1.1. From the Publisher.

Ideal theory is important not only for the intrinsic interest and purity of its logical structure but because it is a necessary tool in many branches of mathematics. In this introduction to the modern theory of ideals, Professor Northcott assumes a sound background of mathematical theory but no previous knowledge of modern algebra. After a discussion of elementary ring theory, he deals with the properties of Noetherian rings and the algebraic and analytical theories of local rings. In order to give some idea of deeper applications of this theory the author has woven into the connected algebraic theory those results which play outstanding roles in the geometric applications.

1.2. From the Preface.

See THIS LINK.

1.3. Review by: H Davenport
Science Progress 43 (169) (1955), 118.

The mathematical theory of ideals was first developed by Dedekind in the 1870's in his work on algebraic numbers, and the need for such a theory arose from the fact that the integers of an algebraic number-field, unlike the ordinary integers, can generally be factorised into "prime" integers in more than one way. In the last thirty years the theory of ideals has been greatly developed by algebraists in a much more general and abstract form. Ideals are defined in any ring, a ring being a system of elements (of any nature) which are capable of combination by two operations analogous to addition and multiplication, the two operations satisfying all the purely formal laws of ordinary algebra except the law that $ab = 0$ implies $a = 0$ or $b = 0$. This general theory has proved to be of great importance for algebraic geometry, and the present tract gives an excellent account of it, mainly from the point of view of such applications. The exposition is exceedingly clear and postulates a minimum of knowledge on the part of the reader. It necessarily postulates an ability for mastering purely abstract reasoning.

1.4. Review by: I S Cohen.
Mathematical Reviews MR0058575 (15,390f).

This well-written book provides a self-contained treatment of certain portions of the modern theory of ideals in Noetherian rings, including the elements of the theory of local rings. No previous knowledge whatsoever of ring theory is assumed, and beginners to the subject will find here a very readable account. There is very little overlapping with McCoy's book, Rings and ideals [1948], which is concerned mainly with rings not necessarily satisfying any chain condition. Indeed, to the reviewer's knowledge, there exists to date no other book which covers the same material ab initio.

After rings have been defined in a preliminary section, Chapter I begins with the stipulation that "ring" shall always mean commutative ring with identity. In this chapter are defined the elementary operations on ideals, the radical of an ideal, prime and primary ideals, isolated components of an ideal with respect to a multiplicative set. For ideals which are intersections of primary ideals the customary uniqueness theorems are proved and the representation of the radical is obtained. There follows a discussion of Noetherian rings, including the proof of the primary decomposition theorem and of the Hilbert basis theorem.

Chapter II is concerned with the standard properties of homomorphisms, residue class rings, and quotient rings, the last named including the generalised Chevalley-Uzkov quotient rings. Chapter III contains a further development of the ideal theory of Noetherian rings: theorems on the intersection of all the powers of an ideal, and of all the symbolic powers of a prime ideal; proof of the uniqueness of the length of the composition series for a primary ideal; the principal ideal theorem; some theorems on the rank of an ideal.

The final two chapters deal mainly with local rings. The topics covered in the first half of Chapter IV include the dimension of a local ring, systems of parameters and their analytical independence, regular local rings. Integral dependence (for arbitrary rings) is now defined, and it is proved that every proper principal ideal in an integrally closed Noetherian integral domain is an intersection of symbolic powers of minimal prime ideals. With every minimal prime ideal there is associated a valuation of the quotient field of $R$, and some properties of these valuations are obtained. In point of fact, valuations are not mentioned by name at all, the author preferring to speak of "divisors" (formal integral linear combinations of minimal prime ideals). In addition, we note that in most of this theory of integrally closed rings the author has not eschewed zero divisors, but we have assumed $R$ to be an integral domain for simplicity of statement.

Chapter V, "The analytic theory of local rings", deals, as its name implies, with those properties of a local ring in which convergence plays a role. The discussion here includes some properties of complete local rings, rings of formal power series, existence and uniqueness of the completion $\bar{Q}$ of a local ring $Q$, and some results on the relationship between the ideal theory in $Q$ and $\bar{Q}$.

In connection with each chapter there is a supplementary note giving some historical remarks as well as some indication of further developments. There is also a bibliography at the end of the book.

The author has been hampered by lack of space, and some topics have had to be omitted entirely. It is rather unfortunate that one of these omissions is the dimension theory of polynomial rings, particularly since the hope is expressed in the preface that the book may be of assistance to those interested in modern algebraic geometry. The reading of current papers requires also some familiarity with a certain amount of technical apparatus, such as the language and elementary theory of modules, or of relatively prime ideals; neither topic is mentioned (except in the supplementary notes). It is, of course, true that inclusion of additional material of this sort might result, for reasons of space, in the displacement of some of the more interesting developments of ideal theory, but in a basic introductory book to a broad subject it is perhaps inevitable that a certain amount of moderately dull foundation underlie the more attractive superstructure. The fact remains, though, as indicated above, that the presentation is clear and makes pleasant reading, and that the book will encourage many who would not otherwise have done so to study ideal theory and algebraic geometry.

1.5. Review by: Maxwell Rosenlicht.
Bull. Amer. Math. Soc. 60 (3) (1954), 282-283.

This is an excellent exposition of the basic facts about Noetherian rings, that is, commutative rings with unit element and ascending chain condition for ideals. Starting from scratch (no knowledge of modern algebra is assumed) the author proceeds clearly and efficiently up to the deeper parts of ideal theory that are used today in algebraic geometry. The transparency of style makes the book excellent for use as a text or for the practicing mathematician who wishes to absorb the essentials of the theory quickly and painlessly. The expert will be especially grateful for the unified presentation of results which, except for the more elementary parts already available in several texts, have thus far appeared only in Ergebnisse-style monographs or in the many widely scattered original papers of the last 25 years.

The chapter headings are: I. The primary decomposition. II. Residue rings and rings of quotients. III. Some fundamental properties of Noetherian rings. (This includes Krull's intersection theorem, symbolic powers of prime ideals, composition series for primary ideals, and dimension theory.) IV. The algebraic theory of local rings. (Including accounts of regular local rings and the quasi-gleichheit theory.) V. The analytic theory of local rings. (The existence of a local ring's completion and some transition properties.) The book ends with brief notes indicating the history and applications of the theory. The applications indicated are of course to algebraic geometry, but the author quite naturally refrains from identifying the two subjects.

Polynomial rings appear only incidentally or as examples and no mention is made of their unique factorisation. Very little is said about rings of dimension one and even the theory of Dedekind rings, whose full development would have required about one extra page at the appropriate place, is omitted. Field theory, valuation theory, the structure theory of complete local rings, the important relations existing between the ideals of a ring and an integrally dependent over-ring, the Hubert function, and unmixedness theorems are either missing entirely (presumably for reasons of space, connectedness and essentiality) or are mentioned in passing, in the notes at the end. There is a brief bibliography, most of whose items are made mainly of historical interest by the present tract. A brief list of the best (as opposed to the earliest) references would have been of more help to the unguided reader wishing to pursue the subject further.

2.1. From the Publisher.

Homological algebra, because of its fundamental nature, is relevant to many branches of pure mathematics, including number theory, geometry, group theory and ring theory. Professor Northcott's aim is to introduce homological ideas and methods and to show some of the results which can be achieved. The early chapters provide the results needed to establish the theory of derived functors and to introduce torsion and extension functors. The new concepts are then applied to the theory of global dimensions, in an elucidation of the structure of commutative Noetherian rings of finite global dimension and in an account of the homology and cohomology theories of monoids and groups. A final section is devoted to comments on the various chapters, supplementary notes and suggestions for further reading. This book is designed with the needs and problems of the beginner in mind, providing a helpful and lucid account for those about to begin research, but will also be a useful work of reference for specialists. It can also be used as a textbook for an advanced course.

2.2. From the Preface.

The past ten years or so have seen the emergence of a new mathematical subject which now bears the name Homological Algebra. To begin with, it was the concern of a few enthusiasts in certain specialised fields but since the publication of Cartan and Eilenberg's now famous book, its importance for several of the main branches of pure mathematics has been generally recognised.

The young mathematician, about to start on research, will be anxious to learn about homological ideas and methods, and one of the aims of this book is to help him to get started. In trying to cater for his needs, I have imagined such a reader as being familiar with the notions of group, ring and field but still relatively inexperienced in modern algebra. For him, the account given here is self-contained save in a small number of particulars which are mentioned below, and which need not discourage him.

An introduction to homological algebra must, of necessity, be an introduction to the book of Cartan and Eilenberg, for the student who wishes to go further will need to read their work; but much of great interest and value has been achieved even more recently, and some of this later work has been given a place in the following pages. The list of contents gives a fairly detailed picture of the main topics treated, but a few additional comments may be a help.

Chapters 1-6 develop, in a leisurely manner, the results that are needed to establish and illustrate the theory of derived functors, after which follows an account of torsion and extension functors. These are the most important ones which are obtainable by the process of derivation and, in a sense, the remainder of the book is concerned with their applications. Such an application is the theory of global dimension given at the end of Chapter 7, and here are included some important results of M Auslander on Noetherian rings that have previously been available only in the original research paper.

Chapter 9 deals with the structure of commutative Noetherian rings of finite global dimension and represents one of the most satisfying achievements of homological methods. This, too, appears in a text-book for the first time. Here, it must be admitted, the account is not completely self-contained, but considerable care has been taken in explaining the results of Ideal Theory which are needed to supplement the purely homological arguments. This is the most ambitious chapter, and the author hopes that it will help to stimulate interest in commutative algebra. The treatment given here was found successful in a course of lectures in which the audience had no specialised knowledge of classical Ideal Theory.

