Lemma 91.16.5 (0DIX)—The Stacks project

Lemma 91.16.5. Let $\mathcal{C}$ be a site. Let $\mathcal{O} \to \mathcal{O}_0$ be a surjection of sheaves of rings. Assume given the following data

a complex of $\mathcal{O}$ -modules $\mathcal{F}^\bullet$ ,
a complex $\mathcal{K}_0^\bullet$ of $\mathcal{O}_0$ -modules,
a quasi-isomorphism $\mathcal{K}_0^\bullet \to \mathcal{F}^\bullet \otimes _\mathcal {O} \mathcal{O}_0$ ,

Then there exist a quasi-isomorphism $\mathcal{G}^\bullet \to \mathcal{F}^\bullet$ such that the map of complexes $\mathcal{G}^\bullet \otimes _\mathcal {O} \mathcal{O}_0 \to \mathcal{F}^\bullet \otimes _\mathcal {O} \mathcal{O}_0$ factors through $\mathcal{K}_0^\bullet$ in the homotopy category of complexes of $\mathcal{O}_0$ -modules.

Proof. Set $\mathcal{F}_0^\bullet = \mathcal{F}^\bullet \otimes _\mathcal {O} \mathcal{O}_0$ . By Derived Categories, Lemma 13.9.8 there exists a factorization

$\mathcal{K}_0^\bullet \to \mathcal{L}_0^\bullet \to \mathcal{F}_0^\bullet$

of the given map such that the first arrow has an inverse up to homotopy and the second arrow is termwise split surjective. Hence we may assume that $\mathcal{K}_0^\bullet \to \mathcal{F}_0^\bullet$ is termwise surjective. In that case we take

$\mathcal{G}^ n = \mathcal{F}^ n \times _{\mathcal{F}^ n_0} \mathcal{K}_0^ n$

and everything is clear. $\square$

The Stacks project

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