is c, then f - c s f ' s ~. . . s? has a lexicographic leading term

d$ x p . . .

xk ,

P1 2 P 2 2 ... 2 Pn

which comes after c ~ y ' .x. .~ x:n~ in the ordering. Since only a finite number of monomials x ~ ' . x. . 22 ~ in f satisfying y1 2 7 2 2 . . . 2 7, follow ~ ~ '. .I. xgn C lexicographically, ; ~ a finite number of repetitions of the above 0 process reduce f to a polynomial in s1, ..., s, .

Example (i) The symmetric polynomial

Preliminaries

31

is written lexicographically. And by the method given in the proof we may derive that f = s l s 2 + 3 s 3 . Similarly, ( I C 1 + 2 2 ) ( z l + x 3 ) ( 2 2 + 2 3 ) = s 1 s 2 - s 3 . An application of symmetric polynomials to field extension is given as follows. If K C L is a field extension, a,xn a , - ~ x " - ~ . . ao = f(x) E K [ x ] with degf(z) = n, and f ( ~ i = ) 0 with T I , ...,T~ E L , then, f(x) factors in L [ x ]as

+

f(x) = a,(z

- T1)(Z - T Z ) .

+

= an(xn

where ci = ( - ~ ) Z S ~ ( T ~ , T ~ , of both sides, we have

+. +

.,.,T " ) , i

C2xn-2

+ . . . + cn)

= 1,..., n. After comparing coefficients

...,T,)

(-1)iUnSi(T1,T2,

+

c1xn-l

(x - r,)

= a,-< E

K,

2

= 1,...,n.

+

+

Corollary Let K C L be a field extension, a,xn an-lxn-l E K [ x ] with degf = n, and f ( ~ i )= 0 with T I , ..., T, E L. If h ( x 1 , ...,2,) E K [ q , ...,x,] is a symmetric polynomial, then h ( r l , ~ 2 ..., , T,) E K , i.e., { T I , ...,T,} defines a function 4.2.

...

+ ao = f

K[s1,

Example (ii) Suppose that T I , in C. Find T: T ; T: and T:

+ +

-

..., s,]

7-2, 7-3

are the zeros of f(z)= s 3 + x 2- x + 1

+ + T;

K

T:.

Solution Since f(x) = (x - T ~ ) ( z - ~ 2 ) ( x- ~ 7-1

+ + 7.2

T3

3 ) it ,

= -1,

follows that

Commutative Algebra

32

More generally, the following recurrence relations, called Newton's for. . mulas, can be used to establish formulas for pi = (-1)%(z2, xi . . . zk), i 2 1, in terms of s1,s2, ...,sn.

+ + +

+ s1 = 0, Pz + SlPl + 2sz = 0, Pl

... Pn

+ SlPn-1~2pn-2 + . . . + sn-lpi+

TLS,

= 0.

We close with an application to polynomial building,

Example (iii) Let r1,rz,r3 be the zeros of f(x) = x3 - z the polynomial g(x) that has zeros rf ,r;, rg .

+ 2 in C. Find

Solution Suppose the desired polynomial is of the form g(z) = x3 + Ax2 + Ba: + C. Then

A=-(rf+r; +rg) = - ~ 2 ( r l , r 2 , ~ 3 ) = -s1(r1,r2,r3)2 =0

+ 2(-1)

+ 2sz(r1,r2,r3)

= -2,

= (-1)2 - 2(0)(-2) = 1,

C = - T ~ T ; T ~= - ~ 3 ( ~ 1~ ,2 ~ , Hence g(z)= x3 - 2rc2

3 =)-4. ~

+ z - 4.

Exercises 1. Express the product (xf x;)(z? zi)(zz xg) in terms of sl,s 2 , s3. 2. Let T1,TZlT3be the zeros of f ( z ) = x3 - 6z 11 - 6 in C.Determine the polynomial g(a:) that has zeros r: r;, T: r i , r; T : . 3. Let T I ,r2,r3,7-4 be the zeros of f ( z ) = a d z 4 a3x3 1322' alx a0 in

+

+

+ + + + + + + +

+

Preliminaries

33

@, where ai E Q. Suppose a4 = -5 and the elementary polynomials in 3 1 T I , T Z , T ~ are , ~ ~s1 = %, s 2 = 16, s3 = -8, s4 = -E. Find a3,a2,a1,ao. 4. Let R be a ring. A polynomial belonging to R[xl,..,xn] is said t o be antisymmetric if it is invariant under even permutations of the variables, but changes sign under odd permutations. Let

i<j

Show that (a) A is antisymmetric, and (b) if 2r = 0 implies T = 0 for r E R, then any antisymmetric polynomial f is expressible as a polynomial in the elementary symmetric polynomials, together with A. (Hint: Note that f(x11x21x3, ...,xn) = -f(x2,xl,x3,

...,xn)r

2 f ( x l , x l , x 3 , . . . , % )=

0. Thus, f vanishes when 21 = 5 2 . So a division on f by x1 - x2 in R[Q, ...,xn][xl] yields (21 -xz)lf. Similarly, (z1-xi)lf, i = 3, ..., n. Writing f = niZ1(x1- xi)fi, where f l E R[x2,...,x,] and is antisymmetric. Now an induction on n finishes the proof.)

5 . Trace and Norm

s

Throughout this section we let K L be a simple algebraic field extension, that is, L = K ( 6 ) , 6 E L. If p(x) E K [ x ]is the minimal polynomial of 6 over K, we may set a tower of field extensions

K c L c E such that E contains all distinct zeros of p ( x ) ,say dl = d , 8 2 , ...,6,, where m 5 n = [ L : K] = degp(x), that is, E contains the splitting field of p(x). 5.1. Proposition With notation as above, there are exactly m distinct ring monomorphisms L -+ E that are K-linear. Moreover each K-linear monomorphism L --f E is given by 6 4 6i, 1 5 i 5 m.

Proof If a: L t E is a monomorphism as described, then 0 = a ( p ( 6 ) )= p ( a ( d ) ) , i.e., ~ ( 1 9 )is a zero of p(x). Note that the elements of L are of the form EX,@. It follows that if L Z E and L"2.E are two K-linear ring

Commutative Algebra

34

monomorphisms such that a1(6)= a2(6),then 01 = 02. Conversely, each 6i defines a desired monomorphism

ai:

L CXjSj

H

E cxj79;

because all t9i's have the same minimal polynomial p ( z ) .

0

With the help of Proposition 5.1 we may determine, for every a E L , the minimal polynomial pa(.) of a over K and the splitting field of p a ( % ) . To see this, let a1, ...,em be all distinct monomorphisms L 4 E defined by ai(6) = t9i, a = 1,..., rn. Suppose that each 6i has multiplicity ei 2 1, that is, p ( z ) = - 6i)ei in E [ z ] .Then

nzl(z

el

+ e2 + . . . + em = n = degp(s),

and each a E L = K ( 8 ) is associated to a monic polynomial in E [ z ]that , is,

For convenience, we call fa(z)the total polynomial of a. 5.2. Proposition Let K K ( 6 ) ,the following hold:

L = K ( 6 ) c E be as above. For any a E L =

( 9 f a ( z ) E Wzl. (ii) Let pa(.) E K [ z ]be the minimal polynomial of a over K . Then fa(z)= pa(z)' for some s 2 1. (iii) E contains the splitting field of the minimal polynomial pa(.) of a over K .

Proof (i) Since a = ~ ( 2 9 ) = Cyi: A@, where we have

i=1

T(X)

=

CyL;

Xixi E K [ z ] ,

Preliminaries

35

Note that all X i E K . After expanding the latter product we see that the coefficients of fa(z)are given by symmetric polynomials in the n zeros of p ( z ) . By Corollary 4.2, fa(z)E K [ z ] . (ii) By part (i), fa(.) E K [ z ]and f a ( a )= 0. It follows that fa(z)= p a ( s ) s h ( z )where , h ( z ) E K [ z ]and p a ( z ) , h ( z ) are coprime and both are monic. If h ( z ) is not a constant, then some C ~ ( C Yis) a zero of h ( z ) . Let CY = r ( 6 ) = Eyz: Xi62 with r ( z ) = CyI; Xizz E K [ z ] .Then ai(a) = r(6i) and h(ai(a)) = h(r(6i))= 0. Set g(z) = h ( r ( z ) )E K [ z ] .Then g(Si) = h(r(6i))= 0 implies p(z)lg(z),for p ( z ) is the minimal polynomial of 6 and hence the minimal polynomial of each 6i. It follows that 0 = g(6) = h ( r ( 6 ) )= h ( a ) and p a ( z ) l h ( z )a, contradiction. This shows that h ( z ) is a constant and h(s)= 1 because it is monic. Thus, f a ( z )= p a ( z ) ' . (iii) By parts (i) and (ii), fa(.) E K [ z ]and fa(z)= -ai(~~= ))"~ p , ( ~ for ) ~ some s 2 1. So pa(.) factors into linear divisors over E.

nEl(z

We now introduce two functions on L that will play important roles in Chapter 3 section 3 and throughout Chapter 4. Let Q E L = K ( 6 ) . By Proposition 5.2(i), the total polynomial f a ( z ) of CY belongs to K [ z ] .By the definition, fa(z) = - o ~ ( c Y )where )"~, el + .. . em = n = [ L : K ] = degp(z). If we check the expanded = zn &-1zn-' ... c1z co with expression of fa(z),then fa(.) m eiai(a) and co = o ~ ( Q ) ~Thus, + . we have obc++l = tained two well-defined functions:

nEl(z

+

+

nzl

TLIK: L

+ +

+

K

+ m.

i= 1

i= I

5.3. Definition For Q E L , TL/K(Q)is called the trace of NL,K(Q) is called the norm of CY in K . 5.4. Proposition For

Q,P E L , X,p E K , the following hold:

+

(i) TL/K(XCY + PP) = XTL/K(CY)/-LTL/K(P).

Q

in K and

Commutative Algebra

36

Knowledge on bilinear forms needed by the next theorem and Chapter 3 Theorem 3.2 is given as an appendix a t the end of this section. Viewing L as an n-dimensional K-vector space, Proposition 5.4 enables us t o define a symmetric bilinear form on L:

LxL-

K TL/K(QP)

5.5. Theorem For K C L = K(6) with [L : K] = n, if the minimal polynomial p ( z ) of 6 over K has n distinct zeros 61 = 6,192,..., 8,, then the bilinear form defined above is nondegenerate.

I=[ I*

Proof Set r ( k ) = (6!,..., d i ) , Ic = 0, ..., n - 1, and write V for the Vandermonde matrix

[ ;;;; r(0)

V=

r ( n - 1)

1

1

...

1

:; :; ::: ::

6:-'

q - 1

... . . . 6;-1

Let ~ 1 ..., , on be all the n distinct K-linear monomorphisms from L t o E as described in Proposition 5.1 such that ai(6) = 6i,i = l,...,n. If we of L, then since consider the standard K-basis {1,6, ...,

c

cs p t

i= 1

i=l

n

T,/,(6k@)

=

Ci(6k6j)=

n

= d(Ic)(6(j))t,

the matrix of the bilinear form is given by ( T L / K ( d W ) ) = VVt

and hence det ( T ~ / ~ ( s ~= d j(det(V))2. )) Since all the 6i are distinct, det(V) = n,,,(6i - 6j) # 0. This shows that 0 the bilinear form is nondegenerate.

Preliminaries

37

5.6. Corollary If K & L is a finite dimensional separable field extension, then Theorem 5.5 hold.

cl Appendix. Bilinear forms

Let U and V be two vector spaces over a field K , U x V = { ( u , v ) [ u E U , v E V ) the Cartesian product of U and V . A bilinear form on U x V is a mapping

< , >: U X V (u,v)

++

K

< u,v >

satisfying

< xu1 + p u 2 , v >

=

x < u1,v > 3-p < U 2 , V >

< u,xv1 + p v 2 > = x < u,v1 > -I-p < u,v2 > for all u1,u2 E U , 211,212 E V and A,p E K . A bilinear form < , > on V x V is called a bilinear form o n V . A bilinear form < , > on V is said t o be symmetric if < x,y >=< y , x > for all z,y E V. If < , > is a bilinear form on U x V which satisfies

< u,v’ >= 0 for all u E U implies v’ = 0, and

< u’,v >= 0 for all ‘u E V implies u’ then < ,

= 0,

> is called a nondegenerate bilinear form.

Let U and V be finite dimensional K-spaces. Given a basis { u l ,..., um} of U and a basis {VI, ..,vn} of V , if < , > is any bilinear form on U x V , then there is an associated m x n matrix A = ( a i j ) with aij

=< ui, vj >, i = 1, ..., m, j

= 1,...,n,

and < , > is completely determined by A , that is, given

i= 1

j=1

38

Commutative Algebra

itf follows from the bilinear property of

< , > that

< U,Y > = < c x i u i , c p j v j >

Conversely, any m x n matrix over K yields a bilinear form on U x V in this way. Now, suppose that {ui, ...,u&} and {Y;, ..., v i } are new bases for U and V respectively, and that m.

n

i=1

j=1

Then it follows from a change of bases and (I) that

(Al, ...,) ,A (PI, ' " 7 p n )

= (A;,

..., A&)P,

(pi, . . . t ~ i ) Q

1

for some invertible matrices P = PmX,, Q = Qnxn. Thus,

< u , Y >= (A;, ..., A ~ ) P A Q ~ and consequently, the matrix referred to the new bases is PAQt. 5.7. Theorem (i) Let U and V be finite dimensional vector spaces over a field K , where dimKU = m and dimKV = n. If < , > is any nondegenerate bilinear form on U x V, then m = n, and for any basis {ul, ..., un}of U there exists a unique basis {vl, ..., vn} of V such that

< U i , Y j >= 6 . . -

0, if i # j , 1,if i = j.

Preliminaries

39

(ii) A bilinear form < , > on a finite n-dimensional K-space V is nondegenerate if and only if the associated matrix A = ( a i j ) , where aij =< Vi,v j >, is invertible for any basis {q, ...,v,} of V.

Proof (i) Let { u ~..., ,u,} be any basis of U . Consider the linear mapping induced by < , > 0

:

V

-

Km = {(Ail ...,)A,

I X i E K}

Then 0 is injective because < , > is nondegenerate. Thus, n = dimKV 5 dimKU = m. Similarly we also have m 5 n. Hence m = n. Note that 0 is now an isomorphism. If we use the standard basis {el, ..., e m } of Km, where ej

= (0,...,0,1,0,...,0), j = l,...,m,

+ j-1

and write vj for the inverse image of ej under 0,then it is clear that { w ~ , . . . , ~ ~ is} a basis for V and < u i l v j >= S i j , i , j = l,...,m. If {w;,...,&} is another basis of V with this property, then a(vi - v:) = 0 implies zli = v:, i = 1,...,m, because a is injective. (ii) This follows from part (i) and previous discussion on the associated matrices of < , > with respect to given bases. Exercises 1. Complete the proof of Proposition 5.4. 2. Let d E Z be square-free, K = Q(&). Find all Q-linear ring monomorphisms K 4 C. 3. Let K = be as in exercise 2 above, a = r s&. Show that T K , Q ( ~ ) = 2 r and N K I Q ( ~ = ) r2 - s2d. (Compare with section 3, exercise 6.) 4. Let 6 = 4 K = Q(6). Find the minimal polynomial p ( z ) of 6. (Hint: Note that ()(A), c K . Any Q-linear ring monomorphism u: K C induces a Q-linear ring monomorphism GI: Q1 and a Q-linear ring monomorphism 0 2 : Q1, such that ~ ( 6=) a l ( 4 ) 02(&). The answer is p ( z ) = x4 - lox2 1.) Can you generalize this result to 6 = fi 4 for arbitrary squarefree p # q?

+

a(&)

+ a,

a(&)

Q(a) --f

-+

+

a(&)

-+

+

+

Commutative Algebra

40

6. Free Abelian Groups of Finite Rank

+

Let G be an abelian group with the binary additive operation and the identity element 0. For g E G , we write Z g for the cyclic subgroup of G generated by g , and consequently, we write C g E n Z g for the subgroup of G generated by a nonempty subset R C G. A subset S2 = {gi}iE J of G is said to be Z-linearly independent if for any finitely many gill gi2, ...,gin E 0, there do not exist s1, ..., sn E Z, not all zero, such that slgil s2gi2 . . . sngin = 0. If R is not Zlinearly independent, then it is Zlinearly dependent.

+

+ +

6.1. Definition Let = {gi}iGJ C G. If G = C g i E n Z g i and R is Zlinearly independent, then G is called a free abelian group and $2 is called a Zbasis of G, or just a basis of G.

Below we focus on free abelian groups with finite Zbasis. 6.2. Proposition If a free abelian group G has two bases (91, ...,gn} and { h l , ...,hm},then m = n.

Proof Suppose m < n. Then, as dealing with vector bases over a field in linear algebra, after expressing each gi as a Zlinear combination of hi's, we may derive, by passing to Q, that (91, ...,gn} is Zlinearly dependent, a contradiction. Hence, m 2 n. By symmetry, n 2 m. Thus, m = n as desired. 0 6.3. Definition An abelian group G with a basis of n elements is called a free abelian group of Z-rank n , or just a free abelian group of rank n.

Z is a free abelian group of rank 1 and { 1) is a basis for Z. The direct sum

of n copies of Z, where

( k l , .., kn)

+ ( e l ,...,en) = ( k l + e,, ..., k, + en>,

m ( k 1 , ..., kn) = ( m k l , ..., mkn), m E Z,

Preliminaries

,..., O , I , O ,..., 0 )

ei=(O

41

i = l , ..., n

6.4. Proposition Any finitely generated abelian group G = Cy=lZgi is a homomorphic image of some free abelian group of rank n. If G is free and (91, ..., g n } is a basis of G, then G E Z".

Proof Exercise. In view of Proposition 6.4, from now on we write n i= 1

for the free abelian group G with basis (gl, ...,gn}. Let M,(Z) be the set of all n x n matrices over Z. If A E M n ( Z ) and det(A) = f l , then we say that A is unimodular. If A E M n ( Z ) is unimodular, then A is invertible and

A-l = -A* =&A* det(A) where A* is the adjoint matrix of A. Clearly, the construction of A* implies A* E M,(Z). It follows that A-' E M,(Z). 6.5. Lemma If { u l ,.,.,u,} is a basis for the free abelian group G, then (~11,..., vn}, where n

vi = C a i j u j , aij E Z,i = 1,..., n, j

is a basis for G if and only if A

= ( a i j ) is

unimodular.

Proof With the help of the above remark, this is an easy exercise. 6.6. Theorem Let G be a free abelian group of rank n and H a nonzero subgroup of G. Then the following hold: (i) H is free of rank s I n.

Commutative Algebra

42

(ii) There exist a basis (91, ...,gn} for G and integers that { e l g l , ...,e,gs} is a basis for H .

[I,

..., ls E Zfsuch

Proof We prove the theorem by induction on the rank of G. If rankG = 1, then G 2 Z and the conclusions (i) and (ii) are clear. Suppose the assertions (i) and (ii) are true for any free abelian group of rank < n. Let G be a free abelian group of rank n and let H be a nonzero subgroup of G. Then, with respect to a fixed basis {el, ...,en} of G, H contains elements (1)

h

= klel

+ . . . + knen, some ki's are positive.

Choose a basis { X I , ...,x n } of G such that I! is the smallest positive coefficient with respect to (l),and rearrange the members of this basis (if necessary) so that H contains an element of the form

On division by

el,

write

If we define

92 = 5 2

then

[f] Sn

=

[;:::: ;][:) Xn

0 0 ...

where the square matrix is clearly unimodular. By Lemma 6.5, (91,x2, ..., x,}

43

Preliminaries

is a basis of G. With respect to this new basis and previous formula (2), f l = C l x l + 7712x2

= C l g l i 7-252

+ . . . + mnx,

+ . . . + r,x,.

By the choice of C l , we must have

(3)

r2

= . . . = r, = 0. It follows that

fl = a m .

Now, with respect t o the new basis expression

h =clgl+ Write c1 = C 1 q contains

{g1,22,

+ . . . + GX,,

~2x2

+ r with q,r E Z, 0 5 r
Iand PnS=0. (ii) Every prime ideal contains a nonzero minimal prime ideal (with respect t o G on SpecR).

