Algebraic Geometry and Commutative Algebra

During the last few years we have witnessed a steady flow of books on algebraic geometry, at all levels from advanced monographs to introductory textbooks. Styles range from from classically motivated with plenty of pictures and diagrams to more sober, no need to draw pictures, just commutative diagrams and straight to the core of the subject, sheaves, schemes and cohomology. Well, algebraic geometry deserves all these approaches and more. Someone once wrote that having taught algebraic geometry many times, every time seemed like a new beginning, and with a high probability of no repetition!

The book under review takes as its main goal to build the foundations upon which an interested reader can move to more advanced topics, and at the same time keeping the book almost self-contained. Let me say from the start, that the approach taken by the author is completely à la Grothendieck. Hence, the book is naturally partitioned in two halves. The first part covers the basics of commutative algebra as a prerequisite for the second part, devoted to the category of schemes.

The first part of the book, chapters one to five are planned to be used for a Commutative Algebra course. This part, starts in chapter one with the definition and elementary properties of (commutative) rings, ideals and modules, including the chain conditions and localization. Primary decomposition, Noetherian and Artinian rings are treated in detail in chapter two, while integral dependence and the Noether normalization theorem, Hilbert’s Nullstellensatz and the going-up and going-down theorems of Cohen-Seidenberg are studied in chapter three. Chapter four is devoted to tensor products, flatness, extension of coefficients and descent properties. Homological methods are treated in chapter five.

The second part of the book, chapters six to nine, is an introduction to the theory of schemes. Chapter six treats affine schemes: the prime spectrum of a ring, its structure sheaf and its functorial properties. We also find in this chapter a detailed introduction to the category of sheaves: presheaves, sheafification, sheaves of modules, quasi-coherent sheaves and direct and inverse images of sheaf of modules.

Chapter seven is devoted to some techniques for the construction of global schemes and morphisms between them (gluing, fiber products, subschemes) and some of their global properties (separated schemes, Noetherian schemes, and dimension). This chapter ends with an introduction to sheaf and Čech cohomologies. Chapter eight is devoted to some of the various types of morphisms that are available in algebraic geometry: finite, of finite type, finitely presentated, unramified, flat, étale and smooth. This chapter includes the necessary requisites on sheaves of differential forms. The final chapter is devoted to some important morphisms and schemes, including proper and projective morphisms and schemes. Thus, we find in this chapter invertible sheaves, divisors, ample and very ample invertible sheaves and, as an application, a proof that Abelian varieties are projective.

The book is carefully written, with good examples, detailed proofs, and plenty of exercises at the end of every one of its sections. Each chapter has an introduction where the author discusses informally and motivates the contents of the chapter. I liked the book and I believe that it can be used either as textbook for a two-semester introduction to algebraic geometry or for self-study by a motivated student.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx.