  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 $Um(P)\neq \varnothing$.
