Tight closure/Graded ring/Generic results/Fact

Let ${}d\geq 1$ and ${}a_{1},\ldots ,a_{n}$ be natural numbers, ${}n\geq d+1$. Let ${}K[x_{0},x_{1},\ldots ,x_{d}]\subseteq R$ be a finite extension of standard-graded domains (a graded Noether normalization). Suppose that there exist ${}n$ homogeneous polynomials ${}g_{1},\ldots ,g_{n}$ in ${}P=K[x_{0},x_{1},\ldots ,x_{d}]$ with ${}\operatorname {deg} \,(g_{i})=a_{i}$ such that ${}P_{\geq m}\subseteq {\left(g_{1},\ldots ,g_{n}\right)}$.

Then

1. ${}R_{m+d}\subseteq {\left(f_{1},\ldots ,f_{n}\right)}^{*}$ holds in the generic point of the parameter space of homogeneous elements ${}f_{1},\ldots ,f_{n}$ in ${}R$ of this degree type (the coefficients of the ${}f_{i}$ are taken as indeterminates).
2. If ${}R$ is normal, then ${}R_{m+d+1}\subseteq {\left(f_{1},\ldots ,f_{n}\right)}^{F}\subseteq {\left(f_{1},\ldots ,f_{n}\right)}^{*}$ holds for (open) generic choice of homogeneous elements ${}f_{1},\ldots ,f_{n}$ in ${}R$ of this degree type.

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