# Topologization and Functional Analytification I: Intrinsic Morphisms of
Commutative Algebras

Xin Tong

###### Abstract.

Eventually after Dieudonné-Grothendieck, we give intrinsic definitions of
étale, lisse and non-ramifié morphisms for general adic rings and general
locally convex rings. And we investigate the corresponding étale-like, lisse-
like and non-ramifié-like morphisms for general $\infty$-Banach,
$\infty$-Borné and $\infty$-ind-Fréchet $\infty$-rings and $\infty$-functors
into $\infty$-groupoid (as in the work of Bambozzi-Ben-Bassat-Kremnizer) in
some intrinsic way by using the corresponding infinitesimal stacks and
crystalline stacks. The two directions of generalization will intersect at
Huber’s book in the strongly noetherian situation.

11footnotetext: Version: Feb 21 2021.22footnotetext: 2020 Mathematics Subject
Classification Number: 14G45, 14G99.33footnotetext: Keywords and Phrases:
Intrinsic Morphisms, Topologization, Functional Analysis of Locally Convex
spaces, Geometric Grothendieck Sites, Stackization.

###### Contents

  1. 1 Introduction
    1. 1.1 Main Consideration
    2. 1.2 Further Consideration
  2. 2 Affinoid Morphisms of Huber Rings
  3. 3 Affinoid Morphisms of Banach Rings
  4. 4 Naive Étale Morphisms and Intrinsic Étale Morphisms
  5. 5 Properties
  6. 6 Étale-Like Morphisms of $\infty$-Banach Rings and the $\infty$-Analytic Stacks
    1. 6.1 Approach through De Rham Stacks
    2. 6.2 Approach through Crystalline Stack and PD-morphisms
  7. 7 Lisse-Like and Non-Ramifié-Like Morphisms of $\infty$-Banach Rings and the $\infty$-Analytic Stacks
    1. 7.1 Approach through De Rham Stacks
    2. 7.2 Approach through Crystalline Stack and PD-morphisms
  8. 8 Perfectization and Fontainisation of $\infty$-Analytic Stacks Situation
    1. 8.1 Perfectization, Fontainisation and Crystalline Stacks
    2. 8.2 Perfectization, Fontainisation and Robba Stacks

##  1\. Introduction

###  1.1. Main Consideration

Scholze’s diamond is actually very general notion beyond the corresponding
perfectoid spaces, partially because it contains the corresponding diamantine
spaces after Hansen-Kedlaya. This point of view certainly gives the motivation
for this notion from Hansen-Kedlaya. Therefore one could regard to some extent
diamantine spaces as giving some (maybe better to say more ring theoretic)
analogs of the corresponding Scholze’s diamonds, moreover they behave as if
they are perfectoid spaces. Similar discussion could be made to suosperfectoid
space, which should be more ’perfectoid’ generalization.

Hansen-Kedlaya [HK] have given the definition of naive étale morphisms among
any Tate Huber pairs namely these are locally composites of the rational
localizations and finite étale morphisms (very importantly with strongly
sheafy domains and targets). This is because one definitely believes that
correct notion of étale morphism should admit such admissible decomposition
and factorization. However what should be the correct intrinsic one has not
been given in full detail yet. In the significant strongly sheafy situation,
we are going to try to answer this question as proposed in [Ked1, Appendix 5].
In this paper, we try to study the corresponding properties of the
corresponding naive étale morphisms along the ideas of [Ked1, Appendix 5] and
[HK]. The goal is to accurately characterize the corresponding naive étale
morphisms in some intrinsic way.

Sheafiness plays a very crucial role in the discussion above. However suppose
we do not have to worry about the sheafiness at all (in fact in some sense we
really do not have to worry about this at all by the work of Clausen-Scholze
[CS] and Bambozzi-Kremnizer [BK]), then one might want to believe that the
robust definitions could be made even more robust. Therefore we investigate
the corresponding morphisms of the corresponding ring objects where sheafiness
could be replaced by $\infty$-sheafiness (namely sheafiness up to higher
homotopy) after [BBBK] and [BK]. One should be able to consider Clausen-
Scholze’s foundation [CS] as well, however we will mainly focus on the
$\infty$-locally convex objects in [BBBK] and [BK], as in the corresponding
schematic situation in [Lu1], [Lu2], [TV1] and [TV2]. We consider the
corresponding interesting approaches through the corresponding formal and PD
completions just as in the $\infty$-schematic situation in [R] which is very
related to the corresponding Drinfeld’s stacky construction [Dr1] and [Dr2] by
using the Čech-Alexander complex on the corresponding crystalline cohomology
and prismatic cohomology.

The current list of definitions will be established for discrete
$\mathbb{E}_{\infty}$-ring objects and $\mathbb{E}_{\infty}$-ring objects in
suitable locally convex $\infty$-categories after after Bambozzi-Ben-Bassat-
Kremnizer [BBBK]:

D1. Localized intrinsic étale morphisms of open mapping Huber rings;
D2. Localized intrinsic étale morphisms of open mapping adic Banach rings;
D3. Localized intrinsic lisse morphisms of open mapping Huber rings;
D4. Localized intrinsic lisse morphisms of open mapping adic Banach rings;
D5. Localized intrinsic non-ramifié morphisms of open mapping Huber rings;
D6. Localized intrinsic non-ramifié morphisms of open mapping adic Banach
rings;
$\infty$1\. De Rham intrinsic étale-like morphisms of $\infty$-analytic
functors;
$\infty$2\. De Rham intrinsic lisse-like morphisms of $\infty$-analytic
functors;
$\infty$3\. De Rham intrinsic non-ramifié-like morphisms of $\infty$-analytic
functors;
$\infty$4\. PD (crystalline) intrinsic étale-like morphisms of
$\infty$-analytic functors;
$\infty$5\. PD (crystalline) intrinsic lisse-like morphisms of
$\infty$-analytic functors;
$\infty$6\. PD (crystalline) intrinsic non-ramifié-like morphisms of
$\infty$-analytic functors.

Certainly for general locally convex spaces producing nice ring structures we
really have to be very precise and accurate in any sorts of characterization.
However we have not unfortunately achieve this due to some very subtle issues,
mainly coming from the corresponding issues in very general functional
analytification. That being otherwise all said, we still actually could
literally talk about the desired definitions for simplicial noetherian Banach
rings in certain situations.

###  1.2. Further Consideration

Our ultimate goal is certainly to study the corresponding geometric sites
(étale, pro-étale, crystalline and prismatic
[SGAIV],[Gro1],[Sch1],[KL1],[KL2],[BS], [Dr1],[Dr2]) and the corresponding
cohomologies (étale, pro-étale, crystalline and prismatic) for really general
$\infty$-analytic spaces (possibly also noncommutative analogs of those in
[KR1]) over $\mathbb{F}_{1}$ and try to apply to the locally noetherian
situations, the strongly noetherian situations (such as in [G1], [GL]), the
strongly sheafy situations under the foundation of $\infty$-locally convex
spaces (as in [HK], [KL1] and more general situations), although our very
beginning corresponding motivation for this article is an attempt to answer
some questions in [Ked1, Appendix A5].

##  2\. Affinoid Morphisms of Huber Rings

We start with the discussion on the corresponding intrinsic definition of
étale morphisms.

######  Setting 2.1.

We start with an analytic uniform Huber pair $(A,A^{+})$. And we will consider
the category of all such rings. We assume the corresponding completeness for
the Huber pairs.

######  Definition 2.2.

(Hansen-Kedlaya [HK, Definition 5.1]) We call a map of Huber rings
$(A,A^{+})\rightarrow(B,B^{+})$ naive étale after [HK, Definition 5.1] if it
admit a factorization into rational localizations and finite étale morphisms.
Here we assume $(A,A^{+})$ is strong sheafy and we assume that $(B,B^{+})$ is
strongly sheafy. 111Certainly one needs to be careful since we are now
considering more general context than [HK] without assuming the corresponding
Tateness.

######  Definition 2.3.

(Kedlaya [Ked1, Definition A5.2]) Recall from [Ked1, Definition A5.2], we have
the corresponding affinoid morphism from any strongly sheafy Huber ring $A$,
namely a morphism $A\rightarrow B$, such that $B$ admits some surjective
covering from $A\left<T_{1},...,T_{d}\right>$ and through this map we have
that $B$ is a stably-pseudocoherent sheaf over $A\left<T_{1},...,T_{d}\right>$
and the corresponding ring $B$ is assumed to be sheafy 222Cetainly one needs
to be more careful since this is also slightly different from the original
definition [Ked1, Definition A5.2], thanks Professor Kedlaya for telling me
this should be better. We want to mention that this is a quite subtle point
around the sheafiness (see [Ked1, Theorem 1.4.20]), the point here is that we
do not know the kernel of an affinoid morphism is closed or not, if it is
closed then we could keep the knowledge that $B$ being stably-pseudocoherent
is equivalent to $B$ being sheafy. However if this is not closed, then we do
not have this sort of equivalence to our knowledge..

The belief (as proposed in [Ked1, Problem A5.3, Problem A5.4]) is that somehow
the corresponding affinoid morphisms in the definition should be directly used
in the corresponding definitions of lisse morphisms and unramified morphisms,
as well as certainly the étale morphisms. To investigate this kind of idea, we
are going to first investigate the corresponding naive étale morphisms along
this idea.

######  Lemma 2.4.

Let $f_{1}:\Gamma_{1}\rightarrow\Gamma_{2}$ and
$f_{2}:\Gamma_{2}\rightarrow\Gamma_{3}$ be two affinoid morphisms, then the
composition $f_{2}\circ f_{1}$ is also affinoid.

