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.
Best Answer
Since your question might interest other readers, allow me to expand it.
Given a scheme $T$, you can associate to it the contravariant functor $h_T: \mathcal{ Schemes}^\text{opp} \to \mathcal{Sets}$. In a nutshell, Eivind's request is for documents showing how you can study the scheme $T$ by studying the functor $h_T$.
Not all functors
$$h: \mathcal{ Schemes}^\text{opp} \to \mathcal{Sets}$$ 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.
Edit Mumford and Oda have published in 2015 the rest of Mumford's course mentioned in b) as the book Algebraic Geometry II at the Hindustan Book Agency.
The link I gave in b) fortunately still works, and we should be grateful to the authors for this act of generosity.