* The Frobenius $\varphi\colon \mathcal A \to \mathcal A$ induces the structure of an analytic $\sigma$-ring $(\mathcal A, \varphi)$ over $(V,\mm)$. This construction is functorial in $\mathcal A$ as follows: Let $\AnRing^p_{(-)} \subset \AnRing_{(-)}$ be the full subcategory spanned by the analytic rings lying over an almost setup $(V,\mm)$ with $p = 0$ in $V$. The forgetful functor $\sigobj{\AnRing^p_{(-)}} \to \AnRing^p_{(-)}$ induces an equivalence of $\AnRing^p_{(-)}$ with the full subcategory of $\sigobj{\AnRing^p_{(-)}}$ spanned by the objects $(\mathcal A, \varphi)$. Inverting this equivalence yields a fully faithful functor

\begin{align*}
\AnRing^p_{(-)} \injto \sigobj{\AnRing^p_{(-)}}, \qquad \mathcal A \mapsto (\mathcal A, \varphi).
\end{align*}

We say that $\mathcal A$ is perfect if $(\mathcal A, \varphi)$ is perfect. Since the induced morphism $\varphi_V\colon (V,\mm) \to (V,\mm)$ is always strict (by <cit.>), $\mathcal A$ is perfect if and only if the map $\varphi\colon \mathcal A \to \varphi_{V*} \mathcal A$ is an isomorphism.

* A $\varphi$-module over $\mathcal A$ is an object of $\D(\mathcal A)^\varphi$. A lax $\varphi$-module over $\mathcal A$ is an object of $\D(\mathcal A[T_\varphi])$.

We first study $\varphi$-modules in the local setting, i.e. on affinoid perfectoid spaces. The main result of this investigation is a local Riemann-Hilbert correspondence (see <ref>), which lies at the heart of the proof of the global Riemann-Hilbert correspondence. In the local Riemann-Hilbert correspondence we want to relate the $\infty$-category $\D(\Fld_p(X))_\omega$ of discrete $\Fld_p(X)$-modules (where $\Fld_p(X)$ is the ring of continuous maps $\abs X \to \Fld_p$) to the $\infty$-category $\Dqcohri(A^+/\pi)^\varphi$ of $\varphi$-modules over $(A^+/\pi)^a_\solid$, for any strictly totally disconnected space $X = \Spa(A, A^+)$ of characteristic $p$ and any pseudouniformizer $\pi \in A^+$. The general strategy is as follows:

* Given a $\varphi$-module $M \in \Dqcohri(A^+/\pi)^\varphi$, we can use the $\varphi$-structure on $M$ to “lift” $M$ to an object in $\Dqcohri(A^+)^\varphi$. Roughly this operation takes $M$ to $\varprojlim_{\varphi_M} M$. For example, if $M = A^{+a}/\pi$ then the associated $\varphi$-module in $\Dqcohri(A^+)^\varphi$ is $A^{+a}$.

* The functor $\D(\Fld_p(X))_\omega \to \Dqcohri(A^+/\pi)$ is given by $- \tensor (A^{+a}/\pi)$ and it admits the right adjoint $(-)^\varphi_\omega$ (cf. <ref>). By the previous step $(-)^\varphi$ factors over $\Dqcohri(A^+)^\varphi$, which makes the situation more pleasent as $A^+$ is perfect. However, the functor $(-)^\varphi\colon \Dqcohri(A^+)^\varphi \to \D(\Fld_p(X))$ is still a bit more subtle than one might expect, because it entails the functor $(-)_*\colon \Dqcohri(A^+) \to \D_\solid(A^+)$, passing from almost modules to honest modules. The functor $(-)_*$ does not preserves colimits, so it is a priori not clear that $(-)^\varphi$ does. Here the following observation comes to the rescue: We have $M^\varphi = (M \tensor_{A^+} A)^\varphi$. This is helpful because $\pi$ is invertible in $A$ so that the almost world over $A$ collapses to the non-almost world.

* The previous step will eventually allow us to reduce the local Riemann-Hilbert correspondence to an understanding of the relation between $\D(\Fld_p(X))_\omega$ and $\D(A_\omega)^\varphi_\omega$. After reducing to connected components (so that $A = C$ is an algebraically closed field) we end up in the classical situation of comparing $\varphi$-modules on $C$-vector spaces to $\Fld_p$-vector spaces.

In the following we carry out the above agenda. Step (1) is handled by the following results:

Let $\Spa(A, A^+)$ be an affinoid perfectoid space of characteristic $p$ and let $\pi \in A^+$ be a pseudouniformizer. Then the functor

\begin{align*}
- \tensor_{(A^+)^a_\solid} (A^+/\pi)^a_\solid\colon \D^a_\solid(A^+)^\varphi_{\hat\pi} \isoto \D^a_\solid(A^+/\pi)^\varphi
\end{align*}

is an equivalence of $\infty$-categories. Here $\D^a_\solid(A^+)^\varphi_{\hat\pi} \subset \D^a_\solid(A^+)^\varphi$ denotes the full subcategory of those $\varphi$-modules whose underlying $A^+$-module is $\pi$-adically complete.

We first claim that $\D^a_\solid(A^+)^\varphi_{\hat\pi} = \varprojlim_n \D^a_\solid(A^+/\pi^n)^\varphi$ via the natural functor. Namely, for lax $\varphi$-modules this can be checked on underlying modules (where all pushforward functors behave as expected) and thus follows from <ref>. It then remains to check that if $M \in \D((A^+)^a_\solid[T_\varphi])$ satisfies $M \tensor_{(A^+)^a_\solid} (A^+/\pi^n)^a_\solid \in \D^a_\solid(A^+/\pi^n)^\varphi$ then $M \in \D^a_\solid(A^+)^\varphi$. But this follows easily from $M = \varprojlim_n M/\pi^n$.

It is now enough to show that all the transition functors $\D^a_\solid(A^+/\pi^{n+1})^\varphi \to \D^a_\solid(A^+/\pi^n)^\varphi$ are equivalences. In fact it is enough to check that the base-change functor $\D^a_\solid(A^+/\pi^p)^\varphi \to \D^a_\solid(A^+/\pi)^\varphi$ is an equivalence (then replace $\pi$ by $\pi^p$ and repeat). Using that the Frobenius on $A^+/\pi^p$ factors as $A^+/\pi^p \to A^+/\pi \xto{\varphi} A^+/\pi^p$, this follows immediately from <ref>.

Let $X = \Spa(A, A^+)$ be an affinoid perfectoid space of characteristic $p$ with pseudouniformizer $\pi \in A^+$. Then every dualizable object in $\D^a_\solid(A^+)^\varphi_{\hat\pi}$ (equipped with the $\pi$-completed tensor product) is bounded and dualizable as an object in $\D^a_\solid(A^+)^\varphi$.

Let $P \in \D^a_\solid(A^+)^\varphi_{\hat\pi}$ be dualizable. We only need to show that $P$ is bounded; then it is automatically dualizable in $\D^a_\solid(A^+)^\varphi$ because its dual $P^*$ is then also bounded and hence by <ref> the $\pi$-completed tensor product $P \hat\tensor P^*$ coincides with the solid tensor product $P \tensor P^*$ in $\D^a_\solid(A^+)^\varphi$.

Since $P$ is dualizable, so is $P/\pi \in \D^a_\solid(A^+/\pi)$. By <ref> there is some $N \ge 0$ such that for all integers $n$ with $\abs n \ge N$, $\pi_n(P/\pi)$ is killed by $\pi^{1/p}$. By considering the long exact homology sequence associated to the exact triangle $P \xto\pi P \to P/\pi$ in $\D^a_\solid(A^+)$ it follows that for $\abs n > N$ both kernel and cokernel of the map $\pi_n(P) \xto\pi \pi_n(P)$ are annihilated by $\pi^{1/p}$. We claim that this implies that this map is actually an isomorphism. Namely, let $P_n := \pi_0(\pi_n(P)_*)$, which is a static $\varphi$-module over $A^+_\solid$ such that $P_n^a = \pi_n(P)$. Choose any $x \in P_n$ (more precisely, any $x \in P_n(S)$ for any profinite set $S$) such that $\pi x = 0$. Then $\pi^{1/p} x = 0$ by the above observation, hence $\pi \varphi_P(x) = \varphi_P(\pi^{1/p} x) = 0$. But then $\varphi_P(x)$ lies in the kernel of multiplication by $\pi$ and thus $\pi^{1/p} \varphi_P(x) = 0$. Applying $\varphi_P^{-1}$ we deduce $\pi^{1/p^2} x = 0$. Thus $\pi \varphi_P^2(x) = 0$, which implies $\pi^{1/p} \varphi_P^2(x) = 0$ and therefore $\pi^{1/p^3} x = 0$. Repeating this argument we deduce that $\varepsilon x = 0$ for all $\varepsilon \in \mm_A$, hence $x = 0$. This shows that multiplication by $\pi$ is injective on $P_n$. We can now similarly show that it is surjective: Given any $y \in P_n$, we can find some $x_0 \in P_n$ such that $\pi x_0 = \pi^{1/p} y$. Then $\pi^{1/p^2} \varphi_P^{-1}(y) = \pi^{1/p} \varphi_P^{-1}(x_0)$, hence there is some $x_1 \in P_n$ such that $\pi x_1 = \pi^{1/p^2} \varphi_P^{-1}(y)$. It follows that $\pi^p \varphi_P(x_1) = \pi^{1/p} y$ and since multiplication by $\pi$ is injective on $P_n$ we deduce $y = \pi \cdot (\pi^{p - 1 - 1/p} \varphi_P(x_1))$, i.e. $y$ lies in the image of multiplication by $\pi$.

We have now shown that for $\abs n \gg 0$ the map $\pi_n(P) \xto\pi \pi_n(P)$ is an isomorphism. But by <ref> $\pi_n(P)$ is $\pi$-adically complete, so by definition we have $0 = \varprojlim(\dots \xto\pi \pi_n(P) \xto\pi \pi_n(P) \xto\pi \pi_n(P))$. Thus $\pi_n(P) = 0$, as desired.

We now come to step (2) of the above proof strategy. The following result shows the promised $M^\varphi = (M \tensor_{A^+} A)^\varphi$, while the subsequent result deduces that the functor $(-)^\varphi\colon \Dqcohri(A^+/\pi)^\varphi \to \D_\solid(\Fld_p(X))$ preserves small colimits.

Let $X = \Spa(A, A^+)$ be an affinoid perfectoid space of characteristic $p$ and let $R := \Fld_p(X)$ (i.e. the ring of continuous maps $\abs X \to \Fld_p$). Then for all $M \in \D^a_\solid(A^+)^\varphi$ the natural morphism

\begin{align*}
M^\varphi \isoto (M \tensor_{(A^+)^a_\solid} (A, A^+)_\solid)^\varphi
\end{align*}

is an isomorphism in $\D_\solid(R)$.

Both sides commute with the fully faithful inclusion from the solid theory to all topological modules, so it is enough to prove the claimed isomorphism in the latter setting. Moreover, both sides can be computed on sections on extremally disconnected sets $S$, so we reduce to showing the claim in the discrete setting: Given $M \in \D(A^{+a}_\omega)_\omega$ we need to see that the natural map $M^\varphi \isoto (M \tensor_{A^{+a}_\omega} A_\omega)^\varphi$ is an isomorphism in $\D(R)_\omega$. From now on we drop the subscript $\omega$ and simply assume that everything is discrete.

Note that $\D(A^{+a})^\varphi$ has a left-complete $t$-structure given by the $t$-structure on $\D(A^{+a})$. In fact, by <ref> it can be identified with $\D(A^{+a}[T_\varphi^{\pm}])$, where $A^{+a}[T_\varphi^\pm]$ is the associative $A^{+a}$-algebra given by the rule $T_\varphi a = \varphi(a) T_\varphi$. Both of the described functors commute with the limit $M = \varprojlim_n \tau_{\le n} M$, so we can assume that $M$ is left-bounded. By the above description of $\D(A^{+a})^\varphi$ we note that this $\infty$-category is the derived $\infty$-category of its heart, which itself is a Grothendieck abelian category. In particular we can resolve $M$ by injective objects to reduce to the case that $M$ itself lies in the heart and is injective. In this case the $R$-module $M^\varphi$ is also concentrated in degree $0$ (the functor $(-)^\varphi$ is a derived functor); it can be described explicitly by

\begin{align*}
M^\varphi = \{ x \in M_* \setst \varphi_M(x) = x \},
\end{align*}

where $\varphi_M\colon M \to \varphi_* M$ is the $\varphi$-linear map belonging to the $\varphi$-module $M$. We now observe that $(M \tensor_{A^{+a}} A)^\varphi$ is also concentrated in degree $0$: It is computed as $(M_* \tensor_{A^+} A^+[1/\pi])^\varphi$ for any pseudouniformizer $\pi \in A^+$, so the claim follows from <cit.>. We get

\begin{align*}
(M \tensor_{A^{+a}} A)^\varphi = \{ a \tensor x \in A \tensor_{A^+} M_* \setst a^p \tensor \varphi_M(x) = a \tensor x \}.
\end{align*}

It remains to check that the map of classical $R$-modules $\alpha\colon M^\varphi \to (M \tensor_{A^{+a}} A)^\varphi$ is an isomorphism, which amounts to showing that it is both injective and surjective. Let us also denote $\alpha'\colon M_* \to (M_* \tensor_{A^+} A)$ the natural map (so that $\alpha$ is the restriction of $\alpha'$) and $\varphi_M'$ the Frobenius action on $M \tensor_{A^+} A$.

We first show that $\alpha$ is injective. Suppose we have $x \in M^\varphi$ such that $\alpha(x) = 0$. Then there is some pseudouniformizer $\pi \in A^+$ such that $\pi x = 0$ (in $M_*$). But then also $0 = \varphi_M^{-1}(\pi x) = \pi^{1/p} x$. We inductively deduce that $\varepsilon x = 0$ for all $\varepsilon \in \mm_A$. But this implies $x = 0$ in $M_* = \IHom_{A^+}(\mm_A, M_*)$, as desired.

It remains to show that $\alpha$ is surjective, so suppose we are given some $y \in (M \tensor_{A^+} A)^\varphi$. Then there is some pseudouniformizer $\pi \in A^+$ and some $x' \in M_*$ such that $y = \pi^{-1} \alpha'(x')$. We deduce $\alpha'(\varphi_M(x') - \pi^{p-1} x') = \varphi_{M'}(\pi y) - \pi^p y = \pi^p(\varphi_{M'}(y) - y) = 0$, hence there is some pseudouniformizer $\varpi \in A^+$ such that $\pi \divides \varpi$ and $\frac{\varpi^p}{\pi^p} (\varphi_M(x') - \pi^{p-1} x') = 0$. Let us rewrite the latter condition as

\begin{align*}
\frac\varpi\pi x' = \varphi_M^{-1}(\frac{\varpi^p}\pi x').
\end{align*}

We now define an element $x \in M_* = \IHom_{A^+}(\mm_A, M_*)$ by requiring

\begin{align*}
x(\varpi_n) = \varphi_M^{-n}(\frac\varpi\pi x')
\end{align*}

for all $n \ge 0$, where $\varpi_n := \varpi^{1/p^n}$ For this to define an element of $M_*$ we need to verify the following compatibility: For all $n \ge 0$ we have

\begin{align*}
&\frac{\varpi_n}{\varpi_{n+1}} x(\varpi_{n+1}) = \frac{\varpi_n}{\varpi_{n+1}} \varphi_M^{-n-1}(\frac\varpi\pi x') = \varphi_M^{-n-1}(\frac{\varpi^p}{\varpi} \frac\varpi\pi x') = \varphi_M^{-n}(\varphi_M^{-1}(\frac{\varpi^p}\pi x')) =\\&\qquad= \varphi_M^{-n}(\frac\varpi\pi x') = x(\varpi_n).
\end{align*}

It is clear that $x \in M_*$ is $\varphi_M$-invariant: By definition $\varphi_M$ acts on $x$ by $(\varphi_M(x))(\varepsilon) = \varphi_M(x(\varepsilon^{1/p}))$ for all $\varepsilon \in \mm_A$; thus for $\varepsilon = \varpi_n$ we have $(\varphi_M(x))(\varepsilon) = x(\varepsilon)$ and for all other $\varepsilon$ it follows from that case. One also checks immediately that $\alpha(x) = y$, as desired.

Let $X = \Spa(A, A^+)$ be an affinoid perfectoid space of characteristic $p$ and let $R := \Fld_p(X)$. Then the functor

\begin{align*}
(-)^\varphi\colon \D^a_\solid(A^+/\pi)^\varphi \to \D_\solid(R), \qquad M \mapsto M^\varphi
\end{align*}

preserves all small colimits.

It is clear that $(-)^\varphi\colon \D_\solid(A^+/\pi)^\varphi \to \D_\solid(R)$ preserves all small colimits, since it is given by an equalizer and hence a finite limit. If we work with almost coefficients instead then it is less clear that $(-)^\varphi$ preserves small colimits because it factors as $\D^a_\solid(A^+/\pi)^\varphi \xto{(-)_*} \D_\solid(A^+/\pi)^\varphi \xto{(-)^\varphi} \D_\solid(R)$ and the functor $M \mapsto M_*$ does not preserve all small colimits. To circumvent this issue we make use of <ref> which allows us to move from the almost world to the non-almost world in a colimit-preserving way.

More concretely, we can argue as follows: Let $(M_i)_i$ be a small family of objects in $\D^a_\solid(A^+/\pi)^\varphi$. We need to show that $(\bigdsum_i M_i)^\varphi = \bigdsum_i M_i^\varphi$. By <ref> we can lift $M_i$ to $M'_i \in \D^a_\solid(A^+)^\varphi$ and since $(-)^\varphi\colon \D^a_\solid(A^+) \to \D_\solid(R)$ preserves all small colimits by <ref>, the claimed identity of $\varphi$-invariants is equivalent to the identity $(\bigdsum_i M'_i)^\varphi = (\widehat\bigdsum_i M'_i)^\varphi$ (where $\widehat\bigdsum$ denotes the $\pi$-completed direct sum). Let us denote $N := \fib(\bigdsum_i M'_i \to \widehat\bigdsum_i M'_i) \in \D^a_\solid(A^+)^\varphi$. Then the claimed identity on $\varphi$-invariants is equivalent to $N^\varphi = 0$. Now reverse the above equivalences in the non-almost world, with $M_{i*} \in \D_\solid(A^+/\pi)^\varphi$ in place of $M_i$, $M'_{i*} \in \D_\solid(A^+)^\varphi$ in place of $M'_i$ and $N' := \fib(\bigdsum_i M'_{i*} \to \widehat\bigdsum_i M'_{i*})$ in place of $N$ (note that <ref> holds in the non-almost setting by the same proof). From the fact that $(-)^\varphi\colon \D_\solid(A^+/\pi)^\varphi \to \D_\solid(R)$ preserves all small colimits we can therefore deduce that $N'^\varphi = 0$. There is a natural almost isomorphism $N' \to N_*$. Moreover, by the proof of <ref> the underlying $(A^+)_\solid$-module of $N_*$ is $\varprojlim(\dots \xto\pi (\bigdsum_i M_i)_* \xto\pi (\bigdsum_i M_i)_*)$ and a similar description holds for $N'$. In particular multiplication by $\pi$ is an isomorphism on $N_*$ and on $N'$ so that the almost isomorphism $N' \to N_*$ is actually an isomorphism. It follows that $N^\varphi = (N_*)^\varphi = N'^\varphi = 0$, as desired.

