# Mechanism, measurement, and quantification of stress in decision process: a
model based systematic-review protocol

CSChang Su  XLXiaoyuan Li  LYLin Yang  YZYong Zeng  Concordia Institute for
Information Systems Engineering, Concordia University, H3G 1M8 Montreal,
Canada  School of Electrical Engineering, Zhengzhou University, Zhengzhou,
China  Department of Cancer Epidemiology and Prevention Research, Alberta
Health Service, Calgary, Canada  Departments of Oncology and Community Health
Sciences, University of Calgary, Calgary, Canada

###### Abstract

Background: Every human action begins with decision-making. Stress is a
significant source of biases that can influence human decision-making. In
order to understand the relationship between stress and decision-making,
stress quantification is fundamental. Different methods of measuring and
quantifying stress in decision-making have been described in the literature
while an up-to-date systematic review of the existing methods is lacking.
Moreover, mental stress, mental effort, cognitive workload, and workload are
often used interchangeably but should be distinguished to enable in-depth
investigations of decision-mechanisms. Our objectives are to clarify stress
related concepts and review the measurement, quantification, and application
of stress during decision making activities.

Methods: We developed and followed a systematic reviews protocol to analyze
the literature related to stress in decision-making. We systematically
searched Web of Science, Scopus, PubMed, and ERIC (EBSCO) between 1990 and
2020 with English language. We will include any literature reporting
measurable stress-related outcomes, including stress, workload, cognitive
workload, mental effort during the decision-making process. All research
designs investigating the quantification and measurement of stress for healthy
adults are eligible for this review. Research postulates are proposed based on
the stress-effort model. Two reviewers will independently screen the articles
for inclusion and assess for study quality using the Quality Assessment of
Diagnostic Accuracy Studies-2 tool. We will extract data from each study
including research objective, research method, research domain, triggering
method, objective measure, subjective measure, and research results for
knowledge syntheses.

Discussion: The physiological responses to stress are proposed for
verification. This systematic review will provide knowledge on decision
mechanisms under stress and stress related concepts. The improved
understanding on stress measurement, quantification, and application in
specific research context will advance methods to study and optimize decision
making.

systematic review,

protocol, stress,

acute stress,

workload,

cognitive workload,

mental effort,

stress measures,

stress quantification,

###### keywords:

Research

## Background

Every human action begins with decision-making, which is an established
discipline that identifies and selects alternatives based on the values and
preferences of decision-makers [1]. Decision-making can be characterized as
the organism’s ability to choose the most adaptive action and outcomes from
all possible alternatives [2].

Stress is a significant source of biases and can influence human decision-
making [3, 4, 5]. Stress can be defined as ’the nonspecific result of any
demand upon the body’ [6]. This definition indicates that stressor, as the
load or stimulus, triggers stress response [7]. A stressor can be a physical
trigger that threatens homeostasis [8, 9, 10], such as injury, noise, and
extreme temperature. A stressor can also be a psychological trigger, such as
time-pressured tasks, trauma events [11], and unexpected events. The stress
response represents the body’s attempt to counteract the stressor and re-
establish homeostasis [8], such as increased blood pressure, the increased
cortisol levels, and the decreased heart rate variability. Hence, these
physiological responses can be proxy measures for stress. The entire process,
from the stimulus activation to the individual response, is a decision-making
process. Stress can bias decision-making strategies, affecting the individual
decision-making ability to perform actions, and may cause mistakes and
incorrect decisions [12, 13].

Different levels of stress have different impacts on decision-making. Stress
can be classified into ”eustress” (positive stress) and ”distress” (negative
stress) [14]. Eustress is considered as constructive stress because it will
not cause any issues in solving and pursuing goals. In contrast, it can
stimulate the individual to feel happy or motivated, which improves
productivity and performance. On the contrary, distress is considered as
destructive stress because it is more challenging than eustress to deal with
and can become a source of barriers to achieving goals [15] . Distress can
damage cognitive abilities and reduce individual performance. The Yerkes-
Dodson law illustrates the relationship between stress and performance in an
inverse U-shaped curve [16]. However, the Yerkes-Dodson law did not specify
factors that influence stress. Furthermore, for simple problems, people can
perform excellently with minimal stress. Nguyen and Zeng thus revised the
Yerkes-Dodson law and proposed the stress-effort model [17], as shown in
Figure 1. Given that both the presence and the level of stress could influence
decision-making, it is critical to measure and quantify stress.

Figure 1:  The stress-effort model. Source: Nguyen and Zeng [17]

Likewise, the stress-effort model shows a bell-shaped curve relationship
between stress and effort. The effort may enhance with stress, up to a point,
then it will decline with continuously increased stress. Nguyen and Zeng also
developed a conceptual stress equation to describe the relationship of stress
with the perceived workload, knowledge, skills, and affect [18], as shown in a
mathematical form below:

$\displaystyle\sigma=\frac{w_{p}}{(k+S)*\alpha}$ (1)

where $\sigma$ represents stress, and $W_{p}$,$K$,$S$, and $\alpha$ represents
perceived workload, knowledge, skills, and affect, respectively.

To measure stress is to measure the stress perceived by individuals [19].
Researchers measures the difference from stressful situation to the normal
situation (baseline) of individuals. The process of stress activation and the
process of stress recovery can make this difference. Moreover, how quickly
people recover from the episode of stress can determine whether it is acute
stress (short-term stress) or chronic stress (long-term stress) [20, 21].
Acute stress will develop rapidly and last for a short time through a
particularly stressful event. Chronic stress can bring depression, personality
disorders, and diseases [22, 23, 24]. Our review will focus on acute stress,
which has been the central focus of decision-making [25].

It can be seen from Equation 1 that stress is related to workload, knowledge,
emotion, perception, cognitive and affective capabilities. Previous research
on the relationship between stress and decision-making had involved concepts,
such as mental effort, workload, cognitive workload [26, 27, 28, 29, 30, 31].
These stress-related concepts sometimes are used interchangeably to describe
the same phenomenon [32, 33] and sometimes the same concept had been used to
refer to different phenomena [34, 35]. Correctly distinguishing these concepts
can improve the interpretations of findings from stress-related experiments or
applications. Therefore, this review will integrate these stress-related
concepts to elucidate accurate stress measurement and quantification through a
decision-making mechanism.

Stress measures can be categorized into subjective and objective measures.
Subjective measures assess stress using self-report protocols or
questionnaires [1, 2, 3, 4, 36, 37]. In contrast, objective measures use
devices to assess physiological, physical, or behavioural responses [38, 39,
40, 41]. Both measures have their unique advantages and disadvantages.
Subjective measures are perceived to have high reliability and user trust
[42]. However, as self-report are retrospective measures, subjective measures
can induce recall bias [19], which may cause errors and biases that could
compromise their validity. Meanwhile, objective measures could have high
validity [42]. Taking advantage of the rapid development of sensor
technologies and data science, objective measures have become increasingly
popular in recent years [17, 38, 43, 44]. While the adoption of objective
measures brings significant benefits to stress measurement and quantification,
advanced interdisciplinary knowledge is critical to their successful
applications. Furthermore, both subjective and objective measures are combined
to achieve a comprehensive stress assessment [17, 45]. Therefore, this review
will summarize the experiment protocols of stress measures, categorizes
measures according to their functions and applications, and survey and assess
the stress quantification algorithms and processing methods.

## Methods

### Research postulates

The foundation of this review comes from the stress-effort model shown in
Figure 1 [17]. Herein, we propose a general decision mechanism, (as shown in
Figure 2), which to integrates stress and stress-related concepts into the
stress-effort model.

The general decision mechanism consists of three components specified below:

  * •

Input: The input of the decision mechanism is a stressor.

  * •

Process: The decision-making process involves four distinctive stages,
including high-level cognitive activities [46] such as: “perception and
engagement”, “understanding”, “analysis”, and “decision making”.

  * •

Output: The output consists of everything produced by the decision-making
process, such as behaviours, stress, and cognitive workload [47].

A stressor, as the input, stimulates an individual to start the decision-
making process. Guided by the individual’s knowledge, skills, and affect, each
stage of the decision-making process leads to certain behaviours while taking
corresponding mental effort and triggering consequential mental stress [47].
In Figure 2, behaviours contains physical actions associated with individual
psychomotor skills and abilities [48]. In addition, a cognitive workload will
be created and becomes the input for the next stage. As the process continues
to evolve, cognitive workload, stress, and behaviour are recursively updated.

Figure 2: General decision mechanism. $K$ represents knowledge, $S$ represents
skills, and $\alpha$ represents affect.

While decision-making can be a controlled, rational, and effortful process, it
can also be automatic, unconscious, emotional, and effortless. During the
rational decision-making process, people can reason with their knowledge and
skills to deal with their cognitive workload. In contrast, automatic decision-
making has no self-awareness or control of the mind [49]. The former occurs
when the cognitive workload is mild, whereas the latter kicks in when the
stress levels are excessively heavy or lacking [18]. Figure 2 describes the
rational decision-making process, and Figure 3 illustrates the emotional
(automatic) process of decision-making.

Figure 3:  Automatic decision mechanism under extreme stresses

Like the rational decision-making mechanism shown in Figure 2, the automatic
decision-making mechanism also includes three components: input, process, and
output but with only two stages, which are “perception and engagement” and
“decision making”. Thus, people would make their decisions based only on
perception and engagement while skipping the understanding and analysis
stages.

Several physiological responses to stress are involved during decision-making.
A stressor will stimulate many organisms and various systems in the human
body, including the automatic nervous system (ANS), the hypothalamic-
pituitary-adrenal axis (HPA axis), and the brain network [9]. Figure 4
illustrates the physiological responses to stress, which serves as the
foundation for stress measurements.

Figure 4: The physiological responses to stress

ANS contains the sympathetic nervous system and the parasympathetic nervous
system. Stressor stimulates the sympathetic nervous system and modulates
control on the Adrenal medulla, which creates the synthesis and secretion of
norepinephrine and epinephrine. The ANS triggers physiological responses, such
as heart rate, HRV, blood pressure, eye movements, and skin conductance.

Stress can also induce an increased cortisol output via the HPA axis
activation. When the stressor is applied to the HPA axis, the hypothalamus
creates corticotropin-releasing factor (CRF), stimulating the pituitary. Then
the pituitary secretes adrenocorticotropic hormone (ACTH), which stimulates
the adrenal. The adrenal cortex is stimulated and secretes glucocorticoid
(GC), which signals negative feedback to CRF and ACTH. Cortisol is the most
critical human glucocorticoid, which is known as the stress hormone. It
elevates blood sugar levels, enhances the brain’s use of glucose under stress
conditions.

Research over the past years has clarified that the brain network related to
stress is involved in responding to stressors. With brain imaging technology
development in mammals and the remarkable progress in genetic studies, the
understanding of stress networks is rapidly growing in recent years. Stress
networks are a set of highly connected brain structures that are activated
when the animals perceive their surroundings or are exposed to various
stressful events [50]. The brain network as shown in Figure 4 indicates EEG
and brain region activation under stress.

