On $\mathfrak{m}$-adic Continuity of $F$-Splitting Ratio
Maria Akter
TL;DR
This paper investigates the $\mathfrak{m}$-adic continuity of Frobenius splitting data for divisor pairs $(R,\Delta)$ in $F$-finite local rings of prime characteristic $p>0$. Under the assumptions that $R$ is $F$-finite, $\mathbb{Q}$-Gorenstein, and Cohen–Macaulay, the Frobenius splitting numbers $a^{\Delta}_e(R)$ remain invariant under sufficiently small perturbations of a defining hypersurface, i.e. $a^{\Delta}_{te_0}(R/(f))=a^{\Delta}_{te_0}(R/(f+\varepsilon))$ for perturbations $\varepsilon$ in a high $\mathfrak{m}$-adic power. Moreover, a general inequality shows that the Frobenius splitting dimension does not decrease under perturbations: $\dim (R/\mathcal{P}(R/(f),\Delta|_{f})) \le \dim (R/\mathcal{P}(R/(f+\varepsilon),\Delta|_{(f+\varepsilon)}))$, with examples illustrating that equality need not hold and strict improvements can occur. The approach combines Inversion of Adjunction for $F$-purity, analysis of splitting ideals $I_e(R,\Delta)$, and local-cohomological techniques to compare splitting data between $R/(f)$ and $R/(f+\varepsilon)$, shedding light on how perturbations affect the splitting prime and associated dimensions.
Abstract
We investigate the $\mathfrak{m}$-adic continuity of Frobenius splitting dimensions and ratios for divisor pairs $(R,Δ)$ in an $F$-finite local ring $(R,\mathfrak{m},k)$ of prime characteristic $p>0$. Our main result states that if $R$ is an $F$-finite, $\mathbb{Q}$-Gorenstein, Cohen-Macaulay local ring of prime characteristic $p>0$, the Frobenius splitting numbers $a^Δ_e(R)$ remain unchanged under a suitable small perturbation. Moreover, we establish a desirable inequality of Frobenius splitting dimensions under general perturbations. That is, $\dim (R/(\mathcal{P}(R/(f),Δ|_{f})))\leq \dim (R/(\mathcal{P}(R/(f+\varepsilon),Δ|_{(f+\varepsilon)})))$ for all $\varepsilon \in \mathfrak{m}^{N\gg0}$, providing an example that demonstrates strict improvement can occur.