# Arithmetic duality for $p$-adic pro-étale cohomology of analytic curves

Pierre Colmez  CNRS, IMJ-PRG, Sorbonne Université, 4 place Jussieu, 75005
Paris, France<EMAIL_ADDRESS>,  Sally Gilles  Max Plack Institute
for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany  gilles.mpim-bonn.mpg.de
and  Wiesława Nizioł  CNRS, IMJ-PRG, Sorbonne Université, 4 place Jussieu,
75005 Paris, France<EMAIL_ADDRESS>
###### Abstract.

We prove a Poincaré duality for arithmetic $p$-adic pro-étale cohomology of
smooth dagger curves over finite extensions of ${\bf Q}_{p}$. We deduce it,
via the Hochschild-Serre spectral sequence, from geometric comparison theorems
combined with Tate and Serre dualities. The compatibility of all the products
involved is checked via reduction to the ghost circle, for which we also prove
a Poincaré duality (showing that it behaves like a proper smooth analytic
variety of dimension $1/2$). Along the way we study functional analytic
properties of arithmetic $p$-adic pro-étale cohomology and prove that the
usual cohomology is nuclear Fréchet and the compactly supported one – of
compact type.

The authors’ research was supported in part by the grant NR-19-CE40-0015-02
COLOSS and by the grant No. DMS-1928930 of the National Science Foundation
while P.C. and W.N. were in residence at the Mathematical Sciences Research
Institute in Berkeley, California, during the Spring 2023 semester.

###### Contents

  1. 1 Introduction
  2. 2 Functional analysis
  3. 3 Galois cohomology of $K$
  4. 4 Pro-étale cohomology
  5. 5 Pro-étale cohomology with compact support
  6. 6 Trace maps and pairings
  7. 7 Poincaré duality for a ghost circle
  8. 8 Arithmetic Poincaré duality

##  1\. Introduction

This paper is a contribution to the study of dualities in $p$-adic pro-étale
cohomology of analytic varieties.

###  1.1. Statement of the main theorem

Let $p$ be a prime and let $K$ be a finite extension of ${\bf Q}_{p}$. Our
main theorem is the following duality result.

######  Theorem 1.1.

(Arithmetic Poincaré duality) Let $X$ be a smooth, geometrically irreducible,
dagger variety111Dagger varieties, introduced by Grosse-Klönne in [22], are
rigid analytic varieties with overconvergent structure sheaves. of dimension
$1$ over $K$. Assume that $X$ is proper, Stein, or affinoid. Then:

  1. (1)

There is a natural trace map isomorphism of solid ${\bf Q}_{p}$-vector
spaces222See Chapter 5 for a definition of compactly supported pro-étale
cohomology.

(1.2) ${\rm Tr}_{X}:\,H^{4}_{\operatorname{pro\acute{e}t},c}(X,{\bf
Q}_{p}(2))\stackrel{{\scriptstyle\sim}}{{\to}}{\bf Q}_{p}.$

  2. (2)

The pairing

$H^{i}_{\operatorname{pro\acute{e}t}}(X,{\bf Q}_{p}(j))\otimes^{\Box}_{{\bf
Q}_{p}}H^{4-i}_{\operatorname{pro\acute{e}t},c}(X,{\bf
Q}_{p}(2-j)){\to}H^{4}_{\operatorname{pro\acute{e}t},c}(X,{\bf
Q}_{p}(2))\xrightarrow[\sim]{{\rm Tr}_{X}}{\bf Q}_{p}$

is a perfect duality, i.e., we have induced isomorphisms of solid ${\bf
Q}_{p}$-vector spaces

$\displaystyle\gamma_{X,i}:H^{i}_{\operatorname{pro\acute{e}t}}(X,{\bf
Q}_{p}(j))\stackrel{{\scriptstyle\sim}}{{\to}}H^{4-i}_{\operatorname{pro\acute{e}t},c}(X,{\bf
Q}_{p}(2-j))^{*},$
$\displaystyle\gamma_{X,i}^{c}:H^{i}_{\operatorname{pro\acute{e}t},c}(X,{\bf
Q}_{p}(j))\stackrel{{\scriptstyle\sim}}{{\to}}H^{4-i}_{\operatorname{pro\acute{e}t}}(X,{\bf
Q}_{p}(2-j))^{*},$

where $(-)^{*}$ denotes $\underline{{\rm{Hom}}}_{{\bf Q}_{p}}(-,{\bf Q}_{p})$.

######  Remark 1.3.

(i) If the curve $X$ is proper then the cohomology groups
$H^{i}_{\operatorname{pro\acute{e}t}}(X,{\bf Q}_{p}(j))$ are of finite rank
over ${\bf Q}_{p}$; this is also the case if $j\neq 1$ and if $X$ is affinoid
(or Stein with $H^{1}_{\rm dR}(X)$ finite dimensional).

(ii) If $X$ is Stein, $H^{i}_{\operatorname{pro\acute{e}t}}(X,{\bf Q}_{p}(j))$
is a nuclear333Nuclear in the classical sense not in the solid sense. Fréchet
and $H^{i}_{\operatorname{pro\acute{e}t},c}(X,{\bf Q}_{p}(j))$ is a space of
compact type; if $X$ is a dagger affinoid – it is the other way around.

######  Remark 1.4.

If $X$ is proper or Stein we have a more general derived duality: the duality
map

(1.5)
${\gamma_{{X}}}:{\mathrm{R}}\Gamma_{\operatorname{pro\acute{e}t}}({X},{\bf
Q}_{p}(j))\xrightarrow{}{\mathbb{D}}({\mathrm{R}}\Gamma_{\operatorname{pro\acute{e}t},c}({X},{\bf
Q}_{p}(2-j))[4]),$

where ${\mathbb{D}}(-)={\mathrm{R}}\underline{{\rm{Hom}}}_{{\bf Q}_{p}}(-,{\bf
Q}_{p})$, is a quasi-isomorphism in ${\mathscr{D}}({\bf Q}_{p,\Box})$. The
isomorphisms $\gamma_{X,i}$ are obtained from it because all the higher
$\operatorname{Ext}$ groups involved vanish since
$H^{i}_{\operatorname{pro\acute{e}t},c}(X,{\bf Q}_{p}(j))$ is of compact type.
The isomorphisms $\gamma^{c}_{X,i}$ are obtained from $\gamma_{X,i}$ by
dualizing using the fact that the spaces
$H^{i}_{\operatorname{pro\acute{e}t},c}(X,{\bf Q}_{p}(j))$ are (solid)
reflexive.

We venture to conjecture an arithmetic Poincaré duality in any dimension:

######  Conjecture 1.6.

Let $X$ be a smooth Stein dagger variety over $K$, geometrically irreducible,
of dimension $d$. Then:

  1. (1)

The cohomology groups $H^{i}_{\operatorname{pro\acute{e}t}}(X,{\bf Q}_{p}(j))$
and $H^{i}_{\operatorname{pro\acute{e}t},c}(X,{\bf Q}_{p}(j))$ are nuclear
Fréchet and of compact type, respectively.

  2. (2)

We have (quasi-)isomorphisms in ${\mathscr{D}}({\bf Q}_{p,\Box})$

$\displaystyle{\mathrm{R}}\Gamma_{\operatorname{pro\acute{e}t}}(X,{\bf
Q}_{p}(j))$
$\displaystyle\simeq{\mathbb{D}}({\mathrm{R}}\Gamma_{\operatorname{pro\acute{e}t},c}(X,{\bf
Q}_{p}(d+1-j))[2d+2]),$ $\displaystyle
H^{i}_{\operatorname{pro\acute{e}t}}(X,{\bf Q}_{p}(j))$ $\displaystyle\simeq
H^{2d+2-i}_{\operatorname{pro\acute{e}t},c}(X,{\bf Q}_{p}(d+1-j))^{*},$
$\displaystyle H^{i}_{\operatorname{pro\acute{e}t},c}(X,{\bf Q}_{p}(j))$
$\displaystyle\simeq H^{2d+2-i}_{\operatorname{pro\acute{e}t}}(X,{\bf
Q}_{p}(d+1-j))^{*}.$

######  Example 1.7.

The starting point of our study of dualities for $p$-adic pro-étale cohomology
of analytic spaces was the computation for $X=D$, the open unit disc. For
example, we computed that:

$\displaystyle H^{1}_{\operatorname{pro\acute{e}t}}(X,{\bf Q}_{p}(1))$
$\displaystyle\simeq({\mathscr{O}}(D)/K)\oplus H^{1}({\mathscr{G}}_{K},{\bf
Q}_{p}(1)),$ $\displaystyle H^{3}_{\operatorname{pro\acute{e}t},c}(X,{\bf
Q}_{p}(1))$ $\displaystyle\simeq({\mathscr{O}}(\partial
D)/{\mathscr{O}}(D))\oplus H^{1}({\mathscr{G}}_{K},{\bf Q}_{p}),$

where ${\mathscr{G}}_{K}=\operatorname{Gal}(\overline{K}/K)$ and $\partial
D:=\varprojlim_{n}D\mathbin{\fgebackslash}D_{n}$ is the “ghost circle”,
boundary of the unit disk (here, $D_{n}$ is the closed ball
$\\{v_{p}(z)\geq\frac{1}{n}\\}$). The splitting is not canonical but it is
compatible with products. This computation showed that there is a duality of
the form stated in Theorem 1.1 and, moreover, that it is induced by Galois
duality and coherent duality (i.e., Serre duality):

$\displaystyle H^{i}({\mathscr{G}}_{K},{\bf Q}_{p})$ $\displaystyle\simeq
H^{2-i}({\mathscr{G}}_{K},{\bf Q}_{p}(1))^{*},$ $\displaystyle
H^{0}(D,\Omega^{1}_{D})$ $\displaystyle\simeq
H^{1}_{c}(D,{\mathscr{O}}_{D})^{*}.$

We used here the isomorphisms (the first one is almost the definition of
$H^{1}_{c}(D,{\mathscr{O}}_{D})$, taking into account vanishing of
$H^{2}(D,{\mathscr{O}}_{D})$; the second is induced by $f\mapsto df$: that it
is an isomorphism is equivalent to the fact that $H^{1}_{\rm dR}(D)=0$):

(1.8)
$H^{1}_{c}(D,{\mathscr{O}}_{D})\stackrel{{\scriptstyle\sim}}{{\leftarrow}}{\mathscr{O}}(\partial
D)/{\mathscr{O}}(D),\quad
H^{0}(D,\Omega^{1}_{D})\stackrel{{\scriptstyle\sim}}{{\leftarrow}}{\mathscr{O}}(D)/K.$

Since $[K:{\bf Q}_{p}]<\infty$, the above coherent $K$-duality can be turned
into a ${\bf Q}_{p}$-duality by composing with ${\rm Tr}_{K/{\bf Q}_{p}}$.

######  Remark 1.9.

The above results for the open disk do not involve the solid formalism, and we
attempted at first to write this paper using classical functional analysis as
in [15]. We passed to solid formalism because of two reasons. One was a need
for a derived dual that would work with Mayer-Vietoris sequences and which
does not seem to exist in the classical world in the generality we wanted. The
second one was a construction of topological Hochschild-Serre spectral
sequences which we did not know how to do in the classical set-up.

###  1.2. The proof of Theorem 1.1

An analog of Theorem 1.1 holds for schemes and the proof uses Galois descent
from geometric, i.e., over $C:=\widehat{\overline{K}}$, Poincaré duality and
Galois duality (the Hochschild-Serre spectral sequence does not degenerate at
$E_{2}$ but it does degenerate at $E_{3}$). An analogous argument, starting
with geometric Poincaré duality due to Zavyalov [34], Gabber, and Mann [25],
yields Theorem 1.1 for proper rigid analytic varieties (in fact, in any
dimension).

We use here a similar strategy. In our setting we do not yet have the
geometric duality so we replace it with comparison isomorphisms. They allow us
to pass from geometric $p$-adic pro-étale cohomology to de Rham data and the
terms of the Hochschild-Serre spectral sequence can be identified with
coherent cohomology and Galois cohomology of ${\bf Q}_{p}$-vector spaces with
some finiteness properties. Our Poincaré duality is now deduced, via this
spectral sequence, from coherent and Galois dualities.

$\bullet$ Geometric comparison results. Recall that, for smooth Stein rigid
analytic varieties over $K$, we have comparison theorems between geometric
$p$-adic pro-étale cohomology and geometric syntomic cohomology. In the case
of usual cohomology they are due to Colmez-Dospinescu-Nizioł [15] and Colmez-
Nizioł [18]; in the case of compactly supported cohomology – to Achinger-
Gilles-Nizioł [1].

In the case where $X$ is a smooth, geometrically irreducible, Stein curve over
$K$, they yield:

  1. (1)

Vanishings:

(1.10) $\displaystyle H^{i}_{\operatorname{pro\acute{e}t}}(X_{C},{\bf
Q}_{p})=0,\quad\mbox{ for }i\neq 0,1,$ $\displaystyle
H^{i}_{\operatorname{pro\acute{e}t},c}(X_{C},{\bf Q}_{p})=0,\quad\mbox{ for
}i\neq 1,2.$

  2. (2)

Isomorphisms:

$H^{0}_{\operatorname{pro\acute{e}t}}(X_{C},{\bf Q}_{p})\simeq{\bf
Q}_{p},\quad H^{1}_{\operatorname{pro\acute{e}t},c}(X_{C},{\bf
Q}_{p}(1))\xrightarrow{\sim}{\rm HK}^{1}_{c}(X_{C},1),$

where444Here $\breve{F}$ is the completion of the maximal unramified extension
of the fraction field $F$ of the Witt vectors of the residue field of $K$ and
$\widehat{{\bf B}}^{+}_{\operatorname{st}}$ is the semistable period ring of
Fontaine in its Banach form. ${\rm HK}^{j}_{*}(X_{C},i):=(H^{j}_{{\rm
HK},*}(X_{C}){\otimes}^{\Box}_{\breve{F}}\widehat{{\bf
B}}^{+}_{\operatorname{st}})^{N=0,\varphi=p^{i}}$ and $H^{i}_{{\rm
HK},*}(X_{C})$, for $*=[\phantom{x}],c$, is the Hyodo-Kato cohomology.

  3. (3)

Exact sequences:

(1.11) $\displaystyle 0\to{\mathscr{O}}(X_{C})/C\to$ $\displaystyle
H_{\operatorname{pro\acute{e}t}}^{1}(X_{C},{\bf Q}_{p}(1))\to{\rm
HK}^{1}(X_{C},1)\to 0$ $\displaystyle{\rm HK}^{1}_{c}(X_{C},2)\to H^{1}{\rm
DR}_{c}(X_{C},2)\to$ $\displaystyle
H^{2}_{\operatorname{pro\acute{e}t},c}(X_{C},{\bf Q}_{p}(2))\to{\bf
Q}_{p}(1)\to 0,$

where we set

$\displaystyle{\rm
DR}_{c}(X_{C},i):=(H^{1}_{c}(X,{\mathscr{O}}_{X}){\otimes}^{\Box}_{K}({\bf
B}^{+}_{\operatorname{dR}}/F^{i})\to
H^{1}_{c}(X,\Omega^{1}_{X}){\otimes}^{\Box}_{K}({\bf
B}^{+}_{\operatorname{dR}}/F^{i-1}))[-1],$

The last sequence yields the definition of the geometric trace map

${\rm Tr}_{X_{C}}:H^{2}_{\operatorname{pro\acute{e}t},c}(X_{C},{\bf
Q}_{p}(1))\to{\bf Q}_{p}.$

The arithmetic trace map (1.2) is defined as the composition

${\rm Tr}_{X}:H^{4}_{\operatorname{pro\acute{e}t},c}(X,{\bf Q}_{p}(2))\simeq
H^{2}({\mathscr{G}}_{K},H^{2}_{\operatorname{pro\acute{e}t},c}(X,{\bf
Q}_{p}(2)))\smash{\mathop{\relbar\joinrel\longrightarrow}\limits^{{\rm
Tr}_{X_{C}}(1)}}H^{2}({\mathscr{G}}_{K},{\bf
Q}_{p}(1))\smash{\mathop{\relbar\joinrel\longrightarrow}\limits^{{\rm
Tr}_{K}}}{\bf Q}_{p}.$

$\bullet$ Galois descent. For $s\in{\bf Z}$, we have Hochschild-Serre spectral
sequences

(1.12)
$E^{i,j}_{2}=H^{i}({\mathscr{G}}_{K},H^{j}_{\operatorname{pro\acute{e}t},*}(X_{C},{\bf
Q}_{p}(s)))\Rightarrow H^{i+j}_{\operatorname{pro\acute{e}t},*}(X,{\bf
Q}_{p}(s)).$

The computations done above show that to understand the terms of the spectral
sequences (1.12) we need to understand Galois cohomology of

(1.13) $\displaystyle H^{0}(X,\Omega^{1}_{X})\otimes^{\Box}_{K}C(s),\quad
H^{1}_{c}(X,{\mathscr{O}}_{X}){\otimes}^{\Box}_{K}C(s),\quad{\rm
HK}^{1}_{c}(X_{C},1)(s).$

We used here the isomorphisms (1.8). We claim that, in the case $H^{1}_{{\rm
HK}}(X_{C})$ is of finite rank over $\breve{F}$, this cohomology is or trivial
or isomorphic to

(1.14) $H^{0}(X,\Omega^{1}_{X}),\quad H^{1}_{c}(X,{\mathscr{O}}_{X}),\quad
H^{i}({\mathscr{G}}_{K},V),$

where $V$ is a finite rank ${\bf Q}_{p}$-Galois representation. Indeed, note
that the last group in (1.13) is an almost $C$-representation [20] (a finite
rank $C$-vector space plus/minus a finite rank ${\bf Q}_{p}$-vector space).
Our claim then follows easily from a generalization of Tate’s computations:

$H^{i}({\mathscr{G}}_{K},C(j))\simeq\begin{cases}K&\mbox{ if }j=0,i=0,1;\\\
0&\mbox{ otherwise.}\end{cases}$

By the vanishing results (1.10), the spectral sequences (1.12) have only two
nontrivial columns, hence degenerate at $E_{2}$. Looking at the terms of the
spectral sequences in (1.14), we see that if we knew that the arithmetic pro-
étale product is compatible with the coherent and Galois products we would
have our duality (at least in the case the Hyodo-Kato cohomology is of finite
rank but we can always reduce to that case). However, we have found it very
difficult to check this compatibility for a general curve $X$ as above: the
main difficulty arises from the exact sequence (1.11) that does not behave
well with respect to products. We decided thus to pass from $X$ to simpler
curves.

Nevertheless, the above computations imply the following functional analytic
result:

######  Theorem 1.15.

Let $X$ be a smooth Stein dagger curve over $K$, geometrically irreducible.
Then the cohomology groups $H^{i}_{\operatorname{pro\acute{e}t}}(X,{\bf
Q}_{p}(j))$ and $H^{i}_{\operatorname{pro\acute{e}t},c}(X,{\bf Q}_{p}(j))$ are
nuclear Fréchet and of compact type, respectively.

This is because we know these properties for $H^{0}(X,\Omega^{1}_{X})$ and
$H^{1}_{c}(X,{\mathscr{O}}_{X})$, respectively, hence for the terms of the
Hochschild-Serre spectral sequences, and the extension problems arising from
the spectral sequences can be solved.

$\bullet$ Reductions. By a simple limit argument, to prove the derived
Poincaré duality (1.5), we can pass from a general Stein curve to a wide open
curve, i.e., to a complement of a finite number of closed discs in a proper
curve (we allow for a finite field extension here). For wide open curves we
use a Mayer-Vietoris argument to reduce to proper curves, open discs, and open
annuli. For proper curves we know the result, for open discs and annuli we
reduce the computation to the one of their boundaries.

$\bullet$ Arithmetic duality for ghost circle. We show that, in duality
theory, the boundary of an open disc, a ghost circle, behaves like a proper
rigid analytic variety over $K$ of dimension $1/2$.

