# Higher Coleman Theory

George Boxer<EMAIL_ADDRESS>Institut de mathématique
d’Orsay, Université Paris-Saclay, Bâtiment 307, rue Michel Magat, F-91405
Orsay Cedex, France  and  Vincent Pilloni  vincent.pilloni@universite-paris-
saclay.fr Institut de mathématique d’Orsay, Université Paris-Saclay, Bâtiment
307, rue Michel Magat, F-91405 Orsay Cedex, France

###### Abstract.

We develop local cohomology techniques to study the finite slope part of the
coherent cohomology of Shimura varieties. The local cohomology groups we
consider are a generalization of overconvergent modular forms, and they are
defined by using a stratification on the Shimura variety obtained from the
Bruhat stratification on a flag variety via the Hodge-Tate period map. We
construct a spectral sequence from the local cohomologies to the classical
cohomology and use it to obtain classicality and vanishing results. We also
develop a theory of $p$-adic families and construct eigenvarieties. As an
application, we prove some new properties of Galois representations arising
from certain non-regular algebraic cuspidal automorphic representations.

###### Contents

  1. 1 Introduction
    1. 1.1 Setup
    2. 1.2 Overconvergent cohomologies
    3. 1.3 Slope bounds and classicality
    4. 1.4 Locally analytic cohomology, interpolation, and the eigenvariety
    5. 1.5 A toy model: the cohomology of flag varieties and the Cousin complex of Kempf
  2. 2 Cohomological preliminaries
    1. 2.1 Cohomology with support in a closed subspace
    2. 2.2 Cup products
    3. 2.3 The spectral sequence of a filtered topological space
    4. 2.4 The category of projective Banach modules
      1. 2.4.1 Modules
      2. 2.4.2 Banach modules
      3. 2.4.3 The derived category
      4. 2.4.5 Projectives
      5. 2.4.7 Limits of Banach spaces
      6. 2.4.8 Compact operators
    5. 2.5 The cohomology of Banach sheaves
      1. 2.5.1 Sheaves of Banach modules over adic spaces
      2. 2.5.5 Cohomological properties of Banach sheaves
      3. 2.5.13 Acyclicity of quasi-Stein spaces
      4. 2.5.18 Cohomology complexes
      5. 2.5.22 Compact morphisms in the cohomology of adic spaces
    6. 2.6 Integral structures on Banach sheaves
    7. 2.7 Duality for analytic adic spaces
  3. 3 Flag variety
    1. 3.1 Bruhat decomposition
    2. 3.2 Interlude: the cohomology of the flag variety
    3. 3.3 Analytic geometry
      1. 3.3.1 The Iwahori decomposition
      2. 3.3.4 Tubes of Bruhat cells
      3. 3.3.6 Analytic subspaces of $\mathcal{U}_{w}^{an}$
      4. 3.3.10 Orbits of Cells
      5. 3.3.18 Intersections and unions of cells
    4. 3.4 Dynamics
    5. 3.5 Dynamics of correspondences
      1. 3.5.1 Certain compact open subgroups
      2. 3.5.5 Change of group
      3. 3.5.7 Hecke algebras
      4. 3.5.9 Action of correspondences on the flag variety
  4. 4 Shimura varieties
    1. 4.1 Automorphic vector bundles and their cohomology
      1. 4.1.1 Automorphic vector bundles
      2. 4.1.6 The cohomology of automorphic vector bundles
    2. 4.2 Action of the Hecke algebra
      1. 4.2.1 Action of Hecke correspondences on cohomology
      2. 4.2.2 Composition of Hecke correspondences
      3. 4.2.8 Serre duality
      4. 4.2.11 The finite slope part of classical cohomology
    3. 4.3 Jacquet Modules
    4. 4.4 The Hodge-Tate period morphism
      1. 4.4.1 Inverse limit of adic spaces
      2. 4.4.8 Siegel Shimura varieties
      3. 4.4.10 Perfectoid Siegel Shimura varieties
      4. 4.4.12 Formal models of perfectoid Siegel Shimura varieties
      5. 4.4.24 The Hodge type case
      6. 4.4.26 Perfectoid Hodge type Shimura varieties
      7. 4.4.27 Compactifications in the Hodge case
      8. 4.4.31 Integral models in the Hodge case
      9. 4.4.38 General Shimura varieties
      10. 4.4.41 Abelian type Shimura varieties
      11. 4.4.48 Proof of theorem 4.4.45
    5. 4.5 The truncated Hodge-Tate period map
    6. 4.6 Reductions of the torsor $\mathcal{M}^{an}_{dR}$
      1. 4.6.1 Preparations
      2. 4.6.2 Integral structure
      3. 4.6.8 Further reductions of the group structure
      4. 4.6.18 Maps between torsors
  5. 5 Overconvergent cohomologies and the spectral sequence
    1. 5.1 The finite slope part
      1. 5.1.1 Slope decomposition
      2. 5.1.3 Compact operators
      3. 5.1.5 Action of an algebra
    2. 5.2 Correspondences and cohomology with support
      1. 5.2.1 Action of the correspondences on subsets of $\mathcal{X}$
      2. 5.2.3 Action of the correspondence on cohomology
      3. 5.2.5 Representing the correspondence by a map of complexes
    3. 5.3 A formal analytic continuation result
      1. 5.3.1 The diamond of a correspondence
      2. 5.3.5 The infinite diamond
    4. 5.4 Overconvergent cohomologies
      1. 5.4.1 First definition
      2. 5.4.2 Existence of finite slope cohomology
      3. 5.4.4 The finite slope cohomology and the cohomology of dagger spaces
      4. 5.4.9 Change of support condition
      5. 5.4.13 Change of level
    5. 5.5 The spectral sequence associated with the Bruhat stratification
      1. 5.5.2 Construction of a filtration
      2. 5.5.5 Proof of Theorem 5.5.1
      3. 5.5.6 Cousin complexes
    6. 5.6 Cohomological vanishing
    7. 5.7 Duality
    8. 5.8 Interior cohomology
    9. 5.9 Lower bounds on slopes
      1. 5.9.8 Proof of theorem 5.9.6
    10. 5.10 Comparisons with slope bounds on classical cohomology
      1. 5.10.1 Some combinatorics
      2. 5.10.6 Slope bounds on classical coherent cohomology
      3. 5.10.10 Slope bounds on Betti cohomology
      4. 5.10.13 Connection with [FP19]
    11. 5.11 Slopes and small slope conditions
      1. 5.11.1 Small slope conditions
      2. 5.11.13 The strongly small slope conditions
    12. 5.12 Small slopes, classicality, and vanishing
      1. 5.12.1 Coherent cohomology
      2. 5.12.10 Betti cohomology
      3. 5.12.15 Small slope conditions and Jacquet modules
    13. 5.13 De Rham and rigid cohomology
    14. 5.14 Explicit formulas in the symplectic case
      1. 5.14.2 $GL_{2}/\mathbb{Q}$
      2. 5.14.3 $GSp_{4}/\mathbb{Q}$
  6. 6 $p$-adic families of overconvergent cohomology
    1. 6.1 Relative spectral theory and slope decompositions
      1. 6.1.1 Slope decomposition over Tate algebras
      2. 6.1.8 Finite slope decomposition for an algebra of operators
      3. 6.1.10 Relative spectral theory
      4. 6.1.13 Relative spectral theory for an algebra of operators
      5. 6.1.16 The case of projective or inductive limits of complexes
    2. 6.2 Locally analytic inductions and the locally analytic BGG resolution
      1. 6.2.2 Algebraic inductions
      2. 6.2.3 Analytic weights
      3. 6.2.4 Analytic inductions
      4. 6.2.9 Twist by a finite order character
      5. 6.2.11 Locally algebraic induction
      6. 6.2.12 The BGG complex
      7. 6.2.20 Distributions
    3. 6.3 $p$-adic families of sheaves
      1. 6.3.1 Definition of the sheaves
      2. 6.3.2 First properties
      3. 6.3.5 Interpolation sheaves and locally algebraic weights
      4. 6.3.9 Definition of the Hecke action
    4. 6.4 Locally analytic overconvergent cohomology
      1. 6.4.1 First definition
      2. 6.4.2 Existence of finite slope cohomology
      3. 6.4.4 Change of analyticity radius
      4. 6.4.6 Change of support condition
      5. 6.4.9 Change of level
    5. 6.5 Two spectral sequences
      1. 6.5.1 The Tor spectral sequence
      2. 6.5.4 The spectral sequence from locally analytic overconvergent to overconvergent cohomology
    6. 6.6 Cohomological vanishing
    7. 6.7 Cup products and Serre duality
    8. 6.8 Slope estimates and control theorem
    9. 6.9 Eigenvarieties
    10. 6.10 Improved slope bounds for interior cohomology and applications
    11. 6.11 Application: construction of Galois representations and local-global compatibility $p$

##  1\. Introduction

In the 90’s, Coleman proved his classicality theorem for overconvergent
$p$-adic modular forms [Col96] and developed the theory of $p$-adic families
of finite slope overconvergent modular forms [Col97]. Subsequently, Coleman
and Mazur constructed the eigencurve [CM98]. Since then, these works have been
generalized to a wide class of Shimura varieties.

These Coleman theories concern classical and overconvergent modular forms, or
zeroth cohomology of automorphic vector bundles on Shimura varieties. Recently
we have come to expect [Pil20], [BCGP18], [LPSZ19], [BP20] that there should
be some analogous “higher Coleman theories” concerning the higher coherent
cohomology of Shimura varieties. The goal of this paper is to develop such
theories for a rather general class of Shimura varieties.

###  1.1. Setup

Let $(G,X)$ be a Shimura datum of abelian type, so by definition
$G/\mathbb{Q}$ is a reductive group and $X$ is a complex analytic space with
an action of $G(\mathbb{R})$ satisfying a list of axioms ([Del79]). There are
two opposite parabolic subgroups of $G$ attached to $(G,X)$, called $P_{\mu}$
and $P_{\mu}^{std}$, with common Levi $M_{\mu}$. The space $X$ embeds
$G(\mathbb{R})$-equivariantly as an open subspace of
$FL_{G,\mu}^{std}(\mathbb{C})=G/P_{\mu}^{std}(\mathbb{C})$. This is the Borel
embedding.

For any neat compact open subgroup $K\subseteq G(\mathbb{A}_{f})$, we let
$S_{K}(\mathbb{C})=G(\mathbb{Q})\backslash X\times G(\mathbb{A}_{f})/K$ be the
corresponding Shimura variety over $\mathbb{C}$. This is a finite disjoint
union of arithmetic quotients of $X$.

Any representation of $P_{\mu}^{std}$ defines a $G$-equivariant vector bundle
over $FL_{G,\mu}^{std}$. Let $Z_{s}\subseteq G$ be the maximal torus of the
center of $G$ which splits over $\mathbb{R}$ but contains no
$\mathbb{Q}$-split subtorus. We let $M_{\mu}^{c}=M_{\mu}/Z_{s}$. By pull back
to $X$ and descent to $S_{K}(\mathbb{C})$, we obtain a functor from the
category of representations of $M^{c}_{\mu}$ to the category of vector bundles
on $S_{K}(\mathbb{C})$, called (totally decomposed) automorphic vector
bundles. We make a choice of Borel subgroup $B$ contained in $P_{\mu}$, let
$T$ be a maximal torus contained in this Borel, and let $T^{c}=T/Z_{s}$. We
label irreducible representations of $M^{c}_{\mu}$ by their highest weight in
$X^{\star}(T^{c})^{M_{\mu},+}$. For any weight $\kappa\in
X^{\star}(T^{c})^{M_{\mu},+}$ we let $\mathcal{V}_{\kappa}$ be the
corresponding vector bundle over $S_{K}(\mathbb{C})$.

The Shimura variety $S_{K}(\mathbb{C})$ has a structure of an algebraic
variety $S_{K}$ defined over a number field $E$, called the reflex field. For
a combinatorial choice $\Sigma$ of cone decomposition, there are projective
compactifications $S_{K,\Sigma}^{tor}$ whose (reduced) boundary
$D_{K,\Sigma}=S_{K,\Sigma}^{tor}\setminus S_{K}$ is a Cartier divisor. The
vector bundles $\mathcal{V}_{\kappa}$ admit models over $S_{K}$ and canonical
extensions $\mathcal{V}_{\kappa,\Sigma}$ to $S_{K,\Sigma}^{tor}$ ([Mil90],
[Har90a]). We also have $\mathcal{V}_{\kappa,\Sigma}(-D_{K,\Sigma})$, the so
called sub-canonical extension (its sections are holomorphic cusp forms).

This paper is devoted to the study of the coherent cohomology of weight
$\kappa$: $\mathrm{H}^{i}(S^{tor}_{K,\Sigma},\mathcal{V}_{\kappa,\Sigma})$,
$\mathrm{H}^{i}(S^{tor}_{K,\Sigma},\mathcal{V}_{\kappa,\Sigma}(-D_{K,\Sigma}))$
as well as the interior cohomology:

$\overline{\mathrm{H}}^{i}(S^{tor}_{K,\Sigma},\mathcal{V}_{\kappa,\Sigma})=\mathrm{Im}({\mathrm{H}}^{i}(S^{tor}_{K,\Sigma},\mathcal{V}_{\kappa,\Sigma}(-D_{K,\Sigma}))\rightarrow{\mathrm{H}}^{i}(S^{tor}_{K,\Sigma},\mathcal{V}_{\kappa,\Sigma})).$

They are independent of the cone decomposition $\Sigma$ and we denote them
simply by $\mathrm{H}^{i}(K,\kappa)$, $\mathrm{H}^{i}(K,\kappa,cusp)$, and
$\overline{\mathrm{H}}^{i}(K,\kappa)$.

We now fix a rational prime $p$ and pass to the $p$-adic theory. We assume
throughout this paper that $G_{\mathbb{Q}_{p}}$ is quasi-split (this
assumption is necessary to have finite slope families over all of weight
space). We fix a finite extension $F$ of $\mathbb{Q}_{p}$ over which
$G_{\mathbb{Q}_{p}}$ splits, and we assume that we have chosen $T\subseteq
B\subseteq P_{\mu}$ so that $T$ and $B$ are defined over $\mathbb{Q}_{p}$ and
$P_{\mu}$ is defined over $F$. We now pass to analytic geometry and let
$\mathcal{S}_{K,\Sigma}^{tor}/\mathrm{Spa}(F,\mathcal{O}_{F})$ be the adic
Shimura variety. We assume for the rest of the introduction that
$K=K^{p}K_{p}$ where $K_{p}\subset G(\mathbb{Q}_{p})$ is a certain congruence
subgroup (denoted $K_{p,1,0}$ in 3.5.1) possessing an Iwahori factorization

$K_{p}=(\overline{U}(\mathbb{Q}_{p})\cap K_{p})\cdot
T(\mathbb{Z}_{p})\cdot(U(\mathbb{Q}_{p})\cap K_{p})$

where $U$ and $\overline{U}$ denote the unipotent radicals of $B$ and the
opposite Borel. When $G_{\mathbb{Q}_{p}}$ is unramified, $K_{p}$ will be an
Iwahori subgroup.

Our study of the $p$-adic properties of coherent cohomology is based on the
Hodge-Tate period map introduced by Scholze [Sch15] and refined in [CS17] and
[DLLZ19]. More specifically we will use the existence of a diagram

where $\mathcal{FL}_{G,\mu}=P_{\mu}\backslash G$ is the adic space associated
to the Hodge-Tate flag space $FL_{G,\mu}=P_{\mu}\backslash G$. Here we only
view $\mathcal{FL}_{G,\mu}/K_{p}$ as a topological space, and the “truncated”
Hodge-Tate period map $\pi_{HT,K_{p}}$ as a continuous map. The map $\pi_{HT}$
is equivariant for the right actions of the group $G(\mathbb{Q}_{p})$, while
the map $\pi_{HT,K_{p}}$ is equivariant for the action of the Hecke algebra
$\mathcal{H}_{p}=F[K_{p}\backslash G(\mathbb{Q}_{p})/K_{p}]$ by
correspondences. It also extends to a map on the toroidal compactification
$\pi_{HT,K_{p}}^{tor}:\mathcal{S}_{K^{p}K_{p},\Sigma}^{tor}\to\mathcal{FL}_{G,\mu}/K_{p}$.
The basic idea in this paper is to use $\pi_{HT,K_{p}}^{tor}$ to define
certain support conditions for coherent cohomology on the Shimura variety, and
to study the dynamical properties of certain Hecke operators at $p$.

We introduce some further notation. For a choice of $+$ or $-$ which we denote
from now on by $\pm$, we consider the monoids

$T^{\pm}=\\{t\in T(\mathbb{Q}_{p})\mid v(\alpha(t))\geq
0,\forall\alpha\in\Phi^{\pm}\\}$

where $\Phi^{+}$ denotes the positive roots of $G$ with respect to $T\subseteq
B$, and also the associated commutative subalgebras of the Iwahori Hecke
algebra

$\mathcal{H}_{p}^{\pm}=\mathrm{span}\\{[K_{p}tK_{p}]\mid t\in
T^{\pm}\\}\subseteq\mathcal{H}_{p}.$

These should be thought of as the algebra of “$U_{p}$-type” Hecke operators.
The reason we consider both the $+$ and $-$ algebras is in order to study
duality (we note that $[K_{p}tK_{p}]$ and $[K_{p}t^{-1}K_{p}]$ are transposes
as correspondences), but the reader might assume that all $\pm$’s are just $+$
on first reading.

We let $W$ be the (absolute) Weyl group of $G$ and $W_{M}\subseteq W$ the Weyl
group of $M_{\mu}$. We write $\ell:W\to\mathbb{Z}$ for the length function and
we write $w_{0}\in W$ and $w_{0,M}\in W_{M}$ for the longest elements. In this
paper a crucial role will be played by the set ${}^{M}W\subseteq W$ of minimal
length coset representatives for $W_{M}\backslash W$. It has a unique longest
element $w_{0}^{M}=w_{0,M}w_{0}$ of length $d=\dim S_{K}=\dim FL_{G,\mu}$. We
define $\ell_{\pm}:{}^{M}W\to[0,d]$ by $\ell_{+}=\ell$ and
$\ell_{-}(w)=d-\ell(w)$.

The set ${}^{M}W$ will index the “higher Coleman theories” constructed in this
paper. We recall how it arises naturally in the study of coherent cohomology,
both from the archimedean and $p$-adic points of view.

The coherent cohomologies $\mathrm{H}^{i}(K,\kappa)$ for $\kappa\in
X^{\star}(T^{c})^{M_{\mu},+}$ can be computed in terms of automorphic forms on
$G$ ([Har90a], [Su18]). Cuspidal automorphic representations
$\pi=\pi_{\infty}\otimes\pi_{f}$ contribute to coherent cohomology according
to the archimedean component $\pi_{\infty}$. The (essentially) tempered
$\pi_{\infty}$ which contribute to coherent cohomology have been completely
classified: according to work of Blasius-Harris-Ramakrishnan, Mirkovich,
Schmid, and Williams, they are exactly the so called non-degenerate limits of
discrete series (see [Har90a], theorems 3.4 and 3.5). According to Harish-
Chandra, for each element of ${}^{M}W$ there is a corresponding family of
(limits of) discrete series representations, parameterized by the
infinitesimal character (more accurately, this would be true if
$G(\mathbb{R})$ were semisimple and simply connected).

We recall how the limits of discrete series contribute to coherent cohomology.
Beginning with a (dominant) “C-algebraic” infinitesimal character $\nu+\rho\in
X^{\star}(T^{c})_{\mathbb{R}}^{+}$ with $\nu\in X^{\star}(T^{c})$, then for a
choice of $w\in{}^{M}W$, if $\kappa_{w}=-w_{0,M}w(\nu+\rho)-\rho$ is
$M_{\mu}$-dominant, then there is a (limit of) discrete series with
infinitesimal character $\nu+\rho$ contributing to coherent cohomology in
weight $\kappa_{w}$ and degree $\ell(w)$. We recall some features of this
description:

  * •

The case of $w=1$ corresponds to the family of holomorphic (limits of)
discrete series. They contribute to degree zero coherent cohomology.

  * •

When $\nu+\rho$ is regular, $\kappa_{w}$ is automatically $M_{\mu}$-dominant,
and the corresponding representations of $G(\mathbb{R})$ are discrete series.
Moreover a $\kappa\in X^{\star}(T^{c})^{M_{\mu},+}$ with $\kappa+\rho$ regular
will be of the form $\kappa_{w}$ for a unique $\nu$ and $w\in{}^{M}W$, and so
cusp forms which are tempered at $\infty$ contribute to cohomology in weight
$\kappa$ in only a single degree.

