Set Theoretic Complete Intersection for Curves in a Smooth Affine Algebra

Let A be a regular ring of dimension d ( d ≥ 3 ) containing an infinite field k. Let n be an integer such that 2n ≥ d+3. Let I be an ideal in A of height n and P be a projective Amodule of rank n. Suppose P ⊕A ≈ A and there is a surjection α: P → I. It is proved in this note that I is a set theoretic complete intersection ideal. As a consequence, a smooth curve in a smooth affine C-algebra with trivial conormal bundle is a set theoretic complete intersection if its corresponding class in the Grothendieck group is torsion.