# Tangent spaces on the trianguline variety at companion points

Seginus Mowlavi  Université Paris-Saclay, CNRS, Laboratoire de mathématiques
d’Orsay, 91405, Orsay, France

###### Abstract.

Many results about the geometry of the trianguline variety have been obtained
by Breuil-Hellmann-Schraen. Among them, using geometric methods, they have
computed a formula for the dimension of the tangent space of the trianguline
variety at dominant crystalline generic points, which has a conjectural
generalisation to companion (_i.e._ non-dominant) points. In an earlier work,
they proved a weaker form of this formula under the assumption of modularity
using arithmetic methods. We prove a generalisation of a result of Bellaïche-
Chenevier in $p$-adic Hodge theory and use it to extend the arithmetic methods
of Breuil-Hellmann-Schraen to a wide class of companion points.

###### Contents

  1. 1 Introduction
  2. 2 Crystalline trianguline representations
    1. 2.1 Refinements of a crystalline representation $V$ and flags on $\mathbf{D}_{\mathrm{dR}}(V)$
    2. 2.2 Triangulations and refinements
  3. 3 The trianguline variety and companion points on the patched eigenvariety
    1. 3.1 The trianguline variety
    2. 3.2 The patched eigenvariety
    3. 3.3 Strong linkage and companion points
  4. 4 Tangent spaces on the trianguline variety
    1. 4.1 Crystalline generic points and pairs of permutations
    2. 4.2 Crystalline tangent directions
    3. 4.3 Trianguline tangent directions
    4. 4.4 Proof of Theorem 4.1.4

##  1\. Introduction

Let $p$ be a prime number. Inspired by Kisin’s study [27] of $p$-adic
overconvergent eigenforms, and building on the notion of trianguline
representation from Colmez [15], Hellmann [22] followed by Hellmann-Schraen
[23] introduced a rigid analytic variety parametrising triangulations of
framed deformations of a fixed residual representation $\bar{r}$, called the
trianguline variety $X_{\mathrm{tri}}^{\square}(\bar{r})$. Hellmann used this
construction to advance the study of eigenvarieties by Bellaïche-Chenevier
[1]. Breuil-Hellmann-Schraen focused on the geometry of the trianguline
variety in a series of papers [12], [11], [13], especially the local geometry
at crystalline generic points. They derived a number of consequences, among
which results concerning classicality of overconvergent forms, existence of
certain eigenforms called companion eigenforms, and the existence of companion
constituents (we do not focus on automorphic representations in this work, but
this features prominently in the $p$-adic local Langlands program; see for
instance [10], [2], [16] and [17]).

In particular, Breuil-Hellmann-Schraen prove an upper bound for the dimension
of the tangent space of $X_{\mathrm{tri}}^{\square}(\bar{r})$ at a crystalline
generic point, and conjecture that it is exact. They also show, for strictly
dominant points (_i.e._ “classical” points), that this conjecture is implied
by classical modularity lifting conjectures.

In this paper, we generalise a result of Bellaïche-Chenevier [1] about
refinements of crystalline representations and associated triangulations. This
allows us to adapt the method of Breuil-Hellmann-Schraen [11] to prove their
conjecture about the dimension of the tangent space of
$X_{\mathrm{tri}}^{\square}(\bar{r})$, assuming the modularity lifting
conjectures, at many points which are not classical (_i.e._ “companion”
points).

To state our results, we briefly introduce the relevant objects. We fix an
integer $n>0$ and a representation
$\bar{r}\colon\mathcal{G}_{K}\longrightarrow\operatorname{GL}_{n}(k_{L})$ from
the absolute Galois group of $K$ to the residual field of $L$, where $K$ and
$L$ are finite extensions of $\mathbb{Q}_{p}$. We assume that $p\nmid 2n$ and
that $\Sigma\coloneqq\operatorname{Hom}_{\mathbb{Q}_{p}}(K,L)$ has
$[K:\mathbb{Q}_{p}]$ elements.

Let $X_{\mathrm{tri}}^{\square}(\bar{r})$ be the trianguline variety: it is a
rigid analytic variety over $L$ defined as the Zariski-closure of the set
$U_{\mathrm{tri}}^{\square}(\bar{r})$ of points $(r,\underline{\delta})$ where
$r$ is a framed deformation of $\bar{r}$ and $\underline{\delta}$ is a locally
$\mathbb{Q}_{p}$-analytic character of $(K^{\times})^{n}$ such that $r$ is
trianguline of parameter $\underline{\delta}$ in the sense of Colmez (see
_e.g._[13, Dfn. 3.3.8]). We fix a crystalline point
$x=(r,\underline{\delta})\in X_{\mathrm{tri}}^{\square}(\bar{r})$, _i.e._ a
point such that $r$ is a crystalline representation.

We assume that $r$ is generic, which means that (i) for each $\tau\in\Sigma$,
the $\tau$-Hodge-Tate weights of $r$ are all distinct, and (ii)
$\varphi/\varphi^{\prime}\notin\\{1,p^{f(K/\mathbb{Q}_{p})}\\}$ for any two
eigenvalues $\varphi,\varphi^{\prime}$ of the linearised Frobenius $\Phi$ on
$\mathbf{D}_{\mathrm{crys}}(r)$ (with multiplicities), where
$f(K/\mathbb{Q}_{p})$ is the inertia degree. For each $\tau\in\Sigma$, we
write the $\tau$-Hodge-Tate weights of $r$ in increasing order
$h_{\tau,1}<\ldots<h_{\tau,n}$. Then $\underline{\delta}$ is determined by the
data of (i) an ordering $\underline{\varphi}=(\varphi_{1},\ldots,\varphi_{n})$
of the eigenvalues of $\Phi$ and (ii) for each $\tau\in\Sigma$, a permutation
$w_{\tau}$ in the symmetric group $\mathcal{S}_{n}$. More precisely, one can
write
$\underline{\delta}=z^{w(\mathbf{h})}\operatorname{unr}(\underline{\varphi})$,
where the character
$\operatorname{unr}(\underline{\varphi})=(\operatorname{unr}(\varphi_{i}))_{i}$
of $(K^{\times})^{n}$ is defined by
$\operatorname{unr}(\varphi_{i})|_{\mathcal{O}_{K}^{\times}}=1$ and
$\operatorname{unr}(\varphi_{i})(\varpi)=\varphi_{i}$ for a uniformiser
$\varpi$ of $K$, and where $w\coloneqq(w_{\tau})\in(\mathcal{S}_{n})^{\Sigma}$
and
$w(\mathbf{h})\coloneqq\big{(}h_{\tau,w_{\tau}^{-1}(i)}\big{)}_{\tau,i}\in(\mathbb{Z}^{n})^{\Sigma}$.
Furthermore, there is a unique $w_{\mathrm{sat}}\in(\mathcal{S}_{n})^{\Sigma}$
such that the point
$x_{\mathrm{sat}}\coloneqq\left(r,z^{w_{\mathrm{sat}}(\mathbf{h})}\operatorname{unr}(\underline{\varphi})\right)$
lies in $U_{\mathrm{tri}}^{\square}(\bar{r})$.

Writing $G\coloneqq\operatorname{GL}_{n,\mathbb{Q}_{p}}$, note that
$(\mathcal{S}_{n})^{\Sigma}$ is the Weyl group of the reductive group
$\prod_{\tau\in\Sigma}G$ over $\mathbb{Q}_{p}$. Set $B\subseteq G$ to be the
Borel subgroup of to upper triangular matrices; it determines a product Bruhat
order $\preceq$ on $(\mathcal{S}_{n})^{\Sigma}$. It is known that
$w_{\mathrm{sat}}\preceq w$ [13, Thm. 1.7].

The main result of this paper is the following:

######  Theorem 1 (Theorem 4.1.4).

Assume Conjecture 3.2.2. Let $x\in X_{\mathrm{tri}}^{\square}(\bar{r})$ be
crystalline generic, such that the associated pair $(w_{\mathrm{sat}},w)$ in
$(\mathcal{S}_{n})^{\Sigma}$ is good in the sense of Definition 4.1.1. Then

(1.1) $\dim_{k(x)}T_{X_{\mathrm{tri}}^{\square}(\bar{r}),x}=\dim
X_{\mathrm{tri}}^{\square}(\bar{r})-d_{ww_{\mathrm{sat}}^{-1}}+\sum_{\tau\in\Sigma}\dim_{\mathbb{Q}_{p}}T_{\overline{(Bw_{\tau}B/B)},w_{\mathrm{sat},\tau}B}-\operatorname{lg}(w_{\mathrm{sat}})$

where $k(x)$ is the residue field of $X_{\mathrm{tri}}^{\square}(\bar{r})$ at
$x$, $\overline{Bw_{\tau}B/B}$ is the Schubert variety associated to
$w_{\tau}$ in the flag variety $G/B$ and we refer to the paragraph preceding
Definition 4.1.1 for the definition of $d_{ww_{\mathrm{sat}}^{-1}}$.

In the case where $w=w_{0}$ is the maximal element of
$(\mathcal{S}_{n})^{\Sigma}$ for the Bruhat order, (1.1) is proved in [13]
(without assuming Conjecture 3.2.2). Our proof of Theorem 1 is based on a
method from [11], where a weak version (also for $w=w_{0}$ only) is proved.
This method consists in finding two subspaces $T_{1}$ and $T_{2}$ of
$T_{X_{\mathrm{tri}}^{\square}(\bar{r}),x}$ and computing
$\dim(T_{1}+T_{2})=\dim{T_{1}}+\dim{T_{2}}-\dim(T_{1}\cap T_{2})$ in the
following way.

We first use the following construction due to [11, §2.2] and [28]. Denoting
by
$\mathbf{h}=(h_{\tau,1}<\ldots<h_{\tau,n})_{\tau}\in(\mathbb{Z}^{n})^{\Sigma}$
the Hodge-Tate weights of $r$, let
$\widetilde{W}_{\bar{r}}^{\mathbf{h}-\mathrm{cr}}$ be the reduced rigid
analytic space over $L$ parametrising the data of a framed deformation
$r^{\prime}$ of $\bar{r}$ which is crystalline generic with Hodge-Tate weights
$\mathbf{h}$, together with a refinement
$\underline{\varphi^{\prime}}=(\varphi^{\prime}_{1},\ldots,\varphi^{\prime}_{n})$
of $r^{\prime}$ (_i.e._ an ordering of the eigenvalues of the linearised
Frobenius $\Phi^{\prime}$ of $r^{\prime}$). There is a smooth map of $L$-rigid
analytic spaces
$h\colon\widetilde{W}_{\bar{r}}^{\mathbf{h}-\mathrm{cr}}\longrightarrow\prod_{\tau\in\Sigma}(G/B)^{\mathrm{rig}}$
which sends a point $(r^{\prime},\underline{\varphi^{\prime}})$ of
$\widetilde{W}_{\bar{r}}^{\mathbf{h}-\mathrm{cr}}$ to the Hodge filtration
seen as a flag on $\mathbf{D}_{\mathrm{dR}}(r^{\prime})$ relative to the basis
induced by the refinement $\underline{\varphi^{\prime}}$. Recall the Schubert
decomposition of the flag variety
$\prod_{\tau\in\Sigma}G/B=\coprod_{w^{\prime}\in(\mathcal{S}_{n})^{\Sigma}}\prod_{\tau\in\Sigma}Bw^{\prime}_{\tau}B/B$.
For $w^{\prime}\in(\mathcal{S}_{n})^{\Sigma}$, we consider the closed rigid
subspace $\widetilde{W}_{\bar{r},w^{\prime}}^{\mathbf{h}-\mathrm{cr}}$ of
$\widetilde{W}_{\bar{r}}^{\mathbf{h}-\mathrm{cr}}$ defined by
$\widetilde{W}_{\bar{r},w^{\prime}}^{\mathbf{h}-\mathrm{cr}}\coloneqq
h^{-1}\big{(}\prod_{\tau\in\Sigma}\overline{Bw^{\prime}_{\tau}B/B}^{\mathrm{rig}}\big{)}$.
There is a closed immersion of rigid analytic spaces
$\iota_{\mathbf{h},w^{\prime}}\colon\widetilde{W}_{\bar{r},w^{\prime}}^{\mathbf{h}-\mathrm{cr}}\longrightarrow
X_{\mathrm{tri}}^{\square}(\bar{r})$ defined by
$\iota_{\mathbf{h},w^{\prime}}(r^{\prime},\underline{\varphi^{\prime}})=\left(r^{\prime},z^{w^{\prime}(\mathbf{h})}\operatorname{unr}(\underline{\varphi^{\prime}})\right)$.
We have
$(r,\underline{\varphi})\in\widetilde{W}_{\bar{r},w}^{\mathbf{h}-\mathrm{cr}}$
and $\iota_{\mathbf{h},w}(r,\underline{\varphi})=x$; we then set $T_{1}$ to be
the image of the injection
$T_{\widetilde{W}_{\bar{r},w}^{\mathbf{h}-\mathrm{cr}},(r,\underline{\varphi})}\lhook\joinrel\relbar\joinrel\rightarrow
T_{X_{\mathrm{tri}}^{\square}(\bar{r}),x}$ induced by $\iota_{\mathbf{h},w}$.

We next define a rigid automorphism
$\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}\colon(r^{\prime},\underline{\delta^{\prime}})\longmapsto\left(r^{\prime},z^{w(\mathbf{h})-w_{\mathrm{sat}}(\mathbf{h})}\underline{\delta^{\prime}}\right)$
which maps $x_{\mathrm{sat}}$ to $x$, and we want
$\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}$ to map a locally closed subvariety of
$U_{\mathrm{tri}}^{\square}(\bar{r})$ containing $x_{\mathrm{sat}}$ into
$X_{\mathrm{tri}}^{\square}(\bar{r})$. We use the weight morphism
$\omega\colon
X_{\mathrm{tri}}^{\square}(\bar{r})\longrightarrow\mathcal{W}_{L}^{n}$ defined
by
$\omega(r^{\prime},\underline{\delta^{\prime}})=\underline{\delta^{\prime}}|_{(\mathcal{O}_{K}^{\times})^{n}}$,
where $\mathcal{W}_{L}$ is the rigid analytic space over $L$ parametrising
continuous characters of $\mathcal{O}_{K}^{\times}$, and cut out a closed
subspace of characters
$\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}\subset\mathcal{W}_{L}^{n}$
with the equations
$\operatorname{wt}_{\tau}(\eta_{w_{\mathrm{sat},\tau}(i)}\eta_{w_{\tau}(i)}^{-1})=h_{\tau,i}-h_{\tau,w_{\mathrm{sat},\tau}^{-1}w_{\tau}(i)}$.
Then, using the patched eigenvariety $X_{p}(\bar{\rho})$ of Breuil-Hellmann-
Schraen [12, §3] (where $\bar{\rho}$ is a globalisation of $\bar{r}$ given by
a result of Emerton-Gee [21, Theorem 1.2.3]) and assuming Conjecture 3.2.2, we
show that if $(w_{\mathrm{sat}},w)$ is a good pair of
$(\mathcal{S}_{n})^{\Sigma}$, then there exists an open neighborhood
$U_{x_{\mathrm{sat}}}$ of $x_{\mathrm{sat}}$ in
$U_{\mathrm{tri}}^{\square}(\bar{r})$ such that
$\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}$ induces a Zariski-closed embedding of
reduced rigid $L$-analytic spaces
$\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}\colon\overline{U_{x_{\mathrm{sat}}}\times_{\mathcal{W}^{n}_{L}}\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}}\lhook\joinrel\relbar\joinrel\rightarrow
X_{\mathrm{tri}}^{\square}(\bar{r})$. We then set $T_{2}$ to be the image of
the injection
$T_{U_{x_{\mathrm{sat}}}\times_{\mathcal{W}^{n}_{L}}\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L},x_{\mathrm{sat}}}\lhook\joinrel\relbar\joinrel\rightarrow
T_{X_{\mathrm{tri}}^{\square}(\bar{r}),x}$ induced by
$\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}$.

It is quite straightforward to compute
$\dim{T_{2}}=\dim{X_{\mathrm{tri}}^{\square}(\bar{r})}-d_{ww_{\mathrm{sat}}^{-1}}$.
The computation of $\dim{T_{1}}$ and $\dim(T_{1}\cap T_{2})$ is where our
approach differs from that of [11]. We prove, using Berger’s dictionary
between cristalline $(\varphi,\Gamma_{K})$-modules over the Robba ring and
filtered $\varphi$-modules ([4, Théorème A], [3, Théorème 3.6]), the following
generalisation of [1, Prop. 2.4.1].

######  Proposition 2 (Proposition 2.2.4).

Let $A$ be a local artinian $L$-algebra with residue field $L$. Let
$r_{A}\colon\mathcal{G}_{K}\longrightarrow\operatorname{GL}_{n}(A)$ be a
regular crystalline representation of rank $n\in\mathbb{Z}_{>0}$ over $A$,
with Hodge-Tate weights $\mathbf{h}\in(\mathbb{Z}^{n})^{\Sigma}$. Assume that
there exist $\mathbf{k}\in\mathbb{Z}^{\Sigma}$ and
$\underline{\varphi}_{A}\in(A^{\times})^{n}$, with the $\varphi_{A,i}$ having
pairwise distinct reductions in $L$, such that $r_{A}$ is trianguline of
parameter $z^{\mathbf{k}}\operatorname{unr}(\underline{\varphi}_{A})$. Then
there exists $w_{A}\in(\mathcal{S}_{n})^{\Sigma}$ such that
$\mathbf{k}=w_{A}(\mathbf{h})$, and the linearised Frobenius $\Phi_{A}$ on
$\mathbf{D}_{\mathrm{crys}}(r_{A})$ admits $\underline{\varphi}_{A}$ as a set
of eigenvalues. Furthermore, the flag
$\operatorname{Fil}_{\bullet}\mathbf{D}_{\mathrm{dR}}(r_{A})$ relative to the
basis of $\mathbf{D}_{\mathrm{dR}}(r_{A})$ induced by the refinement
$\underline{\varphi}_{A}$, seen as a point of $\prod_{\tau\in\Sigma}(G/B)(A)$,
lies in the cell $\prod_{\tau\in\Sigma}(Bw_{A,\tau}B/B)(A)$.

