Going-up theorem

Remarks

The ring $B/\mathfrak{q}_1$ is integral over its subring $A/\mathfrak{p}_1$. The lying over theorem gives a prime of $B/\mathfrak{q}_1$ over $\mathfrak{p}_2/\mathfrak{p}_1$, and its preimage in $B$ is $\mathfrak{q}_2$.

By induction, every chain $\mathfrak{p}_1 \subseteq \cdots \subseteq \mathfrak{p}_n$ of primes of $A$ lifts to a chain $\mathfrak{q}_1 \subseteq \cdots \subseteq \mathfrak{q}_n$ of primes of $B$ with $\mathfrak{q}_i \cap A = \mathfrak{p}_i$. So the Krull dimension of $B$ is at least that of $A$.

Example: for $\Z \subseteq \Z[i]$, the chain $0 \subseteq 5\Z$ lifts to $0 \subseteq (2 + i)$.

Integrality is needed: for $k[x] \subseteq k[x, x^{-1}]$, the prime $0$ lies under $0$, but no prime of $k[x, x^{-1}]$ lies over $(x)$, since $x$ is a unit there.

Going deeper

For a student

Going-up and incomparability together give $\dim B = \dim A$ for every integral extension $A \subseteq B$. This is the key step in computing the dimension of a finitely generated algebra over a field through Noether normalization. The companion result in the other direction, going-down, needs extra hypotheses.