\documentclass[12pt]{amsart} \usepackage[margin=1.05in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools,amscd} \usepackage[hidelinks]{hyperref} \usepackage{microtype} \usepackage{enumitem} \usepackage{array} \usepackage{needspace,etoolbox} \setlength{\emergencystretch}{1.5em} \allowdisplaybreaks[2] \setlist[enumerate]{topsep=3pt,itemsep=2pt,parsep=0pt,partopsep=0pt} \AtBeginDocument{% \setlength{\abovedisplayskip}{6pt plus 2pt minus 1pt}% \setlength{\belowdisplayskip}{6pt plus 2pt minus 1pt}% \setlength{\abovedisplayshortskip}{3pt plus 1pt}% \setlength{\belowdisplayshortskip}{4pt plus 1pt minus 1pt}% \setlength{\jot}{2pt}% } % Keep the AMS title and headings, with less space around the front matter. \makeatletter \patchcmd{\@maketitle}{\global\topskip42\p@\relax} {\global\topskip20\p@\relax}{}{} \patchcmd{\@maketitle}{\dimen@34\p@} {\dimen@22\p@}{}{} \patchcmd{\@setauthors}{\@topsep30\p@\relax} {\@topsep22\p@\relax}{}{} \patchcmd{\@starttoc}{\addvspace{32\p@\@plus14\p@}} {\addvspace{12\p@\@plus3\p@}}{}{} \makeatother \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem*{introclassical}{Theorem~\ref{thm:classical-main}} \newtheorem*{introstable}{Theorem~\ref{thm:stable-main}} \newtheorem*{introboundary}{Theorem~\ref{thm:boundary-pushforward}} \newcommand{\Dqc}{\mathrm D_{\mathrm{qc}}} \newcommand{\Dpc}{\mathrm D_{\mathrm{pc}}} \newcommand{\Dpcbox}{\mathsf D_{\mathrm{pc}}^{\square}} \newcommand{\Dbcoh}{\mathrm D^b_{\mathrm{coh}}} \newcommand{\Cbox}{\mathcal C_{\square}} \newcommand{\Dsquare}{\mathsf D_{\square}} \newcommand{\Dcond}{\mathrm D_{\mathrm{cond}}} \newcommand{\Perf}{\operatorname{Perf}} \newcommand{\Coh}{\operatorname{Coh}} \newcommand{\PCoh}{\operatorname{PCoh}} \newcommand{\Vect}{\operatorname{Vect}} \newcommand{\Map}{\operatorname{Map}} \newcommand{\colim}{\operatorname*{colim}} \newcommand{\liminv}{\operatorname*{lim}} \newcommand{\An}{\mathrm{an}} \newcommand{\Alg}{\mathrm{alg}} \newcommand{\ad}{\mathrm{ad}} \newcommand{\AnSpec}{\operatorname{AnSpec}} \newcommand{\Spa}{\operatorname{Spa}} \newcommand{\Spec}{\operatorname{Spec}} \newcommand{\cO}{\mathcal O} \newcommand{\cX}{\mathcal X} \newcommand{\cY}{\mathcal Y} \newcommand{\cZ}{\mathcal Z} \newcommand{\cU}{\mathcal U} \newcommand{\cV}{\mathcal V} \newcommand{\cA}{\mathcal A} \newcommand{\cI}{\mathcal I} \newcommand{\cL}{\mathcal L} \newcommand{\Cond}{\operatorname{Cond}} \newcommand{\RHom}{\operatorname{RHom}} \title[Bootstrapping GAGA Theorems]{Bootstrapping GAGA Theorems} \hypersetup{pdftitle={Bootstrapping GAGA},pdfauthor={Aniruddh Venkatesan}} \author{Aniruddh Venkatesan} \email{aniruddhv@berkeley.edu} \date{} \begin{document} \begin{abstract} Let $X$ be a noetherian scheme with affine diagonal, and let $\lambda:\cX\to X$ be a morphism of locally ringed spaces from a quasi-compact quasi-separated locally strongly noetherian analytic adic space. We prove that an equivalence on coherent sheaves, together with comparison of their cohomology, extends to every proper $X$-scheme. As an application, we obtain GAGA for proper schemes over the Fargues--Fontaine curve in equal and mixed characteristic. \end{abstract} \maketitle \raggedbottom \setcounter{tocdepth}{1} \begingroup \small \setlength{\baselineskip}{12pt} \tableofcontents \endgroup \section{Introduction}\label{sec:introduction} We prove that, in the adic setting below, a GAGA comparison on a base extends to every proper scheme over it. Given an equivalence on coherent sheaves and a comparison of their cohomology, . For the schematic and adic Fargues--Fontaine curves, this gives GAGA in equal and mixed characteristic. \subsection{The relative problem}\label{subsec:GAGA-background}\label{subsec:setup} Let $X$ be a noetherian scheme with affine diagonal, let $\cX$ be a quasi-compact quasi-separated locally strongly noetherian analytic adic space, and let $\lambda:\cX\to X$ be a morphism of locally ringed spaces. Assume that pullback gives an exact equivalence \[ \lambda^*:\Coh(X)\xrightarrow{\sim}\Coh(\cX) \] and that the canonical maps \[ H^i(X,\mathcal F)\longrightarrow H^i(\cX,\lambda^*\mathcal F) \] are isomorphisms for every coherent $\mathcal F$ and every $i\geq0$. We keep these hypotheses throughout Sections~\ref{sec:base}--\ref{sec:classical}. For a proper $X$-scheme $Y$, write $c_Y:Y^{\ad}_{\cX}\to Y$ for its relative Huber analytification. Over an affinoid $V=\Spa(R,R^+)\subset\cX$ mapping into an affine open of $X$, this is the analytification of $Y\times_X\Spec R$ over $V$. The local spaces agree under rational base change and hence glue \cite[Proposition~3.8]{Huber1994}. \begin{introclassical} For every proper $f:Y\to X$, analytification induces equivalences \begin{align*} \Coh(Y)&\xrightarrow{\sim}\Coh(Y^{\ad}_{\cX}),\\ \Dbcoh(Y)&\xrightarrow{\sim}\Dbcoh(Y^{\ad}_{\cX}),\\ \Perf(Y)&\xrightarrow{\sim}\Perf(Y^{\ad}_{\cX}),\\ \Vect(Y)&\xrightarrow{\sim}\Vect(Y^{\ad}_{\cX}). \end{align*} For $\mathcal F\in\Coh(Y)$ and $i\geq0$, the canonical maps are isomorphisms: \begin{align*} (R^if_*\mathcal F)^{\ad}&\xrightarrow{\sim}R^if_*^{\ad}(\mathcal F^{\ad}),\\ H^i(Y,\mathcal F)&\xrightarrow{\sim}H^i(Y^{\ad}_{\cX},\mathcal F^{\ad}). \end{align*} \end{introclassical} \subsection{The solid comparison}\label{subsec:conventions}\label{subsec:strategy} Write $\Dqc(T)$ for the category of quasicoherent complexes and $\Dpc(T)$ for its pseudocoherent subcategory; on a noetherian scheme, $\Dpc(T)=\mathrm D^-_{\mathrm{coh}}(T)$. On the analytic side, let $\Cbox(\cZ)$ denote solid quasicoherent complexes, and let $\Dpcbox(\cZ)$ be the subcategory defined by nuclearity and pseudocompactness on bounded charts. On $\Spa(R,R^+)$ this subcategory is $\Cond_R(\Dpc(R))$ \cite[Theorem~5.50]{Andreychev}. The definitions and comparison functors are recalled in Appendix~\ref{app:analytic-foundations}. A separated finitely presented $T/X$ has an algebraic realization $T^{\Alg}_{\cX}$ and an analytic locus $T^{\An}_{\cX}$. They are analytic stacks and coincide for proper $T/X$ \cite[A.9--A.13]{LiHuerta}. The comparison on their solid quasicoherent categories gives the stronger statement below. \begin{introstable} For proper $f:Y\to X$, the comparison \[ \alpha_Y:\Dqc(Y)\longrightarrow\Cbox(Y^{\An}_{\cX}) \] is fully faithful, colimit-preserving, and strong symmetric monoidal. It takes perfect complexes to compact objects and induces a symmetric monoidal equivalence \[ \Dpc(Y)\xrightarrow{\sim}\Dpcbox(Y^{\An}_{\cX}). \] For every $M\in\Dqc(Y)$, the canonical comparisons are equivalences: \begin{align*} \alpha_X(\mathbf Rf_*M)&\xrightarrow{\sim} \mathbf Rf_*^{\An}\alpha_Y(M),\\ \mathbf R\Gamma(Y,M)&\xrightarrow{\sim} \mathbf R\Gamma_\square(Y^{\An}_{\cX},\alpha_YM). \end{align*} \end{introstable} \subsection{The boundary calculation}\label{subsec:intro-boundary} The main calculation is a relative localization formula. Let $f:Y\to X$ be proper, let $j:U\hookrightarrow Y$ be quasi-compact open, and put $q=f\circ j$. Blowing up an ideal cutting out $Y\setminus U$ gives $p:\widetilde Y\to Y$, an isomorphism over $U$, with effective Cartier boundary $D$. Put $\widetilde f=f\circ p$. For $C\in\Cbox(Y^{\An}_{\cX})$, write $\widetilde C=(p^{\An})^*C$ and set \[ E_n(C)=\mathbf R\widetilde f_*^{\An} \bigl(\widetilde C\otimes\cO_{\widetilde Y}(nD)\bigr). \] The canonical section of $\cO(D)$ gives the transition maps. \begin{introboundary} With this notation, there is a canonical equivalence, natural in $C$, \[ \mathbf Rq_*^{\Alg}j^{\Alg,*}C \simeq \colim_{n\geq0}\mathbf R\widetilde f_*^{\An} \bigl(\widetilde C\otimes\cO_{\widetilde Y}(nD)\bigr). \] If $C$ is pseudocoherent, the sequence $E_\bullet(C)$ lifts functorially to a sequence $M_\bullet(C)$ in $\Dpc(X)$. Its colimit satisfies \[ \alpha_X\!