Chapter 10 is an introduction to the homology and cohomology theories of monoids and groups. This, by itself, has a considerable literature and was one of the earliest branches of our subject to be developed. The chapter can be read, if desired, before Chapter 9 and does not require any specialised knowledge of Group Theory. In deciding how far to go with this topic, I had in mind the student who might wish to acquire some general background before proceeding to the applications in some specialised field such as Class Field Theory.

Nearly all the topics covered in the following pages were included in a course of lectures given at Sheffield University. When lecturing, it is possible to digress at some length in order to explain the general plan of development and the connexions with other branches of mathematics. Also one likes to mention important results connected with what one is discussing even if there is no time for a full treatment. Some of this supplementary material, which I hope will add to the enjoyment and interest of the main text, will be found in the Notes which follow Chapter 10.

The final chapter has been much improved as the result of suggestions of J Tate with whom I had an opportunity of discussing it. At Sheffield, I have been aided, at all stages, by my colleague H K Farahat. Of particular value has been his willingness to discuss points of detail and to make helpful criticisms. This work owes a great deal to his continued interest. I am also indebted to Sir William Hodge, who, when I first had the idea of writing an introduction to homo-logical methods, encouraged me to go ahead.

Writing a book takes up much time and energy, and this one could never have been completed without the generous help of J J Kiely who typed the first draft from notes taken at lectures. I am also greatly indebted to my secretary, Mrs M Ludbrook, for the great care and patience with which she cut innumerable exquisite stencils. To both of these I wish to express my thanks. Their strenuous efforts made it unthinkable not to finish a work to which they had contributed so much.

D G Northcott
Sheffield
July 1958

2.3. Review by: D Buchsbaum.
Mathematical Reviews MR0118752 (22 #9523).

The author has written this book in order to acquaint the student of mathematics with the ideas and methods of homological algebra. He has assumed that the reader is familiar with the notions of group, ring, and field, but otherwise the presentation is self-contained.

Chapters 1-6 develop the theory of derived functors, and the remaining chapters are devoted to some applications. In particular, there is a chapter dealing with Noetherian rings and modules, and the homological characterisation of regular local rings. Another chapter discusses the homology and cohomology theory of groups and monoids, and some attention is given to finite groups.

There are two inferences that can be drawn from the author's having written this book. One is that the book Homological algebra by H Cartan and S Eilenberg [1956] is too difficult for the beginner to read, and the other is that some of the recent applications of homological techniques to commutative algebra should be presented to the student in order to stimulate his interest in homology theory. Although it is generally conceded that the Cartan-Eilenberg book is more encyclopaedic than inspiring, it is not clear that the first six chapters of the book under review remedy this deficiency. Although a general theory of categories and functors is discussed (Chapter 3), the author later restricts himself exclusively to categories of modules and to the functors, tensor product and Hom. For all but one of the applications considered in this book, these are the only categories and functors that are needed, so that it might have been more instructive to the reader if inessential generalities had been omitted. It seems that an attempt to give a watered-down version of Cartan-Eilenberg avoids none of the dryness of that work, and provides even less inspiration.

As for the problem of presenting stimulating applications, on the one hand it is difficult to believe that the reader of this book, who is assumed to know so little algebra, can appreciate the applications of homology to the theory of local rings. On the other hand, for instance, the failure to point out that the canonical complex constructed over the residue field of a local ring has for its components the exterior products of a finitely generated free module will probably make it difficult for the reader to fully understand some of the more recent work, using the Koszul complex in dealing with multiplicity theory and codimension.

Another criticism of this book is that the presentation of the material is neither new nor different. As mentioned earlier, the development of derived functors follows pretty much the line of development of Cartan-Eilenberg. In flirting with the notion of category, the author fails to point out that his diagrams are covariant functors and that his translations of diagrams are natural transformations of functors. In presenting the recent applications to Noetherian ring theory, there seems to be little attempt to be selective about the proofs. For example, for Auslander's important theorem about the global dimension of Noetherian rings being determined by the cyclic modules, the original lengthy proof is given, rather than the two-or three-line proof that is now fairly standard. The chapter on cohomology theory of groups goes no further than showing the existence of complete resolutions for finite groups, and gives no inkling as to how it comes up in class field theory.

Of course, all the foregoing remarks pertain more to the question of the need for such a book as this, than to an evaluation of the book on its own merits. Since the level of the book is supposed to be quite elementary, it might have been useful to have included some exercises. However, the author does leave some proofs to the reader and these (together with the exercises in Cartan-Eilenberg) should serve to give more facility with homological technique. It should be pointed out that the author has used his talent for expository writing, and the book is very clear and easy to read.

2.4. Review by: Davis Rees.
The Mathematical Gazette 45 (354) (1961), 376-377.

The branch of algebra known as homological algebra is of very recent origin, although its methods have been used by topologists for the last two decades. Its results first achieved book-form with the publication of Cartan and Eilenberg's book "Homological Algebra" in 1956 and it was in this book that a systematic development of the subject first appeared. Cartan and Eilenberg's account of the subject is not an easy one to read, since it contains a great number of results packed into a relatively small number of pages. Professor Northcott's book has a more modest aim, that of developing the basic ideas of the subject in a leisurely manner and illustrating the use of the methods with a few applications. Most of, but not all, the material is already available in Cartan and Eilenberg.

The book contains 10 chapters, the first two being, respectively, introductory chapters on modules and on tensor products and groups of homomorphisms. The accounts of both these topics are clear but offer no surprises. Chapter 3 commences the study of Homological Algebra since it is concerned with the basic notions of a category and of a functor. Next the author considers the theory of homology in Chapter 4 and here an innovation appears since Northcott considers homology as a function of a three-term sequence
           (A) $A_2 \overline{\lambda}$ → $A_1 \overline{\mu}$ → $A_{0}$
with $\lambda\mu = 0$, $H(A)$ being defined as the quotient group Ker $\lambda$/Im $\mu$. This idea is due to Yoneda, and to the reviewer seems to offer considerable advantages. Chapter 5 contains an account of projective and injective modules, with a very clear proof of the theorem that any module can be imbedded in an injective module. Further, the chapter also deals with the basic properties of projective and injective resolutions. Chapter 6 gives the definition and development of derived functors and the related theory of connected series of functors which are the fundamental notions underlying homological algebra. In Chapter 7, the Torsion and Extension functors are introduced as examples of derived functors and they are applied to the theory of global dimension of rings. Chapter 8 is devoted to certain auxiliary results to be used in the last two chapters. These two chapters are concerned with applications. Chapter 9 is concerned with modules over Noether rings and its pièce de resistance is Serre's proof that regular local rings are the only local rings of finite global dimension. Chapter 10 is concerned with the homology and cohomology of groups.

This book is a very much more elementary account of Homological Algebra than has hitherto been available. As such, its appearance is greatly to be welcomed. Its value is further enhanced by the notes at the end of the book. The book can be strongly recommended to those algebraists, geometers and topologists who wish to understand what homological algebra is about.

2.5. Review by: Albert Newhouse.
The American Mathematical Monthly 68 (8) (1961), 827.

Since this is the first book on a more or less introductory level on the relatively new subject of homological algebra, it is indeed a welcome addition to the mathematical literature.

The ten chapter headings will only give a rough idea of the wealth of material covered in this volume. 1. Generalities Concerning Modules, 2. Tensor Products and Groups of Homomorphisms, 3. Categories and Functors, 4. Homology Functors, 5. Projective and Injective Modules, 6. Derived Functors, 7. Torsion and Extension Functors, 8. Some Useful Identities, 9. Commutative Noetherian Rings of Finite Global Dimension, 10. Homology and Cohomology Theories of Groups and Monoids.

An appendix contains twelve pages of notes "meant to give the reader some help in getting his orientation."

The book is written in pleasing and leisurely style. Whereas the notes to each chapter are of great help, the reviewer feels that the complete lack of examples and exercises are extremely detrimental to the understanding of the subject matter for a beginner.

2.6. Review by: C Racine.
Current Science 30 (3) (1961), 117.

Professor Northcott, author of a remarkable Cambridge tract on Ideal Theory, is now to be thanked for a no less remarkable Introduction to Homological Algebra. From the time of its creation by H Poincaré until recently, the discipline called first Combinatorial Topology, then Algebraic Topology, grew as a mixed structure in the sense of Bourbaki, its purpose being to derive topological invariants from the analysis of certain algebraic structures. These had to be linked up in a certain manner to manifolds, or to subsets of topological spaces and, to realise such an association, various, mostly disconnected, methods were available. For a long time the need of a more unified treatment was keenly felt but it is only during the last twenty years or so that great progress was made to achieve it. As a consequence the conviction gradually gained ground that Algebraic Topology was more than a mere combination of two well known structures; that it was pregnant with an altogether new branch of Algebra. In 1953, H Cartan and S Eilenberg published their celebrated Homological Algebra, a first treatise on this new branch and an astonishing feat because, dealing with a newly born theory, it was setting it forth in a state of remarkable maturity.

