Relative crystalline representations and $p$-divisible groups in the small ramification case

TL;DR

This paper proves that in the case of small ramification, every crystalline Galois representation with Hodge-Tate weights in [0,1] over a certain base ring originates from a $p$-divisible group, extending the understanding of such representations.

Contribution

It establishes a correspondence between crystalline representations and $p$-divisible groups in the small ramification case, under mild conditions on the base ring.

Findings

Crystalline representations with weights in [0,1] are from $p$-divisible groups when ramification degree e < p-1.

The result applies to a broad class of relative base rings over $W(k)$.

Provides a link between Galois representations and algebraic groups in the small ramification setting.

Abstract

Let $k$ be a perfect field of characteristic $p > 2$, and let $K$ be a finite totally ramified extension over $W(k)[\frac{1}{p}]$ of ramification degree $e$. Let $R_0$ be a relative base ring over $W(k)\langle t_1^{\pm 1}, \ldots, t_m^{\pm 1}\rangle$ satisfying some mild conditions, and let $R = R_0\otimes_{W(k)}\mathcal{O}_K$. We show that if $e < p-1$, then every crystalline representation of $\pi_1^{\text{\'et}}(\mathrm{Spec}R[\frac{1}{p}])$ with Hodge-Tate weights in $[0, 1]$ arises from a $p$-divisible group over $R$.

Equations

$${\alpha }_{\mathrm{cris}}:D_{\mathrm{cris}}(V){\otimes }_{R_0[\frac{1}{p}]}{B}_{\mathrm{cris}}(R)\to V^{\vee }{\otimes }_{{\mathbf{Q}}_p}{B}_{\mathrm{cris}}(R)$$

$${\tilde{R}}_{\infty }:=W({\tilde{E}}_{R_{\infty }}^{+}){\otimes }_{W(K_{\infty }^{♭\circ }),\theta }{\mathcal{O}}_{K_{\infty }}.$$

$${\mathfrak{M}}^{\ast }:\{p\text{mbox}-divisiblegroupsover\mathrm{Spec}R\}\to {\mathrm{Kis}}^1(\mathfrak{S},\nabla ).$$

$${\mathfrak{M}}_n:={\mathcal{M}}_n\cap {\mathfrak{M}}_{{\mathcal{O}}_L,n}$$

$$1\otimes \phi :\mathfrak{S}{\otimes }_{\phi ,\mathfrak{S}}{\mathfrak{M}}_n\to {\mathfrak{M}}_n$$

$${\mathfrak{M}}_n{\otimes }_{\mathfrak{S}}{\mathcal{O}}_{\mathcal{E}}\cong {\mathcal{M}}_n$$

$${\mathfrak{M}}_n{\otimes }_{\mathfrak{S}}{\mathfrak{S}}_{{\mathcal{O}}_L}\cong {\mathfrak{M}}_{{\mathcal{O}}_L,n}.$$

$$(f_1,\dots ,f_d{)}^t=A\cdot (e_1,\dots ,e_d{)}^t$$

$$(b_1,\dots ,b_d)\cdot A=(c_1,\dots ,c_d)$$

$${\mathfrak{M}}_n{\otimes }_{\mathfrak{S}}{\mathcal{O}}_{\mathcal{E}}\cong {\mathfrak{M}}_n[\frac{1}{u}]\cong {\mathcal{M}}_n\cap {\mathcal{M}}_{{\mathcal{O}}_L,n}={\mathcal{M}}_n$$

$$\begin{array}{rl}{\mathfrak{M}}_n{\otimes }_{\mathfrak{S}}{\mathfrak{S}}_{{\mathcal{O}}_L}& \cong ({\mathcal{M}}_n{\otimes }_{\mathfrak{S}}{\mathfrak{S}}_{{\mathcal{O}}_L})\cap ({\mathfrak{M}}_{{\mathcal{O}}_L,n}{\otimes }_{\mathfrak{S}}{\mathfrak{S}}_{{\mathcal{O}}_L})={\mathcal{M}}_{{\mathcal{O}}_L,n}\cap ({\mathfrak{M}}_{{\mathcal{O}}_L,n}{\otimes }_{\mathfrak{S}}{\mathfrak{S}}_{{\mathcal{O}}_L})\\ & \cong ({\mathfrak{M}}_{{\mathcal{O}}_L,n}{\otimes }_{{\mathfrak{S}}_n}{\mathfrak{S}}_n[\frac{1}{u}])\cap ({\mathfrak{M}}_{{\mathcal{O}}_L,n}{\otimes }_{{\mathfrak{S}}_n}{\mathfrak{S}}_{{\mathcal{O}}_L,n})\cong {\mathfrak{M}}_{{\mathcal{O}}_L,n}\end{array}$$

$$y_1\in (\mathfrak{S}{\otimes }_{\phi ,\mathfrak{S}}{\mathcal{M}}_n)\cap (\mathfrak{S}{\otimes }_{\phi ,\mathfrak{S}}{\mathfrak{M}}_{{\mathcal{O}}_L,n})\cong \mathfrak{S}{\otimes }_{\phi ,\mathfrak{S}}({\mathcal{M}}_n\cap {\mathfrak{M}}_{{\mathcal{O}}_L,n})=\mathfrak{S}{\otimes }_{\phi ,\mathfrak{S}}{\mathfrak{M}}_n.$$

$${\phi }^{\ast }\mathfrak{N}\stackrel{1\otimes \phi }{⟶}\mathfrak{N}\stackrel{\psi }{\to }{\phi }^{\ast }\mathfrak{N}$$

$${\mathfrak{S}}_{\mathfrak{q}}{\otimes }_{\phi ,{\mathfrak{S}}_{\mathfrak{q}}}{\mathfrak{N}}_n\stackrel{1\otimes \phi }{⟶}{\mathfrak{N}}_n\stackrel{\psi }{\to }{\mathfrak{S}}_{\mathfrak{q}}{\otimes }_{\phi ,{\mathfrak{S}}_{\mathfrak{q}}}{\mathfrak{N}}_n$$

$$k_{\mathfrak{q}}[[u]]{\otimes }_{\phi ,k_{\mathfrak{q}}[[u]]}{\overline{\mathfrak{N}}}_n\stackrel{1\otimes \phi }{⟶}{\overline{\mathfrak{N}}}_n\stackrel{\psi }{\to }{k}_{\mathfrak{q}}[[u]]{\otimes }_{\phi ,k_{\mathfrak{q}}[[u]]}{\overline{\mathfrak{N}}}_n$$

$$k_{\mathfrak{q}}[[u]]{\otimes }_{\phi ,k_{\mathfrak{q}}[[u]]}{\overline{\mathfrak{N}}}_{n,\mathrm{tor}}\stackrel{1\otimes \phi }{⟶}{\overline{\mathfrak{N}}}_{n,\mathrm{tor}}\stackrel{\psi }{\to }{k}_{\mathfrak{q}}[[u]]{\otimes }_{\phi ,k_{\mathfrak{q}}[[u]]}{\overline{\mathfrak{N}}}_{n,\mathrm{tor}}$$

$$k_{\mathfrak{q}}[[u]]/(u^{pa})\stackrel{1\otimes \phi }{⟶}{k}_{\mathfrak{q}}[[u]]/(u^a)\stackrel{\psi }{\to }{k}_{\mathfrak{q}}[[u]]/(u^{pa})$$

$$f:\underset{i=1}{\overset{d}{⨁}}{\mathfrak{S}}_{\mathfrak{q},n}\cdot e_i\twoheadrightarrow {\mathfrak{N}}_n$$

$$p{\mathfrak{M}}_{n+1}\cong {\mathfrak{M}}_n.$$

$$\mathfrak{M}:=\underset{n}{\underleftarrow{\lim }}{\mathfrak{M}}_{\mathfrak{n}},$$

$$\underline{x}=B{\underline{x}}^{[p]}+C$$

$$A_1:=A_0[x_1,\dots ,x_n]/(x_1-c_1-\sum _{i=1}^n{b}_{1i}{x}_i^p,\dots ,x_n-c_n-\sum _{i=1}^n{b}_{ni}{x}_i^p),$$

$$x_1^p=B^{-1}{x}_1-B^{-1}C,$$

$${\underline{x}}^{[p]}=B^{-1}\underline{x}-B^{-1}C.$$

$$\sum _{i=1}^n{e}_i{B}^{(i)}=0$$

$$x_n=-\sum _{i=1}^{n-1}{e}_i{x}_i+c_n+\sum _{i=1}^{n-1}{c}_i{e}_i.$$

$${\underline{x}}^{\prime }=B^{\prime }{\underline{x}}^{\prime [p]}+C^{\prime }$$

$${\nabla }_{A,n}:A/(p^n){\otimes }_{R_0}\mathcal{N}\to (A/(p^n){\otimes }_{R_0}\mathcal{N}){\otimes }_A{\hat{\Omega }}_A$$

$$\begin{array}{ccc}A/(p^n){\otimes }_A\tilde{\mathcal{N}}& \stackrel{{\phi }^{\ast }({\nabla }_{A,n})}{\to }& A/(p^n){\otimes }_A\tilde{\mathcal{N}}{\otimes }_A{\hat{\Omega }}_A\\ 1\otimes \phi \downarrow & & \downarrow (1\otimes \phi )\otimes {\mathrm{id}}_{{\hat{\Omega }}_A}\\ A/(p^n){\otimes }_A\mathcal{N}& \stackrel{{\nabla }_{A,n}}{\to }& A/(p^n){\otimes }_A\mathcal{N}{\otimes }_A{\hat{\Omega }}_A\end{array}$$

$${\nabla }_{{\mathfrak{M}}_S}:(S_0{\otimes }_{\phi ,{\mathfrak{S}}_S}{\mathfrak{M}}_S)\to (S_0{\otimes }_{\phi ,{\mathfrak{S}}_S}{\mathfrak{M}}_S){\otimes }_{S_0}{\hat{\Omega }}_{S_0}$$

Full text

1 Introduction

Let $k$ be a perfect field of characteristic $p>2$, and let $W(k)$ be its ring of Witt vectors. Let $K$ be a finite totally ramified extension over $W(k)[\frac{1}{p}]$ with ramification degree $e$, and denote by $\mathcal{O}_{K}$ its ring of integers. If $G$ is a $p$-divisible group over $\mathcal{O}_{K}$, then it is well-known that its Tate module $T_{p}(G)$ is a crystalline $\mathrm{Gal}(\overline{K}/K)$-representation with Hodge-Tate weights in $[0,1]$. Conversely, Kisin showed the following result in [Kis06].

Theorem 1.1**.**

(cf. [Kis06, Corollary 2.2.6])* Let $T$ be a crystalline $\mathrm{Gal}(\overline{K}/K)$-representation finite free over $\mathbf{Z}_{p}$ whose Hodge-Tate weights lie in $[0,1]$. Then there exists a $p$-divisible group $G$ over $\mathcal{O}_{K}$ such that $T_{p}(G)\cong T$ as $\mathrm{Gal}(\overline{K}/K)$-representations.*

The goal of this paper is to study the analogous statement in the relative case.

Recently, relative $p$-adic Hodge theory generalizing the classical theory has been developed by Brinon [Bri08], Scholze [Sch13], Kedlaya-Liu [KL15], and Diao-Lan-Liu-Zhu [DLLZ18]. It aims to understand certain $p$-adic étale local systems, namely de Rham local systems on a smooth (rigid analytic) variety $X$ over a $p$-adic field. As in the classical case, one can expect that studying the full subcategory of crystalline local systems on $X$, when it is well-defined, would exhibit a lot of useful information.

Let $R_{0}$ be a base ring over $W(k)\langle t_{1}^{\pm 1},\ldots,t_{m}^{\pm 1}\rangle$ given as in Section 2.1, and let $R=R_{0}\otimes_{W(k)}\mathcal{O}_{K}$. In this paper, we will work in the setting $X=\mathrm{Spec}(R[\frac{1}{p}])$. For this case, the category of $p$-adic étale local systems on $X$ is equivalent to that of $\mathbf{Q}_{p}$-representations of the étale fundamental group $\mathcal{G}_{R}$ of $X$ as $X$ is connected. Moreover, for representations of $\mathcal{G}_{R}$, the condition being crystalline is well-defined by [Bri08]. If $G_{R}$ is a $p$-divisible group over $R$, its Tate module $T_{p}(G_{R})$ is a crystalline $\mathcal{G}_{R}$-representation with Hodge-Tate weights in $[0,1]$ (cf. [Kim15]). A natural question is whether the converse statement analogous to Theorem 1.1 will hold in the relative case. We prove that the answer is affirmative if the ramification index $e$ is small.

