Commutative algebra with a view toward algebraic geometry /

In principle, commutative algebra stands in relation to algebraic geometry as does local to global, and one cannot master the latter before mastering the former. To see which sophisticated ideas from commutative algebra bear spectacular fruit in the context of algebraic geometry, one must think of the role played by Gorenstein rings in Wiles's recent proof of Fermat's Last Theorem. Unfortunately, in practice, commutative algebra tends to have its own practitioners, language, problems, and culture; students of algebraic geometry, a central and burgeoning field, often have difficulty finding what they need from it. These days, all serious students of algebraic geometry read R. Hartshorne's Algebraic Geometry (1977), where theorems from commutative algebra are quoted constantly without proof. It will hardly do justice to Eisenbud's monograph merely to say that it is the one book everyone will always want open next to Hartshorne, a one-stop shop for all(!) the proofs Hartshorne omits, but this is certainly a major selling point. Running longer by half than Hartshorne, Eisenbud's book is indeed a magnum opus from one of the foremost masters of the subject. To read it is to hear the author's voice taking care with every essential detail, all the while pointing out the best ways to understand the ideas. Essential for college libraries. Undergraduate through faculty. D. V. Feldman; University of New Hampshire