---
abstract: 'Vasiu proved that the level torsion $\ell_{\mathcal{M}}$ of an $F$-crystal $\mathcal{M}$ over an algebraically closed field of characteristic $p>0$ is a non-negative integer that is an effectively computable upper bound of the isomorphism number $n_{\mathcal{M}}$ of $\mathcal{M}$ and expected that in fact one always has $n_{\mathcal{M}} = \ell_{\mathcal{M}}$. In this paper, we prove that this equality holds.' 
address: 'Mathematics Department, Utica College, 1600 Burrstone Road, Utica, NY 13502'
author:
- Xiao Xiao
bibliography:
- 'references.bib'
title: 'Subtle invariants of $F$-crystals'
---

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

Notations
---------

Let $p$ be a prime number and $k$ an algebraically closed field of characteristic $p$. For every $k$-algebra $R$, let $W(R)$ be the ring of $p$-typical Witt vectors with coefficients in $R$. For every integer $s \geq 1$, let $W_s(R)$ be the ring of truncated $p$-typical Witt vectors of length $s$ with coefficients in $R$. Let $\sigma_{R}$ be the Frobenius of $W(R)$ and $W_s(R)$. Let $\theta_{R}$ be the Verschiebung of $W(R)$ and $W_s(R)$. Recall that $\sigma_R \theta_R = \theta_R \sigma_R = p$. When there is no confusion of the base ring, we also denote $\sigma_R$ by $\sigma$ and $\theta_R$ by $\theta$. Set $B(R) = W(R)[1/p]$. When $R = k$, $B(k)$ is the field of fractions of $W(k)$. An $F$-crystal $\mathcal{M}$ over $k$ is a pair $(M, \varphi)$ where $M$ is a free $W(k)$-module of finite rank and $\varphi : M \to M$ is a $\sigma$-linear monomorphism.  Unless mentioned otherwise, all $F$-crystals in this paper are over $k$. We denote by $\mathcal{M}_R$ the pair $(M \otimes_{W(k)} W(R), \varphi \otimes \sigma_R)$. For every $W(k)$-linear automorphism $g$ of $M$, we denote by $\mathcal{M}(g)$ the $F$-crystal $(M, g\varphi)$ over $k$.

Aim and scope
-------------

The isomorphism number $n_{\mathcal{M}}$ of an $F$-crystal $\mathcal{M} = (M, \varphi)$ is the smallest non-negative integer such that for every $W(k)$-linear automorphism $g$ of $M$ with the property that $g \equiv 1_M$ modulo $p^{n_{\mathcal{M}}}$, the $F$-crystal $\mathcal{M}(g)$ is isomorphic to $\mathcal{M}$. This is the generalization of the isomorphism number $n_D$ of a $p$-divisible group $D$ over $k$, which is defined to be the smallest non-negative integer such that for every $p$-divisible group $D'$ over $k$ with the same dimension and codimension as $D$, $D'[p^{n_{D}}]$ and $D[p^{n_{D}}]$ are isomorphic if and only if $D'$ is isomorphic to $D$. The isomorphism numbers of $p$-divisible groups are known to exist as early as in [@Manin:formalgroups], as a consequence of Theorems 3.4 and 3.5 of the loc.  cit.  Recently, the isomorphism numbers of $F$-crystals are known to exist by [@Vasiu:CBP Main Theorem A]. 

Traverso proved that $n_D \leq cd+1$ in [@Traverso:pisa Theorem 3], where $c$ and $d$ are the codimension and the dimension (respectively) of the $p$-divisible group $D$. He later conjectured that $n_D \leq \min\{c, d\}$ in [@Traverso:specializations Section 40, Conjecture 4].  In search of optimal upper bounds of $n_D$, the following theorem plays an important role:

\[theorem:maintheoremspecial\] If $D$ is a non-ordinary $p$-divisible group over an algebraically closed field $k$, then its isomorphism number $n_D$ is equal to its level torsion $\ell_D$.

For the definition of $\ell_D$, see [@Vasiu:reconstructing Subsection 1.4] and [@Vasiu:traversosolved Definition 8.3].  We point out that the two definitions are slightly different.  In the case when $D$ is a direct sum of two or more isoclinic ordinary $p$-divisible groups of different Newton slopes, we get $\ell_D = 1$ by the definition in [@Vasiu:reconstructing Subsection 1.4]; on the other hand, we get $\ell_D=0$ by [@Vasiu:traversosolved Definition 8.3].  If we assume that $D$ is non-ordinary, then the two definitions coincide. 

Vasiu proved that $n_D \leq \ell_D$ in [@Vasiu:reconstructing Main Theorem A], and that $n_{D} = \ell_{D}$ provided $D$ is a direct sum of isoclinic $p$-divisible groups, that is, of $p$-divisible groups whose Newton polygons are straight lines.  Later Lau, Nicole and Vasiu proved the equality $n_D = \ell_D$ in [@Vasiu:traversosolved] for all $p$-divisible groups $D$ over $k$. Theorem \[theorem:maintheoremspecial\] builds a bridge between the isomorphism number $n_D$ and other invariants of $D$, such as the level torsion $\ell_D$, the endomorphism number $e_D$, and the coarse endomorphism number $f_D$, which turn out to be all equal by [@Vasiu:traversosolved Theorem 8.11]; see [@Vasiu:traversosolved Definitions 2.2 and 7.2] for their definitions.  Using Theorem \[theorem:maintheoremspecial\], Lau, Nicole and Vasiu were able to find the optimal upper bound of $n_D \leq \lfloor 2cd/(c+d) \rfloor$ (see [@Vasiu:traversosolved Theorem 1.4]), which provides a corrected version of Traverso’s conjecture. 

The level torsion $\ell_{\mathcal{M}}$ of an $F$-crystal $\mathcal{M}$ is well-defined; see [@Vasiu:reconstructing Section 1.2] or Subsection \[subsection:leveltorsion\] for its definition.  Therefore it is natural to ask if the similar equality $n_{\mathcal{M}} = \ell_{\mathcal{M}}$ holds or not in general.  As mentioned before, Vasiu has already proved that $n_{\mathcal{M}} \leq \ell_{\mathcal{M}}$ and the equality holds when $\mathcal{M}$ is a direct sum of isoclinic $F$-crystals.  He expressed the expectation that the equality is true in general; see the paragraph after [@Vasiu:reconstructing 1.3 Main Theorem A].  In this paper, we confirm this expectation. 

\[theorem:maintheorem\] If $\mathcal{M}$ is a non-ordinary $F$-crystal over an algebraically closed field $k$, then its isomorphism number $n_{\mathcal{M}}$ is equal to its level torsion $\ell_{\mathcal{M}}$.

See Theorem \[theorem:maintheorem3\] for its proof.  The definition of the level torsion $\ell_{\mathcal{M}}$ in our paper is slightly different from the definition in [@Vasiu:reconstructing Subsection 1.2]; see Remark \[remark:leveltorsiondiff\].  When $\mathcal{M}$ is a non-ordinary $F$-crystal, the two definitions are exactly the same just as in the case of $p$-divisible groups. 

On the proof of the Main Theorem
--------------------------------

The proof of the Main Theorem uses many ideas from [@Vasiu:traversosolved], [@Vasiu:levelm], and [@Vasiu:dimensions].  It involves two major steps: 0.1in [**Step (1)**]{}: Generalize the level torsion $\ell_{\mathcal{M}}$, the homomorphism number $e_{\mathcal{M}}$, and the coarse homomorphism number $f_{\mathcal{M}}$ to $F$-crystals $\mathcal{M}$ over $k$. Then prove that they are all equal via a sequence of inequalities $f_{\mathcal{M}} \leq e_{\mathcal{M}} \leq \ell_{\mathcal{M}} \leq f_{\mathcal{M}}$ that are the generalization of the inequalities $f_D \leq e_D \leq \ell_D \leq f_D$ obtained in [@Vasiu:traversosolved]. 

The main difficulty in Step (1) is to have the right generalizations of $\ell_{\mathcal{M}}$, $e_{\mathcal{M}}$ and $f_{\mathcal{M}}$ so that they remain unchanged under extensions of algebraically closed fields.  This requires the constructions of suitable groups schemes $\mathbf{End}_s(\mathcal{M})$ (resp.  $\mathbf{Aut}_s(\mathcal{M})$) whose $k$-valued points are the endomorphisms (resp.  automorphisms) of $F$-truncations modulo $p^s$ of $\mathcal{M}$ for all $s \geq 1$. The $F$-truncations modulo $p^s$ of $F$-crystals are the generalization of truncated Barsotti–Tate groups of level $s$ associated to $p$-divisible groups.  They are first introduced by Vasiu in [@Vasiu:CBP] and will be recalled in Section \[section:ftruncation\]; see Definition \[definition:ftruncation\].  We will show that $\ell_{\mathcal{M}}$, $e_{\mathcal{M}}$ and $f_{\mathcal{M}}$ are invariant under extensions of algebraically closed fields.  This allows us to generalize the proof in [@Vasiu:traversosolved Section 8] to our case.  0.1in [**Step (2)**]{}: Prove that $f_{\mathcal{M}} = n_{\mathcal{M}}$ by showing that both $f_{\mathcal{M}}$ and $n_{\mathcal{M}}$ are equal to the smallest number $m$ defined by the property that the image of the natural reduction homomorphism $\pi_{s,1}: \mathbf{End}_s(\mathcal{M}) \to \mathbf{End}_1(\mathcal{M})$ has zero dimension if and only if $s-1 \geq m$.

In Step (2), the main result (see Theorem \[theorem:monotonicitygamma\]) is to show that $n_{\mathcal{M}}$ is the place where the non-decreasing sequence $(\mathrm{dim}(\mathbf{Aut}_s(\mathcal{M})))_{s \geq 1}$ stabilizes, which generalizes a similar result for $p$-divisible groups in [@Vasiu:dimensions].  In order to show this, we construct a group action for each $s \geq 1$ whose orbits parametrize isomorphism classes of $F$-truncations modulo $p^s$; see Subsection \[subsection:groupaction\].  It turns out that the dimension of the stabilizer of the identity element of this action is equal to the dimension of $\mathbf{Aut}_s(\mathcal{M})$ (Lemma \[lemma:connectiondimension\]).  This allows us to use the machinery of group actions to work with the sequence $(\mathrm{dim}(\mathbf{Aut}_s(\mathcal{M})))_{s \geq 1}$ in a way similar to [@Vasiu:dimensions] and [@Vasiu:levelm]. 

We note that the proof of our Main Theorem does not rely on the known fact that $n_{\mathcal{M}} \leq \ell_{\mathcal{M}}$ proved in [@Vasiu:reconstructing].  0.1in [*Notes.*]{}  After this manuscript was finished, we learned that Sian Nie had a proof of the fact $\ell_{\mathcal{M}} \leq n_{\mathcal{M}}$ where $\mathcal{M}$ is defined over the ring $k[[\epsilon]]$ of formal power series instead of over $W(k)$; see [@Nie:Vasiuconj].  He expressed the hope that the same strategy might be used to prove Theorem \[theorem:maintheorem\]. 

$F$-truncations of $F$-crystals {#section:ftruncation}
===============================

In this section, we recall $F$-truncations modulo $p^s$ of an $F$-crystal $\mathcal{M}$ over $k$ and provide several equivalent descriptions of homomorphisms and isomorphisms between them. 

Filtrations of $F$-crystals {#subsection:filtrationfcry}
---------------------------

Let $r$ be the rank of $\mathcal{M}$. Throughout this paper, the integers $e_1 \leq \dots \leq e_r$ will always be the Hodge slopes of $\mathcal{M}$ and the integers $f_1 < \dots < f_{t}$ will always be all the distinct Hodge slopes of $\mathcal{M}$; thus $\{f_1, \dots, f_t\} = \{e_1, \dots, e_r\}$ as sets.  Clearly $f_1 = e_1$ and $f_t=e_r$. For each integer $s \geq 0$, let $h_s$ be the Hodge number of $\mathcal{M}$, that is, $h_s = \#\{e_i \; | \; e_i = s, 1 \leq i \leq r\}$. Clearly, $h_{f_i} \geq 1$ for all $1 \leq i \leq t$. We say that a $W(k)$-basis $\{v_1, v_2, \dots, v_r\}$ of $M$ is an *$F$-basis* of $\mathcal{M}$ if $\{p^{-e_1}\varphi(v_1), p^{-e_2}\varphi(v_2), \dots, p^{-e_r}\varphi(v_r)\}$ is as well a $W(k)$-basis of $M$. Every $F$-basis of $\mathcal{M}$ is also an $F$-basis of $\mathcal{M}(g)$ for all $g \in \mathrm{GL}_M(W(k))$. For each isomorphism of $F$-crystals $h : \mathcal{M}_1 \to \mathcal{M}_2$ and an $F$-basis $\mathcal{B}$ of $\mathcal{M}_1$, it is easy to see that $h(\mathcal{B})$ is an $F$-basis of $\mathcal{M}_2$.

For each positive integer $1 \leq j\leq t$, we define $I_j = \{i \; | \; e_i = f_j, 1 \leq i \leq r\}$. For an $F$-basis $\mathcal{B}$ of $\mathcal{M}$, let $\widetilde{F}^j_{\mathcal{B}}(M)$ be the free $W(k)$-submodule of $M$ generated by all $v_i$ with $i \in I_j$. We obtain two direct sum decompositions of $M$ that depend on $\mathcal{B}$ (and thus on $\mathcal{M}$): $$M = \bigoplus_{j=1}^{t} \widetilde{F}^j_\mathcal{B}(M) = \bigoplus_{j=1}^{t} \frac{1}{p^{f_j}} \varphi(\widetilde{F}^j_\mathcal{B}(M)).$$ For each $1 \leq i \leq t$, by letting $F^i_\mathcal{B}(M) := \bigoplus_{j=i}^t \widetilde{F}^j_\mathcal{B}(M)$, we get a decreasing and exhaustive filtration of $M$ $$F^{\bullet}_\mathcal{B}(M)\; : \; \widetilde{F}^t_\mathcal{B}(M) = F^t_\mathcal{B}(M) \subset F^{t-1}_\mathcal{B}(M) \subset \cdots \subset F^1_\mathcal{B}(M) = M.$$ For each $F^i_\mathcal{B}(M)$, let $\varphi_{F^i_\mathcal{B}(M)} : F^i_\mathcal{B}(M) \to M$ be the restriction of $p^{-f_i}\varphi$ to $F^i_{\mathcal{B}}(M)$. For every integer $s>0$, let $F^{\bullet}_\mathcal{B}(M)_s$ be the reduction modulo $p^s$ of the filtration $F^{\bullet}_\mathcal{B}(M)$, namely $$F^t_\mathcal{B}(M)/p^sF^t_\mathcal{B}(M)  \subset F^{t-1}_\mathcal{B}(M)/p^sF_\mathcal{B}^{t-1}(M) \subset \cdots \subset F^1_\mathcal{B}(M)/p^sF^1_\mathcal{B}(M).$$ For each $1 \leq i \leq t$, we denote by $\varphi_{F^i_\mathcal{B}(M)}[s]$ the $\sigma$-linear monomorphism $\varphi_{F^i_\mathcal{B}(M)}$ modulo $p^s$, and by $\varphi_{F^{\bullet}_\mathcal{B}(M)}[s]$ the sequence of the $\sigma$-linear monomorphisms $\varphi_{F^i_\mathcal{B}(M)}[s]$ with $1 \leq i \leq t$. By a *filtered $F$-crystal modulo $p^s$* of an $F$-crystal $\mathcal{M}$, we mean a triple of the form $$(M/p^sM, F^{\bullet}_\mathcal{B}(M)_s, \varphi_{F_\mathcal{B}^{\bullet}(M)}[s]).$$ Let $\mathcal{M}_1$ and $\mathcal{M}_2$ be two $F$-crystals with the same Hodge polygons as $\mathcal{M}$, $\mathcal{B}_1$ and $\mathcal{B}_2$ two $F$-bases of $\mathcal{M}_1$ and $\mathcal{M}_2$ respectively.  By an *isomorphism of filtered $F$-crystals modulo $p^s$* from a filtered $F$-crystal modulo $p^s$ of $\mathcal{M}_1$ to a filtered $F$-crystal modulo $p^s$ of $\mathcal{M}_2$, we mean a $W_s(k)$-linear isomorphism $f: M_1/p^sM_1 \to M_2/p^sM_2$ such that for all $1 \leq i \leq t$ we have $f(F^i_{\mathcal{B}_1}(M_1)/p^sF^i_{\mathcal{B}_1}(M_1)) = F^i_{\mathcal{B}_2}(M_2)/p^sF^i_{\mathcal{B}_2}(M_2)$ and $\varphi_{F^i_{\mathcal{B}_2}(M_2)}[s]  f = f  \varphi_{F^i_{\mathcal{B}_1}(M_1)}[s]$.

$F$-truncations
---------------

In this subsection, we recall the $F$-truncation modulo $p^s$ of an $F$-crystal defined in [@Vasiu:CBP Sect.  3.2.9].  It is the generalization of the $D$-truncation $(M/p^sM, \varphi[s], \theta[s])$ of a Dieudonné module $(M, \varphi, \theta)$; see [@Vasiu:CBP Sect.  3.2.1] for the definition of $D$-truncations. 

\[definition:ftruncation\] For every integer $s>0$, the *$F$-truncation modulo $p^s$* of an $F$-crystal $\mathcal{M}$ is the set $F_s(\mathcal{M})$ of isomorphism classes of filtered $F$-crystals modulo $p^s$ of $\mathcal{M}$ as $\mathcal{B}$ varies among all possible $F$-bases of $\mathcal{M}$. Let $\mathcal{M}_1$ and $\mathcal{M}_2$ be two $F$-crystals with the same Hodge polygon.  A $W_s(k)$-linear isomorphism $f : M_1/p^sM_1 \to M_2/p^sM_2$ is an *isomorphism of $F$-truncations modulo $p^s$* from $F_s(\mathcal{M}_1)$ to $F_s(\mathcal{M}_2)$ if for every $F$-basis $\mathcal{B}_1$ of $\mathcal{M}_1$, there exists an $F$-basis $\mathcal{B}_2$ of $\mathcal{M}_2$ such that $$f: (M_1/p^sM_1, F^{\bullet}_{\mathcal{B}_1}(M_1)_s, \varphi_{F^{\bullet}_{\mathcal{B}_1}(M_1)}[s]) \to (M_2/p^sM_2, F^{\bullet}_{\mathcal{B}_2}(M_2)_s, \varphi_{F^{\bullet}_{\mathcal{B}_2}(M_2)}[s])$$ is an isomorphism of filtered $F$-crystals modulo $p^s$.

Suppose $f$ is an isomorphism of $F$-truncations modulo $p^s$ from $F_s(\mathcal{M})$ to $F_s(\mathcal{M}(g))$. Define a set function $\Gamma_{f,s} : F_s(\mathcal{M}) \to F_s(\mathcal{M}(g))$ as follows: the image of the isomorphism class represented by $(M/p^sM, F^{\bullet}_{\mathcal{B}_1}(M)_s, \varphi_{F^{\bullet}_{\mathcal{B}_1}(M)}[s])$ under $\Gamma_{f,s}$ is the isomorphism class represented by $(M/p^sM, F^{\bullet}_{\mathcal{B}_2}(M)_s, (g\varphi)_{F^{\bullet}_{\mathcal{B}_2}(M)}[s])$ if $$f: (M/p^sM, F^{\bullet}_{\mathcal{B}_1}(M)_s, \varphi_{F^{\bullet}_{\mathcal{B}_1}(M)}[s]) \to (M/p^sM, F^{\bullet}_{\mathcal{B}_2}(M)_s, (g\varphi)_{F^{\bullet}_{\mathcal{B}_2}(M)}[s])$$ is an isomorphism of filtered $F$-crystals modulo $p^s$. It is easy to see that this function is well-defined and we shall prove that $\Gamma_{f,s}$ is a bijection of sets in Corollary \[corollary:h\_sbij\]. 

The following lemma is a generalization of [@Vasiu:CBP Lemma 3.2.2] to $F$-crystals for $G = \mathbf{GL}_M$.

\[lemma:F-truncations same isonumber\] For each $F$-crystal $\mathcal{M}$ and every $g \in \mathrm{GL}_M(W(k))$, the following two statements are equivalent:

1.   There exist $h \in \mathrm{GL}_M(W(k))$, $F$-bases $\mathcal{B}_1$ and $\mathcal{B}_2$ of $\mathcal{M}$ and $\mathcal{M}(g)$ respectively, such that the reduction $h[s]$ of $h$ modulo $p^s$ induces an isomorphism $$\label{equation:homftruncationdef}
    \mbox{\small $h[s] : (M/p^sM, F^{\bullet}_{\mathcal{B}_1}(M)_s, \varphi_{F_{\mathcal{B}_1}^{\bullet}(M)}[s]) \to (M/p^sM, F^{\bullet}_{\mathcal{B}_2}(M)_s, (g\varphi)_{F^{\bullet}_{\mathcal{B}_2}(M)}[s])$}$$ of filtered $F$-crystals modulo $p^s$.

2.   There exists an element $g_s \in \mathrm{GL}_M(W(k))$ with the property that it is congruent to $1_M$ modulo $p^s$ such that $\mathcal{M}(g_s)$ is isomorphic to $\mathcal{M}(g)$.

To prove that (2) implies (1), suppose $h \in \mathrm{GL}_M(W(k))$ is an isomorphism from $\mathcal{M}(g_s)$ to $\mathcal{M}(g)$. For every $F$-basis $\mathcal{B}$ of $\mathcal{M}(g_s)$, there is an $F$-basis $h(\mathcal{B})$ of $\mathcal{M}(g)$, and the reduction of $h$ modulo $p^s$ is an isomorphism of filtered $F$-crystals modulo $p^s$: $$h[s]: (M/p^sM, F_{\mathcal{B}}^{\bullet}(M)_s, (g_s\varphi)_{F^{\bullet}_{\mathcal{B}}(M)}[s]) \to (M/p^sM, F_{h(\mathcal{B})}^{\bullet}(M)_s, (g\varphi)_{F_{h(\mathcal{B})}^{\bullet}(M)}[s]).$$ As $g_s \equiv 1_M$ modulo $p^s$ and $\mathcal{B}$ is also an ordered $F$-basis of $\mathcal{M}$, we have a canonical identification of filtered $F$-crystals modulo $p^s$: $$\mathrm{id}[s]: (M/p^sM, F_\mathcal{B}^{\bullet}(M)_s, \varphi_{F^{\bullet}_\mathcal{B}(M)}[s]) \cong (M/p^sM, F^{\bullet}_\mathcal{B}(M)_s, (g_s\varphi)_{F^{\bullet}_\mathcal{B}(M)}[s]).$$ Composing the two isomorphisms $h[s] \circ \mathrm{id}[s] = h[s]$, we get the desired isomorphism by taking $\mathcal{B}_1 = \mathcal{B}$ and $\mathcal{B}_2 = h(\mathcal{B})$.