######  Theorem 1.16.

(Arithmetic duality for ghost circle) Let $D$ be an open disc over $K$. Let
$Y:=\partial D$ be the boundary of $D$. Then:

  1. (1)

There is a natural trace map isomorphism of solid ${\bf Q}_{p}$-vector spaces

${\rm Tr}_{Y}:\,H^{3}_{\operatorname{pro\acute{e}t}}(Y,{\bf
Q}_{p}(2))\stackrel{{\scriptstyle\sim}}{{\to}}{\bf Q}_{p}.$

  2. (2)

The pairing

(1.17) $H^{i}_{\operatorname{pro\acute{e}t}}(Y,{\bf
Q}_{p}(j))\otimes^{\Box}_{{\bf
Q}_{p}}H^{3-i}_{\operatorname{pro\acute{e}t}}(Y,{\bf Q}_{p}(2-j))\to
H^{3}_{\operatorname{pro\acute{e}t}}(Y,{\bf
Q}_{p}(2))\stackrel{{\scriptstyle\sim}}{{\to}}{\bf Q}_{p}$

is a perfect duality, i.e., we have the induced isomorphism of solid ${\bf
Q}_{p}$-vector spaces

$\displaystyle\gamma_{Y}:\quad H^{i}_{\operatorname{pro\acute{e}t}}(Y,{\bf
Q}_{p}(j))\stackrel{{\scriptstyle\sim}}{{\to}}H^{3-i}_{\operatorname{pro\acute{e}t}}(Y,{\bf
Q}_{p}(2-j))^{*}.$

We will now sketch the proof of this theorem. We define an ascending
filtration on $H^{i}_{\operatorname{pro\acute{e}t}}(Y,{\bf Q}_{p}(j))$:

$F^{2}_{i,j}=H^{i}_{\operatorname{pro\acute{e}t}}(Y,{\bf Q}_{p}(j))\supset
F^{1}_{i,j}\supset F^{0}_{i,j}\supset F^{-1}_{i,j}=0,$

such that

$\displaystyle F^{2}_{i,j}/F^{1}_{i,j}\simeq H^{i-1}({\mathscr{G}}_{K},{\bf
Q}_{p}(j-1)),$ $\displaystyle F^{1}_{i,j}/F^{0}_{i,j}\simeq
H^{i-1}({\mathscr{G}}_{K},{\mathscr{O}}(Y_{C})/C(j-1)),$ $\displaystyle
F^{0}_{i,j}\simeq H^{i}({\mathscr{G}}_{K},{\bf Q}_{p}(j)).$

It is helpful to visualize this filtration in the following way (with
$H^{i}(-{\bf Q}_{p}(j)):=H^{i}_{\operatorname{pro\acute{e}t}}(-{\bf
Q}_{p}(j))$):

---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{0}_{i,j}:=H^{i}({\mathscr{G}}_{K},{\bf
Q}_{p}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{1}_{i,j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{i-1}({\mathscr{G}}_{K},\frac{{\mathscr{O}}(Y_{C})}{C}(j-1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{i}({\mathscr{G}}_{K},{\bf
Q}_{p}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{2}_{i,j}:=H^{i}(Y,{\bf
Q}_{p}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{i-1}({\mathscr{G}}_{K},H^{1}(Y_{C},{\bf
Q}_{p}(j)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{H^{i-1}({\mathscr{G}}_{K},{\bf
Q}_{p}(j-1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{i-1}({\mathscr{G}}_{K},{\bf
Q}_{p}(j-1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$

The middle exact row comes from the filtration induced by the Hochschild-Serre
spectral sequence; the right exact column is induced by the syntomic
filtration from the analog555Take this sequence for an annuli and go to the
limit towards the boundary. of the exact sequence (1.11). The term
$F^{1}_{i,j}$ is defined as a pullback of the top right square.

Now, the key computation is the following:

######  Theorem 1.18.

(Explicit Reciprocity Law) We have:

  1. (1)

The pairing (1.17) is compatible with the above filtration.

  2. (2)

On the associated grading the pairing (1.17) yields a pairing induced by the
Galois cohomology pairing and coherent pairing.

The proof of this Theorem interpolates between syntomic and
$(\varphi,\Gamma)$-techniques.

This finishes the proof of Theorem 1.16 and hence of Theorem 1.1, the main
theorem of this paper.

###  1.3. Conjectural geometric duality

We finish the introduction with a discussion of what we think happens over
$C$. Let us start with the example that guided our study.

######  Example 1.19.

Let $D$ be the open unit disc over $C$. The nontrivial cohomology groups are:

$\displaystyle H^{0}_{\operatorname{pro\acute{e}t}}(D,{\bf
Q}_{p}(j))\simeq{\bf Q}_{p}(j),\quad$ $\displaystyle
H^{1}_{\operatorname{pro\acute{e}t}}(D,{\bf
Q}_{p}(j))\simeq({\mathscr{O}}(D)/C)(j-1),$ $\displaystyle
H^{2}_{\operatorname{pro\acute{e}t},c}(D,{\bf Q}_{p}(j))$
$\displaystyle\simeq{\bf Q}_{p}(j-1)\oplus({\mathscr{O}}(\partial
D)/{\mathscr{O}}(D))(j-1).$

It looks like we have the right groups for a duality (for appropriate choices
of $j$): ${\mathscr{O}}(D)/C$ and ${\mathscr{O}}(\partial D)/{\mathscr{O}}(D)$
are $C$-vector spaces in duality via coherent duality and ${\bf Q}_{p}(j)$ and
${\bf Q}_{p}(j-1)$ can be put in duality. But the degrees are wrong for a
Poincaré-type duality to work: coherent duality adds up to degree $3$ but
${\bf Q}_{p}$-vector space duality adds up to degree $2$. Also, since $[C:{\bf
Q}_{p}]=\infty$, it is impossible to turn a $C$-duality into a ${\bf
Q}_{p}$-duality.

To find a set-up in which a duality could be restored, we turned to the
category of Vector Spaces, i.e., $v$-sheaves of topological ${\bf
Q}_{p}$-vector spaces on the category ${\rm Perf}_{C}$ of perfectoid spaces
over $C$ (that it is not unreasonnable to do so is due to the fact that there
is a natural way [17] to turn pro-étale geometric cohomology groups into
VS’s). There we have the following computation [11, Prop. 10.16] of
$\operatorname{Ext}$-groups:

$\displaystyle{\rm{Hom}}_{\rm VS}({\bf Q}_{p},{\bf Q}_{p}(1))\simeq{\bf
Q}_{p}(1),\quad\operatorname{Ext}^{1}_{\rm VS}({\bf Q}_{p},{\bf Q}_{p}(1))=0,$
$\displaystyle{\rm{Hom}}_{\rm VS}({\mathbb{G}}_{a},{\bf
Q}_{p}(1))=0,\quad\operatorname{Ext}^{1}_{\rm VS}({\mathbb{G}}_{a},{\bf
Q}_{p}(1))\simeq C.$

Moreover, $\operatorname{Ext}^{i}$, for $i\geq 2$, vanish by [2, Th. 3.8]. The
nontrivial $\operatorname{Ext}$-group in the second row is generated by the
fundamental exact sequence of Banach-Colmez spaces

$0\to{\bf
Q}_{p}(1)\to{\mathbb{B}}_{\operatorname{cr}}^{+,\varphi=p}\to{\mathbb{G}}_{a}\to
0$

The above computations, ignoring functional analytic questions, yield a
Verdier duality isomorphisms (with $H^{i}_{*}(-{\bf
Q}_{p}(j)):=H^{i}_{{\operatorname{pro\acute{e}t}},*}(-{\bf Q}_{p}(j))$):

$\displaystyle\operatorname{Ext}^{1}_{\rm VS}(H^{2}_{c}(D,{\bf
Q}_{p}(2-j)),{\bf Q}_{p}(1))\stackrel{{\scriptstyle\sim}}{{\to}}$
$\displaystyle H^{1}(D,{\bf Q}_{p}(j)),$ $\displaystyle
0\to\operatorname{Ext}^{1}_{\rm VS}(H^{1}(D,{\bf Q}_{p}(2-j)),{\bf
Q}_{p}(1))\to$ $\displaystyle H^{2}_{c}(D,{\bf Q}_{p}(j))\to{\rm{Hom}}_{\rm
VS}(H^{0}(D,{\bf Q}_{p}(2-j)),{\bf Q}_{p}(1))\to 0$

Guided by this, we venture a conjectural statement of geometric duality for
$p$-adic pro-étale cohomology in any dimension (again, ignoring topology):

######  Conjecture 1.20.

(Geometric Verdier duality) Let $X$ be a smooth Stein rigid analytic variety
over $C$, connected, of dimension $d$. There is a natural quasi-isomorphism

${\mathrm{R}}\Gamma_{\operatorname{pro\acute{e}t}}(X,{\bf
Q}_{p}(j))\simeq{\mathrm{R}}{\rm{Hom}}_{\rm
VS}({\mathrm{R}}\Gamma_{\operatorname{pro\acute{e}t},c}(X,{\bf
Q}_{p}(d+1-j))[2d],{\bf Q}_{p}(1)).$

The interested reader can find more details in [14].

######  Acknowledgments.

We would like to thank Guido Bosco, Juan Esteban Rodriguez Camargo, Matthew
Emerton, Akhil Mathew, and Peter Scholze for help concerning solid
mathematics. Special thanks go to Gabriel Dospinescu for many helpful
discussions involving functional analysis in general. We also thank Guido
Bosco and Zhenghui Li for helpful comments on an earlier version of this
paper.

This material is based upon work supported by the National Science Foundation
under Grant No. 1440140, while the first and the third authors were in
residence at the Mathematical Sciences Research Institute in Berkeley,
California, during the Spring semester of 2023. Parts of this paper were
written during the first and third author’s stay at the Hausdorff Research
Institute for Mathematics in Bonn, in the Summer of 2023. We would like to
thank the Institute for support and hospitality.

The second author was partially supported by the ERC under the European
Union’s Horizon 2020 research and innovation programme (grant agreement No.
804176) and through the fellowship from the MPIM in Bonn.

#### Notation and conventions.

Let $K$ be a finite extension of ${\bf Q}_{p}$, with ring of integers
${\mathscr{O}}_{K}$ and residue field $k$. Let $\overline{K}$ be an algebraic
closure of $K$ and let ${\mathscr{O}}_{\overline{K}}$ denote the integral
closure of ${\mathscr{O}}_{K}$ in $\overline{K}$. Set
${\mathscr{G}}_{K}=\operatorname{Gal}(\overline{K}/K)$. Let
$C=\widehat{\overline{K}}$ be the $p$-adic completion of $\overline{K}$. Let
$W(k)$ be the ring of Witt vectors of $k$ with fraction field $F$ (i.e.,
$W(k)={\mathscr{O}}_{F}$); let $e=e_{K}$ be the ramification index of $K$ over
$F$. Let $\breve{F}:=W(\overline{k})[\frac{1}{p}]$ denote the completion of
the maximal unramified extension of $F$ and let $\varphi$ be the absolute
Frobenius on $W(\overline{k})$.

We will denote by ${\bf A}_{\rm inf},{\bf A}_{\operatorname{cr}},{\bf
B}_{\operatorname{cr}},\widehat{{\bf B}}_{\operatorname{st}},{\bf
B}_{\operatorname{dR}}$ the Witt, crystalline, semistable, and de Rham period
rings of Fontaine, respectively.

All rigid analytic spaces and dagger spaces considered will be over $K$ or
$C$. We assume that they are separated, taut, and countable at infinity.

Since the only cohomology with ${\bf Q}_{p}(r)$-coefficients that we consider
is the pro-étale cohomology, we will remove “${\operatorname{pro\acute{e}t}}$”
from the notations, i.e. write:

$H^{i}(-,{\bf Q}_{p}(r)):=H^{i}_{\operatorname{pro\acute{e}t}}(-,{\bf
Q}_{p}(r)),\quad H^{i}_{c}(-,{\bf
Q}_{p}(r)):=H^{i}_{{\operatorname{pro\acute{e}t}},c}(-,{\bf Q}_{p}(r)).$

##  2\. Functional analysis

###  2.1. Classical functional analysis

We gather here some basic facts from classical $p$-adic functional analysis
that we use in the paper.

####  2.1.1. Locally convex vector spaces

Our cohomology groups will be equipped with a canonical topology. To talk
about it in a systematic way, we will work in the category $C_{K}$ of locally
convex $K$-vector spaces. For details the reader may consult [15, Sec. 2.1,
2.3]. To summarize quickly: $C_{K}$ is a quasi-abelian category. We will
denote the category of left-bounded complexes of $C_{K}$ by $C(C_{K})$ and the
associated derived $\infty$-category by ${\mathscr{D}}(C_{K})$. A morphism of
complexes that is a quasi-isomorphism in ${\mathscr{D}}(C_{K})$, i.e., its
mapping cofiber is strictly exact, will be called a strict quasi-isomorphism.
The associated cohomology objects are denoted by666$LH$ stands for “left
heart”. $\widetilde{H}^{n}(E)\in{LH}(C_{K})$:

(2.1) $\widetilde{H}^{n}(E):=\tau_{\leq n}\tau_{\geq n}(E)=({\rm
coim}(d_{n-1})\to\ker(d_{n})),\quad E\in C(C_{K});$

they are called classical if the canonical map
$\widetilde{H}^{n}(E)\to{H}^{n}(E)$ is an isomorphism777In our situations this
is usually equivalent to $H^{n}(E)$ being separated.. Classical objects are
closed under extensions in ${\rm LH}(C_{K})$.

####  2.1.2. Hausdorff locally convex vector space

Our main reference here is [28, Sec. 3.1]. We will denote by $C^{H}_{K}$ the
category of Hausdorff locally convex $K$-vector spaces. It is stable under
direct sums and direct products and if $V\subset W$ are two locally convex
$K$-vector spaces then $W/V$ is Hausdorff if and only if $V$ is closed in $W$.
The category $C^{H}_{K}$ is quasi-abelian: kernels and coimages are defined as
in $C_{K}$; cokernels and images are defined using closures of images in
$C_{K}$. A sequence in $C^{H}_{K}$ is strictly exact if and only if it is
strictly exact in $C_{K}$.

####  2.1.3. Duality

If $V,W\in C_{K}$, we denote by ${\mathscr{L}}(V,W)$ the space of continuous
linear maps from $V$ to $W$. We write ${\mathscr{L}}_{s}(V,W)$ and
${\mathscr{L}}_{b}(V,W)$ for the vector space ${\mathscr{L}}(V,W)$ equipped
with the weak and strong topologies (i.e., the topology of pointwise
convergence and the topology of bounded convergence), respectively. For $V\in
C_{K}$, we set $V^{\prime}_{s}:={\mathscr{L}}_{s}(V,K)$ and
$V^{*}:=V^{\prime}_{b}:={\mathscr{L}}_{b}(V,K)$. Both of these dual spaces are
Hausdorff. If $V$ is a Hausdorff space it is called reflexive if the duality
map $V\to(V^{\prime}_{b})^{\prime}_{b}$ is a topological isomorphism.

We will also use stereotype dual $V^{\star}$ of $V\in C_{K}$. It is defined as
${\mathscr{L}}(V,K)$ equipped with the topology of compactoid
convergence888See Section 2.1.5 for a definition of compactoids. of $V$. We
have continuous maps

$V^{\prime}_{b}\to V^{\star}\to V^{\prime}_{s}.$

If $V$ is a Banach space then they are not topological isomorphisms unless $V$
has finite dimension. For any Banach space, its stereotype dual is a Smith
space999A Smith space is a complete compactly generated locally convex vector
space $V$ having a universal compact set, i.e., a compact set $K$, which
absorbs every other compact set $T\subseteq V$ (i.e., $T\subseteq\lambda\cdot
K$, for some $\lambda>0$). , and vice versa, for any Smith space, its
stereotype dual is a Banach space, and $(V^{\star})^{\star}=V$ if $V$ is
Banach or Smith. If we write a Banach space $V\in C_{K}$ as
$V\simeq(\widehat{\oplus_{I}{\mathscr{O}}_{K}})[1/p]$ then its stereotype dual
is the Smith space $V^{\star}\simeq(\prod_{I}{\mathscr{O}}_{K})[1/p]$. A
Banach space is a Smith space if and only if it is finite dimensional.

####  2.1.4. Hausdorff compactly generated locally convex vector spaces

Recall that a topological space $T$ is compactly generated if a map $f:T\to
T^{\prime}$ to another topological space is continuous as soon as the
composite $S\to T\to T^{\prime}$ is continuous for all compact Hausdorff
spaces $S$ mapping to $T$. The inclusion of compactly generated spaces into
all topological spaces admits a right adjoint $T\mapsto T^{\rm cg}$ that sends
a topological space $T$ to its underlying set equipped with the quotient
topology for the map $\coprod_{S\to T}S\to T$, where the disjoint union runs
over all compact Hausdorff spaces $S$ (alternatively, profinite sets $S$)
mapping to $T$.

Any first-countable space (in particular, any metrizable topological space) is
compactly generated (see [8, Remark 1.6] for a proof). So, for example,
Fréchet spaces are compactly generated. The category of compactly generated
spaces is closed under taking coproducts, closed subspaces and quotients by
closed subspaces. Hence a colimit of Fréchet spaces is compactly generated if
it is Hausdorff; this applies, in particular, to locally convex vector spaces
of compact type101010See Section 2.1.5 below for the definition of spaces of
compact type. since they are Hausdorff and can be written as a countable
colimit of Banach spaces.

The category $C^{\rm Hcg}_{K}$ of Hausdorff compactly generated locally convex
vector spaces over $K$ is quasi-abelian: kernels and coimages are defined as
in $C_{K}$; cokernels and images are defined using closures of images in
$C_{K}$. A sequence in $C^{\rm Hcg}_{K}$ is strictly exact if and only if it
is strictly exact in $C_{K}$.

####  2.1.5. Spaces of compact type and nuclear spaces

Our references for this section are [30, IV.19], [31, Ch. 1], [27, Ch. 8].

######  Definition 2.2.

Let $V,W\in C_{K}$ be Hausdorff.

  1. (1)

A subset $B\subset V$ is called compactoid if for any open lattice $L\subset
V$ there are finitely many vectors $v_{1},\ldots,v_{m}\in V$ such that
$B\subset L+{\mathscr{O}}_{K}v_{1}+\cdots+{\mathscr{O}}_{K}v_{m}$. Compactoids
are preserved by maps in $C_{K}$.

  2. (2)

A continuous linear map $f:V\to W$ is called compact if there is an open
lattice $L$ in $V$ such that the closure of $f(L)$ in $W$ is compactoid and
complete. If $W$ is quasi-complete (in particular, if it is complete) then
this is equivalent to $f(L)$ being compactoid.

  3. (3)

$V$ is called of compact type if it is the inductive limit of a sequence

$V_{1}\stackrel{{\scriptstyle\iota_{1}}}{{\to}}V_{2}\stackrel{{\scriptstyle\iota_{2}}}{{\to}}V_{3}\to\cdots$

of Hausdorff spaces $V_{n}\in C_{K}$, for $n\in{\bf N}$, with injective
compact linear maps. By the proof of [30, Prop. 16.10], we may assume the
$V_{n}$’s to be Banach spaces.

We will use often the following facts (see [30, Rem. 16.7]).

######  Lemma 2.3.

Let $g:V\to W$ be a compact map. Then

  1. (1)

If $h:V_{1}\to V$ and $f:W\to W_{1}$ are arbitrary continuous linear maps then
the map $fgh:V_{1}\to W_{1}$ is compact.

  2. (2)

If the image of $g$ is contained in a closed subpace $W_{0}\subset W$ then the
induced map $g:V\to W_{0}$ is compact.

We will denote by $C_{c,K}$ the full subcategory of $C_{K}$ consisting of
spaces of compact type. Spaces of compact type are Hausdorff, complete, and
reflexive. Their strong duals are Fréchet and satisfy
$V^{\prime}_{b}=\mathop{\vtop{\halign{#\cr$\hfill{\lim}\hfil$\crcr\kern
1.0pt\nointerlineskip\cr\leftarrowfill\crcr\kern-3.0pt\cr}}}\limits_{n}(V^{\prime}_{n})_{b}$.
Moreover, a closed subspace of a space of compact type is also of compact type
and so is the relevant quotient; $C_{c,K}$ is closed under countable direct
sums.

######  Definition 2.4.

The space $V\in C_{K}$ is called nuclear if for any open lattice $L\subset V$
there exists another open lattice $M\subset L$ such that the canonical map
between completions $\widehat{V}_{M}\to\widehat{V}_{L}$ is compact (i.e., the
image of $M\to L/p^{L}$ is, for any $n$, contained in a module of finite type
over ${\mathscr{O}}_{K}$).

Nuclear Banach spaces are finite dimensional. Nuclear Fréchet spaces are
reflexive. Moreover, a subspace of a nuclear space is nuclear and if this
subspace is closed the relevant quotient is nuclear. A countable inductive
limit of nuclear spaces is nuclear (see [27, Th. 8.5.7]). By loc. cit.,
projective limits of nuclear spaces are nuclear. Any compact projective or
inductive limit111111In the sense of [30, Cor. 16.6, Prop. 16.10] . of locally
convex $K$-vector spaces is nuclear. A Fréchet space is the strong dual of a
space of compact type if and only if it is nuclear.

The following lemma will be essential for us.

######  Lemma 2.5.

  1. (1)

Every strict exact sequence of spaces from $C_{K}$

$0\to V\to W\to W^{\prime}\to 0$

such that $V$ is of finite rank and $W^{\prime}$ is Hausdorff splits (and
$W\simeq V\oplus W^{\prime}$).

  2. (2)

In a strict extension of spaces from $C_{K}$

(2.6)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where $V$ is of finite rank over $K$ and $W^{\prime}$ is nuclear Fréchet, $W$
is also nuclear Fréchet. Same holds for spaces of compact type.

