# 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 ${\mathbb{G}}_{m}$. 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 $\mathrm{Spec}\mathbb{Z}$, 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 | ${\mathbb{G}}_{m}$ |

Related topic | GroupScheme |

Defines | group scheme of multiplicative units |