###### Proof.

Straightforward. ∎

######  Lemma 2.5.

(Kedlaya) For any standard binary rational localization of $A$ with respect to
$f,g\in A$, suppose we know that there are two surjective morphisms:

(2.1) $\displaystyle
s_{1}:A\left<\frac{f}{g}\right>\left<T_{1},...,T_{n}\right>\rightarrow
B\left<\frac{f}{g}\right>,$ (2.2) $\displaystyle
s_{2}:A\left<\frac{g}{f}\right>\left<T_{1},...,T_{n}\right>\rightarrow
B\left<\frac{g}{f}\right>.$

Then we have that there is a surjective morphism:

(2.3) $\displaystyle s:A\left<T_{1},...,T_{n^{\prime}}\right>\rightarrow B.$

###### Proof.

The following argument is due to Kedlaya 111Thanks Professor Kedlaya for
mentioning the similarity of this to the corresponding locality of morphisms
of finite type as in Grothendieck’s EGA I and II., we work out it for the
convenience of the readers. First, we have the following short exact sequence:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\left<\frac{f}{g}\right>\bigoplus
B\left<\frac{g}{f}\right>\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\left<\frac{f}{g},\frac{g}{f}\right>\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Take any $b\in B$, and use the notation $(b_{1},b_{2})$ for the image in the
middle. By the surjectivity of the maps $s_{1},s_{2}$ we have that there exist
some element $a_{1}\in A\left<\frac{f}{g}\right>\left<T_{1},...,T_{n}\right>$
and some element $a_{2}\in
A\left<\frac{f}{g}\right>\left<T_{1},...,T_{n}\right>$ such that we have:

(2.4) $\displaystyle s_{1}(a_{1})=b_{1},$ (2.5) $\displaystyle
s_{2}(a_{2})=b_{2}.$

With more explicit expression we have the following:

(2.6) $\displaystyle
s_{1}(\sum_{i_{1},...,i_{n}}\sum_{i}a_{1}^{i,i_{1},...,i_{n}}u^{i}T_{1}^{i_{1}}...T_{n}^{i_{n}})=\sum_{i}b_{1}^{i}u^{i},$
(2.7) $\displaystyle
s_{2}(\sum_{i_{1},...,i_{n}}\sum_{i}a_{2}^{i,i_{1},...,i_{n}}v^{i}T_{1}^{i_{1}}...T_{n}^{i_{n}})=\sum_{i}b_{2}^{i}v^{i},$

under the corresponding presentations up to liftings:

(2.8) $\displaystyle B\left<\frac{f}{g}\right>=B\left<u\right>/(gu-f),$ (2.9)
$\displaystyle B\left<\frac{g}{f}\right>=B\left<v\right>/(fv-g).$

Then to finish we only have to take some finite sum in the summation to make
approximation. We first claim that such finite sum approximation and
modification will not change the corresponding surjectivity of the map $s_{1}$
and $s_{2}$. Namely for each $k=1,2$ the map $s_{k}$ will maintain surjective
once we modify the image of $T_{1},...,T_{n}$ infinitesimally around some
neighbourhood $U$ of $0$, in other words it will maintain to be surjective
even if we set $s_{k}(T_{1}),...,s_{k}(T_{n})$ to be $x_{1},...,x_{n}$
whenever $x_{1}-s_{k}(T_{1}),...,x_{n}-s_{k}(T_{n})$ lives in the
neighbourhood $U$, and moreover we have that the corresponding modification
could be assumed to take $T_{i}$ to $x_{i}$ with $i=1,...,n$. By open mapping,
we have that the corresponding lifts of the corresponding differences
$x_{1}-s_{k}(T_{1}),...,x_{n}-s_{k}(T_{n})$ could be made to be living in some
arbitrarily chosen neighbourhood $V$ of $0$. Then we only have to consider the
following map factoring through the corresponding map $s_{k}$:

(2.11) $\displaystyle h:A_{k}\left<T_{1},...,T_{n}\right>$
$\displaystyle\rightarrow A_{k}\left<T_{1},...,T_{n}\right>$ (2.12)
$\displaystyle T_{i}$ $\displaystyle\mapsto
T_{i}+\mathrm{lifts~{}of}~{}x_{i}-s_{k}(T_{k})$

where $A_{1}$ is the ring $A\left<\frac{f}{g}\right>$ while we have $A_{2}$ is
the ring $A\left<\frac{g}{f}\right>$, which basically proves the claim. Then
this will indicate that one can find some joint finite subset
$T:=\\{T_{1},...,T_{n^{\prime}}\\}$ for $B\left<\frac{f}{g}\right>$ and
$B\left<\frac{f}{g}\right>$ such that the modified

(2.13) $\displaystyle
s_{1}:A\left<\frac{f}{g}\right>\left<T_{1},...,T_{n^{\prime}}\right>\rightarrow
B\left<\frac{f}{g}\right>,$ (2.14) $\displaystyle
s_{2}:A\left<\frac{g}{f}\right>\left<T_{1},...,T_{n^{\prime}}\right>\rightarrow
B\left<\frac{g}{f}\right>,$

are basically surjective and they fit into the following commutative diagram:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\left<T_{1},...,T_{n^{\prime}}\right>\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\left<\frac{f}{g}\right>\left<T_{1},...,T_{n^{\prime}}\right>\bigoplus
A\left<\frac{g}{f}\right>\left<T_{1},...,T_{n^{\prime}}\right>\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\left<\frac{f}{g},\frac{g}{f}\right>\left<T_{1},...,T_{n^{\prime}}\right>\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\left<\frac{f}{g}\right>\bigoplus
B\left<\frac{g}{f}\right>\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\left<\frac{f}{g},\frac{g}{f}\right>\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where the middle and the rightmost vertical arrows are surjective. Then claim
is then that the left vertical one is also surjective. The kernels
$K_{1}\oplus K_{2}$ in the middle is mapped surjectively to the kernel
$K_{12}$ of the rightmost vertical map. So the snake lemma will force the
cokernel of the left vertical arrow to be zero which shows the corresponding
exactness at the corresponding location $?$ in the following commutative
diagram:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}\bigoplus
K_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{12}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\left<T_{1},...,T_{n^{\prime}}\right>\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\left<\frac{f}{g}\right>\left<T_{1},...,T_{n^{\prime}}\right>\bigoplus
A\left<\frac{g}{f}\right>\left<T_{1},...,T_{n^{\prime}}\right>\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\left<\frac{f}{g},\frac{g}{f}\right>\left<T_{1},...,T_{n^{\prime}}\right>\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}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{?}$$\textstyle{B\left<\frac{f}{g}\right>\bigoplus
B\left<\frac{g}{f}\right>\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\left<\frac{f}{g},\frac{g}{f}\right>\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}$$\textstyle{0}$$\textstyle{0}$

where $K_{1},K_{2},K_{12}$ are pseudocoherent, which implies that the
corresponding module $K$ is also pseudocoherent.

∎

######  Proposition 2.6.

Any naive étale morphism is affinoid.

###### Proof.

First let $f:(A,A^{+})\rightarrow(B,B^{+})$ be any naive étale morphism. Then
locally this is basically composition of the corresponding rational
localizations and finite étale maps. Locally rational localizations involved
are actually affinoid, and locally the corresponding finite étale maps from
strongly sheafy rings will have strongly sheafy target, which will imply that
locally finite étale maps are affinoid. Then this could be globalized to force
the global map $f$ to be affinoid. The properties of factoring through a
surjection globally could be proved by glueing local ones through lemma 3.6,
by considering [KL1, Proposition 2.4.20]. And globally the corresponding ring
$B$ is stably-pseudocoherent over $A\left<T_{1},...,T_{n}\right>$ for some $n$
since this is a local property. ∎

Therefore we have proved that the corresponding étale maps in the
corresponding naive sense is actually affinoid in the above sense. Therefore
it is now natural to try to find the corresponding properties which may
completely characterize the corresponding naive étale morphisms which are
affinoid.

Certainly we may have the corresponding conjectures that all the naive étale
morphisms will satisfy the corresponding properties of algebraically étale
ones (such as in [EGAIV4, Chapitre 17], [SP, Tag 00U1]). We now discuss the
corresponding completed cotangent complex after Huber [Hu1, 1.6.2]. Recall for
our current $B$ the corresponding completed differential
$\Omega^{1}_{B/A,\mathrm{topo}}$ (see [Hu1, 1.6.2] for the construction for
any $f$-adic rings in the noetherian setting). Therefore we consider the
corresponding topological naive cotangent complex:

(2.15) $\displaystyle\tau_{\leq 1}\mathbb{L}_{B/A,\mathrm{topo}}$

for any naive étale map $f:(A,A^{+})\rightarrow(B,B^{+})$. We now discuss the
construction without the corresponding strongly noetherian requirement in our
current situation. First we know that $B$ is of topologically finite type over
$A$:

(2.16) $\displaystyle B=A\left<X_{1},...,X_{n}\right>_{T_{1},...,T_{n}}/I.$

Then we could first define the topological free differentials:

(2.17)
$\displaystyle\Omega^{1}:=A\left<X_{1},...,X_{n}\right>_{T_{1},...,T_{n}}dX_{1}+...+A\left<X_{1},...,X_{n}\right>_{T_{1},...,T_{n}}dX_{n}.$

Then we have:

(2.18) $\displaystyle\Omega^{1}_{B/A,\mathrm{topo}}:=\Omega^{1}/(I\bigcup
d(I))\Omega^{1}.$

Here everything is assumed to be basically complete with respect to the
corresponding natural topology. Namely we need to take the corresponding
completion always with respect to the corresponding induced topology.
Certainly here $\Omega^{1}$ is already complete due to the fact that it is
finitely projective. Recall that a map $f:\Gamma_{1}\rightarrow\Gamma_{2}$ is
called étale in the scheme theory if the naive cotangent complex (truncated
and could be regarded as an $\infty$-module spectrum) is quasi-isomorphic to
zero. The corresponding underlying complex reads:

$\textstyle{[I/I^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{1}_{\Gamma_{2}/\Gamma_{1},\mathrm{topo}}].}$

In the situation where we consider $A\rightarrow B$ is affinoid, the
corresponding ideal $I$ is actually stably-pseudocoherent over
$A\left<X_{1},...,X_{n}\right>$. It is peudocoherent by the corresponding two
out of three property. The stability holds locally, so we have the case. And
if the morphism if furthermore naive étale then we have $I/I^{2}$ is also
stably-pseudocoherent, see lemma 4.8.

######  Remark 2.7.

Note that we are considering the very general and complicated non-noetherian
situation, modules will need to be endowed with the natural topology and
complete, although finite projective modules are complete automatically. This
will have nontrivial things to do with the corresponding definition of
$\Omega^{1}_{B/A,\mathrm{topo}}$.

