The Stacks project

Definition 66.20.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$.

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

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 66.20.2).