# The 6-Functor Formalism for ${\mathbb{Z}}_{\ell}$\- and
$\mathbb{Q}_{\ell}$-Sheaves on Diamonds

Lucas Mann

###### Abstract

For every nuclear ${\mathbb{Z}}_{\ell}$-algebra $\Lambda$ and every small
v-stack $X$ we construct an $\infty$-category
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$ of nuclear
$\Lambda$-modules on $X$. We then construct a full 6-functor formalism for
these sheaves, generalizing the étale 6-functor formalism for
$\Lambda=\mathbb{F}_{\ell}$. Prominent choices for $\Lambda$ are
${\mathbb{Z}}_{\ell}$, $\mathbb{Q}_{\ell}$ and $\overline{\mathbb{Q}_{\ell}}$
and especially in the latter two cases, no satisfying 6-functor formalism has
been found before. Applied to classifying stacks we obtain a theory of nuclear
representations, i.e. continuous representations on filtered colimits of
Banach spaces.

###### Contents

  1. 1 Introduction
  2. 2 $\omega_{1}$-Solid Sheaves
  3. 3 Nuclear Sheaves
  4. 4 Pushforward and Base-Change
  5. 5 The 6-Functor Formalism
  6. 6 Perfect, Dualizable and Overconvergent Sheaves
  7. 7 Relatively Dualizable Sheaves
  8. 8 Cohomological Smoothness
  9. 9 Cohomological Properness
  10. 10 Classifying Stacks and Representations
  11. References

##  1 Introduction

Fix two primes $\ell\neq p$. In this paper we construct and study a full
6-functor formalism for certain sheaves of modules over nuclear
${\mathbb{Z}}_{\ell}$-algebras $\Lambda$ like ${\mathbb{Z}}_{\ell}$,
$\mathbb{F}_{\ell}$, $\mathbb{Q}_{\ell}$, $\overline{\mathbb{Q}_{\ell}}$ or
$\mathbb{C}_{\ell}$. The main difference to previous attempts at such a
6-functor formalism is that we never impose any finiteness conditions or pass
to any isogeny categories – everything is completely natural and embedds fully
faithfully into the category of pro-étale sheaves (even on the derived level).
For simplicity this whole paper will work with diamonds and small v-stacks
over ${\mathbb{Z}}_{p}$ [15], i.e. in the realm of rigid-analytic geometry. We
expect that a similar 6-functor formalism can be constructed on schemes,
albeit somewhat harder to implement.

Recall that for torsion coefficients, e.g. $\Lambda=\mathbb{F}_{\ell}$, a full
6-functor formalism for étale sheaves on diamonds has been worked out in [15]
and has been used extensively in [3] to gain new insights into the $\ell$-adic
geometric Langlands program. The present paper generalizes this étale
6-functor formalism to non-discrete nuclear ${\mathbb{Z}}_{\ell}$-algebras
like the ones mentioned above. Note that even in the case that
$\Lambda={\mathbb{Z}}_{\ell}$ we allow non-complete
${\mathbb{Z}}_{\ell}$-sheaves in our 6-functor formalism (unlike the somewhat
ad-hoc definition in [15, §26]), which becomes especially important when
working with non-qcqs maps, see Example 1.4 below.

Without further ado, let us state the main definitions and results. In the
following, we say that a spatial diamond (e.g. qcqs rigid-analytic variety) is
_$\ell$ -bounded_ if it has finite cohomological dimension for étale
$\mathbb{F}_{\ell}$-sheaves. We can then define the following category of
nuclear sheaves:

######  Definition 1.1 (cf. 3.1.(b)).

Let $X$ be a small v-stack. A _nuclear ${\mathbb{Z}}_{\ell}$-sheaf_ on $X$ is
a (derived) v-sheaf
$\mathcal{M}\in\operatorname{\mathcal{D}}(X_{\mathrm{v}},{\mathbb{Z}}_{\ell})$
with the following property: For every map $f\colon Y\to X$ from an
$\ell$-bounded spatial diamond $Y$, the sheaf
$f^{*}\mathcal{M}\in\operatorname{\mathcal{D}}(Y_{\mathrm{v}},{\mathbb{Z}}_{\ell})$
is a filtered colimit of $\ell$-adically complete sheaves, all of which are
étale modulo $\ell$. We denote by

$\displaystyle\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell})\subseteq\operatorname{\mathcal{D}}(X_{\mathrm{v}},{\mathbb{Z}}_{\ell})$

the full subcategory of nuclear ${\mathbb{Z}}_{\ell}$-sheaves. For any nuclear
${\mathbb{Z}}_{\ell}$-algebra $\Lambda$ we denote by
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)\subseteq\operatorname{\mathcal{D}}(X_{\mathrm{v}},\Lambda)$
the full subcategory of those $\Lambda$-sheaves whose underlying
${\mathbb{Z}}_{\ell}$-sheaf is nuclear.

If $X$ is a geometric point then
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell})=\operatorname{\mathcal{D}}_{\mathrm{nuc}}({\mathbb{Z}}_{\ell})\subseteq\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}({\mathbb{Z}}_{\ell})$
recovers the $\infty$-category of nuclear ${\mathbb{Z}}_{\ell}$-modules as
defined in [1, §8] and [16, Definition 13.10]. In particular, a _nuclear
${\mathbb{Z}}_{\ell}$-algebra_ is a condensed ${\mathbb{Z}}_{\ell}$-algebra
whose underlying ${\mathbb{Z}}_{\ell}$-module is a filtered colimit of Banach
${\mathbb{Z}}_{\ell}$-algebras. If $\Lambda$ is discrete then
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)=\operatorname{\mathcal{D}}_{\mathrm{et}}(X,\Lambda)$
(see Proposition 3.20), so in this case we recover the classical étale theory.

Perhaps somewhat surprisingly, the $\infty$-category of nuclear sheaves
satisfies v-descent. This observation lies at the heart of this paper.

######  Theorem 1.2 (cf. 3.9.(ii) and 3.16.(i)).

The $\infty$-category of nuclear $\Lambda$-modules is stable under pullback
and satisfies v-descent. More precisely, the assignment

$\displaystyle X\mapsto\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$

defines a hypercomplete sheaf of $\infty$-categories on the v-site of all
small v-stacks.

Let us now come to the 6-functor formalism for nuclear $\Lambda$-modules. The
six functors can roughly be constructed as follows. Fix a map $f\colon Y\to X$
of small v-stacks; then:

  1. 1.

The $\infty$-category $\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$
comes naturally equipped with a symmetric monoidal structure

$\displaystyle-\otimes-\colon\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)\times\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)\to\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda),$

which agrees with the solid tensor product from [3, Proposition VII.2.2] (note
that clearly
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)\subseteq\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$).

  2. 2.

Since $\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$ is presentable,
the symmetric monoidal structure is closed and hence allows an internal hom

$\displaystyle\operatorname{\underline{\operatorname{Hom}}}(-,-)\colon\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)^{\mathrm{op}}\times\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)\to\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$

given as the right adjoint of $\otimes$.

  3. 3.

The pullback functor

$\displaystyle
f^{*}\colon\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)\to\operatorname{\mathcal{D}}_{\mathrm{nuc}}(Y,\Lambda)$

is simply the pullback of v-sheaves (it preserves nuclear sheaves by
definition).

  4. 4.

The pushforward functor

$\displaystyle
f_{*}\colon\operatorname{\mathcal{D}}_{\mathrm{nuc}}(Y,\Lambda)\to\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$

is the right adjoint of the pullback functor. In general it does not agree
with the v-pushforward, but if $f$ is qcqs then this is true under mild
assumptions (see Section 4).

  5. 5.

If $f$ is “$\ell$-fine” (see Definition 5.8) then we can construct a lower
shriek functor

$\displaystyle
f_{!}\colon\operatorname{\mathcal{D}}_{\mathrm{nuc}}(Y,\Lambda)\to\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$

as follows. If $f$ is proper then $f_{!}=f_{*}$. If $f$ is étale then $f_{!}$
is the left adjoint of $f^{*}$ (this agrees with the functor $f_{!}$ on
v-sheaves, see Lemma 5.2). If $f$ is compactifiable and qcqs then we can
define $f_{!}$ by composing the previous two examples along a relative
compactification of $f$. If $f$ is only locally compactifiable then we can
filter $Y$ by qcqs open subsets on which $f$ becomes compactifiable and then
define $f_{!}$ as the colimit over this filtration. Finally, one can extend
the construction of $f_{!}$ even to certain “stacky” maps. For more details on
the construction of $f_{!}$ see Remark 5.13.

  6. 6.

If $f$ is $\ell$-fine then we define

$\displaystyle
f^{!}\colon\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)\to\operatorname{\mathcal{D}}_{\mathrm{nuc}}(Y,\Lambda)$

as the right adjoint of $f_{!}$.

######  Theorem 1.3 (cf. Theorem 5.11).

The above six functors $\otimes$,
$\operatorname{\underline{\operatorname{Hom}}}$, $f^{*}$, $f_{*}$, $f_{!}$ and
$f^{!}$ provide a 6-functor formalism. In particular, they satisfy the
following properties:

  1. (i)

(Functoriality) For composable maps $f$, $g$ of small v-stacks we have natural
isomorphisms $(f\mathbin{\circ}g)^{*}=g^{*}\mathbin{\circ}f^{*}$ and
$(f\mathbin{\circ}g)_{*}=f_{*}\mathbin{\circ}g_{*}$. If $f$ and $g$ are
$\ell$-fine then also $(f\mathbin{\circ}g)_{!}=f_{!}\mathbin{\circ}g_{!}$ and
$(f\mathbin{\circ}g)^{!}=g^{!}\mathbin{\circ}f^{!}$.

  2. (ii)

(Special Cases) If $j\colon U\to X$ is an étale map of small v-stacks then
$j^{!}=j^{*}$. If $f\colon Y\to X$ is a proper $\ell$-fine map of small
v-stacks then $f_{!}=f_{*}$.

  3. (iii)

(Projection Formula) Let $f\colon Y\to X$ be an $\ell$-fine map of small
v-stacks. Then for all
$\mathcal{M}\in\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$ and
$\mathcal{N}\in\operatorname{\mathcal{D}}_{\mathrm{nuc}}(Y,\Lambda)$ there is
a natural isomorphism

$\displaystyle f_{!}(\mathcal{N}\otimes
f^{*}\mathcal{M})=(f_{!}\mathcal{N})\otimes\mathcal{M}.$

  4. (iv)

(Proper Base-Change) Let

${Y^{\prime}}$${Y}$${X^{\prime}}$${X}$$\scriptstyle{g^{\prime}}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{g}$

be a cartesian diagram of small v-stacks such that $f$ is $\ell$-fine. Then
there is a natural equivalence

$\displaystyle g^{*}f_{!}=f^{\prime}_{!}g^{\prime*}$

of functors
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(Y,\Lambda)\to\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X^{\prime},\Lambda)$.

Before we continue, let us briefly mention how the just constructed 6-functor
formalism behaves in the classical rigid-analytic world.

######  Example 1.4.

Suppose $f\colon Y\to X$ is a map of rigid-analytic varieties over some fixed
non-archimedean base field $K$. Then $f$ is $\ell$-fine. The pushforward
functor
$f_{*}\colon\operatorname{\mathcal{D}}_{\mathrm{nuc}}(Y,\Lambda)\to\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$
computes the nuclearized derived pushforward of pro-étale $\Lambda$-sheaves;
in particular if $X=\operatorname{Spa}K$ then $f_{*}$ computes nuclearized
pro-étale cohomology. If $f$ is qcqs then $f_{*}$ preserves all small
colimits, so for example we have
$f_{*}\mathbb{Q}_{\ell}=(\varprojlim_{n}f_{*}({\mathbb{Z}}/\ell^{n}{\mathbb{Z}}))[1/\ell]$;
this is not true if $f$ is not qcqs.

With $f$ as before, let us explain how to compute
$f_{!}\colon\operatorname{\mathcal{D}}_{\mathrm{nuc}}(Y,\Lambda)\to\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$.
By the projection formula $f_{!}$ commutes with a change of coefficients, e.g.
we have $f_{!}\mathbb{Q}_{\ell}=(f_{!}{\mathbb{Z}}_{\ell})[1/\ell]$. This
often reduces the computation of $f_{!}$ to the case of
${\mathbb{Z}}_{\ell}$-sheaves. If $f$ is qcqs then $f_{!}$ preserves
$\ell$-adically complete sheaves and so in this case we have
$f_{!}{\mathbb{Z}}_{\ell}=\varprojlim_{n}f_{!}({\mathbb{Z}}/\ell^{n}{\mathbb{Z}})$,
where each $f_{!}({\mathbb{Z}}/\ell^{n}{\mathbb{Z}})$ is the usual lower
shriek on étale sheaves. If $f$ is not qcqs then this is not true; instead one
can then filter $Y$ by qcqs open subsets and compute $f_{!}$ as the colimit of
the lower shrieks on these subsets. In particular $f_{!}$ agrees with previous
definitions of “${\mathbb{Z}}_{\ell}$-cohomology with compact support”, cf.
[14, Remark 5.5].

With our full 6-functor formalism at hand, we study so-called _relatively
dualizable_ sheaves in this 6-functor formalism. This concept is not new and
is known as universally locally acyclic sheaves in the case of discrete
$\Lambda$. With the power of the magical 2-category found by Lu-Zheng [7] we
show that relatively dualizable sheaves satisfy all the expected properties
and that this is completely formal. As an important special case of this
theory we deduce that the following definition of $\ell$-cohomologically
smooth maps is sensible:

######  Definition 1.5 (cf. Definition 8.1).

An $\ell$-fine map $f\colon Y\to X$ is _$\ell$ -cohomologically smooth_ if
$f^{!}\mathbb{F}_{\ell}\in\operatorname{\mathcal{D}}_{\mathrm{et}}(Y,\mathbb{F}_{\ell})$
is invertible and its formation commutes with base-change along $f$.

We emphasize that our definition of $\ell$-cohomological smoothness only
depends on the étale theory with $\mathbb{F}_{\ell}$-coefficients and is
therefore compatible with previously defined notions of $\ell$-cohomological
smoothness in [15, 3]. This may seem very surprising, in particular because we
still obtain the expected Poincaré duality (see below). This fact was
independently observed by Bogdan Zavyalov, whose suggestions led to our
pursuit of these ideas and which will also be used in Zavyalov’s upcoming work
on $p$-adic Poincaré duality in rigid-analytic geometry [19].

######  Proposition 1.6 (cf. 8.5.(i)).

Let $f\colon Y\to X$ be an $\ell$-cohomologically smooth map of small
v-stacks. Then
$f^{!}\Lambda\in\operatorname{\mathcal{D}}_{\mathrm{nuc}}(Y,\Lambda)$ is
invertible and the natural morphism

$\displaystyle f^{!}\Lambda\otimes f^{*}\mathbin{\xrightarrow{\sim}}f^{!}$

is an isomorphism of functors
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)\to\operatorname{\mathcal{D}}_{\mathrm{nuc}}(Y,\Lambda)$.

