Semistable abelian varieties and $\text{Crys}_{1}$]Semistable abelian varieties and maximal torsion 1-crystalline submodules
Cody Gunton
Cody Gunton

Department of Mathematical Sciences

Universitetsparken 5

University of Copenhagen

2100 Copenhagen Ø

Supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).

Let $p$ be a prime, let $K$ be a discretely valued extension of $\Qp$, and let $A_{K}$ be an abelian $K$-variety with semistable reduction. Extending work by Kim and Marshall from the case where $p>2$ and $K/\Qp$ is unramified, we prove an $l=p$ complement of a Galois cohomological formula of Grothendieck for the $l$-primary part of the Néron component group of $A_{K}$. Our proof involves constructing, for each $m\in \Z_{\geq 0}$, a finite flat $\OK$-group scheme with generic fiber equal to the maximal 1-crystalline submodule of $A_{K}[p^{m}]$. As a corollary, we have a new proof of the Coleman-Iovita monodromy criterion for good reduction of abelian $K$-varieties.

§ INTRODUCTION

Let $p$ be any rational prime, let $K/\Qp$ be a discretely valued extension of $\Qp$, let $A_{K}$ be an abelian $K$-variety, and write $A$ for the Néron model of $A_{K}$. Fix an embedding $K\subset \Kbar$ into an algebraic closure and let $\Gamma_{K} = \Gal(\Kbar/K)$. Assume that $A_{K}$ has semistable reduction. We prove a formula for the $p$-primary part $\Phi_{A_{K}}[p^{\oo}]$ of the finite abelian group of connected components $\Phi_{A_{K}}$  of the special fiber of $A$. The formula has a much older $l\neq p$ analogue due to Grothendieck. He established the following result without any restriction on the ramification of $K/\Qp$.

Let $l$ be a prime distinct from $p$. Writing $T_{l}A_{K}$ for the $l$-adic Tate module of $A_{K}$, and $I_{K}\subset \Gamma_{K}$ for the inertia subgroup, there is a canonical isomorphism of unramified $\bZ_{l}[\Gamma_{K}]$-modules

\[
\Phi_{A_{K}}[l^{\oo}] \simeq (H^{1}_{\cts}(\Gamma_{K}, T_{l}A_{K}))_{\tors},
\]

where $(-)_{\tors}$ gives the maximal torsion submodule.

Loosely speaking, the formula breaks down when $l=p$ because maximal unramified submodules tend to be too small in that case. The following idea is pursued in the work [24] of Kim and Marshall: to `repair' Grothendieck's formula in the case $l=p$, one should replace the condition of unramifiedness with a weaker condition from $p$-adic Hodge theory. This is the condition of torsion 1-crystallinity, i.e., the condition of arising as a torsion quotient of a $\Gamma_{K}$-stable $\Zp$-lattice in some crystalline $\Qp[\Gamma_{K}]$-module with Hodge-Tate weights in the interval $[0,1]$. Defining $r$-crystallinity analogously for any $r\in \Z_{\geq 0}$, it can be shown that unramifiedness is the same as $0$-crystallinity, so that $1$-crystallinity is a weakening of unramifiedness.

For any $\Zp[\Gamma_{K}]$-module $M$ of finite-type over $\Zp$, the notion of a maximal $r$-crystalline submodule $\Crys_{r}(M)$ is well defined. When $M$ is torsion free, it can happen that $\Crys_{r}(M)$ stabilizes to a proper submodule of $M$ as $r\to \oo$. In contrast, it is a recent result that if $M$ is finite torsion then $\Crys_{r}(M) = M$ for $r \gg 0$ (see [14], Theorem 3.3.2). The following result further motivates our interest in $1$-crystalline subrepresentations.

Let $M$ be a finite torsion $\Z_{p}[\Gamma_{K}]$-module. Then $M$ is the generic fiber of a finite flat $\OK$-group if and only if $M$ is $1$-crystalline.

The proofs of this result are inexplicit, in a way, proceeding via Raynaud's embedding theorem (see <Ref>) and abstract $p$-adic Hodge theory. In particular, in the case where $M = \Crys_{1}(A_{K}[p^{m}])$ with $A_{K}$ as above, one knows that $M$ is the generic fiber of a finite flat $\OK$-group $Q_{p^{m}}$, but the above result and its proof do not provide a useful relationship between $Q_{p^{m}}$ and the geometry of the abelian variety $A_{K}$. Despite this complication, Kim and Marshall were able to use this result in the case where $p>2$ and $K/\Qp$ is unramified to construct a finite flat $\OK$-group with generic fiber $\Crys_{1}(A_{K}[p^{m}])$. Using this, they proved the following result, which we prove for any $p$ and any discretely valued $K/\Qp.$

Suppose $A_{K}$ has semistable reduction. Then there is an isomorphism of unramified $\Zp[\Gamma_{K}]$-modules

\[
\Phi_{A_{K}}[p^{\oo}] \simeq (R^{1}\Crys_{1}(T_{p}A_{K}))_{\tors}.
\]

The semistability assumption in <Ref> is needed to ensure that the identity component $A^{\circ}_{\kappa}$ of the special fiber of $A$ is $p$-divisible, a fact that is used in the result of Grothendieck invoked in <Ref>. Note that, for $l\neq p$, $A^{\circ}_{\kappa}$ is always $l$-divisible; this accounts for the fact that <Ref> does not make any restriction on the reduction type of $A$. The semistability assumption is essential in our methods and those of Kim-Marshall, and it is not at all clear how to make an $l=p$ analogue of Grothendieck's result without such an assumption.

Using Fontaine-Laffaille theory, Kim and Marshall choose a finite flat $\OK$-group $Q_{p^{m}}$ with generic fiber isomorphic to $\Crys_{1}(A_{K}[p^{m}])$, and then use Raynaud's result to produce an embedding of the identity component $Q_{p^{m}}^{\circ}$ into the Néron model $A$. They then used Raynaud's result to deduce that this embedding gives an isomorphism of $Q_{p^{m}}$ with the finite flat $\OK$-group that we will denote ${}^{*}\sQo[p^{m}]$. Their work is in terms of finite parts of quasi-finite flat $\OK$-groups, which allows them to deduce from results of [15] to construct the key isomorphisms

\begin{equation}
\label{eq:intro-phi-and-crys-1}
\Phi_{A_{K}}[p^{m}] \simeq \frac{\Crys_{1}(T_{p}A_{K}\otimes \Z/p^{m}\Z)}{\Crys_{1}(T_{p}A_{K})\otimes \Z/p^{m}\Z}
\qquad (m \geq 1).
\end{equation}

We note that a related expression for $\Phi_{A_{K}}[p^{m}]$ is claimed without proof in Theorem 4 of the appendix of [19].

Replacing Kim and Marshall's use of Fontaine-Laffaille's result with <Ref>, their proof extends immediately to the case where the ramification index $e$ of $K/\Qp$ satisfies $e<p-1$ (in particular, $p>2$). This ramification restriction is essential to their approach, which relies on the following result of Raynaud.

Suppose $e<p-1$. Then every finite torsion $\Z_{p}[\Gamma_{K}]$-module arises as the generic fiber of at most one finite flat $\OK$-group, up to isomorphism. Let $H', H$ be finite flat $\OK$-groups.

* Every morphism $H'_{K}\to H_{K}$ prolongs uniquely to a morphism $f: H'\to H$. Moreover, the kernel and cokernel of $f$ formed in the category of group schemes over $\OK$ are finite flat over $\OK$.

* The category of finite flat $\OK$-groups is an abelian category.

We take a different approach. Rather than choosing $Q_{p^{m}}$ by an existence result and demonstrating that it has a certain relationship to $A$, we construct a finite flat $\OK$-group with generic fiber $\Crys_{1}(A_{K}[p^{m}])$ directly from the Néron model $A$ of $A_{K}$. For this, we will use a semiabelian $K$-variety, the generic fiber of the Raynaud extension $\widetilde A$ attached to $A_{K}$, that covers $A_{K}$ in the category of rigid analytic spaces over $K$.  From this space and the monodromy pairing, we will construct a certain finite flat $\OK$-group with monodromy. An isomorphism as in (<ref>) can then be constructed using results on the monodromy pairing.

Fix a uniformizer $\varpi$ of $\OK$. Let $A_{K}$ be a semistable abelian $K$-variety, and let $n\in \Z_{\geq 2}$, be a power of $p$. Let $(\sQon, N_{n})$ denote the finite flat $\OK$-group with monodromy of <Ref>. There is an isomorphism of $\Zp[\Gamma_{K}]$-modules

\[
A_{K}[n]\times_{\sQo[n]^{\et}_{K}} \ker N_{n,K}(-1) \simeq \Crys_{1}(A_{K}[n]).
\]

Moreover, $\Crys_{1}(A_{K}[n])$ is the generic fiber of the finite flat $\OK$-group

\begin{align*}
{}^{*}\sQo[n] :=
\sQo[n]
\times_{\sQo[n]^{\et}}
\ker N_{n}(-1).
\end{align*}

To show that ${}^{*}\sQo[n]$ actually has the claimed generic fiber, we use a recent technical result from $p$-adic Hodge theory, the Torsion Fully Faithfulness Theorem of Ozeki, Breuil and others (<Ref>).

From <Ref>, it is straightforward to prove the following result.

With assumptions as in <Ref>, we have

\[
\Crys_{1}(T_{p}A_{K}) \simeq T_{p}\widetilde A_{K},
\]

where $\widetilde A_{K}$ is the Raynaud extension attached to $A_{K}$.

The approach of Coleman and Iovita in [11] is markedly different from ours, at least in that it takes the perspective of $p$-adic integration. Their Corollary 4.6 is used in [11] to prove their main result, an $l=p$ analogue of the Ogg-Néron-Shafarevich formula. We show how this follows also from our work in <Ref>.

Using <Ref> and <Ref>, we have an expression

\[
\frac{
\widetilde A_{K}[p^{m}]
\simeq
\frac{
\Crys_{1}(T_{p}A_{K}\otimes \Z/p^{m}\Z)
}{\Crys_{1}(T_{p}A_{K})\otimes \Z/p^{m}\Z
\]

for the right-hand side of (<ref>). From the construction of ${}^{*}\sQo[p^{m}]_{K}$ and results about the monodromy pairing that can be found in [15], it is straightforward to deduce that there is an isomorphism

\[
\frac{
\widetilde A_{K}[p^{m}]
\simeq
\Phi_{A_{K}}[p^{m}]
\]

so we have (<ref>) without assuming any bound on the ramification $e$ of $K/\Qp$. From there, we may reuse the homological algebra of the proof of <Ref>, which were repurposed by [24], to obtain <Ref>.

We hope we have made our debt to Kim and Marshall apparent. The idea of studying derived functors of $\Crys_{1}$ originates in their work [24], as does their definition of these derived functors in the spirit of Jannsen's definition of continuous étale cohomology. Marshall's thesis [28], was an invaluable resource to us in study of these derived functors and the general categories of (e.g., non-finite type torsion) $r$-crystalline $\Gamma_{K}$-modules that arise in their construction. Each of these two papers proves interesting results which are not generalized in the present article.

We further motivate the choice of methods in this article. As noted above, the work of Kim and Marshall makes essential use of the restriction $e<p-1$ in its use of Raynaud's fully faithfulness result. Conditions weaker than fullness of the generic fiber functor which are valid without any condition on $e$ have been proved by Bondarko [7] and Liu [26]. These results take the following form: there exists a constant $\fc\in \Z_{\geq 0}$ depending only on $e$ such that, for any two finite flat $\OK$-groups $H'$, $H$ and any morphism $f_{K}: H'_{K}\to H_{K}$, there exist a morphism $H'\to H$ inducing $p^{\fc}f_{K}$ on generic fibers. In the cases where $\fc>0$, we were unable to use these results to generalize the work of Kim and Marshall.

Kisin's proof that torsion 1-crystallinity is the same as finite flatness loses contact with the representation $T_{p}A_{K}$. The proof uses a functor $\fM(-): \RepcrysoneZp\to \Mod^{1,\fr}_{\fS}$, which is not exact. While one can make[See [27]; [21] does consider categories of $\fS$-modules with monodromy, but the monodromy maps are only defined after inverting $p$.] a variant of $\fM(-)$ amenable to the study of lattices in 1-semistable representations by enriching $\Mod^{1,\fr}_{\fS}$ with monodromy operators, the non-exactness of $\fM(-)$ obstructs some natural approaches to proving <Ref>. Kisin has shown in [21] that there exists a $p$-divisible group $G_{n}[p^{\oo}]$ over $\OK$ such that $G_{n}[n]_{K}$ is isomorphic to $\Crys_{1}(A_{K}[n])$. However, there is no immediately useful relationship between the $G_{n}[p^{\oo}]$ for different values of $n$.

Our approach is to work with a category $\finpN$ of finite flat group schemes (equivalently, torsion Kisin modules of height 1) with monodromy. Such objects were introduced long ago by Kato in his study of log finite flat group schemes; see the unpublished preprint [18], where his result Lemma 3.8 (2) amounts to a criterion for the torsion in a log finite flat group scheme to be $1$-crystalline. The reader may wonder why we chose not to use the framework of log schemes. One reason is that this would require translating results from the theory of degenerations of abelian varieties to the logarithmic context, a major undertaking. While this has been done to some extent in [23] (see, especially, Section 18), this paper was not available when we completed much of the present work. It also seems likely that an approach using log schemes would still go through the Torsion Fully Faithfulness Theorem, and that the category $\fin^{p, N}_{\OK}$ would intervene at this point in some guise. The use of log schemes also suffers from the fact that important results on the Kummer log flat topology and on logarithmic group schemes are either unpublished or scattered in the literature. The wide availability of comprehensive references was one important factor in our choice of approach.

The present work is derived from the author's PhD thesis, which was conducted under the guidance of Bryden Cais. We sincerely thank him for suggesting the project of removing the ramification restriction in the work of Kim and Marshall, and for his consistent support and help. We thank Keerthi Madapusi Pera for sharing his unpublished preprint [27]. We also thank Lars Halvard Halle for suggesting some minor corrections and clarifications on an earlier draft.

§.§ Notation

Let $p$ be any prime and let $K/\Qp$ be a discretely valued field extension. Write $\fm_{K}\subset \OK$ be the maximal ideal, let $\kappa$ be the residue field of $\OK$, and let $v_{K}$ be the valuation of $\OK$ satisfying $v_{K}(K^{\times}) = \bZ$. Fix an embedding $K\subset \Kbar$ into an algebraic closure, let $\Gamma_{K} := \Gal(\Kbar/K)$, and let $C$ be the completion of $\Kbar$ under the absolute value extending the absolute value of $K$.

Fix a uniformizer $\varpi\in \OK$. Let $\sO_{C}^{\flat} := \lim_{x\mapsto x^{p}}\sO_{C}$, let $\varpi^{\flat} = (\varpi^{\flat,(i)})_{i\geq 0}\in \sO_{C}^{\flat}$ be an element with $\varpi^{\flat, (0)}=\varpi$, and let $\epsilon = (\epsilon^{(i)})_{i\geq 0}\in \sO_{C}^{\flat}$ be an element with $\epsilon^{(0)}= 1 \neq \epsilon^{(1)}$. Then $\varpi^{\flat}$ is a system of $p$-power roots of $\varpi$ and $\epsilon$ is a nontrivial system of $p$-power roots of 1. Let $\Gamma_{\varpi^{\flat}}\subset \Gamma_{K}$ be the (non-normal) subgroup consisting of those automorphisms that fix each of the $\varpi^{\flat, (i)}$.

§.§ Example: Tate curves

We describe our results in the special case of the Tate curve. While this very simple example could be treated in a more streamlined and direct way, we give a presentation that aims to motivate our more general constructions. To that end, we use notation and terminology that will not be properly defined until later.

Let $q$ be an element of $K^{\times}$ satisfying $v_{K}(q)>0$, and let $E_{q,K}$ be the associated Tate curve. It is well known that the homomorphism $\iota_{q,K}: \bZ_{K}\longto \GmK$ of $K$-group schemes sending $1$ to $q$ fits into a short exact sequence of $K$-rigid spaces

\begin{align}
\label{eq:q-tate-unif-ses}
\ses{\bZ_{K}^{\an}}{\iota_{q,K}}{
\GmK^{\an}}{\psi_{K}}{E_{q,K}^{\an}},
\end{align}

where $\psi_{K}$ is given by explicit series over $K$. Let $n\geq 2$ be a power of $p$. The short exact sequence (<ref>)  gives rise to the short exact sequence of finite $K$-group schemes

\begin{align}
\label{eq:q-tate-mot-ses}
\ses{\mu_{n,K}}{\psi_{K}}{E_{q,K}[n]}{}{\Z_{K}/n\Z_{K}}.
\end{align}

Since the $p$-adic cyclotomic character is 1-crystalline, we see that $\Crys_{1}(E_{q,K}[n])$, the sum in $E_{q, K}[n]$ of all of its 1-crystalline subobjects, is at least as large as $\Z/n\Z(1)$. The map $\iota_{q,K}$ can be regarded a log 1-motive over $K$

\begin{align}
\label{eq:q-tate-1-motive}
\sT_{q,K} = [\Z_{K}\overset{\iota_{q,K}}\longto \GmK],
\end{align}

which is, in particular, a complex of $K$-groups concentrated in degrees $-1$ and $0$. The integer $n$ gives a morphism of complexes $[n]: \sT_{q,K}\to \sT_{q,K}$. Defining $\sT_{q,K}[n] := H^{-1}(\cone([n]))$, $\psi_{K}$ induces an isomorphism (cf (<ref>))

\begin{align*}
\sT_{q,K}[n]\simeq E_{q,K}[n].
\end{align*}

As a torsion $\Gamma_{K}$-module, $E_{q,K}[n]$ is 1-semistable. To describe the maximal 1-crystalline subobject $\Crys_{1}(E_{q,K}[n])$, we will use Raynaud's decomposition of  $\sT_{q,K}$ (see <Ref>). Use the uniformizer $\varpi\in \OK$ to write $q = u\varpi^{v}$ with $u\in \OK^{\times}$ and $v\in \Z_{\geq 1}$. This allows us to define two log 1-motives

:= [_Kι_q,K,ϖ^1],
ι_q,K,ϖ^1(1) = u

:= [_Kι_q,K,ϖ^2],
ι_q,K,ϖ^2(1) = ϖ^v.

Then $\iota_{q,K} = \iota_{q, K, \varpi}^{1} \cdot \iota_{q, K, \varpi}^{2}$ using the group law in $\GmK$. It is clear that $\sT_{q,K,\varpi}^{1}$ extends to an $\OK$-1-motive $\sT_{q,\varpi}^{1}$. Defining torsion in 1-motives as we did above, we have a finite flat $\OK$-group scheme
$\sT_{q,\varpi}^{1}[n]$ with generic fiber $\sT_{q,K,\varpi}^{1}[n]$, so that the $\Gamma_{K}$-module $\sT_{q,K,\varpi}^{1}[n]$ is 1-crystalline.

Under the Kummer isomorphism $c_{(-),n}: K^{\times}/(K^{\times})^{n}\to H^{1}(\Gamma_{K}, \Z/n\Z(1))$, the class $\eta(\sT_{q,K}, n)$ (resp., $\eta(\sT_{q,K,\varpi}^{1}, n)$, resp., $\eta(\sT_{q,K,\varpi}^{2}, n)$) of $\sT_{q,K}$ (resp., $\sT_{q,K,\varpi}^{1}$, resp., $\sT_{q,K,\varpi}^{2}$) in the abelian group $\Ext^{1}_{\ModZpGK}(\Z/n\Z, \Z/n\Z(1))$  arises from $q$ (resp., $u$, resp., $\varpi^{v}$). The equation $c_{q,n} = c_{u,n} + c_{\varpi^{v},n}$ gives a Baer sum decomposition

\begin{align}
\label{eq:q-tate-tors-dec}
\eta(\sT_{q,K}, n)
= \eta(\sT_{q,K,\varpi}^{1}, n)
+ \eta(\sT_{q,K,\varpi}^{2}, n).
\end{align}

By construction of the Baer sum, $\sT_{q,K}[n]$ is a subquotient of $\sT_{q,K,\varpi}^{1}[n] \oplus \sT_{q,K,\varpi}^{2}[n]$. Since finite direct sums and subquotients of 1-crystalline $\Gamma_{K}$-modules are again $1$-crystalline, taking a difference in (<ref>) allows us to conclude that $E_{q,K}[n]$ is 1-crystalline if and only if $\sT_{q,K,\varpi}^{2}[n]$ is 1-crystalline.

