# Period Rings with Big Coefficients and Applications III

Xin Tong

###### Abstract.

We continue our study on the corresponding period rings with big coefficients,
with the corresponding application in mind on relative $p$-adic Hodge theory
and noncommutative analytic geometry. In this article, we extend the
discussion of the corresponding noncommutative descent over étale topology to
the corresponding noncommutative descent over pro-étale topology in both Tate
and analytic setting.

11footnotetext: Version: Mar 7 2021.22footnotetext: Keywords and Phrases:
Noncommutative Deformation, Descent.

###### Contents

  1. 1 Introduction
    1. 1.1 Introduction and Main Results
    2. 1.2 Future Work
  2. 2 Descent over Analytic Adic Banach Rings in the Rational Setting
    1. 2.1 Noncommutative Functional-Analytic Pseudocoherence over Pro-Étale Topology
  3. 3 Descent over Analytic Huber Rings in the Integral Setting
    1. 3.1 Noncommutative Topological Pseudocoherence over Étale Topology
    2. 3.2 Noncommutative Topological Pseudocoherence over Pro-Étale Topology
  4. 4 Descent in General Setting for Analytic Huber Pairs
    1. 4.1 Étale Topology
    2. 4.2 Pro-étale Topology
  5. 5 Descent in General Setting for Analytic Adic Banach Rings
    1. 5.1 Étale Topology
    2. 5.2 Pro-étale Topology
    3. 5.3 Quasi-Stein Spaces
  6. 6 Applications
    1. 6.1 Application to Noetherian Cases
    2. 6.2 Application to Descent over Noncommutative $\infty$-Analytic Prestacks after Bambozzi-Ben-Bassat-Kremnizer

##  1\. Introduction

###  1.1. Introduction and Main Results

In our previous work on the corresponding period rings with big coefficients
and applications, we essentially studied many very general deformations of the
corresponding pseudocoherent sheaves over adic Banach rings in the sense of
[KL1] and [KL2]. We studied the corresponding glueing in the fashion of [KL1]
and [KL2] carrying sufficiently large coefficients. Although we have some
conditions on both the corresponding adic spaces we are considering and the
corresponding coefficients, but the descent results on their own are already
general enough to tackle some specific situations in the Hodge-Iwasawa
theoretic consideration and noncommutative analytic geometry such as in [TX3]
and [TX4].

First in our current scope of discussion we will first consider the
corresponding extension of our paper [TX2] to the corresponding pro-étale
topology setting. In the Tate setting, this will follow the corresponding
unrelative situation in [KL2] while in the analytic situation the
corresponding will following the corresponding unrelative situation in [Ked1]
(in the context of section 2 and section 3):

######  Theorem 1.1.

(See proposition 2.10, proposition 3.17) Over sheafy Tate adic Banach rings or
analytic Huber rings, the corresponding descent of stably pseudocoherent
modules (with respect to the underlying spaces) carrying the corresponding
noncommutative coefficients holds over étale sites and pro-étale sites. Here
for pro-étale sites we will consider only perfectoid rings.

We then consider some application to the corresponding ’complete’ noetherian
situation (this will mean that we consider the noetherian deformation of
noetherian rings) in the context of section 6:

######  Theorem 1.2.

(See proposition 6.9) Over noetherian sheafy Tate adic Banach rings, the
corresponding descent of finitely presented modules (with respect to the
underlying spaces) carrying the corresponding noncommutative coefficients
(such that the products of rings are noetherian as well) holds over étale
sites.

We also discussed the corresponding possible application to the descent in
certain cases over the corresponding $\infty$-analytic stacks after Bambozzi-
Ben-Bassat-Kremnizer [BBBK] and Bambozzi-Kremnizer [BK], where the latter is
really more related to the adic geometry we considered after [KL1] and [KL2].
Our approach certainly is definitely parallel to [KL1] and [KL2], which is not
actually parallel to more $\infty$-method considered by Ben-Bassat-Kremnizer
in [BBK].

###  1.2. Future Work

We expect more interesting applications. For instance partial noetherian
spaces could admit some desired descent carrying general noncommutative
coefficients (certainly not need to be noetherian coefficients), such as
different types of eigenvarieties or Shimura varieties.

##  2\. Descent over Analytic Adic Banach Rings in the Rational Setting

###  2.1. Noncommutative Functional-Analytic Pseudocoherence over Pro-Étale
Topology

Now we consider the corresponding discussion of the corresponding glueing
deformed pseudocoherent sheaves over étale topology which generalize the
corresponding discussion in [Ked1].

######  Setting 2.1.

Let $(W,W^{+})$ be Tate adic Banach uniform ring which is defined over
$\mathbb{Q}_{p}$. And we assume that the ring $V$ over $\mathbb{Q}_{p}$ is a
Banach ring over the $\mathbb{Q}_{p}$. Assume that the ring $W$ is sheafy.

######  Remark 2.2.

The corresponding glueing of the corresponding pseudocoherent sheaves over the
corresponding pro-étale site does not need the corresponding notions on the
corresponding stability beyond the corresponding étale-stably pseudocoherence.

######  Lemma 2.3.

(After Kedlaya-Liu [KL2, Proposition 3.4.3]) Suppose that we are taking
$(W,W^{+})$ to be Fontaine perfectoid (note that we are considering the
corresponding context of Tate adic Banach situation). Then we have that over
the corresponding étale site, the corresponding group
$H^{i}(\mathrm{Spa}(W,W^{+})_{\text{p\'{e}t}},\mathcal{O}\widehat{\otimes}V)$
vanishes for each $i>0$, while we have that the corresponding group
$H^{0}(\mathrm{Spa}(W,W^{+})_{\text{p\'{e}t}},\mathcal{O}\widehat{\otimes}V)$
is just isomorphic $W\widehat{\otimes}V$.

###### Proof.

See [KL2, Proposition 3.4.3], where the corresponding [KL2, Corollary 3.3.20]
applies. ∎

Now we proceed to consider the corresponding pro-étale topology. First we
recall the following result from [KL2, Lemma 3.4.4] which does hold in our
situation since we did not change much on the corresponding underlying adic
spaces.

######  Lemma 2.4.

(Kedlaya-Liu [KL2, Lemma 3.4.4]) Consider the ring $W$ as above which is
furthermore assumed to be Fontaine perfectoid in the sense of [KL2, Definition
3.3.1], and consider any direct system of faithfully finite étale morphisms
as:

$\textstyle{W_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...,}$

where $A_{0}$ is just the corresponding base $A$. Then the corresponding
completion of the infinite level could be decomposed as:

(2.1) $\displaystyle
W_{0}\oplus\widehat{\bigoplus}_{k=0}^{\infty}W_{k}/W_{k-1}.$

As in [KL2, Lemma 3.4.4] we endow the corresponding completion mentioned above
with the corresponding seminorm spectral. And for the corresponding quotient
we endow with the corresponding quotient norm.

Then we consider the corresponding deformation by the ring $V$ over
$\mathbb{Q}_{p}$:

######  Lemma 2.5.

(After Kedlaya-Liu [KL2, Lemma 3.4.4]) Consider the ring $A$ as above which is
furthermore assumed to be Fontaine perfectoid in the sense of [KL2, Definition
3.3.1], and consider any direct system of faithfully finite étale morphisms
as:

$\textstyle{W_{0,V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{1,V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{2,V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...,}$

where $W_{0}$ is just the corresponding base $W$. Then the corresponding
completion of the infinite level could be decomposed as:

(2.2) $\displaystyle
W_{0,V}\oplus\widehat{\bigoplus}_{k=0}^{\infty}W_{k,V}/W_{k-1,V}.$

######  Corollary 2.6.

(After Kedlaya-Liu [KL2, Corollary 3.4.5]) Starting with an adic Banach ring
which is as in [KL2, Corollary 3.4.5] assumed to be Fontaine perfectoid in the
sense of [KL2, Definition 3.3.1]. And as in the previous two lemmas we
consider an admissible infinite direct system:

$\textstyle{W_{0,V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{1,V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{2,V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...,}$

whose infinite level will be assumed to take the corresponding spectral
seminorm as in the [KL2, Corollary 3.4.5]. Then carrying the corresponding
coefficient $V$, we have in such situation the corresponding 2-pseudoflatness
of the corresponding embedding map:

(2.3) $\displaystyle W_{0,V}\rightarrow W_{\infty,V}.$

Here $W_{\infty,V}$ denotes the corresponding completion of the limit
$\varinjlim_{k\rightarrow\infty}W_{k,V}$.

###### Proof.

See [KL2, Corollary 3.4.5]. ∎

Now recall that from [KL2] since we do not have to modify the corresponding
underlying spatial context, so we will also only have to consider the
corresponding stability with respect to étale topology even although we are
considering the corresponding profinite étale site in our current section. We
first generalize the corresponding Tate acyclicity in [KL2] to the
corresponding $V$-relative situation in our current situation:

######  Remark 2.7.

In the following we only consider the corresponding perfectoid ring $W$.

