Introduction

Let $R$ be a normal complete noetherian local ring with perfect residue field $k$ of positive characteristic $p$ and with fraction field $K$ of characteristic zero. For a $p$ -divisible group $G$ over $R$ , the Tate module $T_{p}(G)$ is a free $\mathbb{Z}_{p}$ -module of finite rank with a continuous action of the absolute Galois group ${\mathcal{G}}_{K}$ . We want to describe the Tate module in terms of the Dieudonné display $\mathscr{P}=(P,Q,F,F_{1})$ associated to $G$ in [Reference ZinkZi2, Reference LauLa3], and relate this to other descriptions of the Tate module when $R$ is a discrete valuation ring.

Let us recall the notion of a Dieudonné display. The Zink ring $\mathbb{W}(R)$ is a certain subring of the ring of Witt vectors $W(R)$ which is stable under the Frobenius endomorphism $f$ of $W(R)$ . The components of $\mathscr{P}$ are $\mathbb{W}(R)$ -modules $Q\subseteq P$ where $P$ is finite free and $P/Q$ is a free $R$ -module, and $f$ -linear maps $F:P\rightarrow P$ and $F_{1}:Q\rightarrow P$ such that $F_{1}(Q)$ generates $P$ and $F_{1}(v(u_{0}a)x)=aF(x)$ for $x\in P$ and $a\in \mathbb{W}(R)$ . Here $v$ is the Verschiebung of $W(R)$ , and $u_{0}\in W(R)$ is the unit defined by $u_{0}=1$ if $p\geqslant 3$ and by $v(u_{0})=2-[2]$ if $p=2$ . The twist by $u_{0}$ is necessary since $v$ does not stabilize $\mathbb{W}(R)$ when $p=2$ .

To state the main result we need the following scalar extension of $\mathscr{P}$ . Let $\hat{R}^{\operatorname{nr}}$ be the completion of the strict Henselization of $R$ , let $\tilde{K}$ be an algebraic closure of the fraction field $\hat{K}^{\operatorname{nr}}$ of $\hat{R}^{\operatorname{nr}}$ , and let $\tilde{R}\subset \tilde{K}$ be the integral closure of $\hat{R}^{\operatorname{nr}}$ . We define

$$\begin{eqnarray}\mathbb{W}(\tilde{R})=\mathop{\varinjlim }\nolimits_{E}\mathbb{W}(R_{E})\end{eqnarray}$$

where $E$ runs through the finite extensions of $\hat{K}^{\operatorname{nr}}$ in $\tilde{K}$ and where $R_{E}=\tilde{R}\cap E$ . Let $\tilde{R}^{\wedge }$ and $\hat{\mathbb{W}}(\tilde{R})$ be the $p$ -adic completions of $\tilde{R}$ and $\mathbb{W}(\tilde{R})$ . We define

$$\begin{eqnarray}\hat{P}_{\tilde{R}}=\hat{\mathbb{W}}(\tilde{R})\otimes _{\mathbb{W}(R)}P\end{eqnarray}$$

and

$$\begin{eqnarray}\hat{Q}_{\tilde{R}}=\operatorname{Ker}(\hat{P}_{\tilde{R}}\rightarrow \tilde{R}^{\wedge }\otimes _{R}P/Q).\end{eqnarray}$$

Let $\bar{K}$ be the algebraic closure of $K$ in $\tilde{K}$ and let $\tilde{{\mathcal{G}}}_{K}$ be the group of automorphisms of $\tilde{K}$ whose restriction to $\bar{K}\hat{K}^{\operatorname{nr}}$ is induced by an element of ${\mathcal{G}}_{K}$ . The natural map $\tilde{{\mathcal{G}}}_{K}\rightarrow {\mathcal{G}}_{K}$ is surjective, and bijective when $R$ is one-dimensional since then $\tilde{K}=\bar{K}\hat{K}^{\operatorname{nr}}$ . The following is the main result of this note; see Proposition 4.1.

Theorem A. There is an exact sequence of $\tilde{{\mathcal{G}}}_{K}$ -modules

$$\begin{eqnarray}0\longrightarrow T_{p}(G)\longrightarrow \hat{Q}_{\tilde{R}}\xrightarrow[{}]{F_{1}-1}\hat{P}_{\tilde{R}}\longrightarrow 0.\end{eqnarray}$$

Here $F_{1}$ is a natural extension of $F_{1}:Q\rightarrow P$ . If $G$ is connected, a similar description of $T_{p}(G)$ in terms of the nilpotent display of $G$ is part of Zink’s theory of displays. In this case $k$ need not be perfect; see [Reference MessingMe2, Proposition 4.4]. The proof is recalled in Proposition 2.1 below.

The one-dimensional case

Assume now in addition that $R$ is a discrete valuation ring. Then Theorem A can be related to the descriptions of $T_{p}(G)$ in terms of $p$ -adic Hodge theory and in terms of Breuil–Kisin modules as follows.

Relation with the crystalline period homomorphism

Let $M_{\operatorname{cris}}$ be the value of the covariant Dieudonné crystal of $G$ over $A_{\operatorname{cris}}(\bar{R})$ . It carries a filtration and a Frobenius, and by [Reference FaltingsFa] there is a period homomorphism

$$\begin{eqnarray}T_{p}(G)\rightarrow \operatorname{Fil}^{1}M_{\operatorname{ cris}}^{F=p}\end{eqnarray}$$

which is bijective if $p\geqslant 3$ , and injective with cokernel annihilated by $p$ if $p=2$ . The $v$ -stabilized Zink ring $\mathbb{W}^{+}(R)=\mathbb{W}(R)[v(1)]$ studied in [Reference LauLa3] induces an extension $\hat{\mathbb{W}}^{+}(\tilde{R})$ of the ring $\hat{\mathbb{W}}(\tilde{R})$ defined above, which is the trivial extension when $p\geqslant 3$ . The universal property of $A_{\operatorname{cris}}(\bar{R})$ gives a ring homomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D718}_{\operatorname{cris}}:A_{\operatorname{cris}}(\bar{R})\rightarrow \hat{\mathbb{W}}^{+}(\tilde{R}).\end{eqnarray}$$

Using the crystalline description of Dieudonné displays of [Reference LauLa3], one obtains an $A_{\operatorname{cris}}(\bar{R})$ -linear map

$$\begin{eqnarray}\unicode[STIX]{x1D70F}:M_{\operatorname{cris}}\rightarrow \hat{\mathbb{W}}^{+}(\tilde{R})\otimes _{\hat{\mathbb{ W}}(\tilde{R})}\hat{P}_{\tilde{R}}\end{eqnarray}$$

compatible with Frobenius and filtration. We will show that $\unicode[STIX]{x1D70F}$ induces the identity on $T_{p}(G)$ , viewed as a submodule of $\operatorname{Fil}^{1}M_{\operatorname{cris}}$ by the period homomorphism and as a submodule of $\hat{Q}_{\tilde{R}}\subseteq \hat{P}_{\tilde{R}}$ by Theorem A; see Proposition 6.2.

Relation with Breuil–Kisin modules

Let $\unicode[STIX]{x1D70B}\in R$ generate the maximal ideal. Let $\mathfrak{S}=W(k)[[t]]$ and let $\unicode[STIX]{x1D70E}:\mathfrak{S}\rightarrow \mathfrak{S}$ extend the Frobenius automorphism of $W(k)$ by $t\mapsto t^{p}$ ; see below for the case of more general Frobenius lifts. We consider pairs $M=(M,\unicode[STIX]{x1D719})$ where $M$ is an $\mathfrak{S}$ -module of finite type and where $\unicode[STIX]{x1D719}:M\rightarrow M^{(\unicode[STIX]{x1D70E})}=\mathfrak{S}\otimes _{\unicode[STIX]{x1D70E},\mathfrak{S}}M$ is an $\mathfrak{S}$ -linear map with cokernel annihilated by the minimal polynomial of $\unicode[STIX]{x1D70B}$ over $W(k)$ . Following [Reference Vasiu and ZinkVZ], $M$ is called a Breuil window if $M$ is free over $\mathfrak{S}$ , and $M$ is called a Breuil module if $M$ is a $p$ -power torsion $\mathfrak{S}$ -module of projective dimension at most one. These notions are dual to the classical Breuil–Kisin modules.

It is known that $p$ -divisible groups over $R$ are equivalent to Breuil windows. This was conjectured by Breuil [Reference BreuilBr] and proved by Kisin [Reference KisinKi1, Reference KisinKi2] if $p\geqslant 3$ , and for connected groups if $p=2$ . The general case is proved in [Reference LauLa3] by showing that Breuil windows are equivalent to Dieudonné displays. (This equivalence holds when $R$ is regular of arbitrary dimension, with appropriate definition of $\mathfrak{S}$ . For $p\geqslant 3$ this equivalence is already proved in [Reference Vasiu and ZinkVZ] for some regular rings, including all discrete valuation rings.) As a corollary, commutative finite flat $p$ -group schemes over $R$ are equivalent to Breuil modules. Other proofs for $p=2$ , more closely related to Kisin’s methods, were obtained independently by Kim [Reference KimK] and Liu [Reference LiuLi].

Let $K_{\infty }$ be the extension of $K$ generated by a chosen system of successive $p$ th roots of $\unicode[STIX]{x1D70B}$ . For a $p$ -divisible group $G$ over $R$ let $T(G)$ be its Tate module, and for a commutative finite flat $p$ -group scheme $G$ over $R$ let $T(G)=G(\bar{K})$ . The results of Kisin, Liu, and Kim include a description of $T(G)$ as a ${\mathcal{G}}_{K_{\infty }}$ -module in terms of the Breuil window or module $(M,\unicode[STIX]{x1D719})$ corresponding to $G$ . In the covariant theory used here it takes the form of an isomorphism of ${\mathcal{G}}_{K_{\infty }}$ -modules $T(G)\cong T^{\operatorname{nr}}(M)$ where

$$\begin{eqnarray}T^{\operatorname{nr}}(M)=\{x\in M^{\operatorname{nr}}\mid \unicode[STIX]{x1D719}(x)=1\otimes x\text{ in }\mathfrak{S}^{\operatorname{nr}}\otimes _{\unicode[STIX]{x1D70E},\mathfrak{S}^{\operatorname{nr}}}M^{\operatorname{nr}}\}\end{eqnarray}$$

with $M^{\operatorname{nr}}=\mathfrak{S}^{\operatorname{nr}}\otimes _{\mathfrak{S}}M$ ; the ring $\mathfrak{S}^{\operatorname{nr}}$ is recalled in Section 7.

To complete the approach via Dieudonné displays, we will show how the isomorphism $T(G)\cong T^{\operatorname{nr}}(M)$ can be deduced from Theorem A; see Corollary 8.6. It suffices to consider the case where $G$ is a $p$ -divisible group. The equivalence between Breuil windows and Dieudonné displays over $R$ is induced by a ring homomorphism $\unicode[STIX]{x1D718}:\mathfrak{S}\rightarrow \mathbb{W}(R)$ , which extends to a ring homomorphism $\unicode[STIX]{x1D718}^{\operatorname{nr}}:\mathfrak{S}^{\operatorname{nr}}\rightarrow \hat{\mathbb{W}}(\tilde{R}).$ Using Theorem A, this allows to define a homomorphism of ${\mathcal{G}}_{K_{\infty }}$ -modules

$$\begin{eqnarray}\unicode[STIX]{x1D70F}:T^{\operatorname{nr}}(M)\rightarrow T(G),\end{eqnarray}$$

and we show in Proposition 8.5 that $\unicode[STIX]{x1D70F}$ is bijective. The verification is easy if $G$ is étale, and the general case follows quite formally using a duality argument.

Other lifts of Frobenius

The equivalence between Breuil windows and $p$ -divisible groups requires only a Frobenius lift $\unicode[STIX]{x1D70E}:\mathfrak{S}\rightarrow \mathfrak{S}$ which stabilizes the ideal $t\mathfrak{S}$ such that $p^{2}$ divides the linear term of the power series $\unicode[STIX]{x1D70E}(t)$ . In this case, let $K_{\infty }$ be the extension of $K$ generated by a system $\unicode[STIX]{x1D70B}^{(n)}\in \bar{K}$ of successive $\unicode[STIX]{x1D70E}(t)$ -roots of $\unicode[STIX]{x1D70B}$ , which means that $\unicode[STIX]{x1D70B}^{(0)}=\unicode[STIX]{x1D70B}$ and $\unicode[STIX]{x1D70E}(t)(\unicode[STIX]{x1D70B}^{(n+1)})=\unicode[STIX]{x1D70B}^{(n)}$ . Then we obtain an isomorphism of ${\mathcal{G}}_{K_{\infty }}$ -modules $T(G)\cong T^{\operatorname{nr}}(M)$ as before; here the ring $\mathfrak{S}^{\operatorname{nr}}$ depends on $\unicode[STIX]{x1D70E}$ as well.

1 Notation

All rings are commutative and unitary unless the contrary is stated. For the convenience of the reader we recall the notion of frames, windows, and displays.

A frame $\mathscr{F}=(S,I,R,\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}_{1})$ in the sense of [Reference LauLa2] consists of a pair of rings $S$ and $R=S/I$ with $I+pS\subseteq \operatorname{Rad}(S)$ , a ring endomorphism $\unicode[STIX]{x1D70E}:S\rightarrow S$ that lifts the Frobenius of $S/pS$ , and a $\unicode[STIX]{x1D70E}$ -linear map $\unicode[STIX]{x1D70E}_{1}:I\rightarrow S$ with $\unicode[STIX]{x1D70E}_{1}(I)S=S$ .