Meanwhile, the body and nervous system’s organization and interactions reflect
a high degree of complexity and multidirectional communication. Although
stress involves many factors and various systems of the body, a stress
reaction is a significant activation of stress response systems that result in
a cascade of neuroendocrine changes [50]. Stress and related concepts, such as
mental effort, workload, and cognitive workload, can be monitored and measured
using physiological responses. Several physiological responses can be used as
measurement metrics for stress, such as HRV, EEG, and NASA-TLX. Given the
availability of multiple measurement methods, interests are increasing to
identify the best methods and algorithms to measure and quantify stress.
Researchers proposed a number of different measures. However, how to select
the appropriate measures and metrics to measure and quantify stress remains a
challenge.

Therefore, the literature review is based on the following four research
postulates of stress in decision making:

  1. 1)

there is Bell-shaped relationship between stress and effort;

  2. 2)

stress is proportional to perceived workload, and inversely proportional to
knowledge, skills, and affect;

  3. 3)

there are two decision mechanisms: one is rational while the other is
automatic. Both decision mechanism starts with a stressor as the input and
ends with the output containing behaviours, stress, and cognitive workload.
The output will recursively influence the decision process; and

  4. 4)

responses in the automatic nervous system (ANS), the hypothalamic-pituitary-
adrenal axis (HPA axis), and the brain network can be used to measure stresses
during a decision process.

### Protocol registration and reporting

The systematic review will identify available measures for measuring stress,
and evaluate and summarize the quantification algorithms and processing
methods of different measures. This review will be conducted following the
systematic review methodology and guideline by Normadhi, et al. [51]. The
Preferred Reporting Items for Systematic Reviews and Meta-Analysis (PRISMA)
framework will be used to guide the formulation of this systematic review
protocol (see Additional file 1) [52]. The completed PRISMA-P checklist is
provided in Additional file 2.

### Inclusion criteria

#### Population

We will identify studies investigating the effect of acute stress in disease-
free adults. We will exclude experiments conducted among pediatrics
participants or population with illness or pregnancies.

#### Outcomes

The outcome of interest in this review is to explore the measurement,
quantification, and application of emotional and rational decision process.
This review focuses on studies investigating the theoretical or conceptual
background of stress-related concepts. It also focuses on studies
investigating the psychological measurement properties of stress-related
concepts.

#### Type of studies

We will select studies published in peer-reviewed scientific journals and
related conferences. Studies need to present empirical data and assessed acute
stress. The language restriction will be enforced on the condition that the
English abstract is provided. Moreover, studies that reported bio-signal
measures with sufficient quality will be included.

### Information sources and search strategy

We will conduct comprehensive literature search in the following databases:

  * •

Web of Science, 1990-2020

  * •

Scopus, 1990-2020

  * •

PubMed,1990-2020

  * •

ERIC (EBSCO), 1990-2020

Search terms related to stress, stress-related concepts, stress measurements,
stress quantification, and stress application will be used. Searching uses a
combination of keywords from different search terms. The search strategy is
detailed in Additional file 3. We use the topic (including title, abstract,
and keywords) search in the search design to retrieve the results. This method
is used to extract the most relevant research and reduce the number of
articles that do not match the search. When the topic results are
insufficient, other search structures (for example, title and abstract search)
are performed.

### Screening and selection procedure

Two reviewers will screen titles, abstracts, and keywords independently.
Conflicts will be resolved through discussion. A third reviewer will resolve
any conflicts through consensus. The full text of all remaining articles will
be obtained and assessed independently for inclusion by two reviewers.
Discrepancies will be resolved by discussion, and any disagreement will be
evaluated by a third reviewer. Articles selected for inclusion in the review
must meet the predefined inclusion criteria.

### Data extraction and synthesis

For each included article, two reviewers will independently extract the data
and verify the extracted data. If there is more than one publication reporting
data from the same study, the first publication will be chosen to avoid double
reporting data on the same participants. Two reviewers will also make a
summary assessment of study validity and verify the assessment of validity.
Any discrepancies will be resolved by discussion.

Data on the following factors will be extracted for each study: research
objective, research method, research domain, triggering method, objective
measure, subjective measure, and research results; the identification
techniques and data analysis techniques. A standard information form to
extract data from the selected articles [51]. Reviewers will use Mendeley to
extract the essential information (e.g., title, authors, date of publication).
Specific data will be extracted from each article based on the initial study
categorization. Excel spreadsheet will be created and finalized after the two
reviewers reviewed the data extraction. Reviewers will check the consistency
of the information extracted from each article to achieve high-quality data
extraction. Data extracted from the included articles will be synthesized
separately for stress measurement and quantification. The application, i.e.,
the research domain and research question, of stress will also be summarized.

### Quality assessment

The methodological quality of each included study will be assessed using the
Quality Assessment of Diagnostic Accuracy Studies-2 (QUADAS-2) tool [53]. The
QUADAS-2 is used to assess the risk of bias and applicability concerns in the
study. Each study will be assessed from 4 domains: participant/measure
selection, index test, reference standard, and flow and timing [54].
Classification of studies as low risk and high risk of bias and applicability
will be based on these four domains. Studies with missing information for any
of the domains were classified as having an unclear risk of bias or
applicability for that specific domain. Other quality assessment tools will
also be considered depending on the nature of final included studies.

## Discussion

Our systematic review aims to synthesize previous evidence on the definition
of stress, and the measurement, quantification, and application of stress. We
propose that human effort in decision-making is related to individual stress
through a Bell-shaped curve. The decision mechanisms present that decision
making is a dynamic process under stress. According to the stress level, the
decision process can be classified into rational and automatic processes. We
also aim to map the physiological responses to stress during the decision
process to guide the identification of stress measures.

To the best of our knowledge, this will be the first systematic review to
investigate measures for stress and all stress-related concepts. Performing an
inclusive search in significant databases across a 30-year timescale will
cover related research studies on this topic. We have provided a comprehensive
overview of stress and stress-related concepts, decision mechanisms and the
physiological responses to stress in the protocol. Based on the relationship
among stress and stress-related concepts, the decision mechanisms during
rational and automatic decision processes are provided.

The systematic review will focus on investigating the measurement,
quantification, and application of stress and stress-related concepts,
including workload, cognitive workload, and mental effort. This review will
aid technicians and researchers in selecting suitable stress measures for
specific purpose. More importantly, appropriate stress measure selection will
improve experiment objectivity, which will positive affect stress
quantification. Clearly, there is a wide range of stressors and stress
responses. We seek to identify the effective methods to measure and quantify
stress and stress-related concepts, which will provide technical support and
theoretical foundation for stress-related applications. Moreover, we will
identify research trends from the current literature, which aims to establish
a solid base for stress measurement and quantification and to guide further
research. We expect our results to have significant implications in research
and practice under stress such as ergonomics, human-computer interactions,
design, learning and training, and so forth.

## Competing interests

The authors declare that they have no competing interests.

## Abbreviations

EEG: Electroencephalogram; HRV: Heart rate variability; NASA-TLX: NASA Task
Load Index; DOI: Digital Object Identifier; URL: Uniform Resource Locator;
ANS: Automatic nervous system; SNS: Sympathetic nervous system; PNS:
Parasympathetic nervous system; HPA: Hypothalamic–pituitary–adrenal; CRF:
Corticotrophin-releasing factor; ACTH: Adrenocorticotropic hormone; GC:
Glucocorticoid; PRISMA-P: Preferred Reporting Items for Systematic Reviews and
Meta-Analysis Protocol; QUADAS-2: Quality Assessment of Diagnostic Accuracy
Studies-2.

## Funding

Funding was supported by NSERC Discovery Grant (RGPIN-2019-07048), NSERC CRD
Project (CRDPJ 514052-17), and NSERC Design Chairs Program (CDEPJ 485989-14).

## Author’s contributions

CS: manuscript writing. LY: critical revision of the manuscript for crucial
conceptual improvement. XL: draft the manuscript and participate in early
research. YZ: supervised research, design and develop the research framework,
critical revision of the manuscript. All authors read and approved the final
manuscript.

## Author’s information

YZ has published substantially in design and stress. Specific topics of his
research include the science of design, methodology for innovative and
creative design, and neurocognitive model of design creativity. His academic
program at Concordia University, Canada, focused on design and stress, which
has led to the stress-effort model and its applications to multiple
applications to design, learning, education, management, and health. CS is a
doctoral student working on quantification of stress. LY is an epidemiologist,
whose research program focuses on biological mechanisms between human movement
and health, behavioural clinical trials, and implementation science. XL
focuses on the neural signal processing.

## Acknowledgements

Not applicable

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## Additional Files

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# Properties of Breuil-Kisin modules inherited by $p$-divisible groups

Absos Ali Shaikh  Department of Mathematics,
The University of Burdwan,
Burdwan-713101, India<EMAIL_ADDRESS>and  MABUD ALI SARKAR
Department of Mathematics
The University of Burdwan
Burdwan-713101, India<EMAIL_ADDRESS>
###### Abstract.

In this paper, by assuming a faithful action of a finite flat
$\mathbb{Z}_{p}$-algebra $\mathscr{R}$ on a $p$-divisible group $\mathcal{G}$
defined over the ring of $p$-adic integers $\mathscr{O}_{K}$, we have
constructed a new Breuil-Kisin module $\mathfrak{M}$ defined over the ring
$\mathfrak{S}:=W(\kappa)[[u]]$ and studied freeness and projectiveness
properties of such a module. Finally we have lifted the Breuil-Kisin module to
étale level following the $\mathscr{R}$-action and showed that if the $p$-adic
Tate module of a $p$-divisible group $\mathcal{G}$ over $\mathscr{O}_{K}$ is
free then the corresponding lifted Breuil-Kisin module is free.

###### Key words and phrases:

p-adic number field, Breuil-Kisin module, p-divisible group, p-adic Tate
module

###### 2020 Mathematics Subject Classification:

11F80, 11F85, 14L05

##  1\. Introduction and motivation

For a fixed prime numebr $p$, consider the $p$-adic field $\mathbb{Q}_{p}$
with ring of integers $\mathbb{Z}_{p}$ and residue field $\mathbb{F}_{p}$. Let
$\pi\in\mathscr{O}_{K}$ be an uniformizer of the ring of integers
$\mathscr{O}_{K}$ in a finite extension $K$ of $\mathbb{Q}_{p}$ with
$\kappa\cong\mathscr{O}_{K}/\pi\mathscr{O}_{K}$ being the residue field of
$K$. Let $E(u)$ be the minimal (Eisenstein) polynomial of $\pi$ over
$W(\kappa)$, where $W(\kappa)$ is the ring of witt vectors of the residue
field $\kappa$. We denote by $\mathfrak{S}:=W(\kappa)[[u]]$ as the ring of
power series over $W(\kappa)$. Let $\varphi_{\mathfrak{S}}:W(\kappa)[[u]]\to
W(\kappa)[[u]]$ be the ring endomorphism which extends the Frobenious on
$W(\kappa)$ satisfying $\varphi_{\mathfrak{S}}(u)=u^{p}$.

