A_{j^{\prime\prime}}$ such that $\alpha_{1}$ and $\alpha_{2}$ become equal on
$A_{j^{\prime\prime}}$. This proves that $J_{\pi}$ is indeed filtered.

Now let $B^{\prime}$ be the (derived) $\pi$-adic completion of
$\varinjlim_{j\in J_{\pi}}\hat{A}_{j}$. There is a natural map $B^{\prime}\to
B^{+}_{\mathrm{wl}}$ of $\pi$-adic rings which we need to show to be an
isomorphism. This can be checked modulo $\pi$, so we need so show that the
natural map

$\displaystyle\varinjlim_{j}\hat{A}_{j}/\pi=\varinjlim_{j}A_{j}/\pi\to
B^{+}_{\mathrm{wl}}/\pi=\varinjlim_{i}B^{+}_{i}/\pi$

is an isomorphism of discrete rings (here the tensor product $A_{j}/\pi$ is a
priori derived!). For this it is enough to show the stronger claim that the
natural map $\varinjlim_{j}A_{j}\to\varinjlim_{i}B^{+}_{i}$ of classical, non-
condensed rings is an isomorphism of static discrete rings. This map is
clearly surjective, because every $b\in B^{+}_{i}$ is hit by some $A_{j}\to
B^{+}_{i}$, e.g., for $A_{j}=A^{+}[x]$ and $x\mapsto b$. The map is injective
by claim (a). This proves the desired isomorphism and thus also claim (b).

We have now shown that the map $A^{+}\to B^{+}_{\mathrm{wl}}$ is the
$\pi$-completed filtered colimit of $\pi$-complete finitely presented
$A^{+}$-algebras whose underived reduction modulo $\pi$ (i.e.
$\pi_{0}(A_{j}/\pi)$) is flat over $A^{+}/\pi$. Since we only talk about
underived reductions modulo $\pi$, we cannot immediately conclude the desired
descendability statement: Since we have no control over the $\pi$-torsion of
$A_{j}\in J_{\pi}$, we cannot deduce from the flatness of $\pi_{0}(A_{j}/\pi)$
that also $\pi_{0}(A_{j}/\pi^{n})$ is flat. We therefore need the following
additional argument.

From now on we denote $J_{n}:=J_{\pi^{n}}$ for some fixed pseudouniformizer
$\pi\in A$. Then $J_{n+1}\subseteq J_{n}$ is a full subcategory for every
$n\geq 1$ and as we noted after the proof of (a) this subcategory is even
cofinal. Now let $J$ be the following category: The objects of $J$ are the
functors $j\colon\mathbb{N}\to J_{1}$ such that $j(n)\in J_{n}$ for all $n\geq
1$. For $j,j^{\prime}\in J$ we define

$\displaystyle\operatorname{Hom}_{J}(j,j^{\prime})=\varinjlim_{n\in\mathbb{N}}\operatorname{Hom}({j}|_{\mathbb{N}_{\geq
n}},{j^{\prime}}|_{\mathbb{N}_{\geq n}}).$

To every $j\in J$ we associate the $A^{+}$-algebra
$A_{j}:=\varinjlim_{n}A_{j(n)}$. It follows from (a) that $J$ is
$\omega_{1}$-filtered888Given any countable diagram $j_{k},k\in K,$ in $J$, we
can write the category $K$ as a filtered union of finite subgraphs
$K_{n},n\in\mathbb{N},$ and set $j^{\prime}(n)\in J_{n}$ as an object with a
map from each $j_{k}(l),k\in K_{n}$ and $l\leq n$ such that these morphisms to
$j^{\prime}(n)$ equalize all morphisms $j_{k}(l)\to j_{k^{\prime}}(l)$ induced
by a morphism in $K_{n}$. Inductively, we can even construct compatible
morphisms $j^{\prime}(n)\to j^{\prime}(n+1)$, and set $j_{k}\to j^{\prime}$
for $k\in K_{n}$ as the morphism, which for $m\geq n$ is the chosen morphism
$j_{k}(m)\to j^{\prime}(m)$.  and it follows as in (b) that
$B^{+}_{\mathrm{wl}}$ is the $\pi$-completed colimit of $(\hat{A}_{j})_{j\in
J}$. Thus, to finish the proof it is now enough to show the following claim:

  * (c)

For every $j\in J$ the map
$(A^{+})_{\scalebox{0.5}{$\square$}}\to(\hat{A}_{j})_{\scalebox{0.5}{$\square$}}$
is adically descendable of index $\leq 4$.

To prove (c), fix $j\in J$ and $n\geq 1$. We need to show that the map
$(A^{+}/\pi^{n})_{\scalebox{0.5}{$\square$}}\to(A_{j}/\pi^{n})_{\scalebox{0.5}{$\square$}}$
is descendable of index $\leq 4$. This map is the filtered colimit of the maps
$(A^{+}/\pi^{n})_{\scalebox{0.5}{$\square$}}\to(A_{j(k)}/\pi^{n})_{\scalebox{0.5}{$\square$}}$,
so by [16, Proposition 2.7.2] it is enough to show that for $k\gg 0$ the map
$(A^{+}/\pi^{n})_{\scalebox{0.5}{$\square$}}\to(A_{j(k)}/\pi^{n})_{\scalebox{0.5}{$\square$}}$
is descendable of index $\leq 2$. We claim that this works for $k\geq n$.
Indeed, we have the factorization

$\displaystyle\alpha\colon(A^{+}/\pi^{n})_{\scalebox{0.5}{$\square$}}\to(A_{j(k)}/\pi^{n})_{\scalebox{0.5}{$\square$}}\to\pi_{0}(A_{j(k)}/\pi^{n})_{\scalebox{0.5}{$\square$}},$

and by definition of $J_{n}$ the composed map $\alpha$ is flat and finitely
presented. The map $\alpha$ is even surjective on spectra, and thus an fppf
cover, because its composition with $\pi_{0}(A_{j(k)}/\pi^{n})\to
B^{+}_{\mathrm{wl}}/\pi^{n}$ is $A^{+}/\pi^{n}\to
B^{+}_{\mathrm{wl}}/\pi^{n}$, which is surjective on spectra (as
$Y^{\mathrm{wl}}\to X$ is a $v$-cover of totally disconnected spaces). Thus by
[16, Lemma 2.10.6] the map $\alpha$ is descendable of index $\leq 2$, so by
[16, Lemma 2.6.10.(ii)] the same is true for the first map
$(A^{+}/\pi^{n})_{\scalebox{0.5}{$\square$}}\to(A_{j(k)}/\pi^{n})_{\scalebox{0.5}{$\square$}}$,
as desired. ∎

###### Proof of Theorem 3.7.

It is sufficient to construct
$X\mapsto\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})$
for $X\in\mathrm{Perfd}^{\mathrm{aff}}$ as it will formally extend via
analytic descent. We define the presheaf $X=\operatorname{Spa}(A,A^{+})\mapsto
F(X):=\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}^{a}(A^{+})$
on $\mathrm{Perfd}^{\mathrm{aff}}$. We first observe that for every v-cover
$f\colon Y=\operatorname{Spa}(B,B^{+})\to X=\operatorname{Spa}(A,A^{+})$ of
totally disconnected spaces, the presheaf $F$ satisfies descent along $f$.
Indeed, by Proposition 3.9 and Lemma 3.6 the presheaf $F$ satisfies descent
along the map $Y^{\mathrm{wl}}\to X$, even after any base change
$X^{\prime}\to X$ with $X^{\prime}$ any affinoid, not necessarily totally
disconnected, perfectoid space.999More precisely, we as well use that descent
of $\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(-)$ for
$(A^{+})_{\scalebox{0.5}{$\square$}}\to(B^{+})_{\scalebox{0.5}{$\square$}}$
implies descent of
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}^{a}(-)$ by
embedding the almost category (via $j_{!}$) into
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(-)$ and using
Remark 2.36 to see that this embedding is compatible with pullback.  Hence, by
[11, Lemma 3.1.2.(3)] the same is true for $f\colon Y\to X$. Using Lemma 3.6
we can argue as in [16, Proposition 3.1.9] to show that there is a
(necessarily unique) quasi-pro-étale sheaf
$X\mapsto\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})$
on $\mathrm{Perfd}^{\mathrm{aff}}$ which has the desired form on those spaces
which are quasi-pro-étale over some totally disconnected space. We recall the
argument for the convenience of the reader: first, the presheaf $F$ defines a
quasi-pro-étale sheaf on the pro-étale site101010We note that a qcqs
perfectoid space $Y$ with a quasi-pro-étale map $Y\to X$ to a strictly totally
disconnected perfectoid space $X$ is itself again strictly totally
disconnected. This need not be true if $X=\operatorname{Spa}(K,K^{+})$ is
connected and totally disconnected as the valuation ring $K^{+}$ need not be
henselian. of strictly totally disconnected perfectoid spaces, i.e., those
totally disconnected spaces whose connected components are of the form
$\operatorname{Spa}(C,C^{+})$ with $C$ algebraically closed and
$C^{+}\subseteq C$ an open and bounded valuation subring. Formally, this
quasi-pro-étale sheaf extends to the quasi-pro-étale sheaf
$X\mapsto\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})$
on $\mathrm{Perfd}^{\mathrm{aff}}$. Assume that
$X=\operatorname{Spa}(A,A^{+})$ is affinoid perfectoid with a quasi-pro-étale
map $X\to Z$ to a totally disconnected space $Z$. If $Y\to Z$ is a quasi-pro-
étale cover with $Y$ strictly totally disconnected, then $Y\to Z$ satisfies
universal descent for $F$ as we have seen above. In particular, $F$ satisfies
descent for the cover $Y^{\prime}:=Y\times_{Z}X\to X$. As the terms of the
Čech nerve for $Y^{\prime}\to X$ are strictly totally disconnected, we can
therefore conclude that
$F(X)=\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})$
as claimed.

It remains to show that the sheaf
$X\mapsto\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})$
satisfies v-hyperdescent. This can be done in the same way as in the proof of
[16, Theorem 3.1.27]: One first shows the analog of [16, Lemma 3.1.23] for
adic coefficients, where the two crucial inputs are [16, Lemma 3.1.22] (which
can be replaced by Lemma 3.10) and the finite Tor dimension (which can be
lifted to the adic level by Corollary 2.14). Then the adic version of [16,
Corollary 3.1.24] is a formal corollary and consequently we deduce the adic
version of [16, Proposition 3.1.25]; here the only non-formal input is the
fact that $+$-modules, i.e.,
$\operatorname{\mathcal{D}}^{+}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+})$ for
$X=\operatorname{Spa}(A,A^{+})$, descend along v-covers of totally
disconnected spaces, as shown above. In other words, we have now shown that
$X\mapsto\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})$
is a v-sheaf. To get v-hyperdescent it remains to prove the adic version of
[16, Proposition 3.1.26]. But here the only non-formal input is the finite Tor
dimension, which can again be deduced from the mod-$\pi$ version via Corollary
2.14. ∎

The following result was used in the proof of Theorem 3.7 above. It is an
interesting statement on its own, so we extract it here:

######  Lemma 3.10.

Let $f\colon Y\to X$ be an étale cover in $\mathrm{Perfd}^{\mathrm{aff}}$.
Then $+$-modules descend along $j$, i.e., the functor
$X=\operatorname{Spa}(A,A^{+})\mapsto\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+})$
is an étale sheaf on $\mathrm{Perfd}^{\mathrm{aff}}$.

###### Proof.

After inverting a pseudo-uniformizer $\pi\in A$, i.e., for the functor
$X=\operatorname{Spa}(A,A^{+})\mapsto\mathrm{Mod}_{A}(\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+}))=\mathrm{Mod}_{A}(\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+}))$
this follows from Theorem 2.38. Using the fiber sequence $A^{+}\to A\to
A/A^{+}=\varinjlim\limits_{n}A^{+}/\pi^{n}$ reduces therefore to the descent
for the functor
$X=\operatorname{Spa}(A,A^{+})\mapsto\mathrm{Mod}_{A^{+}/\pi^{n}}(\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+}))=\mathrm{Mod}_{A^{+}/\pi^{n}}(\operatorname{\mathcal{D}}^{a}_{{\scalebox{0.5}{$\square$}}}(A^{+}))$.
Moreover, using induction and the fiber sequence $A^{+}/\pi^{n-1}\to
A^{+}/\pi^{n}\to A^{+}/\pi$ reduces to the case $n=1$. As $\pi$ was arbitrary,
we may even assume that $p|\pi$, and then (by tilting) that
$X=\operatorname{Spa}(A,A^{+}),Y=\operatorname{Spa}(B,B^{+})$ are perfectoid
spaces of characteristic $p$. As $f\colon Y\to X$ is étale, there exists by
[22, Proposition 6.4.(iv)] some étale morphism $f_{0}\colon
Y_{0}=\operatorname{Spa}(B_{0},B_{0}^{+}\to
X_{0}=\operatorname{Spa}(A_{0},A_{0}^{+})$ of affinoid perfectoid spaces,
which are weakly of perfectly finite type over
$\operatorname{Spa}(\mathbb{F}_{p}((\pi^{1/p^{\infty}})))$ (in the sense of
[16, Definition 3.1.13.]), and morphisms $g\colon X\to X_{0},Y\to Y_{0}$ such
that $Y\cong Y_{0}\times_{X_{0}}X$. By [16, Theorem 3.1.17] the natural
functor

$\operatorname{\mathcal{D}}^{a}_{{\scalebox{0.5}{$\square$}}}(A_{0}^{+}/\pi)\to\varprojlim\limits_{[n]\in\Delta}\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}((B_{0}^{n/A_{0}})^{+}/\pi)$

is an equivalence (here $(B_{0}^{n/A_{0}})^{+}$ denotes the $+$-ring for the
$n$-th stage of the Čech nerve of $Y_{0}\to X_{0}$). Using the arguments from
Lemma 2.40, we see that this implies that the functor

$\operatorname{\mathcal{D}}^{a}_{{\scalebox{0.5}{$\square$}}}(A^{+}/\pi)\to\varprojlim\limits_{[n]\in\Delta}\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}((B^{n/A})^{+}/\pi)$

is an equivalence as well. More precisely, the critical statement to check is
that for any morphism $g\colon X=\operatorname{Spa}(A,A^{+})\to
X_{0}=\operatorname{Spa}(A_{0},A_{0}^{+})$ of affinoid perfectoid spaces over
$\operatorname{Spa}(\mathbb{F}_{p}((\pi^{1/p^{\infty}})))$ the pushforward
$g_{\ast}^{a}\colon\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}(A^{+}/\pi)\to\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}(A^{+}_{0}/\pi)$
on almost categories is conservative. This however follows formally from
conservativity of
$g_{\ast}\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(A^{+}/\pi)\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(A^{+}_{0}/\pi)$
by embedding the almost categories via $\ast$-pushforward, i.e., the functors
of almost elements over $A^{+}/\pi$ resp. $A^{+}_{0}/\pi$, into
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(A^{+}/\pi)$ resp.
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(A^{+}_{0}/\pi)$. This
finishes the proof. ∎

######  Remark 3.11.

The same argument as in Theorem 3.7 can be used to show the following: Let
$X\to S$ be a map in $\mathrm{Perfd}^{\mathrm{aff}}$. Then there is a unique
hypercomplete v-sheaf

$\displaystyle(\mathrm{Perfd}_{/S})^{\mathrm{op}}\to\mathcal{C}at_{\infty},\qquad
T\mapsto\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X_{T}})$

such that if $T$ admits a quasi-pro-étale map to some totally disconnected
space then
$\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X_{T}})=\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(B^{+})$,
where $X\times_{S}T=\operatorname{Spa}(B,B^{+})$. Indeed, Proposition 3.9
implies descent after base change.

We note that logically we did not use Theorem 1.2 in the proof of Theorem 3.7,
only in its disguise through the key ingredient Lemma 3.8 and the étale
descent from [16, Lemma 3.1.22].

######  Remark 3.12.

Let $X=\operatorname{Spa}(A,A^{+})\in\mathrm{Perfd}^{\mathrm{aff}}$ with
pseudo-uniformizer $\pi\in A$. It follows from Theorem 3.7 and Lemma 2.10 that

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

with the right hand side defined in [16, Definition 3.1.3]. Indeed, both sides
satisfy $v$-descent and if $X$ is totally disconnected we can apply Lemma
2.10.

######  Definition 3.13.

Let $f\colon Y\to X$ be a map of perfectoid spaces.

  1. (a)

We let
$f^{\ast}\colon\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}^{a}(\mathcal{O}^{+}_{X})\to\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{Y})$
be the restriction map of the sheaf
$Z\mapsto\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{Z})$
from Theorem 3.7.

  2. (b)

We let $f_{\ast}$ be the right adjoint to $f^{\ast}$.111111It follows by [13,
Proposition 5.5.3.13]
that$\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{Z})$
is presentable for any $Z\in\mathrm{Perfd}$. By reduction to the totally
disconnected case it follows that $f^{\ast}$ preserves colimits, and hence
admits a right adjoint.

  3. (c)

If $X=\operatorname{Spa}(A,A^{+})$ is affinoid, we denote by

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

the natural functor, and by $\Gamma(X,-)$ its right adjoint.

###  3.3 Boundedness conditions

Let $X=\operatorname{Spa}(A,A^{+})$ be an affinoid perfectoid space. In the
rest of this paper, we will be concerned with the question whether the functor

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

from Definition 3.13 is an equivalence. This will turn out to be true under
certain finiteness assumptions. The critical definition of $+$-boundedness is
the following.

######  Definition 3.14.

  1. (a)

A map $f\colon Y^{\prime}\to Y$ of affinoid perfectoid spaces in
characteristic $p$ is called $+$-bounded if it is $p$-bounded in the sense of
[16, Definition 3.5.5.(b)].

