Commutative Algebra [1 ed.] 9780805370249, 0805370242

Table of contents :
Commutative Algebra, Hideyuki Matsumura,1st ed, 1970, 262p, W. A. Benjamin, 978-0805370249
A Note from the Publisher
Preface
Conventions
Contents
Index
Part I
Chapter 1. ELEMENTARY RESULTS
1. General Rings
2. Noetherian Rings and Artinian Rings
Chapter 2. FLATNESS
3. Flatness
4. Faithful Flatness
5. Going-up and Going-down
6. Constructable Sets
Chapter 3. ASSOCIATED PRIMES
7. Ass(M)
8. Primary Decomposition
9. Homomorphisms and Ass
Chapter 4. GRADED RINGS
10. Graded Rings and Modules
11. Artin-Rees Theorem
Chanpter 5. DIMENSION
12. Dimension
13. Homomorphism and Dimension
14. Finitely Generated Extensions
Chapter 6. DEPTH
15. M-regular Sequences
16. Cohen-Macaulay Rings
Chapter 7. NORMAL RINGS and REGULAR RINGS
17. Classical Theory
18. Homological Theory
19. Unique Factorization
Chapter 8. FLATNESS II
20. Local Criteria of Flatness
21. Fibres of Flat Morphisms
22. Theorem of Generic Flatness
Chapter 9. COMPLETION
23. completion
24. Zariski Rings
PART II
Chapter 10. DERIVATION
25. Extension of a Ring by a Module
26. Derivations and DifferentiaIs
27. Separability
Chapter 11. FORMAL SMOOTHNESS
28. Formal Smoothness I
29. Jacobian Criteria
30. FormaI Smoothness II
Chapter 12. NAGATA RINGS
31. Nagata Rings
Chapter 13. EXCELLENT RINGS
32. Closedness of the Singular Locus
33. FormaI Fibres and G-Rings
34. Excellent Rings

Citation preview

COMMUTATIVE ALGEBRA

HIDEYUKI MATSUMURA Nagoya University, Nagoya, Japan

W.A. BENJAMIN, INC. New York

1970

COMMUTATIVE ALGEBRA

Copyright© 1970 by W. A. Benjamin, Ilic. All rights reserved Standard Book Number 8053-7024-2 (Clothbound) 8053-7025-0 (Paperback) Library of Congress Catalog Card Number: 68-59193 Manufactured in the United States of America -

12345R32109

The manuscript was put into production on December 15, 1969. this volume was published on January 30, 1970.

W. A. BENJAMIN, INC. New York, New York 10016

COMMUTATIVE ALGEBRA

MATHEMATICS LECTURE NOTE SERIES J. Frank Adams

LECTURES ON LIE GROUPS

E. Artin and J. Tate

CLASS FIELD THEORY

Michael Atiyah Jacob Barshay

K-THEORY TOPICS IN RING THEORY

Hyman Bass Melvyn S. Berger Marion S. Berger

ALGEBRAIC K-THEORY PERSPECTIVES IN NONLINEARITY

Armand Borel

LINEAR ALGEBRA GROUPS

Raoul Bott

LECTURES ON K (X)

Andrew Browder

INTRODUCTION TO FUNCTION ALGEBRAS

Gustave Choquet

LECTURES ON ANALYSIS I. INTEGRATION AND TOPOLOGICAL VECTOR SPACES II. REPRESENTATION THEORY III. INFINITE DIMENSIONAL MEASURES AND PROBLEM SOLUTIONS