Classification of $p$-divisible groups or Barsotti-Tate groups is an useful
tool in the study of arithmetic geometry and $p$-adic Hodge theory. For
example, given an abelian scheme $S$, consider the inductive system
$\\{(S[p^{n}]),S[p^{n}]\to S[p^{n+1}])\\}_{n}$, where $S[p^{n}]$ denotes the
$p^{n}$-power torsion points on $S$. This gives rise the $p$-divisible group
$S[p^{\infty}]$, which inherits lots of information about the abelian scheme.
That is why it is important to study $p$-divisible groups. There are several
classification of $p$-divisible groups through linear algebraic data. One such
category of important linear algebraic objects that classifies the
$p$-divisible groups is the category of Breuil-Kisin modules defined over the
ring of witt vectors $\mathfrak{S}=W(\kappa)[[u]]$. In fact, the category of
$p$-divisible groups $\text{BT}(\mathscr{O}_{K})$ over $\mathscr{O}_{K}$ is
equivalent to the category of Breuil-Kisin modules
$\text{BT}_{/\mathfrak{S}}^{\varphi}$, which was conjectured by Breuil [Bre]
and proved by Kisin [Kis]. More precisely, Breuil-Kisin module is an important
tool in studying Galois representations.

Let us consider the Breuil-Kisin module $\mathfrak{M}$ over $\mathfrak{S}$.
For a finite flat $\mathbb{Z}_{p}$-algebra $\mathscr{R}$, consider the
$p$-divisible group $\mathcal{G}$ defined over $\mathscr{O}_{K}$ endowed with
a faithful $\mathscr{R}$-action. Since the category of such modules
$\mathfrak{M}$ is equivalent to the category of $p$-dvisible groups
$\mathcal{G}$ over $\mathscr{O}_{K}$, the above $\mathcal{G}$ equipped with
$\mathscr{R}$-action results a newly constructed
$\mathfrak{S}_{\mathscr{R}}:=(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))[[u]]$-module
$\mathfrak{M}$. Therefore, we have studied freeness and projectiveness
properties of this newly produced Breuil-Kisin module over
$(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))[[u]]$. More interestingly, we
have also studied to what extent the freeness property of $p$-adic Tate module
$T_{p}(\mathcal{G})$ of the $p$-divisible group $\mathcal{G}$ affects the
freeness of the Breuil-Kisin module. Let us denote this new category of
modules by $\text{Mod}_{/\mathfrak{S}_{\mathscr{R}}}^{\varphi}$ which consists
of objects of $\text{Mod}_{/\mathfrak{S}}^{\varphi}$ equipped with
$\mathscr{R}$-action. However, the projectiveness and freeness property of the
$(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))[[u]]$-module $\mathfrak{M}$
does not follow for general $\mathscr{R}$. For example, consider the ring
$\mathscr{R}=\mathbb{Z}_{p}[u]/(u^{2})$. This acts as $0$ on any Breuil-Kisin
module, and so the Breuil-Kisin module would not be free or even projective
over $(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))[[u]]$ for such
$\mathscr{R}=\mathbb{Z}_{p}[u]/(u^{2})$. This suggests us $\mathscr{R}$ to be
a regular local ring. Another obstruction is that spectra $Spec(\mathscr{R})$
of $\mathscr{R}$ might not be connected. In that case, $\mathfrak{M}$ could be
projective, but have different ranks on different components of spectra and
hence $\mathfrak{M}$ can not be free. Further the ring
$(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))[[u]]$ is not necessarily
local, for example $W(\kappa)\otimes_{\mathbb{Z}_{p}}W(\kappa)$ decomposes as
a direct product of $[\kappa:\mathbb{F}_{p}]$ copies of $W(\kappa)$.
Therefore, the natural question arises:
When is the Breuil-Kisin module $\mathfrak{M}$ projective and free over
$(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))[[u]]$ ?
The present paper provides the answer of the above question in the next two
sections.

##  2\. Projectiveness and freeness of the objects in
$\text{Mod}_{/\mathfrak{S}_{\mathscr{R}}}^{\varphi}$

Before we start our investigations on the projectiveness and freeness
properties of the modules
$\mathfrak{M}\in\text{Mod}_{/\mathfrak{S}_{\mathscr{R}}}^{\varphi}$, we recall
some definitions:

######  Definition 2.1.

A Breuil-Kisin module (BK-module for brevity) of height $h$ (positive integer)
is a finitely generated and free $\mathfrak{S}$-module $\mathfrak{M}$ endowed
with a $\varphi_{\mathfrak{S}}$-semilinear endomorphism
$\varphi_{\mathfrak{M}}:\mathfrak{M}\to\mathfrak{M}$ such that $E(u)^{h}$
kills the cokernel
$\mathfrak{M}/(1\otimes\varphi_{\mathfrak{M}})(\mathfrak{M})$ (i.e.,
$\varphi_{\mathfrak{M}}$ is injective) of the $\mathfrak{S}$-linearization
$1\otimes\varphi_{\mathfrak{M}}:\varphi_{\mathfrak{S}}^{*}(\mathfrak{M})\to\mathfrak{M}$
of $\varphi_{\mathfrak{M}}$, where we denote
$\varphi_{\mathfrak{S}}^{*}(\mathfrak{M}):=\mathfrak{S}\otimes_{\varphi_{\mathfrak{S}},\mathfrak{S}}\mathfrak{M}$.
We also denote the category of such modules by
$\text{Mod}_{/\mathfrak{S}}^{\varphi}$. For $h=1$, we call such modules as
height $1$ Breuil-Kisin modules and such a category is denoted by
$\text{BT}_{/\mathfrak{S}}^{\varphi}$ as mentioned above. In other words,
$\text{BT}_{/\mathfrak{S}}^{\varphi}$ is a full subcategory of
$\text{Mod}_{/\mathfrak{S}}^{\varphi}$.

######  Remark 1.

In literature there are also slightly different versions of Breuil-Kisin
modules used by different authors (e.g., [Cai], [Gao], [Gee]) in different
persepectives. Some authors preferred only either the freeness property or the
finiteness property. Some authors assumed $\mathfrak{M}$ to be $u$-torsion
free while some authors inverted $E(u)$ in defining
$\mathfrak{S}[1/E(u)]$-linearization map
$\mathfrak{S}\otimes_{\varphi,\mathfrak{S}}\mathfrak{M}[1/E(u)]\to\mathfrak{M}[1/E(u)]$.
Simply, we can think of Breuil-Kisin module as a module over the ring
$\mathfrak{S}$ with some impositions. The best way to understand the Breuil-
Kisin modules (linear algebra data) is to look at the corresponding objects in
the Galois representations such as crystalline Galois representations with
different Hodge-Tate weights or torsion Galois representations.

######  Definition 2.2.

[Tate] Let $p$ be a prime number, and $d$ be a non-negative integer. A
$p$-divisible group $\mathcal{G}$ over the ring of integers $\mathscr{O}_{K}$
of height $d$ is the inductive system

$\mathcal{G}=(\mathcal{G}_{m},i_{m}),\ m\geq 0$

with group scheme homomorphism $i_{m}:\mathcal{G}_{m}\to\mathcal{G}_{m+1}$,
and satisfying the following criterions:

  1. (i)

$\mathcal{G}_{m}$ is a finite flat group scheme over the ring of integers
$\mathscr{O}_{K}$ of order $p^{md}$ for all $m$,

  2. (ii)

the sequence

$0\to\mathcal{G}_{m}\xrightarrow{i_{m}}\mathcal{G}_{m+1}\xrightarrow{p^{m}}\mathcal{G}_{m+1}$

is exact. We usually denote by $\text{BT}(\mathscr{O}_{K})$ the category of
$p$-divisible groups over $\mathscr{O}_{K}$.

######  Definition 2.3.

[Mor] Let $\mathcal{G}$ be a $p$-divisible group over the ring of integers
$\mathscr{O}_{K}$. Let

$\cdots\xrightarrow{i_{3}}\mathcal{G}_{3}\xrightarrow{i_{2}}\mathcal{G}_{2}\xrightarrow{i_{1}}\mathcal{G}_{1}$

be the inverse system associated to the $p$-divisible group $\mathcal{G}$.
Then the $p$-adic Tate module of $\mathcal{G}$ is

$T_{p}(\mathcal{G}):=\varprojlim_{m}\mathcal{G}_{m}(K^{alg}),$

where we take the inverse limit over positive integers $m$ with the transition
maps $i_{m}$ and $K^{alg}$ is the separable closure of $K$. Note that
$T_{p}(\mathcal{G})$ is free $\mathbb{Z}_{p}$-module and it encodes all the
$p$-power torsion points of $\mathcal{G}$ in $K^{alg}$.

Firstly, we start our investigation on the freeness property of
$\mathfrak{M}\in\text{Mod}_{/\mathfrak{S}_{\mathscr{R}}}^{\varphi}$.

######  Theorem 2.1.

If $\mathscr{R}$ be the regular local ring $\mathscr{O}_{K}$ in finite
extension $K$ of $\mathbb{Q}_{p}$, then $\mathfrak{M}$ is
$(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))[[u]]$-free module.

###### Proof.

Let $\mathscr{R}=\mathscr{O}_{K}$ be the ring of integers of the finite
extension $K$ of $\mathbb{Q}_{p}$. Then
$\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa)$ is a direct product of rings of
integers of finite extensions of $\mathbb{Q}_{p}$. For example, different
types of unramified extensions of $K$ has such property for the factor fields
$(\mathscr{R}/\mathfrak{m}_{\mathscr{R}}\otimes_{\mathbb{F}_{p}}\kappa)$,
where $\mathfrak{m}_{\mathscr{R}}$ is the maximal ideal of $\mathscr{R}$.
Hence it is sufficient to show that if $L$ is a finite extension of
$\mathbb{Q}_{p}$ with ring of integers $\mathscr{O}_{L}$ then the
$\mathscr{O}_{L}[[u]]$-module $\mathfrak{M}$ which is finite free
$\mathbb{Z}_{p}[[u]]$-module is also a free as $\mathscr{O}_{L}[[u]]$-module.
When $\mathfrak{M}=0$, the statement is trivial. Let $\mathfrak{M}$ be non-
zero. Since $\mathscr{O}_{L}[[u]]$ is a regular local ring, all finitely
generated modules have finite projective dimension. So by Auslander-Buchsbaum
formula, we have

$\text{projdim}(\mathfrak{M})+\text{depth}(\mathfrak{M})=\text{depth}(\mathscr{O}_{L}[[u]])=2,$
(1)

where “projdim” and “depth” respectively denotes the projective dimension and
depth of the module $\mathfrak{M}$. Now $\mathfrak{M}$ being a finite free
$\mathbb{Z}_{p}[[u]]$-module, clearly $\\{u,p\\}$ is a regular sequence for
$\mathfrak{M}$ as $\mathbb{Z}_{p}[[u]]$-module, and so $\\{u,p\\}$ is a
regular sequence for $\mathfrak{M}$ as a $\mathscr{O}_{L}[[u]]$-module.
Therefore the depth of $\mathfrak{M}$ as $\mathscr{O}_{L}[[u]]$-module is
equal to $2$ i.e., $\text{depth}(\mathfrak{M})=2$. So from equation
$\eqref{e1}$, we get $\text{projdim}(\mathfrak{M})=0$. This means that
$\mathfrak{M}$ is $\mathscr{O}_{L}[[u]]$-free module and hence $\mathfrak{M}$
is $(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))[[u]]$-free module. This
completes the proof. ∎

######  Remark 2.

The above fact was slightly covered by Kisin in the proof of Prop. 1.6.4 of
his paper [Kis1], however the above proof is different.

