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 -schemes that I alluded to last time; these are basically just schemes over a fixed scheme . Formally, if is a scheme then an -scheme is a morphism of schemes

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

is just a morphism of schemes such that . Thus if we denote the category of schemes by then the category of -schemes, or **schemes over **, is just the slice category . So this category encodes all the information about morphisms into a fixed scheme . If for some ring then we call the **category of schemes over ** or -schemes.

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

we know is the terminal object in the category of schemes. Taking slice categories over terminal objects yields an equivalent category.

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

The **set of sections** of is denoted . If for some ring then we write for the set of sections of .

It’s easy to see by the bijections between morphisms of schemes and morphisms of rings (in the opposite directions) that equipping with the structure of a scheme over a ring is the same as giving a sheaf of -algebras on . 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 be a scheme over a field (i.e. a -scheme). Then there is a bijection

where denotes the residue field at .

**Proof**: Pick a section , so is a morphism of schemes with . Let denote the morphism of structure sheaves, and let be the image of the unique point in under . Then the stalk homomorphism

induces a field homomorphism

because since every homomorphism takes units to units. Now is a -algebra i.e. there is a field homomorphism ; therefore since field homomorphisms are injective it follows that .

Conversely, suppose that satisfies . Then the natural map

induces a morphism of schemes

Now take an open set containing . The canonical inclusion homomorphism into the colimit induces a morphism of schemes

Composing this with the open immersion gives a morphism . Then the composition of all these maps:

sends the unique point of to , and thus since we get a unique section i.e. an element . 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 is a variety over a field and is a point. Then adjoining the coordinates of to produces a new field , and precisely when is a -rational point. Since adjoining the coordinates of to is the same as specifying the *residue field* at , this links everything together.

So we should think of rational points on a variety over a field as *sections* , or as points whose residue field is equal to . In terms of the functor of points approach, therefore, the functor restricts to the subcategory of fields in the category of rings in several different ways: