Hilbert-Kunz/Relative two-dimensional situation/Limit and interpretation/Fact

Let ${}\mathbb {Z} \subseteq S$ be a finitely generated graded domain of relative dimension two with normal fibers and let ${}I\subseteq S$ be a homogeneous ideal such that ${}I_{p}$ is an ${}(S\otimes _{\mathbb {Z} }\mathbb {Z} /(p))_{+}$-primary ideal for all prime reductions.

Then the limit

$\lim _{p\rightarrow \infty }e_{HK}(I_{p})$

exists and equals the Hilbert-Kunz multiplicity ${}e_{HK}^{0}(I_{0})$ in characteristic zero.

Write a proof