We assume that $S$ is a local ring. Then an $\mathscr{F}$ -window $\mathscr{P}=(P,Q,F,F_{1})$ consists of a finite free $S$ -module $P$ , a submodule $Q\subseteq P$ with $IP\subseteq Q$ such that $P/Q$ is free over $R$ , and a pair of $\unicode[STIX]{x1D70E}$ -linear maps $F:P\rightarrow P$ and $F_{1}:Q\rightarrow P$ with $F_{1}(ax)=\unicode[STIX]{x1D70E}_{1}(a)F(x)$ for $a\in I$ and $x\in P$ , such that $F_{1}(Q)$ generates $P$ . Then there is a unique $S$ -linear map $V^{\sharp }:P\rightarrow S\otimes _{\unicode[STIX]{x1D70E},S}P=P^{(\unicode[STIX]{x1D70E})}$ with $V^{\sharp }(F_{1}(x))=1\otimes x$ for $x\in Q$ . A sequence $0\rightarrow \mathscr{P}\rightarrow \mathscr{P}^{\prime }\rightarrow \mathscr{P}^{\prime \prime }\rightarrow 0$ of $\mathscr{F}$ -windows will be called exact if the resulting sequences of $P$ ’s and of $Q$ ’s are exact.

A frame homomorphism $\unicode[STIX]{x1D6FC}:\mathscr{F}\rightarrow \mathscr{F}^{\prime }=(S^{\prime },I^{\prime },R^{\prime },\unicode[STIX]{x1D70E}^{\prime },\unicode[STIX]{x1D70E}_{1}^{\prime })$ is a ring homomorphism $\unicode[STIX]{x1D6FC}:S\rightarrow S^{\prime }$ with $\unicode[STIX]{x1D6FC}(I)\subseteq I^{\prime }$ such that $\unicode[STIX]{x1D70E}^{\prime }\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70E}$ and $\unicode[STIX]{x1D70E}_{1}^{\prime }\unicode[STIX]{x1D6FC}=u\cdot \unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70E}_{1}$ for a unit $u\in S^{\prime }$ , which then is unique. If $u=1$ then $\unicode[STIX]{x1D6FC}$ is called strict. There is a base change functor

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{\ast }:(\mathscr{F}\text{-windows})\rightarrow (\mathscr{F}^{\prime }\text{-windows})\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}_{\ast }(\mathscr{P})=(P^{\prime },Q^{\prime },F^{\prime },F_{1}^{\prime })$ is determined by $P^{\prime }=S^{\prime }\otimes _{S}P$ and $P^{\prime }/Q^{\prime }=(P/Q)\otimes _{R}R^{\prime }$ with $F^{\prime }(1\otimes x)=1\otimes F(x)$ for $x\in P$ and $F_{1}^{\prime }(1\otimes x)=u\otimes F_{1}(x)$ for $x\in Q$ .

For a not necessarily unitary ring $R$ let $W(R)$ be the ring of $p$ -typical Witt vectors. If $R$ is $p$ -adic and unitary, we have a frame

$$\begin{eqnarray}\mathscr{W}(R)=(W(R),I_{R},R,f,f_{1})\end{eqnarray}$$

where $I_{R}$ is the image of the Verschiebung $v:W(R)\rightarrow W(R)$ , where $f$ is the Frobenius, and $f_{1}$ is the inverse of $v$ . Windows over $\mathscr{W}(R)$ are the displays over $R$ of [Reference ZinkZi1]. A display is called $V$ -nilpotent if the map $V^{\sharp }$ becomes nilpotent over $R/pR$ . A homomorphism $R\rightarrow R^{\prime }$ gives a strict frame homomorphism $\mathscr{W}(R)\rightarrow \mathscr{W}(R^{\prime })$ , and we write $\mathscr{P}\mapsto \mathscr{P}\otimes _{R}R^{\prime }$ for the resulting base change of displays.

If $N$ is a nilpotent nonunitary ring, ${\hat{W}}(N)\subseteq W(N)$ denotes the subgroup of all Witt vectors with only finitely many nonzero coefficients. If $A$ is a local Artin ring with perfect residue field $k=A/\mathfrak{m}$ of characteristic $p$ , there is a unique ring homomorphism $s:W(k)\rightarrow W(A)$ that lifts the projection $W(A)\rightarrow W(k)$ , and the Zink ring $\mathbb{W}(A)={\hat{W}}(\mathfrak{m})\oplus s(W(k))$ is a subring of $W(A)$ . There is a frame $\mathscr{D}_{A}=(\mathbb{W}(A),\mathbb{I}(A),A,f,f_{1})$ with an injective frame homomorphism $\mathscr{D}_{A}\rightarrow \mathscr{W}_{A}$ , which is strict when $p\geqslant 3$ ; see [Reference LauLa3, Section 2.C]. Windows over $\mathscr{D}_{A}$ are called Dieudonné displays over $A$ .

2 The case of connected $p$ -divisible groups

Let $R$ be a normal complete noetherian local ring with (not necessarily perfect) residue field $k$ of positive characteristic $p$ , with fraction field $K$ of characteristic zero, and with maximal ideal $\mathfrak{m}$ . In this section, we recall how the Tate module of a connected $p$ -divisible group over $R$ is expressed in terms of its nilpotent display.

We fix an algebraic closure $\bar{K}$ of $K$ and write ${\mathcal{G}}_{K}=\operatorname{Gal}(\bar{K}/K)$ . Let $\bar{R}\subset \bar{K}$ be the integral closure of $R$ , and for a finite extension $E/K$ in $\bar{K}$ let $R_{E}=\bar{R}\cap E$ . Then $R_{E}$ is finite over $R$ , and $R_{E}$ is a complete noetherian local ring. Thus $\bar{R}$ is a local ring. Let $\bar{\mathfrak{m}}\subset \bar{R}$ and $\mathfrak{m}_{E}\subset R_{E}$ be the maximal ideals. We write

$$\begin{eqnarray}{\hat{W}}(\mathfrak{m}_{E})=\mathop{\varprojlim }\nolimits_{n}{\hat{W}}(\mathfrak{m}_{E}/\mathfrak{m}_{E}^{n});\qquad {\hat{W}}(\bar{\mathfrak{m}})=\mathop{\varinjlim }\nolimits_{E}{\hat{W}}(\mathfrak{m}_{E}).\end{eqnarray}$$

Let $\bar{W}(\bar{\mathfrak{m}})$ be the $p$ -adic completion of ${\hat{W}}(\bar{\mathfrak{m}})$ and let $\bar{\mathfrak{m}}^{\wedge }$ be the $p$ -adic completion of $\bar{\mathfrak{m}}$ . The natural map $\bar{W}(\bar{\mathfrak{m}})\rightarrow \bar{\mathfrak{m}}^{\wedge }$ is surjective. For a display $\mathscr{P}=(P,Q,F,F_{1})$ over $R$ let

$$\begin{eqnarray}\bar{P}_{\bar{\mathfrak{m}}}=\bar{W}(\bar{\mathfrak{m}})\otimes _{W(R)}P;\qquad \bar{Q}_{\bar{\mathfrak{m}}}=\operatorname{Ker}(\bar{P}_{\bar{\mathfrak{m}}}\rightarrow \bar{\mathfrak{m}}^{\wedge }\otimes _{R}P/Q).\end{eqnarray}$$

We call $\mathscr{P}$ nilpotent if the reduction $\mathscr{P}\otimes _{R}k$ is $V$ -nilpotent in the usual sense, or equivalently if $\mathscr{P}\otimes _{R}R/\mathfrak{m}_{R}^{n}$ is $V$ -nilpotent for all $n$ ; cf. [Reference ZinkZi1, Definition 13]. The functor $\operatorname{BT}$ of [Reference ZinkZi1] induces an equivalence of categories between nilpotent displays over $R$ and connected $p$ -divisible groups over $R$ ; this follows from [Reference ZinkZi1, Theorem 9] applied to the rings $R/\mathfrak{m}_{R}^{n}$ , using that $V$ -nilpotent displays and $p$ -divisible groups over $R$ are equivalent to compatible systems of such objects over $R/\mathfrak{m}_{R}^{n}$ for all $n$ . A variant of the following result is stated in [Reference MessingMe2, Proposition 4.4].

Proposition 2.1. (Zink)

Let $\mathscr{P}$ be a nilpotent display over $R$ and let $G=\operatorname{BT}(\mathscr{P})$ be the associated connected $p$ -divisible group over $R$ . There is a natural exact sequence of ${\mathcal{G}}_{K}$ -modules

$$\begin{eqnarray}0\longrightarrow T_{p}(G)\longrightarrow \bar{Q}_{\bar{\mathfrak{m}}}\xrightarrow[{}]{F_{1}-1}\bar{P}_{\bar{\mathfrak{m}}}\longrightarrow 0.\end{eqnarray}$$

Here $T_{p}(G)=\operatorname{Hom}(\mathbb{Q}_{p}/\mathbb{Z}_{p},G(\bar{K}))$ is the Tate module of $G$ , and ${\mathcal{G}}_{K}$ acts on $\bar{P}_{\bar{\mathfrak{m}}}$ and $\bar{Q}_{\bar{\mathfrak{m}}}$ by its natural action on $\bar{W}(\bar{\mathfrak{m}})$ .

The proof of Proposition 2.1 uses the following standard facts.

Proof. The group $\operatorname{Hom}(\mathbb{Q}_{p}/\mathbb{Z}_{p},A)$ is isomorphic to  $\varprojlim \operatorname{Hom}(\mathbb{Z}/p^{n}\mathbb{Z},A)$ with transition maps induced by $p:\mathbb{Z}/p^{n}\mathbb{Z}\rightarrow \mathbb{Z}/p^{n+1}\mathbb{Z}$ . If the abelian group $A$ is injective, the projective system $\operatorname{Hom}(\mathbb{Z}/p^{n}\mathbb{Z},A)$ has surjective transition maps and thus its $\varprojlim ^{1}$ vanishes. Hence there is a Grothendieck spectral sequence for the functor $A\mapsto \operatorname{Hom}(\mathbb{Z}/p^{n},A)_{n}$ from abelian groups to projective systems of abelian groups, composed with the functor $\varprojlim$ ,

(2.1) $$\begin{eqnarray}\varprojlim ^{i}(\operatorname{Ext}^{j}(\mathbb{Z}/p^{n},A))\Rightarrow \operatorname{Ext}^{i+j}(\mathbb{Q}_{p}/\mathbb{Z}_{p},A).\end{eqnarray}$$

The projective system of groups $\operatorname{Ext}^{1}(\mathbb{Z}/p^{n}\mathbb{Z},A)$ is isomorphic to the system $A/p^{n}A$ with transition maps induced by $\operatorname{id}_{A}$ . Thus the exact sequence of low degree terms (see for example, [Reference WeibelWe, Theorem 5.8.3]) associated to (2.1) gives an exact sequence

$$\begin{eqnarray}0\rightarrow \varprojlim ^{1}\operatorname{Hom}(\mathbb{Z}/p^{n}\mathbb{Z},A)\rightarrow \operatorname{Ext}^{1}(\mathbb{Q}_{p}/\mathbb{Z}_{p},A)\rightarrow \varprojlim A/p^{n}A\rightarrow 0.\end{eqnarray}$$

If $A$ has no $p$ -torsion then $\operatorname{Hom}(\mathbb{Z}/p^{n}\mathbb{Z},A)=0$ , and (i) follows. If $pA=A$ then the projective system $\operatorname{Hom}(\mathbb{Z}/p^{n}\mathbb{Z},A)$ has surjective transition maps, thus its $\varprojlim ^{1}$ is zero, moreover $A/p^{n}A=0$ . This proves (ii).◻

For a $p$ -divisible group $G$ over $R$ and for $E$ as above we write

$$\begin{eqnarray}{\hat{G}}(R_{E})=\mathop{\varprojlim }\nolimits_{n}G(R_{E}/\mathfrak{m}_{E}^{n});\qquad {\hat{G}}(\bar{R})=\mathop{\varinjlim }\nolimits_{E}{\hat{G}}(R_{E}).\end{eqnarray}$$

Proof of Proposition 2.1.

For a finite Galois extension $E/K$ in $\bar{K}$ we write

$$\begin{eqnarray}\hat{P}_{E,n}={\hat{W}}(\mathfrak{m}_{E}/\mathfrak{m}_{E}^{n})\otimes _{W(R)}P\end{eqnarray}$$

and define $\hat{Q}_{E,n}$ by the exact sequence of ${\mathcal{G}}_{K}$ -modules

$$\begin{eqnarray}0\rightarrow \hat{Q}_{E,n}\rightarrow \hat{P}_{E,n}\rightarrow \mathfrak{m}_{E}/\mathfrak{m}_{E}^{n}\otimes _{R}P/Q\rightarrow 0.\end{eqnarray}$$

The definition of the functor $\operatorname{BT}$ in [Reference ZinkZi1, Theorem 81] gives an exact sequence of ${\mathcal{G}}_{K}$ -modules

$$\begin{eqnarray}0\longrightarrow \hat{Q}_{E,n}\xrightarrow[{}]{F_{1}-1}\hat{P}_{E,n}\longrightarrow G(R_{E}/\mathfrak{m}_{E}^{n})\longrightarrow 0;\end{eqnarray}$$

note that in [Reference ZinkZi1] a formal group $G$ is viewed as a functor $G^{\prime }$ on nilpotent algebras, and $G(R_{E}/\mathfrak{m}_{E}^{n})=G^{\prime }(\mathfrak{m}_{E}/\mathfrak{m}_{E}^{n})$ under this identification. The modules $\hat{Q}_{E,n}$ form a projective system with respect to $n$ with surjective transition maps. Indeed, using a normal decomposition of $\mathscr{P}$ as in the paragraph before [Reference ZinkZi1, Theorem 81], this is reduced to the assertion that ${\hat{W}}(\mathfrak{m}_{E}/\mathfrak{m}_{E}^{n+1})\rightarrow {\hat{W}}(\mathfrak{m}_{E}/\mathfrak{m}_{E}^{n})$ is surjective, which is clear. Thus taking $\mathop{\varinjlim }\nolimits_{E}\mathop{\varprojlim }\nolimits_{n}$ of the preceding two sequences gives exact sequences of ${\mathcal{G}}_{K}$ -modules

