Note that our definition of the index of descendable maps is analogous to <cit.>. One nice feature of descendable morphisms is that they enjoy many stability properties. We now discuss several of them. Later in <ref> we investigate how descendable morphisms behave under filtered colimits of analytic rings.

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

X' [r]     X

be a Cartesian square of analytic spaces over $(V,\mm)$. If $f$ is descendable (of index $n$) then $f'$ is descendable (of index $\le n$).

Assume that $f$ is descendable of index $n$, so in particular it is steady and qcqs. By base-change for the $\natural$-functors (see <ref>) the map $\ri_X \to f_\natural \ri_Y$ is transformed under $g^\natural$ to $\ri_{X'} \to f'_\natural \ri_{Y'}$. It follows that the map $\mathcal K_f \to \ri_X$ is transformed under $g^\natural$ to $\mathcal K_{f'} \to \ri_{X'}$. Since $g^\natural$ is monoidal with respect to the composition monoidal structure (see <ref>) it follows that the map $\mathcal K_f^n \to \ri_X$ is transformed under $g^\natural$ to $\mathcal K_{f'}^n \to \ri_{X'}$. The claim follows.

Let $g\colon Z \to Y$ and $f\colon Y \to X$ be two morphisms of analytic spaces over $(V,\mm)$.

*  If $f$ is descendable (of index $n$) and $g$ is descendable (of index $m$) then $f \comp g$ is descendable (of index $\le nm$).

*  If $f \comp g$ is descendable (of index $n$) and $f$ is qcqs and steady, then $f$ is descendable (of index $\le n$).

Let $h := f \comp g$. Then $h_\natural \ri_Z = f_\natural (g_\natural \ri_Z)$, so there is a cofiber sequence $f_\natural \mathcal K_g \to f_\natural \ri_X \to h_\natural \ri_Z$. Moreover, the counit $\ri_X \to h_\natural \ri_Z$ factors as $\ri_X \to f_\natural \ri_Y \to h_\natural \ri_Z$, where $\ri_X \to f_\natural \ri_Y$ is the counit and $f_\natural \ri_Y \to h_\natural \ri_Z$ is obtained via $f_\natural$ from the counit $\ri_Y \to g_\natural \ri_Z$. Completing this factorization to cofiber sequences, we obtain the following homotopy coherent diagram in $\mathcal E(X)$:

𝒦_f [r] [d]     𝒦_f [r] [d]     0 [d]

𝒦_h [r] [d,dashed]     _X [r] [d]     h_♮_Z [d]

f_♮𝒦_g [r]     f_♮_Y [r]     h_♮_Z

(All horizontal and vertical lines in this diagram are cofiber sequences and the dashed morphism is obtained through the diagram.) It follows that the morphism $\mathcal K_f \to \ri_X$ factors as $\mathcal K_f \to \mathcal K_h \to \ri_X$. Consequently, $\mathcal K_f^n \to \ri_X$ factors as $\mathcal K_f^n \to \mathcal K_h^n \to \ri_X$ for all $n \ge 0$, which implies (ii).

To prove (i), let the indices $n$ and $m$ be as in the claim. Consider the following diagram in $\mathcal E(X)$:

𝒦_f [d]

𝒦_h^m [r] [d]     _X [d]

(f_♮𝒦_g)^m [r]     f_♮_Y

Here the morphism $\mathcal K_h^m \to (f_\natural \mathcal K_g)^m$ is obtained from the morphism $\mathcal K_h \to f_\natural \mathcal K_g$ in the previous diagram and the morphism $(f_\natural \mathcal K_g)^m \to f_\natural \ri_Y$ is obtained as the composition of the obvious morphism $(f_\natural \mathcal K_g)^m \to (f_\natural \ri_Y)^m$ and the morphism $(f_\natural \ri_Y)^m \to f_\natural \ri_Y$ (given by adjunction).
Note that the natural map $\ri_X \to f_\natural \ri_Y$ factors as $\ri_X = \ri_X^m \to (f_\natural \ri_Y)^m \to f_\natural \ri_Y$ (e.g. use that $(f_\natural\ri_Y)^m = (f_\natural f^\natural)^m \ri_Y$ via <ref> and then use the properties of the monad $T = f_\natural f^\natural$). Using also the homotopy coherence of the lower left square of the first diagram, it follows that the second diagram is homotopy coherent.

One checks easily that the map $(f_\natural \mathcal K_g)^m \to f_\natural \ri_Y$ factors as $(f_\natural \mathcal K_g)^m \to f_\natural(\mathcal K_g^m) \to f_\natural \ri_Y$, where the second morphism is the one induced from the morphism $\mathcal K_g^m \to \ri_Y$ in $\mathcal E(Y)$ by applying $f_\natural$. By the descendability of $g$ this second morphism is zero, consequently $(f_\natural \mathcal K_g)^m \to f_\natural \ri_Y$ is zero. It follows that the composition $\mathcal K_h^m \to \ri_X \to f_\natural \ri_Y$ is zero and because the vertical column in the second diagram is a cofiber sequence we deduce that the map $\mathcal K_h^m \to \ri_X$ factors as $\mathcal K_h^m \to \mathcal K_f \to \ri_X$. This implies that the map $\mathcal K_h^{mn} \to \ri_X$ factors as $\mathcal K_h^{mn} \to \mathcal K_f^n \to \ri_X$ and is therefore zero by the descendability of $f$.

Let $f\colon \bigdunion_{i=1}^n U_i \surjto X$ be a finite qcqs cover of an analytic space $X$ over $(V,\mm)$ by steady subspaces $U_i \subset X$. Then $f$ is descendable with index bounded by a constant $c(n)$ depending only on $n$.

For each nonempty subset $I \subset \{ 1, \dots, n \}$ let $U_I := \bigisect_{i\in I} U_i$ and let $f_I\colon U_I \to X$ denote the embedding. Then the natural functor $\mathcal E(X) \to \varprojlim_I \mathcal E(U_I)$, $\mathcal F \mapsto (f_I^\natural \mathcal F)_I$ is an equivalence (this is just saying that $\mathcal E(-)$ satisfies descent along steady covers, which is true by <ref>). This equivalence has a right adjoint $\varprojlim_I \mathcal E(U_I) \to \mathcal E(X)$ given by $(\mathcal F_I)_I \mapsto \varprojlim_I f_{I\natural} \mathcal F_I$ (cf. <ref>). In particular the natural morphism $\ri_X \to \varprojlim_I f_{I\natural} \ri_{U_I}$ is an isomorphism. Now each $f_{\{i\}\natural} \ri_{U_{\{i\}}}$ is a retract of $f_\natural \ri_{\bigdunion_i U_i}$ and any $f_{I\natural} \ri_{U_I}$ is obtained from that by composition, which proves that $f$ is descendable. Moreover, the structure of the limit $\ri_X = \varprojlim_I f_{I\natural} \ri_{U_I}$ does not depend on $X$ and the $U_i$'s, which implies that the index of descendability can be bounded by a constant only depending on $n$.

Let $f\colon Y \to X$ be a map of analytic spaces over $(V,\mm)$.

*  Let $\bigdunion_{i=1}^n U_i \to X$ be a finite qcqs covering by steady subspaces $U_i \subset X$. If each restricted map $f^{-1}(U_i) \to U_i$ is descendable of index $\le m$, then $f$ is descendable of index $\le c(n) m$.

* Let $\bigdunion_{j\in J} V_j \to Y$ be a covering by steady subspaces $V_j \subset Y$. If the composed map $\bigdunion_{j\in J} V_j \to Y \to X$ is descendable (of index $n$) and $f$ is qcqs then $f$ is descendable (of index $\le n$).

Part (ii) is just a special case of <ref> (using <ref> to see that $f$ is steady). To prove (i), note that it follows from the assumptions that the map $\bigdunion_i f^{-1}(U_i) \to \bigdunion_i U_i$ is descendable of index $\le m$. Composing this map with the map $\bigdunion_i U_i \to X$ and using <ref> and <ref> we are reduced to (ii).

Let $G$ be a geometry blueprint over $(V,\mm)$ (see <ref>). Then all of the above definitions and results can be adapted to $G$-analytic spaces via the analytification functor from <ref>. For example we say that a qcqs steady map $f\colon Y \to X$ of $G$-analytic spaces is descendable (of index $n$) if this is true for $f^\an\colon Y^\an \to X^\an$.

We close this subsection by discussing how descendability behaves under a change of almost setup $(V,\mm)$. In fact, this works as nicely as one can hope:

Let $\varphi\colon (V,\mm) \to (V',\mm')$ be a morphism of almost setups and let $f\colon \mathcal A \to \mathcal B$ be a morphism of analytic rings over $(V,\mm)$. If quasicoherent sheaves descend along $f$, then they also descend along $\varphi^* f$.

Assume that quasicoherent sheaves descend along $f$ and denote $\mathcal A \to \mathcal B^\bullet$ the co-Čech nerve of $f$. We have natural pairs of adjoint functors

\begin{align*}
L\colon \D(\mathcal A) \rightleftarrows \varprojlim_{n\in\Delta} \D(\mathcal B^n) \noloc R, \qquad L'\colon \D(\varphi^*\mathcal A) \rightleftarrows \varprojlim_{n\in\Delta} \D(\varphi^*\mathcal B^n) \noloc R'.
\end{align*}

Here $L(M) = (M \tensor_{\mathcal A} \mathcal B^\bullet)$ and $R(M^\bullet) = \varprojlim_{n\in\Delta} M = \Tot(M^\bullet)$ (and similarly for $L'$ and $R'$). By assumption both $L$ and $R$ are equivalences, and we need to show that the same is true for $L'$ and $R'$. This can be checked separately in the cases that $\varphi$ is strict and that $\varphi$ is localizing.

First assume that $\varphi$ is strict. Then $\varphi$ corresponds to the morphism $V^a \to V'^a$ of analytic rings over $(V,\mm)$ and $\varphi^* \mathcal A$ is the induced analytic ring structure on $\underline{\mathcal A} \tensor_V V'$ from $\mathcal A$ (and similarly for $\mathcal B^n$ in place of $\mathcal A)$. In particular, $\varphi^*\colon \D(\mathcal A) \to \D(\varphi^*\mathcal A)$ has a right adjoint $\varphi_*$ (similarly for $\mathcal B^n$), which induces a functor

\begin{align*}
\varphi_*\colon \varprojlim_{n\in\Delta} \D(\varphi^*\mathcal B^n) \to \varprojlim_{n\in\Delta} \D(\mathcal B^n), \qquad (M^\bullet) \mapsto (\varphi_* M^\bullet).
\end{align*}

(For this to work one needs to check that $\varphi_*$ commutes with pullbacks in the diagram $\D(\varphi^* \mathcal B^\bullet)$ which is obvious by the above description of $\mathcal B^\bullet$ in terms of the induced analytic ring structure.) Note that $\varphi_* L' = L \varphi_*$ and $\varphi_* R' = R \varphi_*$. In particular $\varphi_* L'R' = LR \varphi_* = \varphi_*$ and $\varphi_* R'L' = \varphi_*$. Since $\varphi_*$ is conservative this implies $L'R' = \id$ and $R'L' = \id$, hence $L'$ and $R'$ are equivalences.

Now assume that $\varphi$ is localizing. Then by <ref> the functor $\varphi^*\colon \D(\mathcal A) \to \D(\varphi^* \mathcal A)$ admits a fully faithful left adjoint $\varphi_!$ (and similarly for $\mathcal B^n$ in place of $\mathcal A$). By passing to left adjoints in <ref> we deduce that $\varphi_!$ commutes with pullbacks in the diagram $\D(\varphi^* \mathcal B^\bullet)$ and hence defines a functor

\begin{align*}
\varphi_!\colon \varprojlim_{n\in\Delta} \D(\varphi^*\mathcal B^n) \to \varprojlim_{n\in\Delta} \D(\mathcal B^n), \qquad (M^\bullet) \mapsto (\varphi_! M^\bullet).
\end{align*}

Then $\varphi_! L' = L \varphi_!$ and $R' = \varphi^* R \varphi_!$. Therefore $R'L' = \varphi^* R \varphi_! L' = \varphi^* RL \varphi_! = \varphi^* \varphi_! = \id$. By functoriality of pullbacks (see <ref>) we also have $L' \varphi^* = \varphi^* L$ and hence $L' R' = L' \varphi^* R \varphi_! = \varphi^* LR \varphi_! = \varphi^* \varphi_! = \id$. This shows that $L'$ and $R'$ are equivalences, as desired.

Let $\varphi\colon (V,\mm) \to (V',\mm')$ be a morphism of almost setups and let $f\colon Y \to X$ be a map of analytic spaces over $(V,\mm)$. If $f$ is descendable (of index $n$) then $\varphi^* f$ is descendable (of index $\le n$).

Assume that $f$ is descendable of some index $n$. By <ref> $\varphi^* f$ is qcqs and steady. By definition of descendability the morphism $\mathcal K_f^n \to \ri_X$ in $\mathcal E(X)$ is zero. By <ref> the morphism $\mathcal K_{\varphi^* f}^n = \varphi^\natural \mathcal K_f^n \to \varphi^\natural \ri_X = \ri_{\varphi^* X}$ in $\mathcal E(\varphi^* X)$ is zero, hence $\varphi^* f$ is descendable of index $\le n$.

Let $f\colon \mathcal A \to \mathcal B$ be a map of analytic rings over $V$. If quasicoherent sheaves descend along $f$ then they also descend along $f^a\colon \mathcal A^a \to \mathcal B^a$. Moreover, if $f$ is descendable (of index $n$) then $f^a$ is descendable (of index $\le n$).

§.§ Descendability of Filtered Colimits

As before we fix an almost setup $(V,\mm)$. In our applications we are often interested in the descendability of filtered colimits of maps of analytic rings over $(V,\mm)$ each of which is of a nice simple form; most prominently, pro-étale maps are locally given as a filtered colimit of étale maps. Unfortunately descendability is in general not preserved under filtered colimits (cf. <cit.>). However, quasicoherent sheaves still descend along a filtered colimit of descendable maps of bounded index. Proving this statement is the main goal of the present subsection (see <ref>). The proof consists of two steps: The first step is to show that countable filtered colimits of descendable maps of bounded index are still descendable (similar to <cit.>) and the second step is to show that the descent of quasicoherent sheaves is stable under $\omega_1$-filtered colimits of analytic rings.

We start with the first step, i.e. that descendability is preserved by countable filtered colimits. This will be the content of <ref> below, for which we first need to do a straightforward computation in the $\infty$-category of spectra. Recall that this $\infty$-category comes equipped with a natural $t$-structure whose heart is the category of abelian groups. Moreover, the $\infty$-category of spectra is symmetric monoidal via the smash product $\tensor$.

Let $I$ be a countable filtered category.

*  Let $(X_i)_{i\in I}$ be a diagram of spectra. Then for every $n \in \Z$ there is a natural short exact sequence of abelian groups

\begin{align*}
0 \to R^1 \varprojlim_i \pi_{n+1} X_i \to \pi_n \varprojlim_i X_i \to \varprojlim_i \pi_n X_i \to 0.
\end{align*}

Here the limit on the right denotes the non-derived limit of abelian groups.

*  Let $(X_i)_{i\in I}$ and $(X'_i)_{i\in I}$ be diagrams of spectra. Then for all $n, k \in \Z$ the natural map

\begin{align*}
(R^1 \varprojlim_i \pi_{n+1} X_i) \tensor (R^1 \varprojlim_i \pi_{k+1} X'_i) \to (\pi_n \varprojlim_i X_i) \tensor (\pi_k \varprojlim_i X'_i) \to \pi_{n+k} \varprojlim_i (X_i \tensor X_i')
\end{align*}

is zero. Here the second map comes from the natural map $\varprojlim_i X_i \tensor \varprojlim_i X'_i \to \varprojlim_i X_i \tensor X'_i$.

We first prove (i), so let $(X_i)_i$ and $n \in \Z$ be given. There is a natural cofiber sequence $(\tau_{\ge n+1} X_i)_i \to (X_i)_i \to (\tau_{\le n} X_i)_i$ of diagrams of spectra. By applying $\varprojlim$ and taking the associated homology long exact sequence we obtain an exact sequence of abelian groups

\begin{align*}
&\pi_{n-1} \varprojlim_i \tau_{\le n} X_i \to \pi_n \varprojlim_i \tau_{\ge n+1} X_i \to \pi_n \varprojlim_i X_i \to \pi_n \varprojlim_i \tau_{\le n} X_i \to \pi_n \varprojlim_i \tau_{\ge n+1} X_i \to\\&\qquad\to \pi_{n-1} \varprojlim_i \tau_{\ge n+1} X_i
\end{align*}

Now $\varprojlim_i \tau_{\le n} X_i$ is concentrated in homological degrees $\le n$, so the first term of the above sequence is $0$. Moreover, since $I$ is countable, $\varprojlim_i \tau_{\ge n+1} X_i$ is concentrated in homological degrees $\ge n$ (this boils down to the fact that $R^2\varprojlim_i$ vanishes on systems of abelian groups), hence the last term of the above sequence is $0$. This proves (i).

For (ii), first note that the natural map $\varprojlim_i X_i \tensor \varprojlim_i X'_i \to \varprojlim_i X_i \tensor X'_i$ is obtained by noticing that $\varprojlim_i$ is right adjoint to the functor $X \mapsto (X)_i$ from spectra to $I$-indexed diagrams of spectra (indeed, $\varprojlim_i$ is just a right Kan extension). Since the functor $X \mapsto (X)_i$ is clearly symmetric monoidal, its right adjoint is lax symmetric monoidal. Now apply the functors $\varprojlim$ and $\tensor$ to the cofiber sequence

\begin{align*}
(\tau_{\ge n+1} X_i, \tau_{\ge k+1} X'_i)_i \to (X_i, X'_i)_i \to (\tau_{\le n} X_i, \tau_{\le k} X'_i)_i
\end{align*}

of $I \cprod I$-indexed diagrams of spectra. Looking at the associated homology long exact sequences we see that the described map in (ii) factors over $\pi_{n+k} \varprojlim_i (\tau_{\ge n+1} X_i \tensor \tau_{\ge k+1} X'_i)$. But $\tau_{\ge n+1} X_i \tensor \tau_{\ge k+1} X'_i$ is concentrated in homological degrees $\ge n + k + 2$, hence the $\varprojlim$ of that lies in homological degrees $\ge n + k + 1$ (again because $I$ is countable and therefore $R^2\varprojlim_i = 0$). Thus the map in (ii) factors over $0$, as desired.

