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.
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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
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38 of 40 people found the following review helpful
Format: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.
38 of 45 people found the following review helpful
Format: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.
13 of 16 people found the following review helpful
Format: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.
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.
Most Recent Customer Reviews
1.0 out of 5 stars
Paper and print
These springer print on demand books should be labelled low quality for their paper and print. Pages are very thick printed with the lowest quality setting.
Published 8 months ago by E. D.
3.0 out of 5 stars
The text itself is great. The paperback book isn't.
The text itself is great -- it fills in all the background Hartshorne inexplicably assumes you'll be familiar with in a clear and well-motivated way. Read more
Published 10 months ago by Daniel McLaury
5.0 out of 5 stars
Everything one could want in such a book.
Eisenbud's book is remarkable in that it is written as a text to learn commutative algebra and yet it also functions as a reference book on algebra to a professional working in... Read more
Published 12 months ago by Adamg
5.0 out of 5 stars
Excellent book
I started reading it only recently and made past first 3 chapters. So this review is less about the specifics of the content, but about the general style. Read more
Published 21 months ago by Arnab
4.0 out of 5 stars
Not for the beginner
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. Read more
Published on February 26, 2007 by Loopspace
4.0 out of 5 stars
Good book of reference
I purchased this as a book of reference. When I want to know something about Commutative Algebra (while reading Hartshorne's Algebraic Geometrry), I like a standard book of... Read more
Published on March 17, 2006 by Tim Wouters
5.0 out of 5 stars
very good, but should be read slowly
Some proofs are somewhat abstract to the beginner. Although you are forced to check them on the paper, I think it is very good for the study. Read more
Published on September 25, 2004 by Jadeson
2.0 out of 5 stars
Not Good For A First Read.
I found this book to be too wordy. Theorems are sometimes stated without proofs then followed by several corollaries with proof, followed finally by the proof of the original... Read more
Published on April 7, 2000
Citations (learn more)
This book cites 100
books:
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100
books
cite this book:
See all 100 books citing this book
- Cyclotomic Fields (Graduate Texts in Mathematics) by S. Lang in Front Matter, Back Matter (1), and Back Matter (2)
- Algebraic Geometry (Graduate Texts in Mathematics) by Robin Hartshorne in Front Matter (1), Front Matter (2), and Back Matter
- Dimension Theory (Princeton Mathematical Ser.; Vol 4) by Witold Hurewicz on page 215, page 217, and Back Matter
- Algebraic Number Theory (Cambridge Studies in Advanced Mathematics) by A. Fröhlich in Back Matter (1), and Back Matter (2)
- Intersection Theory, 2nd Edition by William Fulton in Back Matter (1), and Back Matter (2)
See all 100 books this book cites
- Some Tapas of Computer Algebra (Algorithms and Computation in Mathematics) by Arjeh M. Cohen on page 32, and page 64
- Symbolic and Numerical Scientific Computation: Second International Conference, SNSC 2001, Hagenberg, Austria, September 10-11, 2001, Revised Papers (Lecture Notes in Computer Science) by Franz Winkler on page 37, and page 85
- Algebraic Functions and Projective Curves (Graduate Texts in Mathematics) by David Goldschmidt in Back Matter (1), and Back Matter (2)
- Applications of Computational Algebraic Geometry: American Mathematical Society Short Course January 6-7, 1997 San Diego, California (Proceedings of Symposia in Applied Mathematics) by David A. Cox on page 21, and page 37
- Geometric And Combinatorial Aspects Of Commutative Algebra (Lecture Notes in Pure and Applied Mathematics) by Jurgen Herzog on page 110, and page 197
See all 100 books citing this book
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