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Commutative algebra. With a view toward algebraic geometry. (English) Zbl 0819.13001
Graduate Texts in Mathematics. 150. Berlin: Springer-Verlag. xvi, 785 p. (1995).
Although being one of the comparatively recent creations in mathematics, abstract commutative algebra has a long and fascinating genesis. Its development into a beautiful, deep and widely applied mathematical discipline, in its own right, must be understood as a function of that of algebraic number theory and algebraic geometry, both of which essentially gave birth to it.
In the second half of the 19th century, two concrete classes of commutative rings (and their ideal theory) marked the beginning of what is now called commutative algebra: rings of integers of algebraic number fields, on the one hand, and polynomial rings occurring in classical algebraic geometry and invariant theory, on the other hand. In the first half of the 20th century, after the basics of abstract algebra had been established, commutative algebra grew into an independent subject, mainly under the influence of E. Noether, E. Artin, W. Krull, B. L. van der Waerden, and others. In the 1940s this abstract and general framework was applied, in turn, to give classical algebraic geometry both a completely new footing and a new toolkit for further investigations. In this context, the epoch-making innovations by Chevalley, Zariski, and Weil not only created a revolution in algebraic geometry, but also had a very strong impact on the quickened growing of commutative algebra itself in the following decades.
The 1950s and 1960s saw the development of the structural theory of local rings, the foundations of algebraic multiplicity theory, Nagata’s counter-examples to Hilbert’s 14th problem, the introduction of homological methods into commutative algebra, and other pioneering achievements. However, the indisputably most characteristic mark of this period was A. Grothendieck’s creation of the theory of schemes, the (till now) ultimate revolution of algebraic geometry. His foundational work culminated in a far-reaching alliance of commutative algebra and algebraic geometry, which also made is possible, in turn, to apply geometric methods as a tool in commutative algebra.
In the following decades, geometric and homological methods have substantially engraved the vigorous research activities in commutative algebra. At present, commutative algebra is an independent, abstract, deep, and smoothly polished subject for its own sake, on the one hand, and an indispensible conceptual, methodical, and technical resource for modern algebraic and complex analytic geometry, on the other hand. Studying or actively pursuing research in geometry or number theory requires today a profound knowledge of commutative algebra; however, most textbooks on algebraic or complex analytic geometry usually assume such a knowledge from the beginning, often refer to the (undoubtedly excellent) great standard texts on abstract commutative algebra, or survey a minimal account of the basic results, mostly in a series of appendices. The reason for that is quite clear, for the fusion of the two subjects is close to such an extent that the necessary prerequisites from commutative algebra would occupy a considerable part of the text, thereby possibly discouraging the reader who is mainly interested in the geometric aspects of the subject. Conversely, most textbooks on commutative algebra present the material in its purely algebraic and perfectly polished abstract form, with at most a few elementary hints and applications to the related geometry. This situation really creates a dilemma for both students and teachers of algebraic geometry or commutative algebra.
The present book under review aims at toning down this traditional, nevertheless somewhat artificial discrepancy. The author, in person one of the leading experts in both fields, has tried to write on commutative algebra in a way that makes the heritage, the geometric character, and the geometric applications of the subject as apparent as possible. A first attempt in this direction, namely to offer a textbook that mixes algebra and geometry in an organic manner, has been successfully carried out by E. Kunz in 1980. His textbook “Einführung in die kommutative Algebra und algebraische Geometrie.” Braunschweig etc.: Friedr. Vieweg (1980; Zbl 0432.13001) and the English translation “Introduction to commutative algebra and algebraic geometry.” Boston etc.: Birkhäuser (1985; Zbl 0563.13001), (reprint 2013; Zbl 1263.13001) provided an exposition of the basic definitions and results in both commutative algebra and algebraic geometry, centered around the (at this time) recent solution of Serre’s problem on projective modules over polynomial rings or, respectively, Kronecker’s longstanding problem on complete intersections in projective spaces. Along this road, E. Kunz gave a very natural introduction to commutative algebra and algebraic geometry, especially emphasizing the concrete elementary nature of the objects which were at the beginning of both subjects.