To prove that (1) implies (2), let $g_s = h^{-1}g\varphi h \varphi^{-1}$. We claim that $g_s$ belongs to $\mathrm{GL}_M(W(k))$, which is equivalent to $h(\varphi^{-1}(M)) \subset \varphi^{-1}(M)$. As $M = \bigoplus_{j=1}^t p^{-f_j} \varphi(\widetilde{F}_{\mathcal{B}_1}^j(M))$, it is enough to show that $$h (\widetilde{F}_{\mathcal{B}_1}^j(M)) \subset \bigoplus_{i=1}^t p^{\mathrm{max}(0,f_j-f_i)}\widetilde{F}_{\mathcal{B}_1}^i(M) = \varphi^{-1}(p^{f_j}M) \cap M.$$ Indeed, for each $v \in \widetilde{F}_{\mathcal{B}_1}^j(M) \subset F_{\mathcal{B}_1}^j(M)$, we have $h\varphi_{F_{\mathcal{B}_1}^j(M)}(v) - g\varphi_{F_{\mathcal{B}_2}^j(M)} h(v) \in p^sM$, therefore $h \varphi(v) - g \varphi h (v) \in p^{s+f_j}M$. As $v \in \widetilde{F}^j_{\mathcal{B}_1}(M)$, we know that $\varphi(v) \in p^{f_j}M$ and thus $h \varphi(v) \in p^{f_j}M$. By the last two sentences, we know that $g\varphi h(v) \in p^{f_j}M$, whence $\varphi h(v) \in p^{f_j}M$. This implies that $h(v) \in \varphi^{-1}(p^{f_j}M) \cap M$. As $$h^{-1} : (M, g\varphi) \cong (M, h^{-1}g\varphi h) = (M, g_s \varphi),$$ it remains to prove that $g_s$ is congruent to $1_M$ modulo $p^s$. As $\mathcal{B}_2$ is an $F$-basis of $\mathcal{M}(g)$, $h^{-1}(\mathcal{B}_2)$ is an $F$-basis of $\mathcal{M}(g_s)$. We have an isomorphism of filtered $F$-crystals modulo $p^s$ as follows: $$\mbox{\footnotesize $h^{-1}[s] : (M/p^sM, F^{\bullet}_{\mathcal{B}_2}(M)_s, (g\varphi)_{F^{\bullet}_{\mathcal{B}_2}(M)}[s]) \longrightarrow (M/p^sM, F^{\bullet}_{h^{-1}(\mathcal{B}_2)}(M)_s, (g_s\varphi)_{F^{\bullet}_{h^{-1}(\mathcal{B}_2)}(M)}[s]).$}$$ Composing the isomorphism with the last isomorphism, we have an isomorphism $$\mbox{\footnotesize $\mathrm{id}[s] : (M/p^sM, F^{\bullet}_{\mathcal{B}_1}(M)_s, \varphi_{F^{\bullet}_{\mathcal{B}_1}(M)}[s]) \longrightarrow (M/p^sM, F^{\bullet}_{h^{-1}(\mathcal{B}_2)}(M)_s, (g_s\varphi)_{F^{\bullet}_{h^{-1}(\mathcal{B}_2)}(M)}[s]).$}$$ For every $1 \leq j \leq t$, and for each $v \in \widetilde{F}^j_{\mathcal{B}_1}(M)$, we have $$(g_s \varphi)_{F^j_{h^{-1}(\mathcal{B}_2)}(M)}(v) - \varphi_{F_{\mathcal{B}_1}^j(M)}(v) \in p^sM.$$ This means that $g_s(p^{-f_j}\varphi(v)) - p^{-f_j}\varphi(v) \in p^{s}M$, that is, $g_s$ fixes every element of $p^{-j}\varphi(\widetilde{F}^j_{\mathcal{B}_1}(M))$ modulo $p^s$. Because $M = \bigoplus_{j=1}^t p^{-f_j} \varphi(\widetilde{F}^j_{\mathcal{B}_1}(M))$, we know that $g_s$ fixes every element of $M$ modulo $p^s$, whence $g_s \equiv 1_M$ modulo $p^s$.

\[proposition:homftruncation\] For all $g,h \in \mathrm{GL}_M(W(k))$, the reduction of $h$ modulo $p^s$ is an isomorphism from $F_s(\mathcal{M})$ to $F_s(\mathcal{M}(g))$ if and only if $h^{-1}g\varphi h \varphi^{-1} \equiv 1_M$ modulo $p^s$.

For every $h \in \mathrm{GL}_M(W(k))$, if the reduction of $h$ modulo $p^s$ is an isomorphism from $F_s(\mathcal{M})$ to $F_s(\mathcal{M}(g))$, then $h: \mathcal{M}(g_s) \to \mathcal{M}(g)$ is an isomorphism of $F$-crystals where $g_s \equiv h^{-1}g\varphi h\varphi^{-1} \equiv 1_M$ modulo $p^s$ by Lemma \[lemma:F-truncations same isonumber\]. 

If $h^{-1}g\varphi h \varphi^{-1} \equiv 1_M$ modulo $p^s$, then there exists $g_s \equiv h^{-1}g\varphi h \varphi^{-1}$ congruent to $1_M$ modulo $p^s$ such that $h$ induces an isomorphism from $\mathcal{M}(g_s)$ to $\mathcal{M}(g)$. For every $F$-basis $\mathcal{B}$ of $\mathcal{M}$, which is also an $F$-basis of $\mathcal{M}(g_s)$, we get an isomorphism of filtered $F$-crystals modulo $p^s$: $$h[s] : (M/p^sM, F^{\bullet}_\mathcal{B}(M)_s, \varphi_{F_\mathcal{B}^{\bullet}(M)}[s]) \to (M/p^sM, F^{\bullet}_{h(\mathcal{B})}(M)_s, (g\varphi)_{F_{h(\mathcal{B})}^{\bullet}(M)}[s]).  \qedhere$$

\[corollary:h\_sbij\] Let $s$ be a positive integer.  We recall that $\Gamma_{f,s} : F_s(\mathcal{M}) \to F_s(\mathcal{M}(g))$ is the function defined by an isomorphism $f$ of $F$-truncations modulo $p^s$ from $F_s(\mathcal{M})$ to $F_s(\mathcal{M}(g))$ (see the paragraph after Definition \[definition:ftruncation\] for its definition).  Then the function $\Gamma_{f,s}$ is a bijection. 

Let $h \in \mathrm{GL}_M(W(k))$ be a preimage of $f \in \mathrm{GL}_M(W_s(k))$ via the canonical surjection $\mathrm{GL}_M(W(k)) \to \mathrm{GL}_M(W_s(k))$. By Proposition \[proposition:homftruncation\], we have $h^{-1}g\varphi h \varphi^{-1} \equiv 1_M$ modulo $p^s$. Taking inverses on both hand sides, we have $\varphi h^{-1} \varphi^{-1} g^{-1} h \equiv1_M$ modulo $p^s$. After multiplying $h$ on the left and $h^{-1}$ on the right on both hand sides, we get $h\varphi h^{-1} \varphi^{-1} g^{-1} \equiv 1$ modulo $p^s$, that is, $h\varphi h^{-1} (g\varphi)^{-1} \equiv 1_M$ modulo $p^s$. Hence $h^{-1}$ defines an isomorphism of $F$-truncations modulo $p^s$ from $F_s(\mathcal{M}(g))$ to $F_s(\mathcal{M})$. This implies that $\Gamma_{f,s}$ is a bijection. 

The next corollary justifies that the isomorphism number of $F$-crystals is the right generalization of the isomorphism number of $p$-divisible groups. 

\[corollary:correctgeneralization\] Let $t_{\mathcal{M}}$ be the smallest integer such that for all $g \in \mathrm{GL}_M(W(k))$, if $F_{t_{\mathcal{M}}}(\mathcal{M})$ is isomorphic to $F_{t_{\mathcal{M}}}(\mathcal{M}(g))$, then $\mathcal{M}$ is isomorphic to $\mathcal{M}(g)$. We have $t_{\mathcal{M}} = n_{\mathcal{M}}$.

If $F_{n_{\mathcal{M}}}(\mathcal{M})$ is isomorphic to $F_{n_{\mathcal{M}}}(\mathcal{M}(g))$, then by Lemma \[lemma:F-truncations same isonumber\], there exists $g_{n_{\mathcal{M}}} \in \mathrm{GL}_M(W(k))$ with the property that $g_{n_{\mathcal{M}}} \equiv 1_M$ modulo $p^{n_{\mathcal{M}}}$ such that $\mathcal{M}(g_{n_{\mathcal{M}}})$, which is isomorphic to $\mathcal{M}$ by the definition of isomorphism numbers, is isomorphic to $\mathcal{M}(g)$. Thus $t_{\mathcal{M}} \leq n_{\mathcal{M}}$.

Let $g_{t_{\mathcal{M}}} \equiv 1_M$ modulo $p^{t_{\mathcal{M}}}$. By Proposition \[proposition:homftruncation\], $1_M[t_{\mathcal{M}}] \in \mathrm{GL}_M(W_{t_{\mathcal{M}}}(k))$ is an isomorphism from $F_{t_{\mathcal{M}}}(\mathcal{M})$ to $F_{t_{\mathcal{M}}}(\mathcal{M}({g_{t_{\mathcal{M}}}}))$. By definition of $t_{\mathcal{M}}$, $\mathcal{M}$ is isomorphic to $\mathcal{M}(g_{t_{\mathcal{M}}})$. Thus $n_{\mathcal{M}} \leq t_{\mathcal{M}}$.

Proposition \[proposition:homftruncation\] motivates the following definition of a homomorphism modulo $p^s$ between two $F$-crystals. 

\[definition:homftruncation\] A $W_s(k)$-linear map $h[s]: M_1/p^sM_1 \to M_2/p^sM_2$ is a homomorphism from $F_s(\mathcal{M}_1)$ to $F_s(\mathcal{M}_2)$ if a preimage $h \in \mathrm{Hom}_{W(k)}(M_1, M_2)$ of $h[s]$ under the canonical surjection $\mathrm{Hom}_{W(k)}(M_1, M_2) \to \mathrm{Hom}_{W_s(k)}(M_1/p^sM_1, M_2/p^sM_2)$ satisfies $\varphi_2 h \varphi_1^{-1} \equiv h$ modulo $p^s$. We call $h$ a lift of $h[s]$ and $h[s]$ a homomorphism modulo $p^s$ from $\mathcal{M}_1$ to $\mathcal{M}_2$.

A homomorphism $h[s]$ modulo $p^s$ between $\mathcal{M}_1$ and $\mathcal{M}_2$ implicitly implies that there exists a lift $h$ of $h[s]$ in $\mathrm{Hom}_{W(k)}(M_1, M_2)$ such that $\varphi_2 h \varphi^{-1}_1$ is also an element in $\mathrm{Hom}_{W(k)}(M_1, M_2)$. Note that $h[s]$ is *not* just a $W_s(k)$-linear homomorphism $h[s] : M_1/p^sM_1 \to M_2/p^sM_2$ such that $h\varphi_1 \equiv \varphi_2 h$ modulo $p^s$, although this is a consequence of the definition but it is not equivalent to the definition. 

Note that the definition of an isomorphism between two filtered $F$-crystals modulo $p^s$ requires that the two $F$-crystals have the same Hodge polygon described in Subsection \[subsection:filtrationfcry\].  In Proposition \[proposition:homftruncation\] we also require that the two $F$-crystals have the same Hodge polygon.  On the other hand, in Definition \[definition:homftruncation\], we do not require that the two $F$-crystals have the same Hodge polygon.  It is reasonable to ask if $h[s] \in \mathrm{GL}_M(W_s(k))$ and there exists a lift $h \in \mathrm{GL}_M(W(k))$ of $h[s]$ such that $\varphi_2 h \varphi_1^{-1} \equiv h$ modulo $p^s$, do $(M, \varphi_1)$ and $(M, \varphi_2)$ have the same Hodge polygon so that $h[s]$ induces an isomorphism between $F_s(\mathcal{M}_1)$ and $F_s(\mathcal{M}_2)$?  The answer is yes because if $\varphi_2 h \varphi_1^{-1} \equiv h$ modulo $p^s$, then we know that $\varphi_2 h \varphi_1^{-1} \in \mathrm{GL}_M(W(k))$. Thus $\varphi_2 h \varphi_1^{-1}(\varphi_1(M)) = \varphi_2 h(M) = \varphi_2(M)$. As a result, $\varphi_2 h \varphi_1^{-1}$ induces an isomorphism from $M/\varphi_1(M)$ to $M/\varphi_2(M)$ and thus $(M, \varphi_1)$ and $(M, \varphi_2)$ have the same Hodge polygon.  Therefore if $\mathcal{M}_1$ and $\mathcal{M}_2$ have different Hodge polygons, then $F_s(\mathcal{M}_1)$ and $F_s(\mathcal{M}_2)$ are not isomorphic modulo $p^s$.

\[proposition:anotherdefinitionofftruncation\] Let $s \geq 1$ be an integer.  A homomorphism $h[s] : M_1/p^sM_1 \to M_2/p^sM_2$ is a homomorphism from $F_s(\mathcal{M}_1)$ to $F_s(\mathcal{M}_2)$ if and only if there exists a lift $h$ of $h[s]$ in $\mathrm{Hom}_{W(k)}(M_1, M_2)$ such that for every $x \in M_1\; \backslash\; pM_1$, if $\varphi_1(x) \in p^iM_1\; \backslash\; p^{i+1}M_1$, then $h\varphi_1(x) \equiv \varphi_2h(x)$ modulo $p^{s+i}$. Moreover, if we fix an $F$-basis $\mathcal{B}_1 = \{v_1, v_2, \dots, v_r\}$ of $\mathcal{M}_1$, then the condition “for every $x\in M_1 \; \backslash \; pM_1$" in the prior sentence can be strengthen to “for all $x \in \mathcal{B}_1$". 

Let $h[s]$ be a homomorphism from $F_s(\mathcal{M}_1)$ to $F_s(\mathcal{M}_2)$, then there exists $h \in \mathrm{Hom}_{W(k)}(M_1, M_2)$ such that $\varphi_2h\varphi_1^{-1} \equiv h$ modulo $p^s$. Let $x \in M_1 \; \backslash \; pM_1$ be such that $\varphi_1(x) \in p^iM_1 \; \backslash \; p^{i+1}M_1$, whence $\frac{1}{p^i}\varphi_1(x) \in M_1 \; \backslash \; pM_1$. Plugging $\frac{1}{p^i}\varphi_1(x)$ into $\varphi_2 h \varphi_1^{-1} \equiv h$ modulo $p^s$ gives the desired congruence $h\varphi_1(x) = \varphi_2h(x)$ modulo $p^{s+i}$.

Suppose $h[s] \in \mathrm{Hom}(M_1/p^sM_1, M_2/p^sM_2)$ satisfies that for every $x \in M_1\; \backslash\; pM_1$, if $\varphi_1(x) \in p^iM_1\; \backslash\; p^{i+1}M_1$, then $h\varphi_1(x) \equiv \varphi_2h(x)$ modulo $p^{s+i}$. For every $x \in M_1 \; \backslash \; \{0\}$, there exists $l \geq 0$ such that $x \in p^lM_1 \; \backslash \; p^{l+1}M_1$. We write $\varphi_1^{-1}(x) = p^jx'$ for some $j \in \mathbb{Z}$ and $x' \in M_1 \; \backslash \; pM_1$. Therefore $\varphi_1(x') \in p^{l-j}M_1 \backslash p^{l-j+1}M_1$. Plugging $x' = p^{-j}\varphi_1^{-1}(x)$ into the congruence $\varphi_2 h \equiv h \varphi_1$ modulo $p^{s+l-j}$, we get $\varphi_2 h \varphi_1^{-1}(x) \equiv h (x)$ modulo $p^s$ as $l \geq 0$.

To prove the strengthening part, for all $x \in M_1 \; \backslash \; pM_1$, $x = \sum_{i=1}^r x_iv_i$ for some $x_i \in W(k)$, we have $\varphi_1(x) = \sum_{i=1}^r p^{e_i} \sigma(x_i)w_i$ for some $F$-basis $\{w_1, w_2, \dots, w_r\}$ of $\mathcal{M}_2$. Let $i = \mathrm{min}\{e_i+\mathrm{ord}_p(x_i) \; | \; 1 \leq i \leq r\}$. Then $\varphi_1(x) \in p^iM_1 \; \backslash \; p^{i+1}M_1$. Suppose for every $1 \leq i \leq r$, $h\varphi_1(v_i) - \varphi_1h(v_i) = p^{s+e_i}v_i'$ for some $v_i' \in  M_2 \; \backslash \; pM_2$, we conclude the proof by considering the difference $$\begin{aligned}
h\varphi_1(x) - \varphi_1h(x) &= \sum_{i=1}^r\sigma(x_i)h(\varphi_1(v_i)) - \sum_{i=1}^r \sigma(x_i)\varphi_1(h(v_i)) \\
&= \sum_{i=1}^r p^{s+e_i}\sigma(x_i)v_i' \in p^{s+i} M_2.  \qedhere\end{aligned}$$

\[corollary:anotherdefinitionofftruncation\] Let $\mathcal{M}$ be an $F$-crystal over $k$ and let $\mathcal{B} = \{v_1, v_2, \dots, v_r\}$ be an $F$-basis of $\mathcal{M}$. For all $g,h \in \mathrm{GL}_M(W(k))$, the reduction of $h$ modulo $p^s$ is an isomorphism between $F_s(\mathcal{M})$ and $F_s(\mathcal{M}(g))$ if and only if for all $v_i \in \mathcal{B}$ we have $h\varphi_1(v_i) \equiv \varphi_2h(v_i)$ modulo $p^{s+e_i}$ where $e_1 \leq e_2 \leq \cdots \leq e_r$ are the Hodge slopes of $\mathcal{M}$.

We denote by $\mathrm{Hom}_s(\mathcal{M}_1, \mathcal{M}_2)$ the (additive) group of all homomorphisms modulo $p^s$ from $\mathcal{M}_1$ to $\mathcal{M}_2$, that is, all homomorphisms from $F_s(\mathcal{M}_1)$ to $F_s(\mathcal{M}_2)$. For $i = 1, 2$, if $h_i[s] \in \mathrm{GL}_M(W_s(k))$ is an automorphism of $F_s(\mathcal{M})$, and $h_i \in \mathrm{GL}_M(W(k))$ is a lift of $h_i[s]$ such that $\varphi h_i \varphi^{-1} \equiv h_i$ modulo $p^s$, then $(h_1h_2)[s]$ is also an automorphism of $F_s(\mathcal{M})$ as $h_1h_2 \in \mathrm{GL}_M(W(k))$ is a lift of $(h_1h_2)[s]$ that satisfies $$(h_1h_2)^{-1} \varphi (h_1h_2) \varphi^{-1} \equiv h_2^{-1} (h_1^{-1} \varphi h_1 \varphi^{-1}) \varphi h_2 \varphi^{-1} \equiv h_2^{-1} \varphi h_2 \varphi^{-1} \equiv 1 \; \mathrm{modulo} \; p^s.$$ Thus all automorphisms of $F_s(\mathcal{M})$ form an abstract group $\mathrm{Aut}_s(\mathcal{M})$ under composition. 

$\mathbb{W}_s$ functor {#section:wsfunctor}
----------------------

For every affine scheme ${\bf X}$ over $\mathrm{Spec}\,W(k)$, there is a functor $\mathbb{W}_s({\bf X})$ from the category of affine schemes over $k$ to the category of sets defined as follows: For every affine scheme $\mathrm{Spec}\,R$, $$\mathbb{W}_s({\bf X})(\mathrm{Spec}\,R) := {\bf X}(W_s(R)).$$ If ${\bf X}$ is of finite type over $W(k)$, it is known that this functor is representable by an affine $k$-scheme of finite type (see [@Greenberg:Schemata p. 639 Corollary 1]), which will be denoted by $\mathbb{W}_s({\bf X})$. If in addition ${\bf X}$ is smooth over $\mathrm{Spec}\,W(k)$, then $\mathbb{W}_s({\bf X})$ is smooth.  Indeed, for every $k$-algebra $R$ and an ideal $I$ of $R$ such that $I^2=0$, the kernel of $W_s(R) \to W_s(R/I)$ is of square zero.  As ${\bf X}$ is smooth, we get that $$\mathbb{W}_s({\bf X})(R) = {\bf X}(W_s(R)) \to {\bf X}(W_s(R/I)) = \mathbb{W}_s({\bf X})(R/I)$$ is surjective by [@Neronmodels Ch.  2, Sec.  2, Prop.  6], whence $\mathbb{W}_s({\bf X})$ is smooth by the loc.  cit.  Suppose $\mathbf{X}$ is a smooth affine group scheme over $\mathrm{Spec}\,W(k)$, then $\mathbb{W}_s(\mathbf{X})$ is a smooth affine group scheme over $k$. The reduction epimorphism $W_{s+1}(R) \to W_s(R)$ naturally induces a smooth epimorphism of affine group schemes over $k$ $$\mathrm{Red}_{s+1,{\bf X}}: \mathbb{W}_{m+1}({\bf X}) \to \mathbb{W}_m({\bf X}).$$ The kernel of $\mathrm{Red}_{s+1,{\bf X}}$ is a unipotent commutative group isomorphic to $\mathbb{G}_a^{\mathrm{dim}({\bf X}_k)}$. Identifying $\mathbb{W}_1({\bf X}) = {\bf X}_k$, an inductive argument shows that $\mathrm{dim}(\mathbb{W}_s({\bf X})) = s \cdot \mathrm{dim}({\bf X}_k)$ and $\mathbb{W}_s({\bf X})$ is connected if and only if ${\bf X}_k$ is connected. 

Group schemes pertaining to $F$-truncations modulo $p^s$ {#subsection:groupscheme}
--------------------------------------------------------

In this subsection, we construct a smooth (additive) group scheme $\mathbf{Hom}_s(\mathcal{M}_1, \mathcal{M}_2)$ of finite type over $k$ such that its group of $k$-valued points is $\mathrm{Hom}_s(\mathcal{M}_1, \mathcal{M}_2)$, and a smooth (multiplicative) group scheme $\mathbf{Aut}_s(\mathcal{M})$ of finite type over $k$ such that its group of $k$-valued points is $\mathrm{Aut}_s(\mathcal{M})$.

Fix $s \geq 1$. Let $\mathcal{M}_1$ and $\mathcal{M}_2$ be two $F$-crystals over $k$. Let $r_1$ and $r_2$ be the ranks of $M_1$ and $M_2$ respectively.  We fix $W(k)$-bases $\mathcal{B}_1$ of $M_1$ and $\mathcal{B}_2$ of $M_2$ (they are not necessarily $F$-bases.)  Thus a $W(k)$-linear homomorphism $h: M_1 \to M_2$ corresponds to an $r_2 \times r_1$ matrix $X =  [h]_{\mathcal{B}_1}^{\mathcal{B}_2} = (x_{ij})_{1 \leq i \leq r_2, 1 \leq j \leq r_1}$ with respect to $\mathcal{B}_1$ and $\mathcal{B}_2$. Here and in all that follows we adopt the following convention: for any $v \in M_1$, $[h(v)]_{\mathcal{B}_2} = X[v]_{\mathcal{B}_1}$. The Frobenius of $\mathcal{M}_1$ corresponds to an $r_1 \times r_1$ matrix $U= [\varphi_1]_{\mathcal{B}_1}^{\mathcal{B}_1} = (u_{ij})_{1 \leq i, j \leq r_1}$ with respect to $\mathcal{B}_1$, and the Frobenius of $\mathcal{M}_2$ corresponds to an $r_2 \times r_2$ matrix $V=[\varphi_2]_{\mathcal{B}_2}^{\mathcal{B}_2} = (v_{ij})_{1 \leq i, j \leq r_2}$ with respect to $\mathcal{B}_2$.

Let $W = (w_{ij})_{1 \leq i, j \leq r_1}$ be the transpose of the cofactor matrix of $U$. We have $w_{ij} \in W(k)$. The matrix representation of $\varphi_2 h \varphi_1^{-1}$ with respect to $\mathcal{B}_1$ and $\mathcal{B}_2$ is $V \sigma(X) \sigma(W/\mathrm{det}(U))$. We would like to find conditions on $X$ so that the reduction of $h$ modulo $p^s$, denoted by $h[s]$, is a homomorphism from $F_s(\mathcal{M}_1)$ to $F_s(\mathcal{M}_2)$. By definition, the condition $\varphi_2h\varphi_1^{-1} \equiv h$ modulo $p^s$ is equivalent to the system of equations $$\label{equation:solvehom}
\frac{1}{\sigma(\mathrm{det}(U))}\sum_{m=1}^{r_1}\sum_{n=1}^{r_2} v_{in} \sigma(x_{nm})\sigma(w_{mj}) \equiv x_{ij} \quad \mathrm{modulo} \; \; p^s,$$ for all $1 \leq i \leq r_1$ and $1 \leq j \leq r_2$. Let $l := \mathrm{ord}_p(\mathrm{det}(U))$, and $\mathrm{det}(U)^{-1} = p^{-l}d$ where $d \in W(k) \; \backslash \; pW(k)$. Then the system of equations is equivalent to $$\label{equation:solvehom2}
\sum_{m=1}^{r_1}\sum_{n=1}^{r_2} \sigma(d)v_{in} \sigma(x_{nm})\sigma(w_{mj}) \equiv p^l x_{ij} \equiv \theta^l (\sigma^l (x_{ij})) \quad \mathrm{modulo} \; p^{s+l}.$$

If $R$ is a perfect ring, two elements $u = (u^{(0)}, u^{(1)}, \dots)$ and $w = (w^{(0)}, w^{(1)}, \dots)$ of $W(R)$ are congruent modulo $p^s$ if and only if $u^{(i)} \equiv w^{(i)}$ for all $0 \leq i \leq s-1$. This is true because $p^s = (\sigma_R \theta_R)^s = \sigma^s_R \theta^s_R$, and $\sigma_R$ is an automorphism of $W(R)$ when $R$ is perfect.  Thus over perfect rings, the system of equations is equivalent to $$\label{equation:solvehom3}
\sum_{m=1}^{r_1}\sum_{n=1}^{r_2} \sigma(d)v_{in} \sigma(x_{nm})\sigma(w_{mj}) \equiv \theta^l (\sigma^l (x_{ij})) \quad \mathrm{modulo} \; \; \theta^{s+l}(W(R)).$$

Let $x_{nm} = (x_{nm}^{(0)}, x_{nm}^{(1)}, \dots)$ and $P_{r,q}$ the polynomial with integral coefficients that computes the $q$-th coordinate of the $p$-typical Witt vector which is a product of $r$ $p$-typical Witt vectors.  Then the system of equations is equivalent to $$\label{equation:solvehomwitt1}
\sum_{m=1}^{r_1} \sum_{n=1}^{r_2} P_{4, q+l}(\sigma(d), v_{in}, \sigma(x_{nm}), \sigma(w_{mj})) - (x_{ij}^{(q)})^{p^l} = 0$$ for all $1 \leq i \leq r_1$, $1 \leq j \leq r_2$, and $0 \leq q \leq s-1$, and the equations $$\label{equation:solvehomwitt2}
\sum_{m=1}^{r_1} \sum_{n=1}^{r_2} P_{4, q}(\sigma(d), v_{in}, \sigma(x_{nm}), \sigma(w_{mj}))=0$$ for all $1 \leq i \leq r_1$, $1 \leq j \leq r_2$, and $0 \leq q \leq l-1$.

