The Stacks project

Lemma 21.18.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. A bounded above complex of flat $\mathcal{O}$-modules is K-flat.

Proof. Let $\mathcal{K}^\bullet $ be a bounded above complex of flat $\mathcal{O}$-modules. Let $\mathcal{L}^\bullet $ be an acyclic complex of $\mathcal{O}$-modules. Note that $\mathcal{L}^\bullet = \mathop{\mathrm{colim}}\nolimits _ m \tau _{\leq m}\mathcal{L}^\bullet $ where we take termwise colimits. Hence also

\[ \text{Tot}(\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{L}^\bullet ) = \mathop{\mathrm{colim}}\nolimits _ m \text{Tot}( \mathcal{K}^\bullet \otimes _\mathcal {O} \tau _{\leq m}\mathcal{L}^\bullet ) \]

termwise. Hence to prove the complex on the left is acyclic it suffices to show each of the complexes on the right is acyclic. Since $\tau _{\leq m}\mathcal{L}^\bullet $ is acyclic this reduces us to the case where $\mathcal{L}^\bullet $ is bounded above. In this case the spectral sequence of Homology, Lemma 12.22.6 has

\[ {}'E_1^{p, q} = H^ p(\mathcal{L}^\bullet \otimes _ R \mathcal{K}^ q) \]

which is zero as $\mathcal{K}^ q$ is flat and $\mathcal{L}^\bullet $ acyclic. Hence we win. $\square$


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