Scheme-functors

Question

I am currently reading Mumford-Oda book on Algebraic Geometry. On Chapter 1, Definition 6.6 they talk about scheme-functors. Afterwards, it says that scheme-functors are precisely those given by $$F(R)=\text{Mor}(\text{Spec}(R),X)$$ for some scheme X. I tried to prove it, but I could not. What is the main idea?

Answer 1

In one direction, you need to show that given a scheme $X$, the functor $R \mapsto \text{Mor}(\text{Spec}(R),X)$ satisfies the properties of a scheme-functor.

In the other direction, you need to find a scheme $X$ representing the scheme functor. For this, there is a "geometric realization functor", described in detail (as Mumford and Oda note in their book) by Demazure and Gabriel in their book Groupes Algébriques. Its definition, for any functor $F$, is

$$|F| = \varinjlim_{(R,\rho)} \text{Spec}(R),$$

where the colimit is taken over the category of pairs $(R,\rho)$ for $R$ a commutative ring, and $\rho \in F(R)$, with morphisms $(R,\rho) \to (R',\rho')$ being morphisms $f: R \to R'$ such that $F(f)(\rho) = \rho'$. Note by Yoneda's Lemma, an element $\rho \in F(R)$ corresponds to a morphism of functors $\hat\rho: h_R \to F$, where $h_R$ is the functor representing the commutative ring $R$. The intuition here is that the functor of points $R \mapsto \text{Mor}(\text{Spec}(R),X)$ of a scheme $X$ is basically a way to separate out the various types of "points" of a scheme, with the notion of "point" depending on a specific ring. The geometric realization glues these points back together "in the most natural way".

Demazure and Gabriel show in detail that the functor taking a scheme to its functor of points is inverse to the functor taking a scheme functor to its geometric realization, up to isomorphism.

The trick is the second condition in the definition of a scheme-functor, which basically stipulates an open covering by affine schemes. To see this, note affine schemes correspond to representable functors on CRing. In other words, the scheme-functor corresponding to $\text{Spec}(R)$ is the functor

$$S \mapsto \text{Mor}_\text{Schemes}(\text{Spec}(S),\text{Spec}(R)) = \text{Mor}_{\text{CRing}^\text{op}}(S,R) = \text{Mor}_{\text{CRing}}(R,S) = h_R(S)$$

It is a good exercise to check the equivalence of categories between schemes and scheme-functors for affine schemes.