Definition 17.17.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. An $\mathcal{O}_ X$-module $\mathcal{F}$ is *flat* if the functor

\[ \textit{Mod}(\mathcal{O}_ X) \longrightarrow \textit{Mod}(\mathcal{O}_ X), \quad \mathcal{G} \mapsto \mathcal{G} \otimes _\mathcal {O} \mathcal{F} \]

is exact.

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