The Stacks project

Lemma 59.31.4. Let $S$ be a scheme. Let $\mathcal{F}$ be an abelian sheaf on $S_{\acute{e}tale}$. Let $U \in \mathop{\mathrm{Ob}}\nolimits (S_{\acute{e}tale})$ and $\sigma \in \mathcal{F}(U)$.

1. The support of $\sigma$ is closed in $U$.

2. The support of $\sigma + \sigma '$ is contained in the union of the supports of $\sigma , \sigma ' \in \mathcal{F}(U)$.

3. If $\varphi : \mathcal{F} \to \mathcal{G}$ is a map of abelian sheaves on $S_{\acute{e}tale}$, then the support of $\varphi (\sigma )$ is contained in the support of $\sigma \in \mathcal{F}(U)$.

4. The support of $\mathcal{F}$ is the union of the images of the supports of all local sections of $\mathcal{F}$.

5. If $\mathcal{F} \to \mathcal{G}$ is surjective then the support of $\mathcal{G}$ is a subset of the support of $\mathcal{F}$.

6. If $\mathcal{F} \to \mathcal{G}$ is injective then the support of $\mathcal{F}$ is a subset of the support of $\mathcal{G}$.