Acyclicity criteria for complexes associated with an alternating map

When $I$ is a Gorenstein ideal of grade $3$ in a local ring $R$, results of Boffi and Sánchez, and of Kustin and Ulrich show that for each $t\ge 1$one can construct in a canonical way a finite free complex $\mathcal{D}^{t}$ that is ``approximately" a resolution for the ideal $I^{t}$. Kustin and Ulrich also provide a sufficient condition that $\mathcal{D}^{t}$ is acyclic, and a sufficient condition that $\mathcal{D}^{t}$ is a resolution of $I^{t}$. We complete these two acyclicity criteria by showing that the corresponding sufficient conditions are also necessary.