  2. (b)

$Y\in\mathrm{Perfd}^{\mathrm{aff}}_{\mathbb{F}_{p}}$ is called $+$-bounded if
there exists a $+$-bounded map $Y\to Z$ with $Z$ totally disconnected.

  3. (c)

Let $\ell$ be a prime, e.g., $\ell=p$. A spatial diamond $Y$, e.g., an
affinoid perfectoid space in characteristic $p$, is called $\ell$-bounded if
there exists some integer $d\geq 0$, such that for all (static)
$\mathbb{F}_{\ell}$-sheaves $\mathcal{M}$ on $Y_{\mathrm{et}}$ we have
$H^{i}(Y_{\mathrm{et}},\mathcal{M})=0$ for $k>d$. Here, $Y_{\mathrm{et}}$ is
the étale site of $Y$ as defined in [22, Definition 14.1.(i)].

  4. (d)

An affinoid perfectoid space $X\in\mathrm{Perfd}$ is $+$-bounded (resp.
$\ell$-bounded) if its tilt $X^{\flat}\in\mathrm{Perfd}_{\mathbb{F}_{p}}$ is
$+$-bounded (resp. $\ell$-bounded).

Most notably, $X=\operatorname{Spa}(A,A^{+})$ $+$-bounded implies by Theorem
1.2 (and Remark 3.12) that for any pseudo-uniformizer $\pi\in A$

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

and this will be a critical ingredient in our proof of Theorem 1.1. We note
that the property of being $+$-bounded (resp. $\ell$-bounded) depends only on
the tilt. This is crucial for our major application.

######  Remark 3.15.

Roughly, a morphism $f\colon Y^{\prime}\to Y$ is $+$-bounded if $f_{\ast}$ has
finite “cohomological dimension on $\mathcal{O}^{+}/\pi$-modules” and $f$ has
$\mathrm{dim.trg}(f)<\infty$. In [16, Definition 3.5.5] the definition of
$+$-bounded from Definition 3.14 is phrased for all (morphisms of) small
$v$-stacks. This extra generality is not necessary for this paper. We note
that [16, Definition 3.5.5.] uses the terminology “$p$-bounded” for what we
call “$+$-bounded” here, while our use of “$p$-bounded” refers to the
different (but related) notion considered in [15, Definition 2.1].

######  Example 3.16.

Let $f\colon Y^{\prime}\to Y$ be a morphism of affinoid perfectoid spaces in
characteristic $p$. If $f$ is quasi-pro-étale or weakly of perfectly finite
type, then $f$ is $+$-bounded ([16, Lemma 3.5.10.(v)], [16, Lemma 3.5.13]).

######  Lemma 3.17.

Let $X$ be an $\ell$-bounded spatial diamond, and $f\colon X^{\prime}\to X$ a
qcqs quasi-pro-étale map. Then $X^{\prime}$ is $\ell$-bounded.

###### Proof.

We note that
$f_{\ast}\colon\operatorname{\mathcal{D}}^{+}(X^{\prime}_{\mathrm{et}},\mathbb{F}_{\ell})\to\operatorname{\mathcal{D}}^{+}(X_{\mathrm{et}},\mathbb{F}_{\ell})$
is $t$-exact by [22, Remark 21.14] if $f$ is separated. This implies that the
assertion is local on $X^{\prime}$, and thus we may furthermore assume that
$f$ is separated. Then again [22, Remark 21.14] reduces to the
$\ell$-boundedness of $X$. ∎

The notions of $+$-boundedness and $p$-boundedness are closely related. The
next result shows that the former implies the latter. We have not investigated
to what extend the other implication is valid.

######  Theorem 3.18.

Let $X$ be a $+$-bounded affinoid perfectoid space. Then $X$ is $p$-bounded.
In fact, if $f\colon X\to Z$ is a $+$-bounded map to a totally disconnected
space $Z$, then $H^{i}(X_{\mathrm{et}},\mathcal{F})=0$ for
$i>3+\mathrm{dim.trg}(f)$ and any (static) $\mathbb{F}_{p}$-sheaf
$\mathcal{F}$ on $X_{\mathrm{et}}$.121212As the proof shows it is sufficient
that $f$ has $\mathrm{dim.trg}(f)<\infty$.

###### Proof.

We may replace $X$ by $X^{\flat}$ and hence assume that $X$ is of
characteristic $p$. We first deal with the case that $X=Z$ is totally
disconnected. Let $\pi\colon X_{\mathrm{et}}\to\pi_{0}(X)_{\mathrm{et}}$ be
the natural morphism of sites. As $\pi_{0}(X)_{\mathrm{et}}$ has cohomological
dimension $0$ ([27, Lemma 0A3F]), it suffices to show that
$R^{i}\pi_{\ast}(\mathcal{F})=0$ for $i>2$ and any static
$\mathbb{F}_{p}$-sheaf on $X_{\mathrm{et}}$. This can be checked on stalks,
and hence we may assume that $X=\operatorname{Spa}(K,K^{+})$ is a perfectoid
affinoid field in characteristic $p$, i.e., $K$ is a perfectoid field of
characteristic $p$ and $K^{+}\subseteq K$ an open and bounded valuation
subring. We may write $(K,K^{+})$ as a (completed) filtered colimit of
perfectoid affinoid fields $(L,L^{+})$ and therefore reduce (via [22,
Proposition 14.9]) to the case that $K^{+}$ has finite Krull dimension. Let
$j\colon U\to X$ be the complement of the closed point. Then
$U=\operatorname{Spa}(K,\widetilde{K}^{+})$ for some valuation ring
$K^{+}\subseteq\widetilde{K}^{+}$. By [22, Lemma 21.13] the fiber
$\mathcal{G}:=[\mathcal{F}\to Rj_{\ast}j^{\ast}\mathcal{F}]$ lies in
$\operatorname{\mathcal{D}}^{[0,1]}(X_{\mathrm{et}},\mathbb{F}_{p})$. By [22,
Proposition 21.15] we can conclude that $H^{i}(X_{\mathrm{et}},\mathcal{G})=0$
for $i\geq 3$ because $\operatorname{Spa}(K,\mathcal{O}_{K})$ has
$p$-cohomological dimension $\leq 1$ (using
$\operatorname{Spa}(K,\mathcal{O}_{K})_{\mathrm{et}}\cong\operatorname{Spec}(K)_{\mathrm{et}}$
and [20, Proposition 6.5.10]). This implies that
$H^{i}(X_{\mathrm{et}},\mathcal{F})\cong
H^{i}(U_{\mathrm{et}},\mathcal{F}_{|U})$ for $i\geq 3$. By induction this
reduces to the case $X=\operatorname{Spa}(K,\mathcal{O}_{K})$, in which case
$H^{i}(X_{\mathrm{et}},\mathcal{F})=0$ for $i\geq 2$.

Now assume that $X=\operatorname{Spa}(A,A^{+})$ is a general affinoid
perfectoid space with a $+$-bounded morphism $f\colon X\to Z$ to a totally
disconnected space $Z=\operatorname{Spa}(R,R^{+})$ (it is actually sufficient
to assume that $\mathrm{dim.trg}(f)<\infty$). We claim that
$R^{i}f_{\ast}(\mathcal{F})=0$ for any $i>\mathrm{dim.trg}(f)+1$ and any
static $\mathbb{F}_{p}$-sheaf $\mathcal{F}$ on $X_{\mathrm{et}}$. We may write
$f$ as a cofiltered inverse limits of morphisms $f_{j}\colon
X_{j}=\operatorname{Spa}(A,A^{+}_{j})\to Z$ and $A_{j}^{+}$ finitely generated
over $B^{+}$. By [16, Lemma 3.5.10.(iv)] the morphisms $f_{j}$ are $+$-bounded
again as the relative compactifications
$\overline{X}^{/Z}=\overline{X_{j}}^{/Z}$ agree. By definition
$\mathrm{dim.trg}(f)=\mathrm{dim.trg}(f_{j})$ for any $j$. In particular, we
may replace $f$ by $f_{j}$ and assume that $f\colon X\to Z$ is compactifiable,
i.e., $X\to\overline{X}^{/Z}$ is a (necessarily quasi-compact) open immersion.
Note that $\overline{X}^{/Z}$ is an affinoid perfectoid space, and thus in
particular a spatial diamond. Thus, the claim follows from Theorem 3.19. ∎

The next result Theorem 3.19 is a mod $p$ analog of [22, Theorem 22.5]. In
[22, Theorem 22.5] $f$ is assumed to be a compactifiable morphism of general
small $v$-stacks, which is representable in spatial diamonds. We have to add
the stronger assumption that $Y^{\prime},Y$ are diamonds as the base change
results for mod $p$-coefficients are weaker than in the prime-to-$p$-case.
However, we follow the strategy of proof of [22, Theorem 22.5] quite closely.

######  Theorem 3.19.

Let $f\colon Y^{\prime}\to Y$ be a compactifiable morphism of spatial
diamonds, with canonical compactification
$Y^{\prime}\mathbin{\xrightarrow{j}}Y^{\prime\prime}:=\overline{Y^{\prime}}^{/Y}\mathbin{\xrightarrow{\overline{f}}}Y$.
Then $R^{i}f_{\ast},R^{i}\overline{f}_{\ast}$ vanish on static étale
$\mathbb{F}_{p}$-sheaves for $i>2\mathrm{dim.trg}(f)+1$. If $\overline{f}$ is
representable in spatial diamonds, they vanish for $i>\mathrm{dim.trg}(f)+1$.

###### Proof.

We note that $j_{\ast}$ is exact by [22, Lemma 21.13]. Hence, it suffices to
prove the assertion about $R^{i}\overline{f}_{\ast}$. Let $X\to Y$ be a quasi-
pro-étale cover with $X$ strictly totally disconnected. Then by [22, Corollary
16.10.(ii)] we may replace $f$ by its base change $Y^{\prime}\times_{Y}X\to
X$, and thus reduce to the case that $Y$ is strictly totally disconnected. As
the statement can be checked on stalks we may even assume that
$Y=\operatorname{Spa}(C,C^{+})$ is an affinoid field. Let $s\in Y$ be the
closed point, and $U:=Y\setminus\\{s\\}$ with pullback
$V:=\overline{f}^{-1}(U)\mathbin{\xrightarrow{j}}Y^{\prime\prime}$. Using
Theorem 3.20 we see that for every static $\mathbb{F}_{p}$-sheaf $\mathcal{F}$
on $Y^{\prime\prime}$ we have

$R\Gamma(Y^{\prime\prime},j_{!}j^{\ast}\mathcal{F})=0,$

which allows us to reduce to the case that $\mathcal{F}_{|U}=0$. If
$Y^{\prime\prime}$ is a spatial diamond, then [22, Proposition 21.11] applies
and yields the cohomological bound $\mathrm{dim.trg}(f)+1$ (as
$\mathrm{cd}_{p}(y)\leq 1$ for all maximal points $y$ of $Y^{\prime\prime}$
and $\mathrm{dim.trg}(f)=\mathrm{dim.trg}(\overline{f})$). It
$Y^{\prime\prime}$ is not a spatial diamond, then the arguments of [22,
Theorem 22.5] go through without change and yield the cohomological bound
$2\mathrm{dim.trg}(f)+1$. ∎

The following is a mod $p$-analog of [22, Theorem 19.2]. Again the stronger
assumption that $Y^{\prime},Y$ are spatial diamonds avoids the failure of
strong base change results mod $p$.

######  Theorem 3.20.

Let $f\colon Y^{\prime}\to Y$ be a proper morphism of spatial diamonds, and
let $j\colon U\to Y$ be an open immersion with pullback $g\colon
U^{\prime}:=Y^{\prime}\times_{Y}U\to U$, $j^{\prime}\colon U^{\prime}\to
Y^{\prime}$. Then for any
$A\in\operatorname{\mathcal{D}}^{+}(U^{\prime},\mathbb{F}_{p})$ the natural
morphism $j_{!}Rg_{\ast}(A)\to Rf_{\ast}(j^{\prime}_{!}A)$ is an isomorphism.

###### Proof.

The proof of [22, Theorem 19.2] goes through with a minor change: To avoid the
use of [22, Proposition 17.6] take a quasi-pro-étale cover $X\to Y$ with $X$
strictly totally disconnected (this is possible as $Y$ is a diamond) and apply
instead [22, Corollary 16.10.(ii)] to reduce to the case $Y$ strictly totally
disconnected. We note that the reduction in [22, Theorem 19.2] to the case
that $Y^{\prime}=\overline{X^{\prime}}^{/Y}$ for some strictly totally
disconnected space $X^{\prime}$ over $Y=X$ does not need [22, Corollary
16.8.(ii)] (or something similar): the $v$-site of $Y^{\prime}$ is replete,
and $R\Gamma(Y^{\prime}_{\mathrm{et}},B)\cong
R\Gamma(Y^{\prime}_{v},\varepsilon^{\ast}B)$ by [22, Proposition 14.10] for
$B\in\operatorname{\mathcal{D}}^{+}(Y_{\mathrm{et}},\mathbb{F}_{p})$, where
$\varepsilon\colon Y^{\prime}_{v}\to Y_{\mathrm{et}}$ is the natural morphism
of sites. This implies the desired cohomological descent. The remaining steps
in the proof, e.g., [22, Lemma 19.4], go through without change. ∎

The major goal of this paper is to find a large class of affinoid perfectoid
spaces $X=\operatorname{Spa}(A,A^{+})$ for which we have
$\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})=\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+})$.
The next two results provide useful abstract criteria for when this
identification is valid.

######  Lemma 3.21.

Let $X=\operatorname{Spa}(A,A^{+})\in\mathrm{Perfd}^{\mathrm{aff}}$. Assume
that there exists a quasi-pro-étale cover $Y\to X$ with
$Y=\operatorname{Spa}(B,B^{+})$ totally disconnected almost $+$-modules (in
the sense of Definition 3.2) descent along
$(A^{+})_{{\scalebox{0.5}{$\square$}}}\to(B^{+})_{\scalebox{0.5}{$\square$}}$.
Then the natural functor

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

is an equivalence.

###### Proof.

Let $Y_{\bullet}\to X$ be the Čech nerve of $Y\to X$ and write
$Y_{n}=\operatorname{Spa}(B_{n},B_{n}^{+})$. Note that each $Y_{n},n\geq 0$,
is quasi-pro-étale over the totally disconnected space $Y=Y_{0}$. Then

$\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{Y})\cong\varprojlim\limits_{[n]\in\Delta}\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{Y_{n}})\cong\varprojlim\limits_{[n]\in\Delta}\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(B_{n}^{+})\cong\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+}),$

using Theorem 3.7 in the second isomorphism and that almost $+$-modules
descent along
$(A^{+})_{\scalebox{0.5}{$\square$}}\to(B^{+})_{\scalebox{0.5}{$\square$}}$ in
the third. ∎

######  Lemma 3.22.

Let $X=\operatorname{Spa}(A,A^{+})\in\mathrm{Perfd}^{\mathrm{aff}}$ be
$+$-bounded, and let $\pi\in A$ be a pseudo-uniformizer. Then the functor
$\widetilde{(-)}\colon\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+})\to\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})$
is an equivalence if and only if
$\Gamma(X,-)\colon\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})\to\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+})$
preserves colimits and
$\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})$
is generated under colimits by $\pi$-complete objects $\mathcal{N}$ with
$\Gamma(X,\mathcal{N})$ bounded to the right.

###### Proof.

The necessity is clear (as
$\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+})$ is
generated by $\pi$-complete objects), so let us check the converse. There is
the natural adjunction

$\displaystyle\widetilde{(-)}\colon\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+})\rightleftarrows\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})\nobreak\mskip
6.0mu plus 1.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 2.0mu\Gamma(X,-)$

and we need to show that these functors are inverse to each other. We first
show that $\widetilde{(-)}$ is fully faithful, i.e. that for every
$M\in\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+})$
we have $\Gamma(X,\widetilde{M})=M$. Since $\Gamma(X,-)$ preserves colimits,
we can assume that $M=P^{a}$ for a compact generator
$P\in\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+})$, in
particular $M$ is $\pi$-complete and bounded to the right. We claim that
$\widetilde{M}\in\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})$
is $\pi$-complete. This can be checked on a $v$-cover $Y\to X$ with $Y$
totally disconnected. By naturality of $\widetilde{(-)}$ and Lemma 2.12, this
reduces to the case that $Y=X$, where $\widetilde{(-)}$ is an equivalence.
Thus $\Gamma(X,\widetilde{M})$ and $M$ are $\pi$-complete and hence the
identity $\Gamma(X,\widetilde{M})=M$ can be checked modulo $\pi$. But then it
follows from [16, Theorem 3.5.21] (note that it is formal that
$\widetilde{(-)},\Gamma(X,-)$ are compatible with passage to
$A^{+}/\pi$-modules).

To finish the proof, it is now enough to show that $\widetilde{(-)}$ is
essentially surjective. Since by assumption
$\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})$
is generated under colimits by $\pi$-complete objects, it is enough to show
that every $\pi$-complete object lies in the essential image of
$\widetilde{(-)}$. Given a $\pi$-complete
$\mathcal{N}\in\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})$
we need to see that the counit
$\widetilde{\Gamma(X,\mathcal{N})}\mathbin{\xrightarrow{\sim}}\mathcal{N}$ is
an isomorphism. By assumption we may even assume that $\Gamma(X,\mathcal{N})$
is bounded to the right. Then by the same argument as above both sides of the
claimed isomorphism are $\pi$-complete, so we can check the claimed
isomorphism modulo $\pi$. Then it reduces to again [16, Theorem 3.5.21]. ∎

###  3.4 Descent for relative compactifications of totally disconnected spaces

In this section we want to identify
$\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X})$
for certain relative compactifications of totally disconnected spaces. The
precise statement is the following.

