The F-rational signature and drops in the Hilbert–Kunz multiplicity

Abstract

Let $(R,\mathfrak{𝔪})$ be a Noetherian local ring of prime characteristic $p$. We define the F-rational signature of $R$, denoted by $\text{r}(R)$, as the infimum, taken over pairs of ideals $I⊊J$ such that $I$ is generated by a system of parameters and $J$ is a strictly larger ideal, of the drops ${\text{e}}_{\text{HK}}(I,R))-{\text{e}}_{\text{HK}}(J,R)$ in the Hilbert–Kunz multiplicity. If $R$ is excellent, then $R$ is F-rational if and only if $\text{r}(R)>0$. The proof of this fact depends on the following result in the sequel: Given an $\mathfrak{𝔪}$-primary ideal $I$ in $R$, there exists a positive ${\delta }_I\in {ℝ}^{+}$ such that, for any ideal $J⊋I$, ${\text{e}}_{\text{HK}}(I,R)-{\text{e}}_{\text{HK}}(J,R)$ is either $0$ or at least ${\delta }_I$. We study how the F-rational signature behaves under deformation, flat local ring extension, and localization.