Ideals of smooth functions and residue currents

Let f=( f1,…,fm) be a holomorphic mapping in a neighborhood of the origin in Cn. We find sufficient condition, in terms of residue currents, for a smooth function to belong to the ideal in C∞ (or Ck) generated by f. If f is a complete intersection the condition is necessary. More generally we give a sufficient condition for an element of class C∞ (or Ck) in the Koszul complex induced by f to be exact. For the proofs we introduce explicit homotopy formulas for the Koszul complex induced by f.