Since $\sT_{q,K,\varpi}^{2}[n]$ corresponds to the 1-cocycle $c_{\varpi^{v},n} = v c_{\varpi,n}:\Gamma_{K}\to \Z/n\Z(1)$, we see that $\sT_{q,K,\varpi}^{2}[n]$ is split, i.e., $\sT_{q,K,\varpi}^{2}[n] \simeq \Z/n\Z(1)\oplus \Z/n\Z$ as a $\Gamma_{K}$-module, if $n\mid v$. In particular, in this case $\sT_{q,K,\varpi}^{2}[n]$ is 1-crystalline. Conversely, by the Torsion Fully Faithfulness Theorem of Breuil, Ozeki and others (see <Ref>), if $\sT_{q,K,\varpi}^{2}[n]$ is 1-crystalline and $V$ is another 1-crystalline $\Gamma_{K}$-module, then any isomorphism
\[\Resoo(\sT_{q,K,\varpi}^{2}[n]) \simeq \Resoo V\]

induces an isomorphism $\sT_{q,K,\varpi}^{2}[n]\simeq V$. But $c_{\varpi, n}(\Gamma_{\varpi^{\flat}}) = 0$, so that any choice of an adapted basis for $\sT_{q,K,\varpi}^{2}[n]$ induces an isomorphism

\[\Resoo \sT_{q,K,\varpi}^{2} [n]
\simeq \Resoo
\Big(\bZ/n\bZ(1)\oplus \bZ/n\bZ\Big).
\]

We conclude that $E_{q,K}[n]$ is 1-crystalline if and only if $v_{p}(n) \leq v_{p}(v_{K}(q))$.

We push this argument to describe $\Crys_{1}(E_{q,K}[n])$ in general. Suppose $v_{p}(n)\geq v_{p}(v_{K}(q))$. With reference to <ref>, form a basis of $E_{q,K}[n]$ by choosing an element $x_{1}$ that spans the image of $\psi_{K}$ and letting $x_{2}\in E_{q,K}[n]$ be a lift of $1\in \bZ/n\bZ$. Then $\Crys_{1}(E_{q,K}[n])$ is generated by $x_{1}$ and $p^{w}x_{2}$ for some integer $w$ satisfying $0 \leq w \leq v_{p}(n)$. In particular, $p^{w}E_{q,K}[n]$ is contained in $\Crys_{1}(E_{q,K}[n])$. Since 1-crystallinity is inherited by submodules, we see that $E_{q,K}[p^{v_{p}(n)-w}]$ is 1-crystalline. Our above work shows that $v_{p}(n)-w\leq v_{p}(v_{K}(q))$, and the maximality property of $\Crys_{1}(E_{q,K}[n])$ implies that $w = v_{p}(n)-v_{p}(v_{K}(q))$. Therefore, for $v_{p}(n) \geq v_{K}(q)$, we have $\Crys_{1}(E_{q,K}[n]) = \bZ/n\bZ x_{1} + p^{v_{p}(n)-v_{p}(v_{K}(q))}\bZ/n\bZ x_{2}$.

The subspace $\im \psi_{K}\simeq \bZ/n\bZ(1)$ is distinguished by the fact that it arises from a 1-crystalline submodule of $T_{p}E_{q,K}$. A direct calculation with the period ring $B_{\crys}$ show that $T_{p}E_{q,K}$ is not crystalline. Being that $T_{p}E_{q,K}$ is of rank 2 over $\Zp$, we see that $\Crys_{1}(T_{p}E_{q,K})$ is equal to the copy $\Zp(1)$ obtained in the limit from the maps $\psi_{K}$ for varying $n$. It is well known that the Néron component group $\Phi_{E_{q}}$ attached to $E_{q,K}$ is cyclic of order $v_{K}(q)$, hence that the $p$-Sylow subgroup $\Phi_{E_{q}}[p^{\oo}]$ is cyclic of order $p^{v_{p}(v_{K}(q))}$. Altogether, we have the following result.

Let $q\in K^{\times}$ satisfy $v_{K}(q) > 0$, let $m\in \bZ_{\geq 1}$, and let $v = v_{p}(\# \Phi_{E_{q}}[p^{\oo}])$. Then

\begin{equation}
\label{eq:q-tate-crys-1-cases}
\Crys_{1}(E_{q,K}[p^{m}]) =
\begin{cases}
E_{q,K}[p^{m}] & \text{ if } m\leq v \\
\bZ/p^{m}\bZ(1) + p^{m-v}E_{q,K}[p^{m}]
& \text{ if } m > v
\end{cases}
\end{equation}

We will see later that there is an isomorphism of finite abelian groups

\begin{align*}
\frac{
\Crys_{1}(T_{p}E_{q,K}\otimes_{\Zp} \bZ/p^{m}\bZ)
}{\Crys_{1}(T_{p}E_{q,K})\otimes_{\Zp} \bZ/p^{m}\bZ}
\simeq \Phi_{E_{q}}[p^{m}].
\end{align*}

We end this example by noting that the cases in (<ref>) can be described uniformly. Let $N_{p^{m}}^{*}$ be the endomorphism of the finite flat $\OK$-group scheme $\bZ/p^{m}\bZ$ given by multiplication by $v_{K}(q)$. Let ${}^{*}\sT^{1}_{\varpi}[p^{m}]$ the finite flat $\OK$-group obtained by pulling the class of $\sT^{1}_{\varpi}[p^{m}]$ in $\Ext(\bZ/p^{m}\bZ, \bZ/p^{m}\bZ(1))$ back under the inclusion $\ker N_{p^{m}}^{*}\subset \bZ/p^{m}\bZ$. Then, for all $m\in \bZ_{\geq 1}$, there is an isomorphism

\[{}^{*}\sT^{1}_{\varpi}[p^{m}]_{K}\simeq \Crys_{1}(E_{q,K}[p^{m}]).\]

§ FINITE FLAT GROUP SCHEMES WITH MONODROMY AND LOG 1-MOTIVES

Throughout this section we adopt the notation of <Ref>.

We describe some known results about semistable degenerations of abelian varieties; nothing is original beyond our presentation. Our main goals are to show that the torsion in a semistable abelian $K$-variety is the torsion in a certain log 1-motive; to describe Raynaud decomposition of log 1-motives; and to record the relationship between the monodromy pairing and the Néron component group.

§.§ Degenerations of abelian varieties

Let $A_{K}$ be a semistable abelian $K$-variety. The Néron model $A$ of $A_{K}$ is a smooth $\OK$-group. Its special fiber $A_{\kappa}$ need not be connected. We are concerned with $A_{\kappa}/A_{\kappa}^{\circ}$, which ([29], 1.1.33) is a finite étale $\kappa$-group. We define two $\OK$-groups.

Define $A^{\circ}$ to be the open subgroup scheme $A^{\circ}\subset A$ which is the disjoint union of $A_{K}$ and the identity component $A^{\circ}_{\kappa}$ of the special fiber.

Define the Néron component group of $A_{K}$ to be the unique étale $\OK$-group $\Phi_{A_{K}}$ with special fiber $\Phi_{A_{K},\kappa} = A_{\kappa}/A_{\kappa}^{\circ}$.

Since $\kappa$ is perfect, the identity component $A_{\kappa}^{\circ}$ lies in a canonical short exact sequence ([29] 8.27 and 16.15)

\begin{align}
\label{chevalley-seq-kappa}
\ses{T_{\kappa}\times_{\kappa} U_{\kappa}}{}{A^{\circ}_{\kappa}}{}{B_{\kappa}}
\end{align}

where $B_{\kappa}$, $U_{\kappa}$ and $T_{\kappa}$ are algebraic $\kappa$-group which are, respectively, abelian, unipotent, and a torus. We say that $A_{K}$ has semistable reduction (or simply that $A_{K}$ is semistable) if $U_{\kappa}$ is trivial. By [15] 2.2.1, $A_{K}$ has semistable reduction if and only if, for all $n\in \Z_{\geq 1}$, the multiplication-by-$n$ map on $A^{\circ}$ is quasi-finite, flat, and surjective. When these conditions are satisfied, the natural injection

\[
\longto \Phi_{A_{K},\kappa}[n]
\]

is surjective, opening up the possibility of studying the Néron component group (a finite abelian group with unramified $\Gamma_{K}$-action) via the quasi-finite flat $\OK$-group $A[n]$. Rather than to work directly with $A$, we will work with $A^{\circ}$, the Mumford degeneration data formed from this, and Grothendieck's monodromy pairing.

The theory of degenerations developed by Mumford and Faltings-Chai is exposed in great detail in [25]. We give only a summary of background we will need, omitting many technical issues relating to, for example, independence of certain choices that are made in order to define the functors $F$ and $M$ below. We note, as Lan does, that most of the results we cite can be found in the older works of Raynaud [32], [30], [12] and [5].  Since we only work with degenerations over the ring $\OK$, we could have exclusively used the rigid analytic theory. We use the more general framework, in part, because Lan's text is quite self contained and is freely available on his website.

The Raynaud extension attached to $A_{K}$ is obtained by successively lifting (<ref>) (in the case $U_{\kappa}=0$) to the rings $\sO_{K}/\fm_{K}^{i+1}$ or all $i\geq 0$, and applying Grothendieck's result that the resulting semiabelian formal $\OK$-group scheme is algebraizable (see [25], 3.3.3.6 and 3.3.3.9), producing the Raynaud extension[Letting $G = A^{\circ}$, Lan uses the notation $G^{\natural}$. This is what Grothendieck would call $G^{\natural\circ}$, reserving $G^{\natural}$ for an associated $G^{\natural\circ}$-torsor over the component group $\Phi_{A_{K}}$. To avoid confusion, since we will study the component group, we use the notation of Faltings-Chai, though we adopt the abuse of writing $\widetilde A$ for what should be called $\widetilde{A^{\circ}}$.] $\widetilde A$. As just described, this fits into a short exact sequence

\begin{align}
\label{tilde-A-ses}
\ses{T}{}{\widetilde A}{}{B}
\end{align}

with $B$ an abelian $\OK$-scheme and $T$ an $\OK$-torus. Let $Y^{D}$ be the character group of $T$. Replacing $A_{K}$ by its dual in the above discussion gives the dual Raynaud extension $\widetilde A^{D}$, which is an extension of the dual $B^{D}$ of $B$ by a torus $T^{D}$. Let $Y$ be the character group of $T^{D}$, a twisted constant group which becomes free of finite rank after finite étale base change on $\OK$. From the rigid analytic perspective, $A_{K}$ can be recovered as the cokernel of the analytification of a homomorphism of $K$-group schemes $Y_{K}\to \widetilde A_{K} := (\widetilde A)_{K}$. An important, concrete consequence (known in a different guise since [15], at the latest) is that $A_{K}[n]$ gives a class in $\Ext(Y_{K}/nY_{K}, \widetilde A_{K}[n])$. It was shown in [30] (in the case $B_{\kappa}=0$) and [12] (in the general case) that construction of $A_{K}$ from $Y_{K}\to \widetilde A_{K}$ and of a resulting extension structure on $A_{K}[n]$ can be accomplished using only algebraic constructions which are valid for degenerations over rings more general than $\OK$. For this, we introduce a category of degeneration data.

By [25] 3.1.5.1, the $T$-torsor (resp., $T^{D}$-torsor) $\widetilde A$ (resp., $\widetilde A^{D}$) is the same information as a homomorphism $c^{D}:Y \to B$ (resp., $c: Y^{D}\to B^{D}$). The pullback of the Poincaré sheaf $\sP_{B}^{-1}$ on $B\times_{\OK} B^{D}$ to $Y\times_{\OK}Y^{D}$ under $c^{D} \times c$ need not admit a trivialization, but its restriction to the generic fiber must.

Let $\DEG(\OK,\fm_{K})$ denote the category of $\OK$-schemes whose objects are the semiabelian $\OK$-schemes $G$ such that $G_{K}$ is an abelian variety.

Let $\DD(\OK,\fm_{K})$ denote the category of triples $(B, Y, Y^{D}, c, c^{D}, \tau)$, such that

* $B$ is a semiabelian $\OK$-scheme;

* $Y$ and $Y^{D}$ are twisted constant $\OK$-groups  which are isomorphic to the constant group $\Z^{t}$ after a finite étale base change on $\OK$, for some $t\in \Z_{\geq 0}$;

* $c^{D}:Y\to B$ and $c: Y^{D}\to B^{D}$ are homomorphisms; and

* $\tau$ is a trivialization of the $\GmK$-biextension $(c^{D}\times c)^{*}\sP_{B,K}^{-1}$;

and such that $\tau$ satisfies the following positivity condition:

$(\ast)$ there exists an injection with finite cokernel $\phi:Y\to Y^{D}$ such that, for all $y\in Y_{K}$, the section $\tau(y,\phi(y))\in (c^{D}(y)\times c(\phi(y)))^{*}\sP_{B,K}^{\otimes -1}$ extends to a section of $(c^{D}(y)\times c(\phi(y)))^{*}\sP_{B}^{\otimes -1}$, and for $y\neq 0$, the morphism $(c^{D}(y)\times c(\phi(y)))^{*}\sP_{B}^{\otimes}\to \sO_{\Spec \OK}$ induced by $\tau(y, \chi)$ factors through $\fm_{K}$.

By [25] 4.2.1.7, such a $\tau$ determines and is determined by a period homomorphism, i.e., a homomorphism $\iota_{K}:Y_{K}\to \widetilde A_{K}$ lifting $c^{D}:Y_{K}\to B_{K}$ under the surjection $\widetilde A\to B$ of (<ref>).

Let $\tau$ be as in <Ref>. Indexing the components of $Y_{K}\times_{K}Y^{D}_{K}$ by $K$-points $(y,\chi)$, the datum of $\tau$ is equivalent to a collection of sections $\tau(y,\chi) \in (c^{D}(y)\times c(\chi))^{*}\sP_{B, K}^{-1}$ over the components of $Y_{K}\times_{K}Y^{D}_{K}$. The image of

\[
(c^{D}(y)\times c(\chi))^{*} \sP_{B}
\subset (c^{D}(y)\times c(\chi))^{*}\sP_{B, K}
\overset{\tau(y,\chi)}\longisoto \sO_{\Spec K}
\]

is the same information as an $\OK$-submodule

\begin{align}
\label{eq:I-y-chi-def}
I_{y,\chi}\subset K
\end{align}

Writing $I_{y,\chi} = \fm_{K}^{\mu(y,\chi)}$
defines a canonical map $\mu: Y(\Kbar) \times  Y^{D}(\Kbar)\longto \Z$ which is $\Gamma_{K}$-equivariant. Since $\tau$ is a trivialization of $\sP_{B,K}^{-1}$ as a $\GmK$-biextension, one obtains a bilinear morphism

\begin{equation}
\label{eq:mu-def}
\mu: Y \times_{\OK} X\longto \Z_{\OK}.
\end{equation}

By studying enrichments of the categories $\DEG(\OK, \fm_{K})$ and $\DD(\OK, \fm_{K})$, it can be shown ([25] 4.5.5.5) that there are inverse equivalences of categories
\[F(\OK,\fm_{K}):
\DEG(\OK, \fm_{K})\longto \DD(\OK, \fm_{K})\]

\[M(\OK,\fm_{K}):
\DD(\OK, \fm_{K})\longto \DEG(\OK, \fm_{K}).\]

In order to apply results of [34] and [4], we  will view $F(\OK,\fm_{K})(A^{\circ})$ as a log 1-motive. We introduce this notion now.

Let $S$ be a scheme. An $S$-1-motive is a two-term complex $\sM = [\iota:Y\to C]$ in degrees $-1,0$ of commutative $S$-group schemes such that

* $Y$ is a twisted constant group which is étale-locally modeled on a free abelian group of finite rank;

* $C$ is an extension of an abelian $S$-scheme by an $S$-torus;

* $\iota$ is an homomorphism of $S$-groups.

Suppose now $S = \Spec \OK$. A log 1-motive $\sQ$ over $\OK$ is a triple $(Y, C,\iota_{K})$ where $Y$ and $C$ are commutative group schemes over $S$ satisfying (i) and (ii) above and $\iota_{K}$ is a homomorphism $\iota_{K}:Y_{K}\to C_{K}$. In this case $\sQ_{K} := [\iota_{K}:Y_{K}\to C_{K}]$ is a $K$-1-motive.

The sort of semiabelian $S$-schemes that can give a term in a 1-motive over $S$ have constant toric rank. In particular, if $A$ is the Néron model of an abelian $K$-variety $A_{K}$ with bad reduction, then $A^{\circ}$ is not a term in any 1-motive over $\OK$.

Let $S^{\log}$ be the log scheme obtained by equipping $\Spec \OK$ with its canonical (divisorial) log structure. The definition of a log 1-motive does not involve any log schemes. However, from a log 1-motive it is possible to build a complex of logarithmic group schemes defined over $S^{\log}$ by taking a “logarithmic enhancement” of the semiabelian scheme $C$. For more on this, see [23] 1.4.

A fundamental example of a log 1-motive is given in (<ref>).

We are interested in the “torsion” in the $K$-1-motives underlying log 1-motives, which is defined in the following proposition.

Let $S$ be a scheme. Let $\sM = [\iota:Y\to C]$ be a 1-motive over $S$. Each positive integer $n$ gives a morphism of complexes of $S$-groups $[n]:\sM\to \sM$ which is multiplication by $n$ on each term. Let $c(\sM, n)$ be the cone of this morphism, and let $\sM[n]:= H^{-1}(c(\sM, n))$. Then $\sM[n]$ is a finite locally free $S$-group scheme, and $(\sM[p^{m}])_{m\geq 1}$ forms a $p$-divisible group $\sM[p^{\oo}]$.

Continuing the notation of the proposition, in local sections we have

\begin{align}
\label{eq:1-mot-n-tors-isom}
\sM[n] = \frac{\{
(g,y)\in C \times Y:\: ng = -\iota(y)
\}}{\{
(-\iota(y), ny) :\: y\in Y
\}}.
\end{align}

By divisibility of semiabelian schemes ([6], page 180),
the natural homomorphism $\sM[n] \to Y/nY$ is surjective. This fits into a canonical short exact sequence

\begin{align}
\label{eq:eta-M-n}
\tikzses{\eta(\sM, n):}{C[n]}{}{\sM[n]}{}{Y/nY}.
\end{align}

Continue notation, so that $A_{K}$ is an abelian $K$-variety with semistable reduction, and $A$ is its Néron model. Let

\[
\sQ_{K}:= [Y_{K}\overset{\iota_{K}}\to \widetilde A_{K}]
\]

be the log 1-motive attached to $A_{K}$ via the functor $F(\OK, \fm_{K})$.
There is a canonical isomorphism $\sQ_{K}[n] \isoto A_{K}[n]$. Therefore, the short exact sequence $\eta(\sQ_{K},n)$ gives rise to canonical short exact sequence

\begin{align*}
\tikzses{\eta(A_{K}, n):}{\widetilde A_{K}[n]}{}{A_{K}[n]}{}{Y_{K}/nY_{K}}.
\end{align*}

In this case, (<ref>) reads

\begin{align*}
\sQ_{K}[n] = \frac{\{
(g,y)\in \widetilde A_{K} \times Y_{K}:\: ng = -\iota_{K}(y)
\}}{\{
(-\iota_{K}(y), ny) :\: y\in Y_{K}
\}}.
\end{align*}

Granting rigid analytic uniformization, comparing this with $\coker(\iota_{K}^{\an})[n]$ makes it clear that there is a canonical isomorphism $\sQ_{K}[n] \simeq A_{K}[n]$. We check this isomorphism holds in a different way.

Mumford's construction involves a so-called “partial compactification”, an $\OK$-scheme $P^{\natural}$ containing $\widetilde A$, and which carries an action of $Y$ compatible with the action of $Y_{K}$ on $\widetilde A_{K}$ through $\iota_{K}$. Assume $Y$ is constant; the general case follows by finite étale descent. For $y\in Y_{K}$, let $S_{y}$ denote the action of the connected component of $Y$ containing $y$. Let $\widetilde A^{*} := \bigcup_{y\in Y_{K}}S_{y}(\widetilde A)$. For $y\in Y_{K}$, define $\sigma_{y}:\Spec \OK\to \widetilde A$ to be the morphism determined by the condition that its restriction to $\Spec K$ factors as $\iota_{K}(y):\Spec K\to \widetilde A_{K} \subset \widetilde A$. Let ${}^{(n)}Z_{y}:= \Spec \OK\times_{\sigma_{y}, \widetilde A^{*}, [n]}\widetilde A^{*}$ and let $Z_{n} := \bigsqcup_{y\in Y_{K}/nY_{K}} {}^{(n)}Z_{y}$. It is shown in [25] that $Z_{n}$ is an $\OK$-group scheme and (Theorem 4.5.3.10) that there is a canonical isomorphism $Z_{n}\to A^{\circ}[n]$.

Since the action of $Y$ on $P^{\natural}$ preserves the generic fiber of $\widetilde A$, we have $(\widetilde A^{*})_{K} = \widetilde A_{K}$. Then ${}^{(n)}Z_{y}(\Kbar) = \{g\in \widetilde A_{K}:\: ng = \iota_{K}(y)\}$ and $Z_{n}(\Kbar) = \{g\in \widetilde A_{K}:\: ng \in \iota_{K}(Y_{K})\}$. There is a morphism $\sQ_{K}[n]\to Z_{n,K}$ given on geometric points by projecting the class of $(g,y)$ in $\sQ_{K}[n](\Kbar)$ onto the first coordinate. This is clearly a surjection. To see that this is an injection, it suffices to show that $\iota_{K}$ is injective. This fact is noted in [25], proof of 4.5.5.2, but we give an alternative argument in <Ref>.