######  Proposition 2.8.

(After Kedlaya-Liu [KL2, Theorem 3.4.6]) Now we consider the corresponding
pro-étale site of $X$ which we denote it by $X_{\text{p\'{e}t}}$. We assume we
have a stable basis $\mathbb{H}$ of the corresponding perfectoid subdomains
for this profinite étale site where each morphism therein will be étale
pseudoflat. We consider as in [KL2, Theorem 3.4.6] a corresponding module over
$W\widehat{\otimes}V$ which is assumed to be étale-stably pseudocoherent. Then
consider the corresponding presheaf $\widetilde{G}$ attached to this étale-
stably pseudocoherent module with respect to in our situation the
corresponding chosen well-defined basis $\mathbb{H}$, we have that the
corresponding sheaf over some element $Y\in\mathbb{H}$ (that is to say over
$\mathcal{O}_{Y_{\text{p\'{e}t}}}\widehat{\otimes}V$) is acyclic and is
acyclic with respect to some Čeck covering coming from elements in
$\mathbb{H}$.

###### Proof.

The corresponding proof could be made as in the corresponding nonrelative
setting as in [KL2, Theorem 3.4.6]. As in [KL2, Theorem 3.4.6] the
corresponding first step is to check the statement is true with respect to
some specific basis consisting of all the perfectoid subdomains which are
faithfully finite étale over some specific perfectoid subdomain $Y$ (along
towers). Then the corresponding statement could be reduced to those with
respect to covering which is simple Laurent rational and the corresponding
covering in the sense of the previous corollary. Then actually in this first
scope of consideration, the statement holds by previous corollary and [TX2,
Theorem 2.12]. Then in general for some general $Y$ we consider the
corresponding covering of $Y$ consisting of elements satisfying the previous
situation, then apply the previous argument to the corresponding basis in
current situation satisfying the previous situation Then we consider the
corresponding sheafification of the following chosen short exact sequence:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

with $G^{\prime}$ finite projective. Then the idea is to consider the
corresponding diagram factoring through the corresponding pro-étale
sheafification of this short exact sequence down to the corresponding étale
site. Then the five lemma will finish the corresponding proof as one considers
the following commutative diagram:

$\textstyle{G^{\prime\prime}_{V_{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{\prime}_{V_{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{V_{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma(Y_{\text{\'{e}t}},\widetilde{G^{\prime\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma(Y_{\text{\'{e}t}},\widetilde{G^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma(Y_{\text{\'{e}t}},\widetilde{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Here the corresponding ring $V_{Y}$ is the adic Banach ring such that
$Y=\mathrm{Spa}(V_{Y},V_{Y}^{+})$. ∎

######  Definition 2.9.

(After Kedlaya-Liu [KL2, Definition 3.4.7]) Consider the pro-étale site of $X$
attached to the adic Banach ring $(W,W^{+})$. We will define a sheaf of module
$G$ over $\widehat{\mathcal{O}}_{\text{p\'{e}t}}\widehat{\otimes}V$ to be
$V$-pseudocoherent if locally we can define this as a sheaf attached to a
$V$-étale-stably pseudocoherent module. As in [KL2, Definition 3.4.7], we do
not have to consider the corresponding notion of $V$-profinite-étale-stably
pseudocoherent module.

######  Proposition 2.10.

(After Kedlaya-Liu [KL2, Theorem 3.4.8]) Taking the corresponding global
section will realize the corresponding equivalence between the following two
categories. The first one is the corresponding one of all the
$V$-pseudocoherent sheaves over
$\widehat{\mathcal{O}}_{\text{p\'{e}t}}\widehat{\otimes}V$, while the second
one is the corresponding one of all the $V$-étale-stably pseudocoherent
modules over $W\widehat{\otimes}V$.

###### Proof.

As in [KL2, Theorem 3.4.8], the corresponding [KL1, Proposition 9.2.6] applies
in the way that the corresponding conditions of [KL1, Proposition 9.2.6] hold
in our current situation. ∎

Obviously we have the following analog of [KL2, Corollary 3.4.9]:

######  Corollary 2.11.

(After Kedlaya-Liu [KL2, Corollary 3.4.9]) The following two categories are
equivalent. The first is the corresponding category of all $V$-pseudocoherent
sheaves over $\widehat{\mathcal{O}}_{\text{p\'{e}t}}\widehat{\otimes}V$. The
second is the corresponding category of all $V$-pseudocoherent sheaves over
${\mathcal{O}}_{\text{\'{e}t}}\widehat{\otimes}V$. The corresponding functor
is the corresponding pullback along the corresponding morphism of sites
$X_{\text{p\'{e}t}}\rightarrow X_{\text{\'{e}t}}$.

##  3\. Descent over Analytic Huber Rings in the Integral Setting

###  3.1. Noncommutative Topological Pseudocoherence over Étale Topology

Now we consider the corresponding discussion of the corresponding glueing
deformed pseudocoherent sheaves over étale topology which generalize the
corresponding discussion in [Ked1]. And we will consider the corresponding
situation as assumed in the following setting:

######  Setting 3.1.

Let $(W,W^{+})$ be analytic Huber uniform pair which is defined over
$\mathbb{Z}_{p}$. And we assume that the ring $V$ over $\mathbb{Z}_{p}$ is a
topological ring (complete) and splitting over the $\mathbb{Z}_{p}$. We now
fix a corresponding stable basis $\mathbb{H}$ for the corresponding étale site
of the adic space $\mathrm{Spa}(W,W^{+})$, locally consisting of the
corresponding compositions of rational localizations and the corresponding
finite étale morphisms. And as in [Ked1, Hypothesis 1.10.3] we need to assume
that the corresponding basis is made up of the corresponding adic spectrum of
sheafy rings. Assume the corresponding sheafiness of the Huber ring $W$.

######  Definition 3.2.

(After Kedlaya-Liu [KL2, Definition 2.5.9]) We define a $V$-étale stably
pseudocoherent module over the corresponding Huber ring $W$ with respect to
the corresponding basis $\mathbb{H}$ chosen above for the corresponding étale
site of the analytic adic space $X$. We define a module over
$W\widehat{\otimes}V$ is a $V$-étale stably pseudocoherent module if it is
algebraically pseudocoherent (namely formed by the corresponding possibly
infinite length resolution of finitely generated and projective modules) and
at least complete with respect to the corresponding natural topology and also
required to be also complete with respect to the natural topology along some
base change with respect to any morphism in $\mathbb{H}$.

######  Definition 3.3.

(After Kedlaya-Liu [KL2, Definition 2.5.9]) Along the previous definition we
have the corresponding notion of $V$-étale-pseudoflat left (or right
respectively) modules with respect to the corresponding chosen basis
$\mathbb{H}$. A such module is defined to be a topological module $G$ over
$W\widehat{\otimes}V$ complete with respect to the natural topology and for
any right (or left respectively) $V$-étale stably pseudocoherent module, they
will jointly give the $\mathrm{Tor}_{1}$ vanishing.

######  Lemma 3.4.

(After Kedlaya-Liu [KL2, Lemma 2.5.10]) One could find another basis
$\mathbb{H}^{\prime}$ which is in our situation contained in the corresponding
original basis $\mathbb{H}$ such that any morphism in $\mathbb{H}$ could be
$V$-étale-pseudoflat with respect $\mathbb{H}^{\prime}$ or just $\mathbb{H}$
itself.

###### Proof.

The derivation of such new basis is actually along the same way as in [KL2,
Lemma 2.5.10] since the corresponding compositions of rational localizations
and the finite étales are actually satisfying the corresponding conditions in
the statement by [TX2, Theorem 2.12] over the analytic topology. In general
one just show any general morphism will also decompose in the same way, which
one can proves certainly as in [KL2, Lemma 2.5.10], where one instead consider
in the current context the corresponding basis spreading result in [Ked1,
Lemma 1.10.4]. ∎

######  Proposition 3.5.

(After Kedlaya-Liu [KL2, Theorem 2.5.11]) Consider the site
$X_{\text{\'{e}t}}$ and consider the basis $\mathbb{H}$. Take any $V$-étale
stably pseudocoherent module $M$ over $W\widehat{\otimes}V$. Consider the
corresponding presheaf by taking the inverse limit throughout all the
corresponding base change along morphisms in $\mathbb{H}$. Then we have the
corresponding acyclicity of the presheaf over any element in $\mathbb{H}$ and
any covering of this element by elements in $\mathbb{H}$.

###### Proof.

As in [KL2, Theorem 2.5.11] apply the corresponding [KL1, Proposition 8.2.21].
∎

######  Proposition 3.6.

(After Kedlaya-Liu [KL2, Lemma 2.5.13]) The corresponding glueing of
$V$-étale-stably pseudocoherent modules holds in this current situation over
$W$ along binary morphisms (namely along binary rational decomposition).

