2.1 Basic notations

Let $k$ be a perfect field of characteristic $p$ , $W(k)$ the ring of Witt vectors with coefficients in $k$ , $K_{0}=W(k)[1/p]$ , $K$ a finite totally ramified extension of $K_{0}$ of degree $e$ , $\overline{K}$ a fixed algebraic closure of $K$ . Throughout this paper, we fix a uniformizer $\unicode[STIX]{x1D70B}$ of $K$ . Let $E(u)$ be the minimal polynomial of $\unicode[STIX]{x1D70B}$ over $K_{0}$ . Then $E(u)$ is an Eisenstein polynomial. For any integer $n\geqslant 0$ , we fix a system $(\unicode[STIX]{x1D70B}_{n})_{n\geqslant 0}$ of $p^{n}$ th roots of $\unicode[STIX]{x1D70B}$ in $\overline{K}$ such that $\unicode[STIX]{x1D70B}_{n+1}^{p}=\unicode[STIX]{x1D70B}_{n}$ . Let $R=\varprojlim {\mathcal{O}}_{\overline{K}}/p$ , where ${\mathcal{O}}_{\overline{K}}$ is the integer ring of $\overline{K}$ and the transition maps are given by the $p$ th power map. For any integer $s\geqslant 0$ , we write $\text{}\underline{\unicode[STIX]{x1D70B}_{s}}:=(\unicode[STIX]{x1D70B}_{s+n})_{n\geqslant 0}\in R$ and $\text{}\underline{\unicode[STIX]{x1D70B}}:=\text{}\underline{\unicode[STIX]{x1D70B}_{0}}\in R$ . Note that we have $\text{}\underline{\unicode[STIX]{x1D70B}_{s}}^{p^{s}}=\text{}\underline{\unicode[STIX]{x1D70B}}$ .

Let $L$ be the completion of an unramified algebraic extension of $K$ with residue field $k_{L}$ . Then $\unicode[STIX]{x1D70B}_{s}$ is a uniformizer of $L_{(s)}:=L(\unicode[STIX]{x1D70B}_{s})$ and $L_{(s)}$ is a totally ramified degree $ep^{s}$ extension of $L_{0}:=W(k_{L})[1/p]$ . We set $L_{\infty }:=\bigcup _{n\geqslant 0}L_{(n)}$ . We put $G_{L,s}:=G_{L_{(s)}}=\text{Gal}(\overline{L}/L_{(s)})$ and $G_{L,\infty }:=G_{L_{\infty }}=\text{Gal}(\overline{L}/L_{\infty })$ . By definitions, we have $L=L_{(0)}$ and $G_{L,0}=G_{L}$ . Put $\mathfrak{S}_{L,s}=W(k_{L})[\![u_{s}]\!]$ (resp.  $\mathfrak{S}_{L}=W(k_{L})[\![u]\!]$ ) with an indeterminate $u_{s}$ (resp.  $u$ ). We equip a Frobenius endomorphism $\unicode[STIX]{x1D711}$ of $\mathfrak{S}_{L,s}$ (resp.  $\mathfrak{S}_{L}$ ) by $u_{s}\mapsto u_{s}^{p}$ (resp.  $u\mapsto u^{p}$ ) and the Frobenius on $W(k_{L})$ . We embed the $W(k_{L})$ -algebra $W(k_{L})[u_{s}]$ (resp. $W(k_{L})[u]$ ) into $W(R)$ via the map $u_{s}\mapsto [\text{}\underline{\unicode[STIX]{x1D70B}_{s}}]$ (resp.  $u\mapsto [\text{}\underline{\unicode[STIX]{x1D70B}}]$ ), where $[\ast ]$ stands for the Teichmüller representative. This embedding extends to an embedding $\mathfrak{S}_{L,s}{\hookrightarrow}W(R)$ (resp.  $\mathfrak{S}_{L}{\hookrightarrow}W(R)$ ). By identifying $u$ with $u_{s}^{p^{s}}$ , we regard $\mathfrak{S}_{L}$ as a subalgebra of $\mathfrak{S}_{L,s}$ . It is readily seen that the embedding $\mathfrak{S}_{L}{\hookrightarrow}\mathfrak{S}_{L,s}{\hookrightarrow}W(R)$ is compatible with the Frobenius endomorphisms. If we denote by $E_{s}(u_{s})$ the minimal polynomial of $\unicode[STIX]{x1D70B}_{s}$ over $K_{0}$ , with indeterminate $u_{s}$ , then we have $E_{s}(u_{s})=E(u_{s}^{p^{s}})$ . Therefore, we have $E_{s}(u_{s})=E(u)$ in $\mathfrak{S}_{L,s}$ . We note that the minimal polynomial of $\unicode[STIX]{x1D70B}_{s}$ over $L_{0}$ is $E_{s}(u_{s})$ .

Let $S_{L_{0},s}^{\text{int}}$ (resp.  $S_{L_{0}}^{\text{int}}$ ) be the $p$ -adic completion of the divided power envelope of $W(k_{L})[u_{s}]$ (resp.  $W(k_{L})[u]$ ) with respect to the ideal generated by $E_{s}(u_{s})$ (resp.  $E(u)$ ). There exist a unique Frobenius map $\unicode[STIX]{x1D711}:S_{L_{0},s}^{\text{int}}\rightarrow S_{L_{0},s}^{\text{int}}$ (resp.  $\unicode[STIX]{x1D711}:S_{L_{0}}^{\text{int}}\rightarrow S_{L_{0}}^{\text{int}}$ ) and monodromy $N:S_{L_{0},s}^{\text{int}}\rightarrow S_{L_{0},s}^{\text{int}}$ defined by $\unicode[STIX]{x1D711}(u_{s})=u_{s}^{p}$ (resp.  $\unicode[STIX]{x1D711}(u)=u^{p}$ ) and $N(u_{s})=-u_{s}$ (resp.  $N(u)=-u$ ). Put $S_{L_{0},s}=S_{L_{0},s}^{\text{int}}[1/p]=L_{0}\otimes _{W(k_{L})}S_{L_{0},s}^{\text{int}}$ (resp. $S_{L_{0}}=S_{L_{0}}^{\text{int}}[1/p]=L_{0}\otimes _{W(k_{L})}S_{L_{0}}^{\text{int}}$ ). We equip $S_{L_{0},s}^{\text{int}}$ and $S_{L_{0},s}$ (resp.  $S_{L_{0}}^{\text{int}}$ and $S_{L_{0}}$ ) with decreasing filtrations $\text{Fil}^{i}S_{L_{0},s}^{\text{int}}$ and $\text{Fil}^{i}S_{L_{0},s}$ (resp.  $\text{Fil}^{i}S_{L_{0},s}^{\text{int}}$ and $\text{Fil}^{i}S_{L_{0},s}$ ) by the $p$ -adic completion of the ideal generated by $E_{s}^{j}(u_{s})/j!$ (resp.  $E^{j}(u)/j!$ ) for all $j\geqslant 0$ . The inclusion $W(k_{L})[u_{s}]{\hookrightarrow}W(R)$ (resp. $W(k_{L})[u]{\hookrightarrow}W(R)$ ) via the map $u_{s}\mapsto [\text{}\underline{\unicode[STIX]{x1D70B}_{s}}]$ (resp.  $u\mapsto [\text{}\underline{\unicode[STIX]{x1D70B}}]$ ) induces $\unicode[STIX]{x1D711}$ -compatible inclusions $\mathfrak{S}_{L,s}{\hookrightarrow}S_{L_{0},s}^{\text{int}}{\hookrightarrow}A_{\text{cris}}$ and $S_{L_{0},s}{\hookrightarrow}B_{\text{cris}}^{+}$ (resp. $\mathfrak{S}_{L}{\hookrightarrow}S_{L_{0}}^{\text{int}}{\hookrightarrow}A_{\text{cris}}$ and $S_{L_{0}}{\hookrightarrow}B_{\text{cris}}^{+}$ ). By these inclusions, we often regard these rings as subrings of $B_{\text{cris}}^{+}$ . By identifying $u$ with $u_{s}^{p^{s}}$ as before, we regard $S_{L_{0}}^{\text{int}}$ (resp.  $S_{L_{0}}$ ) as a $\unicode[STIX]{x1D711}$ -stable (but not $N$ -stable) subalgebra of $S_{L_{0},s}^{\text{int}}$ (resp. $S_{L_{0},s}$ ). By definitions, we have $\mathfrak{S}_{L}=\mathfrak{S}_{L,0},S_{L_{0},0}^{\text{int}}=S_{L_{0}}^{\text{int}}$ and $S_{L_{0},0}=S_{L_{0}}$ (cf. Figure 1).

Convention: For simplicity, if $L=K$ , then we often omit the subscript “ $L$ ” from various notations (e.g., $G_{K_{s}}=G_{s}$ , $G_{K_{\infty }}=G_{\infty }$ , $\mathfrak{S}_{K}=\mathfrak{S},\mathfrak{S}_{K,s}=\mathfrak{S}_{s}$ ).

Figure 1. Ring extensions.

2.2 Kisin modules

Let $r,s\geqslant 0$ be integers. A $\unicode[STIX]{x1D711}$ -module over $\mathfrak{S}_{L,s}$ is an $\mathfrak{S}_{L,s}$ -module $\mathfrak{M}$ equipped with a $\unicode[STIX]{x1D711}$ -semilinear map $\unicode[STIX]{x1D711}:\mathfrak{M}\rightarrow \mathfrak{M}$ . A morphism between two $\unicode[STIX]{x1D711}$ -modules $(\mathfrak{M}_{1},\unicode[STIX]{x1D711}_{1})$ and $(\mathfrak{M}_{2},\unicode[STIX]{x1D711}_{2})$ over $\mathfrak{S}_{L,s}$ is an $\mathfrak{S}_{L,s}$ -linear map $\mathfrak{M}_{1}\rightarrow \mathfrak{M}_{2}$ compatible with $\unicode[STIX]{x1D711}_{1}$ and $\unicode[STIX]{x1D711}_{2}$ . Denote by $^{\prime }\text{Mod}_{/\mathfrak{S}_{L,s}}^{r}$ the category of $\unicode[STIX]{x1D711}$ -modules $(\mathfrak{M},\unicode[STIX]{x1D711})$ over $\mathfrak{S}_{L,s}$ of height ${\leqslant}r$ in the sense that $\mathfrak{M}$ is of finite type over $\mathfrak{S}_{L,s}$ and the cokernel of $1\otimes \unicode[STIX]{x1D711}:\mathfrak{S}_{L,s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{L,s}}\mathfrak{M}\rightarrow \mathfrak{M}$ is killed by $E_{s}(u_{s})^{r}$ .

Let $\text{Mod}_{/\mathfrak{S}_{L,s}}^{r}$ be the full subcategory of $^{\prime }\text{Mod}_{/\mathfrak{S}_{L,s}}^{r}$ consisting of finite free $\mathfrak{S}_{L,s}$ -modules. We call an object of $\text{Mod}_{/\mathfrak{S}_{L,s}}^{r}$ a free Kisin module of height ${\leqslant}r$ (over $\mathfrak{S}_{L,s}$ ).

Let $\text{Mod}_{/\mathfrak{S}_{L,s,\infty }}^{r}$ be the full subcategory of $^{\prime }\text{Mod}_{/\mathfrak{S}_{L,s}}^{r}$ consisting of finite $\mathfrak{S}_{L,s}$ -modules which are killed by some power of $p$ and have projective dimension $1$ in the sense that $\mathfrak{M}$ has a two term resolution by finite free $\mathfrak{S}_{L,s}$ -modules. We call an object of $\text{Mod}_{/\mathfrak{S}_{L,s,\infty }}^{r}$ a torsion Kisin module of height ${\leqslant}r$ (over $\mathfrak{S}_{L,s}$ ).

For any free or torsion Kisin module $\mathfrak{M}$ over $\mathfrak{S}_{L,s}$ , we define a $\mathbb{Z}_{p}[G_{L,\infty }]$ -module $T_{\mathfrak{S}_{L,s}}(\mathfrak{M})$ by

$$\begin{eqnarray}T_{\mathfrak{S}_{L,s}}(\mathfrak{M}):=\left\{\begin{array}{@{}ll@{}}\text{Hom}_{\mathfrak{S}_{L,s},\unicode[STIX]{x1D711}}(\mathfrak{M},W(R))\quad & \text{if}~\mathfrak{M}~\text{is}~\text{free},\\ \text{Hom}_{\mathfrak{S}_{L,s},\unicode[STIX]{x1D711}}(\mathfrak{M},\mathbb{Q}_{p}/\mathbb{Z}_{p}\otimes _{\mathbb{Z}_{p}}W(R))\quad & \text{if}~\mathfrak{M}~\text{is}~\text{torsion}.\end{array}\right.\end{eqnarray}$$

Here a $G_{L,\infty }$ -action on $T_{\mathfrak{S}_{L,s}}(\mathfrak{M})$ is given by $(\unicode[STIX]{x1D70E}.g)(x)=\unicode[STIX]{x1D70E}(g(x))$ for $\unicode[STIX]{x1D70E}\in G_{L,\infty },g\in T_{\mathfrak{S}}(\mathfrak{M}),x\in \mathfrak{M}$ .

Convention: For simplicity, if $L=K$ , then we often omit the subscript “ $L$ ” from various notations (e.g., $\text{Mod}_{/\mathfrak{S}_{K,s,\infty }}^{r}=\text{Mod}_{/\mathfrak{S}_{s,\infty }}^{r}$ , $T_{\mathfrak{S}_{K,s}}=T_{\mathfrak{S}_{s}}$ ). Also, if $s=0$ , we often omit the subscript “ $s$ ” from various notations (e.g., $\text{Mod}_{/\mathfrak{S}_{L,0,\infty }}^{r}=\text{Mod}_{/\mathfrak{S}_{L,\infty }}^{r}$ , $T_{\mathfrak{S}_{L,0}}=T_{\mathfrak{S}_{L}}$ , $\text{Mod}_{/\mathfrak{S}_{K,0,\infty }}^{r}=\text{Mod}_{/\mathfrak{S}_{\infty }}^{r}$ , $T_{\mathfrak{S}_{K,0}}=T_{\mathfrak{S}}$ ).

2.3 $(\unicode[STIX]{x1D711},{\hat{G}})$ -modules

The notion of $(\unicode[STIX]{x1D711},{\hat{G}})$ -modules is introduced by Liu in [Reference LiuLi2] to classify lattices in semistable representations. We recall definitions and properties of them. We continue to use same notations as above.

Let $L_{p^{\infty }}$ be the field obtained by adjoining all $p$ -power roots of unity to $L$ . We denote by $\hat{L}$ the composite field of $L_{\infty }$ and $L_{p^{\infty }}$ . We define $H_{L}:=\text{Gal}(\hat{L}/L_{\infty })$ , $H_{L,\infty }:=\text{Gal}(\overline{K}/\hat{L})$ $G_{L,p^{\infty }}:=\text{Gal}(\hat{L}/L_{p^{\infty }})$ and ${\hat{G}}_{L}:=\text{Gal}(\hat{L}/L)$ (cf. Figure 2). Furthermore, putting $L_{(s),p^{\infty }}=L_{(s)}L_{p^{\infty }\!}$ , we define ${\hat{G}}_{L,s}=\text{Gal}(\hat{L}/L_{(s)})$ and $G_{L,s,p^{\infty }}:=\text{Gal}(\hat{L}/L_{(s),p^{\infty }})$ .

Figure 2. Galois groups of field extensions.

Since $p>2$ , it is known that $L_{(s),p^{\infty }}\cap L_{\infty }=L_{(s)}$ and thus ${\hat{G}}_{L,s}=G_{L,s,p^{\infty }}\rtimes H_{L,s}$ (cf. [Reference LiuLi1, Lemma 5.1.2]). Furthermore, $G_{L,s,p^{\infty }}$ is topologically isomorphic to $\mathbb{Z}_{p}$ .

We fix a topological generator $\unicode[STIX]{x1D70F}$ of $G_{K,p^{\infty }}$ . We also denote by $\unicode[STIX]{x1D70F}$ the preimage of $\unicode[STIX]{x1D70F}\in G_{K,p^{\infty }}$ under the bijection $G_{L,p^{\infty }}\simeq G_{K,p^{\infty }}$ of the above lemma. Note that $\unicode[STIX]{x1D70F}^{p^{s}}$ is a topological generator of $G_{L,s,p^{\infty }}$ .

For any $g\in G_{K}$ , we put $\text{}\underline{\unicode[STIX]{x1D700}}(g)=g(\text{}\underline{\unicode[STIX]{x1D70B}})/\text{}\underline{\unicode[STIX]{x1D70B}}\in R$ , and define $\text{}\underline{\unicode[STIX]{x1D700}}:=\text{}\underline{\unicode[STIX]{x1D700}}(\tilde{\unicode[STIX]{x1D70F}})$ . Here, $\tilde{\unicode[STIX]{x1D70F}}\in G_{K}$ is any lift of $\unicode[STIX]{x1D70F}\in {\hat{G}}_{K}$ and then $\text{}\underline{\unicode[STIX]{x1D700}}(\tilde{\unicode[STIX]{x1D70F}})$ is independent of the choice of the lift of $\unicode[STIX]{x1D70F}$ . With these notations, we also note that we have $g(u)=[\text{}\underline{\unicode[STIX]{x1D700}}(g)]u$ (recall that $\mathfrak{S}$ is embedded in $W(R)$ ). An easy computation shows that $\unicode[STIX]{x1D70F}(\text{}\underline{\unicode[STIX]{x1D70B}})/\text{}\underline{\unicode[STIX]{x1D70B}}=\unicode[STIX]{x1D70F}^{p^{s}}(\text{}\underline{\unicode[STIX]{x1D70B}_{s}})/\text{}\underline{\unicode[STIX]{x1D70B}_{s}}=\text{}\underline{\unicode[STIX]{x1D700}}$ . Therefore, we have $\unicode[STIX]{x1D70F}(u)/u=\unicode[STIX]{x1D70F}^{p^{s}}(u_{s})/u_{s}=[\text{}\underline{\unicode[STIX]{x1D700}}]$ .

We put $t=-\text{log}([\text{}\underline{\unicode[STIX]{x1D700}}])\in A_{\text{cris}}$ . Denote by $\unicode[STIX]{x1D708}:W(R)\rightarrow W(\overline{k})$ the unique lift of the projection $R\rightarrow \overline{k}$ , which extends to a map $\unicode[STIX]{x1D708}:B_{\text{cris}}^{+}\rightarrow W(\overline{k})[1/p]$ . For any subring $A\subset B_{\text{cris}}^{+}$ , we put $I_{+}A=\text{Ker}(\unicode[STIX]{x1D708}~\text{on}~B_{\text{cris}}^{+})\cap A$ . For any integer $n\geqslant 0$ , let $t^{\{n\}}:=t^{r(n)}\unicode[STIX]{x1D6FE}_{\tilde{q}(n)}(\frac{t^{p-1}}{p})$ where $n=(p-1)\tilde{q}(n)+r(n)$ with $\tilde{q}(n)\geqslant 0,~0\leqslant r(n)<p-1$ and $\unicode[STIX]{x1D6FE}_{i}(x)=\frac{x^{i}}{i!}$ is the standard divided power. We define a subring ${\mathcal{R}}_{L_{0},s}$ (resp.  ${\mathcal{R}}_{L_{0}}$ ) of $B_{\text{cris}}^{+}$ as below:

$$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{R}}_{L_{0},s}:=\Bigl\{\mathop{\sum }_{i=0}^{\infty }f_{i}t^{\{i\}}\mid f_{i}\in S_{L_{0},s}~\text{and}~f_{i}\rightarrow 0~\text{as}~i\rightarrow \infty \Bigr\} & \displaystyle \nonumber\\ \displaystyle & \displaystyle \Bigl(\text{resp}.\quad {\mathcal{R}}_{L_{0}}:=\Bigl\{\mathop{\sum }_{i=0}^{\infty }f_{i}t^{\{i\}}\mid f_{i}\in S_{L_{0}}~\text{and}~f_{i}\rightarrow 0~\text{as}~i\rightarrow \infty \Bigr\}\Bigr). & \displaystyle \nonumber\end{eqnarray}$$

Put $\widehat{{\mathcal{R}}}_{L,s}={\mathcal{R}}_{L_{0},s}\cap W(R)$ (resp. $\widehat{{\mathcal{R}}}_{L}={\mathcal{R}}_{L_{0}}\cap W(R)$ ) and $I_{+,L,s}=I_{+}\widehat{{\mathcal{R}}}_{L,s}$ (resp. $I_{+,L}=I_{+}\widehat{{\mathcal{R}}}_{L}$ ). By definitions, we have ${\mathcal{R}}_{L_{0},0}={\mathcal{R}}_{L_{0}}$ , $\widehat{{\mathcal{R}}}_{L,0}=\widehat{{\mathcal{R}}}_{L}$ and $I_{+,L,0}=I_{+,L}$ . Lemma 2.2.1 in [Reference LiuLi2] shows that $\widehat{{\mathcal{R}}}_{L,s}$ (resp. ${\mathcal{R}}_{L_{0},s}$ ) is a $\unicode[STIX]{x1D711}$ -stable $\mathfrak{S}_{L,s}$ -subalgebra of $W(R)$ (resp.  $B_{\text{cris}}^{+}$ ), and $\unicode[STIX]{x1D708}$ induces ${\mathcal{R}}_{L_{0},s}/I_{+}{\mathcal{R}}_{L_{0},s}\simeq L_{0}$ and $\widehat{{\mathcal{R}}}_{L,s}/I_{+,L,s}\simeq S_{L_{0},s}^{\text{int}}/I_{+}S_{L_{0},s}^{\text{int}}\simeq \mathfrak{S}_{L,s}/I_{+}\mathfrak{S}_{L,s}\simeq W(k_{L})$ . Furthermore, $\widehat{{\mathcal{R}}}_{L,s},I_{+,L,s},{\mathcal{R}}_{L_{0},s}$ and $I_{+}{\mathcal{R}}_{L_{0},s}$ are $G_{L,s}$ -stable, and $G_{L,s}$ -actions on them factors through ${\hat{G}}_{L,s}$ . For any torsion Kisin module $\mathfrak{M}$ over $\mathfrak{S}_{L,s}$ , we equip $\widehat{{\mathcal{R}}}_{L,s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{L,s}}\mathfrak{M}$ with a Frobenius by $\unicode[STIX]{x1D711}_{\widehat{{\mathcal{R}}}_{L,s}}\otimes \unicode[STIX]{x1D711}_{\mathfrak{M}}$ . It is known that the natural map $\mathfrak{M}\rightarrow \widehat{{\mathcal{R}}}_{L,s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{L,s}}\mathfrak{M}$ given by $x\mapsto 1\otimes x$ is an injection (cf. [Reference OzekiOz1, Corollary 2.12]). By this injection, we regard $\mathfrak{M}$ as a $\unicode[STIX]{x1D711}(\mathfrak{S}_{L,s})$ -stable submodule of $\widehat{{\mathcal{R}}}_{L,s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{L,s}}\mathfrak{M}$ .

A morphism between two $(\unicode[STIX]{x1D711},{\hat{G}}_{L,s})$ -modules $\hat{\mathfrak{M}}_{1}=(\mathfrak{M}_{1},\unicode[STIX]{x1D711}_{1},{\hat{G}})$ and $\hat{\mathfrak{M}}_{2}=(\mathfrak{M}_{2},\unicode[STIX]{x1D711}_{2},{\hat{G}})$ is a morphism $f:\mathfrak{M}_{1}\rightarrow \mathfrak{M}_{2}$ of $\unicode[STIX]{x1D711}$ -modules over $\mathfrak{S}_{L,s}$ such that $\widehat{{\mathcal{R}}}_{L,s}\otimes f:\widehat{{\mathcal{R}}}_{L,s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{L,s}}\mathfrak{M}_{1}\rightarrow \widehat{{\mathcal{R}}}_{L,s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{L,s}}\mathfrak{M}_{2}$ is ${\hat{G}}_{L,s}$ -equivariant. We denote by $\text{Mod}_{/\mathfrak{S}_{L,s}}^{r,{\hat{G}}_{L,s}}$ (resp. $\text{Mod}_{/\mathfrak{S}_{L,s,\infty }}^{r,{\hat{G}}_{L,s}}$ ) the category of free (resp. torsion) $(\unicode[STIX]{x1D711},{\hat{G}}_{L,s})$ -modules of height ${\leqslant}r$ over $\mathfrak{S}_{L,s}$ . We often regard $\widehat{{\mathcal{R}}}_{L,s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{L,s}}\mathfrak{M}$ as a $G_{L,s}$ -module via the projection $G_{L,s}{\twoheadrightarrow}{\hat{G}}_{L,s}$ .

For any free or torsion $(\unicode[STIX]{x1D711},{\hat{G}}_{L,s})$ -module $\hat{\mathfrak{M}}$ over $\mathfrak{S}_{L,s}$ , we define a $\mathbb{Z}_{p}[G_{L,s}]$ -module $\hat{T}_{L,s}(\hat{\mathfrak{M}})$ by

$$\begin{eqnarray}\hat{T}_{L,s}(\hat{\mathfrak{M}})=\left\{\begin{array}{@{}ll@{}}\text{Hom}_{\widehat{{\mathcal{R}}}_{L,s},\unicode[STIX]{x1D711}}(\widehat{{\mathcal{R}}}_{L,s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{L,s}}\mathfrak{M},W(\!R))\quad & \text{if}~\mathfrak{M}~\text{is}~\text{free},\\ \text{Hom}_{\widehat{{\mathcal{R}}}_{L,s},\unicode[STIX]{x1D711}}(\widehat{{\mathcal{R}}}_{L,s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{L,s}}\mathfrak{M},\mathbb{Q}_{p}/\mathbb{Z}_{p}\otimes _{\mathbb{Z}_{p}}\!W(\!R))\quad & \text{if}~\mathfrak{M}~\text{is}~\text{torsion}.\end{array}\right.\end{eqnarray}$$

Here, $G_{L,s}$ acts on $\hat{T}_{L,s}(\hat{\mathfrak{M}})$ by $(\unicode[STIX]{x1D70E}.f)(x)=\unicode[STIX]{x1D70E}(f(\unicode[STIX]{x1D70E}^{-1}(x)))$ for $\unicode[STIX]{x1D70E}\in G_{L,s},~f\in \hat{T}_{L,s}(\hat{\mathfrak{M}}),~x\in \widehat{{\mathcal{R}}}_{L,s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{L,s}}\mathfrak{M}$ . Then, there exists a natural $G_{L,\infty }$ -equivariant map

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{L,s}:T_{\mathfrak{S}_{L,s}}(\mathfrak{M})\rightarrow \hat{T}_{L,s}(\hat{\mathfrak{M}})\end{eqnarray}$$

defined by $\unicode[STIX]{x1D703}(f)(a\otimes x)=a\unicode[STIX]{x1D711}(f(x))$ for $f\in T_{\mathfrak{S}_{L,s}}(\mathfrak{M}),~a\in \widehat{{\mathcal{R}}}_{L,s},x\in \mathfrak{M}$ . We have the following theorem.

Convention: For simplicity, if $L=K$ , then we often omit the subscript “ $L$ ” from various notations (e.g., “a $(\unicode[STIX]{x1D711},{\hat{G}}_{K,s})$ -module” = “a $(\unicode[STIX]{x1D711},{\hat{G}}_{s})$ -module”, $\text{Mod}_{/\mathfrak{S}_{K,s}}^{r,{\hat{G}}_{K,s}}=\text{Mod}_{/\mathfrak{S}_{s}}^{r,{\hat{G}}_{s}}$ , $\text{Mod}_{/\mathfrak{S}_{K,s,\infty }}^{r,{\hat{G}}_{K,s}}=\text{Mod}_{/\mathfrak{S}_{s,\infty }}^{r,{\hat{G}}_{s}}$ , $\hat{T}_{K,s}=\hat{T}_{s}$ , $\unicode[STIX]{x1D703}_{K,s}=\unicode[STIX]{x1D703}_{s}$ ). Furthermore, if $s=0$ , we often omit the subscript “ $s$ ” from various notations (e.g., $\text{Mod}_{/\mathfrak{S}_{L,0}}^{r,{\hat{G}}_{L,0}}=\text{Mod}_{/\mathfrak{S}_{L}}^{r,{\hat{G}}_{L}},\text{Mod}_{/\mathfrak{S}_{L,0,\infty }}^{r,{\hat{G}}_{L,0}}=\text{Mod}_{/\mathfrak{S}_{L,\infty }\!}^{r,{\hat{G}}_{L}}$ , $\hat{T}_{L,0}=\hat{T}_{L}$ , $\text{Mod}_{/\mathfrak{S}_{K,0}}^{r,{\hat{G}}_{K,0}}=\text{Mod}_{/\mathfrak{S}}^{r,{\hat{G}}}$ , “a $(\unicode[STIX]{x1D711},{\hat{G}}_{K,0})$ -module” = “a $(\unicode[STIX]{x1D711},{\hat{G}})$ -module”, $\hat{T}_{K,0}=\hat{T}$ , $\unicode[STIX]{x1D703}_{K,0}=\unicode[STIX]{x1D703}$ ).

We denote by $\text{Rep}_{\mathbb{Q}_{p}}^{r,\text{st}}(G_{L,s})$ (resp. $\text{Rep}_{\mathbb{Q}_{p}}^{r,\text{cris}}(G_{L,s})$ , resp. $\text{Rep}_{\mathbb{Z}_{p}}^{r,\text{st}}(G_{L,s})$ , resp. $\text{Rep}_{\mathbb{Z}_{p}}^{r,\text{cris}}(G_{L,s})$ ) the category of semistable $\mathbb{Q}_{p}$ -representations of $G_{L,s}$ with Hodge–Tate weights in $[0,r]$ (resp. the category of crystalline $\mathbb{Q}_{p}$ -representations of $G_{L,s}$ with Hodge–Tate weights in $[0,r]$ , resp. the category of lattices in semistable $\mathbb{Q}_{p}$ -representations of $G_{L,s}$ with Hodge–Tate weights in $[0,r]$ , resp. the category of lattices in crystalline $\mathbb{Q}_{p}$ -representations of $G_{L,s}$ with Hodge–Tate weights in $[0,r]$ ).

There exists $\mathfrak{t}\in W(R)\smallsetminus pW(R)$ such that $\unicode[STIX]{x1D711}(\mathfrak{t})=pE(0)^{-1}E(u)\mathfrak{t}$ . Such $\mathfrak{t}$ is unique up to units of $\mathbb{Z}_{p}$ (cf. [Reference LiuLi2, Example 2.3.5]). Now we define the full subcategory $\text{Mod}_{/\mathfrak{S}}^{r,{\hat{G}},\text{cris}}$ (resp. $\text{Mod}_{/\mathfrak{S}_{\infty }}^{r,{\hat{G}},\text{cris}}$ ) of $\text{Mod}_{/\mathfrak{S}}^{r,{\hat{G}}}$ (resp. $\text{Mod}_{/\mathfrak{S}_{\infty }}^{r,{\hat{G}}}$ ) consisting of objects $\hat{\mathfrak{M}}$ which satisfy the following condition; $\unicode[STIX]{x1D70F}(x)-x\in u^{p}\unicode[STIX]{x1D711}(\mathfrak{t})(W(R)\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M})$ for any $x\in \mathfrak{M}$ .

The following results are important properties for the functor $\hat{T}_{L,s}$ .

2.4 $(\unicode[STIX]{x1D711},{\hat{G}})$ -modules, Breuil modules and filtered $(\unicode[STIX]{x1D711},N)$ -modules

We recall some relations between Breuil modules and $(\unicode[STIX]{x1D711},{\hat{G}})$ -modules. Here we give a rough sketch only. For more precise information, see [Reference BreuilBr1, Section 6], [Reference LiuLi1, Section 5] and the proof of [Reference LiuLi2, Theorem 2.3.1(2)].

Let $\hat{\mathfrak{M}}$ be a free $(\unicode[STIX]{x1D711},{\hat{G}}_{L,s})$ -module over $\mathfrak{S}_{L,s}$ . If we put ${\mathcal{D}}:=S_{L_{0},s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{L,s}}\mathfrak{M}$ , then ${\mathcal{D}}$ has a structure of a Breuil module over $S_{L_{0},s}$ which corresponds to the semistable representation $\mathbb{Q}_{p}\otimes _{\mathbb{Z}_{p}}\hat{T}_{L,s}(\hat{\mathfrak{M}})$ of $G_{L,s}$ (for the definition and properties of Breuil modules, see [Reference BreuilBr1]). Thus ${\mathcal{D}}$ is equipped with a Frobenius $\unicode[STIX]{x1D711}_{{\mathcal{D}}}(=\unicode[STIX]{x1D711}_{S_{L_{0},s}}\otimes \unicode[STIX]{x1D711}_{\mathfrak{M}})$ , a decreasing filtration $(\text{Fil}^{i}{\mathcal{D}})_{i\geqslant 0}$ of $S_{L_{0},s}$ -submodules of ${\mathcal{D}}$ and a $L_{0}$ -linear monodromy operator $N:{\mathcal{D}}\rightarrow {\mathcal{D}}$ which satisfy certain properties (for example, Griffiths transversality).

Putting $D={\mathcal{D}}/I_{+}S_{L_{0},s}{\mathcal{D}}$ , we can associate a filtered $(\unicode[STIX]{x1D711},N)$ -module over $L_{(s)}$ as following: $\unicode[STIX]{x1D711}_{D}:=\unicode[STIX]{x1D711}_{{\mathcal{D}}}~\text{mod}~I_{+}S_{L_{0},s}{\mathcal{D}}$ , $N_{D}:=N_{{\mathcal{D}}}~\text{mod}~I_{+}S_{L_{0},s}{\mathcal{D}}$ and $\text{Fil}^{i}D_{L_{(s)}}:=f_{\unicode[STIX]{x1D70B}_{s}}(Fil^{i}({\mathcal{D}}))$ . Here, $f_{\unicode[STIX]{x1D70B}_{s}}:{\mathcal{D}}\rightarrow D_{L_{(s)}}$ is the projection defined by ${\mathcal{D}}{\twoheadrightarrow}{\mathcal{D}}/\text{Fil}^{1}S_{L_{0},s}{\mathcal{D}}\simeq D_{L_{(s)}}$ . Proposition 6.2.1.1 of [Reference BreuilBr1] implies that there exists a unique $\unicode[STIX]{x1D711}$ -compatible section $s:D{\hookrightarrow}{\mathcal{D}}$ of ${\mathcal{D}}{\twoheadrightarrow}D$ . By this embedding, we regard $D$ as a submodule of ${\mathcal{D}}$ . Then we have $N_{{\mathcal{D}}}|_{D}=N_{D}$ and $N_{{\mathcal{D}}}=N_{S_{L_{0},s}}\otimes \text{Id}_{D}+\text{Id}_{S_{L_{0},s}}\otimes N_{D}$ under the identification ${\mathcal{D}}=S_{L_{0},s}\otimes _{L_{(s)}}D$ .

The $G_{L,s}$ -action on $\widehat{{\mathcal{R}}}_{L,s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{L,s}}\mathfrak{M}$ extends to $B_{\text{cris}}^{+}\otimes _{\widehat{{\mathcal{R}}}_{L,s}}(\widehat{{\mathcal{R}}}_{L,s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{L,s}}\mathfrak{M})\simeq B_{\text{cris}}^{+}\otimes _{S_{L_{0},s}}{\mathcal{D}}$ . This action is in fact explicitly written as follows:

(1) $$\begin{eqnarray}\displaystyle g(a\otimes x) & = & \displaystyle \mathop{\sum }_{i=0}^{\infty }g(a)\unicode[STIX]{x1D6FE}_{i}\Bigl(-\text{log}\Bigl(\frac{g[\text{}\underline{\unicode[STIX]{x1D70B}_{s}}]}{[\text{}\underline{\unicode[STIX]{x1D70B}_{s}}]}\Bigr)\Bigr)\otimes N_{{\mathcal{D}}}^{i}(x)\nonumber\\ \displaystyle & & \displaystyle \quad \text{for}~g\in G_{L,s},a\in B_{\text{cris}}^{+},x\in {\mathcal{D}}.\end{eqnarray}$$

