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

We investigate the $\mathfrak{m}$-adic continuity of Frobenius splitting dimensions and ratios for divisor pairs $(R,\mathrm{\Delta })$ 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 $a_e^{\mathrm{\Delta }}(R)$ remain unchanged under a suitable small perturbation. Moreover, we establish a desirable inequality of Frobenius splitting dimensions under general perturbations. That is, $\dim (R\mathrm{/}(\mathcal{P}(R\mathrm{/}(f),\mathrm{\Delta }{\mathrm{\mid }}_f)))\le \dim (R\mathrm{/}(\mathcal{P}(R\mathrm{/}(f+\epsilon ),\mathrm{\Delta }{\mathrm{\mid }}_{(f+\epsilon )})))$ for all $\epsilon \in {\mathfrak{m}}^{N\gg 0}$, providing an example that demonstrates strict improvement can occur.