Lemma 17.17.2. Let (X, \mathcal{O}_ X) be a ringed space. An \mathcal{O}_ X-module \mathcal{F} is flat if and only if the stalk \mathcal{F}_ x is a flat \mathcal{O}_{X, x}-module for all x \in X.
Proof. Assume \mathcal{F}_ x is a flat \mathcal{O}_{X, x}-module for all x \in X. In this case, if \mathcal{G} \to \mathcal{H} \to \mathcal{K} is exact, then also \mathcal{G} \otimes _{\mathcal{O}_ X} \mathcal{F} \to \mathcal{H} \otimes _{\mathcal{O}_ X} \mathcal{F} \to \mathcal{K} \otimes _{\mathcal{O}_ X} \mathcal{F} is exact because we can check exactness at stalks and because tensor product commutes with taking stalks, see Lemma 17.16.1. Conversely, suppose that \mathcal{F} is flat, and let x \in X. Consider the skyscraper sheaves i_{x, *} M where M is a \mathcal{O}_{X, x}-module. Note that
again by Lemma 17.16.1. Since i_{x, *} is exact, we see that the fact that \mathcal{F} is flat implies that M \mapsto M \otimes _{\mathcal{O}_{X, x}} \mathcal{F}_ x is exact. Hence \mathcal{F}_ x is a flat \mathcal{O}_{X, x}-module. \square
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Comment #779 by Anfang Zhou on