Derived internal Hom

Definition 6.6.33 (Derived hom). Let \(\Ch ^-_{\proj }(\Aa )\) denote the full subcategory of bounded below projective chain complexes. The internal hom restricts to a functor \[ \uHom (-,-)\colon \Ch ^-_{\proj }(\Aa )\catop \times \Ch (\Aa )\to \Ch (\Aa ). \] It preserves quasi-isomorphisms in both variables: in the first variable this follows from Proposition 6.6.11, while in the second it follows from Lemma 6.6.31. Consequently, localization and Proposition 6.6.16 produce a functor \[ \bR \uHom (-,-)\colon \D ^-(\Aa )\catop \times \D (\Aa ) \to \D (\Aa ). \] For any projective resolution \(P\to C\), it is computed by \[ \bR \uHom (C,D)\simeq \uHom (P,D). \]