# On The rationalization of the $K({n})$-local sphere

Tobias Barthel  Max Planck Institute for Mathematics, Vivatsgasse 7, 53111
Bonn, Germany<EMAIL_ADDRESS>,  Tomer M. Schlank  The Hebrew
University of Jerusalem<EMAIL_ADDRESS>,  Nathaniel Stapleton
University of Kentucky<EMAIL_ADDRESS>and  Jared Weinstein  Boston
University Department of Mathematics and Statistics, 665 Commonwealth Avenue,
Boston, MA, USA<EMAIL_ADDRESS>
###### Abstract.

We compute the rational homotopy groups of the $K(n)$-local sphere for all
heights $n$ and all primes $p$, verifying a prediction that goes back to the
pioneering work of Morava in the early 1970s. More precisely, we show that the
inclusion of the Witt vectors into the Lubin–Tate ring induces a split
injection on continuous stabilizer cohomology with torsion cokernel of bounded
exponent, thereby proving Hopkins’ chromatic splitting conjecture and the
vanishing conjecture of Beaudry–Goerss–Henn rationally. The key ingredients
are the equivalence between the Lubin–Tate tower and the Drinfeld tower due to
Faltings and Scholze–Weinstein, integral $p$-adic Hodge theory, and an
integral refinement of a theorem of Tate on the Galois cohomology of non-
archimedean fields.

René Magritte, Memory of a Journey, 1955.

###### Contents

  1. 1 Introduction
  2. 2 Chromatic homotopy theory
  3. 3 Arithmetic prerequisites
  4. 4 The Galois cohomology of $\mathcal{O}_{C}$
  5. 5 Pro-étale cohomology of rigid-analytic spaces
  6. 6 Proof of B

##  1\. Introduction

A central problem in homotopy theory is to understand the homotopy groups of
spheres $\pi_{k+d}S^{k}$, i.e., the group of continuous maps $S^{k+d}\to
S^{k}$ up to homotopy. It follows from the Freudenthal suspension theorem that
$\pi_{k+d}S^{k}$ stabilizes for $k>d+1$, so one may first seek to determine
the stable homotopy groups of spheres,
$\pi_{d}S^{0}\coloneqq\operatorname{lim}_{k\to\infty}\pi_{k+d}S^{k}$. These
are abelian groups. We have $\pi_{d}S^{0}=0$ for $d<0$ and an isomorphism
$\pi_{0}S^{0}\cong\mathbb{Z}$ encoding the degree of a map. For $d>0$, Serre
showed that $\pi_{d}S^{0}$ is finite.

Early attempts at understanding $\pi_{*}S^{0}$ through explicit calculations
in small degrees only provided limited information about the large-scale
structure. Chromatic homotopy theory begins with the deep observation that the
elements of $\pi_{*}S^{0}$ may be organized into certain periodic families of
increasing periodicity, which depend on a prime number $p$. From a modern
perspective, these periodic families are the homotopy groups of localizations
$L_{n}S^{0}$ of $S^{0}$. These localizations provide successive approximations
to the sphere spectrum

$S^{0}\rightarrow\ldots\rightarrow L_{n}S^{0}\rightarrow\ldots\rightarrow
L_{1}S^{0}\rightarrow L_{0}S^{0}\simeq S_{\mathbb{Q}}^{0}$

whose associated graded pieces are given by the $K(n)$-local spheres
$L_{K(n)}S^{0}$, implicitly depending on $p$. The homotopy limit of this tower
recovers $S_{(p)}^{0}$, the $p$-localization of $S^{0}$. Since
$\pi_{*}S_{(p)}^{0}$ is the $p$-localization of $\pi_{*}S^{0}$, and since
these taken together determine $\pi_{*}S^{0}$, a fundamental problem in the
field is to understand the homotopy groups $\pi_{*}L_{K(n)}S^{0}$.

In the case $n=0$, we have $L_{K(0)}S^{0}\cong S^{0}_{\mathbb{Q}}$, the
rational sphere spectrum. As an immediate consequence of the aforementioned
theorem of Serre,
$\pi_{*}S^{0}_{\mathbb{Q}}\cong\mathbb{Q}\otimes\pi_{*}S^{0}$ is $\mathbb{Q}$
in degree $0$ and $0$ otherwise. In the case $n=1$, $\pi_{*}L_{K(1)}S^{0}$ was
calculated in the 1970s by Adams–Baird (unpublished) and Ravenel [Rav84]. The
case $n=2$ took many years of work by many people and was only recently
resolved (see [SY95, SW02b, SW02a, GHMR05, Beh12, Koh13, BGH22] for an
incomplete list); even stating the answer is very involved. Consequently, a
full computation of $\pi_{*}L_{K(n)}S^{0}$ for $h>2$ seems to be out of reach.

In light of this, attention over the last few decades has gradually turned
towards understanding structural features of $\pi_{*}L_{K(n)}S^{0}$. Since the
1970s and motivated by the work of Lazard and Morava, a guiding problem has
been to determine the free $\mathbb{Z}_{p}$-summands in $\pi_{*}L_{K(n)}S^{0}$
or, equivalently, to understand $\mathbb{Q}\otimes\pi_{*}L_{K(n)}S^{0}$.
Through the full force of the computations mentioned above, this is now known
for $n\leq 2$ and all primes $p$. In this paper, we resolve this question
completely for all $n$ and all primes $p$:

######  Theorem A.

There is an isomorphism of graded $\mathbb{Q}$-algebras

$\mathbb{Q}\otimes\pi_{*}L_{K(n)}S^{0}\cong\Lambda_{\mathbb{Q}_{p}}(\zeta_{1},\zeta_{2},\dots,\zeta_{n}),$

where the latter is the exterior $\mathbb{Q}_{p}$-algebra on generators
$\zeta_{i}$ in degree $1-2i$.

In particular, this result confirms the rational part of Hopkins’ chromatic
splitting conjecture [Hov95] for all primes $p$ and all heights $n$.
Previously, this was only known for $n\leq 2$ through the explicit computation
in the works listed above. As such, A constitutes the first general result in
the direction of this conjecture since the construction of the class
$\zeta\in\pi_{-1}L_{K(n)}S^{0}$ by Devinatz and Hopkins [DH04] in the early
2000s. We will give a more thorough explanation of the homotopical context for
our results in Section 2.

In order to explain our approach, we recall that the homotopy groups of
$L_{K(n)}S^{0}$ can be approached algebraically through Lubin and Tate’s
deformation theory of formal groups. Let $\Gamma_{n}$ be a formal group of
dimension 1 and height $n$ over $\overline{\mathbb{F}}_{p}$, and let
$\mathbb{G}_{n}=\operatorname{Aut}(\Gamma_{n},\overline{\mathbb{F}}_{p})$ be
the so-called Morava stabilizer group, defined as the group of automorphisms
of $\Gamma_{n}$ that lie over an automorphism of $\overline{\mathbb{F}}_{p}$.
Then $\mathbb{G}_{n}$ is an extension of
$\operatorname{Gal}(\overline{\mathbb{F}}_{p}/\mathbb{F}_{p})\cong\widehat{\mathbb{Z}}$
by $\operatorname{Aut}_{\overline{\mathbb{F}}_{p}}(\Gamma_{n})$. Let
$W=W(\overline{\mathbb{F}}_{p})$ be the ring of $p$-typical Witt vectors. By
Lubin–Tate theory, there is a complete local ring $A\cong W\llbracket
u_{1},\ldots,u_{n-1}\rrbracket$, corepresenting deformations of $\Gamma_{n}$,
which admits a continuous action by $\mathbb{G}_{n}$. Further, the invariant
differentials $\omega$ of the universal deformation of $\Gamma_{n}$ form an
invertible $A$-module, and the natural actions of $\mathbb{G}_{n}$ on $A$ and
$\omega$ extend to an action of $\mathbb{G}_{n}$ on the graded ring
$A_{*}=\bigoplus_{t\in 2\mathbb{Z}}\omega^{\otimes t/2}$ (i.e., $A_{*}$ is
evenly concentrated). The Devinatz–Hopkins spectral sequence [DH04] takes the
form

$H^{s}_{\mathrm{cts}}(\mathbb{G}_{n},A_{t})\implies\pi_{t-s}L_{K(n)}S^{0},$

where $H^{s}_{\operatorname{cts}}$ refers to continuous cohomology. Thus
understanding the bi-graded ring $H^{s}_{\mathrm{cts}}(\mathbb{G}_{n},A_{t})$
is of great importance in chromatic homotopy theory. However, the action of
$\mathbb{G}_{n}$ on $A_{*}$ is difficult to describe, see [DH95].

Consider instead the problem of computing
$H^{*}_{\operatorname{cts}}(\mathbb{G}_{n},W)$, where $\mathbb{G}_{n}$ acts on
$W$ through its quotient
$\operatorname{Gal}(\overline{\mathbb{F}}_{p}/\mathbb{F}_{p})$. A classical
theorem of Lazard [Laz65] states that the cohomology of a $p$-adic Lie group
$G$ with $\mathbb{Q}_{p}$-coefficients can be computed in terms of Lie algebra
cohomology. Applied to the $p$-adic Lie group
$\operatorname{Aut}_{\overline{\mathbb{F}}_{p}}(\Gamma_{n})$, Lazard’s theorem
provides an isomorphism of graded $\mathbb{Q}_{p}$-algebras:

$\textstyle{H^{\ast}_{\operatorname{cts}}(\mathbb{G}_{n},W)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}\cong\Lambda_{\mathbb{Q}_{p}}(x_{1},x_{2},\ldots,x_{n})}$
(1.0.1)

Here, the right hand side is the exterior $\mathbb{Q}_{p}$-algebra on
generators $x_{i}$ of degree $2i-1$.

Remarkably, and verified by extensive calculations for heights $n\leq 2$ over
the last 40 years, work of Morava [Mor85] from the early 1970s suggests that
the natural map of $\mathbb{Z}_{p}$-modules

$\textstyle{H^{\ast}_{\operatorname{cts}}(\mathbb{G}_{n},W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{\ast}_{\operatorname{cts}}(\mathbb{G}_{n},A)}$
(1.0.2)

is a rational isomorphism; i.e., it becomes an isomorphism after inverting
$p$. The main result of this paper establishes a refinement of this
conjecture:

######  Theorem B.

For every integer $s\geq 0$, the natural map $W\hookrightarrow A$ induces a
split injection

$H^{s}_{\mathrm{cts}}(\mathbb{G}_{n},W)\hookrightarrow
H^{s}_{\mathrm{cts}}(\mathbb{G}_{n},A)$

whose complement is killed by a power of $p$. In particular,

$H^{s}_{\operatorname{cts}}(\mathbb{G}_{n},W)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}\to
H^{s}_{\operatorname{cts}}(\mathbb{G}_{n},A)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$

is an isomorphism.

We have not attempted to make explicit the power of $p$ which kills the
complement of $H^{s}_{\operatorname{cts}}(\mathbb{G}_{n},W)$ in
$H^{s}_{\operatorname{cts}}(\mathbb{G}_{n},A)$, though this should be possible
by our methods. In fact, at heights $n=1,2$ this complement is zero,
suggesting that this might be the case in general; this is known as the
“chromatic vanishing conjecture”.

The proof of B is summarized in Section 3.9. It relies upon recent advances in
$p$-adic geometry. Ultimately we draw much of our power from the isomorphism,
due to Faltings [Fal02a] and clarified by Scholze and Weinstein [SW13],
between the Lubin–Tate and Drinfeld towers. Faltings’ isomorphism may be
regarded as an equivalence of stacks:

$[\operatorname{LT}/\mathbb{G}_{n}]\simeq[\mathcal{H}/\operatorname{GL}_{n}(\mathbb{Z}_{p})].$
(1.0.3)

The cohomology of the stack on the left accesses
$H^{*}_{\operatorname{cts}}(\mathbb{G}_{n},A)$. The main idea is to use this
isomorphism to replace the opaque action of $\mathbb{G}_{n}$ on Lubin–Tate
space $\operatorname{LT}$ with the far more transparent action of the group
$\operatorname{GL}_{n}(\mathbb{Z}_{p})$ on Drinfeld’s symmetric space
$\mathcal{H}$. A is then readily deduced from B via the Devinatz–Hopkins
spectral sequence.

The groups appearing in (1.0.3) are profinite, and accordingly the stacks
appearing there must be construed as living on the pro-étale topology
$X_{\mathrm{pro\acute{e}t}}$ on rigid-analytic spaces. Thereby the pro-étale
cohomology of rigid-analytic spaces, first considered in [Sch13a] and expanded
in [BMS18], enters the picture as an indispensable tool. Much of this article
is concerned with controlling the pro-étale cohomology
$H^{*}(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})$ of the completed
structure sheaf $\hat{\mathcal{O}}^{+}$ on $X_{\mathrm{pro\acute{e}t}}$ for a
rigid-analytic space $X$ over a local field $K$ of characteristic $(0,p)$, for
example $X=\operatorname{LT}$ or $X=\mathcal{H}$.

The results we obtain are new even for the case when $X=\operatorname{Spa}K$
is a single point. In that case,
$H^{*}(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})$ is the continuous
cohomology
$H^{*}_{\operatorname{cts}}(\operatorname{Gal}(\overline{K}/K),\mathcal{O}_{C})$,
where $C$ is the completion of an algebraic closure $\overline{K}$ of $K$. One
of our results (Theorem 4.0.4) is a refinement of a classical theorem of Tate.

######  Theorem C.

Let $K$ be a local field of characteristic $(0,p)$, and let $C$ be the
completion of an algebraic closure $\overline{K}$ of $K$. Let
$\mathcal{O}_{K}\subset K$ and $\mathcal{O}_{C}\subset C$ denote the valuation
rings. There is an isomorphism of graded $\mathcal{O}_{K}$-modules:

$H^{*}_{\operatorname{cts}}(\operatorname{Gal}(\overline{K}/K),\mathcal{O}_{C})\cong\mathcal{O}_{K}[\varepsilon]\oplus
T,$

where $\varepsilon$ is a degree 1 element with $\varepsilon^{2}=0$ and $T$ is
$p^{N}$-torsion with $N$ an absolute constant (in fact, $N=6$ suffices).

Our results on the pro-étale cohomology of rigid-analytic spaces $X$ apply
under certain hypotheses regarding the existence of semi-stable models for $X$
after extension of scalars. The cleanest result we have (5.6.9) applies when
$X$ admits a semistable model over $\mathcal{O}_{K}$.

######  Theorem D.

Let $K$ be a local field of characteristic $(0,p)$, and let $\mathfrak{X}$ be
a quasi-separated semistable formal scheme over $\mathcal{O}_{K}$, with rigid-
analytic generic fiber $X$. There is a natural isomorphism of graded
$K$-vector spaces:

$H^{*}(\mathfrak{X},\mathcal{O})\otimes_{\mathcal{O}_{K}}K[\varepsilon]\cong
H^{*}(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})\otimes_{\mathcal{O}_{K}}K$

Here $H^{*}(\mathfrak{X},\mathcal{O})$ is coherent cohomology, and
$\varepsilon$ is a degree 1 element with $\varepsilon^{2}=0$.

### Outline of the document

Since this paper is written with an audience of both arithmetic geometers and
homotopy theorists in mind, we have chosen to include additional background
material that might be familiar for one group but not necessarily the other.
In that spirit, we begin in Section 2 with a rapid review of the key players
in chromatic homotopy theory, working towards stating the chromatic splitting
conjecture. We then construct a splitting on the level of cohomology via power
operations and deduce A from B. Section 3 then collects preliminary material
from arithmetic geometry, including the notion of an adic space, a short
treatment of continuous cohomology from the condensed perspective, and the
pro-étale topology. The section concludes with an outline of the proof of B,
see Section 3.9, which is then fleshed out in the rest of the paper. In
Section 4, we establish integral refinements of a theorem of Tate, including
C, by determining bounds on the torsion exponents in the Galois cohomology of
$\mathcal{O}_{C}$ and its Tate twists, where $C$ is the completion of an
algebraic closure of a local field $K$ of mixed characteristic. We then
globalize this result in Section 5 to the pro-étale cohomology of the generic
fiber of a semistable formal scheme over $\mathcal{O}_{K}$ with coefficients
in the sheaf of bounded functions, and prove D. Finally, in Section 6, we put
all the pieces together to prove B, by first applying methods separately to
the Drinfeld tower and the Lubin–Tate tower, and then deducing our main
theorem via the isomorphism of towers.

### Acknowledgements

We thank Agnès Beaudry, Pierre Colmez, Paul Goerss, Mike Hopkins, Kiran
Kedlaya, Eric Peterson, Jay Pottharst, Joaquín Rodrigues Jacinto, Juan Esteban
Rodríguez Camargo, Peter Scholze, Ehud de Shalit, and Padmavathi Srinivasan
for many helpful discussions. We are also grateful to Pierre Colmez, Jack
Morava, Ningchuan Zhang, and especially Peter Scholze for comments on a
preliminary draft of this paper.

TB is supported by the European Research Council (ERC) under Horizon Europe
(grant No. 101042990) and would like to thank the Max Planck Institute for its
hospitality. NS and TS were supported by the US-Israel Binational Science
Foundation under grant 2018389. NS was supported by a Sloan research
fellowship, NSF Grants DMS-2304781 and DMS-1906236, and a grant from the
Simons Foundation (MP-TSM-00002836, NS).

##  2\. Chromatic homotopy theory

The goal of this section is to place B in the context of stable homotopy
theory and deduce A from B. To this end, we begin with a rapid review of some
relevant material from chromatic homotopy theory, before turning to the
applications. We refer the reader interested in a more thorough introduction
to the subject to the following sources: [Lur10, BB20, BGH22]. The readers
familiar with chromatic homotopy theory can safely skip ahead to the new
results, beginning in Section 2.5.

###  2.1. Chromatic characteristics

Our starting point is the chromatic perspective on the category of spectra as
envisioned by Morava [Mor85] and Ravenel [Rav84] and established by Devinatz,
Hopkins, and Smith [DHS88, HS98]. This story has been told by many, and we
take a revisionistic approach following [BB20], also freely using the language
of higher algebra as developed by Lurie in [Lur09, Lur17].

In order to motivate the homotopical constructions, let us first recast some
familiar concepts from algebra in more category-theoretic language; the
resulting definitions can then be transported more easily to higher algebra.
Let $\operatorname{Mod}(\mathbb{Z})$ be the symmetric monoidal abelian
category of abelian groups. A non-trivial unital associative ring
$A\in\operatorname{Mod}(\mathbb{Z})$ is said to be a division algebra (or skew
field) if any module over $A$ is free. Two division algebras $A$ and $B$ are
said to be of the same characteristic if $A\otimes_{\mathbb{Z}}B\neq 0$. It is
straightforward to verify that this notion induces an equivalence relation on
the collection of all division algebras in $\operatorname{Mod}(\mathbb{Z})$.

