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

Let $(Q,\mathfrak{m},\mathsf{k})$ 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 $U{\otimes }_{Q}Q\mathrm{/}(f,g)$ and use the mapping cone of the operator to study two exact sequences that relate homology over $Q$ to homology over $Q\mathrm{/}(f)$. Several applications are given.