For any three non-negative integers $n_1, n_2$ and $n_3$, let $R_{n_1, n_2, n_3}$ be the polynomial $k$-algebra with variables $x_{ij}^{(q)}$ where $1 \leq i \leq n_1$, $1 \leq j \leq n_2$ and $0 \leq q \leq n_3$. Let $\mathfrak{I}$ be the ideal of $R_{r_1,r_2,s+l-1}$ generated by equations and .  Let $\mathbf{Y}_s$ be the scheme theoretic closure of $\mathbf{X}_s = \mathrm{Spec}\,R_{r_1,r_2,s+l-1}/\mathfrak{I}$ under the canonical morphism $\mathrm{Spec}\,R_{r_1,r_2,s+l-1} \to \mathrm{Spec}\,R_{r_1,r_2,s-1}$ induced by the natural inclusion $i:R_{r_1,r_2,s-1} \hookrightarrow R_{r_1,r_2,s+l-1}$. Thus $\mathbf{Y}_s$ is affine and is isomorphic to $\mathrm{Spec}\,R_{r_1,r_2,s-1}/i^{-1}(\mathfrak{I}) =: \mathrm{Spec}\,R_s$.

-   If $s \leq l$, then $i^{-1}(\mathfrak{I})$ is generated by equations for all $1 \leq i \leq r_1$, $1 \leq j \leq r_2$ and $0 \leq q \leq s-1$.

-   If $s > l$, then $i^{-1}(\mathfrak{I})$ is generated by equations for all $1 \leq i \leq r_1$, $1 \leq j \leq r_2$ and $0 \leq q \leq l-1$, and also equations for all $1 \leq i \leq r_1$, $1 \leq j \leq r_2$ and $0 \leq q \leq s-l-1$.

For each $k$-algebra $R$ (not necessarily perfect), the set of $R$-valued points $\mathbf{Y}_s(R)$ is set of all $W_s(R)$-linear maps $$h[s] :M_1 \otimes_{W_s(k)} W_s(R) \to M_2 \otimes_{W_s(k)} W_s(R)$$ with the property that there exists a lift $$h: M_1 \otimes_{W(k)} W(R) \to M_2 \otimes_{W(k)} W(R))$$ such that for each $x \in M$, if $\varphi_1(x) \in p^iM \; \backslash \; p^{i+1}M$, then we have $$h \circ (\varphi_1 \otimes_{W(k)} \sigma_{R}) (x \otimes 1_{W(R)}) \equiv (\varphi_2 \otimes_{W(k)} \sigma_{R}) \circ h(x \otimes 1_{W(R)})$$ modulo $M \otimes_{W(k)} \theta^{i+s}(W(R))$. It is clear that $\mathbf{Y}_s(R)$ has a functorial group structure under addition, and thus $\mathbf{Y}_s$ is a group scheme.  Let $\mathbf{Hom}_s(\mathcal{M}_1, \mathcal{M}_2) := (\mathbf{Y}_s)_{\mathrm{red}}$. If no confusions can occur, we denote $\mathbf{Hom}_s(\mathcal{M}_1, \mathcal{M}_2)$ by $\mathbf{H}_s$. From the construction of $(R_s)_{\mathrm{red}}$, it is clear that $\mathbf{H}_s$ is a smooth group scheme of finite type over $k$, and $\mathbf{H}_s(k) = \mathrm{Hom}_s(\mathcal{M}_1, \mathcal{M}_2)$.

The definition of $\mathbf{H}_s$ would not be very useful if it would depend on the choices of $\mathcal{B}_1$ and $\mathcal{B}_2$. We now show that $\mathbf{H}_s$ does not depend on the choices of $\mathcal{B}_1$ and $\mathcal{B}_2$. Let $\mathcal{B}_1'$ and $\mathcal{B}_2'$ be other $W(k)$-bases of $M_1$ and $M_2$ respectively.  Let $T = (t_{ij})$ be the change of basis matrix from $\mathcal{B}_1$ to $\mathcal{B}_1'$ and $T^{-1} = (t'_{ij})$ be its inverse.  Let $S = (s_{ij})$ be the change of basis matrix from $\mathcal{B}_2$ to $\mathcal{B}_2'$ and $S^{-1} = (s'_{ij})$ be its inverse.  Let $U' = [\varphi_1]_{\mathcal{B}'_1}^{\mathcal{B}'_1}$ and $V' = [\varphi_2]_{\mathcal{B}'_2}^{\mathcal{B}'_2}$ be the matrix representations of $\varphi_1$ and $\varphi_2$ with respect to $\mathcal{B}_1'$ and $\mathcal{B}_2'$ respectively.  We get that $T^{-1}U'\sigma(T) = U$, $S^{-1}V'\sigma(S) = V$ and $TU^{-1}\sigma^{-1}(T^{-1}) = U'^{-1}$. Let $W'$ be the transpose of the cofactor matrix of $U'$, then $W'/\mathrm{det}(U') = U'^{-1}$. Let $Y$ be the $r_2 \times r_1$ matrix $[h]_{\mathcal{B}'_1}^{\mathcal{B}'_2} = (y_{ij})_{1 \leq i \leq r_2, 1 \leq j \leq r_1}$ representing $h$ with respect to $\mathcal{B}'_1$ and $\mathcal{B}'_2$. Therefore we have $X = S^{-1} Y T$. By solving $V'\sigma(Y)\sigma(W'/\mathrm{det}(U')) \equiv Y$ modulo $p^s$, we get a similar system of equations like and , with $d$ replaced by $d'$, $v_{in}$ replaced by $v'_{in}$, $x_{nm}$ replaced by $y_{nm}$, and $w_{mj}$ replaced by $w'_{mj}$. They generate an ideal $\mathfrak{I}'$ of a polynomial algebra $R'_{r_1,r_2,s+l-1}$ with variables $y_{ij}^{(q)}$. We now construct an isomorphism $\iota : R_{r_1,r_2,s+l-1} \to R'_{r_1,r_2,s+l-1}$ induced by the equality $X = S^{-1}YT$. More precisely, as the $(i,j)$-entry of $S^{-1}YT$ is $\sum_{l,m} s_{il}'y_{lm}t_{mj}$, we define $\iota(x_{ij}^{(q)}) = \sum_{l,m} P_{3,q}(s'_{il},y_{lm},t_{mj})$. It is easy to see that $\iota$ is an isomorphism as its inverse $\eta$ can be constructed by the equality $Y = SXT^{-1}$ in a similar way.  Now we show that $\iota$ induces a well-defined homomorphism $\iota: R_{r_1,r_2,s+l-1}/\mathfrak{I} \to R'_{r_1,r_2,s+l-1}/\mathfrak{I}'$. Suppose that $f \in \mathfrak{I}$, then we want to show that $\iota(f) \in \mathfrak{I}'$. This is equivalent to show that if $V\sigma(X)\sigma(U^{-1}) = X$, then $V'\sigma(Y)\sigma(U'^{-1}) = Y$, assuming that $X = S^{-1}YT$. Indeed, we have $TU^{-1}\sigma^{-1}(T^{-1}) = U'^{-1}$, and $S^{-1}V'\sigma(S) = V$, we get $$\begin{gathered}
Y = SXT^{-1} = SV\sigma(X)\sigma(U^{-1})T^{-1}= S(S^{-1}V'\sigma(S))\sigma(X)\sigma(U^{-1})T^{-1} =\\
V'\sigma(S)\sigma(X)\sigma(T^{-1}U'^{-1}\sigma^{-1}(T)) T^{-1}= 
V'\sigma(Y)\sigma(U'^{-1})TT^{-1}= V'\sigma(Y)\sigma(U'^{-1}).\end{gathered}$$ Thus the induced $\iota$ is well-defined.  By the same token, we can show that the inverse $\eta$ also induces a well-defined homomorphism at the level of quotient $k$-algebras.  As $\iota$ and $\eta$ are inverses of each other, we know that $R_{r_1,r_2,s+l-1}/\mathfrak{I} \cong R'_{r_1,r_2,s+l-1}/\mathfrak{I}'$. Let $\mathbf{Y}'_s$ be the scheme theoretic closure of $\mathrm{Spec}\,R'_{r_1,r_2,s+l-1}/\mathfrak{I}'$ under the canonical morphism $\mathrm{Spec}\,R'_{r_1,r_2,s+l-1} \to \mathrm{Spec}\,R'_{r_1,r_2,s-1}$ induced by the natural inclusion $i':R'_{r_1,r_2,s-1} \hookrightarrow R'_{r_1,r_2,s+l-1}$. It is clear that $\mathbf{Y}_s$ is isomorphic to $\mathbf{Y}'_s$ as $k$-schemes.  To see that they are also isomorphic as $k$-group schemes under addition, it is enough to see that the definition $\iota$ and $\eta$ respect addition because if $X_1 =S^{-1}Y_1T$ and $X_2 = S^{-1}Y_2T$, then $X_1+X_2 = S^{-1}(Y_1+Y_2)T$. Thus the definition of $\mathbf{H}_s$ does not depend on the choice of basis. 

If $\mathcal{M}_1 = \mathcal{M}_2 = \mathcal{M}$, then $r_1=r_2=r$. In this case, we denote $\mathbf{Hom}_s(\mathcal{M}_1, \mathcal{M}_2)$ by $\mathbf{End}_s(\mathcal{M})$ or for simplicity $\mathbf{E}_s$ if no confusions can occur. 

Now we assume that $\mathcal{M}_1 = \mathcal{M}_2 = \mathcal{M}$ (thus $r_1=r_2=r$) and construct a group scheme $\mathbf{Aut}_s(\mathcal{M})$ whose $k$-valued points is $\mathrm{Aut}_s(\mathcal{M})$. Here we make use of a simple fact of Witt vectors: for any $k$-algebra $R$, an element $x \in W(R)$ is invertible if and only if $x^{(0)}$ is a unit in $R$. Put $$T_s := R_{r,r,s-1}[\frac{1}{\mathrm{det}(x_{ij}^{(0)})}]/i^{-1}(\mathfrak{I}).$$ Then $\mathrm{Spec}\,T_s(R)$ contains all the multiplicative invertible elements in $\mathbf{Y}_s(R)$. It is the set of all $W_s(R)$-linear automorphisms $$h[s] :M \otimes_{W_s(k)} W_s(R) \to M \otimes_{W_s(k)} W_s(R)$$ with the property that there exists a lift $h \in \mathrm{GL}_{M \otimes_{W(k)} W(R)}(W(R))$ of $h[s]$ such that for each $x \in M$, if $\varphi(x) \in p^iM \; \backslash \; p^{i+1}M$, then we have $$h \circ (\varphi \otimes_{W(k)} \sigma_{R}) (x \otimes 1_{W(R)}) \equiv (\varphi \otimes_{W(k)} \sigma_{R}) \circ h(x \otimes 1_{W(R)})$$ modulo $M \otimes_{W(k)} \theta^{i+s}(W(R))$. If $h_1[s], h_2[s] \in \mathrm{Spec}\, T_s(R)$, then $(h_1h_2)[s]$ is in $\mathrm{Spec}\, T_s(R)$. Here $(h_1h_2)[s]$ is $h_1h_2$ modulo $\theta^s$. It coincides with the notation that $h[s]$ is $h$ modulo $p^s$ when $R$ is perfect.  Hence $\mathrm{Spec}\, T_s(R)$ has a functorial group structure under composition and thus $\mathrm{Spec}\, T_s$ is a group scheme.  Let $\mathbf{A}_s = \mathbf{Aut}_s(\mathcal{M}) := \mathrm{Spec}\,(T_s)_{\mathrm{red}}$. Then $\mathbf{A}_s(k) = \mathrm{Aut}_s(\mathcal{M})$ is the group under composition of automorphisms of $F$-truncations modulo $p^s$ of $\mathcal{M}$. From the construction of $(T_s)_{\mathrm{red}}$, it is clear that $\mathbf{A}_s$ is a smooth group scheme of finite type over $k$ and, as a scheme, it is an open subscheme of $\mathbf{E}_s$. We now study an important invariant $\gamma_{\mathcal{M}}(s) := \mathrm{dim}(\mathbf{Aut}_s(\mathcal{M}))$ associated to $\mathcal{M}$. As $\mathbf{E}_s$ is smooth, all connected components of $\mathbf{E}_s$ have the same dimension.  Therefore $\gamma_{\mathcal{M}}(s) = \mathrm{dim}(\mathbf{E}_s)$.

\[proposition:nonincreasingdelta\] For every $l \geq 0$, the sequence $(\gamma_{\mathcal{M}}(s+l) - \gamma_{\mathcal{M}}(s))_{s \geq 1}$ is a non-increasing sequence of non-negative integers.  Therefore, we have a chain of inequalities $0 \leq \gamma_{\mathcal{M}}(1) \leq \gamma_{\mathcal{M}}(2) \leq \cdots.$

For each pair of integers $t \geq s$, there is a canonical reduction homomorphism $\pi_{t,s}: \mathbf{E}_t \to \mathbf{E}_s$. For every perfect $k$-algebra $R$, and every $h[s] \in \mathbf{E}_s(R)$, there is a lift $h$ of $h[s]$ such that $\varphi h \varphi^{-1} \equiv h$ modulo $p^s$, then $\varphi p^{t-s}h \varphi^{-1} \equiv p^{t-s}h$ modulo $p^t$. Hence we get a monomorphism $p^{t-s} : \mathbf{E}_s \to \mathbf{E}_t$ that sends $h[s]$ to $p^{t-s}h[s]$ at the level of $R$-valued points.  For every perfect $k$-algebra $R$ and every $h[t] \in \mathbf{E}_t(R)$, $h[t] = p^{t-s}(h'[s])$ for some $h'[s] \in \mathbf{E}_s(R)$ if and only if $h[s]$ belongs to the kernel of $\pi_{t,t-s}$. Hence we have an exact sequence on the level of $R$-valued points $$0  \xrightarrow{\; \; \; \; \; \; \;} \mathbf{E}_s(R) \xrightarrow{\; \; \; p^{t-s} \; \; \;} \mathbf{E}_t(R) \xrightarrow{\pi_{t,t-s}} \mathbf{E}_{t-s}(R).$$ The dimension of $\mathrm{Im}(\pi_{t,t-s})$ is equal to $\gamma_{\mathcal{M}}(t) - \gamma_{\mathcal{M}}(s) \geq 0$. Because $\pi_{s+1+l, l} = \pi_{s+l, l } \circ \pi_{s+1+l, s+l}$, $\mathrm{Im}(\pi_{s+1+l,l})$ is a subgroup scheme of $\mathrm{Im}(\pi_{s+1,l})$. Hence the dimension of $\mathrm{Im}(\pi_{s+1+l,l})$, which is $\gamma_{\mathcal{M}}(s+1+l) - \gamma_{\mathcal{M}}(s+1)$, is less than or equal to the dimension of $\mathrm{Im}(\pi_{s+l,l})$, which is $\gamma_{\mathcal{M}}(s+l) - \gamma_{\mathcal{M}}(s)$.

Recall an $F$-crystal $\mathcal{M}$ is ordinary if its Hodge polygon and Newton polygon coincide.  It is well known that the ordinary $F$-crystals over $k$ are precisely those $F$-crystals over $k$ which are direct sums of $F$-crystals of rank $1$.

\[proposition:ordinaryimpliesdimensionzero\] Let $\mathcal{M}$ be an ordinary $F$-crystal, then $\gamma_{\mathcal{M}}(s) = 0$ for all $s \geq 1$.

If $\mathcal{M}$ is ordinary, then $\mathcal{M} = \oplus_{i=1}^t \mathcal{M}_i$ where $\mathcal{M}_i$ are isoclinic ordinary $F$-crystals.  Thus there exists an $F$-basis $\mathcal{B} = \{v_1, v_2, \dots, v_r\}$ of $M$ such that $\varphi(v_i) = p^{e_i}v_i$ for $1 \leq i \leq r$. The ideal $\mathfrak{I}$ that defines the representing $k$-algebra of $\mathbf{E}_s(\mathcal{M})$ is generated by equations of the following two types:

1.   $\sigma(x_{ij}^{(r)}) - x_{ij}^{(r)}$ for all $r$ and $i, j \in I_l$ for all $1 \leq l \leq t$;

2.   $x^{(r)}_{ij}$ for all $r$ and $i, j$ that don’t belong to the same $I_l$.

It is clear now that representing $k$-algebra is finite dimensional over $k$. Thus $\mathbf{E}_s$ is of dimension zero, so is $\mathbf{A}_s$.

Isomorphism classes of $F$-truncations
======================================

In this section, we follow the ideas of [@Vasiu:dimensions] and [@Vasiu:levelm] to define a group action for each $s \geq 1$ whose orbits parametrize the isomorphism classes of $F_s(\mathcal{M}(g))$ for all $g \in \mathrm{GL}_M(W(k))$. We show that the stabilizer of the identity element of this action has the same dimension as $\mathbf{Aut}_s(\mathcal{M})$, which allows us to study the non-decreasing sequence $(\gamma_{\mathcal{M}}(i))_{i\geq 1}$ via the orbits and the stabilizers of the action.  The main result of this section is Theorem \[theorem:monotonicitygamma\], which is a partial generalization of [@Vasiu:dimensions Theorem 1].  It will play an important role in the proof of the Main Theorem in Section \[section:proofmaintheorem\]. 

Group schemes
-------------

In this subsection, we will introduce some affine group schemes that are necessary to define the group actions in order to study isomorphism classes of $F$-truncations. 

Let $\mathcal{M} = (M, \varphi)$ be an $F$-crystal over $k$. Recall $\mathrm{\bf GL}_M$ is the group scheme over $\mathrm{Spec}\,W(k)$ with the property that for every $W(k)$-algebra $S$, $\mathrm{\bf GL}_M(S)$ is the group of $S$-linear automorphism of $M \otimes_{W(k)} S$. Put $V = M \otimes_{W(k)} B(k)$, then we have canonical identifications $$\mathrm{\bf GL}_{V} = \mathrm{\bf GL}_M \times_{W(k)} \mathrm{Spec}\,B(k) = \mathrm{\bf GL}_{\varphi^{-1}(M)} \times_{W(k)}\mathrm{Spec}\,B(k).$$ Let ${\bf G}$ be the scheme theoretic closure of $\mathrm{\bf GL}_V$ in $\mathrm{\bf GL}_M \times \mathrm{\bf GL}_{\varphi^{-1}(M)}$ embedded via the composite homomorphism $$\mathrm{\bf GL}_V \xrightarrow{\Delta} \mathrm{\bf GL}_V \times \mathrm{\bf GL}_V \to \mathrm{\bf GL}_M \times \mathrm{\bf GL}_{\varphi^{-1}(M)}.$$ For any flat $W(k)$-algebra $S$, $\mathbf{G}(S)$ contains all $h \in \mathbf{GL}_{M \otimes_{W(k)} S}(S)$ with the property that $h(\varphi^{-1}(M) \otimes_{W(k)} S) = \varphi^{-1}(M) \otimes_{W(k)} S$. Let $P_{\bf G}: {\bf G} \to \mathrm{\bf GL}_M$ be the composition of the inclusion and the first projection $\mathbf{GL}_M \times \mathbf{GL}_{\varphi^{-1}(M)} \to \mathbf{\bf GL}_M$.

Let $\mathcal{B} = \{v_1, v_2, \dots, v_r\}$ be an $F$-basis of $\mathcal{M}$. There are two direct sum decompositions of $M = \bigoplus_{j=1}^{t} \widetilde{F}^j_{\mathcal{B}}(M) =  \bigoplus_{j=1}^{t} p^{-f_j} \varphi(\widetilde{F}^j_{\mathcal{B}}(M))$, which implies that $\varphi^{-1}(M) = \bigoplus_{j=1}^tp^{-f_i}\widetilde{F}^j_{\mathcal{B}}(M)$. With respect to $\mathcal{B}$, the representing $k$-algebras of the following affine group schemes are clear:

-   $\displaystyle \mathrm{\bf GL}_V = \mathrm{Spec}\,B(k)[x_{ij} \; | \; 1 \leq i, j \leq r][\frac{1}{\mathrm{det}(x_{ij})}]$;

-   $\displaystyle \mathrm{\bf GL}_M = \mathrm{Spec}\,W(k)[x_{ij} \; | \; 1 \leq i , j \leq r][\frac{1}{\mathrm{det}(x_{ij})}]$;

-   $\displaystyle \mathrm{\bf GL}_{\varphi^{-1}(M)} = \mathrm{Spec}\,W(k)[p^{\delta_{ij}}x_{ij} \; | \; 1 \leq i, j \leq r][\frac{1}{\mathrm{det}(x_{ij})}],$ where $\delta_{ij} = f_l-f_m$ if $i \in I_l$ and $j \in I_m$; see Subsection \[subsection:filtrationfcry\] for the definition of $I_l$ and $I_m$. Note that $\mathrm{det}(p^{\delta_{ij}}x_{ij}) = \mathrm{det} (x_{ij})$ as for each permutation $\pi$ of $\{1, 2, \dots, r\}$, we have $\prod_{i=1}^r p^{\delta_{i\pi(i)}}x_{i\pi(i)} = \prod_{i=1}^r x_{i\pi(i)}$.

-   $\displaystyle {\bf G} = \mathrm{Spec}\,W(k)[p^{\epsilon_{ij}}x_{ij} \; | \; 1 \leq i, j \leq r][\frac{1}{\mathrm{det}(x_{ij})}],$ where $\epsilon_{ij} = \mathrm{min}(0, \delta_{ij})$. For any affine scheme $\mathbf{H}$, let $R_{\mathbf{H}}$ be the ring such that $\mathbf{H} = \mathrm{Spec} \, R_{\mathbf{H}}$. Let $\mathcal{K}$ be the kernel of the composition $$R_{\mathbf{GL}_M} \otimes R_{\mathbf{GL}_{\varphi^{-1}(M)}} \to R_{\mathbf{GL}_V} \otimes R_{\mathbf{GL}_V} \to R_{\mathbf{GL}_V}.$$ Then $R_{\mathbf{G}} \cong R_{\mathbf{GL}_M} \otimes R_{\mathbf{GL}_{\varphi^{-1}(M)}} / \mathcal{K}$. It is easy to see that the natural homomorphism $$R_{\mathbf{GL}_M} \otimes R_{\mathbf{GL}_{\varphi^{-1}(M)}} / \mathcal{K} \to W(k)[p^{\epsilon_{ij}}x_{ij} \; | \; 1 \leq i, j \leq r][\frac{1}{\mathrm{det}(x_{ij})}]$$ is an isomorphism of $W(k)$-algebras. 

The scheme ${\bf G}$ is a connected smooth, affine group scheme over $\mathrm{Spec}\,W(k)$ of relative dimension $r^2$.

As $G$ is a principal open subscheme of the affine space $\mathrm{Spec} \, W(k)[p^{\epsilon_{ij}}x_{ij} \; | \; 1 \leq i, j \leq r]$ over $W(k)$, it is affine, smooth, integral and of relative dimension $r^2$. From this the lemma follows. 