Applied to the algebra $A=k(x)[\varepsilon]/(\varepsilon^{2})$ of
infinitesimal numbers over the local residue field of $x$, Proposition 2 shows
that the commutative diagram

${T_{\widetilde{W}_{\bar{r},w_{\mathrm{sat}}}^{\mathbf{h}-\mathrm{cr}},(r,\underline{\varphi})}}$${T_{\overline{U_{x_{\mathrm{sat}}}\times_{\mathcal{W}^{n}_{L}}\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}},x_{\mathrm{sat}}}}$${T_{\widetilde{W}_{\bar{r},w}^{\mathbf{h}-\mathrm{cr}},(r,\underline{\varphi})}}$${T_{X_{\mathrm{tri}}^{\square}(\bar{r}),x}\,.}$$\scriptstyle{\mathop{}\mathopen{}\mathrm{d}\iota_{\mathbf{h},w_{\mathrm{sat}}}}$$\scriptstyle{\subseteq}$$\scriptstyle{\mathop{}\mathopen{}\mathrm{d}\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}}$$\scriptstyle{\mathop{}\mathopen{}\mathrm{d}\iota_{\mathbf{h},w}}$

is a cartesian square. This implies that $\dim(T_{1}\cap
T_{2})=\dim{T_{\widetilde{W}_{\bar{r},w_{\mathrm{sat}}}^{\mathbf{h}-\mathrm{cr}},(r,\underline{\varphi})}}$,
from which we finally get (1.1) without difficulty.

#### Notation

We conclude this introduction with the main notation of the paper.

We fix a prime number $p$ and a finite extension $K$ of $\mathbb{Q}_{p}$,
equipped with the $p$-adic norm $\lvert\cdot\rvert_{K}$ normalised so that
$\lvert p\rvert_{K}=p^{-[K:\mathbb{Q}_{p}]}$. We write $\mathcal{G}_{K}$ for
the absolute Galois group of $K$ and
$\Gamma_{K}\coloneqq\operatorname{Gal}(K(\mu_{p^{\infty}})/K)$. We also write
$K_{0}$ for the maximal unramified extension of $\mathbb{Q}_{p}$ inside $K$
and $\varphi$ for the Frobenius endomorphism of $K_{0}$, normalised so that it
induces $x\longmapsto x^{p}$ on the residue field. If $L$ is another finite
extension of $\mathbb{Q}_{p}$, we say that $L$ splits $K$ when the set
$\operatorname{Hom}_{\mathbb{Q}_{p}}(K,L)$ of homomorphisms of fields over
$\mathbb{Q}_{p}$ has cardinality $[K:\mathbb{Q}_{p}]$; in that case we write
$\Sigma\coloneqq\operatorname{Hom}_{\mathbb{Q}_{p}}(K,L)$.

We write $\mathbf{Ar}_{\mathbb{Q}_{p}}$ for the category of local artinian
$\mathbb{Q}_{p}$-algebras whose residue field is a finite extension of
$\mathbb{Q}_{p}$ (_i.e._ finite-dimensional local $\mathbb{Q}_{p}$-algebras).
For $A\in\mathbf{Ar}_{\mathbb{Q}_{p}}$, we write $\mathfrak{m}_{A}$ for its
maximal ideal.

For a ring $R$, we write $\mathbf{Fil}^{\bullet}_{R}$ for the category of
filtered $R$-modules, where objects are $R$-modules equipped with a decreasing
exhaustive separated filtration by $R$-submodules, and morphisms are
compatible families of $R$-linear maps in each degree. For
$A\in\mathbf{Ar}_{\mathbb{Q}_{p}}$, we write
$\mathbf{Rep}_{A}(\mathcal{G}_{K})$ for the category of representations of
$\mathcal{G}_{K}$ over $A$, where objects are free $A$-modules of finite type
equipped with a continuous linear action of $\mathcal{G}_{K}$, and morphisms
are $\mathcal{G}_{K}$-equivariant $A$-linear maps. We use the usual functors
of Fontaine
$V\mapsto\left(\mathbf{D}_{\mathrm{dR}}(V),\operatorname{Fil}^{\bullet}\mathbf{D}_{\mathrm{dR}}(V)\right)$
from $\mathbf{Rep}_{A}(\mathcal{G}_{K})$ to
$\mathbf{Fil}^{\bullet}_{K\otimes_{\mathbb{Q}_{p}}A}$ and
$V\mapsto(\mathbf{D}_{\mathrm{crys}}(V),\varphi)$ from
$\mathbf{Rep}_{A}(\mathcal{G}_{K})$ to the category of $K_{0}$-vector spaces
equipped with a $\varphi$-semilinear endomorphism (called Frobenius and still
noted $\varphi$). For $V\in\mathbf{Rep}_{A}(\mathcal{G}_{K})$, we write
$\Phi\coloneqq\varphi^{[K_{0}:\mathbb{Q}_{p}]}$ for the linearised Frobenius,
which is a linear endomorphism of $\mathbf{D}_{\mathrm{crys}}(V)$ (note that
depending on the context, $\Phi$ can also denote a root system).

If $A\in\mathbf{Ar}_{\mathbb{Q}_{p}}$ is an algebra over its residue field
$L\coloneqq A/\mathfrak{m}_{A}A$ and $L$ splits $K$, for
$V\in\mathbf{Rep}_{A}(\mathcal{G}_{K})$ which is de Rham and $\tau\in\Sigma$,
we call _$\tau$ -Hodge-Tate weights_ the integers $h_{\tau}$ such that
$\operatorname{rk}_{A}\left(\operatorname{Fil}^{-h_{\tau}}\mathbf{D}_{\mathrm{dR}}(V)/\operatorname{Fil}^{-h_{\tau}+1}\mathbf{D}_{\mathrm{dR}}(V)\right)\neq
0$. We count them with multiplicities, so that there are exactly
$\operatorname{rk}_{A}(V)$ $\tau$-Hodge-Tate weights. We say that $V$ is
_regular_ when the $\tau$-Hodge-Tate weights are distinct for each
$\tau\in\Sigma$, _i.e._ all $\tau$-Hodge-Tate weights have multiplicity 1.

With the same hypotheses on $A$, given
$\mathbf{k}=(k_{\tau})_{\tau\in\Sigma}\in\mathbb{Z}^{\Sigma}$, we write
$z^{\mathbf{k}}$ for the character $K^{\times}\rightarrow
A^{\times},\,z\mapsto\prod_{\tau\in\Sigma}\tau(z)^{k_{\tau}}$. For $a\in
A^{\times}$, we write $\operatorname{unr}(a)$ for the character
$K^{\times}\rightarrow A^{\times},\,z\mapsto a^{v_{K}(z)}$ where $v_{K}$ is
the valuation on $K$ normalised to have image $\mathbb{Z}$ (for instance,
$\operatorname{unr}(p^{[K_{0}:\mathbb{Q}_{p}]})(p)=p^{[K:\mathbb{Q}_{p}]}$;
note that $v_{K}$ is not the valuation of the norm $\lvert\cdot\rvert_{K}$).
If $\delta\colon K^{\times}\longrightarrow A^{\times}$ is any continuous
character, then $\delta$ is locally $\mathbb{Q}_{p}$-analytic so it is a
morphism of $p$-adic Lie groups; we write
$\mathop{}\mathopen{}\mathrm{d}\delta\colon K\longrightarrow A$ for its
differential, which is a morphism of vector spaces over $\mathbb{Q}_{p}$. Note
that the $\tau\colon K\lhook\joinrel\relbar\joinrel\rightarrow L$ composed
with $L\longrightarrow A$, for $\tau\in\Sigma$, form a basis of
$\operatorname{Hom}_{\mathbb{Q}_{p}\text{-vect. sp.}}(K,A)$ over $A$, which
induces $\operatorname{Hom}_{\mathbb{Q}_{p}\text{-vect.\ sp.}}(K,A)\simeq
A^{\Sigma}$; under this isomorphism, we write
$\operatorname{wt}_{\tau}(\delta)\in A$ for the weights of $\delta$ defined by
$\mathop{}\mathopen{}\mathrm{d}\delta=(\operatorname{wt}_{\tau}(\delta))_{\tau\in\Sigma}$.
If $\delta^{\prime}$ is a continuous character
$\mathcal{O}_{K}^{\times}\longrightarrow A^{\times}$, we write
$\operatorname{wt}_{\tau}(\delta^{\prime})\coloneqq\operatorname{wt}_{\tau}(\delta)$
where $\delta\colon K^{\times}\longrightarrow A^{\times}$ is any character
such that $\delta|_{\mathcal{O}_{K}^{\times}}=\delta^{\prime}$.

For any $A\in\mathbf{Ar}_{\mathbb{Q}_{p}}$, we write $\mathcal{R}_{A,K}$ for
the relative Robba ring over $A$ for $K$ (see [26, Definition 2.2.2], where it
is written $\mathcal{R}_{A}(\pi_{K})$), which is equipped with a Frobenius
endomorphism written $\varphi$ and a $\Gamma_{K}$-action. We write
$\mathbf{Mod}^{\varphi,\Gamma_{K}}_{A}$ for the category of
$(\varphi,\Gamma_{K})$-modules over $A$; its objects are
$\mathcal{R}_{A,K}$-modules $D$ which are finitely-generated over
$\mathcal{R}_{A,K}$ and free over $\mathcal{R}_{K}$, equipped with semilinear
actions of $\varphi$ and $\Gamma_{K}$ which commute and are continuous (as
actions of free $\mathcal{R}_{K}$-modules), and such that
$\mathcal{R}_{K}\varphi(D)=D$; its morphisms are $\mathcal{R}_{A,K}$-linear
maps which commute with the respective actions of $\varphi$ and $\Gamma_{K}$.
We use the usual functor $\mathbf{D}_{\mathrm{rig}}$ from
$\mathbf{Rep}_{A}(\mathcal{G}_{K})$ to $\mathbf{Mod}^{\varphi,\Gamma_{K}}_{A}$
(see [26, Theorem 2.2.17]). To a continuous character
$\delta\colon\mathbb{Q}_{p}^{\times}\longrightarrow A^{\times}$, we associate
a $(\varphi,\Gamma_{K})$-module $\mathcal{R}_{A,K}(\delta)$ of rank one over
$\mathcal{R}_{A,K}$, defined in [26, Construction 6.2.4].

#### Acknowledgements

The results of this paper are part of the author’s PhD work at the Laboratoire
de Mathématiques d’Orsay. The author would like to thank his PhD advisor
Christophe Breuil for his enlightening supervision, Ariane Mézard and Benjamin
Schraen for valuable exchanges, as well as Gabriel Dospinescu and Ruochuan Liu
for their careful review of the author’s PhD dissertation. Finally, the
author’s debt to the work of Breuil-Hellmann-Schraen will be obvious to the
reader.

##  2\. Crystalline trianguline representations

We recall the notion of refinement of a crystalline representation $V$ over
$A\in\mathbf{Ar}_{\mathbb{Q}_{p}}$ in relation with that of triangulation of
the associated $(\varphi,\Gamma)$-module. For $V$ regular, we characterise the
refinement arising from a triangulation, relatively to the Hodge filtration,
as a point of a Schubert cell of the flag variety of
$\mathbf{D}_{\mathrm{dR}}(V)$.

###  2.1. Refinements of a crystalline representation $V$ and flags on
$\mathbf{D}_{\mathrm{dR}}(V)$

We begin with a lemma about the functor of points of a Schubert cell of the
flag variety.

######  Definition 2.1.1.

Let $R$ be a ring and let $M$ be a free $R$-module. A _complete flag_ on $M$
is an increasing sequence $\operatorname{Fil}_{\bullet}M$ of $R$-submodules of
$M$

$\operatorname{Fil}_{0}M\coloneqq\\{0\\}\subset\ldots\subset\operatorname{Fil}_{i}M\subset\ldots\subset\operatorname{Fil}_{n}M\coloneqq
M$

with $n=\operatorname{rk}_{R}(M)$, such that
$\operatorname{Fil}_{i}M/\operatorname{Fil}_{i-1}M$ is free of rank $1$ over
$R$ for all $1\leq i\leq n$.

Fix an integer $n\in\mathbb{Z}_{>0}$, a ring $R$ and a free module $M$ of rank
$n$ over $R$. Equip $M$ with a complete flag
$\operatorname{Fil}_{\bullet}M=(\operatorname{Fil}_{i}(M))_{0\leq i\leq n}$,
and fix a basis basis $(e_{i})_{1\leq i\leq n}$ of
$\operatorname{Fil}_{\bullet}M$, _i.e._ a family of elements of $M$ such that
$e_{i}\in\operatorname{Fil}_{i}M$ and
$e_{i}\,\mathrm{mod}\,\operatorname{Fil}_{i-1}M$ generates
$\operatorname{Fil}_{i}M/\operatorname{Fil}_{i-1}M$ for all $1\leq i\leq n$.
Let $G\coloneqq\operatorname{GL}_{n,\mathbb{Q}_{p}}$ be the general linear
group over $\mathbb{Q}_{p}$ and let $B$ be its Borel subgroup of upper
triangular matrices; the Weyl group of $G$ is the group $\mathcal{S}_{n}$ of
bijections of $\\{1,\ldots,n\\}$ to itself.

The map $G(R)\longrightarrow\\{\text{complete flags of }M\\}$ which sends a
matrix $(a_{ij})_{1\leq i,j\leq n}\in G(R)$ to the flag
$\left(\bigoplus_{j\leq k}Rb_{j}\right)_{1\leq k\leq n}$, where
$b_{j}=\sum_{1\leq i\leq n}a_{ij}e_{j}$, factors through $B(R)$ and induces an
bijection
$G(R)/B(R)\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}\\{\text{complete
flags of }M\\}$, hence an injection

(2.1) $\\{\text{complete flags of
}M\\}\lhook\joinrel\relbar\joinrel\rightarrow(G/B)(R)$

(see [25, §5.4, §5.5]). We call the image under (2.1) of a given flag the
_position of the flag relative to $\operatorname{Fil}_{\bullet}M$_.

Recall the Schubert decomposition on the flag variety $G/B$. We embed
$\mathcal{S}_{n}$ in $G$ by mapping each $w\in\mathcal{S}_{n}$ to the matrix
$\dot{w}=(w_{ij})_{1\leq i,j\leq n}$ defined by $w_{ij}=1$ if $i=w(j)$ and
$w_{ij}=0$ if $i\neq w(j)$. Let $w\in\mathcal{S}_{n}$. The left-action of $B$
on $G$ by multiplication induces an action on $G/B$; write $BwB/B$ for the
orbit of $\dot{w}B/B$ under this action. Then $BwB/B$ is a smooth locally
closed subvariety of $G/B$ called a _Schubert cell_ , and we have the
decomposition

(2.2) $G/B=\coprod_{w\in W}BwB/B$

(see [31, Theorem 21.73]). The cells $BwB/B$ are stable under the left action
of $B$, thus their Zariski closure $\overline{BwB/B}$ too, and this action is
transitive on $BwB/B$. Thus, a closed cell $\overline{BwB/B}$ intersects an
open cell $Bw^{\prime}B/B$ if and only if
$Bw^{\prime}B\subseteq\overline{BwB/B}$, which happens if and only if
$w^{\prime}\preceq w$ for the Bruhat order (see [6, Theorem 2.11]). Therefore
(2.2) gives a good stratification of $G/B$, and we have a more general
decomposition into locally closed strata

$\overline{BwB/B}=\coprod_{w^{\prime}\preceq w}Bw^{\prime}B/B$

for all $w\in W$.

For flags, Schubert cells are elementarily described as such: given a flag
$\mathcal{F}$ of $M$, its position relative to $\operatorname{Fil}_{\bullet}M$
is in the closed cell $(\overline{BwB/B})(R)$ if and only if $\mathcal{F}$ can
be represented by a matrix $(a_{ij})_{1\leq i,j\leq n}\in G(R)$ such that
$a_{ij}=0$ for all $i>w(j)$; and it is in the open cell $(BwB/B)(R)$ if and
only if we can impose the additional condition $a_{w(j),j}\in R^{\times}$ for
all $j$. (Thus we see that the Schubert decomposition induces a partition of
the $R$-points if and only if $R$ is a field).

######  Lemma 2.1.2.

Let $R$ be a ring, let $M$ be a free $R$-module of rank $n\in\mathbb{Z}_{>0}$
equipped with a complete flag $\operatorname{Fil}_{\bullet}M$. Let
$x\in(G/B)(R)$ be a point corresponding to a flag $(\mathcal{F}_{j})_{1\leq
j\leq n}$ of $M$ under (2.1). Then $x\in(BwB/B)(R)$ for some
$w\in\mathcal{S}_{n}$ if and only if
$(\mathcal{F}_{j}\cap\operatorname{Fil}_{w(j)})/(\mathcal{F}_{j}\cap\operatorname{Fil}_{w(j)-1})$
is free of rank $1$ over $R$ for all $1\leq j\leq n$.

###### Proof.

Let $(e_{i})_{1\leq i\leq n}$ be a basis of $\operatorname{Fil}_{\bullet}M$.

If $x\in(BwB/B)(R)$, then there is a basis $(b_{j})_{1\leq j\leq n}$ of
$\mathcal{F}_{\bullet}$ such that each $b_{j}$ can be written
$b_{j}=\sum_{i\leq w(j)}r_{ij}e_{i}$ with $r_{w(j),j}\in R^{\times}$. Then,
for all $1\leq j\leq n$, the injective morphism

$(\mathcal{F}_{j}\cap\operatorname{Fil}_{w(j)})/(\mathcal{F}_{j}\cap\operatorname{Fil}_{w(j)-1})\lhook\joinrel\relbar\joinrel\rightarrow\operatorname{Fil}_{w(j)}/\operatorname{Fil}_{w(j)-1}$

sends
$r_{w(j),j}^{-1}b_{j}\,\mathrm{mod}\,(\mathcal{F}_{j}\cap\operatorname{Fil}_{w(j)-1})$
to $e_{w(j)}\,\mathrm{mod}\,\operatorname{Fil}_{w(j)-1}$, so it is an
isomorphism.

