# Derived analytic geometry for $\mathbb{Z}$-valued functions Part I -
Topological properties

Federico Bambozzi and Tomoki Mihara

###### Abstract

We study the Banach algebras $\text{\rm C}(X,R)$ of continuous functions from
a compact Hausdorff topological space X to a Banach ring $R$ whose topology is
discrete. We prove that the Berkovich spectrum of $\text{\rm C}(X,R)$ is
homeomorphic to $\zeta(X)\times\mathscr{M}(R)$, where $\zeta(X)$ is the
Banaschewski compactification of $X$ and $\mathscr{M}(R)$ is the Berkovich
spectrum of $R$. We study how the topology of the spectrum of $\text{\rm
C}(X,R)$ is related to the notion of homotopy Zariski open embedding used in
derived geometry. We find that the topology of $\zeta(X)$ can be easily
reconstructed from the homotopy Zariski topology associated to $\text{\rm
C}(X,R)$. We also prove some results about the existence of Schauder bases on
$\text{\rm C}(X,R)$ and a generalisation of the Stone–Weierstrass Theorem,
under suitable hypotheses on $X$ and $R$.

###### Contents

  1. 0 Introduction
  2. 1 Preliminaries
  3. 2 Berkovich spectrum for functions valued in discrete Banach rings
  4. 3 Homotopy epimorphisms for functions valued in discrete Banach rings
  5. 4 Tate Acyclicity for functions valued in discrete Banach rings
  6. 5 Effective descent for modules over Banach algebras of functions valued in discrete Banach rings
  7. 6 Homological algebra for discrete Banach modules
  8. 7 Schauder bases and $\mathbb{Z}$-forms

##  0 Introduction

Background

The theory of commutative $\text{\rm C}^{*}$-algebras, i. e. algebras of
continuous functions from compact Hausdorff topological spaces $X$ to
$\mathbb{C}$, is one of the most important topics of functional analysis. The
richness of this theory comes from the structure of $\mathbb{C}$ as a complete
valued field with respect to the standard absolute value. In the authors’
recent work [BM21], it has been shown that the theory of $\text{\rm
C}^{*}$-algebras has a nice description in terms of the derived analytic
geometry introduced in [BK17], [BB16], [BBK19] and [BK20]. Therefore, it is
natural to ask what happens when the continuous $\mathbb{C}$-valued functions
on $X$ are replaced by the Banach algebra $\text{\rm C}(X,R)$ of continuous
functions valued on other Banach rings. Already in [BM21], the case when $R$
is a complete non-Archimedean valued field has been considered and it has been
checked that from the derived analytic geometry of the Banach algebra
$\text{\rm C}(X,R)\cong\text{\rm C}(\zeta(X),R)$, one can reconstruct the
topological space $\zeta(X)$, where $\zeta(X)$ is the Banaschewski
compactification of $X$.

Main results

In this work, we consider a case that is at the opposite extreme of the ones
considered in [BM21]. We study the Banach modules $\text{\rm C}(X,M)$ for a
Banach module $M$ whose topology induced by the norm is discrete. We call such
an $M$ a Banach module isolated at $0$. In particular, we are interested in a
Banach ring isolated at $0$. The main example of such a Banach ring is
$\mathbb{Z}_{\infty}=(\mathbb{Z},{\mathchoice{\left|\cdot\right|}{|\cdot|}{|\cdot|}{|\cdot|}}_{\infty})$,
i. e. $\mathbb{Z}$ equipped with the Euclidean absolute value. Such rings have
been already considered, for example in [Nöb68], and more recently in Lecture
5 of [Sch19], where they play a key role in the study of the notion of solid
modules, but they have been usually considered just as algebras without
considering the canonical Banach ring structure induced by the norm(s) on
$\mathbb{Z}$. We will argue that this additional structure is a fundamental
ingredient in the investigation of the algebras $\text{\rm C}(X,R)$ and that
these algebras should be considered to be objects whose study belongs both to
geometry and to functional analysis as $\text{\rm C}^{*}$-algebras do.
Moreover, the ring $R$ can be more complicated than the complete valued fields
considered in [BM21], and this has a strong impact on the properties of the
algebras $\text{\rm C}(X,R)$. Indeed, in the first five sections of this
paper, we explain how to generalise the results of [BM21] to the case of
continuous functions valued in a Banach ring isolated at $0$. The strategies
of the proofs are the same as the ones in [BM21], but almost all the lemmata
have to be proved and almost all the basic computations have to be done in
different ways. Whereas the last two sections prove some further properties of
the Banach algebras $\text{\rm C}(X,R)$. This paper is the first part of a
two-part work. In the second part, we will study the morphisms $\text{\rm
C}(X,R)\to\text{\rm C}(X,S)$ induced by a morphism of Banach ring $R\to S$
(where $S$ does not necessarily need to be isolated at $0$).

We notice that when the base ring $R$ is non-Archimedean, all our results can
be presented in two different contexts. One context is when only non-
Archimedean Banach $R$-modules are considered, and the other is when all
Banach modules are allowed. We denote such a choice by the choice of the base
category $\mathscr{C}=\text{\rm Ab}^{\text{\rm A}}$, in the first case, and
$\mathscr{C}=\text{\rm Ab}^{\text{\rm nA}}$, in the second case. We will
explain the reasons for this notation in §1. Although most of the basic
results of this paper are not influenced by this choice, the results in the
last two sections are strongly influenced. More extreme is the situation of
the results presented in the second part of this work, where we will be
interested in understanding when a homotopy epimorphism $R\to S$ of Banach
rings induces a homotopy epimorphism $\text{\rm C}(X,R)\to\text{\rm C}(X,S)$.
We will see that not only that the Archimedean projective tensor product
rarely satisfies the property that the canonical morphism $\text{\rm
C}(X,R)\widehat{\otimes}{}_{R}S\to\text{\rm C}(X,S)$ is an isomorphism, but
also that, unexpectedly, the most basic localisations of the Banach ring
$\mathbb{Z}_{\infty}$ are not homotopy epimorphisms. This is in stark contrast
with the case of non-Archimedean Banach rings where one can prove that all the
localisation used in analytic geometry are homotopy epimorphisms, even when
they are not defined by a Koszul-regular sequence. See [BK20] for more
information on this topic and in particular loc. cit. Proposition 5.8 where
examples of such non-Koszul-regular homotopy epimorphisms are considered in
relation to the sheafiness problem for Banach rings. So, in these situations,
these two choices lead to very different outcomes.

Structure of the paper

In §1, we recall basic notions from the theory of Banach rings and the theory
of quasi-Abelian categories. The main purpose of this section is to fix the
notation used in the rest of the paper that follows the notation used in
[BM21]. The reader is referred to [BM21] §1 for a more detailed exposition of
this fundamental material. In §2, we define Banach rings and Banach modules
isolated at $0$, provide some fundamental examples, and prove some basic
properties of the algebras of continuous functions from a topological space to
a Banach module isolated at $0$. It is proved in Proposition 2.6 that for any
topological space $X$ and any Banach module $M$ isolated at $0$, the Banach
module $\text{\rm C}_{\text{\rm fin}}(X,M)$ of continuous functions $X\to M$
with finite image is naturally isomorphic to the space $\text{\rm
C}(\zeta(X),M)$ of continuous functions $\zeta(X)\to M$. From this, we deduce
in Theorem 2.12 Gel’fand’s duality between the category of totally
disconnected compact Hausdorff topological spaces $X$ and the category of
Banach $R$-algebras $\text{\rm C}(X,R)$ for any Banach ring $R$ isolated at
$0$ whose Berkovich spectrum $\mathscr{M}(R)$ is connected. In §3, we
characterise closed topological embeddings between totally disconnected
compact Hausdorff topological spaces in terms of homotopy epimorphisms, where
the latter is one of the fundamental notions of derived geometry. More
precisely, by Gel’fand’s duality over $R$, the datum of a continuous map
$f\colon X\to Y$ between totally disconnected compact Hausdorff topological
spaces is equivalent to the datum of a morphism of Banach $R$-algebras
$f^{*}\colon\text{\rm C}(Y,R)\to\text{\rm C}(X,R)$. In Theorem 3.1, we show
that $f$ is a closed topological embedding if and only if $f^{*}$ is a
homotopy epimorphism. This generalises one of the main results of [BM21] and
shows how the geometry of $X$ can be studied using the methods of derived
geometry. In §4, we generalise in Theorem 4.1 Tate’s Acyclicity Theorem for
the morphism of $\text{\rm C}(X,R)$ induced by a finite covering by closed
subsets. In more detail, we prove that a finite family of closed topological
embeddings into the totally disconnected compact Hausdorff topological space
$X$ set-theoretically covers $X$ if and only if the associated
Tate–$\check{\text{C}}$ech complex (resp. derived Tate–$\check{\text{C}}$ech
complex) is strictly exact (resp. a zero object). We notice that, since $R$ is
isolated at $0$, Banach’s Open Mapping Theorem does not necessarily hold for
Banach $R$-modules. Therefore, in this situation, the strict exactness of a
complex of Banach $R$-modules cannot be checked by proving only the algebraic
exactness. This is one of the biggest difficulties which did not occur in the
classical setting. In §5, we prove from these results effective descent both
for Banach modules over $\text{\rm C}(X,R)$ and for objects of the derived
category of Banach modules over $\text{\rm C}(X,R)$ with respect to the covers
induced by closed topological embeddings of the Banaschewski compactification
of $X$.

The results discussed up to this point are parallel to the results proved in
[BM21] in the case when $R$ is a complete non-trivially valued field. Although
the proofs in the case when $R$ is isolated at $0$ differ in many details due
to the fact that many fundamental theorems of functional analysis do not hold
for Banach modules over $R$, the basic strategy is similar. On the other hand,
the algebras $\text{\rm C}(X,R)$ have a much richer structure than the
algebras $\text{\rm C}(X,k)$, for already the underlying ring of $R$ itself
can be a quite complicated ring instead of a field. Indeed, we prove in §2
that the Berkovich spectrum of $\text{\rm C}(X,R)$ is homeomorphic to
$\zeta(X)\times\mathscr{M}(R)$. We note that $\mathscr{M}(R)$ can be a quite
complicated compact Hausdorff topological space in our setting, while the
Berkovich spectrum of a complete valuation field is just a point. Therefore,
in the last two sections, we study further properties of the algebras
$\text{\rm C}(X,R)$ that do not have analogues when $R$ is a complete valued
field.

In §6, we prove some basic properties of the category of Banach $R$-modules
isolated at $0$, which will be denoted by $\text{\rm Mod}^{\text{\rm
iso}}_{\mathscr{C}}(R)$. The main result of this section, Theorem 6.11, gives
sufficient conditions on $(M_{1},M_{2})\in\text{\rm Mod}^{\text{\rm
iso}}_{\mathscr{C}}(R)^{2}$ that ensure that the canonical map $\text{\rm
C}(X,M_{1})\widehat{\otimes}{}_{R}M_{2}\to\text{\rm
C}(X,M_{1}\widehat{\otimes}{}_{R}M_{2})$ is an isomorphism and also
counterexamples to this statement. A direct consequence of this theorem is
that if $x\in\mathscr{M}(R)$ is a point such that ${\mathchoice{\left\\{c\in
R\middle|x(c)=0\right\\}}{\\{c\in R\mid x(c)=0\\}}{\\{c\in R\mid
x(c)=0\\}}{\\{c\in R\mid x(c)=0\\}}}$ is a maximal ideal, then $\text{\rm
C}(X,M)\widehat{\otimes}{}_{R}\kappa(x)\cong\text{\rm
C}(X,M\widehat{\otimes}{}_{R}\kappa(x))$ for all $M\in\text{\rm
Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$, where $\kappa(x)$ denotes the residue
field at $x$. In §7, we deal with the existence of $R$-linear Schauder bases
for the algebras $\text{\rm C}(X,R)$. We recall that it is classically known
for the case where $R$ is a complete valuation field that $\text{\rm C}(X,R)$
admits $R$-linear Schauder bases if $R$ is non-Archimedean. As an analogue, we
show in Proposition 7.1 for the case where $R$ is isolated at $0$ that
$\text{\rm C}(X,R)$ admits $R$-linear Schauder bases if $R$ is non-
Archimedean. We then discuss the existence of a particular kind of Schauder
bases that we call generalised van der Put bases and the difference between
them and the Mahler bases. We conclude with a generalisation of the
Stone–Weierstrass Theorem in Theorem 7.11.

##  1 Preliminaries

We follow the conventions and the terminology on Banach rings and quasi-
Abelian category theory from [BM21]. For example, we always assume rings to be
unital and commutative, and monoid objects of a symmetric monoidal category to
be commutative. Although we briefly recall the conventions and the terminology
in this section, it is better for the reader to refer to [BM21] §1 for the
detailed explanations on omitted facts such as the relation between Banach
ring theory and the monoidal categories $\text{\rm Ab}^{\text{\rm A}}$ and
$\text{\rm Ab}^{\text{\rm nA}}$ of Banach Abelian groups and non-Archimedean
Banach Abelian groups that we introduce below. Moreover, when we consider a
functor between two additive categories we will always tacitly assume that the
functor is additive unless stated otherwise. We also use the same symbol to
denote a category and its class of objects. Therefore, if $\mathscr{C}$ is a
category, the formula $X\in\mathscr{C}$ means that $X$ is an object of
$\mathscr{C}$. We formally set $0^{-1}\coloneqq\infty$ and $\infty\cdot
0=0\cdot\infty=0$ in order to avoid a redundant case classification when we
take the inverse of the operator norm of a bounded homomorphism which might be
$0$.

A normed set is a set $S$ equipped with a map
${\mathchoice{\left|\cdot\right|}{|\cdot|}{|\cdot|}{|\cdot|}}_{S}\colon
S\to[0,\infty)$. For normed sets $S_{0}$ and $S_{1}$, we denote by $S_{0}\odot
S_{1}$ the normed set given as the set $S_{0}\times S_{1}$ equipped with the
map ${\mathchoice{\left|\cdot\right|}{|\cdot|}{|\cdot|}{|\cdot|}}_{S_{0}\odot
S_{1}}\colon S_{0}\times S_{1}\to[0,\infty)$ assigning
${\mathchoice{\left|s_{0}\right|}{|s_{0}|}{|s_{0}|}{|s_{0}|}}_{S_{0}}{\mathchoice{\left|s_{1}\right|}{|s_{1}|}{|s_{1}|}{|s_{1}|}}_{S_{1}}$
to each $(s_{0},s_{1})\in S_{0}\times S_{1}$. For normed sets $S_{0}$ and
$S_{1}$, a map $\phi\colon S_{0}\to S_{1}$ is said to be bounded if its
operator norm
$\sup{\mathchoice{\left\\{{\mathchoice{\left|s\right|}{|s|}{|s|}{|s|}}_{S_{0}}^{-1}{\mathchoice{\left|\phi(s)\right|}{|\phi(s)|}{|\phi(s)|}{|\phi(s)|}}_{S_{1}}\middle|s\in
S_{0}\right\\}}{\\{{\mathchoice{\left|s\right|}{|s|}{|s|}{|s|}}_{S_{0}}^{-1}{\mathchoice{\left|\phi(s)\right|}{|\phi(s)|}{|\phi(s)|}{|\phi(s)|}}_{S_{1}}\mid
s\in
S_{0}\\}}{\\{{\mathchoice{\left|s\right|}{|s|}{|s|}{|s|}}_{S_{0}}^{-1}{\mathchoice{\left|\phi(s)\right|}{|\phi(s)|}{|\phi(s)|}{|\phi(s)|}}_{S_{1}}\mid
s\in
S_{0}\\}}{\\{{\mathchoice{\left|s\right|}{|s|}{|s|}{|s|}}_{S_{0}}^{-1}{\mathchoice{\left|\phi(s)\right|}{|\phi(s)|}{|\phi(s)|}{|\phi(s)|}}_{S_{1}}\mid
s\in S_{0}\\}}}\in[0,\infty]$ is finite. We denote by NSet the category of
normed sets and bounded maps, which naturally forms a symmetric monoidal
category with respect to $\odot$.

A complete norm on an Abelian group $M$ is a map
${\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}\colon
M\to[0,\infty)$ such that the map $M^{2}\to[0,\infty),\
(m_{0},m_{1})\mapsto{\mathchoice{\left\|m_{0}-m_{1}\right\|}{\|m_{0}-m_{1}\|}{\|m_{0}-m_{1}\|}{\|m_{0}-m_{1}\|}}$
is a complete metric on $M$. A Banach Abelian group is an Abelian group $M$
equipped with a complete norm
${\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{M}$ on
it. A Banach Abelian group is said to be non-Archimedean if it is an
ultrametric space with respect to the metric induced by the norm. We denote by
$\text{\rm Ab}^{\text{\rm A}}$ the category of Banach Abelian groups and
bounded group homomorphisms, and by $\text{\rm Ab}^{\text{\rm
nA}}\subset\text{\rm Ab}^{\text{\rm A}}$ the full subcategory of non-
Archimedean Banach Abelian groups.

Let $\mathscr{C}$ denote $\text{\rm Ab}^{\text{\rm A}}$ (resp. $\text{\rm
Ab}^{\text{\rm nA}}$). We equip $\mathscr{C}$ with the symmetric monoidal
structures given by the completed tensor product assigning to each
$(M_{0},M_{1})\in\mathscr{C}^{2}$ the completion
$M_{0}\widehat{\otimes}{}M_{1}$ of $M_{0}\otimes_{\mathbb{Z}}M_{1}$ with
respect to the metric associated to the tensor seminorm
$M_{0}\otimes_{\mathbb{Z}}M_{1}\to[0,\infty)$ assigning to each $m\in
M_{0}\otimes_{\mathbb{Z}}M_{1}$ the infimum of
$\sum_{j=0}^{n}{\mathchoice{\left\|(m_{0,j},m_{1,j})\right\|}{\|(m_{0,j},m_{1,j})\|}{\|(m_{0,j},m_{1,j})\|}{\|(m_{0,j},m_{1,j})\|}}_{M_{0}\odot
M_{1}}$ (resp.
$\max_{j=0}^{n}{\mathchoice{\left\|(m_{0,j},m_{1,j})\right\|}{\|(m_{0,j},m_{1,j})\|}{\|(m_{0,j},m_{1,j})\|}{\|(m_{0,j},m_{1,j})\|}}_{M_{0}\odot
M_{1}}$) for $(m_{0,j},m_{1,j})_{j=0}^{n}\in(M_{0}\odot M_{1})^{n+1}$ with
$n\in\mathbb{N}$ and $\sum_{j=0}^{n}m_{0,j}\otimes m_{1,j}=m$.

Let $R$ be a monoid object of $\mathscr{C}$. We denote by $\text{\rm
Mod}_{\mathscr{C}}(R)$ the quasi-Abelian category of $R$-module objects of
$\mathscr{C}$ and $R$-linear bounded homomorphisms in $\mathscr{C}$ (cf.
[BB16] Proposition 3.15 and [BB16] Proposition 3.18). We equip $\text{\rm
Mod}_{\mathscr{C}}(R)$ with the closed symmetric monoidal structure
$\widehat{\otimes}{}_{R}$ induced by $\otimes$ and the internal hom functor
$\underline{\text{\rm Hom}}{}_{\text{\rm Mod}_{\mathscr{C}}(R)}$ given by
$\text{\rm Hom}_{\text{\rm Mod}_{\mathscr{C}}(R)}$ equipped with the pointwise
operations and the operator norm. The reader should be careful that the unitor
$R\widehat{\otimes}{}_{R}M\stackrel{{\scriptstyle\cong}}{{\to}}M$ is not
necessarily an isometry for an $M\in\text{\rm Mod}_{\mathscr{C}}(R)$, because
the scalar multiplication $R\widehat{\otimes}{}M\to M$ is not necessarily
contracting. We denote by $\text{\rm Alg}_{\mathscr{C}}(R)$ the category of
monoid objects of $\text{\rm Mod}_{\mathscr{C}}(R)$.

For an $M\in\text{\rm Mod}_{\mathscr{C}}(R)$, we say that $M$ is projective if
the external hom functor $\text{\rm Hom}_{\text{\rm
Mod}_{\mathscr{C}}(R)}(M,\cdot)\colon\text{\rm
Mod}_{\mathscr{C}}(R)\to\text{\rm Ab}$ is exact, or equivalently by the left
exactness of $\text{\rm Hom}_{\text{\rm Mod}_{\mathscr{C}}(R)}(M,\cdot)$,
sends any strict epimorphism to a surjective map (cf. [Sch99] Definition
1.3.18), and is strongly flat if the functor
$(\cdot)\widehat{\otimes}{}_{R}M\colon\text{\rm
Mod}_{\mathscr{C}}(R)\to\text{\rm Mod}_{\mathscr{C}}(R)$ is strongly exact, or
equivalently by the strong right exactness of
$(\cdot)\widehat{\otimes}{}_{R}M$, preserves the kernel of any morphism (cf.
[BK20] Definition 3.2).

##  2 Berkovich spectrum for functions valued in discrete Banach rings

In this section, we introduce Banach rings and Banach modules isolated at $0$,
and we study the basic properties of the modules of continuous functions from
a topological space to such modules. We prove in Proposition 2.6 that for any
topological space $X$, any Banach ring $R$, and any Banach $R$-module $M$
isolated at $0$, the Banach $R$-module $\text{\rm C}_{\text{\rm fin}}(X,M)$ of
continuous functions $X\to M$ with finite image is canonically isometrically
isomorphic to the Banach $R$-module $\text{\rm C}(\zeta(X),M)$ of continuous
functions $\zeta(X)\to M$, where $\zeta(X)$ denotes the Banaschewski
compactification of $X$. Then, we deduce in Theorem 2.10 that the Berkovich
spectrum of $\text{\rm C}(\zeta(X),M)$ is homeomorphic to
$\zeta(X)\times\mathscr{M}(R)$, where $\mathscr{M}(R)$ denotes the Berkovich
spectrum of $R$. We conclude this section by proving Gel’fand’s duality
between the category of totally disconnected compact Hausdorff topological
spaces $X$ and the category of Banach $R$-algebras $\text{\rm C}(X,R)$ in
Theorem 2.12, under the hypothesis that $\mathscr{M}(R)$ is connected.

