Lemma 6.6.32. Let \(P_{\bullet }\) be a bounded below projective chain complex and let \(X_{\bullet }\) be any chain complex. Then the localization functor induces an isomorphism \[ \Hom _{\Kk (\Aa )}(P_{\bullet },X_{\bullet }) \iso \Hom _{\D (\Aa )}(P_{\bullet },X_{\bullet }). \]
Proof. By Lemma 6.6.31, the functor \(\uHom (P_{\bullet },-)\) preserves quasi-isomorphisms. Since \[ \pi _k\Hom _{\Kk (\Aa )}(P_{\bullet },X_{\bullet }) \cong H_k(\uHom (P_{\bullet },X_{\bullet })) \] for every \(k\geq 0\), the functor \(\Hom _{\Kk (\Aa )}(P_{\bullet },-)\) also preserves quasi-isomorphisms. Viewing \(\D (\Aa )\) as the localization of \(\Kk (\Aa )\) at the quasi-isomorphisms, the claim follows from Lemma 6.2.10. □
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