The book under review aims at the same goal; however, it does so under much wider aspects. In fact, the author strives for a rather complete and up-to-date exposition of the present state of commutative algebra, with all its old and new links to modern algebraic geometry. As he says in the preface, his precise goal has been, from the beginning, to cover at least all the material that graduate students in algebraic geometry should have at their disposal, in particular those studying R. Hartshorne’s matchless modern textbook “Algebraic geometry” [New York etc.: Springer-Verlag (1977; Zbl 0367.14001)] and, perhaps subsequently, A. Grothendieck’s and J. Dieudonné’s “Éléments de géométrie algébrique” [Publ. Math., Inst. Haut. Étud. Sci. I–IV (1960- 1967; Zbl 0118.36206; Zbl 0122.16102; Zbl 0136.15901; Zbl 0135.39701; Zbl 0144.19904; Zbl 0153.02202)].
According to this strategy, the text is subdivided into three major parts and six appendices.
After a beautiful introduction, providing readers of different background knowledge or expertise with instructions for both self-teaching and teaching, and after a brief synopsis of the elementary definitions concerning rings, ideals and modules, part I of the book under review discusses the “Basic constructions” in commutative algebra. This first part consists of seven separate chapters: Chapter 1 is still introductory and surveys some of the history of commutative algebra in number theory, algebraic curve theory, one-dimensional complex analysis, and invariant theory. It also explains the dictionary “Commutative algebra – projective algebraic geometry”, including Hilbert’s basis theorem, Hilbert’s syzygy theorem, Hilbert’s Nullstellensatz, graded rings, and the Hilbert polynomials.
Chapter 2 deals with the localization principle in commutative algebra, with an analysis of zero-dimensional rings, and chapter 3 turns to associated prime ideals and the primary decomposition in noetherian rings. Chapter 4 discusses Bourbaki’s proof of Hilbert’s Nullstellensatz, Nakayama’s lemma, integral dependence, and the normalization process, whereas chapter 5 goes back to graded rings and modules, including the construction of the blow-up algebra, Krull’s intersection theorem, and their geometric interpretation. Chapter 6 introduces the concept of flatness, the Tor-functor, and derives the most important flatness criteria. Completions of rings and Hensel’s lemma are presented in chapter 7, together with their geometric significance, and Cohen’s structure theorems are also found here. – Each chapter comes with a large number of well-prepared exercises, most of which concern additional theoretical material and results. The same holds for the following chapters, whereby hints and solutions for selected exercises are given at the end of the book. By this method, the author manages to cover even more interesting and recent material, at least in outlines.
Part II of the book is entitled “Dimension theory”. It consists of nine more chapters. Chapter 8 illustrates the history of dimension theory in its topological, geometric, and algebraic aspects. In this complexity, it is much more advanced than the following ones, and (as the author says) it is actually meant to be read for motivation and “for culture only”. Chapter 9 gives then the fundamental definitions of algebraic dimension theory (Krull dimension of a ring), with special emphasis on the case of dimension zero. Chapter 10 covers Krull’s principal ideal theorem, regular sequences, parameter systems, regular local rings, and their algebro-geometric meanings. Chapter 11 is entitled “Dimension and codimension one” and treats normal rings, discrete valuation rings, Serre’s criterion, Dedekind domains, and the ideal class group. Hilbert-Samuel functions and polynomials, together with their natural appearance in multiplicity theory are discussed in chapter 12. The geometric aspects of dimension theory, above all Noether normalization and the finiteness of the integral closure of an affine ring, are the subject of chapter 13, and the following chapter 14 is devoted to both classic and modern elimination theory. – Chapter 15 gives, for the first time in a textbook on commutative algebra, an account of the fast growing computational part of the subject. Gröbner bases, initial ideals, and their applications to constructive module theory and projective algebraic geometry are the principal items here. The rather mathematical approach, relative to the usual more computational ones, is particularly convenient for algebraic geometers, and a set of seven computer algebra projects, at the end of the chapter, shows how the computational possibilities of this approach lead to new (at least conjectural) insights. – Chapter 16 is concerned with the differential calculus in commutative algebra, that is with modules of differentials, tangent and cotangent bundles, smoothness and generic smoothness, the Jacobi criterion, infinitesimal automorphisms, and some deformation theory.