(2.2) $$\begin{eqnarray}0\rightarrow \hat{Q}_{\bar{\mathfrak{m}}}\rightarrow \hat{P}_{\bar{\mathfrak{m}}}\rightarrow \bar{\mathfrak{m}}\otimes _{R}P/Q\rightarrow 0\end{eqnarray}$$

and

(2.3) $$\begin{eqnarray}0\longrightarrow \hat{Q}_{\bar{\mathfrak{m}}}\xrightarrow[{}]{F_{1}-1}\hat{P}_{\bar{\mathfrak{m}}}\longrightarrow {\hat{G}}(\bar{R})\longrightarrow 0\end{eqnarray}$$

with $\hat{Q}_{\bar{\mathfrak{m}}}=\mathop{\varinjlim }\nolimits_{E}\mathop{\varprojlim }\nolimits_{n}\hat{Q}_{E,n}$ and $\hat{P}_{\bar{\mathfrak{m}}}={\hat{W}}(\bar{\mathfrak{m}})\otimes _{W(R)}P$ . Since $\bar{\mathfrak{m}}\otimes _{R}P/Q$ has no $p$ -torsion, the $p$ -adic completion of (2.2) remains exact, moreover the $p$ -adic completion of the second and third terms are $\bar{P}_{\bar{\mathfrak{m}}}$ and $\bar{\mathfrak{m}}^{\wedge }\otimes _{R}P/Q$ . Thus the $p$ -adic completion of $\hat{Q}_{\bar{\mathfrak{m}}}$ is $\bar{Q}_{\bar{\mathfrak{m}}}$ . Moreover $\hat{P}_{\bar{\mathfrak{m}}}$ has no $p$ -torsion since ${\hat{W}}(\bar{\mathfrak{m}})$ is contained in the $\mathbb{Q}$ -algebra $W(\bar{K})$ . Using Lemmas 2.3 and 2.2, the $\operatorname{Ext}$ -sequence of $\mathbb{Q}_{p}/\mathbb{Z}_{p}$ with (2.3) reduces to the short exact sequence

$$\begin{eqnarray}0\longrightarrow \operatorname{Hom}(\mathbb{Q}_{p}/\mathbb{Z}_{p},{\hat{G}}(\bar{R}))\longrightarrow \bar{Q}_{\bar{\mathfrak{m}}}\xrightarrow[{}]{F_{1}-1}\bar{P}_{\bar{\mathfrak{m}}}\longrightarrow 0.\end{eqnarray}$$

Lemma 2.4 gives an isomorphism $\operatorname{Hom}(\mathbb{Q}_{p}/\mathbb{Z}_{p},{\hat{G}}(\bar{R}))\cong T_{p}(G)$ of ${\mathcal{G}}_{K}$ -modules.◻

3 Module of invariants

Before we proceed we introduce a formal definition. Let $\mathscr{F}=(S,I,R,\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}_{1})$ be a frame in the sense of [Reference LauLa2] such that $S$ is a $\mathbb{Z}_{p}$ -algebra and $\unicode[STIX]{x1D70E}$ is $\mathbb{Z}_{p}$ -linear; see Section 1. For an $\mathscr{F}$ -window $\mathscr{P}=(P,Q,F,F_{1})$ we consider the module of invariants

$$\begin{eqnarray}T(\mathscr{P})=\{x\in Q\mid F_{1}(x)=x\};\end{eqnarray}$$

this is a $\mathbb{Z}_{p}$ -module. Let us record some of its formal properties.

Functoriality in $\mathscr{F}$

Let $\unicode[STIX]{x1D6FC}:\mathscr{F}\rightarrow \mathscr{F}^{\prime }=(S^{\prime },I^{\prime },R^{\prime },\unicode[STIX]{x1D70E}^{\prime },\unicode[STIX]{x1D70E}_{1}^{\prime })$ be a $u$ -homomorphism of frames; see Section 1. Assume that a unit $c\in S^{\prime }$ with $c\unicode[STIX]{x1D70E}^{\prime }(c)^{-1}=u$ is given. For an $\mathscr{F}$ -window $\mathscr{P}$ as above, one verifies that the $S$ -linear map $P\rightarrow S^{\prime }\otimes _{S}P$ , $x\mapsto c\otimes x$ induces a $\mathbb{Z}_{p}$ -linear map

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D70F}(\mathscr{P})=\unicode[STIX]{x1D70F}_{c}(\mathscr{P}):T(\mathscr{P})\rightarrow T(\unicode[STIX]{x1D6FC}_{\ast }\mathscr{P}).\end{eqnarray}$$

Duality

A bilinear form of $\mathscr{F}$ -windows $\unicode[STIX]{x1D6FE}:\mathscr{P}\times \mathscr{P}^{\prime }\rightarrow \mathscr{P}^{\prime \prime }$ is a bilinear map of $S$ -modules $\unicode[STIX]{x1D6FE}:P\times P^{\prime }\rightarrow P^{\prime \prime }$ that restricts to $Q\times Q^{\prime }\rightarrow Q^{\prime \prime }$ such that for $x\in Q$ and $x^{\prime }\in Q^{\prime }$ we have

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}(F_{1}(x),F_{1}^{\prime }(x^{\prime }))=F_{1}^{\prime \prime }(\unicode[STIX]{x1D6FE}(x,x^{\prime }));\end{eqnarray}$$

see [Reference LauLa2, Section 2]. It induces a bilinear map of $\mathbb{Z}_{p}$ -modules $T(\mathscr{P})\times T(\mathscr{P}^{\prime })\rightarrow T(\mathscr{P}^{\prime \prime })$ and a $\mathbb{Z}_{p}$ -linear map $T(\mathscr{P})\rightarrow \operatorname{Hom}(\mathscr{P}^{\prime },\mathscr{P}^{\prime \prime })$ . Let us denote the $\mathscr{F}$ -window $(S,I,\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}_{1})$ by $\mathscr{F}$ again. For each $\mathscr{F}$ -window $\mathscr{P}$ there is a well-defined dual $\mathscr{F}$ -window $\mathscr{P}^{t}=(P^{t},Q^{t},F^{t},F_{1}^{t})$ with a perfect bilinear form $\mathscr{P}\times \mathscr{P}^{t}\rightarrow \mathscr{F}$ ; see [Reference LauLa2, Section 2]. Explicitly, $P^{t}=\operatorname{Hom}_{S}(P,S)$ and $Q^{t}=\{g\in P^{t}\mid g(Q)\subseteq I\}$ ; the maps $F_{1}^{t}$ and $F^{t}$ are determined by (3.2) and the window axioms. The resulting homomorphism

(3.3) $$\begin{eqnarray}T(\mathscr{P})\rightarrow \operatorname{Hom}(\mathscr{P}^{t},\mathscr{F})\end{eqnarray}$$

is bijective, which can be verified as follows: We have $\mathscr{F}^{t}=(S,S,\unicode[STIX]{x1D70E}_{-1},\unicode[STIX]{x1D70E})$ for some $\unicode[STIX]{x1D70E}$ -linear map $\unicode[STIX]{x1D70E}_{-1}$ ,Footnote ¹ thus $T(\mathscr{P})\cong \operatorname{Hom}(\mathscr{F}^{t},\mathscr{P})$ , which identifies (3.3) with the duality isomorphism $\operatorname{Hom}(\mathscr{F}^{t},\mathscr{P})\cong \operatorname{Hom}(\mathscr{P}^{t},\mathscr{F})$ .

Functoriality of duality

Let $\unicode[STIX]{x1D6FC}:\mathscr{F}\rightarrow \mathscr{F}^{\prime }$ be a $u$ -homomorphism of frames, and let $c$ be as above. For a bilinear form of $\mathscr{F}$ -windows $\unicode[STIX]{x1D6FE}:\mathscr{P}\times \mathscr{P}^{\prime }\rightarrow \mathscr{P}^{\prime \prime }$ , the base change of $\unicode[STIX]{x1D6FE}$ multiplied by $c^{-1}$ is a bilinear form of $\mathscr{F}^{\prime }$ -windows $\unicode[STIX]{x1D6FC}_{\ast }\mathscr{P}\times \unicode[STIX]{x1D6FC}_{\ast }\mathscr{P}^{\prime }\rightarrow \unicode[STIX]{x1D6FC}_{\ast }\mathscr{P}^{\prime \prime }$ , which we denote by $\unicode[STIX]{x1D6FC}_{\ast }(\unicode[STIX]{x1D6FE})$ ; see [Reference LauLa2, Lemma 2.14] and its proof. By passing to the modules of invariants we obtain a commutative diagram

This will be applied to the bilinear form $\mathscr{P}\times \mathscr{P}^{t}\rightarrow \mathscr{F}$ .

4 The case of perfect residue fields

Let $R,K,k,\mathfrak{m}$ be as in Section 2, and assume in addition that the residue field $k$ is perfect. As in [Reference LauLa3, Sections 2.C and 2.G] we consider the frame

$$\begin{eqnarray}\mathscr{D}_{R}=\mathop{\varprojlim }\nolimits_{n}\mathscr{D}_{R/\mathfrak{m}^{n}}=(\mathbb{W}(R),\mathbb{I}_{R},R,f,f_{1}).\end{eqnarray}$$

Windows over $\mathscr{D}_{R}$ , called Dieudonné displays over $R$ , are equivalent to $p$ -divisible groups $G$ over $R$ by [Reference ZinkZi2] if $p\geqslant 3$ and by [Reference LauLa3, Theorem A] in general. The Tate module $T_{p}(G)$ can be expressed in terms of the Dieudonné display of $G$ by a variant of Proposition 2.1 as follows.

Let $R^{\operatorname{nr}}$ be the strict Henselization of $R$ . This is a normal local domain, which is excellent by [Reference GrecoGre, Corollary 5.6] or [Reference SeydiSe], and thus its completion $\hat{R}^{\operatorname{nr}}$ is a normal complete noetherian local ring; see EGA IV, (7.8.3.1). Let $K^{\operatorname{nr}}\subset \hat{K}^{\operatorname{nr}}$ be the fraction fields of the rings $R^{\operatorname{nr}}\subset \hat{R}^{\operatorname{nr}}$ , let $\tilde{K}$ be an algebraic closure of $\hat{K}^{\operatorname{nr}}$ , and let $\tilde{R}$ be the integral closure of $\hat{R}^{\operatorname{nr}}$ in $\tilde{K}$ . For each finite extension $E/\hat{K}^{\operatorname{nr}}$ in $\tilde{K}$ the ring $R_{E}=\tilde{R}\cap E$ is finite over $\hat{R}^{\operatorname{nr}}$ , and $R_{E}$ is a normal complete noetherian local ring. We define a frame

$$\begin{eqnarray}\mathscr{D}_{\tilde{R}}=\mathop{\varinjlim }\nolimits_{E}\mathscr{D}_{R_{E}}=(\mathbb{W}(\tilde{R}),\mathbb{I}_{\tilde{R}},\tilde{R},f,f_{1})\end{eqnarray}$$

where the direct limit is taken componentwise. Here $\mathbb{W}(\tilde{R})$ is a local ring since all $\mathbb{W}(R_{E})$ are local with local homomorphisms in between. Since $\tilde{R}$ has no $p$ -torsion, the componentwise $p$ -adic completion of $\mathscr{D}_{\tilde{R}}$ is a frame

$$\begin{eqnarray}\hat{\mathscr{D}}_{\tilde{R}}=(\hat{\mathbb{W}}(\tilde{R}),\hat{\mathbb{I}}_{\tilde{R}},\tilde{R}^{\wedge },f,f_{1}).\end{eqnarray}$$

There are natural strict frame homomorphisms $\mathscr{D}_{R}\rightarrow \mathscr{D}_{\tilde{R}}\rightarrow \hat{\mathscr{D}}_{\tilde{R}}$ .

Let $\bar{K}$ be the algebraic closure of $K$ in $\tilde{K}$ and let ${\mathcal{G}}_{K}=\operatorname{Gal}(\bar{K}/K)$ . The tensor product $\bar{K}\otimes _{K^{\operatorname{nr}}}\hat{K}^{\operatorname{nr}}$ is a subfield of $\tilde{K}$ . Indeed, this ring is algebraic over $\hat{K}^{\operatorname{nr}}$ , and it is a localization of $\bar{K}\otimes _{R^{\operatorname{nr}}}\hat{R}^{\operatorname{nr}}$ , which is an integral domain by [Reference RaynaudRa, Chapitre XI, Théorème 3]. If $R$ is one-dimensional, then $\bar{K}\otimes _{K^{\operatorname{nr}}}\hat{K}^{\operatorname{nr}}=\tilde{K}$ because for every $R$ , the étale coverings of the complements of the maximal ideals in $\operatorname{Spec}R^{\operatorname{nr}}$ and $\operatorname{Spec}\hat{R}^{\operatorname{nr}}$ coincide by [Reference ArtinAr, Part II, Theorem 2.1] or by [Reference ElkikEl, Théorème 5]. Let $\tilde{{\mathcal{G}}}_{K}$ be the group of automorphisms of $\tilde{K}$ whose restriction to $\bar{K}\hat{K}^{\operatorname{nr}}$ is induced by an element of ${\mathcal{G}}_{K}$ . This group acts naturally on $\mathscr{D}_{\tilde{R}}$ and on $\hat{\mathscr{D}}_{\tilde{R}}$ . By the above, the projection $\tilde{{\mathcal{G}}}_{K}\rightarrow {\mathcal{G}}_{K}$ is surjective, and bijective if $R$ is one-dimensional.

Proposition 4.1. Let $G$ be a $p$ -divisible group over $R$ and let $\mathscr{P}=\unicode[STIX]{x1D6F7}_{R}(G)$ be the Dieudonné display over $R$ associated to $G$ in [Reference LauLa3]. Let $\hat{\mathscr{P}}_{\!\tilde{R}}=(\hat{P}_{\tilde{R}},\hat{Q}_{\tilde{R}},F,F_{1})$ be the base change of $\mathscr{P}$ to $\hat{\mathscr{D}}_{\tilde{R}}$ . There is a natural exact sequence of $\tilde{{\mathcal{G}}}_{K}$ -modules

$$\begin{eqnarray}0\longrightarrow T_{p}(G)\longrightarrow \hat{Q}_{\tilde{R}}\xrightarrow[{}]{F_{1}-1}\hat{P}_{\tilde{R}}\longrightarrow 0.\end{eqnarray}$$

In particular, we have an isomorphism of ${\mathcal{G}}_{K}$ -modules

$$\begin{eqnarray}\operatorname{per}_{G}:T_{p}(G)\xrightarrow[{}]{{\sim}}T(\hat{\mathscr{P}}_{\!\tilde{R}})\end{eqnarray}$$

which we call the period isomorphism in display theory.

