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 $k$, $T_1,\ldots,T_r$ be variables over $A$ and $P$ be a projective $A[T_1,\ldots,T_r]$-module of rank $n$. Assume one of the following conditions hold. (1) $2n\geq d+3$ and $P$ is extended from $A$. (2) $2n\geq d+2$, $A$ is an affine $\overline {\mathbb F}_p$-algebra and $P$ is extended from $A$. (3) $2n\geq d+3$ and singular locus of $Spec(A)$ is a closed set $V(\mathcal J)$ with ht $\mathcal J\geq d-n+2$. Assume $Um(P_f)\neq \varnothing$ for some monic polynomial $f(T_r)\in A[T_1,\ldots,T_r]$. Then $Um(P)\neq \varnothing$.