Finitely generated projectives are locally free

Finitely generated projectives are locally free: Let $R$ be a commutative ring and let $P$ be a finitely generated projective $R$-module. Then for every prime ideal $\mathfrak p\subset R$, the localization $P_{\mathfrak p}$ is a free $R_{\mathfrak p}$-module of finite rank.

This is a fundamental structure theorem for projective modules over a commutative ring: after localizing at any prime ideal, a finitely generated projective becomes free with well-defined rank.