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

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.

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