  * •

When $\nu+\rho$ is not regular, there will be some subset of ${}^{M}W$
(possibly empty) for which $\kappa_{w}$ is $M_{\mu}$-dominant (we emphasize in
particular that this set may be nonempty but not contain $1$, and this is a
major motivation for studying higher coherent cohomology of Shimura
varieties). In this case the corresponding representations of $G(\mathbb{R})$
are limits of discrete series. Beginning with a $\kappa\in X^{\star}(T)^{+}$
with $\kappa+\rho$ irregular, we will have $\kappa=\kappa_{w}$ for multiple
$w\in{}^{M}W$ (with a range of lengths), and so the cohomology in weight
$\kappa$ will have contributions from cuspidal automorphic representations
tempered at $\infty$ in a range of degrees.

We note that in this paper, our study of coherent cohomology is purely
geometric, and makes no use of the theory of automorphic forms or the results
recalled above (except in the application to local-global compatibility for
irregular automorphic representations in section 6.11). Nonetheless, it is
useful to keep in mind, for interpreting the results that follow.

Now we pass to $p$-adic geometry, beginning with the Hodge-Tate flag space
$FL_{G,\mu}=P_{\mu}\backslash G$. The set ${}^{M}W$ indexes the Schubert
stratification of $FL_{G,\mu}$:

$FL_{G,\mu}=P_{\mu}\backslash G=\coprod_{w\in WM}P_{\mu}\backslash P_{\mu}wB$

Passing to a more dynamical point of view, the set of fixed points for the
action of $T$ on $FL_{G,\mu}$ is exactly

$\mathrm{Fix}_{T}(FL_{G,\mu})=\\{P_{\mu}w\mid w\in{}^{M}W\\}$

as indeed there is one $T$ fixed point in each Schubert cell. A closely
related fact is that the set of points of
$\mathrm{FL}_{G,\mu}(\mathbb{C}_{p})/K_{p}$ fixed by all of the “$U_{p}$-type”
Hecke correspondences $[K_{p}tK_{p}]$ for $t\in T^{\pm}$ is

$\\{P_{\mu}wK_{p}\mid w\in{}^{M}W\\}.$

(Here we say that a point $xK_{p}\in\mathrm{FL}_{G,\mu}(\mathbb{C}_{p})/K_{p}$
is fixed by $[K_{p}tK_{p}]$ means that $x\in xK_{p}tK_{p}$.)

According to Scholze-Weinstein [SW13], $FL_{G,\mu}(\mathbb{C}_{p})/K_{p}$
should be thought of as the set of $p$-divisible groups “with $G$-structure”
over $\mathcal{O}_{C_{p}}$, equipped with a $K_{p}$ level structure (on the
generic fiber). In particular, the points $P_{\mu}wK_{p}$ for $w\in{}^{M}W$
correspond to certain very special such $p$-divisible groups, which are fixed
points for the action of the “$U_{p}$-type” Hecke correspondences. We can now
pass to the Shimura variety via the Hodge-Tate period map. According to
Caraiani-Scholze [CS17], the fibers of $\pi_{HT}$ are perfectoid Igusa
varieties. In this paper we will consider the “Igusa varieties”
$(\pi_{HT,K_{p}}^{tor})^{-1}(P_{\mu}wK_{p})\subset\mathcal{S}_{K,\Sigma}^{tor}$
at finite level. We view them as some kind of fixed subspaces for the
“$U_{p}$-type” Hecke correspondences. We emphasize that in contrast to the
situation at infinite level, they are not Zariski closed, and we do not try to
give them any geometric structure (in fact, we will only consider
neighborhoods of them, constructed using the Hodge-Tate period map).

We give two illustrative examples:

######  Example 1.1.1.

If $(G,X)=(\mathrm{GSp}_{2g},\mathcal{H}_{g}^{\pm})$ is the Siegel Shimura
datum and $K_{p}$ is Iwahori, then the corresponding Shimura varieties are
moduli spaces of polarized abelian varieties of dimension $g$. The polarized
$p$-divisible groups with Iwahori level structures
$(\mathcal{G},\lambda,\\{H_{i}\\}_{0\leq i\leq 2g})$ corresponding to the
elements of ${}^{M}W$ are all ordinary (i.e.
$\mathcal{G}=\mu_{p^{\infty}}^{g}\times(\mathbb{Q}_{p}/\mathbb{Z}_{p})^{g}$)
and the element $w\in{}^{M}W$ measures the relative position of the canonical
subgroup $H_{can}=\mu_{p}^{g}$ with the symplectic flag

$0=H_{0}\subset H_{1}\subset\cdots\subset H_{2g}=\mathcal{G}[p]$

giving the Iwahori level structure (alternatively, the $2^{g}$ elements of
${}^{M}W$ correspond to the $2^{g}$ possibilities that $H_{i}/H_{i-1}$ for
$i=1,\ldots,g$ is either multiplicative or étale). Passing to the Shimura
variety, the “Igusa varieties” $\pi_{HT,K_{p}}^{-1}(P_{\mu}wK_{p})$ are simply
the (closures of the) corresponding components of the (quasi-compact, open)
ordinary locus.

######  Example 1.1.2.

If
$(G,X)=(\mathrm{Res}_{F/\mathbb{Q}}\mathrm{GL}_{2},(\mathcal{H}_{1}^{\pm})^{[F:\mathbb{Q}]})$
is a Hilbert Shimura datum for $F$ a totally real field in which $p$ remains
prime (for simplicity), then the corresponding Shimura varieties are (coarse)
moduli spaces of abelian varieties of dimension $[F:\mathbb{Q}]$ with an
action of $\mathcal{O}_{F}$. In this case,
${}^{M}W=W=\\{1,w\\}^{\mathrm{Hom}(F,\mathbb{C}_{p})}$. For
$I\subseteq\mathrm{Hom}(F,\mathbb{C}_{p})$ we write $w_{I}\in W$ for the
element which is $1$ in the factors corresponding to the elements of $I$. Then
the $p$-divisible group with $\mathcal{O}_{F}$-action and Iwahori level
structure corresponding to $w_{I}$ is
$(\mathrm{LT}_{I}\times\mathrm{LT}_{I^{c}},\mathrm{LT}_{I^{c}}[p])$ where
$I^{c}=\mathrm{Hom}(F,\mathbb{C}_{p})\setminus I$, and
$\mathrm{LT}_{I}/\mathcal{O}_{\mathbb{C}_{p}}$ is the unique $p$-divisible
group with $\mathcal{O}_{F}$-action of height $[F:\mathbb{Q}]$ and such that
$\mathrm{Lie}_{LT_{I}}=\oplus_{\tau\in I}\mathbb{C}_{p}(\tau)$ as
$F\otimes\mathbb{C}_{p}$ modules, where $\mathbb{C}_{p}(\tau)$ denotes the
$F\otimes\mathbb{C}_{p}$ module on which $F$ acts via $\tau$. The isogeny
corresponding to quotienting by $\mathrm{LT}_{I}[p]$ witnesses these
$p$-divisible groups as fixed points for the “$U_{p}$-type” Hecke
correspondences. In particular, in contrast to the previous example, only
$1=w_{\mathrm{Hom}(F,\mathbb{C}_{p})}$ and $w_{0}=w_{\emptyset}$ correspond to
ordinary $p$-divisible groups.

###  1.2. Overconvergent cohomologies

The first aim of this paper is the construction, for all $w\in{}^{M}W$,
weights $\kappa\in X^{\star}(T^{c})^{M_{\mu},+}$, and choices $\pm$ of $+$ or
$-$, of some finite slope, overconvergent cohomologies
$\mathrm{H}_{w}^{i}(K^{p},\kappa)^{\pm,fs}$ and
$\mathrm{H}_{w}^{i}(K^{p},\kappa,cusp)^{\pm,fs}$. These spaces carry actions
of the algebras $\mathcal{H}_{p}^{\pm}$ for which the “$U_{p}$-type” operators
$[K_{p}tK_{p}]$ for $t\in T^{\pm}$ act invertibly (hence “finite slope”) as
well as actions of the prime to $p$ Hecke operators.

We explain the rough idea of the construction: we start with the complex
$\mathrm{R}\Gamma_{\mathcal{U}\cap\mathcal{Z}}(\mathcal{U},\mathcal{V}_{\kappa,\Sigma})$
of cohomology with support, where
$\mathcal{U}\cap\mathcal{Z}\subset\mathcal{S}_{K,\Sigma}^{tor}$ is a suitably
chosen, locally closed neighborhood of the “Igusa variety”
$\pi_{HT,K_{p}}^{-1}(w\cdot K_{p})$. The support conditions are chosen in
order to have an action of the Hecke operators $[K_{p}tK_{p}]$ for $t\in
T^{\pm}$, and so that furthermore sufficiently regular Hecke operators act
compactly. Then we use the spectral theory of compact operators to take the
finite slope part, i.e. pass to the part where the Hecke operators
$[K_{p}tK_{p}]$ for $t\in T^{\pm}$ act invertibly. The space
$\mathrm{H}^{0}_{1}(K^{p},\kappa)^{+,fs}$ is nothing but the usual space of
finite slope overconvergent modular forms of weight $\kappa$.

The following theorem summarizes the basic results about the finite slope
overconvergent cohomologies.

######  Theorem 1.2.1.

  1. (1)

(Spectral sequence to classical cohomology) There is a
$\mathcal{H}_{p}^{\pm}$-equivariant spectral sequence

$\mathbf{E}_{1}^{p,q}=\bigoplus_{w\in{}^{M}W,\ell_{\pm}(w)=p}\mathrm{H}_{w}^{p+q}(K^{p},\kappa)^{\pm,fs}\Rightarrow\mathrm{H}^{p+q}(K,\kappa)^{\pm,fs}$

and similarly for cuspidal cohomology.

  2. (2)

(Analytic continuation) The finite slope, overconvergent cohomologies
$\mathrm{R}\Gamma_{w}(K^{p},\kappa)^{\pm,fs}$ and
$\mathrm{R}\Gamma_{w}(K^{p},\kappa,cusp)^{\pm,fs}$, together with their Hecke
action, can be computed on arbitrarily small neighborhoods of the “Igusa
variety” $\pi_{HT,K_{p}}^{-1}(w\cdot
K_{p})\subseteq\mathcal{S}_{K,\Sigma}^{tor}$. (See sections 5.3 and 5.4 for a
precise statement.)

  3. (3)

(Duality) There is a perfect pairing

$\langle,\rangle:\mathrm{H}^{i}_{w}(K^{p},\kappa,cusp)^{\pm,fs}\times\mathrm{H}^{d-i}_{w}(K^{p},\kappa^{\vee})^{\mp,fs}\to
F$

such that for all $t\in T^{\pm}$, the Hecke operators $[K_{p}tK_{p}]$ on the
left and $[K_{p}t^{-1}K_{p}]$ on the right are adjoint. Moreover, these
pairings are compatible, via the spectral sequence of part (1), with the
classical Serre duality pairings

$\mathrm{H}^{i}(K,\kappa,cusp)\times\mathrm{H}^{d-i}(K,\kappa^{\vee})\to F.$

(Here $\kappa^{\vee}:=-2\rho_{nc}-w_{0,M}\kappa$ where $2\rho_{nc}\in
X^{\star}(T)$ is the sum of the positive roots of $G$ which are not roots of
$M_{\mu}$. The canonical bundle of $S^{tor}_{K,\Sigma}$ is
$\mathcal{V}_{-2\rho_{nc},\Sigma}(-D_{K,\Sigma})$, and the dual of
$\mathcal{V}_{\kappa,\Sigma}$ is $\mathcal{V}_{-w_{0,M}\kappa,\Sigma}$.)

  4. (4)

(Vanishing) We have

$\displaystyle\mathrm{H}^{i}_{w}(K^{p},\kappa)^{\pm,fs}$ $\displaystyle=$
$\displaystyle 0\quad\mathrm{for}\quad i<\ell_{\pm}(w),$
$\displaystyle\mathrm{H}^{i}_{w}(K^{p},\kappa,cusp)^{\pm,fs}$ $\displaystyle=$
$\displaystyle 0\quad\mathrm{for}\quad i>\ell_{\pm}(w),$

and so in particular the interior cohomology
$\overline{\mathrm{H}}_{w}^{i}(K^{p},\kappa)^{\pm,fs}$ vanishes except when
$i=\ell_{\pm}(w)$.

The spectral sequence is simply the finite slope part of the spectral sequence
of a filtration on $S_{K,\Sigma}^{tor}$ defined using $\pi_{HT}$ and the
Bruhat stratification of the flag variety. We note that if $K_{p}$ is Iwahori,
then the “$U_{p}$-type” operators $[K_{p}tK_{p}]$ for $t\in T^{\pm}$ are
already invertible in the Iwahori Hecke algebra $\mathcal{H}_{p}$, and hence
the classical cohomology at Iwahori level is already “finite slope”, i.e.
$\mathrm{H}^{i}(K,\kappa)^{\pm,fs}=\mathrm{H}^{i}(K,\kappa)$. The “analytic
continuation” result follows from a dynamical study of the Hecke operators an
their interactions with the support conditions. The vanishing theorem is
ultimately deduced from the affineness of the Hodge-Tate period map.

If the Shimura variety is compact, then the vanishing theorem implies that the
overconvergent cohomologies $\mathrm{H}^{\star}_{w}(K^{p},\kappa)^{\pm,fs}$
are concentrated in the single degree $\ell_{\pm}(w)$, and so the spectral
sequence takes the particularly simple form of a single complex

$\mathrm{H}^{0}_{Id/w_{0}^{M}}(K^{p},\kappa)^{\pm,fs}\rightarrow\cdots\rightarrow\bigoplus_{w\in{}^{M}W,\ell_{\pm}(w)=i}\mathrm{H}^{i}_{w}(K^{p},\kappa)^{\pm,fs}\to\cdots\rightarrow\mathrm{H}^{d}_{w_{0}^{M}/Id}(K^{p},\kappa)^{\pm,fs}$

whose cohomology is the classical cohomology $\mathrm{H}^{i}(K,\kappa)$. We
call this the Cousin complex of the Shimura variety, in analogy with the work
of Kempf [Kem78] on the cohomology of the flag variety.

###  1.3. Slope bounds and classicality

The key input for proving the classicality theorem is a lower bound for the
slopes (i.e. $p$-adic valuations of eigenvalues) of the Hecke operators acting
on finite slope, overconvergent cohomology. More precisely we prove the
following (see theorems 5.9.6 and 6.10.1).

######  Theorem 1.3.1 (Slope bounds).

For any $t\in T^{\pm}$ and any eigenvalue $\lambda$ of $[K_{p}tK_{p}]$ acting
on $\overline{\mathrm{H}}_{w}^{i}(K^{p},\kappa)^{\pm,fs}$ we have

$\displaystyle v(\lambda)\geq v((w^{-1}w_{0,M}(\kappa+\rho)+\rho)(t))$ in the
$+$ case $\displaystyle v(\lambda)\geq v((w^{-1}(\kappa+\rho)-\rho)(t))$ in
the $-$ case

Here $\rho\in X^{\star}(T)_{\mathbb{Q}}$ is half the sum of the positive roots
of $G$ and, given an algebraic character $\mu\in X^{\star}(T)$, we may view it
as a character $\mu:T(F)\to F^{\times}$, so that it makes sense to evaluate it
on $t\in T(\mathbb{Q}_{p})$ and take the valuation. We remark that an
eigenclass is “ordinary” if the inequalities of the theorem are in fact
equalities.

Actually we expect (conjecture 5.9.2) that the same bound holds for the full
overconvergent cohomologies $\mathrm{H}^{i}_{w}(K^{p},\kappa)^{\pm,fs}$ and
$\mathrm{H}^{i}_{w}(K^{p},\kappa,cusp)^{\pm,fs}$, but we were only able to
prove a slightly weaker bound (theorem 5.9.6). The improved bound on interior
cohomology above is proved by a bootstrapping trick using $p$-adic
interpolation.

From the slope bound we can immediately deduce a classicality theorem in
regular weight. The idea is that if we fix $\kappa$, the lower bounds vary
with $w\in{}^{M}W$, and so there will be a certain range of “small slopes”
which can only occur for a single $w$. Then by passing to the small slope part
of the spectral sequence we will obtain an isomorphism between the small slope
parts of the $w$ overconvergent and classical cohomologies.

######  Theorem 1.3.2 (Classicality theorem).

Let $\kappa\in X^{\star}(T^{c})^{M,+}$ be a weight such that $\kappa+\rho$ is
$G$-regular. Let $w_{+}$ (resp. $w_{-}$) be the unique $w\in{}^{M}W$ such that
$-w^{-1}w_{0,M}(\kappa+\rho)$ (resp. $w^{-1}(\kappa+\rho)$) is $G$-dominant.

Then we have a quasi-isomorphism:

$\mathrm{R}\Gamma_{w_{\pm}}(K^{p},\kappa)^{\pm,sss^{M}(\kappa)}=\mathrm{R}\Gamma(K^{p},\kappa)^{\pm,sss^{M}(\kappa)}$

and similarly for cuspidal cohomology. Here the superscript is the “strongly
small slope part”, i.e. the part where the eigenvalues of the Hecke operators
$[K_{p}tK_{p}]$ for $t\in T^{\pm}$ satisfy certain upper bounds (see section
5.11 for the precise definition).

If the Shimura variety is compact, then “strongly small slope” can be replaced
by the weaker “small slope” condition $\pm,ss^{M}(\kappa)$ (again see section
5.11). Even if the Shimura variety is not compact it is still true that a
small slope overconvergent cusp form (i.e. an element of
$H^{0}_{1}(K^{p},\kappa,cusp)^{+,ss^{M}(\kappa)})$ is classical.

######  Remark 1.3.3.

Many cases of this theorem for the degree $0$ cohomology of PEL Shimura
varieties were already proven, see for example [Col96], [Kas06], [Pil11],
[BPS16], [TX16], [Bij17].

######  Remark 1.3.4.

Even in cases of degree 0 cohomology where classicality theorems were already
known, this theorem often improves on the slope bound. For example in the
Siegel case, with the usual labelling of weights $\kappa=(k_{1},\cdots,k_{g})$
with $k_{1}\geq\cdots\geq k_{g}$, and for the usual “minuscule” Hecke operator
$U_{p}$, the theorem states that any finite slope overconvergent cusp form of
weight $\kappa$ and $U_{p}$-slope $<k_{g}-g$ is classical. The bound
previously proven in [Pil11] and [BPS16] was $<k_{g}-\frac{g(g+1)}{2}$.
Holomorphic discrete series contribute to the $\mathrm{H}^{0}$ in weights with
$k_{g}>g$, while holomorphic limits of discrete series contribute when
$k_{g}=g$. Thus we cannot expect any “numerical” classicality results when
$k_{g}=g$, while the theorem is non-vacuous for all $k_{g}>g$.

######  Remark 1.3.5.

We conjecture that the theorem should hold with “small slope” instead of
“strongly small slope” for all Shimura varieties. This would follow from the
conjectural improved slope bound 5.9.2. Moreover we believe that the “small
slope” condition is optimal. We remark that an “ordinary” cohomology class is
small slope in all weights, but it may only be strongly small slope under an
additional regularity condition.

######  Remark 1.3.6.

We cannot expect to have any “numerical” classicality criteria when
$\kappa+\rho$ is not $G$-regular. In this case, there are multiple
$w\in{}^{M}W$ for which $-w^{-1}w_{0,M}(\kappa+\rho)$ (or
$w^{-1}(\kappa+\rho)$) are $G$-dominant, and the lower bound on the slopes are
the same for these $w$. Thus we can impose some small slope condition to kill
the contributions of the remaining $w\in{}^{M}W$ to the spectral sequence, but
there will still be further cancellation among those that remain.

######  Example 1.3.7.

We explicate all the results so far in the case of the modular curve, where
everything was already done in [BP20]. Then ${}^{M}W=W=\\{1,w\\}$ has only two
elements, and the spectral sequence reduces to a four term exact sequence

$0\rightarrow\mathrm{H}^{0}(K,k)\rightarrow\mathrm{H}^{0}_{1}(K^{p},k)^{+,fs}\rightarrow\mathrm{H}^{1}_{w}(K^{p},k)^{+,fs}\rightarrow\mathrm{H}^{1}(K,k)\to
0$

relating the the classical coherent cohomology of the modular line bundle
$\omega^{k}$, $k\in\mathbb{Z}$, to the two overconvergent cohomologies, and
moreover $H^{0}_{1}(K,k)^{+,fs}$ is nothing but the usual space of finite
slope overconvergent modular forms of weight $k$.

Then the slope bound says that the “unnormalized $U_{p}$ operator”
$U_{p}^{naive}$ has slopes $\geq 1$ on $\mathrm{H}^{0}_{1}(K,k)^{+,fs}$, and
$\geq k$ on $\mathrm{H}^{1}_{w}(K,k)^{+,fs}$. Motivated by this we renormalize
the $U_{p}$ operator by setting $U_{p}=p^{-\min(1,k)}U_{p}^{naive}$, and then
from the four term exact sequence above and the slope bounds, we immediately
deduce Coleman’s classicality theorem

$\mathrm{H}^{0}_{1}(K^{p},k)^{<k-1}=\mathrm{H}^{0}(K,k)^{<k-1}$

in weights $k\geq 2$, as well as its “$\mathrm{H}^{1}$-analog”

$\mathrm{H}^{1}_{w}(K^{p},k)^{<1-k}=\mathrm{H}^{1}(K,k)^{<1-k}$

in weights $k\leq 0$. (Here the superscripts $<\star$ mean we take the slope
$<\star$ part for the normalized $U_{p}$ operator.)

