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$$ X = \Spec k[X_1,\ldots,X_n]/\left<f_1,\ldots,f_m\right>,$$ with the structure morphism $X\to\Spec k$ induced from the natural embedding $k\to k[X_1,\ldots,X_n]$ .

Let $k'$ be some field extension of $k$ . What are the $k'$ -points of $X$ ? Recall that a $k'$ -point of $X$ is by definition a morphism $\Spec k' \to X$ (observe that since we have an embedding $k\to k'$ we have a morphism $\Spec k' \to \Spec k$ , so $\Spec k'$ is natuarlly a $k$ -scheme). Since $X$ is affine, this must come from a ring homomorphism$$ k[X_1,\ldots,X_n]/\left<f_1,\ldots,f_m\right> \to k'$$ which takes elements of $k$ to themselves inside $k'$ . 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 $k'$ -point on $X$ is identified with an element of $(k')^n$ satisfying all the polynomials $f_i$ .

If $k'$ 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 $k'$ 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 $X=\Spec \mathbb{R}[X]/\left<X^2+1\right>$ . 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 $k'$ (or in fact $k$ ) was a field. One often chooses $k'$ as something other than a field in order to solve a problem. For example, one can take $k' = k[\epsilon]/\left<\epsilon^2\right>$ . Then specifying a $k'$ -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.