###### Proof.

In the first claim, we may assume that $V$ is of rank $1$. Note that $W$ is
Hausdorff. Take a continuous linear form $\lambda:W\to K$, not identically $0$
on $V$ (such a form exists by [30, Cor. 9.3]). It suffices to show that the
canonical map $V\oplus{\rm Ker}\,\lambda\to W$ is a topological isomorphism.

It is an algebraic isomorphism and it is continuous. Hence we just need to
check that it is open. So, let $U_{\lambda}$ be an open lattice in ${\rm
Ker}\,\lambda$ and let $V_{0}$ be an open lattice in $V$. We need to show that
$U_{\lambda}+V_{0}$ is open in $W$ and it is enough to do it for open sub-
lattices of $U_{\lambda}$ and $V_{0}$.

Any neighborhood of $0$ in ${\rm Ker}\,\lambda$ contains an open of the form
$U\cap({\rm Ker}\,\lambda)$, where $U$ is an open lattice in $W$. Hence we can
assume $U_{\lambda}$ to be of this form. By construction, we have an exact
sequence $0\to U_{\lambda}\to U\to\lambda(U)\to 0$.

Now, $U$ contains $p^{N}V_{0}$ for $N$ big enough, since it is a lattice. Then
$U_{\lambda}+p^{N}V_{0}$ is the inverse image of $p^{N}\lambda(V_{0})$ in $U$
by $\lambda$, hence it is open as $\lambda$ is continuous and
$p^{N}\lambda(V_{0})$ is open in $K$. We proved what we wanted, up to
replacing $V_{0}$ by its sub-lattice $p^{N}V_{0}$.

The second claim follows immediately from the first one. ∎

######  Lemma 2.7.

If we have a map of strict exact sequences of complete Hausdorff spaces from
$C_{K}$

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{V}}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{W}}$$\textstyle{W^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{W^{\prime}}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{1}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

such that $V,V_{1}$ are of finite rank over $K$, the bottom sequence splits,
and the map $f_{W^{\prime}}$ is compact then the map $f_{W}$ is compact as
well.

###### Proof.

Take an open lattice $L_{W^{\prime}}$ in $W^{\prime}$ such that
$f_{W^{\prime}}(L_{W^{\prime}})$ is compactoid. Let $L_{W}$ be the preimage of
$L_{W^{\prime}}$ in $W$. Choose a section $s:W_{1}^{\prime}\to W_{1}$ of the
projection $\pi:W_{1}\to W_{1}^{\prime}$ and consider the continuous map

$g:W\to V_{1},\quad g(w):=f_{W}(w)-s\pi(f_{W}(w)).$

Take a compact lattice $L_{V_{1}}$ in $V_{1}$, its preimage (via $g$) in $W$,
and then change $L_{W}$ to its intersection with that preimage.

Now, we see that $f_{W}(L_{W})$ is compactoid: we have

$f_{W}(L_{W})\subset g(L_{W})+sf_{W^{\prime}}(L_{W^{\prime}})$

and both $g(L_{W})$ and $sf_{W^{\prime}}(L_{W^{\prime}})$ are compactoid in
$W_{1}$. This concludes the proof of our lemma. ∎

Let $C_{nF,K}$ denote the full subcategory of $C_{K}$ consisting of nuclear
Fréchet spaces. The functor

$C_{c,K}\to C_{nF,K},\quad V\mapsto V^{\prime}_{b},$

is an anti-equivalence of categories. For any two $V,W\in C_{c,K}$ the natural
linear map
${\mathscr{L}}_{b}(V,W)\to{\mathscr{L}}_{b}(W^{\prime}_{b},V^{\prime}_{b})$ is
a topological isomorphism.

######  Lemma 2.8.

If $X\in C_{K}$ is nuclear Fréchet or of compact type then the canonical map
$X^{\prime}_{b}\to X^{\star}$ is a topological isomorphism.

###### Proof.

In both cases the space $X$ is reflexive, hence Montel (see [27, Cor.
8.4.22]). But in Montel spaces, by [27, Th. 8.4.5], every bounded subset is
compactoid (and vice versa, of course) thus the strong topology on the
algebraic dual of $X$ coincides with the topology of compactoid convergence,
as wanted. ∎

###  2.2. Solid functional analysis

We will review here briefly results from solid functional analysis that we
will need. Our main references are [5], [29], [8], [9].

####  2.2.1. Basic properties of condensed sets

Let Cond denote the category of condensed sets, i.e., sheaves of sets on the
site of pro-finite sets with coverings given by finite families of jointly
surjective maps121212We refer the reader to [8, Lecture III] for a discussion
of set theoretical issues involved in this definition. or, equivalently, on
the pro-étale site $*_{\operatorname{pro\acute{e}t}}$ of a geometric point. We
define similarly condensed groups, rings, etc.

We will denote by ${\rm CondAb}$ the category of condensed abelian groups. It
is an abelian category [8, Thm. 2.2]. It has all limits and colimits.
Arbitrary products, arbitrary direct sums and filtered colimits are exact. It
is generated by compact projective objects. For a condensed commutative ring
$A$, we will write ${\rm Mod}^{\rm cond}_{A}$ for the category of $A$-modules
in ${\rm CondAb}$ and $\underline{{\rm{Hom}}}_{A}(-,-)$ for its internal
${\rm{Hom}}$ (in the case $A={\bf Z}$, we will often omit the subscript ${\bf
Z}$).

######  Remark 2.9.

Because of set theoretical issues this definition of the category ${\rm Cond}$
is not sensu stricto correct. The correct definition of ${\rm Cond}$ is given
in [8, Lecture II]. In this paper we will use the latter though we find it
helpful to keep in mind its simplified version given above.

For a condensed set $X$, we think of $X(*)$ as the underlying set of $X$ and
about $X(S)$ as the continuous maps from $S$ to $X$. A quasi-separated131313A
condensed set $X$ is called quasi-compact if there is a profinite set $S$ and
a surjective map ${S}\to X$; it is called quasi-separated if, for any pair of
profinite sets $S$ and $S^{\prime}$ over $X$ the fiber product
${S}\times_{X}{S}^{\prime}$ is quasi-compact. condensed set $X$ is trivial as
soon as $X(*)$ is trivial (a fact that is false for a general $X$, see [9,
Lecture I]).

Let ${\rm Top}$ denote the category of T1 topological spaces141414A
topological space is called T1 if all its points are closed.. We have a
functor

$(\underline{\phantom{x}}):{\rm Top}\to{\rm Cond},\quad
T\mapsto\underline{T}={\mathscr{C}}(S,T),$

where ${\mathscr{C}}(S,T)$ denotes the set of continuous functions from $S$ to
$T$. Condensed sets that come from topological spaces we will call classical.
We quote:

######  Proposition 2.10.

(Clausen-Scholze, [9, Prop. 1.2]). The functor $(\underline{\phantom{x}})$:

  1. (1)

has a left adjoint $X\mapsto X(*)_{\rm top}$ sending any condensed set $X$ to
the set $X(*)$ equipped with the quotient topology arising from the map

$\coprod\nolimits_{S,a\in X(S)}S\overset{a}{\to}X(*).$

  2. (2)

restricted to compactly generated topological spaces is fully faithful.

  3. (3)

induces an equivalence between the category of compact Hausdorff spaces and
qcqs (i.e., quasi-compact quasi-separated) condensed sets.

  4. (4)

induces a fully faithful functor from the category of compactly generated weak
Hausdorff151515A topological space is called weak Hausdorff if the image of
every continuous map from a compact Hausdorff space into the space is closed.
In particular, every Hausdorff space is weak Hausdorff. Every weak Hausdorff
space is a T1 space. spaces, to quasi-separated condensed sets. The category
of quasi-separated condensed sets is equivalent to the category of ind-compact
Hausdorff spaces “$\operatorname{colim}_{n}$”$X_{n}$, where all transition
maps are closed immersions. If $\\{X_{n}\\}_{n\in{\bf N}}$ is a ind-system of
compact Hausdorff spaces with closed immersions and
$X=\operatorname{colim}_{n}X_{n}$ as topological spaces, then the canonical
map

$\operatorname{colim}_{n}\underline{X_{n}}\to\underline{X}$

is an isomorphism of condensed sets. In particular,
$\operatorname{colim}_{n}\underline{X_{n}}$ is classical, i.e., it comes from
a topological space.

######  Remark 2.11.

(1) For $T\in{\rm Top}$, the counit $\underline{T}(*)_{\rm top}\to T$ of the
adjunction agrees with the counit $T^{\rm cg}\to T$ of the adjunction between
compactly generated spaces and all topological spaces by [8, Prop. 1.7]. In
particular, $\underline{T}(*)_{\rm top}\simeq T^{\rm cg}$.

(2) For a finite extension $K$ of ${\bf Q}_{p}$, we will abbreviate the
notation ${\rm Mod}^{\rm cond}_{\underline{K}}$ to ${\rm Mod}^{\rm cond}_{K}$;
we will call the elements of this category condensed $K$-vector spaces.

####  2.2.2. Solid modules

We will briefly review basic facts concerning solid modules.

($\bullet$) Analytic rings. We start with analytic rings.

(i) (Clausen-Scholze, [8, Th. 5.8]) The analytic ring ${\bf Z}_{\Box}=({\bf
Z},{\mathscr{M}}_{{\bf Z}})$ is defined as the ring ${\bf Z}$ equipped with
the functor of measures ${\mathscr{M}}_{{\bf Z}}$ sending an extremally
disconnected set $S=\lim_{i}S_{i}$, where each $S_{i}$ is a finite set, to the
condensed abelian group ${\bf Z}_{\Box}:=\lim_{i}{\bf Z}[S_{i}]$.

(ii) (Clausen-Scholze, [8, Prop. 7.9]) Let $K$ be a finite extension of ${\bf
Q}_{p}$. There is an analytic structure on the condensed rings
${\mathscr{O}}_{K}$ and $K$ given by sending an extremally disconnected set
$S=\lim_{i}S_{i}$, where each $S_{i}$ is finite, to

${\mathscr{O}}_{K,\Box}[S]:=\lim_{i}{\mathscr{O}}_{K}[S_{i}],\quad
K_{\Box}[S]:=K\otimes_{{\mathscr{O}}_{K}}{\mathscr{O}}_{K,\Box}[S].$

The first analytic ring structure is induced from the analytic ring structure
of ${\bf Z}_{\Box}$ by base change to ${\mathscr{O}}_{K}$.

($\bullet$) Solid modules. Now we pass to solid modules.

######  Proposition 2.12.

(Clausen-Scholze [8, Prop. 7.5]) Let $A=(A,{\mathscr{M}}_{A})$ be one of the
analytic rings above.

  1. (1)

The full subcategory of solid $A$-modules

(2.13) ${\rm Mod}^{\rm solid}_{A}\subset{\rm Mod}^{\rm cond}_{A}$

consists of all $A$-modules $M$ such that, for all extremally disconnected
sets $S$, the maps

${\rm{Hom}}_{A}({\mathscr{M}}_{A}[S],M)\to{\rm{Hom}}_{A}(A[S],M)$

are isomorphisms. It is an abelian subcategory, stable under all limits,
colimits, and extensions. The inclusion (2.13) admits a left adjoint

(2.14) ${\rm Mod}^{\rm cond}_{A}\to{\rm Mod}^{\rm solid}_{A}:\quad M\mapsto
M\otimes_{A}(A,{\mathscr{M}}_{A}),$

which preserves all colimits and is symmetric monoidal.

  2. (2)

The functor

(2.15) ${\mathscr{D}}({\rm Mod}^{\rm solid}_{A})\to{\mathscr{D}}({\rm
Mod}^{\rm cond}_{A})$

is fully faithful. Its essential image is stable under all limits and
colimits. It is given by complexes $M\in{\mathscr{D}}({\rm Mod}^{\rm
cond}_{A})$ such that the map

${\mathrm{R}}{\rm{Hom}}_{A}({\mathscr{M}}_{A}[S],M)\to{\mathrm{R}}{\rm{Hom}}_{A}(A[S],M)$

is a quasi-isomorphism for all extremally disconnected sets $S$.

A complex $M\in{\mathscr{D}}({\rm Mod}^{\rm cond}_{A})$ is in
${\mathscr{D}}({\rm Mod}^{\rm solid}_{A})$ if and only if $H^{i}(M)$ is in
${\rm Mod}^{\rm solid}_{A}$, for all $i$. The functor (2.15) admits a left
adjoint

(2.16) ${\mathscr{D}}({\rm Mod}^{\rm cond}_{A})\to{\mathscr{D}}({\rm Mod}^{\rm
solid}_{A}):M\mapsto M\otimes^{\rm L}_{A}(A,{\mathscr{M}}_{A}),$

which is the left derived functor of (2.14). It is symmetric monoidal.

  3. (3)

For $M,N\in{\mathscr{D}}({\rm Mod}^{\rm solid}_{A})$, we have the derived
internal Hom

${\mathrm{R}}\underline{{\rm{Hom}}}_{A}(M,N)\in{\mathscr{D}}({\rm Mod}^{\rm
solid}_{A}).$

The natural map
${\mathrm{R}}\underline{{\rm{Hom}}}_{(A,{\mathscr{M}}_{A})}(M,N)\to{\mathrm{R}}\underline{{\rm{Hom}}}_{A}(M,N)$
is a quasi-isomorphism.

######  Notation 2.17.

(1) We write ${\rm Solid}:={\rm Mod}^{\rm solid}_{({\bf Z},{\mathscr{M}}_{\bf
Z})}$ for the category of solid abelian groups; we write ${\rm CondAb}\to{\rm
Solid}:M\mapsto M^{\Box}$ for the functor (5.22) of Proposition 2.12, and call
it solidification, and we denote by $\otimes^{\Box}_{\bf Z}$ the unique
symmetric monoidal tensor product making the solidification functor symmetric
monoidal.

(2) For a finite extension $K$ of ${\bf Q}_{p}$, we write ${\rm Mod}^{\rm
solid}_{{\mathscr{O}}_{K}}$ and ${\rm Mod}^{\rm solid}_{K}$ for the categories
of solid ${\mathscr{O}}_{K}$-modules and $K$-vector spaces, respectively. We
will denote by ${\mathscr{D}}({\mathscr{O}}_{K,\Box})$ and
${\mathscr{D}}(K_{\Box})$ the corresponding derived $\infty$-categories.

(3) For a finite extension $K$ of ${\bf Q}_{p}$ and a commutative solid
$K$-algebra $A$, we write $\otimes^{\Box}_{A}$ for the symmetric monoidal
tensor product $\otimes_{(A,{\mathscr{M}}_{A})}$.

####  2.2.3. Locally convex and condensed vector spaces

Consider the functor

${\rm CD}:=(\underline{\phantom{x}}):\quad C^{\rm Hcg}_{K}\to{\rm Mod}^{\rm
cond}_{{K}},\quad V\mapsto\underline{V}.$

We will denote in the same way its extension to the category of complexes. By
Lemma 2.18 below, the functor ${\rm CD}$ preserves (strict) quasi-isomorphisms
of complexes of Fréchet spaces or spaces of compact type and if $V$ is such a
complex then

${\rm CD}(\widetilde{H}^{i}(V))\simeq H^{i}({\rm CD}(V)).$

######  Lemma 2.18.

The functor $(\underline{\phantom{x}})$ maps strict exact sequences of Fréchet
spaces over $K$ to exact sequences of condensed ${K}$-vector spaces.
Similarly, for strict exact sequences of spaces of compact type over $K$.

###### Proof.

Since the functor $V\mapsto\underline{V}$ is left exact, it suffices to show
that a strict surjection $V\to W$ of Fréchet spaces or of spaces of compact
type is carried to a surjection $\underline{V}\to\underline{W}$. For that, it
suffices to show that, for an extremally disconnected set $S$, we have
${\mathscr{C}}(S,V)\twoheadrightarrow{\mathscr{C}}(S,W)$, i.e., given
$g\in{\mathscr{C}}(S,W)$, there exists ${g}^{\prime}\in{\mathscr{C}}(S,V)$
making the following diagram commute

$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{{g}^{\prime}}$$\textstyle{W}$

Assume first that both $V$ and $W$ are Fréchet. In that case we have the
following argument of Guido Bosco [5, Lemma 1.A.33]: Since $g(S)$ is compact
in $W$, by [33, Lemma 45.1], it is the image $f(H)$ of a compact subset $H$ of
$V$. We conclude by recalling that the extremally disconnected sets are the
projective objects of the category of compact Hausdorff topological spaces.

Assume now that both $V$ and $W$ are of compact type. Since $g(S)$ is compact
in $W$, if we write $W=\operatorname{colim}_{n}W_{n}$, for an ind-system
$\\{W_{n}\\}_{n\in{\bf N}}$ of Banach spaces over $K$ with compact injective
transition maps, then $g(S)\subset W_{m}$, for some $m\in{\bf N}$ (see [30,
Lemma 16.9]). By [29, Lemma 3.39], we have a commutative diagram of solid
arrows

$\textstyle{V_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{m}}$$\textstyle{W_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{W}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{g}$$\scriptstyle{g_{W_{m}}}$$\scriptstyle{{g}^{\prime}}$

for a $K$\- Banach space $V_{m}$. As above, it follows that the dashed arrow
exists, which concludes our proof of the lemma. ∎

####  2.2.4. Solid Fréchet spaces and solid spaces of compact type

We will define solid Fréchet spaces and solid spaces of compact type as the
images in the category of solid $K$-vector spaces of their classical analogs.
See [29, Ch. 3] for alternative definitions.

(i) Solid Fréchet spaces. A solid Banach space is a solid $K$-vector space of
the form $\underline{V}$, for a classical Banach space $V$. A solid Fréchet
space is a solid $K$-vector space of the form $\underline{V}$, for a classical
Fréchet space $V$. It is called nuclear if $V$ is nuclear. The functor
$(\underline{\quad})$ identifies the categories of classical and solid Fréchet
spaces (identifying also exact sequences by Lemma 2.18). If this does no
confusion we will use the term "Fréchet spaces" for elements of both of these
categories.

Warning. The definition of nuclear used here is stronger than the definition
of nuclear in the solid formalism. To distinguish, we will call the latter
"solid nuclear".

Let $V$ be a solid Fréchet space written as a limit $V=\lim_{n}V_{n}$ of
Banach spaces $V_{n}$ with dense transition maps. Then:

  1. (1)

(Topological Mittag-Leffler) Then

${\mathrm{R}}^{j}\lim_{n}V_{n}=0,\quad j\geq 1.$

In particular, $V\simeq{\mathrm{R}}\underline{{\rm{Hom}}}_{K}(V^{*},K)$.

  2. (2)

If $W$ is a Banach space, then
$\underline{{\rm{Hom}}}_{K}(V,W)=\operatorname{colim}_{n}\underline{{\rm{Hom}}}_{K}(V_{n},W)$.

(ii) Solid spaces of compact type. A solid space of compact type is a solid
$K$-vector space of the form $\underline{V}$, for a classical space of compact
type $V$. The functor $(\underline{\quad})$ identifies the categories of
classical and solid spaces of compact type (identifying also exact sequences
by Lemma 2.18). If this does no confusion we will use the term "spaces of
compact type" for elements of both of these categories.

The following two lemmas will be needed later.

######  Lemma 2.19.

Let $V\in C_{K}$ be of compact type. Write it as
$V\simeq\operatorname{colim}_{n}V_{n}$ with $V_{n}$, $n\in{\bf N}$, Hausdorff
and injective compact transition maps. Then we have a canonical isomorphism of
condensed $K$-modules

$\operatorname{colim}_{n}\underline{V_{n}}\stackrel{{\scriptstyle\sim}}{{\to}}\underline{V}.$

In particular, the condensed $K$-module
$\operatorname{colim}_{n}\underline{V_{n}}$ is classical.

###### Proof.

Since compact sets are bounded, this follows immediately from [30, Lemma
16.9], which states that any bounded subset of $V$ comes from some $V_{n}$. ∎

######  Lemma 2.20.

If a sequence of solid Fréchet spaces over $K$ is exact then the classical
sequence is strictly exact. Similarly, for a sequence of solid spaces of
compact type over $K$.

