A noncommutative analogue of the Peskine--Szpiro Acyclicity Lemma

Abstract

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; establishing a new result about quasi-free structures of free multi-derivations of hyperplane arrangements.