By this explicit formula, we can obtain an easy relation between $N_{{\mathcal{D}}}$ and $\unicode[STIX]{x1D70F}^{p^{s}}$ -action on $\hat{\mathfrak{M}}$ as follows: first we recall that $t=-\text{log}(\unicode[STIX]{x1D70F}([\text{}\underline{\unicode[STIX]{x1D70B}}])/[\text{}\underline{\unicode[STIX]{x1D70B}}])=-\text{log}(\unicode[STIX]{x1D70F}^{p^{s}}([\text{}\underline{\unicode[STIX]{x1D70B}_{s}}])/[\text{}\underline{\unicode[STIX]{x1D70B}_{s}}])$ . By the formula, for any $n\geqslant 0$ and $x\in {\mathcal{D}}$ , an induction on $n$ shows that we have

$$\begin{eqnarray}(\unicode[STIX]{x1D70F}^{p^{s}}-1)^{n}(x)=\mathop{\sum }_{m=n}^{\infty }\Bigl(\mathop{\sum }_{\substack{ i_{1}+\cdots i_{n}=m \\ i_{j}\geqslant 0}}\frac{m!}{i_{1}!\cdots i_{n}!}\Bigr)\unicode[STIX]{x1D6FE}_{m}(t)\otimes N_{{\mathcal{D}}}^{m}(x)\in B_{\text{cris}}^{+}\otimes _{S_{L_{0},s}}{\mathcal{D}}\end{eqnarray}$$

and in particular we see $\frac{(\unicode[STIX]{x1D70F}^{p^{s}}-1)^{n}}{n}(x)\rightarrow 0$ $p$ -adically as $n\rightarrow \infty$ . Hence we can define

$$\begin{eqnarray}\text{log}(\unicode[STIX]{x1D70F}^{p^{s}})(x):=\mathop{\sum }_{n=1}^{\infty }(-1)^{n-1}\frac{(\unicode[STIX]{x1D70F}^{p^{s}}-1)^{n}}{n}(x)\in B_{\text{cris}}^{+}\otimes _{S_{L_{0},s}}{\mathcal{D}}.\end{eqnarray}$$

It is not difficult to check the equation

(2) $$\begin{eqnarray}\text{log}(\unicode[STIX]{x1D70F}^{p^{s}})(x)=t\otimes N_{{\mathcal{D}}}(x).\end{eqnarray}$$

2.5 Base changes for Kisin modules

Let $\mathfrak{M}$ be a free or torsion Kisin module of height ${\leqslant}r$ over $\mathfrak{S}_{L}$ (resp. over $\mathfrak{S}$ ). We put $\mathfrak{M}_{L,s}=\mathfrak{S}_{L,s}\otimes _{\mathfrak{S}_{L}}\mathfrak{M}$ (resp. $\mathfrak{S}_{L}=\mathfrak{S}_{L}\otimes _{\mathfrak{S}}\mathfrak{M}$ ) and equip $\mathfrak{M}_{L,s}$ (resp. $\mathfrak{M}_{L}$ ) with a Frobenius by $\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{\mathfrak{S}_{L,s}}\otimes \unicode[STIX]{x1D711}_{\mathfrak{M}}$ (resp.  $\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{\mathfrak{S}_{L}}\otimes \unicode[STIX]{x1D711}_{\mathfrak{M}}$ ). Then it is not difficult to check that $\mathfrak{M}_{L,s}$ (resp.  $\mathfrak{M}_{L}$ ) is a free or torsion Kisin module of height ${\leqslant}r$ over $\mathfrak{S}_{L,s}$ (resp. over $\mathfrak{S}_{L}$ ) (here we recall that $E_{s}(u_{s})=E(u_{s}^{p^{s}})=E(u)$ ). Hence we obtained natural functors

$$\begin{eqnarray}\displaystyle & \displaystyle \text{Mod}_{/\mathfrak{S}_{L}}^{r}\rightarrow \text{Mod}_{/\mathfrak{S}_{L,s}}^{r}\qquad \text{and}\qquad \text{Mod}_{/\mathfrak{S}_{L,\infty }}^{r}\rightarrow \text{Mod}_{/\mathfrak{S}_{L,s,\infty }}^{r} & \displaystyle \nonumber\\ \displaystyle & \displaystyle \Big(\text{resp.}~\text{Mod}_{/\mathfrak{S}}^{r}\rightarrow \text{Mod}_{/\mathfrak{S}_{L}}^{r}~\text{and}~\text{Mod}_{/\mathfrak{S}_{\infty }}^{r}\rightarrow \text{Mod}_{/\mathfrak{S}_{L,\infty }}^{r}\Big). & \displaystyle \nonumber\end{eqnarray}$$

By definition, we immediately see that we have $T_{\mathfrak{S}_{L}}(\mathfrak{M})\simeq T_{\mathfrak{S}_{L,s}}(\mathfrak{M}_{L,s})$ (resp. $T_{\mathfrak{S}}(\mathfrak{M})|_{G_{L_{\infty }}}\simeq T_{\mathfrak{S}_{L}}(\mathfrak{M}_{L})$ ). In particular, it follows from Proposition 2.1(1) that the following holds:

2.6 Base changes for $(\unicode[STIX]{x1D711},{\hat{G}})$ -modules

Let $\hat{\mathfrak{M}}$ be a free or torsion $(\unicode[STIX]{x1D711},{\hat{G}}_{L})$ -module (resp.  $(\unicode[STIX]{x1D711},{\hat{G}})$ -module) of height ${\leqslant}r$ over $\mathfrak{S}_{L}$ (resp. over $\mathfrak{S}$ ). The $G_{L,s}$ action on $\widehat{{\mathcal{R}}}_{L}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{L}}\mathfrak{M}$ (resp. the $G_{L}$ action on $\widehat{{\mathcal{R}}}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M}$ ) extends to $\widehat{{\mathcal{R}}}_{L,s}\otimes _{\widehat{{\mathcal{R}}}_{L}}(\widehat{{\mathcal{R}}}_{L}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{L}}\mathfrak{M})\simeq \widehat{{\mathcal{R}}}_{L,s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{L,s}}\mathfrak{M}_{L,s}$ (resp.  $\widehat{{\mathcal{R}}}_{L}\otimes _{\widehat{{\mathcal{R}}}}(\widehat{{\mathcal{R}}}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M})\simeq \widehat{{\mathcal{R}}}_{L}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{L}}\mathfrak{M}_{L}$ ), which factors through ${\hat{G}}_{L,s}$ (resp.  ${\hat{G}}_{L}$ ). Then it is not difficult to check that $\mathfrak{M}_{L,s}$ (resp.  $\mathfrak{M}_{L}$ ) has a structure of a $(\unicode[STIX]{x1D711},{\hat{G}}_{L,s})$ -module (resp.  $(\unicode[STIX]{x1D711},{\hat{G}}_{L})$ -module). Hence we obtained natural functors

$$\begin{eqnarray}\displaystyle & \displaystyle \text{Mod}_{/\mathfrak{S}_{L}}^{r,{\hat{G}}_{L}}\rightarrow \text{Mod}_{/\mathfrak{S}_{L,s}}^{r,{\hat{G}}_{L,s}}\qquad \text{and}\qquad \text{Mod}_{/\mathfrak{S}_{L,\infty }}^{r,{\hat{G}}_{L}}\rightarrow \text{Mod}_{/\mathfrak{S}_{L,s,\infty }}^{r,{\hat{G}}_{L,s}} & \displaystyle \nonumber\\ \displaystyle & \displaystyle \Big(\text{resp.}~\text{Mod}_{/\mathfrak{S}}^{r,{\hat{G}}}\rightarrow \text{Mod}_{/\mathfrak{S}_{L}}^{r,{\hat{G}}_{L}}~\text{and}~\text{Mod}_{/\mathfrak{S}_{\infty }}^{r,{\hat{G}}}\rightarrow \text{Mod}_{/\mathfrak{S}_{L,\infty }}^{r,{\hat{G}}_{L}}\Big). & \displaystyle \nonumber\end{eqnarray}$$

By definition, we immediately see that we have $\hat{T}_{L}(\hat{\mathfrak{M}})|_{G_{L,s}}\simeq \hat{T}_{L,s}(\hat{\mathfrak{M}}_{L,s})$ (resp. $\hat{T}(\hat{\mathfrak{M}})|_{G_{L}}\simeq \hat{T}_{L}(\hat{\mathfrak{M}}_{L})$ ). Similar to Proposition 2.6, we can prove the following.

The proposition immediately follows from the full faithfulness property of Theorem 2.5(1) and the lemma below.

3.1 Definitions

We start with some definitions which are our main concern in this and the next section.

For any object $\hat{\mathfrak{M}}$ of $\text{Mod}_{/\mathfrak{S}_{L}}^{r,{\hat{G}}_{L,s}}$ or $\text{Mod}_{/\mathfrak{S}_{L,\infty }}^{r,{\hat{G}}_{L,s}}$ , we define a $\mathbb{Z}_{p}[G_{L,s}]$ -module $\hat{T}_{L,s}(\hat{\mathfrak{M}})$ by

$$\begin{eqnarray}\hat{T}_{L,s}(\hat{\mathfrak{M}})=\left\{\begin{array}{@{}ll@{}}\text{Hom}_{\widehat{{\mathcal{R}}}_{L},\unicode[STIX]{x1D711}}(\widehat{{\mathcal{R}}}_{L}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{L}}\mathfrak{M},W(R))\quad & \text{if}~\mathfrak{M}~\text{is}~\text{free},\\ \text{Hom}_{\widehat{{\mathcal{R}}}_{L},\unicode[STIX]{x1D711}}(\widehat{{\mathcal{R}}}_{L}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{L}}\mathfrak{M},\mathbb{Q}_{p}/\mathbb{Z}_{p}\otimes _{\mathbb{Z}_{p}}W(R))\quad & \text{if}~\mathfrak{M}~\text{is}~\text{torsion}.\end{array}\right.\end{eqnarray}$$

Here, $G_{L,s}$ acts on $\hat{T}_{L,s}(\hat{\mathfrak{M}})$ by $(\unicode[STIX]{x1D70E}.f)(x)=\unicode[STIX]{x1D70E}(f(\unicode[STIX]{x1D70E}^{-1}(x)))$ for $\unicode[STIX]{x1D70E}\in G_{L,s},~f\in \hat{T}_{L,s}(\hat{\mathfrak{M}}),~x\in \widehat{{\mathcal{R}}}_{L}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{L}}\mathfrak{M}$ . Similar to the above, for any object $\hat{\mathfrak{M}}$ of $\widetilde{\text{Mod}}_{/\mathfrak{S}_{L}}^{r,{\hat{G}}_{L,s}}$ or $\widetilde{\text{Mod}}_{/\mathfrak{S}_{L,\infty }}^{r,{\hat{G}}_{L,s}}$ , we define a $\mathbb{Z}_{p}[G_{L,s}]$ -module $\hat{T}_{L,s}(\hat{\mathfrak{M}})$ by

$$\begin{eqnarray}\hat{T}_{L,s}(\hat{\mathfrak{M}})=\left\{\begin{array}{@{}l@{}}\text{Hom}_{\widehat{{\mathcal{R}}}_{L,s},\unicode[STIX]{x1D711}}(\widehat{{\mathcal{R}}}_{L,s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{L}}\mathfrak{M},W(R))\hspace{59.75078pt}\text{if}~\mathfrak{M}~\text{is}~\text{free},\quad \\ \text{Hom}_{\widehat{{\mathcal{R}}}_{L,s},\unicode[STIX]{x1D711}}(\widehat{{\mathcal{R}}}_{L,s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{L}}\mathfrak{M},\mathbb{Q}_{p}/\mathbb{Z}_{p}\otimes _{\mathbb{Z}_{p}}W(R))\hspace{5.69054pt}\text{if}~\mathfrak{M}~\text{is}~\text{torsion}.\quad \end{array}\right.\end{eqnarray}$$

On the other hand, we have natural functors $\text{Mod}_{/\mathfrak{S}_{L}}^{r,{\hat{G}}_{L}}\rightarrow \text{Mod}_{/\mathfrak{S}_{L}}^{r,{\hat{G}}_{L,s}}\rightarrow \widetilde{\text{Mod}}_{/\mathfrak{S}_{L}}^{r,{\hat{G}}_{L,s}}\rightarrow \text{Mod}_{/\mathfrak{S}_{L,s}}^{r,{\hat{G}}_{L,s}}$ and

$$\begin{eqnarray}\text{Mod}_{/\mathfrak{S}_{L,\infty }}^{r,{\hat{G}}_{L}}\rightarrow \text{Mod}_{/\mathfrak{S}_{L,\infty }}^{r,{\hat{G}}_{L,s}}\rightarrow \widetilde{\text{Mod}}_{/\mathfrak{S}_{L,\infty }}^{r,{\hat{G}}_{L,s}}\rightarrow \text{Mod}_{/\mathfrak{S}_{L,s,\infty }}^{r,{\hat{G}}_{L,s}}\end{eqnarray}$$

and it is readily seen that these functors are compatible with $\hat{T}_{L}$ and $\hat{T}_{L,s}$ . In particular, the functors $\hat{T}_{L,s}$ on $\text{Mod}_{/\mathfrak{S}_{L}}^{r,{\hat{G}}_{L,s}}$ and $\widetilde{\text{Mod}}_{/\mathfrak{S}_{L}}^{r,{\hat{G}}_{L,s}}$ take their values in $\text{Rep}_{\mathbb{Z}_{p}}^{r,\text{st}}(G_{L,s})$ since we have an equivalence of categories $\hat{T}_{L,s}:\text{Mod}_{/\mathfrak{S}_{L,s}}^{r,{\hat{G}}_{L,s}}\overset{{\sim}}{\rightarrow }\text{Rep}_{\mathbb{Z}_{p}}^{r,\text{st}}(G_{L,s})$ by Theorem 2.5.

