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 $\mathcal{G}$.

Proof. Part (1) holds by definition. Parts (2) and (3) hold because they holds for the restriction of $\mathcal{F}$ and $\mathcal{G}$ to $U_{Zar}$, see Modules, Lemma 17.5.2. Part (4) is a direct consequence of Lemma 59.31.2 part (3). Parts (5) and (6) follow from the other parts. $\square$

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