A noncommutative analogue of the Peskine--Szpiro Acyclicity Lemma

TL;DR

This paper introduces a noncommutative version of the Peskine--Szpiro Acyclicity Lemma, providing a new method to verify the exactness of complexes over Auslander regular rings, with applications in D-module theory and hyperplane arrangements.

Contribution

It develops a noncommutative analogue of a classical lemma, enabling exactness certification in broader algebraic contexts and demonstrating its utility in several geometric and algebraic problems.

Findings

Recovered results related to Bernstein--Sato polynomials

Established a new result on quasi-free structures of hyperplane arrangements

Validated the lemma's effectiveness through multiple applications

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.