#  $p$-adic sheaves on classifying stacks, and the $p$-adic Jacquet-Langlands
correspondence

David Hansen and Lucas Mann

###### Abstract.

We establish several new properties of the $p$-adic Jacquet-Langlands functor
defined by Scholze in terms of the cohomology of the Lubin-Tate tower. In
particular, we reprove Scholze’s basic finiteness theorems, prove a duality
theorem, and show a kind of partial Künneth formula. Using these results, we
deduce bounds on Gelfand-Kirillov dimension, together with some new vanishing
and nonvanishing results.

Our key new tool is the six functor formalism with solid almost
$\mathcal{O}^{+}/p$-coefficients developed recently by the second author
[Man22]. One major point of this paper is to extend the domain of validity of
the $!$-functor formalism developed in [Man22] to allow certain “stacky” maps.
In the language of this extended formalism, we show that if $G$ is a $p$-adic
Lie group, the structure map of the classifying small v-stack $B\underline{G}$
is $p$-cohomologically smooth.

###### Contents

  1. 1 Introduction
    1. 1.1 Prologue
    2. 1.2 Context and duality
    3. 1.3 The stacky 6-functor formalism
    4. 1.4 Applications to representation theory
    5. 1.5 A word on notation
    6. 1.6 Acknowledgments
  2. 2 Geometric preparations
    1. 2.1 Recollections on the solid six functors
    2. 2.2 Six functors for stacky morphisms
    3. 2.3 Cohomologically smooth maps
  3. 3 Representation-theoretic preparations
    1. 3.1 Smooth representations
    2. 3.2 Sheaves on classifying stacks
    3. 3.3 Admissible representations
  4. 4 Duality and applications
    1. 4.1 The main result
    2. 4.2 Applications to Gelfand-Kirillov dimension
    3. 4.3 The case of $\mathrm{GL}_{2}$
      1. 4.3.1 A non-generic case
  5. 5 A partial Künneth formula

##  1\. Introduction

###  1.1. Prologue

This paper has two goals. Our immediate goal is to prove some new properties
of the $p$-adic Jacquet-Langlands functor defined by Scholze. Unlike the
classical Jacquet-Langlands correspondence, which is defined by a simple
character identity, Scholze’s functor is defined inexplicitly, and has turned
out to be rather difficult to study. Scholze himself proved a basic finiteness
theorem, and established a local-global compatibility result in some cases. We
reprove Scholze’s finiteness theorem, and prove a duality theorem and a kind
of “partial” Künneth formula. As consequences, we deduce bounds on Gelfand-
Kirillov dimension and prove some new vanishing and nonvanishing results for
Scholze’s functor.

To explain our second goal, let us briefly recall some recent history. The
study of the classical local Langlands correspondence has recently been given
new life through the geometrization program proposed by Fargues [Far16]. This
program has already seen spectacular success, most notably in the work of
Fargues-Scholze [FS21]. A key technical ingredient in this program is a
sufficiently flexible six-functor formalism for $\ell$-adic étale cohomology
of diamonds and v-stacks. This formalism was developed by Scholze [Sch18], and
then extended further in [GHW22].

Our second goal is to lay some technical foundations for the geometrization of
the $p$-adic Langlands correspondence. In particular, we extend the $p$-adic
six functor formalism developed in [Man22] to allow certain stacky maps, in
parallel with the extension provided in [GHW22], and show that classifying
stacks of $p$-adic Lie groups fit perfectly into this formalism. This stacky
six-functor machinery allows for an elegant formulation of the proof of our
main results. With it in hand, one can also begin a meaningful study of
$p$-adic sheaves on $\mathrm{Bun}_{G}$. We will pursue this in a subsequent
paper.

###  1.2. Context and duality

Let us begin by recalling the classical Jacquet-Langlands correspondence. Fix
a finite extension $F/\mathbf{Q}_{p}$ and an integer $n>1$. Set
$G=\mathrm{GL}_{n}(F)$, and let $D/F$ be the central division algebra of
invariant $1/n$.

######  Theorem 1.1 (Jacquet-Langlands, Deligne-Kazhdan-Vignéras, Rogawski).

There is a natural bijection

$\mathrm{JL}:\mathrm{Irr}_{L^{2}}(G)\overset{\sim}{\to}\mathrm{Irr}(D^{\times})$

from the irreducible essentially square-integrable complex representations of
$G$ to the irreducible representations of $D^{\times}$, characterized by the
requirement that $\chi_{\mathrm{JL}(\pi)}(g^{\prime})=(-1)^{n-1}\chi_{\pi}(g)$
for all pairs of matching elliptic elements $g\in G$ and $g^{\prime}\in
D^{\times}$.

This bijection is compatible with twisting and passing to contragredient
representations.

A very similar statement holds with
$\overline{\mathbf{F}_{\ell}}$-coefficients instead of complex coefficients,
cf. work of Dat and Vignéras.

In recent years, a $p$-adic version of the Langlands program has emerged, many
aspects of which are still conjectural. In the context of this program, a
candidate for a $p$-adic Jacquet-Langlands functor has been proposed by
Scholze [Sch18a]. Let us begin by recalling the definition of Scholze’s
functor. Set $C=\widehat{\overline{F}}$, and let $\mathcal{M}_{\infty}$ be the
infinite level Lubin-Tate space over $\mathrm{Spa}\,C$. Recall that this is a
perfectoid space with commuting continuous actions of $G$ and $D^{\times}$,
and with a natural surjective period map
$\pi_{\mathrm{GH}}:\mathcal{M}_{\infty}\to\mathbf{P}_{C}^{n-1}$. The period
map $\pi_{\mathrm{GH}}$ is a $G$-torsor, and is $D^{\times}$-equivariant for a
suitable action of $D^{\times}$ on projective space.

Using this geometry, Scholze defines an exact functor, denoted
$\pi\mapsto\mathcal{F}_{\pi}$, from the category
$\mathrm{Rep}_{\mathbf{F}_{p}}^{\mathrm{sm}}(G)$ of smooth $G$-representations
towards the category of étale $\mathbf{F}_{p}$-sheaves on
$\mathbf{P}_{C}^{n-1}$. This functor is defined by a simple and explicit
recipe: given any étale map $U\to\mathbf{P}_{C}^{n-1}$, one defines
$\mathcal{F}_{\pi}(U)=\mathcal{C}_{G}(|U\times_{\mathbf{P}_{C}^{n-1}}\mathcal{M}_{\infty}|,\pi)$.
Since $\pi_{\mathrm{GH}}$ is $D^{\times}$-equivariant, the sheaf
$\mathcal{F}_{\pi}$ is also $D^{\times}$-equivariant, and the étale cohomology
groups

$\mathcal{J}^{i}(\pi):=H^{i}(\mathbf{P}_{C}^{n-1},\mathcal{F}_{\pi})$

are naturally smooth $D^{\times}$-representations which also carry a natural
action of the Weil group $W_{F}$. By general vanishing theorems, it is easy to
see that $\mathcal{J}^{i}(\pi)=0$ for all $i>2n-2$, and it is also easy to
compute $\mathcal{J}^{0}(\pi)$ explicitly. However, these are the only obvious
properties of $\mathcal{J}^{i}.$ One of the main results of [Sch18a] is a
highly nonobvious finiteness theorem: if $\pi$ is an _admissible_
$G$-representation, then also each $\mathcal{J}^{i}(\pi)$ is an admissible
$D^{\times}$-representation. As a key step in the proof of this finiteness
result, Scholze also proved a form of the primitive comparison theorem: if
$\pi$ is admissible, the natural map

$H^{i}(\mathbf{P}_{C}^{n-1},\mathcal{F}_{\pi})\otimes\mathcal{O}_{C}/p\to
H^{i}(\mathbf{P}_{C}^{n-1},\mathcal{F}_{\pi}\otimes\mathcal{O}^{+}/p)$

is an almost isomorphism.

Consideration of simple examples shows that in general, $\mathcal{J}^{i}(\pi)$
can be nonzero for multiple values of $i$.111This already happens for the
simplest example $\pi=\mathbf{1}$, in which case
$\mathcal{F}_{\pi}=\mathbf{F}_{p}$ and
$\mathcal{J}^{i}(\pi)=H^{i}(\mathbf{P}_{C}^{n-1},\mathbf{F}_{p})$ is the mod
$p$ étale cohomology of projective space, which by the usual computation is
nonzero in degrees $0,2,4,\dots,2n-2$. As such, we prefer to package the
individual cohomologies $\mathcal{J}^{i}$ into a single operation on the
derived category. More precisely, by the exactness of the association
$\pi\mapsto\mathcal{F}_{\pi}$, it naturally upgrades to a functor
$\operatorname{\mathcal{D}}(G):=\operatorname{\mathcal{D}}(\mathrm{Rep}_{\mathbf{F}_{p}}^{\mathrm{sm}}(G))\to\operatorname{\mathcal{D}}(\mathbf{P}^{n-1}_{C,\mathrm{\acute{e}t}},\mathbf{F}_{p})$,
and we write $\mathcal{J}$ for the composite functor

$\operatorname{\mathcal{D}}(G)\overset{V\mapsto\mathcal{F}_{V}}{\longrightarrow}\operatorname{\mathcal{D}}(\mathbf{P}^{n-1}_{C,\mathrm{\acute{e}t}},\mathbf{F}_{p})\overset{R\Gamma}{\to}\operatorname{\mathcal{D}}(D^{\times}).$

By the finiteness theorems mentioned above, the functor $\mathcal{J}$
preserves bounded complexes with admissible cohomology.

Now, one would guess that $\mathcal{J}$ intertwines some natural duality
operation on $G$-representations and $D^{\times}$-representations. However, in
stark contrast with the case of complex or $\ell$-adic representations, the
naive functor of passing to smooth duals on mod-$p$ representations of
$p$-adic Lie groups is essentially useless. To remedy this, Kohlhaase has
introduced a family of “derived” smooth duality functors $\mathcal{S}^{i}_{H}$
on the smooth mod-$p$ representations of any $p$-adic Lie group $H$, which can
be packaged into a single endofunctor
$\mathcal{S}_{H}:\operatorname{\mathcal{D}}(H)\to\operatorname{\mathcal{D}}(H)$.
This functor has excellent properties, which we review in §3.1 below.

Our first main result can now be stated as follows.

######  Theorem 1.2.

There is a natural equivalence
$\mathcal{S}_{D^{\times}}\circ\mathcal{J}\cong(\mathcal{J}\circ\mathcal{S}_{G})[2n-2](n-1)$
of functors
$\operatorname{\mathcal{D}}_{\mathrm{adm}}(G)\to\operatorname{\mathcal{D}}_{\mathrm{adm}}(D^{\times})$.

In other words, Scholze’s functor $\mathcal{J}$ intertwines Kohlhaase’s
derived duality on $G$-representations and $D^{\times}$-representations, up to
an explicit shift and twist. This result was conjectured by the first author,
after observing that for certain $V$’s arising from the completed cohomology
of Shimura varieties, both $\mathcal{S}_{D^{\times}}\circ\mathcal{J}(V)$ and
$(\mathcal{J}\circ\mathcal{S}_{G})(V)[2n-2](n-1)$ can be computed by hand, and
turn out to be abstractly isomorphic.

In the remainder of this section, we give a detailed sketch of the proof of
Theorem 1.2. The first key ingredient is a more conceptual interpretation of
Kohlhaase’s derived duality functor.

######  Proposition 1.3.

For any $p$-adic Lie group $G$, the functor
$\mathcal{S}_{G}(-):\operatorname{\mathcal{D}}(G)\to\operatorname{\mathcal{D}}(G)$
is naturally isomorphic to the functor
$\operatorname{\underline{\operatorname{Hom}}}_{G}(-,\mathbf{1})$. Here
$\operatorname{\underline{\operatorname{Hom}}}_{G}$ denotes the natural
internal Hom in $\operatorname{\mathcal{D}}(G)$.

Be aware that in this paper, $\operatorname{\underline{\operatorname{Hom}}}$
will always denote the _derived_ internal hom (which is often denoted by
$R\operatorname{\underline{\operatorname{Hom}}}$ in the literature) – see
Section 1.5 for more remarks on that choice. In the case of mod-$\ell$
coefficients with $\ell\neq p$,
$\operatorname{\underline{\operatorname{Hom}}}_{G}(-,\mathbf{1})$ is actually
$t$-exact and therefore restricts to a functor on the hearts, where it
coincides with the usual functor of smooth duality (3.5). In this light,
Proposition 1.3 puts Kohlhaase’s duality on the same footing as classical
smooth duality. We also note that Proposition 1.3 was independently discovered
by Schneider-Sorensen [SS22].

Next, we invoke a slightly different formula for the functor $\mathcal{J}$.
For this, we briefly return to the geometry of the Lubin-Tate tower. Along
with the surjective period map
$\pi_{\mathrm{GH}}:\mathcal{M}_{\infty}\to\mathbf{P}_{C}^{n-1}$ mentioned
earlier, there is a second period map
$\pi_{\mathrm{HT}}:\mathcal{M}_{\infty}\to\Omega_{C}^{n-1}$, presenting
$\mathcal{M}_{\infty}$ as a $D^{\times}$-torsor over Drinfeld space. Both
period maps are $G\times D^{\times}$ -equivariant, with $G$ (resp.
$D^{\times}$) acting trivially on $\mathbf{P}_{C}^{n-1}$ (resp. on
$\Omega_{C}^{n-1}$). In particular, we can pass to the stack quotient $X$ of
$\mathcal{M}_{\infty}$ by the whole $G\times D^{\times}$-action, obtaining a
small v-stack $X$ which admits two natural presentations

$X\cong[\mathbf{P}_{C}^{n-1}/\underline{D^{\times}}]\cong[\Omega_{C}^{n-1}/\underline{G}].$

We think of this stack as sitting in a diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{B\underline{G}=[\mathrm{Spa}\,C/\underline{G}]}$$\textstyle{B\underline{D^{\times}}=[\mathrm{Spa}\,C/\underline{D^{\times}}]}$

where $f$ is smooth, and $g$ is smooth and proper, with both maps being of
pure relative dimension $n-1$. Let us call this the fundamental diagram. Now,
for any small v-stack $X$, the formalism in [Sch18] defines an associated
category $\mathcal{D}_{{\mathrm{\acute{e}t}}}(X,\mathbf{F}_{p})$ of étale
$\mathbf{F}_{p}$-sheaves. We will see below (Proposition 3.9) that there is a
natural identification
$\operatorname{\mathcal{D}}(H)=\mathcal{D}_{{\mathrm{\acute{e}t}}}(B\underline{H},\mathbf{F}_{p})$
of symmetric monoidal categories for any $p$-adic Lie group $H$. Under this
identification, it is not hard to show that $\mathcal{J}$ is simply given by
the operation $g_{{\mathrm{\acute{e}t}}\ast}f_{{\mathrm{\acute{e}t}}}^{\ast}$
(where by $g_{{\mathrm{\acute{e}t}}\ast}$ we denote the _derived_ pushforward,
often denoted $Rg_{{\mathrm{\acute{e}t}}\ast}$ – cf. Section 1.5). At this
point, it is heuristically clear how the argument should go: $f$ is smooth and
$g$ is proper smooth, so the operation
$g_{{\mathrm{\acute{e}t}}\ast}f_{{\mathrm{\acute{e}t}}}^{\ast}$ should commute
with internal hom towards the monoidal unit, up to an explicit shift and
twist. However, it is entirely unclear how to make this heuristic rigorous,
because the $!$-functors associated with the maps $f$ and $g$ are
conspicuously absent from the
$\mathcal{D}_{{\mathrm{\acute{e}t}}}$-formalism.222Of course, this absence is
a special feature of the $\ell=p$ setting of the present paper; one of the
main points of [Sch18] is to construct the $!$-functors in the
$\mathcal{D}_{{\mathrm{\acute{e}t}}}$-formalism when the coefficient ring is
killed by an integer prime to $p$.

The key new tool which lets us circumvent this issue is the formalism of
_solid almost $\mathcal{O}^{+}/p$-sheaves_ recently developed by the second
author [Man22]. This is a theory which attaches to any small v-stack $X$ over
$\operatorname{Spa}C$ a symmetric monoidal stable $\infty$-category
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X}/p)$,
whose objects are (to first approximation) complexes of solid quasicoherent
almost $\mathcal{O}^{+}_{X}/p$-sheaves on the étale site of $X$. There is also
a variant
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X}/p)^{\varphi}$
where the objects are equipped with a suitable $\varphi$-module structure,
which comes with an evident forgetful functor
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X}/p)^{\varphi}\to\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X}/p)$.
Crucially, the sheaf theories
$X\mapsto\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X}/p)$
and
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X}/p)^{\varphi}$
underlie _full_ six functor formalisms, and all six functors are compatible
with the operation of forgetting the $\varphi$-module structure. In
particular, for any map of small v-stacks $f:X\to Y$ over
$\operatorname{Spa}C$, we have functors $f_{\ast}$ and $f^{\ast}$, and if $f$
is a so-called bdcs333“$p$-bounded, locally compactifiable, and representable
in locally spatial diamonds” map we also have functors $f_{!}$ and $f^{!}$
with all expected properties. Moreover, there is a notion of
$p$-cohomologically smooth maps, for which $f^{!}$ is an invertible twist of
$f^{\ast}$. Finally, if $H$ is a $p$-adic Lie group, there is a natural
identification
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{B\underline{H}}/p)=\operatorname{\mathcal{D}}^{{\mathrm{sm}},a}_{\scalebox{0.5}{$\square$}}(H,\mathcal{O}_{C}/p)$,
where the right-hand side denotes the derived $\infty$-category of smooth
$H$-representations on complexes of solid almost $\mathcal{O}_{C}/p$-modules
(and similarly with $\varphi$-structures).

Returning to the fundamental diagram, we can define a solid variant
$\mathcal{J}_{{\scalebox{0.5}{$\square$}}}$ of the functor $\mathcal{J}$ by
the formula

$\mathcal{J}_{{\scalebox{0.5}{$\square$}}}=g_{\ast}f^{\ast}:\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{B\underline{G}}/p)^{\varphi}\to\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{B\underline{D^{\times}}}/p)^{\varphi}.$

Using the formalism developed in [Man22], it is entirely straightforward to
prove a solid variant of our goal, i.e. an isomorphism
$\mathcal{S}_{D^{\times},{\scalebox{0.5}{$\square$}}}\circ\mathcal{J}_{\scalebox{0.5}{$\square$}}\cong(\mathcal{J}_{\scalebox{0.5}{$\square$}}\circ\mathcal{S}_{G,{\scalebox{0.5}{$\square$}}})[2n-2](n-1)$,
where (for any $p$-adic Lie group $H$)
$\mathcal{S}_{H,{\scalebox{0.5}{$\square$}}}$ denotes the internal hom towards
the monoidal unit in
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{B\underline{H}}/p)^{\varphi}$.
However, we have potentially lost sight of our goal of understanding duality
for the functor $\mathcal{J}$.

