Lemma 30.9.4. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_ X$-modules. The $\mathcal{O}_ X$-modules $\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G}$ and $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ are coherent.

**Proof.**
It is shown in Modules, Lemma 17.22.5 that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ is coherent. The result for tensor products is Modules, Lemma 17.16.6
$\square$

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## Comments (2)

Comment #5854 by Takagi Benseki（高城 辨積） on

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