The final claim about the existence of $\eta(A_{K},n)$ is clear. We note that the existence of this short exact sequence is also recorded in [25] 4.5.3.12. The compatibility with the sequence there and our sequence is immediately verified from Lan's proof, but we won't need this.

With notation as in <Ref>, $\ker \iota_{K} = 0$.

We treat the case where $Y$ is split. The general case follows by étale descent. Let $y\in Y(\OK)$. Using $\mid-\mid$ to represent total spaces of invertible sheaves, for each $\chi\in Y^{D}(\OK)$, define a line bundle by pullback as in

\begin{equation*}
\begin{tikzcd}
\mid\sO_{\chi}(c^{D}(y))\ \mid \ar[d]\ar[r] &
\mid\sO_{\chi}\mid \ar[d]\ar[r] &
\mid\sP_{B}\mid \ar[d]
\\
\Spec \OK \ar[r, "(c^{D}(y){, } \id)"] &
B \times_{\OK} \Spec \OK \ar[r,"\id\times c(\chi)"] &
B \times_{\OK}B^{D}
\end{tikzcd}
\end{equation*}

Then $\sO_{\widetilde A} = \bigoplus_{\chi}\sO_{\chi}$ as an $\sO_{B}$-algebra by [25] 4.5.2.3. By the proof of [25] 3.1.4.4, the identity section of $\widetilde A$ is characterized by the condition that, for each $\chi$, it restricts to the generic fiber of the morphism $e_{\chi,K}^{*}:\sO_{\chi}\to e_{B}^{*}\sO_{\chi}\simeq \sO_{\Spec \OK}$, where the isomorphism is the rigidification arising from the natural rigidification of $\sP_{B}$ along the map $B\to B\times_{\OK} B^{D}$. Each $\tau(y,\chi)\in \sO_{\chi}(c^{D}(y))^{\otimes -1}$ is a morphism $\sO_{\chi}(c^{D}(y))\to \sO_{\Spec K}$. Using the natural maps $\sO_{\chi}\to \sO_{\chi}(c^{D}(y))$ of the above diagram, the map $\iota_{K}(y)$ is defined to be the composition

\[
\bigoplus_{\chi}\sO_{\chi}
\longto \bigoplus_{\chi}\sO_{\chi}(c^{D}(y))
\overset{\sum_{\chi}\tau(y, \chi)}\longto \sO_{\Spec K}.
\]

Therefore, if $y\in \ker\iota_{K}$, then for all $\chi$ we have a commuting diagram

\begin{equation*}
\begin{tikzcd}
\sO_{\chi, \Spec K} \ar[r]
\ar[rr, bend left=20, "\tau(y{,} \chi)"] &
e_{B}^{*}\sO_{\chi, \Spec K} \ar[r,"\sim"] &
\sO_{\Spec K}
\\
\sO_{\chi} \ar[r] \ar[u] &
e_{B}^{*}\sO_{\chi} \ar[r,"\sim"] \ar[u] &
\sO_{\Spec \OK} \ar[u]
\end{tikzcd}
\end{equation*}

Since the diagram commutes, the $\OK$-module $I_{y,\chi}$ defined in (<ref>) satisfies $I_{y,\chi} = \OK$. In particular, $\tau(y,\chi)$ does not factor through the ideal sheaf $\fm_{K}$. Taking $\phi$ as in <Ref> and setting $\chi = \phi(y)$, the positivity condition of the definition implies that $y = 0$.

In the case where $A_{K} = E_{q,K}$ is the $q$-Tate curve over $K$, the log 1-motive $\sQ_{K} = \sT_{q,K}$ is defined in (<ref>) and $\eta(E_{q, K},n)$ coincides with the short exact sequence (<ref>).

§.§ Raynaud decomposition and monodromy

As we shall see, $\eta(A_{K}, n)$ exhibits $A_{K}$ as an extension of a $0$-crystalline representation by a $1$-crystalline representation, hence as an extension of generic fibers of finite flat $\OK$-group schemes. If $A_{K}$ has bad reduction, then Coleman-Iovita's monodromy criterion for good reduction (<Ref>) coupled with work of Liu (<Ref>) shows that, for $n\gg 0$ a power of $p$, $A_{K}[n]$ is not the generic fiber of a finite flat $\OK$-group. This is related to the fact that, though the individual terms in $\sQ_{K}$ do extend to group schemes over $\OK$, the morphism $\iota_{K}$ does not, in general, extend to a morphism of $\OK$-group schemes. As noted in [34] 2.4.1, the failure of $\iota_{K}$ to extend over $\OK$ is due to nontriviality of the pullback $(c^{D}\times c)^{*}\sP_{B}$ as a $\Gm$-torsor on the $\OK$-scheme $Y\times_{\OK}Y^{D}$. This nontriviality is, in turn, captured by the nontriviality of the monodromy pairing $\mu$. Raynaud's decomposition theorem trivializes the monodromy pairing to get an $\OK$-1-motive, and records the lost information in a $K$-1-motive.

Let $\sQ_{K} = F(\OK,\fm_{K})(A^{\circ}) = [Y_{K}\overset{\iota_{K}}\longto \widetilde A_{K}]$ be as in <Ref>. Let $\varpi$ be a uniformizer of $\OK$. Then there homomorphisms $\iota_{K,\varpi}^{1},\,\iota_{K,\varpi}^{2}:Y_{K}\to \widetilde A_{K}$ such that, using addition in the common target $\widetilde A_{K}$, we have

\begin{align*}
\iota_{K} = \iota_{K,\varpi}^{1} + \iota_{K,\varpi}^{2}.
\end{align*}

These homomorphisms satisfy

* $\iota_{K,\varpi}^{1}$ is the generic fiber of an $\OK$-1-motive $\sQo:= [Y\overset{\iota^{1}_{\varpi}}\longto \widetilde A]$; and

* $\iota_{K,\varpi}^{2}$ factors through the inclusion $ T_{K}\to \widetilde A_{K}$.

Let $\sQ_{K,\varpi}^{1}$ and $\sQ_{K,\varpi}^{2}$ be the $K$-1-motives attached to $\iota_{K,\varpi}^{1}$ and $\iota_{K,\varpi}^{2}$, respectively. Then we have an equality in $\Ext(Y_{K}/nY_{K}, \widetilde A_{K}[n])$

\begin{equation}
\label{eq:raynaud-baer-decomp}
\eta(\sQ_{K}, n) = \eta(\sQ_{K,\varpi}^{1}, n) + \eta(\sQ_{K,\varpi}^{2}, n).
\end{equation}

In the case of a Tate curve, this is the decomposition of (<ref>).

Since $\eta(\sQ_{K,\varpi}^{1}, n)$ is the generic fiber of a sequence of finite flat $\OK$-group schemes $\eta(\sQ_{K}^{1}, n)$, roughly speaking, (<ref>) describes how to isolate the non-1-crystallinity of $A_{K}[n]$ in the term $\eta(\sQ_{K,\varpi}^{2},n)$. In the next section, we will develop some of Kato's theory of finite flat group scheme with monodromy, which develops this perspective. We now recall the relationship between the monodromy pairing and the component group (valid only under the semistability assumption) first articulated in [15].

Recall ((<ref>) above) that we have a bilinear morphism

\[
\mu: Y\times_{\OK} Y^{D}\longto \Z_{\OK}.
\]

Let $Y^{D,\vee} := \Hom(Y^{D}, \Z_{\OK})$. This is a twisted constant $\OK$-group. We define a homomorphism

\begin{align}
\label{eq:mu1}
\mu_{Y}: Y\longto Y^{D,\vee}
\end{align}

on local sections by $\mu(z):\chi\mapsto \mu(z,\chi)$. Twisting as in <Ref> gives a homomorphism of finite $\OK$-groups of multiplicative type

\begin{align}
\label{eq:twist-of-muY}
Y/nY(1) \longto Y^{D,\vee}/nY^{D,\vee}(1).
\end{align}

Recall that $Y^{D}$ is defined so that the torus $T$ in $\widetilde A$ has the form $T = \Hom(Y^{D},\Gm)$. Using the isomorphisms of abelian sheaves

\[
T = \underline{\Hom}(Y^{D},\Z)\otimes \Gm = Y^{D,\vee}\otimes\Gm
\]

and writing $T[n] = \ker(Y ^{D,\vee}\otimes \Gm \overset{n}\to Y^{D,\vee}\otimes \Gm)$, we see that

\begin{align}
\label{eq:T-n-as-quotient-twist}
= Y^{D,\vee}/nY^{D,\vee}\otimes \mu_{n}
\simeq Y^{D,\vee}\otimes \mu_{n}
= T[n].
\end{align}

Composing (<ref>) and the identification (<ref>) with the canonical closed immersion $T[n]\hooklongrightarrow \widetilde A[n]$ gives a homomorphism.

\begin{align}
\label{eq:nu-n-def}
\nu_{n}: Y/nY(1) \longto \widetilde A[n].
\end{align}

In the case of a Tate curve, where $Y/nY = \Z_{\OK}/n\Z_{\OK}$ and $\widetilde A = \GmOK$, the morphism $\nu_{n}$ is the Cartier dual of the morphism $N_{n}^{*}$ defined at the end of <Ref>.

The following result is needed for the proof of <Ref>.

There is a canonical isomorphism

\[
\ker \nu_{n}(-1) \simeq \Phi_{A_{K}}[n].
\]

By [12] Corollary 8.2 (see also [15], 11.5), there is an isomorphism $\Phi_{A_{K}}(\Kbar)\simeq \coker \mu_{Y}$, where $\mu_{Y}: Y\to Y^{D\vee}$ is the morphism of (<ref>). The groups $Y_{K}(\Kbar)$, $Y^{D\vee}_{K}(\Kbar)$ are free of rank $\rk T = \rk T^{D}$. Since the homomorphism $\mu_{Y}: Y_{K}(\Kbar)\to Y_{K}^{D\vee}(\Kbar)$ has finite cokernel $\Phi_{A_{K}}(\Kbar)$, it is injective, and it becomes an isomorphism after inverting $c := \#\Phi_{A_{K}}(\Kbar)$.  Therefore, $\mu_{Y}$ gives rise to a surjection $(1/c)Y(\Kbar)\to Y^{D\vee}(\Kbar)$. Post-composing with the quotient map $Y^{D\vee}(\Kbar) \to Y^{D\vee}(\Kbar)/cY^{D\vee}(\Kbar)$, we obtain a surjection
\[\mu_{Y, c}(-1):(1/c)Y(\Kbar)/Y(\Kbar) \to Y^{D\vee}(\Kbar)/cY^{D\vee}(\Kbar).\]
Consider the diagram of $\Gamma_{K}$-equivariant homomorphisms of abelian groups

\[
\begin{tikzcd}
& Y_{K}(\Kbar)\ar[d,"\mu_{Y}"] \ar[r]
& \frac{1}{c} Y_{K}(\Kbar) \ar[d, "c\mu_{Y}"] \ar[r]
& \frac{1}{c}Y_{K}(\Kbar)/Y_{K}(\Kbar)
\ar[d, "\mu_{Y, c}(-1)"] \ar[r]
& 0
\\
0 \ar[r]
&  Y_{K}^{D\vee}(\Kbar) \ar[r, "c"]
& Y_{K}^{D\vee}(\Kbar) \ar[r]
& Y_{K}^{D\vee}(\Kbar)/cY_{K}^{D\vee}(\Kbar)
\end{tikzcd}.
\]

As noted above, the vertical map $\mu_{Y}$ is injective with cokernel $\Phi_{A_{K}}(\Kbar)$. Therefore, the vertical map $c\mu_{Y}$ also is injective with cokernel $\Phi_{A_{K}}(\Kbar)$. The snake lemma gives an exact sequence

\[
0 \longto 0 \longto \ker\mu_{Y, c}(-1) \longto \Phi_{A_{K}}(\Kbar)
\overset{c}\longto \Phi_{A_{K}}(\Kbar)\longto 0.
\]

Since $c$ is the order of $\Phi_{A_{K}}(\Kbar)$, we have $\ker\mu_{Y, c}(-1) = \Phi_{A_{K}}(\Kbar )$. The proposition follows upon passing to the $n$-torsion subgroups. Specifically, restricting $\mu_{Y,c}(-1)$ to $\frac{1}{n}Y_{K}(\Kbar)/Y_{K}(\Kbar)$, we obtain a morphism whose image is killed by $n$. This induces a morphism $(1/n)Y/Y \to Y^{D\vee}/nY^{D\vee}$. Unwinding the definitions, and making the identification $(1/n)Y/Y = Y/nY$, we see that this is exactly the morphism $\nu_{n}(-1)$.

§ MAXIMAL 1-CRYSTALLINE SUBMODULES

In this section we develop results from integral and $p$-adic Hodge theory, and we define the functor $\Crys_{r}$ of maximal $r$-crystalline submodules and its continuous derived functors. Beyond some innovations in presentation, including our use of the result <Ref>, which was not available to Kim and Marshall when they wrote [24], we have added little to their work, which we recommend (along with Marshall's more expansive thesis [28]) as a reference.

Throughout the section, we adopt the notation of <Ref>.

§.§ Categories of representations and fully faithfulness theorems

To define the continuous derived functors $R^{i}\Crys_{r}$ of the maximal $r$-crystalline sub-representation functor as in [24] and [28], we first define various categories of $\Gamma_{K}$-modules that can be constructed from $\Zp$-lattices in crystalline $\Qp$-representations. We do this now, but we  do not review Fontaine's theory of period rings, which are described in many sources (for example, [3]). We normalize Hodge-Tate weights so that the $p$-adic cyclotomic character has 1 as its only Hodge-Tate weight.

Let $r\in \Z_{\geq 0}$. Let $\ast\in\{\dR, \stbl, \crys\}$.

* Let $\ModZpGK$ denote the category of topological $\Zp[\Gamma_{K}]$-modules, i.e., the category of topological spaces $M$ equipped with a continuous morphism $\Zp[\Gamma_{K}]\times M\to M$ giving $M$ the structure of a $\Zp[\Gamma_{K}]$-module.

* Let $\RepZp$ denote the full subcategory of $\ModZpGK$ consisting of the objects that are finite and free over $\Zp$ and that are equipped with the $p$-adic topology.

* Let $\ModtorsZpGK$ denote the full subcategory of $\ModZpGK$ consisting of torsion objects with the discrete topology.

* Let $\RepastrQp$ denote the full subcategory of the $B_{\ast}$-admissible representation of $\Gamma_{K}$ on finite-dimensional $\Qp$-vector spaces having all Hodge-Tate weights in the interval $[0,r]$.

* Let $\Repastrfree$ denote the full subcategory of $\RepZp$ consisting of those objects $L$ such that $L[1/p] := L\otimes_{\Zp} \Qp$ is an object of $\RepastrQp$.

* Let $\Repastrtors$ denote the full subcategory of $\ModtorsZpGK$ consisting of those objects $V$ such that there exists an object $L$ of $\Repastrfree$ and a surjective morphism $L \to  V$ in the category $\ModZpGK$.

* Let $\Repastrft$ denote the full subcategory of $\ModZpGK$ consisting of those $M$ such that $M$ is finite type over $\Zp$ and $M$ is isomorphic to the inverse limit of a system of objects of $\Repastrtors$.

Any unramified representation is crystalline with $0$ as its only Hodge-Tate weight. In particular, the trivial character $\Gamma_{K}\to \{\id\}\subset \GL_{1}(\Qp)$ is crystalline. Conversely, any de Rham representation with $0$ as its only Hodge-Tate weight is unramified (see [3] 8.3.5).

The $p$-adic cyclotomic character $\Zp(1)$ is 1-crystalline, hence each of its torsion quotients, $\Z/p^{m}\Z(1)$ is 1-crystalline. In general, any crystalline $\Qp$-character of $\Gamma_{K}$ is the tensor product of an unramified $\Qp$-character with an integer power of $\chi$. In particular, a character of $\Gamma_{K}$ is crystalline if and only if it is de Rham.

Let $A_{K}$ be an abelian variety over $K$. Then
\[V_{p}A_{K} := T_{p}A_{K}\otimes_{\Zp}\Qp\]

is a $2\dim (A_{K})$-dimensional de Rham representation with Hodge-Tate weights in $[0,1]$. When $A_{K}$ has semistable reduction, $V_{p}A_{K}$ is semistable; this fact, which is well known, follow from <Ref> and <Ref>. When $A_{K}$ has good reduction, $V_{p}A_{K}$  is crystalline; see <Ref>.

The following classification result summarizes work of many people completed over the course of fifty years.

One has the following relationships between $1$-crystalline $\Gamma_{K}$-modules and finite flat $\OK$-groups. Let $\pdiv_{\OK}$ denote the category of $p$-divisible groups over $\OK$.

* The Tate module of any object of $\pdiv_{\OK}$ is an object of $\RepcrysoneZp$.

* The generic fiber of any object of $\fin^{p}_{\OK}$ is an object of $\Repcrysonetors$.

* Every object of $\RepcrysoneZp$ is the Tate module an object of $\pdiv_{\OK}$.

* Every object of $\Repcrysonetors$ is the generic fiber of an object of $\fin^{p}_{\OK}$.

Part (i) was proved by Tate in different terms in [36], Section 4, Corollary 2. Part (ii) is attributed to Oort in [3], but it also follows from Raynaud's stronger result that every object of $\fin^{p}_{\OK}$ is the kernel of an isogeny of abelian $\OK$-schemes (see [2], 3.1.1). Parts (iii) and (iv) were proven by Kisin in [21] for $p>2$ and for $p=2$ in some cases (see also [22]), and was proved unconditionally by Wansu Kim in the case for $p=2$. See [20] 2.3 for an explanation of Kisin's work, and 4.1 for the extension to the case $p=2$.

We are primarily concerned with objects of the categories $\Repastrfree$ and $\Repastrtors$. A key relationship between these was conjectured by Breuil and proved by Liu.

Let $r\in \Z_{\geq 1}$. Let $\ast\in \{\stbl, \crys\}$. Let $L$ be an object of $\RepZp$. Then $L$ is an object of $\Repcrysrfree$ (resp., $\Repsstrfree$) if and only if, for each $m\geq 1$, $L/p^{m}L$ is an object of $\Repcrysrtors$ (resp.  $\Repsstrtors$).

Let $L\in \Repsstrfree$ and assume $L$ is not crystalline. By <Ref>, we know that $L/p^{m}L$ is not an object of $\Repcrysrtors$ for all $m$ sufficiently large. However, it can happen that $L/p^{m}L$ is $r$-crystalline for small values of $m$, as we see in <Ref>.

Recall from <Ref> the subgroup $\Gamma_{\varpi^{\flat}}$ of $\Gamma_{K}$. Let $\Rep_{\free}(\Gamma_{\varpi^{\flat}})$ (resp., $\Rep_{\tors}(\Gamma_{\varpi^{\flat}})$) denote the category of finite-type free (resp., torsion) $\Zp$-modules equipped with an action of $\Gamma_{\varpi^{\flat}}$. The following result is essential in the study of crystalline $\Qp$-representations.

For any $r\in \Z_{\geq 0}$, the restriction functor

\[
\Res^{\Gamma_{K}}_{\Gamma_{\varpi^{\flat}}}:
\Repcrysrfree
\longto
\Rep_{\free}(\Gamma_{\varpi^{\flat}})
\]

is fully faithful.

In general, the analogous statement for torsion $\Gamma_{K}$-modules fails for reasons which are related to the bound in Raynaud's result recorded in

Let $r \in \Z_{\geq 0}$ and let $e$ be the ramification index of $K/\Qp$. Suppose $e(r-1)<p-1$. Then the restriction functor

\[
\Res^{\Gamma_{K}}_{\Gamma_{\varpi^{\flat}}}: \Repcrysrtors \longto \Reptors(\Gamma_{\varpi^{\flat}})
\]

is fully faithful.

The utility of this result was illustrated in <Ref>.

In <Ref> and <Ref> we establish some basic closure properties of the categories of $\Gamma_{K}$-modules defined in <Ref>. We have given our own expanded proofs, but everything in these results was known already to Marshall and Kim.

Let $\sC$ be any of the categories defined in  <Ref>. By definition, $\sC$ is a full subcategory of $\ModZpGK$.

* A $\ModZpGK$-subobject of an object of $\sC$ is also an object of $\sC$.

* Suppose $\sC$ is one of the categories containing a nontrivial torsion object.  If $M$ is a $\ModZpGK$-quotient of an object of $\sC$, then $M$ is an object of $\sC$.

* If $M$ is a $\ModZpGK$-direct sum of finitely-many objects of $\sC$, then $M$ is an object of $\sC$.