Of course, $\ell$-cohomologically smooth maps also satisfy all the other
expected properties for nuclear sheaves, see Section 8; here the magic of Lu-
Zheng’s construction really shines. We can use similar ideas to get a robust
notion of $\ell$-cohomologically proper maps (which are more subtle to define
than one might think at first), which we study in Section 9.

In the final part of this paper we apply the 6-functor formalism to
classifying stacks, in which case we recover a category of representations.
More concretely, suppose that $G$ is a locally profinite group which has
locally finite $\ell$-cohomological dimension (e.g. $G$ is locally pro-$p$).
Then there is a natural equivalence

$\displaystyle\operatorname{\mathcal{D}}_{\mathrm{nuc}}(*/G,\Lambda)=\operatorname{\mathcal{D}}_{\mathrm{nuc}}(\Lambda)^{BG},$

where the right-hand side denotes the $\infty$-category of _nuclear
$G$-representations_, i.e. continuous $G$-representations on nuclear
$\Lambda$-modules. If $\Lambda$ is concentrated in degree $0$ (which is
probably always true in practice) then
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(\Lambda)^{BG}$ is the derived
$\infty$-category of its heart. For example, if $\Lambda=\mathbb{Q}_{\ell}$
then this heart is the abelian category of continuous $G$-representations on
filtered colimits of $\mathbb{Q}_{\ell}$-Banach spaces.

We get the following result on the interaction of the 6-functor formalism with
classifying stacks:

######  Theorem 1.7 (cf. Theorem 10.13).

Let $G$ be a locally pro-$p$ group. Then the map $*/G\to*$ is $\ell$-fine and
$\ell$-cohomologically smooth. If $G$ is pro-$p$ then this map is additionally
$\ell$-cohomologically proper.

Note that the pushforward along $*/G\to*$ computes nuclearized continuous
$G$-cohomology, so the above theorem implies a Poincaré duality statement for
continuous $G$-cohomology on nuclear $\Lambda$-modules. With this knowledge at
hand, one can also introduce a good notion of _admissible nuclear
$G$-representations_ (see Definition 10.14) and deduce some basic properties
for them (see Proposition 10.15). If $\Lambda$ is discrete then this recovers
the usual notion of admissible representations. If $\Lambda$ is not discrete
then the notion of admissible representations seems less common in the
literature and has probably not been defined in the generality presented here.

### Background and Motivation.

In recent years, 6-functor formalisms have proven to be an extremely powerful
tool in the study of various cohomology theories. With recent advances in the
geometric Langlands program, 6-functor formalisms have also found tremendous
success in applications to representation theory. In the case of rigid-
analytic geometry, there have been introduced two 6-functor formalisms: A
6-functor formalism for étale $\Lambda$-sheaves for any discrete ring
$\Lambda$ which is killed by some power of $\ell$ [15] and an analog for
mod-$p$ coefficients [10]. These 6-functor formalisms have then been extended
to certain stacky maps in [4, 5], which makes them applicable to
representation theory. However, as far as we know there has not yet been
introduced a satisfying 6-functor formalism for sheaves of modules over non-
discrete rings like ${\mathbb{Z}}_{\ell}$, $\mathbb{Q}_{\ell}$ or
$\overline{\mathbb{Q}_{\ell}}$. There have been the following attempts:

  1. 1.

The most naive way of defining a 6-functor formalism for
${\mathbb{Z}}_{\ell}$-sheaves is by formally taking $\ell$-adic completions,
i.e. associating to every small v-stack $X$ the $\infty$-category
$\operatorname{\mathcal{D}}_{\mathrm{naive}}(X,{\mathbb{Z}}_{\ell}):=\varprojlim_{n}\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}/\ell^{n}{\mathbb{Z}})$
(cf. [15, §26]). This works well for most purposes: The $\infty$-category
$\operatorname{\mathcal{D}}_{\mathrm{naive}}(X,{\mathbb{Z}}_{\ell})$ satisfies
v-descent and embedds fully faithfully into
$\operatorname{\mathcal{D}}(X_{\mathrm{v}},{\mathbb{Z}}_{\ell})$, even into
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell})$. One also
gets a full 6-functor formalism for these sheaves. In the case of qcqs spaces
this 6-functor formalism mostly recovers the nuclear 6-functor formalism. For
non-qcqs spaces and maps, the naive 6-functor formalism forces $\ell$-adic
completions and thus loses information compared to the nuclear version. This
is especially relevant in rigid-analytic geometry, where one is often
interested in cohomologies of non-qcqs spaces like those appearing in the
Drinfeld tower.

The main issue with the naive 6-functor formalism for
${\mathbb{Z}}_{\ell}$-sheaves is that it does not produce a good theory for
$\mathbb{Q}_{\ell}$-sheaves. One can attempt to define
$\operatorname{\mathcal{D}}_{\mathrm{naive}}(X,\mathbb{Q}_{\ell}):=\operatorname{\mathcal{D}}_{\mathrm{naive}}(X,{\mathbb{Z}}_{\ell})\otimes\mathbb{Q}$,
i.e. by formally inverting $\ell$. This roughly amounts to studying those
$\mathbb{Q}_{\ell}$-sheaves which admit a ${\mathbb{Z}}_{\ell}$-model, but
that is not precisely correct. In fact,
$\operatorname{\mathcal{D}}_{\mathrm{naive}}(X,\mathbb{Q}_{\ell})$ does not
even embedd into
$\operatorname{\mathcal{D}}(X_{\mathrm{v}},\mathbb{Q}_{\ell})$!

  2. 2.

In [3, §VII] an $\infty$-category
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})\subseteq\operatorname{\mathcal{D}}(X_{\mathrm{v}},{\mathbb{Z}}_{\ell})$
of _solid ${\mathbb{Z}}_{\ell}$-sheaves_ on $X$ is defined. This
$\infty$-category satisfies v-descent and one can easily define
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ for every
solid ${\mathbb{Z}}_{\ell}$-algebra $\Lambda$, so in particular for every
nuclear ${\mathbb{Z}}_{\ell}$-algebra $\Lambda$. The theory of solid sheaves
has excellent geometric properties; in fact it forms the foundation upon which
we build our nuclear theory. Fargues-Scholze provide a “5-functor formalism”
for $\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$, which
in many applications is sufficiently strong to replace an actual 6-functor
formalism. Still,
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ does not
have a 6-functor formalism (it is far too big for that) and hence does not
generalize the étale theory for discrete $\Lambda$. See below for more
information on the comparison between the solid 5-functor formalism and our
nuclear 6-functor formalism.

  3. 3.

In [3, §VII.6] a full subcategory
$\operatorname{\mathcal{D}}_{\mathrm{lis}}(X,\Lambda)\subseteq\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$
is defined in order to remedy some of the downsides of
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$. While
$\operatorname{\mathcal{D}}_{\mathrm{lis}}(X,\Lambda)$ seems to work very well
with regards to the Langlands conjecture, it feels rather unnatural from a
purely geometric point of view. Some of the main issues are that it is not
stable under various operations and does not satisfy v-descent.

The nuclear 6-functor formalism has all the expected formal properties of a
6-functor formalism and generalizes the étale version for discrete
coefficients. We therefore believe it to be the “right” way of working with
sheaves over nuclear ${\mathbb{Z}}_{\ell}$-algebras. While the existence of
the nuclear 6-functor formalism is a very neat abstract result and has many
formal consequences (e.g. Poincaré duality for
$\overline{\mathbb{Q}_{\ell}}$-local systems), we do not provide any actual
applications of the 6-functor formalism in our paper. From our point of view,
the main scientific value of this paper comes from the following motivations:

  1. (a)

We found the definition of $\operatorname{\mathcal{D}}_{\mathrm{lis}}$ rather
unsatisfying and were therefore looking for different ways of understanding
it. It seems plausible that one can describe
$\operatorname{\mathcal{D}}_{\mathrm{lis}}\subseteq\operatorname{\mathcal{D}}_{\mathrm{nuc}}$
in terms of some abstract properties, like $\operatorname{Ind}$-compact or
$\operatorname{Ind}$-perfect objects.

  2. (b)

With $\operatorname{\mathcal{D}}_{\mathrm{nuc}}$ appearing so naturally in
geometry, we would be surprised if it does not play some role in the local
Langlands program. For example, one could hope that the categorical Langlands
conjecture generalizes to a conjecture describing
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(\operatorname{Bun}_{G},\Lambda)$.
So far $\ell$-adic Banach representations have not received much attention in
the Langlands program, but in [18] it was shown that they seem to have some
connection to Langlands.

  3. (c)

We hope that the nuclear 6-functor formalism sheds some light on the $p$-adic
Langlands program. In fact, we have previously been trying to generalize our
mod-$p$ 6-functor formalism to a ${\mathbb{Z}}_{p}$\- or
$\mathbb{Q}_{p}$-version without much success. The nuclear $\ell$-adic
6-functor formalism provides a lot of insight on how its $p$-adic analog
should work.

  4. (d)

A large part of this paper is rather formal. One can therefore use this paper
as a general recipe on how to construct a 6-functor formalism and study its
basic properties like smoothness and representations.

### Why Nuclear Sheaves.

Our definition of nuclear sheaves, while being rather explicit, comes somewhat
out of nowhere. Let us therefore explain why one should expect such a
definition to appear in a 6-functor formalism for
${\mathbb{Z}}_{\ell}$-sheaves. There are two major motivations:

  1. (i)

In condensed mathematics, the $\infty$-category
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}({\mathbb{Z}}_{\ell})$ of nuclear
${\mathbb{Z}}_{\ell}$-modules should be seen as an analog of “discrete”
modules over ${\mathbb{Z}}_{\ell}$ (since ${\mathbb{Z}}_{\ell}$ is equipped
with a topology, the actual category of discrete ${\mathbb{Z}}_{\ell}$-modules
is rather small). Since étale $\mathbb{F}_{\ell}$-sheaves on a small v-stack
$X$ are a relative version of discrete $\mathbb{F}_{\ell}$-modules, it is
believable that the correct ${\mathbb{Z}}_{\ell}$-analog should be a relative
version of nuclear ${\mathbb{Z}}_{\ell}$-modules.

  2. (ii)

One of the main reasons why the solid 5-functor formalism cannot be upgraded
to a 6-functor formalism is that for proper maps $f\colon Y\to X$ the functor
$f_{*}\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(Y,\Lambda)\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$
often does not satisfy the projection formula. A geometric example of this
phenomenon is given in [3, Warning VII.2.5]. Let us discuss a different
example, which also demonstrates where the nuclearity condition comes from:
Suppose $X=\operatorname{Spa}C$ is a geometric point and $Y=\underline{S}$ is
the pro-étale space over $X$ given by some profinite set $S$. Then
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)=\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\Lambda)$
is the $\infty$-category of solid condensed $\Lambda$-modules. If $f_{*}$
satisfied the projection formula then in particular for all solid
$\Lambda$-modules $M$ we would have

$\displaystyle f_{*}f^{*}M=f_{*}\Lambda\otimes_{\Lambda}M,$

where the tensor product on the right is the solid tensor product. Note that
$f_{*}\Lambda=\operatorname{\mathcal{C}}(S,\Lambda)=\Lambda_{\scalebox{0.5}{$\square$}}[S]^{\vee}$
and
$f_{*}f^{*}M=\operatorname{\underline{\operatorname{Hom}}}(\Lambda_{\scalebox{0.5}{$\square$}}[S],M)$.
Thus for fixed $M$ the above condition (for all $S$) amounts precisely to the
condition that $M$ is nuclear (as defined in [16, Definition 13.10]).

### Relation to the Solid 5-Functor Formalism.

In [3, §VII] Fargues-Scholze define a “5-functor formalism” for the
$\infty$-category
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$ of solid
$\Lambda$-sheaves on any small v-stack $X$. Apart from the usual four functors
$\otimes$, $\operatorname{\underline{\operatorname{Hom}}}$, $f^{*}$ and
$f_{*}$ they introduce a functor $f_{\natural}$ defined to be the left adjoint
of $f^{*}$. The goal of the solid 5-functor formalism is to approximate a
6-functor formalism for $\Lambda$-sheaves, but there has been some confusion
as to what extent this is possible. With the nuclear 6-functor formalism at
hand we can shed some light on this and in particular try to answer the
questions raised in [3, Remark VII.3.6].

Namely, in Proposition 8.10 we show the following: Suppose $f\colon Y\to X$ is
an $\ell$-fine and $\ell$-cohomologically smooth map of small v-stacks which
is representable in locally spatial diamonds (this latter condition can be
relaxed, cf. Remark 8.11). Then
$f_{\natural}\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(Y,\Lambda)\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)$
preserves nuclear sheaves and there is a natural isomorphism

$\displaystyle f_{\natural}=f_{!}(-\otimes f^{!}\Lambda)$

of functors
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(Y,\Lambda)\to\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$.
In other words, along $\ell$-cohomologically smooth maps $f$, $f_{\natural}$
computes $f_{!}$ up to a twist. In particular this allows one to simulate
$f_{!}$ by $f_{\natural}$ whenever one is in a geometric situation that can be
built out of smooth maps, which explains why the 5-functor formalism is so
useful in [3]. Note that it also follows that
$\operatorname{\mathcal{D}}_{\mathrm{lis}}(X,\Lambda)\subseteq\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$.

Of course, for non-smooth $f$ the functors $f_{\natural}$ and $f_{!}$ are very
different and $f_{\natural}$ does usually not preserve nuclear sheaves.

### Structure of the Paper.

This paper is roughly structured into four parts:

  * •

Part I consists of Sections 2 and 3 and introduces the $\infty$-category of
nuclear sheaves. These sections are very specific to the concrete setting at
hand. In Section 2 we introduce $\omega_{1}$-solid sheaves as a particularly
nice full subcategory of all solid sheaves. Their main advantage is that it is
easier to control their compact objects (which is crucial for studying nuclear
sheaves later on) while still maintaining most of the nice properties of solid
sheaves. In Section 3 we then introduce nuclear sheaves. We provide different
characterizations of nuclearity (thereby also motivating the terminology) and
show that they satisfy v-descent.

  * •

Part II consists of Sections 4 and 5 and constructs the 6-functor formalism
for nuclear sheaves. While not being fully formal, most of the ideas in these
sections should be applicable to many different geometric settings. It can
therefore be used as a general recipe for constructing 6-functor formalisms.

  * •

Part III consists of Sections 6, 7, 8 and 9 and is almost completely formal.
Here we introduce dualizable and relatively dualizable sheaves (the latter
being also known as universally locally acyclic sheaves in the étale context)
and use them to define cohomologically smooth and proper maps. The main
insight is that by using the magic of Lu-Zheng’s ideas one can show that
cohomological smoothness is a condition that only depends on very little data
of the 6-functor formalism (one only needs to understand its behavior on the
monoidal unit). Since all of the sections in part III are very formal, they
should apply to every 6-functor formalism.

  * •

Part IV consists of Section 10, where we apply the nuclear 6-functor formalism
to classifying stacks in order to obtain a 6-functor formalism for
representation theory. The main result here is that under mild assumptions on
the locally profinite group $G$ the classifying stack $*/G$ is
$\ell$-cohomologically smooth.

