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

Let $X$ be a reduced Cohen-Macaulay complex germ of codimension $k\ge2$ in a smooth ambient germ $Y$ of dimension $n$. Generalizing Saito's logarithmic forms along a hypersurface, Aleksandrov introduced a module of multi-logarithmic differential $q$-forms on $Y$ along $X$. We describe its dual module of multi-logarithmic $q$-vector fields on $Y$ along $X$. Aleksandrov and Pol showed that, as in the hypersurface case, the module of multi-logarithmic $q$-forms fits into a short exact multi-logarithmic residue sequence on $Y$ whose cokernel is the module $\omega_X^{q-k}$ of regular differential $(q-k)$-forms on $X$. We describe a short exact sequence on $Y$ that can be considered as the dual multi-logarithmic residue sequence. Its cokernel is an $(n-q)$th Jacobian module $\mathcal{J}_X^{n-q}$ on $X$ whose dual on $X$ identifies with $\omega_X^{q-k}$. If $X$ is a complete intersection (or Gorenstein), then $\mathcal{J}_X^{n-k}$ is the ($\omega$-)Jacobian ideal of $X$. We show that $\mathcal{J}_X^{n-q}$ is maximal Cohen-Macaulay and hence reflexive if an only if the module of multi-logarithmic $q$-forms has (the minimal) projective dimension $k-1$. This generalizes results of Aleksandrov and Terao in the hypersurface case and of Pol for $q=k$. Reducing the hypotheses to the essential we develop a residual duality over Gorenstein rings that yields our result by specialization.