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 $$a$$.