$\left\lvert\alpha\right\rvert\leq\left\lvert p\right\rvert^{1/(p-1)}$.

Sen’s theorem does not imply that
$H^{1}_{\operatorname{cts}}(\operatorname{Gal}(L/K),\mathcal{O}_{\hat{L}})$ is
torsion for $L/K$ infinite. Indeed this $H^{1}_{\operatorname{cts}}$ is not
torsion when $L=\overline{K}$, as the following classical result of Tate
shows.

######  Theorem 4.0.3 ([Tat67]).

Let $K$ be a local field of characteristic $(0,p)$, and let $C$ be the
completion of an algebraic closure $\overline{K}/K$. Let
$\Gamma_{K}=\operatorname{Gal}(\overline{K}/K)$. Then:

  1. (1)

$H^{i}_{\operatorname{cts}}(\Gamma_{K},C)\cong K$ for $i=0,1$,

  2. (2)

$H^{i}_{\operatorname{cts}}(\Gamma_{K},C)=0$ for $i\geq 2$.

For $j\in\mathbb{Z}$, let $C(j)$ be the $j$th Tate twist: this is $C$ as a
topological group, but the $\Gamma_{K}$-action has been twisted by the $j$th
power of the cyclotomic character. For $j\neq 0$ and all $i\geq 0$ we have
$H^{i}_{\operatorname{cts}}(\Gamma_{K},C(j))=0$.

Our contribution to this story is the following refinement of Tate’s theorem,
which controls torsion in the cohomology of $\mathcal{O}_{C}$. We call a local
field $K$ of characteristic $(0,p)$ tame if $p$ does not divide its absolute
ramification index $e_{K}$ (defined in the following subsection).

######  Theorem 4.0.4.

Let $K$ be a local field of characteristic $(0,p)$. Let $C$ be the completion
of an algebraic closure $\overline{K}/K$. Let
$\Gamma_{K}=\operatorname{Gal}(\overline{K}/K)$. Then:

  1. (1)

$H^{0}(\Gamma_{K},\mathcal{O}_{C})=\mathcal{O}_{K}.$

  2. (2)

There is an isomorphism
$H^{1}_{\operatorname{cts}}(\Gamma_{K},\mathcal{O}_{C})\cong\mathcal{O}_{K}\oplus
T$, where $T$ is $p^{4}$-torsion (resp., $p^{6}$-torsion) as $p$ is odd or
even, respectively. If $K$ is tame, or if it is a cyclotomic extension of a
tame field, this bound can be improved to $p^{3}$ (resp., $p^{5}$).

  3. (3)

For $i>1$, $H^{i}_{\operatorname{cts}}(\Gamma_{K},\mathcal{O}_{C})$ is
$p$-torsion (resp., $p^{2}$-torsion) as $p$ is odd or even, respectively.

Theorem 4.0.4 appears later in this section as Theorem 4.3.6.

We also obtain an explicit bound on the continuous cohomology of the
nontrivial Tate twists $\mathcal{O}_{C}(j)$.

######  Theorem 4.0.5.

With notation as in Theorem 4.0.4, let $j\neq 0$ be an integer, and let
$\mathcal{O}_{C}(j)$ be the $j$th Tate twist of $\mathcal{O}_{C}$ as a
$\operatorname{Gal}(\overline{K}/K)$-module. Then:

  1. (1)

$H^{0}(\Gamma_{K},\mathcal{O}_{C}(j))=0$.

  2. (2)

$H^{1}_{\operatorname{cts}}(\Gamma_{K},\mathcal{O}_{C}(j))$ is
$p^{M+v(j)}$-torsion. Here $v(j)$ is the $p$-adic valuation of $j$, and
$M=M_{K}$ is a constant which only depends on $K$ and which is insensitive to
passage to a tamely ramified extension of $K$. If $p\nmid e_{K}$ we may take
$M=2$ if $p$ is odd and $M=5$ if $p=2$.

  3. (3)

For $i\geq 2$, $H^{i}_{\operatorname{cts}}(\Gamma_{K},\mathcal{O}_{C}(j))$ is
$p$-torsion if $p$ is odd and $p^{3}$-torsion if $p=2$.

The theorem appears later as Theorem 4.4.3.

###  4.1. Ramification theory for extensions of local fields

All material in this subsection is taken from Serre’s book [Ser79], to which
we refer the reader for more details.

Suppose $K$ is a local field of characteristic $(0,p)$. We fix here some
notation regarding valuations. The ring of integers $\mathcal{O}_{K}$ is a
discrete valuation ring with maximal ideal $\pi_{K}\mathcal{O}_{K}$, where
$\pi_{K}$ is a uniformizing parameter for $K$. Any element $\alpha\in
K^{\times}$ can be written as $\alpha=\pi_{K}^{v_{K}(\alpha)}u$, where $u$ is
a unit in $\mathcal{O}_{K}$. The map

$v_{K}\colon K^{\times}\to\mathbb{Z}$

is then a surjective homomorphism not depending on the choice of $\pi$; this
is the relative valuation map. For any real number $0<a<1$, the absolute value
$\left\lvert\alpha\right\rvert=a^{v_{K}(\alpha)}$ defines the topology on $K$.
Note that $p\in K^{\times}$ (since $K$ has characteristic $0$) and
$v_{K}(p)>0$ (since the residue field $\mathcal{O}_{K}/\pi_{K}\mathcal{O}_{K}$
has characteristic $p$); relatedly, $K$ contains $\mathbb{Q}_{p}$ as a
subfield. The number $e_{K}=v_{K}(p)$ is the absolute ramification index of
$K$. We define the absolute valuation map

$v\colon K^{\times}\to\mathbb{Q}$

by $v(\alpha)=v_{K}(\alpha)/e_{K}$, so that $v(p)=1$; thus $v$ extends the
valuation $v_{\mathbb{Q}_{p}}$ on $\mathbb{Q}_{p}$.

If $L/K$ is a field extension of finite degree, let $\mathcal{O}_{L}$ be the
integral closure of $\mathcal{O}_{K}$ in $L$. Then $\mathcal{O}_{L}$ is also a
complete discrete valuation ring [Ser79, Chapter 2, §2, Proposition 3], so
that $L$ is a local field of characteristic $(0,p)$. The number
$e_{L/K}=v_{L}(\pi_{K})$ is the relative ramification index of $L/K$; the
extension $L/K$ is totally ramified if $e_{L/K}=[L:K]$, ramified if
$e_{L/K}>1$ and unramified if $e_{L/K}=1$. Note that for $\alpha\in
K^{\times}$ we have $v_{L}(\alpha)=e_{L/K}v_{K}(\alpha)$. Applied to
$\alpha=p$, this last relation gives $e_{L/K}=e_{L}/e_{K}$. Thus for
$\alpha\in K^{\times}$ we have $v_{K}(\alpha)/e_{K}=v_{L}(\alpha)/e_{L}$. As a
result, the absolute valuation map $v$ defined above extends unambiguously to
$L^{\times}$. Indeed, if $\overline{K}$ is an algebraic closure of $K$, we
continue to write $v\colon\overline{K}^{\times}\to\mathbb{Q}$ for the absolute
valuation map.

Besides the ramification index $e_{L/K}$, there are two other quantities which
measure ramification in the extension $L/K$: the discriminant
$\mathfrak{d}_{L/K}$ and the different $\mathcal{D}_{L/K}$. To define these,
we need the trace map $\operatorname{tr}_{L/K}\colon L\to K$. The discriminant
$\mathfrak{d}_{L/K}$ is the ideal of $\mathcal{O}_{K}$ generated by the
determinant of the matrix $(\operatorname{tr}_{L/K}(e_{i}e_{j}))$, where
$e_{i}$ runs through a basis for $\mathcal{O}_{L}$ as an
$\mathcal{O}_{K}$-module. The different $\mathcal{D}_{L/K}$ is the ideal of
$\mathcal{O}_{L}$ whose inverse is the dual of $\mathcal{O}_{L}$ under the
trace pairing $L\times L\to K$ defined by
$(x,y)\mapsto\operatorname{tr}_{L/K}(xy)$. The discriminant and different are
related by $N_{L/K}(\mathcal{D}_{L/K})=\mathfrak{d}_{L/K}$, where $N_{L/K}$ is
the norm map [Ser79, Chapter 3, §3, Proposition 6]. An extension $L/K$ is
unramified if and only if $\mathcal{D}_{L/K}=\mathcal{O}_{L}$ [Ser79, Chapter
3, §5, Theorem 1], which in turn is true if and only if
$\mathfrak{d}_{L/K}=\mathcal{O}_{K}$.

Now suppose $L/K$ is a finite Galois extension. The Galois group
$\operatorname{Gal}(L/K)$ admits an exhaustive decreasing filtration
$\operatorname{Gal}(L/K)_{u}$ by real numbers $u\geq-1$, defined by

$\operatorname{Gal}(L/K)_{u}=\left\\{\sigma\in\operatorname{Gal}(L/K)\biggm{|}v_{L}(\sigma(\alpha)-\alpha)\geq
u+1\text{ for all $\alpha\in\mathcal{O}_{L}$}\right\\}.$

Since $v_{L}$ takes integer values, we have
$\operatorname{Gal}(L/K)_{u}=\operatorname{Gal}(L/K)_{\left\lceil
u\right\rceil}$, where $\left\lceil u\right\rceil$ is the ceiling function of
$u$.

This filtration of $\operatorname{Gal}(L/K)$ is known as the lower numbering
filtration. The first three subgroups appearing in this filtration are:

$\displaystyle\operatorname{Gal}(L/K)_{-1}$ $\displaystyle=$
$\displaystyle\operatorname{Gal}(L/K),$
$\displaystyle\operatorname{Gal}(L/K)_{0}$ $\displaystyle=$
$\displaystyle\text{the inertia group of $L/K$},$
$\displaystyle\operatorname{Gal}(L/K)_{1}$ $\displaystyle=$
$\displaystyle\text{the maximal $p$-subgroup of
$\operatorname{Gal}(L/K)_{0}$},$

see [Ser79, Chapter IV, §2, Corollary 1 and Corollary 3]. The cardinality of
the inertia group $\operatorname{Gal}(L/K)_{0}$ equals the relative
ramification index $e_{L/K}$. The extension $L/K$ is tame if $p\nmid e_{L/K}$;
this condition is equivalent to $\operatorname{Gal}(L/K)_{1}=1$. Otherwise,
$L/K$ is wild.

The lower numbering filtration of $\operatorname{Gal}(L/K)$ recovers the
different $\mathcal{D}_{L/K}$ by the relation [Ser79, Chapter IV, §1,
Proposition 4]:

$v_{L}(\mathcal{D}_{L/K})=\sum_{i=0}^{\infty}\left(\\#\operatorname{Gal}(L/K)_{i}-1\right)$
(4.1.1)

The lower number filtration is adapted to subgroups of
$\operatorname{Gal}(L/K)$, in the sense that if $K^{\prime}/K$ is a
subextension of $L/K$, then
$\operatorname{Gal}(L/K^{\prime})_{u}=\operatorname{Gal}(L/K)_{u}\cap\operatorname{Gal}(L/K^{\prime})$.
There is an alternative descending filtration $\operatorname{Gal}(L/K)^{u}$ by
real numbers $u\geq-1$, the upper numbering filtration. This filtration is
adapted to quotients of $\operatorname{Gal}(L/K)$, in the sense that if
$K^{\prime}/K$ is a Galois subextension of $L/K$, then
$\operatorname{Gal}(K^{\prime}/K)^{u}$ is the image of
$\operatorname{Gal}(L/K)^{u}$ under the quotient map
$\operatorname{Gal}(L/K)\to\operatorname{Gal}(K^{\prime}/K)$. To define the
upper numbering filtration, one needs the Herbrand function
$\varphi_{L/K}\colon[-1,\infty)\to[-1,\infty)$, a continuous, increasing, and
bijective function defined by the rules:

$\varphi_{L/K}(u)=\begin{cases}u,&{-1\leq u\leq 0},\\\
\int_{0}^{u}[\operatorname{Gal}(L/K)_{0}:\operatorname{Gal}(L/K)_{t}]^{-1}dt,&u>0.\end{cases}$

The inverse function to $\varphi_{L/K}$ is denoted $\psi_{L/K}$. The upper
numbering filtration on $\operatorname{Gal}(L/K)$ is now defined by:

$\operatorname{Gal}(L/K)^{u}=\operatorname{Gal}(L/K)_{\psi_{L/K}(u)}$

for all $u\geq-1$. Note that
$\operatorname{Gal}(L/K)^{0}=\operatorname{Gal}(L/K)_{0}$ is the inertia group
of $L/K$.

In terms of the upper numbering filtration, the formula for the different in
(4.1.1) becomes:

$v_{L}(\mathcal{D}_{L/K})=e_{L/K}\int_{-1}^{\infty}\left(1-\frac{1}{\\#\operatorname{Gal}(L/K)^{u}}\right)\;du$
(4.1.2)

The property claimed above about the upper numbering filtration being adapted
to quotients is [Ser79, Chapter IV, §3, Proposition 14]. This property allows
for the definition of $\operatorname{Gal}(L/K)^{u}$ when $L/K$ is an infinite
Galois extension. Indeed, the $\operatorname{Gal}(L_{i}/K)^{u}$ form a
projective system as $L_{i}$ ranges over Galois subextensions of $L/K$, and
one defines
$\operatorname{Gal}(L/K)^{u}=\varprojlim\operatorname{Gal}(L_{i}/K)^{u}$, a
closed subgroup of $\operatorname{Gal}(L/K)$.

Suppose $L/K$ is a possibly infinite Galois extension. A real number $u$ is a
jump for $L/K$ if
$\operatorname{Gal}(L/K)^{u+\varepsilon}\subsetneq\operatorname{Gal}(L/K)^{u}$
for every $\varepsilon>0$. The Hasse–Arf theorem states that if
$\operatorname{Gal}(L/K)$ is abelian, the jumps for $L/K$ are all integers.
For a proof, see [Ser79, Chapter V, §7].

Finally we recall some of the main theorems of local class field theory, which
furnishes a description of the abelianization
$\operatorname{Gal}(\overline{K}/K)^{\operatorname{ab}}$. Assume first that
the residue field of $K$ is finite. For an integer $i\geq 0$, let $U_{K}^{i}$
equal $\mathcal{O}_{K}^{\times}$ if $i=0$, and $1+\pi_{K}^{i}\mathcal{O}_{K}$
if $i>0$. The $U_{K}^{i}$ are profinite topological groups.

######  Theorem 4.1.1.

Let $K^{\operatorname{ab}}/K$ be the maximal abelian extension, so that
$\operatorname{Gal}(\overline{K}/K)^{\operatorname{ab}}\cong\operatorname{Gal}(K^{\operatorname{ab}}/K)$.
Let $\hat{K}^{\times}$ be the profinite completion of the group $K^{\times}$,
so that
$\hat{K}^{\times}=\pi_{K}^{\hat{\mathbb{Z}}}\times\mathcal{O}_{K}^{\times}$.
There exists an isomorphism (the reciprocity map)

$\operatorname{rec}_{K}\colon\hat{K}^{\times}\to\operatorname{Gal}(K^{\operatorname{ab}}/K)$

which carries $U_{K}^{i}$ onto
$\operatorname{Gal}(K^{\operatorname{ab}}/K)^{i}$ for all integers $i\geq 0$.

For a finite extension $L/K$, the inclusion $K^{\operatorname{ab}}\subset
L^{\operatorname{ab}}$ induces a homomorphism
$\operatorname{Gal}(L^{\operatorname{ab}}/L)\to\operatorname{Gal}(K^{\operatorname{ab}}/K)$.
The following diagram commutes:

$\textstyle{L^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{rec}_{L}\;\;}$$\scriptstyle{N_{L/K}}$$\textstyle{\operatorname{Gal}(L^{\operatorname{ab}}/L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{rec}_{K}\;\;}$$\textstyle{\operatorname{Gal}(K^{\operatorname{ab}}/K)}$

For a proof, see [Ser67].

For an abelian group $A$ written multiplicatively, write $A^{(p)}$ for the
image of $x\mapsto x^{p}$ on $A$.

######  Corollary 4.1.2.

Let $L/K$ be a (possibly infinite) abelian extension. For $u>e_{K}/(p-1)$, we
have $(\operatorname{Gal}(L/K)^{u})^{(p)}=\operatorname{Gal}(L/K)^{u+e_{K}}$.

###### Proof.

By the Hasse–Arf theorem, the jumps of $L/K$ are integers, and we are reduced
to the case that $u=i$ is an integer. By Theorem 4.1.1, specifically the
equality
$\operatorname{rec}_{K}(U_{K}^{i})=\operatorname{Gal}(K^{\operatorname{ab}}/K)^{i}$,
it is enough to prove the relation $(U_{K}^{i})^{(p)}=U_{K}^{i+e_{K}}$
whenever $i>e_{K}/(p-1)$. For $(U_{K}^{i})^{(p)}\subset U_{K}^{i+e_{K}}$:
Suppose $\alpha\in\mathcal{O}_{K}$. Expanding $(1+\pi^{i}_{K}\alpha)^{p}$
gives $1+\pi_{K}^{ip}\alpha^{p}$ plus terms divisible by $p\pi_{K}^{i}$. We
have $v_{K}(p\pi_{K}^{i})=i+e_{K}$, and $v_{K}(\pi_{K}^{ip})=ip>i+e_{K}$ by
hypothesis.

For $U_{K}^{i+e_{K}}\subset(U_{K}^{i})^{(p)}$: Again suppose
$\alpha\in\mathcal{O}_{K}$. We claim that the binomial expansion of
$(1+p\pi_{K}^{i}\alpha)^{1/p}$ converges to an element of $U_{K}^{i}$. For
$n\geq 1$, the $n$th term in the expansion is $p^{n}\begin{pmatrix}1/p\\\
n\end{pmatrix}\pi_{K}^{ni}\alpha^{n}$, which equals an element of
$\mathcal{O}_{K}$ times $\pi_{K}^{ni}/n!$. Using the inequality
$v(n!)\leq(n-1)/(p-1)$ (a standard inequality), we have:

$v_{K}\left(\frac{\pi_{K}^{ni}}{n!}\right)-i=ni-
e_{K}v(n!)-i\geq(n-1)(i-\frac{e_{k}}{p-1}).$

This shows that the terms of the expansion of $(1+p\pi_{K}^{i}\alpha)^{1/p}$
beyond the initial 1 lie in $\pi^{i}\mathcal{O}_{K}$ and converge to 0. ∎

We remark that 4.1.2 holds even when the residue field of $K$ is infinite. One
reduces to the case that the residue field is algebraically closed (since the
higher ramification groups are insensitive to unramified base change). In that
case Serre [Ser61] proves an analogue of Theorem 4.1.1 in which the
$U^{i}_{K}$ are replaced by pro-algebraic groups $\mathcal{U}^{i}_{K}$. It is
still true that the $p$th power map carries $\mathcal{U}^{i}_{K}$ surjectively
onto $\mathcal{U}^{i+e_{K}}_{K}$ for $i$ sufficiently large, which is all that
is necessary for 4.1.2.

Some of the results in this section pertain to the stability of some property
under base change by a tame extension $L/K$. We list some properties of tame
extensions in the following proposition.

######  Proposition 4.1.3.

Suppose $L/K$ is a finite Galois extension which is tame (that is, $p\nmid
e_{L/K}$). Let $e=e_{L/K}$.

  1. (1)

For $u\geq 0$ we have $\varphi_{L/K}(u)=u/e$ and $\psi_{L/K}(u)=eu$.

  2. (2)

For $u>0$ we have $N_{L/K}(U_{L}^{eu})=U_{K}^{u}$.

  3. (3)

Let $M/K$ be a (possibly infinite) abelian extension. For $u>0$, the image of
$\operatorname{Gal}(LM/L)^{eu}$ under
$\operatorname{Gal}(LM/L)\to\operatorname{Gal}(M/K)$ is
$\operatorname{Gal}(M/K)^{u}$.

###### Proof.

Part (1) follows from the definition of $\varphi_{L/K}$, using the facts that
$\\#\operatorname{Gal}(L/K)_{0}=e$ and $\operatorname{Gal}(L/K)_{u}$ is
trivial for all $u>0$.

For part (2): The claim is transitive in the sense that if it is true for
$L/K$ and $M/L$, then it is true for $M/K$. Using the fact that inertia groups
of local fields are solvable [Ser79, Chapter IV, §2, Corollary 5], we can
reduce to the case that $L/K$ is either unramified or totally ramified and
cyclic of prime degree $\ell\neq p$. The proof of the claim in those cases is
found in [Ser79, Chapter V, §2, Proposition 1] and [Ser79, Chapter V, §3,
Corollary 4], respectively.