Theorem 1.2**.**

Suppose $e<p-1$. Let $T$ be a crystalline $\mathcal{G}_{R}$-representation finite free over $\mathbf{Z}_{p}$ whose Hodge-Tate weights lie in $[0,1]$. Then there exists a $p$-divisible group $G_{R}$ over $R$ such that $T_{p}(G_{R})\cong T$ as $\mathcal{G}_{R}$-representations.

As an immediate corollary using the results in [Moo18], when $R$ has Krull dimension $2$, we obtain the following result on the geometry of the locus of crystalline $\mathcal{G}_{R}$-representations with Hodge-Tate weights in $[0,1]$. For a fixed absolutely irreducible $\mathbf{F}_{p}$-representation $V_{0}$ of $\mathcal{G}_{R}$, there exists a universal deformation ring which parametrizes the deformations of $V_{0}$ ([SL97]). By [Moo18, Theorem 5.7], we deduce:

Corollary 1.3**.**

Suppose $R$ has Krull dimension $2$ and $e<p-1$. Then the locus of crystalline representations of $\mathcal{G}_{R}$ with Hodge-Tate weights in $[0,1]$ cuts out a closed subscheme of the universal deformation scheme.

We give a more precise statement of Corollary 1.3 in Section 6.

There are three major ingredients for the proof of Theorem 1.2. Firstly, Brinon and Trihan proved in [BT08] the generalization of Theorem 1.1 for the case when the base is a complete discrete valuation ring whose residue field has a finite $p$-basis. We use this result together with the fact that the $p$-adic completion of $R_{0,(p)}$ is an example of such rings studied loc. cit. Secondly, Kim generalized the Breuil-Kisin classification in the relative setting in [Kis06], and showed that the category of $p$-divisible groups over $R$ is anti-equivalent to the category of Kisin modules of height $1$ over $R_{0}[\![u]\!]$. Using the classification, we reduce our problem to constructing desired Kisin modules. We remark that our method of constructing the appropriate Kisin modules relies on the assumption that $e<p-1$. Lastly, to show the statement for the special case when $R$ is a formal power series ring of dimension $2$, we use the purity result for $p$-divisible groups proved in [VZ10] when the ramification index is small.

1.1 Notations

We will reserve $\varphi$ for various Frobenius. To be more precise, let $A$ be an $W(k)$-algebra on which the arithmetic Frobenius $\varphi$ on $W(k)$ extends, and $M$ an $A$-module. We denote $\varphi_{A}:A\to A$ for such an extension. Let $\varphi_{M}:M\to M$ be a $\varphi_{A}$-semi-linear map. This is equivalent to having an $A$-linear map $1\otimes\varphi_{M}:\varphi_{A}^{*}M\to M$, where $\varphi_{A}^{*}M$ denotes $A\otimes_{\varphi_{A},A}M$. We always drop the subscripts $A$ and $M$ from $\varphi$ if no confusion arises. Let $f:A\to B$ be a ring map compatible with Frobenius, that is, $f\circ\varphi_{A}=\varphi_{B}\circ f$. Then $\varphi_{M}$ naturally extends to $\varphi_{M_{B}}:M_{B}\to M_{B}$ for $M_{B}:=B\otimes_{A}M$. It is easy to check that $\varphi^{*}_{B}M_{B}=B\otimes_{A}\varphi^{*}_{A}M$ and $1\otimes\varphi_{M_{B}}:\varphi^{*}_{B}M_{B}\to M_{B}$ is equal to $B\otimes_{A}(1\otimes\varphi_{M})$.

2 Relative $p$-adic Hodge theory and étale $\varphi$-modules

2.1 Base ring and crystalline period ring in the relative case

We follow the same notations as in the Introduction. We recall the assumptions on the base rings and the construction of crystalline period ring in relative $p$-adic Hodge theory in [Bri08] (and in [Kim15] for Breuil-Kisin classification). We also impose some mild additional assumptions which will be needed later. Let $R_{0}$ be a ring obtained from $W(k)\langle t_{1}^{\pm 1},\ldots,t_{m}^{\pm 1}\rangle$ by a finite number of iterations of the following operations:

• •

$p$-adic completion of an étale extension;

• •

$p$-adic completion of a localization;

• •

completion with respect to an ideal containing $p$.

We assume that either $W(k)\langle t_{1}^{\pm 1},\ldots,t_{m}^{\pm 1}\rangle\rightarrow R_{0}$ has geometrically regular fibers or $R_{0}$ has Krull dimension less than $2$, and that $k\rightarrow R_{0}/pR_{0}$ is geometrically integral. In addition, we suppose that $R_{0}$ is an integral domain containing a Cohen ring $W$ over $W(k)$ such that $R_{0}$ is formally finite type over $W$, and that $R_{0}/pR_{0}$ is a unique factorization domain.

$R_{0}/pR_{0}$ has a finite $p$-basis given by $\{t_{1},\ldots,t_{m}\}$ in the sense of [DJ95, Definition 1.1.1]. The Witt vector Frobenius on $W(k)$ extends (not necessarily uniquely) to $R_{0}$, and we fix such a Frobenius endomorphism $\varphi:R_{0}\rightarrow R_{0}$. Let $\widehat{\Omega}_{R_{0}}\coloneqq\varprojlim_{n}\Omega_{(R_{0}/p^{n})/W(k)}$ be the module of $p$-adically continuous Kähler differentials. By [Bri08, Proposition 2.0.2], $\widehat{\Omega}_{R_{0}}\cong\bigoplus_{i=1}^{m}R_{0}\cdot dt_{i}$. We work over the base ring $R$ given by $R\coloneqq R_{0}\otimes_{W(k)}\mathcal{O}_{K}$.

Let $\overline{R}$ denote the union of finite $R$-subalgebras $R^{\prime}$ of a fixed separable closure of $\mathrm{Frac}(R)$ such that $R^{\prime}[\frac{1}{p}]$ is étale over $R[\frac{1}{p}]$. Then $\mathrm{Spec}\overline{R}[\frac{1}{p}]$ is a pro-universal covering of $\mathrm{Spec}R[\frac{1}{p}]$, and $\overline{R}$ is the integral closure of $R$ in $\overline{R}[\frac{1}{p}]$. Let $\mathcal{G}_{R}\coloneqq\mathrm{Gal}(\overline{R}[\frac{1}{p}]/R[\frac{1}{p}])=\pi_{1}^{\text{\'{e}t}}(\mathrm{Spec}R[\frac{1}{p}])$. By a representation of $\mathcal{G}_{R}$, we always mean a finite continuous representation.

The crystalline period ring $B_{\mathrm{cris}}(R)$ is constructed as follows. Let $\displaystyle\overline{R}^{\flat}=\varprojlim_{\varphi}\overline{R}/p\overline{R}$. There exists a natural $W(k)$-linear surjective map $\theta:W(\overline{R}^{\flat})\rightarrow\widehat{\overline{R}}$ which lifts the projection onto the first factor. Here, $\widehat{\overline{R}}$ denotes the $p$-adic completion of $\overline{R}$. Let $\theta_{R_{0}}:R_{0}\otimes_{W(k)}W(\overline{R}^{\flat})\rightarrow\widehat{\overline{R}}$ be the $R_{0}$-linear extension of $\theta$. Define the integral crystalline period ring $A_{\mathrm{cris}}(R)$ to be the $p$-adic completion of the divided power envelope of $R_{0}\otimes_{W(k)}W(\overline{R}^{\flat})$ with respect to $\mathrm{ker}(\theta_{R_{0}})$. Choose compatibly $\epsilon_{n}\in\overline{R}$ such that $\epsilon_{0}=1,~{}\epsilon_{n}=\epsilon_{n+1}^{p}$ with $\epsilon_{1}\neq 1$, and let $\widetilde{\epsilon}=(\epsilon_{n})_{n\geq 0}\in\overline{R}^{\flat}$. Then $\tau\coloneqq\log{[\widetilde{\epsilon}]}\in A_{\mathrm{cris}}(R)$. Define $B_{\mathrm{cris}}(R)=A_{\mathrm{cris}}(R)[\frac{1}{\tau}]$. $B_{\mathrm{cris}}(R)$ is equipped naturally with $\mathcal{G}_{R}$-action and Frobenius endomorphism, and $B_{\mathrm{cris}}(R)\otimes_{R_{0}[\frac{1}{p}]}R[\frac{1}{p}]$ is equipped with a natural filtration by $R[\frac{1}{p}]$-submodules. Furthermore, we have a natural integrable connection $\nabla:B_{\mathrm{cris}}(R)\rightarrow B_{\mathrm{cris}}(R)\otimes_{R_{0}}\widehat{\Omega}_{R_{0}}$ such that Frobenius is horizontal and Griffiths transversality is satisfied.

For a $\mathcal{G}_{R}$-representation $V$ over $\mathbf{Q}_{p}$, let $D_{\mathrm{cris}}(V)\coloneqq\mathrm{Hom}_{\mathcal{G}_{R}}(V,B_{\mathrm{cris}}(R))$. The natural morphism

[TABLE]

is injective. We say $V$ is crystalline if $\alpha_{\mathrm{cris}}$ is an isomorphism. When $V$ is crystalline, then $D_{\mathrm{cris}}(V)$ is a finite projective $R_{0}[\frac{1}{p}]$-module, and $D_{\mathrm{cris}}(V)\otimes_{R_{0}[\frac{1}{p}]}R[\frac{1}{p}]$ has the filtration induced by that on $B_{\mathrm{cris}}(R)\otimes_{R_{0}[\frac{1}{p}]}R[\frac{1}{p}]$. We define the Hodge-Tate weights similarly as in the classical $p$-adic Hodge theory. Frobenius and connection on $B_{\mathrm{cris}}(R)$ induce those structures on $D_{\mathrm{cris}}(V)$; for the Frobenius endomorphism on $D_{\mathrm{cris}}(V)$, $1\otimes\varphi:\varphi^{*}D_{\mathrm{cris}}(V)\rightarrow D_{\mathrm{cris}}(V)$ is an isomorphism, and the connection $\nabla:D_{\mathrm{cris}}(V)\rightarrow D_{\mathrm{cris}}(V)\otimes_{R_{0}}\widehat{\Omega}_{R_{0}}$ is integrable and topologically quasi-nilpotent. Furthermore, Griffiths transversality is satisfied and $\varphi$ is horizontal. For a $\mathcal{G}_{R}$-representation $T$ which is free over $\mathbf{Z}_{p}$, we say it is crystalline if $T[\frac{1}{p}]$ is crystalline.

Suppose $S_{0}$ is another relative base ring over $W(k)\langle t_{1}^{\pm 1},\ldots,t_{m}^{\pm 1}\rangle$ satisfying the above conditions and equipped with a choice of Frobenius, and let $b:R_{0}\rightarrow S_{0}$ be a $\varphi$-equivariant $W(k)\langle t_{1}^{\pm 1},\ldots,t_{m}^{\pm 1}\rangle$-algebra map. We also denote $b:R=R_{0}\otimes_{W(k)}\mathcal{O}_{K}\rightarrow S\coloneqq S_{0}\otimes_{W(k)}\mathcal{O}_{K}$ the map induced $\mathcal{O}_{K}$-linearly. By choosing a common geometric point, this induces a map of Galois groups $\mathcal{G}_{S}\rightarrow\mathcal{G}_{R}$, and also a map of crystalline period rings $B_{\mathrm{cris}}(R)\rightarrow B_{\mathrm{cris}}(S)$ compatible with all structures. If $V$ is a crystalline representation of $\mathcal{G}_{R}$ with certain Hodge-Tate weights, then via these maps $V$ is also a crystalline representation of $\mathcal{G}_{S}$ with the same Hodge-Tate weights, and the construction of $D_{\mathrm{cris}}(V)$ is compatible with the base change.

