The Stacks project

Lemma 17.23.4. Let $(X,\mathcal{O}_ X)$ be a ringed space. If $\mathcal{O}_ X$ and $\mathcal{F}$ are coherent, then so is $\text{Ann}_{\mathcal{O}_ X}(\mathcal{F})$.

Proof. Since $\text{Ann}_{\mathcal{O}_ X}(\mathcal{F})$ is the kernel of $\mathcal{O}_ X \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{F})$ by Lemma 17.12.4 it suffices to show that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{F})$ is coherent. This follows from Lemma 17.22.6 and the fact that $\mathcal{F}$ is coherent and a fortiori finitely presented (Lemma 17.12.2). $\square$

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