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

David Eisenbud (Author)

Book Description

ISBN-10: 0387942696 | ISBN-13: 978-0387942698 | Publication Date: March 30, 1995

This is a comprehensive review of commutative algebra, from localization and primary decomposition through dimension theory, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. The book gives a concise treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Many exercises included.

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 Details

• Paperback: 813 pages
• Publisher: Springer (March 30, 1995)
• 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: 4.4 out of 5 stars  See all reviews (8 customer reviews)
• Amazon Best Sellers Rank: #435,812 in Books (See Top 100 in Books)

More About the Author

Discover books, learn about writers, read author blogs, and more.

Customer Reviews

Most Helpful Customer Reviews

37 of 38 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 review is from: Commutative Algebra: with a View Toward Algebraic Geometry (Graduate Texts in Mathematics) (Paperback)

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.

36 of 42 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
(VINE VOICE)    (HALL OF FAME REVIEWER)    (REAL NAME)   

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

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.

9 of 11 people found the following review helpful:

5.0 out of 5 stars Excellent Book...very atypical for a math book, but I like it!, December 16, 2007

By 

Alexander C. Zorach "Alex Zorach" (Lancaster, PA) - See all my reviews
(REAL NAME)   

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

People tend to have strong feelings about this book. In my opinion, the people who dislike it are those who expect it to be like a typical graduate-level math book. This book is extremely atypical for a math book; it's not meant to be read linearly, and the topics in it do not follow a typical logical dependency. Personally, I find it to be outstanding; my only complaint about it is that I wish there were more books like it!

Commutative algebra and algebraic geometry are extremely difficult subjects requiring a great deal of background. This book is written as a sort of intermediary text between introductory abstract algebra books with a full and exposition of algebraic structures, and advanced, highly technical texts that can be difficult to follow and grasp on a technical level. As such, this book focuses on developing intuition, and discussing the history and motivation behind the various mathematical structures presented. It assumes that most of the other aspects of the subject, including both the elementary expositions, and the more advanced technical details, can be found elsewhere (although, believe me, this book certainly has its share of both elementary expositions and advanced technical details!)

I think this book is actually better for self-study than for use as a textbook. Most of the people I have known who have used it as a textbook have been frustrated with it. Either way, it needs to be supplemented by other books. Personally, on algebra, I like the Dummit and Foote, Isaacs, and Lang books. Those three books have very little overlap with each other, and very little overlap with this book, and they offer a very useful difference of perspectives where they do overlap! I also would recommend reading the more elementary book by Cox, Little, and O'Shea, which can help you get a feel for the subject of algebraic geometry. Many people see this book's primary purpose as preparation for Robin Hartshorne's "Algebraic Geometry". I can't say, however, how effective it is at that purpose, as no matter how far I get in this book, all but a few sections from that book still remain quite far beyond my grasp.