On $\mathfrak{m}$-adic Continuity of $F$-Splitting Ratio

TL;DR

This paper studies the stability and continuity of Frobenius splitting ratios in certain algebraic rings under small perturbations, revealing conditions for invariance and inequalities that describe their behavior.

Contribution

It proves the $rak{m}$-adic continuity of Frobenius splitting numbers in $F$-finite, $Q$-Gorenstein Cohen-Macaulay rings and establishes inequalities for their dimensions under perturbations.

Findings

Frobenius splitting numbers are invariant under small perturbations.

Established inequalities for Frobenius splitting dimensions under perturbations.

Provided examples of strict improvements in splitting dimensions.

Abstract

We investigate the $\mathfrak{m}$-adic continuity of Frobenius splitting dimensions and ratios for divisor pairs $(R,\Delta)$ 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 $a^{\Delta}_e(R)$ remain unchanged under a suitable small perturbation. Moreover, we establish a desirable inequality of Frobenius splitting dimensions under general perturbations. That is, $\dim (R/(\mathcal{P}(R/(f),\Delta|_{f})))\leq \dim (R/(\mathcal{P}(R/(f+\varepsilon),\Delta|_{(f+\varepsilon)})))$ for all $\varepsilon \in \mathfrak{m}^{N\gg0}$, providing an example that demonstrates strict improvement can occur.