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.

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.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.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.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.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.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.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.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 $(\mathcal{C}/U, \mathcal{O}_ U)$.

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