INTRODUCTION TO COMMUTATIVE ALGEBRA

This book is in theADDISON-WESLEY SERIES IN MATHEMATICS

Consulting Editor: l y n n h . l o o m i s

Introduction to Commutative Algebra

M. F. ΑΤΊΥΑΗ, f r s

I. G. MACDONALDU N IV E R SITY OF OXFORD

The Advanced Book Program

First published 1969 by Westview Press

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Introduction.....................................................................vii

Notation and Terminology.................................................. ix

Chapter 1 Rings and Ideals................................................................ 1

Rings and ring homomorphisms......................................... 1Ideals. Quotient rings....................................................... 2Zero-divisors. Nilpotent elements. Units................................ 2Prime ideals and maximal ideals......................................... 3Nilradical and Jacobson radical . . ........................... 5Operations on id ea ls ....................................................... 6Extension and contraction.................................................. 9E xercises.....................................................................10

Chapter 2 Modules......................................................................... 17

Modules and module homomorphisms................................ 17Submodules and quotient modules.....................................ISOperations on submodules...................................................19Direct sum and product....................... ........................... 20Finitely generated m odu les..............................................21Exact sequences................................................................ 22Tensor product of m odules..............................................24Restriction and extension of scalars.....................................27Exactness properties of the tensor product............................28Algebras..........................................................................29Tensor product of algebras.............................................. 30Exercises..........................................................................31

Chapter 3 Rings and Modules of Fractions......................................... 36

Local properties................................................................ 40Extended and contracted ideals in rings of fractions . . . . 41Exercises......................................................................... 43

Chapter 4 Primary Decomposition...................................................50

Exercises......................................................................... 55

Contents

V

VI CONTENTS

Chapter 5 Integral Dependence and Valuations..................................... 59

Integral dependence........................................................59The going-up theorem........................................................ 61Integrally closed integral domains. The going-down theorem. . 62Valuation rings................................................................. 65Exercises.......................................................................... 67

Chapter 6 Chain Conditions.............................................................74

Exercises.......................................................................... 78

Chapter 7 Noetherian R in g s .............................................................80

Primary decomposition in Noetherian r in g s ....................... 82Exercises...........................................................................84

Chapter 8 Artin R ings...................................................................... 89

Exercises.......................................................................... 91

Chapter 9 Discrete Valuation Rings and Dedekind Domains................... 93

Discrete valuation r in g s ................................................... 94Dedekind domains............................................................ 95Fractional id e a ls .............................................................96Exercises.......................................................................... 99

Chapter 10 Completions......................................................................100

Topologies and completions.............................................. 101Filtrations .............................................................105Graded rings and modules................................................... 106The associated graded r i n g .............................................. I llExercises...........................................................................113

Chapter 11 Dimension Theory............................................................ 116

Hilbert functions............................................................ 116Dimension theory of Noetherian local r in g s ....................... 119Regular local rings.............................................................123Transcendental dimension................................................... 124Exercises...........................................................................125

Index................................................................................................ 127

Introduction

Commutative algebra is essentially the study of commutative rings. Roughly speaking, it has developed from two sources: (1) algebraic geometry and (2) algebraic number theory. In (1) the prototype of the rings studied is the ring fc[xl9. . xn] o f polynomials in several variables over a field k; in (2) it is the ring Z of rational integers. Of these two the algebro-geometric case is the more far-reaching and, in its modern development by Grothendieck, it embraces much of algebraic number theory. Commutative algebra is now one of the foundation stones of this new algebraic geometry. It provides the complete local tools for the subject in much the same way as differential analysis provides the tools for differential geometry.

This book grew out of a course o f lectures given to third year under-graduates at Oxford University and it has the modest aim of providing a rapid introduction to the subject. It is designed to be read by students who have had a first elementary course in general algebra. On the other hand, it is not intended as a substitute for the more voluminous tracts on commutative algebra such as Zariski-Samuel [4] or Bourbaki [1]. We have concentrated on certain central topics, and large areas, such as field theory, are not touched. In content we cover rather more ground than Northcott [3] and our treatment is substantially different in that, following the modern trend, we put more emphasis on modules and localization.

The central notion in commutative algebra is that of a prime ideal. This provides a common generalization of the primes of arithmetic and the points of geometry. The geometric notion of concentrating attention “near a point” has as its algebraic analogue the important process of localizing a ring at a prime ideal. It is not surprising, therefore, that results about localization can usefully be thought of in geometric terms. This is done methodically in Grothendieck’s theory of schemes and, partly as an introduction to Grothendieck’s work [2], and partly because of the geometric insight it provides, we have added schematic versions of many results in the form of exercises and remarks.

The lecture-note origin o f this book accounts for the rather terse style, with little general padding, and for the condensed account of many proofs. We have resisted the temptation to expand it in the hope that the brevity of our presenta-tion will make clearer the mathematical structure of what is by now an elegant

vii

Vlll INTRODUCTION

and attractive theory. Our philosophy has been to build up to the main theorems in a succession of simple steps and to omit routine verifications.

Anyone writing now on commutative algebra faces a dilemma in connection with homological algebra, which plays such an important part in modem developments. A proper treatment o f homological algebra is impossible within the confines o f a small book: on the other hand, it is hardly sensible to ignore it completely. The compromise we have adopted is to use elementary homological methods—exact sequences, diagrams, etc.—but to stop short o f any results requiring a deep study o f homology. In this way we hope to prepare the ground for a systematic course on homological algebra which the reader should under-take if he wishes to pursue algebraic geometry in any depth.

We have provided a substantial number o f exercises at the end o f each chapter. Some o f them are easy and some of them are hard. Usually we have provided hints, and sometimes complete solutions, to the hard ones. We are indebted to Mr. R. Y. Sharp, who worked through them all and saved us from error more than once.

We have made no attempt to describe the contributions o f the many mathematicians who have helped to develop the theory as expounded in this book. We would, however, like to put on record our indebtedness to J.-P. Serre and J. Tate from whom we learnt the subject, and whose influence was the determining factor in our choice of material and mode of presentation.

REFERENCES

1. N. Bourbaki, Algèbre Commutative, Hermann, Paris (1961-65).2. A. Grothendœck and J. Dœ udonné, Éléments de Géometrie Algébrique,

Publications Mathématiques de n.H.E.S., Nos. 4, 8,11, .., Paris (1960- ).3. D. G. Northcott, Ideal Theory, Cambridge U n iversity Press (1953).4. O. ZARisia and P. Samuel, Commutative Algebra I, Π, Van Nostrand, Princeton

(1958,1960).

Notation and Terminology

Rings and modules are denoted by capital italic letters, elements of them by small italic letters. A field is often denoted by k. Ideals are denoted by small German characters. Z, Q, R, C denote respectively the ring of rational integers, the field o f rational numbers, the field of real numbers and the field of complex numbers.

Mappings are consistently written on the left, thus the image of an element x under a mapping/is written f (x ) and not (x ) f The composition of mappings /: X -+ Y,g: Z is therefore g °f , not fog .

A mapping/: Y is injective if/ fo ) = f ( x 2) implies xx = x2; surjectiveif f { X ) = Y; bijective if both injective and suijective.

The end of a proof (or absence of proof) is marked thus ■.

Inclusion of sets is denoted by the sign £ . We reserve the sign c= for strict inclusion. Thus A

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1

Rings and Ideals

We shall begin by reviewing rapidly the definition and elementary properties of rings. This will indicate how much we are going to assume o f the reader and it will also serve to fix notation and conventions. After this review we pass on to a discussion of prime and maximal ideals. The remainder o f the chapter is devoted to explaining the various elementary operations which can be performed on ideals. The Grothendieck language o f schemes is dealt with in the exercises at the end.

RINGS AND RING HOMOMORPHISMS

A ring A is a set with two binary operations* (addition and multiplication) such that

1) A is an abelian group with respect to addition (so that A has a zero element, denoted by 0, and every x e A has an (additive) inverse, — x).

2) Multiplication is associative ((xy)z = x(yz)) and distributive over addition (x (y + z) * xy + xz, (y + z)x = yx + zx\

We shall consider only rings which are commutative:

3) xy « yx for all jc, y e A,

and have an identity element (denoted by 1):

4) 31 e A such that jcl == Ijc = jc for all jc € A.The identity element is then unique.

Throughout this book the word “ring” shall mean a commutative ring with an identity element, that is, a ring satisfying axioms (1) to (4) above.

Remark. We do not exclude the possibility in (4) that 1 might be equal to 0. If so, then for any x e A we have

X as x\ — xO = 0

and so A has only one element, 0. In this case A is the zero ringy denoted by 0 (by abuse of notation).

1

2 RINGS AND IDEALS

A ring homomorphism is a mapping/of a ring A into a ring B such that

i) f (x + y) = f ( x ) + /0 ) (so that / is a homomorphism o f abelian groups, and therefore also/(x - y) * / (*) - f ( y ) , f ( —x) - - / (x ),/ (0) - 0),

ii)/ (a ? )= / (* )/ 0 0 ,iii)/(l) = I-

In other words,/respects addition, multiplication and the identity element.

A subset S o f a ring A is a subring o f A if S is closed under addition and multiplication and contains the identity element o f A. The identity mapping of S into A is then a ring homomorphism.

If/: A -*■ B,g: B —*■ Care ring homomorphisms then so is their composition g «/: A —*■ C.

IDEALS. QUOTIENT RINGS

An ideal a o f a ring A is a subset o f A which is an additive subgroup and is such that Aa s a (i.e., x e A and y e a imply xy e a). The quotient group A/a inherits a uniquely defined multiplication from A which makes it into a ring, called the quotient ring (or residue-class ring) A/a. The elements o f Afa are the cosets of a in A, and the mapping φ :Α -* · A/a which maps each x e A to its coset x + o is a surjective ring homomorphism.

We shall frequently use the following fact:

Proportion 1.1. There is a one-to-one order-preserving correspondence between the ideals b o f A which contain a, and the ideals 6 of A/a, given by b · ■

If/: A -*■ B is any ring homomorphism, the kernel of/ (“ / " 1(0)) is an ideal a of A, and the image o f/ (= / (Λ )) is a subring C of B; and/induces a ring isomorphism A/a ~ C.

We shall sometimes use the notation x s y (mod a); this means that x — yea.

ZERO-DIVISORS. NILPOTENT ELEMENTS. UNITS

A zero-divisor in a ring A is an element x which “divides 0” , i.e., for which there exists y Φ 0 in A such that xy = 0. A ring with no zero-divisors #0 (and in which 1 Φ 0) is called an integral domain. For example, Z and k[xly (k a field, x, indeterminates) are integral domains.