Conversely, assume that
$(\mathcal{F}_{j}\cap\operatorname{Fil}_{w(j)})/(\mathcal{F}_{j}\cap\operatorname{Fil}_{w(j)-1})$
is free of rank $1$ over $R$ for all $1\leq j\leq n$; it has a basis element
that lifts to some $b_{j}\in\mathcal{F}_{j}\cap\operatorname{Fil}_{w(j)}$.
Then each $b_{j}$ can be written $b_{j}=\sum_{i\leq w(j)}r_{ij}e_{i}$ in
$\mathcal{F}_{j}$ with $r_{w(j),j}\in R^{\times}$. Therefore it is enough to
prove that $(b_{j})_{1\leq j\leq n}$ forms a basis of $\mathcal{F}_{\bullet}$,
_i.e._ that $b_{j}\,\mathrm{mod}\,\mathcal{F}_{j-1}$ is a basis of
$\mathcal{F}_{j}/\mathcal{F}_{j-1}$ for all $1\leq j\leq n$. We proceed by
induction on $j$. Let $b=\sum_{1\leq i\leq n}r_{i}e_{i}\in\mathcal{F}_{j}$
such that $b\,\mathrm{mod}\,\mathcal{F}_{j-1}$ is a basis of
$\mathcal{F}_{j}/\mathcal{F}_{j-1}$. As $b_{j}\in\mathcal{F}_{j}$, one can
write

(2.3) $b_{j}=\sum_{k<j}\lambda_{k}b_{k}+\lambda b$

for some $\lambda_{k},\lambda\in R$. Let $I$ be the ideal of $R$ generated by
$\lambda$. If $\lambda_{k}\notin I$ for some $k<j$, take such a $\lambda_{k}$
with $w(k)$ maximal. Then taking the coordinate along $e_{w(k)}$ of (2.3)
gives

$r_{w(k),j}=\lambda_{k}r_{w(k),k}+\sum_{k^{\prime}<j\text{ s.t.\
}w(k^{\prime})>w(k)}\lambda_{k^{\prime}}r_{w(k),k^{\prime}}+\lambda
r_{w(k)}\notin I$

since $r_{w(k),k}\in R^{\times}$. In particular $r_{w(k),j}\neq 0$, so
$w(k)\leq w(j)$. This proves that $\lambda_{k}\in I$ for all $k<j$ such that
$w(k)>w(j)$. Then (2.3) along $e_{w(j)}$ gives

$r_{w(j),j}=\sum_{k<j\text{ s.t.\ }w(k)>w(j)}\lambda_{k}r_{w(j),k}+\lambda
r_{w(j)}\in I\,.$

As $r_{w(j),j}\in R^{\times}$, we deduce $I=R$. Hence $\lambda\in R^{\times}$,
so $b_{j}\,\mathrm{mod}\,\mathcal{F}_{j-1}$ is a basis of
$\mathcal{F}_{j}/\mathcal{F}_{j-1}$; this concludes the proof. ∎

We are mainly concerned with the situation of $R=K\otimes_{\mathbb{Q}_{p}}A$
and $M=\mathbf{D}_{\mathrm{dR}}(V)$, where we fix an algebra
$A\in\mathbf{Ar}_{\mathbb{Q}_{p}}$ and a de Rham representation
$V\in\mathbf{Rep}_{A}(\mathcal{G}_{K})$. More specifically, we make the
following two assumptions: (i) $A$ is an algebra over a finite extension $L$
of $\mathbb{Q}_{p}$ which is the residue field of $A$ and which splits $K$,
and (ii) $V$ is regular.

Then we define a complete flag
$\operatorname{Fil}_{\bullet}\mathbf{D}_{\mathrm{dR}}(V)$ on the free
$(K\otimes_{\mathbb{Q}_{p}}A)$-module $\mathbf{D}_{\mathrm{dR}}(V)$ by
$\operatorname{Fil}_{0}\mathbf{D}_{\mathrm{dR}}(V)\coloneqq 0$ and

$\operatorname{Fil}_{i}\mathbf{D}_{\mathrm{dR}}(V)\coloneqq\bigoplus_{\tau\in\Sigma}\operatorname{Fil}^{-h_{\tau,i}}\mathbf{D}_{\mathrm{dR}}(V)_{\tau},\quad
1\leq i\leq n$

where $n=\operatorname{rk}_{A}(V)$ and $h_{\tau,1}<\ldots<h_{\tau,n}$ are the
$\tau$-Hodge-Tate weights of $V$.

Next, we further assume that $V$ is crystalline. Recall the following
definition from [1, §2.4].

######  Definition 2.1.3.

Let $A\in\mathbf{Ar}_{\mathbb{Q}_{p}}$, let
$V\in\mathbf{Rep}_{A}(\mathcal{G}_{K})$ be any crystalline representation. A
_refinement_ of $V$ is a complete flag on $\mathbf{D}_{\mathrm{crys}}(V)$
which is stable by the linearised Frobenius $\Phi$.

A refinement $\mathcal{F}_{\bullet}$ of $V$ induces a complete flag
$K\otimes_{K_{0}}\mathcal{F}_{\bullet}$ on $\mathbf{D}_{\mathrm{dR}}(V)$, and
the position of $\operatorname{Fil}_{\bullet}\mathbf{D}_{\mathrm{dR}}(V)$
relative to $K\otimes_{K_{0}}\mathcal{F}_{\bullet}$ determines an $A$-point of

$\left(\operatorname{Res}_{K/\mathbb{Q}_{p}}(G/B)_{K}\right)\times_{\mathbb{Q}_{p}}L\simeq\prod_{\tau\in\Sigma}(G/B)_{L}$

where $G=\operatorname{GL}_{n,\mathbb{Q}_{p}}$ and $B$ is the Borel subgroup
of upper triangular matrices.

######  Remark 2.1.4.

A refinement of $V$ determines an ordered sequence of eigenvalues of $\Phi$,
_i.e._ a sequence
$(\varphi_{1},\ldots,\varphi_{n})\in(K_{0}\otimes_{\mathbb{Q}_{p}}A)^{n}$ such
that the characteristic polynomial of $\Phi$ can be written
$P_{\Phi}(\lambda)=\prod_{i=1}^{n}(\lambda-\varphi_{i})\in(K_{0}\otimes_{\mathbb{Q}_{p}}A)[\lambda]$.
The following proposition is a partial converse of this fact.

######  Proposition 2.1.5.

Let $\varphi_{1},\ldots,\varphi_{n}\in A$ be such that
$(1\otimes\varphi_{i})_{1\leq i\leq n}$ is a sequence of eigenvalues of the
Frobenius $\Phi$ on $\mathbf{D}_{\mathrm{crys}}(V)$ in the sense of Remark
2.1.4. Assume that the $\varphi_{i}$ have pairwise distinct reductions modulo
$\mathfrak{m}_{A}$. Then there is a unique refinement of $V$ associated to
$(1\otimes\varphi_{i})_{1\leq i\leq n}$.

The proof uses the following classic result of linear algebra.

######  Lemma 2.1.6.

Let $R$ be a commutative ring with unity, let $M$ be an $R$-module and
$f\in\operatorname{End}_{R}(M)$.

  1. _(1)_

Assume that there are $\lambda_{1},\ldots,\lambda_{m}\in R$ for some
$m\in\mathbb{Z}_{>0}$, with $\lambda_{i}-\lambda_{j}\in R^{\times}$ for all
$1\leq i<j\leq m$, such that
$(f-\lambda_{1})\cdots(f-\lambda_{m})=0\in\operatorname{End}_{R}(M)$. Then $M$
is the direct sum of eigenspaces

(2.4) $M=\ker(f-\lambda_{1})\oplus\cdots\oplus\ker(f-\lambda_{m}).$

  2. _(2)_

Assume that $R$ is local with maximal ideal $\mathfrak{m}$, and that $M$ is
free of rank $n\in\mathbb{Z}_{>0}$. Also assume that the characteristic
polynomial of $f$ can be written
$P_{f}(\lambda)=\prod_{i=1}^{n}(\lambda-\lambda_{i})\in R[\lambda]$ with the
$\lambda_{i}$ having pairwise distinct reductions $\overline{\lambda_{i}}$
modulo $\mathfrak{m}$. Then the eigenspaces $\ker(f-\lambda_{i})$, $1\leq
i\leq n$, are free of rank $1$ over $R$ and any choice of generators of these
eigenspaces form a basis of $M$.

###### Proof.

We first prove (1). One can find polynomials $P_{1},\ldots,P_{m}\in
R[\lambda]$ such that

(2.5) $\sum_{1\leq i\leq m}P_{i}(\lambda)\prod_{j\neq
i}(\lambda-\lambda_{j})=1\in R[\lambda]\,;$

we show this by induction on $m$. The case $m=1$ is trivial with $P_{1}=1$;
for $m>1$, let $Q_{1},\ldots,Q_{m-1}\in R[\lambda]$ satisfy

$\sum_{1\leq i\leq m-1}Q_{i}(\lambda)\prod_{j\neq
i,m}(\lambda-\lambda_{j})=1\in R[\lambda]\,.$

For $l\coloneqq\prod_{1\leq i<m}(\lambda_{m}-\lambda_{i})\in R^{\times}$,
there is $Q\in R[\lambda]$ such that $\prod_{1\leq
i<m}(\lambda-\lambda_{i})=Q(\lambda)(\lambda-\lambda_{m})+l$. Then taking
$P_{i}\coloneqq-l^{-1}Q_{i}Q\in R[\lambda]$ for $1\leq i<m$ and
$P_{m}\coloneqq l^{-1}$ gives (2.5).

Applying (2.5) to $\lambda=f$, we see that any $v\in M$ decomposes as
$v=v_{1}+\ldots+v_{m}$, where $v_{i}\coloneqq\prod_{j\neq
i}(f-\lambda_{j})P_{i}(f)v$ for $1\leq i\leq m$, and obviously
$v_{i}\in\ker(f-\lambda_{i})$ by the assumption. Now to check that the sum
$M=\ker(f-\lambda_{1})+\ldots+\ker(f-\lambda_{m})$ is direct, let
$v_{1}+\ldots+v_{m}=0$, with $v_{i}\in\ker(f-\lambda_{i})$. We show by
induction on the number of nonzero $v_{i}$ that $v_{i}=0$ for all $i$. Take
$i_{0}$ such that $v_{i_{0}}\neq 0$; applying $f-\lambda_{i_{0}}$ gives
$\sum_{i\neq i_{0}}(\lambda_{i}-\lambda_{i_{0}})v_{i}=0$, so by induction
$(\lambda_{i}-\lambda_{i_{0}})v_{i}=0$ for all $1\leq i\leq m$. We get
$v_{i}=0$ for $i\neq i_{0}$ since $\lambda_{i}-\lambda_{i_{0}}$ is a unit,
then $v_{i_{0}}=0$. This proves (1).

We now prove (2). The reduction of $f$ to the $(R/\mathfrak{m})$-vector space
$M/\mathfrak{m}M$ has pairwise distinct eigenvalues $\overline{\lambda_{i}}$;
choose a corresponding diagonalisation basis $(\overline{v_{i}})_{1\leq i\leq
n}$. Choose a lift $w_{i}\in M$ of each $\overline{v_{i}}$. By (1) and Cayley-
Hamilton’s theorem, we can write $w_{i}=v_{i,1}+\ldots+v_{i,n}$ with
$v_{i,j}\in\ker(f-\lambda_{j})$ for all $1\leq i,j\leq n$. Then, in
$M/\mathfrak{m}M$,

$\sum_{1\leq j\leq
n}(\overline{\lambda_{j}}-\overline{\lambda_{i}})\overline{v_{i,j}}=f(\overline{v_{i}})-\overline{\lambda_{i}}\overline{v_{i}}=0$

for all $1\leq i\leq n$. Applying (1) to $M/\mathfrak{m}M$, we see that the
eigenvectors
$(\overline{\lambda_{j}}-\overline{\lambda_{i}})\overline{v_{i,j}}$ are zero
for all $1\leq i,j\leq n$, which means, since the $\overline{\lambda_{i}}$ are
pairwise distinct, that $v_{i,j}\in\mathfrak{m}$ for all $i\neq j$. Therefore
$v_{i}\coloneqq v_{i,i}=w_{i}-\sum_{j\neq i}v_{i,j}$ lifts $\overline{v_{i}}$.
By Nakayama’s lemma, the $v_{i}$, $1\leq i\leq n$, generate $M$. Since $M$ is
free of rank $n$, the $v_{i}$ actually form a basis of $M$. Since
$Rv_{i}\subseteq\ker(f-\lambda_{i})$ for each $i$, (2.4) gives
$Rv_{i}=\ker(f-\lambda_{i})$. This proves (2). ∎

###### Proof of Proposition 2.1.5.

Write $K_{0}\otimes_{\mathbb{Q}_{p}}A=\bigoplus_{\tau\in\Sigma_{0}}Ae_{\tau}$,
where $e_{\tau}^{2}=e_{\tau}$ for
$\tau\in\Sigma_{0}\coloneqq\operatorname{Hom}_{\mathbb{Q}_{p}}(K_{0},L)$. Then
$\mathbf{D}_{\mathrm{crys}}(V)=\bigoplus_{\tau\in\Sigma_{0}}\mathbf{D}_{\mathrm{crys}}(V)_{\tau}$
where each $\mathbf{D}_{\mathrm{crys}}(V)_{\tau}\coloneqq
e_{\tau}\mathbf{D}_{\mathrm{crys}}(V)$ is a free $A$-module of rank $n$. Each
$\mathbf{D}_{\mathrm{crys}}(V)_{\tau}$ is stabilised by $\Phi$, and the
induced $\Phi_{\tau}$ is an endomorphism of $A$-modules with sequence of
eigenvalues $(\varphi_{i})_{1\leq i\leq n}$. Applying Lemma 2.1.6(2), we get a
diagonalisation basis $(b_{\tau,i})_{1\leq i\leq n}$ of
$\mathbf{D}_{\mathrm{crys}}(V)_{\tau}$ for $(\varphi_{i})_{1\leq i\leq n}$.
Setting $b_{i}\coloneqq\sum_{\tau\in\Sigma_{0}}b_{\tau,i}$, the family
$(b_{i})_{1\leq i\leq n}$ is a diagonalisation basis of
$\mathbf{D}_{\mathrm{crys}}(V)$ for $(1\otimes\varphi_{i})_{1\leq i\leq n}$.

Let $\mathcal{F}_{\bullet}$ be any refinement of $V$ associated to
$(1\otimes\varphi_{i})_{1\leq i\leq n}$. Let $(a_{i})_{1\leq i\leq n}$ be a
basis of $\mathcal{F}_{\bullet}$, and write $a_{j}=\sum_{j=1}^{n}a_{ij}b_{i}$
for all $1\leq j\leq n$. Using an induction on the columns of the matrix
$(a_{ij})_{1\leq i,j\leq n}$, we see that it is upper triangular, hence has
invertible diagonal entries. Therefore $(b_{i})_{1\leq i\leq n}$ is a basis of
$\mathcal{F}_{\bullet}$; this proves unicity of $\mathcal{F}_{\bullet}$.
Conversely, the flag arising from $(b_{i})_{1\leq i\leq n}$ induces the
ordered sequence of eigenvalues $(1\otimes\varphi_{i})_{1\leq i\leq n}$; this
proves existence. ∎

###  2.2. Triangulations and refinements

We recall the notion of triangular $(\varphi,\Gamma)$-module from [1,
Definition 2.3.2] and [13, Definition 3.3.8] in the following definition.

######  Definition 2.2.1.

Let $A\in\mathbf{Ar}_{\mathbb{Q}_{p}}$. Let $D$ be a
$(\varphi,\Gamma_{K})$-module over $\mathcal{R}_{A,K}$ which is free as an
$\mathcal{R}_{A,K}$-module. A _triangulation_ of $D$ is a strictly increasing
sequence $\operatorname{Fil}_{\bullet}D$ of $(\varphi,\Gamma_{K})$-submodules
over $\mathcal{R}_{A,K}$

$\operatorname{Fil}_{0}D\coloneqq\\{0\\}\subsetneq\ldots\subsetneq\operatorname{Fil}_{i}D\subsetneq\ldots\subsetneq\operatorname{Fil}_{d}D\coloneqq
D$

with $d=\operatorname{rk}_{\mathcal{R}_{A,K}}(D)$, such that each
$\operatorname{Fil}_{i}(D)$ is a direct summand of $D$ as
$\mathcal{R}_{A,K}$-modules. We say that $D$ is _triangulable_ if it can be
equipped with a triangulation, and the data of $D$ with a triangulation is
called a _triangular_ $(\varphi,\Gamma_{K})$-module. A representation
$V\in\mathbf{Rep}_{A}(\mathcal{G}_{K})$ is called _trianguline_ if
$\mathbf{D}_{\mathrm{rig}}(V)$ is triangulable.

Given a triangular $(\varphi,\Gamma_{K})$-module
$\left(D,\operatorname{Fil}_{\bullet}D\right)$ over $\mathcal{R}_{A,K}$, the
graded pieces

$\operatorname{gr}_{i}D\coloneqq\operatorname{Fil}_{i}D/\operatorname{Fil}_{i-1}D\,,\quad
1\leq i\leq d$

of the triangulation are free of rank 1 as $\mathcal{R}_{A,K}$-modules. When
$A$ is a $K_{0}$-algebra, according to Proposition 2.2.2 below, for each
$1\leq i\leq d$ there is a unique continuous character $\delta_{i}\colon
K^{\times}\longrightarrow A^{\times}$ such that
$\operatorname{gr}_{i}D\simeq\mathcal{R}_{A,K}(\delta_{i})$.

######  Proposition 2.2.2 (Colmez, Kedlaya-Pottharst-Xiao).

Let $A\in\mathbf{Ar}_{\mathbb{Q}_{p}}$ be a $K_{0}$-algebra. Given any
$(\varphi,\Gamma_{K})$-module $D$ over $\mathcal{R}_{A,K}$ which is free of
rank $1$ as $\mathcal{R}_{A,K}$-module, there is a unique continuous character
$\delta\colon K^{\times}\longrightarrow A^{\times}$ such that
$D\simeq\mathcal{R}_{A,K}(\delta)$ as $(\varphi,\Gamma_{K})$-modules.

###### Proof.

This is [26, Lemma 6.2.13]; see also [15, Proposition 3.1] over
$\mathcal{R}_{L,\mathbb{Q}_{p}}$ and [1, Proposition 2.3.1] over
$\mathcal{R}_{A,\mathbb{Q}_{p}}$. ∎

######  Definition 2.2.3.

Let $A\in\mathbf{Ar}_{\mathbb{Q}_{p}}$ be a $K_{0}$-algebra. Let
$\left(D,\operatorname{Fil}_{\bullet}D\right)$ be a triangular
$(\varphi,\Gamma_{K})$-module of rank $d$ over $\mathcal{R}_{A,K}$. The
_parameter_ of the triangulation $\operatorname{Fil}_{\bullet}D$ is the unique
continuous character

$\underline{\delta}=(\delta_{1},\ldots,\delta_{d})\colon(K^{\times})^{d}\longrightarrow
A^{\times}$

such that $\operatorname{gr}_{i}D\simeq\mathcal{R}_{A,K}(\delta_{i})$ for all
$1\leq i\leq d$.

The following result somewhat generalises [1, Proposition 2.4.1].

######  Proposition 2.2.4.

Let $A\in\mathbf{Ar}_{\mathbb{Q}_{p}}$ be an algebra over its residue field
$L\coloneqq A/\mathfrak{m}_{A}A$ such that $L$ splits $K$. Let
$V\in\mathbf{Rep}_{A}(\mathcal{G}_{K})$ be a regular crystalline
representation of rank $n\in\mathbb{Z}_{>0}$ and of Hodge-Tate weights
$(h_{\tau,1}<\ldots<h_{\tau,n})_{\tau\in\Sigma}$.

Assume that there exist $\mathbf{k}=(k_{\tau,i})_{\tau\in\Sigma,1\leq i\leq
n}\in(\mathbb{Z}^{n})^{\Sigma}$ and $\varphi_{i}\in A^{\times}$ for $1\leq
i\leq n$ with pairwise distinct reductions in $L$, such that $V$ is
trianguline of parameter

$\underline{\delta}=z^{\mathbf{k}}\operatorname{unr}(\underline{\varphi})
\coloneqq\left(z^{\mathbf{k}_{1}}\operatorname{unr}(\varphi_{1}),\ldots,z^{\mathbf{k}_{n}}\operatorname{unr}(\varphi_{n})\right).$

Then:

  1. _(1)_

There exists $w=(w_{\tau})_{\tau\in\Sigma_{n}}\in(\mathcal{S}_{n})^{\Sigma}$
such that $k_{\tau,i}=h_{\tau,w_{\tau}^{-1}(i)}$ for all $\tau\in\Sigma$ and
$1\leq i\leq n$.

