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The tensor product of globally generated sheaves of modules is globally generated.

Lemma 17.4.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$, $\mathcal{G}$ be sheaves of $\mathcal{O}_ X$-modules. If $\mathcal{F}$ and $\mathcal{G}$ are generated by global sections then so is $\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G}$.

Proof. Omitted. $\square$

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