It turns out that we can classify all such characteristics: Indeed, the
minimal representative of the equivalence classes of characteristics of
division algebras in $\operatorname{Mod}(\mathbb{Z})$ are given by the prime
fields $\mathbb{F}_{p}$ for primes $p$ and $\mathbb{Q}$. This fact is
essentially a translation of the basic classification of prime fields in
classical algebra.

Stable homotopy theory is the study of the category $\operatorname{Sp}$ of
spectra, which forms a higher analogue of the category of abelian groups;
Waldhausen and May coined the term ‘brave new algebra.’ The role of the
integers is then played by the sphere spectrum $S^{0}$, and the tensor product
is replaced by the smash product written as $\otimes$ or $\otimes_{S^{0}}$ for
emphasis. Equipped with this structure, $\operatorname{Sp}$ forms a symmetric
monoidal stable $\infty$-category; we may therefore speak of rings and their
modules in this setting. Interpreted in this context, formally there is an
identification $\operatorname{Sp}=\operatorname{Mod}(S^{0})$. The next
definition is then the natural higher algebraic counterpart to the concept of
characteristic as discussed above:

######  Definition 2.1.1.

A _division algebra_ in $\operatorname{Sp}$ is a unital associative ring
spectrum $A$ such that every $A$-module $M$ splits into a direct sum of shifts
of free rank 1 modules. Two division algebras $A,B\in\operatorname{Sp}$ are of
the same _(chromatic) characteristic_ if and only if $A\otimes_{S^{0}}B\neq
0$.

We again obtain an equivalence relation on the collection of all division
algebras in $\operatorname{Sp}$, so we are naturally led to ask if we can
understand the equivalence classes. This question has been answered completely
in the aforementioned seminal work of Devinatz, Hopkins, and Smith. Stating
their classification in the form we want requires a short detour. Let $p$ be a
prime, let $\kappa$ be a perfect field of characteristic $p$, let $\Gamma$ be
a 1-dimensional height $n$ commutative formal group over $\kappa$, and finally
let $n\in\mathbb{N}\cup\\{\infty\\}$. It is an insight of Morava, based on
earlier work of Quillen, that this data lifts to $\operatorname{Sp}$: there
exists a multiplicative cohomology theory $K(\Gamma,\kappa)^{*}$ with the
following properties:

  1. (1)

The value of $K(\Gamma,\kappa)^{*}$ on a point is given by

$K(\Gamma,\kappa)^{*}(\mathrm{pt})\cong\begin{cases}\mathbb{Q}&\text{if
}n=0\\\ \kappa[v_{n}^{\pm 1}]&\text{if }0<n<\infty\\\ \kappa&\text{if
}n=\infty,\end{cases}$

where $v_{n}$ is a formal variable in degree $2p^{n}-2$.

  2. (2)

$K(\Gamma,\kappa)^{*}$ is complex oriented, and the
$K(\Gamma,\kappa)^{*}$-cohomology of complex projective space represents the
formal group $\Gamma$:

$\operatorname{Spf}(K(\Gamma,\kappa)^{*}(\mathbf{CP}^{\infty}))\cong\Gamma.$

  3. (3)

$K(\Gamma,\kappa)^{*}$ satisfies the Künneth formula for any two spectra
$X,Y$:

$K(\Gamma,\kappa)^{*}(X\otimes Y)\cong
K(\Gamma,\kappa)^{*}(X)\otimes_{K(\Gamma,\kappa)^{*}}K(\Gamma,\kappa)^{*}(Y).$

By Brown representability, $K(\Gamma,\kappa)^{*}$ is represented in the
category of spectra by a (unital and associative) ring spectrum
$K(\Gamma,\kappa)$, known as _Morava $K$-theory_ (at height $n$ and over the
field $\kappa$). Since the ring of coefficients
$K(\Gamma,\kappa)^{*}(\mathrm{pt})$ is a graded field, $K(\Gamma,\kappa)$
itself must be a division algebra in the sense of 2.1.1. For example, if
$\hat{\mathbb{G}}_{a}$ is the formal additive group, then
$K(\hat{\mathbb{G}}_{a},\mathbb{Q})$ and
$K(\hat{\mathbb{G}}_{a},\mathbb{F}_{p})$ represent singular cohomology with
coefficients in $\mathbb{Q}$ and $\mathbb{F}_{p}$, respectively. If
$\hat{\mathbb{G}}_{m}$ is the formal multiplicative group, then
$K(\hat{\mathbb{G}}_{m},\mathbb{F}_{p})$ is a summand of mod $p$ complex
topological $K$-theory. Generalizing the last example, for any prime $p$ and
any height $n\in\mathbb{N}$, there exists a formal group $\Gamma_{n}$ of
height $n$ over $\mathbb{F}_{p}$. Mildly abusing notation, we set
$K(n,p)\coloneqq K(\Gamma_{n},\mathbb{F}_{p})$, keeping the choice of field
implicit. (After base change to an algebraically closed field, $\Gamma_{n}$ is
unique up to isomorphism by a theorem of Lazard [Laz75].) We also set
$K(0,p)=K(\hat{\mathbb{G}}_{a},\mathbb{Q})$ and
$K(\infty,p)=K(\hat{\mathbb{G}}_{a},\mathbb{F}_{p})$.

Armed with a good collection of division algebras, we can now return to the
classification of characteristics in $\operatorname{Sp}$ to state:

######  Theorem 2.1.2 (Devinatz–Hopkins–Smith).

The collection of Morava $K$-theories $K(n,p)$ for $p$ ranging through the
primes and $n\in\mathbb{N}\cup\\{\infty\\}$ forms a complete and pairwise
distinct set of representatives for the characteristics of division algebras
in $\operatorname{Sp}$. Moreover, the $K(n,p)$s are minimal in the sense that
any division algebra of the same characteristic as $K(n,p)$ is a module over
$K(n,p)$.

In other, more plain terms: the Morava $K$-theories $K(n,p)$ provide precisely
the prime fields of the category of spectra. A couple of remarks are in order.

  * •

(Non-commutativity) In contrast to the situation in classical algebra, the
Morava $K$-theories in intermediate characteristic $0<n<\infty$ cannot be made
commutative. In fact, they do not even afford the structure of an
$\mathbb{E}_{2}$-ring spectrum, see for instance [ACB19]. This is the main
reason to work with division algebras in the definition of characteristic.

  * •

(Interrelation) A finitely generated abelian group $M$ with
$M\otimes\mathbb{Q}\neq 0$ also satisfies $M\otimes\mathbb{F}_{p}\neq 0$ for
all primes $p$. This statement has a chromatic refinement: If $X$ is a finite
spectrum, i.e., a compact object in $\operatorname{Sp}$, then
$K(n,p)^{*}(X)\neq 0$ implies $K(n+1,p)^{*}(X)\neq 0$. Both the classical
algebraic statement and the chromatic statement are false in general for non-
compact objects.

To access and isolate the part of $\operatorname{Sp}$ that is visible to a
fixed Morava $K$-theory $K(n,p)$, we need another important tool from stable
homotopy theory, namely the theory of Bousfield localization. These form a
suitable generalization of localizations and (derived) completions familiar
from commutative algebra.

Fix an arbitrary spectrum $M\in\operatorname{Sp}$. A spectrum $X$ is said to
be _$E$ -acyclic_ if $M_{*}(X)=0$; a spectrum $Y$ is then called _$M$ -local_
if any map $X\to Y$ from an $M$-acyclic spectrum $X$ is null, i.e., factors
through a zero object. Intuitively, we wish to quotient $\operatorname{Sp}$ by
the ideal of all $M$-acyclic spectra to focus on those spectra which are
“seen” by $M$. Bousfield [Bou79] rigorously proved that this works, thereby
constructing a localization functor
$L_{M}\colon\operatorname{Sp}\to\operatorname{Sp}$ with the following
properties:

  1. (1)

$L_{M}X=0$ if and only if $X$ is $M$-acyclic;

  2. (2)

$L_{M}$ is idempotent and has essential image spanned by the $M$-local
spectra;

  3. (3)

for any $Z\in\operatorname{Sp}$, there is a natural map $Z\to L_{M}Z$ which
exhibits $L_{M}Z$ as the initial map out of $Z$ to an $M$-local spectrum.

These properties characterize $L_{M}$ uniquely up to homotopy. Finally, we
denote the full subcategory of $M$-local spectra by $\operatorname{Sp}_{M}$;
alternatively, this category is obtained from $\operatorname{Sp}$ by inverting
all $M$-equivalences, i.e., those maps which induce isomorphisms in
$M_{*}$-homology. Via localization, $\operatorname{Sp}_{M}$ inherits a
structure of symmetric monoidal $\infty$-category from $\operatorname{Sp}$,
with tensor product given by the _localized_ smash product.

Some examples might be illuminating. If $M=\mathbb{Q}$, then $L_{\mathbb{Q}}$
is rationalization, whose effect on homotopy groups of any spectrum is
tensoring with $\mathbb{Q}$. The element $p\in\mathbb{Z}\cong\pi_{0}S^{0}$ is
represented by a map $p\colon S^{0}\to S^{0}$, whose cofiber we denote by
$S^{0}/p$, the _mod $p$ Moore spectrum_. On the one hand, the local category
$\operatorname{Sp}_{S^{0}/p}$ is the category of $p$-complete spectra, and
localization at $S^{0}/p$ has the effect of derived $p$-completion on homotopy
groups. It is customary to write $X_{p}\coloneqq L_{S^{0}/p}X$ and
$\operatorname{Sp}_{p}\coloneqq\operatorname{Sp}_{S^{0}/p}$. On the other
hand, localizing at $M=S^{0}[1/p]$, the colimit over multiplication by $p$ on
$S^{0}$, has the effect of inverting $p$ on $S^{0}$. Similarly, we can
construct spectral analogues of $p$-localization, by inverting all primes but
$p$, to obtain the category $\operatorname{Sp}_{(p)}$ of $p$-local spectra.
For more information about various localizations on $\operatorname{Sp}$, we
refer to [Bou79].

The main example of interest to us is $\operatorname{Sp}_{K(n,p)}$, the so-
called $K(n,p)$-local category. In light of Theorem 2.1.2, it forms an
irreducible piece of the category of spectra, as we will explain momentarily,
and it is one of the key objects of study in chromatic homotopy theory. Note
that the functor $L_{K(n,p)}$ and thus $\operatorname{Sp}_{K(n,p)}$ only
depend on the characteristic and are in particular independent of the choice
of $\Gamma_{n}$.

###  2.2. Chromatic divide and conquer

From now on, we will restrict attention to the category of $p$-local spectra
for a fixed prime $p$, and usually drop the prime from the notation.
Intuitively speaking, the idea of the chromatic approach to stable homotopy
theory is to filter the category of $p$-local spectra
$\operatorname{Sp}_{(p)}$ by its subcategories of mixed chromatic
characteristics $(0,1,\ldots,n)$ for $n\to\infty$. Here we say that a spectrum
has mixed chromatic characteristic $(0,1,\ldots,n)$ if it is local with
respect to the direct sum $K(0)\oplus K(1)\oplus\ldots\oplus K(n)$, and we
write $L_{n}$ for the associated Bousfield localization. There is then a
sequence of Bousfield localization functors and natural transformations,

$\ldots\to L_{n}\to L_{n-1}\to\ldots\to L_{1}\to L_{0}$ (2.2.1)

the so-called _chromatic tower_. Note that the bottom layer is rationalization
$L_{0}=L_{\mathbb{Q}}$. Also note that the infinite height has been omitted, a
curiosity justified by the chromatic convergence theorem of Hopkins–Ravenel
[Rav92]: If $X$ is a $p$-local finite spectrum, then it can be recovered from
its chromatic tower (2.2.1):

$X\simeq\operatorname{lim}_{n}L_{n}X.$

Given $X$, it is then sensible to ask for the graded pieces of its chromatic
filtration, i.e., the difference between $L_{n}X$ and $L_{n-1}X$. This is
captured by the _chromatic fracture square_ , taking the form of a (homotopy)
pullback square that exists for any height $n>0$ and an arbitrary spectrum
$X$:

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
14.95149pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{L_{n}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
47.47939pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-30.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
47.47939pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{L_{K(n)}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
64.85365pt\raise-30.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{L_{n-1}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
38.95149pt\raise-40.33331pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
38.95149pt\raise-40.33331pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{L_{n-1}L_{K(n)}X.}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$
(2.2.2)

Geometrically, one should think of this square as being analogous to the
gluing square for a sheaf over an open-closed decomposition of a space. In
this picture, $L_{K(n)}X$ corresponds to the sheaf over a formal neighborhood
of a point, while $L_{n-1}X$ is the restriction to the open complement of the
point. The term $L_{n-1}L_{K(n)}X$ along with the maps pointing to it then
control the gluing process.

The weakest form of Hopkins’ _chromatic splitting conjecture_ stipulates that
the bottom horizontal map in (2.2.2) is split for $X=S^{0}$ (and hence for any
finite spectrum $X$), so that the chromatic assembly process takes a
particularly simple form. The strong form of the conjecture (2.4.1 below)
gives a complete description of $L_{n-1}L_{K(n)}S^{0}$ in terms of the
$L_{i}S^{0}$ for $i\leq n$. Assuming it, one could inductively reduce the
study of the $L_{n}S^{0}$ to that of the $L_{K(n)}S^{0}$, whose homotopy
groups are identified after inverting $p$ by our A.

###  2.3. Morava $E$-theory and the cohomology of the stabilizer group

Just as a height $n$ formal group over $\mathbb{F}_{p}$ gives rise to the
spectrum $K(n)$, the Lubin–Tate ring also admits a spectral incarnation. Let
$\Gamma$ be a height $n$ formal group over a perfect field $\kappa$ of
characteristic $p$. Let $A(\Gamma,\kappa)$ denote its ring of deformations, so
that $A(\Gamma,\kappa)\cong W(\kappa)\llbracket
u_{1},\dots,u_{n-1}\rrbracket$. An unpublished theorem of
Goerss–Hopkins–Miller, revisited and extended by Lurie in [Lur18], lifts this
data to a commutative algebra in $\operatorname{Sp}_{K(n)}$, called
$E(\Gamma,\kappa)$, with the property that

$\pi_{*}E(\Gamma,\kappa)\cong
A(\Gamma,\kappa)[\beta,\beta^{-1}],\,\,|\beta|=2.$

The commutative ring spectrum $E(\Gamma,\kappa)$ is known as _Morava
$E$-theory_ or _Lubin–Tate theory_.

In fact, Goerss, Hopkins, and Miller prove something stronger. Consider the
1-category of formal groups over perfect fields $\operatorname{FG}$. The
objects of $\operatorname{FG}$ are given by pairs $(\Gamma,\kappa)$ as above,
and a morphism $(\Gamma,\kappa)\to(\Gamma^{\prime},\kappa^{\prime})$ in
$\operatorname{FG}$ consists of a ring map $i\colon\kappa\to\kappa^{\prime}$
together with an isomorphism of formal groups
$i^{*}\Gamma\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\Gamma^{\prime}$.
Goerss, Hopkins, and Miller produce a fully faithful functor
$E(-,-)\colon\operatorname{FG}\to\operatorname{CAlg}(\operatorname{Sp})$. It
is important to note that the source is a $1$-category and that the target is
an $\infty$-category. Thus this theorem identifies a very rigid portion of
$\operatorname{CAlg}(\operatorname{Sp})$, in which the mapping spaces are
homotopy equivalent to sets.

The underlying spectrum of $E(\Gamma,\kappa)$ is easily constructed as a
consequence of the Landweber exact functor theorem. That theorem produces
cohomology theories out of the complex cobordism spectrum with specified
formal groups, so long as these satisfy a certain hypothesis. The universal
deformation of $\Gamma$ over $A(\Gamma,\kappa)$ satisfies the hypothesis, and
this gives rise to Morava $E$-theory. However, producing Morava $E$-theory as
a commutative algebra in $\operatorname{Sp}_{K(n)}$ is quite a bit more
difficult and requires either obstruction theory or a derived deformation
theory of formal groups—this is the content of the Goerss–Hopkins–Miller
theorem.

Let $\Gamma_{n}$ be any formal group of height $n$ over
$\overline{\mathbb{F}}_{p}$; then $\Gamma_{n}$ is unique up to isomorphism. We
write $E_{n}=E(\Gamma,\overline{\mathbb{F}}_{p})$. We let
$\mathbb{G}_{n}=\operatorname{Aut}(\Gamma_{n},\overline{\mathbb{F}}_{p})\cong\operatorname{Aut}E_{n}$,
the _Morava stabilizer group_. As a topological group, $\mathbb{G}_{n}$ may be
identified with the profinite completion $\widehat{D^{\times}}$ of
$D^{\times}$, where $D$ is the central simple algebra of invariant
$\frac{1}{n}$ over $\mathbb{Q}_{p}$, with ring of integers
$\mathcal{O}_{D}=\operatorname{End}_{\overline{\mathbb{F}}_{p}}(\Gamma_{n})$.

It turns out that $L_{E_{n}}=L_{n}$, so $E_{n}$ does not provide us with a new
localization functor. Furthermore, there is a close relationship between
$E_{n}$ and $K(n)$, akin to the one between a local ring and its residue
field. Since the ideal $I_{n}=(p,u_{1},\ldots,u_{n-1})$ of $\pi_{0}E_{n}$ is
generated by a regular sequence, we may form a (not necessarily commutative)
ring spectrum $E_{n}/I_{n}$ by iterated cofibers. This will have the property
that $\pi_{*}(E_{n}/I_{n})\cong\pi_{*}(E_{n})/I_{n}=\kappa[\beta,\beta^{-1}]$.
The ring spectrum obtained in this way is equivalent as a spectrum to a finite
direct sum of suspensions of $K(\Gamma_{n},\kappa)$:

$E_{n}/I_{n}\simeq\bigoplus_{0\leq i\leq
p^{n}-2}\Sigma^{2i}K(\Gamma_{n},\kappa).$

Here, the direct sum accounts for the fact that $E_{n}$ is 2-periodic, while
the periodicity of $K(\Gamma_{n},\kappa)$ is $2p^{n}-2$.