In the rest of this section, we study free cases. We leave studies for torsion cases to the next section.

Convention: For simplicity, if $L=K$ , then we often omit the subscript “ $L$ ” from various notations (e.g., $\text{Mod}_{/\mathfrak{S}_{K}}^{r,{\hat{G}}_{K,s}}=\text{Mod}_{/\mathfrak{S}}^{r,{\hat{G}}_{s}},\widetilde{\text{Mod}}_{/\mathfrak{S}_{K}}^{r,{\hat{G}}_{K,s}}=\widetilde{\text{Mod}}_{/\mathfrak{S}}^{r,{\hat{G}}_{s}}$ ). Furthermore, if $s=0$ , we often omit the subscript “ $s$ ” from various notations (e.g., $\text{Mod}_{/\mathfrak{S}_{L}}^{r,{\hat{G}}_{L,0}}=\text{Mod}_{/\mathfrak{S}_{L}}^{r,{\hat{G}}_{L}},\widetilde{\text{Mod}}_{/\mathfrak{S}_{L,0}}^{r,{\hat{G}}_{L,0}}=\widetilde{\text{Mod}}_{/\mathfrak{S}_{L}}^{r,{\hat{G}}_{L}}$ ).

3.2 The functors $\text{Mod}_{/\mathfrak{S}}^{r,{\hat{G}}}\rightarrow \text{Mod}_{/\mathfrak{S}}^{r,{\hat{G}}_{s}}\rightarrow \widetilde{\text{Mod}}_{/\mathfrak{S}}^{r,{\hat{G}}_{s}}\rightarrow \text{Mod}_{/\mathfrak{S}_{s}}^{r,{\hat{G}}_{s}}$

Now we consider the functors $\text{Mod}_{/\mathfrak{S}}^{r,{\hat{G}}}\rightarrow \text{Mod}_{/\mathfrak{S}}^{r,{\hat{G}}_{s}}\rightarrow \widetilde{\text{Mod}}_{/\mathfrak{S}}^{r,{\hat{G}}_{s}}\rightarrow \text{Mod}_{/\mathfrak{S}_{s}}^{r,{\hat{G}}_{s}}$ . At first, by Proposition 2.6, we see that the functor $\widetilde{\text{Mod}}_{/\mathfrak{S}}^{r,{\hat{G}}_{s}}\rightarrow \text{Mod}_{/\mathfrak{S}_{s}}^{r,{\hat{G}}_{s}}$ is fully faithful. It follows from this fact and Theorem 2.5(1) that the functor $\hat{T}_{s}:\widetilde{\text{Mod}}_{/\mathfrak{S}}^{r,{\hat{G}}_{s}}\rightarrow \text{Rep}_{\mathbb{Z}_{p}}^{r,\text{st}}(G_{s})$ is fully faithful. It is clear that the functor $\text{Mod}_{/\mathfrak{S}}^{r,{\hat{G}}_{s}}\rightarrow \widetilde{\text{Mod}}_{/\mathfrak{S}}^{r,{\hat{G}}_{s}}$ is fully faithful and thus so is $\hat{T}_{s}:\text{Mod}_{/\mathfrak{S}}^{r,{\hat{G}}_{s}}\rightarrow \text{Rep}_{\mathbb{Z}_{p}}^{r,\text{st}}(G_{s})$ . Combining this with Theorem 2.5(1) and Lemma 2.8, we obtain that the functor $\text{Mod}_{/\mathfrak{S}}^{r,{\hat{G}}}\rightarrow \text{Mod}_{/\mathfrak{S}}^{r,{\hat{G}}_{s}}$ is also fully faithful. Furthermore, we prove the following.

Summary, we obtained the following commutative diagram.

Before starting the proof of Proposition 3.3, we give an explicit formula such as (1) for an object of $\widetilde{\text{Mod}}_{/\mathfrak{S}}^{r,{\hat{G}}_{s}}$ . The argument below follows the method of [Reference LiuLi2]. Let $\hat{\mathfrak{M}}$ be an object of $\widetilde{\text{Mod}}_{/\mathfrak{S}}^{r,{\hat{G}}_{s}}$ . Let $\hat{\mathfrak{M}}_{s}$ be the image of $\hat{\mathfrak{M}}$ for the functor $\widetilde{\text{Mod}}_{/\mathfrak{S}}^{r,{\hat{G}}_{s}}\rightarrow \text{Mod}_{/\mathfrak{S}_{s}}^{r,{\hat{G}}_{s}}$ . Put ${\mathcal{D}}=S_{K_{0}}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M}$ and also put ${\mathcal{D}}_{s}=S_{K_{0},s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{s}}\mathfrak{M}_{s}=S_{K_{0},s}\otimes _{S_{K_{0}}}{\mathcal{D}}$ . Then ${\mathcal{D}}_{s}$ has a structure of a Breuil module and also $D={\mathcal{D}}_{s}/I_{+}S_{K_{0},s}{\mathcal{D}}_{s}$ has a structure of a filtered $(\unicode[STIX]{x1D711},N)$ -module corresponding to $\mathbb{Q}_{p}\otimes _{\mathbb{Z}_{p}}\hat{T}_{s}(\hat{\mathfrak{M}}_{s})$ (see Section 2.4), which is isomorphic to ${\mathcal{D}}/I_{+}S_{K_{0}}{\mathcal{D}}$ as a $\unicode[STIX]{x1D711}$ -module over $K_{0}$ . By [Reference LiuLi1, Lemma 7.3.1], we have a unique $\unicode[STIX]{x1D711}$ -compatible section $D{\hookrightarrow}{\mathcal{D}}$ and we regard $D$ as a submodule of ${\mathcal{D}}\subset {\mathcal{D}}_{s}$ by this section. Then we have ${\mathcal{D}}=S_{K_{0}}\otimes _{K_{0}}D$ and ${\mathcal{D}}_{s}=S_{K_{0},s}\otimes _{K_{0}}D$ . By the explicit formula (1) for $\hat{\mathfrak{M}}_{s}$ , we know that

$$\begin{eqnarray}{\hat{G}}_{s}(D)\subset (K_{0}[\![t]\!]\cap {\mathcal{R}}_{K_{0},s})\otimes _{K_{0}}D.\end{eqnarray}$$

(Note that ${\mathcal{R}}_{K_{0},s}$ can be regarded as a subring of $K_{0}[\![t,u_{s}]\!]$ via [Reference LiuLi1, Lemma 7.1.2 ].) Hence, taking any $K_{0}$ -basis $e_{1},\ldots ,e_{d}$ of $D$ , there exist $A_{s}(t)\in M_{d\times d}(K_{0}[\![t]\!])$ such that $\unicode[STIX]{x1D70F}^{p^{s}}(e_{1},\ldots ,e_{d})=(e_{1},\ldots ,e_{d})A_{s}(t)$ . Since $A_{s}(0)=\text{I}_{d}$ , we see that $\text{log}(A_{s}(t))\in M_{d\times d}(K_{0}[\![t]\!])$ is well defined. On the other hand, choose $g_{0}\in G_{s}$ such that $\unicode[STIX]{x1D712}_{p}(g_{0})\not =1$ , where $\unicode[STIX]{x1D712}_{p}$ is the $p$ -adic cyclotomic character. Since $g_{0}\unicode[STIX]{x1D70F}^{p^{s}}=(\unicode[STIX]{x1D70F}^{p^{s}})^{\unicode[STIX]{x1D712}_{p}(g_{0})}g_{0}$ , we have $A_{s}(\unicode[STIX]{x1D712}_{p}(g_{0})t)=A_{s}(t)^{\unicode[STIX]{x1D712}_{p}(g_{0})}$ and thus we also have $\text{log}(A_{s}(\unicode[STIX]{x1D712}_{p}(g_{0})t))=\unicode[STIX]{x1D712}_{p}(g_{0})\text{log}(A_{s}(t))$ . Since $\text{log}(A_{s}(0))=\text{log}(I_{d})=0$ , we can write $\text{log}(A_{s}(t))$ as $tB(t)$ for some $B(t)\in M_{d\times d}(K_{0}[\![t]\!])$ . Then we have $\unicode[STIX]{x1D712}_{p}(g_{0})tB(\unicode[STIX]{x1D712}_{p}(g_{0})t)=\unicode[STIX]{x1D712}_{p}(g_{0})tB(t)$ , that is, $B(\unicode[STIX]{x1D712}_{p}(g_{0})t)=B(t)$ . Hence the assumption $\unicode[STIX]{x1D712}_{p}(g_{0})\not =1$ implies that $B(t)$ is a constant. Putting $N_{s}=B(t)\in M_{d\times d}(K_{0})$ , we obtain

$$\begin{eqnarray}\unicode[STIX]{x1D70F}^{p^{s}}(e_{1},\ldots ,e_{d})=(e_{1},\ldots ,e_{d})\Bigl(\mathop{\sum }_{i=0}^{\infty }N_{s}^{i}\unicode[STIX]{x1D6FE}_{i}(t)\Bigr).\end{eqnarray}$$

Now we define $N_{D}:D\rightarrow D$ by $N(e_{1},\ldots ,e_{d})=(e_{1},\ldots ,e_{d})p^{-s}N_{s}$ and also define $N_{{\mathcal{D}}}:=N_{S_{K_{0}}}\otimes \text{Id}_{D}+\text{Id}_{S_{K_{0}}}\otimes N_{D}$ . (Note that we have $N_{D}\unicode[STIX]{x1D711}_{D}=p\unicode[STIX]{x1D711}_{D}N_{D}$ and thus $N_{D}$ is nilpotent.) It is a routine work to check the following:

(3) $$\begin{eqnarray}g(a\otimes x)=\mathop{\sum }_{i=0}^{\infty }g(a)\unicode[STIX]{x1D6FE}_{i}(-\text{log}([\text{}\underline{\unicode[STIX]{x1D700}}(g)]))\otimes N_{D}^{i}(x)\quad \text{for}~g\in G_{s},a\in B_{\text{cris}}^{+},x\in D.\end{eqnarray}$$

Since we have

(4) $$\begin{eqnarray}g(f)=\mathop{\sum }_{i=0}^{\infty }\unicode[STIX]{x1D6FE}_{i}(-\text{log}([\text{}\underline{\unicode[STIX]{x1D700}}(g)]))N_{S_{K_{0}}}^{i}(f)\end{eqnarray}$$

for any $g\in G_{K}$ and $f\in S_{K_{0}}$ , we obtain the following explicit formula:

(5) $$\begin{eqnarray}g(a\otimes x)=\mathop{\sum }_{i=0}^{\infty }g(a)\unicode[STIX]{x1D6FE}_{i}(-\text{log}([\text{}\underline{\unicode[STIX]{x1D700}}(g)]))\otimes N_{{\mathcal{D}}}^{i}(x)\quad \text{for}~g\in G_{s},a\in B_{\text{cris}}^{+},x\in {\mathcal{D}}.\end{eqnarray}$$

In particular, as in Section 2.4, we can show that

(6) $$\begin{eqnarray}\text{log}(\unicode[STIX]{x1D70F}^{p^{s}})(x)=p^{s}t\otimes N_{{\mathcal{D}}}(x)\end{eqnarray}$$

for any $x\in {\mathcal{D}}$ .

Proof of Proposition 3.3.

