Let $R\subseteq S$ be a finite ring extension such that $R$ and $S$ are both complete noetherian local domains. Let $\mathfrak{p}$ be a prime ideal of $R$ and denote by $\mathfrak{q}_1,\ldots, \mathfrak{q}_n$ all the prime ideals of $S$ that lie over $\mathfrak{p}$.
Assume that $R$ is regular and the local rings $S_{\mathfrak{q_1}}, \ldots, S_{\mathfrak{q}_n}$ are all of finite global dimension, can we deduce that $S_\mathfrak{p}$ is of finite global dimension?
Any comments are welcome. Thanks a lot!