The unit in $\operatorname{Sp}_{K(n)}$ is the $K(n)$-local sphere, denoted
$L_{K(n)}S^{0}$. Being the unit, $L_{K(n)}S^{0}$ is the initial object in
$\operatorname{CAlg}(\operatorname{Sp}_{K(n)})$, so there is a canonical map
of commutative algebras $L_{K(n)}S^{0}\to E_{n}$. Since $\mathbb{G}_{n}$ acts
on $E_{n}$ through commutative algebra automorphisms, this map is
$\mathbb{G}_{n}$-equivariant for the trivial action on $L_{K(n)}S^{0}$. A
result of Devinatz and Hopkins [DH04], reinterpreted in Rognes’ framework of
spectral Galois extensions [Rog08], says that the unit map $L_{K(n)}S^{0}\to
E_{n}$ exhibits $E_{n}$ as a pro-Galois extension of $L_{K(n)}S^{0}$ in
$\operatorname{Sp}_{K(n)}$, with Galois group $\mathbb{G}_{n}$. Concretely,
this means that we have canonical equivalences of commutative ring spectra

$L_{K(n)}S^{0}\simeq E_{n}^{h\mathbb{G}_{n}}\quad\text{and}\quad
L_{K(n)}(E_{n}\otimes E_{n})\simeq
C_{\operatorname{cts}}(\mathbb{G}_{n},E_{n}),$ (2.3.1)

where $C_{\operatorname{cts}}(\mathbb{G}_{n},E_{n})$ denotes the ring spectrum
of continuous functions on $\mathbb{G}_{n}$ with coefficients in $E_{n}$. This
enables us to run Galois descent along $L_{K(n)}S^{0}\to E_{n}$. Form the
associated $K(n)$-local cosimplicial Amitsur complex

$L_{K(n)}S^{0}\to E_{n}^{\hat{\otimes}\bullet+1}\coloneqq
L_{K(n)}\left(E_{n}\rightrightarrows E_{n}\otimes
E_{n}\mathrel{\begin{subarray}{c}\textstyle\rightarrow\\\\[-2.58334pt]
\textstyle\rightarrow\\\\[-2.58334pt]
\textstyle\rightarrow\end{subarray}}E_{n}^{\otimes 3}\ldots\right),$ (2.3.2)

where we have omitted the degeneracy maps from the display. Applying the
homotopy groups to the resolution (2.3.2) and using (2.3.1) to identify the
abutment and $E_{2}$-page, we obtain a Bousfield–Kan spectral sequence of
signature

$E_{2}^{s,t}\cong
H_{\operatorname{cts}}^{s}(\mathbb{G}_{n},\pi_{t}E_{n})\implies\pi_{t-s}L_{K(n)}S^{0},$
(2.3.3)

where $H^{*}_{\operatorname{cts}}$ denotes cohomology with continuous
cocycles.

Since $\mathbb{G}_{n}$ has finite virtual cohomological dimension, this
spectral sequence provides an excellent approximation to the homotopy groups
of $L_{K(n)}S^{0}$: it converges strongly with a finite horizontal vanishing
line on some finite page, i.e., there exists $r\geq 2$ and some $N>0$ such
that $E_{r}^{s,t}=0$ for all $s>N$. In fact, if $p$ is odd and $2(p-1)\geq
n^{2}$, then we may take $r=2$ and $N=n^{2}$. The spectral sequence (2.3.3),
often simply referred to as “the” descent spectral sequence in chromatic
homotopy theory, provides a gateway between stable homotopy theory and
$p$-adic geometry.

###  2.4. The chromatic splitting conjecture and the vanishing conjecture

Computational evidence at low heights $n\leq 2$ suggests that the continuous
cohomology of the action of $\mathbb{G}_{n}$ on $W\subseteq\pi_{0}E_{n}$
largely controls the behavior of $\pi_{*}L_{K(n)}S^{0}$, as we shall now
explain. Recall the isomorphism (1.0.1)

$H^{*}_{\operatorname{cts}}(\mathbb{G}_{n},W)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}\cong\Lambda_{\mathbb{Q}_{p}}(x_{1},x_{2},\dots,x_{n}),$

where the latter is the exterior $\mathbb{Q}_{p}$-algebra on generators
$x_{i}$ in degree $2i-1$. The relevance of these classes for stable homotopy
theory was first realized by Morava, who also exhibited explicit cocycle
representatives for the $x_{i}$, see [Mor85, Remark 2.2.5].

Each of the classes $x_{i}$ can be lifted111In the chromatic practice, there
are certain preferred choices of lifts, but these will not matter for our
purposes here. For details, we refer to [Hov95]. to a class $\tilde{x}_{i}$ in
the integral cohomology ring $H^{*}_{\operatorname{cts}}(\mathbb{G}_{n},W)$.
Let

$\varphi\colon H^{*}_{\operatorname{cts}}(\mathbb{G}_{n},W)\to
H^{*}_{\operatorname{cts}}(\mathbb{G}_{n},\pi_{0}E_{n})\cong E_{2}^{*,0}$
(2.4.1)

be the natural map induced from the inclusion $W\hookrightarrow
A\cong\pi_{0}E_{n}$.

Hopkins’ chromatic splitting conjecture, recorded in [Hov95], predicts that
the classes $\tilde{x}_{i}$ carry all the relevant information about the
homotopy groups of $L_{K(n)}S^{0}$; more precisely:

######  Conjecture 2.4.1 (Chromatic splitting conjecture).

For $p$ odd and each $i=1,\dots,n$, we have the following behavior.

  1. (1)

For each $i=1,\dots,n$, the class $\varphi(\tilde{x}_{i})$ survives the
spectral sequence (2.3.3), thereby giving rise to a homotopy class

$e_{i}\in\pi_{1-2i}L_{K(n)}S^{0},$

or all the same, a map $e_{i}\colon S^{1-2i}_{p}\to L_{K(n)}S^{0}$.

  2. (2)

The composition $S^{1-2i}_{p}\stackrel{{\scriptstyle
e_{i}}}{{\to}}L_{K(n)}S^{0}\to L_{n-1}L_{K(n)}S^{0}$ factors through a map
$\overline{e}_{i}\colon L_{n-i}S^{1-2i}_{p}\to L_{n-1}L_{K(n)}S^{0}$.

  3. (3)

The $\overline{e}_{i}$ induce an equivalence of spectra

$L_{n-1}L_{K(n)}S^{0}\cong\bigwedge_{i=1}^{n}(L_{n-i}S_{p}^{1-2i})\coloneqq\bigoplus_{0\leq
j\leq n\atop 1\leq i_{1}<\ldots<i_{j}\leq
n}\left(\bigotimes_{k=1}^{j}L_{n-i_{k}}S_{p}^{1-2i_{k}}\right),$

where the right hand side is indexed on the $\mathbb{Z}_{p}$-module generators
of the exterior algebra
$\Lambda_{\mathbb{Z}_{p}}(\bar{e}_{1},\bar{e}_{2},\ldots,\bar{e}_{n})$.

A more refined formulation of the conjecture as well as the necessary
modifications for the prime 2 can be found in [BGH22]. It has been verified by
explicit computation of both sides for heights $n\leq 2$ and all primes $p$.
After tensoring with $\mathbb{Q}$, the chromatic splitting conjecture predicts
that

$\mathbb{Q}\otimes\pi_{*}L_{K(n)}S^{0}\cong\Lambda_{\mathbb{Q}_{p}}(\bar{e}_{1},\bar{e}_{2},\ldots,\bar{e}_{n}),\;\left\lvert
e_{i}\right\rvert=1-2i.$ (2.4.2)

Indeed, there are natural equivalences $\mathbb{Q}\otimes
L_{i}X\simeq\mathbb{Q}\otimes X$ for all spectra $X$ and all $i\geq 0$, so (3)
of 2.4.1 rationalizes to (2.4.2). Further chromatic consequences of 2.4.1 can
be found in [BB20] and [Mor14].

A related question concerns the map $\varphi$ appearing in (2.4.1). The
following conjecture222The name comes from the equivalent formulation that
$H_{\operatorname{cts}}^{*}(\mathbb{G}_{n},\pi_{0}E_{n}/W)$ = 0. was
formulated by Beaudry, Goerss, and Henn in [BGH22, Page 3].

######  Conjecture 2.4.2 (Vanishing conjecture).

The inclusion of coefficients $W\hookrightarrow\pi_{0}E_{n}$ induces an
isomorphism $H_{\operatorname{cts}}^{*}(\mathbb{G}_{n},W)\cong
H_{\operatorname{cts}}^{*}(\mathbb{G}_{n},\pi_{0}E_{n})$.

2.4.2 has been verified for all $n\leq 2$ and for all primes $p$; moreover, in
these cases, it does not need to be modified for $p=2$, but rather accounts
for the additional complications witnessed there. If correct, this conjecture
would substantially simplify the task of understanding $\pi_{*}L_{K(n)}S^{0}$.
Our goal in this paper is to prove (2.4.2) for all heights $n$ and all primes
$p$.

###  2.5. Power operations and the splitting of $W\to A$

Fix a prime $p$ and a height $n\geq 1$, and let $E=E_{n}$ be Morava
$E$-theory. Thus $A=E^{0}\cong W\llbracket u_{1},\dots,u_{n-1}\rrbracket$,
where $W=W(\overline{\mathbb{F}}_{p})$. For the first time in the article, we
introduce new material.

Our proof of Theorem A begins with the following proposition.

######  Proposition 2.5.1.

The inclusion $W\hookrightarrow A$ admits a continuous
$\mathbb{G}_{n}$-equivariant (additive) splitting. In other words, there is a
$\mathbb{G}_{n}$-equivariant decomposition of topological abelian groups

$A\cong W\oplus A^{c}.$

Our proof of Proposition 2.5.1 uses input from homotopy theory, namely the
power operations on Morava $E$-theory. We will briefly recall this theory
here:

Power operations on Morava $E$-theory are a consequence of the
$\mathbb{E}_{\infty}$-ring structure on $E$, which endows $E$ with
multiplication maps

$(E^{\otimes m})_{h\Sigma_{m}}\to E.$

This structure is essentially unique by a theorem of Goerss, Hopkins, and
Miller [GH04]. For $m\in\mathbb{N}$, there are natural multiplicative
operations

$P^{m}\colon E^{0}\to E^{0}(B\Sigma_{m}).$

These are defined to be the composite
$[S^{0},E]\to[(S^{0})_{h\Sigma_{m}},(E^{\otimes
m})_{h\Sigma_{m}}]\to[(S^{0})_{h\Sigma_{m}},E]$, where the first map is given
by applying the $m$th tensor power (recalling that $S^{0}$ is the unit) and
applying homotopy orbits for the resulting $\Sigma_{m}$-action, and the second
map makes use of the $\mathbb{E}_{\infty}$-ring structure on $E$.

The operation $P^{0}$ is the constant function $1$ and the operation $P^{1}$
is the identity map on $E^{0}$. Since the $E$-cohomology of a symmetric group
is a free $E^{0}$-module, there is a Kunneth isomorphism

$E^{0}(B\Sigma_{i}\times B\Sigma_{j})\cong
E^{0}(B\Sigma_{i})\otimes_{E^{0}}E^{0}(B\Sigma_{j}).$

Power operations have the property that, if $i+j=m$, then the composite

$E^{0}\xrightarrow{P^{m}}E^{0}(B\Sigma_{m})\to E^{0}(B\Sigma_{i}\times
B\Sigma_{j})\cong E^{0}(B\Sigma_{i})\otimes_{E^{0}}E^{0}(B\Sigma_{j})$

is $P^{i}\otimes P^{j}$.

Although $P^{m}$ is not additive, by [BMMS86, Chapter VIII], the ideal
$I_{\mathrm{tr}}\subseteq E^{0}(B\Sigma_{m})$ generated by the images of the
transfer maps along $\Sigma_{i}\times\Sigma_{j}\subseteq\Sigma_{m}$ has the
property that

$P^{m}/I_{\mathrm{tr}}\colon E^{0}\to E^{0}(B\Sigma_{m})/I_{\mathrm{tr}}$

is a ring map. A theorem of Strickland’s [Str98] proves that
$E^{0}(B\Sigma_{m})/I_{\mathrm{tr}}$ is a finitely generated free
$E^{0}$-module and that there is a canonical isomorphism of formal schemes
over Lubin–Tate space

$\operatorname{Spf}(E^{0}(B\Sigma_{m})/I_{\mathrm{tr}})\cong\mathrm{Sub}_{m}(\bar{\Gamma}),$

where $\bar{\Gamma}$ is the universal deformation of the formal group $\Gamma$
and $\mathrm{Sub}_{m}(\bar{\Gamma})$ is the formal scheme classifying subgroup
schemes of order $m$ in $\bar{\Gamma}$. Note that no such subgroup exists
unless $m=p^{k}$ for some $k$ and thus
$\mathrm{Sub}_{m}(\bar{\Gamma})=\emptyset$ if $m\neq p^{k}$. Ando, Hopkins,
and Strickland [AHS04, Section 3] proved that the map $P^{m}/I_{\mathrm{tr}}$
classifies the deformation associated to the quotient of $\bar{\Gamma}$ by the
universal subgroup of order $m$. In particular, they show that
$P^{m}/I_{\mathrm{tr}}$ is a continuous ring map for all $m\in\mathbb{N}$.

######  Lemma 2.5.2.

The power operations on Morava $E$-theory are continuous with respect to the
$I_{n}$-adic topology on $E^{0}$ and $E^{0}(B\Sigma_{m})$.

###### Proof.

The proof makes use of the fact that $P^{m}/I_{\mathrm{tr}}$ is continuous as
well as an application of Hopkins–Kuhn–Ravenel character theory [HKR00].

Assume that $m=\sum_{i=0}^{j}a_{i}p^{i}$ is the base $p$ expansion of $m$. The
restriction map

$E^{0}(B\Sigma_{m})\to\bigotimes_{i=0}^{j}E^{0}(B\Sigma_{p^{i}})^{\otimes
a_{i}}$

is injective as $E$ is $p$-local and $\prod_{i=0}^{j}\Sigma_{p^{k}}^{\times
a_{i}}$ contains the Sylow $p$-subgroup of $\Sigma_{m}$. All of the tensor
products are over $E^{0}$. We are reduced to proving that $P^{p^{k}}$ is
continuous.

For $0\leq i\leq k$, consider the $E^{0}$-algebra map

$E^{0}(B\Sigma_{p^{k}})\to(E^{0}(B\Sigma_{p^{i}})/I_{\mathrm{tr}})^{\otimes
p^{k-i}}$

given by restriction to $\Sigma_{p^{i}}^{\times p^{k-i}}$, applying the
Kunneth isomorphism, and then taking the quotient by the ideal
$I_{\mathrm{tr}}$. The composite of the power operation $P^{p^{k}}$ with this
map is continuous since it may be identified with the map
$(P^{p^{i}}/I_{\mathrm{tr}})^{\otimes p^{k-i}}$.

Taking the product of these maps for all $0\leq i\leq k$, we get a map

$E^{0}(B\Sigma_{p^{k}})\to\prod_{i=0}^{k}(E^{0}(B\Sigma_{p^{i}})/I_{\mathrm{tr}})^{\otimes
p^{k-i}}.$

Hopkins–Kuhn–Ravenel character theory implies that this map is injective.
Since the composite of the $p^{k}$th power operation with this map is
continuous, the $p^{k}$th power operation is continuous. ∎

###### Proof of Proposition 2.5.1.

Recall that $A=E^{0}$. Let $P^{m}\colon E^{0}\to E^{0}(B\Sigma_{m})$ be the
$m$th power operation, determined by the $\mathbb{E}_{\infty}$-ring structure
on $E$. Let

$\beta_{m}\colon
E^{0}\xrightarrow{P_{m}}E^{0}(B\Sigma_{m})\xrightarrow{\mathrm{Tr}_{\Sigma_{m}}^{e}}E^{0}$

be the composite of the $m$th power operation with the $K(n)$-local transfer
map along the surjection from $\Sigma_{m}$ to the trivial group.

The standard relations among the power operations implies that the formal sum
$\beta(x)=\sum_{m\geq 0}\beta_{m}x^{m}$, considered as map $E^{0}\to
E^{0}\llbracket x\rrbracket$, satisfies $\beta(x+y)=\beta(x)\beta(y)$. Since
$\beta_{0}(a)=1$ and $\beta_{1}(a)=a$ for all $a\in E^{0}$, the map $\beta$
factors through a homomorphism from the additive group $E^{0}$ to the subgroup
$1+xE^{0}\llbracket x\rrbracket$ of $E^{0}\llbracket x\rrbracket^{\times}$.
Now we may quotient the target by the maximal ideal in $E^{0}$ to obtain a map
$E^{0}\to\overline{\mathbb{F}}_{p}\llbracket x\rrbracket$ that sends addition
to multiplication. The big Witt vectors
$W_{\mathrm{big}}(\overline{\mathbb{F}}_{p})$ may be canonically identified
(additively) with the abelian group of units in
$\overline{\mathbb{F}}_{p}\llbracket x\rrbracket$ with constant coefficient
$1$ under multiplication. Further, the $p$-typical Witt vectors
$W=W(\overline{\mathbb{F}}_{p})$ splits off of the big Witt vectors. The
quotient map $(1+x\overline{\mathbb{F}}_{p}\llbracket x\rrbracket)\cong
W_{\mathrm{big}}(\overline{\mathbb{F}}_{p})\to\overline{\mathbb{F}}_{p}$ is
given by reading off the coefficient of $x$, and this quotient map factors
through $W$. The maps constructed so far fit into a diagram:

$\textstyle{E^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{E^{0}\llbracket
x\rrbracket\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathbb{F}}_{p}\llbracket
x\rrbracket\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1+xE^{0}\llbracket
x\rrbracket\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1+x\overline{\mathbb{F}}_{p}\llbracket
x\rrbracket\cong
W_{\mathrm{big}}(\overline{\mathbb{F}}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathbb{F}}_{p}}$

Let $\gamma\colon E^{0}\to W$ be the composition of the maps appearing in the
diagram. Precomposing with the inclusion $W\to E^{0}$, we obtain an additive
endomorphism $f$ of $W$. This is a map between $p$-adically complete modules
and it is the identity modulo $p$; therefore $f$ is an automorphism. The map
$\alpha:=f^{-1}\circ\gamma$ is therefore a section of $W\to E^{0}$.

Further, the maps that go into the construction of $\alpha$ are
$\mathbb{G}_{n}$-equivariant. For $P_{m}$, this follows from the fact that
$\mathbb{G}_{n}$ acts on $E$ via $\mathbb{E}_{\infty}$-ring maps. The transfer
map is $\mathbb{G}_{n}$-equivariant as it is given by restriction along a map
of spectra (alternatively by the formula for this transfer and the action of
$\mathbb{G}_{n}$ on the level of characters). ∎

######  Remark 2.5.3.

We do not know if the map $f\colon W\to W$ appearing in the proof above is the
identity.

######  Corollary 2.5.4.

The inclusion $W\hookrightarrow A$ induces a split injection
$H_{\operatorname{cts}}^{*}(\mathbb{G}_{n},W)\to
H_{\operatorname{cts}}^{*}(\mathbb{G}_{n},A)$ with cokernel
$H_{\operatorname{cts}}^{*}(\mathbb{G}_{n},A^{c})$.

B has thus been reduced to the statement that
$H^{*}_{\operatorname{cts}}(\mathbb{G}_{n},A^{c})$ is $p$-power torsion. This
will be established in the course of the next sections.