One can actually generalize the corresponding full cotangent complexes and
derived de Rham complexes to this topological context following [III1], [III2]
and [B1]. First for the corresponding topological cotangent complex we
consider the following definition (note that we have to assume the
corresponding topologically finite type condition). We start with the
corresponding algebraic ones for
$B^{h}=A[X_{1},...,X_{n}]_{T_{1},...,T_{n}}/I$, under the topologization we
have the corresponding derived cotangent complex:

(2.19) $\displaystyle\mathbb{L}_{B^{h}/A,\mathrm{alg}},$

by taking the usual algebraic one. Then we take the corresponding completion
with respect to the corresponding topologization which gives rise to the
following topological one:

(2.20) $\displaystyle\mathbb{L}_{B/A,\mathrm{topo}}.$

We define the corresponding de Rham complex in the following parallel way.
What is happen is that consider the presentation
$B^{h}=A[X_{1},...,X_{n}]_{T_{1},...,T_{n}}/I$ which gives rise to the
corresponding algebraic de Rham complex:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{1}_{B^{h}/A,\mathrm{alg}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{2}_{B^{h}/A,\mathrm{alg}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{\bullet}_{B^{h}/A,\mathrm{alg}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...,}$

which will give rise to the corresponding topological one if we take the
corresponding completion induced from the subset $T_{1},...,T_{n}$:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\Omega}^{1}_{B^{h}/A,\mathrm{topo}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\Omega}^{2}_{B^{h}/A,\mathrm{topo}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\Omega}^{\bullet}_{B^{h}/A,\mathrm{topo}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{....}$

From our construction for ${\Omega}^{1}_{B^{h}/A,\mathrm{topo}}$, one can
actually define:

(2.21)
$\displaystyle{\Omega}^{\bullet,\mathrm{f}}_{B^{h}/A,\mathrm{topo}}:=\bigoplus_{i_{1},...,i_{\bullet}\in\\{1,...,n\\}}A\left<X_{1},...,X_{n}\right>_{T_{1},...,T_{n}}dX_{i_{1}}\wedge
dX_{i_{2}}\wedge...\wedge dX_{i_{\bullet}}$

and then define:

(2.22)
$\displaystyle{\Omega}^{\bullet}_{B/A,\mathrm{topo}}:=\left(\bigoplus_{i_{1},...,i_{\bullet}\in\\{1,...,n\\}}A\left<X_{1},...,X_{n}\right>_{T_{1},...,T_{n}}dX_{i_{1}}\wedge
dX_{i_{2}}\wedge...\wedge dX_{i_{\bullet}}\right)/$ (2.23)
$\displaystyle\left((I\bigcup dI\bigcup
d^{\bullet}I)\bigoplus_{i_{1},...,i_{\bullet}\in\\{1,...,n\\}}A\left<X_{1},...,X_{n}\right>_{T_{1},...,T_{n}}dX_{i_{1}}\wedge
dX_{i_{2}}\wedge...\wedge dX_{i_{\bullet}}\right),$

after taking suitable completion when needed 111As in [B1] and [GL] where one
takes the corresponding derived $p$-completion out from the algebraic
cotangent complex and the corresponding derived algebraic de Rham complex..

##  3\. Affinoid Morphisms of Banach Rings

We now consider the parallel situation of Banach rings.

######  Setting 3.1.

We start with a uniform adic Banach ring $(A,A^{+})$ in the general sense of
[KL1] and [KL2] (without assumption on the topologically nilpotent units being
existing, but we assume this is open mapping). And we will consider the
category of all such rings. We assume the corresponding completeness as well.

######  Definition 3.2.

(Hansen-Kedlaya [HK, Definition 5.1]) We call a map of adic Banach rings
$(A,A^{+})\rightarrow(B,B^{+})$ naive étale after [HK, Definition 5.1] if it
admit a factorization into rational localizations and finite étale morphisms.
Here we assume $(A,A^{+})$ is strong sheafy and we assume that $(B,B^{+})$ is
sheafy.

######  Remark 3.3.

Certainly this is in more general situation than the corresponding context of
[HK].

######  Definition 3.4.

(Kedlaya [Ked1, Definition A5.2]) Recall from [Ked1, Definition A5.2], we have
the corresponding affinoid morphism from any strongly sheafy adic Banach ring
$A$, namely a morphism $A\rightarrow B$, such that $B$ admits some surjective
covering from $A\left<T_{1},...,T_{d}\right>$ and through this map we have
that $B$ is a stably-pseudocoherent sheaf over $A\left<T_{1},...,T_{d}\right>$
and we assume that $(B,B^{+})$ is strongly sheafy.

######  Remark 3.5.

Of course the corresponding foundation is not the same but parallel to such
situation we are considering now, however it is definitely reasonable and
parallel to follow [Ked1, Appendix A5] to give the definition here.

The belief (as proposed in [Ked1, Problem A5.3, Problem A5.4]) is that somehow
the corresponding affinoid morphisms in the definition should be directly used
in the corresponding definitions of lisse morphisms and unramified morphisms,
as well as certainly the étale morphisms. To investigate this kind of idea, we
are going to first investigate the corresponding naive étale morphisms along
this idea.

######  Lemma 3.6.

(Kedlaya) For any standard binary rational localization of $A$ with respect to
$f,g\in A$, suppose we know that there are two surjective morphisms:

(3.1) $\displaystyle
s_{1}:A\left<\frac{f}{g}\right>\left<T_{1},...,T_{n}\right>\rightarrow
B\left<\frac{f}{g}\right>,$ (3.2) $\displaystyle
s_{2}:A\left<\frac{g}{f}\right>\left<T_{1},...,T_{n}\right>\rightarrow
B\left<\frac{g}{f}\right>.$

Then we have that there is a surjective morphism:

(3.3) $\displaystyle s:A\left<T_{1},...,T_{n^{\prime}}\right>\rightarrow B.$

###### Proof.

The following argument is due to Kedlaya, we work out it for the convenience
of the readers. And this is the corresponding Banach analog of corresponding
parallel result in the Huber ring situation. First, we have the following
short exact sequence:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\left<\frac{f}{g}\right>\bigoplus
B\left<\frac{g}{f}\right>\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\left<\frac{f}{g},\frac{g}{f}\right>\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Take any $b\in B$, and use the notation $(b_{1},b_{2})$ for the image in the
middle. By the surjectivity of the maps $s_{1},s_{2}$ we have that there exist
some element $a_{1}\in A\left<\frac{f}{g}\right>\left<T_{1},...,T_{n}\right>$
and some element $a_{2}\in
A\left<\frac{f}{g}\right>\left<T_{1},...,T_{n}\right>$ such that we have:

(3.4) $\displaystyle s_{1}(a_{1})=b_{1},$ (3.5) $\displaystyle
s_{2}(a_{2})=b_{2}.$

With more explicit expression we have the following:

(3.6) $\displaystyle
s_{1}(\sum_{i_{1},...,i_{n}}\sum_{i}a_{1}^{i,i_{1},...,i_{n}}u^{i}T_{1}^{i_{1}}...T_{n}^{i_{n}})=\sum_{i}b_{1}^{i}u^{i},$
(3.7) $\displaystyle
s_{2}(\sum_{i_{1},...,i_{n}}\sum_{i}a_{2}^{i,i_{1},...,i_{n}}v^{i}T_{1}^{i_{1}}...T_{n}^{i_{n}})=\sum_{i}b_{2}^{i}v^{i},$

under the corresponding presentations up to liftings:

(3.8) $\displaystyle B\left<\frac{f}{g}\right>=B\left<u\right>/(gu-f),$ (3.9)
$\displaystyle B\left<\frac{g}{f}\right>=B\left<v\right>/(fv-g).$

Then to finish we only have to take some finite sum in the summation to make
approximation. We first claim that such finite sum approximation and
modification will not change the corresponding surjectivity of the map $s_{1}$
and $s_{2}$. Namely for each $k=1,2$ the map $s_{k}$ will maintain surjective
once we modify the image of $T_{1},...,T_{n}$ infinitesimally around some
neighbourhood $U$ of $0$, in other words it will maintain to be surjective
even if we set $s_{k}(T_{1}),...,s_{k}(T_{n})$ to be $x_{1},...,x_{n}$
whenever
$\|x_{1}-s_{k}(T_{1})\|\leq\delta,...,\|x_{n}-s_{k}(T_{n})\|\leq\delta$ for
some prescribed constant $\delta<1$ and moreover we have that the
corresponding modification could be assumed to take $T_{i}$ to $x_{i}$ with
$i=1,...,n$. By open mapping in this current context, we have that the
corresponding lifts of the corresponding differences
$x_{1}-s_{k}(T_{1}),...,x_{n}-s_{k}(T_{n})$ could be made to be living in some
arbitrarily chosen neighbourhood $V$ of $0$ namely, we can find lifts
$y_{1},...,y_{n}$ of these differences such that

(3.11) $\displaystyle\|y_{1}\|<1,...,\|y_{n}\|<1.$

Then we only have to consider the following map factors through the
corresponding map $s_{k}$:

(3.12) $\displaystyle h:A_{k}\left<T_{1},...,T_{n}\right>$
$\displaystyle\rightarrow A_{k}\left<T_{1},...,T_{n}\right>$ (3.13)
$\displaystyle T_{i}$ $\displaystyle\mapsto
T_{i}+\mathrm{lifts~{}of}~{}x_{i}-s_{k}(T_{k})$

where $A_{1}$ is the ring $A\left<\frac{f}{g}\right>$ while we have $A_{2}$ is
the ring $A\left<\frac{g}{f}\right>$, which basically proves the claim. Then
this will indicate that one can find some joint finite subset
$T:=\\{T_{1},...,T_{n^{\prime}}\\}$ for $B\left<\frac{f}{g}\right>$ and
$B\left<\frac{f}{g}\right>$ such that the modified

(3.14) $\displaystyle
s_{1}:A\left<\frac{f}{g}\right>\left<T_{1},...,T_{n^{\prime}}\right>\rightarrow
B\left<\frac{f}{g}\right>,$ (3.15) $\displaystyle
s_{2}:A\left<\frac{g}{f}\right>\left<T_{1},...,T_{n^{\prime}}\right>\rightarrow
B\left<\frac{g}{f}\right>,$

are basically surjective and they fit into the following commutative diagram:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\left<T_{1},...,T_{n^{\prime}}\right>\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\left<\frac{f}{g}\right>\left<T_{1},...,T_{n^{\prime}}\right>\bigoplus
A\left<\frac{g}{f}\right>\left<T_{1},...,T_{n^{\prime}}\right>\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\left<\frac{f}{g},\frac{g}{f}\right>\left<T_{1},...,T_{n^{\prime}}\right>\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\left<\frac{f}{g}\right>\bigoplus
B\left<\frac{g}{f}\right>\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\left<\frac{f}{g},\frac{g}{f}\right>\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where the middle and the rightmost vertical arrows are surjective. Then claim
is then that the left vertical one is also surjective. The kernels
$K_{1}\oplus K_{2}$ in the middle is mapped surjectively to the kernel
$K_{12}$ of the rightmost vertical map. So the snake lemma will force the
cokernel of the left vertical arrow to be zero which shows the corresponding
exactness at the corresponding location $?$ in the following commutative
diagram:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}\bigoplus
K_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{12}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\left<T_{1},...,T_{n^{\prime}}\right>\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\left<\frac{f}{g}\right>\left<T_{1},...,T_{n^{\prime}}\right>\bigoplus
A\left<\frac{g}{f}\right>\left<T_{1},...,T_{n^{\prime}}\right>\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\left<\frac{f}{g},\frac{g}{f}\right>\left<T_{1},...,T_{n^{\prime}}\right>\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}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{?}$$\textstyle{B\left<\frac{f}{g}\right>\bigoplus
B\left<\frac{g}{f}\right>\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\left<\frac{f}{g},\frac{g}{f}\right>\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}$$\textstyle{0}$$\textstyle{0}$

where $K_{1},K_{2},K_{12}$ are pseudocoherent, which implies that the
corresponding module $K$ is also pseudocoherent.

