Lemma 21.17.7. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Let 0 \to \mathcal{K}_1^\bullet \to \mathcal{K}_2^\bullet \to \mathcal{K}_3^\bullet \to 0 be a short exact sequence of complexes such that the terms of \mathcal{K}_3^\bullet are flat \mathcal{O}-modules. If two out of three of \mathcal{K}_ i^\bullet are K-flat, so is the third.
Proof. By Modules on Sites, Lemma 18.28.9 for every complex \mathcal{L}^\bullet we obtain a short exact sequence
0 \to \text{Tot}(\mathcal{L}^\bullet \otimes _\mathcal {O} \mathcal{K}_1^\bullet ) \to \text{Tot}(\mathcal{L}^\bullet \otimes _\mathcal {O} \mathcal{K}_1^\bullet ) \to \text{Tot}(\mathcal{L}^\bullet \otimes _\mathcal {O} \mathcal{K}_1^\bullet ) \to 0
of complexes. Hence the lemma follows from the long exact sequence of cohomology sheaves and the definition of K-flat complexes. \square
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Comment #9529 by nkym on