Proof (i) Consider the partially ordered set R = {ideals J with J 2 I and J n S = 0) with the inclusion ordering 2. By Zorn’s lemma, R contains a maximal element P. We claim that P E SpecR. For if a , b E R, a, b # P , then (bR+ P ) n S # 0,

( a R + P ) n S # 0,

+

+

because P is properly contained in both aR P and bR P. Thus, there a r e a s + p l E ( a R + P ) n S , by+p2 E ( b R + P ) n S , and (as+pl)(by+p2) = p’+abxy E S since S is a multiplicative set, where p’ E P. This shows that ab # P by the choice of P. (ii) Exercise. Let R be a ring and a E R. If an = 0 for some n 2 1, then a is called a nilpotent element. Set

I

r ( ~=) { a E R a is nilpotent

1

and call r(R) the nilradical of R. R is said to be reduced if r ( R ) = ( 0 )

1.7. Theorem (i) r(R) = npESpecRP. Consequently r ( R ) is an ideal of R.

Local Rings, DVRs, and Localization

57

(ii) r ( R ) = nQwhere Q runs over all nonzero minimal prime ideals of R.

Proof (i) The inclusion r(R) C nP is clear. The inclusion 2 follows from the fact that if a E R is not nilpotent then there is some prime P not containing a. To see this, consider the multiplicative set S = (1, a,a2,...,an, ...}. Then 0 @ S , for a is not nilpotent. By Proposition 1.6(i) (taking I = {0}), there is a prime ideal P with P n S = 8, as desired. (ii) This follows from the proof of part (i) and Proposition 1.6(ii). 0 For a ring R , we also set the intersection

n

J(R)= ME

M

m-SpecR

and call J ( R ) the Jacobson radical of R. 1.8. Theorem With notation as above,

J(R)

=

{r E R

I

1 - y r E U(R) for a l l y E R .

}

Proof By Proposition 1.5(i), 1 - y r is a unit if and only if 1 - y r is not contained in any maximal ideal of R. Now the assertion is clear. 1.9. Theorem Let R be a Noetherian ring and J ( R ) its Jacobson radical. If I is an ideal of R and I C_ J ( R ) , then np==,InM= (0) holds for any finitely generated R-module M .

Proof This follows from Chapter 1 (section 7, Theorem 7.7 and exercise 6). Exercises 1. Complete the proof of Proposition 1.6(ii). (Hint: Apply Zorn’s lemma to the prime ideals contained in a prime ideal P by defining P2 < PI if Pl c P2.) 2. Let R be a PID. Show that SpecR = m-SpecR. 3. Let R be a UFD. (a) Show that every minimal nonzero prime ideal of R is principal. (b) Show that, without counting associates, there is a one-to-one and onto correspondence between prime elements of R and minimal nonzero prime ideals of R.

Commutative Algebra

58

4. Let R be a ring and I an ideal of R. Define

fi= {a E R and call

4 the radical of I

I

an E I for s o m e n

21

1

in R. Show that

n

J7=

p.

PEspeCR, P > I

5. Let I and J be ideals of a ring R. Show that = fi n 6. 6. Let R be a ring. Two ideals I , J of R are said to be comaximal if I J = R (for instance, if one of them is maximal). (a) Show that if I and J are comaximal then I . J = I n J , I J 2 = R, and I m J” = R for all integers m, n 2 1. (b) If I1, ..., I N are ideals of R, and Ii and Ji = n j f i I j are comaximal for a l l i = l,...,N, show that I p n . . . n I ; = ( I l . . . I N ) n= ( I I n . . . n IN)^ for all integers n 2 1. 7. Show that in Z[&] the ideal P = ( 2 , &) is a maximal ideal but not a principal ideal. (Hint: Z [ & ] / P E Z2) 8. For f = 1 x2,g = y2(x x3) (y - 1)x2 y 2 in the polynomial ring R[x,y], is the ideal (f,g)a prime ideal or a maximal ideal?

+

+

+

+

+

+

+ +

2. Local Rings and DVRs 2.1. Definition Let R be a ring. If R has only one maximal ideal, then R is called a local ring.

By definition, all fields are local rings, for (0) is the only maximal ideal in these rings. 2.2. Theorem The following statements are equivalent for a ring R. (i) R is a local ring. (ii) All nonunits of R form an ideal.

Proof This follows from Proposition 1.5.

I?

Example The following examples, which are from different aspects of mathematics, may perhaps help to qualify the name “local ring”.

59

Local Rings, D V R s , and Localization

(i) (Foundation of modern scheme theory) Let V be the real line (or a topological space, or a differentiable manifold), and P E V a point. Consider the set E of real-valued continuous functions (differentiable functions in case V is a differentiable manifold) on some open interval (open neighborhood) around P. Then, two functions f ,g E E may be “locally” identified by the equivalence relation: f g if and only if they agree on some open neighborhood of P. The quotient set E p = ( E x E ) / forms a ring with the addition and multiplication induced by that on functions. Elements of E p are called function germs a t P. It is easy to see that the subset

-

{

mp = f

-

~

I

f (~p ) =p ~ }

forms an ideal of E p . If g @ m, then there is an open neighborhood U of P on which g is nonzero. So h = is defined on U and gh = 1 on U . By Theorem 2.2, E p is a local ring. (ii) Let R[z] be the polynomial ring in z over R, and R(x) its field of

fractions. For any a E R,

is a local ring with the unique maximal ideal ma = { fo E R(z) I g ( a ) # 9(X) 0, f ( a ) = 0). R [ z ] ,is called the local ring of the point a, due to the fact that each rational function in R[z], is “locally defined” on some Zariski open neighborhood of a. (See Chapter 5 section 3 about this topic in a more general setting.) (iii) Let p be a prime in Z. Then

is a local ring with the unique maximal ideal mp = {

E Z ( p )[ Ic

E

Z}.

(iv) Let K be a field and R the ring of formal series in one variable, i.e.,

If f = a0 g E R and

+ ala: + u2z2 + . . . with a0 # 0, then f

= ao(1

ao(l+zg)’a;l(l-xg+x2g2-~-)=

1.

+ zg) for some

60

Commutative Algebra

Conversely, if fh = 1 for some h E R, then a0 # 0. Thus, R is a local ring with the unique maximal ideal m = XR = (x). The next easy but powerful lemma enables us to see the link between finitely generated modules over a local ring R and finite dimensional vector spaces over the field R/m, where m is the unique maximal ideal of R. 2.3. Lemma (Nakayama) Let R be a ring and I an ideal contained in the Jacobson radical J ( R ) of R (see section 1). If M is a finitely generated

R-module and I M = M, then M = (0).

Proof If I M = M and M # {0}, then let M = CB1R O , p , / ' b =(p).

(vi) The only discrete valuations on Q are p-adic valuations. To see this, let v: Q 4 Z U { ca} be a nonzero discrete valuation on Q and R its valuation ring with the maximal ideal m. Note that Z c R because 1 E R. Then

mnZ=

(p) for some prime number p E

Z.

Since w is onto, it follows from the proof of Theorem 2.6(i) that m nZ = (p) with k E Z, for some prime number p. Thus, if 0 # 2 E Q then f = pk

($)

gcd(p, a') = 1, and gcd(p, b') = 1. Hence a', b' $! m, or in other words, a' and b' are units of R. It follows that v ( f ) = k v ( p ) . Suppose .(A) = 1. Then .(A) = mv(p) for some m E Zimplies v ( p ) = 1 because p E rn. This shows that v is the p a d i c valuation. (vii) Let K [ z ]be the polynomial ring in z over a field K and K ( z ) its field of fractions. Consider all discrete valuations v on K ( z ) such that .(A) = 0 for X E K x . Then there is only one such v which is not a p a d i c valuation. To see this, let w: K ( z ) 4 ZU{ca} be a nonzero discrete valuation on K ( z ) and R its valuation ring with the maximal ideal m. Below we consider two cases. Case I z E R. In this case K [ z ]c R and

m n K [ z ]=

(p(z)) for some irreducible p(z) E K [ z ] .

Local Rings, DVRs, and Localization

63

Arguing as in Example (vi), we may conclude that v is a p(x)-adic valuation. Case I1 x $2 R. It follows from the proof of Theorem 2.6(i) that y = $ E m. Hence, K(y] c R. Note that K[y] E K[x] and y E K[y]n m. So K[y]n m = (y) (why?). Thus, if f = a,xn . . . alx a0 with a, # 0, then

+ +

+

This shows that v(q(y)) = 0 and consequently v(f(x)) = -nv(y). Now, for E K ( x ) , if degf(x) = n and degg(x) = m, then 'u = any 0 #

#

(#)

(rn - n)v(y). As in Example (v) we may derive from y So eventually we have

E m that v(y) = 1.

that is, v is actually defined by the degree of polynomials. 2.8. Proposition Let R be a UFD and K its field of fractions. Then

R=

nv,

where V, runs over all p-adic valuation rings in K .

Proof By the definition of a p a d i c valuation on K , R 2 nv,. If 0 nv,, say, x = %,a, b E R, then v ( x ) 2 0 for any prime p . Suppose

#x

E

where pi's and q j ' s are primes and ai,& > 0. If some q j does not appear in the numerator of x, then vgj(x) < 0; if q j = p j but P j > a j , then v(qj) < 0. It follows that m 5 n, pj = q j for 1 5 j 5 m (up to a necessary re-arrangement of prime divisors), and ,OjI cyj for 1 5 j 5 m. This shows that x E R. Therefore R S nv, 2 R. To see under what condition a local domain is a DVR, we need the following result that is a special case of Chapter 1 Corollary 7.8.

Commutative Algebra

64

2.9. Lemma Let R be a Noetherian domain and I an ideal of R. Then 00

n I - = (0). n=l

0 2.10. Theorem Let R be a local domain and m its maximal ideal. Suppose m = ( t ) for some t # 0. If nr==lmn = {0}, then the following statements hold: (i) Every 0 # a E R is of the form a = tnu for some n 2 0 and some unit u. (ii) Let K be the field of fractions of R. Define .(a) = n for a = tnu as in part (i), and

K-

u {I.

Then v is a discrete valuation on K with R its associated valuation ring. (iii) Every nonzero ideal I of R is of the form I = (t") for some n 2 0.

Proof (i) By the assumption, if 0 # a E R is not a unit then a E me- me+' for some e 2 1, and it follows that a = tnu for some n 2 1 and some unit. (ii) By part (i), v is well-defined and surjective. Moreover, for s E K , v(s) 2 0 if and only if s E R. To verify that v is a discrete valuation on K , it is sufficient to note that for a , b E R with b # 0, if v(u) 2 v(b) then = f 1 E R and v(a + b ) 2 v(b); if v ( a ) > v(b) then = + 1 E R is a unit and v(a b) = v(b). (iii) By part (i), let n be the smallest integer such that t" E I where u is some unit. Then it is easy t o see that I = (t").

+

9

9 +

A full structural characterization of a DVR is given in Chapter 3 Theorem 4.5.

Appendix. General valuation rings Let R be the associated discrete valuation ring of a discrete valuation w on some field K . If z E K , z @ R, then z # 0 and v(xU1) > 0, i.e., z-' E R.

Local Rings, D VRs, and Localization

65

In a general valuation theory (cf. [Jac], [Coh]), all (not necessarily discrete) valuation rings are defined in this way, 2.11. Definition Let K be a field and R a subring of K . If for every E K X ,either x or x-' E R, then R is called a valuation ring of K .

x

If R is a valuation ring of some field K , then it is easy to see that K = Q ( R ) ,that is, K is necessarily the field of fractions of R.

To see why the above definition is really more general than a discrete valuation ring, we need to introduce general valuation functions. By an ordered abelian group (G,+; 3 )we mean an abelian group (G, +) equipped with a total ordering 3 which is compatible with the binary opa 3 b and c 4 d implies a c 4 b d, For example, (Z, +; I), eration +; I)and (R, 5 ) are ordered abelian groups. Let K be a field and (G, +; 5 ) an ordered abelian group. Define in G U {a) the rule co a , 03 a = co,for all a E G, and 00 00 = co. If there is an onto function

(a,

+:

+

+;

+

+

+

v

: K

+

-

G U {co}

2.12. Proposition Let v be a valuation on the field K . Then

is a valuation ring in K . Moreover, R is a local ring with maximal ideal m = {z E R I w ( ~ )+ 0 ) . Proof Exercise.

0

Let R be the valuation ring associated to a valuation w on the field K .

66

Commutative Algebra

For x , y E K X ,suppose u(x)

= g, u ( y ) = h.

Then

4 E R if and only if v ( 4 ) = u ( y ) - V(X)= h - g k 0 if and only if g 5 h. This, indeed, induces a total order on the abelian group K '. Motivated by the above idea, let R be a subring of the field K . If we write the binary operation of the abelian group G = K X / U ( R additively, ) that is, 3: + i j = 3:. i j , for x, y E K X, then a partial ordering 5 on G may be defined as

Y E R, x , y E K X . 3: 5 7 if and only if X

2.13. Proposition Let R following hold: (i) The correspondence

c K and ( G l + ; 5 ) be as above. Then the

u: K -

x

G U { 4

H

u ( . )

00,

if x = O;

3,

ifx#O,

=

defines a valuation function with R as its associated valuation ring if and only if 5 is a total ordering on G. (ii) If ( H ,+; a) is some ordered abelian group and z1: K 3 H U {cm}is a valuation with the associated valuation ring R, then 'p:

G?H

as ordered abelian groups, i.e., g 5 h implies q ( g ) 9 cp(h) for g, h E G. Proof Exercise.

0

2.14. Theorem Let R be a valuation ring. Then R is a DVR if and only if R is Noetherian.

Proof One direction is known by Theorem 2.6. Suppose R is Noetherian. Then the maximal ideal m of R is finitely generated, say m = ( a l ,a2, ..., a,) with all ai # 0. Thus, we may assume E R, and so m = (u2,a3, ..., an). Repeating this procedure will finally yield m = (ui), a principal ideal. Hence R is a DVR by Theorem 2.10.

Local Rings, D VRs, and Localization

67

Exercises 1. Let K [ z )be the polynomial ring in z over a field K . Show that R = K [ z ] / ( z nis) a local ring for any n E Z+. What is the maximal ideal of

R? 2. Let R be a ring and M a maximal ideal of R. Show that R is a local ring with the unique maximal ideal M if and only if 1 a is a unit in R for every a E M . 3. Show that Nakayama’s lemma holds for any ideal I C r ( R ) . 4. Consider the local ring V, given in previous Example (v). Show that K = V, $Vp $Vp . . . , and that K = p K . But K # (0). This illustrates that Nakayama’s lemma doesn’t work if the module considered is not finitely generated. (See also Chapter 1 (section 7, exercise

+

+

+

+

3).)

5. Let R be a local ring and m its maximal ideal. If m = (t) is principal-, show that either R is a DVR or tm = 0 for some m > 1. (Hint: Similar to the proof of Theorem 2.10.) 6. Complete the proof of Propositions 2.12 and 2.13.

3. The Ring of Fractions and Localization Recall from (section 2, Examples (ii)-(iii)) that in the local ring q p , = { + /

all b E

S 1=Z

-

P D } ,

( p ) , and in the local ring

all g(z) E S’z = R(z] - (z - a ) . 0bserve that 0

every s E Sl is invertible in Z(,), and every s’ E Sz is invertible in R[z],.

We may say that Z(,)is the ring of fractions of Zwith denominators in 5’1, and that R[x:], is the ring of fractions of R[x] with denominators in Sa. Also observe that

Commutative Algebra

68

the set of denominators S1,respectively tion in Z, respectively in R[rc].

is closed under multiplica-

5’2,

Considering any domain R and its field of fractions K , if S is a multiplicative set of R and 0 6 S (for instance, S = (1,r, r 2 , ...}, 0 # r E R), then a direct verification shows that

Rs

=

{:

I

a E R, s E S }

cK

is a subring of K containing R. We may call Rs the ring of fractions with denominators in S. Our aim in this section is to demonstrate that the above described idea of establishing the ring of fractions may be carried out in a more general setting including the case where the ring R may have divisors of zero. Let R be a ring and S a multiplicative set in R. Define on R x S the relation ( a l ,s1) ( a 2 , s2) if and only if there is s E S such that ( s l a 2 - s 2 a l ) s = 0.

(*I

N

-

Then one easily sees that is reflexive, symmetric. If (allsl) and ( a 2 , ~ 2 ) (a3, %), then

-

-

-

-

(a2, s2)

and thus, 0 = S ~ U ~ S ’ S-~ S ~ U ~ S S ~=S (’ S l a 3 - S ~ U I ) S S ~ S ’i.e., , (allsl) (as, s 3 ) . This shows that is also transitive, and hence is an equivalence relation on R x S. Write Rs for the quotient set R x S/ N , and write for the equivalence class of ( a , s ) E R x S. Define the addition and multiplication on Rs as follows: a2 -a 1+ - = s1

s2

S2al f s 1 a 2 1

SlS2

N

a 1- . a2 = - a1a2

s1

32

SlS2

Then a direct verification shows that Rs is an associative commutative ring with zero 0 = and identity 1 = f.

4

3.1 Definition The ring Rs constructed above is called the ring of fractions with denominators in S.

Local Rings, DVRs, and Localization

-

69

Remark Recall that if R is a domain and S = R - {0}, then the field K of fractions of R is given by R x S/ where is defined on R x S by

(**I

( U I ,SI)

-

(UZ,SZ)

N

if and only if

51u2 - S Z U I

= 0.

-

The reason why we did not define N in the foregoing (*) exactly as in (**) above will be clear if one checks the transitivity of defined in (**). (Also see Example (i) below.)

3.2. Proposition Let Rs be the ring of fractions with denominators in S . (i) If 0 E S , then Rs is the zero ring. (ii) The correspondence XR : R

-+

Rs X

2 -

-

1

defines a ring homomorphism (called the canonical homomorphism) such that (a) XR(S) = f is invertible in Rs for every s E S; and (b) KerXR = { u E R I as = 0 for some s E S } . Hence XR is injective if and only if S does not contain nontrivial divisor of zero (note that this is compatible with the case where R is a domain). (iii) XR has the universal property: If f : R + R' is any ring homomorphism ' f ( s ) is invertible in R' for every s E S , then such that f ( 1 R ) = 1 ~and = there exists a unique ring homomorphism f: Rs -+ R' with 7 that yields the commutative diagram

(5)

%

R h R s f]

Jr

foXR=f

R' Proof Exercise.

0

Example (i) Let K [ z ,y] be the polynomial ring in x and y over a field K . Consider the ring R = K[z,y]/(xy) and S = {1,Z,Z2,...}. Then (a) the previous (**) itself cannot define an equivalence relation on R x S ; and (b) since ?@= 0 and T is invertible in Rs, it follows that Rs K [ t ,t-'] c K ( t ) ,where K [ t ]is the polynomial ring in t over K .

Commutative Algebra

70

Note that R has nontrivial divisors of zero, but Rs is a domain. (ii) Let R be a ring and

f E R. Consider S

= (1,f,f 2 , ...}. Then

where R[x]is the polynomial ring in x over R. (This trick plays a very important role in algebraic geometry, for example, in proving the Nullstellensatz and in obtaining the open afine covering of an algebraic set, etc.)

Proof Note that 7 is invertible in R[x]/(xf- 1). By the universal property of Rs, the diagram

R

3Rs

is commutative, where a is the natural ring homomorphism R R[x]/(sf1). On the other hand, under the ring homomorphism P: R[z]-+ Rs with P(Crixi) = crib, P(xf - 1) = 0. By the first isomorphism theorem (or see section 0), there is a ring homomorphism R[x]/(xf- 1) 4 Rs that yields the commutative diagram --f

p:

Tr

Rfzl 01

R[Xl ( x f - 1)

JP

J.

RS Now one checks that h and

are inverses to each other.

The next proposition will be generalized in Chapter 3 Theorem 4.1.

3.3. Proposition Let R be a UFD, S a multiplicative set of R and 0 $ S. Then Rs is a UFD.

Proof View R as a subring via the canonical mapping XR: R 4 Rs. We show that factorization in Rs is completely determined by that in R. First note that if p is a prime in R then p [ s ,for some s E S, if and only if p is a unit in Rs. Secondly, note that if p is a prime of R but not a unit in Rs

Local Rings, DVRs, and Localization

71

then p is a prime in Rs. (Indeed, using the factorization in R it is easy to see that every prime in Rs is of the form p u for some prime p of R and some unit in Rs). Finally, note that if E Rs with r # 0 then

5

(21

(22

Pgm . u ,

- = P l P2 S

where the pi are primes of R but not units in Rs while u is a unit in Rs. 0 It follows that Rs is a UFD by Chapter 1 Theorem 2.9. Let R be a ring, Rs the ring of fractions of R with denominators in a multiplicative set S , and XR: R + Rs the canonical ring homomorphism. We now consider the relation between ideals of R and ideals of Rs. If I is an ideal of R, then ~e = x

{ a I a E I, s E S}

~ ( I )= R ~S

is an ideal of Rs and is called the extension of I in Rs. If J is an ideal of Rs, then the preimage

is an ideal of R and is called the contraction of J in R.