We prove the statements (a)-(c) one category at time.
For $\ModZpGK$, $\RepZp$ and $\ModtorsZpGK$, the claimed statements are tautological or trivial to verify. For the categories $\RepastrQp$, the claims are standard (see, e.g., [3] 5.2.1). For $\Repastrfree$, statement (a) follows from the results for the preceding categories, using the fact that $\Zp$-submodules of objects of $\Repastrfree$ are themselves free. We don't claim statement (b) for this category, and statement (c) follows from the closure properties of $\RepastrQp$.

We now consider the category $\Repastrtors$. Statement (c) is trivial to verify. We prove (a). Let $V$ be an object of $\Repastrtors$, and choose some surjection $\pi: L \to V$ with $L$ in $\Repastrfree$. Suppose $V'\subset V$ is a $\Zp[\Gamma_{K}]$-submodule and let $L' = \pi^{-1}(V')\subset L$. Then $L'$ is a $\Zp[\Gamma_{K}]$-submodule of $L$. Equipping $L'$ with the induced topology, we have that $L'$ is an object of $\RepZp$. Since $L'[1/p]\subset L[1/p]$, we see that $L'$ is an object of $\Repastrfree$. This shows that $V'$, being a quotient of $L'$, is an object of $\Repastrtors$. We prove (b) for $\Repastrtors$. Adopt the notation of the previous paragraph. Let $U$ be an object of $\Repastrtors$ and suppose there is a surjection $V\to U$ of objects of $\ModZpGK$. Composing this with $\pi$ gives a map $\bar\pi: L \to U$. Arguing as before, we see that $L / \ker(\bar \pi) = U$ realizes $U$ as an object of $\Repastrtors$.

We now consider the category  $\Repastrft$. Note that the proof of <Ref> relies on the result we are presently proving, but only in the case of the category $\Repastrtors$, whose proof is already complete, Hence, to prove (a)-(b) for $\Repastrft$, we may use <Ref>. Write $M = M_{\free}\oplus M_{\tors}$ where $M_{\tors}$ is the maximal torsion submodule and $M_{\free}$ is torsion-free. By <Ref>, $M_{\free}$ and $M_{\tors}$ are, respectively, objects of $\Repastrfree$ and $\Repastrtors$.

We prove (a). Let $M'$ be a $\ModZpGK$-subobject of $M$. Then $M' = M'_{\free}\oplus M'_{\tors}$. By (a) for the categories $\Repastrfree$ and $\Repastrtors$, $M'_{\free}$ is an object of $\Repastrfree$ and $M'_{\tors}$ is an object of $\Repastrtors$. It is easy to see that $M'$ is an object of $\Repastrft$.

We prove (c). Suppose $M_{1}$ and $M_{2}$ are objects of $\Repastrft$, say with $M_{i} = \lim_{j\geq 1}M_{i,j}$ where $M_{i,j}$ is an object of $\Repastrtors$ for all $i,j$. Then $M_{1}\oplus M_{2}$ is the inverse limit of the natural inverse system formed by the modules $M_{1,j}\oplus M_{2,j}$. By (c) for the category $\Repastrtors$, $M_{1,j}\oplus M_{2,j}$ is an object of $\Repastrtors$, so we see that $M_{1}\oplus M_{2}$ is an object of $\Repastrft$.

We prove (b). First, let $L$ and $L'$ be objects of $\Repastrft$ that are free. Then, by <Ref>, $L$ and $L'$ are objects of $\Repastrfree$, in particular, they are each $p$-adically complete. By definition of $\Repastrtors$, $L/p^{m}L$ and $L'/p^{m}L'$ are objects of $\Repastrtors$. For each $m\in \Z_{\geq 1}$ we have a short exact sequence

\[\ses{L'/p^{m}L'}{}{L/p^{m}L}{}{L''_{m}}\]

with $L''_{m}$ an object of $\Repastrtors$ by the claim (c) for $\Repastrtors$. These short exact sequences are compatible with the quotient maps defining the inverse systems $(L'/p^{m}L')_{m\geq 1}$ and $(L'/p^{m}L')_{m\geq 1}$. The Mittag-Leffler condition is satisfied (see 03CA), so we have a short exact sequence

\[\ses{L'}{}{L}{}{\lim_{m\geq 1}L''_{m}}.\]

This has shown that $L/L'$ is an object of $\Repastrft$.

For the general case of (b), let $M$ be an object of $\Repastrft$ and let $M'$ be a subobject of $M$ in that category. Then

\[
= (M_{\free}/M'_{\free})\oplus(M_{\tors}/M'_{\tors}).
\]

We have just shown that $M_{\free}/M'_{\free}$ is an object of $\Repastrft$, and we saw above that $M_{\tors}/M'_{\tors}$ is an object of $\Repastrtors$, hence an object of $\Repastrft$.

Let $\ast\in \{\stbl, \crys\}$. Let $M$ be an object of $\Repastrft$ and let $M'$ be a $\ModZpGK$-subobject of $M$.

* If $M'$ is free, then $M'$ is an object of $\Repastrfree$.

* If $M'$ is torsion, then $M'$ is an object of $\Repastrtors$.

* Write $M = M_{\free}\oplus M_{\tors}$ for the decomposition of $M$ as a direct sum of a free $\Zp$-module and its maximal torsion $\Zp$-submodule. Then $M_{\free}$ (resp., $M_{\tors}$) is an object of $\Repastrfree$ (resp., $\Repastrtors$).

We may assume $M = \lim_{i\in \Z_{\geq 1}}M_{i}$ with $M_{i}$ in $\Repastrtors$.

Consider claim (i). Assume the subobject $M'\subset M$ is nonzero and free over $\Zp$. For each $i\in \bZ_{\geq 0}$, choose $k(i)\in \Z_{\geq 1}$ such that $p^{k(i)}M_{i}=0$. Therefore $p^{k(i)}M'\subset \ker(M\to M_{i})$, so $M'/p^{k(i)}M'\subset M_{i}$. Since $M_{i}$ is in $\Repastrtors$, <Ref> for $\Repastrtors$ implies that its quotient $M'/p^{k(i)}M'$ is in $\Repastrtors$. Since $M$ is not torsion, the $k(i)$ tend to infinity as $i$ tends to infinity. We see that in fact each $M'/p^{m}M'$ ($m\in \Z_{\geq 1}$) is an object of $\Repastrtors$. Since $M'$ is assumed $p$-adically complete and free over $\Zp$, <Ref> implies that $M'$ is an object of $\Repastrfree$.

We now prove (ii). By <Ref>, it suffices to build a finite direct sum of the $M_{i}$'s into which $M'$ injects. Using  for $\Repastrtors$, we will conclude that $M$ is in $\Repastrtors$. Since $M'$ is torsion and $M$ is of finite type over $\Zp$, $M'$ is a finite set, say $M' = \{m_{1}, \ldots, m_{d}\}.$ For $j\in \{1,\ldots, d\}$ , write $m_{j} = (m_{j,i})_{i\geq 1}\in M\subset \prod_{i\in \Z_{\geq 1}} M_{i}$. For each $j'\in \{1,\ldots, d\}$ with $j'\neq j$, choose an index $i(j;j')$ such that the image of $m_{j}-m_{j'}$ in $M_{i(j;j')}$ is nonzero. Define

\[
:= \{i(j,j')
:\: j'\in \{1,\ldots, d\},\, j'\neq j\},
\qquad I := \coprod_{j = 1}^{d} I(j).
\]

By construction, the natural map $M\to \bigoplus_{\alpha\in I} M_{\alpha}$ is injective. This proves (ii).

Claim (iii) follows from (i) and (ii) using the decomposition $M = M_{\free}\oplus M_{\tors}$.

Our proof of the key result <Ref> relies on <Ref> below. Before giving the result, we recall the construction of the Baer sum.

Let $\sA$ be an abelian category, and let $\eta^{1}$ and $\eta^{-1}$ be objects of $\sA$. Then the set $\Ext_{\sA}(\eta^{1},\eta^{-1})$ of isomorphism classes of Yoneda extensions of $\eta^{1}$ by $\eta^{-1}$ is an abelian group under Baer sum. The Baer sum of two classes

\begin{align*} \tikzses{\eta^{\bullet}_{1}:}{\eta^{-1}}{d^{0}_{1}}{\eta^{0}_{1}}{d^{1}_{1}}{\eta^{1}} \\
\tikzses{\eta^{\bullet}_{2}:}{\eta^{-1}}{d^{0}_{2}}{\eta^{0}_{2}}{d^{1}_{2}}{\eta^{1}}
\end{align*}

in $\Ext_{\Sh_{\Ab}(\sC)}(\eta^{1},\eta^{-1})$ is given explicitly by the extension

\begin{align*} \tikzses{\eta^{\bullet}_{1}+ \eta^{\bullet}_{2}:}{\eta^{-1}}{d^{0}_{+}}{\eta^{0}_{1}+ \eta^{0}_{2}}{d^{1}_{+}}{\eta^{1}}   \end{align*}

defined as follows. The sheaf $(\eta^{\bullet}_{1}+ \eta^{\bullet}_{2})^{0}$ is defined by

\begin{align}
\label{eq:eta-Baer-subquo-desc}
(\eta^{\bullet}_{1}+ \eta^{\bullet}_{2})^{0} := \frac{\eta^{0}_{1}\times_{\eta^{1}}\eta^{0}_{2}}{\im((d^{0}_{1}, -d^{0}_{2}): \eta^{-1}\to \eta^{0}_{1}\times_{\eta^{1}}\eta^{0}_{2})}.
\end{align}

The morphism $d^{0}_{+}: \eta^{-1}\to (\eta^{\bullet}_{1}+ \eta^{\bullet}_{2})^{0}$ is given in local sections, using $[-,-]$ to represent classes of pairs, by

\begin{align}
\label{eq:inj-into-baer} \twoardef{d^{0}_{+}}{\eta^{-1}}{(\eta^{\bullet}_{1}+ \eta^{\bullet}_{2})^{0}}{a}{
[d^{0}_{1}(a), 0] = [0, d^{0}_{2}(a)].}
\end{align}

The morphism $d^{1}_{+}:  \eta^{0}_{1}+ \eta^{0}_{2}\to \eta^{1}$ is given on local sections by

\begin{align}
\label{eq:surj-from-baer}
\twoardef{d^{1}_{+}}{(\eta^{\bullet}_{1}+ \eta^{\bullet}_{2})^{0}}{\eta^{1}}{[e_{1}, e_{2}]}{
d^{1}_{1}(e_{1}) = d^{1}_{2}(e_{2}).}
\end{align}

Let $\ast\in \{\dR, \stbl, \crys\}$. Suppose we have two short exact sequences

\begin{align*}
\tikzses{\eta(V_{i})}{V'}{\alpha_{i}}{V_{i}}{\beta_{i}}{V''} \qquad i=1,2
\end{align*}

of objects of $\ModtorsZpGK$. Let $V_{+}$ be the middle term in the Baer sum $\eta(V_{1}) + \eta(V_{2})$

* If $V_{1}$ and $V_{2}$ are objects of $\Repastrtors$, then so is $V_{+}$.

* If any two of $V_{1}$, $V_{2}$ and $V_{+}$ is an object of $\Repcrysrtors$, then every one of these is an object of $\Repcrysrtors$.

By <Ref>, $\Repastrtors$ is stable under finite direct sums and passage to subquotients. Therefore, since $U_{1}$ and $U_{2}$ are visibly $\Gamma_{K}$-stable inside of $V_{1}\oplus V_{2}$, they are both objects of $\Repastrtors$, so $V_{+}$ is in $\Repastrtors$.

Taking a Baer difference, (ii) follows from (i).

§.§ Maximal crystalline subobjects and continuous derived functors

Following Marshall's thesis [28], we define functors $\Crys_{r}$ and construct their derived functors. As we shall see, the derived functors are constructed using inverse systems of torsion $\Gamma_{K}$-modules. As was noted in [28], this technique for defining derived functors has its origin in Jannsen's work [17]. We first define maximal $r$-crystalline subobjects and a category where such things lie. <Ref> contains nothing original.

For $M$ in $\ModZpGK$ and for $r\in \Z_{\geq 0}$, define $\Crys_{r}(M)$ to be the sum in $M$ of all $C\subset M$ such that $C$ is an object $\Repcrysrft$.

Let $\ModcrysrZpGK$ denote the full subcategory of $\ModZpGK$ consisting of those $M$ such that $M$ is isomorphic to a filtered colimit of objects of $\Repcrysrft$.

Let $M$ be an object of $\ModZpGK$. Let $r\in \Z_{\geq 0}$.

* The object $\Crys_{r}(M)$ of $\ModZpGK$ is an object of $\ModcrysrZpGK$.

* If $M$ is finite type and torsion with the discrete topology, then $\Crys_{r}(M)$ is an object of $\Repcrysrtors$.

* If $M$ is an object of $\RepZp$, then $\Crys_{r}(M)$ is an object of $\Repcrysrfree$.

* If $M$ is an object of $\RepQp$, then $\Crys_{r}(M)$ is an object of $\RepcrysrQp$.

Statement (i) is verified by noting that internal sums are filtered colimits. When $M$ is finite type, the $\Zp$-submodule $\Crys_{r}(M)\subset M$ is finite type and torsion with the discrete topology with $\Zp$-rank bounded by that of $M$. It is therefore equal to one of the summands defining $\Crys_{r}(M)$ as a sum of finite type submodules of $M$. If $M$ is torsion (resp., free), then $\Crys_{r}(M)$ is torsion (resp., free). Therefore, (ii) and (iii) follow from <Ref>, and (iv) follows from (iii).

For a Tate curve $E_{q,K}$, $\Crys_{1}(E_{q,K}[n])$ was described in <Ref>.

Let $r\in \Z_{\geq 0}$. There exists a functor $\Crys_{r}: \ModZpGK \to \ModcrysrZpGK$ that sends an object $M$ of $\ModZpGK$ to $\Crys_{r}(M)$ as defined in <Ref>. This functor is right adjoint to the forgetful functor $\ModcrysrZpGK\to \ModZpGK$, and hence it is left exact and commutes with limits.

We have shown in <Ref> that $\Crys_{r}$ is well-defined on objects. For any morphism $f: M \to M'$ of objects of $\ModZpGK$, define $\Crys_{r}(f) := f|_{\Crys_{r}(M)}. $ It is immediate from the definition of $\Crys_{r}(M')$ that $f|_{\Crys_{r}(M)}$ factors through $\Crys_{r}(M')$.

Let $L\in \RepZp$. Then $\Crys_{r}(L[1/p])\cap L$ is a $\Zp$-lattice in $\Crys_{r}(L[1/p])$ and $\Crys_{r}(L)[1/p] = \Crys_{r}(L[1/p])$.

We wish to study the derived functors of $\Crys_{r}$, but to define these using the usual formalism of Grothendieck we must restrict $\Crys_{r}$ to some category with enough injectives. Since injective objects must be $p$-divisible, to compute the derived functors of $\Crys_{r}$ on $L\in \RepZp$, we are lead to consider replacing  $L$ by its Pontryagin dual $L\otimes \Qp/\Zp$.

Let $L\in \RepZp$. Then $L\otimes \Qp/\Zp$ is an object of $\ModtorsZpGK$. The natural morphism

\[
\colim_{m\geq 1} \Crys_{r}(L/p^{m}L)\longto
\Crys_{r}(L\otimes\Qp/\Zp),
\]
where the colimit is with respect to maps $L/p^{m}L\to L/p^{m+1}L$ given by multiplication by $p$, is an isomorphism.
By definition of $\Crys_{r}$, we have $\Crys_{r}(L\otimes\Qp/\Zp) = \colim M_{i}$ with $M_{i}\in \Repcrysrft$. Since $L\otimes \Qp/\Zp$ is torsion and each $M_{i}$ is of finite type and $r$-crystalline, we see that for each $i\in I$ there is some $k(i)\in \Z_{\geq 1}$ such that $M_{i}\to \Crys_{r}(L\otimes \Qp/\Zp)$ factors through $\Crys_{r}(L/p^{k(i)}L)$. This induces an inverse to the map $\colim_{m\geq 1} \Crys_{r}(L/p^{m}L) \longto \Crys_{r}(L\otimes\Qp/\Zp)$.

Following Kim-Marshall and Jannsen, we work with inverse systems of torsion $\Gamma_{K}$-modules, since the category of such inverse systems has enough injectives.

The categories $\ModZpGK$, $\ModtorsZpGK$, and the category of $\Inv\ModtorsZpGK$ of inverse systems of objects of $\ModtorsZpGK$ each have enough injectives.

The claim for $\ModZpGK$ follows from 04JE. Let $M$ be an object of $\ModtorsZpGK$. Let $M\to I$ be an injection of $M$ into an injective object of $\ModZpGK$. Since $M$ is torsion, this injection factors through $I_{\tors}$, so we need only show that $I_{\tors}$ is an injective object in the category $\ModtorsZpGK$. Suppose we have morphisms $f:T\to I_{\tors}$ and $g:T\to T^{'}$ in $\ModtorsZpGK$. Then $f$ and $g$ are also morphisms in the category $\ModZpGK$, so we get a morphism $f':T'\to I$ such that $f = f'\circ g : T\to I$. But $T'$ is torsion, so $f'$ factors through $I_{\tors}$. This shows that $I_{\tors}$ is injective, finishing the proof of the claim for $\ModtorsZpGK$. The claim for $\Inv\ModtorsZpGK$ is immediate from what we have.

Let $M$ be an object of $\ModtorsZpGK$. Define objects of $\Inv\ModtorsZpGK$ as follows.

* $T(M) := (T_{m})_{m\geq 1}$ with $T_{m} = M[p^{m}]$ and transition maps $T_{m}\to T_{m'}$, for $m'\leq m$, given by multiplication by $p^{m-m'}$.

* $V(M) := (V_{m})_{m\geq 1}$ with $V_{m} = M$ and transition maps $V_{m}\to V_{m'}$, for $m'\leq m$, given by multiplication by $p^{m-m'}$.

* $\underline{M} := (M_{m})_{m\geq 1}$ the constant system with $M_{m} = M$ and each transition morphism is the identity.

Let $L\in \RepZp$, $j\geq 0$, $r\geq 0$. Then $R^{i}\Crys_{r}(L[1/p])$ has no $p$-torsion.

Every element of $V(L \otimes \Qp/\Zp)$ is sequence of the form $(a_{j}/p^{m+j})_{j\geq 0}$, where $m\in \Z_{\geq 1}$, $a_{j}\in L$ for all $j\in \Z_{\geq 0}$, and $a_{j+1} \equiv a_{j}$ modulo $p^{m+j}$. From this we see that the morphism $p: V(L \otimes \Qp/\Zp) \to V(L \otimes \Qp/\Zp)$ is an isomorphism. Taking the long exact sequence associated to this isomorphism using the usual derived functor cohomology, we find that $p$ is an isomorphism on $R^{i}\Crys_{r}(V(L \otimes \Qp/\Zp))$ for each $j\geq 0$. In particular, $R^{i}\Crys_{r}(L[1/p])$ has no $p$-torsion.

Let $L$ be an object of $ \RepZp$ and let $r\in \Z_{\geq 0}$. Then there are canonical isomorphisms

\begin{align*}
\lim\Crys_{r}(T(L \otimes\Qp/\Zp))
&\simeq \Crys_{r}(L)
\\
\lim\Crys_{r}(V(L \otimes\Qp/\Zp))
&\simeq \Crys_{r}(L[1/p])
\\
\lim\Crys_{r}\left(\underline{L \otimes\Qp/\Zp}\right)
&\simeq \Crys_{r}(L\otimes\Qp/\Zp).
\end{align*}

For $i\geq 1$, define

\begin{align*}
&:= R^{i}(\lim\Crys_{r})(T(L \otimes\Qp/\Zp))
\\
&:= R^{i}(\lim\Crys_{r})(V(L \otimes\Qp/\Zp))
\\
R^{i}\Crys_{r}(L\otimes \Qp/\Zp)
&:= R^{i}(\lim\Crys_{r})\left(\underline{L \otimes\Qp/\Zp}\right).
\end{align*}

The morphisms

\[
\pi_{m} = \id\otimes p^{m}: L \otimes \Qp/\Zp \longto L\otimes \Qp/\Zp
\]

defined for $m\in \Z_{\geq 1}$ give rise to a short exact sequence of objects of $\Inv\ModtorsZpGK$

\begin{align}
\label{eq:ses-of-inv-systems}
\ses{T(L\otimes \Qp/\Zp)}{}{V(L\otimes \Qp/\Zp)}{}{\underline{L\otimes \Qp/\Zp}},
\end{align}

hence a long exact sequence in $\ModcrysrZpGK$

\begin{equation*}
\begin{tikzcd}[column sep=small]
0 \ar[r]
& \Crys_{r}(L) \ar[r]
& \Crys_{r}(L[1/p]) \ar[r]
& \Crys_{r}(L\otimes\Qp/\Zp) \ar[lld] \\
% nothing
& R^{1}\Crys_{r}(L) \ar[r]
& R^{1}\Crys_{r}(L[1/p]) \ar[r]
& R^{1}\Crys_{r}(L\otimes\Qp/\Zp) \ar[lld] \\
% nothing
& R^{2}\Crys_{r}(L) \ar[r]
& \cdots
& % nothing
\end{tikzcd}
\end{equation*}

The first three claimed equalities are immediately verified, where for the second we use the fact from <Ref> that $\Crys_{r}$ preserves limits.

