Going down for flat extensions

Statement

Suppose $\varphi :R\to S$ 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