Let $R \subseteq A$ be two $\mathbb{C}$-algebras, $P$ a prime ideal of $R$, $Q$ a prime ideal of $A$, and $Q \cap R = P$. Assume that $(A_Q,QA_Q)$ is flat over $(R_P,PR_P)$.
'When' $A$ is flat over $R$?
I do not mind to further assume that the above local rings are regular; by Auslander-Buchsbaum theorem, such rings are UFD's.
Perhaps this question may help.
Remark: I have edited the original question, according to the comments below.
Thank you very much!