We will consider the following base change maps in later sections. Let $\mathcal{O}_{L_{0}}$ be the $p$-adic completion of $R_{0,(p)}$, and let $b_{L}:R_{0}\rightarrow\mathcal{O}_{L_{0}}$ be the natural $\varphi$-equivariant map. This induces $b_{L}:R\rightarrow\mathcal{O}_{L}\coloneqq\mathcal{O}_{L_{0}}\otimes_{W(k)}\mathcal{O}_{K}$. Note that $L=\mathcal{O}_{L}[\frac{1}{p}]$ is an example of a complete discrete valuation field with a residue field having a finite $p$-basis, studied in [BT08]. On the other hand, for each maximal ideal $\mathfrak{q}\in\mathrm{mSpec}R_{0}$, let $\widehat{R_{0,\mathfrak{q}}}$ be the $\mathfrak{q}$-adic completion of $R_{0,\mathfrak{q}}$. By the structure theorem of complete regular local rings, we have $\widehat{R_{0,\mathfrak{q}}}\cong\mathcal{O}_{\mathfrak{q}}[\![s_{1},\ldots,s_{l}]\!]$ where $\mathcal{O}_{\mathfrak{q}}$ is a Cohen ring with the maximal ideal $(p)$ and $l\geq 0$ is an integer ($\widehat{R_{0,\mathfrak{q}}}$ is understood to be $\mathcal{O}_{\mathfrak{q}}$ when $l=0$). We consider the natural $\varphi$-equivariant morphism $b_{\mathfrak{q}}:R_{0}\rightarrow\widehat{R_{0,\mathfrak{q}}}$, which induces $b_{\mathfrak{q}}:R\rightarrow R_{\mathfrak{q}}\coloneqq\widehat{R_{0,\mathfrak{q}}}\otimes_{W(k)}\mathcal{O}_{K}$.

2.2 Étale $\varphi$-modules

We study étale $\varphi$-modules and associated Galois representations. Most of the material in this section is a review of [Kim15] Section 7, and the underlying geometry is based on perfectoid spaces as in [Sch12].

Let $R_{0}$ be a relative base ring over $W(k)\langle t_{1}^{\pm 1},\ldots,t_{m}^{\pm 1}\rangle$ and let $R=R_{0}\otimes_{W(k)}\mathcal{O}_{K}$ as above. Choose a uniformizer $\varpi\in\mathcal{O}_{K}$. For integers $n\geq 0$, we choose compatibly $\varpi_{n}\in\overline{K}$ such that $\varpi_{0}=\varpi$ and $\varpi_{n+1}^{p}=\varpi_{n}$, and let $K_{\infty}$ be the $p$-adic completion of $\bigcup_{n\geq 0}K(\varpi_{n})$. Then $K_{\infty}$ is a perfectoid field and $(\widehat{\overline{R}}[\frac{1}{p}],\widehat{\overline{R}})$ is a perfectoid affinoid $K_{\infty}$-algebra. Let $K_{\infty}^{\flat}$ denote the tilt of $K_{\infty}$ as defined in [Sch12], and let $\underline{\varpi}\coloneqq(\varpi_{n})\in K_{\infty}^{\flat}$.

Let $\mathfrak{S}\coloneqq R_{0}[\![u]\!]$ equipped with the Frobenius extending that on $R_{0}$ by $\varphi(u)=u^{p}$. Let $E_{R_{\infty}}^{+}=\mathfrak{S}/p\mathfrak{S}$, and let $\tilde{E}_{R_{\infty}}^{+}$ be the $u$-adic completion of $\varinjlim_{\varphi}E_{R_{\infty}}^{+}$. Let $E_{R_{\infty}}=E_{R_{\infty}}^{+}[\frac{1}{u}]$ and $\tilde{E}_{R_{\infty}}=\tilde{E}_{R_{\infty}}^{+}[\frac{1}{u}]$. By [Sch12, Proposition 5.9], $(\tilde{E}_{R_{\infty}},\tilde{E}_{R_{\infty}}^{+})$ is a perfectoid affinoid $L^{\flat}$-algebra, and we have the natural injective map $(\tilde{E}_{R_{\infty}},\tilde{E}_{R_{\infty}}^{+})\hookrightarrow(\overline{R}^{\flat}[\frac{1}{\underline{\varpi}}],\overline{R}^{\flat})$ given by $u\mapsto\underline{\varpi}$.

Let

[TABLE]

By [Sch12, Remark 5.19], $(\tilde{R}_{\infty}[\frac{1}{p}],\tilde{R}_{\infty})$ is a perfectoid affinoid $K_{\infty}$-algebra whose tilt is $(\tilde{E}_{R_{\infty}},\tilde{E}_{R_{\infty}}^{+})$. Furthermore, it is shown in [Kim15] that we have a natural injective map $(\tilde{R}_{\infty}[\frac{1}{p}],\tilde{R}_{\infty})\hookrightarrow(\widehat{\overline{R}}[\frac{1}{p}],\widehat{\overline{R}})$ whose tilt is $(\tilde{E}_{R_{\infty}},\tilde{E}_{R_{\infty}}^{+})\hookrightarrow(\overline{R}^{\flat}[\frac{1}{\underline{\varpi}}],\overline{R}^{\flat})$. For $\mathcal{G}_{\tilde{R}_{\infty}}\coloneqq\pi_{1}^{\text{\'{e}t}}(\mathrm{Spec}\tilde{R}_{\infty}[\frac{1}{p}])$, we then have a continuous map of Galois groups $\mathcal{G}_{\tilde{R}_{\infty}}\rightarrow\mathcal{G}_{R}$, which is a closed embedding by [GR03, Proposition 5.4.54]. By the almost purity theorem in [Sch12], $\overline{R}^{\flat}[\frac{1}{\underline{\varpi}}]$ can be canonically identified with the $\underline{\varpi}$-adic completion of the affine ring of a pro-universal covering of $\mathrm{Spec}\tilde{E}_{R_{\infty}}$, and letting $\mathcal{G}_{\tilde{E}_{R_{\infty}}}$ be the Galois group corresponding to the pro-universal covering, there exists a canonical isomorphism $\mathcal{G}_{\tilde{E}_{R_{\infty}}}\cong\mathcal{G}_{\tilde{R}_{\infty}}$.

Lemma 2.1**.**

Consider the map of Galois groups $\mathcal{G}_{\mathcal{O}_{L}}\rightarrow\mathcal{G}_{R}$ induced by choosing a common geometric point for the base change map $b_{L}:R\rightarrow\mathcal{O}_{L}$ in Section 2.1. Then the images of $\mathcal{G}_{\mathcal{O}_{L}}$ and $\mathcal{G}_{\tilde{R}_{\infty}}$ inside $\mathcal{G}_{R}$ generate the group $\mathcal{G}_{R}$.

Proof.

$E_{R_{\infty}}^{+}$ has a finite $p$-basis given by $\{t_{1},\ldots,t_{m},u\}$. Note that for any element of $g\in\mathcal{G}_{R}$, there exists an element $h\in\mathcal{G}_{\mathcal{O}_{L}}$ whose image in $\mathcal{G}_{R}$ induces the same actions on $t_{1}^{\frac{1}{p^{\infty}}},\ldots,t_{m}^{\frac{1}{p^{\infty}}},\varpi^{\frac{1}{p^{\infty}}}$. Since $\tilde{R}_{\infty}=W(\tilde{E}_{R_{\infty}}^{+})\otimes_{W(K_{\infty}^{\flat\circ}),~{}\theta}\mathcal{O}_{K_{\infty}}$, the actions of $g$ and $h$ are the same on the elements of $\tilde{R}_{\infty}$. Hence, the assertion follows. ∎

Now, let $\mathcal{O}_{\mathcal{E}}$ be the $p$-adic completion of $\mathfrak{S}[\frac{1}{u}]$. Note that $\varphi$ on $\mathfrak{S}$ extends naturally to $\mathcal{O}_{\mathcal{E}}$.

Definition 2.2**.**

An étale $(\varphi,\mathcal{O}_{\mathcal{E}})$-module is a pair $(\mathcal{M},\varphi_{\mathcal{M}})$ where $\mathcal{M}$ is a finitely generated $\mathcal{O}_{\mathcal{E}}$-module and $\varphi_{\mathcal{M}}:\mathcal{M}\rightarrow\mathcal{M}$ is a $\varphi$-semilinear endomorphism such that $1\otimes\varphi_{\mathcal{M}}:\varphi^{*}\mathcal{M}\rightarrow\mathcal{M}$ is an isomorphism. We say that an étale $(\varphi,\mathcal{O}_{\mathcal{E}})$-module is projective (resp. torsion) if the underlying $\mathcal{O}_{\mathcal{E}}$-module $\mathcal{M}$ is projective (resp. $p$-power torsion).

Let $\mathrm{Mod}_{\mathcal{O}_{\mathcal{E}}}$ denote the category of étale $(\varphi,\mathcal{O}_{\mathcal{E}})$-modules whose morphisms are $\mathcal{O}_{\mathcal{E}}$-module maps compatible with Frobenius. Let $\mathrm{Mod}_{\mathcal{O}_{\mathcal{E}}}^{\mathrm{pr}}$ and $\mathrm{Mod}_{\mathcal{O}_{\mathcal{E}}}^{\mathrm{tor}}$ respectively denote the full subcategories of projective and torsion objects. Note that we have a natural notion of a subquotient, direct sum, and tensor product for étale $(\varphi,\mathcal{O}_{\mathcal{E}})$-modules, and duality is defined for projective and torsion objects.

Lemma 2.3**.**

Let $\mathcal{M}\in\mathrm{Mod}_{\mathcal{O}_{\mathcal{E}}}^{\mathrm{tor}}$ be a torsion étale $\varphi$-module annihilated by $p$. Then $\mathcal{M}$ is a projective $\mathcal{O}_{\mathcal{E}}/p\mathcal{O}_{\mathcal{E}}$-module.

Proof.

This follows from essentially the same proof as in [And06, Lemma 7.10]. ∎

We consider $W(\overline{R}^{\flat}[\frac{1}{\underline{\varpi}}])$ as an $\mathcal{O}_{\mathcal{E}}$-algebra via mapping $u$ to the Teichmüller lift $[\underline{\varpi}]$ of $\underline{\varpi}$, and let $\mathcal{O}_{\mathcal{E}}^{\mathrm{ur}}$ be the integral closure of $\mathcal{O}_{\mathcal{E}}$ in $W(\overline{R}^{\flat}[\frac{1}{\underline{\varpi}}])$. Let $\widehat{\mathcal{O}}_{\mathcal{E}}^{\mathrm{ur}}$ be its $p$-adic completion. Since $\mathcal{O}_{\mathcal{E}}$ is normal, we have $\mathrm{Aut}_{\mathcal{O}_{\mathcal{E}}}(\mathcal{O}_{\mathcal{E}}^{\mathrm{ur}})\cong\mathcal{G}_{E_{R_{\infty}}}\coloneqq\pi_{1}^{\text{\'{e}t}}(\mathrm{Spec}E_{R_{\infty}})$, and by [GR03, Proposition 5.4.54] and the almost purity theorem, we have $\mathcal{G}_{E_{R_{\infty}}}\cong\mathcal{G}_{\tilde{E}_{R_{\infty}}}\cong\mathcal{G}_{\tilde{R}_{\infty}}$. This induces $\mathcal{G}_{\tilde{R}_{\infty}}$-action on $\widehat{\mathcal{O}}_{\mathcal{E}}^{\mathrm{ur}}$. The following is proved in [Kim15].

