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Commutative Algebra I

Oscar Zariski and Pierre Samuel
Springer Verlag
Publication Date: 
Number of Pages: 
Graduate Texts in Mathematics 28
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[Reviewed by
Felipe Zaldivar
, on

When the first editions of these books appeared (D. Van Nostrand, Vol. 1, 1958, Vol. 2, 1960) there were no other books on the same level devoted to commutative algebra, except for Krull’s Idealtheorie (Springer, 1935). Shortly thereafter, Bourbaki’s treatise on commutative algebra (Hermann, 1960–1961) was published, but this is an encyclopedic work, good for reference but hardly a textbook for the newcomer. A very successful extract of Bourbaki was also published in 1969, Atiyah and MacDonald’s Introduction to Commutative Algebra (Addison-Wesley, 1969). And from there on, there has not been a shortage of books on commutative algebra, at all levels: From introductory textbooks to research monographs, from theorem-proof approaches to combinatorial and computational emphasis. The Zariski-Samuel books on commutative algebra helped put the subject within reach of anyone interested on it. Generations of algebraic geometry students learned their subject from them, with Bourbaki as a handy reference.

What can we find in Zariski-Samuel that may be absent in the other books? Content-wise, very little. But in addition to the style and masterly treatment, perhaps one should also mention the detailed explanations of concepts and proofs, contrasting with the terse, almost telegraphic, style found in some other important texts (Nagata’s Local Rings or Matsumura’s Commutative Algebra, to mention two well-known examples).

What is missing in Zariski-Samuel? Well, to begin with, exercises are totally absent, and so when using this as a textbook the lecturer must provide them. Homological methods are mostly absent: in the late 1950s they were just starting to percolate into most parts of algebra, so they are just mentioned in chapters 7 and 8. One should also point out that what one usually means by commutative algebra starts properly in chapter 3 of the first volume. The first two chapters on groups, rings, and fields are usually covered in the (now) standard undergraduate abstract algebra course, with the possible exception of the last three sections of chapter two (linear disjointness, irreducibility and derivations).

Chapter three gives a detailed introduction to commutative rings and modules: prime, maximal and primary ideals, tensor products of rings and algebras. (In case you are wondering why it is sometimes necessary to emphasize the detailed exposition in Zariski-Samuel, notice that in page 183 of chapter three we have the correct algebra structure for the tensor product of two algebras, as a push-out of the two structure morphisms. Contrast this with the discussion in Atiyah-Macdonald, page 31 line 15, where the given map is not a ring morphism. However, I believe this is the only mathematical error in that fine book.)

Chapter four is devoted to Noetherian rings. Here we find Hilbert’s basis theorem, the decomposition of ideals in Noetherian rings as the intersection of primary ideals, uniqueness theorems for these decompositions, and Krull’s intersection and prime ideal theorems, to mention some of the highlights. Chapter five is devoted to integral dependence and Dedekind domains; in contrast with most other books it includes the arithmetic aspects of these topics, e.g., decomposition of prime ideals in extensions of Dedekind domains, the decomposition, inertia and ramification groups, the different and the discriminant, with examples given by rings of integers in quadratic or cyclotomic extensions of number fields.

Chapter six gives a thorough treatment of valuation theory, including a discussion of ramification theory and divisors in function fields. Chapter seven is a gem of exposition: here after a quick but detailed overview of affine and projective algebraic varieties, we find the classical theory of polynomial and formal power series rings, proving the main results which are so essential in algebraic geometry, for example Hilbert’s Nullstellensatz, which is given two proofs: one using (what else?) Zariski’s lemma (page 165 of the second volume) and the other using Noether’s normalization lemma. It is in this chapter that we also find the basic results of dimension theory.

The last chapter, on completions, includes an introduction to the theory of regular local rings. There are seven small appendices covering some topics that extend some of the results found on the main text, for example, in Appendix 7 it is proved that every local ring is a unique factorization domain. The proof, a variant of the Auslander and Buchsbaum original one, uses some tools from homological algebra. It is a nice way to end this fine book, showing that the subject had opened up: homological methods had shown some of their power and would eventually lead to new deep developments in commutative algebra.

It should by now be obvious: I like these books. Reading them is a pleasure. The leisurely style makes them appropriate for self-study, perhaps complementing the textbook being used. The cross-references in these volumes are handled with ease (you don’t have to consult a different volume of the encyclopedia, as in Bourbaki, where instead of giving you a summary of the result being quoted you are sent to a different volume, say on Set Theory). Any mathematical library would be incomplete if these two volumes were not present.



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