Let $I$ be a countable filtered category and let $(f_i\colon \mathcal A_i \to \mathcal B_i)_{i \in I}$ be a diagram of maps of analytic rings over $(V,\mm)$. Assume that there is some $n \ge 0$ such that each $f_i$ is descendable of index $\le n$. Then the colimit

\begin{align*}
f := \varinjlim_i f_i\colon \varinjlim_i \mathcal A_i \to \varinjlim_i \mathcal B_i
\end{align*}

is descendable of index $\le 2n$.

Let $\mathcal A := \varinjlim_i \mathcal A_i$ and let $\mathcal B'_i := \mathcal B_i \tensor_{\mathcal A_i} \mathcal A$ with morphism $f'_i \colon \mathcal A \to \mathcal B'_i$ for every $i \in I$. Then each $f'_i$ is descendable of index $\le n$ by <ref> and $f = \varinjlim_i f'_i$. We can thus replace $\mathcal A_i$ by $\mathcal A$ and $\mathcal B_i$ by $\mathcal B'_i$ to assume that all $f_i$ have the same source $\mathcal A$.

The rest of the proof is analogous to <cit.>: Note first that since $\mathcal E(\mathcal A)$ is stable, it is naturally enriched over the $\infty$-category of spectra (cf. <cit.>), so in the following we can view all $\Hom$'s as spectra. We have $f_\natural \mathcal B = \varinjlim_i f_{i\natural} \id_{\mathcal B_i}$ in $\mathcal E(\mathcal A)$ and consequently $\mathcal K_f = \varinjlim_i \mathcal K_{f_i}$. By assumption the morphisms $\mathcal K_{f_i}^n \to \id_{\mathcal A}$ are zero for all $i$. We will show that $\mathcal K_f^{2n} \to \id_{\mathcal A}$ is zero. By <ref> we have a short exact sequence of abelian groups

\begin{align*}
0 \to R^1 \varprojlim_i \pi_1\Hom(\mathcal K_{f_i}^n, \id_{\mathcal A}) \to \pi_0 \Hom(\mathcal K_f^n, \id_{\mathcal A}) \to \varprojlim_i \pi_0 \Hom(\mathcal K_{f_i}^n, \id_{\mathcal A}) \to 0.
\end{align*}

It follows that the map $\mathcal K_f^n \to \id_{\mathcal A}$ lies in the subgroup

\begin{align*}
R^1 \varprojlim_i \pi_1\Hom(\mathcal K_{f_i}^n, \id_{\mathcal A}) \subset \pi_0 \Hom(\mathcal K_f^n, \id_{\mathcal A}).
\end{align*}

In order to conclude, it is enough to observe that map on $\pi_0$ induced by the obvious map

\begin{align*}
\Hom(\mathcal K_f^n, \id_{\mathcal A}) \cprod \Hom(\mathcal K_f^n, \id_{\mathcal A}) \to \Hom(\mathcal K_f^{2n}, \id_{\mathcal A})
\end{align*}

kills the subgroups $R^1 \varprojlim_i \pi_1\Hom(\mathcal K_{f_i}^n, \id_{\mathcal A})$ (note that the morphism $\mathcal K_f^{2n} \to \id_{\mathcal A}$ is the image of the morphism $\mathcal K_f^n \to \id_{\mathcal A}$ under the above map). This follows from the fact that the above map pairs the $R^1\varprojlim$'s to an $R^2\varprojlim = 0$, see <ref>.

We now come to the second step in our study of descent along filtered colimits. While descendability is not retained under filtered colimits, we now show that descent of quasicoherent sheaves is retained under $\omega_1$-filtered colimits. This is mostly a formal consequence of the fact that $\omega_1$-filtered colimits commute with countable limits in $\infcatinf$ (see <ref>) and that countable limits of $\omega_1$-compactly generated $\infty$-categories are still $\omega_1$-compactly generated (see <ref>). We could not find these two results in the literature, so we included proofs in the appendix. With these results at hand, we can argues as follows (see <ref> for the final result).

Let $\mathcal C = \varinjlim_{i\in I} \mathcal C_i$ be a filtered colimit in $\infcatinf$. Then up to equivalence, $\mathcal C$ can be described as follows:

* The objects of $\mathcal C$ are given by $\Obj(\mathcal C) = \varinjlim_i \Obj(\mathcal C_i)$.

* Given two objects $X, Y \in \mathcal C$, choose $i \in I$ so that $X, Y$ lie in the image of $\mathcal C_i \to \mathcal C$. For every morphism $i \to j$ in $I$ let $X_j$ and $Y_j$ be the images of $X$ and $Y$ along $\mathcal C_i \to \mathcal C_j$. Then

\begin{align*}
\Hom_{\mathcal C}(X, Y) = \varinjlim_{j\in I_{i/}} \Hom_{\mathcal C_j}(X_j, Y_j).
\end{align*}

A close inspection of the construction in the proof of <ref> reveals that the functor $\infcatinf \equiv S^{-1} \mathcal P(\Delta) \injto \mathcal P(\Delta)$ is given by sending $\mathcal C \in \infcatinf$ to the functor $[n] \mapsto \Hom_{\infcatinf}([n], \mathcal C)$. As we showed that the morphisms in $S$ are between compact objects, the proof of <ref> shows that $\infcatinf \subset \mathcal P(\Delta)$ is stable under filtered colimits. As colimits in $\mathcal P(\Delta)$ are computed pointwise, the claim follows.

Let $\kappa$ be a regular cardinal such that $(V,\mm)$ is $\kappa$-compact. Let $\mathcal A = \varinjlim_{i\in I} \mathcal A_i$ be a $\kappa$-filtered colimit of analytic rings over $(V,\mm)$. Then there is a natural equivalence

\begin{align*}
\D(\mathcal A)^\kappa = \varinjlim_i \D(\mathcal A_i)^\kappa.
\end{align*}

Let us abbreviate $\mathcal C_i = \D(\mathcal A_i)$ and $\mathcal C = \D(\mathcal A)$. Let further $\mathcal C' := \varinjlim_i \mathcal C_i^\kappa$. We will implicitly make use of the description from <ref> for $\mathcal C'$.

First note that since pullback functors preserve $\kappa$-compact objects, there is a natural functor $f\colon \mathcal C' \to \mathcal C^\kappa$. We first show that this functor is fully faithful, so let $X', Y' \in \mathcal C'$ be given. Choose $i \in I$ large enough so that $X'$ and $Y'$ come from objects $X_i, Y_i \in \mathcal C_i^\kappa$. For each $j \in I_{i/}$ let $X_j, Y_j \in \mathcal C_j^\kappa$ be the pullback of $X_i$ and $Y_i$ to $\mathcal C_j$. Finally, let $X$ and $Y$ be the images of $X'$ and $Y'$ in $\mathcal C^\kappa$. By the adjunction of pullback and pushforward (the latter being a forgetful functor) and the $\kappa$-compactness of $X_i$ we get

\begin{align*}
&\Hom_{\mathcal C}(X, Y) = \Hom_{\mathcal C_i}(X_i, Y) = \Hom_{\mathcal C_i}(X_i, \varinjlim_{j \in I_{i/}} Y_j) = \varinjlim_{j \in I_{i/}} \Hom_{\mathcal C_i}(X_i, Y_j) =\\&\qquad= \varinjlim_{j \in I_{i/}} \Hom_{\mathcal C_j}(X_j, Y_j) = \Hom_{\mathcal C'}(X, Y).
\end{align*}

This proves full faithfulness of the functor $f\colon \mathcal C' \injto \mathcal C^\kappa$. To show essential surjectivity, first note that all $\mathcal A[S] \in \mathcal C^\kappa$ lie in the image of $f$, where $S$ runs through extremally disconnected sets (because they come via pullback from $\mathcal A_i[S] \in \mathcal C_i^\kappa$ for any $i \in I$). By the full faithfulness of $f$ we know that the essential image of $f$ is stable under retracts and $\kappa$-small colimits (for the latter use <cit.>). But by <ref> every $X \in \mathcal C^\kappa$ is a retract of an iterated $\kappa$-small colimit of $\mathcal A[S]$'s (more precisely, apply <ref> to the presentable $\infty$-categories $\D(\mathcal A)_{\kappa'}$ for increasing strong limit cardinals $\kappa'$). This finishes the proof.

Let $\kappa$ be an uncountable regular cardinal such that $(V,\mm)$ is $\kappa$-compact and let $(f_i\colon \mathcal A \to \mathcal B_i)_{i\in I}$ be a $\kappa$-filtered family of morphisms of analytic rings over $(V,\mm)$. If quasicoherent sheaves descend along each $f_i$, then they also descend along the colimit $f\colon \mathcal A \to \varinjlim_i \mathcal B_i$.

Let $\mathcal B = \varinjlim_i \mathcal B_i$. For each $i \in I$ and $n \ge 0$ let $\mathcal C_i^n := \D(\mathcal B_i^{\tensor n+1})$, where $\mathcal B_i^{\tensor k}$ is the $k$-fold tensor product of $\mathcal B_i$ over $\mathcal A$. Similarly define $\mathcal C^n := \D(\mathcal B^{\tensor n+1})$ and let also $\mathcal C := \D(\mathcal A)$. By assumption the functor

\begin{align*}
\mathcal C \isoto \varprojlim_{n\in\Delta} \mathcal C_i^n
\end{align*}

is an equivalence for all $i \in I$. We have to show that $\mathcal C \to \varprojlim_{n\in\Delta} \mathcal C^n$ is an equivalence as well.

Let us denote $\mathcal C' := \varprojlim_{n\in\Delta} \mathcal C^n$. Working with $\D(-)_{\kappa'}$ instead of $\D(-)$ in all of the above definitions (where $\kappa'$ is a suitably large strong limit cardinal), all $\mathcal C_i^n$ and $\mathcal C^n$ are $\kappa$-compactly generated, so that by <ref> all $\infty$-categories introduced in this proof are $\kappa$-compactly generated and the functor $\mathcal C \to \mathcal C'$ preserves $\kappa$-compact objects. Now working with $\D(-)$ instead of $\D(-)_{\kappa'}$ again, it is enough to show that the induced functor $\mathcal C^\kappa \to \mathcal C'^\kappa$ is an equivalence.

From <ref> we get $\mathcal C'^\kappa = \varprojlim_n (\mathcal C^n)^\kappa$ and for fixed $i$, $\mathcal C^\kappa = \varprojlim_n (\mathcal C_i^n)^\kappa$. On the other hand, for fixed $n$ we have $(\mathcal C^n)^\kappa = \varinjlim_i (\mathcal C_i^n)^\kappa$ by <ref>. By <ref> countable limits commute with $\kappa$-filtered colimits in $\infcatinf$, so we get

\begin{align*}
\mathcal C'^\kappa = \varprojlim_n (\mathcal C^n)^\kappa = \varprojlim_n \varinjlim_i (\mathcal C_i^n)^\kappa = \varinjlim_i \varprojlim_n (\mathcal C_i^n)^\kappa = \varinjlim_i \mathcal C^\kappa = \mathcal C^\kappa,
\end{align*}

as desired.

We have prove the promised stability of descent under $\kappa$-filtered colimits. This suggests the following definition, which generalizes the notion of descendability while keeping many of its stability properties:

A morphism $\mathcal A \to \mathcal B$ of analytic rings over $(V,\mm)$ is called weakly descendable if it is an iterated $\kappa$-filtered colimit of descendable maps, where $\kappa$ is any uncountable regular cardinal such that $(V,\mm)$ is $\kappa$-compact.

One could introduce an index of descendability on weakly descendable morphisms by adding a bound on the index of descendability in the $\kappa$-filtered family of morphisms. However, the main motivation to introduce the index of descendability was to show descendability of countable colimits (cf. <ref>) and this motivation does not apply to weakly descendable morphisms.

We now arrive at the main result of the present subsection, which summarizes the above work by showing that weakly descendable morphisms form a rather nice class of morphisms for studying descent:

* Quasi-coherent sheaves descend along weakly descendable morphisms.

* Every descendable morphism is weakly descendable and weakly descendable morphisms are stable under base-change and $\kappa$-filtered colimits, where $\kappa$ is any uncountable regular cardinal such that $(V,\mm)$ is $\kappa$-compact.

*  Suppose that $(V,\mm)$ is $\omega_1$-compact (e.g. $\mm$ is countably generated). Then a filtered colimit of descendable morphisms of bounded index is weakly descendable.

Part (i) follows from <ref>. The only non-trivial part of (ii) is the stability under base-change, which follows from <ref>.

It remains to prove part (iii). Note that if $\mm$ is countably generated then $(V,\mm)$ is indeed $\omega_1$-compact by <ref>. Let $(f_i\colon \mathcal A \to \mathcal B_i)_{i\in I}$ be a filtered family of morphisms of analytic rings over $(V,\mm)$ and assume that there is some integer $d \ge 0$ such that all $f_i$ are descendable of index $\le d$. We want to show that the colimit $f\colon \mathcal A \to \varinjlim_i \mathcal B_i$ is weakly descendable. By <cit.> we can assume that $I$ is a filtered partially ordered set. Let $J$ be the filtered partially ordered set of all functors $j\colon \N \to I$, where we define $j_1 \le j_2$ iff $j_1(k) \le j_2(k)$ for $k \gg 0$. Then $J$ is in fact $\omega_1$-filtered, i.e. for every countable subset $\{ j_1, j_2, \dots \} \subset J$ there is some $j \in J$ with $j_k \le j$ for all $k$; just choose $j(k)$ inductively to be greater than $j_1(k), \dots, j_k(k)$ and $j(k-1)$. For every $j \in J$ let $\mathcal B_j := \varinjlim_{k\in\N} \mathcal B_{j(k)}$ with associated morphism $f_j\colon \mathcal A \to \mathcal B_j$. We then obtain the $\omega_1$-filtered system $(f_j)_{j\in J}$ of morphisms in $\AnRing_{(V,\mm)}$. By <ref> each $f_j$ is descendable. This implies the claim.

§.§ Descendability over Flat Covers

As before we fix an almost setup $(V,\mm)$. In the present subsection we study a weaker version of descendability which still implies descent of quasicoherent sheaves. Roughly, instead of asking for a map $f\colon \mathcal A \to \mathcal B$ of analytic rings to be descendable, we only require that $f$ be descendable (of bounded index) over a “flat cover” of $\AnSpec \mathcal A$. In our application this “flat cover” will be given by the connected components of an affine scheme $\Spec A$. Note that this cover is not an fpqc cover in general – we make no assumption on quasicompactness.

The main ingredient for the weaker version of descendability is a thorough study of convergence speeds. In the proof of <ref> the crucial input from the descendability condition on a map $f\colon Y \to \AnSpec \mathcal A$ of analytic spaces over $(V,\mm)$ was the fact that if a cosimplicial object $M^\bullet$ in $\D(\mathcal A)$ becomes split in $\D(Y)$ after applying $f^*$ then the associated $\Tot$-tower $(\Tot_n(M^\bullet))_n$ is pro-constant. In the following we will refine this statement by showing that the index of descendability of $f$ gives a bound on the speed of convergence of $(\Tot_n(M^\bullet))_n$.

The following definitions and results are similar to <cit.> but we keep close attention to the speed of convergence throughout.

Let $\mathcal C$ be a stable $\infty$-category.

* Let $\Tow(\mathcal C) := \Fun((\Z_{\ge0})^\opp, \mathcal C)$ denote the category of towers in $\mathcal C$.

* A tower $(X_n)_n \in \Tow(\mathcal C)$ is called nilpotent of index $\le h$ if for all $n \ge h$ the map $X_{n+h} \to X_n$ is zero.

* A tower $(X_n)_n \in \Tow(\mathcal C)$ is called strongly constant of index $h$ if $X := \varprojlim_n X_n$ exists and the cofiber of $X \to (X_n)_n$ is nilpotent of index $h$ (where we identify $X$ with the constant tower with value $X$).

* A cosimplicial object $M^\bullet$ in $\mathcal C$ is called strongly constant of index $h$ if the associated tower $(\Tot_n(M^\bullet))_n$ is strongly constant of index $h$.

* A strongly constant tower is automatically pro-constant, but the converse is far from true (cf. <cit.>). Evidently a tower $(X_n)_n$ is nilpotent (of index $h$) if and only if it is strongly constant (of index $h$) and $\varprojlim_n X_n \isom 0$.

*  One could refine the definition of nilpotent towers by saying that $(X_n)_n$ is nilpotent of index $(h, n_0)$ if for all $n \ge n_0$ the map $X_{n+h} \to X_n$ is zero. However, one is usually only concerned with upper bounds on both $h$ and $n_0$ so that we can savely assume $n_0 = h$ to simplify the notation.

Let $f\colon Y \to X$ be a morphism of analytic spaces over $(V,\mm)$ which is descendable of index $m$ and let $(\mathcal M_n)_n \in \Tow(\D(X))$. If $(f^*M_n)_n$ is strongly constant of index $h$ then $(\mathcal M_n)_n$ is strongly constant of index $s(m) h$, where $s(m)$ is the constant from <ref>.

Suppose that $(f^* \mathcal M_n)_n \in \Tow(\D(Y))$ is strongly constant of index $h$. Then the same is true for $(f_* f^* \mathcal M_n)_n = ((f_\natural \ri_Y)(\mathcal M_n))_n \in \Tow(\D(X))$ (where we use the evaluation functor $\mathcal E(X) \cprod \D(X) \to \D(X)$ from <ref>). By definition of descendability, $\ri_X \in \mathcal E(X)$ can be obtained as a finite sequence of compositions, finite (co)limits and retracts from $f_\natural \ri_Y$. One checks easily that strongly constant towers are stable under these operations as well, proving that $(\mathcal M_n)_n$ is indeed strongly constant. To get the bound on the index, we can argue as follows: We can reduce to the case that $((f_\natural \ri_Y)(\mathcal M_n))_n$ is nilpotent of index $h$, i.e. for every $n \ge h$ the map $(f_\natural \ri_Y)(\mathcal M_{n+h}) \to (f_\natural \ri_Y)(\mathcal M_n)$ is zero. For every $\mathcal G \in \mathcal E(X)$ let $\mathcal I_{\mathcal G}$ be the collection of morphisms $\mathcal M \to \mathcal N$ in $\D(X)$ such that $\mathcal G(\mathcal M) \to \mathcal G(\mathcal N)$ is zero. Arguing as in the proof of <ref> we see that $\mathcal I_{f_\natural \ri_Y}^{s(m)} = 0$. Since each map $\mathcal M_{n+h} \to \mathcal M_n$ lies in $\mathcal I_{f_\natural \ri_Y}$ we deduce that $M_{n+s(m)h} \to M_n$ is zero and hence that $(\mathcal M_n)_n$ is nilpotent of index $s(m) h$.