At this point we use another key feature of the formalism developed in
[Man22], namely the _$p$ -torsion Riemann-Hilbert correspondence_, which gives
a fully faithful embedding
$-\otimes\mathcal{O}^{+a}_{X}/p:\operatorname{\mathcal{D}}_{{\mathrm{\acute{e}t}}}(X,\mathbf{F}_{p})^{\dagger}\mathrel{\hookrightarrow}\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X}/p)^{\varphi}$
for any small v-stack $X$. Here the superscripted $\dagger$ denotes the
restriction to the full subcategory of overconvergent sheaves. This embedding
comes with a right adjoint $M\to M^{\varphi}$ which is (morally) the functor
of $\varphi$-invariants. Using the basic properties of these functors, it is
formal to check that $\mathcal{J}$ can be recovered from
$\mathcal{J}_{{\scalebox{0.5}{$\square$}}}$, in the sense that
$\mathcal{J}(A)\cong\mathcal{J}_{{\scalebox{0.5}{$\square$}}}(A\otimes\mathcal{O}^{+a}_{B\underline{G}}/p)^{\varphi}$
for any
$A\in\operatorname{\mathcal{D}}(G)=\mathcal{D}_{{\mathrm{\acute{e}t}}}(B\underline{G},\mathbf{F}_{p})^{\dagger}$.
We now apply a key new result, Proposition 3.26, which gives a practical
criterion for an object
$M\in\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{B\underline{H}}/p)^{\varphi}$
to arise from a bounded admissible complex in $\operatorname{\mathcal{D}}(H)$
via the Riemann-Hilbert embedding, and shows that in this case the solid dual
$\mathcal{S}_{H,{\scalebox{0.5}{$\square$}}}$ is the Riemann-Hilbert embedding
of the smooth dual $\mathcal{S}_{H}$. It turns out that in order to apply this
criterion to $\mathcal{J}_{{\scalebox{0.5}{$\square$}}}$, we only need to know
two things:

  1. (1)

$\mathcal{J}_{{\scalebox{0.5}{$\square$}}}$ preserves discrete objects, and

  2. (2)

for any admissible $A\in\operatorname{\mathcal{D}}(G)$,
$\mathcal{J}_{{\scalebox{0.5}{$\square$}}}(A\otimes\mathcal{O}^{+a}_{B\underline{G}}/p)$
is isomorphic to its solid double dual.

We now conclude by noting that (1) is formal, and (2) is a consequence of the
solid duality theorem. For the details of this argument, see Theorem 4.3 and
its proof.

Let us emphasize that in the course of running this argument, we will reprove
Scholze’s finiteness and primitive comparison theorems for $\mathcal{J}$. Just
as in Scholze’s proof, the key point here is the properness of
$\mathbf{P}^{n-1}_{C}$, or equivalently the properness of the map $g$.
However, Scholze exploits this properness in a very indirect way, by a Cartan-
Serre style argument with shrinking covers and an elaborate use of spectral
sequences. For us, the properness of $g$ is used entirely through the
identification $g_{!}=g_{\ast}$.

###  1.3. The stacky 6-functor formalism

In the process of proving our main results we also extend the 6-functor
formalism from [Man22] to “stacky” maps and study the behavior of this
extended 6-functor formalism in the setting of classifying stacks. More
concretely, we say that a map $f\colon Y\to X$ of small v-stacks is _$p$
-fine_ if it admits a cover $g\colon Z\to Y$ such that $g$ admits universal
$p$-codescent and $f\mathbin{\circ}g$ is bdcs (see 2.4). Using the results on
abstract 6-functor formalisms in [Man22], it is formal that the 6-functor
formalism extends to $p$-fine maps, i.e. to any $p$-fine map $f$ we get
associated functors $f_{!}$ and $f^{!}$ with all the expected properties. The
typical example of a map satisfying universal $p$-codescent is a
$p$-cohomologically smooth cover, but it is important to not restrict to
smooth covers in the definition of $p$-fine maps.

Now, if $G$ is a profinite group which has finite $p$-cohomological dimension,
we prove that the structure map $f\colon*/G\to*$ is $p$-fine and that
$f_{!}=f_{*}$ (see 3.12). The key point here is to show that the obvious cover
$*\to*/G$, which is not $p$-cohomologically smooth, nevertheless admits
universal $p$-codescent (see Lemma 3.11); the argument for this crucially uses
Mathew’s notion of descendability. This result easily implies that for all
_locally_ profinite groups $G$ which _locally_ have finite $p$-cohomological
dimension, the map $*/G\to*$ is $p$-fine. Moreover, we provide a precise
criterion on $G$ determining whether or not $*/G\to*$ is $p$-cohomologically
smooth (see 3.16). If this is the case then we say that $G$ is _virtually
$p$-Poincaré_. Using classical results of Lazard [Laz65], we show that every
$p$-adic Lie group $G$ is virtually $p$-Poincaré (see 3.18), so that for these
groups the associated map $*/G\to*$ is $p$-cohomologically smooth. This
applies to groups like $\operatorname{GL}_{n}(\mathbf{Q}_{p})$ and
$D^{\times}$. With these results in hand, we deduce a $p$-adic version of the
$\ell$-adic smoothness criterion [Sch18, Proposition 24.2]:

######  Proposition 1.4 (see 3.17).

Let $G$ be a locally profinite group which is virtually $p$-Poincaré (e.g. $G$
could be any $p$-adic Lie group). Suppose $G$ acts on a small v-stack $Y$ and
$f\colon Y\to X$ is a $G$-equivariant separated bdcs map of small v-stacks,
where $G$ acts trivially on $X$. Then the induced map $Y/G\to X$ is $p$-fine,
and it is $p$-cohomologically smooth as soon as $f$ is $p$-cohomologically
smooth.

Note that in the $\ell$-adic case, one proves [Sch18, Proposition 24.2]
directly, and then deduces that classifying stacks of $p$-adic Lie groups are
$\ell$-cohomologically smooth. Such a direct argument seems very unlikely in
the $p$-adic setting, due to the nonexistence of Haar measures.

The above results on the stacky 6-functor formalism and classifying stacks are
useful for proving the crucial 3.26. However, all we really need in that proof
is the more classical statement that smooth group cohomology of $p$-adic Lie
groups satisfies Poincaré duality (as proved by Lazard). The real power of the
stacky 6-functor formalism will reveal itself when we apply it to
$\mathrm{Bun}_{G}$ in subsequent papers.

###  1.4. Applications to representation theory

One of our main motivations for proving the duality theorem is that
Kohlhaase’s duality functors are intimately related to the _Gelfand-Kirillov
dimension_ $\dim_{G}\pi$ of an admissible representation $\pi$. Recall that
this is a nonnegative integer measuring the asymptotic size of $\pi$; for
instance, $\dim_{G}\pi=0$ exactly when $\dim_{\mathbf{F}_{p}}\pi<\infty$.

Computing the dimension of admissible representations arising in nature is
generally quite difficult. However, our duality theorem gives a powerful tool
for understanding how the functor $\mathcal{J}$ interacts with Gelfand-
Kirillov dimension. Our first application in this direction is the following
result, which gives a quantitative form of Scholze’s finiteness theorem.

######  Theorem 1.5.

Let $\pi$ be an admissible smooth $G=\mathrm{GL}_{n}(F)$-representation, and
let $N_{\pi}\geq 0$ be the least integer such that $\mathcal{J}^{i}(\pi)=0$
for all $i>N_{\pi}$. Then

$\dim_{D^{\times}}\mathcal{J}^{i}(\pi)\leq\dim_{G}(\pi)+N_{\pi}$

for all $i$. In particular, since $N_{\pi}\leq 2n-2$, we have

$\dim_{D^{\times}}\mathcal{J}^{i}(\pi)\leq\dim_{G}(\pi)+2n-2$

for all $\pi$ and all $i$.

Note that the possible increase of dimension by $2n-2$ is quite small compared
to the maximal Gelfand-Kirillov dimension in this setting, which is $\dim
G=\dim D^{\times}=n^{2}[F:\mathbf{Q}_{p}]$. Nevertheless, we conjecture that
Theorem 1.5 is not optimal, and that in fact one always has an inequality

$\dim_{D^{\times}}\mathcal{J}^{i}(\pi)\leq\dim_{G}(\pi),$

cf. Conjecture 4.7 below. This conjecture in turn implies that
$\mathcal{J}^{2n-2}(\pi)$ is always a finite-dimensional vector space, which
can be computed explicitly (Proposition 4.9).

Under a stronger assumption on $\pi$, we can also prove a bound in the other
direction, namely that for some $i$ the association
$\pi\rightsquigarrow\mathcal{J}^{i}(\pi)$ cannot decrease the dimension by
very much. Here is a precise statement.

######  Theorem 1.6.

Let $\pi$ be an admissible smooth $G$-representation which is Cohen-Macaulay
in the sense that $\mathcal{S}_{G}^{i}(\pi)$ is nonzero for a single value of
$i$. Then either $\mathcal{J}(\pi)=0$ or

$\max_{i}\dim_{D^{\times}}\mathcal{J}^{i}(\pi)\geq\dim_{G}\pi-(2n-2).$

We note that the Cohen-Macaulay condition is often satisfied in practice, and
is preserved by parabolic induction.

The duality theorem is even more useful when combined with nontrivial
vanishing theorems for $\mathcal{J}$. To illustrate this, we now specialize to
the case $n=2$, but with $F/\mathbf{Q}_{p}$ still arbitrary. Set
$d=[F:\mathbf{Q}_{p}]$. Let $B\subset G=\mathrm{GL}_{2}(F)$ be the upper-
triangular Borel subgroup, with $T=F^{\times}\times F^{\times}\subset B$ the
diagonal maximal torus. For any smooth characters
$\chi_{1},\chi_{2}:F^{\times}\to\mathbf{F}_{p}^{\times}$, we have the usual
parabolic induction
$\pi(\chi_{1},\chi_{2})=\mathrm{Ind}_{B}^{G}(\omega\chi_{1}\otimes\chi_{2})$,
where $\omega$ is the mod-$p$ cyclotomic character. We assume that
$\chi_{1}/\chi_{2}\neq\omega^{\pm 1}$. Then $\pi(\chi_{1},\chi_{2})$ and
$\pi(\chi_{2},\chi_{1})$ are irreducible, and are exchanged by the functor
$\mathcal{S}_{G}^{d}(-)\otimes(\chi_{1}\chi_{2}\omega\circ\det)$. Moreover,
$\mathcal{S}_{G}^{i}(-)$ applied to either representation vanishes for all
$i\neq d$.

By irreducibility, it is easy to see that
$\mathcal{J}^{0}(\pi(\chi_{1},\chi_{2}))=0$, and likewise with the characters
exchanged. Much less obviously, the main theorem of [JLH21] shows also that
$\mathcal{J}^{2}(\pi(\chi_{1},\chi_{2}))=0$. With these vanishing results in
mind, we define

$\tau(\chi_{1},\chi_{2})=\mathcal{J}^{1}(\pi(\chi_{1},\chi_{2})).$

Our duality theorem then implies the following dichotomy.

######  Theorem 1.7.

Notation and assumptions as above, exactly one of the following is true.

i. $\tau(\chi_{1},\chi_{2})=\tau(\chi_{2},\chi_{1})=0.$

ii. Both $\tau(\chi_{1},\chi_{2})$ and $\tau(\chi_{2},\chi_{1})$ are nonzero,
and are exchanged by the functor
$\mathcal{S}_{D^{\times}}^{d}(-)\otimes(\chi_{1}\chi_{2}\omega\circ\mathrm{Nm})$.
In particular, they are both of dimension $d$. Moreover,
$\mathcal{S}_{D^{\times}}^{i}(-)$ applied to either representation vanishes
for all $i\neq d$.

We conjecture that in fact scenario i. in this theorem _never_ occurs. When
$F=\mathbf{Q}_{p}$ and $p>3$, this follows from work of Paškūnas [Paš22], who
showed by global methods that $\tau(\chi_{1},\chi_{2})$ and
$\tau(\chi_{2},\chi_{1})$ cannot _both_ vanish. Combining his results with the
previous theorem yields the following corollary.

######  Corollary 1.8.

Maintain the notation and assumptions above, and assume moreover that
$F=\mathbf{Q}_{p}$ and $p>3$. Then $\tau(\chi_{1},\chi_{2})$ and
$\tau(\chi_{2},\chi_{1})$ are nonzero, of dimension one, and are exchanged by
the functor
$\mathcal{S}_{D^{\times}}^{1}(-)\otimes(\chi_{1}\chi_{2}\omega\circ\mathrm{Nm})$.
Moreover, $\mathcal{S}_{D^{\times}}^{0}(-)$ applied to either representation
vanishes.

We note that the final vanishing statement here has a more concrete
interpretation: it simply means that $\tau(\chi_{1},\chi_{2})$ does not admit
any nonzero $H$-stable quotient $\sigma$ with
$\dim_{\mathbf{F}_{p}}\sigma<\infty$, for any open subgroup $H\subset
D^{\times}$ (and similarly for $\tau(\chi_{2},\chi_{1})$). Intuitively, this
means that $\tau(\chi_{1},\chi_{2})$ is “maximally non-semisimple”. It would
be very interesting to find an explicit construction of one-dimensional
admissible representations of $D^{\times}$ with this property. We also note
that a similar but more complicated theorem holds in the setting of
_reducible_ principal series representations of
$\mathrm{GL}_{2}(\mathbf{Q}_{p})$; see §4.3.1 for details.

Our last result is an intriguing alternative formula for $\mathcal{J}(\pi)$ in
the case where $\pi$ is parabolically induced. To state this result, we
momentarily change our notation slightly, to emphasize the dependence of the
Jacquet-Langlands functor on the chosen integer $n$. More precisely, we write
$\mathcal{J}_{n}$ for the Jacquet-Langlands functor from
$\operatorname{\mathcal{D}}(\mathrm{GL}_{n}(F))$ towards
$\operatorname{\mathcal{D}}(D_{1/n}^{\times})$. Choose an integer $0<d<n$, and
let $P_{n-d,d}\subset\mathrm{GL}_{n}(F)$ be the usual upper-triangular
parabolic with Levi $\mathrm{GL}_{n-d}(F)\times\mathrm{GL}_{d}(F)$. If
$\sigma_{1}\boxtimes\sigma_{2}$ is any smooth representation of the Levi, we
write
$\mathrm{Ind}_{P_{n-d,d}}^{\mathrm{GL}_{n}(F)}(\sigma_{1}\boxtimes\sigma_{2})$
for the usual parabolic induction to a smooth representation of
$\mathrm{GL}_{n}(F)$. In §5 below, we define a certain auxiliary spatial
diamond $W_{n,d}$ over $\operatorname{Spd}C$, which comes equipped with a
natural $D_{1/n}^{\times}$-action as well as natural
$D_{1/n}^{\times}$-equivariant maps $h_{1}:W_{n,d}\to
B\underline{\mathrm{GL}_{n-d}(F)}$ and $h_{2}:W_{n,d}\to
B\underline{D_{1/d}^{\times}}$. In the special case $d=1$, $W_{n,1}$ is
exactly the quotient
$\mathcal{M}_{P}=\mathcal{M}_{\infty}/\underline{P_{n-1,1}}$ studied in
[JLH21].

######  Theorem 1.9.

Notation as above, there is a natural isomorphism

$\mathcal{J}_{n}(\mathrm{Ind}_{P_{n-d,d}}^{\mathrm{GL}_{n}(F)}(\sigma_{1}\boxtimes\sigma_{2}))\cong
R\Gamma(W_{n,d},h_{1,{\mathrm{\acute{e}t}}}^{\ast}\sigma_{1}\otimes
h_{2,{\mathrm{\acute{e}t}}}^{\ast}\mathcal{J}_{d}(\sigma_{2})).$

We like to think of this as a “partial” Künneth formula. Although this theorem
is logically independent of our other main results, the key point in the proof
is still the new machinery provided by [Man22].

Just as in [JLH21], this alternative formula leads to new vanishing results.
Here we state the simplest such corollary.

######  Corollary 1.10.

Let $\pi$ be an irreducible admissible representation of $\mathrm{GL}_{3}(F)$
which is not supersingular, and whose underlying vector space is infinite-
dimensional. Then $\mathcal{J}^{4}(\pi)=0$.

###  1.5. A word on notation

This paper depends heavily on the mathematics developed in [Man22]. However,
since DH is less derived than LM, there are some small notational differences
between [Man22] and the present paper. In particular, we will write
$R\mathrm{Hom}$ for the derived external Hom in any sheaf theory, and
$R\Gamma$ for derived global sections. These are notated as $\mathrm{Hom}$ and
$\Gamma$ in [Man22]. On the other hand, we follow [Man22] in writing
$\operatorname{\underline{\operatorname{Hom}}}$ for the derived internal Hom,
$f_{\ast}$ for derived pushforward, etc. We hope this does not cause any
confusion.

###  1.6. Acknowledgments

The idea that $\mathcal{J}$ should satisfy some form of duality was suggested
to DH by Judith Ludwig during a visit to Heidelberg in January 2020. We
heartily thank Judith for this key initial suggestion. We also sincerely thank
Peter Scholze for some very helpful conversations about this material, and for
suggesting that the authors might profitably collaborate. Some of these
results were discovered during the Oberwolfach workshop “Nonarchimedean
geometry and applications” in February 2022, and we sincerely thank the
organizers for the opportunity to participate in this event. We also thank
Gabriel Dospinescu, Eugen Hellmann, Yongquan Hu, Christian Johansson, Vytautas
Paškūnas, Claus Sorensen, and Bogdan Zavyalov for some helpful conversations.

##  2\. Geometric preparations

In this section we recall the $p$-adic 6-functor formalism constructed in
[Man22] and extend it to a context where it can be applied to classifying
stacks of $p$-adic Lie groups. In particular we extend the 6-functor formalism
to incorporate shriek functors along “stacky” maps, similar to the stacky
extension of the $\ell$-adic formalism provided in [GHW22].