Proof of Proposition 4.1.

For a finite extension $E/\hat{K}^{\operatorname{nr}}$ in $\tilde{K}$ let $\mathfrak{m}_{E}$ be the maximal ideal of $R_{E}$ . For a $p$ -divisible group $G$ over $R$ we set

$$\begin{eqnarray}{\hat{G}}(R_{E})=\mathop{\varprojlim }\nolimits_{n}G(R_{E}/\mathfrak{m}_{E}^{n});\qquad {\hat{G}}(\tilde{R})=\mathop{\varinjlim }\nolimits_{E}{\hat{G}}(R_{E}).\end{eqnarray}$$

The group $\tilde{{\mathcal{G}}}_{K}$ acts on the system ${\hat{G}}(R_{E})$ for varying $E$ and thus on ${\hat{G}}(\tilde{R})$ . The latter can be described using [Reference LauLa3, Section 9] as follows.

Following [Reference LauLa3, Definition 9.1] let ${\mathcal{J}}_{n}={\mathcal{J}}_{R/\mathfrak{m}^{n}}$ be the category of all $R/\mathfrak{m}^{n}$ -algebras $A$ such that the nilradical ${\mathcal{N}}_{A}$ is bounded nilpotent and such that $A/{\mathcal{N}}_{A}$ is a union of finite-dimensional $k$ -algebras. Let $\mathscr{P}_{n}$ be the base change of $\mathscr{P}$ to $R/\mathfrak{m}^{n}$ , and for $A\in {\mathcal{J}}_{n}$ let $\mathscr{P}_{A}=(P_{A},Q_{A},F,F_{1})$ be the base change of $\mathscr{P}$ to $A$ . As in [Reference LauLa3, (9–2)] we define a complex of presheaves $Z^{\prime }(\mathscr{P}_{n})$ on ${\mathcal{J}}_{n}^{\operatorname{op}}$ whose value on $A$ is the complex of abelian groups

$$\begin{eqnarray}[Q_{A}\xrightarrow[{}]{F_{1}-1}P_{A}]\otimes [\mathbb{Z}\rightarrow \mathbb{Z}[1/p]]\end{eqnarray}$$

in degrees $-1,0,1$ . By [Reference LauLa3, Proposition 9.4], $Z^{\prime }(\mathscr{P}_{n})$ is a complex of pro-étale sheaves on ${\mathcal{J}}_{n}^{\operatorname{op}}$ , which is acyclic outside degree zero, and the middle cohomology sheaf $H^{0}(Z^{\prime }(\mathscr{P}))$ is represented by a well-defined $p$ -divisible group $\operatorname{BT}(\mathscr{P})$ over $R$ . By [Reference LauLa3, Proposition 9.7] there is a canonical isomorphism $G\cong \operatorname{BT}(\mathscr{P})$ .

The ring $R_{E,n}=R_{E}/\mathfrak{m}_{E}^{n}$ is a local Artin ring with residue field $\bar{k}$ , and thus it lies in ${\mathcal{J}}_{n}$ . Every pro-étale covering of $\operatorname{Spec}R_{E,n}$ has a section since every étale covering of $\operatorname{Spec}R_{E,n}$ has a nonempty finite set of sections, and the projective limit of a projective system of nonempty finite sets is nonempty by [SP, Tag 086J]. Hence evaluating pro-étale sheaves at $R_{E,n}$ is an exact functor. It follows that the complex of abelian groups

$$\begin{eqnarray}C_{E,n}=[Q_{R_{E,n}}\xrightarrow[{}]{F_{1}-1}P_{R_{E,n}}]\otimes [\mathbb{Z}\rightarrow \mathbb{Z}[1/p]]\end{eqnarray}$$

in degrees $-1,0,1$ is acyclic outside degree zero, and there is a canonical isomorphism $G(R_{E,n})\cong H^{0}(C_{E,n})$ . For varying $n$ and $E$ these are preserved by the action of $\tilde{{\mathcal{G}}}_{K}$ . Let

$$\begin{eqnarray}C_{E}=\mathop{\varprojlim }\nolimits_{n}C_{E,n};\qquad C=\mathop{\varinjlim }\nolimits_{E}C_{E},\end{eqnarray}$$

where $E$ runs through the finite extensions of $\hat{K}^{\operatorname{nr}}$ in $\tilde{K}$ . The group $\tilde{{\mathcal{G}}}_{K}$ acts on the complex $C$ . Since the groups $G(R_{E,n})$ and the components of $C_{E,n}$ form surjective systems with respect to $n$ , the complex $C$ is acyclic outside degree zero, and we have an isomorphism of $\tilde{{\mathcal{G}}}_{K}$ -modules ${\hat{G}}(\tilde{R})\cong H^{0}(C)$ . We will verify the following chain of isomorphisms $\cong$ and quasi-isomorphisms $\simeq$ of complexes of $\tilde{{\mathcal{G}}}_{K}$ -modules, where $\operatorname{Hom}$ , $R\operatorname{Hom}$ , and $\operatorname{Ext}^{1}$ are taken in the category of abelian groups using a projective resolution of $\mathbb{Q}_{p}/\mathbb{Z}_{p}$ , in particular $\operatorname{Ext}^{1}$ is taken componentwise with respect to the second argument.

$$\begin{eqnarray}\displaystyle T_{p}(G) & \underset{(1)}{\cong } & \displaystyle \operatorname{Hom}(\mathbb{Q}_{p}/\mathbb{Z}_{p},{\hat{G}}(\tilde{R}))\underset{(2)}{\simeq }R\operatorname{Hom}(\mathbb{Q}_{p}/\mathbb{Z}_{p},{\hat{G}}(\tilde{R}))\nonumber\\ \displaystyle & \underset{(3)}{\simeq } & \displaystyle R\operatorname{Hom}(\mathbb{Q}_{p}/\mathbb{Z}_{p},C)\underset{(4)}{\simeq }\operatorname{Ext}^{1}(\mathbb{Q}_{p}/\mathbb{Z}_{p},C[-1])\underset{(5)}{\cong }[\hat{Q}_{\tilde{R}}\xrightarrow[{}]{F_{1}-1}\hat{P}_{\tilde{R}}]\nonumber\end{eqnarray}$$

This will prove the proposition.

The torsion subgroups of $G(\bar{K})$ and of ${\hat{G}}(\tilde{R})$ coincide by Lemma 2.4 applied over $\hat{R}^{\operatorname{nr}}$ , and (1) follows. Multiplication by $p$ is surjective on ${\hat{G}}(\tilde{R})$ by Lemma 2.3 applied over $\hat{R}^{\operatorname{nr}}$ , thus Lemma 2.2 gives $\operatorname{Ext}^{1}(\mathbb{Q}_{p}/\mathbb{Z}_{p},{\hat{G}}(\tilde{R}))=0$ , which proves (2). Since the cohomology of $C$ is ${\hat{G}}(\tilde{R})$ in degree zero and zero otherwise, we obtain (3). To prove (4) we choose an exact sequence of abelian groups $0\rightarrow F_{1}\rightarrow F_{0}\rightarrow \mathbb{Q}_{p}/\mathbb{Z}_{p}\rightarrow 0$ with free $F_{i}$ . This gives an exact sequence of complexes of $\tilde{{\mathcal{G}}}_{K}$ -modules

$$\begin{eqnarray}0\rightarrow \operatorname{Hom}(\mathbb{Q}_{p}/\mathbb{Z}_{p},C)\rightarrow \operatorname{Hom}(F_{0},C)\xrightarrow[{}]{u}\operatorname{Hom}(F_{1},C)\rightarrow \operatorname{Ext}^{1}(\mathbb{Q}_{p}/\mathbb{Z}_{p},C)\rightarrow 0.\end{eqnarray}$$

We claim that $\operatorname{Hom}(\mathbb{Q}_{p}/\mathbb{Z}_{p},C)$ is zero. Then the complex $\operatorname{Ext}^{1}(\mathbb{Q}_{p}/\mathbb{Z}_{p},C)[-1]$ is quasi-isomorphic to the cone of $u$ , which represents $R\operatorname{Hom}(\mathbb{Q}_{p}/\mathbb{Z}_{p},C)$ , and (4) follows. Let $(P_{\tilde{R}},Q_{\tilde{R}},F,F_{1})$ be the base change of $\mathscr{P}$ to $\mathscr{D}_{\tilde{R}}$ and let $P_{\bar{k}}=W(\bar{k})\otimes _{\mathbb{W}(R)}P$ . We have $Q_{R_{E,n}}[1/p]=P_{R_{E,n}}[1/p]=P_{\bar{k}}[1/p]$ because the cokernel of the inclusion $Q_{R_{E,n}}\rightarrow P_{R_{E,n}}$ is an $R_{E,n}$ -module and thus $p$ -power torsion, and the kernel of the surjective map $P_{R_{E,n}}\rightarrow P_{\bar{k}}$ is $p$ -power torsion by [Reference ZinkZi3, Lemma 2.2]. Thus the complex $C$ can be identified with the cone of the map of complexes

$$\begin{eqnarray}[Q_{\tilde{R}}\xrightarrow[{}]{F_{1}-1}P_{\tilde{R}}]\longrightarrow [P_{\bar{k}}[1/p]\xrightarrow[{}]{F_{1}-1}P_{\bar{k}}[1/p]].\end{eqnarray}$$

Since $\tilde{R}$ is a domain of characteristic zero, the ring $W(\tilde{R})$ has no $p$ -torsion. Since $\mathbb{W}(\tilde{R})$ is a subring of $W(\tilde{R})$ the projective $\mathbb{W}(\tilde{R})$ -module $P_{\tilde{R}}$ and its submodule $Q_{\tilde{R}}$ have no $p$ -torsion. Clearly $P_{\bar{k}}[1/p]$ has no $p$ -torsion. Hence $\operatorname{Hom}(\mathbb{Q}_{p}/\mathbb{Z}_{p},C)$ is zero, and (4) is proved. The $p$ -adic completions of $P_{\tilde{R}}$ and $Q_{\tilde{R}}$ are $\hat{P}_{\tilde{R}}$ and $\hat{Q}_{\tilde{R}}$ , while the $p$ -adic completion of $P_{\bar{k}}[1/p]$ is zero. Thus Lemma 2.2 gives (5).◻

5 A variant for the prime $2$

We keep the notation and assumptions of Section 4 and assume that $p=2$ . One can ask what the preceding constructions give when $\mathbb{W}$ and $\mathscr{D}$ are replaced by their $v$ -stabilized variants $\mathbb{W}^{+}$ and $\mathscr{D}^{+}$ defined in [Reference LauLa3, Sections 1.D, 2.E]. This will be used in Section 6. We recall that $\mathbb{W}(R)\subset \mathbb{W}^{+}(R)\subset W(R)$ where the ring $\mathbb{W}^{+}(R)$ is stable under $v$ , and we have a frame

$$\begin{eqnarray}\mathscr{D}_{R}^{+}=\varprojlim \mathscr{D}_{R/\mathfrak{m}^{n}}^{+}=(\mathbb{W}^{+}(R),\mathbb{I}_{R}^{+},R,f,f_{1})\end{eqnarray}$$

where $f_{1}$ is the inverse of $v$ . As earlier we put

$$\begin{eqnarray}\mathscr{D}_{\tilde{R}}^{+}=\mathop{\varinjlim }\nolimits_{E}\mathscr{D}_{R_{E}}^{+}=(\mathbb{W}^{+}(\tilde{R}),\mathbb{I}_{\tilde{R}}^{+},\tilde{R},f,f_{1})\end{eqnarray}$$

where $E$ runs through the finite extensions of $\hat{K}^{\operatorname{nr}}$ in $\tilde{K}$ as in Section 4, and we denote the componentwise $2$ -adic completion of $\mathscr{D}_{\tilde{R}}^{+}$ by

$$\begin{eqnarray}\hat{\mathscr{D}}_{\tilde{R}}^{+}=(\hat{\mathbb{W}}^{+}(\tilde{R}),\hat{\mathbb{I}}_{\tilde{R}}^{+},\tilde{R}^{\wedge },f,f_{1}).\end{eqnarray}$$

For a $2$ -divisible group $G$ over $R$ let $G^{m}$ and $G^{u}$ be the multiplicative and unipotent parts of $G$ and define $G^{+}$ as a pushout of fppf sheaves in the following diagram.

(5.1)

The rows of (5.1) are exact, so $G^{+}$ is a $2$ -divisible group by [Reference MessingMe1, Chapter I, (2.4.3)]. On the level of Tate modules (5.1) gives a commutative diagram with exact rows

(5.2)

which shows that $T_{2}(G^{+})$ is the pushout in the left hand square as a Galois module.

Proposition 5.1. Let $G$ be a $2$ -divisible group over $R$ with associated Dieudonné display $\mathscr{P}=\unicode[STIX]{x1D6F7}_{R}(G)$ . Let $\hat{\mathscr{P}}_{\!\tilde{R}}^{+}=(\hat{P}_{\tilde{R}}^{+},\hat{Q}_{\tilde{R}}^{+},F,F_{1}^{+})$ be the base change of $\mathscr{P}$ to $\hat{\mathscr{D}}_{\tilde{R}}^{+}$ . There is a natural exact sequence of $\tilde{{\mathcal{G}}}_{K}$ -modules

$$\begin{eqnarray}0\longrightarrow T_{2}(G^{+})\longrightarrow \hat{Q}_{\tilde{R}}^{+}\xrightarrow[{}]{F_{1}^{+}-1}\hat{P}_{\tilde{R}}^{+}\longrightarrow 0.\end{eqnarray}$$

In particular, we have an isomorphism of ${\mathcal{G}}_{K}$ -modules

$$\begin{eqnarray}\operatorname{per}_{G}^{+}:T_{2}(G^{+})\xrightarrow[{}]{{\sim}}T(\hat{\mathscr{P}}_{\!\tilde{R}}^{+}).\end{eqnarray}$$

Proof. Let $\bar{P}_{\bar{k}}=\bar{k}\otimes _{\mathbb{W}(R)}P$ . We will construct the following commutative diagram with exact rows, where $\bar{F}$ is induced by $F$ .

