The Euler class groups of polynomial rings and unimodular elements in projective modules

Let A be a commutative noetherian ring of dimension n. Let P be a projective A[T ]-module. Plumstead ([P]) proved that if rankP > dimA then P splits off a free summand of rank one. It is natural to ask what happens when rankP = dimA. In this paper we investigate this question when P has trivial determinant. Let α : P I be a generic surjection (i.e. I ⊂ A[T ] is an ideal of height n). It is proved in ([D]) that if P splits off a free summand of rank one then I is generated by n elements. It is natural to ask whether the converse holds, i.e., if I is generated by n elements then whether P has a free summand of rank one. This has been proved in ([B-RS 4]) if A is an affine domain over an algebraically closed field. But the following example shows that this converse is false in general. Let A be the coordinate ring of the even dimensional real sphere and P̃ be the tangent bundle. It can be shown that P̃ does not have a free summand of rank one whereas there is a generic surjection β : P̃ J such that J is generated by n elements. Tensoring with A[T ] we obtain a generic surjection β ⊗ A[T ] : P̃ [T ] J [T ] showing that the converse is false. This leads us to the following