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.

Proof. In each case we have (a) $\Rightarrow (b)$. In each of the cases (1) - (6) condition (b) implies condition (c) by axiom (2) of a site (see Sites, Definition 7.6.2) and the definition of the local types of modules. Suppose $\{ X_ i \to X\}$ is a covering. Then for every object $U$ of $\mathcal{C}$ we get an induced covering $\{ X_ i \times _ X U \to U\}$. Moreover, the global property for $\mathcal{F}|_{\mathcal{C}/X_ i}$ in part (c) implies the corresponding global property for $\mathcal{F}|_{\mathcal{C}/X_ i \times _ X U}$ by Lemma 18.17.2, hence the sheaf has property (a) by definition. We omit the proof of (b) $\Rightarrow$ (a) in case (7). $\square$

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