Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 101 / 4 / b / i / Solution

The Going-up theorem states: if $B$ is integral over $A$, ${\mathfrak{p}}_1\subseteq {\mathfrak{p}}_2$ are primes of $A$, and ${\mathfrak{q}}_1$ lies over ${\mathfrak{p}}_1$, then some prime ${\mathfrak{q}}_2\supseteq {\mathfrak{q}}_1$ lies over ${\mathfrak{p}}_2$.

Pass to $A/{\mathfrak{p}}_1\subseteq B/{\mathfrak{q}}_1$, which remains integral, and localize at the complement of ${\mathfrak{p}}_2/{\mathfrak{p}}_1$. The lying-over theorem supplies a prime of the localized upper ring over the maximal ideal of the localized lower ring. Contracting it to $B/{\mathfrak{q}}_1$, and then pulling it back to $B$, gives the required ${\mathfrak{q}}_2$.