We now show that the morphisms $\pi_{m}$ give rise to a short exact sequence (<ref>). Let $i = (i_{m})_{m\geq 1} :T(L\otimes \Qp/\Zp) \to V(L\otimes \Qp/\Zp)$ be the morphism of inverse systems obtained by taking $i_{m}: p^{-m}L/L = (L\otimes \Qp/\Zp)[p^{m}] \to L\otimes \Qp/\Zp$ to be the inclusion. Then the cokernel of $i_{m}$ is isomorphic $L\otimes \Qp/\Zp$, so we have a short exact sequence

\[\ses{\frac{1}{p^{m}}L/L}{[p^{m}]\circ i_{m}}{L \otimes \Qp/\Zp}{\pi_{m}}{L \otimes \Qp/\Zp}\]

The transition maps in the systems $T(L\otimes \Qp/\Zp)$ and $V(L\otimes \Qp/\Zp)$ are both multiplication by $p$, so they commute with the maps $[p^{m}]\circ i_{m}$. A direct verification shows that $\pi_{m+1} = p\circ\pi_{m}$. Therefore, we have the short exact sequence (<ref>). As we saw in <Ref>, the category $\Inv\ModtorsZpGK$ has enough injectives. With this, the existence of the long exact sequence is standard.

§ FINITE FLAT GROUP SCHEMES, MONODROMY, AND 1-SEMISTABLE MODULES

In this section, we develop the theory of finite flat group schemes with monodromy structures, which was initiated by initiated by Kato (see, e.g., [18]). For the relationship between this perspective and the perspective of log finite flat group schemes, the interested reader should refer to the introduction <Ref> and Remarks <ref>, <ref> and <ref>. After discussing some standard results and describing some important examples, we introduce a category $\ExtMonR$ that provides a useful, flexible generalization of the category $\fin^{p,N}_{\OK}$ introduced by Kato. We then develop results on Kato's generic fiber functor. This part of the work amounts to making explicit calculations with Baer sums and certain pushouts. We then describe the relationship to the theory of degenerations of abelian varieties and 1-motives developed in <Ref>. In the final part of the section, we describe a monodromy criterion for 1-crystallinity. This result uses the Torsion Fully Faithfulness Theorem, which can also be viewed as an explicit calculation involving certain Kummer 1-cocycles of $\Gamma_{K}$

Throughout this section, we adopt the notation of <Ref>.

§.§ Finite flat group schemes with monodromy

Until <Ref>, let $R$ be as in the following definition.

Let $R$ be a field or a Henselian discrete valuation ring. Let $\fin^{p}_{R}$ denote the category of commutative $\Spec R$-group schemes of $p$-power order that are finite and flat over $\Spec R$, and where morphisms are homomorphisms of $R$-group schemes.

It is well known that $\fin^{p}_{R}$ inherits the structure of an exact category by embedding it in $\ShAbfppf{R}$. With this structure, a short exact sequence of objects of $\fin^{p}_{R}$ is a sequence

\begin{align*}
\ses{\eta^{-1}}{i}{\eta^{0}}{\pi}{\eta^{1}}
\end{align*}

in $\ShAbfppf{R}$, where $i$ is given by a closed immersion of $R$-schemes and $\pi$ is given by a faithfully flat morphism of $R$-schemes.

The full subcategory of $\fin^{p}_{R}$ consisting of the étale objects is equivalent to the category of continuous representations of the étale fundamental group of $\Spec R$ on discrete torsion $\Zp$-modules. In particular, since $K$ has characteristic 0, $\ModtorsZpGK$ is equivalent to $\fin^{p}_{K}$. However, it is known (for instance, by <Ref>) that not every object of $\fin^{p}_{K}$ is the generic fiber of an object of $\fin^{p}_{\OK}$. The objects of  $\fin^{p}_{K}$ that do prolong over $\OK$ have traditionally been called “finite flat”. As we saw in <Ref>, the condition of being finite flat in this sense is the same the condition of being 1-crystalline.

Suppose $G$ is a semiabelian $R$-scheme that sits in a short exact sequence

\begin{align}
\label{eq:TGB-ses}
\ses{T}{}{G}{}{B}
\end{align}

where $T$ is an $\OK$-torus and $B$ is an abelian $\OK$-scheme. It can be shown ([16] Example III 6.4) that, for any $n\in \bZ_{\geq 2}$ a power of $p$, there is a short exact sequence in $\fin^{p}_{\OK}$

\begin{align}
\label{eq:global-semiab-torsion-ses}
\ses{T[n]}{}{G[n]}{}{B[n]}.
\end{align}

Note that, for a general semiabelian $R$-scheme, the torsion $G[n]$ need not be an object of $\fin^{p}_{R}$. For $G$ as in (<ref>), however, the finite flat groups $G[n]$ fit into a $p$-divisible group $G[p^{\oo}]$.

In the case where $G$ is the Raynaud extension $G = \widetilde A$ attached to a $K$-variety with semistable reduction, we have the Tate module $T_{p}\widetilde A_{K}$. This will play an important role in <Ref>.

For $\sM$ an $R$-1-motive and $n\in \Z_{\geq 2}$ a power of $p$, we have the short exact sequence $\eta(\sM, n)$ in $\fin^{p}_{R}$ as shown in (<ref>).

Aside from the short exact sequences of the form (<ref>), the short exact sequences of finite flat $R$-groups of greatest interest to us are those described by the following proposition.

Let $H$ be an object of $\fin^{p}_{R}$. Then the connected component $H^{\circ}$ through which the identity section factors is an open subgroup. The quotient $H^{\et} := H/H^{\circ }$ is the maximal étale quotient of $H$. The canonical short exact sequence

\[
\tikzses{\ceseq(H):}{H^{\circ}}{}{H}{}{H^{\et}}
\]

is functorial in $H$.

In <Ref>, though $Y/nY$ is étale, the sequence $\eta(\sM,n)$ need not be the connected-étale sequence of $\sM[n]$. We can say more. Taking étale parts is an exact functor. Applying this functor to (<ref>) and noting that $T[n]$ is connected, we find that $G[n]^{\et} = B[n]^{\et}$. Since $Y/nY$ is étale, taking étale parts in $\eta(\sM, n)$ gives a canonical short exact sequence

\begin{align}
\label{eq:R-1-motive-et-ses}
\ses{B[n]^{\et}}{}{\sM[n]^{\et}}{}{Y/nY}.
\end{align}

It follows that $\eta(\sM, n)$ is the connected-étale sequence of $\sM[n]$ if and only if $B[n]^{\et}$ is trivial.

To describe Kato's enrichment of $\fin^{p}_{R}$ with monodromy maps, we first give a simple but non-standard definition of Tate twisting. While Tate twisting makes sense in a broader context from the perspective of Kisin modules, for Tate twisting to preserve 1-crystallinity one must restrict to certain types of twists. The proof of the following proposition is elementary.

Let $H$ be an object of $\fin^{p}_{R}$ and suppose that $n$ is a power of $p$ such that $nH = 0$. Write $(-)^{*}$ for Cartier duality. Pontryagin duality of finite abelian groups with $\pi_{1}(\Spec R)$-action induces an endofunctor $(-)^{\wedge}$ on the full subcategory of $\fin^{p}_{R}$ consisting of the objects étale over $R$.

* Suppose $H$ is étale. Define

\[H(1) := (H^{\wedge})^{*}.\]

Then $H(1)$ is a multiplicative-type object of $\fin^{p}_{R}$ and there is a canonical isomorphism of sheaves of $\Z/n\Z_{R}$-modules on $\Sh_{\Ab}((\Sch_{R})_{\fppf})$

\begin{align*}
H(1) = H\otimes\mu_{n}.
\end{align*}

* Suppose $H$ is multiplicative. Define

\[H(-1) := (H^{*})^{\wedge}.\]

Then $H(-1)$ is an étale object of $\fin^{p}_{R}$ and there is a canonical isomorphism of sheaves of $\Z/n\Z_{R}$-modules on $\Sh_{\Ab}((\Sch_{R})_{\fppf})$

\begin{align*}
H(-1) = \underline{\Hom}(\mu_{n}, H).
\end{align*}

We now begin to develop Kato's enrichment of $\fin^{p}_{R}$.

Let $R$ be as in <Ref>.
Let $\ExtMonR$ denote the category of pairs $(\eta^{\bullet}, \nu)$ such that

* $\eta^{\bullet} = [\eta^{-1}\to \eta^{0}\to \eta^{1}]$ is a short exact sequence of objects of $\fin^{p}_{R}$ regarded as a cochain complex concentrated in degrees $-1$, $0$ and $1;$

* the term $\eta^{1}$ is an étale $R$-group;

* $\nu \in \Hom_{\fin^{p}_{R}}(\eta^{1}(1), \eta^{-1})$.

A morphism $f: (\eta_{1}^{\bullet}, \nu_{1}) \to (\eta_{2}^{\bullet}, \nu_{2})$ of objects of $\ExtMonR$ is a morphism $f: \eta_{1}^{\bullet} \to \eta_{2}^{\bullet}$ such that the following diagram[Note that $f^{-1}$ is $f^{\bullet}$ with $\bullet=-1$, not the inverse of $f$ which, if it existed, would point in the opposite direction. We will be careful to avoid confusing uses of this notation.]

\[
\begin{tikzcd}
\eta_{1}^{1}(1)\ar[r, "\nu_{1}"]\ar[d, "f^{1}(1)"]
& \eta_{1}^{-1} \ar[d, "f^{-1}"] \\
\eta_{2}^{1}(1)\ar[r, "\nu_{2}"] & \eta_{2}^{-1}
\end{tikzcd}
\]

commutes. Define $\fin^{p, N}_{R}$ to be the category of pairs $(H, N)$ such that $(\ceseq(H), N)$ is an object of $\ExtMon$, and where a morphism $(H_{1}, N_{1})\to (H_{2}, N_{2})$ of objects of $\fin^{p, N}_{\OK}$ is the same as a morphism $(\ceseq(H_{1}), N_{1})\to (\ceseq(H_{2}), N_{2})$.

It is shown in [18] that a choice of uniformizer of $\OK$ identifies $\fin^{p, N}_{\OK}$ with a category of log finite flat $\OK$-group schemes. This category consists of certain representable sheaves on the big Kummer log flat site of the fs log scheme $S^{\log}$ defined in <Ref>.

The categories $\ExtMonR$ and $\fin^{p, N}_{R}$ each inherit the structure of an exact category from that of $\fin^{p}_{R}$. There is a fully faithful functor $\fin^{p}_{R}\to \fin^{p, N}_{R}$ given on objects by $H\mapsto (H, 0)$. In the context of <Ref>, this functor amounts to equipping $H$ with the induced log structure.

For the rest of <Ref>, we will specialize to the case were $R$ is $K$ or $\OK$.

Let $(\eta^{\bullet},\nu)\in \ExtMon$. Let $n$ be the order of $\eta^{1}$, so that $\eta^{1}$ is a $\Z/n\Z_{\OK}$-module. Recall that for any Tate curve $E_{q,K}$, the finite $K$-group of $n$-torsion points $E_{q,K}[n]$ lies in a short exact sequence

\begin{align*}
\tikzses{\theta^{q}_{n,K}:}{\mu_{n,K}}{}{E_{q,K}[n]}{}{\Z/n\Z_{K}}
\end{align*}

Consider the special case where $q$ is a uniformizer $\varpi$. Tensor the sequence $\theta^{\varpi}_{n,K}$ of free $\Z/n\Z$-modules with $\eta^{1}_{K}$ and push out by $\nu$ to obtain

\begin{align*}
\MP(\eta^{\bullet}, \nu)_{K} :=
\nu_{*}(\theta^{\varpi}_{n,K}\otimes \eta^{-1}_{K})
\in \Ext_{\fin^{p}_{K}}(\eta^{-1}_{K}, \eta^{1}_{K}).
\end{align*}

By definition, $\MP(\eta^{\bullet}, \nu)_{K}$ is the bottom row in the pushout diagram

\[
\begin{tikzcd}
% \theta^{\varpi}_{n,K}\otimes \eta_{-1,K}
% &
& \eta^{1}_{K}(1)\ar[r]\ar[d, "\nu"]
& \eta^{1}_{K}\otimes E_{\varpi, K}[n]\ar[r]\ar[d]
& \eta^{1}_{K}\ar[r]\ar[d, equals]
& 0
\\
% \nu_{*}(\theta^{\varpi}_{n,K}\otimes \eta_{-1, K})
% &
& \eta^{-1}_{K}\ar[r]
& \MP(\eta^{\bullet}, \nu)_{K}^{0} \ar[r]
& \eta^{1}_{K} \ar[r]
& 0.
\end{tikzcd}
\]

By definition, we have

\begin{align}
\label{eq:nu-star-eta-0-explicit}
\MP(\eta^{\bullet}, \nu)_{K}^{0}
= \frac{(\eta^{1}_{K}\otimes E_{\varpi, K}[n])\oplus \eta^{-1}_{K}}{\{(y,0)-(0, \nu(y)):\: y\in \eta^{1}_{K}(1)\}}.
\end{align}

Let $n\in \Z_{\geq 2}$ be a power of $p$ and let $S^{\log}$ be as in <Ref>. Kato has constructed a log finite flat $S^{\log}$-group that has generic fiber isomorphic to $E_{\varpi, K}[n]$. Using this, one can mimic the construction in <Ref> integrally, i.e., in a certain category of fs log schemes. With this in mind, we sometimes carry a subscript of $K$ in our notation even in situations where we do not define an $\OK$-model.

We will use the following lemma, which holds in much greater generality, in our definition of the generic fiber functor on $\ExtMonR$.

For $i=1,2,3$, let $V_{i}^{\bullet}$ be cochain complexes of objects of $\fin^{p}_{K}$ concentrated in degrees $-1$, $0$ and $1$. Suppose we have two morphisms

\begin{align}
\label{eq:eta-chain-complex}
\overset{f_{2}^{\bullet}}\longto V_{2}^{\bullet}
\overset{f_{1}^{\bullet}}\longto  V_{1}^{\bullet}
\end{align}

giving short exact sequences in degrees $1$ and $-1$.

* Then $f_{2}^{0}$ is injective and $f_{1}^{0}$ is surjective.

* If $\ker f_{1}^{0}\subset \im f_{2}^{0}$, then in fact $\ker f_{1}^{0}= \im f_{2}^{0}$. Therefore, by (i), (<ref>) gives a short exact sequence in degree $0$.

The morphisms of (<ref>) fit into the following commutative diagram with exact rows and where the outer two columns are short exact:

\begin{equation*}
\begin{tikzcd}
% V_{3}^{\bullet} \ar[d, "f_{2}^{\bullet}"]
% &
0 \ar[r]
& V_{3}^{-1}  \ar[r] \ar[d, "f_{2}^{-1}"]
& V_{3}^{0}  \ar[r] \ar[d, "f_{2}^{0}"]
& V_{3}^{1} \ar[r] \ar[d, "f_{2}^{1}"]
& 0 \\
% V_{2}^{\bullet} \ar[d, "f_{1}^{\bullet}"]
% &
0 \ar[r]
& V_{2}^{-1}  \ar[r] \ar[d, "f_{1}^{-1}"]
& V_{2}^{0}  \ar[r] \ar[d, "f_{1}^{0}"]
& V_{2}^{1} \ar[r] \ar[d, "f_{1}^{1}"]
& 0 \\
% V_{1}^{\bullet}
% &
0 \ar[r]
& V_{1}^{-1}  \ar[r]
& V_{1}^{0}  \ar[r]
& V_{1}^{1} \ar[r]
& 0
\end{tikzcd}
\end{equation*}

Applying the Snake Lemma to the top two rows and using the hypotheses, we obtain an exact sequence

\begin{align*}
0\longto \ker f_{2}^{0} \longto 0 \longto V_{1}^{-1} \longto \coker f_{2}^{0} \longto V_{1}^{1}.
\end{align*}

This shows that $\ker f_{2}^{0} = 0$. Similarly, applying the Snake Lemma to the bottom two rows gives an exact sequence

\begin{align}
\label{eq:snake-lemma-2}
V_{3}^{-1} \longto \ker f_{1}^{0} \longto V_{3}^{1} \longto 0 \longto \coker f_{1}^{0} \longto 0,
\end{align}

which shows that $\coker f_{1}^{0} = 0$. This has proven (i).

We prove (ii). Note that the morphism $d_{2}:\ker f_{1}^{-1} \longto \ker f_{1}^{0}$ is injective by assumption, so (<ref>) gives a short exact sequence

\begin{align*}
0\longto V_{3}^{-1} \longto \ker f_{1}^{0} \longto V_{3}^{1} \longto 0.
\end{align*}

We therefore have equations relating orders of finite abelian groups

\[
\#\ker f_{1}^{0}
= (\#V_{3}^{1})(\#V_{3}^{-1})
= \# V_{3}^{0}.
\]

By (i), $f_{2}^{0}$ is injective, so $\#\im f_{2}^{0} = \# V_{3}^{0}$. We see that $\ker f_{1}^{0}$ and $\im f_{2}^{0}$ are both of order $\# V_{3}^{0}$. Therefore, if $\ker f_{1}^{0}\subset \im f_{2}^{0}$, then $\ker f_{1}^{0}= \im f_{2}^{0}$.

The association $(\eta^{\bullet},\nu)\mapsto \MP(\eta^{\bullet},\nu)_{K}$ gives rise to an exact functor

\begin{align}
\label{eq:pushout-functor}
\ExtMon \longto \Ext_{\fin^{p}_{K}}.
\end{align}

We define the functor on morphisms. Let

\[
f_{2}^{\bullet}: (\eta_{1}^{\bullet}, \nu_{1})
\longto (\eta_{2}^{\bullet}, \nu_{2})
\]

be a morphism of objects of $\ExtMon$. Let $n$ be the maximum of the orders of $\eta_{1}^{1}$ and $\eta_{2}^{1}$. Using the universal property of $\MP(\eta_{1}^{\bullet}, \nu_{1})_{K}^{0}$ as a pushout, we obtain a morphism $\MP f_{2, K}^{0}$ making the diagram

\begin{equation}
\label{eq:two-etas-cube}
\begin{tikzcd}
\eta_{1, K}^{1}(1) \ar[rr]
\ar[dr, "f_{2}^{1}(1)"] \ar[dd, near start, "\nu_{1}"]
& \eta_{1, K}^{1}\otimes E_{\varpi, K}[n]
\ar[dr, "f_{2}^{1}\otimes \id"] \ar[dd]
\\
& \eta_{2, K}^{1}(1) \ar[rr] \ar[dd, near start, "\nu_{2}"]
& \eta_{2, K}^{1} \otimes E_{\varpi, K}[n] \ar[dd]
\\
\eta_{1, K}^{-1} \ar[rr] \ar[dr, "f_{2}^{-1}"]
& \MP(\eta_{1}^{\bullet}, \nu_{1})_{K}^{0} \ar[dr, dashed, "\MP f_{2, K}^{0}"]
\\
& \eta_{2, K}^{-1} \ar[rr]
& \MP(\eta_{2}^{\bullet}, \nu_{2})_{K}^{0}
\end{tikzcd}
\end{equation}

commute. The morphism $\MP f_{2, K}^{0}$ fits into a morphism of complexes

\[\MP f_{2, K}^{\bullet}: \MP(\eta_{1}^{\bullet}, \nu_{1})_{K} \longto \MP(\eta_{2}^{\bullet}, \nu_{2})_{K}\]

with $\MP f_{2, K}^{-1} = f_{2, K}^{-1}$ and $\MP f_{2, K}^{1} = f_{2, K}^{1}$.
This has defined the functor (<ref>). We now show this functor is exact. Let

\begin{equation}
\label{eq:etas-ses}
\ses{
(\eta_{1}^{\bullet}, \nu_{1})}{f_{2}^{\bullet}}{
(\eta_{2}^{\bullet}, \nu_{2})}{f_{3}^{\bullet}}{
(\eta_{3}^{\bullet}, \nu_{3})
\end{equation}

be a short exact sequence of objects of $\ExtMon$, which means simply that (<ref>) is a short exact sequence of cochain complexes

\begin{align}
\label{eq:eta-ses-no-push}
\ses{\eta_{1, K}^{\bullet}}{f_{2}^{\bullet}}{
\eta_{2, K}^{\bullet}}{f_{3}^{\bullet}}{
\eta_{3, K}^{\bullet}}
\end{align}

that is compatible with the monodromy maps. We must show that the sequence

\begin{equation}
\label{eq:nu-star-etas-ses}
\MP(\eta_{1, K}^{\bullet}, \nu_{1})_{K}
\overset{\MP f_{2, K}^{\bullet}}\longto
\MP(\eta_{2, K}^{\bullet}, \nu_{2})_{K}
\overset{\MP f_{3, K}^{\bullet}}\longto
\MP(\eta_{3, K}^{\bullet}, \nu_{3})_{K}
\end{equation}

is short exact in each degree $\bullet \in \{-1, 0, 1\}$. For degrees $\bullet\in \{-1, 1\}$, (<ref>) agrees with (<ref>),
which is exact by the assumption. Therefore, (<ref>) is short exact in degrees $-1$ and $1$. By <Ref>, to prove short exactness in degree $0$, it is enough to show that $\ker \MP f_{3, K}^{0}\subset \im \MP f_{2, K}^{0}$. We do this now.