###  2.6. The proof of A assuming B

We finish this section by explaining how to deduce A from B. The key point is
that, rationally, the cohomology of the stabilizer group action on the
homotopy groups of Morava $E$-theory simplifies dramatically in non-zero
degrees:

######  Lemma 2.6.1.

For all $t\neq 0$ and all $s\in\mathbb{Z}$, we have
$H^{s}_{\operatorname{cts}}(\mathbb{G}_{n},\mathbb{Q}\otimes\pi_{t}E_{n})=0$.

###### Proof.

Recall that the (extended) Morava stabilizer group $\mathbb{G}_{n}$ can be
described naturally as a semidirect product

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{D}^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{G}_{n}\simeq\mathcal{O}_{D}^{\times}\rtimes\operatorname{Gal}(\overline{\mathbb{F}}_{p}/\mathbb{F}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Gal}(\overline{\mathbb{F}}_{p}/\mathbb{F}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1,}$
(2.6.1)

where $\mathcal{O}_{D}^{\times}$ is isomorphic to the automorphism group of
our chosen formal group law $\Gamma_{n}$ over $\overline{\mathbb{F}}_{p}$. The
center of $\mathcal{O}_{D}^{\times}$ is isomorphic to
$\mathbb{Z}_{p}^{\times}$ and we may consider the central subgroup
$\mathbb{Z}_{p}\subset\mathbb{Z}_{p}^{\times}\trianglelefteqslant\mathcal{O}_{D}^{\times}$,
which we can take to be generated by the element
$1+p\in\mathbb{Z}_{p}^{\times}$. Fixing some integer $t$, the associated
convergent Lyndon–Hochschild–Serre spectral sequence for continuous cohomology
(e.g., [Ser02, Section I.2.6(b)]) has signature

$H^{p}_{\operatorname{cts}}(\mathcal{O}_{D}^{\times}/\mathbb{Z}_{p},H_{\operatorname{cts}}^{q}(\mathbb{Z}_{p},\mathbb{Q}\otimes\pi_{t}E_{n}))\implies
H^{p+q}_{\operatorname{cts}}(\mathcal{O}_{D}^{\times},\mathbb{Q}\otimes\pi_{t}E_{n}).$

It is thus enough to show that

$H^{q}_{\operatorname{cts}}(\mathbb{Z}_{p},\mathbb{Q}\otimes\pi_{t}E_{n})=0$

for $t\neq 0$. To this end, we use that the generator of $\mathbb{Z}_{p}$ acts
by multiplication by $(1+p)^{t}$, see for example [BB20, Secton 3.3.2(c)]. The
continuous $\mathbb{Z}_{p}$-cohomology of $\mathbb{Q}\otimes\pi_{t}E_{n}$ is
thus computed via the complex

$\mathbb{Q}\otimes\pi_{t}E_{n}\xrightarrow{(1+p)^{t}-1}\mathbb{Q}\otimes\pi_{t}E_{n}.$

Since $\mathbb{Q}\otimes\pi_{t}E_{n}$ is a $\mathbb{Q}_{p}$-vector space, when
$t\neq 0$ the action by $(1+p)^{t}-1$ is invertible, so the complex is
acyclic. We then conclude by another application of the
Lyndon–Hochschild–Serre spectral sequence, this time for the extension
(2.6.1). ∎

######  Proposition 2.6.2.

B implies A.

###### Proof.

We use the Devinatz–Hopkins spectral sequence (2.3.3):

$E_{2}^{s,t}\cong
H_{\operatorname{cts}}^{s}(\mathbb{G}_{n},\pi_{t}E_{n})\implies\pi_{t-s}L_{K(n)}S^{0}.$

This spectral sequence converges strongly and collapses on a finite page with
a horizontal vanishing line. On the one hand, it follows that rationalization
yields another strongly convergent spectral sequence

$\mathbb{Q}\otimes E_{2}^{s,t}\cong
H_{\operatorname{cts}}^{s}(\mathbb{G}_{n},\mathbb{Q}\otimes\pi_{t}E_{n})\implies\mathbb{Q}\otimes\pi_{t-s}L_{K(n)}S^{0}.$

Here, the identification of the $E_{2}$-term uses that $\mathbb{G}_{n}$ is a
compact group so that rationalization commutes with taking continuous
cohomology; see for example [BP21, Corollary 12.9]. On the other hand, Lemma
2.6.1 implies that the rationalized Devinatz–Hopkins spectral sequence
collapses on the $E_{2}$-page, resulting in a graded isomorphism

$H_{\operatorname{cts}}^{*}(\mathbb{G}_{n},\mathbb{Q}\otimes\pi_{0}E_{n})\cong\mathbb{Q}\otimes\pi_{-*}L_{K(n)}S^{0}.$

Since the spectral sequence is multiplicative, this is in fact an isomorphism
of graded rings. B combined with Lazard’s theorem [Laz65] as stated in Lemma
3.8.2 identifies the left hand side as

$H_{\operatorname{cts}}^{*}(\mathbb{G}_{n},\mathbb{Q}\otimes\pi_{0}E_{n})\cong
H_{\operatorname{cts}}^{*}(\mathbb{G}_{n},\mathbb{Q}\otimes
W)\cong\Lambda_{\mathbb{Q}_{p}}(x_{1},x_{2},\ldots,x_{n}),$

with $x_{i}$ in cohomological degree $2i-1$. This gives A. ∎

##  3\. Arithmetic prerequisites

To complete the proof of B, we must show that the complement of the split
injection

$H_{\operatorname{cts}}^{*}(\mathbb{G}_{n},W)\rightarrow
H_{\operatorname{cts}}^{*}(\mathbb{G}_{n},A)$

is $p^{N}$-torsion for some $N\geq 0$. Our proof lies entirely within the
domain of $p$-adic geometry. In this section, we review some fundamentals of
$p$-adic geometry and condensed mathematics and how these interact with each
other. At the end of the section, we offer a summary of the proof of B.

###  3.1. Adic spaces

We offer the reader a brief summary of the necessary techniques from non-
archimedean analytic geometry, starting with Huber’s category of adic spaces
[Hub94]. This category contains all formal schemes, rigid-analytic varieties,
and perfectoid spaces. For a more leisurely exposition, see the last named
author’s chapter in [BCKW19].

A topological ring $A$ is a Huber ring if it contains an open subring $A_{0}$
whose topology is induced by a finitely generated ideal $I\subset A_{0}$. A
subset $S$ of a Huber ring $A$ is bounded if for every $n\geq 0$ there exists
$N\geq 0$ such that $I^{N}S\subset I^{n}$. A single element $f\in A$ is power-
bounded if $\left\\{f^{n}\right\\}_{n\geq 1}$ is bounded. A Huber pair is a
pair $(A,A^{+})$ consisting of a Huber ring $A$ and an open and integrally
closed subring $A^{+}\subset A$ whose elements are power-bounded. A continuous
valuation on a Huber ring $A$ is a continuous multiplicative function
$\left\lvert\;\right\rvert\colon A\to H\cup\left\\{0\right\\}$, where $H$ is a
totally ordered abelian group (written multiplicatively). The adic spectrum
$\operatorname{Spa}(A,A^{+})$ is the set of equivalence classes of continuous
valuations satisfying $\left\lvert A^{+}\right\rvert\leq 1$. It is endowed
with the topology generated by rational subsets of the form

$U=U\left(\frac{f_{1},\dots,f_{r}}{g}\right)=\left\\{\left\lvert\;\right\rvert\in\operatorname{Spa}(A,A^{+})\biggm{|}\left\lvert
f_{i}\right\rvert\leq\left\lvert g\right\rvert\neq 0,\;i=1,\dots,r\right\\}$

for $f_{1},\dots,f_{r}\in A$ generating an open ideal and $g\in A$. Then
$\operatorname{Spa}(A,A^{+})$ is quasi-compact.

One defines presheaves of rings $\mathcal{O}_{X}^{+}\subset\mathcal{O}_{X}$ on
$X=\operatorname{Spa}(A,A^{+})$ as follows. For the rational subset $U$ above,
we declare that $\mathcal{O}_{X}(U)$ is the completion of $A[f_{i}/g]$ with
respect to the topology in which $A_{0}[f_{i}/g]$ (with its $I$-adic topology)
is an open subring, and $\mathcal{O}_{X}^{+}(U)$ is the completion of the
integral closure of $A^{+}[f_{i}/g]$ in $A[f_{i}/g]$. Then
$(\mathcal{O}_{X}(U),\mathcal{O}_{X}^{+}(U))$ is another Huber pair. For
certain classes of Huber pairs $(A,A^{+})$ (including all which are considered
in this article), the presheaves $\mathcal{O}_{X}$ and $\mathcal{O}_{X}^{+}$
are sheaves. In such cases, the triple
$(X,\mathcal{O}_{X},\mathcal{O}_{X}^{+})$ is an affinoid adic space. A general
adic space is a triple $(X,\mathcal{O}_{X},\mathcal{O}_{X}^{+})$ which is
locally isomorphic to an affinoid adic space. The sheaves $\mathcal{O}_{X}$
and $\mathcal{O}_{X}^{+}$ are the structure sheaf and integral structure
sheaf, respectively.

As a basic example, if $X=\operatorname{Spa}(W,W)$, then
$\mathcal{O}_{X}=\mathcal{O}_{X}^{+}$ and the ringed space
$(X,\mathcal{O}_{X})$ is isomorphic to $\operatorname{Spec}W$. In particular
it has two points: a generic point lying in $\operatorname{Spa}(K,W)$ (which
extends to the usual absolute value on $K$) and a special point (which
satisfies $\left\lvert p\right\rvert=0$).

A rigid-analytic space over $K$ is an adic space over
$\operatorname{Spa}(K,W)$ that is locally isomorphic to an affinoid adic space
of the form $\operatorname{Spa}(A,A^{+})$, where $(A,A^{+})$ obeys a certain
finiteness condition. Namely, $A$ is isomorphic to a ring quotient of a Tate
algebra $K\left<T_{1},\dots,T_{d}\right>$, and $A^{+}$ is equal to the subring
of power-bounded elements of $A$. A special case is
$A=K\left<T_{1},\dots,T_{d}\right>$, in which case
$\operatorname{Spa}(A,A^{+})$ is the closed ball of radius 1.
(Chronologically, Tate’s theory of rigid-analytic spaces long predates Huber’s
theory of adic spaces; rigid-analytic spaces as Tate defined them sit inside
of adic spaces over $\operatorname{Spa}(K,W)$ as a full subcategory, so there
is no harm in thinking of them this way.)

Fix a continuous real-valued valuation $\left\lvert\;\right\rvert$
representing the sole point of $\operatorname{Spa}(K,W)$, so as to fix a value
of $\left\lvert p\right\rvert$.

######  Example 3.1.1 (The rigid-analytic open ball).

Let $A=W\llbracket T_{1},\dots,T_{d}\rrbracket$, and let
$\mathfrak{B}^{d}=\operatorname{Spa}(A,A)$, the formal $d$-dimensional unit
ball over $\operatorname{Spa}(W,W)$. Let $B^{d,\circ}$ be the fiber of
$\mathfrak{B}^{d}$ over the generic point of $\operatorname{Spa}(W,W)$; i.e.,
$B^{d,\circ}$ is the locus where $\left\lvert p\right\rvert\neq 0$. Let us
observe that $B^{d,\circ}$ is exhausted by affinoid (closed) balls over $K$.
For each real number $r$ of the form $r=\left\lvert p\right\rvert^{1/n}$ with
$n=1,2,3,\dots$, let $B_{r}^{d}$ be the following rational subset of
$\mathfrak{B}^{d}$:

$B_{r}^{d}=U\left(\frac{T_{1}^{n},\dots,T_{d}^{n},p}{p}\right).$

Then $B_{r}^{d}$ is an affinoid rigid-analytic space over $K$. We have

$B^{d,\circ}=\varinjlim_{r<1}B_{r}^{d},$

since for each continuous valuation on $A$ with $\left\lvert p\right\rvert\neq
0$, we must have $\left\lvert T_{i}\right\rvert^{n}\leq\left\lvert
p\right\rvert$ for $n$ sufficiently large. Therefore $B^{d,\circ}$ is a rigid-
analytic space over $K$.

For a rigid-analytic space $X/K$, there is an important distinction between
$H^{0}(X,\mathcal{O}_{X})$ and $H^{0}(X,\mathcal{O}_{X}^{+})[1/p]$. When $X$
is affinoid, these agree, but in general they are quite different. In the
situation of the rigid-analytic open ball, we have

$H^{0}(B^{d,\circ},\mathcal{O}_{X}^{+})[1/p]=W\llbracket
T_{1},\dots,T_{d}\rrbracket[1/p]$

is the ring of power series in $K\llbracket T_{1},\dots,T_{d}\rrbracket$ which
are bounded on the open unit ball. Whereas,
$H^{0}(B^{d,\circ},\mathcal{O}_{X})$ is the much larger ring of power series
in $K\llbracket T_{1},\dots,T_{d}\rrbracket$ which converge on the open unit
ball.

###  3.2. Condensed mathematics

When $G$ is a topological group and $M$ is a topological abelian group
admitting a continuous $G$-action, continuous cohomology groups
$H^{i}_{\operatorname{cts}}(G,M)$ are defined using the complex of continuous
cocycles. At this level of generality, however, one has no abelian category of
topological abelian groups, and so $M\mapsto H^{i}_{\operatorname{cts}}(G,M)$
has no interpretation as the $i$th derived functor of fixed points $M\mapsto
M^{G}$.

The language of condensed mathematics [CS] is well-suited to resolve this
issue in all contexts which will arise in this article. We quickly review the
main points of the theory, ignoring set-theoretic issues throughout.333They
can be dealt with as in [CS, Lecture I]. The pro-étale site of a point
$\ast_{\mathrm{pro\acute{e}t}}$ is the category of profinite sets with jointly
surjective continuous maps as the covers. A condensed set is a sheaf of sets
on $\ast_{\mathrm{pro\acute{e}t}}$. Similarly there are condensed groups,
rings, etc. There is a functor $X\mapsto\underline{X}$ from topological
spaces/groups/rings to condensed sets/groups/rings, via
$\underline{X}(S)=C_{\operatorname{cts}}(S,X)$, meaning continuous maps $S\to
X$. This functor is fully faithful when restricted to compactly generated
topological spaces [CS, Proposition 1.7].

Let $\operatorname{Cond}(\operatorname{Ab})$ be the category of condensed
abelian groups. Then $\operatorname{Cond}(\operatorname{Ab})$ is an abelian
category containing all limits and colimits [CS, Theorem 1.10]. It has a
symmetric monoidal tensor product $\mathcal{M}\otimes\mathcal{N}$ and an
internal Hom-functor $\underline{\operatorname{Hom}}(\mathcal{M},\mathcal{N})$
related by the adjunction

$\operatorname{Hom}(\mathcal{P},\underline{\operatorname{Hom}}(\mathcal{M},\mathcal{N}))\cong\operatorname{Hom}(\mathcal{P}\otimes\mathcal{M},\mathcal{N}).$

The forgetful functor
$\operatorname{Cond}(\operatorname{Ab})\to\operatorname{Cond}(\mathrm{Set})$
has a left adjoint $\mathcal{X}\mapsto\mathbb{Z}[\mathcal{X}]$, the “free
condensed abelian group on $\mathcal{X}$”. In the case
$\mathcal{X}=\underline{S}$ for $S$ profinite, we abuse notation and write
$\mathbb{Z}[S]$ for $\mathbb{Z}[\underline{S}]$.

The category $\operatorname{Cond}(\operatorname{Ab})$ has enough projectives
[CS, Theorem 2.2], so we can form its derived category
$D(\operatorname{Cond}(\operatorname{Ab}))$. Then
$D(\operatorname{Cond}(\operatorname{Ab}))$ admits a derived tensor product
$\otimes$ and a derived internal hom functor $\operatorname{RHom}$ satisfying
the usual adjunction relation.

In the language of condensed mathematics, the notion of completeness goes by
the term “solid”. For a profinite set $S=\varprojlim S_{i}$ with each $S_{i}$
finite, the free solid abelian group on $S$ is defined as

$\mathbb{Z}[S]^{\blacksquare}=\varprojlim\underline{\mathbb{Z}[S_{i}]},$

where $\mathbb{Z}[S_{i}]$ is the free abelian group on $S_{i}$, considered as
a discrete topological group. A solid abelian group is a condensed abelian
group $\mathcal{M}$ such that for all profinite sets $S$, any morphism
$\underline{S}\to\mathcal{M}$ extends uniquely to a morphism
$\mathbb{Z}[S]^{\blacksquare}\to\mathcal{M}$. Let $\operatorname{Solid}$
denote the category of solid abelian groups; by [CS, Theorem 5.8],
$\operatorname{Solid}$ is closed under all limits and colimits in
$\operatorname{Cond}(\operatorname{Ab})$. Then $\operatorname{Solid}$ is an
abelian subcategory of $\operatorname{Cond}(\operatorname{Ab})$. The functor
$D(\operatorname{Solid})\to D(\operatorname{Cond}(\operatorname{Ab}))$ is
fully faithful. For an object $\mathcal{C}$ of
$D(\operatorname{Cond}(\operatorname{Ab}))$, the following are equivalent:

  1. (1)

$\mathcal{C}$ lies in the essential image of $D(\operatorname{Solid})\to
D(\operatorname{Cond}(\operatorname{Ab}))$.

  2. (2)

$H^{i}(\mathcal{C})$ is a solid abelian group for all $i\in\mathbb{Z}$.

  3. (3)

For all profinite sets $S$, the natural map

$\operatorname{RHom}(\mathbb{Z}[S]^{\blacksquare},\mathcal{C})\to\operatorname{RHom}(\mathbb{Z}[S],\mathcal{C})$

is an isomorphism.

The inclusion
$\operatorname{Solid}\subset\operatorname{Cond}(\operatorname{Ab})$ admits a
left adjoint $\mathcal{M}\mapsto\mathcal{M}^{\blacksquare}$, the
“solidification functor”; this is the unique colimit-preserving extension of
$\mathbb{Z}[S]\mapsto\mathbb{Z}[S]^{\blacksquare}$ [CS, Theorem 5.8]. There is
a unique symmetric monoidal tensor product
$\mathcal{M}\otimes^{\blacksquare}\mathcal{N}$ such that
$\mathcal{M}\mapsto\mathcal{M}^{\blacksquare}$ is symmetric monoidal [CS,
Theorem 6.2].

The following lemma shows how solidity and completeness are related.

######  Lemma 3.2.1.

Let $M$ be a topological abelian group which is separated and complete for a
linear topology. Then $\underline{M}$ is a solid abelian group.

###### Proof.

The hypothesis means there is a directed system of open subgroups
$M_{n}\subset M$ inducing the topology on $M$, such that the map
$M\to\varprojlim M/M_{n}$ is an isomorphism.