###### Proof.

We start with an exact sequence of solid Fréchet spaces over $K$

$0\to\underline{V}_{1}\to\underline{V}_{2}\to\underline{V}_{3}\to 0.$

We need to show that the sequence of classical Fréchet spaces

(2.21) $0\to{V}_{1}\stackrel{{\scriptstyle f_{1}}}{{\to}}{V}_{2}\to V_{3}\to
0$

is strictly exact. But the sequence

(2.22) $0\to\underline{V}_{1}(*)_{\rm top}\to\underline{V}_{2}(*)_{\rm
top}\to\underline{V}_{3}(*)_{\rm top}\to 0$

maps, via topological isomorphisms $\underline{V}_{i}(*)_{\rm
top}\stackrel{{\scriptstyle\sim}}{{\to}}V_{i}$ (since $V_{i}$ is compactly
generated), to the sequence (2.21). Since $\underline{V_{3}}$ is quasi-
separated, the map $f_{1}$ is a closed immersion. In particular we have a
strict exact sequence of Fréchet spaces

$0\to{V}_{1}\stackrel{{\scriptstyle f_{1}}}{{\to}}{V}_{2}\to V^{\prime}_{3}\to
0$

and a continuous injection $f_{2}:V^{\prime}_{3}\to V_{3}$. We need to prove
that the map $f_{2}$ is a topological isomorphism. Since $f_{2}$ is a map
between two Fréchet spaces, by the Open Mapping Theorem, it suffices to show
that $f_{2}$ is an algebraic isomorphism. But, by Lemma 2.18, we have the
exact sequence

$0\to\underline{V}_{1}\to\underline{V}_{2}\to\underline{V}^{\prime}_{3}\to 0.$

Hence the canonical map
$\underline{f}_{2}:\underline{V}^{\prime}_{3}\to\underline{V}_{3}$ is an
isomorphism, which yields that so is the map $f_{3}$ (by the faithfullness of
the $(\underline{\phantom{x}})$ functor.

The argument for spaces of compact type is similar (the Open Mapping Theorem
is valid for LB spaces). ∎

####  2.2.5. Solid tensor product

We list properties of the solid tensor product that we will often use.

  1. (1)

([5, Prop. A.68]) Let $V,W$ be Fréchet spaces over $K$. Then we have a natural
isomorphism of solid $K$-vector spaces

$\underline{V}\otimes_{K}^{\Box}\underline{W}\stackrel{{\scriptstyle\sim}}{{\to}}\underline{V\widehat{\otimes}_{K}W},$

where $V\widehat{\otimes}_{K}W$ denotes the projective tensor product in the
category $C_{K}$.

  2. (2)

([5, Cor. A.65]) Any Fréchet space over $K$ is acyclic for the tensor product
$\otimes^{\Box}_{K}$. That is, if $V$ is a Fréchet space over $K$ then
$(-)\otimes^{{\rm L}\Box}_{K}V\simeq(-)\otimes^{\Box}_{K}V$.

  3. (3)

([5, Cor. A.67])

    1. (a)

Let $\\{V_{n}\\}_{n\in{\bf N}}$ be a pro-system of solid nuclear $K$-vector
spaces and let $W$ be a Fréchet vector space over $K$. Then we have an
isomorphism

$\lim_{n}(V_{n}\otimes_{K}^{\Box}W)\stackrel{{\scriptstyle\sim}}{{\leftarrow}}(\lim_{n}V_{n})\otimes_{K}^{\Box}W.$

    2. (b)

Let $\\{V_{n}\\}_{n\in{\bf N}}$ be a pro-system in ${\mathscr{D}}(K_{\Box})$
of complexes of solid nuclear $K$-vector spaces. Let $W$ be a complex of
$K$-Fréchet spaces. Then we have a quasi-isomorphism

${\mathrm{R}}\lim_{n}(V_{n}\otimes_{K}^{{\rm
L}\Box}W)\stackrel{{\scriptstyle\sim}}{{\leftarrow}}({\mathrm{R}}\lim_{n}V_{n})\otimes_{K}^{{\rm
L}\Box}W.$

The above properties also hold if we replace $K$ with $\breve{F}$ (with the
same references).

##  3\. Galois cohomology of $K$

This chapter gathers together a number of properties of Galois cohomology that
we will need later.

###  3.1. Preliminaries

We record here few basic facts about Galois cohomology seen via condensed
formalism.

####  3.1.1. Condensed Galois cohomology

Let $G$ be a condensed group. A condense $G$-module is a condensed abelian
group endowed with a ${\bf Z}[G]$-module structure. The condensed group
cohomology of $G$ with values in a condense $G$-module $V$ is defined as

${\mathrm{R}}\Gamma(G,V):={\mathrm{R}}\underline{{\rm{Hom}}}_{{\bf Z}[G]}({\bf
Z},V)\in{\mathscr{D}}({\rm CondAb}).$

######  Notation 3.1.

Let $G$ be a profinite group.

(a) The Iwasawa algebra of $G$ is the solid ring

$\displaystyle{\mathscr{O}}_{K,\Box}[G]$ $\displaystyle:=\lim_{H\subset
G}{\mathscr{O}}_{K}[G/H]\in{\rm Mod}^{\rm solid}_{{\mathscr{O}}_{K}},$
$\displaystyle K_{\Box}[G]={\mathscr{O}}_{K,\Box}[G][1/p]$
$\displaystyle:=\big{(}\lim_{H\subset
G}{\mathscr{O}}_{K}[G/H]\big{)}[1/p]\in{\rm Mod}^{\rm solid}_{K},$

where $H$ runs over all open and normal subgroups of $G$.

(b) A solid $G$-module over ${\mathscr{O}}_{K}$ (or a solid
${\mathscr{O}}_{K,\Box}[G]$-module) is a solid abelian group endowed with an
${\mathscr{O}}_{K,\Box}[G]$-module structure. The category of solid
${\mathscr{O}}_{K,\Box}[G]$-modules will be denoted by ${\rm Mod}^{\rm
solid}_{{\mathscr{O}}_{K,\Box}[G]}$ and its derived $\infty$-category by
${\mathscr{D}}({\mathscr{O}}_{K,\Box}[G])$. Similarly, for solid
$K_{\Box}[G]$-modules.

We list the following properties of ${\mathrm{R}}\Gamma(G,-)$:

  1. (1)

([5, Prop. B.2]) Let $G$ be a profinite group and let $V$ be a ${G}$-module in
solid abelian groups. Then

    1. (a)

The complex ${\mathrm{R}}\Gamma({G},V)$ is quasi-isomorphic to the complex of
solid abelian groups

(3.2) $V\to\underline{{\rm{Hom}}}({\bf
Z}[{G}^{1}],V)\to\underline{{\rm{Hom}}}({\bf
Z}[{G}^{2}],V)\to\underline{{\rm{Hom}}}({\bf Z}[{G}^{3}],V)\cdots.$

    2. (b)

If $V=\underline{V_{\rm top}}$, with $V_{\rm top}$ a T1 topological $G$-module
over ${\bf Z}$, then, for all $i\geq 0$, we have a natural isomorphism of
abelian groups161616The second cohomology group is the continuous group
cohomology.

${\mathrm{R}}\Gamma({G},V)(*)\simeq{\mathrm{R}}\Gamma({G},V_{\rm top}).$

  2. (2)

([5, Prop. B.3]) For $n\in{\bf N}$, let $\Gamma:={\bf Z}_{p}^{n}$, and let
$\gamma_{1},\ldots,\gamma_{n}$ denote the generators of $\Gamma$. Let $V$ be a
$\Gamma$-module in ${\rm Mod}^{\rm solid}_{{\bf Z}_{p}}$. Then we have a
quasi-isomorphism

${\mathrm{R}}\Gamma(\Gamma,V)\simeq{\rm Kos}_{\gamma}(V):={\rm
Kos}_{V}(\gamma_{1}-1,\ldots,\gamma_{n}-1),$

the Koszul complex of $V$ with respect to the elements
$\gamma_{1}-1,\ldots,\gamma_{n}-1$.

  3. (3)

([29, Lemma 5.2]) Let $G$ be a profinite group. There is a solid projective
resolution of the trivial representation

$\cdots\to K_{\Box}[{G}^{n+1}]\to K_{\Box}[{G}^{n}]\to\cdots\to
K_{\Box}[{G}^{1}]\to K\to 0.$

In particular, if $V$ is a ${G}$-module in solid modules over $K$, one has
that

${\mathrm{R}}\underline{{\rm{Hom}}}_{K_{\Box}[G]}(K,V)\simeq{\mathrm{R}}\Gamma({G},V).$

######  Lemma 3.3.

Let $G$ be a profinite group and let $V$ be a finite rank ${\bf Q}_{p}$-vector
space equipped with a continuous action of $G$. Then

  1. (1)

we have a quasi-isomorphism and isomorphisms

${\rm
CD}({\mathrm{R}}\Gamma({G},V))\simeq{\mathrm{R}}\Gamma({G},\underline{V}),\quad{\rm
CD}(\widetilde{H}^{i}({G},V))\simeq H^{i}({G},\underline{V}),\,i\geq 0.$

  2. (2)

we have a quasi-isomorphism171717Topology on ${\mathrm{R}}\Gamma({G},V)$ is
defined using continuous cochains. See the proof of the lemma for details. in
${\mathscr{D}}(C_{{\bf Q}_{p}})$

${\mathrm{R}}\Gamma({G},\underline{V})(*)_{\rm
top}\simeq{\mathrm{R}}\Gamma({G},V).$

###### Proof.

For claim (1) we compute

$\displaystyle{\mathrm{R}}\Gamma({G},V)\simeq C(G,V),\quad
n\mapsto{\mathscr{C}}(G^{n-1},V);$ $\displaystyle{\rm
CD}({\mathrm{R}}\Gamma({G},V))(S):n\mapsto{\mathscr{C}}(S,{\mathscr{C}}(G^{n-1},V))\simeq{\mathscr{C}}(S\times
G^{n-1},V),$

where $S$ is a profinite set and ${\mathscr{C}}(-,-)$ denotes the space of
continuous maps equipped with compact open topology (note that
${\mathscr{C}}(S\times G^{n-1},V)$, since $S\times G^{n-1}$ is compact, is a
${\bf Q}_{p}$-Banach space). We also have

$\displaystyle{\mathrm{R}}\Gamma({G},\underline{V}):n\mapsto\underline{{\rm{Hom}}}({\bf
Z}[{G}^{n-1}],\underline{V});$ $\displaystyle\underline{{\rm{Hom}}}({\bf
Z}[{G}^{n-1}],\underline{V})(S)={{\rm{Hom}}}({\bf Z}[{S}]\otimes{\bf
Z}[{G}^{n-1}],\underline{V})\simeq{{\rm{Hom}}}({\bf Z}[{S\times
G^{n-1}}],\underline{V}).$

Since ${{\rm{Hom}}}({\bf Z}[{S\times
G^{n-1}}],\underline{V})\simeq{\mathscr{C}}(S\times G^{n-1},{V})$, we get
claim (1) of the lemma.

Claim (2) follows from the fact that the complex of continuous cochains
representing ${\mathrm{R}}\Gamma({G},V)$ is a complex of Banach spaces and
[29, Prop. 3.5]. ∎

####  3.1.2. Poitou-Tate duality

Let $K$ be a finite extension of ${\bf Q}_{p}$ and let $V$ be a continuous,
finite rank ${\bf Q}_{p}$-representation of ${\mathscr{G}}_{K}$. Recall that
the Galois pairing

$H^{i}({\mathscr{G}}_{K},V)\otimes^{\Box}_{{\bf
Q}_{p}}H^{2-i}({\mathscr{G}}_{K},V^{*}(1))\stackrel{{\scriptstyle\cup}}{{\to}}H^{2}({\mathscr{G}}_{K},{\bf
Q}_{p}(1))\xrightarrow[\sim]{{\rm Tr}_{K}}{\bf Q}_{p}$

is a perfect pairing (by Poitou-Tate duality). Hence, for $i\in{\bf N}$,
$H^{i}({\mathscr{G}}_{K},V)$ and $H^{2-i}({\mathscr{G}}_{K},V^{*}(1))$ are
natural duals (via the above pairing). In particular, we have

$\displaystyle H^{0}({\mathscr{G}}_{K},{\bf Q}_{p}(j))$
$\displaystyle\simeq\begin{cases}{\bf Q}_{p}&{\text{if $j=0$,}}\\\
0&{\text{otherwise;}}\end{cases}\quad H^{2}({\mathscr{G}}_{K},{\bf
Q}_{p}(j))\simeq\begin{cases}{\bf Q}_{p}&{\text{if $j=1$,}}\\\
0&{\text{otherwise.}}\end{cases}$

###  3.2. $(\varphi,\Gamma)$-modules and Galois cohomology

In the next two sections, we will briefly recall and refine the relationship
between $(\varphi,\Gamma)$-modules and Galois cohomology.

####  3.2.1. Notations

If $n\geq 1$, let $F_{n}={\bf Q}_{p}({\boldsymbol{\mu}}_{p^{n}})$ and let
$F_{\infty}:=\cup_{n}F_{n}$ be the cyclotomic extension of ${\bf Q}_{p}$. Let
$\chi:{\mathscr{G}}_{{\bf Q}_{p}}\to{\bf Z}_{p}^{\boldsymbol{*}}$ be the
cyclotomic character. Then $\chi$ factors through $\Gamma:={\rm
Gal}(F_{\infty}/{\bf Q}_{p})$ and induces an isomorphism
$\chi:\Gamma\overset{\sim}{\to}{\bf Z}_{p}^{\boldsymbol{*}}$.

If $\Delta$ is the torsion subgroup of $\Gamma$, then $\chi^{|\Delta|}$ takes
values in $1+p{\bf Z}_{p}$ (resp. $1+8{\bf Z}_{2}$) if $p\neq 2$ (resp.
$p=2$). Let $\tau:{\mathscr{G}}_{{\bf Q}_{p}}\to{\bf Z}_{p}$ be defined by

$\tau=\begin{cases}\tfrac{1}{p\,|\Delta|}\log\chi^{|\Delta|}&{\text{if $p\neq
2$,}}\\\ \tfrac{1}{4\,|\Delta|}\log\chi^{|\Delta|}&{\text{if
$p=2$.}}\end{cases}\quad{\text{i.e., }}\tau=\tfrac{1}{p^{c(p)}}\log\chi,\ {\rm
with}\ {c(p)}=\begin{cases}1&{\text{if $p\neq 2$,}}\\\ 2&{\text{if
$p=2$.}}\end{cases}$

Let $F_{\infty}^{\prime}:=F_{\infty}^{\Delta}$ be the cylotomic ${\bf
Z}_{p}$-extension of ${\bf Q}_{p}$. Then $\tau$ factors through
$\Gamma^{\prime}:={\rm Gal}(F_{\infty}^{\prime}/{\bf Q}_{p})$ and induces an
isomorphism $\tau:\Gamma^{\prime}\overset{\sim}{\to}{\bf Z}_{p}$.

Let $K$ be a finite extension of ${\bf Q}_{p}$. If $n\in{\bf N}$, let
$K_{n}=K({\boldsymbol{\mu}}_{p^{n}})$ and let $K_{\infty}:=\cup_{n}K_{n}$ be
the cyclotomic extension of $K$. Let $\Gamma_{K}:={\rm Gal}(K_{\infty}/K)$;
then $\chi$ induces an isomorphism from $\Gamma_{K}$ to an open subgroup of
${\bf Z}_{p}^{\boldsymbol{*}}$. Let $\Delta_{K}$ be the torsion subgroup of
$\Gamma_{K}$, let $K^{\prime}_{\infty}:=K_{\infty}^{\Delta_{K}}$ be the
cylotomic ${\bf Z}_{p}$-extension of $K$. Then $\Gamma_{K}^{\prime}:={\rm
Gal}(K_{\infty}^{\prime}/K)=\Gamma_{K}/\Delta_{K}$ and $\tau$ induces an
isomorphism $\tau:\Gamma_{K}^{\prime}\overset{\sim}{\to}p^{n(K)}{\bf Z}_{p}$
for some $n(K)\in{\bf N}$.

Let $\gamma_{K}\in\Gamma^{\prime}_{K}$ be the element verifying
$\tau(\gamma_{K})=p^{n(K)}$. Then $\gamma_{K}$ has a unique lifting in
$\Gamma_{K}$ whose image by $\chi$ belongs to $1+p^{n(K)+c(p)}{\bf Z}_{p}$; we
denote this lifting also by $\gamma_{K}$. Then $\Gamma_{K}=\gamma_{K}^{{\bf
Z}_{p}}\times\Delta_{K}$.

Let $L=K({\boldsymbol{\mu}}_{p^{c(p)}})$. Then ${\rm Gal}(L/K)=\Delta_{K}$ and
$\Gamma_{L}=\gamma_{K}^{{\bf Z}_{p}}$.

######  Remark 3.4.

(i) Let $F:=K\cap F_{\infty}$. Then $[K_{\infty}:F_{\infty}]=[K:F]$ and
$\Gamma_{F}=\Gamma_{K}=\Delta_{K}\times p^{n(K)}{\bf Z}_{p}$; hence $[F:{\bf
Q}_{p}]=p^{n(K)}\frac{|\Delta|}{|\Delta_{K}|}$ and $[K:{\bf
Q}_{p}]=[K_{\infty}:F_{\infty}]\cdot p^{n(K)}\frac{|\Delta|}{|\Delta_{K}|}$.

(ii) We have $\tau(\gamma_{K})=p^{n(K)}$, hence
$\log\chi(\gamma_{K})=p^{n(K)+c(p)}$.

For $0<u\leq v\in v(K_{\infty}^{\flat})$, let

${\bf
B}_{K_{\infty}}^{[u,v]}:=(W({\mathscr{O}}_{K_{\infty}}^{\flat})[\tfrac{p}{[\alpha]},\tfrac{[\beta]}{p}]^{\wedge_{p}})[\tfrac{1}{p}],\quad
v(\alpha)=\tfrac{1}{v},\ v(\beta)=\tfrac{1}{u}.$

It is a Banach space over ${\bf Q}_{p}$. Let $\varphi$ denote the Frobenius
morphism acting on ${\bf B}_{K_{\infty}}^{[u,v]}$ and $\psi$ its left inverse.
Let $U^{[u,v]}$ be the corresponding open set of the Fargues-Fontaine curve
over $K_{\infty}$. Take $u=\frac{p-1}{p}$, $v=p-1$ if $p\neq 2$, and
$u=\frac{2}{3}$, $v=\frac{4}{3}$ if $p=2$. Then ${\bf
B}^{[u,v]}_{K_{\infty}}/t=\widehat{K}_{\infty}$ and $t$ is a unit in 181818
That is, the intersection of $U^{[u,v]}$ with the orbit under $\varphi$ of the
point $\infty$ of the Fargues-Fontaine curve is reduced to $\\{\infty\\}$ and
$U^{[u,v/p]}$ does not intersect this orbit. ${\bf B}^{[u,v/p]}_{K_{\infty}}$.
We write $\theta$ the canonical map ${\bf
B}_{K_{\infty}}^{[u,v]}\to\widehat{K}_{\infty}$ with ${\rm Ker}(\theta)=t{\bf
B}^{[u,v]}_{K_{\infty}}$.

####  3.2.2. Galois cohomology

For $j\in{\bf Z}$, the theory of $(\varphi,\Gamma)$-modules yields quasi-
isomorphisms

(3.5) $\displaystyle\alpha_{j}:{\rm Kos}_{\varphi,\gamma}({\bf
B}^{[u,v]}_{K_{\infty}}(j))\simeq{\rm R}\Gamma({\mathscr{G}}_{L},{\bf
Q}_{p}(j)),$ $\displaystyle\alpha_{j}:{\rm Kos}_{\varphi,\gamma}({\bf
B}^{[u,v]}_{K_{\infty}}(j))^{\Delta_{K}}\simeq{\rm
R}\Gamma({\mathscr{G}}_{K},{\bf Q}_{p}(j)),$

where the $(\varphi,\gamma)$-Koszul complexes (of Banach spaces over ${\bf
Q}_{p}$) are defined by:

$\displaystyle{\rm Kos}_{\varphi,\gamma}({\bf B}^{[u,v]}_{K_{\infty}}(j))$
$\displaystyle:=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
24.39195pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\big{(}({\bf
B}^{[u,v]}_{K_{\infty}}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
27.26045pt\raise 6.5pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{(\varphi-1,\gamma_{K}-1)}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 67.07108pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
67.07108pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{({\bf
B}^{[u,v/p]}_{K_{\infty}}(j))\oplus({\bf
B}^{[u,v]}_{K_{\infty}}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
164.3748pt\raise 6.5pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{-(\gamma_{K}-1)+(\varphi-1)}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 209.82433pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
209.82433pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{({\bf
B}^{[u,v/p]}_{K_{\infty}}(j))\big{)}}$}}}}}}}\ignorespaces}}}}\ignorespaces$

and the complex ${\rm Kos}_{\varphi,\gamma}({\bf
B}^{[u,v]}_{K_{\infty}}(j))^{\Delta_{K}}$ is obtained by taking fixed points
under $\Delta_{K}$ of each of the terms of the complex.

