The Chow group of oriented cycles and the Euler class of vector bundles

Let k be a field of characteristic not 2 and X be a smooth k-variety of dimension n. To any oriented n-dimensional vector bundle xi, we functorially assign an Euler class e(xi) is an element of H-Zar(n) (X; (J) under bar(n)), where (J) under bar(n) is a certain sheaf. When k is algebraically closed, this cohomology group reduces to the Chow group CHn(X) of 0-cycles and the Euler class to the Chern class c(n) (xi). In this Note we study the following problem: for X affine, is the vanishing of the Euler class the only obstruction for xi to have a free direct summand of rank one. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.