Let $S=\varprojlim S_{i}$ with each $S_{i}$ finite, and let $f\colon S\to M$
be continuous. For each $n$, the map $S\to M/M_{n}$ is locally constant, so it
must factor through $S_{i}\to M/M_{n}$ for some $i=i(n)$. This map can be
extended to a morphism of condensed abelian groups
$\underline{\mathbb{Z}[S_{i}]}\to\underline{M/M_{n}}$. After passage to the
limit in $n$, we obtain a morphism
$\mathbb{Z}[S]^{\blacksquare}\to\underline{M}$. ∎

###  3.3. Solid complexes with $G$-action

Now suppose $\mathcal{G}$ is a condensed group. A $\mathcal{G}$-action on a
condensed abelian group $\mathcal{M}$ is a morphism
$\mathcal{G}\times\mathcal{M}\to\mathcal{M}$ satisfying the usual axioms.

######  Lemma 3.3.1.

Let $M$ be a topological abelian group which is separated and complete for a
linear topology. Let $G$ be a profinite group which acts continuously on $M$.
Then $\underline{M}$ is a solid abelian group with $\underline{G}$-action.

###### Proof.

The claim in the lemma is that the group action $G\times M\to M$ can be
upgraded to an action $\underline{G}\times\underline{M}\to\underline{M}$ on
the level of condensed sets. That is, we need for every profinite
$S=\varprojlim S_{i}$ an action

$C_{\operatorname{cts}}(S,G)\times C_{\operatorname{cts}}(S,M)\to
C_{\operatorname{cts}}(S,M)$

which is functorial in $S$. Therefore let $f\colon S\to G$ and $h\colon S\to
M$ be continuous. Since $M\cong\varprojlim M/M_{n}$ for a directed system of
open subgroups $M_{n}$, it is enough to produce an action with values in
$C_{\operatorname{cts}}(S,M/M_{n})$ compatibly in $n$.

The continuity of the action of $G$ on $M$ means exactly that there exists an
index $N$ and an open subgroup $H\subset G$ such that $HM_{N}\subset M_{n}$.
Since $h$ is continuous, it is locally constant modulo $M_{N}$; that is, the
composition $S\to M\to M/M_{N}$ factors through $h_{i}\colon S_{i}\to M/M_{N}$
for some $i$. After replacing $i$, we may also assume that $f\colon S\to G\to
G/H$ factors through $f_{i}\colon S_{i}\to G/H$. Then the sum $\sum_{s\in
S_{i}}f_{i}(s)h(s)$ is well-defined in $M/M_{N}$; this is the required action
of $f$ on $h$. ∎

For a profinite group $G$, let $\operatorname{Solid}_{G}$ be the category of
solid abelian groups admitting an action of $\underline{G}$. Then
$\operatorname{Solid}_{G}$ is an abelian category.

######  Lemma 3.3.2.

Let $G$ be a profinite group. Consider the functor of fixed points
$\operatorname{Solid}_{G}\to\operatorname{Solid}$ defined by
$\mathcal{M}\mapsto\mathcal{M}^{G}$, i.e., the right adjoint to the trivial
action functor $\operatorname{Solid}\to\operatorname{Solid}_{G}$. Let
$\mathcal{C}\mapsto R\Gamma(G,\mathcal{C})$ be its derived functor
$D(\operatorname{Solid}_{G})\to D(\operatorname{Solid})$, and let
$H^{i}(G,\mathcal{C})=R^{i}\Gamma(G,\mathcal{C})$.

  1. (1)

For an object $\mathcal{C}$ of $D(\operatorname{Solid})$, the object
$R\Gamma(G,\mathcal{C})$ is the totalization of the cosimplicial object

$\mathcal{C}\to\underline{\operatorname{RHom}}(\mathbb{Z}[G],\mathcal{C})\rightrightarrows\underline{\operatorname{RHom}}(\mathbb{Z}[G^{2}],\mathcal{C})\mathrel{\begin{subarray}{c}\textstyle\rightarrow\\\\[-2.58334pt]
\textstyle\rightarrow\\\\[-2.58334pt]
\textstyle\rightarrow\end{subarray}}\cdots,$

where the degeneracy maps are omitted from the notation.

  2. (2)

In particular, if $M$ is an abelian group which is separated and complete for
a linear topology, and $G$ acts continuously on $M$, then

$H^{i}(G,\underline{M})\cong\underline{H^{i}_{\operatorname{cts}}(G,M)}.$

Here $H^{i}_{\operatorname{cts}}(G,M)$ is continuous cohomology.

###### Proof.

The idea is to construct a simplicial resolution of the trivial module
$\mathbb{Z}$ in $\operatorname{Solid}_{G}$. Let

$\mathbb{Z}[G]^{\blacksquare}=\varprojlim_{H}\underline{\mathbb{Z}[G/H]}$

be the “solid Iwasawa algebra”; here $H$ runs over open subgroups of $G$. We
claim that $\mathbb{Z}[G]^{\blacksquare}$ is projective in
$\operatorname{Solid}_{G}$. Indeed, let
$\mathcal{M}\to\mathbb{Z}[G]^{\blacksquare}$ be a surjection; we want to
produce a section. Since $\mathcal{M}$ is solid, it is enough to produce a
morphism $\underline{G}\to\mathcal{M}$ such that the composition
$\underline{G}\to\mathcal{M}\to\mathbb{Z}[G]^{\blacksquare}$ is the natural
map. The required morphism is $g\mapsto gm$, where $m\in\mathcal{M}(\ast)$
lifts the identity section $\ast\to\mathbb{Z}[G]^{\blacksquare}$.

Similarly, the solid Iwasawa algebras $\mathbb{Z}[G^{n}]^{\blacksquare}$ are
projective. Thus we have the usual simplicial resolution of the trivial
$\underline{G}$-module $\underline{\mathbb{Z}}$:

$\cdots\mathrel{\begin{subarray}{c}\textstyle\rightarrow\\\\[-2.58334pt]
\textstyle\rightarrow\\\\[-2.58334pt]
\textstyle\rightarrow\end{subarray}}\mathbb{Z}[G^{2}]^{\blacksquare}\rightrightarrows\mathbb{Z}[G]^{\blacksquare}\to\underline{\mathbb{Z}}.$
(3.3.1)

If $\mathcal{C}$ is an object of $D(\operatorname{Solid}_{G})$, then
$R\Gamma(G,\mathcal{C})$ is the totalization of the cosimplicial object with
terms
$\underline{\operatorname{RHom}}(\mathbb{Z}[G^{n}]^{\blacksquare},\mathcal{C})$,
obtained from (3.3.1) by applying the functor
$\underline{\operatorname{RHom}}(-,\mathcal{C})$. Since $\mathcal{C}$ is
solid, the latter is isomorphic to
$\underline{\operatorname{RHom}}(\mathbb{Z}[G^{n}],\mathcal{C})$. This is (1).

In the case $\mathcal{C}=\underline{M}$ described in (2), we have (since
$\mathbb{Z}[G]$ is a free condensed abelian group)

$\underline{\operatorname{RHom}}(\mathbb{Z}[G^{n}],\underline{M})\cong\underline{\operatorname{Hom}}(G^{n},\underline{M})\cong\underline{C_{\operatorname{cts}}(G^{n},M)},$

so that $R\Gamma(G,\underline{M})$ is the totalization of the condensed
version of the cosimplicial resolution which computes continuous cohomology.
Therefore $H^{i}(G,\underline{M})$ is the condensed version of
$H^{i}_{\operatorname{cts}}(G,M)$. ∎

Lemma 3.3.2 shows that as long as $M$ is separated and complete for a linear
topology, the continuous cohomology $H^{i}_{\operatorname{cts}}(G,M)$ really
is the derived functor of $G$-fixed points in an abelian category. This allows
us to seamlessly use the language of derived categories to compute
$H^{i}_{\operatorname{cts}}(G,M)$.

######  Notation 3.3.3 (Continuous homotopy fixed points).

Let $G$ be a profinite group. We write

$\displaystyle D(\operatorname{Solid}_{G})$ $\displaystyle\to$ $\displaystyle
D(\operatorname{Solid})$ $\displaystyle\mathcal{C}$ $\displaystyle\mapsto$
$\displaystyle\mathcal{C}^{hG}$

for the functor $R\Gamma(G,\mathcal{C})$ described in Lemma 3.3.2; i.e., the
derived functor of $\mathcal{M}\mapsto\mathcal{M}^{G}$. Similarly, if $M$ is a
topological abelian group which is separated and complete for a linear
topology, and $G$ acts continuously on $M$, then we write $M^{hG}$ for
$R\Gamma(G,\underline{M})$.

By Lemma 3.3.2, $M^{hG}$ is a complex of solid abelian groups which computes
$H^{i}_{\operatorname{cts}}(G,M)$; it would also be appropriate to use the
notation $R\Gamma_{\operatorname{cts}}(G,M)$ for $M^{hG}$. If $H\subset G$ is
a closed normal subgroup, then $M^{hH}$ is an object of
$D(\operatorname{Solid}_{G/H})$, and then the quasi-isomorphism

$M^{hG}\cong\left(M^{hH}\right)^{h(G/H)}$

formally implies the Hochschild–Serre spectral sequence in continuous
cohomology:

$H^{i}_{\operatorname{cts}}(G/H,H^{j}_{\operatorname{cts}}(H,M))\implies
H^{i+j}_{\operatorname{cts}}(G,M).$

Crucially, we can apply this picture to the example where $M=A$ is the
Lubin–Tate ring and $G=\mathbb{G}_{n}$ is the Morava stabilizer group. Another
important example we will encounter occurs when $K$ is a nonarchimedean local
field, $G=\operatorname{Gal}(\overline{K}/K)$, and $M=\mathcal{O}_{C}$, the
valuation ring of the completion $C$ of an algebraic closure $\overline{K}$ of
$K$.

We will find the following “projection formula” useful, and omit its proof.

######  Lemma 3.3.4.

Let $M,N\in D(\operatorname{Solid}_{G})$ be objects such that the action of
$G$ on $M$ is trivial. There is a canonical isomorphism in
$D(\operatorname{Solid})$:

$M\otimes^{\blacksquare}N^{hG}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}(M\otimes^{\blacksquare}N)^{hG}.$

###  3.4. $p$-adic solid complexes

We gather some results here regarding the notion of derived $p$-completeness
(for $p$ a prime number), in the contexts of both $D(\operatorname{Ab})$ and
$D(\operatorname{Cond}(\operatorname{Ab}))$. Many of these results could be
extended to the setting of derived $I$-completeness of $R$-modules, where $I$
is a finitely generated ideal in a ring $R$. The main result Lemma 3.4.8 below
is that $D(\operatorname{Solid})$ contains the category
$D(\operatorname{Ab})_{p}$ of derived $p$-complete complexes as a fully
faithful subcategory.

First we discuss the notion of derived $p$-completeness in
$D(\operatorname{Ab})$, for which some standard results are found in [Aut, Tag
091N].

######  Definition 3.4.1.

An object $A$ of $D(\operatorname{Ab})$ is $p$-complete (the word “derived”
being understood) if $\underline{\operatorname{Hom}}(B,A)\cong 0$ for all
$B\in D(\operatorname{Ab})$ such that $B\otimes\mathbb{Z}/p\mathbb{Z}\cong 0$.
Let $D(\operatorname{Ab})_{p}$ be the full subcategory of
$D(\operatorname{Ab})$ consisting of $p$-complete objects. (Here and elsewhere
we write $\otimes$ instead of $\otimes^{\mathbb{L}}$ for the symmetric
monoidal operation on $D(\operatorname{Ab})$.)

######  Remark 3.4.2.

This definition makes it clear that $D(\operatorname{Ab})_{p}$ is exactly the
Bousfield localization of $D(\operatorname{Ab})$ at the object
$\mathbb{Z}/p\mathbb{Z}$, as described in Section 2. It suffices to check the
condition in the definition with the object $B=\mathbb{Z}[1/p]$.

The inclusion $D(\operatorname{Ab})_{p}\subset D(\operatorname{Ab})$ has a
left adjoint $A\mapsto\hat{A}$, namely:

$\hat{A}=\varprojlim A/p^{n}.$

(Here $A/p^{n}$ is the cofiber of $p^{n}\colon A\to A$.) There is a unique
symmetric monoidal operation $\hat{\otimes}$ on $D(\operatorname{Ab})_{p}$
which makes $A\mapsto\hat{A}$ a symmetric monoidal functor.

It is straightforward to extend these notions to
$D(\operatorname{Cond}(\operatorname{Ab}))$.

######  Definition 3.4.3.

An object $A$ of $D(\operatorname{Cond}(\operatorname{Ab}))$ is $p$-complete
if $\underline{\operatorname{Hom}}(B,A)=0$ for all $B\in
D(\operatorname{Cond}(\operatorname{Ab}))$ with
$B\otimes\mathbb{Z}/p\mathbb{Z}\cong 0$. Let
$D(\operatorname{Cond}(\operatorname{Ab}))_{p}$ be the full subcategory of
$D(\operatorname{Cond}(\operatorname{Ab}))$ consisting of $p$-complete
objects.

######  Lemma 3.4.4.

Regarding the category $D(\operatorname{Cond}(\operatorname{Ab}))_{p}$ of
$p$-adically complete complexes of condensed abelian groups, we have:

  1. (1)

$D(\operatorname{Cond}(\operatorname{Ab}))_{p}$ is closed under limits.

  2. (2)

Suppose $A\in D(\operatorname{Cond}(\operatorname{Ab}))_{p}$. If $A/p\cong 0$,
then $A\cong 0$.

  3. (3)

The inclusion $D(\operatorname{Cond}(\operatorname{Ab}))_{p}\to
D(\operatorname{Cond}(\operatorname{Ab}))$ admits a left adjoint $A\mapsto
A^{\wedge}_{p}$, namely:

$A^{\wedge}_{p}=\varprojlim A/p^{n}.$

###### Proof.

For part (1): if $A_{i}$ is a diagram in
$D(\operatorname{Cond}(\operatorname{Ab}))_{p}$, and $B/p\cong 0$, then
$\underline{\operatorname{Hom}}(B,\operatorname{lim}A_{i})\cong\operatorname{lim}\underline{\operatorname{Hom}}(B,A_{i})\cong
0$, so that $\operatorname{lim}A_{i}$ lies in
$D(\operatorname{Cond}(\operatorname{Ab}))_{p}$.

For part (2): $A/p\cong 0$ means that multiplication by $p$ is an isomorphism
on $A$, so that
$\underline{\operatorname{Hom}}(\underline{\mathbb{Z}[1/p]},A)\to\underline{\operatorname{Hom}}(\underline{\mathbb{Z}},A)\cong
A$ is an isomorphism. On the other hand,
$\underline{\operatorname{Hom}}(\underline{\mathbb{Z}[1/p]},A)\cong 0$ by
definition of derived $p$-completeness.

For part (3): it is enough to show that (a) $A^{\wedge}_{p}$ is $p$-adically
complete, and (b) if $A$ is $p$-adically complete, then $A\to A^{\wedge}_{p}$
is an isomorphism.

For (a): We first claim $A/p^{n}$ is $p$-complete. Indeed, if $B/p\cong 0$,
then multiplication by $p^{n}$ is an isomorphism on $B$, and thus on
$\underline{\operatorname{Hom}}(B,A)$. As
$D(\operatorname{Cond}(\operatorname{Ab}))_{p}$ is closed under limits, it
must contain $A^{\wedge}_{p}$ as well.

For (b): Suppose $A$ is $p$-adically complete, and let $F$ be the fiber of
$A\to A^{\wedge}_{p}$. Then $F$ is also $p$-adically complete (being a limit
in $D(\operatorname{Cond}(\operatorname{Ab}))_{p}$), and so by part (2) it
suffices to show that $F/p\cong 0$. But this is clear, as $F$ is the limit of
multiplication by $p$ on $A$. ∎

At this point we start relating the notions of $p$-completeness in
$D(\operatorname{Ab})$ and $D(\operatorname{Cond}(\operatorname{Ab}))$. First
we need a notion of discreteness for objects of the latter category.

######  Definition 3.4.5.

Let
$\Gamma_{*}\colon\operatorname{Cond}(\operatorname{Ab})\to\operatorname{Ab}$
be the functor which evaluates an object on the point: $\Gamma_{*}(A)=A(*)$.
Then $\Gamma_{*}$ is right adjoint to the pullback functor $\Gamma^{*}$ which
sends an abelian group $A$ to the constant condensed abelian group
$\underline{A}$. Both $\Gamma^{*}$ and $\Gamma_{*}$ are exact (the latter
because all covers of $\ast$ have a section), so we continue to write
$\Gamma^{*}$ and $\Gamma_{*}$ for their derived functors.

For $A\in D(\operatorname{Cond}(\operatorname{Ab}))$ we let
$A^{\delta}=\Gamma^{*}\Gamma_{*}(A)$ be the “discretization” of $A$. The
counit of the adjunction gives rise to a natural map $A^{\delta}\to A.$ If
this map is an isomorphism we say that $A$ is _discrete_.

######  Lemma 3.4.6.

  1. (1)

Discretization is $t$-exact. That is, if $A$ lies in the heart
$D(\operatorname{Cond}(\operatorname{Ab}))^{\heartsuit}\simeq\operatorname{Cond}(\operatorname{Ab})$,
then $A^{\delta}\in D(\operatorname{Cond}(\operatorname{Ab}))^{\heartsuit}$ as
well.

  2. (2)

The subcategory of discrete objects in
$D(\operatorname{Cond}(\operatorname{Ab}))$ is closed under fibers, cofibers
and retracts.

  3. (3)

If $A\in D(\operatorname{Ab})$, then $\Gamma^{*}A$ is discrete.

  4. (4)

An object $A\in D(\operatorname{Cond}(\operatorname{Ab}))$ has the property
that, for all $i\in\mathbb{Z}$, $H^{i}(A)$ is discrete if and only if $A$ is
discrete.

###### Proof.

(1) Since $\Gamma^{*}$ is given by pulling back sheaves, it is $t$-exact.
Since $\Gamma_{*}$ is corepresented by $\underline{\mathbb{Z}}$, which is
projective as a condensed abelian group, $\Gamma_{*}$ is also $t$-exact. For
(2), this follows from the fact that both $\Gamma^{*}$ and $\Gamma_{*}$ are
exact, i.e., they both commute with fibers, cofibers and retracts. For (3),
this is equivalent to the statement that $\Gamma^{*}$ is fully faithful. This
is in turn the same as the statement that the unit of the adjunction is an
equivalence, and this is true. The final statement follows from the previous
items. ∎

######  Definition 3.4.7.

An object $A\in D(\operatorname{Cond}(\operatorname{Ab}))$ is called _$p$
-adic_ if it is $p$-complete and $A\otimes\mathbb{Z}/p$ is discrete.

######  Lemma 3.4.8.

Consider the functor

$\displaystyle D(\operatorname{Ab})$ $\displaystyle\to$ $\displaystyle
D(\operatorname{Cond}(\operatorname{Ab}))$ $\displaystyle A$
$\displaystyle\mapsto$
$\displaystyle(\Gamma^{*}A)^{\wedge}_{p}=\varprojlim\Gamma^{*}(A)/p^{n}$

It factors through an equivalence between $D(\operatorname{Ab})_{p}$ and the
full subcategory of $D(\operatorname{Cond}(\operatorname{Ab}))_{p}$ consisting
of the $p$-adic objects.