Lemma 2.4**.**

(cf. [Kim15, Lemma 7.5 and 7.6])* We have $(\widehat{\mathcal{O}}_{\mathcal{E}}^{\mathrm{ur}})^{\mathcal{G}_{\tilde{R}_{\infty}}}=\mathcal{O}_{\mathcal{E}}$ and the same holds modulo $p^{n}$. Furthermore, there exists a unique $\mathcal{G}_{\tilde{R}_{\infty}}$-equivariant ring endomorphism $\varphi$ on $\widehat{\mathcal{O}}_{\mathcal{E}}^{\mathrm{ur}}$ lifting the $p$-th power map on $\widehat{\mathcal{O}}_{\mathcal{E}}^{\mathrm{ur}}/(p)$ and extending $\varphi$ on $\mathcal{O}_{\mathcal{E}}$. The inclusion $\widehat{\mathcal{O}}_{\mathcal{E}}^{\mathrm{ur}}\hookrightarrow W(\overline{R}^{\flat}[\frac{1}{\underline{\varpi}}])$ is $\varphi$-equivariant where the latter ring is given the Witt vector Frobenius.*

Let $\mathrm{Rep}_{\mathbf{Z}_{p}}(\mathcal{G}_{\tilde{R}_{\infty}})$ be the category of $\mathbf{Z}_{p}$-representations of $\mathcal{G}_{\tilde{R}_{\infty}}$, and let $\mathrm{Rep}_{\mathbf{Z}_{p}}^{\mathrm{pr}}(\mathcal{G}_{\tilde{R}_{\infty}})$ and $\mathrm{Rep}_{\mathbf{Z}_{p}}^{\mathrm{tor}}(\mathcal{G}_{\tilde{R}_{\infty}})$ respectively denote the full subcategories of free and torsion objects. For $\mathcal{M}\in\mathrm{Mod}_{\mathcal{O}_{\mathcal{E}}}$ and $T\in\mathrm{Rep}_{\mathbf{Z}_{p}}(\mathcal{G}_{\tilde{R}_{\infty}})$, we define $T(\mathcal{M})\coloneqq(\mathcal{M}\otimes_{\mathcal{O}_{\mathcal{E}}}\widehat{\mathcal{O}}_{\mathcal{E}}^{\mathrm{ur}})^{\varphi=1}$ and $\mathcal{M}(T)\coloneqq(T\otimes_{\mathbf{Z}_{p}}\widehat{\mathcal{O}}_{\mathcal{E}}^{\mathrm{ur}})^{\mathcal{G}_{\tilde{R}_{\infty}}}$. For a torsion étale $\varphi$-module $\mathcal{M}\in\mathrm{Mod}_{\mathcal{O}_{\mathcal{E}}}^{\mathrm{tor}}$, we define its length to be the length of $\mathcal{M}\otimes_{\mathcal{O}_{\mathcal{E}}}(\mathcal{O}_{\mathcal{E}})_{(p)}$ as an $(\mathcal{O}_{\mathcal{E}})_{(p)}$-module.

Proposition 2.5**.**

(cf. [Kim15, Proposition 7.7])* The assignments $T(\cdot)$ and $\mathcal{M}(\cdot)$ are exact equivalences (inverse of each other) of $\otimes$-categories between $\mathrm{Mod}_{\mathcal{O}_{\mathcal{E}}}$ and $\mathrm{Rep}_{\mathbf{Z}_{p}}(\mathcal{G}_{\tilde{R}_{\infty}})$. Moreover, $T(\cdot)$ and $\mathcal{M}(\cdot)$ restrict to rank-preserving equivalence of categories between $\mathrm{Mod}_{\mathcal{O}_{\mathcal{E}}}^{\mathrm{pr}}$ and $\mathrm{Rep}_{\mathbf{Z}_{p}}^{\mathrm{pr}}(\mathcal{G}_{\tilde{R}_{\infty}})$ and length-preserving equivalence of categories between $\mathrm{Mod}_{\mathcal{O}_{\mathcal{E}}}^{\mathrm{tor}}$ and $\mathrm{Rep}_{\mathbf{Z}_{p}}^{\mathrm{tor}}({\mathcal{G}}_{\tilde{R}_{\infty}})$. In both cases, $T(\cdot)$ and $\mathcal{M}(\cdot)$ commute with taking duals.*

Proof.

This is [Kim15, Proposition 7.7]. We remark here for some additional details. Note that $E_{R_{\infty}}$ is a normal domain and $\pi_{1}^{\text{\'{e}t}}(\mathrm{Spec}\mathbf{E}_{R_{\infty}})\cong\mathcal{G}_{\tilde{R}_{\infty}}$. Given Lemma 2.3, the assertion therefore follows from the usual dévissage and [Katz, Lemma 4.1.1]. Note that both functors $T(\cdot)$ and $\mathcal{M}(\cdot)$ are a priori left exact by definition, and exactness can be proved by the same argument as in the proof of [And06, Theorem 7.11]. ∎

Suppose $S_{0}$ is another relative base ring over $W(k)\langle t_{1}^{\pm 1},\ldots,t_{m}^{\pm 1}\rangle$ as in Section 2.1 equipped with a choice of Frobenius, and suppose $b:R_{0}\hookrightarrow S_{0}$ be a $\varphi$-equivariant $W(k)\langle t_{1}^{\pm 1},\ldots,t_{m}^{\pm 1}\rangle$-algebra map which is injective. Let $b:R=R_{0}\otimes_{W(k)}\mathcal{O}_{K}\hookrightarrow S\coloneqq S_{0}\otimes_{W(k)}\mathcal{O}_{K}$ be the induced injective map. By choosing a common geometric point we have an injective map $\overline{R}\hookrightarrow\overline{S}$, and this induces an embedding $\tilde{R}_{\infty}\hookrightarrow\tilde{S}_{\infty}$ by the constructions given in equation (2.1). Hence, the corresponding map of Galois groups $\mathcal{G}_{S}\rightarrow\mathcal{G}_{R}$ restricts to $\mathcal{G}_{\tilde{S}_{\infty}}\rightarrow\mathcal{G}_{\tilde{R}_{\infty}}$. Let $\mathfrak{S}_{S}=S_{0}[\![u]\!]$ and let $\mathcal{O}_{\mathcal{E},S}$ be the $p$-adic completion of $\mathfrak{S}_{S}[\frac{1}{u}]$. Let $\mathcal{M}_{S}(\cdot)$ be the functor for the base ring $S$ constructed similarly as above. Let $T\in\mathrm{Rep}_{\mathbf{Z}_{p}}^{\mathrm{pr}}(\mathcal{G}_{\tilde{R}_{\infty}})$. Then $T$ is also a $\mathcal{G}_{\tilde{S}_{\infty}}$-representation via the map $\mathcal{G}_{\tilde{S}_{\infty}}\rightarrow\mathcal{G}_{\tilde{R}_{\infty}}$, and we have the natural isomorphism $\mathcal{M}(T)\otimes_{\mathcal{O}_{\mathcal{E}}}\mathcal{O}_{\mathcal{E},S}\cong\mathcal{M}_{S}(T)$ as étale $(\varphi,\mathcal{O}_{\mathcal{E},S})$-modules by the definition of the functors $\mathcal{M}(\cdot)$ and $T(\cdot)$ and by Proposition 2.5.

3 Relative Breuil-Kisin classification

We now explain the classification of $p$-divisible groups over $\mathrm{Spec}R$ via Kisin modules, which is proved in [Kis06] when $R=\mathcal{O}_{K}$ and generalized in [Kim15] for the relative case. Denote by $E(u)$ the Eisenstein polynomial for the extension $K$ over $W(k)[\frac{1}{p}]$, and let $\mathfrak{S}=R_{0}[\![u]\!]$ as above.

Definition 3.1**.**

Denote by $\mathrm{Kis}^{1}(\mathfrak{S})$ the category of pairs $(\mathfrak{M},\varphi_{\mathfrak{M}})$ where

• •

$\mathfrak{M}$ is a finitely generated projective $\mathfrak{S}$-module;

• •

$\varphi_{\mathfrak{M}}:\mathfrak{M}\rightarrow\mathfrak{M}$ is a $\varphi$-semilinear map such that $\mathrm{coker}(1\otimes\varphi_{\mathfrak{M}})$ is annihilated by $E(u)$.

The morphisms are $\mathfrak{S}$-module maps compatible with Frobenius.

Note that for $(\mathfrak{M},\varphi_{\mathfrak{M}})\in\mathrm{Kis}^{1}(\mathfrak{S})$, $1\otimes\varphi_{\mathfrak{M}}:\varphi^{*}\mathfrak{M}\rightarrow\mathfrak{M}$ is injective since $\mathfrak{M}$ is finite projective over $\mathfrak{S}$ and $\mathrm{coker}(1\otimes\varphi_{\mathfrak{M}})$ is killed by $E(u)$. Consider the composite $\mathfrak{S}\twoheadrightarrow\mathfrak{S}/u\mathfrak{S}=R_{0}\stackrel{{\scriptstyle\varphi}}{{\rightarrow}}R_{0}$.

Definition 3.2**.**

A Kisin module of height $1$ is a tuple $(\mathfrak{M},\varphi_{\mathfrak{M}},\nabla_{\mathfrak{M}})$ such that

• •

$(\mathfrak{M},\varphi_{\mathfrak{M}})\in\mathrm{Kis}^{1}(\mathfrak{S})$;

• •

Let $\mathcal{N}\coloneqq\mathfrak{M}\otimes_{\mathfrak{S},\varphi}R_{0}$ equipped with the Frobenius $\varphi_{\mathfrak{M}}\otimes\varphi_{R_{0}}$. Then $\nabla_{\mathfrak{M}}:\mathcal{N}\rightarrow\mathcal{N}\otimes_{R_{0}}\widehat{\Omega}_{R_{0}}$ is a topologically quasi-nilpotent integrable connection commuting with Frobenius.

Here, $\nabla_{\mathfrak{M}}$ being topologically quasi-nilpotent means that the induced connection on $\mathcal{N}/p\mathcal{N}$ is nilpotent. Denote by $\mathrm{Kis}^{1}(\mathfrak{S},\nabla)$ the category of Kisin modules of height $1$ whose morphisms are $\mathfrak{S}$-module maps compatible with Frobenius and connection.

The following theorem classifying the $p$-divisible groups is proved in [Kim15].

Theorem 3.3**.**

(cf. [Kim15, Corollary 6.7 and Remark 6.9])* There exists an exact anti-equivalence of categories*

[TABLE]

Let $S_{0}$ be another base ring satisfying the condition as in Section 2.1 and equipped with a Frobenius, and let $b:R_{0}\rightarrow S_{0}$ be a $\varphi$-equivariant map. Then the formation of $\mathfrak{M}^{*}$ commutes with the base change $R\rightarrow S\coloneqq S_{0}\otimes_{W(k)}\mathcal{O}_{K}$ induced $\mathcal{O}_{K}$-linearly from $b$.

Note that if $(\mathfrak{M},\varphi_{\mathfrak{M}})\in\mathrm{Kis}^{1}(\mathfrak{S})$, then $(\mathfrak{M}\otimes_{\mathfrak{S}}\mathcal{O}_{\mathcal{E}},\varphi_{\mathfrak{M}}\otimes\varphi_{\mathcal{O}_{\mathcal{E}}})$ is a projective étale $(\varphi,\mathcal{O}_{\mathcal{E}})$-module since $1\otimes\varphi_{\mathfrak{M}}$ is injective and its cokernel is killed by $E(u)$ which is a unit in $\mathcal{O}_{\mathcal{E}}$. If $G_{R}$ is a $p$-divisible group over $R$, its Tate module is given by $T_{p}(G_{R})\coloneqq\mathrm{Hom}_{\overline{R}}(\mathbf{Q}_{p}/\mathbf{Z}_{p},G_{R}\times_{R}\overline{R})$, which is a finite free $\mathbf{Z}_{p}$-representation of $\mathcal{G}_{R}$. By [Kim15, Corollary 8.2], we have a natural $\mathcal{G}_{\tilde{R}_{\infty}}$-equivariant isomorphism $T^{\vee}(\mathfrak{M}^{*}(G_{R})\otimes_{\mathfrak{S}}\mathcal{O}_{\mathcal{E}})\cong T_{p}(G_{R})$ where $T^{\vee}(\mathfrak{M}^{*}(G_{R})\otimes_{\mathfrak{S}}\mathcal{O}_{\mathcal{E}})$ denotes the dual of $T(\mathfrak{M}^{*}(G_{R})\otimes_{\mathfrak{S}}\mathcal{O}_{\mathcal{E}})$.