######  Corollary 2.1.1.

Let $\mathscr{R}$ be the ring of integers $\mathscr{O}_{K}$ in the finite
extension $K$ of $\mathbb{Q}_{p}$. If there is a finite flat morphism
$\mathscr{R}\to\mathscr{R}^{\prime}$, where $\mathscr{R}^{\prime}$ is also a
regular local ring, and if $\mathfrak{M}$ is an
$(\mathscr{R}^{\prime}\otimes_{\mathbb{Z}_{p}}W(\kappa))[[u]]$-module which is
finite free over $(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))[[u]]$, then
$\mathfrak{M}$ is a finite free
$(\mathscr{R}^{\prime}\otimes_{\mathbb{Z}_{p}}W(\kappa))[[u]]$-module.

###### Proof.

It is sufficient to show that if $\mathscr{R}\to\mathscr{R}^{\prime}$ is a
finite flat morphism between regular local rings and if $\mathfrak{M}$ is an
$\mathscr{R}^{\prime}$-module that is finite free over $\mathscr{R}$, then
$\mathfrak{M}$ is a finite free module over $\mathscr{R}^{\prime}$ and this
follows using the same argument as in Theorem 2.1. ∎

Now we investigate the projectiveness property of
$\mathfrak{M}\in\text{Mod}_{/\mathfrak{S}_{\mathscr{R}}}^{\varphi}$. For this
we need help of the following three results:

######  Theorem 2.2.

[Mat, Theorem 22.3] Let $\mathcal{A}$ be a ring with an ideal $\mathcal{I}$
and an $\mathcal{A}$-module $\mathcal{M}$ such that either $\mathcal{I}$ is a
nilpotent ideal or $\mathcal{A}$ is a Noetherian ring and $\mathcal{M}$ is
$\mathcal{I}$-adically ideal-separated. Then the following statements are
equivalent.

  1. (i)

$\mathcal{M}$ is flat over $\mathcal{A}$;

  2. (ii)

$\text{Tor}_{1}^{\mathcal{A}}(\mathcal{M},\mathcal{N})=0$ for every
$\mathcal{A}/\mathcal{I}$-module $\mathcal{N}$;

  3. (iii)

$\mathcal{M}/\mathcal{I}\mathcal{M}$ is flat over $\mathcal{A}/\mathcal{I}$
and $\mathcal{I}\otimes_{\mathcal{A}}\mathcal{M}=\mathcal{I}\mathcal{M}$;

  4. (iv)

$\mathcal{M}/\mathcal{I}\mathcal{M}$ is flat over $\mathcal{A}/\mathcal{I}$
and $\text{Tor}_{1}^{\mathcal{A}}(\mathcal{A}/\mathcal{I},\mathcal{M})=0$;

  5. (v)

$\mathcal{M}/\mathcal{I}\mathcal{M}$ is flat over $\mathcal{A}/\mathcal{I}$
and
$\gamma_{n}:(\mathcal{I}^{n}/\mathcal{I}^{n+1})\otimes_{\mathcal{A}/\mathcal{I}}\mathcal{M}/\mathcal{I}\mathcal{M}\to\mathcal{I}^{n}\mathcal{M}/\mathcal{I}^{n+1}\mathcal{M}$
is an isomorphism for every $n\geq 0$;

  6. (vi)

$\mathcal{M}/\mathcal{I}\mathcal{M}$ is flat over $\mathcal{A}/\mathcal{I}$
and $\gamma:\oplus_{n\geq
0}\mathcal{I}^{n}/\mathcal{I}^{n+1}\otimes_{\mathcal{A}/\mathcal{I}}\mathcal{M}/\mathcal{I}\mathcal{M}\to\oplus_{n\geq
0}\mathcal{I}^{n}/\mathcal{M}\mathcal{I}^{n+1}\mathcal{M}$ is an isomorphism;

  7. (vii)

$\mathcal{M}/\mathcal{I}^{n+1}\mathcal{M}$ is flat over
$\mathcal{A}/\mathcal{I}^{n+1}$ for every $n\geq 0$.

######  Lemma 2.3.

[TSP, Tag 0561] Let $\mathcal{A}\to\mathcal{B}$ be a ring map of finite type
and let $\mathcal{M}$ be a $\mathcal{B}$-module. If $M$ is finitely generated
$\mathcal{A}$-module, then it is finitely generated $\mathcal{B}$-module.

######  Theorem 2.4.

[Rot, Theorem 3.56] Let $\mathcal{A}$ be a commutative ring and $\mathcal{M}$
be a finitely generated $\mathcal{A}$-module, then $\mathcal{M}$ is flat if
and only if it is projective.

######  Theorem 2.5.

For an arbitrary $\mathbb{Z}_{p}$-algebra $\mathscr{R}$, if
$\mathfrak{M}/u\mathfrak{M}$ is finitely generated projective
$\mathscr{R}$-module such that $\mathfrak{M}$ is $u$-adically complete,
separated and $u$-torsion free, then $\mathfrak{M}$ is finitely generated and
projective as $(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))[[u]]$-module .

###### Proof.

Since $\mathfrak{M}$ is a Breuil-Kisin module over $W(\kappa)$, $\mathfrak{M}$
is projective module over $W(\kappa)[[u]]$. For the given
$\mathbb{Z}_{p}$-algebra $\mathscr{R}$, the
$(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))[[u]]$-module $\mathfrak{M}$
will be finitely generated and projective if and only if
$\mathfrak{M}/u\mathfrak{M}$ is finitely generated and projective as an
$\mathscr{R}$-module. Further, $\mathfrak{M}/u\mathfrak{M}$ is projective
module over $(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))[[u]]$ if and only
if it is projective over $\mathscr{R}[[u]]$. On the other hand, the
$\mathscr{R}[[u]]$-module $\mathfrak{M}$ is finitely generated and projective
once it is $u$-torsion free, $u$-adically complete and separated as well as
$\mathfrak{M}/u\mathfrak{M}$ is a finitely generated and projective
$\mathscr{R}$-module. But it is equivalent to show that
$\mathfrak{M}/u^{i}\mathfrak{M},\ i\in\mathbb{N}$ is a projective
$\mathscr{R}[u]/u^{i}$-module. For, if $\mathfrak{M}$ is $u$-adically
complete, $u$-adically separated, and $\mathfrak{M}/u^{i}\mathfrak{M}$ is
projective module over $\mathscr{R}[u]/u^{i}$ for each $i$, and if
additionally $\mathfrak{M}/u\mathfrak{M}$ is finitely generated and projective
$\mathscr{R}$-module, then $\mathfrak{M}$ is finitely generated and projective
over $\mathscr{R}[[u]]$, shown as follows:

Since $\mathfrak{M}$ is $u$-adically complete by assumption and
$\mathfrak{M}/u\mathfrak{M}$ is finitely generated, applying local version of
Nakayama lemma, any basis that generates $\mathfrak{M}/u\mathfrak{M}$ lifts to
a basis that generates $\mathfrak{M}$, and thus $\mathfrak{M}$ is finitely
generated $\mathscr{R}[[u]]$-module. So we get a surjection
$\mathscr{R}[[u]]^{r}\to\mathfrak{M}$ for some $r$. This induces a surjection
$\left(\mathscr{R}[u]u^{i}\right)^{r}\to\mathfrak{M}/u^{i}\mathfrak{M}$ for
each $i$. Since the target $\mathfrak{M}/u^{i}\mathfrak{M}$ of these
surjections are projective, these surjections split. Such splitings are
compatible as $i$ changes. Thus we get a spliting of the original surjection
$\mathscr{R}[[u]]^{r}\to\mathfrak{M}$. Hence $\mathfrak{M}$ is projective over
$\mathscr{R}[[u]]$.

Since $\mathfrak{M}$ is $u$-torsion free, the multiplication map
$\mathfrak{M}\to\mathfrak{M}$ by $u$ is injectve. In other word, there exists
a short exact sequence of $R$-modules

$0\to\mathfrak{M}/u^{i}\mathfrak{M}\xrightarrow{u^{j}}\mathfrak{M}/u^{i+j}\mathfrak{M}\to\mathfrak{M}/u^{j}\mathfrak{M}\to
0.$

Using the conditions (i) and (vii) of Theorem 2.2, we conclude that
$\mathfrak{M}/u^{i}\mathfrak{M}$ is a flat module over $\mathscr{R}[u]/u^{i}$.
But $\mathscr{R}[u]/u^{i}$ is isomorphic to $\mathscr{R}[[u]]/u^{i}$ and hence
$\mathfrak{M}/u^{i}\mathfrak{M}$ is a flat $\mathscr{R}[[u]]/u^{i}$-module.
Now we already know that the $\mathscr{R}$-module $\mathfrak{M}/u\mathfrak{M}$
is finitely generated, projective and hence of finite presentation. For $i=2$,
$\mathfrak{M}/u^{2}\mathfrak{M}$ is an $\mathscr{R}$-module of finite
presentation using that fact that $\mathfrak{M}/u\mathfrak{M}$ is an
$\mathscr{R}$-module of finite presentation. So by induction on $i$, in the
above exact sequence, we get $\mathfrak{M}/u^{i}\mathfrak{M}$ is an
$\mathscr{R}$-module of finite presentation. As $\mathscr{R}[[u]]/u^{i}$ is a
free $\mathscr{R}$-module of rank $i\in\mathbb{N}$, it is finitely generated
$\mathscr{R}$-module and hence the ring map
$\mathscr{R}\to\mathscr{R}[[u]]/u^{i}$ is of finite type. Thus
$\mathfrak{M}/u^{i}\mathfrak{M}$ is an $\mathscr{R}[[u]]/u^{i}$-module of
finite presentation by Lemma 2.3. Since $\mathfrak{M}/u^{i}\mathfrak{M}$ is a
flat as well as finitely presented $\mathscr{R}[u]/u^{i}$-module, by Theorem
2.4 it is projective $\mathscr{R}[[u]]/u^{i}$-module. Hence
$\mathfrak{M}/u^{i}\mathfrak{M}$ is projective $\mathscr{R}[u]/u^{i}$-module.
This completes the proof. ∎

######  Theorem 2.6.

If $\mathscr{R}$ is a ring such that every finitely generated and $p$-torsion
free $\mathscr{R}$-module is projective, then $\mathfrak{M}$ is projective as
$(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))[[u]]$-module.

