Forcing algebra/Relation to tight closure/Local cohomology/Characterization/Fact

Let ${}R$ be a normal excellent local domain with maximal ideal ${}{\mathfrak {m}}$ over a field of positive characteristic. Let ${}f_{1},\ldots ,f_{n}$ generate an ${}{\mathfrak {m}}$-primary ideal ${}I$ and let ${}f$ be another element in ${}R$.

Then

${}f\in I^{*}$ if and only if ${}H_{\mathfrak {m}}^{\dim(R)}(A)\neq 0$, where ${}A=R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}+f\right)}$

denotes the forcing algebra of these elements.

Write a proof