The Stacks project

Lemma 13.31.7. Let $\mathcal{A}$ be an abelian category. Assume every complex has a quasi-isomorphism towards a K-injective complex. Then any exact functor $F : K(\mathcal{A}) \to \mathcal{D}'$ of triangulated categories has a right derived functor

\[ RF : D(\mathcal{A}) \longrightarrow \mathcal{D}' \]

and $RF(I^\bullet ) = F(I^\bullet )$ for K-injective complexes $I^\bullet $.

Proof. To see this we apply Lemma 13.14.15 with $\mathcal{I}$ the collection of K-injective complexes. Since (1) holds by assumption, it suffices to prove that if $I^\bullet \to J^\bullet $ is a quasi-isomorphism of K-injective complexes, then $F(I^\bullet ) \to F(J^\bullet )$ is an isomorphism. This is clear because $I^\bullet \to J^\bullet $ is a homotopy equivalence, i.e., an isomorphism in $K(\mathcal{A})$, by Lemma 13.31.2. $\square$

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