### Notation and Conventions.

The whole paper is written in the modern language of $\infty$-categories, as
this is the most natural and clean way of describing our ideas (for example
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$ does not have a
$t$-structure and is therefore not that easily accessible with conventional
methods; also the 6-functor formalism for stacky maps requires an
$\infty$-categorical framework). In particular every functor, sheaf, ring,
module, representation etc. is always assumed to be derived if not explicitly
specified differently. In the presence of a $t$-structure on a stable
$\infty$-category we usually denote by $\pi_{n}M=H^{-n}(M)$ the homology
objects of $M$. We say that $M$ is _static_ if it is concentrated in degree 0,
i.e. has no derived structure. For a pro-étale map $j\colon U\to X$ of
diamonds and a pro-étale sheaf $\mathcal{M}$ on $U$ we denote
$\mathcal{M}[U]:=j_{!}\mathcal{M}$ (in the sense of sites), which is a pro-
étale sheaf on $X$. All sheaves will always be considered as part of the
derived $\infty$-category of v-sheaves. For example by an étale sheaf we mean
a sheaf in
$\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}})\subseteq\operatorname{\mathcal{D}}(X_{\mathrm{v}},{\mathbb{Z}})$.
We denote by
$(-)_{\mathrm{et}}\colon\operatorname{\mathcal{D}}(X_{\mathrm{v}},{\mathbb{Z}})\to\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}})$
the right adjoint to the inclusion. For a ${\mathbb{Z}}_{\ell}$-module $M$ we
denote $M/\ell^{n}M:=\operatorname{cofib}(M\mathbin{\xrightarrow{\ell^{n}}}M)$
and similarly for sheaves of ${\mathbb{Z}}_{\ell}$-modules. We warn the reader
that by $\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})$ we
denote the $\infty$-category of those pro-étale ${\mathbb{Z}}_{\ell}$-sheaves
on $X$ whose underlying sheaf of abelian groups is étale (this is different
from the notation in [15, §26]).

We are aware of the fact that $\infty$-categories can be intimidating to the
uninitiated, so we refer the reader to [10, §1.5] for a quick down-to-earth
introduction to $\infty$-categories with a special focus on the terminology
used in the algebraic setting.

### Acknowledgements.

I heartily thank David Hansen for many very helpful discussions regarding this
material and for reading preliminary versions of it. I similarly thank Peter
Scholze for useful discussions (and some clarification regarding dualizable
objects…) and for checking whether the main results of this paper are
plausible. Special thanks go to Johannes Anschütz, Arthur-César Le Bras,
Alexander Ivanov and Bogdan Zavyalov who provided valuable input and feedback
to some of the ideas appearing in this paper.

##  2 $\omega_{1}$-Solid Sheaves

Fix a prime $\ell\neq p$. The $\infty$-category
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$
constructed in [3, §VII] lacks some formal properties which we need for our
construction of nuclear sheaves, most prominently the unit object is usually
not compact. We will remedy this by considering a much smaller subcategory
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}\subseteq\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$
of _$\omega_{1}$ -solid sheaves_. This subcategory is generated under small
colimits by _countable_ limits of qcqs étale sheaves. Since countable limits
have finite cohomological dimension, this gives us a lot of control. In
particular, the $\infty$-category of $\omega_{1}$-solid sheaves has similar
properties as
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$,
but under mild assumptions on $X$ the unit is compact. This “mild assumption”
on $X$ is the following:

######  Definition 2.1.

A spatial diamond $X$ is called _$\ell$ -bounded_ if there is some integer $d$
such that for all static étale $\mathbb{F}_{\ell}$-modules
$\mathcal{M}\in\operatorname{\mathcal{D}}_{\mathrm{et}}(X,\mathbb{F}_{\ell})$
on $X$ we have $H^{k}(X,\mathcal{M})=0$ for $k>d$.

######  Definition 2.2.

Let $X$ be a spatial diamond. A quasi-pro-étale map $U\to X$ is called _basic_
if it can be written as a cofiltered limit $U=\varprojlim_{i}U_{i}$ such that
all $U_{i}\to X$ are étale, quasicompact and separated.

######  Remark 2.3.

Suppose that $X$ is an $\ell$-bounded spatial diamond and let
$\mathcal{M}\in\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}})$ be
any static étale sheaf on $X$ which is killed by some power of $\ell$. Then
for the same $d$ as in the definition of $\ell$-boundedness and all basic
$U\in X_{\mathrm{proet}}$ we have $H^{k}(U,\mathcal{M})=0$ for $k>d+1$.
Namely, if $U=X$ then this follows by writing
$\mathcal{M}=\varprojlim_{n}\mathcal{M}/\ell^{n}\mathcal{M}$ and noting that
for each $\mathcal{M}/\ell^{n}\mathcal{M}$ the result follows by repeatedly
applying fiber sequences of the form
$\mathcal{M}/\ell^{n-1}\mathcal{M}\to\mathcal{M}/\ell^{n}\mathcal{M}\to\mathcal{M}/\ell\mathcal{M}$.
If $U$ is étale over $X$ then the result follows from the observation that the
étale pushforward along $U\to X$ is $t$-exact (this can be checked after
pullback to a strictly totally disconnected space, in which case $U$ is also
strictly totally disconnected and therefore has vanishing étale cohomology).
For general $U=\varprojlim_{i}U_{i}$ use that
$H^{k}(U,\mathcal{M})=\varinjlim_{i}H^{k}(U_{i},\mathcal{M})$. In the
following we will implicitly make use of these facts.

Let us now come to the definition of $\omega_{1}$-solid sheaves. Here by
$\omega_{1}$ we mean the first uncountable cardinal.

######  Definition 2.4.

Let $X$ be an $\ell$-bounded spatial diamond. A solid
${\mathbb{Z}}_{\ell}$-module
$\mathcal{M}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$
on $X$ is called _$\omega_{1}$ -solid_ if for every $\omega_{1}$-cofiltered
limit $U=\varprojlim_{i}U_{i}$ of basic objects in $X_{\mathrm{proet}}$ the
natural map

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

is an isomorphism. We denote by

$\displaystyle\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}\subseteq\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$

the full subcategory spanned by the $\omega_{1}$-solid sheaves.

The basic properties of
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
are summarized by the following result, which appeared to be surprisingly
subtle.

######  Proposition 2.5.

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

  1. (i)

Let
$\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})\subseteq\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$
denote the full subcategory spanned by those sheaves whose underlying abelian
sheaf lies in $\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}})$. Then
$\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})$ is compactly
generated and the compact objects are generated under finite (co)limits and
retracts by the objects $\mathbb{F}_{\ell}[U]$ for quasicompact separated
$U\in X_{\mathrm{et}}$. Moreover, the $t$-structure on
$\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})$ restricts to
a $t$-structure on
$\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})^{\omega}$.

  2. (ii)

$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
is compactly generated. The compact objects are $\ell$-adically complete and
generated under finite (co)limits and retracts by the objects

$\displaystyle{\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U]=\varprojlim_{n}{\mathbb{Z}}_{\ell}[U_{n}]=\varprojlim_{n}({\mathbb{Z}}/\ell^{n}{\mathbb{Z}})[U_{n}]$

for sequential limits $U=\varprojlim_{n}U_{n}$ with all $U_{n}\to X$ being
étale, quasicompact and separated. This identifies
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})^{\omega}_{\omega_{1}}$
with a full subcategory

$\displaystyle\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}^{\omega}\subseteq\operatorname{Pro}(\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})^{\omega}).$

  3. (iii)

$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
is stable under all colimits and countable limits in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$
and contains
$\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})$. It admits a
complete $t$-structure and a symmetric monoidal structure by restricting the
ones on
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$.
Moreover, the compact objects in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
are stable under tensor product.

###### Proof.

We first prove (i). We claim that the forgetful functor
$\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})\to\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}})$
is fully faithful. Indeed, the right adjoint of the forgetful functor
$\operatorname{\mathcal{D}}(X_{\mathrm{proet}},{\mathbb{Z}}_{\ell})\to\operatorname{\mathcal{D}}(X_{\mathrm{proet}},{\mathbb{Z}})$
is given by
$\operatorname{\underline{\operatorname{Hom}}}_{\mathbb{Z}}({\mathbb{Z}}_{\ell},-)$
and one observes that this functor preserves étale sheaves; in fact, if
$\mathcal{M}\in\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}})$ then
$\operatorname{\underline{\operatorname{Hom}}}_{\mathbb{Z}}({\mathbb{Z}}_{\ell},-)=\varinjlim_{n}\operatorname{\underline{\operatorname{Hom}}}_{\mathbb{Z}}({\mathbb{Z}}/\ell^{n}{\mathbb{Z}},\mathcal{M})$
by the usual Breen resolution argument (cf. the proof of [3, Proposition
VII.1.12]). One deduces from the same formula that if
$\mathcal{M}\in\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})$
then
$\operatorname{\underline{\operatorname{Hom}}}_{\mathbb{Z}}({\mathbb{Z}}_{\ell},\mathcal{M})=\mathcal{M}$,
proving the desired fully faithfulness.

The embedding
$\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})\mathrel{\hookrightarrow}\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}})$
is $t$-exact. Moreover, if
$\mathcal{P}\in\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})$
is perfect constructible as an object in
$\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}})$ then it is compact
as an object in
$\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})$: By the
proof of [15, Proposition 20.17] this reduces to showing that
$\operatorname{Hom}({\mathbb{Z}}[U],-)=\Gamma(U,-)$ preserves small colimits
of objects in
$\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})$, which
follows from $\ell$-boundedness of $X$ (using the fact that every
$\mathcal{M}\in\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})$
can be written as
$\mathcal{M}=\varinjlim_{n}\operatorname{\underline{\operatorname{Hom}}}({\mathbb{Z}}/\ell^{n}{\mathbb{Z}},\mathcal{M})$
by the previous paragraph). We now prove that conversely every compact object
in $\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})$ is
perfect construcible as an object in
$\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}})$. First observe that
for every quasicompact separated $U\in X_{\mathrm{et}}$,
$\mathbb{F}_{\ell}[U]\in\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})$
is compact because it is perfect constructible in
$\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}})$. We now claim that
the objects $\mathbb{F}_{\ell}[U]$ generate
$\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})$. Note that
these objects generate $({\mathbb{Z}}/\ell^{n}{\mathbb{Z}})[U][k]$ for all
integers $n\geq 1$ and $k$; by abstract nonsense it is therefore enough to
show that the family of functors
$\operatorname{Hom}(({\mathbb{Z}}/\ell^{n}{\mathbb{Z}})[U][k],-)$ is
conservative. We can ignore the shifts by $k$ by instead taking spectra-
enriched $\operatorname{Hom}$’s. We now claim that for every
$\mathcal{M}\in\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})$
the natural map

$\displaystyle\varinjlim_{n}\operatorname{Hom}(({\mathbb{Z}}/\ell^{n}{\mathbb{Z}})[U],\mathcal{M})\mathbin{\xrightarrow{\sim}}\operatorname{Hom}({\mathbb{Z}}_{\ell}[U],\mathcal{M})=\Gamma(U,\mathcal{M}),$

is an isomorphism of spectra. Using the $\ell$-boundedness of $X$, a standard
Postnikov limit argument reduces this claim to the case that $\mathcal{M}$ is
left-bounded. Since both sides commute with colimits in $\mathcal{M}$ we can
further reduce to the case that $\mathcal{M}$ is static. But then the usual
Breen-Deligne resolution works (cf. the proof of [3, Proposition VII.1.12]).
We are thus reduced to showing that the family of functors $\Gamma(U,-)$ is
conservative; but this is clear. In particular we deduce the claimed
description of compact objects in
$\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})$ and identify
them as precisely those objects which are perfect constructible in
$\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}})$. By [15,
Proposition 20.12] the $t$-structure on
$\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}})$ restricts to a
$t$-structure on perfect constructible sheaves; this finishes the proof of
(i).

We now prove (ii) and (iii). The fact that
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
is stable under countable limits in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$
follows immediately from the definition by using the fact that countable
limits commute with $\omega_{1}$-filtered colimits in spectra. Also, it is
obvious that
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
contains $\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})$
because étale sheaves $\mathcal{M}$ on $X$ satisfy
$\Gamma(U,\mathcal{M})=\varinjlim_{i}\Gamma(U_{i},\mathcal{M})$ for all
cofiltered limits $U=\varprojlim_{i}U_{i}$ (this holds for unbounded
$\mathcal{M}$ by the $\ell$-boundedness of $X$).

Let now $U=\varprojlim_{n}U_{n}$ be given as in (ii). We first check that
${\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U]$ is $\omega_{1}$-solid.
Since $\omega_{1}$-solid sheaves are stable under countable limits we reduce
to showing this for ${\mathbb{Z}}_{\ell}[U_{n}]$ for all $n$. Since étale
sheaves are $\omega_{1}$-solid, we further reduce to showing that the natural
map
${\mathbb{Z}}_{\ell}[U_{n}]\mathbin{\xrightarrow{\sim}}\varprojlim_{n}({\mathbb{Z}}/\ell^{n}{\mathbb{Z}})[U_{n}]$
is an isomorphism. Both sides of this claimed isomorphism are static, so we
can check this by applying Yoneda in the heart, i.e. we need to see that for
every
$\mathcal{M}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})^{\heartsuit}$
the natural map

$\displaystyle\operatorname{Hom}(\varprojlim_{n}({\mathbb{Z}}/\ell^{n}{\mathbb{Z}})[U_{n}],\mathcal{M})\mathbin{\xrightarrow{\sim}}\operatorname{Hom}({\mathbb{Z}}_{\ell}[U_{n}],\mathcal{M})$

is an isomorphism. Since both ${\mathbb{Z}}_{\ell}[U_{n}]$ and
${\mathbb{Z}}_{\ell}/\ell^{n}{\mathbb{Z}}_{\ell})[U_{n}]$ are compact objects
in the heart (by the description of finitely presented, i.e. compact, objects
in [3, Theorem VII.1.3]), so we can reduce to $\mathcal{M}$ being of the form
$\mathcal{M}=\varprojlim_{i}\mathcal{M}_{i}$ for qcqs étale sheaves
$\mathcal{M}_{i}$. Pulling out limits from both sides of the claimed
$\operatorname{Hom}$-identity, we reduce to the case that $\mathcal{M}$ is
étale. But then by the proof of [3, Theorem VII.1.3] we have

$\displaystyle\operatorname{Hom}(\varprojlim_{n}({\mathbb{Z}}/\ell^{n}{\mathbb{Z}})[U_{n}],\mathcal{M})=\varinjlim_{n}\operatorname{Hom}(({\mathbb{Z}}/\ell^{n}{\mathbb{Z}})[U_{n}],\mathcal{M})=\varinjlim_{n}\operatorname{Hom}({\mathbb{Z}}/\ell^{n}{\mathbb{Z}},{\mathcal{M}}|_{U})$
$\displaystyle\qquad=\operatorname{Hom}({\mathbb{Z}}_{\ell},{\mathcal{M}}|_{U})=\operatorname{Hom}({\mathbb{Z}}_{\ell}[U],\mathcal{M}),$

as desired. This finishes the proof that
${\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U]$ is $\omega_{1}$-solid.
Now let
$\mathcal{C}\subseteq\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
be the full subcategory generated under finite (co)limits and retracts by the
objects ${\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U]$ for
$U=\varprojlim_{n}U_{n}$ as in (ii). This induces a natural functor