Combining the slope bounds with the vanishing theorem for overconvergent
cohomology, we obtain vanishing theorems for the small slope part of classical
cohomology. For any $\kappa\in X_{\star}(T^{c})^{M_{\mu},+}$, there is a range
$[\ell_{\min}(\kappa),\ell_{\max}(\kappa)]\subseteq[0,d]$ where cuspidal
automorphic representations which are (limits of) discrete series at $\infty$
contribute. We recall a combinatorial description of this range: let
$C(\kappa)^{+}=\\{w\in W,-w^{-1}w_{0,M}(\kappa+\rho)\in
X^{\star}(T)_{\mathbb{Q}}^{+}\\}$ and then let $\ell_{\min}(\kappa)=\min_{w\in
C(\kappa)^{+}}\ell(w)$, $\ell_{\max}(\kappa)=\max_{w\in
C(\kappa)^{+}}\ell(w)$. We note that when $\kappa+\rho$ is regular, then
$C(\kappa)^{+}=\\{w_{+}\\}$ where $w_{+}$ is as in theorem 1.3.2 above, and so
$\ell_{\min}(\kappa)=\ell_{\max}(\kappa)=\ell(w_{+})$ and hence the range is
reduced to a single degree.

Then we have the following vanishing theorem for the small slope part of
classical cohomology (theorems 5.12.20 and 6.10.2).

######  Theorem 1.3.8 (Vanishing for classical cohomology).

For any $\kappa\in X_{\star}(T^{c})^{M_{\mu},+}$,

  1. (1)

$\overline{\mathrm{H}}^{i}(K,\kappa)^{ss^{M}(\kappa)}$ is concentrated in the
range $[\ell_{\min}(\kappa),\ell_{\max}(\kappa)]$,

  2. (2)

${\mathrm{H}}^{i}(K,\kappa,cusp)^{sss^{M}(\kappa)}$ is concentrated in the
range $[0,\ell_{\max}(\kappa)]$,

  3. (3)

${\mathrm{H}}^{i}(K,\kappa)^{sss^{M}(\kappa)}$ is concentrated in the range
$[\ell_{\min}(\kappa),d]$.

Again we conjecture that strongly small slope can be replaced by small slope
in the second and third points, and this would follow from conjecture 5.9.2.
We also prove a similar result at deeper levels at $p$, and give a more
representation theoretic statement (see definition 5.12.17).

######  Remark 1.3.9.

This result should be compared to vanishing theorems in [Lan16] or [BHR94]
where there is no small slope condition but instead a regularity condition on
the weight $\kappa$.

Using Faltings’s dual BGG spectral sequence, we can also deduce vanishing
theorems for de Rham and hence Betti cohomology. Let $\nu\in
X^{\star}(T^{c})^{+}$ and let $W_{\nu}$ be the corresponding irreducible
representation of $G$ and $W_{\nu}^{\vee}$ its contragredient. We can attach
to it a local system $\mathcal{W}^{\vee}_{\nu}$ on $S_{K}(\mathbb{C})$. We
have the Betti cohomology groups
$\mathrm{H}^{\star}(S_{K}(\mathbb{C}),\mathcal{W}_{\nu}^{\vee})$,
$\mathrm{H}^{\star}_{c}(S_{K}(\mathbb{C}),\mathcal{W}_{\nu}^{\vee})$ and the
interior cohomology
$\overline{\mathrm{H}}^{\star}(S_{K}(\mathbb{C}),\mathcal{W}_{\nu}^{\vee})=\mathrm{Im}(\mathrm{H}^{\star}_{c}(S_{K}(\mathbb{C}),\mathcal{W}^{\vee}_{\nu})\rightarrow\mathrm{H}^{\star}(S_{K}(\mathbb{C}),\mathcal{W}^{\vee}_{\nu}))$.
We have the following (Theorems 5.12.20 and 6.10.2):

######  Theorem 1.3.10 (Vanishing of Betti cohomology).

For any $\nu\in X_{\star}(T^{c})^{+}$,

  1. (1)

$\overline{\mathrm{H}}^{i}(K,\mathcal{W}_{\nu}^{\vee})^{ss_{b}(\nu)}$ is
concentrated in the middle degree $d$,

  2. (2)

${\mathrm{H}}_{c}^{i}(K,\mathcal{W}_{\nu}^{\vee})^{sss_{b}(\nu)}$ is
concentrated in the range $[0,d]$,

  3. (3)

${\mathrm{H}}^{i}(K,\mathcal{W}_{\nu}^{\vee})^{sss_{b}(\nu)}$ is concentrated
in the range $[d,2d]$.

######  Remark 1.3.11.

The conditions $ss_{b}(\nu)$ and $sss_{b}(\nu)$ are the union of all
conditions $ss^{M}(\kappa)$ and $sss^{M}(\kappa)$ respectively where
$\kappa\in X_{\star}(T^{c})^{M,+}$ runs through the set
$\\{-ww_{0}(\nu+\rho)-\rho,w\in{}^{M}W\\}$.

In [CS17] and [CS19], Caraiani and Scholze proved a similar concentration
result for the Betti cohomology of unitary Shimura varieties under a
genericity condition for the action of the spherical Hecke algebra at a prime
number $\ell$. Their result is much more powerful because it also applies to
the cohomology with coefficients in a $p$-torsion local system for a prime
$p\neq\ell$. In this section we have seen three kinds of conditions that can
be used to kill cohomology outside of the expected range of degrees: small
slope at $p$, genericity at some prime $\ell$, and tempered at $\infty$. We
give the simplest example of a cohomology class outside of the expected
degree, and explain how all of these conditions fail.

######  Example 1.3.12.

Consider a compact Shimura curve $S_{K}$ associated to a quaternion algebra
over $\mathbb{Q}$ split at $\infty$ and $p$. Consider the constant function
$1\in\mathrm{H}^{0}(S_{K},\mathscr{O}_{S_{K}})$, which comes from the trivial
automorphic representation. The degree $0$ cohomology is the “wrong” degree in
this weight, in the sense that the interesting cohomology of
$\mathscr{O}_{S_{K}}=\mathcal{V}_{0}$ sits in degree $1$. We observe that the
cohomology class $1$ is:

  1. (1)

Non tempered at $\infty$, since
$\mathrm{Vol}(\mathrm{PGL}_{2}(\mathbb{R}))=\infty$.

  2. (2)

Not small slope at $p$, because the $U_{p}$ eigenvalue of the trivial
representation is $p$, and $v(p)=1$. The small slope condition in weight $0$
is having $U_{p}$-eigenvalue of slope $<1$.

  3. (3)

Not generic at any prime $\ell$, because the semi-simple conjugacy class
attached to the trivial representation at $\ell$ via the local Langlands
correspondence is $\mathrm{diag}(\ell^{\frac{1}{2}},\ell^{-\frac{1}{2}})$.

###  1.4. Locally analytic cohomology, interpolation, and the eigenvariety

Our next results concern the $p$-adic interpolation of the overconvergent
cohomologies over weight space, and the construction of the eigenvariety. In
the classical work [CM98] (revisited in [AIS14] and [Pil13]), the eigencurve
was constructed by interpolating the Hecke action on the spaces of
overconvergent modular forms $\mathrm{H}^{0}_{1}(K^{p},k)^{+,fs}$ in the
weight $k$. This approach of interpolating overconvergent modular forms was
generalized to Siegel varieties in [AIP15] (see also [CHJ17], [Bra16],
[Her19], [Bra20], [BHW20] for further generalizations). In another direction,
$p$-adic interpolation of the Betti cohomologies
$\mathrm{H}^{i}(S_{K}(\mathbb{C}),\mathcal{W}_{\nu}^{\vee})$ was considered
(and in this setting, one considers general arithmetic quotients of locally
symmetric space, not only Shimura varieties). See for example [AS08], [Urb11],
[Han17], [Eme06]. We also mention the case of algebraic modular forms where
the group $G(\mathbb{R})$ is compact modulo center, which has been studied
intensively, and for which there is no distinction between Betti and coherent
cohomology (see for example [Buz07], [Che04], [Loe11]).

In this paper, we construct eigenvarieties by interpolating the local
cohomologies $\mathrm{H}^{i}_{w}(K^{p},\kappa)^{\pm,fs}$. For the case of
overconvergent modular forms $\mathrm{H}^{0}_{1}(K^{p},\kappa)^{+,fs}$, we
recover certain of the constructions recalled above. In this case, the
improvement is that we are not assuming that the group $G_{\mathbb{Q}_{p}}$ is
unramified or that there is a nonempty ordinary locus.

Let $\Lambda=\mathbb{Z}_{p}[[T^{c}(\mathbb{Z}_{p})]]$ be the Iwasawa algebra
and let
$\mathcal{W}=\mathrm{Spa}(\Lambda,\Lambda)\times_{\mathrm{Spa}(\mathbb{Z}_{p},\mathbb{Z}_{p})}\mathrm{Spa}(\mathbb{Q}_{p},\mathbb{Z}_{p})$
be the weight space of continuous characters of $T^{c}(\mathbb{Z}_{p})$. In
order to construct families, we must first replace the finite slope,
overconvergent cohomologies $\mathrm{H}^{i}_{w}(K^{p},\kappa)^{\pm,fs}$, which
are defined for algebraic weights $\kappa\in X^{\star}(T^{c})^{+,M_{\mu}}$,
with some finite slope, locally analytic overconvergent cohomologies
$\mathrm{H}^{i}_{w,an}(K^{p},\nu)^{\pm,fs}$, which are defined for any
$p$-adic weight $\nu:T^{c}(\mathbb{Z}_{p})\rightarrow\mathbb{C}_{p}^{\times}$.

To define these cohomologies, we use the same support conditions, but we
replace the automorphic vector bundles $\mathcal{V}_{\kappa}$ with certain
Banach sheaves $\mathcal{V}_{\nu}^{an}$ and $\mathcal{D}_{\nu}^{an}$. These
Banach sheaves are defined on neighborhoods of
$(\pi_{HT,K_{p}}^{tor})^{-1}(w\cdot K_{p})$, using a reduction of the
$M_{\mu}^{c}$-torsor which is used to define the sheaves
$\mathcal{V}_{\kappa}$. This reduction is defined using the Hodge-Tate period
map (see section 4.6). Locally, the sheaves $\mathcal{V}_{\nu}^{an}$ (resp.
$\mathcal{D}_{\nu}^{an}$) are modeled on a locally analytic induction (resp.
locally analytic distributions) of a certain profinite subgroup of
$M_{\mu}(F)$ (actually with a fixed “radius of analyticity” which we suppress
from the notation, as it will not matter when we pass to the finite slope part
of cohomology). We define $\mathrm{H}^{i}_{w,an}(K^{p},\nu)^{+,fs}$ using
$\mathcal{V}_{\nu}^{an}$ and $\mathrm{H}^{i}_{w,an}(K^{p},\nu)^{-,fs}$ using
$\mathcal{D}_{\nu}^{an}$. We must consider both of the sheaves
$\mathcal{V}_{\nu}^{an}$ and $\mathcal{D}_{\nu}^{an}$ in order to have a
duality theory, and we note that here we break the symmetry between the $+$
and $-$ theories.

The finite slope, locally analytic overconvergent cohomologies
$\mathrm{H}^{i}_{w,an}(K^{p},\nu)^{\pm,fs}$ carry a Hecke action of the monoid
$T^{\pm}$ (see section 6.3.9 for the definition), where one should think of
the action of $t\in T^{\pm}$ as something like the “$U_{p}$-type” Iwahori
Hecke operator $[K_{p}tK_{p}]$, except that one does not actually get an
action of $\mathcal{H}_{p}^{\pm}$, as the “diamond operators” $t\in
T(\mathbb{Z}_{p})$ do not act trivially: in fact they act by $\nu(t)^{\pm}$.
The finite slope, locally analytic overconvergent cohomologies enjoy similar
properties to parts (2),(3),(4) of theorem 1.2.1 (see section 6.7 for duality
and theorems 6.6.1 and 6.7.3 for vanishing).

We now explain the relation between the two cohomologies. Fix $w\in{}^{M}W$
and let $\kappa\in X^{\star}(T^{c})^{M_{\mu},+}$ be an algebraic weight. We
may associate a $p$-adic weight $\nu:T(\mathbb{Z}_{p})\to F^{\times}$, as
follows: we first consider $\nu_{alg}=-w^{-1}w_{0,M}(\kappa+\rho)-\rho\in
X^{\star}(T^{c})$, take the associated character $T(F)\to F^{\times}$, and
define $\nu$ to be its restriction to $T(\mathbb{Z}_{p})$. We emphasize that
when $G$ does not split over $\mathbb{Q}_{p}$, neither $\kappa$ nor $w$ need
be defined over $\mathbb{Q}_{p}$, and the (absolute) Weyl group $W$ need not
act on $T^{c}(\mathbb{Z}_{p})$ nor on $\mathcal{W}$.

From the construction we have maps of sheaves in a neighborhood of
$\pi_{HT,K_{p}}^{-1}(w\cdot K_{p})$

$\mathcal{V}_{\kappa}\hookrightarrow\mathcal{V}_{\nu}^{an}\quad\mathrm{and}\quad\mathcal{D}_{\nu}^{an}\twoheadrightarrow\mathcal{V}_{\kappa^{\vee}}$

which are locally modeled on the inclusion of an algebraic induction into a
locally analytic induction and its dual. Passing to cohomology with supports,
we obtain maps

$\mathrm{H}^{i}_{w}(K^{p},\kappa)^{+,fs}\rightarrow\mathrm{H}^{i}_{w,an}(K^{p},\nu)^{+,fs}(-\nu_{alg})$
$\mathrm{H}^{i}_{w,an}(K^{p},\nu)^{-,fs}(\nu_{alg})\to\mathrm{H}^{i}_{w}(K^{p},\kappa^{\vee})^{-,fs}$

where we have twisted the Hecke action of $T^{\pm}$ on the analytic cohomology
in order to make these maps Hecke equivariant (the twist of $(-\nu_{alg})$
means that we multiply the action of $t\in T^{+}$ by $\nu_{alg}(t)^{-1}$. Note
that with this twist, $T(\mathbb{Z}_{p})$ acts trivially.)

Then we can state our second classicality theorem (see corollary 6.8.4).

######  Theorem 1.4.1 (Classicality at the level of the sheaf).

The two maps above become isomorphisms after passing to the
$+,sss_{M,w}(\kappa)$ part in the $+$ case and the
$-,sss_{M,w}(\kappa^{\vee})$ part in the $-$ case. If the Shimura variety is
compact, the same statement is true with weaker conditions
$+,ss_{M,w}(\kappa)$ and $-,ss_{M,w}(\kappa^{\vee})$ (again see section 5.11
for the precise definitions of these small slope conditions).

######  Remark 1.4.2.

We try to clarify the switch from $\kappa$ to $\nu$, and why it is really
$\nu$ and not $\kappa$ that is the natural weight parameter on the
eigenvariety. The situation will become much clearer if we start by fixing a
$G$-dominant $\nu_{alg}\in X^{\star}(T^{c})^{+}$. Then associated to each
$w\in{}^{M}W$ we can consider $\kappa_{w}=-w_{0,M}w(\nu_{alg}+\rho)-\rho$.
Combining the two classicality theorems we have an isomorphism

$\mathrm{H}^{i}_{w,an}(K^{p},\nu)^{+,sss_{w}(\nu_{alg})}=\mathrm{H}^{i}(K,\kappa_{w})^{+,sss_{w}(\nu_{alg})}$

where the small slope condition $+,sss_{w}(\nu_{alg})$ is simply the
combination of $+,sss^{M}(\kappa_{w})$ and $+,sss_{M,w}(\kappa_{w})$ (if the
Shimura variety is compact, we can use the condition $ss(\nu_{alg})$, which is
independent of $w$ and is the usual small slope condition in the algebraic
modular form setting).

Now a stable $L$-packet where the archimedean components are discrete series
with infinitesimal character $\nu_{alg}+\rho$ should give the same
contribution to the cohomologies $\mathrm{H}^{\ell(w)}(K,\kappa_{w})$ for
$w\in{}^{M}W$. If the contributions to the
$\mathrm{H}^{\ell(w)}(K,\kappa_{w})^{+,sss_{w}(\nu_{alg})}$ are nonempty, then
we will see eigenclasses with the same system of Hecke eigenvalues in all of
the $\mathrm{H}^{\ell(w)}_{w,an}(K^{p},\nu)^{+,fs}$. These eigenclasses should
all correspond to the same point of the eigenvariety, and thus we should use
$\nu$, which is nothing but a $\rho$ shift of the dominant representative of
the infinitesimal character, rather than the $\kappa_{w}$, which vary with
$w\in{}^{M}W$.

We finally turn to results about the eigenvariety
$\pi:\mathcal{E}\rightarrow\mathcal{W}$. The eigenvariety comes with a map
$T(\mathbb{Q}_{p})\rightarrow\mathscr{O}_{\mathcal{E}}$ (compatible with the
map
$T(\mathbb{Z}_{p})\rightarrow\mathbb{Z}_{p}[[T^{c}(\mathbb{Z}_{p})]]\stackrel{{\scriptstyle\pi^{\star}}}{{\rightarrow}}\mathscr{O}_{\mathcal{E}}$)
and a map $\mathcal{H}^{S}\rightarrow\mathscr{O}_{\mathcal{E}}$ where
$\mathcal{H}^{S}$ is the spherical Hecke algebra away from the finite set of
primes $S$ which contains $p$ and all primes $\ell$ where $K_{\ell}$ is not
hyperspecial. Points of the eigenvariety are pairs $(\lambda_{p},\lambda^{S})$
where $\lambda_{p}$ is a character of $T(\mathbb{Q}_{p})$, and $\lambda^{S}$
is a system of eigenvalues for the prime to $S$ spherical Hecke algebra. The
projection to $\mathcal{W}$ is given by
$\lambda_{p}\mapsto\nu=\lambda_{p}|_{T^{c}(\mathbb{Z}_{p})}$. When $\nu$ is
associated to an algebraic weight $\nu_{alg}\in X^{\star}(T)$, we let
$\lambda_{p}^{sm}=\lambda_{p}\nu_{alg}^{-1}$ be the smooth part of
$\lambda_{p}$ (it is trivial on $T(\mathbb{Z}_{p})$). We have the following
theorem on the existence and properties of the eigenvariety (Theorems 6.9.3
and 6.9.6):

######  Theorem 1.4.3.

The eigenvariety $\pi:\mathcal{E}\rightarrow\mathcal{W}$ is locally quasi-
finite and partially proper. It carries coherent sheaves

$\mathrm{H}^{k}_{w,an}(K^{p},\nu^{un})_{\mathcal{Z}}^{\pm,fs},\qquad\mathrm{H}^{k}_{w,an}(K^{p},\nu^{un},cusp)_{\mathcal{Z}}^{\pm,fs}$

for all $w\in{}^{M}W$ and $k\in\mathbb{Z}$ and they satisfy the following
properties:

  1. (1)

(Any classical, finite slope eigenclass gives a point of the eigenvariety) For
any $\kappa\in X^{\star}(T^{c})^{M,+}$ and any system of Hecke eigenvalues
$(\lambda^{sm}_{p},\lambda^{S})$ occurring in
$\mathrm{H}^{i}(K,\kappa)^{+,fs}$ there is a $w=w_{M}w^{M}\in W$ with
$w_{M}\in W_{M}$, $w^{M}\in{}^{M}W$, so that if $\nu:T^{c}(\mathbb{Z}_{p})\to
F^{\times}$ is associated with
$\nu_{alg}=-w^{-1}w_{0,M}(\kappa_{alg}+\rho)-\rho$, then
$(\lambda^{sm}_{p}\nu_{alg},\lambda^{S})$ is a point of the eigenvariety
$\mathcal{E}$ which lies in the support of
$\mathrm{H}^{k}_{w^{M},an}(K^{p},\nu^{un})_{\mathcal{Z}}^{+,fs}$ for some $k$.
The analogous result hold for the cuspidal and $-$ cohomologies.

