## Zariski and Samuel: *Commutative Algebra*

In 1958

Professor of Mathematics, Harvard University

and

Professor of Mathematics, University of Clermont-Ferrand

with the cooperation of I S Cohen

We give below a version of the Preface to the first volume of Zariski and Samuel's book

Le juge: Accusé, vous tâcherez d'être bref.

L'accusi: Je tacherai d'être clair.

G COURTELINE

This book is the child of an unborn parent. Some years ago the senior author began the preparation of a Colloquium volume on algebraic geometry, and he was then faced with the difficult task of incorporating in that volume the vast amount of purely algebraic material which is needed in abstract algebraic geometry. The original plan was to insert, from time to time, algebraic digressions in which concepts and results from commutative algebra were to be developed in full as and when they were needed. However, it soon became apparent that such a parenthetical treatment of the purely algebraic topics, covering a wide range of commutative algebra, would impose artificial bounds on the manner, depth, and degree of generality with which these topics could be treated. As is well known, abstract algebraic geometry has been recently not only the main field of applications of commutative algebra but also the principal incentive of new research in commutative algebra. To approach the underlying algebra only in a strictly utilitarian, auxiliary, and parenthetical manner, to stop short of going further afield where the applications of algebra to algebraic geometry stop and the general algebraic theories inspired by geometry begin, impressed us increasingly as being a program scientifically too narrow and psychologically frustrating, not to mention the distracting effect that repeated algebraic digressions would inevitably have had on the reader, vis-à-vis the central algebro-geometric theme. Thus the idea of a separate book on commutative algebra was born, and the present book - of which this is the first of two volumes is a realization of this idea, come to fruition at a time when its parent - a treatise on abstract algebraic geometry-has still to see the light of the day.

In the last twenty years commutative algebra has undergone an intensive development. However, to the best of our knowledge, no systematic account of this subject has been published in book form since the appearance in 1935 of the valuable

The algebro-geometric origin and motivation of the book will become more evident in the second volume (which will deal with valuation theory, polynomial and power series rings, and local algebra; more will be said of that volume in its preface) than they are in this first volume. Here we develop the elements of commutative algebra which we deem to be of general and basic character. In chapter I we develop the introductory notions concerning groups, rings, fields, polynomial rings, and vector spaces. All this, except perhaps a somewhat detailed discussion of quotient rings with respect to multiplicative systems, is material which is usually given in an intermediate algebra course and is often briefly reviewed in the beginning of an advanced graduate course. The exposition of field theory given in chapter II is fairly complete and follows essentially the lines of standard modern accounts of the subject. However, as could be expected from algebraic geometers, we also stress treatment of transcendental extensions, especially of the notions of separability and linear disjointness (the latter being due to A Weil). The study of maximally algebraic subfields and regular extensions has been postponed, however, to Volume II (chapter VII), since that study is so closely related to the question of ground field extension in polynomial rings.

Chapter III contains classical material about ideals and modules in arbitrary commutative rings. Direct sum decompositions are studied in detail. The last two sections deal respectively with tensor products of rings and free joins of integral domains. Here we introduce the notion of quasi-linear disjointness, and prove some results about free joins of integral domains which we could not readily locate in the literature.

With chapter IV, devoted to noetherian rings, we enter commutative algebra proper. After a preliminary section on the Hilbert basis theorem and a side trip to the rings satisfying the descending chain condition, the first part of the chapter is devoted mostly to the notion of a primary representation of an ideal and to applications of that notion. We then give a detailed study of quotient rings (as generalized by Chevalley and Uzkov). The end of the chapter contains miscellaneous complements, the most important of which is Krull's theory of prime ideal chains in noetherian rings. An appendix generalizes some properties of the primary representation to the case of noetherian modules.

Chapter V begins with a study of integral dependence (a subject which is nowadays an essential prerequisite for almost everything in commutative algebra) and includes the so-called "going-up" and "going-down" theorems of Cohen-Seidenberg and the normalization theorem. (Other variations of that theorem will be found in Volume II, in the chapter on polynomial and power series rings.) With Matusita we then define a Dedekind domain as an integral domain in which every ideal is a product of prime ideals and derive from that definition the usual characterization of Dedekind domains and their properties. An important place is given to the study of finite algebraic field extensions of the quotient field of a Dedekind domain, and the degree formula ∑

The preparation of the first volume of this book began as a collaboration between the senior author and our former pupil and friend, the late Irving S Cohen. We extend a grateful thought to the memory of this gifted young mathematician.