Part (3) is proved by combining compatibility of $\operatorname{rec}_{K}$ with
the norm map (Theorem 4.1.1) with part (2). ∎

###  4.2. Ramified $\mathbb{Z}_{p}$-extensions

Keep the assumption that $K$ is a local field of characteristic $(0,p)$.

Let $K_{\infty}/K$ be a Galois extension with
$\operatorname{Gal}(K_{\infty}/K)\cong\mathbb{Z}_{p}$. We assume that
$K_{\infty}/K$ is ramified, in the sense that the inertia group
$\operatorname{Gal}(K_{\infty}/K)^{0}$ is nontrivial. The inertia group is
always closed, so this condition is equivalent to
$\operatorname{Gal}(K_{\infty}/K)^{0}$ being of finite index in
$\operatorname{Gal}(K_{\infty}/K)$, or equivalently, that $K_{\infty}$
contains elements of arbitrarily small positive valuation.

Such an extension always exists. For instance, let $L/K$ be the extension
obtained by adjoining a primitive $p^{n}$th root of unity $\zeta_{p^{n}}$ for
all $n\geq 1$. Since $v(\zeta_{p^{n}}-1)=1/p^{n}$ gets arbitrarily small,
$L/K$ is ramified. Its Galois group is an open subgroup of
$\mathbb{Z}_{p}^{\times}$, and so admits a maximal quotient isomorphic to
$\mathbb{Z}_{p}$. The extension $K_{\infty}/K$ corresponding to this quotient
is a ramified $\mathbb{Z}_{p}$-extension of $K$, which we call the cyclotomic
$\mathbb{Z}_{p}$-extension.

The Galois group $\operatorname{Gal}(K_{\infty}/K)\cong\mathbb{Z}_{p}$ has an
obvious filtration by subgroups $p^{n}\mathbb{Z}_{p}$ for $n=0,1,\dots$. We
write $K_{n}/K$ for the finite extension corresponding to the subgroup
$p^{n}\mathbb{Z}_{p}\subseteq\mathbb{Z}_{p}$.

On the other hand we have the upper numbering ramification filtration
$\operatorname{Gal}(K_{\infty}/K)^{u}$ for real numbers $u\geq-1$. By the
Hasse–Arf theorem, the jumps of this filtration occur at rational integers.
That is, there is a sequence of integers $-1=u_{0}\leq u_{1}\leq
u_{2}\leq\dots$ tending to $\infty$ such that (as a subgroup of
$\operatorname{Gal}(K_{\infty}/K)\cong\mathbb{Z}_{p})$:

$\operatorname{Gal}(K_{\infty}/K)^{u}\cong p^{n}\mathbb{Z}_{p}\,\text{
whenever }u_{n}<u\leq u_{n+1}$

for all $n=0,1,\dots$. Let us call the sequence $u_{0},u_{1},u_{2},\dots$ the
jumps of $K_{\infty}/K$.

The two filtrations on $\operatorname{Gal}(K_{\infty}/K)$ are “eventually
compatible” in the following sense.

######  Lemma 4.2.1.

Let $u_{0},u_{1},u_{2},\dots$ be the sequence of jumps for $K_{\infty}/K$
defined above.

  1. (1)

There exists $N$ such that $u_{N}\geq 0$ and $u_{n+1}=u_{n}+e_{K}$ for all
$n\geq N$.

  2. (2)

If $e_{K}=1$ and $K_{\infty}/K$ is the cyclotomic $\mathbb{Z}_{p}$-extension,
then the sequence $u_{0},u_{1},u_{2},\dots$ is $-1,1,2,3,\dots$.

  3. (3)

Suppose $L/K$ is a finite Galois extension which is tame, and let $e=e_{L/K}$.
Then $L_{\infty}=LK_{\infty}$ is a ramified $\mathbb{Z}_{p}$-extension of $L$.
The sequence of jumps for $L_{\infty}/L$ is $-1,eu_{1},eu_{2},eu_{3},\dots$.

  4. (4)

Suppose $e_{K}=1$ and $L/K$ is a finite Galois extension which is tame. If
$L_{\infty}/L$ is the cyclotomic $\mathbb{Z}_{p}$-extension, then the sequence
of jumps for $L_{\infty}/L$ is $-1,e_{L},2e_{L},3e_{L},\dots$.

###### Proof.

For part (1): Let $N$ be large enough so that $u_{N}>e_{K}/(p-1)$. By 4.1.2,
for $u\geq u_{N}$ we have
$(\operatorname{Gal}(K_{\infty}/K)^{u})^{(p)}=\operatorname{Gal}(K_{\infty}/K)^{u+e_{K}}$.
Now suppose $n\geq N$. If $u_{n}<u\leq u_{n+1}$, this relation means
$\operatorname{Gal}(K_{\infty}/K)^{u+e_{K}}\cong p^{n+1}\mathbb{Z}_{p}$. This
means precisely that $u_{n+1}=u_{n}+e_{K}$.

For part (2): The computation of the higher ramification groups of the
cyclotomic $\mathbb{Z}_{p}$-extension over $\mathbb{Q}_{p}$ (or any unramified
extension thereof) is standard, see for instance [Ser79, Chapter IV, §4].

For part (3): By Proposition 4.1.3, for all $u>0$ the image of
$\operatorname{Gal}(L_{\infty}/L)^{eu}$ under
$\operatorname{Gal}(L_{\infty}/L)\to\operatorname{Gal}(K_{\infty}/K)$ is
$\operatorname{Gal}(K_{\infty}/K)^{u}$, so that the jumps of $L_{\infty}/L$
are exactly $e$ times the jumps of $K_{\infty}/K$.

For part (4): Combine parts (2) and (3). ∎

######  Lemma 4.2.2.

Let $K_{\infty}/K$ be a ramified $\mathbb{Z}_{p}$-extension with jumps
$u_{0},u_{1},u_{2},\dots$. Then for each $n\geq 1$ large enough so that
$K_{n}/K_{n-1}$ is ramified, the valuation of the different of $K_{n}/K_{n-1}$
is:

$v_{K_{n}}(\mathcal{D}_{K_{n}/K_{n-1}})=\frac{e_{K_{n}/K}}{p^{n}}\sum_{k=0}^{n-1}(u_{k+1}-u_{k})(p^{k+1}-p^{k}).$

###### Proof.

Applying the formula for the different (4.1.2) to $K_{n}/K$, we find:

$v_{K_{n}}(\mathcal{D}_{K_{n}/K})=e_{K_{n}/K}\sum_{k=0}^{n-1}(u_{k+1}-u_{k})\left(1-\frac{1}{p^{n-k}}\right)$

By the transitivity of the different [Ser79, Chapter III, §4, Proposition 8],
we have
$\mathcal{D}_{K_{n}/K}=\mathcal{D}_{K_{n}/K_{n-1}}\mathcal{D}_{K_{n-1}/K}$.
Applying $v_{K_{n}}$ (and using the relation $v_{K_{n-1}}=p^{-1}v_{K_{n}}$,
true by our assumption that $K_{n}/K_{n-1}$ is ramified) gives the result. ∎

######  Lemma 4.2.3.

Let $K_{\infty}/K$ be a ramified $\mathbb{Z}_{p}$-extension. There exists an
integer $N\geq 0$ such that for all $n\geq N+1$ we have:

$v_{K_{n}}(\mathcal{D}_{K_{n}/K_{n-1}})\geq p-1+e_{K_{n}}(1-p^{N-n}).$

###### Proof.

Take for $N$ the integer appearing in Lemma 4.2.1(1), so that $u_{N}\geq 0$
(this means that $K_{\infty}/K_{N}$ is totally ramified) and
$u_{n}=u_{n-1}+e_{K}$ for all $n\geq N+1$. For such an $n$, $K_{n}/K_{n-1}$ is
totally ramified, and so Lemma 4.2.2 applies:

$\displaystyle v_{K_{n}}(\mathcal{D}_{K_{n}/K_{n-1}})$ $\displaystyle=$
$\displaystyle\frac{e_{K_{n}/K}}{p^{n}}\sum_{k=0}^{n-1}(u_{k+1}-u_{k})(p^{k+1}-p^{k})$
$\displaystyle=$
$\displaystyle\frac{e_{K_{n}/K}}{p^{n}}\sum_{k=0}^{N-1}(u_{k+1}-u_{k})(p^{k+1}-p^{k})+\frac{e_{K_{n}/K}}{p^{n}}\sum_{k=N}^{n-1}e_{K}(p^{k+1}-p^{k})$
$\displaystyle=$
$\displaystyle\frac{e_{K_{n}/K}}{p^{n}}\sum_{k=0}^{N-1}(u_{k+1}-u_{k})(p^{k+1}-p^{k})+e_{K_{n}}(1-p^{N-n})$

Suppose that the inertia group $\operatorname{Gal}(K_{\infty}/K)^{0}\cong
p^{r}\mathbb{Z}_{p}$ as a subgroup of
$\operatorname{Gal}(K_{\infty}/K)\cong\mathbb{Z}_{p}$; this means exactly that
$u_{0}=u_{1}=\cdots=u_{r}=-1$ and $u_{r+1}\geq 0$, and also that
$p^{r}e_{K_{n}/K}=p^{n}$. Since $K_{n}/K_{n-1}$ is ramified, we must have
$r<N$. Therefore the first sum above is bounded below by
$(e_{K_{n}/K}/p^{n})(p^{r+1}-p^{r})=p-1$. ∎

In the context of Lemma 4.2.3, $K_{\infty}/K_{N}$ is a totally ramified
$\mathbb{Z}_{p}$-extension. If it is relabeled so that $K=K_{N}$ is the base
of the extension, then each $K_{n}/K$ is totally ramified, and the inequality
in that lemma becomes

$v_{K_{n}}(\mathcal{D}_{K_{n}/K_{n-1}})\geq e_{K}(p^{n}-1)+(p-1).$ (4.2.1)

######  Definition 4.2.4.

Let $K_{\infty}/K$ be a ramified $\mathbb{Z}_{p}$-extension. We call
$K_{\infty}/K$ sufficiently ramified if the inequality in (4.2.1) holds for
each $n\geq 1$.

In light of the above considerations and Lemma 4.2.1, there are plenty of
examples of sufficiently ramified $\mathbb{Z}_{p}$-extensions.

######  Lemma 4.2.5.

Sufficiently ramified $\mathbb{Z}_{p}$-extensions exist in the following
situations:

  1. (1)

Let $K$ be a local field of characteristic $(0,p)$ such that $p\nmid e_{K}$.
Then $K$ admits a sufficiently ramified $\mathbb{Z}_{p}$-extension.

  2. (2)

Let $K$ be as in part (1), and let $L/K$ be a finite subextension of the
cyclotomic $\mathbb{Z}_{p}$-extension over $L$. Then $L$ admits a sufficiently
ramified $\mathbb{Z}_{p}$-extension.

  3. (3)

Let $K$ be any local field of characteristic $(0,p)$, and let $K_{\infty}/K$
be a ramified $\mathbb{Z}_{p}$-extension. Then for sufficiently large $n$,
$K_{\infty}/K_{n}$ is a sufficiently ramified $\mathbb{Z}_{p}$-extension.

  4. (4)

Let $K_{\infty}/K$ be a sufficiently ramified $\mathbb{Z}_{p}$-extension, and
let $L/K$ be a finite Galois extension which is tame. Let
$L_{\infty}=K_{\infty}L$. Then $L_{\infty}/L$ is a sufficiently ramified
$\mathbb{Z}_{p}$-extension.

We need the following result on traces.

######  Lemma 4.2.6 ([Ser79, Chapter V Lemma 4]).

Let $L/K$ be a cyclic extension of order $p$. Then for any $\alpha\in L$ we
have:

$v_{K}(\operatorname{tr}_{L/K}(\alpha))\geq\left\lfloor\frac{v_{L}(\alpha)+v_{L}(\mathcal{D}_{L/K})}{p}\right\rfloor,$

where $\left\lfloor x\right\rfloor$ is the floor function.

In the following lemma, $\left\lvert x\right\rvert=a^{v(x)}$ for $x\in
K_{\infty}$ and an arbitrary real number $0<a<1$. Then $\left\lvert
x\right\rvert$ is an absolute value inducing the topology on $K_{\infty}$.

######  Lemma 4.2.7.

Assume $K_{\infty}/K$ is sufficiently ramified. We have the following
inequalities.

  1. (1)

For $x\in K_{n}$ we have:

$\left\lvert\operatorname{tr}_{K_{n}/K_{n-1}}(x)\right\rvert\leq\left\lvert
p\right\rvert^{1-p^{-n}}\left\lvert x\right\rvert$

  2. (2)

For $x\in K_{n}$ we have:

$\left\lvert\operatorname{tr}_{K_{n}/K}(x)\right\rvert\leq\left\lvert
p\right\rvert^{n-(p-1)^{-1}}\left\lvert x\right\rvert$

###### Proof.

For part (1): Let $d=v_{K_{n}}(\mathcal{D}_{K_{n}/K_{n-1}})$. By definition of
sufficiently ramified we have $d\geq e_{K}(p^{n}-1)+p-1$. Applying Lemma 4.2.6
to $K_{n+1}/K_{n}$, we find:

$\displaystyle v_{K_{n-1}}(\operatorname{tr}_{K_{n}/K_{n-1}}(x))$
$\displaystyle\geq$
$\displaystyle\left\lfloor\frac{v_{K_{n}}(x)+d}{p}\right\rfloor$
$\displaystyle\geq$ $\displaystyle\frac{v_{K_{n}}(x)+d-(p-1)}{p}$
$\displaystyle=$ $\displaystyle
e_{K}p^{n-1}-\frac{e_{K}}{p}+\frac{v_{K_{n}}(x)}{p}.$

Translating this inequality in terms of the absolute valuation gives:

$v(\operatorname{tr}_{K_{n}/K_{n-1}}(x))\geq 1-\frac{1}{p^{n}}+v(x),$

which (recalling that $v(p)=1$) means exactly the inequality in part (1).

For part (2), we apply part (1) inductively on $x\in K_{n}$ to obtain:

$\displaystyle\left\lvert\operatorname{tr}_{K_{n}/K}(x)\right\rvert$
$\displaystyle=$
$\displaystyle\left\lvert\operatorname{tr}_{K_{1}/K}\operatorname{tr}_{K_{2}/K_{1}}\cdots\operatorname{tr}_{K_{n}/K_{n-1}}(x)\right\rvert$
$\displaystyle\leq$ $\displaystyle\left\lvert
p\right\rvert^{n-p^{-1}-p^{-2}-\cdots-p^{-n}}\left\lvert x\right\rvert$
$\displaystyle\leq$ $\displaystyle\left\lvert
p\right\rvert^{n-(p-1)^{-1}}\left\lvert x\right\rvert$

∎

Keep the assumption that $K_{\infty}/K$ is sufficiently ramified. Define the
normalized trace $t\colon K_{\infty}\to K$ by
$t(x)=p^{-n}\operatorname{tr}_{K_{n}/K}(x)$ whenever $x\in K_{n}$ (note that
this is unambiguous). For $n\geq 1$ we define $t_{n}\colon K_{n}\to K_{n-1}$
by $t_{n}(x)=p^{-1}\operatorname{tr}_{K_{n}/K_{n-1}}(x)$. Then Lemma 4.2.7
translates into:

$\left\lvert t(x)\right\rvert\leq\left\lvert
p\right\rvert^{-(p-1)^{-1}}\left\lvert x\right\rvert,\;x\in K_{\infty}$
(4.2.2)

and

$\left\lvert t_{n}(x)\right\rvert\leq\left\lvert
p\right\rvert^{-p^{-n}}\left\lvert x\right\rvert.$ (4.2.3)

Finally note that $t(t_{n}(x))=t(x)$ for all $x\in K_{n}$.

Let $\sigma$ be a topological generator of
$\operatorname{Gal}(K_{\infty}/K)\cong\mathbb{Z}_{p}$. Elements of the
polynomial ring $K[\sigma]$ act on $K_{\infty}$ in the evident manner. For
$x\in K_{n}$ we record the relation

$t_{n}(x)=p^{-1}(1+\sigma^{p^{n-1}}+\sigma^{2p^{n-1}}+\cdots+\sigma^{(p-1)p^{n-1}})(x).$
(4.2.4)

######  Lemma 4.2.8.

Assume that $K_{\infty}/K$ is sufficiently ramified. For $x\in K_{\infty}$ we
have:

$\left\lvert x-t(x)\right\rvert\leq\left\lvert
p\right\rvert^{-1-\frac{1}{p(p-1)}}\left\lvert\sigma(x)-x\right\rvert$

###### Proof.

For each $x\in K_{n}$, (4.2.4) implies:

$\displaystyle x-t_{n}(x)$ $\displaystyle=$ $\displaystyle
p^{-1}\sum_{i=1}^{p-1}(1-\sigma^{ip^{n-1}})(x)$ $\displaystyle=$
$\displaystyle
p^{-1}\sum_{i=1}^{p-1}(1+\sigma^{p^{n-1}}+\cdots+\sigma^{(i-1)p^{n-1}})(1-\sigma^{p^{n-1}})(x).$

Noting that $\left\lvert\alpha(x)\right\rvert\leq\left\lvert x\right\rvert$
whenever $\alpha\in\mathcal{O}_{K}[\sigma]$, we find:

$\left\lvert x-t_{n}(x)\right\rvert\leq\left\lvert
p\right\rvert^{-1}\left\lvert(1-\sigma^{p^{n-1}})x\right\rvert\leq\left\lvert
p\right\rvert^{-1}\left\lvert\sigma(x)-x\right\rvert.$ (4.2.5)

We will prove by induction on $n\geq 1$ the following statement which implies
the lemma: for $x\in K_{n}$, we have

$\left\lvert x-t(x)\right\rvert\leq\left\lvert
p\right\rvert^{-1-\frac{1}{p^{2}}-\cdots-\frac{1}{p^{n}}}\left\lvert\sigma(x)-x\right\rvert.$
(4.2.6)

(The exponent of $\left\lvert p\right\rvert$ is meant to be $-1$ in the case
$n=1$.) The base case $n=1$ is an instance of (4.2.5). Now suppose $n\geq 2$.
Assume (4.2.6) for $n-1$, and then for $x\in K_{n}$ we have:

$\left\lvert x-t(x)\right\rvert\leq\max\left\\{\left\lvert
x-t_{n}(x)\right\rvert,\left\lvert t_{n}(x)-t(x)\right\rvert\right\\}$

Treating each quantity on the right side in turn, we have

$\left\lvert x-t_{n}(x)\right\rvert\leq\left\lvert
p\right\rvert^{-1}\left\lvert\sigma(x)-x\right\rvert$

by (4.2.5); this is $\leq$ the right side of (4.2.6). The other quantity is

$\displaystyle\left\lvert t_{n}(x)-t(x)\right\rvert$ $\displaystyle=$
$\displaystyle\left\lvert t_{n}(x)-t(t_{n}(x))\right\rvert$
$\displaystyle\leq$ $\displaystyle\left\lvert
p\right\rvert^{-1-\frac{1}{p^{2}}-\cdots-\frac{1}{p^{n-1}}}\left\lvert(\sigma-1)t_{n}(x)\right\rvert$

by applying the inductive hypothesis to $t_{n}(x)\in K_{n-1}$. Noting that
$\sigma$ commutes with $t_{n}$, we have

$\left\lvert(\sigma-1)t_{n}(x)\right\rvert=\left\lvert
t_{n}(\sigma-1)(x)\right\rvert\leq\left\lvert
p\right\rvert^{-p^{-n}}\left\lvert\sigma(x)-x\right\rvert$

by (4.2.3) and so

$\left\lvert t_{n}(x)-t(x)\right\rvert\leq\left\lvert
p\right\rvert^{-1-\frac{1}{p^{2}}-\cdots-\frac{1}{p^{n}}}\left\lvert\sigma(x)-x\right\rvert$

as required. ∎

The bounds in (4.2.2) and Lemma 4.2.8 show that $t\colon K_{\infty}\to K$ is
continuous and extends uniquely to a $K$-linear map
$t\colon\hat{K}_{\infty}\to K$ satisfying the same bounds for all
$x\in\hat{K}_{\infty}$:

$\displaystyle\left\lvert t(x)\right\rvert$ $\displaystyle\leq$
$\displaystyle\left\lvert p\right\rvert^{-\frac{1}{p-1}}\left\lvert
x\right\rvert$ (4.2.7) $\displaystyle\left\lvert x-t(x)\right\rvert$
$\displaystyle\leq$ $\displaystyle\left\lvert
p\right\rvert^{-1-\frac{1}{p(p-1)}}\left\lvert\sigma(x)-x\right\rvert$ (4.2.8)

We turn our attention now to the continuous cohomology
$H^{i}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K),\mathcal{O}_{\hat{K}_{\infty}})$
when $K_{\infty}/K$ is a sufficiently ramified $\mathbb{Z}_{p}$-extension.
Since $\operatorname{Gal}(K_{\infty}/K)\cong\mathbb{Z}_{p}$, the continuous
cohomology of $\operatorname{Gal}(K_{\infty}/K)$ acting continuously on a
$p$-adically complete abelian group $M$ is computed by the complex

$M^{h\operatorname{Gal}(K_{\infty}/K)}:\;\;\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
8.39583pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
12.47922pt\raise 5.25555pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.25555pt\hbox{$\scriptstyle{\sigma-1}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 32.39583pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
32.39583pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{M}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

which is to say that $H^{0}_{\operatorname{cts}}(M)=M^{\sigma}$,
$H^{1}_{\operatorname{cts}}(M)=M/(\sigma-1)M$, and
$H^{i}_{\operatorname{cts}}(M)=0$ for $i\geq 2$. In the case of
$M=\mathcal{O}_{\hat{K}_{\infty}}$, the $H^{0}$ is $\mathcal{O}_{K}$ by Ax’s
theorem (Theorem 4.0.1), so we are left with describing the $H^{1}$.

