The F-rational signature and drops in the Hilbert-Kunz multiplicity (Q2104864)

Let \((R, \mathfrak{m})\) be a Noetherian local ring of prime characteristic \(p\). The ring \(R\) is called F-rational if all parameter ideals are tightly closed. In the paper under review the authors define the following new invariant: For any \(\mathfrak{m}\)-primary ideal \(I\), consider the Hilbert-Kunz multiplicity as \(e_{HK}(I,R)\). The F-rational signature of \(R\), denoted \(r(R)\), is \[ r(R)=\inf\{e_{HK}(I,R)-e_{HK}(J,R)\mid I \text{ is generated by an s.o.p. and }I\subsetneq J\}. \] Among other results, it is shown that, under mild conditions, \(r(R)\) can be defined via one single choice of ideal generated by a system of parameters. Then it is proved that when \(R\) is excellent, \(r(R)>0\) if and only if \(R\) is F-rational. The authors establish results on how the F-rational signature behaves under deformation, local flat extension and localization. In addition, they calculate the F-rational signature of some specific rings of interest.