*250*
*45*
*15MB*

*English*
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*Year 2006*

*Table of contents : PrefaceIntroductionConventions and terminology1 Commutative rings and modules1 Ideals2 Modules3 Chain conditions2 Prime ideals4 Localisation and Spec of a ring5 The Hilbert Nullstellensatz and first steps in dimension theory6 Associated primes and primary decompositionAppendix to §6. Secondary representations of a module3 Properties of extension rings7 FlatnessAppendix to §7. Pure submodules8 Completion and the Artin-Rees lemma9 Integral extensions4 Valuation rings10 General valuations11 DVRs and Dedekind rings12 Krull rings5 Dimension theory13 Graded rings, the Hilbert function and the Samuel functionAppendix to §13. Determinantal ideals14 Systems of parameters and multiplicity15 The dimension of extension rings6 Regular sequences16 Regular sequences and the Koszul complex17 Cohen-Macaulay rings18 Gorenstein rings7 Regular rings19 Regular rings20 UFDs21 Complete intersection ringsVllixxiii11614202030374245455355647171788692921031041161231231331391531531611698 Flatness revisited22 The local flatness criterion23 Flatness and fibres24 Generic freeness and open loci results9 Derivations25 Derivations and differentials26 Separability27 Higher derivations10 I-smoothness28 /-smoothness29 The structure theorems for complete local rings30 Connections with derivations11 Applications of complete local rings31 Chains of prime ideals32 The formal fibre33 Some other applicationsAppendix A. Tensor products, direct and inverse limitsAppendix B. Some homological algebraAppendix C. The exterior algebraSolutions and hints for exercisesReferencesIndex173173178185190190198207213213223230246246255261266274283287298315*

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Commutative ring theory

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W.M.L. Holcombe Algebraic automata theory K. Petersen Ergodic theory P.T. Johnstone Stone spaces W.H. Schikhof Ultrametric calculus J.-P. Kahane Some random series of functions, 2nd edition H. Cohn Introduction to the construction of class fields J. Lambek & P.J. Scott Introduction to higher-order categorical logic H. Matsumura Commutative ring theory C.B. Thomas Characteristic classes and the cohomology of finite groups M. Aschbacher Finite group theory J.L. Alperin Local representation theory P. Koosis The logarithmic integral I A. Pietsch Eigenvalues and s-numbers S.J. Patterson An introduction to the theory of the Riemann zeta-function H.J. Baues Algebraic homotopy V.S. Varadarajan Introduction to harmonic analysis on semisimple Lie groups W. Dicks &; M. Dunwoody Groups acting on graphs L.J. Corwin &: F.P. Greenleaf Representations of nilpotent Lie groups and their applications R. Fritsch &: R. Piccinini Cellular structures in topology H Klingen Introductory lectures on Siegel modular forms M.J. Collins Representations and characters of finite groups H. Kunita Stochastic flows and stochastic differential equations P. Wojtaszczyk Banach spaces for analysts J.E. Gilbert & M.A.M. Murray Clifford algebras and Dirac operators in harmonic analysis A. Frohlich &c M.J. Taylor Algebraic number theory K. Goebel & W.A. Kirk Topics in metric fixed point theory J.F. Humphreys Reflection groups and Coxeter groups D.J. Benson Representations and cohomology I D.J. Benson Representations and cohomology II C. Allday &: V. Puppe Cohomological methods in transformation groups C. Soul£ et al Lectures on Arakelov geometry A. Ambrosetti & G. Prodi A primer of nonlinear analysis J. Palis &: F. Takens Hyperbolicity, stability and chaos at homoclinic bifurcations Y. Meyer Wavelets and operators I C. Weibel An introduction to homological algebra W. Bruns & J. Herzog Cohen-Macaulay rings V. Snaith Explicit Brauer induction G. Laumon Cohomology of Drinfield modular varieties I E.B. Davies Spectral theory and differential operators J. Diestel, H. Jarchow & A. Tonge Absolutely summing operators P. Mattila Geometry of sets and measures in Euclidean spaces R. Pinsky Positive harmonic functions and diffusion G. Tenenbaum Introduction to analytic and probabilistic number theory C. Peskine An algebraic introduction to complex protective geometry I I. Porteous Clifford algebras and the classical groups M. Audin Spinning Tops J. Le Potier Lectures on Vector bundles

Commutative ring theory

HIDEYUKI MATSUMURA Department of Mathematics, Faculty of Sciences Nagoya University, Nagoya, Japan

Translated by M. Reid

| CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521367646 Originally published in Japanese as Kakan kan ron, Kyoritsu Shuppan Kabushiki Kaisha, Kyoritsu texts on Modern Mathematics, 4, Tokyo, 1980 and © H. Matsumura, 1980. English translation © Cambridge University Press 1986 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in English by Cambridge University Press 1986 as Commutative ring theory First paperback edition (with corrections) 1989 Ninth printing 2006 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Matsumura, Hideyuki, 1930Commutative ring theory. Translation of: Kakan kan ron. Includes index. 1. Cummutative rings. I. Title. QA251.3.M37213

1986 512'.4 86-11691

ISBN 978-0-521-36764-6 paperback Transferred to digital printing 2008

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface Introduction Conventions and terminology 1

Commutative rings and modules

1 Ideals 2 Modules 3 Chain conditions 2

Prime ideals

4 Localisation and Spec of a ring 5 The Hilbert Nullstellensatz and first steps in dimension theory 6 Associated primes and primary decomposition Appendix to §6. Secondary representations of a module 3

Properties of extension rings

7 Flatness Appendix to §7. Pure submodules

8 Completion and the Artin-Rees lemma 9 Integral extensions 4

Valuation rings

10 General valuations 11 DVRs and Dedekind rings 12 Krull rings 5

Dimension theory

13 Graded rings, the Hilbert function and the Samuel function Appendix to §13. Determinantal ideals

14 Systems of parameters and multiplicity 15 The dimension of extension rings 6

Regular sequences

16 Regular sequences and the Koszul complex 17 Cohen-Macaulay rings 18 Gorenstein rings 7

Regular rings

19 Regular rings 20 UFDs 21 Complete intersection rings

Vll

ix

xiii 1 1 6 14 20 20 30 37 42 45 45 53 55 64 71 71 78 86 92 92 103 104 116 123 123 133 139 153 153 161 169

vi

Contents

8 Flatness revisited 22 The local flatness criterion 23 Flatness and fibres 24 Generic freeness and open loci results 9 Derivations 25 Derivations and differentials 26 Separability 27 Higher derivations 10 I-smoothness 28 /-smoothness 29 The structure theorems for complete local rings 30 Connections with derivations 11 Applications of complete local rings 31 Chains of prime ideals 32 The formal fibre 33 Some other applications Appendix A. Tensor products, direct and inverse limits Appendix B. Some homological algebra Appendix C. The exterior algebra Solutions and hints for exercises References Index

173 173 178 185 190 190 198 207 213 213 223 230 246 246 255 261 266 274 283 287 298 315

Preface

In publishing this English edition I have tried to make a rather extensive revision. Most of the mistakes and insufficiencies in the original edition have, I hope, been corrected, and some theorems have been improved. Some topics have been added in the form of Appendices to individual sections. Only Appendices A, B and C are from the original. The final section, §33, of the original edition was entitled 'Kunz' Theorems' and did not substantially differ from a section in the second edition of my previous book Commutative Algebra (Benjamin, 2nd edn 1980), so I have replaced it by the present §33. The bibliography at the end of the book has been considerably enlarged, although it is obviously impossible to do justice, to all of the ever-increasing literature. Dr Miles Reid has done excellent work of translation. He also pointed out some errors and proposed some improvements. Through his efforts this new edition has become, I believe, more readable than the original. To him, and to the staff of Cambridge University Press and Kyoritsu Shuppan Co., Tokyo, who cooperated to make the publication of this English edition possible, I express here my heartfelt gratitude. Hideyuki Matsumura Nagoya

Vll

Introduction

In addition to being a beautiful and deep theory in its own right, commutative ring theory is important as a foundation for algebraic geometry and complex analytic geometry. Let us start with a historical survey of its development. The most basic commutative rings are the ring Z of rational integers, and the polynomial rings over a field. Z is a principal ideal ring, and so is too simple to be ring-theoretically very interesting, but it was in the course of studying its extensions, the rings of integers of algebraic number fields, that Dedekind first introduced the notion of an ideal in the 1870s. For it was realised that only when prime ideals are used in place of prime numbers do we obtain the natural generalisation of the number theory of Z. Meanwhile, in the second half of the 19th century, polynomial rings gradually came to be studied both from the point of view of algebraic geometry and of invariant theory. In his famous papers of the 1890s on invariants, Hilbert proved that ideals in polynomial rings are finitely generated, as well as other fundamental theorems. After the turn of the present century had seen the deep researches of Lasker and Macaulay on primary decomposition of polynomial ideals came the advent of the age of abstract algebra. A forerunner of the abstract treatment of commutative ring theory was the Japanese Shozo Sono (On congruences, I-IV, Mem. Coll Sci. Kyoto, 2 (1917), 3(1918-19)); in particular he succeeded in giving an axiomatic characterisation of Dedekind rings. Shortly after this Emmy Noether discovered that primary decomposition of ideals is a consequence of the ascending chain condition (1921), and gave a different system of axioms for Dedekind rings (1927), in work which was to have a decisive influence on the direction of subsequent development of commutative ring theory. The central position occupied by Noetherian rings in commutative ring theory became evident from her work. However, the credit for raising abstract commutative ring theory to a substantial branch of science belongs in the first instance to Krull (1899— 1970). In the 1920s and 30s he established the dimension theory of Noetherian rings, introduced the methods of localisation and completion, ix