4 Construction of Kisin modules

Throughout this section, we assume $e<p-1$. We denote $\mathfrak{S}_{n}\coloneqq\mathfrak{S}/p^{n}\mathfrak{S}$ for positive integers $n\geq 1$. Let $T$ be a crystalline $\mathcal{G}_{R}$-representation which is free over $\mathbf{Z}_{p}$ of rank $d$ with Hodge-Tate weights in $[0,1]$. Let $\mathcal{M}\coloneqq\mathcal{M}^{\vee}(T)$ be the associated étale $(\varphi,\mathcal{O}_{\mathcal{E}})$-module, where $\mathcal{M}^{\vee}(T)$ denotes the dual of $\mathcal{M}(T)$. For each integer $n\geq 1$, denote $\mathcal{M}_{n}=\mathcal{M}/p^{n}\mathcal{M}$. Note that $\mathcal{M}_{n}\cong\mathcal{M}^{\vee}(T/p^{n}T)$. On the other hand, consider the map $b_{L}:R\rightarrow\mathcal{O}_{L}$ as in Section 2.1. $T$ is also a crystalline $\mathcal{G}_{\mathcal{O}_{L}}$-representation with Hodge-Tate weights in $[0,1]$, so by [BT08, Theorem 6.10], there exists a $p$-divisible group $G_{\mathcal{O}_{L}}$ over $\mathcal{O}_{L}$ such that $T_{p}(G_{\mathcal{O}_{L}})\cong T$ as $\mathcal{G}_{\mathcal{O}_{L}}$-representations. Let $(\mathfrak{M}_{\mathcal{O}_{L}},\nabla_{\mathfrak{M}_{\mathcal{O}_{L}}})\coloneqq\mathfrak{M}^{*}(G_{\mathcal{O}_{L}})\in\mathrm{Kis}^{1}(\mathfrak{S}_{\mathcal{O}_{L}},\nabla)$ be the associated Kisin module over $\mathfrak{S}_{\mathcal{O}_{L}}$. Denote $\mathfrak{M}_{\mathcal{O}_{L},n}=\mathfrak{M}_{\mathcal{O}_{L}}/p^{n}\mathfrak{M}_{\mathcal{O}_{L}}$. The map between the Galois groups $\mathcal{G}_{\mathcal{O}_{L}}\rightarrow\mathcal{G}_{R}$ restricts to $\mathcal{G}_{\widetilde{\mathcal{O}}_{L},\infty}\rightarrow\mathcal{G}_{\tilde{R}_{\infty}}$. Hence, we have the natural isomorphism $\mathcal{M}\otimes_{\mathcal{O}_{\mathcal{E}}}\mathcal{O}_{\mathcal{E},\mathcal{O}_{L}}\cong\mathfrak{M}_{\mathcal{O}_{L}}\otimes_{\mathfrak{S}_{\mathcal{O}_{L}}}\mathcal{O}_{\mathcal{E},\mathcal{O}_{L}}$ of étale $(\varphi,\mathcal{O}_{\mathcal{E},\mathcal{O}_{L}})$-modules. Let $\mathcal{M}_{\mathcal{O}_{L}}\coloneqq\mathcal{M}\otimes_{\mathcal{O}_{\mathcal{E}}}\mathcal{O}_{\mathcal{E},\mathcal{O}_{L}}$ and $\mathcal{M}_{\mathcal{O}_{L},n}\coloneqq\mathcal{M}_{\mathcal{O}_{L}}/p^{n}\mathcal{M}_{\mathcal{O}_{L}}$.

For each $n\geq 1$, we define

[TABLE]

where the intersection is taken as $\mathfrak{S}$-submodules of $\mathcal{M}_{\mathcal{O}_{L},n}$. The Frobenius endomorphisms on $\mathcal{M}_{n}$ and $\mathfrak{M}_{\mathcal{O}_{L},n}$ induce a Frobenius endomorphism on $\mathfrak{M}_{n}$. Since the Frobenius on $\mathcal{M}_{\mathcal{O}_{L},n}$ is injective, we have the injective $\mathfrak{S}$-module morphism

[TABLE]

for each $n$.

Lemma 4.1**.**

$\mathfrak{M}_{n}$* is a finitely generated $\mathfrak{S}_{n}$-module. Furthermore, we have $\varphi$-equivariant isomorphisms*

[TABLE]

and

[TABLE]

Proof.

We first prove that $\mathfrak{M}_{n}$ is finite over $\mathfrak{S}_{n}$. Note that $\mathfrak{M}_{\mathcal{O}_{L},n}$ is free over $\mathfrak{S}_{\mathcal{O}_{L},n}$ of rank $d$, and choose a basis $\{e_{1},\ldots,e_{d}\}$ of $\mathfrak{M}_{\mathcal{O}_{L},n}$. On the other hand, since $\mathcal{M}_{n}$ is projective over $\mathfrak{S}_{n}[\frac{1}{u}]$ of rank $d$, there exists a non-zero divisor $g\in\mathfrak{S}_{n}$ such that $\mathcal{M}_{n}[\frac{1}{g}]$ is free of rank $d$ over $\mathfrak{S}_{n}[\frac{1}{u}][\frac{1}{g}]$. Since $\mathcal{M}_{n}$ is finite over $\mathfrak{S}_{n}[\frac{1}{u}]$, we can choose a basis $\{f_{1},\ldots,f_{d}\}$ of $\mathcal{M}_{n}[\frac{1}{g}]$ over $\mathfrak{S}_{n}[\frac{1}{u}][\frac{1}{g}]$ such that letting $\mathfrak{N}$ to be the $\mathfrak{S}_{n}$-submodule of $\mathcal{M}_{n}[\frac{1}{g}]$ generated by $f_{1},\ldots,f_{d}$, we have $\mathcal{M}_{n}\subset\mathfrak{N}[\frac{1}{u}]$ as $\mathfrak{S}_{n}[\frac{1}{u}]$-modules. It suffices to show that $\mathfrak{M}_{n}\subset\frac{1}{u^{h}}\cdot\mathfrak{N}$ as $\mathfrak{S}_{n}$-modules for some integer $h\geq 1$. We have

[TABLE]

where $A$ is an invertible $d\times d$ matrix with entries in $\mathfrak{S}_{\mathcal{O}_{L},n}[\frac{1}{u}][\frac{1}{g}]$. Consider the intersection $\mathfrak{N}[\frac{1}{u}]\cap\mathfrak{M}_{\mathcal{O}_{L},n}$ as submodules of $\mathfrak{M}_{\mathcal{O}_{L},n}[\frac{1}{u}][\frac{1}{g}]$. For an element $x=b_{1}f_{1}+\cdots+b_{d}f_{d}\in\mathfrak{N}[\frac{1}{u}]$ with $b_{1},\ldots,b_{d}\in\mathfrak{S}_{n}[\frac{1}{u}]$, we have $x\in\mathfrak{M}_{\mathcal{O}_{L},n}$ if and only if

[TABLE]

for some $c_{1},\ldots,c_{d}\in\mathfrak{S}_{\mathcal{O}_{L},n}$. Then $(b_{1},\ldots,b_{d})=A^{-1}(c_{1},\ldots,c_{d})$, which implies that $\mathfrak{N}[\frac{1}{u}]\cap\mathfrak{M}_{\mathcal{O}_{L},n}\subset\frac{1}{u^{h}}\cdot\mathfrak{N}$ as $\mathfrak{S}_{n}$-modules for some integer $h\geq 1$. Since $\mathfrak{M}_{n}\subset\mathfrak{N}[\frac{1}{u}]\cap\mathfrak{M}_{\mathcal{O}_{L},n}$, this shows the first statement.

We have

[TABLE]

and hence the first isomorphism. On the other hand, since $\mathfrak{S}\rightarrow\mathfrak{S}_{\mathcal{O}_{L}}$ is flat and $\mathfrak{M}_{\mathcal{O}_{L},n}$ is finite free over $\mathfrak{S}_{\mathcal{O}_{L},n}$, we have

[TABLE]

by $\mathfrak{S}_{n}[\frac{1}{u}]\cap\mathfrak{S}_{\mathcal{O}_{L},n}=\mathfrak{S}_{n}$. ∎

Lemma 4.2**.**

The cokernel of the $\mathfrak{S}$-module map $1\otimes\varphi:\mathfrak{S}\otimes_{\varphi,\mathfrak{S}}\mathfrak{M}_{n}\rightarrow\mathfrak{M}_{n}$ is killed by $E(u)$.

Proof.

Let $x\in\mathfrak{M}_{n}$. There exists a unique $y_{1}\in\mathcal{O}_{\mathcal{E}}\otimes_{\varphi,\mathcal{O}_{\mathcal{E}}}\mathcal{M}_{n}\cong\mathfrak{S}\otimes_{\varphi,\mathfrak{S}}\mathcal{M}_{n}$ such that $(1\otimes\varphi)(y_{1})=E(u)x$. On the other hand, there exists a unique $y_{2}\in\mathfrak{S}_{\mathcal{O}_{L}}\otimes_{\varphi,\mathfrak{S}_{\mathcal{O}_{L}}}\mathfrak{M}_{\mathcal{O}_{L},n}$ such that $(1\otimes\varphi)(y_{2})=E(u)x$. Then we have $y_{1}=y_{2}\in(\mathfrak{S}\otimes_{\varphi,\mathfrak{S}}\mathcal{M}_{n})\cap(\mathfrak{S}_{\mathcal{O}_{L}}\otimes_{\varphi,\mathfrak{S}_{\mathcal{O}_{L}}}\mathfrak{M}_{\mathcal{O}_{L},n})$.

Since $\mathcal{O}_{L_{0}}/p\mathcal{O}_{L_{0}}$ has a finite $p$-basis given by $t_{1},\ldots,t_{m}\in R_{0}/pR_{0}$ which also gives a $p$-basis of $R_{0}/pR_{0}$, the natural map $\mathfrak{S}\otimes_{\varphi,\mathfrak{S}}\mathfrak{M}_{\mathcal{O}_{L},n}\rightarrow\mathfrak{S}_{\mathcal{O}_{L}}\otimes_{\varphi,\mathfrak{S}_{\mathcal{O}_{L}}}\mathfrak{M}_{\mathcal{O}_{L},n}$ is an isomorphism. Hence,

[TABLE]

since $\varphi:\mathfrak{S}\rightarrow\mathfrak{S}$ is flat by [Bri08, Lemma 7.1.8]. This proves the assertion. ∎

For any finite $\mathfrak{S}$-module $\mathfrak{N}$ equipped with a $\varphi$-semilinear endomorphism $\varphi:\mathfrak{N}\rightarrow\mathfrak{N}$, say $\mathfrak{N}$ has $E(u)$-height $\leq 1$ if there exists a $\mathfrak{S}$-module map $\psi:\mathfrak{N}\rightarrow\varphi^{*}\mathfrak{N}=\mathfrak{S}\otimes_{\varphi,\mathfrak{S}}\mathfrak{N}$ such that the composite

[TABLE]

is $E(u)\cdot\mathrm{Id}_{\varphi^{*}\mathfrak{N}}$. By Lemma 4.2, $\mathfrak{M}_{n}$ has $E(u)$-height $\leq 1$.

For each maximal ideal $\mathfrak{q}\in\mathrm{mSpec}R_{0}$, consider $b_{\mathfrak{q}}:R\rightarrow R_{\mathfrak{q}}$ as in Section 2.1. By choosing a common geometric point, we have the induced map of Galois groups $\mathcal{G}_{R_{\mathfrak{q}}}\rightarrow\mathcal{G}_{R}$ which ristricts to $\mathcal{G}_{\widetilde{R}_{\mathfrak{q},\infty}}\rightarrow\mathcal{G}_{\tilde{R},\infty}$, and $T$ is a crystalline $\mathcal{G}_{R_{\mathfrak{q}}}$-representation with Hodge-Tate weights in $[0,1]$. Denote $\mathfrak{S}_{\mathfrak{q}}\coloneqq\widehat{R_{0,\mathfrak{q}}}[\![u]\!]$.

Proposition 4.3**.**

For each integer $n\geq 1$, $\mathfrak{M}_{n}$ is projective over $\mathfrak{S}_{n}$ of rank $d$.

Proof.

