Going-down theorem

Remarks

The proof shows that $\mathfrak{p}_2 B_{\mathfrak{q}_1} \cap A = \mathfrak{p}_2$; a prime of $B_{\mathfrak{q}_1}$ over $\mathfrak{p}_2$ then exists, and it contracts to the wanted $\mathfrak{q}_2$ in $B$. The key step uses that $A$ is integrally closed. If an element of $B$ is integral over the ideal $\mathfrak{p}_2$, then the coefficients of its minimal polynomial over the fraction field of $A$ lie in $\mathfrak{p}_2$. Here one also uses that $B$ is a domain, so that nonzero elements of $A$ are not zero divisors on $B$.

Example: $\Z \subseteq \Z[i]$ satisfies the hypotheses, and for $\mathfrak{p}_1 = 5\Z \supseteq \mathfrak{p}_2 = 0$ and $\mathfrak{q}_1 = (2 + i)$, the prime $\mathfrak{q}_2 = 0$ works.

The hypothesis that $A$ is integrally closed cannot be dropped: there are integral extensions of domains in which going-down fails. A different sufficient condition is flatness: going-down holds for every flat ring extension.

Going deeper

For a student

Going-down is used to prove that heights behave well in integral extensions over a normal base: $\operatorname{ht} \mathfrak{q} = \operatorname{ht}(\mathfrak{q} \cap A)$ for every prime $\mathfrak{q}$ of $B$. It is a key ingredient in dimension theory of finitely generated algebras over a field, for example to show that such domains are catenary. In geometry it gives that a finite surjective map from an irreducible variety onto a normal variety is an open map.