Characterizations of freeness for equidimensional subspaces

The purpose of this paper is to investigate properties of the minimal free resolution of the modules of multi-logarithmic forms along a reduced equidimensional subspace. We first consider a notion of freeness for reduced complete intersections, and more generally for reduced equidimensional subspaces embedded in a smooth manifold, which generalizes the notion of Saito free divisors. The first main result is a characterization of freeness in terms of the projective dimension of the module of multi-logarithmic k -forms, where k is the codimension. We also prove that there is a perfect pairing between the module of multi-logarithmic differential k -forms and the module of multi-logarithmic k -vector fields which generalizes the duality between the corresponding modules in the hypersurface case. We deduce from this perfect pairing a duality between the Jacobian ideal and the module of multi-residues of multi-logarithmic k -forms. In the last part of this paper, we investigate logarithmic modules along some examples of free singularities. The main result in this section is an explicit computation of the minimal free resolution of the module of multi-logarithmic forms and multi-residues for quasi-homogeneous complete intersection curves which uses our first main theorem.