Let $A\subset B$ be an integral extension of rings and let $\mathfrak{p}\in\text{Spec}(A)$. Setting $S=A\setminus\mathfrak{p}$ and $A_\mathfrak{p}=S^{-1}A$, if $\mathfrak{q}\subset B$ lies over $\mathfrak{p}$, is it necessarily true that there is some $Q\in\text{Spec}(S^{-1}B)$ which lies over $\mathfrak{p}A_\mathfrak{p}\in\text{Spec}(A_\mathfrak{p})$?
I'm having tremendous difficulty parsing this. I know that the inclusion goes the 'other way', though. I know that if there is $Q\in\text{Spec}(S^{-1}B)$ that lies over $\mathfrak{p}A_{\mathfrak{p}}$, then $Q\cap B$ lies over $\mathfrak{p}$. (because we know that $\text{Spec}(S^{-1}B)\to\text{Spec}(B)$ is injective).