---
abstract: 'Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$ with perfect residue field. Let $(\pi_n)_{n\ge 0}$ be a system of $p$-power roots of a uniformizer $\pi=\pi_0$ of $K$ with $\pi^p_{n+1}=\pi_n$, and define $G_s$ (resp. $G_{\infty}$) the absolute Galois group of $K(\pi_s)$ (resp. $K_{\infty}:=\bigcup_{n\ge 0} K(\pi_n)$). In this paper, we study $G_s$-equivatiantness properties of $G_{\infty}$-equivariant homomorphisms between torsion (potentially) crystalline representations.'
author:
- 'Yoshiyasu Ozeki[^1]'
title: On Galois equivariance of homomorphisms between torsion potentially crystalline representations
---

Introduction
============

Let $p$ be a prime number and $r\ge 0 $ an integer. Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$ with perfect residue field and absolute ramification index $e$. Let $\pi=\pi_0$ be a uniformizer of $K$ and $\pi_n$ a $p^n$-th root of $\pi$ such that $\pi^p_{n+1}=\pi_n$ for all $n\ge 0$. For any integer $s\ge 0$, we put $K_{(s)}=K(\pi_s)$. We also put $K_{\infty}=\bigcup_{n\ge 0}K_{(n)}$. We denote by $G_K, G_s$ and $G_{\infty}$ absolute Galois groups of $K$, $K_{(s)}$ and $K_{\infty}$, respectively. By definition we have the following decreasing sequence of Galois groups: $$G_K=G_0\supset G_1\supset G_2\supset\cdots \supset G_{\infty}.$$ Since $K_{\infty}$ is a strict APF extension of $K$, the theory of fields of norm implies that $G_{\infty}$ is isomorphic to the absolute Galois group of some field of characteristic $p$. Therefore, representations of $G_{\infty}$ has easy interpretations via Fontaine’s étale ${\varphi}$-modules. Hence it seems natural to have the following question:

\[que1\] Let $T$ be a representation of $G_K$. For a “small” integer $s\ge 0$, can we reconstruct various information of the $G_s$-action on $T$ from that of the $G_{\infty}$-action?

Nowadays there is an interesting insight of Breuil; he showed that representations of $G_K$ arising from finite flat group schemes or $p$-divisible groups over the integer ring of $K$ is “determined” by its restriction to $G_{\infty}$. Moreover, for ${\mathbb}{Q}_p$-representations, Kisin proved the following theorem in [@Kis] (which was a conjecture of Breuil): the restriction functor from the category of crystalline ${\mathbb}{Q}_p$-representations of $G_K$ into the category of ${\mathbb}{Q}_p$-representations of $G_{\infty}$ is fully faithful.

In this paper, we give some partial answers to Question \[que1\] for [*torsion crystalline representations*]{}, moreover, [*torsion potentially crystalline representations*]{}. Our first main result is as follows. Let ${\mathrm}{Rep}^{r,{\mathrm}{ht},{\mathrm}{pcris}(s)}_{{\mathrm}{tor}}(G_K)$ be the category of torsion ${\mathbb}{Z}_p$-representations $T$ of $G_K$ which satisfy the following: there exist free ${\mathbb}{Z}_p$-representations $L$ and $L'$ of $G_K$, of height $\le r$, such that

-   $L|_{G_s}$ is a subrepresentation of $L'|_{G_s}$. Furthermore, $L|_{G_s}$ and $L'|_{G_s}$ are lattices in some crystalline ${\mathbb}{Q}_p$-representation of $G_s$ with Hodge-Tate weights in $[0,r]$;

-   $T|_{G_s} \simeq (L'|_{G_s})/(L|_{G_s})$.

\[Main1\] Suppose that $p$ is odd and $e(r-1)<p-1$. Let $T$ and $T'$ be objects of ${\mathrm}{Rep}^{r,{\mathrm}{ht},{\mathrm}{pcris}(s)}_{{\mathrm}{tor}}(G_K)$. Then any $G_{\infty}$-equivariant homomorphism $T\to T'$ is in fact $G_s$-equivariant.

We should remark that the condition $e(r-1)<p-1$ in the above does not depend on $s$. We put ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)={\mathrm}{Rep}^{r, {\mathrm}{ht}, {\mathrm}{pcris}(0)}_{{\mathrm}{tor}}(G_K)$. By definition, a torsion ${\mathbb}{Z}_p$-representation $T$ of $G_K$ is contained in this category if and only if it can be written as the quotient of lattices in some crystalline ${\mathbb}{Q}_p$-representation of $G_K$ with Hodge-Tate weights in $[0,r]$. We call the objects in ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ [*torsion crystalline representations with Hodge-Tate weights in $[0,r]$*]{}. In the case $r=1$, such representations are equivalent to finite flat representations. (Here, a torsion ${\mathbb}{Z}_p$-representation of $G_K$ is finite flat if it arises from the generic fiber of some $p$-power order finite flat commutative group scheme over the integer ring of $K$.) Combining Theorem \[Main1\] with results of [@Kim], [@La], [@Li3] (the case $p=2$) we obtain the following full faithfulness theorem for torsion crystalline representations.

\[FFTHMtorcris\] Suppose $e(r-1)<p-1$. Then the restriction functor ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$ is fully faithful.

Before this work, some results are already known. First, the full faithfulness theorem was proved by Breuil for $e=1$ and $r<p-1$ via the Fontaine-Laffaille theory ([@Br2], the proof of Théorèm 5.2). He also proved the theorem under the assumptions $p>2$ and $r\le 1$ as a consequence of a study of the category of finite flat group schemes ([@Br3 Theorem 3.4.3]). Later, his result was extended to the case $p=2$ in [@Kim], [@La], [@Li3] (proved independently). In particular, the full faithfulness theorem for $p=2$ is a consequence of their works. On the other hand, Abrashkin proved the full faithfulness in the case where $p>2, r<p$ and $K$ is a finite unramified extension of ${\mathbb}{Q}_p$ ([@Ab2 Section 8.3.3]). His proof is based on calculations of ramification bounds for torsion crystalline representations. In [@Oz2], a proof of Corollary \[FFTHMtorcris\] under the assumption $er<p-1$ is given via $({\varphi},\hat{G})$-modules (which was introduced by Tong Liu [@Li2] to classify lattices in semi-stable representations).

Our proof of Theorem \[Main1\] is similar to the proof for the main result of [@Oz2], but we need more careful considerations for $({\varphi},\hat{G})$-modules. Indeed we need special base change arguments to study some potential crystalline representations. In fact, we prove a full faithfulness theorem for torsion representations arising from certain classes of $({\varphi},\hat{G})$-modules (cf. Theorem \[FFTHM\]), which immediately gives our main theorem. In addition, our study gives a result as below which is the second main result of this paper (here, we define ${\mathrm}{log}_p(x):=-\infty$ for any real number $x\le 0$).

\[Main2\] Suppose that $p$ is odd and $s> n-1 + {\mathrm}{log}_p(r-(p-1)/e)$. Let $T$ and $T'$ be objects of ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ which are killed by $p^n$. Then any $G_{\infty}$-equivariant homomorphism $T\to T'$ is in fact $G_s$-equivariant.

For torsion semi-stable representations, a similar result was shown in Theorem 3 of [@CL2], which was a consequence of a study of ramification bounds. The bound appearing in their theorem was $n-1 + {\mathrm}{log}_p(nr)$. By applying our arguments given in this paper, we can obtain a generalization of their result; our refined condition is $n-1 + {\mathrm}{log}_pr$ (see Theorem \[Main3\]). Some other consequences of our study are described in subsection \[consequences\]. Motivated by the full faithfulness theorem (= Corollary \[FFTHMtorcris\]) and Theorem \[Main2\], we raise the following question.

Is any $G_{\infty}$-equivariant homomorphism in the category ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ in fact $G_s$-equivariant for $s>{\mathrm}{log}_p(r-(p-1)/e)$?

On the other hand, there exist counter examples of the full faithfulness theorem when we ignore the condition $e(r-1)< p-1$. Let ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_1)$ be the category of torsion ${\mathbb}{Z}_p$-representations of $G_1$.

\[nonfull\] Suppose that $K$ is a finite extension of ${\mathbb}{Q}_p$, and also suppose $e\mid (p-1)$ or $(p-1)\mid e$. If $e(r-1)\ge p-1$, the restriction functor ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)\to 
{\mathrm}{Rep}_{{\mathrm}{tor}}(G_1)$ is not full $($in particular, the restriction functor ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)\to 
{\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$ is not full$)$.

In particular, if $p=2$, then the full faithfulness never hold for any finite extension $K$ of ${\mathbb}{Q}_2$ and any $r\ge 2$. Theorem \[nonfull\] implies that the condition “$e(r-1)<p-1$” in Corollary \[FFTHMtorcris\] is the best possible for many finite extensions $K$ of ${\mathbb}{Q}_p$.

Now we describe the organization of this paper. In Section 2, we setup notations and summarize facts we need later. In Section 3, we define variant notions of $({\varphi},\hat{G})$-modules and give some basic properties. They are needed to study certain classes of potentially crystalline representations and restrictions of semi-stable representations. In Section 4, we study technical torsion $({\varphi},\hat{G})$-modules which are related with torsion (potentially) crystalline representations. The key result in this section is the full faithfulness result Theorem \[FFTHM\] on them, which allows us to prove our main results immediately. Finally, in Section 5, we calculate the smallest integer $r$ for a given torsion representation $T$ such that $T$ admits a crystalline lift with Hodge-Tate weights in $[0,r]$. We mainly study the rank two case. We use our full faithfulness theorem to assure the non-existence of crystalline lifts with small Hodge-Tate weights. Theorem \[nonfull\] is a consequence of studies of this section.

The author would like to thank Shin Hattori, Naoki Imai and Yuichiro Taguchi who gave him many valuable advice. This work was supported by JSPS KAKENHI Grant Number 25$\cdot$173.

[**Notation and convention:**]{} Throughout this paper, we fix a prime number $p$. Except Section 5, we always assume that [*$p$ is odd*]{}.

For any topological group $H$, we denote by ${\mathrm}{Rep}_{{\mathrm}{tor}}(H)$ (resp. ${\mathrm}{Rep}_{{\mathbb}{Z}_p}(H)$, resp. ${\mathrm}{Rep}_{{\mathbb}{Q}_p}(H)$) the category of torsion ${\mathbb}{Z}_p$-representations of $H$ (resp. the category of free ${\mathbb}{Z}_p$-representations of $H$, resp. the category of ${\mathbb}{Q}_p$-representations of $H$). All ${\mathbb}{Z}_p$-representations (resp. ${\mathbb}{Q}_p$-representations) in this paper are always assumed to be finitely generated over ${\mathbb}{Z}_p$ (resp. ${\mathbb}{Q}_p$).

For any field $F$, we denote by $G_F$ the absolute Galois group of $F$ (for a fixed separable closure of $F$).

Preliminaries
=============

In this section, we recall definitions and basic properties for Kisin modules and $({\varphi},\hat{G})$-modules. Throughout Section 2, 3 and 4, we always assume that $p$ is an odd prime.

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 $\pi$ of $K$. Let $E(u)$ be the minimal polynomial of $\pi$ over $K_0$. Then $E(u)$ is an Eisenstein polynomial. For any integer $n\ge 0$, we fix a system $(\pi_n)_{n\ge 0}$ of a $p^n$-th root of $\pi$ in $\overline{K}$ such that $\pi^p_{n+1}=\pi_n$. 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 $s\ge 0$, we write $\underline{\pi_s}:=(\pi_{s+n})_{n\ge 0}\in R$ and $\underline{\pi}:=\underline{\pi_0}\in R$. Note that we have $\underline{\pi_s}^{p^s}=\underline{\pi}$.

Let $L$ be the completion of an unramified algebraic extension of $K$ with residue field $k_L$. Then $\pi_s$ is a uniformizer of $L_{(s)}:=L(\pi_s)$ and $L_{(s)}$ is a totally ramified degree $ep^s$ extension of $L_0:=W(k_L)[1/p]$. We set $L_{\infty}:=\bigcup_{n\ge 0}L_{(n)}$. We put $G_{L,s}:=G_{L_{(s)}}={\mathrm}{Gal}(\overline{L}/L_{(s)})$ and $G_{L,\infty}:=G_{L_{\infty}}={\mathrm}{Gal}(\overline{L}/L_{\infty})$. 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 $\varphi$ 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 $u_s\mapsto [\underline{\pi_s}]$ (resp. $u\mapsto [\underline{\pi}]$), 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 $\pi_s$ 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 $\pi_s$ over $L_0$ is $E_s(u_s)$.

Let $S^{{\mathrm}{int}}_{L_0,s}$ (resp. $S^{{\mathrm}{int}}_{L_0})$) 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 exists a unique Frobenius map ${\varphi}\colon S^{{\mathrm}{int}}_{L_0,s}\to S^{{\mathrm}{int}}_{L_0,s}$ (resp. ${\varphi}\colon S^{{\mathrm}{int}}_{L_0}\to S^{{\mathrm}{int}}_{L_0}$) and monodromy $N\colon S^{{\mathrm}{int}}_{L_0,s}\to S^{{\mathrm}{int}}_{L_0,s}$ defined by $\varphi(u_s)=u_s^p$ (resp. $\varphi(u)=u^p$) and $N(u_s)=-u_s$ (resp. $N(u)=-u$). Put $S_{L_0,s}=S^{{\mathrm}{int}}_{L_0,s}[1/p]=L_0\otimes_{W(k_L)} S^{{\mathrm}{int}}_{L_0,s}$ (resp. $S_{L_0}=S^{{\mathrm}{int}}_{L_0}[1/p]=L_0\otimes_{W(k_L)} S^{{\mathrm}{int}}_{L_0}$). We equip $S^{{\mathrm}{int}}_{L_0,s}$ and $S_{L_0,s}$ (resp. $S^{{\mathrm}{int}}_{L_0}$ and $S_{L_0}$) with decreasing filtrations ${\mathrm}{Fil}^iS^{{\mathrm}{int}}_{L_0,s}$ and ${\mathrm}{Fil}^iS_{L_0,s}$ (resp. ${\mathrm}{Fil}^iS^{{\mathrm}{int}}_{L_0,s}$ and ${\mathrm}{Fil}^iS_{L_0,s}$) by the $p$-adic completion of the ideal generated by $E^j_s(u_s)/j!$ (resp. $E^j(u)/j!$) for all $j\ge 0$. The inclusion $W(k_L)[u_s]\hookrightarrow W(R)$ (resp. $W(k_L)[u]\hookrightarrow W(R)$) via the map $u_s\mapsto [\underline{\pi_s}]$ (resp. $u\mapsto [\underline{\pi}]$) induces ${\varphi}$-compatible inclusions ${\mathfrak{S}}_{L,s}\hookrightarrow S^{{\mathrm}{int}}_{L_0,s}\hookrightarrow A_{{\mathrm}{cris}}$ and $S_{L_0,s}\hookrightarrow B^+_{{\mathrm}{cris}}$ (resp. ${\mathfrak{S}}_L\hookrightarrow S^{{\mathrm}{int}}_{L_0}\hookrightarrow A_{{\mathrm}{cris}}$ and $S_{L_0}\hookrightarrow B^+_{{\mathrm}{cris}}$). By these inclusions, we often regard these rings as subrings of $B^+_{{\mathrm}{cris}}$. By identifying $u$ with $u_s^{p^s}$ as before, we regard $S^{{\mathrm}{int}}_{L_0}$ (resp. $S_{L_0}$) as a ${\varphi}$-stable (but not $N$-stable) subalgebra of $S^{{\mathrm}{int}}_{L_0,s}$ (resp. $S_{L_0,s}$). By definitions, we have ${\mathfrak{S}}_L={\mathfrak{S}}_{L,0},\ S^{{\mathrm}{int}}_{L_0,0}=S^{{\mathrm}{int}}_{L_0}$ and $S_{L_0,0}= S_{L_0}$.\
[**Convention:**]{} For simplicity, if $L=K$, then we often omit the subscript “$L$” from various notations (e.g. $G_{K_s}=G_s$, $G_{K_{\infty}}=G_{\infty}$, ${\mathfrak{S}}_K={\mathfrak{S}}, {\mathfrak{S}}_{K,s}={\mathfrak{S}}_s$).\

$\displaystyle \xymatrix{
& W(R) \ar@{-}[rr]
& 
& A_{{\mathrm}{cris}} \ar@{-}[rr]
& 
& B^+_{{\mathrm}{cris}}
\\
&
{\mathfrak{S}}_{L,s} \ar@{-}[rr] \ar@{-}[u]
& 
& S^{{\mathrm}{int}}_{L_0,s}\ar@{-}[rr] \ar@{-}[u]
& 
& S_{L_0,s} \ar@{-}[u]
\\
{\mathfrak{S}}_L \ar@{-}[rr] \ar@{-}[ru]
& \ar@{-}[u]
& S^{{\mathrm}{int}}_{L_0}\ar@{-}[rr] \ar@{-}[ru]
& \ar@{-}[u]
& S_{L_0} \ar@{-}[ru]
& 
\\
&
{\mathfrak{S}}_s \ar@{-}[r] \ar@{-}[u]
& \ar@{-}[r]
& S^{{\mathrm}{int}}_{K_0,s}\ar@{-}[r] \ar@{-}[u]
& \ar@{-}[r]
& S_{K_0,s} \ar@{-}[uu]
\\
{\mathfrak{S}}\ar@{-}[rr] \ar@{-}[uu] \ar@{-}[ru]
& 
& S^{{\mathrm}{int}}_{K_0}\ar@{-}[rr] \ar@{-}[uu] \ar@{-}[ru]
& 
& S_{K_0}  \ar@{-}[uu] \ar@{-}[ru]
&
}$

Kisin modules
-------------

Let $r, s\ge 0$ be integers. A [*${\varphi}$-module*]{} [*over ${\mathfrak{S}}_{L,s}$*]{} is an ${\mathfrak{S}}_{L,s}$-module ${\mathfrak{M}}$ equipped with a ${\varphi}$-semilinear map ${\varphi}\colon {\mathfrak{M}}\to {\mathfrak{M}}$. A morphism between two ${\varphi}$-modules $({\mathfrak{M}}_1,{\varphi}_1)$ and $({\mathfrak{M}}_2,{\varphi}_2)$ over ${\mathfrak{S}}_{L,s}$ is an ${\mathfrak{S}}_{L,s}$-linear map ${\mathfrak{M}}_1\to {\mathfrak{M}}_2$ compatible with ${\varphi}_1$ and ${\varphi}_2$. Denote by $'{\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}}$ the category of ${\varphi}$-modules $({\mathfrak{M}},{\varphi})$ over ${\mathfrak{S}}_{L,s}$ [*of height $\le r$*]{} in the sense that ${\mathfrak{M}}$ is of finite type over ${\mathfrak{S}}_{L,s}$ and the cokernel of $1\otimes {\varphi}\colon {\mathfrak{S}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}}{\mathfrak{M}}\to {\mathfrak{M}}$ is killed by $E_s(u_s)^r$.

Let ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}}$ be the full subcategory of $'{\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}}$ consisting of finite free ${\mathfrak{S}}_{L,s}$-modules. We call an object of ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}}$ a [*free Kisin module*]{} [*of height $\le r$ $($over ${\mathfrak{S}}_{L,s})$*]{}.

