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Definition 16.10.1. 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 {\it 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 $\textit{QCoh}(\mathcal{O}_X)$.
\begin{definition}
\label{definition-quasi-coherent}
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 {\it 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 $\textit{QCoh}(\mathcal{O}_X)$.
\end{definition}
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