Let $\mathscr{C}$ be either $\text{\rm Ab}^{\text{\rm A}}$ or $\text{\rm
Ab}^{\text{\rm nA}}$, $R$ a monoid object of $\mathscr{C}$, and $M\in\text{\rm
Mod}_{\mathscr{C}}(R)$. We say that $M$ is isolated at $0$ if
${\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}\subset M$ is an
open subset for the topology induced by the norm. We say that $R$ is isolated
at $0$ if so is the regular $R$-module object. Although we focus on the case
where $M$ is the regular $R$-module object in §3 – §5, we will deal with the
general case in §6 and §7.

######  Example 2.1.

  * (i)

Let $O\subset K$ be the ring of algebraic integers of a number field $K$. We
denote by
${\mathchoice{\left|\cdot\right|}{|\cdot|}{|\cdot|}{|\cdot|}}_{\infty}\colon
O\to[0,\infty)$ the norm given by the maximum of the restrictions of the norm
of
$(\mathbb{C},{\mathchoice{\left|\cdot\right|}{|\cdot|}{|\cdot|}{|\cdot|}}_{\infty})$
to $K$ via all infinite places $K\to\mathbb{C}$. For any $\epsilon\in[0,1]$,
$(O,{\mathchoice{\left|\cdot\right|}{|\cdot|}{|\cdot|}{|\cdot|}}_{\infty}^{\epsilon})$
is a monoid of $\text{\rm Ab}^{\text{\rm A}}$ isolated at $0$. In particular,
$\mathbb{Z}_{\infty}\coloneqq(\mathbb{Z},{\mathchoice{\left|\cdot\right|}{|\cdot|}{|\cdot|}{|\cdot|}}_{\infty})$
is a monoid of $\text{\rm Ab}^{\text{\rm A}}$ isolated at $0$.

  * (ii)

Let $K$ be a number field. We denote by
${\mathchoice{\left|\cdot\right|}{|\cdot|}{|\cdot|}{|\cdot|}}_{\text{\rm
ad}}\colon K\to[0,\infty)$ the norm given by the supremum of the restrictions
of the absolute values corresponding to all the places on $K$, where the
absolute values
${\mathchoice{\left|\cdot\right|}{|\cdot|}{|\cdot|}{|\cdot|}}_{v}$
corresponding to finite places $v$ over a prime number $p$ are normalised by
${\mathchoice{\left|p\right|}{|p|}{|p|}{|p|}}_{v}=p^{-1}$. For any
$\epsilon\in[0,1]$,
$(K,{\mathchoice{\left|\cdot\right|}{|\cdot|}{|\cdot|}{|\cdot|}}_{\text{\rm
ad}}^{\epsilon})$ is a monoid of $\text{\rm Ab}^{\text{\rm A}}$ isolated at
$0$. In particular,
$(\mathbb{Q},{\mathchoice{\left|\cdot\right|}{|\cdot|}{|\cdot|}{|\cdot|}}_{\text{\rm
ad}})$ is a monoid of $\text{\rm Ab}^{\text{\rm A}}$ isolated at $0$.

  * (iii)

Let $A$ be a ring, and $N$ be an $A$-module. We denote by
${\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{\text{\rm
triv}}$ the trivial norms on $A$ and $N$, which assign $1$ to each non-zero
element and $0$ to the zero elements. Then, $A_{\text{\rm
triv}}\coloneqq(A,{\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{\text{\rm
triv}})$ is a monoid of $\text{\rm Ab}^{\text{\rm nA}}$ isolated at $0$, and
$N_{\text{\rm
triv}}\coloneqq(N,{\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{\text{\rm
triv}})$ is an object of $\text{\rm Mod}_{\text{\rm Ab}^{\text{\rm
nA}}}(A_{\text{\rm triv}})$ isolated at $0$. In particular,
$\mathbb{Z}_{\text{\rm triv}}$ is a monoid of $\text{\rm Ab}^{\text{\rm nA}}$
isolated at $0$.

  * (iv)

Let $A$ be a monoid of $\mathscr{C}$. We denote by
${\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{\text{\rm
hyb}}$ the hybrid norm on $A$, which assigns
$\max{\mathchoice{\left\\{{\mathchoice{\left\|a\right\|}{\|a\|}{\|a\|}{\|a\|}}_{A},{\mathchoice{\left\|a\right\|}{\|a\|}{\|a\|}{\|a\|}}_{\text{\rm
triv}}\right\\}}{\\{{\mathchoice{\left\|a\right\|}{\|a\|}{\|a\|}{\|a\|}}_{A},{\mathchoice{\left\|a\right\|}{\|a\|}{\|a\|}{\|a\|}}_{\text{\rm
triv}}\\}}{\\{{\mathchoice{\left\|a\right\|}{\|a\|}{\|a\|}{\|a\|}}_{A},{\mathchoice{\left\|a\right\|}{\|a\|}{\|a\|}{\|a\|}}_{\text{\rm
triv}}\\}}{\\{{\mathchoice{\left\|a\right\|}{\|a\|}{\|a\|}{\|a\|}}_{A},{\mathchoice{\left\|a\right\|}{\|a\|}{\|a\|}{\|a\|}}_{\text{\rm
triv}}\\}}}$ to each $a\in A$. Then, $A_{\text{\rm
hyb}}\coloneqq(A,{\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{\text{\rm
hyb}})$ is a monoid of $\mathscr{C}$ isolated at $0$. In particular,
$\mathbb{C}_{\text{\rm hyb}}$ is a monoid of $\text{\rm Ab}^{\text{\rm A}}$
isolated at $0$.

We study elementary functional analysis on Banach modules isolated at $0$.

######  Proposition 2.2.

The topology of $M$ is discrete if and only if $M$ is isolated at $0$.

###### Proof.

If the topology of $M$ is discrete, then every subset of $M$ (and hence
${\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}\subset M$) is
open. Suppose that $M$ is isolated at $0$. Since
${\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}\subset M$ is
open, there exists an $\epsilon\in(0,\infty)$ such that
${\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}={\mathchoice{\left\\{m\in
M\middle|{\mathchoice{\left\|m\right\|}{\|m\|}{\|m\|}{\|m\|}}_{M}<\epsilon\right\\}}{\\{m\in
M\mid{\mathchoice{\left\|m\right\|}{\|m\|}{\|m\|}{\|m\|}}_{M}<\epsilon\\}}{\\{m\in
M\mid{\mathchoice{\left\|m\right\|}{\|m\|}{\|m\|}{\|m\|}}_{M}<\epsilon\\}}{\\{m\in
M\mid{\mathchoice{\left\|m\right\|}{\|m\|}{\|m\|}{\|m\|}}_{M}<\epsilon\\}}}$.
For any $m\in M$, we have
${\mathchoice{\left\\{m\right\\}}{\\{m\\}}{\\{m\\}}{\\{m\\}}}={\mathchoice{\left\\{m+m^{\prime}\middle|m^{\prime}\in
M\land{\mathchoice{\left\|m^{\prime}\right\|}{\|m^{\prime}\|}{\|m^{\prime}\|}{\|m^{\prime}\|}}<\epsilon\right\\}}{\\{m+m^{\prime}\mid
m^{\prime}\in
M\land{\mathchoice{\left\|m^{\prime}\right\|}{\|m^{\prime}\|}{\|m^{\prime}\|}{\|m^{\prime}\|}}<\epsilon\\}}{\\{m+m^{\prime}\mid
m^{\prime}\in
M\land{\mathchoice{\left\|m^{\prime}\right\|}{\|m^{\prime}\|}{\|m^{\prime}\|}{\|m^{\prime}\|}}<\epsilon\\}}{\\{m+m^{\prime}\mid
m^{\prime}\in
M\land{\mathchoice{\left\|m^{\prime}\right\|}{\|m^{\prime}\|}{\|m^{\prime}\|}{\|m^{\prime}\|}}<\epsilon\\}}}={\mathchoice{\left\\{m^{\prime}\in
M\middle|{\mathchoice{\left\|m^{\prime}-m\right\|}{\|m^{\prime}-m\|}{\|m^{\prime}-m\|}{\|m^{\prime}-m\|}}_{M}<\epsilon\right\\}}{\\{m^{\prime}\in
M\mid{\mathchoice{\left\|m^{\prime}-m\right\|}{\|m^{\prime}-m\|}{\|m^{\prime}-m\|}{\|m^{\prime}-m\|}}_{M}<\epsilon\\}}{\\{m^{\prime}\in
M\mid{\mathchoice{\left\|m^{\prime}-m\right\|}{\|m^{\prime}-m\|}{\|m^{\prime}-m\|}{\|m^{\prime}-m\|}}_{M}<\epsilon\\}}{\\{m^{\prime}\in
M\mid{\mathchoice{\left\|m^{\prime}-m\right\|}{\|m^{\prime}-m\|}{\|m^{\prime}-m\|}{\|m^{\prime}-m\|}}_{M}<\epsilon\\}}}$,
and hence ${\mathchoice{\left\\{m\right\\}}{\\{m\\}}{\\{m\\}}{\\{m\\}}}$ is an
open subset of $M$. ∎

Throughout this paper, let $X$ denote a topological space. We denote by
$\text{\rm C}(X,M)$ the $R$-module of continuous maps $X\to M$, by $\text{\rm
C}_{\text{\rm bd}}(X,M)\subset\text{\rm C}(X,M)$ the $R$-submodule of
continuous maps whose images are bounded, and by $\text{\rm C}_{\text{\rm
fin}}(X,M)\subset\text{\rm C}_{\text{\rm bd}}(X,M)$ the $R$-submodule of
continuous maps whose images are finite sets. Then, $\text{\rm C}_{\text{\rm
bd}}(X,M)$ forms an object of $\text{\rm Mod}_{\mathscr{C}}(R)$ with respect
to the supremum norm, while $\text{\rm C}_{\text{\rm fin}}(X,M)$ is not
necessarily closed in $\text{\rm C}_{\text{\rm bd}}(X,M)$. When $X$ is
compact, then $\text{\rm C}_{\text{\rm bd}}(X,M)$ coincides with $\text{\rm
C}(X,M)$, and hence we also regard $\text{\rm C}(X,M)$ as an object of
$\text{\rm Mod}_{\mathscr{C}}(R)$. Applying this construction to the regular
$R$-module object, we obtain $R$-algebras $\text{\rm C}_{\text{\rm
fin}}(X,R)\subset\text{\rm C}_{\text{\rm bd}}(X,R)\subset\text{\rm C}(X,R)$.
Then, $\text{\rm C}_{\text{\rm fin}}(X,M)$ forms a $\text{\rm C}_{\text{\rm
fin}}(X,R)$-module, and $\text{\rm C}_{\text{\rm bd}}(X,R)$ forms an object of
$\text{\rm Alg}_{\mathscr{C}}(R)$.

######  Proposition 2.3.

  * (i)

If $M$ is isolated at $0$, then every $R$-submodule of $\text{\rm
C}_{\text{\rm bd}}(X,M)$ is closed, and is isolated at $0$ as an object of
$\text{\rm Mod}_{\mathscr{C}}(R)$.

  * (ii)

If $X$ is compact and $M$ is isolated at $0$, then $\text{\rm C}_{\text{\rm
fin}}(X,M)$ coincides with $\text{\rm C}(X,M)$.

###### Proof.

The assertion (i) follows from the fact that every Cauchy sequence in
$\text{\rm C}_{\text{\rm bd}}(X,M)$ is eventually constant because $M$ is
isolated at $0$. The assertion (ii) immediately follows from Proposition 2.2.
∎

We denote by $\text{\rm CO}(X)$ the set of clopen subsets of $X$. For each
$U\in\text{\rm CO}(X)$, we denote by $1_{X,R,U}\in\text{\rm C}_{\text{\rm
fin}}(X,R)$ the characteristic function of $U$. For each $(f,m)\in\text{\rm
C}_{\text{\rm fin}}(X,R)\times M$, we denote by $fm\in\text{\rm C}_{\text{\rm
fin}}(X,M)$ the scalar multiplication of $f$ and the constant map $X\to M$
with value $m$. We show a finite image analogue of [Mih14] Theorem 4.12.

######  Proposition 2.4.

  * (i)

The set ${\mathchoice{\left\\{1_{X,R,U}m\middle|(U,m)\in\text{\rm CO}(X)\times
M\right\\}}{\\{1_{X,R,U}m\mid(U,m)\in\text{\rm CO}(X)\times
M\\}}{\\{1_{X,R,U}m\mid(U,m)\in\text{\rm CO}(X)\times
M\\}}{\\{1_{X,R,U}m\mid(U,m)\in\text{\rm CO}(X)\times M\\}}}$ generates the
underlying $R$-module of $\text{\rm C}_{\text{\rm fin}}(X,M)$.

  * (ii)

For any $f\in\text{\rm C}_{\text{\rm fin}}(X,M)$, there exists a pair
$(\mathscr{U},e)$ of a finite subset $\mathscr{U}\subset\text{\rm
CO}(X)\setminus{\mathchoice{\left\\{\emptyset\right\\}}{\\{\emptyset\\}}{\\{\emptyset\\}}{\\{\emptyset\\}}}$
and a map $e\colon\mathscr{U}\to M$ such that $\mathscr{U}$ is a disjoint
cover of $X$ and $\sum_{U\in\mathscr{U}}1_{X,R,U}e(U)=f$.

###### Proof.

The assertion (i) follows from assertion (ii). We show assertion (ii). By the
finiteness of $f(X)$, $U_{m}\coloneqq
f^{-1}({\mathchoice{\left\\{m\right\\}}{\\{m\\}}{\\{m\\}}{\\{m\\}}})$ is a
clopen subset of $X$ for any $m\in M$. Put
$\mathscr{U}\coloneqq{\mathchoice{\left\\{U_{m}\middle|m\in
f(X)\right\\}}{\\{U_{m}\mid m\in f(X)\\}}{\\{U_{m}\mid m\in
f(X)\\}}{\\{U_{m}\mid m\in f(X)\\}}}\subset\text{\rm
CO}(X)\setminus{\mathchoice{\left\\{\emptyset\right\\}}{\\{\emptyset\\}}{\\{\emptyset\\}}{\\{\emptyset\\}}}$.
We denote by $e$ the map $\mathscr{U}\to M$ assigning to each
$U\in\mathscr{U}$ the unique element of the singleton $f(U)\subset f(X)$.
Then, $(\mathscr{U},e)$ satisfies the desired property. ∎

Let $X_{0}$ be a subset of $X$. We put
$I_{X,R,X_{0}}\coloneqq{\mathchoice{\left\\{f\in\text{\rm C}_{\text{\rm
fin}}(X,R)\middle|f|_{X_{0}}=0\right\\}}{\\{f\in\text{\rm C}_{\text{\rm
fin}}(X,R)\mid f|_{X_{0}}=0\\}}{\\{f\in\text{\rm C}_{\text{\rm fin}}(X,R)\mid
f|_{X_{0}}=0\\}}{\\{f\in\text{\rm C}_{\text{\rm fin}}(X,R)\mid
f|_{X_{0}}=0\\}}}$. We show a finite image analogue of [BM21] Lemma 2.21.

######  Proposition 2.5.

Every finitely generated $\text{\rm C}_{\text{\rm fin}}(X,R)$-submodule of
$I_{X,R,X_{0}}$ is contained in a principal ideal generated by $1_{X,R,U}$ for
some $U\in\text{\rm CO}(X)$ with $X_{0}\cap U=\emptyset$.

###### Proof.

Since ${\mathchoice{\left\\{U\in\text{\rm CO}(X)\middle|X_{0}\cap
U=\emptyset\right\\}}{\\{U\in\text{\rm CO}(X)\mid X_{0}\cap
U=\emptyset\\}}{\\{U\in\text{\rm CO}(X)\mid X_{0}\cap
U=\emptyset\\}}{\\{U\in\text{\rm CO}(X)\mid X_{0}\cap U=\emptyset\\}}}$ is
directed by inclusions, it is reduced to the case of an ideal generated by a
single element. Therefore it suffices to show that for any $f\in
I_{X,R,X_{0}}$, there exists a $U\in\text{\rm CO}(X)$ such that $1_{X,R,U}f=f$
and $X_{0}\cap U=\emptyset$. Put $U\coloneqq
f^{-1}(R\setminus{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}})\subset
X$. We have $X_{0}\cap U=\emptyset$ by $f|_{X_{0}}=0$. By the finiteness of
$f(X)$, we have $U\in\text{\rm CO}(X)$. For any $x\in U$, we have
$(1_{X,R,U}f)(x)=1_{X,R,U}(x)f(x)=1\cdot f(x)=f(x)$. For any $x\in X\setminus
U$, we have $(1_{X,R,U}f)(x)=1_{X,R,U}(x)f(x)=0=f(x)$. Therefore, we obtain
$1_{X,R,U}f=f$. ∎

We abbreviate $(\mathscr{C},R)$ to $\mathscr{D}_{2}$. We denote by Top the
category of topological spaces and continuous maps, by $\text{\rm
CH}\subset\text{\rm Top}$ the full subcategory of compact Hausdorff
topological spaces, by $\text{\rm TDCH}\subset\text{\rm CH}$ the full
subcategory of totally disconnected compact Hausdorff topological spaces, by
$F_{\text{\rm TDCH}}^{\text{\rm Top}}$ the inclusion functor $\text{\rm
TDCH}\hookrightarrow\text{\rm Top}$, by $\zeta$ the Banaschewski
compactification functor $\text{\rm Top}\to\text{\rm TDCH}$ (cf. [Mih14]
Theorem 1.3), which is left adjoint to $F_{\text{\rm TDCH}}^{\text{\rm Top}}$,
by $\iota_{\zeta,\bullet}$ the unit adjunction $\text{\rm id}_{\text{\rm
Top}}\Rightarrow F_{\text{\rm TDCH}}^{\text{\rm Top}}\circ\zeta$, and by
$\mathscr{M}_{\mathscr{D}_{2}}\colon\text{\rm
Alg}_{\mathscr{C}}(R)\to\text{\rm CH}$ the Berkovich spectrum functor (cf.
[Ber90] 1.2.1 Theorem). We recall that $\zeta(X)$ is explicitly described by
using ultrafilters, and can be described as $\mathscr{M}_{\mathscr{D}_{2}}$
for the case where $R$ is a complete valued field by [Mih14] Corollary 2.3.
The correspondence assigning $\text{\rm C}_{\text{\rm fin}}(X,R)$ to each
$X\in\text{\rm Top}$ gives a functor $\Gamma_{\mathscr{D}_{2}}\colon\text{\rm
Top}^{\text{\rm op}}\to\text{\rm Alg}_{\mathscr{C}}(R)$ by Proposition 2.3. We
show an isolated analogue of [BM21] Lemma 3.7 (ii).

######  Proposition 2.6.

If $M$ is isolated at $0$, then the pre-composition $\text{\rm
C}(\zeta(X),M)\to\text{\rm C}_{\text{\rm fin}}(X,M)$ by $\iota_{\zeta,X}\colon
X\to\zeta(X)$ is an isometric $R$-linear isomorphism.

###### Proof.

Since $\iota_{\zeta,X}(X)$ is dense in $\zeta(X)$ by [Mih14] Lemma 1.8, the
pre-composition by $\iota_{\zeta,X}$ is an isometry. Let $f\in\text{\rm
C}_{\text{\rm fin}}(X,R)$. Since $f(X)$ is a finite subset of the Hausdorff
topological space $M$, $f(X)$ is a totally disconnected compact Hausdorff
topological space. Therefore, by the universality of Banaschewski
compactification, there exists a unique $\tilde{f}{}\in\text{\rm
C}(\zeta(X),M)$ such that $\tilde{f}{}\circ\iota_{\zeta,X}=f$. This implies
the assertion. ∎

For each $U\in\text{\rm CO}(X)$, we denote by $\text{\rm
cl}_{\zeta,X}(U)\subset\zeta(X)$ the closure of $\iota_{\zeta,X}(U)$.
Proposition 2.6 gives an alternative proof of the following well-known fact:

######  Corollary 2.7.

For any $U\in\text{\rm CO}(X)$, $\text{\rm cl}_{\zeta,X}(U)$ is the unique
$\overline{U}{}\in\text{\rm CO}(\zeta(X))$ satisfying
$\iota_{\zeta,X}^{-1}(\overline{U}{})=U$.

###### Proof.

The assertion follows from Proposition 2.6 applied to
$R=M=(\mathbb{F}_{2})_{\text{\rm triv}}$, because $\text{\rm
CO}\colon\text{\rm Top}\to\text{\rm Set}$ is represented by
$(\mathbb{F}_{2})_{\text{\rm triv}}$. ∎

We abbreviate $(\mathscr{C},X,R)$ to $\mathscr{D}_{3}$. For each
$\mathscr{F}\in\zeta(X)$, we denote by $\lim^{\mathscr{D}_{3}}_{\mathscr{F}}$
the $R$-algebra homomorphism $\text{\rm C}_{\text{\rm fin}}(X,R)\to R$, which
is bounded with respect to the restriction of
${\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{\text{\rm
C}_{\text{\rm bd}}(X,R)}$, assigning to each $f\in\text{\rm C}_{\text{\rm
fin}}(X,R)$ the limit $\lim_{\mathscr{F}}f$, i.e. the unique $a\in R$
satisfying the property that there exists a $U\in\mathscr{F}$ such that
$f|_{U}$ is the constant map with value $a$. We denote by $F_{\text{\rm
Top}}^{\text{\rm Set}}$ the forgetful functor $\text{\rm Top}\to\text{\rm
Set}$. We show an isolated analogue of [Mih14] Corollary 3.6 (iv), which gives
a non-Archimedean counterpart of Gel’fand’s transform.

######  Proposition 2.8.

If $R$ is isolated at $0$, then the correspondence assigning to each
$Y\in\text{\rm Top}$ the map
$\lim^{(\mathscr{C},Y,R)}\colon\zeta(Y)\to\text{\rm Hom}_{\text{\rm
Alg}_{\mathscr{C}}(R)}(\text{\rm C}_{\text{\rm fin}}(Y,R),R),\
\mathscr{F}\mapsto\lim^{(\mathscr{C},Y,R)}_{\mathscr{F}}$ is a natural
transformation $\lim^{(\mathscr{C},\bullet,R)}\colon F_{\text{\rm
Top}}^{\text{\rm Set}}\circ\zeta\Rightarrow\text{\rm Hom}_{\text{\rm
Alg}_{\mathscr{C}}(R)}(\Gamma_{\mathscr{D}_{2}},R)$, which consists of
injective maps unless
$R={\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$.