###  2.1. Recollections on the solid six functors

We recall the $p$-adic 6-functor formalism on small v-stacks defined in
[Man22]. Let us first introduce the correct category of geometric objects to
work with. In order to make sense of the sheaf $\mathcal{O}^{+}_{X}/\pi$ on a
small v-stack, we need to choose an “untilt” $X^{\sharp}$ of $X$ (this is
automatically chosen if $X$ comes from an adic space over $\mathbf{Z}_{p}$)
and a pseudouiformizer $\pi$ on $X^{\sharp}$. To make things more precise, we
define an _untilted small v-stack_ $X^{\sharp}$ to be a small v-stack $X$
together with a chosen map $X\to\operatorname{Spd}\mathbf{Z}_{p}$. For every
map $Y\to X$ from a perfectoid space $Y$ to $X$, the untilt $X^{\sharp}$
induces an untilt $Y^{\sharp}$ (in the sense of perfectoid spaces) of $Y$. The
untilted small v-stack $X^{\sharp}$ comes equipped with a structure sheaf
$\mathcal{O}_{X^{\sharp}}$ and an integral structure sheaf
$\mathcal{O}^{+}_{X^{\sharp}}\subset\mathcal{O}_{X^{\sharp}}$ on
$X_{\mathrm{v}}$ such that on every affinoid perfectoid $Y\in X_{\mathrm{v}}$
we have $\mathcal{O}_{X^{\sharp}}(Y)=\mathcal{O}_{Y^{\sharp}}(Y^{\sharp})$
(and similarly for $\mathcal{O}^{+}_{X^{\sharp}}$). A _pseudouniformizer_ on
$X^{\sharp}$ is a sheaf of ideals $\pi\subset\mathcal{O}^{+}_{X^{\sharp}}$
such that for every affinoid perfectoid $Y=\operatorname{Spa}(B,B^{+})\in
X_{\mathrm{v}}$ the restriction of $\pi$ to $Y$ is a principal ideal and
generated by a pseudouniformizer in $B^{\sharp+}$. We denote by
$\operatorname{vStack^{\sharp}_{\pi}}$ the category of pairs
$(X^{\sharp},\pi)$, where $X^{\sharp}$ is an untilted small v-stack and $\pi$
is a pseudouniformizer on $X^{\sharp}$. A map
$(Y^{\sharp},\pi_{Y})\to(X^{\sharp},\pi_{X})$ in
$\operatorname{vStack^{\sharp}_{\pi}}$ is a map of small v-stacks $Y\to X$
over $\operatorname{Spd}\mathbf{Z}_{p}$ such that $\pi_{X}|_{Y}=\pi_{Y}$.

The first main result of [Man22] is the following:

######  Theorem 2.1 (see [Man22, Theorem 3.1.27]).

There is a unique hypercomplete v-sheaf

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

such that for every untilted affinoid perfectoid space
$X^{\sharp}=\operatorname{Spa}(A^{\sharp},A^{\sharp+})$ with pseudouniformizer
$\pi\in A^{\sharp+}$, where $X$ is weakly of perfectly finite type over a
totally disconnected space, we have

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

Here $\mathcal{C}at_{\infty}^{\otimes}$ denotes the $\infty$-category of
symmetric monoidal $\infty$-categories and
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(A^{\sharp+}/\pi)$
denotes the derived $\infty$-category of _solid almost
$A^{\sharp+}/\pi$-modules_. The term “almost” refers to the fact that we care
about everything only up to $\mathfrak{m}_{A^{\sharp}}$-torsion. The term
“solid” was introduced by Clausen-Scholze in [Sch19] and roughly means
“complete topological”. See also [Man22, §2.2, §2.9] for the precise
definition of
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(A^{\sharp+}/\pi)$.

In the following we abstract a little further in order to simplify notation:
We denote $\operatorname{vStack_{\Lambda}}$ the category of pairs
$(X,\Lambda)$, where $X$ is a small v-stack and $\Lambda$ is a system of
integral torsion coefficients on $X$ (see [Man22, Definition 3.2.10]). The
main example for $\Lambda$ is $\Lambda=\mathcal{O}^{+}_{X^{\sharp}}/\pi$ for
an untilt $X^{\sharp}$ of $X$ and a pseudouniformizer $\pi$ on $X^{\sharp}$ –
the reader is invited to always assume this case in the following. To every
$(X,\Lambda)\in\operatorname{vStack_{\Lambda}}$ one can then define the
$\infty$-category
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)$
in the same way as in 2.1.

The second main result of [Man22] is the construction of a 6-functor formalism
for the $\infty$-categories
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X}/\pi)$.
In order to formulate it, let us recall the definition of an abstract
6-functor formalism from [Man22, §A.5]: Suppose we are given an
$\infty$-category $\mathcal{C}$ of geometric objects and a collection of edges
$E$ in $\mathcal{C}$ which contains all isomorphisms and is stable under
composition and pullback. To the pair $(\mathcal{C},E)$ one can then associate
an $\infty$-operad $\operatorname{Corr}(\mathcal{C})_{E,all}^{\otimes}$ which
roughly has the following shape. The underlying $\infty$-category
$\operatorname{Corr}(\mathcal{C})_{E,all}$ has as objects the same objects as
$\mathcal{C}$ and as morphisms the diagrams of the form
$Y\mathbin{\leftarrow}Y^{\prime}\to X$ such that $[Y^{\prime}\to X]\in E$,
where the composition of morphisms is given by the obvious pullback diagrams.
The operadic structure on $\operatorname{Corr}(\mathcal{C})_{E,all}$ is
roughly given by $X\otimes Y=X\times Y$ (if $\mathcal{C}$ admits products then
this is literally true and $\operatorname{Corr}(\mathcal{C})_{E,all}$ is a
symmetric monoidal $\infty$-category). With this notation at hand, a
_pre-6-functor formalism_ on $(\mathcal{C},E)$ is defined to be a map of
$\infty$-operads
$\operatorname{\mathcal{D}}\colon\operatorname{Corr}(\mathcal{C})_{E,all}^{\otimes}\to\mathcal{C}at_{\infty}^{\times}$,
where $\mathcal{C}at_{\infty}^{\times}$ denotes the cartesian structure on
$\mathcal{C}at_{\infty}$. Such a functor $\operatorname{\mathcal{D}}$ encodes
the following data: For every $X\in\mathcal{C}$ we get a symmetric monoidal
$\infty$-category $\operatorname{\mathcal{D}}(X)$, for every morphism $f\colon
Y\to X$ in $\mathcal{C}$ we get a symmetric monoidal functor
$f^{*}\colon\operatorname{\mathcal{D}}(X)\to\operatorname{\mathcal{D}}(Y)$ and
for every morphism $[f\colon Y\to X]\in E$ we get a functor
$f_{!}\colon\operatorname{\mathcal{D}}(Y)\to\operatorname{\mathcal{D}}(X)$
such that the functors $\otimes$, $f^{*}$ and $f_{!}$ satisfy the usual
compatibilities. We say that $\operatorname{\mathcal{D}}$ is a _6-functor
formalism_ if the functors $\otimes$, $f^{*}$ and $f_{!}$ admit right
adjoints.

In the setting of the desired $p$-adic 6-functor formalism, we take
$\mathcal{C}=\operatorname{vStack_{\Lambda}}$ and $E=bdcs$, the class of those
maps $f\colon Y\to X$ of small v-stacks which are $p$-bounded (this roughly
means that the map has finite $p$-cohomological dimension, see [Man22, §3.5]),
locally compactifiable and representable in locally spatial diamonds. The main
result of [Man22] now reads as follows:

######  Theorem 2.2 (see [Man22, Theorem 3.6.12]).

There is a 6-functor formalism

$\displaystyle\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}\colon\operatorname{Corr}(\operatorname{vStack_{\Lambda}})^{\otimes}_{bdcs,all}\to\mathcal{C}at_{\infty}^{\times},\qquad(X,\Lambda)\mapsto\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)$

with the following properties:

  1. (i)

The symmetric monoidal $\infty$-categories
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)$
and pullback functors $f^{*}$ coincide with the version in 2.1.

  2. (ii)

For every bdcs morphism $f\colon Y\to X$ in $\operatorname{vStack_{\Lambda}}$
the functor
$f_{!}\colon\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(Y,\Lambda)\to\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)$
has the following properties: If $f$ is proper then $f_{!}=f_{*}$ is right
adjoint to $f^{*}$. If $f$ is étale then $f_{!}$ is left adjoint to $f^{*}$.

###  2.2. Six functors for stacky morphisms

In order to efficiently apply the above 6-functor formalism to representation
theory, i.e. sheaves on classifying stacks, it is very useful to generalize
the definition of the shriek functors $f_{!}$ and $f^{!}$ to a larger class of
maps $f$. Namely, we need to allow certain “stacky” maps $f$ as well, similar
to what is done in [GHW22].

######  Definition 2.3.

We say that a bdcs map $f\colon Y\to X$ of small v-stacks admits _universal
$p$-codescent_ if is satisfies the following property: Given any
$(X^{\prime},\Lambda)\in\operatorname{vStack_{\Lambda}}$ which admits a map
(in $\operatorname{vStack_{\Lambda}}$) to some strictly totally disconnected
space and given any map $X^{\prime}\to X$ with base-change $f^{\prime}\colon
Y^{\prime}\to X^{\prime}$ and Čech nerve $Y^{\prime}_{\bullet}\to X^{\prime}$
the natural functor

$\displaystyle\operatorname{\operatorname{\mathcal{D}}^{a!}_{\scalebox{0.5}{$\square$}}}(X^{\prime},\Lambda)\mathbin{\xrightarrow{\sim}}\varprojlim_{n\in\Delta}\operatorname{\operatorname{\mathcal{D}}^{a!}_{\scalebox{0.5}{$\square$}}}(Y^{\prime}_{n},\Lambda)$

is an equivalence. Here
$\operatorname{\operatorname{\mathcal{D}}^{a!}_{\scalebox{0.5}{$\square$}}}$
denotes the functor
$(Z,\Lambda)\mapsto\operatorname{\operatorname{\mathcal{D}}^{a!}_{\scalebox{0.5}{$\square$}}}(Z,\Lambda)$,
$f\mapsto f^{!}$.

######  Definition 2.4.

A map $f\colon Y\to X$ of small v-stacks is called _$p$ -fine_ if there is a
map $g\colon Z\to Y$ such that $f\mathbin{\circ}g$ is bdcs and $g$ is bdcs and
admits universal $p$-codescent.

######  Lemma 2.5.

  1. (i)

The condition of being $p$-fine is étale local on source and target.

  2. (ii)

The collection of $p$-fine maps is stable under composition and base-change.

  3. (iii)

Every bdcs map is $p$-fine. In particular every étale map and every map of
rigid varieties over a fixed base field is $p$-fine.

  4. (iv)

Let $f\colon Y\to X$ and $g\colon Z\to Y$ be maps of small v-stacks. If $f$
and $f\mathbin{\circ}g$ are $p$-fine, then so is $g$.

###### Proof.

It follows from [LZ17, Lemma 3.1.2.(4)] that bdcs maps admitting universal
$p$-codescent are stable under composition. Part (ii) of the claim follows
immediately and part (iii) is obvious (see also [Man22, Lemma 3.6.10, Example
3.6.11]). Part (i) follows from the observation that étale maps admit
universal $p$-codescent.

It remains to prove (iv), so let $f$ and $g$ be given as in the claim. Choose
a bdcs map $Y^{\prime}\to Y$ admitting universal $p$-codescent such that the
composed map $Y^{\prime}\to X$ is bdcs. Denote $g^{\prime}\colon Z^{\prime}\to
Y^{\prime}$ the base-change of $g$ along $Y^{\prime}\to Y$. Since the map
$Z^{\prime}\to Z$ is bdcs, it follows from (ii) that the composed map
$Z^{\prime}\to X$ is $p$-fine. We can therefore choose a bdcs map
$Z^{\prime\prime}\to Z^{\prime}$ admitting universal $p$-codescent such that
$Z^{\prime\prime}\to X$ is bdcs. Note that the map $Z^{\prime\prime}\to X$
factors as $Z^{\prime\prime}\to Y^{\prime}\to X$ and since $Y^{\prime}\to X$
is bdcs it follows from [Man22, Lemma 3.6.10.(iv)] that $Z^{\prime\prime}\to
Y^{\prime}$ is bdcs. In particular $Z^{\prime\prime}\to Y$ is bdcs, so the map
$Z^{\prime\prime}\to Z$ shows that $g$ is $p$-fine. ∎

It is completely formal that the $p$-adic 6-functor formalism extends to all
$p$-fine maps (in fact a similar extension can be performed in any 6-functor
formalism):

######  Proposition 2.6.

The 6-functor formalism in 2.2 extends uniquely to a 6-functor formalism

$\displaystyle\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}\colon\operatorname{Corr}(\operatorname{vStack_{\Lambda}})_{pfine,all}\to\mathcal{C}at_{\infty},\qquad(X,\Lambda)\mapsto\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda),$

where $pfine$ denotes the class of $p$-fine maps in
$\operatorname{vStack_{\Lambda}}$.

###### Proof.

Let $\mathcal{C}\subset\operatorname{vStack_{\Lambda}}$ be the full
subcategory spanned by those $(X,\Lambda)$ which admit a map (in
$\operatorname{vStack_{\Lambda}}$) to some strictly totally disconnected
space. Then $\mathcal{C}$ is a basis of $\operatorname{vStack_{\Lambda}}$,
hence by [Man22, Proposition A.5.16] and [Man22, Proposition A.3.11.(i)] it is
enough to construct the desired extension of the 6-functor formalism on
$\mathcal{C}$, i.e. we need to construct a 6-functor formalism
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}\colon\operatorname{Corr}(\mathcal{C})_{pfine,all}\to\mathcal{C}at_{\infty}$
extending the one from 2.2. We now apply [Man22, Proposition A.5.14] with
$E=bdcs$, $E^{\prime}=pfine$ and $S\subset E$ being the subset of those maps
which admit universal $p$-codescent. Condition (a) requires
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)$
to be presentable, which can be achieved by temporarily restricting to
$\kappa$-condensed objects (as in the proof of [Man22, Theorem 3.6.12]).
Condition (b) follows from the definition of universal $p$-codescent,
condition (c) follows from the definition of $p$-fine maps and condition (d)
follows from [Man22, Lemma 3.6.10] (see [Man22, Remark A.5.15.(ii)]. ∎

######  Remark 2.7.

More concretely, the shriek functor
$f_{!}\colon\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(Y,\Lambda)\to\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)$
along a $p$-fine map $f\colon Y\to X$ in $\operatorname{vStack_{\Lambda}}$ is
computed as follows. Via v-descent we can reduce to the case that
$(X,\Lambda)$ admits a map to some strictly totally disconnected space. Now
choose a bdcs map $g\colon Z\to Y$ such that $g$ admits universal
$p$-codescent and $f\mathbin{\circ}g$ is bdcs. Let $g_{\bullet}\colon
Z_{\bullet}\to Y$ denote the Čech nerve of $g$. It follows from the
$p$-codescent property of $g$ that for every
$\mathcal{M}\in\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(Y,\Lambda)$
the natural map
$\varinjlim_{n\in\Delta}g_{n!}g_{n}^{!}\mathcal{M}\mathbin{\xrightarrow{\sim}}\mathcal{M}$
is an isomorphism. Consequently we have

$\displaystyle
f_{!}\mathcal{M}=f_{!}\varinjlim_{n\in\Delta}g_{n!}g_{n}^{!}\mathcal{M}=\varinjlim_{n\in\Delta}(f\mathbin{\circ}g_{n})_{!}g_{n}^{!}\mathcal{M}.$

Note that the maps $f\mathbin{\circ}g_{n}\colon Z_{n}\to X$ are bdcs, hence
$(f\mathbin{\circ}g_{n})_{!}$ was already defined in 2.2.

In order to make use of 2.6 we need a good amount of examples of bdcs maps
admitting universal $p$-codescent. Arguably the most important class of such
maps is formed by the $p$-cohomologically smooth maps (see [Man22, Definition
3.8.1]). The following is an analog of [GHW22, Proposition 3.16]:

######  Lemma 2.8.

Every bdcs and $p$-cohomologically smooth map of small v-stacks admits
universal $p$-codescent.

