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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