Now, although carefully and clearly written, this treatise was assuming on the part of the reader a good deal of familiarity with Modern Algebra and, to a certain extent, with Algebraic Topology. That is why the book under review should be most welcome by students about to start on research for it does not take for granted more than an elementary knowledge of groups and rings. In fact the first two chapters are concerned only with fundamentals of General Algebra, in particular with tensor products, (O)- and exact sequences. Homological Algebra proper begins only with Chapter 3 which is about categories and functors. The important notions of Homology Functor and of Connected Homomorphisms are dealt with in Chapter 4. The author does not derive them, as it is usually done, from the theory of modules with differential operators. Following Yoheda he prefers to start with the notions of diagrams over a ring and of functors on translation categories; the case of a particular diagram yields the homology functor. In spite of its theoretical interest such an approach may fail to satisfy inexperienced algebraists. The chapter ends with a brief mention of homotopy. One may regret that such an important notion, and also that of homotopy operator, should receive so little attention. Chapter 5 treats of complexes and resolutions, particularly of projective and injective resolutions, of modules and of sequences of modules. Now the ground having been fully prepared Chapter 6 introduces the central notions of Homological Algebra: functors of complexes and derived functors. Chapter 7 makes a detailed study of torsion and extension functors, the most important derived functors.

It was likely that the author's intention was not to burden the student with too many new notions. This would explain why no mention is made of spectral sequences. Fortunately those who wish to acquire a good grasp of them may now consult the Théorie des Faisceaux by R Godement (Hermann, Paris) published almost at the same time as the book under review.

Chapter 8 is about various refinements and the last two chapters are devoted to two noteworthy applications: the theory of homology and of cohomology of groups or monoids and the theory of homological dimension, as well as of global dimension of Noetherian rings. Chapter 9 gives an excellent account of the research work carried out recently, mostly after 1956, regarding the latter.

This is a lucid and scholarly Introduction to Homological Algebra and it cannot but be too warmly recommended. Its admirable get-up does great credit to the Cambridge University Press.

2.7. Review by: Alex Rosenberg.
Bull. Amer. Math. Soc. 67 (5) (1961), 440-442.

This book is a leisurely and detailed introduction to homological algebra. Very little background is assumed and the account is essentially self-contained.

The author begins by explaining the basic ideas concerning modules over a ring, functors and categories, as well as treating the tensor product and Horn in some detail. Next the homology functor is introduced, the connecting homomorphism is defined and its standard basic properties are given. After introducing projective and injective modules, resolutions of modules are discussed and then the theory of derived functors is developed. Next, the general theory of Ext and Tor is treated, the various homological dimensions are defined and their basic properties stated. This whole development is carried out only for the category of modules over rings.

The material sketched occupies about 140 pages. It seems to this reviewer that the author has fallen into the trap of believing that the more detail given, the more intelligible a subject becomes. For example, it takes four theorems with separate proofs for the author to say that $\text{Tor}^A (A, B)$ may be computed from a projective resolution of $A$ or from a projective resolution of $B$, or from using resolutions of both. The analogous four theorems are also given for Ext. The ideas of the proofs are essentially those to be found in Cartan-Eilenberg's Homological algebra. While the present treatment is much more detailed than that to be found there, it does not seem to this reviewer that it renders the subject more accessible. Indeed, because of the length of the text, it will be difficult for the novice to pick out those points that are really basic.

The next part of the book deals in the main with material not to be found in Cartan-Eilenberg: M Auslander's result that l.gl.dim. is the supremum of the projective dimensions of cyclic left modules is given. Homological dimensions in general noetherian rings are discussed and the connection for these between a ring and its various rings of quotients are carefully worked out. The high point of this part of the book is undoubtedly the homological theory of local rings culminating in the Serre, Auslander-Buchsbaum theorem that a local ring is regular if and only if its global dimension is finite. The author has made this theory fairly self-contained by carefully stating all the ideal theoretic background necessary and supplying references to missing proofs.

The exposition of all these topics follows very closely that of the original research papers. No mention is made of the simpler proof of M. Auslander's theorem due to Eilenberg and to be found in Matlis' paper, Applications of duality (1960). Nor is Kaplansky's shorter proof of the Serre, Auslander-Buchsbaum theorem (1959) referred to.

The final chapter of the book provides a brief introduction to the homology of groups. The standard resolution, interpretation of the first and second groups, the case of free abelian and nonabelian groups and monoids, and the complete derived sequence for finite groups are given very much as in Cartan-Eilenberg.

Thus, this volume treats a rather circumscribed area of homo- logical algebra, the exposition being, however, not very different from that already available. Moreover, the usefulness of this book would have been increased by including a discussion of several points well within its scope. For example, although the text hints strongly at the fact that in general l.gl.dim. ≠ r.gl.dim. this point is never made explicit and Kaplansky's example of a ring $R$ with r.gl.dim. $R$ = l, l.gl.dim. $R$ = 2 (1958) is not cited. More serious, in this reviewer's opinion, is the lack of a complete bibliography; the inexperienced reader may remain unaware of much material in the literature, especially the kind that should certainly be given as suitable for further study. Thus, for example, no mention is made of Hochschild's relative theory, nor is most of the theory dealing with abstract categories and leading to a cohomology theory of sheaves mentioned. Since neither Godement's book nor Grothendieck's Tohoku journal papers are referred to, there is no indication of the many applications of homological algebra to topology and algebraic geometry.

Despite these drawbacks, it should be noted that in the topics treated, the author has given a very careful treatment of a relatively new subject. His work will certainly serve to disseminate these new ideas to a wide public.

3.1. From the Publisher.

This volume provides a clear and self-contained introduction to important results in the theory of rings and modules. Assuming only the mathematical background provided by a normal undergraduate curriculum, the theory is derived by comparatively direct and simple methods. It will be useful to both undergraduates and research students specialising in algebra. In his usual lucid style the author introduces the reader to advanced topics in a manner which makes them both interesting and easy to assimilate. As the text gives very full explanations, a number of well-ordered exercises are included at the end of each chapter. These lead on to further significant results and give the reader an opportunity to devise his own arguments and to test his understanding of the subject.

3.2. From the Preface.

This book has grown out of lectures and seminars held at the University of Sheffield in recent years. Its purpose is to give a virtually self-contained introduction to certain parts of Modern Algebra and to provide a bridge between undergraduate and postgraduate study.

The title, Lessons on rings, modules and multiplicities, was chosen partly because a certain emphasis has been placed on instruction. I have long been interested in problems involving the introduction of young mathematicians to relatively advanced topics and, in this book, I have endeavoured to present the chosen material in a manner which will not only make it interesting but also easy to assimilate.

One fact of general interest has emerged which I did not foresee when I started. It was my intention to write about Commutative Algebra, but the contents of the first chapter are of such generality that it seemed wrong to exclude non-commutative rings at that particular stage. From then on the question continually arose as to the proper place at which to assume commutativity, and, indeed, the precise form the assumption should take. The outcome has been that this book, particularly in its later stages, is often concerned with Quasi-commutative Algebra. By this I mean that non-commutative rings are allowed but the emphasis is on the behaviour of central elements. In fact much that one normally regards as belonging to Commutative Algebra can be accommodated comfortably within this framework. For example, this is true of considerable areas of Multiplicity Theory and the theory of Hilbert Functions. It is also true of the theory of I-adic Completions and, to some extent, the theory of Primary Decompositions, though the latter fact gets only a passing mention in the exercises. Other instances where this observation is valid will doubtless occur to the reader as he proceeds.

It is with pleasure that I take this opportunity to acknowledge many sources of help and information. Since the subject matter of the book is strongly slanted in the direction of Commutative Algebra it was inevitable that the writings of N Bourbaki, M Nagata, P Samuel and O Zariski should have a persistent influence. Those who are familiar with the literature will also recognise that the chapter dealing with the Koszul Complex owes much to the classic paper on Codimension and Multiplicity by M Auslander and D A Buchsbaum. In a similar manner, the chapter describing the properties of Hilbert Rings is based on papers by O Goldman and W Krull.

This book has also profited from the research investigations of recent postgraduate students at Sheffield University. In particular, some ideas involving Multiplicity Theory and Hilbert Functions, made use of here, first appeared in the doctoral theses of K Blackburn, D J Wright and W R Johnstone though they are now more widely available in standard mathematical journals.

An author is very fortunate if he has someone who is willing to read his manuscript with care and make detailed comments. D W Sharpe has performed this labour for me with a thoroughness which is familiar to those who know him well. His observations and constructive criticisms ranged from matters of punctuation and assistance with proof-correcting to comments on the organisation of whole chapters. A number of sections have been rewritten to incorporate improvements which he has suggested. In the later stages, P Vámos also helped me in a similar way and the final version has gained by being modified to take account of his observations.

Finally my thanks go to my secretary, Mrs E Benson, who typed the manuscript and remained cheerful when I changed my mind and asked to have considerable proportions done again. Without her help this book would have taken very much longer to complete.

D G Northcott
Sheffield
March 1968

3.3. Review by: B L Osofsky.
Mathematical Reviews MR0231816 (38 #144).

The author states that the purpose of the book is "to give a virtually self-contained introduction to certain parts of modern algebra and to provide a bridge between undergraduate and postgraduate study". The area of study is commutative algebra, although some aspects of non-commutative ring theory are introduced but not developed or worked with. Proofs are given in great detail with very little left to the reader. Each chapter has exercises following it, averaging about 16 per chapter. The scope of the material is essentially that of portions of M Nagata's Local rings [1962], done in a much more leisurely way. Thus, Chapters 2 through 4 essentially cover the first 30 pages of Nagata's book, with primary decomposition and localisation done for modules, so that the results for rings are special cases. Chapters 5 through 9 deal with rather specialised aspects of commutative algebra, although some of their results are stated in terms of central elements rather than requiring all elements to be central.