###### Proof.

Let $f\colon Y\to X$ be a bdcs and $p$-cohomologically smooth map in
$\operatorname{vStack_{\Lambda}}$ with Čech nerve $f_{\bullet}\colon
Y_{\bullet}\to X$; we need to show that the natural functor
$f_{\bullet}^{!}\colon\operatorname{\operatorname{\mathcal{D}}^{a!}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)\mathbin{\xrightarrow{\sim}}\varprojlim_{n\in\Delta}\operatorname{\operatorname{\mathcal{D}}^{a!}_{\scalebox{0.5}{$\square$}}}(Y_{n},\Lambda)$.
This functor admits a left adjoint

$\displaystyle
f_{\bullet!}\colon\varprojlim_{n\in\Delta}\operatorname{\operatorname{\mathcal{D}}^{a!}_{\scalebox{0.5}{$\square$}}}(Y_{n},\Lambda)\to\operatorname{\operatorname{\mathcal{D}}^{a!}_{\scalebox{0.5}{$\square$}}}(X,\Lambda),\qquad\mathcal{M}_{\bullet}\mapsto\varinjlim_{n\in\Delta}f_{n!}\mathcal{M}_{n}.$

Note that $f_{\bullet}^{!}$ is conservative: This follows from the fact that
$f_{0}^{!}$ is conservative, which in turn is implied by the fact that
$f_{0}^{!}$ is just a twist of $f_{0}^{*}$ by definition of the
$p$-cohomological smoothness of $f_{0}$. Thus in order to show that
$f_{\bullet}^{!}$ is an equivalence, it is enough to show that $f_{\bullet!}$
is fully faithful, i.e. for given
$\mathcal{M}_{\bullet}\in\varprojlim_{n\in\Delta}\operatorname{\operatorname{\mathcal{D}}^{a!}_{\scalebox{0.5}{$\square$}}}(Y_{n},\Lambda)$
we need to see that the natural map
$f_{\bullet}^{!}f_{\bullet!}\mathcal{M}_{\bullet}\mathbin{\xrightarrow{\sim}}\mathcal{M}_{\bullet}$
is an isomorphism. This boils down to showing that the map
$f_{0}^{!}f_{\bullet!}\mathcal{M}_{\bullet}\to\mathcal{M}_{0}$ is an
isomorphism. Since $f_{0}$ is $p$-cohomologically smooth, $f_{0}^{!}$
preserves small colimits, so we end up with the claim that the natural map

$\displaystyle\varinjlim_{n\in\Delta}f_{0}^{!}f_{n!}\mathcal{M}_{n}\mathbin{\xrightarrow{\sim}}\mathcal{M}_{0}$

is an isomorphism. Let us denote by $f^{\prime}_{\bullet}\colon
Y^{\prime}_{\bullet}\to Y_{0}$ the base-change of $Y_{\bullet}\to X$ along
$f_{0}\colon Y_{0}\to X$. Denote $g_{\bullet}\colon Y^{\prime}_{\bullet}\to
Y_{\bullet}$ the natural projections. Then for every $n\geq 0$ the natural
morphism
$f^{\prime}_{n!}g_{n}^{!}\mathcal{M}_{n}\mathbin{\xrightarrow{\sim}}f_{0}^{!}f_{n!}\mathcal{M}_{n}$
is an isomorphism (by $p$-cohomological smoothness of $f_{0}$ and $g_{n}$ this
reduces to proper base-change). On the other hand
$g_{\bullet}^{!}\mathcal{M}_{\bullet}$ defines an object of
$\varprojlim_{n\in\Delta}\operatorname{\operatorname{\mathcal{D}}^{a!}_{\scalebox{0.5}{$\square$}}}(Y^{\prime}_{n},\Lambda)$
so the above claimed isomorphism reduces to showing that the natural functor
$\operatorname{\operatorname{\mathcal{D}}^{a!}_{\scalebox{0.5}{$\square$}}}(Y_{0},\Lambda)\mathbin{\xrightarrow{\sim}}\varprojlim_{n\in\Delta}\operatorname{\operatorname{\mathcal{D}}^{a!}_{\scalebox{0.5}{$\square$}}}(Y^{\prime}_{n},\Lambda)$
is an equivalence. This means that we can replace $Y_{\bullet}\to X$ by
$Y^{\prime}_{\bullet}\to Y_{0}$. But the latter augmented simplicial object is
split, hence the desired equivalence of $\infty$-categories is formal. ∎

Let
$\operatorname{vStack^{\sharp}_{\pi|p}}\subset\operatorname{vStack^{\sharp}_{\pi}}$
be the full subcategory spanned by those pairs $(X^{\sharp},\pi)$ such that
$\pi|p$. Recall the definition of the $\infty$-category
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X^{\sharp}}/\pi)^{\varphi}$
from [Man22, Definition 3.9.10.(b)]: Roughly, this is the $\infty$-category of
pairs $(\mathcal{M},\varphi_{\mathcal{M}})$ where
$\mathcal{M}\in\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X^{\sharp}}/\pi)$
and $\varphi_{\mathcal{M}}$ is an isomorphism
$\varphi_{\mathcal{M}}\colon\varphi^{*}\mathcal{M}\mathbin{\xrightarrow{\sim}}\mathcal{M}$;
here $\varphi$ denotes the Frobenius $x\mapsto x^{p}$ on
$\mathcal{O}^{+}_{X^{\sharp}}/\pi$. The above constructed 6-functor formalism
extends to $\varphi$-modules:

######  Proposition 2.9.

The 6-functor formalism from 2.6 extends to a 6-functor formalism

$\displaystyle\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(-)^{\varphi}\colon\operatorname{Corr}(\operatorname{vStack_{\Lambda}})_{pfine,all}\to\mathcal{C}at_{\infty},\qquad(X,\Lambda)\mapsto\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)^{\varphi}.$

The forgetful functor
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)^{\varphi}\to\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)$
preserves all small limits and colimits and commutes with the 6 functors
$\otimes$, $\operatorname{\underline{\operatorname{Hom}}}$, $f^{*}$, $f_{*}$,
$f_{!}$ and $f^{!}$.

###### Proof.

By [Man22, Proposition 3.9.13] the same is true for the 6-functor formalism on
bdcs maps. To get this result also on $p$-fine maps, we note that the same
extension procedure as in the proof of 2.6 applies verbatim to the
$\varphi$-module case so that we get the desired 6-functor formalism. By
[Man22, Proposition 3.9.13] and functoriality of Kan extensions the forgetful
functor preserves small limits and colimits and commutes with all the 6
functors except possibly with $f^{!}$. To see that it also commutes with
$f^{!}$, note first that by the same argument as in the proof of [Man22, Lemma
3.9.12.(v)] we can reduce to the case that the target of $f\colon Y\to X$ is a
strictly totally disconnected space. Choose also a bdcs cover $g\colon
Z\mathrel{\twoheadrightarrow}Y$ such that $g$ admits universal $p$-codescent
and $f\mathbin{\circ}g$ is bdcs. Then by universal $p$-codescent the functor
$g^{!}$ is conservative, hence to check that $f^{!}$ commutes with the
forgetful functor it is enough to check that $g^{!}f^{!}$ commutes with the
forgetful functor. This follows from the bdcs case. ∎

In the following we will often omit the explicit mention of untilt and
pseudouniformizer and simply write
$X\in\operatorname{vStack^{\sharp}_{\pi|p}}$. The main reason for introducing
$\varphi$-modules is that they have a tight connection to actual étale
$\mathbf{F}_{p}$-sheaves: The $p$-torsion Riemann-Hilbert correspondence (see
[Man22, Theorem 3.9.23]) tells us that for
$X\in\operatorname{vStack^{\sharp}_{\pi|p}}$ there is a natural fully faithful
embedding

$\displaystyle\operatorname{\mathcal{D}}_{\mathrm{\acute{e}t}}(X,\mathbf{F}_{p})^{\mathrm{oc}}\mathrel{\hookrightarrow}\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X}/\pi)^{\varphi},\qquad\mathcal{F}\mapsto\mathcal{F}\otimes_{\mathbf{F}_{p}}\mathcal{O}^{+a}_{X}/\pi$

which induces an equivalence of dualizable objects on both sides. Here
$\operatorname{\mathcal{D}}_{\mathrm{\acute{e}t}}(X,\mathbf{F}_{p})^{\mathrm{oc}}$
denotes the (derived) $\infty$-category of overconvergent
$\mathbf{F}_{p}$-sheaves. The Riemann-Hilbert functor admits a right adjoint
$(-)^{\varphi}\colon\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X}/\pi)^{\varphi}\to\operatorname{\mathcal{D}}_{\mathrm{\acute{e}t}}(X,\mathbf{F}_{p})^{\mathrm{oc}}$.

###  2.3. Cohomologically smooth maps

The notion of $p$-cohomologically smooth maps of small v-stacks generalizes in
a straightforward way from bdcs maps to $p$-fine maps:

######  Definition 2.10.

A $p$-fine map $f\colon Y\to X$ of small v-stacks is called _$p$
-cohomologically smooth_ if for every map $X^{\prime}\to X$ from a strictly
totally disconnected perfectoid space
$X^{\prime}=\operatorname{Spa}(A^{\prime},A^{\prime+})$ there is a
pseudouniformizer $\pi\in A^{\prime+}$ such that the map $f^{\prime}\colon
Y^{\prime}:=Y\times_{X}X^{\prime}\to X^{\prime}$ satisfies the following
property: The natural morphism $f^{\prime*}\otimes
f^{\prime!}(\mathcal{O}^{+a}_{X^{\prime}}/\pi)\mathbin{\xrightarrow{\sim}}f^{\prime!}$
is an isomorphism of functors
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X^{\prime}}/\pi)\to\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{Y^{\prime}}/\pi)$
and $f^{\prime!}(\mathcal{O}^{+a}_{X^{\prime}}/\pi)$ is invertible.

The basic results on bdcs and $p$-cohomologically smooth maps extend to the
$p$-fine setting, as follows.

######  Proposition 2.11.

  1. (i)

Let $f\colon Y\to X$ be a $p$-fine and $p$-cohomologically smooth map in
$\operatorname{vStack_{\Lambda}}$. Then the natural morphism

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

is an isomorphism of functors
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)\to\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(Y,\Lambda)$
and $f^{!}\Lambda^{a}$ is invertible.

  2. (ii)

Let

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

be a cartesian square in $\operatorname{vStack_{\Lambda}}$.

    1. (a)

Assume that $f$ is $p$-fine and that either $f$ or $g$ is $p$-fine and
$p$-cohomologically smooth. Then the natural morphism

$\displaystyle g^{\prime*}f^{!}\mathbin{\xrightarrow{\sim}}f^{\prime!}g^{*}$

is an isomorphism of functors
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)\to\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(Y^{\prime},\Lambda)$.

    2. (b)

Assume that $g$ is $p$-fine and $p$-cohomologically smooth. Then the natural
morphism

$\displaystyle
g^{*}f_{*}\mathbin{\xrightarrow{\sim}}f^{\prime}_{*}g^{\prime*}$

is an isomorphism of functors
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(Y,\Lambda)\to\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X^{\prime},\Lambda)$.

  3. (iii)

Let $f\colon Y\to X$ be a $p$-fine and $p$-cohomologically smooth map in
$\operatorname{vStack_{\Lambda}}$. Then for all
$\mathcal{M},\mathcal{N}\in\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)$
the natural morphism

$\displaystyle
f^{*}\operatorname{\underline{\operatorname{Hom}}}(\mathcal{M},\mathcal{N})\mathbin{\xrightarrow{\sim}}\operatorname{\underline{\operatorname{Hom}}}(f^{*}\mathcal{M},f^{*}\mathcal{N})$

is an isomorphism.

###### Proof.

We first note that part (i) is formal if $X$ is strictly totally disconnected
and follows for general $X$ once we have shown (ii).(a) (see the proof of
[Man22, Proposition 3.8.4.(i)]). Moreover, once we have shown (ii).(a), also
(iii) is formal (see the proof of [Man22, Proposition 3.8.7]) and (ii).(b) is
formal (see the proof of [Man22, Proposition 3.8.6.(ii)]. Therefore, the whole
claim boils down to showing (ii).(a). By the proof of [Man22, Proposition
3.8.6.(iii)] the case that $g$ is $p$-cohomologically smooth follows formally
from the case that $f$ is $p$-cohomologically smooth. Thus from now on we can
assume that we are given a cartesian square as in (ii) with $f$ being $p$-fine
and $p$-cohomologically smooth and we need to show that the morphism
$g^{\prime*}f^{!}\mathbin{\xrightarrow{\sim}}f^{\prime!}g^{*}$ is an
isomorphism. As in the proof of [Man22, Proposition 3.8.4.(ii)] we can
formally reduce to the case that $X$ and $X^{\prime}$ are strictly totally
disconnected. Then by (i) (which holds unconditionally on (ii).(a) in the
strictly totally disconnected case) we are reduced to showing that the natural
map $\alpha\colon
g^{\prime*}f^{!}\Lambda^{a}\mathbin{\xrightarrow{\sim}}f^{\prime!}\Lambda^{a}$
is an isomorphism. We claim that $g^{\prime}_{*}$ is conservative on bounded
objects (in the sense of [Man22, Definition 3.3.1.(a)]): Pick any cover
$Z^{\prime}\mathrel{\twoheadrightarrow}X^{\prime}$ such that the composition
$Z\to X$ is bdcs (this can be done by the definition of $p$-fine maps). Then
$Z^{\prime}$ is a locally spatial diamond, so we can find a pro-étale cover of
$Z^{\prime}$ by totally disconnected spaces $Z_{i}$. Letting
$W_{i}:=Z_{i}\times_{X^{\prime}}Y^{\prime}=Z_{i}\times_{X}X^{\prime}$ and
denoting $g_{i}^{\prime}\colon W_{i}\to Z_{i}$ the base-change of
$g^{\prime}$, we note that each $W_{i}$ is a $p$-bounded affinoid perfectoid
space. Hence by [Man22, Theorem 3.5.21] each $g^{\prime}_{i}$ is a forgetful
functor of module categories and in particular conservative. Using also
bounded base-change (see [Man22, Proposition 3.3.7]) we deduce that
$g^{\prime}$ is conservative on bounded objects. Now note that
$f^{!}\Lambda^{a}$ is bounded: If $\Lambda=\mathcal{O}^{+}_{X}/\pi$ for some
$\pi|p$ then this follows from the Riemann-Hilbert correspondence (see [Man22,
Theorem 3.9.23]); for general $\Lambda$ one can make the usual reductions.
Similarly $f^{\prime!}\Lambda^{a}$ is bounded. Thus altogether we deduce that
it is enough to show that $g^{\prime}_{*}\alpha\colon
g^{\prime}_{*}g^{\prime*}f^{!}\Lambda^{a}\mathbin{\xrightarrow{\sim}}g^{\prime}_{*}f^{\prime!}\Lambda^{a}$
is an isomorphism. The remaining proof of [Man22, Proposition 3.8.4.(ii)]
applies verbatim. ∎

######  Lemma 2.12.

  1. (i)

The condition of being $p$-fine and $p$-cohomologically smooth is analytically
local on both source and target.

  2. (ii)

Among $p$-fine maps, the condition of being $p$-cohomologically smooth is
v-local on the target and $p$-cohomologically smooth local on the source.

  3. (iii)

$p$-fine and $p$-cohomologically smooth maps are stable under composition and
base-change.

###### Proof.

This is formal, see [Man22, Lemma 3.8.5]. ∎

##  3\. Representation-theoretic preparations

In this section, we apply the 6-functor formalism to study representations of
$p$-adic Lie groups. Since a number of categories will appear in what follows,
we begin by introducing them. For the moment, we allow $G$ to be any locally
profinite group with locally finite $p$-cohomological dimension.

First of all, we will need the $\infty$-category of solid $G$-representations
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(G,\mathbf{F}_{p})=\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\mathbf{F}_{p})^{BG}$,
as defined in [Man22, Definition 3.4.3]. This can be identified with the
derived $\infty$-category of the category of continuous $G$-representations on
solid $\mathbf{F}_{p}$-vector spaces (in the obvious sense). We also have the
derived category
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}^{{\mathrm{sm}}}(G,\mathbf{F}_{p})$
of smooth $G$-representations on solid $\mathbf{F}_{p}$-vector spaces [Man22,
Definition 3.4.11]. There is a natural functor
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}^{{\mathrm{sm}}}(G,\mathbf{F}_{p})\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(G,\mathbf{F}_{p})$,
which is not fully faithful. Most classically, we have the usual derived
category $\operatorname{\mathcal{D}}^{{\mathrm{sm}}}(G,\mathbf{F}_{p})$ of the
abelian category of smooth $G$-representations on discrete
$\mathbf{F}_{p}$-vector spaces. There is a natural identification
$\operatorname{\mathcal{D}}^{{\mathrm{sm}}}(G,\mathbf{F}_{p})=\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}^{{\mathrm{sm}}}(G,\mathbf{F}_{p})_{\omega}$,
identifying $\operatorname{\mathcal{D}}^{{\mathrm{sm}}}(G,\mathbf{F}_{p})$ as
the full subcategory of
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}^{{\mathrm{sm}}}(G,\mathbf{F}_{p})$
consisting of those smooth representations whose underlying
$\mathbf{F}_{p}$-vector space is discrete (in the topological, i.e. condensed,
sense), which we will use without comment. We also note that the composite
functor
$\operatorname{\mathcal{D}}^{{\mathrm{sm}}}(G,\mathbf{F}_{p})\to\operatorname{\mathcal{D}}_{{\scalebox{0.5}{$\square$}}}(G,\mathbf{F}_{p})$
is fully faithful.

Now fix an algebraically closed nonarchimedean field $C$ of residue
characteristic $p$, and a pseudouniformizer $\pi|p$. Let
$B\underline{G}=\operatorname{Spd}(C)/\underline{G}$ denote the classifying
stack of $G$. We then have the $\infty$-category
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{B\underline{G}}/\pi)^{\varphi}$,
which (by [Man22, Lemma 3.4.26.(ii)]) can be identified with the
$\infty$-category of smooth $G$-representations on solid almost
$\varphi$-modules over $\mathcal{O}_{C}^{+}/\pi$. This category can be
directly accessed by the 6-functor formalism, as we will see. On the other
hand, Proposition 3.9 gives a natural identification
$\operatorname{\mathcal{D}}^{{\mathrm{sm}}}(G,\mathbf{F}_{p})=\operatorname{\mathcal{D}}_{{\mathrm{\acute{e}t}}}(B\underline{G},\mathbf{F}_{p})$.
The Riemann-Hilbert functor then gives a natural fully faithful symmetric
monoidal embedding

$\operatorname{\mathcal{D}}^{{\mathrm{sm}}}(G,\mathbf{F}_{p})\to\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{B\underline{G}}/\pi)^{\varphi},$

which factors through the full subcategory
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{B\underline{G}}/\pi)^{\varphi}_{\omega}$
of discrete objects.

###  3.1. Smooth representations

We continue to fix a locally profinite group $G$ of locally bounded
$p$-cohomological dimension. In this section, we recall some results on the
derived $\infty$-category
$\operatorname{\mathcal{D}}^{{\mathrm{sm}}}(G,\mathbf{F}_{p})$. However, to
place our results in context, we momentarily allow our coefficients to be any
field $k$ (still assuming that $G$ has locally bounded
$\mathrm{char}k$-cohomological dimension). Thus, let
$\mathrm{Rep}_{k}^{\mathrm{sm}}(G)$ denote the category of smooth
$G$-representations on $k$-vector spaces, and let
$\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,k)$ denote its derived
$\infty$-category. When $k$ is clear from context, we just write
$\operatorname{\mathcal{D}}(G)=\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,k)$.
We sometimes write $\mathbf{1}\in\mathrm{Rep}_{k}^{\mathrm{sm}}(G)$ for $k$
with the trivial $G$-action.

The assertions in the following proposition are well-known.

######  Proposition 3.1.

1\. $\mathrm{Rep}_{k}^{\mathrm{sm}}(G)$ is a Grothendieck abelian category. In
particular, unbounded complexes have $K$-injective resolutions.

2\. The derived category $\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,k)$ is
left-complete and has a natural symmetric monoidal structure given by
$k$-linear tensor product of cochain complexes.

3\. If $H\subset G$ is an open subgroup, the restriction functor
$r=r_{G}^{H}:\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,k)\to\operatorname{\mathcal{D}}^{\mathrm{sm}}(H,k)$
is $t$-exact and symmetric monoidal.

Let $\operatorname{\underline{\operatorname{Hom}}}_{G}(A,B)$ be the internal
hom in $\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,k),$ defined as a formal
adjoint satisfying
$\mathrm{Hom}_{\operatorname{\mathcal{D}}(G)}(C\otimes_{k}A,B)=\mathrm{Hom}_{\operatorname{\mathcal{D}}(G)}(C,\operatorname{\underline{\operatorname{Hom}}}_{G}(A,B))$.

