On homotopy classes of cochain maps (Q1070031)

The author proves the following theorem. Let \(A\) and \(B\) be cochain complexes of modules over a commutative ring R. Let \(A_{\#}\) be the complex with \(A^ n_{\#}=A^{n+1}\) and \(\delta_{\#}=-\delta\). Let \([A,B]\) denote the abelian group of homotopy classes of cochain maps from \(A\) to \(B\) and \(E_ R(A_{\#},B)\) the abelian group of equivalence classes of weakly splitting extensions of \(B\) by \(A_{\#}\). Then there is an isomorphism \(\Phi\) : [A,B]\(\cong E_ R(A_{\#},B)\) given by \(\Phi ([f])=\{C(f)\}\), where \(C(f)\) is the mapping cone of \(f\).