I'm trying to prove Mori-Nagata theorem.

Let $A$ be a noetherian domain, $K=\operatorname{Frac}(A)$ and $B$ be the integral closure of $A$ in $K$.

I want to show that for every nonzero $x\in B$, there are finitely many primes of $B$ of height $1$ containing $x$.

Replacing $A$ by $A[x]$, we may assume that $x\in A$.

It suffices to show that for every prime ideal $P$ of $B$ of height $1$ containing $x$, $P\cap A\in \operatorname{Ass}_A(A/xA)$.

(I know that the corresponding morphism $\operatorname{Spec}B\rightarrow \operatorname{Spec}A$ has finite fibres.)

This claim is equivalent to the following:

Let $P$ be a prime ideal of $B$ of height $1$, and put $p=P\cap A$.

Then $\operatorname{depth}A_p =1$.

How can I prove this? Any advice would be helpful.

Thank you.


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