Set $[\Delta_{K}]:=\sum_{\sigma\in\Delta_{K}}\sigma$. The following
commutative diagram allows to deduce results for $K$ from results for $L$,
i.e. we can often assume that $\Delta_{K}=1$ in the proofs,

$\textstyle{{\rm Kos}_{\varphi,\gamma}({\bf
B}^{[u,v]}_{K_{\infty}}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{[\Delta_{K}]}$$\textstyle{{\rm
R}\Gamma({\mathscr{G}}_{L},{\bf
Q}_{p}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
cor}_{L}^{K}}$$\textstyle{{\rm Kos}_{\varphi,\gamma}({\bf
B}^{[u,v]}_{K_{\infty}}(j))^{\Delta_{K}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\rm
id}$$\textstyle{{\rm R}\Gamma({\mathscr{G}}_{K},{\bf
Q}_{p}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
res}_{K}^{L}}$

######  Remark 3.6.

If $n$ is big enough so that $K_{n}$ has enough roots of unity in the sense of
[16], there exist normalized trace maps ${\rm Res}_{p^{-n}{\bf Z}_{p}}:{\bf
B}^{[u,v]}_{K_{\infty}}\to{\bf B}_{K_{n}}^{[u,v]}$ where ${\bf
B}_{K_{n}}^{[u,v]}:=\varphi^{-n}({\bf B}_{K}^{[u/p^{n},v/p^{n}]})$ which
commute with $\Gamma_{K}$ and verify $\varphi\circ{\rm Res}_{p^{-n-1}{\bf
Z}_{p}}={\rm Res}_{p^{-n}{\bf Z}_{p}}\circ\varphi$. These decompletion maps
play a big role in the proofs because ${\bf B}_{K_{n}}^{[u,v]}$ is a much
nicer ring than ${\bf B}^{[u,v]}_{K_{\infty}}$ (it is a ring of analytic
functions on an annulus, whereas the later is a ring of analytic functions on
a perfectoid annulus).

Applying ${\rm Res}_{p^{-n}{\bf Z}_{p}}$ to each of the terms of the above
complex produces a quasi-isomorphic complex with rings related to $K_{n}$
instead of $K_{\infty}$, which is closer to the complexes used in the theory
of $(\varphi,\Gamma)$-modules, like in Herr’s thesis [23] or in [6]. Going
from these rings to the usual rings uses the standard techniques of
$(\varphi,\Gamma)$-modules, i.e., normalized trace ${\rm Res}_{p^{-n}{\bf
Z}_{p}}$, bijectivity of $\gamma_{K}-1$ on $({\bf
B}^{[u/p^{n},v/p^{n}]}_{K}(j))^{\psi=0}$, etc., as in [16].

####  3.2.3. Examples

Denote by

(3.7) $h^{i}_{K}:Z^{i}({\rm Kos}_{\varphi,\gamma}({\bf
B}^{[u,v]}_{K_{\infty}}(j))^{\Delta_{K}})\to H^{i}({\mathscr{G}}_{K},{\bf
Q}_{p}(j))$

the map induced by quasi-isomorphism (3.5); it factorizes as

$h^{i}_{K}:Z^{i}({\rm Kos}_{\varphi,\gamma}({\bf
B}^{[u,v]}_{K_{\infty}}(j))^{\Delta_{K}})\to H^{i}({\rm
Kos}_{\varphi,\gamma}({\bf B}^{[u,v]}_{K_{\infty}}(j))^{\Delta_{K}})\simeq
H^{i}({\mathscr{G}}_{K},{\bf Q}_{p}(j)).$

If $(a,b)\in Z^{1}({\rm Kos}_{\varphi,\gamma}({\bf
B}_{K_{\infty}}^{[u,v]}(j))^{\Delta_{K}})$ and $u\in{\bf
B}_{\overline{K}}^{[u,v]}(j)$ satisfy $(\varphi-1)u=a$, then

(3.8) $h^{1}_{K}(a,b)={\rm
cl}\big{(}\sigma\mapsto\tfrac{\sigma-1}{\gamma_{K}-1}b-(\sigma-1)u\big{)}.$

(In this expression, $\sigma$ acts through its image in $\Gamma^{\prime}_{K}$
on $b$, and we think of $\tfrac{\sigma-1}{\gamma_{K}-1}$ as an element of
${\bf Z}_{p}[[\Gamma^{\prime}_{K}]]$; the expression between parenthesis is a
$1$-cocycle on ${\mathscr{G}}_{K}$ with values in ${\bf Q}_{p}(j)$, and ${\rm
cl}$ denotes its image in $H^{1}({\mathscr{G}}_{K},{\bf Q}_{p}(j))$.)

$\bullet$ The case $j=0$. We can apply the formula (3.8) to $j=0$ and cocycle
$(1,0)$. Then the corresponding ${\mathscr{G}}_{K}$-cocycle factors through
${\rm Gal}(\overline{\bf F}_{p}/k_{K})$ since the solution of $(\varphi-1)u=1$
belongs to $W(\overline{\bf F}_{p})$, and sends the relative Frobenius to
$f:=f(K/{\bf Q}_{p})$ (because $(\sigma-1)u$ is equal to $(\varphi^{f}-1)u=f$
since $\varphi(u)=u+1$). Since the relative Frobenius corresponds to a
uniformizer of $K^{\boldsymbol{*}}$, it follows that the element $\lambda$ of
${\rm Hom}(K^{\boldsymbol{*}},{\bf Q}_{p})$ which corresponds to
$h^{1}_{K}(1,0)$ is $f(K/{\bf Q}_{p})v_{K}=v_{p}\circ{\rm N}_{K/{\bf Q}_{p}}$
(where $v_{K}$ is the valuation on $K$ with image ${\bf Z}$).

Starting with $(0,p^{n(K)+c(p)})$, formula (3.8) with $u=0$ yields a group
morphism ${\mathscr{G}}_{K}\to{\bf Z}_{p}$ which factors through
$\Gamma^{\prime}_{K}$ and has value $\log\chi(\gamma_{K})$ at $\gamma_{K}$; it
follows that this morphism is $\log\chi$.

$\bullet$ The case $j=1$. Let $n=n(K)+c(p)$. We have
$(\varphi-1)\tfrac{1}{\pi}\in({\bf B}_{{\bf Q}_{p}}^{(0,v]})^{\psi=0}$ and
there exists $a\in({\bf B}_{{\bf
Q}_{p}}^{(0,v/p^{n}]})^{\psi=0}\otimes\tfrac{d\pi}{1+\pi}$ such that
$(\gamma_{K}-1)a=((\varphi-1)\tfrac{1}{\pi})\otimes\tfrac{d\pi}{1+\pi}$. Then,

$(\varphi^{-n}(a),\varphi^{-n}(\tfrac{1}{\pi}\otimes\tfrac{d\pi}{1+\pi}))\in
Z^{1}({\rm Kos}_{\varphi,\gamma}({\bf B}_{F_{n}}^{[u,v]}(1)))\quad{\rm
and}\quad
h^{1}_{F_{n}}(\varphi^{-n}(a),\varphi^{-n}(\tfrac{1}{\pi}\otimes\tfrac{d\pi}{1+\pi}))={\rm
cl}(\zeta_{p^{n}}-1).$

(This follows from point iii) of [6, Prop. V.3.2], using the constructions
leading to [6, th. II.1.3] and the (obvious) fact that Coleman power series
attached to $(\zeta_{p^{n}}-1)_{n\geq 1}$ is just $T$.)

####  3.2.4. Cup-products

We define compatible cup products ($\alpha\in{\bf Q}_{p}$)

$\displaystyle\cup_{\alpha}:\quad{\rm Kos}_{\varphi,\gamma}({\bf
B}^{[u,v]}_{K_{\infty}}(j_{1}))^{\Delta_{K}}$
$\displaystyle\otimes^{\Box}_{{\bf Q}_{p}}{\rm Kos}_{\varphi,\gamma}({\bf
B}^{[u,v]}_{K_{\infty}}(j_{2}))^{\Delta_{K}}\to{\rm Kos}_{\varphi,\gamma}({\bf
B}^{[u,v]}_{K_{\infty}}(j_{1}+j_{2}))^{\Delta_{K}}$

using (twice) the formulas from Section 7.3.2.

######  Lemma 3.9.

On the level of cohomology, the quasi-isomorphism (3.5) is compatible with
products.

###### Proof.

This is easy to check for the pairing of $0$\- and $2$-cocycles. For the
pairing of two $1$-cocycles, this amounts to checking that

(3.10) $h^{1}_{K}(a,b)\cup
h^{1}_{K}(a^{\prime},b^{\prime})=h^{2}_{K}(b\otimes\gamma_{K}(a^{\prime})-a\otimes\varphi(b^{\prime})).$

(For the product on the Koszul complexes, we used here (twice) the formulas
from Section 7.3.2 with $\alpha=1$.) But equality (3.10) is standard and was
checked by Herr and Benois in [23, 3]. ∎

###  3.3. Residues and duality

####  3.3.1. The trace map

We will describe the trace isomorphism

${\rm Tr}_{K}:H^{2}({\mathscr{G}}_{K},{\bf Q}_{p}(1))\overset{\sim}{\to}{\bf
Q}_{p}$

in terms of $(\varphi,\Gamma)$-modules (cf. [23], [3]). For this, we start
with the description of ${\rm Tr}_{K}$ given by local class field theory. We
have isomorphisms

${\rm cl}:{\rm Hom}(K^{\boldsymbol{*}},{\bf
Q}_{p})\overset{\sim}{\to}H^{1}({\mathscr{G}}_{K},{\bf Q}_{p})\quad{\rm
and}\quad{\bf
Q}_{p}\widehat{\otimes}K^{\boldsymbol{*}}\overset{\sim}{\to}H^{1}({\mathscr{G}}_{K},{\bf
Q}_{p}(1))$

and ${\rm Tr}_{K}$ is given by the formula

${\rm Tr}_{K}({\rm cl}(\tau)\cup{\rm cl}(\alpha))=\tau(\alpha),\quad{\text{ if
$\tau\in{\rm Hom}(K^{\boldsymbol{*}},{\bf Q}_{p})$ and $\alpha\in
K^{\boldsymbol{*}}$.}}$

####  3.3.2. Trace map and residues

Let $\pi:=[\varepsilon]-1\in{\bf B}_{F_{\infty}}^{[u,v]}$. We have
$\gamma(\pi)=(1+\pi)^{\chi(\gamma)}-1$ if $\gamma\in\Gamma$, and we make
$\varphi$ and $\Gamma$ act on $\frac{d\pi}{1+\pi}$ by
$\varphi(\frac{d\pi}{1+\pi})=\frac{d\pi}{1+\pi}$ and
$\gamma(\frac{d\pi}{1+\pi})=\chi(\gamma)\frac{d\pi}{1+\pi}$, respectively
(this allows us to identify $\Lambda(1)$ with
$\Lambda\otimes\tfrac{d\pi}{1+\pi}$). Then there exists a unique continuous
linear map

${\rm res}_{\pi}:{\bf B}_{F_{\infty}}^{[u,v]}\otimes\tfrac{d\pi}{1+\pi}\to{\bf
Q}_{p}$

which commutes with $\varphi$ and $\Gamma$, and sends $\sum_{k\in{\bf
Z}}a_{k}\pi^{k}\,d\pi$ to $a_{-1}$. (See [13, Prop. IV.3.3] for the existence
of ${\rm res}_{\pi}$.)

Set

${\rm Tr}_{K_{\infty}/F_{\infty}}:=\sum\nolimits_{\sigma\in
G_{F_{\infty}}/G_{K_{\infty}}}\sigma$

(well defined on modules on which ${\mathscr{G}}_{K_{\infty}}$ acts
trivially).

######  Proposition 3.11.

We have

$\displaystyle{\rm Tr}_{K}\circ h^{2}_{K}=\tfrac{1}{|\Delta_{K}|}\,{\rm
res}_{\pi}\circ{\rm Tr}_{K_{\infty}/F_{\infty}}\quad{\rm on}\ ({\bf
B}_{K_{\infty}}^{[u,v/p]}\otimes\tfrac{d\pi}{1+\pi})^{\Delta_{K}}$

###### Proof.

If $\alpha={\rm res}_{\pi}\circ{\rm Tr}_{K_{\infty}/F_{\infty}}$, then
$\alpha(\varphi(x))=\alpha(x)$ and $\alpha(\sigma(x))=\alpha(x)$, for all
$\sigma\in{\mathscr{G}}_{{\bf Q}_{p}}$. In particular, $\alpha$ factors
through $H^{2}({\rm Kos}_{\varphi,\gamma}({\bf B}_{K_{\infty}}^{[u,v]}(1)))$,
which is of dimension $1$ over ${\bf Q}_{p}$. This proves the result up to
multiplication by a constant and to show that this constant is $1$, it
suffices to show that the two terms coincide on an element on which one of the
two is nonzero.

Moreover, we have a commutative diagram

$\textstyle{{\bf
B}_{K_{\infty}}^{[u,v/p]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{2}_{L}}$$\scriptstyle{[\Delta_{K}]}$$\textstyle{H^{2}({\mathscr{G}}_{L},{\bf
Q}_{p}(1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
cor}_{L/K}}$$\scriptstyle{{\rm Tr}_{L}}$$\textstyle{{\bf
Q}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({\bf
B}_{K_{\infty}}^{[u,v/p]})^{\Delta_{K}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{2}_{K}}$$\textstyle{H^{2}({\mathscr{G}}_{K},{\bf
Q}_{p}(1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
Tr}_{K}}$$\textstyle{{\bf Q}_{p}}$

Hence ${\rm Tr}_{L}\circ h^{2}_{L}=|\Delta_{K}|\,{\rm Tr}_{K}\circ h^{2}_{K}$
on $({\bf B}_{K_{\infty}}^{[u,v/p]})^{\Delta_{K}}$. It follows that the result
holds for $K$ if and only if it holds for $L$, and we can assume that
$\Delta_{K}=1$, i.e. $K=K_{n}$, with $n=n(K)+c(p)$.

Let $\lambda=v_{p}\circ{\rm N}_{K/{\bf Q}_{p}}$ as in the case $j=0$ in
section 3.2.3. It follows, using the identity ${\rm N}_{F_{n}/{\bf
Q}_{p}}(\zeta_{p^{n}}-1)=p$ (resp. $=-2$ if $p=2$), that

$\displaystyle{\rm cl}(\lambda)\cup{\rm cl}(\zeta_{p^{n}}-1)$
$\displaystyle=v_{p}({\rm N}_{K/{\bf Q}_{p}}(\zeta_{p^{n}}-1))$
$\displaystyle=[K_{n}:F_{n}]\,v_{p}({\rm N}_{F_{n}/{\bf
Q}_{p}}(\zeta_{p^{n}}-1))=[K_{n}:F_{n}]=[K_{\infty}:F_{\infty}]$

But formula (3.10) gives us

${\rm cl}(\lambda)\cup{\rm cl}(\zeta_{p^{n}}-1)={\rm Tr}_{K}\circ
h^{2}_{K}\big{(}\varphi^{1-n}\big{(}\tfrac{1}{\pi}\otimes\tfrac{d\pi}{1+\pi}\big{)}\big{)}$

Since ${\rm Tr}_{K_{\infty}/F_{\infty}}$ and $x\mapsto{\rm
res}_{\pi}x\frac{d\pi}{1+\pi}$ commute with $\varphi$ and $\Gamma_{K}$, we
have

${\rm res}_{\pi}\circ{\rm
Tr}_{K_{\infty}/F_{\infty}}\big{(}\varphi^{1-n}\big{(}\tfrac{1}{\pi}\otimes\tfrac{d\pi}{1+\pi}\big{)}\big{)}=[K_{\infty}:F_{\infty}]\,{\rm
res}_{\pi}(\tfrac{1}{\pi}\otimes\tfrac{d\pi}{1+\pi}\big{)}=[K_{\infty}:F_{\infty}]$

Our proposition follows. ∎

####  3.3.3. An alternative description of the trace map

Let

${\rm Tr}:\overline{\bf Q}_{p}\to{\bf Q}_{p}$

be the unique ${\mathscr{G}}_{{\bf Q}_{p}}$-equivariant projection (if
$[L:{\bf Q}_{p}]<\infty$, the restriction of ${\rm Tr}$ to $L$ coincides with
$\frac{1}{[L:{\bf Q}_{p}]}{\rm Tr}_{L/{\bf Q}_{p}}$). Then ${\rm Tr}$ extends
by continuity to $\widehat{K}_{\infty}$ (normalized Tate trace; it does not
extend to ${\bf C}_{p}$), and ${\rm Tr}\circ\theta$ gives a well defined map
${\bf B}_{K_{\infty}}^{[u,v]}\to{\bf Q}_{p}$.

######  Proposition 3.12.

If $\alpha\in{\bf B}^{[u,v]}_{K_{\infty}}$,

${\rm Tr}_{K}\circ
h^{2}_{K}((\varphi-1)\tfrac{\alpha}{t}\otimes\tfrac{d\pi}{1+\pi})=-\tfrac{[K:{\bf
Q}_{p}]}{\log\chi(\gamma_{K})}{\rm Tr}\circ\theta(\alpha)$

###### Proof.

Denote by $\delta_{K}:{\bf B}^{[u,v]}_{K_{\infty}}\to{\bf Q}_{p}$ the map
$\alpha\mapsto\delta_{K}(\alpha):={\rm Tr}_{K}\circ
h^{2}_{K}((\varphi-1)\tfrac{\alpha}{t}\otimes\tfrac{d\pi}{1+\pi})$. Then
$\delta_{K}$ is identically $0$ on ${\rm Ker}\,\theta=t{\bf
B}^{[u,v]}_{K_{\infty}}$ (because $(\varphi-1)\tfrac{\alpha}{t}$ is then a
coboundary), hence it factors through ${\bf
B}^{[u,v]}_{K_{\infty}}/t=\widehat{K}_{\infty}$. It commutes with the action
of ${\mathscr{G}}_{{\bf Q}_{p}}$ (i.e.,
$\delta_{\sigma(K)}\circ\sigma=\delta_{K}$, for all
$\sigma\in{\mathscr{G}}_{{\bf Q}_{p}}$). So there exists $c(K)$ such that
$\delta_{K}=c(K){\rm Tr}\circ\theta$.

To determine $c(K)$, it is enough to compute the value of the term on the
left-hand side for $\alpha=1$, which can be done using Proposition 3.11. We
have $(\varphi-1)\frac{1}{t}=\frac{1-p}{p\,t}$, and ${\rm
Tr}_{K_{\infty}/F_{\infty}}$ is multiplication by $[K_{\infty}:F_{\infty}]$ on
${\bf B}^{[u,v/p]}_{F_{\infty}}$ (which contains $\frac{1}{t}$). Finally,
$\frac{1}{t}\frac{d\pi}{1+\pi}=(\frac{1}{\pi}+\sum_{n\geq
0}a_{n}\pi^{n})\,d\pi$ in ${\bf B}^{[u,v/p]}_{{\bf
Q}_{p}}\otimes\frac{d\pi}{1+\pi}$, and ${\rm
res}_{\pi}(\frac{1}{t}\frac{d\pi}{1+\pi})=1$.

We get $c(K)=\frac{1-p}{p\,|\Delta_{K}|}\,[K_{\infty}:F_{\infty}]$, and we use
(i) and (ii) of Remark 3.4 to get $c(K)=-\tfrac{[K:{\bf
Q}_{p}]}{\log\chi(\gamma_{K})}$ which concludes the proof. ∎

###  3.4. Tate’s formulas

We will present now a generalization of Tate’s computations of the Galois
cohomology of $C$.

####  3.4.1. Classical Tate’s formulas

We will often use the following well-known isomorphisms:

(3.13) $\displaystyle H^{i}({{\mathscr{G}}}_{{K}},{C}(j))$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\leftarrow}}\begin{cases}{K}&{\text{if
$j=0$ and $i=0,1,$}}\\\ 0&{\text{otherwise.}}\end{cases}$

For $i=0$ the above isomorphism is given by the canonical map ${K}\to
H^{0}({{\mathscr{G}}}_{K},{C})$; for $i=1$ – by the ${K}$-linear map sending
$1$ to the $1$-cocycle $\log\chi$, where $\chi$ is the cyclotomic character.

######  Remark 3.14.

The following rescaling of the map from (3.13) will be useful later. Consider
the composition

$\displaystyle\alpha_{0}:{\rm
Kos}_{\gamma}({{K}})\stackrel{{\scriptstyle}}{{\to}}{\rm
Kos}_{\gamma}({\widehat{K}}_{\infty})\xleftarrow[\sim]{\lambda}C(\Gamma^{\prime}_{K},\widehat{K}^{\prime}_{\infty})\xrightarrow[\sim]{}C(\Gamma_{K},\widehat{K}_{\infty})\xrightarrow[\sim]{}C({\mathscr{G}}_{K},C),$

where the last three complexes are complexes of nonhomogeneous condensed
cochains191919Nonhomogeneous version of (3.2)., and we have

${\rm
Kos}_{\gamma}({\widehat{K}}_{\infty})=[{\widehat{K}}^{\prime}_{\infty}\smash{\mathop{\relbar\joinrel\longrightarrow}\limits^{\gamma_{K}-1}}{\widehat{K}}^{\prime}_{\infty}],\quad{\rm
Kos}_{\gamma}({K}):=[{K}\smash{\mathop{\relbar\joinrel\longrightarrow}\limits^{\gamma_{K}-1}}{K}]=[{K}\smash{\mathop{\relbar\joinrel\longrightarrow}\limits^{0}}{K}].$

Map $\lambda$ is given by the identity in degree $0$, evaluation on
$\gamma_{K}$ in degree $1$, and $0$ in higher degrees. On cohomology level
$\alpha_{0}$ yields: for $i=0$, the canonical map ${K}\to
H^{0}({{\mathscr{G}}}_{K},{C})$; for $i=1$, the ${K}$-linear map sending $1$
to the $1$-cocycle $\frac{\log\chi}{\log\chi(\gamma_{K})}$.