  2. (2)

(Small slope points of the eigenvariety in regular, locally algebraic weights
are classical) Conversely if $\nu_{alg}\in X^{\star}(T^{c})^{+}$ is a
$G$-dominant weight and $(\lambda_{p},\lambda^{S})$ is a point of
$\mathcal{E}$ in weight $\nu$ and in the support of
$\mathrm{H}^{k}_{w,an}(K^{p},\nu^{un})_{\mathcal{Z}}^{+,fs}$ for some
$w\in{}^{M}W$, and if $\lambda_{p}^{sm}$ satisfies $+,sss_{w}(\nu)$ then
$(\lambda^{sm}_{p},\lambda^{S})$ occurs in $\mathrm{H}^{i}(K,\kappa)^{+,fs}$
for $\kappa=-w_{0,M}w(\nu_{alg}+\rho)-\rho$ and some $i$. The analogous result
hold for the cuspidal and $-$ cohomologies.

  3. (3)

(Serre duality interpolates over the eigenvariety) We have pairings:

$\mathrm{H}^{k}_{w,an}(K^{p},\nu^{un})_{\mathcal{Z}}^{\pm,fs}\otimes\mathrm{H}^{d-k}_{w,an}(K^{p},\nu^{un},cusp)_{\mathcal{Z}}^{\mp,fs}\rightarrow\pi^{-1}\mathscr{O}_{\mathcal{W}}.$

and these pairings are compatible with Serre duality under the classicality
theorem.

  4. (4)

(Interior cohomology classes deform over the weight space) Any interior
cohomology eigenclass $c\in\overline{\mathrm{H}}^{\star}(K,\kappa)^{\pm,fs}$
defines a point on a component of the eigenvariety of dimension equal to the
dimension of the weight space.

######  Remark 1.4.4.

It is plausible that the eigenvariety $\mathcal{E}$ coincides with an
eigenvariety constructed via Betti cohomology interpolation, as in [Han17].
One can also believe that there is a $p$-adic Eichler-Shimura theory relating
both constructions. See [AIS15], as well as some forthcoming work of Juan
Esteban Rodriguez.

######  Remark 1.4.5.

We can define certain $\pi^{-1}\mathscr{O}_{\mathcal{W}}$-torsion free sheaves
$\overline{\mathrm{H}}^{\ell_{\pm}(w)}_{w,an}(K^{p},\nu^{un})_{\mathcal{Z}}^{\pm,fs}$
over the eigenvariety. This sheaves interpolate the various modules of the
interior Cousin complex (which can be used to compute the interior
cohomology). For all $w\in{}^{M}W$, we let $\mathcal{E}_{w}^{!}$ be the
support of
$\overline{\mathrm{H}}^{\ell_{\pm}(w)}_{w,an}(K^{p},\nu^{un})_{\mathcal{Z}}^{\pm,fs}$,
which is a union of irreducible components of the eigenvariety of dimension
equal to $\dim~{}\mathcal{W}$. Any interior cohomology class lifts to a point
on $\cup_{w\in{}^{M}W}\mathcal{E}^{!}_{w}$.

######  Remark 1.4.6.

For $\mathrm{GSp}_{4}/L$ with $L$ a totally real field, variants or special
cases of this theory are considered in [Pil20], [BCGP18], [LPSZ19].

######  Remark 1.4.7.

The point $(4)$ of the theorem is an advantage of the method we use to
construct the eigenvarieties. Such a result was only available in a limited
number of cases (e.g. Shimura sets and automorphic forms contributing to
cuspidal coherent $\mathrm{H}^{0}$).

Finally, using point $(4)$ we can give a new construction of Galois
representations associated to certain automorphic representations realizing in
the coherent cohomology of Shimura varieties, but not in the Betti cohomology.
This construction is via $p$-adic interpolation, and yields new results on
local-global compatibility at $p$. In [FP19], section 9, we defined a certain
class of cuspidal automorphic representations for the group
$\mathrm{GL}_{n}/L$ where $L$ is a totally real or CM number field. These are
called weakly regular, odd, algebraic, essentially (conjugate) self dual,
cuspidal automorphic representations. Informally, the weak regularity
condition means that we allow the weights of the infinitesimal character to
repeat at most twice.

######  Theorem 1.4.8.

Let $\pi$ be a weakly regular, odd, algebraic, essentially (conjugate) self
dual, cuspidal automorphic representation of $\mathrm{GL}_{n}/L$ with
$\pi^{c}=\pi^{\vee}\otimes\chi$ and infinitesimal character
$\lambda=((\lambda_{1,\tau},\ldots,\lambda_{n,\tau})_{\tau\in\mathrm{Hom}(L,\mathbb{C})})$
with $\lambda_{1,\tau}\geq\cdots\geq\lambda_{n,\tau}$. Then for each
isomorphism $\iota:\overline{\mathbb{Q}}_{p}\simeq\mathbb{C}$ there is a
continuous Galois representation
$\rho_{\pi,\iota}:G_{L}\rightarrow\mathrm{GL}_{n}(\overline{\mathbb{Q}}_{p})$
such that:

  1. (1)

$\rho_{\pi,\iota}^{c}\simeq\rho^{\vee}\otimes\epsilon_{p}^{1-n}\otimes\chi_{\iota}$
where $\chi_{\iota}$ is the $p$-adic realization of $\chi$.

  2. (2)

$\rho_{\pi,\iota}$ is unramified at all finite places $v\nmid p$ for which
$\pi_{v}$ is unramified and one has

$\iota
WD(\rho_{\pi,\iota}|_{G_{L_{v}}})^{F-ss}\simeq\mathrm{rec}(\pi_{v}\otimes|\det|_{v}^{\frac{1-n}{2}}).$

  3. (3)

$\rho_{\pi,\iota}$ has generalized $\iota^{-1}\circ\tau$-Hodge–Tate weights
$(-\lambda_{n,\tau}+\frac{n-1}{2},\cdots,-\lambda_{1,\tau}+\frac{n-1}{2})$.

  4. (4)

Let $v\mid p$ be a place of $L$ and assume that $\pi_{v}$ is a regular
principal series. Then $\rho_{\pi,\iota}|_{G_{L_{v}}}$ is potentially
crystalline and

$\iota
WD(\rho_{\pi,\iota}|_{G_{L_{v}}})^{F-ss}\simeq\mathrm{rec}(\pi_{v}\otimes|\det|_{v}^{\frac{1-n}{2}}).$

Here the hypothesis that $\pi_{v}$ is a regular principal series in part (4)
is a “$p$-distinguished” condition, which in particular implies that
$\mathrm{rec}(\pi_{v}\otimes|\det|_{v}^{\frac{1-n}{2}})$ is a sum of $n$
distinct characters, so that we can prove that $\rho_{\pi,\iota}|G_{L_{v}}$ is
potentially crystalline by finding its periods one at a time.

When $\pi$ is regular rather than just weakly regular and odd, then a stronger
form of the theorem holds according to results of Bellaiche, Caraiani,
Chenevier, Clozel, Harris, Kottwitz, Labesse, Shin, Taylor, etc. (see [CH13],
[BLGGT14]) including purity and local-global compatibility at all places. The
above theorem is deduced from these results by $p$-adic interpolation.
Moreover even in the irregular case, the Galois representations in the theorem
have already been constructed by different methods in [PS16] (and in special
cases in [Box15], [GK19]). The novelty in the above theorem is the results
towards local global compatibility at $p$ in points (3) and (4).

The theorem is proved by first descending $\pi$ to a cuspidal automorphic
representation on a quasi-split unitary group over $L$, which then contributes
to the interior coherent cohomology of the corresponding Shimura variety. Then
we apply point (4) of theorem 1.4.3 to get a point on the eigenvariety which
is a limit of regular weight classical points, and conclude using results of
Kisin [Kis03] on the interpolation of crystalline periods in analytic
families, as in the work of Jorza and Mok [Jor12], [Mok14].

######  Remark 1.4.9.

As explained, our proof of this theorem makes use of the results of [Mok15] to
descend certain automorphic representations to quasi-split unitary groups, and
is therefore conditional on the results announced in [Art13].

###  1.5. A toy model: the cohomology of flag varieties and the Cousin complex
of Kempf

There is a strong analogy between the coherent cohomology of Shimura
varieties, and the much simpler coherent cohomology of flag varieties. In
[Kem78], Kempf introduced a local cohomology method for computing the coherent
cohomology of flag varieties, which may be viewed as a “toy model” for the
methods of this paper. We briefly review this computation and some of the
analogies (see section 3.2 for more details).

For this section of the introduction, which is independent of the rest of the
paper, we let $G$ be a split reductive group over a field $F$ and let
$T\subset B\subset G$ be a Borel and maximal torus. Let $FL=B\backslash G$ be
the flag variety for $G$ and write $\pi:G\rightarrow FL$ for the projection.
Let $d$ be the dimension of $FL$. Let $W$ be the Weyl group of $G$ with length
function $\ell$, and let $w_{0}\in W$ be the longest element. For any weight
$\kappa\in X^{\star}(T)$, we associate a $G$-equivariant line bundle
$\mathcal{L}_{\kappa}$ over $FL$, whose sections on an open $U\hookrightarrow
FL$ are

$\mathcal{L}_{\kappa}(U)=\\{f:\pi^{-1}(U)\rightarrow\mathbb{A}^{1}\mid
f(bg)=(w_{0}\kappa)(b)f(g),\quad\forall b\in B,g\in\pi^{-1}(U)\\}.$

The right action of $G$ on $\mathrm{FL}$ induces a left action on the
cohomology groups $\mathrm{H}^{i}(FL,\mathcal{L}_{\kappa})$. If $\kappa$ is
dominant, then $\mathrm{H}^{0}(FL,\mathcal{L}_{\kappa})$ contains a highest
weight representation of weight $\kappa$.

The classical Borel-Weil-Bott theorem describes the cohomology of the sheaves
$\mathcal{L}_{\kappa}$ over $FL$ when the characteristic of $F$ is $0$:

######  Theorem 1.5.1 ([Jan03], 5.5, corollary).

Assume that $\mathrm{char}(F)=0$. Let $\kappa\in X^{\star}(T)$ then:

  1. (1)

If there exists no $w\in W$ such that $w(\kappa+\rho)-\rho$ is dominant (i.e.
$\kappa+\rho$ is irregular) then $\mathrm{H}^{i}(FL,\mathcal{L}_{\kappa})=0$
for all $i$.

  2. (2)

If there exists $w\in W$ such that $w(\kappa+\rho)-\rho$ is dominant, then
there is a unique such $w$, and $\mathrm{H}^{i}(FL,\mathcal{L}_{\kappa})=0$ if
$\ell(w)\neq i$, while $\mathrm{H}^{\ell(w)}(FL,\mathcal{L}_{\kappa})$ is the
highest weight $w(\kappa+\rho)-\rho$ representation.

Following section 12 of [Kem78], one can study the cohomology of the sheaves
$\mathcal{L}_{\kappa}$ over $FL$ with the help of the Bruhat stratification
$FL=\cup_{w\in W}B\backslash BwB$. Given $w\in W$, let $C_{w}=B\backslash
BwB$, and let $U_{w}\subseteq\mathrm{FL}$ be an open so that $C_{w}\subseteq
U_{w}$ is closed (for example one can take $U_{w}=B\backslash
Bw_{0}Bw_{0}^{-1}w$). Then one can introduce the local cohomology groups:

$\mathrm{H}_{w}^{i}(\kappa):=\mathrm{H}^{i}_{C_{w}}(U_{w},\mathcal{L}_{\kappa}).$

Unlike the cohomology of the full flag variety, these local cohomologies do
not carry an action of $G$. However they do still carry an action of $B$, as
well as of the Lie algebra $\mathfrak{g}$ of $G$.

Then we have the following results:

  * •

(Vanishing) The local cohomologies $H^{i}_{w}(\kappa)$ vanish unless
$i=d-\ell(w)$.

  * •

(Cousin complex) For each weight $\kappa\in X^{\star}(T)$ there is a
Grothendieck-Cousin complex

$\mathrm{H}^{0}_{w_{0}}(\kappa)\to\bigoplus_{w\in
W,\ell(w)=d-1}\mathrm{H}^{1}_{w}(\kappa)\to\cdots\to\bigoplus_{w\in
W,\ell(w)=d-i}\mathrm{H}^{i}_{w}(\kappa)\to\cdots\to\mathrm{H}^{d}_{1}(\kappa)$

computing the cohomology $\mathrm{H}^{\star}(FL,\mathcal{L}_{\kappa})$. This
complex arises as the $E_{1}$ page of the spectral sequence of the following
filtration of $FL$ associated to the Bruhat stratification:

$Z^{i}=\bigcup_{w\in W,\ell(w)\leq d-i}C_{w}.$

We also remark that this complex is closely related to the BGG resolution. For
example, if the characteristic of $F$ is zero $\kappa$ is dominant, then this
complex is exactly the dual of the BGG resolution of the irreducible
representation of highest weight $-w_{0}\kappa$.

  * •

(Weight bounds) The analog of the Hecke action of $\mathcal{H}_{p}^{\pm}$ on
the finite slope, overconvergent cohomologies is the action of the torus $T$
on the local cohomologies $\mathrm{H}^{i}_{w}(\kappa)$ (there is no analog of
taking the “finite slope part”, as we already have the action of a group).
Then the analog of the slope bound is the following “weight bound”: any weight
$\lambda\in X^{\star}(T)$ occurring in $H_{w}^{d-\ell(w)}$ satisfies

$\lambda\leq w^{-1}w_{0}(\kappa+\rho)-\rho$

where we write $\lambda\leq\lambda^{\prime}$ if
$\lambda^{\prime}-\lambda=\sum_{\alpha\in\Delta}c_{\alpha}\alpha$ with
$c_{\alpha}\geq 0$ where $\Delta$ denotes the simple roots of $G$ associated
to $T\subseteq B$. Actually this weight bound is an immediate consequence of a
much better result, which we lack an analog of in the Shimura variety setting:
the character of $T$ acting on $\mathrm{H}^{i}_{w}(\kappa)$ equals that of the
Verma module of highest weight $w^{-1}w_{0}(\kappa+\rho)-\rho$.

  * •

(“Classicality” and vanishing for the “big weight” part) In analogy with
passing to the small slope part in the Shimura variety setting, we can try to
impose a “big weight” condition in order to kill most (ideally all but one) of
the terms in the Cousin complex.

We make some combinatorial definitions: let $C(\kappa)=\\{w\in W\mid
w^{-1}w_{0}(\kappa+\rho)\in X^{\star}(T)_{\mathbb{R}}^{+}\\}$ be the subset of
$W$ for which the weight bound is as big as possible. We say that a weight
$\lambda\in X^{\star}(T)^{+}$ is a big weight (depending on $\kappa$) if
$\lambda\not\leq w^{-1}w_{0}(\kappa+\rho)-\rho$. The point of this definition
is exactly that by the weight bound, $\mathrm{H}_{w}^{d-\ell(w)}(\kappa)$ will
not contain any big weights unless $w\in C(\kappa)$. Then from this and the
Cousin complex we have the following immediate consequences, which can be
viewed as a weak form of the Borel-Weil-Bott theorem (as well as toy models
for theorems 1.3.2, 1.3.8):

    * –

$\mathrm{H}^{i}(FL,\mathcal{L}_{\kappa})^{bw}=0$ unless $i\in[\min_{w\in
C(\kappa)}d-\ell(w),\max_{w\in C(\kappa)}d-\ell(w)]$.

    * –

If $\kappa+\rho$ is regular, so that $C(\kappa)=\\{w\\}$ consists of a single
element, then we have the following “classicality theorem”:

$\mathrm{H}^{i}_{w}(\kappa)^{bw}=\mathrm{H}^{i}(FL,\mathcal{L}_{\kappa})^{bw}$

and both sides vanish when $i\not=d-\ell(w)$.

Although these results are not as strong as the Borel-Weil-Bott theorem, we
emphasize that we have not assumed that $\mathrm{char}(F)=0$. Moreover, in
characteristic zero it is not too hard to deduce the full Borel-Weil-Bott
theorem using some basic properties of the BGG category $\mathcal{O}$ (see
section 3.2).

  * •

(Interpolation) We assume $F$ has characteristic 0. Although the line bundles
$\mathcal{L}_{\kappa}$ and their cohomologies only exist for integral weights
$\kappa\in X^{\star}(T)$, the local cohomologies
$\mathrm{H}^{i}_{w}(X,\kappa)$, together with their $\mathfrak{g}$ action can
be defined for all $\kappa\in X^{\star}(T)^{+}\otimes F=\mathfrak{t}^{\vee}$,
where $\mathfrak{t}$ is the Lie algebra of $T$. The families of
$\mathfrak{g}$-modules obtained in this way are called twisted Verma modules
(see [AL03]).

Acknowledgements: We would like to thank D. Loeffler and S. Zerbes for asking
about the possibility to generalize the results of [Pil20] to more general
situations and for many discussions about this paper. A first generalization
was found in our joint work with C. Skinner [LPSZ19]. We further thank D.
Loeffler for his careful reading of a first draft of the manuscript, and his
many useful comments which in particular lead us to prove theorem 5.7.2 and
theorem 5.7.3 which where stated as conjectures in the draft. It is also a
pleasure to thank F. Calegari and T. Gee, some of the ideas were discovered
while working on [BCGP18]. We also thank F. Andreatta, A. Iovita, J. Rodrigues
and J.E. Rodriguez for many interesting exchanges in Lyon. We thank A.
Caraiani and the London number theory group for their interest in our work
(and their patience), as well as the Montréal number theory group for giving
us the opportunity to give lectures on this topic. We finally acknowledge the
financial support of the ERC-2018-COG-818856-HiCoShiVa.

##  2\. Cohomological preliminaries

###  2.1. Cohomology with support in a closed subspace

We recall the notion and the basic properties of the cohomology of an abelian
sheaf on a topological space, with support in a closed subspace. A reference
for this material is [Gro05], chapter I. Let $X$ be a topological space. We
let $Ab_{X}$ be the category of abelian sheaves over $X$. If $X$ is a point,
$Ab_{X}$ is simply $Ab$ the category of abelian groups. We let
$\mathcal{D}(Ab_{X})$ be the derived category of $Ab_{X}$. Let
$i:Z\hookrightarrow X$ be a closed subspace. For an object $\mathscr{F}$ of
$Ab_{X}$, we let $\Gamma_{Z}(X,\mathscr{F})$ be the subgroup of
$\mathrm{H}^{0}(X,\mathscr{F})$ of sections whose support is included in $Z$.
We let
$\mathrm{R}\Gamma_{Z}(X,-):\mathcal{D}(Ab_{X})\rightarrow\mathcal{D}(Ab)$ be
the right derived functor of $\Gamma_{Z}(X,-)$ (see [Sta13], Tag 079V, in
particular for the unbounded version). Let $U=X\setminus Z$ and let
$\mathscr{F}$ be an object of $\mathcal{D}(Ab_{X})$. We have an exact triangle
in $\mathcal{D}(Ab)$ ([Gro05], I, corollaire 2.9):

$\mathrm{R}\Gamma_{Z}(X,\mathscr{F})\rightarrow\mathrm{R}\Gamma(X,\mathscr{F})\rightarrow\mathrm{R}\Gamma(U,\mathscr{F})\stackrel{{\scriptstyle+1}}{{\rightarrow}}$

We have the classical pushforward functor $i_{\star}:Ab_{Z}\rightarrow Ab_{X}$
and it admits a right adjoint $i^{!}:Ab_{X}\rightarrow Ab_{Z}$ which can be
described as follows: Let $W\subseteq Z$ be an open subset. Let
$W^{\prime}\subseteq X$ be an open subset of $X$ such that $W=W^{\prime}\cap
Z$. For any object $\mathscr{F}$ of $Ab_{X}$, we have
$i^{!}\mathscr{F}(W)=\Gamma_{W}(W^{\prime},\mathscr{F}|_{W^{\prime}})$. It
follows that
$\Gamma_{Z}(X,\mathscr{F})=\mathrm{H}^{0}(X,i_{\star}i^{!}\mathscr{F})$. The
functor $i^{!}$ has a right derived functor
$\mathrm{R}i^{!}:\mathcal{D}(Ab_{X})\rightarrow\mathcal{D}(Ab_{Z})$. Moreover
$\mathrm{R}\Gamma_{Z}(X,\mathscr{F})=\mathrm{R}\Gamma(X,i_{\star}\mathrm{R}i^{!}\mathscr{F})$.

Some properties of the cohomology with support are:

  1. (1)

(corestriction)[[Gro05], I, Proposition 1.8] If $Z\subseteq
Z^{\prime}\subseteq X$ are closed subsets, there is a corestriction map
$\mathrm{R}\Gamma_{Z}(X,\mathscr{F})\rightarrow\mathrm{R}\Gamma_{Z^{\prime}}(X,\mathscr{F})$.

  2. (2)

(pull-back) If we have a continuous map $f:X\to X^{\prime}$, closed subsets
$Z\subset X$, $Z^{\prime}\subset X^{\prime}$ satisfying
$f^{-1}(Z^{\prime})\subseteq Z$, and a sheaf $\mathscr{F}$ on $X^{\prime}$,
then there is a map
$\mathrm{R}\Gamma_{Z^{\prime}}(X^{\prime},\mathscr{F})\rightarrow\mathrm{R}\Gamma_{Z}(X,f^{-1}\mathscr{F})$.

