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Maths > Algebraic geometry > Functor of points

Functors of points and base ring

Posted by Martin Orr on Thursday, 08 October 2009 at 09:58

So far in my series on functors of points, I have considered functors for some fixed field . We begin by observing that we may allow to be any ring. Then I consider whether it is possible to relate functors with base ring to functors with base ring , with only partial success.

Functors over rings

So far, I have only defined the functor of points of an algebraic-geometric object defined over to be a functor where is some field. (For convenience, I shall call this a -functor in this post.) However, if you look carefully, you will see that we have never used the fact that is a field. Hence we may consider a functor where is any ring, and view it as the functor of points of an object defined over .

Recall that the basic example of a -functor is an affine -scheme . This tells you the sets of solutions in a varying -algebra to some fixed polynomials with coefficients in . The same thing is true for -functors: an affine -scheme is defined by some polynomials with coefficients in , and the functor of points tells you their solutions over all -algebras.

This is not just generalisation for the sake of it. For example, suppose we have a polynomial with integer coefficients. In order to understand integer solutions to , it is often useful to consider mod for various (or over the -adics), as well as looking at over or extensions thereof. This means that it is not enough to consider the corresponding affine -scheme , because the functor of points of is only defined on and is not a -algebra in any way. However is a -algebra, so the functor of points of the corresponding affine -scheme contains all this useful information.

Changing the base ring

In the locally-ringed spaces approach to algebraic geometry, you define a -scheme to be a -scheme together with a morphism . So the question arises, is this description valid for functors? More precisely, is a -functor the same as a -functor together with a natural transformation to ?

Because functors are much more general than schemes, there is no particular reason to assume that this should be true. In fact there is a simple map { -functors with morphisms to } { -functors }, but I cannot find any map in the reverse direction.

To construct this map, suppose we have a functor and a natural transformation . We want to define a functor .

Suppose that is a -algebra (where is the defining homomorphism ). The easy way to see what should be is to view -points of as morphisms , as in the last post. Then -points of are those morphisms such that the following diagram commutes:

This gives the formula

The other way

In an attempt to go the other way, I looked for a map { -functors } { -functors with morphisms to }, but have been unable to find one. Let us more modestly restrict attention to affine schemes. Then there exists such a map, because the affine subcategories of both sides are dual to the category of -algebras, but I can't find a way to describe the map in terms of functors of points.

Of course you could describe the map by first saying "construct the -algebra such that " (you can do this by ) but this is not very elegant, and doesn't generalise to non-affine schemes, where I hope the correspondence still holds.

A serious obstacle to a nicer description of this map seems to be that has a single -point for every -algebra , so getting the complicated functor out of this is not going to be straightforward.