Fix an $F$-basis $\mathcal{B}$ of $\mathcal{M}$. If $l \neq m$, let ${\bf G}_{l,m}$ be the maximal subgroup scheme of $\mathbf{GL}_M$ that fixes both $$\widetilde{F}^1_\mathcal{B}(M) \oplus \cdots \oplus \widetilde{F}^{m-1}_\mathcal{B}(M)\oplus \widetilde{F}^{m+1}_\mathcal{B}(M) \oplus \cdots \oplus \widetilde{F}^t_\mathcal{B}(M) \; \, \textrm{and} \; \, \widetilde{F}^l_\mathcal{B}(M) \oplus \widetilde{F}^m_\mathcal{B}(M)/\widetilde{F}^l_\mathcal{B}(M).$$ With respect to the $F$-basis $\mathcal{B}$, the (multiplicative) group scheme $\mathbf{G}_{l,m}$ is isomorphic to $\mathrm{Spec}\,W(k)[x_{ij} \; | \; i \in I_l, j \in I_m]$. If $R$ is a $W(k)$-algebra, then $${\bf G}_{l,m}(R) = 1_{M \otimes_{W(k)} R} + \mathrm{Hom}(\widetilde{F}^m_{\mathcal{B}}(M),\widetilde{F}^l_{\mathcal{B}}(M)) \otimes_{W(k)} R,$$ and thus ${\bf G}_{l,m} \cong \mathbb{G}_a^{h_{f_l}h_{f_m}}$. If $l = m$, let ${\bf G}_{l,l}$ be $\mathbf{GL}_{\widetilde{F}^l_{\mathcal{B}}(M)}$. With respect to the $F$-basis $\mathcal{B}$, ${\bf G}_{l,l}$ is isomorphic to $ \mathrm{Spec}\,W(k)[x_{ij} \; | \; i, j \in I_l][\frac{1}{\mathrm{det}(x_{ij})}].$ Put $$\mathbf{G}_+ = \prod_{1 \leq m < l \leq t} \mathbf{G}_{l,m} = \mathbf{G}_{t,t-1} \times \mathbf{G}_{t,t-2} \times \mathbf{G}_{t-1, t-2} \cdots \times \mathbf{G}_{3,2}   \times \mathbf{G}_{t,1} \times \cdots \times \mathbf{G}_{3,1} \times \mathbf{G}_{2,1},$$ $$\mathbf{G}_- = \prod_{1 \leq l < m \leq t} \mathbf{G}_{l,m} = \mathbf{G}_{1,2} \times \mathbf{G}_{1,3} \times \cdots \times \mathbf{G}_{1,t} \times  \mathbf{G}_{2,3} \times \cdots \mathbf{G}_{t-2,t-1} \times \mathbf{G}_{t-2,t} \times \mathbf{G}_{t-1,t},$$ $${\bf G}_0 := \prod_{l=1}^t \mathbf{G}_{l,l}, \qquad \mathrm{and} \qquad \widetilde{\bf G} := \mathbf{G}_+ \times \mathbf{G}_0 \times \mathbf{G}_-.$$ With respect to the $F$-basis $\mathcal{B}$, $$\widetilde{\bf G} = \mathrm{Spec}\,W(k)[x_{ij} \; | \; 1 \leq i,j \leq r][\frac{1}{\prod_{l=1}^t\mathrm{det}(x_{ij})_{i,j\in I_l}}].$$ Let $P_m : \widetilde{\bf G} \to \mathbf{GL}_M$ be the natural product morphism, and let $P_{{\widetilde{\bf G}}}$ be the composition $$P_m \circ \big (1_{\bf G_+} \times 1_{\bf G_0}\times \prod_{1 \leq l < m \leq t} (\bullet )^{p^{f_m-f_l}} \big ): \widetilde{\bf G} \to \widetilde{\bf G} \to \mathrm{\bf GL}_M.$$

For any morphism $Q: \mathbf{H}_1 \to \mathbf{H}_2$ of affine schemes, let $Q' : R_{\mathbf{H}_2} \to R_{\mathbf{H}_1}$ be the natural homomorphism induced by $Q$.

There is a unique morphism $P: \widetilde{\bf G} \to {\bf G}$ such that $P_{\bf G} \circ P = P_{\widetilde{\bf G}}$.

The morphism $P_{\bf G}: {\bf G} \to \mathbf{GL}_M$ at the level of $W(k)$-algebras $$P_{\bf G}' : W(k)[x_{ij} \; | \; 1 \leq i, j \leq r][\frac{1}{\mathrm{det}(x_{ij})}] \longrightarrow W(k)[p^{\epsilon_{ij}}x_{ij} \; | \; 1 \leq i, j \leq r][\frac{1}{\mathrm{det}(x_{ij})}]$$ is such that $P_{\bf G}'(x_{ij}) = x_{ij}$; see the coordinate description of $\mathbf{G}$ for the definition of $\epsilon_{ij}$. Note that $\epsilon_{ij} \leq 0$. The morphism $P_{\widetilde{\bf G}} : \widetilde{\bf G} \to \mathbf{GL}_M$ at the level of $W(k)$-algebras $$P_{\widetilde{\bf G}}': W(k)[x_{ij} | 1 \leq i,j \leq r][\frac{1}{\mathrm{det}(x_{ij})}] \longrightarrow W(k)[x_{ij} | 1 \leq i,j \leq r][\frac{1}{\prod_{l=1}^t\mathrm{det}(x_{ij})_{i,j \in I_l}}]$$ is such that $P_{\widetilde{\bf G}}'(x_{ij}) = P_ m'(p^{-\epsilon_{ij}}x_{ij})$. It is easy to check (at the level of $R$-valued points) that $P'_{\widetilde{\bf G}}(\mathrm{det}(x_{ij})) = \prod_{l=1}^t\mathrm{det}(x_{ij})_{i,j \in I_l}$. This forces $P:\widetilde{\bf G} \to {\bf G}$ to satisfy $P'(p^{\epsilon_{ij}}x_{ij}) = P'_m(x_{ij})$ and it indeed defines a $W(k)$-algebra homomorphism $$\mbox{\small $\displaystyle P': W(k)[p^{\epsilon_{ij}}x_{ij} \, | \, 1 \leq i, j \leq r][\frac{1}{\mathrm{det}(x_{ij})}] \to W(k)[x_{ij} \, | \, 1 \leq i, j \leq r][\frac{1}{\prod_{l=1}^t \mathrm{det}(x_{ij})_{i,j \in I_l}}],$}$$ as $$P'(\mathrm{det}(x_{ij})) = \mathrm{det}(P'(x_{ij})) = \mathrm{det}(P'_m(p^{-\epsilon_{ij}}x_{ij})) = \mathrm{det}(P'_{\widetilde{\bf G}}(x_{ij})) = \prod_{l=1}^t \mathrm{det}(x_{ij})_{i,j \in I_l}.  \qedhere$$

\[lemma:matrixmanipulation\] For every $k$-algebra $R$, the morphism $P$ induces a bijection on $W_s(R)$-valued points for all positive integer $s$.

We first show that $P$ induces a bijection on $W(R)$-valued points. 

We start by showing that the image of $P_{\widetilde{\bf G}}(W(R))$ is the same as the image $P_{\bf G}(W(R))$ in $\mathbf{GL}_M(W(R))$, which is $$S := \{(p^{-\epsilon_{ij}}r_{ij})_{1 \leq i,j \leq r} \; | \; r_{ij} \in W(R),\, \mathrm{det}(r_{ij}) \in W(R)^*\}.$$ As $t \times t$ block matrices, these are matrices of the type $$N = \begin{pmatrix}
N_{11} & p^{f_2-f_1}N_{12} & p^{f_3-f_1}N_{13} & \cdots & p^{f_t-f_1}N_{1t} \\
N_{21} & N_{22} & p^{f_3-f_2}N_{23} & \cdots & p^{f_t-f_2}N_{2t} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
N_{t1} & N_{t2} & N_{t3} & \cdots & N_{tt}
\end{pmatrix}$$ where $N_{lm}$ is an arbitrary $h_{f_l} \times h_{f_m}$ matrix for $1 \leq l, m \leq t$ with entries in $W(R)$, and $\mathrm{det}(N) \in W(R)^*$. We claim that $N_{ii}$ are invertible for $1 \leq i \leq t$. After reduction modulo $\theta(W(R))$, the matrix $N$ is a lower triangular block matrix.  The determinant of $N$ modulo $\theta(W(R))$ is $\prod_{i=1}^t \mathrm{det}(N_{ii})$ modulo $\theta(W(R))$, which is a unit in $R$, this implies that $\mathrm{det}(N_{ii})$ modulo $\theta(W(R))$ is a unit in $R$ and hence $\mathrm{det}(N_{ii})$ is a unit in $W(R)$.

Let $X$ be an arbitrary $t \times t$ block matrix in $\mathbf{G}_0(W(R))$ so that the diagonal blocks are denoted by $X_{ii}$. If $l>m$, let $Y_{lm}$ be an arbitrary $t \times t$ block matrix in $\mathbf{G}_{l,m}(W(R))$ with $\widetilde{Y}_{lm}$ at $(l,m)$ block entry and $0$ at everywhere else.  If $l<m$, let $Z_{lm}$ be an arbitrary $t \times t$ block matrix in $\mathbf{G}_{l,m}(W(R))$ with $p^{f_m-f_l}\widetilde{Z}_{lm}$ at $(l,m)$ block entry and $0$ at everywhere else.  We need to show that the set $$\{\prod_{1 \leq m < l \leq t} Y_{lm} X \prod_{1 \leq l <m \leq t} Z_{lm} \; | \; X, Y_{lm}, Z_{lm} \; \textrm{satisfy the conditions stated above}\}$$ is equal to the set $S$ of all $t \times t$ matrices $N$ as described above.  Here the order of the product $\prod_{1 \leq m < l \leq t} Y_{lm}$ is the same as the order in the definition of $\mathbf{G}_+$. The order of the product $\prod_{1 \leq l <m \leq t} Z_{lm}$ is the same as the order in the definition of $\mathbf{G}_-$.

We use induction on $t$. The base case when $t=1$ is trivial.  Suppose it is true for $t-1$. Then $$\prod_{1 \leq m < l \leq t-1} Y_{lm} X \prod_{1 \leq l <m \leq t-1} Z_{lm}
= \begin{pmatrix}
X_{11} & p^{f_2-f_1}X_{12} & \cdots & p^{f_{t-1}-f_1}A_{1,t-1} & 0 \\
X_{21} & X_{22} & \cdots & p^{f_{t-1}-f_2}X_{2,t-1} & 0 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
X_{t-1,1} & X_{t-1,2} & \cdots & X_{t-1,t-1} & 0 \\
0 & 0  & \cdots & 0 & X_{tt} 
\end{pmatrix}$$ satisfies $\mathrm{det}(X_{ii}) \in W(R)^*$, and each $X_{lm}$ is an arbitrary $h_{f_l} \times h_{f_m}$ matrix if $l \neq m$. We abbreviate this matrix by $\begin{pmatrix} \widetilde{X} & 0 \\ 0 & X_{tt} \end{pmatrix}$, then $$\mbox{\small $\displaystyle \prod_{1 \leq m \leq t-1} Y_{tm} \begin{pmatrix} \widetilde{X} & 0 \\ 0 & X_{tt} \end{pmatrix} \prod_{1 \leq l \leq t-1} Z_{lt}= \begin{pmatrix} 1 & 0 \\ \widetilde{Y} & 1 \end{pmatrix} \begin{pmatrix} \widetilde{X} & 0 \\ 0 & X_{tt} \end{pmatrix} \begin{pmatrix} 1 & \widetilde{Z} \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} \widetilde{X} & \widetilde{X} \widetilde{Z} \\ \widetilde{Y} \widetilde{X} & X_{tt}+\widetilde{Y}\widetilde{X}\widetilde{Z} \end{pmatrix}.$}$$ Here the matrix $\widetilde{Y} = (\widetilde{Y}_{t1}, \widetilde{Y}_{t2}, \dots, \widetilde{Y}_{t,t-1})$ has size $h_{f_t} \times (r-h_{f_t})$ and the matrix $\widetilde{Z} = (p^{f_t-f_1}\widetilde{Z}_{1t},\, p^{f_t-f_2}\widetilde{Z}_{2t},\, \dots,\, p^{f_t-f_{t-1}}\widetilde{Z}_{t-1,t})^T$ has size $(r-h_{f_t}) \times h_{f_t}$. As $\widetilde{X}$ is invertible, the right multiplication of $\widetilde{X}$ induces a bijection from the set of all $h_{f_t} \times (r-h_{f_t})$ matrices to itself.  Thus $\widetilde{Y}\widetilde{X}$ can be any $h_{f_t} \times (r-h_{f_t})$ matrix with $\widetilde{X}$ fixed and $\widetilde{Y}$ varied.  Multiplying $\widetilde{X}$ and $\widetilde{Z}$, we get $$\begin{pmatrix}
X_{11} & p^{f_2-f_1}X_{12} & \cdots & p^{f_{t-1}-f_1}X_{1,t-1} \\
X_{21} & X_{22} & \cdots & p^{f_{t-1}-f_2}X_{2,t-1}\\
\vdots & \vdots & \ddots & \vdots \\
X_{t-1,1} & X_{t-1,2} & \cdots & X_{t-1,t-1}
\end{pmatrix} \begin{pmatrix} p^{f_t-f_1}\widetilde{Z}_{1t} \\ p^{f_t-f_2}\widetilde{Z}_{2t} \\ \vdots \\  p^{f_t-f_{t-1}}\widetilde{Z}_{t-1,t} \end{pmatrix} =$$ $$\begin{pmatrix}p^{f_t-f_1}(X_{11}\widetilde{Z}_{1t} + \cdots + X_{1,t-1}\widetilde{Z}_{t-1,t}) \\ p^{f_t-f_2}(p^{f_2-f_1}X_{21}\widetilde{Z}_{1t} + \cdots + X_{2,t-1}\widetilde{Z}_{t-1,t}) \\ \vdots \\ p^{f_t-f_{t-1}}(p^{f_{t-1}-f_1}X_{t-1,1}\widetilde{Z}_{1t} + \cdots + X_{t-1,t-1}\widetilde{Z}_{t-1,t})\end{pmatrix}.$$ To show that $\widetilde{X}\widetilde{Z}$ can be any matrix of the type $$(p^{f_t-f_1}N_{1t}, p^{f_t-f_2}N_{2t}, \dots, p^{f_t-f_{t-1}}N_{t-1,t})^T$$ with $\widetilde{X}$ fixed and $\widetilde{Z}$ varied, it is enough to show that the matrix $$\begin{pmatrix}
X_{11} & X_{12} & \cdots & X_{1,t-1}\\
p^{f_2-f_1}X_{21} & X_{22} & \cdots & X_{2,t-1}\\
\vdots & \vdots & \ddots & \vdots \\
p^{f_{t-1}-f_1}X_{t-1,1} & p^{f_{t-1}-f_2}X_{t-1,2} & \cdots & X_{t-1,t-1} \\
\end{pmatrix}$$ is invertible.  But this is so because $X_{ii}$ are invertible.  When $\widetilde{X}$, $\widetilde{Y}$ and $\widetilde{Z}$ are fixed, $X_{tt}+\widetilde{Y}\widetilde{X}\widetilde{Z}$ can be an arbitrary invertible $h_t \times h_t$ matrix with $X_{tt}$ varied because $\widetilde{Y}\widetilde{X}\widetilde{Z}$ modulo $p$ is zero.  Thus we have shown that $P_{\widetilde{\bf G}}(W(R))$ and $P_{\bf G}(W(R))$ are the same in $\mathbf{GL}_M(W(R))$.

To show that $P(W(R))$ is a bijection, it is enough to show that $P_{\widetilde{\bf G}}(W(R))$ is an injection.  If this is true, as $P_{\bf G}(W(R))$ is an injection and the image of $P_{\widetilde{\bf G}}(W(R))$ and $P_{\bf G}(W(R))$ are the same, then $P(W(R))$ is a bijection.  Suppose $$\label{equation:productmatrixequal}
\prod_{1 \leq m < l \leq t} Y_{lm} X \prod_{1 \leq l < m \leq t} Z_{lm} = \prod_{1 \leq m < l \leq t} Y'_{lm} X' \prod_{1 \leq l < m \leq t} Z'_{lm},$$ and we want to show that $Y_{lm} = Y'_{lm}$ for all $1 \leq m < l \leq t$, $X=X'$ and $Z_{lm} = Z'_{lm}$ for all $1 \leq l < m \leq t$. By the definition of $Y_{lm}$ and $Z_{lm}$ it suffices to show that $\prod_{1 \leq m < l \leq t} Y_{lm} = \prod_{1 \leq m < l \leq t} Y'_{lm}$ and $\prod_{1 \leq l < m \leq t} Z_{lm} =  \prod_{1 \leq l < m \leq t} Z'_{lm}$. Equality is equivalent to $$\label{equation:productmatrixequalreduced}
(\prod_{1 \leq m < l \leq t} Y'_{lm})^{-1}\prod_{1 \leq m < l \leq t} Y_{lm}X = X \prod_{1 \leq l < m \leq t} Z'_{lm} ( \prod_{1 \leq l < m \leq t} Z_{lm})^{-1}.$$ Let $(\prod_{1 \leq m < l \leq t} Y'_{lm})^{-1}\prod_{1 \leq m < l \leq t} Y_{lm} = I + Y$ where $Y$ is strictly lower triangular and $\prod_{1 \leq l < m \leq t} Z'_{lm} ( \prod_{1 \leq l < m \leq t} Z_{lm})^{-1} = I+Z$ where $Z$ is strictly upper triangular.  The equality is equivalent to $YX = XZ$. It is easy to see that $Y = Z = 0$ as $X$ is a diagonal block matrix with invertible blocks $X_{ii}$. This completes the proof that $P(W(R))$ is a bijection. 

To show that $P(W_s(R))$ is injective, let $\bar{f}_1, \bar{f}_2 \in \widetilde{\mathbf{G}}(W_s(R))$ with lifts $f_1, f_2 \in \widetilde{\mathbf{G}}(W(R))$ respectively such that $P(W_s(R))(\bar{f}_1) = P(W_s(R))(\bar{f}_2)$. The images of $P(W(R))(f_1)$ and $P(W(R))(f_2)$ under the reduction epimorphism $\mathbf{G}(W(R)) \to \mathbf{G}(W_s(R))$ are the same.  Hence $P(W(R))(f_1)$ and $P(W(R))(f_2)$ are congruent modulo $\theta^s$. As $P(W(R))$ is a bijection, $f_1$ and $f_2$ are also congruent modulo $\theta^s$ as well.  Hence $\bar{f}_1 = \bar{f}_2$.

To show that $P(W_s(R))$ is surjective, let $\bar{f} \in {\bf G}(W_s(R))$, a lift $f \in {\bf G}(W(R))$ of $\bar{f}$ has a preimage $g \in \widetilde{\bf G}(W(R))$ such that $P(W(R))(g) = f$ because $P(W(R))$ is surjective.  Thus the image of $g$ in $\widetilde{\bf G}(W_s(R))$ is a preimage of $\bar{f}$. This shows that $P(W_s(R))$ is bijective and thus completes the proof the lemma. 

\[corollary:natidg\] The morphism $P: \widetilde{\bf G} \to {\bf G}$ induces an isomorphism $P_{W_s(k)} : \widetilde{\bf G}_{W_s(k)} \to {\bf G}_{W_s(k)}$ for each $s \geq 1$.

If $s = 1$, then $P_{W_1(k)} = P_k$. It is an isomorphism by Lemma \[lemma:matrixmanipulation\].  Suppose that $s>1$. As $R_{\bf G}$ and $R_{\widetilde{\bf G}}$ are $W_s(k)$-flat algebras, we get that $p^{s-1}R_{\bf G} / p^{s}R_{\bf G} \cong R_{\bf G}/pR_{\bf G}$ and $p^{s-1}R_{\widetilde{{\bf G}}} / p^{s}R_{\widetilde{{\bf G}}} \cong R_{\widetilde{\bf G}}/pR_{\widetilde{\bf G}}$ by the local criteria on flatness.  As a result, $p^{s-1}R_{\bf G} / p^{s}R_{\bf G} \cong p^{s-1}R_{\widetilde{{\bf G}}} / p^{s}R_{\widetilde{{\bf G}}}$. We have the following commutative diagram: $$\xymatrix{
0 \ar[r] & p^{s-1}R_{\bf G} / p^{s}R_{\bf G} \ar[r] \ar[d]^{\cong} & R_{\bf G}/p^{s}R_{\bf G} \ar[r] \ar[d]^{P'_{W_{s}(k)}} & R_{\bf G}/p^{s-1}R_{\bf G} \ar[r] \ar[d]^{P'_{W_{s-1}(k)}} & 0 \\
0 \ar[r] &  p^{s-1}R_{\widetilde{{\bf G}}} / p^{s}R_{\widetilde{{\bf G}}} \ar[r] & R_{\widetilde{\bf G}}/p^{s}R_{\widetilde{\bf G}} \ar[r] & R_{\widetilde{\bf G}}/p^{s-1}R_{\widetilde{\bf G}} \ar[r] & 0}.$$ An easy induction on $s$ using the five lemma concludes the proof of the lemma. 

Let $\mathcal{B}$ be an $F$-basis of $\mathcal{M}$. The $\mathrm{Spec}\,W(k)$-scheme $\mathbf{GL}_M$ is represented by the $W(k)$-algebra $W(k)[x_{ij} \; | \; 1 \leq i, j \leq r][1/\mathrm{det}(x_{ij})]$. We construct the cocharacter $\mu : \mathbb{G}_m \to \mathbf{GL}_M$ (that depends on $\mathcal{B}$) defined by the $k$-algebra homomorphism $$\mu' : W(k)[x_{ij} \; | \; 1 \leq i, j \leq r][1/\mathrm{det}(x_{ij})] \to W(k)[x,1/x]$$ with the property that $\mu'(x_{ij}) = 0$ if $i \neq j$ and $\mu'(x_{ii}) = (1/x)^{e_{i}}$ for $1 \leq i \leq r$ where $e_1, e_2, \dots, e_r$ are the Hodge slopes of $(M, \varphi)$. Put $\sigma_{\mathcal{M}} := \varphi \mu(B(k))(p)$. It is a $\sigma$-linear isomorphism of $M$ defined by the rule $\sigma_{\mathcal{M}}(x) = p^{-f_j}\varphi(x)$ for $x \in \widetilde{F}^j_{\mathcal{B}}(M)$. It is well known [@Fontaine:padicrep1 A.1.2.6] that there is a $\mathbb{Z}_p$-submodule $M_0 = \{ x \in M \; | \; \sigma_{\mathcal{M}}(x) = x\}$ of $M$, whose rank is the same as the rank of $M$ and such that $M = M_0 \otimes_{\mathbb{Z}_p} W(k)$. Note that the construction of $M_0$ also depends on $\mathcal{B}$. We fix a $\mathbb{Z}_p$-basis of $\mathcal{B}_0 = \{w_1, w_2, \dots, w_r\}$ of $M_0$. It induces a $\mathbb{Z}_p$-basis $\mathcal{B}_0^* = \{e_{ij} \; | \; 1 \leq i,j \leq r\}$ of $\mathrm{End}_{\mathbb{Z}_p}(M_0)$ such that $e_{ij}(w_j) = w_i$. Note that $$\mathrm{End}_{W(R)}(M \otimes_{W(k)} W(R)) = \mathrm{End}_{\mathbb{Z}_p}(M_0) \otimes_{\mathbb{Z}_p} W(R).$$ Let $h = \sum_{i,j}a_{ij}e_{ij} \in \mathrm{End}_{W(R)} (M \otimes_{W(k)} W(R))$, $a_{ij} \in W(R)$ for all $1 \leq i, j \leq r$. Define $$\bar{\sigma}_{\mathcal{M}} : \mathrm{End}_{W(R)}(M \otimes_{W(k)} W(R)) \to \mathrm{End}_{W(R)}(M \otimes_{W(k)} W(R))$$ by the formula $\bar{\sigma}_{\mathcal{M}}(h) = \sum_{i,j} \sigma_R(a_{ij})e_{ij}$ and similarly, define $$\bar{\sigma}_{\mathcal{M}} : \mathrm{End}_{W_s(R)}(M \otimes_{W(k)} W_s(R)) \to \mathrm{End}_{W_s(R)}(M \otimes_{W(k)} W_s(R))$$ by the formula $\bar{\sigma}_{\mathcal{M}}(h[s]) = (\sum_{i,j} \sigma_R(a_{ij})e_{ij})[s]$, where $h[s]$ is $h$ modulo $\theta^s$ (again this does not contradict to the previous convention that $h[s]$ is $h$ modulo $p^s$ when $R$ is perfect.)  One can easily show that the definition of $\bar{\sigma}_{\mathcal{M}}$ does not depend on $\mathcal{B}_0$ and $\mathcal{B}^*_0$ but does depend on $\mathcal{B}$. If $R$ is a perfect field, then $\overline{\sigma}_{\mathcal{M}}$ satisfies $\bar{\sigma}_{\mathcal{M}}(h) = \sigma_{\mathcal{M}} h \sigma_{\mathcal{M}}^{-1}$, which is a formula that does not depend on the choice of $\mathcal{B}_0$ or $\mathcal{B}_0^*$ but does depend on $\mathcal{B}$ since $\sigma_{\mathcal{M}}$ does. 

