Algebraic geometry. 1. From algebraic varieties to schemes. Transl. from the Japanese by Goro Kato.

*(English)*Zbl 0937.14001
Iwanami Series in Modern Mathematics. Translations. Series 2. American Mathematical Society (AMS). 185. Providence, RI: American Mathematical Society (AMS). xix, 154 p. (1999).

The present book is the first of three volumes of an introductory text on the theory of algebraic schemes. These three volumes, the original Japanese versions of which have gained a widespread acceptance as a standard introduction to this subject in Japan, are mainly designed for non-specialists in the field of algebraic geometry, and are aimed to provide the interested reader with the basic ideas and techniques of Grothendieck’s theory of schemes, the most general, natural and rigorously elaborated form of algebraic geometry.

There is certainly no lack of good texts on algebraic geometry in its scheme-theoretic setting, but most of them are written for students and mathematicians who already have a fundamental, profound knowledge of (and experience in) both the classical theory of algebraic varieties and the allied commutative, homological, and categorical algebra. In contrast to this situation, the author of the present introduction to algebraic geometry has attempted to develop the subject in the modern scheme-theoretic language, basically so from the very beginning on, and to keep the prerequisites from algebra and classical algebraic geometry to an absolute minimum, just in order to provide a nearly self-contained introduction to scheme theory and its applications as a primer for the very beginner in the field, without leaving out the most crucial theorems and their complete proofs.

Writing such a textbook, with that ambitious methodological program as the basic framework, is a very delicate task, which the author seems to have surpassed in a brilliant manner. Already the English translation of the first volume, which is being reviewed here, indicates the outstanding character of the entire text. – The major part of this first volume is devoted to carefully preparing the concept of a scheme in the sense of Grothendieck. The first chapter of the book discusses the basics from the classical theory of algebraic varieties over an algebraically closed field, and that just to the extent that is needed to motivate and understand the concept of a scheme treated in the subsequent second chapter. More precisely, chapter 1 treats, in a brief but sufficiently detailed exposition, the following fundamental elementary topics: algebraic sets, Hilbert’s Nullstellensatz, affine algebraic varieties, multiplicities and local intersection multiplicities, projective varieties, and an outlook to the ideal-theoretic and geometric aspects of polynomial equations over an arbitrary ring of coefficients.

This last section leads over to chapter 2, entitled “Schemes”, which explains the prime spectrum of a commutative ring, together with its Zariski topology and sheaf-theoretic aspects, the algebraic principle of localization, the necessary toolkit from general sheaf theory, the concept of a general ringed space and, concludingly, arbitrary schemes, subschemes, and morphisms of schemes.

Chapter 3, the final chapter of this first volume, deals with the indispensible categorical interpretation of scheme theory. Along with a brief introduction to categories, functors, representable functors, and fibre products in categories, scheme-valued points are discussed, the existence of fibre products in the category of schemes is fully proved, and separated morphisms are analyzed.

This first volume ends with the last-mentioned topic, and the author points out in the preface that the subsequent volumes “Algebraic Geometry. 2” and “Algebraic Geometry. 3” will provide the reader with a unified understanding of algebraic geometry from the scheme-theoretic viewpoint, including sheaf cohomology and its applications to the theory of algebraic curves and surfaces as well as the complex-analytic aspects of algebraic geometry.

In this first volume under review, each chapter comes with a suggestive summary of the main results covered by it, and with a set of complementary exercises, the solutions of which are compiled at the end of the book. There is no bibliography, but perhaps there will be one at the end of the concluding third volume, as all three volumes are to be understood as an integrated treatise. Certainly, some hints to the existing standard texts for further or parallel reading would not have been displaced, but in regard of the author’s goal to provide the non-expert reader, and originally the Japanese reader, with a coherent, self-contained textbook on the subject, the missing bibliography is not a serious deficiency whatsoever. This masterly written text, now also available to the international mathematical audience, tells its own tale and represents a highly welcome addition to the great standard textbooks on algebraic geometry. The author’s hope that his book in three volumes could find the same benevolent acceptance by the international mathematical community as it did in Japan seems to be absolutely justified and realistic.