Chapter 1 is entitled "Introduction to some basic ideas". It includes sections on General remarks concerning rings, Modules, Homomorphisms and isomorphisms, Submodules, Factor modules, Isomorphism theorems, Composition series, The maximal and minimal conditions, Direct sums, The ring of endomorphisms, Simple and semi-simple rings, Exact sequences and commutative diagrams, Free modules, and Change of rings. Although this is presumably aimed at a beginning algebra student, the reader must supply his own examples. Mention of the integers, reals, and quaternions is left for the exercises. In general, almost all of the examples in the book are left to the exercises. When matrices are used, their multiplication is assumed known. Polynomials and power series are done with coefficients in a module. No categorical or universal mapping properties are introduced. Thus, in the discussion of endomorphism rings, we have a statement that $R$ is the direct sum of the rings $R_{1}, R_{2}, ... , R_{p}$ but "It is necessary, however, to add a word of warning. The mapping $R_{i}$ → $R$ ... is not a ring homomorphism ...". Direct products are not defined. A semi-simple ring is defined as one expressible as a direct sum of minimal left ideals. If all the simple ideals occurring in this direct sum are isomorphic, the ring is by definition simple. Such rings are shown to be ring direct sums (products) of matrix rings over division rings. It is only in the exercises that one is asked to show that a simple ring has no non-trivial two-sided ideals. No mention is made of complete reducibility, except for one theorem that a finitely generated module over a semi-simple ring is a direct sum of simple ones. In the discussion of free modules, no mention is made of projectivity properties.

Chapter 2, "Prime ideals and primary submodules", contains sections entitled Zorn's lemma, Prime ideals and integral domains, Minimal prime ideals, Subrings and extension rings, Integral extensions, Prime ideals and integral dependence, Further operations with ideals and modules, Primary submodules, Submodules which possess a primary decomposition, Existence of primary decompositions, Graded rings and modules, Torsionless grading monoids, and Homogeneous primary decompositions. To extract a bit of extra generality, arbitrary grading monoids are discussed, although the main results are on graded rings and modules where the grading monoid is totally ordered, and the later applications use the non-negative integers as grading monoid. This leads to a section showing that torsionless monoids can be totally ordered, which the author says "do not help one to understand the applications [and] it is suggested that they be omitted at first reading". Sometimes striving for generality does not pay. A similar comment might well have been made in other places in the book.

Chapter 3, entitled "Rings and modules of fractions", contains the following sections: Formation of fractions, Rings of fractions, The full ring of fractions, Functors, Further properties of fraction functors, Fractions and prime ideals, Primary submodules and fractions, Localisation, and Some uses of localisation. This is a leisurely discussion of modules $E_{S}$ as equivalence classes of elements $[e/s], e \in E, s \in S$, where $S$ is a multiplicatively closed subset of $R$, assumed commutative. There is no mention of tensor products or of any universal mapping property of $R \rightarrow R_{S}$. The only functor mentioned is $E \rightarrow E_{S}$, and the section on functors (in which the word "category" is undefined but used to describe appropriate collections of $R$-modules and $R$-homomorphisms) gives assurances that functors are important, but no examples except $E \rightarrow E_{S}$. Its major results are about exact functors. Natural transformations are defined but not used. In the exercises a second functor (also an exact tensor product) is defined for polynomial rings.

Chapter 4 is entitled "Noetherian rings and modules". Its sections include Further consideration of the maximal and minimal conditions, Noetherian modules and Artin rings, Noetherian graded modules, Submodules of a Noetherian module, Lengths and primary submodules, The intersection theorem, The Artin-Rees lemma, Rank and dimension, and Local and semi-local rings. Except for the first section, all rings are commutative. Artin rings are defined as having both chain conditions, and for a commutative ring, d.c.c. is shown to imply a.c.c. and the structure theory of such rings is worked out. In this chapter, the usual grading on polynomial rings $R[x_{1} ... x_{n}]$ is introduced and used to prove the Artin-Rees lemma. Rank and dimension correspond to height and depth, respectively, in Nagata's book.

Chapter 5 is entitled "The theory of grade". Its sections include The concept of grade, The theory of grade for semi-local rings, Semi-regular rings, General properties of polynomial rings, and Semi-regular polynomial rings. No homological algebra is used. Semi-regular rings are Cohen-Macaulay rings. Polynomial rings are discussed in some detail, and if the coefficients are in a semi-regular ring (such as a field), they are shown to be semi-regular.

Chapter 6, entitled "Hilbert rings and the zeros theorem", has sections entitled Hilbert rings, Polynomials with coefficients in a field, and The zeros theorem.

Chapter 7 is on "Multiplicity theory". Its section are Preliminary considerations, Key theorems on central ideals, Multiplicity systems, The multiplicity symbol, The limit formula of Lech, Hilbert functions, The limit formula of Samuel, Localisation and extension, The associative law for multiplicities, and Further consideration of Hilbert functions. Multiplicity symbols $e(\gamma_{1}, ... , \gamma_{s}| E)$ are initially defined inductively on $s$ for $\gamma_{1}, ... , \gamma_{s}$ central and $E/(\gamma_{1}, ... , \gamma_{s})E$ of finite length and then later tied up with Hilbert polynomials. In the exercises, a regular local ring is defined as one of dimension $d$ whose maximal ideal is generated by $d$ elements$\{u_{1} ... u_{d}\}$and the reader is asked to use multiplicities to show the ideal $(u_{1}, ... ,u_{s})$ is prime for $1 ≤ s ≤ d$. This appears to be the only reference to regular local rings, although one exercise in Chapter 8 connects this definition with the Koszul complex.

Chapter 8, "The Koszul complex", has the following sections: Complexes, Construction of the Koszul complex, Properties of the Koszul complex, Connections with multiplicity theory, and Connections with the theory of grade.

Chapter 9, "Filtered rings and modules", has sections entitled Topological prerequisites, Filtered modules, Continuous, compatible and strict homomorphisms, Complete filtered modules, The completion of a filtered module, The existence of completions, Filtered rings, Filtered modules over filtered rings, Multiplicative filtrations, $I-$adic filtrations, and $I$-adic completions of commutative Noetherian rings. This final chapter, except for one exercise, does not concern itself with any of the material on multiplicities, and so could easily be read after Chapter 5. Another example of a functor, namely, the completion functor, is introduced here. Of course, as is true of the only other functors mentioned in the book, it is exact.

3.4. Review by: Michael Rosen.
Quarterly of Applied Mathematics 28 (4) (1971), 612-613.

Although the title of Professor Northcot's latest book has a general flavour to it, it is really rather specialised. For the most part the subject matter concerns Noetherian rings and modules, and most of the early part of the book is concerned with building up elementary background for the study of the concepts of grade and multiplicity.

Chapter 1 is concerned with basic ideas, Chapter 2 with prime ideals and primary decomposition for modules, Chapter 3 with rings and modules of fractions, and Chapter 4 with Noetherian rings and modules. This last chapter contains proofs of the Krull intersection theorem, the Artin-Rees theorem, and the principal ideal theorem. It also introduces the notion of rank and dimension for ideals, and begins the study of local and semi-local rings.

Chapter 5 is about the notion of grade. This notion has, in the previous literature, gone by the name of homological co-dimension. The present treatment goes quite far into the theory without the use of any homological algebra. The theory is applied to the study of semi-regular rings (usually called Cohen-Macaulay rings). For example, it is proved that if R is semi-regular, then so is R[x]. Surprisingly, no mention is made of regular local rings.

Chapter 6 is devoted to the study of Hilbert rings. $R$ is a Hilbert ring if every prime ideal is the intersection of the maximal ideals containing it. Among other things, it is shown that if $R$ is a Hilbert ring, then so is $R[x]$. This rapidly yields the Hilbert Nullstellensatz.

Chapter 7 deals with the notion of multiplicity. Let $E$ be a module over a ring $R$. If $E$ has a composition series, let $L_{R}(E)$ be the length of the composition series. A sequence of elements $\gamma_{1}, ..., \gamma_{s} \in R$ is said to be a multiplicity system on an $R$ module $E$ if $E/\{\gamma_{1} E\} + ... + \gamma_{s} E$ has a composition series. Given such a system, a multiplicity symbol $e_R(\gamma_{1}, \gamma_{2}, .... \gamma_{s} | E)$ is defined. The remainder of the chapter is devoted to developing properties of this symbol. If $E$ is Noetherian, it is shown that

          $0 ≤ e(\gamma_{1}, ..., \gamma_{s} | E) ≤ L_{R}(E/(\gamma_{1} E + ... + \gamma_{s} E)$.

An investigation is made into the question of when the right-hand equality holds.

Two important limit formulae are proved. The first is due to Lech:

          $e_R(\gamma_1, \gamma_2, .... \gamma_s | E) = lim \frac { L_R(E/{{\gamma_1}^n_1 E} + ... + {\gamma_s}^n_s E)}{n_1n_2...n_s}$.

The limit is taken as min$(n_{i}) \rightarrow ∞$.

The second limit formula is due to Samuel. Its proof comes after a study of Hilbert functions and polynomials. If $A = (\gamma_{1}, ... , \gamma_{s})$ and $E$ is Noetherian, then

          $e_R(\gamma_1, ..., \gamma_s | E) = lim_{n \rightarrow \infty} \frac {L_R(E/A^nE)}{n^s/s!}$.