An element x e A is nilpotent if jc* = 0 for some n > 0. A nilpotent element is a zero-divisor (unless A » 0), but not conversely (in general).

A unit in A is an element x which "divides 1” , i.e., an element x such that xy = 1 for some y e A. The dement y is then uniquely determined by x, and is written x ' 1. The units in A form a (multiplicative) abelian group.

PRIME IDEALS AND MAXIMAL IDEALS 3

The multiples ax o f an element x e A form a principal ideal, denoted by (x) or Αχ. X is a unit o (x) = A = (1). The zero ideal (0) is usually denoted by 0.

A field is a ring A in which 1 ^ 0 and every non-zero element is a unit. Every field is an integral domain (but not conversely: Z is not a field).

Proposition 1J2. Let A be a ring Φ 0. Then the following are equivalent: i ) A is a field;ii) the only ideals in A are 0 and (1);iii) every homomorphism o f A into a non-zero ring B is injective.

Proof i) => ii). Let α Φ 0 be an ideal in A. Then a contains a non-zero element x; x is a unit, hence a 2 (x) = (1), hence a = ( 1).

ii) => iii). Let φ: A B be a ring homomorphism. Then Ker \φ) is an ideal φ (1) in A, hence Ker (φ) = 0, hence φ is injective.

iii) => i). Let x be an element of A which is not a unit. Then (x) φ (1), hence B = A/(x) is not the zero ring. Let φ: A -+ B be the natural homo-morphism of A onto 2?, with kernel (x). By hypothesis, φ is injective, hence (x) = 0, hence x = 0. ■

PRIME IDEALS AND MAXIMAL IDEALS

An ideal p in A is prime if p Φ (1) and if xyep => x e p or yep.An ideal m in A is maximal if τη Φ (1) and if there is no ideal a such that

w c û c ( l ) (strict inclusions). Equivalently:

p is prime o A/p is an integral domain; m is maximal o A/m is a field (by (1.1) and (1.2)).

Hence a maximal ideal is prime (but not conversely, in general). The zero ideal ,is prime o A is an integral domain.

If /: A -> B is a ring homomorphism and q is a prime ideal in B, then / ’ \q) is a prime ideal in A, for A/f~\q) is isomorphic to a subring of B/q and hence has no zero-divisor φ 0. But if n is a maximal ideal o f B it is not neces-sarily true that/" *(n) is maximal in A ; all we can say for sure is that it is prime. (Example: A = Z, B = Q, n = 0.)

Prime ideals are fundamental to the whole of commutative algebra. The following theorem and its corollaries ensure that there is always a sufficient supply of them.

Theorem 1.3. Every ring A Φ Qhas at least one maximal ideal. (Remember that “ ring” means commutative ring with 1.)

Proof This is a standard application of Zorn’s lemma.* Let Σ be the set of all ideals φ (1) in A. Order Σ by inclusion. Σ is not empty, since 0 € Σ. To apply

* Let S be a non-empty partially ordered set (i.e., we are given a relation x < y on 5 which is reflexive and transitive and such that x < y and y ^ x together imply

4 RINGS AND IDEALS

Zorn’s lemma we must show that every chain in Σ has an upper bound in Σ; let then (αβ) be a chain of ideals in Σ, so that for each pair of indices «, β we have either aa g afi or a0 c ασ. Let o — (Je α*· Then a is an ideal (verify this) and 1 $ a because 1 φ ae for all a. Hence a e Σ, and a is an upper bound o f the chain. Hence by Zorn’s lemma Σ has a maximal element. ■

Corollary I A. I f a Φ (1) is an ideal o f A , there exists a maximal ideal o f A containing a.

Proof Apply (1.3) to A/ay bearing in mind (1.1). Alternatively, modify the proof o f (1.3). ■

Corollary 7.5. Every non-unit o f A is contained in a maximal ideal. ■Remarks. 1) I f A is Noetherian (Chapter 7) we can avoid the use of Zorn’s lemma: the set o f all ideals Φ ( 1) has a maximal element.

2) There exist rings with exactly one maximal ideal, for example fields. A ring A with exactly one maximal ideal m is called a local ring. The field k = A/m is called the residue field of A.

Proposition L6 . i) Let A be a ring and m Φ (1) an ideal o f A such that everyxeA — m is a unit in A. Then A is a local ring and m its maximal ideal.ii) Let Abe a ring and m a maximal ideal o f A, such that every element o f 1 + m (i.e., every 1 + x, where x s m ) is a vaut in A . Then A is a local ring.

Proof, i) Every ideal # (1) consists of non-units, hence is contained in m. Hence m is the only maximal ideal of A.

ii) Let x e A — m. Since m is maximal, the ideal generated by x and m is(1), hence there exist y e A and te m such that xy + t — 1 ; hence xy « 1 - t belongs to 1 + tn and therefore is a unit. Now use i). ■

A ring with only a finite number of maximal ideals is called semi-local.

Examples. 1) A = k[xl t ..., x„], k a field. Let fe A be an irreducible poly-nomial. By unique factorization, the ideal (/ ) is prime.

2) A — Z. Every ideal in Z is o f the form (m) for some m ^ 0. The ideal (m) is prime o m = 0 or a prime number. A ll the ideals (/>), where p is a prime number, are maximal: Zftp) is the field o f p elements.

The same holds in Example 1) for n = 1, but not for n > 1. The ideal m of all polynomials in A * k[xu ..., jcn] with zero constant term is maximal (since

x « y). Asubset Γ of 5 is a chain if either x < yo iy < x for every pair of elements x^ymT. Then Zorn’s lemma may be stated as follows: if every chain T of S has an upper bound in S (i.e., if there exists x e S such that / < x for all / e 7) then S has at least one maximal element.

For a proof of the equivalence of Zorn’s lemma with the axiom of choice, the well-ordering principle, etc., see for example P. R. Halmos, Naive Set Theory, Van Nostrand (1960).

N1LRAD1CAL AND JACOBSON RADICAL 5

it is the kernel of the homomorphism A - + k which maps f e A to /(0)). But if n > 1, m is not a principal ideal: in fact it requires at least n generators.

3) A principal ideal domain is an integral domain in which eveiy ideal is principal. In such a ring every non-zero prime ideal is maximal. For if (x ) Φ 0 is a prime ideal and (y) => (x), we have x e (y )9 say x * yz9 so that yz e (x ) and y $ (x), hence z e (x) : say z = tx. Then x = yz — ytx, so that yt — 1 and therefore (y ) = (1).

NILRADICAL AND JACOBSON RADICAL

Proposition 1.7. The set 91 of all nilpotent elements in a ring A is an ideal9 and A/îfl has no nilpotent element Φ 0.

Proof\ I f x e % clearly ax e for all a e A. Let x 9y e fft: say je"1 = 0, yn = 0. By the binomial theorem (which is valid in any commutative ring), (x + j )m+n~1 is a sum of integer multiples o f products xry*9 where r + 5 = m + « - 1 ; we cannot have both r < m and s < n> hence each of these products vanishes and therefore (x + ^)m+n“ 1 = 0. Hence x + yefft and therefore 91 is an ideal.

Let x € Afâ be represented by jc e A. Then x* is represented by xn9 so that jcn = 0=>xnG9f2=> (xn)k = 0 for some k > 0 = > x e i f t = > x = Q. ■

The ideal is called the nilradical o f A. The following proposition gives an alternative definition of

Proposition 1.8. The nilradical o f A is the intersection o f all the prime ideals o f A,

Proof. Let 91' denote the intersection of all the prime ideals o f A. I f fe A is nilpotent and if £ is a prime ideal, then /n * 0 € p for some n > 0, hence fe p (because p is prime). Hence/ e 9fc'.

Conversely, suppose that / is not nilpotent. Let Σ be the set o f ideals a with the property

n > 0 a.

Then Σ is not empty because 0 6 Σ. As in (1.3) Zorn’s lemma can be applied to the set Σ, ordered by inclusion, and therefore Σ has a maximal element Let p be a maximal element of Σ. We shall show that p is a prime ideal. Let x,y^p. Then the ideals p + (x), p + (y) strictly contain p and therefore do not belong to Σ; hence

/ * € * + (*), /“ € * + 0 0

for some m, n. It follows that/m+n e p + (xy), hence the ideal *> + (xy) is not inS and therefore xy$p. Hence we have a prime ideal p such that/£ p, so that / * * '. ' ■

The Jacobson radical 31 of A is defined to be the intersection o f all the maxi-mal ideals of A. It can be characterized as follows:

6 RINGS AND IDEALS

Proposition 1.9. x e S H o l — xy is a unit in A for a lly e A.Proof. => : Suppose 1 — xy is not a unit. By (1.5) it belongs to some maximal ideal m; but x e 81 ε tn, hence xy e m and therefore 1 e tn, which is absurd.

0) is the ideal generated by all products ΧιΧ2· ■ -xn in which each factor xt belongs to a.

Examples. 1) I f A « Z, a «* (m), b = (r ) then e + b is the ideal generated by the h.c.f. o f m and η; β n b is the ideal generated by their l.c.m. ; and ab = ( ir r ). Thus (in this case) ab = a n i o m,n are coprime.

2) A - * [* ! , . . . , x j, a = (* ! , . . . , x„) = ideal generated by xu .. . ,x n. Then am is the set o f all polynomials with no terms of degree < m.

The three operations so far defined (sum, intersection, product) are all commutative and associative. Also there is the distributive law

a(b + c) = ab + ac.

In the ring Z, π and + are distributive over each other. This is not the case in general, and the best we have in this direction is the modular law

a n (b + c) = c n b + a n c i f a 2 b o r a 2 c.

Again, in Z, we have (α + &Xa π b) = ab; but in general we have only (a + b)(a π b) £ ab (since (a + b)(a n b) = a(a π b) + b(a n b) £ ab). Clearly ab ς a n 6, hence

α n b = ab provided a + b = (I).

Two ideals û, b are said to be coprime (or comaximal) if a + b « (1). Thus for coprime ideals we have a n b = ab. Clearly two ideals a, b are coprime if and only if there exist x e a and y e b such that x + y = 1.

Let Aif. . An be rings. Their direct product

Α = Υ \ Α >:

Proposition 1.11. i) Let pl9. . pn be prime ideals and let a be an ideal contained in U ?«i Pt- Then a £ pi for some Lii) Let al9. . an be ideals and let p be a prime ideal containing Π?*ι α*·Then £ 2 ûj for some i. Ifp = Π then p = cafor some i.