For every $ h \in \mathrm{End}_{W(R)}(M \otimes_{W(k)} W(R))$, define $$\varphi(h) = \bar{\sigma}_{\mathcal{M}} (\mu(B(R))(1/p) \circ h \circ \mu(B(R))(p)).$$ A priori, the definition of $\varphi(h)$ depends on the choice of the $F$-basis $\mathcal{B}$ of $\mathcal{M}$ as $\bar{\sigma}_{\mathcal{M}}$ and $\mu$ do.  As $h = \sum_{i,j} a_{ij}e_{ij}$, $a_{ij} \in W(R)$, we get $$\begin{aligned}
\varphi(h) = & \; \bar{\sigma}_{\mathcal{M}} (\mu(B(R))(1/p) \circ h \circ \mu(B(R))(p)) \\
= &\;  \bar{\sigma}_{\mathcal{M}} (\sum_{i,j} (\mu(B(k))(1/p) \circ e_{ij} \circ \mu(B(k))(p)) \otimes a_{ij}) \\
= &\; \sum_{i,j} \bar{\sigma}_{\mathcal{M}} ((\mu(B(k))(1/p) \circ e_{ij} \circ \mu(B(k))(p)) \otimes \sigma_R(a_{ij}) \\
= &\; \sum_{i,j} \sigma_{\mathcal{M}} \mu(B(k))(1/p) \circ e_{ij} \circ \mu(B(k))(p) \sigma^{-1}_{\mathcal{M}} \otimes \sigma_R(a_{ij}) \\
= &\; \sum_{i,j} (\varphi \circ e_{ij} \circ \varphi^{-1}) \otimes \sigma_R(a_{ij}).\end{aligned}$$ Thus $\varphi(h)$ is a $B(R)$-linear endomorphism of $M \otimes_{W(k)} B(R)$ defined by the following rule: let $h = \sum_i h_i \otimes c_i$ under the natural identification (basis free) $$\mathrm{End}_{W(R)}(M \otimes_{W(k)} W(R)) = \mathrm{End}_{W(k)}(M) \otimes_{W(k)} W(R).$$ For any $m \otimes b \in M \otimes_{W(k)} W(R)$, we have $\varphi(h)(m \otimes b) = \sum_i (\varphi \circ h_i \circ \varphi^{-1})(m) \otimes \sigma_R(c_i)b \in (M \otimes_{W(k)} B(k)) \otimes_{B(k)} B(R) = M \otimes_{W(k)} B(R)$. Thus the definition $\varphi(h)$ does not depend on the choice of $\mathcal{B}$. Note that $\varphi(h)$ might not be an element in $\mathrm{End}_{W(R)}(M \otimes_{W(k)} W(R))$ in general, but it is always an element in $\mathrm{End}_{W(R)}(M \otimes_{W(k)} B(R))$.

For simplicity, set $\mu(B(R))(p) = \mu(p)$ and $\mu(B(R))(1/p) = \mu(1/p)$. For every $g \in {\bf G}_{l,m}(W(R))$, the following three formulae hold. 

1.   If $m<l$, then $\mu(p)g^{p^{f_l-f_m}} \mu(1/p) = g$.

2.   If $m=l$, then $\mu(p)g \mu(1/p) = g$.

3.   If $m>l$, then $\mu(p)g\mu(1/p) = g^{p^{f_m-f_l}}$.

We first prove (1) when $m<l$. By definition, $g \in {\bf G}_{l,m}(W(R))$ if and only if $g = 1_{M \otimes W(R)}+e$ where $e \in \mathrm{Hom}(\widetilde{F}^m_{\mathcal{B}}(M),\widetilde{F}^l_{\mathcal{B}}(M)) \otimes_{W(k)} W(R)$. If $m<l$, then $$\begin{aligned}
\mu(p)g^{p^{f_l-f_m}}\mu(1/p) &= \mu(p)(1_{M \otimes_{W(k)} W(R)}+p^{f_l-f_m}e)\mu(1/p)  \\ &= 1_{M \otimes_{W(k)} W(R)} + p^{f_l-f_m}\mu(p)e\mu(1/p).\end{aligned}$$ Thus it suffices to show that $p^{f_l-f_m}\mu(p)e\mu(1/p) = e$.

As $e \in \mathrm{Hom}(\widetilde{F}^m_{\mathcal{B}}(M),\widetilde{F}^l_{\mathcal{B}}(M)) \otimes_{W(k)}W(R)$, $\mu(1/p)$ acts on $\widetilde{F}^m_{\mathcal{B}}(M) \otimes_{W(k)} W(R)$ as $p^{f_m}$, and $\mu(p)$ acts on $\widetilde{F}^l_{\mathcal{B}}(M) \otimes_{W(k)} W(R)$ as $p^{-f_l}$, we get the desired equality. 

The cases when $m=l$ and $m>l$ are similar and are left to the reader. 

For every $g \in {\bf G}_{l,m}(W(R))$, the following three formulae hold. 

1.   If $m<l$, then $\overline{\sigma}_{\mathcal{M}}(g^{p^{f_l-f_m}}) = \varphi(g)$.

2.   If $m=l$, then $\overline{\sigma}_{\mathcal{M}}(g) = \varphi(g)$.

3.   If $m>l$, then $\overline{\sigma}_{\mathcal{M}}(g) = \varphi(g^{p^{f_m-f_l}})$.

The group action $\mathbb{T}_s$ {#subsection:groupaction}
-------------------------------

Set ${\bf G}_s = \mathbb{W}_s({\bf G})$ and ${\bf D}_s = \mathbb{W}_s(\mathrm{\bf GL}_M)$. As ${\bf G}_{W_s(k)} = \widetilde{\bf G}_{W_s(k)}$, we have $\widetilde{\bf G}_s :=\mathbb{W}_s(\widetilde{\bf G}) = {\bf G}_s$. The group action $$\mathbb{T}_s : {\bf G}_s \times_k {\bf D}_s \to {\bf D}_s$$ is defined on $R$-valued points as follows: For every $h[s] \in {\bf G}_s(R)$, $g[s] \in {\bf D}_s(R)$, let $h \in \mathbf{G}(W(R))$ be a lift of $h[s]$ under the reduction epimorphism $\mathbf{G}(W(R)) \to \mathbf{G}(W_s(R))$ and $g \in \mathbf{GL}_M(W(R))$ be a lift of $g[s]$ under the reduction epimorphism $\mathbf{GL}_M(W(R)) \to \mathbf{GL}_M(W_s(R))$. Define $$\mathbb{T}_s(R)(h[s], g[s]) := (hg\varphi(h^{-1}))[s].$$ It is clear that the definition does not depend on the choices of lifts of $h[s]$ and $g[s]$ and does not depend on choice of basis. 

To see that $(hg\varphi(h^{-1}))[s] \in {\bf D}_s(R)$, let us first recall the identification ${\bf G}_{W_s(k)} = \widetilde{\bf G}_{W_s(k)}$ from Corollary \[corollary:natidg\], thus $h[s] \in \mathbf{G}_s(R) = \mathbf{G}(W_s(R)) = \mathbf{G}_{W_s(k)}(W_s(R)$ is an element of $\widetilde{\mathbf{G}}_{W_s(k)}(W_s(R))$. We can g (non-uniquely) $h[s]$ as a product $$\prod_{1 \leq m < l \leq t} h_{lm}[s]  h_0[s] \prod_{1 \leq l < m \leq t} h^{p^{f_m-f_l}}_{lm}[s],$$ where $\prod_{1 \leq m < l \leq t} h_{lm}[s] \in ({\bf G}_-)_{W_s(k)}(W_s(R))$, $h_0[s] \in (\mathbf{G}_0)_{W_s(k)}(W_s(R))$, and $\prod_{1 \leq l < m \leq t} h_{lm}[s] \in ({\bf G}_+)_{W_s(k)}(W_s(R))$. Therefore $(hg\varphi(h^{-1}))[s]$ is equal to $$\begin{aligned}
 & (\prod_{1 \leq m < l \leq t} h_{lm}  h_0 \prod_{1 \leq l < m \leq t} h^{p^{f_m-f_l}}_{lm} g\varphi( \prod_{1 \leq l < m \leq t} h^{p^{f_l-f_m}}_{lm} h^{-1}_{0} \prod_{1 \leq m < l \leq t} h^{-1}_{lm} ))[s]\\
= &(\prod_{1 \leq m < l \leq t} h_{lm} h_{0} \prod_{1 \leq l < m \leq t} h^{p^{f_m-f_l}}_{lm} g \prod_{1 \leq l < m \leq t} \varphi(h^{p^{f_l-f_m}}_{lm})\varphi(h^{-1}_{0}) \prod_{1 \leq m < l \leq t} \varphi(h^{-1}_{lm}))[s]\\
=&(\prod_{1 \leq m < l \leq t} h_{lm} h_{0} \prod_{1 \leq l < m \leq t} h^{p^{f_m-f_l}}_{lm} g \prod_{1 \leq l < m \leq t} \bar{\sigma}_{\mathcal{M}}(h^{-1}_{lm})\bar{\sigma}_{\mathcal{M}}(h^{-1}_{0}) \prod_{1 \leq m < l \leq t} \bar{\sigma}_{\mathcal{M}} ((h^{-1}_{lm})^{p^{f_l-f_m}}))[s]\\
=&\mbox{\small $\displaystyle \prod_{1 \leq m < l \leq t} h_{lm}[s] h_{0}[s] \prod_{1 \leq l < m \leq t} h^{p^{f_m-f_l}}_{lm}[s] g[s] \prod_{1 \leq l < m \leq t} \bar{\sigma}_{\mathcal{M}}(h^{-1}_{lm}[s])\bar{\sigma}_{\mathcal{M}}(h^{-1}_{0}[s]) \prod_{1 \leq m < l \leq t} \bar{\sigma}_{\mathcal{M}} ((h^{-1}_{lm}[s])^{p^{f_l-f_m}})$}\end{aligned}$$ which is in $\mathbf{D}_s(R)$. The above formula proves that $\mathbb{T}_s$ is a morphism. 

For later use, we record the following formula when $R=k$ and $s=1$. $$\label{equation:actionlevel1}
\mathbb{T}_1(k)(h[1],g[1]) = \prod_{1 \leq m < l \leq t} h_{lm}[1] h_{0}[1] g[1] \prod_{1 \leq l < m \leq t} (\bar{\sigma}_{\mathcal{M}}(h^{-1}_{lm}[1]))(\bar{\sigma}_{\mathcal{M}}(h^{-1}_{0}[1]))$$

Orbits and Stabilizers of $\mathbb{T}_s$
----------------------------------------

Let $1_M[s] \in \mathbf{D}_s(k)$. The image of the morphism $$\Psi := \mathbb{T}_s \circ (1_{\mathbf{G}_s} \times_k 1_M[s]) : \mathbf{G}_s \cong \mathbf{G}_s \times_k \mathrm{Spec}\,k \to \mathbf{G}_s \times_k \mathbf{D}_s \to \mathbf{D}_s .$$ is the orbit of $1_M[s]$, which we denoted by $\mathbf{O}_s$. Its Zariski closure $\overline{\mathbf{O}}_s$ is a closed integral subscheme of $\mathbf{D}_s$. The orbit $\mathbf{O}_s$ is a smooth connected open subscheme of $\overline{\mathbf{O}}_s$.

\[proposition:descriptionoforbits\] Let $g_1, g_2 \in \mathrm{\bf GL}_M(W(k))$. Then $g_1[s], g_2[s] \in \mathrm{\bf GL}_M(W_s(k)) = {\bf D}_s(k)$ belong to the same orbit of the action $\mathbb{T}_s$ if and only if $F_s(\mathcal{M}(g_1))$ is isomorphic to $F_s(\mathcal{M}(g_2))$.

We know that $g_1[s]$ and $g_2[s]$ belong to the same orbit of the action $\mathbb{T}_s$ if and only if there exists $h[s] \in {\bf G}_s(k)$ such that $\mathbb{T}_s(h[s],g_1[s]) = (hg_1\varphi h^{-1}\varphi^{-1})[s] = g_2[s]$. This implies that $h[s]$ is an isomorphism from $F_s(\mathcal{M}( g_1))$ to $F_s(\mathcal{M}(g_2))$.

If $h[s]$ is an isomorphism from $F_s(\mathcal{M}(g_1))$ to $F_s(\mathcal{M}(g_2))$, then $hg_1\varphi h^{-1} \varphi^{-1} \equiv g_2$ modulo $p^s$. To conclude the proof, it is enough to show that $h \in \mathbf{G}_s(k)$, but this is clear from the facts that $h(M) = M$ and $h(\varphi^{-1}(M)) \subset \varphi^{-1}(M)$.

The set of orbits of the action $\mathbb{T}_s$ is in natural bijection to the set of isomorphism classes of $F$-truncations modulo $p^s$ of $\mathcal{M}(g)$ for all $g \in \mathbf{GL}_M(W(k))$.

Let $\mathbf{S}_s$ be the fibre product defined by the following commutative diagram: $$\xymatrix{
\mathbf{S}_s \ar[r] \ar[d] & \mathrm{Spec}\,k \ar[d]^{1_M[s]} \\
\mathbf{G}_s  \ar[r]^{\Psi} & \mathbf{D}_s
}$$ It is the stabilizer of $1_M[s]$ and is a subgroup scheme of $\mathbf{G}_s$. We denote by $\mathbf{C}_s$ the reduced scheme $(\mathbf{S}_s)_{\mathrm{red}}$, and $\mathbf{C}^0_s$ the identity component of $\mathbf{C}_s$. Clearly, $$\label{equation:gammases}
\mathrm{dim}(\mathbf{S}_s) = \mathrm{dim}(\mathbf{C}_s) = \mathrm{dim}(\mathbf{C}^0_s) = \mathrm{dim}(\mathbf{G}_s) - \mathrm{dim}(\mathbf{O}_s).$$

\[example:stabilizerunipotent\] In this example, we follow the ideas of [@Vasiu:levelm Section 2.3] to discuss $\mathbb{T}_1(k)$. As a result, we will see that $\mathbf{C}_1^0$ is a unipotent group scheme over $k$. Let $(M, \varphi)$ be an $F$-crystal over $k$ such that $e_1 = 0$. By [@Vasiu:modp Section 1.8] or [@Viehmann:truncation1 Theorem 1.1], there exist an element $g \in \mathbf{GL}_M(W(k))$ with the property that $g \equiv 1_M$ modulo $p$, an $F$-basis $\mathcal{B} = \{v_1, v_2, \dots, v_r\}$ of $\mathcal{M}$, and a permutation $\pi$ on the set $I = \{1, 2, \dots, r\}$ that defines a $\sigma$-linear monomorphism $\varphi_{\pi} : M \to M$ with the property that $\varphi_{\pi}(v_i)  = p^{e_i} v_{\pi(i)}$ for all $i \in I$, such that $\mathcal{M}$ is isomorphic to $(M, g\varphi_{\pi})$. Let $\mu$ be the cocharacter defined with respect to $\mathcal{B}$ and let $\bar{\sigma}_{\mathcal{M}}$ be the $\sigma$-linear endomorphism of $\mathrm{End}_{W(R)}(M \otimes_{W(k)} W(R))$ defined with respect to $\mu$. Set $$\begin{aligned}
I_+ &= \{(i,j) \in I \times I \; | \; i \in I_l,\, j \in I_m, \; \textrm{where $m > l$} \}; \\
I_0 &= \{(i,j) \in I \times I  \; | \; i, j \in I_l \; \textrm{for some $l$}\}; \\
I_- &= \{(i,j) \in I \times I \; | \; i \in I_l,\, j \in I_m, \; \textrm{where $m < l$}\}.\end{aligned}$$ See Subsection \[subsection:filtrationfcry\] for the definition of $I_l$ and $I_m$. For $1 \leq i, j \leq r$, let $f_{i,j} \in \mathrm{End}(M)$ be such that $f_{i,j} (v_j) = v_i$ and $f_{i,j}(v_l) = 0$ for $l \neq j$. For every $1+ \bar{f}_{i,j} \in \mathbf{GL}_M(k)$, where $\bar{f}_{i,j}$ is $f_{i,j}$ modulo $p$, $\bar{\sigma}_{\mathcal{M}}(1+\bar{f}_{i,j}) = \varphi_{\pi}(1+\bar{f}_{i,j})\varphi_{\pi^{-1}}  = 1+\bar{f}_{\pi(i),\pi(j)}$. For every $h = h_+h_0h_- \in {\bf G}(k) = \widetilde{\bf G}^*(k)$, where $h_{\dagger} \in \mathbf{G}_{\dagger} $ for $\dagger \in \{+, 0, -\}$, we know that $h[1] \in \mathbf{S}_1(k) = \mathbf{C}_1(k)$ if and only if $$h_+[1]h_0[1] = \bar{\sigma}_{\mathcal{M}}(h_0[1])\bar{\sigma}_{\mathcal{M}}(h_-[1])$$ by .  This is equivalent to $$\label{equation:examplet1stabilizer}
(h_+h_0)[1] = \bar{\sigma}_{\mathcal{M}}((h_0h_-)[1])$$ Let $$(h_+h_0)[1] = 1_M[1] + \sum_{(i,j) \in I_+ \cup I_0} x_{i,j}\bar{e}_{i,j}$$ and $$(h_0h_-)[1] = 1_M[1] + \sum_{(i,j) \in I_0 \cup I_-} x_{i,j}\bar{e}_{i,j}.$$ Then can be rewritten as $$\label{equation:example1stabilizerneed}
\sum_{(i,j) \in I_+ \cup I_0} x_{i,j}\bar{e}_{i,j} = \sum_{(i,j) \in I_0 \cup I_-} x^p_{i,j} \bar{e}_{\pi(i),\pi(j)}$$ This is equivalent to three types of equalities: $$\begin{gathered}
x_{\pi(i),\pi(j)} = x^p_{i,j} \; \qquad \textrm{if} \; (i,j) \in I_- \cup I_0 \; \textrm{and} \; (\pi(i),\pi(j)) \in I_+ \cup I_0,  \label{equation:example1important1} \\
x_{\pi(i),\pi(j)} = 0 \; \qquad \textrm{if} \; (i,j) \in I_+ \; \textrm{and} \; (\pi(i),\pi(j)) \in I_+ \cup I_0, \label{equation:example1important2} \\
x^p_{i,j} = 0 \; \qquad \textrm{if} \; (i,j) \in I_- \cup I_0 \; \textrm{and} \; (\pi(i),\pi(j)) \in I_-.  \label{equation:example1important3}\end{gathered}$$ We decompose the permutation $\pi \times \pi$ on $I \times I$ into a product of disjoint cycles $\prod_u (\pi \times \pi)_u$. To ease language, we say that a pair $(i,j) \in I \times I$ is *in* $(\pi \times \pi)_u$ if $(\pi \times \pi)_u(i,j) \neq (i,j)$. To study the system of equations defined by to we consider the following three cases:

1.   Consider $(\pi \times \pi)_u$ such that all $(i,j)$ in $(\pi \times \pi)_u$ are in $I_0$. By , $x_{i,j} = x^{p^{\mathrm{ord}((\pi \times \pi)_u)}}_{i,j}$. Thus there are finitely many solutions for $x_{i,j}$.

2.   Consider $(\pi \times \pi)_u$ such that all $(i,j)$ in $(\pi \times \pi)_u$ are in $I_0 \cup I_+$ and at least one $(i,j)$ is in $I_+$. By , $x_{i,j} = 0$ for all $(i,j)$ in $(\pi \times \pi)_u$.

3.   Consider $(\pi \times \pi)_u$ such that at least one $(i,j)$ in $(\pi \times \pi)_u$ is in $I_-$. Let $\nu_{\pi}(i,j)$ be the smallest positive integer such that $$(\pi^{\nu_{\pi}(i,j)}(i), \pi^{\nu_{\pi}(i,j)}(j)) \in I_+ \cup I_-.$$ By , $x_{\pi^m(i),\pi^m(j)} = x^{p^m}_{i,j}$ for all $1 \leq m < \nu_{\pi}(i,j)$.

    -   If $(\pi^{\nu_{\pi}(i,j)}(i), \pi^{\nu_{\pi}(i,j)}(j)) \in I_-$, then $x_{\pi^m(i),\pi^m(j)} = 0$ for all $0 \leq m \leq \nu_{\pi}(i,j)$.

    -   If $(\pi^{\nu_{\pi}(i,j)}(i), \pi^{\nu_{\pi}(i,j)}(j)) \in I_+$, then $x_{\pi^m(i),\pi^m(j)} = x^{p^m}_{i,j}$ for all $1 \leq m \leq \nu_{\pi}(i,j)$.

    Thus $x_{\pi^m(i),\pi^m(j)}$ for all $1 \leq m < \nu_{\pi}(i,j)$ has finitely many solutions. 

    -   If $(\pi^{\nu_{\pi}(i,j)+1}(i),\pi^{\nu_{\pi}(i,j)+1}(j)) \in I_+ \cup I_0$, then $x_{\pi^{\nu_{\pi}(i,j)+1}(i),\pi^{\nu_{\pi}(i,j)+1}(j)}$ equals $0$ by . 

    -   If $(\pi^{\nu_{\pi}(i,j)+1}(i),\pi^{\nu_{\pi}(i,j)+1}(j)) \in I_-$, then $x_{\pi^{\nu_{\pi}(i,j)+1}(i),\pi^{\nu_{\pi}(i,j)+1}(j)}$ is not related to $x_{i,j}$.

Let $I_-^{\pi}$ be a subset of $I_-$ that contains pairs $(i,j)$ such that $(\pi^{\nu_{\pi}(i,j)}(i), \pi^{\nu_{\pi}(i,j)}(j)) \in I_+$. We conclude that $h[1] \in \mathbf{C}_1^0(k)$ if and only if the following equations hold: $$\begin{aligned}
h_+[1]h_0[1] &= 1_M[1] + \sum_{(i,j) \in I_-^{\pi}} \sum_{l=1}^{\nu_{\pi}(i,j)} x_{i,j}^{p^l}\bar{e}_{\pi^l(i),\pi^l(j)},\\
h_0[1] &=  1_M[1] + \sum_{(i,j) \in I_-^{\pi}} \sum_{l=1}^{\nu_{\pi}(i,j)-1} x_{i,j}^{p^l}\bar{e}_{\pi^l(i),\pi^l(j)},\\
h_0[1]h_-[1]  &= 1_M[1] + \sum_{(i,j) \in I_-^{\pi}} \sum_{l=0}^{\nu_{\pi}(i,j)-1} x_{i,j}^{p^l}\bar{e}_{\pi^l(i),\pi^l(j)},\end{aligned}$$ where $x_{i,j} \in I_-^{\pi}$ can take independently all values in $k$ such that $h_0[1] \in \mathbf{G}_1(k)$. This shows that $\mathrm{Lie}(\mathbf{C}^0_1) = \bigoplus_{(i,j) \in I_-^{\pi}} k \bar{e}_{i,j}$, which contains no non-zero semi-simple elements.  Thus $\mathbf{C}_1^0$ has no subgroup isomorphic to $\mathbb{G}_m$ and hence it is unipotent.  We also get that the dimension of $\mathbf{C}_1^0$ is equal to the cardinality of $I_-^{\pi}$. Therefore the dimension of $\mathbf{O}_1$ is equal to the cardinality of the set $I^2 - I_-^{\pi}$.

For every $s \geq 1$, the connected smooth group scheme $\mathbf{C}^0_s$ is unipotent. 

We proceed by induction.  The base case $s=1$ is checked in Example \[example:stabilizerunipotent\].  Suppose $\mathbf{C}^0_{s-1}$ is unipotent.  The image of $\mathbf{C}^0_{s}$ under the reduction map $\mathrm{Red}_{s,\mathbf{G}} : \mathbf{G}_s \to \mathbf{G}_{s-1}$ is in $\mathbf{C}^0_{s-1}$, and thus is unipotent.  The kernel of $\mathbf{C}^0_s \to \mathbf{C}^0_{s-1}$ is in the kernel of $\mathrm{Red}_{s,\mathbf{G}}$, and thus is unipotent.  Therefore $\mathbf{C}^0_s$ is an extension of unipotent group schemes, and thus is unipotent; see [@SGA3II Exp.  XVII, Prop 2.2]. 