######  Theorem 3.23.

Let $X\in\mathrm{Perfd}^{\mathrm{aff}}$ whose tilt admits a map $f\colon
X^{\flat}\to Z$ to some totally disconnected space $Z$ with
$\mathrm{dim.trg}f<\infty$. Let $X^{\prime}\to X$ be a quasi-pro-étale map,
where $X^{\prime}$ is a totally disconnected space. Then for
$Y:=\overline{X}^{\prime/X}=\operatorname{Spa}(B,B^{+})$ the natural functor

$\displaystyle\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(B^{+})\to\operatorname{\mathcal{D}}^{a}_{\hat{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{Y})$

is an equivalence.

###### Proof.

Set $Z^{\prime}:=(X^{\prime})^{\flat}$. We let $X^{\prime}\to Y^{\prime}$ be
the unique map of perfectoid spaces whose tilt is the map
$Z^{\prime}\to\overline{Z^{\prime}}^{/Z}$ (note that
$\mathcal{O}(X^{\prime})=\mathcal{O}(Y^{\prime})$ as this holds for the tilt).
We note that there exists a natural inclusion $Y\to Y^{\prime}$ whose tilt is
the natural map $\overline{Z^{\prime}}^{/X}\to\overline{Z^{\prime}}^{/Z}$. In
fact, $Y$ is an intersection of rational open subsets in $Y^{\prime}$ as this
holds on the tilt. By Proposition 3.24 there is a pro-étale cover
$W^{\prime}\to Y^{\prime}$ which is of universal almost $+$-descent, where
$W^{\prime}$ is a totally disconnected space, i.e., almost $+$-modules (in the
sense of Definition 3.2) descend along $W^{\prime}\to Y^{\prime}$ and all its
base changes to affinoid perfectoid spaces. Then $W:=Y\times_{Y^{\prime}}W\to
Y$ is a pro-étale cover with $+$-descent for $Y$, and $W$ is still totally
disconnected (as it is an intersection of rational open subsets in the totally
disconnected space $W^{\prime}$). By Lemma 3.21 this finishes the proof. ∎

In the next proposition, an untilt of a perfectoid space $Z$ of characteristic
$p$ is a perfectoid space $X$ together with an identification $X^{\flat}\cong
Z$. Proposition 3.24 is similar to [16, Proposition 3.1.11]. Again the
essential point is to show that open covers of relative compactifications are
descendable with uniformly bounded index. This will as in [16, Proposition
3.1.11] be reduced to open covers of (families of) Riemann–Zariski spaces. The
necessary results on these kind of adic spaces will be discussed in Section
3.5.

######  Proposition 3.24.

Let $f\colon Z^{\prime}\to Z$ be a map of totally disconnected spaces in
$\mathrm{Perfd}^{\mathrm{aff}}_{\mathbb{F}_{p}}$ with
$\mathrm{dim.trg}f<\infty$. Let $X$ be an untilt of $\overline{Z}^{\prime/Z}$.
Then there is a pro-étale cover $Y\to X$ by a totally disconnected space $Y$
such that almost $+$-modules descend along $Y\times_{X}X^{\prime}\to
X^{\prime}$ for any morphism $X^{\prime}\to X$ of affinoid perfectoid spaces.

###### Proof.

Let $Y:=X^{\mathrm{wl}}$ be the w-localization of $X$, which by [22, Lemma
7.13] is a totally disconnected space of the form
$Y=\varprojlim_{i}Y_{i}\mathrel{\twoheadrightarrow}X$ such that each $Y_{i}$
is a disjoint union of rational open subsets of $X$. We will show that the map
$Y\to X$ has the claimed descent properties and similarly any base change.
Write $X=\operatorname{Spa}(A,A^{+})$, $Y=\operatorname{Spa}(C,C^{+})$. It
suffices to check that the map
$(A^{\circ},A^{+})_{\scalebox{0.5}{$\square$}}\to(C^{\circ},C^{+})_{\scalebox{0.5}{$\square$}}$
is weakly adically descendable of an index $\leq c(d)$, where $c(d)$ is a
constant only depending on $d:=\mathrm{dim.trg}f$. Indeed, then almost
$+$-modules will descent universally along $Y\to X$ because
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(A^{\circ},A^{+})$
and $\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(A^{+})$
define the same almost category using $(A^{\circ})^{a}=(A^{+})^{a}$ (and the
similar statement for
$\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(C^{\circ},C^{+})$
and $\operatorname{\mathcal{D}}_{\hat{\scalebox{0.5}{$\square$}}}(C^{+})$). By
2.33.(iii) it is enough to show that for each map
$Y_{i}=\operatorname{Spa}(C_{i},C_{i}^{+})\to X$ the map
$(A^{\circ},A^{+})_{\scalebox{0.5}{$\square$}}\to(C_{i}^{\circ},C_{i}^{+})_{\scalebox{0.5}{$\square$}}$
of adic analytic rings is adically descendable with an index $\leq c(d)$. By
definition of adic descendability it is enough to prove this statement modulo
$\pi$ for every pseudouniformizer $\pi$ on $X$. Now we can argue as in [16,
Proposition 3.1.11], using Lemma 3.26 in place of [16, Proposition 2.10.10] in
order to get descendability (and not just fs-descendability) of the covering.
In the following we provide the details.

We are given a map $U:=Y_{i}=\bigsqcup_{j=1}^{n}U_{j}\to X$ for a cover of $X$
by qcqs open subsets $U_{j}\subseteq X$. Write
$X=\operatorname{Spa}(A,A^{+})$, $Z=\operatorname{Spa}(B,B^{+})$ and
$Z^{\prime}=\operatorname{Spa}(B^{\prime},B^{\prime+})$, where we note that
$B^{\prime}=A^{\flat}$ and that $A^{\flat+}$ is the completed integral closure
of $B^{+}+B^{\prime\circ\circ}$ in $B^{\prime+}$. Let
$\overline{B}^{\prime}:=B^{\prime\circ}/B^{\prime\circ\circ}$,
$\overline{B}^{+}:=B^{+}/B^{\circ\circ}$ and
$\overline{X}:=\operatorname{Spa}(\overline{B}^{\prime},\overline{B}^{+})$.
Note that $\overline{B}^{\prime}=A^{\flat\circ}/A^{\flat\circ\circ}$, that
$A^{\flat+}/A^{\flat\circ\circ}$ is the integral closure of the image of
$\overline{B}^{+}$ in $\overline{B}^{\prime}$ and that $X^{\flat}$ satisfies
the hypothesis of Lemma 3.27. Hence by Lemma 3.27 we obtain a canonical
homeomorphism $\lvert X\rvert=\lvert
X^{\flat}\rvert\cong\lvert\overline{X}\rvert$. In particular, the open cover
$X=\bigcup_{j}U_{j}$ corresponds to an open cover
$\overline{X}=\bigcup_{j}\overline{U}_{j}$ and we define
$\overline{U}:=\bigsqcup_{j}\overline{U}_{j}$.

We now claim that $\overline{X}$ satisfies the condition of Lemma 3.26.
Indeed, the connected components of $Z^{\prime}$ are of the form
$\operatorname{Spa}(K^{\prime},K^{\prime+})$ for a perfectoid field
$K^{\prime}$, hence the connected components of
$\operatorname{Spec}\overline{B}^{\prime}$ are of the form
$\operatorname{Spec}(\mathcal{O}_{K^{\prime}}/\mathfrak{m}_{K^{\prime}})$ (we
note for later that this implies that
$|\operatorname{Spec}(\overline{B}^{\prime})|$ is a profinite set). Similarly,
if a connected component of $Z$ is of the form $\operatorname{Spa}(K,K^{+})$
then the corresponding connected component of
$\operatorname{Spec}\overline{B}^{+}$ is of the form $K^{+}/\mathfrak{m}_{K}$,
which is a valuation ring. This implies that the condition (a) of Lemma 3.26
is satisfied. Similarly, condition (b) is satisfied using the same $d$ ([4,
VI.10.3.Corollaire 1]). Altogether we deduce that the map
$\overline{U}\to\overline{X}$ is descendable of index bounded by a constant
only depending on $d$. More precisely, by the proof of Lemma 3.26 there is a
reduced projective $\overline{B}^{+}$-scheme $\overline{S}$ together with a
dominant map $\operatorname{Spec}\overline{B}^{\prime}\to\overline{S}$ such
that the map $\overline{U}\to\overline{X}$ comes via base-change from an open
covering of $\overline{S}$ along the map $\overline{X}\to\overline{S}$.
Moreover, $\overline{S}$ can be covered by $d+1$ open affine subschemes
$\overline{W}_{i}=\operatorname{Spec}\overline{R}_{i}\subseteq\overline{S}$.
As $\overline{S}$ is separated, the morphism
$\operatorname{Spec}\overline{B}^{\prime}\to\overline{S}$ is affine. Note that
we have a surjective map
$A^{\circ}/\pi\mathrel{\twoheadrightarrow}A^{\circ}/A^{\circ\circ}=A^{\flat\circ}/A^{\flat\circ\circ}=\overline{B}^{\prime}$
with locally nilpotent kernel. By [27, Lemma 07RT] we can therefore form the
pushout
$S:=\operatorname{Spec}(A^{\circ}/\pi)\cup_{\operatorname{Spec}(\overline{B}^{\prime})}\overline{S}$
in the category of schemes. By the construction in [27, Lemma 07RT], the
scheme $S$ is covered by the $d+1$ open affine subschemes
$\operatorname{Spec}(A^{\prime}_{i})\cup_{\operatorname{Spec}(\overline{B}^{\prime}_{i})}\overline{W}_{i}$,
where
$\overline{Z}_{i}=\operatorname{Spec}(\overline{B}^{\prime}_{i})\subseteq\operatorname{Spec}(\overline{B}^{\prime})$
is the affine open preimage of $W_{i}$ along
$\operatorname{Spec}(\overline{B}^{\prime})\to\overline{S}$, and
$\operatorname{Spec}(A^{\prime}_{i})\subseteq\operatorname{Spec}(A^{\circ}/\pi)$
is the unique affine open subscheme with underlying topological space
$\overline{Z}_{i}$.

The morphism $\overline{X}\to\overline{S}$ lifts to a morphism
$X_{\pi}:=\operatorname{Spa}(A^{\circ}/\pi,A^{+}/\pi)\to S$ which is the same
map on underlying topological spaces. Now the given open covering
$U=\operatorname{Spa}(C_{i},C_{i}^{+})\to X$ induces an open covering
$U_{\pi}:=\operatorname{Spa}(C_{i}^{\circ}/\pi,C_{i}^{+}/\pi)\to X_{\pi}$
(using that Lemma 3.27 applies to $U$ as well), and $U_{\pi}\to X_{\pi}$
reduces to the covering $\overline{U}\to\overline{X}$. Thus the covering
$U_{\pi}\to X_{\pi}$ is an open covering of discrete adic spaces which comes
via base-change from an open covering of $S$ (as can be checked after base
change to $\overline{X}$). Since $S$ is covered by $d+1$ open affine subsets,
the map $U_{\pi}\to X_{\pi}$ is therefore descendable of index bounded by a
constant only depending on $d$ ([16, Corollary 2.10.7]). This finishes the
proof. ∎

###  3.5 Families of Riemann–Zariski spaces

In this section we provide the necessary results on families of
Riemann–Zariski spaces, which entered in the proof of Proposition 3.24. Unless
stated otherwise, each Huber pair is classical in this section. A related
discussion can be found in [26], but we need to ensure that the relative
Riemann–Zariski space is a cofiltered inverse limits of projective schemes
(and not merely of proper schemes) in order to ensure the uniform bound on
descendability in Lemma 3.26.

######  Lemma 3.25.

Let $A^{+}\to A$ be a map of classical, discrete rings such that each
connected component of $X:=\operatorname{Spa}(A,A^{+})$ is of the form
$\operatorname{Spa}(K,A_{x}^{+})$ for some field $K$ and some subring
$A_{x}^{+}\subseteq K$. Let $I$ be the category of factorizations
$\operatorname{Spec}A\to Y_{i}\to\operatorname{Spec}A^{+}$ such that $Y_{i}$
is a reduced projective $A^{+}$-scheme and the map $\operatorname{Spec}A\to
Y_{i}$ is dominant. Then:

  1. (i)

$I$ is cofiltered and for every $i\in I$ the map $\operatorname{Spec}A\to
Y_{i}$ factors uniquely over a map $X\to Y_{i}$ over
$\operatorname{Spec}(A^{+})$. The induced map $\lvert X\rvert\to\lvert
Y_{i}\rvert$ is closed.

  2. (ii)

The map $\lvert X\rvert\mathbin{\xrightarrow{\sim}}\varprojlim_{i}\lvert
Y_{i}\rvert$ is a homeomorphism of topological spaces.

We note that under the assumptions of this lemma the natural map
$\operatorname{Spa}(A,A)\to\operatorname{Spec}(A)$ is an isomorphism of
locally ringed spaces, which makes condition (i) well-defined.

###### Proof.

Let us first show that $I$ is cofiltered. Given any $i,i^{\prime}\in I$ there
can be at most one map $Y_{i}\to Y_{i^{\prime}}$ in $I$. Indeed, this follows
immediately from the fact that the map $\operatorname{Spec}A\to Y_{i}$ is an
epimorphism (in general dominant morphisms are epimorphisms in the category of
reduced separated schemes, as one easily checks by a standard diagonal
argument). Thus to prove that $I$ is cofiltered we only need to check that
there is some $i^{\prime\prime}\in I$ such that there are maps
$Y_{i^{\prime\prime}}\to Y_{i}$ and $Y_{i^{\prime\prime}}\to Y_{i^{\prime}}$.
But note that we can take $Y_{i^{\prime\prime}}$ to be the scheme-theoretic
image of the natural map $\operatorname{Spec}A\to
Y_{i}\times_{\operatorname{Spec}A^{+}}Y_{i^{\prime}}$.

Now fix some $i\in I$. Note that $X$ is the relative compactification of the
map $\operatorname{Spec}A\to\operatorname{Spec}A^{+}$ (see [16, Example
2.9.27]), so since $Y_{i}\to\operatorname{Spec}A^{+}$ is proper (and thus its
own relative compactification) we deduce that the map $\operatorname{Spec}A\to
Y_{i}$ extends canonically to a map $X\to Y_{i}$ over
$\operatorname{Spec}(A^{+})$. The uniqueness of the extension can be checked
on connected components, where it follows from the valuative criterion for
properness. We now check that the morphism $X\to Y_{i}$ is closed. As $A$ is
discrete, the morphism $X\to Y_{i}$ is a quasi-compact morphism of spectral
spaces. Because it is as well specializing (by the valuative criterion for
properness) this implies that the morphism $X\to Y_{i}$ is closed. Indeed, by
quasi-compactness of $X\to Y_{i}$ the image of a closed subsets is pro-
constructible and stable under specialization, hence closed ([27, Lemma
0903]). This proves (i).

It remains to prove (ii). By (i) we know that the map $\lvert
X\rvert\to\varprojlim_{i}\lvert Y_{i}\rvert$ is closed and continuous, so it
is enough to show that this map is bijective. Let us first check that it is
injective, so let $y,y^{\prime}\in\lvert X\rvert$ be two given points which
get mapped to the same point in each $\lvert Y_{i}\rvert$. Let
$\widetilde{A^{+}}$ be the integral closure of the image of $A^{+}$ in $A$.
Then
$\pi_{0}(\operatorname{Spec}(\widetilde{A^{+}}))=\pi_{0}(\operatorname{Spec}(A))$
and writing $\widetilde{A^{+}}$ as a filtered colimit of finite, reduced
$A^{+}$-algebras, we see that $y$ and $y^{\prime}$ lie in the same connected
component $x=\operatorname{Spa}(K,A^{+}_{x})$ of $X$ and hence correspond to
valuation rings $V_{y},V_{y^{\prime}}\subseteq K$ containing $A^{+}_{x}$.
Assuming $y\neq y^{\prime}$ we can w.l.o.g. assume that there is some
$\overline{a}\in V_{y}\setminus V_{y^{\prime}}$. Let $a\in A$ be a lift of
$\overline{a}$, which we can assume to be a unit (note that $K$ is the
filtered colimit of the ring of functions of clopen neighbourhoods of
$\operatorname{Spec}K\subseteq\operatorname{Spec}A$). Then the pair
$\\{1,a\\}$ determines a map
$f\colon\operatorname{Spec}A\to\mathbb{P}^{1}_{A^{+}}$ and we let $Y_{a}$ be
the scheme theoretic image of $f$. Then $Y_{a}\in I$ and we claim that the
induced map $X\to Y_{a}$ separates $y$ and $y^{\prime}$. To see this, we can
assume that the map $A^{+}\to A$ is injective, because otherwise the
surjection from $A^{+}$ to its image in $A$ induces a closed immersion on
schemes and in particular an injective map. Now $Y_{a}$ can be described
explicitly: It is obtained by gluing $\operatorname{Spec}A^{+}[a]$ and
$\operatorname{Spec}A^{+}[1/a]$ along $\operatorname{Spec}A^{+}[a,1/a]$.
Moreover, the map $X\to Y_{a}$ sends a valuation ring $V\subseteq K$ to the
unique point on $Y_{a}$ such that $V$ dominates the associated local ring.
Thus $V_{y^{\prime}}$ is sent to a point in $\operatorname{Spec}A^{+}[1/a]$
which does not lie in $\operatorname{Spec}A^{+}[a,1/a]$, whereas $V_{y}$ is
sent to a point in $\operatorname{Spec}A^{+}[a]$.