######  Lemma 3.2.

If $H\subset G$ is an open subgroup, then there is a natural isomorphism
$r\operatorname{\underline{\operatorname{Hom}}}_{G}(A,B)\cong\operatorname{\underline{\operatorname{Hom}}}_{H}(rA,rB)$
for any $A,B\in\operatorname{\mathcal{D}}(G)$. Here
$r:\operatorname{\mathcal{D}}(G)\to\operatorname{\mathcal{D}}(H)$ denotes the
restriction functor.

###### Proof.

The functor $r$ has an left adjoint, given by the (exact) functor
$\mathrm{ind}_{H}^{G}(-)=-\otimes_{k[H]}k[G]$ of compact induction. We then
compute

$\displaystyle\mathrm{Hom}_{\operatorname{\mathcal{D}}(H)}(C,r\operatorname{\underline{\operatorname{Hom}}}_{G}(A,B))$
$\displaystyle\cong\mathrm{Hom}_{\operatorname{\mathcal{D}}(G)}(\mathrm{ind}_{H}^{G}C,\operatorname{\underline{\operatorname{Hom}}}_{G}(A,B))$
$\displaystyle\cong\mathrm{Hom}_{\operatorname{\mathcal{D}}(G)}(A\otimes\mathrm{ind}_{H}^{G}C,B)$
$\displaystyle\cong\mathrm{Hom}_{\operatorname{\mathcal{D}}(G)}(\mathrm{ind}_{H}^{G}(rA\otimes
C),B)$
$\displaystyle\cong\mathrm{Hom}_{\operatorname{\mathcal{D}}(H)}(rA\otimes
C,rB)$
$\displaystyle\cong\mathrm{Hom}_{\operatorname{\mathcal{D}}(H)}(C,\operatorname{\underline{\operatorname{Hom}}}_{H}(rA,rB))$

naturally in $C\in\operatorname{\mathcal{D}}(H)$, so the result follows from
Yoneda. ∎

If $A\in\operatorname{\mathcal{D}}(G)$ is any object and $H\subset G$ is any
open subgroup, we write

$R\Gamma_{{\mathrm{sm}}}(H,A)=R\mathrm{Hom}_{\operatorname{\mathcal{D}}(H)}(\mathbf{1},rA)=R\mathrm{Hom}_{\operatorname{\mathcal{D}}(G)}(\mathrm{ind}_{H}^{G}\mathbf{1},A).$

To actually compute this, let $A\overset{\sim}{\to}I^{\bullet}$ be a
$K$-injective resolution by injectives in $\mathrm{Rep}_{k}^{\mathrm{sm}}(G)$;
then $R\Gamma_{{\mathrm{sm}}}(H,A)\cong(I^{\bullet})^{H}$. When $H$ is a
$p$-adic Lie group, $R\Gamma_{{\mathrm{sm}}}(H,-)$ coincides with continuous
group cohomology, by the main result in [Fus21].

######  Lemma 3.3.

For any $A\in\operatorname{\mathcal{D}}(G)$, we have
$A\cong\mathrm{colim}_{H\subset G\,\mathrm{open}}R\Gamma_{\mathrm{sm}}(H,A)$.

###### Proof.

If $A\overset{\sim}{\to}I^{\bullet}$ is a $K$-injective resolution by
injectives, then $R\Gamma_{\mathrm{sm}}(H,A)\cong(I^{\bullet})^{H}$ for any
$H$; here we use that restriction along
$\mathrm{Rep}_{k}^{\mathrm{sm}}(G)\to\mathrm{Rep}_{k}^{\mathrm{sm}}(H)$
preserves $K$-injective complexes of injectives. But then

$\displaystyle\mathrm{colim}_{H\subset
G\,\mathrm{open}}R\Gamma_{\mathrm{sm}}(H,A)$
$\displaystyle\cong\mathrm{colim}_{H\subset
G\,\mathrm{open}}(I^{\bullet})^{H}\cong I^{\bullet}\cong A,$

using the fact that each $I^{n}$ is a smooth representation. ∎

######  Lemma 3.4.

For any $A,B\in\operatorname{\mathcal{D}}(G)$, we have
$\operatorname{\underline{\operatorname{Hom}}}_{G}(A,B)\cong\mathrm{colim}_{H}R\mathrm{Hom}_{\operatorname{\mathcal{D}}(H)}(A,B).$

Here (and in what follows) we drop the restriction functors $r$ from the
notation.

###### Proof.

There are natural isomorphisms

$\displaystyle\mathrm{colim}_{H}R\mathrm{Hom}_{\operatorname{\mathcal{D}}(H)}(A,B)$
$\displaystyle\cong\mathrm{colim}_{H}R\Gamma_{\mathrm{sm}}(H,\operatorname{\underline{\operatorname{Hom}}}_{H}(A,B))$
$\displaystyle\cong\mathrm{colim}_{H}R\Gamma_{\mathrm{sm}}(H,\operatorname{\underline{\operatorname{Hom}}}_{G}(A,B))$
$\displaystyle\cong\operatorname{\underline{\operatorname{Hom}}}_{G}(A,B),$

where the second line follows from Lemma 3.2, and the third line follows from
Lemma 3.3. ∎

######  Proposition 3.5.

When $G$ is locally pro-$p$ and $\mathrm{char}k\neq p$,
$\operatorname{\underline{\operatorname{Hom}}}_{G}(-,\mathbf{1})$ is the usual
exact smooth duality functor, i.e. it sends any
$\pi\in\mathrm{Rep}_{k}^{\mathrm{sm}}(G)$ to the smooth vectors in
$\mathrm{Hom}_{k}(\pi,k)$.

###### Proof.

Easy, using the previous lemma together with the fact that $\mathbf{1}$ is an
injective object of $\mathrm{Rep}_{k}^{\mathrm{sm}}(H)$ for any pro-$p$ open
compact $H\subset G$. (Use that any injective map $V\to W$ in
$\mathrm{Rep}_{k}^{\mathrm{sm}}(H)$ admits an $H$-equivariant retraction, by
averaging any $k$-linear retraction against a $k$-valued Haar measure.) ∎

When $G$ is a $p$-adic Lie group and $k=\mathbf{F}_{p}$, the naive smooth
duality functor on $\mathrm{Rep}_{\mathbf{F}_{p}}^{\mathrm{sm}}(G)$ is rather
useless, because the functor of passage to smooth vectors is no longer exact.
To remedy this, Kohlhaase has introduced a family of derived smooth duality
functors $\mathcal{S}_{G}^{i}(-)$ on
$\mathrm{Rep}_{\mathbf{F}_{p}}^{\mathrm{sm}}(G)$, which are essentially the
right-derived functors of the functor $V\mapsto(V^{\ast})^{\mathrm{sm}}$. More
precisely, Kohlhaase defines

$\mathcal{S}_{G}^{i}(V)=\mathrm{colim}_{H\subset
G\,\mathrm{open}\,\mathrm{compact}}\mathrm{Ext}_{\mathbf{F}_{p}[[H]]}^{i}(k,V^{\ast}).$

Here $V^{\ast}=\mathrm{Hom}_{\mathbf{F}_{p}}(V,\mathbf{F}_{p})$ denotes the
Pontryagin dual of $V$ with its natural pseudocompact $\mathbf{F}_{p}$-module
structure; this is naturally a pseudocompact $\mathbf{F}_{p}[[H]]$-module for
any open compact $H\subset G$. We prefer to package the $\mathcal{S}^{i}$’s
into a single functor on the derived category. In other words, we consider the
endofunctor of $\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,\mathbf{F}_{p})$
defined by

$V\mapsto\mathcal{S}_{G}(V)=\mathrm{colim}_{H\subset
G\,\mathrm{open}\,\mathrm{compact}}R\mathrm{Hom}_{\mathbf{F}_{p}[[H]]}(\mathbf{F}_{p},V^{\ast}),$

where
$V^{\ast}\in\operatorname{\mathcal{D}}(\mathbf{F}_{p}[[H]]-\mathrm{mod})$ is
computed termwise. Here $R\mathrm{Hom}_{\mathbf{F}_{p}[[H]]}$ denotes
$R\mathrm{Hom}$ in the derived category of $\mathbf{F}_{p}[[H]]$-modules. Of
course, $H^{i}(\mathcal{S}(V))=\mathcal{S}^{i}(V)$ when $V$ is concentrated in
degree zero.

The following proposition was noted independently by Schneider-Sorensen
[SS22].

######  Proposition 3.6.

As endofunctors of
$\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,\mathbf{F}_{p})$, the functor
$\operatorname{\underline{\operatorname{Hom}}}_{G}(-,\mathbf{1})$ coincides
with Kohlhaase’s derived smooth duality functor $\mathcal{S}_{G}(-)$.

###### Proof.

As noted above, $\mathcal{S}_{G}(V)$ is defined by the formula

$\mathcal{S}_{G}(V)=\mathrm{colim}_{H\subset
G\,\mathrm{open}\,\mathrm{compact}}R\mathrm{Hom}_{\mathbf{F}_{p}[[H]]}(\mathbf{F}_{p},V^{\ast}).$

To actually compute this, choose a uniformly powerful open pro-$p$ subgroup
$G_{0}\subset G$. By work of Lazard [Laz65], we can choose a finite length
resolution $P_{\bullet}\to\mathbf{F}_{p}\to 0$ of $\mathbf{F}_{p}$ by finite
free $\mathbf{F}_{p}[[G_{0}]]$-modules. Then we can express
$\mathcal{S}_{G}(-)$ more concretely as

$\mathcal{S}_{G}(V)=\mathrm{colim}_{H\subset
G_{0}\,\mathrm{open}}\mathrm{Hom}_{\mathbf{F}_{p}[[H]]}(P_{\bullet},V^{\ast}).$

Set
$I^{n}=\mathrm{Hom}_{\mathbf{F}_{p}}^{\mathrm{cts}}(P_{n},\mathbf{F}_{p})$, so
$0\to\mathbf{F}_{p}\to I^{\bullet}$ is a finite resolution of $\mathbf{1}$ by
injective objects in $\mathrm{Rep}_{\mathbf{F}_{p}}^{\mathrm{sm}}(G_{0})$.
Since Pontryagin duality gives an exact anti-equivalence between smooth
$H$-representations over $\mathbf{F}_{p}$ and pseudocompact
$\mathbf{F}_{p}[[H]]$-modules, we have natural identifications

$\mathrm{Hom}_{\mathbf{F}_{p}[[H]]}(P_{\bullet},V^{\ast})\cong\mathrm{Hom}_{\mathrm{Rep}_{\mathbf{F}_{p}}^{\mathrm{sm}}(H)}(V,I^{\bullet})\cong
R\mathrm{Hom}_{\operatorname{\mathcal{D}}(H)}(V,\mathbf{1}).$

Putting things together, we get

$\displaystyle\mathcal{S}_{G}(V)$ $\displaystyle\cong\mathrm{colim}_{H\subset
G_{0}\,\mathrm{open}}\mathrm{Hom}_{\mathrm{Rep}_{\mathbf{F}_{p}}^{\mathrm{sm}}(H)}(V,I^{\bullet})$
$\displaystyle\cong\mathrm{colim}_{H\subset
G_{0}\,\mathrm{open}}R\mathrm{Hom}_{\operatorname{\mathcal{D}}(H)}(V,\mathbf{1})$
$\displaystyle\cong\operatorname{\underline{\operatorname{Hom}}}_{G}(V,\mathbf{1}),$

where the last line follows by Lemma 3.4. ∎

In the remainder of this subsection, we continue to assume that $G$ is a
$p$-adic Lie group. Recall that the _dimension_ of an admissible smooth
representation $V\in\mathrm{Rep}_{\mathbf{F}_{p}}^{\mathrm{sm}}(G)$ is the
quantity

$\mathrm{dim}_{G}V=\mathrm{dim}\,G\cdot\lim_{n\to\infty}\frac{\log\dim_{\mathbf{F}_{p}}V^{K_{n}}}{\log\,[K_{n}:K_{0}]}=\lim_{n\to\infty}\frac{\log\dim_{\mathbf{F}_{p}}V^{K_{n}}}{n\log
p}.$

Here $K_{0}\subset G$ is a fixed choice of a uniformly powerful open pro-$p$
subgroup, and $K_{n}=K^{p^{n}}\subset K_{0}$ is the closed normal subgroup
generated by the $p^{n}$th powers of elements of $K$. (One automatically has
that $[K_{n}:K_{0}]=p^{n\cdot\dim G}$, which explains the second equality
above.) By [EP20, Proposition 2.18], this limit exists independently of the
choice of $K_{0}$, and defines an integer in the interval $[0,\dim G]$.
Moreover, $\mathrm{dim}_{G}V=0$ iff $\mathrm{dim}_{\mathbf{F}_{p}}V<\infty$.
Additionally, if $0\to V^{\prime}\to V\to V^{\prime\prime}\to 0$ is an exact
sequence of admissible smooth representations, then
$\mathrm{dim}_{G}V=\max(\mathrm{dim}_{G}V^{\prime},\dim_{G}V^{\prime\prime})$.
In particular, any subquotient $W$ of a given $V$ satisfies
$\mathrm{dim}_{G}W\leq\mathrm{dim}_{G}V$.

A key point for us is that $\dim_{G}$ is closely related to the behavior of
the functors $\mathcal{S}_{G}^{i}$.

######  Proposition 3.7.

  1. (i)

If $V$ is an admissible smooth $G$-representation, then each
$\mathcal{S}_{G}^{i}(V)$ is admissible smooth, with
$\mathrm{dim}_{G}\mathcal{S}_{G}^{i}(V)\leq i$ for all $i$. Moreover, if
$d=\mathrm{dim}_{G}V$ then $\mathrm{dim}_{G}\mathcal{S}_{G}^{d}(V)=d$ and
$\mathcal{S}_{G}^{i}(V)=0$ for all $i>d$.

  2. (ii)

For any $A\in\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,\mathbf{F}_{p})$ whose
cohomologies $H^{i}(A)$ are all admissible, the complex $\mathcal{S}_{G}(A)$
has admissible cohomologies, and the biduality map
$A\to(\mathcal{S}_{G}\circ\mathcal{S}_{G})(A)$ is an isomorphism.

  3. (iii)

If $V$ is an admissible smooth representation with $\dim_{G}V=0$, so
$\dim_{\mathbf{F}_{p}}V<\infty$, then $\mathcal{S}(V)\cong\mathcal{S}^{0}(V)$
is the naive linear dual of $V$.

###### Proof.

The first part is essentially a summary of results from [Koh17]. The second
part is Pontryagin-dual to the biduality discussed in the proof of [Koh17,
Theorem 3.5]; see also [SS22, Proposition 3.4]. The third part is immediate
from the first part. ∎

Note that part (i) here implies that
$\dim_{G}V=\sup\left\\{i\mid\mathcal{S}_{G}^{i}(V)\neq 0\right\\}$. We also
observe that $\mathcal{S}^{0}(-)$ can be completely understood.

######  Proposition 3.8.

Let $V$ be any admissible smooth $G$-representation. Then
$\mathcal{S}_{G}^{0}(V)\neq 0$ if and only if $V$ admits a nonzero $G$-stable
quotient $V\twoheadrightarrow W$ with $\mathrm{dim}_{\mathbf{F}_{p}}W<\infty$.

###### Proof.

“If”: Given such a $V\to W$, we get a nonzero map
$\mathcal{S}(W)\to\mathcal{S}(V)$. But the previous proposition shows that
$\mathcal{S}(W)=\tau^{\leq 0}\mathcal{S}(W)=\mathcal{S}^{0}(W)$, so we get a
nonzero map $\mathcal{S}^{0}(W)\to\mathcal{S}(V)$ which necessarily factors
over a nonzero map $\mathcal{S}^{0}(W)\to\mathcal{S}^{0}(V)$.

“Only if”: Suppose $\mathcal{S}^{0}(V)\neq 0$. By the previous proposition,
this is zero-dimensional as a $G$-representation, so
$\mathcal{S}(\mathcal{S}^{0}(V))\cong\mathcal{S}^{0}(\mathcal{S}^{0}(V))$. In
particular, applying $\mathcal{S}$ to the map $\mathcal{S}^{0}(V)=\tau^{\leq
0}\mathcal{S}(V)\to\mathcal{S}(V)$ and using biduality, we get a nonzero map
$V\cong\mathcal{S}(\mathcal{S}(V))\to\mathcal{S}(\mathcal{S}^{0}(V))\cong\mathcal{S}^{0}(\mathcal{S}^{0}(V))$,
and $\mathcal{S}^{0}(\mathcal{S}^{0}(V))$ is finite-dimensional over
$\mathbf{F}_{p}$. ∎

We conclude this section by giving a geometric interpretation of
$\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,\mathbf{F}_{p})$. Let $C$ be any
algebraically closed nonarchimedean field of residue characteristic $p$.

######  Proposition 3.9.

Notation as above, there are natural symmetric monoidal equivalences

$\operatorname{\mathcal{D}}_{{\mathrm{\acute{e}t}}}([\operatorname{Spd}(C)/G],\mathbf{F}_{p})\cong\operatorname{\mathcal{D}}([\operatorname{Spd}(C)/G]_{{\mathrm{\acute{e}t}}},\mathbf{F}_{p})\cong\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,\mathbf{F}_{p}).$

In particular, the first two categories are left-complete and canonically
independent of $C$.

###### Proof.

This follows from the proof of Theorem V.1.1 in [FS21], noting that that
$[\operatorname{Spd}(C)/G]_{{\mathrm{\acute{e}t}}}$ locally has uniformly
bounded $p$-cohomological dimension by Lazard-Serre. ∎

###  3.2. Sheaves on classifying stacks

We now study the stacky 6-functor formalism introduced above in the context of
the classifying stack $*/G$ associated to a locally profinite group $G$.
Recall that if $G$ has locally finite $p$-cohomological dimension then for
every totally disconnected space $X=\operatorname{Spa}(A,A^{+})$ with
pseudouniformizer $\pi$ the $\infty$-category
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{X/G}/\pi)$
can be identified with the $\infty$-category of smooth $G$-representations on
solid almost $A^{+}/\pi$-modules (see [Man22, Lemma 3.4.26.(ii)]). In the
following we will show that if $G$ has locally finite $p$-cohomological
dimension then $*/G\to*$ is $p$-fine, and we will deduce a precise criterion
for when the map $*/G\to*$ is $p$-cohomologically smooth. In particular, this
will hold in all practical situations, see Theorem 3.18 below.

To start things off, we first translate the finite $p$-cohomological dimension
condition into a vanishing of Ext-groups. Also recall the definition of the
$p$-cohomological dimension $\operatorname{cd}_{p}G$ from [Man22, Definition
3.4.20] (see [Man22, Proposition 3.4.22] to relate this to more classical
versions).

