The structure sheaf can be computed as the intersection of stalks

Question

Let $X$ be an integral scheme. Let $U\subset X$ be a non-empty open set. There is a canonical identification $\mathcal{O}_{U, \eta}\approx \mathcal{O}_{X, \eta}$ where $\eta$ is the generic point. Are the subrings $\mathcal{O}_X(U)\subseteq\mathcal{O}_{X, \eta}$ and $\bigcap_{p\in U}\mathcal{O}_{X, p}\subseteq\mathcal{O}_{X, \eta}$ equal?

If $X$ is affine it holds for $U=X$ (An integral domain $A$ is exactly the intersection of the localisations of $A$ at each maximal ideal). In the case $X=\mathrm{Spec}\:R$ where $R$ is a DVR it is true for $U=\{\eta\}$ because $\mathcal{O}_X(U)$ is the fraction field.

Answer 1

In the case of an integral scheme $X$, for any non-empty open subsets $U \subset V \subset X$, we have that the maps

$$ \mathcal{O}_X (V) \xrightarrow{\text{restriction}} \mathcal{O}_X (U) \xrightarrow{s \mapsto s_{\eta}} \mathcal{O}_{X,\eta} $$

are injective.

Since $\mathcal{O}_X$ is a sheaf, it follows from the above that for any non-empty open subset $U \subset X$ and any affine open covering $U = \bigcup_i U_i$, we have

$$ \mathcal{O}_X (U) = \bigcap_i \mathcal{O}_X (U_i) = \bigcap_{p \in U} \mathcal{O}_{X,p}.$$