x

Introduction

and the notion of a regular local ring, and went beyond the framework of Noetherian rings to create the theory of general valuation rings and Krull rings. The contribution of Akizuki in the 1930s was also considerable; in particular, a counter-example, which he obtained after a year's hard struggle, of an integral domain whose integral closure is not finite as a module was to become the model for many subsequent counter-examples. In the 1940s Krull's theory was applied to algebraic geometry by Chevalley and Zariski, with remarkable success. Zariski applied general valuation theory to the resolution of singularities and the theory of birational transformations, and used the notion of regular local ring to give an algebraic formulation of the theory of simple (non-singular) point of a variety. Chevalley initiated the theory of multiplicities of local rings, and applied it to the computation of intersection multiplicities of varieties. Meanwhile, Zariski's student I.S. Cohen proved the structure theorem for complete local rings [1], underlining the importance of completion. The 1950s opened with the profound work of Zariski on the problem of whether the completion of a normal local ring remains normal (Sur la normalite analytique des varietes normales, Ann. Inst. Fourier 2 (1950)), taking Noetherian ring theory from general theory deeper into precise structure theorems. Multiplicity theory was given new foundations by Samuel and Nagata, and became one of the powerful tools in the theory of local rings. Nagata, who was the most outstanding research worker of the 1950s, also created the theory of Hensel rings, constructed examples of noncatenary Noetherian rings and counter-examples to Hilbert's 14th problem, and initiated the theory of Nagata rings (which he called pseudogeometric rings). Y. Mori carried out a deep study of the integral closure of Noetherian integral domains. However, in contrast to Nagata and Mori's work following the Krull tradition, there was at the same time a new and completely different movement, the introduction of homological algebra into commutative ring theory by Auslander and Buchsbaum in the USA, Northcott and Rees in Britain, and Serre in France, among others. In this direction, the theory of regular sequences and depth appeared, giving a new treatment of CohenMacaulay rings, and through the homological characterisation of regular local rings there was dramatic progress in the theory of regular local rings. The early 1960s saw the publication of Bourbaki's Algebre commutative, which emphasised flatness, and treated primary decomposition from a new angle. However, without doubt, the most characteristic aspect of this decade was the activity of Grothendieck. His scheme theory created a fusion of commutative ring theory and algebraic geometry, and opened up ways of applying geometric methods in ring theory. His local cohomology

Introduction

xi

is an example of this kind of approach, and has become one of the indispensable methods of modern commutative ring theory. He also initiated the theory of Gorenstein rings. In addition, his systematic development, in Chapter IV of EGA, of the study of formal fibres, and the theory of excellent rings arising out of it, can be seen as a continuation and a final conclusion of the work of Zariski and Nagata in the 1950s. In the 1960s commutative ring theory was to receive another two important gifts from algebraic geometry. Hironaka's great work on the resolution of singularities [1] contained an extremely original piece of work within the ideal theory of local rings, the ring-theoretical significance of which is gradually being understood. The theorem on resolution of singularities has itself recently been used by Rotthaus in the study of excellent rings. Secondly, in 1969 M. Artin proved his famous approximation theorem; roughly speaking, this states that if a system of simultaneous algebraic equations over a Hensel local ring A has a solution in the completion A, then there exist arbitrarily close solutions in A itself. This theorem has a wide variety of applications both in algebraic geometry and in ring theory. A new homology theory of commutative rings constructed by M. Andre and Quillen is a further important achievement of the 1960s. The 1970s was a period of vigorous research in homological directions by many workers. Buchsbaum, Eisenbud, Northcott and others made detailed studies of properties of complexes, while techniques discovered by Peskine and Szpiro [1] and Hochster [H] made ingenious use of the Frobenius map and the Artin approximation theorem. Cohen-Macaulay rings, Gorenstein rings, and most recently Buchsbaum rings have been studied in very concrete ways by Hochster, Stanley, Kei-ichi Watanabe and S. Goto among others. On the other hand, classical ideal theory has shown no sign of dying off, with Ratliff and Rotthaus obtaining extremely deep results. To give the three top theorems of commutative ring theory in order of importance, I have not much doubt that Krull's dimension theorem (Theorem 13.5) has pride of place. Next perhaps is I.S. Cohen's structure theorem for complete local rings (Theorems 28.3, 29.3 and 29.4). The fact that a complete local ring can be expressed as a quotient of a wellunderstood ring, the formal power series ring over a field or a discrete valuation ring, is something to feel extremely grateful for. As a third, I would give Serre's characterisation of a regular local ring (Theorem 19.2); this grasps the essence of regular local rings, and is also an important meeting-point of ideal theory and homological algebra. This book is written as a genuine textbook in commutative algebra, and is as self-contained as possible. It was also the intention to give some

xii

Introduction

thought to the applications to algebraic geometry. However, both for reasons of space and limited ability on the part of the author, we are not able to touch on local cohomology, or on the many subsequent results of the cohomological work of the 1970s. There are readable accounts of these subjects in [G6] and [H], and it would be useful to read these after this book. This book was originally to have been written by my distinguished friend Professor Masao Narita, but since his tragic early death through illness, I have taken over from him. Professor Narita was an exact contemporary of mine, and had been a close friend ever since we met at the age of 24. Wellrespected and popular with all, he was a man of warm character, and it was a sad loss when he was prematurely called to a better place while still in his forties. Believing that, had he written the book, he would have included topics which were characteristic of him, UFDs, Picard groups, and so on, I have used part of his lectures in §20 as a memorial to him. I could wish for nothing better than to present this book to Professor Narita and to hear his criticism. Hideyuki Matsumura Nagoya

Conventions and terminology

(1) Some basic definitions are given in Appendixes A-C. The index contains references to all definitions, including those of the appendixes. (2) In this book, by a ring we always understand a commutative ring with unit; ring homomorphisms A —• B are assumed to take the unit element of A into the unit element of B. When we say that A is a subring of B it is understood that the unit elements of A and B coincide. (3) If f:A —>B is a ring homomorphism and J is an ideal of B, then f~l{J) is an ideal of B, and we denote this by An J; if A is a subring of B and / is the inclusion map then this is the same as the usual set-theoretic notion of intersection. In general this is not true, but confusion does not arise. Moreover, if / is an ideal of A, we will write IB for the ideal f(I)B of B. (4) If A is a ring and ax,..., an elements of A, the ideal of A generated by these is written in any of the following ways: axA + a2A + —h anA,Y,ai^ (5) The sign c is used for inclusion of a subset, including the possibility of equality; in [M] the sign ^ was used for this purpose. However, when we say that iMl c M 2 A natural map; then ideals J of A and ideals J = f ~l (7) of A containing / are in one-to-one correspondence, with / = J/I and A/J ^ A/7. Hence, when we just want to think about ideals of A containing /, it is convenient to shift attention to A/1. (If V is any ideal of A then /(/') is an ideal of A,

with / " W ) ) = / + /', and /(/') = (/ + /')//.) A is itself an ideal of A9 often written (1) since it is generated by the identity element 1. An ideal distinct from (1) is called a proper ideal. An element aeA which has an inverse in A (that is, for which there exists a'eA with aa! = 1) is called a unit (or invertible element) of A; this holds if and only if the principal ideal (a) is equal to (1). If a is a unit and x is nilpotent then a + x is again a unit: indeed, if xn = 0 then setting y = — a~*x, we have yn = 0; now so that a + x = a (1 — y) has an inverse. In a ring A we are allowed to have 1 = 0 , but if this happens then it follows that a=la = 0a = 0 for every aeA, so that A has only one element 0; in this case we write A = 0. In definitions and theorems about 1

2

Commutative rings and modules

rings, it may sometimes happen that the condition A ^ 0 is omitted even when it is actually necessary. A ring A is an integral domain (or simply a domain) if A / 0, and if A has no zero-divisors other than 0. If A is an integral domain and every non-zero element of A is a unit then A is a field. A field is characterised by the fact that it is a ring having exactly two ideals (0) and (1). An ideal which is maximal among all proper ideals is called a maximal ideal; an ideal m of A is maximal if and only if A/m is a field. Given a proper ideal /, let M be the set of ideals containing / and not containing 1, ordered by inclusion; then Zorn's lemma can be applied to M. Indeed, IeM so that M is non-empty, and if LaM is a totally ordered subset then the union of all the ideals belonging to L is an ideal of A and obviously belongs to M, so is the least upper bound of L in M. Thus by Zorn's lemma M has got a maximal element. This proves the following theorem. Theorem 1.1. If / is a proper ideal then there exists at least one maximal ideal containing /. An ideal P of A for which A/P is an integral domain is called a prime ideal. In other words, P is prime if it satisfies

(i) P # A and (ii) x,y$P=>xy$P for x,yeA A field is an integral domain, so that a maximal ideal is prime. If / and J are ideals and P a prime ideal, then Indeed, taking xel and yeJ with x,y$P, we have xyelJ but xy$P. A subset S of A is multiplicative if it satisfies (i) x,)>eS=>x)>eS, and (ii) leS; (here condition (ii) is not crucial: given a subset S satisfying (i), there will usually not be any essential change on replacing S by 5 u { l } ) . If / is an ideal disjoint from S, then exactly as in the proof of Theorem 1 we see that the set of ideals containing / and disjoint from S has a maximal element. If P is an ideal which is maximal among ideals disjoint from S then P is prime. For if x$P, y$P, then since P + xA and P-\-yA both meet S, the product (P + xA) (P + yA) also meets S. However, (P + xA) (P + yA) cz P + xyA, so that we must have xy$P. We have thus obtained the following theorem. Theorem 1.2. Let S be a multiplicative set and / an ideal disjoint from S; then there exists a prime ideal containing / and disjoint from 5. If / is an ideal of A then the set of elements of A, some power of which belongs to /, is an ideal of A (for xnel and ymel=>(x + y)n+m~1el and

§1

Ideals

(axfel). This set is called the radical of /, and is sometimes written

y/l = {aeA\aneI for some n>0}. If P is a prime ideal containing / then xnel c P implies that xeP, and hence y/l>=l then both {1} and {x,y} are minimal bases of A. However, if A is a local ring then the situation is clear: Theorem 2.3. Let (A, m, k) be a local ring and M a finite A-module; set M = M/xnM. Now A? is a finite-dimensional vector space overfe,and we

§2

Modules

9

write n for its dimension. Then: (i) If we take a basis {ul9...9uH} for M over fc, and choose an inverse image uteM of each ui9 then {ul9...9un} is a minimal basis of M; (ii) conversely every minimal basis of M is obtained in this way, and so has n elements. (iii) If {ul9...9uH} and {vu...,vn} are both minimal bases of Af, and v i = Y,aijuj w ^ aveA then det(a 0 ) is a unit of A, so that (atj) is an invertible matrix. Proof, (i) M = £ 4Mf -f mM, and M is finitely generated (hence also M/^i4wf), so that by the above corollary M = £y4M,. If {uu...,uH} is not minimal, so that, for example, {u2,...,un} already generates Af then {M 2 ,...,«„} generates Af, which is a contradiction. Hence {ti!,...,!4). The descending chain / 3 1 2 => ••• stops after finitely many steps, so that there is an s such that Is = Is+ x. If we set (0:7s) = J then (J:7) = ((O:7S):7) = (O:7S+1) = J; let's prove that J = A. By contradiction, suppose that J ^ A; then there exists an ideal J' which is minimal among all ideals strictly bigger than J. For any xeJ'-J we have J' = Ax + J. Now 7 = rad (A) and J # J', so that by NAK J' ^ Ix + J, and hence by minimality of J' we have Ix + J = J, and this gives 7x c J. Thus xe(J:I) = J, which is a contradiction. Therefore J = A, so that 7s = 0. Now consider the chain of ideals ^ => Pi => P1P2 => - =3 Pi - P r - 1 => / => 'Pi => W 2 D-D/2D/2PID-D/S =

0.