(5.3)

Assume that (5.3) is constructed and functorial in $G$ . Since $\bar{P}_{\bar{k}}=\bar{k}\otimes _{\mathbb{W}(R)}P$ is the reduction mod $2$ of the covariant Dieudonné module of $G_{\bar{k}}$ , the Frobenius-linear endomorphism $\bar{F}$ is nilpotent if $G$ is unipotent, and is given by an invertible matrix if $G$ is of multiplicative type. Thus $\bar{F}-1$ is surjective with kernel an $\mathbb{F}_{2}$ -vector space of dimension equal to the height of $G^{m}$ . Hence Proposition 4.1 implies that $F_{1}^{+}-1$ is surjective and gives an exact sequence

$$\begin{eqnarray}0\rightarrow T_{2}(G)\rightarrow T(\hat{\mathscr{P}}_{\!\tilde{R}}^{+})\rightarrow \operatorname{Ker}(\bar{F}-1)\rightarrow 0.\end{eqnarray}$$

The ring $W(\tilde{R})$ and its subring $\mathbb{W}^{+}(\tilde{R})$ are torsion free, which carries over to the $2$ -adic completion, hence $T(\hat{\mathscr{P}}_{\!\tilde{R}}^{+})$ is torsion free. It follows that $T(\hat{\mathscr{P}}_{\!\tilde{R}}^{+})=T_{2}(G)$ if $G$ is unipotent, and multiplication by $2$ gives an isomorphism $T(\hat{\mathscr{P}}_{\!\tilde{R}}^{+})\rightarrow T_{2}(G)$ if $G$ is multiplicative type. Hence there is a pushout diagram (5.2) with $T(\hat{\mathscr{P}}_{\!\tilde{R}}^{+})$ in place of $T_{2}(G^{+})$ , which gives an isomorphism between these modules as required.

Let us construct (5.3). [Reference LauLa3, Lemma 1.10] implies that the inclusion map $\mathbb{W}(R_{E}/\mathfrak{m}_{E}^{n})\rightarrow \mathbb{W}^{+}(R_{E}/\mathfrak{m}_{E}^{n})$ is bijective when $2\in \mathfrak{m}_{E}^{n}$ , and its cokernel is $\bar{k}\cdot v(1)$ as a $\mathbb{W}(R_{E})$ -module when $2\not \in \mathfrak{m}_{E}^{n}$ . It follows that the natural map $\unicode[STIX]{x1D704}:\hat{\mathbb{W}}(\tilde{R})\rightarrow \hat{\mathbb{W}}^{+}(\tilde{R})$ is injective with cokernel

(5.4) $$\begin{eqnarray}\hat{\mathbb{W}}^{+}(\tilde{R})/\hat{\mathbb{W}}(\tilde{R})=\hat{\mathbb{I}}_{\tilde{R}}^{+}/\hat{\mathbb{I}}_{\tilde{R}}=\bar{k}\cdot v(1).\end{eqnarray}$$

Moreover $\unicode[STIX]{x1D704}$ is a $u_{0}$ -homomorphism of frames $\hat{\mathscr{D}}_{\tilde{R}}\rightarrow \hat{\mathscr{D}}_{\tilde{R}}^{+}$ where the unit $u_{0}\in \mathbb{W}^{+}(\mathbb{Z}_{2})$ is defined by $v(u_{0})=2-[2]$ ; see [Reference LauLa3, Section 2.E]. Since $u_{0}$ maps to $1$ in $W(\mathbb{F}_{2})$ there is a unique unit $c_{0}$ of $\mathbb{W}^{+}(\mathbb{Z}_{2})$ which maps to $1$ in $W(\mathbb{F}_{2})$ such that $c_{0}f(c_{0}^{-1})=u_{0}$ , namely $c_{0}=u_{0}f(u_{0})f^{2}(u_{0})\cdots \,$ ; see the proof of [Reference LauLa2, Proposition 8.7].

We extend the operator $f_{1}$ of $\hat{\mathscr{D}}_{\tilde{R}}$ to $\hat{\mathscr{D}}_{\tilde{R}}^{+}$ by $f_{1}=u_{0}^{-1}f_{1}$ . Then $f_{1}$ induces an $f$ -linear endomorphism $\bar{f}_{1}$ of $\bar{k}\cdot v(1)$ . We claim that $\bar{f}_{1}(v(1))=v(1)$ . It suffices to prove this formula in $\mathbb{W}^{+}(\mathbb{Z}_{2})/\mathbb{W}(\mathbb{Z}_{2})\cong \mathbb{F}_{2}$ , and thus it suffices to show that $f_{1}(v(1))\not \in \mathbb{W}(\mathbb{Z}_{2})$ . But $\mathbb{W}(\mathbb{Z}_{2})$ is stable under $x\mapsto v(x)=v(u_{0}x)$ , and the element $v(f_{1}(v(1)))=v(1)$ does not lie in $\mathbb{W}(\mathbb{Z}_{2})$ . This proves the claim.

Similarly, we extend the operator $F_{1}$ of $\hat{\mathscr{P}}_{\!\tilde{R}}$ to $\hat{\mathscr{P}}_{\!\tilde{R}}^{+}$ by $F_{1}=u_{0}^{-1}F_{1}^{+}$ . Then we have $c_{0}(F_{1}-1)=(F_{1}^{+}-1)c_{0}$ as homomorphisms $\hat{Q}_{\tilde{R}}^{+}\rightarrow \hat{P}_{\tilde{R}}^{+}$ , and it suffices to construct the desired diagram with $F_{1}$ in place of $F_{1}^{+}$ . Now (5.4) implies that $\hat{Q}_{\tilde{R}}^{+}/\hat{Q}_{\tilde{R}}=\hat{P}_{\tilde{R}}^{+}/\hat{P}_{\tilde{R}}=\bar{P}_{\bar{k}}\cdot v(1)$ , which gives the exact rows. Clearly the left hand square of (5.3) commutes. The relation $F_{1}(ax)=f_{1}(a)F(x)$ for $x\in \hat{P}_{\tilde{R}}^{+}$ and $a\in \hat{\mathbb{I}}_{\tilde{R}}^{+}$ applied with $a=v(1)$ , together with $\bar{f}_{1}(v(1))=v(1)$ , shows that the right hand square of (5.3) commutes.◻

6 The relation with $A_{\operatorname{cris}}$

Let $R$ be a complete discrete valuation ring with perfect residue field $k$ of characteristic $p$ and fraction field $K$ of characteristic zero. In this case the ring $\tilde{R}^{\wedge }$ is equal to $\bar{R}^{\wedge }$ , the $p$ -adic completion of the integral closure of $R$ in $\bar{K}$ . Let $A_{\operatorname{cris}}=A_{\operatorname{cris}}(\bar{R})$ , this is the $p$ -adic completion of the divided power envelope of the kernel of the canonical homomorphism $\unicode[STIX]{x1D703}:A_{\inf }\rightarrow \bar{R}^{\wedge }$ , where $A_{\inf }=W({\mathcal{R}})$ , and where ${\mathcal{R}}$ is the projective limit of $\bar{R}/p\bar{R}$ under Frobenius. We have a frame

$$\begin{eqnarray}{\mathcal{A}}_{\operatorname{cris}}=(A_{\operatorname{cris}},\operatorname{Fil}^{1}A_{\operatorname{ cris}},\bar{R}^{\wedge },\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}_{1})\end{eqnarray}$$

with $\unicode[STIX]{x1D70E}_{1}=p^{-1}\unicode[STIX]{x1D70E}$ . Footnote ²

For a $p$ -divisible group $G$ over $R$ let $\mathbb{D}(G)$ be its covariant Dieudonné crystal. The free $A_{\operatorname{cris}}$ -module $M=\mathbb{D}(G_{\bar{R}^{\wedge }})_{A_{\operatorname{cris}}}$ carries a filtration $\operatorname{Fil}^{1}M$ and a $\unicode[STIX]{x1D70E}$ -linear endomorphism $F$ . The operator $F_{1}=p^{-1}F$ is well defined on $\operatorname{Fil}^{1}M$ , and we get an ${\mathcal{A}}_{\operatorname{cris}}$ -window ${\mathcal{M}}=(M,\operatorname{Fil}^{1}M,F,F_{1})$ ; see [Reference KisinKi1, Lemma A.2] or [Reference LauLa3, Proposition 3.17]. The window associated to $\mathbb{Q}_{p}/\mathbb{Z}_{p}$ in this way is ${\mathcal{A}}_{\operatorname{cris}}^{t}=(A_{\operatorname{cris}},A_{\operatorname{cris}},p\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E})$ .

Following [Reference FaltingsFa, Section 6] one defines a period homomorphism

$$\begin{eqnarray}\operatorname{per}_{G,\operatorname{cris}}:T_{p}(G)\rightarrow T({\mathcal{M}})\end{eqnarray}$$

as follows. An element of $T_{p}(G)$ corresponds to a homomorphism $\mathbb{Q}_{p}/\mathbb{Z}_{p}\rightarrow G$ over $\bar{R}^{\wedge }$ , and the resulting map of ${\mathcal{A}}_{\operatorname{cris}}$ -windows ${\mathcal{A}}_{\operatorname{cris}}^{t}\rightarrow {\mathcal{M}}$ corresponds to an element of $T({\mathcal{M}})$ .Footnote ³ By [Reference FaltingsFa, Theorem 7], $\operatorname{per}_{G,\operatorname{cris}}$ is bijective when $p\geqslant 3$ , and injective with cokernel annihilated by $p$ when $p=2$ . More precisely, for $p=2$ the cokernel of $\operatorname{per}_{G,\operatorname{cris}}$ is zero if $G$ is unipotent by [Reference KisinKi2, Proposition 1.1.10], but the cokernel is an $\mathbb{F}_{2}$ -vector space of dimension equal to the height of $G$ if $G$ is of multiplicative type; this can be verified for the multiplicative group $G=\unicode[STIX]{x1D707}_{p^{\infty }}$ and then follows from the fact that Fontaine’s element $t\in A_{\operatorname{cris}}$ satisfies $t^{p-1}\in pA_{\operatorname{cris}}$ ; see [Reference FontaineFo2, (2.3.4)]. As in the proof of Proposition 5.1 it follows that for $p=2$ , the homomorphism $\operatorname{per}_{G,\operatorname{cris}}$ extends to an isomorphism $T_{p}(G^{+})\cong T({\mathcal{M}})$ with $G^{+}$ as in Section 5.

We want to relate this with the period isomorphisms of Sections 4 and 5. For the sake of uniformity, for $p\geqslant 3$ we write $\mathbb{W}^{+}=\mathbb{W}$ etc. Then $\hat{\mathbb{W}}^{+}(\tilde{R})\rightarrow \bar{R}^{\wedge }$ is a divided power thickening of $p$ -adic rings for every $p$ .

Lemma 6.1. There are unique homomorphisms $\unicode[STIX]{x1D718}_{\inf }$ and $\unicode[STIX]{x1D718}_{\operatorname{cris}}$ of thickenings of $\bar{R}^{\wedge }$ as below. They commute with Frobenius, and the diagram commutes.

Since $\hat{\mathbb{W}}^{+}(\tilde{R})$ has no $p$ -torsion it follows that $\unicode[STIX]{x1D718}_{\operatorname{cris}}$ is a $\tilde{{\mathcal{G}}}_{K}$ -equivariant strict frame homomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D718}_{\operatorname{cris}}:{\mathcal{A}}_{\operatorname{cris}}\rightarrow \hat{\mathscr{D}}_{\tilde{R}}^{+}.\end{eqnarray}$$

For $G$ and ${\mathcal{M}}$ as above let $\mathscr{P}=\unicode[STIX]{x1D6F7}_{R}(G)$ be the Dieudonné display associated to $G$ and let $\unicode[STIX]{x1D6F7}_{R}^{+}(G)$ be its base change under the inclusion $\unicode[STIX]{x1D704}:\mathscr{D}_{R}\rightarrow \mathscr{D}_{R}^{+}$ , which is the identity when $p\geqslant 3$ . The $\mathscr{D}_{R}^{+}$ -window $\unicode[STIX]{x1D6F7}_{R}^{+}(G)$ can be defined by evaluating the crystal $\mathbb{D}(G)$ at $\mathbb{W}^{+}(R)$ ; see [Reference LauLa3, Theorem 3.19] if $p\geqslant 3$ and [Reference LauLa3, Proposition 3.24 & Theorem 4.9] if $p=2$ . By the functoriality of $\mathbb{D}(G)$ we get an isomorphism $\hat{\mathscr{P}}_{\tilde{R}}^{+}\cong \unicode[STIX]{x1D718}_{\operatorname{cris}\ast }({\mathcal{M}})$ of $\hat{\mathscr{D}}_{\tilde{R}}^{+}$ -windows, which induces a homomorphism of ${\mathcal{G}}_{K}$ -modules

$$\begin{eqnarray}\unicode[STIX]{x1D70F}:T({\mathcal{M}})\rightarrow T(\hat{\mathscr{P}}_{\!\tilde{R}}^{+})\end{eqnarray}$$

as defined in (3.1) with $c=1$ .

Proposition 6.2. The following diagram of ${\mathcal{G}}_{K}$ -modules commutes, and $\unicode[STIX]{x1D70F}$ is an isomorphism.

7 The ring $\mathfrak{S}^{\operatorname{nr}}$

Let us recall the ring $\mathfrak{S}^{\operatorname{nr}}$ of [Reference KisinKi1], which is denoted by $A_{S}^{+}$ in [Reference FontaineFo1]. One starts with a two-dimensional complete regular local ring $\mathfrak{S}$ of characteristic zero with perfect residue field $k$ of characteristic $p$ equipped with a Frobenius lift $\unicode[STIX]{x1D70E}:\mathfrak{S}\rightarrow \mathfrak{S}$ .

