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

Proof

By localization, there exists ${}h\in D$, ${}h\neq 0$, such that ${}hf\in I^{*}{\text{ in }}R$. By persistence of tight closure (under a ring homomorphism), we get

$hf\in I^{*}{\text{ in }}R_{\kappa ({\mathfrak {p}})}.$

The element ${}h$ does not belong to ${}{\mathfrak {p}}$ for almost all ${}{\mathfrak {p}}$, so ${}h$ is a unit in ${}R_{\kappa ({\mathfrak {p}})}$ and hence

$f\in I^{*}{\text{ in }}R_{\kappa ({\mathfrak {p}})}$

for almost all ${}{\mathfrak {p}}$.

To fact