Definition 17.10.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. We say that $\mathcal{F}$ is a quasi-coherent sheaf of $\mathcal{O}_ X$-modules if for every point $x \in X$ there exists an open neighbourhood $x\in U \subset X$ such that $\mathcal{F}|_ U$ is isomorphic to the cokernel of a map
\[ \bigoplus \nolimits _{j \in J} \mathcal{O}_ U \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O}_ U \]
The category of quasi-coherent $\mathcal{O}_ X$-modules is denoted $\mathit{QCoh}(\mathcal{O}_ X)$.
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