Paul J. Cohen

SET THEORY AND THE CONTINUUM HYPOTHESIS

Eldon Dyer

COHOMOLOGY THEORIES

Robert Ellis

LECTURES ON TOPOLOGICAL DYNAMICS

Walter Feit

CHARA.CrERS OF FINITE GROUPS

John Fogarty

INVARIANT THEORY

William Fulton

ALGEBRAIC CURVES

Marvin J. Greenberg

LECTURES ON ALGEBRAIC TOPOLOGY

Marvin J. Greenberg

LECTURES ON FORMS IN MANY VARIABLES

Robin Hartshorne

FOUNDATIONS OF PROJECTIVE GEOMETRY

J. F. P. Hudson

PIECEWISE LINEAR TOPOLOGY

K. Kapp and H. Schneider

RINGS OF OPERATORS COMPLETELY 0-SIMPLE SEMIGROUPS

Joseph B. Keller

BIFURCATION THEORY AND

Stuart Antman

NONLINEAR EIGENVALUE PROBLEMS

Serge Lang Serge Lang

ALGEBRAIC FUNCTIONS RAPPORT SUR LA COHOMOLOGIE DES GROUPES

Ottmar Loos

SYMMETRIC SPACES

Irving Kaplansky

I. GENERAL THEORY II. COMPACT SPACES AND CLASSIFICATIONS I. G. Macdonald

ALGEBRAIC GEOMETRY: INTRODUCTION TO SCHEMES

GeorgeW.Mackey

INDUCED REPRESENTATIONS OF GROUPS AND QUANTUM MECHANICS

Andrew Ogg

MODULAR FORMS AND DIRICHLET SERIES

Richard Palais

FOUNDATIONS OF GLOBAL NON-LINEAR ANALYSIS ENTROPY AND GENERATORS IN ERGODIC THEORY

William Parry D. S: Passman

PERMUTATION GROUPS

Walter Rudin

FUNCTION THEORY IN POLYDISCS

Jean-Pierre Serre

ABELIAN l-ADIC REPRESENTATIONS AND ELLIPTIC CURVES

Jean-Pierre Serre

ALGEBRES DE LIE SEMI-SIMPLE COMPLEXES

Jean-Pierre Serre

LIE ALGEBRAS AND LIE GROUPS

Shlomo Sternberg

CELESTIAL MECHANICS PART I

Shlomo Sternberg

CELESTIAL MECHANICS PART II

Moss E. Sweedler

HOPF ALGEBRAS

A Note from the Publisher This volume was printed directly from a typescript prepared by the author, who takes full responsibility for its content and appearance. The Publisher has not performed his usual functions of reviewing, editing, typesetting, and proofreading the material prior to publication. The Publisher fully endorses this informal and quick method of publishing lecture notes at a moderate price, and he wishes to thank the author for preparing the material for publication.

To my teacher, Yasuo Akizuki

Preface

This book has evolved out of a graduate course in algebra I gave at Brandeis University during the academic year of 1967-1968. At that time M. Auslauder taught algebraic geometry to the same group of students, and so I taught commutative algebra for use in algebraic geometry. Teaching a course in geometry and a course in commutative algebra in parallel seems to be a good way to introduce students to algebraic geometry. Part I is a self-contained exposition of bas.ic concepts such as flatness, dimension, depth, normal rings, and regular local rings. Part II deals with the finer structure theory of noetherian rings, which was initiated by Zariski (Sur la normalite analytique des varietes normales, Ann. Inst. Fourier 2, 1950) and developed by Nagata and Grotheudieck. Our purpose is to lead the reader as quickly as poss:ble to Nagata's theory of pseudo-geometric rings (here called Nagata rings) and to Grotheudieck's theory of excellent rings. The interested reader should advance to Nagata's book LOCAL RINGS and to Grotheudiee::k's EGA, chapter IV. The theory of multiplicity was omitted because one has little to add on this subject to the lucid exposition of Serre's lecture notes (Algebre Local Multiplicite, Springer Verlag). Due to lack of space some important results on formal smoothness (especially its relation to flatness) had to be omitted also. For these, see EGA. We assume that the reader is familiar with the elements of algebra (rings, modules, and Galois theory) and of homological algebra (Tor and Ext). Also, it is desirable but not indispensable to have some knowledge of scheme theory. I thank my students at Brandeis, especially Robin Hur, for helpful comments. Nagoya, Japan November 1969

Hideyuki Matsumura

Conventions

1. All rings and algebras are tacitly assumed to be commutative. 2. If f:A--,B is a homomorphism of rings and if I is an ideal of B, then the ideal f'(I) is denoted by I A. 3. Cmeans proper inclusion. 4. We sometimes use the old-fashioned notation I= (a,, . . . an) for an ideal I generated by the elements a,. 5. By a finite A-module we mean a finitely generated A-module; by a finite A-algebra, we mean an algebra which is a finite A-module. By an A-algebra of finite type, we mean an algebra which is finitely generated as a ring over the canonical image of A.

