# Non-regular trianguline representations

Martina Fruttidoro

###### Abstract.

Let $G_{\mathbb{Q}_{p}}$ be the absolute Galois group of $\mathbb{Q}_{p}$ and
let $L$ be a finite extension of $\mathbb{Q}_{p}$. Moreover let
$\bar{\rho}:G_{\mathbb{Q}_{p}}\rightarrow GL_{n}(k_{L})$ be a continous
representation of $G_{\mathbb{Q}_{p}}$, where $k_{L}$ is the residue field of
$L$. We find sufficient conditions for which a trianguline representation
$\rho:G_{\mathbb{Q}_{p}}\rightarrow GL_{n}(L)$ lifting $\bar{\rho}$ is a point
of the trianguline variety associated to $\bar{\rho}$.

###### Contents

  1. 1 Preliminaries on $(\phi,\Gamma)$-modules
    1. 1.1 The Robba ring
    2. 1.2 $(\phi,\Gamma)$-modules
    3. 1.3 The $(\phi,\Gamma)$-cohomology
  2. 2 Trianguline representations
    1. 2.1 Definitions and basic properties
    2. 2.2 The trianguline variety
  3. 3 Strategy
  4. 4 Trianguline representations with no $\mathcal{T}^{+}$-type irregularity
    1. 4.1 Constructing extensions
    2. 4.2 Proof of Theorem A
  5. 5 Some preparation
    1. 5.1 The Beauville-Laszlo Theorem
    2. 5.2 $\phi$-modules over $\mathcal{R}_{A}^{r}$
    3. 5.3 From $(\phi,\Gamma)$-modules over $\mathcal{R}$ to $(\phi,\Gamma)$-modules over $\mathcal{R}[1/t]$
    4. 5.4 The $(\phi,\Gamma)$-cohomology of $(\phi,\Gamma)$-modules over $\mathcal{R}[1/t]$
    5. 5.5 The affine Grassmannian
    6. 5.6 Almost de Rham representations
    7. 5.7 $B_{\mathrm{dR}}$-representations of non-integral weights
  6. 6 Trianguline representations of dimension 2
    1. 6.1 Construction of space of extensions
    2. 6.2 Density of the regular locus
  7. 7 Construction of family of extensions for trianguline  representations of higher dimension
    1. 7.1 Construction of space of extensions
    2. 7.2 Density of the regular locus

## Introduction

The $p$-adic local Langlands programme (for $p$ a prime number) conjectures a
correspondence between $p$-adic Galois representations and certain $p$-adic
representations of reductive groups over $p$-adic local fields. The theory of
$p$-adic automorphic forms allows us to attach a $p$-adic Galois
representation to a certain $p$-adic automorphic form; thus, this approach
provides a plausible candidate for the correspondence between $p$-adic Galois
representations that arise in this way and $p$-adic automorphic
representations of $GL_{n}$. One nice aspect of this theory is that $p$-adic
automorphic forms come in families parametrised by geometric objects called
_eigenvarieties_. In [7, §3.2], Breuil, Hellmann and Schraen construct an
analogue of an eigenvariety called _patched eigenvariety_ using the _patching
module_ constructed by Caraiani, Emerton, Gee, Geraghty, Paškūnas and Shin in
[10]. In [7], the authors also construct an analogue of an eigenvariety in the
world of Galois representations, called _trianguline variety_. Both the
patched eigenvariety and the trianguline variety are defined as subspaces of
$\mathfrak{X}_{\bar{\rho}}^{\square}\times\mathcal{T}^{n}$ (see below for
definitions of these spaces) and conjecturally they coincide (up to power
series-rings over deformation rings at auxiliary places away from $p$). One of
the main results of this thesis is the description of the underlying set of
points of the trianguline variety in some special cases: we will show that in
these cases, for a trianguline representation $\rho$, the representation
$\rho$ is a point of the trianguline variety. Despite what the name might
suggest, the above statement is highly non-trivial: as we will see, the
trianguline variety is by definition the closure of the set of trianguline
representations with a regularity condition on the parameters and it is
unclear whether such closure adds all trianguline representations. In the
following, I will give a more precise summary of the main results.

Let $L$ be a finite extension of $\mathbb{Q}_{p}$ and let $A$ be an affinoid
$L$-algebra. Let us denote by $G_{\mathbb{Q}_{p}}$ the absolute Galois group
of $\mathbb{Q}_{p}$ and let us fix a natural number $n\geqslant 1$. One of the
main results in the theory of $p$-adic Galois representations is that there
exists a fully faithful and exact functor

$\mathbb{D}^{\dagger}_{\mathrm{rig}}:\left\\{\begin{array}[]{c}\text{continous
representations }\\\ \rho:G_{\mathbb{Q}_{p}}\rightarrow
GL_{n}(A)\end{array}\right\\}\rightarrow\left\\{\begin{array}[]{c}\text{\'{e}tale
}(\phi,\Gamma)\text{-modules}\\\ \text{over }\mathcal{R}_{A}\text{ of rank
}n\end{array}\right\\}.$

A continous representation $\rho:G_{\mathbb{Q}_{p}}\rightarrow GL_{n}(L)$ is
_trianguline_ if $\mathbb{D}^{\dagger}_{\mathrm{rig}}(\rho)$ is _triangulable_
, i.e. it is a successive extension of $(\phi,\Gamma)$-modules of rank 1. This
kind of representations constitute a very important category of
representations in the framework of the $p$-adic local Langlands programme:
they first appeared in the work of Colmez [12], who showed that 2-dimensional
trianguline representations are Zariski dense in the deformation space of
2-dimensional $p$-adic Galois representations. This result was crucial for
proving the $p$-adic local Langlands correspondence for
$GL_{2}(\mathbb{Q}_{p})$.

In [22], Kedlaya, Pottharst and Xiao prove that it is possible to classify
$(\phi,\Gamma)$-modules of rank 1 in a nice way: they show that there exists a
bijection

$\displaystyle\\{\text{continous characters
}\delta:\mathbb{Q}_{p}^{\times}\rightarrow A^{\times}\\}$
$\displaystyle\leftrightarrow\\{\text{free }(\phi,\Gamma)\text{-modules of
rank }1\text{ over }\mathcal{R}_{A}\\}$ $\displaystyle\delta$
$\displaystyle\mapsto\mathcal{R}_{A}(\delta).$

The functor $\mathrm{Hom}_{\mathrm{cont}}(\mathbb{Q}_{p}^{\times},-)$ of
continous characters of $\mathbb{Q}_{p}^{\times}$ is representable by a rigid
space $\mathcal{T}$, which constitutes then a nice moduli space for
$(\phi,\Gamma)$-modules of rank 1. Now let us fix a residual representation
$\bar{\rho}:G_{\mathbb{Q}_{p}}\rightarrow GL_{n}(k_{L})$ (where $k_{L}$ is the
residue field of $L$) and let us denote by
$\mathfrak{X}_{\bar{\rho}}^{\square}$ the rigid analytic generic fibre of its
universal framed deformation space. Let us define

$U_{\mathrm{tri}}^{\square}(\bar{\rho})\coloneqq\left\\{(\rho,\delta_{1},\dots,\delta_{n})\in\mathfrak{X}_{\bar{\rho}}^{\square}\times\mathcal{T}^{\mathrm{reg}}_{n}\colon\begin{array}[]{c}\rho\text{
is trianguline}\\\ \text{with graded pieces}\\\ \text{of a filtration of
}\mathbb{D}_{\mathrm{rig}}^{{\dagger}}(\rho)\\\ \text{ given by
}(\delta_{1},\dots,\delta_{n})\end{array}\right\\}\subset\mathfrak{X}_{\bar{\rho}}^{\square}\times\mathcal{T}^{n},$

where $\mathcal{T}^{\mathrm{reg}}_{n}$ is a subspace of $\mathcal{T}^{n}$ of
tuples of characters satisfying a certain regularity condition (see (1.4) in
the body of the paper). The _trianguline variety_ is the Zariski closure of
$U_{\mathrm{tri}}^{\square}(\bar{\rho})$ inside
$\mathfrak{X}_{\bar{\rho}}^{\square}\times\mathcal{T}^{n}$ and is denoted by
$X_{\mathrm{tri}}^{\square}(\overline{\rho})$. This space was first introduced
by Hellmann in [17] and its geometry has been studied in [6],[7] and [8], yet
there still are some properties to be investigated. Consider for example a
trianguline representation $\rho:G_{\mathbb{Q}_{p}}\rightarrow GL_{n}(L)$ and
let $(\delta_{1},\dots,\delta_{n})\in\mathcal{T}^{n}(L)$ be _parameters_ for
$\rho$, which means that $\mathbb{D}_{\mathrm{rig}}^{{\dagger}}(\rho)$ has a
filtration with graded pieces $\mathcal{R}_{L}(\delta_{i})$; it is well-known
that up to conjugation by a matrix in $GL_{n}(L)$, we can assume $\rho$ has
values in $GL_{n}(\mathcal{O}_{L})$. Let
$\bar{\rho}:G_{\mathbb{Q}_{p}}\rightarrow GL_{n}(k_{L})$ be the reduction of
$\rho$ to the residue field of $L$. We can then ask ourselves whether
$(\rho,\delta_{1},\dots,\delta_{n})$ is a point of
$X_{\mathrm{tri}}^{\square}(\overline{\rho})(L)$ or not.

######  Conjecture.

Let $\rho:G_{\mathbb{Q}_{p}}\rightarrow GL_{n}(L)$ be a trianguline
representation with parameters
$\underline{\delta}\coloneqq(\delta_{1},\dots,\delta_{n})\in\mathcal{T}^{n}(L)$
and let $\bar{\rho}:G_{\mathbb{Q}_{p}}\rightarrow GL_{n}(k_{L})$ be the
reduction of $\rho$ to the residue field of $L$. Then
$(\rho,\underline{\delta})\in X_{\mathrm{tri}}^{\square}(\overline{\rho})(L)$.

Of course if $\underline{\delta}\in\mathcal{T}_{n}^{\mathrm{reg}}(L)$, then we
have that $(\rho,\underline{\delta})\in
U_{\mathrm{tri}}^{\square}(\bar{\rho})(L)$, hence in particular
$(\rho,\underline{\delta})\in X_{\mathrm{tri}}^{\square}(\overline{\rho})(L)$.
But as remarked before, the above conjecture is not obvious at all in case
$(\delta_{1},\dots,\delta_{n})\notin\mathcal{T}_{n}^{\mathrm{reg}}(L)$. We
will prove that $(\rho,\underline{\delta})\in
X_{\mathrm{tri}}^{\square}(\overline{\rho})$ if either $n\leqslant 4$, or the
number of non-regularities in $\underline{\delta}$ is at most 5, or if
$\underline{\delta}$ doesn’t present non-regularities of type
$\mathcal{T}^{+}$ (see 1.3 for a definition of $\mathcal{T}^{+}$). More
precisely, we will prove the following Theorems.

######  Theorem A.

Let $\rho:G_{\mathbb{Q}_{p}}\rightarrow GL_{n}(\mathcal{O}_{L})$ be a
trianguline representation of parameters
$(\delta_{1},\dots,\delta_{n})\in\mathcal{T}^{n}(L)$. Moreover let
$\bar{\rho}:G_{\mathbb{Q}_{p}}\rightarrow GL_{n}(k_{L})$ be the reduction of
$\rho$ to the residue field of $L$. If
$\delta_{i}/\delta_{j}\notin\mathcal{T}^{+}(L)$ for all $i<j$, then
$(\rho,\delta_{1},\dots,\delta_{n})\in
X_{\mathrm{tri}}^{\square}(\overline{\rho})(L)$.

######  Theorem B.

Let $\rho:G_{\mathbb{Q}_{p}}\rightarrow GL_{n}(\mathcal{O}_{L})$ be a
trianguline representation of parameters
$(\delta_{1},\dots,\delta_{n})\in\mathcal{T}^{n}(L)$. Moreover let
$\bar{\rho}:G_{\mathbb{Q}_{p}}\rightarrow GL_{n}(k_{L})$ be the reduction of
$\rho$ to the residue field of $L$.

  1. (1)

If $n\leqslant 4$, then $(\rho,\delta_{1},\dots,\delta_{n})\in
X_{\mathrm{tri}}^{\square}(\overline{\rho})(L)$.

  2. (2)

If
$|\\{\delta_{i}/\delta_{j}\in\mathcal{T}\setminus\mathcal{T}^{\mathrm{reg}}\colon
i<j\\}|\leqslant 5$, then $(\rho,\delta_{1},\dots,\delta_{n})\in
X_{\mathrm{tri}}^{\square}(\overline{\rho})(L)$.

For $(\phi,\Gamma)$-modules there is a notion of $(\phi,\Gamma)$-cohomology,
which coincides with Galois cohomology in case the $(\phi,\Gamma)$-module
comes from a family of Galois representations. In the statement of Theorem A,
the space $\mathcal{T}^{+}$ is the subspace of $\mathcal{T}$ consisting of
characters $\delta$ such that the second $(\phi,\Gamma)$-cohomology is non-
trivial.

The strategy used to prove the above results is the following: given a
trianguline representation $\rho:G_{\mathbb{Q}_{p}}\rightarrow GL_{n}(L)$ and
$\underline{\delta}\in\mathcal{T}^{n}(L)$ characters associated to a
filtration of $\mathbb{D}_{\mathrm{rig}}^{{\dagger}}(\rho)$, we will show that
$(\rho,\underline{\delta})$ is in the Zariski closure of some “regular
family”. More precisely, for each given $\rho$, we will construct a rigid
space $X_{n}$ with a family of triangulable $(\phi,\Gamma)$-modules
$\mathbb{D}_{n}$ over $X_{n}$ such that

  * •

the family $\mathbb{D}_{n}$ specializes to
$\mathbb{D}_{\mathrm{rig}}^{{\dagger}}(\rho)$ at some point $x\in X_{n}(L)$;

  * •

the triangulation of $\mathbb{D}_{n}$ gives rise to characters
$(\widetilde{\delta}_{1},\dots,\widetilde{\delta}_{n})\in\mathcal{T}^{n}(X_{n})$
such that there exists an open dense $U\xhookrightarrow{j}X_{n}$ for which
$j^{*}\widetilde{\underline{\delta}}\in\mathcal{T}^{\mathrm{reg}}_{n}(U)$.

For a detailed explanation of how the strategy implies the desired results,
refer to Theorem 3.3.

Following a construction in [13], when two $(\phi,\Gamma)$-modules satisfy
some condition on their $(\phi,\Gamma)$-cohomology, we will be able to
construct a space with a family of extensions of the two given
$(\phi,\Gamma)$-modules. More precisely, given two $(\phi,\Gamma)$-modules
$D_{1}$ and $D_{2}$ over $\mathcal{R}_{A}$ such that
$H^{2}_{\phi,\Gamma}(D_{1}\otimes D_{2}^{\vee})=0$, in Section 4.1 we explain
how to construct a vector bundle $X$ over $\mathrm{Sp}(A)$ with a family of
$(\phi,\Gamma)$-modules $\mathbb{D}$ over $X$ such that for any any rigid
space $Y\xrightarrow{g}\mathrm{Sp}(A)$ over $\mathrm{Sp}(A)$, we have a
surjective map

$\displaystyle\Psi_{Y}:X(Y)$
$\displaystyle\twoheadrightarrow\mathrm{Ext}^{1}_{\phi,\Gamma}(g^{*}D_{2},g^{*}D_{1})(Y)$
$\displaystyle f$ $\displaystyle\mapsto f^{*}\mathbb{D}$

which is functorial in $Y$ (see Theorem 4.2). It follows from this that if we
are in the hypothesis of Theorem A, which means that
$H^{2}_{\phi,\Gamma}(\mathcal{R}_{L}(\delta_{i})\otimes\mathcal{R}_{L}(\delta_{j})^{\vee})=0$
(i.e. $\delta_{i}/\delta_{j}\notin\mathcal{T}^{+}(L)$) for all $i<j$, then
there exists a vector bundle $X_{n}$ over an appropriate neighbourhood
$\mathcal{U}\subset\mathcal{T}^{n}$ of $\underline{\delta}$ with a family of
$(\phi,\Gamma)$-modules $\mathbb{D}_{n}$ specializing to
$\mathbb{D}_{\mathrm{rig}}^{{\dagger}}(\rho)$, as stated in Proposition 4.5.
Since the regular locus is open and dense inside $\mathcal{T}^{n}$, we have
that the pullback of such locus to $X_{n}$ is open and dense, as $X_{n}$ is
simply a vector bundle over $\mathcal{U}$ (in fact the preimage of a dense
subspace is dense if the map is open).

The above is enough to conclude the argument in the hypothesis of Theorem A,
but things complicate a great deal when
$\delta_{i}/\delta_{j}\in\mathcal{T}^{+}(L)$ for at least one pair $(i,j)$
with $i<j$. Indeed in this case, we cannot use Theorem 4.2: in fact the fiber
dimension of the second $(\phi,\Gamma)$-cohomology is not locally constant,
hence we cannot find a family such that the base change for the first
$(\phi,\Gamma)$-cohomology is surjective. In order to get around this problem,
we will make use of an equivalence of categories which is an analogue for
$(\phi,\Gamma)$-modules of the Theorem of Beauville-Laszlo; more precisely,
the equivalence is between $(\phi,\Gamma)$-modules over $\mathcal{R}_{A}$ and
the data consisting of pairs of $(\phi,\Gamma)$-modules over
$\mathcal{R}_{A}[1/t]$ (where $t$ is a particular element of
$\mathcal{R}_{A}$) and $\Gamma$-stable lattices of $A((t))$. It should be
emphasized that the above statement is a simplification (and false as stated)
and the reader should refer to Section 5.1 for the correct statement and a
detailed explanation. The main point in using this equivalence is that passing
from $(\phi,\Gamma)$-modules over $\mathcal{R}_{A}$ to $(\phi,\Gamma)$-modules
over $\mathcal{R}_{A}[1/t]$ annihilates the second $(\phi,\Gamma)$-cohomology;
thus it is possible to use Theorem 4.2 again in order to construct a rigid
space $X_{n}^{\prime}$ with a family of $(\phi,\Gamma)$-modules
$\mathbb{D}_{n}^{\prime}$ specializing to
$\mathbb{D}_{\mathrm{rig}}^{{\dagger}}(\rho)[1/t]$ at some point
$x^{\prime}\in X_{n}^{\prime}(L)$. As a remark, this is not enough for our
purpose, as we want a family of $(\phi,\Gamma)$-modules specializing to
$\mathbb{D}_{\mathrm{rig}}^{{\dagger}}(\rho)$. We will then define the space
$X_{n}$ inside $X_{n}^{\prime}\times\mathrm{Gr}_{n}$ (where $\mathrm{Gr}_{n}$
is the affine Grassmannian) consisting of pairs $(D^{\prime},\Lambda)$, where
$D^{\prime}$ is given by the universal $(\phi,\Gamma)$-module
$\mathbb{D}_{n}^{\prime}$ over $X_{n}^{\prime}$ and $\Lambda\subset
D^{\prime}[1/t]\otimes_{\mathcal{R}_{A}^{r}[1/t]}A((t))$ is a $\Gamma$-stable
lattice (see Theorem 6.8 and Theorem 7.6). If we let $\mathbb{D}_{n}$ be the
sheaf on $X_{n}$ corresponding to the pullback to $X_{n}$ of the pair of
$\mathbb{D}_{n}^{\prime}$ and of the universal lattice over $\mathrm{Gr}_{n}$,
we have that $\mathbb{D}_{n}$ specializes to
$\mathbb{D}_{\mathrm{rig}}^{{\dagger}}(\rho)$ at some point $x\in X_{n}(L)$,
by virtue of the equivalence written above. We still need to show that the
family $\mathbb{D}_{n}$ is densely regular, i.e. there exists a Zariski open
dense of $X_{n}$ such that the pullback of $\mathbb{D}_{n}$ to such open has
regular parameters. Now the main issue is that $X_{n}$ is not a vector bundle
over an open $\mathcal{U}$ of $\mathcal{T}^{n}$, hence the pullback of the
regular locus $\mathcal{T}_{n}^{\mathrm{reg}}$ to $X_{n}$ might not be dense
in $X_{n}$: indeed $X_{n}$ is not even flat over $\mathcal{T}^{n}$, as we will
see in Remark 6.11. In Section 6.2 and Section 7.2, we will study the geometry
of the space $X_{n}$ and give a bound to the dimension of the non-regular
locus inside $X_{n}$ when $n\leqslant 4$ or
$|\\{\delta_{i}/\delta_{j}\in\mathcal{T}\setminus\mathcal{T}^{\mathrm{reg}}\colon
i<j\\}|\leqslant 5$ as in the hypothesis of Theorem B. This will allow us to
conclude that $X_{n}$ is irreducible and that the pullback of the regular
locus $\mathcal{T}_{n}^{\mathrm{reg}}$ is dense inside $X_{n}$, concluding the
proof of Theorem B.

Structure of the paper: In section 1 we recollect basics and well-known
results concerning $(\phi,\Gamma)$-modules. section 2 is devoted to defining
trianguline representations and the trianguline variety. The outline of the
strategy we use to prove Theorem A and Theorem B is explained in section 3. In
section 4 we first explain how to construct a family of extensions of two
$(\phi,\Gamma)$-modules (§4.1) and in §4.2 we prove Theorem A. The rest of the
paper is dedicated to the proof of Theorem B: section 5 provides all the tools
that we will need in the course of the proof; section 6 gives a proof of
Theorem B when $n=2$ and finally section 7 finishes the proof of Theorem B
generalizing (when possible) the results in section 6.

## Acknowledgements

My first thanks go to my supervisor Prof. Eugen Hellmann for introducing me to
the topics of the thesis and for his invaluable guidance, continuous support
and patience during my PhD study.

I would like to offer my special thanks to Dr. Zhixiang Wu for patiently
explaining me some results from his PhD thesis, which are essential for this
dissertation.

I am very grateful to Luisa Meinke for the mathematical and emotional support
and for the fun we had together.

Finally my thanks go to the most important people of all: my friends and
family. I want to thank Stefano, Andrea, Margherita, Beatrice, Rafaela and my
parents for encouraging and supporting me during these years, but mostly just
for being present in my life.

##  1\. Preliminaries on $(\phi,\Gamma)$-modules

One of the most powerful tools in the study of the theory of $p$-adic Galois
representations is the theory of $(\phi,\Gamma)$-modules, providing a tool of
investigating such representations in terms of semi-linear algebra objects.
The category of these modules (with some additional condition) is in fact
equivalent to the category of $p$-adic Galois representations, as will be more
precisely stated in this section. In this section we discuss some basic
notions on $(\phi,\Gamma)$-modules and their cohomology as well as their
variation in families. Indeed, in [1] Berger and Colmez introduce the notion
of families of overconvergent $(\phi,\Gamma)$-modules associated to families
of $p$-adic Galois representations. The connection between these two
categories will be better investigated in the second section.

From now on, let $p$ be a prime number. Let $L$ be a finite extension of
$\mathbb{Q}_{p}$, $\mathcal{O}_{L}$ its ring of integers and $k_{L}$ its
residue field.