We now prove surjectivity of the map $\lvert X\rvert\to\varprojlim_{i}\lvert
Y_{i}\rvert$, so let any $(y_{i})_{i}\in\varprojlim_{i}\lvert Y_{i}\rvert$ be
given. Replacing $A^{+}$ by $\widetilde{A^{+}}$ we may assume that
$\pi_{0}(\operatorname{Spec}(A))=\pi_{0}(\operatorname{Spec}(A^{+}))$. Then
all $y_{i}$ live over the same connected component of $X$, so by base-changing
to this connected component we can assume that $X$ is connected, i.e. of the
form $X=\operatorname{Spa}(K,A^{+})$ for a field $K$. For each $i$ the
dominant map $\operatorname{Spec}K\to Y_{i}$ induces an injection
$\mathcal{O}_{Y_{i},y_{i}}\subseteq K$. Let $R\subseteq K$ be the (filtered)
union of all $\mathcal{O}_{Y_{i},y_{i}}$. Then $R$ is a local ring and hence
dominated by some valuation ring $V\subseteq K$. This valuation ring
corresponds to a point $y\in\lvert X\rvert$ which maps to $(y_{i})_{i}$ (by
uniqueness), as desired. ∎

The next lemma yields the crucial uniform descendability bound for Proposition
3.24. It is the analog of [16, Proposition 2.10.10] for families of
Riemann–Zariski spaces.

######  Lemma 3.26.

Let $A^{+}\to A$ be a map of classical, discrete rings and
$X=\operatorname{Spa}(A,A^{+})$ the associated discrete adic space. Assume
that:

  1. (a)

Every connected component of $\operatorname{Spec}A$ is of the form
$\operatorname{Spec}K^{\prime}$ for some field $K^{\prime}$ and every
connected component of $\operatorname{Spec}A^{+}$ is of the form
$\operatorname{Spec}V$ for some valuation ring $V$.

  2. (b)

There is an integer $d\geq 0$ such that for every map of connected components
$\operatorname{Spec}K^{\prime}\to\operatorname{Spec}V$ induced by $A^{+}\to
A$, the associated map $V\mathrel{\hookrightarrow}K^{\prime}$ is injective and
the transcendence degree of $K^{\prime}$ over the fraction field of $V$ is
$\leq d$.

Let $U=\bigsqcup_{j=1}^{n}U_{j}\mathrel{\twoheadrightarrow}X$ be an open
covering by quasicompact open subsets $U_{j}\subseteq X$. Then $U\to X$ is
descendable of index bounded by a constant only depending on $d$.

###### Proof.

Let $I$ be as in Lemma 3.25. Then by Lemma 3.25 we have $\lvert
X\rvert=\varprojlim_{i}\lvert Y_{i}\rvert$ and all the maps $\lvert
X\rvert\to\lvert Y_{i}\rvert$ are closed, which together implies that there is
some $i\in I$ and an open covering $Y_{i}=\bigcup_{j=1}^{m}W_{ij}$ by qcqs
open subsets $W_{ij}\subseteq Y_{i}$ such that $U\to X$ is the base-change of
the map $W_{i}:=\bigsqcup_{j=1}^{m}W_{ij}\mathrel{\twoheadrightarrow}Y_{i}$
along the map $X\to Y_{i}$. By [16, Lemma 2.6.9] it is thus enough to show
that the map $W_{i}\to Y_{i}$ is descendable of bounded index.

To simplify notation let us write $Y:=Y_{i}$ and $W:=W_{i}$. By [16, Corollary
2.10.7] it is enough to show that $Y$ can be covered by $d+1$ affine open
subschemes. The map
$\pi_{0}(\operatorname{Spec}A)\to\pi_{0}(\operatorname{Spec}A^{+})$ is a map
of compact Hausdorff spaces and in particular has closed image. This image
thus corresponds to an affine subscheme
$\operatorname{Spec}A^{\prime+}\subseteq\operatorname{Spec}A^{+}$ (the limit
over all clopen neighbourhoods of the set-theoretic image inside
$\operatorname{Spec}A^{+}$) by definition of $I$ we know that $Y$ is supported
over $\operatorname{Spec}A^{\prime+}$. We can thus replace $A^{+}$ by
$A^{\prime+}$ in order to assume that the map
$\pi_{0}(\operatorname{Spec}A)\to\pi_{0}(\operatorname{Spec}A^{+})$ is
surjective. Now pick a connected component
$x=\operatorname{Spec}V\in\pi_{0}(\operatorname{Spec}A^{+})$ and let $Y_{x}$
be the fiber of $Y$ over $x$. Denote by $\eta$ and $s$ the open and closed
point of $V$ and let $Y_{x,\eta}$ and $Y_{x,s}$ be the the generic and special
fiber of $Y_{x}$, respectively. We claim:

  * ($*$)

$Y_{x,s}$ has dimension $\leq d$.

To see this, we first claim that there are finitely many connected components
$y_{1}=\operatorname{Spec}K_{1}^{\prime}$, …,
$y_{n}=\operatorname{Spec}K_{n}^{\prime}$ over $x$ such that the map
$y_{1}\mathbin{\sqcup}\dots\mathbin{\sqcup}y_{n}\to Y_{x}$ is dominant. To see
this, fix any irreducible component $Z\subseteq Y_{x,\eta}$. For every (qcqs)
open subset $W\subseteq Z$, let $S_{W}\subseteq\operatorname{Spec}A$ be the
preimage of $W$ under the map $\operatorname{Spec}A\to Y$. Then $S_{W}$ is a
quasicompact open subset and since the topological space of
$\operatorname{Spec}A$ is profinite it follows that $S_{W}$ is closed. We
claim that the intersection of all $S_{W}$, for $W$ ranging through non-empty
subsets of $Z$, is non-empty. Otherwise, by compactness there would be some
$W_{1},\dots,W_{m}\subseteq Z$ such that
$S_{W_{1}}\mathbin{\cap}\dots\mathbin{\cap}S_{W_{m}}=\varnothing$. Since $Z$
is irreducible, $W:=W_{1}\mathbin{\cap}\dots\mathbin{\cap}W_{m}$ is still a
non-empty open subset of $Z$ and we deduce $S_{W}=\varnothing$; but this
contradicts the fact that $\operatorname{Spec}A\to Y$ is dominant. Pick any
$y$ in the intersection of all $S_{W}$; then the map $y\to Z$ is dominant.
Repeating this procedure for all irreducible components of $Y_{x,\eta}$, we
find $y_{1},\dots,y_{n}\subseteq\operatorname{Spec}A$ such that the map
$y_{*}:=y_{1}\mathbin{\sqcup}\dots\mathbin{\sqcup}y_{n}\to Y_{x,\eta}$ is
dominant. By the assumption that the induced maps
$V\mathrel{\hookrightarrow}K^{\prime}$ are injective (see (b)) we see that
every point of $Y_{x}$ that is hit by some $y\in\operatorname{Spec}A$
necessarily lies in $Y_{x,\eta}$. Consequently, since the map
$\operatorname{Spec}A\to Y$ is dominant, we deduce that $y_{*}\to Y_{x}$ must
be dominant.

With $y_{1},\dots,y_{n}$ as in the previous paragraph, let
$Z_{1},\dots,Z_{n}\subseteq Y_{x}$ be the scheme-theoretic images of these
points under the map $\operatorname{Spec}A\to Y$. Then
$Y_{x}=Z_{1}\mathbin{\cup}\dots\mathbin{\cup}Z_{n}$, so to prove claim ($*$)
it is enough to show that each $Z_{i,s}$ has dimension $\leq d$. Since $Z_{i}$
is integral, by [27, Lemma 0B2J] it is enough to show that the generic fiber
$Z_{i,\eta}$ has dimension $\leq d$. But this follows immediately from the
fact that $Z_{i}$ is a projective $K$-variety and there is a dominant map
$\operatorname{Spec}K^{\prime}_{i}\to Z_{i,\eta}$ such that $K^{\prime}_{i}$
has transcendence degree $\leq d$ over $K$. This finishes the proof of claim
($*$).

With claim ($*$) proved, we now proceed with the proof that $Y$ is covered by
$d+1$ affine open subschemes. Let $\kappa=\kappa(s)$ be the residue field of
$s$. By the proof of [16, Lemma 2.10.9] we can find some $n\geq 0$ such that
$Y\subseteq\mathbb{P}^{n}_{A^{+}}$ and homogeneous polynomials
$\overline{f}_{0},\dots,\overline{f}_{d}\in\kappa[t_{1},\dots,t_{n}]$ such
that the intersection of the vanishing loci of the $\overline{f}_{j}$’s in
$\mathbb{P}^{n}_{\kappa}$ is disjoint from $Y_{x,s}$. We can assume that each
$\overline{f}_{j}$ has a coefficient $1$ and we can then lift each
$\overline{f}_{j}$ to a homogeneous polynomial $f_{j}\in
A^{+}[t_{1},\dots,t_{n}]$, each of which has one coefficient $1$ (first lift
the coefficients to $V$ and then use that $V$ is the colimit of the functions
on clopen neighbourhoods of $x\subseteq\operatorname{Spec}A^{+}$). Let
$H_{j}\subseteq\mathbb{P}^{n}_{A^{+}}$ be the zero locus of $f_{j}$. As in the
proof of [16, Lemma 2.10.9] each $\mathbb{P}^{n}_{A^{+}}\setminus H_{j}$ is
affine. Now let $Y_{j}:=Y\mathbin{\cap}(\mathbb{P}^{n}_{A^{+}}\setminus
H_{j})$. This is a closed subscheme of the affine scheme
$\mathbb{P}^{n}_{A^{+}}\setminus H_{j}$ and hence affine. Let
$Y^{\prime}:=\bigcup_{j=0}^{d}Y_{j}$. This is an open subscheme of $Y$ and by
the argument in [16, Lemma 2.10.9] it contains $Y_{x}$.

Since the map $Y\to\operatorname{Spec}A^{+}$ is closed, the image
$Z^{\prime}\subseteq\operatorname{Spec}A^{+}$ of $Y\setminus Y^{\prime}$ under
this map is closed. By construction we have $x\not\in Z^{\prime}$. By the
argument in [22, Lemma 7.5] there exists a clopen neighborhood
$U\subseteq\operatorname{Spec}A^{+}$ of $x$ contained in
$\operatorname{Spec}(A^{+}\setminus Z^{\prime}$. Hence
$Y_{U}=Y\times_{\operatorname{Spec}A^{+}}U$ is covered by the affine opens
$(Y_{0})_{U},\dots,(Y_{d})_{U}$.

The above proof shows that for every $x\in\pi_{0}(X)$ there is a clopen
neighbourhood $U_{x}\subseteq\operatorname{Spec}A^{+}$ of $x$ such that
$Y_{U}$ can be covered by $d+1$ affine open subsets. By passing to a finite
disjoint cover of $\operatorname{Spec}A^{+}$ by $U_{x}$’s we can deduce that
$Y$ can be covered be $d+1$ affine open subsets, as desired. ∎

In Proposition 3.24 we used as well the following isomorphism of topological
spaces (similar to [16, Lemma 3.6.1]). Note that for any Tate-Huber pair
$(A,A^{+})$ there exists a natural specialization map

$\mathrm{sp}=\mathrm{sp}_{(A,A^{+})}\colon|\operatorname{Spa}(A,A^{+})|\to|\operatorname{Spa}(A^{\circ}/A^{\circ\circ},A^{+}/A^{\circ\circ})|,$

which is uniquely determined by naturality and the requirement that if
$(A,A^{+})=(K,K^{+})$ for a non-archimedean valued field $K$ and an open and
bounded valuation subring $K^{+}\subseteq K$, then $\mathrm{sp}_{(K,K^{+})}$
sends a point $x\in\operatorname{Spa}(K,K^{+})$ corresponding to a prime ideal
$\mathfrak{p}_{x}$ containing $K^{\circ\circ}$ to the valuation on
$\mathcal{O}_{K}/K^{\circ\circ}$ corresponding to the prime ideal
$\mathfrak{p}_{x}/K^{\circ\circ}$.

######  Lemma 3.27.

Let $\operatorname{Spa}(A,A^{+})$ be an affinoid Tate adic space such that
every connected component is of the form $\operatorname{Spa}(K,A^{+}_{K})$,
where $K$ is a non-archimedean field and $A^{+}_{K}\subseteq K$ is a subring.
Then the specialiation map

$\displaystyle\mathrm{sp}\colon|\operatorname{Spa}(A,A^{+})|\to|\operatorname{Spa}(A^{\circ}/A^{\circ\circ},A^{+}/A^{\circ\circ})|$

is a homeomorphism.

###### Proof.

Without loss of generality $A^{+}\subseteq A$ is an open and integrally closed
subring in $A^{\circ}$. We denote $X=\operatorname{Spa}(A,A^{+})$,
$\overline{A}=A^{\circ}/A^{\circ\circ}$,
$\overline{A}^{+}=A^{+}/A^{\circ\circ}$ and
$\overline{X}=\operatorname{Spa}(\overline{A},\overline{A}^{+})$. Then the
claim is that the specialization map induces a homeomorphism $\lvert
X\rvert\cong\lvert\overline{X}\rvert$.

First note that there is a canonical homeomorphism
$\pi_{0}(X)=\pi_{0}(\overline{X})$, as both can be derived from the idempotent
elements in $A^{+}$. Fix a connected component
$c=\operatorname{Spa}(K,A^{+}_{K})$ of $X$ and let $\overline{c}$ be the
corresponding connected component of $\overline{X}$. Let us write
$\overline{c}=\operatorname{Spa}(\overline{K},\overline{A}^{+}_{K})$. Then
$K^{\circ}$ is the completed filtered colimit of $\mathcal{O}^{\circ}_{X}(W)$
for clopen neighbourhoods $W\subseteq X$ of $c$ as the clopen neighborhoods of
$c$ are cofinal among all neighborhoods of $c$ (by the argument in [22, Lemma
7.5]). Similarly, $\overline{K}$ is the filtered colimit of
$\mathcal{O}_{\overline{X}}(\overline{W})$ for clopen neighbourhoods
$\overline{W}\subseteq\overline{X}$ of $\overline{c}$. By the homeomorphism of
spaces of connected components, the $W$ correspond to $\overline{W}$ and under
this correspondence we have
$\mathcal{O}_{\overline{X}}(\overline{W})=\mathcal{O}^{\circ}_{X}(W)/\mathcal{O}^{\circ\circ}_{X}(W)$.
It follows that $\overline{K}=K^{\circ}/K^{\circ\circ}$, which is a discrete
field.

With $c$ and $\overline{c}$ as above, we note that the points of $c$ are
precisely the open and bounded valuation rings $V$ of $K$ such that
$A^{+}_{K}\subseteq V$. Similarly, the points of $\overline{c}$ are the
valuation rings $\overline{V}$ of $\overline{K}$ such that
$\overline{A}^{+}_{K}\subseteq\overline{V}$. These two sets agree as each open
and bounded valuation ring $V$ of $K$ must contain $K^{\circ\circ}$. As $c$
was arbitrary (and $\pi_{0}(X)=\pi_{0}(\overline{X})$) this shows that the
specialization map is bijective, and as well specializing. It remains to show
that this bijection is a homeomorphism. For this it suffices to see that the
specialization map is continuous and quasi-compact, e.g., using [22, Lemma
2.5]. Note that $\overline{X}$ (and $X$) satisfy the assumption of Lemma 3.28.
Hence the topology of $X$ has a basis given by
$U_{\overline{f},\overline{W}}=\\{|\overline{f}|\leq 1\\}\mathbin{\cap}W$ for
$f\in A^{\circ}$ (with reduction $\overline{f}\in A^{\circ}/A^{\circ\circ}$)
and clopen $\overline{W}\subseteq\overline{X}$. Now,
$\mathrm{sp}^{-1}(\\{|\overline{f}|\leq 1\\})=\\{|f|\leq 1\\}$ as can be
checked by reducing to the case that $(A,A^{+})$ is an affinoid field, and
$\mathrm{sp}^{-1}(\overline{W})=W$ where $W\subseteq X$ is the clopen subset
corresponding to $\overline{W}$ under the homeomorphism
$\pi_{0}(X)\cong\pi_{0}(\overline{X})$. In particular, we can conclude that
$\mathrm{sp}^{-1}(U_{\overline{f},\overline{W}})$ is open and quasi-compact.
This finishes the proof. ∎

The proof of Lemma 3.27 made use of the following slightly technical result
about the topology on families of Zariski-Riemann spaces:

######  Lemma 3.28.

Let $X=\operatorname{Spa}(A,A^{+})$ be an affinoid adic space such that every
connected component is of the form $\operatorname{Spa}(K,A^{+}_{K})$, where
$K$ is a field and $A^{+}_{K}\subseteq K$ is a subring. Then $\lvert X\rvert$
has a basis of open subsets given by $\\{\lvert f\rvert\leq
1\\}\mathbin{\cap}W$ for varying $f\in A^{\circ}$ and clopen subsets
$W\subseteq X$.

###### Proof.

Suppose first that $X$ is connected, so that $A$ is a field. A basis of open
subsets of $X$ is given by the rational subsets, i.e. subsets of the form
$\\{\lvert f_{1}\rvert\leq\lvert g\rvert\neq 0,\dots,\lvert
f_{n}\rvert\leq\lvert g\rvert\neq 0\\}\subseteq X$ for certain
$f_{1},\dots,f_{n},g\in A$. We can assume that $g\neq 0$, so that $g$ is
invertible. Then the above subset agrees with
$\\{\lvert\frac{f_{1}}{g}\rvert\leq 1,\dots\lvert\frac{f_{n}}{g}\rvert\leq
1\\}$ and hence is an intersection of subsets of the form $\\{\lvert
f\rvert\leq 1\\}$ for varying $f\in A$. Note that if $f$ is not in $A^{0}$
(i.e. is not powerbounded) then $\\{\lvert f\rvert\leq 1\\}=\varnothing$.
Indeed, $A$ is either discrete or non-archimedean. If $A$ is discrete, then
$A=A^{\circ}$, and if $A$ is non-archimedean, then $\\{\lvert f\rvert\leq
1\\}\neq\varnothing$ implies that the point given by the non-trivial rank $1$
valuation on $A$ lies in $\\{\lvert\leq\rvert 1\\}$, which in turn yields
$f\in A^{\circ}$. This finishes the proof in the case that $X$ is connected.