###### Proof.

The first assertion immediately follows from the construction of
$\lim^{(\mathscr{C},Y,R)}$ for a $Y\in\text{\rm Top}$. We show the second
assertion. By Proposition 2.6, it suffices to show that if
$R\neq{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$, then for
any $Y\in\text{\rm TDCH}$, the map $\text{\rm ev}\colon Y\to\text{\rm
Hom}_{\text{\rm Alg}_{\mathscr{C}}(R)}(\text{\rm C}(Y,R),R),\
y\mapsto\text{\rm ev}_{y}$ is injective, where $\text{\rm ev}_{y}$ denotes the
evaluation at $y$. Let $(y_{0},y_{1})\in Y^{2}$ with $y_{0}\neq y_{1}$. Since
$Y\in\text{\rm TDCH}$, there exists a $U\in\text{\rm CO}(Y)$ such that
$y_{0}\in U$ and $y_{1}\notin U$. Then, we have $\text{\rm
ev}_{y_{0}}(1_{X,R,U})=1\neq 0=\text{\rm ev}_{y_{1}}(1_{X,R,U})$ by
$R\neq{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$. This
implies $\text{\rm ev}_{y_{0}}\neq\text{\rm ev}_{y_{1}}$. ∎

As we announced at the beginning of this section, we will show Gel’fand’s
duality in Theorem 2.12 under the assumption that $\mathscr{M}(R)$ is
connected, and this implies that $\lim^{(\mathscr{C},\bullet,R)}$ is a natural
isomorphism under the same assumption. On the other hand,
$\lim^{(\mathscr{C},\bullet,R)}$ is not a natural isomorphism if $R$ does not
have precisely one non-zero idempotent element, as the following example
shows:

######  Example 2.9.

Let $Y\in\text{\rm TDCH}$ denote the discrete set
${\mathchoice{\left\\{0,1\right\\}}{\\{0,1\\}}{\\{0,1\\}}{\\{0,1\\}}}$. If $R$
has no non-zero idempotent element, i.e.
$R={\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$, then both of
$\lim^{(\mathscr{C},Y,R)}(0)$ and $\lim^{(\mathscr{C},Y,R)}(1)$ are zero
morphisms $\text{\rm C}_{\text{\rm fin}}(Y,R)\to R$, and hence
$\lim^{(\mathscr{C},Y,R)}$ is not injective. If $R$ admits a non-zero
idempotent $e\neq 1$, then the map $\text{\rm C}_{\text{\rm fin}}(Y,R)\to R,\
f\mapsto ef(0)+(1-e)f(1)$ is an element of $\text{\rm Hom}_{\text{\rm
Alg}_{\mathscr{C}}(R)}(\text{\rm C}_{\text{\rm fin}}(Y,R),R)$ which does not
belong to the image of $\lim^{(\mathscr{C},Y,R)}$, and hence
$\lim^{(\mathscr{C},Y,R)}$ is not surjective.

Suppose that $R$ is isolated at $0$. For each
$x\in\mathscr{M}_{\mathscr{D}_{2}}(\text{\rm C}_{\text{\rm fin}}(X,R))$, we
denote by $\mathscr{F}_{X,x}$ the subset ${\mathchoice{\left\\{U\in\text{\rm
CO}(X)\middle|x(1_{X,R,U})\neq 0\right\\}}{\\{U\in\text{\rm CO}(X)\mid
x(1_{X,R,U})\neq 0\\}}{\\{U\in\text{\rm CO}(X)\mid x(1_{X,R,U})\neq
0\\}}{\\{U\in\text{\rm CO}(X)\mid x(1_{X,R,U})\neq 0\\}}}$ of $\text{\rm
CO}(X)$, which is an ultrafilter of $\text{\rm CO}(X)$ by an argument
completely parallel to that in [Mih14] p. 10 for the case where $R$ is a
complete valued field, and by $x|_{R}$ the composite of $x$ and the canonical
ring homomorphism $R\to\text{\rm C}_{\text{\rm fin}}(X,R)$. We denote by
$G_{\mathscr{D}_{3}}$ the map $\mathscr{M}_{\mathscr{D}_{2}}(\text{\rm
C}_{\text{\rm fin}}(X,R))\to\zeta(X)\times\mathscr{M}_{\mathscr{D}_{2}}(R),\
x\mapsto(\mathscr{F}_{X,x},x|_{R})$. We show an isolated analogue of [Mih14]
Theorem 2.1.

######  Theorem 2.10.

If $R$ is isolated at $0$, then $G_{\mathscr{D}_{3}}$ is a homeomorphism.

###### Proof.

An open basis of $\zeta(X)\times\mathscr{M}_{\mathscr{D}_{2}}(R)$ is given by
the set of subsets of the form

$\displaystyle{\mathchoice{\left\\{\mathscr{F}\in\zeta(X)\middle|U\in\mathscr{F}\right\\}}{\\{\mathscr{F}\in\zeta(X)\mid
U\in\mathscr{F}\\}}{\\{\mathscr{F}\in\zeta(X)\mid
U\in\mathscr{F}\\}}{\\{\mathscr{F}\in\zeta(X)\mid
U\in\mathscr{F}\\}}}\times{\mathchoice{\left\\{\overline{x}{}\in\mathscr{M}_{\mathscr{D}_{2}}(R)\middle|r_{0}<\overline{x}{}(a)<r_{1}\right\\}}{\\{\overline{x}{}\in\mathscr{M}_{\mathscr{D}_{2}}(R)\mid
r_{0}<\overline{x}{}(a)<r_{1}\\}}{\\{\overline{x}{}\in\mathscr{M}_{\mathscr{D}_{2}}(R)\mid
r_{0}<\overline{x}{}(a)<r_{1}\\}}{\\{\overline{x}{}\in\mathscr{M}_{\mathscr{D}_{2}}(R)\mid
r_{0}<\overline{x}{}(a)<r_{1}\\}}}$

with $(U,a,r_{0},r_{1})\in\text{\rm CO}(X)\times R\times(0,\infty)^{2}$, and
its preimage by $G_{\mathscr{D}_{3}}$ coincides with

$\displaystyle{\mathchoice{\left\\{x\in\mathscr{M}_{\mathscr{D}_{2}}(\text{\rm
C}_{\text{\rm fin}}(X,R))\middle|x(1_{X,R,U})\neq 0\land
r_{0}<x(a)<r_{1}\right\\}}{\\{x\in\mathscr{M}_{\mathscr{D}_{2}}(\text{\rm
C}_{\text{\rm fin}}(X,R))\mid x(1_{X,R,U})\neq 0\land
r_{0}<x(a)<r_{1}\\}}{\\{x\in\mathscr{M}_{\mathscr{D}_{2}}(\text{\rm
C}_{\text{\rm fin}}(X,R))\mid x(1_{X,R,U})\neq 0\land
r_{0}<x(a)<r_{1}\\}}{\\{x\in\mathscr{M}_{\mathscr{D}_{2}}(\text{\rm
C}_{\text{\rm fin}}(X,R))\mid x(1_{X,R,U})\neq 0\land r_{0}<x(a)<r_{1}\\}}},$

which is an open subset of $\mathscr{M}_{\mathscr{D}_{2}}(\text{\rm
C}_{\text{\rm fin}}(X,R))$. This implies the continuity of
$G_{\mathscr{D}_{3}}$. Since $G_{\mathscr{D}_{3}}$ is a continuous map between
compact Hausdorff topological spaces, it suffices to show its bijectivity.

First, let $(x_{0},x_{1})\in\mathscr{M}_{\mathscr{D}_{2}}(\text{\rm
C}_{\text{\rm fin}}(X,R))^{2}$ with
$G_{\mathscr{D}_{3}}(x_{0})=G_{\mathscr{D}_{3}}(x_{1})$. In order to show
$x_{0}=x_{1}$, it suffices to show $x_{0}(f)=x_{1}(f)$ for any $f\in\text{\rm
C}_{\text{\rm fin}}(X,R)$. By Proposition 2.4 (ii), there exists a pair
$(\mathscr{U},e)$ of a finite subset $\mathscr{U}\subset\text{\rm
CO}(X)\setminus{\mathchoice{\left\\{\emptyset\right\\}}{\\{\emptyset\\}}{\\{\emptyset\\}}{\\{\emptyset\\}}}$
and a map $e\colon\mathscr{U}\to R$ such that $\mathscr{U}$ is a disjoint
cover of $X$ and $\sum_{U\in\mathscr{U}}e(U)1_{X,R,U}=f$. Put
$\mathscr{F}\coloneqq\mathscr{F}_{X,x_{0}}=\mathscr{F}_{X,x_{1}}$. If
$\mathscr{U}\cap\mathscr{F}=\emptyset$, then the images of $1_{X,R,U}$ in the
completed residue fields at $x_{0}$ and $x_{1}$ are $0$ for any
$U\in\mathscr{F}$, and hence so are the images of $f$. If $U\in\mathscr{F}$
for some $U\in\mathscr{U}$, then there is no other $U^{\prime}\in\mathscr{U}$
such that $U^{\prime}\in\mathscr{F}$ because $\mathscr{F}$ is an ultrafilter,
and hence we obtain

$\displaystyle
x_{0}(f)=x_{0}(e(U)1_{X,R,U})=x_{0}|_{R}(e(U))=x_{1}|_{R}(e(U))=x_{1}(e(U)1_{X,R,U})=x_{1}(f).$

Next, let
$(\mathscr{F},\overline{x}{})\in\zeta(X)\times\mathscr{M}_{\mathscr{D}_{2}}(R)$.
We denote by $x\in\mathscr{M}(\text{\rm C}_{\text{\rm fin}}(X,R))$ the bounded
multiplicative seminorm assigning
$\overline{x}{}(\lim^{\mathscr{D}_{3}}_{\mathscr{F}}(f))\in[0,\infty)$ to each
$f\in\text{\rm C}_{\text{\rm fin}}(X,R)$. We have

$\displaystyle\mathscr{F}_{X,x}={\mathchoice{\left\\{U\in\text{\rm
CO}(X)\middle|x(1_{X,R,U})\neq 0\right\\}}{\\{U\in\text{\rm CO}(X)\mid
x(1_{X,R,U})\neq 0\\}}{\\{U\in\text{\rm CO}(X)\mid x(1_{X,R,U})\neq
0\\}}{\\{U\in\text{\rm CO}(X)\mid x(1_{X,R,U})\neq
0\\}}}={\mathchoice{\left\\{U\in\text{\rm
CO}(X)\middle|\overline{x}{}\left(\textstyle\lim^{\mathscr{D}_{3}}_{\mathscr{F}}\displaystyle(1_{X,R,U})\right)\neq
0\right\\}}{\\{U\in\text{\rm
CO}(X)\mid\overline{x}{}\left(\textstyle\lim^{\mathscr{D}_{3}}_{\mathscr{F}}\displaystyle(1_{X,R,U})\right)\neq
0\\}}{\\{U\in\text{\rm
CO}(X)\mid\overline{x}{}\left(\textstyle\lim^{\mathscr{D}_{3}}_{\mathscr{F}}\displaystyle(1_{X,R,U})\right)\neq
0\\}}{\\{U\in\text{\rm
CO}(X)\mid\overline{x}{}\left(\textstyle\lim^{\mathscr{D}_{3}}_{\mathscr{F}}\displaystyle(1_{X,R,U})\right)\neq
0\\}}}$ $\displaystyle=$ $\displaystyle{\mathchoice{\left\\{U\in\text{\rm
CO}(X)\middle|\textstyle\lim^{\mathscr{D}_{3}}_{\mathscr{F}}\displaystyle(1_{X,R,U})=1\right\\}}{\\{U\in\text{\rm
CO}(X)\mid\textstyle\lim^{\mathscr{D}_{3}}_{\mathscr{F}}\displaystyle(1_{X,R,U})=1\\}}{\\{U\in\text{\rm
CO}(X)\mid\textstyle\lim^{\mathscr{D}_{3}}_{\mathscr{F}}\displaystyle(1_{X,R,U})=1\\}}{\\{U\in\text{\rm
CO}(X)\mid\textstyle\lim^{\mathscr{D}_{3}}_{\mathscr{F}}\displaystyle(1_{X,R,U})=1\\}}}={\mathchoice{\left\\{U\in\text{\rm
CO}(X)\middle|\exists V\in\mathscr{F},V\subset U\right\\}}{\\{U\in\text{\rm
CO}(X)\mid\exists V\in\mathscr{F},V\subset U\\}}{\\{U\in\text{\rm
CO}(X)\mid\exists V\in\mathscr{F},V\subset U\\}}{\\{U\in\text{\rm
CO}(X)\mid\exists V\in\mathscr{F},V\subset U\\}}}=\mathscr{F}.$

For any $a\in R$, we have
$x|_{R}(a)=x(a1_{X,R,X})=\overline{x}{}(\lim^{\mathscr{D}_{3}}_{\mathscr{F}}(a1_{X,R,X}))=\overline{x}{}(a)$.
This implies $x|_{R}=\overline{x}{}$. We obtain
$G_{\mathscr{D}_{3}}(x)=(\mathscr{F},\overline{x}{})$. ∎

By Theorem 2.10, we obtain an isolated analogue of [Mih14] Corollary 2.3.

######  Corollary 2.11.

If $R$ is isolated at $0$, then the correspondence assigning
$G_{(\mathscr{C},Y,R)}$ to each $Y\in\text{\rm Top}$ gives a natural
isomorphism
$G_{(\mathscr{C},\bullet,R)}\colon\mathscr{M}_{\mathscr{D}_{2}}\circ\Gamma_{\mathscr{D}_{2}}\Rightarrow\zeta\times\mathscr{M}_{\mathscr{D}_{2}}(R)$

We conclude this section by proving Gel’fand’s duality for algebras of
continuous functions over Banach rings isolated at $0$ with connected
Berkovich spectrum.

######  Theorem 2.12.

Let $R$ be a Banach ring isolated at $0$ such that $\mathscr{M}(R)$ is
connected. Then, the pair $(\zeta\circ\mathscr{M},\Gamma_{\mathscr{D}_{2}})$
of functors realises a duality, i.e. a contravariant equivalence, between TDCH
and the essential image of $\Gamma_{\mathscr{D}_{2}}$.

We note that in our context, a topological space is said to be connected if it
has precisely one connected component, and hence $\emptyset$ is not connected.

###### Proof.

We denote by $F_{\text{\rm L}}$ the contravariant functor from the essential
image of $\Gamma_{\mathscr{D}_{2}}$ to TDCH induced by
$\zeta\circ\mathscr{M}$, and by $F_{\text{\rm R}}$ the contravariant functor
from TDCH to the essential image of $\Gamma_{\mathscr{D}_{2}}$ induced by
$\Gamma_{\mathscr{D}_{2}}$. First, we construct a natural isomorphism
$F_{\text{\rm L}}(F_{\text{\rm R}}(Y))\to Y$ for a $Y\in\text{\rm TDCH}$. By
Corollary 2.11,
$G_{(\mathscr{C},Y,R)}\colon\mathscr{M}(\Gamma_{\mathscr{D}_{2}}(Y))\to\zeta(Y)\times\mathscr{M}(R)$
is an isomorphism in Top. Since the first projection
$\zeta(Y)\times\mathscr{M}(R)\to\zeta(Y)$ is a quotient map whose fibres are
connected, it satisfies the universality of the Banaschewski compactification.
This implies that the natural morphisms

in Top are isomorphisms, and hence there exists a unique dotted filler, which
is a natural isomorphism in TDCH.

Next, we construct a natural isomorphism $\Gamma_{\mathscr{D}_{2}}(Y)\to
F_{\text{\rm R}}(F_{\text{\rm L}}(\Gamma_{\mathscr{D}_{2}}(Y)))$ for a
$Y\in\text{\rm Top}$. By Corollary 2.11,
$G_{(\mathscr{C},Y,R)}\colon\mathscr{M}(\Gamma_{\mathscr{D}_{2}}(Y))\to\zeta(Y)\times\mathscr{M}(R)$
is an isomorphism in Top. Since the first projection
$\pi\colon\zeta(Y)\times\mathscr{M}(R)\to\zeta(Y)$ is a quotient map whose
fibres are homeomorphic to the connected topological space $\mathscr{M}(R)$,
it satisfies the universality of the Banaschewski compactification. This
implies that the natural morphisms

in $\text{\rm Alg}_{\mathscr{C}}(R)$ are isomorphisms by Proposition 2.6, and
there exists a unique dotted filler, which is a natural isomorphism in
$\text{\rm Alg}_{\mathscr{C}}(R)$. ∎

######  Remark 2.13.

In the case where
$R=(\mathbb{Z},{\mathchoice{\left|\cdot\right|}{|\cdot|}{|\cdot|}{|\cdot|}}_{\infty})$,
and under the hypothesis that $X$ is the Stone–$\check{\text{C}}$ech
compactification of a discrete topological space, Theorem 2.12 has been
recently independently obtained by Kobi Kremnizer and André Henriques (private
communication with the first listed author).

We say that $R$ satisfies Shilov idempotent theorem if every $U\in\text{\rm
CO}(\mathscr{M}(R))$ coincides with
${\mathchoice{\left\\{x\in\mathscr{M}(R)\middle|x(e)=1\right\\}}{\\{x\in\mathscr{M}(R)\mid
x(e)=1\\}}{\\{x\in\mathscr{M}(R)\mid x(e)=1\\}}{\\{x\in\mathscr{M}(R)\mid
x(e)=1\\}}}$ for some idempotent $e\in R$. By the definition, Shilov
idempotent theorem is a weak variant of Tate’s acyclicity focusing only on a
disjoint clopen covering. For example, every Banach algebra over a complete
valued field, e.g. a trivially valued field, satisfies Shilov idempotent
theorem by [Ber90] 7.4.1 Theorem.

######  Corollary 2.14.

Let $R$ be a Banach ring isolated at $0$ satisfying Shilov idempotent theorem.
Then, the pair $(\zeta\circ\mathscr{M},\Gamma_{\mathscr{D}_{2}})$ of functors
realises a duality between TDCH and the essential image of
$\Gamma_{\mathscr{D}_{2}}$ if and only if $R$ has precisely one non-zero
idempotent element.

###### Proof.

We follow the convention in the proof of Theorem 2.12. If $R$ does not have
precisely one non-zero idempotent element, then $F_{\text{\rm R}}$ is not
fully faithful, because Example 2.9 implies that $F_{\text{\rm R}}(\pi)$ is
not bijective, where $\pi$ denotes the unique continuous map
${\mathchoice{\left\\{0,1\right\\}}{\\{0,1\\}}{\\{0,1\\}}{\\{0,1\\}}}\to{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$,
where ${\mathchoice{\left\\{0,1\right\\}}{\\{0,1\\}}{\\{0,1\\}}{\\{0,1\\}}}$
is equipped with the discrete topology. If $R$ has precisely one non-zero
idempotent element, then $\mathscr{M}(R)$ is connected by Shilov idempotent
theorem, and hence the assertion follows from Theorem 2.12. ∎

##  3 Homotopy epimorphisms for functions valued in discrete Banach rings

By Theorem 2.12, which is Gel’fand’s duality over Banach rings isolated at
$0$, morphisms $\text{\rm C}(X,R)\to\text{\rm C}(Y,R)$ of Banach $R$-algebras
correspond to continuous maps $Y\to X$ between totally disconnected compact
Hausdorff topological spaces . In this section, we characterise the morphisms
between such algebras corresponding to closed topological embeddings as
homotopy epimorphisms. The notion of homotopy epimorphism is the most basic
notion of localisation of (simplicial) rings used in derived geometry. We
refer the reader to [BM21] §1.3 for a more detailed discussion of this notion
and its use in the context of the theory of Banach rings and in analytic
geometry.

For an additive category $\mathcal{E}$, we denote by $\text{\rm
Ch}^{-}(\mathcal{E})$ the additive category of chain complexes bounded above
associated to $\mathcal{E}$ and chain morphisms. We put $\text{\rm
Ch}^{-}_{\mathscr{C}}(R)\coloneqq\text{\rm Ch}^{-}(\text{\rm
Mod}_{\mathscr{C}}(R))$. We denote by $\text{\rm Der}^{-}_{\mathscr{C}}(R)$
the derived category of $\text{\rm Ch}^{-}_{\mathscr{C}}(R)$, i.e. the
localisation of the homotopy category of $\text{\rm Ch}^{-}_{\mathscr{C}}(R)$
by the multiplicative system of morphisms represented by quasi-isomorphisms
(cf. [BM21] §1.2 for more details about this construction). We recall that the
completed tensor product $(\cdot)\widehat{\otimes}{}_{R}(\cdot)\colon\text{\rm
Mod}_{\mathscr{C}}(R)^{2}\to\text{\rm Mod}_{\mathscr{C}}(R)$ can be derived to
a functor $(\cdot)\widehat{\otimes}{}_{R}^{\mathbb{L}}(\cdot)\colon\text{\rm
Der}^{-}_{\mathscr{C}}(R)^{2}\to\text{\rm Der}^{-}_{\mathscr{C}}(R)$ in a way
similar to the algebraic tensor product. A morphism $\pi\colon A_{0}\to A_{1}$
in $\text{\rm Alg}_{\mathscr{C}}(R)$ is said to be a homotopy epimorphism in
$\text{\rm Alg}_{\mathscr{C}}(R)$ if the multiplication
$A_{1}\widehat{\otimes}{}_{A_{0}}^{\mathbb{L}}A_{1}\to A_{1}$ is an
isomorphism in $\text{\rm Der}^{-}_{\mathscr{C}}(A_{0})$.