###  1.1. The Robba ring

For $r=p^{-a},s=p^{-b}\in p^{\mathbb{Q}}$ with $0\leqslant r\leqslant s<1$,
let

$\mathbb{B}^{[r,s]}=\\{T:r\leqslant|T|\leqslant
s\\}=\mathrm{Spa}(\mathbb{Q}_{p}\langle
p^{-b}T,p^{a}T^{-1}\rangle,\mathbb{Z}_{p}\langle p^{-b}T,p^{a}T^{-1}\rangle)$

be the closed annulus of inner radius $r$ and outer radius $s$ over
$\mathbb{Q}_{p}$. Let $\mathcal{R}^{[r,s]}$ denote the ring of analytic
functions of $\mathbb{B}^{[r,s]}$ and let $\mathcal{A}^{[r,s]}$ denote the
subring of functions whose valuation has maximal value 1. For $0\leqslant
r\leqslant s^{\prime}\leqslant s<1$, we have
$\mathcal{R}^{[r,s]}\subset\mathcal{R}^{[r,s^{\prime}]}$ simply defined by
restriction. We let

$\mathcal{R}^{r}=\bigcap_{r\leqslant s<1}\mathcal{R}^{[r,s]}$

be the ring of analytic functions converging on the half-open annulus

$\mathbb{B}^{r}=\\{T:r\leqslant|T|<1\\}$

and $\mathcal{A}^{r}$ the subring of $\mathcal{R}^{r}$ consisting of functions
whose valuation has maximal value 1. As in [20], for $r\leqslant s<1$, we can
define the valuation

$w_{s}\left(\sum_{n\in\mathbb{Z}}a_{n}T^{n}\right):=\mathrm{min}_{n\in\mathbb{Z}}\\{v_{p}(a_{n})+ns\\};$

then $\mathcal{R}^{r}$ is complete with respect to the Fréchet topology
defined by these valuations.

######  Definition 1.1.

The _Robba ring over $\mathbb{Q}_{p}$_ is the ring

$\displaystyle\mathcal{R}$ $\displaystyle=\bigcup_{0\leqslant
r<1}\mathcal{R}^{r}$
$\displaystyle=\left\\{f(T)=\sum_{n\in\mathbb{Z}}a_{n}T^{n}\colon\begin{array}[]{l}a_{n}\in\mathbb{Q}_{p},a_{n}r(f)^{n}\xrightarrow[n\rightarrow-\infty]{}0\text{
for some }0\leqslant r(f)<1\\\ \text{ and
}a_{n}s^{n}\xrightarrow[n\rightarrow+\infty]{}0\text{ for all }r(f)\leqslant
s<1\end{array}\right\\}$

of power series converging on some annulus. We equip the Robba ring with the
direct limit of the Fréchet topologies on $\mathcal{R}^{r}$. Moreover let
$\mathcal{A}^{\dagger}\coloneqq\bigcup_{0\leqslant r<1}\mathcal{A}^{r}$ and
equip it with the subspace topology.

We can extend the definition of the Robba ring in order to have coefficients
in some affinoid $\mathbb{Q}_{p}$-algebra $A$, so we define

$\begin{array}[]{lr}\mathcal{R}_{A}\coloneqq\mathcal{R}\widehat{\otimes}_{\mathbb{Q}_{p}}A,&\mathcal{A}^{\dagger}_{A}\coloneqq\mathcal{A}^{\dagger}\widehat{\otimes}_{\mathbb{Z}_{p}}A,\end{array}$

where the completed tensor product is the completion of the ordinary tensor
product with respect to the $p$-adic topology on $A$ and the topology on
$\mathcal{R}$ and $\mathcal{A}^{\dagger}$ respectively. For more precise
definitions, we refer to [18, §6.1].

We want to equip the Robba ring with a Frobenius and some action of
$\Gamma:=\mathbb{Z}_{p}^{\times}$. This will be useful later to define the
semi-linear structure of $(\phi,\Gamma)$-modules over the Robba ring. We
define the $A$-linear endomorphism

$\phi:\mathcal{R}_{A}\rightarrow\mathcal{R}_{A},T\mapsto(1+T)^{p}-1$ (1.1)

and an $A$-linear action

$\Gamma\times\mathcal{R}_{A}\rightarrow\mathcal{R}_{A},(\gamma,T)\mapsto(1+t)^{\gamma}-1=\sum_{n\geqslant
1}\binom{\gamma}{n}T^{n}.$

The two operators commute with each other and they are both continuous.

Moreover we set

$t\coloneqq\mathrm{log}(1+T)=\sum_{n=1}^{\infty}(-1)^{n+1}\frac{T^{n}}{n}\in\mathcal{R}.$

Then $\phi(t)=pt$ and $\gamma\cdot t=\gamma t$ for all $\gamma\in\Gamma$.

More generally, one can define the notion of _relative Robba ring_ over some
rigid space $X$ in the following way: for $r,s\in p^{\mathbb{Q}}$ with
$0\leqslant r\leqslant s<1$, let
$\mathbb{B}_{X}^{[r,s]}=X\times_{\mathbb{Q}_{p}}\mathbb{B}^{[r,s]}$ and
$\mathbb{B}_{X}^{r}=X\times_{\mathbb{Q}_{p}}\mathbb{B}^{r}$; then define
$\mathcal{R}_{X}^{[r,s]}=(\mathrm{pr}_{X})_{*}\mathcal{O}_{\mathbb{B}_{X}^{[r,s]}}$
and
$\mathcal{R}^{r}_{X}=(\mathrm{pr}_{X})_{*}\mathcal{O}_{\mathbb{B}_{X}^{r}}$.
The relative Robba ring over $X$ is the sheaf

$\mathcal{R}_{X}\coloneqq\lim_{r\rightarrow 1}\mathcal{R}^{r}_{X}.$

The sheaf $\mathcal{R}_{X}$ is endowed with a continuous
$\mathcal{O}_{X}$-linear endomorphism
$\phi:\mathcal{R}_{X}\rightarrow\mathcal{R}_{X}$ and a continuous
$\mathcal{O}_{X}$-linear action
$\Gamma\times\mathcal{R}_{X}\rightarrow\mathcal{R}_{X}$. In the same way we
can define
$\mathcal{A}_{X}^{[r,s]}=(\mathrm{pr}_{X})_{*}\mathcal{O}^{+}_{\mathbb{B}_{X}^{[r,s]}}$
and
$\mathcal{A}^{r}_{X}=(\mathrm{pr}_{X})_{*}\mathcal{O}^{+}_{\mathbb{B}_{X}^{r}}$.
Finally, let

$\mathcal{A}^{\dagger}_{X}\coloneqq\lim_{r\rightarrow 1}\mathcal{A}^{r}_{X}.$

###  1.2. $(\phi,\Gamma)$-modules

Now we introduce the notion of $(\phi,\Gamma)$-modules as in [11].

######  Definition 1.2.

Let $L$ be a finite extension of $\mathbb{Q}_{p}$.

  1. (i)

A _$(\phi,\Gamma)$ -module over $\mathcal{R}_{L}$_ is a locally free
$\mathcal{R}_{L}$-module $D$ of finite presentation endowed with

     * •

a $\mathcal{R}_{L}$-semilinear endomorphism $\Phi:D\rightarrow D$ such that
$\mathcal{R}_{L}\cdot\Phi(D)=D$;

     * •

a continuous $\mathcal{R}_{L}$-semilinear $\Gamma$-action commuting with
$\Phi$.

  2. (ii)

A _morphism of $(\phi,\Gamma)$-modules_ is a morphism of
$\mathcal{R}_{L}$-modules respecting the actions of $\Gamma$ and $\Phi$.

######  Remark 1.3.

Let us explain in more detail what we mean by _continuous_ action of $\Gamma$
on $D$ in the above definition. We ask that there exists an
$\mathcal{R}_{L}$-basis $\mathcal{E}=\\{e_{1},\dots,e_{d}\\}$ of $D$ and
$r\in[0,1]$ such that the map

$\displaystyle\Gamma$
$\displaystyle\rightarrow\mathrm{Mat}_{d}(\mathcal{R}_{L})$
$\displaystyle\gamma$ $\displaystyle\mapsto\mathrm{Mat}_{\mathcal{E}}(\gamma)$

has values in $\mathrm{Mat}_{d}(\mathcal{R}_{L}^{r})$ and is continuous.

The motivation behind the study of $(\phi,\Gamma)$-modules is the following
well-known Theorem due to Fontaine, Cherbonnier, Colmez and Berger.

######  Theorem 1.4.

There is a fully faithful functor

$\mathbb{D}_{\mathrm{rig}}^{{\dagger}}:\\{L\text{-linear continuous
representations of
}G_{\mathbb{Q}_{p}}\\}\rightarrow\\{(\phi,\Gamma)\text{-modules over
}\mathcal{R}_{L}\\}$

commuting with base change.

For a reference look at [5, Theorem 2.2.2].

######  Definition 1.5.

The $(\phi,\Gamma)$-modules in the essential image of the funtor
$\mathbb{D}_{\mathrm{rig}}^{{\dagger}}$ are called _étale
$(\phi,\Gamma)$-modules_.

The consequence of Theorem 1.4 is that the functor

$\mathbb{D}_{\mathrm{rig}}^{{\dagger}}:\left\\{\begin{array}[]{c}L\text{-linear
continuous}\\\ \text{representations of
}G_{\mathbb{Q}_{p}}\end{array}\right\\}\rightarrow\\{\text{\'{e}tale
}(\phi,\Gamma)\text{-modules over }\mathcal{R}_{L}\\}$ (1.2)

is an equivalence of categories. Let us remark that the above functor is not
an equivalence when we work with families of Galois representations and
families of $(\phi,\Gamma)$-modules because it is not essentially surjective,
as we will see in §2.1.

As mentioned before, the theory of $(\phi,\Gamma)$-modules is compatible with
its variation in analytic families.

######  Definition 1.6.

Let $X$ be a rigid space. A _family of $(\phi,\Gamma)$-modules over $X$_ is an
$\mathcal{R}_{X}$-module $\mathbb{D}$ which is locally on $X$ finite
projective over $\mathcal{R}_{X}$ with

  * •

a $\mathcal{R}_{X}$-semilinear endomorphism
$\Phi:\mathbb{D}\rightarrow\mathbb{D}$ such that
$\Phi:\phi^{*}\mathbb{D}\rightarrow\mathbb{D}$ is an isomorphism;

  * •

a continuous $\mathcal{R}_{X}$-semilinear $\Gamma$-action commuting with
$\Phi$.

A family of $(\phi,\Gamma)$-modules $\mathbb{D}$ over $X$ is said to be
_étale_ if there exists a covering $\\{U_{i}\\}_{i\in I}$ of $X$ and free
$\mathcal{A}_{U_{i}}^{\dagger}$-submodules $D_{i}$ of $\mathbb{D}_{|U_{i}}$
such that

$D_{i}\otimes_{\mathcal{A}_{U_{i}}^{\dagger}}\mathcal{R}_{U_{i}}=\mathbb{D}_{|U_{i}}$

and $\Phi$ induces an isomorphism $\phi^{*}D_{i}\rightarrow D_{i}$.

In [18, Corollary 6.11] it is proved that étaleness is an open condition and
that it can be checked fiber-wise. More precisely:

######  Remark 1.7.

  1. (i)

Given a family of $(\phi,\Gamma)$-modules $\mathbb{D}$ over a rigid space $X$,
the subset of $X$ where the specialization of $\mathbb{D}$ is étale is open.

  2. (ii)

A family of $(\phi,\Gamma)$-modules over a rigid space $X$ is étale if and
only if all fibers are étale.

###  1.3. The $(\phi,\Gamma)$-cohomology

The $(\phi,\Gamma)$-cohomology was first introduced by Herr in order to
compute the Galois cohomology of $p$-adic Galois representations on the side
of $(\phi,\Gamma)$-modules and it is a very useful tool in the study of these
objects.

######  Definition 1.8.

Let $X$ be a rigid space and let $D$ be a family of $(\phi,\Gamma)$-modules
over $X$.

  * •

If $p\neq 2$, the _Herr complex_ of $D$ is the complex of
$\mathcal{O}_{X}$-modules

$0\rightarrow D\xrightarrow{(\Phi-1,\gamma-1)}D\oplus
D\xrightarrow{(\gamma-1)\oplus(1-\Phi)}D\rightarrow 0,$

where $\gamma$ is a topological generator of $\Gamma$.

  * •

If $p=2$, the _Herr complex_ of $D$ is the complex of
$\mathcal{O}_{X}$-modules

$0\rightarrow D^{\Delta}\xrightarrow{(\Phi-1,\gamma-1)}D^{\Delta}\oplus
D^{\Delta}\xrightarrow{(\gamma-1)\oplus(1-\Phi)}D^{\Delta}\rightarrow 0,$

where $\Delta\subset\Gamma$ is the torsion subgroup and $\gamma$ is a
topological generator of $\Gamma/\Delta$.

We will refer to the Herr complex of $D$ as $\mathcal{C}^{\bullet}(D)$.
Moreover we denote by $H^{\bullet}_{\phi,\Gamma}(D)$ the cohomology of the
Herr complex, called the _$(\phi,\Gamma)$ -cohomology_ of $D$.

######  Proposition 1.9.

Let $X$ be a rigid space and $D$ a family of $(\phi,\Gamma)$-modules over $X$.
The $\mathcal{O}_{X}$-modules $H^{i}_{\phi,\Gamma}(D)$ are coherent for all
$i=0,1,2$.

The proof of this Proposition can be found in [22, Theorem 4.4.2].

One of the most important properties of the $(\phi,\Gamma)$-cohomology is that
it computes extensions:

######  Lemma 1.10.

Let $X$ be a rigid space and let $D_{1}$ and $D_{2}$ be two families of
$(\phi,\Gamma)$-modules over $X$. Then for $i=0,1$ there are natural
isomorphisms

$H^{i}_{\phi,\Gamma}(D_{2}^{\vee}\otimes
D_{1})\cong\mathrm{Ext}^{i}_{\phi,\Gamma}(D_{2},D_{1}).$

###### Proof.

Notice that

$H^{0}_{\phi,\Gamma}(D_{2}^{\vee}\otimes
D_{1})\cong\mathrm{Hom}_{\phi,\Gamma}(\mathcal{R}_{X},D_{2}^{\vee}\otimes
D_{1})\cong\mathrm{Hom}_{\phi,\Gamma}(D_{2},D_{1})=\mathrm{Ext}^{0}_{\phi,\Gamma}(D_{2},D_{1}).$

As for $i=1$, we refer to [11, Lemme 2.2]. ∎

It is possible to classify all the $(\phi,\Gamma)$-modules of rank 1 in a very
nice way. Let $\mathcal{T}$ be the rigid space representing the functor

$\displaystyle\\{\text{rigid analytic spaces over }\mathbb{Q}_{p}\\}$
$\displaystyle\rightarrow\underline{Sets}$ $\displaystyle X$
$\displaystyle\mapsto\mathrm{Hom}_{\mathrm{cont}}(\mathbb{Q}_{p}^{\times},\Gamma(X,\mathcal{O}_{X}^{\times})).$

We also denote by $\mathcal{W}$ the rigid space representing the functor

$\displaystyle\\{\text{rigid analytic spaces over }\mathbb{Q}_{p}\\}$
$\displaystyle\rightarrow\underline{Sets}$ $\displaystyle X$
$\displaystyle\mapsto\mathrm{Hom}_{\mathrm{cont}}(\mathbb{Z}_{p}^{\times},\Gamma(X,\mathcal{O}_{X}^{\times})).$

The representability of the above functors is proved in [22, Proposition
6.1.1]. There is a natural projection $\mathcal{T}\rightarrow\mathcal{W}$
given by the restriction of a character to $\mathbb{Z}_{p}^{\times}$; this map
has a section, identifying $\mathcal{T}$ with
$\mathcal{W}\times\mathbb{G}_{m}^{\mathrm{an}}$ via
$\delta\mapsto(\delta_{|\mathbb{Z}_{p}^{\times}},\delta(p))$.

######  Definition 1.11.

Let $\delta\in\mathcal{T}(X)$ for some rigid space $X$.

  * •

We define $\mathcal{R}_{X}(\delta)$ to be the $(\phi,\Gamma)$-module of rank 1
over $\mathcal{R}_{X}$ with basis $e$ such that $\Phi(e)=\delta(p)e$ and
$\gamma\cdot e=\delta(\gamma)e$ for all $\gamma\in\Gamma$.

  * •

If $X$ is an affinoid $L$-rigid space $\mathrm{Sp}(A)$, then we define the
_weight of $\delta$_ following [22, Notation 5.3.2.] as the element

$\mathrm{wt}(\delta)\coloneqq\frac{\mathrm{log}(\delta(a))}{\mathrm{log}(a)}\in
A$

for non-torsion $a\in\mathbb{Z}_{p}^{\times}$ close enough to 1.

The following Theorem gives a classification of all $(\phi,\Gamma)$-modules of
rank 1 (up to twist by a unique line bundle).

######  Theorem 1.12.

Let $D$ be a $(\phi,\Gamma)$-module of rank 1 over $\mathcal{R}_{X}$ for some
rigid space $X$. We have that
$D\cong\mathcal{R}_{X}(\delta)\otimes_{\mathcal{O}_{X}}H^{0}_{\phi,\Gamma}(D(\delta^{-1}))$
for a uniquely determined $\delta\in\mathcal{T}(X)$. Moreover we have that
$H^{0}_{\phi,\Gamma}(D(\delta^{-1}))$ is a line bundle over $X$.

###### Proof.

For the proof look [22, Construction 6.2.4, Theorem 6.2.14]. ∎

The $(\phi,\Gamma)$-cohomology of $(\phi,\Gamma)$-modules of rank 1 does not
behave always in the same way. In the following we partition the space
$\mathcal{T}$ keeping in mind this behaviour. Let $x$ be the identity
character and $\chi=x|x|$ the cyclotomic character in $\mathcal{T}$. Let us
define the following subspaces of $\mathcal{T}$:

$\begin{split}\mathcal{T}^{+}&\coloneqq\\{\chi x^{n}\colon
n\in\mathbb{N}\\}\\\ \mathcal{T}^{-}&\coloneqq\\{x^{-n}\colon
n\in\mathbb{N}\\}\\\
\mathcal{T}^{\mathrm{reg}}&\coloneqq\mathcal{T}\setminus(\mathcal{T}^{+}\cup\mathcal{T}^{-}).\end{split}$
(1.3)

The subsets $\mathcal{T}^{+}$, $\mathcal{T}^{-}$ and $\\{x^{n}\colon
n\in\mathbb{Z}\\}$ are closed in $\mathcal{T}$, making
$\mathcal{T}^{\mathrm{reg}}$ open. Moreover $\mathcal{T}^{\mathrm{reg}}$ is
dense in $\mathcal{T}$.

######  Definition 1.13.

We say that a character $\delta\in\mathcal{T}$ is _regular_ if
$\delta\in\mathcal{T}^{\mathrm{reg}}$.

######  Theorem 1.14.

Let $\delta\in\mathcal{T}(L)$. We have that

  1. (i)

$\mathrm{dim}_{L}H^{1}(\mathcal{R}_{L}(\delta))=\begin{cases}1\text{ if
}\delta\in\mathcal{T}^{\mathrm{reg}}(L)\\\ 2\text{ if
}\delta\in(\mathcal{T}^{+}\cup\mathcal{T}^{-})(L)\end{cases}$

  2. (ii)

$\mathrm{dim}_{L}H^{0}(\mathcal{R}_{L}(\delta))=\begin{cases}0\text{ if
}\delta\in(\mathcal{T}\setminus\mathcal{T}^{-})(L)\\\ 1\text{ if
}\delta\in\mathcal{T}^{-}(L)\end{cases}$

  3. (iii)

$\mathrm{dim}_{L}H^{2}(\mathcal{R}_{L}(\delta))=\begin{cases}0\text{ if
}\delta\in(\mathcal{T}\setminus\mathcal{T}^{+})(L)\\\ 1\text{ if
}\delta\in\mathcal{T}^{+}(L)\end{cases}$

For a proof see [12, Proposition 2.1, Theorem 2.9] and [23, Corollary 2.13]

More generally, we have:

######  Theorem 1.15.

Let $A$ be an affinoid $\mathbb{Q}_{p}$-algebra and let
$\delta\in\mathcal{T}(A)$.

  * (i)

$H^{0}_{\phi,\Gamma}(\mathcal{R}_{A}(\delta))\neq 0$ if and only if there
exists $i\in\mathbb{N}$ and $0\neq a\in A$ such that $a\cdot(\delta-x^{i})$ is
the constant zero function on $\mathbb{Q}_{p}^{\times}$.

  * (ii)

$H^{2}_{\phi,\Gamma}(\mathcal{R}_{A}(\delta))\neq 0$ if and only if the closed
subspace of $\mathrm{Sp}(A)$ given by the equation $\delta=\chi x^{i}$ is non-
empty for some $i\in\mathbb{N}$.

  * (iii)

If $\delta\in\mathcal{T}^{\mathrm{reg}}(A)$, then
$H^{0}_{\phi,\Gamma}(\mathcal{R}_{A}(\delta))=H^{2}_{\phi,\Gamma}(\mathcal{R}_{A}(\delta))=0$
and $H^{1}_{\phi,\Gamma}(\mathcal{R}_{A}(\delta))$ is free of rank 1 over $A$.

The reader can find the proof of this Theorem in [11, Corollaire 2.11,
Théorème 2.29].

Let us moreover define

$\mathcal{T}_{n}^{\mathrm{reg}}\coloneqq\\{(\delta_{1},\dots,\delta_{n})\in\mathcal{T}^{n}\colon\delta_{i}/\delta_{j}\in\mathcal{T}^{\mathrm{reg}}\text{
for all }i<j\\}\subset\mathcal{T}^{n},$ (1.4)

which is an open dense subspace of $\mathcal{T}^{n}$ and

$\mathcal{W}_{n}^{\mathrm{reg}}\coloneqq\\{(\delta_{1},\dots,\delta_{n})\in\mathcal{W}^{n}\colon\delta_{i}/\delta_{j}\notin
x^{\mathbb{Z}}\text{ for all }i<j\\}\subset\mathcal{W}^{n},$

which is a dense subspace of $\mathcal{W}^{n}$.

######  Lemma 1.16.

For every $\underline{\delta}\in\mathcal{T}^{n}(K)$ (for some extesion $K$ of
$\mathbb{Q}_{p}$), there exists an open neighbourhood
$\widetilde{\mathcal{U}}_{n}\subset\mathcal{T}^{n}$ of $\underline{\delta}$
such that the following properties are satisfied. Let
$(\widetilde{\delta}_{1},\dots,\widetilde{\delta}_{n})$ be the characters
corresponding to the open $\widetilde{\mathcal{U}}_{n}$. For all $1\leqslant
i<j\leqslant n$

  * •

for all $u\in\widetilde{\mathcal{U}}_{n}$, we have
$\widetilde{\delta}_{i}/\widetilde{\delta}_{j}\widehat{\otimes}k(u)\notin\\{x^{m}\colon
m\in\mathbb{Z}\\}$, unless $\delta_{i}/\delta_{j}=x^{a_{ij}}$ for some
$a_{ij}\in\mathbb{Z}$; in this case, we have
$\widetilde{\delta}_{i}/\widetilde{\delta}_{j}\widehat{\otimes}k(u)\in\\{x^{m}\colon
m\in\mathbb{Z}\\}$ only if
$\widetilde{\delta}_{i}/\widetilde{\delta}_{j}\widehat{\otimes}k(u)=x^{a_{ij}}$.

