  We prove a uniqueness result for Coleff-Herrera currents which in particular
means that if $f=(f_1,..., f_m)$ 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
$\delta_f-\dbar$, 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.