######  Proposition 4.2.9.

Let $K_{\infty}/K$ be a ramified $\mathbb{Z}_{p}$-extension. Let $N\geq 0$ be
large enough so that $K_{\infty}/K_{N}$ is sufficiently ramified. Define a
$p$-adically complete abelian group $X$ with continuous
$\operatorname{Gal}(K_{\infty}/K)$-action by the exact sequence:

$0\to\mathcal{O}_{K}\to\mathcal{O}_{\hat{K}_{\infty}}\to X\to 0.$

Then $H^{i}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K),X)=0$ for
all $i\neq 1$, and
$H^{1}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K),X)$ is
$p^{N+2}$-torsion for $p\neq 2$ and $p^{N+3}$-torsion for $p=2$.

######  Remark 4.2.10.

Proposition 4.2.9 may be stated in terms of complexes this way: Let
$\mathcal{X}=X^{h\operatorname{Gal}(K_{\infty}/K)}$, so that $\mathcal{X}$
sits in an exact triangle

$\mathcal{O}_{K}^{h\operatorname{Gal}(K_{\infty}/K)}\to\mathcal{O}_{\hat{K}_{\infty}}^{h\operatorname{Gal}(K_{\infty}/K)}\to\mathcal{X}.$

Then we have $H^{i}(\mathcal{X})=0$ for $i\neq 1$, and
$H^{1}(\mathcal{X})=H^{1}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K),X)$
is a torsion group as claimed in the proposition. Note that since
$\operatorname{Gal}(K_{\infty}/K)$ acts trivially on $\mathcal{O}_{K}$, a
choice of isomorphism $\operatorname{Gal}(K_{\infty}/K)\cong\mathbb{Z}_{p}$
induces a quasi-isomorphism
$\mathcal{O}_{K}^{h\operatorname{Gal}(K_{\infty}/K)}\cong\mathcal{O}_{K}[\varepsilon]$.

###### Proof.

Since $\mathbb{Z}_{p}$ has cohomological dimension 1, it is enough to consider
$H^{0}$ and $H^{1}_{\operatorname{cts}}$.

We first consider the case $N=0$, meaning that $K_{\infty}/K$ is sufficiently
ramified. Let $\sigma$ be a topological generator for
$\operatorname{Gal}(K_{\infty}/K)$. The claim that
$H^{0}(\operatorname{Gal}(K_{\infty}/K),X)=0$ is equivalent to the claim that
$\mathcal{O}_{K}\to
H^{1}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K),\mathcal{O}_{\hat{K}_{\infty}})$
is injective. If $\alpha\in\mathcal{O}_{K}$ lies in the kernel, then
$\alpha=(\sigma-1)\alpha^{\prime}$ for
$\alpha^{\prime}\in\mathcal{O}_{\hat{K}_{\infty}}$, but then
$\alpha=t(\alpha)=t(\sigma-1)\alpha^{\prime}$. We claim that $t(\sigma-1)$
acts as $0$ on $\hat{K}_{\infty}$. Indeed, $t(\sigma-1)$ is zero on each
$K_{n}$, hence on $K_{\infty}$, and so by continuity it is zero on all of
$\hat{K}_{\infty}$.

We now turn to
$H^{1}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K),\mathcal{O}_{\hat{K}_{\infty}})$.
Let us write the superscript $t=0$ to mean the kernel of the normalized trace
$t$ wherever this is defined. For each $n\geq 1$, the $K$-linear operator
$\sigma-1$ is injective on the finite-dimensional $K$-vector space
$K_{n}^{t=0}$, since the kernel is $K\cap K_{n}^{t=0}=0$. (The operator $t$ is
the identity on $K$.) Therefore the restriction of $\sigma-1$ to $K_{n}^{t=0}$
is an isomorphism; write $(\sigma-1)^{-1}$ for its inverse. The inequality
(4.2.8) shows that $(\sigma-1)^{-1}$ is bounded on $K_{\infty}^{t=0}$ with
operator norm $\leq\left\lvert p\right\rvert^{-1-1/p(p-1)}$, so it extends to
an operator $(\sigma-1)^{-1}$ on $\hat{K}_{\infty}^{t=0}$ with the same
operator norm. Therefore

$H^{1}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K),\mathcal{O}^{t=0}_{\hat{K}_{\infty}})\cong\frac{\mathcal{O}^{t=0}_{\hat{K}_{\infty}}}{(\sigma-1)\mathcal{O}^{t=0}_{\hat{K}_{\infty}}}$

is $\alpha$-torsion for any $\alpha\in\mathcal{O}_{K}$ with $v(\alpha)\geq
1+1/p(p-1)$.

Consider the following diagram of $p$-adically complete groups, in which both
rows are exact:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{\hat{K}_{\infty}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{K}\oplus\mathcal{O}_{\hat{K}_{\infty}}^{t=0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{\hat{K}_{\infty}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

The inequality (4.2.7) says that $W$ is $\beta$-torsion for any
$\beta\in\mathcal{O}_{K}$ with $v(\beta)\geq 1/(p-1)$, because for any
$x\in\mathcal{O}_{\hat{K}_{\infty}}$, $\beta t(x)\in\mathcal{O}_{K}$, and then
$\beta x=\beta t(x)+\beta(x-t(x))$ lies in
$\mathcal{O}_{K}\oplus\mathcal{O}_{\hat{K}_{\infty}}^{t=0}$.

The long exact sequences in cohomology associated to the above exact sequences
read in part:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\operatorname{cts}}(\mathcal{O}_{K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\operatorname{cts}}(\mathcal{O}_{\hat{K}_{\infty}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{H^{1}_{\operatorname{cts}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(H^{1}_{\operatorname{cts}}(\mathcal{O}_{K})\oplus
H^{1}_{\operatorname{cts}}(\mathcal{O}_{\hat{K}_{\infty}}^{t=0})\right)/H^{0}(W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\operatorname{cts}}(\mathcal{O}_{\hat{K}_{\infty}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\operatorname{cts}}(W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

By the snake lemma, there is an exact sequence

$0\to
H^{1}_{\operatorname{cts}}(\mathcal{O}_{\hat{K}_{\infty}}^{t=0})/H^{0}(W)\to
H^{1}_{\operatorname{cts}}(X)\to H^{1}_{\operatorname{cts}}(W)\to 0$

As $H^{1}_{\operatorname{cts}}(\mathcal{O}_{\hat{K}_{\infty}}^{t=0})$ is
killed by any element $\alpha\in\mathcal{O}_{K}$ with $v(\alpha)\geq
1+1/p(p-1)$, and $H^{1}_{\operatorname{cts}}(W)$ is killed by any element
$\beta$ with $v(\beta)\geq 1/(p-1)$, we find that
$H^{1}_{\operatorname{cts}}(X)$ is killed by any product $\alpha\beta$ of such
elements.

If $e_{K}$ is sufficiently large (to wit $e_{K}\geq p(p-1)$), then any
$\gamma\in\mathcal{O}_{K}$ with $v(\gamma)\geq
1+1/p(p-1)+1/(p-1)=1+(p+1)/p(p-1)$ can be factored into such a product
$\alpha\beta$, and the lemma would follow for $K$. But if for instance
$e_{K}=1$, we could only conclude from the above that
$H^{1}_{\operatorname{cts}}(X)$ is $p^{3}$-torsion.

To improve the result as stated in the proposition, let $L/K$ be a
sufficiently ramified tame Galois extension with group $G$. (For example, $L$
could be the splitting field of $x^{n}-\pi_{K}$, where $\pi_{K}$ is a
uniformizer of $K$ and $n$ is sufficiently large and prime to $p$.) Since each
$K_{n}/K$ is totally wildly ramified, $K_{n}$ and $L$ are linearly disjoint
over $K$. Let $L_{\infty}=K_{\infty}L$, so that $L_{\infty}/L$ is a
sufficiently ramified $\mathbb{Z}_{p}$-extension by Lemma 4.2.1(3). We abuse
notation by referring to both $\operatorname{Gal}(K_{\infty}/K)$ and
$\operatorname{Gal}(L_{\infty}/L)$ as $\mathbb{Z}_{p}$, and we identify
$\operatorname{Gal}(L_{\infty}/K)=G\times\mathbb{Z}_{p}$. Define a complex
$\mathcal{X}_{L}$ with $G$-action by the exact triangle

$\mathcal{O}_{L}^{h\mathbb{Z}_{p}}\to\mathcal{O}_{\hat{L}_{\infty}}^{h\mathbb{Z}_{p}}\to\mathcal{X}_{L}.$

Applying derived $G$-invariants, we obtain an exact triangle

$\mathcal{O}_{L}^{h(G\times\mathbb{Z}_{p})}\to\mathcal{O}_{\hat{L}_{\infty}}^{h(G\times\mathbb{Z}_{p})}\to\mathcal{X}_{L}^{hG}$

Since $L_{n}/K_{n}$ is a tame extension with group $G$, Noether’s theorem
states that $\mathcal{O}_{L_{n}}$ is a free $\mathcal{O}_{K_{n}}[G]$-module.
For every $m$, $\mathcal{O}_{L_{n}}/p^{m}$ is a free
$(\mathcal{O}_{K_{n}}/p^{m})[G]$-module, and so
$H^{i}(G,\mathcal{O}_{L_{n}}/p^{m})=0$ for all $i>0$. Therefore

$H^{i}(G,\mathcal{O}_{\hat{L}_{\infty}})=\varprojlim_{m}\varinjlim_{n}H^{i}(G,\mathcal{O}_{L_{n}}/p^{m})=0,$

meaning that
$\mathcal{O}_{\hat{L}_{\infty}}^{G}=H^{0}(G,\mathcal{O}_{\hat{L}_{\infty}})=\mathcal{O}_{\hat{K}_{\infty}}$.
Our exact triangle now reads

$\mathcal{O}_{K}^{h\mathbb{Z}_{p}}\to\mathcal{O}_{\hat{K}_{\infty}}^{h\mathbb{Z}_{p}}\to\mathcal{X}_{L}^{hG},$

so that $\mathcal{X}=\mathcal{X}_{L}^{hG}$. By the argument above,
$H^{i}(\mathcal{X}_{L})=0$ for all $i\neq 1$, and $H^{1}(\mathcal{X}_{L})$ is
$p^{2}$\- or $p^{3}$-torsion as $p$ is odd or even. By Hochschild-Serre we
have $H^{1}(\mathcal{X})=H^{0}(G,H^{1}(\mathcal{X}_{L}))$, which is subject to
the same bounds.

Now consider the general case: Let $N$ be large enough so that
$K_{\infty}/K_{N}$ is sufficiently ramified. Let
$X=\mathcal{O}_{\hat{K}_{\infty}}/\mathcal{O}_{K}$ and
$X_{N}=\mathcal{O}_{\hat{K}_{\infty}}/\mathcal{O}_{K_{N}}$, so that we have an
exact sequence of $p$-adically complete abelian groups with
$\operatorname{Gal}(K_{\infty}/K)$-action:

$0\to\mathcal{O}_{K_{N}}/\mathcal{O}_{K}\to X\to X_{N}\to 0.$ (4.2.9)

By the above argument we have that
$H^{0}(\operatorname{Gal}(K_{\infty}/K_{N}),X_{N})=0$ and
$H^{1}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K_{N}),X_{N})$ is
$p^{2}$-torsion or $p^{3}$-torsion, as $p$ is odd or even.

Consider the $H^{0}$ part of the long exact sequence associated to (4.2.9):

$H^{0}(\operatorname{Gal}(K_{\infty}/K),\mathcal{O}_{K_{N}}/\mathcal{O}_{K})\to
H^{0}(\operatorname{Gal}(K_{\infty}/K),X)\to
H^{0}(\operatorname{Gal}(K_{\infty}/K),X_{N}).$

We have
$H^{0}(\operatorname{Gal}(K_{\infty}/K),\mathcal{O}_{K_{N}}/\mathcal{O}_{K})=H^{0}(\operatorname{Gal}(K_{N}/K),\mathcal{O}_{K_{N}}/\mathcal{O}_{K})=0$
(for instance, because it injects into
$H^{1}(\operatorname{Gal}(K_{N}/K),\mathcal{O}_{K})=0$). Also
$H^{0}(\operatorname{Gal}(K_{\infty}/K),X_{N})\subset
H^{0}(\operatorname{Gal}(K_{\infty}/K_{N}),X_{N})=0$. Therefore
$H^{0}(\operatorname{Gal}(K_{\infty}/K),X)=0$.

Now consider the $H^{1}$ part of the long exact sequence associated to
(4.2.9), relative to the group $\operatorname{Gal}(K_{\infty}/K_{N})$:

$H^{1}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K_{N}),\mathcal{O}_{K_{N}}/\mathcal{O}_{K})\to
H^{1}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K_{N}),X)\to
H^{1}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K_{N}),X_{N})$
(4.2.10)

The term
$H^{1}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K_{N}),\mathcal{O}_{K_{N}}/\mathcal{O}_{K})$
is isomorphic to $\mathcal{O}_{K_{N}}/\mathcal{O}_{K}$ itself. The terms in
(4.2.10) carry a residual action of $\operatorname{Gal}(K_{N}/K)$; taking
invariants reveals that
$H^{1}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K_{N}),X)^{\operatorname{Gal}(K_{N}/K)}$
injects into
$H^{1}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K_{N}),X_{N})$ and
is therefore $p^{2}$\- or $p^{3}$-torsion.

Finally, consider the inflation-restriction sequence which calculates
$H^{1}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K),X)$:

$H^{1}(\operatorname{Gal}(K_{N}/K),X^{\operatorname{Gal}(K_{\infty}/K_{N})})\to
H^{1}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K),X)\to
H^{1}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K_{N}),X)^{\operatorname{Gal}(K_{N}/K)}$

The term on the left is $p^{N}$-torson (this being the order of
$\operatorname{Gal}(K_{N}/K)$). Therefore the middle term is $p^{N+2}$\- or
$p^{N+3}$-torsion as $p$ is odd or even. ∎

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

Keep the assumption that $K$ is a local field of characteristic $(0,p)$. Let
$\overline{K}$ be an algebraic closure, and let $C$ be the metric completion
of $\overline{K}$. The Galois group
$\Gamma_{K}=\operatorname{Gal}(\overline{K}/K)$ acts continuously on
$\overline{K}$ and so extends to a continuous action on $C$ and on
$\mathcal{O}_{C}$. We are interested in $\mathcal{O}_{C}^{h\Gamma_{K}}$, the
complex which computes the continuous cohomology
$H^{\ast}_{\operatorname{cts}}(\Gamma_{K},\mathcal{O}_{C})$. Tate’s idea is to
use a ramified $\mathbb{Z}_{p}$-extension $K_{\infty}/K$ as an intermediary:

$\mathcal{O}_{C}^{h\Gamma_{K}}\cong\left(\mathcal{O}_{C}^{h\operatorname{Gal}(\overline{K}/K_{\infty})}\right)^{h\operatorname{Gal}(K_{\infty}/K)}.$
(4.3.1)

First we deal with the inner term on the right side of (4.3.1), relating to
the continuous cohomology
$H^{*}_{\operatorname{cts}}(\operatorname{Gal}(\overline{K}/K_{\infty}),\mathcal{O}_{C})$.
Since $\operatorname{Gal}(\overline{K}/K_{\infty})$ acts trivially on
$\mathcal{O}_{\hat{K}_{\infty}}\subset\mathcal{O}_{C}$, this cohomology has
the structure of a $\mathcal{O}_{\hat{K}_{\infty}}$-module. For a module $M$
over a non-discretely valued valuation ring such as
$\mathcal{O}_{\hat{K}_{\infty}}$, one says that $M$ is almost zero if it is
$\alpha$-torsion for any $\alpha\in\mathcal{O}_{\hat{K}_{\infty}}$ with
$v(\alpha)>0$.

######  Theorem 4.3.1.

We have

$H^{i}_{\operatorname{cts}}(\operatorname{Gal}(\overline{K}/K_{\infty}),\mathcal{O}_{C})=\begin{cases}\mathcal{O}_{\hat{K}_{\infty}},&i=0\\\
\text{almost zero},&i>0\end{cases}$

###### Proof.

We only sketch the proof, focusing only on the case $i>0$ (the $i=0$ case
being an instance of Ax’s theorem anyway) and referring the reader to [Tat67,
§3, Corollary 1] for details. We have
$\mathcal{O}_{C}\cong\varprojlim_{n}\varinjlim_{L/K_{\infty}}\mathcal{O}_{L}/p^{n}$,
as $L/K_{\infty}$ ranges over finite Galois extensions. On the level of
continuous cohomology we find:

$H^{i}_{\operatorname{cts}}(\operatorname{Gal}(\overline{K}/K_{\infty}),\mathcal{O}_{C})\cong\varprojlim_{n}\varinjlim_{L/K_{\infty}}H^{i}(\operatorname{Gal}(L/K_{\infty}),\mathcal{O}_{L}/p^{n}\mathcal{O}_{L}),$

so that we are reduced to proving that
$H^{i}(\operatorname{Gal}(L/K_{\infty}),\mathcal{O}_{L}/p^{n}\mathcal{O}_{L})$
is almost zero for every $i>0$.

Therefore let $L/K_{\infty}$ be a finite Galois extension. $L$ descends to
$K_{n}$ for some $n\geq 0$, which is to say $L=L_{n}K_{\infty}$ for some
finite extension $L_{n}/K_{n}$. Let $L_{N}=L_{n}K_{N}$ for $N\geq n$. The
essential phenomenon is that the ramification in $L_{N}/K_{N}$ decays as
$N\to\infty$, in the sense that the valuation of the discriminant
$v(\mathfrak{d}_{L_{N}/K_{N}})$ tends to $0$. Knowing the valuation of the
discriminant in a Galois extension of local fields can be used to control the
image of the ring of integers under the trace map (see once again [Ser79,
Chapter V, §3, Lemma 4]); the result is that for any
$\alpha\in\mathcal{O}_{K_{\infty}}$ with $v(\alpha)>0$, we have
$\alpha=\operatorname{tr}_{L/K_{\infty}}(\beta)$ for some
$\beta\in\mathcal{O}_{L}$.

Finally, for all $i>0$ Tate defines a cup product operation $c\mapsto\beta\cup
c$ from $i$-cochains to $(i-1)$-cochains (on the group
$\operatorname{Gal}(L/K_{\infty})$ and valued in
$\mathcal{O}_{L}/p^{n}\mathcal{O}_{L}$), which has the property that $\alpha
c-d(\beta\cup c)=\beta\cup dc$, where $d$ is the differential. From this
relation one obtains the result that if $c$ is a cocycle, then $\alpha c$ is a
coboundary for all $\alpha\in\mathcal{O}_{K_{\infty}}$ with $v(\alpha)>0$.
This is exactly the statement that
$H^{i}(\operatorname{Gal}(L/K),\mathcal{O}_{L}/p^{n}\mathcal{O}_{L})$ is
almost zero for $i>0$. ∎

######  Remark 4.3.2.

The phenomena appearing in the proof of Theorem 4.3.1 are hallmarks of the
theory of perfectoid algebras [Sch12], of which $K_{\infty}$ (or strictly
speaking its completion) is an example. For such a perfectoid algebra $R$ with
subring of bounded elements $R^{\circ}$, Scholze’s generalization of Faltings’
purity theorem [Sch12, Theorem 7.9] states that whenever $S$ is a finite étale
$R$-algebra, $S^{\circ}$ is almost finite étale over $R^{\circ}$. Applied to
the case of a finite extension $L/K_{\infty}$, one can deduce from this
theorem the essential claim that
$\operatorname{tr}_{L/K_{\infty}}(\mathcal{O}_{L})$ contains the maximal ideal
of $\mathcal{O}_{K}$.