  3. (3)

(Change of ambient space)[[Gro05], I, Proposition 2.2] If we have $Z\subset
U\subset X$ for some open $U$ of $X$, then the pull back map
$\mathrm{R}\Gamma_{Z}(X,\mathscr{F})\rightarrow\mathrm{R}\Gamma_{Z}(U,\mathscr{F})$
is a quasi-isomorphism.

The above properties imply easily the following lemma:

######  Lemma 2.1.1.

Let $Z_{1},Z_{2}\subseteq X$ be two disjoint closed subsets. Then the natural
map given by pushforward:

$\mathrm{R}\Gamma_{Z_{1}}(X,\mathscr{F})\oplus\mathrm{R}\Gamma_{Z_{2}}(X,\mathscr{F})\rightarrow\mathrm{R}\Gamma_{Z_{1}\cup
Z_{2}}(X,\mathscr{F})$

is a quasi-isomorphism.

###### Proof.

For $i\in\\{1,2\\}$, we let $i_{i}:Z_{i}\hookrightarrow X$. We finally let
$i:Z_{1}\cup Z_{2}\hookrightarrow X$. The lemma follows from the claim that
for any $\mathscr{F}\in Ob(Ab_{X})$, the natural map
$(i_{1})_{\star}i_{1}^{!}\mathscr{F}\oplus(i_{2})_{\star}i_{2}^{!}\mathscr{F}\rightarrow
i_{\star}i^{!}\mathscr{F}$ is an isomorphism of sheaves. This is a local
computation. Since $X=Z_{1}^{c}\cup Z_{2}^{c}$ and the claim holds true over
any open subset of $Z_{1}^{c}$ or $Z_{2}^{c}$, we are done. ∎

We now discuss the construction of the trace map in the context of schemes or
adic spaces and finite locally free morphisms ([Hub96], Sect. 1.4.4).

######  Lemma 2.1.2.

Consider a commutative diagram of topological spaces:

with $X$ and $X^{\prime}$ schemes (resp. adic spaces), $f$ a finite locally
free morphism of schemes (resp. adic spaces), $Z^{\prime}\rightarrow
X^{\prime}$ and $Z\rightarrow X$ closed subspaces. Let $\mathscr{F}$ be a
sheaf of $\mathscr{O}_{X^{\prime}}$-modules. Then there is a map
$\mathrm{R}\Gamma_{Z}(X,f^{\star}\mathscr{F})\rightarrow\mathrm{R}\Gamma_{Z^{\prime}}(X^{\prime},\mathscr{F})$.

###### Proof.

We first recall that the category of sheaves of $\mathscr{O}_{T}$-modules on a
ringed space $(T,\mathscr{O}_{T})$ has enough injectives ([Sta13], Tag 01DH).
It follows that it is enough to construct a functorial map
$\Gamma_{Z}(X,f^{\star}\mathscr{F})\rightarrow\Gamma_{Z^{\prime}}(X^{\prime},\mathscr{F})$
for sheaves $\mathscr{F}$ of $\mathscr{O}_{X}$-modules. If we let
$Z^{\prime\prime}=f^{-1}(Z^{\prime})$, then we have a map
$\Gamma_{Z}(X,f^{\star}\mathscr{F})\rightarrow\Gamma_{f^{-1}(Z^{\prime})}(X,f^{\star}\mathscr{F})$.
Therefore, it suffices to consider the case where $Z=f^{-1}(Z^{\prime})$. We
have a trace map
$\mathrm{Tr}:f_{\star}\mathscr{O}_{X}\rightarrow\mathscr{O}_{X^{\prime}}$.
Moreover, the natural morphism
$f_{\star}\mathscr{O}_{X}\otimes_{\mathscr{O}_{X^{\prime}}}\mathscr{F}\rightarrow
f_{\star}f^{\star}\mathscr{F}$ is an isomorphism. We therefore have a trace
map $\mathrm{Tr}:f_{\star}f^{\star}\mathscr{F}\rightarrow\mathscr{F}$. Let us
complete the above diagram into:

where $U^{\prime}=X^{\prime}\setminus Z^{\prime}$ and $U=X\setminus Z$. We
have a commutative diagram:

We deduce that the trace map
$\Gamma(X,f^{\star}\mathscr{F})\rightarrow\Gamma(X^{\prime},\mathscr{F})$
induces a trace map
$\Gamma_{Z}(X,f^{\star}\mathscr{F})\rightarrow\Gamma_{Z^{\prime}}(X^{\prime},\mathscr{F})$.
∎

###  2.2. Cup products

Let $(X,\mathscr{O}_{X})$ be a ringed space. Let
$K=\Gamma(X,\mathscr{O}_{X})$.

######  Proposition 2.2.1.

Let $Z_{1},Z_{2}\subseteq X$ be two closed subsets. Let $\mathscr{F}$ and
$\mathscr{G}$ be two flat sheaves of $\mathscr{O}_{X}$-modules. There is a
natural map, functorial in $\mathscr{F}$ and $\mathscr{G}$, as well as the
support $Z_{1}$ and $Z_{2}$:

$\mathrm{R}\Gamma_{Z_{1}}(X,\mathscr{F})\otimes_{K}^{L}\mathrm{R}\Gamma_{Z_{2}}(X,\mathscr{G})\rightarrow\mathrm{R}\Gamma_{Z_{1}\cap
Z_{2}}(X,\mathscr{F}\otimes^{L}_{\mathscr{O}_{X}}\mathscr{G}).$

###### Proof.

Let $Z_{3}=Z_{1}\cap Z_{2}$. For $1\leq j\leq 3$, we let
$i_{j}:Z_{i}\hookrightarrow X$ be the inclusion. For $\mathscr{F}$ and
$\mathscr{G}$ sheaves of $\mathscr{O}_{X}$-modules we have a map
$(i_{1})_{\star}i_{1}^{!}\mathscr{F}\otimes_{\mathscr{O}_{X}}(i_{2})_{\star}i_{2}^{!}\mathscr{G}\rightarrow(i_{3})_{\star}i_{3}^{!}(\mathscr{F}\otimes\mathscr{G})$.
We claim that there is a map in the derived category:

$(i_{1})_{\star}\mathrm{R}i_{1}^{!}\mathscr{F}\otimes^{L}_{\mathscr{O}_{X}}(i_{2})_{\star}\mathrm{R}i_{2}^{!}\mathscr{G}\rightarrow(i_{3})_{\star}\mathrm{R}i_{3}^{!}(\mathscr{F}\otimes\mathscr{G}).$

Indeed, taking the Godement resolution ([Sta13], Tag 0FKR) gives quasi-
isomorphisms $\mathscr{F}\rightarrow\mathscr{F}^{\bullet}$ and
$\mathscr{G}\rightarrow\mathscr{G}^{\bullet}$ where $\mathscr{F}^{\bullet}$
and $\mathscr{G}^{\bullet}$ are bounded below complexes of flasque sheaves of
$\mathscr{O}_{X}$-modules and for all $x\in X$, the maps
$\mathscr{F}_{x}\rightarrow\mathscr{F}_{x}^{\bullet}$ and
$\mathscr{G}_{x}\rightarrow\mathscr{G}_{x}^{\bullet}$ are homotopy
equivalences in the category of $\mathscr{O}_{X,x}$-modules. In particular,
$\mathscr{F}^{\bullet}$ and $\mathcal{G}^{\bullet}$ are $K$-flat (by [Sta13],
Lemma 06YB and the property that $\mathscr{F}_{x}$ and $\mathscr{G}_{x}$ are
flat $\mathscr{O}_{X}$-modules). Since $\mathscr{F}^{\bullet}$ and
$\mathscr{G}^{\bullet}$ are $K$-flat, the map
$\mathscr{F}\otimes\mathscr{G}\rightarrow\mathrm{Tot}\big{(}\mathscr{F}^{\bullet}\otimes\mathscr{G}^{\bullet}\big{)}$
is a quasi-isomorphism. We see that
$(i_{1})_{\star}\mathrm{R}i_{1}^{!}\mathscr{F}$ is computed by
$(i_{1})_{\star}i_{1}^{!}\mathscr{F}^{\bullet}$ and
$(i_{2})_{\star}\mathrm{R}i_{2}^{!}\mathscr{G}$ is computed by
$(i_{2})_{\star}i_{2}^{!}\mathscr{G}^{\bullet}$. Taking $K$-flat resolutions
$A^{\bullet}\rightarrow(i_{1})_{\star}i_{1}^{!}\mathscr{F}^{\bullet}$ and
$B^{\bullet}\rightarrow(i_{2})_{\star}i_{2}^{!}\mathscr{G}^{\bullet}$, we see
that $\mathrm{Tot}(A^{\bullet}\otimes_{\mathscr{O}_{X}}B^{\bullet})$ computes

$(i_{1})_{\star}\mathrm{R}i_{1}^{!}\mathscr{F}\otimes^{L}_{\mathscr{O}_{X}}(i_{2})_{\star}\mathrm{R}i_{2}^{!}\mathscr{G}$

and there is a composite map

$\mathrm{Tot}(A^{\bullet}\otimes_{\mathscr{O}_{X}}B^{\bullet})\rightarrow\mathrm{Tot}\big{(}(i_{1})_{\star}i_{1}^{!}\mathscr{F}^{\bullet}\otimes_{\mathscr{O}_{X}}(i_{2})_{\star}i_{2}^{!}\mathscr{G}^{\bullet})$
$\rightarrow(i_{3})_{\star}i_{3}^{!}(\mathrm{Tot}\big{(}\mathscr{F}^{\bullet}\otimes\mathscr{G}^{\bullet}\big{)}).$

Taking a $K$-injective resolution
$\mathrm{Tot}(\mathscr{F}^{\bullet}\otimes\mathscr{G}^{\bullet})\rightarrow
C^{\bullet}$ we finally find that $(i_{3})_{\star}i_{3}^{!}(C^{\bullet})$
computes $(i_{3})_{\star}\mathrm{R}i_{3}^{!}(\mathscr{F}\otimes\mathscr{G})$
and we have a morphism

$(i_{3})_{\star}i_{3}^{!}(\mathrm{Tot}\big{(}\mathscr{F}^{\bullet}\otimes\mathscr{G}^{\bullet}\big{)})\rightarrow(i_{3})_{\star}i_{3}^{!}(C^{\bullet}).$

There is also a usual cup-product map by [Sta13], Tag 0FPJ:

$\mathrm{R}\Gamma(X,(i_{1})_{\star}\mathrm{R}i_{1}^{!}\mathscr{F})\otimes^{L}_{K}\mathrm{R}\Gamma(X,(i_{2})_{\star}\mathrm{R}i_{2}^{!}\mathscr{G})\rightarrow\mathrm{R}\Gamma(X,(i_{1})_{\star}\mathrm{R}i_{1}^{!}\mathscr{F}\otimes^{L}_{\mathscr{O}_{X}}(i_{2})_{\star}\mathrm{R}i_{2}^{!}\mathscr{G}).$

Combining the two maps gives the map of the proposition. ∎

###  2.3. The spectral sequence of a filtered topological space

Let $X$ be a topological space, $\mathscr{F}$ a sheaf of abelian groups, and
let $W\subseteq Z$ be two closed subspaces of $X$. We can define
$\mathrm{R}\Gamma_{Z/W}(X,\mathscr{F})=\mathrm{R}\Gamma_{Z\setminus
W}(X\setminus W,\mathscr{F})$. If $Z\subseteq Z^{\prime}$ and $W\subseteq
W^{\prime}$ we have a map
$\mathrm{R}\Gamma_{Z/W}(X,\mathscr{F})\rightarrow\mathrm{R}\Gamma_{Z^{\prime}/W^{\prime}}(X,\mathscr{F})$.

If we have $Z_{3}\subseteq Z_{2}\subseteq Z_{1}$, then there is an exact
triangle ([Kem78], lemma 7.6):
$\mathrm{R}\Gamma_{Z_{2}/Z_{3}}(X,\mathscr{F})\rightarrow\mathrm{R}\Gamma_{Z_{1}/Z_{3}}(X,\mathscr{F})\rightarrow\mathrm{R}\Gamma_{Z_{1}/Z_{2}}(X,\mathscr{F})\stackrel{{\scriptstyle+1}}{{\rightarrow}}$

Assume that there is a filtration by closed subsets $X=Z_{0}\supseteq
Z_{1}\supseteq\cdots Z_{r}=\emptyset$. Then there is a spectral sequence of
filtered topological space ([Har66], p. 227):

$E_{1}^{pq}=\mathrm{H}^{p+q}_{Z_{p}/Z_{p+1}}(X,\mathscr{F})\Rightarrow\mathrm{H}^{p+q}(X,\mathscr{F})$

which we can visualize as follows:

The differential
$d_{1}^{p,q}:\mathrm{H}^{p+q}_{Z_{p}/Z_{p+1}}(X,\mathscr{F})\rightarrow\mathrm{H}^{p+q+1}_{Z_{p+1}/Z_{p+2}}(X,\mathscr{F})$
is the boundary map in the long exact sequence associated with the triangle:

$\mathrm{R}\Gamma_{Z_{p+1}/Z_{p+2}}(X,\mathcal{F})\rightarrow\mathrm{R}\Gamma_{Z_{p}/Z_{p+2}}(X,\mathcal{F})\rightarrow\mathrm{R}\Gamma_{Z_{p}/Z_{p+1}}(X,\mathcal{F})\stackrel{{\scriptstyle+1}}{{\rightarrow}}.$

###  2.4. The category of projective Banach modules

In this work we will consider cohomologies that will be naturally represented
by complexes of Banach modules (or projective, inductive limits of such
complexes). We therefore recall the basics of this theory. Our discussion
follows [Urb11], section 2. Let $(A,A^{+})$ be a complete Tate algebra over a
non-archimedean field $(F,\mathcal{O}_{F})$. We let $\varpi\in\mathcal{O}_{F}$
be a pseudo-uniformizer.

####  2.4.1. Modules

We let $\mathbf{Mod}(A)$ be the abelian category of $A$-modules,
$\mathcal{C}(A)$ be the category of complexes of $A$-modules, $\mathcal{K}(A)$
its homotopy category and $\mathcal{D}(A)$ the derived category. We let
$\mathcal{K}^{perf}(A)$ be the homotopy category of the category of bounded
complexes of finite projective $A$-modules. These are called perfect complex.
The category $\mathcal{K}^{perf}(A)$ is a full subcategory of the category
$\mathcal{D}(A)$.

####  2.4.2. Banach modules

A Banach $A$-module $M$ is a topological $A$-module whose topology can be
described as follows: Let $A_{0}$ be an open and bounded subring of $A$. Then
$M$ contains an open and bounded sub $A_{0}$-module $M_{0}$ which is
$\varpi$-adically complete and separated (and $M=M_{0}[\frac{1}{\varpi}]$). We
let $\mathbf{Ban}(A)$ be the category whose objects are Banach $A$-modules,
and whose morphisms are continuous $A$-linear morphisms. This is a quasi-
abelian category in the sense of [Sch99], def. 1.1.3. This means that
$\mathbf{Ban}(A)$ is an additive category with kernels and cokernels, strict
epimorphisms are stable under pullback and strict monomorphisms are stable
under pushout. Recall that strict morphisms $f$ are those for which the
natural morphism $\mathrm{coim}(f)\rightarrow\mathrm{im}(f)$ is an isomorphism
in $\mathbf{Ban}(A)$.

####  2.4.3. The derived category

We let $\mathcal{C}(\mathbf{Ban}(A))$ be the category of complexes of Banach
modules, and $\mathcal{K}(\mathbf{Ban}(A))$ be its homotopy category. A
complex $M^{\bullet}$ in $\mathcal{C}(\mathbf{Ban}(A))$ is strictly exact if
each differential $d_{i}:M_{i}\rightarrow M_{i+1}$ is strict and the morphism
$\mathrm{Im}(d_{i})\rightarrow\mathrm{ker}(d_{i+1})$ is an isomorphism. A
morphism $f:M^{\bullet}\rightarrow N^{\bullet}$ in
$\mathcal{C}(\mathbf{Ban}(A))$ is a strict quasi-isomorphism if its cone is
strictly exact. It is a general property of quasi-abelian categories that if a
morphism is strictly exact, so is any other morphism in the same homotopy
class. One can consider the derived category $\mathcal{D}(\mathbf{Ban}(A))$
obtained by inverting in $\mathcal{K}(\mathbf{Ban}(A))$ the strict quasi-
isomorphisms ([Sch99], def. 1.2.16, [Urb11], sect. 2.1.3). In the case of the
category $\mathbf{Ban}(A)$ we have the following lemma which helps recognize
strict quasi-isomorphisms:

######  Lemma 2.4.4.

Let $M^{\bullet}$ be a complex in $\mathcal{C}(\mathbf{Ban}(A))$. Then
$M^{\bullet}$ is exact as a complex of $A$-modules if and only if it is
strictly exact. A morphism $f:M^{\bullet}\rightarrow N^{\bullet}$ in
$\mathcal{C}(\mathbf{Ban}(A))$ is a quasi-isomorphism of complexes in
$\mathbf{Mod}(A)$ if and only if it is a strict quasi-isomorphism.

###### Proof.

It follows from the open mapping theorem that any epimorphism in
$\mathbf{Ban}(A)$ is open and therefore strict. The first claim follows. The
second claim is deduced by considering the cone of $f$. ∎

Because of this lemma, strict quasi-isomorphisms will simply be called quasi-
isomorphisms.

####  2.4.5. Projectives

Let $I$ be a set. Denote by $A(I)$ the submodule of $A^{I}$ of sequences of
elements of $A$ indexed by $I$ converging to $0$ according to the filter in
$I$ of the complement of the finite subsets of $I$. This module can also be
described as follows. Let $A_{0}$ be an open and bounded subring of $A$. Let
$A_{0}(I)$ be the $\varpi$-adic completion of the free $A_{0}$-module with
basis $I$. Then $A(I)=A_{0}(I)[\frac{1}{\varpi}]$. We see that $A(I)$ is a
Banach $A$-module. A Banach $A$-module $M$ is orthonormalizable if there
exists a set $I$ and an isomorphism $M\simeq A(I)$. A Banach $A$-module is
called projective if it is a direct factor of an orthonormalizable Banach
$A$-module. We let $\mathcal{C}^{proj}(\mathbf{Ban}(A))$ be the category of
bounded complexes of projective Banach $A$-modules and
$\mathcal{K}^{proj}(\mathbf{Ban}(A))$ be the homotopy category. There is a
natural functor
$\mathcal{K}^{proj}(\mathbf{Ban}(A))\rightarrow\mathcal{D}(\mathbf{Ban}(A))$
and it is fully faithful ([Sch99], prop. 1.3.22, [Urb11], lem. 2.1.8). In
particular we deduce the following lemma which will be used in this work:

######  Lemma 2.4.6.

Let $f:M^{\bullet}\rightarrow N^{\bullet}$ be a quasi-isomorphism in
$\mathcal{C}^{proj}(\mathbf{Ban}(A))$. Then $f$ admits and inverse
$g:N^{\bullet}\rightarrow M^{\bullet}$ up to homotopy.

Recall that finite type $A$-modules are canonically Banach $A$-modules.
Therefore we can view $\mathcal{K}^{perf}(A)$ as a full subcategory of
$\mathcal{K}^{proj}(\mathbf{Ban}(A))$.

####  2.4.7. Limits of Banach spaces

We let $\mathrm{Pro}_{\mathbb{N}}(\mathbf{Ban}(A))$ be the category of
countable projective systems of $\mathbf{Ban}(A)$. Its objects are
contravariant functors $\mathbb{N}\rightarrow\mathbf{Ban}(A)$. An object of
this category will be denoted $``\lim_{i\in\mathbb{N}}"K_{i}$ with $K_{i}\in
Ob(\mathbf{Ban}(A))$. We let $\mathrm{Ind}_{\mathbb{N}}(\mathbf{Ban}(A))$ be
the category of countable inductive systems of $\mathbf{Ban}(A)$. Its objects
are covariant functors $\mathbb{N}\rightarrow\mathbf{Ban}(A)$. An object of
this category will be denoted $``\mathrm{colim}_{i\in\mathbb{N}}"K_{i}$ with
$K_{i}\in Ob(\mathbf{Ban}(A))$. We have a limit functor
$\lim:\mathrm{Pro}_{\mathbb{N}}(\mathbf{Ban}(A))\rightarrow\mathbf{Mod}(A)$,
and a colimit functor
$\mathrm{colim}:\mathrm{Ind}_{\mathbb{N}}(\mathbf{Ban}(A))\rightarrow\mathbf{Mod}(A)$.
Of course, these functors completely forget the topology. We could consider
instead limit and colimit functors in the category of locally convex
$A$-modules instead, but this turns out to be unnecessary to us. We extend
$\lim$ and $\mathrm{colim}$ to functors between the category of complexes:
$\lim:\mathcal{C}(\mathrm{Pro}_{\mathbb{N}}(\mathbf{Ban}(A)))\rightarrow\mathcal{C}(A)$
and
$\mathrm{colim}:\mathcal{C}(\mathrm{Ind}_{\mathbb{N}}(\mathbf{Ban}(A)))\rightarrow\mathcal{C}(A)$.