We have now proved step (2) of the above agenda. In order for it to be very useful we still need a better understanding of the functor $- \tensor_{A^+} A$ in order to reduce everything to $\varphi$-modules over $A$ (as in step (3)). The following result comes in very handy:

Let $\Spa(A, A^+)$ be an affinoid perfectoid space of characteristic $p$. Then the base-change functor

\begin{align*}
- \tensor_{(A^+)^a_\solid} A_\solid\colon \D^a_\solid(A^+)^{a\omega,\varphi} \to \D_\solid(A, A^+)^\varphi
\end{align*}

is conservative. Here $\D^a_\solid(A^+)^{a\omega,\varphi} \subset \D^a_\solid(A^+)^\varphi$ denotes the full subcategory of those $\varphi$-modules whose underlying $(A^+)^a_\solid$-module is almost compact.

Fix a pseudouniformizer $\pi \in A^+$ and any $P \in \D^a_\solid(A^+)^{a\omega,\varphi}$ such that $P \tensor_{(A^+)^a_\solid} (A, A^+)_\solid = 0$; we need to see that $P = 0$. The condition $P \tensor_{(A^+)^a_\solid} (A, A^+)_\solid = 0$ means that $\varinjlim(P \xto\pi P \xto\pi P \xto\pi \dots) = 0$. Since $P$ is almost compact this implies that

\begin{align*}
\varinjlim_\pi \pi_0 \IHom(P, P)_\omega = \pi_0 \IHom(P, \varinjlim_\pi P)_\omega = 0,
\end{align*}

from which we deduce that $\varinjlim_\pi \pi_0 \Hom(P, P)$ is almost zero (where we view $\Hom$ as enriched over $A^+_\omega$). This means that for every $f \in \pi_0 \Hom(P, P)$ there is some $n \ge 0$ such that $\pi^n f = 0$. Applying this to $f = \id$ we deduce that the map $P \xto{\pi^n} P$ is zero. Via the isomorphism $\varphi^* P \isoto P$ we deduce that multiplication by $\pi^{n/p}$ is also zero on $P$. Inductively it follows that multiplication by any $\varepsilon \in \mm_A$ is zero on $P$, hence $P = 0$.

Before we can come to the proof of the local Riemann-Hilbert correspondence, we observe that the whole situation commutes with passage to connected components:

Let $X = \Spa(A, A^+)$ be a totally disconnected space of characteristic $p$ with pseudouniformizer $\pi$, $x = \Spa(K, K^+) \in \pi_0(X)$ a connected component and $R := \Fld_p(X)$. The map $x \to X$ induces a map $R \to \Fld_p(x) = \Fld_p$ and the following diagrams commute

[column sep = huge]
_(_p) [r,"- (K^+/π)^a_"]     ^a_(K^+/π)^φ

_(R) [u,"- _p"] [r, "- (A^+/π)^a_"]     ^a_(A^+/π)^φ[u,swap,"- K^+a/π"]

[column sep = large]
_(_p)     ^a_(K^+/π)^φ[l,swap,"(-)^φ"]

_(R) [u,"- _p"]     ^a_(A^+/π)^φ[u,swap,"- K^+a/π"] [l,swap,"(-)^φ"]

The commutativity of the left diagram is clear (both paths compute $- \tensor_{R_\solid} (K^+/\pi)^a_\solid$, cf. <ref>). It remains to prove that the right diagram commutes, so let $M \in \D^a_\solid(A^+/\pi)^\varphi$ be given. By the commutativity of the left diagram we get a natural morphism $M^\varphi \tensor \Fld_p \to (M \tensor K^{+a}/\pi)^\varphi$ which we claim to be an isomorphism. Now write $x = \bigisect_i U_i$ for a filtered system $(U_i)_i$ of clopen neighborhoods $x \in U_i \subset X$. Write $U_i = \Spa(A_i, A^+_i)$ and $R_i = \Fld_p(U_i)$. Then $A^+_i/\pi$ is a retract of $A^+/\pi$ and $R_i$ is a retract of $R$ in a compatible way, which easily implies that the map $M^\varphi \tensor R_i \isoto (M \tensor A^{+a}/\pi)^\varphi$ is an isomorphism for all $i$. Note that $\Fld_p = \varinjlim_i R_i$ and $K^+/\pi = \varinjlim_i A^+_i/\pi$, so since $(-)^\varphi$ preserves all small colimits (see <ref>) we can pull the colimits out of $M^\varphi \tensor \Fld_p \to (M \tensor K^{+a}/\pi)^\varphi$ to conclude that this map is an isomorphism.

Finally, we are in the position to prove the promised local Riemann-Hilbert correspondence. The combination of the above results leads to the following:

Let $X = \Spa(A, A^+)$ be a strictly totally disconnected space of characteristic $p$ with pseudouniformizer $\pi \in A^+$ and denote $R := \Fld_p(X)$. Then the symmetric monoidal functor

\begin{align*}
- \tensor_R (A^{+a}/\pi)\colon \D(R)_\omega \injto \D^a_\solid(A^+/\pi)^\varphi
\end{align*}

is fully faithful and induces an equivalence of the full subcategories of dualizable objects on both sides.

The given functor has the right adjoint $(-)^\varphi\colon \D^a_\solid(A^+/\pi)^\varphi \to \D(R)_\omega$ (where we implicitly pass to the discretization). Thus in order to prove the claimed full faithfulness we need to show that for every discrete $R$-module $M \in \D(R)_\omega$ the natural map $M \isoto (M \tensor_R A^{+a}/\pi)^\varphi$ is an isomorphism. This can be checked on stalks on $\Spec R$. Now $R$ is the ring of continuous maps from the profinite set $\pi_0(X)$ to $\Fld_p$, so the the topological space $\Spec R$ can naturally be identified with $\pi_0(X)$ (e.g. write $\pi_0(X) = \varprojlim S_i$ for finite $S_i$, so that $\Spec R = \varprojlim_i \Spec \Fld_p(S_i) = \varprojlim_i S_i = \pi_0(X)$). By <ref> $(M \tensor_R A^{+a}/\pi)^\varphi$ commutes with passage to stalks, so we can reduce to the case that $X$ is connected and thus of the form $X = \Spa(C, C^+)$ for some algebraically closed field $C$ and $R = \Fld_p$. Writing $M \in \D(\Fld_p)_\omega$ as a colimit of copies of $\Fld_p$ and using that $(-)^\varphi$ preserves colimits (see <ref>) we reduce to the case $M = \Fld_p$. Then $(M \tensor (C^{+a}/\pi))^\varphi = (C^{+a}/\pi)^\varphi$, which we need to show to be isomorphic to $\Fld_p$. These $\varphi$-invariants can also be computed after applying the right adjoint of $\D^a_\solid(C^+)^\varphi \to \D^a_\solid(C^+/\pi)^\varphi$, which by <ref> maps $C^{+a}/\pi$ to $C^{+a}$. By <ref> we have $(C^{+a})^\varphi = C^\varphi$. We thus need to check that $C^\varphi = \Fld_p$. We obviously have $\pi_0 C^\varphi = \Fld_p$ (these are just the $\varphi$-invariant elements in $C$), so it remains to verify that $\pi_1 C^\varphi = 0$, i.e. that the map $\varphi - 1\colon C \to C$ is surjective. This follows immediately from the fact that $C$ is algebraically closed. We have thus finished the proof that $- \tensor_R (A^{+a}/\pi)$ is fully faithful.

We now prove the second part of the claim, i.e. that $- \tensor_R (A^{+a}/\pi)$ induces an equivalence of dualizable objects on both sides. By the full faithfulness of this functor it only remains to check that its essential image contains all dualizable objects of $\D^a_\solid(A^+/\pi)^\varphi$. Fix a dualizable object $P \in \D^a_\solid(A^+/\pi)^\varphi$. We need to see that the natural map $P^\varphi \tensor_R (A^{+a}/\pi) \to P$ is an isomorphism. This can be checked after pullback to the connected components of $\Spec A^+/\pi$ (see <ref>) which correspond to $\pi_0(X)$. By <ref> we can thus reduce to the case that $X = \Spa(C, C^+)$ is connected (and hence $R = \Fld_p$), as in the proof of full faithfulness. By <ref> we can identify $P$ with a dualizable object $P' \in \D^a_\solid(C^+)^\varphi_{\hat\pi}$ (where the tensor product is the $\pi$-completed tensor product in $\D^a_\solid(C^+)^\varphi$). Then by <ref> $P'$ is bounded and dualizable as an object in $\D^a_\solid(C^+)^\varphi$ (where the tensor product is the solid tensor product). Moreover, it follows immediately from the definitions that $P'^\varphi = P^\varphi$, so it is now enough to show that the natural map $P'^\varphi \tensor_R C^{+a} \to P'$ is an isomorphism. By <ref> this boils down to showing that $P'^\varphi$ is dualizable (and hence $P'^\varphi \tensor_R C^{+a}$ is almost compact) and that for $P'' := P' \tensor_{C^{+a}} C \in \D_\solid(C, C^+)^\varphi$ the natural map $P''^\varphi \tensor_R C \to P''$ is an isomorphism. Clearly $P''$ is dualizable in $\D_\solid(C, C^+)^\varphi$ and by <ref> we have $P'^\varphi = P''^\varphi$, so we are reduced to showing the following claim: The adjoint pair of functors

\begin{align*}
- \tensor_R C\colon \D(R)_\omega \rightleftarrows \D_\solid(C, C^+)^\varphi \noloc (-)^\varphi
\end{align*}

restricts to quasi-inverse equivalences on the full subcategories of dualizable objects. By <cit.> the symmetric monoidal $\infty$-category of dualizable objects in $\D_\solid(C, C^+)$ is naturally equivalent to the symmetric monoidal $\infty$-category of dualizable objects in $\D(C_\omega)_\omega$ (and hence the same follows for $\varphi$-modules in the respective $\infty$-categories by <ref>), so we can replace $\D_\solid(C, C^+)^\varphi$ by $\D(C_\omega)^\varphi_\omega$ in the above claim. Given any dualizable $Q \in \D(C_\omega)^\varphi_\omega$ then the underlying $C_\omega$-module is perfect and hence a finite direct sum of copies of $C_\omega[n]$ for varying $n \in \Z$. We need to verify that such a decomposition also exists as $\varphi$-module. This boils down to showing that every $\varphi$-module structure on a finite-dimensional $C_\omega$-vector space admits a basis of $\varphi$-fixed vectors. This is a classical result, see e.g. <cit.>; alternatively, one can deduce the equivalence of dualizable objects in $\D(R)_\omega$ and $\D(C_\omega)_\omega^\varphi$ directly from <cit.>.

Having understood the Riemann-Hilbert correspondence in the local setting, we now want to globalize it in order to get a similar result on every small v-stack $X$. This is a formal gluing procedure, but in order for the obtained result to be really useful we need to better understand the involved $\infty$-categories of sheaves on $X$. We start by introducing $\varphi$-modules over $\ri^+_X/\pi$.

Let $X^\sharp$ be an untilted small v-stack with pseudouniformizer $\pi$ such that $p \in \pi$. Then the Frobenius $\varphi\colon \ri^+_{X^\sharp}/\pi \to \ri^+_{X^\sharp}/\pi$ is a morphism of integral torsion coefficients and in particular induces a symmetric monoidal functor

\begin{align*}
\varphi^*\colon \DqcohriX{X^\sharp} \to \DqcohriX{X^\sharp}.
\end{align*}

By <cit.> for every almost setup $(V,\mm)$ with $p = 0$ in $V$ the Frobenius $\varphi\colon V \to V$ defines a strict morphism of almost setups $(V,\mm) \to (V,\mm)$. Thus using <ref> it only remains to show that for every map $\Spa(B, B^+) \to \Spa(A, A^+)$ of affinoid perfectoid spaces with pseudouniformizer $\pi \in A^+$ such that $\pi \divides p$ the diagram

(A^+/π)^a_[d,"φ"] [r]     (B^+/π)^a_[d,"φ"]

(A^+/π)^a_[r]     (B^+/π)^a_

is a pushout square of analytic rings. This question only depends on the tilts of $\Spa(A, A^+)$ and $\Spa(B, B^+)$, so we can assume that both are of characteristic $p$. Then $\varphi\colon (A^+/\pi)^a_\solid \to (A^+/\pi)^a_\solid$ factors as the projection $(A^+/\pi)^a_\solid \to (A^+/\pi^{1/p})^a_\solid$ followed by an isomorphism $(A^+/\pi^{1/p})^a_\solid \isoto (A^+/\pi)^a_\solid$ (and similarly for $B^+$ in place of $A^+$). Clearly the isomorphisms form a pushout square, so it remains to check that

\begin{align*}
(A^+/\pi^{1/p})^a_\solid \tensor_{(A^+/\pi)^a_\solid} (B^+/\pi)^a_\solid = (B^+/\pi^{1/p})^a_\solid.
\end{align*}

By <ref> this follows from the observation that $A^+/\pi^{1/p} \tensor_{A^+/\pi} B^+/\pi = B^+/\pi^{1/p}$ (in the derived sense, as always).

* We denote $\vStackspip \subset \vStacksCoeff$ the full subcategory of those pairs $(X, \Lambda)$ where $\Lambda = \ri^+_{X^\sharp}/\pi$ for some untilt $X^\sharp$ of $X$ and some pseudouniformizer $\pi$ on $X^\sharp$ such that $p \in \pi$. We denote the objects of $\vStackspip$ by $(X^\sharp, \pi)$ or often simply by $X$ (with the untilt and $\pi$ being implicit).

*  Let $X \in \vStackspip$. A $\varphi$-module over $\ri^+_X/\pi$ is an object $\mathcal M \in \DqcohriX X$ together with an isomorphism $\varphi^* \mathcal M \isoto \mathcal M$ (here $\varphi^*$ comes from <ref>). We denote

\begin{align*}
\DqcohriX X^\varphi := \equalizer{\DqcohriX X}{\DqcohriX X}\id{\varphi^*}
\end{align*}

the symmetric monoidal $\infty$-category of $\varphi$-modules over $\ri^+_X/\pi$.

By commuting limits with limits one sees immediately that the assignment $X \mapsto \DqcohriX X^\varphi$ defines a hypercomplete v-sheaf of symmetric monoidal $\infty$-categories on $\vStackspip$. In fact this sheaf is just the equalizer of the two morphism $\id, \varphi^*\colon \DqcohriX{(-)} \to \DqcohriX{(-)}$ of sheaves of symmetric monoidal $\infty$-categories.

Before we continue, let us investigate how the six functors from our 6-functor formalism extend to $\varphi$-modules. In fact, this works as nicely as one could hope:

Let $f\colon Y \to X$ be a morphism in $\vStackspip$.

* The forgetful functor $\DqcohriX X^\varphi \to \DqcohriX X$ is symmetric monoidal, conservative and preserves all small limits, colimits and internal homs.

* There is a natural symmetric monoidal pullback functor $f^*\colon \DqcohriX X^\varphi \to \DqcohriX Y^\varphi$ which acts as $f^*$ on the underlying $\ri^+_X/\pi$-modules.

*  The functor from (ii) admits a right adjoint $f_*\colon \DqcohriX Y^\varphi \to \DqcohriX X^\varphi$ which acts as $f_*$ on the underlying $\ri^+_Y/\pi$-modules.

* If $f$ is bdcs then there is a natural functor $f_!\colon \DqcohriX Y^\varphi \to \DqcohriX X^\varphi$ which acts as $f_!$ on the underlying $\ri^+_Y/\pi$-modules.

*  If $f$ is bdcs then the functor from (iv) admits a right adjoint $f^!\colon \DqcohriX X^\varphi \to \DqcohriX Y^\varphi$ which acts as $f^!$ on the underlying $\ri^+_X/\pi$-modules.

We start the proof with some general observations. Let $Y_\bullet \to X$ be a hypercover of $X$ such that each $Y_n = \bigdunion_{i\in I_n} Y_{n,i}$ is a disjoint union of totally disconnected spaces $Y_{n,i} = \Spa(B_{n,i}, B^+_{n,i})$. Let $\DqcohriX{Y_\bullet} \to \Delta$ be the coCartesian fibration classifying the functor $\Delta \to \infcatinf$, $n \mapsto \DqcohriX{Y_n}$ and let $\mathcal C$ be the $\infty$-category of sections $\Delta \to \DqcohriX{Y_\bullet}$. Then $\DqcohriX X$ is the full subcategory of $\mathcal C$ spanned by the coCartesian sections. Let $r\colon \mathcal C \to \DqcohriX X$ be a right adjoint to the inclusion (it exists by the proof of <ref>). We can similarly construct $\DqcohriX{Y_\bullet}^\varphi$ and the $\infty$-category $\mathcal C^\varphi$ of sections $\Delta \to \DqcohriX{Y_\bullet}^\varphi$, together with the right adjoint $r^\varphi\colon \mathcal C^\varphi \to \DqcohriX X^\varphi$ of the inclusion. Here we can identify $\mathcal C^\varphi$ with the equalizer of the two maps $\id, \varphi^*\colon \mathcal C \to \mathcal C$, so in particular there is a natural forgetful functor $\mathcal C^\varphi \to \mathcal C$. We claim that the diagram

𝒞^φ[r,"r^φ"] [d]     X^φ[d]

𝒞 [r,"r"]     X

commutes, where the vertical maps are the forgetful functors. To see this, we first observe that each forgetful functor $\Dqcohri(B_{n,i}^+/\pi)^\varphi \to \Dqcohri(B_{n,i}^+/\pi)$ admits a left adjoint $\ell$; namely, this essentially boils down to showing that this functor preserve small limits, which follows easily from the comparison with $\pi$-adically complete $B_{n,i}^{+a}$-modules (see <ref>) on which the forgetful functor evidently preserves limits (see <ref>). Thus the commutativity of the above diagram can be checked after passing to left adjoints, which reduces the claim to checking that the functors $\ell$ preserve coCartesian edges in $\DqcohriX{Y_\bullet}$. This boils down to showing that for a map $g\colon \Spa(A', A'^+) \to \Spa(A, A^+)$ of totally disconnected spaces in $\vStackspip$ the diagram

(A^+/π)^φ[r,"g^*"]     (A'^+/π)^φ

(A^+/π) [u,"ℓ"] [r,"g^*"]     (A'^+/π) [u,"ℓ"]

commutes. Passing to right adjoint in this diagram, the claim reduces to showing that the forgetful functor from $\varphi$-modules to the underlying modules commutes with the forgetful functor $\Dqcohri(A'^+/\pi) \to \Dqcohri(A^+/\pi)$. This in turn follows from the fact that the latter forgetful functor commutes with $\varphi^*$ by <ref>. This finally proves that the first diagram above commutes.

