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.

Proof. This is true because $f^*$ commutes with arbitrary colimits (Lemma 18.14.3) and $f^*\mathcal{O}_\mathcal {D} = \mathcal{O}_\mathcal {C}$. $\square$

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View Lemma 18.17.2 as pdf