Let now $X$ be general. We argue as in the proof of [16, Lemma 3.6.1]. Fix
some open subset $U\subseteq X$. Pick a point $x\in U$ and let
$c=\operatorname{Spa}(K,A^{+}_{K})\in\pi_{0}(X)$ denote the connected
component of $X$ containing $x$. By what we have shown above, there is some
$\overline{f}\in A^{+}_{K}$ such that $x\in\\{\lvert\overline{f}\rvert\leq
1\\}\subseteq U\mathbin{\cap}c$. Note that $c$ is the cofiltered limit of its
clopen neighbourhoods in $X$, which implies that $A^{+}_{K}$ is the completed
filtered colimit of $\mathcal{O}^{+}_{X}(V)$ for the clopen neighbourhoods
$W\subseteq X$ of $x$. Therefore, after potentially modifying $\overline{f}$
by some topologically nilpotent element in $A^{+}_{K}$ (which does not affect
$\\{\lvert\overline{f}\rvert\leq 1\\}$) we can extend $\overline{f}$ to some
$f_{W}\in\mathcal{O}^{+}_{X}(W)$ for some $W$ as before. Then

$\displaystyle\bigcap_{c\subseteq W^{\prime}\subseteq
W}(W^{\prime}\mathbin{\cap}X\setminus U\mathbin{\cap}\\{\lvert f_{W}\rvert\leq
1\\})=\varnothing,$

where $W^{\prime}$ ranges through all clopen neighbourhoods of $c$ contained
in $W$. Each term in the above intersection is closed in the construcible
topology on $X$, so by compactness of the constructible topology, already a
finite intersection must be empty. This implies that one of the terms must be
empty, i.e. there is some clopen neighbourhood $c\subseteq W^{\prime}\subseteq
W$ such that $W^{\prime}\mathbin{\cap}X\setminus U\mathbin{\cap}\\{\lvert
f_{W}\rvert\leq 1\\}=\varnothing$. Now let $f\in A^{+}$ be the element which
restricts to $f_{W}$ on $W^{\prime}$ and to $0$ outside of $W^{\prime}$. Let
furthermore $f^{\prime}\in A^{+}$ be the elment which is $0$ on $W^{\prime}$
and $1$ outside of $W^{\prime}$. Then $\\{\lvert f\rvert\leq
1\\}\mathbin{\cap}W^{\prime}\subseteq U$. Note that the left-hand side
contains $x$ by construction, so we have constructed an open neighbourhood of
$x$ inside $U$ which belongs to the claimed basis for the topology. This
finishes the proof. ∎

##  4 Overconvergent and nuclear sheaves

In the following we discuss a version of the nuclear
${\mathbb{Z}}_{\ell}$-sheaves defined in [15] in the case $\ell=p$. In fact,
the basic definitions and results all work in the same way (similar to the
fact that the basic results on étale $\mathbb{F}_{p}$-shaeves work the same as
for $\mathbb{F}_{\ell}$), but the theory is lacking the required base-change
result (for solid sheaves) to produce a good 6-functor formalism.131313More
precisely, [9, Proposition VII.2.1] needs the assumption $\ell\neq p$.
Nevertheless, nuclear ${\mathbb{Z}}_{p}$-sheaves will play an important role
in the proof of descent for $+$-bounded affinoid perfectoid spaces in Theorem
1.1.

###  4.1 $\omega_{1}$-solid sheaves

In the following we fix a prime $\ell$, e.g., $\ell=p$, and transport the main
definitions and results from [15] to our situation of interest. As in
Definition 3.14 resp. [15, Definition 2.1] we say that a spatial diamond $X$
is $\ell$-bounded if there is some integer $d\geq 0$ such that for all static
$\mathbb{F}_{\ell}$-modules
$\mathcal{M}\in\operatorname{\mathcal{D}}_{\mathrm{et}}(X,\mathbb{F}_{\ell})$
we have $H^{k}(X,\mathcal{M})=0$ for $k>d$.

From now on we fix a static ring $\Lambda$, which is an adic and profinite
${\mathbb{Z}}_{\ell}$-algebra. Then $\Lambda\cong\varprojlim_{n}\Lambda/I^{n}$
for finite rings $\Lambda/I^{n}$ and a finitely generated ideal of definition
$I\subseteq\Lambda$ containing $\ell$. We will abuse notation and write again
$\Lambda$ for the pro-étale sheaf $\varprojlim_{n}\Lambda/I^{n}$ with inverse
limit formed on the quasi-pro-étale site of a locally spatial diamond.

Before we can define nuclear $\Lambda$-sheaves on $\ell$-bounded spatial
diamonds, we need to study solid and $\omega_{1}$-solid $\Lambda$-sheaves as
considered in [9, Section VII.1] and [15, Section 2]. In the following, a
quasi-pro-étale map $U\to X$ of spatial diamonds is called basic if it can be
written as a cofiltered inverse limit of étale, quasi-compact and separated
maps $U_{i}\to X$ ([15, Definition 2.2]). We note that by [22, Definition
10.1.(i)] quasi-pro-étale maps of diamonds are locally separated by
definition, and hence the basic quasi-pro-étale maps form a basic for
$X_{\mathrm{qproet}}$.

######  Definition 4.1.

Let $X$ be a spatial diamond.

  1. (a)

For every basic quasi-pro-étale map $U=\varprojlim_{i}U_{i}\to X$ (see [15,
Definition 2.2]) we denote
$\Lambda_{\scalebox{0.5}{$\square$}}[U]:=\varprojlim_{i}\Lambda[U_{i}]\in\operatorname{\mathcal{D}}(X_{{\mathrm{qproet}}},\Lambda)$.141414The
quasi-pro-étale site is defined in [22, Definition 14.1.(ii)].

  2. (b)

A quasi-pro-étale sheaf
$\mathcal{M}\in\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},\Lambda)$ is
called _solid_ if for all basic quasi-pro-étale $U\to X$ the natural map

$\displaystyle\operatorname{\underline{\operatorname{Hom}}}(\Lambda_{\scalebox{0.5}{$\square$}}[U],\mathcal{M})\mathbin{\xrightarrow{\sim}}\operatorname{\underline{\operatorname{Hom}}}(\Lambda[U],\mathcal{M})$

is an isomorphism. We denote by
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)\subseteq\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},\Lambda)$
the full subcategory spanned by the solid sheaves.

  3. (c)

Assume $X$ is $\ell$-bounded. A solid sheaf
$\mathcal{M}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$
is called _$\omega_{1}$ -solid_ if for every $\omega_{1}$-filtered colimit
$U=\varprojlim_{i}U_{i}$ of basic objects in $X_{\mathrm{qproet}}$ the natural
map

$\displaystyle\varinjlim_{i}\Gamma(U_{i},\mathcal{M})\mathbin{\xrightarrow{\sim}}\Gamma(U,\mathcal{M})$

is an isomorphism. We denote by
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
the full subcategory spanned by the $\omega_{1}$-solid objects.

  4. (d)

An object $M\in\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},\Lambda)$ is
called complete if it is $I$-adically complete for some finitely generated
ideal of definition $I\subseteq\Lambda$.

######  Proposition 4.2.

Let $X$ be a spatial diamond.

  1. (i)

The category
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ is stable
under limits and colimits in
$\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},\Lambda)$ and contains all
étale sheaves.

  2. (ii)

$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ is
compactly generated and a collection of compact generators is given by
$\Lambda_{\scalebox{0.5}{$\square$}}[U]$ for w-contractible basic $U\in
X_{\mathrm{qproet}}$. Moreover, for each basic quasi-pro-étale $U\to X$ the
object $\Lambda_{\scalebox{0.5}{$\square$}}[U]$ is static.

  3. (iii)

The $t$-structure on $\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},\Lambda)$
restricts to a $t$-structure on
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$.

  4. (iv)

The composition
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)\to\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},\Lambda)\to\operatorname{\mathcal{D}}(X_{v},\Lambda)$
is fully faithful, and
$X\mapsto\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ is
a $v$-sheaf of $\infty$-categories.

  5. (v)

The inclusion
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)\to\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},\Lambda)$
admits a left adjoint $(-)_{\scalebox{0.5}{$\square$}}$ such that
$(\Lambda[U])_{\scalebox{0.5}{$\square$}}=\Lambda_{\scalebox{0.5}{$\square$}}[U]$
for $U\to X$ basic quasi-pro-étale.

  6. (vi)

There exists a unique closed symmetric monoidal structure
$-\otimes^{\scalebox{0.5}{$\square$}}_{\Lambda}-$ on
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ such that
$(-)_{\scalebox{0.5}{$\square$}}$ is symmetric monoidal. We have
$\Lambda_{\scalebox{0.5}{$\square$}}[U]\otimes^{\scalebox{0.5}{$\square$}}_{\Lambda}\Lambda_{\scalebox{0.5}{$\square$}}[U^{\prime}]\cong\Lambda_{\scalebox{0.5}{$\square$}}[U\times_{X}U^{\prime}]$
for $U,U^{\prime}\to X$ basic quasi-pro-étale.

  7. (vii)

If
$M,N\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}^{-}(X,\Lambda)$
are complete, then $M\otimes_{\Lambda}^{\scalebox{0.5}{$\square$}}N$ is
complete.

  8. (viii)

For $M\in\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},\Lambda)$ and
$N\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ we
have
$\operatorname{\underline{\operatorname{Hom}}}_{\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},\Lambda)}(M,N)\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$.

  9. (ix)

The natural functor
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)\to\mathrm{Mod}_{\Lambda}(\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell}))$
is an equivalence.

In [9, Defintion VII.1.17],
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ is defined
as $\Lambda$-modules in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$
for any solid ${\mathbb{Z}}_{\ell}$-algebra $\Lambda$. If $\Lambda$ is an adic
and profinite ${\mathbb{Z}}_{\ell}$-algebra, then Proposition 4.2 shows that
the neat characterization Definition 4.1 holds.

###### Proof.

We first assume that $\Lambda=\mathbb{Z}_{\ell}$. We first note that our (a
priori different) definition Definition 4.1 agrees with [9, Definition
VII.1.10]. Indeed, if
$M\in\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},{\mathbb{Z}}_{\ell})$ has
solid cohomology objects in the sense of [9, Definition VII.1.1] then by [9,
VII.1.12] $M$ is solid in the sense of Definition 4.1. Moreover, the full
subcategory
$\mathcal{C}\subseteq\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},{\mathbb{Z}}_{\ell})$
defined in [9, Definition VII.1.10] is stable under all colimits and limits in
$\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},{\mathbb{Z}}_{\ell})$ (by [9,
Theorem VII.1.3]) and contains ${\mathbb{Z}}_{\ell}[U]$ for any basic quasi-
pro-étale $U\to X$. Let
$N\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$.
Then $N=\varinjlim_{i}\Lambda[U_{i}]$ is a colimit in
$\operatorname{\mathcal{D}}(X,{\mathbb{Z}}_{\ell})$ for basic quasi-pro-étale
morphisms $U_{i}\to X$ with $U_{i}$ $w$-contractible, as these form a basis of
$X_{\mathrm{qproet}}$. This implies that $N$ is a retract of
$\varinjlim_{i}\Lambda_{{\scalebox{0.5}{$\square$}}}[U_{i}]$, and thus
contained in $\mathcal{C}$. This finishes the claim that
$\mathcal{C}=\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$.
Now, the assertions follow from [9, Theorem VII.1.3], [9, Proposition
VII.1.13] and [9, Proposition VII.1.14], [9, Proposition VII.1.8], [9,
Proposition VII.1.11] and [15, Proposition 2.8] (which does not use the
assumption $\ell\neq p$). This finishes the case that
$\Lambda={\mathbb{Z}}_{\ell}$.

Let now $\Lambda$ be a general adic and profinite
${\mathbb{Z}}_{\ell}$-algebra. We claim that the natural map

${\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U]\otimes^{\scalebox{0.5}{$\square$}}\Lambda\to\Lambda_{\scalebox{0.5}{$\square$}}[U]$

is an isomorphism for any basic quasi-pro-étale $U\to X$. Now, $\Lambda=U\to
X$ is represented (as a sheaf on $X_{\mathrm{qproet}}$) by a basic quasi-pro-
étale morphism over $X$ because $\Lambda$ is adic and profinite. In
particular, we see that $\Lambda$ (being solid) is a retract of
${\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[\Lambda]$. We know that

${\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U]\otimes^{\scalebox{0.5}{$\square$}}{\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[\Lambda]\cong{\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U\times_{X}\Lambda],$

which implies the claim by the passage to a retract (and it implies that
$\Lambda_{\scalebox{0.5}{$\square$}}[U]$ is static). From here it is formal
that an object $C\in\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},\Lambda)$
is solid if and only if its underlying object
$C_{|{\mathbb{Z}}_{\ell}}\in\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},{\mathbb{Z}}_{\ell})$
is solid. Indeed, if $C$ is solid, then by the same argument as before $C$ is
a retract of a colimit of $\Lambda_{\scalebox{0.5}{$\square$}}[U]$’s with
$U\to X$ basic quasi-pro-étale, and each of the
$\Lambda_{\scalebox{0.5}{$\square$}}[U]$ is solid. Conversely, if
$C_{|{\mathbb{Z}}_{\ell}}$ is solid, then

$\operatorname{\underline{\operatorname{Hom}}}_{\Lambda}(\Lambda_{\scalebox{0.5}{$\square$}}[U],C)\cong\operatorname{\underline{\operatorname{Hom}}}_{\Lambda}(\Lambda\otimes^{\scalebox{0.5}{$\square$}}{\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U],C)\cong\operatorname{\underline{\operatorname{Hom}}}_{{\mathbb{Z}}_{\ell}}({\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U],C_{|{\mathbb{Z}}_{\ell}})\cong\Gamma(U,C)$

as desired.

We can conclude that
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)\subseteq\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},\Lambda)$
is stable under all colimits and limits, and that it contains all (static)
sheaves of $\Lambda$-modules, which are pulled back from $X_{\mathrm{et}}$.
Similarly, the claim on the $t$-structure on
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ follows as
well as the existence of the desired left adjoint
$(-)_{\scalebox{0.5}{$\square$}}$. To get the symmetric monoidal structure
$-\otimes^{\scalebox{0.5}{$\square$}}_{\Lambda}-$ it suffices to show that if
$C,D\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ then
$\operatorname{\underline{\operatorname{Hom}}}_{\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},\Lambda)}(C,D)$
is contained in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$. By
stability of
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ under
limits it suffices to handle the case that
$C=\Lambda\otimes^{\scalebox{0.5}{$\square$}}_{{\mathbb{Z}}_{\ell}}C^{\prime}$
for some
$C^{\prime}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\mathbb{Z}_{\ell})$.
Then it follows from the same assertion for $\Lambda={\mathbb{Z}}_{\ell}$
(implicitly proven in [9, Proposition VII.1.14]).

Assume that
$M,N\in\operatorname{\mathcal{D}}^{-}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$
are complete. From the case $\Lambda={\mathbb{Z}}_{\ell}$ we can conclude that
for each $n\geq 0$ the tensor product
$M\otimes_{{\mathbb{Z}}_{\ell}}^{\scalebox{0.5}{$\square$}}\Lambda^{\otimes
n}\otimes_{{\mathbb{Z}}_{\ell}}N$ is $\ell$-adically complete. This implies
that $M\otimes^{\scalebox{0.5}{$\square$}}_{\Lambda}N$ is $\ell$-adically
complete as the $\ell$-adic completion commutes with uniformly right-bounded
geometric realizations. Using a similar argument for preservation of
completedness under geometric realizations, we can reduce first to the case
that $\Lambda=\mathbb{F}_{\ell}[[x_{1},\ldots,x_{r}]]$ for some $r\geq 0$, and
then to the case that $r=1$. For $\Lambda=\mathbb{F}_{\ell}[[x]]$ the same
argument as in [15, Proposition 2.8] applies.

The final claim
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)\cong\mathrm{Mod}_{\Lambda}(\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell}))$
follows from the observation that both embed fully faithfully into
$\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},\Lambda)$ with generators
${\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U]\otimes^{\scalebox{0.5}{$\square$}}\Lambda\cong\Lambda_{\scalebox{0.5}{$\square$}}[U]$.
Now, we can conclude that
$X\mapsto\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ is
a $v$-sheaf of $\infty$-categories on spatial diamonds, e.g., using [12,
Corollary 5.2.2.37]. ∎

We now turn to the better behaved subcategory of $\omega_{1}$-solid sheaves,
which was introduced (for $\ell\neq p$) in [15]. We note that the compact
generators of
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
are much simpler than the ones of
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$. In
particular,
$\Lambda\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
is compact, while in general
$\Lambda\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$
is not.

######  Proposition 4.3.

Let $X$ be an $\ell$-bounded spatial diamond.

  1. (i)

The category
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
is stable under colimits and countable limits in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ and
contains all étale sheaves.

  2. (ii)

$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
is compactly generated and a collection of compact generators is given by
$\Lambda_{\scalebox{0.5}{$\square$}}[U]$ for sequential limits
$U=\varprojlim_{n}U_{n}$ with all $U_{n}\to X$ being étale, quasicompact and
separated.

  3. (iii)

The $t$-structure on
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ restricts
to a $t$-structure on
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$.

  4. (iv)

The tensor product $-\otimes^{\scalebox{0.5}{$\square$}}_{\Lambda}-$ in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ restricts
to a closed symmetric monoidal structure on
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$.

