On $\fm$-adic Continuity of $F$-Splitting Ratio

We investigate the $\fm$-adic continuity of Frobenius splitting dimensions and ratio for divisor pairs $(R,\Delta)$ in an $F$-finite ring $(R,\fm,k)$ of prime characteristic $p>0$. Our main result states that if $R$ is an $F$-finite $\QQ$-Gorenstein Cohen-Macaulay local ring of prime characteristics $p>0$, 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, $\dim (R/(\PP(R/(f)),\Delta|_{f}))\leq \dim (R/(\PP(R/(f+\epsilon)),\Delta|_{(f+\epsilon)}))$ for all $\epsilon \in \fm^{N>>0}$, providing an example that demonstrates strict improvement can occur.