The Stacks project

Lemma 17.23.2. Let $(X, \mathcal{O}_ X)$ be a ringed space and let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. If $\mathcal{F}$ is of finite type, then $(\text{Ann}_{\mathcal{O}_ X}(\mathcal{F}))_ x = \text{Ann}_{\mathcal{O}_{X, x}}(\mathcal{F}_ x)$.

Proof. By Lemma 17.22.4 the map

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{F})_ x \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X,x}}(\mathcal{F}_ x, \mathcal{F}_ x) \]

is injective. Thus any section $f$ of $\mathcal{O}_ X$ over an open neighbourhood $U$ of $x$ which acts as zero on $\mathcal{F}_ x$ will act as zero on $\mathcal{F}|_ V$ for some $U \supset V \ni x$ open. Hence the inclusion ( is an equality. $\square$

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