Let M and Mpt be any two consecutive terms in this chain; then M/Mpt is a vector space over the field A/ph and since it is Artinian, it must be finitedimensional. Hence, l(M/Mpi) < 00, and therefore the sum l(A) of these terms is also finite. • Remark. This theorem is sometimes referred to as Hopkins' theorem, but it was proved in the above form by Akizuki [2] in 1935. It was rediscovered four years later by Hopkins [1], and he proved it for non-commutative rings (a left-Artinian ring with unit is also left-Noetherian).

Theorem 33. If A is Noetherian then so are A[X~] and Proof. The statement for A[X~\ is the well-known Hilbert basis theorem (see, for example Lang, Algebra, or [AM], p. 81), and we omit the proof. We now briefly run through the proof for A{XJ. Set B = A{X}9 and let 7 be an ideal of B\ we will prove that 7 is finitely generated. Write 7(r) for the ideal of A formed by the leading coefficients ar of / = arXr + o r+1 A Pr+1 + - - - a s / runs through InX'B; then we have Since A is Noetherian, there is an s such that I(s) = I(s + 1) = •••; moreover, each 7(f) is finitely generated. For each i with 0 < i < s we take finitely many elements aiveA generating 7(i), and choose giveInXlB having aiv as the coefficient of XK These giv now generate 7. Indeed, for fel we can take a linear combination g0 of the gOv with coefficients in A such that

§3

Chain conditions

17

f-goeInXB, then take a linear combination gt of the glv with coefficients in A such that f — go — g1eInX2B9 and proceeding in the same way we get Now I(s + 1) = /(s), so we can take a linear combination gs+1 of the with coefficients in A such that We now proceed in the same way to get 0s+2>— • For i ^ s , each g{ is a linear combination of the giv with coefficients in A, and, for / > s , a combination of the elements Xl~sgsv. For each i ^ s we write 0i = Zvaiv^l~s0sv> a nd then for each v we set K = YJT=sai^i~s'^ K *s a n element of B, and

A ring Albl9...,bn] which is finitely generated as a ring over a Noetherian ring A is a quotient of a polynomial ring >4[A\,..., JfJ, and so by the Hilbert basis theorem is again Noetherian. We now give some other criteria for a ring to be Noetherian. Theorem 3.4 (I. S. Cohen). If all the prime ideals of a ring A are finitely generated then A is Noetherian. Proof. Write F for the set of ideals of A which are not finitely generated. If F ^ 0 then by Zorn's lemma T contains a maximal element /. Then / is not a prime ideal, so that there are elements x, yeA with x^J, y$I but xyel. Now I + Ay is bigger than /, and hence is finitely generated, so that we can choose ul9...9uHeI such that I + Ay = (ul9...,un,y). Moreover, l\y= {aeA\ayeI} contains x, and is thus bigger than /, so it has a finite system of generators {vi9...,vm}. Finally, it is easy to check that I = (ul9...9un9v1y9...9vmy); hence, / ^ F , which is a contradiction. Therefore F = 0 . • Theorem 3.5. Let A be a ring and M an /4-module. Then if M is a Noetherian module, 4/ann (M) is a Noetherian ring. Proof. If we set A = A/ann (M) and view M as an .4-module, then the submodules of M as an 4-module or ^-module coincide, so that M is also Noetherian as an ^-module. We can thus replace A by A, and then ann(Af) = (0). Now letting M = Acol -f ••• + Aoni we can embed A in Mn by means of the map at-+(aa)u..., acon). By Theorem 1, M" is a Noetherian module, so that its submodule A is also Noetherian. (This theorem can be expressed by saying that a ring having a faithful Noetherian module is Noetherian.) •

18

Commutative rings and modules

Theorem 3.6 (E. Formanek [1]). Let A be a ring, and B an >4-module which is finitely generated and faithful over A. Assume that the set of submodules of B of the form IB with / an ideal of A satisfies the a.c.c; then A is Noetherian. Proof. It will be enough to show that B is a Noetherian A-module. By contradiction, suppose that it is not; then the set f I is an ideal of A and B/IB is 1 } non-Noetherian as A-module J contains {0} and so is non-empty, so that by assumption it contains a maximal element. Let IB be one such maximal element; then replacing B by B/IB and A by A/ann (£//£) we see that we can assume that £ is a non-Noetherian A-module, but for any non-zero ideal I of A the quotient B/IB is Noetherian. Next we set r = {N\N is a submodule of B and B/N is a faithful A-module}. If B = Abx + • •• 4- Abn then for a submodule N of B,

NeToVaeA-0,

{ably...,abn} £N.

From this, one sees at once that Zorn's lemma applies to T; hence there exists a maximal element No of T. If B/No is Noetherian then A is a Noetherian ring, and thus B is Noetherian, which contradicts our hypothesis. It follows that on replacing B by B/No we arrive at a module B with the following properties: (1) B is non-Noetherian as an 4-module; (2) for any ideal / ^(0) of A9 B/IB is Noetherian; (3) for any submodule N # (0) of B9 B/N is not faithful as an A-module. Now let N be any non-zero submodule of B. By (3) there is an element aeA with a^O such that a{B/N) = 0y that is such that aBaN. By (2) B/aB is a Noetherian module, so that N/aB is finitely generated; but since B is finitely generated so is aB, and hence N itself is finitely generated. Thus, B is a Noetherian module, which contradicts (1). • As a corollary of this theorem we get the following result. Theorem 3.7. (i) (Eakin-Nagata theorem). Let £ be a Noetherianring,and A a subring of B such that B is finite over A; then A is also a Noetherian ring. (ii) Let B be a non-commutative ring whose right ideals have the a.c.c, and let A be a commutative subring of B. If B is finitely generated as a left /4-module then A is a Noetherian ring. (iii) Let B be a non-commutative ring whose two-sided ideals have the a.c.c, and let A be a subring contained in the centre of B; if B is finitely generated as an /4-module then A is a Noetherian ring.

§3

Chain conditions

19

Proof. B has a unit, so is faithful as an ^-module. Hence it is enough to apply the previous theorem. • Remark. Part (i) of Theorem 7 was proved in Eakin's thesis [1] in 1968, and the same result was obtained independently by Nagata [9] a little later. Subsequently many alternative proofs and extensions to the noncommutative case were published; the most transparent of these seems to be Formanek's result [1], which we have given above in the form of Theorem 6. However, this also goes back to the idea of the proofs of Eakin and Nagata. Exercises to §3. Prove the following propositions. 3.1. Let It ,...,/„ be ideals of aringA such that Ix n • •• n /„ = (0); if each A//, is a Noetherian ring then so is A. 3.2. Let A and B be Noetherian rings, and f:A—>C and g:B—>C ring homomorphisms. If both / and g are surjective then the fibre product AxcB (that is, the subring of the direct product AxB given by {(a,b)eA x B\f(a) = g{b)} is a Noetherian ring. 3.3. Let A be a local ring such that the maximal ideal m is principal and n » > o m " = (0)- Then A is Noetherian, and every non-zero ideal of A is a power of m. 3.4. Let A be an integral domain withfieldof fractions K. A fractional ideal I of A is an 4-submodule / of K such that / # 0 and a/ c A for some 0 ^ aeK. The product of two fractional ideals is defined in the same way as the product of two ideals. If / is a fractional ideal of A we set I~l = {oceK\(xI czA}; this is also a fractional ideal, and II'1 cz A. In the particular case that / / " * = A we say that / is invertible. An invertible fractional ideal of A isfinitelygenerated as an A -module. 3.5. If A is a UFD, the only ideals of A which are invertible as fractional ideals are the principal ideals. 3.6. Let A be a Noetherian ring, and (p.Ac—• A a homomorphism of rings. Then if cp is surjective it is also injective, and hence an automorphism of A. 3.7. If A is a Noetherian ring then anyfiniteA-module is offinitepresentation, but if A is non-Noetherian then A must have finite /1-modules which are not of finite presentation.

