Derived internal Hom computes Ext

Observation 6.6.34. Let \(R\) be a commutative ring and let \(M,N \in \RMod _R\), regarded as complexes concentrated in degree \(0\). Choosing a projective resolution \(P_{\bullet } \to M\), the complex \(\uHom (P,N)\) has \(\uHom (P,N)_{-n} = [P_n,N]\) with the differential induced by that of \(P_{\bullet }\); it is thus the cochain complex \(\Hom _R(P_{\bullet },N)\), placed in non-positive degrees. Taking homology, we obtain \[ H_{-n}\bigl (\bR \uHom _R(M,N)\bigr ) \; \cong \; H^n\bigl (\Hom _R(P_{\bullet },N)\bigr ) \; \cong \; \Ext ^n_R(M,N), \] where the second isomorphism is Proposition 6.4.14. In particular \(\bR \uHom _R(M,N)\) is coconnective, with \(H_0 = \Hom _R(M,N)\).