We let $\mathrm{Pro}_{\mathbb{N}}(\mathcal{K}^{proj}(\mathbf{Ban}(A)))$ be the
category whose objects are projective systems of complexes $\\{K_{i}\in
Ob(\mathcal{K}^{proj}(\mathbf{Ban}(A)))\\}_{i\in\mathbb{N}}$ and the $K_{i}$’s
have non-zero cohomology in a uniformly bounded range of degrees. We denote an
object of this category by $``\lim_{i\in\mathbb{N}}"K_{i}$ (we are not
evaluating the limit). Let $``\lim_{i\in\mathbb{N}}"K_{i}\in
Ob(\mathrm{Pro}_{\mathbb{N}}(\mathcal{K}^{proj}(\mathbf{Ban}(A))))$. Then
there exists an object (unique up to a non unique isomorphism)
$\lim_{i\in\mathbb{N}}K_{i}\in\mathcal{D}(A)$ by [Sta13], Tag 08TB. And
similarly, for any morphism
$``\lim_{i\in\mathbb{N}}"K_{i}\rightarrow``\lim_{i\in\mathbb{N}}"K^{\prime}_{i}\in\mathrm{Pro}_{\mathbb{N}}(\mathcal{K}^{proj}(\mathbf{Ban}(A)))$,
there is a morphism $\lim K_{i}\rightarrow\lim K^{\prime}_{i}$. If
$``\lim_{i\in\mathbb{N}}"K_{i}\in
Ob(\mathrm{Pro}_{\mathbb{N}}(\mathcal{K}^{proj}(\mathbf{Ban}(A))))$ can be
represented by an object in
$\mathcal{C}(\mathrm{Pro}_{\mathbb{N}}(\mathbf{Ban}(A)))$, then $\lim K_{i}$
is obtained by taking the degreewise projective limit.

We also let $\mathrm{Ind}_{\mathbb{N}}(\mathcal{K}^{proj}(\mathbf{Ban}(A)))$
be the category whose objects are inductive systems of complexes $\\{K_{i}\in
Ob(\mathcal{K}^{proj}(A))\\}_{i\in\mathbb{N}}$ and the $K_{i}$’s have non-zero
cohomology in a uniformly bounded range of degrees. We denote an object of
this category by $``\mathrm{colim}_{i\in\mathbb{N}}"K_{i}$ (we are not
evaluating the limit either).

Let $``\mathrm{colim}_{i\in\mathbb{N}}"K_{i}\in
Ob(\mathrm{Ind}_{\mathbb{N}}(\mathcal{K}^{proj}(\mathbf{Ban}(A))))$. Then
there exists an object (unique up to a non unique isomorphism)
$\mathrm{colim}_{i\in\mathbb{N}}K_{i}\in\mathcal{D}(A)$ by [Sta13], Tag 0A5K.
And similarly, for any morphism
$``\mathrm{colim}_{i\in\mathbb{N}}"K_{i}\rightarrow``\mathrm{colim}_{i\in\mathbb{N}}"K^{\prime}_{i}\in\mathrm{Ind}_{\mathbb{N}}(\mathcal{K}^{proj}(\mathbf{Ban}(A)))$,
there is a morphism
$\mathrm{colim}K_{i}\rightarrow\mathrm{colim}K^{\prime}_{i}$. If
$``\mathrm{colim}_{i\in\mathbb{N}}"K_{i}\in
Ob(\mathrm{Ind}_{\mathbb{N}}(\mathcal{K}^{proj}(\mathbf{Ban}(A))))$ can be
represented by an object in
$\mathcal{C}(\mathrm{Ind}_{\mathbb{N}}(\mathbf{Ban}(A)))$, then
$\mathrm{colim}K_{i}$ is obtained by taking the degreewise inductive limit.

####  2.4.8. Compact operators

Recall that a morphism $T:M\rightarrow N$ in $\mathbf{Ban}(A)$ is called
compact if it is a limit of finite rank operators (for the supremum norm of
operators, or equivalently the strong topology on
$\mathrm{Hom}_{\mathbf{Ban}(A)}(M,N)$). A morphism $T:M^{\bullet}\rightarrow
N^{\bullet}$ in $\mathcal{C}(\mathbf{Ban}(A))$ is called compact if it is
compact in each degree.

######  Definition 2.4.9.

A morphism $T:M^{\bullet}\rightarrow N^{\bullet}$ in
$\mathcal{K}^{proj}(\mathbf{Ban}(A))$ is compact if it has a compact
representative in $\mathcal{C}^{proj}(\mathbf{Ban}(A))$.

We need to extend these definitions to the case of objects in
$\mathrm{Pro}_{\mathbb{N}}(\mathcal{K}^{proj}(\mathbf{Ban}(A)))$ or
$\mathrm{Ind}_{\mathbb{N}}(\mathcal{K}^{proj}(\mathbf{Ban}(A)))$. Let
$T:``\lim_{i}"M_{i}^{\bullet}\rightarrow``\lim_{i}"N_{i}^{\bullet}$ be a
morphism in $\mathrm{Pro}_{\mathbb{N}}(\mathcal{K}^{proj}(\mathbf{Ban}(A)))$.
We say that $T$ is compact if there exists a compact operator
$T^{\prime}:M^{\bullet}\rightarrow
N^{\bullet}\in\mathcal{K}^{proj}(\mathbf{Ban}(A))$, and a commutative diagram:

######  Lemma 2.4.10.

Let $``\lim_{i}"M_{i}^{\bullet}\in
Ob(\mathrm{Pro}_{\mathbb{N}}(\mathcal{K}^{proj}(\mathbf{Ban}(A))))$ and let
$T$ be a compact endomorphism of $``\lim_{i}"M_{i}^{\bullet}$. Then $T$
induces canonically a compact endomorphism $T_{i}$ of $M^{\bullet}_{i}$ for
$i$ large enough and there are factorization diagrams:

###### Proof.

By definition, the map $``\lim_{i}"M_{i}^{\bullet}\rightarrow M^{\bullet}$
factors into $``\lim_{i}"M_{i}^{\bullet}\rightarrow M_{i}^{\bullet}\rightarrow
M^{\bullet}$ for some $i$ large enough. The map
$N^{\bullet}\rightarrow``\lim_{i}"M_{i}^{\bullet}$ is given by a collection of
compatible maps $N^{\bullet}\rightarrow M_{i}^{\bullet}$. The lemma follows. ∎

Let
$T:``\mathrm{colim}_{i}"M_{i}^{\bullet}\rightarrow``\mathrm{colim}_{i}"N_{i}^{\bullet}$
be a morphism in
$\mathrm{Ind}_{\mathbb{N}}(\mathcal{K}^{proj}(\mathbf{Ban}(A)))$. We say that
$T$ is compact if there exists a compact operator
$T^{\prime}:M^{\bullet}\rightarrow
N^{\bullet}\in\mathcal{K}^{proj}(\mathbf{Ban}(A))$, and a commutative diagram:

######  Lemma 2.4.11.

Let $``\mathrm{colim}_{i}"M_{i}^{\bullet}\in
Ob(\mathrm{Ind}_{\mathbb{N}}(\mathcal{K}^{proj}(\mathbf{Ban}(A))))$ and let
$T$ be a compact endomorphism of $``\mathrm{colim}_{i}"M_{i}^{\bullet}$. Then
$T$ induces canonically a compact endomorphism $T_{i}$ of $M^{\bullet}_{i}$
for $i$ large enough and there are factorization diagrams:

###### Proof.

By definition, the map
$N^{\bullet}\rightarrow``\mathrm{colim}_{i}"M_{i}^{\bullet}$ factors into
$N^{\bullet}\rightarrow
M_{i}^{\bullet}\rightarrow``\mathrm{colim}_{i}"M_{i}^{\bullet}$ for some $i$
large enough. The map $``\mathrm{colim}_{i}"M_{i}^{\bullet}\rightarrow
M^{\bullet}$ is given by a collection of compatible maps
$M_{i}^{\bullet}\rightarrow M^{\bullet}$. The lemma follows. ∎

######  Definition 2.4.12.

Let $T:N^{\bullet}\rightarrow M^{\bullet}$ be a map in $\mathcal{D}(A)$. We
say that $T$ is compact if it can be represented by a compact map in
$\mathrm{Pro}_{\mathbb{N}}(\mathcal{K}^{proj}(\mathbf{Ban}(A)))$ or
$\mathrm{Ind}_{\mathbb{N}}(\mathcal{K}^{proj}(\mathbf{Ban}(A)))$.

######  Definition 2.4.13.

Let $M^{\bullet}\in Ob(\mathcal{D}(A))$ and let
$T\in\mathrm{End}_{\mathcal{D}(A)}(M^{\bullet})$. We say that $T$ is potent
compact if for some $n\geq 0$, ${T}^{n}$ is compact.

###  2.5. The cohomology of Banach sheaves

In this section we explain how we can obtain complexes of Banach modules from
the cohomology of rigid analytic varieties. We use the theory of adic spaces
described in [Hub96] and [Hub94] for example.

####  2.5.1. Sheaves of Banach modules over adic spaces

In this section we recall some material from [AIP15], appendix A. Let $F$ be a
non archimedean field with ring of integers $\mathcal{O}_{F}$. Let $\varpi\in
F$ be a topologically nilpotent unit.

We recall that there is a good theory of coherent sheaves on finite type adic
spaces over $\mathrm{Spa}(F,\mathcal{O}_{F})$: If
$\mathcal{X}=\mathrm{Spa}(A,A^{+})$ is affinoid and $\mathscr{F}$ is a
coherent sheaf on $\mathcal{X}$, then
$\mathrm{H}^{i}(\mathcal{X},\mathscr{O}_{\mathcal{X}})=0$ for $i\neq 0$,
$M=\mathrm{H}^{0}(\mathcal{X},\mathscr{O}_{\mathcal{X}})$ is an $A$-module of
finite type and the canonical map
${M}\otimes_{A}\mathscr{O}_{\mathcal{X}}\rightarrow\mathscr{F}$ is an
isomorphism ([Hub94], thm. 2.5). Moreover, $M$ is canonically a Banach
$A$-module ([Hub94], lem. 2.4). It follows that a coherent sheaf $\mathscr{F}$
over a finite type adic space $\mathcal{X}$ is a sheaf of topological
$\mathscr{O}_{\mathcal{X}}$-modules. In this paper we will have to manipulate
topological sheaves which are not coherent.

######  Definition 2.5.2.

Let $\mathcal{X}$ be a finite type adic space over
$\mathrm{Spa}(F,\mathcal{O}_{F})$. A sheaf $\mathscr{F}$ of topological
$\mathscr{O}_{\mathcal{X}}$-modules is called a Banach sheaf if:

  1. (1)

For any quasi-compact open $\mathcal{U}\hookrightarrow\mathcal{X}$,
$\mathscr{F}(\mathcal{U})$ is a Banach
$\mathscr{O}_{\mathcal{X}}(\mathcal{U})$-module,

  2. (2)

There is an affinoid covering $\mathcal{X}=\cup_{i}\mathcal{U}_{i}$, such that
for any affinoid $\mathcal{V}\subset\mathcal{U}_{i}$, the continuous
restriction map
$\mathscr{F}(\mathcal{U}_{i})\rightarrow\mathscr{F}(\mathcal{V})$ induces a
topological isomorphism:
$\mathscr{O}_{\mathcal{X}}(\mathcal{V})\hat{\otimes}_{\mathscr{O}_{\mathcal{X}}(\mathcal{U}_{i})}\mathscr{F}(\mathcal{U}_{i})\rightarrow\mathscr{F}(\mathcal{V})$,

A Banach sheaf $\mathscr{F}$ is called locally projective if there is a
covering as in $(2)$ such that $\mathscr{F}(\mathcal{U}_{i})$ is a projective
Banach $\mathscr{O}_{\mathcal{X}}(\mathcal{U}_{i})$-module.

Any coherent sheaf on $\mathcal{X}$ is therefore a Banach sheaf and a coherent
sheaf is a locally projective Banach sheaf if and only if it is locally free.
Banach sheaves over $\mathcal{X}$ form a full subcategory of the category of
topological $\mathscr{O}_{\mathcal{X}}$-modules. We introduce compact
morphisms in this context.

######  Definition 2.5.3.

Let $\mathcal{X}$ be an adic space of finite type over
$\mathrm{Spa}(F,\mathcal{O}_{F})$. Let $\mathscr{F}$ and $\mathscr{G}$ be two
locally projective Banach sheaves. Let
$\phi:\mathscr{F}\rightarrow\mathscr{G}$ be a continuous morphism of
$\mathscr{O}_{\mathcal{X}}$-modules. We say that the map $\phi$ is compact if
there is a covering $\mathcal{X}=\cup_{i}\mathcal{U}_{i}$ satisfying the point
$(2)$ of definition 2.5.2 for both $\mathscr{F}$ and $\mathscr{G}$, such that
the map
$\phi:\mathscr{F}(\mathcal{U}_{i})\rightarrow\mathscr{G}(\mathcal{U}_{i})$ is
a compact map of $\mathscr{O}_{\mathcal{X}}(\mathcal{U}_{i})$-modules.

######  Remark 2.5.4.

Note that if $\mathscr{G}$ is coherent, any morphism to $\mathscr{G}$ is
compact.

####  2.5.5. Cohomological properties of Banach sheaves

We warn the reader that Banach sheaves which are not coherent sheaves are
pathological in general. In particular it is not true that on an affinoid
$\mathcal{X}$, a Banach sheaf has trivial higher cohomology groups, nor is it
true that a Banach sheaf is the sheaf associated with its global sections.
Here is nevertheless a simple situation where this holds. Let
$\mathcal{X}=\mathrm{Spa}(A,A^{+})$ be an affinoid and let $M$ be a projective
Banach $A$-module. Let
$\mathscr{F}=M\hat{\otimes}_{A}\mathscr{O}_{\mathcal{X}}$ be the pre-sheaf
whose value on an affinoid open $\mathcal{U}=\mathrm{Spa}(B,B^{+})$ of
$\mathcal{X}$ is $M\hat{\otimes}_{A}B$.

######  Lemma 2.5.6.

The pre-sheaf $\mathscr{F}$ is a sheaf and
$\mathrm{H}^{i}(\mathcal{X},\mathscr{F})=0$ for all $i>0$.

###### Proof.

We reduce to the case that $M$ is orthonormalizable, and everything follows
from the known properties of $\mathscr{O}_{\mathcal{X}}$. ∎

We now introduce a certain class of Banach sheaves that have better
cohomological properties, following [AIP15], appendix A. These are Banach
sheaves admitting formal models which can be controlled in a certain sense. We
thus begin by discussing formal Banach sheaves over formal schemes.

######  Definition 2.5.7.

Let $\mathfrak{X}\rightarrow\mathrm{Spf}~{}\mathcal{O}_{F}$ be a finite type
formal scheme over $\mathrm{Spf}(\mathcal{O}_{F})$. A sheaf $\mathfrak{F}$ of
$\mathscr{O}_{\mathfrak{X}}$-modules is called a formal Banach sheaf if
$\mathfrak{F}$ is flat as an $\mathcal{O}_{F}$-module,
$\mathscr{F}_{n}:=\mathfrak{F}/\varpi^{n}$ is a quasi-coherent sheaf, and
$\mathfrak{F}=\lim_{n}\mathscr{F}_{n}$.

A formal Banach sheaf is called _flat_ if $\mathscr{F}_{n}$ is a flat
$\mathscr{O}_{\mathfrak{X}}/\varpi^{n}$-module for all $n$. It is called
_locally projective_ if $\mathscr{F}_{n}$ is a locally projective
$\mathscr{O}_{\mathfrak{X}}/\varpi^{n}$-module for all $n$. A formal Banach
sheaf is called _small_ if there exists a coherent sheaf $\mathscr{G}$ over
$\mathfrak{X}$ with the property that $\mathscr{F}_{1}$ is the inductive limit
of coherent sub sheaves $\mathscr{F}_{1}=\mathrm{colim}_{j\in\mathbb{Z}_{\geq
0}}\mathscr{F}_{1,j}$ and $\mathscr{F}_{1,j}/\mathscr{F}_{1,j-1}$ is a direct
summand of $\mathscr{G}$ for all $j\geq 0$.

The relevance of the smallness assumption is given by the following theorem:

######  Theorem 2.5.8.

Let $\mathfrak{X}\rightarrow\mathrm{Spf}~{}\mathcal{O}_{F}$ be a finite type
formal scheme and let $\mathfrak{F}$ be a small formal Banach sheaf. Assume
that $\mathfrak{X}$ has an ample invertible sheaf, and that the generic fiber
$\mathcal{X}$ of $\mathfrak{X}$ is affinoid. Then
$\mathrm{H}^{i}(\mathfrak{X},\mathfrak{F})\otimes_{\mathcal{O}_{F}}F=0$ for
all $i>0$.

###### Proof.

This is [AIP15], theorem A.1.2.2. In the reference, the formal scheme
$\mathfrak{X}$ is assumed to be normal and quasi-projective, but the only
property needed in the proof is the existence of an ample sheaf on
$\mathfrak{X}$. ∎

Let $\mathfrak{X}\rightarrow\mathrm{Spf}~{}\mathcal{O}_{F}$ be a finite type
formal scheme, and let $\mathcal{X}\rightarrow\mathrm{Spa}(F,\mathcal{O}_{F})$
be the generic fiber of $\mathfrak{X}$. Thus
$(\mathcal{X},\mathscr{O}_{\mathcal{X}}^{+})=\lim_{\mathfrak{X}^{\prime}}(\mathfrak{X}^{\prime},\mathscr{O}_{\mathfrak{X}^{\prime}})$
where the limit runs over all admissible blow-ups of $\mathfrak{X}$. Let
$\mathfrak{F}$ be a Banach sheaf over $\mathfrak{X}$. For any admissible blow-
up $f:\mathfrak{X}^{\prime}\rightarrow\mathfrak{X}$, we let
$\mathfrak{F}_{\mathfrak{X}^{\prime}}=\lim_{n}f^{\star}\mathscr{F}_{n}$. We
let
$\mathscr{F}=\lim_{\mathfrak{X}^{\prime}}\mathfrak{F}_{\mathfrak{X}^{\prime}}[1/\varpi]$.
This is a sheaf over $\mathcal{X}$ that we call the generic fiber of
$\mathfrak{F}$.

######  Theorem 2.5.9.

Let $\mathfrak{X}\rightarrow\mathrm{Spf}~{}\mathcal{O}_{F}$ be a finite type
formal scheme, with generic fiber $\mathcal{X}$. We have the following
properties:

  1. (1)

There is a “generic fiber” functor going from the category of flat formal
Banach sheaves over $\mathfrak{X}$ to the category of Banach sheaves over
$\mathcal{X}$, described by the procedure $\mathfrak{F}\mapsto\mathscr{F}$.

  2. (2)

If $\mathcal{U}\hookrightarrow\mathcal{X}$ is a quasi-compact open subset and
$\mathfrak{U}^{\prime}\hookrightarrow\mathfrak{X}^{\prime}$ is a formal model
for the map $\mathcal{U}\hookrightarrow\mathcal{X}$,
$\mathscr{F}(\mathcal{U})=\mathfrak{F}_{\mathfrak{X}^{\prime}}(\mathfrak{U}^{\prime})[1/\varpi]$.

  3. (3)

The property $(2)$ of definition 2.5.2 holds over the generic fiber of any
affine covering of $\mathfrak{X}$.

  4. (4)

The generic fiber functor sends locally projective formal Banach sheaves to
locally projective Banach sheaves.

  5. (5)

Let $\mathscr{F}$ be a Banach sheaf arising from a flat small formal Banach
sheaf. Then for any affinoid open subset
$\mathcal{U}\hookrightarrow\mathcal{X}$ we have
$\mathrm{H}^{i}(\mathcal{U},\mathscr{F})=0$ for all $i>0$.