######  Proposition 3.10.

Let $G$ be a profinite group and let
$M,N\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\mathbf{F}_{p})^{BG}$
be static continuous $G$-representations on solid $\mathbf{F}_{p}$-vector
spaces. If $M$ is discrete then

$\displaystyle\operatorname{Ext}^{i}_{G}(M,N)=0\qquad\text{for
$i>\operatorname{cd}_{p}G$}.$

###### Proof.

The $\infty$-category
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\mathbf{F}_{p})^{BG}$
comes naturally equipped with a symmetric monoidal structure such that the
forgetful functor
$q^{*}\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\mathbf{F}_{p})^{BG}\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\mathbf{F}_{p})$
is symmetric monoidal (e.g. one can construct
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\mathbf{F}_{p})^{BG}$
as certain sheaves on the classifying stack $*/G$ in a suitable geometric
setup). The forgetful functor $q^{*}$ has a left adjoint
$q_{\natural}\colon\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\mathbf{F}_{p})\to\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\mathbf{F}_{p})^{BG}$
which is given by
$q_{\natural}(M)=\mathbf{F}_{p{\scalebox{0.5}{$\square$}}}[G]\otimes_{\mathbf{F}_{p{\scalebox{0.5}{$\square$}}}}M$.
It is straightforward to check that $q_{\natural}$ satisfies the projection
formula, i.e. that for all
$M\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\mathbf{F}_{p})^{BG}$
and
$N\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\mathbf{F}_{p})$
the natural morphism $q_{\natural}(N\otimes
q^{*}M)\mathbin{\xrightarrow{\sim}}q_{\natural}N\otimes M$ is an isomorphism.
The symmetric monoidal structure on
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\mathbf{F}_{p})^{BG}$
is closed, i.e. admits internal hom’s
$\operatorname{\underline{\operatorname{Hom}}}$ (this is clear if we restrict
to $\kappa$-condensed objects for some cardinal $\kappa$ and then follows for
unrestricted condensed objects by the following computation). It follows
formally from the projection formula for $q_{\natural}$ that for all
$M,N\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\mathbf{F}_{p})^{BG}$
the natural morphism

$\displaystyle
q^{*}\operatorname{\underline{\operatorname{Hom}}}(M,N)\mathbin{\xrightarrow{\sim}}\operatorname{\underline{\operatorname{Hom}}}(q^{*}M,q^{*}N)$

is an isomorphism – in other words, the internal hom in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\mathbf{F}_{p})^{BG}$
is computed on the underlying solid $\mathbf{F}_{p}$-vector spaces. In
particular, if $M$ and $N$ are static and $M$ is discrete (and thus
$q^{*}M\cong\bigoplus_{I}\mathbf{F}_{p}$ for some set $I$) then
$\operatorname{\underline{\operatorname{Hom}}}(M,N)$ is static. Therefore,
from

$\displaystyle\operatorname{Ext}^{i}_{G}(M,N)=\pi_{0}R\operatorname{Hom}(M,N[i])=\pi_{0}\operatorname{Hom}(\mathbf{F}_{p},\operatorname{\underline{\operatorname{Hom}}}(M,N[i]))=H^{i}(G,\operatorname{\underline{\operatorname{Hom}}}(M,N))$

we immediately deduce the claim. ∎

We will now show that $*/G\to*$ is $p$-fine. This reduces to showing that the
canonical cover $*\to*/G$ admits universal $p$-codescent, which will be
established by adapting Mathew’s ideas on descendable algebras [Mat16].

######  Lemma 3.11.

Let $G$ be a profinite group with $\operatorname{cd}_{p}G<\infty$. Then the
map $*\to*/G$ of small v-stacks is bdcs and admits universal $p$-codescent.

###### Proof.

It is clear that the map $*\to*/G$ is bdcs (e.g. the pullback along itself is
$G\to*$, which is pro-étale). Now pick any
$(X,\Lambda)\in\operatorname{vStack_{\Lambda}}$ admitting a map to a strictly
totally disconnected space, pick any map $X\to*/G$ and denote $q\colon Y\to X$
the base-change of $*\to*/G$. We need to show that $q$ admits $p$-codescent,
i.e. if $q_{\bullet}\colon Y_{\bullet}\to X$ denotes the Čech nerve of $q$
then the natural functor

$\displaystyle\operatorname{\operatorname{\mathcal{D}}^{a!}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)\mathbin{\xrightarrow{\sim}}\varprojlim_{n\in\Delta}\operatorname{\operatorname{\mathcal{D}}^{a!}_{\scalebox{0.5}{$\square$}}}(Y_{n},\Lambda)$

is an equivalence. To see this we employ Lurie’s Beck-Chevalley condition, see
[Lur, Corollary 4.7.5.3]. Part (2) of this condition is easily satisfied:
Since all the maps in the diagram $Y_{\bullet}\to X$ are proper, the required
left adjointability for the upper shriek functors reduces to proper base-
change (by passing to left adjoints). Thus in order to apply Lurie’s result it
remains to verify the following:

  1. (a)

The functor
$q^{!}\colon\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)\to\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(Y,\Lambda)$
is conservative.

  2. (b)

The functor $q^{!}$ preserves geometric realizations of $q^{!}$-split
simplicial objects in
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)$.

To prove (a) and (b) we apply ideas by Mathew [Mat16] (which were also used
extensively in the setup of the $p$-adic 6-functor formalism, cf. [Man22,
§2.6]). More concretely, we will prove the following claim:

  * ($\ast$)

Let $\langle
q_{*}q^{!}\rangle\subset\operatorname{Fun}(\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda),\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda))$
be the full subcategory generated by the endofunctor $q_{*}q^{!}$ under
compositions, retracts and finite (co)limits. Then $\langle q_{*}q^{!}\rangle$
contains the identity functor.

Let us first see how ($\ast$) implies (a) and (b), so assume that ($\ast$)
holds for the moment. To prove (a), let $\mathcal{M}\to\mathcal{N}$ be a given
map in
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)$
such that $q^{!}\mathcal{M}\mathbin{\xrightarrow{\sim}}q^{!}\mathcal{N}$ is an
isomorphism. Then also
$q_{*}q^{!}\mathcal{M}\mathbin{\xrightarrow{\sim}}q_{*}q^{!}\mathcal{N}$ is an
isomorphism and one deduces easily that
$F(\mathcal{M})\mathbin{\xrightarrow{\sim}}F(\mathcal{N})$ is an isomorphism
for all $F\in\langle q_{*}q^{!}\rangle$; taking $F=\operatorname{id}$ shows
that $\mathcal{M}\mathbin{\xrightarrow{\sim}}\mathcal{N}$ is an isomorphism,
as desired. To prove (b), let $\mathcal{M}_{\bullet}$ be a $q^{!}$-split
simplicial object in
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)$.
Then $q_{*}q^{!}\mathcal{M}_{\bullet}$ is a constant
$\operatorname{Ind}$-object (i.e. lies in the essential image of
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)\mathrel{\hookrightarrow}\operatorname{Ind}(\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda))$)
and it follows immediately that the same is true for
$F(\mathcal{M}_{\bullet})$ for all $F\in\langle q_{*}q^{!}\rangle$. Taking
$F=\operatorname{id}$ shows that $\mathcal{M}_{\bullet}$ is a constant
$\operatorname{Ind}$-object, so that its colimit commutes with any exact
functor; in particular it commutes with $q^{!}$.

It now remains to verify ($\ast$). We compute (using [Man22, Corollary
3.6.17.(i)])

$\displaystyle
q_{*}q^{!}=q_{*}\operatorname{\underline{\operatorname{Hom}}}(\Lambda^{a},q^{!})=\operatorname{\underline{\operatorname{Hom}}}(q_{!}\Lambda^{a},-)=\operatorname{\underline{\operatorname{Hom}}}(q_{*}\Lambda^{a},-).$

Therefore ($\ast$) can be reduced to showing the following: Let $\langle
q_{*}\Lambda^{a}\rangle\subset\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)$
be the full subcategory generated by $q_{*}\Lambda^{a}$ under tensor product,
retracts and finite (co)limits; then $\langle q_{*}\Lambda^{a}\rangle$
contains the tensor unit $\Lambda^{a}$. By [Mat16, Proposition 3.20] this is
equivalent to the following claim:

  * ($\ast^{\prime}$)

The algebra object
$q_{*}\Lambda^{a}\in\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)$
admits descent.

It remains to prove that ($\ast^{\prime}$) is true. We first perform some
reductions. By assumption on $(X,\Lambda)$ there is a map
$(X,\Lambda)\to(Z,\Lambda)$ with $Z=\operatorname{Spa}(A,A^{+})$ being
strictly totally disconnected. We thus get an induced map $X\to
Z\times*/G=Z/G$ such that $q$ is the pullback of the projection $q_{Z}\colon
Z\to Z/G$. The induced pullback functor
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(Z/G,\Lambda)\to\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)$
is exact, symmetric monoidal and sends $q_{Z*}\Lambda^{a}$ to
$q_{*}\Lambda^{a}$ (by proper base-change). It is therefore enough to prove
that ($\ast$’) holds for $q_{Z}$ in place of $q$, so from now on we assume
$X=Z/G$ and hence $Y=Z$.

Given a morphism $\Lambda^{\prime}\to\Lambda$ of integral torsion coefficients
on $Z$, the induced base-change functor
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(Z,\Lambda^{\prime})\to\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(Z,\Lambda)$
preserves tensor products, retracts and finite (co)limits and sends
$q_{*}\Lambda^{\prime a}$ to $q_{*}\Lambda^{a}$ (by [Man22, Proposition
3.6.16]), hence if ($\ast^{\prime}$) holds for $\Lambda^{\prime}$ then it also
holds for $\Lambda$. By definition of integral torsion coefficients we can
thus reduce to the case that $\Lambda=\mathcal{O}^{+a}_{Z^{\sharp}}/\pi$ for
some untilt $Z^{\sharp}$ of $Z$ and some pseudouniformizer $\pi$ on
$Z^{\sharp}$. Pick some pseudouniformizer $\varpi$ on $Z^{\sharp}$ such that
$\varpi|p$ and some integer $n>0$ such that $\pi|\varpi^{n}$. By [Mat16,
Proposition 3.35] the map of discrete rings $A^{\sharp+}/\pi\to
A^{\sharp+}/\varpi$ admits descent, which after pullback along $Z/G\to Z$
implies that the map
$\mathcal{O}_{Z^{\sharp}/G}^{+a}/\pi\to\mathcal{O}_{Z^{\sharp}/G}^{+a}/\varpi$
in
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{Z^{\sharp}/G}/\pi)$
admits descent. Therefore, if we can show that ($\ast^{\prime}$) holds for
$\Lambda=\mathcal{O}^{+}_{Z^{\sharp}/G}/\varpi$ then by [Mat16, Proposition
3.24] and [BS17, Lemma 11.10] it also follows for
$\Lambda=\mathcal{O}^{+}_{Z^{\sharp}/G}/\pi$. Thus from now on we can assume
that $\pi=\varpi$, i.e. we now have $\pi|p$.

Now recall the Riemann-Hilbert functor
$RH\colon\operatorname{\mathcal{D}}_{\mathrm{\acute{e}t}}(Z/G,\mathbf{F}_{p})^{\mathrm{oc}}\to\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{Z^{\sharp}/G}/\pi)$
from [Man22, Definition 3.9.21.(a)] (where we ignore the $\varphi$-module
structure). Then $RH$ is symmetric monoidal and exact and satisfies
$RH(q_{*}\mathbf{F}_{p})=q_{*}(\mathcal{O}_{Z^{\sharp}}^{+a}/\pi)$. To verify
the latter, note that this identity can be checked after applying $q^{*}$,
where we get (using base-change)

$\displaystyle
q^{*}RH(q_{*}\mathbf{F}_{p})=RH(q^{*}q_{*}\mathbf{F}_{p})=RH(\operatorname{\mathcal{C}}(G,\mathbf{F}_{p}))=\operatorname{\mathcal{C}}(G,\mathbf{F}_{p})\otimes_{\mathbf{F}_{p}}A^{\sharp+,a}/\pi=\operatorname{\mathcal{C}}(G,A^{\sharp+,a}/\pi)$
$\displaystyle\qquad=q^{*}q_{*}(A^{\sharp+,a}/\pi)$

(here we implicitly use the identity
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{Z^{\sharp}}/\pi)=\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(A^{\sharp+}/\pi)$).
All in all this means that it is enough to show that
$q_{*}\mathbf{F}_{p}\in\operatorname{\mathcal{D}}_{\mathrm{\acute{e}t}}(Z/G,\mathbf{F}_{p})^{\mathrm{oc}}$
is descendable. It follows easily from [Man22, Lemma 3.9.18.(iii)] that
$\operatorname{\mathcal{D}}_{\mathrm{\acute{e}t}}(Z/G,\mathbf{F}_{p})^{\mathrm{oc}}=\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,\mathbf{F}_{p}(Z))_{\omega}$
is the derived $\infty$-category of discrete smooth $G$-representations on
$\mathbf{F}_{p}(Z)$-modules (use the same argument as in the proof of [Man22,
Lemma 3.4.26.(iii)]). Our goal is to show that the $G$-representation
$\operatorname{\mathcal{C}}(G,\mathbf{F}_{p}(Z))$ admits descent in this
$\infty$-category. By considering base-change along
$\mathbf{F}_{p}\to\mathbf{F}_{p}(Z)$ we reduce the claim to the following:

  * ($\ast^{\prime\prime}$)

The algebra object
$\operatorname{\mathcal{C}}(G,\mathbf{F}_{p})\in\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,\mathbf{F}_{p})_{\omega}$
admits descent.

Note that
$\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,\mathbf{F}_{p})_{\omega}$ is
equivalent to the full subcategory of
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\mathbf{F}_{p})^{BG}$
spanned by the discrete $G$-representations (e.g. both $\infty$-categories can
be constructed using the same descent diagram), hence by 3.10 it follows that
for all static
$M,N\in\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,\mathbf{F}_{p})_{\omega}$ we
have $\pi_{0}\operatorname{Hom}(M,N[i])=0$ for $i>d:=\operatorname{cd}_{p}G$.
Now consider the triangle

$\displaystyle\mathbf{F}_{p}\to\operatorname{Tot}_{d}(\operatorname{\mathcal{C}}(G,\mathbf{F}_{p})^{\otimes\bullet+1})\to
X$

in $\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,\mathbf{F}_{p})_{\omega}$,
where $\operatorname{Tot}_{d}$ denotes the $d$-truncated totalization of a
cosimplicial object (i.e. the limit over $\Delta_{\leq d}$). We know that
$\operatorname{Tot}(\operatorname{\mathcal{C}}(G,\mathbf{F}_{p})^{\otimes\bullet+1})=\mathbf{F}_{p}$
(by descent) and by the dual version of [Lur, Proposition 1.2.4.5] we have
$\pi_{i}\operatorname{Tot}_{d}=\pi_{i}\operatorname{Tot}$ for $i>-d$. It
follows that $X$ is concentrated in homological degrees $\leq-d$, so that by
the above vanishing of hom’s the connecting map $X\to\mathbf{F}_{p}[1]$ must
be zero. It follows that
$\operatorname{Tot}_{d}(\operatorname{\mathcal{C}}(G,\mathbf{F}_{p})^{\otimes\bullet+1})\cong\mathbf{F}_{p}\oplus
X$, which immediately implies that
$\operatorname{\mathcal{C}}(G,\mathbf{F}_{p})$ admits descent. ∎

######  Corollary 3.12.

Let $G$ be a profinite group with $\operatorname{cd}_{p}G<\infty$. Then:

  1. (i)

The map $*/G\to*$ is $p$-fine.

  2. (ii)

For every $(X,\Lambda)\in\operatorname{vStack_{\Lambda}}$ with associated map
$f\colon X/G\to X$ there is a natural isomorphism $f_{!}=f_{*}$ of functors
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X/G,\Lambda)\to\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)$.

###### Proof.

Part (i) follows from 3.11 by using the factorization
$\operatorname{id}=[*\to*/G\to*]$. To prove (ii) let $(X,\Lambda)$ be given
and assume first that $(X,\Lambda)$ admits a map to some strictly totally
disconnected space. Let $q\colon X\to X/G$ denote the projection and let
$q_{\bullet}\colon X_{\bullet}\to X/G$ be the Čech nerve of $q$. By 3.11 we
have
$\operatorname{\operatorname{\mathcal{D}}^{a!}_{\scalebox{0.5}{$\square$}}}(X/G,\Lambda)=\varprojlim_{n\in\Delta}\operatorname{\operatorname{\mathcal{D}}^{a!}_{\scalebox{0.5}{$\square$}}}(X_{n},\Lambda)$,
which means that for all
$\mathcal{M}\in\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X/G,\Lambda)$
the natural map
$\varinjlim_{n\in\Delta}q_{n!}q_{n}^{!}\mathcal{M}\mathbin{\xrightarrow{\sim}}\mathcal{M}$
is an isomorphism. Since both $f_{*}$ and $f_{!}$ preserve small colimits (for
$f_{*}$ this follows from $\operatorname{cd}_{p}G<\infty$) and all $q_{n}$ and
$f\mathbin{\circ}q_{n}$ are proper we deduce

$\displaystyle
f_{*}=f_{*}\varinjlim_{n}q_{n!}q_{n}^{!}=\varinjlim_{n}(f\mathbin{\circ}q_{n})_{*}q_{n}^{!}=\varinjlim_{n}(f\mathbin{\circ}q_{n})_{!}q_{n}^{!}=f_{!}.$

The case of general $(X,\Lambda)$ can be deduced via descent (more precisely,
do the above computation on sections of the cocartesian fibrations over
$\Delta$). ∎

Having established that $*/G\to*$ is $p$-fine in many cases, we now determine
criteria for when this map is $p$-cohomologically smooth. When $G$ has finite
$p$-cohomological dimension, the precise answer is captured by the following
definition.

######  Definition 3.13.

Let $G$ be a profinite group with $\operatorname{cd}_{p}G<\infty$. We say that
$G$ is _$p$ -Poincaré of dimension $d$_ if it satisfies the following
conditions:

  1. (i)

$H^{i}(G,M)$ is finite for all $i\geq 0$ and all $G$-representations $M$ on
finite $\mathbf{F}_{p}$-vector spaces.

  2. (ii)

The solid $\mathbf{F}_{p}$-vector space
$R\Gamma(G,\mathbf{F}_{p{\scalebox{0.5}{$\square$}}}[G])$ is invertible and
concentrated in cohomological degree $d$ (i.e. it is isomorphic to
$\mathbf{F}_{p}[-d]$).