  5. (v)

If
$M\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
is $\omega_{1}$-compact, and
$N\in\operatorname{\mathcal{D}}_{{\scalebox{0.5}{$\square$}}}(X,\Lambda)_{\omega_{1}}$
compact, then
$\operatorname{\underline{\operatorname{Hom}}}_{\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}}(M,N)=\operatorname{\underline{\operatorname{Hom}}}_{\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)}(M,N)$.

  6. (vi)

$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}\cong\mathrm{Mod}_{\Lambda}(\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}})$.

###### Proof.

If $\Lambda={\mathbb{Z}}_{\ell}$ then this - except for the claim on internal
Hom’s - is proven in [15, Proposition 2.5] (note that the proof of that result
uses nowhere that $\ell\neq p$). Thus, let
$M\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
be $\omega_{1}$-compact, and let
$N\in\operatorname{\mathcal{D}}_{{\scalebox{0.5}{$\square$}}}(X,\Lambda)_{\omega_{1}}$
be compact. Then $M$ is a countable colimit of compact objects, and by
stability of
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
under countable limits we may therefore assume that
$M=\Lambda_{\scalebox{0.5}{$\square$}}[U]$ is compact. As $N$ is compact, we
may reduce to the case that it is a countable inverse limit of étale sheaves,
and thus we may assume that $N$ is an étale sheaf. In this case, one checks
directly that
$\operatorname{\underline{\operatorname{Hom}}}_{\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)}(M,N)$
is $\omega_{1}$-solid. This finishes the case $\Lambda={\mathbb{Z}}_{\ell}$.

Back in the case that $\Lambda$ is a static adic and profinite
${\mathbb{Z}}_{\ell}$-algebra we observe that
$\Lambda\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
and that an object
$C\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ is
$\omega_{1}$-solid if and only if
$C_{|\mathbb{Z}_{\ell}}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$
is $\omega_{1}$-solid. Moreover,
$\Lambda_{\scalebox{0.5}{$\square$}}[U]={\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U]\otimes_{{\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}}\Lambda$
by the proof of Proposition 4.2. It follows that
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
is the same as the category of $\Lambda$-modules in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
by Proposition 4.2. All assertions now follow from the case
$\Lambda={\mathbb{Z}}_{\ell}$. ∎

We note the following stability of $(\omega_{1}-)$solid sheaves under pullback
and pushforward along maps of spatial diamonds.

######  Lemma 4.4.

Let $f\colon X^{\prime}\to X$ be a morphism spatial diamonds, with associated
morphisms $f_{\mathrm{qproet}}\colon X^{\prime}_{\mathrm{qproet}}\to
X_{\mathrm{qproet}},f_{v}\colon X^{\prime}_{v}\to X_{v}$ on sites.

  1. (i)

$f^{\ast}_{\mathrm{qproet}}\colon\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},\Lambda)\to\operatorname{\mathcal{D}}(X^{\prime}_{\mathrm{qproet}},\Lambda),f^{\ast}_{v}\colon\operatorname{\mathcal{D}}(X_{v},\Lambda)\to\operatorname{\mathcal{D}}(X_{v},\Lambda)$
restrict to the same $t$-exact, symmetric monoidal functor

$f^{\ast}\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X^{\prime},\Lambda),$

which preserves $\omega_{1}$-solid objects (if $X^{\prime},X$ are
$\ell$-bounded).

  2. (ii)

If $f$ is quasi-pro-étale, then
$f_{{\mathrm{qproet}},\ast}\colon\operatorname{\mathcal{D}}(X^{\prime}_{\mathrm{qproet}},\Lambda)\to\operatorname{\mathcal{D}}(X,\Lambda)$
restricts to a colimit-preserving functor

$f_{\ast}\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X^{\prime},\Lambda)\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda),$

which preserves $\omega_{1}$-solid sheaves (if $X^{\prime},X$ are
$\ell$-bounded).

  3. (iii)

If $\ell\neq p$, then
$f_{{\mathrm{qproet}},\ast}\colon\operatorname{\mathcal{D}}(X^{\prime}_{\mathrm{qproet}},\Lambda)\to\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},\Lambda),f_{v,\ast}\colon\operatorname{\mathcal{D}}(X^{\prime}_{v},\Lambda)\to\operatorname{\mathcal{D}}(X_{v},\Lambda)$
restrict to the same colimit-preserving functor

$f_{v,\ast}\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X^{\prime},\Lambda)\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda),$

which preserves $\omega_{1}$-solid sheaves (if $X^{\prime},X$ are
$\ell$-bounded).

###### Proof.

The first assertion follows from [9, Proposition VII.1.8] (and the proof of
[15, Proposition 2.6] for the preservation of $\omega_{1}$-solid sheaves). The
third assertion (for $f_{v,\ast}$) is [9, Proposition VII.2.1]. Using [22,
Corollary 16.8], [22, Corollary 16.9] and [22, Corollary 16.10] the same
argument implies the second assertion and the case
$f_{{\mathrm{qproet}},\ast}$ in the third assertion. The preservation of
$\omega_{1}$-solid sheaves follows as in [15, Proposition 2.6] (replacing the
use of [22, Corollary 16.8.(ii)] by [22, Corollary 16.8.(i)] for the second
part). ∎

######  Corollary 4.5.

  1. (i)

The functors
$X\mapsto\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$,
$X\mapsto\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$
are hypercomplete quasi-pro-étale sheaves on the site of $\ell$-bounded
spatial diamonds.151515By Lemma 3.17 being $\ell$-bounded ascents along under
quasi-compact, separated quasi-pro-étale maps

  2. (ii)

If $\ell\neq p$, then
$X\mapsto\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$,
$X\mapsto\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$
are hypercomplete $v$-sheaves on the site of $\ell$-bounded spatial diamonds.

###### Proof.

The case for
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ is clear
by [9, Proposition VII.1.8]. Thanks to Lemma 4.4 the proof of [15, Corollary
2.7] implies both assertions for $\omega_{1}$-solid sheaves. ∎

###  4.2 Overconvergent solid sheaves

As in Section 4.1 we fix a prime $\ell$ and a static adic and profinite
${\mathbb{Z}}_{\ell}$-algebra $\Lambda$.

We now discuss the definition of overconvergent objects in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ following
[15, Section 6], although with some differences. Most notably, we don’t
require that overconvergent sheaves are nuclear (in the sense of [15,
Definition 3.1]). Note that if $T$ is a profinite set, then we can consider
its quasi-pro-étale site $T_{\mathrm{qproet}}$161616For example, defined as
the slice $\ast_{\mathrm{proet}}/T$ with $\ast_{\mathrm{proet}}$ the pro-étale
site of a point, [3, Section 4.3]., and the full subcategories
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(T,\Lambda)_{\omega_{1}}\subseteq\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(T,\Lambda)\subseteq\operatorname{\mathcal{D}}(T_{\mathrm{qproet}},\Lambda)$
similarly defined as in Definition 4.1. These categories satisfy the same
properties as in Proposition 4.2 and Proposition 4.3 (this can be checked,
e.g., by taking the product of $T$ with
$\operatorname{Spa}(C,\mathcal{O}_{C})$ for $C$ a perfectoid algebraically
closed field in characteristic $p$). In particular, we can call a morphism
$S\to T$ of profinite sets “basic quasi-pro-étale” if
$S=\varprojlim_{i}S_{i}\to T$ with $S_{i}\to T$ a local isomorphism for
$S_{i}$ a profinite set.

We recall that if $X$ is a spatial diamond, then there exists a natural
morphism $\pi=\pi_{X}\colon
X_{\mathrm{qproet}}\to\pi_{0}(X)_{\mathrm{qproet}}$ of sites such that
$\pi^{-1}(T)=T\times_{\pi_{0}(X)}X$ ([17, Definition 5.5]).

######  Lemma 4.6.

Let $X$ be a spatial diamond with morphism of sites $\pi\colon
X_{\mathrm{qproet}}\to\pi_{0}(X)_{\mathrm{qproet}}$.

  1. (i)

The functor
$\pi^{-1}\colon\widetilde{\pi_{0}(X)_{\mathrm{qproet}}}\to\widetilde{X_{\mathrm{qproet}}}$
commutes with limits, and
$\pi_{\ast}\circ\pi^{-1}=\mathrm{Id}_{\widetilde{\pi_{0}(X)_{\mathrm{qproet}}}}$.
In fact, if $V\in X_{{\mathrm{qproet}}}$ is qcqs, then
$\pi^{-1}(\mathcal{F})(V)=\mathcal{F}(\pi_{0}(V))$ for any sheaf of sets
$\mathcal{F}$ on $\pi_{0}(X)$.

  2. (ii)

The functor
$\pi^{-1}\colon\widetilde{\pi_{0}(X)_{\mathrm{qproet}}}\to\widetilde{X_{\mathrm{qproet}}}$
admits a left adjoint
$\pi_{\natural}\colon\widetilde{X_{\mathrm{qproet}}}\to\widetilde{\pi_{0}(X)_{\mathrm{qproet}}}$
such that $\pi_{\natural}(V\to X)=(\pi_{0}(V)\to\pi_{0}(X))$ if $V\in
X_{\mathrm{qproet}}$ is qcqs.

###### Proof.

The first assertion is [17, Lemma 5.7], and the second is implied by [17,
Lemma 5.4]. ∎

In the strictly totally disconnected case stronger properties hold true
(generalizing the results in [15, Lemma 6.7]).

######  Lemma 4.7.

Let $X$ be a strictly totally disconnected perfectoid space.

  1. (i)

If $X$ is strictly totally disconnected, then
$\pi^{\ast}\colon\operatorname{\mathcal{D}}(\pi_{0}(X)_{\mathrm{qproet}},\Lambda)\to\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},\Lambda)$
is fully faithful, and preserves ($\omega_{1}$-)solid sheaves and all colimits
and limits. Its restriction,
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\pi_{0}(X)_{\mathrm{qproet}},\Lambda)\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X_{\mathrm{qproet}},\Lambda)$
is symmetric monoidal.

  2. (ii)

The functor
$\pi_{\ast}\colon\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},\Lambda)\to\operatorname{\mathcal{D}}(\pi_{0}(X)_{\mathrm{qproet}},\Lambda)$
is $t$-exact and preserves ($\omega_{1}$-)solid sheaves.

  3. (iii)

The functor
$\pi^{\ast}\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\pi_{0}(X),\Lambda)\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$
admits a left adjoint $\pi_{\natural}$, which is symmetric monoidal and
preserves $\omega_{1}$-solid sheaves. Moreover, if $U\to X$ is basic quasi-
pro-étale, then
$\pi_{\natural}(\Lambda_{\scalebox{0.5}{$\square$}}[U])\cong\Lambda_{\scalebox{0.5}{$\square$}}[\pi_{0}(U)]$.

  4. (iv)

For $M\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$
and
$N\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\pi_{0}(X),\Lambda)$
the natural map

$\operatorname{\underline{\operatorname{Hom}}}_{\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)}(M,\pi^{\ast}N)\to\pi^{\ast}\operatorname{\underline{\operatorname{Hom}}}_{\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\pi_{0}(X),\Lambda)}(\pi_{\natural}M,N)$

is an isomorphism. Similarly, with
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(-,\Lambda)$ replaced
by
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(-,\Lambda)_{\omega_{1}}$.

###### Proof.

Note that Lemma 4.6 implies that $\Lambda$ is pulled back from
$\pi_{0}(X)_{\mathrm{qproet}}$, and thus $\pi^{\ast}=\pi^{-1}$ (on static
sheaves of $\Lambda$-modules). Hence, for fully faithfulness in the first
assertion it suffices (by Lemma 4.6) to see that $\pi_{\ast}$ is exact on
static sheaves of $\Lambda$-modules. This can be checked on extremally
disconnected profinite sets $T$ over $\pi_{0}(X)$. For such $T$ the pullback
$Z:=\pi^{-1}(T)=T\times_{\pi_{0}(X)}X$ is $w$-contractible in
$X_{\mathrm{qproet}}$, i.e., each pro-étale cover of $Z$ splits. Preservation
of ($\omega_{1}$-)solid sheaves by $\pi^{\ast}$ follows from commutation of
$\pi^{-1}$ with limits and colimits by checking on the generators of
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\pi_{0}(X),\Lambda)$
resp.
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\pi_{0}(X),\Lambda)_{\omega_{1}}$
provided by Proposition 4.2 resp. Proposition 4.3 (translated to
$\pi_{0}(X)$).

That $\pi^{\ast}$ is symmetric monoidal can be checked on generators where it
follows by Proposition 4.2 from the fact that

$(T\to\pi_{0}(X))\mapsto T\times_{\pi_{0}(X)}X$

commutes with fiber products. Here, $T$ is assumed to be profinite.

For (ii) we have to check that $\pi_{\ast}$ preserves ($\omega_{1}$-)solid
sheaves. Let
$N\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ and
$T\to\pi_{0}(X)$ profinite, written as $T=\varprojlim_{i}T_{i}$ with $T_{i}$
profinite and $T_{i}\to\pi_{0}(X)$ a local isomorphism. Set
$\Lambda_{\scalebox{0.5}{$\square$}}[T]:=\varprojlim_{i}\Lambda[T_{i}]$ on
$\pi_{0}(X)_{\mathrm{qproet}}$. Note that
$\pi^{\ast}(\Lambda_{\scalebox{0.5}{$\square$}}[T])=\Lambda_{\scalebox{0.5}{$\square$}}[\pi^{-1}(T)]$
as $\pi^{-1}$ preserves limits. Then we can conclude

$\displaystyle\operatorname{\underline{\operatorname{Hom}}}(\Lambda_{\scalebox{0.5}{$\square$}}[T],\pi_{\ast}N)$
$\displaystyle=$
$\displaystyle\pi_{\ast}(\operatorname{\underline{\operatorname{Hom}}}(\Lambda_{\scalebox{0.5}{$\square$}}[\pi^{-1}T]),N))$
$\displaystyle=$
$\displaystyle\pi_{\ast}(\operatorname{\underline{\operatorname{Hom}}}(\Lambda[\pi^{-1}T],N))$
$\displaystyle=$
$\displaystyle\pi_{\ast}(\operatorname{\underline{\operatorname{Hom}}}(\pi^{-1}(\Lambda[T]),N))$
$\displaystyle=$
$\displaystyle\operatorname{\underline{\operatorname{Hom}}}(\Lambda[T],\pi_{\ast}(N))$

using that $N$ is solid and the formal identity
$\pi^{-1}(\Lambda[T])\cong\Lambda[\pi^{-1}T]$. For for any cofiltered limit of
maps $T_{j}\to\pi_{0}(X)$ with $T_{j}$ profinite we have

$\Gamma(\varprojlim_{j}T_{j},\pi_{\ast}(N))\cong\Gamma(\pi^{-1}(\varprojlim_{j}T_{j}),N)\cong\Gamma(\varprojlim_{j}\pi^{-1}T_{j},N)$

as $\pi^{-1}$ preserves limits. From here it follows that $\pi_{\ast}$
preserves $\omega_{1}$-solid objects.

The existence of $\pi_{\natural}$ follows as $\pi^{-1}$ preserves limits (or
from [17, Lemma 5.4]). It follows formally from the formula
$\pi^{-1}\mathcal{F}(V)=\mathcal{F}(\pi_{0}(V))$ that
$\pi_{\natural}(\Lambda_{\scalebox{0.5}{$\square$}}[U])=\Lambda_{\scalebox{0.5}{$\square$}}[\pi_{0}(U)]$
for $U\to X$ basic quasi-pro-étale. Using [15, Lemma 5.8] we can conclude that
$\pi_{0}(U\times_{X}V)=\pi_{0}(U)\times_{\pi_{0}(X)}\pi_{0}(V)$ for $U,V$
basic quasi-pro-étale over $X$. This implies that $\pi_{\natural}$ is
symmetric monoidal by Proposition 4.2. We need to see that $\pi_{\natural}$
preserves $\omega_{1}$-solid sheaves. The explicit formula for
$\pi_{\natural}\Lambda_{\scalebox{0.5}{$\square$}}[U]$ and the fact that
$\pi_{0}(\varprojlim_{i}U_{i})\cong\varprojlim_{i}\pi_{0}(U_{i})$ for étale,
separated maps $U_{i}\to X$ in a cofiltered system $\\{U_{i}\\}_{i}$ shows
that it suffices to show that $\pi_{\natural}$ maps
$\Lambda_{\scalebox{0.5}{$\square$}}[U]$ to an $\omega_{1}$-solid sheaf of
$U\to X$ is étale quasi-compact and separated. This proven in Lemma 4.8. The
assertion (iv) follows formally from symmetric monoidality of
$\pi_{\natural}$. In fact, the left hand side is right adjoint as a functor in
$N$ to $\pi_{\natural}(M\otimes^{\scalebox{0.5}{$\square$}}_{\Lambda}(-))$
while the right hand side is right adjoint to
$\pi_{\natural}(M)\otimes^{\scalebox{0.5}{$\square$}}_{\Lambda}\pi_{\natural}(-)$.
∎

The proof of Lemma 4.7 made use of following result about
$\omega_{1}$-solidness of $\pi_{\natural}\Lambda[U]$. Be aware that even if
$U\mathrel{\hookrightarrow}X$ is an open immersion, the map
$\pi_{0}(U)\to\pi_{0}(X)$ is in general quite far from being an open immersion
itself, which makes the $\omega_{1}$-solidness of
$\pi_{\natural}\Lambda[U]=\Lambda_{\scalebox{0.5}{$\square$}}[\pi_{0}(U)]$
more subtle than one might think at first.

######  Lemma 4.8.

Let $j\colon U\to X$ be a quasi-compact, separated, étale morphism to a
strictly totally disconnected space $X$. Then
$\pi_{\natural}\Lambda[U]=\Lambda_{\scalebox{0.5}{$\square$}}[\pi_{0}(U)]\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\pi_{0}(X),\Lambda)$
is $\omega_{1}$-solid.