In the following we will provide an analogue of <ref> in terms of the convergence of homotopy groups and will relate this to the associated spectral sequence.

Let $\mathcal C$ be a stable $\infty$-category with $t$-structure and let $(X_n)_n \in \Tow(\mathcal C)$. After passing to the dual category $\mathcal C^\opp$, <cit.> associates to $(X_n)_n$ a homological spectral sequence $E^r_{p,q}$ in $(\mathcal C^\heartsuit)^\opp$, which is the same as a cohomological spectral sequence $\tilde E_r^{p,q}$ in $\mathcal C^\heartsuit$. As we want to keep homological notation in $\mathcal C$ it is convenient to flip the $q$-coordinate and define $E_r^{p,q} := \tilde E_r^{p,-q}$ to be the spectral sequence associated to $(X_n)_n$. Be aware that the differentials $d_r\colon E_r^{*,*} \to E_r^{*,*}$ have bidegree $(r, r-1)$.

The spectral sequence associated to a tower has the following alternative description in terms of exact couples:

Let $\mathcal C$ be a stable $\infty$-category with $t$-structure and let $(X_n)_n \in \Tow(\mathcal C)$. For $p, q \in \Z$ let $D_1^{p,q} := \pi_{q-p}(X_p)$ and let $E_1^{p,q} := \pi_{q-p}(\fib(X_p \to X_{p-1}))$. There is an exact couple

D_1^*,* [rr,"g"]        D_1^*,* [ld,"h"]

E_1^*,* [ul,"f"]

Here $f$ is induced by $\fib(X_p \to X_{p-1}) \to X_p$ and has bidegree $(0, 0)$, $g$ is induced by $X_p \to X_{p-1}$ and has bidegree $(-1, -1)$, and $h$ is induced by $X_p \to \fib(X_{p+1} \to X_p)[1]$ and has bidegree $(1, 0)$.

Moreover, the spectral sequence associated to the above exact couple is naturally isomorphic to the spectral sequence associated to $(X_n)_n$ as defined in <ref>.

It follows immediately from the long exact sequence associated to $\fib(X_p \to X_{p-1}) \to X_p \to X_{p-1}$ that the given maps $f$, $g$ and $h$ define an exact couple in $\mathcal C^\heartsuit$. Let $C_r\colon D_r^{*,*} \to E_r^{*,*} \to D_r^{*,*}$ denote the $r$-th derived exact couple, so that $E_*^{*,*}$ is the spectral sequence associated to the exact couple. Let us temporarily denote by $\tilde E_*^{*,*}$ the spectral sequence associated to $(X_n)_n$ as in <ref>. We want to show that $E_*^{*,*} \isom \tilde E_*^{*,*}$.

By definition (see <cit.>) we have

\begin{align*}
\tilde E_r^{p,q} = \img(\pi_{q-p} X(p+r-1, p-1) \to \pi_{q-p} X(p, p-r)),
\end{align*}

where $X(i, j) \in \mathcal C$ is functorial in $i, j \in (\Z \union \{ -\infty \})^\opp$ such that $X(i,-\infty) = X_i$ for all $i \ge 0$, $X(i, -\infty) = 0$ for $i < 0$ and $X(i, j) \to X(i, k) \to X(j, k)$ is a cofiber sequence for all $i \le j \le k$. In particular we have $X(i, j) \isom \fib(X_i \to X_j)$.

For all $p$, $q$ and $r$ we have a commutative diagram

π_q-p+1 X_p-1 [r] [d]     π_q-p X(p+r-1, p-1) [r] [d]     π_q-p X_p+r-1 [d]

π_q-p+1 X_p-r [r]     π_q-p X(p, p-r) [r]     π_q-p X_p

Note that $\img(\pi_{q-p+1} X_{p-1} \to \pi_{q-p+1} X_{p-r}) = D_r^{p-r,q-r+1}$ and $\img(\pi_{q-p} X_{p+r-1} \to \pi_{q-p} X_p) = D_r^{p, q}$, so that by taking images along the vertical maps in the above diagram we obtain a couple $\tilde C_r\colon D_r^{*,*} \xto{\tilde h_r} \tilde E_r^{*,*} \xto{\tilde f_r} D_r^{*,*}$. To finish the proof it is enough to show that $C_r \isom \tilde C_r$. One checks easily that $C_1 = \tilde C_1$, so it is enough to show the following, for every $r \ge 1$: Assuming that $\tilde C_r$ is an exact couple, then $\tilde C_{r+1}$ is the derived couple of $\tilde C_r$. This boils down to the following claims:

* For every $r \ge 1$ the differential $\tilde d_r\colon \tilde E_r^{*,*} \to \tilde E_r^{*,*}$ is given by $\tilde d_r = \tilde h_r \comp \tilde f_r$. This follows easily from the functoriality of $X(*,*)$, e.g. the map $\delta\colon \pi_{q-p} X(p, p-r) \to \pi_{q-p-1} X(p+r, p)$ factors as $\pi_{q-p} X(p, p-r) \to \pi_{q-p} X_p \to \pi_{q-p-1} X(p+r, p)$, where the first map is used to define $\tilde f_r$ and the second one is used to define $\tilde h_r$.

* For every $r \ge 1$, the map $\tilde f_{r+1}$ is equal to $\tilde f_r$ after identifying $\tilde E_{r+1}^{*,*} \isom \ker(\tilde d_r) / \img(\tilde d_r)$. To see this, we use the following commutative diagram:

π_q-p+1 X_p-1 [r] [d,equal]     π_q-p X(p+r, p-1) [r] [d]     π_q-p X_p+r [d]

π_q-p+1 X_p-1 [r] [d]     π_q-p X(p+r-1, p-1) [r] [d]     π_q-p X_p+r-1 [d]

π_q-p+1 X_p-r [r] [d,"g"]     π_q-p X(p, p-r) [r] [d]     π_q-p X_p [d,equal]

π_q-p+1 X_p-r-1 [r]     π_q-p X(p, p-r-1) [r]     π_q-p X_p

The image of the middle vertical map is $\tilde E_r^{p,q}$ and the image of the vertical composition of maps is $E_{r+1}^{p,q}$. The diagram provides the desired identification $\tilde E_{r+1}^{*,*} \isom \ker(\tilde d_r) / \img(\tilde d_r)$ (cf. the proof of <cit.>). The maps $\tilde f_r$ and $\tilde f_{r+1}$ are obtained from the right horizontal maps by taking images along the middle vertical maps and the composition of vertical maps, respectively. The claim about $\tilde f_{r+1}$ and $\tilde f_r$ thus follows directly from the fact that the lower right vertical map in the diagram is the identity.

* For every $r \ge 1$, assuming that $\tilde C_r$ is exact, the map $\tilde h_{r+1}$ is equal to $\tilde h_r \comp g^{-1}$ after identifying $\tilde E_{r+1}^{*,*} \isom \ker(\tilde d_r) / \img(\tilde d_r)$. Here $g^{-1}$ denotes the process of choosing any preimage under $g$, using the fact that the choice does not matter. To prove this one can equivalently show that $\tilde h_r$ is equal to $\tilde h_{r+1} \comp g$. This follows from the above diagram in a similar way as for $\tilde f_{r+1}$ and $\tilde f_r$.

This finishes the proof by induction on $r$.

Let $\mathcal C$ be a stable $\infty$-category with $t$-structure and let $(X_n)_n \in \Tow(\mathcal C)$ with associated spectral sequence $E_*^{*,*}$ as defined in <ref>. For $r \ge 1$ and $p, q \in \Z$ let $D_r^{p,q} := \img(\pi_{q-p} X_{p+r-1} \to X_p)$ (where we set $X_p = 0$ for $p < 0$). Then for all $r$, $p$ and $q$ there is a long exact sequence

\begin{align*}
\dots \to E_r^{p,q+1} \to D_r^{p+1,q+1} \xto{g} D_r^{p,q} \to E_r^{p+r-1,q+r-1} \to \dots
\end{align*}

where the map $g$ is induced from the maps $X_p \to X_{p-1}$ of the tower.

This is just rephrasing that $E_r^{*,*}$ fits into an exact couple with $D_r^{*,*}$ as shown in <ref>.

Before we can proceed further, we need a short detour on systems of objects in abelian categories (cf. <cit.>).

Let $\mathcal A$ be an abelian category and let $(A_n)_n = (\dots \to A_n \to \dots \to A_1 \to A_0)$ be an inverse system of objects of $\mathcal A$.

* For any $r \ge 0$, the $r$-th derived inverse system of $(A_n)_n$ is the inverse system $(\img(A_{n+r} \to A_n))_n$.

* We say that $(A_n)_n$ is constant if all transition maps $A_{n+1} \to A_n$ are constant. We say that $(A_n)_n$ is constant from index $n_0 \ge 0$ if the inverse system $(A_n)_{n\ge n_0}$ is constant. In this case the value of $(A_n)_n$ is $A := \varprojlim_n A_n$, which is naturally isomorphic to $A_n$ for $n \ge n_0$. We say $(A_n)_n$ is eventually constant if $(A_n)_n$ is constant from index $n_0$ for some $n_0 \ge 0$.

Let $\mathcal A$ be an abelian category and let $(A_n)_n \to (B_n)_n \to (C_n)_n$ be an exact sequence of inverse systems in $\mathcal A$. Fix integers $r, n_0 \ge 0$.

* Suppose that $0 \to (A_n)_n \to (B_n)_n$ is exact and that both the $r$-th derived systems of $(B_n)_n$ and $(C_n)_n$ are constant from index $n_0$. Then the $r$-th derived system of $(A_n)_n$ is constant from index $n_0$.

* Suppose that $(B_n)_n \to (C_n)_n \to 0$ is exact and that both the $r$-th derived systems of $(A_n)_n$ and $(B_n)_n$ are constant from index $n_0$. Then the $r$-th derived system of $(C_n)_n$ is constant from index $n_0$.

* Suppose that $0 \to (A_n)_n \to (B_n)_n \to (C_n)_n \to 0$ is exact and that both the $r$-th derived systems of $(A_n)_n$ and $(C_n)_n$ are constant from index $n_0$. Then the $2r$-th derived system of $(B_n)_n$ is constant from index $n_0$.

Apart from the claims about the index $n_0$ this is precisely <cit.>. However, the proof in the reference immediately provides the additional claims about $n_0$ as well.

Let $\mathcal A$ be an abelian category and let

\begin{align*}
(A_n)_n \to (B_n)_n \to (C_n)_n \to (D_n)_n \to (E_n)_n
\end{align*}

be an exact sequence of inverse systems in $\mathcal A$. Suppose that the $r$-th derived systems of $(A_n)_n$, $(B_n)_n$, $(D_n)_n$ and $(E_n)_n$ are constant from index $n_0 \ge 0$. Then the $2r$-th derived system of $(C_n)_n$ is constant from index $n_0$.

This follows by applying <ref> three times.

We are now in the position to show that strongly constant towers automatically have a fast converging spectral sequence. In fact, a weaker version of “strongly constant” is sufficient. Let us make the following definitions:

Let $\mathcal C$ be a stable $\infty$-category with $t$-structure.

* A tower $(X_n)_n \in \Tow(\mathcal C)$ is called $t$-nilpotent of index $\le h$ if for all $n \ge h$ the map $\pi_* X_{n+h} \to \pi_* X_n$ is zero.

* A tower $(X_n)_n \in \Tow(\mathcal C)$ is called strongly $t$-constant of index $h$ if $X := \varprojlim_n X_n \in \mathcal C$ exists and $\cofib(X \to (X_n)_n) \in \Tow(\mathcal C)$ is $t$-nilpotent of index $h$.

* A cosimplicial object $M^\bullet$ in $\mathcal C$ is called strongly $t$-constant of index $h$ if the associated tower $(\Tot_n(M^\bullet))_n$ is strongly $t$-constant of index $h$.

Let $E_*^{*,*}$ be a spectral sequence in an abelian category $\mathcal A$. We say that $E_*^{*,*}$ has a horizontal vanishing line at height $h \ge 0$ if for all $p$, $q$ and $r$ with $p \ge h$ and $r \ge h+1$ we have $E_r^{p,q} = 0$.

* In general, being strongly $t$-constant of index $h$ is much weaker than being strongly constant of index $h$. However, as we will see below, under mild assumptions on $\mathcal C$ a strongly $t$-constant tower $(X_n)_n$ still satisfies $\pi_*(\varprojlim_n X_n) = \varprojlim_n \pi_* X_n$. Note also that if there is a constant $d$ such that the composition of $d$ phantom maps in $\mathcal C$ is zero then if a tower $(X_n)_n \in \Tow(\mathcal C)$ is strongly $t$-constant of index $h$ it is automatically strongly constant of index $\le hd$. For example this is true if $\mathcal C$ is the category of spectra and $d = 2$.

* Similar to <ref> one can define a refined version of the horizontal vanishing line by making the page at which it occurs independent of $h$. We decided not to do that for the same reasons as for nilpotent towers.

The next proposition shows that for a tower to be strongly $t$-constant is equivalent to its associated spectral sequence having a horizontal vanishing line.

Let $\mathcal C$ be a stable $\infty$-category with $t$-structure such that countable products exist in $\mathcal C$ and are compatible with the $t$-structure. Let $(X_n)_n \in \Tow(\mathcal C)$ with associated spectral sequence $E_*^{*,*}$ as defined in <ref>.

* If $(X_n)_n$ is strongly $t$-constant of index $h$ then $E_*^{*,*}$ has a horizontal vanishing line at height $3h$.

* If $E_*^{*,*}$ has a horizontal vanishing line at height $h$ then $(X_n)_n$ is strongly $t$-constant of index $2h$.

Moreover, if $(X_n)_n$ is strongly $t$-constant of index $h$ then there is a natural isomorphism

\begin{align*}
\pi_*(\varprojlim_n X_n) = \varprojlim_n \pi_* X_n = \img(\pi_* X_{3h} \to \pi_* X_{h}).
\end{align*}

Let $X := \varprojlim_n X_n$ (which exists because countable products in $\mathcal C$ exist) and let $(N_n)_n := \cofib(X \to (X_n)_n) \in \Tow(\mathcal C)$. Note that there is an exact sequence of graded inverse systems in $\mathcal C^\heartsuit$,

\begin{align*}
(\pi_* N_n)_n \to (\pi_* X)_n \to (\pi_* X_n)_n \to (\pi_* N_n)_n \to (\pi_* X)_n \to (\pi_* X_n)_n,
\end{align*}

where the first and the second-to-last map lower the grading by $1$ each. Obviously $(\pi_* X)_n$ is constant, as is any of its derived systems.

We first prove (i), so let us assume that $(X_n)_n$ is strongly $t$-constant of index $h$, i.e. that $(N_n)_n$ is $t$-nilpotent of index $h$. By definition of $t$-nilpotent towers the $h$-th derived system of $(\pi_* N_n)_n$ is constant from index $h$. Thus by <ref> the above exact sequence implies that the $2h$-th derived system of $(\pi_* X_n)_n$ is constant from index $h$. From <ref> we deduce that $E_{2h+1}^{p,q} = 0$ for $p \ge 3h$. It follows that $E_r^{p,q} = 0$ for $r \ge 2h+1$ and $p \ge 3h$, finishing the proof of (i).

Let us now prove the last part of the claim, so we again assume that $(X_n)_n$ is strongly $t$-constant of some index $h$. By the proof of (i) we deduce that the $2h$-th derived system of $(\pi_* X_n)_n$ is constant from index $h$. This already implies the second identity. It remains to prove the first one. As both limits in the claim are invariant under the transformation $(X_n)_n \leadsto (X_{cn+d})_n$ for some constants $c \ge 1$ and $d \ge 0$, we can replace $(X_n)_n$ by $(X_{2hn+h})_n$ and hence assume that the first derived system of $(\pi_* X_n)_n$ is constant, say of value $A_*$. Denoting $f_n\colon X_n \to X_{n-1}$ we deduce that for all $n \ge 1$, the map $X_n \to \img f_n \isoto \img f_{n+1}$ is split surjective, so that $X_n \isom \img f_{n+1} \dsum \ker f_n$. Now note that the inverse limit $X = \varprojlim_n X_n$ fits into a cofiber sequence $X \to \prod_n X_n \to \prod_n X_n$ where the second map is given by $(\id - f_{n+1})_n$. As countable products are compatible with the $t$-structure in $\mathcal C$ we have $\pi_*(\prod_n X_n) = \prod_n \pi_*X_n$. We thus obtain a long exact sequence

\begin{align*}
\dots \to \prod_n \pi_{k+1}X_n \to \prod_n \pi_{k+1}X_n \to \pi_k X \to \prod_n \pi_k X_n \to \prod_n \pi_k X_n \to \dots
\end{align*}

From the above splitting property of $(\pi_* X_n)_n$ it follows easily that $\prod_n \pi_{k+1} X_n \to \prod_n \pi_{k+1} X_n$ is surjective. Hence $\pi_k X \injto \prod_n \pi_k X_n$ is injective, so that $\pi_k X$ is isomorphic to the kernel of $\prod_n \pi_k X_n \to \prod_n \pi_k X_n$. Using that the first derived inverse system of $(\pi_k(X_n))_n$ is constant, this kernel is easily seen to be isomorphic to $A_*$.

It remains to prove (ii), so assume that $E_*^{*,*}$ has a horizontal vanishing line at height $h$. By <ref> we deduce that for all $p \ge h$ the map

\begin{align*}
g\colon \img(\pi_* X_{p+h+1} \to \pi_* X_{p+1}) \isoto \img(\pi_* X_{p+h} \to \pi_* X_p)
\end{align*}

is an isomorphism. In other words the $h$-th derived system of $(\pi_* X_n)_n$ is constant from index $h$. From the exact sequence of inverse systems at the beginning of the proof and from <ref> we deduce that the $2h$-th derived system of $(\pi_* N_n)_n$ is constant from index $h$. Clearly $\varprojlim_n N_n = 0$, so from the proof of the last part of the claim we deduce $\varprojlim_n \pi_*(N_n) = \pi_*(\varprojlim_n N_n) = 0$. Hence the constant value of the $2h$-th derived system of $(\pi_* N_n)_n$ must be $0$, i.e. for all $n \ge h$, $\pi_* N_{n+2h} \to \pi_* N_n$ is the zero map. Thus $(N_n)_n$ is $t$-nilpotent and hence $(X_n)_n$ is strongly $t$-constant.