2 Prime ideals

The notion of prime ideal is central to commutative ring theory. The set Spec 4 of prime ideals of a ring A is a topological space, and the 'localisation' of rings and modules with respect to this topology is an important technique for studying them. These notions are discussed in §4. Starting with the topology of Spec A9 we can define the dimension of A and the height of a prime ideal as notions with natural geometrical content. In §5 we treat elementary dimension theory using onlyfieldtheory, developing especially the dimension theory of ideals in polynomialrings,including the Hilbert Nullstellensatz. We also discuss, as example of an application of the notion of dimension, the theory of Forster and Swan on estimates for the number of generators of a module. (Dimension theory will be the subject of a detailed study in Chapter 5 using methods of ring theory). In §6 we discuss the classical theory of primary decomposition as modernised by Bourbaki. 4

Localisation and Spec of a ring Let A be a ring and S c A a multiplicative set; that is (as in §1), suppose that (i) x, yeS=>xyeS, and (ii) leS. Definition. Suppose that / : A —• B is a ring homomorphism satisfying the two conditions (1) f(x) is a unit of B for all xeS; (2) if g:A —• C is a homomorphism of rings taking every element of S to a unit of C then there exists a unique homomorphism h: B —• C such that g = hf; then B is uniquely determined up to isomorphism, and is called the localisation or the ring of fractions of A with respect to S. We write B = S~ lA or As, and call f:A —• As the canonical map. We prove the existence of B as follows: define the relation ~ on the set A x S by (a,s) ~(b9sf)o3tsS such that t(sfa - sb) = 0; 20

§4

Localisation and Spec of a ring

21

it is easy to check that this is an equivalence relation (if we just had s'a = sb in the definition, the transitive law would fail when S has zero-divisors). Write a/s for the equivalence class of (a, s) under ~ , and let B be the set of these; sums and products are defined in B by the usual rules for calculating with fractions: a/s + b/s' = {as' + bs)/ss\ (a/s)-(b/sf) = ab/ss'. This makes B into a ring, and defining f:A —• B by f(a) = a/I we see that / is a homomorphism of rings satisfying (1) and (2) above. Indeed, if seS then f(s) = s/\ has the inverse 1/s; and if g:A —• C is as in (2) then we just have to set h(a/s) = g(a)g(s)~l (the reader should check that a/s = b/sf implies g(a)g(s)~1 =g(b)g(s')~1). From this construction we see that the kernel of the canonical map f:A —• As is given by

Ker/ = {aeA\sa = 0 for some

seS}.

Hence / is injective if and only if S does not contain any zero-divisors of A. In particular, the set of all non-zero-divisors of A is a multiplicative set; the ring of fractions with respect to S is called the total ring of fractions of A. If A is an integral domain then its total ring of fractions is the same thing as its field of fractions. In general, let f:A —>B be any ring homomorphism, / an ideal of A and J an ideal of B. According to the conventions at the beginning of the book, we write IB for the ideal f(I)B of B. This is called the extension of / to B, or the extended ideal, and is sometimes also written F. Moreover, we write JnA for the ideal /"*(/) of A. This is called the contracted ideal of J, and is sometimes also written Jc. In this notation, the inclusions

Iec^I

and

JctaJ

follow immediately from the definitions; from the first inclusion we get jtcc ^ je^ b u t substituting J = / c in the second gives / ece cz Jc, and hence (•)

/cce = /c,

and similarly

Jcec = Jc.

This shows that there is a canonical bijection between the sets {IB\I is an ideal of A} and {JnA\ J is an ideal of B}. If P is a prime ideal of B then B/P is an integral domain, and since A/Pc can be viewed as a subring of B/P it is also an integral domain, so that Pc is a prime ideal of A. (The extended ideal of a prime ideal does not have to be prime.) An ideal J of B is said to be primary if it satisfies the two conditions: (1) 1£J, and (2) for x,yeB, if xyeJ and x$J then / e J for some n > 0; in other words, all zero-divisors of B/J are nilpotent. The property that all zero-divisors are nilpotent passes to subrings, so that just as for prime ideals we see that the contraction of a primary ideal remains primary. If J is primary then Jj is a prime ideal (see Ex. 4.1).

22

Prime ideals

The importance of rings of fractions for ring theory stems mainly from the following theorem. Theorem 4.1. (i) All the ideals of As are of the form IAS, with / an ideal of A. (ii) Every prime ideal of As is of the form pAs with p a prime ideal of A disjoint from S, and conversely, pAs is prime in As for every such p; exactly the same holds for primary ideals. Proof, (i) If J is an ideal of As, set I = JnA. If x = a/seJ then

X'f(s) = f(a)eJ, so that ael, and then x = (l/s)f(a)eIAs.

The converse

inclusion IAS cz J is obvious, so that J = L4S. (ii) If P is a prime ideal of As and we set p = PnA, then p is a prime ideal of A, and from the above proof P = pAs. Moreover, since P does not contain units of ASi we have pnS = 0. Conversely, if p is a prime ideal of A disjoint from S then ab --epAs s t

with

s,teS=>rabep

for some

reS,

and since r$p we must have aep or bep, so that a/s or b/tepAs. One also sees easily that l$pAs, so that pA s is a prime ideal of y4s. For primary ideals the argument is exactly the same: if p is a primary ideal of A disjoint from S and if rabep with reS, then since no power of r is in p we have abep. From this we get either a/sepAs or (b/t)nepAs for some n. • Corollary. If A is Noetherian (or Artinian) then so is As. Proof. This follows from (i) of the theorem. • We now give examples of rings of fractions As for various multiplicative sets S. Example 1. Let a e A be an element which is not nilpotent, and set S = {1, a, a 2 ,...}. In this case we sometimes write Aa for As. (The reason for not allowing a to be nilpotent is so that O^S. In general if OeS then from the construction of As it is clear that As = 0, which is not very interesting.) The prime ideals of Aa correspond bijectively with the prime ideals of A not containing a. Example 2. Let p be a prime ideal of A, and set S = A — p. In this case we usually write Ap for As. (Writing Ap and A(A_V) to denote the same thing is totally illogical notation, and the Bourbaki school avoids As, writing S~XA instead; however, the notation As does not lead to any confusion.) The localisation Ap is a localringwith maximal ideal pAp. Indeed, as we saw in Theorem 1, pAp is a prime ideal of Ap9 and furthermore, if J c Ap is any

§4

Localisation and Spec of a ring

23

proper ideal then / = J n A is an ideal of A disjoint from A - p, and so / c p, giving J = IAp cz pAp. The prime ideals of Ap correspond bijectively with the prime ideals of A contained in p. Example 3. Let / be a proper ideal of A and set S = l + / = {1 + x|xe/}. Then 5 is a multiplicative set, and the prime ideals of As correspond bijectively with the prime ideals p of A such that Example4. Let S be a multiplicative set, and set S= {aEA\abeS for some be A}. Then S is also a multiplicative set, called the saturation of S. Since quite generally a divisor of a unit is again a unit, we see from the definition of the ring of fractions that As = As, and Sis maximal among multiplicative sets T such that As = AT. Indeed, one sees easily that S = {aeA\a/l is a unit in As}. The multiplicative set S = A-p of Example 2 is already saturated. Theorem 4.2. Localisation commutes with passing to quotients by ideals. More precisely, let A be a ring, SaA a. multiplicative set, / an ideal of A and S the image of S in A/I; then AS/IAS~(A/I)S. Proof. Both sides have the universal property for ring homomorphisms g:A —>C such that (1) every element of S maps to a unit of C, and (2) every element of / maps to 0; the isomorphism follows by the uniqueness of the solution to a universal mapping problem. In concrete terms the isomorphism is given by a/s mod IAsA S the canonical map. If B is a ring, with ring homomorphisms g:A—>B and h:B—>AS satisfying (1) f=hg, and (2) for every beB there exists SES such that g(s)beg(A\ then As can also be regarded as a ring of fractions of B. More precisely, As = BgiS) = BT, where T= {t e B \ h(t) *, is a unit of As}. Proof. We can factorise h as B—>BT—>AS; write oc:BT—>AS for

24

Prime ideals

the second of these maps. Now g(S) a T, so that the composite A —>B —• BT factorises as A —• As —• BT\ write fl'.As —• BT for the second of these maps. Then

affia/s)) = «fo(4-linear map M —• Ms is given by mt->m/l; the kernel is {meM\sm = 0 for some seS}. If S = A —pis the complement of a prime ideal p of A we write Mp for Ms. The set {pe Spec A \ Mp ^ 0} is called the support of M, and written Supp (M). If M is

26

Prime ideals

finitely generated, and we let M = /let), + ••• + Ao)m then p e S u p p ( M ) o M p ^ 0 o 3 i such that o^O o3i

in Mp

such that ann(co l )czpoann(M)= f] ann (cof) c p,

so that Supp(M) coincides with the closed subset K(ann(M)) of Spec A. Theorem 4.4. MS~M®AAS. Proof. The map M x As—>MS defined by {m,a/s)t-+am/s is ^-bilinear, so that there exists a linear map a:M ® As —• Ms such that a(m ® a/s) — am/s. Conversely we can define fi:Ms—>M®AS by /?(m/s) = m®(l/s); indeed, if m/s — m'/s' then ts'm = tsm' for some teS, and so m® (l/ s ) = m® (tsytss') = ts'm® (1/tss') = tsm' ® (1/tss') Now it is easy to check that a and /? are mutually inverse As-linear maps. Hence, Ms and M ®yly4s are isomorphic as y4s-modules. Theorem 4.5. M\-*MS is an exact (covariant) functor from the category of i4-modules to the category of /ls-modules. That is, for a morphism of A -modules f:M —• N there is a morphism fs'.Ms —• Ns of As -modules such that (id)s = id (where id is the identity map of M or M s ), and such that an exact sequence 0-*M'—•M—>M"->0 goes into an exact sequence 0 -> M's —• Ms —• MJ -• 0. Proof. To prove the exactness of 0 -> M5 —• M s on the last line, view M' as a submodule of M; then for xeM' and seS,

x/s = 0 in Msotx = 0 for some teS ox/s = 0 in M'S, as required. The remaining assertions follow frorii the properties of the tensor product (see Appendix A) and from the previous theorem. (Of course they can easily be proved directly.) • It follows from this that localisation commutes with ® and with Tor, and we will treat all this together in the section on flatness in §7. Let A be a ring, M an 4-module and peSpecA. There are at least two interpretations of what it should mean that some property of A or M holds 'locally at p\ Namely, this could mean that Ap (or Mp) has the property, or it could mean that Aq (or Mq) has the property for all q in some neighbourhood U of p in Spec A. The first of these is more commonly used, but there are cases when the two interpretations coincide. In any case, we now prove a number of theorems which assert that a local property implies a global one.

