group scheme of multiplicative units
What are the -points of ? Recall that an -point of a scheme is a morphism ; if we are working in the category of schemes over , then the morphism is expected to commute with the structure morphisms. So, here, we seek homomorphisms . Such a homomorphism must take to an invertible element, and it must take to its inverse. Therefore there are two, one taking to and one taking to . One recognizes these as the multiplicative units of , and indeed if is any ring, then the -points of are exactly the multiplicative units of . For this reason, this scheme is often denoted . 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 , we can ask about the fibres of this morphism. If we select a point of , we have two choices. Such a point must be a prime ideal of , and there are two kinds: ideals generated by a prime number, and the zero ideal. If we select a point with residue field , then the fiber of this morphism will be , which is the same as . But looking at the definition of , we see that this is , which is just the scheme whose points are the nonzero elements of .
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|
|Date of creation||2013-03-22 14:09:01|
|Last modified on||2013-03-22 14:09:01|
|Last modified by||mathcam (2727)|
|Defines||group scheme of multiplicative units|