Suppose that $R$ is isolated at $0$. Let $(K,j)\in\text{\rm Top}/X$, where
$\text{\rm Top}/X$ denotes the slice category of Top over $X$. We regard
$\text{\rm C}_{\text{\rm fin}}(K,R)$ as an object of $\text{\rm
Alg}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(X,R))$ through
$\Gamma_{\mathscr{D}_{2}}(j)$. We say that $j$ is a $\zeta$-embedding if it is
an $\text{\rm Ab}^{\text{\rm nA}}$-embedding in the sense of [BM21] §3.1, i.e.
if $\zeta(j)\colon\zeta(K)\to\zeta(X)$ is a homeomorphism onto the image. We
give an isolated analogue of [BM21] Lemma 3.9.

######  Theorem 3.1.

If $R$ is isolated at $0$, then the following are equivalent:

  * (i)

The morphism $\Gamma_{\mathscr{D}_{2}}(j)$ is a homotopy epimorphism in
$\text{\rm Alg}_{\mathscr{C}}(R)$.

  * (ii)

The morphism $\Gamma_{\mathscr{D}_{2}}(j)$ is a strict epimorphism in
$\text{\rm Mod}_{\mathscr{C}}(R)$.

  * (iii)

The map $j$ is a $\zeta$-embedding or
$R={\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$.

In order to verify Theorem 3.1, we show several lemmata. First, we show an
isolated analogue of [BM21] Lemma 2.9 and [BM21] Lemma 2.22.

######  Lemma 3.2.

Every finitely generated $\text{\rm C}_{\text{\rm fin}}(X,R)$-submodule of
${\mathchoice{\left\\{f\in\text{\rm C}_{\text{\rm fin}}(X,R)\middle|f\circ
j=0\right\\}}{\\{f\in\text{\rm C}_{\text{\rm fin}}(X,R)\mid f\circ
j=0\\}}{\\{f\in\text{\rm C}_{\text{\rm fin}}(X,R)\mid f\circ
j=0\\}}{\\{f\in\text{\rm C}_{\text{\rm fin}}(X,R)\mid f\circ j=0\\}}}$ is
contained in a principal ideal generated by $1_{X,R,U}$ for some
$U\in\text{\rm CO}(X)$ with $j(K)\cap U=\emptyset$.

###### Proof.

The assertion follows from Proposition 2.5 applied to the subset $j(K)$ and
the corresponding ideal ${\mathchoice{\left\\{f\in\text{\rm C}_{\text{\rm
fin}}(X,R)\middle|f\circ j=0\right\\}}{\\{f\in\text{\rm C}_{\text{\rm
fin}}(X,R)\mid f\circ j=0\\}}{\\{f\in\text{\rm C}_{\text{\rm fin}}(X,R)\mid
f\circ j=0\\}}{\\{f\in\text{\rm C}_{\text{\rm fin}}(X,R)\mid f\circ
j=0\\}}}={\mathchoice{\left\\{f\in\text{\rm C}_{\text{\rm
fin}}(X,R)\middle|\forall x\in K,f(j(x))=0\right\\}}{\\{f\in\text{\rm
C}_{\text{\rm fin}}(X,R)\mid\forall x\in K,f(j(x))=0\\}}{\\{f\in\text{\rm
C}_{\text{\rm fin}}(X,R)\mid\forall x\in K,f(j(x))=0\\}}{\\{f\in\text{\rm
C}_{\text{\rm fin}}(X,R)\mid\forall x\in K,f(j(x))=0\\}}}=I_{X,R,j(K)}$. ∎

We denote by $\mathscr{C}_{\leq 1}\subset\mathscr{C}$ the symmetric monoidal
subcategory with the same class of objects and the class of morphisms
consisting of morphisms of operator norm $\leq 1$. We show an isolated
analogue of [BM21] Lemma 2.13 and [BM21] Lemma 2.24.

######  Lemma 3.3.

Suppose that $R$ is isolated at $0$ and $j$ is a $\zeta$-embedding.

  * (i)

For any $f\in\text{\rm C}_{\text{\rm fin}}(K,R)$, there exists an
$\tilde{f}{}\in\text{\rm C}_{\text{\rm fin}}(X,R)$ such that $\tilde{f}{}\circ
j=f$ and
${\mathchoice{\left\|\tilde{f}{}\right\|}{\|\tilde{f}{}\|}{\|\tilde{f}{}\|}{\|\tilde{f}{}\|}}_{\text{\rm
C}_{\text{\rm
fin}}(X,R)}={\mathchoice{\left\|f\right\|}{\|f\|}{\|f\|}{\|f\|}}_{\text{\rm
C}_{\text{\rm fin}}(K,R)}$.

  * (ii)

The morphism $\Gamma_{\mathscr{D}_{2}}(j)$ is a strict epimorphism in
$\text{\rm Mod}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(X,R))$.

  * (iii)

If $R$ is a monoid object of $\mathscr{C}_{\leq 1}$, then the morphism
$\Gamma_{\mathscr{D}_{2}}(j)$ is a strict epimorphism in $\text{\rm
Mod}_{\mathscr{C}_{\leq 1}}(\text{\rm C}_{\text{\rm fin}}(X,R))$.

###### Proof.

The assertions (ii) and (iii) immediately follow from the assertion (i). We
show the assertion (i). Take an $\epsilon\in(0,\infty)$ with
${\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}={\mathchoice{\left\\{a\in
R\middle|{\mathchoice{\left\|a\right\|}{\|a\|}{\|a\|}{\|a\|}}_{R}<\epsilon\right\\}}{\\{a\in
R\mid{\mathchoice{\left\|a\right\|}{\|a\|}{\|a\|}{\|a\|}}_{R}<\epsilon\\}}{\\{a\in
R\mid{\mathchoice{\left\|a\right\|}{\|a\|}{\|a\|}{\|a\|}}_{R}<\epsilon\\}}{\\{a\in
R\mid{\mathchoice{\left\|a\right\|}{\|a\|}{\|a\|}{\|a\|}}_{R}<\epsilon\\}}}$.
By Proposition 2.6, there exists a unique $g\in\text{\rm C}(\zeta(K),R)$ such
that $g\circ\iota_{\zeta,K}=f$ and
${\mathchoice{\left\|g\right\|}{\|g\|}{\|g\|}{\|g\|}}_{\text{\rm
C}(\zeta(K),R)}={\mathchoice{\left\|f\right\|}{\|f\|}{\|f\|}{\|f\|}}_{\text{\rm
C}_{\text{\rm fin}}(K,R)}$. Since $j$ is a $\zeta$-embedding, there exists a
$\tilde{g}{}\in\text{\rm C}(\zeta(X),R)$ such that
$\tilde{g}{}\circ\zeta(j)=g$ and
${\mathchoice{\left\|\tilde{g}{}\right\|}{\|\tilde{g}{}\|}{\|\tilde{g}{}\|}{\|\tilde{g}{}\|}}_{\text{\rm
C}(\zeta(X),R)}\leq{\mathchoice{\left\|g\right\|}{\|g\|}{\|g\|}{\|g\|}}_{\text{\rm
C}(\zeta(K),R)}+\epsilon$ by [BM21] Lemma 2.13. By the strictness of
$\Gamma_{\mathscr{D}_{2}}(\zeta(j))$ and the arbitrariness of $\epsilon$, we
obtain
${\mathchoice{\left\|\tilde{g}{}\right\|}{\|\tilde{g}{}\|}{\|\tilde{g}{}\|}{\|\tilde{g}{}\|}}_{\text{\rm
C}(\zeta(X),R)}={\mathchoice{\left\|g\right\|}{\|g\|}{\|g\|}{\|g\|}}_{\text{\rm
C}(\zeta(K),R)}$.

Put $\tilde{f}{}\coloneqq\tilde{g}{}\circ\iota_{\zeta,X}\in\text{\rm
C}_{\text{\rm fin}}(X,R)$. For any $x\in K$, we have
$\tilde{f}{}(j(x))=\tilde{g}{}(\iota_{\zeta,X}(j(x)))=\tilde{g}{}(\zeta(j)(\iota_{\zeta,K}(x)))=g(\iota_{\zeta,K}(x))=f(x)$
because $\iota_{\zeta,\bullet}$ is a natural transform $\text{\rm
id}_{\text{\rm CH}}\Rightarrow\text{\rm TDCH}$. This implies $\tilde{f}{}\circ
j=f$ and
${\mathchoice{\left\|\tilde{f}{}\right\|}{\|\tilde{f}{}\|}{\|\tilde{f}{}\|}{\|\tilde{f}{}\|}}_{\text{\rm
C}_{\text{\rm
fin}}(X,R)}\geq{\mathchoice{\left\|f\right\|}{\|f\|}{\|f\|}{\|f\|}}_{\text{\rm
C}_{\text{\rm fin}}(K,R)}$. By
${\mathchoice{\left\|\tilde{f}{}\right\|}{\|\tilde{f}{}\|}{\|\tilde{f}{}\|}{\|\tilde{f}{}\|}}_{\text{\rm
C}_{\text{\rm
fin}}(X,R)}\leq{\mathchoice{\left\|\tilde{g}{}\right\|}{\|\tilde{g}{}\|}{\|\tilde{g}{}\|}{\|\tilde{g}{}\|}}_{\text{\rm
C}(\zeta(X),R)}={\mathchoice{\left\|g\right\|}{\|g\|}{\|g\|}{\|g\|}}_{\text{\rm
C}(\zeta(K),R)}={\mathchoice{\left\|f\right\|}{\|f\|}{\|f\|}{\|f\|}}_{\text{\rm
C}_{\text{\rm fin}}(K,R)}$, we obtain
${\mathchoice{\left\|\tilde{f}{}\right\|}{\|\tilde{f}{}\|}{\|\tilde{f}{}\|}{\|\tilde{f}{}\|}}_{\text{\rm
C}_{\text{\rm
fin}}(X,R)}={\mathchoice{\left\|f\right\|}{\|f\|}{\|f\|}{\|f\|}}_{\text{\rm
C}_{\text{\rm fin}}(K,R)}$. ∎

From now on and until the end of this section, we use the notation
$I\coloneqq\ker(\Gamma_{\mathscr{D}_{2}}(j))$ and $i\colon
I\hookrightarrow\text{\rm C}_{\text{\rm fin}}(X,R)$ for the canonical
inclusion.

######  Lemma 3.4.

Suppose that $R$ is isolated at $0$. Let $M\in\text{\rm
Mod}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(X,R))$. For any
$(\tilde{m}{},\epsilon)\in(I\widehat{\otimes}{}_{\text{\rm C}_{\text{\rm
fin}}(X,R)}M)\times(0,\infty)$, there exists a $(U,m)\in\text{\rm CO}(X)\times
M$ such that $j(K)\cap U=\emptyset$ and
${\mathchoice{\left\|\tilde{m}{}-1_{X,R,U}\otimes
m\right\|}{\|\tilde{m}{}-1_{X,R,U}\otimes m\|}{\|\tilde{m}{}-1_{X,R,U}\otimes
m\|}{\|\tilde{m}{}-1_{X,R,U}\otimes m\|}}_{I\widehat{\otimes}{}_{\text{\rm
C}_{\text{\rm fin}}(X,R)}M}<\epsilon$.

###### Proof.

By the definition of $\widehat{\otimes}{}_{\text{\rm C}_{\text{\rm
fin}}(X,R)}$, there exists a tuple $(k,f,n)$ of a $k\in\mathbb{N}$, an
$f=(f_{h})_{h=0}^{k}\in I^{k+1}$, and an $n=(n_{h})_{h=0}^{k}\in M^{k+1}$ such
that ${\mathchoice{\left\|\tilde{m}{}-\sum_{h=0}^{k}f_{h}\otimes
n_{h}\right\|}{\|\tilde{m}{}-\sum_{h=0}^{k}f_{h}\otimes
n_{h}\|}{\|\tilde{m}{}-\sum_{h=0}^{k}f_{h}\otimes
n_{h}\|}{\|\tilde{m}{}-\sum_{h=0}^{k}f_{h}\otimes
n_{h}\|}}_{I\widehat{\otimes}{}_{\text{\rm C}_{\text{\rm
fin}}(X,R)}M}<\epsilon$. By Lemma 3.2, there exists a $U\in\text{\rm CO}(X)$
such that $1_{X,R,U}f_{h}=f_{h}$ and $j(K)\cap U=\emptyset$ for any $0\leq
h\leq k$. Put $m\coloneqq\sum_{h=0}^{k}f_{h}n_{h}\in M$. We have

$\displaystyle\sum_{h=0}^{k}f_{h}\otimes
n_{h}=\sum_{h=0}^{k}1_{X,R,U}f_{h}\otimes n_{h}=\sum_{h=0}^{k}1_{X,R,U}\otimes
f_{h}n_{h}=1_{X,R,U}\otimes\sum_{h=0}^{k}f_{h}n_{h}=1_{X,R,U}\otimes m$

in $I\otimes_{\text{\rm C}_{\text{\rm fin}}(X,R)}M$, and hence
${\mathchoice{\left\|\tilde{m}{}-1_{X,R,U}\otimes
m\right\|}{\|\tilde{m}{}-1_{X,R,U}\otimes m\|}{\|\tilde{m}{}-1_{X,R,U}\otimes
m\|}{\|\tilde{m}{}-1_{X,R,U}\otimes m\|}}_{I\widehat{\otimes}{}_{\text{\rm
C}_{\text{\rm fin}}(X,R)}M}<\epsilon$. ∎

######  Lemma 3.5.

Suppose that $R$ is isolated at $0$. Let $M\in\text{\rm
Mod}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(X,R))$. Then, the morphism
$i_{M}\colon I\widehat{\otimes}{}_{\text{\rm C}_{\text{\rm fin}}(X,R)}M\to M$
in $\text{\rm Mod}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(X,R))$ induced
by the inclusion $i\colon I\hookrightarrow\text{\rm C}_{\text{\rm fin}}(X,R)$
is a strict monomorphism.

###### Proof.

We abbreviate $I\widehat{\otimes}{}_{\text{\rm C}_{\text{\rm fin}}(X,R)}M$ to
$M_{I}$. It suffices to show
${\mathchoice{\left\|\tilde{m}{}\right\|}{\|\tilde{m}{}\|}{\|\tilde{m}{}\|}{\|\tilde{m}{}\|}}_{M_{I}}\leq{\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}{\mathchoice{\left\|i_{M}(\tilde{m}{})\right\|}{\|i_{M}(\tilde{m}{})\|}{\|i_{M}(\tilde{m}{})\|}{\|i_{M}(\tilde{m}{})\|}}_{M}$
for any $\tilde{m}{}\in M_{I}$. Assume
${\mathchoice{\left\|\tilde{m}{}\right\|}{\|\tilde{m}{}\|}{\|\tilde{m}{}\|}{\|\tilde{m}{}\|}}_{M_{I}}>{\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}{\mathchoice{\left\|i_{M}(\tilde{m}{})\right\|}{\|i_{M}(\tilde{m}{})\|}{\|i_{M}(\tilde{m}{})\|}{\|i_{M}(\tilde{m}{})\|}}_{M}$.
Put $\epsilon\coloneqq
2^{-1}({\mathchoice{\left\|\tilde{m}{}\right\|}{\|\tilde{m}{}\|}{\|\tilde{m}{}\|}{\|\tilde{m}{}\|}}_{M_{I}}-{\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}{\mathchoice{\left\|i_{M}(\tilde{m}{})\right\|}{\|i_{M}(\tilde{m}{})\|}{\|i_{M}(\tilde{m}{})\|}{\|i_{M}(\tilde{m}{})\|}}_{M})\in(0,\infty)$.
We denote by $C_{i_{M}}$ the operator norm of $i_{M}$. By Lemma 3.4, there
exists a $(U,m)\in\text{\rm CO}(X)\times M$ such that
${\mathchoice{\left\|\tilde{m}{}-1_{X,R,U}\otimes
m\right\|}{\|\tilde{m}{}-1_{X,R,U}\otimes m\|}{\|\tilde{m}{}-1_{X,R,U}\otimes
m\|}{\|\tilde{m}{}-1_{X,R,U}\otimes
m\|}}_{M_{I}}<({\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}C_{i_{M}}+1)^{-1}\epsilon$
and $j(K)\cap U=\emptyset$. Put $\delta\coloneqq\tilde{m}{}-1_{X,R,U}\otimes
m\in M_{I}$. We have

$\displaystyle{\mathchoice{\left\|\tilde{m}{}\right\|}{\|\tilde{m}{}\|}{\|\tilde{m}{}\|}{\|\tilde{m}{}\|}}_{M_{I}}={\mathchoice{\left\|1_{X,R,U}\otimes
m+\delta\right\|}{\|1_{X,R,U}\otimes m+\delta\|}{\|1_{X,R,U}\otimes
m+\delta\|}{\|1_{X,R,U}\otimes
m+\delta\|}}_{M_{I}}={\mathchoice{\left\|1_{X,R,U}\otimes
1_{X,R,U}m+\delta\right\|}{\|1_{X,R,U}\otimes
1_{X,R,U}m+\delta\|}{\|1_{X,R,U}\otimes
1_{X,R,U}m+\delta\|}{\|1_{X,R,U}\otimes 1_{X,R,U}m+\delta\|}}_{M_{I}}$
$\displaystyle\leq$ $\displaystyle{\mathchoice{\left\|1_{X,R,U}\otimes
1_{X,R,U}m\right\|}{\|1_{X,R,U}\otimes 1_{X,R,U}m\|}{\|1_{X,R,U}\otimes
1_{X,R,U}m\|}{\|1_{X,R,U}\otimes
1_{X,R,U}m\|}}_{M_{I}}+{\mathchoice{\left\|\delta\right\|}{\|\delta\|}{\|\delta\|}{\|\delta\|}}_{M_{I}}\leq{\mathchoice{\left\|1_{X,R,U}\right\|}{\|1_{X,R,U}\|}{\|1_{X,R,U}\|}{\|1_{X,R,U}\|}}_{I}{\mathchoice{\left\|1_{X,R,U}m\right\|}{\|1_{X,R,U}m\|}{\|1_{X,R,U}m\|}{\|1_{X,R,U}m\|}}_{M}+{\mathchoice{\left\|\delta\right\|}{\|\delta\|}{\|\delta\|}{\|\delta\|}}_{M_{I}}$
$\displaystyle\leq$
$\displaystyle{\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}{\mathchoice{\left\|1_{X,R,U}m\right\|}{\|1_{X,R,U}m\|}{\|1_{X,R,U}m\|}{\|1_{X,R,U}m\|}}_{M}+{\mathchoice{\left\|\delta\right\|}{\|\delta\|}{\|\delta\|}{\|\delta\|}}_{M_{I}}={\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}{\mathchoice{\left\|i_{M}(1_{X,R,U}\otimes
m)\right\|}{\|i_{M}(1_{X,R,U}\otimes m)\|}{\|i_{M}(1_{X,R,U}\otimes
m)\|}{\|i_{M}(1_{X,R,U}\otimes
m)\|}}_{M}+{\mathchoice{\left\|\delta\right\|}{\|\delta\|}{\|\delta\|}{\|\delta\|}}_{M_{I}}$
$\displaystyle=$
$\displaystyle{\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}{\mathchoice{\left\|i_{M}(\tilde{m}{}-\delta)\right\|}{\|i_{M}(\tilde{m}{}-\delta)\|}{\|i_{M}(\tilde{m}{}-\delta)\|}{\|i_{M}(\tilde{m}{}-\delta)\|}}_{M}+{\mathchoice{\left\|\delta\right\|}{\|\delta\|}{\|\delta\|}{\|\delta\|}}_{M_{I}}\leq{\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}({\mathchoice{\left\|i_{M}(\tilde{m}{})\right\|}{\|i_{M}(\tilde{m}{})\|}{\|i_{M}(\tilde{m}{})\|}{\|i_{M}(\tilde{m}{})\|}}_{M}+{\mathchoice{\left\|i_{M}(\delta)\right\|}{\|i_{M}(\delta)\|}{\|i_{M}(\delta)\|}{\|i_{M}(\delta)\|}}_{M})+{\mathchoice{\left\|\delta\right\|}{\|\delta\|}{\|\delta\|}{\|\delta\|}}_{M_{I}}$
$\displaystyle\leq$
$\displaystyle{\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}{\mathchoice{\left\|i_{M}(\tilde{m}{})\right\|}{\|i_{M}(\tilde{m}{})\|}{\|i_{M}(\tilde{m}{})\|}{\|i_{M}(\tilde{m}{})\|}}_{M}+({\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}C_{i_{M}}+1){\mathchoice{\left\|\delta\right\|}{\|\delta\|}{\|\delta\|}{\|\delta\|}}_{M_{I}}<{\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}{\mathchoice{\left\|i_{M}(\tilde{m}{})\right\|}{\|i_{M}(\tilde{m}{})\|}{\|i_{M}(\tilde{m}{})\|}{\|i_{M}(\tilde{m}{})\|}}_{M}+\epsilon,$

but this contradicts the definition of $\epsilon$. ∎

We show an isolated analogue of [BM21] Lemma 2.22.

######  Lemma 3.6.

If $R$ is isolated at $0$, then the objects $I$ and $\text{\rm C}_{\text{\rm
fin}}(X,R)$ of $\text{\rm Mod}_{\mathscr{C}}(\text{\rm C}_{\text{\rm
fin}}(X,R))$ are strongly flat.