\left(\colim_nM_n(C)\right) \simeq\mathbf Rq_*^{\Alg}j^{\Alg,*}C \] and, with this comparison, is canonically independent of the blowup. \end{introboundary} Locally, the formula is the identity $A[t^{-1}]=\colim(A\xrightarrow{t}A\xrightarrow{t}\cdots)$. For pseudocoherent $C$, each proper direct image $E_n(C)$ is pseudocoherent and can therefore be algebraized on $X$. We take the colimit of the resulting sequence in $\Dqc(X)$. When $q$ is affine, full faithfulness lifts the $\alpha_X(q_*\cO_U)$-action to this colimit and so produces a complex on $U$. Its pseudocoherence follows by faithfully flat descent. \subsection{Outline of the proof}\label{subsec:intro-argument} The first step is to extend the comparison on the base to complexes. Exact coherent pullback makes the chart maps flat, so scalar extension followed by condensification defines $\alpha_X$. Coherent-cohomology comparison gives full faithfulness on perfect complexes. Their analytic images are compact by finite open-cover descent, which allows us to pass to filtered colimits and obtain full faithfulness on $\Dqc(X)$. We then algebraize bounded coherent complexes by their truncation triangles. For a pseudocoherent complex, the bounded truncations form a tower whose cohomology is eventually constant in each degree; taking its inverse limit on ordinary affine module categories gives the algebraization. Over the general base, the module description gives full faithfulness on affine schemes, and Zariski descent extends it to relative schemes. For a proper scheme, the boundary formula supplies algebraic preimages on affine opens. A finite faithfully flat family detects pseudocoherence, while full faithfulness recovers the gluing data. Applying the comparison to the algebraic counit defines the proper-direct-image map. We prove that it is an equivalence first on pseudocoherent complexes and then on all quasicoherent complexes by passage to colimits. Finally, sheafification of the ordinary affinoid representatives identifies the coherent and bounded categories with their Huber counterparts. For the application, let $E$ be a nonarchimedean local field with residue field $\mathbf F_q$, and let $F$ be a complete perfectoid field of characteristic $p$ containing $\mathbf F_q$. The schematic and adic curves $X^{\mathrm{sch}}_{F,E}$ and $X^{\ad}_{F,E}$ satisfy the base hypotheses \cite[Theorems~4.7.3--4.7.5]{KedlayaLiu}. Section~\ref{sec:ff} recalls their constructions and verifies these hypotheses, giving Theorem~\ref{thm:ff-main}. \subsection{Comparison with the work Clausen--Scholze}\label{subsec:proof-comparison} Our proof follows the boundary-localization argument of \cite[Theorem~13.10]{ClausenScholze}, in the Tate-base form of \cite[A.13 and A.16--A.18]{LiHuerta}. There the complex-field case begins with the comparison over $\Spec\mathbf C$. Here the base may be nonaffine, and the direct images of the boundary twists must be algebraized as complexes of sheaves on $X$. The comparison on the base supplies these complexes, their transition maps, and the algebra action on their colimit. We use the affinoid results of \cite{Andreychev}; other GAGA theorems and criteria appear in \cite{Serre,FujiwaraKato,Hall,Wang}. Section~\ref{sec:base} proves the comparison on the base. Sections~\ref{sec:relative}--\ref{sec:grauert} construct the relative realizations and establish proper direct image and localization, and Section~\ref{sec:stable-proof} proves the bootstrapping theorem. Section~\ref{sec:classical} recovers ordinary GAGA and compares proper direct images between proper $X$-schemes in Corollary~\ref{cor:proper-relative-pushforward}. The Fargues--Fontaine application occupies Section~\ref{sec:ff}. The appendices recall the analytic categories and give the descent arguments. \subsection*{Notation and conventions}\label{subsec:notation} Derived categories are stable $\infty$-categories, with cohomological indexing $H^i(K[n])=H^{i+n}(K)$. A subscript $\mathrm{conn}$ means cohomologically nonpositive. We write $\Map$ for mapping spectra in stable categories and mapping spaces otherwise. Tensor products are derived, except on coherent hearts; in complete-module categories they include analytic completion. A finite module means a finitely generated module. Over a coherent ring, coherent cohomology means finitely presented cohomology modules. For a topological ring $R$, $R_{\mathrm{disc}}$ and $\underline R$ denote its discrete and topological condensed rings. We write $\Dcond(B)$ for the derived category of condensed $B_{\mathrm{disc}}$-modules, $\mathsf D(\mathcal A)$ for complete modules over an analytic ring, and $\Dsquare(R,R^+)=\mathsf D((R,R^+)^\square)$. Condensification is $\Cond_R(M)=\underline R\otimes_{R_{\mathrm{disc}}}^{\mathbf L}M_{\mathrm{disc}}$. An affinoid representative of $C$ on $V=\Spa(R,R^+)$ is a pseudocoherent $R$-complex $M$ with $\Cond_R(M)\simeq C|_V$. The symbols $\Alg$, $\An$, and $\ad$ distinguish the two analytic-stack realizations from ordinary Huber analytification. We use $\alpha_T$ for the comparison on algebraic realizations and $\alpha_T^{\An}$ for its restriction to the analytic locus; for proper $T$ they agree. Algebraization includes a specified equivalence with the analytic image. For an adic space $Z$, $\Cbox(Z)$ abbreviates $\Cbox(Z^\square)$. The notation $\operatorname{Cart}_V\mathcal C_V$ means families with compatible pullback equivalences. In Section~\ref{sec:ff}, standard affines refer to the Proj presentation of the curve. We write $K(nD)=K\otimes\cO(nD)$. \subsection{Acknowledgements}\label{subsec:acknowledgements} This project was conceived in the summer of 2025 while the author was a Summer Undergraduate Research Fellow at UC Berkeley. We thank the organizers and other participants of the program for helping to foster a productive research environment. We thank Zeyu Liu for mentoring us on the initial portions of this work. We thank Andrew Cheng, Sug Woo Shin, and Avi Zeff for discussions regarding this work. \Needspace{5\baselineskip} \subsection{AI Usage Statement}\label{subsec:ai-usage} AI agents (specifically GPT 6 Pro) were used in the preparation of this manuscript for brainstorming, proofreading, and consistency-checking. In particular, the AI suggested that a separatedness hypothesis in an earlier version of the main result could be weakened to merely requiring an affine diagonal, and suggested references useful for the writing of Appendix \ref{sec:preliminaries}. The author takes full responsibility for the completeness and correctness of all of its content, mathematical and otherwise. \section{The comparison on the base}\label{sec:base} Throughout this section, $\lambda:\cX\to X$ is as in Section~\ref{subsec:setup}. We construct $\alpha_X$ on quasicoherent complexes and prove full faithfulness. The coherent equivalence will then allow us to recover pseudocoherent complexes from their bounded truncations. We use affine opens of $X$ and their intersections to describe scalar extension. These intersections are affine \cite[Tag~0BDX]{Stacks}, and every morphism from an affine scheme to $X$ is affine \cite[Lemma~29.11.12]{Stacks}. The same applies to a separated $X$-scheme $T$, since its diagonal factors as \[ T\longrightarrow T\times_XT\longrightarrow T\times T. \] Here the first map is a closed immersion and the second is a base change of $\Delta_X$. Thus $T$ has affine diagonal, and its affine opens also have affine intersections. \subsection{Affinoid charts and scalar extension}\label{subsec:base-GAGA}\label{subsec:base-comparison} Let $\mathcal B=\operatorname{RatAff}_X(\cX)$ be the basis of strongly noetherian affinoid opens $V\subset\lambda^{-1}(W)$, where $W\subset X$ is affine. Its restriction maps are rational inclusions. We use finite covers from this basis, refining intersections by common rational subdomains. \begin{lemma}\label{lem:base-flat} Let $W=\Spec A\subset X$ be affine and let $V=\Spa(R,R^+)\subset\lambda^{-1}(W)$ be a strongly noetherian affinoid open. Then $A\to R$ is flat, and $\lambda:\cX\to X$ is flat on local rings. \end{lemma} \begin{proof} By the ideal criterion for flatness, it suffices to show that \[ I\otimes_A R\longrightarrow R \] is injective for every finitely generated ideal $I\subset A$. The open immersion $W\hookrightarrow X$ is quasi-compact, so the coherent ideal $\widetilde I\subset\cO_W$ extends to a coherent ideal $\mathcal J\subset\cO_X$ with $\mathcal J|_W=\widetilde I$ \cite[Section~28.23, especially Lemma~28.23.2]{Stacks}. Exactness of $\lambda^*$ preserves the injection $\mathcal J\hookrightarrow\cO_X$. On $V$ we therefore have \[ (\lambda^*\mathcal J)|_V\longrightarrow\cO_V. \] Pull back a finite presentation of $I$ to $V$. The same matrix over $R$ has cokernel $I\otimes_A R$, so the affinoid coherent-sheaf theorem \cite[Theorem~4.2]{Zavyalov} identifies the induced map on sections with $I\otimes_A R\to R$. Taking sections preserves the injection above, and the ideal criterion gives flatness. For $x\in V$ with $\mathfrak p=\lambda(x)$, the map $R\to\cO_{V,x}$ is flat by \cite[Lemma~6.7]{Zavyalov}. Its composite with $A\to R$ remains flat. Localizing at $\mathfrak p$ gives the asserted flatness on local rings. \end{proof} Let $M\in\Dqc(X)$. For an affine open $W=\Spec A$, put \[ M_W=\mathbf R\Gamma(W,M|_W)\in\mathrm D(A). \] On $V=\Spa(R,R^+)\subset\lambda^{-1}(W)$, define \[ \alpha_{W,V}(M)=\Cond_R(R\otimes_A^{\mathbf L}M_W). \] Thus the local comparison extends scalars from $A$ to $R$ and then condensifies. On a complex of finite free modules it retains the differential matrices, now with entries in $\underline R$. Suppose $V$ maps into two affine opens $W_i=\Spec A_i$, and write their affine intersection as $W_{12}=\Spec A_{12}$. Quasicoherent restriction gives \[ A_{12}\otimes_{A_i}^{\mathbf L}M_{W_i}\simeq M_{W_{12}}. \] Since each $A_i\to R$ factors through $A_{12}$, tensoring with $R$ identifies both local definitions with \[ \Cond_R(R\otimes_{A_{12}}^{\mathbf L}M_{W_{12}}). \] Next let $V'=\Spa(R',R'^+)\subset V$ be rational. Condensification commutes with scalar extension, so \begin{align*} \alpha_{W,V}(M)|_{V'} &\simeq\Cond_{R'}\bigl(R'\otimes_R^{\mathbf L} (R\otimes_A^{\mathbf L}M_W)\bigr)\\ &\simeq\Cond_{R'}(R'\otimes_A^{\mathbf L}M_W) =\alpha_{W,V'}(M). \end{align*} For successive rational inclusions, the comparisons are the associativity equivalences of the same tensor product. The comparisons obtained by changing the schematic affine have the same compatibility. Thus the local objects form a cartesian section, which descends to $\alpha_X(M)$. The construction is functorial in $M$. Its tensor-product and unit comparisons commute with restriction, so descent makes $\alpha_X$ strong symmetric monoidal. On each chart it is exact and preserves filtered colimits \cite[Theorem~5.9 and Lemma~5.10]{Andreychev}. It therefore preserves arbitrary coproducts, expressed as filtered colimits of finite partial sums, and hence all small colimits. Since colimits of cartesian sections are computed on charts, $\alpha_X$ has the same properties. \subsection{Finite descent and full faithfulness}\label{subsec:base-full-faithfulness} For perfect complexes, duality expresses the mapping spectrum as global sections of a tensor product. We first prove that these sections commute with filtered colimits; this will allow us to pass from perfect to arbitrary quasicoherent complexes. \begin{lemma}\label{lem:compact-unit-base} For every quasi-compact quasi-separated analytic adic space $\cZ$, the tensor unit of $\Cbox(\cZ)$ is compact. Thus $\mathbf R\Gamma_\square(\cZ,-)$ preserves filtered colimits. \end{lemma} \begin{proof} Choose a finite affinoid cover $(V_i)$ of $\cZ$, and put $V_I=\bigcap_{i\in I}V_i$ for nonempty $I$. Each $V_I$ is quasi-compact and admits a finite cover by rational subdomains of an affinoid containing it. The intersections in this cover are again rational affinoids. On an analytic affine, the unit is the measure module for the one-point set and is compact \cite[Theorem~3.28 and Proposition~2.11]{Andreychev}. Proposition~\ref{prop:finite-descent} therefore shows that $\mathbf R\Gamma_\square(V_I,-)$ preserves filtered colimits. Let $P$ be the poset of nonempty subsets of the covering indices. Finite descent gives \[ \mathbf R\Gamma_\square(\cZ,K) \simeq\lim_{I\in P}\mathbf R\Gamma_\square(V_I,K|_{V_I}) \] by Lemma~\ref{lem:finite-Cech-descent}. The nerve of $P$ has finitely many nondegenerate simplices, so the limit is finite. Restrictions preserve colimits, and filtered colimits of spectra commute with finite homotopy limits. It follows that $\mathbf R\Gamma_\square(\cZ,-)$ preserves filtered colimits, or equivalently, that the unit is compact. \end{proof} Duality reduces the comparison of maps between perfect complexes to the following comparison of global sections. \begin{lemma}\label{lem:solid-ordinary-cohomology} For every bounded coherent complex $K$ on $X$, the natural maps \[ \mathbf R\Gamma_\square(\cX,\alpha_XK) \xrightarrow{\sim} \mathbf R\Gamma_{\mathrm{adic}}(\cX,\lambda^*K) \xleftarrow{\sim} \mathbf R\Gamma(X,K) \] are equivalences. \end{lemma} \begin{proof} We extend the coherent-cohomology comparison to $K$ by applying it to successive truncation triangles. Choose $a\leq b$ such that $\mathcal H^i(K)=0$ outside $[a,b]$, and put $K_r=\tau_{\leq r}K$. The successive triangles are \[ K_{r-1}\longrightarrow K_r\longrightarrow \mathcal H^r(K)[-r]\longrightarrow K_{r-1}[1]. \] By Lemma~\ref{lem:base-flat}, pullback is flat and preserves these triangles. Applying the two section functors gives a comparison of triangles whose third term is the coherent-sheaf comparison of Section~\ref{subsec:setup}. Beginning with $K_{a-1}=0$, induction on $r$ gives the comparison for $K_b=K$. To compare solid and ordinary analytic sections, take $V=\Spa(R,R^+)\subset\lambda^{-1}(W)$ with $W=\Spec A$, and set \[ K_R=R\otimes_A^{\mathbf L}\mathbf R\Gamma(W,K|_W). \] Flatness gives finite $R$-cohomology in $[a,b]$. The restriction of $\alpha_XK$ is $\Cond_R(K_R)$, so full faithfulness of condensification gives \begin{align*} \mathbf R\Gamma_\square(V,\alpha_XK) &\simeq\Map_{\Dsquare(R,R^+)}(\Cond_RR,\Cond_RK_R)\\ &\simeq K_R \end{align*} \cite[Theorem~5.9]{Andreychev}. Ordinary analytic sections give the same complex: for a finite module this is coherent affinoid acyclicity, and the bounded case follows from the truncation triangles above. Both local identifications commute with rational restriction, since both are induced by scalar extension of $K_R$. Choose a finite affinoid cover subordinate to affine opens of $X$, and cover each intersection by finitely many rational subdomains of one of its members. Taking limits of the local comparisons, first on these intersections and then on $\cX$, gives the assertion. Solid sections are computed by Lemma~\ref{lem:finite-Cech-descent}, and ordinary sections by the finite alternating \v Cech complex \cite[Tag~08BZ]{Stacks}. \end{proof} Combining the section comparison with compactness gives full faithfulness on quasicoherent complexes. \begin{theorem}\label{thm:enhanced-base} The functor \(\alpha_X:\Dqc(X)\longrightarrow\Cbox(\cX)\) is fully faithful. It sends perfect complexes to compact objects, and its essential image is closed under arbitrary colimits. The same conclusions hold if the coherent-cohomology hypothesis is replaced by the canonical comparison \[ \mathbf R\Gamma(X,P)\xrightarrow{\sim} \mathbf R\Gamma(\cX,\lambda^*P) \] for every $P\in\Perf(X)$. \end{theorem} \begin{proof} Let $P,Q$ be perfect complexes. Strong monoidality identifies $\alpha_X(P^\vee)$ with the dual of $\alpha_XP$, so \begin{align*} \Map_{\Cbox(\cX)}(\alpha_XP,\alpha_XQ) &\simeq\mathbf R\Gamma_\square \bigl(\cX,\alpha_X(P^\vee\otimes Q)\bigr)\\ &\simeq\mathbf R\Gamma(X,P^\vee\otimes Q)\\ &\simeq\Map_{\Dqc(X)}(P,Q). \end{align*} The middle equivalence is Lemma~\ref{lem:solid-ordinary-cohomology} applied to $P^\vee\otimes Q$. Hence $\alpha_X$ is fully faithful on perfect complexes. For a filtered diagram $(N_j)$ in $\Cbox(\cX)$, duality gives \[ \Map(\alpha_XP,N_j) \simeq\mathbf R\Gamma_\square (\cX,\alpha_X(P^\vee)\otimes N_j). \] Tensor product preserves colimits, and global sections preserve filtered colimits by Lemma~\ref{lem:compact-unit-base}. Thus \[ \colim_j\Map(\alpha_XP,N_j) \simeq\Map(\alpha_XP,\colim_jN_j). \] This proves compactness of $\alpha_XP$. By \cite[Theorem~36.15.3, Tag~09IS, and Proposition~36.17.1, Tag~09M0]{Stacks}, $\Dqc(X)\simeq\operatorname{Ind}(\Perf(X))$. Write $M\simeq\colim_iP_i$ and $N\simeq\colim_jQ_j$ as filtered colimits of perfect complexes. Then \begin{align*} \Map(\alpha_XM,\alpha_XN) &\simeq\lim_i\colim_j\Map(\alpha_XP_i,\alpha_XQ_j)\\ &\simeq\lim_i\colim_j\Map(P_i,Q_j)\\ &\simeq\Map(M,N). \end{align*} The first equivalence uses colimit preservation and compactness of the perfect images; the second is the comparison just proved. The last uses compactness of perfect complexes on $X$. This proves full faithfulness on $\Dqc(X)$. By full faithfulness and Lemma~\ref{lem:functor-descent}, every diagram in the essential image lifts, with its maps and homotopies, to a diagram $(M_i)$ on $X$. Colimit preservation gives \[ \colim_i\alpha_X(M_i)\simeq\alpha_X(\colim_iM_i). \] Thus the essential image is closed under colimits. Now suppose that cohomology comparison is given for perfect complexes. Apply it to $P^\vee\otimes Q$, and use the affinoid section calculation in Lemma~\ref{lem:solid-ordinary-cohomology} to compare solid and ordinary analytic sections. This gives the same mapping-spectrum calculation for $P,Q$, and compactness again extends full faithfulness to $\Dqc(X)$. Mapping from the structure sheaf then gives, for every coherent $\mathcal F$, \[ \mathbf R\Gamma(X,\mathcal F) \simeq\Map_{\cX}(\cO_{\cX},\alpha_X\mathcal F) \simeq\mathbf R\Gamma(\cX,\lambda^*\mathcal F). \] The second equivalence is the local section comparison. Both maps are induced by pullback, so this identifies the canonical coherent-cohomology map. \end{proof} \subsection{Pseudocoherent complexes} We turn to essential surjectivity. The bounded coherent truncations of an analytic pseudocoherent complex can be algebraized one at a time. To recover the whole complex, we first express analytic descent in terms of ordinary complexes on affinoids. Put $\mathcal B=\operatorname{RatAff}_X(\cX)$ and $R_V=\Gamma(V,\cO_V)$. Rational restriction gives the diagram \begin{align*} \mathcal B^{\mathrm{op}}&\longrightarrow\operatorname{Cat}_\infty,\\ V&\longmapsto\Dpc(R_V), \end{align*} with transition functor $M\mapsto R_{V'}\otimes_{R_V}^{\mathbf L}M$ for $V'\subset V$. Write $\operatorname{Cart}_{V\in\mathcal B}\Dpc(R_V)$ for its limit. A cartesian section consists of complexes $M_V\in\Dpc(R_V)$ and equivalences \[ \theta_{V',V}:R_{V'}\otimes_{R_V}^{\mathbf L}M_V \xrightarrow{\sim}M_{V'}. \] For $V''\subset V'\subset V$, these satisfy $\theta_{V'',V}\simeq\theta_{V'',V'}\circ(1\otimes\theta_{V',V})$ under tensor associativity, together with the unit and higher compatibility homotopies. A morphism is a family $M_V\to N_V$ respecting these data. Condensification turns the scalar-extension equivalences into analytic gluing data. On $V=\Spa(R,R^+)$, we identify $\Dpc(R)$ with the nuclear and pseudocompact subcategory of $\Dsquare(R,R^+)$ through condensification. \begin{proposition}\label{prop:base-pc-descent} Restriction induces an equivalence \[ \Dpcbox(\cX)\xrightarrow{\sim} \operatorname{Cart}_{V\in\operatorname{RatAff}_X(\cX)}\Dpc(R_V), \] independent of the rational basis and bounded-chart presentations. \end{proposition} \begin{proof} On each $V=\Spa(R,R^+)$, the affinoid comparison gives \[ \Cond_R:\Dpc(R)\xrightarrow{\sim}\Dpcbox(V) \subset\Dsquare(R,R^+) \] \cite[Theorem~5.50]{Andreychev} \cite[A.15]{LiHuerta}. These equivalences also identify the transition functors. If $u:V'\hookrightarrow V$ is rational, then \[ u^*\Cond_R(M) \simeq\Cond_{R'}(R'\otimes_R^{\mathbf L}M) \] by \cite[Lemma~5.10]{Andreychev}. The scalar extension of a bounded-above finite-projective resolution is again such a resolution, so the object on the right is pseudocoherent. Tensor associativity makes these identifications compatible with successive restrictions. Given a family $(M_V)$, condensify its terms. The scalar-extension equivalences become pullback equivalences, so Proposition~\ref{prop:analytic-stack-Cbox-descent} glues them to $C\in\Cbox(\cX)$ with $C|_V\simeq\Cond_R(M_V)$. These restrictions show that $C$ belongs to $\Dpcbox(\cX)$. Conversely, choose ordinary representatives for the restrictions of $C\in\Dpcbox(\cX)$. Full faithfulness of condensification lifts the restriction equivalences and their homotopies to a cartesian family. This inverts the preceding construction on objects; the limit of the local mapping-space equivalences gives the assertion on morphisms. Common rational refinements and \cite[Theorem~5.3]{Andreychev} identify the descent descriptions for different bases. On each affinoid, the subcategory is $\Cond_R(\Dpc(R))$, independently of the bounded-chart presentation. The global subcategory is therefore independent of both choices. \end{proof} We will use the following inverse-limit calculation. Recall that a ring is coherent if every finitely generated ideal is finitely presented. For such a ring $A$, the notation $\mathrm D^b_{\mathrm{coh}}(A)$ refers to complexes with finitely presented cohomology in a finite interval. \begin{lemma}\label{lem:postnikov-reconstruction} Let $A$ be a coherent ring and $b\in\mathbf Z$. Let $(K_n)_{n\geq0}$ be an inverse system with $K_n\in\mathrm D^b_{\mathrm{coh}}(A)^{[b-n,b]}$ whose transitions induce equivalences $\tau_{\geq b-n}K_{n+1}\simeq K_n$. Then $K=\mathbf R\!\lim_nK_n$ is pseudocoherent and $\tau_{\geq b-n}K\simeq K_n$. For every ring map $A\to A'$, the canonical map \[ A'\otimes_A^{\mathbf L}K \longrightarrow\mathbf R\!\lim_n(A'\otimes_A^{\mathbf L}K_n) \] is an equivalence. \end{lemma} \begin{proof} Exactness of products of ordinary $A$-modules gives the derived-limit exact sequence \[ 0\longrightarrow\lim\nolimits_n^1H^{j-1}(K_n) \longrightarrow H^j(K) \longrightarrow\lim_nH^j(K_n)\longrightarrow0 \] \cite[Tag~08U5]{Stacks}. For fixed $j$, the transition maps on $H^j(K_n)$ are isomorphisms once $b-n\leq j$. Both systems in this sequence are eventually constant, so the $\lim^1$ term vanishes and $H^j(K)\simeq H^j(K_n)$ for sufficiently large $n$. These modules are finitely presented and vanish for $j>b$. Since $A$ is coherent, this makes $K$ pseudocoherent \cite[Tag~0EWZ]{Stacks}. For fixed $n$, the projection to $K_n$ induces isomorphisms in degrees at least $b-n$, proving the truncation assertion. Put $F_n=\operatorname{fib}(K\to K_n)$. The truncation assertion gives \[ F_n\in\mathrm D^{\leq b-n-1}(A). \] Derived scalar extension along any ring map $A\to A'$ preserves upper cohomological bounds. Indeed, choose a bounded-above free resolution with the same upper bound and tensor it with $A'$ \cite[Section~15.60]{Stacks}. It follows that \[ A'\otimes_A^{\mathbf L}F_n\in\mathrm D^{\leq b-n-1}(A'). \] The long exact sequence of the tensored fiber triangle now shows that \[ H^j(A'\otimes_A^{\mathbf L}K) \longrightarrow H^j(A'\otimes_A^{\mathbf L}K_n) \] is an isomorphism once $b-n\leq j$. These maps respect the transitions, so the base-changed cohomology systems are eventually constant. The derived-limit exact sequence identifies their limit with $H^j(A'\otimes_A^{\mathbf L}K)$, proving that the canonical comparison is an equivalence. Naturality in the tower and compatibility with successive scalar extensions follow from the corresponding properties of the tower maps and tensor associativity. \end{proof} \begin{proposition}\label{prop:base-pc-GAGA} The ordinary affinoid $t$-structures descend to $\Dpcbox(\cX)$, and comparison induces a $t$-exact equivalence \[ \alpha_X:\Dpc(X)\xrightarrow{\sim}\Dpcbox(\cX). \] \end{proposition} \begin{proof} On each $V=\Spa(R,R^+)$, choose ordinary representatives $C_V\in\Dpc(R)$. Flat rational localization preserves their degree conditions and truncation triangles, which glue by Proposition~\ref{prop:base-pc-descent}. Orthogonality follows by taking limits of the contractible local mapping spaces from nonpositive objects to objects in degrees at least one. These constructions define the $t$-structure, whose heart is $\Coh(\cX)$ by the coherent affinoid theorem and sheaf descent. On charts, $\alpha_X$ is flat scalar extension followed by condensification, so it preserves pseudocoherence and truncations. It is fully faithful by Theorem~\ref{thm:enhanced-base} and induces the given equivalence on coherent hearts. We first prove essential surjectivity for bounded complexes. Let $C$ have cohomology in $[a,b]$. We argue by induction through the truncation triangles \[ \tau_{\leq r-1}C\longrightarrow\tau_{\leq r}C \longrightarrow\mathcal H^r(C)[-r] \longrightarrow\tau_{\leq r-1}C[1], \] starting with $\tau_{\leq a-1}C=0$. If the first term has been algebraized, then so has the third, by coherent GAGA. Full faithfulness lifts the connecting map between their preimages. Its fiber algebraizes $\tau_{\leq r}C$, and the long exact cohomology sequence makes this preimage bounded coherent. This proves the bounded equivalence. Since $\alpha_X$ commutes with cohomology and the coherent equivalence detects vanishing, the bounded equivalence also preserves and reflects cohomological bounds. For $C\in\Dpcbox(\cX)$, choose a common upper bound $b$ on a finite affinoid cover and put \[ C_n=\tau_{\geq b-n}C. \] The bounded equivalence lets us algebraize these objects together with the maps $C_{n+1}\to C_n$. By Lemma~\ref{lem:functor-descent}, we obtain a tower $(M_n)$ of bounded coherent complexes on $X$ with $\alpha_X(M_\bullet)\simeq C_\bullet$. Its transitions satisfy \[ \tau_{\geq b-n}M_{n+1}\xrightarrow{\sim}M_n, \] and $M_n$ has cohomology in $[b-n,b]$. For each affine $W=\Spec A\subset X$, set \[ M_W=\mathbf R\!