References for the functor of points vision of schemes.

Question

I recently discovered the idea of the functor of points. I would like to find a reference where the different visions of scheme are presented.

It seems to me that the classical texts emphasize the classical vision of locally ringed spaces. I don't know the reason for this, isn't the functorial definition supposed to make some constructions easier?

Answer 1

You can read The Geometry of Schemes by Eisenbud and Harris, whose last chapter is about functorial viewpoint, or Introduction to algebraic geometry and algebraic groups by Demazure and Gabriel which is written essentially with functorial viewpoint.

The classical vision of locally ringed spaces is generally emphasized because it may seem less abstract at first sight for a beginner which knows classical varieties : indeed, schemes can be seen as "enriched varieties", which seems better suited for geometric intuition.

Answer 2

Maybe you can look at the construction of Hilbert and Quot schemes from FGA Explained by Nitin Nitsure (also available in https://arxiv.org/abs/math/0504590 ). It has some concrete examples such as realizing Projective spaces and Grassmannians as objects that represents some particular kind of Quot functors.