Tight closure of certain submodules of the top local cohomology (Q2860915)

This paper concerns the tight closure of finitely generated submodules of the top local cohomology module of a \(d\)-dimensional Cohen-Macaulay local ring \(R\) with respect to the maximal ideal \(\mathfrak{m}\). The main results include showing such a ring \(R\) is weakly \(F\)-regular if and only if every finitely generated submodule of \(H^d_{\mathfrak{m}}(R)\) is tightly closed and that two finite submodules \(N \subseteq M \subseteq H^d_{\mathfrak{m}}(R)\) have the same tight closure if and only if their Hilbert-Kunz multiplicities are equal. The author also determines the tight closure of \((0:_{H^d_{\mathfrak{m}}(R)}(x_1, \ldots, x_{n}))\) in \(H^d_{\mathfrak{m}}(R)\) for \(x_1,\ldots,x_{n}\) any proper subset of a system of parameters of \(R\) and further shows for any \(F\)-rational ring \(R\), any submodule \(M\) of \(H^d_{\mathfrak{m}}(R)\), and any ideal \(I\) of \(R\), both Ann\(_R(M)\) and \((0:_{H^d_{\mathfrak{m}}(R)}I)\) are tightly closed.