The Stacks project

Lemma 18.17.2. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal {D}$-module.

1. If $\mathcal{F}$ is free then $f^*\mathcal{F}$ is free.

2. If $\mathcal{F}$ is finite free then $f^*\mathcal{F}$ is finite free.

3. If $\mathcal{F}$ is generated by global sections then $f^*\mathcal{F}$ is generated by global sections.

4. Given $r \geq 0$ if $\mathcal{F}$ is generated by $r$ global sections, then $f^*\mathcal{F}$ is generated by $r$ global sections.

5. If $\mathcal{F}$ is generated by finitely many global sections then $f^*\mathcal{F}$ is generated by finitely many global sections.

6. If $\mathcal{F}$ has a global presentation then $f^*\mathcal{F}$ has a global presentation.

7. If $\mathcal{F}$ has a finite global presentation then $f^*\mathcal{F}$ has a finite global presentation.