Functor of Points Versus Locally Ringed Spaces

A year ago, the Secret Blogging Seminar had a long thread on how to teach algebraic geometry, one that I never managed to read in its entirety before now. The thread was very interesting. The original post started with the premise that the best way to introduce the idea of an affine scheme is to use the set of maximal ideals, rather than the set of prime ideals as is standard. This is sufficient for classical algebraic geometry.

The thread then wanders off into a different question, of whether the best way to define schemes is as locally ringed space, or rather in terms of a scheme’s functor of points. In the functor of points view, you think of a scheme S as being given by the functor Hom(Spec A, S), as A varies of all rings. You can think of “Spec A” as representing a generalized point — true points correspond to fields. The idea of a generalized point never appealed to me, but still the functor of points view seems more natural to me. What do we know about a general scheme? We know how to map affine schemes into it, and how these pieces are glued together. I remember when I first tried to read Hartshorne, and I learned about sheaves, and then ringed spaces, and then when I got to the requirement that the induced map between stalks had to be local homomorphisms, I got frustrated that after all that machinery, you still needed a weird extra condition to get the right definition. With the functor of points, we start with affine schemes as the dual of the category of rings, and then say what we mean by “gluing” affine pieces together. The Secret Blogging Seminar thread, though, has several practicing algebraic geometers arguing that some notions, such as proper morphism, are much clearer from the locally-ringed space point of view.