Statement: Let $R$ be a commutative Noetherian ring and let $P$ be a finitely generated $R$-module, then $P$ is projective if and only if $P℘$ is a free $R℘$-module for all $℘$ in $Spec(R)$.
For the converse part, I have used for $F$ free module any exact sequence $0→M→N→F→0$ splits and the definition of projective modules $P$ as
Every exact sequence $0 → N → M → P → 0$, of $R$-modules and $R$-linear maps splits.
But for $(\implies way)$ I am unable to approach,