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

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.