On a question of Moshe Roitman and Euler class of stably free module

Abstract

Let A be a ring of dimension d containing an infinite field k, T1,…,Tr be variables over A and P be a projective A[T1,…,Tr]-module of rank n. Assume one of the following conditions holds.(1)2n≥d+3 and P is extended from A.(2)2n≥d+2, A is an affine F‾p-algebra and P is extended from A.(3)2n≥d+3 and singular locus of Spec(A) is a closed set V(J) with ht J≥d−n+2. Assume Um(Pf)≠∅ for some monic polynomial f(Tr)∈A[T1,…,Tr]. Then Um(P)≠∅ (see 6.1).