We now come back to the promised analogue of <ref> in terms of homotopy groups. In contrast to strongly constant towers, the property of being strongly $t$-constant is not preserved by exact functors in general. Therefore the following lemma requires additional hypotheses compared to <ref>.

Let $f\colon Y \to X$ be a morphism of affine analytic spaces over $(V,\mm)$ which is descendable of index $m$ and let $(M_n)_n \in \Tow(\D(X))$. Assume that there is a constant $h \ge 0$ such that $((f_*f^*)^k M_n)_n \in \Tow(\D(X))$ is strongly $t$-constant of index $h$ for all $k$. Then $(M_n)_n$ is strongly $t$-constant of index $s(m) h$, where $s(m)$ is the constant from <ref>.

We can argue as in the proof of <ref>, noting that the tower $(M_n)_n$ can be constructed from the towers $((f_\natural \ri_Y)^k(M_n))_n$ using finite limits and retracts. More precisely, we have the following:

* Let $(R_n)_n, (S_n)_n \in \Tow(\D(X))$ such that $(R_n)_n$ is a retract of $(S_n)_n$. If $(S_n)_n$ is strongly $t$-constant of index $h$ then $(R_n)_n$ is strongly $t$-constant of index $\le h$. To see this we can easily reduce to the case that that $(S_n)_n$ is $t$-nilpotent of index $h$. From $(R_n)_n$ being a retract of $(S_n)_n$ we deduce that $R_{n+h} \to R_n$ factors over $S_{n+h} \to S_n$ for all $n \ge 0$, which implies the claim.

* Let $(R_n)_n \to (S_n)_n \to (T_n)_n$ be a cofiber sequence in $\Tow(\D(X))$. If $(R_n)_n$ and $(T_n)_n$ are strongly $t$-constant of index $h_1$ and $h_2$ respectively then $(S_n)_n$ is strongly $t$-constant of index $\le h_1 + h_2$. To see this we can again reduce to the case that $(R_n)_n$ and $(T_n)_n$ are $t$-nilpotent. Then the claim follows similarly as in the proof of <cit.> (this is essentially the same argument as in the proof of <ref>).

By construction of $s(m)$ this finishes the proof.

It is often the case that the assumptions of <ref> can be deduced from a stronger property of $(f^*M_n)_n \in \Tow(\D(Y))$. One example of this is when $(f^*M_n)_n$ is strongly constant, because this property is preserved by exact functors (however, this gives nothing new compared to <ref>). Another case is the following, which is also the only case we are interested in:

Let $f\colon Y \to X$ be a morphism of affine analytic spaces over $(V,\mm)$ which is descendable of index $m$ and let $M^\bullet$ be a cosimplicial object in $\D(X)$. Assume that $f^*M^\bullet$ is split. Then:

* $M^\bullet$ is strongly $t$-constant of index $2s(m)$.

* The spectral sequence associated to $M^\bullet$ has a horizontal vanishing line at height $6s(m)$.

Here $s(m)$ is the constant from <ref>.

Being a split cosimplicial object is preserved under any functor, so in particular $(f_\natural \ri_Y)^k(M^\bullet)$ is a split cosimplicial object of $\D(X)$ for all $k \ge 1$. Using <ref> and <ref> it is therefore enough to show the following: If $N^\bullet$ is a split simplicial object in $\D(X)$ then the associated spectral sequence $E_*^{*,*}$ has a horizontal vanishing line at height $1$. By the dual version of <cit.>, for every $q \in \Z$, $E_1^{*,q}$ is the normalized chain complex associated to the simplicial object $\pi_q N^\bullet$ of $\D(X)^\heartsuit$. By <cit.> this normalized chain complex is quasi-isomorphic to the unnormalized chain complex

\begin{align*}
\pi_q N^0 \to \pi_q N^1 \to \pi_q N^2 \to \dots
\end{align*}

where the differentials are given by the alternating sums over the coface maps. Since $N^\bullet$ is split by assumption, this unnormalized chain complex is split exact and hence has vanishing cohomology in degree greater zero. On the other hand, this cohomology is precisely $E_2^{*,q}$ so that $E_2^{p,q} = 0$ for $p \ge 1$. Therefore $E_*^{*,*}$ has a horizontal vanishing line at height $1$, as desired.

One can also attempt to show that every split simplicial object is strongly constant, which would simplify the proof of <ref>.

We now come to the promised version of descendability. As mentioned above, we want to work with “flat covers” of $\AnSpec \mathcal A$ for an analytic ring $\mathcal A$ over $(V,\mm)$. Here is the precise definition of that notion:

Let $f\colon \mathcal A \to \mathcal B$ be a map of analytic rings over $(V,\mm)$.

* A map $\mathcal A \to \mathcal A'$ of analytic rings is called an fs-morphism if it is flat (i.e. $- \tensor_{\mathcal A} \mathcal A'$ is $t$-exact) and steady. An fs-cover of $\mathcal A$ is a family $(\mathcal A \to \mathcal A_i)_i$ of fs-morphisms such that the family of functors $- \tensor_{\mathcal A} \mathcal A_i$ is conservative.

* $f$ is called fs-descendable of index $\le d$ if there is an fs-cover $(\mathcal A \to \mathcal A_i)_i$ such that all the induced maps $\mathcal A_i \to \mathcal B \tensor_{\mathcal A} \mathcal A_i$ are descendable of index $\le d$. We say that $f$ is fs-descendable if it is fs-descendable of some index.

* $f$ is called weakly fs-descendable (of index $\le d$) if it is an iterated $\kappa$-filtered colimit of fs-descendable maps (of index $\le d$), where $\kappa$ is any uncountable regular cardinal such that $(V,\mm)$ is $\kappa$-compact.

*  Quasi-coherent sheaves descend along weakly fs-descendable morphisms.

*   (Weakly) fs-descendable morphisms are stable under base-change, fs-descendable morphisms are stable under composition, and weakly fs-descendable morphisms are stable under $\kappa$-filtered colimits, where $\kappa$ is any uncountable regular cardinal such that $(V,\mm)$ is $\kappa$-compact.

*  Suppose that $(V,\mm)$ is $\omega_1$-compact (e.g. $\mm$ is countably generated). Let $(\mathcal A \to \mathcal B_i)_{i\in I}$ be a filtered diagram of maps of analytic rings over $(V,\mm)$ which are fs-descendable of bounded index $\le d$ with respect to a common fs-cover of $\mathcal A$. Then $\mathcal A \to \varinjlim_i \mathcal B_i$ is weakly fs-descendable of index $\le 2d$.

Since fs-morphisms are stable under base-change, it follows easily from <ref> that fs-descendable morphisms are stable under base-chage. It is also clear that $\kappa$-filtered colimits of weakly fs-descendable morphisms are weakly fs-descendable. This proves (ii). Part (iii) follows from <ref> (as in the proof of <ref>).

It remains to prove (i). By <ref> we are reduced to showing that quasicoherent sheaves descend along fs-descendable morphisms. Let $f\colon \mathcal A \to \mathcal B$ be such a morphism and let $d$ be an integer and $(\mathcal A \to \mathcal A_i)_{i\in I}$ an fs-cover such that all $f_i\colon \mathcal A_i \to \mathcal B_i := \mathcal B \tensor_{\mathcal A} \mathcal A_i$ are descendable of index $\le d$. It follows easily that $f$ is steady (note that the family of functors $- \tensor_{\mathcal B} \mathcal B_i$ is conservative, now apply any criterion for steadiness, e.g. <ref>). As in the proof of <ref> we can thus apply Lurie's <cit.> to see that we only have to show the following two properties of the base-change functor $f^*\colon \D(\mathcal A) \to \D(\mathcal B)$:

* $f^*$ is conservative.

* Given a cosimplicial object $M^\bullet \in \D(\mathcal A)$ such that $f^* M^\bullet \in \D(\mathcal B)$ is split, then $f^*$ preserves the totalization of $M^\bullet$, i.e. $\Tot(f^* M^\bullet) = f^* \Tot(M^\bullet)$.

For every $i \in I$ denote $g_i\colon \mathcal A \to \mathcal A_i$ and $g'_i\colon \mathcal B \to \mathcal B_i$. Then we have a commuting diagram

(ℬ_i)     [l,swap,"g_i'^*"] (ℬ)

(𝒜_i) [u,"f_i^*"]     [l,swap,"g_i^*"] (𝒜) [u,swap,"f^*"]

We first show that $f^*$ satisfies (a). Suppose we are given a morphism $M \to N$ in $\D(\mathcal A)$ such that $f^* M \isoto f^* N$ is an isomorphism in $\D(\mathcal B)$. We need to show that $M \to N$ is an isomorphism. It is enough to show that for all $i \in I$, $g_i^* M \to g_i^* N$ is an isomorphism. But we have $f_i^* g_i^* M = g_i'^* f^* M$ (and similarly for $N$), so that $f_i^* g_i^* M \isoto f_i^* g_i^* N$ is an isomorphism. The claim thus follows from the fact that all $f_i^*$ are conservative (because $f_i$ is descendable).

It remains to prove that $f^*$ satisfies (b), so let $M^\bullet \in \D(\mathcal A)$ be such that $f^* M^\bullet \in \D(\mathcal B)$ is split. Then, for every $i \in I$, $f_i^* g_i^* M = g_i'^* f^* M^\bullet \in \D(\mathcal B)$ is split. Since $f_i$ is descendable of index $\le d$, <ref> implies that $g_i^* M^\bullet$ is strongly $t$-constant of index $h := 2s(d)$ (where $s(d)$ is a constant defined in <ref> and only depends on $d$). In other words, the $h$-th derived system of $(\pi_* \Tot_n(g_i^* M^\bullet))_n$ is constant from index $h$.

By assumption $g_i^*$ is $t$-exact and in particular commutes with homology. It follows that $g_i^*$ maps the $h$-th derived system of $(\pi_* \Tot_n(M^\bullet))_n$ to the $h$-th derived system of $(\pi_* \Tot_n(g_i^* M^\bullet))_n$. In other words, the $h$-th derived system of $(\pi_* \Tot_n(M^\bullet))_n$ is mapped to a system which is constant from index $h$, for all $i$. Since the functors $g_i^*$ form a conservative family, it follows that the $h$-th derived system of $(\pi_* \Tot_n(M^\bullet))_n$ is constant from index $h$, i.e. that $M^\bullet$ is strongly $t$-constant of index $h$. By <ref> we have $\pi_* \Tot(M^\bullet) = \varprojlim_n \pi_* \Tot_n(M^\bullet)$, and since $g_i^*$ commutes with finite limits and with homology groups we deduce

\begin{align*}
g_i^* \Tot(M^\bullet) = \Tot(g_i^* M^\bullet)
\end{align*}

for every $i \in I$.

It is now easy to deduce that $f^*$ preserves the totalization of $M^\bullet$: There is a natural morphism $f^* \Tot(M^\bullet) \to \Tot(f^* M^\bullet)$ and since the forgetful functor $f_*$ is conservative, it is enough to to show that the induced map $g_i^* f_* f^* \Tot(M^\bullet) \to g_i^* f_* \Tot(f^* M^\bullet)$ is an isomorphism for all $i \in I$. By steady base-change (see <ref>) we have

\begin{align*}
g_i^* f_* f^* \Tot(M^\bullet) &= f_{i*} g_i'^* f^* \Tot(M^\bullet) = f_{i*} f_i^* g_i^* \Tot(M^\bullet) = f_{i*} f_i^* \Tot(g_i^* M^\bullet) =\\
&= f_{i*} \Tot(f_i^* g_i^* M^\bullet),\\
g_i^* f_* \Tot(f^* M^\bullet) &= f_{i*} g_i'^* \Tot(f^* M^\bullet) = f_{i*} \Tot(g_i'^* f^* M^\bullet) = f_{i*} \Tot(f_i^* g_i^* M^\bullet).
\end{align*}

(In the first line we used that $f_i^*$ preserves $f_i^*$-split totalizations and in the second line we used that $f^* M^\bullet$ is split so that its totalization commutes with any exact functor, in particular with $g_i'^*$.) This finishes the proof.

Let $f\colon \mathcal A \to \mathcal B$ be a map of analytic rings over $(V, \mm)$ and $d \ge 0$ an integer. Then:

* If $f$ is fs-descendable of index $d$ then every $f^*$-split cosimplicial object $M^\bullet$ of $\D(\mathcal A)$ is strongly $t$-constant of index $\le 2s(d)$.

*  There is a constant $b(d) \in \Z$ only depending on $d$ with the following property: If $f$ is weakly fs-descendable of index $d$ and $M \in \D(\mathcal A)$ satisfies $f^* M \in \D_{\ge0}(\mathcal B)$, then $M \in \D_{\ge b(d)}(\mathcal A)$.

Part (i) follows from the proof <ref>. We now prove (ii), so assume that $f$ is weakly fs-descendable of index $d$ and suppose $M \in \D(\mathcal A)$ satisfies $f^* M \in \D_{\ge0}(\mathcal B)$. By <ref> we have $\D(\mathcal A) = \varprojlim_{n\in\Delta} \D(\mathcal B^{\tensor_{\mathcal A} n+1})$, so for the cosimplicial object $M^\bullet = M \tensor_{\mathcal A} \mathcal B^{\tensor_{\mathcal A} \bullet+1}$ we have $M = \Tot(M^\bullet)$. Writing $f$ as a an iterated $\kappa$-filtered colimit of fs-descendable maps $f_i\colon \mathcal B_i \to \mathcal A$, we can similarly write $M^\bullet$ as an iterated $\kappa$-filtered colimit of the associated cosimplicial objects $M_i^\bullet$. Then each $M_i^\bullet$ is $f_i^*$-split, hence by (i) each $M_i^\bullet$ is strongly $t$-constant of index $\le h := 2s(d)$. Since $\pi_*$ and $\varprojlim_n$ commute with $\kappa$-filtered colimits, it follows directly from the definitions that $M^\bullet$ is strongly $t$-constant of index $\le h$. In particular, by <ref> we have

\begin{align*}
\pi_*(M) = \pi_*(\Tot(M^\bullet)) = \img(\pi_*\Tot_{3h}(M^\bullet) \to \pi_*\Tot_h(M^\bullet)))
\end{align*}

But each $M^n$ lies in $\D_{\ge0}$ and $\Tot_h$, $\Tot_{3h}$ are finite limits of fixed form (only depending on $h$), which implies the claim.

§.§ Schemes and Discrete Adic Spaces

In the previous subsections we have developed a very general theory of analytic rings and descent. Now we finally come to some examples of that theory by discussing how schemes (and discrete adic spaces) fit into that setting. To every discrete ring $A$ we will associate an analytic ring $A_\solid$, called the solid analytic ring associated to $A$. In fact we can more generally associate a solid analytic ring $(A, A^+)_\solid$ to every discrete Huber pair $(A, A^+)$. The basic construction of $(A, A^+)_\solid$ follows <cit.>, but we expand upon it by proving several additional properties of the $\infty$-category of solid analytic rings. We also generalize the construction slightly by combining it with almost mathematics. With a good theory of solid analytic rings at hand, we can then glue them to obtain the $\infty$-categories of schemes and of discrete adic spaces. At the end of this subsection we discuss a 6-functor formalism for (solid) quasicoherent sheaves on schemes and discrete adic spaces, following <cit.>.

Let us start with the basic definitions. In order to make the exposition easier to follow, we first work exclusively in the non-almost setting and only later introduce an almost version (see <ref>) below. Most of the results about non-almost solid rings generalize directly to the almost world by viewing a solid almost ring as the almostification of an honest solid ring.

* Let $A$ be a classical ring. We associate to $A$ the pre-analytic ring $A_\solid$ with

\begin{align*}
A_\solid[S] := \varinjlim_{A' \to A} \varprojlim_i A'[S_i]
\end{align*}

for every profinite set $S = \varprojlim_i S_i$ (with all $S_i$ finite), where $A'$ ranges through all finite-type classical $\Z$-algebras with a map to $A$.

For every classical ring $A$, the pre-analytic ring $A_\solid$ defined above is an analytic ring. If $A$ is of finite type over $\Z$ then for every profinite set $S$ there is some set $I$ such that $A_\solid[S] = \prod_I A$.

The second claim follows from <cit.> (as in <cit.>). For the first claim, by definition we have $A_\solid = \varinjlim_{A' \to A} A'_\solid$, where $A'$ ranges over finite-type classical $\Z$-algebras with a map to $A$. Hence we can w.l.o.g. assume that $A$ is of finite type over $\Z$. But then the claim is shown in <cit.> (use also <cit.>).

* A discrete Huber pair is a pair $(A, A^+)$, where $A$ is a discrete ring and $A^+$ is an integrally closed subring of $\pi_0 A$. A morphism $(A, A^+) \to (B, B^+)$ of discrete Huber pairs is a morphism $A \to B$ of rings such that the image of $A^+$ under the induced map $\pi_0 A \to \pi_0 B$ lies in $B^+$.

* Let $(A, A^+)$ be a discrete Huber pair. We let

\begin{align*}
(A, A^+)_\solid \in \AnRing
\end{align*}

denote the following analytic ring structure on $A$: If $A$ is static then there is a map $A^+ \injto A$ and we let $(A, A^+)_\solid := A_{A^+_\solid/}$ be the induced analytic ring. For general $A$ we employ <ref> to define $(A, A^+)_\solid$ as the analytic ring structure induced from $(\pi_0 A, A^+)_\solid$.

*  Given discrete rings $A$ and $A^+$ and a map $A^+ \to \pi_0 A$ we also denote $(A, A^+)_\solid := (A, \tilde A^+)_\solid$, where $\tilde A^+ \subset \pi_0 A$ is the integral closure of the image of the map $\pi_0 A^+ \to \pi_0 A$. We further abbreviate, for every discrete ring $A$,

\begin{align*}
A_\solid := (A, \pi_0 A)_\solid \in \AnRing.
\end{align*}

* We abbreviate $\D_\solid(A) := \D(A_\solid)$ and $\D_\solid(A, A^+) = \D((A, A^+)_\solid)$. A solid $(A, A^+)$-module is an object of $\D_\solid(A, A^+)$.