Part III of the book is devoted to the homological methods in commutative algebra. Chapter 17 deals with regular sequences by means of the Koszul complex, and with applications of the Koszul complex to the study of the cotangent bundle of $$\mathbb{P}^n$$. Chapter 18 discusses the notion of depth and the Cohen-Macaulay property. The significance of the Cohen- Macaulay property is illustrated from various viewpoints, ranging from Hartshorne’s theorem on connectedness in codimension one to the theorem on flatness over a regular base to primeness criteria using Serre’s characterization of normality. – The homological theory of regular local rings occupies chapter 19. This chapter contains, apart from the standard material on projective dimension, minimal resolutions, global dimension, and the Auslander-Buchsbaum formula, also an application to the factoriality of local rings via stably free modules. Chapter 20 concerns free resolutions and their role in algebra and algebraic geometry. The author, who has contributed to this topic by a good deal of his own research in the past, presents here various criteria of exactness, mainly based on the approach via Fitting ideals and Fitting invariants, and he gives some very instructive applications, e.g., the Hilbert-Burch theorem characterizing ideals of projective dimension one, and, at the end, an algebraic treatment of Castelnuovo-Mumford regularity. The concluding chapter 21 gives an account of duality theory for local Cohen-Macaulay rings, and some parts of the theory of Gorenstein rings. This includes the discussion of the canonical module and its properties, maximal Cohen- Macaulay modules and their duality theory, and the theory of linkage à la Peskine and Szpiro from the algebraic point of view. An interesting feature in this chapter, among many others in the text, is the treatment of the canonical module via reduction to the case of an Artinian ring. This makes the whole topic pleasantly concrete and lucid, at least for the beginner.
The main text is followed by seven appendices, in which the author provides both some more technical material from algebra, as it is needed in the course of the text, and some furthergoing topics related to it. In brevity, these appendices are the following:
1. Field theory (transcendency degree, separability, $$p$$-bases);
2. Multilinear algebra (including divided powers and Schur functors);
3. Homological algebra (projective modules, injective modules, complexes, homology, derived functors, Ext and Tor, double complexes, spectral sequences, and derived categories);
4. Local cohomology (local and global cohomology, local duality, depth and dimension via local cohomology);
5. Category theory (categories, functors, natural transformations, adjoint functors, limits, representable functors, and Yoneda’s lemma);
6. Limits and colimits (flat modules as limits of free modules);
7. Where next? (hints for further reading).
The book ends with a section containing hints and solutions for more than one hundred selected exercises spread over the entire text. The bibliography is extremely rich and carefully selected, being a true help for both the reader and the interested expert.
Altogether, the book under review has filled a longstanding need for a text on commutative algebra which thoroughly reflects the naturally grown relations to algebraic geometry. Containing numerous novel results and presentations, the book is still fairly self-contained, accessible for beginners, and a treasure for teachers and researchers in both fields. The consequent mixing of algebra and geometry, from the beginning to the end, has made it impossible to present commutative algebra in its most systematic and abstract perfection, as it has been done, for example, in the great standard textbooks of O. Zariski and P. Samuel [cf. “Commutative algebra”, Vol. I. Princeton etc.: D. van Nostrand Company (1958; Zbl 0081.26501) and II (1960; Zbl 0121.27801); reprints, respectively, New York: Springer-Verlag (1975; Zbl 0313.13001) and 1976 (Zbl 0322.13001)], N. Bourbaki [“Commutative algebra”, Chapters 1–7. Paris: Hermann (1972; Zbl 0279.13001), 2nd printing Berlin etc.: Springer-Verlag (1989; Zbl 0666.13001)], and H. Matsumura [“Commutative algebra.” New York: W. A. Benjamin (1970; Zbl 0211.06501); 2nd edition (1980; Zbl 0441.13001)]. However, the book under review, apart from having the compensating advantage of combining algebra and geometry in a natural manner, at least touches upon numerous recent development and results not yet contained in any other textbook. In this sense, it should be regarded as an unique and excellent enrichment of the existing literature in commutative algebra and algebraic geometry, just as a new standard text among the celebrated others, and as a highly welcome supplement to them.

##### MSC:
 13-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra 14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry 13-03 History of commutative algebra 13Axx General commutative ring theory 13Cxx Theory of modules and ideals in commutative rings 13A50 Actions of groups on commutative rings; invariant theory 13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
##### MathOverflow Questions:
When does prime elements remain prime in certain integral extension
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