3.4. Proposition With notation as above, the following hold: (i) For any ideal J Rs, J"" = J . (ii) For any ideal I c R,

c

rs E I for some s E S (iii) If P is a prime ideal and P

n S = 8, then Pe is a prime ideal of Rs.

Proof (i) If f E J then T E J " , and hence 5 E J"". This shows that J J C e .The inclusion J C e J is clear. (ii) If r E I"" then 7 = $ E Rs for some r' E I and s E S. Thus, S'ST = S'T' E I for some s' E S . This proves the inclusion C.The inclusion 2 is again clear.

0

(iii) This follows from (ii) immediately.

3.5. Corollary (i) Let I be an ideal of R. Then I"" (0)

=I

rs E I implies r E I for r E R and s E S.

if and only if

72

Commutative Algebra

(ii) There is a one-to-one and onto correspondence defined by extension and contraction of ideals: {ideals of R satisfying ( 0 ) in part (i)}

-

{ideals of Rs} .

It follows that if R is Noetherian then so is Rs. In particular, if R is a principal ideal ring then so is Rs. (iii) iec= R if and only if ie= R s if and only if I n S # 0. (iv) If P E SpecR such that P n S = 0, then (*) holds for P , and Pec= P. It follows that there is a one-to-one and onto correspondence defined by extension and contraction of ideals: { P E SpecR

IPn

5’ = 0

1

+-+

SpecRs.

Proof Exercise.

Let R be a ring and P E SpecR. Then Sp = R - P is a multiplicative set of R. Write Rp for the ring of fractions of R with denominators in Sp.

3.6. Proposition With notation as above, the following hold: (i) Rp is a local ring with the unique maximal ideal m = P R p = Pe. (ii) There is a one-to-one and onto correspondence defined by extension and contraction of ideals:

{Q E

1

SpecR Q C P }

H

SpecRp.

Proof Note that 5 E Rp is a nonunit if and only if (ii) follows from Corollary 3.5(iv).

T

E P. So (i) holds.

3.7. Definition With notation as above, Rp is called the localization of R a t P. The above definition comes from the “local study” of points in algebraic geometry (see (section 2, Example (ii)) and Chapter 5 section 3). For instance, if R = K [ z l ,..., xn] and K is algebraically closed, then the localization RM of R at a maximal ideal M is the ring of all rational functions in K(z1,..., 2,) that are defined on some (open) neighborhood of the point determined by M (Chapter 5 Proposition 3.6). Furthermore, the next theorem says that the set of “global rational functions” that are defined at every point is nothing but the ring R (Chapter 5 Theorem 3.4).

Local Rings, DVRs, and Localization

73

3.8. Theorem Let R be a domain. Then

R = nRM = n R p , where M runs over all maximal ideals of R, and P runs over all prime ideals of R.

Proof The inclusion R C nRp is clear. Let x E n R p , and set Iz = { a E R I ax E R}. Then it is easy to see that I , is an ideal of R and x E R if and only if 1 E I,. If 1 # Iz,then there is a prime ideal P with P 2 I , by Proposition 1.5. But then x # Rp, for if x = f E Rp, b @ P , then bx E R and b E I, C P. This shows that x E R and hence R = nRp.

1. For a , a l , a2

Exercises E R, s, s' E S , verify in Rs s'a - a s's s'

a1 f a 2 = S

5 s

+

s. s

2. Let R be a domain and S a multiplicative set of R, where 0 # S. Show that R and Rs have the same field of fractions. 3. Complete the proof of Proposition 3.2. 4. Let S be a multiplicative set in Z, 0 # S. Find all primes in Zs. 5. Give a detailed proof of Corollary 3.5. 6. Let R = K[x,y]/(xy) be as in previous Example (i). (a) For every c E K , show that P = (Z - c, y) is a maximal ideal of R. (b) For every P in part (a) with c # 0, show that Rp is a DVR. What happens if c = O? (Hint: 7 j = ?(Z - c)?j.) 7. Let R = K [ z ,y]/(y2 - x3),where K [ x ,y] is the polynomial in x,y over a field K . Show that Rp is not a DVR, where P = ( Z , j j ) . (Hint: See Chapter 3 (section 3, Example (iii)).) 8. Let R be a PID. Show that if P E SpecR then Rp is a DVR.

4. The Module of Fractions

Let R be a ring and Rs the ring of fractions of R with denominators coming from a multiplicative set S , where 0 @ S. In this section we consider the relation between modules over R and modules over Rs.

Commutative Algebra

74

Let % be an Rs-module. Then also has an R-module structure via the canonical mapping X R : R 4 Rs:

r 1

r .z= XR(r)z= -2, r E R , z E

x.

In view of Chapter 1 section 7, there is a ring homomorphism p : R + EndzM, where p ( r ) = pr with p T ( y ) = r . y for y E %. Moreover, we observe that 0

for every s E S , ps is invertible in E n d z x , that is, every ps is an isomorphism.

Conversely, let M be an R-module. Then there is a ring homomorphism R + EndzM, where 'p(r)= pT with p,(m) = rm for r E R and m E M . Suppose that the property mentioned in the above ( 0 ) holds for M , i.e., ps is invertible for all s E S . Then, by the universal property of Rs,there is a unique ring homomorphism R s 4 EndzM that yields the commutative diagram 'p:

v:

3R s

R

J s

FoXR=P

End&' Consequently, M is equipped with an Rs-module structure. The above remark leads to the following construction. Let M be an R-module. Define the equivalence relation (check it!): ( m l ,s1)

-

(m2,s2)

N

on M x S

if and only if s(slm2 - s2ml) = 0 for some s E S,

for the equivalence class of (m,s)E and set M S = M x S/ N . Write M x S , and then define the R-module structure by

ml

m2

rm s2ml+slm2 , r . -m =-.

F+G=

SlS2

S

S

A direct verification shows that MS is an R-module and every s E S defines an isomorphism ps in EndzMs. It follows from the foregoing remark that M s is an Rs-module on which the Rs-action is given by m-r. - = s1

'92

rm s1s2

Local Rings, D VRs, and Localization

75

4.1. Definition The Rs-module MS constructed above is called the module of fractions with denominators in S. If S = R - P for some P E SpecR, then M s is called the localization of M at P and is denoted by M p .

4.2. Proposition Let M be an R-module and M S the Rs-module of fractions with denominators in the multiplicative set S. (i) If 0 E S , then M s is the zero module. (ii) The correspondence p

~ M: + M s

m m w 1 defines an R-module homomorphism (called the canonical homomorphism) with KerpM = { m E M I sm = 0 for some s E S } . (iii) p~ has the universal property: If is an Rs-module and f: M + is an R-module homomorphism, then there is a unique Rs-module homomorphism M S -+ with ):( = f . f (m)that yields the commutative diagram

u

7:

=

7

M 3 Ms I

Proof Exercise. Let $: M + N be an R-module homomorphism with K = Ker$ and W = Im$. Then K is a submodule of M and W is a submodule of N . It is easy to check that

llts: M s

-

Ns

is an Rs-module homomorphism. 4.3. Theorem With notation as above, the following hold: (i) Ker$s = K s , Im$s = Ws.

76

Commutative Algebra

+

(ii) If is injective then so is +s; if (iii) ( M / K ) s E M s / K s .

+ is onto then so is +s.

Proof (i)-(ii) It is clear that K s = 0, then Ker$s. If 4s ( 7 )= s’$(m) = 0 for some s’ E S. Since $ is an R-module homomorphism, 0 = s’+(m)= +(s’m),that is, s’m E Ker$ = K . It follows that = E Ks.

7 2

This shows that K s = Ker$s. Similarly, one checks that Im$s = Ws. (iii) This is just a consequence of applying parts (i)-(ii) t o the natural onto R-module homomorphism M -+ MIK. 0 Let R be a ring and I an ideal of R. In view of R 3 R s and Theorem - 4.3, one may verify that IS= I R s , and moreover, that the rule = defines a ring multiplication on the Rs-module ( R / I ) s .

:. 5 5

4.4. Corollary (i) Let I be an ideal of R. Then there is a ring isomorphism

&/Is E ( R / I ) s . (ii) If P E SpecR, then K = R p / P p is isomorphic to the field of fractions of the domain R I P . Proof This follows from Theorem 4.3 and the above remark.

Exercises 1. Verify that the relation N on M x S is an equivalence relation. 2. Verify that the Rs-module structure on Ms is well-defined. 3. For m, ml, m2 E M , s, s‘ E S , verify in MS s’m s’s

-

m

ml+m2

s

S

-

ml S

+-.mS2

4. Complete the proof of Proposition 4.2. 5. Consider the prime ideal P = ( p ) in Z, where p is a prime number. Let m E Zf, m = p ~ ’ p ~ ..pgm *. the factorization of m into primes, and M = Z/(m). (a) If p Xm, show that M p = ( 0 ) . (b) If p = pi for some i, show that Mp E Z/(pq’). (Hint: Write m = pgim,. Then ml E Z - P. Moreover, if k E Z, then k = p q i q r , 0 5 r < pqi.) 6. Let S be a multiplicative set of the ring R, and let M be a finitely generated R-module. Show that Ms is a finitely generated Rs-module. (Hint: Apply exercise 3 above to a direct verification, or use Proposition 4.3 and Chapter 1 Proposition 7.3.)

+

Local Rings, DVRs, and Localization

77

7. Let S be a multiplicative set of the ring R, and let I be an ideal of R. Viewing I as an R-module, show that I s = I e via I s L) R s , where I e is the extension of I in Rs in the sense of section 3. Furthermore, if J is another ideal of R, verify that the following hold: (a) ( I J ) s = I S Js. (b) ( I J ) s= I S J S . (c) ( I n J ) s = Is n Js . 8. State and verify a module-version of properties (a) and (c) above for submodules.

+

+

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Chapter 3

Integral Extensions and Norrnalizat ion 1. Integral Extensions In commutative algebra a natural generalization of field extension is the ring extension. Note that field extensions may be studied in terms of vector space. Here the structure of an R-algebra is convenient for the study of ring extensions. Let R and B be rings. B is said to be an R-algebra if there is a nonzero ring homomorphism 'p: R B. Recalling the convention ' p ( 1 ~= ) lg,it follows that B forms an R-module with the module structure: --f

(*> If R is a field, then R

ab = 'p(a)b, a E R, b E B. E 'p(R) c

B and B becomes an R-vector space.

An R-subalgebra of B is a subring B' B with l g ~ = lg, which is also an R-module with respect to the action defined by the above (*). If B is an R-algebra, then it is clear that any ideal of B is an R-module. Let B1 and B2 be R-algebras. An R-algebra homomorphism $: B1 4 B2 is simultaneously a ring homomorphism and an R-module homomorphism.

Let B be an R-algebra defined by the ring homomorphism 'p: R 4 B . If cp is a ring monomorphism, then R is naturally viewed as an R-subalgebra of B . In this case we say that B is an extension ring or extension algebra of R. Bearing this in mind, if no confusion arises, we simply refer R C B t o a ring extension.

The polynomial ring R [ z l ,...,z,]

in variables 2 1 , ..., z, over a ring R is 79

80

Commutative Algebra

called a polynomial R-algebra. 1.1. Definition Let R 5 B be a ring extension. (i) If B = R[S]for some nonempty subset S c B, where R[S]is the subring of B generated by S over R (see Chapter 1 section 0 for the definition), we say that B is an R-algebra generated by S and call S a set of generators of B or a generating set of B . If S = { b l , ...,b,} is a finite set, then B is called a finitely generated R-algebra and is denoted by B = R [ b l , ...,b,]. In the latter case, we also say that B is a finitely generated extension ring (algebra) of R. (ii) If B is a finitely generated R-module, then we say that B is a modulefinite extension ring (algebra) of R or B is module-finite over R.

From definition it is clear that a n y module-finite extension ring (algebra) i s a finitely generated extension ring (algebra). Example (i) Let R be a ring and B = R[x1,...,x,] the polynomial ring in variables xl,...,x, over R. Then B is a finitely generated extension ring (algebra) of R but B is not module-finite over R. (ii) Let d E Z be square-free (i.e., n2 ,/'d for any n E Z). Then Z Z + Z& is a module-finite extension.

c Z[&]

=

(iii) Using the fact that Z is a UFD and there are infinitely many primes in Z, we conclude that Q is not a finitely generated extension ring of Z. (See also Chapter 1 (section 7, exercise 3).) 1.2. Definition (Compare with an algebraic element in a field extension.) Let B be an R-algebra defined by the ring homomorphism cp: R -+ B . Write R' = cp(R).An element b E B is said to be integral over R if there is .~+Xlx+Xo, a monic polynomial f (x)E R'[x],say f (x)= xn+X,-1xn-'+. X i E R', such that

If every b E B is integral over R, we say that B is integral over R. If R C B and B is integral over R, then we call B an integral extension ring (algebra) of R and refer R C B to an integral extension. 1.3. Theorem Let B be an R-algebra defined by the ring homomorphism

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81

R --+ B. Write R’= p(R). For b E B , the following are equivalent: (i) b is integral over R. (ii) The R-subalgebra R’[b] C B is a finitely generated R’-module (or equivalently, a finitely generated R-module). (iii) There exists an R-subalgebra B’ C B with the property that R’[b] 5 B’ and B’ is a finitely generated Rl-module (or equivalently, a finitely generated R-module).

‘p:

Proof (i)

+ (ii) b is integral over R implies bn

+ X,-1bn-’ + . . . + X l b + Xo = 0, Xi E R’.

Inductively we derive that n- 1

R’bj, i E N.

bnii E j=O

It follows that R’[b] = CYZ,’R‘bj. (ii) + (iii) Take B’ = R’[b]. (iii) + (i) Let R’[b] C B’ C B , where B’ is the R-subalgebra as described. Suppose B‘ = Cy=l R’Ji, Ji E B’. Then bJi E B‘, say

In a matrix form we have

Write M for the above n x n matrix. If we multiply the above equality by the adjoint matrix of M , then det(M)Ji = 0, i = 1,..., n. This shows that det(M)B’ = 0. But l g = lgl E B‘ (by our convention for subrings). Hence det(M) = 0, and consequently b is a zero of some monic polynomial over R’.

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82

1.4. Corollary Let B be an R-algebra. If B is a finitely generated Rmodule, then B is integral over A . (One may compare this with a finite dimensional field extension.)

0 Example (iv) Let d E Z be square-free. By previous Example (ii), Z c Z [ d ] is a module-finite extension and hence an integral extension. (v) Let K be a field and B = K[z] the polynomial ring in one variable z over K . Then for n 2 1, K[z"] K [ z ]is an integral extension. Indeed, that one may check that B = K[zn]+K[zn]z++[s"]s2+...+K[zn]zn-1, is, B is module-finite over K [ z n ] . (vi) Let R be a UFD and K its field of fractions. Then any y E K - R is not integral over R. To see this, let y = f # R, where gcd(a, b) = d E U(R). If yn X,-ly"-l . . . Xly XO = 0 , X i E R, then, multiplying by b", we have

+

+ +

an = -b (Xn-1un-'

+

+ Xn-2ban-' + . . . + X1bn-'u + b"-lXo)

E R.

This shows that if a is a unit in R , then b is a unit in R and hence y E R. If a is not a unit in R, then b must be a unit in R and y = ab-I E R; otherwise a and b would have a nonunit common divisor. (vii) Let K [ z ,y] be the polynomial ring in z, y over a field K . Consider the ring R = K [ z ,y]/(y - f(z)), where f is a monic polynomial with degf 2 1. Then, Z is integral over K[V]and R is module-finite over K[y]. (viii) Let K [ z ,y] be the polynomial ring in 2, y over a field K . Then, though 0 in the ring R = K [ z , y ] / ( y z - l ) , T is not integral over K[V]. Indeed, by Chapter 2 (section 3, Example (ii)), with S = {l,y,y2,...},

zy - 1 =

while K[y][l/y] is not a finitely generated K[y]-module by Chapter 1 (section 7, exercise 3).

1.5. Proposition Let B be an R-algebra defined by the ring homomorphism p: R + B. Write R' = cp(R).The following hold: (i) The subset -

{

1

R = b E B b is integral over R

Integral Extensions and Normalization

is an R-subalgebra of B . (ii) If b E B is integral over R, then b E

Proof (i) Let b l , b2 E ring extension

R.

83

-

3. Hence R = R.

We show that

bl

f b2,

blb2

E

2. Consider

the

R'[bi] C_ R'(bi][h] = R'[h1b2]. By Theorem 1.3 (ii),

n

R'[bl]

=

CR'V~,E ~j

R'[bl].

j=1

Thus,

and it follows from Theorem 1.3(iii) that (ii) Let b E B be such that

b"

bl

ib 2 ,

blbz

+ . * . + X l b + A* = 0,

+ X,-lb"-l

xi

E R. E R.

Then, inductively we derive that

c

n-1

bn+Z E

R'[Xo, X I , .", X n - l ] b j ,

2

E N.

j=O

Hence R'[b]C CyIi R'[Xo, X I , ..., X,_l]bj. Note that the X i are integral over R. A similar argument as for part (i) shows that R'[Xo,X I , ..., A,-] is contained in an R-subalgebra of B which is also a finitely generated R'-module. By Theorem 1.3(iii), b E R. 13 1.6. Definition The subalgebra obtained in Proposition 1.5 is called the integral closure of R in B. If R C B and R = R C B, then we say that R is integrally closed in B. (Compare this with the algebraic closure in a field extension defined in Chapter 1 (section 3, exercise 3).)

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84

Combined with the forthcoming Noether normalization of section 2, the next proposition will play an important role in algebraic geometry (see the proof of Chapter 5 (Theorems 5.7 and 6.3)). 1.7. Proposition Let R C B be a module-finite extension and M E mSpecR. Then MB # B and there exists a maximal ideal Q E m-SpecB such that Q n R = M . (A generalization of this result is given in exercise 8 below.) Proof Suppose B =

El=,

C:==, R&, &

E B. If B = MB =

c:=, M&, then

= rij&, rij E M I i = 1,..., s . Arguing as in the proof ((iii) (i)) of Theorem 1.3, there would be 1 6 M . Hence MB # B. Let Q be a maximal ideal of B that contains M B (Chapter 2 Proposition 1.5). Note that R n Q # R (otherwise 1 E Q). Thus, M c R n Q implies M = R n Q.

0 Finally, let us see how integral extension is closely related to field extensions (see the proof of later Theorem 2.5, Chapter 5 (Theorems 1.5 and 5.3) for applications). 1.8. Theorem Let R C B be an integral extension, where R and B are domains. Then R is a field if and only if B is a field.

Proof Suppose R is a field. If 0 # b E B then

b"

+ Xn-lb" + . . . + X l b + Xo = 0, X i E R,

2 1.

Assume that n is the smallest degree. Then XO # 0 and b-l = -x-l0

(b-1

+ X,-,b"-2

+ . . . + X 2 b + A,)

E

B.

Conversely, if B is a field and 0 # a E R, then a-1 E B and hence a-1 is integral over R. It follows that there is a relation of the form

+

(u-')~

Xn-1(~-')"-'

Consequently, a-'

= -Xn-l

+ + X ~ U - ' + Xo = 0, Xi E R. *

*

- Xn-2a - . .

. - Aoan-l, i.e., a-1

E R.

Exercises 1. Let R = Z[@ 1 pi's are distinct prime numbers, i 2 11. Show that R is neither module-finite nor finitely generated as an algebra over 25. (See also previous Example (iii) and Chapter 1 (section 7, exercise 3).)

Integral Extensions and Normalization

85

2. By previous Example (v), K [ s 2 ]c K [ s ]is an integral extension. For any f E K [ z ]find , a monic polynomial g ( t ) E k [ s 2 ] [ tsuch ] that g(f) = 0. 3. Let A C B G C be ring extensions. Suppose that C is module-finite over B and B is module-finite over A. Show that C is module-finite over

A. 4. Let R C B be a ring extension. Show that if b l , ..., b, E B are integral over R then R[bl,..., b,] is module-finite over R. Hence, B is modulefinite over R if and only if B is finitely generated as an R-algebra and

5. 6.

7.

