Projective Modules and Complete Intersections

Let A be a Noetherian ring of dimension n and P be a projective A module of rank n having trivial determinant. It is proved that if n is even and the image of a generic element g ∈ P^* is a complete intersection, then [P] = [Q ⊕ A] in K₀(A) for some projective A-module Q of rank n – 1. Further, it is proved that if n is odd, A is Cohen–Macaulay and [P] = [Q ⊕ A] in K₀(A) for some projective A module Q of rank n – 1, then P has a unimodular element.