The Stacks project

Lemma 59.39.1. Let $S$ be a scheme. Let $\mathcal{F}$ be a sheaf of sets on $S_{\acute{e}tale}$. Let $s, t \in \mathcal{F}(S)$. Then there exists an open $W \subset S$ characterized by the following property: A morphism $f : T \to S$ factors through $W$ if and only if $s|_ T = t|_ T$ (restriction is pullback by $f_{small}$).

Proof. Consider the presheaf which assigns to $U \in \mathop{\mathrm{Ob}}\nolimits (S_{\acute{e}tale})$ the empty set if $s|_ U \not= t|_ U$ and a singleton else. It is clear that this is a subsheaf of the final object of $\mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale})$. By Lemma 59.31.1 we find an open $W \subset S$ representing this presheaf. For a geometric point $\overline{x}$ of $S$ we see that $\overline{x} \in W$ if and only if the stalks of $s$ and $t$ at $\overline{x}$ agree. By the description of stalks of pullbacks in Lemma 59.36.2 we see that $W$ has the desired property. $\square$

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