###### Proof.

Since $\mathfrak{M}/u\mathfrak{M}$ is projective over $W(\kappa)$, it is
$p$-torsion free, and therefore projective over $\mathscr{R}$. Hence
$\mathfrak{M}$ is projective over
$(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))[[u]]$ by Theorem 2.5. ∎

Finally, we conclude this section with the following theorem on the freeness
property of the $(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))[[u]]$-module
$\mathfrak{M}$.

######  Theorem 2.7.

If the finite flat $\mathbb{Z}_{p}$-algebra $\mathscr{R}$ contains all the
embeddings $\sigma$ of $W(\kappa)$, then $\mathfrak{M}$ is free
$(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))[[u]]$-module.

###### Proof.

In this case we need to use the action of Frobenius $\varphi$ to permute the
components of $(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))[[u]]$. Given the
$p$-divisible group $\mathcal{G}$ with $\mathscr{R}$-action, we have
$\mathfrak{M}$ as free $W(\kappa)[[u]]$-module. Since $\mathscr{R}$ contains
all embeddings of $W(\kappa)$, we get

$(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))[[u]]\cong\prod_{\sigma}\mathscr{R}[[u]],$

where $\sigma$ runs over the embeddings of $W(\kappa)$ into $\mathscr{R}$. By
using same reason, we get

$\mathfrak{M}\cong\oplus_{\sigma}\mathfrak{M}_{\sigma}.$

Since $\mathfrak{M}$ is projective, each summand $\mathfrak{M}_{\sigma}$ is
projective and hence free $W(\kappa)[[u]]$-module. By Auslander-Buchsbaum
theorem, we have

$\text{projdim}(\mathfrak{M}_{\sigma})+\text{depth}(\mathfrak{M}_{\sigma})=\text{depth}\left(\mathscr{R}[[u]]\right).$
(2)

Since $\mathfrak{M}_{\sigma}$ is finite free as a $W(\kappa)[[u]]$,
$\\{u,p\\}$ is a regular sequence for $\mathfrak{M}_{\sigma}$, and hence it is
finite free a $\mathscr{R}[[u]]$-module. So depth of $\mathfrak{M}_{\sigma}$
as an $\mathscr{R}[[u]]$-module is $2$. So from equation (2), we get
projdim($\mathfrak{M}_{\sigma})=0$. Equivalently, $\mathfrak{M}_{\sigma}$ is a
projective and hence free $\mathscr{R}[[u]]$-module. Finally, since the
Frobenius $\varphi$ permutes the $\mathfrak{M}_{\sigma}$ transitively or
cyclically, the pullback $\varphi^{*}$ is injective, each
$\mathfrak{M}_{\sigma}$ has the same rank, which implies that $\mathfrak{M}$
is indeed free $(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))[[u]]$-module. ∎

##  3\. p-adic Tate module and lift of objects in
$\text{Mod}_{/\mathfrak{S}_{\mathscr{R}}}^{\varphi}$ to étale level

In this section we focus our attention to the $p$-adic Tate module
$T_{p}(\mathcal{G})$ of the $p$-divisible group $\mathcal{G}$ over
$\mathscr{O}_{K}$. We can recover the $p$-adic Tate module from Breuil-Kisin
module easily by taking the $\varphi$-fixed points of
$\mathfrak{M}\otimes_{\mathfrak{S}}W(\text{Frac}(\mathfrak{R}))$ i.e.,

$T_{p}(\mathcal{G})=\left(\mathfrak{M}\otimes_{\mathfrak{S}}W(\text{Frac}(\mathfrak{R}))\right)^{\varphi=1},$

where $\text{Frac}(\mathfrak{R})$ defined below. From the work of [Kis], we
also have an inverse process. That is, we start with a $p$-adic Tate module
$T_{p}(\mathcal{G})$ of the $p$-divisible group $\mathcal{G}$ and get a
Breuil-Kisin module. Hence the further question remains to check:

Is $(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))$-module $\mathfrak{M}$ free
if the $p$-adic Tate module $T_{p}(\mathcal{G})$ is free as
$\mathscr{R}$-module ?

To answer this question we at first lift the category
$\text{Mod}_{/\mathfrak{S}_{\mathscr{R}}}^{\varphi}$ of Breuil-Kisin module
(with $\mathscr{R}$-action) to étale level and then explain its connection
with $p$-adic Galois representations.

Let $\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu$ be the
algebraic closure of $K$ and denote by $\mathscr{O}_{\mkern
1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}$, the ring of integers
of $\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu$. We set
$G_{K}=\text{Gal}(\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern
1.5mu/K)$. Let us denote the inverse limit $\varprojlim\mathscr{O}_{\mkern
1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}/p\mathscr{O}_{\mkern
1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}$ by the notation
$\mathfrak{R}$, where the transition maps are offered by the Frobenius map.
Note here $\mathfrak{R}$ is not $\mathbb{Z}_{p}$-algebra but it is infact a
$\mathbb{F}_{p}$-algebra. We have the projection
$\Theta_{0}:\mathfrak{R}\to\mathscr{O}_{\mkern
1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}$ onto the first factor
in the inverse limit. Let $\widehat{\mathscr{O}_{\mkern
1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}$ be the $p$-adic
completion of $\mathscr{O}_{\mkern
1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}$. Then there is an
unique surjection $\Theta:W(\mathfrak{R})\to\widehat{\mathscr{O}_{\mkern
1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}$ which lifts the map
$\Theta_{0}$, by the Universal property of witt vectors $W(\mathfrak{R})$.

We now introduce the discrete valuation ring
$\mathscr{O}_{\mathscr{E}}:=\widehat{\mathfrak{S}\left[\frac{1}{u}\right]}$,
the $p$-adic completion of $\mathfrak{S}[\frac{1}{u}]$. The DVR
$\mathscr{O}_{\mathscr{E}}$ has the residue field $\kappa((u))$ which is the
field of Laurent power series. We denote by $\mathscr{E}$ the field of
fraction of the DVR $\mathscr{O}_{\mathscr{E}}$. The inclusion
$\mathscr{E}\hookrightarrow W(\text{Frac}(\mathfrak{R}))[\frac{1}{p}]$ extends
the inclusion $\mathfrak{S}\hookrightarrow W(\mathfrak{R})$, where
$\text{Frac}(\mathfrak{R})$ is the field of fractions of $\mathfrak{R}$. Let
$\mathscr{E}^{ur}$ be the maximal unraified extension of $\mathscr{E}$ that is
contained in $W(\text{Frac}(\mathfrak{R}))[\frac{1}{p}]$ with its ring of
integers $\mathscr{O}_{\mathscr{E}^{ur}}$, the subring of
$W(\text{Frac}(\mathfrak{R}))$. The residue field
$\mathscr{O}_{\mathscr{E}^{ur}}/p\mathscr{O}_{\mathscr{E}^{ur}}$ is the
separable closure of the residue field $\kappa((u))$ of
$\mathscr{O}_{\mathscr{E}}$, because the field $\text{Frac}(\mathfrak{R})$ is
algebraically closed [Fon]. The $p$-adic completion of
$\mathscr{O}_{\mathscr{E}^{ur}}$ is denoted by
$\mathscr{O}_{\widehat{\mathscr{E}^{ur}}}$.

######  Proposition 3.1.

The map
$\mathfrak{S}_{\mathscr{R}}:=\mathfrak{S}\otimes_{\mathbb{Z}_{p}}\mathscr{R}\to\mathscr{O}_{\mathscr{E},\mathscr{R}}:=\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$
is faithfully flat.

###### Proof.

We note that over Noetherian ring, the localization map is flat and completion
map is faithfully flat [TSP]. Since $u\notin(p)$, the localization
$\mathfrak{S}_{(p)}$ identifies the localization of
$\mathfrak{S}\left[\frac{1}{u}\right]$ at the prime ideal $(p)$ and, hence the
map
$\mathfrak{S}\to\mathscr{O}_{\mathscr{E}}:=\widehat{\mathfrak{S}\left[\frac{1}{u}\right]}$
facorizes through the map
$\mathfrak{S}_{(p)}\to\mathscr{O}_{\mathscr{E}}:=\widehat{\mathfrak{S}\left[\frac{1}{u}\right]}$
. The following diagram commutes:

$\mathfrak{S}$$\mathscr{O}_{\mathscr{E}}:=\widehat{\mathfrak{S}[\frac{1}{u}]}$$\mathfrak{S}[\frac{1}{u}]_{(p)}$

=

$\mathfrak{S}_{(p)}$$f$$f_{1}$$f_{2}$

The composition map $f_{2}\circ f_{1}$ is faithfully flat and hence
$f:\mathfrak{S}\to\mathscr{O}_{\mathscr{E}}:=\widehat{\mathfrak{S}\left[\frac{1}{u}\right]}$
is faithfully flat. With the help of this map $f$, we will show that
$\mathfrak{S}_{\mathscr{R}}\to\mathscr{O}_{\mathscr{E},\mathscr{R}}$ is also
faithfully flat. For this consider a morphism
$f^{\prime}:\mathfrak{M}_{\mathscr{R}}\to\mathfrak{M}_{\mathscr{R}}^{{}^{\prime}}$
of $\mathfrak{S}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$-modules. Tensoring, we
get

$\mathfrak{M}_{\mathscr{R}}\otimes_{\mathfrak{S}\otimes_{\mathbb{Z}_{p}}\mathscr{R}}(\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R})\cong\mathfrak{M}_{\mathscr{R}}\otimes_{\mathfrak{S}}\mathscr{O}_{\mathscr{E}}\
\text{and}\
\mathfrak{M}_{\mathscr{R}}^{{}^{\prime}}\otimes_{\mathfrak{S}\otimes_{\mathbb{Z}_{p}}\mathscr{R}}(\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R})\cong\mathfrak{M}_{\mathscr{R}}^{{}^{\prime}}\otimes_{\mathfrak{S}}\mathscr{O}_{\mathscr{E}}.$

Then the map
$f^{\prime\prime}=f^{\prime}\otimes_{\mathfrak{S}\otimes_{\mathbb{Z}_{p}}\mathscr{R}}\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$
acts as
$f^{\prime}\otimes_{\mathfrak{S}}\mathscr{O}_{\mathscr{E}}:\mathfrak{M}_{\mathscr{R}}\otimes_{\mathfrak{S}}\mathscr{O}_{\mathscr{E}}\to\mathfrak{M}_{\mathscr{R}}^{{}^{\prime}}\otimes_{\mathfrak{S}}\mathscr{O}_{\mathscr{E}}$.
Therefore $f^{\prime\prime}$ is injective if and only if $f^{\prime}$ is
injective. Again, $f^{\prime}$ is injective if and only if
$f^{\prime}\otimes_{\mathfrak{S}}\mathscr{O}_{\mathscr{E}}$ injective. Hence
$\mathfrak{S}\otimes_{\mathbb{Z}_{p}}\mathscr{R}\to\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$
is faithfully flat. ∎

######  Definition 3.1.

[Bri] We denote by
$\Phi\mathscr{M}_{\mathscr{O}_{\mathscr{E}}}^{\text{\'{e}t}}$, the category of
étale $\varphi$-module over $\mathscr{O}_{\mathscr{E}}$ consisting of the
pairs $(\mathscr{M},\varphi_{\mathscr{M}})$, where

  1. (i)

$\mathscr{M}$ is a finitely generated $\mathscr{O}_{\mathscr{E}}$-module,

  2. (ii)

$\varphi_{\mathscr{M}}:\mathscr{M}\to\mathscr{M}$ is a
$\varphi_{\mathscr{O}_{\mathscr{E}}}$-semilinear endomorphism such that its
$\mathscr{O}_{\mathscr{E}}$-linearization
$1\otimes\varphi_{\mathscr{M}}:\varphi_{\mathscr{O}_{\mathscr{E}}}^{*}\mathscr{M}=\mathscr{O}_{\mathscr{E}}\otimes_{\mathscr{O}_{\mathscr{E}},\varphi_{\mathscr{O}_{\mathscr{E}}}}\mathscr{M}\to\mathscr{M}$
is an isomorphism.