The first thing we want to do is check that the abbreviated version of $(A, A^+)_\solid$ from <ref> is actually the induced analytic ring structure from $A^+_\solid$ (and not only from $\tilde A^+_\solid$). This allows us to efficiently compute the generators $(A, A^+)_\solid[S]$ in many cases.

Let $A^+ \to A$ be a map of discrete rings and assume that $A$ is static. Then

\begin{align*}
(A, A^+)_\solid = A_{A^+_\solid/}.
\end{align*}

In other words, for every profinite set $S$ we have

\begin{align*}
(A, A^+)_\solid[S] = A \tensor_{A^+} A^+_\solid[S].
\end{align*}

First note that the map $A^+ \to A = \pi_0 A$ factors over $\pi_0 A^+$, so we can easily reduce to the case that $A^+$ is static. Let $\tilde A^+$ be the integral closure of the image of $A^+ \to A$. Then by definition we have $(A, A^+)_\solid = A_{\tilde A^+_\solid/}$, so it is enough to verify that $\tilde A^+_\solid = \tilde A^+_{A^+_\solid/}$. We are thus reduced to the following statement: Let $B \to B'$ be an integral map of classical rings; then $B'_\solid = B'_{B_\solid/}$. To prove this, first note that by writing $B'$ as a filtered colimit of finite-type classical $B$-algebras and pulling out the colimit on both sides of the claimed identity we can reduce to the case that $B \to B'$ is of finite type. By the same argument we can then reduce to the case that $B \to B'$ is of finite presentation. Then it comes via base-change from a finite map $B_0 \to B_0'$ of finite-type classical $\Z$-algebras. Write $B = \varinjlim_i B_i$, where $B_i$ ranges over finite-type classical $\Z$-algebras with maps $B_0 \to B_i \to B$. Let $B' = \varinjlim_i B'_i$ be the base-change along $B_0 \to B_0'$. Then $B_\solid = \varinjlim_i (B_i)_\solid$ and $B'_{B_\solid/} = \varinjlim_i (B'_i)_{(B_i)_\solid/}$, so we can reduce to the case that $B$ and $B'$ are of finite type over $\Z$. Now by <ref> we need to see that for every set $I$ we have $B' \tensor_B \prod_I B = \prod_I B'$. In fact we claim that this is true for any finite classical $B$-module $M$ in place of $B'$: If $M$ is free then this is clear, in general we can resolve $M$ by finite free modules to reduce to the free case.

A general map $A^+ \to A$ of classical rings can be factored into a composition of a finite map and a polynomial algebra (in possibly infinitely many variables). If $A^+ \to A$ is finite (or more generally integral) then $A_\solid = (A, A^+)_\solid$ by definition, thus by <ref> $A_\solid$ possesses the induced analytic ring structure from $A^+_\solid$. In this case many questions about $A_\solid$ can easily be reduced to $A^+_\solid$. Therefore, in order to understand the map $A^+_\solid \to A_\solid$ in general it remains to get a good understanding of polynomial algebras. We have the following result:

Let $A$ be a classical finite-type $\Z$-algebra. Then the functor

\begin{align*}
- \tensor_{A_\solid} A[T]_\solid\colon \D_\solid(A) \to \D_\solid(A[T])
\end{align*}

is $t$-exact and preserves all limits.

Let $f\colon \Spec A[T] \to \Spec A$ be the obvious map. In the following we make use of the shriek functors $f_!$ and $f^!$ defined in <cit.>. For all $M \in \D_\solid(A)$ we have $f^! M = f^*M \tensor_{A[T]_\solid} f^! A$ and $f^! A = A[T][1]$ by (the proof of) <cit.>, hence $f^! M = f^* M[1]$. Therefore

\begin{align*}
M \tensor_{A_\solid} A[T]_\solid = f^*M = f^! M[-1] = \IHom_{A[T]}(A[T], f^! M[-1]) = \IHom_A(f_! A[T][1], M)
\end{align*}

But $f_! A[T] = A((T^{-1}))/A[T][-1]$ (see e.g. <cit.>), which implies

\begin{align*}
M \tensor_{A_\solid} A[T]_\solid = \IHom_A(A((T^{-1}))/A[T], M).
\end{align*}

Clearly this functor preserves limits in $M$ and is left $t$-exact (hence $t$-exact).

We will now prove some useful basic results about the analytic rings $(A, A^+)_\solid$. Namely, we show that the association $(A, A^+) \mapsto (A, A^+)_\solid$ exhibits the $\infty$-category of discrete Huber pairs as a full subcategory of the $\infty$-category of all analytic rings, compatible with all small colimits. Moreover, we show that every map of solid discrete analytic rings is steady, so that unconditional base-change for modules holds, just like one is used to from classical algebra.

*  The assignment $(A, A^+) \mapsto (A, A^+)_\solid$ defines a fully faithful functor from the $\infty$-category of discrete Huber pairs to $\AnRing$.

*  The functor in (i) preserves all non-empty colimits. In particular, for any diagram $(B, B^+) \from (A, A^+) \to (C, C^+)$ of discrete Huber pairs we have

\begin{align*}
(B, B^+)_\solid \tensor_{(A, A^+)_\solid} (C, C^+)_\solid = (B \tensor_A C, B^+ \tensor_{A^+} C^+)_\solid,
\end{align*}

where on the right-hand side we use the abbreviated notation from <ref>.

To prove (i) we need to verify the following: Let $(A, A^+)$ and $(B, B^+)$ be two discrete Huber pairs and let $f\colon A \to B$ be a morphism of rings. Then $f$ is a morphism $(A, A^+)_\solid \to (B, B^+)_\solid$ of analytic rings if and only if it is a morphism of Huber pairs. To prove this, let us first show the “if” direction, so assume that $f$ is a morphism of Huber pairs. Given any $M \in \D_\solid(B, B^+)$ we need to see that the image of $M$ under the forgetful functor $\D(B) \to \D(A)$ lies in $\D_\solid(A, A^+)$. This can be checked on homology groups, so we can assume that $M$ is concentrated in degree $0$. Then we can view $M$ as an object in $\D(\pi_0 B, B^+)$ and it is enough to see that its image in $\D(\pi_0 A)$ lies in $\D_\solid(\pi_0 A, A^+)$; in other words we can assume that both $A$ and $B$ are concentrated in degree $0$. In this case $M$ lies in $\D_\solid(B, B^+)$ if and only if its image under the forgetful functor $\D(B) \to \D(B^+)$ lies in $\D_\solid(B^+)$ (and similarly for $A$). Hence we can assume $B = B^+$ and $A = A^+$ (note that by the assumption on $f$, $f$ restricts to a map $A^+ \to B^+$), i.e. we want to show that $f$ is a morphism of analytic rings $A_\solid \to B_\solid$. Writing $f$ as a filtered colimit of maps of finite-type classical $\Z$-algebras we can reduce to the case that $A$ and $B$ are of finite type over $\Z$. Then for every extremally disconnected set $S$, $B_\solid[S] = \varprojlim_i B[S_i]$ is a limit of discrete (hence solid) $A$-modules, therefore its image under the forgetful functor $\D(B) \to \D(A)$ lies in $\D_\solid(A)$, as desired.

Conversely, assume that $f$ is a morphism $(A, A^+)_\solid \to (B, B^+)_\solid$ of analytic rings. Then $f$ also induces a morphism $(\pi_0 A, A^+)_\solid \to (\pi_0 B, B^+)_\solid$ of analytic rings, so we can assume that $A$ and $B$ are concentrated in degree $0$. By the same argument as in the proof of <cit.>, $f$ maps $A^+$ to $B^+$.

We now prove (ii). It is easy to see that the functor preserves non-empty filtered colimits. We now handle pushouts, so let $(B, B^+) \from (A, A^+) \to (C, C^+)$ be a diagram of discrete Huber pairs. Clearly its pushout in the $\infty$-category of Huber pairs is $(B \tensor_A C, (B^+ \tensor_{A^+} C^+)\tilde{})$, where $(B^+ \tensor_{A^+} C^+)\tilde{}$ denotes the integral closure of the image of $B^+ \tensor_{A^+} C^+ \to \pi_0(B \tensor_A C)$. Thus in order to verify that our functor preserves this pushout, we must show the claimed identity about the pushout of the associated analytic rings. By <ref> this can be checked after passage to the induced structures on $0$-truncated analytic rings, which allows us to assume that $A$, $B$ and $C$ are concentrated in degree $0$. Using <ref> and the explicit computation of pushouts (see <ref>) we can further reduce to the case $A = A^+$, $B = B^+$ and $C = C^+$. Now we want to show

\begin{align*}
B_\solid \tensor_{A_\solid} C_\solid = (B \tensor_A C)_\solid.
\end{align*}

Both sides commute with filtered colimits in $(B \from A \to C)$, so we can assume that $A$, $B$ and $C$ are of finite type over $\Z$. Choose a surjection $B' := A[T_1, \dots, T_n] \to B$ of $A$-algebras. By <ref> we have $B_\solid = (B, B')_\solid$ hence by the same reduction as above we can reduce to the case $B = B'$. By induction on $n$ we reduce to $B = A[T]$. By <ref> $B_\solid \tensor_{A_\solid} C_\solid$ is the analytic ring on $B \tensor_A C$ such that $(B_\solid \tensor_{A_\solid} C_\solid)[S]$ is the colimit of the repeated application of $(- \tensor_C C_\solid) \tensor_B B_\solid$ to $(B \tensor_A C)[S]$. We have

\begin{align*}
((B \tensor_A C)[S] \tensor_C C_\solid) \tensor_B B_\solid = (B \tensor_A C_\solid[S]) \tensor_B B_\solid = B_\solid \tensor_{A_\solid} C_\solid[S].
\end{align*}

By using <ref> (for $B = A[T]$) and the explicit definition of $C_\solid[S]$, we obtain $B_\solid \tensor_{A_\solid} C_\solid[S] = (B \tensor_A C)_\solid[S]$. But this object is both $B$-solid and $C$-solid, hence subsequent applications of $- \tensor_B B_\solid$ and $- \tensor_C C_\solid$ leave this object invariant. We deduce $(B_\solid \tensor_{A_\solid} C_\solid)[S] = (B \tensor_A C)_\solid[S]$. This finishes the case of pushouts.

To finish the proof of (ii) it remains to see that the functor in (i) preserves coproducts. Coproducts in the category of discrete Huber pairs are just tensor products over $(\Z, \Z)$ and hence translate to tensor products over $\Z_\solid$ on the side of analytic rings. Coproducts in $\AnRing$ are tensor products over $\Z$. But by the description of pushouts in <ref>, both tensor products are easily seen to agree.

*  Let $(A, A^+)$ be a discrete Huber pair. Then $P \in \D_\solid(A, A^+)$ is nuclear if and only if it is discrete.

*  Let $(A, A^+)$ and $(B, B^+)$ be discrete Huber pairs. Then every morphism $f\colon (A, A^+)_\solid \to (B, B^+)_\solid$ is steady.

We start with the proof of (i), so let $(A, A^+)$ and $P \in \D_\solid(A, A^+)$ be given. Suppose first that $P$ is nuclear. Pick some extremally disconnected set $S = \varprojlim_i S_i$, where all $S_i$ are finite. Then

\begin{align*}
(A, A^+)_\solid[S]^\vee &= \IHom_A((A, A^+)_\solid[S], A) = \IHom_A(A[S], A) = \IHom(S, A) =\\
&= \varinjlim_i \IHom(S_i, A) = \varinjlim_i A^{S_i}.
\end{align*}

Consequently, by definition of nuclearity we get

\begin{align*}
P(S) &= ((A, A^+)_\solid[S]^\vee \tensor_{(A, A^+)_\solid} P)(*) = (\varinjlim_i A^{S_i} \tensor_A P)(*) = \varinjlim_i P^{S_i}(*) =\\
&= \varinjlim_i P(S_i).
\end{align*}

This immediately implies that $P$ is discrete. Conversely, if $P$ is discrete then it is a colimit of copies of $A$. On the other hand, $A$ is obviously nuclear over $(A, A^+)_\solid$ and nuclear objects are stable under colimits by <ref>, hence $P$ is nuclear.

We now prove (ii), so let $(A, A^+)$ and $(B, B^+)$ be given. In the following we will treat $A^+$ and $B^+$ in the more general way of <ref>. By <ref> $f$ is induced by a morphism $(A, A^+) \to (B, B^+)$ of Huber pairs. We can factor $f$ as $(A, A^+)_\solid \xto{g} (B, A^+)_\solid \xto{h} (B, B^+)_\solid$, so it is enough to show that $g$ and $h$ are steady. For $g$, note that $(B, A^+)_\solid$ has the induced analytic ring structure from $(A, A^+)_\solid$ (by <ref>) and hence $g$ is steady by <ref>, because $B$ is nuclear in $\D_\solid(A, A^+)$. To prove steadiness of $h$, by the “2-out-of-3-property” <ref> we can assume $A^+ = \Z$, i.e. we want to show that the map $(B, \Z)_\solid \to (B, B^+)_\solid$ is steady. Writing $B^+$ as a filtered colimit of finite-type $\Z$-algebras we reduce to the case that $B^+$ is of finite type over $\Z$. Since $(B, B^+)_\solid$ only depends on the image of $B^+$ in $\pi_0 B$ we can assume that $B^+ = \Z[T_1, \dots, T_n]$ for some $n$. Now $h$ is the base-change of the map $(B^+, \Z)_\solid \to (B^+, B^+)_\solid$ along $(B^+, \Z)_\solid \to (B, \Z)_\solid$ (by <ref>), so we reduce to showing that the map

\begin{align*}
(\Z[T_1, \dots, T_n], \Z)_\solid \to (\Z[T_1, \dots, T_n], \Z[T_1, \dots, T_n])_\solid
\end{align*}

is steady. By induction (using preservation of steadiness under base-change and composition) this reduces to the case $n = 1$. In this case the steadiness follows easily from an explicit computation of $- \tensor_{(\Z[T], \Z)_\solid} \Z[T]_\solid$ in terms of a $\IHom$ along the lines of the proof of <ref> (see <cit.>).

Let $A' \from A \to B$ be a diagram of discrete rings and denote $B' := A' \tensor_A B$. Then for every $M \in \D_\solid(B)$ the natural map

\begin{align*}
M \tensor_{A_\solid} A'_\solid \isoto M \tensor_{B_\solid} B'_\solid
\end{align*}

is an isomorphism. In other words, base change along the following diagram holds:

_(B') [d]        [ll,swap,"- _B_ B'_"] [d] _(B)

_(A')        [ll,swap,"- _A_ A'_"] _(A)

By <ref> we have $B'_\solid = A'_\solid \tensor_{A_\solid} B_\solid$ and by <ref> the map $A_\solid \to A'_\solid$ is steady. Hence the claim follows from base-change along steady maps, see <ref>.

We now introduce an almost version of the above definitions and results by reducing them to the non-almost case over $\Z$.

Let $(V,\mm)$ be an almost setup.

* A discrete Huber pair over $(V,\mm)$ is a pair $(A, A^+)$ where $A$ is a discrete $(V,\mm)$-algebra and $A^+$ is an integrally closed subring of $\pi_0 A_{**\omega}$ (where $(-)_\omega$ denotes the discretization functor). A morphism $(A, A^+) \to (B, B^+)$ of discrete Huber pairs over $(V,\mm)$ is a morphism $A \to B$ of $(V,\mm)$-algebras such that the image of $A^+$ under the induced map $\pi_0 A_{**\omega} \to \pi_0 B_{**\omega}$ lies in $B^+$.

* Let $(A, A^+)$ be a discrete Huber pair over $(V,\mm)$. We let

\begin{align*}
(A, A^+)_\solid \in \AnRing_{(V,\mm)}
\end{align*}

denote the following analytic ring structure on $A$: The pair $(A_{**\omega}, A^+)$ is a discrete Huber pair over $V$, and we let $(A, A^+)_\solid := (A_{**\omega}, A^+)^a$.

*  Given a discrete $(V,\mm)$-algebra $A$, a discrete ring $A^+$ and a map $A^+ \to \pi_0 A_*$ we also denote $(A, A^+)_\solid := (A, \tilde A^+)_\solid$, where $\tilde A^+ \subset \pi_0 A_{**\omega}$ is the integral closure of the image of the map $\pi_0 A^+ \to \pi_0 A_{**\omega}$.

* We abbreviate $\D_\solid(A, A^+) = \D((A, A^+)_\solid)$. The objects of $\D_\solid(A, A^+)$ are called the solid $(A, A^+)$-modules.

Let $(V,\mm)$ be an almost setup.

* The assignment $(A, A^+) \mapsto (A, A^+)_\solid$ defines a fully faithful functor from the $\infty$-category of discrete Huber pairs over $(V,\mm)$ to $\AnRing_{(V,\mm)}$.

*  The functor in (i) preserves all non-empty colimits. In particular, for any diagram $(B, B^+) \from (A, A^+) \to (C, C^+)$ of discrete Huber pairs over $(V,\mm)$ we have

\begin{align*}
(B, B^+)_\solid \tensor_{(A, A^+)_\solid} (C, C^+)_\solid = (B \tensor_A C, B^+ \tensor_{A^+} C^+)_\solid,
\end{align*}

where on the right-hand side we use the abbreviated notation from <ref>.

*  Let $(A, A^+)$ be a discrete Huber pair over $(V,\mm)$. Then an object $P \in \D_\solid(A, A^+)$ is nuclear if and only if it is discrete.

*  Let $(A, A^+)$ and $(B, B^+)$ be discrete Huber pairs over $(V,\mm)$. Then every morphism $f\colon (A, A^+)_\solid \to (B, B^+)_\solid$ is steady.

The functor $(A, A^+) \mapsto (A_{**\omega}, A^+)$ defines a fully faithful functor from discrete Huber pairs over $(V,\mm)$ to discrete Huber pairs over $V$. One easily deduces (i) and (ii) from <ref>. Part (iii) can be proved in exactly the same way as <ref>. Part (iv) can be deduced from <ref>.

