How to understand commutative algebra

Question

I'm trying to study commutative algebra, but i cannot understand totally the arguments. I'm afraid to study all mechanically. What can i do to solve this problem? I need some advice!! Thank you

Answer 1

I recently finished a course in commutative algebra, and let me tell you I completely get it. Although I don't really know what you're specifically struggling with let me tell you a little about the problems I faced and the things I did to remedy it after I got completely demolished on the final.

I took commutative algebra in my third year of undergraduate studies. Until then the VAST majority of my classes revolved around analysis -- in fact I had only studied groups and rings in the previous term. I think this contributed to the first problem I faced which was building intuition.

It's unituitive

We spend a huge amount of time emphasizing the analysis side of math especially in our undergraduate careers (at least at my university). I didn't really touch groups and rings in a really rigorous sense until third year. The rest of that time we spent building intuition for calculus and analysis. Personally I always felt that analysis was more "intuitive" than algebra.

So how do you address this problem if you're like me and algebra just doesn't feel intuitive? Unfortunately there are no short cuts for this one. You really need to just work with the objects. What I can offer though is some ways that can help you achieve this goal (although this is pretty widely applicable to any area of math you are struggling with):

1. Work through tons of problems in your textbook (I used Atiyah and MacDonald). It's slow going, and can feel tedious but really nothing helps build intuition like jumping into the fray.
2. Don't be afraid to go back to older courses. I realized that my knowledge of ideals was really insufficient. If you don't have the background yet no amount of slamming your head into the textbook is going to help.
3. Don't just read the proofs, try them out yourself. If you get stuck, then look at the next line in the proof for a hint.
4. Try and build the theory yourself. Once you have the idea of what commutative algebra "is supposed to do" try and build the theory yourself -- spend a few hours working out your own definitions, and your own propositions. It may take you in a completely different direction than your textbook, but the freedom you get by doing this all yourself will help when you come back to compare notes with the textbook. This approach was particularly important for my understanding, it is definitely time consuming -- but if you like research it is also the most fun.
5. Try not to feel the pressure to move through the textbook quickly. It will take you however long it will take you. If you try and rush it you're just going to be back where you started. I made mistakes here before, I'd rush through without really understanding anything, and when I'd go back I would have so much trouble paying attention because I already "know" what the author is going to do next without really understanding it -- but the predictive nature of my memory built up this illusory false sense of confidence which ultimately led to my demise on the final exam!

Particularly for commutative algebra, look into algebraic geometry. A lot of the developments in commutative were built to study geometry problems. Studying commutative algebra in this diluted form can help build not only intuition (especially if you are interested in geometry) but can also go a long way in solving the other problem I faced when I took commutative.

It feels unmotivated

The other the big problem I faced in my course was that the types of proofs we were looking at felt so unmotivated. Why do we care about the going up theorem? What's the point of studying the radical of an ideal? Why do we care about the characterization of finitely generated modules over PIDs? So what if a local ring has a unique maximal ideal?

Coming from a much stronger analysis background, I could see why studying something like the curvature of manifolds, or proving Stoke's theorem is important and a worthwhile mathematical exercise. I had a much harder time justifying why we should care about these concepts in commutative algebra. Building your intuition like above definitely helps with this, but the best thing you can do in my opinion is to pick a problem you are interested in that you know would require some commutative algebra to solve. Algebraic geometry is a great place to start, but if you're having trouble finding a problem ask a professor -- most would be more than happy to help you come up with a good research project.

There is some really good benefits do doing this. The first is that the motivation to solve the problem you're interested in is clear -- and so for any part of your solution that needs commutative algebra you'll be able to build that motivation by proxy. The definitions which had "obscure" properties the first time you encountered them within a purely commutative algebraic context will now have a specific example where it is important they are defined the way they are.