Algebraic universal coefficient theorem

Theorem 6.7.3 (Algebraic universal coefficient theorem). Let \(R\) be a principal ideal domain, let \(A \in \D ^-(R)\), and let \(M\) be an \(R\)-module, regarded as a complex concentrated in degree zero. For each \(n \in \Z \), there is a natural short exact sequence \[ 0 \to \Ext ^1_R(H_{n-1}(A),M) \to H_{-n}(\bR \uHom _R(A,M[0])) \to \Hom _R(H_n(A),M) \to 0, \] and this sequence splits non-naturally.