example of functor of points of a scheme

Let $X$ be an affine scheme of finite type over a field $k$ . Then we must have

$X=\operatorname{Spec}k[X_{1},\ldots,X_{n}]/\left<f_{1},\ldots,f_{m}\right>,$

with the structure morphism $X\to\operatorname{Spec}k$ induced from the natural embedding $k\to k[X_{1},\ldots,X_{n}]$ .

Let $k^{\prime}$ be some field extension of $k$ . What are the $k^{\prime}$ -points of $X$ ? Recall that a $k^{\prime}$ -point of $X$ is by definition a morphism $\operatorname{Spec}k^{\prime}\to X$ (observe that since we have an embedding $k\to k^{\prime}$ we have a morphism $\operatorname{Spec}k^{\prime}\to\operatorname{Spec}k$ , so $\operatorname{Spec}k^{\prime}$ is natuarlly a $k$ -scheme). Since $X$ is affine, this must come from a ring homomorphism

$k[X_{1},\ldots,X_{n}]/\left<f_{1},\ldots,f_{m}\right>\to k^{\prime}$

which takes elements of $k$ to themselves inside $k^{\prime}$ . Such a homomorphism is completely specified by specifying the images of $X_{1},\ldots,X_{n}$ ; for it to be a homomorphism, these images must satisfy $f_{1},\ldots,f_{m}$ . In other words, a $k^{\prime}$ -point on $X$ is identified with an element of $(k^{\prime})^{n}$ satisfying all the polynomials $f_{i}$ .

If $k^{\prime}$ is an algebraically closed field, a point on $X$ corresponds uniquely to a point on an affine variety defined by the same equations as $X$ . If $k^{\prime}$ is just any extension of $k$ , then we have simply found which new points belong on $X$ when we extend the base field. T

For an example of why schemes contain much more information than the list of points over their base field, take $X=\operatorname{Spec}\mathbb{R}[X]/\left<X^{2}+1\right>$ . Then $X$ has no points over $\mathbb{R}$ , its natural base field. Over $\mathbb{C}$ , it has two points, corresponding to $i$ and $-i$ .

This suggests that schemes may be the appropriate adaptation of varieties to deal with non-algebraically closed fields.

Observe that we never used the fact that $k^{\prime}$ (or in fact $k$ ) was a field. One often chooses $k^{\prime}$ as something other than a field in order to solve a problem. For example, one can take $k^{\prime}=k[\epsilon]/\left<\epsilon^{2}\right>$ . Then specifying a $k^{\prime}$ -point on $X$ amounts to choosing an image $\kappa_{i}+\lambda_{i}\epsilon$ for each $X_{i}$ . It is clear that the $\kappa_{i}$ must satisfy the $f_{j}$ . But upon reflection, we see that the $\lambda_{i}$ must specify a tangent vector to $X$ at the point specified by the $\kappa_{i}$ . So the $k[\epsilon]/\left<\epsilon^{2}\right>$ -points tell us about the tangent bundle to $X$ . Observe that we made no assumption about the field $k$ — we can extract these “tangent vectors” in positive characteristic or over a non-complete field.

The ring $k[\epsilon]/\left<\epsilon^{2}\right>$ and rings like it (often any Artinian ring) can be used to define and study infinitesimal deformations of schemes, as a simple case of the study of families of schemes.

Title	example of functor of points of a scheme
Canonical name	ExampleOfFunctorOfPointsOfAScheme
Date of creation	2013-03-22 14:11:07
Last modified on	2013-03-22 14:11:07
Owner	archibal (4430)
Last modified by	archibal (4430)
Numerical id	4
Author	archibal (4430)
Entry type	Example
Classification	msc 14A15