# Algebraic machinery for algebraic geometry

Hello everybody,

I'm a math student who has just got his first degree, and I am studying algebraic geometry since a few months. Something I have noticed is the (to my eyes) huge amount of commutative algebra one needs to push himself some deeper than the elementary subjects. This can be seen just counting the "A's lemmas" on Hartshorne's book or even more by looking in other books where many purely algebraic results are given with references to the proof.

The problem arises more evidently when I try to do things on my own and in particular in the exercises. Many exercises I saw required some fact about commutative algebra which, it happened, I either didn't know or I knew just "randomly" by experience or from my university class about commutative algebra (I attended the most advanced one in my university, but unfortunately it is not enough).

My question is, how should one do to feel "free" from any reasonably elementary gap about commutative algebra? Just take the references books like those by Eisenbud, Zariski-Samuel or even Bourbaki, and study them from the beginning to the end, sounds quite huge a mission. However I am actually skeptical about just looking at the results one by one, since there is the risk to never learn those background notions properly.

Hope my question is proper on this forum,

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– Craig Feinstein Apr 15 '13 at 16:59
Then check out this: jmilne.org/math – Craig Feinstein Apr 15 '13 at 17:27
I find the stacks project, stacks.math.columbia.edu, very useful for commutative algebra results. I would definitely recommend against reading a commutative algebra book from beginning if your primary interest in algebraic geometry. – dhagbert Apr 15 '13 at 18:00
Eisenbud, Commutative Algebra with a view towards Algebraic Geometry? – Timo Keller Apr 15 '13 at 18:55
There are some nice answers at mathoverflow.net/questions/31650/… . – Noah Stein Apr 16 '13 at 1:36

I think this is a very good question, because studying commutative algebra on its own is hard, it is much better to do it with some idea of what all that means geometrically.

In my opinion the best entry to commutative algebra is provided by Miles Reid's Undergraduate Commutative Algebra. Miles Reid is an algebraic geometer so when he writes about commutative algebra, it is with geometry in mind. I would say that this book has everything that you need to be able to start in algebraic geometry except dimension theory which is done excellently in Atiyah-MacDonald.

I would suggest that you read this book, which is brief so you don't lose sight of your ultimate goal and you can already start feeling that you're actually reading about geometry. When you're done start reading algebraic geometry. For instance Hartshorne. In that book as you discovered there are a lot of algebra results quoted and even more is needed for the exercises which you absolutely have to do. More than half of the important material is in the exercises!

When you get stuck in a problem, ask yourself if you can translate the problem or part of it to an algebra problem and then see if you can find anything related to that in one of the standard commutative algebra books such as Eisenbud or Matsumura or for that matter the stacks project.

As you discovered you will also need homological algebra, but not just any general homological algebra, but the kind that is used in commutative algebra. There is a great book for that: Bruns-Herzog: Cohen-Macaulay Rings. This is also a big undertaking, but you don't need to read the whole book to get going. Say read the first two chapters, but not even necessarily in one go. Take you time while you're doing some other things. And most importantly, whatever you read in that book (or for that matter in any algebra book) try to see if you can give statements and notions geometric meaning or at least come up with examples that come from geometry. For instance, find your favorite example of a non-Cohen-Macaulay variety. Then find another one.

Of course, as you advance you will need more and more algebra, but after awhile you actually get into the habit of acquiring that knowledge as you go on. It makes more sense to learn these more advanced notions when you get there.

Without trying to be comprehensive, I assume sooner or later you will need to learn about associated primes (this already happens to some extent in Reid's book), integral extensions, going-up, going down theorems, dimension theory, regular sequences, depth, and the big whale: flatness. Flatness is extremely important, but somewhat hard to grasp full depth at first (or even later). Don't despair, you'll start having a feel for it if you keep at it. Anyway, there are many more things to learn, but you didn't ask that.

So for now, I'd say read Reid's book, then read Hartshorne (or something similar) and then try to get the algebra knowledge that you're missing as you go along.