∎

######  Lemma 3.7.

Let $f_{1}:\Gamma_{1}\rightarrow\Gamma_{2}$ and
$f_{2}:\Gamma_{2}\rightarrow\Gamma_{3}$ be two affinoid morphisms, then the
composition $f_{2}\circ f_{1}$ is also affinoid.

###### Proof.

Straightforward. ∎

######  Proposition 3.8.

Any naive étale morphism is affinoid.

###### Proof.

See proposition 3.8. ∎

Therefore as in the corresponding Huber pair situation, we have proved that
the corresponding étale maps in the corresponding naive sense is actually
affinoid in the above sense. Therefore it is now natural to try to find the
corresponding properties which may completely characterize the corresponding
naive étale morphisms which are affinoid.

Certainly we may have the corresponding conjectures that all the naive étale
morphisms will satisfy the corresponding properties of algebraically étale
ones (such as in [EGAIV4, Chapitre 17], [SP, Tag 00U1]). We now discuss the
corresponding Banach completed cotangent complex after Huber [Hu1, 1.6.2].
Recall for our current $B$ the corresponding completed differential
$\Omega^{1}_{B/A,\mathrm{topo}}$ (see [Hu1, 1.6.2] for the construction for
any $f$-adic rings). Certainly in the context of adic Banach rings we have the
parallel completed version of the differentials by taking Banach completion,
which is in some trival way the current situation. Therefore we consider the
corresponding topological naive cotangent complex:

(3.16) $\displaystyle\tau_{\leq 1}\mathbb{L}_{B/A,\mathrm{topo}}$

for any naive étale map $f:(A,A^{+})\rightarrow(B,B^{+})$. We consider the
construction without the corresponding strongly noetherian requirement in our
current situation. First we know that $B$ is of topologically finite type over
$A$:

(3.17) $\displaystyle B=A\left<X_{1},...,X_{n}\right>_{T_{1},...,T_{n}}/I.$

Then we could first define the Banach free differentials:

(3.18)
$\displaystyle\Omega^{1}:=A\left<X_{1},...,X_{n}\right>_{T_{1},...,T_{n}}dX_{1}+...+A\left<X_{1},...,X_{n}\right>_{T_{1},...,T_{n}}dX_{n}.$

Then we have:

(3.19) $\displaystyle\Omega^{1}_{B/A,\mathrm{topo}}:=\Omega^{1}/(I\bigcup
d(I))\Omega^{1}.$

Here everything is assumed to be basically complete with respect to the
corresponding natural topology. Namely we need to take the corresponding
completion always with respect to the corresponding induced norms. Certainly
here $\Omega^{1}$ is already complete due to the fact that it is finitely
projective. Recall that a map $f:\Gamma_{1}\rightarrow\Gamma_{2}$ is called
étale in the scheme theory if the naive cotangent complex (truncated and could
be regarded as an $\infty$-module spectrum) is quasi-isomorphic to zero. The
corresponding underlying complex reads:

$\textstyle{[I/I^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{1}_{\Gamma_{2}/\Gamma_{1},\mathrm{topo}}].}$

In the situation where we consider $A\rightarrow B$ is affinoid, the
corresponding ideal $I$ is actually stably-pseudocoherent over
$A\left<T_{1},...,T_{n}\right>$. It is peudocoherent by the corresponding two
out of three property. The stability holds locally, so we have the case. And
if the morphism if furthermore naive étale then we have $I/I^{2}$ is also
stably-pseudocoherent, see lemma 4.8.

######  Remark 3.9.

Note that we are considering the very general and complicated non-noetherian
situation, modules will need to be endowed with the natural topology coming
from the Banach structures on the Banach rings and complete, although in this
situation as well finite projective modules are complete automatically. This
will have nontrivial things to do with the corresponding definition of
$\Omega^{1}_{B/A,\mathrm{topo}}$.

In the Banach world, one can actually generalize the corresponding full
cotangent complexes and de Rham complex to this context. First for the
corresponding topological cotangent complex we consider the following
definition (note that we have to assume the corresponding topologically finite
type condition). We start with the corresponding algebraic ones for
$B^{h}=A[X_{1},...,X_{n}]_{T_{1},...,T_{n}}/I$, under the topologization we
have the corresponding derived cotangent complex:

(3.20) $\displaystyle\mathbb{L}_{B^{h}/A,\mathrm{alg}},$

by taking the usual algebraic one. Then we take the corresponding completion
with respect to the corresponding topologization which gives rise to the
following topological one:

(3.21) $\displaystyle\mathbb{L}_{B/A,\mathrm{topo}}.$

We define the corresponding de Rham complex in the following way parallely.
What is happen is that consider the presentation
$B^{h}=A[X_{1},...,X_{n}]_{T_{1},...,T_{n}}/I$ which gives rise to the
corresponding algebraic de Rham complex:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{1}_{B^{h}/A,\mathrm{alg}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{2}_{B^{h}/A,\mathrm{alg}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{\bullet}_{B^{h}/A,\mathrm{alg}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...,}$

which will give rise to the corresponding topological one if we take the
corresponding completion under $(\|.\|,\mathrm{Ban})$ induced from the subset
$T_{1},...,T_{n}$:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{h}_{\|.\|,\mathrm{Ban}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{1}_{B^{h}/A,\mathrm{alg},\|.\|,\mathrm{Ban}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{2}_{B^{h}/A,\mathrm{alg},\|.\|,\mathrm{Ban}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{\bullet}_{B^{h}/A,\mathrm{alg},\|.\|,\mathrm{Ban}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...,}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\Omega}^{1}_{B^{h}/A,\mathrm{topo}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\Omega}^{2}_{B^{h}/A,\mathrm{topo}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\Omega}^{\bullet}_{B^{h}/A,\mathrm{topo}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{....}$

From our construction for ${\Omega}^{1}_{B^{h}/A,\mathrm{topo}}$, one can
actually define:

(3.22)
$\displaystyle{\Omega}^{\bullet,\mathrm{f}}_{B^{h}/A,\mathrm{topo}}:=\bigoplus_{i_{1},...,i_{\bullet}\in\\{1,...,n\\}}A\left<X_{1},...,X_{n}\right>_{T_{1},...,T_{n}}dX_{i_{1}}\wedge
dX_{i_{2}}\wedge...\wedge dX_{i_{\bullet}}$

and then define:

(3.23)
$\displaystyle{\Omega}^{\bullet}_{B/A,\mathrm{topo}}:=\left(\bigoplus_{i_{1},...,i_{\bullet}\in\\{1,...,n\\}}A\left<X_{1},...,X_{n}\right>_{T_{1},...,T_{n}}dX_{i_{1}}\wedge
dX_{i_{2}}\wedge...\wedge dX_{i_{\bullet}}\right)/$ (3.24)
$\displaystyle\left((I\bigcup dI\bigcup
d^{\bullet}I)\bigoplus_{i_{1},...,i_{\bullet}\in\\{1,...,n\\}}A\left<X_{1},...,X_{n}\right>_{T_{1},...,T_{n}}dX_{i_{1}}\wedge
dX_{i_{2}}\wedge...\wedge dX_{i_{\bullet}}\right),$

after taking suitable completion under Banach norms when needed. One can also
follow the construction in [III1], [III2], [B1] and [GL] to first consider the
corresponding polynomial resolution $P_{\bullet}$ for $B$, then consider the
corresponding algebraic derived de Rham complex
$\Omega^{*}_{P_{\bullet}/A,\mathrm{alg}}$, then take the corresponding Banach
completion to produce the corresponding topological one
$\Omega^{*}_{P_{\bullet}/A,\mathrm{topo}}$. Then as in [III1], [III2], [GL]
and [B1] around analytic derived $p$-adic de Rham complex we can take the
corresponding suitable derived filtered completion to get the complex
$\widehat{\Omega}^{\bullet}_{B/A,\mathrm{topo}}$ (certainly we need to
consider Banach version of some filtered derived category of simplicial Banach
rings). In the pro-étale site theoretic setting for rigid spaces (namely the
corresponding p-complete context) this recovers the corresponding construction
in [GL]. Recall in [GL] the corresponding analytic derived $p$-adic de Rham
complex is constructed by first define the integral version
${\Omega}^{\bullet}_{B_{0}/A_{0},\mathrm{topo}}$, and then take the colimit
throughout all such rings of definition, and invert $p$, and then take the
filtered completion.