\lim_n\mathbf R\Gamma(W,M_n|_W). \] This is pseudocoherent by Lemma~\ref{lem:postnikov-reconstruction}. For an affine inclusion $W'\subset W$, that lemma identifies the restriction of $M_W$ with $M_{W'}$. The identifications commute with successive restrictions and their associativity homotopies, so Zariski descent gives $M\in\Dpc(X)$. To identify its image, take $V=\Spa(R,R^+)\subset\lambda^{-1}(W)$ and an ordinary representative $P_V\in\Dpc(R)$ of $C|_V$. The bounded comparisons give an equivalence of towers \[ R\otimes_A^{\mathbf L}\mathbf R\Gamma(W,M_n|_W) \simeq\tau_{\geq b-n}P_V. \] Indeed, their condensifications are the restrictions of $\alpha_X(M_n)\simeq C_n$, and full faithfulness lifts this equivalence of diagrams. Lemma~\ref{lem:postnikov-reconstruction} then gives \begin{align*} R\otimes_A^{\mathbf L}M_W &\simeq\mathbf R\!\lim_n \bigl(R\otimes_A^{\mathbf L}\mathbf R\Gamma(W,M_n|_W)\bigr)\\ &\simeq\mathbf R\!\lim_n\tau_{\geq b-n}P_V\\ &\simeq P_V. \end{align*} The last equivalence is the reconstruction of $P_V$ from its truncations. Condensifying gives $\alpha_X(M)|_V\simeq C|_V$. The identifications commute with rational restriction by Lemma~\ref{lem:postnikov-reconstruction}. They glue to $\alpha_X(M)\simeq C$, proving essential surjectivity. Together with the truncation compatibility already established, this gives the asserted $t$-exact equivalence. \end{proof} \subsection{A faithfully flat family}\label{subsec:base-uniformity} The coordinate spectra of a finite affinoid cover form a faithfully flat family over $X$. This will allow us to test pseudocoherence after passing to the affinoid bases. \begin{lemma}\label{lem:finite-fpqc-charts} There is a finite affinoid cover $V_i=\Spa(R_i,R_i^+)$ of $\cX$, with $V_i\subset\lambda^{-1}(W_i)$ for affine opens $W_i\subset X$, such that \[ \coprod_i\Spec R_i\longrightarrow X \] is fpqc. \end{lemma} \begin{proof} Choose a finite affine cover of $X$, refine its inverse image by strongly noetherian affinoids, and take a finite subcover of $\cX$. Discard the empty members and write $W_i=\Spec A_i$. By Lemma~\ref{lem:base-flat}, each $A_i\to R_i$ is flat. The composite $\Spec R_i\to X$ is affine and flat because $W_i\hookrightarrow X$ is affine, and the finite disjoint union is quasi-compact. It remains to prove surjectivity. Let $x\in X$ be closed. Its coherent skyscraper $\kappa(x)$ is nonzero, so coherent GAGA makes its pullback nonzero. This pullback is nonzero on some $V_i$. Hence $x\in W_i$, and on that chart the sheaf is represented by the nonzero ring \[ R_i\otimes_{A_i}\kappa(x). \] A prime of this ring gives a prime of $R_i$ over the maximal ideal defining $x$ in $A_i$. For any $y\in X$, choose a closed point $x\in\overline{\{y\}}$ and a lift of $x$ to some $\Spec R_i$. The open $W_i$ contains $y$, and the corresponding primes satisfy $\mathfrak p_y\subseteq\mathfrak p_x$. Flat going-down gives a prime of $R_i$ below the chosen lift of $x$ and lying over $y$. Thus the morphism is surjective and hence fpqc. \end{proof} \section{Relative full faithfullness}\label{sec:relative} Let $T/X$ be separated and finitely presented. We construct its realizations over $\cX$ by gluing those of $T_R=T\times_X\Spec R$ over affinoid charts. For an affine morphism $q:U\to X$, a quasicoherent complex on $U$ is a complex on $X$ equipped with a $q_*\cO_U$-action. Applying $\alpha_X$ to this description gives the comparison on $U$; we then glue over a finite affine cover of $T$. \subsection{Relative algebraic and analytic realizations}\label{subsec:relative-construction} We use the rational-affinoid basis $\mathcal B=\operatorname{RatAff}_X(\cX)$ introduced in Section~\ref{subsec:base-GAGA}. Choose $V=\Spa(R,R^+)\subset\cX$ mapping into $W=\Spec A\subset X$, and set $T_R=T\times_X\Spec R$. By \cite[Theorem~A.13]{LiHuerta}, there are an algebraic realization $(T_R)^{\Alg}$ over $(\Spec R)^{\Alg}$ and an analytic locus $(T_R)^{\An}$ over $V=(\Spec R)^{\An}$. Define \begin{align*} (T^{\Alg}_{\cX})|_V &=(T_R)^{\Alg}\times_{(\Spec R)^{\Alg}}V, \\ (T^{\An}_{\cX})|_V &=(T_R)^{\An}. \end{align*} \begin{lemma}\label{lem:relative-cartesian} Let $u:V'=\Spa(R',R'^+)\to V=\Spa(R,R^+)$ be a rational localization in $\operatorname{RatAff}_X(\cX)$. For every separated finitely presented $X$-scheme $T$, \begin{align*} u^*\bigl((T_R)^{\Alg}\times_{(\Spec R)^{\Alg}}V\bigr) &\simeq (T_{R'})^{\Alg}\times_{(\Spec R')^{\Alg}}V',\\ u^*(T_R)^{\An}&\simeq(T_{R'})^{\An}. \end{align*} These equivalences are natural in $T$, compatible with Zariski restriction, and coherent under composition of rational localizations. \end{lemma} \begin{proof} If $\Spec R\to X$ factors through two affine neighborhoods, it factors through their intersection. Both descriptions of $T_R$ are therefore the fiber product $T\times_X\Spec R$. If $T_R=\Spec B$, then $T_{R'}=\Spec(B\otimes_RR')$. The first comparison is obtained by applying $\AnSpec$ to Lemma~\ref{lem:bounded-chart-base-change}(i), and the second is part (iii) of that lemma. The same result gives functoriality and compatibility with successive rational localizations. Choose a finite affine cover $(T_a)$ of $T_R$. Its multiple intersections are affine by separatedness. The realizations have the Zariski presentations \cite[Theorem~A.13]{LiHuerta} \begin{align*} (T_R)^{\Alg} &\simeq\colim_{[n]\in\Delta^{\mathrm{op}}} \coprod_{(a_0,\ldots,a_n)}(T_{a_0\ldots a_n})^{\Alg},\\ (T_R)^{\An} &\simeq\colim_{[n]\in\Delta^{\mathrm{op}}} \coprod_{(a_0,\ldots,a_n)}(T_{a_0\ldots a_n})^{\An}. \end{align*} Base change preserves these colimits. Replacing each affine term by its base change therefore gives precisely the two presentations for $T_{R'}$. The affine comparisons commute with the face and degeneracy maps, morphisms of schemes, and successive base changes. These compatibilities pass to the colimit, and a common affine refinement identifies the constructions from different covers. \end{proof} The rational base-change identifications give the required descent data over $\cX$. \begin{proposition}\label{prop:relative-construction} For a separated finitely presented $X$-scheme $T$, the local realizations glue functorially to analytic stacks $T^{\Alg}_{\cX}$ and $T^{\An}_{\cX}$, with a natural map \[ T^{\An}_{\cX}\longrightarrow T^{\Alg}_{\cX} \] that is an equivalence for proper $T/X$. The construction commutes with rational base change and Zariski gluing. For an open immersion $T\hookrightarrow X$, we have $T^{\An}_{\cX}\simeq\lambda^{-1}(T)$. \end{proposition} \begin{proof} Choose a rational-affinoid cover $V_i=\Spa(R_i,R_i^+)$ of $\cX$. On a common rational subdomain of $V_i\cap V_j$, Lemma~\ref{lem:relative-cartesian} identifies the two local realizations. On triple and higher intersections, all comparisons factor through the construction on a common rational chart, with the composition homotopies supplied by the same lemma. These are descent data for $T^{\Alg}_{\cX}$ and $T^{\An}_{\cX}$. The descent data are functorial in $T$ and commute with rational restriction. Zariski compatibility follows by applying