We construct a homomorphism $\Lambda_s : \mathbf{C}_s \to \mathbf{A}_s$ as follows.  For every $k$-algebra $R$, let $h[s] \in \mathbf{C}_s(R)$. Thus $\varphi(h[s]) = h[s]$. Fix a $\mathbb{Z}_p$-basis $\mathcal{B}_0 = \{v_1, \dots, v_r\}$ of $\mathcal{M}_0$. Let $B_0^* = \{e_{ij}\}$ be the standard $\mathbb{Z}_p$-basis of $\mathrm{End}_{\mathbb{Z}_p}(M_0)$ induced by $B_0$. If $h = \sum_{i,j}e_{ij} \otimes a_{ij} \in \mathrm{End}_{\mathbb{Z}_p}(M_0) \otimes W(R) = \mathrm{End}_{W(R)}(M \otimes_{W(k)} W(R))$, where $a_{ij} \in W(R)$, then $\varphi(h[s]) = h[s]$ is equivalent to $$\label{equation:hinstabilizer}
\sum_{i,j} \varphi e_{ij} \varphi^{-1} \otimes \sigma_R(a_{ij}) \equiv \sum_{i,j} e_{ij} \otimes a_{ij}  \quad \textrm{modulo}  \; \; \theta^s(W(R)).$$ Let $C = (c_{ij})$ be the matrix representation of $\varphi$ with respect to $\mathcal{B}_0$ and $C^{-1} = (c'_{ij})$ be its inverse.  Using the matrix notation, can be restated as $$\label{equation:hinstablizer2}
(c_{ij})(\sigma_R(a_{ij}))(\sigma_R(c'_{ij})) \equiv (a_{ij}) \quad \textrm{modulo}  \; \;  \theta^s(W(R)).$$ This implies that a lift $h$ of $h[s]$ satisfies the equation that defines $\mathbf{A}_s$. Thus we can define $\Lambda_s(R)(h[s]) = h[s]$.

\[lemma:connectiondimension\] The homomorphism $\Lambda_s(k) : \mathbf{C}_s(k) \to \mathbf{A}_s(k)$ is an isomorphism.  Therefore, $\Lambda_s$ is a finite epimorphism and thus $\mathrm{dim}(\mathbf{C}_s)  = \gamma_{\mathcal{M}}(s)$.

The group $\mathbf{C}_s(k)$ consists of all $h \in \mathbf{G}_s(k)$ such that $h \equiv \varphi h \varphi^{-1}$ modulo $p^s$, which are exactly all automorphisms of $F_s(\mathcal{M})$. As $\mathbf{A}_s(k)$ is also the group of automorphisms of $F_s(\mathcal{M})$ and $\Lambda_s(k)$ is the identity map, we know that they are isomorphic. 

As $\Lambda_s(k)$ is an isomorphism, $\Lambda_s$ is a finite epimorphism.  Therefore $\mathrm{dim}(\mathbf{C}_s) =\mathrm{dim}(\mathbf{C}^0_s) = \mathrm{dim}(\mathbf{A}^0_s) = \mathrm{dim}(\mathbf{A}_s)$, which by definition is $\gamma_{\mathcal{M}}(s)$.

Let $\mathbf{T}_{s+1}$ be the reduced group of the group subscheme $\mathrm{Red}^{-1}_{s+1, \mathbf{G}}(\mathbf{C}_s)$ of $\mathbf{G}_{s+1}$, and let $\mathbf{T}_{s+1}^0$ be its identity component.  We have a short exact sequence $$\label{equation:sestc}
1 \to \mathrm{Ker}(\mathrm{Red}_{s+1,\mathbf{G}}) \to \mathbf{T}_{s+1}^0 \to \mathbf{C}_s^0 \to 1.$$ Thus $\mathbf{T}^0_{s+1}$ is unipotent as $\mathrm{Ker}(\mathrm{Red}_{s+1,\mathbf{G}})$ and $ \mathbf{C}_s^0$ are.  We have the following equality $$\label{equation:dimtrc}
\mathrm{dim}(\mathbf{T}_{s+1}^0) = \mathrm{dim}(\mathrm{Ker}(\mathrm{Red}_{s+1,\mathbf{G}})) + \mathrm{dim}(\mathbf{C}_s^0) = r^2 + \mathrm{dim}(\mathbf{C}_s^0).$$ By Lemma \[lemma:connectiondimension\] and , we know that $$\mathrm{dim}(\mathbf{T}_{s+1}^0) = r^2 + \gamma_{\mathcal{M}}(s).$$ By and the fact that $\mathrm{Red}_{s+1,\mathbf{G}}$ is an epimorphism whose kernel has dimension $r^2$, we know that $$\label{equation:gammadiff}
\gamma_{\mathcal{M}}(s+1) - \gamma_{\mathcal{M}}(s)  = r^2 - \mathrm{dim}(\mathbf{O}_{s+1}) + \mathrm{dim}(\mathbf{O}_s).$$

Let $\mathbf{V}_{s+1}$ be the inverse image of the point $1_M[s] \in \mathbf{D}_s(k)$. It is isomorphic to the kernel of $\mathrm{Red}_{s+1, \mathbf{D}}$ and thus isomorphic to $\mathbb{A}_{k}^{r^2}$. The inverse image of $\mathbf{O}_s$ under $\mathrm{Red}^{-1}_{s+1, \mathbf{D}}$ in $\mathbf{D}_{s+1}$ is a union of orbits and $\mathbf{O}_{s+1}$ is one of them.  Let $\mathscr{O}_{s+1, s}$ be the set of orbits of the action $\mathbb{T}_{s+1}$ that is contained in $\mathrm{Red}^{-1}_{s+1,\mathbf{D}}(\mathbf{O_{s}})$. Every orbit in $\mathscr{O}_{s+1,s}$ intersects $\mathbf{V}_{s+1}$ nontrivially. 

We now give another description of $\mathscr{O}_{s+1, s}$ in terms of $F$-truncations modulo $p^s$ of $F$-crystals.  Let $\mathscr{I}_{s}$ be the set of all $F$-crystals $\mathcal{M}(g)$ with $g \in \mathbf{GL}_M(W(k))$ up to $F$-truncations modulo $p^{s}$ isomorphisms.  In other words, if $F_{s}(\mathcal{M}(g_1))$ is isomorphic to $F_{s}(\mathcal{M}(g_2))$, then we identify $\mathcal{M}(g_1)$ and $\mathcal{M}(g_2)$ in $\mathscr{I}_s$. By Proposition \[proposition:descriptionoforbits\], we know that there is a bijection between the set of orbits of $\mathbb{T}_s$ and $\mathscr{I}_{s}$.

\[proposition:bijectionbetweenlifting\] There is a bijection between $\mathscr{O}_{s+1,s}$ and the subset of $\mathscr{I}_{s+1}$ that contains all $\mathcal{M}(g)$ (up to $F$-truncation modulo $p^{s+1}$ isomorphism) such that $F_s(\mathcal{M}(g))$ is isomorphic to $F_s(\mathcal{M})$. Therefore, $\mathscr{O}_{s+1,s}$ has only one orbit if $s \geq n_{\mathcal{M}}$.

The first part of the proposition follows from the following fact: for every $g \in \mathbf{GL}_M(W(k)$, an orbit of $\mathbb{T}_{s+1}$ that contains the $F$-truncation modulo $p^{s+1}$ of the $F$-crystal $\mathcal{M}(g)$ is in $\mathrm{Red}^{-1}_{s+1,\mathbf{D}}(\mathbf{O_{s}})$ if and only if $F_s(\mathcal{M}(g))$ is isomorphic to $F_s(\mathcal{M})$.

If $s \geq n_{\mathcal{M}}$, let $\mathcal{M}(g)$ be an $F$-crystal such that $F_s(\mathcal{M}(g))$ is isomorphic to $F_s(\mathcal{M})$. By Corollary \[corollary:correctgeneralization\], $\mathcal{M}(g)$ is isomorphic to $\mathcal{M}$. Thus $\mathscr{O}_{s+1,s}$ contains only one element by the first part of the proposition. 

Monotonicity of $\gamma_{\mathcal{M}}(i)$ {#subsection:monotonicityofgamma}
-----------------------------------------

\[lemma:keylemmastrictseq\] The following two statements are equivalent:

(i) $\mathrm{dim}(\mathbf{O}_{s+1}) = \mathrm{dim}(\mathbf{O}_s) + r^2$;

(ii) $\mathbf{V}_{s+1} \subset \mathbf{O}_{s+1}$.

As $\mathrm{Red}_{s+1, \mathbf{D}}: \mathbf{O}_{s+1} \to \mathbf{O}_s$ is faithfully flat, the fibers of this morphism are equidimensional.  Hence we have $$\label{equation:dimensionorbits}
\mathrm{dim}(\mathbf{O}_{s+1}) = \mathrm{dim}(\mathbf{O}_s) + \mathrm{dim}(\mathbf{V}_{s+1} \cap \mathbf{O}_{s+1}).$$

If (i) holds, as $\mathrm{dim}(\mathbf{V}_{s+1} \cap \mathbf{O}_{s+1}) = \mathrm{dim}(\mathbf{O}_{s+1}) - \mathrm{dim}(\mathbf{O}_s) = r^2 = \mathrm{dim}(\mathbf{V}_{s+1})$, $\mathbf{V}_{s+1} \cap \mathbf{O}_{s+1}$ is open in $\mathbf{V}_{s+1}$.

Consider the action $\mathbb{T}^0_{s+1} : \mathbf{T}^0_{s+1} \times_k \mathbf{V}_{s+1} \to \mathbf{V}_{s+1}$. By [@Springer:linearalgebraicgroups Proposition 2.4.14], we know that all the orbits of $\mathbb{T}^0_{s+1}$ are closed.  As the orbits of the action $\mathbb{T}_{s+1} : \mathbf{T}_{s+1} \times_k \mathbf{V}_{s+1} \to \mathbf{V}_{s+1}$ is a finite union of the orbits of the action $\mathbb{T}^0_{s+1}$, we know that the orbits of the action $\mathbb{T}_{s+1}$ is also closed.  The orbit of $1_M[s+1]$ under the action of $\mathbb{T}_{s+1}$ is $(\mathbf{V}_{s+1} \cap \mathbf{O}_{s+1})_{\mathrm{red}}$. Because it is an open, closed and dense orbit of $\mathbb{T}_{s+1}$, we know that $\mathbf{V}_{s+1} \cap \mathbf{O}_{s+1} = \mathbf{V}_{s+1}$. Hence $\mathbf{V}_{s+1} \subset \mathbf{O}_{s+1}$.

If (ii) holds, as $\mathrm{dim}(\mathbf{V}_{s+1} \cap \mathbf{O}_{s+1}) = \mathrm{dim}(\mathbf{V}_{s+1}) = r^2$, (i) follows from . 

\[corollary:keycorollarystrictseq\] $\gamma_{\mathcal{M}}(s+1) = \gamma_{\mathcal{M}}(s)$ if and only if $\mathscr{O}_{s+1,s}$ has only one element. 

The first part of the Lemma \[lemma:keylemmastrictseq\] is equivalent to $\gamma_{\mathcal{M}}(s+1) = \gamma_{\mathcal{M}}(s)$ and the second part of the Lemma \[lemma:keylemmastrictseq\] is equivalent to $\mathscr{O}_{s+1,s}$ has only one element. 

\[theorem:monotonicitygamma\] For every $F$-crystal $\mathcal{M}$, we have $$0 \leq \gamma_{\mathcal{M}}(1) < \gamma_{\mathcal{M}}(2) < \cdots < \gamma_{\mathcal{M}}(n_{\mathcal{M}}) = \gamma_{\mathcal{M}}(n_{\mathcal{M}}+1) = \gamma_{\mathcal{M}}(n_{\mathcal{M}}+2) = \cdots$$

We first show that for every $1 \leq i \leq n_{\mathcal{M}}-1$, $\gamma_{\mathcal{M}}(i) \neq \gamma_{\mathcal{M}}(i+1)$. Suppose the contrary, then by Proposition \[proposition:nonincreasingdelta\], $\gamma_{\mathcal{M}}(i) = \gamma_{\mathcal{M}}(j)$ for all $j \geq i$. In particular, we have $\gamma_{\mathcal{M}}(n_{\mathcal{M}}) = \gamma_{\mathcal{M}}(n_{\mathcal{M}}-1)$. By Corollary \[corollary:keycorollarystrictseq\], $\mathscr{O}_{n_{\mathcal{M}}, n_{\mathcal{M}}-1}$ contains one element.  Let $\mathcal{M}(g)$ be an $F$-crystal such that $F_{n_{\mathcal{M}}-1}(\mathcal{M}(g))$ is isomorphic to $F_{n_{\mathcal{M}}-1}(\mathcal{M})$, by Proposition \[proposition:bijectionbetweenlifting\], there is a unique $\mathcal{M}(g)$ up to $F$-truncation modulo $p^{n_{\mathcal{M}}}$ such that $F_{n_{\mathcal{M}}-1}(\mathcal{M}(g))$ is isomorphic to $F_{n_{\mathcal{M}}-1}(\mathcal{M})$, thus $F_{n_{\mathcal{M}}}(\mathcal{M}(g))$ is isomorphic to $F_{n_{\mathcal{M}}}(\mathcal{M})$. By Lemma \[lemma:F-truncations same isonumber\], $\mathcal{M}(g)$ is isomorphic to $\mathcal{M}$. Hence we conclude that $n_{\mathcal{M}}-1$ is the isomorphism number of $\mathcal{M}$, which is a contradiction. 

If $s \geq n_{\mathcal{M}}$, then every $F$-crystal $\mathcal{M}(g)$ such that $F_s(\mathcal{M}(g))$ is isomorphic to $F_s(\mathcal{M})$, is isomorphic to $\mathcal{M}$. Therefore $F_{s+1}(\mathcal{M}(g))$ is isomorphic to $F_{s+1}(\mathcal{M})$, whence $\mathscr{O}_{s+1,s}$ has only one element.  By Corollary \[corollary:keycorollarystrictseq\], $\gamma_{\mathcal{M}}(s+1) = \gamma_{\mathcal{M}}(s)$ for all $s \geq n_{\mathcal{M}}$.

We have a converse of Proposition \[proposition:ordinaryimpliesdimensionzero\]. 

\[proposition:dimensionzeroimpliesordinary\] If there exists an $s \geq 1$ such that $\gamma_{\mathcal{M}}(s) = 0$, then $\mathcal{M}$ is ordinary. 

For some $s \geq 1$, if $\gamma_{\mathcal{M}}(s) = 0$, we know that $\gamma_{\mathcal{M}}(1) = 0$ by Theorem \[theorem:monotonicitygamma\].  By Lemma \[lemma:connectiondimension\], we can assume that $\mathrm{dim}(\mathbf{C}_1^0) = 0$. Hence $|I_-^{\pi}| = 0$; see Example \[example:stabilizerunipotent\] for the definition of $I_-^{\pi}$.

As $(M, \varphi)$ is isomorphic to $(M, g\varphi_{\pi})$ for some $g \equiv 1$ modulo $p$ and the isomorphism number of an ordinary $F$-crystal is less than or equal to $1$ (for example, see [@Xiao:computing Section 2.3]), in order to show that $\mathcal{M}$ is ordinary, it is enough to show that $(M, \varphi_{\pi})$ is ordinary.  Write $\pi$ as a product of disjoint cycle $\pi_u$, it is clear that $(M, \varphi_{\pi}) = \bigoplus_u (M, \varphi_{\pi_u})$. To show that $\mathcal{M}$ is ordinary we can assume that $\pi$ is a cycle and show that $(M, \varphi_{\pi})$ is isoclinic ordinary.  Let $r$ be the rank of $M$.

As $\pi$ is an $r$-cycle, we know that $(\pi \times \pi) = \prod_{u=1}^r (\pi \times \pi)_u$ and each $(\pi \times \pi)_u$ is an $r$-cycle (as a permutation of $I \times I$).  Recall a pair $(i,j) \in I \times I$ is said to be in $(\pi \times \pi)_u$ if and only if $(\pi \times \pi)_u(i,j) \neq (i,j)$. It is easy to see that if there is a pair $(i,j) \in I_+$ in $(\pi \times \pi)_u$, then there is also a pair $(i,j) \in I_-$ in $(\pi \times \pi)_u$, and vice versa.  Since $I_-^{\pi}$ is an empty set, we know that there is no $(\pi \times \pi)_u$ such that $(\pi \times \pi)_u$ sends an element in $I_-$ to an element in $I_+$ by an argument used in Example \[example:stabilizerunipotent\].  This means that if there is an element in $I_-$ (or $I_+$ respectively) that is also in $(\pi \times \pi)_u$, then there are elements also elements in $I_0$ and in $I_+$ (or $I_-$ respectively) that are in $(\pi \times \pi)_u$.

The fact that $I_-^{\pi}$ is empty means that for all $(i,j) \in I_-$, if $\nu_{\pi}(i,j)$ is the smallest positive integer such that $(\pi^{\nu_{\pi}(i,j)}(i), \pi^{\nu_{\pi}(i,j)}(j)) \in I_+ \cup I_-$, then it is in $I_-$. Start with an element $(i,j) \in I_-$, and apply this fact recursively.  We can see that for every integer $n$, $(\pi^n(i), \pi^n(j)) \notin I_+$. This is a contradiction as we know that there must be some element in $I_+$ that is in $(\pi \times \pi)_u$. Therefore we conclude that every element in $(\pi \times \pi)_u$ is in $I_0$. This means that all the Hodge slopes of $(M, \varphi_{\pi})$ are equal and hence $(M, \varphi_{\pi})$ is isoclinic ordinary. 

The inequality $\gamma_{\mathcal{M}}(1) \geq 0$ is an equality if and only if $\mathcal{M}$ is ordinary.  When the equality holds, we have $\gamma_{\mathcal{M}}(s)  = 0$ for all $s \geq 1$.

Invariants
==========

In this section, we introduce several invariants of $F$-crystals over $k$. They are the generalizations of the $p$-divisible groups case introduced in [@Vasiu:traversosolved].  It will turn out that these invariants are all equal to the isomorphism number.  They provide a good source of computing the isomorphism number from different points of view.  All the proofs of this section follow closely the ones of [@Vasiu:traversosolved]. 

Notations
---------

Recall that for every $F$-crystal $\mathcal{M} = (M, \varphi)$ and every field extension $k \subset k'$ with $k'$ perfect, we have an $F$-crystal over $k'$ $$\mathcal{M}_{k'} = (M_{k'},\varphi_{k'}) := (M \otimes_{W(k)} W(k'), \varphi \otimes \sigma_{k'}).$$ We denote by $\mathcal{M}^* = (M^*, \varphi)$ the dual of $\mathcal{M}$, where $M^* = \mathrm{Hom}_{W(k)}(M,W(k))$ and $\varphi (f) = \varphi f \varphi^{-1}$ for $f \in M^*$. Note that the pair $(M^*, \varphi)$ is not an $F$-crystal in general, it is just a latticed $F$-isocrystal (meaning that $\varphi$ is an isomorphism after tensored with $B(k)$ but $\varphi(M^*) \not\subset M^*$ in general).  We denote by $H_{\infty} = \mathrm{Hom}(\mathcal{M}_1, \mathcal{M}_2)$ the (additive) group of all homomorphisms of $F$-crystals from $\mathcal{M}_1$ to $\mathcal{M}_2$. It is a finitely generated $\mathbb{Z}_p$-module.  For every integer $s \geq 1$, let $H_s = \mathrm{Hom}_s(\mathcal{M}_1, \mathcal{M}_2) = \mathbf{Hom}_s(\mathcal{M}_1, \mathcal{M}_2)(k)$ be the (additive) group of all homomorphisms from $F_s(\mathcal{M}_1)$ to $F_s(\mathcal{M}_2)$. It is a $\mathbb{Z}_p/p^s\mathbb{Z}_p$-module but not necessarily finitely generated in general.  We denote by $\pi_{\infty, s} : H_{\infty} \to H_s$ and $\pi_{t, s}: H_t \to H_s, t \geq s$ the natural projections.  We have two exact sequences: $$0 \xrightarrow{\; \; \; \; \; \; \;} H_{\infty} \xrightarrow{\; \; \; p^s \; \;\;} H_{\infty} \xrightarrow{\pi_{\infty,s}} H_s,$$ and $$0  \xrightarrow{\; \; \; \; \; \; \;} H_s \xrightarrow{\; \; \; p \; \; \;} H_{s+1} \xrightarrow{\pi_{s+1,1}} H_1.$$ Let $r_1$ and $r_2$ be the ranks of $M_1$ and $M_2$ respectively. 

The endomorphism number
-----------------------

In this subsection, we generalize the endomorphism number defined in [@Vasiu:traversosolved Section 2] for $p$-divisible groups.  The following proposition is a generalization of [@Vasiu:traversosolved Lemma 2.1].  For the sake of generality, we will work with the homomorphism version. 

\[proposition:eexists\] There exists a non-negative integer $e_{\mathcal{M}_1,\mathcal{M}_2}$ which depends only on $\mathcal{M}_1$ and $\mathcal{M}_2$ with the following property: For every positive integer $n$ and every non-negative integer $e$, we have $\mathrm{Im}(\pi_{\infty, n}) = \mathrm{Im}(\pi_{n+e, n})$ if and only if $e \geq e_{\mathcal{M}_1, \mathcal{M}_2}$.

We first prove that $e_{\mathcal{M}_1,\mathcal{M}_2}$ exists for each $n$ and then prove that it does not depend on $n$. Note that $\pi_{\infty, n} = \pi_{n+1,n} \circ \pi_{\infty, n+1}$ and $\pi_{n+e, n} = \pi_{n+1,n} \circ \pi_{n+e,n+1}$. If $\mathrm{Im}(\pi_{\infty,n+1}) = \mathrm{Im}(\pi_{n+e,n+1})$ for all $e-1 \geq e_{\mathcal{M}_1, \mathcal{M}_2}(n+1)$, then $\mathrm{Im}(\pi_{\infty,n}) = \mathrm{Im}(\pi_{n+e,n})$. Thus $e_{\mathcal{M}_1, \mathcal{M}_2}(n) \leq e_{\mathcal{M}_1, \mathcal{M}_2}(n+1)+1$. Hence to show that $e_{\mathcal{M}_1, \mathcal{M}_2}(n)$ exists for all positive integer $n$, it is enough to show that $e_{\mathcal{M}_1, \mathcal{M}_2}(n)$ exists for sufficient large $n$.

Let $H'_n := \mathrm{Hom}_{W_n(k)}((M_1/p^nM_1, \varphi_1), (M_2/p^nM_2, \varphi_2))$. It is the (additive) group of all $W_n(k)$-linear homomorphisms $h:M_1/p^nM_1 \to M_2/p^nM_2$ such that $\varphi_2 h \equiv h \varphi_1$ modulo $p^n$. Thus $H_n$ is a subgroup of $H_n'$.

The existence of $e_{\mathcal{M}_1,\mathcal{M}_2}$ for each $n$ relies on the following commutative diagram: $$\xymatrixcolsep{4pc} \xymatrix{
H_{\infty} \ar[r]^-{\pi_{\infty, n}} \ar[rd]_-{\pi'_{\infty, n}} &  H_{n} \ar@{^{(}->}[d]& H_{n+e} \ar[l]_-{\pi_{n+e, n}} \ar@{^{(}->}[d]\\
& H'_{n} & H'_{n+e}\ar[l]_-{\pi'_{n+e,n}} }$$ where $\pi'_{\infty,n}$ and $\pi'_{n+e,n}$ are the natural projections. 

By [@Vasiu:CBP Theorem 5.1.1(a)], we know that for any sufficient large $n$ (in fact $n \geq n_{12}$), there exists a positive integer $e_{\mathcal{M}_1,\mathcal{M}_2}(n)$ such that for all $e\geq e_{\mathcal{M}_1,\mathcal{M}_2}(n)$, $\mathrm{Im}(\pi'_{\infty, n}) = \mathrm{Im}(\pi'_{n+e, n})$. Therefore the images of $\mathrm{Im}(\pi_{\infty,n})$ and $\mathrm{Im}(\pi_{n+e,n})$ in $H'_n$ are the same.  Thus $\mathrm{Im}(\pi_{n+e, n}) = \mathrm{Im}(\pi_{\infty, n})$ for all $e \geq e_{\mathcal{M}_1,\mathcal{M}_2}(n)$. This proves that $e_{\mathcal{M}_1,\mathcal{M}_2}(n)$ exists for each $n$.