###### Proof.

The strong flatness of $\text{\rm C}_{\text{\rm fin}}(X,R)$ is obvious because
it is a monoidal unit with respect to $\widehat{\otimes}{}_{\text{\rm
C}_{\text{\rm fin}}(X,R)}$. Let $E$ be a strictly exact sequence

$\displaystyle 0\to M\stackrel{{\scriptstyle
s}}{{\to}}N\stackrel{{\scriptstyle t}}{{\to}}L$

of $\text{\rm Mod}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(X,R))$.
Following the convention in Lemma 3.5 and its proof, we put $M_{I}\coloneqq
I\widehat{\otimes}{}_{\text{\rm C}_{\text{\rm fin}}(X,R)}M$, $N_{I}\coloneqq
I\widehat{\otimes}{}_{\text{\rm C}_{\text{\rm fin}}(X,R)}N$, and
$L_{I}\coloneqq I\widehat{\otimes}{}_{\text{\rm C}_{\text{\rm fin}}(X,R)}L$,
and denote by $i_{M}\colon M_{I}\to M$, $i_{N}\colon N_{I}\to N$, $i_{L}\colon
L_{I}\to L$ the morphisms induced by the inclusions. We consider the
commutative diagram

of $\text{\rm Mod}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(X,R))$. By
Lemma 3.5, $i_{M}$ and $i_{N}$ are strict monomorphisms. By the strict
exactness of $E$, $s$ is a strict monomorphism. By the commutativity of the
second square, $I\widehat{\otimes}{}_{\text{\rm C}_{\text{\rm fin}}(X,R)}s$ is
a strict monomorphism. Therefore, it suffices to show that for any
$(\tilde{n}{},\epsilon)\in\ker(I\widehat{\otimes}{}_{\text{\rm C}_{\text{\rm
fin}}(X,R)}t)\times(0,\infty)$, there exists an $\tilde{m}{}\in M_{I}$ such
that ${\mathchoice{\left\|\tilde{n}{}-(I\widehat{\otimes}{}_{\text{\rm
C}_{\text{\rm
fin}}(X,R)}s)(\tilde{m}{})\right\|}{\|\tilde{n}{}-(I\widehat{\otimes}{}_{\text{\rm
C}_{\text{\rm
fin}}(X,R)}s)(\tilde{m}{})\|}{\|\tilde{n}{}-(I\widehat{\otimes}{}_{\text{\rm
C}_{\text{\rm
fin}}(X,R)}s)(\tilde{m}{})\|}{\|\tilde{n}{}-(I\widehat{\otimes}{}_{\text{\rm
C}_{\text{\rm
fin}}(X,R)}s)(\tilde{m}{})\|}}_{N_{I}}<{\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}^{-1}\epsilon$.

We denote by $C_{N}$ the operator norm of the scalar multiplication $\text{\rm
C}_{\text{\rm fin}}(X,R)\widehat{\otimes}{}N\to N$, by $C_{N_{I}}$ the
operator norm of the scalar multiplication $\text{\rm C}_{\text{\rm
fin}}(X,R)\widehat{\otimes}{}(N_{I})\to N_{I}$, and by $D_{i_{N}}$ the
operator norm of the inverse of the isomorphism

$\displaystyle N_{I}\to\text{\rm im}(i_{N})$

in $\text{\rm Mod}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(X,R))$ induced
by the strict monomorphism $i_{N}$. By Lemma 3.4, there exists a
$(U,n)\in\text{\rm CO}(X)\times N$ such that
${\mathchoice{\left\|\tilde{n}{}-1_{X,R,U}\otimes
n\right\|}{\|\tilde{n}{}-1_{X,R,U}\otimes n\|}{\|\tilde{n}{}-1_{X,R,U}\otimes
n\|}{\|\tilde{n}{}-1_{X,R,U}\otimes
n\|}}_{N_{I}}<2^{-1}C_{N_{I}}^{-1}{\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}^{-1}\epsilon$
and $j(K)\cap U=\emptyset$. By the exactness of $E$ and the commutativity of
the diagram, there exists an $m\in M$ such that
${\mathchoice{\left\|i_{N}(\tilde{n}{})-s(m)\right\|}{\|i_{N}(\tilde{n}{})-s(m)\|}{\|i_{N}(\tilde{n}{})-s(m)\|}{\|i_{N}(\tilde{n}{})-s(m)\|}}_{N}<2^{-1}D_{i_{N}}^{-1}C_{N}^{-1}{\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}^{-1}\epsilon$.
Put $\tilde{m}{}\coloneqq 1_{X,R,U}\otimes m\in M_{I}$. We obtain

$\displaystyle{\mathchoice{\left\|\tilde{n}{}-(I\widehat{\otimes}{}_{\text{\rm
C}_{\text{\rm
fin}}(X,R)}s)(\tilde{m}{})\right\|}{\|\tilde{n}{}-(I\widehat{\otimes}{}_{\text{\rm
C}_{\text{\rm
fin}}(X,R)}s)(\tilde{m}{})\|}{\|\tilde{n}{}-(I\widehat{\otimes}{}_{\text{\rm
C}_{\text{\rm
fin}}(X,R)}s)(\tilde{m}{})\|}{\|\tilde{n}{}-(I\widehat{\otimes}{}_{\text{\rm
C}_{\text{\rm
fin}}(X,R)}s)(\tilde{m}{})\|}}_{N_{I}}={\mathchoice{\left\|\tilde{n}{}-1_{X,R,U}\otimes
s(m)\right\|}{\|\tilde{n}{}-1_{X,R,U}\otimes
s(m)\|}{\|\tilde{n}{}-1_{X,R,U}\otimes s(m)\|}{\|\tilde{n}{}-1_{X,R,U}\otimes
s(m)\|}}_{N_{I}}$ $\displaystyle\leq$
$\displaystyle{\mathchoice{\left\|(1-1_{X,R,U})\tilde{n}{}\right\|}{\|(1-1_{X,R,U})\tilde{n}{}\|}{\|(1-1_{X,R,U})\tilde{n}{}\|}{\|(1-1_{X,R,U})\tilde{n}{}\|}}_{N_{I}}+{\mathchoice{\left\|1_{X,R,U}\tilde{n}{}-1_{X,R,U}\otimes
s(m)\right\|}{\|1_{X,R,U}\tilde{n}{}-1_{X,R,U}\otimes
s(m)\|}{\|1_{X,R,U}\tilde{n}{}-1_{X,R,U}\otimes
s(m)\|}{\|1_{X,R,U}\tilde{n}{}-1_{X,R,U}\otimes s(m)\|}}_{N_{I}}$
$\displaystyle=$
$\displaystyle{\mathchoice{\left\|(1-1_{X,R,U})(\tilde{n}{}-1_{X,R,U}\otimes
n)\right\|}{\|(1-1_{X,R,U})(\tilde{n}{}-1_{X,R,U}\otimes
n)\|}{\|(1-1_{X,R,U})(\tilde{n}{}-1_{X,R,U}\otimes
n)\|}{\|(1-1_{X,R,U})(\tilde{n}{}-1_{X,R,U}\otimes
n)\|}}_{N_{I}}+{\mathchoice{\left\|1_{X,R,U}\tilde{n}{}-1_{X,R,U}\otimes
s(m)\right\|}{\|1_{X,R,U}\tilde{n}{}-1_{X,R,U}\otimes
s(m)\|}{\|1_{X,R,U}\tilde{n}{}-1_{X,R,U}\otimes
s(m)\|}{\|1_{X,R,U}\tilde{n}{}-1_{X,R,U}\otimes s(m)\|}}_{N_{I}}$
$\displaystyle\leq$
$\displaystyle{\mathchoice{\left\|(1-1_{X,R,U})(\tilde{n}{}-1_{X,R,U}\otimes
n)\right\|}{\|(1-1_{X,R,U})(\tilde{n}{}-1_{X,R,U}\otimes
n)\|}{\|(1-1_{X,R,U})(\tilde{n}{}-1_{X,R,U}\otimes
n)\|}{\|(1-1_{X,R,U})(\tilde{n}{}-1_{X,R,U}\otimes n)\|}}_{N_{I}}$
$\displaystyle+D_{i_{N}}{\mathchoice{\left\|(i_{N})(1_{X,R,U}\tilde{n}{}-1_{X,R,U}\otimes
s(m))\right\|}{\|(i_{N})(1_{X,R,U}\tilde{n}{}-1_{X,R,U}\otimes
s(m))\|}{\|(i_{N})(1_{X,R,U}\tilde{n}{}-1_{X,R,U}\otimes
s(m))\|}{\|(i_{N})(1_{X,R,U}\tilde{n}{}-1_{X,R,U}\otimes s(m))\|}}_{N}$
$\displaystyle=$
$\displaystyle{\mathchoice{\left\|(1-1_{X,R,U})(\tilde{n}{}-1_{X,R,U}\otimes
n)\right\|}{\|(1-1_{X,R,U})(\tilde{n}{}-1_{X,R,U}\otimes
n)\|}{\|(1-1_{X,R,U})(\tilde{n}{}-1_{X,R,U}\otimes
n)\|}{\|(1-1_{X,R,U})(\tilde{n}{}-1_{X,R,U}\otimes n)\|}}_{N_{I}}$
$\displaystyle+D_{i_{N}}{\mathchoice{\left\|1_{X,R,U}((i_{N})(\tilde{n}{})-s(m))\right\|}{\|1_{X,R,U}((i_{N})(\tilde{n}{})-s(m))\|}{\|1_{X,R,U}((i_{N})(\tilde{n}{})-s(m))\|}{\|1_{X,R,U}((i_{N})(\tilde{n}{})-s(m))\|}}_{N}$
$\displaystyle\leq$ $\displaystyle
C_{N_{I}}{\mathchoice{\left\|1-1_{X,R,U}\right\|}{\|1-1_{X,R,U}\|}{\|1-1_{X,R,U}\|}{\|1-1_{X,R,U}\|}}_{\text{\rm
C}_{\text{\rm fin}}(X,R)}{\mathchoice{\left\|\tilde{n}{}-1_{X,R,U}\otimes
n\right\|}{\|\tilde{n}{}-1_{X,R,U}\otimes n\|}{\|\tilde{n}{}-1_{X,R,U}\otimes
n\|}{\|\tilde{n}{}-1_{X,R,U}\otimes n\|}}_{N_{I}}$
$\displaystyle+D_{i_{N}}C_{N}{\mathchoice{\left\|1_{X,R,U}\right\|}{\|1_{X,R,U}\|}{\|1_{X,R,U}\|}{\|1_{X,R,U}\|}}_{\text{\rm
C}_{\text{\rm
fin}}(X,R)}{\mathchoice{\left\|(i_{N})(\tilde{n}{})-s(m)\right\|}{\|(i_{N})(\tilde{n}{})-s(m)\|}{\|(i_{N})(\tilde{n}{})-s(m)\|}{\|(i_{N})(\tilde{n}{})-s(m)\|}}_{N}$
$\displaystyle\leq$ $\displaystyle
C_{N_{I}}{\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}{\mathchoice{\left\|\tilde{n}{}-1_{X,R,U}\otimes
n\right\|}{\|\tilde{n}{}-1_{X,R,U}\otimes n\|}{\|\tilde{n}{}-1_{X,R,U}\otimes
n\|}{\|\tilde{n}{}-1_{X,R,U}\otimes
n\|}}_{N_{I}}+D_{i_{N}}C_{N}{\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}{\mathchoice{\left\|i_{N}(\tilde{n}{})-s(m)\right\|}{\|i_{N}(\tilde{n}{})-s(m)\|}{\|i_{N}(\tilde{n}{})-s(m)\|}{\|i_{N}(\tilde{n}{})-s(m)\|}}_{N}.$

By the choice of $(U,n)$ and $m$, the right hand side is smaller than
$2^{-1}\epsilon+2^{-1}\epsilon=\epsilon$. ∎

We denote by $\mathscr{P}_{\mathscr{C}}^{R}$ the projective resolution functor
$\text{\rm Ch}^{-}_{\mathscr{C}}(R)\to\text{\rm Ch}^{-}_{\mathscr{C}}(R)$
explicitly introduced in [BM21] at the beginning of p. 13. In the next lemma,
we perform a basic computation of the derived tensor in a way completely
parallel to the algebraic setting.

######  Lemma 3.7.

Let $A$ be a monoid object of $\mathscr{C}$. For any pair of morphisms
$p_{0}\colon P_{0}\to M_{0}$ and $p_{1}\colon P_{1}\to M_{1}$ in $\text{\rm
Ch}^{-}_{\mathscr{C}}(A)$, if either $P_{0}$ or $P_{1}$ is termwise strongly
flat and both of $p_{0}$ and $p_{1}$ are quasi-isomorphisms, then the span

in $\text{\rm Ch}^{-}_{\mathscr{C}}(A)$ represents both an isomorphism
$M_{0}\widehat{\otimes}{}_{A}^{\mathbb{L}}M_{1}\to\text{\rm
Tot}(P_{0}\widehat{\otimes}{}_{A}P_{1})$ in $\text{\rm
Der}^{-}_{\mathscr{C}}(A)$ and its inverse.

###### Proof.

By [BM21] Proposition 1.6 (i), we may assume that $P_{1}$ is termwise strongly
flat. It suffices to show that every arrow in the commutative diagram

of $\text{\rm Ch}^{-}_{\mathscr{C}}(A)$ represents an isomorphism in
$\text{\rm Der}^{-}_{\mathscr{C}}(A)$. Since $P_{1}$ is termwise strongly
flat, the horizontal arrows are quasi-isomorphisms by [BM21] Proposition 1.6
(iii). Since $P_{1}$ is termwise strongly flat and $p_{0}$ is a quasi-
isomorphism, the right vertical arrow is a quasi-isomorphism by [BM21]
Proposition 1.6 (iv). By the commutativity of the top square, the top left
vertical arrow represents an isomorphism in $\text{\rm
Der}^{-}_{\mathscr{C}}(A)$. In order to show that the bottom left vertical
arrow $\text{\rm
Tot}(\mathscr{P}_{\mathscr{C}}^{A}(p_{0})\widehat{\otimes}{}_{A}\mathscr{P}_{\mathscr{C}}^{A}(M_{1}))$
represents an isomorphism in $\text{\rm Der}^{-}_{\mathscr{C}}(A)$, we
consider the commutative diagram

of $\text{\rm Ch}^{-}_{\mathscr{C}}(A)$. By the definition of the projective
resolution functor $\mathscr{P}_{\mathscr{C}}^{A}$, the two horizontal arrows
are quasi-isomorphisms. Since $p_{1}$ is a quasi-isomorphism,
$\mathscr{P}_{\mathscr{C}}^{A}(p_{1})$ represents an isomorphism in $\text{\rm
Der}^{-}_{\mathscr{C}}(A)$ by the commutativity of the diagram. This implies
that $\mathscr{P}_{\mathscr{C}}^{A}(p_{1})$ is a quasi-isomorphism by [BM21]
Corollary 1.5 (i). Since $\mathscr{P}_{\mathscr{C}}^{A}(M_{0})$ is termwise
projective and $\mathscr{P}_{\mathscr{C}}^{A}(p_{0})$ is a quasi-isomorphism,
the bottom left vertical arrow $\text{\rm
Tot}(P_{\mathscr{C}}^{A}(p_{0})\widehat{\otimes}{}_{A}P_{\mathscr{C}}^{A}(M_{1}))$
is a quasi-isomorphism by [BK20] Proposition 3.11 and [BM21] Proposition 1.6
(iv). ∎

######  Lemma 3.8.

Let $M\in\text{\rm Mod}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(X,R))$. If
$R$ is isolated at $0$ and $j$ is a $\zeta$-embedding, then the chain complex

$\displaystyle 0\to I\widehat{\otimes}{}_{\text{\rm C}_{\text{\rm
fin}}(X,R)}M\to M\to\text{\rm C}_{\text{\rm
fin}}(K,R)\widehat{\otimes}{}_{\text{\rm C}_{\text{\rm fin}}(X,R)}M\to 0$

of $\text{\rm Mod}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(X,R))$ is
strictly exact.

###### Proof.

The assertion immediately follows from the strong right exactness of
$(\cdot)\widehat{\otimes}{}_{\text{\rm C}_{\text{\rm fin}}(X,R)}M$ and Lemma
3.5. ∎

######  Lemma 3.9.

Let $M\in\text{\rm Mod}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(X,R))$. If
$R$ is isolated at $0$ and $j$ is a $\zeta$-embedding, then the natural
morphism $\text{\rm C}_{\text{\rm fin}}(K,R)\widehat{\otimes}{}_{\text{\rm
C}_{\text{\rm fin}}(X,R)}^{\mathbb{L}}M\to\text{\rm C}_{\text{\rm
fin}}(K,R)\widehat{\otimes}{}_{\text{\rm C}_{\text{\rm fin}}(X,R)}M$ in
$\text{\rm Der}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(X,R))$ is an
isomorphism.

###### Proof.

We denote by $E\in\text{\rm Ch}^{-}_{\mathscr{C}}(\text{\rm C}_{\text{\rm
fin}}(X,R))$ the chain complex

$\displaystyle\cdots\to 0\to I\to\text{\rm C}_{\text{\rm fin}}(X,R)\to
0\to\cdots$

of $\text{\rm Mod}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(X,R))$, and by
$p$ the morphism $E\to\text{\rm C}_{\text{\rm fin}}(K,R)$ in $\text{\rm
Ch}^{-}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(X,R))$ associated to
$\Gamma_{\mathscr{D}_{2}}(j)$. By Lemma 3.8, $p\widehat{\otimes}{}_{\text{\rm
C}_{\text{\rm fin}}(X,R)}M$ is a quasi-isomorphism. Moreover, $E$ is termwise
strongly flat by Lemma 3.6. Therefore, the assertion follows from Lemma 3.7
applied to $(p,\text{\rm id}_{M})$. ∎

We record the following consequence of Lemma 3.9 for later use.

######  Corollary 3.10.

The object $\text{\rm C}_{\text{\rm fin}}(K,R)$ of $\text{\rm
Mod}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(X,R))$ is flat, in the sense
that the functor $(\cdot)\widehat{\otimes}{}_{\text{\rm C}_{\text{\rm
fin}}(X,R)}\text{\rm C}_{\text{\rm fin}}(K,R)\colon\text{\rm
Mod}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(X,R))\to\text{\rm
Mod}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(K,R))$ is exact.

###### Proof.

Let $E$ be a strictly exact sequence $0\to M\to N\to L\to 0$ of $\text{\rm
Mod}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(X,R))$ regarded as an object
of $\text{\rm Der}^{-}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(X,R))$.
Since $E$ is a zero object of $\text{\rm Der}^{-}_{\mathscr{C}}(\text{\rm
C}_{\text{\rm fin}}(X,R))$ and $(\cdot)\widehat{\otimes}{}_{\text{\rm
C}_{\text{\rm fin}}(X,R)}^{\mathbb{L}}\text{\rm C}_{\text{\rm fin}}(K,R)$ is a
functor, $E\widehat{\otimes}{}_{\text{\rm C}_{\text{\rm
fin}}(X,R)}^{\mathbb{L}}\text{\rm C}_{\text{\rm fin}}(K,R)$ is a zero object
of $\text{\rm Der}^{-}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(K,R))$.
This implies that $E\widehat{\otimes}{}_{\text{\rm C}_{\text{\rm
fin}}(X,R)}\text{\rm C}_{\text{\rm fin}}(K,R)$ is a zero object of $\text{\rm
Der}^{-}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(K,R))$ by Lemma 3.9, and
hence is strictly exact by [BM21] Corollary 1.5 (iii). ∎

Finally, we prove Theorem 3.1 in a way completely parallel to the proof of
[BM21] Lemma 3.9.

###### Proof of Theorem 3.1.

The condition (iii) implies the condition (ii) by Lemma 3.3 (ii). The
condition (ii) implies the condition (i) by Lemma 3.9 applied to $M=\text{\rm
C}_{\text{\rm fin}}(K,R)$ and [BM21] Proposition 1.8. Suppose that the
condition (i) holds and
$R\neq{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$. Then,
$\Gamma_{\mathscr{D}_{2}}(j)$ is an epimorphism in $\text{\rm
Alg}_{\mathscr{C}}(R)$ by [BM21] Proposition 1.7, and hence the map

$\displaystyle\text{\rm Hom}_{\text{\rm Alg}_{\mathscr{C}}(R)}(\text{\rm
C}_{\text{\rm fin}}(K,R),R)$ $\displaystyle\to$ $\displaystyle\text{\rm
Hom}_{\text{\rm Alg}_{\mathscr{C}}(R)}(\text{\rm C}_{\text{\rm fin}}(X,R),R)$
$\displaystyle\phi$ $\displaystyle\mapsto$
$\displaystyle\phi\circ\Gamma_{\mathscr{D}_{2}}(j)$

is injective. This implies the injectivity of $\zeta(j)$ by Proposition 2.8.
Since $\zeta(K)$ and $\zeta(X)$ are compact Hausdorff topological spaces,
$\zeta(j)$ is a homeomorphism onto the image. ∎

##  4 Tate Acyclicity for functions valued in discrete Banach rings

The next step after the algebraic characterisation of closed topological
embeddings is the characterisation of topological covers. We give such a
characterisation in this section by showing that a finite set of closed
topological embeddings between totally disconnected compact Hausdorff
topological spaces is a (set-theoretic) cover if and only if the associated
derived (or equivalently, non-derived) Tate–$\check{\text{C}}$ech complex is
strictly exact. This result, combined with the characterisation of closed
topological embeddings as homotopy epimorphisms given in §3, shows that the
topology of totally disconnected compact Hausdorff topological spaces agree
with the restriction of the formal homotopy Zariski topology on the dual
category of $\text{\rm Alg}_{\mathscr{C}}(R)$ when totally disconnected
compact Hausdorff topological spaces are considered as objects of the dual
category of algebras of continuous functions to $R$ through Gel’fand’s duality
(Theorem 2.12). We refer the reader to [BM21] §1.3 for more details on this.

For an $A\in\text{\rm Alg}_{\mathscr{C}}(R)$, a non-derived (resp. derived)
cover of $A$ in $\mathscr{C}$ is an $S\subset\text{\rm Alg}_{\mathscr{C}}(A)$
satisfying the following:

  * (i)

For any $B\in S$, the structure morphism $A\to B$ is an epimorphism (resp. a
homotopy epimorphism) in $\text{\rm Alg}_{\mathscr{C}}(R)$.

  * (ii)

There exists a finite subset $S_{0}\subset S$ satisfying Tate’s acyclicity
(resp. derived Tate’s acyclicity) (cf. [BK20] Theorem 2.15 and [BM21] pp.
15–16).