§4

Localisation and Spec of a ring

27

Theorem 4.6. Let A be a ring, M an ^-module and xeM. If x = 0 in M for every maximal ideal p of /I, then x = 0. Proo/. x = 0 in Mpsx = 0 for some seA — p o a n n (x) £ p. However, if l^ann(x) then by Theorem 1.1, there must exist a maximal ideal containing ann(x). Therefore leann(x), that is x = 0. • Theorem 4.7. Let A be an integral domain with field of fractions K; set X = m-Spec A. We consider any ring of fractions of A as a subring of K. Then in this sense we have

Proof. For xeK the set I = {aeA\axeA} is an ideal of A. Now xeAp is equivalent to / C-+0 at a prime ideal q, we get an exact sequence yt;—>M q —>C q ->0, and when q = p we get Cq = 0. C is a quotient of M, so is finitely generated, so that the support Supp(C) is a closed set, and hence there is an open neighbourhood V of p such that Cq = 0 for qe V. This means that V c= l/r. (In short, if 4qa>. for every prime ideal q of A Then by Theorem 6, M / £ 4a>i = 0, that is M = Aco^^ + — + Acor (We think of replacing A by >4a as shrinking Spec A down to the neighbourhood D(a) of p.) Now, defining q>:Ar —• M as above, and letting K be its kernel, we get the exact sequence 0->K—>A r —>M->0; moreover, Kp = 0. By Theorem 2.6, X is finitely generated, so that applying

§4

Localisation and Spec of a ring

29

(i) with r = 0, we have that Kq = 0 for every q in a neighbourhood V of p; this gives (AJ ~ M q , so that K c ( / f . • Exercise to §4. Prove the following propositions. 4.1. The radical of a primary ideal is prime; also, if / is a proper ideal containing a power mv of a maximal ideal m then / is primary and y/l = m. 4.2. If P is a prime ideal of a ring A then the symbolic nth power of P is the ideal Pls). Proof. For xeN we have &nnA(x) = 2innAs(x)nA, so that if PeAssAj,N) then PnAeAssA(N). Conversely if peAss^(A0 and we choose xeiV such that p = ann^(x) then x / 0, and hence, pnS = 0 and pAs is a prime ideal of As with pAs = ann^x). For the second part, if peAss(M)n Spec(^ls) then pnS = 0 , and p = ann^(x) for some xeM; if (a/s)x = 0 in Ms then there is a teS such that tax = 0 in M, and t^p, taep gives aep, so that ann^s(x) = pAs and pj4seAss(Ms). Conversely, if Pe Ass(Ms) then without loss of generality we have P = ann^s(x) with xeM. Setting p = PnA we have P = pAs. Now p is finitely generated since A is Noetherian, and it follows that there exists some teS such that p = ann^(tx). Therefore peAssyl(M). • Corollary. For a Noetherian ring A, an /1-module M and a prime ideal P of 4 we have

Theorem 6.3. Let /I be a ring and 0 - > M ' — • M — • M " - ^ 0 an exact sequence of A-modules; then Ass (M) c Ass (AT) u Ass (M"). Proo/. If PeAss(M) then M contains a submodule N isomorphic to A/P. Since P is prime, for any non-zero element x of N we have ann (x) = P.

§6

Associated primes and primary decomposition

39

Therefore if N n W / 0 we have Ps Ass (Af'>. If JV n M' = 0 then the image of N in Af" is also isomorphic to A/P9 so that PeAss(Af"). • Theorem 6.4. Let A be a Noetherian ring and M / O a finite /1-module. Then there exists a chain 0 = M o c M 2 c • • • c Mn = M of submodules of M such that for each i we have MJM^x ^ 4/P, with P,eSpec4. Proo/. Choose any P 1 eAss(M); then there exists a submodule Mx of M with M! ~ y4/Px. If M! # Af and we choose any P2eAss(Af/Af x) then^here exists M 2 c M such that M1/Ml ca A/P2. Continuing in the same way and using the ascending chain condition, we eventually arrive at Af„ = M. • Theorem 6.5. Let A be a Noetherian ring and Af a finite ^-module. (i) Ass (Af) is a finite set. (ii) Ass (Af)c= Supp (Af). (iii) The set of minimal elements of Ass(Af) and of Supp(Af) coincide. Proof, (i) follows from the previous two theorems; we need only note that Ass(A/P) = {P}. For (ii), if 0-+A/P—>Af is exact then so is 0-+ AP/PAP —> AfP , and therefore MP # 0. For (iii) it is enough to show that if P is a minimal element of Supp(Af) then PeAss(M). We have M P / 0 so that by Theorem 2 and (ii), 0 # Ass (AfP) = Ass (Af) n Spec (AP) c Supp (Af) n Spec (AP) = {*}. Therefore we must have PeAss(M). • Let A be a Noetherian ring and M afinite,4-module. Let Pl,..., Pr be the minimal elements of Supp (M); then Supp (Af) =V(P1)v — v V{Pr\ and the V(Pi) are the irreducible components of the closed set Supp(Af) (see Ex. 4.11). The prime ideals P l f . . . , P r are called the isolated associated primes of Af, and the remaining associated primes of Af are called embedded primes. If / is an ideal of A then SuppA(A/I) is the set of prime ideals containing /, and the minimal prime divisors of / (that is the minimal associated primes of the 4-module A/I) are precisely the minimal prime ideals containing /. We have seen in Ex. 4.12 that there are only a finite number of such primes, and Theorem 5 now gives a new proof of this. (For examples of embedded primes see Ex. 6.6 and Ex. 8.9.) Definition. Let A be a ring, Af an ^-module and i V c M a submodule. We say that N is a primary submodule of Af if the following condition holds for all aeA and xeAf: x$N and axeN=>avM a N for some v. This definition in fact only depends on the quotient module M/N. It can be restated as if as A is a zero-divisor for M/N then aGv/(ann(Af/AT)).

40

Prime ideals

A primary ideal is just a primary submodule of the 4-module A. One might wonder about trying to set up a notion of prime submodule generalising prime ideal, but this does not turn out to be useful. Theorem 6.6. Let A be a Noetherian ring and M afinite>4-module. Then a submodule N c M is primary if and only if Ass (M/N) consists of one element only. In this case, if Ass (M/N) = {P} and ann (M/N) = / then / is primary and yjl = P. Proof. If Ass (M/N) = {P} then by the previous theorem Supp (M/N) = K(P), so that P = ^/(ann (M/N)). Now if aeA is a zero-divisor for M/N it follows from Theorem 1 that aeP, so that a e ^ a n n (M/N)); hence, AT is a primary submodule of M. Conversely, if N is a primary submodule and Pe Ass (M/N) then every aeP is a zero-divisor for M/N, so that by assumption aey/l, where J = ann (M/N). Hence Pa^/l, but from the definition of associated prime we obviously have / c P, and hence ^// cz P, so that P = y/l. Thus Ass {M/N) has just one element y/l. We prove that in this case / is a primary ideal: let a, be A with b $ I; if ab el then ab(M/N) = 0, but b(M/N) # 0, so that a is a zero-divisor for M/N, and therefore Definition. If Ass (M/N) = {P} we say that NczM module, or a primary submodule belonging to P.

is a P-primary sub-

Theorem 6.7. If N and Nf are P-primary submodules of M then so is NnN'. Proof. We can embed M/(NnN') as a submodule of (M/N) © (M/N'), so that Ass (Af/(JV n N ' ) ) c Ass (M/N) u Ass (M/Nf) = {P}. • If N 0. A secondary representation of an A-module M is an expression of M as a finite sum of secondary submodules: (•)

M = N 1 + - + iVlt.

The representation is minimal if (1) the prime ideals Pt: = ^/(ann JV,-) are all distinct, and (2) none of the JV, is redundant. It is easy to see that the sum of two P-secondary submodules is again P-secondary, hence if M has a secondary representation then it has a minimal one. A prime ideal P is called an attached prime ideal of M if M has a Psecondary quotient. The set of the attached prime ideals of M is denoted by Att(M). Theorem 6.9. If (*) is a minimal secondary representation of M and Pt = ^(ann Nt\ then Att (M) = {Px,... ,P n }. Proof. Since M/(NX + ••• + Nf_ x + N i + x + ••• + AfJ is a non-zero quotient of Ni9 it is a Prsecondary module. Thus {P 1 ,\..,P n } c Att(M). Conversely, let PeAtt(M) and let W be a P-secondary quotient of M. Then W = Nx 4- — h NB, where N{ is the image of N( in W. From this we obtain a minimal secondary representation W = Nil + -~ + Ffim, and then Att(W)3{P f i ,...,p. s }. On the other hand Att(W) = {pS} since W is P-secondary. Therefore P = P, for some i. • Theorem 6.10. If 0 - ^ M ' — • M — • M " - > 0 is an exact sequence of ^-modules, then Att (AT) c Att(M) c Att(M / )u Att(M"). Proof. The first inclusion is trivial from the definition. For the second, let PeAtt (M) and let N be a submodule such that M/N is P-secondary. If M' + N = M then Af/N is a non-trivial quotient of Af\ hence PGAtt (M'). If M' + N*M then Af/(M' + JV) is a non-trivial quotient of Af* as well as of M/N, hence Af'' has a P-secondary quotient and PG Att (M"). •

44

Prime ideals

An ^-module M is said to be sum-irreducible if it is neither zero nor the sum of two proper submodules. Lemma. If M is Artinian and sum-irreducible, then it is secondary. Proof. Suppose M is not secondary. Then there isaeA such that M # aM and anM # 0 for all n > 0. Since M is Artinian, we have anM = an+lM for some n. Set K = {xeM|a"x = 0}. Then it is immediate that M = K + aM, and so M is not sum-irreducible. • Theorem 6.11. If M is Artinian, then it has a secondary representation. Proof. Similar to the proof of Theorem 6.8, (iii). • The class of modules which have secondary representations is larger than that of Artinian modules. Sharp [8] proved that an injective module over a Noetherian ring has a secondary representation. Exercises to Appendix to §6. 6.8. An y4-module M is coprimary if Ass (A/) has just one element. Show that a finite module M # 0 over a Noetherian ring A is coprimary if and only if the following condition is satisfied: for every aeA, the endomorphism a.M—•M is either injective or nilpotent. In this case AssM = {P}, where P — ^/(ann M). 6.9. Show that if M is an ^-module of finite length then M is coprimary if and only if it is secondary. Show also that such a module M is a direct sum of secondary modules belonging to maximal ideals, and Ass (M) = Att (M).

