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This is theorem 14.C on p.84 of Matsumura's commutative algebra.

Let $A$ be a noetherian domain, and let $B$ be a finitely generated overdomain of $A$. Let $P \in Spec(B)$ and $p = P \cap A$. Then we have $ht(P) \leq ht(p) + tr.d._{A} B - tr.d._{K(p)} K(P)$ with equality holds when $A$ is universally catenary or if $B$ is a polynomial ring over $A$.

Question: How should one understand this formula? I'm hazarding a guess that this factor, $tr.d._{A} B - tr.d._{K(p)}K(P)$, can somehow measure how primes of $B$ will be identified when they are restricted back to $A$. But this sounds woefully wrong and I just want to know how I should view this result or whether there is any (geometric) intuition behind the result.

Thanks!
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Intuition for Nagata's altitude formula?

This is theorem 14.C on p.84 of Matsumura's commutative algebra.

Let $A$ be a noetherian domain, and let $B$ be a finitely generated overdomain of $A$. Let $P \in Spec(B)$ and $p = P \cap A$. Then we have $ht(P) \leq ht(p) + tr.d._{A} B - tr.d._{K(p)} K(P)$

with equality holds when $A$ is universally catenary or if $B$ is a polynomial ring over $A$.

Question: How should one understand this formula? I'm hazarding a guess that this factor, $tr.d._{A} B - tr.d._{K(p)}K(P)$,

can somehow measure how primes of $B$ will be identified when they are restricted back to $A$. But this sounds woefully wrong and I just want to know how I should view this result or whether there is any (geometric) intuition behind the result.

Thanks!