Grothendieck's generic freeness lemma

Statement

Suppose $R$ is a Noetherian ring and $S$ is a finitely generated $R$-algebra. Further, suppose $M$ is a free module over $S$. Then, there exists $0\ne a\in R$ such that $M[a^{-1}]$ is free as a module over $R[a^{-1}]$. Here $R[a^{-1}]$ denotes the localization of $R$ at $a$, and $M[a^{-1}]$ denotes the localization of $M$ at $a$.