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Fawzi Hreiki (Apr 22 2021 at 15:26):I've been trying to learn some algebraic geometry and I've come across some notes which take seriously the functorial POV so I thought they may be worth sharing here (seeing as there aren't many textbooks which fully commit in this direction). Sam Raskin's notes, Jonathan Wise's notes, Ivan Mirkovic's notes. If anyone has any other relevant references (books, notes, papers, whatever) that would be great.
Fawzi Hreiki (Apr 22 2021 at 15:29):It's quite miraculous how much simpler (and even more concrete) the subject becomes when you take this approach. The only hump for some I guess is fully internalising the Yoneda lemma but that's still much simpler than say a whole preliminary chapter on sheaves, locally ringed spaces, etc.. as in Vakil's notes (which are nevertheless still great).
John Baez (Apr 22 2021 at 17:21):Maybe you've read the comments about how to teach algebraic geometry on this blog article? The whole conversation is interesting, but I especially like the part around where James Borger says:
I agree completely with David Ben-Zvi. It seems vastly preferable to me to define the category of schemes (or better, algebraic spaces) as a certain full subcategory of the category of functors from Rings to Sets. Then things simplify greatly. No silly topological spaces, no prime ideals, no axiom of choice, no local rings, no fields. All you need is all you ever used anyway: the category of rings, covers, and descent. Of course, you can use ideals, for instance prime and maximal ideals, if you want– they’re just no longer necessary to set up the theory.
I would love to teach a class from this point of view.
Fawzi Hreiki (Apr 22 2021 at 17:26):Yeah I’ve read that blog post and the comments - definitely worthwhile.
John Baez (Apr 22 2021 at 17:27):I'm glad you're looking around for sources that actually teach algebraic geometry. I'm not very good at algebraic geometry and would like to learn it this way sometime.
Fawzi Hreiki (Apr 22 2021 at 17:30):The thing I’m most surprised with is the concreteness and geometric intuition in those three notes I posted.
I initially expected that the ‘functorial approach’ would come with a sterility and lack of motivation but that’s not the case.
Fawzi Hreiki (Apr 22 2021 at 17:32):All of those notes actually develop a lot of the algebra (normalisation, nullstellensatz, dimension, etc) along with the geometry so the prerequisites are quite low
John Baez (Apr 22 2021 at 17:33):Great! So it's a teachable approach.
David Michael Roberts (Aug 13 2021 at 06:50):This is old, but there's a textbook by Demazure and Gabriel's called _Groupes Algebriques_ that takes a serious functorial approach to the underlying scheme theory, IIRC. There's fair amount in there that is really just algebraic geometry, and one review of the english translation of the first half complains there's not enough material on algebraic groups!
সায়ন্তন রায় (Aug 17 2021 at 12:50):It's very interesting to see that Grothendieck was Demazure's doctoral advisor.
সায়ন্তন রায় (Aug 17 2021 at 12:58):Also, there is a partial english translation of the book by J. Bell which goes under the name Introduction to Algebraic Geometry and Algebraic Groups.
Fawzi Hreiki (Aug 17 2021 at 13:05):Yeah I’ve seen that book. It’s nice although it doesn’t really commit fully to the functor of points approach.
Fawzi Hreiki (Aug 17 2021 at 13:08):I have found though some more references since I first asked the question. My favourite by far are still Wise’s notes but there are the fantastic notes by Dolgachev which actually introduce AG from scratch using the functor of points
Joe Moeller (Aug 17 2021 at 13:10):I'm glad this topic popped back up. I've been thinking recently I want to learn this stuff, but I forgot all the resources I had previously opened.
Fawzi Hreiki (Aug 17 2021 at 13:10):It’s been a bit of a hassle finding sources online but I’ve managed to compile a reasonable list. I should probably link them on the nLab at some point.
David Michael Roberts (Aug 18 2021 at 06:50):@Fawzi Hreiki that's asking me for a password :-(
Fawzi Hreiki (Aug 18 2021 at 07:06):David Michael Roberts said:
Fawzi Hreiki that's asking me for a password :-(
That's strange. Perhaps try this: Introduction-to-Algebraic-Geometry-Igor-V.-Dolgachev.pdf
David Michael Roberts (Aug 18 2021 at 10:39):Yeah, it's weird. Even just his main website, coming from a Google search gets me the password page