Cohen-Macaulay graded ring/Positive characteristic/Parameter/Tight closure/Degree criterion/Inclusion/Fact/Proof

Proof

We assume ${}d\geq 2$, lower dimensions may be treated directly. We consider the Koszul-resolution of the parameter ideal ${}{\left(f_{1},\ldots ,f_{d}\right)}$. A homogeneous element ${}h$ of degree ${}m$ gives rise to the graded cohomology class

${}c=c_{d-1}={\frac {h}{f_{1}\cdots f_{d}}}\in H_{\mathfrak {m}}^{d}(R)\,.$

Under the condition that ${}\operatorname {deg} (h)\geq \sum _{j=1}^{d}\operatorname {deg} (f_{i})$ this cohomology class has nonnegative degree. It is known that ${}H_{\mathfrak {m}}^{d}(R)$ is ${}0$ for sufficiently large degree, i.e. there is a number ${}a$ such that ${}(H_{\mathfrak {m}}^{d}(R))_{i}=0$ for all ${}i>a$. Now choose a homogeneous element ${}z\in R_{>a}$ which does not belong to any minimal prime ideal. Then the degree of

${}zF^{e*}(c)=z{\frac {h^{q}}{f_{1}^{q}\cdots f_{d}^{q}}}\,$

is at least ${}\geq \operatorname {deg} (z)>a$, so this class must be ${}0$ and therefore ${}h$ belongs to the tight closure of the parameter ideal.

To fact