  * •

for all $u\in\widetilde{\mathcal{U}}_{n}$, we have
$\widetilde{\delta}_{i}/\widetilde{\delta}_{j}\widehat{\otimes}k(u)\notin\\{\chi
x^{m}\colon m\in\mathbb{Z}\\}$, unless $\delta_{i}/\delta_{j}=\chi x^{b_{ij}}$
for some $b_{ij}\in\mathbb{Z}$; in this case, we have
$\widetilde{\delta}_{i}/\widetilde{\delta}_{j}\widehat{\otimes}k(u)\in\\{\chi
x^{m}\colon m\in\mathbb{Z}\\}$ only if
$\widetilde{\delta}_{i}/\widetilde{\delta}_{j}\widehat{\otimes}k(u)=\chi
x^{b_{ij}}$.

###### Proof.

For all $1\leqslant i<j\leqslant n$, we will define a neighbourhood
$\mathcal{V}_{ij}\subset\mathcal{T}^{n}$ of $\underline{\delta}$ in the
following way: let

$\widetilde{\delta}_{i}/\widetilde{\delta}_{j}:\mathcal{T}^{n}\rightarrow\mathcal{T}$

be the map sending
$(\widetilde{\delta}_{1},\dots,\widetilde{\delta}_{n})\in\mathcal{T}^{n}(A)$
(for some affinoid algebra $A$) to
$\widetilde{\delta}_{i}/\widetilde{\delta}_{j}$.

  * •

If $\delta_{i}/\delta_{j}\notin\\{x^{m}\colon m\in\mathbb{Z}\\}\cup\\{\chi
x^{m}\colon m\in\mathbb{Z}\\}$, then we define

$\mathcal{V}_{ij}\coloneqq(\widetilde{\delta}_{i}/\widetilde{\delta}_{j})^{-1}(\mathcal{T}\setminus(\\{x^{m}\colon
m\in\mathbb{Z}\\}\cup\\{\chi x^{m}\colon m\in\mathbb{Z}\\})),$

which is open, since $\mathcal{T}\setminus(\\{x^{m}\colon
m\in\mathbb{Z}\\}\cup\\{\chi x^{m}\colon m\in\mathbb{Z}\\})$ is.

  * •

If instead $\delta_{i}/\delta_{j}=x^{a_{ij}}$ for some $a_{ij}\in\mathbb{Z}$,
then we define

$\mathcal{V}_{ij}\coloneqq(\widetilde{\delta}_{i}/\widetilde{\delta}_{j})^{-1}(\mathcal{T}\setminus(\\{x^{m}\colon
m\in\mathbb{Z}\setminus\\{a_{ij}\\}\\}\cup\\{\chi x^{m}\colon
m\in\mathbb{Z}\\})).$

  * •

Finally, if $\delta_{i}/\delta_{j}=\chi x^{b_{ij}}$ for some
$b_{ij}\in\mathbb{Z}$, then we define

$\mathcal{V}_{ij}\coloneqq(\widetilde{\delta}_{i}/\widetilde{\delta}_{j})^{-1}(\mathcal{T}\setminus(\\{x^{m}\colon
m\in\mathbb{Z}\\}\cup\\{\chi x^{m}\colon
m\in\mathbb{Z}\setminus\\{b_{ij}\\}\\})).$

Finally, the open $\bigcap_{(1\leqslant i<j\leqslant n)}\mathcal{V}_{ij}$ is a
neighbourhood of $\underline{\delta}$ and satisfies the wanted properties. ∎

In what follows we state some well-known properties of
$(\phi,\Gamma)$-cohomology.

The following Theorem is proved in [23, Theorem 0.2].

######  Theorem 1.17.

Let $D$ be a $(\phi,\Gamma)$-module over $\mathcal{R}_{L}$. Then

$\sum_{i=0}^{2}(-1)^{i}\dim_{L}H^{i}_{\phi,\Gamma}(D)=\mathrm{rk}(D).$

######  Theorem 1.18.

Let $A$ be an affinoid $\mathbb{Q}_{p}$-algebra and let $D$ be a family of
$(\phi,\Gamma)$-modules over $\mathrm{Sp}(A)$. The Herr complex of $D$ is
quasi-isomorphic to a bounded complex of projective $A$-modules

$M^{0}\rightarrow M^{1}\rightarrow M^{2}$

that universally computes the cohomology of the Herr complex, i.e. for all
affinoid $A$-algebras $B$, the complex $M^{\bullet}\otimes_{A}B$ is quasi-
isomorphic to $\mathcal{C}^{\bullet}(D\widehat{\otimes}_{A}B)$.

This result follows from [22, Theorem 4.4.3,4.4.5].

##  2\. Trianguline representations

Trianguline representations are a certain class of $p$-adic Galois
representations, just like crystalline, semistable and de Rham
representations. They were introduced for the first time by Colmez in order to
study the functoriality of the $p$-adic local Langlands correspondence for
$GL_{2}(\mathbb{Q}_{p})$.

###  2.1. Definitions and basic properties

######  Definition 2.1.

A Galois representation $\rho:G_{\mathbb{Q}_{p}}\rightarrow GL_{n}(L)$ is a
_trianguline representation_ if $\mathbb{D}_{\mathrm{rig}}^{{\dagger}}(\rho)$
is _triangulable_ , which means that there exists a complete filtration

$0\subset\mathrm{Fil}^{1}(\mathbb{D}_{\mathrm{rig}}^{{\dagger}}(\rho))\subset\dots\subset\mathrm{Fil}^{n}(\mathbb{D}_{\mathrm{rig}}^{{\dagger}}(\rho))=\mathbb{D}_{\mathrm{rig}}^{{\dagger}}(\rho)$

of sub-$(\phi,\Gamma)$-modules of
$\mathbb{D}_{\mathrm{rig}}^{{\dagger}}(\rho)$ such that for all $i$

$\mathrm{Fil}^{i}(\mathbb{D}_{\mathrm{rig}}^{{\dagger}}(\rho))/\mathrm{Fil}^{i-1}(\mathbb{D}_{\mathrm{rig}}^{{\dagger}}(\rho))\cong\mathcal{R}_{L}(\delta_{i})$

for some $\delta_{i}\in\mathcal{T}(L)$. We say that
$(\delta_{1},\dots,\delta_{n})$ are the _parameters_ of the triangulation
$\mathrm{Fil}^{\bullet}(\mathbb{D}_{\mathrm{rig}}^{{\dagger}}(\rho))$ of
$\mathbb{D}_{\mathrm{rig}}^{{\dagger}}(\rho)$ and they are said to be
_regular_ if
$(\delta_{1},\dots,\delta_{n})\in\mathcal{T}_{n}^{\mathrm{reg}}(L)$.

Analogously to the theory of _families_ of $(\phi,\Gamma)$-modules, there is
also a theory of _families_ of $p$-adic Galois representations.

######  Definition 2.2.

Let $X$ be a rigid space. A _family of Galois representations over $X$_ is a
vector bundle $\mathcal{V}$ over $X$ endowed with a continuous
$G_{\mathbb{Q}_{p}}$-action.

######  Theorem 2.3.

There is a fully faithful and exact functor

$\mathbb{D}_{\mathrm{rig}}^{{\dagger}}:\left\\{\begin{array}[]{l}\text{families
of Galois}\\\ \text{represetations over
}X\end{array}\right\\}\rightarrow\left\\{\begin{array}[]{l}\text{families of
\'{e}tale }\\\ (\phi,\Gamma)\text{-modules over }X\end{array}\right\\}$

commuting with base change.

For a reference of the above Theorem, look at [22, Theorem 2.2.17.].

###  2.2. The trianguline variety

In the following, let $k_{L}$ be the residue field of $L$ and fix a continuous
Galois representation $\bar{\rho}:G_{\mathbb{Q}_{p}}\rightarrow
GL_{n}(k_{L})$.

######  Definition 2.4.

  * •

Let $A$ be a complete local Artinian ring with residue field $k_{L}$. We say
that a continuous representation $\rho:G_{\mathbb{Q}_{p}}\rightarrow
GL_{n}(A)$ is a _deformation_ of $\bar{\rho}$ if the composition of $\rho$
with the induced map $GL_{n}(A)\rightarrow GL_{n}(k_{L})$ is exactly
$\bar{\rho}$.

  * •

We say that two deformations $\rho_{1},\rho_{2}:G_{\mathbb{Q}_{p}}\rightarrow
GL_{n}(A)$ are _equivalent_ if $\rho_{1}=M\rho_{2}M^{-1}$ for some matrix
$M\in\mathrm{ker}(GL_{n}(A)\rightarrow GL_{n}(k_{L}))$.

The functor that associates to a complete local Artinian ring $A$ the set of
deformations of $\bar{\rho}$ is pro-representable by a complete local
Noetherian $W(k_{L})$-algebra $R_{\bar{\rho}}^{\square}$, called the
_universal framed deformation ring of $\bar{\rho}$_. This was proved by Mazur
in [24]. We let
$\mathfrak{X}_{\bar{\rho}}^{\square}=(\mathrm{Spf}R_{\bar{\rho}}^{\square})^{\mathrm{rig}}$
be the rigid space associated to the formal scheme
$\mathrm{Spf}(R_{\bar{\rho}}^{\square})$. We have that
$\mathfrak{X}_{\bar{\rho}}^{\square}$ represents the functor which associates
to an affinoid rigid space $\mathrm{Sp}(A)$ over $W(k_{L})[1/p]$ the set of
deformations $\rho:G_{\mathbb{Q}_{p}}\rightarrow GL_{n}(A^{+})$ of
$\bar{\rho}$, where $A^{+}$ denotes the subring of power-bounded elements of
$A$ (in particular it is a $W(k_{L})$-algebra).

######  Definition 2.5.

The _trianguline variety_ $X_{\mathrm{tri}}^{\square}(\bar{\rho})$ is the
rigid analytic space over $L$ defined as the Zariski-closure of the rigid
space

$U_{\mathrm{tri}}^{\square}(\bar{\rho}):=\\{(\rho,\underline{\delta})\in\mathfrak{X}_{\bar{\rho}}^{\square}\times\mathcal{T}_{n}^{\mathrm{reg}}\colon\rho\text{
is trianguline of parameters }\underline{\delta}\\}$

inside $\mathfrak{X}_{\bar{\rho}}^{\square}\times\mathcal{T}^{n}$.

This space was defined and studied in [7]. In particular we recall the
following results.

######  Theorem 2.6.

  1. (i)

The space $X_{\mathrm{tri}}^{\square}(\overline{\rho})$ is equidimensional of
dimension $n^{2}+\leavevmode\nobreak\ \frac{n(n+1)}{2}$;

  2. (ii)

the subset $U_{\mathrm{tri}}^{\square}(\bar{\rho})$ is Zariski-open and
Zariski-dense in $X_{\mathrm{tri}}^{\square}(\overline{\rho})$

  3. (iii)

if $(\rho,\underline{\delta})\in X_{\mathrm{tri}}^{\square}(\overline{\rho})$,
then $\rho$ is trianguline.

It is then natural to ask ourselves whether all trianguline representations
"are in" the trianguline variety. This is the question that motivates this
thesis.

##  3\. Strategy

First we show that, given a Galois representation on a rigid space, its
reduction to the residue field of the ring of integers of a point is locally
constant. Recall that, given an affinoid ring $A$, we denote by $A^{+}$ the
subring of power-bounded elements and by $A^{++}\subset A^{+}$ the ideal of
topologically nilpotent elements.

######  Proposition 3.1.

Let $X$ be a rigid space over $L$ and let $\mathcal{V}$ be a family of Galois
representations over $X$, so that we have
$\widetilde{\rho}:G_{\mathbb{Q}_{p}}\rightarrow GL(\mathcal{V})$. Let $x\in
X(L)$ and let $\rho$ be the reduction of $\widetilde{\rho}$ to the point $x$.
After eventually shrinking $X$, we can choose a basis of $\mathcal{V}$ such
that $\rho$ has values in $GL_{n}(\mathcal{O}_{L})$. Let $\bar{\rho}$ be the
reduction of $\rho$ to the residue field $k_{L}$ of $L$. Then there is an open
neighbourhood $U=\mathrm{Sp}(A)\xhookrightarrow{j}X$ of $x$ such that (with
the basis chosen above) we have
$\widetilde{\rho}_{U}:=j^{*}\widetilde{\rho}:G_{\mathbb{Q}_{p}}\rightarrow
GL_{n}(A^{+})$. Moreover $\widetilde{\rho}_{U}$ reduces to $\bar{\rho}$, which
means that there is a commutative diagram

${G_{\mathbb{Q}_{p}}}$${GL_{n}(A^{+})}$${GL_{n}(k_{L})}$${GL_{n}(A^{+}/A^{++}),}$$\scriptstyle{\widetilde{\rho}_{U}}$$\scriptstyle{\overline{\rho}}$

where the lower horizontal arrow is induced by the inclusion
$k_{L}\hookrightarrow A^{+}/A^{++}$, since $A$ is an $L$-algebra.

######  Remark 3.2.

Let us remark that the representation $\overline{\rho}$ depends on the choice
of a basis for the vector bundle $\mathcal{V}$; however the isomorphism class
of the semisimplification of $\overline{\rho}$ is well-defined independently
of the chosen basis, as showed in [16, Proposition 5.11].

###### Proof.

First of all we show that there is an affinoid neighbourhood
$V\coloneqq\mathrm{Sp}(B)\xhookrightarrow{i}X$ of $x$ such that
$\widetilde{\rho}_{V}\coloneqq i^{*}\widetilde{\rho}$ has values in
$GL_{n}(B^{+})$ with respect to the basis of $\mathcal{V}$ fixed above. By
[19] we know that $G_{\mathbb{Q}_{p}}$ is topologically finitely generated,
hence let $\\{g_{1},\dots,g_{t}\\}$ be a set of topological generators of
$G_{\mathbb{Q}_{p}}$. For every $k=1,\dots,t$, let $a_{ij}^{k}$ be the
$(i,j)$-th entry of the matrix $\widetilde{\rho}(g_{k})$ and
$d^{k}\coloneqq\mathrm{det}(\widetilde{\rho}(g_{k}))$. We then have that
$a_{ij}^{k}(x)\in\mathcal{O}_{L}$, so $|a_{ij}^{k}(x)|\leqslant 1$ and
$|d^{k}(x)|=1$ because $d^{k}(x)\in\mathcal{O}_{L}^{\times}$. Let us define
the affinoid open neighbourhoods $V_{ij}^{k}\coloneqq\\{y\in
X:|a_{ij}^{k}(y)|\leqslant 1\\}$ and $V_{d}^{k}\coloneqq\\{y\in
X:|d^{k}(y)|=1\\}$ of $x$, so that we let

$V=\mathrm{Sp}(C)\coloneqq\bigcap_{i,j=1,\dots,n\atop
k=1,\dots,t}V_{ij}^{k}\cap V_{d}^{k}.$

In particular for all $k=1,\dots,t$ and $i,j=1\dots,n$, we have
$|a_{ij}^{k}|=|a_{ij}^{k}(y)|\leqslant 1$ and $|d^{k}|=|d^{k}(y^{\prime})|=1$
for some $y,y^{\prime}\in V$ thanks to the Maximum Principle, so
$a_{ij}^{k}\in B^{+}$ and $d^{k}\in(B^{+})^{\times}$. This means that
$i^{*}\widetilde{\rho}(g_{k})\in GL_{n}(B^{+})$ for all $k=1,\dots,t$. Now we
show that $\widetilde{\rho}_{V}\coloneqq
i^{*}\widetilde{\rho}:G_{\mathbb{Q}_{p}}\rightarrow GL_{n}(B^{+})$. Let $g\in
G_{\mathbb{Q}_{p}}$ and observe that for any open neighbourhood $Z$ of
$\widetilde{\rho}_{V}(g)\in GL_{n}(B)$, there exists $h\in\langle
g_{1},\dots,g_{t}\rangle$ such that $\widetilde{\rho}_{V}(h)\in Z$. So there
is a sequence $(\widetilde{\rho}_{V}(h_{k}))_{k}$ converging to
$\widetilde{\rho}_{V}(g)$, where $(h_{k})_{k}\subset\langle
g_{1},\dots,g_{t}\rangle$. Let $b_{ij}$ be the $(i,j)$-th entry of
$\widetilde{\rho}_{V}(g)$ and $b_{ij,k}$ the $(i,j)$-th entry of
$\widetilde{\rho}_{V}(h_{k})$. For all $i,j=1,\dots,n$ we have a sequence
$(b_{ij,k})_{k}\subset B^{+}$ converging to $b_{ij}$, so $b_{ij}\in B^{+}$. In
the same way, we get a sequence $(d_{k})_{k}\subset(B^{+})^{\times}$
converging to $\mathrm{det}(\widetilde{\rho}_{V}(g))$, which allows us to
conclude that $\widetilde{\rho}_{V}(g)\in GL_{n}(B^{+})$. Finally it is left
to show that $\widetilde{\rho}_{V}$ reduces to $\overline{\rho}$ modulo
$B^{++}$, after eventually shrinking $V$. Let $f_{ij}^{k}\coloneqq
a_{ij}^{k}-a_{ij}^{k}(x)\in B^{+}$; fix $\epsilon\in(0,1)$ and consider
$U_{ij}^{k}\coloneqq\\{y\in\mathrm{Sp}(B):|f_{ij}^{k}(y)|\leqslant\epsilon\\}$
for all $i,j=1\dots,n$ and $k=1,\dots,t$. We have $x\in U_{ij}^{k}$ because
$f_{ij}^{k}(x)=0$, so

$U\coloneqq\mathrm{Sp}(A)\coloneqq\bigcap_{i,j=1,\dots,n\atop
k=1,\dots,t}U_{ij}^{k}$

is an affinoid open neighbourhood of $x$. We then have
$\widetilde{\rho}_{V}:G_{\mathbb{Q}_{p}}\rightarrow GL_{n}(A^{+})$ and
$|f_{ij}^{k}|=|f_{ij}^{k}(y)|\leqslant\epsilon<1$ for some
$y\in\mathrm{Sp}(A)$, thus $f_{ij}^{k}\in A^{++}$ for all $i,j=1,\dots,n$ and
$k=1,\dots,t$ by the same approximation argument as before. In a very similar
way to the one used above, it is possible to conclude that
$\widetilde{\rho}_{U}(g)\equiv\rho(g)\text{ mod }A^{++}$ for all $g\in
G_{\mathbb{Q}_{p}}$. ∎

In what follows we will outline a strategy that we will use in order to show
that trianguline representations appear as points of the trianguline variety.
The underlying idea for what follows is that, given a trianguline
representation $\rho$, we want to construct a rigid space $X$ and a family of
trianguline Galois representations on $X$ reducing to $\rho$ at some pont
$x\in X$ and having regular parameters in a dense subset of a neighbourhood of
$x$. In this way, we can show that $\rho$ is in the closure of a regular
family, hence in the trianguline variety.

The following Theorem makes the above idea more precise in terms of
$(\phi,\Gamma)$-modules.

######  Theorem 3.3.

Let $\rho:G_{\mathbb{Q}_{p}}\rightarrow GL_{n}(\mathcal{O}_{L})$ be a
trianguline representation with parameters
$\underline{\delta}\coloneqq(\delta_{1},\dots,\delta_{n})\in\mathcal{T}^{n}(L)$
and let $\bar{\rho}$ be the reduction of $\rho$ to the residue field of $L$:

$\bar{\rho}:G_{\mathbb{Q}_{p}}\xrightarrow{\rho}GL_{n}(\mathcal{O}_{L})\twoheadrightarrow
GL_{n}(k_{L}).$

Assume that there exist

  * •

a rigid space $X$ over $L$;

  * •

a family of $(\phi,\Gamma)$-modules $\mathbb{D}$ over $X$;

  * •

a map
$(\widetilde{\delta_{1}},\dots,\widetilde{\delta_{n}}):X\rightarrow\mathcal{T}^{n}$

such that there are

  1. (1)

a point $x\in X(L)$ such that
$\underline{\widetilde{\delta}}\widehat{\otimes}k(x)=\underline{\delta}$ and
$\mathbb{D}\widehat{\otimes}k(x)=\mathbb{D}_{\mathrm{rig}}^{{\dagger}}(\rho)$;

  2. (2)

a Zariski-open dense subset $U\xhookrightarrow{j}X$ such that
$j^{*}\mathbb{D}$ has a filtration with graded pieces
$\mathcal{R}_{U}(j^{*}\widetilde{\delta_{i}})$ and
$(j^{*}\widetilde{\delta_{1}},\dots,j^{*}\widetilde{\delta_{n}})\in\mathcal{T}_{n}^{\mathrm{reg}}(U)$.

Then $(\rho,\underline{\delta})\in X_{\mathrm{tri}}^{\square}(\bar{\rho})(L)$.