Suppose that $R$ is isolated at $0$. We keep the abbreviations
$(\mathscr{C},R)\coloneqq\mathscr{D}_{2}$ and
$(\mathscr{C},X,R)\coloneqq\mathscr{D}_{3}$ introduced before. We denote by
$\Gamma_{\mathscr{D}_{3}}$ the functor $(\text{\rm Top}/X)^{\text{\rm
op}}\to\text{\rm Alg}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(X,R))$
induced by $\Gamma_{\mathscr{D}_{2}}$. For an $S\subset\text{\rm Top}/X$, we
put
$\Gamma^{\mathscr{D}_{3}}_{*}(S)\coloneqq{\mathchoice{\left\\{\Gamma_{\mathscr{D}_{3}}(K,j)\middle|(K,j)\in
S\right\\}}{\\{\Gamma_{\mathscr{D}_{3}}(K,j)\mid(K,j)\in
S\\}}{\\{\Gamma_{\mathscr{D}_{3}}(K,j)\mid(K,j)\in
S\\}}{\\{\Gamma_{\mathscr{D}_{3}}(K,j)\mid(K,j)\in S\\}}}$. A $\zeta$-cover of
$X$ is an $S\subset\text{\rm Top}/X$ satisfying the following:

  * (i)

For any $(K,j)\in S$, $j$ is a $\zeta$-embedding.

  * (ii)

There exists a finite subset $S_{0}\subset S$ such that $\bigcup_{(K,j)\in
S_{0}}\zeta(j)(\zeta(K))=\zeta(X)$.

We give an isolated analogue of [BM21] Theorem 3.1, which is itself an
analogue of Tate’s acyclicity in rigid geometry.

######  Theorem 4.1.

Let $S\subset\text{\rm Top}/X$. If $R$ is isolated at $0$, $X\in\text{\rm
TDCH}$, and $S$ consists of closed immersions, then the following are
equivalent:

  * (i)

The set $S$ is a $\zeta$-cover of $X$ or
$R={\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$.

  * (ii)

The set $S$ admits a finite subset $S_{0}$ with $\bigcup_{(K,j)\in S}j(K)=X$
or $R={\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$.

  * (iii)

The set $\Gamma^{\mathscr{D}_{3}}_{*}(S)$ is a derived cover of $\text{\rm
C}(X,R)$ in $\mathscr{C}$.

  * (iv)

The set $\Gamma^{\mathscr{D}_{3}}_{*}(S)$ is a non-derived cover of $\text{\rm
C}(X,R)$ in $\mathscr{C}$.

In order to show Theorem 4.1, we prepare a couple of lemmata. First, we show
an isolated analogue of [BM21] Lemma 3.2. Recall that for any $X_{0}\subset X$
we put $I_{X,R,X_{0}}\coloneqq{\mathchoice{\left\\{f\in\text{\rm C}_{\text{\rm
fin}}(X,R)\middle|f|_{X_{0}}=0\right\\}}{\\{f\in\text{\rm C}_{\text{\rm
fin}}(X,R)\mid f|_{X_{0}}=0\\}}{\\{f\in\text{\rm C}_{\text{\rm fin}}(X,R)\mid
f|_{X_{0}}=0\\}}{\\{f\in\text{\rm C}_{\text{\rm fin}}(X,R)\mid
f|_{X_{0}}=0\\}}}$.

######  Lemma 4.2.

Let $K_{0}$ and $K_{1}$ be subsets of $X$.

  * (i)

The equalities $I_{X,R,K_{0}}I_{X,R,K_{1}}=I_{X,R,K_{0}\cup
K_{1}}=I_{X,R,K_{0}}\cap I_{X,R,K_{1}}$ hold.

  * (ii)

If $K_{0}$ and $K_{1}$ are closed, $X\in\text{\rm TDCH}$, and $R$ is isolated
at $0$, then the addition $I_{X,R,K_{0}}\oplus I_{X,R,K_{1}}\to
I_{X,R,K_{0}\cap K_{1}}$ is a strict epimorphism in $\text{\rm
Mod}_{\mathscr{C}}(\text{\rm C}(X,R))$.

###### Proof.

The relations $I_{X,R,K_{0}\cup K_{1}}=I_{X,R,K_{0}}\cap I_{X,R,K_{1}}$ and
$I_{X,R,K_{0}}I_{X,R,K_{1}}\subset I_{X,R,K_{0}\cup K_{1}}$ follow easily from
the definitions. We show that for any $f\in I_{X,R,K_{0}\cup K_{1}}$, there
exists an $(f_{0},f_{1})\in I_{X,R,K_{0}}\times I_{X,R,K_{1}}$ such that
$f=f_{0}f_{1}$. By Proposition 2.5 applied to $K_{0}$, there exists a
$U\in\text{\rm CO}(X)$ such that $1_{X,R,U}f=f$ and $K_{0}\cap U=\emptyset$.
Put $f_{0}\coloneqq 1_{X,R,U}$ and $f_{1}\coloneqq f$. Then, $(f_{0},f_{1})$
satisfies the desired property. This implies $I_{X,R,K_{0}\cup K_{1}}\subset
I_{X,R,K_{0}}I_{X,R,K_{1}}$, which proves (i).

In order to prove (ii), we show that for any $f\in I_{X,R,K_{0}\cap K_{1}}$,
there exists an $(f_{0},f_{1})\in I_{X,R,K_{0}}\times I_{X,R,K_{1}}$ such that
$f=f_{0}+f_{1}$ and
${\mathchoice{\left\|f_{0}\right\|}{\|f_{0}\|}{\|f_{0}\|}{\|f_{0}\|}}_{\text{\rm
C}(X,R)}+{\mathchoice{\left\|f_{1}\right\|}{\|f_{1}\|}{\|f_{1}\|}{\|f_{1}\|}}_{\text{\rm
C}(X,R)}\leq 2{\mathchoice{\left\|f\right\|}{\|f\|}{\|f\|}{\|f\|}}_{\text{\rm
C}(X,R)}$ under the assumption that $K_{0}$ and $K_{1}$ are closed,
$X\in\text{\rm TDCH}$, and $R$ is isolated at $0$. Notice that the inclusion
$I_{X,R,K_{0}}+I_{X,R,K_{1}}\subset I_{X,R,K_{0}\cap K_{1}}$ follows
immediately from the definition of the ideals. By Proposition 2.3 (ii), we
have
$f^{-1}({\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}})\in\text{\rm
CO}(X)$. Put $K_{2}\coloneqq K_{1}\setminus
f^{-1}({\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}})$. By
$K_{0}\cap K_{1}\subset
f^{-1}({\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}})$, we have
$K_{0}\cap K_{2}=\emptyset$. By [BM21] Lemma 2.6 applied to $K_{0}$ and
$K_{2}$, there exists a $V\in\text{\rm CO}(X)$ such that $K_{0}\subset V$ and
$K_{2}\subset X\setminus V$. By Proposition 2.4, there exists a pair
$(\mathscr{U},e)$ of a finite subset $\mathscr{U}\subset\text{\rm CO}(X)$ and
a map $e\colon\mathscr{U}\to R$ such that
$f=\sum_{U\in\mathscr{U}}e(U)1_{X,R,U}$. Put
$f_{0}\coloneqq\sum_{U\in\mathscr{U}}e(U)1_{X,R,U\setminus V}$ and
$f_{1}\coloneqq\sum_{U\in\mathscr{U}}e(U)1_{X,R,U\cap V}$. Then,
$(f_{0},f_{1})$ satisfies the desired property because
${\mathchoice{\left\|f\right\|}{\|f\|}{\|f\|}{\|f\|}}_{\text{\rm
C}(X,R)}=\max{\mathchoice{\left\\{{\mathchoice{\left\|f_{0}\right\|}{\|f_{0}\|}{\|f_{0}\|}{\|f_{0}\|}}_{\text{\rm
C}(X,R)},{\mathchoice{\left\|f_{1}\right\|}{\|f_{1}\|}{\|f_{1}\|}{\|f_{1}\|}}_{\text{\rm
C}(X,R)}\right\\}}{\\{{\mathchoice{\left\|f_{0}\right\|}{\|f_{0}\|}{\|f_{0}\|}{\|f_{0}\|}}_{\text{\rm
C}(X,R)},{\mathchoice{\left\|f_{1}\right\|}{\|f_{1}\|}{\|f_{1}\|}{\|f_{1}\|}}_{\text{\rm
C}(X,R)}\\}}{\\{{\mathchoice{\left\|f_{0}\right\|}{\|f_{0}\|}{\|f_{0}\|}{\|f_{0}\|}}_{\text{\rm
C}(X,R)},{\mathchoice{\left\|f_{1}\right\|}{\|f_{1}\|}{\|f_{1}\|}{\|f_{1}\|}}_{\text{\rm
C}(X,R)}\\}}{\\{{\mathchoice{\left\|f_{0}\right\|}{\|f_{0}\|}{\|f_{0}\|}{\|f_{0}\|}}_{\text{\rm
C}(X,R)},{\mathchoice{\left\|f_{1}\right\|}{\|f_{1}\|}{\|f_{1}\|}{\|f_{1}\|}}_{\text{\rm
C}(X,R)}\\}}}$. This implies $I_{X,R,K_{0}\cap K_{1}}\subset
I_{X,R,K_{0}}+I_{X,R,K_{1}}$, and that the addition $I_{X,R,K_{0}}\oplus
I_{X,R,K_{1}}\to I_{X,R,K_{0}\cap K_{1}}$ is a strict epimorphism in
$\text{\rm Mod}_{\mathscr{C}}(\text{\rm C}(X,R))$. ∎

We show an isolated analogue of [BM21] Lemma 3.3.

######  Lemma 4.3.

Let $K_{0}$ and $K_{1}$ be closed subsets of $X$. If $X\in\text{\rm TDCH}$ and
$R$ is isolated at $0$, then the natural morphism $\text{\rm
C}(K_{0},R)\widehat{\otimes}{}_{\text{\rm C}(X,R)}^{\mathbb{L}}\text{\rm
C}(K_{1},R)\to\text{\rm C}(K_{0},R)\widehat{\otimes}{}_{\text{\rm
C}(X,R)}\text{\rm C}(K_{1},R)$ in $\text{\rm Der}^{-}_{\mathscr{C}}(\text{\rm
C}(X,R))$ is an isomorphism, and the natural morphism $\text{\rm
C}(K_{0},R)\widehat{\otimes}{}_{\text{\rm C}(X,R)}\text{\rm
C}(K_{1},R)\to\text{\rm C}(K_{0}\cap K_{1},R)$ in $\text{\rm
Alg}_{\mathscr{C}}(\text{\rm C}(X,R))$ is an isomorphism.

###### Proof.

The first assertion follows from Lemma 3.9, and the second assertion follows
from Proposition 2.2, Proposition 2.3 (i), Lemma 3.3 (ii), and Lemma 4.2 (ii).
∎

Finally, we show Theorem 4.1 in a way completely parallel to the proof of
[BM21] Theorem 3.1.

###### Proof of Theorem 4.1.

The equivalence between the conditions (i) and (ii) immediately follows from
the definition of a $\zeta$-cover. The equivalence between the conditions
(iii) and (iv) follows from Lemma 4.3 and [BM21] Proposition 1.10. It remains
to show that (ii) is equivalent to (iii). Let $S_{0}$ be a finite subset of
$S$. We denote by $T_{S_{0}}=(T_{S_{0},n},d_{n})_{n\in\mathbb{Z}}$ the (non-
derived) Tate–$\check{\text{C}}$ech complex

$\displaystyle 0\to\text{\rm C}(X,R)\to\prod_{(K,j)\in S_{0}}\text{\rm
C}(K,R)\to\prod_{(K_{i},j_{i})_{i=0}^{1}\in[S_{0}]^{2}}\text{\rm
C}(K_{0},R)\widehat{\otimes}{}_{\text{\rm C}(X,R)}\text{\rm
C}(K_{1},R)\to\cdots$

associated to $\Gamma^{\mathscr{D}_{3}}_{*}(S_{0})$. Put
$X_{0}\coloneqq\bigcup_{(K,j)\in S_{0}}j(K)$. By Theorem 3.1, it suffices to
show that $T_{S_{0}}$ is strictly exact in $\text{\rm Mod}_{\mathscr{C}}(R)$
if and only if $X_{0}=X$ or
$R={\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$.

First, suppose $X_{0}=X$ or
$R={\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$. If
$R={\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$, then
$T_{S_{0}}$ is a zero object, and hence is strictly exact. Therefore, we may
assume $X_{0}=X$ and $R\neq\\{0\\}$. We show the strict exactness by induction
on the cardinality $c$ of $S_{0}$. If $c\leq 2$, then the strict exactness
immediately follows from the gluing lemma asserting that the openness of a
subset of a topological space can be tested by the pullback by a finite closed
cover. Suppose $c>2$. Fix a $(K,j)\in S_{0}$. Put $S_{1}\coloneqq
S_{0}\setminus{\mathchoice{\left\\{(K,j)\right\\}}{\\{(K,j)\\}}{\\{(K,j)\\}}{\\{(K,j)\\}}}\subset\text{\rm
Top}/X$, and $\check{K}\coloneqq\bigcup_{(K^{\prime},j^{\prime})\in
S_{1}}j^{\prime}(K^{\prime})\subset X$. We denote by $\check{j}$ the inclusion
$\check{K}\hookrightarrow X$. Replacing $(K,j)$ by a closed immersion with the
same image, we may assume $(\check{K},\check{j})\neq(K,j)$. Put
$S_{2}\coloneqq{\mathchoice{\left\\{(K,j)\right\\}}{\\{(K,j)\\}}{\\{(K,j)\\}}{\\{(K,j)\\}}}\subset\text{\rm
Top}/X$,
$S_{3}\coloneqq{\mathchoice{\left\\{(\check{K},\check{j})\right\\}}{\\{(\check{K},\check{j})\\}}{\\{(\check{K},\check{j})\\}}{\\{(\check{K},\check{j})\\}}}\subset\text{\rm
Top}/X$,
$S_{4}\coloneqq{\mathchoice{\left\\{(K,j),(\check{K},\check{j})\right\\}}{\\{(K,j),(\check{K},\check{j})\\}}{\\{(K,j),(\check{K},\check{j})\\}}{\\{(K,j),(\check{K},\check{j})\\}}}\subset\text{\rm
Top}/X$, $C_{0}\coloneqq T_{S_{1}}\widehat{\otimes}{}_{\text{\rm
C}(X,R)}T_{S_{2}}\in\text{\rm Ch}^{-}(\text{\rm
Ch}^{-}_{\mathscr{C}}(\text{\rm C}(X,R)))$, and $C_{1}\coloneqq
T_{S_{3}}\widehat{\otimes}{}_{\text{\rm C}(X,R)}T_{S_{2}}\in\text{\rm
Ch}^{-}(\text{\rm Ch}^{-}_{\mathscr{C}}(\text{\rm C}(X,R)))$. We denote by $j$
the natural morphism $C_{1}\to C_{0}$ in $\text{\rm Ch}^{-}(\text{\rm
Ch}^{-}_{\mathscr{C}}(\text{\rm C}(X,R)))$. Then, $j$ is a termwise quasi-
isomorphism by the induction hypothesis and $(K,\check{K})\in\text{\rm
TDCH}^{2}$ because of the finiteness of $S_{1}$. Therefore, $j$ induces a
quasi-isomorphism $\text{\rm Tot}(C_{1})\to\text{\rm Tot}(C_{0})$ in
$\text{\rm Ch}^{-}_{\mathscr{C}}(\text{\rm C}(X,R))$ by [BM21] Proposition 1.6
(iv). Since $\text{\rm Tot}(C_{1})$ is naturally identified with $T_{S_{4}}$,
which is strictly exact, again by the induction hypothesis, and $\text{\rm
Tot}(C_{0})$ is naturally identified with $T_{S_{0}}$, $T_{S_{0}}$ is strictly
exact.

Next, suppose that $X_{0}\neq X$ and
$R\neq{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$. Take an
$x_{1}\in X\setminus X_{0}$. By the finiteness of $S_{0}$, $X_{0}$ is closed.
By the regularity of $X$, the characteristic function $\chi\colon
X_{0}\cup{\mathchoice{\left\\{x_{1}\right\\}}{\\{x_{1}\\}}{\\{x_{1}\\}}{\\{x_{1}\\}}}\to
R$ of
${\mathchoice{\left\\{x_{1}\right\\}}{\\{x_{1}\\}}{\\{x_{1}\\}}{\\{x_{1}\\}}}$
is continuous. By Lemma 3.3 (i) applied to
$K=X_{0}\cup{\mathchoice{\left\\{x_{1}\right\\}}{\\{x_{1}\\}}{\\{x_{1}\\}}{\\{x_{1}\\}}}$,
there exists a $\tilde{\chi}{}\in\text{\rm C}(X,R)$ such that
$\tilde{\chi}{}|_{X_{0}\cup{\mathchoice{\left\\{x_{1}\right\\}}{\\{x_{1}\\}}{\\{x_{1}\\}}{\\{x_{1}\\}}}}=\chi$.
We have $\tilde{\chi}{}(x_{1})=\chi(x_{1})=1$, and hence $\tilde{\chi}{}\neq
0$ by $R\neq{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$,
while we have $\tilde{\chi}{}|_{j(K)}=\chi|_{j(K)}=0$ for any $(K,j)\in
S_{0}$. Therefore, the map $\text{\rm C}(X,R)\to\prod_{(K,j)\in
S_{0}}\text{\rm C}(K,R)$ in $T_{S_{0}}$ is not injective. This implies that
$T_{S_{0}}$ is not strictly exact. ∎

We show an isolated analogue of [BM21] Theorem 3.6 (i) in a completely
parallel way.

######  Theorem 4.4.

Let $S\subset\text{\rm Top}/X$. If $R$ is isolated at $0$, then the following
are equivalent:

  * (i)

The set $\Gamma^{\mathscr{D}_{3}}_{*}(S)$ is a derived cover of $\text{\rm
C}_{\text{\rm fin}}(X,R)$ in $\mathscr{C}$.

  * (ii)

The set $\Gamma^{\mathscr{D}_{3}}_{*}(S)$ is a non-derived cover of $\text{\rm
C}_{\text{\rm fin}}(X,R)$ in $\mathscr{C}$.

  * (iii)

The set ${\mathchoice{\left\\{(K,\iota_{\zeta,X}\circ j)\middle|(K,j)\in
S\right\\}}{\\{(K,\iota_{\zeta,X}\circ j)\mid(K,j)\in
S\\}}{\\{(K,\iota_{\zeta,X}\circ j)\mid(K,j)\in
S\\}}{\\{(K,\iota_{\zeta,X}\circ j)\mid(K,j)\in S\\}}}$ is a $\zeta$-cover of
$\zeta(X)$ or
$R={\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$.

  * (iv)

The set ${\mathchoice{\left\\{(\zeta(K),\zeta(j))\middle|(K,j)\in
S\right\\}}{\\{(\zeta(K),\zeta(j))\mid(K,j)\in
S\\}}{\\{(\zeta(K),\zeta(j))\mid(K,j)\in
S\\}}{\\{(\zeta(K),\zeta(j))\mid(K,j)\in S\\}}}$ is a $\zeta$-cover of
$\zeta(X)$ or
$R={\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$.

  * (v)

The set $S$ is a $\zeta$-cover of $X$ or
$R={\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$.

###### Proof.

The assertion immediately follows from Proposition 2.6, Theorem 3.1, and
Theorem 4.1. ∎

As an application of Theorem 4.4, we show an isolated analogue of [BM21]
Corollary 3.6, which is itself an analogue of Kiehl’s Theorem B (cf. [Kie66]
Hilfssatz 1.5) in rigid geometry.

######  Corollary 4.5.

Let $S\subset\text{\rm Top}/X$ be a finite set of $\zeta$-embeddings. If $R$
is isolated at $0$, then the following are equivalent:

  * (i)

Either $\bigcup_{(K,j)\in S}\text{\rm im}(\zeta(j))=\zeta(X)$ or
$R={\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$ holds.

  * (ii)

For any $M\in\text{\rm Mod}_{\mathscr{C}}(\text{\rm C}_{\text{\rm
fin}}(X,R))$, the (non-derived) Tate–$\check{\text{C}}$ech complex

$\displaystyle 0\to M$ $\displaystyle\to$ $\displaystyle\prod_{(K,j)\in
S}\text{\rm C}_{\text{\rm fin}}(K,R)\widehat{\otimes}{}_{\text{\rm
C}_{\text{\rm fin}}(X,R)}M$ $\displaystyle\to$
$\displaystyle\prod_{(K_{i},j_{i})_{i=0}^{1}\in[S]^{2}}\text{\rm C}_{\text{\rm
fin}}(K_{0},R)\widehat{\otimes}{}_{\text{\rm C}_{\text{\rm
fin}}(X,R)}\text{\rm C}_{\text{\rm
fin}}(K_{1},R)\widehat{\otimes}{}_{\text{\rm C}_{\text{\rm
fin}}(X,R)}M\to\cdots$

of $M$ associated to $S$ is strictly exact in $\text{\rm
Mod}_{\mathscr{C}}(R)$.

  * (iii)

For any $M\in\text{\rm Der}^{-}_{\mathscr{C}}(\text{\rm C}_{\text{\rm
fin}}(X,R))$, the total complex of the derived Tate–$\check{\text{C}}$ech
bicomplex

$\displaystyle 0\to M$ $\displaystyle\to$ $\displaystyle\prod_{(K,j)\in
S}\text{\rm C}_{\text{\rm fin}}(K,R)\widehat{\otimes}{}_{\text{\rm
C}_{\text{\rm fin}}(X,R)}^{\mathbb{L}}M$ $\displaystyle\to$
$\displaystyle\prod_{(K_{i},j_{i})_{i=0}^{1}\in[S]^{2}}\text{\rm C}_{\text{\rm
fin}}(K_{0},R)\widehat{\otimes}{}_{\text{\rm C}_{\text{\rm
fin}}(X,R)}^{\mathbb{L}}\text{\rm C}_{\text{\rm
fin}}(K_{1},R)\widehat{\otimes}{}_{\text{\rm C}_{\text{\rm
fin}}(X,R)}^{\mathbb{L}}M\to\cdots$

of $M$ associated to $S$ is strictly exact in $\text{\rm
Mod}_{\mathscr{C}}(R)$.