Let ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s,\infty}}$ be the full subcategory of $'{\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}}$ 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 ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s,\infty}}$ a [*torsion Kisin module of height $\le r$ $($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 $$\begin{aligned}
T_{{\mathfrak{S}}_{L,s}}({\mathfrak{M}}):=
\left\{
\begin{array}{ll}
{\mathrm}{Hom}_{{\mathfrak{S}}_{L,s},{\varphi}}({\mathfrak{M}},W(R))\hspace{21mm} {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm free},  
\cr
{\mathrm}{Hom}_{{\mathfrak{S}}_{L,s},{\varphi}}({\mathfrak{M}},{\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p} W(R))\quad {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm torsion}. 
\end{array}
\right.\end{aligned}$$ Here a $G_{L,\infty}$-action on $T_{{\mathfrak{S}}_{L,s}}({\mathfrak{M}})$ is given by $(\sigma.g)(x)=\sigma(g(x))$ for $\sigma\in G_{L,\infty}, g\in T_{{\mathfrak{S}}}({\mathfrak{M}}), x\in {\mathfrak{M}}$.\
[**Convention:**]{} For simplicity, if $L=K$, then we often omit the subscript “$L$” from various notations (e.g. ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{K,s,\infty}}={\mathrm}{Mod}^r_{/{\mathfrak{S}}_{s,\infty}}$, $T_{{\mathfrak{S}}_{K,s}}=T_{{\mathfrak{S}}_s}$ ). Also, if $s=0$, we often omit the subscript “$s$” from various notations (e.g. ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,0,\infty}}={\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,\infty}}$, $T_{{\mathfrak{S}}_{L,0}}=T_{{\mathfrak{S}}_L}$, ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{K,0,\infty}}={\mathrm}{Mod}^r_{/{\mathfrak{S}}_{\infty}}$, $T_{{\mathfrak{S}}_{K,0}}=T_{{\mathfrak{S}}}$ ).\

\[Kisinfunctor\] $(1)$ [([@Kis Corollary 2.1.4 and Proposition 2.1.12])]{}  The functor $T_{{\mathfrak{S}}_{L,s}}\colon {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}}\to 
{\mathrm}{Rep}_{{\mathbb}{Z}_p}(G_{\infty})$ is exact and fully faithful.

$(2)$ [([@CL1 Corollary 2.1.6, 3.3.10 and 3.3.15])]{}  The functor $T_{{\mathfrak{S}}_{L,s}}\colon {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s,\infty}}\to 
{\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$ is exact and faithful. Furthermore, it is full if $er<p-1$.

$({\varphi},\hat{G})$-modules {#Liumodule:section}
-----------------------------

The notion of $({\varphi},\hat{G})$-modules are introduced by Tong Liu in [@Li2] to classify lattices in semi-stable 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 $H_L:={\mathrm}{Gal}(\hat{L}/L_{\infty})$, $H_{L,\infty}:={\mathrm}{Gal}(\overline{K}/\hat{L})$ $G_{L,p^{\infty}}:={\mathrm}{Gal}(\hat{L}/L_{p^{\infty}})$ and $\hat{G}_L:={\mathrm}{Gal}(\hat{L}/L)$. Furthermore, putting $L_{(s),p^{\infty}}=L_{(s)}L_{p^{\infty}}$, we define $\hat{G}_{L,s}={\mathrm}{Gal}(\hat{L}/L_{(s)})$ and $G_{L,s,p^{\infty}}:={\mathrm}{Gal}(\hat{L}/L_{(s), p^{\infty}})$.

$$\xymatrix{
& & & & & \bar{K} \\
& & & & & \hat{L} \ar@{-}[u] \ar@{-}[u] \ar@/_1pc/@{-}[u]_{H_{L,\infty}}
\\
& & & L_{p^{\infty}} 
\ar@/^1pc/@{-}[rru] ^{G_{L,p^{\infty}}}
\ar@{-}[rru] 
& & & & \\
& & & & & 
L_{\infty} \ar@/_1pc/@{-}[uu] _{H_L}
\ar@{-}[uu]
\ar@/_4pc/@{-}[uuu] _{G_{L,\infty}}
& & & & \\  
& & & L \ar@{-}[rru]  
\ar@/^1pc/@{-}[rruuu] ^{\hat{G}_L}
\ar@{-}[rruuu] 
\ar@/^6pc/@{-}[rruuuu] ^{G_L}
\ar@{-}[uu] 
& & & & \\ 
}$$

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

\[easylemma\] A natural map $G_{L,s,p^{\infty}}\to G_{K,s,p^{\infty}}$ defined by $g\mapsto g|_{\hat{K}}$ is bijective.

By replacing $L_s$ with $L$, we may assume $s=0$. It suffices to prove $\hat{K}\cap L_{p^{\infty}}=K_{p^{\infty}}$. Since $G_{K,p^{\infty}}$ is isomorphic to ${\mathbb}{Z}_p$, we know that any finite subextension of $\hat{K}/K_{p^{\infty}}$ is of the form $K_{(s),p^{\infty}}$ for some $s\ge 0$. Assume that we have $\hat{K}\cap L_{p^{\infty}}\not=K_{p^{\infty}}$. Then we have $K_{(1)}\subset \hat{K}\cap L_{p^{\infty}}\subset L_{p^{\infty}}$. Thus $\pi_1$ is contained in $L_{p^{\infty}}\cap L_{\infty}=L$. However, since $L$ is unramified over $K$, this contradicts the fact that $\pi$ is a uniformizer of $L$.

We fix a topological generator $\tau$ of $G_{K,p^{\infty}}$. We also denote by $\tau$ the pre-image of $\tau\in G_{K,p^{\infty}}$ for the bijection $G_{L,p^{\infty}}\simeq G_{K,p^{\infty}}$ of the above lemma. Note that $\tau^{p^s}$ is a topological generator of $G_{L,s,p^{\infty}}$.

For any $g\in G_K$, we put $\underline{{\varepsilon}}(g)=g(\underline{\pi})/\underline{\pi}\in R$, and define $\underline{{\varepsilon}}:=\underline{{\varepsilon}}(\tilde{\tau})$. Here, $\tilde{\tau}\in G_K$ is any lift of $\tau\in \hat{G}_K$ and then $\underline{{\varepsilon}}(\tilde{\tau})$ is independent of the choice of the lift of $\tau$. With these notation, we also note that we have $g(u)=[\underline{{\varepsilon}}(g)]u$ (recall that ${\mathfrak{S}}$ is embedded in $W(R)$). An easy computation shows that $\tau(\underline{\pi})/\underline{\pi}=\tau^{p^s}(\underline{\pi_s})/\underline{\pi_s}=\underline{{\varepsilon}}$. Therefore, we have $\tau(u)/u=\tau^{p^s}(u_s)/u_s=[\underline{{\varepsilon}}]$.

We put $t=-{\mathrm}{log}([\underline{{\varepsilon}}])\in A_{{\mathrm}{cris}}$. Denote by $\nu\colon W(R)\to W(\overline{k})$ a unique lift of the projection $R\to \overline{k}$, which extends to a map $\nu \colon B^+_{{\mathrm}{cris}}\to W(\overline{k})[1/p]$. For any subring $A\subset B^+_{{\mathrm}{cris}}$, we put $I_+A={\mathrm}{Ker}(\nu\ {\mathrm}{on}\  B^+_{{\mathrm}{cris}})\cap A$. For any integer $n\ge 0$, let $t^{\{n\}}:=t^{r(n)}\gamma_{\tilde{q}(n)}(\frac{t^{p-1}}{p})$ where $n=(p-1)\tilde{q}(n)+r(n)$ with $\tilde{q}(n)\ge 0,\ 0\le r(n) <p-1$ and $\gamma_i(x)=\frac{x^i}{i!}$ is the standard divided power. We define a subring ${\mathcal}{R}_{L_0,s}$ (resp. ${\mathcal}{R}_{L_0}$) of $B^+_{{\mathrm}{cris}}$ as below: $${\mathcal}{R}_{L_0,s}:=\{\sum^{\infty}_{i=0} f_it^{\{i\}}\mid f_i\in S_{L_0,s}\
{\mathrm}{and}\ f_i\to 0\ {\mathrm}{as}\ i\to \infty\}$$ $$({\rm resp}.\quad  {\mathcal}{R}_{L_0}:=\{\sum^{\infty}_{i=0} f_it^{\{i\}}\mid f_i\in S_{L_0}\
{\mathrm}{and}\ f_i\to 0\ {\mathrm}{as}\ i\to \infty\}).$$

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 [@Li2] shows that ${\widehat}{{\mathcal}{R}}_{L,s}$ $($resp. ${\mathcal}{R}_{L_0,s})$ is a ${\varphi}$-stable ${\mathfrak{S}}_{L,s}$-subalgebra of $W(R)$ $($resp. $B^+_{{\mathrm}{cris}})$, and $\nu$ induces ${\mathcal}{R}_{L_0,s}/I_+{\mathcal}{R}_{L_0,s}\simeq L_0$ and ${\widehat}{{\mathcal}{R}}_{L,s}/I_{+,L,s}\simeq S^{{\mathrm}{int}}_{L_0,s}/I_+S^{{\mathrm}{int}}_{L_0,s}
\simeq {\mathfrak{S}}_{L,s}/I_+{\mathfrak{S}}_{L,s}\simeq W(k_L)$. 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 ${\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}$ with a Frobenius by ${\varphi}_{{\widehat}{{\mathcal}{R}}_{L,s}}\otimes {\varphi}_{{\mathfrak{M}}}$. It is known that the natural map $
{\mathfrak{M}}\rightarrow {\widehat}{{\mathcal}{R}}_{L,s}\otimes_{{\varphi}, {\mathfrak{S}}_{L,s}} {\mathfrak{M}}$ given by $x\mapsto 1\otimes x$ is an injection (cf. [@Oz1 Corollary 2.12]). By this injection, we regard ${\mathfrak{M}}$ as a ${\varphi}({\mathfrak{S}}_{L,s})$-stable submodule of ${\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}$.

\[Liumod\] A [*free*]{} (resp. [*torsion*]{}) [*$({\varphi}, \hat{G}_{L,s})$-module of height*]{} $\le r$ over ${\mathfrak{S}}_{L,s}$ is a triple $\hat{{\mathfrak{M}}}=({\mathfrak{M}}, {\varphi}_{{\mathfrak{M}}}, \hat{G}_{L,s})$ where

1.  $({\mathfrak{M}}, {\varphi}_{{\mathfrak{M}}})$ is a free (resp. torsion) Kisin module of height $\le r$ over ${\mathfrak{S}}_{L,s}$,

2.  $\hat{G}_{L,s}$ is an ${\widehat}{{\mathcal}{R}}_{L,s}$-semilinear $\hat{G}_{L,s}$-action on ${\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi}, {\mathfrak{S}}_{L,s}} {\mathfrak{M}}$ which induces a continuous $G_{L,s}$-action on $W({\mathrm}{Fr}R)\otimes_{{\varphi}, {\mathfrak{S}}_{L,s}} {\mathfrak{M}}$ for the weak topology[^2],

3.  the $\hat{G}_{L,s}$-action commutes with ${\varphi}_{{\widehat}{{\mathcal}{R}}_{L,s}}\otimes {\varphi}_{{\mathfrak{M}}}$,

4.  ${\mathfrak{M}}\subset ({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}})^{H_L}$,

5.  $\hat{G}_{L,s}$ acts on the $W(k_L)$-module $({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}})/
                I_{+,L,s}({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}})$ trivially.

A morphism between two $({\varphi}, \hat{G}_{L,s})$-modules $\hat{{\mathfrak{M}}}_1=({\mathfrak{M}}_1, {\varphi}_1, \hat{G})$ and $\hat{{\mathfrak{M}}}_2=({\mathfrak{M}}_2, {\varphi}_2, \hat{G})$ is a morphism $f\colon {\mathfrak{M}}_1\to {\mathfrak{M}}_2$ of ${\varphi}$-modules over ${\mathfrak{S}}_{L,s}$ such that ${\widehat}{{\mathcal}{R}}_{L,s}\otimes f\colon {\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}_1\to 
{\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}_2$ is $\hat{G}_{L,s}$-equivariant. We denote by ${\mathrm}{Mod}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s}}$ (resp. ${\mathrm}{Mod}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s,\infty}}$) the category of free (resp. torsion) $({\varphi}, \hat{G}_{L,s})$-modules of height $\le r$ over ${\mathfrak{S}}_{L,s}$. We often regard ${\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}$ as a $G_{L,s}$-module via the projection $G_{L,s}\twoheadrightarrow \hat{G}_{L,s}$.

For any free or torsion $({\varphi}, \hat{G}_{L,s})$-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 $$\begin{aligned}
\hat{T}_{L,s}(\hat{{\mathfrak{M}}})=
\left\{
\begin{array}{ll}
{\mathrm}{Hom}_{{\widehat}{{\mathcal}{R}}_{L,s},{\varphi}}
({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}, W(R))
\hspace{21mm} {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm free},  
\cr
{\mathrm}{Hom}_{{\widehat}{{\mathcal}{R}}_{L,s},{\varphi}}
({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}, {\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p}W(R))
\hspace{3.5mm} {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm torsion}.  
\end{array}
\right.\end{aligned}$$ Here, $G_{L,s}$ acts on $\hat{T}_{L,s}(\hat{{\mathfrak{M}}})$ by $(\sigma.f)(x)=\sigma(f(\sigma^{-1}(x)))$ for $\sigma\in G_{L,s},\ f\in \hat{T}_{L,s}(\hat{{\mathfrak{M}}}),\ 
x\in {\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}}{\mathfrak{M}}$.\
Then, there exists a natural $G_{L,\infty}$-equivariant map $$\theta_{L,s}\colon T_{{\mathfrak{S}}_{L,s}}({\mathfrak{M}})\to \hat{T}_{L,s}(\hat{{\mathfrak{M}}})$$ defined by $\theta(f)(a\otimes x)=a{\varphi}(f(x))$ for $f\in T_{{\mathfrak{S}}_{L,s}}({\mathfrak{M}}),\ a\in {\widehat}{{\mathcal}{R}}_{L,s}, x\in {\mathfrak{M}}$. We have

The map $\theta_{L,s}$ is an isomorphism of ${\mathbb}{Z}_p[G_{L,\infty}]$-modules.

[**Convention:**]{} For simplicity, if $L=K$, then we often omit the subscript “$L$” from various notations (e.g. “a $({\varphi}, \hat{G}_{K,s})$-module” = “a $({\varphi}, \hat{G}_s)$-module”, ${\mathrm}{Mod}^{r,\hat{G}_{K,s}}_{/{\mathfrak{S}}_{K,s}}
       ={\mathrm}{Mod}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}$, ${\mathrm}{Mod}^{r,\hat{G}_{K,s}}_{/{\mathfrak{S}}_{K,s,\infty}}
       ={\mathrm}{Mod}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{s,\infty}}$, $\hat{T}_{K,s}=\hat{T}_s$, $\theta_{K,s}=\theta_s$). Furthermore, if $s=0$, we often omit the subscript “$s$” from various notations (e.g. ${\mathrm}{Mod}^{r,\hat{G}_{L,0}}_{/{\mathfrak{S}}_{L,0}}
       ={\mathrm}{Mod}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}$, ${\mathrm}{Mod}^{r,\hat{G}_{L,0}}_{/{\mathfrak{S}}_{L,0,\infty}}
       ={\mathrm}{Mod}^{r,\hat{G}_L}_{/{\mathfrak{S}}_{L,\infty}}$, $\hat{T}_{L,0}=\hat{T}_L$, ${\mathrm}{Mod}^{r,\hat{G}_{K,0}}_{/{\mathfrak{S}}_{K,0}}
       ={\mathrm}{Mod}^{r,\hat{G}}_{/{\mathfrak{S}}}$, “a $({\varphi}, \hat{G}_{K,0})$-module” = “a $({\varphi}, \hat{G})$-module”, $\hat{T}_{K,0}=\hat{T}$, $\theta_{K,0}=\theta$).\
Let ${\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Q}_p}(G_{L,s})$ (resp. ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathbb}{Q}_p}(G_{L,s})$, resp. ${\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_{L,s})$, resp. ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathbb}{Z}_p}(G_{L,s})$) be the categories of semi-stable ${\mathbb}{Q}_p$-representations of $G_{L,s}$ with Hodge-Tate weights in $[0,r]$ (resp. the categories of crystalline ${\mathbb}{Q}_p$-representations of $G_{L,s}$ with Hodge-Tate weights in $[0,r]$, resp. the categories of lattices in semi-stable ${\mathbb}{Q}_p$-representations of $G_{L,s}$ with Hodge-Tate weights in $[0,r]$, resp. the categories 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 ${\varphi}({\mathfrak{t}})=pE(0)^{-1}E(u){\mathfrak{t}}$. Such ${\mathfrak{t}}$ is unique up to units of ${\mathbb}{Z}_p$ (cf. [@Li2 Example 2.3.5]). Now we define the full subcategory ${\mathrm}{Mod}^{r,\hat{G},{\mathrm}{cris}}_{/{\mathfrak{S}}}$ (resp. ${\mathrm}{Mod}^{r,\hat{G},{\mathrm}{cris}}_{/{\mathfrak{S}}_{\infty}}$) of ${\mathrm}{Mod}^{r,\hat{G}}_{/{\mathfrak{S}}}$ (resp. ${\mathrm}{Mod}^{r,\hat{G}}_{/{\mathfrak{S}}_{\infty}}$) consisting of objects $\hat{{\mathfrak{M}}}$ which satisfy the following condition; $
\tau(x)-x\in u^p{\varphi}({\mathfrak{t}})(W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}})
$ for any $x\in {\mathfrak{M}}$.

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

\[Thm1\] $(1)$ [([@Li2 Theorem 2.3.1 (2)])]{}The functor $\hat{T}$ induces an anti-equivalence of categories between ${\mathrm}{Mod}^{r,\hat{G}}_{/{\mathfrak{S}}}$ and ${\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K)$.

$(2)$ [([@GLS Proposition 5.9], [@Oz2 Theorem 19])]{}  The functor $\hat{T}$ induces an anti-equivalence of categories between ${\mathrm}{Mod}^{r,\hat{G},{\mathrm}{cris}}_{/{\mathfrak{S}}}$ and ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathbb}{Z}_p}(G_K)$.

$(3)$ [([@Oz1 Corollary 2.8 and 5.34])]{}The functor $\hat{T}_{L,s}\colon {\mathrm}{Mod}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s,\infty}}
\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_{L,s})$ is exact and faithful. Furthermore, it is full if $er<p-1$.

$({\varphi}, \hat{G})$-modules, Breuil modules and filtered $({\varphi},N)$-modules {#relations}
-----------------------------------------------------------------------------------

We recall some relations between Breuil modules and $({\varphi},\hat{G})$-modules. Here we give a rough sketch only. For more precise information, see [@Br1 Section 6], [@Li1 Section 5] and the proof of [@Li2 Theorem 2.3.1 (2)].

Let $\hat{{\mathfrak{M}}}$ be a free $({\varphi},\hat{G}_{L,s})$-module over ${\mathfrak{S}}_{L,s}$. If we put ${\mathcal}{D}:=S_{L_0,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}$, then ${\mathcal}{D}$ has a structure of a Breuil module over $S_{L_0,s}$ which corresponds to the semi-stable 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 [@Br1]). Thus ${\mathcal}{D}$ is equipped with a Frobenius ${\varphi}_{{\mathcal}{D}}(={\varphi}_{S_{L_0,s}}\otimes {\varphi}_{{\mathfrak{M}}})$, a decreasing filtration $({\mathrm}{Fil}^i{\mathcal}{D})_{i\ge 0}$ of $S_{L_0,s}$-submodules of ${\mathcal}{D}$ and a $L_0$-linear monodromy operator $N\colon {\mathcal}{D}\to {\mathcal}{D}$ which satisfy certain properties (for example, Griffiths transversality).

