The Stacks project

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.

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 03YW. Beware of the difference between the letter 'O' and the digit '0'.