Note that condition (i) is equivalent to the finiteness of
$H^{i}(H,\mathbf{F}_{p})$ for all $i\geq 0$ and all open subgroups $H\subseteq
G$. Note also that if $G$ is $p$-Poincaré of dimension $d$, then so is every
open subgroup of $G$.

######  Proposition 3.14.

Let $G$ be a profinite group with $\operatorname{cd}_{p}G<\infty$.

  1. (i)

$G$ satisfies condition (i) of 3.13 if and only if for all strictly totally
disconnected spaces $(X,\Lambda)\in\operatorname{vStack_{\Lambda}}$ with
associated map $f\colon X/G\to X$ the morphism $f^{*}\otimes
f^{!}\Lambda^{a}\mathbin{\xrightarrow{\sim}}f^{!}$ is an equivalence of
functors
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)\to\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X/G,\Lambda)$.

  2. (ii)

$G$ is a $p$-Poincaré group (of dimension $d$) if and only if the map
$*/G\to*$ is $p$-cohomologically smooth (of pure dimension $d/2$).

###### Proof.

Fix a strictly totally disconnected space
$(X,\Lambda)\in\operatorname{vStack_{\Lambda}}$ with associated map $f\colon
X/G\to X$. Then
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)=\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\Lambda(X))$
is the $\infty$-category of solid almost $\Lambda(X)$-modules, while
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X/G,\Lambda)$
is the $\infty$-category of smooth $G$-representations on such modules (see
[Man22, Lemma 3.4.26]). Moreover, $f_{*}$ computes smooth $G$-cohomology and
$f^{*}$ associates to every
$M\in\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\Lambda(X))$
the trivial $G$-representation on $M$.

We first prove (i). By [Man22, Lemma 3.4.19] the collection of functors
$R\Gamma_{\mathrm{sm}}(H,-)\colon\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X/G,\Lambda)\to\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)$
for open compact subgroups $H$ is conservative. Therefore the natural map
$f^{*}\otimes f^{!}\Lambda^{a}\to f^{!}$ is an equivalence if and only if for
all compact open subgroups $H\subset G$ the map
$R\Gamma_{\mathrm{sm}}(H,f^{*}\otimes f^{!}\Lambda^{a})\to
R\Gamma_{\mathrm{sm}}(H,f^{!})$ is an equivalence of endofunctors of
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)$.
Let us denote $f_{H}\colon*/H\to*$ the projection, which factors over the
finite étale projection $*/H\to*/G$. Then we see that $f^{*}\otimes
f^{!}\Lambda^{a}\to f^{!}$ is an equivalence if and only if for all compact
open subgroups $H\subset G$ the map

$\displaystyle f_{H*}(f_{H}^{*}\otimes f_{H}^{!}\Lambda^{a})\to
f_{H*}f_{H}^{!}$

is an equivalence. By 3.12 we have $f_{H*}=f_{H!}$, so in particular $f_{H*}$
satisfies the projection formula. We compute

$\displaystyle f_{H*}(f_{H}^{*}\otimes f_{H}^{!}\Lambda^{a})$
$\displaystyle=-\otimes
f_{H*}f_{H}^{!}\Lambda^{a}=-\otimes\operatorname{\underline{\operatorname{Hom}}}(f_{H*}\Lambda^{a},\Lambda^{a}),$
$\displaystyle f_{H*}f_{H}^{!}$
$\displaystyle=f_{H*}\operatorname{\underline{\operatorname{Hom}}}(\Lambda^{a},f_{H}^{!})=\operatorname{\underline{\operatorname{Hom}}}(f_{H!}\Lambda^{a},-)=\operatorname{\underline{\operatorname{Hom}}}(f_{H*}\Lambda^{a},-).$

It follows that $f^{*}\otimes f^{!}\Lambda^{a}\to f^{!}$ is an equivalence if
and only if $R\Gamma_{\mathrm{sm}}(H,\Lambda(X)^{a})=f_{H*}\Lambda^{a}$ is
dualizable in
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)$
for all open compact $H\subset G$. Now assume that $G$ satisfies condition (i)
of 3.13. Then $R\Gamma_{\mathrm{sm}}(H,\mathbf{F}_{p})$ is dualizable in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\mathbf{F}_{p})$ for
all $H$. If $p=0$ in $\Lambda(X)$ then the same follows for
$R\Gamma_{\mathrm{sm}}(H,\Lambda(X)^{a})=R\Gamma_{\mathrm{sm}}(H,\mathbf{F}_{p})\otimes_{\mathbf{F}_{p}}\Lambda(X)^{a}$.
For general $\Lambda$ we can argue as in the proof of [Man22, Lemma
3.8.4.(i)]. Conversely, assume that $f^{*}\otimes
f^{!}\Lambda^{a}\mathbin{\xrightarrow{\sim}}f^{!}$ is an isomorphism for all
$(X,\Lambda)$. Pick $X=\operatorname{Spa}C$ for some algebraically closed
field over $\mathbf{Q}_{p}$ and $\Lambda=\mathcal{O}_{C}/p$. Then the Riemann-
Hilbert correspondence from [Man22, Theorem 3.9.23] implies that the base-
change functor
$RH\colon\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,\mathbf{F}_{p})\mathrel{\hookrightarrow}\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,\mathcal{O}^{a}_{C}/p)^{\varphi}$
is fully faithful and induces an equivalence of dualizable objects. On the
other hand, this base-change functor sends
$R\Gamma_{\mathrm{sm}}(H,\mathbf{F}_{p})$ to
$R\Gamma_{\mathrm{sm}}(H,\mathcal{O}^{a}_{C}/p)$, hence it follows from the
above that $R\Gamma_{\mathrm{sm}}(H,\mathbf{F}_{p})$ is dualizable for all
open compact subgroups $H\subset G$. But every $G$-representation $M$ on a
finite static $\mathbf{F}_{p}$-vector space is a finite colimit of the
generating representations $\mathbf{F}_{p}[G/H]$, hence also
$R\Gamma_{\mathrm{sm}}(G,M)$ is dualizable. In particular all $H^{i}(G,M)$ are
finite (here we implicitly use $R\Gamma(G,M)=R\Gamma_{\mathrm{sm}}(G,M)$ since
$M$ is discrete).

We now prove (ii), so from now on we can assume that $G$ satisfies the
equivalent conditions of (i). We then need to see that $G$ is $p$-Poincaré if
and only if for every strictly totally disconnected
$(X,\Lambda)\in\operatorname{vStack_{\Lambda}}$ with
$\Lambda=\mathcal{O}^{+}_{X^{\flat}}/\pi$ for some pseudouniformizer $\pi$ on
$X^{\flat}$ the object
$f^{!}\Lambda^{a}\in\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X/G,\Lambda)$
is invertible. Given $(X,\Lambda)$, let us denote $q\colon X\to X/G$ the
projection. Since invertibility satisfies descent, $f^{!}\Lambda^{a}$ is
invertible if and only if $q^{*}f^{!}\Lambda^{a}$ is invertible. For every
compact open subgroup $H\subset G$ we denote $f_{H}\colon*/H\to*$ and
$q_{H}\colon*/H\to*/G$ the natural maps and we compute (using [Man22,
Proposition 3.3.9.(iii)] and 3.12)

$\displaystyle
q^{*}f^{!}\Lambda^{a}=f_{*}q_{*}q^{*}f^{!}\Lambda^{a}=\varinjlim_{H}f_{*}q_{H*}q_{H}^{*}f^{!}\Lambda^{a}=\varinjlim_{H}f_{H*}f_{H}^{!}\Lambda^{a}=\varinjlim_{H}\operatorname{\underline{\operatorname{Hom}}}(f_{H*}\Lambda^{a},\Lambda^{a}).$

By the proof of (i) we know that $f_{H*}\Lambda^{a}=R\Gamma(H,\Lambda(X)^{a})$
is dualizable in
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(X,\Lambda)$
and similarly $R\Gamma(H,\mathbf{F}_{p})$ is dualizable in
$\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\mathbf{F}_{p})$. It
follows that

$\displaystyle\operatorname{\underline{\operatorname{Hom}}}(f_{H*}\Lambda^{a},\Lambda^{a})=\operatorname{\underline{\operatorname{Hom}}}(\Gamma(H,\mathbf{F}_{p}),\mathbf{F}_{p})\otimes_{\mathbf{F}_{p}}\Lambda(X)^{a}.$

Therefore we obtain

$\displaystyle
q^{*}f^{!}\Lambda^{a}=\varinjlim_{H}\operatorname{\underline{\operatorname{Hom}}}(f_{H*}\Lambda^{a},\Lambda^{a})=(\varinjlim_{H}\operatorname{\underline{\operatorname{Hom}}}(\Gamma(H,\mathbf{F}_{p}),\mathbf{F}_{p}))\otimes_{\mathbf{F}_{p}}\Lambda(X)^{a}$

We have $R\Gamma(H,\mathbf{F}_{p})=\bigoplus_{i}H^{i}(H,\mathbf{F}_{p})[-i]$
and since there are no higher $\operatorname{Ext}$-groups between discrete
$\mathbf{F}_{p}$-vector spaces we even have a decomposition of diagrams
$(R\Gamma(H,\mathbf{F}_{p})_{H}=\bigoplus_{i}(H^{i}(H,\mathbf{F}_{p}))_{H}$.
On the other hand, we know that if $(M_{i})_{i}$ is any cofiltered system of
finite discrete $\mathbf{F}_{p}$-vector spaces then $\varprojlim_{i}M_{i}$ is
concentrated in degree $0$ and
$\operatorname{\underline{\operatorname{Hom}}}(\varprojlim_{i}M_{i},\mathbf{F}_{p})=\varinjlim_{i}\operatorname{\underline{\operatorname{Hom}}}(M_{i},\mathbf{F}_{p})$.
Altogether this implies

$\displaystyle\varinjlim_{H}\operatorname{\underline{\operatorname{Hom}}}(R\Gamma(H,\mathbf{F}_{p}),\mathbf{F}_{p})$
$\displaystyle=\operatorname{\underline{\operatorname{Hom}}}(\varprojlim_{H}R\Gamma(H,\mathbf{F}_{p}),\mathbf{F}_{p})=\operatorname{\underline{\operatorname{Hom}}}(\varprojlim_{H}\Gamma(G,\mathbf{F}_{p}[G/H]),\mathbf{F}_{p})=$
$\displaystyle=\operatorname{\underline{\operatorname{Hom}}}(R\Gamma(G,\mathbf{F}_{p{\scalebox{0.5}{$\square$}}}[G]),\mathbf{F}_{p}).$

We deduce:

$\displaystyle
q^{*}f^{!}\Lambda^{a}=\operatorname{\underline{\operatorname{Hom}}}(R\Gamma(G,\mathbf{F}_{p{\scalebox{0.5}{$\square$}}}[G]),\mathbf{F}_{p})\otimes_{\mathbf{F}_{p}}\Lambda(X)^{a}.$

In particular, if $G$ is $p$-Poincaré then $q^{*}f^{!}\Lambda^{a}$ is
invertible. Conversely, assume that $*/G\to*$ is $p$-cohomologically smooth.
Choosing $X=\operatorname{Spa}C$ for some algebraically closed field $C$, the
Riemann-Hilbert correspondence (see [Man22, Theorem 3.9.23]) tells us that the
invertibility of $q^{*}f^{!}\Lambda^{a}$ implies that
$\operatorname{\underline{\operatorname{Hom}}}(R\Gamma(G,\mathbf{F}_{p{\scalebox{0.5}{$\square$}}}[G]),\mathbf{F}_{p})$
is invertible. But then also
$R\Gamma(G,\mathbf{F}_{p{\scalebox{0.5}{$\square$}}}[G])$ is invertible and
thus $G$ is $p$-Poincaré. ∎

So far we have only dealt with profinite groups of bounded $p$-cohomological
dimension. However, the above results generalize in a straightforward way to
_locally_ profinite groups satisfying a much weaker cohomological dimension
condition. The precise definition is as follows.

######  Definition 3.15.

Let $G$ be a locally profinite group.

  1. (a)

We say that $G$ has _locally finite $p$-cohomological dimension_ if there is
some open subgroup $H\subseteq G$ such that $\operatorname{cd}_{p}H<\infty$.

  2. (b)

We say that $G$ is _virtually $p$-Poincaré (of dimension d)_ if there is an
open compact subgroup $H\subset G$ such that $\operatorname{cd}_{p}H<\infty$
and $H$ is $p$-Poincaré of dimension $d$ in the sense of Definition 3.13.

In the following, whenever we talk about a “virtual $p$-Poincaré group $G$” we
implicitly assume that $G$ has locally finite $p$-cohomological dimension.

######  Theorem 3.16.

Let $G$ be a locally profinite group which has locally finite
$p$-cohomological dimension. Then the map $*/G\to*$ is $p$-fine, and it is
$p$-cohomologically smooth (of pure dimension $d/2$) if and only if $G$ is
virtually $p$-Poincaré (of dimension $d$).

###### Proof.

This follows immediately from 3.14 by observing that being $p$-fine and being
$p$-cohomologically smooth can both be checked étale locally on the source
(note that if $H\subseteq G$ is an open subgroup then the map
$*/H\mathrel{\twoheadrightarrow}*/G$ is an étale cover). ∎

As a formal consequence of 3.16 we obtain the following $p$-adic analog of
[Sch18, Proposition 24.2]:

######  Corollary 3.17.

Let $G$ be a locally profinite group of locally finite $p$-cohomological
dimension acting on a small v-stack $Y$ and let $f\colon Y\to X$ be a
$G$-equivariant separated bdcs map of small v-stacks, where $G$ acts trivially
on $X$.

  1. (i)

The map $f^{\prime}\colon Y/G\to X$ is $p$-fine.

  2. (ii)

If $f$ is $p$-cohomologically smooth and $G$ is virtually $p$-Poincaré, then
$f^{\prime}$ is $p$-cohomologically smooth.

###### Proof.

Factor $f^{\prime}$ as $Y/G\to X/G\to X$. We claim that the first map, i.e.
$f/G\colon Y/G\to X/G$, is bdcs: Base-changing this map along the pro-étale
cover $X\mathrel{\twoheadrightarrow}X/G$ gives $f$, hence the map $Y/G\to X/G$
is separated by [Sch18, Proposition 10.11.(ii)], compactifiable by [Sch18,
Proposition 22.3.(iii)] (note that $f$ is automatically compactifiable by
[Sch18, Proposition 22.3.(v)]), $p$-bounded by [Man22, Lemma 3.5.10.(ii)] and
representable in locally spatial diamonds by [Sch18, Proposition 13.4.(iv)].
Since $X/G\to X$ is $p$-fine by 3.12 we deduce that $f^{\prime}$ is $p$-fine.
This proves (i).

To prove (ii), assume in addition that $f$ is $p$-cohomologically smooth and
$G$ is virtually $p$-Poincaré. Then it follows from [Man22, Lemma 3.8.5.(ii)]
that $f/G$ is $p$-cohomologically smooth, and it follows from 3.16 that the
map $X/G\to X$ is $p$-cohomologically smooth. Together this implies that
$f^{\prime}$ is $p$-cohomologically smooth, as desired. ∎

We end this subsection by providing examples of virtual $p$-Poincaré groups.
The key result for applications is the following theorem.

######  Theorem 3.18.

Any $p$-adic Lie group is virtually $p$-Poincaré.

By the local nature of Definition 3.15 and standard structural results on
$p$-adic Lie groups, this is an immediate consequence of the following
theorem, which is what we shall actually prove.

######  Theorem 3.19.

Any torsion-free compact $p$-adic analytic group is $p$-Poincaré.

The proof relies on deep results of Lazard [Laz65], which are in some sense
the only “non-formal” input we need. To apply Lazard’s results, we need some
general comparison between condensed group cohomology and classical continuous
group cohomology, which takes into account a natural topological structure on
the latter. To state the relevant result, fix a locally profinite group $G$.
Let $V$ be a continuous representation of $G$ on a Hausdorff locally compact
$\mathbf{F}_{p}$-module. Then $\underline{V}$ is a solid
$\mathbf{F}_{p}$-vector space with continuous $G$-action, and in particular is
an $\mathbf{F}_{p{\scalebox{0.5}{$\square$}}}[G]$-module. Thus we may form the
condensed group cohomology

$R\Gamma(G,\underline{V})=\operatorname{\underline{\operatorname{Hom}}}_{\mathbf{F}_{p{\scalebox{0.5}{$\square$}}}[G]}(\mathbf{F}_{p},\underline{V})\in\mathcal{D}_{{\scalebox{0.5}{$\square$}}}(\mathbf{F}_{p})$

as considered above.

On the other hand, the classical continuous group cohomology
$R\Gamma_{\mathrm{cts}}(G,V)$ is by definition computed by the usual complex

$0\to V\to\mathcal{C}(G,V)\to\mathcal{C}(G^{2},V)\to\cdots$

of inhomogeneous cochains. Giving each $\mathcal{C}(G^{n},V)$ the compact-open
topology, this is naturally a complex of Hausdorff locally compact
$\mathbf{F}_{p}$-modules. In particular, passing termwise to the condensation,
we get a complex

$0\to\underline{V}\to\underline{\mathcal{C}(G,V)}\to\underline{\mathcal{C}(G^{2},V)}\to\cdots$

of solid $\mathbf{F}_{p}$-vector spaces. Let
$\underline{R\Gamma_{\mathrm{cts}}(G,V)}$ denote the image of this complex in
$\mathcal{D}_{{\scalebox{0.5}{$\square$}}}(\mathbf{F}_{p})$.

######  Lemma 3.20.

There is a natural isomorphism

$R\Gamma(G,\underline{V})\simeq\underline{R\Gamma_{\mathrm{cts}}(G,V)}$

in $\mathcal{D}_{{\scalebox{0.5}{$\square$}}}(\mathbf{F}_{p})$.