Recall from (<ref>) that, for $i \in \{1,2,3\}$, we have

\begin{align*}
\MP(\eta_{i}^{\bullet}, \nu_{i})_{K}^{0}
= \frac{(\eta_{i, K}^{1}\otimes E_{\varpi, K}[n])\oplus \eta_{i, K}^{-1}}{\{(y,0)-(0, \nu_{i}(y)):\: y\in \eta_{i, K}^{1}(1)\}}.
\end{align*}

We will represent the class of a pair $(a,b) \in (\eta_{i,K}^{1}\otimes E_{\varpi, K}[n])\oplus \eta_{i,K}^{-1}$ using brackets as in $[a,b]$. Let $[a_{2}, b_{2}]\in \MP(\eta_{2}^{\bullet}, \nu_{2})_{K}^{0}$. Then $[a_{2}, b_{2}]\in \ker \MP f_{3, K}^{0}$ if and only if

\begin{align}
\label{eq:pushout-in-kernel-desc}
((f_{3}^{1}\otimes \id)(a_{2}), f_{3}^{-1}(b_{2})) = (y_{3}, -\nu_{3}(y_{3})) \text{ for some } y_{3}\in \eta_{3, K}^{-1}(1).
\end{align}

Suppose (<ref>) holds. We wish to show that $[a_{2}, b_{2}]\in \im \MP f_{2, K}^{0}$, i.e., we wish to show that there exists a pair $(a_{1}, b_{1})\in (\eta_{1, K}^{1}\otimes E_{\varpi, K}[n]) \oplus \eta_{1, K}^{-1}$ and some $y_{2}\in \eta_{2, K}^{1}(1)$ such that

\begin{align}
\label{eq:im-desideratum-pushforward}
(a_{2}, b_{2})
= ((f_{2, K}^{1}\otimes \id)(a_{1}),
f_{2, K}^{-1}(b_{1})) + (y_{2}, -\nu_{2}(y_{2})).
\end{align}

Choose a lift $y_{2}$ of $y_{3}$ under the surjection $f_{3, K}^{1}(1): \eta_{2, K}^{1}(1) \to \eta_{3, K}^{1}(1)$. Then, by (<ref>), we have

\[
((f_{3, K}^{1}\otimes \id)\times f_{3, K}^{-1})\big(
(a_{2}, b_{2}) - (y_{2}, -\nu_{2}(y_{2}))
\big) = (0,0) \in (\eta_{3, K}^{1}\otimes E_{\varpi, K}[n]) \oplus \eta_{3, K}^{-1}.
\]

By exactness of (<ref>) in degrees $-1$ and $1$ and exactness of tensoring against the free $\Z/n\Z$-module $E_{\varpi, K}[n]$, there exists
\[(a_{1}, b_{1})\in (\eta_{1, K}^{1}\otimes E_{\varpi, K}[n]) \oplus \eta_{1, K}^{-1}\]

such that we have equalities

\begin{align*}
(f_{2, K}^{1}\otimes \id)(a_{1})
&=  a_{2}-y_{2} \\
f_{2, K}^{-1}(b_{1})
&=  b_{2} + \nu_{2}(y_{2}).
\end{align*}

This has shown that we have an equation of the form (<ref>). Since $[a_{2}, b_{2}]$ is arbitrary, we have shown that $\ker \MP f_{3, K}^{0}\subset \im \MP f_{2, K}^{0}$. Using <Ref>, we conclude that the functor (<ref>) is exact.

If $(\eta^{\bullet},\nu)\in \ExtMon$, then $\MP(\eta^{\bullet},\nu)_{K}^{0}$ is an object of $\Repsstonetors$. Indeed, $\eta_{K}^{1}$ is 0-crystalline since it prolongs to an étale $\OK$-group scheme, and $E_{\varpi,K}[n]$ can be shown by a direct calculation with the period ring $B_{\stbl}$ to be 1-semistable, so $\eta_{K}^{1} \otimes E_{\varpi,K}[n]$ is 1-semistable. Since $\eta_{K}^{-1}$ is 1-crystalline, being the generic fiber of $\eta^{-1}$, the direct sum $\eta_{K}^{1} \otimes E_{\varpi,K}[n] \oplus \eta_{K}^{-1}$ is 1-semistable, and therefore its quotient $\MP(\eta^{\bullet},\nu)_{K}^{0}$ is 1-semistable.

We now define the generic fiber functor on $\ExtMon$ and show that it is exact.

There exists an exact functor

\begin{align}
\label{eq:gen-fiber-functor-def}
(-)_{K}: \ExtMon
\longto \Ext_{\fin^{p}_{K}}
\end{align}

which is given on objects by

\[
(\eta^{\bullet}, \nu)_{K}
:= \eta^{\bullet}_{K} + \MP(\eta^{\bullet}, \nu)_{K}.
\]

Let us first address a consequence for notation. Given  $(\eta^{\bullet}, \nu) \in \ExtMon$, the generic fiber $(\eta^{\bullet}, \nu)_{K}$ is another complex concentrated in degrees $-1$, $0$ and $1$. We suppress the index recording these degrees; for instance, the nonzero terms in $(\eta^{\bullet}, \nu)_{K}$ are $(\eta^{\bullet}, \nu)_{K}^{-1}$, $(\eta^{\bullet}, \nu)_{K}^{0}$ and $(\eta^{\bullet}, \nu)_{K}^{1}$, while the expression $(\eta^{0}, \nu)_{K}$ is undefined.

Let $f_{2}^{\bullet}: (\eta_{1}^{\bullet}, \nu_{1}) \longto (\eta_{2}^{\bullet}, \nu_{2})$ be a morphism of objects of $\ExtMon$. We wish to define a morphism

\[
f_{2, K}^{+,\bullet}:
(\eta_{1}^{\bullet}, \nu_{1})_{K}
\longto (\eta_{2}^{\bullet}, \nu_{2})_{K}.
\]

For $\bullet\in \{1,-1\},$ define $f_{2, K}^{+, \bullet} := f_{2, K}^{\bullet}$. We lighten the notation by setting, for $i=1,2$, $\lambda_{i,K}^{\bullet} := \MP(\eta_{i}^{\bullet},\nu_{i})_{K}$. Then the term $(\eta_{i}^{\bullet}, \nu_{i})_{K}^{0} = ((\eta_{i}^{\bullet}, \nu_{i})_{K})^{0}$ in degree zero of the complex $(\eta_{i}^{\bullet}, \nu_{i})_{K}$ is (see <Ref>)

\begin{align}
\label{eq:eta-i-baer-desc}
(\eta_{i}^{\bullet}, \nu_{i})_{K}^{0}
= \frac{\eta_{i, K}^{0}\times_{\eta_{i, K}^{1}}\lambda_{i,K}^{0}}{
\im(\Delta_{i}^{-})} ,
\end{align}

where $\Delta_{i}^{-}$ is the morphism

\begin{align}
\label{eq:Delta-i-minus-def}
\twoardef{\Delta_{i}^{-}}{\eta_{i,K}^{-1}}{
\eta_{i, K}^{0}
\times_{\eta_{i, K}^{1}}
\lambda_{i,K}^{0}
}{x}{(d_{\eta_{i,K}^{\bullet}}(x), -d_{\lambda_{i,K}^{\bullet}}(x)),}
\end{align}

the morphisms $d_{(-)}$ being the differential in each of the two complexes. By the above description of the degree-zero terms in the Baer sums, to define a morphism

\[
(\eta_{1}^{\bullet}, \nu_{1})_{K}^{0}
\longto (\eta_{2}^{\bullet}, \nu_{2})_{K}^{0}
\]

we may give a morphism $\eta_{1, K}^{0} \times_{\eta_{1, K}^{1}} \lambda_{1,K}^{0} \to \eta_{2, K}^{0} \times_{\eta_{2, K}^{1}} \lambda_{2,K}^{0}$ that intertwines the homomorphisms $\Delta_{1}^{-}$, $\Delta_{2}^{-}$. It is straightforward to  check that

\[
f_{2, K}^{\times,0}
:= (f_{2, K}^{0}\times \MP f_{2}^{0}):
\eta_{1,K}^{0}\times_{\eta_{1, K}^{1}}\lambda_{1, K}^{0}
\longto
\eta_{2, K}^{0} \times_{\eta_{2, K}^{1}}\lambda_{2,K}^{0}
\]

is such a morphism. Define

\[
f^{+, 0}_{2, K}:
(\eta_{1}^{\bullet}, \nu_{1})_{K}^{0}
\longto (\eta_{2}^{\bullet}, \nu_{2})_{K}^{0}
\]

to be the morphism induced by $f^{\times, 0}_{2, K}$. We claim that $f^{+, \bullet}_{2, K}$ defines a morphism of complexes, i.e., that the diagram

\begin{equation}
\label{eq:varphi-2-K}
\begin{tikzcd}
(\eta_{1}^{\bullet}, \nu_{1})_{K}
\ar[d, "f_{2,K}^{+, \bullet}"]
& 0\ar[r]
& \eta^{-1}_{1,K}\ar[r] \ar[d, "f_{2}^{-1}"]
& (\eta_{1}^{\bullet}, \nu_{1})_{K}^{0} \ar[r]\ar[d, "f^{+, 0}_{2, K}"]
& \eta^{1}_{1,K} \ar[r]\ar[d,"f_{2}^{1}"]
& 0
\\
(\eta_{2}^{\bullet}, \nu_{2})_{K}
& 0\ar[r]
& \eta^{-1}_{2,K}\ar[r]
& (\eta_{2}^{\bullet}, \nu_{2})_{K}^{0}\ar[r]
& \eta_{K}^{1}\ar[r]
& 0
\end{tikzcd}
\end{equation}

commutes. For $i\in \{1,2\}$, the morphism $\eta_{i,K}^{-1}\to (\eta_{i}^{\bullet}, \nu_{i})_{K}^{0}$  is defined to be the morphism induced by

\[
\twoar{}{\eta_{i,K}^{-1}}{\eta_{i,K}^{0}\times_{\eta_{i,K}^{1}}\lambda_{i,K}^{0}}{x}{(d_{\eta_{i}^{\bullet}}(x),0).}
\]

Since $f_{2, K}^{+, 0}$ is given by $f_{2, K}^{0}$ in the first factor, and since $f_{2, K}^{0}$ is assumed to commute with the differentials, we see that the left square in (<ref>) commutes. The morphism $(\eta_{i}^{\bullet}, \nu_{i})_{K}^{0}\to \eta_{i,K}^{1}$ is induced by

\[
\twoar{}{\eta_{i,K}^{0}\times_{\eta_{i,K}^{1}} \lambda_{i,K}^{0}}{\eta_{i,K}^{1}}{(a, b)}{(d_{\eta_{i}^{\bullet}}(y),0).}
\]

Arguing as before, we see that the right square in (<ref>) commutes. This has shown that $f_{2, K}^{+, 0}$ gives a morphism of complexes $f_{2, K}^{+, \bullet}$.

We now show that the functor $(-)_{K}$ is exact. Let

\begin{equation}
\label{eq:ses-of-complexes-2}
\ses{
(\eta^{\bullet}_{1}, \nu_{1})}{f_{2}^{\bullet}}{
(\eta^{\bullet}_{2}, \nu_{2})}{f_{3}^{\bullet}}{
(\eta^{\bullet}_{3}, \nu_{3})
\end{equation}

be a short exact sequence of objects of $\ExtMon$. Recall that this means that (<ref>) is a short exact sequence of cochain complexes

\begin{align}
\label{eq:eta-ses-no-log-gen-fiber}
\ses{\eta_{1,K}^{\bullet}}{f_{2}^{\bullet}}{\eta_{2,K}^{\bullet}}{f_{3}^{\bullet}}{\eta_{3,K}^{\bullet}}
\end{align}

that is compatible with the monodromy homomorphisms. We wish to show that

\begin{equation}
\label{eq:ses-of-complexes-2-K}
(\eta^{\bullet}_{1}, \nu_{1})_{K}
\overset{f_{2, K}^{+, \bullet}}\longto
(\eta^{\bullet}_{2}, \nu_{2})_{K}
\overset{f_{3, K}^{+, \bullet}}\longto
(\eta^{\bullet}_{3}, \nu_{3})_{K}
\end{equation}

is short exact in each degree. The sequence (<ref>) is short exact in degrees $-1$ and $1$ by the assumption that (<ref>) is short exact. By <Ref>, to show (<ref>) is short exact, it suffices to show that $\ker f_{3, K}^{+, 0}\subset \im f_{2, K}^{+, 0}$.

Expand the notation established above for $i=1,2$ to the case $i=3$.  We will represent the class of a pair $(a,b)\in \eta^{0}_{i, K} \times_{\eta^{1}_{i, K}} \lambda^{0}_{i,K}$ in the quotient $(\eta_{i}^{\bullet}, \nu_{i})_{K}^{0}$ by $[a,b]$. Recall that the morphisms $f_{i, K}^{+, 0}$, for $i\in \{2, 3\}$, are defined using the morphisms

\[
f_{i, K}^{\times, 0}
:= \big(f^{0}_{i,K}\times \MP f^{0}_{i}\big):
\eta^{0}_{i-1, K}\times_{\eta^{1}_{i-1, K}}\lambda^{0}_{i-1, K}
\longto \eta^{0}_{i,K}\times_{\eta^{1}_{i,K}}\lambda^{0}_{i,K}.
\]

Let $[a_{2}, b_{2}]\in (\eta_{2}^{\bullet}, \nu_{2})_{K}^{0}$. By the description of $(\eta_{2}^{\bullet}, \nu_{2})_{K}^{0}$ given in (<ref>), we have $[a_{2},b_{2}]\in \ker f_{3, K}^{+, 0}$ if and only if there exists $x_{3}\in \eta_{3, K}^{-1}$ such that

\[
f_{3, K}^{\times, 0}(a_{2}, b_{2})
= \big(f^{0}_{3,K}(a_{2}),
\MP f^{0}_{3}(b_{2})\big)
= \Delta_{3}^{-}(x_{3}).
\]

By assumption, (<ref>)
is short exact in every degree, so there exists $x_{2}\in \eta^{-1}_{2,K}$ such that $f_{3,K}^{-1}(x_{2}) = x_{3}$. Since the diagram

\begin{equation*}
\begin{tikzcd}
\eta^{0}_{1,K}\times_{\eta^{1}_{1,K}}
\lambda^{0}_{1,K}
\ar[r, "f_{2, K}^{\times, 0}"]
& \eta^{0}_{2,K}\times_{\eta^{1}_{2,K}}
\lambda^{0}_{2,K} \ar[r]
\ar[r, "f_{3, K}^{\times, 0}"]
& \eta^{0}_{3,K}\times_{\eta^{1}_{3,K}}
\lambda^{0}_{3,K}
\\
\eta_{1,K}^{-1}
\ar[u, "\Delta_{1}^{-}"]
\ar[r, "f_{2,K}^{-1}"]
& \eta_{2,K}^{-1}
\ar[u, "\Delta_{2}^{-}"]
\ar[r, "f_{3,K}^{-1}"]
& \eta_{3,K}^{-1}
\ar[u, "\Delta_{3}^{-}"]
\end{tikzcd}
\end{equation*}

commutes, we have

\begin{align*}
f_{3, K}^{\times, 0}
\big((a_{2}, b_{2}) -\Delta_{2}^{-}(x_{2})\big)
&= f_{3, K}^{\times, 0}((a_{2}, b_{2}))
- f_{3, K}^{\times, 0}(\Delta_{2}^{-}(x_{2}))
\\
&= \Delta_{3}^{-}(x_{3}) -
\Delta_{3}^{-}(f_{3,K}^{-1}(x_{2}))
\\
&= \Delta_{3}^{-}(x_{3}) -
\Delta_{3}^{-}(x_{3}) = 0.
\end{align*}

This has shown that $\big(a_{2}-d_{\eta_{2,K}^{\bullet}}(x_{2}),
b_{2}+ d_{\lambda_{2,K}^{\bullet}}(x_{2})\big)
\in \ker f_{3, K}^{\times, 0}$, which is to say that we have

\begin{align*}
&\in \ker f_{3,K}^{0}
\\
b_{2}+ d_{\nu_{2,*}\eta_{2,K}}(x_{2})
&\in \ker \MP f_{3}^{0}.
\end{align*}

By exactness of (<ref>), there exists $a_{1}\in \eta_{1,K}^{0}$ such that

\[
= a_{2}-d_{\eta_{2,K}^{\bullet}}(x_{2}).
\]

By <Ref>, there exists $b_{1}\in \lambda_{1,K}^{0}$ such that

\[
\MP f_{2,K}^{0}(b_{1})
= b_{2}+ d_{\lambda_{2,K}^{\bullet}}(x_{2}).
\]

We claim that $(a_{1},b_{1})$ gives a pair in the fiber product $\eta^{0}_{1,K} \times_{\eta^{1}_{1,K}} \lambda^{0}_{1,K}$, which is to say that the images of $a_{1}$ and $b_{1}$ in $\eta_{1,K}^{1}$ are equal. The diagram

\begin{equation*}
\begin{tikzcd}[column sep=huge]
\eta_{1,K}^{0} \times \lambda_{1,K}^{0}
\ar[r,
"f_{2,K}^{\times, 0}"]
\ar[d, swap,
"d_{\eta_{1,K}^{\bullet}} \times d_{\lambda_{1,K}^{\bullet}}"]
\eta_{2,K}^{0} \times \lambda_{2,K}^{0}
\ar[d,
"d_{\eta_{2,K}^{\bullet}} \times d_{\lambda_{2,K}^{\bullet}}"]
\\
\eta_{1,K}^{1} \times \eta_{1,K}^{1}
\ar[r, "f_{2,K}^{1}\times f_{2,K}^{1}"]
\eta_{2,K}^{1} \times \eta_{2,K}^{1}
\end{tikzcd}
\end{equation*}

commutes because $f_{1,K}^{\bullet}$ and $\MP f_{1,K}^{\bullet}$ are morphisms of complexes. The image of $(a_{1}, b_{1})$ in the bottom right corner of this diagram has the form $(c,c)\in \eta_{2,K}^{1} \times \eta_{2,K}^{1}$ by the assumption that $(a_{2}, b_{2})$ defines a pair in the relevant fiber product. By the assumption that (<ref>) is short exact, $f_{2,K}^{1}$ is injective. Since the diagram commutes, we see that images of $a_{1}$ and $b_{1}$  in $\eta_{1,K}^{1}$ are equal. This has shown that $(a_{1}, b_{1})$ satisfies $(a_{1}, b_{1})\in \eta^{0}_{1,K} \times_{\eta^{1}_{1,K}} \lambda^{0}_{1,K}$. Taking $x_{2}$ as above, we have

\[
f_{2, K}^{\times, 0}((a_{1}, b_{1}))
= (a_{2},b_{2}) + \Delta^{-}_{2}(x_{2}),
\]

so that

\[
f_{2, K}^{+,0}([a_{1}, b_{1}])
= [a_{2},b_{2}].
\]

Since $[a_{2}, b_{2}]$ was an arbitrary element of $\ker f_{3,K}^{+, 0}\subset (\eta_{2}^{\bullet}, \nu_{2})_{K}^{0}$,  we conclude that $\ker f_{3,K}^{+, 0}\subset \im f_{2, K}^{+, 0}$. By <Ref>, which we noted before applies in the current situation (i.e., to the complex (<ref>)), we conclude that the functor $(-)_{K}$ of (<ref>) is exact.

We see from (<ref>) that $\MP(\eta^{\bullet}, \nu)_{K}^{0} = \eta_{K}^{1}\oplus \eta_{K}^{-1}$  if $\nu = 0$. It follows that $(\eta^{\bullet},0)_{K}^{0} = \eta_{K}^{0}.$

Let $(\eta^{\bullet}, \nu)\in \ExtMon$. By <Ref> and <Ref>, $(\eta^{\bullet},\nu)_{K}^{0}$ is in $\Repsstonetors$.

The following key result follows by applying a more general result of Bertapelle, Candilera and Cristante to the specific context of <Ref>.