###### Proof.

We adapt the argument of [KL2, Lemma 2.5.13] to our current situation. Take
the corresponding map to be $W\rightarrow W_{1}\bigoplus W_{2}$, and we denote
the two spaces associated $W_{1}$ and $W_{2}$ by $Y_{1}$ and $Y_{2}$.
Establish a corresponding descent datum of $V$-étale-stably pseudocoherent
modules along this decomposition of $\mathrm{Spa}(W,W^{+})$. One can realize
this by acyclicity a sheaf $G$ over the whole space $\mathrm{Spa}(W,W^{+})$
Then we have for each $k=1,2$ that the corresponding identification of
$H^{0}(Y_{k,\text{\'{e}t}},G)$ with $H^{0}(Y_{k},G)$ and the vanishing of the
$i$-th cohomology for higher $i$. By the corresponding results we have in the
corresponding rational localization situation we vanishing of the
corresponding $H^{i}(\mathrm{Spa}(W,W^{+}),G),i\geq 1$. Now consider a finite
free covering $C$ of $G$ and take the kernel $K$ to form the corresponding
exact sequence:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

which will give rise to the following short exact sequence:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K(\mathrm{Spa}(W,W^{+}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C(\mathrm{Spa}(W,W^{+}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G(\mathrm{Spa}(W,W^{+}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

To finish we have also to show the corresponding resulting global section is
actually $V$-étale-stably pseudocoherent. For this, we refer the readers to
[TX2, Lemma 2.22].

∎

######  Proposition 3.7.

(After Kedlaya-Liu [KL2, Theorem 2.5.14]) Taking the corresponding global
section will realize the equivalence between the following two categories. The
first is the one of all the corresponding $V$-pseudocoherent sheaves over
$\mathcal{O}_{\text{\'{e}t}}\widehat{\otimes}V$. The second is the one of all
the corresponding $V$-étale-stably pseudocoherent modules.

###### Proof.

This is by considering and applying the corresponding [Ked1, Lemma 1.10.4] in
our current context. The refinement comes from the acyclicity, the
transitivity is straightforward, the corresponding binary rational
localization situation is the previous proposition, while the corresponding
last condition will basically comes from the corresponding f.p. descent [SGAI,
Chapitre VIII]. ∎

###  3.2. Noncommutative Topological Pseudocoherence over Pro-Étale Topology

######  Setting 3.8.

Let $(W,W^{+})$ be analytic Huber ring which is defined over $\mathbb{Z}_{p}$.
And we assume that the ring $V$ over $\mathbb{Z}_{p}$ is a complete
topological ring over the $\mathbb{Z}_{p}$ which is completely exact. Assume
that the ring $W$ is sheafy. We assume $p$ is a topologically nilpotent
element.

######  Remark 3.9.

The corresponding glueing of the corresponding pseudocoherent sheaves over the
corresponding pro-étale site does not need the corresponding notions on the
corresponding stability beyond the corresponding étale-stably pseudocoherence.

######  Lemma 3.10.

(After Kedlaya-Liu [KL2, Proposition 3.4.3]) Suppose that we are taking
$(W,W^{+})$ to be perfectoid now in the sense of [Ked1, Definition 2.1.1]
(note that we are considering the corresponding context of analytic Huber pair
situation). Then we have that over the corresponding étale site, the
corresponding group
$H^{i}(\mathrm{Spa}(W,W^{+})_{\text{p\'{e}t}},\mathcal{O}\widehat{\otimes}V)$
vanishes for each $i>0$, while we have that the corresponding group
$H^{0}(\mathrm{Spa}(W,W^{+})_{\text{p\'{e}t}},\mathcal{O}\widehat{\otimes}V)$
is just isomorphic $W\widehat{\otimes}V$.

###### Proof.

See [KL2, Proposition 3.4.3]. ∎

Now we proceed to consider the corresponding pro-étale topology. First we
recall the following result from [KL2, Lemma 3.4.4] which does hold in our
situation since we did not change much on the corresponding underlying adic
spaces.

######  Lemma 3.11.

(Kedlaya-Liu [KL2, Lemma 3.4.4]) Consider the ring $W$ as above which is
furthermore assumed to be perfectoid in the sense of instead [Ked1, Definition
2.1.1], and consider any direct system of faithfully finite étale morphisms
as:

$\textstyle{W_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...,}$

where $A_{0}$ is just the corresponding base $A$. Then the corresponding
completion of the infinite level could be decomposed as:

(3.1) $\displaystyle
W_{0}\oplus\widehat{\bigoplus}_{k=0}^{\infty}W_{k}/W_{k-1}.$

As in [KL2, Lemma 3.4.4] we endow the corresponding completion mentioned above
with the corresponding seminorm spectral. And for the corresponding quotient
we endow with the corresponding quotient norm.

Then we consider the corresponding deformation by the ring $V$ over
$\mathbb{Z}_{p}$:

######  Lemma 3.12.

(After Kedlaya-Liu [KL2, Lemma 3.4.4]) Consider the ring $A$ as above which is
furthermore assumed to be perfectoid in the sense of [Ked1, Definition 2.1.1],
and consider any direct system of faithfully finite étale morphisms as:

$\textstyle{W_{0,V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{1,V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{2,V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...,}$

where $W_{0}$ is just the corresponding base $W$. Then the corresponding
completion of the infinite level could be decomposed as:

(3.2) $\displaystyle
W_{0,V}\oplus\widehat{\bigoplus}_{k=0}^{\infty}W_{k,V}/W_{k-1,V}.$

######  Corollary 3.13.

(After Kedlaya-Liu [KL2, Corollary 3.4.5]) Starting with an analytic Huber
ring which is as in [KL2, Corollary 3.4.5] assumed to be perfectoid in the
sense of [Ked1, Definition 2.1.1]. And as in the previous two lemmas we
consider an admissible infinite direct system:

$\textstyle{W_{0,V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{1,V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{2,V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...,}$

whose infinite level will be assumed to take the corresponding spectral
seminorm as in the [KL2, Corollary 3.4.5]. Then carrying the corresponding
coefficient $V$, we have in such situation the corresponding 2-pseudoflatness
of the corresponding embedding map:

(3.3) $\displaystyle W_{0,V}\rightarrow W_{\infty,V}.$

Here $W_{\infty,V}$ denotes the corresponding completion of the limit
$\varinjlim_{k\rightarrow\infty}W_{k,V}$.

###### Proof.

See [KL2, Corollary 3.4.5]. ∎

Now recall that from [KL2] since we do not have to modify the corresponding
underlying spatial context, so we will also only have to consider the
corresponding stability with respect to étale topology even although we are
considering the corresponding profinite étale site in our current section. We
first generalize the corresponding Tate acyclicity in [KL2] to the
corresponding $V$-relative situation in our current situation:

######  Remark 3.14.

In the following we only consider the corresponding perfectoid ring $W$.

######  Proposition 3.15.

(After Kedlaya-Liu [KL2, Theorem 3.4.6]) Now we consider the corresponding
pro-étale site of $X$ which we denote it by $X_{\text{p\'{e}t}}$. Then we
assume we have a stable basis $\mathbb{H}$ of the corresponding perfectoid
subdomains for this profinite étale site where each morphism therein will be
étale pseudoflat. Then we consider as in [KL2, Theorem 3.4.6] a corresponding
module over $W\widehat{\otimes}V$ which is assumed to be étale-stably
pseudocoherent. Then consider the corresponding presheaf $\widetilde{G}$
attached to this étale-stably pseudocoherent with respect to in our situation
the corresponding chosen well-defined basis $\mathbb{H}$, we have that the
corresponding sheaf over some element $Y\in\mathbb{H}$ (that is to say over
$\mathcal{O}_{Y_{\text{p\'{e}t}}}\widehat{\otimes}V$) is acyclic and is
acyclic with respect to some Čeck covering coming from elements in
$\mathbb{H}$.

###### Proof.

See proposition 2.8. ∎

######  Definition 3.16.

(After Kedlaya-Liu [KL2, Definition 3.4.7]) Consider the pro-étale site of $X$
attached to the adic Banach ring $(W,W^{+})$. We will define a sheaf of module
$G$ over $\widehat{\mathcal{O}}_{\text{p\'{e}t}}\widehat{\otimes}V$ to be
$V$-pseudocoherent if locally we can define this as a sheaf attached to a
$V$-étale-stably pseudocoherent module. As in [KL2, Definition 3.4.7], we do
not have to consider the corresponding notion of $V$-profinite-étale-stably
pseudocoherent module.

######  Proposition 3.17.

(After Kedlaya-Liu [KL2, Theorem 3.4.8]) Taking the corresponding global
section will realize the corresponding equivalence between the following two
categories. The first one is the corresponding one of all the
$V$-pseudocoherent sheaves over
$\widehat{\mathcal{O}}_{\text{p\'{e}t}}\widehat{\otimes}V$, while the second
one is the corresponding one of all the $V$-étale-stably pseudocoherent module
over $W\widehat{\otimes}V$.