We will now prove the actual claims of the lemma. In part (i) everything is obvious except that the forgetful functor preserves small limits and internal homs. To check these two claims, we note that limits and internal homs in $\DqcohriX X^\varphi$ can be computed by first computing them in $\mathcal C^\varphi$ and then applying $r^\varphi$. Now since in $\mathcal C^\varphi$ both limits and internal homs are computed componentwise (in each $\DqcohriX{Y_{n,i}}^\varphi$) and the above diagram commutes, we can reduce the claim to each $Y_{n,i}$, i.e. we can assume that $X = \Spa(A, A^+)$ is totally disconnected. Then by <ref> we reduce the claim to showing that the forgetful functor $\Dqcohri(A^+)^\varphi_{\hat\pi} \to \Dqcohri(A^+)_{\hat\pi}$ and the base-change functor $\Dqcohri(A^+) \to \Dqcohri(A^+/\pi)$ preserve small limits and internal homs. For the former functor this is true since $A^+$ is perfect (so that $\varphi^*$ is an isomorphism) and for the latter functor it follows because the base-change functor is computed by a simple cofiber sequence $M \mapsto \cofib(M \xto\pi M)$.

Part (ii) is obvious from the definitions (cf. <ref>). We now prove (iii). Note first that there is a right adjoint of $f^*$ on the $\varphi$-module $\infty$-categories (this can be constructed as in <ref>; up to set-theoretic nonsense this is just the adjoint functor theorem). As in the proof of (i) we can make use of the commuting diagram involving $r$ and $r^\varphi$ above in order to reduce to the case that $X$ is totally disconnected. By choosing a hypercover of $Y$ by disjoint unions of totally disconnected spaces we see that $f_*$ is computed as a combination of forgetful functors $\Dqcohri(B^+/\pi)^\varphi \to \Dqcohri(A^+/\pi)^\varphi$ along maps $\Spa(A, A^+) \to \Spa(B, B^+)$ and a fixed limit (built out of products and a totalization). Both of these operations commute with the forgetful functor from $\varphi$-modules to the underlying modules (as shown in the proof of the commuting diagram above). Thus $f_*$ commutes with the forgetful functor.

We now prove (iv), so assume that $f$ is bdcs. Then $f_!$ on $\varphi$-modules can be constructed in the same way as it was constructed in <ref>: Essentially we let $f_! = g_* \comp j_!$ for a (local) factorization of $f$ into a proper map $g$ and an étale map $j$. By (iii) it only remains to show that $j_!$ exists on $\varphi$-modules and commutes with the forgetful functor. But this follows immediately from the fact that $j_!$ commutes with $\varphi^*$ by <ref> (namely, apply the usual coCartesian section argument to the equalizer in the definition of $\varphi$-modules).

It remains to prove (v), so again assume that $f$ is bdcs. It follows in the same way as in the proof of <ref> that the functor $f_!$ admits a right adjoint $f^!$ on $\varphi$-modules; we have to show that $f^!$ commutes with the forgetful functor from $\varphi$-modules to the underlying modules. By passing to a hypercover of $X$ by disjoint unions of totally disconnected spaces and replacing $Y$ by the pullback hypercover, we see that $f^!$ can be computed componentwise on the hypercover plus an application of the functor $r$ constructed above but with respect to the hypercover of $Y$ and not of $X$. Thus the commuting diagram at the beginning of the proof allows us to reduce the claim to the case that $X = \Spa(A, A^+)$ is totally disconnected. Then since $f$ is bdcs we know that $Y$ is a locally spatial diamond and locally compactifiable over $X$, so we can pass to an open cover of $Y$ in order to reduce to the case that $f$ is quasicompact and compactifiable. Now choose a pro-étale cover $g\colon U = \varprojlim_i U_i \to Y$ such that $U$ is totally disconnected and all $g_i\colon U_i \to Y$ are separated quasicompact étale (see <cit.>). Then $g^*$ is conservative, so it is enough to show that $g^* f^!$ commutes with the forgetful functor from $\varphi$-modules to modules. Moreover, $f_* g_*$ is a forgetful functor and hence conservative, so we reduce to showing that $f_* g_* g^* f^!$ commutes with the forgetful functor from $\varphi$-modules to modules. By <ref> we have $g_* g^* = \varinjlim_i g_{i*} g_i^*$ and by <ref> $f_*$ preserves this colimit, so we reduce to showing that $f_* g_{i*} g_i^* f^!$ commutes with the forgetful functor from $\varphi$-modules to modules. By replacing $f$ by $f \comp g_i$ we are left with showing that $f_* f^!$ commutes with the forgetful functor from $\varphi$-modules to modules. By factoring $f$ into an open immersion and a $p$-bounded proper map we see that $f_!$ satisfies the projection formula (even for the $\varphi$-module version). It then follows formally from adjunctions that $f_* f^! = \IHom(f_! \ri^+_Y/\pi, -)$. But we know by (i) and (iv) that the right-hand side commutes with the forgetful functor from $\varphi$-modules to modules, as desired.

The six functors $\tensor$, $\IHom$, $f^*$, $f_*$, $f_!$ and $f^!$ extend naturally to $\varphi$-modules over $\vStackspip$. These $\varphi$-module versions also form a 6-functor formalism.

The 6-functor formalism can be constructed in the same way as in <ref>. Here <ref> guarantees that the $\varphi$-module version of each of the six functors coincides with the original version after applying the forgetful functor from $\varphi$-modules to modules.

In view of <ref> and the Riemann-Hilbert correspondence (see <ref> below) one may view $\DqcohriX X^\varphi$ as the central $\infty$-category of this thesis (instead of the version $\DqcohriX X$ without $\varphi$-module structure), at least if $\pi \divides p$.

We now single out a special subcategory of $\DqcohriX X^\varphi$: the perfect sheaves.[In view of the Riemann-Hilbert correspondence we decided to call the dualizable objects in $\DqcohriX X^\varphi$ “perfect” and not “weakly almost perfect”.] Roughly, they are the ones that are locally given by an almost finite amount of data:

Let $X \in \vStackspip$. A $\varphi$-module $\mathcal P \in \DqcohriX X^\varphi$ is called perfect if the underlying object in $\DqcohriX X$ is weakly almost perfect (see <ref>). We denote

\begin{align*}
\DqcohriX X^\varphi_\perf \subset \DqcohriX X^\varphi
\end{align*}

the full subcategory spanned by the perfect $\varphi$-modules.

*  The assignment $X \mapsto \DqcohriX X^\varphi_\perf$ defines a hypercomplete v-sheaf on $\vStackspip$.

* Let $X \in \vStackspip$. Then $\mathcal P \in \DqcohriX X^\varphi$ is perfect if and only if it is dualizable.

Part (i) follows from <ref>. Part (ii) follows from <ref> by the same argument as in <ref>.

With the above definitions and results we understand one side of the Riemann-Hilbert correspondence: the $\varphi$-modules. We now introduce the other side of the correspondence, i.e. the $\Fld_p$-sheaves. The right category here is the $\infty$-category of overconvergent étale $\Fld_p$-sheaves. This contains all $\Fld_p$-local systems and all Zariski constructible sheaves and thus most of the interesting examples of $\Fld_p$-sheaves. The definition is as follows.

Let $X$ be a small v-stack. A sheaf $\mathcal F \in \D_\et(X, \Fld_p)$ is called overconvergent if for all maps $i\colon \Spa(C, C^+) \to X$ from a geometric point $\Spa(C, C^+)$ and induced maps $i'\colon \Spa(C, C^\circ) \to X$, $j\colon \Spa(C, C^\circ) \injto \Spa(C, C^+)$, the natural map $i^* \mathcal F \isoto j_* i'^* \mathcal F$ is an isomorphism. We denote

\begin{align*}
\D_\et(X, \Fld_p)^\oc \subset \D_\et(X, \Fld_p)
\end{align*}

the full subcategory spanned by the overconvergent sheaves.

We get the following basic results on overconvergent $\Fld_p$-sheaves. Notably they imply that $\D_\et(-, \Fld_p)^\oc$ is a hypercomplete v-sheaf which on strictly totally disconnected spaces is of the form $\D(\Fld_p(X))_\omega$, i.e. forms the left-hand side of the local Riemann-Hilbert correspondence from <ref>.

*  The assignment $X \mapsto \D_\et(X, \Fld_p)^\oc$ defines a hypercomplete v-sheaf of symmetric monoidal $\infty$-categories on the site of small v-stacks.

* A sheaf $\mathcal F \in \D_\et(X, \Fld_p)$ is overconvergent if and only if all $\pi_n(\mathcal F)$ are overconvergent.

*  Let $X$ be a strictly totally disconnected perfectoid space. Then there is a natural equivalence $\D_\et(X, \Fld_p)^\oc = \D(\Fld_p(X))_\omega$.

* Let $f\colon Y \to X$ be a qcqs map of locally spatial diamonds. Then the étale pushforward $f_*\colon \D^+_\et(Y, \Fld_p) \to \D^+_\et(X, \Fld_p)$ preserves overconvergent sheaves and hence restricts to a functor $f_*\colon \D^+_\et(Y, \Fld_p)^\oc \to \D^+_\et(X, \Fld_p)^\oc$.

To prove (i) we first note that the assignment $X \mapsto \D_\et(X, \Fld_p)$ is a hypercomplete v-sheaf of $\infty$-categories: By <cit.> this reduces to the claim that the assignment $X \mapsto \D(X_\vsite, \Fld_p)$ is a hypercomplete v-sheaf, which is formal (the required functoriality of the assignment can be constructed as in <cit.>; then v-hyperdescent is formal, see <cit.>). Thus in order to prove (i) we only have to verify the following: Given a v-cover $f\colon Y \surjto X$ of small v-stacks and some $\mathcal F \in \D_\et(X, \Fld_p)$ such that $f^* \mathcal F$ is overconvergent, then $\mathcal F$ is overconvergent. We can immediately reduce this claim to the case that $X = \Spa(C, C^+)$ and $Y = \Spa(C', C'^+)$ are geometric points. In this case the étale sites of $X$ and $Y$ are equivalent to the analytic sites, and $\abs X$ and $\abs Y$ consist of a totally ordered chain of points under specialization. By assumption we have $(f^* \mathcal F)(V) = (f^* \mathcal F)(Y)$ for every open subset $V \subset Y$. One checks that $(f^* \mathcal F)(V) = \mathcal F(f(V))$, hence $\mathcal F(U) = \mathcal F(X)$ for all open subsets $U \subset X$, as desired. The functoriality of the symmetric monoidal structure on $\D_\et(X, \Fld_p)^\oc$ can either be constructed directly or deduced from (iii) and <ref>.

Part (ii) follows easily from the fact that all the functors involved in the definition of overconvergent sheaves are $t$-exact and that isomorphisms can be checked on $\pi_n$.

We now prove (iii), so assume that $X$ is a strictly totally disconnected perfectoid space. Then the étale site of $X$ is equivalent to the analytic site of $X$ and we have $\D_\et(X, \Fld_p) = \D(X_\et, \Fld_p) = \D(\abs X, \Fld_p)$. Let $\rho\colon \abs X \to \pi_0(X)$ be the projection. Then $\rho^*\colon \D(\pi_0(X), \Fld_p) \injto \D(\abs X, \Fld_p)$ is fully faithful: This amounts to the claim that $\id \isoto \rho_* \rho^*$ is an isomorphism, which can be checked on stalks and hence in the case that $X = \Spa(C, C^+)$ is connected; in this case it is obvious. A sheaf $\mathcal F \in \D(\abs X, \Fld_p)$ lies in the essential image of $\rho^*$ if and only if the map $\rho^* \rho_* \mathcal F \to \mathcal F$ is an isomorphism. By passing to connected components of $X$ one sees that this condition is satisfied precisely if $\mathcal F$ is overconvergent. We therefore obtain a natural equivalence $\rho^*\colon \D(\pi_0(X), \Fld_p) \isoto \D_\et(X, \Fld_p)^\oc$ of symmetric monoidal $\infty$-categories. Now the topological space $\pi_0(X)$ admits a basis by clopen subsets, each of which is a retract of $\pi_0(X)$. It follows easily that the natural functor $\D(\Fld_p(X))_\omega \to \D(\pi_0(X), \Fld_p)$ (given by pullback along $\pi_0(X) \to *$) is an equivalence of symmetric monoidal $\infty$-categories.

Finally, part (iv) can be shown in the same way as <ref>, noting that the overconvergence condition can be checked on quasi-pro-étale maps $\Spa(C, C^+) \to X$ by the arguments in (i).

Similarly as for the $\varphi$-module side, we also introduce perfect objects in $\D_\et(X, \Fld_p)^\oc$ and prove some basic descent results.

Let $X$ be a small v-stack. An overconvergent sheaf $\mathcal F \in \D_\et(X, \Fld_p)^\oc$ is called perfect if for every map $f\colon Y \to X$ from a strictly totally disconnected space $Y$ the pullback $f^* \mathcal F \in \D_\et(Y, \Fld_p)^\oc = \D(\Fld_p(Y))$ is perfect. We denote

\begin{align*}
\D_\et(X, \Fld_p)^\oc_\perf \subset \D_\et(X, \Fld_p)^\oc
\end{align*}

the full subcategory of perfect sheaves.

*  The assignment $X \mapsto \D_\et(X, \Fld_p)^\oc_\perf$ is a hypercomplete v-sheaf of symmetric monoidal $\infty$-categories.

* Let $X$ be a small v-stack. Then an object $\mathcal F \in \D_\et(X, \Fld_p)^\oc$ is perfect if and only if it is dualizable.

As in the proof of <ref> the full subcategory of dualizable objects in $\D_\et(X, \Fld_p)^\oc$ satisfies v-hyperdescent. Thus both (i) and (ii) reduce to the following claim: If $X$ is strictly totally disconnected then $\mathcal F \in \D_\et(X, \Fld_p)^\oc = \D(\Fld_p(X))_\omega$ is dualizable if and only if it is perfect. This follows from <ref>, noting that a weakly almost perfect object in $\D(\Fld_p(X))_\omega$ is a retract of a perfect object and hence itself perfect.

We have now introduced all the necessary $\infty$-categories of sheaves appearing in the Riemann-Hilbert correspondence. As a last step before stating the correspondence we need to introduce the Riemann-Hilbert functor:

Let $X \in \vStackspip$.

* We denote

\begin{align*}
- \tensor \ri^{+a}_X/\pi\colon \D_\et(X, \Fld_p)^\oc \to \DqcohriX X^\varphi
\end{align*}

the symmetric monoidal functor which is obtained as follows: If $X = \Spa(A, A^+)$ is a strictly totally disconnected perfectoid space (w.l.o.g. of characteristic $p$) then it is the functor $- \tensor_R A^{+a}/\pi\colon \D(R)_\omega \to \Dqcohri(A^+/\pi)^\varphi$ using the identification $\D_\et(X, \Fld_p)^\oc = \D(R)$ for $R = \Fld_p(X)$ from <ref>. In general we define $- \tensor \ri^{+a}_X/\pi$ by gluing using <ref> (see <ref>).

* By the adjoint functor theorem (see <cit.>) the functor in (i) admits a right adjoint which we denote by

\begin{align*}
(-)^\varphi\colon \DqcohriX X^\varphi \to \D_\et(X, \Fld_p)^\oc.
\end{align*}

Note that the functor $- \tensor \ri^{+a}_X/\pi\colon \D_\et(X, \Fld_p)^\oc \to \DqcohriX X^\varphi$ factors over the full subcategory $\DqcohriX X^\varphi_\omega \subset \DqcohriX X^\varphi$ of discrete $\varphi$-modules.

We are finally in the position to state and prove the $p$-torsion Riemann-Hilbert correspondence on small v-stacks:

Let $X \in \vStackspip$. Then the functor

\begin{align*}
- \tensor \ri^{+a}_X/\pi\colon \D_\et(X, \Fld_p)^\oc \injto \DqcohriX X^\varphi
\end{align*}

is fully faithful and induces an equivalence of symmetric monoidal $\infty$-categories

\begin{align*}
\D_\et(X, \Fld_p)^\oc_\perf \isoto \DqcohriX X^\varphi_\perf.
\end{align*}

By v-hyperdescent (see <ref>, <ref>, <ref> and <ref>) the claim can be checked v-locally on $X$, so we can assume that $X$ is a strictly totally disconnected perfectoid space. Then the claim follows from <ref>.

The Riemann-Hilbert correspondence allows us to relate $\Fld_p$-sheaves and $\Fld_p$-cohomology to our theory of quasicoherent $\ri^{+a}_X/\pi$-modules. More concretely we deduce the following general version of the Primitive Comparison Theorem. Note that the relevant condition on $f\colon Y \to X$ (i.e. that $f^!$ preserves small colimits) can be checked locally on $Y$. It is in particular satisfied by smooth maps of analytic adic spaces over $\Q_p$ by <ref> below, so that we recover <cit.>.

Let $f\colon Y \to X$ be a proper bdcs map of $p$-bounded locally spatial diamonds in $\vStackspip$. Assume that $f^!\colon \DqcohriX X \to \DqcohriX Y$ preserves all small colimits. Then:

* The pushforward $f_*\colon \D^+_\et(Y, \Fld_p)^\oc \to \D^+_\et(X, \Fld_p)^\oc$ preserves perfect objects.

* For every $\mathcal F \in \D_\et(Y, \Fld_p)^\oc_\perf$ the natural morphism

\begin{align*}
f_* \mathcal F \tensor \ri^{+a}_X/\pi \isoto f_*(\mathcal F \tensor \ri^{+a}_Y/\pi)
\end{align*}

is an isomorphism in $\DqcohriX X$.

Both claims can be checked locally on $X$, so we can assume that $X$ and $Y$ are qcqs. Since $f^!$ preserves all small colimits, it follows from <ref> that $f_* = f_!\colon \DqcohriX Y \to \DqcohriX X$ preserves almost compact objects and it follows from <ref> that this functor also preserves left-bounded discrete objects. It thus follows from <ref> that $f_*$ preserves left-bounded weakly almost perfect objects. Combining this with <ref> we deduce that $f_*\colon \DqcohriX Y^\varphi \to \DqcohriX X^\varphi$ preserves left-bounded perfect $\varphi$-modules. By <ref> every perfect $\varphi$-module is bounded, hence $f_*$ preserves perfect $\varphi$-modules.