Properties of extension rings

Flatness was formulated by Serre in the 1950s and quickly grew into one of the basic tools of both algebraic geometry and commutative algebra. This is an algebraic notion which is hard to grasp geometrically. Flatness is defined quite generally for modules, but is particularly important for extensions of rings. The model case is that of completion. Complete local rings have a number of wonderful properties, and passing to the completion of a local ring is an effective technique in many cases; this is analogous to studying an algebraic variety as an analytic space. The theory of integral extension of rings had been studied by Krull, and he discovered the so-called going-up and going-down theorems. We show that the going-down theorem also holds for flat extensions, and gather together flatness, completion and integral extensions in this chapter. We will use more sophisticated arguments to study flatness over Noetherian rings in Chapter 8, and completion in Chapter 10. 7

Flatness

Let A be a ring and M an A-module. Writing Sf to stand for a sequence •••—>JV'—>N—>N"—>••• of A -modules and linear maps, we let Sf®AM, or simply Sf ® M stand for the induced sequence • • • —>N"® A M—••••. Definition. M is flat over A if for every exact sequence & the sequence Sf®AM is again exact. We sometimes shorten this to >4-flat. M is faithfully flat if for every sequence Sf> Sf is e x a c t o ^ O ^ M is exact. Any exact sequence Sf can be broken up into short exact sequences of the form 0-+Nx —>N 2 —>JV 3 ->0, so that in the definition of flatness we need only consider short exact sequences Sf. Moreover, in view of the right-exactness of tensor product (see Appendix A, Formula 8), we can restrict attention to exact sequences 9> of the form §-*Nx—>N, and need only check the exactness of £f®M:0^>Nl®M —> AT ® Af. If f:A —> B is a homomorphism of rings and B is flat as an 4-module, 45

46

Properties of extension rings

we say that / is a flat homomorphism, or that B is a flat A -algebra. For example, the localisation As of A is a flat A-algebra (Theorems 4.4 and 4.5). Transitivity. Let B be an A-algebra and M a B-module. Then the following hold; (1) B is flat over A and M is flat over B=>M is flat over v4; (2) £ is faithfully flat over A and M is faithfully flat over B=>M is faithfully flat over A; (3) M is faithfully flat over B and flat over A=>B is flat over ,4; (4) M is faithfully flat over both A and B=>B is faithfully flat over A Each of these follows easily from the fact that (9®AB)®BM = £f®AB for any sequence of ,4-modules Sf. Change of coefficient ring. Let B be an ^-algebra and M an ^-module. Then the following hold: (1) M is flat over A=>M®AB is flat over B\ (2) M is faithfully flat over A=>M®AB is faithfully flat over B. These follow from that fact that &?®B{B®AM) = Sf®AM for any sequence of B-modules Sf. Theorem 7.1. Let A—• £ be a homomorphism of rings and M a Bmodule. A necessary and sufficient condition for M to be flat over A is that for every prime ideal P of B, the localisation MP is flat over Ap where p = PnA (or the same condition for every maximal ideal P of #). Proof First of all we make the following observation: if S c A is a multiplicative set and M,N are /ls-modules, then M®AsN = M®AN. This follows from the fact that in N®AM we have a ax s>> sx av a -x®y = — ® — = —|xeM, yeiV and beB}.) Assume now that M is A -flat. The map A—>B induces Ap—>BP, and MP is a BF-module, therefore an ,4p-module. Let Sf be an exact sequence of Ap-modules; then, by the above observation, and the right-hand side is an exact sequence, so that MP is ,4p-flat. Next, suppose that MP is y4p-flat for every maximal ideal P of B. Let 0-*N' —• N be an exact sequence of v4-modules, and write K for the kernel of the B-linear map N'®AM—>N®AM, so that 0 - > K — > N ' ® M — > N ® M is an exact sequence of J5-modules. For any Pern-Spec B the localisation

§7

Flatness

47

0->K P — N'®AMP—+

N®AMP

is exact, and since N'®AMP = N'®A(Ap®AvMP) = N'p®A9MP, and similarly N®AMP=Np®AMP9 we have £,, = 0 by hypothesis. Therefore by Theorem 4.6 we have K = 0, and this is what we have to prove. Theorem 7.2. Let A be a ring and M an A-module. Then the following conditions are equivalent: (1) M is faithfully flat over A\ (2) M is 4-flat, and N®AM ^ 0 for any non-zero ^-module N; (3) M is /4-flat, and mM ^ M for every maximal ideal m of A. Proof. (1)=>(2). Let N->0. If N ® M = 0 then 9* ® M is exact, so ¥ is exact, and therefore N = 0. (2)=>(3). This is clear from M/mM = {A/m)®AM. (3)=>(2). If JV#O and O^xeJV then /lx ~ 4/ann(x), so that taking a maximal ideal m containing ann(x), we have M # m M :z>ann(x)-M; hence, 4 x (g) M # 0. By the flatness assumption, Ax ® M —• N ® M is injective, so that N®M^0. (2)=>(1). Consider a sequence of 4-modules If is exact then ^M°/M = ( ^ O / ) M = 0, SO that by flatness, lm(g°f)®M = Im(0 M °/ M ) = 0. By assumption we then have Im(g°f) = 0, that is g°f = 0; hence Ker # => Im / . If we set H = Ker g/ltn f then by flatness, so that the assumption gives H = 0. Therefore £f is exact. • A ring homomorphism f:A —>B induces a map fl/:Spec£ —• Spec4, under which a point peSpec/1 has an inverse image a /" 1 (p) = {PeSpecB\PnA = p} which is homeomorphic to SpectBO^^p)). Indeed, setting C = £®,4K:(p) and S = A — p, and defining g:B —• C by g(b) = b® 1, then since /c(p) = (/4/p)(g),4s, we have C = B f c ^ / l / p ) ® ^ = (B/pB)s = Thus a #:SpecC—•Spec B has the image i

{PeSpec£|P z> pJ5 and P n / ( S ) = 0 } which is af~l(p\ and ag induces a homomorphism of SpecC with l f~ {V>)- For this reason we call SpecC = Spec(fJ®tc(p)) the fibre over p. The inverse map af~ \p) —• SpecC takes Peaf~ \p) into PC = PBs/pBs. a

48

Properties of extension rings

For P*eSpecC we set P = P*nB; then by Theorems 4.2 and 4.3, we have P* = PC and CP* = (Bs/pBs)PC

= BP/pBP =

BP®AK(P)-

Theorem 7.3. Let f:A—>B be a ring homomorphism and M a Bmodule. Then (i) M is faithfully flat over A=>af(Supp(M)) = Spec A (ii) If M is a finite B-module then M is 4-flat and fl/(Supp(M)) => m-Spec,4oM is faithfully flat over A. Proof, (i) For p e Spec /I, by faithful flatness we have M ®A K(P) ^ 0. Hence, if we set C = B®AK(P) and M' = M®AK(P) = M®BC, the C-module M ' # 0 , so that there is a P*eSpecC such that M P *T*0. Now set

P = P*nB; then M p* = M ®BCP* = M ®B (BP(x)BpCp*) = MF®BpCp« so that M p / 0 , that is PeSupp(M). But P*eSpec(5®Ac(p)), so that as we have seen PnA = p. Therefore pefl/(Supp(M)). (ii) It is enough to show that M/mM i=- 0 for any maximal ideal m of A. By assumption there is a prime ideal P of B such that Pn>4 = m and MP # 0. By NAK, since MP is finite over BP we have MP/PMP ^ 0, and a /orfion Mp/mMP = (M/mM)P ^ 0, so that M/mM # 0 . • Let (A,m) and (£,n) be local rings, and f:A —>B a ring homomorphism; / is said to be a local homomorphism if /(m) c: n. If this happens then by Theorem 2, or by Theorem 3, (ii), we see that it is equivalent to say that / is flat or faithfully flat. Let S be a multiplicative set of A. Then it is easy to see that Spec(>4s) —• Spec ,4 is surjective only if S consists of units, that is A = As. Thus from the above theorem, if A # As then As is flat but not faithfully flat over A. Theorem 7.4. (i) Let A be a ring, M a flat ^-module, and Nl9 N2 two submodules of an X-module N. Then as submodules of iV®M we have (ii) Let A —>B be a flat ring homomorphism, and let Ix and I2 be ideals of A. Then (iii) If in addition I2 is finitely generated then Proof (i) Define N —> N/Nl © N/N2 is exact, and hence so is ® M) © (N ® M)/(N2 ® M).

§7

Flatness

49

This is the assertion in (i). (ii) This is a particular case of (i) with N = A, M = B, in view of the fact that for an ideal I of A the subset / ®AB of A ®A B = B coincides with IB. (iii) If 12 = Aal + ••• + Aan then since (Ii'I2)= f]iUi:ai\ w e c a n u s e (ii) to reduce to the case that / 2 is principal. For a eA we have the exact sequence

and tensoring this with B gives the assertion.

