Tight closure/localization problem/geometric deformation/description

Let ${}D\subseteq R$ flat, ${}f$ and ${}I$ in ${}R$. For every ${}D\rightarrow K$, ${}K$ a field, we can consider ${}f$ and ${}I$ in ${}R\otimes _{D}K$. In particular for ${}K=\kappa ({\mathfrak {p}})$, ${}{\mathfrak {p}}\in \operatorname {Spec} \,D$.

How does the property

$f\in I^{*}{\text{ in }}R\otimes _{D}\kappa ({\mathfrak {p}})\,$

vary with ${}{\mathfrak {p}}$?

There are two cases:

• ${}D$ contains ${}\mathbb {Z} /(p)$. This is a geometric or equicharacteristic deformation. Example ${}D=\mathbb {Z} /(p)[t]$.

• ${}D$ contains ${}\mathbb {Z} \subseteq D$. This is an arithmetic or mixed characteristic deformation. Example ${}D=\mathbb {Z}$.