###### Proof.

By [21, Theorem 0.2], there exists a neighbourhood
$V=\mathrm{Sp}(B)\xhookrightarrow{i}X$ of $x$ and a family of representations
on $V$

$\widetilde{\rho}:G_{\mathbb{Q}_{p}}\rightarrow GL_{n}(B^{+})$

such that
$\mathbb{D}_{\mathrm{rig}}^{\dagger}(\widetilde{\rho})=i^{*}\mathbb{D}$. By
Proposition 3.1, eventually shrinking $V$, we can choose a basis of
$\mathcal{V}$ such that the following diagram commutes:

${G_{\mathbb{Q}_{p}}}$${GL_{n}(B^{+})}$${GL_{n}(k_{L})}$${GL_{n}(B^{+}/B^{++}).}$$\scriptstyle{\widetilde{\rho}}$$\scriptstyle{\overline{\rho}}$

Thus $\widetilde{\rho}$ induces a morphism

$\psi\coloneqq(\widetilde{\rho},i^{*}\widetilde{\underline{\delta}}):V\rightarrow\mathfrak{X}^{\square}_{\bar{\rho}}\times\mathcal{T}^{n}.$

We now show that $\psi_{|U\cap V}$ factors through
$U_{\mathrm{tri}}^{\square}(\bar{\rho})^{\mathrm{reg}}$: first of all let
$u:U\cap V\hookrightarrow V$ and notice that it is sufficient to prove that
the family of representations $u^{*}\widetilde{\rho}$ is trianguline with
regular parameters $(i\circ u)^{*}\underline{\widetilde{\delta}}$. By
hypothesis and by Theorem 2.3,
$\mathbb{D}_{\mathrm{rig}}^{{\dagger}}(u^{*}\widetilde{\rho})=u^{*}\mathbb{D}_{\mathrm{rig}}^{{\dagger}}(\widetilde{\rho})\cong(i\circ
u)^{*}\mathbb{D}$. Hence
$\mathbb{D}_{\mathrm{rig}}^{{\dagger}}(u^{*}\widetilde{\rho})$ has parameters
$(i\circ
u)^{*}\underline{\widetilde{\delta}}\in\mathcal{T}^{\mathrm{reg}}_{n}(U\cap
V)$. Thus $\psi(U\cap V)\subset U_{\mathrm{tri}}^{\square}(\bar{\rho})$.
Notice that $U_{\mathrm{tri}}^{\square}(\bar{\rho})\cap\mathrm{im}(\psi)$ is
dense in $\mathrm{im}(\psi)$: let $W$ be an open in $\mathrm{im}(\psi)$, so
that $\psi^{-1}(W)$ is open in $V$. Then $\psi^{-1}(W)\cap U\neq\emptyset$,
which implies that $W\cap
U_{\mathrm{tri}}^{\square}(\bar{\rho})\neq\emptyset$. Thanks to the density of
$U_{\mathrm{tri}}^{\square}(\bar{\rho})\cap\mathrm{im}(\psi)$ in
$\mathrm{im}(\psi)$, we can conclude that

$\mathrm{im}(\psi)\subset\overline{\mathrm{im}(\psi)}=\overline{U_{\mathrm{tri}}^{\square}(\bar{\rho})\cap\mathrm{im}(\psi)}\subset
X_{\mathrm{tri}}^{\square}(\overline{\rho}).$

In particular, $\psi(x)=(\rho,\underline{\delta})\in
X_{\mathrm{tri}}^{\square}(\overline{\rho})$. ∎

######  Lemma 3.4.

Let $\rho:G_{\mathbb{Q}_{p}}\rightarrow GL_{n}(L)$ be a representation and let
$\underline{\delta}\in\mathcal{T}^{n}(L)$. Let us assume that there is a
finite extension $L\xhookrightarrow{i}L^{\prime}$ such that
$(\rho^{\prime},\underline{\delta}^{\prime})\in
X_{\mathrm{tri}}^{\square}(\overline{\rho})(L^{\prime})$, where
$\rho^{\prime}\coloneqq
i^{*}\rho:G_{\mathbb{Q}_{p}}\xrightarrow{\rho}GL_{n}(L)\hookrightarrow
GL_{n}(L^{\prime})$ and $\underline{\delta}^{\prime}\coloneqq
i^{*}\underline{\delta}:\mathbb{Q}_{p}^{\times}\xrightarrow{\underline{\delta}}(L^{\times})^{n}\hookrightarrow(L^{\prime\times})^{n}$.
Then $(\rho,\underline{\delta})\in
X_{\mathrm{tri}}^{\square}(\overline{\rho})(L)$.

###### Proof.

We have that the point
$(\rho^{\prime},\underline{\delta}^{\prime})\in(\mathfrak{X}_{\bar{\rho}}^{\square}\times\mathcal{T}^{n})(L^{\prime})$
corresponds by construction to the map

$(\rho^{\prime},\underline{\delta}^{\prime}):\mathrm{Sp}(L^{\prime})\rightarrow\mathrm{Sp}(L)\xrightarrow{(\rho,\underline{\delta})}\mathfrak{X}_{\bar{\rho}}^{\square}\times\mathcal{T}^{n}.$

The fact that $(\rho^{\prime},\underline{\delta}^{\prime})\in
X_{\mathrm{tri}}^{\square}(\overline{\rho})(L^{\prime})$ implies that
$\mathrm{im}(\rho^{\prime},\underline{\delta}^{\prime})\subset
X_{\mathrm{tri}}^{\square}(\overline{\rho})$; hence in particular we can
deduce that $\mathrm{im}(\rho,\underline{\delta})\subset
X_{\mathrm{tri}}^{\square}(\overline{\rho})$. This means
$(\rho,\underline{\delta})\in X_{\mathrm{tri}}^{\square}(\overline{\rho})(L)$.
∎

##  4\. Trianguline representations with no $\mathcal{T}^{+}$-type
irregularity

In this section we prove Theorem A, which states that any trianguline
representation whose parameters don’t admit quotients in $\mathcal{T}^{+}$ are
in the trianguline variety. More precisely, we show the following:

See A

###  4.1. Constructing extensions

We want to use Theorem 3.3 to prove Theorem A. In order to do so, it is
essential to construct a rigid space $X$ of extensions of
$(\phi,\Gamma)$-modules. In this section we will show how to do so in the case
the second $(\phi,\Gamma)$-cohomology of these $(\phi,\Gamma)$-modules is a
locally free sheaf.

######  Theorem 4.1.

Let $D$ be a $(\phi,\Gamma)$-module over an affinoid ring $A$ and let $B$ be
an affinoid $A$-algebra. If $H^{2}_{\phi,\Gamma}(D)$ is a locally free
$A$-module, then

$H^{1}_{\phi,\Gamma}(D)\otimes_{A}B\cong
H^{1}_{\phi,\Gamma}(D\widehat{\otimes}_{A}B).$

###### Proof.

By Theorem 1.18, there is a complex of projective $A$-modules

$M^{0}\xrightarrow{d^{0}}M^{1}\xrightarrow{d^{1}}M^{2}$

universally computing the cohomology of the Herr complex of $D$. This means
that $H^{1}(D\widehat{\otimes}_{A}B)\cong H^{1}(M^{\bullet}\otimes_{A}B)$ for
any $A$-algebra $B$. Being $H^{2}(M^{\bullet})$ locally free, we have
$M^{2}\cong H^{2}(M^{\bullet})\oplus\mathrm{im}(d^{1})$. Since both $M^{2}$
and $H^{2}(M^{\bullet})$ are projective modules, we can conclude
$\mathrm{im}(d^{1})$ is as well, implying that $M^{1}$ also splits as the
direct sum of $\mathrm{im}(d^{1})$ and $\mathrm{ker}(d^{1})$. Again we can
deduce that $Z^{1}:=\mathrm{ker}(d^{1})$ is a projective $A$-module. Now
notice that the complex

$M^{0}\xrightarrow{d^{0}}Z^{1}$

universally computes $H^{0}_{\phi,\Gamma}(D)$ and $H^{1}_{\phi,\Gamma}(D)$.
Moreover

$M^{0}\xrightarrow{d^{0}}Z^{1}\rightarrow H^{1}(M^{\bullet})\rightarrow 0$

is a projective resolution of $H^{1}(M^{\bullet})$. Therefore

$\displaystyle H^{1}_{\phi,\Gamma}(D)\otimes_{A}B$ $\displaystyle\cong
H^{1}(M^{\bullet})\otimes_{A}B\cong\mathrm{Tor}_{0}(H^{1}(M^{\bullet}),B)\cong(Z^{1}\otimes_{A}B)/\mathrm{im}(d^{0}\otimes\mathrm{id}_{B})$
$\displaystyle\cong H^{1}(M^{\bullet}\otimes_{A}B)\cong
H^{1}_{\phi,\Gamma}(D\widehat{\otimes}_{A}B).$

∎

######  Theorem 4.2.

Let $D_{1},D_{2}$ be two families of $(\phi,\Gamma)$-modules over some rigid
analytic affinoid space $\mathrm{Sp}(A)$. Assume that
$H^{2}_{\phi,\Gamma}(D_{1}\otimes_{\mathcal{R}_{A}}D_{2}^{\vee})$ is locally
free. There exists a vector bundle $X$ over $\mathrm{Sp}(A)$ (of dimension
equal to the minimal number of generators of the module
$H^{1}_{\phi,\Gamma}(D_{1}\otimes_{\mathcal{R}_{A}}D_{2}^{\vee})$ over $A$)
and a family of $(\phi,\Gamma)$-modules $\mathbb{D}$ over $X$ such that for
any rigid space $Y\xrightarrow{g}\mathrm{Sp}(A)$ over $\mathrm{Sp}(A)$, we
have a surjective map

$\displaystyle\Psi_{Y}:X(Y)$
$\displaystyle\twoheadrightarrow\mathrm{Ext}^{1}_{\phi,\Gamma}(g^{*}D_{2},g^{*}D_{1})(Y)$
$\displaystyle f$ $\displaystyle\mapsto f^{*}\mathbb{D}$

which is functorial in $Y$. We moreover can choose $X$ such that the above map
$\Psi_{Y}$ is bijective for all rigid spaces $Y\xrightarrow{g}\mathrm{Sp}(A)$
if the module
$H^{1}_{\phi,\Gamma}(D_{1}\otimes_{\mathcal{R}_{A}}D_{2}^{\vee})$ is free over
$A$.

######  Remark 4.3.

In [13, §5.4] Emerton and Gee explain how to construct a family of extensions
of $(\phi,\Gamma)$-modules with coefficients in characteristic $p$ in case the
second $(\phi,\Gamma)$-cohomology has constant fiber rank. In fact Theorem 4.2
is the analogue in characteristic 0 of that construction.

###### Proof.

By Proposition 1.9, we know
$H^{1}_{\phi,\Gamma}(D_{1}\widehat{\otimes}_{A}D_{2}^{\vee})$ is a coherent
$\mathcal{O}_{\mathrm{Sp}(A)}$-module. Let $n$ be the minimal number of
generators of $H^{1}_{\phi,\Gamma}(D_{1}\widehat{\otimes}_{A}D_{2}^{\vee})$
and let $F$ be a finitely generated free $A$-module of rank $n$; in
particular, we have a surjective map of $A$-modules

$F\twoheadrightarrow
H^{1}_{\phi,\Gamma}(D_{1}\widehat{\otimes}_{A}D_{2}^{\vee}).$

We define the rigid space

$X\coloneqq\mathrm{Sp}(A)\times\mathbb{A}^{n,\mathrm{an}}_{\mathbb{Q}_{p}}\xrightarrow{h}\mathrm{Sp}(A),$

which is a vector bundle over $\mathrm{Sp}(A)$. Let $Y\to X$ be a map of rigid
spaces over $\mathrm{Sp}(A)$ and let $g$ be the map
$Y\rightarrow\mathrm{Sp}(A)$. Let us study the space $X(Y)$:

$\displaystyle X(Y)=\mathrm{Hom}_{\mathrm{Sp}(A)}(Y,X)$
$\displaystyle\cong\mathrm{Hom}_{\mathrm{Sp}(\mathbb{Q}_{p})}(Y,\mathbb{A}_{\mathbb{Q}_{p}}^{n,\mathrm{an}}).$

For any rigid $\mathbb{Q}_{p}$-space $Y$ there is a bijection

$\mathrm{Hom}_{\mathrm{Sp}(\mathbb{Q}_{p})}(Y,\mathbb{A}_{\mathbb{Q}_{p}}^{n,\mathrm{an}})\xrightarrow{\sim}\mathrm{Hom}_{\mathrm{Sp}(\mathbb{Q}_{p})}(Y,\mathbb{A}_{\mathbb{Q}_{p}}^{n})\cong\Gamma(Y,\mathcal{O}_{Y})^{n}$

between the set of rigid morphisms
$Y\rightarrow\mathbb{A}_{\mathbb{Q}_{p}}^{n,\mathrm{an}}$ and the set of
morphisms between locally $G$-ringed spaces
$Y\rightarrow\mathbb{A}_{\mathbb{Q}_{p}}^{n}$. We then have the map

$\displaystyle X(Y)$
$\displaystyle\cong\mathrm{Hom}_{\mathrm{Sp}(\mathbb{Q}_{p})}(Y,\mathbb{A}_{\mathbb{Q}_{p}}^{n,\mathrm{an}})\cong\Gamma(Y,\mathcal{O}_{Y})^{n}$
$\displaystyle\cong\Gamma(Y,g^{*}F)\twoheadrightarrow\Gamma(Y,g^{*}H^{1}_{\phi,\Gamma}(D_{1}\otimes_{\mathcal{R}_{A}}D_{2}^{\vee})).$

Since $H^{2}_{\phi,\Gamma}(D_{1}\otimes_{\mathcal{R}_{A}}D_{2}^{\vee})$ is
locally free by hypothesis, we get as a consequence that

$g^{*}H^{1}_{\phi,\Gamma}(D_{1}\otimes_{\mathcal{R}_{A}}D_{2}^{\vee})\cong
H^{1}_{\phi,\Gamma}(g^{*}(D_{1}\otimes_{\mathcal{R}_{A}}D_{2}^{\vee}))$

by Theorem 4.1. Finally, we have

$\displaystyle\Psi_{Y}:X(Y)\twoheadrightarrow
g^{*}H^{1}_{\phi,\Gamma}(D_{1}\otimes_{\mathcal{R}_{A}}D_{2}^{\vee})(Y)$
$\displaystyle\cong
H^{1}_{\phi,\Gamma}(g^{*}(D_{1}\otimes_{\mathcal{R}_{A}}D_{2}^{\vee}))(Y)$
$\displaystyle\cong\mathrm{Ext}^{1}_{\phi,\Gamma}(g^{*}D_{2},g^{*}D_{1})(Y).$

Let us now define the universal family of $(\phi,\Gamma)$-modules over $X$. We
know we have a surjective map

$\displaystyle\Psi_{X}:\mathrm{Hom}_{\mathrm{Sp}(A)}(X,X)\twoheadrightarrow\mathrm{Ext}^{1}_{\phi,\Gamma}(h^{*}D_{2},h^{*}D_{1})(X).$

Let $\mathbb{D}\in\mathrm{Ext}^{1}_{\phi,\Gamma}(h^{*}D_{2},h^{*}D_{1})$ be
the image of $\mathrm{id}_{X}$ through $\Psi_{X}$. By the Yoneda Lemma it
follows immediately that $\mathbb{D}$ satisfies the universal property
specified above. In case the module
$H^{1}_{\phi,\Gamma}(D_{1}\otimes_{\mathcal{R}_{A}}D_{2}^{\vee})$ is free over
$A$, it is sufficient to take the module $F$ exactly equal to
$H^{1}_{\phi,\Gamma}(D_{1}\otimes_{\mathcal{R}_{A}}D_{2}^{\vee})$; it is then
obvious that all the above maps composing $\Psi_{Y}$ are isomorphisms. ∎

###  4.2. Proof of Theorem A

In order to prove Theorem A, we proceed by induction; hence let us first prove
the 2-dimensional case.

######  Proposition 4.4.

Let $(\delta_{1},\delta_{2})\in\mathcal{T}^{2}(L)$ such that
$\delta_{1}\delta_{2}^{-1}\notin\mathcal{T}^{+}(L)$ and let $D\in
H^{1}_{\phi,\Gamma}(\mathcal{R}_{L}(\delta_{1}/\delta_{2}))$. Then there exist

  * •

a rigid space $X$ over $L$ which is also a vector bundle of dimension

$\dim_{L}(H^{1}_{\phi,\Gamma}(\mathcal{R}_{L}(\delta_{1}/\delta_{2})))$

over an appropriate neighbourhood of $(\delta_{1},\delta_{2})$ in
$\mathcal{T}^{2}$;

  * •

a family of $(\phi,\Gamma)$-modules $\mathbb{D}$ over $X$;

  * •

a map
$(\widetilde{\delta}_{1},\widetilde{\delta}_{2}):X\rightarrow\mathcal{T}^{2}$

such that there are

  1. (1)

$x\in X$ such that $\mathbb{D}\widehat{\otimes}k(x)=D$ and
$(\widetilde{\delta}_{1},\widetilde{\delta}_{2})\widehat{\otimes}k(x)=(\delta_{1},\delta_{2})$;

  2. (2)

a Zariski-open dense subset $U\xhookrightarrow{j}X$ such that
$j^{*}\mathbb{D}$ has a filtration of sub-$(\phi,\Gamma)$-modules with graded
pieces $\mathcal{R}_{U}(j^{*}\widetilde{\delta}_{1})$ and
$\mathcal{R}_{U}(j^{*}\widetilde{\delta}_{2})$ and
$(j^{*}\widetilde{\delta}_{1},j^{*}\widetilde{\delta}_{2})\in\mathcal{T}^{\mathrm{reg}}_{2}(U)$.

###### Proof.

Let $\mathcal{U}=\mathrm{Sp}(A)\subset\mathcal{T}^{2}$ be an open affinoid
neighbourhood of $(\delta_{1},\delta_{2})$. Let
$(\widetilde{\delta_{1}},\widetilde{\delta_{2}})\in\mathcal{T}^{2}(A)$ be the
characters corresponding to $\mathcal{U}\hookrightarrow\mathcal{T}^{2}$. Since
$\mathcal{T}^{+}$ is a closed subspace of $\mathcal{T}$, we can choose
$\mathcal{U}$ in such a way so that
$\widetilde{\delta_{1}}/\widetilde{\delta_{2}}\widehat{\otimes}k(u)\notin\mathcal{T}^{+}(k(u))$
for any $u\in\mathcal{U}$. By Theorem 1.15, we have
$H^{2}_{\phi,\Gamma}(\mathcal{R}_{A}(\widetilde{\delta_{1}}/\widetilde{\delta_{2}}))=0$,
in particular it is a locally free sheaf over $\mathcal{U}$. By Theorem 4.2,
there exists a vector bundle $Y\rightarrow\mathcal{U}$ of rank
$\dim_{L}H^{1}(\mathcal{R}_{L}(\delta_{1}/\delta_{2}))$ and a family of
$(\phi,\Gamma)$-modules $\mathbb{D}^{\prime}$ on $Y$ such that there exists a
surjective map

$\displaystyle Y(k(\delta_{1},\delta_{2}))=Y(L)$
$\displaystyle\twoheadrightarrow
H^{1}_{\phi,\Gamma}(\mathcal{R}_{A}(\widetilde{\delta_{1}}/\widetilde{\delta_{2}})\widehat{\otimes}_{A}L)=\mathrm{Ext}^{1}_{\phi,\Gamma}(\mathcal{R}_{L}(\delta_{2}),\mathcal{R}_{L}(\delta_{1}))$
$\displaystyle y$ $\displaystyle\mapsto y^{*}\mathbb{D}^{\prime}.$

As the $(\phi,\Gamma)$-module $D$ is an extension of
$\mathcal{R}_{L}(\delta_{2})$ by $\mathcal{R}_{L}(\delta_{1})$, there is a
point $y\in Y(L)$ such that $y^{*}\mathbb{D}^{\prime}=D$. Let us define the
rigid $L$-space

${X\coloneqq
Y\times\mathrm{Sp}(L)}$${\mathrm{Sp}(L)}$${Y}$${\mathrm{Sp}(\mathbb{Q}_{p})}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$

and let $\mathbb{D}\coloneqq p_{1}^{*}\mathbb{D}^{\prime}$. Let $x$ be any
point of $X$ such that $p_{1}(x)=y$, so that $x^{*}\mathbb{D}=(p_{1}\circ
x)^{*}\mathbb{D}^{\prime}=y^{*}\mathbb{D}^{\prime}=D$. Now we only have to
show that there is a Zariski-open dense subset $U\xhookrightarrow{j}Y$ such
that $j^{*}\mathbb{D}$ has regular parameters. Let
$\mathcal{U}^{\mathrm{reg}}:=\mathcal{U}\cap\mathcal{T}_{2}^{\mathrm{reg}}$
which is a Zariski-open dense in $\mathcal{U}$. The rigid space

${U\coloneqq
X\times_{\mathcal{U}}\mathcal{U}^{\mathrm{reg}}}$${\mathcal{U}^{\mathrm{reg}}}$${X}$${\mathcal{U}}$$\scriptstyle{q_{2}}$$\scriptstyle{q_{1}}$$\scriptstyle{u}$$\scriptstyle{v}$

is Zariski-open dense in $X$: in fact $Y$ is a vector bundle over
$\mathcal{U}$, so in particular the map $v$ is open and preimages of dense
subsets through an open map are dense. Moreover $q_{1}^{*}\mathbb{D}$ has
parameters
$u^{*}(\widetilde{\delta}_{1},\widetilde{\delta}_{2})\in\mathcal{T}^{2}(\mathcal{U}^{\mathrm{reg}})$,
hence regular. ∎

######  Proposition 4.5.

Let $(\delta_{1},\dots,\delta_{n+1})\in\mathcal{T}^{n+1}(L)$ such that
$\delta_{i}/\delta_{j}\notin\mathcal{T}^{+}(L)$ for $i<j$ and let $D_{n+1}\in
H^{1}_{\phi,\Gamma}(D_{n}(\delta_{n+1}^{-1}))$ where $D_{n}$ is a triangulable
$(\phi,\Gamma)$-module with parameters
$(\delta_{1},\dots,\delta_{n})\in\mathcal{T}^{n}(L)$. Let us denote by $D_{i}$
the $i$-th piece of the filtration with graded pieces
$\mathcal{R}_{L}(\delta_{1}),\dots,\mathcal{R}_{L}(\delta_{n})$. Then there
exist

  * •

a rigid space $X_{2}$ over $L$ which is also a vector bundle over an
appropriate neighbourhood $\mathcal{U}_{1}\times\mathcal{U}_{2}$ of
$(\delta_{1},\delta_{2})$ in $\mathcal{T}^{2}$;

  * •

rigid spaces $X_{i}$ over $L$ which are also vector bundles over
$X_{i-1}\times\mathcal{U}_{i}$, where $\mathcal{U}_{i}$ is an appropriate
neighbourhood of $\delta_{i}$ in $\mathcal{T}$ for all $3\leqslant i\leqslant
n+1$;

  * •

families of $(\phi,\Gamma)$-modules $\mathbb{D}_{i}$ over $X_{i}$ for all
$2\leqslant i\leqslant n+1$;

  * •

maps
$(\widetilde{\delta}_{1},\dots,\widetilde{\delta}_{i}):X_{i}\rightarrow\mathcal{T}^{i}$
for all $2\leqslant i\leqslant n+1$

such that there are

  1. (1)

$x_{i}\in X_{i}$ such that $\mathbb{D}_{i}\widehat{\otimes}k(x_{i})=D_{i}$ and
$(\widetilde{\delta}_{1},\dots,\widetilde{\delta}_{i})\widehat{\otimes}k(x_{i})=(\delta_{1},\dots,\delta_{i})$
for all $2\leqslant i\leqslant n+1$;

  2. (2)

Zariski open dense subsets $U_{i}\xhookrightarrow{j_{i}}X_{i}$ such that
$j_{i}^{*}\mathbb{D}_{i}$ has a filtration of sub-$(\phi,\Gamma)$-modules with
graded pieces
$\mathcal{R}_{U_{i}}(\widetilde{\delta_{1}}),\dots,\mathcal{R}_{U_{i}}(\widetilde{\delta}_{i})$
and
$j_{i}^{*}(\widetilde{\delta_{1}},\dots,\widetilde{\delta_{i}})\in\mathcal{T}^{\mathrm{reg}}_{i}(U_{i})$
for all $2\leqslant i\leqslant n+1$.