In sections 7.8 and 7.9, respectively, the extension theorem and associativity law for multiplicities are proved.

Chapter 8 introduces the Koszul complex and shows how it is related to grade and multiplicity. Chapter 9, the final chapter, is about filtrations and completions. It is elementary in character and independent of the previous four chapters.

I have a number of criticisms. There is little attention paid to the history and background of the subject. More surprising, and even less understandable, is the absence of a bibliography. The ahistorical attitude results in a lack of motivation. Grade and multiplicity are not immediately interesting in themselves. Some account of how these ideas arise naturally in algebraic geometry would have been highly desirable.

The last chapter, though interesting and well written, is somehow out of place. After the developments of Chapters 5 through 8, it would seem natural to conclude with a discussion of regular local rings. At this point in the book, a number of important theorems about such rings are easily accessible, but even the concept of regular local ring is not mentioned except in an exercise on page 350.

For all these criticisms, it remains true that Professor Northcott has written a clear and readable exposition of relatively new, difficult, and important material. It is remarkable that he is able to penetrate so deeply into his subject with such an economy of means. By so doing, he has made some very interesting mathematics, of current research interest, available to a much larger mathematical public.

3.5. Review by: E C Thompson.
Bulletin of the London Mathematical Society 2 (2) (1970), 247-248.

This is an expository work of unusual distinction. The author has a gift for providing all the necessary explanations without being tedious or condescending, and although he does not disdain to give an account of Zorn's Lemma or to state the axioms for a topological space, he also covers much recent work on multiplicity theory which has not previously been presented in a unified treatment. From a smooth start, the exposition acquires an exhilarating momentum, and its value is increased by the well-chosen exercises. Some may consider that the lack of any indication of the geometrical motivation conceals part of the significance of the results; for instance, the reader can gather little idea of why a multiplicity is so named. It must however be admitted that as a subject advances in abstraction and generality, the original motivation ceases to provide illumination for the student, and the terminology which has accreted is accepted without enquiry into its derivation. Few students know why an ideal is so named, but the knowledge would be irrelevant or even misleading in many contexts.

4.1. From the Publisher.

Based on a series of lectures given at Sheffield during 1971-72, this text is designed to introduce the student to homological algebra avoiding the elaborate machinery usually associated with the subject. This book presents a number of important topics and develops the necessary tools to handle them on an ad hoc basis. The final chapter contains some previously unpublished material and will provide additional interest both for the keen student and his tutor. Some easily proven results and demonstrations are left as exercises for the reader and additional exercises are included to expand the main themes. Solutions are provided to all of these. A short bibliography provides references to other publications in which the reader may follow up the subjects treated in the book. Graduate students will find this an invaluable course text as will those undergraduates who come to this subject in their final year.

4.2. From the Preface.

The main part of this book is an expanded version of lectures which I gave at Sheffield University during the session 1971-2. These lectures were intended to provide a first course of Homological Algebra, assuming only a knowledge of the most elementary parts of the theory of modules. The amount of time available was very limited and ruled out any approach which required the elaborate machinery or great generality that is sometimes associated with the subject. The alternative, it seemed to me, was to build the course round a number of topics which I hoped my audience would find interesting, and create the necessary tools by ad hoc constructions. Fortunately it proved rather easy to find topics where the techniques needed to treat one of them could also be used on the others. In the event, the first five chapters were fully covered in the course. The last chapter was added later and it differs from those that precede it by including some material which, so far as I am aware, has not previously appeared in print. This material has to do with what are here called semi-commutative local algebras. It is hoped that it may be of some interest to the specialist as well as to the beginner.

Reference has already been made to one way in which the amount of available time influenced the structure of the course. It had, indeed, a second effect. In order to speed up the presentation, some easily proved results and parts of some demonstrations were left as exercises. Other exercises were included in order to expand the main themes. What actually happened was that two members of the class, Mr A S McKerrow and Mr P M Scott, were good-natured enough to do all the exercises and, in addition, they provided the other participants with copies of their solutions. These solutions, edited so as to remove differences of style, are reproduced here. However the reader will find that his grasp of the subject is much improved if he works out a fair proportion of the problems for himself, rather than merely checks through the details of the arguments provided. The more difficult exercises have been marked with an asterisk.

I am much indebted to other mathematicians who have written on similar or related topics, and the list of references at the end shows the books and papers that I have consulted recently. It is a pleasure to acknowledge the help and benefit that I have derived from these and other sources. I have not attempted to compile a comprehensive bibliography. Naturally the degree of my indebtedness varies from one author to another. I have, for example, made much use of I Kaplansky's treatment of homological dimension. Also I am very conscious of the influence which the writings of H Bass and R G Swan have had on this account.

As on other occasions, I have been very fortunate in the help that has been given to me. Once again my secretary, Mrs E Benson, has converted pages of untidy manuscript into an orderly form where the idea that they might turn into a book no longer seemed un- reasonable. Besides this Mr A S McKerrow checked much of the first draft to see that it was technically correct. Their assistance has been extremely valuable and I am most grateful to them both.

D G Northcott
Sheffield
October 1972

4.3. Review by: David L Johnson.
The Mathematical Gazette 58 (406) (1974), 314-315.

Homological concepts and constructions, together with the language of category theory in terms of which they are expressed, exert an ever-increasing influence on many aspects of pure mathematical research, and also on the teaching of mathematics at university level. The main obstacle encountered in the teaching of this subject is that a fair degree of sophistication is required, if not to understand the ideas involved, at least to motivate them, for the subject per se is a rather dry one. In this book, Professor Northcott has gone a long way towards overcoming this obstacle, and has done so by means of a variety of expedients. First of all, the material has been chosen, in at least four out of the six chapters, from the more stimulating areas of the subject. Secondly, the treatment is carried out at a fairly concrete level and the rarefied abstractions of category theory have largely been avoided. Next, the introduction of many of the ideas is motivated by their subsequent application to problems in the 'real world' (ring theory). Finally, there is the inclusion of a large number of exercises, full solutions to all of which are provided, and this comprises a notable feature of the book. As the author acknowledges, some of the exercises are a bit of a cheat - for example, within the course of a single page, the reader is asked to prove the Hilbert basis theorem and to provide examples of rings which are left Noetherian but not right Noetherian and left Artinian but not right Artinian. On the whole though, I think the exercises are a success and their inclusion greatly enhances the text.

Chapter 1 deals with some basic notions, notably the theory of functors, and is thus rather dull. Things brighten up though in Chapter 2 when we get to projective and injective modules, and the mood persists throughout Chapter 3 which is equally readable and deals with the functor {Ext_A}1 and the concept of homological dimension. Although entitled Polynomial rings and matrix rings, Chapter 4 is chiefly concerned with the theory of equivalence of categories, and I was both surprised and disappointed to find that it did not contain a proof of Wedderburn's structure theorem for Artinian rings. The interest-rating recovers well in Chapter 5 with the study of duality, leading to the theory of quasi-Frobenius rings. The book is nicely rounded off by a chapter on local homological algebra which contains among other things an account of the author's own work on generalising certain results in the theory of commutative rings to the non-commutative case.

As is bound to happen, this book overlaps with others in the same field but as pointed out above contains some novel features as well as some original material. While it would be a bright student who could master this text without help, it is certainly a most suitable book to be read in conjunction with a course of lectures. I could find no errors and an amazingly small number of misprints - the only one I recall is trivial and appears on p. 74. Apart from one or two small points, such as the almost indiscriminate use of Latin and Greek letters for elements and mappings, and the convention of writing functions on the left, it is difficult to find anything in this book to which to take exception. The several topics are dealt with in a masterful and polished fashion and the layout and printing are immaculate.

4.4. Review by: T W Hungerford.
Mathematical Reviews MR0323867 (48 #2220).

In the author's words, this book is designed "for a first course of homological algebra, assuming only a knowledge of the most elementary parts of the theory of modules. The amount of time available... ruled out any approach which required the elaborate machinery or great generality that is sometimes associated with the subject. The alternative was to build the course round a number of [interesting] topics... and create the necessary tools by ad hoc constructions."

The book is aimed at first year graduate students (or well prepared undergraduates). The necessary background in module theory could readily be provided for this audience in half a dozen lectures or so.

The discussion is entirely restricted to categories of unitary modules over rings with identity. Arbitrary categories are not defined, but the language of functors is introduced in Chapter I and used frequently thereafter. Although the Hom functor is discussed at length, tensor products are never mentioned.

The ad hoc constructions are most apparent in the definition of the functor Ext$^{1}(A,B)$, which is done without explicit resort either to resolutions or to equivalence classes of extensions. The extension problem is briefly touched upon. The functors Ext$^{n} (n > 1)$ are not discussed, but the theory of homological dimension of modules and rings is presented in reasonable detail, including the global dimension of polynomial rings.

Equivalence of categories of modules is dealt with nicely, particularly the role of projective generators. A corollary of the main result, for example, is the fact that the category of right $\Lambda$-modules is equivalent to the category of right modules over the ring of $n \times n$ matrices over $\Lambda$.

The last third of the book begins with the standard facts about Noetherian rings and modules, duality, reflexive modules, and quasi-Frobenius rings. In the last chapter projective covers and noncommutative (quasi-) local rings are introduced. After a brief discussion of primary decomposition of ideals and modules over a commutative ring, the book closes with some previously unpublished material on semicommutative local algebras.

