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,
thank you in advance for your opinion!
Best Answer
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.