Proof, i) is proved by induction on n in the form

a $ fc (1 < i ^ λ ) => a £ U Vi-

It is certainly true for « = 1. If n > 1 and the result is true for λ — 1, then foreach i there exists xt e a such that x* $ pf whenever j Φ i. I f for some i we have'X\ Φ Ρ » we are through. I f not, then xt e pt for all i. Consider the element

TV

(=1we have .yea and y φ pt (I < / < n). Hence a φ υ ? «ι £1·

ii) Suppose $ e, for ali i. Then there exist x% e a,, xt φ £ (1 ^ i < «), and therefore Πχ( e Πα; ç f ja , ; but IIjcj φ p (since £ is prime). Hence p $ f l ci·Finally, if = Π ai> then p £ a, and hence p = o, for some i. ■

If a, 6 are ideals in a ring A, their ideal quotient is (a:b) = {xeA :xb £ a}

which is an ideal. In particular, (0:6) is called the annihilator o f b and is also denoted by Ann (b): it is the set o f all x e A such that xl· — 0. In this notation the set of all zero-divisors in A is

D - U Ann (jc).x*0

If 6 is a principal ideal (x), we shall write (a : x ) in place of (a : (x)).Example. I f A = Z, a = (« ) , b = («), where say m - Π ι>Ρ*Ρ, « = ΓίνΡντ* then (a:b) = (q ) where q = tip Pr” anc*

γ, = max 0 , - v„, 0) = μ, - min (μ„, vp).

Hence q = m/(m, »), where (m, n) is the h.c.f. of m and ».

Exercise 1.12. i) a £ (a:6)ii) (a : b)b £ aiii) ((a:b ):c) = (a:be) = ((a:c):b )iv) (Π« β*:&) = Πί(βι:6)v) (α :Σ Λ ) = Π ι (a:b,).

I f a is any ideal o f A, the radical o f a is

r(a) = {x e A :xn e a for some n > 0}.

If φ : A -*■ A/a is the standard homomorphism, then r(a) = ~ 1(5Ry,,a) and hence r(a) is an ideal by (1.7).

8 * RINGS AND IDEALS

EXTENSION AND CONTRACTION 9

Exercise 1.13. i) r(a) 2 a ü) »-(r(e)) = r(e)iii) r(ûb) = r(a n b) = r(û) n r(b)iv) r(a) = (1) o a = (1)v) r(a + b) - r(r(a) + r(b))

vi) z/ p w prime, r(pn) = £ /or û// n > 0.

Proposition 1.14. The radical o f art ideal a & the intersection o f the prime ideals which contain a.

/V00/. Apply (1.8) to A ja. ■

More generally, we may define the radical r(E) of any subset E of A in the same way. It is not an ideal in general. We have r(|Je Ea) = U r(Ea), for any family of subsets Ea of A.

Proposition 1.15. D = set of zero-divisors of A — U**o r(Ann (λ·)).

Proof D * r(D) = r(U **o Ann (x )) « U**o KAnn (* ))· ■

Example. If A = Z, a = (m), let pt (1 < 1 < r) be the distinct prime divisors of ro. Then r(a) = (/V* */?r) * f l i - i (/>«)-

Proposition 1.16. Let a, b fee ideals in a ring A such that r(a), r(b) are coprime. Then a, b are coprime.

Proof r(a + b) = r(r(a) + r(b)) = r(l) = (1), hence a + b = (1) by (1.13).

EXTENSION AND CONTRACTION

Let f : A - + B be a ring homomorphism. I f o is an ideal in the set/(a) is not necessarily an ideal in B (e.g., let / be the embedding of Z in Q, the field of rationals, and take a to be any non-zero ideal in Z.) We define the extension ae of a to be the ideal Bf(a) generated by /(a) in B: explicitly, ae is the set o f all sums 2 yf(xd where xt ea9yt e B.

If b is an ideal of B, then / '2(b) is always an ideal o f A, called the contrac-tion bc of b. If b is prime, then bc is prime. I f a is prime, a* need not be prime (for example,/: Z -> Q, α Φ 0; then ae = Q, which is not a prime ideal).

We can factorize/ as follows:

where p is suijective and j is injective. For p the situation is very simple (1.1): there is a one-to-one correspondence between ideals of f (A ) and ideals of A which contain Ker (/), and prime ideals correspond to prime ideals. For j, on the other hand, the general situation is very complicated. The classical example is from algebraic number theory.

10 RINGS AND IDEALS

Example, Consider Z Z [/], where i = V ^ l. A prime ideal (jp) o f Z may or may not stay prime when extended to Z[i]. In fact Z (i] is a principal ideal domain (because it has a Euclidean algorithm) and the situation is as follows:

i) (2)e = ((1 + i)2), the square o f a prime ideal in Z [i];ii) I f p s 1 (mod 4) then (p )e is the product o f two distinct prime ideals(for example, (5)e = (2 + i)(2 — i));iii) I f p ss 3 (mod 4) then (p)e is prime in Z[i].

Of these, ii) is not a trivial result. It is effectively equivalent to a theorem ofFermat which says that a prime p s 1 (mod 4) can be expressed, essentiallyuniquely, as a sum of two integer squares (thus 5 = 22 + I2, 97 = 92 + 42,etc.).

In fact the behavior of prime ideals under extensions o f this sort is one of the central problems o f algebraic number theory.

Let f : A B, a and b be as before. Then

Proposition 1.17. i) a £ aec, b 2 bce;ii) bc = bcec, ae = a*ce;

iii) J/C « //re .rei q/' contracted ideals in A and if E is the set o f extended ideals in By then C = {a| aec = a}, £ = {b| bce = b}, anrf û ae is a bijective map of C onto £, w/tewe inverse w b bc.

/V00/. i) is trivial, and ii) follows from i).

iii) I f a e C, then a = bc.= bcec = aec; conversely if a = aec then a is the contraction o f ae. Similarly for E. ■

Exercise 1.18. I fa u a2 are ideals o f A and i/bl5 b2 are idea/s ο/ B, then

(ai + ΰβ)β ** aî + a2j (&i + &2)c 2 bf + b|,(αχ n û2)e s aj n a§, (b2 n b2)c = bj n bg,(ûiû2)e s ûîa2> 0>i&2)c S &1&2>( a ^ ) « c (afraj), (b2:b2)c

EXERCISES 11

i) / is a unit in A[x] o a0 is a unit in A and au . . On are nilpotent. [Ifb0 + b±x + ----h bmxT is the inverse of /, prove by induction on r thatc^ xbm. T = 0. Hence show that a* is nilpotent, and then use Ex. 1.]

ii) /is nilpotent aQ, au . . are nilpotent.iii) / is a zero-divisor there exists a Φ 0 in A such that af « 0. [Choose a

polynomials » b0 + bxx + ---- l· of least degree m such thatjÿ = 0.Then anbm = 0, hence e,»* = 0 (because ang annihilates / and has degree < m). N o w show by induction that an-rg = 0 (0 ^ r ^ «).]

iv) /is said to be primitive if (a0, al9..., an) = (1). Prove that if/, ̂ e Λ[χ], then Jfc is primitive o f and £ are primitive.

3. Generalize the results of Exercise 2 to a polynomial ring A[xu in severalindeterminates.

4. In the ring A[x\ the Jacobson radical is equal to the nilradical.5. Let Λ be a ring and let A[[x]] be the ring of formal power series / = 2®-o anxn

with coefficients in A. Show thati) / is a unit in Allx]] o a0 is a unit in A.ii) If / is nilpotent, then an is nilpotent for ail n ^ 0. Is the converse true?

(See Chapter 7, Exercise 2.)iii) / belongs to the Jacobson radical of Allx]] o a0 belongs to the Jacobson

radical of A.iv) The contraction of a maximal ideal m of ̂ [[jc]] is a maximal ideal of A, and

m is generated by mc and x.v) Every prime ideal of A is the contraction of a prime ideal of Allx]].

6. A ring A is such that every ideal not contained in the nilradical contains a non-zero idempotent (that is, an element e such that é2 « e Φ 0). Prove that the nilradical and Jacobson radical of A are equal.

7. Let A be a ring in which every element x satisfies xn = x for some n > 1 (depending on x). Show that every prime ideal in A is maximal.

8. Let A be a ring Φ 0. Show that the set of prime ideals of A has minimal ele-ments with respect to inclusion.

9. Let a be an ideal Φ (1) in a ring A. Show that a = r(a) o a is an intersection of prime ideals.

10. Let A be a ring, % its nilradical. Show that the following are equivalent:i) A has exactly one prime ideal;ii) every element of Λ is either a unit or nilpotent;iii) A/Hfl is a field.

11. A ring A is Boolean if x* » χ for all x e A. In a Boolean ring A, show thati) 2x « 0 for all xeA ;ii) every prime ideal p is maximal, and A/p is a field with two elements;iii) every finitely generated ideal in A is principal.

12. A local ring contains no idempotent Φ 0 ,1.

Construction of an algebraic closure of a field (E. Artin).13. Let K be a field and let Σ be the set of all irreducible monic polynomials/in one

12 RINGS AND IDEALS

indeterminate with coefficients in K. Let A be the polynomial ring over K generated by »determinates xf9 one for each /€ Σ. Let a be the ideal of A generated by the polynomials f (x f) for all/€Σ. Show that α Φ (1).

Let m be a maximal ideal of A containing a, and let Κχ = A/m. Then jKi is an extension field of K in which each /e Σ has a root. Repeat the construction with K\ in place of K, obtaining a field K2i and so on. Let L = U®«i Kn. Then £ is a field in which each/e Σ splits completely into linear factors. Let R be the set of all elements of L which are algebraic over K. Then J? is an algebraic closure of K.

14. In a ring A, let Σ be the set of all ideals in which every element is a zero-divisor. Show that the set Σ has maximal elements and that every maximal element of Σ is a prime ideal. Hence the set of zero-divisors in A is a union of prime ideals.

The prime spectrum of a ring15. Let A be a ring and let X be the set o f all prime ideals of A. For each subset

E of A, let V(E) denote the set of all prime ideals of A which contain E. Prove thati) if û is the ideal generated by E, then V(E) = V(a) = V(r(a)).ii) V(0) = X9 F (l) = 0 .iii) if 05)i67 is any family of subsets of A, then

F( y * ) - o wiv) V(a n b ) = V(ab) - V(a) u V(b) for any ideals a, t of A.

