Example 39.5.2 (Roots of unity). Let $n \in \mathbf{N}$. Consider the functor which associates to any scheme $T$ the subgroup of $\Gamma (T, \mathcal{O}_ T^*)$ consisting of $n$th roots of unity. This is representable by the scheme

\[ \mu _ n = \mathop{\mathrm{Spec}}(\mathbf{Z}[x]/(x^ n - 1)). \]

The morphism giving the group structure is the morphism

\begin{eqnarray*} \mu _ n \times \mu _ n & \to & \mu _ n \\ \mathop{\mathrm{Spec}}( \mathbf{Z}[x]/(x^ n - 1) \otimes _{\mathbf{Z}} \mathbf{Z}[x]/(x^ n - 1)) & \to & \mathop{\mathrm{Spec}}(\mathbf{Z}[x]/(x^ n - 1)) \\ \mathbf{Z}[x]/(x^ n - 1) \otimes _{\mathbf{Z}} \mathbf{Z}[x]/(x^ n - 1) & \leftarrow & \mathbf{Z}[x]/(x^ n - 1) \\ x \otimes x & \leftarrow & x \end{eqnarray*}

Hence we see that $\mu _ n$ is a group scheme over $\mathbf{Z}$. For any scheme $S$ the base change $\mu _{n, S}$ is a group scheme over $S$ whose functor of points is

\[ T/S \longmapsto \mu _{n, S}(T) = \mu _ n(T) = \{ f \in \Gamma (T, \mathcal{O}_ T^*) \mid f^ n = 1\} \]

as before.

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