  2. _(2)_

The linearised Frobenius $\Phi$ on $\mathbf{D}_{\mathrm{crys}}(V)$ admits
$(1\otimes\varphi_{i})_{1\leq i\leq n}$ as eigenvalues, and this ordering
determines a refinement $\mathcal{F}_{\bullet}$ of $V$.

  3. _(3)_

The position $x\in\prod_{\tau\in\Sigma}(G/B)(A)$ of
$\operatorname{Fil}_{\bullet}\mathbf{D}_{\mathrm{dR}}(V)$ relative to
$\mathcal{F}_{\mathrm{dR},\bullet}\coloneqq
K\otimes_{K_{0}}\mathcal{F}_{\bullet}$ lies in the cell
$\prod_{\tau\in\Sigma}(Bw_{\tau}B/B)(A)$.

###### Proof.

Recall from Berger’s work the functor $\mathcal{D}_{\mathrm{crys}}$ from
$\mathbf{Mod}^{\varphi,\Gamma_{K}}_{A}$ to the category of $K_{0}$-modules
equipped with a $\varphi$-semilinear endomorphism (see [3, Théorème 3.6]) and
the functor $\mathcal{D}_{\mathrm{dR}}$ from
$\mathbf{Mod}^{\varphi,\Gamma_{K}}_{A}$ to
$\mathbf{Fil}^{\bullet}_{K\otimes_{\mathbb{Q}_{p}}A}$ (see [4, Théorème A]).
They are characterised by the property that
$\mathcal{D}_{\mathrm{crys}}(\mathbf{D}_{\mathrm{rig}}(V))=\mathbf{D}_{\mathrm{crys}}(V)$
and
$\mathcal{D}_{\mathrm{dR}}(\mathbf{D}_{\mathrm{rig}}(V))=\mathbf{D}_{\mathrm{dR}}(V)$
for all $V\in\mathbf{Rep}_{A}(\mathcal{G}_{K})$. It is also known that
$\mathcal{D}_{\mathrm{crys}}$ and $\mathcal{D}_{\mathrm{dR}}$ are exact
functors on the full subcategory of crystalline
$(\varphi,\Gamma_{K})$-modules, _i.e._ $(\varphi,\Gamma_{K})$-modules $D$ such
that
$\dim_{K_{0}}\mathcal{D}_{\mathrm{crys}}(D)=\operatorname{rk}_{\mathcal{R}_{K}}D$.

Let $(D_{i})_{0\leq i\leq n}$ be a triangulation of
$\mathbf{D}_{\mathrm{rig}}(V)$ of parameter $\underline{\delta}$. For $0\leq
i\leq n$, write $\mathcal{F}_{i}\coloneqq\mathcal{D}_{\mathrm{crys}}(D_{i})$.
The $D_{i}$ are crystalline (as subobjects of $\mathbf{D}_{\mathrm{rig}}(V)$),
hence by exactness of $\mathcal{D}_{\mathrm{crys}}$ each exact sequence

$0\longrightarrow D_{i-1}\longrightarrow
D_{i}\longrightarrow\mathcal{R}_{A,K}\left(z^{\mathbf{k}_{i}}\operatorname{unr}(\varphi_{i})\right)\longrightarrow
0$

induces an exact sequence

$0\longrightarrow\mathcal{F}_{i-1}\longrightarrow\mathcal{F}_{i}\longrightarrow\mathcal{D}_{\mathrm{crys}}\left(\mathcal{R}_{A,K}\left(z^{\mathbf{k}_{i}}\operatorname{unr}(\varphi_{i})\right)\right)\longrightarrow
0$

of $K_{0}\otimes_{\mathbb{Q}_{p}}A[\varphi]$-modules. It is known that
$\mathcal{D}_{\mathrm{crys}}\left(\mathcal{R}_{A,K}\left(z^{\mathbf{k}_{i}}\operatorname{unr}(\varphi_{i})\right)\right)$
is free of rank one over $K_{0}\otimes_{\mathbb{Q}_{p}}A$ and that its
linearised Frobenius acts by multiplication by $\varphi_{i}$ [26, Example
6.2.6]. Therefore $\Phi$ acts on $\mathcal{F}_{i}/\mathcal{F}_{i-1}$ by
multiplication by $1\otimes\varphi_{i}$, thus $(1\otimes\varphi_{i})_{1\leq
i\leq n}$ is a sequence of eigenvalues of $\Phi$ and $(\mathcal{F}_{i})$ is
the unique (see Proposition 2.1.5) refinement of $V$ associated to
$(1\otimes\varphi_{i})_{1\leq i\leq n}$. This proves (2).

Similarly, the exact functor $\mathcal{D}_{\mathrm{dR}}$ induces for each
$1\leq i\leq n$ an exact sequence

$0\longrightarrow\mathcal{F}_{\mathrm{dR},i-1}\longrightarrow\mathcal{F}_{\mathrm{dR},i}\longrightarrow\mathcal{D}_{\mathrm{dR}}\left(\mathcal{R}_{A,K}\left(z^{\mathbf{k}_{i}}\operatorname{unr}(\varphi_{i})\right)\right)\longrightarrow
0$

in $\mathbf{Fil}^{\bullet}_{K\otimes_{\mathbb{Q}_{p}}A}$. It induces, for each
$1\leq i\leq n$, each $\tau\in\Sigma$ and each $k\in\mathbb{Z}$, an
isomorphism

(2.6)
$\operatorname{Fil}^{k}\left(\mathcal{F}_{\mathrm{dR},i}\right)_{\tau}/\operatorname{Fil}^{k}\left(\mathcal{F}_{\mathrm{dR},i-1}\right)_{\tau}\\\
\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}\operatorname{Fil}^{k}\mathcal{D}_{\mathrm{dR}}\left(\mathcal{R}_{A,K}\left(z^{\mathbf{k}_{i}}\operatorname{unr}(\varphi_{i})\right)\right)_{\tau}\simeq\begin{cases}0&\text{if
}k\neq k_{\tau,i}\\\ A&\text{if }k=k_{\tau,i}\end{cases}$

(see [26, Example 6.2.6] for the last isomorphism). Applying (2.6) to
$k=k_{\tau,i}+1$, we get
$\operatorname{Fil}^{k_{\tau,i}+1}\left(\mathcal{F}_{\mathrm{dR},i}\right)_{\tau}/\operatorname{Fil}^{k_{\tau,i}+1}\left(\mathcal{F}_{\mathrm{dR},i-1}\right)_{\tau}\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}0$,
thus

(2.7)
$\operatorname{Fil}^{k_{\tau,i}}\left(\mathcal{F}_{\mathrm{dR},i}\right)_{\tau}/\operatorname{Fil}^{k_{\tau,i}+1}\left(\mathcal{F}_{\mathrm{dR},i}\right)_{\tau}=\operatorname{Fil}^{k_{\tau,i}}\left(\mathcal{F}_{\mathrm{dR},i}\right)_{\tau}/\operatorname{Fil}^{k_{\tau,i}+1}\left(\mathcal{F}_{\mathrm{dR},i-1}\right)_{\tau}\\\
\relbar\joinrel\twoheadrightarrow\operatorname{Fil}^{k_{\tau,i}}\left(\mathcal{F}_{\mathrm{dR},i}\right)_{\tau}/\operatorname{Fil}^{k_{\tau,i}}\left(\mathcal{F}_{\mathrm{dR},i-1}\right)_{\tau}$

The exactness of $\mathbf{D}_{\mathrm{dR}}$ also shows that
$\operatorname{Fil}^{k}\mathcal{F}_{\mathrm{dR},i}=\mathcal{F}_{\mathrm{dR},i}\cap\operatorname{Fil}^{k}\mathbf{D}_{\mathrm{dR}}(V)$
for all $k\in\mathbb{Z}$, therefore (2.7) rephrases as

(2.8)
$\left(\mathcal{F}_{\mathrm{dR},i}\cap\operatorname{Fil}^{k_{\tau,i}}\mathbf{D}_{\mathrm{dR}}(V)\right)_{\tau}/\left(\mathcal{F}_{\mathrm{dR},i}\cap\operatorname{Fil}^{k_{\tau,i}+1}\mathbf{D}_{\mathrm{dR}}(V)\right)_{\tau}\\\
\relbar\joinrel\twoheadrightarrow\operatorname{Fil}^{k_{\tau,i}}\left(\mathcal{F}_{\mathrm{dR},i}\right)_{\tau}/\operatorname{Fil}^{k_{\tau,i}}\left(\mathcal{F}_{\mathrm{dR},i-1}\right)_{\tau}.$

On the other hand, there is an obvious injective morphism

(2.9)
$\left(\mathcal{F}_{\mathrm{dR},i}\cap\operatorname{Fil}^{k_{\tau,i}}\mathbf{D}_{\mathrm{dR}}(V)\right)_{\tau}/\left(\mathcal{F}_{\mathrm{dR},i}\cap\operatorname{Fil}^{k_{\tau,i}+1}\mathbf{D}_{\mathrm{dR}}(V)\right)_{\tau}\\\
\lhook\joinrel\relbar\joinrel\rightarrow\operatorname{Fil}^{k_{\tau,i}}\mathbf{D}_{\mathrm{dR}}(V)_{\tau}/\operatorname{Fil}^{k_{\tau,i}+1}\mathbf{D}_{\mathrm{dR}}(V)_{\tau}.$

The right-hand side of (2.8) is free of rank one over $A$ by (2.6) applied to
$k=k_{\tau,i}$. Therefore the right-hand side of (2.9) is nonzero: this means
that the jumps of the Hodge filtration of $V$ are in degrees
$(k_{\tau,i})_{\tau\in\Sigma,1\leq i\leq n}$, which proves (1).

Furthermore, by regularity of $V$, the right-hand side of (2.9) is actually
free of rank one over $A$. Comparing dimensions over $L$, we deduce that (2.8)
and (2.9) are isomorphisms, hence the $A$-module
$\left(\mathcal{F}_{\mathrm{dR},i}\cap\operatorname{Fil}^{k_{\tau,i}}\mathbf{D}_{\mathrm{dR}}(V)\right)_{\tau}/\left(\mathcal{F}_{\mathrm{dR},i}\cap\operatorname{Fil}^{k_{\tau,i}+1}\mathbf{D}_{\mathrm{dR}}(V)\right)_{\tau}$
is free of rank $1$. Then (3) follows from Lemma 2.1.2 (note that our
computations give the position of $\mathcal{F}_{\mathrm{dR},\bullet}$ relative
to $\operatorname{Fil}_{\bullet}\mathbf{D}_{\mathrm{dR}}(V)$, which is inverse
of the position of the position of
$\operatorname{Fil}_{\bullet}\mathbf{D}_{\mathrm{dR}}(V)$ relative to
$\mathcal{F}_{\mathrm{dR},\bullet}$). ∎

##  3\. The trianguline variety and companion points on the patched
eigenvariety

In this section, we recall the definition of the trianguline variety as well
as some of the results of [12] and [11].

Let $L$ be a finite extension of $\mathbb{Q}_{p}$ which splits $K$, with
integer ring $\mathcal{O}_{L}$ and residue field $k_{L}$; recall the notation
$\Sigma\coloneqq\operatorname{Hom}_{\mathbb{Q}_{p}}(K,L)$. We fix
$n\in\mathbb{Z}_{>0}$ and a continuous representation

$\bar{r}\colon\mathcal{G}_{K}\longrightarrow\operatorname{GL}_{n}(k_{L})$

(where $k_{L}$ has the discrete topology).

###  3.1. The trianguline variety

Let $R_{\bar{r}}^{\square}$ be the _framed deformation ring of $\bar{r}$_ from
the work of Mazur [30] and Kisin [29]; it is a complete local noetherian
$\mathcal{O}_{L}$-algebra with residue field $k_{L}$, which pro-represents the
functor sending a local artinian $\mathcal{O}_{L}$-algebra $A$ with residue
field $k_{L}$ to the set of framed deformations of $\bar{r}$ over $A$. Define
the _framed deformation variety of $\bar{r}$_ as the rigid $L$-analytic space
$\mathfrak{X}^{\square}_{\bar{r}}\coloneqq\operatorname{Spf}(R_{\bar{r}}^{\square})^{\mathrm{rig}}$,
where $\operatorname{Spf}$ is the formal spectrum and $(-)^{\mathrm{rig}}$ is
the rigidification functor of Berthelot [7, §0.2] (see also [18, §7.1]). Using
[18, Proposition 7.1.7] and [29, Proposition 2.3.5], we see that for any local
artinian $L$-algebra $A$ with residue field denoted by $E$, the set of
$A$-points $\mathfrak{X}^{\square}_{\bar{r}}(A)$ is the set of representations
$r_{A}\colon\mathcal{G}_{K}\longrightarrow\operatorname{GL}_{n}(A)$ such that
the composition of $r_{A}$ with
$\operatorname{GL}_{n}(A)\relbar\joinrel\twoheadrightarrow\operatorname{GL}_{n}(E)$
has image in $\operatorname{GL}_{n}(\mathcal{O}_{E})$ and is a framed
deformation of the composition of $\bar{r}$ with
$\operatorname{GL}_{n}(k_{L})\longrightarrow\operatorname{GL}_{n}(k_{E})$
(where $\mathcal{O}_{E}$ and $k_{E}$ are respectively the integer ring and
residue field of $E$).

Let $\mathcal{W}\coloneqq\widehat{\mathcal{O}_{K}^{\times}}$ be the rigid
analytic space over $\mathbb{Q}_{p}$ that parametrises continuous characters
of $\mathcal{O}_{K}^{\times}$, _i.e._ such that
$\mathcal{W}(X)=\operatorname{Hom}_{\mathrm{cont}}\left(\mathcal{O}_{K}^{\times},\Gamma(X,\mathcal{O}_{X})^{\times}\right)$
for all rigid analytic space $X$ over $\mathbb{Q}_{p}$; the existence and
smoothness of $\mathcal{W}$ can be seen using a decomposition
$\mathcal{O}_{K}^{\times}\simeq H\times\mathbb{Z}_{p}^{[K:\mathbb{Q}_{p}]}$ of
topological groups, where $H$ is a finite torsion abelian group (see [33,
Proposition II.5.7]). Similarly, let
$\mathcal{T}\coloneqq\widehat{K^{\times}}$ be the rigid analytic space over
$\mathbb{Q}_{p}$ that parametrises continuous characters of $K^{\times}$; it
is also smooth and any decomposition
$K^{\times}\simeq\mathbb{Z}\times\mathcal{O}_{K}^{\times}$ induces an
isomorphism
$\mathcal{T}\simeq\mathbb{G}_{\mathrm{m}}^{\mathrm{rig}}\times_{\mathbb{Q}_{p}}\mathcal{W}$
of rigid analytic spaces over $\mathbb{Q}_{p}$. The restriction of a
$K^{\times}$-character to $\mathcal{O}_{K}^{\times}$ induces a morphism
$\mathcal{T}\longrightarrow\mathcal{W}$ which corresponds to the projection
$\mathbb{G}_{\mathrm{m}}^{\mathrm{rig}}\times_{\mathbb{Q}_{p}}\mathcal{W}\relbar\joinrel\twoheadrightarrow\mathcal{W}$.

Write $\mathcal{T}_{L}\coloneqq\mathcal{T}\times_{\mathbb{Q}_{p}}L$. The set

$\left\\{(\delta_{i})_{1\leq i\leq
n}\in\mathcal{T}_{L}^{n}(L)\mathrel{}\middle|\mathrel{}\begin{array}[]{l}\delta_{i}\delta_{j}^{-1}=z^{-\mathbf{k}}\text{
or }\delta_{i}\delta_{j}^{-1}=N_{K/\mathbb{Q}_{p}}(z)z^{\mathbf{k}}\lvert
z\rvert_{K}\\\ \text{for some }i\neq j,\mathbf{k}\in\mathbb{Z}_{\geq
0}^{\Sigma}\end{array}\right\\}$

is an analytic subset of $\mathcal{T}_{L}^{n}$ (in the sense of [8, §9.5.2]),
so its complement is a Zariski-open subset of $\mathcal{T}_{L}^{n}$, which we
call $\mathcal{T}^{n}_{\mathrm{reg}}$.

######  Definition 3.1.1 (Hellmann).

Let $U_{\mathrm{tri}}^{\square}(\bar{r})$ be the subset

$U_{\mathrm{tri}}^{\square}(\bar{r})\coloneqq\left\\{(r,\underline{\delta})\in\mathfrak{X}^{\square}_{\bar{r}}\times_{L}\mathcal{T}^{n}_{\mathrm{reg}}\mathrel{}\middle|\mathrel{}r\text{
is trianguline of parameter }\underline{\delta}\right\\}$

of $\mathfrak{X}^{\square}_{\bar{r}}\times_{L}\mathcal{T}^{n}_{\mathrm{reg}}$.
The _trianguline variety_ is the Zariski-closure
$X_{\mathrm{tri}}^{\square}(\bar{r})$ of $U_{\mathrm{tri}}^{\square}(\bar{r})$
in $\mathfrak{X}^{\square}_{\bar{r}}\times_{L}\mathcal{T}_{L}^{n}$, with its
structure of reduced rigid analytic space over $L$. The composition of the
inclusion
$X_{\mathrm{tri}}^{\square}(\bar{r})\lhook\joinrel\relbar\joinrel\rightarrow\mathfrak{X}^{\square}_{\bar{r}}\times_{L}\mathcal{T}_{L}^{n}$,
the projection
$\mathfrak{X}^{\square}_{\bar{r}}\times_{L}\mathcal{T}_{L}^{n}\relbar\joinrel\twoheadrightarrow\mathcal{T}_{L}^{n}$
and the restriction morphism
$\mathcal{T}_{L}^{n}\longrightarrow\mathcal{W}_{L}^{n}$ is a morphism of rigid
analytic varieties over $L$ called the _weight morphism_ and denoted
$\omega\colon
X_{\mathrm{tri}}^{\square}(\bar{r})\longrightarrow\mathcal{W}_{L}^{n}$.

Note that $U_{\mathrm{tri}}^{\square}(\bar{r})$ (resp.
$X_{\mathrm{tri}}^{\square}(\bar{r})$) is also noted
$U_{\mathrm{tri}}^{\square}(\bar{r})^{\mathrm{reg}}$ or
$U_{\mathrm{tri}}(\bar{r})$ (resp. $X_{\mathrm{tri}}(\bar{r})$) in the
literature.