8.

integral over R. Let A 5 B & C be ring extensions. Suppose that C is integral over B and B is integral over A. Show that C is integral over A. Let R B be an integral extension. (a) For u E R, show that u E U(R) if and only if u E U(B). (b) Let H be a proper ideal of B and h = H n R. Show that (up to a ring monomorphism) R / h c B/H is an integral extension. Let R C B be an integral extension, and let S be a multiplicative set of R, 0 @ S. Then clearly S is also a multiplicative set in B . (a) Show that Rs C Bs, and that Bs is integral over Rs. (Hint: Use Chapter 2 Proposition 4.3 and Chapter 2 (section 3, exercise 2 ) . ) (b) Show that if B is module finite over R then Bs is module finite over R s . (Hint: Use Chapter 2 (section 4, exercise 6).) (Lying over theorem) Let R G B be an integral extension. Show that if P E SpecR then there is some Q E SpecB such that Q n R = P. (Hint: First, by arguing as in the proof of Proposition 1.7, show that the assertion is true for every M E m-SpecR. Then, for each P E SpecR, use exercise 7 above, Chapter 2 (Corollary 3.5 and Proposition 4.3) in order to pass to the case R p E Bp.)

c

2 . Noether Normalization

Let R = K [ a l ,..., a,] be a finitely generated K-algebra over the field K . Then by Chapter 1 section 0, R K [ z l ,...,z n ] / I ,where I is an ideal of the polynomial K-algebra K [ z l ,...,sn].Hence R is Noetherian by Chapter 1 (section 1, exercise 2). In this section we show how t o build R by a module-finite extension over some polynomial K-algebra. By the definition of a polynomial ring we know that

C Xa(j)sy31 . . . xnaJn

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86

= 0 in K [ q , ...,zn]if and only if all X,(j) = 0. This property has the following generalization t o an arbitrary K-algebra.

2.1. Definition Let R be a K-algebra over the field K, and r1, ..., rm E R. If there is some polynomial g E K [ q ,...,z,] such that g ( r 1 , ..., rm) = 0, then we say that r1, ...,r, are algebraically dependent over the field K ; otherwise, r1, ..., r, are said to be algebraically independent over the field

K. Clearly, if K L is a field extension and 6 E L is a transcendental element over K (Chapter 1 section 3), then 6 is algebraically independent over K and K [ 8 ]is isomorphic to the polynomial algebra K[z]. More generally, if R is a K-algebra over the field K and r1, ...,r, E R are algebraically independent over K , then the subalgebra K[rl, ...,r,] of R is isomorphic to the polynomial K-algebra K[zl, ..., z.],

To reach the main result of this section, the following technical preliminary is required. 2.2. Lemma Let a(j) = ( a j l..., , q,),~ ( k=) ( f f k l ,...,f f k , ) E Nn. Supffj8= f f k i = m , but lexicographically (see Chapter 1 secpose tion 4) ~ ( k+lez ) a(j),that is, there is some s 5 n such that

xy=l cy=l

f f k l = f f j l , ...,f f k , - l

Then

cy=l

mn-'aki

=B

= f f j s - , , while

ffk,

< CYjs.

< A = Cy=lmn-iaja.

Proof Without loss of generality we may assume

/

n

\

ffkl

n

< aj1. Then

Integral Extensions and Normalization

i=3

87

i=3

i=l

=

A.

F'urthermore, let f E Kjxl, ..., x,] be a polynomial of degf = m 2 1, where K is a field. Then by Chapter 1 (section 7, exercise 8), f may be written uniquely as

where each Fi is a homogeneous polynomial of degree nil i.e.,

with a ( j )= ( a j l ,...,aj,) E Nn,

E K.

Let us call F, the leading homogeneous part of f for convenience. 2.3. Proposition Let R = K [ a l ,..., a,] be a finitely generated K algebra over the field K . Suppose that there is a nonzero polynomial f E K [ x ~...,, x,] such that f ( a 1 , ,.., a,) = 0. Then there exist b l , ..., bn-1 E R such that a, is integral over K[bl,...,bn-l] and R = K [ b l ,..., bn-1][~,].

+

+ +

Proof Write f = F, F,- 1 . . . FO as a sum of homogeneous polynomials, where the leading homogeneous part o f f has the form

Since dlexis a total ordering on

N",we may also assume that

Commutative Algebra

88

Now set

and define

in which the highest degree term in x, is given by

Frn (z; +xg1,...,x;-1

+xK*-',~n)

that is,

Setting

bk = ak - agk, k

= 1,..., n - 1, we see from (3) above that

H(b1,..., bn-l, a,)

= f(a1,..., a,) = 0.

Note that 0 # Xa(l) E K . We conclude that a, is a zero of the monic polynomial X;tllH(bl, ..., b,-l,t) E K[bl, ...,b,-l][t]. Finally, it IS clear that R = K[bl,..., b,-l][a,], and thus, the proof is completed. 0 2.4. Theorem (Noether normalization) Let R = K [ a l ,..., a,] be a finitely generated K-algebra over a field K . Then there exist elements z1,..., zd E R, 0 5 d 5 n, such that (i) 21, ..., z d are algebraically independent over K ; (ii) R is module-finite over the polynomial K-algebra B = K [ z l ,...,z d ] .

Proof We prove by induction on the number n of generators of R. For n = 1, the assertion follows from (section 1, exercise 4). So, For n 2 1, if a l , ..., a, are algebraically independent over K , then the assertion is trivially true. Suppose that a l , ...,a, are algebraically dependent over K , and that f E K [ z l ,...,x,] is such that f(a1,...,a,) = 0. Then, by

Integral Extensions and Normalization

89

Proposition 2.3, there exist b l , ...,b,-l E R such that a, is integral over B = K[bl,..., bn-l] and R = B[a,]. Applying the inductive hypothesis t o B , we may find 21, ...,z d E B , that are algebraically independent over K, such that B is module-finite over A = K[zl, ...,z d ] . Since an is integral over B , it follows that B[a,] is module-finite over B . If we look at each step of the tower A c B c B[a,] = R, then by (section 1, exercise 3), R is module finite over A , which completes the proof. 0 Theorems 1.8 and 2.4 enable us to derive the following wonderful result that is crucial in dealing with the Nullstellensatz in algebraic geometry (Chapter 5 section 1). Theorem (Zariski) Let K L be a field extension. If L = K[al,...,a,] is a finitely generated K-algebra, then L is module-finite over K, i.e., [ L : K] < 00, and hence L is algebraic over K . 2.5.

Proof By the Noether normalization, there are algebraically independent over K, such that

21,

...,Zd

E

L , which are

n

Thus, K [ z l , ...,zd] 2 L is an integral extension. But since L is a field, it follows from Theorem 1.8that K[z1, ..., z d ] is a field. This shows that d = 0, and consequently L is module-finite over K . Exercises 1. Given distinct a(l),a(2), ...,a(m)E Nn with a ( j )= (ajll a j 2 ..., , ajJ1 show that there are nonnegative integers p1, ..., pn-ll 1 = p, such that

(Hint: Use induction on n. If n = 1, it is trivially true. Suppose for a ' ( l ) ..., , a'(m) E N"-l with a'(j) = (ai2,...,aim)we can choose p 2 , ...,pn-1, pn = 1 SO that the conclusion is true. Now the choice

completes the induction process.)

commutative Algebra

90

2. Let R = K [ z ,y]/(y2 - x3),where K [ z ,y] is the polynomial ring in z, y over a field K . Show that Z is algebraically independent over K , y is integral over K [ 4 ,and R = k[Z,jj] is module-finite over K [ Z ] . 3. Let K [ z , y , z ]be the polynomial ring in z,y,z over a field K , R = K [ z ,y, z ] / ( x z ,y2z). Write u = T 2, v = ij. Show that u and v are algebraically independent over K and that R is module-finite over the subring K [ u ,v]. 4. Let K [ z ,y] be the polynomial ring in z, y over a field K . By (section 1, Example (viii)), in the ring R = K [ z ,y]/(yz - l),Z is not integral over K[jj],and similarly is not integral over K [ z ] .Use Noether normalization to find some z E R that is algebraically independent over K , such that R is module-finite over K [ z ] .

+

3. Normal Domains and Normalization

In this section we introduce the notion of a normal domain, and establish the module-finite property of the normalization (integral closure) of a finitely generated domain. 3.1. Definition Let R be a domain, K its field of fractions, and E its integral closure in K (Definition 1.6). (i) If R is integrally closed in K , i.e., R = R c K , then we say that R is a

normal domain. (ii) If R = F [ a l ,..., a,] is a finitely generated F-algebra over some field F , then is called the normalization of R (this name will be qualified by the proof of Corollary 3.3 below and by the behavior of R in Chapter 5 section 5). Example (i) Let R be a domain and K its field of fractions. If R is the integral closure of-R in K , then, since the field of fractions of is also K (why?) and = E c K , by Definition 3.1, R is a normal domain. By (section 1, Example (vi)), any UFD is normal. In particular, any DVR is normal.

a

(ii) Let K = Q(s)be a number field and d K its ring of algebraic integers. Then d K is normal and Noetherian but not necessarily a UFD (see Chapter 4 for details).

Integral Extensions and Normalization

91

(iii) Let K [ x ,y] be the polynomial ring in variables x,y over a field K , and let R = K[rc,y]/(y2 - x 3 ) (see section 2, exercise 2). Then jj2 = I 3 in R. Thus, any element of R has the form

= ~ A n i m l y 1 + 3 m 1y-

+cXnzmzP+3mz.

Let K [ t ]be the polynomial ring in variable t over K . Then the ring , with x H t2 and y H t3 induces a ring homomorphism K [ x ,y] -+ K [ t 2t3] homomorphism (see Chapter 1 section 0): p: R x -

y

-+ K [ t 2 t3] , H

t2

H

t3

By the remark made above, it is not difficult to see that p is an isomorphism. Note that $ = t in the field of fractions K ( t ) of K [ t ] .Identifying R with the subring K[t2,t3]of K [ t ] ,R and K [ t ]have the same field of fractions K ( t ) ,i.e., we have the relation

R c K [ t ]c K ( t ) Clearly, t is integral over R but is not contained in R. Hence, R is not normal. But since K [ t ]is a UFD, the normalization of R is K [ t ]by Example (i) above. A geometric explanation of this example will be given in Chapter 5 section 5. (iv) Let K[x,y,z] be the polynomial ring in x , y , z over a field K , R = K [ x ,y, z]/(zy2 - x 2 ) . If K [ u ,u]is the polynomial ring in u,v over K , then the ring homomorphism K [ x ,y, 21 + K [ u v ,u,u2] with x H U V , y H u,and z H v2 induces a ring homomorphism

$:

R -

X -

H

K[uu,u,v~] 2/21

Y H

U

Z

V2

-

H

A similar argumentation as in Example (iii) above shows that $ is an isomorphism, and the normalization of R is K [ u ,u]# K [ u v ,u,u2].

Commutative Algebra

92

By (section 1, exercise 4), the normalization R of R obtained in Example (iii), respectively in Example (iv) above, is module-finite over R (see also other examples provided by exercises of the current section). Below we will see that this is by no means accidental. 3.2. Theorem Let R be a normal domain and K its field of fractions. If K c L = K(t9) is an n-dimensional separable field extension, then the integral closure of R in L , denoted ?i, is contained in a finitely generated R-submodule of L. If R is Noetherian then i? is module-finite over R.

Proof Choose elements a1, ...,an E R to form a K-basis for L (why is this possible?). Then C:=l R a i C Consider the trace function TL/K on L as defined in Chapter 1 section 5. By Chapter 1 (Theorems 5.5 and 5.7), TL/K(uv) is a nondegenerate bilinear form and there exists a dual K-basis {PI, ...,On} for L such that

x.

x

We claim that & C:=,RPi. To see this, let c l , . . . , c n be all distinct K-linear monomorphisms from L to a splitting field of pe(x) (the minimal polynomial of 6 over K ) . Then for u E R C L , we have ai(u) E XI i = 1, ...,n. But Cy=lei(u) = T L / K ( u )E K, it follows that TL/K(u) E R because R is normal. Now, for y E R, y = CZ1Xipi with Xi E K , we have TL/K(yaj) E R and by the above ( l ) ,

This shows that R C CZl RPi, as claimed. The last assertion follows from Chapter 1 Theorem 7.6.

0

3.3. Corollary Let R = F [ a l ,...,a,] be a finitely generated F-algebra over a field F . If R is a domain and K is its field of fractions, then the normalization of R in K is module-finite over R and hence a Noetherian ring.

Proof By the Noether normalization theorem, there are algebraically in-

Integral Extensions and Normalization

dependent

zl,

..., Zd

E

93

R over F such that n

F C: F [ZI, ...,Z d ] 2 R =

CF

.

[ZI,.., zd]<j, < j E

R.

j=1

Assume that K is a separable extension of F ( z ~..., , z d ) that is the field R is also the integral closure of F [ z l ,...,zd] in K , it follows from Theorem 3.2 that R is module-finite over R (note that R is Noetherian). 0 of fractions of the polynomial ring F [ z l ,..., zd]. Then since

Applications of Theorem 3.2 and Corollary 3.3 are given in Chapter 4 and Chapter 5.

Remark In the proof of Corollary 3.3 we argued by assuming the separability of K over F(z1, ...,z d ) . If the ground field F is of characteristic 0, the separability of K is naturally guaranteed. In general case, the feasibility of this assumption is proved in detail by Zariski and Samuel in Theorem 9 (p. 267) of Commutative Algebra, Vol. I (New York: Springer-Verlag, 1958). We do not quote Zariski-Samuel’s detailed argumentation here because it involves more deeper field theory that is not included in Chapter 1. So the reader may accept Corollary 3.3 and use it without any doubt. Exercises 1. Let R = K [ z ,y]/(y - f(z)), where K [ z ,y] is the polynomial ring in z, y over a field K and degf(x) 2 1. Is R a normal domain? 2. Let R = K [ z ,y]/(y2 - x 2 - x3) where K [ z ,y] is the polynomial ring in z and y over a field K . Prove that the normalization of R is isomorphic to the polynomial ring K[t]. (See also Chapter 5 (section 7, Example (iv>1.1

3. Let R = K[z,y,z]/(y - x 2 , z - z3), where K[z,y, z] is the polynomial ring in z, y, z over a field K . Show that R is a normal domain. (Hint: R 2 K[t] with z H t , y H t 2 , and z H t3.)

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94

4. Normal Domains and DVRs

This section is devoted t o a local study of normal domains in terms of

DVRS. First, a typical example of “global concern versus local solutions”.

4.1. Theorem Let R be a domain and K its field of fractions. The following statements are equivalent. (i) R is normal. (ii) R s is normal for any multiplicative set S of R. (iii) Rp is normal for all P E SpecR. (iv) RM is normal for all M E m-SpecR. Proof (i) 3 (ii) let x E K be integral over R s . Then

zn + Xn-1xn-l

Ti +. . . + Xlx + XO = 0, X i = -, Si

ri E R, si

E

S.

Multiplying the relation by ( S O S .~. . s,-l),, we see that ( ~ 0 .~. .1S,_~)X is integral over R. By the fact that R is normal, (soq . . . s,-~)x = y E R, i.e., 2 = SOSl

*

.

S,-1

€Rs.

(ii) j (iii) + (iv) is obvious. (iv) + (i) let z E K be integral over R. Then Zn

+ X,-lXn--l + . . . + XlX + A0 = 0

holds over both R and RM for each M E m-SpecR (note that R 2 RM). 0 Since RM is normal, x E nRM = R by Chapter 2 Theorem 3.8. Comparing with Chapter 2 Proposition 2.8, our goal is to show that the equality R = n R p holds for a normal and Noetherian domain R , where P runs over all minimal nonzero primes of R and every Rp is a DVR, though a normal and Noetherian domain is not necessarily a UFD (section 3, Example (ii)). To this end, we need the following preparation. Let R be any ring, M an R-module, and x E M . Then the set of annihilators of x in R, denoted AnnRX = {a E R

1 ax

= 0}

Integral Extensions and Normalization

95

is an ideal of R. Indeed, AnnRx coincides with the kernel of the R-module homomorphism px: R-M a H ax

Hence R/Kerpx = R/AnnRx is isomorphic to an R-submodule of M . Note that if 3: # 0 then AnnRx # R, for 1~E R and all modules considered in this book are unitary. If AnnRx = P E SpecR, we say that P is an associated prime of M . Set

P E SpecR P an associated prime of M Example (i) For any P E SpecR, AssR(R/P) = { P } .

(ii) Let R be a PID, and let M = R/(a), where a = p y p y for two primes p l # pa. Then ii = pT-'py E M has AnnRii = ( P I ) , and similarly V = pn , p 2m-1 E M has AnnRij = ( p ~ ) Moreover, . if w E M with AnnRw = ( T ) E SpecR, then, it follows from the first isomorphism theorem that rla. Hence AssRM = { ( P l ) , ( P 2 ) ) . 4.2. Proposition Let R be a ring and M an R-module. The following

hold: (i) Any maximal element (with respect to the inclusion relation) in the set

1

fl= {AnnRx O f x E M} is prime and hence belongs to ASSRM. (ii) If R is Noetherian and M # {0}, then ASSRM#

0.

Proof (i) Let x E M be chosen such that AnnRx is maximal in a. If a, b E R, a b E AnnRx, then abx = 0. It follows that if bx = 0 then b E AnnRx; if bx # 0, then since bx E M and AnnRx C AnnR(bx), we have AnnR(bx) = AnnRx by the choice of AnnRx. Therefore, abx = 0 yields a E AnnRx. (ii) If R is Noetherian and M # {0}, then

R

=

{AnnRx

I

0 # x E M}

and has a maximal element. Hence ASSRM#

#0

8 by

(i).

96

Commutative Algebra

The next proposition provides the key result in the local study of a normal Noetherian domain. 4.3. Proposition Let R be a local domain with the unique maximal ideal m. Assume that R is normal and Noetherian. If m E A s s ~ ( R / ( a ) )for some 0 # a E R, then R is a DVR, and hence m is a minimal nonzero

prime ideal. Proof Let u E R - ( a ) such that mu c ( a ) . Then m: C R. If m: 5 m, then since m is a finitely generated R-module, a similar demonstration as in the proof of Theorem 1.3 ((iii) (i)) shows that is integral over R (note that now R is a domain) and hence in R. Thus, u = ar E ( a ) for some T E R. This contradicts the choice of u. Therefore, we must have m: = R because rn: is an ideal of R. Let 1 = v: for some v E m. Then, for any y E m, y: = E R implies y E (v). This shows that m = (v). It follows from Chapter 2 Theorem 2.10 that R is a DVR and m is a minimal nonzero prime ideal. 0

*

Corollary Let R be a normal and Noetherian domain and P E A s s ~ ( R / ( a )for ) some 0 # a E R. Then P is a minimal nonzero prime ideal of R. 4.4.

Proof Consider the localization (R/(a))p of R / ( a ) at R - P. Then

by Chapter 2 Corollary 4.4. Now P E A s s ~ ( R / ( a ) )implies P R p E A S S R(~R p / a R p ) by later Exercise 4, and the assertion follows from Proposition 4.3 and Chapter 2 Proposition 3.6. 0 Comparing with Chapter 2 (Theorems 2.6 and 2.10), we now have the following characterization of a DVR. 4.5. Theorem Let R be a domain. The following statements are equivalent. (i) R is a DVR. (ii) R is a normal and Noetherian local domain with SpecR = {{O},m} where m is the maximal ideal of R. (Therefore, this theorem may also be viewed as the converse of Chapter 2 Theorem 2.6.)

Integral Extensions and Normalization

97

Proof (i) + (ii) This follows from (section 3, Example (i)) and Chapter 2 Theorem 2.6. (ii) + (i) By Chapter 2 Theorem 2.10, we need only to prove that (ii) implies m is a principal ideal. By Nakayama’s lemma, m # m2. Let 0 # u E m - m2. If (u) is a prime ideal then m = (a) as desired. If (u) is not a prime ideal, then (u) # m and there are y,z E R, y,z # (u) but yz E (u). This shows that AnnRF # {0}, where F is the image of z in M = R/(u). By Proposition 4.2, ASSRM# ((0)) and hence m E ASSRM. It follows from Proposition 4.3 (or its proof) that m is principal and R is a DVR. 0 4.6. Corollary Let R be normal and Noetherian. If P is a minimal nonzero prime ideal of R (i.e., P is minimal in SpecR with respect to the inclusion relation), then Rp is a DVR.

Proof Rp is normal by Theorem 4.1, and Noetherian by Chapter 2 Corollary 3.5. Since P is minimal in SpecR, it follows from Chapter 2 Proposition 3.6 that SpecRp = (0, PRp}. Hence Rp is a DVR by the theorem above.

0 We are ready to prove the previously promised result.

4.7. Theorem Let R be a normal and Noetherian domain, K its field of fractions. Then

where P runs over all minimal nonzero prime ideals of R. In particular, R is an intersection of DVRs. Proof nRp R is clear. To get the inclusion nRp C R, a trick used in the proof of Chapter 2 Theorem 3.8 is employed again. Let x = E K , and consider the ideal of R given by

I

I , = { a E R ax E R} . Then I z = { u E R I ub E (c)} = Ann& where 5 is the class of b in R/(c). If i; = 0, then b E (c) and hence x E R. Suppose # 0. Then O#EER/(c)}#0.