the presentation of \cite[Theorem~A.13]{LiHuerta} on each base chart before gluing over $\cX$. For an open immersion $T\hookrightarrow X$, the local realization is $V_i\cap\lambda^{-1}(T)$, and these opens glue to $\lambda^{-1}(T)$. It remains to compare the two realizations. For an affine scheme, the bounded charts have natural maps to the algebraic realization \cite[A.11--A.13]{LiHuerta}. Lemma~\ref{lem:bounded-chart-base-change} makes these maps compatible with restriction. We glue first over a Zariski cover of $T_R$ and then over the base cover, obtaining \[ T^{\An}_{\cX}\longrightarrow T^{\Alg}_{\cX}. \] If $T/X$ is proper, then each $T_R/R$ is proper. The proper analytic-locus theorem makes the comparison an equivalence on every base chart. These local inverses agree on overlaps, and hence give a global inverse. \end{proof} The pseudocoherent subcategories also glue over the affinoid charts. Put $T_R=T\times_X\Spec R$ and $\mathcal P_T(V)=\Dpcbox((T_R)^{\An})$. Nuclearity and pseudocompactness of a complex on $(T_R)^{\An}$ are tested on the bounded charts of \cite[A.15]{LiHuerta}. For a rational inclusion $u:V'\hookrightarrow V$, Lemma~\ref{lem:relative-cartesian} gives the pullback $u_T^*$, computed on analytic affines by derived scalar extension followed by analytic completion. As before, $\operatorname{Cart}_{V\in\mathcal B}\mathcal P_T(V)$ denotes the limit. Its objects are families $C_V$ on $(T_R)^{\An}$ with equivalences $u_T^*C_V\simeq C_{V'}$ and the unit, composition, and higher homotopies. Its morphisms are maps of families respecting these identifications. \begin{proposition}\label{prop:relative-pc-descent} For a separated finitely presented $X$-scheme $T$, restriction gives \[ \Dpcbox(T^{\An}_{\cX})\xrightarrow{\sim} \operatorname{Cart}_{V\in\operatorname{RatAff}_X(\cX)} \Dpcbox((T_R)^{\An}), \] with rational pullbacks as transition functors, independently of the basis and bounded-chart presentations. For proper $T$, the same description holds on $T^{\Alg}_{\cX}$. \end{proposition} \begin{proof} We show that descent for $\Cbox$ restricts to $\Dpcbox$. Consider scalar extension on an analytic affine, \[ F:\mathsf D(\mathcal A)\longrightarrow\mathsf D(\mathcal A') \] with restriction-of-scalars right adjoint $G$. The right adjoint preserves direct sums and lower cohomological bounds \cite[Proposition~12.4, Remark~12.5, and Proposition~12.6]{ScholzeAnalytic}. For pseudocompact $M$ and a family $(N_a)$ with a common lower bound, adjunction gives \begin{align*} \Map(FM,\bigoplus_aN_a) &\simeq\Map(M,G(\bigoplus_aN_a))\\ &\simeq\Map(M,\bigoplus_aGN_a)\\ &\simeq\bigoplus_a\Map(M,GN_a)\\ &\simeq\bigoplus_a\Map(FM,N_a). \end{align*} This is the mapping criterion for pseudocompactness of $FM$ \cite[A.5]{LiHuerta}. By \cite[A.4]{LiHuerta}, nuclear objects are colimits of sequential colimits of compact objects with trace-class transitions. Since $G$ preserves filtered colimits, $F$ preserves compact objects. The natural map $F(P^\vee)\to F(P)^\vee$ carries a class defining a trace-class map to one defining its scalar extension. Thus $F$ preserves these presentations and, being colimit-preserving, preserves nuclear objects. We apply these observations to $V'\subset V$. Choose a finite affine cover of $T_R$. On each intersection, Lemma~\ref{lem:bounded-chart-base-change} gives maps between the universal bounded charts, and the preceding adjunction calculation proves that their pullbacks preserve pseudocompactness. For nuclearity, the defining charts use the analytic structure $\mathcal Q_R$ induced from $(R,R^+)^\square$ on $R\langle T_1,\ldots,T_m\rangle$ \cite[A.10 and A.15]{LiHuerta}. Consider the ring map \[ R\langle T_1,\ldots,T_m\rangle \longrightarrow R'\langle T_1,\ldots,T_m\rangle. \] It respects these structures: a complete $\mathcal Q_{R'}$-module is complete over $(R',R'^+)^\square$, hence over $(R,R^+)^\square$, which is the completeness condition for $\mathcal Q_R$ \cite[Proposition~12.8]{ScholzeAnalytic} \cite[A.3]{LiHuerta}. Base change gives maps of the chart pushouts for each tuple $(\varpi^nb_i)$, and the preceding argument for $F$ proves preservation of nuclearity. Nuclearity and pseudocompactness descend along bounded charts and Zariski covers \cite[A.15 and Theorem~A.13]{LiHuerta}. Pullback therefore restricts to $\mathcal P_T(V)\to\mathcal P_T(V')$, with compatibility under composition given by associativity of the pushouts. Restriction sends a global pseudocoherent object to a cartesian family. Conversely, regard a family $(C_V)$ as descent data in the solid categories. The equivalence \[ \Cbox(T^{\An}_{\cX}) \simeq\lim_{V\in\mathcal B^{\mathrm{op}}}\Cbox((T_R)^{\An}), \] glues it to $C\in\Cbox(T^{\An}_{\cX})$. Every defining-chart restriction is a restriction of some $C_V$, and hence is nuclear and pseudocompact. Thus $C\in\Dpcbox$. This construction inverts restriction on objects, and the same limit formula identifies the mapping spaces. Common rational and affine refinements identify different choices of base and Zariski covers. Independence of bounded charts follows from \cite[A.15]{LiHuerta} together with the restriction compatibility above. If $T$ is proper, Proposition~\ref{prop:relative-construction} identifies the two realizations chart by chart, so the description applies equally to $T^{\Alg}_{\cX}$. \end{proof} We will also use descent in the scheme variable. \begin{lemma}\label{lem:alg-zariski-descent} Let $(U_i)_{i\in I}$ be a finite Zariski cover of a separated finitely presented $X$-scheme $T$, and put $U_{i_0\ldots i_n}=U_{i_0}\cap\cdots\cap U_{i_n}$. Restriction gives \begin{align*} \Dqc(T)&\simeq\lim_{[n]\in\Delta} \prod_{(i_0,\ldots,i_n)\in I^{n+1}}\Dqc(U_{i_0\ldots i_n}),\\ \Cbox(T^{\Alg}_{\cX})&\simeq\lim_{[n]\in\Delta} \prod_{(i_0,\ldots,i_n)\in I^{n+1}} \Cbox(U_{i_0\ldots i_n}^{\Alg}{}_{\cX}). \end{align*} The comparison functors form a map of these diagrams. Degreewise full faithfulness, equivalences, and colimit preservation descend, as do tensor structures, module objects, and pullback transformations. \end{lemma} \begin{proof} The first formula is ordinary Zariski descent: a complex on each $U_i$, equivalences on overlaps, and compatible homotopies on higher intersections. Allowing ordered tuples with repetitions gives the full \v Cech diagram in the statement. The same intersections present the algebraic realization as a geometric realization \cite[Theorem~A.13]{LiHuerta}, compatibly with base charts by Lemma~\ref{lem:relative-cartesian}. Proposition~\ref{prop:analytic-stack-Cbox-descent} gives the second formula. Refine the cover by affine opens. On the resulting affine intersections, comparison commutes with restriction by associativity of scalar extension. Lemma~\ref{lem:functor-descent} passes full faithfulness and equivalences to the limits. Colimits, tensor products, and module actions are computed componentwise, so they descend with the functors and their pullback comparisons. Passing to common affine refinements gives these assertions for the original cover. \end{proof} \subsection{Ordinary relative Huber analytification}\label{subsec:Huber-comparison} The ordinary adic realization is obtained from Huber's relative analytification on the same base charts. \begin{proposition}\label{cor:global-Huber} Let $T$ be an $X$-scheme locally of finite type. The Huber analytifications $(T_R)^{\ad/V}$, for $V=\Spa(R,R^+)$ and $T_R=T\times_X\Spec R$, glue canonically to a locally strongly noetherian adic space $T^{\ad}_{\cX}/\cX$ and a morphism of locally ringed spaces \[ c_T:T^{\ad}_{\cX}\longrightarrow T. \] The construction commutes with rational base change and Zariski gluing, and preserves properness over the base. \end{proposition} \begin{proof} For $V=\Spa(R,R^+)\in\operatorname{RatAff}_X(\cX)$, the scheme $T_R$ is locally of finite type over $R$, so its relative Huber analytification $(T_R)^{\ad/V}$ exists \cite[Proposition~3.8]{Huber1994}. For rational $V'=\Spa(R',R'^+)\subset V$, the base change \[ (T_R)^{\ad/V}\times_VV' \] has the mapping property of $(T_{R'})^{\ad/V'}$: a compatible map from an analytic test space over $V'$ to $T_R$ factors through $T_{R'}$. We obtain a canonical isomorphism \[ (T_R)^{\ad/V}\times_VV'\xrightarrow{\sim}(T_{R'})^{\ad/V'}. \] For $V''\subset V'\subset V$, both composites represent the identity of the same functor over $V''$, so they agree. Choose a rational-affinoid cover $(V_i)$. The isomorphisms just constructed agree on common rational refinements of $V_i\cap V_j$. They therefore glue over each overlap, and their uniqueness gives the cocycle condition on triple intersections. Gluing the spaces, structure sheaves, and valuations gives $T^{\ad}_{\cX}$, with $c_T$ induced by the local maps to $T$. Each piece is locally of finite type over $V_i$, so the resulting space is locally strongly noetherian. Common refinements compare different base covers, and the same mapping property gives compatibility with Zariski gluing in $T$. Finally, if $T/X$ is proper, then each $(T_{R_i})^{\ad/V_i}\to V_i$ is proper \cite[Remark~6.3]{Zavyalov}. As properness is local on the target, the glued morphism $T^{\ad}_{\cX}\to\cX$ is proper. \end{proof} \subsection{Module categories and relative full faithfulness}\label{subsec:modules} Let $q:U\to X$ be affine and finitely presented, and put $\cA=q_*\cO_U\in\operatorname{CAlg}(\Dqc(X))$. The usual equivalence $\Dqc(U)\simeq\operatorname{Mod}_{\cA}(\Dqc(X))$ has an analytic counterpart, which describes the relative comparison as a functor on modules. \begin{lemma}\label{lem:affine-module-category} There is a canonical equivalence, compatible with rational restriction, \[ \Cbox(U^{\Alg}_{\cX})\simeq \operatorname{Mod}_{\alpha_X\cA}(\Cbox(\cX)). \] It identifies $\mathbf Rq_*^{\Alg}$ with the forgetful functor and $\alpha_U$ with the module functor induced by $\alpha_X$. \end{lemma} \begin{proof} Take a base chart $V=\Spa(R,R^+)\subset\lambda^{-1}(W)$ with $W=\Spec A$, and write $U_W=\Spec B_0$ and $U_R=\Spec B$. Flatness identifies $B=B_0\otimes_A R$ with the derived tensor product. Put $\mathcal R=(R,R^+)^\square$ and \begin{align*} \mathcal R_0&=(R_{\mathrm{disc}},R^+_{\mathrm{disc}})^\square,\\ \mathcal B_0&=(B_{\mathrm{disc}},B^+_{\mathrm{disc}})^\square, \end{align*} where $B^+$ is the integral closure of the image of $R^+$ in $B$. The algebraic realization on this chart is defined by \[ B_R^{\mathrm{alg}}=\mathcal R\otimes^{\mathrm{AnRing}}_{\mathcal R_0}\mathcal B_0 \] \cite[A.9]{LiHuerta}. Passing to module categories gives \begin{align*} \mathsf D(B_R^{\mathrm{alg}}) &\simeq\mathsf D(\mathcal R)\otimes_{\mathsf D(\mathcal R_0)}\mathsf D(\mathcal B_0)\\ &\simeq\mathsf D(\mathcal R)\otimes_{\mathsf D(\mathcal R_0)} \operatorname{Mod}_B(\mathsf D(\mathcal R_0))\\ &\simeq\operatorname{Mod}_{\Cond_R(B)}(\Dsquare(R,R^+)). \end{align*} The first line is the analytic-ring pushout formula \cite[Lemma~A.1]{LiHuerta}. The second uses Lemma~\ref{lem:integral-induced-modules} to regard complete $B$-modules as complete base modules equipped with a $B$-action. The final line extends this module category from $\mathsf D(\mathcal R_0)$ to $\mathsf D(\mathcal R)$ \cite[Section~4.8.5]{LurieHA}; the algebra acting after extension is $\Cond_R(B)$. Under $\Dqc(W)\simeq\mathrm D(A)$, the restriction $\cA|_W$ corresponds to $B_0$. Hence \[ (\alpha_X\cA)|_V =\Cond_R(R\otimes_A^{\mathbf L}B_0) \simeq\Cond_R(B). \] This identifies the acting algebra with $(\alpha_X\cA)|_V$. Under rational base change, Lemma~\ref{lem:integral-base-change} identifies the same calculation with the one for $B\otimes_RR'$. Reassociating tensor products makes the identifications compatible with successive restrictions. The module categories, with their actions, therefore glue by Proposition~\ref{prop:analytic-stack-Cbox-descent}. In this description, structural pullback is the free-module functor $N\mapsto\Cond_R(B)\otimes N$. Its right adjoint forgets the $\Cond_R(B)$-action. These adjoint pairs commute with restriction, so the forgetful functor is $\mathbf Rq_*^{\Alg}$ after gluing. Applying $\alpha_X$ to an algebraic module and its action gives the local comparison and hence, by descent, the global one. \end{proof} We apply the module description on affine opens to extend the comparison to $T$. \begin{proposition}\label{prop:relative-full-faithfulness} For every separated finitely presented $X$-scheme $T$, scalar extension gives a fully faithful, colimit-preserving, strong symmetric monoidal functor \[ \alpha_T:\Dqc(T)\longrightarrow\Cbox(T^{\Alg}_{\cX}), \] compatible with pullback. For proper $T/X$, its target identifies with $\Cbox(T^{\An}_{\cX})$. \end{proposition} \begin{proof} Suppose first that $q:T\to X$ is affine, and put $\cA=q_*\cO_T$. By Lemma~\ref{lem:affine-module-category}, the two categories are $\operatorname{Mod}_{\cA}(\Dqc(X))$ and $\operatorname{Mod}_{\alpha_X\cA}(\Cbox(\cX))$. Theorem~\ref{thm:enhanced-base} and Lemma~\ref{lem:module-image} make the induced functor fully faithful. Colimits are computed on underlying objects, so it preserves them. It preserves relative tensor products as well, because $\alpha_X$ is strong symmetric monoidal and preserves colimits. For general $T$, choose a finite affine cover $(U_i)$. The separated morphism $T\to X$ and the affine diagonal of $X$ give an affine diagonal on $T$. Hence every intersection $U_I$ is affine, and its map to $X$ is affine by \cite[Lemma~29.11.12]{Stacks}. The preceding argument therefore applies on every intersection. Tensor associativity makes the local functors commute with restriction. They therefore define a map of the descent diagrams in Lemma~\ref{lem:alg-zariski-descent}. Taking limits of their mapping-space equivalences proves full faithfulness. Since colimits are computed on the intersections and preserved by each local functor, they are preserved globally as well. The tensor comparisons agree under restriction and supply the monoidal structure. For a morphism $T'\to T$, choose source affines mapping into target affines. Pullback and comparison are scalar extensions on these charts. Their associativity identifications agree on intersections, so they give the required pullback compatibility, including compatibility with composition. \end{proof} On the analytic locus, the comparison preserves pseudocoherence, as one sees on finite-free resolutions. \begin{proposition}\label{prop:forward-pc} Let $T$ be a separated finitely presented $X$-scheme, with analytic-locus map $\jmath_T:T^{\An}_{\cX}\to T^{\Alg}_{\cX}$. Then \[ \alpha_T^{\An}:=\jmath_T^*\alpha_T:\Dpc(T)\longrightarrow \Dpcbox(T^{\An}_{\cX}) \] is well-defined. For proper $T$, it is the restriction of the fully faithful $\alpha_T$ and is denoted by $\alpha_T$. \end{proposition} \begin{proof} Take $M\in\Dpc(T)$, a rational-affinoid base chart, and an affine $\Spec B\subset T_R$. Pulling back a bounded-above finite-projective resolution shows that the pullback of $M$ is pseudocoherent. After a further restriction we may represent it by \[ \cdots\longrightarrow P^{b-2}\longrightarrow P^{b-1} \longrightarrow P^b\longrightarrow0. \] Each $P^i$ is finite free over $B$. On a bounded chart, scalar extension and analytic completion replace the finite free terms by finite sums of the tensor unit, with the same differential matrices. By \cite[A.15 and the paragraph preceding Theorem~A.18]{LiHuerta}, using \cite[Theorem~5.50 and Proposition~5.33]{Andreychev}, the resulting complexes are nuclear and pseudocompact on the respective defining charts. All these local complexes are restrictions of $\jmath_T^*\alpha_T(M)$, with the scalar-extension identifications used in its construction. Descent along the bounded and Zariski covers therefore gives \[ \jmath_T^*\alpha_T(M)\in\Dpcbox(T^{\An}_{\cX}). \] When $T$ is proper, $\jmath_T$ is an equivalence by Proposition~\ref{prop:relative-construction}, so this is the restriction of $\alpha_T$ itself. \end{proof} \subsection{Flatness}\label{subsec:base-flat} Flatness of ordinary analytification follows from the flatness of the base charts and the local Huber comparison. \begin{proposition}\label{prop:global-flat} For every $X$-scheme $T$ locally of finite type, the morphism \(c_T:T^{\ad}_{\cX}\longrightarrow T\) is flat. \end{proposition} \begin{proof} Fix $z\in T^{\ad}_{\cX}$ and