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Fulton's "Algebraic Curves" does an excellent job of introducing commutative algebra in a geometric context, and its selection of exercises does an amazing job at conveying the rich interaction of geometry and algebra beyond what is done in the text. I recommend trying Fulton's book alongside Reid's, and then you can decide for yourself which you prefer. – user30379 Apr 16 '13 at 3:34
I believe that the first edition of Rotman's "An Introduction to Homological Algebra" would be also a good source for homological algebra in addition to the book of Bruns-Herzog. – Youngsu Apr 16 '13 at 5:58
I wonder why still Hartshorne gets recommended these days. The material on the foundatations of sheaf theory and divisors is a big mess, when compared to EGA or other more recent textbooks. – Martin Brandenburg Apr 16 '13 at 13:36
@Martin, EGA is not a realistic introduction to algebraic geometry for most human beings. Which more recent textbook would you recommend. (Notice I added "or something similar" for exactly that reason). – Sándor Kovács Apr 16 '13 at 15:16
@ACL: uhm...I agree, Liu has (almost) no cohomology. But Hartshorne's presentation always left me unsatisfied. He introduces some tools, but I do not have the feeling that then he really uses them to the extent that homological algebra might allow. And I think that after the first 5 chapters in Liu one can easily go into EGA. Most probably, I am begin too subjective for MO, though... – Filippo Alberto Edoardo Apr 19 '13 at 8:52

I want to offer a possibly heretical opinion based on conversations I've had with people who do algebraic geometry, especially Joe Harris. I think that it is not necessary to know very much commutative algebra in order to study and understand algebraic geometry. The fundamental objects algebraic geometry studies are very concrete and intuitive, and even when you encounter non-reduced schemes, or jump into positive characteristic to study number theory using AG tools, there are a few standard techniques of translating results from familiar settings into new ones, and much of the formalism surrounding them can be treated as a black box. If you need to understand some exercise in Hartshorne, for example, then as long as you want to know the geometry, you can safely skip and assume the commutative algebra exercises, or make additional assumptions about your rings that make things easier technically.

There are two situations when it's good to know what's under the hood of a car: one is when it breaks, the other is when you need to make a new car. Neither of them should prevent you from getting behind the wheel and learning to drive. And if you know what a car does and when it breaks, you'll be surprised by how much more sense the fancy technology under the hood will make.

That said, a lot of great math has been done by digging around under the hood. If you want a recommendation book-wise, I'd go for Atiyah-MacDonald.

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I of course agree with your opinion here Dmitry. Once you've been working in the subject for a while you start to see why some of the nitpicky details are important, but it's certainly possible to go a long way without, say, ever just studying Hartshorne for 2 years. And at least for certain people, it's a lot more interesting to start doing geometry than to bash your head against concepts like flatness (which I'm not discrediting--it's extremely important, but at least in nice cases more intuitive than Hartshorne might have you believe) before having any idea what it might be good for. – Jack Huizenga Apr 16 '13 at 8:25
I sympathise with your heresy, so +1. That said, people differ as to their need to understand about the engine. Everybody can drive safely knowing very little, but to be comfortable and happy with it is another thing. It does not seem advisable to start doing algebraic geometry without at least some familiarity with the content of Atiyah-Macdonald, and if the OP feels this is not enough for them, going for more won't hurt. Note that also Sándor's thoughtful answer suggests giving lower priority to CA as one goes deeper into AG. – quim Apr 16 '13 at 8:32
This is not heretical at all, that's what most people did (-: – Hailong Dao Apr 16 '13 at 14:55
I too agree with this heresy. (Maybe enough people have already agreed that it's no longer heresy.) There are plenty of nice problems in algebraic geometry one can work on with a really minimal amount of commutative and homological algebra. I'm not recommending that approach, but I think the idea, which one might get from looking at Hartshorne, that you have to know that depth equals dimension in Cohen-Macaulay rings (or whatever) before you start thinking about curves on del Pezzo surfaces, is not accurate, and holds people back. – user5117 Apr 16 '13 at 15:10

Knowing Sandor, I will heartily second what he said. On second mention, I will say that Mumford's red book on algebraic geometry begins with 5-10 pages (depending on your edition) called "some algebra". This consists of the following subset of Sandor's list: noether's normalization lemma, and cohen - seidenberg's going up lemma, plus corollaries such as the "weak" nullstellensatz. Hence this seems to be a minimum of algebra to know.

I remark that this is also in sync with the post by Dmitry that not that much is necessary.

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I am by no means an algebraic geometer, but have used many of its ideas and results for studying algebraic dynamics. I found that the book "Ideals, Varieties, and Algorithms" by Cox, Little, and O'Shea was a wonderful way to learn both the algebra and geometry, all in the context of concrete computations using Groebner bases. This is a beautifully written text, quite accessible to undergraduates, filled with great examples and problems. Highly recommended.

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