Now we show that $e_{\mathcal{M}_1,\mathcal{M}_2}(n)$ does not depend on $n$. The proof relies on the following commutative diagram: $$\xymatrix{
0 \ar[r] & \mathrm{Im}(\pi_{\infty,n}) \ar[r]^-p \ar[d]^{i_1} & \mathrm{Im}(\pi_{\infty,n+1}) \ar[r] \ar[d]^{i_2} & \mathrm{Im}(\pi_{\infty,1}) \ar[r] \ar[d]^{i_3} & 0 \\
0 \ar[r] & \mathrm{Im}(\pi_{n+e,n}) \ar[r]^-p & \mathrm{Im}(\pi_{n+1+e,n+1}) \ar[r] & \mathrm{Im}(\pi_{1+e,1})
}$$ with horizontal exact sequences and with all vertical maps injective.  By the snake lemma, we have an exact sequence $$0  \to \mathrm{Coker}(i_1) \to \mathrm{Coker}(i_2) \to \mathrm{Coker}(i_3).$$ If we take $e \geq e_{\mathcal{M}_1,\mathcal{M}_2}(n+1)$, then $\mathrm{Coker}(i_2) = 0$. Thus $\mathrm{Coker}(i_1)=0$ and $e_{\mathcal{M}_1,\mathcal{M}_2}(n+1) \geq e_{\mathcal{M}_1,\mathcal{M}_2}(n)$. If we take $e \geq \mathrm{max}(e_{\mathcal{M}_1,\mathcal{M}_2}(n),e_{\mathcal{M}_1,\mathcal{M}_2}(1))$, then $\mathrm{Coker}(i_2)=0$. Thus $e_{\mathcal{M}_1,\mathcal{M}_2}(n+1) \leq \mathrm{max}(e_{\mathcal{M}_1,\mathcal{M}_2}(n),e_{\mathcal{M}_1,\mathcal{M}_2}(1))$. An easy induction on $n \geq 1$ using the sequence of inequalities $$e_{\mathcal{M}_1,\mathcal{M}_2}(n) \leq e_{\mathcal{M}_1,\mathcal{M}_2}(n+1) \leq \mathrm{max}(e_{\mathcal{M}_1,\mathcal{M}_2}(n),e_{\mathcal{M}_1,\mathcal{M}_2}(1))$$ gives that $e_{\mathcal{M}_1,\mathcal{M}_2}(n) = e_{\mathcal{M}_1,\mathcal{M}_2}(1)$ for all $n$.

The non-negative integer $e_{\mathcal{M}_1,\mathcal{M}_2}$ of Proposition \[proposition:eexists\] is called the *homomorphism number* of $\mathcal{M}_1$ and $\mathcal{M}_2$. If $\mathcal{M}_1 = \mathcal{M}_2 = \mathcal{M}$, we denote $e_{\mathcal{M},\mathcal{M}}$ by $e_{\mathcal{M}}$ and call it the *endomorphism number* of $\mathcal{M}$.

The following lemma is a generalization of [@Vasiu:traversosolved Lemma 2.8(c)] and is proved in a similar way. 

Let $k \subset k'$ be an extension of algebraically closed fields.  We have $e_{\mathcal{M}_1,\mathcal{M}_2} = e_{\mathcal{M}_{1,k'},\mathcal{M}_{2,k'}}$.

When $m \geq n$, let $\pi_{m,n} : \mathbf{H}_m \to \mathbf{H}_n$ be the canonical reduction homomorphism and let $\mathbf{H}_{m,n}$ be the scheme theoretic image of $\pi_{m,n}$, which is of finite type over $k$, and whose definition is compatible with base change $k \subset k'$. If $l \geq m$, then $\mathbf{H}_{l,n}$ is a subgroup scheme of $\mathbf{H}_{m,n}$. By the definition of $e_{\mathcal{M}_1,\mathcal{M}_2}$, we have $m-n \geq e_{\mathcal{M}_1,\mathcal{M}_2}$ if and only if $\mathbf{H}_{m,n}(k) = \mathbf{H}_{l,n}(k)$. As $k$ and $k'$ are algebraically closed, we have $\mathbf{H}_{m,n}(k) = \mathbf{H}_{l,n}(k)$ if and only if $\mathbf{H}_{m,n}(k') = \mathbf{H}_{l,n}(k')$. This is further equivalent to $(\mathbf{H}_{m,n})_{k'}(k') = (\mathbf{H}_{l,n})_{k'}(k')$, thus $e_{\mathcal{M}_1,\mathcal{M}_2} = e_{\mathcal{M}_{1,k'},\mathcal{M}_{2,k'}}$.

Coarse endomorphism number {#subsection:cendnum}
--------------------------

In this subsection, we generalize the coarse endomorphism number defined in [@Vasiu:traversosolved Section 7] for $p$-divisible groups.  The following proposition is a generalization of [@Vasiu:traversosolved Lemma 7.1].  Again for the sake of generality, we will work with the homomorphism version. 

\[lemma:cendnumexists\] There exists a non-negative integer $f_{\mathcal{M}_1,\mathcal{M}_2}$ that depends on $\mathcal{M}_1$ and $\mathcal{M}_2$ such that for positive integers $m \geq n$, the restriction homomorphism $\pi_{m,n}: H_m \to H_n$ has finite image if and only if $m \geq n+f_{\mathcal{M}_1,\mathcal{M}_2}$.

As $H_{\infty}$ is a finitely generated $\mathbb{Z}_p$-module, $\mathrm{Im}(\pi_{\infty,n})$ inside the $p^n$-torsion $\mathbb{Z}_p$-module $H_n$ is finite.  By Proposition \[proposition:eexists\], there exists $f_{\mathcal{M}_1,\mathcal{M}_2}(n)$ such that for all $m\geq n+f_{\mathcal{M}_1,\mathcal{M}_2}(n)$, $\mathrm{Im}(\pi_{m,n}) = \mathrm{Im}(\pi_{\infty,n})$ is finite. 

To show that $f_{\mathcal{M}_1, \mathcal{M}_2}(n)$ is independent of $n$, we consider the exact sequence $$\xymatrix{
0 \ar[r] & \mathrm{Im}(\pi_{m,n}) \ar[r]^-p & \mathrm{Im}(\pi_{m,n+1}) \ar[r] & \mathrm{Im}(\pi_{m,1}) 
}.$$ It implies that $$f_{\mathcal{M}_1,\mathcal{M}_2}(n) \leq f_{\mathcal{M}_1,\mathcal{M}_2}(n+1) \leq \mathrm{max}(f_{\mathcal{M}_1,\mathcal{M}_2}(n),f_{\mathcal{M}_1,\mathcal{M}_2}(1))$$ An easy induction on $n \geq 1$ shows that $f_{\mathcal{M}_1,\mathcal{M}_2}(n) = f_{\mathcal{M}_1,\mathcal{M}_2}(1)$ for all $n \geq 1$.

The non-negative integer $f_{\mathcal{M}_1,\mathcal{M}_2}$ of Lemma \[lemma:cendnumexists\] is called the *coarse homomorphism number* of $\mathcal{M}_1$ and $\mathcal{M}_2$. If $\mathcal{M}_1 = \mathcal{M}_2 = \mathcal{M}$, we denote $f_{\mathcal{M},\mathcal{M}}$ by $f_{\mathcal{M}}$ and call it the *coarse endomorphism number* of $\mathcal{M}$.

\[proposition:f&lt;e\] We have an inequality $f_{\mathcal{M}_1,\mathcal{M}_2} \leq e_{\mathcal{M}_1,\mathcal{M}_2}$.

It is clear as $\mathrm{Im}(\pi_{\infty,n})$ is finite. 

\[lemma:fextension\] Let $k \subset k'$ be an extension of algebraically closed fields.  We have $f_{\mathcal{M}_1,\mathcal{M}_2} = f_{\mathcal{M}_{1,k'},\mathcal{M}_{2,k'}}$.

For positive integers $m \geq n$, we have $m-n < f_{\mathcal{M}_1,\mathcal{M}_2}$ if and only if the image of $\pi_{m,n}$ is infinite by definition.  It is further equivalent to the image of $\mathbf{H}_m \to \mathbf{H}_n$ having positive dimension.  This property is invariant under base change from $k$ to $k'$ and hence the lemma follows. 

Level torsion {#subsection:leveltorsion}
-------------

In this subsection, we generalize the level torsion defined in [@Vasiu:traversosolved Section 8.1] for $p$-divisible groups. 

Let $H_{12} $ be the set of all $W(k)$-linear homomorphisms from $M_1$ to $M_2$. We have a latticed $F$-isocrystal $(H_{12}, \varphi_{12})$ where $\varphi_{12}: H_{12} \otimes_{W(k)} B(k) \to H_{12} \otimes_{W(k)} B(k)$ is a $\sigma$-linear isomorphism defined by the rule $\varphi_{12}(h) = \varphi_2 h \varphi^{-1}_1$. By Dieudonné-Manin’s classification of $F$-isocrystals, we have finite direct sum decompositions $$(M_1 \otimes_{W(k)} B(k), \varphi_1) \cong \bigoplus_{\lambda_1 \in J_1} E^{m_{\lambda_1}}_{\lambda_1}, \qquad (M_2 \otimes_{W(k)} B(k), \varphi_2) \cong \bigoplus_{\lambda_2 \in J_2} E^{m_{\lambda_2}}_{\lambda_2}$$ where the simple $F$-isocrystals $E_{\lambda_1}$ and $E_{\lambda_2}$ have Newton slopes equal to $\lambda_1$ and $\lambda_2$ respectively, the multiplicities $m_{\lambda_1}, m_{\lambda_2} \in \mathbb{Z}_{> 0}$ and the finite index sets $J_1, J_2 \subset \mathbb{Q}_{>0}$ are uniquely determined.  From these decompositions, we obtain a direct sum decomposition $$(H_{12} \otimes_{W(k)} B(k), \varphi_{12}) \cong L_{12}^+ \oplus L_{12}^0 \oplus L_{12}^-,$$ where $$\begin{gathered}
L_{12}^+ = \bigoplus_{\lambda_1<\lambda_2}\mathrm{Hom}(E_{\lambda_1}^{m_{\lambda_1}}, E_{\lambda_2}^{m_{\lambda_2}}), \qquad  L_{12}^- =  \bigoplus_{\lambda_1 > \lambda_2} \mathrm{Hom}(E_{\lambda_1}^{m_{\lambda_1}}, E_{\lambda_2}^{m_{\lambda_2}}), \\ L_{12}^0 = \bigoplus_{\lambda_1=\lambda_2} \mathrm{Hom}(E_{\lambda_1}^{m_{\lambda_1}}, E_{\lambda_2}^{m_{\lambda_2}}).\end{gathered}$$ Define $$\begin{gathered}
O_{12}^+ = \bigcap_{i=0}^{\infty} \varphi_{12}^{-i}(H_{12} \cap L_{12}^+),  \qquad O_{12}^- = \bigcap_{i=0}^{\infty} \varphi_{12}^i(H_{12} \cap L_{12}^-),\\
O_{12}^0 = \bigcap_{i=0}^{\infty} \varphi_{12}^{-i}(H_{12} \cap L_{12}^0) = \bigcap_{i=0}^{\infty} \varphi_{12}^i(H_{12} \cap L_{12}^0).\end{gathered}$$ Let $A_{12}^0 = \{x \in H_{12} \; | \; \varphi_{12}(x) = x\}$ be the $\mathbb{Z}_p$-algebra that contains the elements fixed by $\varphi_{12}$. For $\dagger \in \{+, 0, -\}$, each $O_{12}^{\dagger}$ is a lattice of $L_{12}^{\dagger}$. We have the following relations: $$\varphi(O_{12}^+) \subset O_{12}^+, \quad \varphi(O_{12}^0) = O_{12}^0 = A_{12}^0 \otimes_{\mathbb{Z}_p} W(k) = \varphi^{-1}(O_{12}^0), \quad \varphi^{-1}(O_{12}^-) \subset O_{12}^-.$$ Write $O_{12} := O_{12}^+ \oplus O_{12}^0 \oplus O_{12}^-$; it is a lattice of $H_{12} \otimes_{W(k)} B(k)$ inside $H_{12}$. The $W(k)$-module $O_{12}$ is called the *level module* of $\mathcal{M}_1$ and $\mathcal{M}_2$.

The *level torsion* of $\mathcal{M}_1$ and $\mathcal{M}_2$ is the smallest non-negative integer $\ell_{\mathcal{M}_1, \mathcal{M}_2}$ such that $$p^{\ell_{\mathcal{M}_1, \mathcal{M}_2}} H_{12} \subset O_{12} \subset H_{12}.$$ If $\mathcal{M}_1 = \mathcal{M}_2 = \mathcal{M}$, then $\ell_{\mathcal{M}_1, \mathcal{M}_2}$ will be denoted by $\ell_{\mathcal{M}}$.

\[remark:leveltorsiondiff\] The definition of level torsion in this paper is slightly different from the definition in [@Vasiu:reconstructing].  When $\mathcal{M}$ is a direct sum of two or more isoclinic ordinary $F$-crystals of different Newton polygons, its isomorphism number is $n_{\mathcal{M}} = 1$. According to the definition in [@Vasiu:reconstructing], the level torsion $\ell_{\mathcal{M}} = 1$ but the definition in this paper gives $\ell_{\mathcal{M}} = 0$.

For the duals $\mathcal{M}_1^*$ and $\mathcal{M}_2^*$ of $\mathcal{M}_1$ and $\mathcal{M}_2$ respectively, we can define $\ell_{\mathcal{M}_1^*, \mathcal{M}_2^*}$ in a similar way. 

\[lemma:leveltorsionequal\] We have $\ell_{\mathcal{M}_1, \mathcal{M}_2} = \ell_{\mathcal{M}_2, \mathcal{M}_1} = \ell_{\mathcal{M}_1^*, \mathcal{M}_2^*}$.

Write $H_{21} := \mathrm{Hom}(M_2, M_1) \cong \mathrm{Hom}(H_{12}, W(k)) =: H_{12}^*$. There is a direct sum decomposition $$H_{12}^* \otimes_{W(k)} B(k) \cong H_{21} \otimes_{W(k)} B(k) = L_{21}^+ \oplus L_{21}^0 \oplus L_{21}^-.$$ It is easy to see that $$L_{21}^+ \cong \mathrm{Hom}(L_{12}^-,B(k)) =: L_{12}^{-*}, \qquad  L_{21}^- \cong \mathrm{Hom}(L_{12}^+, B(k)) =: L_{12}^{+*},$$ $$\quad L_{21}^0 \cong \mathrm{Hom}(L_{12}^0,B(k)) =: L_{12}^{0*}$$ are isomorphic as $B(k)$-vector spaces.  One can define $O_{21}$ in the same way: $$O_{21} : =O_{21}^+ \oplus O_{21}^0 \oplus O_{21}^-  \cong O_{12}^{-*}\oplus  O_{12}^{0*} \oplus O_{12}^{+*}$$ and thus $O_{21} \cong O_{12}^*$. Therefore $$p^{\ell_{\mathcal{M}_2,\mathcal{M}_1}}H_{21} \subset O_{21} \subset H_{21}\quad \textrm{if and only if} \quad p^{\ell_{\mathcal{M}_1,\mathcal{M}_2}}H_{12} \subset O_{12} \subset H_{12},$$ whence $\ell_{\mathcal{M}_1, \mathcal{M}_2} = \ell_{\mathcal{M}_2,\mathcal{M}_1}$. As $H_{12}^* \cong H_{21}$, we get $\ell_{\mathcal{M}_2,\mathcal{M}_1} = \ell_{\mathcal{M}_1^*,\mathcal{M}_2^*} $. 

\[lemma:leveltorsionrelation\] $\ell_{\mathcal{M}_1 \oplus \mathcal{M}_2} = \mathrm{max}\{\ell_{\mathcal{M}_1},\ell_{\mathcal{M}_2},\ell_{\mathcal{M}_1,\mathcal{M}_2}\}$.

The direct sum decomposition into $W(k)$-modules of $$\mathrm{End}(M_1\oplus M_2) = \mathrm{End}(M_1) \oplus \mathrm{End}(M_2) \oplus \mathrm{Hom}(M_1,M_2) \oplus \mathrm{Hom}(M_2,M_1)$$ gives birth to the direct sum decomposition of the level module of $\mathcal{M}_1 \oplus \mathcal{M}_2$ $$O = O_{11} \oplus O_{22} \oplus O_{12} \oplus O_{21}.$$ Hence $\ell_{\mathcal{M}_1 \oplus \mathcal{M}_2} = \mathrm{max}\{\ell_{\mathcal{M}_1},\ell_{\mathcal{M}_2},\ell_{\mathcal{M}_1,\mathcal{M}_2}, \ell_{\mathcal{M}_2,\mathcal{M}_1}\} = \mathrm{max}\{\ell_{\mathcal{M}_1},\ell_{\mathcal{M}_2},\ell_{\mathcal{M}_1,\mathcal{M}_2}\}$ by Lemma \[lemma:leveltorsionequal\]. 

\[lemma:ellextension\] Let $k \subset k'$ be an extension of algebraically closed fields.  We have $\ell_{\mathcal{M}_1,\mathcal{M}_2} = \ell_{\mathcal{M}_{1,k'},\mathcal{M}_{2,k'}}$.

For $\mathcal{M}_{1,k'}$ and $\mathcal{M}_{2,k'}$, we can define $H'_{12}$ and $O'_{12}$ in an analogous manner.  One can check that $$H'_{12} = H_{12} \otimes_{W(k)} W(k'), \qquad O'_{12} = O_{12} \otimes_{W(k)} W(k').$$ The lemma follows easily. 

Proof of the Main Theorem {#section:proofmaintheorem}
=========================

The proofs of this section follow closely the ones of [@Vasiu:traversosolved Section 8]. 

Notations
---------

For this section, we denote by $H:=H_{12}$ the group of $W(k)$-linear homomorphisms from $M_1$ to $M_2$, and $H_{s}$ the group of homomorphisms from $F_s(\mathcal{M}_1)$ to $F_s(\mathcal{M}_2)$. For simplicity, we denote $O_{12}$ by $O$, $O_{12}^{\dagger}$ by $O^{\dagger}$ for $\dagger \in \{+,0,-\}$, and $A^0_{12}$ by $A^0$.

The inequality $e_{\mathcal{M}_1,\mathcal{M}_2} \leq \ell_{\mathcal{M}_1,\mathcal{M}_2}$ {#subsection:e<f}
----------------------------------------------------------------------------------------

We will follow the ideas of [@Vasiu:traversosolved Section 8.2] and prove that $\mathrm{Im}(\pi_{\infty,1}) = \mathrm{Im}(\pi_{\ell_{\mathcal{M}_1,\mathcal{M}_2}+1,1})$. For any $\bar{h} \in \mathrm{Im}(\pi_{\ell_{\mathcal{M}_1,\mathcal{M}_2}+1,1})$, let $h \in H_{\ell_{\mathcal{M}_1,\mathcal{M}_2}+1}$ be a preimage of $\bar{h}$. Hence $ \varphi_2  h \varphi^{-1}_1 \equiv h$ modulo $p^{\ell_{\mathcal{M}_1,\mathcal{M}_2}+1}$, that is, $ \varphi_2  h \varphi^{-1}_1 - h \in p^{\ell_{\mathcal{M}_1,\mathcal{M}_2}+1} \mathrm{Hom}(M_1, M_2) \subset pO$. By Lemma \[lemma:solveequation\] below, there exists $h'' \in pO$ such that $$\varphi_2  h \varphi^{-1}_1 - h = \varphi_2  h'' \varphi^{-1}_1 - h''.$$ Thus $h' := h-h'' \in H_{\infty}$ is a homomorphism whose image in $H_1$ is exactly $\bar{h}$.

\[lemma:solveequation\] For each $x \in O$, the equation $x = \varphi_{12}(X) - X$ in $X$ has a solution in $O$ that is unique up to the addition of elements in $A^0$. Moreover, if $x \in p^sO$, then there exists a solution $X \in p^sO$.

Writing $x = x^+ + x^0 + x^-$ with $x^{\dagger} \in O^{\dagger}$ for $\dagger \in \{+, 0, -\}$, we will find $y^{\dagger} \in O^{\dagger}$ such that $x^{\dagger} = \varphi_{12}(y^{\dagger}) - y^{\dagger}$ for each $\dagger \in \{+,0,-\}$. Therefore $y = y^++y^0+y^-$ is a solution of the given equation. 

Let $y^+ = -\sum_{i=0}^{+\infty} \varphi_{12}^i(x^+)$, and $y^- = \sum_{i=1}^{+\infty} \varphi_{12}^{-i}(x^-)$. Because $\varphi_{12}(O^+) \subset O^+$ and $\varphi_{12}^{-1}(O^-) \subset O^-$, we have $y^+ \in O^+$ and $y^- \in O^-$. It is easy to check that $x^+ = \varphi_{12}(y^+)-y^+$ and $x^- = \varphi_{12}(y^-)-y^-$.

Let $\{v_1, v_2, \dots, v_r\}$ be a $\mathbb{Z}_p$-basis of $A^0$; it is also a $W(k)$-basis of $O^0$. We also write $x^0  = \sum_{i=1}^d x_iv_i$. For $1 \leq i \leq r$, let $z_i \in W(k)$ be a solution of $\sigma(z_i)-z_i = x_i$ and put $y^0 = \sum_{i=1}^d z_iv_i \in O^0$. Using the fact that $\varphi_{12}(v_i) = v_i$ for all $1 \leq i \leq r$, it is easy to check that $x^0 = \varphi_{12}(y^0)-y^0$.

If $y, y' \in O$ satisfy the equation $x = \varphi_{12}(X) - X$, we have $\varphi_{12}(y) - y = \varphi_{12}(y')-y'$, i.e. $\varphi_{12}(y-y') = y-y'$, whence $y-y' \in A^0$.

If $x  = p^sx' \in p^sO$, then $y = p^sy' \in p^sO$ will be a solution of $x = \varphi_{12}(X) - X$ where $y' \in O$ is a solution of $x' = \varphi_{12}(X) - X$.

The inequality $\ell_{\mathcal{M}_1,\mathcal{M}_2} \leq f_{\mathcal{M}_1,\mathcal{M}_2}$ {#subsection:l<f}
----------------------------------------------------------------------------------------

We follow the ideas of [@Vasiu:traversosolved Section 8.3].  By Lemmas \[lemma:fextension\] and \[lemma:ellextension\], we can assume that $k \supset k'[[\alpha]]=R$ where $k \supset k'$ is an extension of algebraically closed fields and for $i=1,2$, we have $$(M_i,\varphi_i) \cong (M'_{i} \otimes_{W(k')} W(k), \varphi'_{i} \otimes \sigma),$$ where $(M'_{i},\varphi'_{i})$ are $F$-crystals over $k'$. Let $\mathfrak{m}$ be the ideal of $R$ generated by $\alpha$. Let $H'$ and $O'$ be the analogues of $H$ and $O$ obtained from $(M'_{i},\varphi'_{i})$ instead of $(M_i,\varphi_i)$. It is easy to check that $$H = H' \otimes_{W(k')} W(k), \qquad O = O' \otimes_{W(k')} W(k),$$ and $p^jO \cap O' = p^jO'$ for $j \in \mathbb{Z}_{\geq 0}$.

Let $x = x^+ + x^0 + x^- \in O'$ where $x^{\dagger} \in O'^{\dagger}$, $\dagger \in \{+,0,-\}$.

\[lemma:mainproofsolveequation\] For each $\eta \in W(\mathfrak{m})$, the equation $\eta x = \varphi_{12}(X) - X$ has a solution $x_{\eta} \in O$, that is unique up to the addition of an element of $A^{0}$.

Put $$x_{\eta}^{+} = -\sum_{i=0}^{\infty} \varphi_{12}^{i}(\eta x^+) \in O^+,  x^-_{\eta} = \sum_{i=1}^{\infty} \varphi_{12}^{-i}(\eta x^-) \in O^-,  x_{\eta}^{0} = -\sum_{i=0}^{\infty} \varphi_{12}^{i}(\eta x^0) \in O^0.$$ The elements $x_{\eta}^{\pm}$ are well-defined as $\{\varphi_{12}^i(x^+)\}_{i \in \mathbb{Z}_{\geq 0}}$ and $\{\varphi_{12}^{i}(x^-)\}_{i \in \mathbb{Z}_{\geq 0}}$ are $p$-adically convergent in $O'^+$ and $O'^-$ respectively.  As $\{\sigma^i(\eta)\}_{i \in \mathbb{Z}_{\geq 0}}$ is a $\alpha$-adically convergent in $W(R)$, $x_{\eta}^0$ is convergent in $O^0$. One can check that $$x_{\eta} := x_{\eta}^++x_{\eta}^0+x_{\eta}^- \in O$$ satisfies $\eta x = \varphi_{12}(x_{\eta})-x_{\eta}$.

Suppose $\eta x = \varphi_{12}(x_{\eta}) - x_{\eta}$ and $\eta x = \varphi_{12}(x^{\circ}_{\eta}) - x^{\circ}_{\eta}$, we have $x_{\eta} - x^{\circ}_{\eta} = \varphi_{12}(x_{\eta} - x^{\circ}_{\eta})$, hence the lemma. 

We define a homomorphism of abelian groups $\Omega_x:W(\mathfrak{m}) \to H/A^0$ by the formula $\Omega_x (\eta) = x_{\eta} + A^0$ where $x_{\eta} \in O \subset H$ satisfies $\eta x = \varphi_{12}(x_{\eta}) - x_{\eta}$. By Lemma \[lemma:mainproofsolveequation\], it is well-defined. 

