# Descent for solid quasi-coherent sheaves on perfectoid spaces

Johannes Anschütz  Lucas Mann

###### Abstract

We prove $v$-descent for solid quasi-coherent sheaves on perfectoid spaces as
a key technical input for the development of a $6$-functor formalism with
values in solid quasi-coherent sheaves on relative Fargues–Fontaine curves.

###### Contents

  1. 1 Introduction
    1. 1.1 Acknowledgements
    2. 1.2 Notations and conventions
  2. 2 $\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(A)$ for adic rings
    1. 2.1 Adic rings
    2. 2.2 Definition of $\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(A)$
    3. 2.3 Complete descent
    4. 2.4 Globalization for stably uniform adic spaces
    5. 2.5 Universal descent
  3. 3 $\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}^{a}(\mathcal{O}_{X}^{+})$ for perfectoid spaces
    1. 3.1 Definition of $\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}^{a}(A^{+})$
    2. 3.2 Descent on totally disconnected spaces
    3. 3.3 Boundedness conditions
    4. 3.4 Descent for relative compactifications of totally disconnected spaces
    5. 3.5 Families of Riemann–Zariski spaces
  4. 4 Overconvergent and nuclear sheaves
    1. 4.1 $\omega_{1}$-solid sheaves
    2. 4.2 Overconvergent solid sheaves
    3. 4.3 Nuclear sheaves
    4. 4.4 $\mathcal{C}$-valued nuclear sheaves
  5. 5 Main theorem: $\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}^{a}(\mathcal{O}_{X}^{+})$ on $+$-bounded affinoid perfectoid spaces
  6. References

##  1 Introduction

Let $p$ be a prime, and let $\mathrm{Perfd}$ be the category of perfectoid
spaces over ${\mathbb{Z}}_{p}$. We equip $\mathrm{Perfd}$ with the
$v$-topology, i.e., the Grothendieck topology generated by surjections of
affinoid perfectoid spaces. The main result of this paper is the following
theorem:

######  Theorem 1.1 (Theorem 3.7, Theorem 5.6).

There exists a unique hypercomplete $v$-sheaf of $\infty$-categories

$\mathrm{Perfd}^{\mathrm{op}}\to\mathcal{C}at_{\infty},\
X\mapsto\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X}),$

such that for each $+$-bounded affinoid perfectoid space
$X=\operatorname{Spa}(A,A^{+})$ we have
$\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})\cong\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+})$
compatibly with pullback.

Here,
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}^{a}(A^{+})$
refers to a(n almost version of a) slight modification of the category
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(A^{+})$ of solid
$A^{+}$-modules introduced in [23] and [1] (the exact definition is Definition
2.1, Definition 3.2). Roughly the potentially non-complete compact generators
of $\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(A^{+})$ get
replaced in
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+})$ by their
$\pi$-adic completions for a pseudo-uniformizer $\pi\in A$. Theorem 1.1
formally implies $v$-hyperdescent of the functor
$X\mapsto\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}_{X})$
on perfectoid spaces because
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}_{X}):=\mathrm{Mod}_{\mathcal{O}_{X}}(\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X}))$.

The notion of $+$-boundedness (Definition 3.14) is a mild cohomological
finiteness condition on perfectoid spaces, which only depends on the tilt to
characteristic $p$ and is satisfied for affinoid perfectoid spaces of
characteristic $p$ that are quasi-pro-étale or weakly of perfectly finite type
over a totally disconnected space (Example 3.16). In particular, this yields
the uniqueness of the $v$-sheaf
$X\mapsto\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})$
in Theorem 1.1.

Theorem 1.1 will be used in [2] as the essential technical ingredient for
setting up a $6$-functor formalism with values in (modified) solid quasi-
coherent sheaves on the Fargues–Fontaine curve. This $6$-functor formalism
implies finiteness and duality results for the pro-étale cohomology of general
pro-étale $\mathbb{Q}_{p}$-local systems on smooth rigid-analytic varieties,
and we refer to [2] for more details and motivation for the study of these
assertions (and hence of Theorem 1.1). We note that our proof of Theorem 1.1
would not simplify if the pro-étale topology is considered instead of the
$v$-topology. Moreover, the identification of
$\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})$
for $X$ a $+$-bounded affinoid perfectoid space (and not merely in the easier
case if $X$ totally disconnected), is critical to establish existence of the
$6$-functor formalism, e.g., the projection formula, and some of its
properties, e.g., cohomological smoothness for smooth rigid-analytic varieties
over $\mathbb{C}_{p}$.

We now give a rough sketch of the proof of Theorem 1.1, which at the same time
will summarize the content of the different sections in this paper. First of
all, Theorem 1.1 is an $\mathcal{O}^{+}$-analog of a main theorem of [16],
namely [16, Theorem 1.8.3], which we recall due to its importance for this
paper:

######  Theorem 1.2 ([16, Theorem 1.8.2., Theorem 1.8.3., Theorem 3.5.21]).

There exists a unique hypercomplete $v$-sheaf of $\infty$-categories

$\mathrm{Perfd}_{\pi}^{\mathrm{op}}\to\mathcal{C}at_{\infty},\
X\mapsto\operatorname{\mathcal{D}}^{a}_{{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X}/\pi),$

such that for each $+$-bounded affinoid perfectoid space
$X=\operatorname{Spa}(A,A^{+})$ with uniformizer $\pi$ we have
$\operatorname{\mathcal{D}}^{a}_{{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X}/\pi)\cong\operatorname{\mathcal{D}}^{a}_{{\scalebox{0.5}{$\square$}}}(A^{+}/\pi)$
compatibly with pullback.

Here, $\mathrm{Perfd}_{\pi}$ denotes the category of perfectoid spaces $X$
with a choice of a pseudo-uniformizer $\pi$ and morphisms respecting $\pi$. We
note that here
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}^{a}(A^{+}/\pi)$ is (an
almost version of) the solid category for the discrete ring $A^{+}/\pi$
introduced in [23].

Theorem 1.1 and Theorem 1.2 are tightly related. On the one hand, Remark 3.5
implies that

$\mathrm{Mod}_{\mathcal{O}^{+}_{X}/\pi}(\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X}))\cong\operatorname{\mathcal{D}}^{a}_{{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X}/\pi)$

for each perfectoid space $X$ with a pseudo-uniformizer $\pi$, which shows
that Theorem 1.1 implies Theorem 1.2. On the other hand our definition of
$\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+})$ is
made to use Theorem 1.2 in our proof of Theorem 1.1. We note that this
reduction does not seem possible for the usual category
$\operatorname{\mathcal{D}}^{a}_{{\scalebox{0.5}{$\square$}}}(A^{+})$ instead
of $\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+})$ –
we heavily use that the compact generators of the latter are $\pi$-complete.

In Section 2.2 we define the modification
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(A)$ of
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(A)$ for any adic
analytic ring $A$ (in the general sense of Definition 2.1) by $I$-completing
the compact generators of
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(A)$ with respect to
some ideal of definition $I\subseteq\pi_{0}(A)$, and study its basic
properties. In particular, we prove the following comparison statements.

######  Theorem 1.3 (Proposition 2.17, Lemma 2.9, 2.18.(i)).

Let $A$ be an adic analytic ring, and
$\alpha^{\ast}\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(A)\to\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(A)$
the natural functor.

  1. (i)

If $A^{+}$ is of finite type over ${\mathbb{Z}}_{p}$, then $\alpha^{\ast}$ is
an equivalence.

  2. (ii)

The functor $\alpha^{\ast}$ induces an equivalence
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(A)^{\mathrm{nuc}}\cong\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(A)^{\mathrm{nuc}}$
on nuclear objects.

  3. (iii)

$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(A)^{\mathrm{nuc}}\subseteq\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(A)$
is the full subcategory generated under colimits by $I$-complete objects
$M\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(A)$, which are
discrete mod $I$.

In Theorem 2.30 we prove a descent theorem for
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(-)$ as a functor
on adic rings, by reducing the descent question modulo all $I^{n}$:

######  Theorem 1.4 (Theorem 2.30).

Let $A\to B$ be an adic morphism of adic rings, and $I\subseteq\pi_{0}(A)$ an
ideal of definition. Assume that $A/I^{n}\to B/I^{n},n\geq 0,$ is descendable
of index independent of $n$ (in the sense of [16, Definition 2.6.7]). Then
$A\to B$ satisfies descent for
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(-)$.

Again, such a statement does not seem possible for
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(-)$. Morever,
following [1] we globalize
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}$ to stably
uniform analytic adic spaces in Section 2.4 with an eye towards our
applications in [2].

######  Theorem 1.5 (Theorem 2.38).

The functor
$\operatorname{Spa}(B,B^{+})\mapsto\mathrm{Mod}_{B}(\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(B^{\circ},B^{+}))$
satisfies analytic (resp. étale) descent on stably uniform (resp.
sousperfectoid) affinoid analytic adic spaces.

In Section 3.1 we move to perfectoid spaces, where we define
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}^{a}(A^{+})$ for
any affinoid perfectoid space $X=\operatorname{Spa}(A,A^{+})$ (Definition
3.2). Then we apply the complete descent results from Section 2.3 to obtain
$v$-descent of
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}^{a}(A^{+})$ on
totally disconnected perfectoid spaces (Theorem 3.7). From here, we can define
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}^{a}(\mathcal{O}_{X}^{+})$
for any perfectoid space $X$ (or even small $v$-stack on $\mathrm{Perfd}$) by
$v$-descent. This proves the existence part of Theorem 1.1, with the exception
of the identification
$\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})\cong\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+})$
if $X=\operatorname{Spa}(A,A^{+})$ is a $+$-bounded affinoid perfectoid space.
This question we adress next.

There exists an evident functor

$\widetilde{(-)}\colon\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}^{a}(A^{+})\to\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}^{a}(\mathcal{O}_{X}^{+}),$

and this functor admits a right adjoint, denoted by $\Gamma(X,-)$. Let $\pi\in
A$ be a pseudo-uniformizer. Using Theorem 1.2 (and in particular the
assumption that $X$ is $+$-bounded), it is fairly easy to see that
$\widetilde{(-)}$ is an equivalence if (and only if) the abstractly defined
category
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}^{a}(\mathcal{O}_{X}^{+})$
is generated under colimits by (suitably bounded) $\pi$-complete objects, and
$\Gamma(X,-)$ preserves colimits (Lemma 3.22). Both of these properties of
$\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})$
are proven by a detour through a category
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{p}[[\pi]])$ of
(overconvergent) nuclear sheaves of ${\mathbb{Z}}_{p}[[\pi]]$-modules on
$X_{\mathrm{qproet}}$ that we discuss in Section 4 following [15] and [9]. The
discussion of
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{p}[[\pi]])$
involves a rather long and detailed analysis of solid, $\omega_{1}$-solid and
overconvergent objects in
$\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},{\mathbb{Z}}_{p}[[\pi]])$ for
a general $p$-bounded spatial diamond. The criticial assertion in this section
concerns the nuclear objects in the category of $\omega_{1}$-solid objects
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{p}[[\pi]])_{\omega_{1}}\subseteq\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},{\mathbb{Z}}_{p}[[\pi]])$.
We stress that nuclearity here refers to the abstract notion from [6, Lecture
VIII], and not to the more geometric notion introduced in [15].

######  Lemma 1.6 (Lemma 4.15, Lemma 4.20).

Let $\ell$ be a prime, and let $X$ be an $\ell$-bounded affinoid perfectoid
space, and $\Lambda$ an adic profinite ${\mathbb{Z}}_{\ell}$-algebra, e.g.,
$\Lambda={\mathbb{Z}}_{p}[[\pi]]$ if $\ell=p$. Then the category
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$ of nuclear objects in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
is generated under colimits by overconvergent, complete, right-bounded
objects.

Especially the generation by complete objects is non-obvious. Together with
the good cohomological properties of $\omega_{1}$-solid
${\mathbb{Z}}_{p}[[\pi]]$-sheaves on $X_{\mathrm{qproet}}$ this is the source
of the desired properties of
$\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})$
for a $+$-bounded affinoid perfectoid space $X$. The precise implementation of
these properties involves Corollary 5.5, which realizes
$\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})$
as a category of $\mathcal{O}^{+a}$-modules in the abstract
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}({\mathbb{Z}}_{p}[[\pi]])$-linear
category

$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,(A^{+})^{a}_{\hat{\scalebox{0.5}{$\square$}}}):=\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{p}[[\pi]])\otimes_{\operatorname{\mathcal{D}}_{\mathrm{nuc}}({\mathbb{Z}}_{p}[[\pi]])}\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+}).$

This latter category is well-behaved thanks to the rigidity of
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}({\mathbb{Z}}_{p}[[\pi]])$ (Remark
4.22) and dualizability of
$\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+})$ as a
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}({\mathbb{Z}}_{p}[[\pi]])$-module
(Lemma 5.2). However, the equivalence

$\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})\cong\mathrm{Mod}_{\mathcal{O}^{+a}}(\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,(A^{+})^{a}_{\hat{\scalebox{0.5}{$\square$}}})).$

of Corollary 5.5 needs as an input the equivalence
$\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{\overline{Z}^{/X}})\cong\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(B^{+},A^{+})$
for the relative compactification $\overline{Z}^{/X}$ of a strictly totally
disconnected space $Z=\operatorname{Spa}(B,B^{+})$, which is quasi-pro-étale
over $X$. As $\overline{Z}^{/X}$ is not necessarily totally disconnected, this
does not follow from the descent of totally disconnected spaces proven in
Theorem 3.7, and we supply an argument in Theorem 3.23 using an analysis of
families of Riemann-Zariski spaces (Lemma 3.26).

###  1.1 Acknowledgements

We thank Arthur-César Le Bras heartily for all his feedback, and for all
discussions in an early stage of this project. For personal, Monadic reasons
he unfortunately had to leave this project in a later stage. We moreover thank
Gregory Andreychev, Johan de Jong, Juan Esteban Rodriguez Camargo and Peter
Scholze for discussions related to this paper.

###  1.2 Notations and conventions

For technical convenience we fix an implicit cut-off cardinal $\kappa$ (in the
sense of [22, Section 4]), and assume all our perfectoid spaces, and condensed
sets to be $\kappa$-small. In particular, for a Huber pair $(A,A^{+})$ its
associated category
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(A,A^{+})$ ([1, Theorem
3.28]) is generated by a set of compact objects. Passing to the filtered
colimit over all $\kappa$’s implies the descent statement in general.

We work implicitly “condensed” and “animated”. More precisely, a ring $R$ will
mean a condensed animated ring. It is called static if $R\cong\pi_{0}(R)$ (as
a condensed ring), and discrete if
$R\cong\underline{R(\ast)_{\mathrm{disc}}}$, where $R(\ast)$ is the
“underlying animated ring”, or equivalently, $R$ is discrete if the functor
$S\mapsto R(S)$ on profinite sets maps cofiltered inverse limits to filtered
colimits.

Given a ring $R$ and an ideal $I\subseteq\pi_{0}(R(\ast))$ generated by
elements $f_{1},\ldots,f_{n}\in I$, we set
$M/I:={\mathbb{Z}}\otimes_{{\mathbb{Z}}[x_{1},\ldots,x_{n}]}M$ for any
$R$-module $M$ where the tensor product is implicitly derived. Here, the map
${\mathbb{Z}}[x_{1},\ldots,x_{n}]\to R$ making $M$ into an
${\mathbb{Z}}[x_{1},\ldots,x_{n}]$-module is classified by the elements
$f_{1},\ldots,f_{n}$. In particular, the quotient $M/I$ depends on the choice
of elements, and we will carefully make this clear when this may create
confusion. Concretely, the quotient $M/I$ is calculated by the tensor product
of the complexes $M\xrightarrow{f_{i}}M$, $i=1,\ldots,n$.

Given a ${\mathbb{Z}}[x]$-linear category $\mathcal{C}$ with sequential limits
we call $X\in\mathcal{C}$ $x$-complete if the inverse limit

$\ldots\to X\xrightarrow{x}X$

vanishes.

##  2 $\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(A)$ for
adic rings

When proving descent for solid modules over $p$-complete rings, one of the
main challenges is that in general the compact generators of
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(A)$ are not
$p$-complete, even if $A$ itself is so. Indeed, by the very definition the
compact generators of
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(A)$ are colimits over
the category of maps $A^{\prime}\to A$ with $A^{\prime}$ a finitely generated
${\mathbb{Z}}$-algebra ([1, Definition 3.20]). This defect breaks the usual
strategy of reducing descent questions to the reductions mod $p^{n}$ and
therefore makes it hard to get any good general descent results on
$p$-adically complete rings and general solid modules. To overcome this issue,
we present a slight modification of
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(A)$, denoted
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(A)$ in the
following (Definition 2.4), where we replace the compact generators of
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(A)$ by their $p$-adic
completions. This construction may seem a bit artificial, but the results in
this section show that we do not lose much and still have a very tight
relation to $\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(A)$. With
this modified version of solid modules we then explain how descent questions
can indeed be reduced modulo $p^{n}$ (Theorem 2.30).

Of course the construction of
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(A)$ works for
any adically complete rings (instead of just $p$-complete rings), so we
perform it in the natural generality. Hence, we start by discussing adic
analytic rings.

###  2.1 Adic rings

We fix the following terminology. Note that we adhere to our conventions from
Section 1.2.

We let ${\mathbb{Z}}_{\scalebox{0.5}{$\square$}}$ be the solid analytic ring
from [23, Definition 5.1] and ${\mathbb{Z}}[T]_{\scalebox{0.5}{$\square$}}$
the analytic ring from [23, Theorem 8.2]. If $A$ is any solid ring, and
$S\to\pi_{0}(A)(\ast)$ a map of sets, then we denote by
$(A,S)_{\scalebox{0.5}{$\square$}}$ the analytic ring over
${\mathbb{Z}}_{\scalebox{0.5}{$\square$}}$ such that an object
$M\in\mathrm{Mod}_{A}(D({\mathbb{Z}}_{\scalebox{0.5}{$\square$}}))$ is
$(A,S)_{\scalebox{0.5}{$\square$}}$-complete if and only if $M$ is
${\mathbb{Z}}[s]_{\scalebox{0.5}{$\square$}}$-complete for any $s\in S$ with
induced map ${\mathbb{Z}}[s]\to A$ of rings. For $(A,S)=(A,A^{+})$ a classical
complete Huber pair, this analytic ring structure has been analyzed in [1]. We
note that by [24, Appendix to lecture XII] and [1, Proposition 3.32]
$(A,S)_{\scalebox{0.5}{$\square$}}\cong(A,\widetilde{S})_{\scalebox{0.5}{$\square$}}$
if $\widetilde{S}\subseteq\pi_{0}(A)(\ast)$ denotes the smallest integrally
closed subring containing each topologically nilpotent element and the image
of $S$.111By definition, an element $a\in\pi_{0}(A)(\ast)$ is topologically
nilpotent if the associated map ${\mathbb{Z}}[T]\to A$ factors over
${\mathbb{Z}}[[T]]$ as a map of rings. As $A$ is solid and
${\mathbb{Z}}[[T]]\otimes_{({\mathbb{Z}}[T],{\mathbb{Z}})_{\scalebox{0.5}{$\square$}}}{\mathbb{Z}}[[T]]\cong{\mathbb{Z}}[[T]]$
this factorization is unique if it exists.