######  Corollary 4.3.3.

Define a complex $\mathcal{Y}_{0}$ of solid
$\mathcal{O}_{\hat{K}_{\infty}}$-modules by the exact triangle

$\mathcal{O}_{\hat{K}_{\infty}}\to\mathcal{O}_{C}^{h\operatorname{Gal}(\overline{K}/K_{\infty})}\to\mathcal{Y}_{0},$
(4.3.2)

where the first morphism is induced from
$\mathcal{O}_{\hat{K}_{\infty}}\hookrightarrow\mathcal{O}_{C}$. Then
$H^{0}(\mathcal{Y}_{0})=0$, and for all $i>0$, $H^{i}(\mathcal{Y}_{0})$ is
almost zero in the sense that the action of any
$\alpha\in\mathcal{O}_{\hat{K}_{\infty}}$ with $v(\alpha)>0$ on it is zero.

###### Proof.

This is simply a translation of Theorem 4.3.1 in terms of complexes. ∎

The complex $\mathcal{Y}_{0}$ in 4.3.3 admits a residual
$\operatorname{Gal}(K_{\infty}/K)$-action, so we may define the derived
invariants $\mathcal{Y}:=\mathcal{Y}_{0}^{h\operatorname{Gal}(K_{\infty}/K)}$,
which fit into an exact triangle:

$\mathcal{O}_{\hat{K}_{\infty}}^{h\operatorname{Gal}(K_{\infty}/K)}\to\mathcal{O}_{C}^{h\Gamma_{K}}\to\mathcal{Y}.$
(4.3.3)

######  Lemma 4.3.4.

We have $H^{0}(\mathcal{Y})=0$, and for all $i\geq 1$, $H^{i}(\mathcal{Y})$ is
$p$-torsion if $p$ is odd, and $p^{2}$-torsion if $p=2$.

###### Proof.

Consider the spectral sequence

$H^{i}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K),H^{j}(\mathcal{Y}_{0}))\implies
H^{i+j}(\mathcal{Y}).$

The left side only has nonzero terms for $i=0,1$. By 4.3.3, we have
$H^{0}(\mathcal{Y}_{0})=0$ and (as an $\mathcal{O}_{K}$-module)
$H^{i}(\mathcal{Y}_{0})$ is $\pi$-torsion for all $i>0$, where $\pi$ is a
uniformizer for $K$. Therefore $H^{i}(\mathcal{Y})$ is $\pi^{2}$-torsion for
all $i>0$. In particular, $H^{i}(\mathcal{Y})$ is $p^{2}$-torsion.

To improve the result as in the lemma, we may assume that $p$ is odd. Let
$L/K$ be a finite Galois extension such that $e_{L/K}\geq 2$ and
$p\nmid\\#\operatorname{Gal}(L/K)$. For instance, $L/K$ could be the splitting
field of $x^{\ell}-\pi$, where $\ell$ is a prime such that
$p\nmid\ell(\ell-1)$. Let $\mathcal{Y}_{L}$ be the complex defined analogously
to $\mathcal{Y}$, using $L_{\infty}=LK_{\infty}$, so that we have an exact
triangle:

$\mathcal{O}_{\hat{L}_{\infty}}^{h\operatorname{Gal}(L_{\infty}/L)}\to\mathcal{O}_{C}^{h\Gamma_{L}}\to\mathcal{Y}_{L}.$
(4.3.4)

Since $K_{\infty}$ and $L$ are linearly disjoint over $K$, we have
$\operatorname{Gal}(L_{\infty}/K)=\operatorname{Gal}(L_{\infty}/L)\times\operatorname{Gal}(L_{\infty}/K_{\infty})$.
Taking derived $\operatorname{Gal}(L/K)$-invariants in (4.3.4) gives an exact
triangle:

$\left(\mathcal{O}_{\hat{L}_{\infty}}^{h\operatorname{Gal}(L_{\infty}/K_{\infty})}\right)^{h\operatorname{Gal}(K_{\infty}/K)}\to\mathcal{O}_{C}^{h\Gamma_{K}}\to\mathcal{Y}_{L}^{h\operatorname{Gal}(L/K)}.$

By Noether’s theorem,
$\mathcal{O}_{\hat{L}_{\infty}}^{h\operatorname{Gal}(L_{\infty}/K_{\infty})}\cong\mathcal{O}_{\hat{K}_{\infty}}$,
and (comparing with the definition of $\mathcal{Y}$) we find
$\mathcal{Y}\cong\mathcal{Y}_{L}^{h\operatorname{Gal}(L/K)}$. Now consider the
spectral sequence

$H^{i}(\operatorname{Gal}(L/K),H^{j}(Y_{L}))\implies
H^{i+j}(Y_{L}^{h\operatorname{Gal}(L/K)})$

Since $\\#\operatorname{Gal}(L/K)$ is invertible in $\mathcal{O}_{K}$, the
left side is only nonzero for $i=0$, and so $H^{i}(Y)\cong
H^{i}(Y_{L})^{\operatorname{Gal}(L/K)}$. We have just seen that this is
$\pi_{L}^{2}$-torsion, where $\pi_{L}$ is a uniformizer for $L$. Since
$e_{L/K}\geq 2$, we find that $H^{i}(Y)$ is $\pi$-torsion as well. ∎

We have a composition of morphisms:

$\mathcal{O}_{K}^{h\operatorname{Gal}(K_{\infty}/K)}\to\mathcal{O}_{\hat{K}_{\infty}}^{h\operatorname{Gal}(K_{\infty}/K)}\to\mathcal{O}_{C}^{h\Gamma_{K}}$
(4.3.5)

where the first is induced by the inclusion
$\mathcal{O}_{K}\to\mathcal{O}_{\hat{K}_{\infty}}$ and the second appears in
(4.3.3).

######  Theorem 4.3.5.

Let $K_{\infty}/K$ be a ramified $\mathbb{Z}_{p}$-extension, and let
$N=N_{K_{\infty}/K}$ be large enough so that $K_{\infty}/K_{N}$ is
sufficiently ramified. Define $\mathcal{Z}$ by the exact triangle

$\mathcal{O}_{K}^{h\operatorname{Gal}(K_{\infty}/K)}\to\mathcal{O}_{C}^{h\Gamma_{K}}\to\mathcal{Z},$

where the first morphism is the composition in (4.3.5). Then:

$H^{i}(\mathcal{Z})=\begin{cases}0,&i=0,\\\ \text{$p^{N+3}$-torsion (resp.,
$p^{N+5}$-torsion)},&i=1\\\ \text{$p$-torsion (resp., $p^{2}$-torsion)},&i\geq
2\end{cases}$

as $p$ is odd or even, respectively.

If $L/K$ is a tame Galois extension and $L_{\infty}=K_{\infty}L$, then
$L_{\infty}/L$ is a ramified $\mathbb{Z}_{p}$-extension, and we can take
$N_{L_{\infty}/L}=N_{K_{\infty}/K}$.

###### Proof.

Consider the three exact triangles:

$\displaystyle\mathcal{O}_{K}^{h\operatorname{Gal}(K_{\infty}/K)}$
$\displaystyle\to$
$\displaystyle\mathcal{O}_{\hat{K}_{\infty}}^{h\operatorname{Gal}(K_{\infty}/K)}$
$\displaystyle\to$ $\displaystyle\mathcal{X}$
$\displaystyle\mathcal{O}_{\hat{K}_{\infty}}^{h\operatorname{Gal}(K_{\infty}/K)}$
$\displaystyle\to$ $\displaystyle\mathcal{O}_{C}^{h\Gamma_{K}}$
$\displaystyle\to$ $\displaystyle\mathcal{Y}$
$\displaystyle\mathcal{O}_{K}^{h\operatorname{Gal}(K_{\infty}/K)}$
$\displaystyle\to$ $\displaystyle\mathcal{O}_{C}^{h\Gamma_{K}}$
$\displaystyle\to$ $\displaystyle\mathcal{Z}$

Here the first triangle is from Proposition 4.2.9 (more precisely, the remark
following it), the second from (4.3.3), and the third is as in the theorem. By
the octahedral axiom, we have an exact triangle
$\mathcal{X}\to\mathcal{Z}\to\mathcal{Y}$. The bounds in the theorem now
follow by combining Proposition 4.2.9 with Lemma 4.3.4.

The statement about tame extensions follows from Lemma 4.2.5. ∎

Finally, we can prove Theorem 4.0.4, repeated here for convenience.

######  Theorem 4.3.6.

Let $K$ be a local field of characteristic $(0,p)$. Let $C$ be the completion
of an algebraic closure $\overline{K}/K$. Let
$\Gamma_{K}=\operatorname{Gal}(\overline{K}/K)$. Then:

  1. (1)

$H^{0}(\Gamma_{K},\mathcal{O}_{C})=\mathcal{O}_{K}.$

  2. (2)

There exists an isomorphism of $\mathcal{O}_{K}$-modules

$H^{1}_{\operatorname{cts}}(\Gamma_{K},\mathcal{O}_{C})\cong\mathcal{O}_{K}\oplus
T$

where $T$ is $p^{4}$-torsion (resp., $p^{6}$-torsion) as $p$ is odd or even,
respectively. If $K$ is tame, or if it is a cyclotomic extension of a tame
field, this bound can be improved to $p^{3}$ (resp., $p^{5}$).

  3. (3)

For $i>1$, $H^{i}_{\operatorname{cts}}(\Gamma_{K},\mathcal{O}_{C})$ is
$p$-torsion (resp., $p^{2}$-torsion) as $p$ is odd or even, respectively.

###### Proof.

Let $K_{\infty}/K$ be a ramified $\mathbb{Z}_{p}$-extension. Taking the long
exact sequence associated to the exact triangle appearing in Theorem 4.3.5, we
obtain $H^{0}(\Gamma_{K},\mathcal{O}_{C})=\mathcal{O}_{K}$, an exact sequence

$0\to\mathcal{O}_{K}\to
H^{1}_{\operatorname{cts}}(\Gamma_{K},\mathcal{O}_{C})\to
H^{1}(\mathcal{Z})\to 0,$

and isomorphisms $H^{i}_{\operatorname{cts}}(\Gamma_{K},\mathcal{O}_{C})\cong
H^{i}(\mathcal{Z})$ for $i\geq 2$. The cases $i=0$ and $i\geq 2$ of the
theorem follow directly from Theorem 4.3.5. When $K$ admits a sufficiently
ramified $\mathbb{Z}_{p}$-extension, which includes the cases listed in part
(2) of the theorem, then $H^{1}(\mathcal{Z})$ is $p^{3}$\- or $p^{5}$-torsion
as $p$ is odd or even, and then Lemma 4.3.7 below applies to show that
$H^{1}_{\operatorname{cts}}(\Gamma_{K},\mathcal{O}_{C})$ has the structure
claimed by the theorem.

In the general case, the above exact sequence shows (again after applying
Lemma 4.3.7) that
$H^{1}_{\operatorname{cts}}(\Gamma_{K},\mathcal{O}_{C})\cong\mathcal{O}_{K}\oplus
T$, where $T$ is $p^{r}$-torsion for some $r$. We claim that in fact we can
take $r=4$ or $r=6$ as $p$ is odd or even.

Let $N$ be large enough so that $K_{\infty}/K_{N}$ is sufficiently ramified.
There is an injective map $\mathcal{O}_{K_{N}}\to
H^{1}_{\operatorname{cts}}(\Gamma_{K_{N}},\mathcal{O}_{C})$ whose cokernel is
killed by $p^{3}$ (resp., $p^{5}$). Let $G=\operatorname{Gal}(K_{N}/K)$. The
inflation-restriction sequence for the tower $K_{\infty}/K_{n}/K$ reads in
part:

$H^{1}(G,\mathcal{O}_{K_{N}})\to
H^{1}_{\operatorname{cts}}(\Gamma_{K},\mathcal{O}_{C})\to
H^{1}_{\operatorname{cts}}(\Gamma_{K_{N}},\mathcal{O}_{C})^{G}$

By Sen’s theorem (Theorem 4.0.2), the term on the left is $p$-torsion.
Therefore $p^{4}$ (resp., $p^{6}$) kills all torsion in
$H^{1}_{\operatorname{cts}}(\Gamma_{K},\mathcal{O}_{C})$. ∎

######  Lemma 4.3.7.

Given an exact sequence of $\mathcal{O}_{K}$-modules

$0\to M_{1}\to M\to M_{2}\to 0$

with $M_{1}$ finite free of rank $r$ and $M_{2}$ killed by $p^{n}$, there
exists an isomorphism $M\cong\mathcal{O}_{K}^{\oplus r}\oplus T$, where $T$ is
$p^{n}$-torsion.

###### Proof.

Let $T\subset M$ be the submodule consisting of torsion elements, and let
$N=M/T$. We may identify $M_{1}$ and $N$ with $\mathcal{O}_{K}$-submodules of
$M\otimes_{\mathcal{O}_{K}}K\cong K^{r}$ in such a way that $M_{1}\subset
N\subset p^{-n}M_{1}$. Since $\mathcal{O}_{K}$ is Noetherian and $M_{1}$ is
finitely generated, $N$ is finitely generated as well, and therefore (since it
is torsion-free and spans $K^{r}$) it must be free of rank $r$. Let
$M_{0}\subset M$ be the free submodule spanned by lifts of generators of $N$.
Then $M=M_{0}\oplus T$. ∎

###  4.4. Galois cohomology of characters

Once again suppose $K$ is a local field of characteristic $(0,p)$. Let

$\chi\colon\operatorname{Gal}(\overline{K}/K)\to\mathbb{Z}_{p}^{\times}$

be a character; i.e., a continuous homomorphism. We assume that $\chi$ is
infinitely ramified, in the sense that the image of the inertia group under
$\chi$ is infinite (equivalently, this image has finite index). Let
$K_{\infty}$ be the fixed field of the kernel of $\chi$. Let
$\mathbb{Z}_{p}(\chi)$ be $\mathbb{Z}_{p}$ with an action of
$\operatorname{Gal}(\overline{K}/K)$ through $\chi$. Then if $M$ is any
$p$-adically complete $\operatorname{Gal}(\overline{K}/K)$-module, we may
define a new such module by
$M(\chi)=M\otimes_{\mathbb{Z}_{p}}\mathbb{Z}_{p}(\chi)$. The present goal is
to bound the continuous cohomology of $\operatorname{Gal}(K_{\infty}/K)$
acting on $\mathcal{O}_{\hat{K}_{\infty}}(\chi)$.

Let $U\subset\mathbb{Z}_{p}^{\times}$ be the largest subgroup which is
isomorphic to $\mathbb{Z}_{p}$. Thus $U=1+p\mathbb{Z}_{p}$ for $p$ odd and
$U=1+p^{2}\mathbb{Z}_{p}$ for $p=2$. Let $K_{0}$ be the fixed field of
$\chi^{-1}(U)$; then $K_{\infty}/K_{0}$ is a ramified
$\mathbb{Z}_{p}$-extension. As usual, we let $K_{n}/K_{0}$ be the fixed field
of $p^{n}\operatorname{Gal}(K_{\infty}/K_{0})$. Finally, we define an integer
$r\geq 1$ by

$\chi(\operatorname{Gal}(K_{\infty}/K_{0}))=1+p^{r}\mathbb{Z}_{p}.$

######  Lemma 4.4.1.

Let $\chi$ be a non-trivial character and $N\geq 0$ be large enough so that
$K_{\infty}/K_{N}$ is sufficiently ramified (see Lemma 4.2.5). Then

$H^{i}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K_{0}),\mathcal{O}_{\hat{K}_{\infty}}(\chi))=\begin{cases}0,&i=0\\\
(\text{$p^{k}$-torsion}),&i=1.\end{cases}$

Here we may take $k=N+r+1$ if $p$ is odd and $k=N+r+2$ if $p=2$.

###### Proof.

We first prove the result in the case that $K_{\infty}/K_{0}$ is sufficiently
ramified. Write $t\colon\hat{K}_{\infty}\to K_{0}$ for the normalized trace.
As in the proof of Proposition 4.2.9, we make use of the fact that the
inclusion

$\mathcal{O}_{K_{0}}\oplus\mathcal{O}_{\hat{K}_{\infty}}^{t=0}\hookrightarrow\mathcal{O}_{\hat{K}_{\infty}}$

has $\alpha$-torsion cokernel for any $\alpha\in\mathcal{O}_{K_{N}}$ with
$v(\alpha)\geq 1/(p-1)$.

We are therefore reduced to studying the continuous cohomology of
$\operatorname{Gal}(K_{\infty}/K_{0})$ acting on $\mathcal{O}_{K_{0}}(\chi)$
and $\mathcal{O}_{\hat{K}_{\infty}}^{t=0}(\chi)$, respectively. Let
$\sigma\in\operatorname{Gal}(K_{\infty}/K_{0})$ be a topological generator,
and let $\lambda=\chi(\sigma)^{-1}$. Then for any $p$-adically complete
$\operatorname{Gal}(K_{\infty}/K_{0})$-module $M$, the continuous cohomology
of $M(\chi)$ is computed by the complex $\sigma-\lambda\colon M\to M$. Note
that $\lambda=1+p^{r}u$ for a $p$-adic unit $u\in\mathbb{Z}_{p}^{\times}$.

In the case $M=\mathcal{O}_{K_{0}}$, we have
$H^{0}(\operatorname{Gal}(K_{\infty}/K_{0}),\mathcal{O}_{K_{0}}(\chi))=0$,
since $\sigma$ acts on $\mathcal{O}_{K_{0}}(\chi)$ as the scalar $\lambda\neq
1$. On the other hand
$H^{1}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K_{0}),\mathcal{O}_{K_{0}}(\chi))$
is $p^{r}$-torsion, being the cokernel of multiplication by
$\lambda-1=p^{r}u$.

To treat $M=\mathcal{O}_{\hat{K}_{\infty}}^{t=0}$, we apply Lemma 4.2.8 to the
sufficiently ramified extension $K_{\infty}/K_{0}$. We find that
$(\sigma-1)^{-1}$ is defined on $\hat{K}_{\infty}^{t=0}$ and has operator norm
$\leq\left\lvert p\right\rvert^{-1-1/p(p-1)}$.

Let

$\mu=(\sigma-1)^{-1}(\sigma-\lambda)=1-(\lambda-1)(\sigma-1)^{-1}.$

In the case that $r\geq 2$, we have
$\left\lvert\lambda-1\right\rvert\left\lvert\sigma-1\right\rvert^{-1}<1$, and
so $\mu$ has continuous inverse satisfying
$\left\lvert\mu^{-1}\right\rvert\leq 1$. Therefore
$(\sigma-\lambda)^{-1}=(\sigma-1)^{-1}\mu^{-1}$ exists on
$\hat{K}_{\infty}^{t=0}$ and has operator norm bounded by $\left\lvert
p\right\rvert^{-1-1/p(p-1)}$. Thus
$H^{0}(\operatorname{Gal}(K_{\infty}/K_{0}),\mathcal{O}_{\hat{K}_{\infty}}^{t=0}(\chi))=0$
and
$H^{1}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K_{0}),\mathcal{O}_{\hat{K_{\infty}}}^{t=0}(\chi))$
is annihilated by any element $\alpha\in\mathcal{O}_{K_{0}}$ with
$v(\alpha)\geq 1+1/p(p-1)$.

If $r=1$, the idea is to apply the same argument to the sufficiently ramified
extension $K_{\infty}/K_{1}$, noting that $\lambda^{p}=\chi(\sigma^{p})$ now
satisfies
$\left\lvert\lambda^{p}-1\right\rvert\left\lvert\sigma^{p}-1\right\rvert^{-1}<1$.
Thus
$H^{i}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K_{1}),\mathcal{O}_{\hat{K_{\infty}}}^{t=0}(\chi))$
is 0 for $i=0$ and is $p^{1+1/p(p-1)}$-torsion for $i=1$. The inflation-
restriction sequence allows us to deduce the same results for
$K_{\infty}/K_{0}$.

We have found that
$H^{0}(\operatorname{Gal}(K_{\infty}/K_{0}),\mathcal{O}_{\hat{K}_{\infty}}(\chi))=0$
and
$H^{1}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K_{0}),\mathcal{O}_{\hat{K}_{\infty}}(\chi))$
is $\alpha$-torsion for any $\alpha\in\mathcal{O}_{K_{0}}$ with
$v(\alpha)\geq\max\left\\{r,1+1/p(p-1)\right\\}+1/(p-1)$. The latter is $\geq
r+1$ or $\geq r+2$ as $p$ is odd or even, respectively, which implies the
lemma in the case that $K_{\infty}/K_{0}$ is sufficiently ramified. (To handle
the problem that $\mathcal{O}_{K_{0}}$ may not contain elements of
sufficiently small valuation, it may be necessary to pass to a tamely ramified
extension $L/K_{0}$ as in the proof of Proposition 4.2.9.)

