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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.


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up vote 2 down vote accepted

Put dim B=n for the dimension of the variety with coordinates ring B. Then

n-ht P ≥ ((n-tr deg A B)- ht p)+ tr deg k(p) k(P)

The first member of the inequality indicates the dimension of the subvariety definited by P.
The term (n-tr deg A B) in the second member is the dimension of the variety with coordinates ring A: it looses tr deg A B dimensions with respect the other variety with coordinate ring B.
Then ((n-tr deg A B)- ht p) represent the dimension of the subvariety definited by p.
The term tr deg k(p) k(P) is a corrector term because blow up can occur.

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hmm, this looks reasonable for finitely generated k-algebras. Just to be safe, what you have said is not going to be true for an arbitrary noetherian domain right? – Ho Chung Siu Nov 25 '09 at 14:21

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