We continue to use the above notation. It suffices to prove that the $G_{s}$ -action on $\widehat{{\mathcal{R}}}_{s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M}$ preserves $\widehat{{\mathcal{R}}}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M}$ . Note that we have $\widehat{{\mathcal{R}}}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M}=({\mathcal{R}}_{K_{0}}\otimes _{K_{0}}D)\cap (W(R)\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M})$ , $G_{s}(\mathfrak{M})\subset \widehat{{\mathcal{R}}}_{s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M}\subset W(R)\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M}$ and $G_{s}({\mathcal{R}}_{K_{0}})\subset {\mathcal{R}}_{K_{0}}$ . Thus it is enough to show $G_{s}(D)\subset {\mathcal{R}}_{K_{0}}\otimes _{K_{0}}D$ . This quickly follows from (3). In fact, we have

$$\begin{eqnarray}g(x)=\mathop{\sum }_{i=0}^{\infty }\unicode[STIX]{x1D6FE}_{i}(-\text{log}([\text{}\underline{\unicode[STIX]{x1D700}}(g)]))\otimes N_{D}^{i}(x)\in {\mathcal{R}}_{K_{0}}\otimes _{K_{0}}D\quad \text{for}~x\in D,~g\in G_{s}.\square\end{eqnarray}$$

3.3 Relations with crystalline representations

We know that $\mathbb{Q}_{p}\otimes _{\mathbb{Z}_{p}}\hat{T}_{s}(\hat{\mathfrak{M}})$ is semistable over $K_{s}$ for any object $\hat{\mathfrak{M}}$ of $\text{Mod}_{/\mathfrak{S}}^{r,{\hat{G}}_{s}}$ or $\widetilde{\text{Mod}}_{/\mathfrak{S}}^{r,{\hat{G}}_{s}}$ . This subsection is devoted to prove a criterion, for $\hat{\mathfrak{M}}$ , that describes when $\mathbb{Q}_{p}\otimes _{\mathbb{Z}_{p}}\hat{T}_{s}(\hat{\mathfrak{M}})$ becomes crystalline.

Following [Reference FontaineFo2, Section 5] we set $I^{[m]}B_{\text{cris}}^{+}:=\{x\in B_{\text{cris}}^{+}\mid \unicode[STIX]{x1D711}^{n}(x)\in \text{Fil}^{m}B_{\text{cris}}^{+}~\text{for all}~n\geqslant 0\}$ . For any subring $A\subset B_{\text{cris}}^{+}$ , we put $I^{[m]}A=A\cap I^{[m]}B_{\text{cris}}^{+}$ . Furthermore, we also put $I^{[m+]}A=I^{[m]}A.I_{+}A$ (here, $I_{+}A$ is defined in Section 2.3). By [Reference FontaineFo2, Proposition 5.1.3] and the proof of [Reference LiuLi2, Lemma 3.2.2], we know that $I^{[m]}W(R)$ is a principal ideal which is generated by $\unicode[STIX]{x1D711}(\mathfrak{t})^{m}$ .

Now we recall Theorem 2.5(2): if $\mathfrak{M}$ is an object of $\text{Mod}_{/\mathfrak{S}_{s}}^{r,{\hat{G}}_{s}}$ , then $\mathbb{Q}_{p}\otimes _{\mathbb{Z}_{p}}\hat{T}_{s}(\hat{\mathfrak{M}})$ is crystalline if and only if $\unicode[STIX]{x1D70F}^{p^{s}}(x)-x\in u_{s}^{p}(I^{[1]}W(R)\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{s}}\mathfrak{M})$ for any $x\in \mathfrak{M}$ . However, if such $\mathfrak{M}$ descends to a Kisin module over $\mathfrak{S}$ , then we can show the following.

Proof. (1) $\Rightarrow$ (2): The proof here mainly follows that of [Reference Gee, Liu and SavittGLS, Proposition 4.7]. We may suppose $\hat{\mathfrak{M}}$ is an object of $\widetilde{\text{Mod}}_{/\mathfrak{S}}^{r,{\hat{G}}_{s}}$ . Put ${\mathcal{D}}=S_{K_{0}}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M}$ and $D={\mathcal{D}}/I_{+}S_{K_{0}}{\mathcal{D}}$ as in the previous subsection. We fix a $\unicode[STIX]{x1D711}(\mathfrak{S})$ -basis $(\hat{e}_{1},\ldots ,\hat{e}_{d})$ of $\mathfrak{M}\subset {\mathcal{D}}$ and denote by $(e_{1},\ldots ,e_{d})$ the image of $(\hat{e}_{1},\ldots ,\hat{e}_{d})$ for the projection ${\mathcal{D}}\rightarrow D$ . Then $(e_{1},\ldots ,e_{d})$ is a $K_{0}$ -basis of $D$ . As described before the proof of Proposition 3.3, we regard $D$ as a $\unicode[STIX]{x1D711}$ -stable submodule of ${\mathcal{D}}$ , and we have $N_{D}:D\rightarrow D$ and $N_{{\mathcal{D}}}:D_{{\mathcal{D}}}\rightarrow D_{{\mathcal{D}}}$ .

Now we consider a matrix $X\in GL_{d\times d}(S_{K_{0}})$ such that $(\hat{e}_{1},\ldots ,\hat{e}_{d})=(e_{1},\ldots ,e_{d})X$ . We define $\tilde{S}=W(k)[\![u^{p},u^{ep}/p]\!]$ as in [Reference Gee, Liu and SavittGLS, Section 4], which is a sub $W(k)$ -algebra of $S_{K_{0}}^{\text{int}}$ with the property $N_{S_{K_{0}}}(\tilde{S})\subset u^{p}\tilde{S}$ . By an easy computation we have $U=X^{-1}BX+X^{-1}N_{S_{K_{0}}}(X)$ . Here, $B\in M_{d\times d}(K_{0})$ and $U\in M_{d\times d}(S_{K_{0}})$ are defined by $N_{D}(e_{1},\ldots ,e_{d})=(e_{1},\ldots ,e_{d})B$ and $N_{{\mathcal{D}}}(\hat{e}_{1},\ldots ,\hat{e}_{d})=(\hat{e}_{1},\ldots ,\hat{e}_{d})U$ . By the same proof as in the former half part of the proof of [Reference Gee, Liu and SavittGLS, Proposition 4.7], we obtain $X,X^{-1}\in M_{d\times d}(\tilde{S}[1/p])$ . On the other hand, let $\hat{\mathfrak{M}}_{s}$ be the image of $\hat{\mathfrak{M}}$ for the functor $\widetilde{\text{Mod}}_{/\mathfrak{S}}^{r,{\hat{G}}_{s}}\rightarrow \text{Mod}_{\mathfrak{S}_{s}}^{r,{\hat{G}}_{s}}$ . Now we recall that ${\mathcal{D}}_{s}=S_{K_{0},s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}_{s}}\mathfrak{M}_{s}$ has a structure of the Breuil module corresponding to $\mathbb{Q}_{p}\otimes _{\mathbb{Z}_{p}}\hat{T}_{s}(\hat{\mathfrak{M}}_{s})$ Denote by $N_{{\mathcal{D}}_{s}}$ its monodromy operator. By the formula (2) for $\hat{\mathfrak{M}}_{s}$ and the formula (6) for $\hat{\mathfrak{M}}$ , we see that $p^{s}N_{{\mathcal{D}}}=N_{{\mathcal{D}}_{s}}$ on ${\mathcal{D}}$ . Therefore, $\mathbb{Q}_{p}\otimes _{\mathbb{Z}_{p}}\hat{T}_{s}(\hat{\mathfrak{M}})$ is crystalline if and only if $N_{{\mathcal{D}}_{s}}$ mod $I_{+}S_{K_{0},s}{\mathcal{D}}_{s}$ is zero, which is equivalent to say that $N_{D}=(N_{{\mathcal{D}}}~\text{mod}~I_{+}S_{K_{0}}{\mathcal{D}})$ is zero, that is, $B=0$ . Therefore, the latter half part of the proof [Reference Gee, Liu and SavittGLS, Proposition 4.7] gives the assertion (2).

(2) $\Rightarrow$ (3): This is clear.

(3) $\Rightarrow$ (1): Suppose that (3) holds. We denote by $\hat{\mathfrak{M}}_{s}$ the image of $\hat{\mathfrak{M}}$ for the functor $\widetilde{\text{Mod}}_{/\mathfrak{S}}^{r,{\hat{G}}_{s}}\rightarrow \text{Mod}_{\mathfrak{S}_{s}}^{r,{\hat{G}}_{s}}$ as above. We claim that, for any $x\in \mathfrak{M}_{s}$ , we have $\unicode[STIX]{x1D70F}^{p^{s}}(x)-x\in I^{[1+]}W(R)\otimes _{\unicode[STIX]{x1D711}_{s}}\mathfrak{M}_{s}$ . Let $x=a\otimes y\in \mathfrak{M}_{s}=\mathfrak{S}_{s}\otimes _{\mathfrak{S}}\mathfrak{M}$ where $a\in \mathfrak{S}_{s}$ and $y\in \mathfrak{M}$ . Then

$$\begin{eqnarray}\unicode[STIX]{x1D70F}^{p^{s}}(x)-x=\unicode[STIX]{x1D70F}^{p^{s}}(\unicode[STIX]{x1D711}(a))(\unicode[STIX]{x1D70F}^{p^{s}}(y)-y)+(\unicode[STIX]{x1D70F}^{p^{s}}(\unicode[STIX]{x1D711}(a))-\unicode[STIX]{x1D711}(a))y\end{eqnarray}$$

and thus it suffices to show $\unicode[STIX]{x1D70F}^{p^{s}}(\unicode[STIX]{x1D711}(a))-\unicode[STIX]{x1D711}(a)\in I^{[1+]}W(R)$ . This follows from the lemma below and thus we obtained the claim. By the claim and [Reference OzekiOz2, Theorem 21], we know that $\mathbb{Q}_{p}\otimes _{\mathbb{Z}_{p}}\hat{T}_{s}(\hat{\mathfrak{M}}_{s})\simeq \mathbb{Q}_{p}\otimes _{\mathbb{Z}_{p}}\hat{T}_{s}(\hat{\mathfrak{M}})$ is crystalline.◻

4.1 Full faithfulness for $\widetilde{\text{Mod}}_{/\mathfrak{S}_{\infty }}^{r,{\hat{G}}_{s},J}$

For the beginning of a study of $\widetilde{\text{Mod}}_{/\mathfrak{S}_{\infty }}^{r,{\hat{G}}_{s},J}$ , we prove the following full faithfulness result.

Proof. A very similar proof of [Reference OzekiOz2, Lemma 7] proceeds, and hence we only give a sketch here. Let $\hat{\mathfrak{M}}$ and $\hat{\mathfrak{N}}$ be objects of $\widetilde{\text{Mod}}_{/\mathfrak{S}_{\infty }}^{r,{\hat{G}}_{s},J}$ and $\widetilde{\text{Mod}}_{/\mathfrak{S}_{\infty }}^{r^{\prime },{\hat{G}}_{s},J}$ , respectively. Let $f:\mathfrak{M}\rightarrow \mathfrak{N}$ be a morphism of Kisin modules over $\mathfrak{S}$ . Put $\hat{f}=W(R)\otimes f:W(R)\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M}\rightarrow W(R)\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M}$ . Choose any lift of $\unicode[STIX]{x1D70F}\in {\hat{G}}$ to $G_{K}$ ; we denote it also by $\unicode[STIX]{x1D70F}$ . Since the ${\hat{G}}_{s}$ -action for $\hat{\mathfrak{M}}$ is continuous, it suffices to prove that $\unicode[STIX]{x1D6E5}(1\otimes x)=0$ for any $x\in \mathfrak{M}$ where $\unicode[STIX]{x1D6E5}:=\unicode[STIX]{x1D70F}^{p^{s}}\circ \hat{f}-\hat{f}\circ \unicode[STIX]{x1D70F}^{p^{s}}$ . We use induction on $n$ such that $p^{n}\mathfrak{N}=0$ .

Suppose $n=1$ . Since $\unicode[STIX]{x1D6E5}=(\unicode[STIX]{x1D70F}^{p^{s}}-1)\circ \hat{f}-\hat{f}\circ (\unicode[STIX]{x1D70F}^{p^{s}}-1)$ , we obtain the following:

$$\begin{eqnarray}(0):\quad \text{For any }x\in \mathfrak{M},\quad \unicode[STIX]{x1D6E5}(1\otimes x)\in \mathfrak{m}_{R}^{{\geqslant}c(0)}(R\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{N})\end{eqnarray}$$

where $c(0)=c_{J}$ . Since $\mathfrak{M}$ is of height ${\leqslant}r$ , we further obtain the following for any $i\geqslant 1$ inductively:

$$\begin{eqnarray}(i):\quad \text{For any }x\in \mathfrak{M},\quad \unicode[STIX]{x1D6E5}(1\otimes x)\in \mathfrak{m}_{R}^{{\geqslant}c(i)}(R\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{N})\end{eqnarray}$$

where $c(i)=pc(i-1)-pr=(c_{J}-pr/(p-1))p^{i}+pr/(p-1)$ . The condition $c_{J}>pr/(p-1)$ implies that $c(i)\rightarrow \infty$ as $i\rightarrow \infty$ and thus $\unicode[STIX]{x1D6E5}(1\otimes x)=0$ .