######  Definition 2.1.

  1. (a)

We say that a ring $A$ is _adic_ if there is some finitely generated ideal
$I\subseteq\pi_{0}(A)(\ast)$ such that $A/I$ is discrete and $A$ is
$I$-adically complete. We call any such ideal $I$ an _ideal of definition_ of
$A$.

  2. (b)

An _adic analytic ring_ is an analytic ring $\mathcal{A}$ of the form
$\mathcal{A}=(A,A^{+})_{\scalebox{0.5}{$\square$}}$, where $A$ is an adic ring
and $A^{+}\subseteq\pi_{0}(A)(\ast)$ is a subring. We denote by

$\displaystyle\operatorname{AdicRing}\subseteq\operatorname{AnRing}$

the full subcategory spanned by the adic analytic rings. Given an adic
analytic ring $\mathcal{A}$ we usually write
$\underline{\mathcal{A}}=\mathcal{A}[\ast]$ for its underlying adic ring.

  3. (c)

A map $f\colon A\to B$ of adic rings is called _adic_ if for some ideal of
definition $I$ of $A$, $f(I)$ generates an ideal of definition of $B$. A map
of adic analytic rings is called adic if the map of underlying adic rings is
adic.

Given an adic ring $A$ with ideal of definition $I$, we denote by
$\operatorname{\mathcal{D}}(A):=\mathrm{Mod}_{A}(\operatorname{\mathcal{D}}(\mathrm{Cond}(\mathrm{Ab}))$
its (stable $\infty$-)category of modules in condensed abelian groups, and we
let
$\operatorname{\mathcal{D}}(A)_{\hat{I}}\subseteq\operatorname{\mathcal{D}}(A)$
be the full subcategory of $I$-adically complete $A$-modules, i.e., those
which are $x$-complete for any $x\in I$. This notion does not depend on the
choice of $I$ or is generators:

######  Lemma 2.2.

Let $A$ be an adic ring.

  1. (i)

For any two ideals of definition $I$, $I^{\prime}$ of $A$ we have
$\operatorname{\mathcal{D}}(A)_{\hat{I}}=\operatorname{\mathcal{D}}(A)_{\hat{I^{\prime}}}$.

  2. (ii)

A map $f\colon A\to B$ is adic if and only if for every ideal of definition
$I$ of $A$, $f(I)\subseteq\pi_{0}(B)(\ast)$ generates an ideal of definition
of $B$.

###### Proof.

We first prove (i), so let $I$ and $I^{\prime}$ be given. Without loss of
generality we can assume $I\subseteq I^{\prime}$ because the intersection of
two ideals of definition is an ideal of definition. Pick any $a\in
I^{\prime}$. Then $A/I$ is $a$-adically complete (as a finite limit of copies
of $A$’s), and hence the induced map ${\mathbb{Z}}[x]\to A/I$ factors over
${\mathbb{Z}}[[x]]\to A/I$. Since $A/I$ is discrete and in particular nuclear
as a ${\mathbb{Z}}_{\scalebox{0.5}{$\square$}}$-module, we have
$\operatorname{Hom}_{\mathbb{Z}}({\mathbb{Z}}[[x]],A/I)=\varinjlim_{n}\operatorname{Hom}_{\mathbb{Z}}({\mathbb{Z}}[x]/x^{n},A/I)$,
which implies that $a$ is nilpotent in $\pi_{0}(A/I)(\ast)$. But then any
$I$-adically complete $A$-module is also $a$-adically complete. Since $a\in
I^{\prime}$ was arbitrary, we immediately arrive at the claimed identity.

We now prove (ii), so let $f\colon A\to B$ be an adic map of adic rings and
$I$ some ideal of definition of $A$. By assumption there is some ideal of
definition $I^{\prime}$ of $A$ such that $f(I^{\prime})$ generates an ideal of
definition of $B$. If $I^{\prime\prime}$ is any ideal of definition of $A$
containing $I^{\prime}$ then by the proof of (i), every element in
$I^{\prime\prime}$ has some power which lies in $I^{\prime}$; hence also
$f(I^{\prime\prime})$ generates an ideal of definition of $B$. We apply this
to $I^{\prime\prime}=I^{\prime}+I$. But by reversing this argument we also
deduce that $f(I)$ generates an ideal of definition of $B$, as desired. The
converse direction is clear. ∎

######  Definition 2.3.

Let $A$ be an adic ring. We say that an $A$-module
$M\in\operatorname{\mathcal{D}}(A)$ is _adically complete_ if it is
$I$-adically complete for some (equivalently every) ideal of definition $I$ of
$A$. We denote by
$\operatorname{\mathcal{D}}_{\mathrm{cpl}}(A)\subseteq\operatorname{\mathcal{D}}(A)$
the full subcategory spanned by the adically complete $A$-modules and by

$\displaystyle(-)_{\mathrm{cpl}}\colon\operatorname{\mathcal{D}}(A)\to\operatorname{\mathcal{D}}_{\mathrm{cpl}}(A)$

the left-adjoint of the inclusion. For an analytic adic ring $\mathcal{A}$ we
similarly denote
$\operatorname{\mathcal{D}}_{\mathrm{cpl}}(\mathcal{A}):=\operatorname{\mathcal{D}}(\mathcal{A})\mathbin{\cap}\operatorname{\mathcal{D}}_{\mathrm{cpl}}(\underline{\mathcal{A}})$.

Here, the existence of $(-)_{\mathrm{cpl}}$ is guaranteed by the adjoint
functor theorem, which applies here as all categories here are presentable
(recall that we have fixed a cut-off cardinal $\kappa$ in Section 1.2).

###  2.2 Definition of
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(A)$

With a good notion of adic rings and adically complete modules at hand, we can
now construct the promised modification of
$\operatorname{\mathcal{D}}(\mathcal{A})$. The definition is very simple,
albeit somewhat artificial:

######  Definition 2.4.

Given $\mathcal{A}$ an adic analytic ring, let
$\mathcal{C}_{\mathcal{A}}\subseteq\operatorname{\mathcal{D}}_{\mathrm{cpl}}(\mathcal{A})$
be the full subcategory generated under finite (co)limits and retracts from
the essential image of the full subcategory
$\operatorname{\mathcal{D}}(\mathcal{A})^{\omega}$ of compact
$\mathcal{A}$-modules under $(-)_{\mathrm{cpl}}$. Then we define

$\displaystyle\operatorname{\mathcal{D}}(\hat{\mathcal{A}}):=\operatorname{Ind}(\mathcal{C}_{\mathcal{A}}).$

Writing $\mathcal{A}=(A,A^{+})_{\scalebox{0.5}{$\square$}}$, we also use the
notation
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(A,A^{+}):=\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$.

The following results show that
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ is quite close to
$\operatorname{\mathcal{D}}(\mathcal{A})$ and in particular inherits all of
its nice properties:

######  Lemma 2.5.

Let $\mathcal{A}$ be an adic analytic ring.

  1. (i)

$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ is a stable $\infty$-category
that can naturally be equipped with a closed symmetric monoidal structure and
a $t$-structure.

  2. (ii)

There is a natural colimit-preserving, symmetric monoidal and right $t$-exact
functor
$\alpha^{*}\colon\operatorname{\mathcal{D}}(\mathcal{A})\to\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$.

  3. (iii)

The functor $\alpha^{*}$ has a right adjoint
$\alpha_{*}\colon\operatorname{\mathcal{D}}(\hat{\mathcal{A}})\to\operatorname{\mathcal{D}}(\mathcal{A})$
which is $t$-exact, conservative, preserves all small limits and colimits and
commutes with truncations.

  4. (iv)

For every compact $\mathcal{A}$-module
$P\in\operatorname{\mathcal{D}}(\mathcal{A})$ we have
$\alpha_{*}\alpha^{*}P=P_{\mathrm{cpl}}$.

###### Proof.

Let
$\mathcal{C}_{\mathcal{A}}\subseteq\operatorname{\mathcal{D}}_{\mathrm{cpl}}(\mathcal{A})$
be as in Definition 2.4. Recall that there is a symmetric monoidal structure
$\hat{\otimes}_{\mathcal{A}}$ on
$\operatorname{\mathcal{D}}_{\mathrm{cpl}}(\mathcal{A})$ given by the
completed tensor product.222This follows from the fact that
$\underline{\operatorname{Hom}}(M,N)$ is adically complete whenever
$N\in\operatorname{\mathcal{D}}(A)$ is adically complete. Since compact
objects in $\operatorname{\mathcal{D}}(\mathcal{A})$ are stable under
$\otimes_{\mathcal{A}}$ (here we use that $\mathcal{A}$ is an analytic ring
over ${\mathbb{Z}}_{\scalebox{0.5}{$\square$}}$) it follows that
$\mathcal{C}_{\mathcal{A}}$ is stable under $\hat{\otimes}_{\mathcal{A}}$, so
that we naturally get a symmetric monoidal structure on
$\mathcal{C}_{\mathcal{A}}$. By [12, Corollary 4.8.1.14] we can uniquely
extend this symmetric monoidal structure to
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})=\operatorname{Ind}(\mathcal{C}_{\mathcal{A}})$
so that it preserves colimits in each argument.

The functor $\alpha^{*}$ is the natural functor colimit
$\operatorname{\mathcal{D}}(\mathcal{A})=\operatorname{Ind}(\operatorname{\mathcal{D}}(\mathcal{A})^{\omega})\to\operatorname{Ind}(\mathcal{C}_{\mathcal{A}})=\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$
induced by the completion functor
$\operatorname{\mathcal{D}}(\mathcal{A})^{\omega}\to\mathcal{C}_{\mathcal{A}}$.
By [12, Corollary 4.8.1.14] $\alpha^{*}$ can be upgraded uniquely to a
symmetric monoidal functor. The right adjoint $\alpha_{*}$ exists by the
adjoint functor theorem (using that we have fixed a cut-off cardinal). Since
$\alpha^{*}$ preserves compact objects (by definition) it follows that
$\alpha_{*}$ commutes with filtered colimits and hence with all colimits. It
is also easy to see that $\alpha_{*}$ is conservative: Given any
$M\in\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ with $\alpha_{*}M=0$, pick
any compact $P\in\operatorname{\mathcal{D}}(\mathcal{A})$; then
$0=\operatorname{Hom}(P,\alpha_{*}M)=\operatorname{Hom}(\alpha^{*}P,M)$, so it
is enough to show that the family of functors
$\operatorname{Hom}(\alpha^{*}P,-)$ is conservative (with $P$ ranging over
compact $\mathcal{A}$-modules). But this follows immediately from the fact
that the $\alpha^{*}P$ form compact generators of
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ by construction.

We now prove (iv), so fix any compact $\mathcal{A}$-module $P$. Note that
$\alpha^{*}P$ is adically complete: For any ideal of definition $I$ for
$\underline{\mathcal{A}}$ and any $x\in I$ we need to check that
$\varprojlim_{x}\alpha^{*}P=0$ with $\varprojlim$ taken in
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$, and for this it is enough to
check that $\varprojlim_{x}\operatorname{Hom}(Q,\alpha^{*}P)=0$ for every
$Q\in\mathcal{C}_{\mathcal{A}}$; but this statement depends only on
$\mathcal{C}_{\mathcal{A}}$, which is a full subcategory of
$\operatorname{\mathcal{D}}_{\mathrm{cpl}}(\mathcal{A})$. Thus, we have
checked that $\alpha^{\ast}P$ is adically complete. Since $\alpha_{*}$
preserves all small limits, we deduce that also $\alpha_{*}\alpha^{*}P$ is
adically complete, i.e. lies in
$\operatorname{\mathcal{D}}_{\mathrm{cpl}}(\mathcal{A})$. Hence the unit
$P\to\alpha_{*}\alpha^{*}P$ induces a map
$P_{\mathrm{cpl}}\to\alpha_{*}\alpha^{*}P$. To show that this map is an
isomorphism, it is enough to do so after applying $\operatorname{Hom}(Q,-)$
for any compact $Q\in\operatorname{\mathcal{D}}(\mathcal{A})$; we have:

$\displaystyle\operatorname{Hom}(Q,P_{\mathrm{cpl}})=\operatorname{Hom}(Q_{\mathrm{cpl}},P_{\mathrm{cpl}})=\operatorname{Hom}(\alpha^{*}Q,\alpha^{*}P)=\operatorname{Hom}(Q,\alpha_{*}\alpha^{*}P),$

where in the second step we used that
$\mathcal{C}_{\mathcal{A}}\subseteq\operatorname{\mathcal{D}}(\mathcal{A})$ is
a full subcategory.

We have now proved everything apart from the claims about the $t$-structure.
We define $\operatorname{\mathcal{D}}_{\geq
0}(\hat{\mathcal{A}})\subseteq\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$
to be the full subcategory spanned by those objects
$M\in\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ such that
$\alpha_{*}M\in\operatorname{\mathcal{D}}_{\geq 0}(\mathcal{A})$. This
subcategory is clearly stable under colimits and extensions and by (iv) it
contains $\alpha^{*}\mathcal{A}[S]$ for all profinite sets $S$. It follows
that the $\alpha^{*}\mathcal{A}[S]$ are compact generators of
$\operatorname{\mathcal{D}}_{\geq 0}(\hat{\mathcal{A}})$, so by [12,
Proposition 1.4.4.11.(i)] this subcategory defines a $t$-structure on
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$. Since every
$M\in\operatorname{\mathcal{D}}_{\geq 0}(\mathcal{A})$ is a (sifted) colimit
of the free generators $\mathcal{A}[S]$ it follows easily from (iv) that
$\alpha^{*}$ is right $t$-exact (because $(-)_{\mathrm{cpl}}$ is so).
Moreover, $\alpha_{*}$ is right $t$-exact by construction and it is left
$t$-exact because $\alpha^{*}$ is right $t$-exact. Finally, one checks easily
that $\alpha_{*}$ commutes with $\tau_{\geq 0}$ by passing to left adjoints;
from the usual truncation triangle it then follows that $\alpha_{*}$ also
commutes with $\tau_{\leq 0}$. ∎

######  Remark 2.6.

By Lemma 2.5 the adjunction of $\alpha^{*}$ and $\alpha_{*}$ is monadic and
hence identifies $\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ with a
category of modules over some monad on
$\operatorname{\mathcal{D}}(\mathcal{A})$. Thus one may think of
$\hat{\mathcal{A}}$ as some form of generalized analytic ring and
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ as its category of modules.
One may attempt to formalize this idea in order to get a general theory of
generalized analytic rings, but we will not pursue this further. We note that
$\alpha_{\ast}\colon\operatorname{\mathcal{D}}(\hat{\mathcal{A}})\to\operatorname{\mathcal{D}}(\mathcal{A})$
is in general not $\operatorname{\mathcal{D}}(\mathcal{A})$ linear, or
equivalently $\alpha_{\ast}$ does not satisfy the projection formula
(otherwise for each compact object $P\in\mathcal{A}$ one has
$P\otimes\alpha_{\ast}(\alpha^{\ast}(\underline{\mathcal{A}})\cong\alpha_{\ast}(\alpha^{\ast}P)\cong
P_{{\mathrm{cpl}}}$). Thus, modules over the monad
$\alpha_{\ast}(\alpha^{\ast}(-))$ do in general not identify with modules
under some ring object in $\operatorname{\mathcal{D}}(\mathcal{A})$.

Before we continue, let us make everything more functorial. We have the
following straightforward result:

######  Lemma 2.7.

The construction
$\mathcal{A}\mapsto\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ defines a
functor
$\operatorname{AdicRing}^{\mathrm{op}}\to\operatorname{CAlg}(\mathcal{C}at_{\infty})$
and
$\alpha^{*}\colon\operatorname{\mathcal{D}}(\mathcal{A})\to\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$
defines a natural transformation of such functors.

###### Proof.

The assignment $\mathcal{A}\mapsto\operatorname{\mathcal{D}}(\mathcal{A})$
defines a functor
$\operatorname{AdicRing}^{\mathrm{op}}\to\operatorname{CAlg}(\mathcal{C}at_{\infty})$
(see e.g. [16, Proposition 2.3.26]). By looking at the associated cocartesian
family of $\infty$-operads and restricting to a full subcategory of that, we
can construct the functor
$\mathcal{A}\mapsto\operatorname{\mathcal{D}}_{\mathrm{cpl}}(\mathcal{A})$
(see [16, Lemma 2.2.22] for the general strategy of this argument). By
restricting to even small full subcategories we obtain the functor which
associates to each $\mathcal{A}$ the full subcategory
$\mathcal{C}_{\mathcal{A}}\subseteq\operatorname{\mathcal{D}}_{\mathrm{cpl}}(\mathcal{A})$
from Definition 2.4. By the remark right before [12, Proposition 4.8.1.10] the
assignment $\mathcal{C}\mapsto\operatorname{Ind}(\mathcal{C})$ defines an
endofunctor of symmetric monoidal $\infty$-categories. This finishes the
construction of the functor
$\mathcal{A}\mapsto\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$.

The construction of the natural transformation $\alpha^{*}$ works in a very
similar fashion, by making everything relative over $\Delta^{1}$. ∎

######  Definition 2.8.

  1. (a)

For every $\mathcal{A}\in\operatorname{AdicRing}$ we denote by
$-\otimes_{\mathcal{A}}\hat{\mathcal{A}}\colon\operatorname{\mathcal{D}}(\mathcal{A})\to\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$
the functor $\alpha^{*}$ from 2.5.(ii).

  2. (b)

For every map $\mathcal{A}\to\mathcal{B}$ in $\operatorname{AdicRing}$ we
denote by
$-\otimes_{\hat{\mathcal{A}}}\hat{\mathcal{B}}\colon\operatorname{\mathcal{D}}(\hat{\mathcal{A}})\to\operatorname{\mathcal{D}}(\hat{\mathcal{B}})$
the induced functor from Lemma 2.7.

We will write $\alpha^{\ast}_{\mathcal{A}},\alpha_{\mathcal{A},\ast}$ in case
we need to clarify the dependence of $\alpha^{\ast},\alpha_{\ast}$ on
$\mathcal{A}$. If $\mathcal{A}\to\mathcal{B}$ is a morphism of adic analytic
rings, then
$\alpha_{\mathcal{B}}^{\ast}((-)\otimes_{\mathcal{A}}\mathcal{B})\cong\alpha^{\ast}_{\mathcal{A}}(-)\otimes_{\hat{\mathcal{A}}}\hat{\mathcal{B}}$
by Lemma 2.7.