Let us denote $\text{Mod}_{/\mathscr{O}_{\mathscr{E}}}^{\varphi}$ as the
subcategory of $\Phi\mathscr{M}_{\mathscr{O}_{\mathscr{E}}}^{\text{\'{e}t}}$
whose objects are free over $\mathscr{O}_{\mathscr{E}}$, and by
$\text{Mod}_{/\mathscr{O}_{\mathscr{E},\mathscr{R}}}^{\varphi}$ the category
consisting of objects of $\text{Mod}_{/\mathscr{O}_{\mathscr{E}}}^{\varphi}$
equipped with action of $\mathscr{R}$. Also, we have already denoted by
$\text{Mod}_{/\mathfrak{S}_{\mathscr{R}}}^{\varphi}$ the category consisting
of objects of $\text{Mod}_{/\mathfrak{S}}^{\varphi}$ equipped with action of
$\mathscr{R}$.

######  Proposition 3.2.

For any object
$\mathfrak{M}_{\mathscr{R}}:=\mathfrak{M}\otimes_{\mathbb{Z}_{p}}\mathscr{R}\in\text{Mod}_{/\mathfrak{S}_{\mathscr{R}}}^{\varphi}$,
the scalar extension
$\mathscr{M}_{\mathscr{R}}:=\mathfrak{M}_{\mathscr{R}}\otimes_{\mathfrak{S}_{\mathscr{R}}}\mathscr{O}_{\mathscr{E},\mathscr{R}}$
is an object of
$\text{Mod}_{/\mathscr{O}_{\mathscr{E},\mathscr{R}}}^{\varphi}$.

###### Proof.

It follows from Proposition 3.1. ∎

######  Proposition 3.3.

The functor
$\text{Mod}_{/\mathfrak{S}_{\mathscr{R}}}^{\varphi}\to\text{Mod}_{/\mathscr{O}_{\mathscr{E},\mathscr{R}}}^{\varphi}$
given by

$\mathfrak{M}_{\mathscr{R}}\mapsto\mathscr{M}_{\mathscr{R}}:=\mathfrak{M}_{\mathscr{R}}\otimes_{\mathfrak{S}_{\mathscr{R}}}\mathscr{O}_{\mathscr{E},\mathscr{R}}$
(3)

is fully faithful.

###### Proof.

It follows from [Kis] that the functor
$\mathcal{F}:\text{Mod}_{/\mathfrak{S}}^{\varphi}\to\text{Mod}_{/\mathscr{O}_{\mathscr{E}}}^{\varphi}$
defined by
$\mathfrak{M}\mapsto\mathscr{M}:=\mathfrak{M}\otimes_{\mathfrak{S}}\mathscr{O}_{\mathscr{E}}$
is fully faithful. Then by definition of full faithfulness, the functor
$\mathcal{F}$ induces a bijective map

$\mathcal{F}_{\mathfrak{M},\mathfrak{M}^{\prime}}:\text{Hom}_{\text{Mod}_{/\mathfrak{S}}^{\varphi}}(\mathfrak{M},\mathfrak{M}^{\prime})\to\text{Hom}_{\text{Mod}_{/\mathscr{O}_{\mathscr{E}}}^{\varphi}}(\mathcal{F}(\mathfrak{M}),\mathcal{F}(\mathfrak{M}^{\prime})).$
(4)

So any $\text{Mod}_{/\mathfrak{S}}^{\varphi}$-module homomorphism
$h:\mathfrak{M}\to\mathfrak{M}^{\prime}$ corresponds uniquely to
$\text{Mod}_{/\mathscr{O}_{\mathscr{E}}}^{\varphi}$-module homomorphism
$\mathcal{F}_{\mathfrak{M},\mathfrak{M}^{\prime}}(h):\mathcal{F}(\mathfrak{M})\to\mathcal{F}(\mathfrak{M}^{\prime})$.
The product functor:

$\mathcal{F}\otimes
1:\text{Mod}_{/\mathfrak{S}}^{\varphi}\otimes_{\mathbb{Z}_{p}}\mathscr{R}\to\text{Mod}_{/\mathscr{O}_{\mathscr{E}}}^{\varphi}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$
(5)

is defined by

$(\mathcal{F}\otimes
1)(\mathfrak{M}\otimes_{\mathbb{Z}_{p}}\mathscr{R})=\mathcal{F}(\mathfrak{M})\otimes_{\mathbb{Z}_{p}}1(\mathscr{R}),$
(6)

where $1$ denotes the identity functor. We have the following diagram:

$\displaystyle\begin{matrix}\text{Mod}_{/\mathfrak{S}}^{\varphi}&\overset{\mathcal{F}}{\longrightarrow}&\text{Mod}_{/\mathscr{O}_{\mathscr{E}}}^{\varphi}\cr\big{\downarrow}&&\big{\downarrow}\cr\text{Mod}_{/\mathfrak{S}}^{\varphi}\otimes_{\mathbb{Z}_{p}}\mathscr{R}&\overset{\mathcal{F}\otimes
1}{\longrightarrow}&\text{Mod}_{/\mathscr{O}_{\mathscr{E}}}^{\varphi}\otimes_{\mathbb{Z}_{p}}\mathscr{R}\end{matrix}$
(7)

The morphisms between the objects in the category
$\text{Mod}_{/\mathfrak{S}}^{\varphi}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$ are
of the form

$h\otimes
1:\mathfrak{M}\otimes_{\mathbb{Z}_{p}}\mathscr{R}\to\mathfrak{M}^{\prime}\otimes_{\mathbb{Z}_{p}}\mathscr{R},$
(8)

where $1$ is the identity morphism. Note that we have adapted the same
notation $1$ both for the identity functor and the identity mophism, however
its use in different meaning is obvious.

For better understanding, let us denote by “Ob” for objects and by “Hom” for
the hom-sets in the given categories. Then

$\displaystyle\text{Ob}(\text{Mod}_{/\mathfrak{S}}^{\varphi}\otimes_{\mathbb{Z}_{p}}\mathscr{R})=\text{Ob}(\text{Mod}_{/\mathfrak{S}}^{\varphi})\otimes_{\mathbb{Z}_{p}}\mathscr{R}$
$\displaystyle\text{Hom}_{\text{Mod}_{/\mathfrak{S}}^{\varphi}\otimes_{\mathbb{Z}_{p}}\mathscr{R}}(\mathfrak{M}\otimes_{\mathbb{Z}_{p}}\mathscr{R},\mathfrak{M}^{\prime}\otimes_{\mathbb{Z}_{p}}\mathscr{R})=\text{Hom}_{\text{Mod}_{/\mathfrak{S}}^{\varphi}}(\mathfrak{M},\mathfrak{M}^{\prime})\otimes\text{Hom}_{\mathscr{R}}(\mathscr{R},\mathscr{R})$

for
$\mathfrak{M},\mathfrak{M}^{\prime}\in\text{Mod}_{/\mathfrak{S}}^{\varphi}$.

Now consider the action finite flat $\mathbb{Z}_{p}$-algebra $\mathscr{R}$ on
the objects of the both categories $\text{Mod}_{/\mathfrak{S}}^{\varphi}$ and
$\text{Mod}_{/\mathscr{O}_{\mathscr{E}}}^{\varphi}$. The product functor
$\mathcal{F}\otimes 1$ induces the following map on the hom-sets

$(\mathcal{F}\otimes 1)_{\mathfrak{M},\mathfrak{M}^{\prime}}(h\otimes
1)=\mathcal{F}_{\mathfrak{M},\mathfrak{M}^{\prime}}(h)\otimes\mathcal{F}_{\mathfrak{M},\mathfrak{M}^{\prime}}(1)$
(9)

and hence by Proposition 3.1 along with the flatness property of $\mathscr{R}$
induces the bijective map

$\displaystyle(\mathcal{F}\otimes 1)_{\mathfrak{M},\mathfrak{M}^{\prime}}:$
$\displaystyle\text{Hom}_{\text{Mod}_{/\mathfrak{S}}^{\varphi}\otimes\mathscr{R}}(\mathfrak{M}\otimes_{\mathbb{Z}_{p}}\mathscr{R},\mathfrak{M}^{\prime}\otimes_{\mathbb{Z}_{p}}\mathscr{R})$
(10)
$\displaystyle\to\text{Hom}_{\text{Mod}_{/\mathscr{O}_{\mathscr{E}}}^{\varphi}\otimes\mathscr{R}}((\mathcal{F}\otimes
1)(\mathfrak{M}\otimes_{\mathbb{Z}_{p}}\mathscr{R}),(\mathcal{F}\otimes
1)(\mathfrak{M}^{\prime}\otimes_{\mathbb{Z}_{p}}\mathscr{R})).$ (11)

Thus the full faithfullness of the functor
$\mathcal{F}:\text{Mod}_{/\mathfrak{S}}^{\varphi}\to\text{Mod}_{/\mathscr{O}_{\mathscr{E}}}^{\varphi}$
is preserved under the action of the finite flat $\mathscr{R}$-algebra. This
completes the proof.

∎

Let us consider the compatible sequence $\\{\pi_{n}\\}_{n\geq 0}$ of elements
$\pi_{n}\in\mathscr{O}_{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern
1.5mu}$, where $\pi_{0}=\pi\in\mathscr{O}_{K}$, and $\pi_{n}^{p}=\pi_{n-1},\
n\geq 1$. Denote $K_{\infty}=\cup_{n\geq 1}K(\pi_{n-1})$. This is known as
Kummer extension. Note that $K_{\infty}$ is contained in $\mkern
1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu$. We write
$G_{K_{\infty}}=\text{Gal}(\mkern
1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu/K_{\infty})\subset
G_{K}=\text{Gal}(\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern
1.5mu/K)$. We denote the category of objects which are continuous
representations of the Galois group $G_{K_{\infty}}$ on the finite
$\mathbb{Z}_{p}$-modules by $\text{Rep}_{\mathbb{Z}_{p}}(G_{K_{\infty}})$.
Note that $G_{K_{\infty}}$ fixes the DVR $\mathscr{O}_{\mathscr{E}}\subset
W(\text{Frac}(\mathfrak{R}))$, and therefore
$\mathscr{O}_{\widehat{\mathscr{E}^{ur}}}$ is stable under the action of
$G_{K_{\infty}}$.

