Example 39.5.1 (Multiplicative group scheme). Consider the functor which associates to any scheme $T$ the group $\Gamma (T, \mathcal{O}_ T^*)$ of units in the global sections of the structure sheaf. This is representable by the scheme

\[ \mathbf{G}_ m = \mathop{\mathrm{Spec}}(\mathbf{Z}[x, x^{-1}]) \]

The morphism giving the group structure is the morphism

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

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

\[ T/S \longmapsto \mathbf{G}_{m, S}(T) = \mathbf{G}_ m(T) = \Gamma (T, \mathcal{O}_ T^*) \]

as before.

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