######  Theorem 3.1.2 (Breuil-Hellmann-Schraen).

  1. _(1)_

The rigid analytic space $X_{\mathrm{tri}}^{\square}(\bar{r})$ is
equidimensional of dimension $n^{2}+[K:\mathbb{Q}_{p}]\frac{n(n+1)}{2}$.

  2. _(2)_

The subset $U_{\mathrm{tri}}^{\square}(\bar{r})$ of
$X_{\mathrm{tri}}^{\square}(\bar{r})$ is Zariski-open and Zariski-dense.

  3. _(3)_

The open set $U_{\mathrm{tri}}^{\square}(\bar{r})$ is smooth over
$\mathbb{Q}_{p}$, and the restriction of $\omega$ to
$U_{\mathrm{tri}}^{\square}(\bar{r})$ is a smooth morphism.

###### Proof.

This is [12, Théorème 2.6]. ∎

######  Definition 3.1.3.

A point $x=(r,\underline{\delta})\in X_{\mathrm{tri}}^{\square}(\bar{r})$ is
said to be _crystalline_ if the representation
$r\colon\mathcal{G}_{K}\longrightarrow\operatorname{GL}_{n}(k(x))$ is
crystalline ($k(x)$ is the local residue field of
$X_{\mathrm{tri}}^{\square}(\bar{r})$ at $x$).

######  Lemma 3.1.4 (Breuil-Hellmann-Schraen).

Let $x=(r,\underline{\delta})\in X_{\mathrm{tri}}^{\square}(\bar{r})$ be a
crystalline point. Then there exist
$\mathbf{h}=(h_{\tau,i})\in(\mathbb{Z}^{\Sigma})^{n}$ and
$\underline{\varphi}=(\varphi_{1},\ldots,\varphi_{n})\in(k(x)^{\times})^{n}$
such that

$\underline{\delta}=z^{\mathbf{h}}\operatorname{unr}(\underline{\varphi})
\coloneqq\left(z^{\mathbf{h}_{1}}\operatorname{unr}(\varphi_{1}),\ldots,z^{\mathbf{h}_{n}}\operatorname{unr}(\varphi_{n})\right)\,.$

Moreover, the $1\otimes\varphi_{i}\in K_{0}\otimes_{\mathbb{Q}_{p}}k(x)$ are
the eigenvalues of the linearised Frobenius $\Phi$ on
$\mathbf{D}_{\mathrm{crys}}(r)$ and, for each $\tau\in\Sigma$, the multiset of
$\tau$-Hodge-Tate weights of $r$ is
$\left\\{h_{\tau,i}\mathrel{}\middle|\mathrel{}1\leq i\leq n\right\\}$.

If $\varphi_{i}\varphi_{j}^{-1}\notin\\{1,p^{[K_{0}:\mathbb{Q}_{p}]}\\}$ for
$i\neq j$, then there also exists
$\mathbf{h}^{\prime}\in(\mathbb{Z}^{\Sigma})^{n}$ such that
$(r,z^{\mathbf{h}^{\prime}}\operatorname{unr}(\underline{\varphi}))\in
U_{\mathrm{tri}}^{\square}(\bar{r})$.

###### Proof.

This is [11, Lemma 2.1]; the second part is a consequence of its proof. Note
that necessarily $\left\\{h_{\tau,i}\mathrel{}\middle|\mathrel{}1\leq i\leq
n\right\\}=\left\\{h^{\prime}_{\tau,i}\mathrel{}\middle|\mathrel{}1\leq i\leq
n\right\\}$ for all $\tau\in\Sigma$. ∎

###  3.2. The patched eigenvariety

For the rest of the paper, we make the assumption $p\nmid 2n$ (which implies
in particular $p>2$).

We consider the following global setting. Let $F$ be a CM field, _i.e._ a
quadratic totally imaginary extension of a totally real number field $F^{+}$,
and write $S_{p}$ for the set of places of $F^{+}$ dividing $p$. We assume
that $F/F^{+}$ is unramified in all finite places, and that all places $v\in
S_{p}$ split in $F$ and satisfy $F^{+}_{v}\simeq K$. Let $G$ be a unitary
group in $n$ variables over $F^{+}$ quasi-split in all finite places of
$F^{+}$, such that $G\times_{F^{+}}F\simeq\operatorname{GL}_{n,F}$ and
$G(F\otimes_{\mathbb{Q}}\mathbb{R})$ is compact. Let $U^{p}=\prod_{v\notin
S_{p}\text{ finite}}U_{v}\subset G(\mathbb{A}^{p\infty}_{F^{+}})$ be a tame
level, where each $U_{v}$ is a compact subgroup of $F^{+}_{v}$ which is
assumed to be hyperspecial whenever $v$ is inert. We then fix a finite set $S$
of finite places of $F^{+}$ containing $S_{p}$ and all the places $v$ such
that $U_{v}$ is not hyperspecial; in particular all places $v\in S$ split in
$F$. Let
$\bar{\rho}\colon\mathcal{G}_{F^{+}}\longrightarrow\operatorname{GL}_{n}(k_{L})$
be a continuous irreducible representation which is automorphic (see for
example [20, Definition 5.3.1]). In particular, it follows from [20, §5.1,
§5.2, §5.3] or [14, §2.1, §2.3] that $\bar{\rho}$ comes from $p$-adic
automorphic forms on $G(\mathbb{A}_{F^{+}})$ of tame level $U^{p}$; see [12,
§2.4] for a detailed exposition (see also the beginning of [11, §3.1, §3.2]).
Also,
$\bar{\rho}\left(\operatorname{Gal}\left(\overline{F}/F(\zeta_{p})\right)\right)$
is adequate in the sense of [34, Def. 2.3]. Finally, we assume that for each
$v\in S_{p}$ there exists a place $\tilde{v}\mid v$ of $F$ such that
$\bar{\rho}_{\tilde{v}}\coloneqq\bar{\rho}|_{\mathcal{G}_{F_{\tilde{v}}}}$ is
isomorphic to $\bar{r}$ via $F_{\tilde{v}}\simeq F^{+}_{v}\simeq K$. We fix
such a place $\tilde{v}$.

Enlarging $L$ if necessary, the existence of such a globalisation has been
proved by Emerton-Gee using the Emerton-Gee stack [21]; see [21, Theorem
1.2.2] and [20, Corollary A.7].

For $v\in S_{p}$, let $R^{\overline{\square}}_{\bar{\rho}_{\tilde{v}}}$ be the
maximal reduced quotient of $R^{\square}_{\bar{\rho}_{\tilde{v}}}$ without
$p$-torsion. By [18, Lemma 7.1.4.a], the rigid analytic space
$\mathfrak{X}_{\bar{\rho}_{\tilde{v}}}\coloneqq\operatorname{Spf}(R^{\overline{\square}}_{\bar{\rho}_{\tilde{v}}})^{\mathrm{rig}}$
is reduced and has the same underlying topological space as
$\operatorname{Spf}(R^{\square}_{\bar{\rho}_{\tilde{v}}})^{\mathrm{rig}}$. We
also write $\mathfrak{X}_{\bar{\rho}^{p}}\coloneqq\prod_{v\in S\setminus
S_{p}}\mathfrak{X}_{\bar{\rho}_{\tilde{v}}}$ and
$\mathfrak{X}_{\bar{\rho}_{p}}\coloneqq\prod_{v\in
S_{p}}\mathfrak{X}_{\bar{\rho}_{\tilde{v}}}$.

Let $T_{p}\coloneqq\prod_{v\in S_{p}}T_{v}$ where
$T_{v}\coloneqq(F_{\tilde{v}}^{\times})^{n}$ is the $n$-dimensional torus over
$F_{\tilde{v}}$. Then the continuous characters $T_{v}\longrightarrow
L^{\times}$ for each $v\in S_{p}$ (resp. $T_{p}\longrightarrow L^{\times}$)
are parametrised by a rigid analytic space $\widehat{T}_{v,L}$ (resp.
$\widehat{T}_{p,L}\coloneqq\prod_{v\in S_{p}}\widehat{T}_{v,L}$) over $L$.
Note that a character $T_{v}\longrightarrow L^{\times}$ can be seen as a
character $(K^{\times})^{n}\longrightarrow L^{\times}$ through
$F_{\tilde{v}}\simeq K$; this induces a rigid isomorphism
$\widehat{T}_{v,L}\simeq\mathcal{T}_{L}^{n}$ over $L$. We define
$X_{\mathrm{tri}}^{\square}(\bar{\rho}_{\tilde{v}})$ in the same way as
$X_{\mathrm{tri}}^{\square}(\bar{r})$ with $\bar{\rho}_{\tilde{v}}$ instead of
$\bar{r}$ for each $v\in S_{p}$, and
$X_{\mathrm{tri}}^{\square}(\bar{\rho}_{p})\coloneqq\prod_{v\in
S_{p}}X_{\mathrm{tri}}^{\square}(\bar{\rho}_{\tilde{v}})$; again the
isomorphisms $F_{\tilde{v}}\simeq K$ induce an isomorphism
$X_{\mathrm{tri}}^{\square}(\bar{\rho}_{p})\simeq(X_{\mathrm{tri}}^{\square}(\bar{r}))^{\lvert
S_{p}\rvert}$. Also, since $X_{\mathrm{tri}}^{\square}(\bar{\rho}_{p})$ is by
definition reduced, there is a Zariski-closed embedding

$X_{\mathrm{tri}}^{\square}(\bar{\rho}_{p})\lhook\joinrel\relbar\joinrel\rightarrow\mathfrak{X}_{\bar{\rho}_{p}}\times_{L}\widehat{T}_{p,L}$

of rigid analytic spaces over $L$.

We fix $g\in\mathbb{Z}_{\geq 1}$, and we let $R_{\infty}\coloneqq
R^{\overline{\square}}_{\bar{\rho}_{S}}\llbracket
t_{1},\ldots,t_{g}\rrbracket$, where
$R^{\overline{\square}}_{\bar{\rho}_{S}}\coloneqq\widehat{\bigotimes}_{v\in
S}R^{\overline{\square}}_{\bar{\rho}_{\tilde{v}}}$, and
$\mathfrak{X}_{\infty}\coloneqq\operatorname{Spf}(R_{\infty})^{\mathrm{rig}}$.
Then, by [18, Proposition 7.2.4.g], we have

$\mathfrak{X}_{\infty}=\mathfrak{X}_{\bar{\rho}^{p}}\times_{L}\mathfrak{X}_{\bar{\rho}_{p}}\times_{L}\mathbb{U}^{g}$

where $\mathbb{U}\coloneqq\operatorname{Spf}(\mathcal{O}_{L}\llbracket
t\rrbracket)^{\mathrm{rig}}$ is the unit open disk over $L$. By adapting the
patching construction of Caraiani-Emerton-Gee-Geraghty-Paškūnas-Shin [14, §2],
Breuil-Hellmann-Schraen constructed for some $g\in\mathbb{Z}_{\geq 1}$ an
$R_{\infty}$-representation $\Pi_{\infty}$ of $G_{p}\coloneqq\prod_{v\in
S_{p}}G(F_{\tilde{v}})$ over $L$ called the patched representation, satisfying
different properties for which we refer to [12, Théorème 3.5]. They then
consider the subspace $\Pi_{\infty}^{R_{\infty}-\mathrm{an}}$ of locally
$R_{\infty}$-analytic vectors (see [12, Définition 3.2]), and apply Emerton’s
Jacquet module functor $J_{B_{p}}$ (see [19, Definition 3.4.5]) associated to
$B_{p}\coloneqq\prod_{v\in S_{p}}B_{\tilde{v}}$, where $B_{\tilde{v}}$ is the
Borel subgroup of upper triangular matrices in
$\operatorname{GL}_{n}(F_{\tilde{v}})$. This gives a coherent
$\mathcal{O}_{\mathfrak{X}_{\infty}\times_{L}\widehat{T}_{p,L}}$-module
$\mathcal{M}_{\infty}$ whose module of global sections
$\Gamma(\mathfrak{X}_{\infty}\times_{L}\widehat{T}_{p,L},\mathcal{M}_{\infty})$
is isomorphic to $J_{B_{p}}(\Pi_{\infty}^{R_{\infty}-\mathrm{an}})^{\prime}$,
where $(-)^{\prime}$ is the continuous dual.

######  Definition 3.2.1 (Breuil-Hellmann-Schraen).

The _patched eigenvariety_ $X_{p}(\bar{\rho})$ is the support of
$\mathcal{M}_{\infty}$ (in the sense of [8, §9.5.2]), which is an analytic
subset of $\mathfrak{X}_{\infty}\times_{L}\widehat{T}_{p,L}$ (see Proposition
4 of _ibid._), equipped with its structure of reduced rigid analytic space
over $L$. For more details about this construction, we refer to [12, §3.2,
Corollaire 3.20] where $X_{p}(\bar{\rho})$ is called the _Hecke-Taylor-Wiles
variety_.

We next recall the relation between $X_{p}(\bar{\rho})$ and
$X_{\mathrm{tri}}^{\square}(\bar{\rho}_{p})$. For each $v\in S_{p}$, define a
character
$\underline{\delta}_{B_{v}}=(\lvert\cdot\rvert_{F_{\tilde{v}}}^{n+1-2i})_{1\leq
i\leq n}\in\widehat{T}_{v,L}$, where $\lvert\cdot\rvert_{F_{\tilde{v}}}$ is
normalised as usual by $\lvert
p\rvert_{F_{\tilde{v}}}=p^{-[F_{\tilde{v}}:\mathbb{Q}_{p}]}$, and an
automorphism $\iota_{v}$ of $\widehat{T}_{v,L}$ as follows:

$\iota_{v}(\underline{\delta})\coloneqq\underline{\delta}_{B_{v}}\cdot(\delta_{i}\cdot(\chi\circ\theta_{F_{\tilde{v}}}^{-1})^{i-1})_{1\leq
i\leq n}$

where $\chi$ is the cyclotomic character and $\theta_{F_{\tilde{v}}}\colon
W_{F_{\tilde{v}}}^{\mathrm{ab}}\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}F_{\tilde{v}}^{\times}$
is the local reciprocity isomorphism normalised so that geometric Frobeniuses
are sent to uniformisers. The inverse of the automorphism

$\iota\colon\;\begin{aligned}
\mathfrak{X}_{\infty}\times_{L}\widehat{T}_{p,L}\;&\longrightarrow\;\mathfrak{X}_{\infty}\times_{L}\widehat{T}_{p,L}\\\
\left(y,(\delta_{v})_{v\in
S_{p}}\right)\;&\longmapsto\;\left(y,(\iota_{v}(\delta_{v}))_{v\in
S_{p}}\right)\end{aligned}$

induces a Zariski-closed embedding

$\iota^{-1}\colon
X_{p}(\bar{\rho})\lhook\joinrel\relbar\joinrel\rightarrow\mathfrak{X}_{\bar{\rho}^{p}}\times_{L}\left(\mathfrak{X}_{\bar{\rho}_{p}}\times_{L}\widehat{T}_{p,L}\right)\times_{L}\times\mathbb{U}^{g}\,.$

According to [12, Théorème 3.21], $\iota^{-1}$ induces an isomorphism of rigid
analytic spaces over $L$ between $X_{p}(\bar{\rho})$ and a union of
irreducible components of
$\mathfrak{X}_{\bar{\rho}^{p}}\times_{L}X_{\mathrm{tri}}^{\square}(\bar{\rho}_{p})\times_{L}\mathbb{U}^{g}$.
Such irreducible components are of the form
$\mathfrak{X}^{p}\times_{L}Z\times_{L}\mathbb{U}^{g}$, where
$\mathfrak{X}^{p}$ is an irreducible component of
$\mathfrak{X}_{\bar{\rho}^{p}}$ and $Z$ is an irreducible component of
$X_{\mathrm{tri}}^{\square}(\bar{\rho}_{p})$. Therefore, we have an
isomorphism

(3.1) $\iota^{-1}\colon
X_{p}(\bar{\rho})\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}\bigcup_{\mathfrak{X}^{p}\text{
irr.\ comp.\ of
}\mathfrak{X}_{\bar{\rho}^{p}}}\left(\mathfrak{X}^{p}\times_{L}X_{\mathrm{tri}}^{\mathfrak{X}^{p}-\mathrm{aut}}(\bar{\rho}_{p})\times_{L}\mathbb{U}^{g}\right)$

where, for each $\mathfrak{X}^{p}$,
$X_{\mathrm{tri}}^{\mathfrak{X}^{p}-\mathrm{aut}}(\bar{\rho}_{p})$ is a union
of irreducible components of $X_{\mathrm{tri}}^{\square}(\bar{\rho}_{p})$.

The following is [12, Conjecture 3.23].

######  Conjecture 3.2.2.

For any irreducible component $\mathfrak{X}^{p}$ of
$\mathfrak{X}_{\bar{\rho}^{p}}$, the rigid analytic subvariety
$X_{\mathrm{tri}}^{\mathfrak{X}^{p}-\mathrm{aut}}(\bar{\rho}_{p})\subseteq
X_{\mathrm{tri}}^{\square}(\bar{\rho}_{p})$ is equal to
$\widetilde{X}_{\mathrm{tri}}^{\square}(\bar{\rho}_{p})\coloneqq\prod_{v\in
S_{p}}\widetilde{X}_{\mathrm{tri}}^{\square}(\bar{\rho}_{\tilde{v}})$, where
$\widetilde{X}_{\mathrm{tri}}^{\square}(\bar{\rho}_{\tilde{v}})$ is the union
of irreducible components $Z$ of
$X_{\mathrm{tri}}^{\square}(\bar{\rho}_{\tilde{v}})$ such that $Z\cap
U_{\mathrm{tri}}^{\square}(\bar{\rho}_{\tilde{v}})$ contains a crystalline
point. In particular,
$X_{\mathrm{tri}}^{\mathfrak{X}^{p}-\mathrm{aut}}(\bar{\rho}_{p})$ does not
depend on $\mathfrak{X}^{p}$.

###  3.3. Strong linkage and companion points

We recall a tool from [11] to find points on $X_{p}(\bar{\rho})$ (so-called
_companion points_), which is related to the notion of strong linkage of
characters which we now explain.

