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 $\mathcal{F} \otimes _\mathcal {O} \mathcal{G}$.