Lemma 21.17.15. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$ be an $\mathcal{O}$-module. The following are equivalent

$\mathcal{F}$ is a flat $\mathcal{O}$-module, and

$\text{Tor}_1^\mathcal {O}(\mathcal{F}, \mathcal{G}) = 0$ for every $\mathcal{O}$-module $\mathcal{G}$.

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