We remark that $-\otimes_{\mathcal{A}}\hat{\mathcal{A}}$ and
$-\otimes_{\hat{\mathcal{A}}}\hat{\mathcal{B}}$ are symmetric monodial. While
for general adic analytic rings $\mathcal{A}$,
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ differs from
$\operatorname{\mathcal{D}}(\mathcal{A})$, in praxis these two categories are
often the same:

######  Lemma 2.9.

Let $(A,A^{+})_{\scalebox{0.5}{$\square$}}$ be an adic analytic ring. If
either $A$ is discrete or $A^{+}$ is finitely generated over ${\mathbb{Z}}$,
then
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})=\operatorname{\mathcal{D}}(\mathcal{A})$.

###### Proof.

We use the notation from Lemma 2.5. Since $\alpha_{*}$ is conservative, in
order to prove the desired equivalence of categories it is enough to show that
$\alpha^{*}$ is fully faithful, i.e. that for all
$M\in\operatorname{\mathcal{D}}(\mathcal{A})$ the unit of the adjunction
$M\mathbin{\xrightarrow{\sim}}\alpha_{*}\alpha^{*}M$ is an isomorphism. Since
$\alpha_{*}$ and $\alpha^{*}$ preserve colimits, this reduces to the case that
$M=P$ is compact, in which case by 2.5.(iv) the claim reduces to showing that
$P$ is adically complete. This is evident if $A$ is discrete, so it remains to
treat the case where $A^{+}$ is finitely generated.

We can assume that $A^{+}$ is a finitely generated polynomial algebra over
${\mathbb{Z}}$ and that there is a map $A^{+}\to A$. By choosing generators of
an ideal of definition of $A$ we construct a map
$A^{+}[x_{\bullet}]:=A^{+}[x_{1},\dots,x_{n}]\to A$, which then automatically
factors as $A^{+}[[x_{\bullet}]]\to A$. Note that for every profinite set $S$
we have

$\displaystyle(A,A^{+})_{\scalebox{0.5}{$\square$}}[S]=A\otimes_{A^{+}_{\scalebox{0.5}{$\square$}}}A^{+}_{\scalebox{0.5}{$\square$}}[S]=A\otimes_{(A^{+}[[x_{\bullet}]],A^{+})_{\scalebox{0.5}{$\square$}}}(A^{+}[[x_{\bullet}]],A^{+})_{\scalebox{0.5}{$\square$}}[S]$

On the other hand,
$(A^{+}[[x_{\bullet}]],A^{+})_{\scalebox{0.5}{$\square$}}[S]\cong\prod_{I}A^{+}[[x_{\bullet}]]$
for some set $I$ (as $A^{+}$ is finitely generated over ${\mathbb{Z}}$), so in
particular it is $(x_{1},\dots,x_{n})$-adically complete. It follows from [16,
Proposition 2.12.10] that $(A,A^{+})_{\scalebox{0.5}{$\square$}}[S]$ is
$(x_{1},\dots,x_{n})$-adically complete, i.e. adically complete as an
$A$-module. This proves the claim. ∎

######  Lemma 2.10.

Let $\mathcal{A}$ be an adic analytic ring with ideal of definition $I$. Then
$-\otimes_{\mathcal{A}}\hat{\mathcal{A}}$ induces an equivalence of categories

$\displaystyle\operatorname{\mathcal{D}}(\mathcal{A}/I)=\operatorname{Mod}_{\underline{\mathcal{A}}/I}(\operatorname{\mathcal{D}}(\hat{\mathcal{A}})).$

Here, the $\operatorname{\mathcal{D}}(\mathcal{A}/I)$ denotes analytic ring
structure on $\underline{\mathcal{A}}/I$ with its induced analytic ring
structure along $\underline{\mathcal{A}}\to\underline{\mathcal{A}}/I$.

###### Proof.

Let us write $\alpha^{*}:=-\otimes_{\mathcal{A}}\hat{\mathcal{A}}$. We have
$\operatorname{\mathcal{D}}(\mathcal{A}/I)=\operatorname{Mod}_{\underline{\mathcal{A}}/I}(\operatorname{\mathcal{D}}(\mathcal{A}))$,
so since $\alpha^{*}$ is symmetric monoidal, it induces a functor
$\operatorname{\mathcal{D}}(\mathcal{A}/I)\to\operatorname{Mod}_{\underline{\mathcal{A}}/I}(\operatorname{\mathcal{D}}(\hat{\mathcal{A}}))$.
Its right adjoint is given by $\alpha_{*}$, which is conservative, so the
desired equivalence reduces to showing that $\alpha^{*}$ is fully faithful on
$\underline{\mathcal{A}}/I$-modules, i.e. that
$\operatorname{id}\mathbin{\xrightarrow{\sim}}\alpha_{*}\alpha^{*}$ is an
isomorphism. This in turn can be checked on compact generators, i.e. for
$P=(\mathcal{A}/I)[S]$. But such a $P$ is also compact as $\mathcal{A}$-module
(by our conventions Section 1.2), hence by 2.5.(iv) we have
$\alpha_{*}\alpha^{*}P=P_{\mathrm{cpl}}=P$, as desired. ∎

Next we study adic completeness in
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$. Since this category is
$\operatorname{\mathcal{D}}(\mathcal{A})$-linear via the symmetric monoidal
functor $-\otimes_{\mathcal{A}}\hat{\mathcal{A}}$, there is a good notion of
adically complete modules in $\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$:

######  Definition 2.11.

Let $\mathcal{A}$ be an adic analytic ring. An $\hat{\mathcal{A}}$-module
$M\in\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ is called _adically
complete_ if for some (equivalently every) ideal of definition $I$ of
$\underline{\mathcal{A}}$ and every $x\in I$ we have $\varprojlim_{x}M=0$. We
denote by
$\operatorname{\mathcal{D}}_{\mathrm{cpl}}(\hat{\mathcal{A}})\subseteq\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$
the full subcategory spanned by the adically complete modules. Similarly, we
let
$\operatorname{\mathcal{D}}_{\mathrm{cpl}}(\mathcal{A})\subseteq\operatorname{\mathcal{D}}(\mathcal{A})$
be the full subcategory of adically complete objects in
$\operatorname{\mathcal{D}}(\mathcal{A})$.333We show in Lemma 2.15 that
$\operatorname{\mathcal{D}}_{\mathrm{cpl}}(\mathcal{A})$ and
$\operatorname{\mathcal{D}}_{\mathrm{cpl}}(\hat{\mathcal{A}})$ agree.

The following result is a first indicator for why we work with
$\hat{\mathcal{A}}$ in this paper: The complete objects in
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ behave much better with
respect to pullbacks and tensor products than those in
$\operatorname{\mathcal{D}}(\mathcal{A})$.

######  Lemma 2.12.

Let $\mathcal{A}$ be an adic analytic ring.

  1. (i)

An $\hat{\mathcal{A}}$-module is adically complete if and only if the
underlying $\mathcal{A}$-module is adically complete. All the equivalent
characterizations from [16, Lemma 2.12.4] apply to adically complete modules
in $\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$, Most notably, an
$\hat{\mathcal{A}}$-module $M$ is adically complete if and only if all its
homotopy objects $\pi_{i}(M)$, $i\in{\mathbb{Z}}$, are.

  2. (ii)

The $t$-structure on $\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ restricts
to a $t$-structure on
$\operatorname{\mathcal{D}}_{\mathrm{cpl}}(\hat{\mathcal{A}})$.

  3. (iii)

$\operatorname{\mathcal{D}}^{-}_{\mathrm{cpl}}(\hat{\mathcal{A}})$ is stable
under the symmetric monoidal structure $\otimes_{\hat{\mathcal{A}}}$ on
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ from Lemma 2.5.

  4. (iv)

Let $\mathcal{A}\to\mathcal{B}$ be an adic map of adic analytic rings. Then
$-\otimes_{\hat{\mathcal{A}}}\hat{\mathcal{B}}$ restricts to a functor
$\operatorname{\mathcal{D}}^{-}_{\mathrm{cpl}}(\hat{\mathcal{A}})\to\operatorname{\mathcal{D}}^{-}_{\mathrm{cpl}}(\hat{\mathcal{B}})$.

###### Proof.

We use the notation $\alpha^{*}$ and $\alpha_{*}$ from Lemma 2.5. Then (i)
follows easily from 2.5.(iii) and (ii) is just a reformulation of the fact
that $M$ is adically complete if and only if all $\pi_{i}M$ are adically
complete (which is part of (i)).

We now prove (iv), so let
$M\in\operatorname{\mathcal{D}}^{-}_{\mathrm{cpl}}(\hat{\mathcal{A}})$ be
given. We need to show that $M\otimes_{\hat{\mathcal{A}}}\hat{\mathcal{B}}$ is
adically complete. For some $n\geq 0$ we can find an adic map
$\mathcal{A}_{0}:=({\mathbb{Z}}[[x_{1},\dots,x_{n}]],{\mathbb{Z}})_{\scalebox{0.5}{$\square$}}\to\mathcal{A}$.
By using the bar resolution for the monadic adjunction between base-change and
forgetful functor along
$\mathcal{A}_{0}=\hat{\mathcal{A}_{0}}\to\hat{\mathcal{A}}$ (using Lemma 2.9)
we can write $M$ as a uniformly right-bounded geometric realizations of
modules of the form $M_{0}\otimes_{\mathcal{A}_{0}}\hat{\mathcal{A}}$ for some
$M_{0}\in\operatorname{\mathcal{D}}(\mathcal{A}_{0})$. Since adic completion
is bounded and hence commutes with uniformly right-bounded geometric
realizations, adic completeness is stable under uniformly right-bounded
geometric realizations. We can thus reduce the claim to the case
$\mathcal{A}=\mathcal{A}_{0}$. We can assume that $M$ is connective and thus
write it as a geometric realization of objects of the form $\bigoplus_{i\in
I}\mathcal{A}[S_{i}]$ for some profinite sets $S_{i}$. By using again the fact
that adic completeness is stable under uniformly right-bounded geometric
realizations, and also under $\omega_{1}$-filtered colimits, we reduce to the
case that $M=\widehat{\bigoplus}_{k\in\mathbb{N}}\,\mathcal{A}[S_{k}]$ for
some profinite sets $S_{k}$. Note that it is enough to show that
$-\otimes_{\mathcal{A}}\hat{\mathcal{B}}$ preserves $x$-complete objects for
all $x\in I$, so from now on we work only with $x$-completions. Then we can
write $M=\varinjlim_{\alpha}\prod_{k}x^{\alpha_{k}}\mathcal{A}[S_{k}]$, where
the colimit is taken over all monotonous sequences
$\alpha\colon{\mathbb{Z}}_{\geq 0}\to{\mathbb{Z}}_{\geq 0}$ converging to
$\infty$. We now claim that the natural map

$\displaystyle
M\otimes_{\mathcal{A}}\hat{\mathcal{B}}=\varinjlim_{\alpha}((\prod_{k}x^{\alpha_{k}}\mathcal{A}[S_{k}])\otimes_{\mathcal{A}}\hat{\mathcal{B}})\mathbin{\xrightarrow{\sim}}\varinjlim_{\alpha}\prod_{k}x^{\alpha_{k}}(\mathcal{A}[S_{k}]\otimes_{\mathcal{A}}\hat{\mathcal{B}})$

is an isomorphism. The argument is similar to the proof of [15, Lemma
3.7.(ii)], where the crucial input is that the constituents
$(\prod_{k}x^{\alpha_{k}}\mathcal{A}[S_{k}])\otimes_{\mathcal{A}}\hat{\mathcal{B}}$
are $x$-adically complete – this is true because $\prod_{k}\mathcal{A}[S_{k}]$
is compact in $\operatorname{\mathcal{D}}(\mathcal{A})$ and thus
$(\prod_{k}x^{\alpha_{k}}\mathcal{A}[S_{k}])\otimes_{\mathcal{A}}\hat{\mathcal{B}}$
is compact in $\operatorname{\mathcal{D}}(\hat{\mathcal{B}})$ (as the right
adjoint to $-\otimes_{\mathcal{A}}\hat{\mathcal{B}}$ commutes with colimits).
Now compact objects in $\operatorname{\mathcal{D}}(\hat{\mathcal{B}})$ are
adically complete (by 2.5.(iv)). To finish the proof of (iv) it remains to see
that the natural map

$\displaystyle\varinjlim_{\alpha}\prod_{k}x^{\alpha_{k}}(\mathcal{A}[S_{k}]\otimes_{\mathcal{A}}\hat{\mathcal{B}})\mathbin{\xrightarrow{\sim}}\widehat{\bigoplus}_{k}\,(\mathcal{A}[S_{k}]\otimes_{\mathcal{A}}\hat{\mathcal{B}})$

is an isomorphism, or equivalently that the left-hand side is $x$-adically
complete. This can be checked after applying the forgetful functor to
$\operatorname{\mathcal{D}}(\mathcal{A})$ and on homology, where it is a
straightforward computation. This finishes the proof of (iv).

It remains to prove (iii), so let
$M,N\in\operatorname{\mathcal{D}}^{-}_{\mathrm{cpl}}(\hat{\mathcal{A}})$ be
given. By the same bar resolution argument as in the proof of (iv) we can
assume that $M$ and $N$ come via pullback from adically complete right-bounded
$\mathcal{A}_{0}:=({\mathbb{Z}}[[x_{1},\dots,x_{n}]],{\mathbb{Z}})_{\scalebox{0.5}{$\square$}}$-modules
for some $n\geq 0$. But then by (iv) the claim reduces to the case
$\mathcal{A}=\mathcal{A}_{0}$, where it follows from [16, Proposition
2.12.10]. ∎

######  Example 2.13.

The boundedness assumption in 2.12.(iv) is necessary: If
$M=N=\bigoplus\limits_{i\in{\mathbb{Z}}}\underline{\mathcal{A}}[i]$, then
$M\otimes_{\hat{\mathcal{A}}}N$ is not adically complete (if
$\underline{\mathcal{A}}$ is not discrete), even though $M,N$ are adically
complete. Namely, the tensor product contains the direct summand
$\bigoplus\limits_{i\in{\mathbb{Z}}}\underline{\mathcal{A}}$.

Similarly, the preservation of completeness is wrong for the base change
$-\otimes_{\mathcal{A}}\mathcal{B}$ of adic analytic rings with the usual
definition of $D_{\scalebox{0.5}{$\square$}}(-)$.

In the presence of finite Tor dimension, the preservation of adic completeness
can be strengthened to unbounded complexes:

######  Corollary 2.14.

Let $\mathcal{A}\to\mathcal{B}$ be an adic map of adic analytic rings such
that for some ideal of definition $I$ the map $\mathcal{A}/I\to\mathcal{B}/I$
has finite Tor dimension. Then $\hat{\mathcal{A}}\to\hat{\mathcal{B}}$ has
finite Tor dimension and the functor
$-\otimes_{\hat{\mathcal{A}}}\hat{\mathcal{B}}$ preserves adic completeness.

###### Proof.

We first prove the claim about Tor dimension, so let
$M\in\operatorname{\mathcal{D}}^{\heartsuit}(\hat{\mathcal{A}})$ be given.
Then we can write $M$ as a filtered colimit of adically complete modules; by
truncating we can assume that these adically complete modules lie in
$\operatorname{\mathcal{D}}^{\heartsuit}$. We can thus assume that $M$ itself
is adically complete, hence the same is true for
$N:=M\otimes_{\hat{\mathcal{A}}}\hat{\mathcal{B}}$ by 2.12.(iv). But by
assumption $N/I$ is bounded to the left (independent of $M$), hence so is $N$.

The second claim follows from the finite Tor dimension by writing any adically
complete module as a filtered colimit over its truncations $\tau_{\geq n}$ for
$n\to-\infty$. ∎

Unsurprisingly, the difference between $\mathcal{A}$ and $\hat{\mathcal{A}}$
vanishes on complete objects; more precisely we have:

######  Lemma 2.15.

Let $\mathcal{A}$ be an adic analytic ring. Then the functor
$\alpha_{\ast}\colon\operatorname{\mathcal{D}}(\hat{\mathcal{A}})\to\operatorname{\mathcal{D}}(\mathcal{A})$
restricts to an equivalence
$\operatorname{\mathcal{D}}_{\mathrm{cpl}}(\hat{\mathcal{A}})\cong\operatorname{\mathcal{D}}_{\mathrm{cpl}}(\mathcal{A})$.

###### Proof.

By Lemma 2.12 the restriction
$\alpha_{\ast}\colon\operatorname{\mathcal{D}}_{\mathrm{cpl}}(\hat{\mathcal{A}})\to\operatorname{\mathcal{D}}_{\mathrm{cpl}}(\mathcal{A})$
is well-defined. By general theory of completion, it admits as a left adjoint
the functor $\hat{\alpha}^{\ast}:=(-)_{{\mathrm{cpl}}}\circ\alpha^{\ast}$. As
both $\alpha_{\ast},\hat{\alpha}^{\ast}$ commute with the reduction
$\mathcal{A}\to\mathcal{A}/I$ for some (finitely generated) ideal of
definition $I$, the unit/counit of this adjunction are isomorphisms by Lemma
2.10. This finishes the proof. ∎

For any adic analytic ring $\mathcal{A}$,
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ is a compactly generated
closed symmetric monoidal category with compact unit object
$\underline{\mathcal{A}}$, hence the definition of nuclear objects from [6,
Lecture VIII] applies:

######  Definition 2.16.

We denote by
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})^{\mathrm{nuc}}\subseteq\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$
the full subcategory spanned by the nuclear objects, as defined in [6,
Definition 8.5].