Putting $D={\mathcal}{D}/I_+S_{L_0,s}{\mathcal}{D}$, we can associate a filtered $({\varphi},N)$-module over $L_{(s)}$ as following: ${\varphi}_{D}:={\varphi}_{{\mathcal}{D}}\ {\mathrm}{mod}\ I_+S_{L_0,s}{\mathcal}{D}$, $N_D:= N_{{\mathcal}{D}}\ {\mathrm}{mod}\ I_+S_{L_0,s}{\mathcal}{D}$ and ${\mathrm}{Fil}^iD_{L_{(s)}}:=f_{\pi_s}(Fil^i({\mathcal}{D}))$. Here, $f_{\pi_s}\colon {\mathcal}{D}\to D_{L_{(s)}}$ is the projection defined by ${\mathcal}{D}\twoheadrightarrow {\mathcal}{D}/{\mathrm}{Fil}^1S_{L_0,s}{\mathcal}{D}\simeq D_{L_{(s)}}$. Proposition 6.2.1.1 of [@Br1] implies that there exists a unique ${\varphi}$-compatible section $s\colon D\hookrightarrow {\mathcal}{D}$ 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 $N_{{\mathcal}{D}}=N_{S_{L_0,s}}\otimes {\mathrm}{Id}_D + {\mathrm}{Id}_{S_{L_0,s}}\otimes N_D$ under the identification ${\mathcal}{D}=S_{L_0,s} \otimes_{L_{(s)}} D$.

The $G_{L,s}$-action on ${\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}}{\mathfrak{M}}$ extends to $B^+_{{\mathrm}{cris}}\otimes_{{\widehat{\mathcal{R}}}_{L,s}} ({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}}{\mathfrak{M}})
\simeq B^+_{{\mathrm}{cris}}\otimes_{S_{L_0,s}}{\mathcal}{D}$. This action is in fact explicitly written as follows: $$\label{explicit}
g(a\otimes x)=\sum^{\infty}_{i=0}g(a)\gamma_i
(-{\mathrm}{log}(\frac{g[\underline{\pi_s}]}{[\underline{\pi_s}]}))\otimes N^i_{{\mathcal}{D}}(x)\quad
{\rm for}\ g\in G_{L,s}, a\in B^+_{{\mathrm}{cris}}, x\in {\mathcal}{D}.$$ By this explicit formula, we can obtain an easy relation between $N_{{\mathcal}{D}}$ and $\tau^{p^s}$-action on $\hat{{\mathfrak{M}}}$ as follows: first we recall that $t=-{\mathrm}{log}(\tau([\underline{\pi}])/[\underline{\pi}])
  =-{\mathrm}{log}(\tau^{p^s}([\underline{\pi_s}])/[\underline{\pi_s}])$. By the formula, for any $n\ge 0$ and $x\in {\mathcal}{D}$, an induction on $n$ shows that we have $$(\tau^{p^s}-1)^n(x)=\sum^{\infty}_{m=n}(\sum_{i_1+\cdots i_n=m, i_j\ge 0}\frac{m!}{i_1!\cdots i_n!})
\gamma_m(t)\otimes N^m_{{\mathcal}{D}}(x)
\in B^{+}_{{\mathrm}{cris}}\otimes_{S_{L_0,s}} {\mathcal}{D}$$ and in particular we see $\frac{(\tau^{p^s}-1)^n}{n}(x)\to 0$ $p$-adically as $n\to \infty$. Hence we can define $${\mathrm}{log}(\tau^{p^s})(x):=
\sum^{\infty}_{n=1}(-1)^{n-1}\frac{(\tau^{p^s}-1)^n}{n}(x) \in B^{+}_{{\mathrm}{cris}}\otimes_{S_{L_0,s}} {\mathcal}{D}.$$ It is not difficult to check the equation $$\label{eq1}
{\mathrm}{log}(\tau^{p^s})(x)=t\otimes N_{{\mathcal}{D}}(x).$$

Base changes for Kisin modules
------------------------------

Let ${\mathfrak{M}}$ be a free or torsion Kisin module of height $\le r$ 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 ${\varphi}={\varphi}_{{\mathfrak{S}}_{L,s}}\otimes {\varphi}_{{\mathfrak{M}}}$ (resp. ${\varphi}={\varphi}_{{\mathfrak{S}}_L}\otimes {\varphi}_{{\mathfrak{M}}}$). 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 $\le r$ over ${\mathfrak{S}}_{L,s}$ (resp. over ${\mathfrak{S}}_L$) (here we recall that $E_s(u_s)=E(u^{p^s}_s)=E(u)$). Hence we obtained natural functors $${\mathrm}{Mod}^r_{/{\mathfrak{S}}_L}\to {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}} \quad  {\rm and}\quad 
{\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,\infty}}\to {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s,\infty}}$$ $${\rm (resp.}\quad 
{\mathrm}{Mod}^r_{/{\mathfrak{S}}}\to {\mathrm}{Mod}^r_{/{\mathfrak{S}}_L} \quad  {\rm and}\quad 
{\mathrm}{Mod}^r_{/{\mathfrak{S}}_{\infty}}\to {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,\infty}}).$$ 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 \[Kisinfunctor\] (1) that the following holds:

\[basechange1:Kisin\] The functor ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_L}\to {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}}$ is fully faithful.

Base changes for $({\varphi},\hat{G})$-modules
----------------------------------------------

Let $\hat{{\mathfrak{M}}}$ be a free or torsion $({\varphi},\hat{G}_L)$-module (resp. $({\varphi},\hat{G})$-module) of height $\le r$ over ${\mathfrak{S}}_L$ (resp. over ${\mathfrak{S}}$). The $G_{L,s}$ action on ${\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}$ (resp. the $G_L$ action on ${\widehat{\mathcal{R}}}\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$) extends to ${\widehat{\mathcal{R}}}_{L,s}\otimes_{{\widehat{\mathcal{R}}}_L}({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})
\simeq {\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}_{L,s}$ (resp.  ${\widehat{\mathcal{R}}}_L\otimes_{{\widehat{\mathcal{R}}}}({\widehat{\mathcal{R}}}\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}})
\simeq {\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}_L$), 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 $({\varphi},\hat{G}_{L,s})$-module (resp. $({\varphi},\hat{G}_L)$-module). Hence we obtained natural functors $${\mathrm}{Mod}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}\to {\mathrm}{Mod}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s}} \quad  {\rm and}\quad 
{\mathrm}{Mod}^{r,\hat{G}_L}_{/{\mathfrak{S}}_{L,\infty}}\to {\mathrm}{Mod}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s,\infty}}$$ $${\rm (resp.}\quad 
{\mathrm}{Mod}^{r,\hat{G}}_{/{\mathfrak{S}}}\to {\mathrm}{Mod}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L} \quad  {\rm and}\quad 
{\mathrm}{Mod}^{r,\hat{G}}_{/{\mathfrak{S}}_{\infty}}\to {\mathrm}{Mod}^{r,\hat{G}_L}_{/{\mathfrak{S}}_{L,\infty}}).$$ 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 \[basechange1:Kisin\], we can prove the following.

The functor ${\mathrm}{Mod}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}\to {\mathrm}{Mod}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s}} $ is fully faithful.

The proposition immediately follows from the full faithfulness property of Theorem \[Thm1\] (1) and the lemma below.

\[totst\] Let $K'$ is a finite totally ramified extension of $K$. Then the restriction functor from the category of semi-stable ${\mathbb}{Q}_p$-representations of $G_K$ into the category of semi-stable ${\mathbb}{Q}_p$-representations of $G_{K'}$ is fully faithful.

Let $V$ and $V'$ be semi-stable ${\mathbb}{Q}_p$-representations of $G_K$ and let $f\colon V\to V'$ be a $G_{K'}$-equivariant homomorphism. Considering the morphism of filtered $({\varphi}, N)$-modules over $K'$ corresponding to $f$, we can check without difficulty that $f$ is in fact a morphism of filtered $({\varphi}, N)$-modules over $K$. This is because $K'$ is totally ramified over $K_0$ as same as $K$. This gives the desired result.

Variants of free $({\varphi},\hat{G})$-modules
==============================================

In this section, we define some variant notions of $({\varphi},\hat{G})$-modules. We continue to use same notation as in the previous section. In particular, $p$ is odd.

Definitions {#vardef}
-----------

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

\[varLiumod\] We define the category ${{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}$ (resp. ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}$) as follows. An object is a triple $\hat{{\mathfrak{M}}}=({\mathfrak{M}}, {\varphi}_{{\mathfrak{M}}}, \hat{G}_{L,s})$ where

1.  $({\mathfrak{M}}, {\varphi}_{{\mathfrak{M}}})$ is a free Kisin module of height $\le r$ over ${\mathfrak{S}}_L$,

2.  $\hat{G}_{L,s}$ is an ${\widehat}{{\mathcal}{R}}_L$-semilinear $\hat{G}_{L,s}$-action on ${\widehat{\mathcal{R}}}_L\otimes_{{\varphi}, {\mathfrak{S}}_L} {\mathfrak{M}}$ (resp. an ${\widehat}{{\mathcal}{R}}_{L,s}$-semilinear $\hat{G}_{L,s}$-action on ${\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi}, {\mathfrak{S}}_L} {\mathfrak{M}}$) which induces a continuous $G_{L,s}$-action on $W({\mathrm}{Fr}R)\otimes_{{\varphi}, {\mathfrak{S}}_L} {\mathfrak{M}}$ for the weak topology,

3.  the $\hat{G}_{L,s}$-action commutes with ${\varphi}_{{\widehat}{{\mathcal}{R}}_L}\otimes {\varphi}_{{\mathfrak{M}}}$ (resp. ${\varphi}_{{\widehat}{{\mathcal}{R}}_{L,s}}\otimes {\varphi}_{{\mathfrak{M}}}$),

4.  ${\mathfrak{M}}\subset ({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})^{H_L}$ (resp. ${\mathfrak{M}}\subset ({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})^{H_L}$),

5.  $\hat{G}_{L,s}$ acts on the $W(k_L)$-module $({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})/
                I_{+,L}({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})$ (resp. $({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})/
                I_{+,L,s}({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})$) trivially.

Morphisms are defined by the obvious way. By replacing “free” of (1) with “torsion”[^3], we define the category ${{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}$ (resp. ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}$).

For any object $\hat{{\mathfrak{M}}}$ of ${{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}$ or ${{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}$, we define a ${\mathbb}{Z}_p[G_{L,s}]$-module $\hat{T}_{L,s}(\hat{{\mathfrak{M}}})$ by $$\begin{aligned}
\hat{T}_{L,s}(\hat{{\mathfrak{M}}})=
\left\{
\begin{array}{ll}
{\mathrm}{Hom}_{{\widehat}{{\mathcal}{R}}_L,{\varphi}}
({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}, W(R))
\hspace{21mm} {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm free},  
\cr
{\mathrm}{Hom}_{{\widehat{\mathcal{R}}}_L,{\varphi}}
({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}, {\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p}W(R))
\hspace{3.5mm} {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm torsion}.  
\end{array}
\right.\end{aligned}$$ Here, $G_{L,s}$ acts on $\hat{T}_{L,s}(\hat{{\mathfrak{M}}})$ by $(\sigma.f)(x)=\sigma(f(\sigma^{-1}(x)))$ for $\sigma\in G_{L,s},\ f\in \hat{T}_{L,s}(\hat{{\mathfrak{M}}}),\ 
x\in {\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}}$. Similar to the above, for any object $\hat{{\mathfrak{M}}}$ of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}$ or ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}$, we define a ${\mathbb}{Z}_p[G_{L,s}]$-module $\hat{T}_{L,s}(\hat{{\mathfrak{M}}})$ by $$\begin{aligned}
\hat{T}_{L,s}(\hat{{\mathfrak{M}}})=
\left\{
\begin{array}{ll}
{\mathrm}{Hom}_{{\widehat{\mathcal{R}}}_{L,s},{\varphi}}
({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}, W(R))
\hspace{21mm} {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm free},  
\cr
{\mathrm}{Hom}_{{\widehat}{{\mathcal}{R}}_{L,s},{\varphi}}
({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}, {\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p}W(R))
\hspace{3.5mm} {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm torsion}.  
\end{array}
\right.\end{aligned}$$

On the other hand, we obtain functors $
{{\mathrm}{Mod}}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}\to {{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}\to 
{\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}\to {{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s}}
$ and $
{{\mathrm}{Mod}}^{r,\hat{G}_L}_{/{\mathfrak{S}}_{L,\infty}}\to {{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}\to 
{\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}\to {{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s,\infty}}
$ by natural manners and it is readily seen that these functors are compatible with $\hat{T}_L$ and $\hat{T}_{L,s}$. In particular, the essential images of the functors $\hat{T}_{L,s}$ on ${{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}$ and ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}$ has values in ${\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_{L,s})$ since we have an equivalence of categories $\hat{T}_{L,s}\colon {{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s}}
\overset{\sim}{\rightarrow} {\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_{L,s})$ by Theorem \[Thm1\].

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. ${{\mathrm}{Mod}}^{r,\hat{G}_{K,s}}_{/{\mathfrak{S}}_K}={{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}},  
{\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{K,s}}_{/{\mathfrak{S}}_K}={\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$). Furthermore, if $s=0$, we often omit the subscript “$s$” from various notations (e.g. ${{\mathrm}{Mod}}^{r,\hat{G}_{L,0}}_{/{\mathfrak{S}}_L}={{\mathrm}{Mod}}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L},  
{\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,0}}_{/{\mathfrak{S}}_{L,0}}={\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}$).\

The functors $
{{\mathrm}{Mod}}^{r,\hat{G}}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to 
{\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}
$
------------------------------------------------------------------------------------------------------------------

Now we consider the functors $
{{\mathrm}{Mod}}^{r,\hat{G}}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to 
{\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}.
$ At first, by Proposition \[basechange1:Kisin\], we see that the functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}$ is fully faithful. It follows from this fact and Theorem \[Thm1\] (1) that the functor $\hat{T}_s\colon {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}
\to {\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_s)$ is fully faithful. It is clear that the functor ${{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to 
{\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$ is fully faithful and thus so is $\hat{T}_s\colon {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}
\to {\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_s)$. Combining this with Theorem \[Thm1\] (1) and Lemma \[totst\], we obtain that the functor ${{\mathrm}{Mod}}^{r,\hat{G}}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$ is also fully faithful. Furthermore, we prove the following.

\[equal\] The functor ${{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to 
{\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$ is an equivalence of categories.

Summary, we obtained the following commutative diagram.

$\displaystyle \xymatrix{
{{\mathrm}{Mod}}^{r,\hat{G}}_{/{\mathfrak{S}}}
\ar@{^{(}->}[r] \ar[d]_{\wr} \ar^{\hat{T}}[d]
& 
{{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}
\ar^{\sim}[r] \ar@{^{(}->}[rrd] \ar^{\hat{T}_s}[rrd]
&
{\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}} 
\ar@{^{(}->}[r] \ar@{^{(}->}[rd] \ar^{\hat{T}_s}[rd]
&
{{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s} 
\ar[d]_{\wr} \ar^{\hat{T}_s}[d]\\
{\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K) 
\ar@{^{(}->}[rrr]  \ar^{{\mathrm}{restriction}}[rrr]   
& 
&
& 
{\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_s).
}$

The functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\hookrightarrow {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}$ may not be possibly essentially surjective. In fact, under some conditions, there exists a representation of $G_K$ which is crystalline over $K_s$ but not of finite height. For more precise information, see [@Li2 Example 4.2.3].

Before a proof of Proposition \[equal\], we give an explicit formula such as (\[explicit\]) for an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$. The argument below follows the method of [@Li2]. Let $\hat{{\mathfrak{M}}}$ be an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$. Let $\hat{{\mathfrak{M}}}_s$ be the image of $\hat{{\mathfrak{M}}}$ for the functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}$. Put ${\mathcal}{D}=S_{K_0}\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$ and also put ${\mathcal}{D}_s=S_{K_0,s}\otimes_{{\varphi},{\mathfrak{S}}_s} {\mathfrak{M}}_s
=S_{K_0,s}\otimes_{S_{K_0}} {\mathcal}{D}$. 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 $({\varphi},N)$-module corresponding to ${\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_s(\hat{{\mathfrak{M}}}_s)$ (see subsection \[relations\]), which is isomorphic to ${\mathcal}{D}/I_+S_{K_0}{\mathcal}{D}$ as a ${\varphi}$-module over $K_0$. By [@Li1 Lemma 7.3.1], we have a unique ${\varphi}$-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 (\[explicit\]) for $\hat{{\mathfrak{M}}}_s$, we know that $$\hat{G}_s(D)\subset (K_0[\![t]\!]\cap {\mathcal}{R}_{K_0,s})\otimes_{K_0} D.$$ Hence, taking any $K_0$-basis $e_1,\dots ,e_d$ of $D$, there exist $A_s(t)\in M_{d\times d}(K_0[\![t]\!])$ such that $\tau^{p^s}(e_1,\cdots ,e_d)=(e_1,\dots ,e_d)A_s(t)$. Since $A_s(0)={\mathrm}{I}_d$, we see that ${\mathrm}{log}(A_s(t))\in M_{d\times d}(K_0[\![t]\!])$ is well-defined. On the other hand, choose $g_0\in G_s$ such that $\chi_p(g_0)\not=1$, where $\chi_p$ is the $p$-adic cyclotomic character. Since $g_0\tau^{p^s}=(\tau^{p^s})^{\chi_p(g_0)}g_0$, we have $A_s(\chi_p(g_0)t)=A_s(t)^{\chi_p(g_0)}$ and thus we also have ${\mathrm}{log}(A_s(\chi_p(g_0)t))=\chi_p(g_0){\mathrm}{log}(A_s(t))$. Since ${\mathrm}{log}(A_s(0))={\mathrm}{log}(I_d)=0$, we can write ${\mathrm}{log}(A_s(t))$ as $tB(t)$ for some $B(t)\in M_{d\times d}(K_0[\![t]\!])$. Then we have $\chi_p(g_0)tB(\chi_p(g_0)t)=\chi_p(g_0)tB(t)$, that is, $B(\chi_p(g_0)t)=B(t)$. Hence the assumption $\chi_p(g_0)\not= 1$ implies that $B(t)$ is a constant. Putting $N_s=B(t)\in M_{d\times d}(K_0)$, we obtain $$\tau^{p^s}(e_1,\cdots ,e_d)=(e_1,\cdots ,e_d)(\sum_{i\ge 0}N^i_s\gamma_i(t)).$$ Now we define $N_D\colon D\to D$ by $N(e_1,\cdots ,e_d)=(e_1,\cdots ,e_d)p^{-s}N_s$ and also define $N_{{\mathcal}{D}}:=N_{S_{K_0}}\otimes {\mathrm}{Id}_D+{\mathrm}{Id}_{S_{K_0}}\otimes N_D$. (Note that we have $N_D{\varphi}_D=p{\varphi}_D N_D$ and thus $N_D$ is nilpotent.) It is a routine work to check the following: $$\label{explicit''}
g(a\otimes x)=\sum^{\infty}_{i=0}g(a)\gamma_i
(-{\mathrm}{log}([\underline{{\varepsilon}}(g)]))\otimes N^i_D(x)\quad
{\rm for}\ g\in G_s, a\in B^+_{{\mathrm}{cris}}, x\in D.$$ Since we have $$\label{easyeq}
g(f)=\sum_{i\ge 0} \gamma_i(-{\mathrm}{log}([\underline{{\varepsilon}}(g)]))N^i_{S_{K_0}}(f)$$ for any $g\in G_K$ and $f\in S_{K_0}$, we obtain the following explicit formula: $$\label{explicit'}
g(a\otimes x)=\sum^{\infty}_{i=0}g(a)\gamma_i
(-{\mathrm}{log}([\underline{{\varepsilon}}(g)]))\otimes N^i_{{\mathcal}{D}}(x)\quad
{\rm for}\ g\in G_s, a\in B^+_{{\mathrm}{cris}}, x\in {\mathcal}{D}.$$ In particular, as in subsection \[relations\], we can show that $$\label{eq2}
{\mathrm}{log}(\tau^{p^s})(x)=p^st\otimes N_{{\mathcal}{D}}(x)$$ for any $x\in {\mathcal}{D}$.