In general, suppose $N\geq 0$ is large enough so that $K_{\infty}/K_{N}$ is
sufficientlty ramified. Note that $\operatorname{Gal}(K_{\infty}/K_{N})$ is
generated by $\sigma^{p^{N}}$ and that $\chi(\sigma^{p^{N}})\equiv
1\pmod{p^{N+r}}$. Thus the preceding argument shows that
$H^{0}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K_{N}),\mathcal{O}_{\hat{K}_{\infty}}(\chi))=0$
and
$H^{1}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K_{N}),\mathcal{O}_{\hat{K}_{\infty}}(\chi))$
is $p^{N+r+1}$\- or $p^{N+r+2}$-torsion as $p$ is odd or even, respectively.
The inflation-restriction sequence for the tower $K_{\infty}/K_{N}/K_{0}$
reads:

$\displaystyle
H^{1}(\operatorname{Gal}(K_{N}/K_{0}),\mathcal{O}_{\hat{K}_{\infty}}(\chi)^{\operatorname{Gal}(K_{\infty}/K_{N})})$
$\displaystyle\to$ $\displaystyle
H^{1}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K_{0}),\mathcal{O}_{\hat{K}_{\infty}}(\chi)^{\operatorname{Gal}(K_{\infty}/K_{N})})$
$\displaystyle\to$ $\displaystyle
H^{1}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K_{N}),\mathcal{O}_{\hat{K}_{\infty}})^{\operatorname{Gal}(K_{N}/K_{0})}$

By the above argument, the term on the left is zero, and we get the result. ∎

######  Lemma 4.4.2.

Assume that $K_{\infty}/K_{N}$ is sufficiently ramified. Let $a_{p}=0$ if $p$
is odd, and $a_{p}=1$ if $p=2$. We have:

$H^{i}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K),\mathcal{O}_{\hat{K}_{\infty}}(\chi))=\begin{cases}0,&i=0,\\\
\text{\em{($p^{N+r+1+a_{p}}$-torsion)}},&i=1,\\\
\text{\em{($p^{a_{p}}$-torsion)}},&i>1.\end{cases}$

###### Proof.

Combine the spectral sequence

$H^{i}(\operatorname{Gal}(K_{0}/K),H^{j}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K_{0}),\mathcal{O}_{\hat{K}_{\infty}}(\chi)))\implies
H^{i+j}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K),\mathcal{O}_{\hat{K}_{\infty}}(\chi))$

with Lemma 4.4.1. The only terms on the left that contribute occur when $j=1$.
If $p$ is odd, the $i>0$ terms on the left side vanish because
$\\#\operatorname{Gal}(K_{0}/K)$ is invertible in $\mathcal{O}_{K}$. If $p=2$
then $\\#\operatorname{Gal}(K_{0}/K)=2$, and the best we can say is that the
$i>0$ terms on the left side are $2$-torsion. ∎

At this point we specialize to the case of the cyclotomic character

$\chi_{\text{cycl}}\colon\Gamma_{K}\to\mathbb{Z}_{p}^{\times},$

defined by the relation $\tau(\zeta)=\zeta^{\chi_{\text{cycl}}(\tau)}$ for any
$\tau\in\Gamma_{K}$ and any $p$th power root of unity $\zeta$. For any
$j\in\mathbb{Z}$ we write
$\mathbb{Z}_{p}(j)=\mathbb{Z}_{p}(\chi_{\text{cycl}}^{j})$ for the $j$th Tate
twist, and similarly
$\mathcal{O}_{C}(j)=\mathcal{O}_{C}\otimes\mathbb{Z}_{p}(j)$.

We conclude the section with a bound on the cohomology of
$\mathcal{O}_{C}(j)$.

######  Theorem 4.4.3.

With notation as in Theorem 4.0.4, let $j\neq 0$ be an integer, and let
$\mathcal{O}_{C}(j)$ be the $j$th Tate twist of $\mathcal{O}_{C}$ as a
$\operatorname{Gal}(\overline{K}/K)$-module. Then:

  1. (1)

$H^{0}(\Gamma_{K},\mathcal{O}_{C}(j))=0$.

  2. (2)

$H^{1}_{\operatorname{cts}}(\Gamma_{K},\mathcal{O}_{C}(j))$ is
$p^{M+v(j)}$-torsion. Here $M=M_{K}$ is a constant which only depends on $K$
and which is insensitive to passage to a tamely ramified extension of $K$. If
$p\nmid e_{K}$ we may take $M=2$ if $p$ is odd and $M=5$ if $p=2$.

  3. (3)

For $i\geq 2$, $H^{i}_{\operatorname{cts}}(\Gamma_{K},\mathcal{O}_{C}(j))$ is
$p$-torsion if $p$ is odd and $p^{3}$-torsion if $p=2$.

(We recall that $v(j)$ the $p$-adic valuation of $j$, normalized so that
$v(p)=1$.)

###### Proof.

Let $K_{\infty}/K$ be the extension obtained by adjoining all $p$th power
roots of unity. As in the beginning of this subsection, we let $K_{0}/K$ be
the minimal subextension such that $K_{\infty}/K_{0}$ is a
$\mathbb{Z}_{p}$-extension, and we define an integer $r\geq 1$ by
$\chi_{\text{cycl}}(\operatorname{Gal}(K_{\infty}/K_{0}))=1+p^{r}\mathbb{Z}_{p}$.
Then

$\chi^{j}_{\text{cycl}}(\operatorname{Gal}(K_{\infty}/K_{0}))=(1+p^{r}\mathbb{Z}_{p})^{j}=1+p^{r+v(j)}\mathbb{Z}_{p}.$

Lemma 4.4.2 bounds the cohomology
$H^{i}_{\operatorname{cts}}(\operatorname{Gal}(K_{\infty}/K),\mathcal{O}_{\hat{K}_{\infty}}(j))$
in terms of the integer $N=N_{K}$ for which $K_{\infty}/K_{N}$ is infinitely
ramified: we have $H^{0}=0$, $H^{1}$ is $p^{N+r+v(j)+a_{p}}$-torsion, and
$H^{2}$ is $p^{a_{p}}$-torsion, where $a_{p}$ is 0 or 1 as $p$ is odd or even.

As in Lemma 4.3.4, define a complex $\mathcal{Y}_{0}$ by the exact triangle

$\mathcal{O}_{\hat{K}_{\infty}}\to\mathcal{O}_{C}^{h\operatorname{Gal}(\overline{K}/K_{\infty})}\to\mathcal{Y}_{0}.$

Then $H^{0}(\mathcal{Y}_{0})=0$ and $H^{i}(\mathcal{Y}_{0})$ is almost zero
for $i>0$. Twisting by $\chi_{\text{cycl}}^{j}$ and taking derived
$\operatorname{Gal}(K_{\infty}/K)$-invariants, we obtain an exact triangle

$\mathcal{O}_{\hat{K}_{\infty}}(j)^{h\operatorname{Gal}(K_{\infty}/K)}\to\mathcal{O}_{C}(j)^{h\Gamma_{K}}\to\mathcal{Y},$

where $\mathcal{Y}=\mathcal{Y}_{0}(j)^{h\operatorname{Gal}(K_{\infty}/K)}$.
The same method of proof for Lemma 4.3.4 shows that $H^{0}(\mathcal{Y})=0$ and
$H^{i}(\mathcal{Y})$ is $p^{a_{p}+1}$-torsion for $i>0$. Combining these
bounds with those obtained in Lemma 4.4.2, we obtain the bounds appearing in
the theorem. ∎

##  5\. Pro-étale cohomology of rigid-analytic spaces

Let $K$ be a local field of characteristic $(0,p)$, and let $X$ be a smooth
rigid-analytic space over $K$. We consider three topologies on $X$ in order of
coarsest to finest:

  1. (1)

The analytic topology $X_{\mathrm{an}}$, whose opens are simply the open
subsets of the underlying topological space of $X$,

  2. (2)

The étale topology $X_{\mathrm{\acute{e}t}}$, whose opens are étale morphisms
$U\to X$,

  3. (3)

The pro-étale topology $X_{\mathrm{pro\acute{e}t}}$, whose opens are formal
limits $U=\varprojlim U_{i}$, where each $U_{i}\to X$ is étale.

In this section, we present some comparison results relating the cohomology of
the structure sheaf among these sites. For the purposes of exposition, we
might start with the case of the rational structure sheaf $\mathcal{O}_{X}$.
Recall from Section 3 the notation $\hat{\mathcal{O}}_{X}$ for the completed
structure sheaf on $X_{\mathrm{pro\acute{e}t}}$.

The case that $X=\operatorname{Spa}K$ is a single point is instructive. In
this case $H^{*}(X_{\mathrm{an}},\mathcal{O}_{X})=K$ rather trivially, and
$H^{*}(X_{\mathrm{\acute{e}t}},\mathcal{O}_{X})$ is the Galois cohomology
$H^{*}_{\operatorname{cts}}(\operatorname{Gal}(\overline{K}/K),\overline{K})$,
where one must put the discrete topology on $\overline{K}$; by the normal
basis theorem this is $K$ again. By contrast,
$H^{*}(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}_{X})$ may be identified
with $H^{*}_{\operatorname{cts}}(\operatorname{Gal}(\overline{K}/K),C)$, where
$C$ is the $p$-adic completion of $\overline{K}$ (with its $p$-adic topology).
Tate’s theorem (Theorem 4.0.3) states that this is isomorphic to
$K[\varepsilon]$.

In the general case where $X$ is a smooth rigid-analytic variety over $K$, it
is still the case that $H^{*}(X_{\mathrm{an}},\mathcal{O}_{X})$ is isomorphic
to $H^{*}(X_{\mathrm{\acute{e}t}},\mathcal{O}_{X})$ by étale descent, but of
course $H^{*}(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}_{X})$ may be
different. The general pattern we observed is that the difference is accounted
for entirely by the pro-étale cohomology of the base $\operatorname{Spa}K$. In
the next subsection we make this explicit for the rational structure sheaf
before passing to the study of the integral structure sheaf in subsequent
subsections.

###  5.1. The rational comparison isomorphism

The following is a rapid discussion of the sort of comparison result we need,
in the much simpler situation where $p$ has been inverted. We leave precise
details to the subsections that follow.

Let $C$ be an algebraically closed nonarchimedean field of characteristic
$(0,p)$, and let $X$ be a smooth rigid-analytic space over $C$. We recall here
a theorem of Scholze, which relates the difference between
$X_{\mathrm{\acute{e}t}}$ and $X_{\mathrm{pro\acute{e}t}}$ to differential
forms on $X$. Let us write $\Omega^{j}_{X/C}$ for the sheaf of differential
$j$-forms, considered as a sheaf of $\mathcal{O}_{X}$-modules on
$X_{\mathrm{\acute{e}t}}$. Also we write

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

for the projection map between sites.

######  Theorem 5.1.1 ([Sch13b, Proposition 3.23]).

Let $X$ be a smooth rigid-analytic space over $C$. Let $\nu\colon
X_{\mathrm{pro\acute{e}t}}\to X_{\mathrm{\acute{e}t}}$ be the projection. Then
for each $j\geq 0$ there is an isomorphism of
$\mathcal{O}_{X_{\mathrm{\acute{e}t}}}$-modules:

$\Omega^{j}_{X_{\mathrm{\acute{e}t}}/C}(-j)\cong
R^{j}\nu_{*}\hat{\mathcal{O}}_{X}.$

The essential calculation behind Theorem 5.1.1 goes back to Faltings [Fal88].
The assumption that $X$ is smooth means that (locally on
$X_{\mathrm{\acute{e}t}}$) it admits an étale morphism to the $d$-dimensional
torus

$\displaystyle\mathbb{T}^{d}$ $\displaystyle=$
$\displaystyle\operatorname{Spa}(R_{d},R^{+}_{d})$ $\displaystyle R^{+}_{d}$
$\displaystyle=$
$\displaystyle\mathcal{O}_{C}\left<T_{1}^{\pm},\dots,T_{d}^{\pm 1}\right>$
$\displaystyle R_{d}$ $\displaystyle=$ $\displaystyle R_{d}^{+}[1/p].$

Therefore let us explain the proof of Theorem 5.1.1 in the case of
$X=\mathbb{T}^{d}$. There is an affinoid perfectoid pro-étale torsor
$\tilde{\mathbb{T}}^{d}\to\mathbb{T}^{d}$ for the group
$\mathbb{Z}_{p}(1)^{d}$, namely

$\displaystyle\tilde{\mathbb{T}}^{d}$ $\displaystyle=$
$\displaystyle\operatorname{Spa}(\tilde{R}_{d},\tilde{R}_{d}^{+})$
$\displaystyle\tilde{R}_{d}^{+}$ $\displaystyle=$
$\displaystyle\mathcal{O}_{C}\left<T_{1}^{\pm 1/p^{\infty}},\dots,T_{d}^{\pm
1/p^{\infty}}\right>$ $\displaystyle\tilde{R}_{d}$ $\displaystyle=$
$\displaystyle\tilde{R}_{d}^{+}[1/p].$

(Here $\mathbb{Z}_{p}(1)=\varprojlim\mu_{p^{n}}$, where $\mu_{p^{n}}$ is the
group of $p^{n}$th roots of 1.) Affinoid perfectoid covers in
$X_{\mathrm{pro\acute{e}t}}$ are convenient because if $U$ is affinoid
perfectoid, then $H^{i}(U_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}_{U})=0$
is for $i>0$ [Sch13a, Lemma 4.10]. As in the proof of Proposition 3.6.2, this
allows us to compute the pro-étale cohomology of $X$ in terms of group
cohomology:

$H^{*}(\mathbb{T}^{d}_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}_{\mathbb{T}^{d}})\cong
H^{*}_{\operatorname{cts}}(\mathbb{Z}_{p}(1)^{d},\tilde{R}_{d})$

But now an explicit calculation in group cohomology shows that the natural map

$H^{*}_{\operatorname{cts}}(\mathbb{Z}_{p}(1)^{d},R_{d})\to
H^{*}_{\operatorname{cts}}(\mathbb{Z}_{p}(1)^{d},\tilde{R}_{d})$

is an isomorphism. Note that
$H^{i}(\mathbb{Z}_{p}(1)^{d},R_{d})\cong\bigwedge^{i}_{R_{d}}(R_{d})^{d}$.

The upshot is that
$R^{i}\nu_{*}\hat{\mathcal{O}}_{X}\cong\bigwedge^{i}R^{1}\nu_{*}\hat{\mathcal{O}}_{X}$,
and $R^{1}\nu_{*}\hat{\mathcal{O}}_{X}$ is a locally free
$\mathcal{O}_{X_{\mathrm{\acute{e}t}}}$-module of rank $d$. This already
suggests that $R^{j}\nu_{*}\hat{\mathcal{O}}_{X}$ should be related to
differentials. We refer the reader to [Sch13b, Lemma 3.24] for a functorial
construction of the isomorphism in Theorem 5.1.1.

Now suppose once again that $X$ is a rigid-analytic space over a local field
$K$. Write $C$ for the completion of an algebraic closure $\overline{K}$, and
write $\Gamma_{K}=\operatorname{Gal}(\overline{K}/K)$. Let $\overline{X}$ be
the base change of $X$ to $C$. Once again we write $\nu\colon
X_{\mathrm{pro\acute{e}t}}\to X_{\mathrm{\acute{e}t}}$ for the projection; let
$\overline{\nu}\colon\overline{X}_{\mathrm{pro\acute{e}t}}\to\overline{X}_{\mathrm{\acute{e}t}}$
be the corresponding projection for $\overline{X}$. We have

$R\nu_{*}\hat{\mathcal{O}}_{X}=R\nu_{*}(\hat{\mathcal{O}}_{\overline{X}}^{h\Gamma_{K}})=(R(\overline{\nu})_{*}\hat{\mathcal{O}}_{\overline{X}})^{h\Gamma_{K}}.$

Applying Theorem 5.1.1, we find that
$(R(\overline{\nu})_{*}\hat{\mathcal{O}}_{\overline{X}})^{h\Gamma_{K}}$ admits
a filtration with graded pieces
$\Omega_{\overline{X}/C}^{j}(-j)^{h\Gamma_{K}}$. For an affinoid object $U\in
X_{\mathrm{\acute{e}t}}$, the global sections of the $j$th piece on $U$ are

$\displaystyle\Gamma(U,\Omega_{\overline{X}/C}^{j}(-j)^{h\Gamma_{K}})$
$\displaystyle\cong$
$\displaystyle(\Gamma(U,\Omega_{U/K}^{j})\hat{\otimes}_{K}C(-j))^{h\Gamma_{K}}$
$\displaystyle\cong$
$\displaystyle\Gamma(U,\Omega_{U/K}^{j})\hat{\otimes}_{K}C(-j)^{h\Gamma_{K}}.$

(Here the $\hat{\otimes}$ means completed tensor product of Banach
$K$-algebras.) By Theorem 4.0.3, the terms with nonzero $j$ vanish, and the
$j=0$ term is isomorphic to
$\Gamma(U,\mathcal{O}_{X_{\mathrm{\acute{e}t}}})[\varepsilon]$. (We remind the
reader that for an abelian group $Y$, $Y[\varepsilon]$ is the complex $Y\oplus
Y[1]$.) Therefore

$R\nu_{*}\mathcal{O}_{X}\cong\mathcal{O}_{X_{\mathrm{\acute{e}t}}}[\varepsilon].$

We have proved:

######  Theorem 5.1.2.

Let $K$ be a discretely valued nonarchimedean field of characteristic 0 with
perfect residue field. Let $X/K$ be a smooth rigid-analytic space. There is an
isomorphism

$R\Gamma(X_{\mathrm{\acute{e}t}},\mathcal{O}_{X})[\varepsilon]\cong
R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}_{X}).$

###  5.2. Integral $p$-adic Hodge theory

Suppose again that $C$ is an algebraically closed nonarchimedean field of
characteristic $(0,p)$, and that $X$ is a smooth rigid-analytic space over
$C$. When $X$ has a sufficiently nice formal model over $\mathcal{O}_{C}$, the
theorems of [BMS18] and [vK19] can be used to gain control over the integral
pro-étale cohomology
$R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}_{X}^{+})$.

######  Definition 5.2.1 ([vK19]).

Let $K$ be a nonarchimedean field, and let $\mathfrak{X}$ be a formal scheme
over $\operatorname{Spf}\mathcal{O}_{K}$. We say that $\mathfrak{X}$ is
semistable of dimension $d$ if $\mathfrak{X}$ can be covered by affine opens
$\mathfrak{U}$ which admit an étale $\mathcal{O}_{K}$-morphism to a formal
scheme of the form

$\operatorname{Spf}\mathcal{O}_{K}\left<T_{0},\dots,T_{r},T_{r+1}^{\pm},\dots,T_{d}^{\pm}\right>/(T_{0}\cdots
T_{r}-\pi)$ (5.2.1)

where $0\leq r\leq d$, and $\pi\in\mathcal{O}_{K}$ is a nonunit. (The values
of $r$ and $\pi$ may vary with $\mathfrak{U}$.)

Then $\mathfrak{X}$ carries a log structure associated to the subpresheaf
$\mathcal{M}=\mathcal{O}_{\mathfrak{X}_{\mathrm{\acute{e}t}}}\cap\mathcal{O}_{\mathfrak{X}_{\mathrm{\acute{e}t}}}[1/p]^{\times}$
of $\mathcal{O}_{\mathfrak{X}_{\mathrm{\acute{e}t}}}$. Let
$\Omega^{1}_{\mathfrak{X}_{\mathrm{\acute{e}t}},\log}$ be the sheaf of log-
differentials on $\mathfrak{X}_{\mathrm{\acute{e}t}}$; that is, the sheaf
generated by Kähler differentials
$\Omega^{1}_{\mathfrak{X}_{\mathrm{\acute{e}t}}}$ together with logarithmic
differentials $df/f$ for $f\in\mathcal{M}$. (We refer to continuous
differentials throughout; the right way to construct
$\Omega^{1}_{\mathfrak{X}_{\mathrm{\acute{e}t}},\log}$ is to do it over
$\mathcal{O}_{C}/p^{n}$ and then take a limit over $n$.) The formal scheme
$\mathfrak{X}$ is log-smooth, and
$\Omega^{1}_{\mathfrak{X}_{\mathrm{\acute{e}t}},\log}$ is a locally free
$\mathcal{O}_{\mathfrak{X}_{\mathrm{\acute{e}t}}}$-module of rank $d$. (See
[Kat89, §3] for proofs of these claims.) Finally, let
$\Omega^{j}_{\mathfrak{X}_{\mathrm{\acute{e}t}},\log}=\bigwedge^{j}\Omega^{1}_{\mathfrak{X}_{\mathrm{\acute{e}t}},\log}.$

If $\mathfrak{X}$ is affine and semistable, then its adic generic fiber
$X=\mathfrak{X}^{\operatorname{ad}}_{C}$ is a smooth rigid-analytic variety.