######  Proposition 2.17.

For every adic analytic ring $\mathcal{A}$, the functor
$-\otimes_{\mathcal{A}}\hat{\mathcal{A}}$ induces an equivalence of symmetric
monoidal categories

$\displaystyle\operatorname{\mathcal{D}}(\mathcal{A})^{\mathrm{nuc}}=\operatorname{\mathcal{D}}(\hat{\mathcal{A}})^{\mathrm{nuc}}.$

###### Proof.

We use the notation $\alpha^{*}=(-)\otimes_{\mathcal{A}}\hat{\mathcal{A}}$ and
$\alpha_{*}$ from Lemma 2.5. We first show that $\alpha_{*}$ preserves nuclear
modules (for $\alpha^{\ast}$ this follows by symmetric monoidality). Since
$\alpha_{*}$ preserves colimits, this boils down to showing that for any
compact objects
$P^{\prime},Q^{\prime}\in\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ and
any trace-class map $f\colon P^{\prime}\to Q^{\prime}$ in
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$, the induced map
$\alpha_{*}f\colon\alpha_{*}P^{\prime}\to\alpha_{*}Q^{\prime}$ is trace-class.
This follows if we can show that the natural map

$\displaystyle(\alpha_{*}P^{\prime})^{\vee}\otimes\alpha_{*}Q^{\prime}\mathbin{\xrightarrow{\sim}}\alpha_{*}((P^{\prime})^{\vee}\otimes
Q^{\prime})$

is an isomorphism in $\operatorname{\mathcal{D}}(\mathcal{A})$. As this claim
is stable under finite colimits and retracts in $P^{\prime},Q^{\prime}$ we may
assume that $P^{\prime}=\alpha^{\ast}P,Q^{\prime}=\alpha^{\ast}Q$ for two
compact objects $P,Q\in\operatorname{\mathcal{D}}(\mathcal{A})$. By Lemma 2.5
$\alpha^{\ast}\alpha_{\ast}\alpha^{\ast}P\cong\alpha^{\ast}P$ as
$P\in\operatorname{\mathcal{D}}(\mathcal{A})$ is compact. Using adjunctions
and
$\alpha_{\ast}\alpha^{\ast}\underline{\mathcal{A}}\cong\underline{\mathcal{A}}$
shows $(\alpha_{\ast}\alpha^{\ast}P)^{\vee}\cong P^{\vee}$; in particular this
object is right bounded and adically complete.444Here, we are using that
$\mathcal{A}$ is living over ${\mathbb{Z}}_{\scalebox{0.5}{$\square$}}$, and
that the compact projective generators for
${\mathbb{Z}}_{\scalebox{0.5}{$\square$}}$ are internally projective to obtain
the right-boundedness of $P^{\vee}$. Similarly, $(\alpha^{\ast}P)^{\vee}$ is
right-bounded because $\alpha_{\ast}((\alpha^{\ast}P)^{\vee})\cong P^{\vee}$
by adjunctions and symmetric monoidality of $\alpha^{\ast}$ while $P^{\vee}$
is right-bounded. It follows from 2.12.(iii) that the right-hand side of the
above claimed isomorphism is adically complete, and it follows from [16,
Proposition 2.12.10] that the left-hand side is adically complete (here we use
that $(\alpha_{*}\alpha^{*}P)^{\vee}=P^{\vee}$ is discrete modulo any ideal of
definition as can be checked directly). Hence the above isomorphism can be
checked modulo any ideal of definition of $\underline{\mathcal{A}}$, where it
follows from Lemma 2.10.

We have established that $\alpha_{*}$ restricts to a functor
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})^{\mathrm{nuc}}\to\operatorname{\mathcal{D}}(\mathcal{A})^{\mathrm{nuc}}$,
which is automatically conservative. Thus in order to prove the claimed
equivalence of categories it is now enough to show that $\alpha^{*}$ is fully
faithful on nuclear modules, i.e. for any nuclear $M$ the natural map
$M\mathbin{\xrightarrow{\sim}}\alpha_{*}\alpha^{*}M$ is an isomorphism. Since
$\alpha_{*}\alpha^{*}$ preserves colimits, this reduces to showing the
following: Let $(P_{n})_{n}$ be a sequence of compact objects in
$\operatorname{\mathcal{D}}(\mathcal{A})$ with trace-class transition maps;
then $\varinjlim_{n}P_{n}=\varinjlim_{n}(P_{n})_{\mathrm{cpl}}$. To prove
this, it suffices to check that the map $P_{n}\to P_{n+1}$ factors as
$P_{n}\to(P_{n})_{\mathrm{cpl}}\to P_{n+1}$. But this follows easily from the
definition of trace-class maps by observing that
$(P_{n})^{\vee}=((P_{n})_{\mathrm{cpl}})^{\vee}$ as $\underline{\mathcal{A}}$
is complete. ∎

The following result provides a concrete description of nuclear
$\underline{\mathcal{A}}$-modules. Namely, they are exactly the ones that can
be written as colimits of “Banach” modules:

######  Lemma 2.18.

Let $\mathcal{A}$ be an adic analytic ring with ideal of definition $I$.

  1. (i)

If $M\in\operatorname{\mathcal{D}}(\mathcal{A})$ is adically complete and
$M/I$ is discrete, then $M$ is nuclear.

  2. (ii)

The category
$\operatorname{\mathcal{D}}^{\mathrm{nuc}}(\hat{\mathcal{A}})=\operatorname{\mathcal{D}}^{\mathrm{nuc}}(\mathcal{A})$
is generated under colimits by adically complete objects
$M\in\operatorname{\mathcal{D}}(\mathcal{A})$ with $M/I$ discrete.

###### Proof.

Let $M$ as in (i). Writing $M$ as a colimit of its truncations $\tau_{\geq
n}M,n\to-\infty$, we can reduce to the case that $M$ is right-bounded. Let
$P\in\operatorname{\mathcal{D}}(\mathcal{A})$ be compact. We need to see that
the map

$P^{\vee}\otimes M\to\underline{\operatorname{Hom}}(P,M)$

is an isomorphism. The right hand side is adically complete, because $M$ is,
and the left hand side is adically complete by [16, Proposition 2.12.10.(ii)]
as $P^{\vee}=(P_{\mathrm{cpl}})^{\vee}$ and $M$ are adically complete, and
$M/I$ is discrete. Hence, it suffices to check the claim modulo $I$, where the
claim follows from discreteness of $M/I$ as in [15, Proposition 2.9.7.(i)].

Now we show (ii). Let $M\in\operatorname{\mathcal{D}}(\mathcal{A})$ be
nuclear. By [6, Theorem 8.6.(2)] we may assume that $M$ is basic nuclear. Then
it is sufficient (by the last part of the proof of Proposition 2.17) to show
the following: If $f\colon P_{1}\to P_{2}$ is any trace class morphism between
compact objects in $\operatorname{\mathcal{D}}(\mathcal{A})$, then
$f_{{\mathrm{cpl}}}\colon P_{1,{\mathrm{cpl}}}\to P_{2,{\mathrm{cpl}}}$
factors over the completion $Q$ of $\underline{P_{2,{\mathrm{cpl}}}(\ast)}$.
Here, $\underline{(-)}$ denotes the functor from
$\underline{\mathcal{A}}(\ast)$-modules to
$\operatorname{\mathcal{D}}(\mathcal{A})$, which is left adjoint to the
functor $N\mapsto N(\ast)$. This last claim follows if we can show

$\operatorname{Hom}(P_{1},Q)\cong(P_{1}^{\vee}\otimes
Q)(\ast)\cong(P_{1}^{\vee}\otimes P_{2,{\mathrm{cpl}}})(\ast),$

with the first isomorphism implied by nuclearity of $Q$ as shown in (i). As
the functor $(-)(\ast)$ preserves $I$-adic completeness both sides are
adically complete as $A(\ast)$-modules, by arguments as in Proposition 2.17.
Hence, it suffices to check the statement modulo $I$. But $P_{1}^{\vee}$ is
discrete modulo $I$, and more generally for any discrete
$\mathcal{A}/I$-module $D$ the map

$(D\otimes Q)(\ast)\cong(D\otimes P_{2,{\mathrm{cpl}}})(\ast)$

is an isomorphism. Indeed, this statement commutes with colimits in $D$ and is
clear in the case $D=\mathcal{A}/I$. This finishes the proof. ∎

As the full subcategory
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})^{\mathrm{nuc}}\subseteq\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$
is stable under all colimits, it admits a right adjoint
$(-)_{\mathrm{nuc}}\colon\operatorname{\mathcal{D}}(\hat{\mathcal{A}})\to\operatorname{\mathcal{D}}(\hat{\mathcal{A}})^{\mathrm{nuc}}$.
The next result provides an explicit description of $(-)_{\mathrm{nuc}}$,
similar to [15, Proposition 3.12]. Note that it is not clear to us how to
explicitly compute the analogous nuclearization functor for $\mathcal{A}$ in
place of $\hat{\mathcal{A}}$.

######  Proposition 2.19.

Let $\mathcal{A}$ be an adic analytic ring.

  1. (i)

$(-)_{\mathrm{nuc}}\colon\operatorname{\mathcal{D}}(\hat{\mathcal{A}})\to\operatorname{\mathcal{D}}(\hat{\mathcal{A}})^{\mathrm{nuc}}$
preserves all colimits.

  2. (ii)

If $P\in\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ is compact, then
$P_{\mathrm{nuc}}$ is naturally isomorphic to the completion of
$\alpha^{\ast}(\underline{\alpha_{\ast}P(\ast)})$.

###### Proof.

Sending $P\in\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ compact to the
completion $\alpha^{\ast}(\underline{\alpha_{\ast}P(\ast)})^{\wedge}$ extends
uniquely to a colimit preserving functor

$F\colon\operatorname{\mathcal{D}}(\hat{\mathcal{A}})\to\operatorname{\mathcal{D}}(\hat{\mathcal{A}})^{\mathrm{nuc}}$

(using 2.18.(i) to see that value is indeed nuclear). Let
$\iota\colon\operatorname{\mathcal{D}}(\mathcal{A})^{\mathrm{nuc}}\to\operatorname{\mathcal{D}}(\mathcal{A})$
be the fully faithful inclusion. As the compact generators of
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ are complete (by Lemma 2.12)
there exists a natural transformation, $\iota\circ F\to\mathrm{Id}$. Indeed,
it suffices to construct it on compact objects, and there it suffices to
construct a natural map $\alpha^{\ast}(\underline{\alpha_{\ast}P(\ast)})\to
P$, which exists as $\alpha^{\ast}\circ\underline{(-)}$ is left adjoint to
$(\alpha_{\ast}(-))(\ast)$. Applying $(-)_{\mathrm{nuc}}$ to this
transformation yields a natural transformation $F\to(-)_{\mathrm{nuc}}$. Let
$\mathcal{L}=\varinjlim\limits_{i\in J}\mathcal{L}_{i}$ in
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ with $\mathcal{L}_{i}$
compact, and let
$\mathcal{M}\in\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$. Then

$\operatorname{Hom}_{\operatorname{\mathcal{D}}(\hat{\mathcal{A}})}(\mathcal{L},\iota\circ
F(\mathcal{M}))=\varprojlim\limits_{i\in
J}\operatorname{Hom}_{\operatorname{\mathcal{D}}(\hat{\mathcal{A}})}(\mathcal{L}_{i},\iota\circ
F(\mathcal{M}))=\varprojlim\limits_{i\in
J}(\mathcal{L}_{i}^{\vee}\otimes\iota\circ F(\mathcal{M}))(\ast)$

using nuclearity of $\iota\circ F(M)$ (and the notations
$(-)(\ast)=\operatorname{Hom}_{\operatorname{\mathcal{D}}(\hat{\mathcal{A}})}(\underline{\mathcal{A}},-),(-)^{\vee}=\underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}(\hat{\mathcal{A}})}(-,\underline{\mathcal{A}})$).
On the other hand,

$\operatorname{Hom}_{\operatorname{\mathcal{D}}(\hat{\mathcal{A}})}(\mathcal{L},\iota(\mathcal{M}_{\mathrm{nuc}}))=\varprojlim\limits_{i\in
J}\operatorname{Hom}_{\operatorname{\mathcal{D}}(\hat{\mathcal{A}})}(\mathcal{L}_{i},\mathcal{M}).$

We have to see that both inverse limits agree if $\mathcal{L}$ is nuclear, or
even basic nuclear, i.e., we can assume that
$\mathcal{L}=\varinjlim\limits_{n\in\mathbb{N}}\mathcal{L}_{n}$ with
$\mathcal{L}_{n}$ compact and the transition maps trace-class. This assumption
implies that each map

$\operatorname{Hom}_{\operatorname{\mathcal{D}}(\hat{\mathcal{A}})}(\mathcal{L}_{n+1},\mathcal{M})\to\operatorname{Hom}_{\operatorname{\mathcal{D}}(\hat{\mathcal{A}})}(\mathcal{L}_{n},\mathcal{M})$

factors over the space of trace class maps
$(\mathcal{L}_{n}^{\vee}\otimes\mathcal{M})(\ast)$ from $\mathcal{L}_{n}$ to
$\mathcal{M}$ ([6, Lemma 8.2.(3)]). Hence, it suffices to show that, similarly
to the last assertion in the proof of 2.18.(i), the map

$(P^{\vee}\otimes\iota\circ F(\mathcal{M}))(\ast)\to(P^{\vee}\otimes M)(\ast)$

is an isomorphism for any
$P,M\in\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ with $P$ compact. As
both sides commute with colimits in $M$ we may assume that $M$ is compact.
Then both sides are $I$-adically complete over $A(\ast)$ for some ideal of
definition $I\subseteq A(\ast)$, which reduces us to the case that $I=\\{0\\}$
by passing to the reduction mod $I$. In this case
$\alpha^{\ast},\alpha_{\ast}$ are inverse equivalences (by Lemma 2.10), and
the assertion follows from the last assertion in the proof of 2.18.(i). ∎

From Proposition 2.17 and Lemma 2.18 we can produce a weaker form of a
projection formula for $\alpha_{\ast}$.

######  Corollary 2.20.

Let $\mathcal{A}$ be an adic analytic ring. Let
$M\in\operatorname{\mathcal{D}}(\mathcal{A})^{\mathrm{nuc}}$. Then the natural
map

$M\otimes\alpha_{\ast}(N)\to\alpha_{\ast}(\alpha^{\ast}M\otimes N)$

is an isomorphism for any $N\in\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$.

###### Proof.

Both sides commute with colimits in $N$. Hence, we may assume that
$N\in\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ is compact (and hence
complete by Lemma 2.5). As $M\cong M_{\mathrm{nuc}}$ and the nuclearization
commutes with colimits, and sends compacts of bounded complete objects, which
are discrete modulo some ideal of definition $I$, we may assume that $M$ is
complete and discrete modulo $I$. Then the left hand side is complete by [16,
Proposition 2.12.10.(ii)], and it is by reduction to the discrete case enough
to show that the right hand side is complete, too. By Lemma 2.12 it is
sufficient to show that $\alpha^{\ast}M$ is complete (as $N$ is complete and
right bounded). Now,
$\alpha^{\ast}M\in\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ is nuclear,
which implies $\alpha_{\ast}(\alpha^{\ast}M)\cong M$ by Proposition 2.17. In
particular, $\alpha^{\ast}M$ is complete by Lemma 2.12. ∎

We now prove a relative variant of Lemma 2.9. We use the terminology that a
morphism
$\mathcal{A}=(A,A^{+})_{\scalebox{0.5}{$\square$}}\to\mathcal{B}=(B,B^{+})_{\scalebox{0.5}{$\square$}}$
of adic analytic rings is of $+$-finite type, if
$(B,B^{+})_{\scalebox{0.5}{$\square$}}\cong(B,A\cup\\{S\\})_{\scalebox{0.5}{$\square$}}$
for some finite set $S$.

######  Proposition 2.21.

Let
$f\colon\mathcal{A}=(A,A^{+})_{\scalebox{0.5}{$\square$}}\to\mathcal{B}=(B,B^{+})_{\scalebox{0.5}{$\square$}}$
be an adic morphism of adic analytic rings of $+$-finite type.

  1. (i)

The diagram

${{\operatorname{\mathcal{D}}(\hat{\mathcal{A}})}}$${{\operatorname{\mathcal{D}}(\hat{\mathcal{B}})}}$${{\operatorname{\mathcal{D}}(\mathcal{A})}}$${{\operatorname{\mathcal{D}}(\mathcal{B})}}$$\scriptstyle{-\otimes_{\hat{\mathcal{A}}}\hat{\mathcal{B}}}$$\scriptstyle{\alpha_{\mathcal{A},\ast}}$$\scriptstyle{-\otimes_{\mathcal{A}}\mathcal{B}}$$\scriptstyle{\alpha_{\mathcal{B},\ast}}$

naturally commutes, i.e., base change holds for $\alpha$.

  2. (ii)

The natural functor

$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})\otimes_{\operatorname{\mathcal{D}}(\mathcal{A})}\operatorname{\mathcal{D}}(\mathcal{B})\mathbin{\xrightarrow{\sim}}\operatorname{\mathcal{D}}(\hat{\mathcal{B}})$

is an equivalence.

###### Proof.

For (i) we have to see that the natural morphism

$\mathcal{B}\otimes_{\mathcal{A}}\alpha_{\mathcal{A},\ast}(M)\to\alpha_{\mathcal{B},\ast}(\hat{\mathcal{B}}\otimes_{\hat{\mathcal{A}}}M)$

is an isomorphism for any $M\in\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$.
Both sides commute with colimits in $M$, which reduces to the case
$M=\alpha^{\ast}_{\mathcal{A}}P$ for some compact object
$P\in\operatorname{\mathcal{D}}(\mathcal{A})$. Then the left hand side equates
to $\mathcal{B}\otimes_{\mathcal{A}}P_{\mathrm{cpl}}$ by Lemma 2.5 while the
right hand side is

$\alpha_{\mathcal{B},\ast}(\hat{\mathcal{B}}\otimes_{\hat{\mathcal{A}}}\alpha_{A}^{\ast}(P))\cong\alpha_{\mathcal{B},\ast}(\alpha_{\mathcal{B}}^{\ast}(\mathcal{B}\otimes_{\mathcal{A}}P)\cong(\mathcal{B}\otimes_{\mathcal{A}}P)_{\mathrm{cpl}}$

by Lemma 2.7 and Lemma 2.5, and this object is complete. Hence, it is
sufficient to show that $\mathcal{B}\otimes_{\mathcal{A}}P_{\mathrm{cpl}}$ is
complete because then one can check the statement modulo some ideal of
definition, where it is clear. If $S=\varnothing$, then the statement follows
from [16, Proposition 2.12.10]. Using induction one reduces to the case that
$S=\\{s\\}$. Then

$\mathcal{B}\otimes_{\mathcal{A}}N=\underline{\operatorname{Hom}}_{{\mathbb{Z}}[T]}({\mathbb{Z}}((T^{-1}))/{\mathbb{Z}}[T][-1],\underline{\mathcal{B}}\otimes_{\mathcal{A}}N)$

for any $N\in\operatorname{\mathcal{D}}(\mathcal{A})$ by the right adjoint
assertion to [23, Observation 8.11]. Here,
${\mathbb{Z}}[T]\to\underline{\mathcal{B}}$ is the map classified by $s$.
Again using [16, Proposition 2.12.10] we can conclude that

$\begin{matrix}\mathcal{B}\otimes_{\mathcal{A}}N_{\mathrm{cpl}}&\cong&\underline{\operatorname{Hom}}_{{\mathbb{Z}}[T]}({\mathbb{Z}}((T^{-1}))/{\mathbb{Z}}[T][-1],\underline{\mathcal{B}}\otimes_{\mathcal{A}}N_{\mathrm{cpl}})\\\
&\cong&\underline{\operatorname{Hom}}_{{\mathbb{Z}}[T]}({\mathbb{Z}}((T^{-1}))/{\mathbb{Z}}[T][-1],(\underline{\mathcal{B}}\otimes_{\mathcal{A}}N)_{\mathrm{cpl}})\\\
&\cong&(\mathcal{B}\otimes_{\mathcal{A}}N)_{\mathrm{cpl}}\end{matrix}$

for any $N\in\operatorname{\mathcal{D}}(\mathcal{A})$.