###### Proof.

As in [KL2, Theorem 3.4.8], the corresponding analog of [KL1, Proposition
9.2.6] applies in the way that the corresponding conditions of the analog of
[KL1, Proposition 9.2.6] hold in our current situation. Also see [Ked1,
Theorem 2.9.9, Remark 2.9.10 and Lemma 1.10.4]. ∎

Obviously we have the following analog of [KL2, Corollary 3.4.9]:

######  Corollary 3.18.

(After Kedlaya-Liu [KL2, Corollary 3.4.9]) The following two categories are
equivalent. The first is the corresponding category of all $V$-pseudocoherent
sheaves over $\widehat{\mathcal{O}}_{\text{p\'{e}t}}\widehat{\otimes}V$. The
second is the corresponding category of all $V$-pseudocoherent sheaves over
${\mathcal{O}}_{\text{\'{e}t}}\widehat{\otimes}V$. The corresponding functor
is the corresponding pullback along the corresponding morphism of sites
$X_{\text{p\'{e}t}}\rightarrow X_{\text{\'{e}t}}$.

##  4\. Descent in General Setting for Analytic Huber Pairs

We now consider the situation where there is no condition on the base (except
that we will assume certainly the corresponding sheafiness). We will discuss
along [Ked1] the corresponding descent in analytic and étale topology, as well
as the corresponding quasi-Stein spaces. The current situation will not assume
we are working over $\mathbb{Z}_{p}$.

######  Remark 4.1.

The corresponding discussion in this section and in the following section is
actually an exercise proposed by Kedlaya included in [Ked1], where the
corresponding étale situation essentially is proposed by Kedlaya in [Ked1,
Discussion after Lemma 1.10.4] while we give the corresponding exposition in
both étale and pro-étale situations closely after [KL1] and [KL2].

###  4.1. Étale Topology

######  Setting 4.2.

Let $(W,W^{+})$ be analytic Huber uniform pair. We now fix a corresponding
stable basis $\mathbb{H}$ for the corresponding étale site of the adic space
$\mathrm{Spa}(W,W^{+})$, locally consisting of the corresponding compositions
of rational localizations and the corresponding finite étale morphisms. And as
in [Ked1, Hypothesis 1.10.3] we need to assume that the corresponding basis is
made up of the corresponding adic spectrum of sheafy rings. Assume the
corresponding sheafiness of the Huber ring $W$.

######  Remark 4.3.

One should treat our discussion in this section as essentially some
detailization of some exercise predicted in [Ked1].

######  Definition 4.4.

(After Kedlaya-Liu [KL2, Definition 2.5.9]) We define the étale stably
pseudocoherent module over the corresponding Huber ring $W$ with respect to
the corresponding basis $\mathbb{H}$ chosen above for the corresponding étale
site of the analytic adic space $X$. We define a module over $W$ to be an
étale stably pseudocoherent module if it is algebraically pseudocoherent
(namely formed by the corresponding possibly infinite length resolution of
finitely generated and projective modules) and at least complete with respect
to the corresponding natural topology and also required to be also complete
with respect to the natural topology along some base change with respect to
any morphism in $\mathbb{H}$.

######  Definition 4.5.

(After Kedlaya-Liu [KL2, Definition 2.5.9]) Along the previous definition we
have the corresponding notion of étale-pseudoflat left (or right respectively)
modules with respect to the corresponding chosen basis $\mathbb{H}$. A such
module is defined to be a topological module $G$ over $W$ complete with
respect to the natural topology and for any right (or left respectively) étale
stably pseudocoherent module, they will jointly give the $\mathrm{Tor}_{1}$
vanishing.

######  Remark 4.6.

Although in this definition we considered the corresponding left or right
modules, but essentially speaking this is no different from the usual
situation since we are in the corresponding essential site theoretic
situation.

######  Lemma 4.7.

(After Kedlaya-Liu [KL2, Lemma 2.5.10]) One could find another basis
$\mathbb{H}^{\prime}$ which is in our situation contained in the corresponding
original basis $\mathbb{H}$ such that any morphism in $\mathbb{H}$ could be
étale-pseudoflat with respect $\mathbb{H}^{\prime}$ or just $\mathbb{H}$
itself.

###### Proof.

The derivation of such new basis is actually along the same way as in [KL2,
Lemma 2.5.10] since the corresponding compositions of rational localizations
and the finite étales are actually satisfying the corresponding conditions in
the statement by [TX2, Theorem 2.12] over the analytic topology. In general
one just show any general morphism will also decompose in the same way, which
one can proves certainly as in [KL2, Lemma 2.5.10], where one instead consider
in the current context the corresponding basis spreading result in [Ked1,
Lemma 1.10.4]. ∎

######  Proposition 4.8.

(After Kedlaya-Liu [KL2, Theorem 2.5.11]) Consider the site
$X_{\text{\'{e}t}}$ and consider the basis $\mathbb{H}$. Take any étale stably
pseudocoherent module $M$ over $W$. Consider the corresponding presheaf by
taking the inverse limit throughout all the corresponding base change along
morphisms in $\mathbb{H}$. Then we have the corresponding acyclicity of the
presheaf over any element in $\mathbb{H}$ and any covering of this element by
elements in $\mathbb{H}$.

###### Proof.

As in [KL2, Theorem 2.5.11] apply the corresponding [KL1, Proposition 8.2.21].
∎

######  Proposition 4.9.

(After Kedlaya-Liu [KL2, Lemma 2.5.13]) The corresponding glueing of étale-
stably pseudocoherent modules holds in this current situation over $W$ along
binary morphisms (namely along binary rational decomposition).

###### Proof.

We adapt the argument of [KL2, Lemma 2.5.13] to our current situation. Take
the corresponding map to be $W\rightarrow W_{1}\bigoplus W_{2}$, and we denote
the two spaces associated $W_{1}$ and $W_{2}$ by $Y_{1}$ and $Y_{2}$.
Establish a corresponding descent datum of $V$-étale-stably pseudocoherent
modules along this decomposition of $\mathrm{Spa}(W,W^{+})$. One can realize
this by acyclicity a sheaf $G$ over the whole space $\mathrm{Spa}(W,W^{+})$
Then we have for each $k=1,2$ that the corresponding identification of
$H^{0}(Y_{k,\text{\'{e}t}},G)$ with $H^{0}(Y_{k},G)$ and the vanishing of the
$i$-th cohomology for higher $i$. By the corresponding results we have in the
corresponding rational localization situation we have vanishing of the
corresponding $H^{i}(\mathrm{Spa}(W,W^{+}),G),i\geq 1$. Now consider a finite
free covering $C$ of $G$ and take the kernel $K$ to form the corresponding
exact sequence:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

which will give rise to the following short exact sequence:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K(\mathrm{Spa}(W,W^{+}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C(\mathrm{Spa}(W,W^{+}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G(\mathrm{Spa}(W,W^{+}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

To finish we have also to show the corresponding resulting global section is
actually étale-stably pseudocoherent. For this, we refer the readers to [TX2,
Lemma 2.22].

∎

######  Proposition 4.10.

(Kedlaya [Ked1, Below Lemma 1.10.4], After Kedlaya-Liu [KL2, Theorem 2.5.14])
Taking the corresponding global section will realize the equivalence between
the following two categories. The first is the one of all the corresponding
pseudocoherent sheaves over $\mathcal{O}_{\text{\'{e}t}}$. The second is the
one of all the corresponding étale-stably pseudocoherent modules.

###### Proof.

This is by considering and applying the corresponding [Ked1, Lemma 1.10.4] in
our current context. The refinement comes from the acyclicity, the
transitivity is straightforward, the corresponding binary rational
localization situation is the previous proposition, while the corresponding
last condition will basically come from the corresponding f.p. descent [SGAI,
Chapitre VIII]. ∎

###  4.2. Pro-étale Topology

######  Setting 4.11.

Let $(W,W^{+})$ be an analytic Huber ring. Assume that the ring $W$ is sheafy.
We assume $p$ is a topologically nilpotent element.

######  Remark 4.12.

Although it seems that the current discussion is basically outside the
corresponding deformed setting, but the foundation here may be generalized to
the deformed setting by taking the corresponding completed tensor product over
$\mathbb{F}_{1}$ as in [BBBK].

######  Remark 4.13.

The corresponding glueing of the corresponding pseudocoherent sheaves over the
corresponding pro-étale site does not need the corresponding notions on the
corresponding stability beyond the corresponding étale-stably pseudocoherence.

######  Lemma 4.14.

(After Kedlaya-Liu [KL2, Proposition 3.4.3]) Suppose that we are taking
$(W,W^{+})$ to be perfectoid now in the sense of [Ked1, Definition 2.1.1]
(note that we are considering the corresponding context of analytic Huber pair
situation). Then we have that over the corresponding étale site, the
corresponding group
$H^{i}(\mathrm{Spa}(W,W^{+})_{\text{p\'{e}t}},\mathcal{O})$ vanishes for each
$i>0$, while we have that the corresponding group
$H^{0}(\mathrm{Spa}(W,W^{+})_{\text{p\'{e}t}},\mathcal{O})$ is just isomorphic
to $W$.

