Definition 17.11.1. 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 of *finite presentation* if for every point $x \in X$ there exists an open neighbourhood $x\in U \subset X$, and $n, m \in \mathbf{N}$ such that $\mathcal{F}|_ U$ is isomorphic to the cokernel of a map

\[ \bigoplus \nolimits _{j = 1, \ldots , m} \mathcal{O}_ U \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{O}_ U \]

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