Before we continue, we need to deal with some set-theoretic issues. In general, if $\mathcal A$ is an analytic ring then we defined the presentable $\infty$-categories $\D(\mathcal A)_\kappa$ for large enough strong limit cardinals $\kappa$ such that $\D(\mathcal A) = \bigunion_\kappa \D(\mathcal A)_\kappa$. By definition, $\D(\mathcal A)_\kappa$ is generated by the objects $\mathcal A[S]$ for $\kappa$-small extremally disconnected sets $S$. However, it may in general be false that $\mathcal A[S]$ is itself $\kappa$-condensed and therefore $\D(\mathcal A)_\kappa$ may not be a subcategory of $\D(\underline{\mathcal A})_\kappa$. In particular, for a map $f\colon \mathcal A \to \mathcal B$ of analytic rings, the “forgetful functor” $f_{*\kappa}\colon \D(\mathcal B)_\kappa \to \D(\mathcal A)_\kappa$ is in general the composition of the actual forgetful functor $f_*\colon \D(\mathcal B)_\kappa \to \D(\mathcal A)$ and the restriction $\D(\mathcal A) \to \D(\mathcal A)_\kappa$. Therefore $f_{*\kappa}$ may not be conservative in general, even though $f_*$ is. One can often circumvent this issue by juggling strong limit cardinals, but it turns out that in the setting of solid analytic rings the whole situation becomes much easier:

A solid cutoff cardinal is a strong limit cardinal $\kappa$ such that either $\kappa = \omega$ or $\kappa$ satisfies the following property: For all cardinals $\lambda < \kappa$ there is a strong limit cardinal $\kappa_\lambda < \kappa$ such that the cofinality of $\kappa_\lambda$ is larger than $\lambda$.

* There is a cofinal class of solid cutoff cardinals.

* Let $\kappa$ be a solid cutoff cardinal and $(V,\mm)$ an almost setup. Then for every discrete Huber pair $(A, A^+)$ over $(V,\mm)$ we have $\D_\solid(A, A^+)_\kappa \subset \D(A)_\kappa$.

In particular, for every map $f\colon (A, A^+) \to (B, B^+)$ of discrete Huber pairs over $(V,\mm)$, the forgetful functor $f_*\colon \D_\solid(B, B^+) \to \D_\solid(A, A^+)$ restricts to a forgetful functor $f_*\colon \D_\solid(B, B^+)_\kappa \to \D_\solid(A, A^+)_\kappa$.

Part (i) is <cit.>. We now prove (ii), so let $\kappa$ and $(A, A^+)$ be given. If $\kappa = \omega$ then the claim is obvious, so from now on we assume that $\kappa$ is uncountable. Fix a $\kappa$-small extremally disconnected set $S$. We need to show that $(A, A^+)_\solid[S]$ is $\kappa$-condensed, i.e. lies in $\D(V,\mm)_\kappa$. Using that the almost localization functor $(V,\mm)^*\colon \D(V) \to \D(V,\mm)$ maps $\D(V)_\kappa$ to $\D(V,\mm)_\kappa$ we can reduce to the case that $(A, A^+)$ is a discrete Huber pair over $V$; we can further assume $V = \Z$. Since $\D(\Z)_\kappa$ is stable under all small colimits we can reduce to the case that $A = A^+$ and that this ring is of finite type over $\Z$. Then by definition we have $(A, A^+)_\solid[S] = A_\solid[S] = \varprojlim_i A[S_i]$, where $S = \varprojlim_{i\in I} S_i$ with all $S_i$ finite. We can arrange that $I$ is $\kappa$-small (e.g. we can choose $I$ to be the set of disjoint clopen covers of $S$, so that $\abs I \le \omega^{\kappa'} \le 2^{\omega \cprod \kappa'} < \kappa$, where $\kappa' < \kappa$ is the cardinality of $S$). Let $\lambda < \kappa$ denote the cardinality of $I$. By definition of solid cutoff cardinals, we can find a strong limit cardinal $\kappa_\lambda < \kappa$ such that the cofinality of $\kappa_\lambda$ is larger than $\lambda$. By (the proof of) <ref> it follows that $\D(\Z)_{\kappa_\lambda}$ is stable under $\lambda$-small limits in $\D(\Z)$, which immediately implies that $\varprojlim_i A[S_i] \in \D(\Z)_{\kappa_\lambda} \subset \D(\Z)_\kappa$, as desired.

We now glue the solid analytic rings studied above in order to obtain schemes and discrete adic spaces. Using geometry blueprints (see <ref>) the gluing process is completely formal:

Let $(V,\mm)$ be an almost setup.

* We let $G_{\omega-adic}$ denote the geometry blueprint over $\Z$ whose $G_{\omega-adic}$-analytic rings are precisely the analytic rings of the form $(A, A^+)_\solid$ for discrete Huber pairs $(A, A^+)$ and whose $G_{\omega-adic}$-localizations are generated by the two maps

\begin{align*}
(\Z[T], \Z)_\solid \to (\Z[T,T^{-1}], \Z)_\solid, \qquad (\Z[T], \Z)_\solid \to \Z[T]_\solid.
\end{align*}

We let $G^{(V,\mm)}_{\omega-adic}$ denote the geometry blueprint which is generated by the image of $G_{\omega-adic}$ under the almost localization $\AnRing_V \to \AnRing_{(V,\mm)}$.

* A discrete adic space over $(V,\mm)$ is a $G_{\omega-adic}^{(V,\mm)}$-analytic space. For every discrete Huber pair $(A, A^+)$ over $(V,\mm)$ we denote

\begin{align*}
\Spa(A, A^+) := \AnSpec_{G_{\omega-adic}^{(V,\mm)}} (A, A^+)_\solid.
\end{align*}

An open immersion $U \injto X$ of discrete adic spaces over $(V,\mm)$ is a map which exhibits $U$ as a $G_{\omega-adic}^{(V,\mm)}$-analytic subspace of $X$. A discrete adic space is a discrete adic space over $\Z$.

* A discrete adic space is called classical if it can be covered by open sets of the form $\Spa(A, A^+)$ with $A$ static.

* We let $G_{sch}$ denote the geometry blueprint over $\Z$ whose $G_{sch}$-analytic rings are precisely the analytic rings of the form $A_\solid$ for discrete rings $A$ and whose $G_{sch}$-localizations are generated by the map $\Z[T]_\solid \to \Z[T, T^{-1}]_\solid$.

* A scheme is a $G_{sch}$-analytic space. For every discrete ring $A$ we denote

\begin{align*}
\Spec A := \AnSpec_{G_{sch}} A_\solid.
\end{align*}

An open immersion $U \injto X$ of schemes is a map which exhibits $U$ as a $G_{sch}$-analytic subspace of $X$.

* A scheme is called classical if it can be covered by open sets of the form $\Spec A$ with $A$ static.

* By <ref> a scheme (in the above sense) can be identified with a functor $\Alg_{\ge0}(\Z)_\omega \to \Ani$ which satisfies descent along open covers of affine schemes. It follows that the above defined $\infty$-category of schemes is equivalent to the $\infty$-category of derived schemes as mentioned in <cit.> via the Yoneda embedding (use also <ref> to see that Zariski sheaves on the site of all derived schemes are the same as Zariski sheaves on affine derived schemes).

* There is an analytification functor $X \mapsto X^\an$ (see <ref>) associating to every discrete adic space over some almost setup $(V,\mm)$ an analytic space over $(V,\mm)$. By an unpublished argument of Scholze the restriction of this functor to schemes is fully faithful.

* There is a natural functor from schemes to discrete adic spaces. This functor can be shown to be fully faithful by the same argument as for the analytificiation functor.

* For every almost setup $(V,\mm)$ there is an almost localization functor $X \mapsto X^a$ associating to every discrete adic space $X$ over $V$ a discrete adic space $X^a$ over $(V,\mm)$ (see <ref>). It maps $\Spa(A, A^+) \mapsto \Spa(A^a, A^+)$. This allows to deduce many of the following results (which are only stated over $\Z$) also in the almost setting.

It follows from <ref> that every morphism of discrete adic spaces over any almost setup $(V,\mm)$ is steady.

The above defined $\infty$-category of discrete adic spaces is a generalization of the category of classical discrete adic spaces where we additionally allow non-static structure sheaves. More concretely we have the following comparison:

There is a natural fully faithful functor from the category of classical discrete adic spaces to the $\infty$-category of discrete adic spaces. This functor has a right adjoint mapping $\Spa(A, A^+) \mapsto \Spa(\pi_0 A, A^+)$. The right adjoint preserves fiber products and open immersions.

The existence of the desired fully faithful functor essentially boils down to the fact that if $\Spa(B, B^+) \injto \Spa(A, A^+)$ is an open immersion of discrete adic spaces and $\Spa(A, A^+)$ is classical, then $\Spa(B, B^+)$ is classical, which is easy to see. The right adjoint is easily constructed (on the level of the surrounding $\infty$-categories of sheaves it is just a pushforward, which in this case is just a restriction) and has the desired properties (note that preservation of pullbacks is true for any right adjoint).

Let $X$ be a discrete adic space. We call the associated classical discrete adic space from <ref> the underlying classical discrete adic space $X_0$ of $X$.

Next we want to introduce the underlying topological space $\abs X$ of a discrete adic space $X$. This can be done directly using the $\infty$-category of discrete adic spaces as defined above, e.g. the points of $\abs X$ are precisely the equivalence classes of maps $\Spa(K, K^+) \to X$, where $K$ is a field and $K^+ \subset K$ is a valuation ring with fraction field $K$. However, we take a shortcut by referring to the known topological space underlying a classical discrete adic space. To make this work, we need the following result:

A finite family $((A, A^+)_\solid \to (B_i, B_i^+)_\solid)_i$ of $G_{\omega-adic}$-localizations is a $G_{\omega-adic}$-analytic cover if and only if the family $\abs{\Spa(\pi_0 B_i, B_i^+)} \to \abs{\Spa(\pi_0 A, A^+)}$ of open immersions forms a topological cover.

We first prove the “if” direction, so assume that the given maps form a topological cover. We need to show that the family of functors $- \tensor_{(A, A^+)_\solid} (B_i, B_i^+)_\solid$ is conservative. Note that each of the maps $(A, A^+)_\solid \to (B_i, B_i^+)_\solid$ identifies $(B_i, B_i^+)_\solid$ with an $(A, A^+)_\solid$-algebra of the form

\begin{align*}
(A[1/f], A^+[g_1/f, \dots, g_n/f])_\solid
\end{align*}

for some $f, g_1, \dots, g_n \in A$ (one checks easily that the collection of such maps is stable under base-change and compositions and hence provides all $G_{\omega-adic}$-localizations). In other words, each of the maps $(A, A^+)_\solid \to (B_i, B_i^+)_\solid$ is given by a rational open subset of $\abs{\Spa(\pi_0 A, A^+)}$ and by assumption these rational open subsets form a cover of $\abs{\Spa(\pi_0 A, A^+)}$. We can now argue as in the proof of <cit.>; the argument roughly goes as follows: One first refines the given topological cover by a topolocial cover of the form $(A[1/s_i], A^+[s_1/s_i, \dots, s_N/s_i])_\solid$ for certain $s_1, \dots, s_N \in \pi_0 A$ generating the unit ideal. Then one can argue by induction on $N$ which reduces the claim to the case of a topological cover by $(A, A^+[f])_\solid$ and $(A[1/f], A^+[1/f])_\solid$ for some $f \in \pi_0 A$. This cover is the base-change of the standard topological cover of $(\Z[T], \Z)_\solid$ given by $\Z[T]_\solid$ and $(\Z[T, T^{-1}], \Z[T^{-1}])_\solid$. Then a quick computation shows that this last topological cover is indeed a $G_{\omega-adic}$-analytic cover, which proves the claim.

We now prove the “only if” direction. Suppose that the given maps form a $G_{\omega-adic}$-analytic cover but not a topological cover. Then one can find a valuation ring $K^+$ with fraction field $K$ and a map $(A, A^+)_\solid \to (K, K^+)_\solid$ which does not factor over any $(B_i, B_i^+)_\solid$. In particular the maps $(K, K^+)_\solid \to (A, A^+)_\solid \tensor_{(K, K^+)_\solid} (B_i, B_i^+)_\solid$ do not form a topological cover. We can thus reduce to the case that $(A, A^+) = (K, K^+)$. In this case every $(B_i, B_i^+)_\solid$ is of the form $(K, K^+[f_1, \dots, f_n])_\solid$ and two such analytic rings agree precisely if the $f_1, \dots, f_n$ generate the same valuation subring of $K$. Since the valuation rings between $K$ and $K^+$ form a chain under inclusion, we reduce to the case of a single map $\alpha\colon (K, K^+)_\solid \to (K, K'^+)_\solid$ which is a $G_{\omega-adic}$-analytic cover, where $K'^+ \supset K^+$ is another valuation ring with fraction field $K$. But then $\alpha$ is a localization and base-change along $\alpha$ is an equivalence, which shows that $\alpha$ is an isomorphism. This implies $K'^+ = K^+$ (using the full faithfulness of the functor $(A, A^+) \mapsto (A, A^+)_\solid$ from <ref> which we implicitly used throughout the whole proof). Contradiction!

For every discrete adic space $X$ we denote $\abs X$ the topological space glued from the topological spaces $\abs{\Spa(\pi_0 A, A^+)}$ on an open cover of $X$ by affinoid subspaces $\Spa(A, A^+)$. This topological space is well-defined by <ref> and depends only on the underlying classical discrete adic space $X_0$.

Even if one is only interested in classical discrete adic spaces one still needs to show <ref> (at least for classical discrete Huber rings) to verify that our definition of classical discrete adic spaces coincides with the definition by Huber.

With the underlying topological space of a discrete adic space at hand, one could now attempt to characterize discrete adic spaces as certain locally ringed spaces. We do not pursue this idea any further (but cf. <cit.>). However, let us prove the following handy tools motivated by the picture of locally ringed spaces:

Let $X$ be a discrete adic space. Then the assignment $U \mapsto \abs U$ defines a bijection between open subspaces $U \subset X$ and open subsets of $\abs X$.

This can be checked locally on $X$ so we can assume that $X = \Spa(A, A^+)$ is affine. Now every open subset of $\abs X$ can be covered by rational open subsets and hence comes from some open subspace $U \subset X$. Moreover, if $U, U' \subset X$ are two open subspaces such that $\abs U = \abs U'$ then we must have $U = U'$ by <ref>.

Let $f\colon Y \to X$ be a map of discrete adic spaces and let $U \subset X$ be an open subspace such that $\abs f\colon \abs Y \to \abs X$ factors over $\abs U$. Then $f$ factors over $U$.

To show that $f$ factors over $U$, we can equivalently show that the map $U \cprod_X Y \injto Y$ is an isomorphism. By assumption on $f$ we have $\abs{U \cprod_X Y} = \abs Y$, hence the claim follows from <ref>.

Let $f\colon Y \to X$ be a map of discrete adic spaces which satisfies the following properties:

* There is a cover of $Y$ by open subspaces $V \subset Y$ such that each map $V \to X$ is a monomorphism.

* The map $\abs Y \injto \abs X$ is injective.

Then $f$ is a monomorphism.

Let $Z = \Spa(A, A^+)$ be an affine discrete adic space. In order to show that $f$ is a monomorphism we need to see that the induced map $\Hom(Z, X) \to \Hom(Z, Y)$ is a monomorphism of anima. To see this, we can fix a map $h\colon Z \to X$ and now need to show that the anima $\Hom_X(Z, Y)$ is either empty or contractible. Assume that it is non-empty, i.e. that there is a map $g\colon Z \to X$ such that $h = f \comp g$. Using (a) we pick a cover $Y = \bigunion_i V_i$ of open subspaces $V_i \subset Y$ such that each map $V_i \to X$ is a monomorphism. Let $W_i := g^{-1}(V_i) \subset Z$, let $W' := \bigdunion_i W_i$ and let $W'_\bullet \to Z$ be the associated Čech cover. Then $\Hom_X(Z, Y) = \varprojlim_{n\in\Delta} \Hom_X(W'_n, Y)$, so it is enough to show that each $\Hom_X(W'_n, Y)$ is contractible. Each $W'_n$ is a disjoint union of finite intersections of $W_i$'s, so we are reduced to showing the following: Let $V \subset Y$ be an open subset such that the map $V \to X$ is a monomorphism and let $W := g^{-1}(V) \subset Z$; then $\Hom_X(W, Y)$ is contractible. We know that $\Hom_X(W, V)$ is contractible, so we only need to show that $\Hom_X(W, V) \isoto \Hom_X(W, Y)$ is an isomorphism; since this map is a monomorphism this boils down to showing that every $X$-morphism $W \to Y$ factors over $V$. By (b) it factors over $\abs V$, hence the claim follows from <ref>.

One big advantage of discrete adic spaces over schemes is that discrete adic spaces allow canonical compactifications. This in particular provides a useful notion of “affine proper maps”, so that often questions about proper maps can be checked locally on the source. The canonical compactification can be constructed as follows:

Let $X$ be a discrete adic space. Then there is a discrete adic space $\overline X$ which is uniquely characterized by the property

\begin{align*}
\Hom(\Spa(A, A^+), \overline X) = \Hom(\Spa(A, A), X)
\end{align*}

for all discrete Huber pairs $(A, A^+)$. The functor $X \mapsto \overline X$ preserves finite limits, monomorphisms, disjoint unions and covers.

Viewing $X$ as a sheaf on the analytic site of $G_{\omega-adic}$-analytic rings, the given $\Hom$-identity immediately defines $\overline X$ as a presheaf on that site. It follows from <ref> that $\overline X$ satisfies descent along analytic covers: One needs to check that if $(\Spa(B_i, B_i^+) \to \Spa(A, A^+))_i$ is a cover of classical discrete adic spaces by rational open subsets then $(\Spa(B_i, B_i) \to \Spa(A, A))_i$ is still a cover; this is an easy exercise. Hence $\overline X$ is a sheaf on the analytic site of $G_{\omega-adic}$-analytic rings. It is clear that the thus defined functor $X \mapsto \overline X$ (seen as a functor of sheaves on the site of $G_{\omega-adic}$-analytic rings; we do not yet know that $\overline X$ is a discrete adic space) preserves finite limits and monomorphisms. To check that it preserves covers we employ <ref>: Suppose we are given a cover $Y \to X$ of discrete adic spaces and a section $s\colon \Spa(C, C^+) \to \overline X$ for some discrete Huber pair $(C, C^+)$. Then $s$ corresponds to a map $s'\colon \Spa(C, C) \to X$ and thus there are a $G_{\omega-adic}$-analytic cover $((C, C) \to (C_i, C_i))_i$ (by the same argument as above) and sections $t_i'\colon \Spa(C_i, C_i) \to Y$ compatible with $s$. Then each $C_i$ is of the form $C[1/f_i]$ for some $f_i \in \pi_0 C$, so we can form the $G_{\omega-adic}$-analytic cover $((C, C^+) \to (C_i, C_i^+))_i$, where $C_i^+ = C^+[1/f_i]$. Then the sections $t'_i$ correspond to sections $t_i\colon \Spa(C_i, C_i^+) \to \overline Y$, compatible with $s$. This shows that $\overline Y \to \overline X$ is a cover. One can similarly show that the functor $X \mapsto \overline X$ preserves disjoint unions.

