On the duality theorem on an analytic variety

The duality theorem for Coleff-Herrera products on a complex manifold says that if $f=(f_1,\dots ,f_p)$ defines a complete intersection, then the annihilator of the Coleff-Herrera product ${\mu }^f$ equals (locally) the ideal generated by $f$. This does not hold unrestrictedly on an analytic variety $Z$. We give necessary, and in many cases sufficient conditions for when the duality theorem holds. These conditions are related to how the zero set of $f$ intersects certain singularity subvarieties of the sheaf ${\mathcal{O}}_Z$.