###### Proof.

See [KL2, Proposition 3.4.3]. ∎

Now we proceed to consider the corresponding pro-étale topology. First we
recall the following result from [KL2, Lemma 3.4.4] which does hold in our
situation since we did not change much on the corresponding underlying adic
spaces.

######  Lemma 4.15.

(Kedlaya-Liu [KL2, Lemma 3.4.4]) Consider the ring $W$ as above which is
furthermore assumed to be perfectoid in the sense of instead [Ked1, Definition
2.1.1], and consider any direct system of faithfully finite étale morphisms
as:

$\textstyle{W_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...,}$

where $W_{0}$ is just the corresponding base $W$. Then the corresponding
completion of the infinite level could be decomposed as:

(4.1) $\displaystyle
W_{0}\oplus\widehat{\bigoplus}_{k=0}^{\infty}W_{k}/W_{k-1}.$

As in [KL2, Lemma 3.4.4] we endow the corresponding completion mentioned above
with the corresponding seminorm spectral. And for the corresponding quotient
we endow with the corresponding quotient norm.

######  Corollary 4.16.

(After Kedlaya-Liu [KL2, Corollary 3.4.5]) Starting with an analytic Huber
ring which is as in [KL2, Corollary 3.4.5] assumed to be perfectoid in the
sense of [Ked1, Definition 2.1.1]. And as in the previous two lemmas we
consider an admissible infinite direct system:

$\textstyle{W_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...,}$

whose infinite level will be assumed to take the corresponding spectral
seminorm as in the [KL2, Corollary 3.4.5]. Then we have in such situation the
corresponding 2-pseudoflatness of the corresponding embedding map:

(4.2) $\displaystyle W_{0}\rightarrow W_{\infty}.$

Here $W_{\infty}$ denotes the corresponding completion of the limit
$\varinjlim_{k\rightarrow\infty}W_{k}$.

###### Proof.

See [KL2, Corollary 3.4.5]. ∎

Now recall that from [KL2] since we do not have to modify the corresponding
underlying spatial context, so we will also only have to consider the
corresponding stability with respect to étale topology even although we are
considering the corresponding profinite étale site in our current section. We
first generalize the corresponding Tate acyclicity in [KL2] to the
corresponding our current situation.

######  Remark 4.17.

In the following we only consider the base spaces which are perfectoid.

######  Proposition 4.18.

(After Kedlaya-Liu [KL2, Theorem 3.4.6]) Now we consider the corresponding
pro-étale site of $X$ which we denote it by $X_{\text{p\'{e}t}}$. Then we
assume we have a stable basis $\mathbb{H}$ of the corresponding perfectoid
subdomains for this profinite étale site where each morphism therein will be
étale pseudoflat. Then we consider as in [KL2, Theorem 3.4.6] a corresponding
module over $W$ which is assumed to be étale-stably pseudocoherent. Then
consider the corresponding presheaf $\widetilde{G}$ attached to this étale-
stably pseudocoherent with respect to in our situation the corresponding
chosen well-defined basis $\mathbb{H}$, we have that the corresponding sheaf
over some element $Y\in\mathbb{H}$ (that is to say over
$\mathcal{O}_{Y_{\text{p\'{e}t}}}$) is acyclic and is acyclic with respect to
some Čeck covering coming from elements in $\mathbb{H}$.

###### Proof.

See proposition 2.8. ∎

######  Definition 4.19.

(After Kedlaya-Liu [KL2, Definition 3.4.7]) Consider the pro-étale site of $X$
attached to the analytic Huber pair $(W,W^{+})$. We will define a sheaf of
module $G$ over $\widehat{\mathcal{O}}_{\text{p\'{e}t}}$ to be pseudocoherent
if locally we can define this as a sheaf attached to an étale-stably
pseudocoherent module. As in [KL2, Definition 3.4.7], we do not have to
consider the corresponding notion of profinite-étale-stably pseudocoherent
modules.

######  Proposition 4.20.

(Kedlaya [Ked1, Section 3.8], after Kedlaya-Liu [KL2, Theorem 3.4.8]) Taking
the corresponding global section will realize the corresponding equivalence
between the following two categories. The first one is the corresponding one
of all the pseudocoherent sheaves over
$\widehat{\mathcal{O}}_{\text{p\'{e}t}}$, while the second one is the
corresponding one of all the étale-stably pseudocoherent modules over $W$.

###### Proof.

As in [KL2, Theorem 3.4.8], the corresponding analog of [KL1, Proposition
9.2.6] applies in the way that the corresponding conditions of the analog of
[KL1, Proposition 9.2.6] hold in our current situation. Also see [Ked1,
Theorem 2.9.9, Remark 2.9.10 and Lemma 1.10.4]. ∎

Obviously we have the following analog of [KL2, Corollary 3.4.9]:

######  Corollary 4.21.

(After Kedlaya-Liu [KL2, Corollary 3.4.9]) The following two categories are
equivalent. The first is the corresponding category of all pseudocoherent
sheaves over $\widehat{\mathcal{O}}_{\text{p\'{e}t}}$. The second is the
corresponding category of all pseudocoherent sheaves over
${\mathcal{O}}_{\text{\'{e}t}}$. The corresponding functor is the
corresponding pullback along the corresponding morphism of sites
$X_{\text{p\'{e}t}}\rightarrow X_{\text{\'{e}t}}$.

##  5\. Descent in General Setting for Analytic Adic Banach Rings

We now translate the results in previous section to the corresponding analytic
adic Banach context. Again everything is essentially as proposed in [Ked1,
Below Lemma 1.10.4, also see Section 3.8].

###  5.1. Étale Topology

######  Setting 5.1.

Let $(W,W^{+})$ be analytic adic Banach ring. We now fix a corresponding
stable basis $\mathbb{H}$ for the corresponding étale site of the adic space
$\mathrm{Spa}(W,W^{+})$, locally consisting of the corresponding compositions
of rational localizations and the corresponding finite étale morphisms. And as
in [Ked1, Hypothesis 1.10.3] we need to assume that the corresponding basis is
made up of the corresponding adic spectrum of sheafy rings. Assume the
corresponding sheafiness of the Huber ring $W$.

######  Definition 5.2.

(After Kedlaya-Liu [KL2, Definition 2.5.9]) We define the étale-stably
pseudocoherent module over the corresponding adic Banach $W$ with respect to
the corresponding basis $\mathbb{H}$ chosen above for the corresponding étale
site of the analytic adic space $X$. We define a module over $W$ to be an
étale stably pseudocoherent module if it is algebraically pseudocoherent
(namely formed by the corresponding possibly infinite length resolution of
finitely generated and projective modules) and at least complete with respect
to the corresponding natural topology and also required to be also complete
with respect to the natural topology along some base change with respect to
any morphism in $\mathbb{H}$.

######  Definition 5.3.

(After Kedlaya-Liu [KL2, Definition 2.5.9]) Along the previous definition we
have the corresponding notion of étale-pseudoflat left (or right respectively)
modules with respect to the corresponding chosen basis $\mathbb{H}$. A such
module is defined to be a topological module $G$ over $W$ complete with
respect to the natural topology and for any right (or left respectively) étale
stably pseudocoherent module, they will jointly give the $\mathrm{Tor}_{1}$
vanishing.

######  Lemma 5.4.

(After Kedlaya-Liu [KL2, Lemma 2.5.10]) One could find another basis
$\mathbb{H}^{\prime}$ which is in our situation contained in the corresponding
original basis $\mathbb{H}$ such that any morphism in $\mathbb{H}$ could be
étale-pseudoflat with respect $\mathbb{H}^{\prime}$ or just $\mathbb{H}$
itself.

###### Proof.

See lemma 4.7. ∎

######  Proposition 5.5.

(After Kedlaya-Liu [KL2, Theorem 2.5.11]) In our current analytic adic Banach
situation, consider the site $X_{\text{\'{e}t}}$ and consider the basis
$\mathbb{H}$. Take any étale stably pseudocoherent module $M$ over $W$.
Consider the corresponding presheaf by taking the inverse limit throughout all
the corresponding base change along morphisms in $\mathbb{H}$. Then we have
the corresponding acyclicity of the presheaf over any element in $\mathbb{H}$
and any covering of this element by elements in $\mathbb{H}$.

###### Proof.

As in [KL2, Theorem 2.5.11] apply the corresponding [KL1, Proposition 8.2.21].
∎

######  Proposition 5.6.

(After Kedlaya-Liu [KL2, Lemma 2.5.13]) In our current analytic adic Banach
situation, the corresponding glueing of étale-stably pseudocoherent modules
holds in this current situation over $W$ along binary morphisms (namely along
binary rational decomposition).

