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 ?


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