Lemma 15.57.14. Let $R$ be a ring. Let $\alpha : P^\bullet \to Q^\bullet $ be a quasi-isomorphism of K-flat complexes of $R$-modules. For every complex $L^\bullet $ of $R$-modules the induced map

is a quasi-isomorphism.

Lemma 15.57.14. Let $R$ be a ring. Let $\alpha : P^\bullet \to Q^\bullet $ be a quasi-isomorphism of K-flat complexes of $R$-modules. For every complex $L^\bullet $ of $R$-modules the induced map

\[ \text{Tot}(\text{id}_ L \otimes \alpha ) : \text{Tot}(L^\bullet \otimes _ R P^\bullet ) \longrightarrow \text{Tot}(L^\bullet \otimes _ R Q^\bullet ) \]

is a quasi-isomorphism.

**Proof.**
Choose a quasi-isomorphism $K^\bullet \to L^\bullet $ with $K^\bullet $ a K-flat complex, see Lemma 15.57.12. Consider the commutative diagram

\[ \xymatrix{ \text{Tot}(K^\bullet \otimes _ R P^\bullet ) \ar[r] \ar[d] & \text{Tot}(K^\bullet \otimes _ R Q^\bullet ) \ar[d] \\ \text{Tot}(L^\bullet \otimes _ R P^\bullet ) \ar[r] & \text{Tot}(L^\bullet \otimes _ R Q^\bullet ) } \]

The result follows as by Lemma 15.57.4 the vertical arrows and the top horizontal arrow are quasi-isomorphisms. $\square$

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