Let $R$ be a commutative unital ring, and let $P$ be a finitely generated $R$-module. Then $P$ is projective if and only if $P$ is locally free in the Zariski topology: for every prime ideal $\mathfrak p\in\operatorname{Spec}(R)$, there exist an element $f\in R\setminus\mathfrak p$ and an integer $r\ge 0$ such that