These results show that the sets V(E) satisfy the axioms for closed sets in a topological space. The resulting topology is called the Zariski topology· The topological space X is called the prime spectrum of A, and is written Spec 04).

16. Draw pictures of Spec (Z), Spec (R), Spec (C[jc]), Spec (RM ), Spec (Z[x]).17· For each /€ A, let Xf denote the complement of V(f) in X = Spec (A). The

sets Xf are open. Show that they form a basis of open sets for the Zariski topology, and that

i) Xf n X 9 * Xf0;ii) X/ = 0 o f is nilpotent;iii) Xf m X o f isaunit;iv) Xf = X, o r ( ( f ) ) = r ((* ));v) X is quasi-compact (that is, every open covering of X has a finite sub-

covering). \vi) More generally, each Xf is quasi-compact.vii) An open subset of X is quasi-compact if and only if it is a finite union of

sets Xf.The sets Xf are called basic open sets of X = Spec (A).

[To prove (v), remark that it is enough to consider a covering of X by basic open sets Xu (i € /). Show that the/ generate the unit ideal and hence that there is an equation of the form

1 = 2 Sifi (il e Ä)

EXERCISES 13

18. For psychological reasons it is sometimes convenient to denote a prime ideal of A by a letter such as x or y when thinking of it as a point of X « Spec (A). When thinking of x as a prime ideal of A, we denote it by px (logically, of course, it is the same thing). Show thati) the set {x} is closed (we say that x is a “closed point”) in Spec (A) px is

maximal;ii) {* } =_ V(px);iii) y e { x } o p x z py;iv) X is a Γο-space (this means that if x, y are distinct points of X, then either

there is a neighborhood of x which does not contain y, or else there is a neighborhood of y which does not contain jc).

19. A topological space X is said to be irreducible if X Φ 0 and if every pair of non-empty open sets in X intersect, or equivalently if every non-empty open set is dense in X. Show that Spec (A ) is irreducible if and only if the nilradical of A is a prime ideal.

20. Let A" be a topological space.i) I f K is an irreducible (Exercise 19) subspace of X, then the closure F of y

in X is irreducible.ii) Every irreducible subspace of X is contained in a maximal irreducible

subspace.. iii) The maximal irreducible subspaces of X are closed and cover X. They are

called the irreducible components of X. What are the irreducible components of a Hausdorff space?

iv) If A is a ring and X = Spec (A), then the irreducible components of X are the closed sets V(p\ where p is a minimal prime ideal of A (Exercise 8).

21. Let φ: A -*■ B be a ring homomorphism. Let X - Spec (A) and Y * Spec (B). If q e Y9 then ÿ~*()) = V(pc).iv) If φ is surjective, then φ* is a homeomorphism of Y onto the closed subset

K(Ker (Φ)) of X. (In particular, Spec (A ) and Spec (A/ÎR) (where 3£ is the nilradical of A) are naturally homeomorphic.)

v) If φ is injective, then φ*( Y) is dense in X. More precisely, φ*( Y) is dense in X o Ker (φ) s fjft,

vi) Let ψ:Β-+ C be another ring homomorphism. Then (Φ ο φ)* = ψ* ο φ*.vii). Let A be an integral domain with just one non-zero prime ideal and let K

be the field of fractions of A. Let B = (A/p) x K. Define φ :Α -* Β by Φ(χ) = (jc, x), where x is the image of x in A/p. Show that φ* is bijective but not a homeomorphism.

22. Let A = Πί1« ! At be the direct product of tings At. ’ Show that Spec (A) is the disjoint union of open (and closed) subspaces Xi9 where X{ is canonically homeomorphic with Spec (At).

14 RINGS AND IDEALS

Conversely, let A be any ring. Show that the following statements are equivalent:i) X ■ Spec (A) is disconnected.ii) A s Ax x A2 where neither of the rings Au A2 is the zero ring.iii) A contains an idempotent ^ 0,1.

In particular, the spectrum of a local ring is always connected (Exercise12).

23· Let A be a Boolean ring (Exercise 1 \% and let X « Spec (A).i) For each/e A, the set Xf (Exercise 17) is both open and closed in X.ii) Let/!,.. .,/neÆ Show that Xfl u - ' -u Xfn = Xf for some f e A.iii) The sets Xf are the only subsets of X which are both open and closed.

[Let Y £ X be both open and closed. Since Y is open, it is a union of basic open sets Xr. Since Y is closed and X is quasi-compact (Exercise 17), Y is quasi-compact. Hence Y is a finite union of basic open sets; now use (ii) above.]

iv) X is a compact Hausdorff space.

24. Let L be a lattice, in which the sup and inf of two elements a9 b are denoted by a V b and a A b respectively. L is a Boolean lattice (or Boolean algebra) ifi) £ has a least element and a greatest element (denoted by 0,1 respectively).ii) Each of v , a is distributive over the other.iii) Each a eL has a unique “complement” a 'eL such that a v a' « 1 and

β A a' * 0.(For example, the set of all subsets of a set, ordered by inclusion, is a Boolean lattice.)

Let L be a Boolean lattice. Define addition and multiplication in L by the rules

a -f b « (a A b') v (a' a b)9 ab « a a b.

Verify that in this way L becomes a Boolean ring, say A{L).Conversely, starting from a Boolean ring A9 define an ordering on A as

follows: a < b means that a « ab. Show that, with respect to this ordering, A is a Boolean lattice. [The sup and inf are given by a v b = a + b + ab and a A b - ab9 and the complement by a’ * l - a.] In this way we obtain a one-to-one correspondence between (isomorphism classes of) Boolean rings and (isomorphism classes of) Boolean lattices.

25. From the last two exercises deduce Stone’s theorem, that eveiy Boolean lattice is isomoxphic to the lattice of open-and-closed subsets of some compact Haus-dorff topological space.

26. Let A be a ring. The subspace of Spec {A) consisting of the maximal ideals of A9 with the induced topology, is called the maximal spectrum of A and is denoted by Max (A). For arbitrary commutative rings it does not have the nice functorial properties of Spec {A) (see Exercise 21), because the inverse image of a maximal ideal under a ring homomorphism need not be maximal.

Let JIT be a compact Hausdorff space and let C(X) denote the ring of all real-valued continuous functions on X (add and multiply functions by adding

EXERCISES 15

and multiplying their values). For each xe X, let nt* be the set of all f e C(X) such that /(jc) == 0. The ideal m* is maximal, because it is the kernel of the (suijective) homomorphism C(X) —► R which takes / to f(x). If X denotes Max (C(X))t we have therefore defined a mapping μ: X -+ Jt, namely x mx.

We shall show that μ is a homeomorphism of X onto X.i) Let m be any maximal ideal of C(X), and let V = V(m) be the set of com-

mon zeros of the functions in m: that is,

V ^ {xe X:f(x) = 0 for all/e m}.

Suppose that V is empty. Then for each xe X there exists/* e m such that fx(x) Φ 0. Since Λ is continuous, there is an open neighborhood Ux of x in X on which fx does not vanish. By compactness a finite number of the neighborhoods, say £/*,',..., UXn, cover X. Let

/ - A + · · · + /»«·Then / does not vanish at any point of X> hence is a unit in C(X). But this contradicts f e τη, hence V is not empty.

Let jc be a point of V. Then m £ m*, hence τη - mx because m is maximal. Hence μ is suijective.

ii) By Uiysohn’s lemma (this is the only non-trivial fact required in the argu-ment) the continuous functions separate the points of X. Hence x Φ y => mx Φ my, and therefore μ is injective.

iii) Let f e C( X); letUf = {xeX:f(x ) * 0}

and letOf = {m e f :/ im }

Show that μ(υ;) = 0f. The open sets Uf (resp. Uf) form a basis of the top-ology of X (resp. X ) and therefore μ is a homeomorphism.Thus X can be reconstructed from the ring of functions C(X).

Affine algebraic varieties27. Let k be an algebraically closed field and let

fa(ti9 · · · » Λ») = 0

be a set of polynomial equations in n variables with coefficients in k. The set X of all points x = (* i , ..., xn) e kn which satisfy these equations is an affine algebraic variety.

Consider the set of all polynomials g e k[tXi..., rn] with the property that £(x) = 0 for all x € X. This set is an ideal I{X ) in the polynomial ring, and is called the ideal of the variety X. The quotient ring

P{X) = k{tl9. . . t tnyi{X)

is the ring of polynomial functions on X, because two polynomials gt h define the same polynomial function on Xif and only if g — h vanishes at every point of X, that is, if and only if g — h e I(X).

16 RINGS AND IDEALS

Let & be the image of /, in P(X). The ft (1 < x < n) are the coordinate functions on X:\fxe X, then f t(x) is the ith coordinate of x. P (X ) is generated as a fc-algebra by the coordinate functions, and is called the coordinate ring (or affine algebra) of X.

As in Exercise 26, for each x e Xict mx be the ideal of allf e P (X ) such that f (x ) = 0; it is a maximal ideal of P (X ). Hence, if X « Max (P (X ))t we have defined a mapping μ: X -+ namely x κ m*.

It is easy to show that μ is injective: if x & y, we must have Ψ yt for for some / (I ^ î < λ ), and hence £< — is in m* but not in my, so that mx Φ my. What is less obvious (but still true) is that μ is surjective. This is one form of Hilbert’s Nullstellensatz (see Chapter 7).

28· Let/i,. . -,/nbe elements of k[tu . . f j . They determine a polynomial mapping φ: kn km: if x e &n, the coordinates of φ(χ) are fi(x) , .. .,/»»(*).

Let Jf, y be affine algebraic varieties in kn, km respectively. A mapping φ: X y is said to be regular if φ is the restriction to AT of a polynomial map-ping from £* to km.

If η is a polynomial function on Y, then η © φ is a polynomial function on X. Hence φ induces a fc-algebra homomorphism P (Y ) P(X)> namely -η κ v ο φ. Show that in this way we obtain a one-to-one correspondence between the regular mappings X -+ Y and the Ar-algebra homomorphisms P(Y ) P(X).

2

Modules

One of the things which distinguishes the modern approach to Commutative Algebra is the greater emphasis on modules, rather than just on ideals. The extra “elbow-room” that this gives makes for greater clarity and simplicity. For instance, an ideal a and its quotient ring A/a are both examples of modules and so, to a certain extent, can be treated on an equal footing. In this chapter we give the definition and elementary properties o f modules. We also give a brief treatment o f tensor products, including a discussion o f how they behave for exact sequences.