Suppose $n>1$ . Consider the exact sequence $0\rightarrow \text{Ker}(p)\rightarrow \mathfrak{N}\overset{p}{\rightarrow }p\mathfrak{N}\rightarrow 0$ of $\unicode[STIX]{x1D711}$ -modules over $\mathfrak{S}$ . It is not difficult to check that $\mathfrak{N}^{\prime }:=\text{Ker}(p)$ and $\mathfrak{N}^{\prime \prime }:=p\mathfrak{N}$ are torsion Kisin modules of height ${\leqslant}r^{\prime }$ over $\mathfrak{S}$ (cf. [Reference LiuLi1, Lemma 2.3.1]). Moreover, we can check that $\mathfrak{N}^{\prime }$ and $\mathfrak{N}^{\prime \prime }$ have natural structures of objects of $\widetilde{\text{Mod}}_{/\mathfrak{S}_{\infty }}^{r^{\prime },{\hat{G}}_{s}}$ (which are denoted by $\hat{\mathfrak{N}}^{\prime }$ and $\hat{\mathfrak{N}}^{\prime \prime }$ , respectively) such that the sequence $0\rightarrow \mathfrak{N}^{\prime }\rightarrow \mathfrak{N}\overset{p}{\rightarrow }\mathfrak{N}^{\prime \prime }\rightarrow 0$ induces an exact sequence $0\rightarrow \hat{\mathfrak{N}}^{\prime }\rightarrow \hat{\mathfrak{N}}\rightarrow \hat{\mathfrak{N}}^{\prime \prime }\rightarrow 0$ . By the lemma below, we know that $\hat{\mathfrak{N}}^{\prime }$ and $\hat{\mathfrak{N}}^{\prime \prime }$ are in fact contained in $\widetilde{\text{Mod}}_{/\mathfrak{S}_{\infty }}^{r^{\prime },{\hat{G}}_{s},J}$ . By the induction hypothesis, we see that $\unicode[STIX]{x1D6E5}(1\otimes x)$ has values in $(W(R)\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{N}^{\prime })\cap (JW(R)\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{N})$ . By [Reference OzekiOz2, Lemma 6] and the assumption that $J\not \subset pW(R)$ is principal, we obtain that $\unicode[STIX]{x1D6E5}(1\otimes x)\in JW(R)\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{N}^{\prime }$ . Since $p\mathfrak{N}^{\prime }=0$ , an analogous argument in the case $n=1$ proceeds and we have $\unicode[STIX]{x1D6E5}(1\otimes x)=0$ as desired.◻

4.2 The category $\widetilde{\text{Rep}}_{\text{tor}}^{r,{\hat{G}}_{s},J}(G_{s})$

In this subsection, we study some categorical properties of $\widetilde{\text{Rep}}_{\text{tor}}^{r,{\hat{G}}_{s},J}(G_{s})$ .

Let $\hat{\mathfrak{M}}$ be an object of $\widetilde{\text{Mod}}_{/\mathfrak{S}_{\infty }}^{r,{\hat{G}}_{s}}$ . Following [Reference LiuLi2, Section 3.2] (note that arguments in [Reference LiuLi2] is the “free case”), we construct a map $\hat{\unicode[STIX]{x1D704}}_{s}$ which connects $\hat{\mathfrak{M}}$ and $\hat{T}_{s}(\hat{\mathfrak{M}})$ as follows. Observe that there exists a natural isomorphism of $\mathbb{Z}_{p}[G_{s}]$ -modules

$$\begin{eqnarray}\hat{T}_{s}(\hat{\mathfrak{M}})\simeq \text{Hom}_{W(R),\unicode[STIX]{x1D711}}(W(R)\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M},\mathbb{Q}_{p}/\mathbb{Z}_{p}\otimes _{\mathbb{Z}_{p}}W(R))\end{eqnarray}$$

where $G_{s}$ acts on $\text{Hom}_{W(R),\unicode[STIX]{x1D711}}(W(R)\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M},\mathbb{Q}_{p}/\mathbb{Z}_{p}\otimes _{\mathbb{Z}_{p}}W(R))$ by $(\unicode[STIX]{x1D70E}.f)(x)=\unicode[STIX]{x1D70E}(f(\unicode[STIX]{x1D70E}^{-1}(x)))$ for $\unicode[STIX]{x1D70E}\in G_{s}$ ,

$$\begin{eqnarray}\displaystyle & \displaystyle f\in \text{Hom}_{W(R),\unicode[STIX]{x1D711}}(W(R)\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M},\mathbb{Q}_{p}/\mathbb{Z}_{p}\otimes _{\mathbb{Z}_{p}}W(R)), & \displaystyle \nonumber\\ \displaystyle & \displaystyle x\in W(R)\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M}=W(R)\otimes _{\widehat{{\mathcal{R}}}_{s}}(\widehat{{\mathcal{R}}}_{s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M}). & \displaystyle \nonumber\end{eqnarray}$$

Thus we can define a morphism

$$\begin{eqnarray}\hat{\unicode[STIX]{x1D704}}_{s}^{\prime }:W(R)\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M}\rightarrow \text{Hom}_{\mathbb{Z}_{p}}(\hat{T}_{s}(\hat{\mathfrak{M}}),\mathbb{Q}_{p}/\mathbb{Z}_{p}\otimes _{\mathbb{Z}_{p}}W(R))\end{eqnarray}$$

by

$$\begin{eqnarray}x\mapsto (f\mapsto f(x)),\quad x\in W(R)\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M},f\in \hat{T}_{s}(\hat{\mathfrak{M}}).\end{eqnarray}$$

Since $\hat{T}_{s}(\hat{\mathfrak{M}})\simeq \bigoplus _{i\in I}\mathbb{Z}_{p}/p^{n_{i}}\mathbb{Z}_{p}$ as $\mathbb{Z}_{p}$ -modules, we have a natural isomorphism $\text{Hom}_{\mathbb{Z}_{p}}(\hat{T}_{s}(\hat{\mathfrak{M}}),\mathbb{Q}_{p}/\mathbb{Z}_{p}\otimes _{\mathbb{Z}_{p}}W(R))\simeq W(R)\otimes _{\mathbb{Z}_{p}}\hat{T}_{s}^{\vee }(\hat{\mathfrak{M}})$ where $\hat{T}_{s}^{\vee }(\hat{\mathfrak{M}})=\text{Hom}_{\mathbb{Z}_{p}}(\hat{T}_{s}(\hat{\mathfrak{M}}),\mathbb{Q}_{p}/\mathbb{Z}_{p})$ is the dual representation of $\hat{T}_{s}(\hat{\mathfrak{M}})$ . Composing this isomorphism with $\hat{\unicode[STIX]{x1D704}}_{s}^{\prime }$ , we obtain the desired map

$$\begin{eqnarray}\hat{\unicode[STIX]{x1D704}}_{s}:W(R)\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M}\rightarrow W(R)\otimes _{\mathbb{Z}_{p}}\hat{T}_{s}^{\vee }(\hat{\mathfrak{M}}).\end{eqnarray}$$

It follows from a direct calculation that $\hat{\unicode[STIX]{x1D704}}_{s}$ is $\unicode[STIX]{x1D711}$ -equivariant and $G_{s}$ -equivariant. If we denote by $\hat{\mathfrak{M}}_{s}$ the image of $\hat{\mathfrak{M}}$ for the functor $\widetilde{\text{Mod}}_{/\mathfrak{S}_{\infty }}^{r,{\hat{G}}_{s}}\rightarrow \text{Mod}_{/\mathfrak{S}_{s,\infty }}^{r,{\hat{G}}_{s}}$ (cf. Section 3.1), then the above $\hat{\unicode[STIX]{x1D704}}_{s}$ is isomorphic to “ $\hat{\unicode[STIX]{x1D704}}$ for $\hat{\mathfrak{M}}_{s}$ in [Reference OzekiOz1, Section 4.1]”. Hence [Reference OzekiOz1, Lemma 4.2(4)] implies that

$$\begin{eqnarray}\displaystyle & & \displaystyle W(\text{Fr}~R)\otimes \hat{\unicode[STIX]{x1D704}}_{s}:W(\text{Fr}~R)\otimes _{W(R)}(W(R)\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M})\nonumber\\ \displaystyle & & \displaystyle \quad \rightarrow W(\text{Fr}~R)\otimes _{W(R)}(W(R)\otimes _{\mathbb{Z}_{p}}\hat{T}_{s}^{\vee }(\hat{\mathfrak{M}}))\nonumber\end{eqnarray}$$

is bijective.

Let $L$ be as in Section 2, that is, the completion of an unramified algebraic extension of $K$ with residue field $k_{L}$ . We prove the following base change lemma.

By an obvious way, we define a functor $\widetilde{\text{Mod}}_{/\mathfrak{S}_{\infty }}^{r,{\hat{G}}_{s}}\rightarrow \widetilde{\text{Mod}}_{/\mathfrak{S}_{L,\infty }}^{r,{\hat{G}}_{L,s}}$ . The underlying Kisin module of the image of $\hat{\mathfrak{M}}\in \widetilde{\text{Mod}}_{/\mathfrak{S}_{\infty }}^{r,{\hat{G}}_{s}}$ for this functor is $\mathfrak{M}_{L}=\mathfrak{S}_{L}\otimes _{\mathfrak{S}}\mathfrak{M}$ . Lemma 4.6 immediately follows from the lemma below.

4.3 Full faithfulness theorem for $\widetilde{\text{Rep}}_{\text{tor}}^{r,{\hat{G}}_{s},J}(G_{s})$

Our goal in this subsection is to prove the following full faithfulness theorem, which plays an important role in our proofs of main theorems.

First we give a very rough sketch of the theory of maximal models for Kisin modules (cf. [Reference Caruso and LiuCL1]). For any $\mathfrak{M}\in \text{Mod}_{/\mathfrak{S}_{\infty }}^{r}$ , put $\mathfrak{M}[1/u]=\mathfrak{S}[1/u]\otimes _{\mathfrak{S}}\mathfrak{M}$ and denote by $F_{\mathfrak{S}}^{r}(\mathfrak{M}[1/u])$ the (partially) ordered set (by inclusion) of torsion Kisin modules $\mathfrak{N}$ of height ${\leqslant}r$ which are contained in $\mathfrak{M}[1/u]$ and $\mathfrak{N}[1/u]=\mathfrak{M}[1/u]$ as $\unicode[STIX]{x1D711}$ -modules. The set $F_{\mathfrak{S}}^{r}(\mathfrak{M}[1/u])$ has a greatest element (cf. [Reference Caruso and LiuCL1, Corollary 3.2.6]). We denote this element by $\text{Max}^{r}(\mathfrak{M})$ . We say that $\mathfrak{M}$ is maximal of height ${\leqslant}r$ (or, maximal for simplicity) if it is the greatest element of $F_{\mathfrak{S}}^{r}(\mathfrak{M}[1/u])$ . The association $\mathfrak{M}\mapsto \text{Max}^{r}(\mathfrak{M})$ defines a functor “ $\text{Max}^{r}$ ” from the category $\text{Mod}_{/\mathfrak{S}_{\infty }}^{r}$ of torsion Kisin modules of height ${\leqslant}r$ into the category $\text{Max}_{/\mathfrak{S}_{\infty }}^{r}$ of maximal Kisin modules of height ${\leqslant}r$ . The category $\text{Max}_{/\mathfrak{S}_{\infty }}^{r}$ is abelian (cf. [Reference Caruso and LiuCL1, Theorem 3.3.8]). Furthermore, the functor $T_{\mathfrak{S}}:\text{Max}_{/\mathfrak{S}_{\infty }}^{r}\rightarrow \text{Rep}_{\text{tor}}(G_{\infty })$ , defined by $T_{\mathfrak{S}}(\mathfrak{M})=\text{Hom}_{\mathfrak{S},\unicode[STIX]{x1D711}}(\mathfrak{M},\mathbb{Q}_{p}/\mathbb{Z}_{p}\otimes _{\mathbb{Z}_{p}}W(R))$ , is exact and fully faithful (cf. [Reference Caruso and LiuCL1, Corollary 3.3.10]). It is not difficult to check that $T_{\mathfrak{S}}(\text{Max}^{r}(\mathfrak{M}))$ is canonically isomorphic to $T_{\mathfrak{S}}(\mathfrak{M})$ as representations of $G_{\infty }$ for any torsion Kisin module $\mathfrak{M}$ of height ${\leqslant}r$ .

