Lemma 17.17.4. Let $(X, \mathcal{O}_ X)$ be a ringed space. A filtered colimit of flat $\mathcal{O}_ X$-modules is flat. A direct sum of flat $\mathcal{O}_ X$-modules is flat.

Proof. This follows from Lemma 17.16.5, Lemma 17.16.1, Algebra, Lemma 10.8.8, and the fact that we can check exactness at stalks. $\square$

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