MODULES AND MODULE HOMOMORPHISMS

Let Λ be a ring (commutative, as always). An A-module is an abelian group M (written additively) on which A acts linearly: more precisely, it is a pair (Μ ,μ), where M is an abelian group and μ is a mapping o f A x M into M such that, if we write ax for μ{α, x)(a e A, x e M ), the following axioms are satisfied:

a(x + y) — ax + ay,(a + b)x — ax + bx,

(ab)x = a(bx),Ix = x (a, be A; x ,y e M ) .

(Equivalently, M is an abelian group together with a ring homomorphism A -»> E(M), where E (M ) is the ring of endomoiphisms of the abelian group M.)

The notion o f a module is a common generalization of several familiar concepts, as the following examples show:

Examples. 1) An ideal a o f A is an Λ-module. In particular A itself is an (̂•module.

2) I f ii is a field k, then Λ-module « λ-vector space.3) A = Z, then Z-module « abelian group (define rx to be x + ■ · · + x).4) A = £[.*] where k is a field; an ̂ -module is a Λ-vector space with a linear

transformation.5) G = finite group, A = k[G] = group-algebra of G over the field k (thus

A is not commutative, unless G is). Then Λ-module « ^-representation o f G.17

18 MODULES

Let M, AT be /4-modules. A mapping is an A-module homo-morphism (or is A-Iinear) if

/(* + y) = fix) + / ω/(ex) = β·/(χ)

for all a e A and all x, y eM . Thus / is a homomorphism o f abelian groups which commutes with the action o f each a g A. I f A is a field, an ^-module homomorphism is the same thing as a linear transformation o f vector spaces.

The composition of ^-module homomorphisms is again an ,4-module homomorphism.

The set o f all ̂ -module homomorphisms from M toNcan be turned into an Λ-module as follows: we define/ + g and af by the rules

ifl/yJ) = a-fix)

for all x e M. It is a trivial matter to check that the axioms for an Λ-module are satisfied. This Λ-module is denoted by Homx (Μ, N ) (or just Horn (Μ, N ) if there is no ambiguity about the ring A).

Homomorphisms u: M ’ -*■ M and v: N -*■ N " induce mappings

a: Horn (Ai, N ) -*· Horn (Λ/', N ) and ϋ: Horn (Μ, N ) - * Horn (Μ, Ν ' )

defined as follows:

«(/ ) = / » u, e (f ) = v o f

These mappings are Λ-module homomorphisms.For any module M there is a natural isomorphism Horn (A, M ) £ M : any

Λ-module homomorphism /: Λ -»■ Λί is uniquely determined by /(1), which can be any element o f M.

SUBMODULES AND QUOTIENT MODULES

A submodule AT o f JI/is a subgroup o f M which is closed under multiplication by dements o f A. The abelian group M jM ' then inherits an Λ-module structure from M, defined by a(x + Μ 1) = ax + M '. The Λ-module M jM ' is the quotient o f M by M ‘. The natural map o f M onto M jM ' is an Λ-module homo-morphism. There is a one-to-one order-preserving correspondence between submodules o f M which contain A f', and submodules o f M " (just as for ideals; the statement for ideals is a special case).

If/: A/-* AT is an Λ-module homomorphism, the kernel of/is the set

Ker (/ ) = {x e M :f ix ) - 0}

and is a submodule o f M. The image of/is the set

Im (/ ) = / (M )

OPERATIONS ON SUBMODULES 19

and is a submodule of N. The cokernel of / is

Coker (/ ) = AT/Im (/ )

which is a quotient module of N.If A/ 7 is a submodule of M such that M ' £ Ker (/), then/ gives rise to a

homomorphism/: M/M' —► Af, defined as follows: if x e M/M' is the image of xeA/, then/(x) = /(x). The kernel of/is Ker (f ) IM \ The homomorphism/ is said to be induced by / In particular, taking M ' — Ker (/), we have an isomorphism of ̂ -modules

M/lLtv (/ ) £ Im (/).

OPERATIONS ON SUBMODULES

Most of the operations on ideals considered in Chapter 1 have their counter-parts for modules. Let M be an Λ-module and let (A/,) Μ τ + M 2 -*■ (M i 4- M2)/M1 is surjective, and its kernel is M x n M2; hence (ii). ■

We cannot in general define the product of two submodules, but we can define the product aM, where a is an ideal and M an Λ-module; it is the set o f all finite sums 2 with at e a, xi e M, and is a submodule of M.

If N> P are submodules of M, we define (N :P) to be the set of all a e A such that aP £ N; it is an ideal o f A. In particular, (0: M ) is the set o f all a e A such that aM = 0; this ideal is called the annihilator of M and is also denoted by Ann (Af). If a c Ann (Af), we may regard M as an Λ/α-module, as follows: if x € A/a is represented by x e A, define xm to be xm(m e M ): this is independ-ent of the choice of the representative x of x, since aM = 0.

20 MODULES

An Λ-module is faithful if Ann (Af)=0. If Ann (M ) = a, then M is faithful as an Λ/α-module.

Exercise 2.2. i) Ann (M + N ) — Ann (Af ) n Ann (N).ii) (N :P ) = Ann ( (N + P)/AT).

If x is an element o f M , the set of all multiples ax(a e A ) is a submodule o f M, denoted by Ax or (x). I f Af = 2 l6J Axi9 the xt are said to be a set of gen-erators o f Af; this means that every element of M can be expressed (not neces-sarily uniquely) as a finite linear combination of the x{ with coefficients in A. An Λ-module M is said to be finitely generated if it has a finite set of genera-tors.

DIRECT SUM AND PRODUCT

If My N are Λ-modules, their direct sum Μ ® N is the set of all pairs (x, y) with x e M , y e N. This is an Λ-module if we define addition and scalar multiplica-tion in the obvious way:

(x »y i ) + (* 2, y 2) = (* i + * 2, yi + y2) α(χ, y) = (axy ay).

More generally, if (Aft)i€/ is any family of Λ-modules, we can define their direct sum @ j€/ Af, ; its elements are families (xt)ie/ such that x{ e Af» for each ie I and almost all jct are 0. If we drop the restriction on the number of non-zero x's we have the direct product Πιε; Mi· Direct sum and direct product are therefore the same if the index set / is finite, but not otherwise, in general.

Suppose that the ring A is*a direct product ΠΓ=; Λ, (Chapter 1). Then the set of all elements o f A of the form

(0, . . . , 0, a*, 0, . . . , 0)

with a, 6 Ai is an ideal a{ o f A (it is not a subring o f A—except in trivial cases— because it does not contain the identity element o f A). The ring A, considered as an Λ-module, is the direct sum of the ideals al9. . αΛ. Conversely, given a module decomposition

Λ =» Û! © · · -0 cn

of Λ as a direct sum of ideals, we have

A s Y l (A !b {)iml

where b, = @ #jM a,. Each ideal a< is a ring (isomorphic to A[b{). The identity element o f a* is an idempotent in A, and Oj = (et).

FINITELY GENERATED MODULES

A free Λ-module is one which is isomorphic to an Λ-module of the form @ i€/ M h where each AT* £ A (as an Λ-module). The notation Aœ is sometimes used. A finitely generated free Λ-module is therefore isomorphic to A · ·® Λ (n summands), which is denoted by An. (Conventionally, A0 is the zero module, denoted by 0.)

Proposition 2.3. M is a finitely generated A-module ο M is isomorphic to a quotient o f An for some integer n > 0.

Proof =>: Let xly. . xn generate M. Define φ:Αη- + Μ by φ(αΐ9. . an) = QiXj. H------l· anxn. Then φ is an Λ-module homomorphism onto M , and there-fore M £ An!Ker (φ).

*1

where is the Kronecker delta. By multiplying on the left by the adjoint of the matrix (δ,^ - ais) i t follows that det ( ^ - aif) annihilates each x*, hence is the zero endomorphism of M. Expanding out the determinant, we have an equation of the required form. ■

Corollary 2J. Let M be a finitely generated A-module and let a be an ideal of A such that aM = M. Then there exists x s l(mod a) such that xM = 0.

Proof Take φ = identity, x = 1 + ax H------V an in (2.4). ■

Proposition 2.6. (Nakayama's lemma). Let M be a finitely generated A-module and a an ideal o f A contained in the Jacobson radical iftof A. Then aM = M implies M = 0.

First Proof By (2.5) we have xM = 0 for some x ξ 1 (mod $R). By (1.9) jc is a unit in Λ, hence M = x~xxM = 0. ■

FINITELY GENERATED MODULES 21

22 MODULES

Second Proof. Suppose Μ φ 0, and let ux, ... , « » be a minimal set o f gener-ators of M. Then un e aM, hence we have an equation o f the form u„ = ax« i + ---- 1- «„a,, with the a, e a. Hence

(1 - ΰφ η - OjUi + · · · + On-jUn.!-,since an e JR, it follows from (1.9) that 1 — an is a unit in A. Hence u„ belongs to the submodule o f M generated by mx...... u » - i ’. contradiction. ■

Corollary 2.7. Let M be afinitely generated A-module, N a submodule o f M, a £ 8Ran ideal. Then M — aM + N => Μ — N.

Proof. Apply (2.6) to MjN, observing that a(M/N) =* (aM + N)/N. m

Let A be a local ring, nt its maximal ideal, k — Aim its residue field. Let M be a finitely generated Λ-module. M/mM is annihilated by m, hence is naturally an Λ/τη-module, i.e., a fc-vector space, and as such is finite-dimensional.

Proposition 2J8. Let xt (1 < i < n) be elements o f M whose images in M/mM form a basis o f this vector space. Then the xt generate M .

Proof. Let N be the submodule o f M generated by the xt. Then the composite map N -+ M -* · M/mM maps N onto M/mM, lienee N + mM = M, hence N = M by (2.7). ■

EXACT SEQUENCES

A sequence of Λ-modules and Λ-homomorphisms

-------- » M ^ - l + M . m + M ^ -----►··· (0)is said to be exact at M, if Im ( f ) = Ker (f ,+1). The sequence is exact if it is exact at each M,. In particular:

0 -»· A f 'A - M is exact - o f is injective; (1)Μ M " -> 0 is exact

EXACT SEQUENCES 23

ii) LetO-j-W'-V N " (5)

be a sequence of A-modules and homomorphisms. Then the sequence (5) is exact -=> for all A-modules M, the sequence

0 -* Horn (Μ, Ν') * + Horn (M, N) -V Horn (Μ, N") (5')

is exact.All four parts of this proposition are easy exercises. For example, suppose

that (4') is exact for all N. First o f all, since v is injective for all N it follows thatV is suijective. Next, we have ü o v = 0, that is v o u o f 0 for all /: M * N. Taking N to be M ” and / to be the identity mapping, it follows that v © u = 0, hence Im (u) £ Ker (t>). Next take N = Af/Im (u) and let φ : Μ - * Ν be the projection. Then φ e Ker (u), hence there exists ψ : Μ ” N such that φ *= φ ον. Consequently Im (u) = Ker (ÿ) 2 Ker (z?). ■

Proposition 2.10. Le/

Μ - ^ Λ Τ - * 0 Γψ j/"

o N ” -+ 0

6e a commutative diagram of A-modules and homomorphisms, vW/Λ rAe raws exac/. 77ze/i fAere ex/$/.s exatf sequence

0 Ker (/ ') Ker (/ ) A . Ker (/*) UCoker (/ ') 2V Coker (/ ) Coker (/ *)-> 0 (6)

in which ü, v are restrictions o f u, v, and S', ϋ' are induced by ν'.