We continue to use the above notation. It suffices to prove that the $G_s$-action on ${\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$ preserves ${\widehat{\mathcal{R}}}\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}$. Take any $g\in G_s$. We know that $g({\mathfrak{M}})\subset {\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}\subset W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$. Hence it is enough to prove that $g({\mathcal}{D})\in {\mathcal}{R}_{K_0}\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$. Let $s\in S^{{\mathrm}{int}}_{K_0}$ and $y\in D$ and put $x=s\otimes y\in S^{{\mathrm}{int}}_{K_0}\otimes_{W(k)} D={\mathcal}{D}$. By (\[explicit”\]) or (\[explicit’\]), we have $$g(x)=\sum_{i\ge 0}\sum_{0\le j\le i} \gamma_i(-{\mathrm}{log}([\underline{{\varepsilon}}(g)]))
\binom{i}{j}N^{i-j}_{S_{K_0}}(s)\otimes N^j_D(y)$$ On the other hand, we know that $N_D$ is nilpotent, that is, there exists $j_0>0$ such that $N^{j_0}_D=0$. Then we obtain $$g(x)=\sum_{0\le j\le j_0} \sum^{\infty}_{i=j} \gamma_i(-{\mathrm}{log}([\underline{{\varepsilon}}(g)]))
     \binom{i}{j}N^{i-j}_{S_{K_0}}(s)\otimes N^j_D(y).$$ Therefore, it suffices to show that $\sum^{\infty}_{i=j} \gamma_i(-{\mathrm}{log}([\underline{{\varepsilon}}(g)]))
     \binom{i}{j}N^{i-j}_{S_{K_0}}(s)$ is contained in ${\mathcal}{R}_{K_0}$ for each $0\le j\le j_0$. Taking $\alpha(g)\in {\mathbb}{Z}_p$ such that ${\mathrm}{log}([\underline{{\varepsilon}}(g)])=-\alpha(g)t$, we have $$\sum^{\infty}_{i=j} \gamma_i(-{\mathrm}{log}([\underline{{\varepsilon}}(g)]))
     \binom{i}{j}N^{i-j}_{S_{K_0}}(s)
     =
     \sum^{\infty}_{i=j} (\alpha(g))^i\frac{\tilde{q}(i)!p^{\tilde{q}(i)}}{i!}
     \binom{i}{j}N^{i-j}_{S_{K_0}}(s)t^{\{i\}}.$$ Since $\frac{\tilde{q}(i)p^{\tilde{q}(i)}}{i!}\to 0$ ($p$-adically) as $i\to \infty$, we finish a proof.

Relations with crystalline representations
------------------------------------------

We know that ${\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_s(\hat{{\mathfrak{M}}})$ is semi-stable over $K_s$ for any object $\hat{{\mathfrak{M}}}$ of ${{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$ or ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$. 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 [@Fo2 Section 5] we set $I^{[m]}B^+_{{\mathrm}{cris}}:=\{x\in B^+_{{\mathrm}{cris}} \mid {\varphi}^n(x)
\in {\mathrm}{Fil}^mB^+_{{\mathrm}{cris}}\ {\rm for\ all}\ n\ge0 \}$. For any subring $A\subset B^+_{{\mathrm}{cris}}$, we put $I^{[m]}A=A\cap I^{[m]}B^+_{{\mathrm}{cris}}$. Furthermore, we also put $I^{[m+]}A=I^{[m]}A.I_+A$ (here, $I_+A$ is defined in Subsection \[Liumodule:section\]). By [@Fo2 Proposition 5.1.3] and the proof of [@Li2 Lemma 3.2.2], we know that $I^{[m]}W(R)$ is a principal ideal which is generated by ${\varphi}({\mathfrak{t}})^m$.

Now we recall Theorem \[Thm1\] (2): if ${\mathfrak{M}}$ is an object of ${{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}$, then ${\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_s(\hat{{\mathfrak{M}}})$ is crystalline if and only if $\tau^{p^s}(x)-x\in u_s^p(I^{[1]}W(R)\otimes_{{\varphi},{\mathfrak{S}}_s} {\mathfrak{M}})$ 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.

\[cris\] Let $\hat{{\mathfrak{M}}}$ be an object of ${{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$ or ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$. Then the following is equivalent:

1.  ${\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_s(\hat{{\mathfrak{M}}})$ is crystalline,

2.  $\tau^{p^s}(x)-x\in u^p(I^{[1]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}})$ for any $x\in {\mathfrak{M}}$,

3.  $\tau^{p^s}(x)-x\in I^{[1+]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$ for any $x\in {\mathfrak{M}}$.

\(1) $\Rightarrow$ (2): The proof here mainly follows that of [@GLS Proposition 4.7]. We may suppose $\hat{{\mathfrak{M}}}$ is an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$. Put ${\mathcal}{D}=S_{K_0}\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$ and $D={\mathcal}{D}/I_+S_{K_0} {\mathcal}{D}$ as in the previous subsection. We fix a ${\varphi}({\mathfrak{S}})$-basis $(\hat{e}_1,\dots ,\hat{e}_d)$ of ${\mathfrak{M}}\subset {\mathcal}{D}$ and denote by $(e_1,\dots ,e_d)$ the image of $(\hat{e}_1,\dots ,\hat{e}_d)$ for the projection ${\mathcal}{D}\to D$. Then $(e_1,\dots ,e_d)$ is a $K_0$-basis of $D$. As described before the proof of Proposition \[equal\], we regard $D$ as a ${\varphi}$-stable submodule of ${\mathcal}{D}$, and we have $N_D\colon D\to D$ and $N_{{\mathcal}{D}}\colon D_{{\mathcal}{D}}\to D_{{\mathcal}{D}}$.

Now we consider a matrix $X\in GL_{d\times d}(S_{K_0})$ such that $(\hat{e}_1,\dots ,\hat{e}_d)=(e_1,\dots ,e_d)X$. We define $\tilde{S}=W(k)[\![u^p, u^{ep}/p]\!]$ as in Section 4 of [@GLS], which is a sub $W(k)$-algebra of $S^{{\mathrm}{int}}_{K_0}$ 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 $N_D(e_1,\dots ,e_d)=(e_1,\dots ,e_d)B$ and $N_{\mathcal}{D}(\hat{e}_1,\dots ,\hat{e}_d)=(\hat{e}_1,\dots ,\hat{e}_d)U$. By the same proof as in the former half part of the proof of [@GLS 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 ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{{\mathfrak{S}}_s}$. Now we recall that ${\mathcal}{D}_s=S_{K_0,s}\otimes_{{\varphi},{\mathfrak{S}}_s} {\mathfrak{M}}_s$ 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 (\[eq1\]) for $\hat{{\mathfrak{M}}}_s$ and the formula (\[eq2\]) for $\hat{{\mathfrak{M}}}$, we see that $p^sN_{{\mathcal}{D}}=N_{{\mathcal}{D}_s}$ 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 $N_D=(N_{{\mathcal}{D}}$ mod $I_+S_{K_0}{\mathcal}{D})$ is zero, that is, $B=0$. Therefore, the latter half part of the proof [@GLS Proposition] 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 ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{{\mathfrak{S}}_s}$ as above. We claim that, for any $x\in {\mathfrak{M}}_s$, we have $\tau^{p^s}(x)-x\in I^{[1+]}W(R)\otimes_{{\varphi}_s} {\mathfrak{M}}_s$. 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 $$\tau^{p^s}(x)-x=\tau^{p^s}({\varphi}(a))(\tau^{p^s}(y)-y)+(\tau^{p^s}({\varphi}(a))-{\varphi}(a))y$$ and thus it suffices to show $\tau^{p^s}({\varphi}(a))-{\varphi}(a)\in I^{[1+]}W(R)$. This follows from the lemma below and thus we obtained the claim. By the claim and Theorem \[Thm1\] (2), 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.

\[cryslem\] $(1)$ We have $I^{[1]}W(R)\cap u^{\ell}B^+_{{\mathrm}{cris}}
=u^{\ell}I^{[1]}W(R)$ for $\ell\ge 0$.

$(2)$ We have $g(a)-a\in uI^{[1]}W(R)$ for $g\in G$ and $a\in {\mathfrak{S}}$.

This is due to [@GLS the proof of Proposition 7] but we write a proof here.

\(1) Take $x=u^{\ell}y\in I^{[1]}W(R)$ with $y\in B^+_{{\mathrm}{cris}}$. By Lemma 3.2.2 of [@Li3] we have $y\in W(R)$. Now we remark that $uz\in {\mathrm}{Fil}^nW(R)$ with $z\in W(R)$ implies $z\in {\mathrm}{Fil}^nW(R)$ since $u$ is a unit of $B^+_{{\mathrm}{dR}}$. Hence $u^{\ell}y\in I^{[1]}W(R)$ implies $y\in I^{[1]}W(R)$.

\(2) By the relation (\[easyeq\]), we see that $g(a)-a\in I^{[1]}W(R)$. On the other hand, if $i>0$, we can write $N^i_{S_{K_0}}(a)=ub_i$ for some $b_i\in {\mathfrak{S}}$. Thus by the relation (\[easyeq\]) again we obtain $g(a)-a\in uB^+_{{\mathrm}{cris}}$. Then the result follows from (1).

Variants of torsion $({\varphi},\hat{G})$-modules
=================================================

In this section, we mainly study full subcategories of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$ defined below and also study representations associated with them. As a consequence, we prove theorems in Introduction. We use same notation as in Section 2 and 3. In particular, $p$ is odd. In below, let $v_R$ be the valuation of $R$ normalized such that $v_R(\underline{\pi})=1/e$ and, for any real number $x\ge 0$, we denote by ${\mathfrak{m}}^{\ge x}_R$ the ideal of $R$ consisting of elements $a$ with $v_R(a)\ge x$.

Let $J$ be an ideal of $W(R)$ which satisfies the following conditions:

-   $J\not\subset pW(R)$,

-   $J$ is a principal ideal,

-   $J$ is ${\varphi}$-stable and $G_s$-stable in $W(R)$.

By the above first and second assumptions for $J$, the image of $J$ under the projection $W(R)\twoheadrightarrow R$ is of the form ${\mathfrak{m}}^{\ge c_J}_{R}$ for some real number $c_J\ge 0$.

We denote by ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ the full subcategory of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$ consisting of objects $\hat{{\mathfrak{M}}}$ which satisfy the following condition: $$\tau^{p^s}(x)-x\in JW(R)\otimes_{{\varphi}, {\mathfrak{S}}} {\mathfrak{M}}\quad {\rm for\ any}\ x\in {\mathfrak{M}}.$$ Also, we denote by ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s, J}_{{\mathrm}{tor}}(G_s)$ the essential image of the functor $\hat{T}_s\colon {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_s)$ restricted to ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$.

By definition, we have ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}\subset {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J'}_{/{\mathfrak{S}}_{\infty}}$ and ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)
\subset {\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J'}_{{\mathrm}{tor}}(G_s)$ for $J\subset J'$.

Full faithfullness for ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$
------------------------------------------------------------------------------------------------

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

\[FFTHMMOD\] Let $r$ and $r'$ be non-negative integers with $c_J> pr/(p-1)$. Let $\hat{{\mathfrak{M}}}$ and $\hat{{\mathfrak{N}}}$ be objects of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ and ${\widetilde{{\mathrm}{Mod}}}^{r',\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$, respectively. Then we have ${\mathrm}{Hom}(\hat{{\mathfrak{M}}}, \hat{{\mathfrak{N}}})={\mathrm}{Hom}({\mathfrak{M}}, {\mathfrak{N}})$. $($Here, two “${\mathrm}{Hom}$”s are defined by obvious manners.$)$

In particular, if $c_J> pr/(p-1)$, then the forgetful functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}\to {{\mathrm}{Mod}}^r_{/{\mathfrak{S}}_{\infty}}$ is fully faithful.

A very similar proof of [@Oz2 Lemma 7] proceeds, and hence we only give a sketch here. Let $\hat{{\mathfrak{M}}}$ and $\hat{{\mathfrak{N}}}$ be objects of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ and ${\widetilde{{\mathrm}{Mod}}}^{r',\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$, respectively. Let $f\colon {\mathfrak{M}}\to {\mathfrak{N}}$ be a morphism of Kisin modules over ${\mathfrak{S}}$. Put $\hat{f}=W(R)\otimes f\colon W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}\to W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}$. Choose any lift of $\tau\in \hat{G}$ to $G_K$; we denote it also by $\tau$. Since the $\hat{G}_s$-action for $\hat{{\mathfrak{M}}}$ is continuous, it suffices to prove that $\Delta(1\otimes x)=0$ for any $x\in {\mathfrak{M}}$ where $\Delta:=\tau^{p^s}\circ \hat{f}-\hat{f}\circ \tau^{p^s}$. We use induction on $n$ such that $p^n{\mathfrak{N}}=0$.

Suppose $n=1$. Since $\Delta=(\tau^{p^s}-1)\circ \hat{f}-\hat{f}\circ (\tau^{p^s}-1)$, we obtain the following:

$(0)$:For any $x\in {\mathfrak{M}}$, $\Delta(1\otimes x)\in 
{\mathfrak{m}}^{\ge c(0)}_R(R\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{N}})$

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

$(i)$:For any $x\in {\mathfrak{M}}$, $\Delta(1\otimes x)\in 
{\mathfrak{m}}^{\ge c(i)}_R(R\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{N}})$

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 $c(i)\to \infty$ as $i\to \infty$ and thus $\Delta(1\otimes x)=0$.

Suppose $n>1$. Consider the exact sequence $0\to {\mathrm}{Ker}(p)\to {\mathfrak{N}}\overset{p}{\to} p{\mathfrak{N}}\to 0$ of ${\varphi}$-modules over ${\mathfrak{S}}$. It is not difficult to check that ${\mathfrak{N}}':={\mathrm}{Ker}(p)$ and ${\mathfrak{N}}'':=p{\mathfrak{N}}$ are torsion Kisin modules of height $\le r'$ over ${\mathfrak{S}}$ (cf. [@Li1 Lemma 2.3.1]). Moreover, we can check that ${\mathfrak{N}}'$ and ${\mathfrak{N}}''$ have natural structures of objects of ${\widetilde{{\mathrm}{Mod}}}^{r',\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$ (which are denoted by $\hat{{\mathfrak{N}}}'$ and $\hat{{\mathfrak{N}}}''$, respectively) such that the sequence $0\to {\mathfrak{N}}'\to {\mathfrak{N}}\overset{p}{\to} {\mathfrak{N}}''\to 0$ induces an exact sequence $0\to \hat{{\mathfrak{N}}}'\to \hat{{\mathfrak{N}}}\to \hat{{\mathfrak{N}}}''\to 0$. By the lemma below, we know that $\hat{{\mathfrak{N}}}'$ and $\hat{{\mathfrak{N}}}''$ are in fact contained in ${\widetilde{{\mathrm}{Mod}}}^{r',\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$. By the induction hypothesis, we see that $\Delta(1\otimes x)$ has values in $(W(R)\otimes_{{\varphi}, {\mathfrak{S}}} {\mathfrak{N}}')\cap 
(JW(R)\otimes_{{\varphi}, {\mathfrak{S}}} {\mathfrak{N}})$. By Lemma 6 of [@Oz2] and the assumption that $J\not\subset pW(R)$ is principal, we obtain that $\Delta(1\otimes x)\in JW(R)\otimes_{{\varphi}, {\mathfrak{S}}} {\mathfrak{N}}'$. Since $p{\mathfrak{N}}'=0$, an analogous argument in the case $n=1$ proceeds and we have $\Delta(1\otimes x)=0$ as desired.

\[speciallemma\] Let $0\to \hat{{\mathfrak{M}}}'\to \hat{{\mathfrak{M}}}\to \hat{{\mathfrak{M}}}''\to 0$ be an exact sequence in ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$. Suppose that $\hat{{\mathfrak{M}}}$ is an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$. Then $\hat{{\mathfrak{M}}}'$ and $\hat{{\mathfrak{M}}}''$ are also objects of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$.

The fact $\hat{{\mathfrak{M}}}''\in {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ is clear. Take any $x\in {\mathfrak{M}}'$. Then we have $\tau^{p^s}(x)-x\in (JW(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}})\cap
(W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}')$. Since $J$ is a principal ideal which is not contained in $pW(R)$, we obtain $\tau^{p^s}(x)-x\in JW(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}'$ by Lemma 6 of [@Oz2]. This implies $\hat{{\mathfrak{M}}}'\in {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$.

The category ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$
---------------------------------------------------------------------------------

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

Let $\hat{{\mathfrak{M}}}$ be an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$. Following Section 3.2 of [@Li2] (note that arguments in [@Li2] is the “free case”), we construct a map $\hat{\iota}_s$ 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 $$\hat{T}_s(\hat{{\mathfrak{M}}})
\simeq
{\mathrm}{Hom}_{W(R),{\varphi}}(W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}, {\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p} W(R))$$ where $G_s$ acts on ${\mathrm}{Hom}_{W(R),{\varphi}}(W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}},{\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p} W(R))$ by $(\sigma.f)(x)=\sigma(f(\sigma^{-1}(x)))$ for $\sigma\in G_s,
f\in {\mathrm}{Hom}_{W(R),{\varphi}}(W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}},{\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p} W(R)),
x\in W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}=W(R)\otimes_{{\widehat{\mathcal{R}}}_s} ({\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}})$. Thus we can define a morphism $\hat{\iota}'_s\colon W(R)\otimes_{{\varphi}, {\mathfrak{S}}}{\mathfrak{M}}\to 
{\mathrm}{Hom}_{{\mathbb}{Z}_p}(\hat{T}_s(\hat{{\mathfrak{M}}}),{\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p} W(R))$ by $$x\mapsto (f\mapsto f(x)),\quad 
x\in W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}, f\in \hat{T}_s(\hat{{\mathfrak{M}}}).$$ Since $\hat{T}_s(\hat{{\mathfrak{M}}})\simeq \oplus_{i\in I}{\mathbb}{Z}_p/p^{n_i}{\mathbb}{Z}_p$ as ${\mathbb}{Z}_p$-modules, we have a natural isomorphism ${\mathrm}{Hom}_{{\mathbb}{Z}_p}(\hat{T}_s(\hat{{\mathfrak{M}}}),{\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p} W(R))\simeq 
W(R)\otimes_{{\mathbb}{Z}_p}\hat{T}_s^{\vee}(\hat{{\mathfrak{M}}})$ where $\hat{T}_s^{\vee}(\hat{{\mathfrak{M}}})={\mathrm}{Hom}_{{\mathbb}{Z}_p}(\hat{T}_s(\hat{{\mathfrak{M}}}),{\mathbb}{Q}_p/{\mathbb}{Z}_p)$ is the dual representation of $\hat{T}_s(\hat{{\mathfrak{M}}})$. Composing this isomorphism with $\hat{\iota}'_s$, we obtain the desired map $$\hat{\iota}_s\colon W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}\to 
W(R)\otimes_{{\mathbb}{Z}_p}\hat{T}_s^{\vee}(\hat{{\mathfrak{M}}}).$$ It follows from a direct calculation that $\hat{\iota}_s$ is ${\varphi}$-equivariant and $G_s$-equivariant. If we denote by $\hat{{\mathfrak{M}}}_s$ the image of $\hat{{\mathfrak{M}}}$ for the functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}\to 
{{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{s,\infty}}$ (cf. Section \[vardef\]), then the above $\hat{\iota}_s$ is isomorphic to “$\hat{\iota}$ for $\hat{{\mathfrak{M}}}_s$ in Section 4.1 of [@Oz1]”. Hence Lemma 4.2 (4) in [*loc*]{}. [*cit*]{}. implies that $$W({\mathrm}{Fr}\ R)\otimes \hat{\iota}_s\colon 
W({\mathrm}{Fr}\ R)\otimes_{W(R)}(W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}})
\to 
W({\mathrm}{Fr}\ R)\otimes_{W(R)}(W(R)\otimes_{{\mathbb}{Z}_p}\hat{T}_s^{\vee}(\hat{{\mathfrak{M}}}))$$ is bijective.