As we have come to expect from the author the exposition is clear and straightforward. There are many nontrivial exercises, with complete solutions included at the end of each chapter. The chief drawback of the book is the lack of examples, especially in Chapter I. Some readers will be put off by the ad hoc approach and the absence of some standard homological machinery. But anyone who believes that a first homological algebra course should include some "applications" and that a limited, concrete approach is the best way to prepare students for the high level of abstraction and generality of contemporary categorical algebra should seriously consider this book.

4.5. Review by: P M Cohn.
Bulletin of the London Mathematical Society 7 (1) (1975), 107.

The aim of this book is expressed in the title: it is a first introduction to homological algebra (assuming only some elementary module theory) with applications to local rings. In the first half of the book the author introduces us to functors, in particular the hom functor, projective and injective modules and the definition and basic properties of Ext$^{1}$, as far as the notion of homological dimension of a module. The treatment is elementary and self-contained throughout; no category theory is assumed (or needed). The author has been at pains to keep down the number of concepts used, thus Ext$^{1}$ is introduced and studied without any general theorems: this relatively concrete approach gives an insight which might be lost in a more high-powered treatment.

There are three chapters of applications. The first discusses the "polynomial functor" (with a proof of the syzygy theorem) and a special case of Morita equivalence, viz. between a ring $R$ and its matrix ring $M_{n}(R)$. The next topic is duality: the functor Hom$R(-, R)$, especially for Noetherian or more particularly quasi-Frobenius rings. The final chapter is on local rings, culminating in the study of the grade of a module. Some of the results are stated in a more general form than usual; in place of (commutative) local rings the author considers local $K$-algebras $R$ (where $K$ is a commutative coefficient ring) with an ideal $\mathfrak {a}$ in K such that $J^k \subseteq \mathfrak {a} R \subseteq J$ (where $J$ is the unique maximal ideal of $R$). Although these local rings are not the most general type, the results obtained strengthen the case for studying the notion of grade in this more general context.

Readers familiar with Professor Northcott's earlier book will naturally wonder about the connexion between the two: the earlier book, appearing soon after the publication of Cartan and Eilenberg's classic, aimed at giving a broad leisurely account of the subject, with some applications to group rings. The new book is possibly a little more concise in its introduction to homological algebra, without being less elementary; about the same amount of space is devoted to the applications, although a different selection is made. A particularly useful feature of the book under review is the collection of exercises throughout the text, with full solutions at the end of the chapters. This should make the book particularly well suited to the student meeting the subject for the first time

5.1. From the Publisher.

An important part of homological algebra deals with modules possessing projective resolutions of finite length. This goes back to Hilbert's famous theorem on syzygies through, in the earlier theory, free modules with finite bases were used rather than projective modules. The introduction of a wider class of resolutions led to a theory rich in results, but in the process certain special properties of finite free resolutions were overlooked. D A Buchsbaum and D Eisenbud have shown that finite free resolutions have a fascinating structure theory. This has revived interest in the simpler kind of resolution and caused the subject to develop rapidly. This Cambridge Tract attempts to give a genuinely self-contained and elementary presentation of the basic theory, and to provide a sound foundation for further study. The text contains a substantial number of exercises. These enable the reader to test his understanding and they allow the subject to be developed more rapidly. Each chapter ends with the solutions to the exercises contained in it.

5.2. From the Preface.

This Cambridge Tract originated in a seminar given by J A Eagon while he was visiting Sheffield University during the session 1972/3. The aim of the seminar was to report on some recent discoveries of D A Buchsbaum and D Eisenbud concerning finite free resolutions. I found myself fascinated by the subject and Eagon and I had many discussions on different aspects of it. In the end we were able to construct what we considered to be a simplified treatment of certain parts of the theory, and our ideas appeared subsequently in a joint paper. I continued to think about these matters after Eagon had left Sheffield, and during 1973/4 and 1974/5 gave seminars covering an enlarged range of topics, but still using what I regard as elementary methods. The elementary approach was based on the belief that, for those sections of the theory I was considering, Noetherian conditions were never really necessary; consequently

I was committed to showing that, where such considerations had previously been used, a way of getting rid of them could be found. Now the parts of the theory in which Noetherian properties had originally played an apparently vital role were, to a considerable extent, concerned with applications of the concept of grade; and I had, for some time, known of M Hochster's approach to a theory of grade in which it was not necessary to restrict oneself to Noetherian modules. Thus the programme I had set myself hinged on adapting Hochster's ideas in order to produce a simplified theory of grade applicable to general modules over an arbitrary commutative ring. At the same time the theory had to be rich enough for the applications that I had in mind.

The results of the attempt to construct an appropriate theory of grade will be found in Chapter 5, and with the aid of what is contained there the original programme was carried through. One advantage that has emerged is that the account in the following pages demands remarkably little in the way of prerequisites. All that is required is a knowledge of the basic properties of modules and linear mappings, and, in a few places, a facility in working with tensor products is presupposed. Otherwise, with the exception of the Appendices, the account is quite self-contained. In particular I have resisted any temptation to use the theory of exterior algebras in the main text because, at this level, it is not particularly helpful and there are some key places where its use seems to have definite disadvantages. Of course it is to be expected that, as the theory grows, general structural considerations will come to play a more dominant role and that ad hoc computational arguments will be increasingly hard to find. I have tried, at appropriate places, to indicate how the subject of finite free resolutions has grown and to mention the names of those who have contributed to its development. Here I would like to thank those who came to my lectures and who encouraged me by their continued interest. I must add a special word of thanks to P Vamos whose wide knowledge proved most helpful and whose comments on particular points led to many improvements; and to D W Sharpe for discussions on the role of grade in the theory of linear equations, for bringing various errors to my attention, and for help with reading the proofs. Also I am much indebted to C J Knight who invariably assists me with queries that have to do with Topology. But above all I wish to place on record my appreciation of the work done by my secretary, Mrs E Benson, who, by producing the whole typescript with her characteristic skill and good judgement, has again enabled me to turn the notes for a seminar into a book.

5.3. Review by: S G.
The American Mathematical Monthly 84 (1) (1977), 74.

A "self-contained" introduction to the study of modules possessing free resolutions of finite length. Includes basic material on matrices, localisation, free modules as well as discussion of fitting to MacRae invariants, stability, latent non-zerodivisors, theory of grade, multiplicative structures. Includes some exercises with solutions.

5.4. Review by: Melvin Hochster.
Mathematical Reviews MR0460383 (57 #377).

This well-written book explores the theory of finite free resolutions assuming a very minimal algebra background from its readers. Moreover, there are problems dispersed throughout the text which are completely solved at the ends of the chapters. This attractive feature makes the book quite suitable for those who are studying the subject on their own. The book begins by treating some basic facts about matrices and determinants, about modules and localisation. Then, bypassing the usual homological methods as well as the machinery of exterior algebra, the author develops both the basic (Auslander-Buchsbaum) theory of finite free resolutions and the more recent Buchsbaum-Eisenbud theory. The former includes the fact that if $E$ is a finitely generated module over a Noetherian local ring $(R, m)$ then pd$_{R}E$ + grade$_{m}E$ = grade$_{m}R$. (Here, pd denotes projective dimension, while grade$_{I}M$ denotes the length of any maximal sequence $x_{1}, ... , x_{n} \in I$ such that for each $i, 1 ≤ i ≤ n, x_{i}$ is not a zero divisor on $M/\Sigma _{j<i} x_{j}M$.)

The Buchsbaum-Eisenbud theory gives down-to-earth and quite useful criteria, derived from the Peskine-Szpiro acyclicity lemma, for complexes to be acyclic, as well as some rather interesting results on factoring exterior powers of matrices occurring in finite free resolutions which generalise the Hilbert-Burch theorem, and yield yet another proof of unique factorisation in regular local rings.

There is also a treatment of Fitting invariants and MacRae's invariant. The author makes use of a modified notion of grade in the case where the ring is not necessarily Noetherian which enables him to obtain the main results without Noetherian restrictions on the ring. (When $R$ is not Noetherian, let $x_{1}, ... , x_{n}$ be indeterminates, let $R_{n} = R[x_{1}, ... , x_{n}]$, and use sup grade$_{I}R_{n}E \otimes _{\mathbb{R}} R_{n}$ instead of grade$_{I}E$. This agrees with grade$_{I}E$ in the Noetherian case.) It is this notion, together with the very elementary form of the treatment, which permits the generalisation.
5.5. Review by: Melvin Hochster.
Bull. Amer. Math. Soc. 84 (4) (1978), 652-656.

This book gives a beautifully self-contained treatment of the recent Buchsbaum-Eisenbud theory of finite free resolutions over a commutative ring with identity, as well as of a number of related topics (e.g. MacCrae's invariant). There are two features in which the author's treatment differs from existing accounts of the subject: first, he confines himself almost entirely to elementary methods, avoiding Ext, Tor, and even exterior powers (we shall do likewise), and, second, he exploits a new notion of grade (or depth) in the non-Noetherian case which permits him to dispense entirely with the Noetherian restrictions on the ring. The very elementary form of the treatment enables the author to make accessible some fancy results from the homological theory of rings to readers with virtually no background in algebra.
...
The book is extremely well written and there is a large set of Exercises dispersed through the text which are completely solved at the ends of the various chapters.

This book should be of great value to anyone with little background in algebra who wishes to plunge directly into the homological theory of modules over commutative rings.

