  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.

Algebraic Geometry