Existence of unimodular elements in a projective module
Journal of Pure and Applied Algebra
(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[T1,⋯,Tk]-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⊕An−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).
Keshari, Manoj K. and Ali Zinna, Md, "Existence of unimodular elements in a projective module" (2017). Journal Articles. 2373.