Lemma 59.31.2. Let $S$ be a scheme. Let $\mathcal{F}$ be an abelian sheaf on $S_{\acute{e}tale}$. Let $\sigma \in \mathcal{F}(U)$ be a local section. There exists an open subset $W \subset U$ such that

$W \subset U$ is the largest Zariski open subset of $U$ such that $\sigma |_ W = 0$,

for every $\varphi : V \to U$ in $S_{\acute{e}tale}$ we have

\[ \sigma |_ V = 0 \Leftrightarrow \varphi (V) \subset W, \]for every geometric point $\overline{u}$ of $U$ we have

\[ (U, \overline{u}, \sigma ) = 0\text{ in }\mathcal{F}_{\overline{s}} \Leftrightarrow \overline{u} \in W \]where $\overline{s} = (U \to S) \circ \overline{u}$.

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