$\displaystyle\alpha\colon\operatorname{Ind}(\mathcal{C})\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell}).$

Our goal will be to show that $\alpha$ induces an equivalence of
$\operatorname{Ind}(\mathcal{C})$ and
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$.
As a first step towards seeing this, let us prove that the $t$-structure on
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$
restricts to a $t$-structure on the essential image of $\alpha$. Namely, this
reduces to showing that for every $\mathcal{P}\in\mathcal{C}$, all
$\pi_{k}\mathcal{P}$ lie in the essential image of $\alpha$. By [3, Theorem
VII.1.3], $\pi_{k}\mathcal{P}$ is finitely presented and can be written as a
countable limit $\pi_{k}\mathcal{P}=\varprojlim_{n}\mathcal{P}_{n,k}$ of qcqs
étale sheaves. The latter property implies that $\pi_{k}\mathcal{P}$ is
$\omega_{1}$-solid, which in turn implies that the sections of $\pi_{k}P$ on
any basic $U\in X_{\mathrm{proet}}$ are determined by the sections of
$\pi_{k}P$ on the $U=\varprojlim_{n}U_{n}$ as in (ii). But note that basic $U$
form a basis of $X_{\mathrm{proet}}$ (this is clear if $X$ is a perfectoid
space and follows from [15, Proposition 11.24, 11.23.(iii)] in general), hence
there is a surjection of the form
$\bigoplus_{i}{\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U_{i}]\mathrel{\twoheadrightarrow}\pi_{k}\mathcal{P}$
with all $U_{i}$ as in (ii). Note that this direct sum is still
$\omega_{1}$-solid, hence so is the kernel of the map to $\pi_{k}\mathcal{P}$
so that we get a two-term resolution
$\bigoplus_{j}{\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U_{j}]\to\bigoplus_{i}{\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U_{i}]\mathrel{\twoheadrightarrow}\pi_{k}\mathcal{P}$.
By writing this map of infinite direct sums as a filtered colimit of maps of
finite direct sums and using that $\pi_{k}\mathcal{P}$ is finitely presented,
we deduce that one arrange that both direct sums are finite, i.e. lie in
$\mathcal{C}$. Continuing this argument, we obtain a resolution of
$\pi_{k}\mathcal{P}$ by objects in $\mathcal{C}$ which implies that
$\pi_{k}\mathcal{P}$ is a geometric resolution of objects in $\mathcal{C}$ and
in particular a filtered colimit of objects in $\mathcal{C}$ (as every finite
step of the geometric resolution lies in $\mathcal{C}$). This implies that
$\pi_{k}\mathcal{P}$ lies in the essential image of $\alpha$, as desired.

Next we claim that for $d$ as in the definition of $\ell$-boundedness, for
every $\mathcal{M}$ in the heart of the essential image of $\alpha$ and for
every basic $U$ we have $H^{k}(U,\mathcal{M})=0$ for $k>d+2$. Indeed, since
$H^{k}(U,\mathcal{M})$ commutes with filtered colimits of static
$\mathcal{M}$, we can reduce to the case that $\mathcal{M}=\pi_{m}\mathcal{P}$
for some $\mathcal{P}\in\mathcal{C}$. As seen in the previous paragraph this
implies that $\mathcal{M}$ is a sequential limit
$\mathcal{M}=\varprojlim_{n}\mathcal{M}_{n}$ for qcqs static
$\mathcal{M}_{n}\in\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})$.
By factoring the countable limit out of $H^{k}(U,-)$ and noting that countable
limits have cohomological dimension 1 in spectra, we reduce to showing that
$H^{k}(U,\mathcal{M}_{n})=0$ for $k>d$, which follows immediately from the
definition of $\ell$-boundedness.

We are now ready to prove that $\alpha$ is fully faithful. This amounts to
showing the following: Given $U$ as in (ii) and any filtered diagram
$(\mathcal{P}_{i})_{i}$ of objects in $\mathcal{C}$, the natural map

$\displaystyle\varinjlim_{i}\operatorname{Hom}({\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U],\mathcal{P}_{i})\mathbin{\xrightarrow{\sim}}\operatorname{Hom}({\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U],\varinjlim_{i}\mathcal{P}_{i})$

is an isomorphism (here the colimit $\varinjlim_{i}\mathcal{P}_{i}$ is formed
in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$).
Using the identification
$\operatorname{Hom}({\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U],-)=\Gamma(U,-)$
and hence the fact that this functor has finite cohomological dimension on the
essential image of $\alpha$, we can employ a standard Postnikov limit argument
to reduce to the claim that $\Gamma(U,-)$ preserves filtered colimits of
uniformly left-bounded sheaves.

We have established that
$\alpha\colon\operatorname{Ind}(\mathcal{C})\mathrel{\hookrightarrow}\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$
is an embedding. We now show that the essential image is precisely
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$.
First observe that the essential image of $\alpha$ is contained in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$.
Indeed, this follows from the fact that every $\mathcal{P}\in\mathcal{C}$ is
$\omega_{1}$-solid and that for all basic $U$ the functor $\Gamma(U,-)$
preserves filtered colimits in the essential image of $\alpha$ because it has
finite cohomological dimension on that image (as used in the previous
paragraph). It remains to see that every $\omega_{1}$-solid
${\mathbb{Z}}_{\ell}$-module on $X$ lies in the essential image of $\alpha$.
By standard arguments (e.g. looking at the right adjoint to the embedding
$\operatorname{Ind}(\mathcal{C})\mathrel{\hookrightarrow}\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$)
this reduces to showing that the family of functors
$\operatorname{Hom}({\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U],-)=\Gamma(U,-)$,
for $U$ as in (ii), is conservative on
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$.
But by definition of $\omega_{1}$-solid objects, these functors determine
$\Gamma(U,-)$ for _all_ basic $U\in X_{\mathrm{proet}}$, and as the basic $U$
form a basis of $X_{\mathrm{proet}}$ (as noted above), this family of functors
is indeed conservative.

We have finally shown the equivalence
$\operatorname{Ind}(\mathcal{C})\cong\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$.
The rest of (ii) goes as follows: Everything is clear except that the compact
objects of
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
are $\ell$-adically complete and embed into
$\operatorname{Pro}(\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})^{\omega})$.
The $\ell$-adic completeness was shown for ${\mathbb{Z}}_{\ell}[U]$ with qcqs
$U\in X_{\mathrm{et}}$ above and follows immediately for all compact objects
because $\ell$-adic completeness is stable under limits. To get the embedding
into the $\operatorname{Pro}$-category, we use the following more general
statement: Equip
$\operatorname{Pro}(\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})^{\omega})$
with the natural $t$-structure induced from the one on
$\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})^{\omega}$
(cf. [9, Lemma C.2.4.3]); then the natural functor

$\displaystyle\operatorname{Pro}(\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})^{\omega})^{b}\mathrel{\hookrightarrow}\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$

is fully faithful. Namely, the functor
$\operatorname{Pro}(\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})^{\omega})\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$
is $t$-exact by [3, Proposition VII.1.6] and so the claimed fully faithfulness
reduces to showing that
$\operatorname{Hom}(\varprojlim_{i}\mathcal{P}_{i},\mathcal{Q})=\varinjlim_{i}\operatorname{Hom}(\mathcal{P}_{i},\mathcal{Q})$
for compact static
$\mathcal{P}_{i},\mathcal{Q}\in\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})$;
this follows from the usual Breen resolution argument (as in [3, Proposition
VII.1.12]).

The rest of (iii) is easy: We have already seen that
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
is stable under countable limits and contains all étale sheaves. By the proof
of (ii) it follows that it is also stable under all small colimits and that
the $t$-structure on
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$
restricts to a $t$-structure on
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$.
It remains to see that
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
is stable under tensor product: This can be checked on compact generators and
thus follows from the observation
${\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U]\otimes{\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[V]={\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U\times_{X}V]$;
this also proves that compact objects are stable under tensor product. ∎

In order to work with $\omega_{1}$-solid sheaves it is important to understand
how they behave under a change of spatial diamond:

######  Proposition 2.6.

Let $f\colon Y\to X$ be a map of $\ell$-bounded spatial diamonds.

  1. (i)

The pullback
$\operatorname{\mathcal{D}}(X_{\mathrm{v}},{\mathbb{Z}}_{\ell})\to\operatorname{\mathcal{D}}(Y_{\mathrm{v}},{\mathbb{Z}}_{\ell})$
restricts to a $t$-exact symmetric monoidal functor

$\displaystyle
f^{*}\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(Y,{\mathbb{Z}}_{\ell})_{\omega_{1}}$

which preserves all small colimits and all countable limits.

  2. (ii)

The pushforward
$f_{{\mathrm{v}}*}\colon\operatorname{\mathcal{D}}(Y_{\mathrm{v}},{\mathbb{Z}}_{\ell})\to\operatorname{\mathcal{D}}(X_{\mathrm{v}},{\mathbb{Z}}_{\ell})$
restricts to a functor

$\displaystyle
f_{{\scalebox{0.5}{$\square$}}*}\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(Y,{\mathbb{Z}}_{\ell})_{\omega_{1}}\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$

which has finite cohomological dimension, is right adjoint to $f^{*}$ and
preserves all small limits and colimits.

###### Proof.

Since the pullback on the v-site is $t$-exact and preserves all small limits
and colimits, part (i) follows immediately from Proposition 2.5 and the fact
that $f^{*}{\mathbb{Z}}_{\ell}[U]={\mathbb{Z}}_{\ell}[U\times_{X}Y]$ for every
quasicompact separated $U\in X_{\mathrm{et}}$ (for the proof that $f^{*}$ is
symmetric monoidal see e.g. [3, Proposition VII.2.2]). This proves (i).

To prove (ii), first note that the v-pushforward has finite cohomological
dimension when restricted to
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(Y,{\mathbb{Z}}_{\ell})_{\omega_{1}}$:
Recall that for a static v-sheaf $\mathcal{M}$ on $X$,
$H^{k}(f_{*}\mathcal{M})$ is the sheafification of the presheaf $U\mapsto
H^{k}(U\times_{X}Y,\mathcal{M})$; but if $\mathcal{M}$ is $\omega_{1}$-solid
then $H^{k}(U,\mathcal{M})=0$ for $k>d+1$ with $d$ as in the definition of
$\ell$-boundedness for $Y$ (see the proof of Proposition 2.5). It follows that
$f_{{\mathrm{v}}*}$ preserves small colimits on
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(Y,{\mathbb{Z}}_{\ell})_{\omega_{1}}$,
hence to show that it maps this $\infty$-category to
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
we are reduced to the compact generators, i.e. we need to show that
$f_{{\mathrm{v}}*}{\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U]\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
for all $U=\varprojlim_{n}U_{n}$ as in 2.5.(ii). But
$f_{{\mathrm{v}}*}{\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U]=\varprojlim_{n}f_{{\mathrm{v}}*}({\mathbb{Z}}/\ell^{n}{\mathbb{Z}})[U_{n}]$
so since
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
is stable under countable limits it is enough to show that
$f_{{\mathrm{v}}*}({\mathbb{Z}}/\ell^{n}{\mathbb{Z}})[U_{n}]$ is étale – this
follows from [15, Corollary 16.8.(ii)]. ∎

######  Corollary 2.7.

The assignment

$\displaystyle
X\mapsto\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$

defines a hypercomplete sheaf of symmetric monoidal $\infty$-categories on the
v-site of $\ell$-bounded spatial diamonds.

###### Proof.

By using the embedding
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})\subseteq\operatorname{\mathcal{D}}(X_{\mathrm{v}},{\mathbb{Z}}_{\ell})$
and using that the right-hand category satisfies hypercomplete v-descent for
formal reasons, we are left with showing that for any v-cover $f\colon Y\to X$
of $\ell$-bounded spatial diamonds and any
$\mathcal{M}\in\operatorname{\mathcal{D}}(X_{\mathrm{v}},{\mathbb{Z}}_{\ell})$
we have
$f^{*}\mathcal{M}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(Y,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
if and only if
$\mathcal{M}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$.
The “if” part follows immediately from 2.6.(i). It remains to prove the “only
if” part, so assume that $f^{*}\mathcal{M}$ is $\omega_{1}$-solid. Extend $f$
to a v-hypercover $f_{\bullet}\colon Y_{\bullet}\to X$ such that all $Y_{n}$
are $\ell$-bounded spatial diamonds. Then by v-hyperdescent for v-sheaves we
have
$\mathcal{M}=\varprojlim_{n\in\Delta}f_{n{\mathrm{v}}*}f_{n}^{*}\mathcal{M}$,
which we want to show to be $\omega_{1}$-solid. Since $\omega_{1}$-solid
sheaves are stable under countable limits by 2.5.(iii) it is enough to show
that each $f_{n{\mathrm{v}}*}f_{n}^{*}\mathcal{M}$ is $\omega_{1}$-solid. But
this follows immediately from Proposition 2.6 since $f_{0}^{*}\mathcal{M}$ is
$\omega_{1}$-solid by assumption. ∎

It is convenient to know that the tensor product of $\ell$-adically complete
sheaves is again $\ell$-adically complete:

######  Proposition 2.8.

Let $X$ be an $\ell$-bounded spatial diamond and let
$\mathcal{M},\mathcal{N}\in\operatorname{\mathcal{D}}^{-}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$
be $\ell$-adically complete. Then $\mathcal{M}\otimes\mathcal{N}$ is
$\ell$-adically complete.

###### Proof.

By choosing projective resolutions we can write $\mathcal{M}$ and
$\mathcal{N}$ as geometric realizations of direct sums of copies of the
compact projective generators
${\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U]$ for w-contractible $U\in
X_{\mathrm{proet}}$. Since the $\ell$-adic completion functor on
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$
has homological dimension $1$ and therefore preserves geometric realizations
of uniformly left-bounded sheaves, we can reduce to the case that
$\mathcal{M}$ and $\mathcal{N}$ are completed direct sums of copies of
${\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U]$ for w-contractible $U\in
X_{\mathrm{proet}}$; in particular both $\mathcal{M}$ and $\mathcal{N}$ are
static (note that ${\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U]$ is
$\ell$-adically complete by the proof of Proposition 2.5). Since $\ell$-adic
completion commutes with $\omega_{1}$-filtered colimits, we can further reduce
to the case of _countable_ completed direct sums, i.e. we have
$\mathcal{M}=\widehat{\bigoplus}_{n}\,{\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U_{n}]$
and
$\mathcal{N}=\widehat{\bigoplus}_{m}\,{\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[V_{m}]$
for w-contractible $U_{n},V_{m}\in X_{\mathrm{proet}}$ and integers $m,n>0$.
Note that we can write $\mathcal{M}$ as a union of subsheaves of the form
$\mathcal{M}_{\alpha}=\prod_{n}\ell^{\alpha_{n}}{\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U_{n}]$
for all sequences $\alpha_{n}\in{\mathbb{Z}}_{\geq 0}$ converging to $\infty$
(and similarly for $\mathcal{N}$ in terms of $\mathcal{N}_{\beta}$ for
sequences $\beta$). Using [3, Proposition VII.2.3] we deduce

$\displaystyle\mathcal{M}\otimes\mathcal{N}=\varinjlim_{\alpha,\beta}\mathcal{M}_{\alpha}\otimes\mathcal{N}_{\beta}=\varinjlim_{\alpha,\beta}\prod_{m,n}(\ell^{\alpha_{n}}{\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U_{n}]\otimes\ell^{\beta_{m}}{\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[V_{m}])$
$\displaystyle\qquad=\varinjlim_{\alpha,\beta}\prod_{m,n}\ell^{\alpha_{n}+\beta_{m}}{\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U_{n}\times_{X}V_{m}].$

By a simple cofinality argument one checks that this is just the completed
direct sum of
${\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U_{n}\times_{X}V_{m}]$ for
all $n,m>0$. In particular this sheaf is $\ell$-adically complete, as desired.
∎

##  3 Nuclear Sheaves

Fix the prime $\ell\neq p$. In the following we will define a full subcategory
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell})\subseteq\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$
of nuclear sheaves on any small v-stack $X$. In general this $\infty$-category
will be defined by v-descent, but on $\ell$-bounded spatial diamonds it admits
a much more explicit description:

######  Definition 3.1.

Let $X$ be an $\ell$-bounded spatial diamond and let
$\mathcal{M}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$
be a solid ${\mathbb{Z}}_{\ell}$-sheaf on $X$.