We will now make use of the connection between the entire category of $p$-adic
Galois representations $\text{Rep}_{\mathbb{Z}_{p}}(G_{K_{\infty}})$ and the
semilinear data defined over the scalar extension $\mathscr{O}_{\mathscr{E}}$
of $\mathfrak{S}$, and which is exactly the fundamental theory of Fontaine
studied in [Fon]. Indeed, the following functor

$\displaystyle\mathcal{V}_{\mathscr{E}}:\Phi\mathscr{M}_{\mathscr{O}_{\mathscr{E}}}^{\text{\'{e}t}}\to\text{Rep}_{\mathbb{Z}_{p}}(G_{K_{\infty}})$
$\displaystyle\mathscr{M}\mapsto(\mathscr{O}_{\widehat{\mathscr{E}^{ur}}}\otimes_{\mathscr{O}_{\mathscr{E}}}\mathscr{M})^{\varphi=1}=\\{x\in\mathscr{O}_{\widehat{\mathscr{E}^{ur}}}\otimes_{\mathscr{O}_{\mathscr{E}}}\mathscr{M}~{}|~{}\varphi
x=x\\}$

is an equivalence of the abelian categories due to [Fon]. The quasi-inverse is
given by the functor

$\displaystyle\mathcal{D}_{\mathscr{E}}:\text{Rep}_{\mathbb{Z}_{p}}(G_{K_{\infty}})\to\Phi\mathscr{M}_{\mathscr{O}_{\mathscr{E}}}^{\text{\'{e}t}}$
$\displaystyle
V\mapsto(\mathscr{O}_{\widehat{\mathscr{E}^{ur}}}\otimes_{\mathbb{Z}_{p}}V)^{G_{K_{\infty}}}.$

Now consider the finite flat $\mathbb{Z}_{p}$-algebra $\mathscr{R}$ and denote
by $\Phi\mathscr{M}_{\mathscr{O}_{\mathscr{E}},\mathscr{R}}^{\text{\'{e}t}}$
the category consisting of objects of
$\Phi\mathscr{M}_{\mathscr{O}_{\mathscr{E}}}^{\text{\'{e}t}}$ equipped with
the action of $\mathscr{R}$. We also denote the category of continuous
representations of $G_{K_{\infty}}$ on finite $\mathscr{R}$-modules by
$\text{Rep}_{\mathscr{R}}(G_{K_{\infty}})$.

At these point we prove the following two lemmas:

######  Lemma 3.4.

Let $\mathfrak{m}_{\mathscr{R}}$ be the maximal ideal of a finite flat
$\mathbb{Z}_{p}$-algebra $\mathscr{R}$. If $\mathscr{M}$ is
$\mathscr{R}$-flat, then $\mathscr{M}$ is free over
$\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$ if and only if
$\mathscr{M}\otimes_{\mathscr{R}}\mathscr{R}/\mathfrak{m}_{\mathscr{R}}$ is
free over
$\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}/\mathfrak{m}_{\mathscr{R}}$.

###### Proof.

Note that
$\mathscr{M}\otimes_{\mathscr{R}}\mathscr{R}/\mathfrak{m}_{\mathscr{R}}\cong\mathscr{M}/\mathfrak{m}_{\mathscr{R}}\mathscr{M}$.
We recall from Definition 3.1 that $\mathscr{M}$ is finitely generated
$\mathscr{O}_{\mathscr{E}}$-module.

Now assume that $a_{i}\in\mathscr{M}$ be such that their reductions mod
$\mathfrak{m}_{\mathscr{R}}\mathscr{M}$ forms a basis of
$\mathscr{M}/\mathfrak{m}_{\mathscr{R}}\mathscr{M}$ over
$\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}/\mathfrak{m}_{\mathscr{R}}$.
If $(b_{j})$ is a basis of $\mathscr{O}_{\mathscr{E}}$ over $\mathbb{Z}_{p}$,
then $e_{ij}=(b_{j}\otimes 1)a_{i}$ spans $\mathscr{M}$ mod
$\mathfrak{m}_{\mathscr{R}}\mathscr{M}$, and hence by local version of
Nakayama lemma $(e_{ij})$ spans $\mathscr{M}$. Therefore

$\mathscr{M}=\sum_{i,j}\mathscr{R}e_{ij}=\sum_{i}(\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R})a_{i}.$

Since $\mathscr{M}$ is finitely generated and flat over the local ring
$\mathscr{R}$, it is free $\mathscr{R}$-module. Further, we have
$\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}/\mathfrak{m}_{\mathscr{R}}=\oplus_{j}b_{j}\otimes_{\mathbb{Z}_{p}}\mathscr{R}/\mathfrak{m}_{\mathscr{R}}$
and it follows that $(e_{ij})$ is a basis of
$\mathscr{M}/\mathfrak{m}_{\mathscr{R}}\mathscr{M}$ over
$\mathscr{R}/\mathfrak{m}_{\mathscr{R}}$. As $\mathscr{M}$ is free
$\mathscr{R}$-module and $(e_{ij})$ is a basis of $\mathscr{M}$ mod
$\mathfrak{m}_{\mathscr{R}}\mathscr{M}$, it is a basis of $\mathscr{M}$ over
$\mathscr{R}$. Thus $(a_{i})$ is a basis of $\mathscr{M}$ over
$\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$. Hence the
proof follows. ∎

######  Lemma 3.5.

If $\mathscr{R}$ is a finite field then
$\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$ is a finite
product of fields.

###### Proof.

If $L=\mathbb{Q}_{p}(\zeta_{q-1})$ is the maximal unramified extension of
$\mathbb{Q}_{p}$ contained in $K$, then
$W(\kappa)=\mathscr{O}_{L}=\mathbb{Z}_{p}[\zeta_{q-1}]$ for $q=|\kappa|$.
Therefore,

$\mathfrak{S}[\frac{1}{u}]=\mathscr{O}_{L}[[u]][\frac{1}{u}]=\\{\sum_{n\geq-
n_{0}}c_{n}u^{n},\ c_{n}\in\mathscr{O}_{L}\\},$

for some natural number $n_{0}$. Taking $p$-adic completion
$\mathfrak{S}[\frac{1}{u}]$, we get

$\mathscr{O}_{\mathscr{E}}=\widehat{\mathfrak{S}[\frac{1}{u}]}=\\{\sum_{n=-\infty}^{\infty}c_{n}u^{n},\
c_{n}\in\mathscr{O}_{L},\ \lim_{n\to-\infty}v_{p}(c_{n})=\infty\\}.$

Dividing the discrete valuation ring $\mathscr{O}_{\mathscr{E}}$ by the prime
ideal $(p)$, we get the function field

$\mathscr{O}_{\mathscr{E}}/(p)=\mathbb{F}_{q}((u)).$

If $\text{char}(\mathscr{R})\neq p$, then $\text{char}(\mathscr{R})\otimes
1=1\otimes\text{char}(\mathscr{R})=1\otimes 0=0$. Hence taking quotient of
$\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$ by the ideal
generated by $\text{char}(\mathscr{R})\otimes 1$, we have

$\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}=\mathscr{O}_{\mathscr{E}}/(\text{char}(\mathscr{R}))\otimes_{\mathbb{Z}_{p}}\mathscr{R}=0$

and hence it is trivial case.
If $\text{char}(\mathscr{R})=p$, then we must have
$\mathscr{R}\cong\mathbb{F}_{p}[t]/(z(t))$ for an irreducible polynomial
$z(t)\in\kappa[t]$. In this case, we have

$\displaystyle\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}=\mathscr{O}_{\mathscr{E}}/(p)\otimes_{\mathbb{Z}_{p}}\mathbb{F}_{p}[t]/(z(t))$
$\displaystyle=\mathbb{F}_{q}((u))\otimes_{\mathbb{F}_{p}}\mathbb{F}_{p}[t]/(z(t))$
$\displaystyle=\mathbb{F}_{q}((u))[t]/(z(t))$
$\displaystyle=\mathbb{F}_{q}((u))[t]/(\prod_{j}z_{j}(t))$
$\displaystyle=\prod_{j}\mathbb{F}_{q}((u))[t]/(z_{j}(t))$
$\displaystyle=\prod_{j}\mathbb{F}_{q^{\text{deg}(z_{j})}}((u)),$

where $z_{j}(t)$ are irreducible factors of $z(t)$ over $\mathbb{F}_{q}$.
Hence, we are done. ∎

We note the following remark:

######  Remark 3.

In the following theorem,
$\mathscr{M}\in\Phi\mathscr{M}_{\mathscr{O}_{\mathscr{E},\mathscr{R}}}$ is
uniquely associated (étale lift of $\mathfrak{M}$) with the corresponding
$(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))[[u]]$-module $\mathfrak{M}$ in
the category $\text{Mod}_{/\mathfrak{S}_{\mathscr{R}}}^{\varphi}$.

######  Theorem 3.6.

For a finite local $\mathbb{Z}_{p}$-algebra $\mathscr{R}$, if the $p$-adic
Tate module $T_{p}(\mathcal{G})$ of a $p$-divisible group $\mathcal{G}$ of
height $d$ over $\mathscr{O}_{K}$ is free over $\mathscr{R}$, then
$T_{p}(\mathcal{G})$ corresponds to a free module
$\mathscr{M}\in\Phi\mathscr{M}_{\mathscr{O}_{\mathscr{E},\mathscr{R}}}$ over
$\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$ of rank $d$.