######  Remark 3.10.

Certainly after this previous discussion we can construct the corresponding
Banach derived de Rham complex, Banach cotangent complex and Banach André-
Quillen homology for any morphism $A\rightarrow B$ of Banach rings admissible
in our situation. Recall from [III1], [III2], [B1] we have the corresponding
algebraic $p$-adic derived cotangent complex:

(3.25) $\displaystyle\Omega^{1}_{A[B]^{\bullet}/A}\otimes_{A[B]^{\bullet}}B$

where $A[B]^{\bullet}$ is just the corresponding standard cofibrant
replacement (in the topology theoretic language) of $B/A$. Then we take the
corresponding derived completion 333The derived completion in our Banach
situation is the derived Banach completion which for instance could happen by
using the ’completion’ functor from
$\mathrm{Simp}(\mathrm{Ind}(\mathrm{NormSets}))$ to
$\mathrm{Simp}(\mathrm{Ind}(\mathrm{BanachSets}))$ literally in [BBBK]. under
the induced norm to achieve the corresponding topological one:

(3.26)
$\displaystyle\mathbb{L}_{B/A,\mathrm{topo}}:=(\Omega^{1}_{A[B]^{\bullet}/A}\otimes_{A[B]^{\bullet}}B)^{\wedge}_{\|.\|}.$

We have the corresponding algebraic $p$-adic derived de Rham complex:

(3.27) $\displaystyle\Omega^{\bullet}_{A[B]^{\bullet}/A}$

where $A[B]^{\bullet}$ is just the corresponding standard cofibrant
replacement (in the topology theoretic language) of $B/A$. Then we take the
corresponding completion under the induced norm to achieve the corresponding
topological one:

(3.28)
$\displaystyle\mathbb{\mathrm{dR}}_{B/A,\mathrm{topo}}:=(\Omega^{\bullet}_{A[B]^{\bullet}/A})^{\wedge}_{\|.\|},$

carrying certain filtration
$\mathrm{Fil}^{*}_{\mathbb{\mathrm{dR}}_{B/A,\mathrm{topo}}}$, which allows
one take the corresponding filtered completion to achieve the corresponding
final object:

(3.29)
$\displaystyle\mathbb{\mathrm{dR}}^{\wedge}_{B/A,\mathrm{topo}}:=(\Omega^{\bullet}_{A[B]^{\bullet}/A})^{\wedge}_{\|.\|,\mathrm{Fil}^{*}_{\mathbb{\mathrm{dR}}_{B/A,\mathrm{topo}}}}.$

##  4\. Naive Étale Morphisms and Intrinsic Étale Morphisms

As discussed above we now study the corresponding properties of naive étale
morphisms aiming at the corresponding characterization of the corresponding
correct definitions of intrinsic étale morphisms. We now assume the
corresponding analyticity of the adic rings.

######  Lemma 4.1.

Let $f:(A,A^{+})\rightarrow(B,B^{+})$ be any rational localization map. Then
we have that $B$ is of topologically finite type.

###### Proof.

Straightforward. ∎

######  Lemma 4.2.

Let $f:(A,A^{+})\rightarrow(B,B^{+})$ be any rational localization map. Then
we have that $B$ is affinoid over $A$ in the sense definition 3.4.

###### Proof.

See the proof of proposition 3.8. ∎

######  Lemma 4.3.

Let $f:(A,A^{+})\rightarrow(B,B^{+})$ be any rational localization map. Then
we have

(4.1) $\displaystyle\tau_{\leq 1}\mathbb{L}_{B/A,\mathrm{topo}}$

is quasi-isomorphic to zero.

###### Proof.

It suffices to reduce to standard binary localization such as simple Laurent
or balanced localization, where we give a proof in the case of simple Laurent
one:

(4.2) $\displaystyle A\rightarrow A\\{T\\}/(T-f)$

for some $f\in A$. Then we have that actually the corresponding topological
cotangent complex will be the corresponding completion of the corresponding
algebraic ones (see [Hu1, Proposition 1.6.3]). The corresponding quasi-
isomorphism could be defined directly. One just considers the following
algebraic differential map:

(4.3) $\displaystyle I=(T-f)\rightarrow A[T]/(T-f)dT$

which is actually surjective since for any:

(4.5) $\displaystyle\sum_{i\geq 0}a_{i}T^{i}+g(T)(T-f)dT\in A[T]/(T-f)dT$

one takes the corresponding integration of:

(4.6) $\displaystyle\int_{f}^{T}\sum_{i\geq 0}a_{i}T^{i}+g(T)(T-f)dT$
$\displaystyle=\int_{f}^{T}\sum_{i\geq 0}a_{i}T^{i}+(\sum_{i\geq
0}g_{i}T^{i})(T-f)dT$ (4.7) $\displaystyle=\int_{f}^{T}\sum_{i\geq
0}a_{i}T^{i}+\int_{f}^{T}(\sum_{i\geq 0}g_{i}T^{i})(T-f)dT$ (4.8)
$\displaystyle=\int_{f}^{T}\sum_{i\geq 0}a_{i}T^{i}+\int_{f}^{T}\sum_{i\geq
0}g_{i}T^{i+1}-f\int_{f}^{T}\sum_{i\geq 0}g_{i}T^{i}$ (4.9)
$\displaystyle=\sum_{i\geq 0}a_{i}\frac{1}{i+1}T^{i+1}|_{f}^{T}+\sum_{i\geq
0}g_{i}\frac{1}{i+2}T^{i+2}|_{f}^{T}-f\sum_{i\geq
0}g_{i}\frac{1}{i+1}T^{i+1}|_{f}^{T}$ (4.10) $\displaystyle=(*)(T-f),$

where we only have finite sums here since we are considering the corresponding
topologized polynomial. For this algebraic map, we have that kernel is
$(T-f)^{2}$, for instance consider:

(4.11) $\displaystyle d((T-f)h(T))=0,$

we will have:

(4.12) $\displaystyle h(T)+(T-f)h^{\prime}(T)dT=0,$

which implies that the image of $(T-f)h(T)$ lives in the corresponding
quotient $A[T]/(T-f)dT$. Therefore we have that the topologized (not complete
yet) cotangent complex:

$\textstyle{[(T-f)/(T-f)^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A[T]/(T-f)dT],}$

which is quasi-isomorphic to zero. Then we take the corresponding completion
with respect to the corresponding topology induced from $A$ we have the
desired result.

∎

######  Lemma 4.4.

Let $f:(A,A^{+})\rightarrow(B,B^{+})$ be any finite étale map. Then we have
that $B$ is of topologically finite type.

###### Proof.

Since we have that $B$ is affinoid over $A$ by proposition 3.8. ∎

######  Lemma 4.5.

Let $f:(A,A^{+})\rightarrow(B,B^{+})$ be any finite étale map. Then we have
that $B$ is affinoid over $A$ in the sense definition 3.4.

###### Proof.

See the proof of proposition 3.8. ∎

######  Lemma 4.6.

Let $f:(A,A^{+})\rightarrow(B,B^{+})$ be any finite étale map. Then we have

(4.13) $\displaystyle\tau_{\leq 1}\mathbb{L}_{B/A,\mathrm{topo}}$

is quasi-isomorphic to zero.

###### Proof.

This is basically nontrivial due to the fact that we are discussing the
corresponding topological cotangent complex. However, one takes the
corresponding finite presentation $B=A[T_{1},...,T_{n}]/(f_{1},...,f_{p})$
(note that in fact that we have that $B$ is finite over $A$) since we are
considering a finite étale map, which will realize the corresponding desired
algebraic cotangent complex:

$\textstyle{[I/I^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{1}_{B/A}].}$

For the topological situation we have that
$B=A\\{T_{1},...,T_{n}\\}/(f_{1},...,f_{p})$ by taking the corresponding
completion. Again note that this means that actually the corresponding
$A$-algebra $B$ is still finite over $A$ (since that is the very assumption).
Therefore we could have the chance to right $B$ as just $\oplus_{i}Ae_{i}$
this is basically inducing the same differential module
$\bigoplus_{j}(\bigoplus_{i}Ae_{i})dT_{j}$ in both the corresponding
topological setting and the algebraic setting. Then one could get the
corresponding desired topological cotangent complex which is quasi-isomorphic
to zero. ∎

Then we consider the corresponding local composition:

######  Lemma 4.7.

Let $f:(A,A^{+})\rightarrow(B,B^{+})$ be any naive étale map. Then we have
that $B$ is of topologically finite type.

###### Proof.

Since we have that $B$ is affinoid over $A$ by proposition 3.8. ∎

######  Lemma 4.8.

Let $f:(A,A^{+})\rightarrow(B,B^{+})$ be any naive étale morphism. Then we
have

(4.14) $\displaystyle\tau_{\leq 1}\mathbb{L}_{B/A,\mathrm{topo}}$

is quasi-isomorphic to zero locally with respect to the corresponding rational
localization.

###### Proof.

Locally we have that any naive étale morphism takes the corresponding
truncated cotangent complex to be trivialized, by the corresponding
composition properties of the cotangent complex [SP, Tag 08PN]. ∎

In order to globalize the picture one has to work harder. First we have the
following:

######  Proposition 4.9.

Let $f:(A,A^{+})\rightarrow(B,B^{+})$ be any naive étale morphism. Then $f$ is
affinoid, of topologically finite type.

###### Proof.

This is by proposition 3.8 for the affinoidness, which implies the
corresponding second property. ∎

######  Definition 4.10.

We now define localized 444However this could actually be globalized easily.
intrinsic étale morphism to be a morphism $f:(A,A^{+})\rightarrow(B,B^{+})$
which is affinoid with strongly sheafy target, and locally (with respect to
the rational localization) the corresponding truncated topological cotangent
complex is quasi-isomorphic to zero.

We now consider the corresponding intrinsic étale morphisms of the
corresponding special adic spaces after [HK]:

######  Setting 4.11.

We now consider the three special adic spaces after [HK], they are the
corresponding strongly sheafy adic spaces, the corresponding sousperfectoid
adic spaces and the corresponding diamantine adic spaces. We will use the
notations $T,S,D$ to denote them in general respectively.

The corresponding categories of strongly sheafy adic spaces, sousperfectoid
spaces and diamantine adic spaces are nice enough since at least we have well-
defined notion of naive étale morphisms (which is certainly the correct one)
and furthermore well-defined étale sites.

######  Definition 4.12.

For strongly sheafy adic spaces, a morphism $T_{1}\rightarrow T_{2}$ is called
localized intrinsic étale if locally on $T_{1}$ this is localized intrinsic
étale, namely for any neighbourhood $U\subset T_{1}$ we have that the morphism
$(\mathcal{O}_{T_{2}}(U^{\prime}),\mathcal{O}^{+}_{T_{2}}(U^{\prime}))\rightarrow(\mathcal{O}_{T_{1}}(U),\mathcal{O}^{+}_{T_{1}}(U))$
is localized intrinsic étale.

