The Stacks project

Definition 18.23.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. We will freely use the notions defined in Definition 18.17.1.

1. We say $\mathcal{F}$ is locally free if for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} _{i \in I}$ of $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/U_ i}$ is a free $\mathcal{O}_{U_ i}$-module.

2. We say $\mathcal{F}$ is finite locally free if for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} _{i \in I}$ of $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/U_ i}$ is a finite free $\mathcal{O}_{U_ i}$-module.

3. We say $\mathcal{F}$ is locally generated by sections if for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} _{i \in I}$ of $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/U_ i}$ is an $\mathcal{O}_{U_ i}$-module generated by global sections.

4. Given $r \geq 0$ we sat $\mathcal{F}$ is locally generated by $r$ sections if for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} _{i \in I}$ of $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/U_ i}$ is an $\mathcal{O}_{U_ i}$-module generated by $r$ global sections.

5. We say $\mathcal{F}$ is of finite type if for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} _{i \in I}$ of $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/U_ i}$ is an $\mathcal{O}_{U_ i}$-module generated by finitely many global sections.

6. We say $\mathcal{F}$ is quasi-coherent if for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} _{i \in I}$ of $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/U_ i}$ is an $\mathcal{O}_{U_ i}$-module which has a global presentation.

7. We say $\mathcal{F}$ is of finite presentation if for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} _{i \in I}$ of $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/U_ i}$ is an $\mathcal{O}_{U_ i}$-module which has a finite global presentation.

8. We say $\mathcal{F}$ is coherent if and only if $\mathcal{F}$ is of finite type, and for every object $U$ of $\mathcal{C}$ and any $s_1, \ldots , s_ n \in \mathcal{F}(U)$ the kernel of the map $\bigoplus _{i = 1, \ldots , n} \mathcal{O}_ U \to \mathcal{F}|_ U$ is of finite type on $(\mathcal{C}/U, \mathcal{O}_ U)$.