Proof. If $\dim (R) = c$ , then $R/(f)$ is artinian and, thus, has a splitting if an only if it is a field. If we take $\underline {\epsilon } \in ({\mathfrak {m}^{2}})^{\oplus \,c}$ , then $(\underline {f}) = \mathfrak {m}$ if and only if $(\underline {f}+ \underline {\epsilon }) = \mathfrak {m}$ . So the Frobenius splitting numbers of $R/(\underline {f})$ and $R/(\underline {f}+\underline {\epsilon })$ will be the same, either $0$ or $[k : k^{p^{e}}]$ , the latter of which occurs if and only if $(\underline {f}) = \mathfrak {m}$ .

Suppose $\dim (R)=d+c> c$ . Let $x_{1},\ldots , x_{d}$ be a system of parameters for $R/(\underline {f})$ and let $u\in R$ generate the socle mod $(\underline {f},x_{1},\ldots , x_{d})$ . By the proof of [Reference Polstra and SmirnovPS, Corollary 3.2], there exists an integer N so that for each $\underline {\epsilon }\in (\mathfrak {m}^{N})^{\oplus \,c}$ one has that $(\underline {f}+\underline {\epsilon }, x_{1},\ldots ,x_{d})=(\underline {f},x_{1},\ldots ,x_{d})$ and $(\underline {f}+\underline {\epsilon }, x^{p^{e}}_{1},\ldots ,x^{p^{e}}_{d})=(\underline {f},x^{p^{e}}_{1},\ldots ,x^{p^{e}}_{d})$ . For such choices of $\underline {\epsilon }$ the sequence $x_{1},\ldots ,x_{d}$ is a full system of parameters for $R/(\underline {f}+\underline {\epsilon })$ , u still generates the socle of $R/(\underline {f}+\underline {\epsilon },x_{1},\ldots ,x_{d})$ , and by multiple applications of Theorem 2.1

$$ \begin{align*} a_{e}(R/(\underline{f}))&=[k:k^{p^{e}}]{\mathrm{\lambda}}(R/((\underline{f},x_{1}^{p^{e}},\ldots,x_{d}^{p^{e}}):u^{p^{e}}))\\ &=[k:k^{p^{e}}]{\mathrm{\lambda}}(R/((\underline{f}+\underline{\epsilon},x_{1}^{p^{e}},\ldots,x_{d}^{p^{e}}):u^{p^{e}}))=a_{e}(R/(\underline{f}+\underline{\epsilon})).\\[-40pt] \end{align*} $$

Proof. We may assume R is complete. By the Cohen–Gabber theorem [Reference Gabber and OrgogozoGO08], we may choose parameters $x_{1},\ldots , x_{d}\in R$ such that $x_{1},\ldots , x_{d}$ is a system of parameters for $R/(\underline {f})$ and $R/(\underline {f})$ is module finite and generically separable over the regular local ring $A:=k[[x_{1},\ldots ,x_{d}]]$ . Thus we have short exact sequence

$$ \begin{align*} 0 \to R/(\underline{f})[A^{1/p}] \to (R/(\underline{f}))^{1/p} \to M \to 0 \end{align*} $$

and $0 \neq c = {\mathrm {D}}_{A} (R/(\underline {f}))$ annihilates M.

Let M and $M_{\underline {\epsilon }}$ be as in the proof of [Reference Polstra and SmirnovPS, Theorem 3.5], so that there are isomorphisms $R/(\underline {f} + \underline {\epsilon })[A^{1/p}] \cong \oplus ^{p^{d} [k : k^{p}]} (R/(\underline {f} + \underline {\epsilon }))$ and short exact sequences

$$ \begin{align*} 0 \to R/(\underline{f} + \underline{\epsilon})[A^{1/p}] \to (R/(\underline{f} + \underline{\epsilon}))^{1/p} \to M_{\underline{\epsilon}} \to 0. \end{align*} $$

Apply the exact functor $(-)^{1/p^{e}}$ to the above to get the exact sequence

$$ \begin{align*} 0 \to \bigoplus^{p^{d} [k : k^{p}]}\left ( (R/(\underline{f} + \underline{\epsilon})^{1/p^{e}}) \right) \to (R/(\underline{f} + \underline{\epsilon}))^{1/p^{e+1}} \to M_{\underline{\epsilon}}^{1/p^{e}} \to 0. \end{align*} $$

By [Reference Polstra and TuckerPT18, Lemma 2.1]

$$ \begin{align*} {\mathrm{freerank}} \left ( (R/(\underline{f} + \underline{\epsilon}))^{1/p^{e+1}} \right ) \leq p^{d}[k : k^{p}] {\mathrm{freerank}} \left ( (R/(\underline{f} + \underline{\epsilon}))^{1/p^{e}} \right) + \mu ((M_{\underline{\epsilon}})^{1/p^{e}}), \end{align*} $$

or, equivalently, $ a_{e+1}(R/(\underline {f}+\underline {\epsilon }))\leq p^{d}[k : k^{p}]a_{e}(R/(\underline {f}+\underline {\epsilon }))+\mu ((M_{\underline {\epsilon }})^{1/p^{e}}). $ The proof of [Reference Polstra and SmirnovPS, Theorem 3.5] gives the existence of a constant C such that for all $\underline {\epsilon }\in ({\mathfrak {m}^{N}})^{\oplus \,c}$

$$ \begin{align*} \mu ((N_{\underline{\epsilon}})^{1/p^{e}}) = {\mathrm{\lambda}} (N_{\underline{\epsilon}}/\mathfrak{m}^{[p^{e}]}N_{\underline{\epsilon}}) \leq Cp^{e(d-1)}. \end{align*} $$

Let L and $L_{\underline {\epsilon }}$ be as in the proof [Reference Polstra and SmirnovPS, Theorem 3.5]. Similarly, we can also bound

$$ \begin{align*} p^{d}[k : k^{p}] {\mathrm{freerank}} \left ( (R/(\underline{f} + \underline{\epsilon}))^{1/p^{e}} \right) - {\mathrm{freerank}} \left ( (R/(\underline{f} + \underline{\epsilon}))^{1/p^{e + 1}} \right ) \leq {\mathrm{\lambda}} (L_{\underline{\epsilon}}/\mathfrak{m}^{[p^{e}]}L_{\underline{\epsilon}}) \end{align*} $$

and once again obtain constant C, independent of $\underline {\epsilon }$ , such that

$$ \begin{align*}|a_{e+ 1} (R/(\underline{f} + \underline{\epsilon})) - p^{d}[k : k^{p}] a_{e} (R/(\underline{f} + \underline{\epsilon}))| < Cp^{e(d-1)}.\end{align*} $$

Since $ {\mathrm {rank}} R^{1/p^{e}} = p^{ed} [k : k^{p^{e}}]$ [Reference KunzKun76, Proposition 2.3], as explained in [Reference Polstra and SmirnovPS, Corollary 3.6], this gives us a uniform convergence statement: there exist $D, N> 0$ such that for all $\underline {\epsilon } \in ({\mathfrak {m}^{N}})^{\oplus \,c}$

$$ \begin{align*} \left |{\mathrm{s}} (R/(\underline{f}+ \underline{\epsilon})) - \frac 1{{\mathrm{rank}} R^{1/p^{e}}} a_{e} (R/(\underline{f} + \underline{\epsilon})) \right| < \frac{D}{p^{e}}. \end{align*} $$

The statement now follows by employing Corollary 2.2 and following the proof of [Reference Polstra and SmirnovPS, Corollary 3.7].