Let $A$ be a Noetherian ring and let $I$ be an ideal in $A$. Let $\mathcal{ F} = \{ J_n \}_{ n \geq 0}$ be a multiplicative filtration of ideals in $A$ such that $\mathcal{ R}(\mathcal{ F}) = \bigoplus_{ n \geq 0} J_n$ is a finitely generated $A$-algebra. Let $\mathcal{ R} = A[It]$ and assume $I^n \subseteq J_n$ for all $n \geq 1$. We show the following two assertions are equivalent: (1) For all $i \geq 0$ we have $H^i_{ \mathcal{ R}_+}(\mathcal{ R}(\mathcal{ F}))_n = 0$ for all $n \gg 0$. (2) $J_n \subseteq \overline{ I^n}$ for all $n \geq 1$. Here $\overline{ I^n}$ is the integral closure of $I^n$.
Categories: math.AC