Commutative Algebra: with a View Toward Algebraic Geometry (Graduate Texts in Mathematics) (Paperback)

by David Eisenbud (Author) "It has seemed to me for a long time that commutative algebra is best practiced with knowledge of the geometric ideas that played a great..." (more)
Key Phrases: conormal sequence, cotangent sequence, finite colength, Emmy Noether, Exercises Exercise, Hilbert's Nullstellensatz (more...)

Editorial Reviews

Review
D. Eisenbud Commutative Algebra with a View Toward Algebraic Geometry "This text has personality—Those familiar with Eisenbud"s own research will recognize its traces in his choice of topics and manner of approach. The book conveys infectious enthusiasm and the conviction that research in the field is active and yet accessible."—MATHEMATICAL REVIEWS

Product Description
Commutative Algebra is best understood with knowledge of the geometric ideas that have played a great role in its formation, in short, with a view towards algebraic geometry. The author presents a comprehensive view of commutative algebra, from basics, such as localization and primary decomposition, through dimension theory, differentials, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. Many exercises illustrate and sharpen the theory and extended exercises give the reader an active part in complementing the material presented in the text. One novel feature is a chapter devoted to a quick but thorough treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Applications of the theory and even suggestions for computer algebra projects are included. This book will appeal to readers from beginners to advanced students of commutative algebra or algebraic geometry. To help beginners, the essential ideals from algebraic geometry are treated from scratch. Appendices on homological algebra, multilinear algebra and several other useful topics help to make the book relatively self- contained. Novel results and presentations are scattered throughout the text.

Product Details

• Paperback: 797 pages
• Publisher: Springer (February 11, 1999)
• Language: English
• ISBN-10: 0387942696
• ISBN-13: 978-0387942698
• Product Dimensions: 9.2 x 6.1 x 1.8 inches
• Shipping Weight: 2.5 pounds (View shipping rates and policies)
• Average Customer Review: 4.3 out of 5 stars See all reviews (7 customer reviews)
• Amazon.com Sales Rank: #236,637 in Books (See Bestsellers in Books)

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Customer Reviews

Most Helpful Customer Reviews

31 of 32 people found the following review helpful:

5.0 out of 5 stars The standard text, July 27, 2000

By	Michael Spertus (Chicago, IL USA) - See all my reviews (REAL NAME)

This is often referred to as the standard text on commutative algebra.

It is an exceptionally good book on a subject that is normally difficult to get a handle on. Eisenbud's readable book gives intuitive and motivated proofs of even very technical results in commutative algebra, often illustrated with instructive examples, such as the useful figures illustrating embedded primes. A very nice feature is that he gives proofs to all the results in commutative algebra used by Robin Hartshorne's popular "Algebraic Geometry," making them a nice pair of books to read together.

I found this to be useful as a reference as well as a text. Most sections are fairly self-contained and many important topics are included in depth. I almost always find that it is the best place to learn any of the material covered.

This book belongs on the shelf of anyone learning algebraic geometry, although it will spend plenty of time off the shelf as well.

33 of 37 people found the following review helpful:

5.0 out of 5 stars Superb, September 3, 2001

By	Dr. Lee D. Carlson (Baltimore, Maryland USA) - See all my reviews (TOP 100 REVIEWER)    (REAL NAME)

If one is interested in taking on a thorough study of algebraic geometry, this book is a perfect starting point. The writing is excellent, and the student will find many exercises that illustrate and extend the results in each chapter. Readers are expected to have an undergraduate background in algebra, and maybe some analysis and elementary notions from differential geometry. Space does not permit a thorough review here, so just a brief summary of the places where the author has done an exceptional job of explaining or motivating a particular concept:

(1) The history of commutative algebra and its connection with algebraic geometry, for example the origin of the concept of an "ideal" of a ring as generalizing unique factorization.

(2) The discussion of the concept of localization, especially its origins in geometry. A zero dimensional ring (collection of "points") is a ring whose primes are all maximal, as expected.

(3) The theory of prime decomposition as a generalization of unique prime factorization. Primary decomposition is given a nice geometric interpretation in the book.

(4) Five different proofs of the Nullstellensatz discussed, giving the reader good insight on this important result.

(5) The geometric interpretation of an associated graded ring corresponding to the exceptional set in the blowup algebra.

(6) The notion of flatness of a module as a continuity of fibers and a test for this using the Tor functor.

(7) The characterization of Hensel's lemma as a version of Newton's method for solving equations. The geometric interpretation of the completion as representing the properties of a variety in neighborhoods smaller than Zariski open neighborhoods.

(8) The characterization of dimension using the Hilbert polynomial.

(9) The fiber dimension and the proof of its upper semicontinuity.

(10) The discussion of Grobner bases and flat families. Nice examples are given of a flat family connecting a finite set of ideals to their initial ideals.

(11) Computer algebra projects for the reader using the software packages CoCoA and Macaulay.

(12) The theory of differentials in algebraic geometry as a generalization of what is done in differential geometry.

(13) The discussion of how to construct complexes using tensor products and mapping cones in order to study the Koszul complex.

(14) The connection of the Koszul complex to the cotangent bundle of projective space.

(15) The geometric interpretation of the Cohen-Macauley property as a map to a regular variety.

4.0 out of 5 stars Not for the beginner, February 26, 2007

By	Loopspace (Norway) - See all my reviews

Well, the strength of this book lies in where it takes you. There is so much material here that when finished, you'll be prepared for a lot. Personally I think it is too wordy (my preferance is Atiyah & MacDonald) and the typesetting overall isn't all that impressive, so read up or consult other texts before/during your first encounter. M.Reids book is a better place to start.