  1. (a)

$\mathcal{M}$ is called a _Banach sheaf_ if $\mathcal{M}$ is $\ell$-adically
complete and $\mathcal{M}/\ell\mathcal{M}$ is étale, i.e. lies in
$\operatorname{\mathcal{D}}_{\mathrm{et}}(X,\mathbb{F}_{\ell})$.

  2. (b)

$\mathcal{M}$ is called _nuclear_ if it is a filtered colimit of Banach
sheaves. We denote by
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell})\subseteq\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$
the full subcategory spanned by the nuclear sheaves.

######  Remarks 3.2.

  1. (i)

It follows immediately from 2.5.(iii) that every nuclear
${\mathbb{Z}}_{\ell}$-sheaf on an $\ell$-bounded spatial diamond $X$ is
$\omega_{1}$-solid, i.e. we have
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell})\subseteq\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$.

  2. (ii)

The $t$-structure on
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})$
does usually not restrict to a $t$-structure on
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell})$. The
problem is that for a Banach sheaf $\mathcal{M}$ it is generally not true that
$\tau_{\geq 0}\mathcal{M}$ is still a Banach sheaf: While it is true that
$\tau_{\geq 0}\mathcal{M}$ is still $\ell$-adically complete, it may happen
that it is not étale mod $\ell$ anymore. As a counterexample, suppose that
$X={\mathbb{Z}}_{\ell}$ (viewed as a pro-finite étale space over some
algebraically closed non-archimedean field) and let $\mathcal{M}$ be the
$\ell$-adic completion of the following étale sheaf $\mathcal{N}$:
$\mathcal{N}$ is the étale product of sheaves $\mathcal{N}_{n}$ for $n\geq 1$,
where $\mathcal{N}_{n}$ is the constant sheaf
${\mathbb{Z}}/\ell^{n}{\mathbb{Z}}$ supported on
$X\setminus\ell^{n}{\mathbb{Z}}_{\ell}$. Then $\mathcal{M}$ is a Banach sheaf
concentrated in homological degrees $0$ and $1$, but by a direct computation
one checks that $\pi_{1}M/\ell\pi_{1}M$ is not étale: It does not have any
étale sections, but it has a non-trivial value on the pro-étale map
$\\{0\\}\to X$.

######  Warning 3.3.

In [1, Definition 8.5] there is a very general definition of nuclear objects
in a compactly generated symmetric monoidal $\infty$-category. However, our
3.1.(b) differs from that general definition. In fact, if we apply the general
definition from [1] to
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
then we end up with a subcategory
$\mathcal{C}\subseteq\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell})$
spanned only by the overconvergent sheaves. There are still good reasons to
call our sheaves nuclear, as we will see in the following.

The above definition of nuclear sheaves may seem a bit ad-hoc and it is not at
all clear from this definition why the $\infty$-category of nuclear sheaves
satisfies v-descent. Our first goal will therefore be to provide an equivalent
definition in terms of _trace-class_ maps. The general definition of trace-
class maps provided in [1, §8] does not work in our setting, as it is far too
restrictive. We will therefore provide our own definition tailored to the
specific setting at hand:

######  Definition 3.4.

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

  1. (a)

We denote by
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(-,-)$
the internal hom functor in the symmetric monoidal $\infty$-category
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
(this exists because this $\infty$-category is presentable).

  2. (b)

We define a functor

$\displaystyle\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(-,-)\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}^{\mathrm{op}}\times\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$

as follows: This functor will preserve limits in the first argument and for
compact first argument it will preserve colimits in the second argument, so
that it is enough to construct its restriction to
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}^{\omega,{\mathrm{op}}}\times\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}^{\omega}$.
In this case we define it as the $\ell$-adic completion

$\displaystyle\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(-,-)=\widehat{\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\prime}(-,-)},$

where
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\prime}(-,-)$
is the following composition of functors:

$\displaystyle\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}^{\omega,{\mathrm{op}}}\times\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}^{\omega}\mathrel{\hookrightarrow}\operatorname{Ind}(\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})^{\omega,{\mathrm{op}}})\times\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}^{\omega}\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$

Here the first functor is induced by 2.5.(ii) and the second functor is the
unique colimit-preserving functor which restricts to
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(\mathcal{P},\mathcal{Q})$
for
$\mathcal{P}\in\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})^{\omega}$
and
$\mathcal{Q}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}^{\omega}$.

  3. (c)

There is a natural transformation

$\displaystyle\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(-,-)\to\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(-,-)$

of functors
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}^{\mathrm{op}}\times\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$.
Namely, it is enough to construct this on compact arguments, where it boils
down to a natural transformation
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\prime}\to\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}$,
which comes from the fact that
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\prime}$
is a left Kan extension of the restriction of
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}$ to
$\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})^{\omega,{\mathrm{op}}}$
in the first argument.

  4. (d)

For all
$\mathcal{M},\mathcal{N}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
we denote

$\displaystyle\operatorname{Hom}^{\mathrm{tr}}(\mathcal{M},\mathcal{N}):=\Gamma(X,\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{M},\mathcal{N})).$

A map $\mathcal{M}\to\mathcal{N}$ is called _trace-class_ if it lies in the
image of the map
$\operatorname{Hom}^{\mathrm{tr}}(\mathcal{M},\mathcal{N})\to\operatorname{Hom}(\mathcal{M},\mathcal{N})$
induced by the natural transformation above.

######  Remark 3.5.

The definition of
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(-,-)$
may seem a bit intimidating, so let us provide a more intuitive (albeit less
formal) description: Pick any compact objects
$\mathcal{P},\mathcal{Q}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}^{\omega}$.
By 2.5.(ii) we can write $\mathcal{P}=\varprojlim_{n}\mathcal{P}_{n}$ for
compact étale sheaves
$\mathcal{P}_{n}\in\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})^{\omega}$.
We then define

$\displaystyle\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},\mathcal{Q})=(\varinjlim_{n}\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(\mathcal{P}_{n},\mathcal{Q}))\,\widehat{}\,,$

where $\widehat{}$ denotes $\ell$-adic completion. For general
$\mathcal{M},\mathcal{N}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
we write $\mathcal{M}=\varinjlim_{i}\mathcal{P}_{i}$ and
$\mathcal{N}=\varinjlim_{j}\mathcal{Q}_{j}$ for compact $\mathcal{P}_{i}$ and
$\mathcal{Q}_{j}$ and then define

$\displaystyle\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{M},\mathcal{N})=\varprojlim_{i}\varinjlim_{j}\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P}_{i},\mathcal{Q}_{j}).$

Here the limit is taken in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$.

Before getting back to nuclear sheaves, let us study some of the properties of
trace-class maps. It turns out that they enjoy almost the same formal
properties as the abstract trace-class maps studied in [1, §8]:

######  Lemma 3.6.

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

  1. (i)

Let $f\colon\mathcal{M}\to\mathcal{N}$,
$g\colon\mathcal{M}^{\prime}\to\mathcal{M}$ and
$h\colon\mathcal{N}\to\mathcal{N}^{\prime}$ be maps in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$.
If $f$ is trace-class then so is $h\mathbin{\circ}f\mathbin{\circ}g$.

  2. (ii)

If $f\colon\mathcal{M}\to\mathcal{N}$ and
$f^{\prime}\colon\mathcal{M}^{\prime}\to\mathcal{N}^{\prime}$ are trace-class
maps in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$,
then so is $f\otimes
f^{\prime}\colon\mathcal{M}\otimes\mathcal{M}^{\prime}\to\mathcal{N}\otimes\mathcal{N}^{\prime}$.

  3. (iii)

Let $f\colon\mathcal{M}\to\mathcal{N}$ be a trace-class map in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$.
Then for every
$\mathcal{L}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
the commutative square

${\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{L},\mathcal{M})}$${\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{L},\mathcal{N})}$${\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(\mathcal{L},\mathcal{M})}$${\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(\mathcal{L},\mathcal{N})}$

admits a diagonal map
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(\mathcal{L},\mathcal{M})\to\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{L},\mathcal{N})$
making both triangles commute.

  4. (iv)

Let $f\colon\mathcal{P}\to\mathcal{M}$ be a trace-class map in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
with $\mathcal{P}$ compact. Then there is a compact object $\mathcal{Q}$ in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
such that $f$ factors as $\mathcal{P}\to\mathcal{Q}\to\mathcal{M}$, where
$\mathcal{P}\to\mathcal{Q}$ is also trace-class.

###### Proof.

We first make the following general observation: For any
$\mathcal{M},\mathcal{N},\mathcal{L}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
the natural maps

$\displaystyle\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(\mathcal{M},\mathcal{N})\otimes\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{N},\mathcal{L})$
$\displaystyle\to\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(\mathcal{M},\mathcal{L}),$
$\displaystyle\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{M},\mathcal{N})\otimes\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(\mathcal{N},\mathcal{L})$
$\displaystyle\to\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(\mathcal{M},\mathcal{L})$

factor over
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{M},\mathcal{L})$.
To see this, consider the $\infty$-category $\mathcal{C}$ of morphisms
$g\colon\mathcal{N}\to\mathcal{N}^{\prime}$ in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$,
where morphisms $g_{1}\to g_{2}$ in $\mathcal{C}$ are given by commuting
squares

${\mathcal{N}_{1}}$${\mathcal{N}^{\prime}_{1}}$${\mathcal{N}_{2}}$${\mathcal{N}^{\prime}_{2}}$$\scriptstyle{g_{1}}$$\scriptstyle{g_{2}}$

(This can easily be constructed as simplicial sets.) Then we can view all of
the above expressions as functors in $\mathcal{M}$,
$g\colon\mathcal{N}\to\mathcal{N}^{\prime}$ and $\mathcal{L}$ (where we
replace the second appearing $\mathcal{N}$ in the above expression by
$\mathcal{N}^{\prime}$) and the above morphisms as natural transformations of
such functors. Since both
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(\mathcal{M},\mathcal{L})$
and
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{M},\mathcal{L})$
transform colimits in $\mathcal{M}$ into limits, we can formally reduce the
construction of the desired natural transformation to the case that
$\mathcal{M}$ is compact (use that right Kan extension is a right adjoint
functor). Then all functors preserve colimits in $\mathcal{N}$, so we can
reduce to the case that $\mathcal{N}$ is compact as well. By factoring out
colimits in $\mathcal{N}^{\prime}$ (like for $\mathcal{M}$) we can reduce to
the case that $\mathcal{N}^{\prime}$ is compact. Then we can finally also
factor out colimits in $\mathcal{L}$ to reduce to the case that all of
$\mathcal{M}$, $\mathcal{N}$, $\mathcal{N}^{\prime}$ and $\mathcal{L}$ are
compact. In this case
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{M},\mathcal{L})$
is $\ell$-adically complete, so we can ignore $\ell$-adic completions. We
therefore end up with constructing natural transformations

$\displaystyle\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(\mathcal{M},\mathcal{N})\otimes\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\prime}(\mathcal{N}^{\prime},\mathcal{L})$
$\displaystyle\to\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\prime}(\mathcal{M},\mathcal{L}),$
$\displaystyle\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\prime}(\mathcal{M},\mathcal{N})\otimes\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(\mathcal{N}^{\prime},\mathcal{L})$
$\displaystyle\to\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\prime}(\mathcal{M},\mathcal{L}),$

functorial in $\mathcal{M}$, $g\colon\mathcal{N}\to\mathcal{N}^{\prime}$ and
$\mathcal{L}$. For the second natural transformation, we can write
$\mathcal{M}=\varprojlim_{n}\mathcal{M}_{n}$ for qcqs étale $\mathcal{M}_{n}$
and pull out this limit on both sides (by definition of
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\prime}$),
transforming it into colimits. This reduces the construction of the natural
transformation to the case that $\mathcal{M}$ is qcqs étale, in which case
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\prime}(\mathcal{M},-)=\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(\mathcal{M},-)$,
so we are done. For the first natural transformation we can similarly reduce
to the case that $\mathcal{N}^{\prime}$ is qcqs étale. Then if we write
$\mathcal{N}=\varprojlim_{n}\mathcal{N}_{n}$ for qcqs étale $n$ we know that
$g$ factors over some $\mathcal{N}_{n}$ and is thus a limit of maps
$\mathcal{N}_{n^{\prime}}\to\mathcal{N}^{\prime}$ in $\mathcal{C}$. By the
same argument as above we can pull out this limit and thus assume that
$\mathcal{N}$ is also qcqs étale. But then we can further pull out a limit in
$\mathcal{M}$ to reduce to the case that $\mathcal{M}$ is qcqs étale, in which
case the desired natural transformation is evident.

With the above preparations at hand, let us now prove the actual claims. Part
(i) follows immediately from the just constructed natural transformations by
applying $\Gamma(X,-)$. To prove (iii), note that the given map $f$ provides a
map
${\mathbb{Z}}_{\ell}\to\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{M},\mathcal{N})$,
so that the desired diagonal map can be constructed as follows:

$\displaystyle\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(\mathcal{L},\mathcal{M})\to\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(\mathcal{L},\mathcal{M})\otimes\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{M},\mathcal{N})\to\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{L},\mathcal{N}),$

where the second map is the one constructed above. To prove (ii) we need to
show that the natural map

$\displaystyle\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{M},\mathcal{N})\otimes\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{M}^{\prime},\mathcal{N}^{\prime})\to\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(\mathcal{M}\otimes\mathcal{M}^{\prime},\mathcal{N}\otimes\mathcal{N}^{\prime})$

factors over
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{M}\otimes\mathcal{M}^{\prime},\mathcal{N}\otimes\mathcal{N}^{\prime})$.
This can be done in a similar manner as above by reducing to the case that all
sheaves are qcqs étale; we leave the details to the reader. To prove (iv),
write $\mathcal{M}$ as a filtered colimit
$\mathcal{M}=\varinjlim_{i}\mathcal{Q}_{i}$ with compact $\mathcal{Q}_{i}$ and
note that
$\operatorname{Hom}^{\mathrm{tr}}(\mathcal{P},\mathcal{M})=\varinjlim_{i}\operatorname{Hom}^{\mathrm{tr}}(\mathcal{P},\mathcal{Q}_{i})$,
which easily implies the claim. ∎

In order to relate trace-class maps to nuclear sheaves (as defined in 3.1.(b))
we need further properties of
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}$:

######  Lemma 3.7.

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

  1. (i)

The functor

$\displaystyle\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},-)\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$

is bounded, i.e. there are integers $a\leq b$ such that for every static
$\mathcal{M}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
the sheaf
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},\mathcal{M})$
is concentrated in homological degrees $[a,b]$.

