A noncommutative analogue of the Peskine–Szpiro Acyclicity Lemma

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>𝒟</mml:mi> <mml:mi>X</mml:mi> </mml:msub> </mml:math> -modules, for example <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>𝒟</mml:mi> <mml:mi>X</mml:mi> </mml:msub> <mml:mrow> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>s</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>s</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mo>]</mml:mo> </mml:mrow> </mml:mrow> </mml:math> -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.