Lemma 21.17.16. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Let \mathcal{K}^\bullet be a K-flat, acyclic complex with flat terms. Then \mathcal{F} = \mathop{\mathrm{Ker}}(\mathcal{K}^ n \to \mathcal{K}^{n + 1}) is a flat \mathcal{O}-module.
Proof. Observe that
\ldots \to \mathcal{K}^{n - 2} \to \mathcal{K}^{n - 1} \to \mathcal{F} \to 0
is a flat resolution of our module \mathcal{F}. Since a bounded above complex of flat modules is K-flat (Lemma 21.17.8) we may use this resolution to compute \text{Tor}_ i(\mathcal{F}, \mathcal{G}) for any \mathcal{O}-module \mathcal{G}. On the one hand \mathcal{K}^\bullet \otimes _\mathcal {O}^\mathbf {L} \mathcal{G} is zero in D(\mathcal{O}) because \mathcal{K}^\bullet is acyclic and on the other hand it is represented by \mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{G}. Hence we see that
\mathcal{K}^{n - 3} \otimes _\mathcal {O} \mathcal{G} \to \mathcal{K}^{n - 2} \otimes _\mathcal {O} \mathcal{G} \to \mathcal{K}^{n - 1} \otimes _\mathcal {O} \mathcal{G}
is exact. Thus \text{Tor}_1(\mathcal{F}, \mathcal{G}) = 0 and we conclude by Lemma 21.17.15. \square
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