The Stacks project

Definition 59.31.3. Let $S$ be a scheme. Let $\mathcal{F}$ be an abelian sheaf on $S_{\acute{e}tale}$.

1. The support of $\mathcal{F}$ is the set of points $s \in S$ such that $\mathcal{F}_{\overline{s}} \not= 0$ for any (some) geometric point $\overline{s}$ lying over $s$.

2. Let $\sigma \in \mathcal{F}(U)$ be a section. The support of $\sigma $ is the closed subset $U \setminus W$, where $W \subset U$ is the largest open subset of $U$ on which $\sigma $ restricts to zero (see Lemma 59.31.2).