Learning Algebraic Geometry by EGA

Question

Does it make sense to study algebraic geometry by Grothendieck's EGA?

I know French and I want to know whether I can read a treatise Grothendieck to explore this area.

I am familiar with the abstract algebra, commutative algebra, algebraic topology (in the amount of Bourbaki's books), and differential geometry.

Answer 1

If you are thinking in reading all pages from the first one, I don't know. But if you skip some parts, I think yes. Even to study commutative algebra, EGA chapters $0_{III}, 0_{IV}$ are a good sequel to Atiyah's (even today with Bourbaki's Commutative Algebra chapter X, I like much more the exposition of EGA $0_{IV}$). Does it consume more time than other texts? Probably yes, but it depends on the reader. I think it is not so unusual to move from Hartshorne to EGA I and II when learning scheme theory in order to avoid so many noetherian hypothesis.

Shall I understand that you have studied these topics from Bourbaki's books? In some sense EGA is close to Bourbaki in style, and since I think it is mainly a matter of taste to start with EGA, Harshorne, Liu, Mumford-Oda, etc., if you like Bourbaki, probably EGA is a good choice. In this case I would recommend you to start with chapter I (after learning basic sheaf theory from any short source), and only read chapter $0_I$ as needed.

Having said this, I have not yet read but a few pages of Vakil's book, but with this caution, Vakil's book seems to be a good text, in some sense not far from EGA in style, and it was written as a textbook, so it is probably a better choice to start. I hope it will be soon in print form, but at least we have an e-reader version.

Sorry for writing all these comments as an answer, but it was too long for a comment.

Answer 2

I would not recommend that to you. First it has a great generality which you propably will not need (and will consume time). Second it does not have exercises which are essential for learning math (not only, but at least at that level). Here are my personal recommendations: 1) Hartshorne's book. It teaches you the interesting questions and important questions in short time and it has tons of exercises. If you need greater generality of statements later (apart from algebraic geometry over fields) you can easily look them up elsewhere. It is my recommendation if you want to do geometry. 2) Görtz Wedhorns book. It has a great generality and lots of exercises. It is very long and you shouldn't expect that you come to the hot topics soon. 3) Ravi Vakil's online course. It is a unusual way to learn algebraic geometry but it is also fun. One big advantage is that everything is written down in a modern way (e.g. using spectral sequences). 4) Quing Liu's book: If you want to do arithmetic Liu's book is a good start. Unfortunately you have to learn about sheaf cohomology elsewhere.