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

Beweis

We assume ${}d\geq 2$, lower dimensions may be treated directly. Because of ${}I^{*}\subseteq \bigcap _{n\in \mathbb {N} }(I+{\mathfrak {m}}^{n})^{*}$, we can also reduce to the case of a primary ideal ${}I$. Suppose that ${}f\notin I$, and let ${}c_{d-1}\in H^{d-1}(D({\mathfrak {m}}),{\mathcal {O}}_{U})^{\beta _{d}}=H_{\mathfrak {m}}^{d}(R)^{\beta _{d}}$ be the corresponding non-zero class arising from a finite free resolution. At least one component, say ${}c'\in H_{\mathfrak {m}}^{d}(R)$, is then also non-zero, and we can write it in terms of Čech-cohomology as

${}c'={\frac {h}{x_{1}^{n_{1}}\cdots x_{d}^{n_{d}}}}\,,$

where ${}x_{1},\ldots ,x_{n}$ is a regular system of parameters of ${}R$ and ${}n_{j}\geq 1$. We have to show that there is no ${}z\neq 0$ such that ${}zF^{e*}(c)=0$ for all ${}e\in \mathbb {N}$. Multiplying the class with some element of ${}R$ we may assume that ${}h$ is a unit. We have (with ${}q=p^{e}$)

${}F^{e*}(c')={\frac {h^{q}}{x_{1}^{qn_{1}}\cdots x_{d}^{qn_{d}}}}\,$

and its annihilator is ${}{\left(x_{1}^{qn_{1}},\ldots ,x_{d}^{qn_{d}}\right)}$. But then

${}\bigcap _{e\in \mathbb {N} }{\left(x_{1}^{qn_{1}},\ldots ,x_{d}^{qn_{d}}\right)}\subseteq \bigcap _{e\in \mathbb {N} }{\left(x_{1}^{n_{1}},\ldots ,x_{d}^{n_{d}}\right)}^{q}=0\,.$

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