Let $A_{K}$ be a semistable abelian $K$-variety, let $n\geq 2$ be a power of $p$. Let $\sQ_{K}$ be the log 1-motive of <Ref>, let $\sQ^{1}_{\varpi}$ be the  $\OK$-1-motive of <Ref>, and let $\nu_{n}$ be the monodromy homomorphism (<ref>). There exists an isomorphism

\[
(\eta(\sQ^{1}_{\varpi}, n),\nu_{n})_{K} \simeq \eta(\sQ_{K},n)
\]

of objects of $\Ext_{\fin^{p}_{K}}(Y_{K}/nY_{K}, \widetilde A_{K}[n])$. In particular, the degree-zero term in the generic fiber $(\eta(\sQ^{1}_{\varpi}, n),\nu_{n})_{K}$ is isomorphic to $A_{K}[n]$.

Let $(\eta^{\bullet}, \nu)\in \ExtMon$ and let $H = \eta^{0}$. Since $\eta^{1}$ is étale, there is a canonical morphism of complexes $\ceseq(H) \to \eta^{\bullet}$. In fact, as we shall see, $\nu$ induces a monodromy structure on $\ceseq(H)$. While it is not strictly necessary for us to work in $\fin^{p, N}_{\OK}$, it is desirable to do so because it allows us to frame our later results in terms of connected-étale sequences of objects, which are commonly used, and which are the context of Kato's work on log finite flat group schemes.

We call the functor $(-)_{K}$ of <Ref> the generic fiber functor on $\ExtMon$. This defines an exact functor on $\fin^{p,N}_{\OK}$ by sending $(H,N)\in \fin^{p,N}_{\OK}$ to

\[
(H, N)_{K} := (\ceseq(H), N)_{K}^{0}.
\]

The following result lets us compare generic fibers of objects of $\ExtMon$ to generic fibers of objects of $\fin^{p, N}_{\OK}$.

Let $(\eta_{2}^{\bullet}, \nu_{2})\in \ExtMon$ and let  $\eta_{1}^{\bullet}$ be a cochain complex of objects of $\fin^{p}_{\OK}$ concentrated in degrees $-1$, $0$ and $1$, with $\eta_{1}^{-1}\subset \eta_{2}^{1},$ $\eta_{1}^{0}=\eta_{2}^{0}$, and with $\eta_{2}^{0}$ étale. Then there exists a morphism $\nu_{1}$ making $(\eta_{1}^{\bullet}, \nu_{1})$ an object of $\ExtMon$, and the natural morphism $f^{\bullet}: \eta_{1}^{\bullet}\to \eta_{2}^{\bullet}$ is in fact a morphism $(\eta_{1}^{\bullet}, \nu_{1}) \to (\eta_{2}^{\bullet}, \nu_{2})$ inducing an isomorphism

\[
f^{+,0}_{K}: (\eta_{1}^{\bullet}, \nu_{1})_{K}^{0}
\longisoto (\eta_{2}^{\bullet}, \nu_{2})_{K}^{0}.
\]

We first define $\nu_{1}$. Recall that $\nu_{2}$ is a map $\nu_{2}:\eta_{2}^{1}(1)\to \eta_{2}^{-1}$. Since $\eta_{2}^{1}(1)$ is of multiplicative type and of $p$-power order, $\nu_{2}$ factors through the identity component $(\eta_{2}^{-1})^{\circ} = (\eta_{1}^{-1})^{\circ}$. Pre-composing $\nu_{2}$ with $f^{1}(1): \eta_{1}^{1}(1) \to \eta_{2}^{1}(1)$, we obtain a morphism $\nu_{1}: \eta_{1}^{1}(1)\to \eta_{1}^{-1}$. By construction of $\eta_{1}$, $f^{\bullet}$ gives a morphism $(\eta_{1}^{\bullet}, \nu_{1}) \to (\eta_{2}^{\bullet}, \nu_{2})$.

We now show that $f_{K}^{+,0}$ is an isomorphism. We reuse our conventions and notation from the proof of <Ref>, while also letting $E_{n} := E_{\varpi, K}[n]$.  Then we have diagrams like (<ref>) that fit into a larger commutative diagram

\begin{equation}
\label{eq:big-diagram}
\begin{tikzcd}[column sep=tiny]
& & \eta_{1, K}^{1}(1) \ar[rr]   \ar[dr, two heads, "f_{K}^{1} \otimes \id"]
\ar[dd, near start, "\nu_{1}"]
& & \eta_{1, K}^{1}\otimes E_{n}
\ar[rr] \ar[dr, two heads, "f_{K}^{1}\otimes \id"]
\ar[dd, near start, "\nu_{1}'"]
& & \eta_{1, K}^{1}              \ar[rr] \ar[dr, two heads, "f_{K}^{1}"]
\ar[dd, equals]
& & 0
& \\
&   0 \ar[rr]
& &  \eta_{2, K}^{1}(1)   \ar[rr] \ar[dl, near start, "\nu_{2}"]
& & \eta_{2, K}^{1}\otimes E_{n}
\ar[rr] \ar[dd, near start, "\nu_{2}'"]
& & \eta_{2, K}^{1}\ar[rr]       \ar[dd, equals]
& & 0
\\
0                         \ar[rr]
& & \eta_{1, K}^{-1}             \ar[rr]\ar[dr, hook]
& & \lambda_{1, K}^{0}                 \ar[rr]\ar[dr, "\MP f^{0}_{K}"]
& & \eta_{1, K}^{1}             \ar[rr]\ar[dr, two heads]
& & 0
& \\
&   0\ar[rr]
& &  \eta_{2, K}^{-1}           \ar[rr]
& & \lambda_{2, K}^{0}               \ar[rr]
& & \eta_{2, K}^{1}             \ar[rr]
& & 0.
\end{tikzcd}
\end{equation}

The homomorphism $\MP f^{0}_{K}$ is induced by $\eta_{1, K}^{1}\otimes E_{n} \to \eta_{2, K}^{1}\otimes E_{n} \to \lambda_{2, K}^{0}$ and $\eta_{1, K}^{-1}\to \eta_{2, K}^{-1} \to \lambda_{2, K}^{0}$ via the universal property of $\lambda_{1, K}^{0}$ as a cofiber product. We note also that it need not be the case that $\MP f^{0}_{K}$ is an isomorphism, as can be seen in the example where $\eta_{2,K}^{1} = 0$.

We now show that $f_{K}^{+, 0}$ is an isomorphism of finite $K$-groups. For each $i\in \{1,2\}$, $\eta_{i, K}^{0}$ and $(\eta_{i}^{\bullet}, \nu_{i})_{K}^{0}$ both are objects of $\Ext_{\fin^{p}_{K}}(\eta_{i,K}^{1}, \eta_{i,K}^{-1})$, hence $\eta_{i}^{0}$ and $(\eta_{i}^{\bullet}, \nu_{i})_{K}^{0}$ each have the same order. Since $f^{0}$ is the identity map, we see that $(\eta_{1}^{\bullet}, \nu_{1})_{K}^{0}$ and $(\eta_{2}^{\bullet}, \nu_{2})_{K}^{0}$ have the same order. Therefore, to show that $f_{K}^{+,0}$ is an isomorphism, it suffices to show that it is an injection. Note that this does not follow from exactness of $(-)_{K}$ since it may be that $f^{1}$ is not injective.

Recall that any element of $(\eta_{1}^{\bullet}, \nu_{1})_{K}^{0}$ can be represented as $[a, [m, n]]$ with

\[
(a, (m, n)) \in
\eta_{1, K}^{0}
\times_{\eta_{1, K}^{-1}} (\eta_{1, K}^{1}\otimes E_{n} \times \eta_{1, k}^{-1}).
\]

$f_{K}^{+,0}([a, [m, n]])
= [f_{K}^{0}(a), g([m, n])]
= [a, [f_{K}^{1}\otimes \id(m), n]]$. Let $[a, [m, n]]\in \ker f_{K}^{+, 0}$. Then there exist $c \in \eta_{2, K}^{-1}$ and $y \in \eta_{2, K}^{1}$ such that

\begin{align}
\label{eq:ker-condition}
(a, (f_{K}^{1}\otimes \id(m), n)) = (c, (y, -c - \nu_{2}(y)))
\in
\eta_{2, K}^{0}
\times_{\eta_{2, K}^{-1}} (\eta_{2, K}^{1}\otimes E_{n} \times \eta_{2, k}^{-1}).
\end{align}

Since $n$ and $\nu_{2}(y)$ both are in $\eta_{1, K}^{-1}$ and $c = -n - \nu_{2}(y)$, we see that $c$ is in $\eta_{1, K}^{-1}$. Then $a = c$ maps to 0 in $\eta_{1, K}^{1}$. Now, $(a, [m, n])$ defines a point in a fiber product over $\eta_{1, K}^{-1}$, so we see that the surjection $\lambda_{1, K}^{0}\to \eta_{1, K}^{1}$ maps to $[m, n]$ to $0$, and from (<ref>) we see that $m\in \eta_{1, K}^{1}(1)$.

Since $m\in \eta_{1, K}^{1}(1)$, $f_{K}^{1}\otimes \id(m)\in \eta_{2, K}(1)$. By (<ref>), $f_{K}^{1}\otimes \id(m) = y$, so we see that from (<ref>) that

\[
\nu_{2}(y) = \nu_{2}(f_{K}^{1}\otimes \id(m)) = \nu_{1}(m).
\]

Using the relationships derived above, we have shown that there exists $c\in \eta_{1, K}^{-1}$ such that, with $\Delta_{1}^{-}$ is defined as in (<ref>), we have

\begin{align*}
[a, [m, n]]
&= [c, [m, -c-\nu_{2}(y)]] \\
&= [c, [0, -c]] + [0, [m, -\nu_{1}(m)]] \\
&= [c, [0, -c]] + [0, [0, 0]] \\
&= \Delta_{1}^{-}(c).
\end{align*}

This has shown that $[a, [m, n]] = 0$.

There exists a canonical, essentially surjective functor $\ExtMon\to \fin^{p, N}_{\OK}$ that sends an object $(\eta^{\bullet}, \nu)$ to $(\eta^{0}, \nu|_{\eta^{0,\et}(1)})$. The natural map $(\eta^{\bullet}, \nu)_{K}^{0} \to (\eta^{0}, \nu|_{\eta^{0,\et}(1)})_{K}^{0} $ is an isomorphism.
This is the special case of <Ref> where $\eta_{1}^{\bullet} = \ceseq(\eta_{2}^{0})$.

Let $A_{K}$ be a semistable abelian $K$-variety, let $n\geq 2$ be a power of $p$. Let $\sQ_{K}$ be the log 1-motive of <Ref>, let $\sQ^{1}_{\varpi}$ be the  $\OK$-1-motive of <Ref>. Attached to our uniformizer $\varpi$ of $\OK$ there is an object $(\sQ^{1}_{\varpi}[n], N_{n})$ of $\fin^{p, N}_{\OK}$ with a canonical isomorphism

\[(\sQ^{1}_{\varpi}[n], N_{n})_{K} \simeq A_{K}[n].\]

§.§ Monodromy criterion for torsion 1-crystallinity

Let $(Q,N)$ be an object of $\finpN$, and let $V = (Q,N)_{K} \in \Reponessttors$. Informed by Fontaine's theory of filtered $(\varphi,N)$-modules, it is natural to seek a monodromy criterion for 1-crystallinity of subobjects of $V$. One might hope to prove such a criterion by first developing an analogous theory of monodromy structures on integral objects, and then showing that $(Q,N)$, comes from such an object. While [21] does develop a theory of Kisin modules with monodromy, the monodromy operators there are only defined after inverting $p$. In an unpublished preprint [27], Madapusi Pera has developed a theory of height-1 Kisin modules with integral monodromy and has shown that every object of $\Reponesst$ is the Tate module of some log $p$-divisible group. It seems possible that one could then prove a strengthening of Raynaud's embedding theorem [33] 2.3.1 for log finite flat $\OK$-group schemes, and then to apply results from Fontaine's theory to prove a monodromy criterion. We take a different approach by working directly with objects of $\ExtMon$ and applying the Torsion Fully Faithfulness Theorem <Ref>.

Fix $(\eta^{\bullet}, \nu) \in \ExtMon$ until <Ref>.

The $\Gamma_{K}$-module $(\eta^{\bullet}, \nu)_{K}^{0}$ is 1-crystalline if and only if $\MP(\eta^{\bullet}, \nu)_{K}^{0}$ is 1-crystalline.

This is an application of <Ref>.

Let $U^{\bullet} = (\eta^{\bullet}, \nu)_{K}$ and let $\nu_{*}U = \MP((\eta^{\bullet}, \nu))_{K}^{0}$. Recall that $\nu_{*}U$ fits into a commuting diagram

\begin{equation}
\label{eq:push-pull}
% \tag{$\star$}
\begin{tikzcd}
0 \ar[r]
& U^{1}(1)\ar[r] \ar[d, "\nu"]
& E_{\varpi, K}[n]\otimes U^{1} \ar[r] \ar[d, "\nu"]
& U^{1}\ar[r] \ar[d, equals]
& 0
\\
0 \ar[r]
& U^{-1} \ar[r]
& \nu_{*}U \ar[r]
& U^{1} \ar[r]
& 0
\end{tikzcd}
\end{equation}

where the leftmost square is cocartesian, where $n$ is the order of $\eta^{1}$, and where the top row is obtained by tensoring $U^{1}$ against the short exact sequence

\[
\tikzses{\theta^{\varpi}_{n,K}:}{\mu_{n,K}}{}{E_{\varpi, K}[n]}{}{\Z/n\Z_{K}}.
\]

Choose an ordered basis $\{x_{1}, x_{2}\}$ of $E_{\varpi, K}[n]$ as in <Ref> (where now we have specified that $q=\varpi$). If $n=p^{m}$, the action of $\Gamma_{K}$ on $x_{2}$ is determined by a $\Z/n\Z$-valued cocycle $c_{\varpi, n}$ defined by the relation

\begin{align}
\label{eq:c-varpi-n-relation}
\sigma(\varpi^{\flat, (m)}) = (\epsilon^{(m)})^{c_{\varpi, n}(\sigma)}\varpi^{\flat, (m)},
\end{align}

for all $\sigma\in \Gamma_{K}$, where $\varpi^{\flat}$ and $\epsilon$ are as in <Ref>. We record the following basic lemma.

There exists $\sigma\in \Gamma_{K}$ such that $c_{\varpi, n}(\sigma)\in (\Z/n\Z)^{\times}$.

Suppose $p \mid c_{\varpi, n}(\sigma)$ for all $\sigma$. Recall that $n=p^{m}$ with $m\geq 1$. Raising the relation defining $c_{\varpi, n}$ to the $p^{m-1}$-th power, we see that $(\varpi^{\flat, (m)})^{p^{m-1}}$ is fixed by all of $\Gamma_{K}$, which is to say that $K$ contains a $p$-th root of $\varpi$. Since $\varpi$ is a uniformizer of $\OK$, this is impossible, so we conclude that $c_{\varpi, n}(\sigma)\in (\Z/n\Z)^{\times}$ for some

With our choice of basis of $E_{\varpi, K}[n]$, we may write elements of the tensor product
$E_{\varpi, K}[n]\otimes U^{1}$ uniquely in the form

\[
x_{1}\otimes y_{1} + x_{2}\otimes y_{2} \qquad (y_{1},y_{2}\in U^{1}).
\]

Using this basis for $E_{\varpi, K}[n]$, the morphisms in the top row of (<ref>) are simply the inclusion

\begin{align*}
\twoar{}{
U^{1}(1) = \Z/n\Z(1) \otimes U^{1}}{
E_{\varpi, K}[n]\otimes U^{1}}{x_{1}\otimes y_{1}}{x_{1}\otimes y_{1}}
\end{align*}

and the map

\begin{align*}
\twoar{}{E_{\varpi, K}[n]\otimes U^{1}}{U^{1}}{x_{1}\otimes y_{1}+x_{2}\otimes y_{2}}{y_{2}.}
\end{align*}

As in (<ref>), $\nu_{*}U$ satisfies

\begin{align}
\label{eq:N-star-W-baer-def}
\nu_{*}U = \frac{(E_{\varpi, K}[n]\otimes U^{1})\oplus U^{-1}}{\{(x_{1}\otimes y_{1}, 0)-(0, \nu(x_{1}\otimes y_{1})):\: y_{1}\in U^{1} \}}.
\end{align}

Given $(x_{1}\otimes y_{1} + x_{2}\otimes y_{2}, z)\in (E_{\varpi, K}[n]\otimes U^{1})\oplus U^{-1}$, we denote its class in $\nu_{*}U$ by $[x_{1}\otimes y_{1} + x_{2}\otimes y_{2}, z]$. Using the relation in (<ref>)), we have

\[
\nu_{*}U = \{[x_{2}\otimes y_{2}, z]:\: y_{2}\in U^{1},\, z\in U^{-1}\}.
\]

There is a homomorphism of finite abelian groups

\begin{align}
\label{eq:s-def} \twoardef{s}{U^{1}}{\nu_{*}U}{u}{
[x_{2}\otimes u,0]}
\end{align}

which is a section of the quotient map $\nu_{*}U\to U^{1}$. The morphism $s$ need not be $\Gamma_{K}$-equivariant, but we have the following result.

The homomorphism $s$ of finite abelian groups is $\Gamma_{\varpi^{\flat}}$-equivariant.
Restrict (<ref>) to $\Gamma_{\varpi^{\flat}}$. From the relation (<ref>) defining $c_{\varpi, n}$, we see that $c_{\varpi, n}|_{\Gamma_{\varpi^{\flat}}} = 0$, so that our choice of basis $\{x_{1}, x_{2}\}$ induces an isomorphism

\begin{align*}
\Resoo(E_{\varpi, K}[n])
\simeq \Resoo\big(\Z/n\Z(1)\oplus \Z/n\Z\big)
\end{align*}

We see that the homomorphism $y_{2}\mapsto x_{2}\otimes y_{2}$ gives a $\Gamma_{\varpi^{\flat}}$-equivariant splitting of the top row of (<ref>). Composing with $\nu$, we obtain the map $s$, which we see is a $\Gamma_{\varpi^{\flat}}$-equivariant splitting of

\begin{align*}
\ses{\Resoo U^{-1}}{}{\Resoo \nu_{*}U}{}{\Resoo U^{1}}.
\end{align*}

The $\Gamma_{K}$-module $\nu_{*}U$ is 1-crystalline if and only if the $\Gamma_{\varpi^{\flat}}$-equivariant section $s$ defined in (<ref>) is $\Gamma_{K}$-equivariant.
Recall that a torsion $\Gamma_{K}$-module is 1-crystalline if and only if it is the generic fiber of a finite flat $\OK$-group. Therefore $U^{-1} = \eta_{1,K}$ and $U^{1} = \eta_{-1,K}$ are 1-crystalline. Suppose first that $s$ is $\Gamma_{K}$-equivariant. Then $\nu_{*}U \simeq U^{-1}\oplus U^{1}$ as $\Zp[\Gamma_{K}]$-modules. Since finite direct sums of 1-crystalline $\Gamma_{K}$-modules are 1-crystalline, we conclude that $\nu_{*}U$ is 1-crystalline.

Now suppose $\nu_{*}U$ is 1-crystalline. Since $U^{1}$ is 1-crystalline, <Ref> implies that the $\Gamma_{\varpi^{\flat}}$-equivariant morphism $s$ is $\Gamma_{K}$-equivariant.

In <Ref> we made non-canonical choices. On the one hand, $\varpi^{\flat}$ gives the subgroup $\Gamma_{\varpi^{\flat}}\subset \Gamma_{K}$ that was used when applying <Ref>. On the other hand, $\varpi = \varpi^{\flat, (0)}$ was used when we formed $E_{\varpi, K}[n]$ and performed Raynaud decomposition. We could have chosen a different uniformizer, say $u\varpi$ where $u\in \OK^{\times}$, to perform Raynaud decomposition. In that case, $E_{u\varpi, K}[n]$ need not be split as a $\Zp[\Gamma_{\varpi^{\flat}}]$-module.

Returning to the situation of the above <Ref>, we now relate the splitness of $\nu_{*}U$ as an extension of $U^{1}$ by $U^{-1}$ to the monodromy morphism. Recall that $\nu(-1):U^{1}\to U^{-1}(-1)$ is a morphism of $\Gamma_{K}$-modules.

The section $s$ is $\Gamma_{K}$-equivariant if and only if $U^{1}\subset \ker \nu(-1)$.

Suppose $U^{1}\subset \ker \nu(-1)$. In the basis $\{x_{1}, x_{2}\}$ for $E_{\varpi, K}[n]\otimes U^{1}$, this means that $\nu(x_{1}\otimes u)=0$ for all $u\in U^{1}$. Let $\sigma\in \Gamma_{K}$. We have

\begin{align*}
\sigma s(u) - s(\sigma u)
&= \sigma [x_{2}\otimes u,0]
- [x_{2}\otimes \sigma u,0] \\
&= [c_{\varpi, n}(\sigma)x_{1}\otimes \sigma u, 0] \\
&= [0, \nu(x_{1}\otimes c_{\varpi, n}(\sigma)\sigma u)]
= [0, 0].
\end{align*}

This shows that $s$ is $\Gamma_{K}$-equivariant.

