I wasn’t entirely satisfied with my closing remarks in my last post; in particular I don’t think I explained the term “functor of points” well in an arithmetic context. Luckily I just read something in Liu’s book which completely cleared this up for me!

We’re going to need to define these S-schemes that I alluded to last time; these are basically just schemes over a fixed scheme S. Formally, if S is a scheme then an S-scheme is a morphism of schemes

\pi: X\rightarrow S

(called the “structural morphism”) and a morphism of S-schemes

f: (\pi: X\rightarrow S)\rightarrow (\rho: Y\rightarrow S)

is just a morphism of schemes f: X\rightarrow Y such that \rho \circ f = \pi. Thus if we denote the category of schemes by \text{Sch} then the category of S-schemes, or schemes over S, is just the slice category \text{Sch}/S. So this category encodes all the information about morphisms into a fixed scheme S. If S = \text{Spec}(R) for some ring R then we call \text{Sch}/S the category of schemes over R or R-schemes.

Note that the category of \mathbb{Z}-schemes is equivalent to the general category of schemes; this is because \mathbb Z is the initial object in the category of rings, and by the adjunction isomorphism

\text{Mor}(X,\text{Spec}(\mathbb{Z})) \cong \text{Hom}(\mathbb{Z}, \mathcal{O}_X (X))

we know \text{Spec}(\mathbb{Z}) is the terminal object in the category of schemes. Taking slice categories over terminal objects yields an equivalent category.

Definition: If \pi: X\rightarrow S is an S-scheme then a section of X is a morphism of S-schemes \sigma: S\rightarrow X; if you unpack this, you get the usual notion of a “section” of \pi i.e. a morphism \sigma: S\rightarrow X such that

\phi \circ \sigma = \text{id}_S

The set of sections of X is denoted X(S). If S = \text{Spec}(A) for some ring A then we write X(A) for the set of sections of X.

It’s easy to see by the bijections between morphisms of schemes and morphisms of rings (in the opposite directions) that equipping X with the structure of a scheme over a ring A is the same as giving a sheaf of A-algebras on X. Now let’s see how this helps us understand “rational points” as in the functor of points approach. We have the following nice identification:

Theorem: Let (X, \pi) be a scheme over a field K (i.e. a \text{Spec}(K)-scheme). Then there is a bijection

X(K) \cong \left\{x\in X: k(x) = K\right\}

where k(x) = \mathcal{O}_{X,x}/\mathfrak{m}_x denotes the residue field at x.

Proof: Pick a section \sigma \in X(K), so \sigma is a morphism of schemes \text{Spec}(K)\rightarrow X with \pi \circ \sigma = \text{id}_\text{Spec}(K). Let \sigma^\# : \mathcal{O}_X \rightarrow K denote the morphism of structure sheaves, and let x be the image of the unique point in \text{Spec}(K) under \sigma. Then the stalk homomorphism

\sigma_x^\#: \mathcal{O}_{X,x} \rightarrow K

induces a field homomorphism

\mathcal{O}_{X,x}/\mathfrak{m}_x = k(x) \rightarrow K

because \sigma_x^\# (\mathfrak{m}_x) = \left\{0\right\} since every homomorphism takes units to units. Now k(x) is a K-algebra i.e. there is a field homomorphism K\rightarrow k(x); therefore since field homomorphisms are injective it follows that k(x) = K.

Conversely, suppose that x\in X satisfies k(x) = K. Then the natural map

\mathcal{O}_{X,x}\rightarrow k(x)

induces a morphism of schemes

\text{Spec}(k(x))\rightarrow \text{Spec}(\mathcal{O}_{X,x})

Now take an open set U\subseteq X containing x. The canonical inclusion homomorphism into the colimit \mathcal{O}_X (U)\rightarrow \mathcal{O}_{X,x} induces a morphism of schemes

\text{Spec}(\mathcal{O}_{X,x})\rightarrow \text{Spec}(\mathcal{O}_X (U)) = U

Composing this with the open immersion U\hookrightarrow X gives a morphism \mathcal{O}_{X,x}\rightarrow X. Then the composition of all these maps:

\text{Spec}(k(x)) \rightarrow \text{Spec}(\mathcal{O}_{X,x}) \rightarrow X

sends the unique point of \text{Spec}(k(x)) to x, and thus since k(x) = K we get a unique section \sigma: \text{Spec}(K)\rightarrow X i.e. an element \sigma \in X(K). Both of these constructions can be seen to be mutually inverse to one another. This concludes the proof.

Nice! So here’s another way to think about rational points on a variety – say X is a variety over a field K and x\in X is a point. Then adjoining the coordinates of x to K produces a new field K(x), and K(x) = K precisely when x is a K-rational point. Since adjoining the coordinates of x to K is the same as specifying the residue field at x, this links everything together.

So we should think of rational points on a variety over a field K as sections \text{Spec}(K)\rightarrow X, or as points whose residue field is equal to K. In terms of the functor of points approach, therefore, the functor h_X restricts to the subcategory of fields in the category of rings in several different ways:

h_X (K) = \left\{ K\text{-points of } X \right\}

= \left\{\text{sections } \text{Spec}(K)\rightarrow X \right\}

= \text{Mor}(\text{Spec}(K), X)