There is a unique ring homomorphism $\unicode[STIX]{x1D6E5}:\mathfrak{S}\rightarrow W(\mathfrak{S})$ with $w_{n}\circ \unicode[STIX]{x1D6E5}=\unicode[STIX]{x1D70E}^{n}$ where $w_{n}$ is the $n$ th Witt polynomial, and then $\unicode[STIX]{x1D6E5}\circ \unicode[STIX]{x1D70E}=f\circ \unicode[STIX]{x1D6E5}$ ; see [Reference LazardLaz, Chapter VII, Proposition 4.12]. The composition $\mathfrak{S}\rightarrow W(\mathfrak{S})\rightarrow W(k)$ is surjective, which implies that $p\not \in \mathfrak{m}_{\mathfrak{S}}^{2}$ . Let $t\in \mathfrak{m}_{\mathfrak{S}}\setminus \mathfrak{m}_{\mathfrak{S}}^{2}$ map to zero in $W(k)$ . Then $\mathfrak{S}=W(k)[[t]]$ and $t$ generates the kernel of $\mathfrak{S}\rightarrow W(k)$ , in particular $\unicode[STIX]{x1D70E}(t)\in t\mathfrak{S}$ .

Let ${\mathcal{O}}_{{\mathcal{E}}}$ be the $p$ -adic completion of $\mathfrak{S}[t^{-1}]$ and let $\mathbb{E}=k((t))$ be its residue field. Fix a maximal unramified extension ${\mathcal{O}}_{{\mathcal{E}}^{\operatorname{nr}}}$ of ${\mathcal{O}}_{{\mathcal{E}}}$ and let ${\mathcal{O}}_{\widehat{{\mathcal{E}}^{\operatorname{nr}}}}$ be its $p$ -adic completion. Let $\mathbb{E}^{\operatorname{sep}}$ be the residue field of ${\mathcal{O}}_{{\mathcal{E}}^{\operatorname{nr}}}$ , let $\bar{\mathbb{E}}$ be an algebraic closure of $\mathbb{E}^{\operatorname{sep}}$ , let ${\mathcal{O}}_{\mathbb{E}}=\mathfrak{S}/p\mathfrak{S}=k[[t]]$ , and let ${\mathcal{O}}_{\bar{\mathbb{E}}}\subset \bar{\mathbb{E}}$ be its integral closure. The Frobenius lift $\unicode[STIX]{x1D70E}$ on $\mathfrak{S}$ extends uniquely to ${\mathcal{O}}_{\widehat{{\mathcal{E}}^{\operatorname{nr}}}}$ and induces a homomorphism

(7.1) $$\begin{eqnarray}{\mathcal{O}}_{\widehat{{\mathcal{E}}^{\operatorname{nr}}}}\xrightarrow[{}]{\unicode[STIX]{x1D6E5}}W({\mathcal{O}}_{\widehat{{\mathcal{E}}^{\operatorname{nr}}}})\rightarrow W(\bar{\mathbb{E}})\end{eqnarray}$$

with $\unicode[STIX]{x1D6E5}$ as above. (7.1) is injective since both sides are discrete valuation rings with prime element $p$ , and the reduction modulo $p$ is injective. One defines $\mathfrak{S}^{\operatorname{nr}}={\mathcal{O}}_{\widehat{{\mathcal{E}}^{\operatorname{nr}}}}\cap W({\mathcal{O}}_{\bar{\mathbb{E}}})$ inside $W(\bar{\mathbb{E}})$ . This ring is stable under $\unicode[STIX]{x1D70E}$ , and $\mathfrak{S}^{\operatorname{nr}}=\varprojlim \mathfrak{S}_{n}^{\operatorname{nr}}$ with $\mathfrak{S}_{n}^{\operatorname{nr}}=({\mathcal{O}}_{{\mathcal{E}}^{\operatorname{nr}}}/p^{n}{\mathcal{O}}_{{\mathcal{E}}^{\operatorname{nr}}})\cap W_{n}({\mathcal{O}}_{\bar{\mathbb{E}}})$ inside $\mathbb{W}_{n}(\bar{\mathbb{E}})$ . By [Reference FontaineFo1, B 1.8.3] we have $\mathfrak{S}_{n}^{\operatorname{nr}}=\mathfrak{S}^{\operatorname{nr}}/p^{n}\mathfrak{S}^{\operatorname{nr}}$ , in particular $\mathfrak{S}^{\operatorname{nr}}$ is $p$ -adically complete.

8 Breuil–Kisin modules

Let $R$ be a complete discrete valuation ring with perfect residue field $k$ of characteristic $p$ and fraction field $K$ of characteristic zero. We recall briefly the classification of commutative finite flat $p$ -group schemes over $R$ following [Reference LauLa3]; see the introduction for a brief discussion of the history of this result.

Let $\mathfrak{S}=W(k)[[t]]$ and let $\unicode[STIX]{x1D70E}:\mathfrak{S}\rightarrow \mathfrak{S}$ be a Frobenius lift that stabilizes the ideal $t\mathfrak{S}$ . We choose a presentation $R=\mathfrak{S}/E\mathfrak{S}$ where $E$ has constant term $p$ . Let $\unicode[STIX]{x1D70B}\in R$ be the image of $t$ , so $\unicode[STIX]{x1D70B}$ generates the maximal ideal of $R$ .

For an $\mathfrak{S}$ -module $M$ let $M^{(\unicode[STIX]{x1D70E})}=\mathfrak{S}\otimes _{\unicode[STIX]{x1D70E},\mathfrak{S}}M$ . We consider pairs $(M,\unicode[STIX]{x1D719})$ where $M$ is an $\mathfrak{S}$ -module of finite type and where $\unicode[STIX]{x1D719}:M\rightarrow M^{(\unicode[STIX]{x1D70E})}$ is an $\mathfrak{S}$ -linear map with cokernel annihilated by $E$ . Following the [Reference Vasiu and ZinkVZ] terminology, $(M,\unicode[STIX]{x1D719})$ is called a Breuil window (respectively a Breuil module) relative to $\mathfrak{S}\rightarrow R$ if the $\mathfrak{S}$ -module $M$ is free (respectively annihilated by a power of $p$ and of projective dimension at most one).

We have a frame in the sense of [Reference LauLa2]

$$\begin{eqnarray}\mathscr{B}=(\mathfrak{S},E\mathfrak{S},R,\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}_{1})\end{eqnarray}$$

with $\unicode[STIX]{x1D70E}_{1}(Ex)=\unicode[STIX]{x1D70E}(x)$ for $x\in \mathfrak{S}$ . Windows $\mathscr{P}=(P,Q,F,F_{1})$ over $\mathscr{B}$ are equivalent to Breuil windows relative to $\mathfrak{S}\rightarrow R$ by the functor $\mathscr{P}\mapsto (Q,\unicode[STIX]{x1D719})$ where $\unicode[STIX]{x1D719}:Q\rightarrow Q^{(\unicode[STIX]{x1D70E})}$ is the composition of the inclusion $Q\rightarrow P$ with the inverse of the isomorphism $Q^{(\unicode[STIX]{x1D70E})}\cong P$ defined by $a\otimes x\mapsto aF_{1}(x)$ ; the inverse functor maps $(Q,\unicode[STIX]{x1D719})$ to $(P,Q,F,F_{1})$ with $P=Q^{(\unicode[STIX]{x1D70E})}$ such that the inclusion $Q\rightarrow P$ is $\unicode[STIX]{x1D719}$ and $F_{1}:Q\rightarrow P$ is $x\mapsto 1\otimes x$ , which also gives $F(x)=F_{1}(Ex)$ ; see [Reference LauLa2, Lemma 8.2].

As in [Reference LauLa3, Section 6] let $\unicode[STIX]{x1D718}$ be the ring homomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D718}:\mathfrak{S}\xrightarrow[{}]{\unicode[STIX]{x1D6E5}}W(\mathfrak{S})\rightarrow W(R).\end{eqnarray}$$

Its image lies in $\mathbb{W}(R)$ if and only if the endomorphism of $t\mathfrak{S}/t^{2}\mathfrak{S}$ induced by $\unicode[STIX]{x1D70E}$ is divisible by $p^{2}$ . In this case, $\unicode[STIX]{x1D718}:\mathfrak{S}\rightarrow \mathbb{W}(R)$ is a $u$ -homomorphism of frames $\mathscr{B}\rightarrow \mathscr{D}_{R}$ for the unit $u=f_{1}(\unicode[STIX]{x1D718}(E))$ of $\mathbb{W}(R)$ , and $\unicode[STIX]{x1D718}$ induces an equivalence between $\mathscr{B}$ -windows and $\mathscr{D}_{R}$ -windows, which are equivalent to $p$ -divisible groups over $R$ . As a consequence, Breuil modules relative to $\mathfrak{S}\rightarrow R$ are equivalent to commutative finite flat $p$ -group schemes over $R$ ; see [Reference LauLa3, Corollary 6.8].

Since $u$ maps to $1$ under $\mathbb{W}(R)\rightarrow W(k)$ , there is a unique invertible element $c\in \mathbb{W}(R)$ which maps to $1$ in $W(k)$ with $c\unicode[STIX]{x1D70E}(c^{-1})=u$ . It is given by $c=u\unicode[STIX]{x1D70E}(u)\unicode[STIX]{x1D70E}^{2}(u)\cdots \,$ ; see the proof of [Reference LauLa2, Proposition 8.7].

8.1 Modules of invariants

For a Breuil module or Breuil window $(M,\unicode[STIX]{x1D719})$ relative to $\mathfrak{S}\rightarrow R$ we write $M^{\operatorname{nr}}=\mathfrak{S}^{\operatorname{nr}}\otimes _{\mathfrak{S}}M$ and $M_{{\mathcal{E}}}^{\operatorname{nr}}={\mathcal{O}}_{\widehat{{\mathcal{E}}^{\operatorname{nr}}}}\otimes _{\mathfrak{S}}M$ and define

$$\begin{eqnarray}\displaystyle & \displaystyle T^{\operatorname{nr}}(M,\unicode[STIX]{x1D719})=\{x\in M^{\operatorname{nr}}\mid \unicode[STIX]{x1D719}(x)=1\otimes x\text{ in }\mathfrak{S}^{\operatorname{nr}}\otimes _{\unicode[STIX]{x1D70E},\mathfrak{S}^{\operatorname{nr}}}M^{\operatorname{nr}}\}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle T_{{\mathcal{E}}}^{\operatorname{nr}}(M,\unicode[STIX]{x1D719})=\{x\in M_{{\mathcal{E}}}^{\operatorname{nr}}\mid \unicode[STIX]{x1D719}(x)=1\otimes x\text{ in }{\mathcal{O}}_{\widehat{{\mathcal{E}}^{\operatorname{nr}}}}\otimes _{\unicode[STIX]{x1D70E},{\mathcal{O}}_{\widehat{{\mathcal{E}}^{\operatorname{nr}}}}}M_{{\mathcal{E}}}^{\operatorname{nr}}\}. & \displaystyle \nonumber\end{eqnarray}$$

For reference we record the following consequence of some results of [Reference FontaineFo1].

Lemma 8.1. The $\mathbb{Z}_{p}$ -module $T_{{\mathcal{E}}}^{\operatorname{nr}}(M,\unicode[STIX]{x1D719})$ is finitely generated, and the natural map

(8.1) $$\begin{eqnarray}{\mathcal{O}}_{\widehat{{\mathcal{E}}^{\operatorname{nr}}}}\otimes _{\mathbb{Z}_{p}}T_{{\mathcal{E}}}^{\operatorname{nr}}(M,\unicode[STIX]{x1D719})\rightarrow M_{{\mathcal{E}}}^{\operatorname{nr}}\end{eqnarray}$$

is bijective. The natural map

(8.2) $$\begin{eqnarray}T^{\operatorname{nr}}(M,\unicode[STIX]{x1D719})\rightarrow T_{{\mathcal{E}}}^{\operatorname{nr}}(M,\unicode[STIX]{x1D719})\end{eqnarray}$$

is bijective as well.

Proof. The homomorphism $\unicode[STIX]{x1D719}:M_{{\mathcal{E}}}^{\operatorname{nr}}\rightarrow (M_{{\mathcal{E}}}^{\operatorname{nr}})^{(\unicode[STIX]{x1D70E})}$ is bijective. If $\unicode[STIX]{x1D713}:M_{{\mathcal{E}}}^{\operatorname{nr}}\rightarrow M_{{\mathcal{E}}}^{\operatorname{nr}}$ is the $\unicode[STIX]{x1D70E}$ -linear map whose linearization is the inverse of $\unicode[STIX]{x1D719}$ , then $T_{{\mathcal{E}}}^{\operatorname{nr}}(M,\unicode[STIX]{x1D719})$ is equal to $\{x\in M_{{\mathcal{E}}}^{\operatorname{nr}}\mid \unicode[STIX]{x1D713}(x)=x\}$ , and [Reference FontaineFo1, A 1.2.6] gives the first part of the lemma.

It remains to show that (8.2) is bijective. Assume first that $(M,\unicode[STIX]{x1D719})$ is a Breuil window, let $M^{\ast }=\operatorname{Hom}_{\mathfrak{S}}(M,\mathfrak{S})$ , and let $\unicode[STIX]{x1D713}:M^{\ast }\rightarrow M^{\ast }$ be the $\unicode[STIX]{x1D70E}$ -linear map whose linearization is the dual of $\unicode[STIX]{x1D719}$ . Then $(M^{\ast },\unicode[STIX]{x1D713})$ is a Kisin module as considered in [Reference KisinKi1, (2.1.3)], and $T^{\operatorname{nr}}(M,\unicode[STIX]{x1D719})$ can be identified with the module of $\mathfrak{S}$ -linear maps $\unicode[STIX]{x1D706}:M^{\ast }\rightarrow \mathfrak{S}^{\operatorname{nr}}$ with $\unicode[STIX]{x1D70E}\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}\unicode[STIX]{x1D713}$ , and similarly for $T_{{\mathcal{E}}}^{\operatorname{nr}}(M,\unicode[STIX]{x1D719})$ . Thus (8.2) is bijective by [Reference KisinKi1, Corollary 2.1.4], which builds on [Reference FontaineFo1, B 1.8.4].

