Let $S \subset R$ be a multiplicative set, and $M$ is an $S^{-1}R$ module. Then I read on stacks project that
$M$ is flat $R$-module iff $M$ is flat $S^{-1}R$ module.
I see one direction. If $M$ is flat over $S^{-1}R$ then it is so over $R$. This is because localization $R \rightarrow S^{-1}R$ is flat. Hence so is composite $R \rightarrow S^{-1}R \rightarrow M$.
I don't see the other direction... Any hints would be appreciated.