Let $\mathfrak{q}$ be a maximal ideal of $R_{0}$, and let $\mathfrak{N}_{n}\coloneqq\mathfrak{M}_{n}\otimes_{\mathfrak{S}}\mathfrak{S}_{\mathfrak{q}}$ equipped with the induced Frobenius endomorphism. Then we have the induced $\mathfrak{S}_{\mathfrak{q}}$-linear map $\psi:\mathfrak{N}_{n}\rightarrow\mathfrak{S}_{\mathfrak{q}}\otimes_{\varphi,\mathfrak{S}_{\mathfrak{q}}}\mathfrak{N}_{n}$ such that the composite

[TABLE]

is $E(u)\cdot\mathrm{Id}$. For the isomorphism $\widehat{R_{0,\mathfrak{q}}}\cong\mathcal{O}_{\mathfrak{q}}[\![s_{1},\ldots,s_{l}]\!]$ as above, consider the projection $\mathfrak{S}_{\mathfrak{q}}\rightarrow\mathfrak{S}_{\mathfrak{q}}/(p,s_{1},\ldots,s_{l})\cong k_{\mathfrak{q}}[\![u]\!]$ where $k_{\mathfrak{q}}\coloneqq\mathcal{O}_{\mathfrak{q}}/(p)$. Denote $\overline{\mathfrak{N}}_{n}=\mathfrak{N}_{n}\otimes_{\mathfrak{S}_{\mathfrak{q}}}k_{\mathfrak{q}}[\![u]\!]$ equipped with the induced Frobenius. Then we have the induced $k_{\mathfrak{q}}[\![u]\!]$-linear map $\psi:\overline{\mathfrak{N}}_{n}\rightarrow k_{\mathfrak{q}}[\![u]\!]\otimes_{\varphi,k_{\mathfrak{q}}[\![u]\!]}\overline{\mathfrak{N}}_{n}$ such that the composite

[TABLE]

is $u^{e}\cdot\mathrm{Id}$. Since $k_{\mathfrak{q}}[\![u]\!]$ is a principal ideal domain, $\overline{\mathfrak{N}}_{n}$ is a direct sum of its free part and $u$-torsion part $\overline{\mathfrak{N}}_{n}\cong\overline{\mathfrak{N}}_{n,\mathrm{free}}\oplus\overline{\mathfrak{N}}_{n,\mathrm{tor}}$ as $k_{\mathfrak{q}}[\![u]\!]$-modules. Furthermore, $\varphi$ maps $\overline{\mathfrak{N}}_{n,\mathrm{tor}}$ into $\overline{\mathfrak{N}}_{n,\mathrm{tor}}$, and hence the above maps induce

[TABLE]

whose composite is $u^{e}\cdot\mathrm{Id}$.

We claim that $\overline{\mathfrak{N}}_{n,\mathrm{tor}}=0$. Suppose otherwise. Then $\overline{\mathfrak{N}}_{n,\mathrm{tor}}\cong\bigoplus_{i=1}^{b}k_{\mathfrak{q}}[\![u]\!]/(u^{a_{i}})$ for some integers $a_{i}\geq 1$, and $k_{\mathfrak{q}}[\![u]\!]\otimes_{\varphi,k_{\mathfrak{q}}[\![u]\!]}\overline{\mathfrak{N}}_{n,\mathrm{tor}}\cong\bigoplus_{i=1}^{b}k_{\mathfrak{q}}[\![u]\!]/(u^{pa_{i}})$. By taking the appropriate wedge product and letting $a=a_{1}+\ldots+a_{b}$, the above maps induce the map of $k_{\mathfrak{q}}[\![u]\!]$-modules

[TABLE]

whose composite is equal to $u^{eb}\cdot\mathrm{Id}$. Let $(1\otimes\varphi)(1)=f(u)\in k_{\mathfrak{q}}[\![u]\!]/(u^{a})$, and $\psi(1)=h(u)\in k_{\mathfrak{q}}[\![u]\!]/(u^{pa})$. Then $u^{pa}\mid u^{a}h(u)$, so $u^{(p-1)a}\mid h(u)$. On the other hand, $f(u)h(u)=u^{eb}$ in $k_{\mathfrak{q}}[\![u]\!]/(u^{pa})$. This implies $u^{(p-1)a}\mid u^{eb}$. But $e<p-1$ and $a\geq b$, so we get a contradiction. Hence, $\overline{\mathfrak{N}}_{n,\mathrm{tor}}=0$ and $\overline{\mathfrak{N}}_{n}$ is free over $k_{\mathfrak{q}}[\![u]\!]$ of rank $d$, since by Lemma 4.1 $\overline{\mathfrak{N}}_{n}[\frac{1}{u}]\cong(\mathcal{M}_{n}\otimes_{\mathfrak{S}}\mathfrak{S}_{\mathfrak{q}})\otimes_{\mathfrak{S}_{\mathfrak{q}}}k_{\mathfrak{q}}[\![u]\!]$ which is projective over $k_{\mathfrak{q}}(\!(u)\!)$ of rank $d$. Let $b_{1},\ldots,b_{d}\in\mathfrak{N}_{n}$ be a lift of a basis elements of $\overline{\mathfrak{N}}_{n}$. By Nakayama’s lemma, we have a surjection of $\mathfrak{S}_{\mathfrak{q,n}}$-modules

[TABLE]

given by $e_{i}\mapsto b_{i}$. Since $\mathfrak{N}_{n}[\frac{1}{u}]\cong\mathcal{M}_{n}\otimes_{\mathfrak{S}}\mathfrak{S}_{\mathfrak{q}}$ is projective over $\mathfrak{S}_{\mathfrak{q},n}[\frac{1}{u}]$ of rank $d$, $f$ is also injective. Thus, $\mathfrak{N}_{n}=\mathfrak{M}_{n}\otimes_{\mathfrak{S}}\mathfrak{S}_{\mathfrak{q}}$ is projective over $\mathfrak{S}_{\mathfrak{q},n}$ of rank $d$. Since this holds for every $\mathfrak{q}\in\mathrm{mSpec}R_{0}$, it proves the assertion. ∎

Lemma 4.4**.**

Let $\mathfrak{N}$ and $\mathfrak{N}^{\prime}$ be finite $u$-torsion free $\mathfrak{S}$-modules equipped with Frobenius endomorphisms such that $\mathfrak{N}[\frac{1}{u}]$ and $\mathfrak{N}^{\prime}[\frac{1}{u}]$ are torsion étale $\varphi$-modules. Suppose that $\mathfrak{N}$ and $\mathfrak{N}^{\prime}$ have $E(u)$-height $\leq 1$ and $\mathfrak{N}[\frac{1}{u}]=\mathfrak{N}^{\prime}[\frac{1}{u}]$ as étale $\varphi$-modules. Then $\mathfrak{N}=\mathfrak{N}^{\prime}$.

Proof.

Consider $\mathfrak{N}$ and $\mathfrak{N}^{\prime}$ as $\mathfrak{S}$-submodules of $\mathfrak{N}[\frac{1}{u}]$. Let $\mathfrak{L}$ be the cokernel of the embedding $\mathfrak{N}\hookrightarrow\mathfrak{N}+\mathfrak{N}^{\prime}$ of $\mathfrak{S}$-modules. Note that $\mathfrak{S}\otimes_{\varphi,\mathfrak{S}}(\mathfrak{N}+\mathfrak{N}^{\prime})\cong\mathfrak{S}\otimes_{\varphi,\mathfrak{S}}\mathfrak{N}+\mathfrak{S}\otimes_{\varphi,\mathfrak{S}}\mathfrak{N}^{\prime}$ since $\varphi:\mathfrak{S}\rightarrow\mathfrak{S}$ is flat. Thus, $\mathfrak{N}+\mathfrak{N}^{\prime}$ has $E(u)$-height $\leq 1$, and $\mathfrak{L}$ has $E(u)$-height $\leq 1$. Since $\mathfrak{L}[\frac{1}{u}]=0$, we deduce similarly as in the proof of Proposition 4.3 that $\mathfrak{L}=0$. So $\mathfrak{N}=\mathfrak{N}+\mathfrak{N}^{\prime}$. Similarly, $\mathfrak{N}^{\prime}=\mathfrak{N}+\mathfrak{N}^{\prime}$. ∎

It is clear that both $p\mathfrak{M}_{n+1}$ and $\mathfrak{M}_{n}$ are $u$-torsion free, have $E(u)$-height $\leq 1$ and $p\mathfrak{M}_{n+1}[\frac{1}{u}]=p\mathcal{M}_{n+1}\cong\mathcal{M}_{n}=\mathfrak{M}_{n}[\frac{1}{u}]$. We conclude the following:

Proposition 4.5**.**

For each $n\geq 1$, we have a $\varphi$-equivariant isomorphism

[TABLE]

By Lemma 4.2, Proposition 4.3 and 4.5, if we define the $\mathfrak{S}$-module

[TABLE]

then $\mathfrak{M}\in\mathrm{Kis}^{1}(\mathfrak{S})$. Note that we have a $\varphi$-equivariant isomorphism $\mathfrak{M}\otimes_{\mathfrak{S}}\mathfrak{S}_{\mathcal{O}_{L}}\cong\mathfrak{M}_{\mathcal{O}_{L}}$ by Lemma 4.1.

Remark 4.6*.*

Analogous statements hold when $T$ is a crystalline $\mathcal{G}_{R}$-representation with Hodge-Tate weights in $[0,r]$ for the case $er<p-1$, since [BT08] constructs more generally a functor from crystalline representations with Hodge-Tate weights in $[0,r]$ to Kisin modules of height $r$ when the base is a complete discrete valuation field whose residue field has a finite $p$-basis.

To study connections for $\mathfrak{M}$, we first consider the following general situation. Let $A_{0}$ be a $k$-algebra which is an integral domain. Consider $n$-variables $x_{1},\ldots,x_{n}$, and denote $\underline{x}=(x_{1},\ldots,x_{n})^{t}$ and $\underline{x}^{[p]}\coloneqq(x_{1}^{p},\ldots,x_{n}^{p})^{t}$. An Artin-Schreier system of equations in $n$ variables over $A_{0}$ is given by

[TABLE]

where $B=(b_{ij})_{1\leq i,j\leq n}\in M_{n\times n}(A_{0})$ is an $n\times n$ matrix with entries in $A_{0}$ and $C=(c_{i})_{1\leq i\leq n}\in M_{n\times 1}(A_{0})$. Let

[TABLE]

which is the $A_{0}$-algebra parametrizing the solutions of equation (4.1). $A_{0}\rightarrow A_{1}$ is étale by [Vas13, Theorem 2.4.1(a)].

Lemma 4.7**.**

There exists a non-zero element $f\in A_{0}$ which depends only on $B$ such that $A_{1}[\frac{1}{f}]$ is finite étale over $A_{0}[\frac{1}{f}]$.

Proof.

We induct on $n$. Suppose $n=1$. If $\det{B}\neq 0$, then equation (4.1) is equivalent to

[TABLE]

so the assertion holds with $f=\det{B}$. If $\det{B}=0$, then $B=0$ and $A_{1}\cong A_{0}$, so the assertion holds trivially.

For $n\geq 2$, if $\det{B}\neq 0$, then equation (4.1) is equivalent to

[TABLE]

Hence, with $f=\det{B}$, $A_{1}[\frac{1}{f}]$ is finite étale over $A_{0}[\frac{1}{f}]$. Suppose $\det{B}=0$. Denote by $B^{(i)}$ the $i$-th row of $B$. Then up to renumbering the index for $x_{i}$’s, we have

[TABLE]

for some non-zero $f_{1}\in A_{0}$ and some $e_{i}\in A_{0}[\frac{1}{f_{1}}]$ with $e_{n}=1$. From equation (4.1), we get

[TABLE]

Hence, denoting $\underline{x}^{\prime}=(x_{1},\ldots,x_{n-1})^{t}$, equation (4.1) is equivalent to an Artin-Schreier system of equations in $n-1$ variables over $A_{0}[\frac{1}{f_{1}}]$

[TABLE]

where $B^{\prime}\in M_{(n-1)\times(n-1)}(A_{0}[\frac{1}{f_{1}}])$ and $C^{\prime}\in M_{(n-1)\times 1}(A_{0}[\frac{1}{f_{1}}])$. Note that $B^{\prime}$ depends only on $B$ and not on $C$. Hence, the assertion follows by induction. ∎

Let $\mathcal{N}\coloneqq\mathfrak{M}\otimes_{\mathfrak{S},\varphi}R_{0}$ equipped with the Frobenius $\varphi_{\mathfrak{M}}\otimes\varphi_{R_{0}}$. From [Kim15, Eq. (6.1), (6.2) and Remark 3.13], we have the $R_{0}$-submodule $\mathrm{Fil}^{1}\mathcal{N}\subset\mathcal{N}$ associated with $\mathfrak{M}\in\mathrm{Kis}^{1}(\mathfrak{S})$ such that $p\mathcal{N}\subset\mathrm{Fil}^{1}\mathcal{N}$, $\mathcal{N}/\mathrm{Fil}^{1}\mathcal{N}$ is projective over $R_{0}/(p)$, and $(1\otimes\varphi)(\varphi^{*}\mathrm{Fil}^{1}\mathcal{N})=p\mathcal{N}$ as $R_{0}$-modules (cf. [Kim15, Definition 3.4 and 3.6] for the frame $(R_{0},pR_{0},R_{0}/(p),\varphi_{R_{0}},\frac{\varphi_{R_{0}}}{p})$). Fix an $R_{0}$-direct factor $\mathcal{N}^{1}\subset\mathcal{N}$ which lifts $\mathrm{Fil}^{1}\mathcal{N}/p\mathcal{N}\subset\mathcal{N}/p\mathcal{N}$, and let $\tilde{\mathcal{N}}\coloneqq R_{0}\otimes_{\varphi,R_{0}}(\mathcal{N}+\frac{1}{p}\mathcal{N}^{1})\subset R_{0}[\frac{1}{p}]\otimes_{\varphi,R_{0}}\mathcal{N}$.