Assume now that $(M,\unicode[STIX]{x1D719})$ is a Breuil module. Using that $M$ is annihilated by a power of $p$ and of projective dimension ${\leqslant}1$ and that $C={\mathcal{O}}_{\widehat{{\mathcal{E}}^{\operatorname{nr}}}}/\mathfrak{S}^{\operatorname{nr}}$ has no $p$ -torsion, we see that $\operatorname{Tor}_{1}^{\mathfrak{S}}(C,M)$ is zero. It follows that $M^{\operatorname{nr}}\rightarrow M_{{\mathcal{E}}}^{\operatorname{nr}}$ is injective, and thus (8.2) is injective. One can find a Breuil window $(M^{\prime },\unicode[STIX]{x1D719}^{\prime })$ and a surjective map $(M^{\prime },\unicode[STIX]{x1D719}^{\prime })\rightarrow (M,\unicode[STIX]{x1D719})$ ; see (b) in the proof of [Reference LauLa2, Theorem 8.5]. Then $T^{\operatorname{nr}}(M^{\prime },\unicode[STIX]{x1D719}^{\prime })\cong T_{{\mathcal{E}}}^{\operatorname{nr}}(M^{\prime },\unicode[STIX]{x1D719}^{\prime })\rightarrow T_{{\mathcal{E}}}^{\operatorname{nr}}(M,\unicode[STIX]{x1D719})$ is surjective, and thus (8.2) is surjective.◻

8.2 The choice of $K_{\infty }$

Let $\bar{\mathfrak{m}}^{\wedge }$ be the maximal ideal of $\bar{R}^{\wedge }$ . The power series $\unicode[STIX]{x1D70E}(t)$ defines a map $\unicode[STIX]{x1D70E}(t):\bar{\mathfrak{m}}^{\wedge }\rightarrow \bar{\mathfrak{m}}^{\wedge }$ . This map is surjective, and the inverse images of algebraic elements are algebraic by the Weierstrass preparation theorem. Choose a system of elements $(\unicode[STIX]{x1D70B}^{(n)})_{n\geqslant 0}$ of $\bar{K}$ with $\unicode[STIX]{x1D70B}^{(0)}=\unicode[STIX]{x1D70B}$ and $\unicode[STIX]{x1D70E}(t)(\unicode[STIX]{x1D70B}^{(n+1)})=\unicode[STIX]{x1D70B}^{(n)}$ , and let $K_{\infty }$ be the extension of $K$ generated by all $\unicode[STIX]{x1D70B}^{(n)}$ . The system $(\unicode[STIX]{x1D70B}^{(n)})$ corresponds to an element $\text{}\underline{\unicode[STIX]{x1D70B}}\in {\mathcal{R}}=\varprojlim \bar{R}/p\bar{R}$ , the limit taken with respect to Frobenius.

We embed ${\mathcal{O}}_{\mathbb{E}}=k[[t]]$ into ${\mathcal{R}}$ by $t\mapsto \text{}\underline{\unicode[STIX]{x1D70B}}$ , and identify $\mathbb{E}^{\operatorname{sep}}$ and $\bar{\mathbb{E}}$ with subfields of $\operatorname{Frac}{\mathcal{R}}$ ; thus $W(\bar{\mathbb{E}})\subset W(\operatorname{Frac}{\mathcal{R}})$ . Then $\mathfrak{S}^{\operatorname{nr}}={\mathcal{O}}_{\widehat{{\mathcal{E}}^{\operatorname{nr}}}}\cap W({\mathcal{R}})$ , and the unique ring homomorphism $\unicode[STIX]{x1D703}:W({\mathcal{R}})\rightarrow \bar{R}^{\wedge }$ which lifts the projection $W({\mathcal{R}})\rightarrow \bar{R}/p\bar{R}$ induces a homomorphism

$$\begin{eqnarray}pr^{\operatorname{nr}}:\mathfrak{S}^{\operatorname{nr}}\rightarrow \bar{R}^{\wedge }.\end{eqnarray}$$

Let us verify that its restriction to $\mathfrak{S}$ is the given projection $\mathfrak{S}\rightarrow R$ .

Lemma 8.2. We have $pr^{\operatorname{nr}}(t)=\unicode[STIX]{x1D70B}$ .

Proof. The lemma is easy if $\unicode[STIX]{x1D70E}(t)=t^{p}$ since then $\unicode[STIX]{x1D6E5}(t)=[t]$ in $W(\mathfrak{S})$ , which maps to $[\text{}\underline{\unicode[STIX]{x1D70B}}]$ in $W({\mathcal{R}})$ , and $\unicode[STIX]{x1D703}([\text{}\underline{\unicode[STIX]{x1D70B}}])=\unicode[STIX]{x1D70B}$ in this case. In general let $\unicode[STIX]{x1D6E5}(t)=(g_{0},g_{1},\ldots )$ with $g_{i}\in \mathfrak{S}$ ; these power series are determined by the relations

$$\begin{eqnarray}g_{0}^{p^{n}}+pg_{1}^{p^{n-1}}+\cdots +p^{n}g_{n}=\unicode[STIX]{x1D70E}^{n}(t)\end{eqnarray}$$

for $n\geqslant 0$ . Let $x=(x_{0},x_{1},\ldots )\in W({\mathcal{R}})$ be the image of $t$ , thus $x_{i}=g_{i}(\text{}\underline{\unicode[STIX]{x1D70B}})$ . Write $x_{i}=(x_{i,0},x_{i,1},\ldots )$ with $x_{i,n}=g_{i}(\text{}\underline{\unicode[STIX]{x1D70B}})_{n}\in \bar{R}/p\bar{R}$ . If $\tilde{x}_{i,n}\in \bar{R}^{\wedge }$ lifts $x_{i,n}$ we have

$$\begin{eqnarray}pr^{\operatorname{nr}}(t)=\unicode[STIX]{x1D703}(x)=\lim _{n\rightarrow \infty }((\tilde{x}_{0,n})^{p^{n}}+p(\tilde{x}_{1,n})^{p^{n-1}}+\cdots +p^{n}\tilde{x}_{n,n}).\end{eqnarray}$$

If we choose $\tilde{x}_{i,n}=g_{i}(\unicode[STIX]{x1D70B}^{(n)})$ , the sum in the limit becomes $\unicode[STIX]{x1D70E}^{n}(t)(\unicode[STIX]{x1D70B}^{(n)})=\unicode[STIX]{x1D70B}$ , and the lemma is proved.◻

The natural action of ${\mathcal{G}}_{K_{\infty }}=\operatorname{Gal}(\bar{K}/K_{\infty })$ on $W(\operatorname{Frac}{\mathcal{R}})$ is trivial on ${\mathcal{O}}_{{\mathcal{E}}}$ , and therefore it stabilizes ${\mathcal{O}}_{\hat{{\mathcal{E}}}^{\operatorname{nr}}}$ and $\mathfrak{S}^{\operatorname{nr}}$ with trivial action on $\mathfrak{S}$ . Thus ${\mathcal{G}}_{K_{\infty }}$ acts on $T^{\operatorname{nr}}(M,\unicode[STIX]{x1D719})$ for each Breuil window or Breuil module $(M,\unicode[STIX]{x1D719})$ .

8.3 From $\mathfrak{S}^{\operatorname{nr}}$ to Zink rings

The composition of the inclusion $\mathfrak{S}^{\operatorname{nr}}\rightarrow W({\mathcal{R}})$ chosen above with the homomorphism $\unicode[STIX]{x1D718}_{\inf }:W({\mathcal{R}})\rightarrow \hat{\mathbb{W}}(\tilde{R})$ from Lemma 6.1 is a ring homomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D718}^{\operatorname{nr}}:\mathfrak{S}^{\operatorname{nr}}\rightarrow \hat{\mathbb{W}}(\tilde{R})\end{eqnarray}$$

that commutes with Frobenius and with the projections to $\bar{R}^{\wedge }$ .

Lemma 8.3. If the image of $\unicode[STIX]{x1D718}:\mathfrak{S}\rightarrow W(R)$ lies in $\mathbb{W}(R)$ , then the following diagram of rings commutes, where the vertical maps are the obvious inclusions.

Proof. The assumption $\unicode[STIX]{x1D718}(\mathfrak{S})\subset \mathbb{W}(R)$ is equivalent to $\unicode[STIX]{x1D6E5}(\mathfrak{S})\subset \mathbb{W}(\mathfrak{S})$ ; see [Reference LauLa3, Proposition 6.2]. As in the proof of Lemma 8.2 we write $\unicode[STIX]{x1D6E5}(t)=(g_{0},g_{1},\ldots )$ with $g_{i}\in \mathfrak{S}$ . Note that $g_{0}=t$ . We have to show that

$$\begin{eqnarray}\unicode[STIX]{x1D718}_{\inf }((g_{0}(\text{}\underline{\unicode[STIX]{x1D70B}}),g_{1}(\text{}\underline{\unicode[STIX]{x1D70B}}),\ldots ))=\unicode[STIX]{x1D704}((g_{0}(\unicode[STIX]{x1D70B}),g_{1}(\unicode[STIX]{x1D70B}),\ldots ))\end{eqnarray}$$

in $\hat{\mathbb{W}}(\tilde{R})$ . Again, if $y_{i,n}\in \hat{\mathbb{W}}(\tilde{R})$ is a lift of $x_{i,n}=g_{i}(\text{}\underline{\unicode[STIX]{x1D70B}})_{n}\in \bar{R}/p\bar{R}$ , the left hand side of this equation is equal to

$$\begin{eqnarray}\lim _{n\rightarrow \infty }((y_{0,n})^{p^{n}}+p(y_{1,n})^{p^{n-1}}+\cdots +p^{n}y_{n,n}).\end{eqnarray}$$

We will choose $y_{i,n}\in \mathbb{W}(\tilde{R})$ (no $p$ -adic completion) such that the sum in the limit is equal to $(g_{0}(\unicode[STIX]{x1D70B}),g_{1}(\unicode[STIX]{x1D70B}),\ldots )$ in $\mathbb{W}(\tilde{R})$ ; this will prove the lemma. In the special case $\unicode[STIX]{x1D70E}(t)=t^{p}$ we have $g_{i}=0$ for $i\geqslant 1$ , and we can take $y_{0,n}=[\unicode[STIX]{x1D70B}^{(n)}]$ and $y_{i,n}=0$ for $i\geqslant 1$ ; then the calculation is trivial. In general, let $\unicode[STIX]{x1D6E5}(g_{i})=(h_{i,0},h_{i,1},\ldots )$ in $\mathbb{W}(\mathfrak{S})$ , so the power series $h_{i,j}$ are determined by the equations

$$\begin{eqnarray}h_{i,0}^{p^{m}}+ph_{i,1}^{p^{m-1}}+\cdots +p^{m}h_{i,m}=\unicode[STIX]{x1D70E}^{m}(g_{i})=g_{i}(\unicode[STIX]{x1D70E}^{m}(t))\end{eqnarray}$$

for $m\geqslant 0$ , and put $y_{i,n}=(h_{i,0}(\unicode[STIX]{x1D70B}^{(n)}),h_{i,1}(\unicode[STIX]{x1D70B}^{(n)}),\ldots )\in \mathbb{W}(\tilde{R})$ . Since the Witt polynomials $w_{m}(X_{0},\ldots ,X_{m})=X_{0}^{p^{m}}+\cdots +p^{m}X_{m}$ for $m\geqslant 0$ define an injective map $\mathbb{W}(\tilde{R})\subset W(\tilde{R})\rightarrow \tilde{R}^{\infty }$ , we have to show that for $n,m\geqslant 0$ the following holds.

$$\begin{eqnarray}w_{m}((y_{0,n})^{p^{n}}+p(y_{1,n})^{p^{n-1}}+\cdots +p^{n}y_{n,n})=w_{m}((g_{0}(\unicode[STIX]{x1D70B}),g_{1}(\unicode[STIX]{x1D70B}),\ldots ))\end{eqnarray}$$

The right hand side is equal to $\unicode[STIX]{x1D70E}^{m}(t)(\unicode[STIX]{x1D70B})$ . Since $w_{m}$ is a ring homomorphism and since $w_{m}(y_{i,n})=g_{i}(\unicode[STIX]{x1D70E}^{m}(t)(\unicode[STIX]{x1D70B}^{(n)}))$ , the left hand side is equal to $\unicode[STIX]{x1D70E}^{n}(t)(\unicode[STIX]{x1D70E}^{m}(t)(\unicode[STIX]{x1D70B}^{(n)}))=\unicode[STIX]{x1D70E}^{n+m}(t)(\unicode[STIX]{x1D70B}^{(n)})=\unicode[STIX]{x1D70E}^{m}(t)(\unicode[STIX]{x1D70B})$ too.◻

We define a frame

$$\begin{eqnarray}\mathscr{B}^{\operatorname{nr}}=(\mathfrak{S}^{\operatorname{nr}},E\mathfrak{S}^{\operatorname{nr}},\mathfrak{S}^{\operatorname{nr}}/E\mathfrak{S}^{\operatorname{nr}},\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}_{1})\end{eqnarray}$$

with $\unicode[STIX]{x1D70E}_{1}(Ex)=\unicode[STIX]{x1D70E}(x)$ for $x\in \mathfrak{S}^{\operatorname{nr}}$ .

Lemma 8.4. The element $u^{\prime }=f_{1}(\unicode[STIX]{x1D718}^{\operatorname{nr}}(E))\in \hat{\mathbb{W}}(\tilde{R})$ is a unit, and the ring homomorphism $\unicode[STIX]{x1D718}^{\operatorname{nr}}:\mathfrak{S}^{\operatorname{nr}}\rightarrow \hat{\mathbb{W}}(\tilde{R})$ is a $u^{\prime }$ -homomorphism of frames $\unicode[STIX]{x1D718}^{\operatorname{nr}}:\mathscr{B}^{\operatorname{nr}}\rightarrow \hat{\mathscr{D}}_{\tilde{R}}$ .