###### Proof.

The first three points are [AIP15], proposition A.2.2.3. For the fourth point,
let $\mathfrak{U}$ be an open affine of $\mathfrak{X}$. Let
$M_{n}=\mathrm{H}^{0}(\mathfrak{U},\mathscr{F}_{n})$,
$A_{n}=\mathrm{H}^{0}(\mathfrak{U},\mathscr{O}_{\mathfrak{X}}/\varpi^{n})$,
$M=\lim_{n}M_{n}$, and $A=\lim_{n}A_{n}$. We claim that $M$ is a direct factor
of the completion of a free $A$-module. Let us pick a surjection
$A_{1}^{I}\rightarrow M_{1}$ and for any $n$, we can lift it successively to
surjections $A_{n}^{I}\rightarrow M_{n}$. We need to prove that we can find a
compatible system of sections $s_{n}:M_{n}\rightarrow A_{n}^{I}$. It suffices
to show that the map
$\mathrm{Hom}_{A}(M_{n},A_{n}^{I})\rightarrow\mathrm{Hom}_{A}(M_{n-1},A_{n-1}^{I})$
is surjective. This follows from the short exact sequence
$0\rightarrow\mathrm{Hom}_{A_{n}}(M_{n},A^{I}_{1})\rightarrow\mathrm{Hom}_{A_{n}}(M_{n},A^{I}_{n})\rightarrow\mathrm{Hom}_{A_{n}}(M_{n},A^{I}_{n-1})\rightarrow
0$.

For the last point, let $\mathfrak{X}$ be a formal model of $\mathcal{X}$ and
$\mathfrak{F}$ be a small flat formal Banach sheaf over $\mathfrak{X}$. Let
$\mathfrak{U}$ be an affine formal model of $\mathcal{U}$. Let
$\cup_{i}\mathcal{U}_{i}=\mathcal{U}$ be a finite affinoid cover of
$\mathcal{U}$. Let $\mathfrak{U}^{\prime}$ be an admissible blow-up of
$\mathfrak{U}$ with the property that $\cup_{i}\mathcal{U}_{i}=\mathcal{U}$ is
the generic fiber of a covering of $\mathfrak{U}^{\prime}$ and there is a map
$\mathfrak{U}^{\prime}\rightarrow\mathfrak{X}$ inducing the map
$\mathcal{U}\rightarrow\mathcal{X}$. Note that $\mathfrak{U}^{\prime}$ has an
ample invertible sheaf, since it is a blow-up of an affine formal scheme. We
can apply theorem 2.5.8 to $\mathfrak{F}_{\mathfrak{U}^{\prime}}$, the pull-
back to $\mathfrak{U}^{\prime}$ of $\mathfrak{F}$, which is still small by
flatness. This shows that the Čech cohomology of $\mathcal{U}$ with respect to
the covering $\cup_{i}\mathcal{U}_{i}$ vanishes. Since this holds for any
finite cover, and $\mathcal{U}$ is quasi-compact, we deduce that
$\mathrm{H}^{i}(\mathcal{U},\mathscr{F})=0$. ∎

######  Definition 2.5.10.

Let $\mathcal{X}$ be a finite type adic space over
$\mathrm{Spa}(F,\mathcal{O}_{F})$. A Banach sheaf $\mathscr{F}$ is called a
small, locally projective Banach sheaf if it arises as the generic fiber of a
small, locally projective formal Banach sheaf.

######  Remark 2.5.11.

A locally free coherent sheaf over $\mathcal{X}$ is a small, locally
projective Banach sheaf by the flattening techniques of [RG71].

######  Remark 2.5.12.

We don’t know if, for $\mathcal{X}=\mathrm{Spa}(A,A^{+})$ affinoid and
$\mathscr{F}$ a small, locally projective Banach sheaf on $\mathcal{X}$, it is
true that $\mathscr{F}(\mathcal{X})$ is a projective Banach $A$-module and the
map
$\mathscr{F}(\mathcal{X})\hat{\otimes}_{A}\mathscr{O}_{\mathcal{X}}\rightarrow\mathscr{F}$
is an isomorphism.

####  2.5.13. Acyclicity of quasi-Stein spaces

In our arguments, it will often be useful to consider not only affinoid covers
of adic spaces, but also some quasi-Stein covers.

######  Definition 2.5.14 ([Kie67], def. 2.3).

We say that an adic space
$\mathcal{X}\rightarrow\mathrm{Spa}(F,\mathcal{O}_{F})$ is quasi-Stein if
$\mathcal{X}=\cup_{i\in\mathbb{Z}_{\geq 0}}\mathcal{X}_{i}$ is a countable
increasing union of finite type affinoid adic spaces
$\mathcal{X}_{i}\rightarrow\mathrm{Spa}(F,\mathcal{O}_{F})$ and
$\mathscr{O}_{\mathcal{X}_{i+1}}\rightarrow\mathscr{O}_{\mathcal{X}_{i}}$ has
dense image.

######  Example 2.5.15.

Here are some examples of quasi-Stein adic spaces:

  * •

An affinoid space, like the unit ball $\mathbb{B}(0,1)$.

  * •

A Stein space like the open unit ball:
${\mathbb{B}}^{o}(0,1)=\cup_{n}\mathbb{B}(0,|p^{\frac{1}{n}}|)$.

  * •

A “mixed” situation like
$\mathbb{B}^{o}(0,1)\times_{\mathrm{Spa}(F,\mathcal{O}_{F})}\mathbb{B}(0,1)$.

We also recall the following classical acyclicity result:

######  Theorem 2.5.16 ([Kie67], Satz 2.4).

Let $\mathcal{X}$ be a quasi-Stein adic space and let $\mathscr{F}$ be a
coherent sheaf over $\mathcal{X}$. Then
$\mathrm{H}^{i}(\mathcal{X},\mathscr{F})=0$ for all $i>0$.

We also have:

######  Proposition 2.5.17.

Let $\mathcal{X}=\mathrm{Spa}(A,A^{+})$ be an affinoid finite type adic space,
let $M$ be a projective Banach $A$-module, and let
$\mathscr{F}=M\hat{\otimes}_{A}\mathscr{O}_{\mathcal{X}}$. Let
$\mathcal{U}\hookrightarrow\mathcal{X}$ be a quasi-Stein open subset. Then
$\mathrm{H}^{i}(\mathcal{U},\mathscr{F})=0$ for all $i>0$.

###### Proof.

We reduce to the case where $M$ is orthonormalizable, and therefore to the
case of $\mathscr{O}_{\mathcal{X}}$ where we can apply theorem 2.5.16. ∎

####  2.5.18. Cohomology complexes

We now illustrate how one can obtain complexes of Banach modules. We denote by
$\mathcal{D}(F)$ the derived category of the category of $F$-vector spaces. We
also let $\mathbf{Ban}(F)$ be the category of Banach $F$-vector spaces. For an
adic space $\mathcal{X}$ over $\mathrm{Spa}(F,\mathcal{O}_{F})$, and a sheaf
$\mathscr{F}$ of $\mathscr{O}_{\mathcal{X}}$-module, the cohomology groups
$\mathrm{R}\Gamma(\mathcal{X},\mathscr{F})$ are objects of the category
$\mathcal{D}(F)$. Nevertheless, they often carry more structure and can be
represented by objects of the category $\mathcal{C}(\mathbf{Ban}(F))$ of
complexes of Banach modules. We formalize this in this section.

######  Remark 2.5.19.

Our approach is rather elementary and we content ourselves with statements
saying that a given cohomology can be represented by certain complexes of
topological $F$-vector spaces. This is enough for our purposes. We simply
point out in this remark a more conceptual approach yielding stronger results.
Let $\mathscr{F}$ be a coherent sheaf over an adic space $\mathcal{X}$ of
finite type over $\mathrm{Spa}(F,\mathscr{O}_{F})$. Since $\mathscr{F}$
carries a canonical topology, we can view $\mathscr{F}$ as sheaf of locally
convex $F$-vector spaces over $\mathcal{X}$. Let $\mathbf{Loc}_{F}$ be the
category of locally convex $F$-vector spaces. This is a quasi-abelian category
with enough injectives ([Pro00]). One can therefore derive the functor
$\mathrm{H}^{0}(\mathcal{X},\mathscr{F})$ in this category and view
$\mathrm{R}\Gamma(\mathcal{X},\mathscr{F})$ as an object of
$\mathcal{D}(\mathbf{Loc}_{F})$.

######  Lemma 2.5.20.

Let $\mathcal{X}$ be a separated finite type adic space over
$\mathrm{Spa}(F,\mathcal{O}_{F})$. Let $\mathscr{F}$ be a locally projective
Banach sheaf over $\mathcal{X}$. Let $\mathcal{U}\subseteq\mathcal{X}$ be a
quasi-compact open subset. Let $\mathcal{Z}\subseteq\mathcal{X}$ be a closed
subset, with quasi-compact complement. Then one can represent
$\mathrm{R}\Gamma_{\mathcal{Z}\cap\mathcal{U}}(\mathcal{U},\mathscr{F})$ by an
object of $\mathcal{K}^{proj}(\mathbf{Ban}(F))$.

###### Proof.

We have an exact triangle
$\mathrm{R}\Gamma_{\mathcal{Z}\cap\mathcal{U}}(\mathcal{U},\mathscr{F})\rightarrow\mathrm{R}\Gamma(\mathcal{U},\mathscr{F})\rightarrow\mathrm{R}\Gamma(\mathcal{U}\setminus(\mathcal{Z}\cap\mathcal{U}),\mathscr{F})\stackrel{{\scriptstyle+1}}{{\rightarrow}}$
and therefore, we are reduced to prove the claim for
$\mathrm{R}\Gamma(\mathcal{U},\mathscr{F})$ and
$\mathrm{R}\Gamma(\mathcal{U}\setminus(\mathcal{Z}\cap\mathcal{U}),\mathscr{F})$.
Finally, it suffices to prove that for a quasi-compact open
$\mathcal{U}\subseteq\mathcal{X}$,
$\mathrm{R}\Gamma(\mathcal{U},\mathscr{F})\in
Ob(\mathcal{K}^{proj}(\mathbf{Ban}(F)))$. We can compute the cohomology by
considering an $\mathscr{F}$-acyclic affinoid covering of $\mathcal{U}$ for
$\mathscr{F}$, and the associated Čech complex by [Hub94], thm. 2.5. Then each
of the terms of the Čech complex carries a canonical structure of Banach
$F$-algebra. ∎

######  Lemma 2.5.21.

Let $\mathcal{X}$ be a separated finite type adic space over
$\mathrm{Spa}(F,\mathcal{O}_{F})$. Let $\mathscr{F}$ be a locally projective
Banach sheaf over $\mathcal{X}$. Let $\mathcal{U}\subseteq\mathcal{X}$ be an
open subset which is a finite union of quasi-Stein spaces. Let
$\mathcal{Z}\subseteq\mathcal{X}$ be a closed subset, whose complement is a
finite union of quasi-Stein spaces. Then one can represent
$\mathrm{R}\Gamma_{\mathcal{Z}\cap\mathcal{U}}(\mathcal{U},\mathscr{F})$ by an
object of $\mathrm{Pro}_{\mathbb{N}}(\mathcal{K}^{proj}(\mathbf{Ban}(F)))$.

###### Proof.

As in the proof of lemma 2.5.20, we are reduced to showing that for an open
$\mathcal{U}\subseteq\mathcal{X}$ which is a finite union of quasi-Stein
spaces, $\mathrm{R}\Gamma(\mathcal{U},\mathscr{F})\in
Ob(\mathcal{C}(\mathrm{Pro}_{\mathbb{N}}(\mathbf{Ban}(F))))$. We let
$\mathcal{U}=\cup_{k}\mathcal{V}_{k}$ be a finite covering of $\mathcal{U}$ by
$\mathscr{F}$-acyclic quasi-Stein spaces $\mathcal{V}_{k}$. We let
$\mathcal{V}_{k}=\cup_{i\geq 0}\mathcal{V}_{k,i}$ where $\mathcal{V}_{k,i}$ is
affinoid. We let $\mathcal{U}_{i}=\cup_{k}\mathcal{V}_{k,i}$. We let
$\mathrm{R}\Gamma(\mathcal{U},\mathscr{F})=``\lim"\mathrm{R}\Gamma(\mathcal{U}_{i},\mathscr{F})$.
∎

####  2.5.22. Compact morphisms in the cohomology of adic spaces

We give some examples of compact morphisms arising from maps between the
cohomology of adic spaces. First, let us fix a standard notation. Let $T$ be a
topological space and let $S$ be a subset of $T$. Then we denote by
$\overline{S}$ the closure of $S$ in $T$ and by $\overset{\circ}{S}$ the
interior of $S$ in $T$.

######  Lemma 2.5.23.

Let $\mathcal{X}\rightarrow\mathrm{Spa}(F,\mathcal{O}_{F})$ be a proper adic
space and let $\mathscr{F}$ be a locally free sheaf of finite rank over
$\mathcal{X}$. Let
$\mathcal{U}^{\prime}\subseteq\mathcal{U}\subseteq\mathcal{X}$ be quasi-
compact open subsets. Assume that
$\overline{\mathcal{U}^{\prime}}\subseteq\mathcal{U}$. Then the map

$\mathrm{R}\Gamma(\mathcal{U},\mathscr{F})\rightarrow\mathrm{R}\Gamma(\mathcal{U}^{\prime},\mathscr{F})$

is compact.

###### Proof.

We claim that there exists a formal model
$\mathfrak{X}\rightarrow\mathrm{Spf}~{}\mathcal{O}_{F}$ of $\mathcal{X}$, and
two opens $\mathfrak{U}$ and $\mathfrak{U}^{\prime}$ of $\mathfrak{X}$ with
generic fibers $\mathcal{U}$ and $\mathcal{U}^{\prime}$, such that
$\overline{\mathfrak{U}^{\prime}}\subseteq\mathfrak{U}$. Remark that
$\overline{\mathfrak{U}^{\prime}}\rightarrow\mathrm{Spf}~{}\mathcal{O}_{F}$ is
proper because $\mathcal{X}$ was assumed to be proper ([L9̈0], Th. 3.1). We
deduce that $\mathcal{U}^{\prime}$ is relatively compact in $\mathcal{U}$
([L9̈0], lem. 2.5). We prove the claim. Recall that the ringed space
$(\mathcal{X},\mathscr{O}_{\mathcal{X}}^{+})$ is the inverse limit of the
ringed spaces $(\mathfrak{X},\mathscr{O}_{\mathfrak{X}})$ where $\mathfrak{X}$
runs over all the formal models of $\mathcal{X}$. For a cofinal subset of
$\mathfrak{X}$, we have opens $\mathfrak{U}_{\mathfrak{X}}$ and
$\mathfrak{U}^{\prime}_{\mathfrak{X}}$ of $\mathfrak{X}$ with generic fiber
$\mathcal{U}$ and $\mathcal{U}^{\prime}$. We let
$\overline{\mathcal{U}^{\prime}}_{\mathfrak{X}}$ be the generic fiber of
$\overline{\mathfrak{U}^{\prime}}_{\mathfrak{X}}$. Then
$\overline{\mathcal{U}^{\prime}}=\cap_{\mathfrak{X}}\overline{\mathcal{U}^{\prime}}_{\mathfrak{X}}$.
The topological space $\mathcal{X}$ equipped with the constructible topology
is compact (in fact this is a profinite set). We have that
$(\mathcal{U})^{c}=\cup_{\mathfrak{X}}(\overline{\mathcal{U}^{\prime}}_{\mathfrak{X}})^{c}\cap(\mathcal{U})^{c}$.
Since $(\mathcal{U})^{c}$ is compact, we deduce that there is a model
$\mathfrak{X}$ such that
$\overline{\mathcal{U}^{\prime}}_{\mathfrak{X}}\subseteq\mathcal{U}$, and
therefore
$\overline{\mathfrak{U}^{\prime}}_{\mathfrak{X}}\subseteq\mathfrak{U}_{\mathfrak{X}}$.
This finishes the proof of the claim.

Let $\mathcal{U}^{\prime}=\cup_{i\in I}\mathcal{U}^{\prime}_{i}$ be a finite
affinoid cover of $\mathcal{U}^{\prime}$. By [L9̈0], thm. 5.1, for each $i$,
there exists an affinoid
$\mathcal{U}^{\prime}_{i}\subseteq\mathcal{U}_{i}\subseteq\mathcal{U}$ such
that $\mathcal{U}^{\prime}_{i}$ is relatively compact in $\mathcal{U}_{i}$
(equivalently $\overline{\mathcal{U}^{\prime}_{i}}\subseteq\mathcal{U}_{i}$).
Let $\mathcal{U}^{\prime\prime}=\cup_{i\in I}\mathcal{U}_{i}$. We claim that
the map

$\mathrm{R}\Gamma(\mathcal{U}^{\prime\prime},\mathscr{F})\rightarrow\mathrm{R}\Gamma(\mathcal{U}^{\prime},\mathscr{F})$

is compact. This will prove that the map of the lemma is compact because it
factors over a compact map. These last cohomologies can be represented by the
Čech complex with respect to $\cup_{i}\mathcal{U}_{i}$ and
$\cup_{i}\mathcal{U}_{i}^{\prime}$. We now prove that the maps
$\mathscr{F}(\cap_{J\subseteq
I}\mathcal{U}_{i})\rightarrow\mathscr{F}(\cap_{J\subseteq
I}\mathcal{U}^{\prime}_{i})$ are compact. We reproduce the argument of [KL05],
prop. 2.4.1. We have just seen that we can find a formal model
$\mathfrak{V}^{\prime}\hookrightarrow\mathfrak{V}\hookrightarrow\mathfrak{X}$
(with maps being Zariski open immersions) for the maps $\cap_{J\subseteq
I}\mathcal{U}^{\prime}_{i}\rightarrow\cap_{J\subseteq
I}\mathcal{U}_{i}\rightarrow\mathcal{X}$ with the property that the map
$\mathfrak{V}^{\prime}\hookrightarrow\mathfrak{V}$ factors into
$\mathfrak{V}^{\prime}\hookrightarrow\overline{\mathfrak{V}^{\prime}}\hookrightarrow\mathfrak{V}$.
We also let $\mathfrak{F}$ be a flat coherent sheaf over $\mathfrak{X}$ with
generic fiber $\mathscr{F}$. Let $\mathfrak{F}_{n}=\mathfrak{F}/\varpi^{n}$.
We have that $\mathscr{F}(\cap_{J\subseteq
I}\mathcal{U}_{i})=\mathfrak{F}(\mathfrak{V})[\frac{1}{\varpi}]$, and the
image of $\mathfrak{F}(\mathfrak{V})$ in $\mathscr{F}(\cap_{J\subseteq
I}\mathcal{U}_{i})$ is an open and bounded submodule. This also applies to the
′ situation. We claim that the image of $\mathfrak{F}(\mathfrak{V})$ in
$\mathfrak{F}(\mathfrak{V}^{\prime})/\varpi^{n}$ is a finite
$\mathcal{O}_{F}$-module. This will prove that the map
$\mathscr{F}(\cap_{J\subseteq
I}\mathcal{U}_{i})\rightarrow\mathscr{F}(\cap_{J\subseteq
I}\mathcal{U}^{\prime}_{i})$ is compact. The image of
$\mathfrak{F}(\mathfrak{V})$ in
$\mathfrak{F}(\mathfrak{V}^{\prime})/\varpi^{n}$ is contained (inside
$\mathfrak{F}_{n}(\mathfrak{V}^{\prime})$) in the image of
$\mathfrak{F}_{n}(\mathfrak{V})$. But the map
$\mathfrak{F}_{n}(\mathfrak{V})\rightarrow\mathfrak{F}_{n}(\mathfrak{V}^{\prime})$
factors over
$\mathfrak{F}_{n}(\mathfrak{V})\rightarrow\mathfrak{F}_{n}(\overline{\mathfrak{V}^{\prime}})\rightarrow\mathfrak{F}_{n}(\mathfrak{V}^{\prime})$.
The $\mathcal{O}_{F}$-module
$\mathfrak{F}_{n}(\overline{\mathfrak{V}^{\prime}})$ is finite since
$\overline{\mathfrak{V}^{\prime}}$ is proper.

∎

######  Lemma 2.5.24.

Let $\mathcal{X}\rightarrow\mathrm{Spa}(F,\mathcal{O}_{F})$ be a proper adic
space and let $\mathscr{F}$ be a locally free sheaf of finite rank over
$\mathcal{X}$. Let $\mathcal{U}^{\prime}\subseteq\mathcal{U}$ be two quasi-
compact open, and $\mathcal{Z}\subseteq\mathcal{Z}^{\prime}$ be two closed
subspaces, with quasi-compact complements. Assume that
$\overline{\mathcal{U}}^{\prime}\subseteq\mathcal{U}$ and
$\mathcal{Z}\subseteq\overset{\circ}{\mathcal{Z}^{\prime}}$. Then the map

$\mathrm{R}\Gamma_{\mathcal{Z}\cap\mathcal{U}}(\mathcal{U},\mathscr{F})\rightarrow\mathrm{R}\Gamma_{\mathcal{Z}^{\prime}\cap\mathcal{U}^{\prime}}(\mathcal{U}^{\prime},\mathscr{F})$

is compact.