###### Proof.

We adapt the argument of [KL2, Lemma 2.5.13] to our current situation over
analytic adic Banach rings. Take the corresponding map to be $W\rightarrow
W_{1}\bigoplus W_{2}$, and we denote the two spaces associated $W_{1}$ and
$W_{2}$ by $Y_{1}$ and $Y_{2}$. Establish a corresponding descent datum of
$V$-étale-stably pseudocoherent modules along this decomposition of
$\mathrm{Spa}(W,W^{+})$. One can realize this by acyclicity a sheaf $G$ over
the whole space $\mathrm{Spa}(W,W^{+})$ Then we have for each $k=1,2$ that the
corresponding identification of $H^{0}(Y_{k,\text{\'{e}t}},G)$ with
$H^{0}(Y_{k},G)$ and the vanishing of the $i$-th cohomology for higher $i$. By
the corresponding results we have in the corresponding rational localization
situation we vanishing of the corresponding
$H^{i}(\mathrm{Spa}(W,W^{+}),G),i\geq 1$. Now consider a finite free covering
$C$ of $G$ and take the kernel $K$ to form the corresponding exact sequence:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

which will give rise to the following short exact sequence:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K(\mathrm{Spa}(W,W^{+}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C(\mathrm{Spa}(W,W^{+}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G(\mathrm{Spa}(W,W^{+}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

To finish we have also to show the corresponding resulting global section is
actually étale-stably pseudocoherent. For this, we refer the readers to [TX2,
Lemma 2.22].

∎

######  Proposition 5.7.

(After Kedlaya-Liu [KL2, Theorem 2.5.14]) In our current analytic adic Banach
situation, taking the corresponding global section will realize the
equivalence between the following two categories. The first is the one of all
the corresponding pseudocoherent sheaves over $\mathcal{O}_{\text{\'{e}t}}$.
The second is the one of all the corresponding étale-stably pseudocoherent
modules.

###### Proof.

This is by considering and applying the corresponding [Ked1, Lemma 1.10.4] in
our current context. The refinement comes from the acyclicity, the
transitivity is straightforward, the corresponding binary rational
localization situation is the previous proposition, while the corresponding
last condition will basically comes from the corresponding f.p. descent [SGAI,
Chapitre VIII]. ∎

###  5.2. Pro-étale Topology

######  Setting 5.8.

Let $(W,W^{+})$ be an analytic perfectoid adic Banach ring. Assume that the
ring $W$ is sheafy. We assume $p$ is a topologically nilpotent element.

######  Remark 5.9.

As in the corresponding analytic Huber ring situation, the corresponding
glueing of the corresponding pseudocoherent sheaves over the corresponding
pro-étale site does not need the corresponding notions on the corresponding
stability beyond the corresponding étale-stably pseudocoherence.

We translate the corresponding notions of perfectoid rings to the current
context from [Ked1, Definition 2.1.1]:

######  Definition 5.10.

(Kedlaya [Ked1, Definition 2.1.1]) Consider a general analytic adic Banach
ring $(W,W^{+})$, we will call it perfectoid if there exists a definition
ideal $d\subset W^{+}$ such that $d^{p}$ contains $p$ and we have the
corresponding Frobenius map from $W^{+}/d$ to $W^{+}/d^{p}$ is required to be
surjective.

######  Lemma 5.11.

(After Kedlaya-Liu [KL2, Proposition 3.4.3]) Suppose that we are taking
$(W,W^{+})$ to be perfectoid now in the sense of definition 5.10 (note that we
are considering the corresponding context of analytic adic Banach situation).
Then we have that over the corresponding étale site, the corresponding group
$H^{i}(\mathrm{Spa}(W,W^{+})_{\text{p\'{e}t}},\mathcal{O})$ vanishes for each
$i>0$, while we have that the corresponding group
$H^{0}(\mathrm{Spa}(W,W^{+})_{\text{p\'{e}t}},\mathcal{O})$ is just isomorphic
$W$.

###### Proof.

See [KL2, Proposition 3.4.3]. ∎

Now we proceed to consider the corresponding pro-étale topology. First we
recall the following result from [KL2, Lemma 3.4.4].

######  Lemma 5.12.

(Kedlaya-Liu [KL2, Lemma 3.4.4]) Consider the ring $W$ as above which is
furthermore assumed to be perfectoid in the sense of instead definition 5.10,
and consider any direct system of faithfully finite étale morphisms as:

$\textstyle{W_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...,}$

where $W_{0}$ is just the corresponding base $W$. Then the corresponding
completion of the infinite level could be decomposed as:

(5.1) $\displaystyle
W_{0}\oplus\widehat{\bigoplus}_{k=0}^{\infty}W_{k}/W_{k-1}.$

As in [KL2, Lemma 3.4.4] we endow the corresponding completion mentioned above
with the corresponding seminorm spectral. And for the corresponding quotient
we endow with the corresponding quotient norm.

######  Corollary 5.13.

(After Kedlaya-Liu [KL2, Corollary 3.4.5]) Starting with an analytic adic
Banach ring which is as in [KL2, Corollary 3.4.5] assumed to be perfectoid in
the sense of definition 5.10. And as in the previous two lemmas we consider an
admissible infinite direct system:

$\textstyle{W_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...,}$

whose infinite level will be assumed to take the corresponding spectral
seminorm as in the [KL2, Corollary 3.4.5]. Then we have in such situation the
corresponding 2-pseudoflatness of the corresponding embedding map:

(5.2) $\displaystyle W_{0}\rightarrow W_{\infty}.$

Here $W_{\infty}$ denotes the corresponding completion of the limit
$\varinjlim_{k\rightarrow\infty}W_{k}$.

###### Proof.

See [KL2, Corollary 3.4.5]. ∎

Now recall that from [KL2] since we do not have modified the corresponding
underlying spatial context, so we will also only have to consider the
corresponding stability with respect to étale topology even although we are
considering the corresponding profinite étale site in our current section. We
first generalize the corresponding Tate acyclicity in [KL2] to the
corresponding our current situation:

######  Proposition 5.14.

(After Kedlaya-Liu [KL2, Theorem 3.4.6]) Now we consider the corresponding
pro-étale site of $X$ which we denote it by $X_{\text{p\'{e}t}}$. Then we
assume we have a stable basis $\mathbb{H}$ of the corresponding perfectoid
subdomains for this profinite étale site where each morphism therein will be
étale pseudoflat. Then we consider as in [KL2, Theorem 3.4.6] a corresponding
module over $W$ which is assumed to be étale-stably pseudocoherent. Then
consider the corresponding presheaf $\widetilde{G}$ attached to this étale-
stably pseudocoherent with respect to in our situation the corresponding
chosen well-defined basis $\mathbb{H}$, we have that the corresponding sheaf
over some element $Y\in\mathbb{H}$ (that is to say over
$\mathcal{O}_{Y_{\text{p\'{e}t}}}$) is acyclic and is acyclic with respect to
some Čeck covering coming from elements in $\mathbb{H}$.

###### Proof.

See proposition 2.8. ∎

######  Definition 5.15.

(After Kedlaya-Liu [KL2, Definition 3.4.7]) Consider the pro-étale site of $X$
attached to the adic Banach ring $(W,W^{+})$. We will define a sheaf of module
$G$ over $\widehat{\mathcal{O}}_{\text{p\'{e}t}}$ to be pseudocoherent if
locally we can define this as a sheaf attached to an étale-stably
pseudocoherent module. As in [KL2, Definition 3.4.7], we do not have to
consider the corresponding notion of profinite-étale-stably pseudocoherent
module.

######  Proposition 5.16.

(After Kedlaya-Liu [KL2, Theorem 3.4.8]) Taking the corresponding global
section will realize the corresponding equivalence between the following two
categories. The first one is the corresponding one of all the pseudocoherent
sheaves over $\widehat{\mathcal{O}}_{\text{p\'{e}t}}$, while the second one is
the corresponding one of all the étale-stably pseudocoherent modules over $W$.

###### Proof.

As in [KL2, Theorem 3.4.8], the corresponding analog of [KL1, Proposition
9.2.6] applies in the way that the corresponding conditions of the analog of
[KL1, Proposition 9.2.6] hold in our current situation. Also see [Ked1,
Theorem 2.9.9, Remark 2.9.10 and Lemma 1.10.4]. Certainly in this situation we
need to translate the corresponding statemens in [Ked1, Theorem 2.9.9, Remark
2.9.10 and Lemma 1.10.4] to the corresponding adic Banach setting. ∎

Obviously we have the following analog of [KL2, Corollary 3.4.9]:

######  Corollary 5.17.

(After Kedlaya-Liu [KL2, Corollary 3.4.9]) The following two categories are
equivalent. The first is the corresponding category of all pseudocoherent
sheaves over $\widehat{\mathcal{O}}_{\text{p\'{e}t}}$. The second is the
corresponding category of all pseudocoherent sheaves over
${\mathcal{O}}_{\text{\'{e}t}}$. The corresponding functor is the
corresponding pullback along the corresponding morphism of sites
$X_{\text{p\'{e}t}}\rightarrow X_{\text{\'{e}t}}$.

###  5.3. Quasi-Stein Spaces

We now generalize our discussion in [TX2] around pseudocoherent sheaves over
quasi-Stein spaces to the general base situation without the corresponding
restrictive assumption that we are working over $\mathbb{Z}_{p}$.