Let $(R) \colon 0\to T'\to T\to T''\to 0$ be an exact sequence in ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_s)$. Assume that there exists $\hat{{\mathfrak{M}}}\in {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ such that $\hat{T}_s(\hat{{\mathfrak{M}}})\simeq T$. Then there exists an exact sequence $(M) \colon 0\to \hat{{\mathfrak{M}}}''\to \hat{{\mathfrak{M}}}\to \hat{{\mathfrak{M}}}'\to 0$ in ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ such that $\hat{T}_s((M))\simeq (R)$.

The same proof as [@Oz1 Theorem 4.5], except using not $\hat{\iota}$ in the proof of [*loc.*]{} [*cit.*]{} but $\hat{\iota}_s$ as above, gives an exact sequence $(M) \colon 0\to \hat{{\mathfrak{M}}}''\to \hat{{\mathfrak{M}}}\to \hat{{\mathfrak{M}}}'\to 0$ in ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$ such that $\hat{T}_s((M))\simeq (R)$. Therefore, Lemma \[speciallemma\], gives the desired result.

\[stability\] The full subcategory ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ of ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_s)$ is stable under subquotients.

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.

\[bc\] Assume that $J\supset u^pI^{[1]}W(R)$ or $L$ is a finite unramified extension of $K$. If $T$ is an object of ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$, then $T|_{G_{L,s}}$ is an object of ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_{L,s},J}_{{\mathrm}{tor}}(G_{L,s})$.

By an obvious way, we define a functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}\to 
{\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}$. The underlying Kisin module of the image of $\hat{{\mathfrak{M}}}\in {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$ for this functor is ${\mathfrak{M}}_L={\mathfrak{S}}_L\otimes_{{\mathfrak{S}}} {\mathfrak{M}}$. Lemma \[bc\] immediately follows from the lemma below.

Assume that $J\supset u^pI^{[1]}W(R)$ or $L$ is a finite unramified extension of $K$. Then the functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}\to {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}$ induces a functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}\to {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s},J}_{/{\mathfrak{S}}_{L,\infty}}$.

Let $\hat{{\mathfrak{M}}}$ be an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$ and let $\hat{{\mathfrak{M}}}_L$ be the image of $\hat{{\mathfrak{M}}}$ for the functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}\to {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}$. In the rest of this proof, to avoid confusions, we denote the image of $x\in {\mathfrak{M}}_L$ in $W(R)\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}_L$ by $1\otimes x$. Recall that we abuse notations by writing $\tau$ for the pre-image of $\tau\in G_{K,p^{\infty}}$ via the bijection $G_{L,p^{\infty}}\simeq G_{K,p^{\infty}}$ of lemma \[easylemma\]. Then $\tau^{p^s}$ is a topological generator of $G_{L,s,p^{\infty}}$. It suffices to show the following: if $\hat{{\mathfrak{M}}}$ is an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$, then we have $\tau^{p^s}(1\otimes x)-(1\otimes x)\in JW(R)\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}_L$ for any $x\in {\mathfrak{M}}_L$. Now we suppose $\hat{{\mathfrak{M}}}\in {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$. Take any $a\in {\mathfrak{S}}_L$ and $x\in {\mathfrak{M}}$. Note that we have $\tau^{p^s}(1\otimes ax)-(1\otimes ax)
=\tau^{p^s}({\varphi}(a))(\tau^{p^s}(1\otimes x)-(1\otimes x))
+(\tau^{p^s}({\varphi}(a))-{\varphi}(a))(1\otimes x)$ in $W(R)\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}_L$. Since $\hat{{\mathfrak{M}}}$ is an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$, we have $\tau^{p^s}({\varphi}(a))(\tau^{p^s}(1\otimes x)-(1\otimes x))\in 
JW(R)\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}_L$. Therefore, it is enough to show $(\tau^{p^s}({\varphi}(a))-{\varphi}(a))(1\otimes x)\in JW(R)\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}_L$. This follows from Lemma \[cryslem\] immediately in the case where $J\supset u^pI^{[1]}W(R)$. Next we consider the case where $L$ is a finite unramified extension of $K$. Let $c_1,\dots ,c_{\ell}\in W(k_L)$ be generators of $W(k_L)$ as a $W(k)$-module. Then we have ${\mathfrak{S}}_L=\sum^{\ell}_{j=1} c_j {\mathfrak{S}}$ and thus we can write $a=\sum^{\ell}_{j=1}a_jc_j$ for some $a_j\in {\mathfrak{S}}$. Hence it suffices to show $(\tau^{p^s}({\varphi}(a_j))-{\varphi}(a_j))(1\otimes x)\in JW(R)\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}_L$ but this in fact immediately follows from the equation $(\tau^{p^s}({\varphi}(a_j))-{\varphi}(a_j))(1\otimes x)
=(\tau^{p^s}(1\otimes a_jx)-(1\otimes a_jx))-
(\tau^{p^s}({\varphi}(a_j))(\tau^{p^s}(1\otimes x)-(1\otimes x)))$.

Full faithfulness theorem for ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$
--------------------------------------------------------------------------------------------------

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

\[FFTHM\] Assume that $J\supset u^pI^{[1]}W(R)$ or $k$ is algebraically closed. If $p^{s+2}/(p-1)\ge c_J>pr/(p-1)$, then the restriction functor ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$ is fully faithful.

First we give a very rough sketch of the theory of maximal models for Kisin modules (cf. [@CL1]). For any ${\mathfrak{M}}\in {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{\infty}}$, put ${\mathfrak{M}}[1/u]={\mathfrak{S}}[1/u]\otimes_{{\mathfrak{S}}} {\mathfrak{M}}$ and denote by $F^r_{{\mathfrak{S}}}({\mathfrak{M}}[1/u])$ the (partially) ordered set (by inclusion) of torsion Kisin modules ${\mathfrak{N}}$ of height $\le r$ which are contained in ${\mathfrak{M}}[1/u]$ and ${\mathfrak{N}}[1/u]={\mathfrak{M}}[1/u]$ as ${\varphi}$-modules. The set $F^r_{{\mathfrak{S}}}({\mathfrak{M}}[1/u])$ has a greatest element (cf. [*loc*]{}. [*cit*]{}., Corollary 3.2.6). We denote this element by ${\mathrm{Max}}^r({\mathfrak{M}})$. We say that ${\mathfrak{M}}$ is [*maximal of height $\le r$*]{} (or, [*maximal*]{} for simplicity) if it is the greatest element of $F^r_{{\mathfrak{S}}}({\mathfrak{M}}[1/u])$. The implication ${\mathfrak{M}}\mapsto {\mathrm{Max}}^r({\mathfrak{M}})$ defines a functor “${\mathrm{Max}}^r$” from the category ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{\infty}}$ of torsion Kisin modules of height $\le r$ into the category ${\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}$ of maximal Kisin modules of height $\le r$. The category ${\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}$ is abelian (cf. [*loc*]{}. [*cit*]{}., Theorem 3.3.8). Furthermore, the functor $T_{{\mathfrak{S}}}\colon {\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$, defined by $T_{{\mathfrak{S}}}({\mathfrak{M}})={\mathrm}{Hom}_{{\mathfrak{S}},{\varphi}}({\mathfrak{M}},{\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p} W(R))$, is exact and fully faithful (cf. [*loc*]{}. [*cit*]{}., Corollary 3.3.10). It is not difficult to check that $T_{{\mathfrak{S}}}({\mathrm{Max}}^r({\mathfrak{M}}))$ is canonically isomorphic to $T_{{\mathfrak{S}}}({\mathfrak{M}})$ as representations of $G_{\infty}$ for any torsion Kisin module ${\mathfrak{M}}$ of height $\le r$.

\[Def1\] Let $d$ be a positive integer. Let ${\mathfrak{n}}=(n_i)_{i\in {\mathbb}{Z}/d{\mathbb}{Z}}$ be a sequence of non-negative integers of smallest period $d$. We define a torsion Kisin module ${\mathfrak{M}}({\mathfrak{n}})$ as below:

-   as a ${k[\![u]\!]}$-module, ${\mathfrak{M}}({\mathfrak{n}})=\bigoplus_{i\in {\mathbb}{Z}/d{\mathbb}{Z}} {k[\![u]\!]}e_i$;

-   for all $i\in {\mathbb}{Z}/d{\mathbb}{Z}$, ${\varphi}(e_i)=u^{n_i}e_{i+1}$.

We denote by ${\mathcal}{S}^r_{{\mathrm}{max}}$ the set of sequences ${\mathfrak{n}}=(n_i)_{i\in {\mathbb}{Z}/d{\mathbb}{Z}}$ of integers $0\le n_i\le {\mathrm}{min}\{er, p-1\}$ 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 $\le r$ for any ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$. Putting $r_0={\mathrm}{max}\{r'\in {\mathbb}{Z}_{\ge 0};e(r'-1)<p-1 \}$, we also see that ${\mathfrak{M}}({\mathfrak{n}})$ is of height $\le r_0$ for any ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$. It is known that ${\mathfrak{M}}({\mathfrak{n}})$ is maximal for any ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$ ([@CL1 Proposition 3.6.7]). If $k$ is algebraically closed, then ${\mathfrak{M}}({\mathfrak{n}})$ is simple in ${\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}$ for any ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$ (cf. [*loc. cit.*]{}, Propositions 3.6.7 and 3.6.12) and furthermore, the converse holds; any simple object in ${\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}$ is of the form ${\mathfrak{M}}({\mathfrak{n}})$ for some ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$ (cf. [*loc. cit.*]{}, Propositions 3.6.8 and 3.6.12).

\[Lem1\] Assume that $p^{s+2}/(p-1)\ge c_J$. Let $d$ be a positive integer. Let ${\mathfrak{n}}=(n_i)_{i\in {\mathbb}{Z}/d{\mathbb}{Z}}$ be a sequence of non-negative integers of smallest period $d$. If ${\mathfrak{M}}({\mathfrak{n}})$ is of height $\le r$, then ${\mathfrak{M}}({\mathfrak{n}})$ has a structure of an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$.

Choose any $(p^d-1)$-th root $\eta\in R$ of $\underline{{\varepsilon}}$. Since $[\eta]\cdot {\mathrm}{exp}(t/(p^d-1))$ is a $(p^d-1)$-th root of unity, it is of the form $[a]$ for some $a\in {\mathbb}{F}^{\times}_{p^d}$. Replacing $\eta a^{-1}$ with $\eta$, we obtain $[\eta]={\mathrm}{exp}(-t/(p^d-1))\in {\widehat{\mathcal{R}}}^{\times}$. Put $x_i=[\eta]^{m_i}\in {\widehat{\mathcal{R}}}^{\times}$ and $\bar{x}_i=\eta^{m_i}\in ({\widehat{\mathcal{R}}}/p{\widehat{\mathcal{R}}})^{\times}\subset R^{\times}$ for any $i\in {\mathbb}{Z}/d{\mathbb}{Z}$, where $m_i=\sum^{d-1}_{j=0}n_{i+j}p^{d-j}$. 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 ${\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}({\mathfrak{n}})\subset R\otimes_{{\varphi},k[\![u]\!]} {\mathfrak{M}}({\mathfrak{n}})$ by $1\otimes x$. Now we define a $\hat{G}_s$-action on ${\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}({\mathfrak{n}})$ by $\tau^{p^s}(1\otimes e_i):=x^{p^s}_i(1\otimes e_i)$ for the basis $\{e_i\}_{i\in {\mathbb}{Z}/d{\mathbb}{Z}}$ of ${\mathfrak{M}}({\mathfrak{n}})$ as in Definition \[Def1\]. It is not difficult to check that ${\mathfrak{M}}({\mathfrak{n}})$ has a structure of an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$ 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 ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$. Recall that $v_R$ is the valuation of $R$ normalized such that $v_R(\underline{\pi})=1/e$. Define $\tilde{{\mathfrak{t}}}={\mathfrak{t}}\ {\mathrm}{mod}\ pW(R)$ an element of $R$. We denote by $v_p$ the usual $p$-adic valuation normalized by $v_p(p)=1$. Note that we have $v_R(\underline{{\varepsilon}}-1)=p/(p-1)$ and $v_R(\tilde{{\mathfrak{t}}})=1/(p-1)$ (here, the latter equation follows from the relation ${\varphi}({\mathfrak{t}})=pE(0)^{-1}E(u){\mathfrak{t}}$). We see that $$v_R(\bar{x}^{p^s}_i-1)=p^{s+v_p(m_i)}\cdot p/(p-1)\ge p^{s+2}/(p-1).$$ Since $p^{s+2}/(p-1)\ge c_J$ and the image of $J$ in $R$ is ${\mathfrak{m}}_R^{\ge c_J}$, we obtain $$\tau^{p^s}(1\otimes e_i)-(1\otimes e_i)\in 
{\mathfrak{m}}_R^{\ge c_J}R\otimes_{{\varphi},k[\![u]\!]} {\mathfrak{M}}({\mathfrak{n}})
\simeq JW(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}({\mathfrak{n}}).$$ Finally we have to show that $\tau^{p^s}(1\otimes ae_i)-(1\otimes ae_i)\in 
{\mathfrak{m}}_R^{\ge c_J}R\otimes_{{\varphi},k[\![u]\!]} {\mathfrak{M}}({\mathfrak{n}})$ for any $a\in k[\![u]\!]$. Since $\tau^{p^s}(1\otimes ae_i)-(1\otimes ae_i)=\tau^{p^s}({\varphi}(a))
(\tau^{p^s}(1\otimes e_i)-(1\otimes e_i))+
(\tau^{p^s}({\varphi}(a))-{\varphi}(a))(1\otimes e_i)$, it suffices to show $\tau^{p^s}({\varphi}(a))-{\varphi}(a)\in {\mathfrak{m}}_R^{\ge c_J}$. Write ${\varphi}(a)=\sum_{i\ge 0}a_iu^{pi}$ for some $a_i\in k$. Then we have $\tau^{p^s}({\varphi}(a))-{\varphi}(a)=
\sum_{i\ge 1}a_i(\underline{{\varepsilon}}^{p^{s+1}i}-1)u^{pi}$. Since we have $$v_R((\underline{{\varepsilon}}^{p^{s+1}i}-1)u^{pi})
=p^{s+1}v_R(\underline{{\varepsilon}}^i-1)+v_R(u^{pi})
> p^{s+2}/(p-1)\ge c_J$$ for any $i\ge 1$, we have done.

Recall that $r_0={\mathrm}{max}\{r'\in {\mathbb}{Z}_{\ge 0}; e(r'-1)<p-1\}$. Put $r_1:={\mathrm}{min}\{r,r_0\}$.

\[Cor1\] Assume that $p^{s+2}/(p-1)\ge c_J$. If ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$, then ${\mathfrak{M}}({\mathfrak{n}})$ has a structure of an object of ${\widetilde{{\mathrm}{Mod}}}^{r',\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ for any $r'\ge r_1$. Furthermore, if $c_J>pr_1/(p-1)$, it is uniquely determined. We denote this object by $\hat{{\mathfrak{M}}}({\mathfrak{n}})$.

We should remark that ${\mathfrak{M}}({\mathfrak{n}})$ is of height $\le r_1$ for any ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$. The uniqueness assertion follows from Proposition \[FFTHMMOD\].

\[tameres\] The functor from tamely ramified ${\mathbb}{Z}_p$-representations of $G_K$ to ${\mathbb}{Z}_p$-representations of $G_{\infty}$, obtained by restricting the action of $G_K$ to $G_{\infty}$, is fully faithful.

The result immediately follows from the fact that $G_K$ is topologically generated by $G_{\infty}$ and the wild inertia subgroup of $G_K$.

We remark that any semi-simple ${\mathbb}{F}_p$-representation of $G_K$ is automatically tame.

\[FFLEM\] Assume that $J\supset u^pI^{[1]}W(R)$ or $k$ is algebraically closed. Let $T\in {\mathrm}{Rep}_{{\mathrm}{tor}}(G_s)$ and $T'\in {\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$. Suppose that $T$ is tame, $pT=0$ and $T|_{G_{\infty}}\simeq T_{{\mathfrak{S}}}({\mathfrak{M}})$ for some ${\mathfrak{M}}\in {{\mathrm}{Mod}}^r_{/{\mathfrak{S}}_{\infty}}$. Furthermore, we suppose $p^{s+2}/(p-1)\ge c_J>pr/(p-1)$. Then all $G_{\infty}$-equivariant homomorphisms $T\to T'$ are in fact $G_s$-equivariant.

Let $L$ be the completion of the maximal unramified extension $K^{{\mathrm}{ur}}$ of $K$. By identifying $G_L$ with $G_{K^{{\mathrm}{ur}}}$, we may regard $G_L$ as a subgroup of $G_K$. Note that $L_{(s)}=K_{(s)}L$ is the completion of the maximal unramified extension of $K_{(s)}$, and $G_s$ is topologically generated by $G_{L,s}$ and $G_{\infty}$. Consider the following commutative diagram:

$\displaystyle \xymatrix{
{\mathrm}{Hom}_{G_{L,s}}(T,T')\ar@{^{(}->}[rr] & &   
{\mathrm}{Hom}_{G_{L,\infty}}(T,T') \\
{\mathrm}{Hom}_{G_s}(T,T') \ar@{^{(}->}[u] \ar@{^{(}->}[rr] & &   
{\mathrm}{Hom}_{G_{\infty}}(T,T'). \ar@{^{(}->}[u]
}$

Since $T'|_{G_{L,s}}$ is contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_{L,s},J}_{{\mathrm}{tor}}(G_{L,s})$ if $J\supset u^pI^{[1]}W(R)$ (cf. Lemma \[bc\]), the above diagram allows us to reduce a proof to the case where $k$ is algebraically closed. In the rest of this proof, we assume that $k$ is algebraically closed. Under this assumption, an ${\mathbb}{F}_p$-representation of $G_s$ is tame if and only if it is semi-simple by Maschke’s theorem. Thus we may also assume that $T$ is irreducible (here, we remark that any subquotient of $T$ is tame and, also remark that the essential image of $T_{{\mathfrak{S}}}\colon {{\mathrm}{Mod}}^r_{/{\mathfrak{S}}_{\infty}}\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$ is stable under subquotients in ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$). We claim that $T|_{G_{\infty}}$ is also irreducible. If not, there exists a non-zero irreducible ${\mathbb}{F}_p[G_{\infty}]$-submodule $W$ of $T|_{G_{\infty}}$. Let $K^{{\mathrm}{t}}_{(s)}$ be the maximal tamely ramified extension of $K_{(s)}$ and $I_{p,s}:={\mathrm}{Gal}(\overline{K}/K^{{\mathrm}{t}}_{(s)})$ the wild inertia subgroup of $G_s$. We see that $K^{{\mathrm}{t}}_{(s)}\cap K_{\infty}=K_{(s)}$. Since $G_{\infty}\cap I_{p,s}$ acts on $W$ trivially, the $G_{\infty}$-action on $W$ extends to $G_s$ via the composition map $G_s\twoheadrightarrow {\mathrm}{Gal}(K^{{\mathrm}{t}}_{(s)}/K_{(s)})
\simeq G_{\infty}/(G_{\infty}\cap I_{p,s})$. Thus we can regard $W$ as an irreducible ${\mathbb}{F}_p[G_s]$-module. By Lemma \[tameres\], we see that $W$ is a sub ${\mathbb}{F}_p[G_s]$-module of $T$. This contradicts the irreducibility of $T$ and the claim follows.

