Definition 10.40.1. Let $\varphi : R \to S$ be a ring map.

We say a $\varphi : R \to S$ satisfies

*going up*if given primes $\mathfrak p \subset \mathfrak p'$ in $R$ and a prime $\mathfrak q$ in $S$ lying over $\mathfrak p$ there exists a prime $\mathfrak q'$ of $S$ such that (a) $\mathfrak q \subset \mathfrak q'$, and (b) $\mathfrak q'$ lies over $\mathfrak p'$.We say a $\varphi : R \to S$ satisfies

*going down*if given primes $\mathfrak p \subset \mathfrak p'$ in $R$ and a prime $\mathfrak q'$ in $S$ lying over $\mathfrak p'$ there exists a prime $\mathfrak q$ of $S$ such that (a) $\mathfrak q \subset \mathfrak q'$, and (b) $\mathfrak q$ lies over $\mathfrak p$.

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