(Lying over and going up)
Let \(\phi: A \to B\) be a ring map with \(B\) integral over \(\phi(A)\), and \(f: \Spec B \to \Spec A\) the induced map, \(f(\mfq) = \phi^{-1}(\mfq)\).
(Lying over) If \(\phi\) is injective, then for every prime \(\mfp \subseteq A\) there is a prime \(\mfq \subseteq B\) with \(\phi^{-1}(\mfq) = \mfp\); so \(f\) is surjective.
(Going up) If \(\mfp_1 \subseteq \cdots \subseteq \mfp_n\) is a chain of primes of \(A\) and \(\mfq_1 \subseteq \cdots \subseteq \mfq_m\), \(m < n\), is a chain of primes of \(B\) with \(\phi^{-1}(\mfq_i) = \mfp_i\), then it extends to a chain \(\mfq_1 \subseteq \cdots \subseteq \mfq_n\) with \(\phi^{-1}(\mfq_i) = \mfp_i\) for all \(i\).
In particular \(f\) is a closed map. [@AM18]
(Going down)
Let \(A \subseteq B\) be domains with \(A\) integrally closed and \(B\) integral over \(A\). If \(\mfp_1 \supseteq \cdots \supseteq \mfp_n\) is a chain of primes of \(A\) and \(\mfq_1 \supseteq \cdots \supseteq \mfq_m\), \(m < n\), is a chain of primes of \(B\) with \(\mfq_i \cap A = \mfp_i\), then it extends to a chain \(\mfq_1 \supseteq \cdots \supseteq \mfq_n\) with \(\mfq_i \cap A = \mfp_i\). Going down also holds for every flat ring map \(A \to B\). [@AM18]
The translation is the content of the answer: going up says that a chain of primes in \(A\) ascending from one that is hit can be lifted, which is exactly the statement that the image of a closed set is closed. So going up is the algebraic form of “a finite morphism is closed”, hence of “a finite morphism is proper”.
Going up together with incomparability makes \(\dim B = \dim A\) for an integral extension, which is why a finite surjective morphism cannot drop dimension. Going down is what makes heights match, \(\operatorname{ht} \mfq = \operatorname{ht}(\mfq \cap A)\), under its hypotheses: geometrically, every generization of \(f(\mfq)\) in \(\Spec A\) lifts to a generization of \(\mfq\).
The two are the reason finite morphisms behave like branched covers: closed, surjective, with finite fibres, and preserving dimension.
Dependencies
D-VKR54$\Spec A$ as a locally ringed spaceuses