6.1. From the Publisher.

In these notes, first published in 1980, Professor Northcott provides a self-contained introduction to the theory of affine algebraic groups for mathematicians with a basic knowledge of communicative algebra and field theory. The book divides into two parts. The first four chapters contain all the geometry needed for the second half of the book which deals with affine groups. Alternatively the first part provides a sure introduction to the foundations of algebraic geometry. Any affine group has an associated Lie algebra. In the last two chapters, the author studies these algebras and shows how, in certain important cases, their properties can be transferred back to the groups from which they arose. These notes provide a clear and carefully written introduction to algebraic geometry and algebraic groups.

6.2. From the Introduction.

The topics treated in the following pages were largely covered in two seminars, both given at Sheffield University, one during the session 1976/7 and the other during 1978/9. I had noted sometime earlier that M Hochster and J A Eagon had established a connection between Determinantal Ideals and Invariant Theory. However in order to understand what was involved I had first to acquaint myself with the relevant parts of the theory of Algebraic Groups. With this in mind, I began to read J Fogarty's book on Invariant Theory.

Almost at once my interest broadened. It had been my experience to see Commutative Algebra develop out of attempts to provide classical Algebraic Geometry with proper foundations, but it had been a matter of regret that the algebraic machinery created for this purpose tended to conceal the origins of the subject. Fogarty's book helped me to see how one could look at Geometry from a readily accessible modern standpoint that was still not too far removed from the kind of Coordinate Geometry which now belongs to the classical period of the subject.

When my own ideas had reached a sufficiently advanced stage I decided to try and develop them further by committing myself to giving a seminar. With a subject such as this, and in circumstances where the time available was very limited, it was necessary to assume a certain amount of background knowledge. Indeed most accounts of aspects of the theory of Algebraic Groups assume a very great deal in the way of prerequisites. In my case the audience could be assumed to be knowledgeable about Commutative Noetherian rings and I planned the lectures with this in mind. The outcome is that the following treatment is very nearly self-contained if one presupposes a knowledge of field theory, tensor products and the more familiar parts of Commutative Algebra including, of course, the famous Basis Theorem and Zeros Theorem of Hilbert. It is true that there are a few places where additional background knowledge is required, but the reader who knows the topics mentioned above will find that other results are rarely used, and that where they are it will usually require little time and effort to fill in the gaps. To help him I have provided suitable references wherever they are likely to be needed.

The book falls naturally into two parts with Chapters 1-4 forming the first part. Here the aim is to show how those loci in classical analytical geometry that are defined by the solutions of simultaneous algebraic equations (together with the appropriate transformations of one such object into another) can be turned into a category. In this context the ambient affine space which makes geometrical thinking possible has to be removed from the picture, but not so far that it cannot be brought back readily when geometrical insight into a situation is needed.

Turning now to points of detail, Chapter 1 is used to explain certain matters that have to do with terminology and notation. It is also used to give a brief survey of the properties of tensor products over a field. The latter enables the discussion of (i) products of affine algebraic sets, and (ii) the consequences of enlarging the ground field, to take place later without an interruption to explain technicalities of a purely algebraic nature

The development of geometrical ideas begins with Chapter 2. Beside describing affine sets, their products and their morphisms, I have also revived the theory of specialisations because, it seems to me, it provides techniques that are both interesting and highly effective. Chapter 3 introduces the concept of the irreducible components of an affine set, the idea of dimension, and the very important topic of almost surjective morphisms. The last of the chapters devoted to Geometry deals with the subject of tangent spaces and simple points.

Part 1 contains all the Geometry that is needed for the reader to be able to understand the rest of the book. But although it was planned with the idea of supporting an introduction to Algebraic Groups it is the author's hope that it will be of interest to those whose main concern is to get an insight into the foundations of Algebraic Geometry. If the first four chapters are read with this more limited end in view, then I would suggest ending with section (4. 5) because the last section of the chapter consists of technical material used later to study the connection between an Affine Group and its associated Lie Algebra. A natural continuation of the geometrical sections would be the theory of tangent bundles.

The second part moves fairly quickly into the study of Affine Groups. These are groups which also have an affine structure and where the group operations are compatible with this structure. After the definitions and a discussion of the relation between connectedness and irreducibility, such topics as rational representations, linearly reductive groups, and the beginnings of invariant theory are considered. Chapter 5 ends with a comparatively elementary proof that every factor group of an affine group, with respect to a closed normal subgroup, has itself a natural structure as an affine group. This is one place where the account draws more heavily on the reader's knowledge of Commutative Algebra than it does elsewhere, but the topic seemed of sufficient importance to justify a departure from the guide lines which I had set myself for the book as a whole. It may help the reader to know that Chapter 5 makes very little use of Chapter 4. If therefore he wishes to get to the second part as quickly as possible, he may prefer to begin Chapter 5 after completing Chapter 3, and then to return to Chapter 4 to fill in certain gaps at a later stage. This in fact is what I did in my seminars.

The theory of Lie Algebras is not introduced until Chapter 6. Here it is shown that with each Affine Group there is associated a Lie Algebra and a detailed study is made of some of the most important examples. It is possible to exploit this connection very effectively in the case of a ground field of characteristic zero. The final chapter provides the theory which in this case enables properties of the associated Lie Algebras to be transferred back to the Affine Groups. It is here that the account stops. To continue further it would be necessary to include a short course on Lie Algebras after which an account of classical affine groups could be given.

I have, of course, made use of the writings of other mathematicians. I have already mentioned the special debt I owe to J Fogarty's book on Invariant Theory. I also made considerable use of C Chevalley's first account of Algebraic Groups, and the presentation, by H Bass, of a course given by A Borel on Linear Algebraic Groups. In the case of Geometry, it was a pleasure to re-read part of A Weil's book on the Foundations of Algebraic Geometry. So far as Commutative Algebra is concerned I have relied principally on the books of O Zariski, P Samuel M Nagata, and my own writings. Detailed references will be found in the text and in the booklist given at the end.

On a more personal level I would like to thank those who came to my seminars and encouraged me by the interest they showed, and once again it is a pleasure to thank Mrs E Benson for doing all the typing with such excellent results.

Finally I would like to express my appreciation to the London Mathematical Society and the Syndics of the Cambridge University Press for agreeing to let this appear in their series of Lecture Notes.

D G Northcott
Sheffield
February 1979

6.3. Review by: T Kambayashi.
Mathematical Reviews MR0569353 (82c:14002).

The author, apparently inspired by J Fogarty's book on invariant theory [1969], wanted to develop a theory of affine algebraic sets and groups sufficiently modern so as to be applicable to invariant theory and yet not too far removed from the classical coordinate geometry. In practice, this means abandonment of algebraically closed ground fields and groups of matrices with entries from such a field, but at the same time not going so far as group schemes and group functors.

Chapters 1 through 4 constitute Part I: Affine sets. The ground field K being arbitrary, a K-algebra R is called an "affine K-algebra" if R is finitely generated over K and if the intersection of all maximal ideals M with$R/M \simeq K$ is reduced to (0). Then the author's "affine set" is essentially the set of maximal ideals of this kind (i.e., $K$-rational points) in an affine $K$-algebra. This setup entails some awkwardness, as for example a polynomial ring over $K$ is not an affine $K$-algebra if $K$ is finite. By and large, though, these definitions are adequate for the stated purpose of the book, and the usual functoriality is preserved: the dual correspondence of affine sets and affine algebras, freedom of base field extensions, a product of affine sets is an affine set, and so on.

The notions of irreducibility, generic point, dimension, dominant (= "almost surjective" in the book) morphism, derivation and tangent space are introduced, and Part I covers bare rudiments about these. There are hardly any examples in this part.

Part II is Chapters 5, 6, and 7, entitled "Affine groups", "The associated Lie algebra", and "Power series and exponentials", respectively. In Chapter 5, after the standard definitions and examples the author discusses rational $G$-modules and linearly reductive groups. (The tori are the only available examples of linearly reductive groups in this book, though.) Some well-known fundamental theorems are proven there: If a linearly reductive affine group $G$ acts on a finitely generated $K$ algebra $A$ then the invariant subring $R/M \simeq K$ is also finitely generated over $K$ (Theorem 42, p. 195); if such a $G$ acts on an affine set $V$ and all orbits are closed, then a strict quotient $V/G$ exists, provided $K$ is algebraically closed (Theorem 43, p. 200). The last portion of Chapter 5 is devoted to proving the existence of the quotient affine group $G/H$ for any affine group $G$ and its closed normal subgroup $H$, again assuming $K$ to be algebraically closed (Theorem 48, p. 220). Clearly this chapter is the high point of the book.

In Chapters 6 and 7 some very basic parts of Lie theory for affine groups are explained over a characteristic zero ground field $K$. The discussion of exponential mappings in Chapter 7 leads up to the theorem establishing a correspondence between Lie subalgebras and closed connected subgroups (Theorem 8, p. 279). The text ends just before introducing semisimple groups and proving their linear reductivity in characteristic 0.

The book is written in a pleasant, leisurely style, and would serve well as a first introduction to the theory of algebraic groups for graduate students.

6.4. Review by: Mircea Becheanu.
Bulletin mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie, Nouvelle Série 31 (79) (2) (1987), 192.

