  Given a coherent ideal sheaf $J$ we construct locally a vector-valued residue
current $R$ whose annihilator is precisely the given sheaf. In case $J$ is a
complete intersection, $R$ is just the classical Coleff-Herrera product. By
means of these currents we can extend various results, previously known for a
complete intersection, to general ideal sheaves. Combining with integral
formulas we obtain a residue version of the Ehrenpreis-Palamodov fundamental
principle.