######  Lemma 5.2.2.

Let $\mathfrak{X}$ be an affine semistable formal scheme, with generic fiber
$X$. The natural map $\mathcal{O}(\mathfrak{X})\to\mathcal{O}^{+}(X)$ is an
isomorphism.

###### Proof.

Following the recipe for calculating the generic fiber of
$\mathfrak{X}\to\operatorname{Spa}(\mathcal{O}_{K},\mathcal{O}_{K})$ (see
[SW20, Proposition 5.1.5], noting that this map is “analytic”), we find that
$\mathcal{O}^{+}(X)$ is the integral closure of $\mathcal{O}(\mathfrak{X})$ in
$\mathcal{O}(\mathfrak{X})\otimes_{\mathcal{O}_{K}}K$. The statement is étale
local on $\mathfrak{X}$, so we may assumine that $\mathfrak{X}$ has the form
(5.2.1), in which case the result is true because the coordinate ring is
normal. ∎

If $\mathfrak{U}\in\mathfrak{X}_{\mathrm{\acute{e}t}}$, then the adic generic
fiber of $\mathfrak{U}$ is an object of $X_{\mathrm{pro\acute{e}t}}$, which is
to say there is a morphism of sites:

$\nu\colon X_{\mathrm{pro\acute{e}t}}\to\mathfrak{X}_{\mathrm{\acute{e}t}}.$

Assume now that the base field is an algebraically closed nonarchimedean field
$C$ of characteristic $(0,p)$. The integral version of Theorem 5.1.1 relates
the difference between $X_{\mathrm{pro\acute{e}t}}$ and
$\mathfrak{X}_{\mathrm{\acute{e}t}}$ in terms of the sheaf of differentials
$\Omega^{j}_{\mathfrak{X}_{\mathrm{\acute{e}t}},\log}$. To make this work
there are two additional ingredients: the Breuil–Kisin twist and the décalage
functor.

The Breuil–Kisin twist $\mathcal{O}_{C}\left\\{1\right\\}$ is a free
$\mathcal{O}_{C}$-module of rank 1 carrying a
$\operatorname{Gal}(\overline{K}/K)$-action. See [BMS18, Definition 8.2] for
its precise definition. There is a canonical Galois-equivariant injection
$\mathcal{O}_{C}(1)\hookrightarrow\mathcal{O}_{C}\left\\{1\right\\}$ whose
cokernel is killed by $(\zeta_{p}-1)$, where $\zeta_{p}$ is a primitive $p$th
root of 1. For $j\in\mathbb{Z}$ we let $\mathcal{O}_{C}\left\\{j\right\\}$ be
the $j$th tensor power of $\mathcal{O}_{C}\left\\{1\right\\}$.

The décalage functor will be reviewed in Section 5.5. In our context it
appears as an endofunctor $L\eta_{(\zeta_{p}-1)}$ on the derived category of
$\mathcal{O}_{\mathfrak{X}_{\mathcal{O}_{C},\mathrm{\acute{e}t}}}$-modules.
For the moment we need two facts concerning $L\eta_{(\zeta_{p}-1)}$: it is a
lax symmetric monoidal functor, and also there is a natural transformation

$a\colon L\eta_{(\zeta_{p}-1)}\mathcal{C}\to\mathcal{C}$ (5.2.2)

whenever $\mathcal{C}$ is bounded below by 0. Let

$\widetilde{\Omega}_{\mathfrak{X}}=L\eta_{(\zeta_{p}-1)}R\nu_{\ast}\hat{\mathcal{O}}_{X}^{+},$

so that $\widetilde{\Omega}_{\mathfrak{X}}$ is a complex of
$\mathcal{O}_{\mathfrak{X}_{\mathrm{\acute{e}t}}}$-modules.

The following theorem is the basis for all the comparison isomorphisms that
follow.

######  Theorem 5.2.3 ([BMS18, Theorem 8.3],[vK19, Theorem 4.11]).

Let $\mathfrak{X}$ be a semistable affine formal scheme over
$\mathcal{O}_{C}$. For each $j\geq 0$ there is a canonical isomorphism of
sheaves of $\mathcal{O}_{\mathfrak{X}_{\mathrm{\acute{e}t}}}$-modules:

$\Omega^{j}_{\mathfrak{X}_{\mathrm{\acute{e}t}},\log}\left\\{-j\right\\}\cong
H^{j}(\widetilde{\Omega}_{\mathfrak{X}}).$

Now assume that $\mathfrak{X}$ is a semistable affine formal scheme over
$\mathcal{O}_{K}$ and denote by $\overline{\mathfrak{X}}$ the base change to
$\mathcal{O}_{C}$. We have

$\Omega^{j}_{\mathfrak{X}_{\mathcal{O}_{C},\mathrm{\acute{e}t}},\log}\cong\Omega^{j}_{\mathfrak{X}_{\mathrm{\acute{e}t}},\log}{\hat{\otimes}}_{\mathcal{O}_{K}}\mathcal{O}_{C}$

(where $\hat{\otimes}$ means $p$-adically completed tensor product). The
theorem above can be applied to $\mathfrak{X}_{\mathcal{O}_{C}}$. The group
$\Gamma_{K}$ acts on
$\Omega^{j}_{\mathfrak{X}_{\mathcal{O}_{C},\mathrm{\acute{e}t}}}$ and
$\tilde{\Omega}$. (Note that $\Gamma_{K}$ preserves the ideal $(\zeta_{p}-1)$
and so its action commutes with $L\eta_{(\zeta_{p}-1)}$.)

We get the following corollary of the theorem:

######  Corollary 5.2.4.

Let $\mathfrak{X}$ is a semistable affine formal scheme over
$\mathcal{O}_{K}$. For each $j\geq 0$, there is a canonical
$\Gamma_{K}$-equivariant isomorphism

$\Omega^{j}_{\mathfrak{X}_{\mathrm{\acute{e}t}},\log}\hat{\otimes}_{\mathcal{O}_{K}}\mathcal{O}_{C}\left\\{-j\right\\}\cong
H^{j}(\widetilde{\Omega}_{\mathfrak{X}_{\mathcal{O}_{C}}}).$

###  5.3. The integral comparison isomorphism for affine semistable formal
schemes

The idea now is to present integral versions of the comparison isomorphism in
Theorem 5.1.2. The first version applies to the setting where $K$ is a local
field of characteristic $(0,p)$, and $X$ is a rigid-analytic space over $K$
admitting an affine semistable model $\mathfrak{X}$ over $\mathcal{O}_{K}$. We
will write $\mathcal{O}$ for the structure sheaf of
$\mathfrak{X}_{\mathrm{\acute{e}t}}$, and $\hat{\mathcal{O}}^{+}$ for the
completed integral structure sheaf on $X_{\mathrm{pro\acute{e}t}}$.

Consider the projection $\nu\colon X_{\mathrm{pro\acute{e}t}}\to\mathfrak{X}$.
It pulls back functions to integral functions, which is to say we have an
$\mathcal{O}_{K}$-algebra homomorphism
$\mathcal{O}\to\nu_{*}\hat{\mathcal{O}}^{+}$. Taking derived global sections
and noting that $\mathcal{O}$ is acyclic (since $\mathfrak{X}$ is affine), we
obtain a homomorphism of $\mathcal{O}_{K}$-algebra objects in
$D(\operatorname{Ab})$:

$\nu^{*}\colon\mathcal{O}(\mathfrak{X})\to
R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})$

Similarly, the structure morphism $s\colon X\to\operatorname{Spa}K$ induces a
pullback map on the level of cohomology (here we confuse $K$ with
$\operatorname{Spa}K$):

$s^{*}\colon R\Gamma(K_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})\to
R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})$

In Example 3.6.3 we have identified
$R\Gamma(K_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})$ with the Galois
cohomology
$R\Gamma_{\operatorname{cts}}(\Gamma_{K},\mathcal{O}_{C})=R\Gamma_{*}\mathcal{O}_{C}^{h\Gamma_{K}}$,
where $\Gamma_{K}=\operatorname{Gal}(\overline{K}/K)$.

Let $K_{\infty}/K$ be a ramified $\mathbb{Z}_{p}$-extension. There is no harm
in assuming that $K_{\infty}/K$ is the cyclotomic $\mathbb{Z}_{p}$-extension
(see Section 4.2). Theorem 4.3.5 states that there is a homomorphism of
$\mathcal{O}_{K}$-algebra objects in $D(\operatorname{Ab})$:

$\alpha_{0}\colon\mathcal{O}_{K}[\varepsilon]\to
R\Gamma(K_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})$ (5.3.1)

whose cofiber has the bounds:

$H^{i}(\operatorname{cof}(\alpha_{0}))=\begin{cases}0,&i=0,\\\
\text{$p^{M_{K}}$-torsion},&i=1,\\\ \text{$p^{2}$-torsion},&i\geq
2.\end{cases}$ (5.3.2)

(Throughout this discussion, we identify $\operatorname{Gal}(K_{\infty}/K)$
with $\mathbb{Z}_{p}$, so that $\mathcal{O}_{K}^{h\mathbb{Z}_{p}}$ is
identified with $\mathcal{O}_{K}[\varepsilon]$.)

Altogether we obtain a homomorphism of $\mathcal{O}_{K}$-algebra objects in
$D(\operatorname{Ab})$:

$\alpha_{\mathfrak{X}}\colon\mathcal{O}(\mathfrak{X})\otimes_{\mathcal{O}_{K}}\mathcal{O}_{K}[\varepsilon]\to
R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})$ (5.3.3)

obtained by sending $x\otimes y$ to $\nu^{*}(x)\otimes s^{*}\alpha_{0}(y)$.

######  Theorem 5.3.1.

Let $K$ be a local field of characteristic $(0,p)$. Let $\mathfrak{X}$ be an
affine formal scheme which is semistable over $\mathcal{O}_{K}$ of dimension
$d$. Let $\alpha_{\mathfrak{X}}$ be the map defined in (5.3.3). Then for all
$i\in\mathbb{Z}$, $H^{i}(\operatorname{cof}(\alpha_{\mathfrak{X}}))$ is killed
by $p^{M_{K^{t}}(d)}$, where $M_{K^{t}}(d)$ is a linear function of $d$ which
only depends on the maximal tame extension $K^{t}$ of $K$. (This means the
function is insensitive to replacing $K$ with any finite tame extension.)

Note that the theorem recovers Theorem 4.3.5 in the case
$X=\operatorname{Spa}K$.

###### Proof.

We will use repeatedly this fact: if $a_{1}$ and $a_{2}$ are composable
morphisms in $D(\operatorname{Ab})$ and integers $n_{ij}$ for each
$i\in\mathbb{Z}$, $j\in\left\\{1,2\right\\}$ such that $p^{n_{ij}}$ kills
$H^{i}(\operatorname{cof}(a_{j}))$ for each $i,j$, then $p^{n_{i1}+n_{i2}}$
kills $H^{i}(\operatorname{cof}(a_{2}\circ a_{1}))$ for all $i$. As a result,
if each of $\operatorname{cof}(a_{1}),\operatorname{cof}(a_{2})$ is bounded
from below and subject to the same sort of bounds as appears in the theorem,
then so is $\operatorname{cof}(a_{2}\circ a_{1})$.

The map $\alpha_{\mathfrak{X}}$ factors as

$\mathcal{O}(\mathfrak{X})\otimes_{\mathcal{O}_{K}}\mathcal{O}_{K}[\varepsilon]\stackrel{{\scriptstyle\text{id}\otimes\alpha_{0}}}{{\to}}\mathcal{O}(\mathfrak{X})\hat{\otimes}_{\mathcal{O}_{K}}\mathcal{O}_{C}^{h\Gamma_{K}}\stackrel{{\scriptstyle\nu^{*}\otimes
s^{*}}}{{\to}}R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}).$

Since $\mathcal{O}(\mathfrak{X})$ is a flat $\mathcal{O}_{K}$-module, the
cofiber of $\text{id}\otimes\alpha_{0}$ is subject to the same bounds as in
(5.3.2). In light of the observation we made in the first paragraph, it is
enough to show that $\operatorname{cof}(\nu^{*}\otimes s^{*})$ is subject to
the same bounds as in the theorem.

Let $C$ be the completion of an algebraic closure of $K$. Let us write
$\overline{\mathfrak{X}}=\mathfrak{X}\hat{\otimes}_{\mathcal{O}_{K}}\mathcal{O}_{C}$,
which is to say it is the formal scheme with coordinate ring
$\mathcal{O}(\overline{\mathfrak{X}})=\mathcal{O}(\mathfrak{X})\hat{\otimes}_{\mathcal{O}_{K}}\mathcal{O}_{C}$.
Then $\overline{\mathfrak{X}}$ is an affine semistable formal scheme over
$\mathcal{O}_{C}$ admitting a semilinear action of
$\Gamma_{K}=\operatorname{Gal}(\overline{K}/K)$, whose rigid-analytic generic
fiber is $\overline{X}=X\times_{\operatorname{Spa}K}\operatorname{Spa}C$. Let
$\overline{\nu}\colon
X_{C,\mathrm{pro\acute{e}t}}\to\overline{\mathfrak{X}}_{\mathrm{\acute{e}t}}$
be the projection.

We have a commutative diagram:

$\textstyle{\mathcal{O}(\mathfrak{X})\hat{\otimes}_{\mathcal{O}_{K}}\mathcal{O}_{C}^{h\Gamma_{K}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\nu^{*}\otimes
s^{*}}$$\textstyle{R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathcal{O}(\overline{\mathfrak{X}})^{h\Gamma_{K}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{R\Gamma(X_{C,\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})^{h\Gamma_{K}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{R\Gamma(\overline{\mathfrak{X}}_{\mathrm{\acute{e}t}},\mathcal{O})^{h\Gamma_{K}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R\Gamma(\overline{\mathfrak{X}}_{\mathrm{\acute{e}t}},R\overline{\nu}_{*}\hat{\mathcal{O}}^{+})^{h\Gamma_{K}}}$

The upper left vertical arrow is the projection formula from Lemma 3.3.4, and
the upper right vertical arrow is descent along the $\Gamma_{K}$ pro-étale
torsor $\overline{X}\to X$, see Proposition 3.6.2.

In light of this diagram, we wish to bound the cofiber of the lower horizontal
morphism, namely
$R\Gamma(\overline{\mathfrak{X}}_{\mathrm{\acute{e}t}},\operatorname{cof}\overline{\nu}^{*})^{h\Gamma_{K}}$,
where $\overline{\nu}^{*}$ refers to the pullack morphism $\mathcal{O}\to
R\overline{\nu}_{*}\hat{\mathcal{O}}^{+}$.

Throughout this discussion we let $L\eta$ mean $L\eta_{\zeta_{p}-1}$. Let
$\tilde{\Omega}=L\eta R\overline{\nu}_{*}\hat{\mathcal{O}}^{+}$ be the sheaf
introduced in Section 5.2. The next key step is to observe that
$\overline{\nu}^{*}\colon\mathcal{O}\to
R\overline{\nu}_{*}\hat{\mathcal{O}}^{+}$ factors as a composition of
$\Gamma_{K}$-equivariant maps:

$\mathcal{O}\stackrel{{\scriptstyle
i}}{{\to}}\tilde{\Omega}\stackrel{{\scriptstyle
a}}{{\to}}R\overline{\nu}_{*}\hat{\mathcal{O}}^{+}$

To see this, apply the naturality of $a$ to the morphism
$\overline{\nu}^{*}\colon\mathcal{O}\to R\nu_{*}\hat{\mathcal{O}}^{+}$ to find
a commutative diagram:

$\textstyle{L\eta\mathcal{O}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{L\eta(\overline{\nu}^{*})}$$\textstyle{L\eta
R\overline{\nu}_{*}\hat{\mathcal{O}}^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{\mathcal{O}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\nu}^{*}}$$\textstyle{R\nu_{*}\hat{\mathcal{O}}^{+}}$

Now note that since $\mathcal{O}$ is concentrated in degree 0 and is torsion-
free, the map $a\colon L\eta\mathcal{O}\to\mathcal{O}$ is an isomorphism, and
we get the desired factorization of $\overline{\nu}^{*}$.

The complex $\tilde{\Omega}$ is quasi-coherent by Theorem 5.2.3; that is, it
is the complex of sheaves on $X_{\mathrm{\acute{e}t}}$ associated to the
complex of $\mathcal{O}(\overline{\mathfrak{X}})$-modules
$\tilde{\Omega}(\overline{\mathfrak{X}}):=R\Gamma(\overline{\mathfrak{X}}_{\mathrm{\acute{e}t}},\tilde{\Omega})$.

The theorem will follow from combining the following two statements:

  1. (1)

Let $\tilde{\Omega}_{\geq 1}$ be the cofiber of
$i\colon\mathcal{O}(\overline{\mathfrak{X}})\to\tilde{\Omega}(\overline{\mathfrak{X}})$.
Then $H^{i}(\tilde{\Omega}_{\geq 1}^{h\Gamma_{K}})$ obeys bounds of the same
shape as in the theorem.

  2. (2)

Regarding $a\colon\tilde{\Omega}\to R\overline{\nu}_{*}\hat{\mathcal{O}}^{+}$,
we have that $H^{i}(\operatorname{cof}(a)^{h\Gamma_{K}})$ is 0 for $i=0$, and
is killed by a uniform power of $p$ for $i\geq 1$, namely $p^{2d+1}$.

These statements appear as Lemmas 5.4.2 and 5.5.5 below. ∎

###  5.4. Controlling $\tilde{\Omega}_{\geq 1}^{h\Gamma_{K}}$

By 5.2.4, $\tilde{\Omega}_{\geq 1}$ admits a filtration whose associated
graded pieces are

$\Omega^{j}_{\log}(\mathfrak{X})\hat{\otimes}_{\mathcal{O}_{K}}\mathcal{O}_{C}\left\\{-j\right\\}.$

Using the projection formula Lemma 3.3.4, we find that $\tilde{\Omega}_{\geq
1}^{h\Gamma_{K}}$ has a finite filtration with associated graded pieces

$\Omega^{j}_{\log}(\mathfrak{X})\hat{\otimes}_{\mathcal{O}_{K}}\mathcal{O}_{C}\left\\{-j\right\\}^{h\Gamma_{K}}$

for $j=1,\dots,d$.

We can use Theorem 4.4.3 to put bounds on the torsion in the cohomology of
$\Gamma_{K}$ with coefficients in the Breuil–Kisin twists.

######  Lemma 5.4.1.

For all $j\geq 0$ we have:

  1. (1)

$H^{0}(\Gamma_{K},\mathcal{O}_{C}\left\\{-j\right\\})=0$.

  2. (2)

$H^{1}_{\operatorname{cts}}(\Gamma_{K},\mathcal{O}_{C}\left\\{-j\right\\})$ is
$p^{B_{K^{t}}+v(j)}$-torsion, where $B_{K^{t}}$ only depends on $K^{t}$. (In
fact we can take $B_{K^{t}}=M_{K^{t}}+1$ for the constant $M_{K^{t}}$
appearing in Theorem 4.0.5.)

  3. (3)

For $i\geq 2$,
$H^{i}_{\operatorname{cts}}(\Gamma_{K},\mathcal{O}_{C}\left\\{-j\right\\})$ is
$p^{B^{\prime}}$-torsion, where $B^{\prime}$ is an absolute constant. (In fact
we can take $B^{\prime}=2$ or $B^{\prime}=4$ as $p$ is odd or even.)

###### Proof.

The key point is that for each $j>0$ there is an injective
$\Gamma_{K}$-equivariant map
$\mathcal{O}_{C}\left\\{-j\right\\}\to\mathcal{O}_{C}(-j)$ with cokernel
killed by $p$.

Recall the definition of the Breuil-Kisin twist [BMS18, Defintion 8.2]: Start
with the module of Kähler differentials
$\Omega^{1}_{\mathcal{O}_{C}/\mathbb{Z}_{p}}$, and let
$\mathcal{O}_{C}\left\\{1\right\\}$ be its Tate module; i.e.,
$\mathcal{O}_{C}\left\\{1\right\\}=\varprojlim\Omega^{1}_{\mathcal{O}_{C}/\mathbb{Z}_{p}}$,
where the transition maps in the limit are multiplication by $p$. Then
$\mathcal{O}_{C}\left\\{1\right\\}$ is a free $\mathcal{O}_{C}$-module of rank
1 carrying a $\Gamma_{K}$-action.