Now we prove (ii). The natural functor

$\Phi\colon\operatorname{\mathcal{D}}(\hat{\mathcal{A}})\otimes_{\operatorname{\mathcal{D}}(\mathcal{A})}\operatorname{\mathcal{D}}(\mathcal{B})\to\operatorname{\mathcal{D}}(\hat{\mathcal{B}})$

commutes with colimits (by definition), and hence has a right adjoint $\Psi$.
As the image of the functor
$\alpha_{\mathcal{B}}^{\ast}\colon\operatorname{\mathcal{D}}(\mathcal{B})\to\operatorname{\mathcal{D}}(\hat{\mathcal{B}})$
generates the target under colimits, the same is true for $\Phi$. This implies
formally that the functor $\Psi$ is conservative. Hence, it is sufficient to
show that $\Phi$ is fully faithful. We may reduce to the cases $B^{+}=A^{+}$
or $B=A$ as the statement is stable under composition of $+$-finite morphisms.
We first deal with the case $B^{+}=A^{+}$. Then
$\operatorname{\mathcal{D}}(\mathcal{B})\cong\mathrm{Mod}_{B}(\operatorname{\mathcal{D}}(\mathcal{A})$,
and hence
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})\otimes_{\operatorname{\mathcal{D}}(\mathcal{A})}\operatorname{\mathcal{D}}(\mathcal{B})\cong\mathrm{Mod}_{\alpha^{\ast}_{A}B}(\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$.
To get fully faithfulness of $\Phi$ it suffices to see that

$\mathrm{Hom}_{\operatorname{\mathcal{D}}(B)}((B\otimes_{A}P)_{\mathrm{cpl}},(B\otimes_{A}Q)_{\mathrm{cpl}})\cong\mathrm{Hom}_{\operatorname{\mathcal{D}}(\hat{\mathcal{A}}),\alpha^{\ast}_{A}B}(\alpha^{\ast}_{A}B\otimes\alpha^{\ast}P,\alpha^{\ast}_{A}B\otimes\alpha^{\ast}Q)$

for any $P,Q\in\operatorname{\mathcal{D}}(\mathcal{A})$ compact. The left hand
side simplifies to
$\operatorname{Hom}_{\operatorname{\mathcal{D}}(A)}(P,(B\otimes_{A}Q)_{\mathrm{cpl}})$,
while the right hand side is by adjunctions isomorphic to
$\operatorname{Hom}_{\operatorname{\mathcal{D}}(A)}(P,\alpha_{A,\ast}(\alpha^{\ast}_{A}B\otimes\alpha^{\ast}_{A}(Q)))$.
Now, $(B\otimes_{A}Q)_{\mathrm{cpl}}\cong B\otimes_{A}Q_{\mathrm{cpl}}$ by
[16, Proposition 2.12.10.(ii)], which agrees with
$\alpha_{A,\ast}(\alpha^{\ast}_{A}B\otimes\alpha^{\ast}_{A}(Q))\cong
B\otimes\alpha_{A,\ast}\alpha^{\ast}_{A}(Q)$ by Corollary 2.20 and Lemma 2.5.
This finishes the proof in the case that $B^{+}=A^{+}$. Hence, assume that
$B=A$. As $f\colon\mathcal{A}\to\mathcal{B}$ is of $+$-finite type this
implies that the functor
$f^{\ast}:=\mathcal{B}\otimes_{\mathcal{A}}(-)\to\operatorname{\mathcal{D}}(\mathcal{A})\to\operatorname{\mathcal{D}}(\mathcal{B})$
admits a fully faithful left adjoint $f_{!}$ satisfying the projection
formula. Indeed, it is sufficient to treat the case $S=\\{s\\}$, where it
follows by base change from the discussion in [23, Lecture 8]. We first check
that
$\hat{f}^{\ast}:=\hat{\mathcal{B}}\otimes_{\hat{\mathcal{A}}}(-)\colon\operatorname{\mathcal{D}}(\hat{\mathcal{A}})\to\operatorname{\mathcal{D}}(\hat{\mathcal{B}})$
commutes with products, so that $\hat{f}^{\ast}$ admits a left adjoint
$\hat{f}_{!}$. As $\alpha_{B,\ast}$ is conservative, it is sufficient to show
that $\alpha_{B,\ast}\hat{f}^{\ast}\cong f^{\ast}\alpha_{A,\ast}$ (by the
proven assertion (i)) commutes with products. This is clear as
$f^{\ast},\alpha_{A,\ast}$ are right adjoints. Applying (i) again shows that
$\hat{f}_{!}\alpha_{A}^{\ast}\cong\alpha_{B}^{\ast}f_{!}$. As the essential
image of $\alpha_{B}^{\ast}$ generates
$\operatorname{\mathcal{D}}(\hat{\mathcal{B}})$ under colimits this implies
that the natural transformation

$\mathrm{Id}\to\hat{f}^{\ast}\hat{f}_{!}$

is an isomorphism because $f^{\ast}f_{!}\cong\mathrm{Id}$. Similarly, one
checks that $\hat{f}_{!}$ satisfies the projection formula. This implies that
$\hat{f}^{\ast}$ is the open localization associated with the idempotent
algebra $\mathrm{cone}(\hat{f}_{!}(1)\to
1)\in\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ ([6, Proposition 6.5]).
However, this algebra is the pullback of the idempotent algebra
$\mathrm{cone}(f_{!}(1)\to 1)$ because
$\alpha^{\ast}_{B}f_{!}\cong\hat{f}_{!}\alpha^{\ast}_{A}$. By Lemma 2.35 this
implies the claim. ∎

###  2.3 Complete descent

Next we discuss base-change in the setting of adic rings, in particular with
our new modified version of modules. As we will show below, base-change holds
in great generality as long as the involved maps are adic.

######  Lemma 2.22.

Let $(A,A^{+})_{\scalebox{0.5}{$\square$}}$ be an adic analytic ring.

  1. (i)

For every element $a\in\pi_{0}A$ with induced map ${\mathbb{Z}}[x]\to A$ we
have

$\displaystyle(A,A^{+})_{\scalebox{0.5}{$\square$}}\otimes_{({\mathbb{Z}}[x],{\mathbb{Z}})_{\scalebox{0.5}{$\square$}}}{\mathbb{Z}}[x]_{\scalebox{0.5}{$\square$}}=(A,A^{+}[a])_{\scalebox{0.5}{$\square$}}.$

  2. (ii)

We have

$\displaystyle(A,A^{+})_{\scalebox{0.5}{$\square$}}\otimes_{{\mathbb{Z}}_{\scalebox{0.5}{$\square$}}}{\mathbb{Z}}[x]_{\scalebox{0.5}{$\square$}}=(A\langle
x\rangle,A^{+}[x])_{\scalebox{0.5}{$\square$}}.$

###### Proof.

Part (i) follows easily from the definitions by using that $A$ is solid over
any discrete ring (because it is a limit of discrete rings). For (ii) we note
that on modules the functor
$-\otimes_{{\mathbb{Z}}_{\scalebox{0.5}{$\square$}}}{\mathbb{Z}}[x]_{\scalebox{0.5}{$\square$}}$
preserves limits (see [16, Lemma 2.9.5]), so in particular it preserves adic
completeness. Then the claim reduces to the observations that the right-hand
side is an analytic ring and that the statement is true modulo $I$. ∎

For adic analytic rings we have the following characterization of steady maps
(see [24, Definition 12.13]) in terms of adicness:

######  Lemma 2.23.

A map of adic analytic rings is steady if and only if it is adic.

###### Proof.

Let
$f\colon(A,A^{+})_{\scalebox{0.5}{$\square$}}\to(B,B^{+})_{\scalebox{0.5}{$\square$}}$
be a map of adic analytic rings. First assume that $f$ is adic. In order to
show that $f$ is steady, it is enough to show that
$(A,A^{+})_{\scalebox{0.5}{$\square$}}\to(B,A^{+})_{\scalebox{0.5}{$\square$}}$
and $(B,A^{+})_{\scalebox{0.5}{$\square$}}\to(B,B^{+})$ are steady. For the
second map this follows from 2.22.(i) and [16, Proposition 2.9.7.(ii)] by
stability of steadiness under base change. The first map is an induced
analytic ring structure, so the statement reduces to showing that $B$ is
nuclear in $\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(A,A^{+})$
(see [16, Corollary 2.3.23]), which follows directly from 2.18.(i) by adicness
of $f$. We now prove the converse, so assume that $f$ is steady and let $I$ be
an ideal of definition for $A$. Consider the map $g\colon(A,A^{+})\to(A\langle
x\rangle,A^{+}[x])$. Then by 2.22.(ii) the base-change of $g$ along $f$ is
given by the map $g^{\prime}\colon(B,B^{+})\to(B\langle x\rangle,B^{+}[x])$.
Note that “$\langle x\rangle$” means adically completed polynomials with
respect to the adic topology on $A$ respectively $B$. By steadiness of $f$ we
have $B\langle x\rangle=A\langle
x\rangle\otimes_{(A,A^{+})_{\scalebox{0.5}{$\square$}}}(B,B^{+})_{\scalebox{0.5}{$\square$}}$.
Now $B$ is adically complete as an $A$-module. Indeed, to see this we may
assume that $B$ is discrete, in which case it is an $A/I$-module for some
ideal of definition $I\subseteq\pi_{0}A(\ast)$ by nuclearity of $B$ as an
$A$-module, and we have shown the completeness of $B$ as an $A$-module. Hence
by [16, Proposition 2.12.10] $A\langle
x\rangle\otimes_{(A,A^{+})_{\scalebox{0.5}{$\square$}}}(B,B^{+})_{\scalebox{0.5}{$\square$}}$
computes the $I$-adic completion of $B[x]$ (note that this tensor product does
not depend on $B^{+}$ because the $I$-adic completion of $B[x]$ is solid for
any choice of $B^{+}$). For this $I$-adic completion to be the same as the
$J$-adic completion of $B[x]$ for some ideal of definition $J$ of $B$ we must
have that $f$ is adic. Indeed, base change to $\mathcal{A}/I$ reduces to the
case $I=\\{0\\}$, and then $B[x]\cong\bigoplus_{n\geq 0}B\cdot x^{n}$ can be
$J$-adically complete if and only if $J$ is nilpotent.555Let $b\in J$. Then
$\pi_{0}(B[x])$ must contain some element $x$ whose coefficient in front of
$x^{n}$ is $b^{n}$ as follows by the universal case of the ring
${\mathbb{Z}}[[b]]$. This forces $b$ to be nilpotent, and thus $J$ as well
because $J$ is finitely generated. ∎

######  Lemma 2.24.

Let
$(B,B^{+})_{\scalebox{0.5}{$\square$}}\mathbin{\leftarrow}(A,A^{+})_{\scalebox{0.5}{$\square$}}\to(C,C^{+})_{\scalebox{0.5}{$\square$}}$
be a diagram of adic analytic rings and adic maps. Then

$\displaystyle(B,B^{+})_{\scalebox{0.5}{$\square$}}\otimes_{(A,A^{+})_{\scalebox{0.5}{$\square$}}}(C,C^{+})_{\scalebox{0.5}{$\square$}}=(B\hat{\otimes}_{A}C,B^{+}\otimes_{A^{+}}C^{+})_{\scalebox{0.5}{$\square$}}.$

In particular this tensor product is still adic.

###### Proof.

We can reduce to the cases $A^{+}=B^{+}$ and $A=B$. The second case follows
easily from 2.22.(i). The first case follows from [16, Proposition 2.12.10]. ∎

######  Remark 2.25.

In general, the single notation $\hat{\mathcal{A}}$ is not well-defined, but
should be seen as a convenient replacement for the datum
$(\mathcal{A},\operatorname{\mathcal{D}}(\hat{\mathcal{A}}))$. As an example,
given a diagram $\mathcal{B}\mathbin{\leftarrow}\mathcal{A}\to\mathcal{C}$ of
adic analytic rings and adic maps, we denote
$\hat{\mathcal{B}}\otimes_{\hat{\mathcal{A}}}\hat{\mathcal{C}}:=(\mathcal{B}\otimes_{\mathcal{A}}\mathcal{C})\,\widehat{}\,$,
i.e., it denotes the datum
$(\mathcal{B}\otimes_{\mathcal{A}}\mathcal{C},\operatorname{\mathcal{D}}((\mathcal{B}\otimes_{\mathcal{A}}\mathcal{C})\,\widehat{}\,))$.

######  Corollary 2.26.

Let $\mathcal{B}\mathbin{\leftarrow}\mathcal{A}\to\mathcal{C}$ be a diagram of
adic analytic rings and adic maps. Then for every
$M\in\operatorname{\mathcal{D}}(\hat{\mathcal{B}})$ the natural morphism

$\displaystyle
M\otimes_{\hat{\mathcal{A}}}\hat{\mathcal{C}}\mathbin{\xrightarrow{\sim}}M\otimes_{\hat{\mathcal{B}}}(\hat{\mathcal{B}}\otimes_{\hat{\mathcal{A}}}\hat{\mathcal{C}})$

is an isomorphism, i.e. the following base-change diagram commutes:

${\operatorname{\mathcal{D}}(\hat{\mathcal{B}}\otimes_{\hat{\mathcal{A}}}\hat{\mathcal{C}})}$${\operatorname{\mathcal{D}}(\hat{\mathcal{B}})}$${\operatorname{\mathcal{D}}(\hat{\mathcal{C}})}$${\operatorname{\mathcal{D}}(\hat{\mathcal{A}})}$

###### Proof.

The existence of the natural morphism is formally implied by adjunctions as

$(-\otimes_{\hat{\mathcal{A}}}\hat{\mathcal{B}})\otimes_{\hat{\mathcal{B}}}(\hat{\mathcal{B}}\otimes_{\hat{\mathcal{A}}}\hat{\mathcal{C}})\cong(-\otimes_{\hat{\mathcal{A}}}\hat{\mathcal{C}})\otimes_{\hat{\mathcal{C}}}(\hat{\mathcal{B}}\otimes_{\hat{\mathcal{A}}}\hat{\mathcal{C}})$

by Lemma 2.7. Both sides of the claimed isomorphism commute with all colimits,
so we can w.l.o.g. assume that $M$ is compact. In particular $M$ is right-
bounded and adically complete, hence by 2.12.(iv) both sides of the claimed
isomorphism are adically complete. But then we can check the claimed
isomorphism modulo any ideal of definition, so by Lemma 2.10 we reduce to the
well-known claim that maps of discrete Huber pairs are steady (see [16,
Proposition 2.9.7.(ii)]). ∎

We are finally in the position to discuss descent in the setting of
$\hat{\mathcal{A}}$-modules. As promised we will see that descendability can
be checked modulo all powers of an ideal of definition.

######  Definition 2.27.

A map $\mathcal{A}\to\mathcal{B}$ of adic analytic rings is called _adically
descendable of index $\leq d$_ if it is adic and for some ideal of definition
$I$ and all $n\geq 1$ the map $\mathcal{A}/I^{n}\to\mathcal{B}/I^{n}$ is
descendable of index $\leq d$ in the sense of [16, Definition 2.6.7].

######  Remark 2.28.

If $\mathcal{A}\to\mathcal{B}$ is a descendable morphism of adic analytic
rings then it is in particular adically descendable because descendability is
stable under the base-change along $\mathcal{A}\to\mathcal{A}/I^{n}$ (see [16,
Lemma 2.6.9]). The converse seems to be wrong unless the compact objects in
$\operatorname{\mathcal{D}}(\mathcal{A})$ are adically complete. This is fixed
by $\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$, as the following result
shows.

######  Example 2.29.

A typical example for an adically descendable map $\mathcal{A}\to\mathcal{B}$
of index $\leq 2$ is an adic $I$-completely faithfully flat and $I$-completely
finitely presented map, i.e., $I$ is an ideal of definition of $\mathcal{A}$
and $\underline{\mathcal{A}}/I^{n}\to\underline{\mathcal{B}}/I^{n}$ is
faithfully flat and finitely presented (on $\pi_{0}$), cf. [16, Lemma 2.10.6]
for the static case, and [19, Theorem 4.15] for the general case.

######  Theorem 2.30.

Let $\mathcal{A}\to\mathcal{B}$ be an adically descendable morphism of adic
analytic rings. Then modules descend along
$\hat{\mathcal{A}}\to\hat{\mathcal{B}}$, i.e. the functor

$\displaystyle\operatorname{\mathcal{D}}(\hat{\mathcal{A}})\mathbin{\xrightarrow{\sim}}\varprojlim_{n\in\Delta}\operatorname{\mathcal{D}}(\hat{\mathcal{B}}\otimes_{\hat{\mathcal{A}}}\dots\otimes_{\hat{\mathcal{A}}}\hat{\mathcal{B}})$

is an equivalence of categories.

###### Proof.

Let $\operatorname{End}^{L}(\operatorname{\mathcal{D}}(\hat{\mathcal{A}}))$
denote the category of
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$-enriched colimit-preserving
functors
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})\to\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$
(cf. [16, §2.5, §A.4]). It comes equipped with the composition monoidal
structure and there is an embedding
$\eta\colon\operatorname{\mathcal{D}}(\hat{\mathcal{A}})\mathrel{\hookrightarrow}\operatorname{End}^{L}(\operatorname{\mathcal{D}}(\hat{\mathcal{A}}))$
via $M\mapsto-\otimes_{\hat{\mathcal{A}}}M$. As in [16, Lemma 2.5.6] the map
$f\colon\mathcal{A}\to\mathcal{B}$ induces a pair of adjoint functors

$\displaystyle
f^{\natural}\colon\operatorname{End}^{L}(\operatorname{\mathcal{D}}(\hat{\mathcal{A}}))\rightleftarrows\operatorname{End}^{L}(\operatorname{\mathcal{D}}(\hat{\mathcal{B}}))\nobreak\mskip
6.0mu plus 1.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip
2.0muf_{\natural},$

where $f_{\natural}$ acts on underlying functors as $F\mapsto
f_{*}\mathbin{\circ}F\mathbin{\circ}f^{*}$; here $f_{*}$ denotes the forgetful
functor and $f^{*}:=-\otimes_{\hat{\mathcal{A}}}\hat{\mathcal{B}}$ is the
base-change. Note that $f^{\natural}$ exists by the adjoint functor theorem in
our case, as we chose a cutoff cardinal $\kappa$ in the beginning so that
$\operatorname{End}^{L}(\operatorname{\mathcal{D}}(\hat{\mathcal{A}}))$ is
presentable. Also, via compatibility with $\eta$ one sees that
$f^{\natural}\operatorname{id}=\operatorname{id}$.

