The following question essentially appeared (https://math.stackexchange.com/questions/931801/preservation-of-localness-among-certain-krull-domains) on math.SE a while ago, but nobody has done anything with it, and I'm interested in the answer, so I'll post it here.

Namely, let $R$ be a local Noetherian normal domain, and let $\mathfrak p$ be a height one prime whose class in the divisor class group is non-torsion. Let $$ S := \bigcap_{\mathfrak q \in \operatorname{Spec} R, \operatorname{ht} \mathfrak q = 1, \mathfrak q \neq \mathfrak p} R_{\mathfrak q}. $$ Is $S$ local? And if not, what can we say about $\mathfrak p S$? Is it a proper ideal of $S$?