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The Stacks project

Lemma 21.17.12. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Let \alpha : \mathcal{P}^\bullet \to \mathcal{Q}^\bullet be a quasi-isomorphism of K-flat complexes of \mathcal{O}-modules. For every complex \mathcal{F}^\bullet of \mathcal{O}-modules the induced map

\text{Tot}(\text{id}_{\mathcal{F}^\bullet } \otimes \alpha ) : \text{Tot}(\mathcal{F}^\bullet \otimes _\mathcal {O} \mathcal{P}^\bullet ) \longrightarrow \text{Tot}(\mathcal{F}^\bullet \otimes _\mathcal {O} \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 21.17.11. Consider the commutative diagram

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

The result follows as by Lemma 21.17.3 the vertical arrows and the top horizontal arrow are quasi-isomorphisms. \square


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