I am looking for help to understand the following:
Let $R$ be a commutative ring and $P$ a projective $R[x]$-module. If $P_{\mathfrak m}$ (localization at $R-{\mathfrak m}$, for $\mathfrak m$ a maximal ideal of $R$) is a free $R_{\mathfrak m}[x]$-module, then $P_{\mathfrak m}$ is extended from $R_{\mathfrak m}$.
Recall that a $R[x]$-module $P$ is called extended from $R$, if $P\cong P'\otimes_R R[x]$ for an $R$-module $P'$.
Thank you!
Unless I'm misunderstanding the question, this has nothing to do with localizations or projectivity of the original $P$. If $R$ is any ring and $P$ is a free $R[X]$-module, say $P\cong\bigoplus_{i\in I}R[X]$ as $R[X]$-modules for some set $I$, then, taking $P^\prime=\bigoplus_{i\in I}R$, $P^\prime\otimes_RR[X]\cong\bigoplus_{i\in I}R[X]\cong P$ as $R[X]$-modules.