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

Proof

We assume , lower dimensions may be treated directly. We consider the Koszul-resolution of the parameter ideal . A homogeneous element of degree gives rise to the graded cohomology class

Under the condition that this cohomology class has nonnegative degree. It is known that is for sufficiently large degree, i.e. there is a number such that for all . Now choose a homogeneous element which does not belong to any minimal prime ideal. Then the degree of

is at least , so this class must be and therefore belongs to the tight closure of the parameter ideal.