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:

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

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

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