98

Commutative Algebra

Since R is Noetherian, I, is contained in a maximal element P of 0. By Proposition 4.2(i), P E A s s ~ ( R / ( c ) ) .It follows from Corollary 4.4 that P is a minimal nonzero prime. However, note that I , C P implies z Rp. This proves the following: z @ R implies 3: 6Rp for some minimal nonzero prime P.

e

Exercises 1. Let M be an R-module and N an R-submodule of M . Show that ASSRM C A s s ~ N uA s s R ( M / N ) . Can you find an example to show that g holds? 2. Show that for any n 2 1, there is a Zmodule M such that AsszM contains exactly n prime ideals. (Hint: Consider m = p y l . . .p,"" with distinct pimes p i . ) 3. Use Proposition 4.2(i) to show that if R is Noetherian and M an Rmodule, then

u

O#XEM

AnnRz=

u

p.

PEASS~M

4. Let M be a nonzero R-module and P E ASSRM. If S is a multiplicative set of R with P n S = 0, show that PRs E A S S RMs. ~

Chapter 4

The Ring & in K = Q ( 8 ) This chapter introduces the normal and Noetherian domain AK that has its location among numbers as indicated by the following Hasse diagram (of the obvious partial order):

A

K

Q

\"r=Knd z

99

Commutative Algebra

100

where

{

A= pEc K

I

g(p) =

o for some monic g(z) E ~ [ z ],}

c C a subfield with [K : Q] < 00.

Consequently, 6 is the algebraic closure of Q in C (Chapter 1 section 3, Exercise 3), A is the integral closure of Z in C (Chapter 3 Definition 1.6), and AK is the integral closure of Z in K (Chapter 3 Definition 1.6). Very soon in section 1 below we will see that AK is the integral closure of Z[8], where K = Q(8). For any /3 E A, since A is integrally closed in C,we have (Chapter 3 Proposition 1.5(ii)). It follows that

fi E

A

holds for all a E A. Note that not every a E A is a unit (for instance 6). We conclude that factorization into irreducible elements in A is impossible, and consequently, A is not Noetherian (Chapter 1 Proposition 2.5). Thus, A is usually not involved in practice. Why AK? First and foremost, AK is a normal and Noetherian domain (section 1, Theorem 1.2 below). So factorization into irreducible elements in AK is always feasible (Chapter 1 Proposition 2.5), and in many cases AK is a UFD (section 3). Perhaps a motive example to this topic should be the following oldest example. Example Find all integer solution (z, y, z ) of the equation

x2 + y2 = z 2

(1)

where gcd(z, y, z ) = 1.

Solution In order to solve this problem, one way is t o factorize the left of equation (1) in the ring of Gaussian integers Z[i]= { a bi I a, b E Z} with i=m

+

(2)

.(

+ yi)(z - yi) = 22 .

The Ring AK in K = Q(8)

101

Note that Z[i]is a UFD (see section 3). The assumption ( x ,y, z ) = 1 entails that x and y cannot be both even or odd. So, z must be odd. Thus, x yi and x - yi must be coprime, and a comparison of the prime factorizations on both sides of equation (2) yields z yi = %a2, where u is a unit of Z[i] and Q = c di which is not a unit. Since V(Z[i]) = {fl, fi} (see section 3), it follows that the desired solutions ( 2 ,y, z ) are given by z = f ( c 2- d 2 ) , y = f 2 c d , z = f ( c 2 d2). In Chapter 5 (section 7, Example (iii)), another way to find the integer solutions of equation (1) will be given in terms of algebraic geometry.

+

+

+

+

In section 2 we will see that actually Z [ i ]= Q(i)n A. A generalization of last example is the famous Fermat’s last theorem which states: 0

For n > 2, the equation xn solutions (z, y, z ) .

+ yn = zn does not have nonzero integer

Using the above example, one may show that the theorem is true for n = 4 and hence (automatically) also for n = 4k. It is therefore sufficient to consider the case where n is an odd prime p , for if no solutions exist when n = p then no solutions exist when n = pe. Now, for p , an odd prime, yp

= z p - x p = (%

-

z ) ( z - W X ) ( % - w”)

. . . ( 2 - wp-l x )

in Z [ w ] ,where w is a primitive pth root of unity, i.e., w = e2?ri/P. Historically in the literature (cf. Borevich and Shafarevich: Number Theory, Academic Press, 1966, pp. 378-381), it was shown that if Z[w] is a UFD, then the theorem is true for p . But unfortunately, Z[w]is not always a UFD (see section 3). Actually, we have Z[w]= Q(w) n A, where w = e2?ri/P,p an odd prime (see section 2). Though Fermat’s last theorem was finally proved by Wiles in a very deep algebraic-geometric way in 1994, it was the ring AK that started algebraic number theory and led t o the ideal structure theory in both commutative and noncommutative algebra.

Commutative Algebra

102

1. AK is Normal and Free of ZRank [ K : Q]

Adopting the classical definition in number theory, if a field extension Q c K , where K c C, is finite dimensional, then K is called a n u m b e r field. By Chapter 1 Theorem 3.11,

K

= Q(Q),Q E

K is algebraic over Q.

Using the notation as in the introduction of this chapter, let d K =A

nK

=

{ p K I f(P)

= 0 for a monk

E

f(z)E Z[z]}.

Elements in d K are called algebraic integers, and so, d K is called the ring of algebraic integers of K . Note that since the generator a: of K is algebraic over Q, there is some 0 # c E Zsuch that CQ E d K (later Exercise 1). It turns out that we may write

Thus,

Z2 2[6]2 d

K

C K = Q(8).

1.1. Lemma With notationas above, d K is equal to the integral closure of Z[8] in K , that is, d K = 2[8]. Hence AK is a normal domain.

Proof This follows from Chapter 3 (section 1, exercise 5).

0

1.2. Theorem (i) AK is a finitely generated abelian group (or Z-module). Therefore, d K is normal and Noetherian. (ii) d K is a free abelian group of Z r a n k [ K : Q].

Proof (i) Since Z is a normal Noetherian domain (a UFD is normal), Z C Q C Q(6),and d K is the integral closure of Z in K , the assertion follows from Chapter 3 Theorem 3.2 and Chapter 1 Theorem 7.6. (ii) By Chapter 1 (section 6, exercise 4), d K is a free abelian group of finite rank. Note that any Zbasis of d K is also a Q-basis for K (exercise 4). Hence the Z-rank of d K must be equal to [ K : Q]. 0 In the literature, a Zbasis of d K is also called an integral basis of K , due to the fact that a Zbasis of d K is necessarily a Q-basis for K (exercise 4).

The Ring AK in K = Q ( 0 )

103

Let K = Q(19) be as before, where 6 E Arc, [ K : Q] = n. From now on we let p ( z ) be the minimal polynomial of 19 over Q. 1.3. Proposition For a E K , let pa(.) be the minimal polynomial of a over Q. Then a E d K if and only ifp,(x) E Z[z]. In particular, p ( z ) E Z[Z].

Proof The (‘if” part is clear, and the “only if” part follows from Chapter 1 Theorem 2.17 (note that a E dK if f(a) = 0 for some monk f(z) E Z[z]).

Remark From Proposition 1.3 we derive immediately that Q n A = Z. This recaptures the fact that Zis integrally closed in Q. Concerning integral bases of K , we first note an easy fact in view of Proposition 1.3.

Observation Let a E dK,then pa(.) E Z[x]. Since pa(.) is monic, a division algorithm by pa(.) in Z[z] plus the ring homomorphism Z[z] -+ Z [ a ] with z H a yields Z [ a ]= Zai, where n = degp,(z). Consequently, Z[a] is a free abelian group of rank n, and {1,a,a2,...,an-1}is a Zbasis of Z [ a ] .So, if Z [ a ]= d K 1 then we have found a Zbasis for AK (in section 2 we will see that many number fields have this property).

xyr:

However, finding an integral basis for an arbitrary number field K is by no means an easy job in algebraic number theory. As t o this topic, a very useful invariant of K is introduced and discussed below. Consider the n distinct zeros of p ( z ) in Q: (Chapter 1 Theorem 3.6), say

..., 6,.

191 = 6,292,

By Chapter 1 Proposition 5.1, there are exactly n distinct Q-linear ring

monomorphisms

-

a2 : K = Q(19)

19

H

c 192

i = 1,..., n. 1.4. Definition Let

01,

...,a,be as above. If {a~,...,a,}is a Q-basis for

Commutative Algebra

104

K = Q(6),then A [ a l , ...,a,] = [det (ai(aj))]' is called the discriminant of the given basis. 1.5. Lemma Let Then

where

Pk

..., a,} and {PI,...,,&} be two Q-bases of K

{al,

= CT=, c j k a j ,

k = 1,..,n, c j k

E

= Q(6).

Q.

Proof Exercise. 1.6. Proposition With notation as above, the following hold: (i) 0 # A [ a l , ...,a,] E Q for any Q-basis { a l , ..., a,} of K = Q(6). (ii) If all gi(6), i = 1,..., n, are real numbers, then A [ a l , ...,a,] > 0 for any Q-basis { a ~...,, a,} of K .

Proof (i) Consider the Q-basis { 1 , 6 ,d 2 ,..., P-'}and note that oi(6)= 6i, i = 1, ..., n. Then

Since applying any permutation 7~ of (61, ..., 6,) to ni.,,(6i - 8 j ) is the same as applying 7~ to det(ai(@)), it interchanges rows of det(oi(6j)) and hence the sign of det(oi(6j)). But then det(ai(z9j))2 is symmetric on 61, ..., 6,. It follows from Chapter 1 Corollary 7.2 that A[1,6,d2,..., Sn-'] E Q. Since all 6i's are distinct, A [ l , 6 , d 2 ,..., P-'] #0 and the conclusion (i) follows from Lemma 1.5. (ii) This follows from part (i) and Lemma 1.5.

The Ring Ax in K = Q(19)

105

1.7. Corollary If (01, ...,a,} is a Q-basis of K = Q(29)in which all cq's are algebraic integers, then 0 # A[al, ...,an]E d n Q = Z.

Proof This follows from Proposition 1.6(i) and the fact that all ai(aj)'s are algebraic integers.

As argued in [ST], there is a direct proof of Theorem 1.2(ii) by using only discriminant. 1.8. Another proof of Theorem 1.2(ii) Since { 1,8,f12, ...,P-'}is a Qbasis of K consisting of algebraic integers and 0 # A [ l , 29, ..., 29"-'] E Z by Corollary 1.7, we may select a Q-basis W = {[I, ..., , J1, J2 E

F(dK).

The reader will be asked in later exercise 4 to check that this operation is well-defined and it makes F ( d K ) into a commutative associative semigroup with the identity element d K . We now proceed to show that F ( d K ) is indeed a group. For each ideal I of d K , set

Then it is clear that d K C I-'. 4.6. Proposition With notation as above, the following statements hold: (i) If I is an ideal of d K then I-' E F(dK). Moreover, IT-' is an ideal of d K.

Commutative Algebra

120

(ii) If M is a maximal ideal of d K , then d K # M-'. Consequently, # 1-l for any proper ideal I of d K . (iii) Let I be an ideal of d K , I # (0). If S I G I for some subset S K , then S C d K . (iv) If M is a maximal ideal of d K , then M M - l = d K . d K

Proof (i) If 0 # c E I then cT-' G d K . So I-' E F ( d K ) . That 11-' is an ideal of d K may be verified directly. (ii) Let 0 # a E M and ( a ) the ideal of d K generated by a. By Proposition 4.3, P I . .. P, C ( a ) C M for some prime ideals P I ,..., P,. Since M is a prime ideal, M = Pi for some i by Theorem 4.2. Assume Pi = PI without loss of generality. Choosing T to be smallest, we have P2 .. . P, 9 ( a ) . Taking b E PZ. . . P,, b @ ( a ) , we see that bM = bP1 C ( a ) = a d K implies a-lb E M - ' , b @ ( a ) = a d K implies a-lb @ d K . Thus, if I is a proper ideal of d K , then I C M for some maximal ideal M . It follows that M - l G I-l and I-' # d K . (iii) Since d K is Noetherian, assume I = d K w i , vi E I . Let s E Then S I 5 I yields

czl

s.

This means that the system of linear equations

1;

+

(a11 - S ) V l a12212 a21211 (a22 - s)v2

+

am1211

+ . . . + a1m21, + . .. + a2m21,

=0 =0

+ am2212 + . . . + (amrn- s)21,

=0

has a nonzero solution (211, ..., vm). Write A for the coefficient matrix of the system. Working over the field K , A is singular and hence det(A) = 0. This shows that s is a zero of a monic polynomial in d K [ 2 ] . Thus, s E d K by Lemma 1.1. (iv) By (i), M M - ' is an ideal of d K . Since d K C M - l , M C M M - l C d K . The maximality of M entails M = M M - l or M M - l = d K . If M = M M - l , then (iii) implies M-' C_ d K , contradicting (ii). It follows 0 that M M - l = d K . 4.7. Theorem With notation as before, the following hold:

The Ring AK in K = Q(8)

121

(i) If I is a nonzero ideal of d K , t,hen 11-' = d K . (ii) Every nonzero fractional ideal J E F ( d K ) has an inverse J - l E i.e., J J - l = d K . Therefore, F(AK) is a commutative multiplicative group.

F(dK),

Proof (i) Suppose that I1-l # d K , and that I is maximal among all such ideals (with respect to C ) . Let M be a maximal ideal of AK containing I . Then d K

c M-'

c 1-l implies I c IM-'

By Proposition 4.6(ii-iii), I By the choice of I ,

# IMP'.

( 1 M - l ) (IM-')-' = d and this shows that d K =

(1M-l) (IM-')-'

K

5 II-' c d K .

Note that IM-' is an ideal of

dK.

implies M-' (IMP1)-' 2 I-'

2 IT-' 2 d K ,

that is, II-' = d K , a contradiction. It follows that we must have 1I-l = dK.

0

(ii) This follows from part (i) and Lemma 4.5(ii).

a(-).

Example (ii) Let K = ( 2 , l - G)of d K has inverse

Then

d K =

Z[fl].

The ideal I =

We are ready to reach the unique factorization theorem of ideals in

dK.

4.8. Theorem Every nonzero ideal of d K can be written as a product of finitely many prime ideals, uniquely up to the order of factors.

Proof If there is some nonzero ideal I of d K which cannot be written as a product of finitely many prime ideals, we may assume I is maximal among all such nonzero ideals (with respect to C).Then I is not a prime and it is properly contained in some maximal ideal M . Thus, M-' 5 I-', I C IMP' d K , and by Proposition 4.6(ii-iii), I # 1 M - l . Since 1M-l is an ideal of AK, by the choice of I we have IM-' = PI . . . P, for some prime ideals P I ,..., P,. But this yields I = M P l . . . P,, a contradiction.

Commutative Algebra

122

Therefore, every nonzero ideal of d K can be written as a product of finitely many prime ideals. Now if I = pl. . . Pr = Q1 . . . Qs = J for prime ideals Pi, ..., Pr, & I , ...,Q S , then I C PI implies Qi PI for some Qi.So PI = Qi by their maximality. Assuming Qi = &I and multiplying I and J by PF', we have P 2 . . . pr = QZ. . . Qs and the proof of uniqueness may be done by an inductive demonstration. 0

a(-),

Example (iii) In d K = Z [ G ] C K = we know that 6 = 2 x 3 = (1 -)(1 - fl). But we have a unique factorization of (6) into prime ideals:

+

Q)(m),

Moreover, in d K = Z [ m ] C K = we know that 18 = 2 x 3 x 3 = (1 (1But we have a unique factorization of (18) into prime ideals:

+ m) m).

The reader is asked t o check that each ideal used in the above factorizations is prime. Finally, with the help of Theorem 4.8 we are ready to see how far is d from being a UFD. 4.9. Theorem d

K

K

is a UFD if and only if it is a PID.

Proof The "if" part is known. Now suppose d K is a UFD. In view of Theorem 4.8 it is sufficient t o show that every prime ideal of d K is principal. If P is a nonzero prime ideal of d K , then by Lemma 4.1 we may pick up some integer 0 # m E P and consider its unique factorization into irreducible elements in d K : m=YlY2"'Ys.

Then some ~i E P or (ri)G P . But each yi is a prime because d K is a UFD (Chapter 1 Theorem 2.9). It follows that (yi) is maximal and hence ( T i ) = P.

0

The next theorem will show that d K is only one pace away from being a PID (just as indicated in the diagram given in the preface of this book!).

The Ring AK in K = Q(8)

123

We demonstrate this result through several technical preliminaries.

+

4.10. Lemma (i) If I and J are ideals of d K such that I J = d K , then I nJ=IJ. (ii) Let P I ,...,P, be distinct prime ideals of d K , then PI . . . P, = nL='=,Pi.

0

Proof Exercise.

4.11. Proposition (i) Let I be a nonzero ideal of d K and P a prime ideal (hence maximal) of JtK. Then for any integer s >_ 1 there exists b E I such that bI-' Ps = d K . (ii) Let I , J be nonzero ideals of d K , and J = P:' . . . PTSTthe unique prime factorization of J in d K , where T 2 1, si 2 1, i = 1,...,T . If there is some 0 # b E n;='(I - IPi), then bl-' J =dK. (iii) Let I and J = Pfl. . . P:T be as in part (ii), then there is some 0 # b E nZ=.=,(I - IPi).

+

+

Proof (i) Note that I P c I but I P d K ) . Let 0 # b E I - I P . Then

bI-'

(1) If bl-' = d K , then bI-' we have,

# I (otherwise P

= I-l(IP) =I-'I=

9 P.

+ P" = d K . If bI-'

# d K , then by the above (1)

bI-l+ P = d K .

(2)

It follows that

bI-lP

+ P2 = P, and by substitution in (2),

bl-'

+P2= d K .

After repeating the same process for s - 2 times we arrive at

bI-'+ Ps = d K . (ii) By the assumption, bI-'Pi 9 Pi, i = 1,..., T . If b1-l = d K , then bI-' J = d K . If b1-l # d K , then a similar argumentation as in the proof of part (i) shows that

+

b1-l It follows that

+ P,"

= d K , i = 1, ...,T .

Commutative Algebra

124

After substitution step by step, we arrive at

bI-l

+J

= bI-'

(iii) For each i = 1, ..., r , let Ii

+ P:'

= IP1

. . . P,PT-'. Then

Ii = IP1...Pi-lPi+l...Pr

(3)

P,? = d K .

1 . .

g IPj, j # i .

Since Ii # IiPi (otherwise Pi = d K ) , there is bi E Ii - IiPi, i = 1, ...,r. Set b = bl

+ . . . + b,.

Then b E I and we claim that b @ IPi, i = 1, ..., r. To see this, suppose that b E IPj for some j . Then by the construction of b and (3) above, b j E IPj would hold. But we know that b j @ I j P j = ( I P i . . . P , P ~ ~ l ) P j = I P l . . ..P ,

By Lemma 4.10(ii), this yields r

bjI-l

(4)

Pl...P,.=

nPi. i= 1

Moreover, (5)

bj

E Ij implies

bjI-'

C I-lIj

= P I . . . Pj-lPj+l

... P,.

Thus, by (5) above, bj E IPj, or equivalently, bjI-l

g Pj

would yield

c pj n p 1 . .. pj-lpj+l... P, = npi, r

bjI-l

i=l

contradicting the foregoing (4). Therefore, we must have b E n;='(I- IPi).

4.12. Theorem Let I be a nonzero ideal of some b E I such that I = a d K b d K .

+

d K ,

0 # a E I . Then there is

Proof For 0 # a E I , set J = aI-'. Then by Proposition 4.ll(ii-iii), there is b E I such that bI-l aI-' = d K , and it follows that I = b d k adK.

+

+

To measure the extent to which ideals in d K are not principal, another topic in algebraic number theory is to study the class-group of K , that is the

The Ring AK in K = Q(8)

125

quotient group of 3 ( d ~by) the (normal) subgroup of principal fractional ideals.

From

d K

to Dedekind domains

Historically in the literature, if a normal Noetherian domain R also has the property that every nonzero prime ideal is maximal, then R is called a Dedekind domain after the mathematicians, such as Kummer, Dedekind in the 19th century and Noether in the 1 9 3 0 who ~ ~ developed the ring theoretic approach t o the study of d K . With no modification, all results about d K given after Theorem 4.2 in this section hold for a Dedekind domain R, in particular, 0 every nonzero ideal of R has a unique factorization into primes; every ideal of R can be generated by two elements; and 0 R is a UFD if and only if it is a PID. Moreover, there is also the class-group theory associated to the ideals of a Dedekind domain that has an intimate connection with the divisor class-group of an algebraic curve in algebraic geometry.