###### Proof.

We abbreviate $\text{\rm C}_{\text{\rm fin}}(X,R)$ to $A$. By Theorem 4.4 and
[BM21] Lemma 2.20 (i), both of the conditions (ii) and (iii) applied to the
case where $M$ is the regular $A$-module object imply the condition (i). We
show that the condition (i) implies the conditions (ii) and (iii). If
$R={\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$, then
$\text{\rm Mod}_{\mathscr{C}}(R)$ consists of zero objects. Therefore, we may
assume $R\neq{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$. By
Proposition 2.6, we may assume that $X$ is totally disconnected compact
Hausdorff and $S$ consists closed immersions. We denote by
$C=(C_{n})_{n\in\mathbb{Z}}$ the $\check{\text{C}}$ech complex

$\displaystyle 0\to A\to\prod_{(K,j)\in S}\text{\rm C}_{\text{\rm
fin}}(K,R)\to\prod_{(K_{i},j_{i})_{i=0}^{1}\in[S]^{2}}\text{\rm C}_{\text{\rm
fin}}(K_{0}\times_{X}K_{1},R)\to\cdots$

associated to $S$, which is naturally isomorphic in $\text{\rm
Ch}^{-}_{\mathscr{C}}(A)$ to (and hence will be identified with) the (non-
derived) Tate–$\check{\text{C}}$ech complex of $A$ associated to $S$ by Lemma
4.3, and hence is strictly exact by the condition (i) and Theorem 4.4. For any
strongly flat object $M$ of $\text{\rm Mod}_{\mathscr{C}}(A)$, the strict
exactness of $C\widehat{\otimes}{}_{A}M$ follows from the strong flatness of
$(\cdot)\widehat{\otimes}{}_{A}M$. This implies the condition (ii) restricted
to the case where $M$ is strongly flat.

First, let $M\in\text{\rm Der}^{-}_{\mathscr{C}}(A)$. We abbreviate
$(C_{n}\widehat{\otimes}{}_{A}^{\mathbb{L}}M)_{n\in\mathbb{Z}}\in\text{\rm
Ch}^{-}(\text{\rm Der}^{-}_{\mathscr{C}}(A))$, i.e. the derived
Tate–$\check{\text{C}}$ech bicomplex of $M$ associated to $S$, to $C(M)$. The
natural morphism $C(M)\to
C\widehat{\otimes}{}_{A}\mathscr{P}_{\mathscr{C}}^{A}(M)$ in $\text{\rm
Ch}^{-}(\text{\rm Der}^{-}_{\mathscr{C}}(A))$ is represented by a morphism in
$\text{\rm Ch}^{-}(\text{\rm Ch}^{-}_{\mathscr{C}}(A))$ which is a termwise
quasi-isomorphism by the termwise strong flatness of
$\mathscr{P}_{\mathscr{C}}^{A}(M)$ (cf. [BK20] Proposition 3.11), and the zero
morphism $C\widehat{\otimes}{}_{A}\mathscr{P}_{\mathscr{C}}^{A}(M)\to 0$ in
$\text{\rm Ch}^{-}(\text{\rm Ch}^{-}_{\mathscr{C}}(A))$ is a termwise quasi-
isomorphism by the condition (ii) for strongly flat objects. This implies that
the zero morphism $\text{\rm Tot}(C(M))\to 0$ in $\text{\rm
Ch}^{-}_{\mathscr{C}}(A)$ is a quasi-isomorphism by Proposition 1.6 (iv).
Therefore, $\text{\rm Tot}(C(M))$ is strictly exact by [BM21] Proposition 1.6
(iii). This implies the condition (iii).

Next, let $M\in\text{\rm Mod}_{\mathscr{C}}(A)$. We denote by
$C(M)\in\text{\rm Ch}^{-}(\text{\rm Ch}^{-}_{\mathscr{C}}(A))$ the canonical
representative

$\displaystyle 0$ $\displaystyle\to$
$\displaystyle\mathscr{P}_{\mathscr{C}}^{A}(M)\to\prod_{(K,j)\in
S}\mathscr{P}_{\mathscr{C}}^{A}(\text{\rm C}_{\text{\rm
fin}}(K,R))\widehat{\otimes}{}_{A}\mathscr{P}_{\mathscr{C}}^{A}(M)$
$\displaystyle\to$
$\displaystyle\prod_{(K_{i},j_{i})_{i=0}^{1}\in[S]^{2}}\mathscr{P}_{\mathscr{C}}^{A}(\mathscr{P}_{\mathscr{C}}^{A}(\text{\rm
C}_{\text{\rm
fin}}(K_{0},R))\widehat{\otimes}{}_{A}\mathscr{P}_{\mathscr{C}}^{A}(\text{\rm
C}_{\text{\rm
fin}}(K_{1},R)))\widehat{\otimes}{}_{A}\mathscr{P}_{\mathscr{C}}^{A}(M)\to\cdots$

of the derived Tate–$\check{\text{C}}$ech bicomplex of $M$ regarded as an
object of $\text{\rm Ch}^{-}(A)$ concentrated in degree $0$. Then, the natural
morphism $p\colon C(M)\to C\widehat{\otimes}{}_{A}M$ in $\text{\rm
Ch}^{-}(\text{\rm Ch}^{-}_{\mathscr{C}}(A))$ represents a morphism in
$\text{\rm Ch}^{-}(\text{\rm Der}^{-}_{\mathscr{C}}(A))$ which is a termwise
isomorphism by Lemma 3.9 and Lemma 4.3, and hence is a termwise quasi-
isomorphism by [BM21] Corollary 1.5 (i). This implies that $\text{\rm
Tot}(p)\colon\text{\rm Tot}(C(M))\to C\widehat{\otimes}{}_{A}M$ is a quasi-
isomorphism by [BM21] Proposition 1.6 (iv). Since we have shown that the
condition (i) implies the condition (iii), $\text{\rm Tot}(C(M))$ is strictly
exact. Therefore, the (non-derived) Tate–$\check{\text{C}}$ech complex
$C\widehat{\otimes}{}_{A}M$ of $M$ is strictly exact. ∎

##  5 Effective descent for modules over Banach algebras of functions valued
in discrete Banach rings

In this section, we prove effective descent for the categories of Banach
modules over the algebras $\text{\rm C}_{\text{\rm fin}}(X,R)$, and their
derived categories, with respect to the topology determined by finite families
of $\zeta$-embeddings. As we have proved the analogues of the main results of
[BM21] for the algebras $\text{\rm C}_{\text{\rm fin}}(X,R)$, for $R$ isolated
at $0$, the proof of effective descent given here will follow closely that
presented in [BM21] §4.

Suppose that $R$ is isolated at $0$. Although $\mathscr{D}_{3}$ in [BM21] §4
was assumed to be an admissible triple, the formulation of derived and non-
derived effective descent works for $\text{\rm C}_{\text{\rm fin}}(X,R)$ in a
completely parallel way. Since we omit overlapping explanations here, except
for definitions and notations, it is better for the reader to refer to [BM21]
§4 for details.

For a $(Y,\Phi)\in\text{\rm Top}/X$, we consider the monads
$\mathscr{T}_{\text{\rm nd}}(\Phi)=\Phi_{*}\Phi^{*}$ on $\text{\rm
Mod}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(X,R))$ and
$\mathscr{T}_{\text{\rm d}}(\Phi)=\Phi_{*}\mathbb{L}\Phi^{*}$ on $\text{\rm
Der}^{-}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(X,R))$ defined by the
adjunctions

$\displaystyle\Phi^{*}\colon\text{\rm Mod}_{\mathscr{C}}(\text{\rm
C}_{\text{\rm fin}}(X,R))$ $\displaystyle\leftrightarrows$
$\displaystyle\text{\rm Mod}_{\mathscr{C}}(\text{\rm C}_{\text{\rm
fin}}(Y,R))\colon\Phi_{*}$ $\displaystyle\mathbb{L}\Phi^{*}\colon\text{\rm
Der}^{-}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(X,R))$
$\displaystyle\leftrightarrows$ $\displaystyle\text{\rm
Der}^{-}_{\mathscr{C}}(\text{\rm C}_{\text{\rm fin}}(Y,R))\colon\Phi_{*},$

where $\Phi^{*}$ is the extension of scalars functor
$(\cdot)\widehat{\otimes}{}_{\text{\rm C}_{\text{\rm fin}}(X,R)}\text{\rm
C}_{\text{\rm fin}}(Y,R)$, $\mathbb{L}\Phi^{*}$ is the derived extension of
scalars functor $(\cdot)\widehat{\otimes}{}_{\text{\rm C}_{\text{\rm
fin}}(X,R)}^{\mathbb{L}}\text{\rm C}_{\text{\rm fin}}(Y,R)$, and $\Phi_{*}$ is
the restriction of scalars functor.

Let $\mathscr{T}$ denote $\mathscr{T}_{\text{\rm nd}}(\Phi)$ (resp.
$\mathscr{T}_{\text{\rm d}}(\Phi)$), and abbreviate $\text{\rm
Mod}_{\mathscr{C}}$ (resp. $\text{\rm Der}^{-}_{\mathscr{C}}$) to M. There is
an adjunction

$\displaystyle F_{\mathscr{T}}\colon\text{\rm M}(\text{\rm C}_{\text{\rm
fin}}(X,R))$ $\displaystyle\leftrightarrows$ $\displaystyle\text{\rm
Mod}(\mathscr{T})\colon U_{\mathscr{T}},$

where $\text{\rm Mod}(\mathscr{T})$ is the category of modules over the monad
$\mathscr{T}$, $F_{\mathscr{T}}$ is the free module functor, and
$U_{\mathscr{T}}$ is the forgetful functor. We denote by $L_{\mathscr{T}}$ the
comonad $F_{\mathscr{T}}\circ U_{\mathscr{T}}$, and by $\text{\rm
Desc}_{\text{\rm M}(\text{\rm C}_{\text{\rm fin}}(X,R))}(\Phi)$ the category
of left $L_{\mathscr{T}}$-comodule objects of $\text{\rm Mod}(\mathscr{T})$.
An object of $\text{\rm Desc}_{\text{\rm M}(\text{\rm C}_{\text{\rm
fin}}(X,R))}(\Phi)$ can be described as a triple $(M,\rho,\sigma)$ of an
$M\in\text{\rm M}(\text{\rm C}_{\text{\rm fin}}(X,R))$ and morphisms
$\rho\colon\mathscr{T}(M)\to M$ and $\sigma\colon M\to\mathscr{T}(M)$ in
$\text{\rm M}(\text{\rm C}_{\text{\rm fin}}(X,R))$ that give respectively the
left action of $\mathscr{T}$ and the left coaction of $L_{\mathscr{T}}$ and
that satisfy some further compatibility properties. See [Bal12] Remark 1.4 for
details.

We define the comparison functor $Q_{\mathscr{T}}$ as the functor

$\displaystyle\text{\rm M}(\text{\rm C}_{\text{\rm fin}}(X,R))\to\text{\rm
Desc}_{\text{\rm M}(\text{\rm C}_{\text{\rm fin}}(X,R))}(\Phi)$

assigning to each $M\in\text{\rm M}(\text{\rm C}_{\text{\rm fin}}(X,R))$ the
tuple $(\mathscr{T}(M),\epsilon_{M},\mathscr{T}(\eta_{M}))$, where $\epsilon$
is the counit $\mathscr{T}\Rightarrow\text{\rm id}_{\text{\rm M}(\text{\rm
C}_{\text{\rm fin}}(X,R))}$ of the adjunction and $\eta$ is the unit
$\text{\rm id}_{\text{\rm M}(\text{\rm C}_{\text{\rm
fin}}(X,R))}\Rightarrow\mathscr{T}$ of the adjunction. We say that $\Phi$
satisfies non-derived (resp. derived) effective descent if the comparison
functor $Q_{\mathscr{T}_{\text{\rm d}}}$ (resp. $Q_{\mathscr{T}_{\text{\rm
nd}}}$) is an equivalence. We will apply the notion of the descent category in
the following specific case.

Let $S\subset\text{\rm Top}/X$. We put $Y\coloneqq\coprod_{(K,j)\in S}K$, and
denote by $\Phi$ the canonical morphism $Y\to X$ induced by the universal
property of the coproduct of $(j)_{(K,j)\in S}$. We say that $S$ satisfies
non-derived (resp. derived) effective descent if so does $\Phi$. As an
isolated analogue of [BM21] Theorem 4.1, which itself is an analogue of the
equivalence between finite Banach modules over an affinoid algebra and
coherent sheaves over the rigid analytic space associated to it (cf. [Kie66]
Theorem 1.2), we give our main result on derived and non-derived effective
descent in a completely parallel way.

######  Theorem 5.1.

If $S$ consists of $\zeta$-embeddings and $R$ is isolated at $0$, then the
following are equivalent:

  * (i)

The set $S$ is a $\zeta$-cover of $X$ or
$R={\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$.

  * (ii)

The functor $\mathbb{L}\Phi^{*}$ is faithful.

  * (iii)

The functor $\mathbb{L}\Phi^{*}$ is conservative.

  * (iv)

The functor $\Phi^{*}$ is faithful.

  * (v)

The functor $\Phi^{*}$ is conservative.

  * (vi)

The set $S$ satisfies derived effective descent.

  * (vii)

The set $S$ satisfies non-derived effective descent.

###### Proof.

The equivalence between the condition (ii) and the condition (vi), the
implication from the condition (ii) to the condition (iii), and the
implication from the condition (vii) to the condition (iv) follow from
arguments completely parallel to those in the first paragraph of the proof of
[BM21] Theorem 4.1.

The implication from the condition (ii) (resp. (iii)) to the condition (iv)
(resp. (v)) follows from an argument completely parallel to that in the second
paragraph of the proof of [BM21] Theorem 4.1 except that we replace the fact
that the natural transform $j_{S}$ is a natural isomorphism based on [BM21]
Lemma 2.26 by the direct application of Lemma 3.9.

The implication from the condition (iv) (resp. (v)) to the condition (i)
follows from an argument completely parallel to that in the third paragraph of
the proof of [BM21] Theorem 4.1 except that we replace [BM21] Lemma 3.3 by its
isolated analogue Lemma 4.3, because a morphism $\text{\rm
C}({\mathchoice{\left\\{x\right\\}}{\\{x\\}}{\\{x\\}}{\\{x\\}}},R)\to\text{\rm
C}({\mathchoice{\left\\{x\right\\}}{\\{x\\}}{\\{x\\}}{\\{x\\}}},R)$ in
$\text{\rm Mod}_{\mathscr{C}}(R)$ for an $x\in X$ cannot be simultaneously the
zero morphism and the identity morphism unless
$R={\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$.

The implications from the condition (i) to the conditions (ii) and (vii)
follow from arguments completely parallel to those in the fourth and rest
paragraphs of the proof of [BM21] Theorem 4.1 except that we replace [BM21]
Corollary 3.4 by its isolated analogue Corollary 4.5. ∎

##  6 Homological algebra for discrete Banach modules

Up to this point, we have proved analogues of results in [Mih14] and [BM21].
From this section on, we provide new results and new properties of the
algebras of continuous functions valued on a Banach ring $R$ isolated at $0$.
We define the category of Banach $R$-modules isolated at $0$. We prove in
Proposition 6.1 that this category is quasi-Abelian, and in Proposition 6.4
that it is a closed symmetric monoidal subcategory of the category of all
Banach $R$-modules. We also describe its bounded above derived category as a
full monoidal subcategory of the category $\text{\rm
Der}^{-}_{\mathscr{C}}(R)$ and determine its generators in Propositions 6.7.
Then, in Theorem 6.11, we prove that the canonical morphism $\text{\rm
C}_{\text{\rm fin}}(X,M_{1})\widehat{\otimes}{}_{R}M_{2}\to\text{\rm
C}_{\text{\rm fin}}(X,M_{1}\widehat{\otimes}{}_{R}M_{2})$ is always an
isomorphism when the non-Archimedean projective tensor product is considered
(and hence all modules are non-Archimedean), and we give counterexamples for
the Archimedean tensor product.

We denote by $\text{\rm Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)\subset\text{\rm
Mod}_{\mathscr{C}}(R)$ and $\text{\rm Alg}^{\text{\rm
iso}}_{\mathscr{C}}(R)\subset\text{\rm Alg}_{\mathscr{C}}(R)$ the full
subcategories of objects isolated at $0$. We notice that Proposition 2.2
implies that these inclusions of categories are also replete, i. e. they are
closed by isomorphisms of objects. By Lemma 2.3 (i), the correspondence
$(X,M)\mapsto\text{\rm C}_{\text{\rm fin}}(X,M)$ gives functors $\text{\rm
C}_{\text{\rm fin}}^{\text{\rm iso}}\colon\text{\rm Top}\times\text{\rm
Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)\to\text{\rm Mod}^{\text{\rm
iso}}_{\mathscr{C}}(R)$ and $\text{\rm C}_{\text{\rm fin}}^{\text{\rm
iso}}\colon\text{\rm Top}\times\text{\rm Alg}^{\text{\rm
iso}}_{\mathscr{C}}(R)\to\text{\rm Alg}^{\text{\rm iso}}_{\mathscr{C}}(R)$. We
denote by $F^{\text{\rm iso}}$ the inclusion functor $\text{\rm
Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)\hookrightarrow\text{\rm
Mod}_{\mathscr{C}}(R)$. First, we study several operations on $\text{\rm
Mod}_{\mathscr{C}}(R)$ restricted to $\text{\rm Mod}^{\text{\rm
iso}}_{\mathscr{C}}(R)$.

######  Proposition 6.1.

  * (i)

Let $M\in\text{\rm Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$. For any
$R$-submodule $N\subset M$, $N$ (resp. $M/N$) forms an object of $\text{\rm
Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$ with respect to the restriction of
${\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{M}$
(resp. the quotient seminorm).

  * (ii)

The functor $F^{\text{\rm iso}}$ preserves and reflects kernels and cokernels.

  * (iii)

The category $\text{\rm Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$ is closed by
finite direct sums in $\text{\rm Mod}_{\mathscr{C}}(R)$.

  * (iv)

The category $\text{\rm Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$ is quasi-
Abelian.

We recall that a functor $G\colon\mathscr{A}\to\mathscr{B}$ between additive
categories is said to _reflect kernels_ (resp.  _reflect cokernels_) if given
a morphism $f$ in $\mathscr{A}$ such that $G(f)$ is a strict monomorphism
(resp. strict epimorphism) then $f$ is a strict monomorphism (resp. strict
epimorphism).

###### Proof.

The assertion (i) follows from Proposition 2.2, the assertion (ii) follows
from the assertion (i). Let us show the assertion (iii). Let $M,N\in\text{\rm
Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$, then $M\oplus N\cong(M\times
N,{\mathchoice{\left|\cdot\right|}{|\cdot|}{|\cdot|}{|\cdot|}}_{\max{}})$,
where $|(m,n)|_{\max{}}=\max\\{|m|,|n|\\}$. Therefore, $M\oplus N\in\text{\rm
Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$. Then, the assertion (iv) follows from
the assertions (ii) and (iii) because $\text{\rm Mod}_{\mathscr{C}}(R)$ is
quasi-Abelian. ∎

If $R$ is isolated at $0$, then the category $\text{\rm Mod}^{\text{\rm
iso}}_{\mathscr{C}}(R)$ is obviously non-trivial. We provide some basic
examples of objects of $\text{\rm Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$ in
order to show that this category can be non-trivial also when $R$ is not
isolated at $0$.

######  Example 6.2.

  * (i)

The fundamental examples of Banach rings isolated at zero are
$\mathbb{Z}_{\infty}$ and $\mathbb{Z}_{\text{\rm triv}}$, having rich
categories of modules isolated at zero.

  * (ii)

If $R$ is a complete (Archimedean or non-Archimedean) non-trivially valued
field, then the category $\text{\rm Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$
consists of only zero objects.

  * (iii)

If $R$ is $\mathbb{Z}_{p}$ equipped with the $p$-adic absolute value, then
$\mathbb{F}_{p}$ equipped with the trivial norm is an object of $\text{\rm
Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$, via the identification
$\mathbb{F}_{p}\cong\mathbb{Z}_{p}/(p)$. Therefore, $\text{\rm Mod}^{\text{\rm
iso}}_{\mathscr{C}}(R)$ contains the category of $\mathbb{F}_{p}$-vector
spaces equipped with the trivial norm.

  * (iv)

If $k$ is an arbitrary field and $k[\\![T]\\!]$ is the ring of formal power-
series over $k$ equipped with the $T$-adic norm, then $k$ equipped with the
trivial norm is a Banach module over $k[\\![T]\\!]$ isolated at $0$ via the
identification $k\cong k[\\![T]\\!]/(T)$.

Suppose that $R$ is isolated at $0$. Then, for any $M\in\text{\rm
Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$, $\text{\rm C}_{\text{\rm
fin}}(X,M)\in\text{\rm Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$ forms an object
of $\text{\rm Mod}^{\text{\rm iso}}_{\mathscr{C}}(\text{\rm C}_{\text{\rm
fin}}(X,R))$ with respect to the pointwise scalar multiplication. The
correspondence $M\mapsto\text{\rm C}_{\text{\rm fin}}(X,M)$ also gives
functors $\text{\rm C}_{\text{\rm fin}}^{\text{\rm
iso}}(X,\cdot)\colon\text{\rm Mod}^{\text{\rm iso}}(R)\to\text{\rm
Mod}^{\text{\rm iso}}(\text{\rm C}_{\text{\rm fin}}(X,R))$ and $\text{\rm
C}_{\text{\rm fin}}^{\text{\rm iso}}(X,\cdot)\colon\text{\rm Alg}^{\text{\rm
iso}}(R)\to\text{\rm Alg}^{\text{\rm iso}}(\text{\rm C}_{\text{\rm
fin}}(X,R))$. We recall that for any $M\in\text{\rm Mod}_{\mathscr{C}}(R)$,
$\text{\rm C}_{\text{\rm fin}}(X,M)$ forms a $\text{\rm C}_{\text{\rm
fin}}(X,R)$-module even when $R$ or $M$ is not isolated at $0$, and hence we
use the algebraic module structure without such an assumption, as we did in
§2.

By Proposition 6.1 (ii), $F^{\text{\rm iso}}$ preserves and reflects the
strict exactness of a null sequence, and hence the class of strict exact
sequences of $\text{\rm Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$ agrees with
the restriction to $\text{\rm Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$ of the
class of strict exact sequences of $\text{\rm Mod}_{\mathscr{C}}(R)$.
Nevertheless, it is meaningful to consider $\text{\rm Mod}^{\text{\rm
iso}}_{\mathscr{C}}(R)$, because, for example, the following proposition does
not make sense for $\text{\rm Mod}_{\mathscr{C}}(R)$.