###### Proof.

Write $X=\operatorname{Spa}(A,A^{+})$. As $X$ is strictly totally
disconnected, each point $u\in U$ has a quasi-compact open neighborhood
$V_{u}$, such that the map $V_{u}\to X$ is an open immersion. Indeed, this
follows from the definition of an étale morphism ([22, Definition 6.2]) using
that each finite étale map onto some quasi-compact open in $X$ will split.
Hence, without loss of generality $j\colon U\to X$ can be assumed to be a
quasi-compact open immersion (using that $\omega_{1}$-solid sheaves are stable
under finite colimits). By [22, Lemma 7.6] $U$ is the intersection of subsets
$\\{|f|\leq 1\\}$ for varying $f\in A$. As $U\subseteq X$ is quasi-compact
open, $X\setminus U$ is quasi-compact in the constructible topology. As any
$\\{|f|\leq 1\\}$ for $f\in A$ is closed in the constructible topology, this
implies that $U$ is the finite intersection of subsets $U_{1}=\\{|f_{1}|\leq
1\\},\ldots,U_{r}=\\{|f_{r}|\leq 1\\}$ with $f_{1},\ldots,f_{r}\in A$. By [17,
Lemma 5.8]
$\pi_{0}(U)=\pi_{0}(U_{1})\times_{\pi_{0}(X)}\ldots\times_{\pi_{0}(X)}\pi_{0}(U_{r})$
and thus Proposition 4.2, Proposition 4.3 imply that we may assume that
$U=U_{1}=\\{|f|\leq 1\\}$ for $f:=f_{1}$. Let $\pi\in A$ be a pseudo-
uniformizer and set

$W_{n}:=\\{|f|<|\pi^{-1/p^{n}}|\\}$

for $n\in\mathbb{N}$. Each $W_{n}$ is a spectral space, being closed in the
constructible topology ([27, 0902]), and hence $\pi_{0}(W_{n})$ is a profinite
set. We claim:

  1. 1)

$\pi_{0}(U),\pi_{0}(W_{n}),n\in\mathbb{N},$ are subspaces of $\pi_{0}(X)$,

  2. 2)

$\bigcap\limits_{n\in\mathbb{N}}W_{n}$ is the closure of $U$,

  3. 3)

$\pi_{0}(U)=\bigcap\limits_{n\in\mathbb{N}}\pi_{0}(W_{n})$,

  4. 4)

for each $n\in\mathbb{N}$ the subspace $\pi_{0}(U)$ lies in the interior of
$\pi_{0}(W_{n})\subseteq\pi_{0}(X)$.

The claims imply that $\Lambda_{\scalebox{0.5}{$\square$}}[\pi_{0}(U)]$ is
$\omega_{1}$-solid. Indeed, we may inductively construct quasi-compact open
subsets $S_{n}\subseteq\pi_{0}(X)$ with $\pi_{0}(U)\subseteq
S_{n}\subseteq\pi_{0}(W_{n})$, and $S_{n+1}\to S_{n}$. Then
$\Lambda_{\scalebox{0.5}{$\square$}}[\pi_{0}(U)]=\varprojlim\limits_{n\in\mathbb{N}}\Lambda[S_{n}]$
is $\omega_{1}$-solid by Proposition 4.3.

To show 1) it suffices (because the $\pi_{0}$ are profinite and the maps
continuous) to show that $\pi_{0}(U)\to\pi_{0}(X),\pi_{0}(W_{n})\to\pi_{0}(X)$
are injective. Now, $U$ is open, and thus stable under generalization while
the $W_{n}$ are closed, hence stable under specializations. This implies that
if $Z\subseteq X$ is a connected component, then $Z\cap U$, $Z\cap W_{n}$ are
connected (if non-empty) because $Z$ is a linearly ordered chain of points.
This in turn implies injectivity.

As each $W_{n}$ is closed, and $U\subseteq W_{n}$ we see
$\overline{U}\subseteq\bigcap\limits_{n\in\mathbb{N}}W_{n}$. The reverse
implication may be checked after intersection with each connected component
$Z$ of $X$ (because $\overline{U}$ is the set of specializations of $U$ by
[27, 0903]). If $Z\cap U\neq$, then $\overline{U}\cap Z=Z$ and similarly for
the intersection with the $W_{n}$’s. Let $z$ be the generic point of $Z$ and
assume that $Z\cap\overline{U}=\varnothing$ or equivalently $Z\cap
U=\varnothing$. Thus, $|f(z)|>1$ and as $z$ is a rank $1$ valuation this
implies that $z\notin\\{|f|\leq|\pi^{-1/p^{n}}|\\}$ for some $n\in\mathbb{N}$.
This implies that $Z\cap W_{n}=\varnothing$ as the set
$\\{|f|\leq|\pi^{-1/p^{n}}|\\}$ is stable under generalizations. Thus
$Z\cap\bigcap\limits_{n\in\mathbb{N}}W_{n}=\varnothing$ as well.

Assertion 3) follows from 2) because
$\pi_{0}(U)=\pi_{0}(\overline{U})=\bigcap\limits_{n\in\mathbb{N}}\pi_{0}(W_{n})$.

We are left with assertion 4). Fix a connected component $Z\subseteq X$ with
$Z\cap U\neq\varnothing$, i.e., the point $z\in\pi_{0}(X)$ defined by $Z$ lies
in $\pi_{0}(U)$. We can write $Z=\bigcap\limits_{i\in I}f^{-1}(T_{i})$ with
$T_{i}\subseteq\pi_{0}(X)$ running through the quasi-compact open
neighborhoods of $z$ and $f\colon|X|\to\pi_{0}(X)$ the natural quotient map.
Note that the $f^{-1}(T_{i})$ are closed and open in $X$. Moreover, let us fix
$n\in\mathbb{N}$. We know that $Z\subseteq W_{n}$ as $\overline{U}\subseteq
W_{n}$ and the generic point of $Z$ lies in $U$. Furthermore, $X\setminus
W_{n}=\\{|\pi^{-1/p^{n}}|\leq|f|\\}$ is a rational open subset of $X$. In
particular, $X\setminus W_{n}$ is compact in the constructible topology. As
$X\setminus W_{n}\cap\bigcap\limits_{i\in I}f^{-1}(T_{i})=X\setminus W_{n}\cap
Z=\varnothing$, we can conclude that there exists some $i_{0}$ such that
$X\setminus W_{n}\cap f^{-1}(T_{i_{0}})=\varnothing$, i.e.,
$f^{-1}(T_{i_{0}})\subseteq W_{n}$. This implies that $z\in
T_{i_{0}}\subseteq\pi_{0}(W_{n})$. In particular, $z$ lies in the interior of
$\pi_{0}(W_{n})$. This finishes the proof. ∎

With the previous results at hand, we can now introduce a well-behaved (see
Lemma 4.10) notion of overconvergent sheaves on general spatial diamonds:

######  Definition 4.9.

  1. (a)

Let $X$ be a strictly totally disconnected space with morphism of sites
$\pi\colon X_{\mathrm{qproet}}\to\pi_{0}(X)_{\mathrm{qproet}}$. An object
$M\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ is
called overconvergent if $M\cong\pi^{\ast}N$ for some
$N\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\pi_{0}(X)_{\mathrm{proet}},\Lambda)$.

  2. (b)

Let $X$ be a spatial diamond. An object
$M\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ is
called overconvergent if $f^{\ast}M$ is overconvergent for any quasi-pro-étale
morphism $f\colon X^{\prime}\to X$ with $X^{\prime}$ strictly totally
disconnected.

We let
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)^{\mathrm{oc}}\subseteq\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$
(resp.
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)^{\mathrm{oc}}_{\omega_{1}}\subseteq\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$)
denote the full subcategories spanned by the overconvergent objects (resp. the
overconvergent and $\omega_{1}$-solid objects).

######  Lemma 4.10.

Let $f\colon X^{\prime}\to X$ be a morphism of strictly totally disconnected
spaces.

  1. (i)

If $M\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ is
overconvergent, then so is $f^{\ast}M$.

  2. (ii)

Conversely, if $f$ is a quasi-pro-étale cover and $f^{\ast}M$ is
overconvergent, then so is $M$.

###### Proof.

The first claim follows by naturality in $X$ of the morphism of sites
$\pi\colon X_{\mathrm{qproet}}\to\pi_{0}(X)_{\mathrm{qproet}}$. For the
converse we note that (using [17, Lemma 5.8]) it suffices to see that
$X\mapsto\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\pi_{0}(X),\Lambda)$
is a hypercomplete quasi-pro-étale sheaf on strictly totally disconnected
spaces. Now Corollary 4.5 implies that
$T\mapsto\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(T,\Lambda)$ is
a quasi-pro-étale sheaf on profinite sets, in other words a condensed sheaf of
$\infty$-categories on $\ast_{\mathrm{qproet}}$. As $X\mapsto\pi_{0}(X)$ sends
quasi-pro-étle covers, even $v$-covers, to quasi-pro-étale covers the claim
follows. ∎

The next result summarizes the very good properties of the full subcategory of
overconvergent objects.

######  Lemma 4.11.

Let $X$ be an $\ell$-bounded spatial diamond.

  1. (i)

Overconvergent objects satisfy quasi-pro-étale hyperdescent on the big site of
$\ell$-bounded spatial diamonds.

  2. (ii)

The inclusion
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)^{\mathrm{oc}}\subseteq\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$
admits a symmetric monoidal left adjoint $M\mapsto M_{\mathrm{oc}}$ and a
right adjoint $M\mapsto M^{\mathrm{oc}}$. In particular,
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)^{\mathrm{oc}}$
is stable under all colimits and limits in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$.

  3. (iii)

The inclusion
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)^{\mathrm{oc}}_{\omega_{1}}\subseteq\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
admits a symmetric monoidal left adjoint $M\mapsto M_{\mathrm{oc}}$ and a
right adjoint $M\mapsto M^{\mathrm{oc}}$. In particular,
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)^{\mathrm{oc}}_{\omega_{1}}$
is stable under all colimits and limits in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$.

  4. (iv)

If $M,N\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$
and $N$ is overconvergent, then
$\operatorname{\underline{\operatorname{Hom}}}_{\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)}(M,N)$
is overconvergent and naturally isomorphic to
$\operatorname{\underline{\operatorname{Hom}}}_{\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)}(M_{\mathrm{oc}},N)$.
Similarly, with
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ replaced
by
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$.

  5. (v)

If $U=\varprojlim\limits_{i\in I}U_{i}\to X$ with $U_{i}\to X$ étale, quasi-
compact and separated, then the natural map
$\Lambda_{\scalebox{0.5}{$\square$}}[U]\to\Lambda_{\scalebox{0.5}{$\square$}}[\overline{U}^{/X}]$
induces an isomorphism
$\Lambda_{\scalebox{0.5}{$\square$}}[U]_{\mathrm{oc}}\cong\Lambda_{\scalebox{0.5}{$\square$}}[\overline{U}^{/X}]$.
In particular, the functor
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda),\
M\mapsto M_{\mathrm{oc}}$ preserves compact objects.

  6. (vi)

The functor
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda),\
M\mapsto M^{\mathrm{oc}}$ preserves colimits and right bounded objects.

###### Proof.

Assertion (i) follows from Lemma 4.10 and Corollary 4.5. For assertion (ii) it
suffices by the adjoint functor theorem to show that
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)^{\mathrm{oc}}$
is stable under all colimits and limits in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$. By
Proposition 4.2 and (i) we may then assume that $X$ is a strictly totally
disconnected perfectoid space. Then the assertion follows from Lemma 4.7. We
note that in this case we have $M_{\mathrm{oc}}=\pi^{\ast}\pi_{\natural}(M)$
and $M^{\mathrm{oc}}=\pi^{\ast}\pi_{\ast}(M)$ for
$M\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$. As in
Lemma 4.7 symmetric monoidality of $(-)_{\mathrm{oc}}$ implies assertion (iv).

For assertion (iii) we may again reduce to the case that $X$ is strictly
totally disconnected. Indeed, for the existence of $(-)^{\mathrm{oc}}$ this
follows by stability of overconvergent sheaves under colimits (which can be
checked on a strictly totally disconnected cover), and the existence of
$(-)_{\mathrm{oc}}$ can be descended from the strictly totally disconnected
case as there $(-)_{\mathrm{oc}}$ commutes with pullback as will be proved in
Lemma 4.12. Then the assertion follows again from Lemma 4.7 as
$\pi_{\natural},\pi^{\ast},\pi_{\ast}$ all preserve $\omega_{1}$-solid
sheaves.

Let us note that if $U\to X$ is quasi-pro-étale, then $\overline{U}^{/X}\to
X=\overline{X}^{/X}$ is quasi-pro-étale by [22, Corollary 18.8.(vii)]. If $X$
is strictly totally disconnected and $U$ qcqs, then the natural map
$\overline{U}^{/X}\to\pi_{0}(U)\times_{\pi_{0}(X)}X$ is an isomorphism as it
is qcqs and bijective if $X$ is connected. To check (v) we may (by Lemma 4.12
below) reduce to the case that $X$ is strictly totally disconnected. Indeed,
in this case
$\Lambda_{\scalebox{0.5}{$\square$}}[\overline{U}^{/X}]\cong\Lambda_{\scalebox{0.5}{$\square$}}[\pi_{0}(U)\times_{\pi_{0}(X)}X]$
is overconvergent, which implies the same for a general $\ell$-bounded spatial
diamond. This produces the natural map
$\Lambda_{\scalebox{0.5}{$\square$}}[U]_{\mathrm{oc}}\to\Lambda[\overline{U}^{/X}]$,
which then can be checked to be an isomorphism again under the assumption that
$X$ is strictly totally disconnected. Here, it follows from Lemma 4.7 as
$(-)_{\mathrm{oc}}\cong\pi^{\ast}\pi_{\natural}(-)$. It follows from the
explicit description that the functor $(-)_{\mathrm{oc}}$ maps compact objects
in $\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ (resp.
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$)
to compact objects in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ (resp.
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$).
This implies that the inclusion of overconvergent into solid sheaves preserves
compact objects, which implies that the right adjoint $(-)^{\mathrm{oc}}$ (for
solid or $\omega_{1}$-solid objects) preserves colimits. Now pick a right
bounded object
$M\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$.
As $X$ is $\ell$-bounded can conclude that there exists some
$a\in{\mathbb{Z}}$ such that for each compact object
$N\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$,
which is supported to the right of $a$ and has terms given by
$\Lambda_{\scalebox{0.5}{$\square$}}[U]$ for $U=\varprojlim\limits_{n}U_{n}\to
X$ with $U_{n}$ étale quasi-compact and separated, we have
$\operatorname{Hom}(N,M)=0$. This implies

$\operatorname{Hom}(N_{\mathrm{oc}},M)=\operatorname{Hom}(N_{\mathrm{oc}},M^{\mathrm{oc}})=0$

for any such $N$ by the explict description of $(-)_{\mathrm{oc}}$ in (v).
This shows that $M^{\mathrm{oc}}$ is again right bounded. ∎

The left adjoint $(-)_{\mathrm{oc}}$ commutes with base change (while the
right adjoint $(-)^{\mathrm{oc}}$ does not, even on strictly totally
disconnected spaces).

######  Lemma 4.12.

Let $f\colon X^{\prime}\to X$ be a morphism of $\ell$-bounded spatial
diamonds.

  1. (i)

For $M\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$
the natural map

$(f^{\ast}M)_{\mathrm{oc}}\to f^{\ast}(M_{\mathrm{oc}})$

is an isomorphism.

  2. (ii)

If $f$ is quasi-pro-étale, then
$f_{\ast}(\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X^{\prime},\Lambda)^{\mathrm{oc}})\subseteq\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)^{\mathrm{oc}}$
and
$f_{\ast}(\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X^{\prime},\Lambda)^{\mathrm{oc}}_{\omega_{1}})\subseteq\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)^{\mathrm{oc}}_{\omega_{1}}$

###### Proof.

By quasi-pro-étale hyperdescent it suffices to treat the case that
$X,X^{\prime}$ are strictly totally disconnected. Let $\pi_{X}\colon
X_{\mathrm{qproet}}\to\pi_{0}(X)_{\mathrm{qproet}},\pi_{X^{\prime}}\colon
X^{\prime}_{\mathrm{qproet}}\to\pi_{0}(X^{\prime})_{\mathrm{qproet}}$ be the
morphisms of sites. It suffices to check that the natural map
$\pi_{X^{\prime},\natural}f^{\ast}M\to\pi_{0}(f)^{\ast}\pi_{X,\natural}(M)$ is
an isomorphism for
$M\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$.
Passing to right adjoints, it suffices to see that for
$N\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\pi_{0}(X^{\prime}),\Lambda)$

$f_{\ast}\pi^{\ast}_{X^{\prime}}N\cong\pi^{\ast}_{X}\pi_{0}(f)_{\ast}(N)$

via the natural map. This follows from [17, Corollary 5.9]. This formula
(together with Lemma 4.4) imply the second assertion as well if $X$ is
strictly totally disconnected. The case for general $X$ follows by quasi-pro-
étale descent (for which $f_{\ast}$ satisfies base change by Lemma 4.4). ∎

###  4.3 Nuclear sheaves

In this section we will study nuclear objects in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
and show that they are overconvergent. Critically, we will moreover show in
Lemma 4.20 that the full subcategory of nuclear objects is generated under
colimits by right bounded complete objects. In order to achieve this, we need
a better understanding of the compact objects of
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}^{\mathrm{oc}}$.

