If $R$ is a Noetherian normal domain, then it is equal to the intersection of its localizations at height one primes.
What is an example of a non-normal domain that is strictly contained in such an intersection?
I'd prefer an example that is a finitely generated $\mathbb{C}$-algebra, and all I want is a candidate suggested. I'd prefer to verify it myself.
There must be an easier example, but here is one I found after searching a bit, Ex. 7.2 in this arXiv paper by Epstein and Shapiro. It is sadly not finitely generated nor noetherian, but it is normal!
Let $k$ be a field and $R = k[x,y,y/x,y/x^2,y/x^3,\ldots] \subset k(x,y)$, which you might have seen as part of Karl Schwede's construction of a non-discrete valuation ring. The ideal $\mathfrak{m} = (x)$ is a height two principal prime ideal by this answer, hence $$ x^{-1} \in \left(\bigcap_{\operatorname{ht}\mathfrak{p} = 1} R_\mathfrak{p}\right) \setminus R. $$