Contents

IX

Preface Conventions

X

PARTI

Chapter 1. ELEMENTARY RESULTS 1. General Rings 2. Noetherian Rings and Artinian Rings Chapter 2. FLATNESS 3. Flatness 4. Faithful Flatness 5. Going-up Theorem and Going-down Theorem 6. Constructable Sets

1 13

17

25 31

38

Chapter 3. ASSOCIATED PRIMES 7. Ass(M) 8. Primary Decomposition 9. Homomorphisms and Ass

49 52 57

Chapter 4. GRADED RINGS 10. Graded Rings and Modules 11. Arlin-Rees Theorem

67

Chapter 5. DIMENSION 12. Dimension 13. Homomorphisms and Dimension 14. Finitely Generated Extensions

78 83

61

71

Chapter 6. DEPTH 15. M-regular Sequences 16. Cohen-Macaulay Rings

95 103

Chapter 7. NORMAL RINGS AND REGULAR RINGS 17. Classical Theory 18. Homological Theory 19. Unique Factorization

115 127 141

XI

xii

Contents

Chapter 8. FLATNESS II 20. Local Criteria of Flatness 21. Fibres of Flat Morphisms 22. Theorem of Generic Flatness

145 152 156

Chapter 9. COMPLETION 23. Completion 24. Zariski Rings

172

161

PART II Chapter 10. DERIVATION 25. Extension of a Ring by a Module 26. Derivations and Differentials 27. Separability

180 190

Chapter 11. FORMAL SMOOTHNESS 28. Formal Smoothness I 29. Jacobian Criteria 30. Formal Smoothness II

197 213 222

Chapter 12. NAGATA RINGS 31. Nagata Rings

231

Chapter 13. EXCELLENT RINGS 32. Closedness of Singular Locus 33. Formal Fibres and G-rings 34. Excellent Rings

245 249 258

177

PART ONE

CHAPTER 1.

ELEMENTARY RESULTS

In this chapter we give some basic definitions, and some elementary results which are mostly well-known.

1, General Rings

(1.A)

Let A be a ring and m,... an ideal of A.

of elements x in A some powers of which lie in of A, called the radical of

Then the set (J(..

is an ideal

Ol,.

An ideal pis called a prime ideal of A if A/pis an integral domain; in other words, if closed under multiplication.

pf

A

and if

A - pis

If pis prime, and if oz. and

7r

are ideals not contained in p, then (11.i ';/; p. An ideal q is called primary if q f A and if the only zero divisors of A/q are nilpotent elements, i.e. x

i

q implies

y

n

£

q for some n.

xy

£

q,

If q is primary then its

radical pis prime (but the converse is not true), and p and

q are said to belong to each other.

If q is an ideal contain-

ing some power _,,,,,,_n of a maximal ideal~, then q is a primary

2

COMMUTATIVE ALGEBRA

ideal belonging to tW, The set of the prime ideals of A is called the spectrum of A and is denoted by Spec(A) ; the set of the maximal ideals of A is called the maximal spectrum of A and we denote it by ~(A),

The set Spec(A) is topologized as follows.

subset M of A, put V(M) = { p E Spec(A)

f

For any

M £ p }, and take

as the closed sets in Spec(A) all subsets of the form V(M). This topology is called the Zariski topology. put

=

D(f)

Spec(A) - V(f)

set of Spec(A).

If

f EA, we

and call it an elementary open

The elementary open sets form a basis of open-

sets of the Zariski topology in Spec(A). Let

f: A+ B

be a ring homomorphism,

To each

PE

Spec(B) we associate the ideal PAA (i.e. f- 1 (P)) of A.

P/"\ A is prime in A, we then get a map which is denoted by af, easily check.

in A.

Spec (B) + Spec (A),

The map af is continuous as one can

It does not necessarily map ~(B) into ~(A),

When PE Spec(B) and

(LB)

p = Pf\A,

we say that Plies over

Let A be a ring, and let

I,

p.

p 1 , ... , pr be ideals

Suppose that all but possibly two of the

prime ideals,

Since

p. 's are l

Then, if I$ p. for each i, the ideal I is not l

contained in the set-theoretical union Ui pi.

