The mapping cone of an Eisenbud operator and applications to exact zero divisors

Let (Q,m,k) be a local ring that admits an exact pair of zero divisors (f,g), M a Q-module with fM=0 and U a free resolution of M over Q. We construct a degree −2 chain map, which we call an Einsenbud operator, on the complex U⊗QQ∕(f,g) and use the mapping cone of the operator to study two exact sequences that relate homology over Q to homology over Q∕(f). Several applications are given.