The Stacks project

Lemma 15.59.2. Let $R$ be a ring. Let $K^\bullet $ be a K-flat complex. Then the functor

\[ K(R) \longrightarrow K(R), \quad L^\bullet \longmapsto \text{Tot}(L^\bullet \otimes _ R K^\bullet ) \]

transforms quasi-isomorphisms into quasi-isomorphisms.

Proof. Follows from Lemma 15.58.4 and the fact that quasi-isomorphisms in $K(R)$ are characterized by having acyclic cones. $\square$

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