Lemma 22.32.1. In the situation above. If the right derived functor R\mathop{\mathrm{Hom}}\nolimits (K^\bullet , -) of \mathop{\mathrm{Hom}}\nolimits (K^\bullet , -) : K(\mathcal{A}) \to D(\textit{Ab}) is everywhere defined on D(\mathcal{A}), then we obtain a canonical exact functor
R\mathop{\mathrm{Hom}}\nolimits (K^\bullet , -) : D(\mathcal{A}) \longrightarrow D(E, \text{d})
of triangulated categories which reduces to the usual one on taking associated complexes of abelian groups.
Proof.
Note that we have an associated functor K(\mathcal{A}) \to K(\text{Mod}_{(E, \text{d})}) by Lemma 22.26.10. We claim this functor is an exact functor of triangulated categories. Namely, let f : A^\bullet \to B^\bullet be a map of complexes of \mathcal{A}. Then a computation shows that
\mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{A})}(K^\bullet , C(f)^\bullet ) = C\left( \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{A})}(K^\bullet , A^\bullet ) \to \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{A})}(K^\bullet , B^\bullet ) \right)
where the right hand side is the cone in \text{Mod}_{(E, \text{d})} defined earlier in this chapter. This shows that our functor is compatible with cones, hence with distinguished triangles. Let X^\bullet be an object of K(\mathcal{A}). Consider the category of quasi-isomorphisms s : X^\bullet \to Y^\bullet . We are given that the functor (s : X^\bullet \to Y^\bullet ) \mapsto \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(K^\bullet , Y^\bullet ) is essentially constant when viewed in D(\textit{Ab}). But since the forgetful functor D(E, \text{d}) \to D(\textit{Ab}) is compatible with taking cohomology, the same thing is true in D(E, \text{d}). This proves the lemma.
\square
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