Cohen-Macaulay Ring/Positive Charakteristik/Kohomologisches Kriterium für tight closure/Fakt/en/Beweis

Beweis

We consider the short exact sheaf sequences

$0\longrightarrow \operatorname {Syz} _{i}\longrightarrow {\mathcal {O}}_{U}^{\beta _{i}}\longrightarrow \operatorname {Syz} _{i-1}\longrightarrow 0$

on ${}U=D({\mathfrak {m}})$ coming from the resolution for ${}i=1,\ldots ,d-1$ ($\operatorname {Syz} _{0}$ is just the structure sheaf). Because all these sheaves are locally free, taking the absolute Frobenius (and all its iterations) is exact, therefore we get short exact sequences

$0\longrightarrow F^{e*}\operatorname {Syz} _{i}\longrightarrow {\mathcal {O}}_{U}^{\beta _{i}}\longrightarrow F^{e*}\operatorname {Syz} _{i-1}\longrightarrow 0$

and cohomology pull-backs ${}F^{e*}(c_{j})\in H^{j}{\left(U,F^{e*}\operatorname {Syz} _{j}\right)}$. Note also that for ${}i=1$ and ${}I={\left(f_{1},\ldots ,f_{n}\right)}$ we get

$0\longrightarrow F^{e*}\operatorname {Syz} _{1}\longrightarrow {\mathcal {O}}_{U}^{n}{\stackrel {f_{1}^{q},\ldots ,f_{n}^{q}}{\longrightarrow }}{\mathcal {O}}_{U}\longrightarrow 0,$

so the image of this map inside ${}R=\Gamma (U,{\mathcal {O}}_{U})$ is exactly ${}I^{[q]}$. By the universal property of the absolute Frobenius and of the connecting homomorphisms in cohomology we have

${}F^{e*}(c_{j})=F^{e*}(\delta ^{j}(f))=\delta ^{j}(F^{e*}f)=\delta ^{j}(f^{q})\,$

and also

${}zF^{e*}(c_{j})=\delta ^{j}(zf^{q})\,.$

Because of the injectivity of ${}\delta$ in the given range we have that ${}zf^{q}$ belongs to the ideal ${}I^{[q]}$ if and only if ${}zF^{e*}(c_{1})=0$ if and only if ${}zF^{e*}(c_{j})=0$.

Zur bewiesenen Aussage