The Stacks project

Lemma 66.4.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $X = U/R$ be a presentation of $X$, see Spaces, Definition 65.9.3. Then the image of $|R| \to |U| \times |U|$ is an equivalence relation and $|X|$ is the quotient of $|U|$ by this equivalence relation.

Proof. The assumption means that $U$ is a scheme, $p : U \to X$ is a surjective, ├ętale morphism, $R = U \times _ X U$ is a scheme and defines an ├ętale equivalence relation on $U$ such that $X = U/R$ as sheaves. By Lemma 66.4.4 we see that $|U| \to |X|$ is surjective. By Lemma 66.4.3 the map

\[ |R| \longrightarrow |U| \times _{|X|} |U| \]

is surjective. Hence the image of $|R| \to |U| \times |U|$ is exactly the set of pairs $(u_1, u_2) \in |U| \times |U|$ such that $u_1$ and $u_2$ have the same image in $|X|$. Combining these two statements we get the result of the lemma. $\square$

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