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.

Title	group scheme of multiplicative units
Canonical name	GroupSchemeOfMultiplicativeUnits
Date of creation	2013-03-22 14:09:01
Last modified on	2013-03-22 14:09:01
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	7
Author	mathcam (2727)
Entry type	Example
Classification	msc 14A15
Synonym	${𝔾}_m$
Related topic	GroupScheme
Defines	group scheme of multiplicative units