  2. (ii)

For every $\ell$-adically complete
$\mathcal{M}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
the sheaf
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},\mathcal{M})$
is $\ell$-adically complete.

###### Proof.

By 2.5.(ii) we can assume that
$\mathcal{P}={\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[U]$ for some
basic $U=\varprojlim_{n}U_{n}\in X_{\mathrm{proet}}$. We first prove (i) in
the case that $\mathcal{M}={\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[V]$
for some basic $\varprojlim_{m}V_{m}\in X_{\mathrm{proet}}$. Then we have

$\displaystyle\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},\mathcal{M})=(\varinjlim_{n}\varprojlim_{m}\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(({\mathbb{Z}}/\ell^{n}{\mathbb{Z}})[U_{n}],{\mathbb{Z}}_{\ell}[V_{m}]))\,\widehat{}\,.$

Thus the desired boundedness boils down to showing that
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(({\mathbb{Z}}/\ell^{n}{\mathbb{Z}})[U_{n}],{\mathbb{Z}}_{\ell}[V_{m}])$
is bounded independent of $V_{m}$ (since countable limits are bounded by $1$),
which follows immediately from the $\ell$-boundedness of $X$. We can now
deduce that the functor
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},-)$
is right-bounded by writing any static $\mathcal{M}$ as a sifted colimit of
static compact objects (i.e. choosing a resolution by direct sums of compact
objects) and using that
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},-)$
preserves this colimit.

Now let $\mathcal{M}$ be general (and still static). By writing $\mathcal{M}$
as a filtered colimit of finitely presented static objects, we can reduce (i)
to the case that $\mathcal{M}$ is finitely presented, i.e. a countable limit
$\mathcal{M}=\varprojlim_{n}\mathcal{M}_{n}$ of qcqs étale sheaves
$\mathcal{M}_{n}\in\operatorname{\mathcal{D}}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})^{\omega}$.
Then
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},\mathcal{M})$
is $\ell$-adically complete: Since $\ell$-adic completion is stable under
uniformly right-bounded geometric realizations and
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},\mathcal{M})$
is right-bounded by the above, we can pick a resolution of $\mathcal{M}$ by
compact objects in order to reduce to the case that $\mathcal{M}$ is compact –
then the $\ell$-adic completeness is clear by definition. It now follows that
the natural map

$\displaystyle(\varinjlim_{n}\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(({\mathbb{Z}}/\ell^{n}{\mathbb{Z}})[U_{n}],\mathcal{M}))\,\widehat{}\,\mathbin{\xrightarrow{\sim}}\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},\mathcal{M})$

is an isomorphism (by again passing to geometric realizations in
$\mathcal{M}$). In the same way as for compact $\mathcal{M}$ we deduce that
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},\mathcal{M})$
is bounded independent of $\mathcal{M}$. This finishes the proof of (i).

We now prove (ii), so let the $\ell$-adically complete sheaf $\mathcal{M}$ be
given. By the boundedness of
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},-)$
this functor commutes with Postnikov limits, so we can assume that
$\mathcal{M}$ is left-bounded. Moreover, since $\ell$-adic completeness can be
checked on homotopy sheaves and
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},-)$
is bounded, we can use $\mathcal{M}=\varinjlim_{n}\tau_{\geq n}\mathcal{M}$ in
order to reduce to the case that $\mathcal{M}$ is bounded. We can then even
assume that $\mathcal{M}$ is static. By choosing a resolution of $\mathcal{M}$
in terms of direct sums of copies of
${\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[V]$ for varying basic
$V=\varprojlim_{n}V_{n}\in X_{\mathrm{proet}}$ and using that both
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},-)$
and $\ell$-adic completion commute with uniformly right-bounded geometric
realizations we can reduce to the case that $\mathcal{M}$ is a completed
direct sum of copies of ${\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[V]$.
Since $\ell$-adic completion commutes with $\omega_{1}$-filtered colimits we
can further reduce to a countable such sum, i.e. we have
$\mathcal{M}=\widehat{\bigoplus}_{k}\,{\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[V_{k}]$
for various basic $V_{k}=\varprojlim_{n}V_{k,n}\in X_{\mathrm{proet}}$ and
integers $k>0$. To abbreviate notation let us denote
$\mathcal{Q}_{k}:={\mathbb{Z}}_{\ell,{\scalebox{0.5}{$\square$}}}[V_{k}]$. For
every sequence of integers $\alpha_{m}\geq 0$ converging to $\infty$ we write
$\mathcal{M}_{\alpha}:=\prod_{k}\ell^{\alpha_{k}}\mathcal{Q}_{k}$, so that
$\mathcal{M}=\varinjlim_{\alpha}\mathcal{M}_{\alpha}$. We now claim that the
natural map

$\displaystyle\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},\mathcal{M})=\varinjlim_{\alpha}\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},\mathcal{M}_{\alpha})\mathbin{\xrightarrow{\sim}}\varinjlim_{\alpha}\prod_{k}\ell^{\alpha_{k}}\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},\mathcal{Q}_{k})$

is an isomorphism. Here by
$\prod_{k}\ell^{\alpha_{k}}\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\dots)$
we mean the object
$\prod_{k}\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\dots)$
but where the $\ell^{\alpha_{k}}$ determine the transitions maps in the
filtered colimit. To prove the claimed isomorphism of the colimits over
$\alpha$, it is enough to find suitable sections. Concretely, fix any sequence
$\alpha$ and choose another sequence $\alpha^{\prime}$ such that
$\alpha^{\prime}\leq\alpha$ and the sequence $\alpha-\alpha^{\prime}$ still
converges to $\infty$. Then we get natural maps

$\displaystyle\prod_{k}\ell^{\alpha_{k}}\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},\mathcal{Q}_{k})\to\widehat{\bigoplus}_{k}\,\ell^{\alpha^{\prime}_{k}}\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},\mathcal{Q}_{k})\to\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},\prod_{k}\ell^{\alpha^{\prime}_{k}}\mathcal{Q}_{k}).$

The first map exists because $\alpha-\alpha^{\prime}$ converges to $\infty$;
it is the one which multiplies the $k$-th part of the product/sum by
$\ell^{\alpha_{k}-\alpha^{\prime}_{k}}$. The second map exists because
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},\prod_{k}\ell^{\alpha^{\prime}_{k}}\mathcal{Q}_{k})$
is $\ell$-adically complete (by the same argument as in the proof of (i) using
that $\prod_{k}\mathcal{Q}_{k}$ is static and finitely presented), so that we
can replace the completed direct sum by an ordinary direct sum for the
construction. One checks that the thus constructed map is indeed the desired
section of the map of filtered systems, proving the above isomorphism. To
finish the proof that
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},\mathcal{M})$
is $\ell$-adically complete, it remains to see that the natural map

$\displaystyle\varinjlim_{\alpha}\prod_{k}\ell^{\alpha_{k}}\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},\mathcal{Q}_{k})\mathbin{\xrightarrow{\sim}}\widehat{\bigoplus}_{k}\,\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},\mathcal{Q}_{k})$

is an isomorphism, or equivalently that the source of this map is
$\ell$-adically complete. This can be checked on homotopy sheaves and by the
usual arguments involving geometric realizations we reduce to showing this
statement after replacing each
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},\mathcal{Q}_{k})$
by a completed direct sum of static compact objects. Then everything is static
and the claimed isomorphism can easily be checked on sections $\Gamma(U,-)$
for w-contractible $U$ (reducing the problem to an easy problem about
classical abelian groups). ∎

We are finally in the position to provide the promised characterization of
nuclear sheaves in terms of trace-class maps. This will immediately enable us
to prove v-descent as well, leading to our first main result on nuclear
sheaves in this paper.

######  Definition 3.8.

Let $X$ be an $\ell$-bounded spatial diamond. A sheaf
$\mathcal{N}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
is called _basic nuclear_ if it can be written as a sequential colimit
$\mathcal{N}=\varinjlim_{n}\mathcal{P}_{n}$ such that all $\mathcal{P}_{n}$
are compact and all transition maps $\mathcal{P}_{n}\to\mathcal{P}_{n+1}$ are
trace-class.

######  Theorem 3.9.

  1. (i)

Let $X$ be an $\ell$-bounded spatial diamond. Then for an $\omega_{1}$-solid
sheaf $\mathcal{M}$ on $X$ the following are equivalent:

    1. (a)

$\mathcal{M}$ is nuclear.

    2. (b)

$\mathcal{M}$ is an $\omega_{1}$-filtered colimit of basic nuclear sheaves.

    3. (c)

For every compact
$\mathcal{P}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
the natural map
$\operatorname{Hom}^{\mathrm{tr}}(\mathcal{P},\mathcal{M})\to\operatorname{Hom}(\mathcal{P},\mathcal{M})$
is an isomorphism.

The $\infty$-category
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell})$ is
$\omega_{1}$-compactly generated with the $\omega_{1}$-compact objects being
precisely the basic nuclear sheaves. Moreover, this $\infty$-category contains
all étale sheaves and is stable under all small colimits and under the tensor
product in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$.

  2. (ii)

The assignment

$\displaystyle
X\mapsto\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell})$

defines a hypercomplete sheaf of presentable symmetric monoidal
$\infty$-categories on the v-site of $\ell$-bounded spatial diamonds.

###### Proof.

We first prove (i), so let $X$ and $\mathcal{M}$ be given. The equivalence of
(b) and (c) follows formally from Lemma 3.6, see [1, Theorem 8.6]. Now assume
that $\mathcal{M}$ is nuclear; we will show that it satisfies condition (c).
Since (c) is stable under colimits in $\mathcal{M}$ we can assume that
$\mathcal{M}$ is a Banach module. Then by 3.7.(ii) both
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},\mathcal{M})$
and
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(\mathcal{P},\mathcal{M})$
are $\ell$-adically complete, so it is enough to show that
$\mathcal{N}:=\mathcal{M}/\ell\mathcal{M}$ satisfies (c). Now $\mathcal{N}$ is
étale, so by 2.5.(i) we can write it as a colimit of copies of
$\mathbb{F}_{\ell}[U]$ for varying basic $U\in X_{\mathrm{et}}$. It is
therefore enough to show that $\mathbb{F}_{\ell}[U]$ satisfies (c), which
follows by explicit computation (using that if we write
$\mathcal{P}=\varprojlim_{n}\mathcal{P}_{n}$ for qcqs étale sheaves
$\mathcal{P}_{n}$ then
$\operatorname{Hom}(\mathcal{P},\mathbb{F}_{\ell}[U])=\varinjlim_{n}\operatorname{Hom}(\mathcal{P}_{n},\mathbb{F}_{\ell}[U])$).

To finish the equivalence of (a), (b) and (c) it remains to show that (b)
implies (a), so from now on assume that
$\mathcal{M}=\varinjlim_{n}\mathcal{P}_{n}$ is basic nuclear; we need to show
that $\mathcal{M}$ is nuclear. This follows if we can show that every map
$\mathcal{P}_{n}\to\mathcal{P}_{n+1}$ factors over some Banach sheaf because
then $\mathcal{M}$ is the colimit of these Banach sheaves. Let
$\mathcal{P}^{\prime}_{n+1}$ be the $\ell$-adic completion of the pushforward
of $\mathcal{P}_{n+1}$ to the étale site. We claim that the natural map

$\displaystyle\operatorname{Hom}(\mathcal{P}_{n},\mathcal{P}^{\prime}_{n+1})=\operatorname{Hom}^{\mathrm{tr}}(\mathcal{P}_{n},\mathcal{P}^{\prime}_{n+1})\mathbin{\xrightarrow{\sim}}\operatorname{Hom}^{\mathrm{tr}}(\mathcal{P}_{n},\mathcal{P}_{n+1})$

is an isomorphism (here the first identity follows from (c) because
$\mathcal{P}^{\prime}_{n+1}$ is clearly nuclear), which provides the desired
factorization
$\mathcal{P}_{n}\to\mathcal{P}^{\prime}_{n+1}\to\mathcal{P}_{n+1}$. To prove
this isomorphism, note that by 3.7.(ii) both sides of the claimed isomorphism
are $\ell$-adically complete, so we can check the isomorphism modulo $\ell$.
Then it boils down to the following claim: Let $\mathcal{M}$
($=\mathcal{P}^{\prime}_{n+1}/\ell\mathcal{P}^{\prime}_{n+1}$) be an
$\mathbb{F}_{\ell}$-module in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
and let $\mathcal{M}_{\mathrm{et}}$ denote its pushforward to the étale site.
Then the natural map

$\displaystyle\operatorname{Hom}(\mathcal{P}_{n},\mathcal{M}_{\mathrm{et}})=\operatorname{Hom}^{\mathrm{tr}}(\mathcal{P}_{n},\mathcal{M}_{\mathrm{et}})\mathbin{\xrightarrow{\sim}}\operatorname{Hom}^{\mathrm{tr}}(\mathcal{P}_{n},\mathcal{M})$

is an isomorphism (here the first identity follows because
$\mathcal{M}_{\mathrm{et}}$ is nuclear and hence satisfies (c) by the above).
Both sides of the claimed isomorphism preserve all small colimits (for the
functor $\mathcal{M}\mapsto\mathcal{M}_{\mathrm{et}}$ this follows from the
$\ell$-boundedness of $X$, as this implies that $\Gamma(U,-)$ is
cohomologically bounded for all basic $U\in X_{\mathrm{proet}}$), so we can
assume that $\mathcal{M}$ is compact. Then we have

$\displaystyle\operatorname{Hom}^{\mathrm{tr}}(\mathcal{P}_{n},\mathcal{M})=\varinjlim_{k}\operatorname{Hom}(\mathcal{P}_{n,k},\mathcal{M})=\varinjlim_{k}\operatorname{Hom}(\mathcal{P}_{n,k},\mathcal{M}_{\mathrm{et}})=\operatorname{Hom}(\mathcal{P}_{n},\mathcal{M}_{\mathrm{et}})$

for any representation $\mathcal{P}_{n}=\varprojlim_{k}\mathcal{P}_{n,k}$ with
all $\mathcal{P}_{n,k}$ being qcqs étale (in the first identity there is no
$\ell$-adic completion required because all terms are killed by $\ell$). This
finishes the proof that (a), (b) and (c) are equivalent.

We finish the proof of (i): It is formal that
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell})$ is
$\omega_{1}$-compactly generated with the $\omega_{1}$-compact generators
being precisely the basic nuclear objects (see [1, Theorem 8.6]). We have
already shown that all étale sheaves are nuclear, and from (c) it follows
immediately that nuclear sheaves are stable under all small colimits in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$.
To show that nuclear sheaves are also stable under tensor products, it is now
enough to see this for basic nuclear objects, where it follows from 3.6.(ii)
(to get the require right-boundedness, note that the above
$\mathcal{P}^{\prime}_{n+1}$ are clearly bounded).

