The Stacks project

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.

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

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

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

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

  5. 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 $s \in \mathcal{F}(U)$.

Proof. This is true because if $s_ x = 0$, then $s$ is zero in an open neighbourhood of $x$ by definition of stalks. Similarly for $f$. Details omitted. $\square$

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