We wish to acknowledge many improvements in this book which are due to John Tate and Jean-Pierre Serre. We also wish to thank heartily Mr T Knapp who has carefully read the manuscript and the galley proofs and whose constructive criticisms have been most helpful.

Thanks are also due to the Harvard Foundation for Advanced Research whose grant to the senior author was used for typing part of the manuscript. Last but not least, we wish to extend our thanks to the D Van Nostrand Company for having generously cooperated with our wishes in the course of the printing of the book.

OSCAR ZARISKI

PIERRE SAMUEL

Cambridge, Massachusetts

Chamalières, France

This second volume of our treatise on commutative algebra deals largely with three basic topics, which go beyond the more or less classical material of volume I and are on the whole of a more advanced nature and a more recent vintage. These topics are: (a) valuation theory; (b) theory of polynomial and power series rings (including generalizations to graded rings and modules); (c) local algebra. Because most of these topics have either their source or their best motivation in algebraic geometry, the algebro-geometric connections and applications of the purely algebraic material are constantly stressed and abundantly scattered throughout the exposition. Thus, this volume can be used in part as an introduction to some basic concepts and the arithmetic foundations of algebraic geometry. The reader who is not immediately concerned with geometric applications may omit the algebro-geometric material in a first reading, but it is only fair to say that many a reader will find it more instructive to find out immediately what is the geometric motivation behind the purely algebraic material of this volume.

The first 8 sections of Chapter VI (including § 5bis) deal directly with properties of places, rather than with those of the valuation associated with a place. These, therefore, are properties of valuations in which the value group of the valuation is not involved. The very concept of a valuation is only introduced for the first time in § 8, and, from that point on, the more subtle properties of valuations which are related to the value group come to the fore. These are illustrated by numerous examples, taken largely from the theory of algebraic function fields (§§ 14, 15). The last two sections of the chapter contain a general treatment, within the framework of arbitrary commutative integral domains, of two concepts which are of considerable importance in algebraic geometry (the Riemann surface of a field and the notions of normal and derived normal models).

The greater part of Chapter VII is devoted to classical properties of polynomial and power series rings (e.g., dimension theory) and their applications to algebraic geometry. This chapter also includes a treatment of graded rings and modules and such topics as characteristic (Hilbert) functions and chains of syzygies. In the past, these last two topics represented some final words of the algebraic theory, to be followed only by deeper geometric applications. With the modern development of homological methods in commutative algebra, these topics became starting points of extensive, purely algebraic theories, having a much wider range of applications. We could not include, without completely disrupting the balance of this volume, the results which require the use of truly homological methods (e.g., torsion and extension functors, complexes, spectral sequences). However, we have tried to include the results which may be proved by methods which, although inspired by homological algebra, are nevertheless classical in nature. The reader will find these results in Chapter VII, §§ 12 and 13, and in Appendices 6 and 7. No previous knowledge of homological algebra is needed for reading these parts of the volume. The reader who wants to see how truly homological methods may be applied to commutative algebra is referred to the original papers of M Auslander, D Buchsbaum, A Grothendieck, D Rees, J-P Serre, etc., to a forthcoming book of D C Northcott, as well, of course, as to the basic treatise of Cartan-Eilenberg.

Chapter VIII deals with the theory of local rings. This theory provides the algebraic basis for the local study of algebraic and analytical varieties. The first six sections are rather elementary and deal with more general rings than local rings. Deeper results are presented in the rest of the chapter, but we have not attempted to give an encyclopaedic account of the subject.

While much of the material appears here for the first time in book form, there is also a good deal of material which is new and represents current or unpublished research. The appendices treat special topics of current interest (the first 5 were written by the senior author; the last two by the junior author), except that Appendix 6 gives a smooth treatment of two important theorems proved in the text. Appendices 4 and 5 are of particular interest from an algebro-geometric point of view.

We have not attempted to trace the origin of the various proofs in this volume. Some of these proofs, especially in the appendices, are new. Others are transcriptions or arrangements of proofs taken from original papers.