Conversely, suppose $s$ is $\Gamma_{K}$-equivariant and let $u\in U^{1}$. For any $\sigma\in \Gamma_{K}$, we see from the above calculation that we have an equality of elements of $\nu_{*}U$

\[[0,0] = \sigma s(u) - s(\sigma u) = [0, \nu(x_{1}\otimes c_{\varpi, n}(\sigma)\sigma u)],\]

which is to say that, for some $y_{1}\in U^{1}$ depending on $u$, we have the equality of elements of $(E_{\varpi, K}[n]\otimes U^{1})\oplus U^{-1}$

\[
(0, \nu(x_{1}\otimes c_{\varpi, n}(\sigma)\sigma u))
= (x_{1}\otimes y_{1}, -\nu(x_{1}\otimes y_{1})).
\]

We see that we must have $y_{1}=0$, hence $\nu(x_{1}\otimes y_{1})=0$, hence

\begin{align}
\label{eq:N-kills-something}
\nu(x_{1}\otimes c_{\varpi, n}(\sigma)\sigma u) = 0.
\end{align}

By <Ref>, we may choose $\sigma$ such that $c_{\varpi, n}(\sigma)\in (\Z/n\Z)^{\times}$. Let $\chi$ be the $p$-adic cyclotomic character on $\Gamma_{K}$. It follows from (<ref>) that

\[
0 = \nu(x_{1}\otimes \sigma u)
= \chi(\sigma^{-1})\sigma \nu(x_{1}\otimes u),
\]

whence $\nu(x_{1}\otimes u)=0$. We conclude that $U^{1}\subset \ker \nu(-1)$.

The $\Gamma_{K}$-module $(\eta^{\bullet}, \nu)_{K}^{0}$ is 1-crystalline if and only if $\nu = 0$.
Combine <Ref>, <Ref>, and <Ref>.

Let $(Q,N)\in \fin^{p, N}_{\OK}$. We close the section by prolonging $\Crys_{1}((Q, N)_{K})$ in $Q$.

Let $(Q, N)$ be an object of $\finpN$ and let $V^{\bullet} = (\ceseq(Q), N)_{K}$. Let $U^{0}\subset V^{0} = (Q,N)_{K}$ be a $\Gamma_{K}$-submodule containing $V^{-1} = Q^{\circ}_{K}$. Let $\eta^{1}$ be the unique étale prolongation of $U^{0}/V^{-1}$, let $i^{1}:\eta^{1}\to Q^{\et}$ be the unique prolongation of the inclusion $U^{0}/V^{-1}\subset V^{1}$, and let $\eta^{\bullet} := (i^{1})^{*}\ceseq(Q)$. Then $\eta^{0}$ is the scheme theoretic closure of $U^{0}$ in $Q$, and there is a natural morphism $i^{\bullet}:\eta^{\bullet}\to \ceseq(Q)$ with the property that $i^{0}$ is an open immersion. Moreover, $(\eta^{\bullet}, N|_{\eta^{1}(1)})$ is a subobject of $(\ceseq(Q), N)$ via $i^{\bullet}$, and the morphism $i_{K}^{0}$ identifies $(\eta^{\bullet}, N|_{\eta^{1}(1)})_{K}^{0}$ with $U^{0}\subset V^{0}$.

Since $\ceseq(Q)^{1} = Q^{\et}$ is étale, $V^{1} = Q^{\et}_{K}$ is unramified, so existence and uniqueness of $i^{1}$ are standard. By definition, $\eta^{0} = Q \times_{Q^{\et}} \eta^{1}$, and $i^{0}$ is simply the projection onto the first factor. Since $i^{1}$ is an open immersion, its base change $i^{0}$ is an open immersion. It is clear that $\eta^{0}$, the union of the components of $Q$ containing the points of the finite étale $K$-group $U^{0}$, is the scheme theoretic closure of $U^{0}$ in $Q$. The claim that $(\eta^{\bullet}, N|_{\eta^{1}(1)})$ is a subobject of $(\ceseq(Q), N)$ was addressed in <Ref>, but the claim is obvious here because $\eta^{-1} = Q^{\circ} = \ceseq(Q)^{-1}$. By exactness of the generic fiber functor on $\finpN$ (<Ref>), $i^{0}$ injects $(\eta^{\bullet}, N|_{\eta^{1}(1)})_{K}^{0}$ a $\Gamma_{K}$-submodule of $(\ceseq(Q), N)_{K}^{0} = (Q, N)_{K} = V^{0}$. By construction, $\im i_{K}^{0}$ and $U^{0}$ each consist of the same cosets of $V^{-1} = Q^{\circ}_{K}$ in $V^{0}$.

Let $(Q, N)$ in $\finpN$, and let ${}^{*}\ceseq(Q) \in
\Ext_{\finp}(\ker N(-1), Q^{-1})$ be the pullback of $\ceseq(Q)$ under the inclusion $\ker N(-1)\subset Q^{\et}$. Then the natural map ${}^{*}\ceseq(Q)\to \ceseq(Q)$ gives rise to an isomorphism of $\Gamma_{K}$-modules $({}^{*}\ceseq(Q))_{K}^{0}\simeq \Crys_{1}((Q,N)_{K})$.
By <Ref>, $({}^{*}\ceseq(Q), N|_{({}^{*}\ceseq(Q))^{1}})_{K} = ({}^{*}\ceseq(Q))_{K}$. Since $({}^{*}\ceseq(Q))^{0}$ is finite flat, exactness of the generic fiber functor implies that the submodule $({}^{*}\ceseq(Q))_{K}$ of $(Q, N)_{K}$  is contained in $\Crys_{1}((Q,N)_{K})$. We show the reverse inclusion. Since $Q^{\circ}$ is a finite flat $\OK$-group, we have $Q^{\circ}_{K} \subset \Crys_{1}((Q,N)_{K})$. Apply <Ref> to obtain a subobject $(\eta^{\bullet}, N|_{\eta^{1}(1)})$ of $(\ceseq(Q), N)$. By <Ref>, $\eta^{1}(1)\subset \ker N$, so $\eta^{\bullet}$ is a subobject of ${}^{*}\ceseq(Q)$. Taking generic fibers, we have $\Crys_{1}((Q,N)_{K}) = \eta^{0}_{K}\subset {}^{*}\ceseq(Q)_{K}$. This completes the proof.

§ RESULTS ON SEMISTABLE ABELIAN VARIETIES

For this section, we adopt the notation of <Ref>. In particular,  $p$ is any
rational prime and $K/\Qp$ a discretely valued extension. Let $A_{K}$ be an abelian $K$-variety with semistable reduction, let $A$ be the Néron model of $A_{K}$ and let $\widetilde A$ be the Raynaud extension. Let $\sQ_{K}$ be the log 1-motive attached to $A_{K}$ via the functor $F(\OK,\fm_{\OK})$ as in <Ref>. Let  $n\geq 2$ be a power of $p$, and let $(\sQon, N_{n})\in \ExtMon$ and $(\sQon, N_{n}) \in \finpN$ be as in <Ref> and <Ref>, respectively.  As we did in <Ref>, we define

\[
{}^{*}\sQon :=
\sQon
\times_{\sQon^{\et}}
(\ker N_{n})(-1).
\]

Since $Y/nY$ is étale, passing to connected parts in $\eta(\sQ^{1}_{\varpi}[n],n)$ (an object of the group $ \Ext_{\fin^{p}_{\OK}}(Y/nY, \widetilde A[n])$) shows that the morphism $\widetilde A[n]\to \sQo[n]$ is an isomorphism on identity components, so that ${}^{*}\sQon$ fits into a diagram

\begin{equation}
\label{eq:star-Q-1-varpi-pullback-diagram}
\begin{tikzcd}
0\ar[r] & \widetilde A[n]^{\circ}  \ar[r]\ar[d, equal] & {}^{*}\sQon \ar[r] \ar[d, hook] & (\ker N_{n})(-1) \ar[r]\ar[d, hook] & 0 \\
0\ar[r] & \widetilde A[n]^{\circ} \ar[r] & \sQon \ar[r] & \sQon^{\et} \ar[r] & 0
\end{tikzcd}
\end{equation}

where the rows are connected-étale sequences and the vertical morphisms are inclusions of open $\OK$-subgroups.

The canonical morphism $({}^{*}\sQon, 0)\to (\sQon, N_{n})$ coming from (<ref>) gives rise to an isomorphism of $\Gamma_{K}$-modules ${}^{*}\sQon_{K}\simeq \Crys_{1}(A_{K}[n])$.

This follows directly from <Ref>.

<Ref> gives rise to a description of $\Crys_{1}(T_{p}A_{K})$. We note that this result about $T_{p}A_{K}$ follows from [11], Corollary 4.6, which is a key ingredient in their proof of their monodromy criterion for good reduction of $A_{K}$.

The isomorphisms  ${}^{*}\sQ^{1}_{\varpi}[p^{m}]_{K}  \simeq \Crys_{1}(A_{K}[p^{m}])$ for varying $m\in \Z_{\geq 1}$ are compatible in $m$, and give rise to an isomorphism

\[T_{p}\widetilde A_{K} \simeq \Crys_{1}(T_{p}A_{K}).\]

Recall that, for $m\in \Z_{\geq 1}$, the morphism $N_{p^{m}}: \sQ^{1}_{\varpi}[p^{m}]^{\et}(1) \to \sQ^{1}_{\varpi}[p^{m}]^{\circ}$ is obtained by pre-composing the morphism $\nu_{p^{m}}$ of (<ref>) with the canonical morphism $\sQ^{1}_{\varpi}[p^{m}]^{\et}(1)\to (Y/p^{m}Y)(1)$. It is clear that $(\ker N_{p^{m}})_{m\geq 1}$ is an inverse system of objects of $\fin^{p}_{\OK}$. The isomorphisms of <Ref> are compatible for varying $p^{m}$; each such isomorphism is the composition of the projection $\pr_{1,K}$ on $\sQ_{K}[p^{m}] \times_{\sQ^{1}_{\varpi}[p^{m}]^{\et}_{K}} \ker N_{p^{m},K}(-1)$ with the isomorphism $\sQ_{K}[p^{m}]\simeq A_{K}[p^{m}]$ of <Ref>, and the isomorphisms of <Ref> can be seen from the proof to be compatible. By <Ref>,

\[\Crys_{1}(T_{p}A_{K})
= \lim_{m}\Crys_{1}(T_{p}A_{K}/p^{m}T_{p}A_{K})
= \lim_{m}\Crys_{1}(A_{K}[p^{m}]),
\]

so by <Ref> we have isomorphisms

\begin{align}
\label{eq:crys-1-TpAK-is-lim}
\Crys_{1}(T_{p}A_{K})
\simeq \lim_{m} {}^{*}\sQo[p^{m}]_{K}.
\end{align}

Since $T_{p}\widetilde A[p^{\oo}]$ is 1-crystalline and is a submodule of $T_{p}A_{K}$ (via the sequences $\eta(A_{K},n)$ of <Ref>), it is contained in $\Crys_{1}(T_{p}A_{K})$. The quotient $\Crys_{1}(T_{p}A_{K})/p^{m}\Crys_{1}(T_{p}A_{K})$ is torsion 1-crystalline, as is $\widetilde A[p^{m}]_{K}$, so we find

\[
\widetilde A[p^{m}]_{K}
\subset \Crys_{1}(T_{p}A_{K})/p^{m}\Crys_{1}(T_{p}A_{K})
\subset \Crys_{1}(A[p^{m}]_{K}) = {}^{*}\sQo[p^{m}]_{K}.
\]

The upper row in <Ref> is evidently the connected-étale sequence of ${}^{*}\sQ[n]$. It follows from (<ref>) that ${}^{*}\sQ[p^{m}]^{\et} = \ker N_{p^{m}}(-1)$ fits into the short exact sequence

\begin{align*}
\ses{\widetilde A[p^{m}]^{\et}}{}{{}^{*}\sQo[p^{m}]^{\et}}{\pi_{n}}{\ker \nu_{p^{m}}(-1)}.
\end{align*}

By <Ref> (since $\Phi_{A_{K}}$ is finite), $T_{p}((\ker \nu_{p^{m}}(-1))_{m\geq 1})=0$. It follows that $\lim_{m}{}^{*}\sQo[p^{m}]_{K} = \lim_{m}\widetilde A_{K}[p^{m}] = T_{p}\widetilde A_{K}$. By (<ref>), we have $T_{p}\widetilde A_{K}\simeq \Crys_{1}(T_{p}A_{K})$. This completes the proof.

An abelian $K$-variety $A_{K}$ has good reduction over $\OK$ if and only if its $p$-adic Tate module $T_{p}A_{K}$ is 1-crystalline.

Our proof essentially follows the proof of Theorem 4.7 of [11]. The primary difference is that we will invoke <Ref> where they invoke their Corollary 4.6, which they prove in quite different terms.

Suppose that $A_{K}$ has good reduction. Then $T_{p}A_{K}$ is the generic fiber of the $p$-divisible group $A[p^{\oo}]$ over $\OK$, which is 1-crystalline by <Ref>.

For the converse, suppose that $T_{p}A_{K}$ is 1-crystalline. In the proof of Theorem 4.7 in [11], it is shown using the results 7.3.3 and 7.5.1 of [13] that we may assume without loss of generality that $A_{K}$ has semistable reduction. Make this assumption.  By <Ref> and <Ref>, there is a short exact sequence

\[
\ses{T_{p}\widetilde A_{K}}{}{T_{p}A_{K}}{}{Y_{K}\otimes \Qp/\Zp},
\]

where $Y_{K}$ has $\Zp$-rank equal to the dimension of the torus in $\widetilde A_{K}$. By <Ref>, the first morphism in this short exact sequence is an isomorphism. Therefore $Y = 0$, so $T = 0$ and $\widetilde A = B$. Since $\widetilde A_{K} = A_{K}$, we see that $B$, which is an abelian $\OK$-scheme, has generic fiber $A_{K}$. By [6] 1.2/8, $B$ is the Néron model of $A_{K}$. Thus we see that $A_{K}$ has good reduction if $T_{p}A_{K}$ is 1-crystalline.

Recall that $\Phi_{A_{K}}$ is the Néron component group of $A_{K}$, an étale $\OK$-group (<Ref>). It is shown in [15] that, using $(-)^{\fpart}$ for the finite part of a quasi-finite flat $\OK$-group,

\begin{align}
\label{eq:phi-p-m-ito-fparts}
\Phi_{A_{K}}[p^{m}] \simeq \frac{(A[p^{m}]^{\fpart})_{K}}{(A^{\circ}[p^{m}]^{\fpart})_{K}}.
\end{align}

This fact is used by Kim and Marshall (in the case where $p>2$ and $K$ is unramified) to prove (<ref>) below. We give a different proof of (<ref>) using <Ref> and <Ref>; for comparison (<ref>), one uses the fact ([25], 3.4.2.1) that $\widetilde A[p^{m}] \simeq A[p^{m}]^{\fpart} $ for all $m\geq 1$.

There is a canonical isomorphism of unramified finite $\Gamma_{K}$-modules

\begin{align}
\label{eq:Phi-pn-ito-crys}
\Phi_{A_{K}}[n] \simeq
\frac{\Crys_{1}(T_{p}A_{K}\otimes \Z/n\Z)}{ \Crys_{1}(T_{p}A_{K})\otimes \Z/n\Z}.
\end{align}

Recall that ${}^{*}\sQon$ is defined by pullback as in

\[
\begin{tikzcd}
0\ar[r] & \widetilde A[n]^{\circ}  \ar[r]\ar[d, equal] & {}^{*}\sQon \ar[r] \ar[d] & (\ker N_{n})(-1) \ar[r]\ar[d,hook] & 0 \\
0\ar[r] & \widetilde A[n]^{\circ} \ar[r] & \sQon \ar[r] & \sQon^{\et} \ar[r] & 0.
\end{tikzcd}
\]

By the universal property of ${}^{*}\sQon$ as a fiber product, we obtain a homomorphism $\widetilde
A[n]\to {}^{*}\sQon$. We claim that the generic fiber of the morphism $\widetilde A[n]\to {}^{*}\sQon$ has cokernel isomorphic to $\Phi_{A_{K}}[n]$. Recall the short exact sequence

\[
\ses{\widetilde A[n]^{\et}}{}{\sQon^{\et}}{\pi_{n}}{Y/nY}
\]

of (<ref>), and recall that $N_{n}$, defined as in <Ref>, factors as

\[
N_{n}: \sQon^{\et}(1)\overset{\pi_{n}(1)}\longto Y/nY(1)\overset{\nu_{n}}\longto T[n] \hooklongrightarrow \sQon^{\circ}.
\]

By restricting $\pi_{n}(1)$ and untwisting, we obtain a short exact sequence

\[
\  \ses{\widetilde{A} [n]^{\et}}{}{(\ker N_{n})(-1)}{
%\pi_{n}|_{\ker N_{n}}(-1)
}{\ker \nu_{n}(-1)},
\]

where the morphism $\widetilde{A} [n]^{\et} \to (\ker N_{n})(-1)$ is the one obtained by descending the canonical homomorphism $\widetilde A[n]\to {}^{*}\sQon$ to étale parts. We see that

\[
{}^{*}\sQon \longto (\ker N_{n})(-1) \longto \frac{(\ker N_{n})(-1)}{\widetilde A[n]^{\et}}
\]

induces an isomorphism

\begin{align}
\label{eq:star-sQon-wtA-cpt-gp-intmdt}
\frac{{}^{*}\sQon}{\widetilde A[n]}
\simeq \frac{(\ker N_{n})(-1)}{\widetilde A[n]^{\et}}
= \ker \nu_{n}(-1).
\end{align}

Via the isomorphism $(\sQ^{1}_{\varpi}[n], N_{n})^{0}_{K}\simeq A_{K}[n]$ of <Ref>, we obtain isomorphisms

\[
{}^{*}\sQon \simeq \Crys_{1}(A_{K}[n])
\quad \text{ and } \quad
T_{p}\tilde A_{K} \simeq \Crys_{1}(T_{p}A_{K})
\]

as in <Ref> and <Ref>, respectively. Using these, we obtain from (<ref>) and the canonical isomorphism $\ker \nu_{n}(-1) \simeq \Phi_{A_{K}}[n]$ of <Ref> an isomorphism

\[  \Phi_{A_{K}}[n] \simeq
\frac{\Crys_{1}(T_{p}A_{K}\otimes \bZ/n\bZ)}{ \Crys_{1}(T_{p}A_{K})\otimes \bZ/n\bZ}.\]

This completes the proof

Let $\Phi_{A_{K}}[p^{\oo}]$ denote the $p$-Sylow subgroup of $\Phi_{A_{K}}$. Then the isomorphisms of <Ref> give rise to an isomorphism

\begin{align}
\label{eq:R1Crys1-torsion}
\Phi_{A_{K}}[p^{\oo}] \simeq (R^{1}\Crys_{1}(T_{p}A_{K}))_{\tors}.
\end{align}

By <Ref>,

\[
\colim_{m\geq 1} \Crys_{1}(T_{p}A_{K}\otimes \Z/p^{m}\Z)
= \Crys_{1}(T_{p}A_{K}\otimes \Qp/\Zp).
\]

Hence, taking colimits, (<ref>) gives

\begin{align}
\label{eq:Phi-ito-crys}
\Phi_{A_{K}}[p^{\oo}] =
\frac{\Crys_{1}(T_{p}A_{K}\otimes \Qp/\Zp)}{ \Crys_{1}(T_{p}A_{K})\otimes \Qp/\Zp}.
\end{align}

On the other hand, consider the long exact sequence

\begin{equation*}
\begin{tikzcd}[column sep=small]
0 \ar[r]
& \Crys_{1}(T_{p}A_{K}) \ar[r]
& \Crys_{1}(T_{p}A_{K})\otimes \Qp \ar[r]
& \Crys_{1}(T_{p}A_{K}\otimes \Qp/\Zp)\ar[lld] \\
% nothing
& R^{1}\Crys_{1}(T_{p}A_{K}) \ar[r]
& R^{1}\Crys_{1}(V_{p}A_{K}) \ar[r]
&  \cdots
\end{tikzcd}
\end{equation*}

attached to $T_{p}A_{K}$ using <Ref>. The kernel of
\[f: R^{1}\Crys_{1}(T_{p}A_{K})
\longto R^{1}\Crys_{1}(V_{p}A_{K})\]

is exactly the right-hand side of (<ref>), which is torsion, so we have

\[\Phi_{A_{K}}[p^{\oo}]\subset (R^{1}\Crys_{1}(T_{p}A_{K}))_{\tors}.\]

On the other hand, $R^{1}\Crys_{1}(V_{p}A_{K})$ is torsion-free (<Ref>), so the kernel of $f$ contains $R^{1}\Crys_{1}(T_{p}A_{K})_{\tors}$. This completes the proof.

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