The boundary homomorphism dis defined as follows : if x’ e Ker (/ '), we have x" = »(x ) for some x e M , and » '(/ (* )) = f" (v (x )) = 0, hence/ (x) e Ker (o') = Im («')» so that/(x) = u (y') for some ÿ e Ν'. Then d(x") is defined to be the image of ÿ in Coker (/ ')· The verification that d is well-defined, and that the sequence (6) is exact, is a straightforward exercise in diagram-chasing which we leave to the reader. ■

Remark. (2.10) is a special case o f the exact homology sequence of homological algebra.

Let C be a class of -4-modules and let Λ be a function on C with values in Z (or, more generally, with values in an abelian group G). The function A is additive if, for each short exact sequence (3) in which all the terms belong to C, we have λ(Μ’) - M.M) + λ(Μ ') = 0.

Example. Let A be a field k, and let C be the class of all finite-dimensional λ-vector spaces V. Then V dim F is an additive function on C.

2 + I.C. A.

24 MODULES

Proposition 2.11. Let 0 M 0 M i ---- ► M n —> 0 an exac/ je-quence o f A-modules in which all the modules M { and the kernels o f all the homomorphisms belong to C. Then for any additive function Λ on C we have

2 ( - I ) ' W = o. 0

(N 0 — Nn+i = 0). Then we have X(Mt) = λ(Ν,) + λ(JV,+X). Now take the alternating sum of the X(Mt), and everything cancels out. ■

TENSOR PRODUCT OF MODULES

Let Μ ,Ν ,Ρ be three ^-modules. A mapping f : Μ x N - * P is said to be A-bilinear if for each x e M the mapping y t-» f ix , y) o f N into P is /(-linear, and for each y e N the mapping x i-» /(jc, y) o f M into P is ̂ -linear.

We shall construct an ^-module T, called the tensor product o f M and N, with the property that the ^-bilinear mappings M x jV —v jP are in a natural one-to-one correspondence with the Λ-linear mappings T-*-P , for all A- modules P. More precisely:

Proposition 2.12. Let Μ, N be A-modules. Then there exists a pair (T, g ) consisting o f an A-module T and an A-bilinear mapping g: Μ x N -*■ T, with the following property:

Given any A-module P and any A-bilinear mapping f :M x N -+ P , there exists a unique A-Unear mapping f ' . T - * P such t/mt f — f ' ° g (in other words, every bilinear function on Μ x N factors through T ).

Moreover, i f (Γ, g ) and (Τ ', g0 are two pairs with this property, then there exists a unique isomorphism j : T - * -T ' such thatj » g = g’.

Proof, i) Uniqueness. Replacing ( P , f ) by (Τ ', g ') we get a unique j : Γ - » 7” suchthat;' = . j o g . Interchanging the roles o f T and Τ ', we %ttj' : T ’ -+ T such that g = j ' o g'. Each o f the compositions j » j must be the identity, andtherefore j is an isomorphism.

ii) Existence. Let C denote the free ^-module AWxK>. The elements o f C are formal linear combinations o f elements o f Μ x N with coefficients in A,i.e. they are expressions o f the form £ “- i a,(xt,yt)(ai s A ,x ,e M, yt e N).

Let D be the submodule o f C generated by all elements o f C o f the follow-ing types:

(x + x \ y) - (x, y) - (*'. y)(x,y + / ) - (x,y) - (* ,/)

(ax, y) - a (x,y )(x, ay) - o (x,y)·

TENSOR PRODUCT OF MODULES 25

Let T = C\D. For each basis dement (x,.y) of C, let x ® y denote its image in T. Then T is generated by the elements o f the form x y, and from our definitions we have

(x + x ') ® y = x ® y + x ' ® y, x ® ( j + ÿ ) = x ® y + x ® ÿ , (ex) ® y = x ® (ay ) = a(x ® y)

Equivalently, the mapping g: M x N —y T defined by g(x, y) = x ® y is ^•bilinear.

Any map / o f Μ x N into an ^-module P extends by linearity to an A- module homomorphism /: C P. Suppose in particular that/is ^-bilinear. Then, from the definitions,/vanishes on all the generators o f D, hence on the whole o f D, and therefore induces a well-defined Λ-homomorphism / ' o f T = C/2) into P such that / '(x ® y) - f (x,y). The mapping/' is uniquely defined by this condition, and therefore the pair (Γ, g) satisfy the conditions of the proposition. ■

Remarks, i) The module T constructed above is called the tensor product o f M and N, and is denoted by Μ ® Λ N, or just M ® JV if there is no ambiguity about the ring A. It is generated as an ^-module by the “ products” x ® y. I f (xt)iei, Ov)j«/ are families of generators of Μ , N respectively, then the elements x{ ® y, generate Μ ® N. In particular, if M and N are finitely generated, so is M ® i V .

ii) The notation x ® y is inherently ambiguous unless we specify the tensor product to which it belongs. Let M N ' be submodules of Μ, N respectively, and let x e M ' and y e Ν '. Then it can happen that x ® y as an dement o f Μ ® N is zero whilst x ® y as an element o f M ' ® N ' is non-zero. For example, take A — Z,A/ = L ,N = Z/2Z, and let M ' be the submodule 2Z of Z, whilst N ' = N. Let x be the non-zero element of N and consider 2 ® x. As an element o f Μ ® N, it is zero because 2 ® x = 1 ® 2 x * = l ® 0 = 0. But as an element of M ' ® N ' it is non-zero. See the example after (2.18).

However, there is the following result:

Corollary 2.13. Let x, e M, y,e N be such that 2 Xi ® yt = 0 in Μ ® N.Then there exist finitely generated submodules M 0 o f M and N0 o f N such that Σ xi ® Λ = 0 w ® N0.

Proof I f 2 Xi ® Λ = 0 in M ® JV, then in the notation of the proof of (2.11) we have 2 (*t. yd € A and therefore 2 (xi> yd is a finite sum of generators o f D. Let M0 be the submodule of M generated by the x, and all the elements o f M which occur as first coordinates in these generators of D, and define N0 simi-larly. Then 2 Xi ® Λ “ 0 as an element o f M 0 ® N0. m

iii) We shall never again need to use the construction o f the tensor product given in (2.12), and the reader may safely forget it if he prefers. What is essential to keep in mind is the defining property of the tensor product.

26 MODULES

iv) Instead of starting with bilinear mappings we could have started with multilinear mappings f : M x x · · · x MT- + P defined in the same way (i.e., linear in each variable). Following through the proof of (2.12) we should end up with a “ multi-tensor product” T = Μχ ® · · · ® M„ generated by all products * 1 ® · · · ® x, (xi e Μχ, 1 < t < r). The details may safely be left to the reader; the result corresponding to (2.12) is

Proposition 2.12*. Let M x, M T be A-modules. Then there exists a pear (Τ’, g ) consisting o f an A~module T and an A-muItilinear mapping g: M l x ■ ■· x M , —»· T with the following property:

Given any A-module P and any A-multilinear mapping f : M x x · · · x M r —> T, there exists a unique A-homomorphism f : T P such that

Γ ' 8 - f .Moreover, i f (T, g ) and (Τ', g ') are two pairs with this property, then there

exists a unique isomorphism j : T - + T ' such that j ° g = g '· ■

There are various so-called “ canonical isomorphisms” , some of which we state here:

Proposition 2.14. Let M, N, P be A-modules. Then there exist unique isomorphismsi) M N-+N y ® xb) ( x ® y ) ® i i - > x ® ( j ' ® z ) H > - x ® ^ ® z

c) (*, y) ® z ·-► (x ® 2, y ® z)d) a ® jc h»· ax.

Proof. In each case the point is to show that the mappings so described are well defined. The technique is to construct suitable bilinear or multilinear mappings, and use the defining property (2.12) or (2.12*) to infer the existence o f homo-morphisms o f tensor products. We shall prove half o f ii) as an example o f the method, and leave the rest to the reader.

We shall construct homomorphisms

(M ® N ) ® P 4 - M ® N ® P *+ {M ® N ) ® P

such that /((jc ®_ jO ®z ) = j c ® .V ® z and g(x ® y ® z) = (x ® y) ® z for all jc e M , y e N, z e P.

To construct /, fix the element ze P. The mapping (x, y) h* x ® y ® z (x e M , y e N ) is bilinear in x and y and therefore induces a homomorphism

RESTRICTION AND EXTENSION OF SCALARS 27

f 9: M ® N M ® N ® P such XhaXf2{x ® y) = x ® y ® z. Next, consider the mapping (t, z) f z(t) of (M ® N ) x P into A/ ® TV ® P. This is bilinear in f and z and therefore induces a homomorphism

/ : ( Λ / ® 7 / ) ® Ρ - * Μ ® Λ Γ ® Ρ

such that /((jc ® j>) ® 2) = x ® ^ ® z.To construct g, consider the mapping (x, y, z) (x ® y) ® z of M x iV

x Pinto (M ® iV) ® P. This is linear in each variable and therefore induces a homomorphism

g: Μ ® N ® P (Λ/ ® Λ0 ® P

such that £(* ® ® z) = (x ® 7) ® z.Clearly/o g and g o/are identity maps, hence/and g are isomorphisms. ■

Exercise 2.15. Let A,£be rings, lei M be an A-module, P a B-module and Nan {A, Bybimodule ( that is, N is simultaneously cm A-module and a B-module and the two structures are compatible in the sense that a(xb) = (ax)b fo r all a eA , beB9x e N ). Then M ® AN is naturally a B-module, N ®* P an A-module, and we have

(M ® a N ) ® s P z M ® x (N ®bP)·

Let f\ M M\ g: N -+ N ' be homomorphisms of Λ-modules. Define h: M x JV -> A f' ® N ' by h(x, y) = / (*) ® £0>). It is easily checked that h is Λ-bilinear and therefore induces an Λ-module homomorphism

f ® g : M ® N - + M ' ® N ’

such that

i f ® g)(* ® >0 = /(*) ® g(y) (xeM, ye N).Let f ' : M ' M ” and g ': Ν ' N* be homomorphisms of Λ-modules.

