Set theoretic complete intersection for curves in a smooth affine algebra

AbstractLet 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 A-module of rank n. Suppose P⊕A≈An+1 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