Introduction to commutative algebra and algebraic geometry. Transl. from the German by Michael Ackerman. With a preface by David Mumford
Ernst Kunz (born 1933) has published several outstanding textbooks in the course of his long academic career as both researcher and teacher in differential algebra, algebraic geometry, and commutative algebra at the University if Regensburg, Germany. Among his most popular books is certainly the unique text [Einführung in die kommutative Algebra und algebraische Geometrie. Vieweg Studium, Bd. 46, Aufbaukurs Mathematik. Braunschweig/Wiesbaden: Friedr. Vieweg \& Sohn. X, 239 S., 18 Fig., 185 Übungsaufg (1980; Zbl 0432.13001)]. Already this German original text came with an English preface by David Mumford for readers in the United States, mainly in order to have the novel flavor of this very special primer emphasized by one of the most renowned authorities in the field. In fact, the author's [loc. cit.] was the first textbook in the literature that developed the basic principles of both commutative algebra and the geometry of algebraic varieties in their close interrelation, thereby strictly following the didactic principle that certain areas of both subjects are best understood together, that is, in their natural combination. An English translation of the German edition of this book was provided in 1985 by Birkhäuser Verlag, with only a few major changes of the well-tried original text [the author, Introduction to commutative algebra and algebraic geometry. Transl. from the German by Michael Ackerman. With a preface by David Mumford. Boston-Basel-Stuttgart: Birkhäuser. X, 238 p. (1985; Zbl 0563.13001)]. The book under review is a reprint of just that English translation from 1985, without any further alterations or complements. Now as before, the main goal of the book is to lead the reader from the classical Kronecker problem of determining the minimal number of equations needed to define special (affine or projective) algebraic varieties to an exposition of Serre's famous problem on projective modules over polynomial rings. The development of the fundamental concepts and results from commutative algebra and the theory of algebraic varieties serves as a highly natural introduction to both problems and their interplay, culminating in an utmost lucid description of the great progress made in this context by the pioneering contributions, of Quillen, Suslin, Evans, Eisenbud, Szpiro, Kumar, and others in the 1970s. Recall from the review of the original German edition [loc. cit. Zbl 0432.13001] that the author's classic concists of seven chapters, each of which is subdivided into several sections. Chapter 1 presents the basics on affine and projective varieties, together with the necessary tools from commutative algebra. Chapter 2 deals with the fundamentals of dimension theory in both commutative ring theory and (affine or projective) algebraic geometry, whereas Chapter 3 is concerned with regular and rational functions on algebraic varieties, on the one hand, and with the underlying localization techniques from commutative algebra, on the other hand. Chapter 4 treats the local-global principle in commutative algebra in greater detail, with a focus on the generation of modules and ideals, projective modules, and the related deep results by Horrocks, Quillen-Suslin, and Eisenbud-Evans. Chapter 5 discusses the classical Kronecker problem on the number of equations needed to describe an algebraic variety in its modern commutative-algebraic setting, including related modern results by Storch, Eisenbud-Evans, M. Kumar, and others. In the subsequent Chapter 6, the reader gets profoundly acquainted with regular points of algebraic varieties, regular rings, Cohen-Macaulay modules and rings, complete intersections, and primary decompositions of finitely generated modules over Noetherian rings. This chapter concludes with R. Hartshorne's connectedness theorem for set-theoretic complete intersections in projective space. Chapter 7, the last chapter of the book, turns to projective resolutions of modules, the concept of projective dimension for modules, the homological characterizations of regular rings (Auslander-Buchsbaum) and of local complete intersection rings, Hilbert's Syzygy Theorem, and the characterization of modules of projective dimension less than one. The last section of this concluding chapter presents, largely as an application of the various foregoing concepts and results, the proof of L. Szpiro's important theorem stating that if a curve \(C\) in affine 3-space \(\mathbb{A}^3\) is locally a complete intersection, then \(C\) is indeed the intersection of two surfaces. Each chapter ends with a section titled ``References'', in which a number of related historical comments as well as hints for additional reading are given. Furthermore, each single section comes with a series of complementing exercises enhancing the main text by a wealth of related material, and the whole book is interspersed with numerous instructive examples illustrating the abstract concepts and results. All together, this book is really a modern classic in the textbook literatur in both commutative algebra and algebraic geometry. More than thirty years after its first appearance, the author's [loc. cit.] is still an utmost useful complement to the many other standard textbooks, in these two fields. Emphasizing the concrete elementary nature of the objects with which both subjects once began, on the one hand, and leading the reader to topics of contemporary research, on the other hand, Professor Kunz has provided a fairly special and unique introductory textbook of timeless value. Further generations of students and instructors will certainly profit a great deal from this masterpiece of mathematical expository writing.
scientific article; zbMATH DE number 6121075