######  Definition 4.13.

For sousperfectoid adic spaces, a morphism $S_{1}\rightarrow S_{2}$ is called
localized intrinsic étale if locally on $S_{1}$ this is localized intrinsic
étale, namely for any neighbourhood $U\subset S_{1}$ we have that the morphism
$(\mathcal{O}_{S_{2}}(U^{\prime}),\mathcal{O}^{+}_{S_{2}}(U^{\prime}))\rightarrow(\mathcal{O}_{S_{1}}(U),\mathcal{O}^{+}_{S_{1}}(U))$
is localized intrinsic étale.

######  Definition 4.14.

For diamantine adic spaces, a morphism $D_{1}\rightarrow D_{2}$ is called
localized intrinsic étale if locally on $D_{1}$ this is localized intrinsic
étale, namely for any neighbourhood $U\subset D_{1}$ we have that the morphism
$(\mathcal{O}_{D_{2}}(U^{\prime}),\mathcal{O}^{+}_{D_{2}}(U^{\prime}))\rightarrow(\mathcal{O}_{D_{1}}(U),\mathcal{O}^{+}_{D_{1}}(U))$
is localized intrinsic étale.

##  5\. Properties

We now study the corresponding properties of the corresponding localized
intrinsic étale morphisms, following [EGAIV4] and [Hu1]. We now assume the
corresponding analyticity of the adic rings.

######  Conjecture 5.1.

Any localized intrinsic étale morphism of strongly sheafy adic spaces is
locally a composition of rational localization and finite étale morphism.

Here is the special situation.

######  Proposition 5.2.

As in [Hu1], namely in the strongly noetherian situation we have the
conjecture holds.

###### Proof.

This is because in that setting our definition in the intrinsic setting
coincides with the more algebraic one in [Hu1]. And note that in this setting
the affinoidness of the morphism reduces to just being admitting surjections
from Tate algebra over the source. ∎

If this is true then we have:

######  Corollary 5.3.

Any localized intrinsic étale morphism of sousperfectoid adic spaces is
locally a composition of rational localization and finite étale morphism. Any
localized intrinsic étale morphism of diamantine adic spaces is locally a
composition of rational localization and finite étale morphism.

######  Proposition 5.4.

Compositions of localized intrinsic étale morphisms of strongly sheafy adic
spaces are again localized intrinsic étale morphism.

###### Proof.

Locally it is the corresponding compositions of topologically finite type
morphism, and locally it is the corresponding compositions of the
corresponding affinoid morphisms, and locally it is the corresponding
compositions of morphisms giving rise to the quasi-isomorphic to zero
truncated cotangent complex. ∎

######  Corollary 5.5.

Compositions of localized intrinsic étale morphisms of sousperfectoid adic
spaces are again localized intrinsic étale morphism. Compositions of localized
intrinsic étale morphisms of diamantine adic spaces are again localized
intrinsic étale morphism.

######  Proposition 5.6.

The localized intrinsic étaleness of any morphism $T_{1}\rightarrow T_{2}$ of
strongly sheafy adic rings is preserved under the base change along any
morphism of $T_{3}\rightarrow T_{2}$.

###### Proof.

The base change of any morphism of topologically finite type is again of
topologically finite type. The affinoidness of morphism is also preserved
under any base change morphism. Finally for the cotangent complex locally, we
definitely have the corresponding result as well. ∎

######  Proposition 5.7.

The étale property of a morphism between strongly sheafy rings could be
detected locally at each point.

###### Proof.

Straightforward. ∎

We now consider some functoriality issue in our current situation. Now we
consider the corresponding localized intrinsic étale morphisms under the
construction of Witt vectors. Now let:

(5.1) $\displaystyle A\rightarrow B$

be a general morphism in positive characteristic. Therefore we can take the
corresponding Witt vector construction:

(5.2) $\displaystyle W(A^{\flat})\rightarrow W(B^{\flat}),$

where we assume that $A^{\flat}\rightarrow B^{\flat}$ is localized intrinsic
étale. Here we take the completion if needed along the corresponding
Fontainisation.

######  Proposition 5.8.

The map

(5.3) $\displaystyle W(A^{\flat})\rightarrow W(B^{\flat}),$

is affinoid if the kernel is closed 111This is again due to the very subtle
point around the sheafiness such as in [Ked1, Theorem 1.4.20]..

###### Proof.

We only need to check this locally, locally we have that there is a lifting:

(5.4) $\displaystyle W(A^{\flat})\\{T_{1},...\\}\rightarrow
W(B^{\flat})\rightarrow 0$

from the corresponding surjection:

(5.5) $\displaystyle A^{\flat}\\{\overline{T}_{1},...\\}\rightarrow
B^{\flat}\rightarrow 0.$

And what we have is that this map on the Witt vector level is also realizing
the target as a stably-pseudocoherent module over the source since we have
that the target is sheafy ([Ked1, Theorem 1.4.20]). ∎

######  Proposition 5.9.

Same holds for the construction of integral Robba ring
$\widetilde{\mathcal{R}}^{r}_{*}$ and Robba ring
$\widetilde{\mathcal{R}}^{[s,r]}_{*}$ with respect to closed intervals as in
[KL1] and [KL2].

##  6\. Étale-Like Morphisms of $\infty$-Banach Rings and the
$\infty$-Analytic Stacks

###  6.1. Approach through De Rham Stacks

We now extend the corresponding discussion to the $\mathrm{E}_{\infty}$
objects in [BBBK, Remark 3.16] by using the ideas as in [R]. Recall from
[BBBK, Theorem 3.14] we have the corresponding categories
$\mathrm{Simp}\mathrm{Ind}^{m}(\mathrm{BanSets}_{H})$ and
$\mathrm{Simp}\mathrm{Ind}(\mathrm{BanSets}_{H})$ which are the corresponding
categories of the corresponding simplicial sets over the corresponding
inductive categories of the corresponding Banach sets over some Banach ring
$H$ 555It is safer to assume the open mapping properties on homotopy groups..

######  Theorem 6.1.

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

######  Corollary 6.2.

The corresponding categories
$\mathrm{Simp}\mathrm{Ind}^{m}(\mathrm{BanSets}_{H})$ and
$\mathrm{Simp}\mathrm{Ind}(\mathrm{BanSets}_{H})$ admit presentations as
$(\infty,1)$-categories. Same holds for
$\mathrm{Simp}\mathrm{Ind}^{m}(\mathrm{NrSets}_{H})$ and
$\mathrm{Simp}\mathrm{Ind}(\mathrm{NrSets}_{H})$.

Then recall from [BBBK, Remark 3.16] we have the corresponding ring objects in
the $\infty$-categories above:

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

and

(6.3)
$\displaystyle\mathrm{sComm}(\mathrm{Simp}\mathrm{Ind}^{m}(\mathrm{NrSets}_{H})),$
(6.4)
$\displaystyle\mathrm{sComm}(\mathrm{Simp}\mathrm{Ind}(\mathrm{NrSets}_{H})).$

Now we use general notation $A$ to denote any object in these categories,
regarding as a general $\mathrm{E}_{\infty}$-ring. We consider the general
morphism $A\rightarrow B$ in the first two categories in the following
discussion.

######  Definition 6.3.

For any general morphism $A\rightarrow B$, we call this affinoid if we have
that that $\pi_{0}(B)$ is affinoid over $\pi_{0}(A)$, namely we have that
there is a surjection map
$\pi_{0}(A)\left<X_{1},...,X_{d}\right>\rightarrow\pi_{0}(B)$. And moreover we
assume that
$\pi_{0}(B)\otimes_{\pi_{0}(A)}\pi_{n}(A)\overset{\sim}{\rightarrow}\pi_{n}(B)$,
for any $n$.

######  Remark 6.4.

Kedlaya’s theorem [Ked1, Theorem 1.4.20] is actually expected to hold in more
general setting, at least in the situation where the definition of the
affinoid morphisms could be made independent from the corresponding stably-
pseudocoherence for open mapping rings (note that we are working over analytic
fields). However in the previous definition, we have been not really exact. To
be really accurate in the characterization of some desired notion of the
affinoidness we think that one has to add certain $\infty$-sheafiness (which
certainly holds in [BK]). To be more precise for any general morphism
$A\rightarrow B$ in [BK] (namely in current situation one considers the
corresponding Banach algebras over the analytic fields in our situation), we
call this affinoid if we have that that $\pi_{0}(B)$ is affinoid over
$\pi_{0}(A)$, namely we have that there is a surjection map
$\pi_{0}(A)\left<X_{1},...,X_{d}\right>\rightarrow\pi_{0}(B)$. And moreover we
assume that
$\pi_{0}(B)\otimes_{\pi_{0}(A)}\pi_{n}(A)\overset{\sim}{\rightarrow}\pi_{n}(B)$,
for any $n$. In this situation we have the nice sheafiness (up to higher
homotopy). Again similar discussion could be made in the context of [CS]. Note
that [Ked1, Theorem 1.4.20] literally says that the sheafiness is equivalent
(in some nice sense but in more flexible derived sense) to the stably-
pseudocoherence.

######  Definition 6.5.

For any general morphism $A\rightarrow B$, we call this localized intrinsic
étale if we have that that $\pi_{0}(B)$ is localized intrinsic étale over
$\pi_{0}(A)$, namely we have that there is a surjection map
$\pi_{0}(A)\left<X_{1},...,X_{d}\right>\rightarrow\pi_{0}(B)$ and we have that
locally the corresponding truncated topological cotangent complex is basically
quasi-isomorphic to zero. And moreover we assume that
$\pi_{0}(B)\otimes_{\pi_{0}(A)}\pi_{n}(A)\overset{\sim}{\rightarrow}\pi_{n}(B)$,
for any $n$.

We now use the corresponding $X=\mathrm{Spec}A$ to denote the corresponding
$\infty$-stack in the opposite categories with respect to the ring $A$. We now
define the corresponding de Rham stack attached to $X$ as in [R, Remark 1.2]:

######  Definition 6.6.

We now define:

(6.5) $\displaystyle X_{\mathrm{dR}}(R):=\varinjlim_{I}X(\pi_{0}(R)/I)$

for any $R$ in

(6.6)
$\displaystyle\mathrm{sComm}(\mathrm{Simp}\mathrm{Ind}^{m}(\mathrm{BanSets}_{H})),$
(6.7)
$\displaystyle\mathrm{sComm}(\mathrm{Simp}\mathrm{Ind}(\mathrm{BanSets}_{H})).$

And the injective limit is taking throughout all nilpotent ideals of
$\pi_{0}(R)$.