It remains to show that $\overline X$ is a discrete adic space over $X$. If $X = \Spa(B, B^+)$ is affine then one checks immediately that $\overline X = \Spa(B, \Z)$, which is indeed a discrete adic space. If $(B, B^+) \to (B', B'^+)$ is a $G_{\omega-adic}$-localization then so is $(B, \Z) \to (B', \Z)$ (e.g. one checks that the class of $G_{\omega-adic}$-localizations which satisfies this property after any base-change is stable under base-change and composition and contains the generating $G_{\omega-adic}$-localizations). Altogether this implies that if $U \subset X$ is an open subspace of the affine discrete adic space $X$ then $\overline U \subset \overline X$ is an open subspace of $\overline X$. We conclude by <ref> that $\overline X$ is a discrete adic space for general $X$.

* Let $X$ be a discrete adic space. We call the discrete adic space $\overline X$ from <ref> the compactification of $X$. There is a canonical map $X \to \overline X$.

* Let $f\colon Y \to X$ be a map of discrete adic spaces. We denote

\begin{align*}
\overline f^{/X}\colon \overline Y^{/X} := \overline Y \cprod_{\overline X} X \to X
\end{align*}

and call it the compactification of $f$. We also call $\overline Y^{/X}$ the relative compactification of $Y$ over $X$. There is a natural map $Y \to \overline Y^{/X}$ of discrete adic spaces.

If $Y = \Spa(B, B^+)$ and $X = \Spa(A, A^+)$ are affine then $\overline Y^{/X} = \Spa(B, A^+)$.

In general we cannot say much about the map $Y \to \overline Y^{/X}$, but properties of this map have a strong connection to the following classical properties of $f$:

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

* We say that $f$ is locally of $+$-finite type if for all $y \in \abs Y$ there are affine open neighbourhoods $V = \Spa(B, B^+) \subset Y$ of $y$ and $U = \Spa(A, A^+) \subset X$ of $f(x)$ such that $f(V) \subset U$ and there exists a collection of finitely many elements $g_1, \dots, g_n \in B^+$ with $(B, B^+)_\solid = (B, A^+[g_1, \dots, g_n])_\solid$. We say that $f$ is of $+$-finite type if $f$ is locally of $+$-finite type and quasicompact.

* We say that $f$ is separated if for all valuation rings $K^+$ with fraction field $K$ and every commutative solid diagram

(K, K) [r] [d]     Y [d]

(K, K^+) [r] [ur,dashed]     X

there is a dashed morphism making the diagram commute.

* We say that $f$ is partially proper if it is separated, locally of $+$-finite type and for every diagram as in (ii), the dashed morphism is unique. We say that $f$ is proper if $f$ is partially proper and quasicompact.

Note that all of the properties in <ref> only depend on the underlying map $f_0\colon Y_0 \to X_0$ of classical discrete adic spaces. In particular if $X$ and $Y$ are schemes then the above notion of separatedness agrees with its counterpart in [25], while our notion of properness is slightly more general, as we allow maps which need not be of finite type (but are so up to an integral extension); see <cit.>. It is easy to see that every map of affine spaces is separated and every separated map is quasiseparated. The notion “of $+$-finite type” seems to be new, but is motivated by the next results.

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

* $f$ is separated if and only if the map $Y \to \overline Y^{/X}$ is a monomorphism.

* $f$ is locally of $+$-finite type if and only if the map $Y \to \overline Y^{/X}$ is locally on $Y$ an open immersion.

*  $f$ is partially proper if and only if the map $Y \to \overline Y^{/X}$ is an isomorphism.

We first prove (i). The map $j\colon Y \to \overline Y^{/X}$ is always locally a monomorphism. Namely, this can be checked in the case that $X = \Spa(A, A^+)$ and $Y = \Spa(B, B^+)$ are affine, in which case $j$ becomes the map $\Spa(B, B^+) \to \Spa(B, A^+)$ which is a monomorphism. It follows directly from the definitions that $f$ is separated if and only if the map $\abs Y \to \abs{\overline Y^{/X}}$ is bijective (use that the points of these topological spaces are precisely the maps from $\Spa(K, K^+)$'s). Hence (i) reduces to <ref>.

Part (ii) can be checked locally on $X$ and $Y$, so we can assume that $X = \Spa(A, A^+)$ and $Y = \Spa(B, B^+)$ are affine. Then $j$ is the map $\Spa(B, B^+) \to \Spa(B, A^+)$. If $\Spa(B, B^+) = \Spa(B, A^+[g_1, \dots, g_n])$ for some $g_1, \dots, g_n \in B^+$ then clearly $j$ is an open immersion. Conversely, if $j$ is an open immersion then clearly $j$ is of $+$-finite type (cover it by rational open subsets), which immediately implies that $f$ is of $+$-finite type.

It remains to prove (iii). By (i) and (ii), $j$ is an isomorphism if and only if $f$ is separated, locally of $+$-finite type and the map $\abs j\colon \abs Y \to \abs{\overline Y^{/X}}$ is a bijection (see <ref>). This is exactly the definition of partial properness for $f$.

Let $f\colon Y \to X$ be a map of discrete adic spaces and let $\overline f^{/X}\colon \overline Y^{/X} \to X$ be its compactification. Then $\overline f^{/X}$ is partially proper. If $f$ is quasicompact then $\overline f^{/X}$ is proper.

The partial properness of $\overline f^{/X}$ follows immediately from its construction. If $f$ is quasicompact then after base-change to any affine open subspace $\Spa(A, A^+) \subset X$, $Y$ can be covered by finitely many affine open subspaces $\Spa(B_i, B_i^+)$. Then $\overline Y^{/X}$ can be covered by the affine open subspaces $\Spa(B_i, A^+)$ and thus is quasicompact. This shows that $\overline f^{/X}$ is quasicompact and thus proper.

In the following we will often make implicit use of <ref> when dealing with relative compactifications.

To every discrete adic space $X$ we have an associated $\infty$-category $\D_\solid(X)$ of solid quasicoherent sheaves on $X$ (see <ref>). If $X = \Spa(A, A^+)$ is affine then $\D_\solid(X) = \D_\solid(A, A^+)$. For every solid cutoff cardinal $\kappa$ we can define a similar $\infty$-category $\D_\solid(X)_\kappa$ glued from $\D_\solid(A, A^+)_\kappa$'s. One of the main results of [36] (and a big motivation for the theory of solid analytic rings) states that there is a full $6$-functor formalism for $\D_\solid(X)$ on schemes $X$. The following result makes this more precise.

Let $f\colon Y \to X$ be a separated $+$-finite-type map of discrete adic spaces. Then there is a natural pair of adjoint functors

\begin{align*}
f_!\colon \D_\solid(Y) \rightleftarrows \D_\solid(X) \noloc f^!
\end{align*}

satisfying the following properties:

* (Functoriality) If $g\colon Z \to Y$ is another separated $+$-finite-type map of discrete adic spaces then $(f \comp g)_! \isom f_! \comp g_!$ and $(f \comp g)^! \isom g^! \comp f^!$.

*  (Special Cases) If $f$ is proper then $f_! \isom f_*$. If $f$ is an étale map of classical schemes then $f^! \isom f^*$.

*  (Projection Formula) For all $\mathcal M \in \D_\solid(X)$ and all $\mathcal N \in \D_\solid(Y)$ there is a natural isomorphism $f_!(\mathcal N \tensor f^* \mathcal M) \isom (f_! \mathcal N) \tensor \mathcal M$.

*  (Proper Base-Change) Let $g\colon X' \to X$ be any map of discrete adic spaces with base-change $Y' := Y \cprod_X X'$ and projections $f'\colon Y' \to X'$ and $g'\colon Y' \to Y$. Then there is a natural equivalence $g^* f_! \isom f'_! g'^*$ of functors $\D_\solid(Y) \to \D_\solid(X')$.

*  (Poincaré Duality) If $f$ is a smooth map of classical schemes and has pure dimension $d$ then $f^! \isom f^* \tensor \omega_{Y/X}[d]$, where $\omega_{Y/X} := \bigwedge^d \Omega^1_{Y/X}$.

*  For every solid cutoff cardinal (see <ref>) $\kappa$ we have $f_!(\D_\solid(Y)_\kappa) \subset \D_\solid(X)_\kappa$.

We will first prove everything in the case that $X = \Spa(A, A^+)$ and $Y = \Spa(B, B^+)$ are affine and $\Spa(B, B^+) = \Spa(A, A^+[g_1, \dots, g_n])$ for some $g_1, \dots, g_n \in B^+$. We factor $f$ into the maps $j\colon Y \to \overline Y^{/X}$ and $\overline f^{/X}\colon \overline Y^{/X} \to X$, where $\overline Y^{/X} = \Spa(B, A^+)$ is the relative compactification of $f$. As in <cit.> we will define $f_! := (\overline f^{/X})_* \comp j_!$, where $j_!\colon \D_\solid(B, B^+) \to \D_\solid(B, A^+)$ is a left adjoint of $j^*$. To make this construction work, we need to show the existence of $j_!$. Fix a solid cutoff cardinal $\kappa$. We claim that $j^*\colon \D_\solid(B, B^+)_\kappa \to \D_\solid(B, A^+)_\kappa$ preserves all small limits. Using the elements $g_1, \dots, g_n$ we see that the map $(B, A^+)_\solid \to (B, B^+)_\solid$ is a base-change of the map

\begin{align*}
(\Z[T_1, \dots, T_n], \Z)_\solid \to (\Z[T_1, \dots, T_n], \Z[T_1, \dots, T_n])_\solid,
\end{align*}

so it suffices to prove that pullback along the latter map preserves all small limits. This readily reduces to showing that the pullback $j^*\colon \D_\solid(\Z[T], \Z)_\kappa \to \D_\solid(\Z[T])_\kappa$ preserves all small limits. This follows from <cit.>, which explicitly constructs a left-adjoint. We have finished the proof that $j^*\colon \D_\solid(B, A^+)_\kappa \to \D_\solid(B, B^+)_\kappa$ preserves small limits. As both $\D_\solid(B, A^+)_\kappa$ and $\D_\solid(B, B^+)_\kappa$ are presentable, this formally implies the existence of a left-adjoint

\begin{align*}
j_{!\kappa}\colon \D_\solid(B, B^+)_\kappa \to \D_\solid(B, A^+)_\kappa.
\end{align*}

It follows formally from steady base-change (see <ref>) by passing to right-adjoints that $j_{!\kappa}$ satisfies base-change with respect to any morphism $(C, C^+)_\solid \to (B, A^+)_\solid$ of discrete Huber pairs. Now pick another solid cutoff cardinal $\kappa' \ge \kappa$. Then for every $\kappa$-small profinite set $S$ we have $j_{!\kappa'} (B, B^+)_\solid[S] \in \D_\solid(B, A^+)_\kappa$. Namely, by base-change for $j_{!\kappa'}$ we can reduce to the case $B = B^+ = \Z[T]$ and $A^+ = \Z$, where we have by <cit.>

\begin{align*}
j_{!\kappa'} \Z[T]_\solid[S] = \Z[T]_\solid[S] \tensor_{(\Z[T], \Z)_\solid} j_{!\kappa'} \Z[T],
\end{align*}

and $j_{!\kappa'} \Z[T]$ lies in $\D_\solid(\Z[T], \Z)_\kappa$ for every solid cutoff cardinal $\kappa$ by the explicit construction in <cit.>. Since $j_{!\kappa'}$ preserves small colimits, we deduce $j_{!\kappa'}(\D_\solid(B, B^+)_\kappa) \subset \D_\solid(B, A^+)_\kappa$, i.e. the computation of $j_{!\kappa}$ does not depend on $\kappa$. This shows that the functors $j_{!\kappa}$ combine to a functor

\begin{align*}
j_!\colon \D_\solid(B, B^+) \to \D_\solid(B, A^+)
\end{align*}

which is left-adjoint to $j^*$. This finishes the construction of $f_! := (\overline f^{/X})_* \comp j_!$ (in the affine case).

It is easy to see that $f_!$ admits a right adjoint $f^!$: This reduces to the case $f = \overline f^{/X}$, where $f_! = f_*$ is just a forgetful functor. This forgetful functor preserves all small colimits and $\D(-)_\kappa$ (by <ref>), so one can easily construct right adjoint $f^{!\kappa}$. Then note that $f_* f^{!\kappa}$ is computed by an $\IHom$ with fixed first argument and can thus be made independent of $\kappa$ (for large enough $\kappa$).

We now verify that the above defined functors $f_!$ and $f^!$ satisfy properties (i) – (vi): Part (iv) follows from base-change for $j_!$ (as discussed above) and base-change for $(\overline f^{/X})_*$ (see <ref>). Part (iii) can be separately shown for $j_!$ and $\overline f^{/X}$; the latter case is easy. For the former case, i.e. the projection formula for $j_!$, note first that there is a natural map $j_!(\mathcal N \tensor j^* \mathcal M) \to j_! \mathcal N \tensor \mathcal M$. Both sides commute with colimits in $\mathcal M$ and $\mathcal N$, so we can reduce to the case $\mathcal M = (B, A^+)_\solid[S]$ and $\mathcal N = (B, B^+)_\solid[T]$ for some profinite sets $S$ and $T$. Now we can use base-change as above to reduce to the case $B = B^+ = \Z[T]$ and $A^+ = \Z$. In this case we can argue as in the proof of the projection formula in <cit.>: The two sides of the projection formula become isomorphic after applying $j^*$, hence their cofiber is a $B_\infty = \Z((T^{-1}))/B$-module. This means that tensoring the cofiber with $B_\infty$ leaves it invariant (see <cit.>); on the other hand, tensoring any side of the projection formula with $B_\infty$ produces $0$ (in general, $j_! M \tensor_{(B, A)_\solid} B_\infty = 0$, which follows from the triangle $j_!M \to M \to M \tensor B_\infty$ and <cit.>). This implies that the cofiber is $0$, hence the projection formula holds. Part (v) follows from <cit.>. The proper part of (ii) is obvious and the étale part of (ii) is a special case of (v).
Part (vi) can be shown separately for $j_!$ (where it is true by the above construction) and for $(\overline f^{/X})_*$ (where it follows from <ref>).

To finish the affine case, it remains to prove (i). Given $f$ and $g$ as in the claim, one easily reduces the claim to showing that the following diagram commutes (cf. the proof of <cit.>):

(Z^/Y) [r,"j'_!"] [d,"h'_*"]     (Z^/X) [d,"h_*"]

(Y) [r,"j_!"]     (Y^/X)

This follows formally from base-change and the projection formulas (cf. the proof of <ref>). This finishes the proof of all claims in the affine case.

Now let $f$ be general. We construct $f_!$ as in the affine case: Factor $f$ as a composition of $j\colon Y \to \overline Y^{/X}$ and $\overline f^{/X}\colon \overline Y^{/X} \to X$ and set $f_! := (\overline f^{/X})_* \comp j_!$. For this to work, we need to check the existence of the left-adjoint $j_!\colon \D_\solid(Y) \to \D_\solid(\overline Y^{/X})$ of $j^*$. We know that $j_!$ exists in the affine case and satisfies arbitrary base-change (and in particular base-change along open immersions), so it can formally be glued to a functor $j_!$ for general $X$ and $Y$ (write the solid quasicoherent sheaf $\infty$-categories as limits over the solid module categories on an affine cover and use the explicit description of limits of $\infty$-covers in terms of coCartesian sections; then observe that the base-change for $j_!$ implies that the componentwise $j_!$ preserves coCartesian morphisms and thus induces a functor on the limits). The existence of $f^!$ follows in a similar way as in the affine case and the claims (i) – (vi) can also easily be reduced to the affine case.

One can try to improve upon <ref> in several ways: Firstly one may ask if a similar statement is true in the almost setting (see <ref> below), secondly one may make the $6$-functor formalism much more functorial (analogous to <ref>) and thirdly one may improve upon the Poincaré duality statement by allowing non-classical schemes and even discrete adic spaces.

Let $(V,\mm)$ be an almost setup and let $f\colon Y \to X$ be a separated $+$-finite type map of discrete adic spaces over $V$ with associated map $f^a\colon Y^a \to X^a$ of discrete adic spaces over $(V,\mm)$. Then there is a natural pair of adjoint functors

\begin{align*}
f^a_!\colon \D_\solid(Y^a) \rightleftarrows \D_\solid(X^a) \noloc f^{a!}
\end{align*}

satisfying $f^a_! \comp (-)^a = (-)^a \comp f_!$ and $f^{a!} \comp (-)^a = (-)^a \comp f^!$.

As in the proof of <ref> the construction of $f^a_!$ and $f^{a!}$ formally reduces to the case that $Y = \Spa(B, B^+)$ and $X = \Spa(A, A^+)$ are affine. Then $f^a_!$ can be constructed via the relative compactification of $f$, which reduces its construction to the case that $A = B$. By <ref> the map $f^*\colon \D_\solid(B, A^+) \to \D_\solid(B, B^+)$ has a left adjoint $f_!$. Note that $f^{a*}$ is the composition $f^{a*} = (-)^a \comp f^* \comp (-)_*$ and all of the occurring functors on the right-hand side admit left adjoints; hence $f^{a*}$ admits a left adjoint $f^a_!$. We now prove that this functor $f^a_!$ satisfies $f^a_! \comp (-)^a = (-)^a \comp f_!$. Passing to right adjoints we need to see that for all $N \in \D^a_\solid(B, A^+)$ we have

\begin{align*}
(N \tensor_{(B^a, A^+)_\solid} (B^a, B^+)_\solid)_* = N_* \tensor_{(B, A^+)_\solid} (B, B^+)_\solid.
\end{align*}

Writing $N = N'^a$ for some $N' \in \D_\solid(B, A^+)$ and using that $N \tensor_{(B^a, A^+)_\solid} (B^a, B^+)_\solid = (N' \tensor_{(B, A^+)_\solid} (B, B^+)_\solid)^a$ this amounts to

\begin{align*}
\IHom_B(\mm \tensor_V B, N' \tensor_{(B, A^+)_\solid} (B, B^+)_\solid) = \IHom_B(\mm \tensor_V B, N') \tensor_{(B, A^+)_\solid} (B, B^+)_\solid.
\end{align*}

Note that on both sides we can pull colimits in $\mm$ out, turning them into limits: On the left-hand side this is clear and on the right-hand side this follows because $- \tensor_{(B, A^+)_\solid} (B, B^+)_\solid$ preserves all small limits by the existence of its left adjoint $f_!$. But $\mm$ is a colimit of copies of $V$ (in $\D_\solid(V)$), hence the above identity reduces to the case $\mm = V$, where it is obvious. This finishes the construction of the functor $f^a_!$.