Proof.

Omitting those p. which are contained in some other l

ELEMENTARY RESULTS

3

p., we may suppose that there are no inclusion relations J

between the

p.l 's, Take

xi

hences+

We use induction on r, x EI - n 2 ,~

and

When r = 2, suppose

s EI - n 1 , /"

Then x

p 1 , therefore both sands+ x must be in

O C,

n ,~1'

P2·

Then x E µ2 and we get a contradiction, When r > 2, assume that p

r

is prime,

Then Ip 1 .,,pr-l

(1):

for any A-module N.

FLATNESS

25

We use the following

LEMMA.

Let B be an A-algebra, P a prime ideal of B, p

and Nan A-module.

Then A

A

(Tori(B, N))p Proof.

Let

Pf"\A

x. : •••

+

x1

lution of the A-module N.

Tor/ (BP, Np) •. +

x0

(

+

N + 0) be a free reso-

We have

A

Tori (B, N) = Hi (X. ®AB), A

Tori(B, N) @BBP = Hi(X. ®AB ®B BP)

and X. 0 A is a free resolution of the A -module N, hence p A p p the last expression is equal to Tor.P(BP' N ). Thus the 1 p lemma is proved. Now, if BP is flat over A

(Tor 1 (B, N))P = 0 A

Tor 1 (B, N) = 0

for all P

€

AP

for all P

€

Q(B), then

Q(B) by the lemma, therefore

by (l.H) as wanted.

4. Faithful Flatness

(4.A)

THEOREM 2.

Let A be a ring and Man A-module.

following conditions are equivalent:

The

26

COMMUTATIVE ALGEBRA (i)

Mis faithfully flat over A;

(ii)

Mis flat over A, and for any A-module N 'f Owe have

'f O;

N0M

(iii)

M is flat over A, and for any maximal ideal #v of A

we have *1-M

'f M.

Proof.

(i) =}(ii): suppose N®M

quence

O + N + O.

N + O.

Therefore N = O.

As

O.

O + N(8)M + 0

(ii)~ (iii): since A/t'tv M/+M--M

=

Let us consider the seis exact, so is

O +

'f O, we have (A/ 41'1--) ® M

'f Oby hypothesis. (iii) 9 (ii): take an element x

E

N, x

'f O.

The sub-

module Ax is a homomorphic image of A as A-module, hence Ax~ A/I

for some ideal I

of A containing I. M/IM

'f O.

N®M

'f O.

Let M1.- be a maximal ideal

Then M:::, ..._M 2 IM, therefore

By flatness

(ii);::::} (i):

'f A.

O + (A/l)®M + N®M

(A/I) 0 M

is exact, hence

let S: N' + N + N" be a sequence of A-

modules, and suppose that

is exact.

As Mis flat, the exact functor ®M transforms

kernel into kernel and image into image. Im(gMofM)

=

Thus

O, and by the assumption we get

i.e. gof = O.

Im(gof) @M

Im(gcf)

=

O,

Hence Sis a complex, and if H(S) denotes its

FLATNESS

27

homology (at N), we have

H(S)0M = H(S®M)

O.

Using again

the assumption (ii) we obtain H(S) = O, which implies that

Q.E.D.

Sis exact.

COROLLARY.

Let A and B be local rings, and~: A+ Ba local

homomorphism.

Let M be a finite B-111odule.

Mis flat over A #

Then

Mis f.f. over A.

In particular, Bis flat over A iff it is f.f. over A. Proof. Let ,#1, and

,i.i,

be the maximal ideals of A and B respec-

Then ,1.t1,M G 111,M since

tively.

~

is local, and -n.M 'f M by

NAK, hence the assertion follows from the theorem.

(4.B)

Just as flatness, faithful flatness is transitive

(B is f.f. A-algebra and M is f.f. B-module ,:::> M is f.f. over A) and is preserved by change of base (Mis f.f. A-module and B is any A-algebra

~

M ®AB is f.f. B-module).

Faithful flatness has, moreover, the following descent property:

if Bis an A-algebra and if Mis a f.f. B-module

which is also f.f. over A, then Bis f.f. over A. Proofs are easy and left to the reader.