###### Proof.

We proceed by induction on $i$. Proposition 4.4 shows that there exist

  * •

a rigid space $X_{2}$ over $L$ which is also a vector bundle over an
appropriate neighbourhood $\mathcal{U}_{1}\times\mathcal{U}_{2}$ of
$(\delta_{1},\delta_{2})$ in $\mathcal{T}^{2}$;

  * •

a family of $(\phi,\Gamma)$-modules $\mathbb{D}_{2}$ over $X_{2}$;

  * •

a map
$(\widetilde{\delta}_{1},\widetilde{\delta}_{2}):X_{2}\rightarrow\mathcal{T}^{2}$

such that there are

  1. (1)

$x_{2}\in X_{2}$ such that $\mathbb{D}_{2}\widehat{\otimes}k(x_{2})=D_{2}$ and
$(\widetilde{\delta}_{1},\widetilde{\delta}_{2})\widehat{\otimes}k(x_{2})=(\delta_{1},\delta_{2})$;

  2. (2)

a Zariski open dense subset $U_{2}\xhookrightarrow{j_{2}}X_{2}$ such that
$j_{2}^{*}\mathbb{D}_{2}$ has a filtration of sub-$(\phi,\Gamma)$-modules with
graded pieces
$\mathcal{R}_{U_{2}}(\widetilde{\delta_{1}}),\mathcal{R}_{U_{2}}(\widetilde{\delta}_{2})$
and
$j_{2}^{*}(\widetilde{\delta_{2}},\widetilde{\delta_{2}})\in\mathcal{T}^{\mathrm{reg}}_{2}(U_{2})$.

Hence the statement is true for $i=2$. Assume now that the statement of the
Proposition is true for any $i<n+1$. Let $X_{n}$ be the vector bundle over
$X_{n-1}\times\mathcal{U}_{n}$ as described in the statement of the
Proposition, which means in particular that it is a vector bundle over the
neighbourhood
$\widetilde{\mathcal{U}}_{n}\coloneqq\mathcal{U}_{1}\times\dots\times\mathcal{U}_{n}$
of $(\delta_{1},\dots,\delta_{n})$ inside $\mathcal{T}^{n}$; let moreover
$\mathbb{D}_{n}$ be the family of $(\phi,\Gamma)$-modules over $X_{n}$
satisfying the properties above. Let $\mathcal{U}_{n+1}$ be a neighbourhood of
$\delta_{n+1}$ in $\mathcal{T}$ and similarly let $\widetilde{\delta}_{n+1}$
be the character corresponding to $\mathcal{U}_{n+1}\rightarrow\mathcal{T}$.
Let us define the open neighbourhood
$\widetilde{\mathcal{U}}_{n+1}\coloneqq\widetilde{\mathcal{U}}_{n}\times\mathcal{U}_{n+1}$
of $(\delta_{1},\dots,\delta_{n+1})$ in $\mathcal{T}^{n+1}$. We intersect
$\widetilde{\mathcal{U}}_{n+1}$ with the open neighbourhood of
$(\delta_{1},\dots,\delta_{n+1})$ defined as in Lemma 1.16. In particular we
can assume that
$\widetilde{\delta}_{i}/\widetilde{\delta}_{n+1}\widehat{\otimes}k(u)\notin\mathcal{T}^{+}(k(u))$
for all $i<n+1$ and for all $u\in\widetilde{\mathcal{U}}_{n+1}$. Let us
consider the space $X_{n}\times\mathcal{U}_{n+1}$, which is a vector bundle
over $\widetilde{\mathcal{U}}_{n+1}$. For simplicity of notation, we denote by
$\mathbb{D}_{n}$ and
$\mathcal{R}_{\mathcal{U}_{n+1}}(\widetilde{\delta}_{n+1})$ the pullback of
these two $(\phi,\Gamma)$-modules to $X_{n}\times\mathcal{U}_{n+1}$. By the
way we have chosen the neighbourhood $\mathcal{U}_{n+1}$, we have that the
sheaf $H^{2}_{\phi,\Gamma}(\mathbb{D}_{n}(\widetilde{\delta}_{n+1}^{-1}))$ is
locally free over $X_{n}\times\mathcal{U}_{n+1}$. Hence by Theorem 4.2, there
is a vector bundle $Y_{n+1}\rightarrow X_{n}\times\mathcal{U}_{n+1}$ and a
family of $(\phi,\Gamma)$-modules $\mathbb{D}^{\prime}_{n+1}$ on $Y_{n+1}$
such that there exists a surjective map

$\displaystyle Y_{n+1}(k(\delta_{1},\dots,\delta_{n+1}))=Y_{n+1}(L)$
$\displaystyle\twoheadrightarrow
H^{1}_{\phi,\Gamma}(\mathbb{D}_{n}(\widetilde{\delta}_{n+1}^{-1})\widehat{\otimes}L)=\mathrm{Ext}^{1}_{\phi,\Gamma}(\mathcal{R}_{L}(\delta_{n+1}),D_{n})$
$\displaystyle y$ $\displaystyle\mapsto y^{*}\mathbb{D}^{\prime}_{n+1}.$

This implies that there is $y_{n+1}\in Y_{n+1}(L)$ such that
$y_{n+1}^{*}\mathbb{D}^{\prime}_{n+1}=D_{n+1}$. It is now sufficient to
condider the space

${X_{n+1}\coloneqq
Y_{n+1}\times\mathrm{Sp}(L)}$${\mathrm{Sp}(L)}$${Y_{n+1}}$${\mathrm{Sp}(\mathbb{Q}_{p})}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$

and let $\mathbb{D}_{n+1}\coloneqq p_{1}^{*}\mathbb{D}^{\prime}_{n+1}$. Any
point $x_{n+1}\in X_{n+1}(L)$ such that $p_{1}(x_{n+1})=y_{n+1}$ has the
property that $x_{n+1}^{*}\mathbb{D}_{n+1}=(p_{1}\circ
x_{n+1})^{*}\mathbb{D}_{n+1}^{\prime}=y_{n+1}^{*}\mathbb{D}_{n+1}^{\prime}=D_{n+1}$.
Finally, let
$\widetilde{\mathcal{U}}_{n+1}^{\mathrm{reg}}:=\widetilde{\mathcal{U}}_{n+1}\cap\mathcal{T}_{n+1}^{\mathrm{reg}}$
which is a Zariski-open dense in $\widetilde{\mathcal{U}}_{n+1}$. The rigid
space

${U_{n+1}\coloneqq
X_{n+1}\times_{\widetilde{\mathcal{U}}_{n+1}}\widetilde{\mathcal{U}}_{n+1}^{\mathrm{reg}}}$${\widetilde{\mathcal{U}}_{n+1}^{\mathrm{reg}}}$${X_{n+1}}$${\widetilde{\mathcal{U}}_{n+1}}$$\scriptstyle{q_{2}}$$\scriptstyle{q_{1}}$$\scriptstyle{u}$$\scriptstyle{v}$

is Zariski-open dense in $X_{n+1}$ (in fact $Y_{n+1}$ is a vector bundle over
$\widetilde{\mathcal{U}}_{n+1}$, making the map $v$ open). Moreover
$q_{1}^{*}\mathbb{D}_{n+1}$ has parameters
$u^{*}(\widetilde{\delta}_{1},\dots,\widetilde{\delta}_{n+1})\in\mathcal{T}^{n+1}(\widetilde{\mathcal{U}}_{n+1}^{\mathrm{reg}})$,
therefore they are regular. ∎

###### Proof of Theorem A.

Using Theorem 3.3, the Propositions 4.4 and 4.5 above prove that for any
trianguline representation $\rho:G_{\mathbb{Q}_{p}}\rightarrow
GL_{n}(\mathcal{O}_{L})$ satisfying the hypothesis of Theorem A, we have
$(\rho,(\delta_{1},\dots,\delta_{n}))\in
X_{\mathrm{tri}}^{\square}(\overline{\rho})(L)$. ∎

##  5\. Some preparation

Before approaching the proof of Theorem B, we need some more preparation.

###  5.1. The Beauville-Laszlo Theorem

The results presented in this section are mainly taken from Section 5.2 in
[27]. Fix $r\in p^{\mathbb{Q}}\cap[0,1)$. Then there exists
$m_{0}=m_{0}(r)\in\mathbb{N}$ (depending on $r$) such that for every
$m\geqslant m_{0}$, the point $1-\zeta_{p^{m}}$ lies in the half open annulus
$\mathbb{B}^{r}$, where $\zeta_{p^{m}}$ is a $p^{m}$-th root of unity. This
means that the minimal polynomial of $1-\zeta_{p^{m}}$ over $\mathbb{Q}_{p}$
defines a point $\alpha_{m}$ in $\mathbb{B}^{r}$ with residue field
$K_{m}\coloneqq\mathbb{Q}_{p}(\zeta_{p^{m}})$. Therefore for all $m\geqslant
m_{0}$, there exists a surjective morphism

$\mathcal{R}_{\mathbb{Q}_{p}}^{r}\twoheadrightarrow K_{m}.$

The element $t\coloneqq\mathrm{log}(1+T)$ is a uniformizer of the completed
local ring at each $\alpha_{m}$; hence we have that
$\widehat{(\mathcal{R}^{r}_{\mathbb{Q}_{p}})}_{\alpha_{m}}$ is isomorphic to a
power series ring in one variable and with coefficients in $K_{m}$. Therefore
this gives a map

$\mathcal{R}_{\mathbb{Q}_{p}}^{r}\rightarrow K_{m}\llbracket t\rrbracket.$

Let us then fix $m\geqslant m_{0}$ and let us assume that $L$ contains
$K_{m}$. For any affinoid $L$-algebra $A$, we have a morphism

$\iota_{r,m}:\mathcal{R}^{r}_{A}=\mathcal{R}^{r}\widehat{\otimes}_{\mathbb{Q}_{p}}A\rightarrow
K_{m}\llbracket
t\rrbracket\otimes_{\mathbb{Q}_{p}}A\cong\prod_{K_{m}\hookrightarrow
L}A\llbracket t\rrbracket.$

Let $r\leqslant s<1$ such that $s\in p^{\mathbb{Q}}\cap[0,1)$.

######  Notation.

From now on for a ring $R$, we denote by $\mathrm{Mod}(R)$ the category of
modules over $R$.

######  Definition 5.1.

We define

$\mathrm{Mod}(\mathcal{R}_{A}^{[r,s]}[1/t])\times_{\mathrm{Mod}(\prod
A((t)))}\mathrm{Mod}(\prod_{K_{m}\hookrightarrow L}A\llbracket t\rrbracket)$

to be the category of glueing data whose objects are tuples
$(M^{[r,s]},\Lambda^{[r,s]},f^{[r,s]})$ where

  * •

$M^{[r,s]}\in\mathrm{Mod}(\mathcal{R}_{A}^{[r,s]}[1/t])$;

  * •

$\displaystyle\Lambda^{[r,s]}\in\mathrm{Mod}(\prod_{K_{m}\hookrightarrow
L}A\llbracket t\rrbracket)$;

  * •

$f^{[r,s]}:M^{[r,s]}\otimes_{\mathcal{R}_{A}^{[r,s]}[1/t]}\prod_{K_{m}\hookrightarrow
L}A((t))\xrightarrow{\sim}\Lambda^{[r,s]}[1/t]$

is an isomorphism of $\displaystyle\prod_{K_{m}\hookrightarrow
L}A((t))$-modules.

The following Theorem can be found in [27, (5.4)] and it is an analogue for
$(\phi,\Gamma)$-modules of the Theorem of Beauville-Laszlo (cf. [9]).

######  Theorem 5.2.

Let us assume that $\alpha_{m}\in\mathbb{B}^{[r,s]}$ and no other point
$\alpha_{n}$ obtained from the minimal polynomial of $1-\zeta_{p^{n}}$ over
$\mathbb{Q}_{p}$ is contained in $\mathbb{B}^{[r,s]}$ (for example
$s=r^{1/p}$). There is an equivalence of categories

$\displaystyle\mathrm{Mod}(\mathcal{R}_{A}^{[r,s]})\xrightarrow{\sim}$
$\displaystyle\mathrm{Mod}(\mathcal{R}_{A}^{[r,s]}[1/t])\times_{\mathrm{Mod}(\prod
A((t)))}\mathrm{Mod}(\prod_{K_{m}\hookrightarrow L}A\llbracket t\rrbracket)$
$\displaystyle D^{[r,s]}\mapsto$
$\displaystyle(D^{[r,s]}[1/t],D^{[r,s]}\otimes_{\mathcal{R}_{A}^{[r,s]}}\prod_{K_{m}\hookrightarrow
L}A\llbracket t\rrbracket,f^{[r,s]}),$

where the isomorphism $f^{[r,s]}$ of $\prod_{K_{m}\hookrightarrow
L}A((t))$-modules is given by

$(D^{[r,s]}\otimes_{\mathcal{R}_{A}^{[r,s]}}\prod_{K_{m}\hookrightarrow
L}A\llbracket t\rrbracket)[1/t]\cong
D^{[r,s]}[1/t]\otimes_{\mathcal{R}_{A}^{[r,s]}[1/t]}\prod_{K_{m}\hookrightarrow
L}A((t))\\\ $ (5.1)

via
$t^{i}(m\otimes_{\mathcal{R}_{A}^{[r,s]}}1)\mapsto(m\otimes_{\mathcal{R}_{A}^{[r,s]}}t^{i})\otimes_{\mathcal{R}_{A}^{[r,s]}[1/t]}1$
for $m\in D^{[r,s]}$ and $i\in\mathbb{Z}$. The inverse of this equivalence is
given by

$\ker(\overline{f^{[r,s]}})\mapsfrom(M^{[r,s]},\Lambda^{[r,s]},f^{[r,s]}),$

where

$\overline{f^{[r,s]}}:M^{[r,s]}\rightarrow
M^{[r,s]}\otimes_{\mathcal{R}_{A}^{[r,s]}[1/t]}\prod_{K_{m}\hookrightarrow
L}A((t))\xrightarrow{f^{[r,s]}}\Lambda^{[r,s]}[1/t]\twoheadrightarrow\Lambda^{[r,s]}[1/t]/\Lambda^{[r,s]}.$

###  5.2. $\phi$-modules over $\mathcal{R}_{A}^{r}$

We’ll see that any $(\phi,\Gamma)$-module over the Robba ring can also be
represented by a $(\phi,\Gamma)$ -module over the subring $\mathcal{R}^{r}$
for some appropriate $r$ depending on the module. In this section we define
the notion of $(\phi,\Gamma)$-modules over $\mathcal{R}^{r}$ and see some
properties. Let $0\leqslant r\leqslant s<1$ such that $r,s\in
p^{\mathbb{Q}}\cap[0,1)$ and let $A$ be an affinoid $L$-algebra. The rings
$\mathcal{R}_{A}^{r}$ and $\mathcal{R}_{A}^{[r,s]}$ are both equipped with a
Frobenius morphism

$\displaystyle\phi:$
$\displaystyle\mathcal{R}_{A}^{[r,s]}\rightarrow\mathcal{R}_{A}^{[r^{1/p},s^{1/p}]},$
$\displaystyle\phi:$
$\displaystyle\mathcal{R}_{A}^{r}\rightarrow\mathcal{R}_{A}^{r^{1/p}}$

which is simply the restriction of the Frobenius on the Robba ring
$\mathcal{R}_{A}$. Let us define the restriction map

$\mathrm{res}^{r}_{r^{1/p}}:\mathbb{B}_{A}^{r^{1/p}}\hookrightarrow\mathbb{B}_{A}^{r}.$

######  Definition 5.3.

  * •

A $\mathcal{R}^{r}_{A}$-module $D$ is called a _$\phi$ -module_ if $D$ is
projective, of finite presentation and there exists an isomorphism

$\phi^{*}D\cong(\mathrm{res}^{r}_{r^{1/p}})^{*}D.$

  * •

A _$(\phi,\Gamma)$ -module over $\mathcal{R}^{r}_{A}$_ is a $\phi$-module $D$
over $\mathcal{R}^{r}_{A}$ endowed with a continuous action of $\Gamma$
commuting with $\phi$.

######  Definition 5.4.

Let $A$ be an affinoid $L$-algebra and let $r_{0}\in p^{\mathbb{Q}}\cap[0,1)$.

  * (i)

A _coherent sheaf_ on $\mathcal{R}_{A}^{r_{0}}$ consists of a family of finite
$\mathcal{R}_{A}^{[r,s]}$-modules $D^{[r,s]}$ for all $r_{0}\leqslant
r\leqslant s<1$ together with isomorphisms

$D^{[r^{\prime},s^{\prime}]}\cong
D^{[r,s]}\otimes_{\mathcal{R}_{A}^{[r,s]}}\mathcal{R}_{A}^{[r^{\prime},s^{\prime}]}$

for all $r_{0}\leqslant r\leqslant r^{\prime}\leqslant s^{\prime}\leqslant
s<1$ satisfying the cocycle conditions.

  * (ii)

A _vector bundle_ on $\mathcal{R}_{A}^{r_{0}}$ is a coherent sheaf
$(D^{[r,s]})_{r_{0}\leqslant r\leqslant s<1}$ on $\mathcal{R}_{A}^{r_{0}}$
such that each $D^{[r,s]}$ is a projective module over
$\mathcal{R}_{A}^{[r,s]}$.

The following Theorem is due to [22, Proposition 2.2.7.].

######  Theorem 5.5.

For every affinoid $L$-algebra $A$ and for any $r_{0}\in
p^{\mathbb{Q}}\cap[0,1)$, there exists an equivalence of categories

$\displaystyle\\{\phi\text{-modules over
}\mathcal{R}_{A}^{r_{0}}\\}\xrightarrow{\sim}$
$\displaystyle\left\\{\begin{array}[]{l}\text{vector bundles
}(D^{[r,s]})_{r_{0}\leqslant r\leqslant s<1}\text{ on
}\mathcal{R}_{A}^{r_{0}}\text{ together with }\\\ \phi^{*}D^{[r,s]}\cong
D^{[r^{1/p},s^{1/p}]}\text{ compatible with restrictions}\end{array}\right\\}$
$\displaystyle D^{r_{0}}\mapsto$
$\displaystyle(D^{r_{0}}\otimes_{\mathcal{R}_{A}^{r_{0}}}\mathcal{R}_{A}^{[r,s]})_{r_{0}\leqslant
r\leqslant s<1}.$

As already mentioned in the beginning of this section, any
$(\phi,\Gamma)$-module $D$ over the Robba ring can be seen as a
$(\phi,\Gamma)$-module over $\mathcal{R}^{r}$ for some $r$ depending on $D$.
More precisely, we have the following Theorem proved in [2, Lemma 2.2.8].

######  Theorem 5.6.

Let $D$ be any $(\phi,\Gamma)$-module over $\mathcal{R}_{L}$. There exists a
unique $r(D)\in p^{\mathbb{Q}}\cap[0,1)$ such that for all $r\geqslant r(D)$,
there is a unique $\phi$-module $D^{r}$ over $\mathcal{R}_{L}^{r}$ such that
$D=D^{r}\otimes_{\mathcal{R}_{L}^{r}}\mathcal{R}_{L}$. Moreover $D^{r}$ is
stable under the action of $\Gamma$ induced by $D$ on
$D^{r}\otimes_{\mathcal{R}_{L}^{r}}\mathcal{R}_{L}$.

More generally, let $\mathbb{D}$ be a family of free $(\phi,\Gamma)$-modules
over a rigid affinoid space $\mathrm{Sp}(A)$ and let $\\{e_{1},\dots,e_{d}\\}$
be a basis of $\mathbb{D}$. Let us consider the matrix $M$ of $\Phi$ in this
basis. Then there exists $r(D)\in p^{\mathbb{Q}}\cap[0,1)$ such that $M\in
GL_{d}(\mathcal{R}_{A}^{r})$ for all $r\geqslant r(D)$. We can then take
$\mathbb{D}^{r}\coloneqq\oplus_{i=1}^{d}\mathcal{R}_{A}^{r}e_{i}$ and we have
the following Remark.

######  Remark 5.7.

Let $\mathbb{D}$ be a family of locally free $(\phi,\Gamma)$-modules over a
rigid space $X$. Then there exists $r\in p^{\mathbb{Q}}\cap[0,1)$ and a
$\phi$-module $\mathbb{D}^{r}$ over $\mathcal{R}_{X}^{r}$ such that
$\mathbb{D}^{r}\otimes_{\mathcal{R}_{X}^{r}}\mathcal{R}_{X}=\mathbb{D}$.

###  5.3. From $(\phi,\Gamma)$-modules over $\mathcal{R}$ to
$(\phi,\Gamma)$-modules over $\mathcal{R}[1/t]$

We will show how to view a $(\phi,\Gamma)$-module over $\mathcal{R}_{L}$ as a
$(\phi,\Gamma)$-module over $\mathcal{R}_{L}[1/t]$ and a lattice without
losing any information.

Let $A$ be an affinoid $L$-algebra. First of all notice that the morphism

$\phi:\mathcal{R}_{A}\rightarrow\mathcal{R}_{A},T\mapsto(1+T)^{p}-1$

defined in (1.1) can be extended to

$\phi:\mathcal{R}_{A}[1/t]\rightarrow\mathcal{R}_{A}[1/t]$

simply by $\frac{1}{t}\mapsto\frac{1}{pt}$.

######  Definition 5.8.

A _$(\phi,\Gamma)$ -module over $\mathcal{R}_{A}[1/t]$_ is a locally free
$\mathcal{R}_{A}[1/t]$-module $M$ of finite presentation endowed with

  * •

a $\mathcal{R}_{A}[1/t]$-semilinear endomorphism $\Phi:M\rightarrow M$ such
that $\Phi:\phi^{*}M\rightarrow M$ is an isomorphism;

  * •

a continuous $\mathcal{R}_{A}[1/t]$-semilinear $\Gamma$-action commuting with
$\Phi$.

Let $(\phi,\Gamma)-\mathrm{Mod}(\mathcal{R}_{A}^{r})$ denote the category of
$(\phi,\Gamma)$-modules over $\mathcal{R}_{A}^{r}$ and
$(\phi,\Gamma)-\mathrm{Mod}(\mathcal{R}_{A}^{r}[1/t])$ the category of
$(\phi,\Gamma)$-modules over $\mathcal{R}_{A}^{r}[1/t]$.

######  Definition 5.9.

A _lattice $\Lambda$ in $A((t))^{n}$_ is a finitely generated projective
$A\llbracket t\rrbracket$-module such that

$\Lambda\otimes_{A\llbracket t\rrbracket}A((t))\cong A((t))^{n}.$

######  Definition 5.10.

For $r\in p^{\mathbb{Q}}\cap[0,1)$ and $m\in\mathbb{N}$ satisfying the
conditions presented in the beginning of §5.1, let us denote by

$(\phi,\Gamma)-\mathrm{Mod}(\mathcal{R}^{r}_{A}[1/t])\times_{\mathrm{Mod}(\prod
A((t)))^{\Gamma}}\mathrm{Mod}(\prod_{K_{m}\hookrightarrow L}A\llbracket
t\rrbracket)^{\Gamma}$

the category whose objects are tuples $(M,\Lambda,f)$ where

  * •

$M\in(\phi,\Gamma)-\mathrm{Mod}(\mathcal{R}^{r}_{A}[1/t])$;

  * •

$\displaystyle\Lambda\in\mathrm{Mod}(\prod_{K_{m}\hookrightarrow L}A\llbracket
t\rrbracket)$;

  * •

$\displaystyle
f:M\otimes_{\mathcal{R}_{A}^{r}[1/t]}\prod_{K_{m}\hookrightarrow
L}A((t))\xrightarrow{\sim}\Lambda[1/t]$ is an isomorphism of $\prod
A((t))$-modules

and $\Lambda$ is stable under the action of $\Gamma$ induced on $\Lambda[1/t]$
under $f$ from the diagonal action of $\Gamma$ on
$M\otimes_{\mathcal{R}_{A}^{r}[1/t]}\prod_{K_{m}\hookrightarrow L}A((t))$.

