Cancellation of projective modules in polynomial rings of prime characteristic

Let $A$ be a commutative Noetherian ring of characteristic $p>0$, such that $\dim (A)=d$. Let $P$ be a projective $A[T_1,\mathrm{.}\mathrm{.}\mathrm{.},T_n]$-module of rank $d$. We show that $P$ is cancellative if and only if $P\mathrm{/}<T_1,\mathrm{.}\mathrm{.}\mathrm{.},T_n>P$ is cancellative. We deduce some applications. In one of the interesting consequences, we show that the Bass-Quillen conjecture has an affirmative answer in dimension three, when $2$ is invertible.