We may be looking forward to the English translation of the subsequent two volumes with great pleasure, students just as well as teachers in the field of algebraic geometry.

There is certainly no lack of good texts on algebraic geometry in its scheme-theoretic setting, but most of them are written for students and mathematicians who already have a fundamental, profound knowledge of (and experience in) both the classical theory of algebraic varieties and the allied commutative, homological, and categorical algebra. In contrast to this situation, the author of the present introduction to algebraic geometry has attempted to develop the subject in the modern scheme-theoretic language, basically so from the very beginning on, and to keep the prerequisites from algebra and classical algebraic geometry to an absolute minimum, just in order to provide a nearly self-contained introduction to scheme theory and its applications as a primer for the very beginner in the field, without leaving out the most crucial theorems and their complete proofs.

Writing such a textbook, with that ambitious methodological program as the basic framework, is a very delicate task, which the author seems to have surpassed in a brilliant manner. Already the English translation of the first volume, which is being reviewed here, indicates the outstanding character of the entire text. – The major part of this first volume is devoted to carefully preparing the concept of a scheme in the sense of Grothendieck. The first chapter of the book discusses the basics from the classical theory of algebraic varieties over an algebraically closed field, and that just to the extent that is needed to motivate and understand the concept of a scheme treated in the subsequent second chapter. More precisely, chapter 1 treats, in a brief but sufficiently detailed exposition, the following fundamental elementary topics: algebraic sets, Hilbert’s Nullstellensatz, affine algebraic varieties, multiplicities and local intersection multiplicities, projective varieties, and an outlook to the ideal-theoretic and geometric aspects of polynomial equations over an arbitrary ring of coefficients.

This last section leads over to chapter 2, entitled “Schemes”, which explains the prime spectrum of a commutative ring, together with its Zariski topology and sheaf-theoretic aspects, the algebraic principle of localization, the necessary toolkit from general sheaf theory, the concept of a general ringed space and, concludingly, arbitrary schemes, subschemes, and morphisms of schemes.

Chapter 3, the final chapter of this first volume, deals with the indispensible categorical interpretation of scheme theory. Along with a brief introduction to categories, functors, representable functors, and fibre products in categories, scheme-valued points are discussed, the existence of fibre products in the category of schemes is fully proved, and separated morphisms are analyzed.

This first volume ends with the last-mentioned topic, and the author points out in the preface that the subsequent volumes “Algebraic Geometry. 2” and “Algebraic Geometry. 3” will provide the reader with a unified understanding of algebraic geometry from the scheme-theoretic viewpoint, including sheaf cohomology and its applications to the theory of algebraic curves and surfaces as well as the complex-analytic aspects of algebraic geometry.

In this first volume under review, each chapter comes with a suggestive summary of the main results covered by it, and with a set of complementary exercises, the solutions of which are compiled at the end of the book. There is no bibliography, but perhaps there will be one at the end of the concluding third volume, as all three volumes are to be understood as an integrated treatise. Certainly, some hints to the existing standard texts for further or parallel reading would not have been displaced, but in regard of the author’s goal to provide the non-expert reader, and originally the Japanese reader, with a coherent, self-contained textbook on the subject, the missing bibliography is not a serious deficiency whatsoever. This masterly written text, now also available to the international mathematical audience, tells its own tale and represents a highly welcome addition to the great standard textbooks on algebraic geometry. The author’s hope that his book in three volumes could find the same benevolent acceptance by the international mathematical community as it did in Japan seems to be absolutely justified and realistic.

We may be looking forward to the English translation of the subsequent two volumes with great pleasure, students just as well as teachers in the field of algebraic geometry.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

14A15 | Schemes and morphisms |

14-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry |