Lemma 20.53.6. In Situation 20.53.2 let $\mathcal{F}^\bullet \to \mathcal{G}^\bullet$ be a map of complexes of $\mathcal{I}$-torsion free $\mathcal{O}_ X$-modules. Then the induced map $\eta _\mathcal {I}\mathcal{F}^\bullet \to \eta _\mathcal {I}\mathcal{G}^\bullet$ is a quasi-isomorphism too.

Proof. This is true because the isomorphisms of Lemma 20.53.5 are compatible with maps of complexes. $\square$

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