###### Proof.

It suffices to see that
$\mathrm{R}\Gamma(\mathcal{U},\mathscr{F})\rightarrow\mathrm{R}\Gamma(\mathcal{U}^{\prime},\mathscr{F})$
and
$\mathrm{R}\Gamma(\mathcal{U}\setminus(\mathcal{Z}\cap\mathcal{U}),\mathscr{F})\rightarrow\mathrm{R}\Gamma(\mathcal{U}^{\prime}\setminus(\mathcal{Z}^{\prime}\cap\mathcal{U}^{\prime}),\mathscr{F})$
are compact. This follows from the fact that
$\overline{\mathcal{U}}^{\prime}\subseteq\mathcal{U}$ and
$\overline{\mathcal{U}^{\prime}\setminus(\mathcal{Z}^{\prime}\cap{\mathcal{U}^{\prime}})}\subseteq\overline{\mathcal{U}}^{\prime}\setminus(\overset{\circ}{\mathcal{Z}^{\prime}}\cap\overline{\mathcal{U}}^{\prime})\subseteq\mathcal{U}\setminus(\mathcal{Z}\cap\mathcal{U})$
and the previous lemma. ∎

We now give a stronger form of the lemma.

######  Lemma 2.5.25.

Assume that $\mathcal{X}$ is proper. Let
$\mathcal{U}^{\prime}\subseteq\mathcal{U}$ be two open subsets which are
finite unions of quasi-Stein spaces, and
$\mathcal{Z}\subseteq\mathcal{Z}^{\prime}$ be two closed subspaces, whose
complements are finite unions of quasi-Stein spaces. Assume that there exists
a quasi-compact open $\mathcal{U}^{\prime\prime}$ such that
${\mathcal{U}}^{\prime}\cap\mathcal{Z}^{\prime}\subseteq\mathcal{U}^{\prime\prime}$
with $\overline{\mathcal{U}^{\prime\prime}}\subseteq\mathcal{U}$, as well as
two closed subset
$\mathcal{Z}^{\prime\prime}\subseteq\mathcal{Z}^{\prime\prime\prime}$ with
quasi-compact complement, such that
$\mathcal{Z}\cap\mathcal{U}\subseteq\mathcal{Z}^{{}^{\prime\prime}}$,
$\mathcal{Z}^{{}^{\prime\prime}}\subseteq\overset{\circ}{\mathcal{Z}^{\prime\prime\prime}}$
and $\mathcal{Z}^{{}^{\prime\prime\prime}}\subseteq{\mathcal{Z}^{\prime}}$.
Then the map

$\mathrm{R}\Gamma_{\mathcal{Z}\cap\mathcal{U}}(\mathcal{U},\mathscr{F})\rightarrow\mathrm{R}\Gamma_{\mathcal{Z}^{\prime}\cap\mathcal{U}^{\prime}}(\mathcal{U}^{\prime},\mathscr{F})$

is compact.

###### Proof.

We can write $\mathcal{U}=\cup_{n}\mathcal{U}_{n}$ with $\mathcal{U}_{n}$
quasi-compact. Since
$\overline{\mathcal{U}}^{\prime\prime}\subseteq\mathcal{U}$ and $\mathcal{X}$
equipped with the constructible topology is compact, we deduce that
$\overline{\mathcal{U}}^{\prime\prime}\subseteq\mathcal{U}_{n}$ for $n$ large
enough. The map
$\mathrm{R}\Gamma_{\mathcal{Z}^{\prime\prime}\cap\mathcal{U}_{n}}(\mathcal{U}_{n},\mathscr{F})\rightarrow\mathrm{R}\Gamma_{\mathcal{Z}^{\prime\prime\prime}\cap\mathcal{U}^{\prime\prime}}(\mathcal{U}^{\prime\prime},\mathscr{F})$
is compact by lemma 2.5.24. There is a restriction-corestriction map
$\mathrm{R}\Gamma_{\mathcal{Z}\cap\mathcal{U}}(\mathcal{U},\mathscr{F})\rightarrow\mathrm{R}\Gamma_{\mathcal{Z}^{\prime\prime}\cap\mathcal{U}_{n}}(\mathcal{U}_{n},\mathscr{F})$.

We also have a restriction-corestriction map

$\mathrm{R}\Gamma_{\mathcal{Z}^{\prime\prime\prime}\cap\mathcal{U}^{\prime\prime}}(\mathcal{U}^{\prime\prime},\mathscr{F})\rightarrow\mathrm{R}\Gamma_{\mathcal{Z}^{\prime}\cap\mathcal{U}^{\prime}}(\mathcal{U}^{\prime}\cap\mathcal{U}^{\prime\prime},\mathscr{F}).$

On the other hand,
$\mathrm{R}\Gamma_{\mathcal{Z}^{\prime}\cap\mathcal{U}^{\prime}}(\mathcal{U}^{\prime},\mathscr{F})=\mathrm{R}\Gamma_{\mathcal{Z}^{\prime}\cap\mathcal{U}^{\prime}}(\mathcal{U}^{\prime}\cap\mathcal{U}^{\prime\prime},\mathscr{F})$.
All together, we deduce that the map

$\mathrm{R}\Gamma_{\mathcal{Z}\cap\mathcal{U}}(\mathcal{U},\mathscr{F})\rightarrow\mathrm{R}\Gamma_{\mathcal{Z}^{\prime}\cap\mathcal{U}^{\prime}}(\mathcal{U}^{\prime},\mathscr{F})$

factors through the compact map
$\mathrm{R}\Gamma_{\mathcal{Z}^{\prime\prime}\cap\mathcal{U}_{n}}(\mathcal{U}_{n},\mathscr{F})\rightarrow\mathrm{R}\Gamma_{\mathcal{Z}^{\prime\prime\prime}\cap\mathcal{U}^{\prime\prime}}(\mathcal{U}^{\prime\prime},\mathscr{F})$
and is compact. ∎

We finally conclude this section with a last lemma where we deal with Banach
sheaves which are not necessarily coherent.

######  Lemma 2.5.26.

Let $\mathcal{X}$, $\mathcal{U}$, $\mathcal{U}^{\prime}$, $\mathcal{Z}$,
$\mathcal{Z}^{\prime}$ be as in lemma 2.5.25. Let $\mathscr{F}$ and
$\mathscr{G}$ be two locally projective Banach sheaves and let
$\phi:\mathscr{F}\rightarrow\mathscr{G}$ be a compact morphism. The morphism

$\mathrm{R}\Gamma_{\mathcal{Z}\cap\mathcal{U}}(\mathcal{U},\mathscr{F})\rightarrow\mathrm{R}\Gamma_{\mathcal{Z}^{\prime}\cap\mathcal{U}^{\prime}}(\mathcal{U}^{\prime},\mathscr{G})$

is compact.

###### Proof.

Easy and left to the reader. ∎

###  2.6. Integral structures on Banach sheaves

We now consider integral structures on Banach sheaves, but this time more in
the spirit of analytic geometry. We let $\mathcal{X}$ be a separated adic
space locally of finite type over $\mathrm{Spa}(F,\mathcal{O}_{F})$. We let
$\mathscr{F}$ be a locally projective Banach sheaf over $\mathcal{X}$. We can
view $\mathscr{F}$ as a sheaf on the étale site of $\mathcal{X}$ by [BG98].

######  Definition 2.6.1.

An integral structure on $\mathscr{F}$ is a sheaf $\mathscr{F}^{+}$ of
$\mathscr{O}_{\mathcal{X}}^{+}$-modules on the étale site of $X$, such that:

  1. (1)

$\mathscr{F}^{+}\hookrightarrow\mathscr{F}$ and
$\mathscr{F}^{+}\otimes_{\mathcal{O}_{F}}F=\mathscr{F}$,

  2. (2)

There is an étale cover $\coprod U_{i}\rightarrow\mathcal{X}$ by affinoid
spaces such that $\mathscr{F}^{+}(U_{i})$ is the completion of a free
$\mathscr{O}_{\mathcal{X}}^{+}(U_{i})$-module, and the canonical map
$\mathscr{F}^{+}(U_{i})\hat{\otimes}_{\mathscr{O}_{\mathcal{X}}^{+}(U_{i})}\mathscr{O}_{U_{i}}^{+}\rightarrow\mathscr{F}^{+}|_{U_{i}}$
is an isomorphism.

######  Remark 2.6.2.

A stronger property would be to ask that the étale cover $\coprod
U_{i}\rightarrow\mathcal{X}$ is in fact an analytic cover. In our
applications, this stronger property will not be satisfied. Indeed, we will
produce sheaves arising from torsors under various groups, and these torsors
are usually only trivial locally for the étale topology. However, we will not
consider the étale cohomology
$\mathrm{H}^{i}_{et}(\mathcal{X},\mathscr{F}^{+})$, but only the analytic
cohomology $\mathrm{H}^{i}_{an}(\mathcal{X},\mathscr{F}^{+})$.

######  Lemma 2.6.3.

In the situation of definition 2.6.1, assume moreover that $\mathcal{X}$ is
reduced. Let $U\rightarrow\mathcal{X}$ be an étale map. Then
$\mathscr{F}^{+}(U)$ is an open and bounded submodule of $\mathscr{F}(U)$.

###### Proof.

Let $I$ be a finite set and let $\coprod_{i\in I}U_{i}\rightarrow U$ be an
étale cover such each $U_{i}$ is affinoid and $\mathscr{F}^{+}|_{U_{i}}$ is
the completion of a free sheaf of $\mathscr{O}_{U_{i}}^{+}$-modules. By the
sheaf property we have an exact sequence:

$0\rightarrow\mathscr{F}(U)\rightarrow\prod_{i}\mathscr{F}(U_{i})\rightarrow\prod_{i,j}\mathscr{F}(U_{i}\times_{U}U_{j})$

and $\mathscr{F}(U)$ is a closed Banach subspace of
$\prod_{i}\mathscr{F}(U_{i})$. Since $\prod_{i}\mathscr{F}^{+}(U_{i})$ is open
and bounded in $\prod_{i}\mathscr{F}(U_{i})$ (using that
$\mathscr{O}^{+}_{\mathcal{X}}(U_{i})$ is bounded by the reducedness
hypothesis),
$\mathscr{F}^{+}(U)=\prod_{i}\mathscr{F}^{+}(U_{i})\cap\mathscr{F}(U)$ is open
and bounded in $\mathscr{F}^{+}(U)$. ∎

We now will elaborate on a result of Bartenwerfer which we first recall.

######  Theorem 2.6.4 ([Bar78]).

Let $\mathcal{X}$ be an affinoid smooth adic space over
$\mathrm{Spa}(F,\mathcal{O}_{F})$. There exists $N\in\mathbb{Z}_{\geq 0}$ such
that $\mathrm{H}^{i}_{an}(\mathcal{X},\mathscr{O}_{\mathcal{X}}^{+})$ is
annihilated by $p^{N}$ for all $i>0$.

###### Proof.

Bartenwerfer’s result is stated for Čech cohomology. By [Pil20] prop. 3.1.1,
this implies the claim for cohomology. ∎

######  Lemma 2.6.5.

Let $\mathcal{X}$ be an affinoid smooth adic space over
$\mathrm{Spa}(F,\mathcal{O}_{F})$. Let $\mathscr{F}$ be a locally projective
Banach sheaf which is assumed to be associated to its global sections (i.e.
satisfies point $(2)$ in definition 2.5.2). Let $\mathscr{F}^{+}$ be an
integral structure on $\mathscr{F}$. There exists $N\in\mathbb{Z}_{\geq 0}$
such that for all $i>0$ the cohomology groups
$\mathrm{H}^{i}_{an}(\mathcal{X},\mathscr{F}^{+})$ are annihilated by $p^{N}$.

###### Proof.

Let $\mathcal{X}=\mathrm{Spa}(A,A^{+})$. By assumption,
$\mathscr{F}=M\hat{\otimes}_{A}\mathscr{O}_{\mathcal{\mathcal{X}}}$ is
associated to a projective Banach $A$-module $M$. Let $I$ be a set such that
$A(I)=M\oplus N$. Let $M^{+}=A^{+}(I)\cap M$ and $N^{+}=A^{+}(I)\cap N$. The
injective map $M^{+}\oplus N^{+}\rightarrow A^{+}(I)$ has cokernel of bounded
torsion. Moreover, if we let $M_{+}$ be the image of $A^{+}(I)$ in $M$ under
the projection orthogonal to $N$, then $M^{+}\hookrightarrow M_{+}$ has
cokernel of bounded torsion. We deduce that there exists an integer $N$ such
that the multiplication $p^{N}:M^{+}\rightarrow M^{+}$ factors through:
$M^{+}\rightarrow A^{+}(I)\rightarrow M^{+}$ (where the first map is the
inclusion, the second map is the orthogonal projection with respect to $N$
composed with multiplication by $p^{N}$ and followed by the inclusion
$p^{N}M_{+}\subseteq M^{+}$). It follows from theorem 2.6.4 that
$\mathrm{H}^{i}(\mathcal{X},\mathscr{O}_{\mathcal{X}}^{+}\hat{\otimes}_{A^{+}}A^{+}(I))$
is of bounded torsion for all $i>0$. We let $\mathscr{M}^{+}$ be the subsheaf
of $\mathscr{O}_{\mathcal{X}}^{+}\hat{\otimes}_{A^{+}}A^{+}(I)\cap\mathscr{F}$
equal to the image of the sheaf associated to the presheaf
$\mathscr{O}_{\mathcal{X}}^{+}\hat{\otimes}_{A^{+}}M^{+}$ (in other words, it
is the subsheaf of $\mathscr{F}$ of sections which can locally be written as
tensors in $\mathscr{O}_{\mathcal{X}}^{+}\hat{\otimes}_{A^{+}}M^{+}$).

We see that multiplication by
$p^{N}:\mathscr{M}^{+}\rightarrow\mathscr{M}^{+}$ factors through
$\mathscr{M}^{+}\rightarrow\mathscr{O}_{\mathcal{X}}^{+}\hat{\otimes}_{A^{+}}A^{+}(I)\rightarrow\mathscr{M}^{+}$.
We deduce that $\mathrm{H}^{i}(\mathcal{X},\mathscr{M}^{+})$ is of bounded
torsion for all $i>0$. After rescaling, we may assume that
$M^{+}\subseteq\mathscr{F}^{+}(\mathcal{X})$, with cokernel of bounded torsion
by lemma 2.6.3. We therefore get a morphism:
$\mathscr{M}^{+}\rightarrow\mathscr{F}^{+}$. We claim that this morphism has
cokernel of bounded torsion. Let $\cup_{i\in I}U_{i}\rightarrow\mathcal{X}$ be
an affinoid étale covering with the property that $\mathscr{F}^{+}|_{U_{i}}$
is associated with its global sections. We may assume that the set $I$ is
finite. It suffices to show that
$\mathscr{O}_{\mathcal{X}}^{+}(U_{i})\hat{\otimes}_{A^{+}}M^{+}\rightarrow\mathscr{F}^{+}(U_{i})$
has cokernel of bounded torsion. This follows since the image of
$\mathscr{O}_{\mathcal{X}}^{+}(U_{i})\hat{\otimes}_{A^{+}}M^{+}$ is open. We
finally deduce that $\mathrm{H}^{i}_{an}(\mathcal{X},\mathscr{F}^{+})$ is of
bounded torsion for all $i>0$. ∎

###  2.7. Duality for analytic adic spaces

In this section we fix $\mathcal{X}$ a proper smooth adic space over
$\mathrm{Spa}(F,\mathcal{O}_{F})$ of pure dimension $d$. Let
$\Omega^{d}_{\mathcal{X}/F}$ be the canonical sheaf, equal to
$\Lambda^{d}\Omega^{1}_{\mathcal{X}/F}$. Let
$\iota:\mathcal{Z}\hookrightarrow\mathcal{X}$ be a closed subset, equal to the
closure of a quasi-compact open subset $\mathcal{U}$ of $\mathcal{X}$. One can
consider the cohomology group
$\mathrm{H}^{d}_{\mathcal{Z}}(\mathcal{X},\Omega^{d}_{\mathcal{X}/F})$.

######  Theorem 2.7.1 ([GK00], [Bey97]).

  1. (1)

There is a trace map
$\mathrm{tr}_{\mathcal{Z}}:\mathrm{H}^{d}_{\mathcal{Z}}(\mathcal{X},\Omega^{d}_{\mathcal{X}/F})\rightarrow
F$.

  2. (2)

If $\mathcal{Z}\subseteq\mathcal{Z}^{\prime}$, there is a factorization

$\mathrm{tr}_{\mathcal{Z}}:\mathrm{H}^{d}_{\mathcal{Z}}(\mathcal{X},\Omega^{d}_{\mathcal{X}/K})\rightarrow\mathrm{H}^{d}_{\mathcal{Z}^{\prime}}(\mathcal{X},\Omega^{d}_{\mathcal{X}/F})\stackrel{{\scriptstyle\mathrm{tr}_{\mathcal{Z}^{\prime}}}}{{\rightarrow}}F.$

  3. (3)

For any coherent sheaf $\mathscr{F}$ defined on a neighborhood of
$\mathcal{Z}$ the map $\mathrm{tr}_{\mathcal{Z}}$ induces a pairing:

$\mathrm{Ext}^{i}_{\iota^{-1}\mathscr{O}_{\mathcal{X}}}(\iota^{-1}\mathscr{F},\iota^{-1}\Omega^{d}_{\mathcal{X}/F})\times\mathrm{H}^{d-i}_{\mathcal{Z}}(\mathcal{X},\mathscr{F})\rightarrow
F$

  4. (4)

When $\mathcal{Z}=\mathcal{X}$ the cohomology groups are finite dimensional
$F$-vector spaces, and the pairing is perfect.

  5. (5)

When $\mathcal{U}$ is affinoid and $\mathscr{F}$ is a locally free sheaf,
$\mathrm{Ext}^{0}_{\iota^{-1}\mathscr{O}_{\mathcal{X}}}(\iota^{-1}\mathscr{F},\iota^{-1}\Omega^{d}_{\mathcal{X}/F})$
is a compact inductive limit of $F$-Banach spaces,
$\mathrm{H}^{d}_{\mathcal{Z}}(\mathcal{X},\mathscr{F})$ is a compact
projective limit of $F$-Banach spaces, and the pairing is a topological
duality between locally convex topological $F$-vector spaces spaces
identifying each space with the strong dual of the other. Moreover,
$\mathrm{Ext}^{i}_{\iota^{-1}\mathscr{O}_{\mathcal{X}}}(\iota^{-1}\mathscr{F},\iota^{-1}\Omega^{d}_{\mathcal{X}/F})$
and $\mathrm{H}^{d-i}_{\mathcal{Z}}(\mathcal{X},\mathscr{F})$ vanish for
$i\neq 0$.

######  Remark 2.7.2.

We give some translations to the language of [GK00]. Let
$\mathcal{U}^{\dagger}$ be the dagger space attached to
$\mathcal{U}\hookrightarrow\mathcal{X}$. Let $\mathscr{F}$ be a coherent sheaf
defined on a neighborhood of $\mathcal{Z}$ in $\mathcal{X}$. Then
$\mathrm{H}^{i}(\mathcal{U}^{\dagger},\mathscr{F})=\mathrm{H}^{i}(\mathcal{Z},\iota^{-1}\mathscr{F})$
and
$\mathrm{H}^{i}_{c}(\mathcal{U}^{\dagger},\mathscr{F})=\mathrm{H}^{i}_{\mathcal{Z}}(\mathcal{X},\mathscr{F})$.

##  3\. Flag variety

###  3.1. Bruhat decomposition

Let $F$ be a non archimedean local field of mixed characteristic with residue
field $k$ of characteristic $p$ and discrete valuation $v$ normalized so that
$v(p)=1$. Let $G\rightarrow\mathcal{O}_{F}$ be a split reductive group with
maximal torus $T$ contained in a Borel $B$ with unipotent radical $U$. Let
$B\subseteq P$ be a parabolic with Levi $M\subseteq P$ containing $T$.

######  Remark 3.1.1.

In our application to Shimura varieties, the unipotent radical of $P$ will be
abelian. Nevertheless, this assumption is not relevant for the moment.

We let $\Phi$ be the set of roots, $\Delta$ and $\Phi^{+}$ the subsets of
simple and positive roots corresponding to our choice of $B$, and
$\Phi^{-}=-\Phi^{+}$. We let $\Phi^{+}_{M}$ be the subset of positive roots
which lie in the Lie algebra of $M$ and
$\Phi^{+,M}=\Phi^{+}\setminus\Phi^{+}_{M}$. We let
$\Phi^{-}_{M}=-\Phi^{+}_{M}$ and $\Phi^{-,M}=-\Phi^{+,M}$.

We let $W$ be the Weyl group of $G$. We denote by
$\ell:W\rightarrow\mathbb{Z}_{\geq 0}$ the length function. We denote by
$w_{0}$ the longest element of $W$. For each $w\in W$, we choose a
representative in $N(T)$ that we still denote $w$. The group $W$ acts on the
left on the cocharacter group $X_{\star}(T)$ and on the character group
$X^{\star}(T)$ on the left as well via the formula