######  Remark 6.7.

As in [R, Definition 1.1, Remark 1.2], one can actually define the
corresponding de Rham and crystalline spaces for any functor from
$\mathrm{sComm}(\mathrm{Simp}\mathrm{Ind}^{m}(\mathrm{BanSets}_{H})))$ and
$\mathrm{sComm}(\mathrm{Simp}\mathrm{Ind}(\mathrm{BanSets}_{H})))$ to
$\underline{s\mathrm{Sets}}$. This means we do not have to consider
$(\infty,1)$-sheaves satisfying certain $\infty$-descent conditions.

######  Definition 6.8.

The corresponding formal completion of any morphism
$X=\mathrm{Spec}B\rightarrow Y=\mathrm{Spec}A$:

(6.8) $\displaystyle Y_{X,\mathrm{dR}}$

is defined to be:

(6.9) $\displaystyle
Y_{X,\mathrm{dR}}(R):=\varinjlim_{I}X(\pi_{0}(R)/I)\times_{Y(\pi_{0}(R)/I)}Y(R),$

for any $R$ in

(6.10)
$\displaystyle\mathrm{sComm}(\mathrm{Simp}\mathrm{Ind}^{m}(\mathrm{BanSets}_{\mathbb{Q}_{p}})),$
(6.11)
$\displaystyle\mathrm{sComm}(\mathrm{Simp}\mathrm{Ind}(\mathrm{BanSets}_{\mathbb{Q}_{p}})).$

And the injective limit is taking throughout all nilpotent ideals of
$\pi_{0}(R)$.

######  Remark 6.9.

Certainly it is actually not clear how really we should deal with the
corresponding ideals here, namely we are not for sure if we need to consider
closed ideals. But for simplicial noetherian rings we really have some nice
definitions, which will certainly be tangential to the corresponding Huber’s
original consideration.

######  Definition 6.10.

We now define the corresponding de Rham intrinsic étale morphism to be an
affinoid morphism $X=\mathrm{Spec}B\rightarrow Y=\mathrm{Spec}A$ which
satisfies the condition:

(6.12)
$\displaystyle\pi_{0}(X(R))\overset{\sim}{\rightarrow}\pi_{0}(Y_{X,\mathrm{dR}}(R)),$

for any $\mathrm{E}_{\infty}$-object $R$.

######  Proposition 6.11.

Compositions of de Rham intrinsic étale morphisms are again $\mathrm{PD}$
intrinsic étale morphisms.

###### Proof.

This is formal. ∎

###  6.2. Approach through Crystalline Stack and PD-morphisms

We now use the corresponding $X=\mathrm{Spec}A$ to denote the corresponding
$\infty$-stack in the opposite categories with respect to the ring $A$. We now
define the corresponding crystalline stack attached to $X$ as in [R,
Definition 1.1]:

######  Definition 6.12.

We now define:

(6.13) $\displaystyle
X_{\mathrm{crys}}(R):=\varinjlim_{I,\gamma}X(\pi_{0}(R)/I)$

for any $R$ in

(6.14)
$\displaystyle\mathrm{sComm}(\mathrm{Simp}\mathrm{Ind}^{m}(\mathrm{BanSets}_{H})),$
(6.15)
$\displaystyle\mathrm{sComm}(\mathrm{Simp}\mathrm{Ind}(\mathrm{BanSets}_{H})).$

And the injective limit is taking throughout all nilpotent ideals of
$\pi_{0}(R)$ and the corresponding PD-structures.

######  Definition 6.13.

The corresponding PD completion of any morphism $X=\mathrm{Spec}B\rightarrow
Y=\mathrm{Spec}A$:

(6.16) $\displaystyle Y_{X,\mathrm{crys}}$

is defined to be:

(6.17) $\displaystyle
Y_{X,\mathrm{crys}}(R):=\varinjlim_{I,\gamma}X(\pi_{0}(R)/I)\otimes_{Y(\pi_{0}(R)/I)}Y(R),$

for any $R$ in

(6.18)
$\displaystyle\mathrm{sComm}(\mathrm{Simp}\mathrm{Ind}^{m}(\mathrm{BanSets}_{H})),$
(6.19)
$\displaystyle\mathrm{sComm}(\mathrm{Simp}\mathrm{Ind}(\mathrm{BanSets}_{H})).$

And the injective limit is taking throughout all nilpotent ideals of
$\pi_{0}(R)$ and all the corresponding PD structures.

######  Remark 6.14.

Certainly it is actually not clear how really we should deal with the
corresponding ideals here and the corresponding PD structures, namely we are
not for sure if we need to consider closed ideals. But for simplicial
noetherian rings we really have some nice definitions, which will certainly be
tangential to the corresponding Huber’s original consideration.

######  Definition 6.15.

We now define the corresponding PD intrinsic étale morphism to be an affinoid
morphism $X=\mathrm{Spec}B\rightarrow Y=\mathrm{Spec}A$ which satisfies the
condition:

(6.20)
$\displaystyle\pi_{0}(X(R))\overset{\sim}{\rightarrow}\pi_{0}(Y_{X,\mathrm{crys}}(R)),$

for any $\mathrm{E}_{\infty}$-object $R$.

######  Proposition 6.16.

We have that any de Rham intrinsic étale morphism is a PD intrinsic étale
morphism.

###### Proof.

This is formal. ∎

######  Proposition 6.17.

Compositions of $\mathrm{PD}$ intrinsic étale morphisms are again
$\mathrm{PD}$ intrinsic étale morphisms.

###### Proof.

This is formal. ∎

##  7\. Lisse-Like and Non-Ramifié-Like Morphisms of $\infty$-Banach Rings and
the $\infty$-Analytic Stacks

###  7.1. Approach through De Rham Stacks

We now define the corresponding lisse-like morphisms along the idea in the
previous section:

######  Definition 7.1.

For any general morphism $A\rightarrow B$, we call this localized intrinsic
lisse if we have that that $\pi_{0}(B)$ is localized intrinsic lisse over
$\pi_{0}(A)$, namely we have that there is a surjection map
$\pi_{0}(A)\left<X_{1},...,X_{d}\right>\rightarrow\pi_{0}(B)$ and we have that
locally the corresponding truncated topological cotangent complex is basically
quasi-isomorphic to $\Omega_{\pi_{0}(B)/\pi_{0}(A)}[0]$. And moreover we
assume that
$\pi_{0}(B)\otimes_{\pi_{0}(A)}\pi_{n}(A)\overset{\sim}{\rightarrow}\pi_{n}(B)$,
for any $n$.

######  Definition 7.2.

We now define the corresponding de Rham intrinsic lisse morphism to be an
affinoid morphism $X=\mathrm{Spec}B\rightarrow Y=\mathrm{Spec}A$ which
satisfies the condition:

(7.1)
$\displaystyle\pi_{0}(X(R))\overset{}{\rightarrow}\pi_{0}(Y_{X,\mathrm{dR}}(R))$

being surjective, for any $\mathrm{E}_{\infty}$-object $R$.

######  Definition 7.3.

For any general morphism $A\rightarrow B$, we call this localized intrinsic
non-ramifié if we have that that $\pi_{0}(B)$ is localized intrinsic non-
ramifié over $\pi_{0}(A)$, namely we have that there is a surjection map
$\pi_{0}(A)\left<X_{1},...,X_{d}\right>\rightarrow\pi_{0}(B)$ and we have that
locally the corresponding truncated topological cotangent complex is basically
quasi-isomorphic to $\Omega_{\pi_{0}(B)/\pi_{0}(A)}[0]$ which vanishes as
well. And moreover we assume that
$\pi_{0}(B)\otimes_{\pi_{0}(A)}\pi_{n}(A)\overset{\sim}{\rightarrow}\pi_{n}(B)$,
for any $n$.

###  7.2. Approach through Crystalline Stack and PD-morphisms

######  Definition 7.4.

We now define the corresponding PD intrinsic lisse morphism to be an affinoid
morphism $X=\mathrm{Spec}B\rightarrow Y=\mathrm{Spec}A$ which satisfies the
condition:

(7.2)
$\displaystyle\pi_{0}(X(R))\overset{}{\rightarrow}\pi_{0}(Y_{X,\mathrm{crys}}(R))$

being surjective, for any $\mathrm{E}_{\infty}$-object $R$666For non-ramifié
situation one considers injectivity..

######  Proposition 7.5.

We have that any de Rham intrinsic lisse morphism is a PD intrinsic lisse
morphism.

##  8\. Perfectization and Fontainisation of $\infty$-Analytic Stacks
Situation

###  8.1. Perfectization, Fontainisation and Crystalline Stacks

Now we consider the corresponding perfectoidization of $\infty$-analytic
stacks after [R] and [Dr1] in the situation where $H$ is assumed to be of
characteristic $p$.

######  Definition 8.1.

For any object in
$\infty-\mathrm{Fun}(\mathrm{sComm}\mathrm{Simp}\mathrm{Ind}(\mathrm{Ban}_{H}),\underline{s\mathrm{Sets}})$,
denoted by $X$, we define the corresponding perfectization $X^{1/p^{\infty}}$
of $X$ to be the corresponding functor such that for any
$R\in\mathrm{sComm}\mathrm{Simp}\mathrm{Ind}(\mathrm{Ban}_{H})$ we have that
$X^{1/p^{\infty}}(R):=X(R^{\flat})$ where we define the corresponding tilting
Fontainisation $R^{\flat}$ of $R$ to be the corresponding derived completion
of:

(8.1)
$\displaystyle\varprojlim\\{...\overset{\mathrm{Fro}}{\longrightarrow}R\overset{\mathrm{Fro}}{\longrightarrow}R\overset{\mathrm{Fro}}{\longrightarrow}R\\}.$

In the situation of the corresponding monomorphically inductive Banach sets,
we have the parallel definition:

######  Definition 8.2.

For any object in
$\infty-\mathrm{Fun}(\mathrm{sComm}\mathrm{Simp}\mathrm{Ind}^{m}(\mathrm{BanSets}_{H}),\underline{s\mathrm{Sets}})$,
denoted by $X$, we define the corresponding perfectization $X^{1/p^{\infty}}$
of $X$ to be the corresponding functor such that for any ring
$R\in\mathrm{sComm}\mathrm{Simp}\mathrm{Ind}^{m}(\mathrm{BanSets}_{H})$ we
have that $X^{1/p^{\infty}}(R):=X(R^{\flat})$ where we define the
corresponding tilting Fontainisation $R^{\flat}$ of $R$ to be the
corresponding derived completion of:

(8.2)
$\displaystyle\varprojlim\\{...\overset{\mathrm{Fro}}{\longrightarrow}R\overset{\mathrm{Fro}}{\longrightarrow}R\overset{\mathrm{Fro}}{\longrightarrow}R\\}.$

######  Remark 8.3.

This is very general notion beyond the corresponding $(\infty,1)$-sheaves
satisfying certain descent with respect to the derived rational localizations
or more general homotopy Zariski topology as in [BK] and [BBBK].

