group scheme of multiplicative units

Let R=[X,Y]/XY-1. Then SpecR is an affine schemeMathworldPlanetmath. The natural homomorphismPlanetmathPlanetmath R makes R into a scheme over Spec, i.e. a -scheme.

What are the -points of SpecR? Recall that an S-point of a scheme X is a morphism SX; 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 homomorphismsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath [X,Y]/XY-1. Such a homomorphism must take X to an invertible element, and it must take Y to its inverseMathworldPlanetmathPlanetmathPlanetmath. Therefore there are two, one taking X to 1 and one taking X to -1. One recognizes these as the multiplicative units of , and indeed if S is any ring, then the S-points of SpecR are exactly the multiplicative units of S. For this reason, this scheme is often denoted 𝔾m. 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 𝔾m, we can ask about the fibres of this morphism. If we select a point x of Spec, we have two choices. Such a point must be a prime idealMathworldPlanetmathPlanetmathPlanetmath of , and there are two kinds: ideals generated by a prime numberMathworldPlanetmath, and the zero idealMathworldPlanetmathPlanetmath. If we select a point x with residue fieldMathworldPlanetmath k(x), then the fiber of this morphism will be SpecR×Speck(x), which is the same as SpecRk(x). But looking at the definition of R, we see that this is Speck(x)[X,Y]/XY-1, 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 varietiesPlanetmathPlanetmath.

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 𝔾m
Related topic GroupScheme
Defines group scheme of multiplicative units