A Course in Commutative Algebra

From coordinate geometry (Descartes) to algebraic varieties and commutative algebras (Hilbert) and schemes and commutative rings (Grothendieck), one of the most fertile ideas in mathematics has been that of a correspondence between algebra and geometry. Such a correspondence suggests that for each concept or result in algebra there is dual concept or result in geometry. The precise formulation of this correspondence is via an equivalence of the associated categories.

For example, Hilbert’s Nullstellensatz shows that there is an equivalence between the category of (affine) algebraic varieties over an algebraically closed field K and the category of reduced, finitely generated commutative K-algebras. Another instance of this correspondence is given by Serre’s theorem that shows that the category of vector bundles over an affine algebraic variety is equivalent to the category of finitely generated projective modules over the algebra of regular functions on the given variety. The late algebraic geometer George Kempf viewed these correspondences as a seesaw that balances two Mediterranean cultures: the (Arabic) algebraic side and the (Greek) geometric side.

Stretching this point of view a little, these correspondences provide a dictionary or lexicon for translating concepts and results in one category to the other one. The book under review puts this idea at the forefront: Its first part is entitled “The Algebra-Geometry Lexicon” and in its four chapters goes right away to establish the equivalence between the category of affine varieties and the category of reduced commutative algebras (over an algebraically closed field). Thus, in page 11 we have a proof of the weak Nullstellensatz and in page 14 we have the strong theorem proved using Rabinowitsch’s trick. Noetherian and Artinian rings and modules are studied in the second chapter with a proof, of course, of Hilbert’s basis theorem. Chapter three goes a little further introducing the Zariski topology on affine varieties and for the prime spectrum of an arbitrary commutative ring.

The second part of the book, chapters 5 to 8, is devoted to dimension theory. Chapter five introduces Krull’s dimension of a ring and in a few pages proves that, for affine K-algebras Krull’s dimension is the transcendence degree. Chapter six deals with the important tool of localization and chapter seven has as its main result Krull’s principal ideal theorem, some generalizations and a converse, ending with an application to de dimension of images and fibers of morphisms. Chapter eight, the last one of the second part of the book, treats the notion of integral ring extensions, proving, in particular, Noether’s normalization theorem.

Thus, in the first two parts of this book, the four columns of Algebraic Geometry have been introduced: Hilbert’s zeros and basis theorems, Krull’s principal ideal theorem and Noether’s normalization theorem. These chapters and the whole book are rich with geometric examples and exercises and can be used for a slow-paced one-term course on commutative algebra.

The third part of the book is devoted to (some) algorithmic aspects of commutative algebra. Chapter nine introduces the notion of Gröbner basis and Buchberger’s algorithm to compute these bases, with an application to computing elimination ideals. Chapter ten returns to the study of images and fibers of morphisms started in chapter seven, now using Gröbner bases to prove Grothendieck’s generic freeness lemma and a constructive version of it. This is then applied to prove the convergence of an algorithm that computes the image of a morphism of spectra. One highlight of this chapter is Chevalley’s theorem on the constructibility of images of morphisms of spectra.

The last chapter of this part is devoted to the Hilbert series of an ideal in a polynomial ring, proving the Hilbert-Serre theorem that the Hilbert series is a rational function and almost all its coefficients are given by a polynomial, the Hilbert polynomial. Later on it is proved, using Noether’s normalization lemma, that the degree of the Hilbert polynomial of an affine algebra is its Krull dimension. These results are used to obtain algorithms to compute the dimension of one such algebra.

The last part of the book has three chapters and is devoted to local rings, whose geometric counterpart is the local behaviour of a global object such as an affine variety. Chapter twelve deals with the notion of dimension for local rings. Chapter thirteen further specializes this to regular local rings with a highlight being the Jacobian criterion for calculating the singular locus of an affine variety. The last chapter treats the case of rings of dimension one, these being motivated by the observation that a Noetherian local ring of dimension one is regular if and only if it is normal. A geometric consequence is the desingularization of affine curves. The last section deals with global domains of dimension one (Dedekind domains) leaving the local case (discrete valuation rings) to the exercises.

This is a well-written book that goes right away to the core of the subject: Commutative algebra as an introduction to algebraic geometry, with a few asides on algebraic number theory. I liked this book for its balance between the classical theorem-proof part and its computational side. However, as the author remarks in the Introduction, some topics were left out. The ones I missed the most are: tensor products and the notion of flatness (thus, we have generic freeness but not generic flatness, for example); secondly, the notion of completion is just barely touched on some exercises; thirdly, primary decomposition is not mentioned at all. Homological methods are absent, but as the author remarks, to introduce and do them justice would require much more space.

This being said, this is a fine book, in the class of many other well established and fine books on the subject: From Zariski-Samuel’s Commutative Algebra (Springer, 1975) to Eisenbud’s Commutative Algebra with a View Towards Algebraic Geometry (Springer, 1995). I am sure it could become a text of choice for an introductory course in commutative algebra. Both the lecturer and the student would benefit from its balanced and novel approach to the subject.

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