######  Theorem 5.11.

Let $r_{0}\in p^{\mathbb{Q}}\cap[0,1)$ such that $\mathbb{B}^{[r_{0},r_{0}]}$
does not contain any zero for $t$ (i.e. $r_{0}\neq|1-\zeta_{p^{m}}|$ for all
$m\in\mathbb{N}$). There is an equivalence of categories

$\displaystyle(\phi,\Gamma)-\mathrm{Mod}(\mathcal{R}_{A}^{r_{0}})\xrightarrow{\sim}$
$\displaystyle(\phi,\Gamma)-\mathrm{Mod}(\mathcal{R}^{r_{0}}_{A}[1/t])\times_{\mathrm{Mod}(\prod
A((t)))^{\Gamma}}\mathrm{Mod}(\prod_{K_{m}\hookrightarrow L}A\llbracket
t\rrbracket)^{\Gamma}$ (5.2) $\displaystyle D^{r_{0}}\mapsto$
$\displaystyle(D^{r_{0}}[1/t],D^{r_{0}}\otimes_{\mathcal{R}_{A}^{r_{0}}}\prod_{K_{m}\hookrightarrow
L}A\llbracket t\rrbracket,f),$

where

$f:D^{r_{0}}[1/t]\otimes_{\mathcal{R}_{A}^{r_{0}}[1/t]}\prod_{K_{m}\hookrightarrow
L}A((t))\xrightarrow{\sim}(D^{r_{0}}\otimes_{\mathcal{R}_{A}^{r_{0}}}\prod_{K_{m}\hookrightarrow
L}A\llbracket t\rrbracket)[1/t]$

via $e_{i}^{\prime}\otimes 1\mapsto e_{i}\otimes 1$ with $e_{1},\dots,e_{n}$
basis of $D^{r_{0}}$ over $\mathcal{R}_{A}^{r_{0}}$ and
$e_{1}^{\prime}=e_{1}\otimes_{\mathcal{R}_{A}^{r_{0}}}1,\dots,e_{n}^{\prime}=e_{n}\otimes_{\mathcal{R}_{A}^{r_{0}}}1$
basis of $D^{r_{0}}[1/t]$ over $\mathcal{R}_{A}^{r_{0}}[1/t]$.

###### Proof.

Let $\Lambda\coloneqq
D^{r_{0}}\otimes_{\mathcal{R}_{A}^{r_{0}}}\prod_{K_{m}\hookrightarrow
L}A\llbracket t\rrbracket$. First of all, notice that the $\Gamma$-action of
$D^{r_{0}}[1/t]$ extends diagonally to

$f:D^{r_{0}}[1/t]\otimes_{\mathcal{R}_{A}^{r_{0}}[1/t]}\prod_{K_{m}\hookrightarrow
L}A((t))\xrightarrow{\sim}\Lambda[1/t]$

and stabilizes $\Lambda$ by definition. In order to show that the above is an
equivalence, we describe the inverse functor: let $(M,\Lambda,f)$ be an object
of the right hand side of (5.2). We denote by $M_{0}$ and $\Lambda_{0}$ the
restriction of $M$ and $\Lambda$ respectively to th annulus
$\mathbb{B}^{[r_{0},r_{1}]}$, where $r_{1}\coloneqq r_{0}^{1/p}$; moreover let
us denote by $\Lambda_{i}\coloneqq\phi^{*}\Lambda_{i-1}$ and
$M_{i}\coloneqq\phi^{*}M_{i-1}$ for $i\geqslant 1$. If we let $r_{i}\coloneqq
r_{0}^{1/p^{i}}$ for $i\geqslant 1$, this yields to a family of objects

$((M_{i},\Lambda_{i},f_{i})\in\mathrm{Mod}(\mathcal{R}_{A}^{[r_{i},r_{i+1}]}[1/t])\times_{\mathrm{Mod}(\prod
A((t)))}\mathrm{Mod}(\prod_{K_{m}\hookrightarrow L}A\llbracket
t\rrbracket))_{i}$

for $i\geqslant 0$ such that $M_{i}$ is a $\phi$-module over
$\mathcal{R}_{A}^{[r_{i},r_{i+1}]}[1/t]$ endowed with an action of $\Gamma$.
Thanks to Theorem 5.2, this corresponds to a family of vector bundles
$(D^{[r_{i},r_{i+1}]})_{i\geqslant 0}$ with a $\Gamma$-action and such that
$\phi^{*}D^{[r_{i},r_{i+1}]}\cong D^{[r_{i+1},r_{i+2}]}$. We will now show
that we can glue $D^{[r_{i-1},r_{i}]}$ and $D^{[r_{i},r_{i+1}]}$ along the
intersection $\mathbb{B}^{[r_{i},r_{i}]}$ for all $i\geqslant 1$. By
assumption, we have that

$D^{[r_{i},r_{i+1}]}_{|\mathbb{B}^{[r_{i},r_{i+1}]}\setminus
V(t)}=M_{i|\mathbb{B}^{[r_{i},r_{i+1}]}\setminus V(t)}$

for all $i\geqslant 0$. By hypothesis, we have that $V(t)$ does not intersect
$\mathbb{B}^{[r_{i},r_{i}]}$ for all $i\geqslant 0$, thus

$D^{[r_{i-1},r_{i}]}_{|\mathbb{B}^{[r_{i},r_{i}]}}\cong
M_{i-1|\mathbb{B}^{[r_{i},r_{i}]}}\cong\phi^{*}M_{i-1|\mathbb{B}^{[r_{i},r_{i}]}}\cong
M_{i|\mathbb{B}^{[r_{i},r_{i}]}}\cong
D^{[r_{i},r_{i+1}]}_{|\mathbb{B}^{[r_{i},r_{i}]}}.$

Therefore we have that the family $(D^{[r_{i},r_{i+1}]})_{i\geqslant 0}$ is
compatible with restrictions and by Theorem 5.5, this gives a
$(\phi,\Gamma)$-module $D^{r_{0}}$ over $\mathcal{R}_{A}^{r_{0}}$. ∎

######  Remark 5.12.

Let $K_{\infty}:=\bigcup_{m}K_{m}$ and let
$\Gamma_{m}:=\mathrm{Gal}(K_{\infty}/K_{m})\subset\Gamma=\mathrm{Gal}(K_{\infty}/\mathbb{Q}_{p})$.
Let $(M,\Lambda,f)$ be an object of
$(\phi,\Gamma)-\mathrm{Mod}(\mathcal{R}^{r}_{A}[1/t])\times_{\mathrm{Mod}(\prod
A((t)))^{\Gamma}}\mathrm{Mod}(\prod_{K_{m}\hookrightarrow L}A\llbracket
t\rrbracket)^{\Gamma}$. Notice that the action of $\Gamma_{m}$ on

$M\otimes_{\mathcal{R}_{A}^{r}[1/t]}\prod_{K_{m}\hookrightarrow
L}A((t))\cong\prod_{K_{m}\hookrightarrow L}A((t))^{n}$

preserves each factor of the product. Moreover if we fix an embedding
$\tau:K_{m}\hookrightarrow L$ and a $\Gamma_{m}$-stable lattice
$\Lambda_{\tau}$, then it is possible to recover $\Lambda$ from it: let
$g_{m}$ be a generator of $\mathrm{Gal}(K_{m}/\mathbb{Q}_{p})$, then we have

$g_{m}^{n}\cdot\Lambda_{\tau}=\Lambda_{\tau_{n}}$

where $\tau_{n}:K_{m}\xrightarrow{g_{m}^{n}}K_{m}\xhookrightarrow{\tau}L$.
Thus

$\Lambda=\prod_{n}g_{m}^{n}\cdot\Lambda_{\tau},$

where the product ranges over
$n\in\\{0,\dots,|\mathrm{Gal}(K_{m}/\mathbb{Q}_{p})|-1\\}$ and
$\Lambda_{\tau}$ is a $\Gamma_{m}$-stable lattice in $A((t))^{n}$. Hence a
$\Gamma$-stable lattice in $\prod A((t))^{n}$ is equivalent to a
$\Gamma_{m}$-stable lattice in one of the factors $A((t))^{n}$, as the
$\Gamma$-action translates the lattice to the other factors.

Notice furthermore that we have the following bijection.

######  Proposition 5.13.

Let $M\in(\phi,\Gamma)-\mathrm{Mod}(\mathcal{R}_{A}[1/t])$ for some affinoid
algebra $A$. Let $r\in[0,1)\cap\mathbb{Q}$ such that $\mathbb{B}^{[r,r]}$ does
not contain any zero for $t$ and such that there exists a
$(\phi,\Gamma)$-module $M^{r}$ over $\mathcal{R}_{A}^{r}[1/t]$ such that
$M^{r}\otimes_{\mathcal{R}_{A}^{r}[1/t]}\mathcal{R}_{A}[1/t]=M$ (this is
always possible by Theorem 5.6) and let us take $m\in\mathbb{N}$ big enough so
that there is a surjective morphism
$\mathcal{R}_{\mathbb{Q}_{p}}^{r}\twoheadrightarrow K_{m}$ as explained at the
beginning of Section 5.1. Let us assume that $L$ is big enough so that
$K_{m}\subset L$ (we can always do that thanks to Lemma 3.4) and let us fix an
embedding $\tau:K_{m}\hookrightarrow L$. Let $\mathrm{Gr}_{M}^{\Gamma}$ be the
set

$\mathrm{Gr}_{M}^{\Gamma}\coloneqq\left\\{\Lambda\colon\Lambda\text{ is a
}\Gamma_{m}\text{-stable lattice of
}M^{r}\otimes_{\mathcal{R}_{A}^{r}[1/t],\tau}A((t))\right\\}$ (5.3)

and let $M^{(\phi,\Gamma)}$ be the set

$M^{(\phi,\Gamma)}\coloneqq\\{D\in(\phi,\Gamma)-\mathrm{Mod}(\mathcal{R}_{A})\colon
D\subset M\text{ and }D[1/t]=M\\}.$ (5.4)

Then the functor described in Theorem 5.11 induces the bijection

$\displaystyle M^{(\phi,\Gamma)}$
$\displaystyle\rightarrow\mathrm{Gr}_{M}^{\Gamma}$ $\displaystyle D$
$\displaystyle\mapsto(D^{r}\otimes_{\mathcal{R}_{A}^{r},\tau}A\llbracket
t\rrbracket),$

where $D^{r}$ is a $(\phi,\Gamma)$-module over $\mathcal{R}_{A}^{r}$ such that
$D=D^{r}\otimes_{\mathcal{R}_{A}^{r}}\mathcal{R}_{A}$.

######  Remark 5.14.

Notice that for a $(\phi,\Gamma)$-module $D\in M^{(\phi,\Gamma)}$, we have a
morphism of $(\phi,\Gamma)$-modules over $\mathcal{R}_{A}$

$\iota:D\hookrightarrow
M\xrightarrow{\mathrm{i}d}M^{r}\otimes_{\mathcal{R}_{A}^{r}[1/t]}\mathcal{R}_{A}[1/t],$

whose image is obviously $D$ and of the form
$D^{r}\otimes_{\mathcal{R}_{A}^{r}}\mathcal{R}_{A}$ for some
sub-$\mathcal{R}_{A}^{r}$-module $D^{r}$ of $M^{r}$. Let $\gamma$ be a
topological generator of $\Gamma$. The fact that $\iota$ is a morphism of
$(\phi,\Gamma)$-modules implies that the image of $D^{r}$ through $\gamma$ is
contained in $D^{r}\otimes_{\mathcal{R}_{A}^{r}}\mathcal{R}_{A}$, therefore
(after choosing a basis for $D^{r}$) the matrix of $\gamma$ has coefficients
in $\mathcal{R}_{A}$. At the same time, the module $M^{r}$ is stable under
$\gamma$ by hypothesis, thus the matrix of $\gamma$ in the same basis has
coefficients in $\mathcal{R}_{A}^{r}[1/t]$. This implies that the coefficients
of the matrix of the action of $\gamma$ on $D^{r}$ has coefficients in
$\mathcal{R}^{r}_{A}$ and therefore $D^{r}$ is $\Gamma$-stable. The fact that
$M^{r}$ is a $(\phi,\Gamma)$-module over $\mathcal{R}^{r}_{A}[1/t]$ implies
that $\phi^{*}D^{r}\cong(\mathrm{res}_{r}^{r^{1/p}})^{*}D^{r}$. Thus we have
that such a $D^{r}$ is a $(\phi,\Gamma)$-module over $\mathcal{R}_{A}^{r}$ and
$D=D^{r}\otimes_{\mathcal{R}_{A}^{r}}\mathcal{R}_{A}$.

###### Proof.

Let

$\mathcal{A}\coloneqq\left\\{(\Lambda,f)\colon\begin{array}[]{c}\Lambda\text{
is a }\Gamma_{m}\text{-stable projective }A\llbracket t\rrbracket\text{-module
and}\\\
f:\Lambda[1/t]\xrightarrow{\sim}M^{r}\otimes_{\mathcal{R}_{A}^{r}[1/t],\tau}A((t))\text{
isomorphism of }A((t))\text{-modules}\end{array}\right\\},$

where the action of $\Gamma_{m}$ on $\Lambda[1/t]$ is induced by the
isomorphism $f$. Let moreover

$\mathcal{B}\coloneqq\left\\{(D^{r},g)\colon\begin{array}[]{c}D^{r}\text{ is a
}(\phi,\Gamma)\text{-module over }\mathcal{R}_{A}^{r}\text{ and}\\\
g:D^{r}[1/t]\xrightarrow{\sim}M^{r}\text{ isomorphism of
}(\phi,\Gamma)\text{-modules over
}\mathcal{R}_{A}^{r}[1/t]\end{array}\right\\}.$ (5.5)

By Remark 5.12 and since the functor of Theorem 5.11 is an equivalence, we
obviously have that the functor induces a bijection between the sets of
isomorphism classes of the above two sets. If we denote by
$\widetilde{\mathcal{A}}$ and $\widetilde{\mathcal{B}}$ the sets of
isomorphism classes of $\mathcal{A}$ and $\mathcal{B}$ respectively, we just
have to show that we have bijections

$\begin{array}[]{lcr}\begin{aligned}
\mathrm{Gr}_{M}^{\Gamma}&\leftrightarrow\widetilde{\mathcal{A}}\\\
\Lambda&\mapsto(\Lambda,\mathrm{id}_{\Lambda[1/t]})\end{aligned}&\text{ and
}&\begin{aligned} M^{(\phi,\Gamma)}&\leftrightarrow\widetilde{\mathcal{B}}\\\
D&\mapsto(D^{r},\mathrm{id}_{D^{r}[1/t]}).\end{aligned}\end{array}$

  * •

Let’s start showing the injectivity of the first map: assume that
$\Lambda,\Lambda^{\prime}\in\mathrm{Gr}_{M}^{\Gamma}$ such that
$(\Lambda,\mathrm{id}_{\Lambda[1/t]})\cong(\Lambda^{\prime},\mathrm{id}_{\Lambda^{\prime}[1/t]})$.
This means that there is an isomorphism
$\xi:\Lambda\xrightarrow{\sim}\Lambda^{\prime}$ of $A\llbracket
t\rrbracket$-modules such that the diagram

${\Lambda}$${\Lambda^{\prime}}$${\Lambda[1/t]}$${M^{r}\otimes_{\mathcal{R}_{A}^{r}[1/t],\tau}A((t))}$${\Lambda^{\prime}[1/t]}$$\scriptstyle{\xi}$${=}$${=}$

commutes. It is then obvious that $\xi$ must be the identity and so
$\Lambda=\Lambda^{\prime}$. As for surjectivity, let
$(\Lambda,f)\in\widetilde{\mathcal{A}}$; notice then that $f(\Lambda)\subset
M^{r}\otimes_{\mathcal{R}_{A}^{r}[1/t],\tau}A((t))$ is a lattice and it is
stable under the action of $\Gamma_{m}$ induced by $M^{r}$ by definition of
the $\Gamma$-action on $\Lambda$ and by the fact that $\Lambda$ is stable
under $\Gamma_{m}$. Finally, we have that
$(\Lambda,f)\xrightarrow{f}(f(\Lambda),\mathrm{id}_{f(\Lambda)[1/t]})$ is an
isomorphism in the category $\mathcal{A}$.

  * •

The bijectivity of the second map is proved in an analogous way. More
precisely, let $D,D^{\prime}\in M^{(\phi,\Gamma)}$ such that
$(D^{r},\mathrm{id}_{D^{r}[1/t]})\cong(D^{\prime r},\mathrm{id}_{D^{\prime
r}[1/t]})$. In other words, we have an isomorphism
$\xi:D^{r}\xrightarrow{\sim}D^{\prime r}$ of $(\phi,\Gamma)$-modules over
$\mathcal{R}_{A}^{r}$ and a commutative diagram

${D^{r}}$${D^{\prime r}}$${D^{r}[1/t]}$${M^{r}}$${D^{\prime
r}[1/t].}$$\scriptstyle{\xi}$${=}$${=}$

Thus $D^{r}=D^{\prime r}$ and
$D=D^{r}\otimes_{\mathcal{R}_{A}^{r}}\mathcal{R}_{A}=D^{\prime
r}\otimes_{\mathcal{R}_{A}^{r}}\mathcal{R}_{A}=D^{\prime}$, hence the map is
injective. On the other hand, let $(D^{r},f)\in\widetilde{\mathcal{B}}$;
notice that the $\mathcal{R}_{A}^{r}$-module $f(D^{r})\subset M^{r}$ is stable
under the action of $\Gamma$ inherited by $M^{r}$: in fact by definition of
morphism of $(\phi,\Gamma)$-modules, we have that the action of $\Gamma$
commutes with $f$. Moreover, the fact that
$\phi^{*}M^{r}\cong(\mathrm{res}_{r}^{r^{1/p}})^{*}M^{r}$ implies that
$\phi^{*}f(D^{r})\cong(\mathrm{res}_{r}^{r^{1/p}})^{*}f(D^{r})$, therefore
$f(D^{r})$ is a $(\phi,\Gamma)$-module over $\mathcal{R}_{A}^{r}$. Obviously
$f(D^{r})[1/t]=M^{r}$, hence
$f(D^{r})\otimes_{\mathcal{R}_{A}^{r}}\mathcal{R}_{A}\in M^{(\phi,\Gamma)}$.
Finally $(D^{r},f)\xrightarrow{f}(f(D^{r}),\mathrm{id}_{f(D^{r})[1/t]})$ is an
isomorphism in the category $\mathcal{B}$, thus the map
$M^{(\phi,\Gamma)}\rightarrow\widetilde{\mathcal{B}}$ is surjective.

∎

###  5.4. The $(\phi,\Gamma)$-cohomology of $(\phi,\Gamma)$-modules over
$\mathcal{R}[1/t]$

In this section we explore some nice results of how to compute the
$(\phi,\Gamma)$-cohomology of $(\phi,\Gamma)$-modules over
$\mathcal{R}_{L}[1/t]$, which is defined using the Herr complex as in
Definition 1.8.

######  Remark 5.15.

Let $D$ be a $(\phi,\Gamma)$-module over $\mathcal{R}_{A}$ for some affinoid
$L$-algebra $A$. Since direct limits are exact, we have

$H^{i}_{\phi,\Gamma}\left(D\left[1/t\right]\right)=\lim_{\stackrel{{\scriptstyle\rightarrow}}{{n}}}H^{i}_{\phi,\Gamma}(t^{-n}D)=\lim_{\stackrel{{\scriptstyle\rightarrow}}{{n}}}H^{i}_{\phi,\Gamma}(D(x^{-n})).$

for $i=0,1,2$.

######  Lemma 5.16.

Let $\delta\in\mathcal{T}(L)$ and let $k\in\mathbb{N}$ such that
$\mathrm{wt}(\delta)\notin\\{1,\dots,k\\}$. Then

  * (i)

the morphism

$H^{1}_{\phi,\Gamma}(\mathcal{R}_{L}(\delta))\xrightarrow{\sim}H^{1}_{\phi,\Gamma}(\mathcal{R}_{L}(\delta
x^{-k}))$

is an isomorphism;

  * (ii)

$H^{0}_{\phi,\Gamma}(t^{-k}\mathcal{R}_{L}(\delta)/\mathcal{R}_{L}(\delta))=0$.

The reader can find Lemma 5.16 and its proof in [12, Théorème 2.22 (i),
Proposition 2.18 (i)].

######  Lemma 5.17.

Let $\delta\in\mathcal{T}(L)$ such that
$\mathrm{wt}(\delta)\notin\\{1,\dots,k\\}$. Then the map

$H^{0}_{\phi,\Gamma}(\mathcal{R}_{L}(\delta))\rightarrow
H^{0}_{\phi,\Gamma}(\mathcal{R}_{L}(\delta\cdot x^{-k}))$

is an isomorphism.

###### Proof.

Consider the short exact sequence

$0\rightarrow\mathcal{R}_{L}(\delta)\rightarrow
t^{-k}\mathcal{R}_{L}(\delta)\rightarrow
t^{-k}\mathcal{R}_{L}(\delta)/\mathcal{R}_{L}(\delta)\rightarrow 0,$

which induces the long exact sequence of cohomologies

$0\rightarrow H^{0}_{\phi,\Gamma}(\mathcal{R}_{L}(\delta))\rightarrow
H^{0}_{\phi,\Gamma}(t^{-k}\mathcal{R}_{L}(\delta))\rightarrow
H^{0}_{\phi,\Gamma}(t^{-k}\mathcal{R}_{L}(\delta)/\mathcal{R}_{L}(\delta))\rightarrow\dots.$

By Lemma 5.16, we have that
$H^{0}_{\phi,\Gamma}(t^{-k}\mathcal{R}_{L}(\delta)/\mathcal{R}_{L}(\delta))=0$,
which implies the wanted isomorphism. ∎

######  Proposition 5.18.

Let $D_{n}$ be a triangulable $(\phi,\Gamma)$-module over $\mathcal{R}_{L}$
with parameters $(\delta_{1},\dots\delta_{n})\in\mathcal{T}^{n}(L)$ and let
$\delta\in\mathcal{T}(L)$ such that
$\mathrm{wt}(\delta_{i}/\delta)\notin\mathbb{N}_{>0}$ for $i=1,\dots,n$. We
have

$H^{1}_{\phi,\Gamma}(D_{n}(\delta^{-1}))\xrightarrow{\sim}H^{1}_{\phi,\Gamma}(D_{n}(\delta^{-1}x^{-k})).$

is an isomorphism for all $k\in\mathbb{N}$.

