Example 39.5.6 (Constant group). Let $G$ be an abstract group. Consider the functor which associates to any scheme $T$ the group of locally constant maps $T \to G$ (where $T$ has the Zariski topology and $G$ the discrete topology). This is representable by the scheme
\[ G_{\mathop{\mathrm{Spec}}(\mathbf{Z})} = \coprod \nolimits _{g \in G} \mathop{\mathrm{Spec}}(\mathbf{Z}). \]
The morphism giving the group structure is the morphism
\[ G_{\mathop{\mathrm{Spec}}(\mathbf{Z})} \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} G_{\mathop{\mathrm{Spec}}(\mathbf{Z})} \longrightarrow G_{\mathop{\mathrm{Spec}}(\mathbf{Z})} \]
which maps the component corresponding to the pair $(g, g')$ to the component corresponding to $gg'$. For any scheme $S$ the base change $G_ S$ is a group scheme over $S$ whose functor of points is
\[ T/S \longmapsto G_ S(T) = \{ f : T \to G \text{ locally constant}\} \]
as before.
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