Let $\mathfrak{g}$ be a Lie algebra over a field $k$ of characteristic $0$,
together with a Cartan subalgebra $\mathfrak{t}\subseteq\mathfrak{g}$. Let
$\Phi\subseteq\mathfrak{t}^{*}\coloneqq\operatorname{Hom}_{k}(\mathfrak{t},\overline{k})$
be the root system of $(\mathfrak{g},\mathfrak{t})$ (where $\overline{k}$ is
an algebraic closure of $k$), let $W\coloneqq W(\mathfrak{g},\mathfrak{t})$ be
the Weyl group which acts on $\mathfrak{t}^{*}$ and $\Phi$. Let
$E\subset\mathfrak{t}^{*}$ be the $\mathbb{Q}$-span of $\Phi$ and equip $E$
with its nondegenerate bilinear form $(\cdot\mid\cdot)$ invariant by $W$ (see
[9, Chap. VI, §1.1, Prop. 3]). Choose a basis $I$ of $\Phi$ and call its
elements _simple roots_ ; this choice induces a partial order on
$\mathfrak{t}^{*}$ by setting $\mu\leq\lambda$ if and only if $\lambda-\mu$ is
a $\mathbb{Z}_{\geq 0}$-linear combination of simple roots. For each root
$\alpha\in\Phi$, let
$\alpha^{\vee}\coloneqq\frac{2\alpha}{(\alpha\mid\alpha)}$ be the coroot
associated to $\alpha$. Then
$\left\\{\alpha^{\vee}\mathrel{}\middle|\mathrel{}\alpha\in I\right\\}$ is a
basis of $E$; it has an orthogonal dual under $(\cdot\mid\cdot)$, whose
elements we call _fundamental weights_. We call _special weight_ the sum of
the fundamental weights; it is an element $\varpi\in E$ that satisfies in
particular $s_{\alpha}\varpi=\varpi-\alpha$ for any simple root $\alpha\in I$,
where $s_{\alpha}\in W$ is the reflection associated to $\alpha$, defined by
$s_{\alpha}\colon\lambda\mapsto\lambda-(\lambda\mid\alpha^{\vee})\alpha$. Note
that $\varpi$ is usually noted $\rho$, _e.g._[24], but we wish to avoid
confusion with global representations.

######  Definition 3.3.1.

The _dot action_ is the action of $W$ on $\mathfrak{t}^{*}$ defined by

$w\cdot\lambda\coloneqq w(\lambda+\varpi)-\varpi$

for all $w\in W$ and $\lambda\in\mathfrak{t}^{*}$.

Let $\lambda,\mu\in\mathfrak{t}^{*}$. We say that $\mu$ is _strongly linked_
to $\lambda$, and write $\mu\uparrow\lambda$, if there exist roots
$\alpha_{1},\ldots,\alpha_{r}\in\Phi$ for some $r\in\mathbb{Z}_{\leq 0}$ such
that $\mu=(s_{\alpha_{r}}\ldots s_{\alpha_{1}})\cdot\lambda$ and
$(s_{\alpha_{i+1}}\ldots s_{\alpha_{1}})\cdot\lambda<(s_{\alpha_{i}}\ldots
s_{\alpha_{1}})\cdot\lambda$ for all $1\leq i\leq r-1$.

We apply this, for each place $v\in S_{p}$, to the context of the split
reductive group
$G_{v,L}\coloneqq\left(\operatorname{Res}_{F_{\tilde{v}}/\mathbb{Q}_{p}}\operatorname{GL}_{n,F_{\tilde{v}}}\right)\times_{\mathbb{Q}_{p}}L\simeq\prod_{\tau\in\Sigma_{v}}\operatorname{GL}_{n,L}$
over $L$ (where
$\Sigma_{v}\coloneqq\operatorname{Hom}_{\mathbb{Q}_{p}}(F_{\tilde{v}},L)\simeq\Sigma$),
with Lie algebra
$\mathfrak{g}_{v,L}=\prod_{\tau\in\Sigma}\operatorname{\mathfrak{gl}}_{n,L}$
and Cartan algebra
$\mathfrak{t}_{v,L}=\prod_{\tau\in\Sigma_{v}}\mathfrak{t}_{L}$ (with
$\mathfrak{t}_{L}\simeq L^{n}$). Then we have
$\mathfrak{t}_{v,L}^{*}\simeq(\overline{L}^{n})^{\Sigma_{v}}$ as vector spaces
over $\overline{L}$, and the Weyl group $W=(\mathcal{S}_{n})^{\Sigma_{v}}$
acts on $\mathfrak{t}_{v,L}^{*}$ in the obvious way. We choose the basis
$I\subset\Phi$ corresponding to the Borel subgroup of upper triangular
matrices; the simple roots are the
$\alpha=(\alpha_{\tau^{\prime},i^{\prime}})_{\tau^{\prime}\in\Sigma_{v},1\leq
i^{\prime}\leq n}\in\mathfrak{t}_{v,L}^{*}$ such that, for some
$\tau\in\Sigma_{v}$ and $1\leq i\leq n-1$, we have
$\alpha_{\tau,i}=-\alpha_{\tau,i+1}=1$ and
$\alpha_{\tau^{\prime},i^{\prime}}=0$ for all $\tau^{\prime}\neq\tau$ and
$i^{\prime}\notin\\{i,i+1\\}$. Then the special weight is
$\varpi=\left(\frac{n+1}{2}-i\right)_{\tau\in\Sigma_{v},1\leq i\leq
n}\in\mathfrak{t}_{v,L}^{*}$, and the dot action expresses as

$w\cdot\lambda=\left(\lambda_{\tau,w_{\tau}^{-1}(i)}+i-w_{\tau}^{-1}(i)\right)_{\tau\in\Sigma_{v},1\leq
i\leq n}$

for $w=(w_{\tau})_{\tau\in\Sigma_{v}}\in W$ and
$\lambda=(\lambda_{\tau,i})_{\tau\in\Sigma_{v},1\leq i\leq
n}\in\mathfrak{t}_{v,L}^{*}$.

If $\underline{\delta}_{v}=(\delta_{v,i})_{1\leq i\leq n}\colon
T_{v,L}\longrightarrow L^{\prime\times}$ is a continuous character, for some
place $v\in S_{p}$ and some finite extension $L^{\prime}$ of $L$, then one can
see
$\mathop{}\mathopen{}\mathrm{d}\underline{\delta}_{v}=(\operatorname{wt}_{\tau}(\delta_{v,i}))_{\tau\in\Sigma_{v},1\leq
i\leq n}$ as an element of $\mathfrak{t}_{v,L}^{*}$.

######  Definition 3.3.2.

Let $\underline{\delta}=(\underline{\delta}_{v})_{v\in
S_{p}},\underline{\epsilon}=(\underline{\epsilon}_{v})_{v\in
S_{p}}\in\widehat{T}_{p,L}$. We say that $\underline{\epsilon}$ is _strongly
linked_ to $\underline{\delta}$, if
$\mathop{}\mathopen{}\mathrm{d}\underline{\epsilon}_{v}\uparrow\mathop{}\mathopen{}\mathrm{d}\underline{\delta}_{v}$
in $\mathfrak{t}_{v,L}^{*}$ for all $v\in S_{p}$, and if
$\underline{\epsilon}\underline{\delta}^{-1}$ is algebraic in the sense that
for all $v\in S_{p}$ there exists $\mathbf{k}_{v}\in\mathbb{Z}^{\Sigma_{v}}$
such that
$\underline{\epsilon}_{v}\underline{\delta}_{v}^{-1}=z^{\mathbf{k}_{v}}$. In
this case we write $\underline{\epsilon}\uparrow\underline{\delta}$.

######  Theorem 3.3.3 (Breuil-Hellmann-Schraen).

Let
$x=(y,\underline{\delta})\in\mathfrak{X}_{\infty}\times_{L}\widehat{T}_{p,L}$
be a point lying in $X_{p}(\bar{\rho})$, and let
$\underline{\epsilon}\in\widehat{T}_{p,L}$ be a character satisfying
$\underline{\epsilon}\uparrow\underline{\delta}$. Then the point
$x^{\prime}=(y,\underline{\epsilon})\in\mathfrak{X}_{\infty}\times_{L}\widehat{T}_{p,L}$
also lies in $X_{p}(\bar{\rho})$.

###### Proof.

This is [11, Theorem 5.5]; see (3.10) of _loc. cit_. ∎

##  4\. Tangent spaces on the trianguline variety

In this section, we prove the main result of the paper: Theorem 4.1.4. It
computes the dimension of $X_{\mathrm{tri}}^{\square}(\bar{r})$ at crystalline
points satisfying certain technical conditions, generalising [11, Corollary
5.17] and [13, Proposition 4.1.5] (see [13, Remark 4.1.6(iii)] for the
latter).

We keep the notation of §3.

###  4.1. Crystalline generic points and pairs of permutations

We fix a crystalline point $x=(r,\underline{\delta})\in
X_{\mathrm{tri}}^{\square}(\bar{r})$ and assume that $x$ is _generic_ in the
sense of [13], _i.e._ $r$ is regular and
$\varphi_{i}\varphi_{j}^{-1}\notin\\{1,p^{[K_{0}:\mathbb{Q}_{p}]}\\}$ for
$i\neq j$, where the $1\otimes\varphi_{i}\in
K_{0}\otimes_{\mathbb{Q}_{p}}k(x)$ are the eigenvalues of the linearised
Frobenius $\Phi$ on $\mathbf{D}_{\mathrm{crys}}(r)$. We write
$\mathbf{h}=(h_{\tau,i})_{\tau\in\Sigma,1\leq i\leq n}$ for its Hodge-Tate
weights, ordered such that $h_{\tau,1}<\ldots<h_{\tau,n}$ for all
$\tau\in\Sigma$.

By Lemma 3.1.4, we can attach two elements
$w,w_{\mathrm{sat}}\in\mathcal{S}_{n}^{\Sigma}$ such that (i) the character
$\underline{\delta}$ is of the form
$\underline{\delta}=z^{w(\mathbf{h})}\operatorname{unr}(\underline{\varphi})$,
where $\underline{\varphi}=(\varphi_{i})_{1\leq i\leq n}$ and
$w(\mathbf{h})\coloneqq(h_{\tau,w_{\tau}^{-1}(i)})_{\tau,i}\in(\mathbb{Z}^{\Sigma})^{n}$,
and (ii) the point
$x_{\mathrm{sat}}\coloneqq(r,\underline{\delta}_{\mathrm{sat}})\in\mathfrak{X}^{\square}_{\bar{r}}\times_{L}\mathcal{T}_{L}^{n}$
defined by
$\underline{\delta}_{\mathrm{sat}}=z^{w_{\mathrm{sat}}(\mathbf{h})}\operatorname{unr}(\underline{\varphi})$
lies in $U_{\mathrm{tri}}^{\square}(\bar{r})$.

Remember that $(\mathcal{S}_{n})^{\Sigma}$ is the Weyl group of the root
system $\coprod_{\tau\in\Sigma}\Phi$ attached to the split reductive group
$\left(\prod_{\tau\in\Sigma}G,\prod_{\tau\in\Sigma}T\right)$ over
$\mathbb{Q}_{p}$ where $G\coloneqq\operatorname{GL}_{n,\mathbb{Q}_{p}}$ and
$T\subseteq G$ is the torus of diagonal matrices. Writing
$X(T)\coloneqq\operatorname{Hom}(T,\mathbb{G}_{\mathrm{m}})$ for the
$\mathbb{Z}$-module of algebraic characters of $T$, one has $\Phi\subset X(T)$
and the canonical isomorphism $X(T)\simeq\mathbb{Z}^{n}$ induces
$\Phi\simeq\left\\{e_{i}-e_{j}\mathrel{}\middle|\mathrel{}1\leq i,j\leq
n\;,\;i\neq j\right\\}$ where $(e_{i})_{1\leq i\leq n}$ is the standard basis
of $\mathbb{Z}^{n}$. For $w^{\prime}\in(\mathcal{S}_{n})^{\Sigma}$, we denote
by $d_{w^{\prime}}$ the rank of the $\mathbb{Z}$-submodule of
$X\left(\prod_{\tau\in\Sigma}T\right)$ generated by
$\left\\{w^{\prime}(\alpha)-\alpha\mathrel{}\middle|\mathrel{}\alpha\in\coprod_{\tau\in\Sigma}\Phi\right\\}$.
We choose the standard basis
$I\coloneqq\left\\{e_{i}-e_{i+1}\mathrel{}\middle|\mathrel{}1\leq i\leq
n-1\right\\}$ of $\Phi$ corresponding to the Borel subgroup $B\subseteq G$ of
upper triangular matrices. It induces on $\mathcal{S}_{n}$ a length function
$\lg$ and a partial Bruhat order $\preceq$; these extend naturally to a length
function and a Bruhat order on $(\mathcal{S}_{n})^{\Sigma}$, still denoted
$\lg$ and $\preceq$, which can also be defined directly by choosing the
corresponding standard basis $\coprod_{\tau\in\Sigma}I$ of the root system
$\coprod_{\tau\in\Sigma}\Phi$. By [13, Theorem 4.2.3], we have
$w_{\mathrm{sat}}\preceq w$.

######  Definition 4.1.1.

Let $(w_{1},w_{2})\in(\mathcal{S}_{n})^{2}$ such that $w_{1}\preceq w_{2}$. We
say that $(w_{1},w_{2})$ is a _good pair_ if there exists a sequence
$(\alpha_{1},\ldots,\alpha_{r})$ of roots of the form $e_{i}-e_{j}$, where
$i,j\in\\{1,\ldots,n\\}$ are in the same orbit under the action of
$w_{1}w_{2}^{-1}$, such that $w_{2}=s_{\alpha_{r}}s_{\alpha_{r-1}}\ldots
s_{\alpha_{1}}w_{1}$ and $s_{\alpha_{i}}\ldots s_{\alpha_{1}}w_{1}\succ
s_{\alpha_{i-1}}\ldots s_{\alpha_{1}}w_{1}$ for all $1\leq i\leq r$.

For $(w_{1},w_{2})\in\left((\mathcal{S}_{n})^{\Sigma}\right)^{2}$ such that
$w_{1}\preceq w_{2}$, we say that $(w_{1},w_{2})$ is a _good pair_ if and only
if $(w_{1,\tau},w_{2,\tau})\in(\mathcal{S}_{n})^{2}$ is a good pair for all
$\tau\in\Sigma$.

######  Example 4.1.2.

For $n\leq 4$, all pairs are good except for
$(w_{1},w_{2})\in(\mathcal{S}_{4})^{2}$ defined by
$(w_{1}(1),w_{1}(2),w_{1}(3),w_{1}(4))=(1,3,2,4)$ and
$(w_{2}(1),w_{2}(2),w_{2}(3),w_{2}(4))=(4,2,3,1)$. In fact, we show in a
subsequent paper [32, Theorem 2.4.10] a result relating good pairs to pattern
avoidance, which implies in particular that “most” pairs are good.

From now on we assume the following:

######  Hypothesis 4.1.3.

With the notation of §3, there exists an irreducible component
$\mathfrak{X}^{p}$ of $\mathfrak{X}_{\bar{\rho}^{p}}$ such that
$(x_{\mathrm{sat},v})_{v\in S_{p}}\in
X_{\mathrm{tri}}^{\mathfrak{X}^{p}-\mathrm{aut}}(\bar{\rho}_{p})$, where
$x_{\mathrm{sat},v}\in X_{\mathrm{tri}}^{\square}(\bar{\rho}_{\tilde{v}})$
corresponds to $x_{\mathrm{sat}}\in X_{\mathrm{tri}}^{\square}(\bar{r})$ under
$\bar{\rho}_{\tilde{v}}\simeq\bar{r}$ for each $v\in S_{p}$.

Note that Hypothesis 4.1.3 is implied by Conjecture 3.2.2 since
$x_{\mathrm{sat},v}\in\widetilde{X}_{\mathrm{tri}}^{\square}(\bar{\rho}_{\tilde{v}})$
by definition.

######  Theorem 4.1.4.

Assume Hypothesis 4.1.3. Let $x\in X_{\mathrm{tri}}^{\square}(\bar{r})$ be
crystalline generic, such that the associated pair $(w_{\mathrm{sat}},w)$ in
$(\mathcal{S}_{n})^{\Sigma}$ is good in the sense of Definition 4.1.1. Then

$\dim_{k(x)}T_{X_{\mathrm{tri}}^{\square}(\bar{r}),x}=\dim
X_{\mathrm{tri}}^{\square}(\bar{r})-d_{ww_{\mathrm{sat}}^{-1}}+\sum_{\tau\in\Sigma}\dim_{\mathbb{Q}_{p}}T_{\overline{(Bw_{\tau}B/B)},w_{\mathrm{sat},\tau}B}-\operatorname{lg}(w_{\mathrm{sat}})$

where $d_{ww_{\mathrm{sat}}^{-1}}$ is as defined in the paragraph preceding
Definition 4.1.1.

The rest of the section is devoted to the proof of Theorem 4.1.4. We use a
similar strategy to [11], as follows. We construct two subspaces of
$T_{X_{\mathrm{tri}}^{\square}(\bar{r}),x}$, one roughly corresponding to
“crystalline tangent directions” and the other to “trianguline tangent
directions”, and compute the dimension of the sum of these subspaces: this
bounds $\dim_{k(x)}T_{X_{\mathrm{tri}}^{\square}(\bar{r}),x}$ from below. The
upper bound is proved in [13, Theorem 4.1.5(ii)].

###  4.2. Crystalline tangent directions

Following the exposition of [11, §2.2] and [13, §4.2], we recall the
construction of a rigid analytic space over $L$ parametrising crystalline
deformations of given Hodge-Tate weights together with an ordering of the
eigenvalues of the Frobenius.

Let $R_{\bar{r}}^{\square,\mathbf{k}-\mathrm{cr}}$ be the quotient of
$R_{\bar{r}}^{\square}$ representing framed deformations of $\bar{r}$ which
are crystalline of Hodge-Tate weights $\mathbf{h}$ (its existence is
guaranteed by [28, Corollary 2.6.2]; it is reduced and flat over
$\mathbb{Z}_{p}$). Namely, for any complete local noetherian algebra $A$ over
$\mathcal{O}_{L}$ with residue field $k_{L}$, a morphism
$R_{\bar{r}}^{\square}\longrightarrow A$ factors through
$R_{\bar{r}}^{\square,\mathbf{k}-\mathrm{cr}}$ if and only if the
corresponding deformation
$r\colon\mathcal{G}_{K}\longrightarrow\operatorname{GL}_{n}(A)$ is crystalline
with Hodge-Tate weights $\mathbf{h}$. Write
$\mathfrak{X}_{\bar{r}}^{\square,\mathbf{h}-\mathrm{cr}}\coloneqq\operatorname{Spf}(R_{\bar{r}}^{\square,\mathbf{k}-\mathrm{cr}})^{\mathrm{rig}}$
for the associated rigid analytic space over $L$. Arguing as in [29,
Proposition 2.3.5] similarly to the case of $\mathfrak{X}^{\square}_{\bar{r}}$
(see §3), we see that for any local artinian algebra $A$ over $L$, the set of
$A$-points $x\in\mathfrak{X}_{\bar{r}}^{\square,\mathbf{h}-\mathrm{cr}}(A)$ is
the set of points $x\in\mathfrak{X}^{\square}_{\bar{r}}(A)$ such that the
associated representation
$r_{x}\colon\mathcal{G}_{K}\longrightarrow\operatorname{GL}_{n}(A)$ is
crystalline with Hodge-Tate weights $\mathbf{h}$.