(4.C)

Faithful flatness is particularly important in the

case of a ring extension.

Let

~: A+ B be a f.f. homomorph-

28

COMMUTATIVE ALGEBRA

ism of rings. (i) x I-+ x01

Then:

For any A-module N, the map

N -+ N0B

defined by

In particular 1/J is injective and A

is injective.

can be viewed as a subring of B. (ii)

IB"'A = I.

For any ideal I of A, we have a

(iii)

1/J: Spec(B)-+ Spec(A) Let O 'f x

Proof.

(i)

N.

S N0B

by flatness of B.

E:

is surjective. O 'f Ax f: N, hence

Then

Ax0 B

Ax0B = (x0l)B, therefore

Then

x 01 'f O by Th.2.

By change of base, B ® A(A/I)

(ii)

B/IB is f.f. over A/I.

Now the assertion follows from (i). Let p

(iii)

E:

'f B. p

A, hence pB

p

tains /vp nB •

Then

p

pAP is maximal. = ..Wt\

A

= (~n

(4.D) rings.

Spec(A).

The ring B = B0A

p

Take a maximal ideal

A ::, nA , therefore "p-'vp

4-W"

#k

w,i. .-,

Putting P = tn,,,B, we get A ) I'\ A p

THEOREM 3.

=

pA

Let

p

I'\

A

=

p

is f.f. over

of B which con-

p

Ap = /VP nA because Pn

A

(tw"B)/\A

p.

Q.E.D.

1/J: A -+ B be a homomorphism of

The following conditions are equivalent. (1)

1/J is faithfully flat;

(2)

1/J is flat, and al/J: Spec(B) -+ Spec(A) is surjective;

(3)

1/J is flat, and for any maximal ideal

exists a maximal ideal m,' of B lying over·A,Uo.

#Ir

of A there

29

FLATNESS (1) ::;> (2) is already proved.

Proof.

(2) ~ (3). with p 1 " A = ,H\.'

p' , we have (3)

E

Spec(B)

If tr~' is any maximal ideal of B containing

•

.j,i'I,' (\

By assumption there exists p'

A

:::::> (1).

Jl4

as

.}n,,-

is maximal.

The existence of ttv' implies 4,l-B 'F B.

Therefore Bis f.f. over A by Th. 2. Remark. X+ Y

In algebraic geometry one says that a morphism

f:

of preschemes is faithfully flat if f is flat (i.e.

0

for all x EX the associated homomorphisms

Y,f(x)

+

0

X,x

are flat) and surjective.

(4. E)

Let A be a ring and Ba faithfully flat A-algebra.

Let M be an A-module. (i)

Then:

Mis flat (resp. f.f.) over A¢::} M®AB is so

over B, (ii)

when A is local and Mis finite over A we have M is A-free

Proof. (i).

~

M®AB is B-free.

The implication (:::}) is nothing but a change of

base ((3.C) and (4.B)), while (B. pl

f p, by (GD) there exists

If PnA

Spec(B) such that P1 r.A = p and P:::>P 1 •

E:

Then

Then p::,p12

pB, contradicting the minimality of P. (GD')::::} (GD):

left to the reader. a

Put X = Spec(A), Y = Spec(B), f = ¢: Y

Remark.

suppose Bis noetherian.

X, and

Then (GD') can be formulated geomet-

let p

rically as follows:

+

X, put X' = V(p)

E:

s; X and let Y'

be an arbitrary irreducible component of f- 1 (X').

Then f

maps Y' generically onto X' in the sense that the generic point of Y' is mapped to the generic point p of X'. *)

EXAMPLE.

(5.C)

field k, and put

Let k[x] be a polynomial ring over a x 1 = x(x - 1),

k(x 1 , x 2 ), and the inclusion

2

x 2 = x (x - 1). k[x 1 , x 2 ] .s; k[x]

Then k(x) induces

a birational morphism f: C = Spec( k[x] )

+

C' = Spec( k[x 1 , x2] )

where C is the affine line and C' is the affine curve xl

3

- x2

x = 0

2

and

+ x 1x 2 = O. Q2 : x = 1

The morphism f maps the points

Ql:

of C to the same point P = (O,O) of

C', which is an ordinary double point of C', and f maps

*) See (6.A) and (6.D) for the definitions of irreducible

component and of generic point.