####  3.4.2. Generalized Tate’s formulas

######  Proposition 3.15.

Let $W\in C_{K}$ be a Banach space, nuclear Fréchet, or a space of compact
type equipped with a trivial action of ${\mathscr{G}}_{K}$. Then we have
isomorphisms:

(3.16) $\displaystyle H^{i}({\mathscr{G}}_{K},W(j){\otimes}^{\Box}_{K}C)$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\leftarrow}}\begin{cases}W&{\text{if
$j=0$ and $i=0,1$},}\\\ 0&{\text{otherwise.}}\end{cases}$

For $j=0$, the maps are defined in the obvious way in the case $i=0$ and as
the cup product with the class of $\frac{\log\chi}{\log\chi(\gamma_{K})}$ in
$H^{1}({\mathscr{G}}_{K},C)$ in the case $i=1$.

###### Proof.

Assume first that $W$ is a Banach space. We will use the fact that, if $X$ is
profinite and $Y$ is a Banach space, all continuous functions from $X$ to
$Y^{0}/p^{n}$ (where $Y^{0}$ is the unit ball of $Y$) are locally constant. We
can apply this to $X={\mathscr{G}}_{K}\times\cdots\times{\mathscr{G}}_{K}$ and
use original Tate’s arguments to deduce that the inflation map
$H^{i}(\Gamma^{\prime}_{K},\widehat{K}^{\prime}_{\infty}\otimes^{\Box}_{K}W)\to
H^{i}({\mathscr{G}}_{K},C\otimes^{\Box}_{K}W)$ is an isomorphism. Then, we use
the fact that $\widehat{K}^{\prime}_{\infty}=K\oplus R$ with $\gamma_{K}-1$
invertible with a continuous inverse on $R$. It follows that
$H^{i}(\Gamma^{\prime}_{K},R\otimes^{\Box}_{K}W)=0$, for all $i$, and that the
canonical map $H^{i}(\Gamma^{\prime}_{K},W)\to
H^{i}(\Gamma^{\prime}_{K},\widehat{K}^{\prime}_{\infty}\otimes^{\Box}_{K}W)$
is an isomorphism. We have proved our proposition in the Banach case.

Assume now that $W$ is a nuclear Fréchet space. Write $W\simeq\lim_{n}W_{n}$,
for a projective system $\\{W_{n}\\}_{n\in{\bf N}}$, of Banach spaces with
compact transition maps and such that the projections $p_{s}:\lim_{n}W_{n}\to
W_{s}$, $s\in{\bf N}$, have dense images (see [30, Ch. 16] for why this is
possible). We have

(3.17) $\displaystyle H^{i}({\mathscr{G}}_{K},W(j)\otimes^{\Box}_{K}C)$
$\displaystyle\simeq
H^{i}({\mathscr{G}}_{K},(\lim_{n}W_{n}(j))\otimes^{\Box}_{K}C)\simeq
H^{i}({\mathscr{G}}_{K},\lim_{n}(W_{n}(j)\otimes^{\Box}_{K}C))$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\to}}H^{i}({\mathscr{G}}_{K},{\mathrm{R}}\lim_{n}(W_{n}(j)\otimes^{\Box}_{K}C))\stackrel{{\scriptstyle\sim}}{{\to}}\lim_{n}H^{i}({\mathscr{G}}_{K},W_{n}(j)\otimes^{\Box}_{K}C).$

The third isomorphism follows from the fact that the pro-system
$\\{W_{n}(j)\otimes^{\Box}_{K}C\\}_{n\in{\bf N}}$ is Mittag-Leffler (it is a
pro-system of $C$-Banach spaces with projection maps $p_{s}{\otimes}{\rm Id}$
having dense images). The fourth isomorphism follows from the fact that
${\mathrm{R}}^{1}\lim_{n}H^{i-1}({\mathscr{G}}_{K},W_{n}(j)\otimes^{\Box}_{K}C)=0$
since the pro-system
$\\{H^{i}({\mathscr{G}}_{K},W_{n}(j)\otimes^{\Box}_{K}C)\\}_{n\in{\bf N}}$ is
Mittag-Leffler: by (3.13), this system is or trivial or isomorphic to the pro-
system $\\{W_{n}\\}_{n\in{\bf N}}$, with projection maps $p_{s}$ having dense
images maps.

Having the topological isomorphisms (3.17), by (3.13) we have, in the
nontrivial cases, topological isomorphisms

$\lim_{n}H^{i}({\mathscr{G}}_{K},W_{n}(j)\otimes^{\Box}_{K}C)\stackrel{{\scriptstyle\sim}}{{\leftarrow}}\lim_{n}W_{n}\stackrel{{\scriptstyle\sim}}{{\leftarrow}}W,$

as wanted.

Finally, assume that $W$ is of compact type. We write
$W\simeq\operatorname{colim}_{n}W_{n}$, for an inductive system
$\\{W_{n}\\}_{n\in{\bf N}}$, of Banach spaces with injective, compact
transition maps. Then

(3.18) $\displaystyle H^{i}({\mathscr{G}}_{K},W(j)\otimes^{\Box}_{K}C)$
$\displaystyle\simeq
H^{i}({\mathscr{G}}_{K},(\operatorname{colim}_{n}W_{n}(j))\otimes^{\Box}_{K}C)\simeq
H^{i}({\mathscr{G}}_{K},\operatorname{colim}_{n}(W_{n}(j)\otimes^{\Box}_{K}C))$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\leftarrow}}\operatorname{colim}_{n}H^{i}({\mathscr{G}}_{K},W_{n}(j)\otimes^{\Box}_{K}C).$

The third isomorphism follows from the fact that ${\bf Z}[{\mathscr{G}}_{K}]$
is a compact object in ${\rm CondAb}$.

Having the isomorphisms (3.18), by (3.13) we have in the nontrivial cases
isomorphisms

$\operatorname{colim}_{n}H^{i}({\mathscr{G}}_{K},W_{n}(j)\otimes^{\Box}_{K}C)\stackrel{{\scriptstyle\sim}}{{\leftarrow}}\operatorname{colim}_{n}W_{n}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}W,$

as wanted. ∎

##  4\. Pro-étale cohomology

In this chapter, we study the properties of pro-étale cohomology of smooth
dagger curves over $K$. Moreover we assume that the curve is either proper, or
Stein, or a dagger affinoid.

###  4.1. Topology on $p$-adic pro-étale cohomology

Let $X$ be a rigid analytic variety over $K$ or $C$.

####  4.1.1. The condensed approach

######  Definition 4.1.

  1. (1)

We define the pro-étale site of $X$ as
$X_{\operatorname{pro\acute{e}t}}:=X^{\Diamond}_{\operatorname{qpro\acute{e}t}},$
where $X^{\Diamond}$ is the diamond associated to $X$ and
$X^{\Diamond}_{\operatorname{qpro\acute{e}t}}$ denotes the quasi-pro-étale
site of $X^{\Diamond}$ [32, Def. 14.1].

  2. (2)

For a sheaf ${\mathscr{F}}$ on $X_{\operatorname{pro\acute{e}t}}$ with values
in ${\mathscr{D}}({\rm CondAb})$, the pro-étale cohomology complex

${\mathrm{R}}\Gamma_{\operatorname{pro\acute{e}t},\Box}(X,{\mathscr{F}})\in{\mathscr{D}}({\rm
CondAb}).$

This is because the category ${\rm CondAb}$ is closed under all limits and
colimits (see Section 2.2.1). The pro-étale cohomology groups
$H^{i}_{\operatorname{pro\acute{e}t},\Box}(X,{\mathscr{F}})$, for $i\geq 0$,
are objects of ${\rm CondAb}$. Similarly, for a sheaf ${\mathscr{F}}$ with
values in ${\mathscr{D}}({\rm Mod}^{\rm cond}_{{\bf Q}_{p}})$.

If a sheaf ${\mathscr{F}}$ in Definition 4.1 has values in ${\mathscr{D}}({\rm
Solid})$ then
${\mathrm{R}}\Gamma_{\operatorname{pro\acute{e}t},\Box}(X,{\mathscr{F}})$ has
values in ${\mathscr{D}}({\rm Solid})$ as well because the category ${\rm
Solid}$ is closed under all limits and colimits (see Section 2.2.2).
Similarly, for the category ${\rm Mod}^{\rm solid}_{{\bf Q}_{p}}$.

####  4.1.2. Comparison with the classical approach

Recall that ${\mathrm{R}}\Gamma(X,{\bf
Q}_{p}):={\mathrm{R}}\Gamma_{\operatorname{pro\acute{e}t}}(X,{\bf
Q}_{p})\in{\mathscr{D}}(C^{\rm Hcg}_{{\bf Q}_{p}})$. Locally it is a complex
of Banach spaces over ${\bf Q}_{p}$; globally – of Fréchet spaces over ${\bf
Q}_{p}$.

######  Lemma 4.2.

  1. (1)

We have a natural quasi-isomorphism in ${\mathscr{D}}({\rm
Mod}^{\operatorname{cond}}_{{\bf Q}_{p}})$

${\rm CD}({\mathrm{R}}\Gamma(X,{\bf
Q}_{p}))\simeq{\mathrm{R}}\Gamma_{\Box}(X,{\bf Q}_{p}).$

  2. (2)

We have a natural isomorphism in ${\mathscr{D}}(C_{{\bf Q}_{p}})$

${\mathrm{R}}\Gamma_{\Box}(X,{\bf Q}_{p})(*)_{\rm
top}\simeq{\mathrm{R}}\Gamma(X,{\bf Q}_{p}).$

  3. (3)

We have a natural quasi-isomorphism in ${\rm Mod}^{\operatorname{cond}}_{{\bf
Q}_{p}}$

${\rm CD}(\widetilde{H}^{i}(X,{\bf Q}_{p}))\simeq H^{i}_{\Box}(X,{\bf
Q}_{p}).$

###### Proof.

Claims (1) and (2) follow from the fact that ${\mathrm{R}}\Gamma(X,{\bf
Q}_{p})$ is represented locally by Galois cohomology of the fundamental group
and we have Lemma 3.3. Claim (3) follows from claim (1) and Section 2.2.3. ∎

######  Remark 4.3.

By the same arguments, the analog of Lemma 4.2 holds for de Rham cohomology
(de Rham complex) as well as for Hyodo-Kato cohomology (see [17, Sec. 4.2] for
the definition of the latter).

###  4.2. Hochschild-Serre spectral sequence

We record here the Hochschild-Serre spectral sequence for pro-étale
cohomology.

######  Lemma 4.4.

(Bosco, [5, Prop. 4.12]) Let $X$ be a rigid analytic variety over $K$. There
is a natural Hochschild-Serre spectral sequence

(4.5) $E^{a,b}_{2}=H^{a}({\mathscr{G}}_{K},H^{b}_{\Box}(X_{C},{\bf
Q}_{p}(j)))\Rightarrow H^{a+b}_{\Box}(X,{\bf Q}_{p}(j)).$

###### Proof.

We pass to the world of diamonds. Since $X_{C}\to X$ is a
${\mathscr{G}}_{K}$-torsor, we have isomorphisms

(4.6) $(X_{C}/X)^{n}\simeq X_{C}\times{\mathscr{G}}_{K}^{n-1},\quad n\geq 1.$

It suffices thus to show that, for any adic space $Y$ over ${\rm
Spa}(K,{\mathscr{O}}_{K})$ and any profinite set $S$, we have a natural quasi-
isomorphism in ${\mathscr{D}}({\rm CondAb})$ (take $Y=X_{C}$ and
$S={\mathscr{G}}_{K}^{n-1}$ in the notation from (4.6))

${\mathrm{R}}\Gamma(Y\times S,{\bf
Q}_{p})\simeq{\mathrm{R}}\underline{{\rm{Hom}}}({\bf
Z}_{\Box}[S],{\mathrm{R}}\Gamma(Y,{\bf Q}_{p})).$

This is local on $Y$, hence we may assume that $Y$ is a w-contractible space
over ${\rm Spa}(K,{\mathscr{O}}_{K})$. Then we have the following natural
quasi-isomorphisms

(4.7) ${\mathrm{R}}\Gamma(Y\times S,{\bf Q}_{p})\simeq\underline{{\bf
Q}_{p}(Y\times S)}\simeq\underline{{\mathscr{C}}(S,{\bf
Q}_{p}(Y))}\simeq\underline{{\rm{Hom}}}({\bf Z}[S],\underline{{\bf
Q}_{p}(Y)})\simeq\underline{{\rm{Hom}}}({\bf Z}_{\Box}[S],\underline{{\bf
Q}_{p}(Y)}).$

To see the first quasi-isomorphism note that, for any profinite set $T$, we
have

${\mathrm{R}}\Gamma(Y\times T,{\bf
Z}/p^{n})\simeq{\mathrm{R}}\Gamma_{\operatorname{\acute{e}t}}(Y\times T,{\bf
Z}/p^{n})\simeq{\bf Z}/p^{n}(Y\times T),$

where the last quasi-isomorphism holds because $Y\times T$ is a strictly
totally disconnected perfectoid space (by [32, Lemma 7.19]), and the pro-
system $\\{{\bf Z}/p^{n}(Y\times T)\\}_{n\in{\bf N}}$ is Mittag-Leffler. The
second quasi-isomorphism in (4.7) follows from the fact that, for any
profinite set $S^{\prime}$,

${\mathscr{C}}(S^{\prime},{\mathscr{C}}(S,{\bf
Q}_{p}(Y)))\simeq{\mathscr{C}}(S^{\prime}\times S,{\bf Q}_{p}(Y)).$

Now, since ${\bf Z}_{\Box}[S]$ is an internally projective object in Solid,
the right-hand side of (4.7) identifies with

${\mathrm{R}}\underline{{\rm{Hom}}}({\bf Z}_{\Box}[S],\underline{{\bf
Q}_{p}(Y)})\simeq{\mathrm{R}}\underline{{\rm{Hom}}}({\bf
Z}_{\Box}[S],{\mathrm{R}}\Gamma(Y,{\bf Q}_{p})),$

as wanted. ∎

###  4.3. Cohomology of Stein curves

We will now discuss arithmetic and geometric pro-étale cohomology of smooth
Stein curves.

####  4.3.1. Geometric cohomology of Stein varieties

We start with geometric pro-étale cohomology. Let $X$ be a smooth Stein
variety over $K$, geometrically irreducible. By [18, Th. 5.14],
${\mathrm{R}}\Gamma(X_{C},{\bf Q}_{p}(j))\in{\mathscr{D}}(C_{{\bf Q}_{p}})$
has classical cohomology. Moreover, $H^{i}(X_{C},{\bf Q}_{p}(j))$, for $i\geq
0$, is Fréchet and we have a Galois equivariant strict map of strictly exact
sequences of Fréchet spaces

(4.8)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{i-1}(X_{C})/\ker
d\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{i}(X_{C},{\bf
Q}_{p}(i))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{(H^{i}_{\rm
HK}(X_{C})\widehat{\otimes}_{\breve{F}}\widehat{{\bf
B}}^{+}_{\operatorname{st}})^{N=0,\varphi=p^{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\rm
HK}\otimes\theta}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{i-1}(X_{C})/\ker
d\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\textstyle{\Omega^{i}(X_{C})^{d=0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{i}_{\operatorname{dR}}(X_{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Warning: These spaces are not nuclear Fréchet over ${\bf Q}_{p}$ ($C$ is not a
nuclear Banach space over ${\bf Q}_{p}$ because it is not finite dimensional
over ${\bf Q}_{p}$) !

Similarly, in the condensed language, we have the following:

######  Lemma 4.9.

The cohomology $H^{i}_{\Box}(X_{C},{\bf Q}_{p}(j))$, for $i\geq 0$, is Fréchet
and we have a Galois equivariant map of exact sequences of Fréchet spaces

(4.10)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{i-1}(X_{C})/\ker
d\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{\Box}^{i}(X_{C},{\bf
Q}_{p}(i))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{(H^{i}_{{\rm
HK},\Box}(X_{C}){\otimes}^{\Box}_{\breve{F}}\widehat{{\bf
B}}^{+}_{\operatorname{st}})^{N=0,\varphi=p^{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\rm
HK}\otimes\theta}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{i-1}(X_{C})/\ker
d\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\textstyle{\Omega^{i}(X_{C})^{d=0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{i}_{\operatorname{dR},\Box}(X_{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

###### Proof.

We apply the functor ${\rm CD}(-)$ to the diagram (4.8). Since all the spaces
in that diagram are Fréchet, by Lemma 2.18, we obtain a map of exact
sequences. By Lemma 4.2 and Remark 4.3, it remains to show that

${\rm CD}((H^{i}_{{\rm HK}}(X_{C})\widehat{\otimes}_{\breve{F}}\widehat{{\bf
B}}^{+}_{\operatorname{st}})^{N=0,\varphi=p^{i}})\simeq(H^{i}_{{\rm
HK},\Box}(X_{C}){\otimes}^{\Box}_{\breve{F}}\widehat{{\bf
B}}^{+}_{\operatorname{st}})^{N=0,\varphi=p^{i}}.$

Or, since the functor ${\rm CD}(-)$ is left exact, that

${\rm CD}(H^{i}_{{\rm HK}}(X_{C})\widehat{\otimes}_{\breve{F}}\widehat{{\bf
B}}^{+}_{\operatorname{st}})\simeq(H^{i}_{{\rm
HK},\Box}(X_{C}){\otimes}^{\Box}_{\breve{F}}\widehat{{\bf
B}}^{+}_{\operatorname{st}}).$

But, since $H^{i}_{{\rm HK}}(X_{C})\simeq\lim_{n}H^{i}_{{\rm HK}}(X_{n,C})$,
where $H^{i}_{{\rm HK}}(X_{n,C})$ are of finite rank over $\breve{F}$, we have

$\displaystyle{\rm CD}(H^{i}_{{\rm
HK}}(X_{C})\widehat{\otimes}_{\breve{F}}\widehat{{\bf
B}}^{+}_{\operatorname{st}})$ $\displaystyle\simeq{\rm CD}(\lim_{n}H^{i}_{{\rm
HK}}(X_{n,C})\widehat{\otimes}_{\breve{F}}\widehat{{\bf
B}}^{+}_{\operatorname{st}})\simeq\lim_{n}{\rm CD}(H^{i}_{{\rm
HK}}(X_{n,C})\widehat{\otimes}_{\breve{F}}\widehat{{\bf
B}}^{+}_{\operatorname{st}})$ $\displaystyle\simeq\lim_{n}({\rm
CD}(H^{i}_{{\rm HK}}(X_{n,C})){\otimes}^{\Box}_{\breve{F}}\widehat{{\bf
B}}^{+}_{\operatorname{st}})\simeq\lim_{n}(H^{i}_{{\rm
HK},\Box}(X_{n,C}){\otimes}^{\Box}_{\breve{F}}\widehat{{\bf
B}}^{+}_{\operatorname{st}})$ $\displaystyle\simeq(H^{i}_{{\rm
HK},\Box}(X_{C}){\otimes}^{\Box}_{\breve{F}}\widehat{{\bf
B}}^{+}_{\operatorname{st}}),$

The second isomorphism follows from the fact that the functor ${\rm CD}(-)$
commutes with limits; the third one – from the fact that $H^{i}_{{\rm
HK}}(X_{n,C})$ is of finite rank. The last one – from the fact that
$\widehat{{\bf B}}^{+}_{\operatorname{st}}$ is Banach and $H^{i}_{{\rm
HK},\Box}(X_{C})$ is a product of spaces of finite rank. ∎

Notation: From now on we will omit the $\Box$ in
${\mathrm{R}}\Gamma_{\Box}(-,-)$ and other cohomologies. This should not cause
confusion.

####  4.3.2. Arithmetic cohomology of Stein curves

We pass now to arithmetic pro-étale cohomology. Let $X$ be a smooth Stein
curve over $K$, geometrically irreducible. We will look at its arithmetic pro-
étale cohomology complex ${\mathrm{R}}\Gamma(X,{\bf
Q}_{p}(j))\in{\mathscr{D}}({\bf Q}_{p,\Box}).$

######  Theorem 4.11.

  1. (1)

The cohomology of ${\mathrm{R}}\Gamma(X,{\bf Q}_{p}(j)),j\in{\bf Z},$ is
nuclear Fréchet.

