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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.

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It's not a book, so I'll just write this as a comment: I have found the lecture notes of Gaitsgory (math.harvard.edu/~gaitsgde/COMALG_2008) to be a useful complement to Atiyah-MacDonald, emphasizing different subjects than AM does. (Look at the homeworks on that page for even more!) Certainly many orders of magnitude less comprehensive than Matsumura or Eisenbud, though. –  Sam Lichtenstein Feb 25 '10 at 17:58
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I don't think it's respectful to describe Eisenbud's algebra book as "boring". (Moreover, I don't find it boring, but that's not my main point.) David Eisenbud is an actual living person -- with internet access. He is a very nice man, one of the world's leading commutative algebraists, and one of the most influential and well connected mathematicians I know. "Boring" is not appropriate language for professionals publicly discussing his work. –  Pete L. Clark Feb 26 '10 at 18:45
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I did not claim the book is boring, only that it is too thick, and that sometimes it deviates too much from the main exposition. –  Andrea Ferretti Feb 26 '10 at 19:07
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@Clark I'm not sure whether Eisenbud's book is boring, however why is it inappropriate to say so if he/she feels it's boring? It has nothing to do with the author being a nice person or not. If a nice mathematician writes a wrong paper, would it be inappropriate to say his work is wrong? (which seems to be more damning than "boring") –  Anonymous Feb 26 '10 at 22:28
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@Anonymous: "wrong" is (in mathematics) an objective insight into someone's work, hence potentially extremely helpful. "Boring" is purely subjective and yields no useful information. It is a well-known principle of reviewing that one does not make negative comments without justification, let alone unjustifiable ones. –  Pete L. Clark Feb 26 '10 at 22:55
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8 Answers

up vote 20 down vote accepted

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.

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While this is an excellent book, I think it definitely counts as more dense and easier to get lost in than either Matsumura or Eisenbud... Eisenbud gives much better geometric pictures of these things than Bruns-Herzog. –  Gwyn Whieldon Apr 2 '10 at 3:37
    
I wish I could upvote again: great edit! –  Alberto García-Raboso May 24 '10 at 23:11
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Yoshino's book on cohen macaulay modules over cohen macaulay rings is a gem. Highly recommended!

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Dear Andrea,

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

http://www.math.uga.edu/~pete/integral.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 jejeune 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...

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You got a right to blow your own horn with these notes/book draft,Pete-it's REALLY nice! Not quite as geometric as Eisenbud,but more so then the "formal" books like Matsumura and Kaplansky. And a whole lot more up-to-date and digestible. Tell you what-I'm looking into beginning my own publishing company in the next five years to publish nice,inexpensive textbooks for students out of recycled paper. If you haven't found a publisher by then,I'll drop you a line if it materializes and I'm in the market for manuscripts.I'd be a REALLY happy man to publish the finished version. –  Andrew L May 24 '10 at 23:33
    
indeed I have already looked into your book for some things! :-D –  Andrea Ferretti May 25 '10 at 17:25
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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.

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That's helpful to know. It's kind of a daunting book to decide to read cover-to-cover... –  Tim Campion Sep 15 '11 at 23:19
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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.

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Yes, I would suggest reading Hochster's notes for 614 along with Bourbaki's commutative algebra. The notes for 615 this semester are on Zariski's main theorem, the structure theory of smooth étale and unramified morphisms of rings, Artin approximation, Henselian rings, and a bunch of other stuff we haven't gotten to in class. 615 is a topics course, so the notes from different semester will yield different things. –  Harry Gindi Feb 25 '10 at 18:15
    
That's how I learned. A-M, first, and then, though not Hochster, I moved on to Eisenbud in a course, and also Cox-Little-O'Shea taught me Groebner everything. –  Charles Siegel Feb 25 '10 at 18:49
    
Hochster lectures seem nice, but a bit too elementary. I had a brief glance, and it doesn't seem that they cover much more ground than Atiyah-MacDonald. –  Andrea Ferretti Feb 25 '10 at 21:42
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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.

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@Bart: for those of us who are, at best, dimly aware that these are not the same book, could you say a little bit about what makes them different? –  Pete L. Clark Nov 17 '10 at 13:29
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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.

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Thanks, Roy. But does the "Algebra" book cover material that does not appear in "Ring Theory"? –  Pete L. Clark Nov 18 '10 at 5:03
    
@Pete: the first one has a lot on excellent rings which are not in the second one. –  Hailong Dao Nov 18 '10 at 5:36
    
Actually, although the 15 page final chapter of Comm. Alg. is entitled "excellent rings", there are only two pages devoted to them there, i.e. excellent rings are defined on the next to last page of that chapter. Aha! there is a 50 page appendix that discusses them further. So this appendix seems to be mostly what is omitted from the later book. However several proofs are omitted also from this appendix and referenced instead to EGA, Bourbaki, and Nagata. This material does not appear to me to be needed say to read Hartshorne. Most of these are topics I myself have seldom encountered. –  roy smith Nov 18 '10 at 6:36
    
Great, thanks. (I noticed that you answered this more expansively in another question, and I upvoted that as well.) –  Pete L. Clark Nov 18 '10 at 7:26
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There is a new (released December 2010) Springer GTM by Kemper, which looks like it might fit the bill. I haven't read it, but I'd be interested to hear from someone who did.

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Thank you for your reference, but from a glance at the TOC, it seems too elementary. –  Andrea Ferretti Jan 23 '11 at 12:03
    
I recently borrowed this book from a library. It looks like an excellent addition to the literature; however, as Andrea says, it covers similar ground to Atiyah and Macdonald. The author himself recommends one to read Eisenbud or Matsumura afterwards if one wishes to further his knowledge in commutative algebra. –  Amitesh Datta Jun 3 '11 at 7:32
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