######  Setting 5.18.

Assume that we are working over some quasi-Stein space $X$ which could be
written as the corresponding direct limit of analytic adic Banach affinoids:
$X:=\varinjlim_{i}X_{i}$ such that $X_{i}$ could be written as the
corresponding analytic affinoid $\mathrm{Spa}(W_{i},W_{i}^{+})$.

######  Lemma 5.19.

(After Kedlaya-Liu [KL2, Lemma 2.6.3]) Consider a compatible family of Banach
modules over the projective system $\mathcal{O}_{X_{i}}(X_{i})$. Suppose that
the corresponding transition map
$p_{i}:\mathcal{O}_{X_{i}}(X_{i})\widehat{\otimes}_{\mathcal{O}_{X_{i+1}}(X_{i+1})}M_{i+1}\overset{}{\rightarrow}M_{i}$
is surjective. Then we have: I. The corresponding global section of $M$
throughout the limit
$\mathcal{O}(X)=\varprojlim_{i}\mathcal{O}_{X_{i}}(X_{i})$ is dense in each
section $M_{i}$ for any $i\geq 0$; II. The corresponding vanishing of
$R^{1}\varprojlim_{i\rightarrow\infty}$ holds in our situation. (Parallel
statement in analytic Huber ring situation holds as well.)

###### Proof.

This is basically the general version of the corresponding result in [KL2,
Lemma 2.6.3]. One could prove this in the same way. ∎

######  Lemma 5.20.

(After Kedlaya-Liu [KL2, Corollary 2.6.4]) Consider a compatible family of
Banach modules over the projective system $\mathcal{O}_{X_{i}}(X_{i})$.
Suppose that each member in the family is stably-pseudocoherent. And suppose
that the corresponding transition map
$p_{i}:\mathcal{O}_{X_{i}}(X_{i})\widehat{\otimes}_{\mathcal{O}_{X_{i+1}}(X_{i+1})}M_{i+1}\overset{}{\rightarrow}M_{i}$
is strictly isomorphism. Then we have in our situation the corresponding
projection from the global section $M=\varprojlim_{i}M_{i}$ to each $M_{i}$
(for each $i\geq 0$) is then isomorphism in our current situation as in [KL2,
Corollary 2.6.4]. (Parallel statement in analytic Huber ring situation holds
as well.)

###### Proof.

This is basically the general version of the corresponding result in [KL2,
Corollary 2.6.4]. One could prove this in the same way. ∎

######  Proposition 5.21.

(After Kedlaya-Liu [KL2, Theorem 2.6.5]) With the corresponding notations as
above, we have that the global section of any pseudocoherent sheaf $M$ over
$X$ will be dense in the section over any quasicompact. (Parallel statement in
analytic Huber ring situation holds as well.)

###### Proof.

See [KL2, Theorem 2.6.5]. ∎

######  Proposition 5.22.

(After Kedlaya-Liu [KL2, Theorem 2.6.5]) With the corresponding notations as
above, we have that the global section of any pseudocoherent sheaf $M$ over
$X$ will finitely generate each fiber $M_{x}$ over $\mathcal{O}_{X,x}$ for any
$x\in X$. (Parallel statement in analytic Huber ring situation holds as well.)

###### Proof.

See [KL2, Theorem 2.6.5]. ∎

######  Proposition 5.23.

(After Kedlaya-Liu [KL2, Theorem 2.6.5]) With the corresponding notations as
above, we have that the corresponding sheaf cohomology of any pseudocoherent
sheaf $M$ over $X$ admits vanishing for degree bigger than 0. (Parallel
statement in analytic Huber ring situation holds as well.)

###### Proof.

See [KL2, Theorem 2.6.5]. ∎

######  Proposition 5.24.

(After Kedlaya-Liu [KL2, Corollary 2.6.6]) For any pseudocoherent sheaf $M$
over the space $X$, we have in our current situation that the exact functor
from the corresponding category of all the pseudocoherent sheaves to the
corresponding one of all the corresponding $\mathcal{O}_{X}(X)$-modules
through the corresponding taking the global section. (Parallel statement in
analytic Huber ring situation holds as well.)

######  Proposition 5.25.

(After Kedlaya-Liu [KL2, Corollary 2.6.8]) For any pseudocoherent sheaf $M$
admitting a structure of vector bundles over the space $X$, we have in our
current situation that the corresponding sufficient and necessary condition
for the global section to be finite projective is exactly that the global
section of $M$ is finitely generated. (Parallel statement in analytic Huber
ring situation holds as well.)

###### Proof.

See [KL2, Corollary 2.6.8], it is straightforward to exact a global splitting
from the local ones over quasi-compacts. ∎

######  Proposition 5.26.

(After Kedlaya-Liu [KL2, Proposition 2.6.17]) Assume that the space $X$ is
basically $m$-uniform in the sense of [KL2]. Then we have the corresponding
finiteness of the global section is equivalent to to the corresponding uniform
finiteness through out all $X_{i},i=0,1,2,...$. (Parallel statement in
analytic Huber ring situation holds as well.)

###### Proof.

See the proof given in [TX2, Theorem 6.27]. ∎

##  6\. Applications

###  6.1. Application to Noetherian Cases

The corresponding noetherian situation is expected to be in some sense better
than just being pseudoflat with respect to the corresponding rational
localization.

######  Setting 6.1.

We now work with the corresponding analytic adic Banach rings over
$\mathbb{Q}_{p}$ or $\mathbb{F}_{p}((t))$ as the corresponding base spaces.
The deformation will happen along some Banach ring over $\mathbb{Q}_{p}$ or
$\mathbb{F}_{p}((t))$. And moreover we have to assume that the corresponding
noetherianness preserves under the corresponding deformation over some ring
$V$. Namely we have to assume that $W\widehat{\otimes}_{*}V$ is always
noetherian, for instance this could be achieved when we have that $V$ over
$\mathbb{Q}_{p}$ comes from truncations of distribution algebra over some
$p$-adic Lie groups by considering some further $p$-adic microlocal analysis.

######  Lemma 6.2.

With the notations in the [KL2, Lemma 2.4.10], we have that the corresponding
morphisms $W\widehat{\otimes}V\rightarrow B_{1}\widehat{\otimes}V$,
$W\widehat{\otimes}V\rightarrow B_{2}\widehat{\otimes}V$ and
$W\widehat{\otimes}V\rightarrow B_{12}\widehat{\otimes}V$ are 2-pseudoflat.
And furthermore in our current context they are actually flat.

###### Proof.

In our current situation, the corresponding 2-pseudoflatness is achieved since
this is already true in the corresponding more general situation. However for
finite presented modules (we do not have to consider the topological issues
since we are in the noetherian situation) this is already flat which implies
for finitely generated modules (we do not have to consider the topological
issues since we are in the noetherian situation) this is already flat as well.
Then for any module which could be written as injective limit of finite ones,
the result holds. ∎

######  Corollary 6.3.

The corresponding rational localization with respect to the adic Banach ring
$(W,W^{+})$ is flat (namely for all the corresponding finitely presented
module over $W\widehat{\otimes}V$).

######  Proposition 6.4.

Then we consider the corresponding presheafification of any finitely generated
module $M$ over $W\widehat{\otimes}V$, to be more precise over
$\mathrm{Spa}(W,W^{+})$ we will define the corresponding presheaf
$\widetilde{M}$ by taking the inverse limit of the base changes of $M$
throughout all the rational localizations of $W$. Then we have that the Tate
glueing property holds in our current situation for such presheaf
$\widetilde{M}$.

###### Proof.

This reduces to the previous lemma by using [KL1, Proposition 2.4.20,
Proposition 2.4.21]. ∎

The corresponding glueing finitely generated modules is also achieved in
current noetherian situation. First we consider the corresponding result:

######  Proposition 6.5.

The descent along the morphism $W\rightarrow B_{1}\bigoplus B_{2}$ is
effective as long as one focuses on the category of all the finitely presented
modules over $W\widehat{\otimes}V$.

###### Proof.

This is the corresponding noetherian implication of the corresponding result
in [TX2, Lemma 2.14]. ∎

######  Proposition 6.6.

The corresponding functor of taking the corresponding global section will give
rise to the corresponding equivalence between the corresponding $V$-coherent
sheaves and the $V$ finitely presented modules.

###### Proof.

This is also the corresponding noetherian implication of the corresponding
result in [TX2, Theorem 2.15]. ∎

Then fixing a stable basis $\mathbb{H}$ for the étale site of the space
$\mathrm{Spa}(W,W^{+})$, then we have the results over the étale site as well:

######  Proposition 6.7.

We consider the corresponding presheafification of any finitely generated
module $M$ over $W\widehat{\otimes}V$, to be more precise over
$\mathrm{Spa}(W,W^{+})$ we will define the corresponding presheaf
$\widetilde{M}$ by taking the inverse limit of the base changes of $M$
throughout all the member in the basis $\mathbb{H}$. Then we have that the
Tate glueing property holds in our current situation for such presheaf
$\widetilde{M}$.