Professor D G Northcott is also well known for his nice and clear texts in mathematics. The book under review is a good example in this sense. It is based on the seminars given by the author at Sheffield University as an introduction to the theory of algebraic groups. The book is divided in two parts. The first part contains an elementary introduction in algebraic geometry. Here are defined and studied the fundamental concepts as: affine sets and affine variety, morphisms of affine varieties, products of affine sets, tangent space. The second part introduces in the theory of affine algebraic groups and the Lie algebra of an affine group. We recommend this book as a very useful tool for the students and all those are interested in this domain.

6.5. Review by: D W Sharpe.
The Mathematical Gazette 64 (430) (1980), 300-301.

Your reviewer was fortunate to be present at a series (or sequence?) of lectures in the University of Sheffield entitled 'Affine sets and affine groups', given by D G Northcott. The volume of lecture notes at present under review is a result of those lectures. Those who are acquainted with Professor Northcott's previous writings will know that he has the ability to present sophisticated mathematical ideas in an intelligible way, a gift not given to all authors! Many professional mathematicians can testify that their first acquaintance with Commutative or Homological Algebra was through the writings of D G Northcott, and will point to well-thumbed copies of his works on their shelves. The same may be said in the future of the foundations of Algebraic Geometry as a result of these notes.

Those who will benefit from this volume will be postgraduate students and research workers in Pure Mathematics, and those who wish to learn the subject from scratch, having a good working knowledge of the basics of the theory of commutative rings, an essential prerequisite. The publishers are perhaps somewhat optimistic in suggesting on the back cover that the volume is also suitable for 'senior undergraduates'; they would have to be very senior, intellectually at any rate! As such, a detailed description of the contents of these notes is probably out of place here.

But the origins of this fascinating subject will be familiar to all of us from our days spent grappling with Coordinate Geometry. For example, we considered the equation $x^{2} + y^{2} = r^{2}$ and found all points in the plane which satisfied it; low and behold, a circle! Now we are in business. Why not consider an arbitrary collection of algebraic equations in $n$ variables and look at the points in $n$-dimensional Euclidean space which satisfy them? This is our geometrical object, which we may call a locus or variety. (Mathematicians never use one term when they can use two!) We can introduce a topology on our Euclidean space by designating these loci as the closed sets, and the sky is the limit.

This is the background of the modern subject of Algebraic Geometry as originated by such a master as David Hilbert around the turn of the century. Mind you, it has come a long way since then, so much so that it has sometimes lost sight of its origins, which is a pity. To quote Professor Northcott (p 35): 'During the twentieth century, Geometry has been the victim of a takeover bid by Algebra'. If you want some idea of how far the subject has developed, then Professor Northcott's book is a very good guide, which will equip you to attempt to discover what research is going on today in Algebraic Geometry. But be warned; you will need a clear head for that!

7.1. From the Publisher.

Multilinear algebra has important applications in many different areas of mathematics but is usually learned in a rather haphazard fashion. The aim of this book is to provide a readable and systematic account of multilinear algebra at a level suitable for graduate students. Professor Northcott gives a thorough treatment of topics such as tensor, exterior, Grassmann, Hopf and co-algebras and ends each chapter with a section entitled 'Comments and Exercises'. The comments contain convenient summaries and discussion of the content whilst the exercises provide an opportunity to test understanding and add extra material. Complete solutions are provided for those exercises that are particularly important or used later in the book. The volume as a whole is based on advanced lectures given by the author at the University of Sheffield.

7.2. From the Preface.

This account of Multilinear Algebra has developed out of lectures which I gave at the University of Sheffield during the session 1981/2. In its present form it is designed for advanced undergraduates and those about to commence postgraduate studies. At this general level the only special prerequisite for reading the whole book is a familiarity with the notion of a module (over a commutative ring) and with such concepts as submodule, factor module and homomorphism.

Multilinear Algebra arises out of Linear Algebra and like its antecedent is a subject which has applications in a great many different fields. Indeed, there are so many reasons why mathematicians may need some knowledge of its concepts and results that any selection of applications is likely to disappoint as many readers as it satisfies. Furthermore, such a selection tends to upset the balance of the subject as well as adding substantially to the required background knowledge. It is my impression that young mathematicians often acquire their knowledge of Multilinear Algebra in a rather haphazard and fragmentary fashion. Here I have attempted to weld the most commonly used fragments together and to fill out the result so as to obtain a theory with an easily recognisable structure.

The book begins with the study of multilinear mappings and the tensor, exterior and symmetric powers of a module. Next, the tensor powers are fitted together to produce the tensor algebra of a module, and a similar procedure yields the exterior and symmetric algebras. Multilinear mappings and the three algebras just mentioned form the most widely used parts of the subject and, in this account, occupy the first six chapters. However, at this point we are at the threshold of a richer theory, and it is Chapter 7 that provides the climax of the book.

Chapter 7 starts with the observation that if we re-define algebras in terms of certain commutative diagrams, then we are led to a dual concept known as a coalgebra. Now it sometimes happens that, on the same underlying set, there exist simultaneously both an algebra-structure and a coalgebra-structure. When this happens, and provided that the two structures interact suitably, the result is called a Hopf algebra. It turns out that exterior and symmetric algebras are better regarded as Hopf algebras.

This approach confers further benefits. By considering linear forms on a coalgebra it is always possible to construct an associated algebra; and, since exterior and symmetric algebras have a coalgebra-structure, this construction may be applied to them. The result in the first case is the algebra of differential forms (the Grassmann algebra) and in the second case it is the algebra of differential operators.

The final chapter deals with graded duality. From every graded module we can construct another graded module known as its graded dual. If the components of the original graded module are free and of finite rank, then this process, when applied twice, yields a double dual that is a copy of the graded module with which we started. For similarly restricted graded algebras, coalgebras and Hopf algebras this technique gives rise to a full duality; algebras become coalgebras and vice versa; and Hopf algebras continue to be Hopf algebras.

Each chapter has, towards its end, a section with the title 'Comments and exercises'. The comments serve to amplify the main theory and to draw attention to points that require special attention; the exercises give the reader an opportunity to test his or her understanding of the text and a chance to become acquainted with additional results. Some exercises are marked with an asterisk. Usually these exercises are selected on the grounds of being particularly interesting or more than averagely difficult; sometimes they contain results that are used later. Where an asterisk is attached to an exercise a solution is provided in the following section. However, to prevent gaps occurring in the argument, a result contained in an exercise is not used later unless a solution has been supplied.

Once the guide-lines for the book had been settled, I found that the subject unfolded very much under its own momentum. Where I had to consult other sources, I found C Chevalley's Fundamental Concepts of Algebra, even though it was written more than a quarter of a century ago, especially helpful. In particular, the account given here of Pfaffians follows closely that given by Chevalley.

Finally I wish to record my thanks to Mrs E Benson and Mrs J Williams of the Department of Pure Mathematics at Sheffield University. Between them they typed the whole book; and their cheerful co-operation enabled the exacting task of preparing it for the printers to proceed smoothly and without a hitch.

7.3. Review by: W V Vasconcelos.
Mathematical Reviews MR0773853 (86m:13001).

This book is a grand tour of the main objects of multilinear algebra, conducted in a spare, yet rich, fashion. It focuses on the construction of the tensor, exterior and symmetric algebras of a module over a commutative ring and, by bringing out some of their relationships, develops the theory of several associated structures.

The exposition - divided into eight short chapters - from its start at multilinear mappings elegantly constructs tensor products and establishes its general properties. Having the tensor algebra of a module as its focal point, it systematically studies the exterior algebra and the symmetric algebras of modules. A major effort is put into exhibiting the wealth of structure carried by these algebras. We are treated to derivations and algebras of differential operators, Pfaffians and Grassmann algebras. The multiplicity of intertwining structures in the same algebra, so beautifully displayed by Hopf algebras, is further enhanced by the examination of the duality in the basic algebras. Here even Koszul complexes and divided power algebras make anonymous appearances.

Several other points are worthy of emphasis. The economy of the categorical constructions - done with a minimum of jargon - and the fact that linear algebra over commutative rings is fun and carries no extra cost, are seen throughout the text. An appealing feature in each chapter is a section of comments and exercises - many of which are solved.

Reading this book requires only linear algebra and a minor exposure to rings and modules. Although its stated audience is advanced undergraduates and beginning graduate students, researchers will find this a valuable reference source. It remarkably encompasses some of the most fruitful objects of algebra and their manifold connections.

7.4. Review by: J F Humphreys.
Bulletin of the London Mathematical Society 17 (3) (1985), 283.

Starting from basic facts about modules over commutative rings with identity, the author gives an account of multilinear algebra which is completely self-contained; indeed, after the Preface, no reference is made to any book or published paper. The first part of the book is of an introductory nature and considers tensor powers, exterior powers and symmetric powers in Chapter 1; properties of tensor products in Chapter 2 and results on associative algebras in Chapter 3. The next three chapters deal in turn with the tensor algebra $T(M)$ of a module $M$, the exterior algebra $E(M)$ and the symmetric algebra $S(M)$. The concepts of Hopf algebras and modified Hopf algebras are introduced in Chapter 7 where it is shown that, for any module $M, S(M)$ is a Hopf algebra and $E(M)$ is a modified Hopf algebra. Finally, in Chapter 8, a condition for the graded dual of a graded algebra to be a graded coalgebra is discussed. The book is written in the clear style which will be familiar to readers of the author's other books. Each chapter contains exercises with complete solutions being provided to many of these. This feature together with the straightforward, concrete approach should be particularly helpful for readers meeting these ideas for the first time.