•

Example. Let k be a field, and consider the subring A = fc[x2, x 3 ] of the polynomial ring B = fc[x] in an indeterminate x. Then x2A nx3A is the set of polynomials made up of terms of degree ^ 5 in x, so that (x2Anx3A)B = x 5 £, but on the other hand x2Bnx3B = x3B. Therefore by the above theorem, B is not flat over A. Theorem 7.5. Let f:A —>B be a faithfully flat ring homomorphism. (i) For any /4-module M, the map M—>M® A B defined by m n m ® 1 is injective; in particular f:A —>B is itself injective. (ii) If / is an ideal of A then IBnA = I. Proof, (i) Let 0 ^ meM. Then (Am) ® B is a £-submodule of M ® £ which can be identified with (m® 1)2*. But by Theorem 2, (4m) x + — + Acon. Then setting JV, = JV'+ ^ c ^ + • • • + Ao)i (for 1 ^ i ^ n), we need only show that each step in the chain N'(g)M—• JV 1 ®M—>N 2 ®M—>•••—>N®M is injective, and finally that if N = N' + Aa) then ATN®M is injective. Now we set / = {aeA\aa)eNf}, and get the exact sequence 0-*AT—>N—>/4//->0. This induces a long exact sequence (see Appendix B, p. 279) •••—Tor?(M,/!//)—>N'®M—*N®M—>(X//)®M->0; hence it is enough to prove that (•) Torf(M,^l//) = 0. For this consider the short exact sequence

0-+/—>A—>A/I->0 and the induced long exact sequence Tor?(M, A) = 0 —* Tor?(M, A/1) —> / ® M — M — • • •; since / ® M —• M is injective, (*) must hold. • From this theorem we can prove the converse of Theorem 6. Indeed, if / = Aax + — 4- Aan is a finitely generated ideal of A then an element ^ of / ® M can be written as ^ = £ " at ® mf with m,e Af. Suppose that £ is 0 in M, that is that Yjaimi= 0- Now if the conclusion of Theorem 6 holds for M, there exist bkjeA and y^M such that = 0 for all j9

and m, = ^fryty

for all L

§7

Flatness

51

Then £ = X a i ® m ; = Z;Z; a Aj®.y/ == O> so that / ® M — • M is injective, and therefore M is flat. Theorem 7.8. Let ,4 be a ring and M an 4-module. The following conditions are equivalent: (1) M is flat; (2) for every ,4-module N we have Torf (M, N) = 0; (3) Tor? (M9A/l) = 0 for every finitely generated ideal /. Proof. (1)=>(2) If we let •••—>L { —>L i . l —>•••—>L 0 —>JV->0 be a projective resolution of N then •••—>Li®M—•L l _ 1 (g)M—• •••—>L 0 ®M is exact, so that Torf (M, AT) = 0 for all i > 0. (2)=>(3) is obvious. (3)=>(1) The short exact sequence ()-•/—>A—>A/I->0 long exact sequence

induces a

Tor ?(M, A/I)= 0 — / ® M —*M—>M®/l/J->0, and hence / ® M —• M is injective; therefore by the previous theorem M is flat. • Theorem 7.9. Let 0->M'—>M—•M"->0 be an exact sequence of Amodules; then if M' and M" are both flat, so is M. Proof. For any /4-module N the sequence Tor^M',^)—•Tor 1 (M,iV) —•Tor1(M",J/V) is exact, and since the first and third groups are zero, also Tor^M, N) = 0. Therefore by the previous theorem M is flat. • A free module is obvious faithfully flat (if F is free and £f is a sequence of y4-modules then Sf ® F is just a sum of copies of $f in number equal to the cardinality of a basis of F). Conversely, over a local ring the following theorem holds, so that for finite modules flat, faithfully flat and free are equivalent conditions. Theorem 7.10. Let (A9m) be a local ring and M a flat ,4-module. If are such that their images xl,...9xn in M = M/mM are linearly independent over the field A/xn then xl9...,xH are linearly independent over A. Hence if M is finite, or if m is nilpotent, then any minimal basis of M (see §2) is a basis of M, and M is a free module. Proof. By induction on n. If n= 1, and aeA is such that axx = 0 then by Theorem 6 there are bl9...9bseA such that abt = 0 and x e ^ ^ M ; by assumption x^ntM, so that among the b( there must be one not contained in m. This b{ is then a unit, so that we must have a = 0. For n > 1, let Ysaixi — 0 »then there are & o e4 and yjsM (for 1 ^ ; ^ s) such that £ #A, = 0 and xf = £ fc^. Now x^mM, so that among the &„, at least one is a unit. Hence an is a linear combination of ax, , an_ t , that X1,...,XWGM

52

Properties of extension rings

is an — Y,"=i 0, and from this we get a commutative diagram 0 -* F(M) —• F(Aq) —• F(AP) 0 -» G(M) —• G(Aq) —> G(AP) having two exact rows. Now F(AP) = NP®B and G(AP) = {N® B)p, so that the right-hand k is an isomorphism, and similarly the middle X is an isomorphism. Thus, as one sees easily, the left-hand X is also an isomorphism. • Corollary. Let A, M and N be as in the theorem, and let p be a prime ideal of A. Then HomA(M, N)®AAp = HomAp(Mp9 N9). Theorem 7.12. Let A be a ring and M an >4-module of finite presentation. Then M is a projective >4-module if and only if Mm is a free >lm-module for every maximal ideal m of A. Proof of "only if. If M is projective it is a direct summand of a free module, and this property is preserved by localisation, so that Mm is projective over Am, and is therefore free by Theorem 2.5. Proof of 'if. Let Nx—•JV2-*0 be an exact sequence of ^-modules. Write C for the cokernel of Hom^M, Nx) —• Homi4(M, N2);

Appendix to §7

53

then for any maximal ideal m of A we have Cm = C o k e r l H o m ^ M ^ C N J J —>Hom Am (M n ,(N 2 ) m )} Hence C = 0 by Theorem 4.6, and this is what we had to prove .

= 0. •

Corollary. If A is a ring and M is an A -module of finite presentation, then M is flat if and only if it is projective. Proof. This follows from Theorems 1,12 and 10 Exercises to §7. Prove the following propositions. 7.1. If B is a faithfully flat A -algebra then for an ,4-module M we have B®AM is B-flatoM is /1-flat, and similarly for faithfully flat. 7.2. Let A and B be integral domains with A C i + l ®N—>C,(g)N—>•••. If N is flat over A then //,(C.)(g)N = //.(C.® N) for all i. 7.7. Let /I be a ring and B a flat /1-algebra; then if A/ and N are /1-modules, Tor,^(M, N)®AB = Tor?(M®B,N®B)

for all i.

If in addition Af is finitely generated and A is Noetherian then ExtjJM, N)®AB = Extj,(M ®^B, N® A B)

for all i.

7.8. Theorem 7.4, (i) does not hold for the intersection of infinitely many submodules; explain why, and construct a counter-example. 7.9. If B is a faithfully flat A -algebra and B is Noetherian then A is Noetherian. Appendix to §7. Pure submodules Let A be a ring and M an 4-module. A submodule N of Mis said to be pure if the sequence 0 -*> N ® E — • M ® E is exact for every /1-module £. Since tensor product and exactness commute with inductive limits, we need only consider /1-modules E of finite presentation.

54

Properties of extension rings

Example 1. If M/N is a flat A-module, then N is a pure submodule of M. This follows from the exact sequence Tor ? (M/N, E ) — > N ® E — M®£. Example 2. Any direct summand of M is a pure submodule. Example 3. If A = Z, a submodule Af of M is pure if and only if NnmM = mN for all m > 0. In fact the condition is equivalent to the exactness of 0 -> N ® Z/mZ —• M ® Z/mZ, and every finitely generated Z-module is a direct sum of cyclic modules. Theorem 7.13. A submodule N of M is pure if and only if the following condition holds: if xt = Zj= i aumj (f°r 1 ^ * ^ r)> W M m^M, x(eN and a^eA, then there exist yjSN (for l ^ y ^ s ) such that X ^ Z J ^ I ^ J X / (for 1 ^ i ^ r). Proo/. Suppose AT is pure in M. Consider the free module Ar with basis ex,... ,er and let D be the submodule of >T generated by Yjaueb ^ ^3 ^ s- Set £ = i4r/A and let et denote the image of et in E. Then in M ® E we have

X xf ® et = X! Z a o m j ® ^ = Z mJ ® Z a i/«- = °> hence Z ^» ® ^i = 0 in N ® £ by purity. But this means that, in N ® ^ir, the element Z/*/® e» ^s °f ^ e f ° r m Z J ^ J ® Z»*ay^ ^or s o m e ^ G ^ Conversely, suppose the condition is satisfied. Let E be an 4-module of finite presentation. Then we can write E = Ar/D with D generated by a finite number of elements of A\ say Yj=\auei> * ^ J ^ 5 - Then reversing the preceding argument we can see that N ® E —• M ® E is injective. • Theorem 7.14. If N is a pure submodule and M/N is of finite presentation, then N is a direct summand of M. Proof. We will prove that O-* N - U M - ^ M / N - ^ O splits, where i and p are the natural maps. For this we need only construct a linear map f\M/N—>M such that pf is the identity map of M/N. Let {tl9...,tr} be a set of generators of M/N, so that M/N ^ /T/7*> where fl is the submodule of relations among the ty, let {(an,...,air)\\ ^ i^s} be a set of generators of R. Choose a pre-image £j of t, in M for each 7. Then set y]i = Yjai£jeN (for 1 < i ^ s ) . By the preceding theorem there exist Z'jeN such that iy£= X f l ^ i ( for 1 ^ ' < 5 ) - T h e n Z f l y « i - ^ ) = 0 Mx z> M r Then taking 3F as a system of neighbourhoods of 0 makes M into a topological group under addition. In this topology, for any x e M a system of neighbourhoods of x is given by {x-f MA}AeA. In M addition and subtraction are continuous, as is scalar multiplication xi—•ax for any aeA. When M — A each Mx is an ideal, so that multiplication is also continuous: (a + Mx)(b + MA) e ab + MA. This type of topology is called a linear topology on M; it is separated (that is, Hausdorff) if and only if f]xMx = 0. Each Mx c M is an open set, each coset x -I- MA is again open, and the complement M — Mx of MA is a union of cosets, so is also open. Hence MA is an open and closed subset; the quotient module M/Mx is then discrete in the quotient topology. M/P\XMX is called the separated module associated with M. Moreover, since for A 0 is an exact sequence, so that taking the inverse limit, we. see that

0-tf

>M

>(M/Nf

is exact. If we view ft as a submodule of id, the condition that £ = (£x)xe\GM belongs to ft is that each £A can be represented by an element of N, or in other words that £e^(N) + MJ for each 1 Hence # is the same thing as the closure of \j/(N) in M. In general it is not clear whether lH —>(M/Nf is surjective, but this holds in the case A = {1,2,...}. In fact then given an element*^' = (ft,£2,...)e(M/Nf, with £'MeM/(N + MJ, let x ^ M be an inverse image of £'i, and y 2 eM an inverse image of !fi—•(M/N)"is an exact sequence, and ft is the closure of ^(N) in J#, where \jj\M —• ift is the natural map.