###### Proof.

Given the action of $\mathscr{R}$, the equivalence functor
$\mathcal{V}_{\mathscr{E}}:\Phi\mathscr{M}_{\mathscr{O}_{\mathscr{E}}}^{\text{\'{e}t}}\to\text{Rep}_{\mathbb{Z}_{p}}(G_{K_{\infty}})$
induces the functor
$\mathcal{V}_{\mathscr{E},\mathscr{R}}:\Phi\mathscr{M}_{\mathscr{O}_{\mathscr{E},\mathscr{R}}}^{\text{\'{e}t}}\to\text{Rep}_{\mathscr{R}}(G_{K_{\infty}})$.
Since $\mathcal{V}_{\mathscr{E}}$ is equivalence, it is fully faithful and by
the similar argument as in Proposition 3.3, we get that the induced functor
$\mathcal{V}_{\mathscr{E},\mathscr{R}}$ is also fully faithful. Now a fully
faithful functor is equivalence if and only it is essentially surjective and,
hence any object $V\in\text{Rep}_{\mathbb{Z}_{p}}(G_{K_{\infty}})$ is
isomorphic to $\mathcal{V}_{\mathscr{E}}(\mathscr{M})$ for some object
$\mathscr{M}\in\Phi\mathscr{M}_{\mathscr{O}_{\mathscr{E}}}^{\text{\'{e}t}}$.
In order to be equivalence, we need to show that
$\mathcal{V}_{\mathscr{E},\mathscr{R}}$ is essentially surjective. The functor
$\mathcal{V}_{\mathscr{E},\mathscr{R}}$ can be thought of as the functor
$\mathcal{V}_{\mathscr{E}}\otimes
1:\Phi\mathscr{M}_{\mathscr{O}_{\mathscr{E}}}^{\text{\'{e}t}}\otimes\mathscr{R}\to\text{Rep}_{\mathbb{Z}_{p}}(G_{K_{\infty}})\otimes\mathscr{R}$,
where $1$ denotes the identity functor. Thus it follows that
$\mathcal{V}_{\mathscr{E},\mathscr{R}}:=\mathcal{V}_{\mathscr{E}}\otimes 1$ is
also essentially surjective and hence $\mathcal{V}_{\mathscr{E},\mathscr{R}}$
is an equivalence functor. For any finite $\mathscr{R}$-module
$\mathscr{M}^{\prime}$, we define the map

$\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M})\otimes_{\mathscr{R}}\mathscr{M}^{\prime}\to\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M}\otimes_{\mathscr{R}}\mathscr{M}^{\prime}).$
(12)

We will show that this is an isomorphism. We note that both the categories are
both complete and cocomplete with finite limits and colimits, and

$\displaystyle\mathscr{M}\otimes_{\mathscr{R}}\mathscr{R}^{\otimes
n}\cong\mathscr{M}\otimes_{\mathscr{R}}\mathscr{N}\cong\mathscr{P}\cong\underbrace{\mathscr{M}\oplus\mathscr{M}\oplus\cdots\oplus\mathscr{M}}_{\text{n
copies}}$ $\displaystyle\cong\mathscr{M}^{\otimes n},$

where
$\mathscr{N}:=\underbrace{\mathscr{R}\oplus\mathscr{R}\oplus\cdots\oplus\mathscr{R}}_{\text{n
copies}}$ and
$\mathscr{P}:=\underbrace{(\mathscr{M}\otimes_{\mathscr{R}}\mathscr{R})\oplus(\mathscr{M}\otimes_{\mathscr{R}}\mathscr{R})\oplus\cdots\oplus(\mathscr{M}\otimes_{\mathscr{R}}\mathscr{R})}_{\text{n
copies}}.$
Now we take the finite presentation of the finite $\mathscr{R}$-module
$\mathscr{M}^{\prime}$

$\mathscr{R}^{\otimes n}\to\mathscr{R}^{\otimes m}\to\mathscr{M}^{\prime}\to
0$ (13)

for some natural number $n,m$; the $\mathscr{R}$-module homomorphism (i.e., a
matrix) $\mathscr{R}^{\otimes n}\to\mathscr{R}^{\otimes m}$ induces an action
$\mathscr{M}^{\otimes n}\to\mathscr{M}^{\otimes m}$. So given the above
presentation of $\mathscr{M}^{\prime}$, we define
$\mathscr{M}\otimes_{\mathscr{R}}\mathscr{M}^{\prime}$ to be the cokernel of
the induced map $\mathscr{M}^{\otimes n}\to M^{\otimes m}$. Now
$\mathcal{V}_{\mathscr{E},\mathscr{R}}$ being right exact additive functor, it
preserves finite direct sum and cokernel and so

$\displaystyle\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M}\otimes_{\mathscr{R}}\mathscr{R}^{\otimes
n})\cong\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M}^{\otimes
n})\cong\mathscr{X}\cong\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M})^{\otimes
n}\cong\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M})\otimes_{\mathscr{R}}\mathscr{R}^{n},$

where
$\mathscr{X}:=\underbrace{\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M})\oplus\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M})\oplus\cdots\oplus\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M})}_{\text{n
copies}}.$
Now apply $\mathcal{V}_{\mathscr{E},\mathscr{R}}$ to the cokernel of
$\mathscr{M}^{\otimes n}\to\mathscr{M}^{\otimes m}$, in other word, we apply
the functor $\mathcal{V}_{\mathscr{E},\mathscr{R}}$ to the exact sequence

$\mathscr{M}^{\otimes n}\to\mathscr{M}^{\otimes
m}\to\mathscr{M}\otimes_{\mathscr{R}}\mathscr{M}^{\prime}\to 0$

to obtain the exact sequence
$\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M}^{\otimes
n})\to\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M}^{\otimes
m})\to\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M}\otimes_{\mathscr{R}}\mathscr{M}^{\prime})\to
0,$ because $\mathcal{V}_{\mathscr{E},\mathscr{R}}$ is exact. Thus by
definition
$\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M}\otimes_{\mathscr{R}}\mathscr{M}^{\prime})$
is the cokernel of the map
$\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M}^{\otimes
n})\to\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M}^{\otimes m})$ which is
isomorphic to $\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M})^{\otimes
n}\to\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M})^{\otimes m}$ as above.
But the cokernel of
$\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M})^{\otimes
n}\to\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M})^{\otimes m}$ is
$\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M})\otimes_{\mathscr{R}}\mathscr{M}^{\prime}$.
Hence the map in (12) is an isomorphism, infact a natural isomorphism because
it doesn’t depend on the finite presentation (13).

Since the map (12) is a natural isomorphism, the exactness of the functor
$\mathcal{V}_{\mathscr{E},\mathscr{R}}$ implies that $\mathscr{M}$ is
$\mathscr{R}$-flat if and only if
$\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M})$ is $\mathscr{R}$-flat.
Let $\mathfrak{m}_{\mathscr{R}}$ be the maximal ideal of $\mathscr{R}$. Then,
once we show that $\mathscr{M}$ is $\mathscr{R}$-flat module, it will be
finitely generated and free
$\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$-module,
because, by [Bri, Lemma. 3.1.7 and Theorem. 3.2.5], the
$\mathscr{R}/\mathfrak{m}_{\mathscr{R}}$-rank of
$\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M}\otimes_{\mathscr{R}}\mathscr{R}/\mathfrak{m}_{\mathscr{R}})$
and the
$\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}/\mathfrak{m}_{\mathscr{R}}$-rank
of $\mathscr{M}$ are equal and hence the $\mathscr{R}$-rank of
$\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M})$ and the
$\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$-rank of
$\mathscr{M}$ must be equal.

Now by Lemma 3.4 if $\mathscr{M}$ is $\mathscr{R}$-flat, then $\mathscr{M}$ is
free module over
$\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$ if and only if
$\mathscr{M}\otimes_{\mathbb{Z}_{p}}\mathscr{R}/\mathfrak{m}_{\mathscr{R}}$ is
free module over
$\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}/\mathfrak{m}_{\mathscr{R}}$,
and hence $\mathscr{R}$ can be taken as a finite field. Therefore by Lemma 3.5
$\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$ is a product of
fields, and hence $\mathscr{M}$ is projective module over
$\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$. Thus
$\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$ has finitely
many maximal ideals $\mathfrak{m}$. We want to show that the ranks of
$\mathscr{M}$ at all such maximal ideals $\mathfrak{m}$ are equal in order to
be free. Let $\mathfrak{m}$ be one of the maximal ideals of
$\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$, and let the
rank of $\mathscr{M}$ at $\mathfrak{m}$ be $d_{\mathfrak{m}}$. We extend the
Frobenius $\varphi$ to an $\mathscr{R}$-linear map
$\varphi:\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}\to\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$.
By the definition of
$\mathscr{M}\in\Phi\mathscr{M}_{\mathscr{O}_{\mathscr{E},\mathscr{R}}}^{\text{\'{e}t}}$,
we know that $1\otimes\varphi:\varphi^{*}\mathscr{M}\to\mathscr{M}$ is an
isomorphism and $(1\otimes\varphi)(\varphi^{*}\mathscr{M})$ is a submodule of
$\mathscr{M}$ and thus we have $d_{\mathfrak{m}}\leq
d_{\varphi(\mathfrak{m})}$ by the injective property of $1\otimes\varphi$. Now
the Frobenius map $\varphi$ permutes the maximal ideals of
$\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$ transitively,
and hence all the ranks $d_{\mathfrak{m}}$ are equal. Hence it follows that
$\mathscr{M}$ is a free
$\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$-module.

So far we have proved that if
$\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M})$ is free
$\mathscr{R}[G_{K_{\infty}}]$-module of rank $d$ then $\mathscr{M}$ is free
$\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$-module of rank
$d$, and indeed the converse also holds as $\mathscr{M}$ is $\mathscr{R}$-flat
if and only if $\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M})$ is
$\mathscr{R}$-flat, where by the notation $\mathscr{R}[G_{K_{\infty}}]$-module
we mean an $\mathscr{R}$-module on which the Galois group $G_{K_{\infty}}$
acts linearly. Now given that the $p$-adic Tate module $T_{p}(\mathcal{G})$ of
the $p$-divisible group over $\mathscr{O}_{K}$ is a free $\mathscr{R}$-module.
Since the height of $\mathcal{G}$ is $d$, $T_{p}(\mathcal{G})$ is of free rank
$d$. Further, $T_{p}(\mathcal{G})$ is equipped with linear continuous action
of the Galois group $G_{K_{\infty}}$. In other word, it is a free
$\mathscr{R}[G_{K_{\infty}}]$-module and hence it lies in the category
$\text{Rep}_{\mathscr{R}}(G_{K_{\infty}})$. Therefore $T_{p}(\mathcal{G})$
identifies itself with some
$\mathcal{V}_{\mathscr{E},\mathscr{R}}(\mathscr{M})$ for
$\mathscr{M}\in\Phi\mathscr{M}_{\mathscr{O}_{\mathscr{E}}}^{\text{\'{e}t}}$.
Thus, if $T_{p}(\mathcal{G})$ is free, then the corresponding $\mathscr{M}$ is
free, by the above implication. This completes the proof. ∎

######  Remark 4.

One can further investigate whether or not the Theorem 3.6 can be framed in
terms of the objects of $\text{Mod}_{/\mathfrak{S}_{\mathscr{R}}}^{\varphi}$
of $\mathfrak{S}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$-modules instead of the
objects of $\text{Mod}_{\mathscr{O}_{\mathscr{E},\mathscr{R}}}^{\varphi}$ of
$\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$-modules.

##  4\. Conclusion

In Section 2, we have deduced the necessary and sufficient conditions such
that the Breuil-Kisin module $\mathfrak{M}$ becomes projective and free under
the action of finite flat $\mathbb{Z}_{p}$-algebra $\mathscr{R}$. Then in
Section 3, we have lifted the Breuil-Kisin module $\mathfrak{M}$ to étale
level following the $\mathscr{R}$-action and showed that if the $p$-adic Tate
module $T_{p}(\mathcal{G})$ is free $\mathscr{R}$-module, then the étale lift
$\mathscr{M}$ of $\mathfrak{M}$ is a free
$\mathscr{O}_{\mathscr{E}}\otimes_{\mathbb{Z}_{p}}\mathscr{R}$-module.

Acknowledgement: The authors are grateful to Professor Olivier Brinon,
University of Bordeaux, for his valuable comments towards the improvement of
the paper. The second author acknowledges The Council Of Scientific and
Industrial Research, Government of India, for the award of Senior Research
Fellowship.

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