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 ⊗Q Q/(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.