Customer Reviews

Introduction To Commutative Algebra

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Some people believe that, for getting into algebraic geometry (by this I mean Grothendieck-like AG, with schemes and all that), one needs a monolithic training in commutative algebra (something like both volumes of Zariski-Samuel, for example). I disagree. This little book seems to be specially suited to those who want to learn AG. It's a bit too brisk, specially at the beginning - if you don't already have an acquaintance with the basics of groups, rings and ideals, you may run into trouble - but very illuminating. Masterful choice of topics, great exercises (as a matter of fact, about half the topics of the book, and more specifically the ones that are directly related to AG, are treated in the exercises, some of them quite challenging) - like one said before, it looks like a "chapter 0" of Hartshorne's book on AG. The authors consciously estabilish relations between the commutative algebra and the modern foundations of AG over and over along the way, illuminating both topics.

For the algebra itself, it also gets on well with Rotman's "Galois Theory" and MacDonald's out-of-print introduction to AG, "Algebraic Geometry - Introduction to Schemes", besides being the perfect preamble in commutative algebra to the books of Mumford and Hartshorne. A gem.

This book is almost everything you need to gain a solid background in commutative algebra. Moreover, it's trimmed down enough so that it doesn't have the things you don't need. If you're not an algebraic geometer or number theorist, it may be the only commutative algebra book you'll need.

This is how mathematics texts SHOULD be written. As in technical writing, the smaller text is the better written text. Everything is clean and direct, with clairity obviously a prime consideration. One never gets mired down. The proofs are always as close to a "THE BOOK" proof as possible, with illuminating examples, and plenty of excercises, many with outlines for solution, which makes the book ideal for self study. This book is a revelation. If I had to take only one math text with me to a desert island, this would be the one.

The strongest aspects of Atiyah & MacDonald's book are its brevity, accessibility to undergraduates, and subtle introduction of more advanced material.

Audience: I think an undergraduate with a solid understanding of material from a first course in abstract algebra (i.e., the chapter on rings--the modules chapter would help, but isn't necessary--from M. Artin's book 'Algebra' is more than sufficient) and some basic point-set topology from an intro real analysis course (or ch1-4 of Munkres) would be sufficient for fully appreciating the material. I think having experience in PS Topology is important for understanding parts of this book well; doing the exercises is possible if you learn it "on the fly," but I hadn't seen Urysohn's Lemma before, and even that caused me some "intuition" hangups; to fully appreciate the material, I would recommend doing a healthy number of problems in topology first.

Material: The material uses concepts from homological algebra, though in a disguised form; students with experience in category theory will find offhanded comments that recast some of the material in that language, but CT is absolutely not essential to understand the material well. It also provides exercises that lead naturally into topics from Algebraic Geometry and Algebraic Number Theory quite readily; a nice set of problems in CH1 walk a student through construction of the Zariski topology, prime spectrum, etc., and some functional properties of morphisms between spectra. Algebraic Number Theory starts showing up after chapter 4 in greater detail, and would lead comfortably into Lang's GTM on ALNT by CH9 (though I only read a bit of Lang, the first chapter felt natural).

The "details left to the reader" are usually reasonably tackled with the tools made available so far, and the book is short enough that one can cover a lot of ideas in a reasonable amount of time; the commentary made by the authors is brief, to the point, and never redundant as far as I can recall, so I consider this a highly efficient book (but not too efficient, it's self contained enough and not uncompromisingly terse).

Exercises: They are quite good, I think. Very few of them follow from "symbol-pushing" or "robotic theorem proving," and usually require some constructive argument. The exercises are mostly chosen to introduce more advanced material, and do a good job in that regard. The longer chapters have 25-30 exercises, and shorter chapters (a few pages) have maybe 10, so there are plenty of problems to do.

Hazards: The material on modules is brisk, the propositions in the first three sections on modules are mostly left without proof; however, the proofs follow from their analogues for rings, and aren't that hard, just be sure to actually do them because they are mentioned only briefly. Also, the book is not typo-free, but this only caused me one major hangup during the semester. After Chapter 3, the proofs are mostly complete, with a spattering of "left to the reader" exercises, which I usually found helpful.

Companion Material: I think Lang's 'Algebra' GTM would make a nice reference for the material on Homological Algebra and other miscellaneous things that come up in the proofs; I remember once a proof in the book required the notion of the adjoint of a matrix over a ring, and so I had to look it up in Lang, and also the basic category theory covered in CH1 of Lang would at least introduce (though in a very rapid way) the "abstract nonsense" mentioned offhandedly here and there. If you have a lot of money, or access to a good library, 'Categories for the Working Mathematician' is a slower and more thorough introduction to that language, and I would recommend at least having a look, though this isn't really central to the material from Commutative Algebra.

This book is what led to me getting a Ph.D. in commutative algebra. I carried it around for an entire summer studying it. I highly recommend it for a graduate student introduction to the subject, after taking a course that at least introduces modules. Too bad there wasn't a volume II.

Now this is clearly *the* best mathematics text I've come upon. It was the syllabus in my first course in commutative algebra. Taking the course and readig the book was a real pain, but I've never learned so much from so few pages.

In addition to introducing "all" necessary subjects in commutative algebra, the book is stuffed with really good exercises. I always come back to this book - looking up terms and definitions and proofs. And of course, taking it with me on holidays so I can work on the exercises :).

Definitely the best first book on commutative algebra. Contains just enough to get you started for Hartshorne.

Pros:
- Short. Really short. Especially when you have Eisenbud next to you.
- Have many gems in the exercises, especially chapter 5.

Cons:
- No differentials. No homological algebra. Need to read them up somewhere else. (Matsumura should be the best here)
- Spec stuff not very motivated. It makes sense since this is not a textbook for algebraic geometry, but those who reads it should at least roughly know what an affine scheme is and what it tries to do, so that you won't miss out the geometry in the process.

I wouldn't be opposed to this book if it wasn't for the fact that each piece of paper cost over a dollar. Add to that the fact that it is a paperback, and the inferior binding process which makes my dollar bills, I mean pages, fall out, and you have a book which I would never buy again. Thankfully all the information contained herein is easily found on the internet for free.