By the assumption on $T$, we have $T|_{G_{\infty}}\simeq T_{{\mathfrak{S}}}({\mathfrak{M}})\simeq T_{{\mathfrak{S}}}({\mathrm{Max}}^r({\mathfrak{M}}))$ for some ${\mathfrak{M}}\in {{\mathrm}{Mod}}^r_{/{\mathfrak{S}}_{\infty}}$. Since $T|_{G_{\infty}}$ is irreducible and $T_{{\mathfrak{S}}}\colon {\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$ is exact and fully faithful, we know that ${\mathrm{Max}}^r({\mathfrak{M}})$ is a simple object in the abelian category ${\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}$. Therefore, since $k$ is algebraically closed, we have ${\mathrm{Max}}^r({\mathfrak{M}})\simeq {\mathfrak{M}}({\mathfrak{n}})$ for some ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$ (cf. [@CL1 Propositions 3.6.8 and 3.6.12]). Let $\hat{{\mathfrak{M}}}({\mathfrak{n}})$ be the object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G},J}_{/{\mathfrak{S}}_{\infty}}$ as in Corollary \[Cor1\]. We recall that $T_{{\mathfrak{S}}}({\mathfrak{M}}({\mathfrak{n}}))$ is isomorphic to $\hat{T}_s(\hat{{\mathfrak{M}}}({\mathfrak{n}}))|_{G_{\infty}}$ (see Theorem \[Thm1\] (1)), and hence we have an isomorphism $T|_{G_{\infty}}\simeq \hat{T}_s(\hat{{\mathfrak{M}}}({\mathfrak{n}}))|_{G_{\infty}}$. Here, we note that $T$ and $\hat{T}_s(\hat{{\mathfrak{M}}}({\mathfrak{n}}))$ are irreducible as representations of $G_s$ (cf. [@CL1 Theorem 3.6.11]). Applying Lemma \[tameres\] again, we obtain an isomorphism $T\simeq \hat{T}_s(\hat{{\mathfrak{M}}}({\mathfrak{n}}))$ as representations of $G_s$. On the other hand, we can take $\hat{{\mathfrak{M}}}'=({\mathfrak{M}}',{\varphi},\hat{G}_s)\in  {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ such that $T'\simeq \hat{T}_s(\hat{{\mathfrak{M}}}')$. We consider the following commutative diagram:

$\displaystyle \xymatrix{
{\mathrm}{Hom}_{G_s}(T,T')\ar@{^{(}->}[rr] & &   
{\mathrm}{Hom}_{G_{\infty}}(T,T') \\
{\mathrm}{Hom}(\hat{{\mathfrak{M}}}',\hat{{\mathfrak{M}}}({\mathfrak{n}})) \ar^{\hat{T}_s}[u] \ar^{{\mathrm}{forgetful}\ }[r] &
{\mathrm}{Hom}({\mathfrak{M}}',{\mathfrak{M}}({\mathfrak{n}})) \ar^{{\mathrm{Max}}^r\quad \ \ }[r] & {\mathrm}{Hom}({\mathrm{Max}}^r({\mathfrak{M}}'),{\mathfrak{M}}({\mathfrak{n}})). 
\ar^{T_{{\mathfrak{S}}}}[u]. 
}$

Here, ${\mathrm}{Hom}(\hat{{\mathfrak{M}}}',\hat{{\mathfrak{M}}}({\mathfrak{n}}))$ (resp. ${\mathrm}{Hom}({\mathfrak{M}}',{\mathfrak{M}}({\mathfrak{n}}))$, resp. ${\mathrm}{Hom}({\mathrm{Max}}^r({\mathfrak{M}}'),{\mathfrak{M}}({\mathfrak{n}}))$) is the set of morphisms $\hat{{\mathfrak{M}}}'\to \hat{{\mathfrak{M}}}({\mathfrak{n}})$ (resp. ${\mathfrak{M}}'\to {\mathfrak{M}}({\mathfrak{n}})$, resp. ${\mathrm{Max}}^r({\mathfrak{M}}')\to {\mathfrak{M}}({\mathfrak{n}})$) in ${\widetilde{{\mathrm}{Mod}}}^{r_1,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ (resp. ${{\mathrm}{Mod}}^{r_1}_{/{\mathfrak{S}}_{\infty}}$, resp. ${\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}$). The first bottom horizontal arrow is bijective by Theorem \[Thm1\] (3) and so is the second (this follows from the fact that ${\mathfrak{M}}({\mathfrak{n}})$ is maximal by [@CL1 Proposition 3.6.7]). Since the right vertical arrow is bijective, the top horizontal arrow must be bijective.

Now we are ready to prove Theorem \[FFTHM\].

Let $T$ and $T'$ be objects of ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$. Take any Jordan-Höllder sequence $0=T_0\subset T_1\subset \cdots \subset T_n=T$ of $T$ in ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_s)$. By Corollary \[stability\], we know that $T_i$ and $T_i/T_{i-1}$ are contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ for any $i$. By Corollary \[stability\] again, the category ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ is an exact category in the sense of Quillen ([@Qu Section 2]). Hence short exact sequences in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ give rise to exact sequences of Hom’s and Ext’s in the usual way. (This property holds for any exact category.) On the other hand, by Lemma \[FFLEM\], if an exact sequence $0\to T'\to V\to T_i/T_{i-1}\to 0$ in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ splits as representation of $G_{\infty}$, then it splits as a sequence of representations of $G_s$. Therefore, comparing exact sequences of Hom’s and Ext’s arising from $0\to T_{i-1}\to T_i\to T_i/T_{i-1}\to 0$ in the category ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ with that in the category ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$, we obtain the following implication (here, use Lemma \[FFLEM\] again): if we have ${\mathrm}{Hom}_{G_s}(T_{i-1}, T')={\mathrm}{Hom}_{G_{\infty}}(T_{i-1}, T')$, then it gives the equality ${\mathrm}{Hom}_{G_s}(T_i, T')={\mathrm}{Hom}_{G_{\infty}}(T_i, T')$. Hence a dévissage argument works and the desired full faithfulness follows.

Proof of Theorem \[Main1\]
--------------------------

Now we are ready to prove our main theorems. First we prove Theorem \[Main1\].

Recall that ${\mathrm}{Rep}^{r,{\mathrm}{ht},{\mathrm}{pcris}(s)}_{{\mathrm}{tor}}(G_K)$ is the category of torsion ${\mathbb}{Z}_p$-representations $T$ of $G_K$ which satisfy the following: there exist free ${\mathbb}{Z}_p$-representations $L$ and $L'$ of $G_K$, of height $\le r$, such that

-   $L|_{G_s}$ is a subrepresentation of $L'|_{G_s}$. Furthermore, $L|_{G_s}$ and $L'|_{G_s}$ are lattices in some crystalline ${\mathbb}{Q}_p$-representation of $G_s$ with Hodge-Tate weights in $[0,r]$;

-   $T|_{G_s} \simeq (L'|_{G_s})/(L|_{G_s})$.

We apply our arguments given in previous subsections with the following $J$: $$J=u^pI^{[1]}W(R)=u^p{\varphi}({\mathfrak{t}})W(R).$$ Then we have $c_J=p/e+p/(p-1)$ and thus the inequalities $p^{s+2}/(p-1)\ge c_J>pr/(p-1)$ are satisfied if $e(r-1)<p-1$. Therefore, Theorem \[Main1\] is an easy consequence of the following proposition and Theorem \[FFTHM\].

\[proofMain1\] If $T$ is an object of ${\mathrm}{Rep}^{r,{\mathrm}{ht},{\mathrm}{pcris}(s)}_{{\mathrm}{tor}}(G_K)$, then $T|_{G_s}$ is contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$.

Take free ${\mathbb}{Z}_p$-representations $L$ and $L'$ of $G_K$, of height $\le r$, such that

-   $L|_{G_s}$ is a subrepresentation of $L'|_{G_s}$. Furthermore, $L|_{G_s}$ and $L'|_{G_s}$ are lattices in some crystalline ${\mathbb}{Q}_p$-representation of $G_s$ with Hodge-Tate weights in $[0,r]$;

-   $T|_{G_s} \simeq (L'|_{G_s})/(L|_{G_s})$.

By Theorem \[Thm1\] (1), there exists an injection $\hat{{\mathfrak{L}}}'\hookrightarrow \hat{{\mathfrak{L}}}$ of $({\varphi},\hat{G}_s)$-modules over ${\mathfrak{S}}_s$ which corresponds to the injection $L|_{G_s}\hookrightarrow L'|_{G_s}$. On the other hand, there exist ${\mathfrak{N}}$ and ${\mathfrak{N}}'$ in ${{\mathrm}{Mod}}^r_{/{\mathfrak{S}}}$ such that $T_{{\mathfrak{S}}}({\mathfrak{N}})\simeq L|_{G_{\infty}}$ and $T_{{\mathfrak{S}}}({\mathfrak{N}}')\simeq L'|_{G_{\infty}}$. Then Proposition \[Kisinfunctor\] (1) implies that ${\mathfrak{S}}_s\otimes_{{\mathfrak{S}}} {\mathfrak{N}}\simeq {\mathfrak{L}}$ and ${\mathfrak{S}}_s\otimes_{{\mathfrak{S}}} {\mathfrak{N}}'\simeq {\mathfrak{L}}'$ as ${\varphi}$-modules over ${\mathfrak{S}}_s$. Therefore, we see that ${\mathfrak{N}}$ and ${\mathfrak{N}}'$ have structures of objects of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$; denote them by $\hat{{\mathfrak{N}}}$ and $\hat{{\mathfrak{N}}}'$, respectively. Since the functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}$ is fully faithful (cf. Section \[vardef\]), the injection $\hat{{\mathfrak{L}}}'\hookrightarrow \hat{{\mathfrak{L}}}$ descends to an injection $\hat{{\mathfrak{N}}}'\hookrightarrow \hat{{\mathfrak{N}}}$. Now we put ${\mathfrak{M}}={\mathfrak{N}}/{\mathfrak{N}}'$. Since $(L'|_{G_s})/(L|_{G_s})$ is killed by a power of $p$, it is an object of ${{\mathrm}{Mod}}^r_{/{\mathfrak{S}}_{\infty}}$. We equip a $\hat{G}_s$-action with ${\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$ by a natural isomorphism ${\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}\simeq 
({\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{N}})/({\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{N}}')$. Then we see that ${\mathfrak{M}}$ has a structure of an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$; denote it by $\hat{{\mathfrak{M}}}$. Moreover, Theorem \[cris\] implies that $\hat{{\mathfrak{M}}}$ is in fact contained in ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$. By a similar argument to the proof of Lemma 3.1.4 of [@CL2], we have an exact sequence $0\to \hat{T}_s(\hat{{\mathfrak{N}}})\to \hat{T}_s(\hat{{\mathfrak{N}}}')\to \hat{T}_s(\hat{{\mathfrak{M}}})\to 0$ in ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_s)$, which is isomorphic to $0\to L|_{G_s}\to L'|_{G_s}\to T|_{G_s}\to 0$. This finishes a proof.

Proof of Theorem \[Main2\]
--------------------------

We give a proof of Theorem \[Main2\]. If $s\ge n-1$, then we put $$J=u^pI^{[p^{s-n+1}]}W(R)=u^p{\varphi}({\mathfrak{t}})^{p^{s-n+1}}W(R).$$ Note that we have $c_J=p/e+p^{s-n+2}/(p-1)$ and thus the inequalities $p^{s+2}/(p-1)\ge c_J>pr/(p-1)$ are satisfied if $s>n-1+{\mathrm}{log}_p(r-e/(p-1))$.

\[Main2Lem\] Suppose $s\ge n-1$. If $T$ is an object of ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ which is killed by $p^n$, then $T|_{G_s}$ is contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$.

Let $L$ be an object of ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathbb}{Z}_p}(G_K)$. Take a $({\varphi},\hat{G})$-module $\hat{{\mathfrak{L}}}$ over ${\mathfrak{S}}$ such that $L\simeq \hat{T}(\hat{{\mathfrak{L}}})$. It is known that $(\tau-1)^i(x)\in u^pI^{[i]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{L}}$ for any $i\ge 1$ and any $x\in {\mathfrak{L}}$ (cf. the latter half part of the proof of [@GLS Proposition 4.7]). Take any $x\in {\mathfrak{L}}$. Since $(\tau^{p^s}-1)(x)=\sum^{p^s}_{i=1}\binom{p^s}{i}(\tau-1)^i(x)$, we obtain that $$\label{relat}
(\tau^{p^s}-1)(x)\in \sum^{p^s}_{i=1} p^{s-v_p(i)}u^pI^{[i]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{L}}.$$ Now let $T$ be an object of ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ which is killed by $p^n$. Take an exact sequence $(R)\colon 0\to L_1\to L_2\to T\to 0$ of ${\mathbb}{Z}_p$-representations of $G_K$ with $L_1,L_2\in {\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathbb}{Z}_p}(G_K)$. By Theorem 3.1.3 and Lemma 3.1.4 of [@CL2], there exists an exact sequence $(M)\colon 0\to \hat{{\mathfrak{L}}}_2\to \hat{{\mathfrak{L}}}_1\to \hat{{\mathfrak{M}}}\to 0$ of $({\varphi},\hat{G})$-modules over ${\mathfrak{S}}$ such that $\hat{T}((M))\simeq (R)$. By (\[relat\]), we see that $$(\tau^{p^s}-1)(x)\in \sum^{p^s}_{i=1} p^{s-v_p(i)}u^pI^{[i]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$$ for any $x\in {\mathfrak{M}}$. Since ${\mathfrak{M}}$ is killed by $p^n$ and $s\ge n-1$, we have $$\begin{aligned}
\sum^{p^s}_{i=1} p^{s-v_p(i)}u^pI^{[i]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}&= \sum_{i=1,\dots ,p^s, s-v_p(i)<n} p^{s-v_p(i)}u^pI^{[i]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}\\
&= \sum^{n-1}_{\ell=0} p^{\ell}u^pI^{[p^{s-\ell}]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}\\
& \subset u^pI^{[p^{s-n+1}]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}.\end{aligned}$$ Therefore, we obtained the desired result.

By Corollary \[FFTHMtorcris\], we may suppose ${\mathrm}{log}_p(r-(p-1)/e)\ge 0$, that is, $e(r-1)\ge p-1$ . Suppose $s>  n-1+{\mathrm}{log}_p(r-(p-1)/e)$. Note that the condition $s\ge n-1$ is now satisfied. Let $T$ and $T'$ be as in the statement of Theorem \[Main2\]. Let $f\colon T\to T'$ be a $G_{\infty}$-equivariant homomorphism. Denote by $L$ the completion of $K^{{\mathrm}{ur}}$ and identify $G_L$ with the inertia subgroup of $G_K$. We note that $T|_{G_L}$ and $T'|_{G_L}$ are object of ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_L)$. By Proposition \[Main2Lem\], $T|_{G_{L,s}}$ and $T'|_{G_{L,s}}$ are objects of ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_{L,s},J}_{{\mathrm}{tor}}(G_{L,s})$. Hence we have that $f$ is $G_{L,s}$-equivariant by Theorem \[FFTHM\]. Since $G_s$ is topologically generated by $G_{L,s}$ and $G_{\infty}$, we see that $f$ is $G_s$-equivariant.

Galois equivariance for torsion semi-stable representations {#torsemi}
-----------------------------------------------------------

In this subsection, we prove a Galois equivariance theorem for torsion semi-stable representations. A torsion ${\mathbb}{Z}_p$-representation $T$ of $G_K$ is [*torsion semi-stable with Hodge-Tate weights in $[0,r]$*]{} if it can be written as the quotient of lattices in some semi-stable ${\mathbb}{Q}_p$-representation of $G_K$ with Hodge-Tate weights in $[0,r]$. We denote by ${\mathrm}{Rep}^{r, {\mathrm}{st}}_{{\mathrm}{tor}}(G_K)$ the category of them. Note that ${\mathrm}{Rep}^{0, {\mathrm}{st}}_{{\mathrm}{tor}}(G_K)=
{\mathrm}{Rep}^{0, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$. Similar to Theorem \[Main2\], we show the following, which is the main result of this subsection.

\[Main3\] Suppose that $s> n-1 + {\mathrm}{log}_pr$. Let $T$ and $T'$ be objects of ${\mathrm}{Rep}^{r, {\mathrm}{st}}_{{\mathrm}{tor}}(G_K)$ which are killed by $p^n$. Then any $G_{\infty}$-equivariant homomorphism $T\to T'$ is in fact $G_s$-equivariant.

If $s\ge n-1$, then we put $$J=I^{[p^{s-n+1}]}W(R)={\varphi}({\mathfrak{t}})^{p^{s-n+1}}W(R).$$ Then we have $c_J=p^{s-n+2}/(p-1)$. To show Theorem \[Main3\], we use similar arguments to those in the proof of Theorem \[Main2\].

\[Main3Lem\] Suppose $s\ge n-1$. If $T$ is an object of ${\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathrm}{tor}}(G_K)$ which is killed by $p^n$, then $T|_{G_s}$ is contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$.

Let $L$ be a lattice in a semi-stable ${\mathbb}{Q}_p$-representation of $G_K$ with Hodge-Tate weights in $[0,r]$. Take a $({\varphi},\hat{G})$-module $\hat{{\mathfrak{L}}}$ over ${\mathfrak{S}}$ such that $L\simeq \hat{T}(\hat{{\mathfrak{L}}})$. It is known that $(\tau-1)^i(x)\in I^{[i]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{L}}$ for any $i\ge 1$ and any $x\in {\mathfrak{L}}$ (cf. the proof of [@Li4 Proposition 2.4.1]). Thus the same proof proceeds as that of Proposition \[Main2Lem\].

We have the equality ${\mathrm}{Rep}^{0, {\mathrm}{st}}_{{\mathrm}{tor}}(G_K)={\mathrm}{Rep}^{0, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ and thus Theorem \[Main2\] for $r=0$ is an easy consequence of Corollary \[FFTHMtorcris\]. Hence we may assume $r\ge 1$. The rest of a proof is similar to the proof of Theorem \[Main2\].

Some consequences {#consequences}
-----------------

In this subsection, we generalize some results proved in Section 3.4 of [@Br3]. First of all, we show the following elementary lemma, which should be well-known to experts, but we include a proof here for the sake of completeness.

\[stability’\] The full subcategories ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ and ${\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathrm}{tor}}(G_K)$ of ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_K)$ are stable under formation of subquotients, direct sums and the association $T\mapsto T^{\vee}(r)$. Here $T^{\vee}={\mathrm}{Hom}_{{\mathbb}{Z}_p}(T,{\mathbb}{Q}_p/{\mathbb}{Z}_p)$ is the dual representation of $T$.

