The Stacks project

Lemma 66.20.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a subsheaf of the final object of the ├ętale topos of $X$ (see Sites, Example 7.10.2). Then there exists a unique open $W \subset X$ such that $\mathcal{F} = h_ W$.

Proof. The condition means that $\mathcal{F}(U)$ is a singleton or empty for all $\varphi : U \to X$ in $\mathop{\mathrm{Ob}}\nolimits (X_{spaces, {\acute{e}tale}})$. In particular local sections always glue. If $\mathcal{F}(U) \not= \emptyset $, then $\mathcal{F}(\varphi (U)) \not= \emptyset $ because $\varphi (U) \subset X$ is an open subspace (Lemma 66.16.7) and $\{ \varphi : U \to \varphi (U)\} $ is a covering in $X_{spaces, {\acute{e}tale}}$. Take $W = \bigcup _{\varphi : U \to S, \mathcal{F}(U) \not= \emptyset } \varphi (U)$ to conclude. $\square$

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