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.

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). $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 03DN. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 18.23.3 as pdf