We prove the statement only for ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$. Let $T\in {\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ be killed by $p^n$ for some $n>0$. Assertions for quotients and direct sums are clear. We prove that $T^{\vee}(r)$ is contained in ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$. There exist lattices $L_1\subset L_2$ in some crystalline ${\mathbb}{Q}_p$-representation of $G_K$ and an exact sequence $0\to L_1\to L_2\to T\to 0$ of ${\mathbb}{Z}_p[G_K]$-modules. This exact sequence induces an exact sequence $0\to T\to L_1/p^nL_1\to L_2/p^nL_2\to T\to 0$ of finite ${\mathbb}{Z}_p[G_K]$-modules. By duality, we obtain an exact sequence $0\to T^{\vee}\to (L_2/p^nL_2)^{\vee}\to 
(L_1/p^nL_1)^{\vee}\to T^{\vee}\to 0$ of finite ${\mathbb}{Z}_p[G_K]$-modules. Then we obtain a $G_K$-equivariant surjection $L_1^{\vee}\twoheadrightarrow T^{\vee}$ by the composite $L_1^{\vee}\twoheadrightarrow L_1^{\vee}/p^nL_1^{\vee}\overset{\sim}{\to} 
(L_1/p^nL_1)^{\vee}\twoheadrightarrow T^{\vee}$ of natural maps (here, for any free ${\mathbb}{Z}_p$-representation $L$ of $G_K$, $L^{\vee}:={\mathrm}{Hom}_{{\mathbb}{Z}_p}(L,{\mathbb}{Z}_p)$ stands for the dual of $L$). Therefore, we obtain $L_1^{\vee}(r)\twoheadrightarrow T^{\vee}(r)$ and thus $T^{\vee}(r)\in {\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$. Finally, we prove the stability assertion for subobjects. Let $T'$ be a $G_K$-stable submodule of $T$. We have a $G_K$-equivariant surjection $f\colon L_1^{\vee}\twoheadrightarrow T^{\vee}\twoheadrightarrow (T')^{\vee}$. Let $L'_2$ be a free ${\mathbb}{Z}_p$-representation of $G_K$ such that its dual is the kernel of $f$. We have an exact sequence $0\to (L'_2)^{\vee}\to L^{\vee}_1\overset{f}{\to} (T')^{\vee}\to 0$ of ${\mathbb}{Z}_p[G_K]$-modules. Repeating the construction of the surjection $L_1^{\vee}\twoheadrightarrow T^{\vee}$, we obtain a $G_K$-equivariant surjection $L'_2=(L'_2)^{\vee \vee}\twoheadrightarrow (T')^{\vee \vee}=T'$ and thus we have $T'\in {\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$.

In the case where $r=1$, the assertion (1) of the following corollary was shown in Theorem 3.4.3 of [@Br3].

\[imagestable\] Let $T$ be an object of ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ which is killed by $p^n$ for some $n>0$. Let $T'$ be a $G_{\infty}$-stable subquotient of $T$.

$(1)$ If $e(r-1)<p-1$, then $T'$ is $G_K$-stable $($with respect to $T)$.

$(2)$ If $s>n-1+{\mathrm}{log}_p(r-(p-1)/e)$, then $T'$ is $G_s$-stable $($with respect to $T)$.

By the duality assertion of Lemma \[stability’\], it is enough to show the case where $T'$ is a $G_{\infty}$-stable submodule of $T$. Take any sequence $T'=T_0\subset T_1\subset \cdots \subset T_m=T$ of torsion $G_{\infty}$-stable submodules of $T$ such that $T_i/T_{i-1}$ is irreducible for any $i$. As explained in the proof of Proposition \[FFLEM\], the $G_{\infty}$-action on $T_i/T_{i-1}$ can be (uniquely) extended to $G_K$. By Theorem \[tamelift\] given in the next section, we know that $T_i/T_{i-1}$ is an object of ${\mathrm}{Rep}^{r_0,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ where $r_0:={\mathrm}{max}\{ r'\in {\mathbb}{Z}_{\ge 0}; e(r'-1)<p-1 \}$.

\(1) We may suppose $r=r_0$. The $G_{\infty}$-equivariant projection $T=T_m\twoheadrightarrow T_m/T_{m-1}$ is $G_K$-stable by the full faithfulness theorem (= Corollary \[FFTHMtorcris\]). Thus we know that $T_{m-1}$ is $G_K$-stable in $T$, and also know that $T_{m-1}$ is contained in ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ by Lemma \[stability’\]. By the same argument for the $G_{\infty}$-equivariant projection $T_{m-1}\twoheadrightarrow T_{m-1}/T_{m-2}$, we know that $T_{m-2}$ is $G_K$-stable in $T$, and also know that $T_{m-2}$ is contained in ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$. Repeating this argument, we have that $T'=T_0$ is $G_K$-stable in $T$.

\(2) Put $J=u^pI^{[p^{s-n+1}]}W(R)$. By (1) we may assume $e(r-1)\ge p-1$. Under this assumption we have $r\ge r_0$ and $s>n-1+{\mathrm}{log}_p(r-(p-1)/e)\ge n-1$. In particular, $T|_{G_s}$ and $(T_i/T_{i-1})|_{G_s}$, for any $i$, are contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ by Proposition \[Main2Lem\]. First we consider the case where $k$ is algebraically closed. By Theorem \[FFTHM\], the $G_{\infty}$-equivariant projection $T=T_m\twoheadrightarrow T_m/T_{m-1}$ is $G_s$-stable. Thus we know that $T_{m-1}$ is $G_s$-stable in $T$, and also know that $T_{m-1}$ is contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ by Corollary \[stability\]. By the same argument for the $G_{\infty}$-equivariant projection $T_{m-1}\twoheadrightarrow T_{m-1}/T_{m-2}$, we know that $T_{m-2}$ is $G_s$-stable in $T$, and also know that $T_{m-2}$ is contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$. Repeating this argument, we have that $T'=T_0$ is $G_s$-stable in $T$. Next we consider the case where $k$ is not necessary algebraically closed. Let $L$ be the completion of the maximal unramified extension $K^{{\mathrm}{ur}}$ of $K$, and we identify $G_L$ with the inertia subgroup of $G_K$. Clearly $T|_{G_L}$ is contained in ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_L)$ and $T'$ is $G_{L_{\infty}}$-stable submodule of $T$. We have already shown that $T'$ is $G_{L,s}$-stable in $T$. Since $G_s$ is topologically generated by $G_{L,s}$ and $G_{\infty}$, we conclude that $T'$ is $G_s$-stable in $T$.

Now let $V$ be a ${\mathbb}{Q}_p$-representation of $G_K$ and $T$ a ${\mathbb}{Z}_p$-lattice of $V$ which is stable under $G_{\infty}$. Then we know that $T$ is automatically $G_s$-stable for some $s\ge 0$. Indeed we can check this as follows. Take any $G_K$-stable ${\mathbb}{Z}_p$-lattice $T'$ of $V$ which contains $T$, and take an integer $n>0$ with the property that $p^nT'\subset T$. Furthermore, we take a finite extension $K'$ of $K$ such that $G_{K'}$ acts trivially on $T'/p^nT'$. Then $T/p^nT'$ is $G_{\infty}$-stable and also $G_{K'}$-stable in $T'/p^nT'$. If we take any integer $s\ge 0$ with the property $K'\cap K_{\infty}\subset K_{(s)}$, we know that $T/p^nT'$ is $G_s$-stable. This implies that $T$ is $G_s$-stable in $T'$.

The following corollary, which was shown in Corollary 3.4.4 of [@Br3] in the case where $r=1$, is related with the above property.

\[stablelattice\] Let $V$ be a crystalline ${\mathbb}{Q}_p$-representation of $G_K$ with Hodge-Tate weights in $[0,r]$ and $T$ a finitely generated ${\mathbb}{Z}_p$-submodule of $V$ which is stable under $G_{\infty}$. If $e(r-1)<p-1$, then $T$ is stable under $G_K$.

We completely follow the method of the proof of [@Br3 Corollary 3.4.4]. Take any $G_K$-stable ${\mathbb}{Z}_p$-lattice $T'$ of $V$ which contains $T$. Since $T'/p^nT'$ is contained in ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ for any $n>0$, Corollary \[imagestable\] (1) implies that any $G_{\infty}$-stable submodule of $T'/p^nT'$ is in fact $G_K$-stable. Thus $(T+p^nT')/p^nT'$ is $G_K$-stable in $T'/p^nT'$. Therefore, we obtain $g(T)\subset \bigcap_{n>0}\ (T+p^nT')=T$ for any $g\in G_K$.

Crystalline lifts and c-weights
===============================

We continue to use the same notation except for that we may allow $p=2$. We remark that a torsion ${\mathbb}{Z}_p$-representation of $G_K$ is torsion crystalline with Hodge-Tate weights in $[0,r]$ if there exist a lattice $L$ in some crystalline ${\mathbb}{Q}_p$-representation of $G_K$ with Hodge-Tate weights in $[0,r]$ and a $G_K$-equivariant surjection $f\colon L\twoheadrightarrow T$. We call $f$ a [*crystalline lift*]{} ([*of $T$*]{}) [*of weight $\le r$*]{}. Our interest in this section is to determine the minimum integer $r$ (if it exists) such that $T$ admits crystalline lifts of weight $\le r$. We call this minimum integer the [*c-weight of $T$*]{} and denote it by $w_c(T)$. If $T$ does not have crystalline lifts of weight $\le r$ for any integer $r$, then we define the c-weight $w_c(T)$ of $T$ to be $\infty$. Motivated by [@CL2 Question 5.5], we raise the following question.

For a torsion ${\mathbb}{Z}_p$-representation $T$ of $G_K$, is the c-weight $w_c(T)$ of $T$ finite? Furthermore, can we calculate $w_c(T)$?

General properties of c-weights
-------------------------------

We study general properties of c-weights. At first, by ramification estimates, it is known that c-weights may have infinitely large values ([@CL2 Theorem 5.4]); for any $c>0$, there exists a torsion ${\mathbb}{Z}_p$-extension $T$ of $G_K$ with $w_c(T)>c$. In this paper, we mainly consider representations with “small” c-weights. If c-weights are “small”, they are closely related with [*tame inertia weights*]{}. Now we recall the definition of tame inertia weights. Let $I_K$ be the inertia subgroup of $G_K$. Let $T$ be a $d$-dimensional irreducible ${\mathbb}{F}_p$-representation of $I_K$. Then $T$ is isomorphic to $${\mathbb}{F}_{p^d}(\theta^{n_1}_{d,1}\cdots \theta^{n_d}_{d,d})$$ for one sequence of integers between $0$ and $p-1$, periodic of period $d$. Here, $\theta_{d,1},\dots , \theta_{d,d}$ are the fundamental characters of level $d$. The integers $n_1/e,\dots ,n_d/e$ are called the tame inertia weights of $T$. For any ${\mathbb}{F}_p$-representation $T$ of $G_K$, the tame inertia weights of $T$ are the tame inertia weights of the Jordan-Hölder quotients of $T|_{I_K}$.

Let $\chi_p\colon G_K\to {\mathbb}{Z}^{\times}_p$ be the $p$-adic cyclotomic character and $\bar{\chi}_p\colon G_K\to {\mathbb}{F}^{\times}_p$ the mod $p$ cyclotomic character. It is well-known that $\bar{\chi}_p|_{I_K}=\theta_1^e$ where $\theta_1\colon I_K\twoheadrightarrow {\mathbb}{F}^{\times}_p$ is the fundamental character of level $1$. In particular, denoting by $K^{{\mathrm}{ur}}$ the maximal unramified extension of $K$, we have $[K^{{\mathrm}{ur}}(\mu_p):K^{{\mathrm}{ur}}]=(p-1)/{\mathrm}{gcd}(e,p-1)$.

\[poly\] $(1)$ Minimum c-weights are invariant under finite unramified extensions of the base field $K$.

$(2)$ The c-weight of an unramified torsion ${\mathbb}{Z}_p$-representation of $G_K$ is $0$.

$(3)$ Put $\nu=(p-1)/{\mathrm}{gcd}(e,p-1)$. If $\nu (s-1)<w_c(T)\le \nu s$, then we have $\nu (s-1)<w_c(T^{\vee})\le \nu s$. In particular, if $(p-1)\mid e$, then we have $w_c(T)=w_c(T^{\vee})$.

$(4)$ Let $T$ be an ${\mathbb}{F}_p$-representation of $G_K$ and $i$ the largest tame inertia weight of $T$. Then we have $w_c(T)\geq i$.

\(1) Let $T$ be a torsion ${\mathbb}{Z}_p$-representations of $G_K$. Let $K'$ be a finite unramified extension of $K$. It suffices to prove that $T$ has crystalline lifts of weight $\le r$ if and only if $T|_{G_{K'}}$ has crystalline lifts of weight $\le r$. The “only if” assertion is clear and thus it is enough to prove the “if” assertion. Let $f\colon L\twoheadrightarrow T|_{G_{K'}}$ be a crystalline lift of $T|_{G_{K'}}$ of weight $\le r$. Since $K'/K$ is unramified, ${\mathrm}{Ind}^{G_K}_{G_{K'}}L$ is a lattice in some crystalline ${\mathbb}{Q}_p$-representation of $G_K$ with Hodge-Tate weights in $[0,r]$. Furthermore, the map $${\mathrm}{Ind}^{G_K}_{G_{K'}}L={\mathbb}{Z}_p[G_K]\otimes_{{\mathbb}{Z}_p[G_{K'}]} L\to T,\quad
\sigma\otimes x\mapsto \sigma(f(x))$$ is a $G_K$-equivariant surjection and hence we have done.

\(2) The result follows from (1) immediately.

\(3) Taking a finite unramified extension $K'$ of $K$ with the property $[K^{{\mathrm}{ur}}(\mu_p):K^{{\mathrm}{ur}}]=[K'(\mu_p):K']$, it follows from Lemma \[stability’\] that we have $\nu (s-1)<w_c(T|_{G_K'})\le \nu s$ if and only if we have $\nu (s-1)<w_c((T^{\vee})|_{G_K'})\le \nu s$. Thus the result follows from the assertion (1).

\(4) If $ew_c(T)\ge p-1$, then there is nothing to prove, and thus we may suppose that $ew_c(T)< p-1$. Let $L\twoheadrightarrow T$ be a crystalline lift of $T$ of weight $\le w_c(T)$. Since the tame inertia polygon of $L$ lies on the Hodge polygon of $L$ ([@CS Théorème 1]), the largest slope of the former polygon is less than or equal to that of the latter polygon. This implies $w_c(T)\geq i$.

\[tamelift\] Let $T$ be a tamely ramified ${\mathbb}{F}_p$-representation of $G_K$. Let $i$ be the largest tame inertia weight of $T$. Then we have $w_c(T)={\mathrm}{min}\{ h\in {\mathbb}{Z}_{\ge 0}; h\ge i \}$.

The proof below is essentially due to Caruso and Liu [@CL2 Theorem 5.7], but we give a proof here for the sake of completeness. Put $i_0={\mathrm}{min}\{ h\in {\mathbb}{Z}_{\ge 0}; h\ge i \}$. By Proposition \[poly\] (4), we have $w_c(T)\ge i_0$. Thus it suffices to show $w_c(T)\le i_0$. We note that $T|_{I_K}$ is semi-simple. Any irreducible component $T_0$ of $T|_{I_K}$ is of the form $
{\mathbb}{F}_{p^d}(\theta^{n_1}_{d,1}\cdots \theta^{n_d}_{d,d})
$ for one sequence of integers between $0$ and $p-1$, periodic of period $d$. We decompose $n_j=em_j+n'_j$ by integers $0\le m_j\le i_0$ and $0\le n'_j<e$. Now we define an integer $k_{j, \ell}$ by $$\begin{aligned}
k_{j, \ell}:=
\left\{
\begin{array}{ll}
e\quad \hspace{2.0mm} {\rm if}\ 1\le \ell\le m_j,  
\cr
n'_j\quad {\rm if}\ \ell=m_j+1, 
\cr
0\quad \hspace{2.0mm} {\rm if}\ \ell>m_j+1. 
\end{array}
\right.\end{aligned}$$ Note that we have $n_j=\sum^{i_0}_{\ell=1} k_{j, \ell}$, and also have an $I_K$-equivariant surjection $$T_0={\mathbb}{F}_{p^d}(\theta^{n_1}_{d,1}\dots \theta^{n_d}_{d,d})
=\bigotimes_{\ell=1,\dots i_0, {\mathbb}{F}_{p^d}} 
{\mathbb}{F}_{p^d}(\theta^{k_{1,\ell}}_{d,1}\dots \theta^{k_{d,\ell}}_{d,d})
\twoheadleftarrow \bigotimes_{\ell=1,\dots i_0, {\mathbb}{F}_p} 
{\mathbb}{F}_{p^d}(\theta^{k_{1,\ell}}_{d,1}\dots \theta^{k_{d,\ell}}_{d,d}).$$ By a classical result of Raynaud, each ${\mathbb}{F}_{p^d}(\theta^{k_{1,\ell}}_{d,1}\cdots \theta^{k_{d,\ell}}_{d,d})$ comes from a finite flat group scheme defined over $K^{{\mathrm}{ur}}$. We should remark that such a finite flat group scheme is in fact defined over a finite unramified extension of $K$. Since any finite flat group scheme can be embedded in a $p$-divisible group, the above observation implies the following: there exist a finite unramified extension $K'$ over $K$, a lattice $L$ in some crystalline ${\mathbb}{Q}_p$-representation of $G_{K'}$ with Hodge-Tate weights in $[0, i_0]$ and an $I_K$-equivariant surjection $f\colon L\twoheadrightarrow T$. The map $f$ induces an $I_K$-equivariant surjection $\tilde{f}\colon L/pL\twoheadrightarrow T$. Since $L/pL$ and $T$ is finite, we see that $\tilde{f}$ is in fact $G_{K''}$-equivariant for some finite unramified extension $K''$ over $K'$, and then so is $f$. Therefore, we obtain $w_c(T|_{G_{K''}})\le i_0$. By Proposition \[poly\] (1), we obtain $w_c(T)\le i_0$.

Rank $2$ cases
--------------

We give some computations of c-weights related with torsion representations of rank $2$. We prove the following lemma by an almost identical method with [@GLS Lemma 9.4].

\[2liftlem\] Let $K$ be a finite extension of ${\mathbb}{Q}_p$. Let $E$ be a finite extension of ${\mathbb}{Q}_p$ with residue field ${\mathbb}{F}$. Let $i$ and $\nu$ be integers such that $\nu$ is divisible by $[K(\mu_p):K]$. Suppose that $T$ is an ${\mathbb}{F}$-representation of $G_K$ which sits in an exact sequence $(\ast)\colon 0\to {\mathbb}{F}(i)\to T\to {\mathbb}{F}\to 0$ of ${\mathbb}{F}$-representations of $G_K$. Then there exist a ramified degree at most $2$ extension $E'$ over $E$, with integer ring ${\mathcal{O}}_{E'}$, and an unramified continuous character $\chi\colon G_K\to {\mathbb}{F}^{\times}$ with trivial reduction such that $(\ast)$ is the reduction of some exact sequence $0\to {\mathcal{O}}_{E'}(\chi \chi_p^{i+\nu})\to \Lambda\to {\mathcal{O}}_{E'}\to 0$ of free ${\mathcal{O}}_{E'}$-representations of $G_K$. Furthermore, we have the followings:

$(1)$ If $i+\nu=1$ or $\bar{\chi}_p^{1-i}\not= 1$, then we can take $E'=E$ and $\chi=1$.

$(2)$ If $i+\nu=0$ and $T$ is unramified, then we can take $E'=E$, $\chi=1$ and $\Lambda$ to be unramified.

Suppose $i+\nu=1$ (resp. $\bar{\chi}_p^{1-i}\not= 1$). Then the map $H^1(K, {\mathcal{O}}_E(i+\nu))\to H^1(K,{\mathbb}{F}(i))$ arising from the exact sequence $0\to {\mathcal{O}}_E(i+\nu)\overset{\varpi}{\to} {\mathcal{O}}_E(i+\nu)\to {\mathbb}{F}(i)\to 0$ is surjective since $H^2(K, {\mathcal{O}}_E(1))\simeq {\mathcal{O}}_E$ (resp. $H^2(K, {\mathcal{O}}_E(i+\nu))=0$), where $\varpi$ is a uniformizer of $E$. Hence we obtained a proof of (1). The assertion (2) follows immediately from the fact that the natural map $H^1(G_K/I_K, {\mathcal{O}}_E)\to H^1(G_K/I_K, {\mathbb}{F})$ is surjective.

