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The Stacks project

Lemma 21.17.3. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Let \mathcal{K}^\bullet be a K-flat complex. Then the functor

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

transforms quasi-isomorphisms into quasi-isomorphisms.

Proof. Follows from Lemma 21.17.1 and the fact that quasi-isomorphisms are characterized by having acyclic cones. \square

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