###### Proof.

We proceed by induction on $n$.

  1. $n=1:$

This case is proved by Lemma 5.16.

  2. $n\geqslant 2:$

We want to show that the map

$H^{1}_{\phi,\Gamma}(D_{n}(\delta^{-1}))\rightarrow
H^{1}_{\phi,\Gamma}(D_{n}(\delta^{-1}\cdot x^{-k}))$

is an isomorphisms for all $k\in\mathbb{N}$. Let $D_{n-1}$ be the
$(\phi,\Gamma)$-module over $\mathcal{R}_{L}$ such that $D_{n}\in
H^{1}_{\phi,\Gamma}(D_{n-1}(\delta_{n}^{-1}))$. This means we have a short
exact sequence

$0\rightarrow D_{n-1}\rightarrow
D_{n}\rightarrow\mathcal{R}_{L}(\delta_{n})\rightarrow 0,$

which induces the long exact horizontal rows of the following diagram (where
all cohomologies are $(\phi,\Gamma)$-cohomologies):

${H^{0}(\mathcal{R}_{L}(\delta_{n}/\delta))}$${H^{1}(D_{n-1}(\delta^{-1}))}$${H^{1}(D_{n}(\delta^{-1}))}$${H^{1}(\mathcal{R}_{L}(\delta_{n}/\delta))}$${0}$${H^{0}(\mathcal{R}_{L}(\delta_{n}\delta^{-1}x^{-k}))}$${H^{1}(D_{n-1}(\delta^{-1}x^{-k}))}$${H^{1}(D_{n}(\delta^{-1}x^{-k}))}$${H^{1}(\mathcal{R}_{L}(\delta_{n}\delta^{-1}x^{-k}))}$${0.}$

$\scriptstyle\sim$

$\scriptstyle\sim$

$\scriptstyle\sim$

Notice first of all that, since
$\mathrm{wt}(\delta_{i}/\delta)\notin\mathbb{N}_{>0}$, we have that
$H^{2}_{\phi,\Gamma}(D_{n-1}(\delta^{-1}))=H^{2}_{\phi,\Gamma}(D_{n-1}(\delta^{-1}x^{-k}))=0$.
This explains why the two long exact sequences above are right-exact.
Furthermore the first vertical arrow is an isomorphism by Lemma 5.17, the
second vertical arrow is an isomorphism by inductive hypothesis and the last
verical arrow is an isomorphism by Lemma 5.16. We can therefore conclude by
the five lemma that the middle vertical arrow is an isomorphism.

∎

###  5.5. The affine Grassmannian

There exists a "space" with a universal family of lattices: the affine
Grassmannian. In this section we recall some basic notions about it.

Recall that a _lattice_ in $A((t))^{n}$ is a finitely generated projective
$A\llbracket t\rrbracket$-module $\Lambda\subset A((t))^{n}$ such that
$\Lambda[1/t]\cong A((t))^{n}$.

Let $n\in\mathbb{N}_{>0}$. The _affine Grassmannian_ is the functor

$\displaystyle\mathrm{Gr}_{n}:\underline{Rings}$
$\displaystyle\rightarrow\underline{Sets}$

sending a ring $A$ to the set of lattices in $A((t))^{n}$.

######  Theorem 5.19.

The affine Grassmannian is representable by an ind-projective scheme. More
precisely, let $\mathrm{Gr}_{n,i}$ be the subfunctor of the affine
Grassmannian sending a ring $A$ to the set consisting of those lattices
$\Lambda$ such that

$t^{i}A\llbracket t\rrbracket^{n}\subset\Lambda\subset t^{-i}A\llbracket
t\rrbracket^{n}$

for $i\in\mathbb{N}$. We have that $\mathrm{Gr}_{n,i}$ is representable by a
projective scheme and the affine Grassmannian is the direct limit of the
subfunctors $\mathrm{Gr}_{n,i}$.

In particular, we have that each $\mathrm{Gr}_{n,i}$ is locally of finite
type, hence we can consider their rigidification through the GAGA functor and
obtain rigid spaces $\mathrm{Gr}_{n,i}^{\mathrm{rig}}$. Hence we define

$\mathrm{Gr}_{n}^{\mathrm{rig}}\coloneqq\lim_{i\in\mathbb{N}}\mathrm{Gr}_{n,i}^{\mathrm{rig}}.$

This is the functor on rigid spaces

$\mathrm{Gr}_{n}^{\mathrm{rig}}:\underline{Rig}\rightarrow\underline{Sets}$

sending an affinoid rigid space $\mathrm{Sp}(A)$ to the set of lattices in
$A((t))^{n}$. Moreover, we denote by $\mathbb{L}$ the universal object of
$\mathrm{Gr}_{n}^{\mathrm{rig}}$.

Another description of the affine Grassmannian is through loop groups: given
an algebraic group $G$, the _loop group of $G$_ is the presheaf $LG$ defined
by $LG(R)\coloneqq G(R((t)))$ and the _positive loop group of $G$_ is the
presheaf $L^{+}G$ defined by $L^{+}G(R)\coloneqq G(R\llbracket t\rrbracket)$.
Then the affine Grassmannian is the fpqc quotient $LGL_{n}/L^{+}GL_{n}$.

In this paragraph we study a bit more the geometry of the affine Grassmannian:
let $B_{n}\subset GL_{n}$ be the Borel subgroup of upper triangular matrices
of $GL_{n}$, $T_{n}\subset B_{n}$ the maximal torus of diagonal matrices and
$U_{n}\subset B_{n}$ the unipotent radical of $B_{n}$. Let $X_{*}(T_{n})$ be
the coweight group of $T_{n}$, so the group of all algebraic homomorphisms
$\mathbb{G}_{m}\rightarrow T_{n}$, which can also be identified with tuples
$\mu=(\mu_{1},\dots,\mu_{n})\in\mathbb{Z}^{n}$. Moreover, let
$X_{*}(T_{n})^{+}$ be the group of dominant coweights of $T_{n}$; recall that
we say that a coweight $(\mu_{1},\dots,\mu_{n})\in\mathbb{Z}^{n}$ is
_dominant_ if $\mu_{1}\leqslant\mu_{2}\leqslant\dots\leqslant\mu_{n}$. For
dominant coweights $\lambda,\mu\in X_{*}(T_{n})^{+}$, we say
$\lambda\leqslant\mu$ if $\lambda-\mu$ is a sum of simple co-roots and this
defines a partial order on $X_{*}(T_{n})^{+}$. For a ring $R$ we have

$\mathrm{Gr}_{n}(R)=\coprod_{\mu\in X_{*}(T_{n})^{+}}L^{+}GL_{n}(R)\cdot
t^{\mu}\cdot L^{+}GL_{n}(R)/L^{+}GL_{n}(R),$

where $t^{\mu}$ is the image of $t$ through the coweight
$\mu:R((t))^{\times}\rightarrow T_{n}(R((t)))$. For a coweight $\mu\in
X_{*}(T_{n})$, we define the _Schubert cell_

$\mathrm{Gr}_{n}^{\mu}\coloneqq LGL_{n}^{+}\cdot t^{\mu}\cdot
LGL_{n}^{+}/LGL_{n}^{+};$

as a consequence, we have the following _Cartan decomposition_ :

$\mathrm{Gr}_{n}=\coprod_{\mu\in X_{*}(T_{n})^{+}}\mathrm{Gr}_{n}^{\mu}.$

Moreover the closure relations of the Schubert cells are given by the partial
order on dominant coweights:

$\mathrm{Gr}_{n}^{\leqslant\mu}\coloneqq\overline{\mathrm{Gr}_{n}^{\mu}}=\bigcup_{\lambda\leqslant\mu}\mathrm{Gr}_{n}^{\lambda}.$

The space $\mathrm{Gr}_{n}^{\leqslant\mu}$ is called _Schubert variety_ and
the Schubert cell $\mathrm{Gr}_{n}^{\mu}$ is open in
$\mathrm{Gr}_{n}^{\leqslant\mu}$. For a coweight $\mu\in X_{*}(T)^{+}$, let
$P_{\mu}$ be the parabolic subgroup of $GL_{n}$ corresponding to $\mu$, i.e.
the parabolic subgroup generated by $B_{n}$ and by the root subgroups
$U_{\alpha}$ for those roots $\alpha$ satisfying
$\langle\alpha,\mu\rangle\leqslant 0$. For a ring $R$, let us fix the standard
lattice $\Lambda_{0}\coloneqq R\llbracket
t\rrbracket^{n}\in\mathrm{Gr}_{n}(R)$ and a basis $e_{1},\dots,e_{n}$ of
$\Lambda_{0}$ over $R\llbracket t\rrbracket$. Denote by
$\mathrm{Fil}^{\bullet}(V)$ the standard filtration of
$V\coloneqq\Lambda_{0}[1/t]$ given by $0\subset R((t))e_{1}\subset
R((t))e_{1}\oplus R((t))e_{2}\subset\dots\subset V$. There is a natural
projection

$\displaystyle\pi_{\mu}:\mathrm{Gr}_{n}^{\mu}=L^{+}GL_{n}\cdot t^{\mu}\cdot
L^{+}GL_{n}/L^{+}GL_{n}$ $\displaystyle\twoheadrightarrow GL_{n}/P_{\mu}$
(5.6) $\displaystyle(g\cdot t^{\mu}\text{ mod }L^{+}GL_{n})$
$\displaystyle\mapsto(\bar{g}\text{ mod }P_{\mu}),$

where $\bar{g}\coloneqq g\text{ mod }t$. Recall that $GL_{n}/P_{\mu}$ is a
flag variety and for a lattice $\Lambda\in\mathrm{Gr}_{n}^{\mu}(R)$, we have
that the flag $\pi_{\mu}(\Lambda)\in GL_{n}/P_{\mu}(R)$ is given by the
filtration $\mathrm{Fil}^{\bullet}(R^{n})$ where

$\mathrm{Fil}^{i}(R^{n})\coloneqq(t^{-\mu_{i}}\Lambda\cap\Lambda_{0})/(t^{-\mu_{i}}\Lambda\cap
t\Lambda_{0}).$ (5.7)

We moreover have a section

$\displaystyle\sigma_{\mu}:GL_{n}/P_{\mu}\rightarrow L^{+}GL_{n}\cdot
t^{\mu}\cdot L^{+}GL_{n}/L^{+}GL_{n}=\mathrm{Gr}_{n}^{\mu},$

which sends a decreasing filtration $\mathrm{Fil}^{\bullet}(R^{n})$ of $R^{n}$
to the lattice

$\sigma_{\mu}(\mathrm{Fil}^{\bullet}(R^{n}))\coloneqq\sum_{i=1}^{n}\mathrm{Fil}^{i}(R^{n})\otimes_{R}t^{\mu_{i}}R\llbracket
t\rrbracket.$ (5.8)

We define the maps

$\mathrm{Gr}_{n}=\coprod_{\mu\in
X_{*}(T_{n})^{+}}\mathrm{Gr}_{n}^{\mu}\mathrel{\mathop{\rightleftarrows}^{\pi}_{\sigma}}\coprod_{\mu\in
X_{*}(T_{n})^{+}}GL_{n}/P_{\mu}$ (5.9)

taking $\Lambda\in\mathrm{Gr}_{n}$ to $\pi_{\mu}(\Lambda)$ if
$\Lambda\in\mathrm{Gr}_{n}^{\mu}$ and taking $gP_{\mu}\in GL_{n}/P_{\mu}$ to
$\sigma_{\mu}(gP_{\mu})$.

Finally, given $\mu\in X_{*}(T_{n})$, we define the locally closed ind-scheme

$Y_{\mu}\coloneqq LU_{n}\cdot t^{\mu}\cdot
L^{+}GL_{n}/L^{+}GL_{n}:\underline{Rings}\rightarrow\underline{Sets}$

of $\mathrm{Gr}_{n}$.

######  Lemma 5.20.

Let $\mu=(\mu_{1},\dots,\mu_{n})\in X_{*}(T_{n})$ and let $\Lambda\in
Y_{\mu}(A)$ for some affinoid algebra $A$. Then
$\mathrm{Fil}^{i}(\Lambda)\coloneqq\Lambda\cap\mathrm{Fil}^{i}(V)$ is a
locally free module over $A\llbracket t\rrbracket$ for all $i=1,\dots,n$.

###### Proof.

Let $\mathbb{M}$ be the universal lattice on $Y_{\mu}$, hence locally on the
affinoid $\mathrm{Sp}(A)$ the lattice $\mathbb{M}$ has a basis of the form

$\\{t^{\mu_{1}}e_{1},t^{\mu_{2}}e_{2}+x_{(1,2)}e_{1},\dots,t^{\mu_{n}}e_{n}+x_{(n-1,n)}e_{n-1}+\dots+x_{(1,n)}e_{1}\\}$

with $x_{(i,j)}\in A((t))$ for all $i<j$. Then we have that
$\mathrm{Fil}^{i}(\mathbb{M})$ is the free $A\llbracket t\rrbracket$-module
generated by the first $i$ vectors of the above basis. ∎

Hence for a lattice $\Lambda\in Y_{\mu}(A)$, let us define a filtration of
$\Lambda$ as
$\mathrm{Fil}^{i}(\Lambda)\coloneqq\Lambda\cap\mathrm{Fil}^{i}(V)$ (in fact we
know this is a filtration by Lemma 5.20). Then we have that

$Y_{\mu}(A)=\\{\Lambda\in\mathrm{Gr}_{n}(A):\mathrm{gr}^{i}(\Lambda)=t^{\mu_{i}}\mathrm{gr}^{i}(\Lambda_{0})\text{
for }i=1,\dots,n\\},$

where the equality
$\mathrm{gr}^{i}(\Lambda)=t^{\mu_{i}}\mathrm{gr}^{i}(\Lambda_{0})$ is intended
as lattices of $\mathrm{gr}^{i}A((t))^{n}$.

######  Lemma 5.21.

We have $\mathrm{Gr}=\coprod_{\mu\in X_{*}(T_{n})}Y_{\mu}$ as sets.

###### Proof.

First of all notice that $\Lambda\in Y_{\mu}\cap Y_{\lambda}$ if and only if
there exist $V,V^{\prime}\in LU$ such that
$Vt^{\mu}L^{+}G=\Lambda=V^{\prime}t^{\lambda}L^{+}G$; hence $\Lambda\in
Y_{\mu}\cap Y_{\lambda}$ if and only if there exists $V\in LU$ such that
$t^{-\lambda}Vt^{\mu}\in L^{+}G$: it is obvious that this is possible only if
$\lambda=\mu$. Now let $\Lambda\in\mathrm{Gr}_{n}(K)$ for some field $K$: we
show that $\Lambda\in Y_{\mu}$ for some $\mu\in X_{*}(T_{n})$. Let
$\Lambda^{i}\coloneqq\Lambda\cap\mathrm{Fil}^{i}(V)$ for all $1\leqslant
i\leqslant n$ (as $K\llbracket t\rrbracket$ is a PID, $\Lambda^{i}$ is a
finite projective $K\llbracket t\rrbracket$-module) and let us consider the
commutative diagram

${0}$${\Lambda^{i}}$${\Lambda^{i+1}}$${\Lambda^{i+1}/\Lambda^{i}}$${0}$${0}$${\mathrm{Fil}^{i}(V)}$${\mathrm{Fil}^{i+1}(V)}$${\mathrm{gr}^{i+1}(V)}$${0.}$

Then $\Lambda^{i+1}/\Lambda^{i}$ is a $K\llbracket t\rrbracket$-lattice of
$\mathrm{gr}^{i+1}(V)=K((t))e_{i+1}$, which means that there exists
$\mu_{i+1}\in\mathbb{Z}$ such that
$\Lambda^{i+1}/\Lambda^{i}=t^{\mu_{i+1}}e_{i+1}\cdot K\llbracket
t\rrbracket=t^{\mu_{i+1}}\mathrm{gr}^{i+1}(\Lambda_{0})$. It is then obvious
that $\Lambda\in Y_{\mu}$, where $\mu=(\mu_{1},\dots,\mu_{n})$ constructed as
described. ∎

######  Lemma 5.22.

Let $\lambda\in X_{*}(T_{n})^{+}$, $\mu\in X_{*}(T_{n})$ and $w_{0}\in W_{n}$
the longest element of the Weyl group. If $\mu=ww_{0}\lambda$ for some $w\in
W_{n}$, we have that the diagram

${B_{n}wP_{\lambda}/P_{\lambda}}$${GL_{n}/P_{\lambda}}$${Y_{\mu}}$${\mathrm{Gr}_{n}}$$\scriptstyle{\sigma_{\lambda}}$$\scriptstyle{\sigma_{\lambda}}$
(5.10)

is a fiber product.

###### Proof.

We will show that

$\sigma_{\lambda}(B_{n}wP_{\lambda}/P_{\lambda})=Y_{\mu}\cap\sigma_{\lambda}(GL_{n}/P_{\lambda}).$

Notice first of all that
$\sigma_{\lambda}(GL_{n}/P_{\lambda})=Gt^{\lambda}L^{+}G/L^{+}G=Gt^{w_{0}\lambda}L^{+}G/L^{+}G$,
where $w_{0}\in W_{n}$ is the longest element; moreover
$\sigma_{\lambda}(B_{n}wP_{\lambda}/P_{\lambda})=B_{n}wt^{\lambda}L^{+}G/L^{+}G$
by definition of $\sigma_{\lambda}$. As
$B_{n}wt^{\lambda}L^{+}G/L^{+}G=U_{n}wt^{\lambda}L^{+}G/L^{+}G$, we prove that
$Y_{\mu}\cap\sigma_{\lambda}(GL_{n}/P_{\lambda})=U_{n}wt^{\lambda}L^{+}G/L^{+}G$.
As both schemes are reduced and locally closed subschemes of $LG/L^{+}G$, it’s
enough to work on $L$-points for some field $L$. Observe that
$B_{n}t^{w_{0}\lambda}L^{+}G/L^{+}G=t^{w_{0}\lambda}L^{+}G/L^{+}G$, in fact:
for any $(b_{ij})_{1\leqslant i,j\leqslant n}\in B_{n}$, we have

$(b_{ij})_{1\leqslant i,j\leqslant n}\cdot
t^{w_{0}\lambda}=t^{w_{0}\lambda}\cdot(b_{ij}t^{\lambda_{n-j+1}-\lambda_{n-i+1}})_{1\leqslant
i,j\leqslant n};$

since $(b_{ij})_{1\leqslant i,j\leqslant n}$ is upper-triangular, we have that
$b_{ij}=0$ for $i>j$. As a consequence of this fact and of the fact that
$\lambda$ is dominant, we have that the matrix
$(b_{ij}t^{\lambda_{n-j+1}-\lambda_{n-i+1}})_{1\leqslant i,j\leqslant n}\in
L^{+}G$. Moreover by the Bruhat decomposition, we have

$GL_{n}=\bigsqcup_{w\in W}U_{n}wB_{n},$

hence

$\displaystyle Gt^{\lambda}L^{+}G/L^{+}G$
$\displaystyle=Gt^{w_{0}\lambda}L^{+}G/L^{+}G=\bigsqcup_{w\in
W_{n}}U_{n}wB_{n}t^{w_{0}\lambda}L^{+}G/L^{+}G$ $\displaystyle=\bigsqcup_{w\in
W_{n}}U_{n}wt^{w_{0}\lambda}L^{+}G/L^{+}G=\bigsqcup_{w\in
W_{n}}U_{n}t^{ww_{0}\lambda}L^{+}G/L^{+}G.$

Obviously we have $U_{n}t^{ww_{0}\lambda}L^{+}G/L^{+}G\subset
LU_{n}t^{ww_{0}\lambda}L^{+}G/L^{+}G$ and

$U_{n}wt^{w_{0}\lambda}L^{+}G/L^{+}G\cap LU_{n}t^{\mu}L^{+}G/L^{+}G=\emptyset$

for all $\mu\neq ww_{0}\lambda$. This shows that

$Y_{\mu}\cap\sigma_{\lambda}(GL_{n}/P_{\lambda})=U_{n}t^{ww_{0}\lambda}L^{+}G/L^{+}G=\sigma_{\lambda}(B_{n}wP_{\lambda}/P_{\lambda}),$

where $ww_{0}\lambda=\mu$. ∎

######  Lemma 5.23.

Let us fix the standard lattice $\Lambda_{0}$ and a standard basis
$\\{b_{1},\dots,b_{n}\\}$ of $\Lambda_{0}$. Let us decompose $\Lambda_{0}$ as
$\Lambda_{0}=\bigoplus_{i=1}^{l}\Lambda_{0,i}$, where $\Lambda_{0,i}$ is a
lattice of rank $n_{i}$; let us moreover assume that $\Lambda_{0,i}$ has a
basis consisting of a subset of the standard basis for $\Lambda_{0}$ and let
us fix the standard basis of $\Lambda_{0,i}$ to be such basis for all $i$.
Notice we can order the basis elements in such a way so that they respect the
order in the standard basis of $\Lambda_{0}$. Let $\mu\in X_{*}(T_{n})$ and
$\lambda_{i}\in X_{*}(T_{n_{i}})^{+}$ for all $1\leqslant i\leqslant l$. We
have

$Y_{\mu}\times_{\mathrm{Gr}_{n}}\prod_{i=1}^{l}GL_{n_{i}}/P_{\lambda_{i}}=\prod_{i=1}^{l}Y_{\mu_{i}}\times_{\mathrm{Gr}_{n_{i}}}GL_{n_{i}}/P_{\lambda_{i}}$

for some $\mu_{i}\in X_{*}(T_{n_{i}})$, where the map
$\prod_{i=1}^{l}GL_{n_{i}}/P_{\lambda_{i}}\rightarrow\mathrm{Gr}_{n}=\prod_{i=1}^{l}\mathrm{Gr}_{n_{i}}$
is given by

$(g_{i}P_{\lambda_{i}})_{i=1,\dots,l}\mapsto(\sigma_{\lambda_{i}}(g_{i}P_{\lambda_{i}}))_{i=1,\dots,l}.$

###### Proof.

We show that

$Y_{\mu}\cap\prod_{i=1}^{l}\sigma_{\lambda_{i}}(GL_{n_{i}}/P_{\lambda_{i}})=\prod_{i=1}^{l}(Y_{\mu_{i}}\cap\sigma_{\lambda_{i}}(GL_{n_{i}}/P_{\lambda_{i}}))$

for some $\mu_{i}\in X_{*}(T_{n_{i}})$. Observe that, as all schemes are
reduced locally closed subspaces of $LG/L^{+}G$, we can work on $L$-points for
$L$ a field. For all $i$, we denote by $\\{e_{1,i},\dots,e_{n_{i},i}\\}$ the
standard basis of $\Lambda_{0,i}$ consisting of elements of the standard basis
$\\{b_{1},\dots,b_{n}\\}$ of $\Lambda_{0}$ such that if
$e_{j,i}=b_{j^{\prime}}$, $e_{k,i}=b_{k^{\prime}}$ and $j\leqslant k$, then
$j^{\prime}\leqslant k^{\prime}$. So let
$\Lambda\in(Y_{\mu}\cap\prod_{i=1}^{l}\sigma_{\lambda_{i}}(GL_{n_{i}}/P_{\lambda_{i}}))(L)$.
This means that
$\Lambda=(\sigma_{\lambda_{i}}(g_{i}P_{\lambda_{i}}))_{i=1,\dots,l}$ for some
$g_{i}P_{\lambda_{i}}\in GL_{n_{i}}/P_{\lambda_{i}}$. We will show that for
all $1\leqslant j\leqslant n$ we have

