Lemma 68.11.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Consider the map

This map is always injective. If $X$ is decent then this map is a bijection.

Lemma 68.11.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Consider the map

\[ \{ \mathop{\mathrm{Spec}}(k) \to X \text{ monomorphism where }k\text{ is a field}\} \longrightarrow |X| \]

This map is always injective. If $X$ is decent then this map is a bijection.

**Proof.**
We have seen in Properties of Spaces, Lemma 66.4.12 that the map is an injection in general. By Lemma 68.5.1 it is surjective when $X$ is decent (actually one can say this is part of the definition of being decent).
$\square$

Your email address will not be published.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.

## Comments (0)