Example 39.5.3 (Additive group scheme). Consider the functor which associates to any scheme $T$ the group $\Gamma (T, \mathcal{O}_ T)$ of global sections of the structure sheaf. This is representable by the scheme

\[ \mathbf{G}_ a = \mathop{\mathrm{Spec}}(\mathbf{Z}[x]) \]

The morphism giving the group structure is the morphism

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

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

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

as before.

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