Commutative Algebra: with a View Toward Algebraic Geometry (Graduate Texts in Mathematics, 150)

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. First, it is a delight to read. The clarity is excellent. It's understood that part of the clarity is based on reader's background! I didn't major in mathematics, and mine is an effort to (try to) learn some aspects of commutative algebra. Even with that sort of limited background of a beginner, I was able to tread through the material - as far as I read. Second, the motivation and historical backgrounds are fantastic. Especially for someone who may not know a lot about connections with other areas, this was a great help in putting things in perspective. One downside is that it makes the chapter a little more verbose than it needs to be for a quick access. But then for me it was a virtue; someone else may not find it so. Finally, the book doesn't have a strict linear flow. So it is somewhat easier, especially for an expert, to just pick a chapter and start reading it. A feature that I enjoy a lot in general, although I am not an expert in this area by any stretch.

I cannot comment much on this book as a first experience to commutative algebra, but as a reference and particularly a companion to Hartshorne it is irreplacable (as noted in the intro, "a view toward algebraic geometry" is meant as a reference to Algebraic Geometry, the title of Hartshorne). In particular, all the algebraic results which are assumed in that title are proven here. As someone who loves seeing the details worked out, this is fantastic. In addition, reading this after having done a little algebraic geometry can be enlightening as Eisenbud likes to work in geometric examples wherever applicable -- albeit at the expense of a first-time reader. The book is extremely comprehensive as well, any time I have a question which can be phrased purely algebraically, Eisenbud is my first reference. On the other hand, this makes it the heaviest book on my shelf even as a paperback. 800 pages! There is one section I did learn from as a first source, the introduction to homological algebra in the appendices which I found to be really well-done. It certainly isn't done in the generality or depth as Weibel, but if you are only interested in modules over a ring or the easy generalizations to quasi-coherent modules on a scheme, this is a great place to learn the essentials of things like derived functors, spectral sequences, and even a cursory intro to derived categories that allowed me to get into Weibel's derived category chapter with ease.

Excellent book. Very clear, thorough, with tons of interesting examples

The book itself is great, but the copy that I received was made using an inferior printing process. It almost seems like my copy was printed in another country. For instance, the front cover looks as though someone tried to upscale an image to make it fit the size required, and because of this, there is a lack of resolution in how it was printed. The second sign that my copy is not "authentic" is due to the fact that I have compared the size of my copy to the size other copies, and mine is far thicker, meaning that the paper that it was printed on is far thicker. Upon closer inspection, it is easy to detect differences in paper texture and quality between my copy and the others that I have seen. This is not a knock on the book, just a knock on the distributor that sold it to Amazon (I bought it via the Amazon prime option). With those points aside, the book is completely readable and there aren't any major issues with my copy. I am however concerned about whether the binding process will hold up. If I have any issues with the binding, I'll update this in the future. P.S. - I have a suspicion that my copy might have been illegally printed by a company and then sold as an authentic copy. If this is not what happened, then I guess Springer (which I has great graduate textbooks) is using an inferior printing process to what they have used in the past.

Great reference with tons of material, explained from the basics. If you've been studying commutative rings for years and want a geometric perspective, this is the book for you. While I'd be hesitant to recommend it for a first course in commutative algebra (for that, use Kaplansky or Atiyah-MacDonald), it puts everything together very nicely.

This was marketed as a used copy, but it came out to be very new and of a very good quality.

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.

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.

Compré el libro porque es la referencia principal de un curso impartido por el matemático Richard E. Borcherds (Youtube). Y me ha gustado hasta el momento la exposición del autor. El libro llegó en buenas condiciones (algunos golpes no significativos). De hecho el libro fue impreso el mismo día que realice la compra.

Very good book

Sehr gut

L'algèbre commutative, si elle est extraite de tout contexte, est très abstraite. Ses plus belles applications sont sans doute en géométrie algébrique et en théorie des nombres. David Eisenbud a choisi de présenter l'algèbre commutative dans la perspective de la géométrie algébrique (pour permettre par exemple au lecteur d'aborder commodément l'extraordinaire livre de Hartshorne "Algebraic Geometry"). Ce but est largement atteint. Tous les grands chapitres de l'algèbre commutative sont exposés. La première partie est consacrée aux "constructions de base": Nullstellensatz, localisation, décomposition primaire, platitude, complétion, etc. La seconde partie est axée sur la théorie de la dimension. Comment y comprendre quelque chose si l'on ne dispose pas du dictionnaire permettant de passer du point de vue algébrique au point de vue géométrique et réciproquement? Ce dictionnaire, Eisenbud l'établit avec maintes illustrations, pour le grand bonheur du lecteur. La troisième partie, enfin, présente les méthodes homologiques. Les annexes présentent des "rappels" utiles sur, notamment, la théorie des corps, l'algèbre homologique et la théorie des catégories. Eisenbud est aussi un adepte de "l'algèbre effective", d'où la place qu'il donne aux outils de calcul, en particulier aux bases de Gröbner. C'est un aspect qu'on ne trouve pas du tout dans le livre encyclopédique de Bourbaki sur l'algèbre commutative (en 4 volumes). Il est difficile de comparer le livre d'Eisenbud et celui de Bourbaki. L'un ne remplace pas l'autre: ils se complètent. Par exemple, la structure des modules de type fini sur les anneaux de Dedekind (théorème si beau, qui généralise parfaitement le résultat classique sur la structure des modules de type fini sur les anneaux principaux) est très bien détaillée dans Bourbaki, alors qu'elle n'est qu'évoquée en exercice dans le livre d'Eisenbud. Le style de Bourbaki et celui d'Eisenbud sont à l'opposé l'un de l'autre. Celui d'Eisenbud est, à mon goût, un peu touffu, et selon moi moins clair que celui de Bourbaki. A contrario, le style de Bourbaki est plus abstrait... Le livre d'Eisenbud est un excellent ouvrage!

It is quite simply a classic