# 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

 $X=\operatorname{Spec}k[X_{1},\ldots,X_{n}]/\left,$

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

 $k[X_{1},\ldots,X_{n}]/\left\to k^{\prime}$

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 $X=\operatorname{Spec}\mathbb{R}[X]/\left$. 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 ExampleOfFunctorOfPointsOfAScheme 2013-03-22 14:11:07 2013-03-22 14:11:07 archibal (4430) archibal (4430) 4 archibal (4430) Example msc 14A15