Exercises 1. Let a E d K be an irreducible element but not a prime. If ( a )= PI . . P, is the unique factorization of ( a )into prime ideals in d K , show that none of Pi is principal. 2. Show that ( 3 , l + G)is a prime ideal in d K = z [ G ] ,and that (1 is a prime ideal in d K = Z [ m ] . 3. Complete the proof of Lemma 4.5. (Hint: To reach (ii), note that if CJ c d K for some 0 # c E d K then CJ is an ideal of d K . ) 4. Show that the operation on 3 ( d ~is)well-defined and it makes 3 ( d ~ ) into a commutative associative semigroup with the identity element d K . 5. Complete the proof of Lemma 4.10. +

+ a)

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Chapter 5

Algebraic Geometry

This chapter demonstrates how Noetherian rings, field extensions, local rings, DVRs, normalization and localization stem naturally from algebraic geometry. Throughout the text K[zl, ..., 2,] denotes the polynomial ring in z1, ...,z , over a field K. If n = 2 we use K[z,y] instead of K[21,22], and similarly we use K [ z ,y, z ] in place of K [ Q ,2 2 , 2 3 1 . Moreover, let

4

A" = A" - P = (a1,...,a,)

,...,a, E K

1

stand for the n-dimensional ufine space (or afine n-space) over K. An element P E A" is called a point, and if P = ( a l ,...,a,) then ui is called the ith coordinate of P . It is clear that in the case where K = R, A' = R is the real affine line, A2 = R2 is the real affine plane, and A3 = R3 is the real affine space.

1. Finite Field Extension and Nullstellensatz Let f E K [ z l ,..., 2 , ] . Then f defines a function

4f:

A"=AE P = ( U l , ...,a,)

+

H

K f(P)= f(a, ..., a,)

If f = c E K is a constant, then $ f ( P ) = c defines a constant function. If f,g E K [ z l ,..., z,], then, as usual functions, ( 4 f + # g ) ( P ) = 4p(P)+4,(P) and (4f49HP) = 4f(P)49(P). 127

Commutative Algebra

128

1.1. Definition Let T be a subset of

{

I

K[zl, ..., z,], and

V(T) = P E A" dP(p)= f ( p )= 0 , for all f

E

1

T ,

that is, V ( T ) is the set of common zeros of all polynomials in T, or equivalently the set of solutions of the system of polynomial equations

We call V(T) an afine algebraic set (or just an algebraic set) of the affine If T = {fl, ..., fs} is a finite subset, we also write n-space A" = A;. V(T) = V(f1, ...7 fs). If f E K [ x l ,...,z,] is not a constant, the algebraic set V(f) is called the hypersurface defined by f;if deg(f) = 1, V ( f ) is called a hyperplane in A". A hypersurface in A2 is called a plane curve; and a hyperplane in A2 is called a line.

Example (i) A" and 0 are algebraic sets (indeed A" = V(O), 0 = V(1) = V(c) for any c E K X ) . (ii) Conics in IR2 defined by quadratic polynomials in R[z,y] are plane curves; while C = V(y2 - x3) c A: is known as the cuspidal curve and C = V(y2 - z3- z2) c A: is known as the nodal curve. Moreover, V = V(y - x2,z - x3) c A; is known as the twisted cubic in IR3. 1.2. Proposition (i) Let T be a nonempty subset of K[xl, ...,zn] and = (T) the ideal generated by T in K [ z l ,...,z,]. Then V ( T ) = V(I) = V(f1, ...,fs) for finitely many fl, ...,fs E T . (ii) If TI C T2 are subsets of K [ z l ,..., z,], then V(T1) 2 V(T2). Y 2 are algebraic sets, then so is Y 1U Y2.Consequently, the union (iii) If Y1, of any finitely many algebraic sets is an algebraic set. (iv) If { Y , } i G ~is a family of algebraic sets, then nicJyZ is an algebraic set.

I

Proof The existence of f l , ..., fs follows from the fact that K[Q, ...,zn] is Noetherian (Chapter 1 Theorem 1.3). All other assertions may be directly I3 verified by means of part (i) and Definition 1.1. Proposition 1.2(i) establishes the initial connection between ideals and algebraic sets that makes the flexibility of finding easier defining equations of a given system of polynomial equations.

Algebraic Geometry

129

Example (iii) Let V = V(fl, f 2 ) and I = ( f l , f 2 ) , where f l = z2 - zy y2 z 2 , f 2 = z2 - y2 z 2 - z E R[z,y,z]. Then since I = ( f l , f 2 ) = (fi, f i c f i ) for any nonzero c E R, it follows that, after setting c = -1, V = V(f1, z - zy), i.e., every point P E V is of the form P = (z,y,zy).

+

+

+

(iv) Since for ai E k, V(z1 - al, ..., z, - a,) = {P = (al, ..., a,)} c A", it follows from Proposition l.a(iii) that any finite subset of A" is an algebraic set.

For ideals in K [ z l ,...,z,], the operations sum, product, and intersection correspond to operations of algebraic sets. 1.3. Proposition Let I and J be ideals in K [ z l ,..., x,]. The following hold: (i) V(I J ) = V(I) n V ( J ) . (ii) V(I . J) = V ( I ) U V(J). (iii) V(I n J) = V ( I )U V ( J ) .

+

Proof Exercise. Now, if we start with a subset Y E A", then Y may be associated to a set of polynomials

It is easily seen that I(Y) is an ideal of K[zl, ...,z,]. I(Y) is called the ideal of Y in K[zl,..., x,]. As we will see from now on, it is this ideal that makes the essential connection between algebraic structure theory and the geometry of algebraic sets. 1.4. Proposition (i) If J is an ideal of K [ z l , ..., z,], then J C_ I ( V ( J ) ) , and V(I(V(J)))= V ( J ) . (ii) For any subset Y g A",V(I(Y)) = B is the smallest affine algebraic set containing Y. (iii) For any two subsets Y1, Y2 of A",we have I(Y1 U Y2) = I(Y1) n I(Y2). (iv) Let V and W be affine algebraic sets in A". Then: (a) V C W if and only if I(V) 2 I(W). (b) V = W if and only if I(V) = I(W).

Proof Exercise.

0

Commutative Algebra

130

Example (v) If K is infinite then I(A") = (0) (see later exercise 2). If P = (a1,a2,..., a,) E A", then I({P})= (z1-al,z2-a2, ..., z,-a,) which is a maximal ideal of K [ z l ,...,z,] (Chapter 2 section 1, Example (iii)). (vi) For V = V(y - x 2 ) c A;, I(V) = (y - z2). This may be verified by noticing that any monomial z*yP in R[z,y] can be written as Ci(yz2)hi za+2P where hi E R[z,y], and any point P E V is of the form P = (z,z2), z E R.

+

(vii) Note that (z2,y2)is properly contained in I(V(z2,y2))= (z,y). So in general J is not necessarily equal to I(V(J)).

So far we have built two mappings {ideals} 3 { affine algebraic sets}

J

H

V(J)

and {affine algebraic sets}

V

A {ideals}

-

I(V)

Note that by Proposition 1.4 the second mapping is injective. The first mapping, however, is not necessarily injective: diflerent ideals can define the same algebraic sets. For example, the ideals (z) and ( x 2 ) are different in K[z], where K is an arbitrary field, but V(x) = V(z2) = ( 0 ) in Ak. Also consider the ideals of R[x]: J1 =

(1) = R[z],

52

= (1

+ z2),

53 =

(1

+ z2 + z4).

We see that Ji # J k if i # k. But V(1l) = V(l2) = V(13)= 0. Similarly, (z3- 1) # (zy2 - y2 z - 1) in R[z, y], but both ideals correspond to the same algebraic set ((1,y) I y E EX}. The wonderful thing is that if the ground field K is algebraically closed then the Hilbert's Nullstellensatz solves the problem entirely.

+

1.5. Theorem (weak Nullstellensatz) If the field K is algebraically closed, then there is a natural one-to-one and onto correspondence A" (a1,...,a,)

-

m-SpecK[xI, ..., zn]

H (21 - a1, "',Z,

- a,)

Algebraic Geometry

131

Proof For each point P = (al, ...,a,) E An, that (51 - a ] , ...,z, - a,) is a maximal ideal of K[zl, ..., z,] follows from Chapter 2 (section 1, Example (iii)). Conversely, let M E m-SpecK[zI, ..., z,] and write K[&,...,%] = K[zl, ...,z,]/M, where each Tiis the image of z i in K[zl, ...,z,]/M. Then the natural K-algebra homomorphisms

K

Lf

-

K [ q ,..., z,] IK [ T l ,..., z,]

yield a finitely generated field extension K c K[T1,...,Cn] (here K and x ( K ) are identified). By Chapter 3 Theorem 2.5, K[31,...,?En] is algebraic over K . But K is algebraically closed by the assumption. Hence K = K [ T l ,...,En].Writing Ti = bi E K , i = 1,..., n, then x is just given by

fb1, ...,4 H f(h, and it follows that Kerx = M = (z1 - bl, ..., z,

*..I

bn)

- b,) by Chapter 2 (section 1, Example (iii)). That the correspondence is one-to-one and onto is easily seen now.

1.6. Theorem (Nullstellensatz) Let K be an algebraically closed field, and let J be an ideal of K [ q ,..., z,]. (i) If J # K [ q ,...,z,], then V ( J ) # 0. (ii) I ( V ( J ) ) = where fl = {f E K [ z l,..., x,] 1 f" E J for some m 2 l} is the radical of J (Chapter 2 section 1, exercise 2 ) .

a,

Proof (i) By the assumption and Theorem 1.5, J & M = (XI- a l l ..., Z,

-

a,)

for some P = ( a l , ...,a,) E An. Thus, P E V ( M ) V ( J ) . (ii) By the definition of the inclusion f i I ( V ( J ) )is clear. To reach the inclusion 2 I ( V ( J ) ) , let J = (fl, ..., fs), and 0 # f E I ( V ( J ) ) . Then S = (1,f,f 2 , ..., f", ...} is a multiplicative set of K [ x l ,..., zn].Consider the ring of fractions K [ z l ,..., z,]~. By Chapter 2 (section 3, Example

a,

(4 )I

Note that K [ z l ,..., zn]s# (0). So (yf - 1) # K[zi, ..., zn][y]. But we claim

Commutative Algebra

132

for, Theorem 1.5 asserts that there is no maximal ideal of K [ z l ,...,zn,y] containing J'. Thus, S

(1)

1 = c P i ( 5 1,...,z,,

Y)fi +q(zl,.-,z,,Y)(l- Y f )

i= 1

for some polynomials p i , q E K [ q ,...,x,,y], NOW, going back to K[x1, ..., x , ] ~via y H $ (see Chapter 2 (section 3, Example (ii))), the relation (1) above implies that

After clearing all denominators of Eq. (2) by a suitable power f m , we obtain S

(3) i=7

for some polynomials gi E K [ z l ,...,xn],as desired.

0

a,

If an ideal J c K[xl, ...,z,] has the property that J = then J is called a radical ideal. It is clear that I(V(J ) ) is radical, and every prime ideal is radical. But not every ideal is radical, as easily illustrated by the ideal (z2,y2) c R[x, y] (also see later exercise 9). Working with radical ideals, the Nullstellensatz establishes the following perfect correspondences.

-

1.7. Theorem If K is algebraically closed, then the mappings {affine algebraic sets} and {radical ideals}

V

I

{radical ideals)

{affine algebraic sets}

are inclusion-reversing bijections which are inverses to each other.

I7 Theorem 1.6 answered when an algebraic set is nonempty, provided the ground field is algebraically closed. The next proposition answers how to algebraically recognize the finiteness of an algebraic set V(J ) .

Algebraic Geomety

133

1.8. Theorem Let J be an ideal of K [ z l ,...,x,], V = V(J), and R = K[Xl,

...,xn]/J.

(i) If dimKR < 00, then V is finite. (ii) If K is algebraically closed and V is finite, then dimKR

< 00.

Proof (i) Let dimKR = rn and Zi the image of xi in R , i = 1,..., n. Then

AmZy

+ X , _ I ? E ~ - ~ + . . . + X1Zi + XO = 0, for some X i E K .

This implies the one-variable polynomial C,"=,xi = f(q) E J . Hence, f (xi)vanishes a t every point of V. Since f (xi) can have only finitely many zeros in K , it follows that the points of V have only finitely distinct ith coordinates, i = 1, ...,n. This shows that V is finite. (ii) Note that K is algebraically closed. If V = 0, then 1 E J by Nullstelensatz, and hence dimKR = 0. Suppose V # 0, and let aij be the distinct ith coordinates of all s points in V , i = 1,...,n, j = 1,..., s. Define the one-variable polynomials S

= n ( x i - aij),

i = 1,...,n.

j=1

Then f i vanishes a t every point of V . By Nullstellensatz, fyi E J for some mi 2 1. Noticing that fi is a monic polynomial in one variable, a formal division on elements of K [ q ,...,xi-1,xi+l, ...,xCn][xi] by fi, i = 1,...,n, shows that dimKR 5 00.

Exercises 1. Prove Propositions 1.3-1.4. 2. Let f be a nonconstant polynomial in K [ x l ,...,x,], where K is algebraically closed. Show that A" - V(f) is infinite if n 2 1, and V ( f ) is infinite if n 2 2. Conclude that the complement of any algebraic set is infinite. (Hint: Note that K is infinite by Chapter 1 (section 3, exercise 5). f defines the zero function on K" if and only if f = 0 in K [ z l ,...,xn].) 3. Show that if K is infinite then I(A") = (0). Is this still true if K is finite? 4. Consider the plane curve C = V ( f ) c A& where f is a polynomial of degree n in K[z,y]. If L is a line in A$ and L $ C , show that L n C is a finite set of no more than n points. (Hint: Suppose L = V (y - ( a x b ) ) , and consider f(x,ax b) E K [ x ] . ) 5. Let V = V(flg) c A:, where f = x2 - y 1, g = y x2 - 5.

+

+

+

+

Commutative Algebra

134

+

(a) Show that V = V ( z 2 - y 1,z2 - 2). (b) Use part (a) t o determine V . 6. This exercise is to demonstrate that factorization of polynomials may help to determine an algebraic set. (a) Show that if g E K [ z ~...,, zn] factors as g = 9192, where K is a field, then for any f,V ( f , g ) = v ( f , 91) u v ( f t 92). (b) Show that in A$, V ( y - x 2 ,z t - y3) = V ( y - x2,xz- z4). (c) Use part (a) to determine the algebraic set in part (b). 7. For V = V ( z y - 1) c A;, show that I(V) = (zy - 1). 8. For the twisted cubic V = V ( y - z2,z - z3)c A:, show that I(V) = (y-x2, z - z 3 ) . (Hint: Any monomial ~ ~ y f l in z ?R[z, y, z] can be written as c i ( y - z2)hi C . ( z - z3)gj za+2fl+3Y, and every point P E V is of the form P = (z, z4 ,z3).)

+

+

+

9. Let J = (z,y3 1) c R[z,y]. Show that J # fi, and that dimRR[z, y ] / a is finite. 10. Let I = (2, y) c K [ z ,y], where K is a field. Show that dimK(K[x, y]/I”) - “(“:l) for all n 2 1. 11. Let J be a proper ideal of K[z1, ..., z,]. If K is algebraically closed, show that fi = I ( V ( J ) ) = nM, where M runs over all maximal ideals containing J .

2. Irreducible V and the Prime I ( V )

To study the points of an algebraic set in an “analytic-like” way, we need irreducible algebraic sets and use only “polynomial functions and rational functions” that respect the Zariski topology on the afine n-space An = An K over a field K (this viewpoint will he clarified in the next section). Some basic notions and examples on topological space are given in the appendix of this section. 2.1. Definition The Zariski topology on A” is defined by taking the open subsets to be the complements of affine algebraic sets. For any subset Y c A”, the Zariski topology on Y is the induced topology.

Example (i) Let us consider the Zariski topology on the affine line A1 = K . Every ideal of K[z] is principal, so every algebraic set is the zero locus of a single polynomial. Thus the algebraic sets in A’ are just the finite subsets

Algebraic Geometry

135

(including the empty set) and the whole space (corresponding t o f = 0). Consequently the open sets are the empty set and the complements of finite subsets. (ii) In general, the Zariski topology on A" (TI 2 1) is not the product topology of the Zariski topology on A'. For example, if we identify A' x A' with A2 and consider any curve C = V(y - f(z)) in A', where f(z)is a polynomial in K [ z ]of degree 2 1, then from (i) above it is easy to see that the open subset U = A2- C in A2 cannot be a product of two open subsets of A'. 2.2. Definition A nonempty subset Y of a topological space X is irreducible if it cannot be expressed as the union Y = Yl U Y2 of two proper subsets, each of which is closed in Y(where Y has the induced topology). The empty subset 0 is not considered to be irreducible. If X is not irreducible then it is reducible.

Example (iii) If K is infinite, then A:, is irreducible, for its only proper closed subsets are finite sets. (iv) Similarly, by Proposition 1.3(ii) and (section 1, exercise 3), An is irreducible provided K is infinite. (v) V = V(32+y-1, y2 - 2 ) is reducible in A;, for V consists of two points. More reducible algebraic sets may be obtained via Proposition 1.3(ii). The following proposition states two basic properties of irreducible subsets in a topological space. The first one plays a key role in algebraic geometry because of previous Example (iv). 2.3. Proposition (i) Any nonempty open subset of an irreducible space is

dense and irreducible. (ii) If Y is an irreducible subset of the topological space X , then its closure Y in X is also irreducible.

Proof (i) Let U , V be nonempty open subsets of the irreducible space X . Then U n V = 0 leads to X = (X - U ) U (X- V), a contradiction. Hence U is dense. If U = U1 U Uz for closed U1, U2 in U , then there are proper closed V1, V2 in X such that V1 n U = U1, V2 n U = U2. But then X = (VI U V2) U ( X - U ) , both V, UV2 and X - U are proper closed subsets.

136

Commutative Algebra

Once again this is a contradiction. (ii) Exercise.

0

To determine the irreducibility of an algebraic set V algebraic way, let us write

c An in a purely

K[V] = K [ q ,...,z,]/I(V). K[V] is called the Coordinate ring of V (a remark on this name is given in the beginning of section 3). 2.4. Theorem Let V C An be an affine algebraic set. Then V is irreducible if and only if I(V) is a prime ideal if and only if its coordinate ring K[V] is a domain.

Proof First, assume that V is irreducible and let f g E I(V). Set V1 = V n V ( f ) and V2 = V n V(g). Then fg E I(V) implies that V = V1 U V2. Since V is irreducible, we have either V = V1 or V = Vz. Say the former holds, so that V = Vl = V n V ( f ) . This implies that f vanishes on V, so that f E I(V). Thus, I(V) is prime. Next, assume that I(V) is prime and let V = V1 U V2. Suppose that V # V1. We claim that I(V) = I(V2). To prove this, note that I(V) c I(V2) since V2 c V. For the opposite inclusion, first note that I(V) is properly contained in I(V1) since V1 is properly contained in V. Thus, we can pick f E I(V1) - I(V). Now take any g E I(V2). Since V = V1 U Vz, it follows that fg vanishes on V, and, hence, fg E I(V). But I(V) is prime, so that f or g lies in I(V). We know that f $ I(V). So, g E I(V). This proves I ( V ) = I(V2), whence V = V2 by Proposition 1.4(iv)(b). Therefore V is an irreducible algebraic set. 0

In the “algebra-geometry” dictionary of section 1, algebraic sets in An are associated with ideals in K [ z l ,...,z,]. Note that the ring homomorphism K[zl,...,z], 4 K[zl,...,z,]/I(V) = k[V] yields the following oneto-one and onto mappings between the given sets: (a) {ideals of K[zl,...,z,] containing I(V)} and {ideals of K[V]}. (b) {radical ideals of K [ z l ,...,x,] containing I(V)} and {radical ideals of K[VI}. (c) {prime ideals of K[x1, ...,z],

containing I(V)} and {prime ideals of

K[VIl. (d) { maximal ideals of K[zl,...,zn]containing I(V)} and {maximal ideals

Algebraic Geometry

137

of K[V]}. Consequently, the following corollary of Theorem 1.7 is obtained. 2.5. Corollary Let K be an algebraically closed field and let

V c A" be

an affine algebraic set. (i) The correspondences

{

affine algebraic subsets

wcv

I

are inclusion-reversing bijections and are inverses to each other, where IV denotes taking ideal I(W) with W as indicated above and VV denotes taking algebraic set V(J ) with J as indicated above; we use IV, respectively Vv,just for the restrictions of I and V respectively. (ii) Under the correspondence given in part (i) , irreducible algebraic subsets, respectively points of V, correspond to prime ideals, respectively correspond to maximal ideals of K [ V ] .

