Lemma 15.57.4. Let $R$ be a ring. Let $K^\bullet$ be a K-flat complex. Then the functor

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

transforms quasi-isomorphisms into quasi-isomorphisms.

Proof. Follows from Lemma 15.57.2 and the fact that quasi-isomorphisms in $K(\text{Mod}_ R)$ and $K(\text{Mod}_ A)$ are characterized by having acyclic cones. $\square$

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