Over any ringed space, sheaves of modules of finite type have closed support.

Lemma 17.9.6. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. If $\mathcal{F}$ is of finite type then support of $\mathcal{F}$ is closed.

Proof. This is a reformulation of Lemma 17.9.5. $\square$

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