Let $x \in p^{\ell_{\mathcal{M}_{1},\mathcal{M}_{2}}}H' \setminus pO'$. For all $\eta \in W(\mathfrak{m})$, $\varphi_{12}(x_{\eta}) - x_{\eta} = \eta x \in p^{\ell_{\mathcal{M}_{1},\mathcal{M}_{2}}}H'$, thus $\varphi_{12}(x_{\eta}) \equiv x_{\eta}$ modulo $p^{\ell_{\mathcal{M}_{1},\mathcal{M}_{2}}}$. This implies that $x_{\eta}$ is a homomorphism modulo $p^{\ell_{\mathcal{M}_{1},\mathcal{M}_{2}}}$ from $\mathcal{M}_1$ to $\mathcal{M}_2$. Hence $x_{\eta} \in H_{\ell_{\mathcal{M}_{1},\mathcal{M}_{2}}}$. Clearly, every homomorphism of $F$-crystals is a homomorphism modulo powers of $p$. Hence $A^0 \subset H_{\ell_{\mathcal{M}_1,\mathcal{M}_2}}$. Thus the image of $\Omega_x$ is in $H_{\ell_{\mathcal{M}_1,\mathcal{M}_2}}/A^0$.

Suppose $f_{\mathcal{M}_{1},\mathcal{M}_{2}} < \ell_{\mathcal{M}_{1},\mathcal{M}_{2}}$, we will show that $x \in pO'$, which is a contradiction!  Let $\bar{\pi}_{\ell_{M_1,M_2},1} : H_{\ell_{\mathcal{M}_1,\mathcal{M}_2}}/A^0 \to H_{1}/A^0$ be the homomorphism induced by ${\pi}_{\ell_{M_1,M_2},1}$. The image of $$\bar{\pi}_{\ell_{M_1,M_2},1} \circ \Omega_x : W(\mathfrak{m}) \to H_{\ell_{\mathcal{M}_1,\mathcal{M}_2}}/A^0 \to H_{1}/A^0$$ takes only finitely many values $H_{1}/A^0$ as $\mathrm{Im}(\pi_{\ell_{\mathcal{M}_{1},\mathcal{M}_{2}},1})$ is finite by the assumption that $f_{\mathcal{M}_{1},\mathcal{M}_{2}} < \ell_{\mathcal{M}_{1},\mathcal{M}_{2}}$. Since $\mathfrak{m}$ is infinite (and thus $W(\mathfrak{m})$ is infinite), the kernel of $\bar{\pi}_{\ell_{M_1,M_2},1} \circ \Omega_x$ is infinite.  There exists $\eta = (\eta_0,\eta_1,\dots) \in W(\mathfrak{m})$ with $\eta_0 \neq 0$ such that $x_{\eta} \in pH$. Thus $x_{\eta} \in O \cap pH =:N$. Let $N' = O' \cap pH'$, we have $N \cong N' \otimes_{W(k')} W(k)$.

\[lemma:mainproofequivalencemap\] An element $\bar{z} \in O/pO$ lies in $N/pO$ if and only if for every $k'$-linear map $\rho:O'/pO' \to k'$ with $\rho(N'/pO') = 0$ we have $(\rho \otimes 1_{k})(\bar{z}) = 0$.

For every $k'$-linear map $\rho: O'/pO' \to k'$ with $\rho(N'/pO')=0$, $$\mathrm{Ker}(\rho \otimes 1_k) = \mathrm{Ker}(\rho) \otimes_{k'} k \supset N'/pO' \otimes_{k'}k = N/pO.$$ Set $S = \{\rho:O'/pO' \to k \; | \; \rho(N'/pO') = 0\}$. We have $\bigcap_{\rho \in S} \mathrm{Ker}(\rho \otimes 1_k) = N/pO$. This concludes the proof. 

By Lemma \[lemma:mainproofequivalencemap\], for every $k'$-linear map $\rho : O'/pO' \to k'$ such that $\rho(N'/pO') = 0$, we have $(\rho \otimes 1_k)(\bar{x}_{\eta}) = 0$. Therefore the following equality holds in $R$ $$\label{equation:mainproofeq1}
\sum_{i=0}^{\infty} \rho(\varphi_{12}^i(\bar{x}^+)) \eta_0^{p^i} + \sum_{i=0}^{\infty}\rho(\varphi_{12}^i(\bar{x}^0)) \eta_0^{p^i} - \sum_{i=1}^{\infty}\rho(\varphi_{12}^{-i}(\bar{x}^-)) \eta_0^{p^{-i}}=(\rho \otimes 1_k)(\bar{x}_{\eta}) = 0.$$ Because the Newton slopes of $(O^+,\varphi_{12})$ and $(O^-,\varphi_{12}^{-1})$ are positive, there exists a big enough $n$ such that $\varphi_{12}^i(x_+) \in pO'^+$ and $\varphi_{12}^{-i}(x_-) \in pO'^-$ for $i> n.$ As $\rho(N'/pO') = 0$, $$\label{equation:mainprooffact1}
\rho(\varphi_{12}^i(\bar{x}^+)) = 0, \quad \rho(\varphi_{12}^{-i}(\bar{x}^-)) = 0,\quad \forall \; i > n.$$ Thus is reduced to $$\label{equation:mainproofeq2}
-\sum_{i=-n}^{-1} \rho(\varphi_{12}^{i}(\bar{x}^-))\eta_0^{p^i} + \sum_{i=0}^n (\rho(\varphi_{12}^i(\bar{x}^+)) + \rho(\varphi_{12}^i(\bar{x}^0)))\eta_0^{p^i}+\sum_{i=n+1}^{\infty}\rho(\varphi_{12}^i(\bar{x}^0))\eta_0^{p^i}=0.$$ Write $$\begin{gathered}
\Phi(\beta) = -\sum_{i=0}^{n-1} \rho(\varphi_{12}^{i-n}(\bar{x}^-)) \beta^{p^i} + \sum_{i=n}^{2n} (\rho(\varphi_{12}^{i-n}(\bar{x}^+)) + \rho(\varphi_{12}^{i-n}(\bar{x}^0)))\beta^{p^i} \\+\sum_{i=2n+1}^{\infty}\rho(\varphi_{12}^{i-n}(\bar{x}^0))\beta^{p^i} \in k'[[\beta]].\end{gathered}$$ Then is equivalent to $\Phi(\eta_0^{p^{-n}}) = 0$ where $\eta_0^{p^{-n}} \in \alpha^{p^{-n}}k'[[\alpha^{p^{-n}}]]$. As $\eta_0^{p^{-n}} \neq 0$, we deduce that $\Phi(\beta) = 0$ by [@Vasiu:traversosolved Lemma 8.9].  Combining , we get $$\label{equation:mainprooffact2}
\begin{split}
\rho(\varphi_{12}^{-i}(\bar{x}^-)) & = 0,  \; \forall \; i \geq 1 ,\qquad \rho(\varphi_{12}^i(\bar{x}^0)) = 0, \quad \rho(\varphi_{12}^i(\bar{x}^+)) = 0, \; \forall \; i>n
 \\
& \rho(\varphi_{12}^i(\bar{x}^+)) + \rho(\varphi_{12}^i(\bar{x}^0)) = 0, \; \forall \; i = 0, \dots, n. \\
\end{split}$$ As $\varphi_{12}$ is bijective on $O'^0$ and thus on $O'^0/pO'^0$, the subspace $V \subset O'^0/pO'^0$ generated by $\{\varphi_{12}^i(\bar{x}^0) \; | \; i \geq 0\}$ satisfies $\varphi_{12}(V) = V$ and thus $\varphi_{12}^{j}(V) = V$ for every $j \geq 0$. This implies that $V$ is generated by $\{\varphi_{12}^i(\bar{x}_0) \; | \; i > n\}$ and hence for $0 \leq i \leq n$, $\varphi_{12}^i(\bar{x}_0)$ is a linear combination of elements in $\{\varphi_{12}^i(\bar{x}_0) \; | \; i > n\}$ whence $\rho(\varphi_{12}^i(\bar{x}_0)) = 0$ for all $i = 0, \dots, n$. This allows us to extend to get $$\label{equation:mainprooffact3}
\rho(\varphi_{12}^{-i}(\bar{x}^-))  = 0,  \; \forall \; i \geq 1 ,\qquad \rho(\varphi_{12}^i(\bar{x}^0)) = 0, \quad \rho(\varphi_{12}^i(\bar{x}^+)) = 0, \; \forall \; i \geq 0.$$ Finally, since $x \in p^{\ell_{\mathcal{M}_{1},\mathcal{M}_{2}}}H'$ and $\ell_{\mathcal{M}_{1},\mathcal{M}_{2}} > f_{\mathcal{M}_{1},\mathcal{M}_{2}} \geq 0$, we have $x \in pH'$ and thus $x \in pH' \cap O' =: N'$. As $\rho(N'/pO') = 0$, we have $0 = \rho(\bar{x}) = \rho(\bar{x}^++\bar{x}^0+\bar{x}^-) = \rho(\bar{x}^-)$. Thus can be further extended to $$\label{equation:mainprooffact4}
\rho(\varphi_{12}^{i}(\bar{x}^+))  = 0, \qquad \rho(\varphi_{12}^i(\bar{x}^0)) = 0, \qquad \rho(\varphi_{12}^{-i}(\bar{x}^-)) = 0, \qquad \forall \; i \geq 0.$$ By Lemma \[lemma:mainproofequivalencemap\] and , we have $\varphi_{12}^{i}(x^+), \varphi_{12}^i(x^0), \varphi_{12}^{-i}(x^-) \in pH$ and thus in $pH'$ for all $i \geq 0$. By the definition of $O'$, we have $x = x^++x^0+x^- \in pO'$. This reaches the desired contradiction. 

The equality $f_{\mathcal{M}} = n_{\mathcal{M}}$ {#subsection:f=n}
------------------------------------------------

In this subsection, we show that $f_{\mathcal{M}} = n_{\mathcal{M}}$ when $\mathcal{M}$ is not an ordinary $F$-crystal.  Thus in this case, $n_{\mathcal{M}}>0$. Recall $E_s$ is the set of all endomorphisms of $F_s(\mathcal{M})$ and $\mathbf{E}_s(k) = E_s$. The restriction homomorphism $\pi_{s,1}: E_s \to E_1$ has finite image if and only if the image of $\pi_{s,1}:\mathbf{E}_s \to \mathbf{E}_1$ has zero dimension, if and only if $s \geq 1 + f_{\mathcal{M}}$ by definition.  The dimension of $\pi_{s,1}$ is $\gamma_{\mathcal{M}}(s) - \gamma_{\mathcal{M}}(s-1)$. It is zero if and only if $s > n_{\mathcal{M}}$ by Theorem \[theorem:monotonicitygamma\].  As $s \geq 1 + f_{\mathcal{M}}$ if and only if $s > n_{\mathcal{M}}$, we conclude that $f_{\mathcal{M}} = n_{\mathcal{M}}$.

Conclusion
----------

By Subsections \[subsection:e&lt;f\], \[subsection:l&lt;f\], \[subsection:f=n\] and Proposition \[proposition:f&lt;e\], we have the following two theorems:

\[theorem:maintheorem2\] We have equalities $f_{\mathcal{M}_1,\mathcal{M}_2} = e_{\mathcal{M}_1,\mathcal{M}_2} = \ell_{\mathcal{M}_1,\mathcal{M}_2}.$

\[theorem:maintheorem3\] If $\mathcal{M}$ is not ordinary, then $n_{\mathcal{M}} = f_{\mathcal{M}} = e_{\mathcal{M}} = \ell_{\mathcal{M}}$.

We have equalities $f_{\mathcal{M}_1,\mathcal{M}_2} = f_{\mathcal{M}_2,\mathcal{M}_1} = f_{\mathcal{M}^*_1,\mathcal{M}^*_2}$ and $e_{\mathcal{M}_1,\mathcal{M}_2} = e_{\mathcal{M}_2,\mathcal{M}_1} = e_{\mathcal{M}^*_1,\mathcal{M}^*_2}$.

This is clear by Theorem \[theorem:maintheorem2\] and Lemma \[lemma:leveltorsionequal\]. 

Application to $F$-crystal of rank $2$
======================================

In [@Xiao:computing Theorem 1.4], we proved that if $\mathcal{M}$ is a non-isoclinic $F$-crystal of rank $2$, and is not a direct sum of two $F$-crystals of rank $1$, then $n_{\mathcal{M}} \leq 2 \lambda_1$ where $\lambda_1$ is the smallest Newton slope of $\mathcal{M}$. Now we show that the inequality is in fact an equality.  For the sake of completeness, we state the theorem of isomorphism number of rank $2$ in all cases. 

\[theorem:rank2\] Let $\mathcal{M}$ be an $F$-crystal of rank $2$ with Hodge slopes $0$ and $e>0$. Let $\lambda_1$ be the smallest Newton slope of $\mathcal{M}$. Then we have the following three disjoint cases:

(i) if $\mathcal{M}$ is a direct sum of two $F$-crystals of rank $1$, then $n_{\mathcal{M}} = 1$;

(ii) if $\mathcal{M}$ is not a direct sum of two $F$-crystals of rank $1$ and is isoclinic, then $n_{\mathcal{M}} = e$;

(iii) if $\mathcal{M}$ is not a direct sum of two $F$-crystals of rank $1$ and is non-isoclinic, then $n_{\mathcal{M}} = 2\lambda_1$.

Parts (i) and (ii) are proved in [@Xiao:computing Theorem 1.4 (i) and (ii)].  In the case of Part (iii), [@Xiao:computing Theorem 1.4 (iii)] proves only the inequality $n_{\mathcal{M}} \leq 2 \lambda_1$. The proof of [@Xiao:computing Theorem 1.4 (iii)] has a minor mistake that can be easily fixed.  In this paper, we will only prove the equality $n_{\mathcal{M}} = 2\lambda_1$ in the case of Part (iii). 

We show that $\lambda_1 >0$ by showing that the assumption that $\lambda_1= 0$ leads to a contradiction.  If $\lambda_1 = 0$, then the Hodge polygon and the Newton polygon of $\mathcal{M}$ coincide.  By [@Katz:slopefiltration Theorem 1.6.1], we can decompose $\mathcal{M}$ into a direct sum of two $F$-crystals of rank $1$. Hence $\lambda_1>0$. Let $\lambda_2$ be the other Newton slope of $\mathcal{M}$. As $\mathcal{M}$ is not isoclinic, $\lambda_1 < \lambda_2$. It is easy to see that $\lambda_1$ and $\lambda_2$ are two positive integers.  Hence there is a $W(k)$-basis $\mathcal{B}_1 = \{x_1, x_2\}$ of $M$ such that $\varphi(x_1) = p^{\lambda_1}x_1$ and $\varphi(x_2) = ux_1+p^{\lambda_2}x_2$ where $u \in W(k)$. If $u$ is a non-unit and belongs to $pW(k)$, then $\varphi(M) \subset pM$ and thus the smallest Hodge slope of $\mathcal{M}$ must be positive.  This contradicts the assumption of the proposition, hence $u$ is a unit.  By solving equations of the form $\varphi(z) = p^{\lambda_1}z$ and $\varphi(z) = p^{\lambda_2}z$, we find a $B(k)$-basis $\mathcal{B}_2 = \{y_1=x_1, y_2 = vx_1+p^{\lambda_1}x_2\}$ of $M[1/p]$ with $v$ a unit in $W(k)$ such that $\sigma(v)+u=p^{\lambda_2-\lambda_1}v$. It is easy to see that there is a unique $v$ satisfying this equation. 

Let $\mathcal{B}_1 \otimes \mathcal{B}_1^*$ be the $W(k)$-basis of $\mathrm{End}(M)$ that contains $x_i \otimes x_j^*$ for all $1 \leq i, j \leq 2$, where $(x_i \otimes x_j^*)(x_j) = x_i$. It is a $B(k)$-basis of $\mathrm{End}(M[1/p])$. We compute the formula of $\varphi :\mathrm{End}(M[1/p]) \to \mathrm{End}(M[1/p])$ with respect to $\mathcal{B}_1$ as follows: $$\begin{aligned}
\varphi(x_1 \otimes x_1^*) &= x_1 \otimes x_1^* - p^{-\lambda_2}ux_1 \otimes x_2^*, \\
\varphi(x_2 \otimes x_1^*) &= p^{-\lambda_1}u x_1 \otimes x_1^* + p^{\lambda_2 - \lambda_1} x_2 \otimes x_1^* - p^{-\lambda_1-\lambda_2}u^2x_1 \otimes x_2^* - p^{-\lambda_1}u x_2 \otimes x_2^*, \\
\varphi(x_1 \otimes x_2^*) &= p^{\lambda_1-\lambda_2}x_1 \otimes x_2^*, \\
\varphi(x_2 \otimes x_2^*) &= p^{-\lambda_2}ux_1 \otimes x_2^* + x_2 \otimes x_2^*.\end{aligned}$$

Similarly the set $\mathcal{B}_2 \otimes \mathcal{B}_2^*$ is another $B(k)$-basis of $\mathrm{End}(M[1/p])$. As $\varphi(y_1)=p^{\lambda_1}y_1$ and $\varphi(y_2) = p^{\lambda_2}y_2$, we compute the formula of $\varphi :\mathrm{End}(M[1/p]) \to \mathrm{End}(M[1/p])$ with respect to $\mathcal{B}_2$ as follows: $$\begin{aligned}
\varphi(y_2 \otimes y^*_1) &= p^{\lambda_2-\lambda_1} y_2 \otimes y^*_1, \quad & \varphi(y_1 \otimes y^*_1) &= y_1 \otimes y^*_1, \\
\varphi(y_2 \otimes y^*_2) &= y_2 \otimes y^*_2, \quad &\varphi(y_1 \otimes y^*_2) &= p^{\lambda_1-\lambda_2} y_1 \otimes y_2^*.\end{aligned}$$ Therefore, we have found $B(k)$-bases for $$L^+ = \langle y_2 \otimes y^*_1 \rangle_{B(k)}, \quad L^0 = \langle y_1 \otimes y^*_1, y_2 \otimes y^*_2 \rangle_{B(k)}, \quad L^- = \langle y_1 \otimes y^*_2 \rangle_{B(k)}.$$ We compute the change of basis matrix from $\mathcal{B}_1 \otimes \mathcal{B}_1^*$ to $\mathcal{B}_2 \otimes \mathcal{B}_2^*$ as follows: $$\begin{aligned}
y_1 \otimes y_1^* &= x_1 \otimes x^*_1-\frac{v}{p^{\lambda_1}}x_1 \otimes x_2^*,\\
y_2 \otimes y_1^* &= vx_1 \otimes x_1^*+p^{\lambda_1}x_2 \otimes x_1^*-\frac{v^2}{p^{\lambda_1}}x_1 \otimes x_2^*-vx_2 \otimes x_2^*, \\
y_1 \otimes y^*_2 &= \frac{1}{p^{\lambda_1}}x_1 \otimes x^*_2,\\
y_2 \otimes y^*_2 &= \frac{v}{p^{\lambda_1}}x_1 \otimes x^*_2 + x_2 \otimes x^*_2.\end{aligned}$$ It is easy to see that $p^{\lambda_1} y_i \otimes y^*_j \in \mathrm{End}(M) \; \backslash \; p\mathrm{End}(M)$ for $i, j \in \{1,2\}$. We get that

(a) $O^+ = \langle p^{\lambda_1}y_2 \otimes y_1 \rangle_{W(k)}$;

(b) $N := \langle y_1 \otimes y^*_1+y_2 \otimes y_2^*, p^{\lambda_1}y_2\otimes y^*_2\rangle_{W(k)} \subset O^0$ is a lattice;

(c) $O^- = \langle p^{\lambda_1}y_1 \otimes y^*_2 \rangle_{W(k)}$.

We now show that in fact $N=O^0$. As $O^0 = A^0 \otimes W(k)$, it is enough to show that $A^0 \subset N$. Suppose $$ax_1 \otimes x_1^* + bx_2 \otimes x_1^* + cx_1 \otimes x_2^* + dx_2 \otimes x_2^* \in A^0,$$ we have $$\begin{aligned}
& \varphi(ax_1 \otimes x_1^* + bx_2 \otimes x_1^* + cx_1 \otimes x_2^* + dx_2 \otimes x_2^*) \\
= & (\sigma(a)-\sigma(b)p^{-\lambda_1}u) x_1 \otimes x_1^* + \sigma(b)p^{\lambda_2-\lambda_1}x_2 \otimes x_1^* +(-\sigma(b)p^{-\lambda_1}u+\sigma(d))x_2 \otimes x_2^*\\
&+(-\sigma(a)p^{-\lambda_2}u-\sigma(b)p^{-\lambda_1-\lambda_2}u^2+\sigma(c)p^{\lambda_1-\lambda_2}+\sigma(d)p^{-\lambda_2}u)x_1\otimes x_2^*\\
=& ax_1 \otimes x_1^* + bx_2 \otimes x_1^* + cx_1 \otimes x_2^* + dx_2 \otimes x_2^* .\end{aligned}$$ Hence $$\begin{aligned}
a &= \sigma(a)-\sigma(b)p^{-\lambda_1}u,  \label{equation:solvinga}\\
b &= \sigma(b)p^{\lambda_2-\lambda_1}, \label{equation:solvingb} \\
c &= -\sigma(a)p^{-\lambda_2}u-\sigma(b)p^{-\lambda_1-\lambda_2}u^2+\sigma(c)p^{\lambda_1-\lambda_2}+\sigma(d)p^{-\lambda_2}u, \label{equation:solvingc}\\
d &= -\sigma(b)p^{-\lambda_1}u + \sigma(d).  \label{equation:solvingd}\end{aligned}$$ By , we know that $b=0$. Hence $a=\sigma(a)$, $d = \sigma(d)$ by and , and $$c = -ap^{-\lambda_2}u+\sigma(c)p^{\lambda_1-\lambda_2}+dp^{-\lambda_2}u,$$ by , namely $$p^{\lambda_1}(p^{\lambda_2-\lambda_1}c-\sigma(c)) = (d-a)u.$$ In order to have a solution for $c$, we need $d-a \in p^{\lambda_1}W(k)$. Let $d-a=p^{\lambda_1}\alpha$ for some $\alpha \in \mathbb{Z}_p$ as $a, d \in \mathbb{Z}_p$. Then we have a unique solution $c$ such that $$p^{\lambda_2-\lambda_1}c-\sigma(c) = \alpha u.$$ As $u = p^{\lambda_2 - \lambda_1}v-\sigma(v)$, we get $c=\alpha v$. It is now easy to see that $$\begin{aligned}
& ax_1 \otimes x_1^* + bx_2 \otimes x_1^* + c x_1 \otimes x_2^* + d x_2 \otimes x_2^* \\
= & a(x_1 \otimes x_1^* + x_2 \otimes x_2^*) + (\alpha v) x_1 \otimes x_2^*+ (d-a)x_2 \otimes x_2^* \\
= & a(y_1 \otimes y_1^* + y_2 \otimes y_2^*)  + \alpha p^{\lambda_1}y_2 \otimes y_2^* \in N.\end{aligned}$$ Hence $N = O^0$.

The change of basis matrix from $\{y_1 \otimes y_1^*+y_2 \otimes y_2^*, p^{\lambda_1}y_2 \otimes y_1^*, p^{\lambda_1}y_1 \otimes y_2^*, p^{\lambda_1}y_2 \otimes y_2^*\}$ to $\mathcal{B}_1 \otimes \mathcal{B}_1^*$ is $$A = \begin{pmatrix} 1 & p^{\lambda_1}v & 0 & 0 \\ 0 & p^{2\lambda_1} & 0 & 0 \\ 0 & -v^2 & 1 & v \\ 1 & -p^{\lambda_1}v & 0 & p^{\lambda_1} \end{pmatrix}$$ To find an upper bound of $\ell_{\mathcal{M}}$, we compute the inverse of $A$: $$A^{-1} = \frac{1}{p^{2\lambda_1}}\begin{pmatrix} p^{2\lambda_1} & -p^{\lambda_1}v & 0 & 0 \\ 0 & 1 & 0 & 0 \\ p^{\lambda_1}v & -v^2 & p^{2\lambda_1} & -p^{\lambda_1}v \\ -p^{\lambda_1} & 2v & 0 & p^{\lambda_1} \end{pmatrix}$$ Thus the smallest number $\ell$ such that all entries of $p^{\ell}A^{-1} \in W(k)$ is $2\lambda_1$. Hence $\ell_{\mathcal{M}} = 2\lambda_1$. By Theorem \[theorem:maintheorem\], we have $n_{\mathcal{M}} = 2\lambda_1$.

Acknowledgement {#acknowledgement .unnumbered}
===============

The author would like to thank Adrian Vasiu for several suggestions and conversations and the anonymous referee for numerous suggestions.