The construction of $f^{a!}$ can also be performed using relative compactifications: In the case $A = B$ we have $f^{a!} = f^{a*}$ by the above construction of $f^a_!$, so this case is easy. It remains to handle the case $A^+ = B^+$ (more precisely, $B^+$ is the integral closure of the image of $A^+$ in $B$, but this distinction does not matter). Now $f^a_! = f^a_*$ and we can define the right adjoint $f^{a!}$ by the adjoint functor theorem (using the usual arguments to reduce to the presentable case, cf. the proof of <ref>). It remains to verify the identity $f^{a!} \comp (-)^a = (-)^a \comp f^!$. By passing to left adjoints this amounts to showing $(-)_! \comp f^a_* = f_* \comp (-)_!$, which is easy to see by <ref>.

We end this section with a result about the flat dimension of the compact projective generators $(A, A^+)_\solid[S] \in \D_\solid(A, A^+)$, for a classical discrete Huber pair $(A, A^+)$. Even though $(A, A^+)_\solid[S]$ is projective, it is in general not flat. Namely, while the functor $- \tensor -$ on $\D_\solid(A, A^+)$ is the left derived functor of its counterpart on the heart, this does not imply the same for the functor $M \tensor -$ for any fixed $M \in \D_\solid(A, A^+)^\heartsuit$. In particular the fact that $(A, A^+)_\solid[S]$ is projective does not imply its flatness. However, if $A^+$ can be generated (over $\Z$) by finitely many elements, then $(A, A^+)_\solid[S]$ does at least have finite flat dimension (see <ref> below). The following argument was suggested by Scholze.

There is a fully faithful colimit-preserving functor

\begin{align*}
j_!\colon \D_\solid(\Z) \injto \D(\Z), \qquad \prod_I \Z \mapsto \prod_I \R/\Z[-1].
\end{align*}

The $t$-structure on $\D(\Z)$ restricts to a $t$-structure on the essential image of $\D_\solid(\Z)^\omega$ under $j_!$. Moreover, $j_!$ is left $t$-exact and maps $\D_{\solid,\ge0}(\Z)$ to $\D_{\ge-1}(\Z)$.

In the following we abbreviate $M^\solid := M \tensor_\Z \Z_\solid$ for a $\Z$-module $M \in \D(\Z)$. We also ignore set-theoretic issues, which can all be resolved by taking filtered colimits over strong limit cardinals $\kappa$.

First note that <cit.> implies that for all $M \in \D_\solid(\Z)$ we have $\IHom(\R, M) = 0$. This shows that $\IHom(\prod_I \R, M) = \IHom_\R(\prod_I \R, \IHom(\R, M)) = 0$ and hence that $(\prod_I \R)^\solid = 0$ for every small set $I$. We deduce that

\begin{align*}
(\prod_I \R/\Z)^\solid = \prod_I \Z[1].
\end{align*}

Now let $\mathcal C^\omega \subset \D(\Z)$ be the full subcategory generated under finite (co)limits by objects of the form $\prod_I \R/\Z$. By the above identity, the solidification functor $(-)^\solid$ restricts to a functor $\mathcal C^\omega \to \mathcal \D_\solid(\Z)^\omega$. This functor is an equivalence: To see this, it is enough to verify that the functor is fully faithful, which boils down to the following computation based on <cit.>:

\begin{align*}
&\IHom(\prod_J \R/\Z, \prod_I \R/\Z) = \prod_I \IHom(\prod_J \R/\Z, \R/\Z) = \prod_I \IHom(\prod_J \R/\Z, \Z)[1] =\\&\qquad= \prod_I \IHom(\prod_J \Z[1], \Z)[1] = \IHom(\prod_J \Z, \prod_I \Z).
\end{align*}

We thus get a functor $j_!\colon \D_\solid(\Z)^\omega \to \D(\Z)$ with essential image $\mathcal C^\omega$ and this functor preserves all finite (co)limits. Since $\D_\solid(\Z)$ is freely generated by $\D_\solid(\Z)^\omega$ under filtered colimits, we immediately obtain a colimit-preserving functor $j_!\colon \D_\solid(\Z) \to \D(\Z)$ extending the functor on $\D_\solid(\Z)^\omega$. It remains to show that $j_!$ has the desired properties.

We first show that $j_!$ is fully faithful. This boils down to showing that for every filtered diagram $(M_j)_{j\in J}$ in $\mathcal C^\omega$ and every small set $I$ we have

\begin{align*}
\IHom(\prod_I \R/\Z, \varinjlim_j M_j) = \varinjlim_j \IHom(\prod_I \R/\Z, M_j).
\end{align*}

Note that $\prod_I \R/\Z$ is pseudocoherent, i.e. admits a resolution by the compact projective generators $\Z[S]$ for extremally disconnected sets $S$ (this is true for every compact abelian group $K$, which by <cit.> reduces to showing that $\Z[K]$ is pseudocoherent, which follows by simplicially resolving $K$ by extremally disconnected sets). In other words we can write $\prod_I \R/\Z$ as a geometric realization of some cosimplicial object in $\D(\Z)$ all of whose terms are of the form $\Z[S]$. Plugging this into the $\IHom$ turns it into a totalization, which commutes with uniformly left-bounded filtered colimits (as it is computed by a spectral sequence). Thus the above identity on $\IHom$'s is true if $(M_j)_j$ is uniformly left-bounded, even for an arbitrary filtered system $(M_j)_j$ in $\D(\Z)$ (i.e. without requiring it to lie in $\mathcal C^\omega$). In general we can now write $(M_j)_j = \varprojlim_k (\tau_{\le k} M_j)_j$. This reduces the above identity to showing that

\begin{align*}
\varinjlim_j \varprojlim_k \IHom(\prod_I \R/\Z, \tau_{\le k} M_j) = \varprojlim_k \varinjlim_j \IHom(\prod_I \R/\Z, \tau_{\le k} M_j).
\end{align*}

Since $J$ is filtered, this holds as soon as we can show that $\IHom(\prod_I \R/\Z, \tau_{\ge k+1} M_j)$ lies in homological degrees $\ge k + d$ for some fixed constant $d$. In fact we claim that one can take $d = 0$. Namely, this boils down to the following claim: For any map $f\colon \prod_{I_1} \R/\Z \to \prod_{I_2} \R/\Z$, the complex $\IHom(\prod_I \R/\Z, \ker f)$ is concentrated in homological degrees $\ge -1$. To see that this is true, note that $K := \ker f$ is a compact abelian group, so by applying Pontrjagin duality (see <cit.>) we get $K = \mathbb D(\mathbb D(K))$ and $\mathbb D(K)$ is discrete. Choosing a $2$-term free resolution of $\mathbb D(K)$ by free $\Z$-modules, we obtain a $2$-term resolution of $K$ in terms of products of $\R/\Z$. But $\IHom(\prod_I \R/\Z, \prod_{I'} \R/\Z)$ is concentrated in degree $0$ (e.g. by the above explicit computation), so the claim follows. This finishes the proof that $j_!$ is fully faithful.

It remains to prove the claim about the $t$-structures. We first show that the $t$-structure on $\D(\Z)$ restricts to a $t$-structure on the essential image $\mathcal C \subset \D(\Z)$ of $j_!$. As every object in $\mathcal C$ is a filtered colimit of objects in $\mathcal C^\omega$, it is enough to show that the $t$-structure on $\D(\Z)$ restricts to a $t$-structure on $\mathcal C^\omega$, i.e. for every $M \in \mathcal C^\omega$ we have $\tau_{\ge0} M, \tau_{\le0} M \in \mathcal C^\omega$. This in turn reduces to showing that for any map $f\colon \prod_I \R/\Z \to \prod_J \R/\Z$, both $\ker f$ and $\coker f$ lie in $\mathcal C^\omega$. To see that this is true for $K := \ker f$, note that $K$ is a compact abelian group, so by the Pontrjagin duality argument employed above we can find a $2$-term resolution of $K$ in terms of products of $\R/\Z$. This shows that $\ker f \in \mathcal C^\omega$. It follows immediately that also $\coker f \in \mathcal C^\omega$ because $\mathcal C^\omega$ is stable under finite limits.

We now show that $j_! \D_{\solid,\ge0}(\Z) \subset \D_{\ge-1}(\Z)$. Given any $M \in \D_{\solid,\ge0}(\Z)$ we can represent $M$ by a complex in homological degree $\ge 0$ all of whose terms are of the form $\bigdsum \prod \Z$. Then $j_!$ maps $M$ to a similar complex, where $\bigdsum \prod \Z$ is transformed into $\bigdsum \prod \R/\Z$ and shifted one to the right. Hence $j_! M \in \D_{\ge-1}(\Z)$, as desired.

We now prove that $j_!$ is left $t$-exact, so let $M \in \D_{\solid,\le0}(\Z)$ be given. It is enough to show that for every $N \in \D_{\ge1}(\mathcal C)$ we have $\Hom(N, j_! M) = 0$. Writing $N = j_! N'$ for some $N' \in \D_\solid(\Z)$ we see that $N' = (j_! N')^\solid$ and thus $N' \in \D_{\solid,\ge1}(\Z)$. But then

\begin{align*}
\Hom(N, j_! M) = \Hom(j_! N', j_! M) = \Hom(N', M) = 0,
\end{align*}

as desired.

Let $S$ be a profinite set. Then the object $\Z_\solid[S] \in \D_\solid(\Z)$ has flat dimension $\le 1$, i.e. the functor

\begin{align*}
- \tensor_{\Z_\solid} \Z_\solid[S]\colon \D_\solid(\Z) \to \D_\solid(\Z)
\end{align*}

maps $\D_{\le0}$ to $\D_{\le1}$.

By <ref> the $t$-structure of $\D(\Z)$ restricts to a $t$-structure on the image $\mathcal C^\omega$ of $\D_\solid(\Z)^\omega$ under the embedding $j_!\colon \D_\solid(\Z) \to \D(\Z)$ and this $t$-structure differs from the $t$-structure on $\D_\solid(\Z)$ only by an error of $1$.

Now let $M \in \D_{\solid,\le0}(\Z)$ be given and pick a small set $I$ such that $\Z_\solid[S] = \prod_I \Z$. Write $j_! M = \varinjlim_j P'_j$ as a filtered colimit of objects $P'_j \in \mathcal C^\omega$, where by <ref> and the above remark about the $t$-structure on $\mathcal C^\omega$ we can assume $P'_j \in \D_{\le0}(\Z)$ for all $j$. We obtain a corresponding colimit $M = \varinjlim_j P_j$ in $\D_\solid(\Z)$, where $P_j = P_j'^\solid$. Then $M \tensor_{\Z_\solid} \Z_\solid[S] = \varinjlim_j (P_j \tensor_{\Z_\solid} \Z_\solid[S])$, so it is enough to show that each $P_j \tensor_{\Z_\solid} \Z_\solid[S]$ lies in $\D_{\solid,\le1}(\Z)$. Since the $t$-structures on $\D_\solid(\Z)^\omega$ and on $\mathcal C^\omega$ differ only by an error of $1$ (via $j_!$), it is enough to show that $j_!(P_j \tensor_{\Z_\solid} \Z_\solid[S])$ lies in $\D_{\le0}(\Z)$. Taking a resolution of $P_j'$ in terms of objects of the form $\prod_J \R/\Z$, $j_!(P_j \tensor_{\Z_\solid} \Z_\solid[S])$ is computed by the same resolution but where we replace $\prod_J \R/\Z$ by $\prod_{I\cprod J} \R/\Z = \prod_I \prod_J \R/\Z$ (see <cit.>), i.e. $j_!(P_j \tensor_{\Z_\solid} \Z_\solid[S]) = \prod_I P'_j$ and this clearly lies in $\D_{\le0}(\Z)$.

Let $A^+ \to A$ be a map of classical rings such that $A^+$ is of finite type over $\Z$. Then for every profinite set $S$, the object $(A, A^+)_\solid[S] \in \D_\solid(A, A^+)$ has finite flat dimension, i.e. the functor

\begin{align*}
- \tensor_{(A, A^+)_\solid} (A, A^+)_\solid[S]\colon \D_\solid(A, A^+) \to \D_\solid(A, A^+)
\end{align*}

maps $\D_{\le0}$ to $\D_{\le n}$ for some $n \ge 0$. In fact, if there is a surjective map $\Z[T_1, \dots, T_m] \surjto A^+$ then one can chose $n = m + 1$.

By <ref> we have $(A, A^+)_\solid = A_{A^+_\solid/}$ which implies that $- \tensor_{(A, A^+)_\solid} (A, A^+)_\solid[S] = - \tensor_{A^+_\solid} A^+_\solid[S]$. Thus we can assume $A = A^+$ from now on. Choose a surjection $\Z[T_1, \dots, Z_m] \surjto A$. Then $A_\solid = (A, \Z[T_1, \dots, T_m])_\solid$, so by the same argument as before we can reduce to the case $A = \Z[T_1, \dots, T_m]$. Using <ref> we are thus reduced to the following claim: Let $B$ be a classical finite-type $\Z$-algebra such that $B_\solid[S] \in \D_\solid(B)$ has flat dimension $\le d$; then $B[T]_\solid[S] \in \D_\solid(B[T])$ has flat dimension $\le d + 1$. To prove this claim, we make use of the fully faithful functor $j_!\colon \D_\solid(B[T]) \injto \D_\solid(B[T], B)$ defined in the proof of <ref>, which is the left adjoint of $j^* = - \tensor_{(B[T], B)_\solid} B_\solid$. Using the projection formula for $j_!$ (which is also part of the proof of <ref>) we get, for every $M \in \D_\solid(B[T])$,

\begin{align*}
& M \tensor_{B[T]_\solid} B[T]_\solid[S] = j^*j_! (M \tensor_{B[T]_\solid} B[T]_\solid[S]) = j^* (M \tensor_{(B[T], B)_\solid} j_! B[T]_\solid[S]).
\end{align*}

We note:

* The functor $j^*\colon \D_\solid(B[T], B) \to \D_\solid(B[T])$ has Tor dimension $\le 1$. This follows directly from the adjunction of $j^*$ and $j_!$ using that $j_! B[T]$ is concentrated in homological degrees $\ge -1$ (cf. <cit.>).

* The object $j_! B[T]_\solid[S] \in \D_\solid(B[T], B)$ has flat dimension $\le d$, i.e. tensoring with it maps $\D_{\le0}$ to $\D_{\le d}$. Namely, using the projection formula for $j_!$ we compute

\begin{align*}
j_! B[T]_\solid[S] &= (B[T], B)_\solid[S] \tensor_{(B[T], B)_\solid} j_! B\\
&= (B[T], B)_\solid[S] \tensor_{(B[T], B)_\solid} B_\infty/B[-1],
\end{align*}

where $B_\infty = B((T^{-1}))$. We therefore have a cofiber sequence

\begin{align*}
j_! B[T]_\solid[S] \to (B[T], B)_\solid[S] \to (B[T], B)_\solid[S] \tensor B_\infty
\end{align*}

in $\D_\solid(B[T], B)$, which reduces the claimed flat dimension of $j_! B[T]_\solid[S]$ to showing that $(B[T], B)_\solid[S]$ has flat dimension $\le d$ and $(B[T], B)_\solid[S] \tensor B_\infty$ has flat dimension $\le d + 1$. The former follows from our assumption on $B$ and the latter follows from the former using the two-term resolution of $B_\infty$ in <cit.>.

Combining the above two observations with the formula for $M \tensor_{B[T]_\solid} B[T]_\solid[S]$ beforehand we deduce that $B[T]_\solid[S]$ has flat dimension $\le d + 1$, as desired.

§.§ Descent on Schemes

In the previous subsection we defined the $\infty$-category of schemes and more generally of discrete adic spaces. We will now prove various descent results on schemes, using the machinery developed on general analytic spaces. For simplicity we state all results only in the non-almost world, but one easily deduces very similar statements for discrete adic spaces over any almost setup $(V,\mm)$ by almost localization (see <ref>).

In <ref> we defined a general notion of descendable morphisms for analytic spaces. This in particular applies to discrete adic spaces (over any almost setup), so that we have a powerful theory of descent at hand. The only thing that is lacking so far is a good amount of examples of descendable morphisms. To provide such examples, our first goal is to carry over similar descent results from the literature to our setting. Namely, given a morphism $f\colon Y \to X$ of schemes, Mathew defines a descendability condition for $f$ in <cit.>, which is refined in <cit.> including a notion of index of descendability analogous to our version. The main difference to our notion of descendability is that Mathew's version caters only to discrete quasicoherent sheaves (and in particular is a weaker condition in general). Let us first recall how discrete quasicoherent sheaves fit in our picture.

To every discrete adic space $X$ we associated a symmetric monoidal $\infty$-category $\D_\solid(X)$ of solid quasicoherent sheaves on $X$ by gluing the $\infty$-categories $\D_\solid(A, A^+)$ on affine pieces $\Spa(A, A^+)$. For every solid cutoff cardinal $\kappa$ (see <ref>) we get a full subcategory $\D_\solid(X)_\kappa \subset \D_\solid(X)$ stable under the symmetric monoidal structure and small colimits. In particular we obtain the $\infty$-category

\begin{align*}
\D(X)_\omega := \D_\solid(X)_\omega
\end{align*}

of discrete quasicoherent sheaves on $X$. If $X$ is a classical scheme then $\D(X)_\omega$ coincides with the derived $\infty$-category of classical quasicoherent sheaves on $X$ (because this is true on affine schemes and thus follows in the general case by gluing).

By <ref> we also have $\D(X)_\omega = \D(X)^\lnuc$, the $\infty$-category of (locally) nuclear quasicoherent sheaves on $X$. We will stick with the intuition of discrete rather than (locally) nuclear objects for now, but on more general spaces $X$ (where the structure sheaves have a non-trivial topology), $\D(X)^\lnuc$ is the correct replacement for $\D(X)_\omega$.