Lemma 21.26.2. In the situation above, let K_1 \to K_2 \to K_3 \to K_1[1] be a distinguished triangle in D(\mathcal{O}). If c^{K_ i}_{X, Z, Y, E} is a quasi-isomorphism for two i out of \{ 1, 2, 3\} , then it is a quasi-isomorphism for the third i.
Proof. By rotating the triangle we may assume c^{K_1}_{X, Z, Y, E} and c^{K_2}_{X, Z, Y, E} are quasi-isomorphisms. Choose a map f : \mathcal{I}^\bullet _1 \to \mathcal{I}^\bullet _2 of K-injective complexes of \mathcal{O}-modules representing K_1 \to K_2. Then K_3 is represented by the K-injective complex C(f)^\bullet , see Derived Categories, Lemma 13.31.3. Then the morphism c^{K_3}_{X, Z, Y, E} is an isomorphism as it is the third leg in a map of distinguished triangles in K(\textit{Ab}) whose other two legs are quasi-isomorphisms. Some details omitted; use Derived Categories, Lemma 13.4.3. \square
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