  We present a variant of the Peskine--Szpiro Acyclicity Lemma, and hence a way
to certify exactness of a complex of finite modules over a large class of
(possibly) noncommutative rings. Specifically, over the class of Auslander
regular rings. In the case of relative $\mathscr{D}_{X}$-modules, for example
$\mathscr{D}_{X}[s_{1}, \dots, s_{r}]$-modules, the hypotheses have geometric
realizations making them easier to authenticate. We demonstrate the efficacy of
this lemma and its various forms by independently recovering some results
related to Bernstein--Sato polynomials.
