Existence of unimodular elements in a projective module

Abstract (1) Let R be an affine algebra over an algebraically closed field of characteristic 0 with dim ( R ) = n . Let P be a projective A = R [ T 1 , ⋯ , T k ] -module of rank n with determinant L . Suppose I is an ideal of A of height n such that there are two surjections α : P ↠ I and ϕ : L ⊕ A n − 1 ↠ I . Assume that either (a) k = 1 and n ≥ 3 or (b) k is arbitrary but n ≥ 4 is even. Then P has a unimodular element (see 4.1 , 4.3 ). (2) Let R be a ring containing Q of even dimension n with height of the Jacobson radical of R ≥ 2 . Let P be a projective R [ T , T − 1 ] -module of rank n with trivial determinant. Assume that there exists a surjection α : P ↠ I , where I ⊂ R [ T , T − 1 ] is an ideal of height n such that I is generated by n elements. Then P has a unimodular element (see 3.4 ).