Let $(\zeta_{p^{r}})_{r\geq 1}$ be a compatible system of primitive $p$th
power roots of 1; then $\omega=(d\zeta_{p^{r}}/\zeta_{p^{r}})_{r\geq 1}$
represents an element of $\mathcal{O}_{C}\left\\{1\right\\}$. For all
$\sigma\in\Gamma_{K}$ we have
$\sigma(\omega)=\chi_{\text{cycl}}(\sigma)\omega$, which is to say that
$\omega$ induces a $\Gamma_{K}$-equivariant map
$\mathcal{O}_{C}(1)\to\mathcal{O}_{C}\left\\{1\right\\}$. The cokernel of this
map is killed by $(\zeta_{p}-1)$.

After dualizing and taking $j$-fold tensor products, we obtain a
$\Gamma_{K}$-equivariant injective map
$\mathcal{O}_{C}\left\\{-j\right\\}\to\mathcal{O}_{C}(-j)$ with cokernel
killed by $(\zeta_{p}-1)^{j}$. Since $v((\zeta_{p}-1)^{j})=\frac{j}{p-1}$ we
can divide this map by $p^{\lfloor\frac{j}{p-1}\rfloor}$ to obtain a
$\Gamma_{K}$-equivariant injective map
$\mathcal{O}_{C}\left\\{-j\right\\}\to\mathcal{O}_{C}(-j)$ with cokernel
killed by $p$.

The claimed bound now follows from applying the long exact sequence in
cohomology together with Theorem 4.0.5. ∎

######  Lemma 5.4.2.

For each $i\geq 0$, the cohomology $H^{i}(\tilde{\Omega}_{\geq
1}^{h\Gamma_{K}})$ is killed by $p^{C_{K^{t}}(d)}$, where $C_{K^{t}}(d)$ is a
linear function of $d$ which only depends on $K^{t}$.

###### Proof.

The filtration of $\operatorname{cof}(\alpha_{2})$ above gives rise to a
spectral sequence

$\Omega^{j}_{\mathfrak{X},\log}(\mathfrak{X})\otimes_{\mathcal{O}_{K}}H^{i}_{\operatorname{cts}}(\Gamma_{K},\mathcal{O}_{C}\left\\{-j\right\\})\implies
H^{i+j}(\operatorname{cof}(\alpha_{2})),$

where the left side is nonzero only for $j=1,\dots,d$ and $i\geq 1$. Therefore
a power of $p$ which annihilates $H^{s}(\operatorname{cof}(\alpha_{2}))$ can
be obtained by summing together the powers of $p$ which annihilate
$H^{i}_{\operatorname{cts}}(\Gamma_{K},\mathcal{O}_{C})$ for
$\min\left\\{s-d,1\right\\}\leq i\leq s-1$. The proposition then follows from
Lemma 5.4.1, noting the crude bound $v(j)\leq d$ whenever $j\leq d$. ∎

###  5.5. Controlling the décalage functor

We record here some lemmas regarding the $L{\eta}$ functor, recalling and
extending slightly the results in §6 of [BMS18]. Let $(T,\mathcal{O}_{T})$ be
a ringed topos, and let $D(\mathcal{O}_{T})$ be the derived category of
$\mathcal{O}_{T}$-modules. Let $\mathcal{I}\subset\mathcal{O}_{T}$ be an
invertible ideal sheaf. We use $L{\eta_{\mathcal{I}}}$ to denote the lax
symmetric monoidal functor $D(\mathcal{O}_{T})\to D(\mathcal{O}_{T})$ as in
[BMS18, Corollary 6.5., Proposition 6.7]. This functor has the effect of
killing the $\mathcal{I}$-torsion in the cohomology of a complex.

The functor $L{\eta_{\mathcal{I}}}$ commutes with truncations and in
particular preserves the subcategories of bounded complexes $D^{\geq
0}(\mathcal{O}_{T})$, $D^{\leq d}(\mathcal{O}_{T})$,
$D^{[0,d]}(\mathcal{O}_{T})$.

######  Lemma 5.5.1.

Let $\mathcal{C}$ be an object in $D(\mathcal{O}_{T})$. Then

  1. (1)

Assume that $\mathcal{C}\in D^{\geq 0}(\mathcal{O}_{T})$ and that
$H^{0}(\mathcal{C})$ is $\mathcal{I}$-torsion free. We have a natural map in
$D(\mathcal{O}_{T})$:

$a\colon L{\eta_{\mathcal{I}}}(\mathcal{C})\to\mathcal{C}.$

  2. (2)

Assume that $\mathcal{C}\in D^{\leq d}(\mathcal{O}_{T})$. We have a natural
map in $D(\mathcal{O}_{T})$

$b\colon\mathcal{C}\otimes\mathcal{I}^{\otimes d}\to
L{\eta_{\mathcal{I}}}(\mathcal{C})$

  3. (3)

Assume that $\mathcal{C}\in D^{[0,d]}(\mathcal{O}_{T})$ and that
$H^{0}(\mathcal{C})$ is $\mathcal{I}$-torsion free. The cofibers of
$b\circ(a\otimes\mathcal{I}^{\otimes d})$ and $a\circ b$ are
$\mathcal{O}_{T}/\mathcal{I}^{\otimes d}$-modules.

###### Proof.

These claims are all part of [BMS18, Lemma 6.9]. ∎

Applying Lemma 5.5.1 to the ringed topos
$(\overline{\mathfrak{X}}_{\mathrm{\acute{e}t}},\mathcal{O})$, the invertible
ideal $\mathcal{I}=(\zeta_{p}-1)$, and the objects
$R\overline{\nu}_{*}\hat{\mathcal{O}}^{+}$ and $\tau^{\leq
d}R\overline{\nu}_{*}\hat{\mathcal{O}}^{+}$, we obtain morphisms:

$\displaystyle a\colon
L{\eta_{\mathcal{I}}}R\overline{\nu}_{*}\hat{\mathcal{O}}^{+}$
$\displaystyle\to$ $\displaystyle R\overline{\nu}_{*}\hat{\mathcal{O}}^{+}$
$\displaystyle\overline{a}\colon L{\eta_{\mathcal{I}}}\tau^{\leq
d}R\overline{\nu}_{*}\hat{\mathcal{O}}^{+}$ $\displaystyle\to$
$\displaystyle\tau^{\leq d}R\overline{\nu}_{*}\hat{\mathcal{O}}^{+}$
$\displaystyle b\colon\tau^{\leq
d}R\overline{\nu}_{*}\hat{\mathcal{O}}^{+}\otimes\mathcal{I}^{\otimes d}$
$\displaystyle\to$ $\displaystyle
L{\eta_{\mathcal{I}}}R\overline{\nu}_{*}\hat{\mathcal{O}}^{+}$

######  Lemma 5.5.2.

The object $\operatorname{cof}(\overline{a})$ is $\mathcal{I}^{2d}$-torsion.

###### Proof.

Consider the diagram of exact triangles constructed from the octahedral axiom:

$\textstyle{\operatorname{cof}(b\circ(\bar{a}\otimes\mathcal{I}^{\otimes
d}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{\operatorname{cof}(b\circ(\bar{a}\otimes\mathcal{I}^{\otimes
d}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{cof}(b)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{cof}(\bar{a}b)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{cof}(\bar{a})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{\mathcal{I}^{\otimes
d}\otimes\operatorname{cof}(\bar{a})[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{cof}(\bar{a}).}$

Here $N$ is the cofiber of the map $\operatorname{cof}(b\circ(\bar{a}\otimes
I^{\otimes d}))\to\operatorname{cof}(\bar{a}b)$. Since
$\operatorname{cof}(b\circ(\bar{a}\otimes\mathcal{I}^{\otimes d}))$ and
$\operatorname{cof}(\bar{a}b)$ are both $\mathcal{I}^{d}$-torsion, $N$ is
$\mathcal{I}^{2d}$-torsion.

By Lemma 5.5.3 below, the map
$\operatorname{cof}(\overline{a})\to\mathcal{I}^{\otimes
d}\otimes\operatorname{cof}(\overline{a})[2]$ is 0, so that

$N\cong\left(\mathcal{I}^{\otimes
d}\otimes\operatorname{cof}(\bar{a})[1]\right)\oplus\operatorname{cof}(\bar{a}).$

It follows that $\operatorname{cof}(\bar{a})$ is $\mathcal{I}^{2d}$-torsion. ∎

######  Lemma 5.5.3.

Let $a_{1},a_{2},a_{3}$ be morphisms in a triangulated category which fit into
a diagram:

$A\stackrel{{\scriptstyle a_{1}}}{{\to}}B\stackrel{{\scriptstyle
a_{2}}}{{\to}}C\stackrel{{\scriptstyle a_{3}}}{{\to}}D.$

Then the composition of the canonical maps

$\operatorname{cof}(a_{3})\to\operatorname{cof}(a_{2})[1]\to\operatorname{cof}(a_{1})[2]$

is 0.

###### Proof.

We have a commutative diagram of fiber sequences:

$\textstyle{\operatorname{cof}(a_{2}a_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{cof}(a_{3}a_{2}a_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{cof}(a_{3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{\operatorname{cof}(a_{2}a_{1})[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{cof}(a_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{cof}(a_{3}a_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{cof}(a_{3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{cof}(a_{2})[1]}$

It follows that the canonical map
$\operatorname{cof}(a_{3})\to\operatorname{cof}(a_{2})[1]$ factors through
$\operatorname{cof}(a_{2}a_{1})[1]$, which is exactly the fiber of the
canonical map $\operatorname{cof}(a_{2})[1]\to\operatorname{cof}(a_{1})[2]$.
Thus the composition
$\operatorname{cof}(a_{3})\to\operatorname{cof}(a_{1})[2]$ is zero. ∎

######  Lemma 5.5.4.

The complexes $\operatorname{cof}(a)$ and $\operatorname{cof}(\bar{a})$ are
almost isomorphic. In particular $\operatorname{cof}(a)$ is
$(\zeta_{p}-1)^{2d+1}$-torsion.

###### Proof.

Let

$t\colon\tau^{\leq d}R\overline{\nu}_{*}\hat{\mathcal{O}}^{+}\to
R\overline{\nu}_{*}\hat{\mathcal{O}}^{+}$

be the natural map. Consider the diagram of cofiber sequences:

$\textstyle{L{\eta_{\mathcal{I}}}\tau^{\leq
d}R\overline{\nu}_{*}\hat{\mathcal{O}}^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L{\eta_{\mathcal{I}}}t}$$\textstyle{\tau^{\leq
d}R\overline{\nu}_{*}\hat{\mathcal{O}}^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t}$$\textstyle{\operatorname{cof}(\bar{a})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L{\eta_{\mathcal{I}}}R\overline{\nu}_{*}\mathcal{O}^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R\overline{\nu}_{*}\mathcal{O}^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{cof}(a)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{cof}(L{\eta_{\mathcal{I}}}t)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{cof}(t)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D.}$

We claim that $t$ and $L{\eta_{\mathcal{I}}}t$ are almost isomorphisms. We
begin with $t$. It suffices to verify that $t$ is an almost isomorphism after
evaluating the source and target complexes of sheaves on an open. Given an
open $U\to\mathfrak{X}_{\mathrm{\acute{e}t}}^{\prime}$,
$(R\overline{\nu}_{*}\mathcal{O}^{+})(U)\cong
R\Gamma((\overline{\nu})^{-1}(U)_{\mathrm{pro\acute{e}t}},\mathcal{O}^{+})$ is
subject to the constraints imposed by the discussion around [vK19, Equation
3.3.1], in which $\Delta$ has cohomological dimension $d$, so the cofiber of
the map $e$ is almost zero. Applied to our situation, this implies that the
cofiber of $t$ is almost zero. A similar argument applying the source and
target of $L{\eta_{\mathcal{I}}}t$ to an open allows us to apply [vK19,
Theorem 3.9] to see that the cofiber of $L{\eta_{\mathcal{I}}}t$ is almost
zero.

Therefore $\operatorname{cof}(L{\eta_{\mathcal{I}}}t)$ and
$\operatorname{cof}(t)$ are both almost zero. This implies that $D$ is almost
zero, and this further implies that
$\operatorname{cof}(\bar{a})\to\operatorname{cof}(a)$ is an almost
isomorphism. Since an almost zero sheaf of complexes is $\mathcal{I}$-torsion,
we have that $\operatorname{cof}(a)$ is $\mathcal{I}^{2d+1}$-torsion. ∎

Since $\operatorname{cof}(a)$ is $(\zeta_{p}-1)^{2d+1}$-torsion, we (rather
crudely) conclude that it is $p^{2d+1}$-torsion. It follows that it is a
module over $\mathbb{Z}/p^{2d+1}\mathbb{Z}$. Since the functor of taking
homotopy fixed points is lax monoidal, $\operatorname{cof}(a)^{h\Gamma_{K}}$
is also a module over $\mathbb{Z}/p^{2d+1}\mathbb{Z}$ and thus so are the
cohomology groups of $\operatorname{cof}(a^{h\Gamma_{K}})$. We have proved:

######  Lemma 5.5.5.

For $i\geq 0$ we have

$H^{i}(\operatorname{cof}(a)^{h\Gamma_{K}})=\begin{cases}0,&i=0\\\
\text{$p^{2d+1}$-torsion},&i\geq 1.\end{cases}$

###  5.6. Tame descent

Theorem 5.3.1 gives control over the pro-étale cohomology of a rigid-analytic
space $X/K$ which admits an affine semistable model over $\mathcal{O}_{K}$. We
now pose the question of whether this control may be obtained for the general
case of a smooth rigid-analytic space $X/K$. The trouble we encounter is that
$X/K$ does not necessarily have a semistable model; it may be necessary to
extend scalars. The best result we know along these lines is the main theorem
of [Har03], which states that $X_{\mathrm{\acute{e}t}}$ admits an open cover
by rigid-analytic generic fibers of semistable formal schemes defined over
finite extensions $L/K$. If $X$ is not quasi-compact then there may not be a
single $L/K$ that suffices for this purpose, and then the constants appearing
in Theorem 5.3.1 could a priori accumulate.

In the special case that all the $L/K$ are tamely ramified extensions (or, at
least, if the wild ramification of the $L/K$ is bounded), then it becomes
possible to get uniform control over the pro-étale cohomology of $X$. We will
see in the next section that this is the case for the open unit ball over $K$.

We gather a few lemmas about descent through tame extensions. For a Galois
extension of local fields $L/K$ with group $G$, let
$\mathcal{O}_{L}\left<G\right>$ denote the associative
$\mathcal{O}_{L}$-algebra whose underlying module is freely generated by
symbols $[g]$, and whose multiplication law is determined by the relations
$[gh]=[g][h]$ and $[g]\alpha=g(\alpha)[g]$ for $\alpha\in\mathcal{O}_{L}$ and
$g,h\in G$. Let $\text{Mod}_{\mathcal{O}_{L}\left<G\right>}$ denote the
category of $\mathcal{O}_{L}\left<G\right>$-modules. An object of this
category is an $\mathcal{O}_{L}$-module $A$ endowed with an action of $G$
which acts semilinearly over the action of $G$ on $\mathcal{O}_{L}$; i.e., a
descent datum along $L/K$. Its module of fixed points $A^{G}$ is an
$\mathcal{O}_{K}$-module.

######  Lemma 5.6.1.

Let $K$ be a local field of characteristic $(0,p)$, and let $L/K$ be a finite
tame Galois extension with group $G$. The functor $A\mapsto A^{G}$ from
$\mathcal{O}_{L}\left<G\right>$-modules to $\mathcal{O}_{K}$-modules is exact.
As a result, if $C\in D(\mathcal{O}_{L}\left<G\right>)$ is a complex whose
cohomology groups are all killed by $p^{k}$ for some $k\geq 1$, then the same
is true for $C^{hG}\in D(\mathcal{O}_{K})$.

###### Proof.

Since a composition of exact functors is exact, we can reduce to the cases
where $L/K$ is unramified or totally tamely ramified.

In the case where $L/K$ is unramified, $\mathcal{O}_{L}$ is an étale
$G$-torsor over $\mathcal{O}_{K}$, and thus by étale descent, the functor
$A\mapsto A^{G}$ is an equivalence of categories.

In the case where $L/K$ is totally tamely ramified, exactness follows from the
fact that the order of $G$ is a unit in $\mathcal{O}_{K}$. ∎

######  Lemma 5.6.2.

Suppose $X/K$ is a rigid-analytic space. Define $\beta_{X}$ as the
composition:

$\mathcal{O}^{+}(X)[\varepsilon]\stackrel{{\scriptstyle\text{unit}}}{{\to}}R\Gamma(X_{\mathrm{\acute{e}t}},\mathcal{O}^{+})[\varepsilon]\stackrel{{\scriptstyle\nu_{X}}}{{\to}}R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}).$

Assume there exists a finite tame Galois extension $L/K$ such that $X_{L}$ is
the generic fiber of an affine semistable formal scheme $\mathfrak{X}$ of
dimension $d$ over $\mathcal{O}_{L}$. Then the cohomology groups
$H^{i}(\beta_{X})$ are all killed by a uniform power of $p$ which only depends
on $K$.

###### Proof.

Let $G=\operatorname{Gal}(L/K)$. Theorem 5.3.1 applied to the affine
semistable formal scheme $\mathfrak{X}$ gives a bound on the cofiber of the
$G$-equivariant map

$\beta_{X_{L}}\colon\mathcal{O}^{+}(X_{L})[\varepsilon]\to
R\Gamma(X_{L,\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}).$

Here we have used Lemma 5.2.2 to identify $\mathcal{O}(\mathfrak{X})$ with
$\mathcal{O}^{+}(X_{L})$. Applying homotopy fixed points under $G$ is exact by
Lemma 5.6.1, so the same bounds apply to the cofiber of $\beta_{X}$. ∎

We now pass to the non-affinoid case. To do this, we consider covers of a
rigid-analytic space by affinoids satisfying the hypotheses of Lemma 5.6.2. At
this point we will start encountering objects in $D(\operatorname{Solid})$
which are limits of $p$-adic objects but which are no longer $p$-adic
themselves.

######  Definition 5.6.3.

Let $X/K$ be a smooth rigid analytic space of dimension $d$. Let
$\mathcal{U}\coloneqq\\{U_{i}\\}_{i\in I}$ be an open cover in
$X_{\mathrm{an}}$. For a finite subset $J\subset I$, let $U_{J}$ be the
intersection of the $\left\\{U_{i}\right\\}_{i\in J}$. We say that
$\mathcal{U}$ is tamely semistable if for every such $J$, there exists a
finite Galois tame extension $L_{J}/K$ such that $(U_{J})_{L_{J}}$ admits an
affine semistable formal model over $\mathcal{O}_{L_{J}}$.

Such a cover appears when there is already a semistable model for $X$:

######  Lemma 5.6.4.

Let $\mathfrak{X}$ be a quasi-separated semistable formal scheme over
$\mathcal{O}_{K}$ with generic fiber $X$. Let
$\left\\{\mathfrak{U}_{i}\right\\}_{i\in I}$ be an affine open cover of
$\mathfrak{X}$. Let $\mathcal{U}$ be the cover of $X$ obtained by pulling this
cover back through the reduction map. Then $\mathcal{U}$ is a tamely
semistable cover for $X$.

###### Proof.

The quasi-separatedness assumption implies that for all finite non-empty
$J\subset I$ the open subset $\mathfrak{U}_{J}\subset\mathfrak{X}$ is affine.
Let

$\mathrm{red}\colon X\to\mathfrak{X}$

be the reduction map. For each $J$ we have
$U_{J}\cong\mathrm{red}^{-1}(\mathfrak{U}_{J})$. Since semistablity is an
étale local condition, we see that $\mathfrak{U}_{J}$ is an affine semistable
model for $U_{J}$. ∎

######  Definition 5.6.5.

Let $X/K$ be a rigid-analytic space. Let $\mathcal{U}$ be an affinoid open
cover in $X_{\mathrm{an}}$, with the same notational conventions as in 5.6.3.
Define the condensed Čech complex by

$\check{C}(\mathcal{U},\mathcal{O}^{+}_{\operatorname{cond}})\coloneqq\operatorname{lim}\limits_{[n]\in\Delta}\prod_{f\in
I^{[n]}}\mathcal{O}^{+}_{\operatorname{cond}}(U_{\mathrm{im}(f)}).$

In this expression:

  * •

$\Delta$ is the simplex category, with objects
$[n]=\left\\{0,1,\dots,n\right\\}$ for $n\geq 0$, and morphisms order
preserving maps,

  * •

$I^{[n]}$ is the set of functions $f\colon[n]\to I$, and $\mathrm{im}(f)$ is
the image of $f$, and

  * •

the limit is computed in $D(\operatorname{Cond}(\operatorname{Ab}))$.

