5
$\begingroup$

I hope this question is viable for this site. I'm sincerely sorry, if you think it isn't.

For a lot of time, "EGA" by Alexander Grothendieck and Jean Dieudonne was "the" reference on the basics of scheme theory and homological methods. Students and teachers got a lot of books following EGA - Hartshorne's "Algebraic Geometry" and the rest. Those are textbooks. Some can say they are more pedagogical than "EGA". Especially, books like Vakil's "Foundations of Algebraic Geometry", which try to convey the intuition on the highly technical and abstract subject for the new learners. But "EGA" had it's pros. It is complete, it has all the proofs, it's as general as possible. Despite what many people said (and they were right, it's just they probably confused "majority" with "everyone"), for some people "EGA" was "the" source. They like it abstract and rigorous. It's how their mind works.

Nowadays, we have "The Stacks Project". It's 5000 pages for modern algebraic geometry - schemes and stacks, and the needed prerequisites. One could say it's "the modern EGA" or "EGA of 21st century". I'm not saying that "EGA" is not valuable anymore, I don't have expertise to claim it even if I wanted to, and I certainly don't want to.

For those who could benefit more from the "handbook-like" presentation, I wonder how a first course from "The Stacks Project" could be taught. It has a lot of information in there, certainly, one, especially, a novice, could get lost in there.

I would like you to propose a roadmap of studying scheme theory from "The Stacks Project". What chapters are relevant for the first fundamental independent study of modern algebraic geometry, and what are reserved for later or simply more specialized.

P.S. For those who are not familiar with "The Stacks Project", here's the link. Also, I would like this not to turn into a discussion of why something like this is a bad idea. I don't propose for everyone to study like that, I only think it can be useful for those whose mind works in a certain way.

$\endgroup$
  • 7
    $\begingroup$ I think there is no good answer to this, because it heavily depends on what a person already knows. In my opinion, the Stacks Project is much more useful for top-down learning (as opposed to bottom-up): find a result you want to understand, and go deeper to read about the things you need. An experienced algebraic geometer will typically only need to go down one or two levels before she finds statements she's familiar with. $\endgroup$ – R. van Dobben de Bruyn Nov 7 '16 at 16:25
  • $\begingroup$ That is to say, if you wanted to use it bottom-up, you can just keep going deeper. For example, do you first need to study the foundations of set theory before studying algebraic geometry? (The Stacks Project does not contain these (except for a brief review), but EGA does go much deeper.) $\endgroup$ – R. van Dobben de Bruyn Nov 7 '16 at 16:29
  • $\begingroup$ @R.vanDobbendeBruyn That's what the question is about. What can, and should be skipped on a first serious comprehensive (not a quick skimming) study from this "book". Of course, I have no doubt that people intended "The Stacks Project" as a reference. But I'm convinced that for some specially-minded students it might serve better than the standard textbooks, like Vakil's notes (which are awesome for some people, too). $\endgroup$ – TavukKaghul Nov 7 '16 at 16:50
  • 1
    $\begingroup$ @TavukKaghul: On what basis are you "convinced" than it is better to start with SP than various other available sources for first learning the substance of the subject? Are you at a place where there is anyone with experience in the field with whom you can discuss this issue in person? $\endgroup$ – nfdc23 Nov 7 '16 at 17:26
  • $\begingroup$ I also don't quite see the point of this question. If you're a self-studying type of person, then you'll find your own way to make good use of this material, in conjunction with other available materials. If you need an expert to guide you, then find an expert who knows you and can tailor something to your needs. I don't see how there can be a one-size-fits-all guide. $\endgroup$ – Timothy Chow Nov 8 '16 at 16:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.