Good books to study prime spectrum of ring

Question

Let me first introduce my background. I was studying commutative algebra from Abstract algebra from dummit and Foote. But in the book most of the topics related to prime spectrum of the ring isn't well written, and I am having a hard time grasping the concept or even trying to know about those more from internet. It will be great if you can suggest me any good book th at deals with prime sorectum of ring such as structural sheaf, affine scheme etc. At first I thought undergraduate commutative algebra from miles reid will be best. But from reviews available in Amazon many people complaint the proof being too tough and not well written. So may be other books that might help. Any help will be appreciated. Thanks,

Answer 1

Let me suggest Andreas Gathmann's lecture notes. He gives nice examples and explanations and starts by treating the case of varieties to ensure that you will have some intuition. For me this is the best to get a first grasp and an introduction to algebraic geometry. The section on affine schemes is not too long though, but treats everything that is really important.

Another great source would be The rising sea by Ravi Vakil. Pretty much the same style as Gathmann, but leaves lots of things as exercises etc. I also really like that he tells you which things are more important to understand and why that is the case. Sort of like a Terry Tao textbook. That is a quite long document though in case that you want to study it completely. Vakil has a longer and maybe more detailed section about affine schemes.

I can highly recommend both of these. Starting with Gathmann's notes and filling in even more details and aspects (if you want to) by using Vakil's notes is what I would do.