  2. (2)

Let $\\{X_{n}\\}_{n\in{\bf N}}$ be a strictly increasing open covering of $X$
by Stein varieties and let $i,j\in{\bf Z}$. Then
${\mathrm{R}}^{1}\lim_{n}H^{i}(X_{n},{\bf Q}_{p}(j))=0$. Hence we have an
isomorphism

$H^{i}(X,{\bf
Q}_{p}(j))\stackrel{{\scriptstyle\sim}}{{\to}}\lim_{n}H^{i}(X_{n},{\bf
Q}_{p}(j)).$

###### Proof.

Claim (1). By (4.5), we have the Hochschild-Serre spectral sequence

(4.12) $E^{a,b}_{2}=H^{a}({\mathscr{G}}_{K},H^{b}(X_{C},{\bf
Q}_{p}(j)))\Rightarrow H^{a+b}(X,{\bf Q}_{p}(j)).$

From diagram (4.10), we know that the only nontrivial cohomology groups of
$X_{C}$ are in degrees 0,1. Hence, from the spectral sequence (4.12), we get
that $H^{i}(X,{\bf Q}_{p}(j))=0$, for $i\geq 4$, and we have the long exact
sequence

(4.13) $\displaystyle 0\to$ $\displaystyle
H^{0}({\mathscr{G}}_{K},H^{0}(X_{C},{\bf Q}_{p}(j)))\to H^{0}(X,{\bf
Q}_{p}(j))\to H^{-1}({\mathscr{G}}_{K},H^{1}(X_{C},{\bf Q}_{p}(j)))$
$\displaystyle\to$ $\displaystyle H^{1}({\mathscr{G}}_{K},H^{0}(X_{C},{\bf
Q}_{p}(j)))\to H^{1}(X,{\bf Q}_{p}(j))\to
H^{0}({\mathscr{G}}_{K},H^{1}(X_{C},{\bf Q}_{p}(j)))$
$\displaystyle\stackrel{{\scriptstyle d_{2}}}{{\to}}$ $\displaystyle
H^{2}({\mathscr{G}}_{K},H^{0}(X_{C},{\bf Q}_{p}(j)))\to H^{2}(X,{\bf
Q}_{p}(j))\to H^{1}({\mathscr{G}}_{K},H^{1}(X_{C},{\bf Q}_{p}(j)))$
$\displaystyle\to$ $\displaystyle H^{3}({\mathscr{G}}_{K},H^{0}(X_{C},{\bf
Q}_{p}(j)))\to H^{3}(X,{\bf Q}_{p}(j))\to
H^{2}({\mathscr{G}}_{K},H^{1}(X_{C},{\bf Q}_{p}(j)))\to 0$

(i) The groups $H^{0}(X,{\bf Q}_{p}(j))$ and $H^{3}(X,{\bf Q}_{p}(j))$.

Diagram (4.13) yields the isomorphisms

(4.14) $\displaystyle H^{0}(X,{\bf Q}_{p}(j))={\bf Q}_{p},$ $\displaystyle
H^{3}(X,{\bf
Q}_{p}(j))\stackrel{{\scriptstyle\sim}}{{\to}}H^{2}({\mathscr{G}}_{K},H^{1}(X_{C},{\bf
Q}_{p}(j))).$

The top line of diagram (4.10) gives the exact sequence

$0\to{\mathscr{O}}(X_{C})/C\to H^{1}(X_{C},{\bf Q}_{p}(1))\to{\rm
HK}^{1}(X_{C},1)\to 0$

Applying Galois cohomology to it we get the exact sequence (we set $s:j-1$)

(4.15) $\displaystyle 0$ $\displaystyle\to
H^{0}({\mathscr{G}}_{K},({\mathscr{O}}(X_{C})/C)(s))\to
H^{0}({\mathscr{G}}_{K},H^{1}(X_{C},{\bf Q}_{p}(1))(s))\to
H^{0}({\mathscr{G}}_{K},{\rm HK}^{1}(X_{C},1)(s))$ $\displaystyle\to
H^{1}({\mathscr{G}}_{K},({\mathscr{O}}(X_{C})/C)(s))\to
H^{1}({\mathscr{G}}_{K},H^{1}(X_{C},{\bf Q}_{p}(1))(s))\to
H^{1}({\mathscr{G}}_{K},{\rm HK}^{1}(X_{C},1)(s))$ $\displaystyle\to
H^{2}({\mathscr{G}}_{K},({\mathscr{O}}(X_{C})/C)(s))\to
H^{2}({\mathscr{G}}_{K},H^{1}(X_{C},{\bf Q}_{p}(1))(s))\to
H^{2}({\mathscr{G}}_{K},{\rm HK}^{1}(X_{C},1)(s))\to 0.$

Using it, the isomorphisms (4.14), and the generalized Tate’s formulas (3.16),
we get the isomorphism

(4.16) $H^{3}(X,{\bf
Q}_{p}(j))\stackrel{{\scriptstyle\sim}}{{\to}}H^{2}({\mathscr{G}}_{K},{\rm
HK}^{1}(X_{C},1)(s)).$

(ii) Key lemma. Claim (1) of Theorem 4.11 follows from the following fact. Let
$\\{X_{n}\\}_{n\in{\bf N}}$ be a strictly increasing covering of $X$ by
adapted naive interiors of affinoids, i.e., there exists a strictly increasing
(Stein) covering $\\{\overline{X}_{n}\\}_{n\in{\bf N}}$ of $X$ such that
$X_{n+1}$ is a naive interior in $\overline{X}_{n+1}$ adapted to $X_{n}$.

######  Remark 4.17.

By definition, a naive interior of a smooth (dagger) affinoid is a Stein
subvariety whose complement is open and quasi-compact. It is easy to see that,
for a pair of (dagger) affinoids $X_{1}\Subset X_{2}$ there exists a naive
interior $X_{2}^{0}\subset X_{2}$ such that $X_{1}\subset X_{2}^{0}\subset
X_{2}$. We will say that $X_{2}$ is adapted to $X_{1}$.

######  Lemma 4.18.

The transition maps

(4.19) $f_{i,n}:\quad H^{i}(X_{n+1},{\bf Q}_{p}(j))\to H^{i}(X_{n},{\bf
Q}_{p}(j)),\quad n\geq 0,$

are compact maps of (nuclear) Fréchet spaces.

###### Proof.

By the computations in (i), which can be applied to each $X_{n}$ since that
variety is Stein, this is clear for $i=0$.

For $i=3$, the isomorphism (4.16) above combined with the fact that ${\rm
HK}^{1}(X_{n,C},1)$ is an almost $C$-representation (because $H^{1}_{\rm
HK}(X_{n,C})$ is of finite rank over $\breve{F}$ since
$H^{1}_{\operatorname{dR}}(X_{n,C})$ is of finite rank over $C$) yields that
$H^{3}(X_{n},{\bf Q}_{p}(j))$ is a finite rank ${\bf Q}_{p}$-vector space.
Hence the maps $f_{3,n}$ are as wanted.

It remains to treat the cases of $i=1,2$. We start with showing that the
spaces

$H^{a}({\mathscr{G}}_{K},H^{b}(X_{n,C},{\bf Q}_{p}(j))),\quad a,b\in{\bf Z},$

appearing the spectral sequence (4.12) are nuclear Fréchet. To see that, we
apply Galois cohomology to the top row of diagram (4.10) for $X_{n}$ and
obtain the exact sequence (we set $s:=j-b$; ${\rm
HK}^{j}(X_{n,C},i):=(H^{j}_{{\rm
HK}}(X_{n,C}){\otimes}^{\Box}_{\breve{F}}\widehat{{\bf
B}}^{+}_{\operatorname{st}})^{N=0,\varphi=p^{i}}$)

(4.20) $\displaystyle\to H^{a-1}({\mathscr{G}}_{K},$ $\displaystyle{\rm
HK}^{b}(X_{n,C},b)(s))\stackrel{{\scriptstyle\partial_{a-1}}}{{\longrightarrow}}H^{a}({\mathscr{G}}_{K},(\Omega^{b-1}(X_{n,C})/\ker
d)(s))$ $\displaystyle\to H^{a}({\mathscr{G}}_{K},H^{b}(X_{n,C},{\bf
Q}_{p}(j)))\to H^{a}({\mathscr{G}}_{K},{\rm HK}^{b}(X_{n,C},b)(s))$
$\displaystyle\stackrel{{\scriptstyle\partial_{a}}}{{\longrightarrow}}H^{a+1}({\mathscr{G}}_{K},(\Omega^{b-1}(X_{n,C})/\ker
d)(s))\to$

We claim that the spaces $H^{i}({\mathscr{G}}_{K},(\Omega^{b-1}(X_{n,C})/\ker
d)(s))$ and $H^{i}({\mathscr{G}}_{K},{\rm HK}^{b}(X_{n,C},b)(s))$ are nuclear
Fréchet. Indeed, for the first one this follows from the generalized Tate’s
isomorphism (3.16): if nontrivial

$H^{i}({\mathscr{G}}_{K},(\Omega^{b-1}(X_{n,C})/\ker
d)(s))\simeq\Omega^{b-1}(X_{n})/\ker d$

since $\Omega^{b-1}(X_{n,C})/\ker d\simeq(\Omega^{b-1}(X_{n})/\ker
d)\otimes_{K}^{\Box}C$; and the fact that $\Omega^{b-1}(X_{n})/\ker d$ is a
nuclear Fréchet. For the second one, we use the fact that
$H^{i-1}({\mathscr{G}}_{K},{\rm HK}^{b}(X_{n,C},b)(s))$ is a finite rank ${\bf
Q}_{p}$-vector space by the isomorphism (3.16) (since ${\rm
HK}^{b}(X_{n,C},b)$ is an almost $C$-representation).

The above computations imply that the maps $\partial_{a-1}$ and $\partial_{a}$
in (4.20) are between nuclear Fréchet spaces hence
$H^{a}({\mathscr{G}}_{K},H^{b}(X_{n,C},{\bf Q}_{p}(j)))$ is an extension of
two nuclear Fréchet spaces. In fact, it is an extension of a finite rank ${\bf
Q}_{p}$-vector space by a nuclear Fréchet space hence a nuclear Fréchet space.

We proceed now to prove Lemma 4.18 for $i=1,2$. From (4.13), we get a long
exact sequence

(4.21) $\displaystyle 0$ $\displaystyle\to H^{1}({\mathscr{G}}_{K},{\bf
Q}_{p}(j))\to H^{1}(X_{n},{\bf Q}_{p}(j))\to
H^{0}({\mathscr{G}}_{K},H^{1}(X_{n,C},{\bf Q}_{p}(j)))$
$\displaystyle\stackrel{{\scriptstyle
d_{2,n}}}{{\to}}H^{2}({\mathscr{G}}_{K},{\bf Q}_{p}(j))\to H^{2}(X_{n},{\bf
Q}_{p}(j))\to H^{1}({\mathscr{G}}_{K},H^{1}(X_{n,C},{\bf Q}_{p}(j)))\to 0$

(a) Case $j\neq 1$. In this case, we have $H^{2}({\mathscr{G}}_{K},{\bf
Q}_{p}(j))=0$. Thus, it suffices to show that the spaces
$H^{i}({\mathscr{G}}_{K},H^{1}(X_{n,C},{\bf Q}_{p}(j)))$, for $i=0,1$, are of
finite rank over ${\bf Q}_{p}$. For that, since the vector spaces
$H^{i}({\mathscr{G}}_{K},{\rm HK}^{1}(X_{n,C},1)(j-1))$, for $i=0,1$, are of
finite rank over ${\bf Q}_{p}$, it suffices to notice that
$H^{i}({\mathscr{G}}_{K},({\mathscr{O}}(X_{n,C})/C)(j-1))=0$, for $i=0,1$, by
the generalized Tate’s formulas (3.16).

(b) Case $j=1$. To start, we claim that the spaces $H^{i}(X_{n},{\bf
Q}_{p}(1))$, $i=1,2$ are nuclear Fréchet. Indeed, we have shown above that the
spaces $H^{i}({\mathscr{G}}_{K},H^{1}(X_{n,C},{\bf Q}_{p}(j)))$ are Fréchet.
We also know that we have exact sequences

$0\to V_{0,i,n}\stackrel{{\scriptstyle g_{i,n}}}{{\to}}V_{1,i,n}\to
H^{i}(X_{n},{\bf Q}_{p}(j))\to 0,$

with $V_{0,i,n},V_{1,i,n}$ solid Fréchet $K$-vector spaces. Moreover,
$H^{i}(X_{n},{\bf Q}_{p}(j))$ is quasi-separated since it is an extension of
quasi-separated solid $K$-vector spaces (see (4.21)). It follows that the map
$g_{i,n}$ is quasi-compact and, hence, the induced map
$g_{i,n}:V_{0,i,n}(*)_{\rm top}\to V_{1,i,n}(*)_{\rm top}$ is a closed
embedding. As a result, $V_{1,i,n}(*)_{\rm top}/V_{0,i,n}(*)_{\rm top}$ is a
(classical) Fréchet space and then this implies that $H^{i}(X_{n},{\bf
Q}_{p}(j))\simeq{\rm CD}(V_{1,i,n}(*)_{\rm top}/V_{0,i,n}(*)_{\rm top})$ is
Fréchet, as wanted. By Lemma 2.5, as an extension of a nuclear Fréchet space
by a finite rank vector space, it is nuclear.

We will show below (in fact (c)) that the pro-systems
$\\{H^{i}({\mathscr{G}}_{K},H^{1}(X_{n,C},{\bf Q}_{p}(j)))\\}_{n\in{\bf N}}$
have compact transition maps (we will call such systems compact). Then the
pro-system $\\{\ker d_{2,n}\\}_{n\in{\bf N}}$ is also compact. And, by Lemma
2.7, the maps $f_{i,n}$, $i=1,2$, from our theorem are compact, as wanted.

(c) The pro-systems $\\{H^{i}({\mathscr{G}}_{K},H^{1}(X_{n,C},{\bf
Q}_{p}(1))(s))\\}_{n\in{\bf N}}$, for $i=0,1$, are compact. To prove that,
note that the vector spaces $H^{i}({\mathscr{G}}_{K},{\rm
HK}^{1}(X_{n,C},1)(s))$, for $i=0,1,$ are of finite rank over ${\bf Q}_{p}$.
Hence, by (4.15), it suffices to show that the pro-systems
$\\{H^{i}({\mathscr{G}}_{K},({\mathscr{O}}(X_{n,C})/C)(s))\\}_{n\in{\bf N}}$
are compact. But this is clear since, by the generalized Tate’s formulas
(3.16) (note that ${\mathscr{O}}(X_{n})/K$ is a nuclear Fréchet), these pro-
systems are or trivial or we have isomorphisms

$\\{H^{i}({\mathscr{G}}_{K},({\mathscr{O}}(X_{n,C})/C)(s))\\}_{n\in{\bf
N}}\simeq\\{{\mathscr{O}}(X_{n})/K\\}_{n\in{\bf N}}.\qed$

This finishes the proof of claim (1) of the theorem.

Claim (2). It suffices to show that

$H^{i}(X,{\bf
Q}_{p}(j))\stackrel{{\scriptstyle\sim}}{{\to}}\lim_{n}H^{i}(X_{n},{\bf
Q}_{p}(j)).$

Or that ${\mathrm{R}}^{1}\lim_{n}H^{i}(X_{n},{\bf Q}_{p}(j))=0$, $i,j\in{\bf
Z}$. Since the spectral sequence (4.12) degenerates at $E_{3}$ it suffices to
show that ${\mathrm{R}}^{1}\lim_{n}E^{a,b}_{3}(X_{n})=0$. Since cohomological
dimension of ${\mathscr{G}}_{K}$ is $2$, we have

$E_{3}^{a,b}(X_{n})=\begin{cases}\ker d_{2}^{0,b}(X_{n})&\mbox{ if }a=0,\\\
E_{2}^{1,b}(X_{n})&\mbox{ if }a=1,\\\
\operatorname{coker}d_{2}^{2,b+1}(X_{n})&\mbox{ if }a=2,\end{cases}$

where $d_{2}^{0,b}(X_{n}):H^{0}({\mathscr{G}}_{K},H^{b}(X_{n,C},{\bf
Q}_{p}(j)))\to H^{2}({\mathscr{G}}_{K},H^{b-1}(X_{n,C},{\bf Q}_{p}(j)))$ is
the only nontrivial differential in the spectral sequence (4.12). Hence, using
the computations from claim (1), we get immediately that
${\mathrm{R}}^{1}\lim_{n}E^{1,b}_{3}(X_{n})=0$ and, since

${\mathrm{R}}^{1}\lim_{n}H^{2}({\mathscr{G}}_{K},H^{b-1}(X_{n,C},{\bf
Q}_{p}(j)))=0,$

that ${\mathrm{R}}^{1}\lim_{n}E^{2,b}_{3}(X_{n})=0$. It remains to show that

${\mathrm{R}}^{1}\lim_{n}E^{0,b}_{3}(X_{n})={\mathrm{R}}^{1}\lim_{n}\ker
d_{2}^{0,b}(X_{n})=0.$

From the exact sequence (4.20), since $H^{0}({\mathscr{G}}_{K},{\rm
HK}^{b}(X_{n,C},b)(s))$ is finite over ${\bf Q}_{p}$ (because ${\rm
HK}^{b}(X_{n,C},b)(s)$ is an almost $C$-representation), it suffices to show
that
${\mathrm{R}}^{1}\lim_{n}H^{0}({\mathscr{G}}_{K},\Omega^{b-1}(X_{n,C})/\ker
d)=0$. But, by the generalized Tate’s isomorphism (3.16),
$H^{0}({\mathscr{G}}_{K},\Omega^{b-1}(X_{n,C})/\ker
d)\simeq\Omega^{b-1}(X_{n})/\ker d$, so it suffices to show that
${\mathrm{R}}^{1}\lim_{n}\Omega^{b-1}(X_{n})=0$ but this is known. ∎

###  4.4. Filtration

Let $X$ be a smooth geometrically irreducible Stein analytic curve over $K$.
Let $i,j\in{\bf Z}$. Under certain conditions, there exists an ascending
filtration on $H^{i}(X,{\bf Q}_{p}(j))$:

$F^{2}_{i,j}=H^{i}(X,{\bf Q}_{p}(j))\supset F^{1}_{i,j}\supset
F^{0}_{i,j}\supset F^{-1}_{i,j}=0,$

such that we have isomorphisms

$\displaystyle F^{2}_{i,j}/F^{1}_{i,j}\simeq H^{i-1}({\mathscr{G}}_{K},{\bf
Q}_{p}(j-1)),$ $\displaystyle F^{1}_{i,j}/F^{0}_{i,j}\simeq
H^{i-1}({\mathscr{G}}_{K},\tfrac{{\mathscr{O}}(X_{C})}{C}(j-1)),$
$\displaystyle F^{0}_{i,j}/F^{-1}_{i,j}\simeq H^{i}({\mathscr{G}}_{K},{\rm
HK}^{1}(X_{C},1)(j-1)),$

where we set ${\rm HK}^{1}(X_{C},1)=(H^{1}_{{\rm
HK}}(X_{C}){\otimes}^{\Box}_{\breve{F}}\widehat{{\bf
B}}^{+}_{\operatorname{st}})^{N=0,\varphi=p}$. We can visualize this
filtration in the following way:

(4.22)
---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{0}_{i,j}:=H^{i}({\mathscr{G}}_{K},{\bf
Q}_{p}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{1}_{i,j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{i-1}({\mathscr{G}}_{K},\tfrac{{\mathscr{O}}(X_{C})}{C}(j-1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{i}({\mathscr{G}}_{K},{\bf
Q}_{p}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{2}_{i,j}:=H^{i}(X,{\bf
Q}_{p}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{i-1}({\mathscr{G}}_{K},H^{1}(X_{C},{\bf
Q}_{p}(j)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{H^{i-1}({\mathscr{G}}_{K},{\rm
HK}^{1}(X_{C},1)(j-1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{i-1}({\mathscr{G}}_{K},{\rm
HK}^{1}(X_{C},1)(j-1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$

The above diagram is a map of exact sequences. The right column is induced by
the syntomic filtration (see diagram (4.10)). The middle row comes from the
Hochschild-Serre spectral sequence (4.5) (we note that ${\bf Q}_{p}(j)\simeq
H^{0}(X_{C},{\bf Q}_{p}(j))$) and the vanishing of geometric cohomologies
obtained from diagram (4.10). We assume that the middle row and the right
column are exact. The term $F^{1}_{i,j}$ is defined as a pullback of the top
right square.

###  4.5. Arithmetic cohomology of dagger affinoids

Let $X$ be a smooth geometrically connected dagger affinoid over $K$. We will
now study its arithmetic pro-étale cohomology. They key tool is the (studied
above) arithmetic pro-étale cohomology of smooth Stein curves.

######  Proposition 4.23.

The cohomology of ${\mathrm{R}}\Gamma(X,{\bf Q}_{p}(j)),j\in{\bf Z},$ is of
compact type.