Now let $\mathcal F \in \D_\et(Y, \Fld_p)^\oc_\perf$ be given. Then by <ref> we have $\mathcal F = (\mathcal F \tensor \ri^{+a}_Y/\pi)^\varphi$. It follows formally from adjunctions that $(-)^\varphi$ commutes with pushforwards, so we deduce

\begin{align*}
(f_* (\mathcal F \tensor \ri^{+a}_Y/\pi))^\varphi = f_* (\mathcal F \tensor \ri^{+a}_Y/\pi)^\varphi = f_* \mathcal F
\end{align*}

On the other hand, by what we have shown above, $\mathcal M' := f_* (\mathcal F \tensor \ri^{+a}_Y/\pi)$ is perfect and thus $\mathcal M'^\varphi \tensor \ri^{+a}_X/\pi = \mathcal M'$ by <ref>. Therefore (ii) follows from the above computation by applying $- \tensor \ri^{+a}_X/\pi$ and (i) follows from <ref> because $f_* \mathcal F = \mathcal M'^\varphi$.

Another corollary of the Riemann-Hilbert correspondence is that the invertible sheaves appearing in the definition of $p$-cohomologically smooth morphisms are rather simple (which is a priori not clear because invertible objects in the almost setting are quite subtle). In fact, they are given by invertible $\Fld_p$-sheaves and are in particular étale locally free:

Let $f\colon Y \to X$ be a bdcs and $p$-cohomologically smooth map of locally spatial diamonds in $\vStackspip$ and let $\mathcal L := f^! \ri^{+a}_X/\pi$. Then there is a unique invertible object $\mathcal L_0 \in \D_\et(Y, \Fld_p)$ such that

\begin{align*}
\mathcal L = \mathcal L_0 \tensor_{\Fld_p} \ri^{+a}_Y/\pi.
\end{align*}

In particular, $\mathcal L$ is étale locally isomorphic to $\ri^{+a}_Y/\pi[n]$ for some integer $n$.

By <ref> $\mathcal L$ is an invertible object of $\DqcohriX Y$. By <ref> it comes naturally equipped with a $\varphi$-action and can thus be seen as an invertible object of $\DqcohriX Y^\varphi$. In particular it is a perfect $\varphi$-module and hence by <ref> corresponds to a perfect object $\mathcal L_0 \in \D_\et(Y, \Fld_p)$. Since <ref> is an equivalence of symmetric monoidal $\infty$-categories, $\mathcal L_0$ is necessarily invertible.

We expect many more applications of the Riemann-Hilbert correspondence to the rigid-analytic world, which we aim to investigate in subsequent papers. We end this subsection with the following neat little observation:

Let $X = \Spa \Q_p$, let $\tilde K := \Q_p(\mu_{p^\infty})$ and denote $\tilde X := \Spa \tilde K$. Then by <ref> the category of $\Gal(\overline{\Q_p}/\Q_p)$-representations on finite discrete $\Fld_p$-vector spaces is equivalent to the category of finite free $\ri^a_{\tilde K}/p$-modules equipped with a compatible pair of a $\varphi$-action and a $\Gamma := \Z_p^\cprod$-action. This is a version of the classical $(\varphi, \Gamma)$-correspondence.

§.§ Poincaré Duality for Rigid-Analytic Varieties

We now apply the formalism developed in the previous subsections to the rigid-analytic setting in order to prove a version of Poincaré duality for $\Fld_p$-cohomology. By definition of $p$-cohomologically smooth morphisms this essentially amounts to saying that if $f\colon Y \to X$ is a smooth map of analytic adic spaces over $\Q_p$ then the associated map of diamonds $Y^\diamond \to X^\diamond$ is $p$-cohomologically smooth. By the stability properties of $p$-cohomological smoothness (see <ref>) we can reduce this question to the case that $X$ is totally disconnected and $Y$ is a torus over $X$ – this is where the 6-functor formalism really simplifies the computation, as it allows us to localize on $Y$. In order to handle the local case we perform a computation involving formal models similar to Faltings' original work [9].

The local computation relies on a good understanding of the interplay of the theory of quasicoherent $\ri^{+a}_X/\pi$-modules with quasicoherent sheaves on a formal model. To motivate the following definitions, recall that for any (qcqs) rigid-analytic variety $X$ over a non-archemedian field $K$, there is a formal model $\mathfrak X$ of $X$ and a morphism

\begin{align*}
(X, \ri^+_X) \to (\mathfrak X, \ri_{\mathfrak X})
\end{align*}

of locally ringed topological spaces. Given a pseudouniformizer $\pi \in K$ we denote $\mathfrak X_\pi := V(\pi) \subset \mathfrak X$. Then the above morphism also induces a morphism

\begin{align*}
t\colon (X_\proet, \ri^+_X/\pi) \to (X, \ri^+_X/\pi) \to (\mathfrak X_\pi, \ri_{\mathfrak X_\pi})
\end{align*}

of locally ringed spaces (cf. <ref> below). Now $\mathfrak X_\pi$ is a scheme and hence admits a good theory of solid quasicoherent sheaves $\D_\solid(\ri_{\mathfrak X_\pi})$. Similarly, on $X^\diamond_\proet$ we defined the category $\DqcohriX{X^\diamond}$ of quasicoherent $\ri^{+a}/\pi$-modules. One would thus expect there to be a pair of adjoint functors

\begin{align*}
t^*\colon \D_\solid(\ri_{\mathfrak X_\pi}) \rightleftarrows \DqcohriX{X^\diamond}\noloc t_*.
\end{align*}

We will now construct this adjunction and study its properties. Of course we want the formation of $t^*$ and $t_*$ to be functorial in $t$ (in a suitable sense) and there is no reason for us to restrict to formal models – any map of ringed spaces as above works. This leads to the following definition:

Let $S = \Spa(R, R^+) \in \AffPerfd_\pi$ be a totally disconnected perfectoid space with pseudouniformizer $\pi$. Let $\StkSch_S$ be the following $2$-category:

* The objects of $\StkSch_S$ are given by the disjoint union of small v-stacks over $S$ and classical schemes over $R^+/\pi$.

* $1$-morphisms $f\colon Y \to X$ of two objects $X, Y \in \StkSch_S$ are given as follows: If both $X$ and $Y$ are small v-stacks or both are schemes then the morphisms $Y \to X$ are the ones in the respective category (over $S$ or over $\Spec R^+/\pi$, respectively). If $Y$ is a scheme and $X$ is not, then there is no morphism $Y \to X$. Finally, if $X$ is a scheme and $Y$ is a small v-stack, then morphisms $Y \to X$ are the morphisms of locally ringed spaces

\begin{align*}
(\abs Y, \ri^+_Y/\pi) \to (X, \ri_X).
\end{align*}

over $(\abs S, \ri^+_S/\pi) = \Spec R^+/\pi$ (cf. <ref>).

* $2$-morphisms in $\StkSch_S$ are given as follows: If all associated objects are small v-stacks then we use the definition of $2$-morphisms of small v-stacks. If one of the associated objects is a scheme then there are no non-trivial $2$-morphisms.

Our aim is to define natural functors $t_*$ and $t^*$ associated to every morphism $t\colon Y \to X$ in $\StkSch_S$. In order to do so, it is useful to first do this on the full subcategory where only those small v-stacks are allowed which are totally disconnected perfectoid spaces and afterwards extend to all small v-stacks. For the second step it is handy to have a site on $\StkSch_S$ which “restricts” to the sites of small v-stacks and discrete adic spaces respectively:

Let $S = \Spa(R, R^+) \in \AffPerfd_\pi$ be totally disconnected. We define a site on $\StkSch_S$ by declaring a sieve $\mathcal C_X$ over an object $X \in \StkSch_S$ to be a covering sieve in the following cases:

* $X$ is a small v-stack and there are small v-stacks $U_i \in \mathcal C_X$ such that the family $(U_i \to X)_i$ defines a v-cover.

* $X$ is a scheme and there are schemes $U_i \in \mathcal C_X$ such that the family $(U_i \to X)_i$ is an open cover.

Note that for every map $Y \to X$ in $\StkSch_S$ and every open immersion $U \injto X$ the fiber product $Y \cprod_X U$ exists and is an open subset of $Y$, namely the preimage of $U$ under $\abs Y \to \abs X$. This implies that the above definition of the site on $\StkSch_S$ is valid.

We are finally in the position to define the functors $t_*$ and $t^*$ promised above. With the category $\StkSch_S$ at hand, this is essentially formal:

Let $S = \Spa(R, R^+) \in \AffPerfd_\pi$ be totally disconnected. There is a unique (up to isomorphism) sheaf of $\infty$-categories

\begin{align*}
\mathcal F\colon \StkSch_S^\opp \to \infcatinf
\end{align*}

with the following properties:

* Restricted to the category of small v-stacks, $\mathcal F$ is the sheaf $(X, \pi) \mapsto \DqcohriX X$.

* Restricted to the category of schemes, $\mathcal F$ is the sheaf $X \mapsto \D^a_\solid(\ri_X)$.

* Let $X = \Spa(A, A^+)$ be a totally disconnected space over $S$, let $\tilde X = \Spec A^+/\pi$ and let $\psi\colon (\abs{X}, \ri^+_X/\pi) \isoto (\abs{\tilde X}, \ri_{\tilde X})$ be the isomorphism of ringed spaces from <ref>. Then

\begin{align*}
\mathcal F(\psi)\colon \D^a_\solid(A^+/\pi) \isoto \Dqcohri(A^+/\pi)
\end{align*}

is the identity.

Let $\StkSch'_S \subset \StkSch_S$ be the full subcategory which contains all (classical) schemes but only those small v-stacks which are totally disconnected perfectoid spaces. Then the functor $\mathcal F\colon (\StkSch'_S)^\opp \to \infcatinf$ is easily constructed using the functoriality of the construction $X \mapsto \D^a_\solid(X)$ for schemes (cf. <ref>). It is clear that $\mathcal F$ is uniquely determined (on $\StkSch'_S$) by the conditions (i), (ii) and (iii). But $\StkSch'_S$ is a basis of $\StkSch_S$ with respect to the site <ref>, so by <ref>, $\mathcal F$ can uniquely be extended to $\StkSch_S$.

Let $S = \Spa(R, R^+) \in \AffPerfd_\pi$ be totally disconnected and let $t\colon Y \to X$ be a morphism in $\StkSch_S$.

* We denote $t^* := \mathcal F(t)$, where $\mathcal F$ is defined as in <ref>. In particular, if $Y$ is a small v-stack and $X$ is a scheme then we obtain the functor

\begin{align*}
t^*\colon \D^a_\solid(\ri_X) \to \DqcohriX Y.
\end{align*}

The existence of $\mathcal F$ makes the construction $t \mapsto t^*$ functorial, in particular we have $t_1^* \comp t_2^* = (t_2 \comp t_1)^*$ for all composable morphisms $t_1$ and $t_2$. It follows easily from the definitions that $t^*$ preserves all small colimits.

* We denote $t_*$ a right adjoint of $t^*$. This is automatically functorial, so in particular we have $t_{1*} \comp t_{2*} = (t_1 \comp t_2)_*$ for all composable morphisms $t_1$ and $t_2$. If $Y$ is a small v-stack and $X$ is a scheme then we obtain the functor

\begin{align*}
t_*\colon \DqcohriX Y \to \D^a_\solid(\ri_X).
\end{align*}

The functors $t_*$ and $t^*$ can be described more explicitly as follows. Let $t\colon Y \to X$ be a morphism in $\StkSch_S$ with $X$ a scheme and $Y$ a small v-stack.

* Given any $\mathcal M \in \D^a_\solid(\ri_X)$, $t^* \mathcal M \in \DqcohriX Y$ is the object whose pullback to every totally disconnected $Z = \Spa(C, C^+) \to Y$ is $t_Z^* \mathcal M$ where $t_Z$ is the induced map of schemes $\Spec C^+/\pi \to X$.

* To describe $t_*$, pick any hypercover $Z_\bullet \to Y$ such that each $Z_n = \bigdunion_{i\in I_n} Z_{n,i}$ is a disjoint union of (untilted) totally disocnnected spaces $Z_{n,i} = \Spa(C^{n,i}, C^{n,i+})$. Let $t_{n,i}\colon \Spec C^{n,i+}/\pi \to X$ denote the induced map of schemes for all $n$ and $i$. Then, given any

\begin{align*}
\mathcal N = (N^{n,i})_{n,i} \in \DqcohriX Y = \varprojlim_{n\in\Delta} \prod_{i\in I_n} \Dqcohri(C^{n,i+}/\pi)
\end{align*}

(cf. <ref>) we have

\begin{align*}
t_* \mathcal N = \varprojlim_{n\in \Delta} \prod_{i\in I_n} t_{n,i*} N^{n,i} \in \D_\solid(\ri_X).
\end{align*}

Both for schemes (see <ref>) and for small v-stacks (see <ref>) we have a six-functor formalism and in particular a good notion of the functors $f_!$ and $f^!$ for suitable morphisms $f$. In order to reduce the computation of these functors from rigid spaces to their formal models we need to show some compatibilities between the scheme 6-functor formalism and the small v-stack 6-functor formalism. We start with some general results:

Let $S = \Spa(R, R^+) \in \AffPerfd_\pi$ be totally disconnected and let

U' [d,"j'",hook] [r,"g'"]     U [d,"j",hook]

X' [r,"g"]     X

be a cartesian diagram in $\StkSch_S$ with $j$ being an open immersion. Then the natural transformation

\begin{align*}
j'_! g'^* \isoto g^* j_!
\end{align*}

of functors $\mathcal F(U) \to \mathcal F(X')$ (with $\mathcal F$ as in <ref>) is an isomorphism.

If all involved spaces are schemes or all involved spaces are small v-stacks then the claim was already proven (for the latter case see <ref>), thus we can assume that $X$ and $U$ are schemes and $X'$ and $U'$ are untilted small v-stacks. Since, as mentioned, $j_!$ satisfies arbitrary base-change in schemes and small v-stacks, we can easily reduce to the case that $X = \Spec A$ is affine and $X' = \Spa(A', A'^+)$ is totally disconnected. In fact, we can even reduce to the case $A = A'^+/\pi$. As in the proof of <ref>, $j_!$ is the colimit of the $j_{i!}$'s for any open cover $U = \bigunion_i U_i$ with maps $j_i\colon U_i \to X$, so since $g^*$ and $g'^*$ preserve colimits we can reduce to the case that $U = U_i$ is a distinguished open subset. Now $g^*$ and $g'^*$ are the identity functors, so the claim is clear.

Let $S = \Spa(R, R^+) \in \AffPerfd_\pi$ be totally disconnected and let

U' [d,"f'"] [r,"j'",hook]     X' [d,"f"]

U [r,"j",hook]     X

be a cartesian diagram in $\StkSch_S$ with $j$ being an open immersion. Then the natural transformation

\begin{align*}
j^* f_* \isoto f'_* j'^*
\end{align*}

of functors $\mathcal F(X') \to \mathcal F(U)$ (with $\mathcal F$ as in <ref>) is an isomorphism.

Pass to right adjoints in <ref>.

Having studied a rather general version of the functors $t_*$ and $t^*$, we will now turn our focus on our main application for this theory: formal models. Let us first make precise what we mean by the morphism $t$ in this scenario. We do not want to deal with subtleties arising from the non-noetherian setting over the totally disconnected space $S$ so we contend ourselves with a somewhat ad-hoc approach by only using affine formal models. The following definitions and results should extend to all suitable formal models (or even “almost adic spaces”) over $S$ in a straightforward way.

Let $S = \Spa(R, R^+)$ be a totally disconnected perfectoid space and let $\mathfrak X = \Spf A^+$ be an affine formal scheme with an adic map to $\mathfrak S := \Spf R^+$ (i.e. $A^+$ is a classically $\pi$-adically complete $R^+$-algebra).

* Let $\mathfrak X^\diamond$ be the untilted small v-sheaf associated to $\mathfrak X$ (see <cit.>). The generic fiber $X$ of $\mathfrak X$ is the small v-sheaf

\begin{align*}
X = \mathfrak X^\diamond \cprod_{\mathfrak S^\diamond} S^\diamond.
\end{align*}

* For a pseudouniformizer $\pi \in R$ we denote

\begin{align*}
\mathfrak X_\pi := V(\pi) \subset \mathfrak X.
\end{align*}

As $\mathfrak X \to \mathfrak S$ is adic by assumption, $\mathfrak X_\pi$ is a (classical) scheme over $\Spec R^+/\pi$.

Let $\mathfrak X$ be an affine formal scheme over $\Spf \Z_p$ and let $\mathfrak X^\diamond$ be the associated untilted small v-sheaf (see <cit.>). Then there is a natural surjective morphism of locally ringed spaces

\begin{align*}
(\abs{\mathfrak X^\diamond}, \ri^+_{X^\diamond}) \to (\abs{\mathfrak X}, \ri_{\mathfrak X}).
\end{align*}

By <cit.> there is a natural continuous surjective map $f\colon \abs{\mathfrak X^\diamond} \to \abs{\mathfrak X}$, so it only remains to find a suitable map $\ri_{\mathfrak X} \to f_* \ri^+_{\mathfrak X^\diamond}$ of sheaves on $\mathfrak X$. Choose any v-hypercover $Y_\bullet \to \mathfrak X^\diamond$ by perfectoid spaces $Y_n$ and denote $g_n\colon Y_n \to \mathfrak X^\diamond$ the projection. Then

\begin{align*}
f_* \ri^+_{\mathfrak X^\diamond} = \eq(f_* g_{0*} \ri^+_{Y_0} \rightrightarrows f_* g_{1*} \ri^+_{Y_1}).
\end{align*}

Hence it is enough to construct compatible maps $\ri_{\mathfrak X} \to f_* g_{n*}\ri^+_{Y_n}$. But by definition of $\mathfrak X^\diamond$ the map $g_n$ is given by a map $g'_n\colon Y_n \to \mathfrak X$ of pre-adic spaces and clearly $g'_n = f \comp g_n$ on underlying topological spaces. This produces the desired morphism.

