group scheme of multiplicative units

Let . Then $\mathrm{Spec}R$ is an affine scheme. The natural homomorphism $\mathbb{Z}\to R$ makes $R$ into a scheme over $\mathrm{Spec}\mathbb{Z}$, i.e. a $\mathbb{Z}$-scheme.

What are the $\mathbb{Z}$-points of $\mathrm{Spec}R$? 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 . 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 $\mathrm{Spec}R$ are exactly the multiplicative units of $S$. For this reason, this scheme is often denoted ${𝔾}_m$. 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 ${𝔾}_m\to \mathbb{Z}$, we can ask about the fibres of this morphism. If we select a point $x$ of $\mathrm{Spec}\mathbb{Z}$, 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 $\mathrm{Spec}R\times \mathrm{Spec}k(x)$, which is the same as $\mathrm{Spec}R\otimes k(x)$. But looking at the definition of $R$, we see that this is , 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.