We wish to acknowledge the assistance which we have received from M Hironaka, T Knapp, S Shatz, and M Schlesinger in the work of checking parts of the manuscript and of reading the galley proofs. Many improvements have resulted from their assistance.

The work on Appendix 5 was supported by a Research project at Harvard University sponsored by the Air Force Office of Scientific Research.

OSCAR ZARISKI

PIERRE SAMUEL

Cambridge, Massachusetts

Chamalières, France

**Zariski**and**Samuel**published the first volume their classic two volume text*Commutative Algebra.*It was recognised immediately for its importance. A reviewer of the first volume wrote:-The information contained on the title page of the first volume was as follows:The reader of "Commutative algebra" will receive a presentation of much of the research in this area over the last twenty years, a good deal of which was inspired by Krull's classic work. In addition, he will receive a leisurely and thorough exposition of the subject matter, suitable not only for the expert, but for the student as well.

### Commutative Algebra

by**Oscar Zariski**

Professor of Mathematics, Harvard University

and

**Pierre Samuel**

Professor of Mathematics, University of Clermont-Ferrand

with the cooperation of I S Cohen

**D VAN NOSTRAND COMPANY, INC.**

**PRINCETON, NEW JERSEY**

**TORONTO, NEW YORK, LONDON**

We give below a version of the Preface to the first volume of Zariski and Samuel's book

*Commutative algebra*which explains the background to the book as well as giving an indication of its contents:

**PREFACE**

Le juge: Accusé, vous tâcherez d'être bref.

L'accusi: Je tacherai d'être clair.

G COURTELINE

This book is the child of an unborn parent. Some years ago the senior author began the preparation of a Colloquium volume on algebraic geometry, and he was then faced with the difficult task of incorporating in that volume the vast amount of purely algebraic material which is needed in abstract algebraic geometry. The original plan was to insert, from time to time, algebraic digressions in which concepts and results from commutative algebra were to be developed in full as and when they were needed. However, it soon became apparent that such a parenthetical treatment of the purely algebraic topics, covering a wide range of commutative algebra, would impose artificial bounds on the manner, depth, and degree of generality with which these topics could be treated. As is well known, abstract algebraic geometry has been recently not only the main field of applications of commutative algebra but also the principal incentive of new research in commutative algebra. To approach the underlying algebra only in a strictly utilitarian, auxiliary, and parenthetical manner, to stop short of going further afield where the applications of algebra to algebraic geometry stop and the general algebraic theories inspired by geometry begin, impressed us increasingly as being a program scientifically too narrow and psychologically frustrating, not to mention the distracting effect that repeated algebraic digressions would inevitably have had on the reader, vis-à-vis the central algebro-geometric theme. Thus the idea of a separate book on commutative algebra was born, and the present book - of which this is the first of two volumes is a realization of this idea, come to fruition at a time when its parent - a treatise on abstract algebraic geometry-has still to see the light of the day.

In the last twenty years commutative algebra has undergone an intensive development. However, to the best of our knowledge, no systematic account of this subject has been published in book form since the appearance in 1935 of the valuable

*Ergebnisse*monograph "Idealtheorie" of W Krull. As to that monograph, it has exercised a great influence on research in the intervening years, but the condensed and sketchy character of the exposition (which was due to limitation of space in the*Ergebnisse*monographs) made it more valuable to the expert than to, the student wishing to study the subject. In the present book we endeavour to give a systematic and - we may even say - leisurely account of commutative algebra, including some of the more recent developments in this field, without pretending, however, to give an encyclopaedic account of the subject matter. We have preferred to write a self-contained book which could be used in a basic graduate course of modern algebra. It is also with an eye to the student that we have tried to give full and detailed explanations in the proofs, and we feel that we owe no apology to the mature mathematician, who can skip the details that are not necessary for him. We have even found that the policy of trading empty space for clarity and explicitness of the proofs has saved us, the authors, from a number of erroneous conclusions at the more advanced stages of the book. We have also tried, this time with an eye to both the student and the mature mathematician, to give a many-sided treatment of our topics, not hesitating to offer several proofs of one and the same result when we thought that something might be learned, as to methods, from each of the proofs.The algebro-geometric origin and motivation of the book will become more evident in the second volume (which will deal with valuation theory, polynomial and power series rings, and local algebra; more will be said of that volume in its preface) than they are in this first volume. Here we develop the elements of commutative algebra which we deem to be of general and basic character. In chapter I we develop the introductory notions concerning groups, rings, fields, polynomial rings, and vector spaces. All this, except perhaps a somewhat detailed discussion of quotient rings with respect to multiplicative systems, is material which is usually given in an intermediate algebra course and is often briefly reviewed in the beginning of an advanced graduate course. The exposition of field theory given in chapter II is fairly complete and follows essentially the lines of standard modern accounts of the subject. However, as could be expected from algebraic geometers, we also stress treatment of transcendental extensions, especially of the notions of separability and linear disjointness (the latter being due to A Weil). The study of maximally algebraic subfields and regular extensions has been postponed, however, to Volume II (chapter VII), since that study is so closely related to the question of ground field extension in polynomial rings.

