Here is the notation. Let $k$ be a perfect field of characteristic $p$, $W=W(k)$ the ring of Witt vector and $\mathfrak{S}:=W[[u]]$, $\mathcal{O}_{\mathcal{E}}$ is the $p$-adic completion of $W[1/u]$ which is DVR. Denote $R=\lim_{\leftarrow}\mathcal{O}_{ \bar{K}}/p$ where the transition map is Frobenius. The ring of Witt vector $W(R)$ has a canonical surjection $W(R) \xrightarrow{\theta} \mathcal{O}_{C_K}$ where $C_K$ is the completion of $\bar{K}$. The ring $A_{cris}$ is a divided power envelope of $W(R)$ with respect to $\ker \theta$.

One can embed $\mathcal{O}_{\mathcal{E}}$ into $W(R)\subset A_{cris}$ by sending $u\mapsto [\underline{\pi}]$ ($\underline{\pi}=(\pi,\pi^{1/p}, \cdots)$). In this paper, Kisin defines $\mathcal{E}^{un}$ as the maximal unramified extension of $\mathcal{E}=Frac(\mathcal{O}_{\mathcal{E}})$ contained in $W(Frac(R))[1/p]$ and $\mathfrak{S}^{un}:=\mathcal{O}_{\mathcal{E}^{un}}\cap W(R)$ (which must be done by Breuil earlier).

Here is my question. Consider $\mathfrak{S}$ as a subring of $A_{cris}$ by composing the Frobenius on $\mathfrak{S}$ and the inclusion to $W(R)$ as above.(In particular, $u\mapsto [\underline{\pi}]^p$) In the same paper of Kisin, it says that $\mathfrak{S}^{un}\cap pA_{cris}=p\mathfrak{S}^{un}$ when $p>2$.(proof of Theorem 2.2.7.) Why is this true? He mentions a paper of Brueil(proof of 3.3.2.), but I don't see why this helps the assertion. Also, the structure of $\mathfrak{S}$ is somewhat mysterious to me. Naively it should be a subring of $\mathcal{O}_{\mathcal{E}}$ consists of elements "without $u$ in the denominator", as $\mathcal{O}_{\mathcal{E}}$ is a DVR with residue field $k((u))^{sep}$. However, as the residue field is not perfect, this description is not concrete.

Here is the notation. Let $k$ be a perfect field of characteristic $p$, $W=W(k)$ the ring of Witt vector and $\mathfrak{S}:=W[[u]]$, $\mathcal{O}_{\mathcal{E}}$ is the $p$-adic completion of $W[1/u]$ which is DVR. Denote $R=\lim_{\leftarrow}\mathcal{O}_{ \bar{K}}/p$ where the transition map is Frobenius. The ring of Witt vector $W(R)$ has a canonical surjection $W(R) \xrightarrow{\theta} \mathcal{O}_{C_K}$ where $C_K$ is the completion of $\bar{K}$. The ring $A_{cris}$ is a divided power envelope of $W(R)$ with respect to $\ker \theta$.

One can embed $\mathcal{O}_{\mathcal{E}}$ into $W(R)\subset A_{cris}$ by sending $u\mapsto [\underline{\pi}]$ ($\underline{\pi}=(\pi,\pi^{1/p}, \cdots)$). In this paper, Kisin defines $\mathcal{E}^{un}$ as the maximal unramified extension of $\mathcal{E}=Frac(\mathcal{O}_{\mathcal{E}})$ contained in $W(Frac(R))[1/p]$ and $\mathfrak{S}^{un}:=\mathcal{O}_{\mathcal{E}^{un}}\cap W(R)$ (which must be done by Breuil earlier).

Here is my question. Consider $\mathfrak{S}^{un}$ as a subring of $A_{cris}$ by composing the Frobenius on $\mathfrak{S}^{un}$ and the inclusion to $W(R)$ as above.(In particular, $u\mapsto [\underline{\pi}]^p$) In the same paper of Kisin, it says that $\mathfrak{S}^{un}\cap pA_{cris}=p\mathfrak{S}^{un}$ when $p>2$.(proof of Theorem 2.2.7.) Why is this true? He mentions a paper of Brueil(proof of 3.3.2.), but I don't see why this helps the assertion. Also, the structure of $\mathfrak{S}^{un}$ is somewhat mysterious to me. Naively it should be a subring of $\mathcal{O}_{\mathcal{E}^{un}}$ consists of elements "without $u$ in the denominator", as $\mathcal{O}_{\mathcal{E}^{un}}$ is a DVR with residue field $k((u))^{sep}$. However, as the residue field is not perfect, this description is not concrete.