Uniqueness and factorization of Coleff-Herrera currents

We prove a uniqueness result for Coleff-Herrera currents which in particular means that if $f=(f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_m)$ defines a complete intersection, then the classical Coleff-Herrera product associated to $f$ is the unique Coleff-Herrera current that is cohomologous to 1 with respect to the operator ${\delta }_f-\text{dbar}$, where ${\delta }_f$ is interior multiplication with $f$. From the uniqueness result we deduce that any Coleff-Herrera current on a variety $Z$ is a finite sum of products of residue currents with support on $Z$ and holomorphic forms.