The Stacks project

Lemma 18.26.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$, $\mathcal{G}$ be sheaves of $\mathcal{O}$-modules.

  1. If $\mathcal{F}$, $\mathcal{G}$ are locally free, so is $\mathcal{F} \otimes _\mathcal {O} \mathcal{G}$.

  2. If $\mathcal{F}$, $\mathcal{G}$ are finite locally free, so is $\mathcal{F} \otimes _\mathcal {O} \mathcal{G}$.

  3. If $\mathcal{F}$, $\mathcal{G}$ are locally generated by sections, so is $\mathcal{F} \otimes _\mathcal {O} \mathcal{G}$.

  4. If $\mathcal{F}$, $\mathcal{G}$ are of finite type, so is $\mathcal{F} \otimes _\mathcal {O} \mathcal{G}$.

  5. If $\mathcal{F}$, $\mathcal{G}$ are quasi-coherent, so is $\mathcal{F} \otimes _\mathcal {O} \mathcal{G}$.

  6. If $\mathcal{F}$, $\mathcal{G}$ are of finite presentation, so is $\mathcal{F} \otimes _\mathcal {O} \mathcal{G}$.

  7. If $\mathcal{F}$ is of finite presentation and $\mathcal{G}$ is coherent, then $\mathcal{F} \otimes _\mathcal {O} \mathcal{G}$ is coherent.

  8. If $\mathcal{F}$, $\mathcal{G}$ are coherent, so is $\mathcal{F} \otimes _\mathcal {O} \mathcal{G}$.

Proof. Omitted. Hint: Compare with Sheaves of Modules, Lemma 17.16.6. $\square$

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