Let us recall from [6, Lecture VIII] that a morphism $f\colon P\to Q$ in a
closed symmetric monoidal $\infty$-category $\mathcal{C}$ with compact unit,
is called of trace class if it lies in the image of the natural morphism

$\pi_{0}((\operatorname{\underline{\operatorname{Hom}}}(P,1)\otimes
Q)(\ast))\to\pi_{0}(\operatorname{Hom}(P,Q)).$

Here, $\otimes,\operatorname{\underline{\operatorname{Hom}}},1$ refer to the
tensor product, internal Hom, unit in $\mathcal{C}$, while
$(-)(\ast):=\operatorname{Hom}(1,-)$. We will use critically the following
property of trace class maps: If $P_{0}\to P_{1}\to\ldots$ is a sequential
diagram with all morphisms $P_{n}\to P_{n+1}$ of trace class, then for any
$Q\in\mathcal{C}$ the natural map

$\varprojlim\limits_{n}(\operatorname{\underline{\operatorname{Hom}}}(P_{n},1)\otimes
Q)(\ast))\to\varprojlim\limits_{n}\operatorname{Hom}(P_{n},Q)$

is an isomorphism. Indeed, each witness in
$(\operatorname{\underline{\operatorname{Hom}}}(P_{n},1)\otimes
P_{n+1})(\ast)$ for $P_{n}\to P_{n+1}$ being trace class yields a
factorization

$\operatorname{Hom}(P_{n+1},Q)\to(\operatorname{\underline{\operatorname{Hom}}}(P_{n},1)\otimes
Q)(\ast)\to\operatorname{Hom}(P_{n},Q).$

Similarly, if $\ldots\to P_{1}\to P_{0}$ is a sequential diagram with all
$P_{n+1}\to P_{n}$ of trace class, then the natural map

$\varinjlim\limits_{n}(\operatorname{\underline{\operatorname{Hom}}}(P_{n},1)\otimes
Q)(\ast))\to\varinjlim\limits_{n}\operatorname{Hom}(P_{n},Q)$ (4.12.1)

is an isomorphism.

Following [6, Definition 8.5] we make the following definition.

######  Definition 4.13.

Let $X$ be an $\ell$-bounded spatial diamond and
$M\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$.

  1. (a)

$M$ is called basic nuclear if $M=\varinjlim(M_{0}\to M_{1}\to\ldots)$ with
all $M_{n}\to M_{n+1}$ of trace class.

  2. (b)

$M$ is called basic conuclear if $M=\varprojlim(\ldots\to M_{1}\to M_{0})$
with all $M_{n+1}\to M_{n}$ of trace class.

  3. (c)

$M$ is called nuclear if for all compact objects
$P\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$,
the map

$(\operatorname{\underline{\operatorname{Hom}}}_{\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}}(P,\Lambda)\otimes^{\scalebox{0.5}{$\square$}}_{\Lambda}M)(\ast)\to\operatorname{Hom}_{\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}}(P,M)$

is an equivalence.

We let
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)\subseteq\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
be the full subcategory of nuclear objects.

######  Remark 4.14.

We warn the reader that the abstract notion of nuclearity from [6, Lecture
VIII] that we employ here is different from the more subtle “geometric
nuclearity” in [15, Definition 3.1] as noted in [15, Warning 3.3]. In Lemma
4.17 below we will prove that the abstract notion of nuclearity singles out
the overconvergent nuclear sheaves from [15, Definition 6.8] (thereby
providing details to [15, Warning 3.3]).

The basic observation for the overconvergence of the nuclear objects in
$\operatorname{\mathcal{D}}_{{\scalebox{0.5}{$\square$}}}(X,\Lambda)_{\omega_{1}}$
is the overconvergence of
$\operatorname{\underline{\operatorname{Hom}}}_{\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}}(\Lambda_{\scalebox{0.5}{$\square$}}[U],\Lambda)$.

######  Lemma 4.15.

Let $X$ be an $\ell$-bounded spatial diamond.

  1. (i)

Each trace class map $P\to Q$ in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
factors as $P\to P_{\mathrm{oc}}\to Q$ with $P_{\mathrm{oc}}\to Q$ of trace
class.

  2. (ii)

Each nuclear object in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
is overconvergent.

  3. (iii)

Each basic conuclear object
$M\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
is overconvergent and complete. In fact, $M\cong\varprojlim\limits_{n}M_{n}$
with $M_{n}$ étale and compact, and $M_{n+1}\to M_{n}$ trace class for each
$n\geq 0$. If the $M_{n}$ are uniformly bounded, then $M$ is compact.

  4. (iv)

Basic conuclear objects are stable under countable limits.

In particular,
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)_{\omega_{1}}$ is
equivalent to the category
$\mathrm{Nuc}(\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}^{\mathrm{oc}})$
of nuclear objects in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}^{\mathrm{oc}}$.

###### Proof.

By [6, Theorem 8.6] each nuclear object is a colimit of basic nuclear object.
By Lemma 4.11 the subcategory of overconvergent objects is stable under limits
and colimits. Hence, it suffices to prove the first assertion to show
overconvergence. Let $P\to Q$ be a trace class morphism in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$.
As $\Lambda$ is overconvergent, we can conclude from Lemma 4.11 that

$\operatorname{\underline{\operatorname{Hom}}}(P,\Lambda)\cong\operatorname{\underline{\operatorname{Hom}}}(P_{\mathrm{oc}},\Lambda)$

as $(-)_{\mathrm{oc}}$ is symmetric monoidal. This implies the desired
factorization.

In order to see that each basic conuclear object
$M=\varprojlim\limits_{n\in\mathbb{N}}M_{n}$ is complete it suffices to note
that each of the trace class morphisms $M_{n+1}\to M_{n}$ factors over a
compact object ([6, Lemma 8.4]), and that compact objects in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
are complete (Proposition 4.3) as is therefore their limit $M$. To prove the
final assertion, we may assume that the $M_{n}$ are compact. Let
$I\subseteq\Lambda$ by an ideal of definition. Then by completeness of $M$ we
get $M\cong\varprojlim\limits_{n}M_{n}/I^{k_{n}}$ for each sequence $k_{n}$
going to $\infty$ if $n\to\infty$. Moreover, the $M_{n}/I^{k_{n}}$ are
compact, while all morphisms $M_{n+1}/I^{k_{n+1}}\to M_{n}/I^{k_{n}}$ are
still of trace class, at least after reindexing the $n$. Indeed, this follows
from the fact that a composition of two cones of trace class morphism is again
trace class ([24, Lecture XIII.Proposition 13.13]). Moreover, we may write
each $M_{n}/I^{k_{n}}$ as a countable limit of compact, étale sheaves
$M_{n,i}$ (this follows from the proof of [15, Proposition 2.5], which works
for general $\Lambda$). By étaleness and compactness of the $M_{n,i}$ each
morphism $M_{n+1}/I^{k{n+1}}\to M_{n}/I^{k_{n}}$ arises from a compatible
system of morphisms $M_{n+1,i}\to M_{n,i}$ (up to reindexing). Then
$M\cong\varprojlim\limits_{n}M_{n,n}$ using that the diagonal
$\mathbb{N}\to\mathbb{N}\times\mathbb{N}$ is cofinal. If the $M_{n,n}$ can be
chosen to be uniformly bounded, then it follows from [22, Theorem VII.1.3]
that $M$ is compact, even in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$. Moreover,
after potentially reindexing the $n$, the morphisms $M_{n+1,n+1}\to M_{n,n}$
can be assumed to be of trace class by the same argument as above. With this
presentation for basic conuclear objects, the proof of [24, Proposition
13.13], [6, Theorem 8.6] dualizes and shows that basic conuclear objects are
stable under countable limits. ∎

In the strictly totally disconnected case, we can be more explicit and
identify nuclear sheaves with nuclear modules over the ring of continuous
functions:

######  Lemma 4.16.

Let $X$ be a strictly totally disconnected perfectoid space and let $T$ be a
profinite set.

  1. (i)

The functors $\pi_{\ast},\pi^{\ast}$ induce a symmetric monoidal equivalence
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)\cong\operatorname{\mathcal{D}}_{\mathrm{nuc}}(\pi_{0}(X),\Lambda)$.

  2. (ii)

Let $\alpha\colon T_{\mathrm{qproet}}\to\ast_{\mathrm{qproet}}$ be the natural
morphism of sites. Then $\alpha_{\ast}$ induces an equivalence

$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(T_{\mathrm{qproet}},\Lambda)\cong\operatorname{\mathcal{D}}_{\mathrm{nuc}}(C(T,\Lambda)),$

where $C(T,\Lambda)\cong\alpha_{\ast}(\Lambda)$ is the nuclear
$\Lambda$-algebra of continuous functions $T\to\Lambda$.

In particular,
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)\cong\operatorname{\mathcal{D}}_{\mathrm{nuc}}(C(X,\Lambda))$.

We recall that here “nuclear” refers to Definition 4.13 (and not [15,
Definition 3.1]). However, for $\pi_{0}(X)$ both notions agree (by Lemma 4.16
and [15, Lemma 6.7.(iv)] or by [15, Remark 3.10] if
$\Lambda={\mathbb{Z}}_{\ell}$).

###### Proof.

As $\pi^{\ast}$ is symmetric monoidal, $\pi^{\ast}$ preserves nuclear objects.
Moreover, the unit $M\to\pi_{\ast}\pi^{\ast}M$ is an isomorphism for any
$M\in\operatorname{\mathcal{D}}_{\mathrm{nuc}}(\pi_{0}(X),\Lambda)$ by Lemma
4.6. Let $N\in\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$ be
nuclear. We need to see that the counit $\pi^{\ast}\pi_{\ast}N\to N$ is an
isomorphism. This follows from Lemma 4.15.

Let us proof the second assertion. Set $A:=C(T,\Lambda)$. As $A$ is a nuclear
$\Lambda$-algebra, e.g., by 2.18.(i),
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(A)$ is equivalent to
$\mathrm{Mod}_{A}(\operatorname{\mathcal{D}}_{\mathrm{nuc}}(\Lambda))$, cf.
[6, Corollary 8.20]. The existence of a left adjoint
$\alpha^{\ast}\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\ast_{\mathrm{qproet}},\Lambda)_{\omega_{1}}\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(T,\Lambda)_{\omega_{1}}$
to $\alpha_{\ast}$ follows as in the proof of [15, Lemma 6.7.(iv)]. Similarly,
it follows (using 2.18.(i)) that $\alpha^{\ast}$ preserves nuclearity, and
that $\alpha_{\ast}\alpha^{\ast}M\to M$ is an isomorphism for
$M\in\operatorname{\mathcal{D}}_{\mathrm{nuc}}(A)$. It remains to see that
$\alpha^{\ast}$ is essentially surjective. Here, we a priori have to offer a
different proof as in [15, Lemma 6.7.(iv)] as we use a different notion of
nuclearity. However, [15, Remark 3.10] shows that both notions agree in this
case as $T$ is profinite. Thus, the assertion follows from [15, Lemma 6.7]. ∎

We note that
$\alpha_{\ast}\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(T,\Lambda)\to\mathrm{Mod}_{C(T,\Lambda)}\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\Lambda)$
is in general not an equivalence (if $T$ is not finite), even if
$\Lambda=\mathbb{F}_{\ell}$. Indeed, for each $t\in T$ the morphism
$\\{t\\}\to T$ is pro-étale, and thus
$\Lambda[\\{t\\}]\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(T,\Lambda)$
is compact. But if $\Lambda$ is discrete, then
$\alpha_{\ast}(\Lambda[\\{t\\}])\cong\Lambda$ viewed as a
$C(T,\Lambda)$-module via evaluation at $t$. If $t\in T$ is not an open point,
then this implies that $\alpha_{\ast}(\Lambda[\\{t\\}])$ is not a (discrete)
finitely presented $C(T,\Lambda)$-module, and hence not compact in
$\mathrm{Mod}_{C(T,\Lambda)}\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\Lambda)$.
Nuclear objects satisfy quasi-pro-étale descent:

######  Lemma 4.17.

The functor $X\mapsto\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$ is
a hypercomplete quasi-pro-étale sheaf on the big quasi-pro-étale site of
$\ell$-bounded spatial diamonds.

###### Proof.

This is formally implied by Corollary 4.5: Pullback of nuclear objects are
nuclear (by symmetric monoidality of pullbacks) and if $f_{\bullet}\colon
X_{\bullet}\to X$ is a quasi-pro-étale hypercover and
$M\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
with $f^{\ast}_{n}M$ nuclear for each $n\in\Delta$, then for
$P\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
compact we have

$\displaystyle\operatorname{Hom}_{\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}}(P,M)=$
$\displaystyle\varprojlim\limits_{n\in\Delta}\operatorname{Hom}_{\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X_{n},\Lambda)_{\omega_{1}}}(f^{\ast}_{n}P,f^{\ast}_{n}M)$
$\displaystyle=$
$\displaystyle\varprojlim\limits_{n\in\Delta}\operatorname{Hom}_{\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X_{n},\Lambda)_{\omega_{1}}}(\Lambda,\operatorname{\underline{\operatorname{Hom}}}_{X^{\prime}_{n}}(f^{\ast}_{n}P,\Lambda)\otimes^{\scalebox{0.5}{$\square$}}_{\Lambda}f^{\ast}_{n}M)$
$\displaystyle=$
$\displaystyle\varprojlim\limits_{n\in\Delta}\operatorname{Hom}_{\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X_{n},\Lambda)_{\omega_{1}}}(\Lambda,f^{\ast}_{n}(\operatorname{\underline{\operatorname{Hom}}}_{X}(P,\Lambda)\otimes^{\scalebox{0.5}{$\square$}}_{\Lambda}M))$
$\displaystyle=$
$\displaystyle\operatorname{Hom}_{\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}}(\Lambda,\operatorname{\underline{\operatorname{Hom}}}(P,\Lambda)\otimes^{\scalebox{0.5}{$\square$}}_{\Lambda}M)$

using (in this order) the description of Hom-spaces in limits of categories,
nuclearity of all the $f_{n}^{\ast}M$, the fact that $f^{\ast}_{n}$ is the
restriction for some slice (plus Proposition 4.3 to control the
$\operatorname{\underline{\operatorname{Hom}}}$ using that $f^{\ast}_{n}P$ is
compact), and again the description of Hom-spaces in limits of categories. ∎

We now analyze complete objects in
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$. More precisely, fix an
ideal of definition $I\subseteq\Lambda$. In the following, we adhere to the
conventions in Section 1.2 (and in particular implicitly fix a finite set of
generators of $I$). Morever, we use the terminology “complete” instead of
“$I$-adically complete”, whenever convenient.

######  Lemma 4.18.

Let $X$ be an $\ell$-bounded spatial diamond and
$M\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
with $I$-adic completion
$\widehat{M}\in\operatorname{\mathcal{D}}_{{\scalebox{0.5}{$\square$}}}(X,\Lambda)_{\omega_{1}}$.

  1. (i)

If $M$ is nuclear, then $\widehat{M}$ is nuclear.

  2. (ii)

If $M\in\operatorname{\mathcal{D}}_{{\mathrm{nuc}}}(X,\Lambda)$ is
$I$-adically complete in
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$, then the natural map
$M\to\widehat{M}$ is an isomorphism.

We note that a priori completeness in
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$ and
$\operatorname{\mathcal{D}}_{{\scalebox{0.5}{$\square$}}}(X,\Lambda)_{\omega_{1}}$
are different.

###### Proof.

We start with (i). By Lemma 4.17 and Lemma 4.4, specifically the commutation
of pullbacks with limits, the claim is quasi-pro-étale local on $X$ and hence
we may assume that $X$ is strictly totally disconnected. By Lemma 4.6 and
Lemma 4.16 the claim reduces to the case that $X$ is a profinite set (realized
as a qcqs perfectoid space over some $\operatorname{Spa}(C,\mathcal{O}_{C})$
with $C$ algebraically closed and perfectoid). More precisely, we use that
$\pi^{\ast}\colon\operatorname{\mathcal{D}}_{{\scalebox{0.5}{$\square$}}}(\pi_{0}(X),\Lambda)_{\omega_{1}}\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
commutes with limits to see that
$\widehat{M}=\pi^{\ast}\pi_{\ast}(\widehat{M})$ is the pullback along
$\pi^{\ast}$ of the completion of $\pi_{\ast}(M)$. Using that
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)=\operatorname{\mathcal{D}}_{\mathrm{nuc}}(C(X,\Lambda))$
by Lemma 4.6, it suffices by 2.18.(i) to see that $\widehat{M}$ is discrete
modulo $I$. But $M/I\cong\widehat{M}/I$, and the claim follows.

Now, (ii) is a consequence of (i): By (i) $M\to\widehat{M}$ is a morphism of
nuclear objects and both are $I$-adically complete in
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$. Indeed, for
$\widehat{M}$ this follows as the nuclearization
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}\to\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$,
i.e., the right adjoint to the inclusion, preserves limits. Now,
$M/I\cong\widehat{M}/I$ is an isomorphism, which implies the claim. ∎

In the next lemma we will use the notation $M_{\mathrm{et}}$ for the
“étalification” of an object
$M\in\operatorname{\mathcal{D}}(X_{\mathrm{qproet}},\Lambda)$ if $X$ is an
$\ell$-bounded spatial diamond, i.e.,
$M_{\mathrm{et}}:=\varepsilon^{\ast}\varepsilon_{\ast}M$ for
$\varepsilon\colon X_{\mathrm{qproet}}\to X_{\mathrm{et}}$ the natural
morphism of sites.

######  Lemma 4.19.

Let $X$ be an $\ell$-bounded spatial diamond. Assume that
$\operatorname{\mathcal{D}}_{{\scalebox{0.5}{$\square$}}}(X,\Lambda)_{\omega_{1}}^{\mathrm{oc}}$
is generated under colimits by basic conuclear objects, which are compact.

  1. (i)

The right adjoint