######  Proposition 6.3.

If $R$ is isolated at $0$, then $\text{\rm C}_{\text{\rm fin}}^{\text{\rm
iso}}(X,\cdot)\colon\text{\rm Mod}^{\text{\rm iso}}(R)\to\text{\rm
Mod}^{\text{\rm iso}}(\text{\rm C}_{\text{\rm fin}}(X,R))$ is a strongly exact
functor.

###### Proof.

Let $E$ be a null sequence $M_{0}\stackrel{{\scriptstyle
s}}{{\to}}M_{1}\stackrel{{\scriptstyle t}}{{\to}}M_{2}$ of $\text{\rm
Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$. It suffices to show that the
following two claims:

  * (i)

If $s$ is a strict monomorphism and $E$ is exact, then $\text{\rm
C}_{\text{\rm fin}}^{\text{\rm iso}}(X,s)$ is a strict monomorphism and
$\text{\rm C}_{\text{\rm fin}}^{\text{\rm iso}}(X,E)$ is exact.

  * (ii)

If $t$ is a strict epimorphism and $E$ is exact, then $\text{\rm C}_{\text{\rm
fin}}^{\text{\rm iso}}(X,t)$ is a strict epimorphism and $\text{\rm
C}_{\text{\rm fin}}^{\text{\rm iso}}(X,E)$ is exact.

The claim (i) follows from the fact that $\ker(\text{\rm C}_{\text{\rm
fin}}^{\text{\rm iso}}(X,t))$ is given as

$\displaystyle{\mathchoice{\left\\{f\in\text{\rm C}_{\text{\rm
fin}}(X,M_{1})\middle|t\circ f=0\right\\}}{\\{f\in\text{\rm C}_{\text{\rm
fin}}(X,M_{1})\mid t\circ f=0\\}}{\\{f\in\text{\rm C}_{\text{\rm
fin}}(X,M_{1})\mid t\circ f=0\\}}{\\{f\in\text{\rm C}_{\text{\rm
fin}}(X,M_{1})\mid t\circ f=0\\}}}={\mathchoice{\left\\{f\in\text{\rm
C}_{\text{\rm fin}}(X,M_{1})\middle|\forall x\in
X,f(x)\in\ker(t)\right\\}}{\\{f\in\text{\rm C}_{\text{\rm
fin}}(X,M_{1})\mid\forall x\in X,f(x)\in\ker(t)\\}}{\\{f\in\text{\rm
C}_{\text{\rm fin}}(X,M_{1})\mid\forall x\in
X,f(x)\in\ker(t)\\}}{\\{f\in\text{\rm C}_{\text{\rm fin}}(X,M_{1})\mid\forall
x\in X,f(x)\in\ker(t)\\}}}$

equipped with the restriction of
${\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{\text{\rm
C}_{\text{\rm fin}}(X,M_{1})}$. We show the claim (ii). Since $t$ is a strict
epimorphism, there exists a $C_{1}\in(0,\infty)$ such that for any $m\in
M_{2}$, there exists an $\tilde{m}{}\in M_{1}$ such that $t(\tilde{m}{})=m$
and
${\mathchoice{\left\|\tilde{m}{}\right\|}{\|\tilde{m}{}\|}{\|\tilde{m}{}\|}{\|\tilde{m}{}\|}}_{M_{1}}\leq
C_{1}{\mathchoice{\left\|m\right\|}{\|m\|}{\|m\|}{\|m\|}}_{M_{2}}$.

Let $f\in\text{\rm C}_{\text{\rm fin}}(X,M_{2})$. By Proposition 2.4 (ii),
there exists a pair $(\mathscr{U},e)$ of a finite subset
$\mathscr{U}\subset\text{\rm
CO}(X)\setminus{\mathchoice{\left\\{\emptyset\right\\}}{\\{\emptyset\\}}{\\{\emptyset\\}}{\\{\emptyset\\}}}$
and a map $e\colon\mathscr{U}\to M_{2}$ such that $\mathscr{U}$ is a disjoint
cover of $X$ and $\sum_{U\in\mathscr{U}}1_{X,R,U}e(U)=f$. Then, we have
${\mathchoice{\left\|f\right\|}{\|f\|}{\|f\|}{\|f\|}}_{\text{\rm C}_{\text{\rm
fin}}(X,M_{2})}=\sup_{U\in\mathscr{U}}{\mathchoice{\left\|e(U)\right\|}{\|e(U)\|}{\|e(U)\|}{\|e(U)\|}}_{M_{2}}$.
By the choice of $C_{1}$, there exists a map $\tilde{e}{}\colon\mathscr{U}\to
M_{1}$ such that for any $U\in\mathscr{U}$, $t(\tilde{e}{}(U))=e(U)$ and
${\mathchoice{\left\|\tilde{e}{}(U)\right\|}{\|\tilde{e}{}(U)\|}{\|\tilde{e}{}(U)\|}{\|\tilde{e}{}(U)\|}}_{M_{1}}\leq
C_{1}{\mathchoice{\left\|e(U)\right\|}{\|e(U)\|}{\|e(U)\|}{\|e(U)\|}}_{M_{2}}$.
Put
$\tilde{f}{}\coloneqq\sum_{U\in\mathscr{U}}1_{X,R,U}\tilde{e}{}(U)\in\text{\rm
C}_{\text{\rm fin}}(X,M_{1})$. Then, we have $\text{\rm C}_{\text{\rm
fin}}^{\text{\rm iso}}(X,t)(\tilde{f}{})=t\circ\tilde{f}{}=f$ and
${\mathchoice{\left\|\tilde{f}{}\right\|}{\|\tilde{f}{}\|}{\|\tilde{f}{}\|}{\|\tilde{f}{}\|}}_{\text{\rm
C}_{\text{\rm
fin}}(X,M_{1})}=\sup_{U\in\mathscr{U}}{\mathchoice{\left\|\tilde{e}{}(U)\right\|}{\|\tilde{e}{}(U)\|}{\|\tilde{e}{}(U)\|}{\|\tilde{e}{}(U)\|}}_{M_{1}}\leq\sup_{U\in\mathscr{U}}C_{1}{\mathchoice{\left\|e(U)\right\|}{\|e(U)\|}{\|e(U)\|}{\|e(U)\|}}_{M_{2}}=C_{1}{\mathchoice{\left\|f\right\|}{\|f\|}{\|f\|}{\|f\|}}_{\text{\rm
C}_{\text{\rm fin}}(X,M_{2})}$. This implies that $\text{\rm C}_{\text{\rm
fin}}^{\text{\rm iso}}(X,t)$ is a strict epimorphism.

Let $f\in\ker(\text{\rm C}_{\text{\rm fin}}^{\text{\rm iso}}(X,t))$. By
$\text{\rm im}(s)=\ker(t)$, there exists a $C_{0}\in(0,\infty)$ such that for
any $(m,\epsilon)\in\ker(t)\times(0,\infty)$, there exists an $\tilde{m}{}\in
M_{0}$ such that
${\mathchoice{\left\|s(\tilde{m}{})-m\right\|}{\|s(\tilde{m}{})-m\|}{\|s(\tilde{m}{})-m\|}{\|s(\tilde{m}{})-m\|}}_{M_{1}}<\epsilon$
and
${\mathchoice{\left\|\tilde{m}{}\right\|}{\|\tilde{m}{}\|}{\|\tilde{m}{}\|}{\|\tilde{m}{}\|}}_{M_{0}}\leq
C_{0}{\mathchoice{\left\|m\right\|}{\|m\|}{\|m\|}{\|m\|}}_{M_{1}}$. By
Proposition 2.4 (ii), there exists a pair $(\mathscr{U},e)$ of a finite subset
$\mathscr{U}\subset\text{\rm
CO}(X)\setminus{\mathchoice{\left\\{\emptyset\right\\}}{\\{\emptyset\\}}{\\{\emptyset\\}}{\\{\emptyset\\}}}$
and a map $e\colon\mathscr{U}\to M_{1}$ such that $\mathscr{U}$ is a disjoint
cover of $X$ and $\sum_{U\in\mathscr{U}}1_{X,R,U}e(U)=f$. By the choice of
$C_{0}$, there exists a map $\tilde{e}{}\colon\mathscr{U}\to M_{0}$ such that
for any $U\in\mathscr{U}$,
${\mathchoice{\left\|s(\tilde{e}{}(U))-e(U)\right\|}{\|s(\tilde{e}{}(U))-e(U)\|}{\|s(\tilde{e}{}(U))-e(U)\|}{\|s(\tilde{e}{}(U))-e(U)\|}}_{M_{0}}<\epsilon$
and
${\mathchoice{\left\|\tilde{e}{}(U)\right\|}{\|\tilde{e}{}(U)\|}{\|\tilde{e}{}(U)\|}{\|\tilde{e}{}(U)\|}}_{M_{0}}\leq
C_{0}{\mathchoice{\left\|e(U)\right\|}{\|e(U)\|}{\|e(U)\|}{\|e(U)\|}}_{M_{1}}$.
Put
$\tilde{f}{}\coloneqq\sum_{U\in\mathscr{U}}1_{X,R,U}\tilde{e}{}(U)\in\text{\rm
C}_{\text{\rm fin}}(X,M_{0})$. Then, we have $\text{\rm C}_{\text{\rm
fin}}^{\text{\rm iso}}(X,s)(\tilde{f}{})=s\circ\tilde{f}{}=f$ and
${\mathchoice{\left\|\tilde{f}{}\right\|}{\|\tilde{f}{}\|}{\|\tilde{f}{}\|}{\|\tilde{f}{}\|}}_{\text{\rm
C}_{\text{\rm
fin}}(X,M_{0})}=\sup_{U\in\mathscr{U}}{\mathchoice{\left\|\tilde{e}{}(U)\right\|}{\|\tilde{e}{}(U)\|}{\|\tilde{e}{}(U)\|}{\|\tilde{e}{}(U)\|}}_{M_{0}}\leq\sup_{U\in\mathscr{U}}C_{0}{\mathchoice{\left\|e(U)\right\|}{\|e(U)\|}{\|e(U)\|}{\|e(U)\|}}_{M_{2}}=C_{0}{\mathchoice{\left\|f\right\|}{\|f\|}{\|f\|}{\|f\|}}_{\text{\rm
C}_{\text{\rm fin}}(X,M_{1})}$. This implies that $\text{\rm C}_{\text{\rm
fin}}^{\text{\rm iso}}(X,E)$ is exact. ∎

Let $R_{0}$ be a ring. We denote by $\text{\rm Mod}(R_{0})$ the category of
$R_{0}$-modules and $R_{0}$-linear homomorphisms. When $R_{0}$ is the
underlying ring of $R$, then we put $\text{\rm Mod}_{\text{\rm
Ab}}(R)\coloneqq\text{\rm Mod}(R_{0})$. We denote by $F^{\text{\rm Ab}}$ the
forgetful functor $\text{\rm Mod}^{\text{\rm
iso}}_{\mathscr{C}}(R)\to\text{\rm Mod}_{\text{\rm Ab}}(R)$.

######  Proposition 6.4.

  * (i)

For any $(M_{0},M_{1})\in\text{\rm Mod}^{\text{\rm
iso}}_{\mathscr{C}}(R)^{2}$, the natural $R$-linear homomorphism
$\varphi\colon M_{0}\otimes_{R}M_{1}\to M_{0}\widehat{\otimes}{}_{R}M_{1}$ is
bijective, and $M_{0}\widehat{\otimes}{}_{R}M_{1}$ is isolated at $0$.

  * (ii)

The category $\text{\rm Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$ is a symmetric
monoidal subcategory of $\text{\rm Mod}_{\mathscr{C}}(R)$.

  * (iii)

The functor $F^{\text{\rm Ab}}$ is a symmetric monoidal functor.

###### Proof.

The assertions (ii) and (iii) immediately follow from the assertion (i). We
show the assertion (i). Since $M_{0}$ and $M_{1}$ are isolated at $0$, there
exists an $\epsilon\in(0,\infty)$ such that for any
$i\in{\mathchoice{\left\\{0,1\right\\}}{\\{0,1\\}}{\\{0,1\\}}{\\{0,1\\}}}$,
the equality ${\mathchoice{\left\\{m\in
M_{i}\middle|{\mathchoice{\left\|m\right\|}{\|m\|}{\|m\|}{\|m\|}}_{M}<\epsilon\right\\}}{\\{m\in
M_{i}\mid{\mathchoice{\left\|m\right\|}{\|m\|}{\|m\|}{\|m\|}}_{M}<\epsilon\\}}{\\{m\in
M_{i}\mid{\mathchoice{\left\|m\right\|}{\|m\|}{\|m\|}{\|m\|}}_{M}<\epsilon\\}}{\\{m\in
M_{i}\mid{\mathchoice{\left\|m\right\|}{\|m\|}{\|m\|}{\|m\|}}_{M}<\epsilon\\}}}={\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$
holds. Then, for any $m\in M_{0}\otimes_{R}M_{1}$, if the tensor seminorm of
$m$ is smaller than $\epsilon^{2}$, then $m=0$. Therefore, the $R$-linear
homomorphism $\varphi$ is injective, and the equality
${\mathchoice{\left\\{m\in
M_{0}\widehat{\otimes}{}_{R}M_{1}\middle|{\mathchoice{\left\|m\right\|}{\|m\|}{\|m\|}{\|m\|}}_{M_{0}\widehat{\otimes}{}_{R}M_{1}}<\epsilon^{2}\right\\}}{\\{m\in
M_{0}\widehat{\otimes}{}_{R}M_{1}\mid{\mathchoice{\left\|m\right\|}{\|m\|}{\|m\|}{\|m\|}}_{M_{0}\widehat{\otimes}{}_{R}M_{1}}<\epsilon^{2}\\}}{\\{m\in
M_{0}\widehat{\otimes}{}_{R}M_{1}\mid{\mathchoice{\left\|m\right\|}{\|m\|}{\|m\|}{\|m\|}}_{M_{0}\widehat{\otimes}{}_{R}M_{1}}<\epsilon^{2}\\}}{\\{m\in
M_{0}\widehat{\otimes}{}_{R}M_{1}\mid{\mathchoice{\left\|m\right\|}{\|m\|}{\|m\|}{\|m\|}}_{M_{0}\widehat{\otimes}{}_{R}M_{1}}<\epsilon^{2}\\}}}={\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$
holds. By Proposition 2.2, $\varphi$ is surjective. ∎

We compare the quotient by a closed ideal and the base change by the quotient
ring for a Banach module isolated at $0$.

######  Proposition 6.5.

Let $M\in\text{\rm Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$. For any closed
ideal $I\subset R$, $M\widehat{\otimes}{}_{R}R/I$ is isolated at $0$, and the
natural morphism $R/I\widehat{\otimes}{}_{R}M\to M/IM$ in $\text{\rm
Mod}_{\mathscr{C}}(R)$ is an isomorphism.

###### Proof.

By Proposition 6.1 (i), $M/IM$ forms an object of $\text{\rm Mod}^{\text{\rm
iso}}_{\mathscr{C}}(R)$, and hence the first claim follows from the second
claim by Proposition 2.2. We denote by $i$ the inclusion $I\hookrightarrow R$,
and by $E$ the strictly coexact sequence $I\stackrel{{\scriptstyle
i}}{{\to}}R\to R/I\to 0$ of $\text{\rm Mod}_{\mathscr{C}}(R)$. By the strong
right exactness of $(\cdot)\widehat{\otimes}{}_{R}M$,
$E\widehat{\otimes}{}_{R}M$ is strictly coexact. By Proposition 2.2, the image
of $i\widehat{\otimes}{}_{R}M$ in $\text{\rm Mod}_{\mathscr{C}}(R)$ coincides
with $IM$. Therefore, the strict coexactness of $E\widehat{\otimes}{}_{R}M$
implies that the morphism in the assertion is an isomorphism. ∎

For an $M\in\text{\rm Mod}_{\mathscr{C}}(R)$, we say that
${\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{M}$ is
submultiplicative over $R$ if the inequality
${\mathchoice{\left\|am\right\|}{\|am\|}{\|am\|}{\|am\|}}_{M}\leq{\mathchoice{\left\|a\right\|}{\|a\|}{\|a\|}{\|a\|}}_{R}{\mathchoice{\left\|m\right\|}{\|m\|}{\|m\|}{\|m\|}}_{M}$
holds for any $(a,m)\in A\times M$. We recall that
${\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{R}$ is
equivalent to the submultiplicative norm over $R$ assigning $\sup_{b\in
R\setminus{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}}\frac{{\mathchoice{\left\|ab\right\|}{\|ab\|}{\|ab\|}{\|ab\|}}_{R}}{{\mathchoice{\left\|b\right\|}{\|b\|}{\|b\|}{\|b\|}}_{R}}$
to each $a\in R$ by [BGR84] Proposition 1.2.1/2, and if
${\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{R}$
itself is submultiplicative over $R$, then for any $M\in\text{\rm
Mod}_{\mathscr{C}}(R)$,
${\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{M}$ is
equivalent to the submultiplicative norm over $R$ assigning $\sup_{a\in
R\setminus{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}}\frac{{\mathchoice{\left\|am\right\|}{\|am\|}{\|am\|}{\|am\|}}_{M}}{{\mathchoice{\left\|a\right\|}{\|a\|}{\|a\|}{\|a\|}}_{R}}$
to each $m\in M$.

A family $(M_{i})_{i\in I}$ of objects in a category is said to be essentially
finite if $M_{i}$ is a zero object for all but finitely many $i\in I$. For any
family of objects $(M_{i})_{i\in I}$ in $\text{\rm Mod}_{\mathscr{C}}(R)$ with
a uniform bound of the operator norm of the scalar multiplication, we defined
in [BM21] p. 10 the it contracting direct sum $\bigoplus_{i\in
I}^{\mathscr{C}_{\leq 1}}M_{i}$ and the it contracting direct product
$\prod_{i\in I}^{\mathscr{C}_{\leq 1}}M_{i}$. In particular, they are defined
for any family of objects in $\text{\rm Mod}_{\mathscr{C}}(R)$ with
submultiplicative norms over $R$. These constructions are not functorial for
bounded homomorphisms but for contracting homomorphisms. The restrictions of
these two assignments define functors from the categories of essentially
finite discrete diagrams of objects of $\text{\rm Mod}_{\mathscr{C}}(R)$ and
they define naturally isomorphic direct sum functors. Therefore, we will drop
the superscript $\mathscr{C}_{\leq 1}$ and denote the direct sum
$\bigoplus_{i\in I}^{\mathscr{C}_{\leq 1}}M_{i}$ as $\bigoplus_{i\in I}M_{i}$
for essentially finite families $(M_{i})_{i\in I}$ of objects of $\text{\rm
Mod}_{\mathscr{C}}(R)$. Notice that all additive functors preserve essentially
finite direct sums.

We recall that a normed set is a set $S$ equipped with a map
${\mathchoice{\left|\cdot\right|}{|\cdot|}{|\cdot|}{|\cdot|}}_{S}\colon
S\to[0,\infty)$, and NSet denotes the category of normed sets and bounded
maps. See [BBK19] §1 for more details about this category. For an
$S\in\text{\rm NSet}$, we put $S_{+}\coloneqq{\mathchoice{\left\\{s\in
S\middle|{\mathchoice{\left\|s\right\|}{\|s\|}{\|s\|}{\|s\|}}_{S}>0\right\\}}{\\{s\in
S\mid{\mathchoice{\left\|s\right\|}{\|s\|}{\|s\|}{\|s\|}}_{S}>0\\}}{\\{s\in
S\mid{\mathchoice{\left\|s\right\|}{\|s\|}{\|s\|}{\|s\|}}_{S}>0\\}}{\\{s\in
S\mid{\mathchoice{\left\|s\right\|}{\|s\|}{\|s\|}{\|s\|}}_{S}>0\\}}}$. For
example, every subset $S$ of an $M\in\text{\rm Mod}_{\mathscr{C}}(R)$ is a
normed set with respect to the norm and satisfies
$S_{+}=S\setminus{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$.
An $S\in\text{\rm NSet}$ is said to be isolated at $0$ if there exists an
$\epsilon\in(0,\infty)$ such that ${\mathchoice{\left\\{s\in
S\middle|{\mathchoice{\left\|s\right\|}{\|s\|}{\|s\|}{\|s\|}}_{S}<\epsilon\right\\}}{\\{s\in
S\mid{\mathchoice{\left\|s\right\|}{\|s\|}{\|s\|}{\|s\|}}_{S}<\epsilon\\}}{\\{s\in
S\mid{\mathchoice{\left\|s\right\|}{\|s\|}{\|s\|}{\|s\|}}_{S}<\epsilon\\}}{\\{s\in
S\mid{\mathchoice{\left\|s\right\|}{\|s\|}{\|s\|}{\|s\|}}_{S}<\epsilon\\}}}=S\setminus
S_{+}$.

######  Example 6.6.

  * (i)

Every subset of an object of $\text{\rm Mod}^{\text{\rm
iso}}_{\mathscr{C}}(R)$ is a normed set isolated at $0$.

  * (ii)

Every set $S$ forms a normed set isolated at $0$ with respect to the constant
map $1_{S}\colon S\to[0,\infty)$ with value $1$.

For an $S\in\text{\rm NSet}$, we denote by
$\ell^{\mathscr{C}}(S,R)\in\text{\rm Mod}_{\mathscr{C}}(R)$ the projective
object introduced in [BM21] p. 12 (cf. [BB16] Lemma 3.27 and [BK17] Lemma
A.39), i.e. the completion of $R^{\oplus S}$ with respect to the seminorm
assigning to each $(r_{s})_{s\in S}\in R^{\oplus S}$ the value $\sum_{s\in
S}{\mathchoice{\left\|r_{s}\right\|}{\|r_{s}\|}{\|r_{s}\|}{\|r_{s}\|}}_{R}{\mathchoice{\left|s\right|}{|s|}{|s|}{|s|}}_{S}\in[0,\infty)$
when $\mathscr{C}=\text{\rm Ab}^{\text{\rm A}}$ and $\max_{s\in