Let $S = \Spa(R, R^+) \in \AffPerfd_\pi$ be totally disconnected and let $\mathfrak X$ be an adic affine formal scheme over $\mathfrak S = \Spf R^+$ with generic fiber $X$. We define the morphism

\begin{align*}
t_{\mathfrak X}\colon (X, \pi) \to \mathfrak X_\pi
\end{align*}

in $\StkSch_S$ as follows: Using <ref> we get a map of locally ringed spaces $(\abs X, \ri^+_X) \to (\abs{\mathfrak X^\diamond}, \ri^+_{\mathfrak X^\diamond}) \to (\abs{\mathfrak X}, \ri_{\mathfrak X})$. Modding out by $\pi$ and noting that $\abs{\mathfrak X} = \abs{\mathfrak X_\pi}$ we arrive at the desired map

\begin{align*}
(\abs X, \ri^+_X/\pi) \to (\abs{\mathfrak X_\pi}, \ri_{\mathfrak X_\pi})
\end{align*}

of locally ringed spaces. In the following we will often write $t$ for $t_{\mathfrak X}$.

If the generic fiber of the formal scheme $\mathfrak X$ (as defined in <ref>) is $p$-bounded over the base then the morphism $t_{\mathfrak X}$ associated to $\mathfrak X$ as in <ref> behaves a lot like a $p$-bounded proper morphism. This will be made more precise in the following results.

We will frequently make use of the following setup: $S = \Spa(R, R^+) \in \AffPerfd_\pi$ is a totally disconnected perfectoid space with pseudouniformizer $\pi$, $\mathfrak X = \Spf A^+$ is an adic affine formal scheme over $\Spf R^+$ and $X = \Spa(A, A^+)^\diamond$ is the generic fiber of $\mathfrak X$. We furthermore assume that $X \to S$ is $p$-bounded.

Let $S$, $\mathfrak X$ and $X$ be as in <ref>. Then for every totally disconnected space $Y = \Spa(B, B^+)$ and map $Y \to X$ with $\dimtrg(Y/X) < \infty$ we have

\begin{align*}
\DqcohriX{\overline Y^{/X}} = \Dqcohri(B^{+a}/\pi, A^+/\pi),
\end{align*}

compatibly with $\Dqcohrip$, $\Dqcohrim$ and $\Dqcohrib$ on both sides.

We have $\overline Y^{/X} = \Spa(B, (A^+ + \mm_B)')$, where $(-)'$ denotes the integral closure in $B$. In particular $\overline Y^{/X}$ is an affinoid perfectoid space and

\begin{align*}
((A^+ + \mm_B)'/\pi)^a_\solid = (B^{+a}/\pi, A^+/\pi)_\solid.
\end{align*}

Note that $\overline Y^{/X} \to \overline Y^{/S}$ is pro-étale, so the claim follows from <ref>.

Let $S$, $\mathfrak X$ and $X$ be as in <ref>.

* $t_*\colon \DqcohriX X \to \D^a_\solid(\ri_{\mathfrak X_\pi})$ preserves all small colimits.

*  $t_*$ satisfies the projection formula, i.e. for all $\mathcal M \in \D^a_\solid(\ri_{\mathfrak X_\pi})$ and $\mathcal N \in \DqcohriX X$ the natural morphism

\begin{align*}
(t_* \mathcal N) \tensor \mathcal M \isoto t_* (\mathcal N \tensor t^* \mathcal M)
\end{align*}

is an isomorphism.

To prove (i), note that the pushforward $f_*\colon \D_\solid(\ri_{\mathfrak X_\pi}) \to \D_\solid(\ri_K/\pi)$ is just a forgetful functor (since $\mathfrak X$ is affine) and in particular it is conservative and preserves small colimits. It is thus enough to show that the composed functor $f_* t_*\colon \DqcohriX X \to \D^a_\solid(\ri_K/\pi)$ preserves small colimits. Looking at the commutative diagram

X [r,"t"] [d,"f'"]     𝔛_π[d,"f"]

(K, _K) [r,"t'"]     (_K/π)

in $\StkSch_S$ we note that it is enough to show that $t'_* f'_*$ preserves small colimits. But $t'_*$ is the identity functor and $f'$ is $p$-bounded by assumption so that $f'_*$ preserves small colimits by <ref>.

To prove (ii) we argue as in <ref>: Since $t_*$ commutes with colimits by (i), both sides of the claimed isomorphism commute with colimits in $\mathcal M$, so we can assume that $\mathcal M = (A^+/\pi)^a_\solid[S]$ for some profinite set $S$. By a similar argument as in (i) we see that the functor $t_*\colon \DqcohriXZ X \to \D^a_\solid(A^+/\pi, \Z)$ (which can be constructed in the same way as $t_*$ by adapting <ref>) also preserves small colimits and has finite cohomological dimension with respect to the canonical $t$-structures on both sides. It follows that both sides of the claimed isomorphism commute with Postnikov limits in $\mathcal N$, so that we can assume that $\mathcal N$ is left-bounded. Choose any hypercover $Y_\bullet \to X$ by totally disconnected spaces $Y_n = \Spa(B^n, B^{n+})$. Then the relative compactifications of $Y_\bullet$ over $X$ still form a hypercover, so by <ref> we have

\begin{align*}
\DqcohriX X &= \varprojlim_{n\in\Delta} \Dqcohri(B^{n+,a}/\pi, A^+/\pi),\\
\DqcohriXZ X &= \varprojlim_{n\in\Delta} \Dqcohri(B^{n+,a}/\pi, \Z).
\end{align*}

Thus $\mathcal N$ is represented by a cosimplicial object $N^\bullet$ in $\Dqcohri(A^+/\pi)$ with $N^n \in \Dqcohri(B^{n+,a}/\pi, A^+/\pi)$ for all $n$, and $t_*$ computes the totalization. Now the same computation as in <ref> implies the claim.

The main purpose of introducing the morphism $t_{\mathfrak X}$ for formal schemes is that it often allows us to reduce the computation of the functor $f_!$ on quasicoherent $\ri^{+a}_X/\pi$-modules on small v-stacks to the computation of the functor $f_!$ in the setting of schemes. This should be true quite generally, but for simplicity we only show it in the case we care about:

Let $S$, $\mathfrak X$ and $X$ be as in <ref> and assume that $A^+ = R^+\langle T^\pm \rangle$, i.e. $X$ is the relative torus over $S$. Then there is a natural isomorphism

\begin{align*}
f_! \isom \mathfrak f_{\pi !} \comp t_{\mathfrak X *}
\end{align*}

of functors $\DqcohriX X \to \Dqcohri(R^+/\pi)$.

We let $\mathfrak Y$ be the relative projective line over $\Spf R^+$, glued out of two copies of $\Spf R^+\langle T \rangle$ along their intersection. Then $\mathfrak Y_\pi := V(\pi) \subset \mathfrak Y$ is the relative projective line $\mathfrak g_\pi\colon \mathfrak Y_\pi \to \Spec R^+/\pi$ and the generic fiber of $\mathfrak Y$ is the relative projective line $g\colon Y \to S$. We get a canonical map $t = t_{\mathfrak Y}\colon Y \to \mathfrak Y_\pi$ in $\StkSch_S$ (even though we have defined $t$ only for affine formal schemes, it is easy to define it for $\mathfrak Y$ as well). We have a natural open immersion $\mathfrak j_\pi\colon \mathfrak X_\pi \injto \mathfrak Y_\pi$, whose pullback along $Y \to \mathfrak Y_\pi$ is the natural open immersion $j\colon X \injto Y$. We get the following diagram of $\infty$-categories:

X [r,"j_!"] [d,"t_*"]     Y [r,"g_*"] [d,"t_*"]     (R^+/π)

(_𝔛_π) [r,swap,"𝔧_π!"]     (_𝔜_π) [ur,swap,"𝔤_π*"]

The composition of the upper horizontal maps computes $f_!$, while the composition of the bottom horizontal map and the diagonal map computes $\mathfrak f_{\pi!}$. It is clear that the triangle on the right commutes, so in order to finish the claim it is enough to show that the left square also commutes. In other words we need to see that the natural morphism $\mathfrak j_{\pi!} \comp t \isoto t \comp j_!$ is an isomorphism. But this is a formal consequence of the projection formulas for $j_!$, $\mathfrak j_{\pi!}$ and $t_*$ (see <ref>) and base-change for all these maps along open immersions (see <ref>), cf. the proof of <ref>.

As mentioned above, <ref> holds in much greater generality, i.e. for arbitrary formal models over $S$ or more generally for certain (almost) adic spaces over $S$. However, stating such a result requires a good understanding of (almost) adic spaces over $\Spa(R^+/\pi)$, which we avoided here for simplicity.

With <ref> at hand we finally have a good way of computing the lower shriek functor on the torus by passage to its canonical formal model. Thus there is nothing in the way of performing Faltings' computation in our setting now:

Let $X = \Spa(A, A^+)$ be a totally disconnected perfectoid space over $\Q_p(\mu_{p^\infty})$. Let $\mathfrak Y := \Spa A^+\langle T^{\pm1} \rangle$ be the relative torus, with generic fiber $Y \to X$. Then for all $N \in \Dqcohri(A^+/\pi[T^{\pm1}])$ there is a functorial isomorphism

\begin{align*}
t_*(\ri^{+a}_Y/\pi) \tensor N \isom \IHom(t_*(\ri^{+a}_Y/\pi), N[-1])
\end{align*}

in $\Dqcohri(A^+/\pi[T^{\pm1}])$.

For ease of notation denote $B^{(+)}_n := A^{(+)}\langle T^{\pm1/p^n}\rangle$ and $\tilde B^{(+)} := A^{(+)}\langle T^{\pm1/p^\infty}\rangle$. Let us further denote $\Delta := \Hom(\Z[1/p]/\Z, \mu_{p^\infty}(\Q_p(\mu_{p^\infty})) = \Z_p(1)$. Then $\Delta$ acts continuously on $\tilde B^+$ by

\begin{align*}
\gamma \times T^n = \gamma(n \bmod \Z) \, T^n \qquad \text{for $\gamma \in \Delta$ and $n \in \Z[1/p]$}
\end{align*}

and this action induces a $\Delta$-torsor $\Spa(\tilde B, \tilde B^+) \to Y$. By <ref>, $\DqcohriX Y$ is equivalent to the $\infty$-category of smooth $\Delta$-representations on $(\tilde B^+/\pi)^a_\solid$-modules and $\Gamma(Y, -)$ computes smooth $\Delta$-cohomology. Note that $\Gamma(Y, -)$ viewed as a functor to $\Dqcohri(A^+/\pi)$ is the composition of $t_*$ and the forgetful functor $\Dqcohri(B_0^+/\pi) \to \Dqcohri(A^+/\pi)$, hence $t_*$ is computed by smooth $\Delta$-cohomology. By <ref> smooth group cohomology agrees with classical continuous group cohomology on discrete objects. Since $\Delta \isom \Z_p$, it is well-known how to compute the cohomology:

\begin{align*}
t_*(\ri^{+a}_Y/\pi) &= \Gamma(\Delta, \tilde B^{+a}/\pi)\\
&\isom [\tilde B^{+a}/\pi \xto{1 - \gamma} \tilde B^{+a}/\pi]\\
&= \bigdsum_{n\in \Z[\frac1p]/\Z} [B^+_0/\pi \xto{1 - \gamma(n)} B^+_0/\pi]
\end{align*}

where $\gamma$ is any fixed generator of $\Delta$ and the complex on the right-hand side sits in cohomological degrees $0$ and $1$.
We now obtain a natural morphism

\begin{align*}
t_*(\ri^{+a}_Y/\pi) \tensor t_*(\ri^{+a}_Y/\pi) \to t_*(\ri^{+a}_Y/\pi) \to H^1(t_*(\ri^{+a}_Y/\pi))[-1] \surjto B^+_0/\pi[-1](-1),
\end{align*}

where the last morphism is the projection to the summand for $n = 0$.
After tensoring with $N$, this induces a morphism

\begin{align*}
\varphi\colon t_*(\ri^{+a}_Y/\pi) \tensor N \to \IHom(t_*(\ri^{+a}_Y/\pi), N[-1](-1))
\end{align*}

in $\Dqcohri(B^+_0/\pi)$, which we claim to be an isomorphism. Assuming for a moment that $N = B^{+a}_0/\pi$, one checks that for each $n \in \Z[1/p]/\Z$, $\varphi$ maps the $n$-part of $t_*(\ri^{+a}_Y/\pi)$ isomorphically to the dual of the $-n$-part of $t_*(\ri^{+a}_Y/\pi)$. Thus $\varphi$ is essentially given by the natural map

\begin{align*}
\bigdsum_{n\in \Z[\frac1p]/\Z} [B^+_0/\pi \xto{1 - \gamma(n)} B^+_0/\pi] \to \prod_{n\in \Z[\frac1p]/\Z} [B^+_0/\pi \xto{1 - \gamma(n)} B^+_0/\pi].
\end{align*}

Note that the cohomology groups of the $n$-part in the above sum and product is killed by pseudouniformizers which get arbitrarily close to $1$ for increasing powers of $p$ in the denominator of $n$. This implies that the above map from sum to product is indeed an (almost) isomorphism, proving that $\varphi$ is an isomorphism.

For general $N$ it is now enough to show that we can pull $N$ out of the $\IHom$. But by the same argument as before, this $\IHom$ is computed as a direct sum of $\IHom$'s for all $n$-parts, and $N$ can obviously be pulled out of each of these $\IHom$'s.

We finally arrive at the first main result of the present subsection, which is also one of the main applications of our 6-functor formalism:

Let $f\colon Y \to X$ be a smooth map of analytic adic spaces over $\Q_p$. Then the induced map $f^\diamond\colon Y^\diamond \to X^\diamond$ of diamonds is $p$-cohomologically smooth.

In the following we will drop the diamond superscript and simply denote by $X$, $Y$ and $f$ their counterparts in the world of diamonds. By <ref> $f$ is $p$-bounded, hence bdcs. We now show that $f$ is $p$-cohomologically smooth. By <ref> this reduces to the case that $X = \Spa(A, A^+)$ is a strictly totally disconnected perfectoid space and then, as $f$ is locally étale over a relative torus, to the case that $f$ is a relative torus. Finally, by stability of $p$-cohomological smoothness under compositions we reduce to the case that $Y = \Spa(A\langle T^{\pm1} \rangle, A^+\langle T^{\pm1} \rangle)$ is a relative torus of dimension $1$. Now pick a pseudouniformizer $\pi$ on $X$. Given any $M \in \Dqcohri(A^+/\pi)$ we will show that the natural morphism

\begin{align*}
f^!(A^{+a}/\pi) \tensor f^* M \to f^! M
\end{align*}

is an isomorphism in $\DqcohriX Y$ and that $f^!(A^{+a}/\pi) \isom \ri^{+a}_Y/\pi[2]$. Note that $f$ is the generic fiber of the map $\mathfrak f\colon \mathfrak Y = \Spf A^+\langle T^{\pm1} \rangle \to \Spf A^+$ of formal schemes. Thus by <ref> we get $f_! = \mathfrak f_{\pi!} \comp t_*$ (with notation as in that result). By adjunctions it follows that providing a map $\ri^{+a}_Y/\pi[2] \to f^!(A^{+a}/\pi)$ is equivalent to providing a map $t_* \ri^{+a}_Y/\pi[2] \to \mathfrak f_\pi^!(A^{+a}/\pi) \isom A^{+a}/\pi[1]$ (where the computation of $\mathfrak f_\pi^!$ follows from <ref>). Such a map exists by the explicit computations in <ref> (it is given by a projection map on the first homology groups). It now remains to show that the induced map

\begin{align*}
f^* M[2] \to f^!(A^{+a}/\pi) \tensor f^* M \to f^! M
\end{align*}

is an isomorphism.

Let $\tilde Y = \varprojlim_n Y_n \to Y$ be the standard pro-étale tower, where $Y_n$ is obtained by adjoining the $p^n$-th root of $T$ (the maps $Y_n \to Y$ are étale because we are working over $\Q_p$). As $\tilde Y \surjto Y$ is surjective, it is enough to verify the above isomorphism after pullback to $\tilde Y$. Then, since $\tilde Y$ is a $p$-bounded affinoid perfectoid space, by <ref> it is enough to verify the isomorphism after applying $\Gamma(\tilde Y, -)$. By <ref> it is even enough to verify the isomorphism after applying $\Gamma(Y_n, -)$ for all $n$. Each $Y_n$ is itself a relative torus of dimension $1$ over $X$, so the claim boils down to showing the isomorphism after applying $\Gamma(Y, -)$, i.e. we need to see that the morphism

\begin{align*}
\Gamma(Y, f^* M[2]) \to \Gamma(Y, f^! M)
\end{align*}

is an isomorphism in $\Dqcohri(A^+/\pi)$. Let us compute the right-hand side. Note that

\begin{align*}
\Gamma(Y, f^! M) = \Hom(\ri^{+a}_Y/\pi, f^! M)^a = \Hom(f_!(\ri^{+a}_Y/\pi), M)^a
\end{align*}

By <ref> we get

\begin{align*}
\Gamma(Y, f^! M) = \Hom(\mathfrak f_{\pi!} t_* (\ri^{+a}_Y/\pi), M)^a = \IHom(t_*(\ri^{+a}_Y/\pi), \mathfrak f_\pi^! M).
\end{align*}

By Poincaré duality in the setting of schemes (see <ref>) we have $\mathfrak f_\pi^! M \isom \mathfrak f_\pi^* M[1]$. Thus altogether we obtain

\begin{align*}
\Gamma(Y, f^! M) \isom \IHom(t_*(\ri^{+a}_Y/\pi), \mathfrak f_\pi^* M)[1].
\end{align*}

We now compute $\Gamma(Y, f^* M[2])$: By the projection formula for $t_*$ (see <ref>) we have

\begin{align*}
\Gamma(Y, f^*M) = t_* f^* M = t_*(\ri^{+a}_Y/\pi) \tensor \mathfrak f_\pi^* M.
\end{align*}

Thus the desired isomorphism $\Gamma(Y, f^*M[2]) \isom \Gamma(Y, f^! M)$ follows from <ref>.

In order to deduce Poincaré duality for $\Fld_p$-cohomology on smooth rigid-analytic varieties from <ref> it remains to compute the “dualizing complex” $f^!(\ri^{+a}_X/p)$. More specifically we need to show that if $f$ has pure dimension $d$ then $f^!(\ri^{+a}_X/p) = \ri^{+a}_Y/p(d)[2d]$. The proof will rely on the following version of <ref>:

Let $X, Y \to S$ be bdcs and $p$-cohomologically smooth maps of locally spatial diamonds in $\vStackspip$ and let $f\colon Y \to X$ be an $S$-morphism.

*  For all invertible $\mathcal L \in \DqcohriX X^\varphi$ the natural morphism

\begin{align*}
f^!(\ri^{+a}_X/\pi) \tensor f^* \mathcal L \isoto f^! \mathcal L
\end{align*}

is an isomorphism and $f^!(\ri^{+a}_X/\pi)$ is invertible.