Chapter III contains classical material about ideals and modules in arbitrary commutative rings. Direct sum decompositions are studied in detail. The last two sections deal respectively with tensor products of rings and free joins of integral domains. Here we introduce the notion of quasi-linear disjointness, and prove some results about free joins of integral domains which we could not readily locate in the literature.

With chapter IV, devoted to noetherian rings, we enter commutative algebra proper. After a preliminary section on the Hilbert basis theorem and a side trip to the rings satisfying the descending chain condition, the first part of the chapter is devoted mostly to the notion of a primary representation of an ideal and to applications of that notion. We then give a detailed study of quotient rings (as generalized by Chevalley and Uzkov). The end of the chapter contains miscellaneous complements, the most important of which is Krull's theory of prime ideal chains in noetherian rings. An appendix generalizes some properties of the primary representation to the case of noetherian modules.

Chapter V begins with a study of integral dependence (a subject which is nowadays an essential prerequisite for almost everything in commutative algebra) and includes the so-called "going-up" and "going-down" theorems of Cohen-Seidenberg and the normalization theorem. (Other variations of that theorem will be found in Volume II, in the chapter on polynomial and power series rings.) With Matusita we then define a Dedekind domain as an integral domain in which every ideal is a product of prime ideals and derive from that definition the usual characterization of Dedekind domains and their properties. An important place is given to the study of finite algebraic field extensions of the quotient field of a Dedekind domain, and the degree formula ∑

*e*_{j}*f*_{j}=*n*is derived under the usual (and necessary) finiteness assumptions concerning the integral closure of the given Dedekind domain in the extension field. This study finds its natural refinement in the Hilbert ramification theory (sections 9 and 10) and in the properties of the different and discriminant (section 11). The chapter closes with some classical number-theoretic applications and a generalization of the theorem of Kummer. The properties of Dedekind domains give us a natural opportunity of introducing the notion of a valuation (at least in the discrete case) but the reader will observe that this notion is introduced by us quite casually and parenthetically, and that the language of valuations is not used in this chapter. We have done that deliberately, for we wished to emphasize the by now well-known fact that while ideals and valuations cover substantially the same ground in the classical case (which, from a geometric point of view, is the case of dimension 1), the domain in which valuations become really significant belongs to the theory of function fields of dimension greater than 1.The preparation of the first volume of this book began as a collaboration between the senior author and our former pupil and friend, the late Irving S Cohen. We extend a grateful thought to the memory of this gifted young mathematician.

We wish to acknowledge many improvements in this book which are due to John Tate and Jean-Pierre Serre. We also wish to thank heartily Mr T Knapp who has carefully read the manuscript and the galley proofs and whose constructive criticisms have been most helpful.

Thanks are also due to the Harvard Foundation for Advanced Research whose grant to the senior author was used for typing part of the manuscript. Last but not least, we wish to extend our thanks to the D Van Nostrand Company for having generously cooperated with our wishes in the course of the printing of the book.

OSCAR ZARISKI

PIERRE SAMUEL

Cambridge, Massachusetts

Chamalières, France

In 1960 Zariski and Samuel published the second volume their classic two volume text

*Commutative Algebra.*It was recognised immediately for its importance. A reviewer of the second volume wrote:-We give below a version of the Preface to the second volume of Zariski and Samuel's bookIt would be an under-statement to say that the second volume of this work lives up to the standards and expectations set by the first, because the scope and style of the final volume could not have been anticipated even though the first volume contained an outline of the authors' total program and ample evidence of their expository skill. Unlike the first, however, the second volume is concerned in large measure with those parts of commutative algebra that are the fruits of its union with algebraic geometry ...

