Suppose $A$ is a Dedekind domain with fraction field $K$ and $L/K$ is Galois, let $B$ be the integral closure of $A$ in $L$.
Let $P$ be a prime ideal in $A$ and let $P_1,...,P_n$ be prime ideals lying over $P$ in $B$. Then, $P=(P_1\cap ... \cap P_n)\cap A$. I was wondering if it's true that $(P_1^i \cap ... \cap P_n ^i)\cap A$ is primary?
PS: I was posting with username BMI earlier. For some reason, every time I posted this question, I got an error "this question could not be submitted because it does not meet our quality standards". I copy pasted it from a new user id and it posted without a hassle.