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

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