Lemma 66.11.1. 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 66.11.1. 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 64.4.11 that the map is an injection in general. By Lemma 66.5.1 it is surjective when $X$ is decent (actually one can say this is part of the definition of being decent).
$\square$

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