Reference book for commutative algebra

Question

I'm looking for a good book in commutative algebra, so I ask here for some advice. My ideal book should be:

• More comprehensive than Atiyah–Macdonald

• More readable than Matsumura (maybe better organized?)

• Less thick than Eisenbud, and more to the point.

To put this in context, I'm an algebraic geometer, so I know enough commutative algebra, but I didn't study it systematically (apart from a first course on A–M which I followed as an undergraduate); rather I learned the things I needed from time to time. So I would like to give me an occasion to get a better grasp on the subject.

EDIT: I will be more specific about the level. As I said I already had a course on Atiyah–Macdonald, and I know that material well, so I'm not interested in books of a comparable level. But I'm not completely familiar with Cohen–Macaulay rings and the relationship between regular sequences and the Koszul complex for example. And I know very little of Gorenstein rings and duality. So I'm looking for something a little bit more sophisticated than what has been already proposed. Yes, I know Eisenbud does these things but it's easy to get lost in that book. Something more to the point would be nice.

Answer 1

For a reference on Cohen-Macaulay and Gorenstein rings, you can try "Cohen-Macaulay rings" by Bruns-Herzog.

Also, Huneke's lecture note "Hyman Bass and Ubiquity: Gorenstein Rings" is a great introduction to Gorenstein rings, very easy to read and to the point, I highly recommend it.

EDIT: Since this question is already bumped up, I will take this opportunity to make a longer list.

There are of course some classic references which are still very useful (I find myself having to look in them quite often despite the new sources available): Bourbaki, EGA IV, Serre's "Local Algebras" (very nice read and culminated in the beautiful Serre intersection formula).

There has been some work done in commutative algebra since the 60s, so here is a more up-to-date list of reference for some currently active topics (Disclaimer: I am not an expert in any of these, the list was formed by randomly looking at my bookself, and put in alphabetical order (-:). This is community-wiki, so feel free to add or edit or suggest things you found missing.

• Cohen-Macaulay modules, from a representation theory perspective: Yoshino is excellent. Another one is being written.

• Combinatorial commutative algebra: Miller-Sturmfels.

• Free resolutions (over non-regular rings): Avramov lecture note

• Geometry of syzygies: Eisenbud, shorter but free version here.

• Homological conjectures: Hochster, Roberts (more connections to intersection theory), Hochster notes.

• Integral closures: Huneke-Swanson, which is available free at the link.

• Intersection theory done in a purely algebraic way: Flenner-O'Carrol-Vogel (for a very interesting story about this, see Eisenbud beautiful reminiscences, especially page 4)

• Local Cohomology: Brodmann-Sharp, Huneke's lecture note (very easy to read), 24 hours of local cohomology (I have been told that this one was a pain to write, which is probably a good sign).

• Tight closure and characteristic $p$ method: Huneke, Karen Smith's lecture note (more geometric, number 24 here), and of course many well-written introductions available on Hochster website.

Answer 2

For the past couple of years I've been currently writing what amounts to a book on commutative algebra:

http://alpha.math.uga.edu/~pete/integral2015.pdf

I say this not because I think that if/when I'm finished, my "book" will be the book you're looking for. Really! Rather, my point is that when I started writing my book I was in your situation: I had picked up "enough" commutative algebra for my research but hadn't studied it systematically since I was an undergraduate taking a 10 week course at the Atiyah-MacDonald level. There were a handful of texts that I owned and got useful information out of — especially, Eisenbud and Matsumura — but none of the texts covered everything that I wanted or only things that I wanted. (Also, and I don't know whether you are in this situation, I had begun to teach graduate courses and wanted to use facts of commutative algebra in my lectures. It doesn't really fly to say, "This holds by some normalization theorem, which is surely somewhere in Matsumura, or if not then in Eisenbud — I think." They'll believe you, but they won't look it up themselves.) So ….

Anyway, returning to the present, I really like my book. I especially like that I can add to it at any time I like, that I can move the sections around if I choose to, that I have free access to it at all times, etc. There is no doubt in my mind that in writing it I have learned an immense amount of material. In particular, I have long since disabused myself of the somewhat jejune notion that I knew "enough" commutative algebra. I no longer believe that such a thing is even possible.

This is not to say that no one else cares about my "great 21st century commutative algebra book". I have gotten a lot of feedback to the contrary, and I do think it — or rather, parts of it — are being read by a worldwide audience. Conversely, I regularly peruse other people's great 21st century commutative algebra books for nuggets of insight. I look forward to reading yours….

Answer 3

Yoshino's book on cohen macaulay modules over cohen macaulay rings is a gem. Highly recommended!

Answer 4

For more recent readers of this thread, it seems that OP has actually written their own book on the subject, so perheaps it is an alternative to the proposed problems on the references cited by them.

Answer 5

I guess it depends on your specific interest also. I certainly agree with Long's suggestion on Bruns and Herzog above.

From a more geometric perspective... A good source for local cohomology / duality stuff is Hartshorne's book "Local cohomology" based on Grothendieck's seminar (I think)

It's a lot more geometric than Bruns and Herzog.

You can also move from there to "Residues and Duality" if you'd like (and there are other sources for that as well, Brian Conrad's book, Lipman's notes, etc.). Coming from a more geometric perspective originally myself, I didn't really get Bruns and Herzog chapter 3 until I did this.

For Eisenbud's book, perhaps you should take it chapter by chapter. Many chapters don't really rely on anything and can be read out of context. This makes it a very valuable reference.

Answer 6

I learned commutative algebra in the same way you describe: Atiyah-MacDonald and then picking things up along the way. I don't know if your ideal book exists or not, but I can give you one nice reference: Mel Hochster's lecture notes for Math 614 and 615, available from his webpage.

Answer 7

Maybe Matsumura's Commutative Algebra --

(when you say "Matsumura" above I assume you mean "Commutative Ring Theory")

Of course this book is somewhat difficult to obtain.

Answer 8

To give the opposite suggestion from Bart, I was going to recommend Matsumura's Commutative ring theory as opposed to his Commutative algebra. I have said why at length on the "unanswered questions" thread asking exactly Pete's question. Briefly, Ring theory is clearer, better organized, argued more fully, with more exercises (and answers), references, with a better index, and easier to read. Probably because Miles Reid rendered it into English, and possibly also because Matsumura got to revise his first book, which was almost a set of (excellent, and advanced) class lecture notes. At least two of us who took Matsumura's class in 1967 (Sevin Recillas and I) seem to like the second book. Sevin owned and recommended it when i complained I had difficulty using the original book. Since I am judging based on what appears on Amazon, I cannot be positive it contains every result I want to reference, but from the table of contents I would guess it does. I also like Zariski and Samuel for clarity, but homological methods were introduced just as that book was finished.