Let $\mathrm{Spf}(A,p)\rightarrow\mathrm{Spf}(R_{0},p)$ be an étale morphism. Note that $A$ is equipped with a unique Frobenius lifting that on $R_{0}$, and $\widehat{\Omega}_{A}\cong A\hat{\otimes}_{R_{0}}\widehat{\Omega}_{R_{0}}\cong\bigoplus_{i=1}^{m}A\cdot dt_{i}$. For a connection

[TABLE]

on $A/(p^{n})\otimes_{R_{0}}\mathcal{N}$, we say that the Frobenius is horizontal if the following diagram commutes:

[TABLE]

Here, $\varphi^{*}(\nabla_{A,n})$ is given by choosing an arbitrary lift of $\nabla_{A,n}$ on $A/(p^{n+1})\otimes_{A}\mathcal{N}$, and $\varphi^{*}(\nabla_{A,n})$ does not depend on the choice of such a lift (cf. [Vas13, Section 3.1.1 Equation (9)]).

Proposition 4.8**.**

There exists $\tilde{f}\in R_{0}$ with $\tilde{f}\notin pR_{0}$ such that the following holds:

Let $S_{0}$ be the $p$-adic completion of $R_{0}[\frac{1}{\tilde{f}}]$ equipped with the induced Frobenius, and let $\mathfrak{S}_{S}=S_{0}[\![u]\!]$. Let $\mathfrak{M}_{S}=\mathfrak{M}\otimes_{\mathfrak{S}}\mathfrak{S}_{S}$ equipped with the induced Frobenius, so $\mathfrak{M}_{S}\in\mathrm{Kis}^{1}(\mathfrak{S}_{S})$. Then there exists a topologically quasi-nilpotent integrable connection

[TABLE]

such that $\varphi$ is horizontal, and thus $(\mathfrak{M}_{S},\nabla_{\mathfrak{M}_{S}})\in\mathrm{Kis}^{1}(\mathfrak{S}_{S},\nabla)$. Furthermore, we can choose $\nabla_{\mathfrak{M}_{S}}$ so that $\mathfrak{M}_{S}\otimes_{\mathfrak{S}_{S}}\mathfrak{S}_{\mathcal{O}_{L}}$ equipped with the induced Frobenius and connection is isomorphic to $(\mathfrak{M}_{\mathcal{O}_{L}},\nabla_{\mathfrak{M}_{\mathcal{O}_{L}}})$ as Kisin modules over $\mathfrak{S}_{\mathcal{O}_{L}}$.

Proof.

Without loss of generality, we may pass to a Zariski open set of $\mathrm{Spf}(R_{0},p)$ if necessary so that $\mathcal{N}^{1}$ and $\mathcal{N}/\mathcal{N}^{1}$ are free over $R_{0}$. Fix an $R_{0}$-basis of $\mathcal{N}$ adapted to the direct factor $\mathcal{N}^{1}$. Let $\mathrm{Spf}(A,p)\rightarrow\mathrm{Spf}(R_{0},p)$ be an étale morphism. Consider a connection

[TABLE]

such that the Frobenius is horizontal. By [Vas13, Section 3.2 Basic Theorem] and its proof, the set of such connections $\nabla_{A,1}$ corresponds to the set of solutions over $A/(p)$ of an Artin-Schreier system of equations

[TABLE]

for $\underline{x}=(x_{1},\ldots,x_{dm})^{t}$, where $B\in M_{dm\times dm}(R_{0}/(p))$ and $C_{1}\in M_{dm\times 1}(R_{0}/(p))$. When $A=\mathcal{O}_{L_{0}}$, it has a solution given by $\nabla_{\mathfrak{M}_{L_{0}}}$. Since $\mathcal{O}_{L_{0}}/(p)\cong\mathrm{Frac}(R_{0}/(p))$ and $R_{0}/(p)$ is a unique factorization domain, the solution lies in $(R_{0}/(p))[\frac{1}{f}]$ for some non-zero $f\in R_{0}/(p)$ depending only on $B$ by Lemma 4.7 and its proof. Let $\tilde{f}\in R_{0}$ be a lift of $f$, and let $S_{0}$ be the $p$-adic completion of $R_{0}[\frac{1}{\tilde{f}}]$.

For $n\geq 1$, suppose we are given a connection

[TABLE]

such that the Frobenius is horizontal and inducing $\nabla_{\mathfrak{M}_{L_{0}}}(\mathrm{mod}~{}p^{n})$ via the natural map $S_{0}\rightarrow\mathcal{O}_{L_{0}}$. By [Vas13, Section 3.2 Basic Theorem] and its proof, for the choice of a basis of $\mathcal{N}$ as above, the set of connections

[TABLE]

such that the Frobenius is horizontal and lifting $\nabla_{S_{0},n}$ corresponds to the set of solutions over $S_{0}/(p)$ of an Artin-Schreier system of equations

[TABLE]

where $B$ is the same matrix as above and $C_{n+1}\in M_{dm\times 1}(S_{0}/(p))$. The solution over $\mathcal{O}_{L_{0}}/(p)$ given by $\nabla_{\mathfrak{M}_{L_{0}}}$ lies in $S_{0}/(p)$ by Lemma 4.7 and its proof. This proves the assertion. ∎

Proposition 4.9**.**

Let $S_{0}$ be a ring as given in Proposition 4.8, and let $S=S_{0}\otimes_{W(k)}\mathcal{O}_{K}$. Then there exists a $p$-divisible group $G_{S}$ over $S$ such that $T_{p}(G_{S})\cong T$ as $\mathcal{G}_{S}$-representations.

Proof.

Let $G_{S}$ be the $p$-divisible group over $S$ given by $(\mathfrak{M}_{S},\nabla_{\mathfrak{M}_{S}})$ in Proposition 4.8. Since $\mathfrak{M}_{S}\otimes_{\mathfrak{S}_{S}}\mathfrak{S}_{\mathcal{O}_{L}}\cong\mathfrak{M}_{\mathcal{O}_{L}}$ as Kisin modules, we have $T_{p}(G_{S})\cong T$ as $\mathcal{G}_{\mathcal{O}_{L}}$-representations. On the other hand, $\mathfrak{M}_{S}\otimes_{\mathfrak{S}_{S}}\mathcal{O}_{\mathcal{E},S}\cong\mathcal{M}\otimes_{\mathcal{O}_{\mathcal{E}}}\mathcal{O}_{\mathcal{E},S}$ as étale $\varphi$-modules. Hence, $T_{p}(G_{S})\cong T$ as $\mathcal{G}_{\tilde{S},\infty}$-representations. Since $\mathcal{G}_{\tilde{S},\infty}$ and $\mathcal{G}_{\mathcal{O}_{L}}$ generate the Galois group $\mathcal{G}_{S}$ by Lemma 2.1, we have $T_{p}(G_{S})\cong T$ as $\mathcal{G}_{S}$-representations. ∎

5 Proof of the main theorem

In this section, we finish the proof of Theorem 1.2. We begin by recalling the following well-known lemma about $p$-divisible groups.

Lemma 5.1**.**

Let $R_{1}$ be an integral domain over $W(k)$ such that $\mathrm{Frac}(R_{1})$ has characteristic [math]. Then via the Tate module functor $T_{p}(\cdot)$, the category of $p$-divisible groups over $R_{1}[\frac{1}{p}]$ is equivalent to the category of finite free $\mathbf{Z}_{p}$-representations of $\mathcal{G}_{R_{1}}=\pi_{1}^{\text{\'{e}t}}(\mathrm{Spec}R_{1}[\frac{1}{p}])$. Furthermore, such an equivalence is functorial in the following sense:

Let $R_{1}\rightarrow R_{2}$ be a map of integral domains over $W(k)$ such that $\mathrm{Frac}(R_{1})$ and $\mathrm{Frac}(R_{2})$ have characteristic [math]. Let $G_{R_{1}}$ be a $p$-divisible group over $R_{1}$. Then $T_{p}(G_{R_{1}})\cong T_{p}(G_{R_{1}}\times_{R_{1}}R_{2})$ as $\mathcal{G}_{R_{2}}$-representations.

We first consider the case when $R$ is a formal power series ring of dimension $2$. Let $T$ be a crystalline $\mathcal{G}_{R}$-representation which is finite free over $\mathbf{Z}_{p}$ and has Hodge-Tate weights in $[0,1]$.

Proposition 5.2**.**

Suppose $R_{0}=\mathcal{O}[\![s]\!]$ for a Cohen ring $\mathcal{O}$, and $e\leq p-1$. Then there exists a $p$-divisible group $G_{R}$ over $R$ such that $T_{p}(G_{R})\cong T$ as $\mathcal{G}_{R}$-representations.

Proof.

Let $G$ be a $p$-divisible group over $R[\frac{1}{p}]$ given by Lemma 5.1 such that $T_{p}(G)\cong T$ as $\mathcal{G}_{R}$-representations. It suffices to show that $G$ extends to a $p$-divisible group $G_{R}$ over $R$.

By [BT08, Theorem 6.10], there exists a $p$-divisible group $G_{\mathcal{O}_{L}}$ over $\mathcal{O}_{L}$ extending $G\times_{R[\frac{1}{p}]}L$. For each integer $n\geq 1$, let $A_{n}$ be the Hopf algebra over $R[\frac{1}{s}][\frac{1}{p}]$ for the finite flat group scheme $(G\times_{R[\frac{1}{p}]}R[\frac{1}{s}][\frac{1}{p}])[p^{n}]$, and let $B_{n}$ be the Hopf algebra over $\mathcal{O}_{L}$ for the finite flat group scheme $G_{\mathcal{O}_{L}}[p^{n}]$. Identify $A_{n}\otimes_{R[\frac{1}{s}][\frac{1}{p}]}L=B_{n}\otimes_{\mathcal{O}_{L}}L$ as Hopf algebras over $L$. Note that the $p$-adic completion of $R[\frac{1}{s}]$ is isomorphic to $\mathcal{O}_{L}$. By [BL95, Main Theorem] and its proof, the $R[\frac{1}{s}]$-subalgebra $C_{n}\coloneqq A_{n}\cap B_{n}\subset B_{n}\otimes_{\mathcal{O}_{L}}L$ is finite flat over $R[\frac{1}{s}]$. Moreover, $C_{n}$ is equipped with the Hopf algebra structure induced from $(A_{n},B_{n})$ such that $C_{n}\otimes_{R[\frac{1}{s}]}R[\frac{1}{s}][\frac{1}{p}]\cong A_{n}$ and $C_{n}\otimes_{R[\frac{1}{s}]}\mathcal{O}_{L}\cong B_{n}$. Hence, the datum of finite flat group schemes $((G\times_{R[\frac{1}{p}]}R[\frac{1}{s}][\frac{1}{p}])[p^{n}],G_{\mathcal{O}_{L}}[p^{n}])$ descends to a finite flat group scheme over $R[\frac{1}{s}]$ (up to a unique isomorphism by [BL95, Main Theorem]).