*  Let $S' \to S$ be a map of locally spatial diamonds and let $f'\colon Y' \to X'$ denote the base-change of $f$ along $S' \to S$, so that the following diagram is cartesian:

Y' [r,"g'"] [d,"f'"]     Y [d,"f"]

X' [r,"g"]     X

Then for all invertible $\mathcal L \in \DqcohriX X^\varphi$ the natural morphism

\begin{align*}
g'^* f^! \mathcal L \isoto f'^! g^* \mathcal L
\end{align*}

is an isomorphism.

We first prove (i). To show that $f^!(\ri^{+a}_X/\pi)$ is invertible, let us denote $h_X\colon X \to S$ and $h_Y\colon Y \to S$ the two projections. Then by <ref> there is an étale cover $(U_i \to X)_{i\in I}$ such that the restriction of $h_X^!(\ri^{+a}_S/\pi)$ to each $U_i$ is free up to a shift. Since the condition of being invertible is étale local we can replace $X$ by $U_i$ and $Y$ by $Y \cprod_X U_i$ in order to reduce to the case that $h_X^!(\ri^{+a}_S/\pi) \isom \ri^{+a}_X/\pi[n]$ for some $n \in \Z$. Then

\begin{align*}
f^!(\ri^{+a}_X/\pi) = f^!(\ri^{+a}_X/\pi[n])[-n] \isom f^! h_X^!(\ri^{+a}_S/\pi)[-n] = h_Y^!(\ri^{+a}_S/\pi)[-n],
\end{align*}

which is invertible because $h_Y$ is $p$-cohomologically smooth. Now let $\mathcal L \in \DqcohriX X^\varphi$ be a given invertible sheaf; we need to show that the map $f^!(\ri^{+a}_X/\pi) \tensor f^* \mathcal L \isoto f^! \mathcal L$ is an isomorphism. By <ref> $\mathcal L$ comes from an invertible $\Fld_p$-sheaf on $X$ and is in particular étale locally free up to shift. Since the claimed isomorphism is compatible with étale pullback we can pass to this étale cover of $X$ (and the induced étale cover of $Y$) to reduce to the case that $\mathcal L = \ri^{+a}_X/\pi$. But then the claimed isomorphism is clear.

We now prove (ii), so let $S'$, $X'$, $Y'$ and $\mathcal L$ be given as in the claim, and denote $h_X\colon X \to S$ and $h_Y\colon Y \to S$ the projections, with pullback $h_{X'}\colon X' \to S'$ and $h_{Y'}\colon Y' \to S'$. As in the proof of (i) we can pass to an étale cover of $X$ in order to assume that $\mathcal L$ and $h_X^!(\ri^{+a}_S/\pi)$ are free up to shift. After applying a shift to $\mathcal L$ (which is harmless) we reduce to the case that $\mathcal L = h_X^!(\ri^{+a}_S/\pi)$. Now consider the following diagram of pullback squares:

Y' [r,"g'"] [d,"f'"] [dd,bend right,"h_Y'",swap]     Y [d,swap,"f"] [dd,bend left,"h_Y"]

X' [r,"g"] [d,"h_X'"]     X [d,swap,"h_X"]

S' [r,"g_S"]     S

Since $h_X$ and $h_Y$ are $p$-cohomologically smooth we can apply <ref> in order to get

\begin{align*}
&g'^* f^! \mathcal L = g'^* f^! h_X^! (\ri^{+a}_S/\pi) = g'^* h_Y^! (\ri^{+a}_S/\pi) = h_{Y'}^! g_S^* (\ri^{+a}_S/\pi) = f'^! h_{X'}^! g_S^* (\ri^{+a}_S/\pi) =\\&\qquad= f'^! g^* h_X^! (\ri^{+a}_S/\pi) = f'^! g^* \mathcal L,
\end{align*}

as desired.

The following lemma provides another situation in which the conclusion of <ref> holds. This computation will be applied in a very specific situation in <ref>.

Let $C$ be an algebraically closed non-archimedean field over $\Q_p$ and let

Y' [r,"g'"] [d,"f'"]     Y [d,"f"]

X' [r,"g"]     X

be a cartesian diagram of connected smooth rigid-analytic varieties over $C$ such that $f$ is split surjective and $\dim X - \dim X' = \dim Y - \dim Y'$. Then for every invertible $\mathcal L \in \Dqcohri(\ri^+_X/p)^\varphi$ the natural morphisms

\begin{align*}
g'^* f^! \mathcal L &\isoto f'^! g^* \mathcal L,\\
f'^* g^! \mathcal L &\isoto g'^! f^* \mathcal L
\end{align*}

are isomorphisms.

We first prove the second isomorphism. By the proof of <ref> we know that for a smooth rigid-analytic variety $Z$ over $C$ the map $Z \to \Spa C$ is $p$-cohomologically smooth and the $p$-adic dualizing complex is concentrated in cohomological degree $-2 \dim Z$. By <ref> both $f'^* g^! \mathcal L$ and $g'^! f^* \mathcal L$ are invertible and from the proof of that statement and the assumption on dimensions it follows that they are concentrated in the same cohomological degree $\dim X - \dim X'$. Employing also <ref> we find that the map $f'^* g^! \mathcal L \to g'^! f^* \mathcal L$ is essentially a map of $\Fld_p$-local systems on $Y'$. Since $Y'$ is connected this map of local systems is automatically an isomorphism as soon as it is non-zero. Assume for a contradiction that this map is zero; then by adjunctions and proper base-change also the natural map $f^* g_! g^! \mathcal L \to f^* \mathcal L$ is zero. Since $f$ is split surjective it follows that the map $g_! g^! \mathcal L \to \mathcal L$ is zero. By adjunction this means that the identity $g^! \mathcal L \to g^! \mathcal L$ is zero, i.e. that $g^! \mathcal L$ is zero. On the other hand we know by <ref> that $g^! \mathcal L$ is invertible; contradiction! This finishes the proof of the second isomorphism.

It remains to prove the first isomorphism. But this follows from the second isomorphism in the same way as in the proof of <ref> using <ref>.

Finally we are in the position to compute the dualizing complex $f^!(\ri^{+a}_X/p)$ of a smooth map $f\colon Y \to X$ of analytic adic spaces over $\Q_p$. The following proof is somewhat technical and may be a bit hard to grasp on a first read, so we refer the reader to the following <ref> for an overview of the proof strategy.

Let $f\colon Y \to X$ be a smooth map of analytic adic spaces over $\Q_p$. If $f$ has pure dimension $d$ then there is a natural isomorphism

\begin{align*}
f^{\diamond!}(\ri^{+a}_X/p) = \ri^{+a}_Y/p(d)[2d].
\end{align*}

To improve the readability of the proof we make the following simplifications to the notation: Firstly, we view every analytic adic space over $\Spa\Q_p$ implicitly as a locally spatial diamond. Secondly, we use <ref> to identify invertible $\ri^{+a}_S/p$-sheaves with invertible $\Fld_p$-sheaves on every such locally spatial diamond $S$. In particular we write expressions like $f^! \Fld_p$ for $f^!(\ri^{+a}_X/p)$. This notation is justified because in all relevant situations the functor $f^!$ restricts to a functor on invertible $\Fld_p$-sheaves by <ref>.

Let $\mathcal C$ be the category whose objects are smooth maps $f\colon Y \to X$ of analytic adic spaces over $\Q_p$ of pure dimension $d$ and whose morphisms are commuting squares

Y' [r,"g'"] [d,"f'"]     Y [d,"f"]

X' [r,"g"]     X

such that the induced map $Y' \to X' \cprod_X Y$ is étale. By (the proof of) <ref> we know that for every $f\colon Y \to X$ in $\mathcal C$ the “dualizing complex” $\omega_f := f^! \Fld_p$ is given by an invertible $\Fld_p$-sheaf concentrated in cohomological degree $-2d$. Using <ref> we observe that the assignment

\begin{align*}
\omega\colon \mathcal C^\opp \to \D(\Fld_p)^\heartsuit, \qquad [f\colon Y \to X] \mapsto H^{-2d}(Y, \omega_f)
\end{align*}

defines a functor. The goal is to show that this functor is naturally isomorphic to the functor

\begin{align*}
\omega'\colon \mathcal C^\opp \to \D(\Fld_p)^\heartsuit, \qquad [f\colon Y \to X] \mapsto H^0(Y, \Fld_p(d)).
\end{align*}

Let $\mathcal C_t \subset \mathcal C$ be the full subcategory of those maps $f\colon Y \to X$ such that $f$ admits an étale map to a $d$-dimensional torus over $X$. Since every $f\colon Y \to X$ in $\mathcal C$ admits an étale covering $(Y_i \to Y)_i$ such that all the induced maps $Y_i \to X$ lie in $\mathcal C_t$, both $\omega$ and $\omega'$ are determined by their restriction to $\mathcal C_t$. In other words, it is enough to show that $\omega$ and $\omega'$ are naturally isomorphic on $\mathcal C_t$. Now let $\mathcal C_{tb} \subset \mathcal C_t$ be the full subcategory of those maps $f\colon Y \to X$ where $X$ is a strictly totally disconnected perfectoid space; it is enough to show that the restrictions of $\omega$ and $\omega'$ to $\mathcal C_{tb}$ are naturally isomorphic.

Let $\mathcal C_{tp} \subset \mathcal C_{tb}$ be the full subcategory of those maps $f\colon Y \to X$ where $X = \Spa(C, C^+)$ for some algebraically closed extension $C$ of $\Q_p$ and some open and bounded valuation subring $C^+ \subset C$. We claim that in order to show that $\omega$ and $\omega'$ are naturally isomorphic on $\mathcal C_{tb}$, it is enough to show that this is true on $\mathcal C_{tp}$. To see this, assume that the latter condition is satisfied. Then given any $f\colon Y \to X$ in $\mathcal C_{tb}$ we need to find an isomorphism $\omega(f) \isoto \omega'(f)$ which is functorial in $f$ and compatible with the given isomorphism of $\omega$ and $\omega'$ on $\mathcal C_{tp}$. By looking at stalks it becomes clear that there is at most one isomorphism $\omega(f) \isoto \omega'(f)$ which satisfies the latter condition, and should it exist then automatically satisfies the former condition. It thus remains to show that for fixed $f$ there is an isomorphism $\omega(f) \isoto \omega'(f)$ compatible with the isomorphism of $\omega$ and $\omega'$ on $\mathcal C_{tp}$. First assume that $Y = T^d_X = T^d \cprod X$ is the $d$-dimensional torus over $X$. Note that the assignment $S \mapsto \omega(T^d_S \to S)$ (for perfectoid $S$ over $\Q_p$) defines an invertible $\Fld_p$-sheaf on $\Spa\Q_p$ (here we use that by the proof of <ref> the dualizing complex on the relative torus is free), which by the isomorphism of $\omega$ and $\omega'$ on $\mathcal C_{tp}$ is isomorphic to the invertible $\Fld_p$-sheaf $S \mapsto \omega'(T^d_S \to S)$. In particular this provides the desired isomorphism $\omega(T^d_X \to X) \isoto \omega'(T^d_X \to X)$. This isomorphism is even $H^0(X, \Fld_p)$-linear (instead of merely being $\Fld_p$-linear) and since $H^0(X, \Fld_p) = H^0(T^d_X, \Fld_p)$ we deduce that we get an isomorphism of free sheaves $\omega_f[-2d] \isoto \Fld_p(d)$ on $T_X^d$. In particular, if now $f\colon Y \to X$ in $\mathcal C_{tb}$ is general, i.e. admits an étale map $Y \to T^d_X$, then we get an induced isomorphism $\omega(Y \to X) \isoto \omega'(Y \to X)$ which is still compatible with the isomorphism of $\omega$ and $\omega'$ on $\mathcal C_{tp}$, as desired.

We have reduced the proof to checking that $\omega$ and $\omega'$ are naturally isomorphic on $\mathcal C_{tp}$. For simplicity we can make two more adjustments: Firstly, from now on there is no point in restricting to those $f\colon Y \to X$ where $Y$ is étale over a relative torus anymore. Secondly, we can even assume that $X = \Spa(C, \ri_C)$ (i.e. no general $C^+$ is required) by a similar argument as in the first part of the proof of <cit.> (alternatively argue directly with tori as in the previous paragraph).

Altogether we have reduced the proof to the following claim: Let $\mathcal C' \subset \mathcal C$ be the full subcategory of those maps $f\colon Y \to X$ such that $X = \Spa(C, \ri_C)$ for some algebraically closed extension $C$ of $\Q_p$; then the functors $\omega$ and $\omega'$ are naturally isomorphic on $\mathcal C'$. In particular we have managed to put ourselves into the realm of classical rigid-analytic geometry. To construct the desired isomorphism, we perform further transformations of the problem at hand. The following argument was suggested by Scholze.

Fix some $f\colon Y \to X$ in $\mathcal C'$ and consider the diagonal $\Delta_f\colon Y \injto Y \cprod_X Y$ and the two projections $g_1, g_2\colon Y \cprod_X Y \to Y$. Using <ref> (which applies to $\Delta_f$, $g_1$ and $g_2$) and <ref> we compute

\begin{align*}
&\omega_f = \Delta_f^! g_1^! \omega_f = \Delta_f^! (g_1^* \omega_f \tensor g_1^!\Fld_p) = \Delta_f^! (g_1^* \omega_f \tensor g_2^* \omega_f) = \Delta_f^* (g_1^* \omega_f \tensor g_2^* \omega_f) \tensor \Delta_f^!\Fld_p =\\&\qquad= \omega_f \tensor \omega_f \tensor \Delta_f^!\Fld_p.
\end{align*}

It follows that there is a natural isomorphism $\omega_f = (\Delta_f^! \Fld_p)^\vee$, so it is enough to compute the right-hand side. Note that $\Delta_f$ is a local complete intersection in the sense of <cit.> because $Y$ and $Y \cprod_X Y$ are smooth over $X$ (e.g. by <cit.> this map is locally a section of a relative ball). Since $\Delta_f$ is also a closed immersion, it follows from <cit.> that $\Delta_f$ is locally given by a regular closed immersion (in the sense of algebraic geometry) of affinoid $C$-algebras. By adapting the standard construction of the “deformation to the normal cone” from algebraic geometry to the rigid-analytic world (to handle blow-ups in the rigid-analytic world see e.g. [39]) we construct the following diagram of smooth rigid-analytic varieties over $X = \Spa C$:

Y _X ℙ^1_X [r,hook,"i"] [d]     Ỹ [dl]

Here the left vertical map is the projection to the second factor and the map $i$ is a closed immersion such that for a classical point $x \in \mathbb P^1_X$ the fiber

\begin{align*}
i_x\colon Y = Y \cprod_X \{ x \} \injto \tilde Y_x := \tilde Y \cprod_{\mathbb P^1_X} \{ x \}
\end{align*}

has the following form: If $x \ne \infty$ then $\tilde Y_x = Y \cprod_X Y$ and $i_x = \Delta_f$, but if $x = \infty$ then $\tilde Y_\infty$ is the normal cone of $Y$ in $Y \cprod_X Y$ (which is a vector bundle over $Y$) and $i_\infty$ is the $0$-section. More informally, $\tilde Y$ is a family of maps parametrized by $\mathbb P^1_X$ which deforms the map $\Delta_f$ into the zero-section map of the normal cone. We will now use this deformation in order to reduce the computation of $\Delta_f^!\Fld_p$ to the computation of $i_\infty^! \Fld_p$. To make this reduction we first observe the following diagram of pullback squares:

Y [d,hook,"i_x"] [r,hook,"j_x"] [dd,"h_2'",bend right,swap]     Y _X ℙ^1_X [d,hook,"i",swap] [dd,"h_2",bend left]

Ỹ_x [r,hook,"j'_x"] [d,"h_1'"]     Ỹ [d,"h_1",swap]

{ x } [r,hook,"j”_x"]     ℙ^1_X

We claim that for every invertible $\Fld_p$-sheaf $\mathcal L$ on $\tilde Y$ the natural morphism $j_x^* i^! \mathcal L \isoto i_x^! j_x'^* \mathcal L$ is an isomorphism. As in the proof of <ref> it is enough to show that a similar claim holds for the lower square and the outer square. For the outer square this follows immediately from <ref> because $h_2$ is smooth. For the lower square we can apply <ref>: After passing to an open cover of $Y$ we can assume that $Y$ is connected, in which case all the spaces occurring in the lower square are connected (for $\tilde Y_x$ use the explicit description provided above); also note that $h_1$ is split surjective, with a section given by $\mathbb P^1_X \to Y \cprod_X \mathbb P^1_X \xto{i} \tilde Y$, where the first map is induced by the choice of any fixed $C$-point in $Y$. This proves the claim $j_x^* i^! \mathcal L = i_x^! j_x'^* \mathcal L$.

Let us now denote $\Omega_f := i^!\Fld_p$, which is an invertible sheaf on $Y \cprod_X \mathbb P^1_X$. By the argument in the previous paragraph we know that for $x \in \mathbb P^1_X$ with induced map $j_x\colon Y \injto Y \cprod_X \mathbb P^1_X$ we have

\begin{align*}
j_x^* \Omega_f = \begin{cases} \Delta_f^!\Fld_p, & \text{if $x \ne \infty$},\\ i_\infty^!\Fld_p, & \text{if $x = \infty$.} \end{cases}.
\end{align*}

On the other hand we note that via the projection $h\colon Y \cprod_X \mathbb P^1_X \to Y$, invertible $\Fld_p$-sheaves on $Y$ and on $Y \cprod_X \mathbb P^1_X$ are identified (the equivalence being the functors $h^*$ and $h_*$) – in fact this is true for any small v-stack $Y$ over $X$: One can check this after pullback to a cover of $Y$ by strictly totally disconnected spaces and then pass to stalks in order to reduce to the case that $Y$ is a geometric point; then the claim follows from the fact that $\mathbb P^1$ is simply connected. It follows that there is a natural isomorphism

\begin{align*}
\Delta_f^!\Fld_p = j_0^* \Omega_f = j_0^* h^* h_* \Omega_f = h_* \Omega_f = j_\infty^* h^* h_* \Omega_f = j_\infty^* \Omega_f = i_\infty^!\Fld_p
\end{align*}

of invertible sheaves on $Y$. Altogether we obtain

\begin{align*}
\omega_f = (\Delta_f^!\Fld_p)^\vee = (i_\infty^!\Fld_p)^\vee.
\end{align*}

In other words, we are left with computing $i_\infty^!\Fld_p$. Now $i_\infty\colon Y \injto \tilde Y_\infty$ is the zero-section of a rank-$d$ vector bundle over $Y$, so it is enough to show the following: Let $\mathcal C''$ be the category fibered in groupoids over the category of locally spatial diamonds in $\Q_p$ which for to every such diamond $S$ associates the groupoid of rank-$d$ vector bundles $E \to S$. We obtain a functor

\begin{align*}
\chi\colon \mathcal C''^\opp \to \D(\Fld_p)^\heartsuit, \qquad [E \to S] \mapsto H^{2d}(S, s_E^!\Fld_p),
\end{align*}

where $s_E\colon S \injto E$ denotes the zero-section (the required functoriality for $\chi$ follows from <ref> by observing that locally every vector bundle is of the form $S \cprod \mathbb A^d$). We need to see that $\chi$ is naturally isomorphic to the functor

\begin{align*}
\chi'\colon \mathcal C''^\opp \to \D(\Fld_p)^\heartsuit, \qquad [E \to S] \mapsto H^0(S, \Fld_p(-d)).
\end{align*}

Note that $\mathcal C''$ corresponds to the classifying stack $\Spa\Q_p/\GL_d$, hence $\chi$ and $\chi'$ define $\Fld_p$-local systems on $\Spa\Q_p/\GL_d$. By the usual arguments involving the dual version of the Barr-Beck theorem (applied to the pullback-pushforward adjunction along the projection $\Spa\Q_p \surjto \Spa\Q_p/\GL_d$) it follows that an $\Fld_p$-local system $\mathcal L$ on $\Spa\Q_p/\GL_d$ is the same as an $\Fld_p$-local system $\mathcal L_0$ on $\Spa \Q_p$ together with a co-action $\mathcal L \to \mathcal L \tensor H^0(\GL_d, \Fld_p)$. Since $\GL_d$ is connected and thus $H^0(\GL_d, \Fld_p) = \Fld_p$ we deduce that the categories of $\Fld_p$-local systems on $\Spa\Q_p$ and $\Spa\Q_p/\GL_d$ are naturally isomorphic. In particular it is enough to find a natural isomorphism of the pullbacks of $\chi$ and $\chi'$ to $\Spa\Q_p$.