Proof Exercise.

0

Example (vi) If we plot the twisted cubic V = V ( y - x2,z - x3) c A; in a limited space, then an intuitive feeling tells that V is irreducible.

To convince ourselves, however, Theorem 2.4 requires proving that I( V )

Commutative Algebra

138

is a prime ideal of R[z, y,z], that is, if fg E I(V), we have to show that f E I ( V ) or g E I(V). For any P = (a, b, c) E V, note that setting X = a yields

a=X,

b=X2,

c=X3.

Conversely, any X E A' = R determines a point of V as defined above. Thus, we have f(X,X2,X3)g(X,X2,X3) = 0 for all X E A'. Consider the one-variable polynomials f l ( t ) = f ( t , t 2 ,t 3 ) , g1(t) = g ( t , t 2 , t3) E Rjt], and let Vj, = { A E A' I f l ( X ) = f(X,X2,X3) = O}. If Vj, = A' then f vanishes on V and hence f E I(V). If Vj, # A', then g1 vanishes on the nonempty open subset U = A' - Vj, of A'. It follows from Proposition 2.3(i) that g1 vanishes on A', and consequently g vanishes on V. This shows that g E I(V), as desired. (Without using the density of U , the proof may also be completed by applying the infiniteness of U to a one-variable polynomial.) Observe that the last example has used two important facts: (a) There is a one-to-one and onto correspondence

A1=K+ X

V

H (XlX2,X3)

(b) A1 is irreducible (so that Proposition 2.3(i) can be used). The above observation shows the correct way to introduce polynomial mappings and rational mappings between algebraic sets, in order to compare the properties of two algebraic sets (such as irreducibility) with respect to the Zariski topology. 2.6. Definition Let V 5 A", W C A" be afine algebraic sets. (i) If there exist polynomials f l , ...,fn E K[zl, ...,zm]that define a mapping

#:

v-

W

then # is called a polynomial mapping from V to W ;if there is also a polynomial mapping $I: W 4 V such that the composite mapping $4 = l v , the identity mapping on V, and $$I = l w , then # is called a (polynomial)

Algebraic Geometry

139

isomorphism with the inverse (this name is qualified by Proposition 2.7 below). If furthermore V = A" then # is called a polynomial parametrization of (ii) If there exist elements ..., E K(x1, ..., x,), where K(x1,..., x,) is the field of fractions of the polynomial ring K [ z ~...,, x,], that define a mapping

w.

k, 5

#:

u= v

n -

i=1

then then

-

UV(g,)

W

# is called a rational mapping from V to W ;if furthermore V # is called a rational parametrization of W .

=

A"

By definition, in Example (vi) above the correspondence X H (A, X2, X3) defines a polynomial parametrization of the twisted cubic V = V ( y - x 2 ,z 23).

2.7. Proposition (i) Polynomial and rational mappings are continuous with respect to Zariski topology. (ii) Let V C Am, W A" be affine algebraic sets, and let # be a (polynomial) rational mapping from V to W . (a) Suppose 4 is onto. If X is a closed subset of W and its preimage d - ' ( X ) in V is irreducible, then X is irreducible. (b) If V = Am and the image of # in W is dense, then W is irreducible. Proof (i) It is sufficient to prove the assertion for rational mappings. Suppose # is defined by n

4

:

u= v - U V ( g 2 ) +

W

i= 1

where ,..., Sn li E K ( z 1 ,..., x,). If WI = V ( h 1 ,..., h,) n W is a closed subset of W , where h l , ..., h, E K [ y l ,...,y n ] , then P E U and # ( P >E WI

Commutative Algebra

140

implies for i

Write hi

=

1, ...,s

(t,..., k) 5,i

= 1, ..., S , where

=

Fi, Gi E K[xl,...,x,]. Then

it follows from equation (1) that Fi(P) = 0, G,(P) # 0, i hence

P

(2)

E V(Fl, ..., F,)

=

1,..., s, and

n U.

On the other hand, if Q E V(F1, ..., F,) n U ,then Fi(Q) = 0 implies - 0, i = 1, ..., s, and hence equation (1) holds for Q, that is, $ ( Q ) E G,oV(hl, ..., h,) n W = W1. This shows that #(V(Fl,...,F,) n U ) = W1, and consequently # is continuous. (ii) The assertion of part (a) follows from part (i) directly. We prove (b) by showing that I(W) is a prime ideal. Let # be defined by F'(Q)

4:

u-

V

L E K(yi, ...,ym), and # ( U ) is where U = A" - (uZlV(gi)), LL gl,..., , 9n dense in W by the assumption. Suppose f g E I(W) where f , g E K[zl,...,x,]. Then 0 = (fg)(Q) = f(Q)g(Q) for all Q E W . But this implies 0 = f ..., (P)g ..., ( P ) for all P E U. Since

(k , 5)

f

(2, k)

(k,..., k ) ,g (k,..., k ) E ~ ( y l...,ym), , we can write j ( k ,..., 5)

=

G > g ( fi g l l " ' l gfik) n = G'l where F, F', G, G' E Kjyl, ..., ym] with G ( P )# 0 and G'(P) # 0 for all P E U . Thus, if we put

v, = { P E u I f ( k ,...)5)( P )= o} vg

=

{ P E u 1 g ( k ,...)5) ( P )= o}

then V, = V(F) n U , V, = V(F') n U . Now, if V, = U,then f ( # ( U ) ) = 0 and hence f ( W ) = 0 because +(U) is dense in W . So f E I(W). If V, # U , then since (fg)(Q) = 0 for all Q E W , V,, and consequently V ( F ' ) must contain the nonempty open subset U - V' = U - V ( F ) . It follows from

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141

Proposition 2.3(i) that V, = U . Therefore] g(+(U)) = 0 and g(W) = 0 by the density of +(V)in W . Consequently, g E I(W) as desired. 0

Remark (i) By Definition 2.6(i) and Proposition 2.7, algebraic sets may be classified in terms of “polynomial isomorphism”. In a general theory of algebraic geometry] rational mappings provide models of abstract rational morphisms that yield classification of (quasi-)varieties in terms of “birational isomorphism”. (ii) Practically, polynomial parametrization and rational parametrization are very useful in 3D-plotting of curves and surfaces, geometric modelling, and CAD (computer aided design). Since K [ q ,...] z], that 0

is Noetherian, it follows from Proposition 1.4(iv)

A” is a Noetherian space in the sense that every collection of algebraic sets in A” has a minimal member with respect to (check it!).

This observation enables us to prove the following decomposition theorem for affine algebraic sets. 2.8. Theorem Let V be an algebraic set in A”. Then V can be expressed uniquely as a union of finitely many irreducible algebraic sets Vl1...] V, that is, V = Vl U . . . U V, and V , @ V, for all i # j .

Proof Let R

=

{algebraic sets V

C An V

is not a union of finitely many irreducible algebraic sets

We claim that R is empty. If not, let V be a minimal member of R. Since V E R, V is not irreducible. Thus, V = Vl U V, where V , are proper closed subsets of V, and V, @ R. Hence V , = V , 1 U . . . U with V , j irreducible. But then V = Ui,j&jl a contradiction. Therefore 0 = 0. So any algebraic set V may be written as V = V1 U . . . U V, where each Q is irreducible. To get the second condition] simply throw away any & such that V , c vj for i # j . To show uniqueness, let V = Wl U . . . U W, be another such decomposition. Then V , = U j ( W j n V,), so V , c Wj(i) for some j ( i ) . Similarly] Wj(i)c V k for some k . But V , c V k implies i = kl so V , = Wj(i). Likewise each Wj is equal to some 17

142

Commutative Algebra

The V , appearing in the theorem are called the irreducible components of V ; V = Vl U . . . U V, is the minimal decomposition (or sometimes, the irredundant union) of V into irreducible components.

Example (vii) Consider the algebraic set V = V(zz,yz). Then V is a union of a line (the z-axis) and a plane (the zy-plane), both of which are irreducible.

fr'f;'

(viii) Let f E K [ z l ,..., zn]and f = . . . f,". the factorization off into irreducible polynomials. If K is algebraically closed, then, by Nullstellensatz, it may be derived that V(f) = V(f1)L.J...UV(fr)is the decomposition of V(f) into irreducible components. Moreover, it is an exercise to check that I(V(f))= ( f l f 2 . . . fr). (ix) Note that any P E V = V(zz - y2, z3 - z5) c A; is either the form (A3, A4, A5) or the form (-A3, -A4, -A5), where A E R. If we plot a part of V , then it seems that V is a union of two irreducible curves.

In later exercise 8 the reader will be asked to prove that it is the case indeed. Theorem 2.8 can also be stated in a purely algebraic way using the one-to-one correspondence between radical ideals and algebraic sets.

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143

2.9. Theorem If K is algebraically closed, then every radical ideal I in K [ x l , ..., z,] may be written uniquely as a finite intersection of prime ideals: I = PI n . . .nP, where Pi@ Pj for i # j . (Such a presentation is called the minimal decomposition or the irredundant intersection of a radical ideal.) Proof This follows immediately from Theorems 1.7 and 2.8.

0

Remark Theorem 2.9 is indeed the geometric model of a general primary decomposition theory in commutative algebra. We complete this section by seeking irreducible hypersurfaces V ( f ) defined by f E K[zl, ..., z,]. I f f is irreducible then the ideal ( f ) is a prime ideal, and hence = (f). (a) If K is algebraically closed, then I ( V ( f ) ) = = ( f ) and hence V(f) is irreducible. (b) If K is not algebraically closed, then exercise 2 of the current section shows that V ( f ) is not necessarily irreducible. (c) If K is not algebraically closed but f E K[x,y] and V ( f ) is infinite, then I ( V ( f ) )= ( f ) and consequently V ( f ) is irreducible by Theorem 2.4. The assertion (c) is argued in detail as follows.

m

m

2.10. Proposition Let f and g be polynomials in K [ z ,y] with no common divisors, and let V = V(f, g) be the algebraic set defined by f and g. Then V = V ( f ) n V(g) is a finite set of points. Proof f and g have no common divisors in K[x][y],so they also have no common divisors in K(z)[y] by Chapter 1 (section 2, exercise 7). Since K(5)[y] is a PID, p f qg = 1 for some p , q E K(x)[y]. Let d E K [ z ]be such that dp = a, dq = b E K [ z ,y]. Then af bg = d. If P = (u, v) E V, then d ( P ) = d(u) = 0. But d has only a finite number of zeros. This shows that only a finite number of x-coordinates appear among the points of V. Since the same reasoning applies to the y-coordinates, there can only be a finite number of points. 0

+

+

2.11. Corollary I f f is an irreducible polynomial in K[x,y], and if V(f) is infinite, then I(V(f)) = ( f ) and V ( f ) is irreducible.

Proof If g E I(V(f)) then V(f,g) is infinite. By the proposition, fig, i.e., 9 E (f). 0

144

Commutative Algebra

2.12. Corollary Suppose K is infinite. The irreducible algebraic subsets of A2 are: A2, points, and irreducible plane curves V(f), where f is an irreducible polynomial and V ( f )is infinite. (Note that we do not view 0 as an irreducible algebraic set, as we indicated in Definition 2.2.)

Proof Let V be an irreducible algebraic set in A2. If V is finite or I(V) = (0), V is of the required type. Otherwise, I(V) contains an irreducible polynomial f because I(V) is now a prime ideal. Then we claim I(V) = (f); for if g E I(V), f A g , then V c V ( f , g ) is finite by Proposition 2.10, contradicting the infiniteness of V.

Appendix. Topological space Given a nonempty set X , a topology on X is a collection T of subsets of X satisfying (a) 0 E T , X E T , (b) the union of any family of sets in T belongs to T , and (c) the intersection of finitely many sets in T is also in T . The sets in T are called open sets of X , and their complements in X are called closed sets. A set X equipped with a topology T is then called a topological space, and the elements of X are usually called points. Given a topological space X , a neighborhood of a point x E X is any set that contains some open set containing x.

Example (i) Any set X can be made into a topological space by giving the discrete topology, in which every subset (including empty set) is open. (ii) The real line R is a topological space, where a set U is open if it is a union of (finitely or infinitely many) open intervals. (iii) The real Euclidean space R" is a topological space, where the open sets are defined as the union of open balls. (iv) Any metric space X is equipped with the topology in which open sets are unions of open balls. (v) Every manifold (if you know what it is) is a topological space. Topological spaces appear in all branches of modern mathematics, in which it is necessary t o study the convergence, connectedness and continuity.

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145

Let X and Y be topological spaces and f : X + Y a mapping. f is said to be a continuous function if the preimage of every open set in Y is an open set in X . Suppose that f is a one-to-one and onto mapping with inverse f - l . If both f and f-' are continuous, then f is called a homeomorphism, and in this case, X and Y are said to be homeomorphic. Homeomorphic topological spaces have essentially the same topological structure. Any subset Y of a topological space X has the induced topology, in which, every open set is of the form Y n U where U is a n open set of X . Given topological spaces X I ,X 2 , ..., X,, the Cartesian product X I x X2 x . . . x X , has the product topology given by the products of open sets. Finally, let X be a topological space, Y a subset of X . The closure of Y in X , denoted Y,is the smallest closed subset in X containing Y , i.e., Y = nw where W runs over all closed subsets of X containing Y . A subset Z of X is said to be dense in X if Z n U # 0 for every nonempty open subset U c X , or equivalently, if = X .

z

1. 2.

3.

4.

5. 6.

7.

8. 9.

Exercises Show that V = V ( y - f (x))c A; is irreducible. (This exercise and the next help us to better understand the Nullstellensatz.) Show that f = y2 x 2 ( x - 1)2 E IR[x,y] is an irreducible polynomial, but that V ( f ) is reducible. Show that if f E @[XI,...,x,] is irreducible, then V ( f )is irreducible. Also show that if V = V(g) is an irreducible hypersurface in A& there is no irreducible algebraic set W such that V c W c A& W # V , W # A;. Show that any linear subspace of K" = A;, where K is a field, is an irreducible algebraic set. (Hint: Use Proposition 2.7.) Use exercise 3 above to show that V(y2-x(x-l)(x-X)) is an irreducible curve in A& where X E @. Let$: AA -+ V =V(y2-x3-x2) bedefinedby$(c) = (c2-1,c(c2-l)). Show that q5 is one-to-one and onto, except that $ ( f l ) = (0,O). Hence V is irreducible by Proposition 2.7 (see also section 6, Example (i)). Find the irreducible components of V(y2 - xy - x2y x3) in A;, and also in A;. Let 1 = (ZZ - y2, z3 - x'). Show that V(1) is a union of two irreducible algebraic sets. Let f E C [ q ,..., z], and let f = .. . f,". be the decomposition of f into irreducible factors. Show that V ( f ) = V(f1) U ... U V ( f r ) is

+

+

fr'

Commutative Algebra

146

the decomposition of V(f) into irreducible components and I(V(f)) = (flfi...fr).

10. Give a detailed proof of Proposition 2.3(ii) and Proposition 2.5, respectively. 11. Let V C A", W C A" be algebraic sets. If (p: V -+ W is a polynomial mapping defined by P H (fl(P), ...,f n ( P ) ) where f i ,...,fn E K[zi, ..., x,], show that (p*: K [ W ] -+ K [ V ]defined by (p*(?j) = g(f1,...fn) is a K-linear ring homomorphism (i.e., a K-algebra homomorphism); if furthermore (p is an isomorphism then (p* is a ring isomorphism. Conversely, if $I: K [ W ] K [ V ]is a K-linear ring homomorphism such that $I(?ji)= i = 1,..., n, show that q!Y: V 4 W defined by $I*(P) = (fl(P), ..., f n ( P ) )is a polynomial mapping. Thus we may conclude that (p is an isomorphism if and only if & is an isomorphism. -+

Ti

3. Point P and the Local Ring

OP,"

By Theorem 2.8, the study of an algebraic set is reduced t o the study of its irreducible components. In this section we demonstrate how a point P of an irreducible algebraic set V determines a unique local ring Op," consisting of rational functions defined at P. Let V C An = A& be an irreducible algebraic set, I ( V ) its ideal, and K [ V ]= K [ x ~ , . . . , ~ ~ ] its / I (coordinate V) ring. For f , g E K [ x ~ , . . . , x ~ ] , write 7,sfor their images in K [ V ] .Then = 3 if and only if f - g E I ( V ) . Thus, every E K [ V ]defines a function

7

7

-

(pf: V + p

K

f(P)

7.

which is independent of the choice of representatives of Moreover, let W be a closed subset of K = A' with respect t o the Zariski topology on A', i.e., W = V(g) for some g E K [ z ]and W is finite. Then one checks that *(V(g(f)) n V ) = W.

This shows that (pf is a continuous function with respect to the Zariski topology. (pf is called a polynomial function on V . Note that if P = (a', ...,a,) E V , then (pei(P)= ai for i = 1,..., n. And since K [ V ]is

Algebraic Geometry

147

generated by all Zi’sover K , this qualifies the name “coordinate ring”

KFI. Also note that K [ V ]is a domain by Theorem 2.4. We seek more continuous functions on V by looking a t the field of fractions of K [ V ] denoted , K ( V ) . For 0 # E K ( V ) ,set

$

1 2.5

-

d E K [ m ,..., z,]

E K[V]}.

Then J z- is an ideal of K [ q ,..., x,] and I ( V ) C J f (check this!). 9

9

(J

3.1. Lemma V J f

=

{P

E

1

v f

=

Proof By definition, this can be verified directly.

0 -

The algebraic set V

is called the pole set of

f

in V .

-

=

Example (i) Let V = V(y3 - xz) c A3. Then for f9 =

E

K(V),

= {(z,y,z) E A3 I y = t = 0 ) .

For

$ E K ( V ) ,let

Note that if V

# 8 then U-j- # 8 (why?). Thus,

$ defines a function

9

-

which is independent of the choice of representatives of f . Moreover, if W = V(h) for some EX& = h E K [ x ] ,and thus h ( f / g ) = f l / g m , then one checks that

Commutative Algebra

148

i.e., 4~ - is continuous on UBz . By Proposition 2.3(i), the open subset U-f is dense in V . Therefore, from a topological point of view,

4~- is

9

“locally” defined “almost everywhere” on

9

V . It follows that K ( V ) is also called the field of rational functions of V . The key property of 4~ - is that it is an L‘analytic-like”function in the sense 9

of the next proposition. 3.2. Proposition Let V and k ( V ) be as before. Two rational functions 42 and 45 of V are equal if and only if they agree on some open subset a

ugcv.

41- and 4~ agree on U . Then U

U -f n U -h , and 4~ ( P )= 9 d 9 &-( P ) implies (gh - d f ) ( P ) = 0 for all P E U . Let s = gh - df and d GS : V -+ K be the polynomial function defined by s. Then U G W , where W is the preimage of (0) under GS. Note that (0) is closed in K = A’ and U is dense in V . It follows that W = V and s E I(V). Thus, in K [ V ] ,

Proof Suppose

9

a

O = S = ~ ~ - ~ , a n d h e n c =e 44 ~~ 9

.

I7

d

The nice “analytic-like” property of a rational function discussed above makes the basis to define abstract regular functions on a quasi-variety in modern algebraic geometry in the schematic language. Now, we can associate to each P E V a subring of K ( V ) as follows. Notation is maintained as before. 3.3. Definition For P E V , we say that

4~- is defined at P if P E U-z . 9

9

Remark In fact, this definition says that $2- is defined at P if there are some h , s E K [ x l ,...,xn]such that s ( P ) # 0 and

-

$

consider P = (0,O) E V = V(y2 - x3) c A%, ( $ ) 2 =

Put

-

=

f.

For instance,

in K ( V ) .

Algebraic Geometry

Then it is clear that Ic

149

c Ic[V]c 0 p , v c Ic(V).

3.4. Theorem (i) Op," is a local ring with the maximal ideal

Moreover, Op,v is uniquely determined by P. (ii) (Compare with Chapter 2 Theorem 3.8.) If K is algebraically closed, then

K I V I=

n~~,~,

PEV

i.e., the rational functions that are defined at every P E V are nothing but all polynomial functions on V .

