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

with the structure morphism $X\to \mathrm{Spec}k$ induced from the natural embedding $k\to k[X_1,\mathrm{\dots },X_n]$.

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 $\mathrm{Spec}{k}^{\prime }\to X$ (observe that since we have an embedding $k\to k^{\prime }$ we have a morphism $\mathrm{Spec}{k}^{\prime }\to \mathrm{Spec}k$, so $\mathrm{Spec}{k}^{\prime }$ is natuarlly a $k$-scheme). Since $X$ is affine, this must come from a ring homomorphism

which takes elements of $k$ to themselves inside $k^{\prime }$. Such a homomorphism is completely specified by specifying the images of $X_1,\mathrm{\dots },X_n$; for it to be a homomorphism, these images must satisfy $f_1,\mathrm{\dots },f_m$. 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 . Then $X$ has no points over $ℝ$, its natural base field. Over $ℂ$, 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 . Then specifying a $k^{\prime }$-point on $X$ amounts to choosing an image ${\kappa }_i+{\lambda }_iϵ$ 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 -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 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.