§8

Completion and the Artin-Rees lemma

57

(ii) If moreover the topology of M is defined by a decreasing chain of submodules Mx => M 2 3 * • *, then 0 - tf — id — (M//Vf->0 is exact. In other words, (M/N) ~~l Now suppose that M and N are two ,4-modules with linear topologies, and let f:M —• N be a continuous linear map. If the topologies of M and N are given by {Mx}XeA and {Ny}^, then for any yeF there exists As A such that Mxa f~l(Ny). Define q>y:lft—>N/Ny as the composite M —• Af/M} —• N/Nr where the first arrow is the natural map, and the second is induced by / ; one sees at once that q>y does not depend on the choice of X for which Mk c f~l(Ny). Also, for y < / if we let $yy. denote the natural map N/Nr—>N/Ny, it is easy to see that (py — ij/yyoq)y,; hence there is a continuous linear map f.lft—>ft defined by the (B is a continuous ring homomorphism, then / induces a continuous ring homomorphism f:A—>8. Among the linear topologies, those defined by ideals are of particular importance. Let / be an ideal of A and M an ,4-module; the topology on M defined by {/"M}n= ia,... is called the l-adic topology. If we also give A the /adic topology, the completions A and !& of A and M are called I-adic completions; it is easy to see that A? is an i4-module: for a = ( a l f a 2 , . . . ) eA with aneA/In and £ = (x x , x 2 *.. .)e A? with xtteM/InM (for all n), we can just set a% — (axxua2x2,.. As one can easily check, to say that M is complete for the /-adic topology is equivalent to saying that for every sequence xt, x 2 ,... of elements of M satisfying xt — xl + 1 e / ' M for all i, there exists a unique x e M such that x — XiElW for all i. We can define a Cauchy sequence in M in the usual way ({x,} is Cauchy if and only if for every positive integer r there is an n0 such that xw + 1 — x n e / r M for n>n0), and completeness can then be expressed as saying that a Cauchy sequence has a unique limit. Theorem 8.2. Let A be a ring, / an ideal, and M an /4-module. (i) If A is /-adically complete then / c rad (A);

58

Properties of extension rings

(ii) If M is /-adically complete and as I, then multiplication by 1 + a is an automorphism of M. Proof, (i) For ael, 1 - a + a2 - a3 + • • • converges in 4, and provides an inverse of 1 + a ; hence 1 -ha is a unit of A. This means (see §1) that /crad(>4). (ii) M is also an A-module, and 1 + a (or rather, its image in A) is a unit in i?, so that this is clear. • The following two results show the usefulness of completeness. Theorem 8.3 (Hensel's lemma). Let (A, m, k) be a local ring, and suppose that A is m-adically complete. Let F(X)eA[X~\ be a monic polynomial, and let Fek[X~\ be the polynomial obtained by reducing the coefficients of F modulo m. If there are monic polynomials g, hsk[X"\ with (g,h) = l and such that F = gh, then there exist monic polynomials G, H with coefficients in A such that F = GH, G = g and H = h. Proof. If we take polynomials Gx, Hx eA[X~] such that g = Gt,h = Hx then F = G ^ modm[Al. Suppose by induction that monic polynomials Gn, Hn have been constructed such that F = G n // n modm"[X], and Gn = g, Hn = /i; then we can write F-GnHn = ^0)^^X1 with o^ern" and deg(/ £ l9...,ajn, and (oteM is an arbitrary inverse image of d)f in M, then M is generated over A by wl,..., con. Proof By assumption M = £ /Ico^ + /M, so that M = £ 4cof + / ( £ /Icy, + IM) = X /lew, + / 2 M, and similarly, M = £ /lew,. + /VM for all v > 0. For any £eM, write f = X^i^i + 5i w ^ ^i e ^^» then ^ = X^M^f + ^2 with a u e J and £2eI2M, and choose successively fliiVe/v and 0

This theorem is extremely handy for proving the finiteness of M. For a Noetherian ring A, the J-adic topology has several more important properties, which are based on the following theorem, proved independently by E. Artin and D. Rees. Theorem 8.5 (the Artin-Rees lemma). Let A be a Noetherian ring, M a finite ,4-module, JV c, we have Proof. The inclusion z> is obvious, so that we only have to prove c Suppose that / is generated by r elements ax,..., ar, and M by s elements co!,...,cos. An element of /"M can be written as £l/,(#)&>,•, where ft(X) = fi(Xu...,Xr) is a homogeneous polynomial of degree n with coefficients in A. Now set y4[X x ,..., X r ] = B, and for each n > 0 set /« a r e homogeneous of degree n

and XI let C c B s b e the B-submodule generated by (J„ > 0 J n . Since B is Noetherian, C is a finite B-module, so that C = Y,)=i ^Mj» where each Uj is a linear combination of elements of [jJn; therefore C is generated by finitely many elements of (J Jw. Suppose C = Bux + • • H- Bwr, where Wj = (un,..., ujs)£jd. for 1 ^ j " ^ r. Set c = max {dx,..., d j . Now if rjeFM n iV, we can write >7 = X fi(a)Wi with ( / i . - s / s ) e i w , and hence ( / i v - . J ^ I P j ^ i , With p j 6B = /l[X 1 ,...,X r ]. The left-hand side is a vector made up of homogeneous polynomials of degree n only, so that the terms of degree other than n on the right-hand side must cancel out to give 0. Hence we can suppose that the Pj(X) are homogeneous of degree n — d}. Then ^ = X/i(fl)coi = ^pJ-(a)^iii(2). (2)=>(3) Since A isflatover A, we need only prove that mA # A for every maximal ideal m of A. By assumption, {0} is closed in A, so that we can assume that A a A, and by Theorem 11, mA is the closure of m in A. However, m is closed in A, so that mA n A = m, and so mA ^ A. (3) =>(1) By Theorem 7.5, xnAnA = m for every maximal ideal m of A. Now mA c /4 is a closed set by Theorems 2, (i) and 10, (i), and since the natural map A —• A is continuous, m = mA nA is closed in A. If / 0. Hence taking the limit we get A = Urn /4// n = (lim A/m\) x ••• x (lim A/mnr). If we set A( for the localisation of A at m,, then, since A/m" is already local, A/m'! = (A/ml!)m=Ai/(miAi)\ and so lim /1/m" can be identified with At. • We now summarise the main points proved in this section for a local Noetherian ring. Let (A,m) be a local Noetherian ring; then we have: (l)f|i.>om- = (0). (2) For M a finite A-module and N c: M a submodule, H (W + mnM) = JV.

§8

Completion and the Artin-Rees lemma

63

(3) The completion A of A is faithfully flat over A; hence Ac A, and I An A = / for any ideal / of A. (4) A is again a Noetherian local ring, with maximal ideal mA, and it has the same residue class field as A; moreover, A/mnA = A/mn for all n > 0. (5) If A is a complete local ring, then for any ideal I ^ A, A/I is again a complete local ring. Remark 1. Even if A is complete, the localisation A p of A at a prime p may not be. Remark 2. An Artinian local ring (/I, m) is complete; in fact, it is clear from the proof of Theorem 3.2 that there exists a v such that mv = 0, so that A = lim A/mn = A. Exercises to §8. Prove the following propositions. 8.1. If A is a Noetherian ring, / and J are ideals of /t, and A is complete both for the /-adic and J-adic topologies, then A is also complete for the (/ + J)adic topology. 8.2. Let A be a Noetherian ring, and / z> J ideals of A; if A is /-adically complete, it is also J-adically complete. 8.3. Let A be a Zariski ring and A its completion. If a c 4 is an ideal such that aA is principal, then a is principal. 8.4. According to Theorem 8.12, if yef]jv

then

Verify this directly in the special case / = eA, where e2 = e. 8.5. Let /4 be a Noetherian ring and / a proper ideal of A; consider the multiplicative set S = 1 + / as in §4, Example 3. Then As is a Zariski ring with ideal of definition IAS, and its completion coincides with the /-adic completion of A. 8.6. If ^4 is /-adically complete then B = A{X] is (IB + XB)-zdka\\y complete. 8.7. Let (A, m) be a complete Noetherian local ring, and a { => a2 3 • • • a chain of ideals of X for which (°)vav = (0); then for each n there exists v(n) for which av(ll) c= m". In other words, the linear topology defined by {a v } v=1>2l ... is stronger than or equal to the m-adic topology (Chevalley's theorem). 8.8. Let A be a Noetherian ring, a!,..., ar ideals of A\ if M is a finite A -module and N c M a submodule, then there exists c> 0 such that nx ^ c , . . . , n r ^ c^>a\l...Q^MniV = a^1 ~ c ...a; r " c (a c i• • • KMnN). 8.9. Let A be a Noetherian ring and PeAss(/l). Then there is an integer c> 0 such that Pe Ass (A/I) for every ideal / c P * (hint: localise at P). 8.10. Show by example that the conclusion of Ex. 8.7. does not necessarily hold if A is not complete.

64

Properties of extension rings 9

Integral extensions

If A is a subring of a ring B we say that B is an extension ring of A. In this case, an element be Bis said to be integral over A ifftis a root of a monic polynomial with coefficients in A, that is if there is a relation of the form bn + 0Xft""lH h an = 0 with a,e A If every element of B is integral over A we say that B is integral over ,4, or that £ is an integral extension of A. Theorem 9.1. Let A be a ring and £ an extension of A. (i) An element be Bis integral over A if and only if there exists a ring C with y l c C c B and beC such that C is finitely generated as an ,4-module. (ii) Let AczB be the set of elements of B integral over A; then A is a subring of B. Proof, (i) If b is a root of f(X) = Xn + a.X^1 + - + an, for any P(X)e A[X~] let r\X) be the remainder of P on dividing by / ; then P(b) = r(ft) and deg r