Let $A\subseteq B$ be two commutative noetherian local domains such that $A$ is complete (and regular), and $B$ is finitely generated as an $A$-module. Can we deduce from this condition that $$B_{\mathfrak{q}\cap A} = B_\mathfrak{q}$$ for any prime ideal $\mathfrak{q}$ of $B$?
Any comments are welcome. Thanks a lot!