Finally, $p$-adic objects in $D(\operatorname{Cond}(\operatorname{Ab}))_{p}$
are solid, and the functor $D(\operatorname{Ab})_{p}\to
D(\operatorname{Cond}(\operatorname{Ab}))_{p}$ is symmetric monoidal with
respect to $\hat{\otimes}$ and $\otimes^{\blacksquare}$.

###### Proof.

We have

$(\Gamma^{*}A^{\wedge}_{p})^{\wedge}_{p}=\varprojlim\underline{A}^{\wedge}_{p}/p^{n}\cong\varprojlim
A/p^{n}\cong(\Gamma^{*}A)^{\wedge}_{p},$

so the functor factors through $A\mapsto\underline{A}^{\wedge}_{p}$. We need
to check that $(\Gamma^{*}A)^{\wedge}_{p}$ is $p$-adic. It is $p$-complete
because

$\underline{\operatorname{Hom}}(\underline{\mathbb{Z}[1/p]},(\Gamma^{*}A)^{\wedge}_{p})=\varprojlim\underline{\operatorname{Hom}}(\underline{\mathbb{Z}[1/p]},\Gamma^{*}(A)/p^{n})\cong
0,$

and $(\Gamma^{*}A)^{\wedge}/p\cong\Gamma^{*}(A/p)$ is discrete; therefore it
is $p$-adic. The quasi-inverse functor carries a $p$-adic object $A\in
D(\operatorname{Cond}(\operatorname{Ab}))$ onto $\Gamma_{*}A$.

The object $(\Gamma^{*}A)^{\wedge}_{p}$ is solid because it is a limit of
discrete objects $\Gamma^{*}A/p^{n}$, which are all solid. The claim about the
functor being symmetric monoidal follows from applying the same fact about
$\Gamma^{*}$ to objects modulo $p^{n}$, and then taking the limit. ∎

Given a derived $p$-complete abelian group $A$ with a continuous action of a
profinite group $\mathbb{G}$, Lemma 3.3.2 (2) implies that we can consider
continuous cohomology as the cohomology groups of the functor
$(-)^{h\mathbb{G}}$ applied to $(\Gamma^{*}A)^{\wedge}_{p}$. We get the same
identification for complexes of derived $p$-complete continuous
$\mathbb{G}$-modules.

###  3.5. Sheaves of condensed abelian groups

Let $X$ be a site, and let $\mathcal{A}$ be a sheaf of topological abelian
groups on $X$. This means in particular that if $\left\\{U_{i}\right\\}$ is a
cover of an object $U$ in $X$, then

$\mathcal{A}(U)\to\prod_{i}\mathcal{A}(U_{i})\rightrightarrows\prod_{i,j}\mathcal{A}(U_{i}\cap
U_{j})$

is an equalizer diagram in the category of topological abelian groups.

In this situation one would like the cohomology $R\Gamma(X,\mathcal{A})$ to
carry a topology, and indeed this is possible using condensed mathematics.
Define a presheaf $\mathcal{A}_{\operatorname{cond}}$ of condensed abelian
groups on $X$ by setting

$\mathcal{A}_{\operatorname{cond}}(U)=\underline{\mathcal{A}(U)}$

for every object $U$ of $X$. Then in fact $\mathcal{A}_{\operatorname{cond}}$
is a sheaf, which we call the condensed enhancement of $\mathcal{A}$. This
follows from the fact that $A\mapsto\underline{A}$ is limit-preserving since
it admits a left adjoint (see [CS, Proposition 1.7]); in particular it
preserves equalizers.

From here we can form the condensed cohomology
$R\Gamma(X,\mathcal{A}_{\operatorname{cond}})\in
D(\operatorname{Cond}(\operatorname{Ab}))$ as the derived global sections of
$\mathcal{A}_{\operatorname{cond}}$. This is a condensed enhancement of the
cohomology $R\Gamma(X,\mathcal{A})$ of the underlying sheaf of abelian groups,
in the sense that $\Gamma_{*}R\Gamma(X,\mathcal{A}_{\operatorname{cond}})\cong
R\Gamma(X,\mathcal{A})$. Indeed, $\Gamma_{*}$ commutes with all limits and
colimits and in particular simplicial limits and filtered colimits. Thus, it
commutes with the computation of derived global sections $R\Gamma$.

An important example for us comes from formal schemes. Recall the notion of an
admissible topological ring from [Aut, Tag0AHY]. For instance, if $A$ is
separated and complete with respect to the topology induced by an ideal $I$,
then $A$ with its $I$-adic topology is an admissible topological ring. An
ideal of definition in an admissible topological ring is an open ideal $I$,
such that every neighborhood of $0$ contains $I^{n}$ for some $I$. An
admissible topological ring always contains an ideal of definition.

For an admissible topological ring $A$ and any ideal of definition $I$, the
map $\operatorname{Spec}A/I\to\operatorname{Spec}A$ factors through a
homeomorphism from $\operatorname{Spec}A/I$ onto the subset
$\operatorname{Spf}A\subset\operatorname{Spec}A$ consisting of open prime
ideals. The structure sheaf $\mathcal{O}_{\operatorname{Spf}A}$ is defined as
$\mathcal{O}_{\operatorname{Spf}A}=\operatorname{lim}_{I}\mathcal{O}_{\operatorname{Spec}A/I}$,
where the limit runs over all ideals of definition for $A$. Here the limit is
taken in the category of sheaves of topological rings, where each
$\mathcal{O}_{\operatorname{Spec}A/I}$ is discrete. Finally, a formal scheme
is a pair $(\mathfrak{X},\mathcal{O})$ consisting of a topological space
$\mathfrak{X}$ and a sheaf of topological rings $\mathcal{O}$ on $X$, such
that $(\mathfrak{X},\mathcal{O})$ is locally isomorphic to
$(\operatorname{Spf}A,\mathcal{O}_{\operatorname{Spf}A})$ for an admissible
topological ring.

For a formal scheme $\mathfrak{X}$, let $\mathcal{O}_{\operatorname{cond}}$ be
the condensed structure sheaf; that is, the condensed enhancement of the
structure sheaf $\mathcal{O}$ on $\mathfrak{X}$. Then
$R\Gamma(\mathfrak{X},\mathcal{O}_{\operatorname{cond}})$ is a ring object in
$D(\operatorname{Cond}(\operatorname{Ab}))$. We will need the condensed
version of a basic acyclicity theorem for affine formal schemes.

######  Lemma 3.5.1.

Let $A$ be an admissible topological ring, so that
$\mathfrak{X}=\operatorname{Spf}A$ is an affine formal scheme. The condensed
structure sheaf is acyclic on $\mathfrak{X}$, in the sense that the natural
map $\underline{A}\to R\Gamma(\mathfrak{X},\mathcal{O}_{\operatorname{cond}})$
is an isomorphism.

###### Proof.

Considering that
$\mathcal{O}_{\operatorname{Spf}A}=\operatorname{lim}_{I}\mathcal{O}_{\operatorname{Spec}A/I}$
is a limit of sheaves of discrete rings, we have that
$\mathcal{O}_{\operatorname{cond}}=\operatorname{lim}_{I}\mathcal{O}_{\operatorname{Spec}A/I}$
is a limit of sheaves of condensed abelian groups which are discrete.
Therefore

$R\Gamma(\mathfrak{X},\mathcal{O}_{\operatorname{cond}})\cong\operatorname{lim}_{I}R\Gamma(\operatorname{Spec}A/I,\mathcal{O}_{\operatorname{Spec}A/I}).$

We claim that
$R\Gamma(\operatorname{Spec}A/I,\mathcal{O}_{\operatorname{Spec}A/I})\cong\underline{A/I}$,
which implies the lemma, since
$\operatorname{lim}_{I}\underline{A/I}\cong\underline{A}$.

In other words, we have reduced the lemma to the case that $A$ is discrete. By
a standard argument [Aut, Tag 01EW], it suffices to show that the Čech
cohomology of $\mathcal{O}_{\operatorname{cond}}$ is acyclic with respect to
the standard open cover of $\operatorname{Spec}A$ by opens $\left\\{f_{i}\neq
0\right\\}$ for elements $f_{1},\dots,f_{n}\in A$ generating the unit ideal.
Recall [Aut, Tag 01X9] the uncondensed version of this statement, which is the
statement that the extended Čech complex

$0\to A\to\prod_{i}A[1/f_{i}]\to\prod_{i,j}A[1/f_{i}f_{j}]\to\cdots$

is exact. To get the condensed version, we need the fact that the functor
$X\mapsto\underline{X}$ from discrete abelian groups to condensed abelian
groups preserves exact complexes. This follows from the formula
$\underline{X}(S)=\varinjlim_{i}X^{S_{i}}$ whenever $X$ is discrete and
$S=\varprojlim S_{i}$ is profinite. ∎

###  3.6. Pro-étale cohomology of rigid-analytic spaces

Let $X$ be a rigid-analytic space over $K$. We swiftly recall some material
from [Sch13a, §3,4] regarding the pro-étale site $X_{\mathrm{pro\acute{e}t}}$.

Objects in $X_{\mathrm{pro\acute{e}t}}$ are formal limits $U=\varprojlim
U_{i}$, where $i$ runs over a cofiltered index set, the $U_{i}$ are rigid-
analytic spaces étale over $X$, and each transition map $U_{i}\to U_{j}$
commutes with the maps to $X$. It is required that $U_{i}\to U_{j}$ is finite
étale and surjective for large $i>j$. For an object $U=\varprojlim U_{i}$, let
$\left\lvert U\right\rvert=\varprojlim\left\lvert U_{i}\right\rvert$ be its
underlying topological space. A covering in $X_{\mathrm{pro\acute{e}t}}$ is a
family of pro-étale morphisms $f_{j}\colon U_{j}\to U$ such that the
underlying topological space $\left\lvert U\right\rvert$ is covered by the
$f_{j}(\left\lvert U_{j}\right\rvert)$.

There is a natural morphism of sites $\nu\colon X_{\mathrm{pro\acute{e}t}}\to
X_{\mathrm{\acute{e}t}}$ from the pro-étale site to the étale site. The
(uncompleted) integral structure sheaf $\mathcal{O}^{+}$ on
$X_{\mathrm{pro\acute{e}t}}$ is
$\mathcal{O}^{+}=\nu^{*}\mathcal{O}_{X_{\mathrm{\acute{e}t}}}$, which is to
say that if $U=\varprojlim U_{i}$ is an object in
$X_{\mathrm{pro\acute{e}t}}$, then

$\mathcal{O}^{+}(U)=\varinjlim\mathcal{O}^{+}(U_{i}).$

The (completed) integral structure sheaf $\hat{\mathcal{O}}^{+}$ on
$X_{\mathrm{pro\acute{e}t}}$ is defined as the $p$-adic completion:

$\hat{\mathcal{O}}^{+}=\varprojlim\mathcal{O}^{+}/p^{n},$

and the structure sheaf is $\hat{\mathcal{O}}:=\hat{\mathcal{O}}^{+}[1/p]$.

The pro-étale cohomology
$H^{i}(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})$ of a rigid-analytic
space $X$ will be of special interest to us. To investigate it, we will make
crucial use of perfectoid spaces. Let us recall the relevant definitions from
[SW20], adapted to the case where all structures live in characteristic 0. A
topological $\mathbb{Q}_{p}$-algebra $R$ is perfectoid if the following
conditions hold:

  1. (1)

$R$ is uniform, meaning its subring $R^{\circ}$ of power-bounded elements is
bounded,

  2. (2)

$R^{\circ}$ is $p$-adically complete,

  3. (3)

There exists an element $\varpi\in R^{\circ}$ such that $\varpi^{p}|p$ holds
in $R^{\circ}$, and such that the $p$th power Frobenius map

$R^{\circ}/\varpi\to R^{\circ}/\varpi^{p}$

is an isomorphism.

In particular there exists for all $n\geq 1$ an element $\varpi_{n}$ whose
$p^{n}$th power is the product of $\varpi$ by a unit in $R^{\circ}$. A Huber
pair $(R,R^{+})$ is a perfectoid affinoid algebra over
$\operatorname{Spa}(\mathbb{Q}_{p},\mathbb{Z}_{p})$ if $R$ is perfectoid. An
adic space over $\operatorname{Spa}(\mathbb{Q}_{p},\mathbb{Z}_{p})$ is
perfectoid if it admits a cover by adic spectra of perfectoid affinoid
algebras.

Suppose that $X$ is a rigid-analytic space over a field containing
$\mathbb{Q}_{p}$. We say that an object $U\in X_{\mathrm{pro\acute{e}t}}$ is
_affinoid perfectoid_ if $U=\varprojlim U_{i}$, where
$U_{i}=\operatorname{Spa}(R_{i},R_{i}^{+})$ is affinoid, and if $R^{+}$ is the
$p$-adic completion of $\varinjlim R_{i}^{+}$, and $R=R^{+}[1/p]$, then $R$ is
a perfectoid $K$-algebra. More generally, an object $U\in
X_{\mathrm{pro\acute{e}t}}$ is perfectoid if it admits an open cover by
perfectoid affinoid subobjects.

Perfectoid objects are useful to the calculation of
$H^{i}(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})$ for the following
two reasons:

  1. (1)

The affinoid perfectoid objects $U\in X_{\mathrm{pro\acute{e}t}}$ form a basis
for the topology [Sch13a, Proposition 4.8].

  2. (2)

Suppose $U\in X_{\mathrm{pro\acute{e}t}}$ is affinoid perfectoid. Then for all
$i\geq 1$, $H^{i}(U_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})$ is almost
zero, in the sense that it is annihilated by $\varpi_{n}$ for all $n$ [Sch13a,
Lemma 4.10]. In particular
$H^{i}(U_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}})=0$ for $i\geq 1$.

The facts above suggest a strategy for computing
$H^{i}(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})$: By (1), there
exists a pro-étale cover $f_{i}\colon U_{i}\to X$ where each $U_{i}$ is
perfectoid affinoid, and by (2),
$R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})$ is “almost”
computed by the Čech complex
$\mathcal{O}^{+}(X)\to\prod_{i}\mathcal{O}^{+}(U_{i})\to\prod_{i,j}\mathcal{O}^{+}(U_{i}\times_{X}U_{j})\to\cdots$.
The precise consequences for the pro-étale cohomology of rigid-analytic spaces
will be reviewed in Section 5.

It will be important to consider the pro-étale cohomology
$R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})$ of a rigid-
analytic space as a complex of condensed abelian groups. Once again, we
proceed by defining a condensed enhancement
$\hat{\mathcal{O}}^{+}_{\operatorname{cond}}$ of $\hat{\mathcal{O}}^{+}$. The
integral structure sheaf $\mathcal{O}^{+}$ on an adic space is already a sheaf
of topological rings; therefore so are
$\mathcal{O}^{+}_{X_{\mathrm{\acute{e}t}}}$, the uncompleted integral
structure sheaf
$\mathcal{O}^{+}=\nu^{*}\mathcal{O}^{+}_{X_{\mathrm{\acute{e}t}}}$, and the
completed structure sheaf
$\hat{\mathcal{O}}^{+}=\varprojlim\mathcal{O}^{+}/p^{n}$. Let
$\hat{\mathcal{O}}^{+}_{\operatorname{cond}}$ be the condensed enhancement of
$\hat{\mathcal{O}}^{+}$. Then
$R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})$
is a condensed enhancement of
$R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})$.

There is another natural way to define a condensed enhancement of
$R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})$. Observe that we
have a morphism of sites:

$\lambda\colon X_{\mathrm{pro\acute{e}t}}\to\ast_{\mathrm{pro\acute{e}t}}$

Indeed, if $S=\varprojlim S_{i}$ is a profinite set, then $X\times
S=\varprojlim X\times S_{i}$ is an object of $X_{\mathrm{pro\acute{e}t}}$.
Consequently, if $\mathcal{F}$ is a sheaf of abelian groups on
$X_{\mathrm{pro\acute{e}t}}$, we may define
$\Gamma_{\operatorname{cond}}(X_{\mathrm{pro\acute{e}t}},\mathcal{F})=\lambda_{*}\mathcal{F}$,
an object of $\operatorname{Cond}(\operatorname{Ab})$, and
$R\Gamma_{\operatorname{cond}}(X_{\mathrm{pro\acute{e}t}},\mathcal{F})=R\lambda_{*}\mathcal{F}$,
an object of $D(\operatorname{Cond}(\operatorname{Ab}))$. Applied to
$\mathcal{F}=\hat{\mathcal{O}}^{+}$, we find a complex
$R\Gamma_{\operatorname{cond}}(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})$.
In fact we have not found anything new:

######  Lemma 3.6.1.

We have an isomorphism in $D(\operatorname{Cond}(\operatorname{Ab}))$:

$R\Gamma_{\operatorname{cond}}(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})\cong
R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})$

###### Proof.

Since $X_{\mathrm{pro\acute{e}t}}$ admits a basis consisting of perfectoid
affinoids, this reduces to the claim that for all perfectoid affinoids
$U=\operatorname{Spa}(R,R^{+})\in X_{\mathrm{pro\acute{e}t}}$, we have a
natural isomorphism in $\operatorname{Cond}(\operatorname{Ab})$:

$\Gamma_{\operatorname{cond}}(U,\hat{\mathcal{O}}^{+})\cong\hat{\mathcal{O}}^{+}_{\operatorname{cond}}(U)$

This in turn reduces to the following calculation, for any profinite set
$S=\varprojlim S_{i}$:

$\displaystyle H^{0}(U\times S,\hat{\mathcal{O}}^{+})$ $\displaystyle\cong$
$\displaystyle\left(\varinjlim H^{0}(U\times
S_{i},\hat{\mathcal{O}}^{+})\right)^{\wedge}_{(p)}$ $\displaystyle\cong$
$\displaystyle\left(\varinjlim
C_{\operatorname{cts}}(S_{i},R^{+})\right)^{\wedge}_{(p)}$
$\displaystyle\cong$ $\displaystyle C_{\operatorname{cts}}(S,R^{+})$
$\displaystyle\cong$
$\displaystyle\hat{\mathcal{O}}^{+}_{\operatorname{cond}}(U)(S)$

In the penultimate step we used the fact that $R^{+}$ is $p$-adically
complete. This follows from the fact that $R^{\circ}$ is $p$-adically complete
(by definition), together with the fact that $R^{\circ}/R^{+}$ is $p$-torsion.
For the claim that $R^{\circ}/R^{+}$ is $p$-torsion: For $f\in R^{\circ}$ the
sequence $(pf)^{N}$ converges to 0 as $N\to\infty$. Since $R^{+}$ is open, we
have $(pf)^{N}\in R^{+}$ for sufficiently large $N$. Since $R^{+}$ is
integrally closed in $R$ we have $pf\in R^{+}$. (In fact $R^{\circ}/R^{+}$ is
almost zero.) ∎

In the context of the proof of Lemma 3.6.1 we have

$C_{\operatorname{cts}}(S,R^{+})\cong\underline{\operatorname{Hom}}(\mathbb{Z}[S],\underline{R^{+}}).$

Applying this over a covering of $X$ by perfectoid affinoids
$U=\operatorname{Spa}(R,R^{+})$ in $X_{\mathrm{pro\acute{e}t}}$, we obtain an
isomorphism in $D(\operatorname{Cond}(\operatorname{Ab}))$:

$R\Gamma((X\times
S)_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})\cong\underline{\operatorname{RHom}}(\mathbb{Z}[S],R\Gamma(X,\hat{\mathcal{O}}^{+}_{\operatorname{cond}})).$
(3.6.1)

Let $G$ be a profinite group. A pro-étale $G$-torsor over $X$ is an object
$Y\to X$ admitting an action $G\times Y\to Y$ lying over the trivial action of
$X$, such that the map $G\times Y\to Y\times_{X}Y$ given by
$(g,y)\mapsto(y,gy)$ is an isomorphism.