We now prove (ii). It is clear that nuclear sheaves are stable under pullback.
As in the proof of Corollary 2.7, the claimed v-descent now reduces to the
following claim: Let $f\colon Y\to X$ be a map of $\ell$-bounded spatial
diamonds and let
$\mathcal{M}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
such that $f^{*}\mathcal{M}$ is nuclear; then $\mathcal{M}$ is nuclear. Let
$f_{\bullet}\colon Y_{\bullet}\to X$ be a hypercover of $\ell$-bounded spatial
diamonds extending $f$. Then all $f_{n}^{*}\mathcal{M}$ are nuclear. By
2.6.(ii) the pushforwards $f_{n{\scalebox{0.5}{$\square$}}*}$ preserve small
colimits and Banach sheaves and thus nuclear sheaves, so that we deduce that
all $f_{n{\scalebox{0.5}{$\square$}}*}f_{n}^{*}\mathcal{M}$ are nuclear.
Moreover, by v-descent for v-sheaves we have
$\mathcal{M}=\varprojlim_{n\in\Delta}f_{n{\scalebox{0.5}{$\square$}}*}f_{n}^{*}\mathcal{M}$.
Fix any compact
$\mathcal{P}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$.
Then by 3.7.(i) the functor $\operatorname{Hom}^{\mathrm{tr}}(\mathcal{P},-)$
is bounded and therefore preserves Postnikov limits and uniformly left-bounded
totalizations. We deduce

$\displaystyle\operatorname{Hom}^{\mathrm{tr}}(\mathcal{P},\mathcal{M})$
$\displaystyle=\varprojlim_{k}\operatorname{Hom}^{\mathrm{tr}}(\mathcal{P},\tau_{\leq
k}\mathcal{M})=\varprojlim_{k}\operatorname{Hom}^{\mathrm{tr}}(\mathcal{P},\varprojlim_{n\in\Delta}f_{n{\scalebox{0.5}{$\square$}}*}f_{n}^{*}\tau_{\leq
k}\mathcal{M})$
$\displaystyle=\varprojlim_{k}\varprojlim_{n\in\Delta}\operatorname{Hom}^{\mathrm{tr}}(\mathcal{P},f_{n{\scalebox{0.5}{$\square$}}*}f_{n}^{*}\tau_{\leq
k}\mathcal{M})=\varprojlim_{n\in\Delta}\varprojlim_{k}\operatorname{Hom}^{\mathrm{tr}}(\mathcal{P},f_{n{\scalebox{0.5}{$\square$}}*}f_{n}^{*}\tau_{\leq
k}\mathcal{M}),$ and since $f_{n{\scalebox{0.5}{$\square$}}*}$ has finite
cohomological dimension by 2.6.(ii) we get
$\displaystyle=\varprojlim_{n\in\Delta}\operatorname{Hom}^{\mathrm{tr}}(\mathcal{P},\varprojlim_{k}f_{n{\scalebox{0.5}{$\square$}}*}f_{n}^{*}\tau_{\leq
k}\mathcal{M})=\varprojlim_{n\in\Delta}\operatorname{Hom}^{\mathrm{tr}}(\mathcal{P},f_{n{\scalebox{0.5}{$\square$}}*}f_{n}^{*}\mathcal{M}),$
and finally by nuclearity of
$f_{n{\scalebox{0.5}{$\square$}}*}f_{n}^{*}\mathcal{M}$,
$\displaystyle=\varprojlim_{n\in\Delta}\operatorname{Hom}(\mathcal{P},f_{n{\scalebox{0.5}{$\square$}}*}f_{n}^{*}\mathcal{M})=\operatorname{Hom}(\mathcal{P},\mathcal{M}),$

proving that $\mathcal{M}$ is indeed nuclear. ∎

######  Remark 3.10.

In the case that $X=\operatorname{Spa}(C)$ is a geometric point, we recover
the classical $\infty$-category
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell})=\operatorname{\mathcal{D}}_{\mathrm{nuc}}({\mathbb{Z}}_{\ell})$
of nuclear condensed ${\mathbb{Z}}_{\ell}$-modules. In fact, for any profinite
set $X$ (viewed as a diamond over $\operatorname{Spa}(C)$) and all
$\mathcal{P},\mathcal{M}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
with $\mathcal{P}$ compact we have

$\displaystyle\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},\mathcal{M})=\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(\mathcal{P},{\mathbb{Z}}_{\ell})\otimes\mathcal{M},$

i.e. we recover the classical notion of nuclearity. Note that for non-compact
$\mathcal{P}$ the above identity is false, because by definition
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},-)$
transforms colimits in $\mathcal{P}$ into limits, whereas
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}(\mathcal{P},{\mathbb{Z}}_{\ell})\otimes-$
does not. This does not affect the notion of nuclear sheaves, because this
only every uses
$\operatorname{\underline{\operatorname{Hom}}}_{\scalebox{0.5}{$\square$}}^{\mathrm{tr}}(\mathcal{P},-)$
for compact $\mathcal{P}$ (some subtleties arise if compact objects are not
stable under tensor products, though).

While many natural constructions of sheaves preserve the nuclear category,
some (like internal hom) do not. We therefore need a way of “nuclearizing”
sheaves, which can be done as follows.

######  Definition 3.11.

Let $X$ be an $\ell$-bounded spatial diamond. By the adjoint functor theorem
the inclusion
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell})\mathrel{\hookrightarrow}\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
admits a right adjoint

$\displaystyle(-)_{\mathrm{nuc}}\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}\to\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell}),\qquad\mathcal{M}\mapsto\mathcal{M}_{\mathrm{nuc}}.$

For every
$\mathcal{M}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
we call $\mathcal{M}_{\mathrm{nuc}}$ the _nuclearization_ of $\mathcal{M}$.

######  Proposition 3.12.

Let $X$ be an $\ell$-bounded spatial diamond. Then the nuclearization functor
on $X$ is characterized by the following properties:

  1. (i)

The functor
$(-)_{\mathrm{nuc}}\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}\to\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell})$
preserves all small colimits.

  2. (ii)

If
$\mathcal{M}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
is $\ell$-adically complete, then $\mathcal{M}_{\mathrm{nuc}}$ is the
$\ell$-adic completion of $\mathcal{M}_{\mathrm{et}}$. In particular, if
$\mathcal{M}$ is killed by some power of $\ell$ then
$\mathcal{M}_{\mathrm{nuc}}=\mathcal{M}_{\mathrm{et}}$.

  3. (iii)

The functor $(-)_{\mathrm{nuc}}$ is bounded, i.e. there are integers $a\leq b$
such that for every static
$\mathcal{M}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
the sheaf $\mathcal{M}_{\mathrm{nuc}}$ lies in homological degrees $[a,b]$
(with respect to the solid $t$-structure).

###### Proof.

We construct a colimit-preserving functor

$\displaystyle
F\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}\to\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell})$

as follows: It is enough to construct it on compact objects and for compact
$\mathcal{Q}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
we define $F(\mathcal{Q})$ to be the $\ell$-adic completion of
$\mathcal{Q}_{\mathrm{et}}$. After composing $F$ with the inclusion
$\iota\colon\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell})\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
we get a natural map $\iota F\to\operatorname{id}$. It follows that there is a
natural map $F\to(-)_{\mathrm{nuc}}$. We claim that this is an isomorphism.
This can be checked on sections from basic nuclear objects, i.e. for every
$\mathcal{M}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
and every sequential colimit $\mathcal{N}=\varinjlim_{n}\mathcal{P}_{n}$ of
compact
$\mathcal{P}_{n}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
with trace-class transition maps, we need to verify that the natural map

$\displaystyle\operatorname{Hom}(\mathcal{N},F(\mathcal{M}))\mathbin{\xrightarrow{\sim}}\operatorname{Hom}(\mathcal{N},\mathcal{M}_{\mathrm{nuc}})$

is an isomorphism. Let us first compute the left-hand side. We claim that for
any
$\mathcal{L}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
the natural map

$\displaystyle\operatorname{Hom}(\mathcal{L},\iota
F(\mathcal{M}))=\operatorname{Hom}^{\mathrm{tr}}(\mathcal{L},\iota
F(\mathcal{M}))\mathbin{\xrightarrow{\sim}}\operatorname{Hom}^{\mathrm{tr}}(\mathcal{L},\mathcal{M})$

is an isomorphism. This can be checked for compact $\mathcal{L}$; then both
sides commute with colimits in $\mathcal{M}$, so we can also reduce to the
case that $\mathcal{M}$ is compact. Factoring out the $\ell$-adic completion
on both sides, we can assume that $\mathcal{M}$ is killed by $\ell$. But then
both sides evaluate to
$\varinjlim_{n}\operatorname{Hom}(\mathcal{L}_{n},\mathcal{M})$ for any
representation $\mathcal{L}=\varprojlim_{n}\mathcal{L}_{n}$ with qcqs étale
$\mathcal{L}_{n}$.

Let us get back to the claimed isomorphism of $F$ and $(-)_{\mathrm{nuc}}$, so
let $\mathcal{M}$ and $\mathcal{N}=\varinjlim_{n}\mathcal{P}_{n}$ be as
before. The claim now reduces to showing that the natural map

$\displaystyle\varprojlim_{n}\operatorname{Hom}^{\mathrm{tr}}(\mathcal{P}_{n},\mathcal{M})=\operatorname{Hom}^{\mathrm{tr}}(\mathcal{N},\mathcal{M})=\operatorname{Hom}(\mathcal{N},F(\mathcal{M}))$
$\displaystyle\qquad\qquad\mathbin{\xrightarrow{\sim}}\operatorname{Hom}(\mathcal{N},\mathcal{M}_{\mathrm{nuc}})=\operatorname{Hom}(\mathcal{N},\mathcal{M})=\varprojlim_{n}\operatorname{Hom}(\mathcal{P}_{n},\mathcal{M})$

is an isomorphism. This follows by constructing sections
$\operatorname{Hom}(\mathcal{P}_{n+1},\mathcal{M})\to\operatorname{Hom}^{\mathrm{tr}}(\mathcal{P}_{n},\mathcal{M})$,
which exist by 3.6.(i).

Having established the isomorphism $F=(-)_{\mathrm{nuc}}$, we can now easily
prove the claims (i)–(iii). Part (i) is obvious since $F$ preserves all small
colimits by construction. For part (ii) note that if $\mathcal{M}$ is killed
by $\ell$ then clearly $F(\mathcal{M})=\mathcal{M}_{\mathrm{et}}$. Since
$(-)_{\mathrm{nuc}}$ preserves limits (as it is a right adjoint functor) we
deduce that $(-)_{\mathrm{nuc}}$ preserves $\ell$-adically complete objects.
For part (iii), we can immediately reduce to the case that $\mathcal{M}$ is
static and finitely presented and hence $\ell$-adically complete. Then the
claim follows from the fact that both $\ell$-adic completion and pushforward
to the étale site are bounded. ∎

We now have a good understanding of nuclear ${\mathbb{Z}}_{\ell}$-sheaves on
$\ell$-bounded spatial diamonds. It is formal to extend this notion to all
small v-stacks and to modules over any nuclear ${\mathbb{Z}}_{\ell}$-algebra
$\Lambda$, where by “nuclear ${\mathbb{Z}}_{\ell}$-algebra” we mean an
(animated condensed) ${\mathbb{Z}}_{\ell}$-algebra whose underlying
${\mathbb{Z}}_{\ell}$-module is nuclear. Prominent examples of nuclear
${\mathbb{Z}}_{\ell}$-algebras are $\mathbb{F}_{\ell}$, ${\mathbb{Z}}_{\ell}$,
$\mathbb{Q}_{\ell}$, $\overline{\mathbb{Q}_{\ell}}$ and $\mathbb{C}_{\ell}$.

######  Definition 3.13.

For any small v-stack $X$ we define

$\displaystyle\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell})\subseteq\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{{\scalebox{0.5}{$\square$}}}\subseteq\operatorname{\mathcal{D}}(X_{\mathrm{v}},{\mathbb{Z}}_{\ell})$

by descent from the case of $\ell$-bounded spatial diamonds using Corollaries
2.7 and 3.9.(ii) (note that every strictly totally disconnected space is an
$\ell$-bounded spatial diamond, so that $\ell$-bounded spatial diamonds form a
basis for the v-site). In other words, a ${\mathbb{Z}}_{\ell}$-sheaf
$\mathcal{M}\in\operatorname{\mathcal{D}}(X_{\mathrm{v}},{\mathbb{Z}}_{\ell})$
lies in $\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell})$
resp.
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
if and only if this is true after pullback to every $\ell$-bounded spatial
diamond.

######  Definition 3.14.

Let $\Lambda$ be a nuclear ${\mathbb{Z}}_{\ell}$-algebra.

  1. (a)

By pullback $\Lambda$ defines a v-sheaf of connective
$\mathbb{E}_{\infty}$-rings on the big v-site of all small v-stacks, which we
still denote $\Lambda$. In particular, for every small v-stack $X$, we get a
stable $\infty$-category $\operatorname{\mathcal{D}}(X_{\mathrm{v}},\Lambda)$
of $\Lambda$-modules on $X$. This $\infty$-category admits a complete
$t$-structure and the forgetful functor
$\operatorname{\mathcal{D}}(X_{\mathrm{v}},\Lambda)\to\operatorname{\mathcal{D}}(X_{\mathrm{v}},{\mathbb{Z}}_{\ell})$
is $t$-exact, conservative and preserves all small limits and colimits.
Moreover, for any map of small v-stacks, the pullback and pushforward functors
on ${\mathbb{Z}}_{\ell}$-modules can be enhanced to functors on
$\Lambda$-modules.

  2. (b)

Let $X$ be a small v-stack. We define

$\displaystyle\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)\subseteq\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}\subseteq\operatorname{\mathcal{D}}(X_{\mathrm{v}},\Lambda)$

to be the full subcategories spanned by those $\Lambda$-modules which lie in
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell})$ resp.
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
after applying the forgetful functor. Equivalently we can view $\Lambda$ as an
$\mathbb{E}_{\infty}$-algebra in
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell})$ and define
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$ as the
$\infty$-category of $\Lambda$-modules in
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell})$. The
objects of $\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$ are called
the _nuclear $\Lambda$-modules on $X$_ and the objects of
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
are called the _$\omega_{1}$ -solid $\Lambda$-modules on $X$_.

We can formally extend most of the results on nuclear and $\omega_{1}$-solid
${\mathbb{Z}}_{\ell}$-modules on $\ell$-bounded spatial diamonds to nuclear
and $\omega_{1}$-solid $\Lambda$-modules on small v-stacks:

######  Proposition 3.15.

Let $\Lambda$ be a nuclear ${\mathbb{Z}}_{\ell}$-algebra.

  1. (i)

The assignment

$\displaystyle
X\mapsto\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$

defines a hypercomplete sheaf of presentable symmetric monoidal stable
$\infty$-categories with complete $t$-structure on the v-site of all small
v-stacks.