Note that for every $x\in\pi_{0}A$ and every
$F\in\operatorname{End}^{L}(\operatorname{\mathcal{D}}(\hat{\mathcal{A}}))$,
the pointwise multiplication by $x$ induces an endomorphism of $F$; indeed,
via the embedding $\eta$ we can define this endomorphism on
$\operatorname{id}$ and then also on $F=\operatorname{id}\mathbin{\circ}F$. It
thus makes sense to speak of adically complete endofunctors in
$\operatorname{End}^{L}(\operatorname{\mathcal{D}}(\hat{\mathcal{A}}))$. We
now observe:

  * (a)

An endofunctor
$F\in\operatorname{End}^{L}(\operatorname{\mathcal{D}}(\hat{\mathcal{A}}))$ is
adically complete if and only if for every compact
$P\in\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$,
$F(P)\in\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ is adically complete.

Indeed, adic completeness of enriched endofunctors can be checked on the
underlying ordinary functors (see [16, Lemma A.4.5.(ii)]), i.e., in the
category
$\operatorname{Fun}^{L}(\operatorname{\mathcal{D}}(\hat{\mathcal{A}}),\operatorname{\mathcal{D}}(\hat{\mathcal{A}}))$
of colimit-preserving functors
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})\to\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$.
If
$\mathcal{C}_{\mathcal{A}}\subseteq\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$
denotes the full subcategory of compact objects, then by the universal
property of $\operatorname{Ind}$-completions we have

$\displaystyle\operatorname{Fun}^{L}(\operatorname{\mathcal{D}}(\hat{\mathcal{A}}),\operatorname{\mathcal{D}}(\hat{\mathcal{A}}))=\operatorname{Fun}^{\mathrm{ex}}(\mathcal{C}_{\mathcal{A}},\operatorname{\mathcal{D}}(\hat{\mathcal{A}})),$

where the right-hand side denotes exact functors
$\mathcal{C}_{\mathcal{A}}\to\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$.
Since limits of exact functors between stable categories are exact, limits on
the right-hand side are computed pointwise. But this immediately implies (a).
From (a) we immediately deduce that
$\operatorname{id}\in\operatorname{End}^{L}(\operatorname{\mathcal{D}}(\hat{\mathcal{A}}))$
is adically complete; here it is crucial to work with
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$ instead of
$\operatorname{\mathcal{D}}(\mathcal{A})$!

Now let

$\displaystyle\mathcal{K}:=\operatorname{fib}(\operatorname{id}\to
f_{\natural}\operatorname{id})\in\operatorname{End}^{L}(\operatorname{\mathcal{D}}(\hat{\mathcal{A}})),$

where the map $\operatorname{id}\to
f_{\natural}\operatorname{id}=f_{\natural}f^{\natural}\operatorname{id}$ comes
from the unit of the adjunction. We now claim:

  * (b)

For some $d\geq 0$ the natural map $\mathcal{K}^{d}\to\operatorname{id}$ is
zero.

To prove this, let $d^{\prime}\geq 0$ be such that each map
$f_{n}\colon\mathcal{A}/I^{n}\to\mathcal{B}/I^{n}$ is descendable of index
$\leq d^{\prime}$. We claim that $d=2d^{\prime}$ works. To see this, denote
$g_{n}\colon\mathcal{A}\to\mathcal{A}/I^{n}$ the projection, so that we get
adjoint functors

$\displaystyle
g_{n}^{\natural}\colon\operatorname{End}^{L}(\operatorname{\mathcal{D}}(\hat{\mathcal{A}}))\rightleftarrows\operatorname{End}^{L}(\operatorname{\mathcal{D}}(\mathcal{A}/I^{n}))\nobreak\mskip
6.0mu plus 1.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip
2.0mug_{n\natural}.$

(Here we implicitly use Lemma 2.10). The adic completeness of
$\operatorname{id}$ implies that the natural map
$\operatorname{id}\mathbin{\xrightarrow{\sim}}\varprojlim_{n}g_{n\natural}\operatorname{id}$
is an isomorphism. We deduce the following identity of spectra:

$\displaystyle\operatorname{Hom}(\mathcal{K}^{d^{\prime}},\operatorname{id})=\varprojlim_{n}\operatorname{Hom}(\mathcal{K}^{d^{\prime}},g_{n\natural}\operatorname{id})=\varprojlim_{n}\operatorname{Hom}(g_{n}^{\natural}\mathcal{K}^{d^{\prime}},\operatorname{id})$

The induced map
$\operatorname{Hom}(\mathcal{K}^{d^{\prime}},\operatorname{id})\to\operatorname{Hom}(g_{n}^{\natural}\mathcal{K}^{d^{\prime}},g_{n}^{\natural}\operatorname{id})$
is the one coming from the functoriality of $g_{n}^{\natural}$. By [16,
Proposition A.4.17] the functor $g_{n}^{\natural}$ is naturally a monoidal
functor, so that if we denote
$\mathcal{K}_{n}:=\operatorname{fib}(\operatorname{id}\to
f_{n\natural}\operatorname{id})\in\operatorname{End}^{L}(\operatorname{\mathcal{D}}(\mathcal{A}/I^{n})$
then $g_{n}^{\natural}\mathcal{K}^{d^{\prime}}=\mathcal{K}_{n}^{d^{\prime}}$.
We deduce that the natural map $\mathcal{K}^{d^{\prime}}\to\operatorname{id}$
gets sent to the natural map
$\mathcal{K}_{n}^{d^{\prime}}\to\operatorname{id}$ under $g_{n}^{\natural}$.
The latter map is zero by choice of $d^{\prime}$. Now (b) follows completely
analogous to the argument in [16, Proposition 2.7.2].

With (b) at hand, the claimed descent of modules is now formal: By the same
procedure as in [16, Proposition 2.6.5] we deduce from (b) that
$\operatorname{id}$ can be obtained from $f_{\natural}\operatorname{id}$ in a
finite number of steps, each of which consists of a composition, a finite
limit or a retract. Using Lemmas 2.24 and 2.26, we can thus apply the argument
from [16, Proposition 2.6.3] verbatim. ∎

We also need a way to pass to filtered colimits, similar to [16, §2.7]. This
is fairly straightforward since adic completions are countable limits and
therefore commute with $\omega_{1}$-filtered colimits (in the following result
the superscript $\omega_{1}$ refers to $\omega_{1}$-compact objects):

######  Lemma 2.31.

Let $(\mathcal{A}_{i})_{j\in J}$ be an $\omega_{1}$-filtered diagram of adic
analytic rings and adic transition maps. Then
$\mathcal{A}:=\varinjlim_{j}\mathcal{A}_{j}$ is an adic analytic ring and the
natural functor

$\displaystyle\varinjlim_{j}\operatorname{\mathcal{D}}(\hat{\mathcal{A}_{j}})^{\omega_{1}}\mathbin{\xrightarrow{\sim}}\operatorname{\mathcal{D}}(\hat{\mathcal{A}})^{\omega_{1}}$

is an equivalence of categories.

###### Proof.

We can assume that $J$ has some initial element $0\in J$. Let $I$ be some
ideal of definition of $\underline{\mathcal{A}}_{0}$. Then each
$\underline{\mathcal{A}}_{j}$ is $I$-adically complete and since countable
limits commute with $\omega_{1}$-filtered colimits in
$\operatorname{\mathcal{D}}(\mathrm{Cond}(\mathrm{Ab}))$, we deduce that
$\underline{\mathcal{A}}=\varinjlim_{j}\underline{\mathcal{A}}_{j}$ is
$I$-adically complete. Clearly, $\underline{\mathcal{A}}$ is discrete mod $I$
as all transition maps are adic. This shows that $\mathcal{A}$ is indeed an
adic analytic ring. By a similar argument we see that for any
$\omega_{1}$-compact
$P\in\operatorname{\mathcal{D}}(\hat{\mathcal{A}_{0}})^{\omega_{1}}$ the
natural map

$\displaystyle\varinjlim_{j}(P\otimes_{\hat{\mathcal{A}_{0}}}\hat{\mathcal{A}_{j}})\mathbin{\xrightarrow{\sim}}P\otimes_{\hat{\mathcal{A}_{0}}}\hat{\mathcal{A}}$

is an isomorphism in $\operatorname{\mathcal{D}}(\hat{\mathcal{A}_{0}})$.
Indeed, this claim is stable under colimits in $P$, so by [16, Lemma A.2.1] we
can w.l.o.g. assume that $P=\mathcal{A}_{0}[S]_{\hat{I}}$ for some profinite
set $S$. Then the claim boils down to showing that the map
$\varinjlim_{j}\mathcal{A}_{j}[S]_{\hat{I}}\mathbin{\xrightarrow{\sim}}\mathcal{A}[S]_{\hat{I}}$
is an isomorphism. This is true without $I$-completions by [16, Proposition
2.3.15.(ii)] and then follows for the $I$-completed version because
$I$-completions commute with $\omega_{1}$-filtered colimits. With the above
isomorphism at hand, the rest of the argument works exactly as in [16, Lemma
2.7.4]. ∎

######  Definition 2.32.

A map $f\colon\mathcal{A}\to\mathcal{B}$ of adic analytic rings is called
_weakly adically descendable_ if it is an iterated $\omega_{1}$-filtered
colimit of adically descendable maps of adic analytic rings along adic
maps.666I.e., $f$ lies in the smallest subcategory of adic morphisms of adic
analytic rings, which contains the adically descendable maps and is stable
under $\omega_{1}$-filtered colimits. If all the maps in this
$\omega_{1}$-filtered colimit are descendable of index $\leq d$, for some
fixed $d\geq 0$, then we say that $f$ has _index $\leq d$_.

######  Proposition 2.33.

  1. (i)

If $\mathcal{A}\to\mathcal{B}$ is weakly adically descendable then it is adic
and modules descend along $\hat{\mathcal{A}}\to\hat{\mathcal{B}}$.

  2. (ii)

Every adically descendable morphism of adic analytic rings is weakly adically
descendable, and weakly adically descendable morphisms are stable under adic
base-change and $\omega_{1}$-filtered colimits.

  3. (iii)

A completed filtered colimit of adically descendable morphisms of index $\leq
d$ along adic maps of adic analytic rings is weakly adically descendable of
index $\leq 2d$.

###### Proof.

Part (i) follows from Theorems 2.30 and 2.31 by the same argument as in [16,
Proposition 2.7.5] (use also Lemma 2.24 in order to reduce to the case that
all maps in the $\omega_{1}$-filtered colimit have the same source). In part
(ii) the only non-trivial claim is the one about base-change, and this follows
from Lemma 2.24 and [16, Lemma 2.6.9]. Part (iii) follows by the same argument
as in [16, Theorem 2.7.8.(iii)]. ∎

###  2.4 Globalization for stably uniform adic spaces

We finish our general discussion of the modified category
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{A})$
for an adic analytic ring $\mathcal{A}$ with a globalization to (classical)
stably uniform analytic adic spaces. Let $(A,A^{+})$ be a classical complete
analytic Huber pair, and $X:=\operatorname{Spa}(A,A^{+})$. Recall that
$(A,A^{+})$ is called uniform if $A^{\circ}$ is a ring of definition, and that
$(A,A^{+})$ is called stably uniform if for each rational open $U\subseteq X$
the analytic Huber pair $(\mathcal{O}_{X}(U),\mathcal{O}_{X}^{+}(U))$ is
uniform, in which case $(A,A^{+})$ is automatically sheafy ([25, Definition
5.2.4., Theorem 5.2.5.]). In [1, Theorem 4.1] Andreychev has proven that if
$(A,A^{+})$ is sheafy (not necessarily uniform), then the functor
$U\mapsto\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\mathcal{O}_{X}(U),\mathcal{O}_{X}^{+}(U))$
on rational opens $U\subseteq X$ satisfies descent for the analytic topology
on $X$.

######  Definition 2.34.

  1. (a)

For any classical complete analytic uniform Huber pair $(B,B^{+})$ set

$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(B,B^{+}):=\mathrm{Mod}_{B}(\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(B^{\circ},B^{+})).$

Here, we implicitly view $B,B^{\circ}$ as condensed static rings (although
$B^{+}$ is only treated as a discrete ring). Note that we need uniformity to
achieve that $B^{\circ}$ is an adic ring in the sense of Definition 2.1.

  2. (b)

Given a map $f\colon Y=\operatorname{Spa}(B,B^{+})\to
X=\operatorname{Spa}(A,A^{+})$ of stably uniform analytic adic spaces, we let
$f^{\ast}:=\widehat{(B^{\circ},B^{+})}_{\scalebox{0.5}{$\square$}}\otimes_{\widehat{(B^{\circ},B^{+})}_{\scalebox{0.5}{$\square$}}}(-)\colon\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(X):=\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(A,A^{+})\to\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(Y):=\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(B,B^{+})$,
$f_{\ast}\colon\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(Y)\to\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(X)$
be the induced pair of adjoint functors. If we want to distinguish these
functors two functors $f^{\ast},f_{\ast}$ from their counterparts
$f^{\ast}=(B,B^{+})_{\scalebox{0.5}{$\square$}}\otimes_{(A,A^{+})_{\scalebox{0.5}{$\square$}}}(-)\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X):=\operatorname{\mathcal{D}}_{{\scalebox{0.5}{$\square$}}}(A,A^{+})\to\operatorname{\mathcal{D}}_{{\scalebox{0.5}{$\square$}}}(Y):=\operatorname{\mathcal{D}}_{{\scalebox{0.5}{$\square$}}}(B,B^{+})$,
we denote them by $\hat{f}^{\ast},\hat{f}_{\ast}$.

  3. (c)

We let
$\alpha_{X}^{\ast}\colon\operatorname{\mathcal{D}}_{{\scalebox{0.5}{$\square$}}}(X)=\mathrm{Mod}_{A}(\operatorname{\mathcal{D}}_{{\scalebox{0.5}{$\square$}}}(A^{\circ},A^{+}))\to\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(X)$
be the functor induced on module categories by the symmetric monoidal functor
$\widehat{(A^{\circ},A^{+})_{\scalebox{0.5}{$\square$}}}\otimes_{(A^{\circ},A^{+})_{\scalebox{0.5}{$\square$}}}(-)$
from Definition 2.4. We let $\alpha_{X,\ast}$ be the right adjoint of
$\alpha_{X}^{\ast}$.

Let $\mathrm{Sym}$ be the $\infty$-category of cocomplete closed symmetric
monoidal stable $\infty$-categories with morphisms given by cocontinuous,
symmetric monoidal functors, cf. [6, Definition 6.3]. We recall from [6,
Proposition 6.5] that a morphism $g^{\ast}\colon C\to D$ in $\mathrm{Sym}$ is
called an open (resp. closed) immersion if $g^{\ast}$ has a fully faithful
left adjoint $g_{!}$ (resp. colimit preserving fully faithful right adjoint
$g_{\ast}$), which satisfies the projection formula. If $g^{\ast}$ is a closed
immersion, then $g_{\ast}(1_{D})\in C$ is an idempotent algebra, and
$D\cong\mathrm{Mod}_{g_{\ast}(1_{D})}(C)$. If $g^{\ast}$ is an open immersion,
then the cofiber $R:=[g_{!}(1_{D})\to 1_{C}]\in C$ is an idempotent algebra,
and $D$ is isomorphic to the quotient $C/\mathrm{Mod}_{R}(C)$ (via
$g^{\ast}$). We record the following stability of open/closed immersions under
base change with respect to the Lurie tensor product.

######  Lemma 2.35.

Let $h^{\ast}\colon C\to C^{\prime}$, $g^{\ast}\colon C\to D$ be morphisms in
$\mathrm{Sym}$. If $g^{\ast}$ is an open (resp. closed) immersion, the same is
true for the base change $k^{\ast}\colon C^{\prime}\to
D^{\prime}:=C^{\prime}\otimes_{C}D$.

###### Proof.

For closed immersions this is clear as
$\mathrm{Mod}_{R}(C)\otimes_{C}C^{\prime}\cong\mathrm{Mod}_{h^{\ast}(R)}(C^{\prime})$
for any monoid object $R\in C$, and $h^{\ast}$ preserves idempotent algebras.
Thus, assume that $g^{\ast}$ is an open immersion, and let
$R:=[g_{!}(1_{D})\to 1_{C}]\in C$ be the associated idempotent algebra. As the
base change $C^{\prime}\otimes_{C}(-)$ preserves adjunctions between colimit
preserving, $C$-linear functors, the base change $k_{!}$ of $g_{!}$ is a fully
left adjoint of $k^{\ast}$. By $C^{\prime}$-linearity one checks that $k_{!}$
satisfies the projection formula. In particular, $k^{\ast}$ is an open
immersion. Moreover, we note that as in the proof of [6, Proposition 6.5] the
projection formula for $k^{\ast}$ implies that the kernel of the functor
$k^{\ast}$ is exactly the category of $h^{\ast}(R)$-modules in $C^{\prime}$
because $h^{\ast}(R)$ is the cofiber of $k_{!}(1_{D^{\prime}})\to
1_{C^{\prime}}$ by exactness of $h^{\ast}$. ∎

######  Remark 2.36.

The last part of the proof of Lemma 2.35 shows that base change preserves the
“excision sequence” $D\mathbin{\xrightarrow{g_{!}}}C\to\mathrm{Mod}_{R}(C)$
for an open immersion $g^{\ast}\colon C\to D$.

We now show that rational localizations yield open immersion in $\mathrm{Sym}$
for the completed solid category:

######  Lemma 2.37.

Let $X=\operatorname{Spa}(A,A^{+})$ be a stably uniform analytic adic space,
and $j\colon U:=\operatorname{Spa}(B,B^{+})\to X$ a rational open. Then
$\hat{j}^{\ast}\colon\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(X)\to\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(U)$
is an open immersion in $\mathrm{Sym}$. In fact, $\hat{j}^{\ast}$ is the base
change of
$j^{\ast}\colon\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(X)\to\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(U)$
along the functor
$\alpha_{X}^{\ast}\colon\operatorname{\mathcal{D}}_{{\scalebox{0.5}{$\square$}}}(X)\to\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(X)$.

Lemma 2.35 and the proof below shows that $\alpha^{\ast}_{X}\circ
j_{!}\cong\hat{j}_{!}\circ\alpha^{\ast}_{U}$ for the left adjoints
$j_{!},\hat{j}_{!}$ of $j^{\ast},\hat{j}^{\ast}$.

