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I want to understand algebraic geometry from the functorial viewpoint. I've found a set of notes (linked below) that develop algebraic geometry from the elementary beginnings in this framework. They go under the name "Introduction to Functorial Algebraic Geometry" (following a summer course held by Grothendieck), and are in parts in an almost unreadable shape. Is there an available, elementary, and readable source which takes the functorial viewpoint?

http://www.math.jussieu.fr/~leila/grothendieckcircle/FuncAlg.pdf

EDIT: Great answers so far, thanks! I would like to add that if anyone were to have a better scan of the document linked to above, or if anyone has vol. 2 of the notes I would be very interested in aquiring a copy.

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Try Demazure and Gabriel. –  Josh Shadlen Jun 27 '11 at 15:34
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As Josh already said, Demazure/Gabriel is the definite source. If you can read german, try to contact Marc Nieper-Wißkirchen. He has written lecture notes in which he developes algebraic geometry from the functorial viewpoint. –  Martin Brandenburg Jun 27 '11 at 15:42
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The question said "available, elementary and readable"... D&G doesn't fit all three points :-) But it's indeed the best answer I know ; the functorial point of view is also mentioned in Eisenbud&Harris "The geometry of schemes" if I remember well, but it isn't the base. –  Julien Puydt Jun 27 '11 at 18:08
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In my memory, EGA does not qualify -- in EGA the schemes/spaces are defined as ringed spaces and not as sheaves on a site which are locally representable. Knutson's "Algebraic spaces" has more of the functorial approach. FGA (and recent "FGA explained" cf. ncatlab.org/nlab/show/FGA+explained ) are more useful for start, complemened by Demazure-Gabriel. –  Zoran Skoda Jun 27 '11 at 20:00
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The functor of points approach is written down with great clarity in the Springer edition, and it is used to construct the Grassmannian and flag varieties, among other things. Now, a functorial approach to the basics of schemes is a different matter, but EGA 1 (the Springer edition) is a reference for all the notions mentioned in the accepted answer. –  Keerthi Madapusi Pera Jun 28 '11 at 22:22
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3 Answers 3

up vote 13 down vote accepted

Dear Eivind, since your question might interest other readers, allow me to expand it.
Given a scheme $T$, you can associate to it the contravariant functor $h_T: \mathcal{ Schemes}^{opp} \to \mathcal{Sets}$. In a nutshell, Eivind's request is for documents showing how you can study the scheme $X$ by studying the functor $h_T$.
Not all functors
$$h: \mathcal{ Schemes}^{opp} \to \mathcal{Sets}$$ come from a scheme $T$ and you have to characterize those that do. A pleasant surprise is that it is enough to look at what your functor does on affine rings, so that in effect you study a functor $$k: \mathcal{ Rings} \to \mathcal{Sets}$$ The characterization is then relatively easy (once Grothendieck has shown us what to do!): the functor must be a sheaf in the Zariski topology and satisfy a condition which translates that a scheme is covered by affines. There is a real aesthetic appeal to the realization that concepts like closed or open immersions, tangent spaces,... can be expressed purely in terms of functors. The appeal is not only aesthetic, but also technical: it is with the functorial method that parameter spaces, like Grassmannians, Hilbert schemes,...are constructed.
And where is all this to be found?

a) In Mumford's Introduction to Algebraic Geometry, affectionately called The Red Book, Chapter Ii, §6: The functor of points of a prescheme. The book has been reprinted by Springer

b) Unfortunately Mumford didn't publish the rest of his course. However there is a set of notes (more than 300 pages) coauthored by Oda, corresponding to Chapters I to VIII, which can be found on-line here

c) Eisenbud and Harris wrote a book on schemes which came out of a common project with Mumford. The last chapter (chapter VI) is called Schemes and Functors and, as the name says, is dedicated to the point of view we are discussing.

d) Brian Osserman has a very pleasant short hand-out here on the use of the functorial point of view to illuminate the construction of the product of two schemes.

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Very nice answer. –  Martin Brandenburg Jun 27 '11 at 22:31
    
Thank you, Martin. –  Georges Elencwajg Jun 27 '11 at 22:47
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The main reference that I'm aware of is

  • Demazure and Gabriel -- Groupes algébriques (1970)

If I remember correctly the first chapter develops the fundamentals of scheme theory from the functorial point of view. Part of the above work (including the relevant first chapter) was translated into English and published as

  • Demazure and Gabriel (translated by J. Bell) -- Introduction to algebraic geometry and algebraic groups (1980)

Another source is

  • Jantzen -- Representations of algebraic groups (2nd ed., 2003)

Jantzen basically copies the approach of Demazure and Gabriel but his presentation is easier to read. Demazure and Gabriel are pretty rigorous with their foundations and talk a lot about Grothendieck universes and such things, whereas Jantzen doesn't go overboard with these kinds of details.

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If you are group-oriented and functiorially minded, then Sancho de Salas's book "Groupos algebraicos y theoria de invariantes" seems great (although I haven't gone through it yet). (ah, the book's in spanish) librarything.com/work/2781283 –  babubba Jun 27 '11 at 21:02
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Toen's master course on stacks is awesome and it starts from scratch. Unfortunately it doesn't take matters too far (although I guess the next for him would have been model sites and $\infty$-stacks!). http://ens.math.univ-montp2.fr/~toen/m2.html

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