Thus, $\check{C}(\mathcal{U},\mathcal{O}^{+}_{\operatorname{cond}})$ is a
condensed enhancement of the usual Čech complex
$\check{C}(\mathcal{U},\mathcal{O}^{+})$.

######  Remark 5.6.6.

Since $\mathcal{O}^{+}_{\operatorname{cond}}$ (in contrast to
$\mathcal{O}_{\operatorname{cond}}$) is not acyclic on affinoids, in general
$\check{C}(\mathcal{U},\mathcal{O}^{+}_{\operatorname{cond}})$ does not just
compute the $\mathcal{O}^{+}_{\operatorname{cond}}$-cohomology of $X$.

######  Lemma 5.6.7.

Let $\mathfrak{X}$ be a quasi-separated semistable formal scheme over
$\mathcal{O}_{K}$ with generic fiber $X$. Let $\mathcal{U}$ be the affinoid
open cover of $X$ obtained by pulling back an affine open cover
$\left\\{\mathfrak{U}_{i}\right\\}_{i\in I}$ of $\mathfrak{X}$. Then
$R\Gamma(\mathfrak{X},\mathcal{O}_{\operatorname{cond}})$ is quasi-isomorphic
to $\check{C}(\mathcal{U},\mathcal{O}^{+}_{\operatorname{cond}})$.

###### Proof.

By Lemma 3.5.1, $\mathcal{O}_{\operatorname{cond}}$ is acyclic on affine
formal schemes. Therefore
$R\Gamma(\mathfrak{X},\mathcal{O}_{\operatorname{cond}})$ is computed by Čech
cohomology with respect to the affine cover
$\left\\{\mathfrak{U}_{i}\right\\}_{i\in I}$:

$R\Gamma(\mathfrak{X},\mathcal{O}_{\operatorname{cond}})\cong\operatorname{lim}_{[n]\in\Delta}\prod_{f\in
I^{[n]}}\mathcal{O}_{\operatorname{cond}}(\mathfrak{U}_{\mathrm{im}(f)}).$

For each finite $J\subset I$ we have an isomorphism of topological rings Lemma
5.2.2 $\mathcal{O}(\mathfrak{U}_{J})\cong\mathcal{O}^{+}(U_{J})$. Applying the
functor $A\mapsto\underline{A}$ from topological abelian groups to condensed
abelian groups, we find that
$R\Gamma(\mathfrak{X},\mathcal{O}_{\operatorname{cond}})$ is quasi-isomorphic
to $\check{C}(\mathcal{U},\mathcal{O}^{+}_{\operatorname{cond}})$. ∎

The next theorem shows that if $\mathcal{U}$ is a tamely semistable cover of
$X$, then the condensed Čech complex associated to $\mathcal{U}$ is a good
approximation for the pro-étale cohomology of $X$.

######  Theorem 5.6.8.

Let $X/K$ be a smooth rigid analytic space of dimension $d$, and let
$\mathcal{U}\coloneqq\\{U_{i}\\}_{i\in I}$ be a tamely semistable cover of
$X$. We have a natural map

$\alpha_{\mathcal{U}}\colon\check{C}(\mathcal{U},\mathcal{O}^{+}_{\operatorname{cond}})[\varepsilon]\to
R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}}).$

The cohomology groups $H^{i}(\operatorname{cof}(\alpha_{\mathcal{U}}))$ are
each killed by a power of $p$. If the dimension of the nerve of $\mathcal{U}$
is finite, the same power of $p$ kills
$H^{i}(\operatorname{cof}(\alpha_{\mathcal{U}}))$ for all $i$.

###### Proof.

For each finite non-empty set of indices $J\subset I$, we have a natural map

$\beta_{J}\colon\mathcal{O}^{+}(U_{J})[\varepsilon]\to
R\Gamma(U_{J,\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})$

as in Lemma 5.6.2. By that lemma, there exists $n\geq 1$ such that $p^{n}$
kills $H^{i}(\operatorname{cof}(\beta_{J}))$ for all $i,J$.

We consider now the condensed enhancement of $\beta_{J}$. Note that $U_{J}$ is
affinoid, and as a topological ring, $\mathcal{O}^{+}(U_{J})$ is $p$-adic, in
the sense that
$\mathcal{O}^{+}(U_{J})\cong\varprojlim\mathcal{O}^{+}(U_{J})/p^{n}$ as a
limit of discrete groups. On the other hand, Lemma 3.7.1 states that
$R\Gamma(U_{J,\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}})$
is $p$-adic as well. Therefore we can apply the functor $X\mapsto\varprojlim
X/p^{n}$ (where the limit is calculated in
$D(\operatorname{Cond}(\operatorname{Ab}))$) to $\beta_{J}$, we obtain a
morphism in $D(\operatorname{Cond}(\operatorname{Ab}))$ which is functorial in
$J$:

$\beta_{J,\operatorname{cond}}\colon\mathcal{O}^{+}_{\operatorname{cond}}(U_{J})[\varepsilon]\to
R\Gamma(U_{J,\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+}_{\operatorname{cond}}).$

The $H^{i}(\operatorname{cof}(\beta_{J,\operatorname{cond}}))$ are $p$-adic
solid abelian groups killed by $p^{n}$ (which implies they are discrete).

Combining the $\beta_{J,\operatorname{cond}}$ together, we find a morphism
$\alpha_{\mathcal{U}}$ as in the theorem. Its cofiber is

$\operatorname{cof}(\alpha_{\mathcal{U}})\cong\operatorname{lim}\limits_{[n]\in\Delta}\prod_{f\in
I^{[n]}}\operatorname{cof}(\beta_{\mathrm{im}(f)})$

Taking the associated cosimplicial limit spectral sequence

$E_{1}^{s,q}=\prod_{f\in
I^{[q]}}H^{s}(\operatorname{cof}(\beta_{\mathrm{im}(f)}))\implies
H^{s+q}(\operatorname{cof}(\alpha_{\mathcal{U}})),$

we find that $H^{i}(\operatorname{cof}(\alpha_{\mathcal{U}}))$ is killed by
$p^{n(i+1)}$. If the dimension of the nerve of the cover is $\delta$, we find
that $H^{i}(\operatorname{cof}(\alpha_{\mathcal{U}}))$ is killed by
$p^{n(\delta+1)}$. ∎

######  Corollary 5.6.9.

Let $\mathfrak{X}$ be a quasi-separated semistable formal scheme over
$\mathcal{O}_{K}$ with generic fiber $X$. There is a natural morphism in
$D(\operatorname{Cond})$:

$\alpha_{\mathfrak{X}}\colon
R\Gamma(\mathfrak{X},\mathcal{O}_{\operatorname{cond}})[\varepsilon]\to
R\Gamma(X_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}_{\operatorname{cond}}^{+}),$

such that the cohomology groups of $\operatorname{cof}(\alpha_{\mathfrak{X}})$
are $p$-power torsion.

###### Proof.

This is immediate from Lemma 5.6.7 and Theorem 5.6.8. ∎

Applying $\Gamma_{*}$ to the morphism in 5.6.9 and inverting $p$ gives the
non-condensed statement appearing in D.

##  6\. Proof of B

Let us recall the main players in B: a prime number $p$, an integer $n\geq 1$,
the ring of $p$-typical Witt vectors $W=W(\overline{\mathbb{F}}_{p})$, and the
Lubin–Tate ring $A\cong W\llbracket u_{1},\dots,u_{n-1}\rrbracket$, which
admits a continuous action of the Morava stabilizer group $\mathbb{G}_{n}$.
Proposition 2.5.1 states that the inclusion $W\hookrightarrow A$ admits a
continuous $\mathbb{G}_{n}$-equivariant additive splitting, say with
complement $A^{c}$. B is the statement that the continuous cohomology
$H^{*}_{\operatorname{cts}}(\mathbb{G}_{n},A^{c})$ is $p$-power torsion.

In Section 3.9 we explained how to reduce B to a statement (Theorem 3.9.2)
controlling the pro-étale cohomology of the open ball and Drinfeld’s symmetric
space. After a detour on the integral cohomology of $p$-adic Lie groups, we
examine these cases in turn. As a general reference for $p$-adic Lie groups,
we recommend Lazard’s treatment [Laz65] or [Sch11].

###  6.1. Continuous cohomology of $p$-adic Lie groups with integral
coefficients

Let $G$ be a $p$-adic Lie group. In Section 3.8 we reviewed Lazard’s
isomorphism $H^{*}_{\operatorname{cts}}(G,\mathbb{Q}_{p})\cong
H^{*}(\operatorname{Lie}G,\mathbb{Q}_{p})$, where $\operatorname{Lie}G$ is
Lazard’s (rational) Lie algebra. For our purposes we will need the integral
refinement of this isomorphism described in [HKN11]. First we recall a
definition from [DdSMS99]. Let $U$ be a pro-$p$ group. We say $U$ is uniform
if it satisfies the conditions:

  1. (1)

$U$ is topologically finitely generated.

  2. (2)

For $p$ odd (resp., $p=2$), $U/U^{p}$ is abelian (resp., $U/U^{4}$ is
abelian). Here $U^{n}$ is the closure of the subgroup of $U$ generated by
$n$th powers.

  3. (3)

Let $U=U_{1}\supset U_{2}\supset\cdots$ be the lower $p$-series. Then
$[U_{i}:U_{i+1}]$ is independent of $i$.

For a uniform pro-$p$ group $U$, [DdSMS99, §8.2] defines the integral Lie
algebra $\mathcal{L}(U)$ over $\mathbb{Z}_{p}$; this is a lattice in
$\operatorname{Lie}U$. The following is [HKN11, Theorem 3.3.3].

######  Theorem 6.1.1.

Let $U$ be a uniform pro-$p$ group acting continuously on a finitely generated
free $\mathbb{Z}_{p}$-module $M$. Assume that the action map
$U\to\operatorname{Aut}M$ factors through $1+p\operatorname{End}M$ if $p$ is
odd (resp., through $1+4\operatorname{End}M$ if $p=2$). Then $\mathcal{L}(U)$
acts on $M$, and there is an isomorphism of graded $\mathbb{Z}_{p}$-modules:

$H^{*}_{\operatorname{cts}}(U,M)\cong H^{*}(\mathcal{L}(U),M)$

If $G$ is a $\mathbb{Q}_{p}$-analytic group acting continuously on $M$, then
$G$ contains a uniform pro-$p$ subgroup $U$ whose action on $M$ satisfies the
hypothesis of Theorem 6.1.1. Therefore the question of computing continuous
cohomology of $G$ with integral coefficients can be reduced to a question
about the integral Lie algebra $\mathcal{L}(G)$.

######  Lemma 6.1.2.

Let $G$ be either of the groups $\operatorname{GL}_{n}(\mathbb{Z}_{p})$ or
$\mathcal{O}_{D}^{\times}$.

  1. (1)

Let $G$ act trivially on $\mathbb{Z}_{p}$. Then

$H^{i}_{\operatorname{cts}}(G,\mathbb{Z}_{p})\cong\mathbb{Z}_{p}^{\oplus
r_{i}}\oplus S_{i},$

where $r_{i}$ is the dimension of the degree $i$ part of
$\Lambda_{\mathbb{Q}}(x_{1},x_{3},\dots,x_{2n-1})$, and where $S_{i}$ is
annihilated by a uniform power of $p$ (that is, the power does not depend on
$i$).

  2. (2)

Let $C$ be a complex of solid $\mathbb{Z}_{p}$-modules admitting an action of
$G$. Assume that $H^{i}(C)=0$ for all $i<0$ and that $H^{*}(C)$ is annihilated
by a uniform power of $p$. Then $H^{*}(C^{hG})$ is also annihilated by a
uniform power of $p$.

###### Proof.

In each case, $G$ is the group of units of a $\mathbb{Z}_{p}$-algebra $A$ such
that $U=1+p^{2}A$ is a uniform pro-$p$ subgroup of $G$, so we are in the
situation of Theorem 6.1.1.

For part (1): Theorem 6.1.1 gives an isomorphism
$H^{*}_{\operatorname{cts}}(U,\mathbb{Z}_{p})\cong
H^{*}(\mathcal{L}(U),\mathbb{Z}_{p})$. After tensoring with $\mathbb{Q}_{p}$,
we have

$H^{*}(\mathcal{L}(U),\mathbb{Z}_{p})\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}\cong
H^{*}(\operatorname{Lie}U,\mathbb{Q}_{p})\cong\Lambda_{\mathbb{Q}_{p}}(x_{1},x_{3},\dots,x_{2n-1}).$

Since each $H^{*}(\mathcal{L}(U),\mathbb{Z}_{p})$ is finitely generated over
$\mathbb{Z}_{p}$, as can be seen from the Chevalley–Eilenberg complex
computing Lie algebra cohomology, we may write

$H^{i}_{\operatorname{cts}}(U,\mathbb{Z}_{p})\cong
H^{i}(\mathcal{L}(U),\mathbb{Z}_{p})\cong\mathbb{Z}_{p}^{r_{i}}\oplus S_{i},$

where $S_{i}$ is finite. Since the cohomology is 0 for $i>n^{2}$, there is a
uniform power of $p$ which annihilates $S_{i}$ for all $i$.

To extend this statement from $U$ to $G$, we use the existence of the
restriction and corestriction maps between
$H^{*}_{\operatorname{cts}}(G,\mathbb{Z}_{p})$ and
$H^{*}_{\operatorname{cts}}(U,\mathbb{Z}_{p})$; their composition is
multiplication by $\\#G/U$.

For part (2): Consider the spectral sequence

$H^{i}_{\operatorname{cts}}(U,H^{j}(C))\implies H^{i+j}(C^{hU}).$

Since $U$ has cohomological dimension $n^{2}$, the left side is nonzero only
when $0\leq i\leq n^{2}$. Therefore if $p^{r}$ annihilates $H^{i}(C)$ for all
$i$, then $p^{rn^{2}}$ annihilates $H^{i}(C^{hU})$ for all $i$. Once again,
the restriction and corestriction maps can be used to extend the result from
$U$ to $G$. ∎

###  6.2. Pro-étale cohomology of the open ball

Let $K$ be a local field of characteristic $(0,p)$. For an integer $d\geq 1$,
recall that we had defined the $d$-dimensional rigid-analytic open ball over
$K$:

$B^{\circ,d}=\left(\operatorname{Spa}\mathcal{O}_{K}\llbracket
T_{1},\dots,T_{d}\rrbracket\right)\backslash\left\\{\left\lvert
p\right\rvert=0\right\\}$

The ball $B^{\circ,d}$ is not quasi-compact. It is exhausted by affinoid
(closed) balls of increasing radius:

$B^{\circ,d}=\varinjlim_{r<1}B^{d}_{r},$ (6.2.1)

where $r$ runs over real numbers in $\left\lvert
p\right\rvert^{\mathbb{Q}_{+}}$, and for each $r=\left\lvert
p\right\rvert^{m/n}$ with $m,n$ relatively prime positive integers,
$B_{r}^{d}$ is the rational subset defined by the inequality $\left\lvert
T\right\rvert^{n}\leq\left\lvert p\right\rvert^{m}$.

The goal of this section is to control
$R\Gamma(B^{\circ,d}_{\mathrm{pro\acute{e}t}},\hat{\mathcal{O}}^{+})$. The
idea is to apply Theorem 5.6.8 to the cover $\mathcal{U}$ given by a
collection of affinoid balls $B^{d}_{r}$ which cover $B^{\circ,d}$. A
convenient choice of radii is $r_{\ell}=\left\lvert p\right\rvert^{1/\ell}$,
where $\ell$ runs over prime numbers $\neq p$. The closed ball
$B^{d}_{r_{\ell}}$ admits a smooth formal model after passage from $K$ to the
tamely ramified extension

$L_{\ell}=K(p^{1/\ell}).$

Indeed, $B^{d}_{r_{\ell}}$ is the rigid-analytic generic fiber of the smooth
formal scheme $\mathfrak{B}_{r_{\ell}}^{d}$, where

$\mathfrak{B}_{r_{\ell}}^{d}=\operatorname{Spf}\mathcal{O}_{L_{\ell}}\left<\frac{T_{1}}{p^{1/\ell}},\dots,\frac{T_{d}}{p^{1/\ell}}\right>.$

Therefore the cover $\mathcal{U}$ is tamely semistable.

The solid Čech complex
$\check{C}(\mathcal{U},\mathcal{O}_{\operatorname{cond}}^{+})$ is quasi-
isomorphic to
$R\varprojlim_{\ell}H^{0}(B_{r_{\ell}}^{d},\mathcal{O}^{+}_{\operatorname{cond}})$,
which could only have an $H^{0}$ and an $H^{1}$. The $H^{0}$ is

$\varprojlim
H^{0}(B_{r_{\ell}}^{d},\mathcal{O}^{+}_{\operatorname{cond}})=\mathcal{O}_{K}\llbracket
T_{1},\dots,T_{d}\rrbracket,$

where the latter must be considered as the solid abelian group associated to
the topological ring $\mathcal{O}_{K}\llbracket T_{1},\dots,T_{d}\rrbracket$
(with its $(p,T_{1},\dots,T_{d})$-adic topology). We claim there is no
$H^{1}$. In order to prove this, we begin with a formal lemma.

######  Lemma 6.2.1.

If $A_{i}$ is a cofiltered diagram of discrete abelian groups satisfying the
Mittag-Leffler condition and $S$ is a profinite set, then the diagram of
abelian groups $C_{\operatorname{cts}}(S,A_{i})$ satisfies the Mittag-Leffler
condition as well.

###### Proof.

This follows from the fact that for a map of discrete sets $f\colon A\to B$
and a profinite set $S$, we have

$\mathrm{Im}\left[C_{\operatorname{cts}}(S,A)\to
C_{\operatorname{cts}}(S,B)\right]=C_{\operatorname{cts}}(S,\mathrm{Im}(f)).$

Indeed, the functions involved here are all locally constant, which reduces us
to the obvious case where $S$ is finite. ∎

######  Lemma 6.2.2.

Let $K$ be a local field of characteristic $(0,p)$ with uniformizer $\varpi$.
Let $j\in\mathbb{N}$ and let $0<r<1$ and set
$M=\lceil\frac{1+j}{\log_{|\varpi|}(r)}\rceil$. Then for every $1>s\geq
r^{\frac{1}{j+1}}$ and $j^{\prime}\geq j$ the image of the map

$H^{0}(B_{s}^{d},\mathcal{O}^{+})/\varpi^{j^{\prime}}\to
H^{0}(B_{r}^{d},\mathcal{O}^{+})/\varpi^{j},$

is the $\mathcal{O}_{K}/\varpi^{j}$-submodule spanned by the $T^{i}$ for
multi-indices $i$ with $|i|<M$.

###### Proof.

Denote by $V\subset H^{0}(B_{r}^{d},\mathcal{O}^{+})/\varpi^{j}$ the
$\mathcal{O}_{K}/\varpi^{j}$-submodule spanned by $T^{i}$ for $|i|<M$. Let
$b=\sum_{i}b_{i}T^{i}\in H^{0}(B_{s}^{d},\mathcal{O}^{+})$ and define two
elements in $H^{0}(B_{s}^{d},\mathcal{O}^{+})$ by

$b^{\prime}=\sum_{|i|<M}b_{i}T^{i}\quad\text{and}\quad
b^{\prime\prime}=\sum_{|i|\geq M}b_{i}T^{i}.$

We have $b=b^{\prime}+b^{\prime\prime}$. It is enough show that

  1. (1)

the image of $b^{\prime}$ in $H^{0}(B_{r}^{d},\mathcal{O}^{+})/\varpi^{j}$
belongs to $V$;

  2. (2)

and the image of $b^{\prime\prime}$ in
$H^{0}(B_{r}^{d},\mathcal{O}^{+})/\varpi^{j}$ is zero.

Thus it is enough to show

  1. (1)

$|b_{i}|\leq 1$ for $|i|<M$;

  2. (2)

and $|\frac{b_{i}}{\varpi^{j}}|r^{|i|}\leq 1$ for $|i|\geq M$.

Since the value group of $K$ is isomorphic to $|\varpi|^{\mathbb{Z}}$,
Condition (1) is equivalent to $|b_{i}|<\frac{1}{|\varpi|}$ for $|i|<M$. Now
since $b\in H^{0}(B_{s}^{d},\mathcal{O}^{+})$, we have for all $i$ that
$|b_{i}|\leq{s}^{-|i|}\leq{r}^{-\frac{|i|}{j+1}}$. Thus we get

$|b_{i}|\leq{r}^{\frac{-(M-1)}{j+1}}$

for $|i|<M$ and