###### Proof.

This follows from Proposition 2.21 and Lemma 2.35: the morphism
$(A^{\circ},A^{+})_{\scalebox{0.5}{$\square$}}\to(B^{\circ},B^{+})_{\scalebox{0.5}{$\square$}}$
is of $+$-finite type, and therefore

$\operatorname{\mathcal{D}}_{\hat{{\scalebox{0.5}{$\square$}}}}(A^{\circ},A^{+})\otimes_{\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(A^{\circ},A^{+})}\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\mathcal{O}_{X}(U)^{\circ},\mathcal{O}_{X}^{+}(U))\mathbin{\xrightarrow{\sim}}\operatorname{\mathcal{D}}_{\hat{{\scalebox{0.5}{$\square$}}}}(\mathcal{O}_{X}(U)^{\circ},\mathcal{O}_{X}^{+}(U)).$
(2.37.1)

We claim that the natural morphism $A\otimes_{A^{\circ}}B^{\circ}\to B$ is an
isomorphism in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(A^{\circ},A^{+})$.
Namely, by [1, Theorem 1.6] the claim is local on
$X=\operatorname{Spa}(A,A^{+})$, which reduces to the case that $A$ is Tate,
where the claim is clear. Given this claim, we can conclude

$\operatorname{\mathcal{D}}_{\hat{{\scalebox{0.5}{$\square$}}}}(X)\otimes_{\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X)}\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(U)\mathbin{\xrightarrow{\sim}}\operatorname{\mathcal{D}}_{\hat{{\scalebox{0.5}{$\square$}}}}(U)$

by tensoring (2.37.1) over
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(A^{\circ},A^{+})$ with
$\mathrm{Mod}_{A}(\operatorname{\mathcal{D}}_{{\scalebox{0.5}{$\square$}}}((A^{\circ},A^{+}))$.
From here Lemma 2.35 implies the claim. ∎

We now establish the following version of Andreychev’s analytic descent
theorem for the completed solid category:

######  Theorem 2.38.

Let $(A,A^{+})$ be a classical stably uniform analytic Huber pair, and
$X:=\operatorname{Spa}(A,A^{+})$.

  1. (i)

The functor
$U\mapsto\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}_{X}(U),\mathcal{O}_{X}^{+}(U))$
on rational open subsets satisfies descent for the analytic topology.

  2. (ii)

If for any rational open $U\subseteq X$ each finite, étale
$\mathcal{O}_{X}(U)$-algebra is stably uniform (e.g., $A$ is sousperfectoid,
[25, Proposition 6.3.3.]), then the functor
$V\mapsto\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}_{V}(V),\mathcal{O}_{V}^{+}(V))$
on the category of étale maps $V\to X$ with $V$ affinoid, satisfies descent
for the étale topology.

Here, a morphism $V\to X$ is étale if it is a composition of open immersions
and finite étale maps (in the sense of [21, Definition 7.1], generalized to
our situation), and the étale topology is defined by jointly surjective étale
morphisms.

###### Proof.

By definition of the étale topology, it suffices to show (i) and descent along
finite étale maps $(A,A^{+})\to(B,B^{+})$. But by definition of the latter,
$(B,B^{+})_{\scalebox{0.5}{$\square$}}=(B,A^{+})_{\scalebox{0.5}{$\square$}}$,
which reduces to descent along the map $A\to B$ of algebras in
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(A,A^{+})$. Now,
$A(\ast)\to B(\ast)$ is descendable as a map of classical rings ([18,
Corollary 3.33]). As it is moreover finite étale, this implies $B\cong
A\otimes_{\underline{A(\ast)}}\underline{B(\ast)}$, and thus $A\to B$ is
descendable in
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(A,A^{+})$
because the functor
$\operatorname{\mathcal{D}}(A(\ast))\to\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(A,A^{+})$
is symmetric monoidal. Hence, (ii) is implied by (i). In order to show (i) we
first note that if $X=\bigcup\limits_{i=1}^{n}U_{i}$ is a finite open cover
with $j_{i}\colon U_{i}\to X$ rational open, then by Lemma 2.37 the functors
$\hat{j}^{\ast}_{i}\colon\operatorname{\mathcal{D}}_{\hat{{\scalebox{0.5}{$\square$}}}}(X)\to\operatorname{\mathcal{D}}_{\hat{{\scalebox{0.5}{$\square$}}}}(U_{i})$
are open immersions in $\mathrm{Sym}$. Moreover, the collection
$\hat{j}^{\ast}_{i},i=1,\ldots,n,$ is jointly conservative. Indeed, this
statement is equivalent to
$\hat{j}_{1,!}(1_{U_{1}})\otimes\ldots\otimes\hat{j}_{n,!}(1_{U_{n}})=0$ and
thus is implied by $j_{1}(1_{U_{1}})\otimes\ldots\otimes j_{n}(1_{U_{n}})=0$
([1, Proposition 4.12.(v)]) using that $\alpha^{\ast}$ commutes with the
$!$-functors (Lemma 2.37). Thus, the assertion follows from the general
descent theorem [6, Proposition 5.5] (or [1, Proposition 4.13] resp. [23,
Proposition 10.5]). ∎

Theorem 2.38 allows us to give a reasonable definition of completed solid
sheaves on every stably uniform analytic adic space:

######  Definition 2.39.

Let $X$ be a (classical) stably uniform analytic adic space. Then we set
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(X)$ as the
inverse limit of
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(A,A^{+})$ over
all affinoid opens $\operatorname{Spa}(A,A^{+})\subseteq X$, with $(A,A^{+})$
a stably uniform analytic Huber pair.

###  2.5 Universal descent

In this section, we show that descent of $\operatorname{\mathcal{D}}(-)$ along
a steady morphism $\mathcal{A}\to\mathcal{B}$ of analytic rings automatically
implies descent after any base change. We discuss variants of this as well,
e.g., for $\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(-)$.

If $f\colon\mathcal{A}\to\mathcal{B}$ is a morphism of analytic rings, we will
denote by
$f^{\ast}(-):=\mathcal{B}\otimes_{\mathcal{A}}(-)\colon\operatorname{\mathcal{D}}(\mathcal{A})\to\operatorname{\mathcal{D}}(\mathcal{B})$
the associated base change functor, and by $f_{\ast}$ its right adjoint. Note
that $f_{\ast}$ is always conservative.

We recall that $f$ is steady if for any morphism
$g\colon\mathcal{A}\to\mathcal{A}^{\prime}$ of analytic rings with base change
$f^{\prime}\colon\mathcal{A}^{\prime}\to\mathcal{B}^{\prime}:=\mathcal{A}^{\prime}\otimes_{\mathcal{A}}\mathcal{B},g^{\prime}\colon\mathcal{B}\to\mathcal{B}^{\prime}$,
the natural transformation

$f^{\ast}g_{\ast}\to g^{\prime}_{\ast}f^{\prime,\ast}$

of functors
$\operatorname{\mathcal{D}}(\mathcal{A}^{\prime})\to\operatorname{\mathcal{D}}(\mathcal{B})$
is an isomorphism ([24, Proposition 12.14]).

######  Lemma 2.40.

Assume that $f\colon\mathcal{A}\to\mathcal{B}$ is a steady morphism of
analytic rings and that modules descend along $f$, i.e., the natural functor
$\operatorname{\mathcal{D}}(A)\to\varprojlim\limits_{[n]\in\Delta}\operatorname{\mathcal{D}}(\mathcal{B}^{n/\mathcal{A}})$
is an equivalence, where $\mathcal{B}^{\bullet/\mathcal{A}}$ denotes the Čech
nerve for $f$. Then for any morphism
$g\colon\mathcal{A}\to\mathcal{A}^{\prime}$ of analytic rings modules descend
along
$f^{\prime}\colon\mathcal{A}^{\prime}\to\mathcal{B}^{\prime}:=\mathcal{A}^{\prime}\otimes_{\mathcal{A}}\mathcal{B}$.

###### Proof.

This follows from [12, Corollary 5.2.2.37.]. We supplement the details using
the notation of [12, Corollary 5.2.2.37]. Thus, we set $\mathcal{C}=\Delta$,
$\chi\colon\Delta^{\vartriangleleft}\to\mathcal{C}at_{\infty},[n]\mapsto\operatorname{\mathcal{D}}((\mathcal{B}^{\prime})^{n/\mathcal{A}^{\prime}})$
with $(\mathcal{B}^{\prime})^{-1/\mathcal{A}^{\prime}}:=\mathcal{A}^{\prime}$,
and similarly,
$\chi^{\prime}([n]):=\operatorname{\mathcal{D}}(\mathcal{B}^{n/\mathcal{A}})$.
To construct the natural transformation $\rho\colon\chi\to\chi^{\prime}$,
given componentwise by $g_{n,\ast}$ for
$g_{n}\colon\mathcal{B}^{n/\mathcal{A}}\to(\mathcal{B}^{\prime})^{n/\mathcal{A}^{\prime}}$
is the natural map, one can argue as follows: clearly, the $g^{\ast}_{n}$
yield a natural transformation $\sigma\colon\chi^{\prime}\to\chi$ (by
naturality of $\operatorname{\mathcal{D}}(-)$ as a functor on the
$\infty$-category of analytic rings). As for each morphism $[n]\to[m]$ in
$\Delta^{\vartriangleleft}$ the induced morphism
$\mathcal{B}^{n/\mathcal{A}}\to\mathcal{B}^{m/\mathcal{A}}$ is steady (using
[24, Proposition 12.15]), we can conclude777Let
$E\to\Delta^{\vartriangleleft},E^{\prime}\to\Delta^{\vartriangleleft}$ be the
cocartesian fibrations classified by $\chi,\chi^{\prime}$. The natural
transformation $\sigma$ unstraightens to a functor $\kappa\colon E^{\prime}\to
E$, which preserves cocartesian arrows. Using [12, 7.3.2.6.] the functor
$\kappa$ admits an adjoint $\lambda\colon E\to E^{\prime}$ (relative to
$\Delta^{\vartriangleleft}$) as this is true over each
$[n]\in\Delta^{\vartriangleleft}$. Now, steadiness of $f$ implies that
$\lambda$ preserves cocartesian arrows, and thus defines a natural
transformation $\rho\colon\chi\to\chi^{\prime}$. By construction of $\lambda$,
$\rho_{[n]}\colon\chi([n])\to\chi^{\prime}([n])$ is given by the pushforward
$g_{n,\ast}$ as desired. that the $g_{n,\ast}$ assemble to a natural
transformation $\rho\colon\chi\to\chi^{\prime}$. Now condition 1) in [12,
5.2.2.37] is the assumption that modules descend along $f$. Condition 2) is
automatic as each $g_{n,\ast}$ is conservative. Condition 3) is clear as
$\operatorname{\mathcal{D}}(\mathcal{A}^{\prime})$ has all limits and
$g_{\ast}$ preserves these. Condition 4) is again automatic because for each
morphism $\alpha\colon[n]\to[m]$ in $\Delta^{\vartriangleleft}$ the functors
$\chi([n])\to\chi([m])$ resp. $\chi^{\prime}([n])\to\chi^{\prime}([m])$ admit
right adjoints given by $\ast$-pushforward, and clearly these right adjoints
commute with $\rho$ (which in component $[n]$ is given by $g_{n,\ast}$). Thus,
[12, 5.2.2.37] is applicable, and yields exactly that modules descend along
$f^{\prime}$. ∎

Next assume that $f\colon\mathcal{A}\to\mathcal{B}$ is a morphism of adic
analytic rings. By Lemma 2.23 we know that $f$ is steady if and only if $f$ is
adic. We get the following analog of Lemma 2.40 for
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$.

######  Lemma 2.41.

Let $f\colon\mathcal{A}\to\mathcal{B}$ be an adic morphism of adic analytic
rings, such that the natural functor
$\operatorname{\mathcal{D}}(\hat{\mathcal{A}})\to\varprojlim\limits_{[n]\in\Delta}\operatorname{\mathcal{D}}(\hat{\mathcal{B}^{n/\mathcal{A}}})$
is an equivalence. Then the same holds true after any base change
$f^{\prime}\colon\mathcal{A}^{\prime}\to\mathcal{A}^{\prime}\otimes_{\mathcal{A}}\mathcal{B}$
of $f$ along an adic morphism $\mathcal{A}\to\mathcal{A}^{\prime}$ of adic
analytic rings.

Note that adicness of $\mathcal{A}\to\mathcal{A}^{\prime}$ implies by Lemma
2.24 that all terms in the Čech nerves are adic analytic rings (and thus the
assertion is well-defined).

###### Proof.

Replacing $\operatorname{\mathcal{D}}((-))$ by
$\operatorname{\mathcal{D}}(\hat{-})$ we can follow the steps in Lemma 2.40.
Using Corollary 2.26 the existence of $\rho$ follows in the same way. The only
statement to check is that
$\hat{f}_{\ast}\colon\operatorname{\mathcal{D}}(\hat{\mathcal{B}})\to\operatorname{\mathcal{D}}(\hat{\mathcal{A}})$
is conservative for any morphism of adic analytic rings
$f\colon\mathcal{A}\to\mathcal{B}$. As $\alpha_{\ast}\hat{f}_{\ast}\cong
f_{\ast}\alpha_{\ast}$ (by Lemma 2.7 and passage to right adjoints) this
follows from conservativity of $f_{\ast}$ and $\alpha_{\ast}$ (Lemma 2.5). ∎

######  Remark 2.42.

Let $f\colon Y\to X$ be any morphism of stably uniform analytic adic spaces.
Assume that all terms $Y^{n/X}$ of the Čech nerve of $f$ are again stably
uniform (this is a serious assumption, cf. Example 2.43), and that
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}$ satisfies
descent for $f$. Then for any morphism $X^{\prime}\to X$ of stably uniform
analytic adic spaces (such that the terms in the Čech nerve of
$f^{\prime}\colon Y\times_{X}X^{\prime}\to X^{\prime}$ are again stably
uniform) the modified quasi-coherent sheaves
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}$ satisfy descent
for $f^{\prime}$ (by reducing to the affinoid case and then using the same
argument as in Lemma 2.40, Lemma 2.41). Note that the steadiness assumption is
automatically satisfied in this case because $X$ is locally Tate.

######  Example 2.43.

Let $\mathbb{Q}_{p}^{\mathrm{cycl}}$ be the completion of
$\mathbb{Q}_{p}(\mu_{p^{\infty}})$. We mention here the well-known example
that the Huber ring
$A:=\mathbb{Q}_{p}^{\mathrm{cycl}}\otimes_{\mathbb{Q}_{p}}\mathbb{Q}_{p}^{\mathrm{cycl}}$
(with $\otimes$ refering to the solid or equivalently the Banach space tensor
product) is not uniform even though each of the factors is. In fact, the
uniform completion of $A$ is the space
$B:=C(\Gamma,\mathbb{Q}_{p}^{\mathrm{cycl}})$ of continuous functions on
$\Gamma={\mathbb{Z}}_{p}^{\times}=\mathrm{Gal}(\mathbb{Q}_{p}^{\mathrm{cycl}}/\mathbb{Q}_{p})$.
The ring
$A_{0}:={\mathbb{Z}}_{p}^{\mathrm{cycl}}\otimes_{{\mathbb{Z}}_{p}}{\mathbb{Z}}_{p}^{\mathrm{cycl}}$
is a ring of definition of $A$ (by definition of the Banach space tensor
product), and the ring $B_{0}:=C(\Gamma,{\mathbb{Z}}_{p}^{\mathrm{cycl}})$ is
a ring of definition of $B$. Moreover, $B_{0}=B^{0}$ because
${\mathbb{Z}}_{p}^{\mathrm{cycl}}$ is the ring of power bounded elements in
$\mathbb{Q}_{p}^{\mathrm{cycl}}$. In particular, $B$ is uniform. There exists
the natural map

$\Phi\colon A_{0}\to B_{0},a\otimes b\mapsto(\gamma\mapsto a\gamma(b)).$

From the relation

$\sum\limits_{\zeta\in\mu_{p^{n}}}\zeta^{x}=\begin{cases}p^{n},\ &x\in
p^{n}{\mathbb{Z}}_{p}\\\ 0,\ &\textrm{ otherwise }\end{cases}$

for $x\in{\mathbb{Z}}_{p}$ one can conclude that for
$a\in{\mathbb{Z}}_{p}^{\times}$ the continuous function

${\mathbb{Z}}_{p}^{\times}\to{\mathbb{Z}}_{p}^{\mathrm{cycl}},\
x\mapsto\frac{1}{p^{h}}\sum\limits_{\zeta\in\mu_{p^{n}}}\zeta^{x-a}$

is the characteristic function $\chi_{a,n}$ of the clopen subset
$a+p^{n}{\mathbb{Z}}_{p}$ of ${\mathbb{Z}}_{p}^{\times}=\Gamma$. We can
conclude that $\Phi$ is injective with image exactly the closure of the
subspace generated by the $p^{n}\chi_{a,n}$ for
$a\in{\mathbb{Z}}_{p}^{\times}$ and $n\geq 0$ (cf. [7, Proposition I.3.4]). In
particular, $\Phi$ extends to an injection $A\to B$ with dense image and this
identifies $B$ as the uniform completion of $A$. Concretely, for
$a\in{\mathbb{Z}}_{p}^{\times}$ and $n\geq 0$ the elements

$\frac{1}{p^{n}}\sum\limits_{\zeta\in\mu_{p^{n}}}\zeta^{-a}\otimes\zeta\in A$

generate $A^{0}\subseteq A$ as an $A_{0}$-module.

##  3
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}^{a}(\mathcal{O}_{X}^{+})$
for perfectoid spaces

In this section we will prove the first part of Theorem 1.1, i.e., $v$-descent
of almost $\mathcal{O}^{+}$-sheaves on perfectoid spaces for our modified
version of $+$-modules from Definition 2.4.

###  3.1 Definition of
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}^{a}(A^{+})$

As in [16, Definition 3.1.1] we denote by $\mathrm{Perfd}^{\mathrm{aff}}$ the
category of affinoid perfectoid spaces over $\mathrm{Spf}\,{\mathbb{Z}}_{p}$.
We note that each object $(A,A^{+})\in\mathrm{Perfd}^{\mathrm{aff}}$ is
classical and static and hence we will often not distinguish between
$(A,A^{+})$ and the classical Tate-Huber pair $(A(\ast),A^{+}(\ast))$.

######  Lemma 3.1.

Let $(A,A^{+})\in\mathrm{Perfd}^{\mathrm{aff}}$, and let
$A^{\circ\circ}\subseteq A^{+}$ be the ideal of topologically nilpotent
elements. Then $A^{+}/A^{\circ\circ}$ is an idempotent algebra in the
classical derived category
$\operatorname{\mathcal{D}}(A^{+})=\operatorname{\mathcal{D}}(A^{+}(\ast))$ of
$A^{+}$-modules.

###### Proof.

This follows from the fact that $A^{\circ\circ}$ is a filtered colimit of
principal ideals generated by non-zero divisors, and that $A^{\circ\circ}\cdot
A^{\circ\circ}=A^{\circ\circ}$. ∎

######  Definition 3.2.

Let $(A,A^{+})\in\mathrm{Perfd}^{\mathrm{aff}}$. Then we define
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}^{a}(A^{+}):=\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}((A^{+},A^{+}))\otimes_{\operatorname{\mathcal{D}}(A^{+})}\operatorname{\mathcal{D}}^{a}(A^{+})$,
where
$\operatorname{\mathcal{D}}(A^{+})\to\operatorname{\mathcal{D}}^{a}(A^{+})$ is
the open immersion defined by the idempotent algebra $A^{+}/A^{\circ\circ}$
from Lemma 3.1.

