Lemma 68.12.5. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$,. 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. Via Lemma 68.12.2 this follows from the result for schemes, see Cohomology of Schemes, Lemma 30.9.4. $\square$

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