Thus, we obtain a system of finite flat group schemes $(G_{U,n})_{n\geq 1}$ over $U\coloneqq\mathrm{Spec}R~{}\backslash~{}\mathrm{pt}$ extending $(G[p^{n}])_{n\geq 1}$. Here, $\mathrm{pt}$ denotes the closed point given by the maximal ideal of $R$. The natural induced sequence of finite flat group schemes

[TABLE]

is exact by fpqc descent. So $(G_{U,n})_{n\geq 1}$ is a $p$-divisible group over $U$ extending $G$. Since $e\leq p-1$, $G_{U}$ extends to a $p$-divisible group $G_{R}$ over $R$ by [VZ10, Theorem 3]. ∎

Now, let $R_{0}$ be a general ring satisfying the assumptions in Section 2.1, and let $R=R_{0}\otimes_{W(k)}\mathcal{O}_{K}$ with $e<p-1$. Let $T$ be a crystalline $\mathcal{G}_{R}$-representation free over $\mathbf{Z}_{p}$ with Hodge-Tate weights in $[0,1]$. Denote by $\mathfrak{M}_{\mathfrak{S}}(T)$ the $\mathfrak{S}$-module in $\mathrm{Kis}^{1}(\mathfrak{S})$ constructed from $T$ as in Section 4. Let $\tilde{f}\in R_{0}$ be an element as in Proposition 4.8, and let $S_{0}$ be the $p$-adic completion of $R_{0}[\frac{1}{\tilde{f}}]$ as in Proposition 4.9. Let $f\in R_{0}/pR_{0}$ be the image of $\tilde{f}$ in the projection $R_{0}\rightarrow R_{0}/(p)$. We only need to consider the case when $f$ is not a unit in $R_{0}/(p)$. Since $R_{0}/(p)$ is a UFD, there exist prime elements $\bar{s}_{1},\ldots,\bar{s}_{l}$ of $R_{0}/(p)$ dividing $f$. Let $s_{1},\ldots,s_{l}\in R_{0}$ be any preimages of $\bar{s}_{1},\ldots,\bar{s}_{l}$ respectively.

For each $i=1,\ldots,l$, consider the prime ideal $\mathfrak{p}_{i}=(p,s_{i})\subset R_{0}$ and let $R_{0}^{(i)}\coloneqq\widehat{R_{0,\mathfrak{p}_{i}}}$ be the $\mathfrak{p}_{i}$-adic completion of $R_{0,\mathfrak{p}_{i}}$. Note that $R_{0}^{(i)}$ is a formal power series ring over a Cohen ring with Krull dimension $2$. We consider the natural $\varphi$-equivariant map $b_{i}:R_{0}\rightarrow R_{0}^{(i)}$, which induces $b_{i}:R\rightarrow R^{(i)}\coloneqq R_{0}^{(i)}\otimes_{W(k)}\mathcal{O}_{K}$. On the other hand, let $k_{c}$ be a field extension of $\mathrm{Frac}(R_{0}/pR_{0})$ which is a composite of the fields $\mathrm{Frac}(R_{0}^{(i)}/(p))$ for $i=1,\ldots,l$, and let $k_{c}^{\mathrm{perf}}=\varinjlim_{\varphi}k_{c}$ be its direct perfection. By the universal property of $p$-adic Witt vectors, there exists a unique $\varphi$-equivariant map $b_{c}:R_{0}\rightarrow W(k_{c}^{\mathrm{perf}})$. Moreover, for each $i=1,\ldots,l$, we have a unique $\varphi$-equivariant embedding $R_{0}^{(i)}\rightarrow W(k_{c}^{\mathrm{perf}})$ whose composite with $b_{i}$ is equal to $b_{c}$. Note that the natural embedding $R_{0}\rightarrow S_{0}\cap\bigcap_{i=1}^{l}R_{0}^{(i)}$ as subrings of $W(k_{c}^{\mathrm{perf}})$ is bijective, since $S_{0}/(p)\cap\bigcap_{i=1}^{l}(R_{0}^{(i)}/(p))=R_{0}/(p)$ inside $k_{c}^{\mathrm{perf}}$.

By Proposition 5.2, there exists a $p$-divisible group $G_{i}$ over $R^{(i)}$ such that $T_{p}(G_{i})\cong T$ as $\mathcal{G}_{R^{(i)}}$-representations. We have

[TABLE]

as étale $(\varphi,\mathcal{O}_{\mathcal{E},R^{(i)}})$-modules. Applying Lemma 4.4, we can deduce that $\mathfrak{M}_{\mathfrak{S}}(T)\otimes_{\mathfrak{S}}\mathfrak{S}_{R^{(i)}}\cong\mathfrak{M}^{*}(G_{i})$ compatibly with Frobenius.

Let $D=D_{\mathrm{cris}}(T[\frac{1}{p}])$, and denote $\mathfrak{M}=\mathfrak{M}_{\mathfrak{S}}(T)$ and $\mathcal{N}=\mathfrak{M}\otimes_{\mathfrak{S},\varphi}R_{0}$. Let $\nabla:D\rightarrow D\otimes_{R_{0}}\widehat{\Omega}_{R_{0}}$ be the connection given by the functor $D_{\mathrm{cris}}(\cdot)$.

Proposition 5.3**.**

There exists a natural $\varphi$-equivariant embedding

[TABLE]

of $R_{0}$-modules. Furthermore, if we consider $\mathcal{N}$ as an $R_{0}$-submodule of $D$ via $h$, then $\nabla$ maps $\mathcal{N}$ into $\mathcal{N}\otimes_{R_{0}}\widehat{\Omega}_{R_{0}}$. Hence, $\mathfrak{M}$ is a Kisin module of height $1$.

Proof.

By [Kim15, Corollary 5.3 and 6.7], there exists a natural $\varphi$-equivariant embedding

[TABLE]

for each $i=1,\ldots,l$ such that the connections given by $\mathfrak{M}^{*}(G_{i})$ and $D$ are compatible, and there exists a natural $\varphi$-equivariant embedding $h_{c}:\mathcal{N}\rightarrow D\otimes_{R_{0},b_{c}}W(k_{c}^{\mathrm{perf}})$. Moreover, by Proposition 4.9, there exists a natural $\varphi$-equivariant embedding $h_{S}:\mathcal{N}\rightarrow D\otimes_{R_{0}}S_{0}$ such that the connections given by $\mathfrak{M}^{*}(G_{S})$ and $D$ are compatible. Since the construction of those natural maps is compatible with $\varphi$-equivariant base changes (cf. [Kim15, Section 5.5]), we deduce that the maps $h_{1},\ldots,h_{l}$ and $h_{S}$ are compatible with one another, in the sense that their composites with the embedding into $D\otimes_{R_{0},b_{c}}W(k_{c}^{\mathrm{perf}})$ are all equal to $h_{c}$. Hence, we obtain a $\varphi$-equivariant embedding

[TABLE]

since $D$ is flat over $R_{0}[\frac{1}{p}]$ and $S_{0}[\frac{1}{p}]\cap\bigcap_{i=1}^{l}R_{0}^{(i)}[\frac{1}{p}]=R_{0}[\frac{1}{p}]$.

Now, identify $\widehat{\Omega}_{R_{0}}=\bigoplus_{j=1}^{m}R_{0}\cdot dt_{j}$. Then $\nabla$ maps $\mathcal{N}$ to $\mathcal{N}\otimes_{R_{0}}(\bigoplus_{j=1}^{m}R_{0}[\frac{1}{p}]\cdot dt_{j})$. On the other hand, by Proposition 4.8, Proposition 5.2, and the compatibility of $D_{\mathrm{cris}}(\cdot)$ with respect to $\varphi$-compatible base changes, we have that $\nabla$ maps $\mathcal{N}$ into $\mathcal{N}\otimes_{R_{0}}(\bigoplus_{j=1}^{m}S_{0}\cdot dt_{j})$ and also into $\mathcal{N}\otimes_{R_{0}}(\bigoplus_{j=1}^{m}R_{0}^{(i)}\cdot dt_{j})$ for each $i=1,\ldots,l$. Since $\mathcal{N}$ is flat over $R_{0}$ and $S_{0}\cap\bigcap_{i=1}^{l}R_{0}^{(i)}=R_{0}$, $\nabla$ maps $\mathcal{N}$ into $\mathcal{N}\otimes_{R_{0}}(\bigoplus_{j=1}^{m}R_{0}\cdot dt_{j})$. ∎

Theorem 5.4**.**

There exists a $p$-divisible group $G_{R}$ over $R$ such that $T_{p}(G_{R})\cong T$ as $\mathcal{G}_{R}$-representations.

Proof.

By Proposition 5.3, we have $\mathfrak{M}\in\mathrm{Kis}^{1}(\varphi,\nabla)$. Furthermore, $\mathfrak{M}\otimes_{\mathfrak{S}}\mathfrak{S}_{\mathcal{O}_{L}}\cong\mathfrak{M}_{\mathcal{O}_{L}}$ as Kisin modules over $\mathfrak{S}_{\mathcal{O}_{L}}$, since the Frobenius and connection structure on $\mathfrak{M}$ agree with those on $D$. Thus, if $G_{R}$ is the $p$-divisible group corresponding to $\mathfrak{M}$, then $T_{p}(G_{R})\cong T$ as $\mathcal{G}_{\mathcal{O}_{L}}$-representations as well as $\mathcal{G}_{\tilde{R}_{\infty}}$-representations. The assertion follows from Lemma 2.1. ∎

6 Barsotti-Tate deformation ring

As an application of Theorem 5.4, we study the geometry of the locus of crystalline representations with Hodge-Tate weights in $[0,1]$ by using the results in [Moo18]. Denote by $\mathcal{C}$ the category of topological local $\mathbf{Z}_{p}$-algebras $A$ satisfying the following conditions:

• •

the natural map $\mathbf{Z}_{p}\rightarrow A/\mathfrak{m}_{A}$ is surjective, where $\mathfrak{m}_{A}$ denotes the maximal ideal of $A$;

• •

the map from $A$ to the projective limit of its discrete artinian quotients is a topological isomorphism.

By the first condition, the residue field of $A$ is $\mathbf{F}_{p}$. The second condition is equivalent to that $A$ is complete and its topology is given by a collection of open ideals $\mathfrak{a}\subset A$ for which $A/\mathfrak{a}$ is aritinian. Morphisms in $\mathcal{C}$ are continuous $\mathbf{Z}_{p}$-algebra morphisms.

For $A\in\mathcal{C}$, we mean by an $A$-representation of $\mathcal{G}_{R}$ a finite free $A$-module equipped with a continuous $A$-linear $\mathcal{G}_{R}$-action. Fix an $\mathbf{F}_{p}$-representation $V_{0}$ of $\mathcal{G}_{R}$ which is absolutely irreducible. For $A\in\mathcal{C}$, a deformation of $V_{0}$ in $A$ is defined to be an isomorphism class of $A$-representations of $V$ of $\mathcal{G}_{R}$ satisfying $V\otimes_{A}\mathbf{F}_{p}\cong V_{0}$ as $\mathbf{F}_{p}[\mathcal{G}_{R}]$-modules. Denote by $\mathrm{Def}(V_{0},A)$ the set of such deformations. A morphism $f:A\rightarrow A^{\prime}$ in $\mathcal{C}$ induces a map $f_{*}:\mathrm{Def}(V_{0},A)\rightarrow\mathrm{Def}(V_{0},A^{\prime})$ sending the class of an $A$-representation $V$ to the class of $V\otimes_{A,f}A^{\prime}$. The following theorem on universal deformation ring is proved in [SL97].

Theorem 6.1**.**

(cf. [SL97, Theorem 2.3])* There exists a universal deformation ring $A_{\mathrm{univ}}\in\mathcal{C}$ and a deformation $V_{\mathrm{univ}}\in\mathrm{Def}(V_{0},A_{\mathrm{univ}})$ such that for all $A\in\mathcal{C}$, we have a bijection*

[TABLE]

given by $f\mapsto f_{*}(V_{\mathrm{univ}})$.

We deduce that when $R$ has dimension $2$ and $e$ is small, the locus of crystalline representations with Hodge-Tate weights in $[0,1]$ cuts out a closed subscheme of $\mathrm{Spec}A_{\mathrm{univ}}$ in the following sense.

Theorem 6.2**.**

Suppose that $e<p-1$ and that the Krull dimension of $R$ is $2$. Then there exists a closed ideal $\mathfrak{a}_{\mathrm{BT}}\subset A_{\mathrm{univ}}$ such that the following holds:

For any finite flat $\mathbf{Z}_{p}$-algebra $A$ equipped with the $p$-adic topology and any continuous $\mathbf{Z}_{p}$-algebra map $f:A_{\mathrm{univ}}\rightarrow A$, the induced representation $V_{\mathrm{univ}}\otimes_{A_{\mathrm{univ}},f}A[\frac{1}{p}]$ of $\mathcal{G}_{R}$ is crystalline with Hodge-Tate weights in $[0,1]$ if and only if $f$ factors through the quotient $A_{\mathrm{univ}}/\mathfrak{a}_{\mathrm{BT}}$.

Proof.

This follows directly from Theorem 5.4 and [Moo18, Theorem 5.7]. ∎