######  Remark 3.3.

By Lemma 2.35 we have
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}^{a}(A^{+})=\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}((A^{+},A^{+}))/\mathrm{Mod}_{A^{+}/A^{\circ\circ}}(\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}((A^{+},A^{+})))$.

######  Remark 3.4.

We note that $A^{+}\cong A^{\circ}$ in
$\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+})$ as
$A^{\circ\circ}\subseteq A^{+}$. Thus,
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}((A^{\circ},A^{+})),\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}((A^{+},A^{+})$
become isomorphic after
$(-)\otimes_{\operatorname{\mathcal{D}}(A)}\operatorname{\mathcal{D}}^{a}(A)$.
However, in general
$\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+})$ and
$\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(A^{\circ})$
are different.

######  Remark 3.5.

Assume that $\pi\in A\cap A^{+}$ is a pseudo-uniformizer. Then
$\mathrm{Mod}_{A^{+}/\pi}(\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+}))\cong\operatorname{\mathcal{D}}^{a}_{{\scalebox{0.5}{$\square$}}}(A^{+}/\pi)$
is naturally equivalent to the category defined in [16, Definition 3.1.2]
because
$\mathrm{Mod}_{A^{+}/\pi}(\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+}))\cong\mathrm{Mod}_{A^{+}/\pi}(\operatorname{\mathcal{D}}_{{\scalebox{0.5}{$\square$}}}(A^{+}))\otimes_{\operatorname{\mathcal{D}}(A^{+})}\operatorname{\mathcal{D}}^{a}(A^{+})\cong\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}(A^{+}/\pi)$
using Lemma 2.10 in the first isomorphism.

The following special property is the basic reason why (almost) $v$-descent
results for $+$-modules on perfectoid spaces are possible while they fail
drastically for non-perfectoid spaces.

######  Lemma 3.6.

Let

${Y^{\prime}}$${Y}$${X^{\prime}}$${X}$

be a cartesian diagram in $\mathrm{Perfd}^{\mathrm{aff}}$. Write
$X=\operatorname{Spa}(A,A^{+})$,
$X^{\prime}=\operatorname{Spa}(A^{\prime},A^{\prime+})$,
$Y=\operatorname{Spa}(B,B^{+})$ and
$Y^{\prime}=\operatorname{Spa}(B^{\prime},B^{\prime+})$. Then the natural
morphism of analytic rings

$\displaystyle(A^{\prime+})_{\scalebox{0.5}{$\square$}}^{a}\otimes_{(A^{+})_{\scalebox{0.5}{$\square$}}^{a}}(B^{+})_{\scalebox{0.5}{$\square$}}^{a}\mathbin{\xrightarrow{\sim}}(B^{\prime+})_{\scalebox{0.5}{$\square$}}^{a}.$

is an isomorphism.

Here, we view $(A^{+})^{a}_{\scalebox{0.5}{$\square$}}$ etc. as analytic rings
over the almost setup $(A^{+},A^{\circ\circ})$ following [16, Section 2.3].
More concretely, the assertion means that the natural functor
$\operatorname{\mathcal{D}}((B^{\prime})^{+}_{\scalebox{0.5}{$\square$}})\to\operatorname{\mathcal{D}}((A^{\prime+})_{\scalebox{0.5}{$\square$}}\otimes_{A^{+}_{\scalebox{0.5}{$\square$}}}B^{+}_{\scalebox{0.5}{$\square$}})$
becomes an equivalence after
$(-)\otimes_{\operatorname{\mathcal{D}}(A^{+})}\operatorname{\mathcal{D}}^{a}(A^{+})$.

###### Proof.

Since $B^{\prime+}$ is the (completed) integral closure of the image of
$\pi_{0}(A^{\prime+}\otimes_{A^{+}}B^{+})$ in
$\pi_{0}(A^{\prime}\otimes_{A}B)$, by Lemma 2.24 it is enough to show that the
map

$\displaystyle
A^{\prime+a}\hat{\otimes}_{A^{+a}}B^{+a}\mathbin{\xrightarrow{\sim}}B^{\prime+a}$

is an isomorphism of almost rings. But both sides of this claimed isomorphism
are $\pi$-adically complete for any pseudouniformizer $\pi$ of $A$, hence the
claim can be checked modulo $\pi$, where it follows from [16, Lemma 3.1.6]. ∎

As a corollary from the proof we see that the derived (completed) tensor
product $A^{\prime+a}\otimes_{A^{+a}}B^{+a}$ is in fact static, i.e.,
concentrated in degre $0$. This statement also holds before passing to the
almost category by reduction to the case of perfect rings.

###  3.2 Descent on totally disconnected spaces

In this section, we prove the first part of Theorem 1.1, i.e., the existence
of the $v$-sheaf
$X\mapsto\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}_{X})$
on perfectoid spaces. More precisely, in this section we prove the following
theorem.

######  Theorem 3.7.

There is a unique hypercomplete v-sheaf of categories

$\displaystyle(\mathrm{Perfd})^{\mathrm{op}}\to\mathcal{C}at,\qquad
X\mapsto\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})$

such that for every $X=\operatorname{Spa}(A,A^{+})$ which admits a quasi-pro-
étale map to a totally disconnected space we have

$\displaystyle\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})=\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+}).$

We recall that a qcqs perfectoid space $X$ is totally disconnected if each
connected component of $X$ has the form $\operatorname{Spa}(K,K^{+})$ for some
perfectoid field $K$ with open and bounded valuation subring $K^{+}\subseteq
K$ ([22, Lemma 7.3]), in which case $X=\operatorname{Spa}(R,R^{+})$ is
necessarily affinoid, and for any pseudo-uniformizer $\pi\in R$ we have
$(|X|,\mathcal{O}^{+}_{X}/\pi)\cong\operatorname{Spec}(R^{+}/\pi)$ as locally
ringed spaces ([16, Lemma 3.6.1]).

Totally disconnected perfectoid spaces enjoy the following surprising flatness
property ([16, Lemma 3.1.7]).

######  Lemma 3.8.

Let $X=\operatorname{Spa}(A,A^{+})$ be a totally disconnected perfectoid
space, and $\pi\in A$ a uniformizer. Let $A^{\prime}$ be a $\pi$-torsion free
$A^{+}$-algebra of finite type. Then for every connected component
$x=\operatorname{Spec}((A^{+}/\pi)_{x})$ of $\operatorname{Spec}(A^{+}/\pi)$
the map $(A^{+}/\pi)_{x}\to A^{\prime}\otimes_{A^{+}}(A^{+}/\pi)_{x}$ is flat
and finitely presented.

###### Proof.

This is [16, Lemma 3.1.7]. ∎

This result strengthens [22, Proposition 7.23] and is critical for Theorem 1.2
as it implies descent for $\mathcal{O}^{+}/\pi$-modules for a $v$-cover
$f\colon Y\to X$ with $X$ totally disconnected ([16, Lemma 3.1.8]). The proof
of loc. cit. uses critically that the (pre)sheaf
$\operatorname{Spa}(R,R^{+})\mapsto R^{+}/\pi$ on affinoid perfectoid spaces
sends cofiltered inverse limits to filtered colimits, roughly to reduce to a
single connected component of $X$ where Lemma 3.8 applies. This argument does
not work for the sheaf $\mathcal{O}^{+}$. Instead, we roughly spread out the
flatness mod $\pi$ on a connected components (provided by Lemma 3.8) to some
neighborhood, at least up to replacing $Y$ by some disjoint union of some
finite open cover. For the precise assertion below, we denote by
$Y^{\mathrm{wl}}\in\mathrm{Perfd}^{\mathrm{aff}}$ the w-localization of any
$Y\in\mathrm{Perfd}^{\mathrm{aff}}$, as defined in [22, Proposition 7.12].

######  Proposition 3.9.

Let $Y=\operatorname{Spa}(B,B^{+})\to X=\operatorname{Spa}(A,A^{+})$ be a
surjective map of totally disconnected spaces in
$\mathrm{Perfd}^{\mathrm{aff}}$. Then the map $Y^{\mathrm{wl}}\to X$ is
$+$-weakly adically descendable of index $\leq 4$, i.e., if
$Y^{\mathrm{wl}}=\operatorname{Spa}(B_{\mathrm{wl}},B_{\mathrm{wl}}^{+})$ the
map
$(A^{+})_{\scalebox{0.5}{$\square$}}\to(B^{+}_{\mathrm{wl}})_{\scalebox{0.5}{$\square$}}$
is weakly adically descendable of index $\leq 4$ in the sense of Definition
2.32.

###### Proof.

We need to show that the map $A^{+}\to B_{\mathrm{wl}}^{+}$ of adic rings is
an $\omega_{1}$-filtered colimit of adically descendable maps of index $\leq
4$. Let $I$ be the category of factorizations $Y^{\mathrm{wl}}\to Y_{i}\to Y$,
where $Y_{i}$ is a finite disjoint union of qcqs open subsets of $Y$. Then by
[22, Lemma 7.13] $I$ is cofiltered and $Y^{\mathrm{wl}}=\varprojlim_{i\in
I}Y_{i}$. We denote $Y_{i}=\operatorname{Spa}(B_{i},B^{+}_{i})$, so that
$B_{\mathrm{wl}}^{+}$ is the completed filtered colimit of the $B^{+}_{i}$.

In the following discussion, all rings and modules will be discrete and static
unless stated otherwise. For some fixed pseudouniformizer $\pi\in A$ we let
$J_{\pi}$ be the following category: An object of $J_{\pi}$ is a factorization
$A^{+}\to A_{j}\to B_{\mathrm{wl}}^{+}$ of $A^{+}$-algebras such that the map
$A^{+}\to A_{j}$ is a finitely presented map of classical rings which is flat
modulo $\pi$ (here we take the _underived_ quotient $A_{j}/\pi$!) and such
that the map $A_{j}\to B_{\mathrm{wl}}^{+}$ factors over some $B^{+}_{i}$. A
morphism in $J_{\pi}$ is a morphism $\alpha\colon A_{j}\to A_{j^{\prime}}$
over $A^{+}$ such that there is some $i\in I$ with the property that both
$A_{j}\to B^{+}_{\mathrm{wl}}$ and $A_{j^{\prime}}\to B^{+}_{\mathrm{wl}}$
factor over $B^{+}_{i}$ and these factorizations are compatible with $\alpha$.
We now claim that $J_{\pi}$ has the following crucial property:

  * (a)

Let $A_{j}\in J_{\pi}$. Pick some $i$ such that $A_{j}\to B^{+}_{\mathrm{wl}}$
factors over $B^{+}_{i}$ and some $a\in A_{j}$ which is sent to $0$ via
$A_{j}\to B^{+}_{i}$. Then there is some morphism $\alpha\colon A_{j}\to
A_{j^{\prime}}$ in $J_{\pi}$ such that $\alpha(a)=0$.

We now prove this claim, so fix $A_{j}$, $i$ and $a$ as in the claim. Let
$A^{\prime}_{j}\subseteq B^{+}_{i}$ be the image of $A_{j}$ and fix some
connected component $x\in\pi_{0}(X)$. Equivalently $x$ is a connected
component of $\operatorname{Spec}A^{+}/\pi$, i.e. of the form
$x=\operatorname{Spec}(A^{+}/\pi)_{x}$, as $X$ is totally disconnected. For
any $A^{+}$-algebra $A^{\prime}$ we denote
$(A^{\prime}/\pi)_{x}:=A^{\prime}\otimes_{A^{+}}(A^{+}/\pi)_{x}$. Then by
Lemma 3.8 the map $(A^{+}/\pi)_{x}\to(A^{\prime}_{j}/\pi)_{x}$ is flat and
finitely presented for $A^{\prime}_{j}$ as above and in particular the map
$\beta_{x,\pi}\colon(A_{j}/\pi)_{x}\to(A^{\prime}_{j}/\pi)_{x}$ is finitely
presented ([27, Tag 00F4]). Pick some generators
$\overline{a}_{1,x},\dots,\overline{a}_{n,x}$ of $\ker(\beta_{x,\pi})$. They
automatically extend over some (pullback of some) clopen neighbourhood of $x$
in $\operatorname{Spec}A^{+}/\pi$ and hence to all of $A_{j}/\pi$ by extending
them by $0$ on the (pullback of the) clopen complement of that neighbourhood.
Moreover, we can guarantee that these extensions
$\overline{a}_{1},\dots,\overline{a}_{n}\in A_{j}/\pi$ lie in the kernel of
the map $A_{j}/\pi\to A^{\prime}_{j}/\pi$ by shrinking the clopen
neighbourhood of $x$ if necessary. Since the map $\beta\colon A_{j}\to
A^{\prime}_{j}$ is surjective, we can find lifts $a_{1},\dots,a_{n}\in A_{j}$
of $\overline{a}_{1},\dots,\overline{a}_{n}$ which lie in the kernel of
$\beta$. We can assume that $a$ is one of the $a_{k}$’s as by assumption $a$
maps to $0$ in $A_{j}^{\prime}\subseteq B_{i}^{+}$. Now denote

$\displaystyle A^{\prime}_{j,x}:=A_{j}/(a_{1},\dots,a_{n}).$

Then $A^{\prime}_{j,x}$ is a finitely presented $A^{+}$-algebra with a map to
$A^{\prime}_{j}$ such that
$(A^{\prime}_{j,x}/\pi)_{x}=(A^{\prime}_{j}/\pi)_{x}$ and such that $a=0$ in
$A^{\prime}_{j,x}$. By [27, Tag 00RC] the locus
$U\subseteq\operatorname{Spec}A^{\prime}_{j,x}/\pi$ which is flat over
$A^{+}/\pi$ is an open subset and by construction we have
$\operatorname{Spec}(A^{\prime}_{j,x}/\pi)_{x}\subseteq U$. We can thus find
elements $\overline{f}_{1},\dots\overline{f}_{m}\in A^{\prime}_{j,x}/\pi$ such
that each localization $(A^{\prime}_{j,x}/\pi)_{\overline{f}_{k}}$ is flat
over $A^{+}/\pi$ and such that the associated schemes cover
$\operatorname{Spec}(A^{\prime}_{j,x}/\pi)_{x}$. By the proof of [16, Lemma
3.6.1] the elements $\overline{f}_{1},\dots,\overline{f}_{m}$, more precisely
their images in $B_{i}^{+}/\pi$, induce qcqs open, even rational, subsets
$V_{1,x},\dots V_{m,x}\subseteq Y_{i}$ such that $\overline{f}_{k}$ is a unit
in $\mathcal{O}^{+}_{Y_{i}}(V_{k,x})$ for $k=1,\ldots,m$ (here we use that $Y$
and hence $Y_{i}$ is totally disconnected in order to apply [16, Lemma
3.6.1]). Pick any lifts $f_{1},\dots,f_{m}\in A^{\prime}_{j,x}$ and denote

$\displaystyle A_{j,x}:=\prod_{k=1}^{m}(A^{\prime}_{j,x})_{f_{k}},\qquad
Y_{i,x}:=\bigsqcup_{k=1}^{m}V_{k,x}.$

Writing $Y_{i,x}=\operatorname{Spa}(B_{i,x},B^{+}_{i,x})$ we get a map
$A_{j,x}\to B^{+}_{i,x}$ of $A^{+}$-algebras which is compatible with the
natural maps $A_{j}\to A_{j,x}$ and $B^{+}_{i}\to B^{+}_{i,x}$. Moreover, by
construction the subsets $Y_{k,x}$ cover the fiber of $Y_{i}$ over $x$
(because the distinguished opens
$D(\overline{f}_{1}),\ldots,D(\overline{f}_{m})$ cover
$\operatorname{Spec}(A^{\prime}_{j,x}/\pi)_{x})$, and hence their pullbacks
cover $\operatorname{Spec}(B^{+}_{i}/\pi)$). Since $Y_{i}$ is qcqs, we can
find finitely many connected components $x_{1},\dots,x_{\ell}$ of $X$ such
that the map
$Y_{i^{\prime}}:=Y_{i,x_{1}}\mathbin{\sqcup}\dots\mathbin{\sqcup}Y_{i,x_{\ell}}\mathrel{\twoheadrightarrow}Y_{i}$
is a cover, so that $Y_{i^{\prime}}\in I$. Let

$\displaystyle A_{j^{\prime}}:=\prod_{k=1}^{\ell}A_{j,x_{k}}.$

Then $A_{j^{\prime}}\in J_{\pi}$ and the natural map $\alpha\colon A_{j}\to
A_{j^{\prime}}$ satisfies the claim (a). This proves (a).

We note that we did not use that $A_{j}/\pi$ is flat over $A^{+}/\pi$ in the
proof, only flatness of $A^{\prime}_{j}/\pi$, which was supplied by Lemma 3.8
as the $A^{\prime}_{j}$ was a finitely generated, $\pi$-torsion free
$A^{+}$-algebra. Replacing in the argument $\pi$ by $\pi^{n},n\geq 1,$ we see
that we can even guarantee that $A_{j^{\prime}}/\pi^{n}$ is flat over
$A^{+}/\pi^{n}$.

From now on, we work with derived condensed rings again. With (a) at hand, we
can now prove the following claim:

  * (b)

The category $J_{\pi}$ is filtered and $B^{+}_{\mathrm{wl}}$ is the (derived)
$\pi$-adically completed colimit of the system $(\hat{A}_{j})_{j\in J_{\pi}}$,
where $\hat{A}_{j}$ denotes the (derived) $\pi$-adic completion of $A_{j}$.

Let us first show that $J_{\pi}$ is filtered. Clearly, $J_{\pi}$ is non-empty
as $A^{+}\in J_{\pi}$. Given $A_{j},A_{j^{\prime}}\in J_{\pi}$, pick any $i\in
I$ such that both $A_{j}\to B^{+}_{\mathrm{wl}}$ and $A_{j^{\prime}}\to
B^{+}_{\mathrm{wl}}$ factor over $B^{+}_{i}$. Then the static, non-condensed
tensor product $A_{j^{\prime\prime}}:=A_{j}\otimes_{A^{+}}A_{j^{\prime}}$ lies
in $J_{\pi}$ and the map $A_{j^{\prime\prime}}\to B^{+}_{\mathrm{wl}}$ factors
over $B^{+}_{i}$. In particular the natural maps $A_{j}\to
A_{j^{\prime\prime}}$ and $A_{j^{\prime}}\to A_{j^{\prime\prime}}$ lie in
$J_{\pi}$. Now suppose we have two maps $\alpha_{1},\alpha_{2}\colon A_{j}\to
A_{j^{\prime}}$ in $J_{\pi}$. By repeatedly applying (a) to the images of
polynomial generators of $A_{j}$ under $\alpha_{1}-\alpha_{2}$ we can
construct some $A_{j^{\prime\prime}}\in J_{\pi}$ with a map $A_{j^{\prime}}\to