######  Proposition 3.6.2.

Let $X$ be a rigid-analytic space, and let $Y\to X$ be a perfectoid pro-étale
$G$-torsor. There is an isomorphism in $D(\operatorname{Solid})$:

$R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})\cong
R\Gamma(Y_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})^{hG}.$

###### Proof.

Since $Y\to X$ is pro-étale, the pro-étale cohomology of $X$ can be computed
by means of the simplicial cover:

$\cdots\mathrel{\begin{subarray}{c}\textstyle\rightarrow\\\\[-2.58334pt]
\textstyle\rightarrow\\\\[-2.58334pt]
\textstyle\rightarrow\end{subarray}}Y\times_{X}Y\rightrightarrows Y$

Namely,
$R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})$
is quasi-isomorphic to the totalization of the corresponding simplicial object

$R\Gamma(Y,\hat{\mathcal{O}}^{+}_{\operatorname{cond}})\to
R\Gamma(Y\times_{X}Y,\hat{\mathcal{O}}^{+}_{\operatorname{cond}})\rightrightarrows\cdots.$

Since $Y\to X$ is a $G$-torsor, the $n$th term in the complex is quasi-
isomorphic to

$R\Gamma((G^{n}\times
Y)_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})\cong
R\underline{\operatorname{Hom}}(\mathbb{Z}[G^{n}],R\Gamma(Y_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}}))$

by (3.6.1). Lemma 3.3.2(1) identifies the latter as the $n$th term in a
cosimplicial resolution whose totalization computes
$R\Gamma(Y_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})^{hG}$.
∎

######  Example 3.6.3.

Let $K$ be a nonarchimedean field of characteristic $(0,p)$. Let
$\overline{K}$ be an algebraic closure, and let $C$ be the metric completion
of $\overline{K}$. Then
$\operatorname{Spa}(C,\mathcal{O}_{C})\to\operatorname{Spa}(K,\mathcal{O}_{K})$
is a pro-étale torsor for the group $\operatorname{Gal}(\overline{K}/K)$.
Furthermore, since $C$ is algebraically closed, every pro-étale cover of
$\operatorname{Spa}(C,\mathcal{O}_{C})$ is split, meaning that
$H^{i}(\operatorname{Spa}(C,\mathcal{O}_{C})_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})=0$
for $i>0$. Therefore by Proposition 3.6.2 we have an isomorphism in
$D(\operatorname{Solid})$:

$R\Gamma(\operatorname{Spa}(K,\mathcal{O}_{K})_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})\cong\mathcal{O}_{C}^{h\operatorname{Gal}(\overline{K}/K)}.$

###  3.7.
$R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})$
for affinoid $X$

The pro-étale cohomology
$R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})$ of a rigid-
analytic space $X$ is a complex of abelian groups. Whereas in applications
like our Theorem 3.9.2, it will be necessary to control
$R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})$
as a complex of condensed abelian groups. The main theorem of this subsection
is that, when $X$ is affinoid,
$R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})$
is $p$-adic in the sense of 3.4.7, and indeed is the $p$-adic completion of
$R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})$ in the sense of
Lemma 3.4.8. This will allow us to prove condensed enhancements of comparison
theorems relating to
$R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})$.

Recall the $p$-adic completion functor $A\mapsto(\Gamma^{*}A)_{p}^{\wedge}$
from $D(\operatorname{Ab})$ to $D(\operatorname{Solid})$, whose image consists
of the $p$-adic solid complexes.

######  Lemma 3.7.1.

Let $X$ be an affinoid rigid-analytic space over a nonarchimedean field of
characteristic $(0,p)$. There is a natural isomorphism in
$D(\operatorname{Solid})$:

$R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})\cong(\Gamma^{*}R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}))^{\wedge}_{p}$

To prove the lemma, we need the notion of a strictly totally disconnected
perfectoid space $Y$ [Sch22, Definition 1.14]: this means that $Y$ is quasi-
compact, and every étale cover of $Y$ has a section.

######  Lemma 3.7.2.

Let $Y$ be a strictly totally disconnected perfectoid space. Then
$H^{i}(Y_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})=0$ for all $i>0$.

###### Proof.

First we claim that the global sections functor
$\mathcal{F}\mapsto\mathcal{F}(Y)$ on sheaves of abelian groups on
$Y_{\mathrm{\acute{e}t}}$ is exact. Indeed, if
$f\colon\mathcal{F}\to\mathcal{G}$ is surjective, and $s\in\mathcal{G}(Y)$,
there exists an étale cover $c\colon Y^{\prime}\to Y$ and
$t\in\mathcal{F}(Y^{\prime})$ such that $f(t)=c^{*}(s)$. Since $Y$ is strictly
totally disconnected, there exists a section $\sigma$ to $c$, i.e.,
$c\sigma=\operatorname{id}_{Y}$. Then
$s=\operatorname{id}_{Y}^{*}(s)=\sigma^{*}c^{*}(s)=\sigma^{*}f(t)=f\sigma^{*}(t)$
lies in the image of $\mathcal{F}(Y)$ under $f$. As a result,
$H^{i}(Y_{\mathrm{\acute{e}t}},\mathcal{F})=0$ for all $i>0$.

Let $\nu\colon Y_{\mathrm{pro\acute{e}t}}\to Y_{\mathrm{\acute{e}t}}$ be the
projection. For a complex $A$ of sheaves of abelian groups on
$Y_{\mathrm{\acute{e}t}}$, the unit of the adjunction $A\mapsto
R\nu_{*}\nu^{*}A$ is a quasi-isomorphism. A proof of this fact is found in
[BS15, Corollary 5.1.6] (in the context of schemes, but the formalism is the
same).

We apply the above observations to the sheaf
$\mathcal{F}=\mathcal{O}^{+}/p^{n}$ on $Y_{\mathrm{\acute{e}t}}$, for each
$n\geq 1$. The pullback of this sheaf to $Y_{\mathrm{pro\acute{e}t}}$,
evaluated on a perfectoid affinoid $U=\varprojlim U_{i}$, is

$\nu^{*}\mathcal{O}^{+}/p^{n}(U)=\varinjlim(\mathcal{O}^{+}/p^{n})(U_{i})=(\hat{\mathcal{O}}^{+}/p^{n})(U),$

which is to say that
$\nu^{*}\mathcal{O}^{+}/p^{n}\cong\hat{\mathcal{O}}^{+}/p^{n}$ as sheaves on
$Y_{\mathrm{pro\acute{e}t}}$. Therefore

$R\Gamma(Y_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}/p^{n})\cong
R\Gamma(Y_{\mathrm{\acute{e}t}},R\nu_{*}\nu^{*}\mathcal{O}^{+}/p^{n})\cong
R\Gamma(Y_{\mathrm{\acute{e}t}},\mathcal{O}^{+}/p^{n})\cong(\mathcal{O}^{+}/p^{n})(Y)$

has vanishing cohomology in degree $>0$. Taking the limit in $n$ shows that
$R\Gamma(Y_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})\cong\mathcal{O}^{+}(Y)$
has no higher cohomology either. ∎

######  Lemma 3.7.3.

Let $X$ be an affinoid rigid-analytic space over a nonarchimedean field of
characteristic $(0,p)$. There exists a pro-étale cover of $X$ by an affinoid
perfectoid space $\tilde{X}$ which is strictly totally disconnected.

###### Proof.

There exists a pro-étale cover $X^{\prime}\to X$ which is affinoid perfectoid
by [Sch13a, Proposition 4.8], and then there exists a pro-étale cover
$\tilde{X}\to X^{\prime}$ with $\tilde{X}$ strictly totally disconnected by
[Sch22, Lemma 7.18]. ∎

######  Lemma 3.7.4.

Let $Y$ be a strictly totally disconnected perfectoid space, and let
$Y^{\prime}\to Y$ be a pro-étale cover, with $Y^{\prime}$ perfectoid affinoid.
Then $Y^{\prime}$ is also strictly totally disconnected.

###### Proof.

Since $Y^{\prime}$ is affinoid, $Y^{\prime}\to Y$ is quasicompact and
separated, in which case [Sch22, Lemma 7.19] applies to conclude that
$Y^{\prime}$ is strictly totally disconnected. ∎

We can now give the proof of Lemma 3.7.1. By Lemma 3.7.3, there exists a pro-
étale cover $\tilde{X}\to X$ with $\tilde{X}$ strictly totally disconnected.
For $i\geq 1$, let $\tilde{X}^{(i)}$ be the $i$-fold fiber product of
$\tilde{X}$ over $X$. Any of the projections $\tilde{X}^{(i)}\to\tilde{X}$,
being a composite of base changes of pro-étale morphisms, is itself pro-étale.
By Lemma 3.7.4, each $\tilde{X}^{(i)}$ is strictly totally disconnected and
hence affinoid. By Lemma 3.7.2,
$H^{j}(\tilde{X}^{(i)}_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}_{\operatorname{cond}}^{+})=0$
for all $j>0$. Therefore
$R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})$
is computed by the Čech complex associated to the simplicial complex
associated to the pro-étale cover $\tilde{X}\to X$. Explicitly,
$R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})$
is quasi-isomorphic to the complex of condensed abelian groups with terms
$\hat{\mathcal{O}}^{+}_{\operatorname{cond}}(\tilde{X}^{(i)})$. By our
definition of $\hat{\mathcal{O}}^{+}_{\operatorname{cond}}$:

$\hat{\mathcal{O}}^{+}_{\operatorname{cond}}(\tilde{X}^{(i)})=\underline{\hat{\mathcal{O}}^{+}(\tilde{X}^{(i)})}\cong\hat{\mathcal{O}}^{+}(\tilde{X}^{(i)})^{\wedge}_{p},$

where the isomorphism holds because $\hat{\mathcal{O}}^{+}(\tilde{X}^{(i)})$
carries the $p$-adic topology (which is in turn true because $\tilde{X}^{(i)}$
is affinoid perfectoid). Since the complex with terms
$\hat{\mathcal{O}}^{+}(\tilde{X}^{(i)})$ computes
$R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})$ (again by Lemma
3.7.2), we find that the $p$-adic completion of
$R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})$ is quasi-
isomorphic to
$R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})$.

###  3.8. Continuous cohomology of $p$-adic Lie groups

As B is a statement about the continuous cohomology of the Morava stabilizer
group $\mathbb{G}_{n}\cong\mathcal{O}_{D}^{\times}\rtimes\hat{\mathbb{Z}}$, it
will be useful to collect some basic results on the continuous cohomology of
$p$-adic Lie groups such as $\mathcal{O}_{D}^{\times}$.

The first systematic study of the continuous cohomology of $p$-adic Lie groups
was undertaken by Lazard [Laz65]. Lazard’s results include comparision
theorems such as [Laz65, Théorème V.2.4.10], which we briefly summarize. If
$G$ is a $\mathbb{Q}_{p}$-analytic group admitting a $p$-valuation, Lazard
defines its Lie algebra $\operatorname{Lie}G$ over $\mathbb{Q}_{p}$. Then if
$V$ is a finite-dimensional $\mathbb{Q}_{p}$-vector space admitting a
continuous action of $G$, then $\operatorname{Lie}G$ acts on $V$, and we have
an isomorphism $H^{*}_{\operatorname{cts}}(G,V)\cong
H^{*}(\operatorname{Lie}G,V)$. (If $\mathbf{G}$ is an algebraic group over
$\mathbb{Q}_{p}$, then any sufficiently small subgroup of
$\mathbf{G}(\mathbb{Q}_{p})$ satisfies Lazard’s hypotheses, with
$\operatorname{Lie}G$ being the usual Lie algebra of $\mathbf{G}$.)

The following is a well-known consequence of Lazard’s theorem, see for example
[Mor85, Remark 2.2.5].

######  Lemma 3.8.1.

Let $G$ be either of the groups $\operatorname{GL}_{n}(\mathbb{Z}_{p})$ or
$\mathcal{O}_{D}^{\times}$. Consider the trivial action of $G$ on
$\mathbb{Q}_{p}$. There is an isomorphism of graded $\mathbb{Q}_{p}$-algebras:

$H^{*}_{\operatorname{cts}}(G,\mathbb{Q}_{p})\cong\Lambda_{\mathbb{Q}_{p}}(x_{1},x_{3},\dots,x_{2n-1}).$

In the case of $G=\mathcal{O}_{D}^{\times}$, the outer automorphism
$\operatorname{ad}\Pi$ (where $\Pi$ is a uniformizer of $D^{\times}$) acts as
the identity on $H^{*}_{\operatorname{cts}}(G,\mathbb{Q}_{p})$.

###### Proof.

By Lazard’s result, $H^{*}_{\operatorname{cts}}(G,\mathbb{Q}_{p})\cong
H^{*}(\operatorname{Lie}G,\mathbb{Q}_{p})$, so we are reduced to calculating
Lie algebra cohomology. In the case of $G=\mathcal{O}_{D}^{\times}$, the outer
automorphism $\operatorname{ad}\Pi$ on $G$ corresponds to an inner
automorphism of $\operatorname{Lie}G$, which acts trivially on
$H^{*}(\operatorname{Lie}G,\mathbb{Q}_{p})$.

The Lie algebra cohomology of a reductive Lie algebra $\mathfrak{g}$ over a
field $k$ of characteristic zero is well-studied [CE48], [Kos50]; we give a
brief exposition. If $k=\mathbb{R}$ and $\mathfrak{g}$ is the Lie algebra of a
compact Lie group $G$, then $H^{*}(\mathfrak{g},\mathbb{R})$ is isomorphic to
the de Rham cohomology ring $H^{*}_{\mathrm{dR}}(G)$. Generally, this is a
graded-commutative $\mathbb{R}$-algebra whose primitive elements live in odd
degree. In the special case $G=U(n)$, the sequence of fibrations $U(n-1)\to
U(n)\to S^{2n-1}$ allows one to identify the rational cohomology
$H^{*}_{\mathrm{dR}}(U(n))$ with the rational cohomology of a product of
spheres $S^{1}\times S^{3}\times\cdots\times S^{2n-1}$. Thereby we obtain an
isomorphism

$H^{*}(\mathfrak{g}\mathfrak{l}_{n}(\mathbb{Q}),\mathbb{Q})\cong\Lambda_{\mathbb{\mathbb{Q}}}(x_{1},x_{3},\dots,x_{2n-1}),$
(3.8.1)

since the two sides become isomorphic after tensoring with $\mathbb{C}$. (See
[Kos50] for an explicit description, due to Dynkin, of the elements
$x_{1},x_{3},\dots,x_{2n-1}$ in terms of cocycles.) The isomorphism (3.8.1)
implies the lemma for $G=\operatorname{GL}_{n}(\mathbb{Z}_{p})$, since
$\operatorname{Lie}G=\mathfrak{g}\mathfrak{l}_{n}(\mathbb{Q}_{p})$.

Now suppose $G=\mathcal{O}_{D}^{\times}$. Then $\operatorname{Lie}G=D$ is a
twist of $\mathfrak{g}\mathfrak{l}_{n}(\mathbb{Q}_{p})$ in the sense that
there is an isomorphism:

$\operatorname{Lie}G\otimes\overline{\mathbb{Q}}_{p}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathfrak{g}\mathfrak{l}_{n}(\overline{\mathbb{Q}}_{p}).$
(3.8.2)

This implies that $H^{*}(\operatorname{Lie}G,\mathbb{Q}_{p})$ is an exterior
algebra as claimed, since it is one after tensoring with
$\overline{\mathbb{Q}}_{p}$. ∎

######  Lemma 3.8.2.

Let $W=W(\overline{\mathbb{F}}_{p})$ and $K=W[1/p]$.

  1. (1)

The continuous cohomology
$H^{i}_{\operatorname{cts}}(\operatorname{Gal}(\overline{\mathbb{F}}_{p}/\mathbb{F}_{p}),W)$
is $\mathbb{Z}_{p}$ if $i=0$, and is $0$ otherwise.

  2. (2)

Let $\mathbb{G}_{n}$ act on $K$ through its quotient
$\operatorname{Gal}(\overline{\mathbb{F}}_{p}/\mathbb{F}_{p})$. There is an
isomorphism of graded $\mathbb{Q}_{p}$-algebras:

$H^{*}_{\operatorname{cts}}(\mathbb{G}_{n},K)\cong\Lambda_{\mathbb{Q}_{p}}(x_{1},x_{3},\dots,x_{2n-1})$

###### Proof.

(1) Since
$\operatorname{Gal}(\overline{\mathbb{F}}_{p}/\mathbb{F}_{p})\cong\hat{\mathbb{Z}}$,
it is enough to show vanishing of cohomology in degree 1. Since $W$ is
$p$-adically complete, this is further reduced to showing that
$H^{1}_{\operatorname{cts}}(\operatorname{Gal}(\overline{\mathbb{F}}_{p}/\mathbb{F}_{p}),\overline{\mathbb{F}}_{p})=0$;
this is true because $x\mapsto x^{p}-x$ is surjective on
$\overline{\mathbb{F}}_{p}$.