FLATNESS

33

C - {Q 1 , Q2 } bijectively onto

C - {P},

Let y be another indeterminate, and put B = k[x, y], A= k[x 1 , x 2 , y]. is

C' x line;

x = 0 by

and

Then Y = Spec(B) is a plane and X = Spec(A)

Xis obtained by identifying the lines L1 :

L2 : x = 1

y = ax, a# 0,

on Y,

Let L 3

c

Y be the line defined

Let g: Y + X be the natural morphism,

Then g(L 3 ) = X' is an irreducible curve on X, and } g -1 (X') =L3V{(o, a), (1, O),

Therefore the going-down theorem does not hold for AC B,

(5,D)

THEOREM 4,

Let¢: A+ B be a flat homomorphism of

rings,

Then the going-down theorem holds for¢.

Proof.

Let

p and

p' be prime ideals in A with p' c p, and

let P be a prime ideal of B lying over

p.

Then BP is flat

over A

by (3,J), hence faithfully flat since A

local.

Therefore Spec(Bp) + Spec(Ap) is surjective,

p

p

be a prime ideal of BP lying over a prime ideal of B lying over

(5. E)

THEOREM 5.

which Bis integral. i)

j'''< =

Then, putting

Q.E.D.

Let A be a ring and Fa closed subset of

Then Fis irreducible iff F

V(p) for some

This pis unique and is called the generic

point of F. Proof,

Suppose that Fis irreducible,

Since it is closed

it can be written F = V(I) with I

() p, If I is not prime pi:::F we would have elements a and b of A - I such that ab EI, Then F rf;.V(a), F ~V(b) and Ft;;;; V(a)VV(b) = V(ab), hence F = (FAV(a)) V(F0V(b)), which contradicts the irreducibility. The converse is proved by noting p E V(p),

The uniqueness

comes from the fact that pis the smallest element of V(p),

LEMMA 2.

Let¢: A+ B be a homomorphism of rings,

Put X

42

COMMUTATIVE ALGEBRA

Spec(A), Y = Spec(B) and f = a¢: Y + X.

Then f(Y) is dense

in X iff Ker(¢)~ nil(A).

If, in particular, A is reduced,

f(Y) is dense in X iff ¢ is injective,

Proof, The closure f(Y) in Spec(A) is the closed set V(I) defined by the ideal I = (\ ¢-l (µ) = ¢-l ( ('\ p), which is equal psY psY to¢

-1

(nil(B)) by (l,E).

f(Y) is dense in X,

Clearly Ker(¢) s; I,

Then V(I) = X, whence I= nil(A) by (l,E),

Therefore Ker(¢) s; nil(A), nil(A),

Suppose that

Conversely, suppose Ker(¢) f;

Then it is clear that I= ¢- 1 (nil(B)) = nil(A),

which means f(Y) = V(I) = X.

(6, E)

THEOREM 6, (Chevalley),

Let A b_e a noetherian ring

and Ban A-algebra of finite type, ical homomorphism; Y + X.

Let¢: A+ B be the canon-

put X = Spec(A), Y = Spec(B) and f = a¢:

Then the image f(Y') of a constructable set .Y' in Y

is constructable in X.

Proof. Y' = Y,

First we show (6,C) can b.e applied to the case when Let X

0

= V(p) for some

be an irreducible closed set in X.

p

€ Spec(A),

Put A' =

Suppose that X {\ f (Y) is dense in X , 0

0

0

A/p, and B' = B/pB. The map ¢': A' + B'

induced by¢ is then injective by Lemma 2,

We want to show

X 0f(Y) contains a non-empty open subset of X 0

Then X

0

By replacing

43

FLATNESS A, Band¢ by A', B' and¢' respectively, it is enough to prove the following assertion: (*)

if A is a noetherian domain, and if Bis a ring which

contains A and which is finitely generated over A, there exists O

I

a EA such that the elementary open set D(a) of

X = Spec(A) is contained in f(Y), where Y = Spec(B) and f: Y +Xis the canonical map. Write B = A[x 1 , ••• , xn], and suppose that x 1 , ••• , xr are algebraically independent over A while each xj (r 0 be an integer,