###### Proof.

Let $\\{X_{h}\\}$ be the dagger presentation of the dagger structure on $X$.
Denote by $X_{h}^{0}$ a naive interior of $X_{h}$ adapted to $\\{X_{h}\\}$.
The canonical quasi-isomorphism

${\mathrm{R}}\Gamma(X,{\bf
Q}_{p}(j))\simeq\operatorname{colim}_{h}{\mathrm{R}}\Gamma(X^{0}_{h},{\bf
Q}_{p}(j)),\quad j\in{\bf Z}$

yields an isomorphism

$H^{i}(X,{\bf
Q}_{p}(j))\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{colim}_{h}H^{i}(X^{0}_{h},{\bf
Q}_{p}(j)),\quad i,j\in{\bf Z}.$

We note that $X^{0}_{h}$, $h\in{\bf N}$, is a smooth Stein variety. By Lemma
4.18, the ind-systems $\\{H^{i}(X^{0}_{h},{\bf Q}_{p}(j))\\}_{h\in{\bf N}}$,
for $i\in{\bf N}$, have compact transition maps (between Fréchet spaces). This
proves our proposition. ∎

###  4.6. Examples

In this section we will compute $p$-adic pro-étale cohomology of open discs,
annuli, and their boundaries – the basic building blocks of analytic curves.

####  4.6.1. Open disc

Let $D$ be an open disc over $K$.

######  Lemma 4.24.

(Geometric cohomology) Let $j\in{\bf Z}$. We have
${\mathscr{G}}_{K}$-equivariant isomorphisms

(4.25) $H^{i}(D_{C},{\bf Q}_{p}(j))\simeq\begin{cases}{\bf Q}_{p}(j)&\mbox{ if
}i=0,\\\ ({\mathscr{O}}(D_{C})/C)(j-1)&\mbox{ if }i=1,\\\ 0&\mbox{ if }i\geq
2.\end{cases}$

Moreover, $d:{\mathscr{O}}(D_{C})/C{\to}\Omega^{1}(D_{C})$ is an isomorphism.

###### Proof.

The Hyodo-Kato isomorphism $\iota_{\rm HK}:H^{i}_{\rm
HK}(D_{C}){\otimes}_{\breve{F}}^{{\Box}}C\simeq
H^{i}_{\operatorname{dR}}(D_{C})$ and the fact that
$H^{i}_{\operatorname{dR}}(D_{C})=0$, for $i\geq 1$, yield that $H^{i}_{\rm
HK}(D_{C})=0$, for $i\geq 1$. Now, since $D$ is Stein, our lemma follows from
diagram (4.10).

The last claim is equivalent to $H^{1}_{\rm dR}(D_{C})=0$. ∎

Consider now the Hochschild-Serre spectral sequence (from Lemma 4.4):

(4.26) $E^{a,b}_{2}=H^{a}({\mathscr{G}}_{K},H^{b}(D_{C},{\bf
Q}_{p}(j)))\Rightarrow H^{a+b}(D,{\bf Q}_{p}(j)).$

By Lemma 4.24, the only nonzero rows are those of degrees $b=0,1$. We get:

######  Lemma 4.27.

(Arithmetic cohomology) Let $i\geq 0,j\in{\bf Z}$. We have exact sequences

$\displaystyle 0\to H^{i}({\mathscr{G}}_{K},H^{0}(D_{C},{\bf Q}_{p}(j)))\to
H^{i}(D,{\bf Q}_{p}(j))\to H^{i-1}({\mathscr{G}}_{K},H^{1}(D_{C},{\bf
Q}_{p}(j)))\to 0,$ $\displaystyle 0\to H^{i}({\mathscr{G}}_{K},{\bf
Q}_{p}(j))\to H^{i}(D,{\bf Q}_{p}(j))\to
H^{i-1}({\mathscr{G}}_{K},({\mathscr{O}}(D_{C})/C)(j-1))\to 0.$

###### Proof.

The second exact sequence is a translation of the first, granted formula
(4.25). The spectral sequence (4.26) yields the exact sequence

(4.28) $\displaystyle 0$ $\displaystyle\to
H^{0}({\mathscr{G}}_{K},H^{0}(D_{C},{\bf Q}_{p}(j)))\to H^{0}(D,{\bf
Q}_{p}(j))\to H^{-1}({\mathscr{G}}_{K},H^{1}(D_{C},{\bf Q}_{p}(j)))$
$\displaystyle{\to}H^{1}({\mathscr{G}}_{K},H^{0}(D_{C},{\bf Q}_{p}(j)))\to
H^{1}(D,{\bf Q}_{p}(j))\to H^{0}({\mathscr{G}}_{K},H^{1}(D_{C},{\bf
Q}_{p}(j)))$ $\displaystyle\stackrel{{\scriptstyle
d_{2}}}{{\to}}H^{2}({\mathscr{G}}_{K},H^{0}(D_{C},{\bf Q}_{p}(j))\to
H^{2}(D,{\bf Q}_{p}(j))\to H^{1}({\mathscr{G}}_{K},H^{1}(D_{C},{\bf
Q}_{p}(j)))\to 0$

First, we prove that it splits into short exact sequences. If $j\neq 1$, this
is trivial as the last terms of the first two lines are $0$.

Let us now assume that $j=1$. We need to show that the differential $d_{2}$ in
(4.28) is $0$ or, equivalently, that the canonical map
$H^{2}({\mathscr{G}}_{K},H^{0}(D_{C},{\bf Q}_{p}(j)))\to H^{2}(D,{\bf
Q}_{p}(j))$ is injective. But this map is induced by the projection $D\to K$
and any rational point in $D$ yields a section (such a point always exists). ∎

######  Remark 4.29.

We note that the groups $H^{i}(D_{C},{\bf Q}_{p}(j))$ and $H^{i}(D,{\bf
Q}_{p}(j))$ are Fréchet and nuclear Fréchet spaces, by Lemma 4.8 and Theorem
4.11, respectively.

####  4.6.2. Open annulus

Let $A$ be an open annulus over $K$.

######  Lemma 4.30.

(Geometric cohomology) Let $j\in{\bf Z}$.

  1. (1)

We have ${\mathscr{G}}_{K}$-equivariant isomorphisms

(4.31) $H^{i}(A_{C},{\bf Q}_{p}(j))\simeq\begin{cases}{\bf Q}_{p}(j)&\mbox{ if
}i=0,\\\ 0&\mbox{ if }i\geq 2.\end{cases}$

  2. (2)

We have a ${\mathscr{G}}_{K}$-equivariant exact sequence

(4.32) $0\to({\mathscr{O}}(A_{C})/C)(j-1)\to H^{1}(A_{C},{\bf
Q}_{p}(j))\to{\bf Q}_{p}(j-1)\to 0$

admitting a ${\mathscr{G}}_{K}$-equivariant ${\bf Q}_{p}$-linear splitting.

###### Proof.

Since $A$ is Stein we can use the Galois equivariant map of strictly exact
sequences (4.8) for $X=A$. From the Hyodo-Kato isomorphism $\iota_{\rm
HK}:H^{i}_{\rm HK}(A_{C}){\otimes}_{\breve{F}}^{{\Box}}C\simeq
H^{i}_{\operatorname{dR}}(A_{C})$ and the fact that
$H^{i}_{\operatorname{dR}}(A_{C})\simeq C$, for $i=0,1$, and
$H^{i}_{\operatorname{dR}}(A_{C})=0$, for $i\geq 2$, we see that

$H^{0}_{\rm HK}(A_{C})\simeq\breve{F},\quad H^{1}_{\rm
HK}(A_{C})\simeq\breve{F},\quad H^{i}_{\rm HK}(A_{C})\simeq 0,\,i\geq 2.$

The group $H^{1}_{\rm HK}(A_{C})$ is generated by the Hyodo-Kato symbol
$c_{1}^{\rm HK}(z)$, for an arithmetic coordinate $z$ of the annulus.
Frobenius acts on $c_{1}^{\rm HK}(z)$ by multiplication by $p$ and monodromy
is trivial. This implies that

$(H^{i}_{\rm HK}(A_{C}){\otimes}^{\Box}_{\breve{F}}\widehat{{\bf
B}}^{+}_{\operatorname{st}})^{N=0,\varphi=p^{i}}\simeq\begin{cases}{\bf
Q}_{p}&\mbox{if $i=0,1,$}\\\ 0&\mbox{if $i\geq 2$.}\end{cases}$

This yields the isomorphisms in our lemma.

Assume now that $i=1$. Then the diagram (4.8) becomes

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{O}}(A_{C})/C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(A_{C},{\bf
Q}_{p}(1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{{\bf
Q}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{can}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{O}}(A_{C})/C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{1}(A_{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

The top row yields the exact sequence (4.32). The term ${\bf Q}_{p}$ comes
from a Hyodo-Kato term and is generated by the Hyodo-Kato symbol $c_{1}^{\rm
HK}(z)$. The term $C$ comes from de Rham cohomology and is generated by the de
Rham symbol $c_{1}^{\rm dR}(z)$. These symbols are compatible with each other
(via the Hyodo-Kato isomorphism $\iota_{\rm HK}$) and are also compatible with
the pro-étale symbol $c_{1}^{\operatorname{pro\acute{e}t}}(z)$. Sending
$c_{1}^{\rm HK}(z)$ to $c_{1}^{\operatorname{pro\acute{e}t}}(z)$ yields the
wanted splitting of the exact sequence (4.32). ∎

Take now the Hochschild-Serre spectral sequence:

(4.33) $E^{a,b}_{2}=H^{a}({\mathscr{G}}_{K},H^{b}(A_{C},{\bf
Q}_{p}(j)))\Rightarrow H^{a+b}(A,{\bf Q}_{p}(j)).$

By (4.31), the only nonzero rows are those of degrees $b=0,1$. We get:

######  Lemma 4.34.

(Arithmetic cohomology) We have exact sequences

$\displaystyle 0\to H^{i}({\mathscr{G}}_{K},H^{0}(A_{C},{\bf Q}_{p}(j)))\to
H^{i}(A,{\bf Q}_{p}(j))\to H^{i-1}({\mathscr{G}}_{K},H^{1}(A_{C},{\bf
Q}_{p}(j)))\to 0,$ $\displaystyle 0\to
H^{i-1}({\mathscr{G}}_{K},({\mathscr{O}}(A_{C})/C)(j-1))\to
H^{i-1}({\mathscr{G}}_{K},H^{1}(A_{C},{\bf Q}_{p}(j)))\to
H^{i-1}({\mathscr{G}}_{K},{\bf Q}_{p}(j-1))\to 0.$

###### Proof.

The second sequence is obtained from the (split) exact sequence (4.32).

For the rest, we argue exactly as in the case of an open disc (see the proof
of Lemma 4.27) with the exception of the triviality of the map $d_{2}$ in the
exact sequence 4.28 when $j=1$: in this case, a rational point in $A$ does not
always exist but it does after taking a base change to a finite extension $L$
of $K$. Then the map $H^{2}({\mathscr{G}}_{L},H^{0}(A_{C},{\bf Q}_{p}(j)))\to
H^{2}(A_{L},{\bf Q}_{p}(j))$ is injective and the triviality of $d_{2}$
follows. ∎

######  Remark 4.35.

We note that the groups $H^{i}(A_{C},{\bf Q}_{p}(j))$ and $H^{i}(A,{\bf
Q}_{p}(j))$ are Fréchet and nuclear Fréchet spaces, by Lemma 4.8 and Theorem
4.11, respectively.

####  4.6.3. Ghost circle

Take now the ghost circle $Y:=\partial D$.

######  Lemma 4.36.

(Geometric cohomology) Let $i\in{\bf N}$, $j\in{\bf Z}$.

  1. (1)

We have ${\mathscr{G}}_{K}$-equivariant isomorphisms

$H^{i}(Y_{C},{\bf Q}_{p}(j))\simeq\begin{cases}{\bf Q}_{p}(j)&\mbox{ if
}i=0,\\\ 0&\mbox{ if }i\geq 2.\end{cases}$

  2. (2)

We have a ${\mathscr{G}}_{K}$-equivariant (split) exact sequence

(4.37) $0\to({\mathscr{O}}(Y_{C})/C)(j-1)\to H^{1}(Y_{C},{\bf
Q}_{p}(j))\to{\bf Q}_{p}(j-1)\to 0.$

###### Proof.

By definition, we have

$\displaystyle{\mathrm{R}}\Gamma(Y_{C},{\bf Q}_{p}(j))$
$\displaystyle=\operatorname{colim}_{0<\varepsilon<1}{\mathrm{R}}\Gamma(D_{C}\mathbin{\fgebackslash}D_{C}(\varepsilon),{\bf
Q}_{p}(j))$
$\displaystyle=\operatorname{colim}_{0<\varepsilon<1}{\mathrm{R}}\Gamma(A_{C}(\varepsilon),{\bf
Q}_{p}(j)),$

where $D_{C}(\varepsilon)$ is the closed disc of radius $\varepsilon$ (over
$C$) and $A_{C}(\varepsilon):=D_{C}\mathbin{\fgebackslash}D_{C}(\varepsilon)$.

Applying Lemma 4.30, we get immediately that ${H}^{i}(Y_{C},{\bf
Q}_{p}(j))=0$, for $i\geq 2$, and

${H}^{1}(Y_{C},{\bf
Q}_{p}(j))\stackrel{{\scriptstyle\sim}}{{\leftarrow}}\operatorname{colim}_{0<\varepsilon<1}{H}^{1}(A_{C}(\varepsilon),{\bf
Q}_{p}(j)).$

From the exact sequence (4.32), we get the exact sequence

$\displaystyle 0$
$\displaystyle\to\operatorname{colim}_{0<\varepsilon<1}({\mathscr{O}}(A_{C}(\varepsilon)/C)(j-1))\to\operatorname{colim}_{0<\varepsilon<1}{H}^{1}(A_{C}(\varepsilon),{\bf
Q}_{p}(j))$ $\displaystyle\to\operatorname{colim}_{0<\varepsilon<1}{\bf
Q}_{p}(j-1)\to 0.$

Since
$\operatorname{colim}_{0<\varepsilon<1}({\mathscr{O}}(A_{C}(\varepsilon)/C)(j-1))\stackrel{{\scriptstyle\sim}}{{\to}}{\mathscr{O}}(Y_{C})/C$,
we get that ${H}^{1}(Y_{C},{\bf Q}_{p}(j))$ fits into the exact sequence
(4.37).

From Lemma 4.30, we also get the isomorphism

$\operatorname{colim}_{0<\varepsilon<1}{H}^{0}(A_{C}(\varepsilon),{\bf
Q}_{p}(j))\stackrel{{\scriptstyle\sim}}{{\to}}{H}^{0}(Y_{C},{\bf Q}_{p}(j)).$

From it and the exact sequence (4.32), we get the isomorphism

${\bf Q}_{p}(j)\stackrel{{\scriptstyle\sim}}{{\to}}{H}^{0}(Y_{C},{\bf
Q}_{p}(j)).$

This finishes the proof of the lemma. ∎

######  Lemma 4.38.

(Arithmetic cohomology) Let $i\in{\bf N},j\in{\bf Z}$. We have exact sequences

$\displaystyle 0\to H^{i}({\mathscr{G}}_{K},H^{0}(Y_{C},{\bf Q}_{p}(j)))\to
H^{i}(Y,{\bf Q}_{p}(j))\to H^{i-1}({\mathscr{G}}_{K},H^{1}(Y_{C},{\bf
Q}_{p}(j)))\to 0,$ $\displaystyle 0\to
H^{i-1}({\mathscr{G}}_{K},({\mathscr{O}}(Y_{C})/C)(j-1))\to
H^{i-1}({\mathscr{G}}_{K},H^{1}(Y_{C},{\bf Q}_{p}(j)))\to
H^{i-1}({\mathscr{G}}_{K},{\bf Q}_{p}(j-1))\to 0.$

###### Proof.

The second exact sequence is obtained from the (split) exact sequence from
Lemma 4.36.

For the first exact sequence, we write

$\displaystyle{\mathrm{R}}\Gamma(Y,{\bf Q}_{p}(j))$
$\displaystyle=\operatorname{colim}_{0<\varepsilon_{K}<1}{\mathrm{R}}\Gamma(D\mathbin{\fgebackslash}D(\varepsilon_{K}),{\bf
Q}_{p}(j))$
$\displaystyle=\operatorname{colim}_{0<\varepsilon_{K}<1}{\mathrm{R}}\Gamma(A(\varepsilon_{K}),{\bf
Q}_{p}(j)),$

where $\varepsilon_{K}$ are chosen so that the annuli $A(\varepsilon_{K})$ are
defined over $K$. By Lemma 4.34, this yields the exact sequence

$\displaystyle 0$
$\displaystyle\to\operatorname{colim}_{0<\varepsilon_{K}<1}H^{i}({\mathscr{G}}_{K},H^{0}(A(\varepsilon_{K})_{C},{\bf
Q}_{p}(j)))\to H^{i}(Y,{\bf Q}_{p}(j))$
$\displaystyle\to\operatorname{colim}_{0<\varepsilon_{K}<1}H^{i-1}({\mathscr{G}}_{K},H^{1}(A(\varepsilon_{K})_{C},{\bf
Q}_{p}(j)))\to 0.$

Hence it suffices to show that

(4.39)
$\displaystyle\operatorname{colim}_{0<\varepsilon_{K}<1}H^{i}({\mathscr{G}}_{K},H^{0}(A(\varepsilon_{K})_{C},{\bf
Q}_{p}(j)))$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\to}}H^{i}({\mathscr{G}}_{K},H^{0}(Y_{C},{\bf
Q}_{p}(j))),$
$\displaystyle\operatorname{colim}_{0<\varepsilon_{K}<1}H^{i-1}({\mathscr{G}}_{K},H^{1}(A(\varepsilon_{K})_{C},{\bf
Q}_{p}(j)))$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\to}}H^{i-1}({\mathscr{G}}_{K},H^{1}(Y_{C},{\bf
Q}_{p}(j))).$

But this is clear since ${\bf Z}[{\mathscr{G}}_{K}]$ is a compact object in
${\rm CondAb}$. ∎

####  4.6.4. Boundary of an open annulus

Let $A$ be an open annulus over $K$.

######  Corollary 4.40.

(Geometric cohomology) Let $i\in{\bf N},j\in{\bf Z}$. There is a
${\mathscr{G}}_{K}$-equivariant canonical isomorphism

$H^{i}(\partial A_{C},{\bf
Q}_{p}(j))\stackrel{{\scriptstyle\sim}}{{\to}}H^{i}(Y_{C},{\bf
Q}_{p}(j))^{\oplus 2}.$

Hence we have ${\mathscr{G}}_{K}$-equivariant isomorphisms

$H^{i}(\partial A_{C},{\bf Q}_{p}(j))\simeq\begin{cases}{\bf Q}_{p}(j)^{\oplus
2}&\mbox{if }i=0,\\\ 0&\mbox{if }i\geq 2,\end{cases}$

and a ${\mathscr{G}}_{K}$-equivariant (split) exact sequence:

$0\to({\mathscr{O}}(Y_{C})/C)(j-1)^{\oplus 2}\to H^{1}(\partial A_{C},{\bf
Q}_{p}(j))\to{\bf Q}_{p}(j-1)^{\oplus 2}\to 0$

###### Proof.

We write $A=\\{z\in K\;:\;|a|<|z|<|b|\\}$ with $a,b\in K$. Then

$\partial
A\simeq\varinjlim_{|a|<\delta\leq\varepsilon<|b|}A\mathbin{\fgebackslash}A(\delta,\varepsilon)=Y_{a}\sqcup
Y_{b},$

where $A(\delta,\varepsilon):=\\{z\in K\;:\;\delta\leq|z|\leq\varepsilon\\}$
and $Y_{a},Y_{b}$ are the two ghost circles at the boundary of $A$. Our
corollary now follows from Lemma 4.36. ∎

######  Corollary 4.41.

(Arithmetic cohomology) We have a canonical isomorphism

$H^{i}(\partial A,{\bf Q}_{p}(j))\simeq H^{i}(Y,{\bf Q}_{p}(j))^{\oplus 2}$

and exact sequences

$\displaystyle 0\to H^{i}({\mathscr{G}}_{K},H^{0}(\partial A_{C},{\bf
Q}_{p}(j)))\to H^{i}(\partial A,{\bf Q}_{p}(j))\to
H^{i-1}({\mathscr{G}}_{K},H^{1}(\partial A_{C},{\bf Q}_{p}(j)))\to 0,$
$\displaystyle 0\to H^{i-1}({\mathscr{G}}_{K},\tfrac{{\mathscr{O}}(\partial
A_{C})}{C^{\oplus 2}}(j-1))\to H^{i-1}({\mathscr{G}}_{K},H^{1}(\partial A,{\bf
Q}_{p}(j)))\to H^{i-1}({\mathscr{G}}_{K},{\bf Q}_{p}^{\oplus 2}(j-1))\to 0.$

###### Proof.

Since $\partial A$ is a disjoint union of two ghost circles, this follows
immediately from Lemma 4.38. ∎

######  Remark 4.42.

The arithmetic pro-étale cohomology $H^{i}(Y,{\bf Q}_{p}(j))$ and
$H^{i}(\partial A,{\bf Q}_{p}(j))$ are direct sums of nuclear Fréchet spaces