The Stacks project

Definition 59.17.2. Let $\mathcal{C}$ be a ringed site, i.e., a site endowed with a sheaf of rings $\mathcal{O}$. A sheaf of $\mathcal{O}$-modules $\mathcal{F}$ on $\mathcal{C}$ is called quasi-coherent if for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ there exists a covering $\{ U_ i \to U\} _{i\in I}$ of $\mathcal{C}$ such that the restriction $\mathcal{F}|_{\mathcal{C}/U_ i}$ is isomorphic to the cokernel of an $\mathcal{O}$-linear map of free $\mathcal{O}$-modules

\[ \bigoplus \nolimits _{k \in K} \mathcal{O}|_{\mathcal{C}/U_ i} \longrightarrow \bigoplus \nolimits _{l \in L} \mathcal{O}|_{\mathcal{C}/U_ i}. \]

The direct sum over $K$ is the sheaf associated to the presheaf $V \mapsto \bigoplus _{k \in K} \mathcal{O}(V)$ and similarly for the other.

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