The Stacks project

Lemma 18.23.3. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. Let $\mathcal{F}$ be an $\mathcal{O}$-module. Assume that the site $\mathcal{C}$ has a final object $X$. Then

1. The following are equivalent

   1. $\mathcal{F}$ is locally free,

   2. there exists a covering $\{ X_ i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is a locally free $\mathcal{O}_{X_ i}$-module, and

   3. there exists a covering $\{ X_ i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is a free $\mathcal{O}_{X_ i}$-module.

2. The following are equivalent

   1. $\mathcal{F}$ is finite locally free,

   2. there exists a covering $\{ X_ i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is a finite locally free $\mathcal{O}_{X_ i}$-module, and

   3. there exists a covering $\{ X_ i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is a finite free $\mathcal{O}_{X_ i}$-module.

3. The following are equivalent

   1. $\mathcal{F}$ is locally generated by sections,

   2. there exists a covering $\{ X_ i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is an $\mathcal{O}_{X_ i}$-module locally generated by sections, and

   3. there exists a covering $\{ X_ i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is an $\mathcal{O}_{X_ i}$-module globally generated by sections.

4. Given $r \geq 0$, the following are equivalent

   1. $\mathcal{F}$ is locally generated by $r$ sections,

   2. there exists a covering $\{ X_ i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is an $\mathcal{O}_{X_ i}$-module locally generated by $r$ sections, and

   3. there exists a covering $\{ X_ i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is an $\mathcal{O}_{X_ i}$-module globally generated by $r$ sections.

5. The following are equivalent

   1. $\mathcal{F}$ is of finite type,

   2. there exists a covering $\{ X_ i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is an $\mathcal{O}_{X_ i}$-module of finite type, and

   3. there exists a covering $\{ X_ i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is an $\mathcal{O}_{X_ i}$-module globally generated by finitely many sections.

6. The following are equivalent

   1. $\mathcal{F}$ is quasi-coherent,

   2. there exists a covering $\{ X_ i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is a quasi-coherent $\mathcal{O}_{X_ i}$-module, and

   3. there exists a covering $\{ X_ i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is an $\mathcal{O}_{X_ i}$-module which has a global presentation.

7. The following are equivalent

   1. $\mathcal{F}$ is of finite presentation,

   2. there exists a covering $\{ X_ i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is an $\mathcal{O}_{X_ i}$-module of finite presentation, and

   3. there exists a covering $\{ X_ i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is an $\mathcal{O}_{X_ i}$-module has a finite global presentation.

8. The following are equivalent

   1. $\mathcal{F}$ is coherent, and

   2. there exists a covering $\{ X_ i \to X\}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is a coherent $\mathcal{O}_{X_ i}$-module.