  2. (ii)

Let $X$ be a small v-stack. The inclusion
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}\mathrel{\hookrightarrow}\operatorname{\mathcal{D}}(X_{\mathrm{v}},\Lambda)$
is $t$-exact and commutes with all pullbacks, colimits and countable limits.

  3. (iii)

Let $X$ be an $\ell$-bounded spatial diamond. Then
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
is compactly generated. The compact objects are generated under finite
(co)limits and retracts by the objects
$\mathcal{P}\otimes_{{\mathbb{Z}}_{\ell}}\Lambda$ for compact
$\mathcal{P}\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$.

###### Proof.

Part (i) is clear by definition for $\Lambda={\mathbb{Z}}_{\ell}$ (for
presentability use that presentable $\infty$-categories are stable under
limits and use (iii)). For general $\Lambda$ it follows by repeating the
argument in Corollary 2.7 and noting that everything commutes with the
forgetful functor along ${\mathbb{Z}}_{\ell}\to\Lambda$. For part (ii) we can
similarly reduce to the case $\Lambda={\mathbb{Z}}_{\ell}$. Then the
commutation with pullbacks is true by design and the commutation with colimits
and countable limits follows from 2.5.(iii) because everything can be reduced
to $\ell$-bounded spatial diamonds (using that pullbacks preserve limits and
colimits). Part (iii) follows easily from 2.5.(ii). ∎

######  Proposition 3.16.

Let $\Lambda$ be a nuclear ${\mathbb{Z}}_{\ell}$-algebra.

  1. (i)

The assignment

$\displaystyle X\mapsto\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$

defines a hypercomplete sheaf of presentable symmetric monoidal
$\infty$-categories on the v-site of all small v-stacks.

  2. (ii)

Let $X$ be a small v-stack. The inclusion
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\mathrm{nuc}}\mathrel{\hookrightarrow}\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$
is symmetric monoidal and commutes with all pullbacks and colimits.

  3. (iii)

Let $X$ be an $\ell$-bounded spatial diamond. Then
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$ is
$\omega_{1}$-compactly generated. The $\omega_{1}$-compact objects are
generated under countable colimits by the objects
$\mathcal{N}\otimes_{{\mathbb{Z}}_{\ell}}\Lambda$ for basic nuclear
$\mathcal{N}\in\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell})$.

###### Proof.

We can argue as in Proposition 3.15 to reduce everything to Theorem 3.9. ∎

######  Proposition 3.17.

Let $\Lambda\to\Lambda^{\prime}$ be a map of nuclear
${\mathbb{Z}}_{\ell}$-algebras and let $X$ be a small v-stack.

  1. (i)

There is a natural pair of adjoint functors

$\displaystyle-\otimes_{\Lambda}\Lambda^{\prime}\colon\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)\rightleftarrows\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda^{\prime})\nobreak\mskip
6.0mu plus 1.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip
2.0mu\mathrm{Forget},$

both of which commute with all pullbacks and colimits. Moreover, the functor
$-\otimes_{\Lambda}\Lambda^{\prime}$ is symmetric monoidal and the forgetful
functor is conservative and preserves all limits.

The same is true for the $\infty$-categories of $\omega_{1}$-solid sheaves
instead of nuclear sheaves, in which case the forgetful functor is
additionally $t$-exact.

  2. (ii)

Both $-\otimes_{\Lambda}\Lambda^{\prime}$ and the forgetful functor commute
with the inclusion of nuclear sheaves into $\omega_{1}$-solid sheaves.

###### Proof.

This follows immediately from the fact that
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$ is the
$\infty$-category of $\Lambda$-modules in
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,{\mathbb{Z}}_{\ell})$ (and
similarly for
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$)
and the fact that pullback functors and the inclusion of nuclear sheaves into
$\omega_{1}$-solid sheaves are symmetric monoidal. ∎

It is convenient to also define the nuclearization functor in the general
setting of nuclear $\Lambda$-modules on small v-stacks, even though on general
small v-stacks it will not have the same nice properties.

######  Definition 3.18.

Let $\Lambda$ be a nuclear ${\mathbb{Z}}_{\ell}$-algebra and $X$ a small
v-stack. We define the _nuclearization_ functor

$\displaystyle(-)_{\mathrm{nuc}}\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}\to\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$

to be the right adjoint of the inclusion.

######  Proposition 3.19.

Let $\Lambda$ be a nuclear ${\mathbb{Z}}_{\ell}$-algebra and let $X$ be a
small v-stack.

  1. (i)

If $X$ is an $\ell$-bounded spatial diamond then the functor
$(-)_{\mathrm{nuc}}\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}\to\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$
preserves all small colimits and is bounded with respect to the $t$-structure
on
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,\Lambda)_{\omega_{1}}$.

  2. (ii)

If $\Lambda\to\Lambda^{\prime}$ is a map of nuclear
${\mathbb{Z}}_{\ell}$-algebras then nuclearization on $X$ commutes with the
forgetful functor along $\Lambda\to\Lambda^{\prime}$.

###### Proof.

Part (ii) follows immediately from 3.17.(ii) by passing to right adjoints.
Part (i) follows from (ii) and Proposition 3.12 (use the forgetful functor
along ${\mathbb{Z}}_{\ell}\to\Lambda$). ∎

In the case that the nuclear ${\mathbb{Z}}_{\ell}$-algebra $\Lambda$ is
discrete (e.g. $\Lambda=\mathbb{F}_{\ell}$), we recover the classical theory
of étale $\Lambda$-modules:

######  Proposition 3.20.

Let $\Lambda$ be a discrete ${\mathbb{Z}}_{\ell}$-algebra and $X$ a small
v-stack. Then we have

$\displaystyle\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)=\operatorname{\mathcal{D}}_{\mathrm{et}}(X,\Lambda)$

as full subcategories of $\operatorname{\mathcal{D}}(X_{\mathrm{v}},\Lambda)$.

###### Proof.

Since both $\infty$-categories are defined by descent, we can assume that $X$
is an $\ell$-bounded spatial diamond. By 3.9.(i) every étale $\Lambda$-module
is nuclear (for any nuclear ${\mathbb{Z}}_{\ell}$-algebra $\Lambda$), so we
have
$\operatorname{\mathcal{D}}_{\mathrm{et}}(X,\Lambda)\subseteq\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$.
To get the other inclusion, note that the discreteness of $\Lambda$ implies
that
$\operatorname{Hom}({\mathbb{Z}}_{\ell},\Lambda)=\varinjlim_{n}\operatorname{Hom}({\mathbb{Z}}/\ell^{n}{\mathbb{Z}},\Lambda)$
(a priori this holds in the $\infty$-category of
${\mathbb{Z}}_{\ell}$-modules, but then also follows in the $\infty$-category
of rings). Applying this to the structure map ${\mathbb{Z}}_{\ell}\to\Lambda$
we deduce that $\Lambda$ is a ${\mathbb{Z}}/\ell^{n}{\mathbb{Z}}$-algebra for
some $n$. In particular every
$\mathcal{M}\in\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$ is killed
by $\ell^{n}$, which implies that $\mathcal{M}$ is étale by 3.12.(ii). ∎

##  4 Pushforward and Base-Change

Fix a prime $\ell\neq p$ and a nuclear ${\mathbb{Z}}_{\ell}$-algebra
$\Lambda$. In the previous section we have constructed the $\infty$-category
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$ of nuclear
$\Lambda$-modules on every small v-stack $X$. We will now introduce the
pushforward of nuclear sheaves and study its behavior.

######  Definition 4.1.

Let $f\colon Y\to X$ be a map of small v-stacks.

  1. (a)

We denote the pullback functor on nuclear sheaves by

$\displaystyle
f^{*}\colon\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)\to\operatorname{\mathcal{D}}_{\mathrm{nuc}}(Y,\Lambda).$

It is symmetric monoidal, preserves all small colimits and coincides with the
v-pullback by design.

  2. (b)

We denote the _nuclear pushforward_ functor by

$\displaystyle
f_{*}\colon\operatorname{\mathcal{D}}_{\mathrm{nuc}}(Y,\Lambda)\to\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda).$

It is defined to be the right adjoint of $f^{*}$, which exists by the adjoint
functor theorem. It may occasionally be necessary to also consider pushforward
functors on $\omega_{1}$-solid sheaves and on all v-sheaves (defined to be the
right adjoints of the pullback functor on the respective $\infty$-categories),
which we will denote $f_{{\scalebox{0.5}{$\square$}}*}$ and
$f_{{\mathrm{v}}*}$, respectively.

In general we cannot expect the nuclear pushforward to coincide with the
v-pushforward because the latter will in general not preserve nuclear sheaves.
However, this usually works under a qcqs assumption, as follows.

######  Definition 4.2.

Let $X$ be a small v-stack.

  1. (a)

We denote by

$\displaystyle\operatorname{\mathcal{D}}^{b}_{\mathrm{nuc}}(X,\Lambda),\operatorname{\mathcal{D}}^{+}_{\mathrm{nuc}}(X,\Lambda),\operatorname{\mathcal{D}}^{-}_{\mathrm{nuc}}(X,\Lambda)\subseteq\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$

the full subcategories of those v-sheaves whose pullback to every
$\ell$-bounded spatial diamond is bounded, resp. left bounded, resp. right
bounded with respect to the $t$-structure on $\omega_{1}$-solid sheaves. The
elements of these subcategories are called the _locally bounded_ , resp.
_locally left-bounded_ , resp. _locally right-bounded_ nuclear
$\Lambda$-modules on $X$.

  2. (b)

Similar definitions as in (a) apply to $\omega_{1}$-solid sheaves and to
v-sheaves in place of nuclear sheaves.

######  Lemma 4.3.

For any $?\in\\{b,+,-\\}$ the assignment
$X\mapsto\operatorname{\mathcal{D}}^{?}_{\mathrm{nuc}}(X,\Lambda)$ defines a
hypercomplete sheaf of stable $\infty$-categories on the v-site of all small
v-stacks.

###### Proof.

This follows immediately from the fact that pullbacks are $t$-exact and that
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(-,\Lambda)$ is a hypercomplete
v-sheaf. ∎

######  Remark 4.4.

We warn the reader that there is no $t$-structure on
$\operatorname{\mathcal{D}}_{\mathrm{nuc}}(X,\Lambda)$ (unless $\Lambda$ is
discrete, see Proposition 3.20), but it still makes sense to speak of
(left/right) bounded nuclear sheaves.

We can now prove the following characterizations of the nuclear pushforward,
generalizing the ones in [15, Proposition 17.6].

######  Proposition 4.5.

Let $f\colon Y\to X$ be a qcqs map of small v-stacks.

  1. (i)

The v-pushforward
$f_{{\mathrm{v}}*}\colon\operatorname{\mathcal{D}}(Y_{\mathrm{v}},\Lambda)\to\operatorname{\mathcal{D}}(X_{\mathrm{v}},\Lambda)$
preserves $\omega_{1}$-solid sheaves, i.e. we have
$f_{{\mathrm{v}}*}=f_{{\scalebox{0.5}{$\square$}}*}$.

  2. (ii)

The v-pushforward preserves locally left-bounded nuclear sheaves, so that we
have $f_{{\mathrm{v}}*}=f_{*}$ on locally left-bounded sheaves.

  3. (iii)

Suppose that the functor
$f_{{\scalebox{0.5}{$\square$}}*}\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(Y,{\mathbb{Z}}_{\ell})_{\omega_{1}}\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
has finite cohomological dimension. Then (ii) also hold for unbounded sheaves.
Moreover, in this case both $f_{{\scalebox{0.5}{$\square$}}*}$ and $f_{*}$
preserve all small colimits.

###### Proof.

All statements can be checked after pullback to any v-cover, so we can assume
that $X$ is an $\ell$-bounded spatial diamond; in particular $Y$ is qcqs.
Moreover, since the forgetful functor along ${\mathbb{Z}}_{\ell}\to\Lambda$
commutes with pushforwards we can further reduce to the case that
$\Lambda={\mathbb{Z}}_{\ell}$. We fix a hypercover $g_{\bullet}\colon
Y_{\bullet}\to Y$ by $\ell$-bounded spatial diamonds, so that
$f_{{\mathrm{v}}*}$ is computed as the totalization
$f_{{\mathrm{v}}*}=\varprojlim_{n\in\Delta}f_{n{\mathrm{v}}*}g_{n}^{*}$.

Part (i) now follows from that fact that all $f_{n{\mathrm{v}}*}$ preserve
$\omega_{1}$-solid sheaves (see 2.6.(ii) and that $\omega_{1}$-solid sheaves
on $X$ are stable under countable limits (see 2.5.(iii)). Part (ii) follows by
the same argument as in the proof of 3.9.(ii) by exploiting the fact that
$\operatorname{Hom}^{\mathrm{tr}}(\mathcal{P},-)$ preserves uniformly left-
bounded totalizations. Part (iii) follows similarly by additionally taking
limits over Postnikov truncations (the fact that
$f_{{\scalebox{0.5}{$\square$}}*}$ and $f_{*}$ commute with colimits follows
easily from the facts that $f$ is qcqs and has bounded cohomological dimension
by the usual Postnikov limit argument). ∎

######  Corollary 4.6.

Let

${Y^{\prime}}$${Y}$${X^{\prime}}$${X}$$\scriptstyle{g^{\prime}}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{g}$

be a cartesian diagram of small v-stacks and assume that $f$ is qcqs.

  1. (i)

The natural morphism
$g^{*}f_{{\scalebox{0.5}{$\square$}}*}\mathbin{\xrightarrow{\sim}}f^{\prime}_{{\scalebox{0.5}{$\square$}}*}g^{\prime*}$
is an isomorphism of functors
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(Y,\Lambda)_{\omega_{1}}\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X^{\prime},\Lambda)_{\omega_{1}}$.

  2. (ii)

The natural morphism
$g^{*}f_{*}\mathbin{\xrightarrow{\sim}}f^{\prime}_{*}g^{\prime*}$ is an
isomorphism of functors
$\operatorname{\mathcal{D}}^{+}_{\mathrm{nuc}}(Y,\Lambda)\to\operatorname{\mathcal{D}}^{+}_{\mathrm{nuc}}(X^{\prime},\Lambda)$.

  3. (iii)

Suppose that the functor
$f_{{\scalebox{0.5}{$\square$}}*}\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(Y,{\mathbb{Z}}_{\ell})_{\omega_{1}}\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(X,{\mathbb{Z}}_{\ell})_{\omega_{1}}$
has finite cohomological dimension. Then (ii) also holds for unbounded
sheaves.

###### Proof.

All claims follows immediately from Proposition 4.5 by using the fact that
v-pushforward satisfies arbitrary base-change (for (iii) note additionally
that the functor $f^{\prime}_{{\mathrm{v}}*}g^{\prime*}$ preserves nuclear
sheaves because so does $g^{*}f_{{\mathrm{v}}*}$). ∎

The condition that $f_{{\scalebox{0.5}{$\square$}}*}$ has finite cohomological
dimension may look a bit hard to grasp at first, but it turns out that this
condition is usually satisfied in practice:

######  Proposition 4.7.

Let $f\colon Y\to X$ be a map of small v-stacks which is locally
compactifiable, representable in spatial diamonds and has locally finite
$\mathrm{dim.trg}$. Then