According to [28, Theorem 2.5.5] (see also [5, Corollaire 6.3.3]), there is a
coherent locally free module $\mathcal{D}$ over
$K_{0}\otimes_{\mathbb{Q}_{p}}\mathcal{O}_{\mathfrak{X}_{\bar{r}}^{\square,\mathbf{h}-\mathrm{cr}}}$
equipped with a $\varphi$-semilinear automorphism $\varphi_{\mathrm{crys}}$
such that, for all
$x\in\mathfrak{X}_{\bar{r}}^{\square,\mathbf{h}-\mathrm{cr}}$ corresponding to
a deformation
$r_{x}\colon\mathcal{G}_{K}\longrightarrow\operatorname{GL}_{n}(k(x))$, there
is an isomorphism

$\mathcal{D}\otimes_{\mathcal{O}_{\mathfrak{X}_{\bar{r}}^{\square,\mathbf{h}-\mathrm{cr}}}}k(x)\simeq\mathbf{D}_{\mathrm{crys}}(r_{x})$

such that
$\varphi_{\mathrm{crys}}\otimes_{\mathcal{O}_{\mathfrak{X}_{\bar{r}}^{\square,\mathbf{h}-\mathrm{cr}}}}k(x)$
induces the $\varphi$-action on $\mathbf{D}_{\mathrm{crys}}(r_{x})$.
Furthermore, for each $\tau\in\operatorname{Hom}_{\mathbb{Q}_{p}}(K_{0},L)$,
the linearised Frobenius
$\Phi_{\mathrm{crys}}\coloneqq(\varphi_{\mathrm{crys}})^{[K_{0}:\mathbb{Q}_{p}]}$
induces an invertible linear action on
$\mathbf{D}_{\mathrm{crys}}(r_{x})\otimes_{K_{0},\tau}L$ which is independent,
up to isomorphism, of the choice of $\tau$. Therefore, taking the coefficients
of the characteristic polynomial of this action gives a morphism of rigid
analytic spaces over $L$

(4.1)
$\mathfrak{X}_{\bar{r}}^{\square,\mathbf{h}-\mathrm{cr}}\longrightarrow\left((\mathbb{G}_{\mathrm{a}}^{\mathrm{rig}})^{n-1}\times\mathbb{G}_{\mathrm{m}}^{\mathrm{rig}}\right)\times_{\mathbb{Q}_{p}}L$

where the image $(a_{n-1},\ldots,a_{1},a)\in k(x)^{n-1}\times k(x)^{\times}$
of a point $x\in\mathfrak{X}_{\bar{r}}^{\square,\mathbf{h}-\mathrm{cr}}$ is
such that the Frobenius on $\mathbf{D}_{\mathrm{crys}}(r_{x})$ has eigenvalues
$1\otimes\varphi_{i}\in K_{0}\otimes_{\mathbb{Q}_{p}}k(x)$, where the
$\varphi_{i}$ for $1\leq i\leq n$ are the roots of the polynomial
$t^{n}+a_{n-1}t^{n-1}+\ldots+a_{1}t+a$. See also [11, §2.2].

As in the paragraph following Definition 2.1.1, let
$G\coloneqq\operatorname{GL}_{n,\mathbb{Q}_{p}}$, $B\subset G$ be the Borel of
upper triangular matrices, $T\subset B$ be the maximal torus inside $B$; the
Weyl group of $(G,T)$ is the symmetric group $\mathcal{S}_{n}$. The
isomorphism
$L[e_{1},\ldots,e_{n},f]/(e_{n}f-1)\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}L[t_{1},\ldots,t_{n},u]^{\mathcal{S}_{n}}/(u\prod
t_{i}-1)$ of the fundamental theorem of symmetric polynomials gives an
isomorphism of schemes

$T_{L}/\mathcal{S}^{n}\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}\left(\mathbb{G}_{\mathrm{a}}^{n-1}\times\mathbb{G}_{\mathrm{m}}\right)\times_{\mathbb{Q}_{p}}L$

where the left-hand side parametrises a polynomial by its roots and the right-
hand side parametrises it by its coefficients. Therefore (4.1) is a morphism
$\mathfrak{X}_{\bar{r}}^{\square,\mathbf{h}-\mathrm{cr}}\longrightarrow
T_{L}^{\mathrm{rig}}/\mathcal{S}^{n}$ of rigid analytic spaces over $L$, which
we use to define

$\widetilde{\mathfrak{X}}_{\bar{r}}^{\square,\mathbf{h}-\mathrm{cr}}\coloneqq\mathfrak{X}_{\bar{r}}^{\square,\mathbf{h}-\mathrm{cr}}\times_{T_{L}^{\mathrm{rig}}/\mathcal{S}^{n}}T_{L}^{\mathrm{rig}}$

which is a reduced rigid analytic space over $L$ by [11, Lemma 2.2].

From the proof of [13, Theorem 4.2.3] (see also [11, Lemma 2.4]), the set

$\widetilde{W}_{\bar{r}}^{\mathbf{h}-\mathrm{cr}}\coloneqq\left\\{(r^{\prime},\underline{\varphi^{\prime}})\in\widetilde{\mathfrak{X}}_{\bar{r}}^{\square,\mathbf{h}-\mathrm{cr}}\mathrel{}\middle|\mathrel{}\varphi^{\prime}_{i}{\varphi^{\prime}_{j}}^{-1}\notin\\{1,p^{[K_{0}:\mathbb{Q}_{p}]}\\}\;\forall\;i\neq
j\right\\}$

is a Zariski-open dense subset of
$\widetilde{\mathfrak{X}}_{\bar{r}}^{\square,\mathbf{h}-\mathrm{cr}}$, and
there is a smooth morphism

(4.2) $h
\colon\widetilde{W}_{\bar{r}}^{\mathbf{h}-\mathrm{cr}}\longrightarrow\left(\operatorname{Res}_{K/\mathbb{Q}_{p}}(G/B)_{K}^{\mathrm{rig}}\right)\times_{\mathbb{Q}_{p}}L\simeq\prod_{\tau\in\Sigma}(G/B)_{L}^{\mathrm{rig}}$

of rigid analytic spaces over $L$ sending a point
$(r^{\prime},\underline{\varphi^{\prime}})\in\widetilde{W}_{\bar{r}}^{\mathbf{h}-\mathrm{cr}}$
to the position of the Hodge flag
$\operatorname{Fil}_{\bullet}\mathbf{D}_{\mathrm{dR}}(r^{\prime})$ relative to
the refinement on $K\otimes_{K_{0}}\mathbf{D}_{\mathrm{crys}}(r^{\prime})$
induced by the ordering $\underline{\varphi^{\prime}}$ (see the discussion
following Definition 2.1.3). Note that we use the notation $r^{\prime}$ and
$\underline{\varphi^{\prime}}$ to avoid confusion with $r$ and
$\underline{\varphi}$ which are defined in §4.1.

Fix a permutation $w=(w_{\tau})\in\mathcal{S}_{n}^{\Sigma}$ and define the
analytic subset

$\widetilde{W}_{\bar{r},w}^{\mathbf{h}-\mathrm{cr}}\coloneqq
h^{-1}\left(\prod_{\tau\in\Sigma}(\overline{Bw_{\tau}B/B})_{L}^{\mathrm{rig}}\right)\subseteq\widetilde{W}_{\bar{r}}^{\mathbf{h}-\mathrm{cr}}$

of $\widetilde{W}_{\bar{r}}^{\mathbf{h}-\mathrm{cr}}$ (note that it is written
$\overline{\widetilde{W}_{\bar{r},w}^{\mathbf{h}-\mathrm{cr}}}$ in [13, §4.2];
we drop the bar for clarity).

######  Proposition 4.2.1 (Breuil-Hellmann-Schraen).

The morphism

$\iota_{\mathbf{h},w}\colon\;\begin{aligned}
\widetilde{W}_{\bar{r},w}^{\mathbf{h}-\mathrm{cr}}\;&\longrightarrow\;\mathfrak{X}^{\square}_{\bar{r}}\times_{L}\mathcal{T}_{L}^{n}\\\
\left(r^{\prime},\underline{\varphi^{\prime}}\right)\;&\longmapsto\;\left(r^{\prime},z^{w(\mathbf{h})}\operatorname{unr}(\underline{\varphi^{\prime}})\right)\end{aligned}$

of rigid analytic spaces over $L$ (where we recall that
$w(\mathbf{h})\coloneqq(h_{\tau,w_{\tau}^{-1}(i)})_{\tau,i}\in(\mathbb{Z}^{\Sigma})^{n}$)
is a closed immersion and factors through the inclusion
$X_{\mathrm{tri}}^{\square}(\bar{r})\subset\mathfrak{X}^{\square}_{\bar{r}}\times_{L}\mathcal{T}_{L}^{n}$.

###### Proof.

See the proof of [13, Theorem 4.2.3]. ∎

Note that $\iota_{\mathbf{h},w}(r,\underline{\varphi})=x$, therefore
$\iota_{\mathbf{h},w}$ induces a morphism
$T_{\widetilde{W}_{\bar{r},w}^{\mathbf{h}-\mathrm{cr}},(r,\underline{\varphi})}\lhook\joinrel\relbar\joinrel\rightarrow
T_{X_{\mathrm{tri}}^{\square}(\bar{r}),x}$.

###  4.3. Trianguline tangent directions

We generalise a construction of [11, §5.2]

Write
$\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}\subset\mathcal{W}^{n}_{L}$
for the analytic subset of characters
$\underline{\eta}=(\eta_{1},\ldots,\eta_{n})$ defined by the equations:

$\operatorname{wt}_{\tau}\big{(}\eta_{w_{\mathrm{sat},\tau}(i)}\eta_{w_{\tau}(i)}^{-1}\big{)}=h_{\tau,i}-h_{\tau,w_{\mathrm{sat},\tau}^{-1}w_{\tau}(i)}\,,\quad
1\leq i\leq n\,,\quad\tau\in\Sigma$

with its structure of reduced rigid analytic space over $L$. Such a character
$\underline{\eta}$ is of the form
$z^{w_{\mathrm{sat}}(\mathbf{h})}\underline{\eta}^{\prime}$ where
$\underline{\eta}^{\prime}=(\eta^{\prime}_{1},\dots,\eta^{\prime}_{n})\in\mathcal{W}_{L}^{n}$
satisfies
$\operatorname{wt}_{\tau}\big{(}\eta^{\prime}_{w_{\tau}w_{\mathrm{sat},\tau}^{-1}(i)}\big{)}=\operatorname{wt}_{\tau}(\eta^{\prime}_{i})$
for all $\tau,i$. In particular, this is the case of
$\underline{\delta}_{\mathrm{sat}}$.

We define an automorphism
$\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}\colon\mathcal{T}^{n}_{L}\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}\mathcal{T}^{n}_{L}$
by

$\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}(\underline{\eta})\coloneqq
z^{w(\mathbf{h})-w_{\mathrm{sat}}(\mathbf{h})}\underline{\eta}$

and still write $\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}$ for the automorphism
$\operatorname{id}_{\mathfrak{X}^{\square}_{\bar{r}}}\otimes\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}\colon\mathfrak{X}^{\square}_{\bar{r}}\times_{L}\mathcal{T}^{n}_{L}\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}\mathfrak{X}^{\square}_{\bar{r}}\times_{L}\mathcal{T}^{n}_{L}$.

By [13, Corollary 3.7.10], the rigid variety
$X_{\mathrm{tri}}^{\square}(\bar{r})$ is locally irreducible at
$x_{\mathrm{sat}}$ so, with the notation of Hypothesis 4.1.3, we can identify
$X_{\mathrm{tri}}^{\mathfrak{X}^{p}-\mathrm{aut}}(\bar{\rho}_{p})$ to
$X_{\mathrm{tri}}^{\square}(\bar{\rho}_{p})$ locally at
$(x_{\mathrm{sat},v})_{v\in S_{p}}$. Therefore, there exists a Zariski-open
neighbourhood

$U_{x_{\mathrm{sat}}}\subseteq U_{\mathrm{tri}}^{\square}(\bar{r})$

of $x_{\mathrm{sat}}$ such that $\prod_{v\in
S_{p}}U_{x_{\mathrm{sat},v}}\subseteq
X_{\mathrm{tri}}^{\mathfrak{X}^{p}-\mathrm{aut}}(\bar{\rho}_{p})$, where each
$U_{x_{\mathrm{sat},v}}\subset
X_{\mathrm{tri}}^{\square}(\bar{\rho}_{\tilde{v}})$ corresponds to
$U_{x_{\mathrm{sat}}}\subset X_{\mathrm{tri}}^{\square}(\bar{r})$ under
$\bar{\rho}_{\tilde{v}}\simeq\bar{r}$.

Since $U_{x_{\mathrm{sat}}}$ is a Zariski-open subset of
$U_{\mathrm{tri}}^{\square}(\bar{r})$, by Theorem 3.1.2 and by base change,
$U_{x_{\mathrm{sat}}}\times_{\mathcal{W}_{L}^{n}}\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}$
is smooth over $\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}$ hence
reduced, and it is a Zariski-open subset of
$X_{\mathrm{tri}}^{\square}(\bar{r})\times_{\mathcal{W}^{n}_{L}}\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}$.
Consider its Zariski-closure
$\overline{U_{x_{\mathrm{sat}}}\times_{\mathcal{W}_{L}^{n}}\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}}$,
with its structure of reduced rigid analytic space over $L$; it fits in a
chain of Zariski-closed embeddings

$\overline{U_{x_{\mathrm{sat}}}\times_{\mathcal{W}^{n}_{L}}\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}}\lhook\joinrel\relbar\joinrel\rightarrow(X_{\mathrm{tri}}^{\square}(\bar{r})\times_{\mathcal{W}^{n}_{L}}\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L})^{\mathrm{red}}\\\
\lhook\joinrel\relbar\joinrel\rightarrow
X_{\mathrm{tri}}^{\square}(\bar{r})\times_{\mathcal{W}^{n}_{L}}\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}\lhook\joinrel\relbar\joinrel\rightarrow
X_{\mathrm{tri}}^{\square}(\bar{r})\subseteq\mathfrak{X}^{\square}_{\bar{r}}\times_{L}\mathcal{T}^{n}_{L}$

and the automorphism
$\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}\colon\mathfrak{X}^{\square}_{\bar{r}}\times_{L}\mathcal{T}^{n}_{L}\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}\mathfrak{X}^{\square}_{\bar{r}}\times_{L}\mathcal{T}^{n}_{L}$
induces a Zariski-closed embedding

(4.3)
$\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}\colon\overline{U_{x_{\mathrm{sat}}}\times_{\mathcal{W}^{n}_{L}}\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}}\lhook\joinrel\relbar\joinrel\rightarrow\mathfrak{X}^{\square}_{\bar{r}}\times_{L}\mathcal{T}^{n}_{L}\,.$

######  Proposition 4.3.1.

We have

$\dim_{k(x)}T_{U_{x_{\mathrm{sat}}}\times_{\mathcal{W}^{n}_{L}}\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L},x_{\mathrm{sat}}}=\dim{X_{\mathrm{tri}}^{\square}(\bar{r})}-d_{ww_{\mathrm{sat}}^{-1}}$

(see the paragraph preceding Definition 4.1.1 for the definition of
$d_{ww_{\mathrm{sat}}^{-1}}$).

###### Proof.

This is the same as [11, Proposition 5.15]. ∎

The following result is adapted from [11, Proposition 5.9].

######  Proposition 4.3.2.

Assume that the pair $(w_{\mathrm{sat}},w)$ in $(\mathcal{S}_{n})^{\Sigma}$ is
good in the sense of Definition 4.1.1. Then (4.3) induces a Zariski-closed
embedding of reduced rigid analytic spaces over $L$

(4.4)
$\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}\colon\overline{U_{x_{\mathrm{sat}}}\times_{\mathcal{W}^{n}_{L}}\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}}\lhook\joinrel\relbar\joinrel\rightarrow
X_{\mathrm{tri}}^{\square}(\bar{r})\,.$

###### Proof.

It is enough to prove that any point
$x^{\prime}=(r^{\prime},\underline{\delta}^{\prime})\in U_{x_{\mathrm{sat}}}$
such that
$\underline{\delta}^{\prime}|_{(\mathcal{O}_{K}^{\times})n}\in\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}$
satisfies $\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}(x^{\prime})\in
X_{\mathrm{tri}}^{\square}(\bar{r})$. By definition, the open subset
$\prod_{v\in S_{p}}U_{x_{\mathrm{sat},v}}$ of
$X_{\mathrm{tri}}^{\square}(\bar{\rho}_{p})$ is in the image of the morphism

(4.5)
$X_{p}(\bar{\rho})\lhook\joinrel\relbar\joinrel\rightarrow\mathfrak{X}_{\bar{\rho}^{p}}\times_{L}X_{\mathrm{tri}}^{\square}(\bar{\rho}_{p})\times_{L}\times\mathbb{U}^{g}\relbar\joinrel\twoheadrightarrow
X_{\mathrm{tri}}^{\square}(\bar{\rho}_{p})$

where the first arrow is (3.1). Let
$y^{\prime}=(s^{\prime},\underline{\epsilon}^{\prime})\in\mathfrak{X}_{\infty}\times\widehat{T}_{p,L}$
be a point of $X_{p}(\bar{\rho})$ whose image by (4.5) is
$(x^{\prime}_{v})_{v\in S_{p}}$, where $x^{\prime}_{v}\in
U_{x_{\mathrm{sat},v}}$ corresponds to $x^{\prime}\in U_{x_{\mathrm{sat}}}$
under $U_{x_{\mathrm{sat},v}}\simeq U_{x_{\mathrm{sat}}}$. We know in
particular that
$\underline{\epsilon}^{\prime}=(\iota_{v}(\underline{\delta}^{\prime}_{v}))_{v\in
S_{p}}$, where $\underline{\delta}^{\prime}_{v}$ is the composition of
$(F_{\tilde{v}}^{\times})^{n}\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}(K^{\times})^{n}$
with $\underline{\delta}^{\prime}$. Consider the point

$\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}(y^{\prime})\coloneqq(s^{\prime},\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}(\underline{\epsilon}^{\prime}))\in\mathfrak{X}_{\infty}\times\widehat{T}_{p,L}$

where
$\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}(\underline{\epsilon}^{\prime})\coloneqq(\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}(\iota_{v}(\underline{\delta}_{v}^{\prime})))_{v\in
S_{p}}$. It is enough to prove that
$\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}(y^{\prime})\in X_{p}(\bar{\rho})$,
since $(\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}(x^{\prime}_{v}))_{v\in S_{p}}$
would then be the image of
$\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}(y^{\prime})$ under (4.5) (as
$\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}$ and $\iota_{v}$ commute), hence a
point of $X_{\mathrm{tri}}^{\square}(\bar{\rho}_{p})=\prod_{v\in
S_{p}}X_{\mathrm{tri}}^{\square}(\bar{\rho}_{\tilde{v}})$.

