# Symbolic Rees Algebra of an ideal in a Noetherian excellent ring

For an ideal $$I$$ in a commutative Noetherian ring $$R$$ and integer $$n\ge 0$$, the $$n$$-th symbolic power of $$I$$ is define as $$I^{(n)}:=\cap_{P\in Ass(R/I)} \phi_P^{-1} (I^nR_P)$$ , where $$\phi_P : R\to R_P$$ is the localization map.

Now assume $$Ass(R/I)=Min(R/I)$$. Then it can be seen that $$I^{(a)}I^{(b)}\subseteq I^{(a+b)},\forall a,b\ge 0$$.

Let $$\mathcal R_s(I):= \oplus_{n\ge 0} I^{(n)}t^n \subseteq R[t]$$ be the symbolic Rees Algebra.

If $$R$$ is an excellent ring and $$I$$ is an ideal with $$Ass(R/I)=Min(R/I)$$ and there exists $$k\ge 1$$ such that $$I^{(nk)}=(I^{(k)})^n,\forall n\ge 0$$ , then how to show that $$\mathcal R_s(I)$$ is a finitely generated $$R$$-algebra ?