The Stacks project

Definition 71.10.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules on $X_{\acute{e}tale}$.

  1. We denote $\mathcal{K}_ X(\mathcal{F})$ the sheaf of $\mathcal{K}_ X$-modules which is the sheafification of the presheaf $U \mapsto \mathcal{S}(U)^{-1}\mathcal{F}(U)$. Equivalently $\mathcal{K}_ X(\mathcal{F}) = \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{K}_ X$ (see above).

  2. A meromorphic section of $\mathcal{F}$ is a global section of $\mathcal{K}_ X(\mathcal{F})$.

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