Lemma 17.5.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. Let $U \subset X$ open.

The support of $s \in \mathcal{F}(U)$ is closed in $U$.

The support of $fs$ is contained in the intersections of the supports of $f \in \mathcal{O}_ X(U)$ and $s \in \mathcal{F}(U)$.

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

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

If $\varphi : \mathcal{F} \to \mathcal{G}$ is a morphism of $\mathcal{O}_ X$-modules, then the support of $\varphi (s)$ is contained in the support of $s \in \mathcal{F}(U)$.

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