The Stacks project

Lemma 20.26.13. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\alpha : \mathcal{P}^\bullet \to \mathcal{Q}^\bullet $ be a quasi-isomorphism of K-flat complexes of $\mathcal{O}_ X$-modules. For every complex $\mathcal{F}^\bullet $ of $\mathcal{O}_ X$-modules the induced map

\[ \text{Tot}(\text{id}_{\mathcal{F}^\bullet } \otimes \alpha ) : \text{Tot}(\mathcal{F}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{P}^\bullet ) \longrightarrow \text{Tot}(\mathcal{F}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{Q}^\bullet ) \]

is a quasi-isomorphism.

Proof. Choose a quasi-isomorphism $\mathcal{K}^\bullet \to \mathcal{F}^\bullet $ with $\mathcal{K}^\bullet $ a K-flat complex, see Lemma 20.26.12. Consider the commutative diagram

\[ \xymatrix{ \text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{P}^\bullet ) \ar[r] \ar[d] & \text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{Q}^\bullet ) \ar[d] \\ \text{Tot}(\mathcal{F}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{P}^\bullet ) \ar[r] & \text{Tot}(\mathcal{F}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{Q}^\bullet ) } \]

The result follows as by Lemma 20.26.3 the vertical arrows and the top horizontal arrow are quasi-isomorphisms. $\square$

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