Then clearly the homomorphisms (/ ' °/ ) ® (g '0 g) and (/ ' ® g') ° (/ ® g) agree on all elements of the form x ® y in Μ ® N. Since these elements generate M ® iV, it follows that

C f */ ) ® (g '° g ) - ^

RESTRICTION AND EXTENSION OF SCALARS

Let/: A -+ B be a homomorphism of rings and let N be a ^-module. Then N has an Λ-module structure defined as follows: if a eA and x eN , then ax is de-fined to be f(a)x. This Λ-module is said to be obtained from N by restriction of scalars. In particular,/ defines in this way an Λ-module structure on B.

28 MODULES

Proposition 2.16. Suppose N is finitely generated as a B-module and that B is finitely generated as an A-module. Then N is finitely generated as an A-module.

Proof. Let yu ..., yn generate N over B, and let xu . . . , x „ generate B as an Λ-module. Then the mn products xtyj generate N over A. ■

Now let M be an Λ-module. Since, as we have just seen, B can be regarded as an Λ-module, we can form the Λ-module M s = B Λ M. In fact M„ carries a 5-module structure such that b(b' x ) = bb' ® x for all b ,b 'eB and all x e M. The B-module MB is said to be obtained from M by extension o f scalars.

Proposition 2.17. I f M is finitely generated as an A-module, then M B is finitely generated as a B-module.

Proof. I f jcm generate M over A, then the 1 0 xt generate MB over B. u

EXACTNESS PROPERTIES OF THE TENSOR PRODUCT

Let f :M x N -*■ P be an Λ-bilinear mapping. For each x e M the mapping y t-> f (x , y) o f N into P is Λ-linear, hence / gives rise to a mapping M -*■ Horn (.N, P ) which is Λ-linear because / is linear in the variable x. Conversely any Λ-homomorphism : M Homx (N ,P ) defines a bilinear map, namely (x, y) 4>(x)(y). Hence the set S of Λ-bilinear mappings Μ x N -*■ P is in natural one-to-one correspondence with Horn (M , Horn (N , P )). On the other hand S is in one-to-one correspondence with Horn (M ® N, P), by the de-fining property of the tensor product. Hence we have a canonical isomorphism

Horn (M ® N, P) z Horn (M, Horn (N, P )) . (1)

Proposition 2.18. Let

M 'U M**· M ' - + 0 (2)be an exact sequence o f A-modules and homomorphisms, and let N be any A-module. Then the sequence

M · ® ff 121+Μ ® N M ” AT-*0 (3)

(where 1 denotes the identity mapping on N ) is exact.Proof. Let E denote the sequence (2), and let E ® N denote the sequence (3). Let P be any Λ-module. Since (2) is exact, the sequence Horn (£, Horn (N , P )) is exact by (2.9); hence by (1) the sequence Horn (£ ® N, P ) is exact. By (2.9) again, it follows that E ® N is exact. ■

Remarks, i) Let T (M ) = M N and let U{P) * Horn (N, P). Then (1) takes the form Horn (T (M ), P ) = Horn (Af, U (P )) for all Λ-modules M and P. In the language of abstract nonsense, the functor T is the left adjoint o f U, and If is the right adjoint o f T. The proof o f (2.18) shows that any functor which is a left adjoint is right exaet. Likewise any functor which is a right adjoint is left exact.

ii) It is not in general true that, if M ' -► M M* is an exact sequence of Λ-modules and homomorphisms, the sequence A i' ® JV-* M ® Af* ® N obtained by tensoring with an arbitrary Λ-module N is exact.

Example. Take A = Z and consider the exact sequence 0 Z A- Z, where (jc) = 2x for all x e Z. I f we tensor with N = Z/2Z, the sequence 0-> Z ®

■— Z ® N is not exact, because for any x ® y e Z ® N we have

( / ® IXjc ® j ) = 2 x ® j ' = * ® 2 y = x ® 0 = 0,

so that / ® 1 is the zero mapping, whereas Z ® N Φ 0.

The functor TN:M\-+ M ® AN on the category of Λ-modules and homo-morphisms is therefore not in general exact. If TN is exact, that is to say if tensoring with N transforms all exact sequences into exact sequences, then N is said to be a flat Λ-module.

Proposition 2.19. The following are equivalent, for an A-module N :i) N is flat.ii) I f 0 ->* M ' M M " 0 is any exact sequence o f A-modules, the tensoredsequence 0 -* M ' ® N M ® N -+ M " ® N -> 0 is exactiii) I f f : M ' M is injective, then/ ® 1: M ' ® N -> M ® N is injective.iv) I f f : M ' M is injective and M, M ' are finitely generated, then / ® 1 : M ' ® N -+ M ® N is injective.

Proof i) o ii) by splitting up a long exact sequence into short exact sequences.ii) o iii) by (2.18).iii) => iv): clear.iv) => iii). Let/: M ‘ M be injective and let u = 2 jc, ® yt e Ker (/ ® 1),

so that 2/(xr!) ® )>i — 0 in M ® N. Let M'0 be the submodule of M ' generated by the x\ and let u0 denote 2 ® ^ as an element o f M i ® N. By (2.14) thereexists a finitely generated submodule M 0 of M containing/(AQ and such that Σ/(*ί) ® y% * 0 as an element o f M0 ® N. I f /0: Μό M 0 is the restriction of/, this means that ( /0 ® ΐχ «0) * 0. Since Af0 and M'0 are finitely generated, /ο ® 1 is injective and therefore u0 * 0, hence w = 0. ■

Exercise 2.20. I f f : A -+ Bis a ring homomorphism and Misaflat A-module, then Μβ = B M is aflat B-module. (Use the canonical isomorphisms (2.14), (2.15).)

ALGEBRAS

Let/: A -> B be a ring homomorphism. If ae A and be B, define a product

ab = f(a)b.

ALGEBRAS 29

30 MODULES

This definition o f scalar multiplication makes the ring B into an Λ-module (it is a particular example o f restriction o f scalars). Thus B has an Λ-module structure as well as a ring structure, and these two structures are compatible in a sense which the reader will be able to formulate for himself. The ring B, equipped with this Λ-module structure, is said to be an A-algebra. Thus an Λ-algebra is, by definition, a ring B together with a ring homomorphism/: A -* B.

Remarks, i) In particular, if A is a field K (and B Φ 0) then / is injective by (1.2) and therefore K can be canonically identified with its image in B. Thus a A-algebra {K a field) is effectively a ring containing K as a subring.

ii) Let A be any ring. Since A has an identity element there is a unique homomorphism o f the ring o f integers Z into A, namely n i-»· n.l. Thus every ring is automatically a Z-algebra.

Let /: A -*■ B, g : A -> C be two ring homomorphisms. An A-algebra homomorphism ft: B -*■ C is a ring homomorphism which is also an Λ-module homomorphism. The reader should verify that ft is an Λ-algebra homomor-phism if and only if A / = g.

A ring homomorphism/: A -* -B is finite, and B is λ finite Λ-algebra, if B is finitely generated as an Λ-module. The homomorphism/is o f finite type, and B is a finitely-generated A-algebra, if there exists a finite set o f elements x i , ... x„in B such that every element of B can be written as a polynomial in Xi.......xnwith coefficients in / (Λ ) ; or equivalently if there is an Λ-algebra homomorphism from a polynomial ring A[tu ... , r„] onto B.

A ring A is said to be finitely generated if it is finitely generated as a Z- algebra. This means that there exist finitely many elements x l t . .. , xn in A such that every element o f A can be written as a polynomial in the xf with rational integer coefficients.

TENSOR PRODUCT OF ALGEBRAS

Let B, C be two Λ-algebras, f : A - + B , g : A -> C the corresponding homo-morphisms. Since B and C are Λ-modules we may form their tensor product D — B C, which is an Λ-module. We shall now define a multiplication on D.

Consider the mapping B x C x B x C - * D defined by

φ, c, b', c’) i—*· bb' ® cc'.

This is Λ-linear in each factor and therefore, by (2.12*), induces an Λ-module homomorphism

hence by (2.14) an Λ-module homomorphism

D ® D —y D

EXERCISES 31

and this in turn by (2.11) corresponds to an ^-bilinear mapping

μ :Ώ x D - + D

which is such that

μφ ® c, V ® c') * bV ® cc\

Of course, we could have written down this formula directly, but without some such argument as we have given there would be no guarantee that μ was well- defined.

We have therefore defined a multiplication on the tensor product D = B φ * C: for elements of the form b ® c it is given by

The reader should check that with this multiplication D is a commutative ring, with identity element 1 ®1 . Furthermore, D is an Λ-algebra: the mapping a h+f(a) ® g(a) is a ring homomorphism Λ -+ D.

In fact there is a commutative diagram of ring homomorphisms

in which u, for example, is defined by u(b) — b ® 1.

EXERCISES

1. Show that (Z/iwZ) ®*(Z/#iZ) = 0 if m, n are coprime.

2. Let A be a ring, a an ideal, M an ^-module. Show that (A/a) M is isomor-phic to MfaM.[Tensor the exact sequenceΌ û -► A A/a 0 with M.]

3. Let A be a local ring, M and N finitely generated Λ-modules. Prove taat if M ® N — 0, then M * 0 or N » 0.[Let m be the maximal ideal, k * Aim the residue field. Let Mk = k M s M/mM by Exercise 2. By Nakayama’s lemma, Mk = 0 => M = 0. But Μ ΛΓ = 0 => (M JV)ic = 0 => Af* A/fc = 0 ^ Λ/* = 0 or AT* = 0, since Mk, Nk are vector spaces over a field.]

(b ® c)(br ® c') = bV ® cc',

and in general by

£

C

4. Let Mi (i € /) be any family of ̂ -modules, and let M be their direct sum. Prove that M is flat o each M< is flat.

2*

32 MODULES

5. Let A[x] be the ring of polynomials in one indeterminate over a ring A. Prove that A[x] is a flat Λ-algebra. [Use Exercise 4.]