Now we follow [R, Proposition 5.3] and [Dr1, Section 1.1] to give the
following discussion around the corresponding Witt crystalline Stack:

######  Definition 8.4.

We define the corresponding Witt Crystalline Stack $X_{W}$ of any $X$ over
$H/\mathbb{F}_{p}$ in
$\infty-\mathrm{Fun}(\mathrm{sComm}\mathrm{Simp}\mathrm{Ind}(\mathrm{Ban}_{H}),\underline{s\mathrm{Sets}})$
or
$\infty-\mathrm{Fun}(\mathrm{sComm}\mathrm{Simp}\mathrm{Ind}^{m}(\mathrm{Ban}_{H}),\underline{s\mathrm{Sets}})$
to be the functor
$(W(\pi_{0}(X)^{\flat})_{\pi_{0}(X),\mathrm{crys}})_{p}^{\wedge}$. And we
define the corresponding pre-crystals to be sheaves of $\mathcal{O}$-modules
over this functors when we have that $X$ is an $\infty$-analytic stack with
reasonable topology.

######  Example 8.5.

For instance if we have that $X=\mathrm{Spa}^{h}(R)$ coming from the
corresponding Bambozzi-Kremnizer spectrum of any Banach ring over
$\mathbb{F}_{p}((t))$ as constructed in [BK]. Then we have that the
corresponding functor is
$(W(\pi_{0}(X)^{\flat})_{\pi_{0}(X),\mathrm{crys}})_{p}^{\wedge}$ is now
admitting structures coming from the corresponding homotopy Zariski topology
from $X$.

######  Example 8.6.

For instance if we have that $X=\mathrm{Spec}(\mathbb{F}_{p}[[t]])$ coming
from the corresponding object in the opposite category of
$\mathbb{F}_{p}[[t]]$. Then we have that the corresponding functor is
$(W(\pi_{0}(X)^{\flat})_{\pi_{0}(X),\mathrm{crys}})_{p}^{\wedge}$ is now
admitting structures coming from the corresponding homotopy Zariski topology
from $X$, is just the same as the corresponding one in the algebraic setting
constructed in [R, Proposition 5.3] and [Dr1, Section 1.1].

###  8.2. Perfectization, Fontainisation and Robba Stacks

Now we contact [KL1] and [KL2] to look at the corresponding Robba Stacks. Now
take any $X$ to be any $\infty$-analytic stack which admits structures of
simplicial complete Bornological rings or ind-Fréchet structures, namely we
have the corresponding complete bornological topology or ind-Fréchet topology
on $\pi_{0}(X)$. We work over $H/\mathbb{F}_{p}$ as well.

######  Example 8.7.

For instance we take that $X=\mathrm{Spa}^{h}(R)$ coming from the
corresponding Bambozzi-Kremnizer spectrum of any Banach ring over
$\mathbb{F}_{p}((t))$ as constructed in [BK].

######  Definition 8.8.

For any $\infty$-analytic stack $X$ as above, we consider the corresponding
Witt vector functor $W_{n}(\pi_{0}(X)^{\flat})$ and then consider
$\varinjlim_{n\rightarrow\infty}W_{n}(\pi_{0}(X)^{\flat})$, namely
$W(\pi_{0}(X)^{\flat})$, then we consider the ring
$W(\pi_{0}(X)^{\flat})[1/p]$. Then we can take the corresponding completion
with respect to the Gauss norm $\|.\|_{\pi_{0}(X),[s,r]}$ coming from the
corresponding norm on $\pi_{0}(X)$ with respect to some interval
$[s,r]\in(0,\infty)$ as in [KL2, Definition 4.1.1]:

(8.3)
$\displaystyle\widetilde{\Pi}(X)_{[s,r]}:=(W(\pi_{0}(X)^{\flat})[1/p])^{\wedge}_{\|.\|_{\pi_{0}(X),[s,r]},\text{Fr\'{e}}}.$

Then following [KL2, Definition 4.1.1] we consider the following:

(8.4)
$\displaystyle\widetilde{\Pi}(X)_{r}:=\varprojlim_{s>0}(W(\pi_{0}(X)^{\flat})[1/p])^{\wedge}_{\|.\|_{\pi_{0}(X),[s,r]},\text{Fr\'{e}}}$

and

(8.5)
$\displaystyle\widetilde{\Pi}(X):=\varinjlim_{r>0}\varprojlim_{s>0}(W(\pi_{0}(X)^{\flat})[1/p])^{\wedge}_{\|.\|_{\pi_{0}(X),[s,r]},\text{Fr\'{e}}}.$

We call these Robba stacks.

######  Example 8.9.

In the situation where $X$ is some $\infty$-analytic stack carrying the
corresponding sheaves of simplicial Banach rings (namely not in general
bornological or ind-Fréchet) we have that the finite projective modules over
the three Robba stacks (for by enough intervals) carrying semilinear Frobenius
action which realizes the corresponding isomorphisms by Frobenius pullbacks
are equivalent. In the noetherian setting we have the same holds for locally
finite presented sheaves as well. This is the main results of [KL2, Theorem
4.6.1].

### Acknowledgements

This is our independent work, however we would like to thank Professor Kedlaya
for helpful discussion, in particular the corresponding nontrivial input on
our understanding of the corresponding affinoid morphisms, Hansen-Kedlaya and
the corresponding AWS lecture notes.

## References

  * [HK] Hansen, David, and Kiran S. Kedlaya. "Sheafiness criteria for Huber rings." (2020).
  * [Ked1] Kedlaya, K. "Sheaves, stacks, and shtukas, lecture notes from the 2017 Arizona Winter School: Perfectoid Spaces." Math. Surveys and Monographs 242.
  * [EGAIV4] Grothendieck, Alexander. "Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné): IV. Étude locale des schémas et des morphismes de schémas, Quatrieme partie." Publications Mathématiques de l’IHÉS 32 (1967): 5-361.
  * [SP] Stacks Project Authors. "Stacks project." (2019).
  * [Hu1] Huber, Roland. Étale cohomology of rigid analytic varieties and adic spaces. Vol. 30. Springer, 2013.
  * [BBBK] Bambozzi, Federico, Oren Ben-Bassat, and Kobi Kremnizer. "Analytic geometry over $\mathbb{F}_{1}$ and the Fargues-Fontaine curve." Advances in Mathematics 356 (2019): 106815.
  * [BK] Bambozzi, Federico, and Kobi Kremnizer. "On the Sheafyness Property of Spectra of Banach Rings." arXiv preprint arXiv:2009.13926 (2020).
  * [BS] Bhatt, Bhargav, and Peter Scholze. "Prisms and prismatic cohomology." arXiv preprint arXiv:1905.08229 (2019).
  * [Dr1] Drinfeld, Vladimir. "A stacky approach to crystals." arXiv preprint arXiv:1810.11853 (2018).
  * [Dr2] Drinfeld, Vladimir. "Prismatization." arXiv preprint arXiv:2005.04746 (2020).
  * [R] Gregoric, Rok. The Crystalline Space and Divided Power Completion. https://web.ma.utexas.edu/users/gregoric/CrystalMath.pdf.
  * [Lu1] Lurie, Jacob. "Higher algebra. 2014." Preprint, available at http://www.math.harvard.edu/ lurie (2016).
  * [Lu2] Lurie, Jacob. "Spectral algebraic geometry." unpublished paper (2018).
  * [TV1] Toën, Bertrand, and Gabriele Vezzosi. "Homotopical algebraic geometry I: Topos theory." Advances in mathematics 193, no. 2 (2005): 257-372.
  * [TV2] Toën, Bertrand, Bertrand Toën, and Gabriele Vezzosi. Homotopical Algebraic Geometry II: Geometric Stacks and Applications: Geometric Stacks and Applications. Vol. 2. American Mathematical Soc., 2008.
  * [CS] Clausen, Dustin, and Peter Scholze. "Lectures on Analytic Geometry". http://www.math.uni-bonn.de/people/scholze/Notes.html (2020).
  * [KL1] Kedlaya, Kiran S., and Ruochuan Liu. "Relative p-adic Hodge theory: foundations." arXiv preprint arXiv:1301.0792 (2013).
  * [KL2] Kedlaya, Kiran S., and Ruochuan Liu. "Relative p-adic Hodge theory, II: Imperfect period rings." arXiv preprint arXiv:1602.06899 (2016).
  * [G1] Haoyang Guo. Crystalline cohomology of rigid analytic spaces. https://sites.google.com/umich.edu/hyg.
  * [GL] Guo, Haoyang, and Shizhang Li. "Period sheaves via derived de Rham cohomology." arXiv preprint arXiv:2008.06143 (2020).
  * [KR1] Kontsevich, Maxim, and Alexander Rosenberg. Noncommutative Spaces. Max-Planck-Inst., 2004.
  * [B1] Bhatt, Bhargav. "$p$-adic derived de Rham cohomology." arXiv preprint arXiv:1204.6560 (2012).
  * [III1] Illusie, Luc. Complexe cotangent et déformations I. Vol. 239. Springer, 2006.
  * [III2] Illusie, L. "Complexe cotangent et déformations II, SLN, 283 (1972)." Zbl0224 13014.
  * [LL] Li, Shizhang, and Tong Liu. "Comparison of prismatic cohomology and derived de Rham cohomology." arXiv preprint arXiv:2012.14064 (2020).
  * [SGAIV] Artin, Michael, Alexandre Grothendieck, and Jean-Louis Verdier. "Théorie des topos et cohomologie étale des schémas, LNM 269, 270, 305." (1972).
  * [Sch1] Scholze, Peter. "$p$-adic Hodge theory for rigid-analytic varieties." In Forum of Mathematics, Pi, vol. 1. Cambridge University Press, 2013.
  * [Gro1] Grothendieck, Alexander. "Crystals and the de Rham cohomology of schemes." Dix exposés sur la cohomologie des schémas 306 (1968): 358.