Lemma 15.58.4. Let $R$ be a ring. Let $K^\bullet $ be a K-flat complex. Then the functor

transforms quasi-isomorphisms into quasi-isomorphisms.

Lemma 15.58.4. 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 15.58.2 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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