Going down for flat extensions

Statement

Suppose $\phi :R\to S$ is a flat extension i.e. $R$ is a subring of a commutative unital ring $S$ and $S$ is flat as a $R$-module. Then, if $P_1\supset P_2$ are prime ideals of $R$, and $Q_1\in Spec(S)$ contracts to $P_1$, there exists $Q_2\in Spec(S)$ such that $Q_1\supset Q_2$, and $Q_2$ contracts to $P_2$.

References

• Book:Eisenbud, Page 239