Then pis the unique minimal prime

over-ideal of pn, therefore the µ-primary component of pn is uniquely determined; this is called then-th symbolic power of p and is denoted by p n

happen that p

.L

r P

(n)

•

(n)

,

Thus p(n)

n

p Ap0A,

It can

Example: let k be a field and B =

ASSOCIATED PRIMES

57

k[x, y] the polynomial ring in the indeterminates x and y,

2 3 Put A= k[x, xy, y, y] and p

p

2 2

2

3

4

5

(x y, xy, y, y ),

yB("IA = (xy, / ,

2

2

of p

2

3

(y , y )

2

f. p ,

is given by p

2

9. Homomorphisms

(9 ,A)

Then

Since y = xy/x s A , we have B

p

2 y ByB"A =

Thus p (Z)

k[x, y] s.;: AP and hence AP y B" A

y3),

An irredundant primary decomposition 2

3

2

3

4

5

= (y , y ) ,-, (x , xy , y , y ) ,

and Ass

PROPOSITION,

Let¢: A+ B be a homomorphism of

noetherian rings and Ma B-module, module by means of¢.

We can view Mas an A-

Then AssA(M)

Proof,

Let P s AssB(M).

Then there exists an element x of

M such that AnnB(x) = P.

Since AnnA(x) = AnnB(x)nA Conversely, let p s Ass A (M) and take

we have P "'A s Ass A (M).

an element x s M such that AnnA(x) =

p.

Put AnnB(x) = I,

let I = Q1 " , . , nQr be an irredundant primary decomposition of the ideal I and let Qi be Pi-primary.

Since M;:? Bx~ B/I

the set Ass(M) contains Ass(B/I) = {P 1 , ••• , Pr}. prove P. ,-, A = p for some i, l

p for all i.

Suppose P. "A l

a. s P. 0 A such that a. l

l

l

rf_

We will

Since Ir. A = p we have P. "A :, l

f. p

for all i.

p, for each i.

all i if mis sufficiently large, hence a

Then there exists Then a. l

m

s Q. for l

II.a.m s IAA = p, l

l

COMMUTATIVE ALGEBRA

58 contradiction,

p for some i and p s a¢(AssB(M)),

Thus Pi"A

THEOREM 12, (Bourbaki),

(9, B)

Let¢: A+ B be a homo-

morphism of noetherian rings, E an A-module and Fa B-module, Suppose Fis flat as an A-module,

Then:

(i) for any prime ideal p of A,

u

if

F/pF 'f 0

if

F/pF

=

0,

AssB(F/pF),

psAss(E) COROLLARY,

Let A and B be as above and suppose Bis A-flat,

Then AssB (B) =

u

AssB (B/pB),

psAss(A) and a¢(AssB(B)) = {p

= Ass(A)

E

Ass(A)

I

pB I- B},

We have a¢(AssB(B))

if Bis faithfully flat over A,

Proof of Theorem 12. (i) The module F/pF is flat over A/p (base change), and A/pis a domain, therefore F/pF is torsionfree as an A/p-module by (3.F). this,

(ii) The inclusion 2

The assertion follows from

is immediate: if p s Ass(E)

then E contains a submodule isomorphic to A/p, whence E:~9F contains a submodule isomorphic to (A/p) (i.'>AF = F /pF by the flatness of F,

Therefore AssB (F /pF)

~

AssB (E :,:;JF),

the other inclusion 2 is more difficult,

To prove

ASSOCIATED PRIMES

59

Step 1, Suppose Eis finitely generated and coprimary with

=

Ass (E) over

p.

{p}.

Then any associated prime P

AssB (El?' F) lies

E

In fact, the elements of pare locally nilpotent

(on E, hence) on E

F, therefore p ~ Pr-. A.

On the other

hand the elements of A - pare E-regular, hence E©F-regular by the flatness of F, that P" A = p,

Therefore A - p does not meet P, so

Now, take a chain of submodules

such that E./E. 1 ~ A/p. for some prime ideal p .• 1 B 1 1 E@F

=

Eo@F 2 El @F 2 ... 2 Er'gF

~ F/p.F, so that AssB(E@F)