Proof. Clearly $\unicode[STIX]{x1D718}^{\operatorname{nr}}$ commutes with the projections to $\bar{R}^{\wedge }$ and with the Frobenius. Lemma 8.2 implies that $pr^{\operatorname{nr}}(E)=0$ , thus $\unicode[STIX]{x1D718}^{\operatorname{nr}}(E)\in \hat{\mathbb{I}}_{\tilde{R}}$ . For $x\in \mathfrak{S}^{\operatorname{nr}}$ we compute $f_{1}(\unicode[STIX]{x1D718}^{\operatorname{nr}}(Ex))=f_{1}(\unicode[STIX]{x1D718}^{\operatorname{nr}}(E))\cdot f(\unicode[STIX]{x1D718}^{\operatorname{nr}}(x))=u^{\prime }\cdot \unicode[STIX]{x1D718}^{\operatorname{nr}}(\unicode[STIX]{x1D70E}_{1}(Ex))$ as required. It remains to show that $u^{\prime }$ is a unit. The projection $\tilde{R}\rightarrow \bar{k}$ induces a local homomorphism of local rings $\hat{\mathbb{W}}(\tilde{R})\rightarrow W(\bar{k})$ that commutes with $f$ and $f_{1}$ . The composition $\mathfrak{S}\rightarrow \mathfrak{S}^{\operatorname{nr}}\rightarrow \hat{\mathbb{W}}(\tilde{R})\rightarrow W(\bar{k})$ commutes with Frobenius and is thus equal to the homomorphism $t\mapsto 0$ . Thus $E$ maps to $p$ in $W(\bar{k})$ , so $u^{\prime }$ maps to $f_{1}(p)=v^{-1}(p)=1$ in $W(\bar{k})$ , and it follows that $u^{\prime }$ is a unit.◻

From now on we assume that the image of $\unicode[STIX]{x1D718}$ lies in $\mathbb{W}(R)$ , so that Lemma 8.3 applies. Then $u^{\prime }$ is the image of $u\in \mathbb{W}(R)$ , and we get a commutative square of frames where the horizontal arrows are $u$ -homomorphisms and the vertical arrows are strict:

Here ${\mathcal{G}}_{K}$ acts on $\hat{\mathscr{D}}_{\tilde{R}}$ and ${\mathcal{G}}_{K_{\infty }}$ acts on $\mathscr{B}^{\operatorname{nr}}$ , and $\unicode[STIX]{x1D718}^{\operatorname{nr}}$ is ${\mathcal{G}}_{K_{\infty }}$ -equivariant.

8.4 Identification of modules of invariants

Now we can state the main result of this section. Let $(M,\unicode[STIX]{x1D719})$ be a Breuil window relative to $\mathfrak{S}\rightarrow R$ with associated $\mathscr{B}$ -window $\mathscr{P}$ , and let $\mathscr{P}^{\operatorname{nr}}$ be the base change of $\mathscr{P}$ to $\mathscr{B}^{\operatorname{nr}}$ . By definition we have $T^{\operatorname{nr}}(M,\unicode[STIX]{x1D719})=T(\mathscr{P}^{\operatorname{nr}})$ as ${\mathcal{G}}_{K_{\infty }}$ -modules. Let $\mathscr{P}_{\mathscr{D}}$ be the base change of $\mathscr{P}$ to $\mathscr{D}_{R}$ and let $\hat{\mathscr{P}}_{\!\tilde{R}}$ be the common base change of $\mathscr{P}^{\operatorname{nr}}$ and $\mathscr{P}_{\mathscr{D}}$ to $\hat{\mathscr{D}}_{\tilde{R}}$ . As in (3.1), multiplication by $c$ induces a ${\mathcal{G}}_{K_{\infty }}$ -invariant homomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D70F}(\mathscr{P}^{\operatorname{nr}}):T(\mathscr{P}^{\operatorname{nr}})\rightarrow T(\hat{\mathscr{P}}_{\!\tilde{R}}).\end{eqnarray}$$

We recall that the ${\mathcal{G}}_{K}$ -module $T(\hat{\mathscr{P}}_{\!\tilde{R}})$ is canonically isomorphic to the Tate module of the $p$ -divisible group associated to $(M,\unicode[STIX]{x1D719})$ ; see Proposition 4.1.

Proposition 8.5. The homomorphism $\unicode[STIX]{x1D70F}(\mathscr{P}^{\operatorname{nr}})$ is bijective.

Proof. Let $h$ be the $\mathfrak{S}$ -rank of $M$ . The source and target of $\unicode[STIX]{x1D70F}(\mathscr{P}^{\operatorname{nr}})$ are free $\mathbb{Z}_{p}$ -modules of rank $h$ which are exact functors of $\mathscr{P}$ . Indeed, for $T(\mathscr{P}^{\operatorname{nr}})=T^{\operatorname{nr}}(M,\unicode[STIX]{x1D719})$ this follows from Lemma 8.1, and it holds for $T(\hat{\mathscr{P}}_{\!\tilde{R}})$ by Proposition 4.1, using that the height of a $p$ -divisible group is equal to the rank of its Dieudonné display; this can be verified over perfect fields, and then the Dieudonné display is the classical Dieudonné module.

Consider first the case where the $p$ -divisible group associated to $\mathscr{P}$ is étale, which means that $\mathscr{P}=(P,Q,F,F_{1})$ has $P=Q$ , and $F_{1}:Q\rightarrow P$ is a $\unicode[STIX]{x1D70E}$ -linear isomorphism. Then $(P,F_{1})$ is an étale $\unicode[STIX]{x1D70E}$ -module over $\mathfrak{S}$ . Since $\mathfrak{S}^{\operatorname{nr}}$ is $p$ -adically complete with $\mathfrak{S}^{\operatorname{nr}}/p={\mathcal{O}}_{\mathbb{E}^{\operatorname{sep}}}$ , a $\mathbb{Z}_{p}$ -basis of $T(\mathscr{P}^{\operatorname{nr}})$ is an $\mathfrak{S}^{\operatorname{nr}}$ -basis of $P^{\operatorname{nr}}$ . Using Lemma 4.3 it follows that a $\mathbb{Z}_{p}$ -basis of $T(\hat{\mathscr{P}}_{\!\tilde{R}})$ is a $\hat{\mathbb{W}}(\tilde{R})$ -basis of $\hat{P}_{\tilde{R}}=\hat{\mathbb{W}}(\tilde{R})\otimes _{\mathfrak{S}^{\operatorname{nr}}}P^{\operatorname{nr}}$ . Thus the homomorphism of $\mathbb{Z}_{p}$ -modules $\unicode[STIX]{x1D70F}(\mathscr{P}^{\operatorname{nr}})$ becomes an isomorphism over $\hat{\mathbb{W}}(\tilde{R})$ . Since $\mathbb{Z}_{p}\rightarrow \hat{\mathbb{W}}(\tilde{R})$ is a local homomorphism it follows that $\unicode[STIX]{x1D70F}(\mathscr{P}^{\operatorname{nr}})$ is bijective.

Consider next the case $\mathscr{P}=\mathscr{B}$ , which corresponds to the $p$ -divisible group $\unicode[STIX]{x1D707}_{p^{\infty }}$ . Assume that the proposition does not hold for $\mathscr{B}$ , i.e., that $\unicode[STIX]{x1D70F}(\mathscr{B}^{\operatorname{nr}})$ is divisible by $p$ . For a perfect extension $k^{\prime }$ of $k$ let $\mathfrak{S}^{\prime }=W(k^{\prime })[[t]]$ and $R^{\prime }=\mathfrak{S}^{\prime }/E\mathfrak{S}^{\prime }$ , and let $\mathscr{B}^{\prime }$ be the corresponding analogue of the frame $\mathscr{B}$ ; note that the Frobenius lift $\unicode[STIX]{x1D70E}$ of $\mathfrak{S}$ extends uniquely to $\mathfrak{S}^{\prime }$ . The natural homomorphism $T(\mathscr{B}^{\operatorname{nr}})\rightarrow T({\mathscr{B}^{\prime }}^{\operatorname{nr}})$ is bijective because it becomes bijective over ${\mathcal{O}}_{\widehat{{{\mathcal{E}}^{\prime }}^{\operatorname{nr}}}}$ by Lemma 8.1. The natural homomorphism $T(\hat{\mathscr{D}}_{\tilde{R}})\rightarrow T(\hat{\mathscr{D}}_{\tilde{R}^{\prime }})$ is bijective since the equivalence between $p$ -divisible groups and Dieudonné displays is compatible with arbitrary base change by [Reference LauLa3, Lemma 9.6]. Hence the homomorphism $\unicode[STIX]{x1D70F}(\mathscr{B}^{\operatorname{nr}})$ can be identified with $\unicode[STIX]{x1D70F}({\mathscr{B}^{\prime }}^{\operatorname{nr}})$ , so $k$ can be replaced by $k^{\prime }$ , which allows to assume that $k$ is uncountable. Let $\mathscr{P}_{0}$ be the étale $\mathscr{B}$ -window that corresponds to $\mathbb{Q}_{p}/\mathbb{Z}_{p}$ . We consider extensions of $\mathscr{B}$ -windows $0\rightarrow \mathscr{B}\rightarrow \mathscr{P}_{1}\rightarrow \mathscr{P}_{0}\rightarrow 0$ , which correspond to extensions in $\operatorname{Ext}_{R}^{1}(\mathbb{Q}_{p}/\mathbb{Z}_{p},\unicode[STIX]{x1D707}_{p^{\infty }})$ . Since $\unicode[STIX]{x1D70F}(\mathscr{P}_{0}^{\operatorname{nr}})$ is bijective and $\unicode[STIX]{x1D70F}(\mathscr{B}^{\operatorname{nr}})$ is divisible by $p$ , the image of $\unicode[STIX]{x1D70F}(\mathscr{P}_{1}^{\operatorname{nr}})$ provides a splitting of the reduction modulo $p$ of the exact sequence of ${\mathcal{G}}_{K_{\infty }}$ -modules

$$\begin{eqnarray}0\rightarrow T(\hat{\mathscr{D}}_{\tilde{R}})\rightarrow T((\hat{\mathscr{P}}_{1})_{\tilde{R}})\rightarrow T((\hat{\mathscr{P}}_{0})_{\tilde{R}})\rightarrow 0.\end{eqnarray}$$

Hence the composite homomorphism

(8.3) $$\begin{eqnarray}\operatorname{Ext}_{R}^{1}(\mathbb{Q}_{p}/\mathbb{Z}_{p},\unicode[STIX]{x1D707}_{p^{\infty }})\rightarrow \operatorname{Ext}_{\mathbb{F}_{p}[{\mathcal{G}}_{K}]}^{1}(\mathbb{Z}/p\mathbb{Z},\unicode[STIX]{x1D707}_{p})\rightarrow \operatorname{Ext}_{\mathbb{F}_{p}[{\mathcal{G}}_{K_{\infty }}]}^{1}(\mathbb{Z}/p\mathbb{Z},\unicode[STIX]{x1D707}_{p})\end{eqnarray}$$

is zero. The first group in (8.3) can be identified with the set of deformations of $\mathbb{Q}_{p}/\mathbb{Z}_{p}\oplus \unicode[STIX]{x1D707}_{p^{\infty }}$ from $k$ to $R$ . The second group is isomorphic to $\operatorname{Ext}_{K}^{1}(\mathbb{Z},\unicode[STIX]{x1D707}_{p})$ , which is isomorphic to the Galois cohomology group $H^{1}({\mathcal{G}}_{K},\unicode[STIX]{x1D707}_{p})\cong K^{\ast }/(K^{\ast })^{p}$ . As in [Reference LauLa1, Lemma 7.2] it follows that the first arrow in (8.3) can be identified with the natural map $1+\mathfrak{m}_{R}\rightarrow K^{\ast }/(K^{\ast })^{p}$ , whose image is uncountable since $k$ is uncountable. Since for a finite extension $K^{\prime }/K$ the homomorphism $H^{1}(K,\unicode[STIX]{x1D707}_{p})\rightarrow H^{1}(K^{\prime },\unicode[STIX]{x1D707}_{p})$ has finite kernel by the inflation-restriction exact sequence, the kernel of the second map in (8.3) is countable. Thus the composition (8.3) cannot be zero, and the proposition is proved for $\mathscr{P}=\mathscr{B}$ .

Finally let $\mathscr{P}$ be arbitrary. Duality gives the following commutative diagram; see the end of Section 3.

(8.4)

The upper line of (8.4) is a bilinear form of free $\mathbb{Z}_{p}$ -modules of rank $h$ , whose scalar extension under $\mathbb{Z}_{p}\rightarrow {\mathcal{O}}_{\widehat{{\mathcal{E}}^{\operatorname{nr}}}}$ is perfect since (8.2) and (8.1) are bijective. Since this is a local homomorphism the upper line of (8.4) is perfect. Proposition 4.1 implies that the lower line of (8.4) is a bilinear form of free $\mathbb{Z}_{p}$ -modules of rank $h$ . We have seen that $\unicode[STIX]{x1D70F}(\mathscr{B}^{\operatorname{nr}})$ is bijective. These properties imply that $\unicode[STIX]{x1D70F}(\mathscr{P}^{\operatorname{nr}})$ is bijective.◻

For a $p$ -divisible group or commutative finite flat $p$ -group scheme $G$ over $R$ let $(M(G),\unicode[STIX]{x1D719})$ be the associated Breuil window or Breuil module. In the first case let $T(G)$ be the Tate module of $G$ , and in the second case let $T(G)=G(\bar{K})$ .

Corollary 8.6. There is an isomorphism of ${\mathcal{G}}_{K_{\infty }}$ -modules $T(G)\cong T^{\operatorname{nr}}(M(G),\unicode[STIX]{x1D719})$ .

Proof. For $p$ -divisible groups this is immediate from Propositions 4.1 and 8.5. The finite case follows from the $p$ -divisible case as in the proof of [Reference LauLa3, Corollary 6.8]. More precisely, a finite $G$ can be written as the kernel of an isogeny of $p$ -divisible groups $G_{0}\rightarrow G_{1}$ , which gives exact sequences $0\rightarrow T(G_{0})\rightarrow T(G_{1})\rightarrow T(G)\rightarrow 0$ and $0\rightarrow M(G_{0})\rightarrow M(G_{1})\rightarrow M(G)\rightarrow 0$ , and the latter gives an exact sequence $0\rightarrow T^{\operatorname{nr}}(M(G_{0}))\rightarrow T^{\operatorname{nr}}(M(G_{1}))\rightarrow T^{\operatorname{nr}}(M(G))\rightarrow 0$ . The resulting isomorphism $T(G)\cong T^{\operatorname{nr}}(M(G))$ is independent of the resolution $G_{0}\rightarrow G_{1}$ of $G$ .◻