Tight closure/Localization/Geometric deformation over one-dimensional domain/Fact

Let ${}\mathbb {Z} /(p)\subset D$ be a one-dimensional domain, ${}D\subseteq R$ of finite type and ${}I$ an ideal in ${}R$. Suppose that localization holds and that

$f\in I^{*}{\text{ holds in }}R\otimes _{D}Q(D)=R_{D^{*}}=R_{Q(D)}$

(${}S=D^{*}=D\setminus \{0\}$ is the multiplicative system).

Then

${}f\in I^{*}$

holds in ${}R\otimes _{D}\kappa ({\mathfrak {p}})$ for almost all ${}{\mathfrak {p}}$ in Spec ${}D$.

Proof, Write another proof