6. For any Λ-module, let M[x] denote the set of all polynomials in x with co-efficients in Af, that is to say expressions of the form

wo + τπχΧ + · · · + mrXr (mi e Af).

Defining the product of an element of A[x] and an element of M ix'] in the obvious way, show that M [x ] is an ΛΜ -module.

Show that Af [x] £ A[x] ®A Af.

7. Let p be a prime ideal in A. Show that p[x) is a prime ideal in A[x], If m is a maximal ideal in A, is tn[jt] a maximal ideal in Α[χ]Ί

8. i) If Af and N are fiat Λ-modules, then so is Af ®Λ N.ii) If B is a flat Λ-algebra and N is a flat 2?-module, then N is flat as an Λ-module.

9. Let 0 Af' A f-»· M ” 0 be an exact sequence of Λ-modules. If Af' andM ” are finitely generated, then so is Af.

10. Let A be a ring, a an ideal contained in the Jacobson radical of Λ ; let Af be an Λ-module and N a finitely generated Λ-module, and let «: Af -► N be a homo-morphism. If the induced homomorphism Af/aAf —► N/aN is suijective, then u is surjective.

11. Let A be a ring Φ 0. Show that Am £ An => m = n.[Let m be a maximal ideal of A and let φ: Am An be an isomorphism. Then1 ® φ: (A/m) ® An —► (Λ/ηi) ® An is an isomorphism between vector spaces of dimensions m and n over the field k = A/m. Hence m = n J (Cf. Chapter 3, Exercise 15.)

If φ:Αη -+ An is suijective, then m > n.If φ: Am An is injective, is it always the case that m < n?

12. Let Af be a finitely generated Λ-module and φ: M -+ An a suijective homo-morphism. Show that Ker (φ) is finitely generated.[Let eu ..., *n T>e a basis of An and choose ute M such that φ Μ = e, (1 < i < n). Show that Af is the direct sum of Ker (φ) and the submodule generated by uu > · »«ni

13. Let/: A -► B be a ring homomorphism, and let be a ̂ -module. Regarding N as an Λ-module by restriction of scalars, form the ^-module Nb = B ®Λ N. Show that the homomorphism g: iST-> Ns which maps y to 1 0 y is injective and that g(N) is a direct summand of NB.[Definep: Ns -*· Nbyp(b ® y) = by, and show that NB = Im (g) Θ Ker (p).]

Direct limits14. A partially ordered set I is said to be a directed set if for each pair i , j in I there

exists k e lsuch that i < k and/ < k.Let A be a ring, let /be a directed set and let (Afi)

EXERCISES 33

(1) fHi is the identity mapping of M t, for all i e /;(2) μ* = pjk° μα whenever / < j ^ k.

Then the modules M< and homomorphisms μί} are said to form a direct system M = (Mit μα) over the directed set J.

We shall construct an Λ-module M called the direct limit of the direct system M. Let C be the direct sum of the Mu and identify each module Mi with its canonical image in C. Let D be the submodule of C generated by all elements of the form jc* — /*u(x

34 MODULES

For each / e / we have a homomorphism μ* ® 1 : M(

EXERCISES 35

26. Let N be an Λ-module. Then N is flat

3

Rings and Modules of Fractions

The formation o f rings of fractions and the associated process of localization are perhaps the most important technical tools in commutative algebra. They correspond in the algebro-geometric picture to concentrating attention on an open set or near a point, and the importance o f these notions should be self- evident. This chapter gives the definitions and simple properties of the formation o f fractions.

The procedure by which one constructs the rational field Q from the ring of integers Z (and embeds Z in Q) extends easily to any integral domain A and produces the field o f fractions of A . The construction consists in taking all ordered pairs (a, s ) where a, s e A and s Φ 0, and setting up an equivalence relation between such pairs:

(a, s) s (ib, t) o at - bs = 0 .

This works only if A is an integral domain, because the verification that the relation is transitive involves canceling, i.e. the fact that A has no zero-divisor Φ 0. However, it can be generalized as follows:

Let A be any ring. A multiplicatively closed subset o f A is a subset S of A such that 1 € S and S is closed under multiplication: in other words S is a sub-semigroup of the multiplicative semigroup of A. Define a relation s o n A x S as follows:

(a, s ) = (b, t ) o (a t - bs)u = 0 for some u s S.

Clearly this relation is reflexive and symmetric. To show that it is transitive, suppose (a, s)- s (b, t ) and (b, /) = (c, «). Then there exist v9 w in S such that (a t — b s)v = 0 and (bit - c t)w = 0. Eliminate b from these two equations and we have (au — cs)tvw = 0. Since S is closed under multiplication, we have* tvw g S', hence (a, s ) s (c, u). Thus we have an equivalence relation. Let a /s denote the equivalence class of (a, s% and let S~*A denote the set of equivalence classes. We put a ring structure on S ' 1 A by defining addition and multiplication of these “ fractions” a /s in the same way as in elementary algebra: that is,

(a/s) + (bit) - (at + bs)jst,(a/sXb/t) * ab/st.

36

Exercise. Verify that these definitions are independent o f the choices o f rep-resentatives (a, s) and (b, t)> and that S~lA satisfies the axioms of a com-mutative ring with identity.

We also have a ring homomorphism/: A S '1 A defined by f (x ) = jc/1.This is not in general injective.

Remark. I f A is an integral domain and S * A — {0}, then S~1A is the field of fractions of A.

The ring S '1 A is called the ring offractions o f A with respect to S. It has a universal property :

Proposition 3.1. Let g : A -> B be a ring homomorphism such that £(.?) is a unit in B for all se S. Then there exists a unique ring homomorphism h: S~XA B such that g = h o f

Proof i) Uniqueness. I f h satisfies the conditions, then h(a/l) = hf(a) = g(a) for all a e A; hence, if seS,

h(l/s) = % / D -1) - K s/l)-1 = g is )-1

and therefore Ha/s) = h(a/i)-h(l/s) = £ (a )g (i)_1, so that A is uniquely determined by g.

ii) Existence. Let h(a/s) = g(a)g(s)~1. Then h will clearly be a ring homo-morphism provided that it is well-defined. Suppose then that a/s = a'/s'; then there exists te S such that (as' — a’s)t = 0, hence

O(a)s0') - gi f (s ) is a unit in S ~XA ;

2) / («) — 0 as = 0 for some se S;3) Every element o f S~1A is of the form /(e)/(i)-1 for some a e A and some

se S.

Conversely, these three conditions determine the ring S~lA up to iso-morphism. Precisely:

Corollary 5J. Jfg: A -* Bis a ring homomorphism such thati) s e S g(s) is a unit in B;

ü) ε( α) — 0 => as * 0 for some s e S;

RINGS AND MODULES OF FRACTIONS 37

38 RINGS AND MODULES OF FRACTIONS

iii) Every element o f B is o f the form g(a)g(s)~1; then there is a uniqueisomorphism h: S~XA -*· B such that g — h°f .

Proof. By (3.1) we have to show that ft: S '1 A - » B, defined by

Κ Φ ) « gfàgis)'1 (this definition uses i)) is an isomorphism. By iii), h is suijective. To show h is injective, look at the kernel o f A: if h(a/s) — 0, then g(a) = 0, hence by ii) we have at = 0 for some t e S, hence (a, s) s (0,1), i.e., a/s = 0 in S '1 A. m

Examples. 1) Let p be a prime ideal of A. Then S — A — is multiplicatively closed (in fact A — p is multiplicatively closed is prime). We write Ap for S ’ 1 A in this case. The elements a/s with a ep form an ideal m in Ap. I f bjt φ m, then b ip , hence be S and therefore bjt is a unit in Ap. It follows that if a is an ideal in Ap and a $ m, then a contains a unit and is therefore the whole ring. Hence nt is the only maximal ideal in Ap; in other words, Ap is a local ring.

The process o f passing from A to Ap is called localization at p.2) S~XA is the zero ring o O e S .3) Let f e A and let S — We write At for S~XA in this case.4) Let a be any ideal in A, and let S — 1 + o = set o f all 1 + x where

x e a. Clearly S is multiplicatively closed.5) Special cases o f 1) and 3):i) A — Z, $> = (p ), p a prime number; Ap - set of all rational numbers

m/n where n is prime to p ; if /6 Z and f Φ 0, then A, is the set of all rational numbers whose denominator is a power o f /.

ii) A m fcfc,. . tn], where fc is a field and the t, are independent indeter-minatos, p a prime ideal in A. Then Ap is the ring o f all rational functions fig , where g φ p. I f Fis the variety defined by the ideal p, that is to say the set o f all x = x „)e£ n such that f ( x ) = 0 whenever f e p , then (provided k isinfinite) Ap can be identified with the ring of all rational functions on k* which are defiled at almost all points o f V; it is the local ring o f kn along the variety V. This is the prototype of the local rings which arise in algebraic geometry.

The construction o f S '1 A can be carried through with an .4-module M in place of the ring A. Define a relation s on M x S as follows:

(m, s) = (m\ s’) o 3 t e S such that t(sm’ — s’m) — 0.As before, this is an equivalence relation. Let m/s denote the equivalence dass o f the pair (m, s), let S~%M denote the set o f such fractions, and make S~1M into an S ~*4-module with the obvious definitions o f addition and scalar multiplication. As in Examples 1) and 3) above, we write Mp instead o f S~lM when S = A — p (p prime) and M f when S — {/*}„> 0-

Let κ: M -*■ N be an Λ-module homomorphism. Then it gives rise to an S~M-module homomorphism S “ lu: S~1M -*■ S _1N, namely S~Hi maps m/s to u(m)/s. We have S ' l(t> » « ) = (5 ~x») ° (S ~l«).

RINGS AND MODULES OF FRACTIONS 39

Proposition 3.3. The operation S '1 is exact, i.e., if M ' 1+ Μ —*· M " is exact at M , then S~1M ' S~lM s~l’ > S~1M " is exact at S~lM.

Proof. We haveg ° f — 0, hence S' -1# ° S~ lf = .S_1(0) = 0, hence Im (£ -1/ ) £ Ker (S -1#)· To prove the reverse inclusion, let m/s eKer (S ~xg), then g(m)/s = 0 in S~1M ", hence there exists te S such that tg(m) = 0 in M ". But tg(m) = g(tm) since £ is an ^-module homomorphism, hence tm e Ker (g ) * Im (/ ) and therefore tm = f(m ') for some m' e M '. Hence in S~lM we have m/s = f(m')/st = (S " ̂ (m '/st) e Im (S’- xf ) . Hence Ker (S ~ lg) £ Im (S -1/).