A residual duality over Gorenstein rings with application to logarithmic differential forms

TL;DR

This paper generalizes the concept of free divisors to Gorenstein rings, providing a new duality framework that links multi-logarithmic differential forms with Cohen-Macaulay modules, extending previous results in singularity theory.

Contribution

It offers a new duality perspective over Gorenstein rings, generalizing Pol's results and connecting multi-logarithmic forms with Cohen-Macaulay modules in a broader algebraic context.

Findings

Generalization of freeness to Gorenstein rings.

Duality between multi-logarithmic forms and Cohen-Macaulay modules.

Conceptual proof and extension of Pol's duality result.

Abstract

Kyoji Saito's notion of a free divisor was generalized by the first author to reduced Gorenstein spaces and by Delphine Pol to reduced Cohen-Macaulay spaces. Starting point is the Aleksandrov-Terao theorem: A hypersurface is free if and only if its Jacobian ideal is maximal Cohen-Macaulay. Pol obtains a generalized Jacobian ideal as a cokernel by dualizing Aleksandrov's multi-logarithmic residue sequence. Notably it is essentially a suitably chosen complete intersection ideal that is used for dualizing. Pol shows that this generalized Jacobian ideal is maximal Cohen-Macaulay if and only if the module of Aleksandrov's multi-logarithmic differential k-forms has (minimal) projective dimension k-1, where k is the codimension in a smooth ambient space. This equivalent characterization reduces to Saito's definition of freeness in case k=1. In this article we translate Pol's duality result in terms of general commutative algebra. It yields a more conceptual proof of Pol's result and a generalization involving higher multi-logarithmic forms and generalized Jacobian modules.