######  Proposition 6.8.

The descent (in the étale topology) along the morphism $W\rightarrow
B_{1}\bigoplus B_{2}$ is effective as long as one focuses on the category of
all the finitely presented modules over $W\widehat{\otimes}V$.

###### Proof.

This is the corresponding noetherian implication of the corresponding result
in [TX2, Lemma 2.22]. ∎

######  Proposition 6.9.

The corresponding functor (in the étale topology) of taking the corresponding
global section will give rise to the corresponding equivalence between the
corresponding $V$-coherent sheaves and the $V$ finitely presented modules.

###### Proof.

This is also the corresponding noetherian implication of the corresponding
result in [TX2, Theorem 2.23]. ∎

###  6.2. Application to Descent over Noncommutative $\infty$-Analytic
Prestacks after Bambozzi-Ben-Bassat-Kremnizer

We now contact with the corresponding derived analytic spaces from [BBBK].
Recall that we have the categories
$\mathrm{Simp}(\mathrm{Ind}^{m}(\mathrm{BanSets}_{E}))$ and
$\mathrm{Simp}(\mathrm{Ind}(\mathrm{BanSets}_{E}))$ of the corresponding
simplicial sets in the corresponding inductive category of Banach sets over
$E=\mathbb{Q}_{p},\mathbb{F}_{p}((t))$.

######  Theorem 6.10.

(Bambozzi-Ben-Bassat-Kremnizer) The categories
$\mathrm{Simp}(\mathrm{Ind}^{m}(\mathrm{BanSets}_{E}))$ and
$\mathrm{Simp}(\mathrm{Ind}(\mathrm{BanSets}_{E}))$ admit symmetric monoidal
model categorical structure.

Therefore based on this nice structure [BBBK] defined the corresponding
$\infty$-categories of $\mathbb{E}_{\infty}$-rings:

(6.1)
$\displaystyle\mathrm{sComm}(\mathrm{Simp}(\mathrm{Ind}^{m}(\mathrm{BanSets}_{E})))$
(6.2)
$\displaystyle\mathrm{sComm}(\mathrm{Simp}(\mathrm{Ind}(\mathrm{BanSets}_{E}))).$

We now consider the corresponding some object $A$ which is a corresponding
$\infty$-locally convex ring in the categories above and we consider the
corresponding object in the corresponding opposite category we call that
$\mathrm{Spec}A$. And we consider the corresponding homotopy Zariski topology,
which allows us to talk about the corresponding $\infty$-analytic stacks.

Now recall that a connective $\mathbb{E}_{1}$-ring is called noetherian if we
have that $\pi_{0}(A)$ is noetherian and $\pi_{n}(A)$ is basically finitely
generated over $\pi_{0}(A)$.

######  Remark 6.11.

The corresponding Koszul simplicial Banach ring considered in [BK] is actually
concentrated in the nonpositive degrees111Thanks for Federico Bambozzi for
reminding us of this., in the cohomological convention, namely
$\mathrm{H}^{-n}(.)=0$ for $n<0$. But as in any abstract homotopy theory we
consider the corresponding conventional transformation:

(6.3) $\displaystyle\pi_{n}:=\mathrm{H}^{-n},n\geq 0.$

Now we consider an $\infty$-analytic stack $X$ as considered in [BBBK]. And we
consider the corresponding ringed site attached to $X$ under the corresponding
homotopy Zariski topology, denoted by $(X,\mathcal{O}_{X})$.

######  Example 6.12.

The examples which are very interesting to us are the corresponding
$\infty$-Huber spectra constructed from any Banach rings over $E$ from [BK] by
using Koszul complex. The corresponding classical sheafiness issue could be
really forgotten.

Now we consider the corresponding $\mathbb{E}_{1}$-ring $A$ in the
corresponding categories
$\mathrm{Simp}(\mathrm{Ind}^{m}(\mathrm{BanSets}_{E}))$ and
$\mathrm{Simp}(\mathrm{Ind}(\mathrm{BanSets}_{E}))$. We make the following
assumption:

######  Assumption 6.13.

We assume that all the ring objects below are noetherian, as
$\mathbb{E}_{1}$-rings.

Suppose now we have four $\mathbb{E}_{1}$ Banach rings
$A,A_{1},A_{2},A_{12}$222Namely all the homotopy groups have to be Banach
instead of Bornological or ind-Banach. in
$\mathrm{Simp}(\mathrm{Ind}^{m}(\mathrm{BanSets}_{E}))$ and
$\mathrm{Simp}(\mathrm{Ind}(\mathrm{BanSets}_{E}))$ respectively. And we
assume that we have the following short strictly exact sequence in the sense
of a glueing square:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{0}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{0}(A_{1})\oplus\pi_{0}(A_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{0}(A_{12})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

and we assume that the image of $\pi_{0}(A_{i})$ in $\pi_{0}(A_{12})$ is dense
for each $i=1,2$. And we assume the corresponding morphism
$\pi_{0}(A_{i})\rightarrow\pi_{0}(A_{12})$ is flat for $i=1,2$. And we assume
that $A,A_{1},A_{2},A_{12}$ form the corresponding derived glueing sequence.

######  Example 6.14.

One can construct the following example. First consider any glueing sequence
of Banach rings (not required to be sheafy or commutative) over
$\mathbb{Q}_{p}$:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Pi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Pi_{1}\oplus\Pi_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Pi_{12}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

This is very natural in the corresponding situation for example where we
deform from a corresponding nice short exact sequence of commutative sheafy
rings of the same type, but in any rate we do not require the corresponding
commutativity or the correponding sheafyness. Then take any derived global
section of some $\mathrm{Spa}^{h}(R)$ from [BK], denoted by $R^{h}$. Over
$\mathbb{Q}_{p}$, we have the following short strictly exact sequence:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Pi\widehat{\otimes}_{\mathbb{Q}_{p}}\pi_{0}(R^{h})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Pi_{1}\widehat{\otimes}_{\mathbb{Q}_{p}}\pi_{0}(R^{h})\oplus\Pi_{2}\widehat{\otimes}_{\mathbb{Q}_{p}}\pi_{0}(R^{h})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Pi_{12}\widehat{\otimes}_{\mathbb{Q}_{p}}\pi_{0}(R^{h})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

but since the corresponding ring $\Pi,\Pi_{1},\Pi_{2},\Pi_{12}$ will actually
realize the situation where
$\Pi_{*}\widehat{\otimes}\pi_{0}(R^{h})\overset{\sim}{\rightarrow}\pi_{0}(\Pi_{*}\widehat{\otimes}^{\mathbb{L}}R^{h})$,
we will have the desired situation as long as restrict to now the noetherian
situation.

######  Conjecture 6.15.

The map $A\rightarrow\prod_{i=1,2}A_{i}$ is an effective descent morphism with
respect to the finitely presented left module spectra (meaning we have
finitely presented $\pi_{0}$).

######  Proposition 6.16.

The operation of taking equalizer along the corresponding map
$\prod_{i=1,2}A_{i}\rightarrow A_{12}$ preserves the property of being
finitely presented.

###### Proof.

The situation where we deform from the corresponding $\mathbb{E}_{\infty}$ by
some noncommutative deformation could be implied by [TX2, Lemma 2.14]. While
in general this follows from [TX4, Lemma 6.83]. Note that we achieve the
finiteness for each $\pi_{n}(M)$ just by application of the results we know in
the classical deformed situation, which is because the connecting homomorphism
vanishes on each level. ∎

What could happen in the commutative setting is actually also interesting. We
have the following example in mind:

######  Example 6.17.

Consider the context of the previous example, but assume that all the rings
involved are Banach commutative. First we consider over $\mathbb{Q}_{p}$:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Pi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Pi_{1}\oplus\Pi_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Pi_{12}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

a strictly exact sequence of sheafy rings. We now deform this along some
spectrum $\mathrm{Spa}^{h}(R)$ in [BK] which will produce a very interesting
situation where we can actually obtain desired situation for the descent of
finitely presented module spectra. And note that in commutative setting we
could also have more geometric contexts to rely on as in [BK] and [BBBK]. This
may have contact with the corresponding derived Galois deformation theory of
[GV] for instance by considering the corresponding simplicial pro-Artin rings.

######  Remark 6.18.

The corresponding machinery from [CS] should definitely reflect similar things
here even in the $\mathbb{E}_{1}$-ring context. Also we would like to mention
in the commutative setting that actually the descent for certain quasi-
coherent modules are also considered extensively in [BBK]. Strikingly the
ideas in [BBK] (although developed in a quite commutative setting) come from
partially work from Kontsevich-Rosenberg [KR] namely essentially the
noncommutative descent. Our feeling is that definitely the descent for certain
quasi-coherent modules in [BBBK] could be established to noncommutative
setting in some form both in the archimedean and nonarchimedean situations.

### Acknowledgements

We would like to thank Professor Kedlaya for helpful discussion on the
corresponding materials in the chapter [Ked1], especially the discussion
beyond the book which helps us have the chance to enrich the presentation
here.

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