*Commutative algebra*which gives an indication of its contents:**PREFACE**

This second volume of our treatise on commutative algebra deals largely with three basic topics, which go beyond the more or less classical material of volume I and are on the whole of a more advanced nature and a more recent vintage. These topics are: (a) valuation theory; (b) theory of polynomial and power series rings (including generalizations to graded rings and modules); (c) local algebra. Because most of these topics have either their source or their best motivation in algebraic geometry, the algebro-geometric connections and applications of the purely algebraic material are constantly stressed and abundantly scattered throughout the exposition. Thus, this volume can be used in part as an introduction to some basic concepts and the arithmetic foundations of algebraic geometry. The reader who is not immediately concerned with geometric applications may omit the algebro-geometric material in a first reading, but it is only fair to say that many a reader will find it more instructive to find out immediately what is the geometric motivation behind the purely algebraic material of this volume.

The first 8 sections of Chapter VI (including § 5bis) deal directly with properties of places, rather than with those of the valuation associated with a place. These, therefore, are properties of valuations in which the value group of the valuation is not involved. The very concept of a valuation is only introduced for the first time in § 8, and, from that point on, the more subtle properties of valuations which are related to the value group come to the fore. These are illustrated by numerous examples, taken largely from the theory of algebraic function fields (§§ 14, 15). The last two sections of the chapter contain a general treatment, within the framework of arbitrary commutative integral domains, of two concepts which are of considerable importance in algebraic geometry (the Riemann surface of a field and the notions of normal and derived normal models).

The greater part of Chapter VII is devoted to classical properties of polynomial and power series rings (e.g., dimension theory) and their applications to algebraic geometry. This chapter also includes a treatment of graded rings and modules and such topics as characteristic (Hilbert) functions and chains of syzygies. In the past, these last two topics represented some final words of the algebraic theory, to be followed only by deeper geometric applications. With the modern development of homological methods in commutative algebra, these topics became starting points of extensive, purely algebraic theories, having a much wider range of applications. We could not include, without completely disrupting the balance of this volume, the results which require the use of truly homological methods (e.g., torsion and extension functors, complexes, spectral sequences). However, we have tried to include the results which may be proved by methods which, although inspired by homological algebra, are nevertheless classical in nature. The reader will find these results in Chapter VII, §§ 12 and 13, and in Appendices 6 and 7. No previous knowledge of homological algebra is needed for reading these parts of the volume. The reader who wants to see how truly homological methods may be applied to commutative algebra is referred to the original papers of M Auslander, D Buchsbaum, A Grothendieck, D Rees, J-P Serre, etc., to a forthcoming book of D C Northcott, as well, of course, as to the basic treatise of Cartan-Eilenberg.

Chapter VIII deals with the theory of local rings. This theory provides the algebraic basis for the local study of algebraic and analytical varieties. The first six sections are rather elementary and deal with more general rings than local rings. Deeper results are presented in the rest of the chapter, but we have not attempted to give an encyclopaedic account of the subject.

While much of the material appears here for the first time in book form, there is also a good deal of material which is new and represents current or unpublished research. The appendices treat special topics of current interest (the first 5 were written by the senior author; the last two by the junior author), except that Appendix 6 gives a smooth treatment of two important theorems proved in the text. Appendices 4 and 5 are of particular interest from an algebro-geometric point of view.

We have not attempted to trace the origin of the various proofs in this volume. Some of these proofs, especially in the appendices, are new. Others are transcriptions or arrangements of proofs taken from original papers.

We wish to acknowledge the assistance which we have received from M Hironaka, T Knapp, S Shatz, and M Schlesinger in the work of checking parts of the manuscript and of reading the galley proofs. Many improvements have resulted from their assistance.

The work on Appendix 5 was supported by a Research project at Harvard University sponsored by the Air Force Office of Scientific Research.

OSCAR ZARISKI

PIERRE SAMUEL

Cambridge, Massachusetts

Chamalières, France