  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.
