Definition 17.12.1. 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 *coherent $\mathcal{O}_ X$-module* if the following two conditions hold:

$\mathcal{F}$ is of finite type, and

for every open $U \subset X$ and every finite collection $s_ i \in \mathcal{F}(U)$, $i = 1, \ldots , n$ the kernel of the associated map $\bigoplus _{i = 1, \ldots , n} \mathcal{O}_ U \to \mathcal{F}|_ U$ is of finite type.

The category of coherent $\mathcal{O}_ X$-modules is denoted $\textit{Coh}(\mathcal{O}_ X)$.

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