Regulärer Ring/Positive Charakteristik/Tight closure trivial/Fakt/en/Beweis

Beweis

We assume , lower dimensions may be treated directly. Because of , we can also reduce to the case of a primary ideal . Suppose that , and let be the corresponding non-zero class arising from a finite free resolution. At least one component, say , is then also non-zero, and we can write it in terms of Čech-cohomology as

where is a regular system of parameters of and . We have to show that there is no such that for all . Multiplying the class with some element of we may assume that is a unit. We have (with )

and its annihilator is . But then