Lemma 20.26.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{K}^\bullet $ be a K-flat complex. Then the functor

transforms quasi-isomorphisms into quasi-isomorphisms.

Lemma 20.26.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{K}^\bullet $ be a K-flat complex. Then the functor

\[ K(\textit{Mod}(\mathcal{O}_ X)) \longrightarrow K(\textit{Mod}(\mathcal{O}_ X)), \quad \mathcal{F}^\bullet \longmapsto \text{Tot}(\mathcal{F}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{K}^\bullet ) \]

transforms quasi-isomorphisms into quasi-isomorphisms.

**Proof.**
Follows from Lemma 20.26.1 and the fact that quasi-isomorphisms are characterized by having acyclic cones.
$\square$

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