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 nth 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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