In the rest of this proof, we always assume that $i+\nu\not=1$ and $\bar{\chi}_p^{1-i}= 1$. Let $L\in H^1(K,{\mathbb}{F}(i))$ be a $1$-cocycle corresponding to $(\ast)$. We may suppose $L \not=0$. For any unramified continuous character $\chi\colon G_K\to {\mathbb}{F}^{\times}$ with trivial reduction, we denote by $$\begin{aligned}
&\delta^1_{\chi}\colon H^1(K,{\mathbb}{F}(i))\to H^2(K, {\mathcal{O}}_E(\chi\chi_p^{i+\nu}))\\
({\mathrm}{resp.}\ &\delta^0_{\chi}\colon 
H^0(K, E/{\mathcal{O}}_E(\chi^{-1}\chi_p^{1-i-\nu}))\to H^1(K, {\mathbb}{F}))\end{aligned}$$ the connection map arising from the exact sequence $0\to {\mathcal{O}}_E(\chi\chi_p^{i+\nu})\overset{\varpi}{\to} 
{\mathcal{O}}_E(\chi\chi_p^{i+\nu}) \to {\mathbb}{F}(i)\to 0$ (resp.  $0\to {\mathbb}{F}\to E/{\mathcal{O}}_E(\chi^{-1}\chi_p^{1-i-\nu})\overset{\varpi}{\to} 
E/{\mathcal{O}}_E(\chi^{-1}\chi_p^{1-i-\nu})\to 0$) of ${\mathcal{O}}_E[G_K]$-modules. Consider the following commutative diagram:

$\displaystyle \xymatrix{
H^1(K,{\mathbb}{F}(i)) \ar_{\delta^1_{\chi}}[d] 
& \times 
& H^1(K,{\mathbb}{F}) \ar[rrr] 
& &
& E/{\mathcal{O}}_E \ar@{=}[d] \\
H^2(K, {\mathcal{O}}_E(\chi\chi_p^{i+\nu})) 
& \times 
& H^0(K, E/{\mathcal{O}}_E(\chi^{-1}\chi_p^{1-i-\nu})) \ar^{\delta^0_{\chi}}[u] \ar[rrr] 
& &
& E/{\mathcal{O}}_E
}$

Since the above two pairings are perfect, we see that $L$ lifts to $H^1(G_K, {\mathcal{O}}_E(\chi\chi_p^{i+\nu}))$ if and only if $H$ is contained in the image of $\delta^0_{\chi}$. Here, $H\subset H^1(K,{\mathbb}{F})$ is the annihilator of $L$ under the local Tate pairing $H^1(K,{\mathbb}{F}(i)) \times H^1(K,{\mathbb}{F}) \to E/{\mathcal{O}}_E$. Let $n\ge 1$ be the largest integer with the property that $\chi^{-1}\chi_p^{1-i-\nu}\equiv 1\ {\mathrm}{mod}\ \varpi^n$ (such $n$ exists since $\bar{\chi}_p^{1-i}=1$ and $1-i-\nu\not=0$). We define $\alpha_{\chi}\colon G_K\to {\mathcal{O}}_E$ by the relation $\chi^{-1}\chi_p^{1-i-\nu}=1+\varpi^n\alpha_{\chi}$, and denote $(\alpha_{\chi}\ {\mathrm}{mod}\ \varpi)\colon G_K\to {\mathbb}{F}$ by $\bar{\alpha}_{\chi}$. By definition, $\bar{\alpha}_{\chi}$ is a non-zero element of $H^1(K,{\mathbb}{F})$, and it is not difficult to check that the image of $\delta^0_{\chi}$ is generated by $\bar{\alpha}_{\chi}$. If $\bar{\alpha}_{\chi}$ is contained in $H$ for some $\chi$, we are done. Suppose this is not the case.

Suppose that $H$ is not contained in the unramified line in $H^1(K,{\mathbb}{F})$. We claim that we can choose $\chi$ such that $\bar{\alpha}_{\chi}$ is ramified. Let $m$ be the largest integer with the property that $(\chi^{-1}\chi_p^{1-i-\nu})|_{I_K}\equiv 1\ {\mathrm}{mod}\ \varpi^n$. Clearly, we have $m\ge n$. If $m=n$, then we are done and thus we may assume $m>n$. Fix a lift $g\in G_K$ of the Frobenius of $K$. We see that $\bar{\alpha}_{\chi}(g)\not= 0$. Let $\chi'$ be the unramified character sending $g$ to $1+\varpi^n\alpha_{\chi}(g)$. Then $\chi'$ has trivial reduction. After replacing $\chi$ with $\chi\chi'$, we reduce the case where $m=n$ and thus the claim follows. Suppose $\bar{\alpha}_{\chi}$ is ramified. Then there exists a unique $\bar{x}\in {\mathbb}{F}^{\times}$ such that $\bar{\alpha}_{\chi}+u_{\bar x}\in H$ where $u_{\bar x}\colon G_K\to {\mathbb}{F}$ is the unramified character sending $g$ to $\bar x$. Denote by $\chi''$ the unramified character sending $g$ to $1+\varpi^n\alpha_{\chi}(g)$. Replacing $\chi$ with $\chi\chi''$, we have done.

Suppose that $H$ is contained in the unramified line in $H^1(K,{\mathbb}{F})$ (thus $H$ and the unramified line coincide with each other). By replacing $E$ with $E(\sqrt{\varpi})$, we may assume that $n>1$. Let $\chi_0$ be a character defined by $\chi$ times the unramified character sending our fixed $g$ to $1+\varpi$. Since $n>1$, we see that $\chi_0^{-1}\chi_p^{1-i-\nu}\equiv 1\ {\mathrm}{mod}\ \varpi$ and $\chi_0^{-1}\chi_p^{1-i-\nu}\not\equiv 1\ {\mathrm}{mod}\ \varpi^2$. We define $\alpha_{\chi_0}\colon G_K\to {\mathcal{O}}_E$ by the relation $\chi_0^{-1}\chi_p^{1-i-\nu}=1+\varpi\alpha_{\chi_0}$, and denote $(\alpha_{\chi_0}\ {\mathrm}{mod}\ \varpi)\colon G_K\to {\mathbb}{F}$ by $\bar{\alpha}_{\chi_0}$. By definition and the assumption $n>1$, $\bar{\alpha}_{\chi_0}$ is a non-zero unramified element of $H^1(K,{\mathbb}{F})$, hence it is contained in $H$. Therefore, we have done.

Let $K$ be a finite extension of ${\mathbb}{Q}_p$, $n\ge 2$ an integer and $\chi\colon G_K\to E^{\times}$ an unramified character. Since any $E$-representation of $G_K$ which is an extension of $E$ by $E(\chi\chi_p^n)$ is automatically crystalline, we obtain the following.

\[rank2\] Suppose $p>2$. Let $K$ be a finite unramified extension of ${\mathbb}{Q}_p$. Let $T\in {\mathrm}{Rep}_{{\mathrm}{tor}}(G_K)$ be killed by $p$ and sit in an exact sequence $0\to {\mathbb}{F}_p(i)\to T\to {\mathbb}{F}_p\to 0$ of ${\mathbb}{F}_p$-representations of $G_K$. Then we have the followings:

$(1)$ If $i=0$ and $T$ is unramified, then we have $w_c(T)=0$.

$(2)$ If $i=0$ and $T$ is not unramified, then we have $w_c(T)=p-1$.

$(3)$ If $i=2,\dots, p-2$, then we have $w_c(T)=i$.

(1), (2) By Lemma \[2liftlem\] (2), it suffices to prove that $T$ is not torsion crystalline with Hodge-Tate weights in $[0,p-2]$ if $T$ is not unramified. Let $K_T$ be the definition field of the representation $T$ of $G_K$ and put $G={\mathrm}{Gal}(K_T/K)$. Let $G^j$ be the upper numbering $j$-th ramification subgroup of $G$ (in the sense of [@Se]). Since $T$ is not unramified and killed by $p$, we see that $K_T$ is a totally ramified degree $p$ extension over $K$. Thus $G^1$ is the wild inertia subgroup of $G$ and $G^1=G$, which does not act on $T$ trivial by the definition of $G$. Thus we obtain the desired result by ramification estimates of [@Fo1] (or [@Ab1]) for torsion crystalline representations with Hodge-Tate weights in $[0,p-2]$: if $T$ is torsion crystalline with Hodge-Tate weights in $[0,p-2]$, then $G^j$ acts on $T$ trivial for any $j>(p-2)/(p-1)$.

\(3) The result follows immediately from Proposition \[poly\] (4) and Lemma \[2liftlem\].

\[2lift\] Let $K$ be a finite unramified extension of ${\mathbb}{Q}_p$. Then any $2$-dimensional ${\mathbb}{F}_p$-representation of $G_K$ is torsion crystalline with Hodge-Tate weights in $[0,2p-2]$.

If $T$ is irreducible, the result follows from Theorem \[tamelift\]. Assume that $T$ is reducible. Since $K$ is unramified over ${\mathbb}{Q}_p$, any continuous character $G_K\to {\mathbb}{F}^{\times}_p$ is of the form $\chi \bar{\chi}^i_p$ for some unramified character $\chi$ and some integer $i$. Replacing $K$ with its finite unramified extension, we may assume that $T$ sits in an exact sequence $0\to {\mathbb}{F}_p(i)\to T\to {\mathbb}{F}_p(j)\to 0$ of ${\mathbb}{F}_p$-representations of $G_K$, where $i$ and $j$ are integers in the range $[0,p-2]$ (we remark that $w_c(T)$ is invariant under unramified extensions of $K$ by Proposition \[poly\] (1)). It follows from Lemma \[2liftlem\] that $w_c(T(-j))\le p$. Therefore, we obtain $w_c(T)=w_c(T(-j)\otimes_{{\mathbb}{F}_p} {\mathbb}{F}_p(j))
\le w_c(T(-j))+ w_c({\mathbb}{F}_p(j))\le p+(p-2)
=2p-2$.

Extensions of ${\mathbb}{F}_p$ by ${\mathbb}{F}_p(1)$ and non-fullness theorems
-------------------------------------------------------------------------------

By Lemma \[2liftlem\], we know that the c-weight $w_c(T)$ of an ${\mathbb}{F}_p$-representation $T$ of $G_K$ which sits in an exact sequence $0\to {\mathbb}{F}_p(1)\to T\to {\mathbb}{F}_p\to 0$ of ${\mathbb}{F}_p$-representations of $G_K$, is less than or equal to $p$. Let us calculate $w_c(T)$ for such $T$ more precisely. We should remark that such $T$ is written as $p$-torsion points of a Tate curve. Hence we consider torsion representations coming from Tate curves.

Let $v_K$ be the valuation of $K$ normalized such that $v_K(K^{\times})={\mathbb}{Z}$, and take any $q\in K^{\times}$ with $v_K(q)>0$. Let $E_q$ be the Tate curve over $K$ associated with $q$ and $E_q[p^n]$ the module of $p^n$-torsion points of $E_q$ for any integer $n>0$. It is well-known that there exists an exact sequence $$(\#)\quad  0\to \mu_{p^n}\to E_q[p^n]\to {\mathbb}{Z}/p^n{\mathbb}{Z}\to 0$$ of ${\mathbb}{Z}_p[G_K]$-modules. Here, $\mu_{p^n}$ is the group of $p^n$-th roots of unity in $\overline{K}$. Let $x_n\colon G_K\to \mu_{p^n}$ be the $1$-cocycle defined to be the image of $1$ for the connection map $H^0(K, {\mathbb}{Z}/p^n{\mathbb}{Z})\to H^1(K, \mu_{p^n})$ arising from the exact sequence $(\#)$. Then $x_n$ corresponds to $q$ mod $(K^{\times})^{p^n}$ via the isomorphism $K^{\times}/(K^{\times})^{p^n}\simeq H^1(K,\mu_{p^n})$ of Kummer theory. Thus the exact sequence $(\#)$ splits if and only if $q\in (K^{\times})^{p^n}$.

First we consider the case $p\mid v_K(q)$ (i.e.  [*peu ramifié*]{} case).

\[easycase\] Let $K$ be a finite extension of ${\mathbb}{Q}_p$. If $p\mid v_K(q)$, then $E_q[p]$ is the reduction modulo $p$ of a lattice in some $2$-dimensional crystalline ${\mathbb}{Q}_p$-representation with Hodge-Tate weights in $[0,1]$.

Since $p\mid v_K(q)$, there exists $q'\in K^{\times}$ such that $v_K(q'-1)>0$ and $q\equiv q'$ mod $(K^{\times})^p$. Considering the exact sequence $0\to {\mathbb}{Z}_p(1)\to L\to {\mathbb}{Z}\to 0$ of ${\mathbb}{Z}_p$-representations of $G_K$ corresponding to $q'$ via the isomorphism $H^1(K,{\mathbb}{Z}_p(1))\simeq {\varprojlim}_{n} K^{\times}/(K^{\times})^{p^n}$ of Kummer theory, we obtain the desired result.

\[minwt\] Suppose that $K$ is a finite extension of ${\mathbb}{Q}_p$, $(p-1)\nmid e$ and $p\mid v_K(q)$. Then we have $w_c(E_q[p])=1$.

By the assumption $(p-1)\nmid e$, we know that the largest tame inertia weight of $E_q[p]$ is positive. Thus Proposition \[poly\] (4) shows $w_c(E_q[p])\ge 1$. The inequality $w_c(E_q[p])\le 1$ follows from Lemma \[easycase\].

Next we consider the case $p\nmid v_K(q)$ (i.e. [*très ramifié*]{} case).

\[Tatecurve\] If $e(r-1)<p-1$ and $p\nmid v_K(q)$, then $E_q[p^n]$ is not torsion crystalline with Hodge-Tate weights in $[0,r]$ for any $n>0$.

If $e=1$, the fact that $E_{\pi}[p^n]$ is not torsion crystalline with Hodge-Tate weights in $[0,p-1]$ immediately follows from the theory of ramification bound as below. We may suppose $n=1$. Suppose $E_{\pi}[p]$ is torsion crystalline with Hodge-Tate weights in $[0,p-1]$. Then the upper numbering $j$-th ramification subgroup $G^j_K$ of $G_K$ (in the sense of [@Se]) acts trivially on $E_{\pi}[p]$ for any $j>1$ ([@Ab1 Section 6, Theorem 3.1]). However, this contradicts the fact that the upper bound of the ramification of $E_{\pi}[p]$ is $1+1/(p-1)$.

We may suppose $n=1$. We choose any uniformizer $\pi'$ of $K$. Putting $v_K(q)=m$, we can write $q=(\pi')^mx$ with some unit $x$ of the integer ring of $K$. Since $m$ is prime to $p$, we have a decomposition $x=\zeta_{\ell}y^m$ in $K^{\times}$ for some $\ell>0$ prime to $p$ and $y\in K$ with $v_K(y-1)>0$. Here $\zeta_{\ell}$ is a (not necessary primitive) $\ell$-th root of unity. Since $\ell$ is prime to $p$, we have $\zeta_{\ell}=\zeta^{ps}_{\ell}$ for some integer $s$. We put $\pi=\pi'y$. This is a uniformizer of $K$. Choose any $p$-th root $\pi_1$ of $\pi$ and put $q_1=\zeta^s_{\ell}\pi^m_1\in K(\pi_1)^{\times}$. Then we have $q=q^p_1\in (K(\pi_1)^{\times})^p$ and in particular, the exact sequence $(\#)$ (for $n=1$) splits as representations of ${\mathrm}{Gal}(\overline{K}/K(\pi_1))$. Now assume that $E_q[p]$ is torsion crystalline with Hodge-Tate weights in $[0,r]$. Then $(\#)$ (for $n=1$) splits as representations of $G_K$ by Corollary \[FFTHMtorcris\]. This contradicts the assumption $p\nmid v_K(q)$ (and hence $q\notin (K^{\times})^p$).

Now we put $r'_0={\mathrm}{min}\{r\in {\mathbb}{Z}_{\ge 0} ; e(r-1)\ge p-1\}$. Recall that we have $[K^{{\mathrm}{ur}}(\mu_p):K^{{\mathrm}{ur}}]=(p-1)/{\mathrm}{gcd}(e,p-1)$.

\[roughbound\] Let $K$ be a finite extension of ${\mathbb}{Q}_p$. Then $E_q[p]$ is torsion crystalline with Hodge-Tate weights in $[0,1+(p-1)/{\mathrm}{gcd}(e,p-1)]$.

Taking a finite unramified extension $K'$ of $K$ such that $[K^{{\mathrm}{ur}}(\mu_p):K^{{\mathrm}{ur}}]=[K'(\mu_p):K']$, we obtain $w_c((E_q[p])|_{G_{K'}})\le 1+(p-1)/{\mathrm}{gcd}(e,p-1)$ by Lemma \[2liftlem\]. Thus we have $w_c(E_q[p])\le 1+(p-1)/{\mathrm}{gcd}(e,p-1)$ by Proposition \[poly\] (1).

\[wctr1\] Suppose that $K$ is a finite extension of ${\mathbb}{Q}_p$, and also suppose $e\mid (p-1)$ or $(p-1)\mid e$. We further suppose that $p\nmid v_K(q)$. Then we have $w_c(E_q[p])=r'_0$.

We have $w_c(E_q[p])\le r'_0$ by Lemma \[roughbound\]. In addition, we also have $w_c(E_q[p])\ge r'_0$ by Proposition \[Tatecurve\].

Lemma \[roughbound\] gives some non-fullness results on torsion crystalline representations.

\[nonfullthm\] Suppose that $K$ is a finite extension of ${\mathbb}{Q}_p$. If $r\ge 1+(p-1)/{\mathrm}{gcd}(e,p-1)$, then the restriction functor ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)\to 
{\mathrm}{Rep}_{{\mathrm}{tor}}(G_1)$ is not full.

Two representations $E_{\pi}[p]$ and ${\mathbb}{F}_p(1)\oplus {\mathbb}{F}_p$ are objects of ${\mathrm}{Rep}^r_{{\mathrm}{tor}}(G_K)$ by Lemma \[roughbound\]. They are not isomorphic as representations of $G_K$ but isomorphic as representations of $G_1$. Thus the desired non-fullness follows.

\[p2\] Suppose that any one of the following holds:

-   $p=2$ and $K$ is a finite extension of ${\mathbb}{Q}_2$ $($in this case $r'_0=2)$;

-   $K$ is a finite unramified extension of ${\mathbb}{Q}_p$ $($in this case $r'_0=p)$;

-   $K$ is a finite extension of ${\mathbb}{Q}_p(\mu_p)$ $($in this case $r'_0=2)$.

Then the restriction functor ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)\to 
{\mathrm}{Rep}_{{\mathrm}{tor}}(G_1)$ is not full.

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[^1]: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, JAPAN. e-mail: [yozeki@kurims.kyoto-u.ac.jp]{} Supported by the JSPS Fellowships for Young Scientists.

[^2]: This continuity condition is needed to assure that the $G_{L,s}$-action on $\hat{T}_{L,s}(\hat{{\mathfrak{M}}})$, defined after Definition \[Liumod\], is continuous (here, note that $\hat{T}_{L,s}(\hat{{\mathfrak{M}}})$ is the dual of $(W({\mathrm}{Fr}R)\otimes_{{\varphi}, {\mathfrak{S}}_{L,s}} {\mathfrak{M}})^{{\varphi}=1}$; see Corollary 3.12 of [@Oz1]).

[^3]: As explained in the footnote of Definition \[Liumod\], we may ignore the continuity condition of (2) in the case where ${\mathfrak{M}}$ is torsion.
