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!



Intuition for Nagata's altitude formula?
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. 


