The Euler class group of a polynomial algebra with coefficients in a line bundle

Abstract

Let \(A\) be a commutative Noetherian ring and \(P\) be a projective \(A\)-module of rank \(=(\text {dim}(A)-1)\). An intriguing open question is to find the precise obstruction for \(P\) to split as: \(P\simeq Q\oplus A\) for some \(A\)-module \(Q\). In this paper we settle this question when \(A=R[T]\) for some ring \(R\) containing the field of rationals and \(P\) is a projective \(A\)-module of rank \(=\text {dim}(R)\).