Cohen-Macaulay ring/Positive characteristic/Cohomological criterion for tight closure/Fact

Suppose that ${}R$ is a Cohen-Macaulay ring of positive characteristic ${}p$ and of dimension ${}d\geq 2$. Let ${}I$ be an ${}{\mathfrak {m}}$-primary ideal. Let ${}F_{\bullet }$ be a free (not necessarily minimal) resolution of ${}I$, let ${}\operatorname {Syz} _{j}$ be the corresponding syzygy sheaves, let ${}f\in R$ be an element and let ${}c_{j}\in H^{j}(D({\mathfrak {m}}),\operatorname {Syz} _{j})$ be the corresponding cohomology classes.

Then for each

${}j$, ${}1\leq j\leq d-1$, we have the equivalence that ${}f\in I^{*}$ if and only if ${}c_{j}$ is tightly ${}0$ in the sense that there exists ${}z$ not in any minimal prime ideal such that

${}zF^{e}(c_{j})=0\,$

holds in ${}H^{j}(D({\mathfrak {m}}),F^{e*}\operatorname {Syz} _{j})$ for all ${}e$.

Proof, Write another proof