By Theorem 3.3.3, all we need to prove is that
$\underline{\epsilon}^{\prime}\uparrow\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}(\underline{\epsilon}^{\prime})$
in the sense of Definition 3.3.2. Since
$\underline{\epsilon}^{\prime}\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}(\underline{\epsilon}^{\prime})^{-1}=z^{w_{\mathrm{sat}}(\mathbf{h})-w(\mathbf{h})}$
is algebraic by definition of $\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}$, it is
equivalent to check that

(4.6) $(\operatorname{wt}_{\tau}(\epsilon_{v,i}))_{\tau\in\Sigma_{v},1\leq
i\leq
n}\uparrow\left(h_{\tau,w_{v}^{-1}(i)}-h_{\tau,w_{\mathrm{sat},v}^{-1}}(i)+\operatorname{wt}_{\tau}(\epsilon_{v,i})\right)_{\tau\in\Sigma_{v},1\leq
i\leq n}$

as elements of $\mathfrak{t}_{v,L}^{*}$, for all $v\in S_{p}$. Viewing
$\underline{\delta}^{\prime}$ as its restriction to
$(\mathcal{O}_{K}^{\times})^{n}$, we can write:

$\underline{\delta}^{\prime}=(z^{\mathbf{h}_{w_{\mathrm{sat}}^{-1}(1)}}\chi_{1},\ldots,z^{\mathbf{h}_{w_{\mathrm{sat}}^{-1}(n)}}\chi_{n})$

where $\underline{\chi}\in\mathcal{W}_{L}^{n}$ satisfies
$\operatorname{wt}_{\tau}(\chi_{w_{\tau}w_{\mathrm{sat},\tau}^{-1}(i)})=\operatorname{wt}_{\tau}(\chi_{i})$
for all $1\leq i\leq n$ and $\tau\in\Sigma$. Unpacking the definitions and
using the fact that $\chi\circ\theta_{K}^{-1}=N_{K/\mathbb{Q}_{p}}\lvert
N_{K/\mathbb{Q}_{p}}\rvert_{\mathbb{Q}_{p}}$ has the same restriction to
$\mathcal{O}_{K}^{\times}$ as $N_{K/\mathbb{Q}_{p}}=z^{\mathbf{1}}$, we see
that (4.6) can be reformulated as

$\left(h_{\tau,w_{\mathrm{sat},\tau}^{-1}(i)}+i-1+\operatorname{wt}_{\tau}(\chi_{i})\right)_{\tau\in\Sigma,1\leq
i\leq
n}\uparrow\left(h_{\tau,w_{\tau}^{-1}(i)}+i-1+\operatorname{wt}_{\tau}(\chi_{i})\right)_{\tau\in\Sigma,1\leq
i\leq n}\,.$

Since
$\operatorname{wt}_{\tau}(\chi_{w_{\tau}w_{\mathrm{sat},\tau}^{-1}(i)})=\operatorname{wt}_{\tau}(\chi_{i})$
for all $i$, this equation can also be written

(4.7) $w_{\mathrm{sat}}\cdot\lambda\uparrow
w\cdot\lambda\quad\text{with}\quad\lambda\coloneqq(h_{\tau,i}+i-1+\operatorname{wt}_{\tau}(\chi_{w_{\tau}(i)}))_{\tau\in\Sigma,1\leq
i\leq n}$

where $\cdot$ is the dot action of Definition 3.3.1.

Since $(w_{\mathrm{sat}},w)$ is a good pair in $(\mathcal{S}_{n})^{\Sigma}$,
there exists a chain $w_{0}\coloneqq w_{\mathrm{sat}}\prec
w_{1}\prec\ldots\prec w_{r}\coloneqq w$ in $(\mathcal{S}_{n})^{\Sigma}$ for
some $r\in\mathbb{N}$ such that for each $1\leq k\leq r$ and $\tau\in\Sigma$,
the permutation $w_{k,\tau}w_{k-1,\tau}^{-1}\in\mathcal{S}_{n}$ is either the
identity or a reflection which is supported in some orbit of
$w_{\tau}w_{\mathrm{sat},\tau}^{-1}$. By the condition on the weights of
$\underline{\chi}$, this means in particular that
$\operatorname{wt}_{\tau}(\chi_{w_{k,\tau}w_{k-1,\tau}^{-1}(i)})=\operatorname{wt}_{\tau}(\chi_{i})$
for each $\tau,k,i$. Thus, by a descending recursion on $k$, we see that

$\operatorname{wt}_{\tau}\left(\chi_{w_{\tau}w_{k,\tau}^{-1}(i)}\right)=\operatorname{wt}_{\tau}\left(\chi_{i}\right)=\operatorname{wt}_{\tau}\left(\chi_{w_{\tau}w_{k-1,\tau}^{-1}(i)}\right)$

for each $\tau,k,i$. We then get, for all $1\leq k\leq n$,

$w_{k}\cdot\lambda-
w_{k-1}\cdot\lambda=\left(h_{\tau,w_{k,\tau}^{-1}(i)}-h_{\tau,w_{k-1,\tau}^{-1}(i)}\right)_{\tau\in\Sigma,1\leq
i\leq n}$

thus $w_{k}\cdot\lambda>w_{k-1}\cdot\lambda$ (remember that $>$ is the partial
order on $\mathfrak{t}_{L}^{*}$ induced by the lattice of simple roots) since
for each $\tau\in\Sigma$, on the one hand $w_{k-1,\tau}^{-1}\prec
w_{k,\tau}^{-1}$ and $w_{k-1,\tau}w_{k,\tau}^{-1}$ is at most a reflection,
and on the other hand $(h_{\tau,i})_{1\leq i\leq n}$ is strictly increasing.
This means that $w_{k-1}\cdot\lambda\uparrow w_{k}\cdot\lambda$ for each
$1\leq k\leq n$, hence by chaining these together
$w_{\mathrm{sat}}\cdot\lambda\uparrow w\cdot\lambda$. This is exactly (4.7). ∎

Note that $\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}(x_{\mathrm{sat}})=x$,
therefore $\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}$ induces an injective
morphism
$T_{\overline{U_{x_{\mathrm{sat}}}\times_{\mathcal{W}^{n}_{L}}\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}},x_{\mathrm{sat}}}\lhook\joinrel\relbar\joinrel\rightarrow
T_{X_{\mathrm{tri}}^{\square}(\bar{r}),x}$.

###  4.4. Proof of Theorem 4.1.4

We now prove Theorem 4.1.4. It is equivalent to proving, assuming Hypothesis
4.1.3 and that $(w_{\mathrm{sat}},w)$ is a good pair in
$(\mathcal{S}_{n})^{\Sigma}$ in the sense of Definition 4.1.1, that

$\dim_{k(x)}T_{X_{\mathrm{tri}}^{\square}(\bar{r}),x}=\dim
X_{\mathrm{tri}}^{\square}(\bar{r})-d_{ww_{\mathrm{sat}}^{-1}}+\sum_{\tau\in\Sigma}\dim_{k(x)}T_{\overline{(Bw_{\tau}B/B)}_{k(x)}^{\mathrm{rig}},x_{\tau}}-\operatorname{lg}(w_{\mathrm{sat}})$

where $(x_{\tau})_{\tau\in\Sigma}\coloneqq
h(r,\underline{\varphi})\in\prod_{\tau\in\Sigma}(Bw_{\mathrm{sat},\tau}B/B)_{L}^{\mathrm{rig}}$
(see (4.2) for the definition of $h$). Indeed, the dimension of
$T_{\overline{Bw_{\tau}B/B},y_{\tau}B}$ is constant as $y_{\tau}$ varies
through $Bw_{\mathrm{sat},\tau}B/B$.

The inequality $(\text{left-hand side})\leq(\text{right-hand side})$ is the
statement of [13, Theorem 4.1.5(ii)] (where our $w_{\mathrm{sat}}$ is noted
$w_{x}$). We prove the inequality in the other direction. Consider the
following commutative diagram:

(4.8)
${\widetilde{W}_{x_{\mathrm{sat}}}}$${\overline{U_{x_{\mathrm{sat}}}\times_{\mathcal{W}^{n}_{L}}\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}}}$${\widetilde{W}_{\bar{r},w}^{\mathbf{h}-\mathrm{cr}}}$${X_{\mathrm{tri}}^{\square}(\bar{r})}$$\scriptstyle{\iota_{\mathbf{h},w_{\mathrm{sat}}}}$$\scriptstyle{\subseteq}$$\scriptstyle{\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}}$$\scriptstyle{\iota_{\mathbf{h},w}}$

where $\iota_{\mathbf{h},w}$ and $\iota_{\mathbf{h},w_{\mathrm{sat}}}$ are
given in Proposition 4.2.1, $\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}$ is given
in Proposition 4.3.2 and
$\widetilde{W}_{x_{\mathrm{sat}}}\coloneqq\iota_{\mathbf{h},w_{\mathrm{sat}}}^{-1}\left(\overline{U_{x_{\mathrm{sat}}}\times_{\mathcal{W}^{n}_{L}}\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}}\right)$
is an analytic subset of
$\widetilde{W}_{\bar{r},w_{\mathrm{sat}}}^{\mathbf{h}-\mathrm{cr}}$, hence an
analytic subset of $\widetilde{W}_{\bar{r},w}^{\mathbf{h}-\mathrm{cr}}$. From
the definitions, we see that
$\iota_{\mathbf{h},w_{\mathrm{sat}}}\colon\widetilde{W}_{\bar{r},w_{\mathrm{sat}}}^{\mathbf{h}-\mathrm{cr}}\longrightarrow
X_{\mathrm{tri}}^{\square}(\bar{r})$ factors through
$X_{\mathrm{tri}}^{\square}(\bar{r})\times_{\mathcal{W}^{n}_{L}}\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}\longrightarrow
X_{\mathrm{tri}}^{\square}(\bar{r})$. Since
$\iota_{\mathbf{h},w_{\mathrm{sat}}}(r,\underline{\varphi})=x_{\mathrm{sat}}\in
U_{x_{\mathrm{sat}}}\times_{\mathcal{W}^{n}_{L}}\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}$
and
$U_{x_{\mathrm{sat}}}\times_{\mathcal{W}^{n}_{L}}\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}$
is a Zariski-open subset of
$X_{\mathrm{tri}}^{\square}(\bar{r})\times_{\mathcal{W}^{n}_{L}}\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}$,
it follows that
$T_{\widetilde{W}_{x_{\mathrm{sat}}},(r,\underline{\varphi})}=T_{\widetilde{W}_{\bar{r},w_{\mathrm{sat}}}^{\mathbf{h}-\mathrm{cr}},(r,\underline{\varphi})}$.
Therefore, (4.8) induces a commutative diagram of tangent spaces

(4.9)
${T_{\widetilde{W}_{\bar{r},w_{\mathrm{sat}}}^{\mathbf{h}-\mathrm{cr}},(r,\underline{\varphi})}}$${T_{\overline{U_{x_{\mathrm{sat}}}\times_{\mathcal{W}^{n}_{L}}\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}},x_{\mathrm{sat}}}}$${T_{\widetilde{W}_{\bar{r},w}^{\mathbf{h}-\mathrm{cr}},(r,\underline{\varphi})}}$${T_{X_{\mathrm{tri}}^{\square}(\bar{r}),x}\,.}$

Let $T_{1},T_{2}\subseteq T_{X_{\mathrm{tri}}^{\square}(\bar{r}),x}$ be the
respective images of the bottom and right arrows of (4.9). Then this diagram
induces a chain of morphisms

(4.10)
$T_{\widetilde{W}_{\bar{r},w_{\mathrm{sat}}}^{\mathbf{h}-\mathrm{cr}},(r,\underline{\varphi})}\longrightarrow
T_{\widetilde{W}_{\bar{r},w}^{\mathbf{h}-\mathrm{cr}},(r,\underline{\varphi})}\times_{T_{X_{\mathrm{tri}}^{\square}(\bar{r}),x}}T_{\overline{U_{x_{\mathrm{sat}}}\times_{\mathcal{W}^{n}_{L}}\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}},x_{\mathrm{sat}}}\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}T_{1}\cap
T_{2}$

of vector spaces.

The first morphism in (4.10) is injective since
$T_{\widetilde{W}_{\bar{r},w_{\mathrm{sat}}}^{\mathbf{h}-\mathrm{cr}},(r,\underline{\varphi})}\longrightarrow
T_{\widetilde{W}_{\bar{r},w}^{\mathbf{h}-\mathrm{cr}},(r,\underline{\varphi})}$
is injective; we then wish to prove its surjectivity. Let
$A=k(x)[\varepsilon]/(\varepsilon^{2})$ be the algebra of infinitesimal
numbers over $k(x)$. An element $v_{\mathrm{sat}}\in
T_{U_{x_{\mathrm{sat}}},x_{\mathrm{sat}}}$ is a pair
$v_{\mathrm{sat}}=(r_{A},\underline{\delta}_{\mathrm{sat},A})$ where
$r_{A}\colon\mathcal{G}_{K}\longrightarrow\operatorname{GL}_{n}(A)$ is a
representation which reduces to $r$ modulo $(\varepsilon)$ and
$\underline{\delta}_{\mathrm{sat},A}$ is a character
$(K^{\times})^{n}\longrightarrow A^{\times}$ which reduces to
$\underline{\delta}_{\mathrm{sat}}$ modulo $(\varepsilon)$. We know from [26,
Corollary 6.3.10] that the triangulation on $r$ globalises in a neighbourhood
of $x_{\mathrm{sat}}$ in $U_{\mathrm{tri}}^{\square}(\bar{r})$, hence $r_{A}$
is triangular with parameter $\underline{\delta}_{\mathrm{sat},A}$; see the
argument in the proof of [11, Proposition 5.16]. Therefore, an element $v$ in
the right-hand side of (4.10) is a triple
$v=(r_{A},\underline{\varphi}_{A},\underline{\delta}_{A})$ where
$r_{A}\colon\mathcal{G}_{K}\longrightarrow\operatorname{GL}_{n}(A)$ is a
crystalline representation of Hodge-Tate weights $\mathbf{h}$ which reduces to
$r$ modulo $(\varepsilon)$, $\underline{\varphi}_{A}\in(A^{\times})^{n}$ is
the ordering of the eigenvalues of $r_{A}$ which reduces to
$\underline{\varphi}$ modulo $(\varepsilon)$, and $\underline{\delta}_{A}$ is
the parameter of a triangulation of $r_{A}$, such that

$z^{w(\mathbf{h})}\operatorname{unr}(\underline{\varphi}_{A})=\jmath_{w,w_{\mathrm{sat}},\mathbf{h}}(\underline{\delta}_{A})=z^{w(\mathbf{h})-w_{\mathrm{sat}}(\mathbf{h})}\underline{\delta}_{A}\,.$

In particular,
$\underline{\delta}=z^{w_{\mathrm{sat}}(\mathbf{h})}\operatorname{unr}(\underline{\varphi})$;
hence Proposition 2.2.4(3) implies that
$(r_{A},\underline{\varphi}_{A})\in\widetilde{W}_{\bar{r},w_{\mathrm{sat}}}^{\mathbf{h}-\mathrm{cr}}$,
which is to say that $v$ is in the image of (4.10). This proves that (4.10) is
a chain of isomorphisms of vector spaces.

Therefore,

$\dim_{k(x)}(T_{1}\cap
T_{2})=\dim_{k(x)}T_{\widetilde{W}_{\bar{r},w_{\mathrm{sat}}}^{\mathbf{h}-\mathrm{cr}},(r,\underline{\varphi})}$

thus

(4.11)
$\dim_{k(x)}T_{X_{\mathrm{tri}}^{\square}(\bar{r}),x}\geq\dim_{k(x)}(T_{1}+T_{2})=\dim_{k(x)}T_{\overline{U_{x_{\mathrm{sat}}}\times_{\mathcal{W}^{n}_{L}}\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}},x_{\mathrm{sat}}}\\\
+\dim_{k(x)}T_{\widetilde{W}_{\bar{r},w}^{\mathbf{h}-\mathrm{cr}},(r,\underline{\varphi})}-\dim_{k(x)}T_{\widetilde{W}_{\bar{r},w_{\mathrm{sat}}}^{\mathbf{h}-\mathrm{cr}},(r,\underline{\varphi})}\,.$

By Proposition 4.3.1, we have

(4.12)
$\dim_{k(x)}T_{\overline{U_{x_{\mathrm{sat}}}\times_{\mathcal{W}^{n}_{L}}\mathcal{W}^{n}_{w,w_{\mathrm{sat}},\mathbf{h},L}},x_{\mathrm{sat}}}=\dim
X_{\mathrm{tri}}^{\square}(\bar{r})-d_{ww_{\mathrm{sat}}^{-1}}\,.$

Also, the cartesian diagram

${\widetilde{W}_{\bar{r},w_{\mathrm{sat}}}^{\mathbf{h}-\mathrm{cr}}
}$${\prod_{\tau\in\Sigma}\overline{(Bw_{\mathrm{sat},\tau}B/B)}_{L}^{\mathrm{rig}}}$${\widetilde{W}_{\bar{r},w}^{\mathbf{h}-\mathrm{cr}}}$${\prod_{\tau\in\Sigma}\overline{(Bw_{\tau}B/B)}_{L}^{\mathrm{rig}}}$$\scriptstyle{h}$$\scriptstyle{\subseteq}$$\scriptstyle{\subseteq}$$\scriptstyle{h}$

and the smoothness of $h$ yield

(4.13)
$\dim_{k(x)}T_{\widetilde{W}_{\bar{r},w}^{\mathbf{h}-\mathrm{cr}},(r,\underline{\varphi})}-\dim_{k(x)}T_{\widetilde{W}_{\bar{r},w_{\mathrm{sat}}}^{\mathbf{h}-\mathrm{cr}},(r,\underline{\varphi})}\\\
=\sum_{\tau\in\Sigma}\dim_{k(x)}T_{\overline{(Bw_{\tau}B/B)}_{k(x)}^{\mathrm{rig}},x_{\tau}}-\sum_{\tau\in\Sigma}\dim_{k(x)}T_{\overline{(Bw_{\mathrm{sat},\tau}B/B)}_{k(x)}^{\mathrm{rig}},x_{\tau}}\,.$

Finally, as $(x_{\tau})$ lies in the smooth cell
$\prod_{\tau\in\Sigma}(Bw_{\mathrm{sat},\tau}B/B)_{L}^{\mathrm{rig}}$, we have

(4.14)
$\sum_{\tau\in\Sigma}\dim_{k(x)}T_{\overline{(Bw_{\mathrm{sat},\tau}B/B)}_{k(x)}^{\mathrm{rig}},x_{\tau}}=\dim\prod_{\tau\in\Sigma}(Bw_{\mathrm{sat},\tau}B/B)_{L}^{\mathrm{rig}}=\operatorname{lg}(w_{\mathrm{sat}})\,.$

Putting together (4.11), (4.12), (4.13), (4.14) gives the desired inequality,
which finishes the proof.

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