The map $\Spa\Q_p \to \Spa\Q_p/\GL_d$ is given by associating to every locally spatial diamond $S$ over $\Spa\Q_p$ the trivial vector bundle $S \cprod \mathbb A^d$. In particular the pullback $\chi_0$ of $\chi$ to $\Spa\Q_p$ is given by the functor $S \mapsto H^{2d}(S, s_{S \cprod \mathbb A^d}^!\Fld_p)$, i.e. we have $\chi_0 = s^!\Fld_p[2d]$, where $s\colon \Spa\Q_p \injto \mathbb A^d_{\Q_p}$ is the embedding of the origin. By choosing a suitable open embedding of the torus $T^d_{\Q_p} = \Spa \Q_p\langle T_1^\pm, \dots, T_d^\pm \rangle$ into affine space we get $\chi_0 = t^!\Fld_p[2d]$, where $t\colon \Spa\Q_p \injto T^d_{\Q_p}$ is the embedding of $1$. Let $f\colon T^d_{\Q_p} \to \Spa\Q_p$ be the projection. Then by <ref> we have

\begin{align*}
\Fld_p = t^! f^! \Fld_p = t^!\Fld_p \tensor t^* f^!\Fld_p = \chi_0[2d] \tensor t^* f^!\Fld_p,
\end{align*}

which implies that

\begin{align*}
\chi_0[2d] = (t^* f^!\Fld_p)^\vee.
\end{align*}

Therefore we have reduced the computation of $f^!\Fld_p$ for general smooth $f\colon Y \to X$ to the special case $f\colon T^d_{\Q_p} \to \Spa\Q_p$. This reduces further to the case $d = 1$. But the computation in the torus case was done in the proof of <ref> over algebraically closed fields and by carefully keeping track of Galois actions we also deduce the desired result over $\Q_p$.

While the proof of <ref> is somewhat lengthy and technical, it is surprisingly formal; in fact, the same proof strategy can be applied in different 6-functor formalisms and even in different geometric situations. For the convenience of the reader we summarize the main line of argument:

* We first make several abstract reductions in order to reduce the computation of $f^!\Fld_p$ to the case that $X = \Spa C$ for some algebraically closed extension of $\Q_p$. This step may not be strictly necessary, but it allows us to work in the comfortable setting of classical rigid-analytic geometry.

* Denoting $\Delta_f\colon Y \injto Y \cprod_X Y$ the diagonal, a quick computation shows that $f^! \Fld_p = (\Delta_f^! \Fld_p)^\vee$, so it is enough to compute $\Delta_f^! \Fld_p$.

* By deforming the regular closed immersion $\Delta_f$ to its normal cone $i_\infty\colon Y \injto \tilde Y_\infty$ we deduce $\Delta_f^! \Fld_p = i_\infty^! \Fld_p$, so it is enough to compute $i_\infty^! \Fld_p$. Here we use the fact that every invertible $\Fld_p$-sheaf on $\mathbb P^1$ is trivial.

* We note that the normal cone $\tilde Y_\infty$ is a vector bundle over $Y$ and $i_\infty$ is the zero-section of that vector bundle. Therefore the computation of $i_\infty^! \Fld_p$ is a special case of computing $s_E^! \Fld_p$ along the zero section $s_E\colon S \injto E$ of a general diamond $S$ over $\Q_p$ with rank-$d$ vector bundle $E \to S$. The vector bundles on diamonds over $\Q_p$ are classified by the stack $\Spa\Q_p/\GL_d$ and one sees easily that the formula $[E \to S] \mapsto H^{2d}(S, s_E^! \Fld_p)$ defines an invertible $\Fld_p$-sheaf $\chi$ on this stack. It remains to compute $\chi$.

* Since $H^0(\GL_d, \Fld_p) = \Fld_p$, a standard argument shows that invertible $\Fld_p$-sheaves on $\Spa\Q_p$ and $\Spa\Q_p/\GL_d$ are naturally equivalent. It is thus enough to determine the pullback $\chi_0$ of $\chi$ to $\Spa\Q_p$. One sees immediately that $\chi_0 = s^! \Fld_p[2d]$, where $s\colon \Q_p \to \mathbb A^d_{\Q_p}$ is the embedding of the origin. The computation of this special case was essentially done in the proof of <ref>.

Combining <ref> with the results of <ref> we deduce the following application to $\Fld_p$-cohomology:

Let $K$ be a non-archimedean field extension of $\Q_p$ and let $f\colon Y \to X$ be a proper smooth map of rigid-analytic varieties over $K$. Assume that $f$ has pure dimension $d$. Then for every $\Fld_p$-local system $\mathcal L$ on $Y$ there is a natural isomorphism

\begin{align*}
f_* \mathcal L^\vee (d)[2d] = (f_* \mathcal L)^\vee
\end{align*}

of perfect $\Fld_p$-sheaves on $X$.

By <ref> $f$ is $p$-cohomologically smooth and $f^! (\ri^{+a}_X/p) = (\ri^{+a}_Y/p)(d)[2d]$. Using the Primitive Comparison Theorem (see <ref>) and <ref> we deduce

\begin{align*}
f_* \mathcal L^\vee (d)[2d] \tensor_{\Fld_p} \ri^{+a}_X/p &= f_*(\IHom(\mathcal L, \Fld_p(d)[2d]) \tensor_{\Fld_p} \ri^{+a}_Y/p)\\
&= f_* \IHom(\mathcal L \tensor_{\Fld_p} \ri^{+a}_Y/p, f^!(\ri^{+a}_X/p))\\
&= \IHom(f_*(\mathcal L \tensor_{\Fld_p} \ri^{+a}_Y/p), \ri^{+a}_X/p)\\
&= \mathcal L^\vee \tensor_{\Fld_p} \ri^{+a}_X/p.
\end{align*}

The above computation is compatible with the natural $\varphi$-module structures, so we can pass to $\varphi$-invariants to conclude by <ref>.

Let $K$ be a non-archimedean field extension of $\Q_p$ and let $X$ be a proper smooth rigid-analytic variety of pure dimension $d$ over $K$. Then for all $i \in \Z$ there is a natural perfect pairing

\begin{align*}
H^i_\et(X, \Fld_p) \tensor_{\Fld_p} H^{2d - i}_\et(X, \Fld_p) \to \Fld_p(-d).
\end{align*}

Apply <ref> to $f\colon X \to \Spa K$ and $\mathcal L = \Fld_p$.

§ APPENDIX: INFINITY-CATEGORIES

In the appendix we gather several results about $\infty$-categories which we have not found in the literature. More concretely, the appendix is structured as follows. In <ref> we show functoriality of the derived $\infty$-category and compare derived descent with abelian descent. In <ref> we prove some additional results on presentable $\infty$-categories and in particular deduce that filtered colimits commute with finite limits of $\infty$-categories. In <ref> we gather a few basic results about sheaves in the $\infty$-categorical setting, including very explicit descriptions for sheaves on nice sites. In <ref> we study enriched $\infty$-categories and prove basic results like enriched Yoneda in that setting. In <ref> we introduce a general formalism for capturing 6-functor formalisms in the $\infty$-categorical setting and provide some tools to construct them.

§.§ Derived Infinity-Categories

We collect a few straightforward results about derived $\infty$-categories. In the following let $\catcat \subset \infcatinf$ denote the full subcategory spanned by the ($1$-)categories. The first result shows that the construction $\mathcal A \mapsto \D^+(\mathcal A)$ is functorial in an appropriate sense:

Let $\catcat^{AbI,ex} \subset \catcat$ be the subcategory of abelian categories with enough injectives and exact functors. Then the construction from <cit.> defines a functor

\begin{align*}
\D^+\colon \catcat^{AbI,ex} \to \infcatinf, \qquad \mathcal A \mapsto \D^+(\mathcal A).
\end{align*}

Let us denote by $\infcatinf^{st,tex}$ the $\infty$-category of stable $\infty$-categories equipped with $t$-structures whose heart has enough injectives, where morphisms are $t$-exact functors. We can easily construct the functor $(-)^\heartsuit\colon \infcatinf^{st,tex} \to \catcat^{AbI,ex}$, $\mathcal C \to \mathcal C^\heartsuit$. Now let $\mathcal X \subset \infcatinf^{st,tex}$ be the full subcategory spanned by those $\infty$-categories which are equivalent to $D^+(\mathcal A)$ for some $\mathcal A \in \catcat^{AbI,ex}$. By <cit.> the restriction of the functor $(-)^\heartsuit$ to $\mathcal X$ is fully faithful, hence an equivalence. But then taking any quasi-inverse produces the desired functor $\D^+$.

We now prove an abstract descent result. It shows that derived descent along exact functors is equivalent to deriving the abelian descent.

Let $\mathcal A^\bullet$ be a cosimplicial object of $\catcat$ with limit $\mathcal A := \varprojlim_{n\in\Delta} \mathcal A^n$ in $\catcat$. Assume that the following hold:

* All $\mathcal A^n$ are Grothendieck abelian categories and all transition functors in the diagram $\mathcal A^\bullet$ are exact and preserve colimits.

* For every morphism $\alpha\colon [n] \to [m]$ in $\Delta$ let $\alpha^*\colon \mathcal A^n \to \mathcal A^m$ denote the associated functor and let $\alpha_*\colon \mathcal A^m \to \mathcal A^n$ be a right adjoint. Then $\alpha_*$ is exact.

* Letting $d_n^i\colon [n] \to [n+1]$ denote the $i$th coface map, then for all $\alpha\colon [n] \to [m]$ the following diagram satisfies base-change:

𝒜^m+1 [d,swap,"d^0_m*"]     [l,swap,"α'^*"] [d,"d^0_n*"] 𝒜^n+1

𝒜^m     [l,"α^*"] 𝒜^n

In other words the natural morphism of functors $\alpha'^* d^0_{n*} \isoto d^0_{m*} \alpha^*$ is an equivalence. (Here $\alpha'\colon [n+1] \to [m+1]$ is defined by $\alpha'(0) = 0$, $\alpha'(k+1) = \alpha(k)+1$ for $k \ge 0$.)

*  The category $\mathcal A$ is abelian and admits the following description: An object of $\mathcal A$ is a pair $(X, \alpha)$, where $X$ is an object of $\mathcal A^0$ and $\alpha$ is an isomorphism $\alpha\colon d_0^{0*}X \isoto d_0^{1*}X$ in $\mathcal A^1$ such that the following diagram in $\mathcal A^2$ commutes:

d_1^0* d_0^0* X d_1^1* d_0^0* X [dr,swap,"d_1^0*α"] [rr,"d_1^1*α"]         d_1^1* d_0^1* X d_1^2* d_0^1* X

d_1^0* d_0^1* X d_1^2* d_0^0* X [ur,swap,"d_1^2*α"]

A morphism $(X_1, \alpha_1) \to (X_2, \alpha_2)$ in $\mathcal A$ is a morphism $f\colon X_1 \to X_2$ such that $d_0^{1*}f \comp \alpha_1 = \alpha_2 \comp d_0^{0*}f$.

* Assume furthermore that $\mathcal A$ has enough injectives. Then there is a natural equivalence of $\infty$-categories

\begin{align*}
\D^+(\mathcal A) \isoto \varprojlim_{n\in\Delta} \D^+(\mathcal A^n),
\end{align*}

where the limit on the right is computed in $\infcatinf$ (use <ref> to produce the diagram $\D^+(\mathcal A^\bullet)$). Moreover, if $\mathcal A$ is Grothendieck and $\D(\mathcal A)$ and all $\D(\mathcal A^n)$ are left complete then the above equivalence extends to an equivalence

\begin{align*}
\D(\mathcal A) \isoto \varprojlim_{n\in\Delta} \D(\mathcal A^n).
\end{align*}

Part (i) can easily be checked by hand: Note first that the explicit description of $\mathcal A$ implies that $\mathcal A$ is abelian, so we only have to show that $\mathcal A$ has the claimed description. To make things easier we note that the full subcategory $\catcat \subset \infcatinf$ of ordinary categories is equivalent to a 2-category (this follows from <cit.> and <cit.>). It follows from the proof of <cit.> that $\varprojlim_{n\in\Delta} \mathcal A^n = \varprojlim_{n\in\Delta^{\le2}} \mathcal A^n$, i.e. the limit depends only on the first three terms. The rest is an explicit computation.

To prove (ii), note first that all transition functors in the augmented cosimplicial category $\mathcal A \to \mathcal A^\bullet$ are exact and admit exact right adjoints. Indeed, we only need to check this for the functor $d_{-1}^{0*}\colon \mathcal A \to \mathcal A^0$. By the explicit description of $\mathcal A$ it is clear that this functor is exact. Its right adjoint $d_{-1*}^0\colon \mathcal A^0 \to \mathcal A$ can be computed as follows: For $X \in \mathcal A^0$ we have $d_{-1*}^0 X = (d_{0*}^0 d_0^{1*}X, \alpha)$, where $\alpha$ is the isomorphism

\begin{align*}
d_0^{0*} (d_{0*}^0 d_0^{1*}X) \isom d_{1*}^0 d_1^{1*} d_0^{1*} X \isom d_{1*}^0 d_1^{2*} d_0^{1*} X \isom d_0^{1*} (d_{0*}^0 d_0^{1*} X).
\end{align*}

Note that in this chain of isomorphisms we used the base-change assumption of (c) and that some of the isomorphisms are not canonical (but can be chosen independently of $X$). It is straightforward to check that this construction does indeed provide a description of $d_{-1*}^0$. From the fact that $d_0^{1*}$ and $d_{0*}^0$ are exact one deduces easily that $d_{-1*}^0$ is exact. It also follows that the base change of (c) holds for $\alpha\colon \emptyset \to [n]$ as well: This immediately reduces to $n = 0$, where it follows directly from the explicit description of $d_{-1*}^0$.

Now apply the functor $\D^+$ from <ref> in order to get an augmented cosimplicial $\infty$-category $\D^+(\mathcal A) \to \D^+(\mathcal A^\bullet)$. By abuse of notation we still denote $\alpha^*\colon \D^+(\mathcal A^n) \to \D^+(\mathcal A^m)$ resp. $\alpha_*\colon \D^+(\mathcal A^m) \to \D^+(\mathcal A^n)$ the transition functor resp. the right adjoint of the transition functor associated to a map $\alpha\colon [n] \to [m]$ in $\Delta_+$. These are the induced functors from the corresponding functors in the augmented cosimplicial category $\mathcal A \to \mathcal A^\bullet$. Since the latter functors are all exact, the functors $\alpha^*$ and $\alpha_*$ can be computed directly on homology. It follows immediately that the base-change of (c) also holds in $\D^+(\mathcal A) \to \D^+(\mathcal A^\bullet)$ (even for $\alpha\colon \emptyset \to [n]$ by the previous paragraph).

To prove that the induced functor $\D^+(\mathcal A) \to \varprojlim_{n\in\Delta} \D^+(\mathcal A^n)$ is an equivalence we employ <cit.> (applied to the opposite categories). Condition (2) of that reference was shown above. It remains to prove that the functor $F := d_{-1}^{0*}\colon \D^+(\mathcal A) \to \D^+(\mathcal A^0)$ is conservative and preserves totalizations of $F$-split cosimplicial objects in $\D^+(\mathcal A)$. We can check that $F$ is conservative directly on homology and hence reduce to showing that the functor $d_{-1}^{0*}\mathcal A \to \mathcal A_0$ is conservative, which is clear by the explicit description of $\mathcal A$. The preservation of totalizations can be proved as in the proof of <cit.>.

Finally the claim about unbounded derived categories is obtained by writing $\D(-) = \varprojlim_k \D_{\le k}(-)$ (by definition of left-completeness) and observing that $\D_{\le k}(A) \isoto \varprojlim_{n\in\Delta} \D_{\le k}(\mathcal A^n)$ is an equivalence by the $\D^+$-case. Passing to the limit over $k$ and commuting limits with limits produces the desired result.

§.§ Presentable Infinity-Categories

In this subsection we prove various results about presentable $\infty$-categories. Among others, we show that the $\infty$-category $\infcatinf$ of (small) $\infty$-categories is compactly generated and that this implies that for any regular cardinal $\kappa$, $\kappa$-small limits of $\infty$-categories commute with $\kappa$-filtered colimits. Moreover, we give criteria for detecting that an $\infty$-category is $\kappa$-compactly generated; most prominently we will show that if $\kappa$ is uncountable then a $\kappa$-small limit of $\kappa$-compactly generated $\infty$-categories is still $\kappa$-compactly generated.

We start with the following criterion for detecting $\kappa$-compactly generated $\infty$-categories.

Let $\mathcal C$ be an $\infty$-category with all small colimits, $\kappa$ a regular cardinal and $(X_i)_i$ a small family of $\kappa$-compact objects of $\mathcal C$ such that the family of functors $\Hom(X_i, -)\colon \mathcal C \to \Ani$ is conservative. Then:

* $\mathcal C$ is $\kappa$-compactly generated (in particular presentable). Every object of $\mathcal C$ is an iterated small colimit of $X_i$'s.

