Tight closure, coherence, and localization at single elements (Q6597140)

This article introduces a new class of \(F\)-singularities, semi \(F\)-regular singularities, which are shown to lie between weakly \(F\)-regular singularities and \(F\)-regular singularities. The author further shows that when \(R\) is a Jacobson ring, these two classes of singularities are the same. It is not known if tight closure commutes with localization for local rings \((R, \mathfrak{m})\). \textit{If} these two classes are always equivalent for any local \((R,\mathfrak{m})\), then this would settle this longstanding conjecture.\N\NKey to the construction of these singularities is developing a theory of openly persistent preclosures on subsheaves as well as a gluing property for preclosures. Using a \(p\)-system of ideals (\(\{\mathfrak{b}_p^e\}_{e=1}^\infty\) satisfying \(\mathfrak{b}_q(\mathfrak{b}_{q'})^{[q]} \subseteq \mathfrak{b}_{qq'}\)), the author defines the \(\mathfrak{b}_\bullet\)-tight closure of \(L\) in \(M\) as the set of \(z \in M\) satisfying \(c\mathfrak{b}_qz^q \subseteq L^{[q]}\) for some \(c\) not contained in the union of minimal prime ideals of \(R\) and all large \(q\). For example, choosing \(\mathfrak{b}_q=R\) for all \(q\), gives us tight closure, whereas \(\mathfrak{b}_q=\mathfrak{a}^{\lceil tq\rceil}\) for all \(q\) gives the \(\mathfrak{a}^t\)-tight closure of Hara and Yoshida. For any affine open subset \(U\) of a locally Noetherian \(\mathbb{F}_p\)-scheme, the author defines a system of quasi-coherent ideal sheaves \(\mathrm{b}_\bullet:=\{\mathrm{b}_{p^e}\}_{e=1}^\infty\) to be a \(p\)-system of ideal sheaves if \(\mathrm{b}(U)_\bullet\) is a \(p\)-system of \(\mathcal{O}_X(U)\)-ideals. He then shows that \(*\mathrm{b}_\bullet\), a preclosure defined on a \(p\)-system \(\mathrm{b}_\bullet\) of ideal sheaves over a locally Noetherian scheme is openly persistent and glueable.