We denote by ${\mathcal{S}}_{\text{max}}^{r}$ the set of sequences $\mathfrak{n}=(n_{i})_{i\in \mathbb{Z}/d\mathbb{Z}}$ of integers $0\leqslant n_{i}\leqslant \text{min}\{er,p-1\}$ with smallest period $d$ for some integer $d$ except the constant sequence with value $p-1$ (if necessary). By definition, we see that $\mathfrak{M}(\mathfrak{n})$ is of height ${\leqslant}r$ for any $\mathfrak{n}\in {\mathcal{S}}_{\text{max}}^{r}$ . Putting $r_{0}=\text{max}\{r^{\prime }\in \mathbb{Z}_{{\geqslant}0};e(r^{\prime }-1)<p-1\}$ , we also see that $\mathfrak{M}(\mathfrak{n})$ is of height ${\leqslant}r_{0}$ for any $\mathfrak{n}\in {\mathcal{S}}_{\text{max}}^{r}$ . It is known that $\mathfrak{M}(\mathfrak{n})$ is maximal for any $\mathfrak{n}\in {\mathcal{S}}_{\text{max}}^{r}$  [Reference Caruso and LiuCL1, Proposition 3.6.7]. If $k$ is algebraically closed, then $\mathfrak{M}(\mathfrak{n})$ is simple in $\text{Max}_{/\mathfrak{S}_{\infty }}^{r}$ for any $\mathfrak{n}\in {\mathcal{S}}_{\text{max}}^{r}$ (cf. [Reference Caruso and LiuCL1, Propositions 3.6.7 and 3.6.12]) and furthermore, the converse holds; any simple object in $\text{Max}_{/\mathfrak{S}_{\infty }}^{r}$ is of the form $\mathfrak{M}(\mathfrak{n})$ for some $\mathfrak{n}\in {\mathcal{S}}_{\text{max}}^{r}$ (cf. [Reference Caruso and LiuCL1, Propositions 3.6.8 and 3.6.12]).

Proof. Choose any $(p^{d}-1)$ th root $\unicode[STIX]{x1D702}\in R$ of $\text{}\underline{\unicode[STIX]{x1D700}}$ . Since $[\unicode[STIX]{x1D702}]\cdot \text{exp}(t/(p^{d}-1))$ is a $(p^{d}-1)$ th root of unity, it is of the form $[a]$ for some $a\in \mathbb{F}_{p^{d}}^{\times }$ . Replacing $\unicode[STIX]{x1D702}a^{-1}$ with $\unicode[STIX]{x1D702}$ , we obtain $[\unicode[STIX]{x1D702}]=\text{exp}(-t/(p^{d}-1))\in \widehat{{\mathcal{R}}}^{\times }$ . Put $x_{i}=[\unicode[STIX]{x1D702}]^{m_{i}}\in \widehat{{\mathcal{R}}}^{\times }$ and $\bar{x}_{i}=\unicode[STIX]{x1D702}^{m_{i}}\in (\widehat{{\mathcal{R}}}/p\widehat{{\mathcal{R}}})^{\times }\subset R^{\times }$ for any $i\in \mathbb{Z}/d\mathbb{Z}$ , where $m_{i}=\sum _{j=0}^{d-1}n_{i+j}p^{d-j}$ . We see that $x_{i}-1$ is contained in $I_{+}\widehat{{\mathcal{R}}}$ . In the rest of this proof, to avoid confusions, we denote the image of $x\in \mathfrak{M}(\mathfrak{n})$ in $\widehat{{\mathcal{R}}}_{s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M}(\mathfrak{n})\subset R\otimes _{\unicode[STIX]{x1D711},k[\![u]\!]}\mathfrak{M}(\mathfrak{n})$ by $1\otimes x$ . Now we define a ${\hat{G}}_{s}$ -action on $\widehat{{\mathcal{R}}}_{s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M}(\mathfrak{n})$ by $\unicode[STIX]{x1D70F}^{p^{s}}(1\otimes e_{i}):=\bar{x}_{i}^{p^{s}}(1\otimes e_{i})$ for the basis $\{e_{i}\}_{i\in \mathbb{Z}/d\mathbb{Z}}$ of $\mathfrak{M}(\mathfrak{n})$ as in Definition 4.10. We claim that $g\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}g$ on $\widehat{{\mathcal{R}}}_{s}\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M}(\mathfrak{n})$ for any $g\in G_{s}$ . For this, it suffices to check that the equality $\unicode[STIX]{x1D70F}^{p^{s}}\unicode[STIX]{x1D711}(1\otimes e_{i})=\unicode[STIX]{x1D711}\unicode[STIX]{x1D70F}^{p^{s}}(1\otimes e_{i})$ holds for any $i$ . Note that we have

$$\begin{eqnarray}\unicode[STIX]{x1D70F}^{p^{s}}\unicode[STIX]{x1D711}(1\otimes e_{i})=\unicode[STIX]{x1D70F}^{p^{s}}(u^{pn_{i}}(1\otimes e_{i+1}))=\bar{x}_{i+1}^{p^{s}}(\text{}\underline{\unicode[STIX]{x1D700}}^{p^{s}}u)^{pn_{i}}(1\otimes e_{i+1})\end{eqnarray}$$

and

$$\begin{eqnarray}\unicode[STIX]{x1D711}\unicode[STIX]{x1D70F}^{p^{s}}(1\otimes e_{i})=\unicode[STIX]{x1D711}(\bar{x}_{i}^{p^{s}}(1\otimes e_{i}))=\bar{x}_{i}^{p^{s+1}}u^{pn_{i}}(1\otimes e_{i+1}).\end{eqnarray}$$

Hence it is enough to check $x_{i}^{p^{s+1}}=x_{i+1}^{p^{s}}[\text{}\underline{\unicode[STIX]{x1D700}}]^{p^{s+1}n_{i}}$ but we can show this equality without difficulty. In fact, we have equivalences

$$\begin{eqnarray}\displaystyle x_{i}^{p^{s+1}} & = & \displaystyle x_{i+1}^{p^{s}}[\text{}\underline{\unicode[STIX]{x1D700}}]^{p^{s+1}n_{i}}\Leftrightarrow \text{exp}\Bigl(-p^{s+1}m_{i}\frac{t}{p^{d}-1}\Bigr)\nonumber\\ \displaystyle & = & \displaystyle \text{exp}\Bigl(-p^{s}m_{i+1}\frac{t}{p^{d}-1}-p^{s+1}n_{i}t\Bigr)\Leftrightarrow pm_{i}\nonumber\\ \displaystyle & = & \displaystyle m_{i+1}+(p^{d+1}-p)n_{i}\nonumber\end{eqnarray}$$

and the last equality can be checked immediately by definition of $m_{i}$ .

By the claim above, we see that $\mathfrak{M}(\mathfrak{n})$ has a structure of an object of $\widetilde{\text{Mod}}_{/\mathfrak{S}_{\infty }}^{r,{\hat{G}}_{s}}$ via this ${\hat{G}}_{s}$ -action; we denote it by $\hat{\mathfrak{M}}(\mathfrak{n})$ . It suffices to prove that $\hat{\mathfrak{M}}(\mathfrak{n})$ is in fact an object of $\widetilde{\text{Mod}}_{/\mathfrak{S}_{\infty }}^{r,{\hat{G}}_{s},J}$ . Recall that $v_{R}$ is the valuation of $R$ normalized such that $v_{R}(\text{}\underline{\unicode[STIX]{x1D70B}})=1/e$ . Define $\tilde{\mathfrak{t}}=\mathfrak{t}~\text{mod}~pW(R)$ an element of $R$ . We denote by $v_{p}$ the usual $p$ -adic valuation normalized by $v_{p}(p)=1$ . Note that we have $v_{R}(\text{}\underline{\unicode[STIX]{x1D700}}-1)=p/(p-1)$ and $v_{R}(\tilde{\mathfrak{t}})=1/(p-1)$ (here, the latter equation follows from the relation $\unicode[STIX]{x1D711}(\mathfrak{t})=pE(0)^{-1}E(u)\mathfrak{t}$ ). Moreover, we have $v_{R}(\text{}\underline{\unicode[STIX]{x1D700}}^{m}-1)=v_{R}(\unicode[STIX]{x1D702}^{m}-1)=p^{v_{p}(m)+1}/(p-1)$ for any $m\in \mathbb{Z}_{p}$ by [Reference Gee, Liu and SavittGLS, Lemma 6.6(1)]. Thus we have

$$\begin{eqnarray}v_{R}(\bar{x}_{i}^{p^{s}}-1)=v_{R}(\unicode[STIX]{x1D702}^{p^{s}m_{i}}-1)=\frac{p^{s+v_{p}(m_{i})+1}}{p-1}\geqslant \frac{p^{s+2}}{p-1}.\end{eqnarray}$$

Since $p^{s+2}/(p-1)\geqslant c_{J}$ and the image of $J$ in $R$ is $\mathfrak{m}_{R}^{{\geqslant}c_{J}}$ , we obtain

$$\begin{eqnarray}\unicode[STIX]{x1D70F}^{p^{s}}(1\otimes e_{i})-(1\otimes e_{i})\in \mathfrak{m}_{R}^{{\geqslant}c_{J}}R\otimes _{\unicode[STIX]{x1D711},k[\![u]\!]}\mathfrak{M}(\mathfrak{n})\simeq JW(R)\otimes _{\unicode[STIX]{x1D711},\mathfrak{S}}\mathfrak{M}(\mathfrak{n}).\end{eqnarray}$$

Finally we have to show that $\unicode[STIX]{x1D70F}^{p^{s}}(1\otimes ae_{i})-(1\otimes ae_{i})\in \mathfrak{m}_{R}^{{\geqslant}c_{J}}R\otimes _{\unicode[STIX]{x1D711},k[\![u]\!]}\mathfrak{M}(\mathfrak{n})$ for any $a\in k[\![u]\!]$ . Since $\unicode[STIX]{x1D70F}^{p^{s}}(1\otimes ae_{i})-(1\otimes ae_{i})=\unicode[STIX]{x1D70F}^{p^{s}}(\unicode[STIX]{x1D711}(a))(\unicode[STIX]{x1D70F}^{p^{s}}(1\otimes e_{i})-(1\otimes e_{i}))+(\unicode[STIX]{x1D70F}^{p^{s}}(\unicode[STIX]{x1D711}(a))-\unicode[STIX]{x1D711}(a))(1\otimes e_{i})$ , it suffices to show $\unicode[STIX]{x1D70F}^{p^{s}}(\unicode[STIX]{x1D711}(a))-\unicode[STIX]{x1D711}(a)\in \mathfrak{m}_{R}^{{\geqslant}c_{J}}$ . Write $\unicode[STIX]{x1D711}(a)=\sum _{i\geqslant 0}a_{i}u^{pi}$ for some $a_{i}\in k$ . Then we have $\unicode[STIX]{x1D70F}^{p^{s}}(\unicode[STIX]{x1D711}(a))-\unicode[STIX]{x1D711}(a)=\sum _{i\geqslant 1}a_{i}(\text{}\underline{\unicode[STIX]{x1D700}}^{p^{s+1}i}-1)u^{pi}$ . Since we have

$$\begin{eqnarray}v_{R}((\text{}\underline{\unicode[STIX]{x1D700}}^{p^{s+1}i}-1)u^{pi})=p^{s+1}v_{R}(\text{}\underline{\unicode[STIX]{x1D700}}^{i}-1)+v_{R}(u^{pi})>\frac{p^{s+2}}{p-1}\geqslant c_{J}\end{eqnarray}$$

for any $i\geqslant 1$ , we have done.◻

Recall that $r_{0}=\text{max}\{r^{\prime }\in \mathbb{Z}_{{\geqslant}0};e(r^{\prime }-1)<p-1\}$ . Put $r_{1}:=\text{min}\{r,r_{0}\}$ .

Before the lemma below, we remark that any semisimple $\mathbb{F}_{p}$ -representation of $G_{K}$ is automatically tame.

Now we are ready to prove Theorem 4.9.