Proof (i) That Op,v is-a subring of K ( V ) may be verified directly. Note that if &(P) # 0 then f is invertible in Op,v. Hence 0 p , v is a local ring B with the described maximal ideal Mp,v. Indeed, this result can also be obtained by using the surjective ring homomorphism

F o r P = ( a l , ..., a,) ~ V , z i - a i ~ M p , ~ , ,..., i = n. l I f Q = ( b l , ..., b,)EV and Q # P , then it is clear that Mp,v # MQ,V and hence Op,v # OQ,V. This proves (i). - = 0, so 1 E J -g by Nullstellensatz, i.e., (ii) If f E npEvOp,v, then V(J-,) -

-

1 . $ = f E Ic[V],as desired.

9

I7

3.5. Definition Op,v is called the local ring of V at P.

3.6. Proposition Op," = K [ V ] Mwhere , the latter is the localization of K [ V ]at the maximal ideal

A4

=

Hence Op,v is Noetherian.

(f E K[V]1 * ( P ) = o}.

Commutative Algebra

150

Proof If P = (all ...,a,)E V , then M = (21

-

a l l ...,zn- a,)(mod I ( V ) )

by Chapter 2 (section 1, Example (iii)). In view of Chapter 2 section 3, K [ V ] Mis a subring of K ( V ) and it is (by definition) nothing but Op,v. Thus, Op,v is Noetherian by Corollary 3.5(ii) of Chapter 2. 0 The last proposition qualifies Definition 3.7 of Chapter 2.

Exercises 1. Verify Lemma 3.1. 2. Let V = V(y2 - x3 - x2) c AZz be the nodal curve, and Z,?j the image of x, y in k [ V ] let ; t = E k ( V ) . Find the pole sets of z and of z2. (Note that V is irreducible by (section 2, Exercise 6).) 3. Let Op,v be the local ring of a n irreducible algebraic set V at a point P. Show that there is a natural one-to-one correspondence between the prime ideals of Op," and the irreducible algebraic subsets of V containing P. (Hint: If I is prime in Op,v, I n k[V]is prime in k [ V ] , and I is generated by I n k [ V ] ;use Theorem 3.4.) 4. Let V be an irreducible algebraic set in A", I = I ( V ) , P E V , and let J be an ideal of K [ Q ,...,x,] which contains I . Let 5 be the image of J in k [ V ] .Show that there is a natural homomorphism p from Op,An/JOp,An to o p , v / J O p , v ,and p is an isomorphism. In particular, Op,An/IOpAn is isomorphic to Op,v. Also show that if V = V ( I )= { P } ,then

(Hint: Use Chapter 2 Corollary 4.4.) C A", W C A" be irreducible algebraic sets. If 4: U -+ W is a rational mapping defined by P H ( a l ( P ) ..., , a,(P)) where a1 = b, ..., a, = J LE K(z1,...,2,) and U = V - u?==,V(gi), show that &: 91 Sn

5. Let V

K ( W ) 4 K ( V ) defined by +*

(5)

=

F ( a l , ...,a,)/ G(a1,...,a,) is a

K-linear ring homomorphism. Conversely, if $I: K ( W ) -+ K ( V ) is a K-linear ring homomorphism such that $I(Zi)= ai = E K ( z 1 ,..., x"), i = 1,...,n, then show that P H ( a l ( P ) ..., , a,(P)) defines a rational mapping $*: U -+ W where

u = v - u;=lv(gi).

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151

4. Nonsingular Points and DVRs This section is devoted to showing how to algebraically recognize singularities in algebraic curves. Let A2 = A$. If f E K[z,y] is a nonconstant polynomial, then we write C = V ( f ) for the plane curve defined by f (and hence by cf for any 0 # c E K ) , and K[C]= K[z,y]/I(C) for its coordinate ring; in the case where C is irreducible we also write K(C) for its field of rational functions. If f has degree n 2 1, we say that the curve C is of degree n. To have a more convenient discussion, the first thing to be explored is that after an (affine) change of coordinates nothing about C = V(f) is “intrinsically” changed. (Although there is a result that applies polynomial isomorphisms to general affine algebraic sets, we do not have space to discuss it.) Let a , b E K . Define the (affine) change of coordinates (as in R2)

4:

A2

-

(z1?4)

and put C’ = V(f(z

A2 (z+a,y+b)

+ a,y+ b ) ) .

4.1. Lemma With notation as above,

4(C’) = C.

Proof Exercise. 4.2. Proposition (i) The foregoing q5 induces a K-linear ring isomorphism

#*: K[C]-K[C’]. (ii) If C is irreducible, then q5 induces a K-linear ring isomorphism K(C)-K(C’) that yields a ring isomorphism O~(Q),C-OQ,C/ with $ * ( M + ( Q ) ,=~ MQ,P ) for every Q E c’.

4*:

Proof (i) Consider the K-linear ring homomorphism

4*

: KbIYl

dz,Y)

-

9(z

K[5,9l a,!/ b)

+

+

It is well-known that $* is an isomorphism (or use (section 2, exercise 11) as 4 is now an isomorphism). Let h E I = I(C). Then & ( h ) = h ( z + a , y b ) . For any Q = ( c , d ) E C’, 0 = h($(Q))= h(c a , d b) = & ( h ) ( Q ) by Lemma 4.1. This shows that 4,(I) I’ = I(C’). Conversely, if g E I’, then

+

+

+

152

Commutative Algebra

since (u,v) E C implies ( ‘ ~ 1 - a,v- b) E C’, we have g ( u - a,v - b) = 0. It follows that g1 = g ( x - a, y - b) E I and &(g1) = g E I’. Therefore, +,(I) = I’ and

(ii) If C is irreducible, by part (i) we have a K-linear ring isomorphism induced by 4* -

4* : K(C)

-

K(C’)

-

4J It is not hard to check that 04(Q),C*0Q,CJ with $*(kf4(Q),c)= MQ,CJ for 0 every Q E C‘.

For a curve C = V(f) c A’, in order to introduce the singularity in C, we first provide an algebraic way to define the tangent line of C at a point P. Let L be a line passing P = ( a ,b) E C , which is parametrized as

x=a+ct y=b+dt

+

+

where t is a variable. Put g ( t ) = f ( a ct, b d t ) . Then t = 0 is a root of g ( t ) . Considering the formal MacLaurin series expansion of g ( t ) at 0, we have

( 00 )for f? < m, then 0 is a root of multiplicity If g(m)(0)# 0, and ~ ( ~ ) = m for g ( t ) , and we say that L intersects C at P = (a,b) with multiplicity m. It is easy to show that this definition is independent of the particular parametrization of the line L. 4.3. Proposition P u t Of

=

(the gradient vector o f f ) .

(i) If V f ( P ) # (O,O), then there is a unique line through P which intersects C with multiplicity 2 2.

Algebraic Geometry

153

(ii) If Of( P ) = (O,O), then e v e y line through P intersects C with multiplicity 2 2.

Proof Note that t = 0 is a root of multiplicity 2 2 if and only if g’(0) if and only if

=0

af

d = 0.) E ( P ) . c + dY- (. P Hence, if V f ( P ) = (0, 0), then L always intersects C at P with multiplicity 3 2. This proves (ii). Now suppose V f ( P ) # (0,O). Then the solution space of the equation (1) with unknowns c and d is 1-dimensional. Thus, there is (CO,do) # (0,O) with the property that ( c , d ) satisfies (1) if and only if ( c , d ) = X(co,do) for some X E k . It follows that the pairs ( c , d ) that make g’(0) = 0 all parametrize the same line. This shows that there is a unique line which intersects C a t P with multiplicity 2 2, and hence (i) is proved. 0 4.4. Definition If Of( P ) # ( O , O ) , then the tangent line of C a t P is defined to be the unique line through P intersecting C with multiplicity 2 2 at P . In this case, P is called a nonsingular point (or a simple point) of If Vf(P) = ( O , O ) , then P is called a singular point of C , or we say that P defines a singularity in C. If every point P E C is nonsingular, then C is called a nonsingular curve (or a smooth curve).

c.

Example (i) Any irreducible conic C c A$ is nonsingular. To see this, let P E C. After a change of coordinates we may assume P = (0,O). If g ( P ) = g ( P ) = 0, then f would be of the form ax2 by2 m y , which can be factorized into a product of linear forms.

+

+

(ii) P = (0,O) is a singular point in the cuspidal curve C = V ( y 2- x3)

154

Commutative Algebra

and P = (0,O) is also a singular point in the nodal curve C

= V(y2-z3--z2).

Let C = V(f) c A2 be a curve of degree n 2 1, P = (0,O) E C. Then f

Algebraic Geometry

155

must be of the form

f=Fm+Fm+l+...+Fqwithm>l, Fj=

XaoxaylP, a+o=j

i.e., every Fj is a homogeneous polynomial of degree j 2 m 2 1. In view of the previous discussion on singularity, this number m is usually called the multiplicity of C a t P = (O,O), denoted m p ( C ) = m.

Observation P = (0,O) is a simple point if and only if m = 1. If C is irreducible, then K E Op,c/Mp,c by the proof of Theorem 3.4(i). So ME,c/Mp,:l may be regarded as a K-vector space for every n 2 0. This enables us t o characterize mp (C) algebraically. 4.5. Theorem Let C = V(f) c A2 be an irreducible curve of degree n 2 1, P = (0,O) E C. Write 0 for the local ring of C a t P and M for the maximal ideal of 0. If C has the multiplicity m p ( C ) = m as defined above, then

m p ( C ) = dimk (M"/M"+l)

for large n.

Proof From the isomorphism

((O/Mnsl) / ( M n / M n f l ) )

(O/Mn)

it follows that it is sufficient to prove that dimK ( O / M n ) = n m p ( C ) +

+s

for some constant s and all n 2 m = m p ( C ) . Since P = (O,O), we have M n = mnO, where m = (2,y) c K[x,y]. Note that V(m") = { P } = V(mn,f ) . By section 3, exercise 5,

where C?p,A2 C K ( z ,y) is the local ring of A2 at P. Thus, the problem is reduced t o the calculation of the dimension of K [ x ,y]/(mn,f ) . Since m = m p ( C ) implies fg E mn whenever g E mn--m,there is a natural ring homomorphism p: K[x,y]/mn + K[x,y]/(mn,f),and a K-linear mapping $: K [ ~ , y ] / m--t ~ K[x, - ~ y]/mn defined by $ ( g ) = It is easy to verify

z.

Commutative Algebra

156

that cp o II, induces the isomorphism of K-vector spaces:

It follows from section 1, Exercise 14 that

0

as desired.

Remark (i) By the theorem, mp(C) depends only on the local ring of P = (0,0), that is, it is an intrinsic property. Considering any point Q = ( a ,b) E C, by Lemma 4.1 and Proposition 4.2, the multiplicity (for singularity) of C at Q, denoted ms(C), may be defined to be mco,o,(C’), where C’ = V ( f ( . a,Y b)). (ii) It can be shown that the function x(n)= dimK (Op,c/M;,,) is a polynomial in n (for large n ) which is called the Hilbert-Samuel polynomial of the local ring Op,c.

+

+

We are ready t o reach the main result of this section. Theorem Let C = V ( f )c A2 be an irreducible curve of degree n 2 1, P = ( a ,b) E C. Then P is a simple point if and only if the local ring Op,c of C at P is a DVR in K(C),and in this case, the maximal ideal Mp,c of Op,c can be generated by where V ( L= a x by c) is any line through P but not tangent t o C at P. 4.6.

+ +

Proof First suppose that P is a simple point in C. Since L is not the tangent line of C at P , after an appropriate (affine) change of coordinates we may assume that P = (O,O), that y is the tangent line, and that L = 2 by Proposition 4.2. Then by Chapter 2 Theorem 2.10, it is sufficient to show that Mp,c is generated by the image of x in Op,c. Write z, for the images of x,y in k [ C ] . Then P = (0,O) implies M ~ , = c @,g)c Op,c, whether P is simple or not. Moreover, since y is the tangent line t o C a t P = (0,O) (recall that the tangent line a t P is defined by the equation g ( P ) x g(P)y = 0), we have

+

f = y + higher terms.

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Grouping together terms involving y, we may write

f = yg - z2h, Thus, j@

=

22h

where g = 1

+ higher terms, and h E K[x]

-

E k[C], so jj = x2hij-’ since g ( P )

#

0. It follows that

MP,C= (T,jj) = ( T ) ,as desired. Conversely, if 0 = 0 p , c is a DVR, then M = Mp,c = ( t )for some prime M . Thus, dimK(Mn/MnS1) = 1 for all n 2 0, and hence r n p ( C ) = 1 by Theorem 4.5. This shows that P is simple.

t

E

Exercises 1. Complete the proof of Lemma 4.1. 2. Find all singular points in C = V ( x 2 y 2 x 2 y2 2zy(x y 1))C A; and C = V ( x y z3 y3) c A;. 3. Show that the elliptic curve y2 = (x- a1)(x - a2)(x- a3) in A;, where a l , a2, a3 are distinct complex numbers, has no singularities. 4. Use the notation as in the proof of Theorem 4.5 to show that the function x(n) = dimK (Op,cjM;,,) is a polynomial in n (for large n).

+ +

+ + +

+ +

5. Normalization of Algebraic Curves In the light of Proposition 3.6, Theorem 4.6 and Chapter 3 Theorem 4.7, we explore in this section the relation between the singularity of an algebraic curve C and the normality of the coordinate ring of C. As a consequence, this leads to the “resolution” of singularities in algebraic curves by means of “normalization” . Let K be a field and let C = V ( f ) c A2 = A& be an irreducible plane curue, where f E K [ x , y ] is a nonconstant polynomial. As before, K[C] = K [ z ,y]/I(C) denotes the coordinate ring of C and K(C) stands for the field of rational functions of C (i.e., the field of fractions of K[C]).P u t

5.1. Theorem With notation as above, S(C) is a finite subset of C. Hence C - S(C), the set of nonsingular points in C, is a dense open subset of C.

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Commutative Algebra

Proof We may assume that f is irreducible and S(C) # 8. Note that

If

% # 0, then since deg( g)< deff

2.10, V

(2,f) and hence S(C)

and f is irreducible, by Proposition

is a finite set. A similar argumentation

%

# 0. Below we show that and cannot both be holds provided identically zero. If chark = 0, then = = 0 only when x and y do not appear in f. Hence f is a constant in K , a contradiction. If chark = p > 0, then = = 0 only when x and y appear in f in the form of xffpand ypp. Hence f can be expressed as f = CuapxPayPp. Assuming that K is algebraically closed, the equation a,p = tP has a solution, say u,p = spap for some sap E K . Thus, putting g = Cs,pxayP we have g p = f , contradicting the irreducibility of f . I?

% 2

In view of the last theorem, almost all points in C are simple. Next, we show that the normalization K[C]of K[C]in K(C) (i.e., the integral closure of K[C]in K(C) in the sense of Chapter 3 (Definitions 1.6 and 3.1)) defines a nonsingular algebraic curve C' (see later Definition 5.5) which projects onto C with respect to the Zariski topology. To this end, we first generalize Chapter 4 Theorem 4.2(i) (concerning Z c & ) t o a more general setting. 5.2. Proposition Let R C B be a module-finite ring extension, where R and B are domains. If every nonzero prime ideal of R is maximal, then every nonzero prime ideal of B is maximal.

Proof Suppose B = C z , RJi where Ji E B. Let P be a nonzero prime ideal of B , 0 # b E P. Then b is integral over R, that is,

Assume that n is the smallest degree. Then 0 # bo E P n R. Thus, P n R is a nonzero prime ideal of R and hence maximal by the assumption. Put K = R / ( P n R). Then B / P may be viewed as an extension ring of K via the embedding K = R / ( P n R ) 4 B / P . By Chapter 3 (section 3, exercise 6(b)), B / P is integral (indeed module-finite) over K . It follows from Chapter 3 Theorem 1.8 that B / P is a field. Therefore, P is maximal.

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5.3. Corollary Let C = V(f) c A2 be an irreducible - plane curve, where f E K [ z ,y] is a nonconstant polynomial, and let K[C]be the normalization respectively every of K[C]in K(C). Then every -nonzero prime ideal of K[C], nonzero prime ideal of K[C],is maximal. In particular, K[C]is a Dedekind domain (in the sense of the final remark given in Chapter 4 section 4).

Proof By Corollary 2.12 we may assume that C is infinite. It follows from Proposition 1.8(i) and the Noether normalization (Chapter 3 Theorem 2.4) that K[C]is module-finiteover a polynomial subring K [ t ] .Furthermore by Chapter 3 Corollary 3.3, K[C]is module-finite over K[C].Now Proposition 0 5.2 and Chapter 2 (section 1, exercise 2) yield the assertion. 5.4. Theorem Let C = V(f) c A2 be an irreducible - plane curve, where f E K [ z ,y] is a nonconstant polynomial, and let K[C]be the normalization of K[C]in K(C). If K is algebraically closed, then the following statements are equivalent. (i) C is nonsingular. (ii) The localring Op,c of C a t every P E C is a DVR. (iii) K[C]= K[C],that is, K[C]is a Dedekind domain.

Proof By Theorem 4.6, the equivalence (i) @ (ii) holds even if K is not algebraically closed. The implication (ii) + (iii) follows from Corollary 2.5(ii), Proposition 3.6, and Chapter 2 Theorem 3.1. And the implication (iii) + (ii) follows from Corollary 2.5(ii), Proposition 3.6, Chapter 2 Proposition 3.6, Chapter 3 (Theorem 4.1 and Corollary 4.6). Example (i) The fact that the cuspidal curve C = V(y2 - z3)c A; has one singular point (0,O) is indicated by R[C] # R[C] where the latter is isomorphic to the polynomial ring R[t] (see Chapter 3 (section 3, Example (iii))). (ii) The fact that the nodal curve C = V(y2 - x3 - z2) c A; has one singular point (0,O) is indicated by later section 7, Example (i) which shows R[C] # R[t].

Iw[cl

Now let V be an algebraic set in the affine n-space An = A;, n 2 1. By the Noetherian normalization theorem, the coordinate ring K [ V ]of V is module-finite over a polynomial subring K [ t l ,...,t d ] , d 5 n. The last theorem leads to the following definition.

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commutative Algebra

5.5. Definition Let V C A" be an algebraic set, n

2 1. If V is irreducible

and its coordinate ring K [ V ]is module-finite over a polynomial subring K [ t ]then , V is called an algebraic curve in A". Let V c A" be an algebraic curve, P E V . If the local ring Op," of V a t P is a DVR then P is called a nonsingular point of V ; otherwise, P is a singular point (or P defines a singularity in V ) . If every point of V is nonsingular then V is called a nonsingular curve.

Example (iii) By later exercise 1-2, the curve C = V ( y - f(z))c A; is nonsingular, where f(x) E K [ 5 ] ;and the twisted cubic curve V = V(y x2,2 - x3) c A; is nonsingular.

c A; be an algebraic curve. If K is algebraically closed, then there exists a plane curve C c A$ such that K ( V ) = K(C). (Note that this qualifies Definition 5.5.)

5.6. P r o p o s i t i o n Let V

Proof By the definition of an algebraic curve, K [ V ]= CEl K[t]& where E K [ V ] . Then it is easy t o K [ t ]is a polynomial subring of K [ V ]and , where each is algebraic over the field see that K ( V ) = K ( t ) [ & ...,Im] K ( t ) . We may assume that all ti's are separable over K ( t ) . It follows from Chapter 1 Theorem 3.12 that K ( V ) = K ( t , 6 ) for some 6 E K ( V ) . Let K [ z l ,221 be the polynomial ring in 21, 22 over K and consider the onto ring homomorphism K [ z l ,Q] -+ K [ t ,61. Then K [ t ,61 2 K[z1,z2]/1 for some ideal I c K [ z l ,221. Note that K is algebraically closed and hence infinite (Chapter 1 section 3, exercise 5). By the Nullstellensatz and Corollary 2.12, K [ t l ,z2]/I is the coordinate ring of the plane curve C = V ( I ) . Hence K ( V ) = K(C). 5.7. Theorem (normalization of a plane curve) Let C = V(f) C A2 be an irreducible - plane curve, where f E K [ x ,y] is a nonconstant polynomial, and let K[C]be the normalization of K[C]in K(C). If K is algebraically closed, then the following statements hold: (i) There exists a nonsingular algebraic curve V c A" for some m 2 1 such that -

K [ V ]= K[C]. (ii) Let V be the algebraic curve obtained in part (i). There exists an onto polynomial mapping 4: V -+ C and an open subset U c V such that 4 ( U ) = C - S(C),the open subset of all nonsingular points in C.

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Proof (i)Assume that C is infinite. It follows from the proof of Corollary 5.3 that a polynomial subring K [ t ] c K[C], -K[C]is module-finite over say K[C] = K[t]&,& E K[C]. Hence K[C] = K [ t , & ,. .,