(2) Consider the Hochschild–Serre spectral sequence:

$H^{i}_{\operatorname{cts}}(\operatorname{Gal}(\overline{\mathbb{F}}_{p}/\mathbb{F}_{p}),H^{j}_{\operatorname{cts}}(\mathcal{O}_{D}^{\times},K))\implies
H^{i+j}_{\operatorname{cts}}(\mathbb{G}_{n},K).$

Consider the action of
$\operatorname{Gal}(\overline{\mathbb{F}}_{p}/\mathbb{F}_{p})$ on
$H^{j}_{\operatorname{cts}}(\mathcal{O}_{D}^{\times},K)=H^{j}_{\operatorname{cts}}(\mathcal{O}_{D}^{\times},\mathbb{Q}_{p})\otimes_{\mathbb{Z}_{p}}W$.
The action on the first factor is trivial by Lemma 3.8.1, and the action on
the second factor has no higher cohomology by (1). Therefore:

$H^{*}_{\operatorname{cts}}(\mathbb{G}_{n},K)\cong
H^{*}_{\operatorname{cts}}(\mathcal{O}_{D}^{\times},\mathbb{Q}_{p}),$

at which point we apply Lemma 3.8.1. ∎

###  3.9. Overview of the proof of B

Let $p$ be a prime number, and let $\Gamma$ be a formal group of dimension 1
and height $n$ over $\overline{\mathbb{F}}_{p}$. Let $\operatorname{LT}$ be
the functor of deformations of $\Gamma$. Then $\operatorname{LT}$ is
representable by a formal scheme $\operatorname{Spf}A$ whose coordinate ring
is isomorphic to a power series ring over $W=W(\overline{\mathbb{F}}_{p})$:

$A\cong W\llbracket u_{1},\dots,u_{n-1}\rrbracket.$

Let $\mathcal{O}_{D}=\operatorname{End}\Gamma$; then $\mathcal{O}_{D}$ is the
ring of integers in a division algebra $D$ over $\mathbb{Q}_{p}$ of invariant
$1/n$. Explicitly, $\mathcal{O}_{D}$ is generated over $W(\mathbb{F}_{p^{n}})$
by an element $\Pi$ satisfying $\Pi^{n}=p$ and $\Pi\alpha=\sigma(\alpha)\Pi$,
where $\sigma\in\operatorname{Aut}W(\mathbb{F}_{p^{n}})$ is the Frobenius
automorphism. The Morava stabilizer group $\mathbb{G}_{n}$ from Section 2 is
the profinite completion of $D^{\times}$; it fits into an exact sequence:

$1\to\mathcal{O}_{D}^{\times}\to\mathbb{G}_{n}\to\operatorname{Gal}(\overline{\mathbb{F}}_{p}/\mathbb{F}_{p})\to
1.$

There is a continuous action of $\mathbb{G}_{n}$ on $A$.

Our main theorem (B) concerns the continuous cohomology ring
$H^{\ast}_{\operatorname{cts}}(\mathbb{G}_{n},A)$. The action of
$\mathbb{G}_{n}$ on $A$ is rather inexplicit, and it seems difficult to
compute $H^{\ast}_{\operatorname{cts}}(\mathbb{G}_{n},A)$ directly in terms of
cocycles. To address this problem, we pass from the formal scheme
$\operatorname{LT}$ to its rigid-analytic generic fiber
$\operatorname{LT}_{K}$, which is isomorphic to the open unit ball over $K$.
The use of $\operatorname{LT}_{K}$ has a precedent in chromatic homotopy
theory, namely in applications of the Gross–Hopkins period morphism [HG94]:

$\pi\colon\operatorname{LT}_{K}\to\mathbb{P}(M(\Gamma)).$

Here $M(\Gamma)\cong K^{n}$ is the rational Dieudonné module of $\Gamma$, and
$\mathbb{P}(M(\Gamma))\cong\mathbb{P}^{n-1}_{K}$ is the corresponding
projective space, considered as a rigid-analytic space over $K$. The morphism
$\pi$ is étale, surjective, equivariant for the action of
$\mathcal{O}_{D}^{\times}$, and fairly explicit in terms of the variables
$u_{1},\dots,u_{n-1}$. Using the period morphism $\pi$, it is possible in
principle to give formulas for the action of $\mathcal{O}_{D}^{\times}$ on
$A$, see [DH95].

For our work we will not use the period morphism but rather a different
structure possessed by $\operatorname{LT}_{K}$: the isomorphism between the
two towers. This was discovered by Faltings [Fal02b], see also [FGL08] for
more details. We review the form given in [SW13, Theorem D], which treats this
phenomenon as an isomorphism between perfectoid spaces.

Let $\mathcal{H}$ be Drinfeld’s symmetric space (also called Drinfeld’s upper
half-space) in dimension $n-1$. This is a rigid-analytic space over
$\mathbb{Q}_{p}$, defined as

$\mathcal{H}=\mathbb{P}^{n-1}_{\mathbb{Q}_{p}}\backslash\bigcup_{H}H$

where $\mathbb{P}^{n-1}_{\mathbb{Q}_{p}}$ is rigid-analytic projective space,
and $H$ runs over all $\mathbb{Q}_{p}$-rational hyperplanes in
$\mathbb{P}^{n-1}_{\mathbb{Q}_{p}}$. Then $\mathcal{H}$ admits an action of
the group $\operatorname{GL}_{n}(\mathbb{Q}_{p})$ and in particular of its
subgroup $\operatorname{GL}_{n}(\mathbb{Z}_{p})$.

######  Theorem 3.9.1.

There exists a perfectoid space $\mathcal{X}$ and a diagram of adic spaces:

$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{GL}_{n}(\mathbb{Z}_{p})}$$\scriptstyle{\mathbb{G}_{n}}$$\textstyle{\operatorname{LT}_{K}}$$\textstyle{\mathcal{H}}$

Here, $\mathcal{X}$ admits commuting actions of $\mathbb{G}_{n}$ and
$\operatorname{GL}_{n}(\mathbb{Z}_{p})$. The morphism to
$\operatorname{LT}_{K}$ is a pro-étale
$\operatorname{GL}_{n}(\mathbb{Z}_{p})$-torsor, which is equivariant for the
action of $\mathbb{G}_{n}$. The morphism to $\mathcal{H}$ is a pro-étale
$\mathbb{G}_{n}$-torsor, which is equivariant for the action of
$\operatorname{GL}_{n}(\mathbb{Z}_{p})$.

###### Proof.

This is an application of a general duality statement [SW13, Theorem E]
between Rapoport-Zink spaces at infinite level. Both $\operatorname{LT}_{K}$
and the base change $\mathcal{H}_{K}$ arise as the generic fiber of a
deformation problem of formal groups. We have already seen that Lubin–Tate
space $\operatorname{LT}$ parametrizes deformations of a formal group $\Gamma$
over $\overline{\mathbb{F}}_{p}$ of dimension 1 and height $n$. Whereas,
$\mathcal{H}_{K}$ is the generic fiber of a formal scheme $\mathfrak{H}_{W}$
over $\operatorname{Spf}W$ which parametrizes deformations of a special formal
$\mathcal{O}_{D}$-module $X$ in the sense of [Dri76]. The relation between $X$
and $\Gamma$ is:

$X=\Gamma\oplus\Gamma^{(p)}\oplus\cdots\oplus\Gamma^{(p^{n-1})},$

where $\Gamma^{(p^{k})}$ is the pullback of $\Gamma$ under the $p^{k}$th power
Frobenius automorphism of $\overline{\mathbb{F}}_{p}$.

Trivialization of the torsion in the universal deformation of $\Gamma$ (resp.,
$X$) produces a pro-étale torsor over $\operatorname{LT}_{K}$ (resp.,
$\mathcal{H}_{K}$) with group $\operatorname{GL}_{n}(\mathbb{Z}_{p})$ (resp.,
$\mathcal{O}_{D}^{\times}$), known as the Lubin–Tate tower (resp., the
Drinfeld tower). Applied to this situation, [SW13, Theorem E] is the statement
that the two towers are isomorphic in the limit to the same perfectoid space
$\mathcal{X}$.

Note that $\mathcal{H}_{K}\to\mathcal{H}$, being a pullback of
$\operatorname{Spa}K\to\operatorname{Spa}\mathbb{Q}_{p}$, is a pro-étale
$\operatorname{Gal}(\overline{\mathbb{F}}_{p}/\mathbb{F}_{p})$-torsor. It
remains to be seen why the composition
$\mathcal{X}\to\mathcal{H}_{K}\to\mathcal{H}$ is a pro-étale torsor for the
group $\mathbb{G}_{n}$. For this, we observe that there is an
$\mathcal{O}_{D}$-equivariant isomorphism
$\Gamma\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\Gamma^{(p^{n})}$ (for
instance, the $p^{n}$th power isogeny divided by $p$). From this one
constructs an isomorphism $i\colon
X\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}X^{(p)}$ which satisfies the
relation $i\circ\alpha=(\Pi\alpha\Pi^{-1})\circ i$ for all
$\alpha\in\mathcal{O}_{D}^{\times}$. The isomorphism $i$ induces an
automorphism of $\mathcal{X}$ lying over the Frobenius automorphism of
$\operatorname{Spa}K$ and satisfying the same relation with respect to
$\mathcal{O}_{D}^{\times}$. This automorphism is exactly the necessary
structure required to extend the action of $\mathcal{O}_{D}^{\times}$ on
$\mathcal{X}$ to an action of $\mathbb{G}_{n}$. ∎

Theorem 3.9.1 suggests a strategy for accessing the cohomology ring
$H^{*}_{\operatorname{cts}}(\mathbb{G}_{n},A)$. The diagram in Theorem 3.9.1
witnesses an isomorphism in $D(\operatorname{Solid})$:

$R\Gamma(\operatorname{LT}_{K,\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})^{h\mathbb{G}_{n}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}R\Gamma(\mathcal{H}_{K,\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})^{h\operatorname{GL}_{n}(\mathbb{Z}_{p})}$
(3.9.1)

Indeed by Proposition 3.6.2, both objects are isomorphic to

$R\Gamma(\mathcal{X}_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})^{h(\mathbb{G}_{n}\times\operatorname{GL}_{n}(\mathbb{Z}_{p}))}.$

The isomorphism in (3.9.1) is helpful because it translates the opaque action
of $\mathcal{O}_{D}^{\times}$ on $\operatorname{LT}_{K}$ into the transparent
action of $\operatorname{GL}_{n}(\mathbb{Z}_{p})$ on $\mathcal{H}$. To
completely leverage (3.9.1), we will have to say something about the
$\hat{\mathcal{O}}^{+}_{\operatorname{cond}}$-cohomology of
$X_{\mathrm{pro\acute{e}t}}$, where $X$ is $\operatorname{LT}_{K}$ or
$\mathcal{H}$, respectively. Let us write $X_{\mathrm{an}}$ for the analytic
topology, to distinguish it from the pro-étale topology. The following
comparison statements appear as Theorem 6.2.3 and Theorem 6.3.3.

######  Theorem 3.9.2.

The pro-étale cohomology of $\hat{\mathcal{O}}^{+}_{\operatorname{cond}}$ on
$\operatorname{LT}_{K}$ and $\mathcal{H}$ can be approximated as follows.

  1. (1)

There is a morphism of differential graded solid $W$-algebras, which is
equivariant for the action of $\mathbb{G}_{n}$:

$A[\varepsilon]\to
R\Gamma(\operatorname{LT}_{K,\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}}).$

  2. (2)

There is a morphism of differential graded solid $\mathbb{Z}_{p}$-algebras,
which is equivariant for the action of
$\operatorname{GL}_{n}(\mathbb{Z}_{p})$:

$\mathbb{Z}_{p}[\varepsilon]\to
R\Gamma(\mathcal{H}_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}}).$

Here $R[\varepsilon]$ is shorthand for the complex
$\underline{R}\stackrel{{\scriptstyle 0}}{{\to}}\underline{R}$ in degrees 0,1.
Let $\mathcal{C}$ be the cofiber of either of the above morphisms in
$D(\operatorname{Solid})$. Then $H^{i}(\mathcal{C})=0$ for $i\leq 0$, and
there exists a single power of $p$ which annihilates $H^{i}(\mathcal{C})$ for
every $i\geq 1$.

The proof of Theorem 3.9.2 requires the full force of the integral $p$-adic
Hodge theory theorems of [BMS18] and [vK19]. We apply these theorems in
Section 5. The effect is to reduce the study of
$R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})$
(where $X$ is either of $\operatorname{LT}_{K}$ or $\mathcal{H}$) to the case
of a point $X=\operatorname{Spa}(K,\mathcal{O}_{K})$. By Example 3.6.3, the
pro-étale cohomology of $\operatorname{Spa}(K,\mathcal{O}_{K})$ agrees with
the Galois cohomology $\mathcal{O}_{C}^{h\operatorname{Gal}(\overline{K}/K)}$,
where $C$ is the completion of an algebraic closure of $K$. This cohomology
was controlled by Tate [Tat67], in a way that is valid for any local field of
characteristic $(0,p)$. Expressed in our language, Tate’s result is that there
is a morphism of differential graded solid $\mathcal{O}_{K}$-algebras

$W[\varepsilon]\to\mathcal{O}_{C}^{h\operatorname{Gal}(\overline{K}/K)}$

whose cofiber has $p^{N}$-torsion cohomology groups, for some absolute
constant $N$; this will be explained in more detail in the next section.

Combining (3.9.1) with Theorem 3.9.2, we obtain a diagram in
$D(\operatorname{Solid})$:

$\displaystyle
A^{h\mathbb{G}_{n}}\otimes_{\mathbb{Z}_{p}}\mathbb{Z}_{p}[\varepsilon]$
$\displaystyle\cong$ $\displaystyle A[\varepsilon]^{h\mathbb{G}_{n}}$
$\displaystyle\to$ $\displaystyle
R\Gamma(\operatorname{LT}_{K,\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})^{h\mathbb{G}_{n}}$
$\displaystyle\cong$ $\displaystyle
R\Gamma(\mathcal{H}_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})^{h\operatorname{GL}_{n}(\mathbb{Z}_{p})}$
$\displaystyle\leftarrow$
$\displaystyle\mathbb{Z}_{p}[\varepsilon]^{h\operatorname{GL}_{n}(\mathbb{Z}_{p})}$
$\displaystyle\cong$
$\displaystyle\mathbb{Z}_{p}^{h\operatorname{GL}_{n}(\mathbb{Z}_{p})}\otimes_{\mathbb{Z}_{p}}\mathbb{Z}_{p}[\varepsilon]$

Here, each of the two arrows not labeled as an isomorphism has cofiber whose
cohomology groups are annihilated by some uniform power of $p$.

We will briefly indicate how this is used to prove B, leaving the details for
Section 6. By Proposition 2.5.1, we have a $\mathbb{G}_{n}$-equivariant
splitting $A=W\oplus A^{c}$. After inverting $p$ in the above diagram, we
arrive at an isomorphism in cohomology:

$\left(H^{*}_{\operatorname{cts}}(\mathbb{G}_{n},K)\oplus
H^{*}_{\operatorname{cts}}(\mathbb{G}_{n},A^{c})\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}\right)\otimes\mathbb{Q}_{p}[\varepsilon]\cong
H^{*}_{\operatorname{cts}}(\operatorname{GL}_{n}(\mathbb{Z}_{p}),\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}}\mathbb{Q}_{p}[\varepsilon]$
(3.9.2)

By Lemmas 3.8.1 and 3.8.2, $H^{*}_{\operatorname{cts}}(\mathbb{G}_{n},K)$ and
$H^{*}_{\operatorname{cts}}(\operatorname{GL}_{n}(\mathbb{Z}_{p}),\mathbb{Q}_{p})$
are isomorphic to the same exterior $\mathbb{Q}_{p}$-algebra. In particular
$\dim_{\mathbb{Q}_{p}}H^{i}_{\operatorname{cts}}(\mathbb{G}_{n},K)=\dim_{\mathbb{Q}_{p}}H^{i}_{\operatorname{cts}}(\operatorname{GL}_{n}(\mathbb{Z}_{p}),\mathbb{Q}_{p})$
for all $i$.

Comparing dimensions of the $\mathbb{Q}_{p}$-vector spaces in (3.9.2) shows
that
$H^{*}_{\operatorname{cts}}(\mathbb{G}_{n},A^{c})\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}=0$,
which is to say that $H^{*}_{\operatorname{cts}}(\mathbb{G}_{n},A^{c})$ is
torsion: this is the assertion of B.

##  4\. The Galois cohomology of $\mathcal{O}_{C}$

Let us fix some definitions. A nonarchimedean field is a field $K$ which is
complete with respect to the topology induced from a nontrivial nonarchimedean
valuation $\left\lvert\;\right\rvert\colon K\to\mathbb{R}_{\geq 0}$. (Some
authors do not require $K$ to be complete, but for our purposes it will be
useful to always assume this.) Let $\mathcal{O}_{K}$ be its valuation ring;
i.e., the subring of elements with $\left\lvert\alpha\right\rvert\leq 1$. Let
$\kappa$ be the residue field of $\mathcal{O}_{K}$. The characteristic of a
nonarchimedean field $K$ refers to the pair
$(\operatorname{char}K,\operatorname{char}\kappa)$.

A local field is a nonarchimedean field satisfying the additional conditions:
(a) the valuation on $K$ is discrete, in other words $\mathcal{O}_{K}$ is a
discrete valuation ring, and (b) the residue field $\kappa$ is perfect. (Some
authors require that the residue field of a local field be finite, which is
identical to $K$ being locally compact. However, we want to allow for fields
such as $W(\overline{\mathbb{F}}_{p})[1/p]$.)

Let $L$ be a (possibly infinite) Galois extension of a nonarchimedean field
$K$. The valuation on $K$ extends uniquely to a valuation on $L$, and the
completion $\hat{L}$ is a nonarchimedean field admitting a continuous action
of $\operatorname{Gal}(L/K)$. (If $L/K$ is finite then it is not necessary to
complete.) It is an interesting problem to compute the continuous cohomology
groups
$H^{i}_{\operatorname{cts}}(\operatorname{Gal}(L/K),\mathcal{O}_{\hat{L}})$,
or at least to approximate these as $\mathcal{O}_{K}$-modules. For simplicity
we will assume throughout that $\operatorname{char}K=0$.

A basic result along these lines is due to Ax, which settles the problem in
degree $i=0$.

######  Theorem 4.0.1 ([Ax70]).

Let $K$ be a nonarchimedean field with $\operatorname{char}K=0$. Let $L/K$ be
a Galois extension of nonarchimedean fields. Then the subfield of $\hat{L}$
fixed by $\operatorname{Gal}(L/K)$ is exactly $K$. Consequently
$H^{0}(\operatorname{Gal}(L/K),\mathcal{O}_{\hat{L}})=\mathcal{O}_{K}$.

Results on higher cohomology tend to require that $K$ be a local field. A
classical result [Frö83, Theorem 3] attributed to Noether states that if $L/K$
is a finite tamely ramified Galois extension of local fields, then
$\mathcal{O}_{L}$ is a free $\mathcal{O}_{K}[\operatorname{Gal}(L/K)]$-module,
and therefore $H^{i}(\operatorname{Gal}(L/K),\mathcal{O}_{L})=0$ for $i>0$.
For arbitrary finite extensions there is the following result of Sen:

######  Theorem 4.0.2 ([Sen69]).

Let $K$ be a local field of characteristic $(0,p)$. Let $L/K$ be a finite
Galois extension. Then $H^{1}(\operatorname{Gal}(L/K),\mathcal{O}_{L})$ is
$\alpha$-torsion for any $\alpha\in\mathcal{O}_{K}$ with