Ideals and Reality: Projective Modules and Number of Generators of Ideals

According to the blurb on the back cover of this book, "[t]his monograph tells the story of a philosophy of J-P. Serre and his vision of relating that philosophy to problems in affine algebraic geometry… The central topic of the book is the question of whether a curve in n-space is as a set an intersection of n – 1 hypersurfaces, depicted by the central theorems of Ferrand, Szpiro, Cowsik-Nori, Mohan Kumar, Boratinsk." This is a rather ambitious undertaking, but the authors do an admirable job.

The (numerous) topics covered include (Chapters 1–3) the theory of commutative algebra, modules, and homological algebra; (4) Serre's Conjecture, along with the proofs of Suslin and Vaserstein; (5) vector bundles and their relation to projective modules; (6) more commutative algebra, with the notions of Noetherian and Artinian rings and Hilbert's Nullstellensatz; (7) Serre's Splitting Theorem; (8) regular rings and Dedekind rings; (9) determining the number of generators of a module over a commutative Noetherian ring, four different proofs of the Forster-Swan Theorem, and a proof of the Eisenbud-Evans Theorem; and (10) the notion of curves which are set-theoretic complete intersections (s.t.c.i.), and the Cowsik-Nori Theorem.

There are several remarkable things about this book. The two biggest are the density and the efficiency. This book covers a lot of ground in its ten chapters. And it's done very concisely. It is accessible to most graduate students with at least some experience in algebra. And, along with the exercises at the end of the book, it can be used to bring these students "up to speed" with many of the contemporary ideas of algebra. In fact, one good use that graduate students might find for this book would be to decide where they might want to begin their own research. And algebraists will find it to be a handy reference.

The only drawback that I see with the book is the presence of many typos and errors. They tend to be minor, but somewhat numerous. However, given the scope of this text, it can be easily forgiven.

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Donald L. Vestal is an Assistant Professor of Mathematics at Missouri Western State College. His interests include number theory, combinatorics, reading, and listening to the music of Rush. He can be reached at vestal@mwsc.edu.