* Every $\kappa$-compact object of $\mathcal C$ is a retract of an iterated $\kappa$-small colimit of $X_i$'s.

Let $\mathcal D(0) \subset \mathcal C$ be the full subcategory spanned by the objects $X_i$. For every ordinal $\alpha$ we define the full subcategory $\mathcal D(\alpha) \subset \mathcal C$ by transfinite induction: For limit ordinals we just take the union of the previous steps; for a successor we let $\mathcal D(\alpha + 1)$ be the full subcategory of $\mathcal C$ spanned by all $\kappa$-small colimits of objects in $\mathcal D(\alpha)$. Then each $\mathcal D(\alpha)$ consists of $\kappa$-compact objects of $\mathcal C$ (by transfinite induction) and $\mathcal D := \mathcal D(\kappa)$ is stable under all $\kappa$-small colimits. By <cit.> the inclusion $\mathcal D \injto \mathcal C$ induces a $\kappa$-accessible fully faithful functor

\begin{align*}
\Ind_\kappa(\mathcal D) \xinjto{F} \mathcal C.
\end{align*}

We claim that $F$ is an equivalence, which proves (i) and (ii) (for (ii) use also <cit.>). By <cit.> the functor $F$ admits a right adjoint $G$. It is enough to show that $G$ is conservative, so suppose we have a map $Y \to Y'$ in $\mathcal C$ such that $G(Y) \isoto G(Y')$ is an isomorphism. For every $X_i \in \mathcal D(0)$ we have

\begin{align*}
\Hom(X_i, Y) = \Hom(F(X_i), Y) = \Hom(X_i, G(Y))
\end{align*}

(and similarly for $Y'$ in place of $Y$), hence the map

\begin{align*}
\Hom(X_i, Y) \isoto \Hom(X_i, Y')
\end{align*}

is an isomorphism. Since the family of functors $\Hom(X_i, -)$ is conservative, this implies that $Y \isoto Y'$ is an isomorphism, as desired.

Our next goal is to show that for uncountable cardinals $\kappa$, $\kappa$-small limits of $\kappa$-compactly generated $\infty$-categories are still $\kappa$-compactly generated, see <ref> below (these limits are still presentable by <cit.>, but we need a more refined version of that statement). We start with several technical results about filtered $\infty$-categories. Recall the definition of $\kappa$-cofinal functors from <cit.>.

Let $\tau > \kappa$ be two regular cardinals. Let $\catfiltcofin\tau\kappa \subset \infcatinf$ be the subcategory whose objects are spanned by the $\tau$-filtered $\infty$-categories which admit $\tau$-small $\kappa$-filtered colimits and whose morphisms are spanned by the $\kappa$-cofinal functors which preserve $\tau$-small $\kappa$-filtered colimits.

Let $\tau > \kappa$ be two regular cardinals, let $f\colon \mathcal J \to \mathcal I$ be a morphism in $\catfiltcofin\tau\kappa$ and let $p\colon K \to \mathcal J$ be a diagram where $K$ is $\kappa$-small. Then the induced functor $f_{p/}\colon \mathcal J_{p/} \to \mathcal I_{fp/}$ lies in $\catfiltcofin\tau\kappa$.

By <cit.> the $\infty$-categories $\mathcal J_{p/}$ and $\mathcal I_{fp/}$ are $\tau$-filtered. By <cit.> the functor $f_{p/}$ is $\kappa$-cofinal.

We now show that $\mathcal J_{p/}$ admits $\tau$-small $\kappa$-filtered colimits (the same argument then applies to $\mathcal I_{fp/}$). This is just a refinement of <cit.>: Let $\mathcal K$ be a $\tau$-small $\kappa$-filtered $\infty$-category and let $q_0\colon \mathcal K \to \mathcal J_{p/}$ be a diagram corresponding to a map $q\colon K \star \mathcal K \to \mathcal J$. Observe that $K \star \mathcal K$ is still $\tau$-small and $\kappa$-filtered, so that there is a colimit diagram $\overline q\colon (K \star \mathcal K)^\triangleright \to \mathcal J$. Then note that the diagram $\overline q$ can also be identified with a colimit diagram for $q_0$.

A similar argument as in the previous paragraph shows that $f_{p/}$ preserves $\tau$-small $\kappa$-filtered colimits, which finishes the proof.

Let $\tau > \kappa$ be two regular cardinals. The $\infty$-category $\catfiltcofin\tau\kappa$ admits all $\tau$-small limits and the inclusion $\catfiltcofin\tau\kappa \subset \infcatinf$ preserves $\tau$-small limits.

By the proof of <cit.> it is enough to prove the claim for $\tau$-small products and fiber products. For $\tau$-small products this is easy to see: Given a $\tau$-small family $(\mathcal I_i)_{i\in I}$ in $\catfiltcofin\tau\kappa$, their product $\mathcal I := \prod_i \mathcal I_i$ is again in $\catfiltcofin\tau\kappa$ (this follows easily from the fact that colimits in $\mathcal I$ are computed componentwise). One checks similarly that a functor $\mathcal J \to \mathcal I$ of $\infty$-categories is $\kappa$-cofinal and preserves $\tau$-small $\kappa$-filtered colimits if and only if all the projections $\mathcal J \to \mathcal I_i$ do (using <ref>, reduce the $\kappa$-cofinality proof to a proof of weak cofinality).

It remains to prove the claim for fiber products, so let us assume that we are given a homotopy Cartesian (with respect to the Joyal model structure) diagram

𝒥' [r,"q'"] [d,"p'"]     𝒥 [d,"p"]

ℐ' [r,"q"]     ℐ

of $\infty$-categories such that $\mathcal J$, $\mathcal I$, $\mathcal I'$, $p$ and $q$ all lie in $\catfiltcofin\tau\kappa$. By <cit.>, $\mathcal J'$, $p'$ and $q'$ all lie in $\catfiltcofin\tau\kappa$ as well. To finish the proof, we have to show the following: Given an object $\mathcal K \in \catfiltcofin\tau\kappa$ and a functor $f\colon \mathcal K \to \mathcal J'$ such that $p' \comp f$ and $q' \comp f$ both lie in $\catfiltcofin\tau\kappa$, then $f$ lies in $\catfiltcofin\tau\kappa$. Given such $f$, it follows from <cit.> that $f$ preserves $\tau$-small $\kappa$-filtered colimits. It remains to show that $f$ is $\kappa$-cofinal.

Let $K$ be a $\kappa$-small and weakly contractible simplicial set and $g\colon K \to \mathcal K$ a diagram. We have to show that the functor $\mathcal K_{g/} \to \mathcal J'_{fg/}$ induced by $f$ is weakly cofinal. By <ref> and <cit.> the induced diagram

𝒥'_fg/ [r,"q'"] [d,"p'"]     𝒥_p'fg/ [d,"p"]

ℐ'_q'fg/ [r,"q"]     ℐ_pq'fg/

of $\infty$-categories is still homotopy cartesian and lies in $\catfiltcofin\tau\kappa$. This allows us to replace $\mathcal K$ by $\mathcal K_{g/}$, $\mathcal J'$ by $\mathcal J'_{fg/}$ etc. in order to reduce to the case $K = \emptyset$. In other words, we want to show that $f$ is weakly cofinal.

Let an object $j' \in \mathcal J'$ be given. We have to find some $k \in \mathcal K$ and a morphism $j' \to f(k)$ in $\mathcal J'$. Since the diagram $\mathcal J' \to \mathcal I' \cprod \mathcal J \from \mathcal K$ lies in $\catfiltcofin\tau\kappa$, it satisfies the hypothesis of <cit.>. The proof of loc. cit. then constructs a commutative diagram

A [r] [d, "α"]     (κ) [d, "γ"]     A' [l] [d, "β"]

𝒥' [r]     ℐ' 𝒥     𝒦 [l]

where $(\kappa)$ is the set of ordinals less than $\kappa$, which splits as a disjoint union $(\kappa) = A \dunion A'$ of even and odd ordinals, respectively. The inductive nature of the proof of loc. cit. allows us to choose $\alpha(0) = j'$. Now let $\overline j'$ be a colimit of $\alpha$ and $k$ a colimit of $\beta$. Then the images of $\overline j'$ and $k$ in $\mathcal I' \cprod \mathcal J$ are equivalent (both are colimits of $\gamma$). We can assume that $p$ and $q$ are categorical fibrations and hence that $\mathcal J' = \mathcal I' \cprod_{\mathcal I} \mathcal J$. It follows that $f(k) \in \mathcal J'$ is equivalent to $\overline j'$. On the other hand there is clearly a map $j' \to \overline j'$. We therefore obtain a map $j' \to f(k)$, as desired.

Let $\mathcal C = \varprojlim_i \mathcal C_i$ be a limit of $\infty$-categories and let $K$ be a simplicial set. Assume that all $\mathcal C_i$ admit colimits indexed by $K$ and that all transition functors in the diagram $(\mathcal C_i)_i$ preserve colimits indexed by $K$. Then:

* $\mathcal C$ admits colimits indexed by $K$.

* A diagram $\overline p\colon K^\triangleright \to \mathcal C$ is a colimit diagram if and only if all the induced diagrams $K^\triangleright \to \mathcal C_i$ are colimit diagrams. In particular, all the functors $\mathcal C \to \mathcal C_i$ preserve colimits indexed by $K$.

By the proof of <cit.> it is enough to prove the claim in the case that the limit $\varprojlim_i \mathcal C_i$ is a small product or a fiber product. The former case is clear because colimits in a product are computed pointwise. The latter case is <cit.>.

For the following, recall the definition of $\catAcc\kappa$ in <cit.> for any regular cardinal $\kappa$: This category is the category of $\kappa$-accessible $\infty$-categories whose functors preserve $\kappa$-filtered colimits and $\kappa$-compact objects. Also recall the definition of $\tau \gg \kappa$ for two regular cardinals $\kappa$ and $\tau$ (see <cit.>).

Let $\kappa$ and $\tau$ be two regular cardinals such that $\tau > \kappa$ and $\tau \gg \kappa$. Let $(\mathcal C_i)_{i\in I} \in \catAcc\tau$ be a $\tau$-small diagram in $\catAcc\tau$. Assume that each $\mathcal C_i$ admits all small $\kappa$-filtered colimits and all transition functors in $(\mathcal C_i)_i$ preserves these colimits. Let $\mathcal C = \varprojlim_i \mathcal C_i$ be the limit in $\infcatinf$. Then $\mathcal C$ is also the limit of $(\mathcal C_i)_i$ in $\catAcc\tau$, i.e.

* $\mathcal C$ lies in $\catAcc\tau$,

* given any $\mathcal D \in \catAcc\tau$ then a functor $\mathcal D \to \mathcal C$ lies in $\catAcc\tau$ if and only if all induced functors $\mathcal D \to \mathcal C_i$ lie in $\catAcc\tau$.

Moreover, we have a natural equivalence $\mathcal C^\tau = \varprojlim_i \mathcal C_i^\tau$.

We generalize the argument of <cit.>. By <ref>, $\mathcal C$ admits small $\tau$-filtered colimits. Thus in order to show that $\mathcal C \in \catAcc\tau$ we only have to show that there is an essentially small full subcategory $\mathcal C' \subset \mathcal C$ such that $\mathcal C'$ consists of $\tau$-compact objects and every object in $\mathcal C$ can be written as a $\tau$-filtered colimit in $\mathcal C'$ (cf. <cit.>). We let $\mathcal C' := \varprojlim_i \mathcal C_i^\tau$ (the limit being taken in $\infcatinf$). Then we have to show the following:

* The natural functor $\mathcal C' \to \mathcal C$ is fully faithful (so we will write $\mathcal C' \subset \mathcal C$) and $\mathcal C'$ consists of $\tau$-compact objects in $\mathcal C$. To see this, by the proof of <cit.> it is enough to check that both claims are stable under fiber products and $\tau$-small products. Full faithfulness is clear. Regarding $\tau$-compactness, for a $\tau$-small product this follows from <cit.> and for fiber products it follows from <cit.>.

* Every object in $\mathcal C$ is the colimit of a $\tau$-filtered diagram in $\mathcal C'$. To see this, let $X \in \mathcal C$ be given and for each $i \in I$ let $X_i \in \mathcal C_i$ be its image in $\mathcal C_i$. Let $\mathcal C'_{/X}$ be the full subcategory of $\mathcal C_{/X}$ consisting of those morphisms $Y \to X$ such that $Y \in \mathcal C'$. We similarly define $\mathcal C_{i,/X_i}^\tau$ for each $i \in I$.

Note that $\mathcal C'_{/X}$ is a limit of $(\mathcal C_{i,/X_i}^\tau)_i$ in $\infcatinf$: This can again be checked separately for ($\tau$-)small products and fiber products; for ($\tau$-)small products this is clear and for fiber products this follows in the same way as in <cit.> (note that the argument of <cit.> also applies to overcategories, cf. <cit.>).

From $\mathcal C_i \in \catAcc\tau$ it follows that $\mathcal C_{i,/X_i}^\tau$ is $\tau$-filtered. Moreover, by <cit.> the categories $\mathcal C_i^\tau$ are stable under $\tau$-small colimits in $\mathcal C_i$ so that from the assumptions on $\mathcal C_i$ it follows that $\mathcal C_i^\tau$ is admits all $\tau$-small $\kappa$-filtered colimits; the same is then true for $\mathcal C_{i,/X_i}^\tau$. All in all we see $\mathcal C_{i,/X_i}^\tau \in \catfiltcofin\tau\kappa$ for all $i$. By our assumption on the diagram $(\mathcal C_i)_i$, all transition functors in the diagram $(\mathcal C_{i,/X_i}^\tau)_i$ preserve $\tau$-small $\kappa$-filtered colimits. Moreover, from <cit.> it follows that all transition functors in the diagram $(\mathcal C_{i,/X_i}^\tau)_i$ are $\kappa$-cofinal. Altogether this implies that the diagram $(\mathcal C_{i,/X_i}^\tau)_i$ factors through $\catfiltcofin\tau\kappa$. From <ref> we deduce that $\mathcal C'_{/X} \in \catfiltcofin\tau\kappa$ and that all the functors $\mathcal C'_{/X} \to \mathcal C_{i,/X_i}^\tau$ are $\kappa$-cofinal.

There is an obvious map of diagrams

(𝒞'_/X)^▹[r,"h'"] [d,"(p_i)_i"]     𝒞 [d,"(q_i)_i"]

((𝒞_i,/X_i^τ)^▹)_i [r,"(h_i)_i"]     (𝒞_i)_i

Since each $\mathcal C_i$ lies in $\catAcc\tau$, all the functors $h_i$ are colimit diagrams. From the above discussion we obtain that the functors $p_i$ are $\kappa$-cofinal, so by <cit.> they are in particular cofinal. It follows that the functors $h_i \comp p_i = q_i \comp h'$ are colimit diagrams. Since $\mathcal C$ is a limit of $(\mathcal C_i)_i$, it follows using <ref> that $h'$ is a colimit diagram. Thus $X$ is a $\tau$-filtered colimit of objects in $\mathcal C_i$, as desired.

We have finished the proof of (i). Part (ii) is now easy to see: Let $\mathcal D \in \catAcc\tau$, let $f\colon \mathcal D \to \mathcal C$ be a functor and let $f_i\colon \mathcal D \to \mathcal C_i$ be the induced functors; the only nontrivial parts of (ii) are the following:

* If all $f_i$ preserve $\tau$-filtered colimits then so does $f$. As before, this problem can be decomposed into the cases of ($\tau$-)small products and fiber products of $\infty$-categories. For the latter case use <cit.>.

* If all $f_i$ preserve $\tau$-compact objects then so does $f$. Again the only interesting case is that of a fiber product, where we apply <cit.>.

It remains to show the last part of the claim, i.e. $\mathcal C' = \mathcal C^\tau$. By what we have shown above we know that $\mathcal C = \Ind_\tau(\mathcal C')$. By <cit.> all $\mathcal C_i^\tau$ are idempotent complete, hence by <ref> the same is true for $\mathcal C'$. Thus, by again applying <cit.> we deduce $\mathcal C' = \mathcal C^\tau$.

We now arrive at the promised result about $\kappa$-small limits of $\kappa$-compactly generated $\infty$-categories. Recall the definition of $\catPrLk\kappa$ in <cit.> for any regular cardinal $\kappa$: It is the $\infty$-category of $\kappa$-compactly generated $\infty$-categories whose functors preserve all small colimits and $\kappa$-compact objects.

Let $\tau$ be an uncountable regular cardinal. Then $\catPrLk\tau$ admits $\tau$-small limits and the forgetful functor $\catPrLk\tau \to \infcatinf$ preserves these limits. Moreover, for every $\tau$-small diagram $(\mathcal C_i)_i$ in $\catPrLk\tau$ we have $\mathcal C^\tau = \varprojlim_i \mathcal C_i^\tau$.

Let $(\mathcal C_i)_i$ be a $\tau$-small diagram in $\catPrLk\tau$ with limit $\mathcal C = \varprojlim_i \mathcal C_i$ in $\infcatinf$. Since $\tau$ is uncountable we have $\tau \gg \omega$ and $\tau > \omega$. Thus we can apply <ref> with $\kappa = \omega$ to see that $\mathcal C$ is $\tau$-accessible. By <ref> it has all small colimits and hence is $\tau$-compactly generated. We also obtain the identity $\mathcal C^\tau = \varprojlim_i \mathcal C_i^\tau$. To finish the proof it remains to show that for any $\tau$-compactly generated $\infty$-category $\mathcal D$ and any functor $f\colon \mathcal D \to \mathcal C$, $f$ lies in $\catPrLk\tau$ if and only all induced functors $f_i\colon \mathcal D \to \mathcal C_i$ lie in $\catPrLk\tau$. By <ref> $f$ preserves $\tau$-compact objects if and only if all $f_i$ do. Moreover, $f$ preserves small colimits if and only if all $f_i$ do: This follows as in the proof of <ref> by using <cit.>.

We now turn our focus to limits and colimits inside a presentable $\infty$-category. It is a classical fact that filtered colimits preserve finite limits in the category of sets. This generalizes to a similar statement in the $\infty$-category of anima by <cit.>. In the following we will prove that the same property also holds in the $\infty$-category of (small) $\infty$-categories. First we note that this property holds in any compactly generated $\infty$-category:

Let $\kappa$ be a regular cardinal and let $\mathcal C$ be a $\kappa$-compactly generated $\infty$-category. Then for every regular cardinal $\tau \gg \kappa$, $\tau$-filtered colimits commute with $\tau$-small limits in $\mathcal C$.

More precisely, for every $\tau$-filtered $\infty$-category $\mathcal I$ the colimit functor $\varinjlim_{\mathcal I}\colon \Fun(\mathcal I, \mathcal C) \to \mathcal C$ constructed in <cit.> preserves $\tau$-small limits.