$\prod_{i=1}^{l}\sigma_{\lambda_{i}}(g_{i}P_{\lambda_{i}})\cap\prod_{k=1}^{j}L((t))b_{k}=\prod_{i=1}^{l}(\sigma_{\lambda_{i}}(g_{i}P_{\lambda_{i}})\cap(L((t))e_{1,i}\oplus\dots\oplus
L((t))e_{k_{(j,i)},i})),$

where
$\\{b_{1},\dots,b_{j}\\}=\bigcup_{i=1}^{l}\\{e_{1,i},\dots,e_{k_{(j,i)},i}\\}$.
Let

$x\in\prod_{i=1}^{l}\sigma_{\lambda_{i}}(g_{i}P_{\lambda_{i}})\cap\prod_{k=1}^{j}L((t))b_{k},$

in particular $x=\sum_{i=1}^{l}x_{i}$ with
$x_{i}\in\sigma_{\lambda_{i}}(g_{i}P_{\lambda_{i}})$. As
$\sigma_{\lambda_{i}}(g_{i}P_{\lambda_{i}})\subset\prod_{k=1}^{n_{i}}L((t))e_{k,i}$,
we have that $x_{i}=\sum_{h=1}^{n_{i}}y_{h}e_{h,i}$ with $y_{h}\in L((t))$. As
a consequence, we have

$x=\sum_{i=1}^{l}\sum_{h=1}^{n_{i}}y_{h}e_{h,i}\in\prod_{k=1}^{j}L((t))b_{k}.$

This proves the first inclusion. The inverse inclusion is obvious. As a
consequence, we have

$\displaystyle\mathrm{Fil}^{j}(\Lambda)$
$\displaystyle=\prod_{i=1}^{l}\sigma_{\lambda_{i}}(g_{i}P_{\lambda_{i}})\cap\prod_{k=1}^{j}L((t))b_{k}$
$\displaystyle=\prod_{i=1}^{l}(\sigma_{\lambda_{i}}(g_{i}P_{\lambda_{i}})\cap(L((t))e_{1,i}\oplus\dots\oplus
L((t))e_{k_{(j,i)},i}))$
$\displaystyle=\prod_{i=1}^{l}\mathrm{Fil}^{k_{(j,i)}}(\sigma_{\lambda_{i}}(g_{i}P_{\lambda_{i}})).$

In particular observe that for all $1\leqslant j\leqslant n$

$\displaystyle\mathrm{gr}^{j}(\Lambda)$
$\displaystyle=\mathrm{Fil}^{j}(\Lambda)/\mathrm{Fil}^{j-1}(\Lambda)$
$\displaystyle=\left(\prod_{i=1}^{l}\mathrm{Fil}^{k_{(j,i)}}(\sigma_{\lambda_{i}}(g_{i}P_{\lambda_{i}}))\right)/\left(\prod_{i=1}^{l}\mathrm{Fil}^{k_{(j-1,i)}}(\sigma_{\lambda_{i}}(g_{i}P_{\lambda_{i}}))\right)$
$\displaystyle=\mathrm{Fil}^{k_{(j,i^{\prime})}}(\sigma_{\lambda_{i^{\prime}}}(g_{i^{\prime}}P_{\lambda_{i^{\prime}}}))/\mathrm{Fil}^{k_{(j-1,i^{\prime})}}(\sigma_{\lambda_{i^{\prime}}}(g_{i^{\prime}}P_{\lambda_{i^{\prime}}}))$
$\displaystyle=\mathrm{gr}^{k_{(j,i^{\prime})}}(\sigma_{\lambda_{i^{\prime}}}(g_{i^{\prime}}P_{\lambda_{i^{\prime}}})),$

where $i^{\prime}$ is such that $b_{j}=e_{k_{j,i^{\prime}}}$. This proves that

$\Lambda\in
Y_{\mu}\cap\prod_{i=1}^{l}\sigma_{\lambda_{i}}(GL_{n_{i}}/P_{\lambda_{i}})\Leftrightarrow\Lambda\in\prod_{i=1}^{l}(Y_{\mu_{i}}\cap\sigma_{\lambda_{i}}(GL_{n_{i}}/P_{\lambda_{i}})),$

where $(\mu_{i}\colon 1\leqslant i\leqslant l)$ is a permutation of $\mu$. ∎

######  Definition 5.24.

Let us fix $n\geqslant 2$ and let $\mathfrak{g}_{n}$ be the Lie algebra of
$GL_{n}$.

  * (i)

We define the closed subspace

$\mathrm{Fil}^{\mathrm{st}}\coloneqq\\{(N,gP_{\mu})\in\mathfrak{g}_{n}\times\coprod_{\mu\in
X_{*}(T_{n})^{+}}GL_{n}/P_{\mu}\colon NgP_{\mu}\subseteq gP_{\mu}\\}$ (5.11)

of $\mathfrak{g}_{n}\times\coprod_{\mu\in X_{*}(T_{n})^{+}}GL_{n}/P_{\mu}$.

  * (ii)

Let us fix $\lambda\in X_{*}(T_{n})^{+}$, we define the fiber product

${\mathrm{Fil}^{\mathrm{st}}_{\lambda}}$${\mathfrak{g}_{n}\times
GL_{n}/P_{\lambda}}$${\mathrm{Fil}^{\mathrm{st}}}$${\mathfrak{g}_{n}\times\coprod_{\mu\in
X_{*}(T_{n})^{+}}GL_{n}/P_{\mu},}$

which is a closed subspace of $\mathfrak{g}_{n}\times GL_{n}/P_{\lambda}$.

######  Proposition 5.25.

The codimension of $\mathrm{Fil}^{\mathrm{st}}_{\lambda}$ inside
$\mathfrak{g}_{n}\times GL_{n}/P_{\lambda}$ is at most $\dim
GL_{n}/P_{\lambda}$.

###### Proof.

Let $(N,gP_{\lambda})\in\mathfrak{g}_{n}\times GL_{n}/P_{\lambda}$. We have
that $(N,gP_{\lambda})\in\mathrm{Fil}^{\mathrm{st}}_{\lambda}$ if and only if
$NgP_{\lambda}\subseteq gP_{\lambda}$ by construction of
$\mathrm{Fil}^{\mathrm{st}}_{\lambda}$; this means that

$(N,gP_{\lambda})\in\mathrm{Fil}^{\mathrm{st}}_{\lambda}\Leftrightarrow
g^{-1}Ng\in P_{\lambda}.$

Then the space $\mathrm{Fil}^{\mathrm{st}}_{\lambda}$ is cut out by the
condition $g^{-1}Ng\in P_{\lambda}$, which corresponds to the equations
annihilating certain entries of the matrix $g^{-1}Ng$; the number of these
equations is exactly $\dim\overline{U}_{P_{\lambda}}$, where
$\overline{U}_{P_{\lambda}}$ is the unipotent radical of the opposite
parabolic of $P_{\lambda}$. Hence we have

$\operatorname{codim}\mathrm{Fil}^{\mathrm{st}}_{\lambda}\leqslant\dim\overline{U}_{P_{\lambda}}=\dim
GL_{n}/P_{\lambda}.$

∎

######  Definition 5.26.

For $w\in W_{n}$ and $\lambda\in X_{*}(T_{n})^{+}$, let us define
$V_{w}^{\lambda}$ to be the fiber product

${V_{w}^{\lambda}}$${\mathfrak{b}_{n}\times
B_{n}wP_{\lambda}/P_{\lambda}}$${\mathrm{Fil}^{\mathrm{st}}_{\lambda}}$${\mathfrak{g}_{n}\times
GL_{n}/P_{\lambda},}$ (5.12)

where $\mathfrak{b}_{n}$ is the Lie algebra of the Borel $B_{n}$ of upper
triangular matrices.

######  Remark 5.27.

The space $V_{w}^{\lambda}$ is smooth of dimension $\dim\mathfrak{b}_{n}$ by
[8][Proposition 2.2.1]. As the space $\mathfrak{b}_{n}\times
B_{n}wP_{\lambda}/P_{\lambda}$ is smooth of dimension
$\dim\mathfrak{b}_{n}+\dim B_{n}wP_{\lambda}/P_{\lambda}$, we have that
$V_{w}^{\lambda}$ inside $\mathfrak{b}_{n}\times
B_{n}wP_{\lambda}/P_{\lambda}$ is cut out by $\dim
B_{n}wP_{\lambda}/P_{\lambda}$ equations by [26, Tag 0H1G].

###  5.6. Almost de Rham representations

In what follows, $B_{\mathrm{dR}}$ and $B_{\mathrm{dR}}^{+}$ are topological
rings for the so called "natural topology" ([14, §3.2]). Let us choose a
compatible system of primitive $p^{n}$-th roots of unity
$(\epsilon^{(n)})_{n\in\mathbb{N}}$; we have $[\epsilon]\in
B_{\mathrm{dR}}^{+}$, where $[\cdot]$ denotes the Teichmüller map. Moreover
$\log[\epsilon]\coloneqq\sum_{n=1}^{\infty}\frac{([\epsilon]-1)^{n}}{n}$
converges in $B_{\mathrm{dR}}^{+}$ and is a uniformizer; we will denote it by
$t$ and we have that $B_{\mathrm{dR}}\coloneqq B_{\mathrm{dR}}^{+}[1/t]$. Let
$C\coloneqq B_{\mathrm{dR}}^{+}/tB_{\mathrm{dR}}^{+}$ be the residue field of
$B_{\mathrm{dR}}^{+}$. Recall that both the rings $B_{\mathrm{dR}}$ and
$B_{\mathrm{dR}}^{+}$ are endowed with a continuous action of
$G_{\mathbb{Q}_{p}}$. We now recall the definition of _almost de Rham
representations_ as defined in [14, §3.7].

######  Definition 5.28.

  * (i)

A _$B_{\mathrm{dR}}$ -representation of $G_{\mathbb{Q}_{p}}$_ is a finite
dimensional free $B_{\mathrm{dR}}$-module with a continuous semi-linear action
of $G_{\mathbb{Q}_{p}}$. We denote by
$\mathrm{Rep}_{B_{\mathrm{dR}}}(G_{\mathbb{Q}_{p}})$ the category of
$B_{\mathrm{dR}}$-represetations of $G_{\mathbb{Q}_{p}}$.

  * (ii)

Let $W\in\mathrm{Rep}_{B_{\mathrm{dR}}}(G_{\mathbb{Q}_{p}})$, we say that $W$
is _almost de Rham_ if it contains a $B_{\mathrm{dR}}^{+}$-lattice $W^{+}$
that is stable under $G_{\mathbb{Q}_{p}}$ and such that the Sen weights of the
$C$-represetation $W^{+}/tW^{+}$ are all integers. We denote by
$\mathrm{Rep}_{\mathrm{pdR}}(G_{\mathbb{Q}_{p}})$ the full subcategory of
almost de Rham representations.

######  Remark 5.29.

  * (i)

By compactness of $G_{\mathbb{Q}_{p}}$, we have that any
$W\in\mathrm{Rep}_{B_{\mathrm{dR}}}(G_{\mathbb{Q}_{p}})$ contains a
$G_{\mathbb{Q}_{p}}$-stable $B_{\mathrm{dR}}^{+}$-lattice.

  * (ii)

If $W$ is a Hodge-Tate representation of $G_{\mathbb{Q}_{p}}$ over $C$, then
the Sen weights coincide with the Hodge-Tate weights.

We have a functor

$\begin{split}D_{\mathrm{pdR}}:\mathrm{Rep}_{B_{\mathrm{dR}}}(G_{\mathbb{Q}_{p}})&\rightarrow\mathrm{Rep}_{\mathbb{Q}_{p}}(\mathbb{G}_{a})\\\
W&\mapsto(B_{\mathrm{dR}}[\log
t]\otimes_{B_{\mathrm{dR}}}W)^{G_{\mathbb{Q}_{p}}}\end{split}$ (5.13)

defined in [14, §4.3] with the property that
$\dim_{\mathbb{Q}_{p}}D_{\mathrm{pdR}}(W)\leqslant\dim_{B_{\mathrm{dR}}}W$.

The following Theorem is proved in [14, Théorème 4.1.] and [8, Proposition
3.1.1].

######  Theorem 5.30.

A represetation $W\in\mathrm{Rep}_{B_{\mathrm{dR}}}(G_{\mathbb{Q}_{p}})$ is
almost de Rham if and only if
$\dim_{\mathbb{Q}_{p}}D_{\mathrm{pdR}}(W)=\dim_{B_{\mathrm{dR}}}W$. Moreover
the functor $D_{\mathrm{pdR}}$ induces an equivalence of categories between
$\mathrm{Rep}_{\mathrm{pdR}}(G_{\mathbb{Q}_{p}})$ and
$\mathrm{Rep}_{\mathbb{Q}_{p}}(\mathbb{G}_{a})$.

######  Remark 5.31.

If $E$ is a field of characteristic 0, we have that the category
$\mathrm{Rep}_{E}(\mathbb{G}_{a})$ is equivalent to the category whose objects
are pairs $(V,N_{V})$ of finite-dimensional $E$-vector spaces $V$ endowed with
an $E$-linear nilpotent endomorphism $N_{V}$ and whose morphisms are linear
maps of $E$-vector spaces commuting with the nilpotent endomorphism $N_{V}$.

The following is [8, Proposition 3.2.1.].

######  Proposition 5.32.

Let $W\in\mathrm{Rep}_{\mathrm{pdR}}(G_{\mathbb{Q}_{p}})$. We have a bijection

$\displaystyle\\{G_{\mathbb{Q}_{p}}\text{-stable
}B_{\mathrm{dR}}^{+}\text{-lattices of }W\\}$
$\displaystyle\leftrightarrow\\{\text{filtrations of }D_{\mathrm{pdR}}(W)\\}$
$\displaystyle W^{+}$
$\displaystyle\mapsto\mathrm{Fil}^{\bullet}_{W^{+}}(D_{\mathrm{pdR}}(W)),$

where
$\mathrm{Fil}^{i}_{W^{+}}(D_{\mathrm{pdR}}(W))\coloneqq(t^{i}B_{\mathrm{dR}}^{+}[\log
t]\otimes_{B_{\mathrm{dR}}^{+}}W^{+})^{G_{\mathbb{Q}_{p}}}$ for
$i\in\mathbb{Z}$. Here a filtration of a representation
$V\in\mathrm{Rep}_{\mathbb{Q}_{p}}(\mathbb{G}_{a})$ is a decreasing,
exhaustive and separated filtration by subobjects in
$\mathrm{Rep}_{\mathbb{Q}_{p}}(\mathbb{G}_{a})$.

Thanks to Remark 5.31 and Proposition 5.32, given
$W\in\mathrm{Rep}_{\mathrm{pdR}}(G_{\mathbb{Q}_{p}})$ we have a bijection

$\begin{split}\\{G_{\mathbb{Q}_{p}}\text{-stable
}B_{\mathrm{dR}}^{+}\text{-lattices of }W\\}&\leftrightarrow\\{N\text{-stable
filtrations of }D_{\mathrm{pdR}}(W)\\}\\\
W^{+}&\mapsto\mathrm{Fil}^{\bullet}_{W^{+}}(D_{\mathrm{pdR}}(W)),\end{split}$
(5.14)

where the filtration $\mathrm{Fil}^{\bullet}_{W^{+}}(D_{\mathrm{pdR}}(W))$ is
defined as in Proposition 5.32.

In [4], Berger defines a topological ring $B_{e}$ endowed with a continuous
$G_{\mathbb{Q}_{p}}$-action and introduces the notion of _$B$ -pairs_, proving
that there exists an equivalence of categories of such objects with
$(\phi,\Gamma)$-modules.

######  Definition 5.33.

A _$B$ -pair_ is a pair $(W_{e},W_{\mathrm{dR}}^{+})$ where $W_{e}$ is a free
$B_{e}$-module of finite rank endowed with a semi-linear and continuous action
of $G_{\mathbb{Q}_{p}}$ and $W_{\mathrm{dR}}^{+}$ is a
$B_{\mathrm{dR}}^{+}$-lattice of $W_{\mathrm{dR}}\coloneqq
B_{\mathrm{dR}}\otimes_{B_{e}}W_{e}$ stable under the action of
$G_{\mathbb{Q}_{p}}$.

The following result is [4, Théorème 2.2.7].

######  Theorem 5.34.

There is an equivalence of categories

$\displaystyle W:(\phi,\Gamma)-\mathrm{Mod}(\mathcal{R})$
$\displaystyle\rightarrow\\{B\text{-pairs}\\}$ $\displaystyle D$
$\displaystyle\mapsto(W_{e}(D),W_{\mathrm{dR}}^{+}(D)).$

Moreover the definition of the map $W_{e}(D)$ actually only depends on
$D[1/t]$.

The functoriality of these constructions make it possible to consider
representations with different coefficients. From now on until the end of the
section, we let $A$ be a finite dimensional algebra over $\mathbb{Q}_{p}$; we
introduce the following notations for the rings $B_{\mathrm{dR}}$ and
$B_{\mathrm{dR}}^{+}$ with coefficients in $A$:

$\begin{array}[]{cc}B_{\mathrm{dR},A}\coloneqq
B_{\mathrm{dR}}\otimes_{\mathbb{Q}_{p}}A,&B_{\mathrm{dR},A}^{+}\coloneqq
B_{\mathrm{dR}}^{+}\otimes_{\mathbb{Q}_{p}}A.\end{array}$

The following notions are defined in [8, Section 3.1 and 3.2].

######  Definition 5.35.

  1. (1)

A _$B_{\mathrm{dR},A}$ -representation of $G_{\mathbb{Q}_{p}}$_ is a
$B_{\mathrm{dR}}$-representation $W$ of $G_{\mathbb{Q}_{p}}$ together with a
morphism of $\mathbb{Q}_{p}$-algebras
$A\rightarrow\mathrm{End}_{\mathrm{Rep}_{B_{\mathrm{dR}}}(G_{\mathbb{Q}_{p}})}(W)$
which makes $W$ a finite free $B_{\mathrm{dR},A}$-module. We denote by
$\mathrm{Rep}_{B_{\mathrm{dR},A}}(G_{\mathbb{Q}_{p}})$ the category of
$B_{\mathrm{dR},A}$-representations.

  2. (2)

A _$B_{\mathrm{dR},A}^{+}$ -representation of $G_{\mathbb{Q}_{p}}$_ is a
$B_{\mathrm{dR}}^{+}$-representation $W^{+}$ of $G_{\mathbb{Q}_{p}}$ together
with a morphism of $\mathbb{Q}_{p}$-algebras
$A\rightarrow\mathrm{End}_{\mathrm{Rep}_{B_{\mathrm{dR}}^{+}}(G_{\mathbb{Q}_{p}})}(W^{+})$
which makes $W^{+}$ a finite free $B_{\mathrm{dR},A}^{+}$-module.

  3. (3)

We say that a $B_{\mathrm{dR},A}$-representation is _almost de Rham_ if the
underlying $B_{\mathrm{dR}}$-representation is. We denote by
$\mathrm{Rep}_{\mathrm{pdR},A}(G_{\mathbb{Q}_{p}})$ the category of almost de
Rham $B_{\mathrm{dR},A}$-representations.

  4. (4)

An _$(A,B)$ -pair of $G_{\mathbb{Q}_{p}}$_ is a pair
$(W_{e},W_{\mathrm{dR}}^{+})$ such that

     * (i)

$W_{e}$ is a finite $B_{e}\otimes_{\mathbb{Q}_{p}}A$-module with a continuous
semi-linear $G_{\mathbb{Q}_{p}}$-action which is free as $B_{e}$-module;

     * (ii)

$W_{\mathrm{dR}}^{+}\subset W_{\mathrm{dR}}\coloneqq
B_{\mathrm{dR}}\otimes_{B_{e}}W_{e}$ is a $G_{\mathbb{Q}_{p}}$-stable
$B_{\mathrm{dR},A}^{+}$\- lattice of $W_{\mathrm{dR}}$.

We denote by $\mathrm{Rep}_{A}(\mathbb{G}_{a})$ the category of pairs
$(V_{A},N_{A})$ where $N_{A}$ is a nilpotent endomorphism of a finite free
$A$-module $V_{A}$.

The following is [8, Lemma 3.1.4].

######  Theorem 5.36.

The functor $D_{\mathrm{pdR}}$ induces an equivalence of categories between
$\mathrm{Rep}_{\mathrm{pdR},A}(G_{\mathbb{Q}_{p}})$ and
$\mathrm{Rep}_{A}(\mathbb{G}_{a})$.

By [8, Lemma 3.2.2], we have that given
$W\in\mathrm{Rep}_{\mathrm{pdR},A}(G_{\mathbb{Q}_{p}})$, there is a bijection

$\begin{split}\left\\{G_{\mathbb{Q}_{p}}\text{-stable
}B_{\mathrm{dR},A}^{+}\text{-lattices of
}W\right\\}&\leftrightarrow\left\\{\begin{array}[]{c}N_{A}\text{-stable
exhaustive}\\\ \text{filtrations of
}D_{\mathrm{pdR}}(W)\end{array}\right\\}\\\
W^{+}&\mapsto\mathrm{Fil}^{\bullet}_{W^{+}}(D_{\mathrm{pdR}}(W)),\end{split}$
(5.15)

where the filtration $\mathrm{Fil}^{\bullet}_{W^{+}}(D_{\mathrm{pdR}}(W))$ is
defined as in Proposition 5.32.

We moreover have the following result.

######  Theorem 5.37.

Let $A$ be a finite dimensional $\mathbb{Q}_{p}$-algebra. There is an
equivalence of categories

$\displaystyle W:(\phi,\Gamma)-\mathrm{Mod}(\mathcal{R}_{A})$
$\displaystyle\rightarrow(A,B)\text{-pairs}$ $\displaystyle D$
$\displaystyle\mapsto(W_{e}(D),W_{\mathrm{dR}}^{+}(D)),$

where the functor $W_{e}(D)$ only depends on $D[1/t]$ and
$W_{\mathrm{dR}}^{+}(D)\coloneqq
B_{\mathrm{dR}}^{+}\otimes_{\mathcal{R}^{r}}D^{r}$ is independent of $r$,
where $r\in p^{\mathbb{Q}}\cap[0,1)$ such that there exists a
$(\phi,\Gamma)$-module $D^{r}$ over $\mathcal{R}_{A}^{r}$ such that
$D^{r}\otimes_{\mathcal{R}_{A}^{r}}\mathcal{R}_{A}=D$.

Nakamura proved the above Theorem when $A$ is a finite extension of
$\mathbb{Q}_{p}$ in [25], but since the same proof holds also for finite