Let $R=\mathbb{Z}[X,Y]/\left<XY-1\right>$ . Then $\Spec R$ is an affine scheme. The natural homomorphism $\mathbb{Z}\to R$ makes $R$ into a scheme over $\Spec \mathbb{Z}$ , i.e. a $\mathbb{Z}$ -scheme.

What are the $\mathbb{Z}$ -points of $\Spec R$ ? Recall that an $S$ -point of a scheme $X$ is a morphism $S\to X$ ; if we are working in the category of schemes over $Y$ , then the morphism is expected to commute with the structure morphisms. So, here, we seek homomorphisms $\mathbb{Z}[X,Y]/\left<XY-1\right> \to \mathbb{Z}$ . Such a homomorphism must take $X$ to an invertible element, and it must take $Y$ to its inverse. Therefore there are two, one taking $X$ to $1$ and one taking $X$ to $-1$ . One recognizes these as the multiplicative units of $\mathbb{Z}$ , and indeed if $S$ is any ring, then the $S$ -points of $\Spec R$ are exactly the multiplicative units of $S$ . For this reason, this scheme is often denoted $\mathbb{G}_m$ . It is an example of a group scheme.

We can regard any morphism as a family of schemes, one for each fibre. Since we have a morphism $\mathbb{G}_m \to \mathbb{Z}$ , we can ask about the fibres of this morphism. If we select a point $x$ of $\Spec \mathbb{Z}$ , we have two choices. Such a point must be a prime ideal of $\mathbb{Z}$ , and there are two kinds: ideals generated by a prime number, and the zero ideal. If we select a point $x$ with residue field $k(x)$ , then the fiber of this morphism will be $\Spec R \times \Spec k(x)$ , which is the same as $\Spec R\otimes k(x)$ . But looking at the definition of $R$ , we see that this is $\Spec k(x)[X,Y]/\left<XY-1\right>$ , which is just the scheme whose points are the nonzero elements of $k(x)$ .

In other words, we have a family of schemes, one in each characteristic. Of course, normally one wants a family to have some additional sort of smoothness condition, but this demonstrates that it is quite possible to have a family of schemes in different characteristics; sometimes one can deduce the behaviour in one characteristic from the behaviour in another. This approach can be useful, for example, when dealing with Hilbert modular varieties.