###### Proof of Theorem 3.19.

Let $G$ be as in the statement of the theorem. Then condition (i) in
Definition 3.13 is classical, see e.g. [SW00, §5]. For condition (ii), recall
that $R\Gamma_{\mathrm{cts}}(G,\mathbf{F}_{p}[[G]])\simeq\mathbf{F}_{p}[-\dim
G]$ by [Laz65, Ch. V] and [SW00, §4-5]. Since
$\underline{\mathbf{F}_{p}[[G]]}=\mathbf{F}_{p{\scalebox{0.5}{$\square$}}}[G]$,
the conclusion now follows from Lemma 3.20. ∎

###### Proof of Lemma 3.20.

Arguing as in the proof of [Man22, Proposition 3.4.6], one sees that
$R\Gamma(G,\underline{V})$ is computed as the cohomology of the complex

$0\to\underline{V}\to\underline{\operatorname{Hom}}(\underline{G},\underline{V})\to\underline{\operatorname{Hom}}(\underline{G}^{2},\underline{V})\to\cdots$

where the internal homs are taken in condensed sets. By the definition of
$\underline{R\Gamma_{\mathrm{cts}}(G,V)}$, it now suffices to show that
$\underline{\operatorname{Hom}}(\underline{S},\underline{V})$ identifies with
$\underline{\operatorname{\mathcal{C}}(S,V)}$ functorially in locally
profinite $S$. This is a special case of the next lemma. ∎

######  Lemma 3.21.

Let $X$ be any locally compact Hausdorff space, and let $Y$ be any compactly
generated T1 topological space. Equip the space of continuous maps
$\operatorname{\mathcal{C}}(X,Y)$ with the compact-open topology. Then
$\operatorname{\mathcal{C}}(X,Y)$ is compactly generated T1, and
$\underline{\operatorname{\mathcal{C}}(X,Y)}=\underline{\operatorname{Hom}}(\underline{X},\underline{Y})$,
where $\underline{\operatorname{Hom}}$ denotes the internal hom in condensed
sets.

###### Proof.

The claim regarding the topology on $\operatorname{\mathcal{C}}(X,Y)$ is easy.
For the remaining claim, let $T$ be any profinite space. We then compute that

$\displaystyle\operatorname{Hom}_{\mathrm{Cond}}(\underline{T},\underline{\operatorname{Hom}}(\underline{X},\underline{Y}))$
$\displaystyle=\operatorname{Hom}_{\mathrm{Cond}}(\underline{T}\times\underline{X},\underline{Y})$
$\displaystyle=\operatorname{Hom}_{\mathrm{Top}}(T\times X,Y)$
$\displaystyle=\operatorname{Hom}_{\mathrm{Top}}(T,\operatorname{\mathcal{C}}(X,Y))$
$\displaystyle=\operatorname{Hom}_{\mathrm{Cond}}(\underline{T},\underline{\operatorname{\mathcal{C}}(X,Y)})$

naturally in $T,X,Y$. The first equality is immediate from the universal
property of $\underline{\operatorname{Hom}}$. The second equality follows from
the fact that on compactly generated T1 spaces, $S\mapsto\underline{S}$ is
fully faithful and commutes with finite products. The fourth equality follows
similarly, using the first claim in the lemma. The third equality follows from
the assumption that $X$ is locally compact Hausdorff, which is exactly the
condition needed to ensure that $\operatorname{\mathcal{C}}(X,-)$ with its
compact-open topology is right-adjoint to $-\times X$. Since $T$ is arbitrary,
the result now follows from Yoneda. ∎

###  3.3. Admissible representations

Given a locally profinite group $G$, an important subclass of all smooth
$G$-representations are the admissible ones. Classically, a smooth
$G$-representation on an $\mathbf{F}_{p}$-vector space $V$ is called
admissible if for a basis of compact open subgroups $H\subseteq G$ the space
$V^{H}$ of $H$-invariants is finite. In the condensed derived world the
correct analogue of this definition seems to be the following:

######  Definition 3.22.

Let $G$ be a locally profinite group which has locally finite
$p$-cohomological dimension. A smooth $G$-representation
$V\in\operatorname{\mathcal{D}}^{\mathrm{sm}}_{\scalebox{0.5}{$\square$}}(G,\mathbf{F}_{p})$
is called _admissible_ if for a basis of compact open subgroups $H\subseteq G$
the smooth group cohomology
$R\Gamma_{\mathrm{sm}}(H,V)\in\operatorname{\mathcal{D}}_{\scalebox{0.5}{$\square$}}(\mathbf{F}_{p})$
is perfect.

######  Remarks 3.23.

  1. (i)

An admissible $G$-representation
$V\in\operatorname{\mathcal{D}}^{\mathrm{sm}}_{\scalebox{0.5}{$\square$}}(G,\mathbf{F}_{p})$
is automatically discrete because all $R\Gamma_{\mathrm{sm}}(H,V)$ are
discrete by assumption and hence so is
$V=\varinjlim_{H}R\Gamma_{\mathrm{sm}}(H,V)$ (cf. [Man22, Lemma 3.4.19]). Thus
$V$ admits a representation as a complex of classical smooth
$G$-representations.

  2. (ii)

Suppose that $G$ is a $p$-adic Lie group. Then by [SS22, Corollary 4.12] a
smooth representation
$V\in\operatorname{\mathcal{D}}^{\mathrm{sm}}_{\scalebox{0.5}{$\square$}}(G,\mathbf{F}_{p})$
is admissible if and only if it is bounded and discrete and all $\pi_{n}V$ are
admissible smooth representations in the classical sense.

Now suppose we are additionally given an algebraically closed perfectoid field
$C$ with pseudouniformizer $\pi|p$ and we consider the classifying stack
$B\underline{G}=\operatorname{Spd}(C)/\underline{G}$. Then
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{B\underline{G}}/\pi)^{\varphi}$
is the $\infty$-category of smooth $G$-representations on solid almost
$\varphi$-modules over $\mathcal{O}^{a}_{C}/\pi$ and by the $p$-torsion
Riemann-Hilbert correspondence this $\infty$-category contains the
$\infty$-category $\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,\mathbf{F}_{p})$
of smooth $G$-representations as a full subcategory. We make the following
definition:

######  Definition 3.24.

Let $G$ be a locally profinite group which has locally finite
$p$-cohomological dimension. Let $C$ be an algebraically closed perfectoid
field with pseudouniformizer $\pi\in\mathcal{O}_{C}$ such that $\pi|p$. A
smooth $G$-representation
$M\in\operatorname{\mathcal{D}}^{{\mathrm{sm}},a}_{\scalebox{0.5}{$\square$}}(G,\mathcal{O}_{C}/\pi)^{\varphi}$
is called _admissible_ if it comes from an admissible
$\mathbf{F}_{p}$-representation via the embedding

$\displaystyle-\otimes_{\mathbf{F}_{p}}\mathcal{O}^{a}_{C}/\pi\colon\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,\mathbf{F}_{p})\mathrel{\hookrightarrow}\operatorname{\mathcal{D}}^{{\mathrm{sm}},a}_{\scalebox{0.5}{$\square$}}(G,\mathcal{O}_{C}/\pi)^{\varphi}$

obtained from the Riemann-Hilbert correspondence on the classifying stack
$B\underline{G}$.

One might also consider defining $M$ to be admissible if the smooth group
cohomology groups $R\Gamma_{\mathrm{sm}}(H,M)$ are perfect for a basis of open
subgroups $H\subseteq G$. In fact, this easily turns out to be equivalent to
3.24, as we will see below. If $G$ is $p$-Poincaré then there is also a
characterization of admissible representations in terms of their behavior
under duality. To show this, we need the following characterization of perfect
(equivalently, dualizable) objects in terms of their duals:

######  Lemma 3.25.

Let $K$ be a perfectoid field with pseudouniformizer $\pi\in\mathcal{O}_{K}$.
Then a $\varphi$-module
$M\in\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}_{K}/\pi)^{\varphi}$
is dualizable if and only if $M$ is bounded, both $M$ and $M^{\vee}$ are
discrete and the natural map $M\to M^{\vee\vee}$ is an isomorphism.

###### Proof.

First assume that $M$ is dualizable. Then by [Man22, Proposition 3.7.5] both
$M$ and $M^{\vee}$ are discrete. It follows from [Man22, Lemmas 3.9.2, 3.9.3]
that $M$ is bounded. The isomorphism
$M\mathbin{\xrightarrow{\sim}}M^{\vee\vee}$ is clear. This proves the “only
if” part of the claim.

We now prove the “if” part, so assume that $M$ is bounded, $M$ and $M^{\vee}$
are discrete and $M\mathbin{\xrightarrow{\sim}}M^{\vee\vee}$ is an
isomorphism. We want to show that $M$ is dualizable, for which we can assume
that $M$ is concentrated in homological degree $\geq 0$. Now consider
$M_{0}:=\pi_{0}M$, which is a static discrete $\mathcal{O}^{a}_{K}$-module.
From the fact that $M^{\vee}$ is discrete it follows that
$\pi_{0}M_{0}^{\vee}=\pi_{0}M^{\vee}$ is discrete. After choosing a surjection
$\bigoplus_{I}\mathcal{O}^{a}_{K}/\pi\mathrel{\twoheadrightarrow}M_{0}$ we get
an injection
$\pi_{0}M_{0}^{\vee}\mathrel{\hookrightarrow}\prod_{I}\mathcal{O}^{a}_{K}/\pi$.
Now apply $\pi_{0}(-)_{*}$ to get a submodule

$\displaystyle
M_{0}^{\prime}:=\pi_{0}(\pi_{0}M_{0}^{\vee})_{*}\subseteq\prod_{I}A,$

where we abbreviate $A:=\pi_{0}(\mathcal{O}^{a}_{K}/\pi)_{*}$. Note that
$M_{0}^{\prime}$ is equipped with the subspace topology since it is a kernel
of a map $\prod_{I}A\to\prod_{J}A$. Moreover, the fact that
$\pi_{0}M_{0}^{\vee}$ is discrete implies that for every
$\varepsilon\in\mathfrak{m}_{K}$ the submodule $T_{\varepsilon}\subset
M_{0}^{\prime}$ of $\varepsilon$-torsion in $M_{0}^{\prime}$ is open. In
particular there is an open subset $U_{\varepsilon}\subset\prod_{I}A$
containing $0$ such that $U_{\varepsilon}\mathbin{\cap}M_{0}^{\prime}\subseteq
T_{\varepsilon}$. In fact we can choose $U_{\varepsilon}$ to be an element of
the standard basis for the product topology, i.e.
$U_{\varepsilon}=U_{\varepsilon}^{\prime}\times\prod_{I\setminus
I_{\varepsilon}}A$ for some finite subset $I_{\varepsilon}\subset I$ and some
open subset $U_{\varepsilon}^{\prime}\subset\prod_{I_{\varepsilon}}A$. This
means that if $(a_{i})_{i},(b_{i})_{i}\in\prod_{I}A$ lie in $M_{0}^{\prime}$
and $a_{i}=b_{i}$ for $i\in I_{\varepsilon}$ then
$\varepsilon(a_{i})_{i}=\varepsilon(b_{i})_{i}$. Now let
$M_{0,\varepsilon}^{\prime}\subset\prod_{I_{\varepsilon}}A$ be the image of
$M_{0}^{\prime}$ under the projection
$\prod_{I}A\mathrel{\twoheadrightarrow}\prod_{I_{\varepsilon}}A$ (equipped
with the subspace topology). Then there is a natural projection map
$f^{\prime}_{\varepsilon}\colon M_{0}^{\prime}\to M_{0,\varepsilon}^{\prime}$.
We can also construct a map $g^{\prime}_{\varepsilon}\colon
M_{0,\varepsilon}^{\prime}\to M_{0}^{\prime}$ as follows: Given $a\in
M_{0,\varepsilon}^{\prime}$ we pick any $a^{\prime}\in M_{0}^{\prime}$ such
that $f^{\prime}_{\varepsilon}(a^{\prime})=a$. Then we set
$g^{\prime}_{\varepsilon}(a):=\varepsilon a^{\prime}$, which by the above
observation is independent of the choice of $a^{\prime}$. Note that
$f^{\prime}_{\varepsilon}g^{\prime}_{\varepsilon}=\varepsilon$ and
$g^{\prime}_{\varepsilon}f^{\prime}_{\varepsilon}=\varepsilon$, hence we have
$M^{\prime}_{0}\approx_{\varepsilon}M^{\prime}_{0,\varepsilon}$. Going back to
the almost world we now see that

$\displaystyle\pi_{0}M_{0}^{\vee}\approx_{\varepsilon}M^{\prime
a}_{0,\varepsilon},\qquad\text{with $M^{\prime
a}_{0,\varepsilon}\subseteq\prod_{i=1}^{n_{\varepsilon}}\mathcal{O}^{a}_{K}/\pi$
a discrete submodule},$

where we took the liberty of naming
$I_{\varepsilon}=\\{1,\dots,n_{\varepsilon}\\}$. By [Man22, Lemma 3.7.15] the
discrete dualization functor $N\mapsto(N^{\vee})_{\omega}$ is $t$-exact on
discrete $N$. By applying this functor to $M^{\vee}$ it follows from the
isomorphism $M\mathbin{\xrightarrow{\sim}}M^{\vee\vee}$ that also
$M_{0}\mathbin{\xrightarrow{\sim}}((\pi_{0}M_{0}^{\vee})^{\vee})_{\omega}$ is
an isomorphism. Letting $M_{0,\varepsilon}:=((M^{\prime
a}_{0,\varepsilon})^{\vee})_{\omega}$ we deduce that
$M_{0}\approx_{\varepsilon}M_{0,\varepsilon}$ and there is a surjection
$\bigoplus_{i=1}^{n_{\varepsilon}}\mathcal{O}^{a}_{K}/\pi\mathrel{\twoheadrightarrow}M_{0,\varepsilon}$.
As this works for all $\varepsilon$, we deduce that $M_{0}$ is almost finitely
generated as an $\mathcal{O}^{a}_{K}/\pi$-module. Then it is also almost
finitely generated as an $\mathcal{O}^{a}_{K}$-module (where we equip
$\mathcal{O}^{a}_{K}$ with the discrete topology in this proof), hence by
[Sch13, Theorem 2.5] we have

$\displaystyle
M_{0}\approx(\mathcal{O}^{a}_{K}/\pi)^{r}\oplus\mathcal{O}^{a}_{K}/I_{\gamma_{1}}\oplus\mathcal{O}^{a}_{K}/I_{\gamma_{2}}\oplus\dots$

for a series of real numbers $1>\gamma_{1}\geq\gamma_{2}\geq\dots\geq 0$
converging to $0$. Here for a real number $\gamma$ we denote by
$I_{\gamma}\subset\mathcal{O}_{K}$ the ideal generated by
$\pi^{\gamma^{\prime}}$ for all $\gamma^{\prime}>\gamma$ (see [Sch13, Example
2.4.(i)]). The $\varphi$-module structure
$\varphi^{*}M\mathbin{\xrightarrow{\sim}}M$ induces an isomorphism
$\pi_{0}\varphi^{*}M_{0}\mathbin{\xrightarrow{\sim}}M_{0}$. On the other hand
for $\gamma<1$ we have

$\displaystyle\pi_{0}\varphi^{*}(\mathcal{O}^{a}_{K}/I_{\gamma})=\begin{cases}\mathcal{O}^{a}_{K}/\pi,\qquad&\text{if
$\gamma\geq 1/p$},\\\
\mathcal{O}^{a}_{K}/I_{p\gamma},\qquad&\text{else},\end{cases}$

as $\mathcal{O}^{a}_{K}/\pi$-modules. This forces all $\gamma_{i}=0$, i.e. we
have $M_{0}\approx(\mathcal{O}^{a}_{K}/\pi)^{r}$. It follows that $M_{0}$ is
dualizable: We need to check that the natural map $\alpha\colon M_{0}\otimes
M_{0}^{\vee}\to\operatorname{\underline{\operatorname{Hom}}}(M_{0},M_{0})$ is
an isomorphism but since the same statement holds for
$(\mathcal{O}^{a}_{K}/\pi)^{r}$ in place of $M_{0}$ one deduces easily that
both kernel and cokernel of $\pi_{i}\alpha$ are killed by $\varepsilon$ for
all $i\in\mathbf{Z}$ and $\varepsilon\in\mathfrak{m}_{K}$ (cf. the proof of
[Man22, Lemma 3.7.17]). In particular $M_{0}$ is flat (which can also be seen
directly), so that $\pi_{0}\varphi^{*}M_{0}=\varphi^{*}M$ and therefore the
$\varphi$-module structure on $M_{0}$ restricts to a $\varphi$-module
structure on $M_{0}$. Since the conditions on $M$ are preserved under finite
limits, we reduce the claimed dualizability of $M$ to that of $\tau_{\geq
1}M$, which satisfies the same assumptions as $M$. Since $M$ is bounded, we
can conclude by induction. ∎

As promised, here come the three equivalent ways of characterizing admissible
representations on a classifying stack. The third characterization is very
useful for showing that admissible representations are stable under various of
the 6 operations, and plays a key role in our analysis of the Jacquet-
Langlands functor.

######  Proposition 3.26.

Let $C$ be an algebraically closed perfectoid field with pseudouniformizer
$\pi\in\mathcal{O}_{C}$ such that $\pi|p$ and let $G$ be a locally profinite
$p$-Poincaré group. Then for
$M\in\operatorname{\mathcal{D}}^{{\mathrm{sm}},a}_{\scalebox{0.5}{$\square$}}(G,\mathcal{O}_{C}/\pi)^{\varphi}$
the following are equivalent:

  1. (i)

$M$ is bounded and admissible.

  2. (ii)

$M$ is bounded and for a basis of compact open subgroups $H\subseteq G$,
$\Gamma_{\mathrm{sm}}(H,M)$ is dualizable in
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}_{C}/\pi)$.

  3. (iii)

Both $M$ and $M^{\vee}$ are bounded and discrete and the natural morphism
$M\mathbin{\xrightarrow{\sim}}M^{\vee\vee}$ is an isomorphism.

If this is the case then $M^{\vee}$ is bounded and admissible and this dual
can unambiguously be computed in any of the following $\infty$-categories:
$\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,\mathbf{F}_{p})$,
$\operatorname{\mathcal{D}}^{\mathrm{sm}}_{\scalebox{0.5}{$\square$}}(G,\mathbf{F}_{p})$,
$\operatorname{\mathcal{D}}^{{\mathrm{sm}},a}_{\scalebox{0.5}{$\square$}}(G,\mathcal{O}_{C}/\pi)^{\varphi}_{\omega}$,
$\operatorname{\mathcal{D}}^{{\mathrm{sm}},a}_{\scalebox{0.5}{$\square$}}(G,\mathcal{O}_{C}/\pi)^{\varphi}$.

###### Proof.

Note first that (i) implies (ii) because
$R\Gamma_{\mathrm{sm}}(H,V)\otimes_{\mathbf{F}_{p}}\mathcal{O}^{a}_{C}/\pi=R\Gamma_{\mathrm{sm}}(H,V\otimes_{\mathbf{F}_{p}}\mathcal{O}^{a}_{C}/\pi)$
for all $V\in\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,\mathbf{F}_{p})$ (e.g.
by the explicit computation of smooth group cohomology in [Man22, Corollary