write $t=c_T(z)$. Choose an affine $W=\Spec A\subset X$ containing the image of $t$, a rational affinoid $V=\Spa(R,R^+)\subset\lambda^{-1}(W)$ containing the image of $z$, and an affine neighborhood $U\subset T\times_XW$ of $t$. On these neighborhoods the map factors as \[ U_R^{\ad/V}\longrightarrow U_R\longrightarrow U, \] where $U_R=U\times_A\Spec R$. The second map is a base change of the flat map from Lemma~\ref{lem:base-flat}. The first is flat on local rings by \cite[Lemma~6.7]{Zavyalov}. Their composite is flat at $z$, which proves the assertion. \end{proof} \section{Localization at the boundary}\label{sec:grauert} We now construct the direct images that will be algebraized on $X$. The main ingredients are proper direct image and localization at a boundary. We establish these first, then identify the analytic-stack realization with Huber analytification. This last comparison supplies the ordinary affinoid representatives used in faithfully flat descent. \Needspace{6\baselineskip} \subsection{Proper direct images of solid quasicoherent complexes}\label{subsec:global-Grauert} \subsubsection{The affinoid case} The affinoid proper-direct-image theorem follows from the adjunction argument of \cite[A.16--A.17]{LiHuerta}. Let $(R,R^+)$ be strongly noetherian and complete Tate. For $S$ locally of finite type over $R$, let $S^{\An}$ be its analytic-stack analytification \cite[Theorem~A.13]{LiHuerta}, with structural map $\pi:S^{\An}\to\AnSpec(R,R^+)^\square$. The notation $\Dpcbox(S^{\An})$ uses the nuclearity and pseudocompactness conditions of \cite[A.15]{LiHuerta}. On the base we use $\Cond_R:\Dpc(R)\hookrightarrow\Dsquare(R,R^+)$ \cite[Theorem~5.9]{Andreychev}. \begin{theorem}\label{thm:local-solid-grauert} Let $(R,R^+)$ be a strongly noetherian complete Tate Huber pair and let $S/R$ be proper and finitely presented. For every $K\in\Dpcbox(S^{\An})$, there is $M\in\Dpc(R)$ with \[ \mathbf R\pi_*K\simeq\Cond_R(M), \] unique up to contractible choice together with this equivalence. \end{theorem} \begin{proof} We apply the exceptional-pullback calculation of \cite[Proposition~A.16]{LiHuerta} on polynomial and quotient charts. We first give the finite resolutions entering that calculation. Write $B=P/I$ with $P=R[T_1,\ldots,T_m]$. Since $P$ is noetherian, a finite free surjection onto $B$ has finitely generated kernel. Repeating this for the successive kernels gives a resolution \[ \cdots\longrightarrow P^{r_2}\longrightarrow P^{r_1} \longrightarrow P^{r_0}\longrightarrow B\longrightarrow0 \] with every $r_i$ finite. In particular, $B\in\Dpc(P)$ \cite[Tag~064T]{Stacks}. Put $D=R\langle T_1,\ldots,T_m\rangle$. For a finitely generated ideal $J\subset P$, pull $\widetilde J\hookrightarrow\cO_{\mathbf A_R^m}$ back to relative analytification and restrict to the closed unit polydisc. Flatness on local rings preserves the injection \cite[Lemma~6.7]{Zavyalov}. A finite presentation of $J$ and the affinoid coherent-sheaf theorem \cite[Theorem~4.2]{Zavyalov} identify the map on sections with \[ J\otimes_PD\lhook\joinrel\longrightarrow D. \] The ideal criterion gives flatness of $P\to D$ \cite[Tag~00HD]{Stacks}. Consequently, \[ D\otimes_P^{\mathbf L}B\simeq(D/ID)[0], \] and the tensored resolution makes $D/ID$ pseudocoherent. Equipped with its complete quotient topology, this module has nuclear and pseudocompact condensification \cite[Theorem~5.9, Lemma~5.23, and Theorem~5.50]{Andreychev}. The substitution $U_i=\varpi^nT_i$ gives the same calculation on any bounded polydisc. It also remains valid after rational localization, which preserves strong noetherianity. On a quotient chart, exceptional pullback is internal derived Hom from $(D/ID)[0]$ \cite[Proposition~A.16]{LiHuerta}. When the targets have a common lower bound, only finitely many terms of the finite-free resolution contribute in each total degree. Pseudocompactness therefore gives the direct-sum comparison, while concentration of the quotient in degree zero gives the lower bound. On a polynomial chart, the cited calculation gives ordinary pullback shifted by $[m]$. Applying these calculations on the bounded charts and a finite affine cover of $S$ shows that, whenever $(M_a)$ has a common lower bound, so does $(\pi^!M_a)$ on the defining charts. Moreover, \[ \bigoplus_a\pi^!M_a\xrightarrow{\sim} \pi^!\!\left(\bigoplus_aM_a\right). \] For proper $S$, the identification $S^{\An}\simeq S^{\Alg}$ gives a finite cover by the induced bounded charts of \cite[A.10 and A.15]{LiHuerta}. The chart algebras are nuclear over the base, and restriction of scalars preserves nuclear objects by \cite[Lemma~A.3 and Proposition~A.10]{LiHuerta}. The finite-cover argument in the proof of \cite[Corollary~A.17]{LiHuerta}, using \cite[Theorem~8.6]{ClausenScholze}, gives nuclearity of $\mathbf R\pi_*K$. The same proof identifies $\mathbf R\pi_*$ with $\mathbf R\pi_!$, so that $\mathbf R\pi_*\dashv\pi^!$ on the full solid categories. For a family $(M_a)$ with a common lower bound, we may therefore compute \begin{align*} \Map(\mathbf R\pi_*K,\bigoplus_aM_a) &\simeq\Map(K,\pi^!\!\bigoplus_aM_a)\\ &\simeq\Map(K,\bigoplus_a\pi^!M_a)\\ &\simeq\bigoplus_a\Map(K,\pi^!M_a)\\ &\simeq\bigoplus_a\Map(\mathbf R\pi_*K,M_a). \end{align*} The outside equivalences are adjunction. The second uses the direct-sum comparison for $\pi^!$, and the third uses pseudocompactness of $K$ and the common lower bound on $(\pi^!M_a)$ \cite[Definition~9.9]{ClausenScholze}. Hence $\mathbf R\pi_*K$ is pseudocompact as well as nuclear. The affinoid comparison identifies these objects with $\Cond_R(\Dpc(R))$ \cite[A.15]{LiHuerta} \cite[Theorem~5.50]{Andreychev}. Thus $M$ exists. Full faithfulness of condensification makes $M$, equipped with its comparison equivalence, unique up to contractible choice. \end{proof} The same adjoint pair gives the colimit comparison. \begin{corollary}\label{cor:local-grauert-colimits} Under the hypotheses of Theorem~\ref{thm:local-solid-grauert}, $\mathbf R\pi_*$ preserves arbitrary colimits on $\Cbox(S^{\An})$. \end{corollary} \begin{proof} For proper $\pi$, the identification $\mathbf R\pi_!\simeq\mathbf R\pi_*$ holds on all solid quasicoherent complexes \cite[proof of Corollary~A.17]{LiHuerta}. Thus $\mathbf R\pi_*$ is also a left adjoint, with right adjoint $\pi^!$. Explicitly, for a diagram $(K_i)$ and a base object $N$, the adjunction gives \begin{align*} \Map(\mathbf R\pi_*\colim_iK_i,N) &\simeq\Map(\colim_iK_i,\pi^!N)\\ &\simeq\lim_i\Map(K_i,\pi^!N)\\ &\simeq\Map(\colim_i\mathbf R\pi_*K_i,N). \end{align*} The composite is induced by the canonical colimit map and is natural in $N$. Yoneda therefore proves the assertion. \end{proof} \subsubsection{Globalization} We pass to $\cX$ by restricting direct images to its affinoid opens. The restriction formula follows from extension by zero. \begin{lemma}\label{lem:open-BC} Let \[ \begin{CD} \mathcal Z' @>{i'}>> \mathcal Z\\ @V{g'}VV @VV{g}V\\ \mathcal V @>{i}>> \mathcal S \end{CD} \] be a cartesian square of analytic stacks. Suppose $i_!\dashv i^*$ and $i'_!\dashv i'^*$ are extension-by-zero adjoint pairs and the canonical map $i'_!(g')^*\to g^*i_!$ is an equivalence. Then the canonical map \[ i^*\mathbf Rg_*\longrightarrow\mathbf Rg'_*i'^* \] is an equivalence on $\Cbox(\mathcal Z)$. \end{lemma} \begin{proof} For $A\in\Cbox(\mathcal Z)$ and $B\in\Cbox(\mathcal V)$, adjunction gives \begin{align*} \Map_{\mathcal V}(B,i^*\mathbf Rg_*A) &\simeq\Map_{\mathcal S}(i_!B,\mathbf Rg_*A)\\ &\simeq\Map_{\mathcal Z}(g^*i_!B,A)\\ &\simeq\Map_{\mathcal Z}(i'_!(g')^*B,A)\\ &\simeq\Map_{\mathcal Z'}((g')^*B,i'^*A)\\ &\simeq\Map_{\mathcal V}(B,\mathbf Rg'_*i'^*A). \end{align*} The middle equivalence is the comparison for extension by zero. The units and counits identify the composite with the canonical base-change map. Since the calculation is natural in $B$, Yoneda proves the lemma. \end{proof} For quasi-compact opens in the class $I$ of \cite[A.2]{LiHuerta}, extension by zero and its pullback comparison are supplied by \cite[Proposition~5.16 and Theorem~5.19]{ScholzeSix}. This includes rational-affinoid inclusions \cite[Proposition~2.3.2]{RodriguezCamargo}. Lemma~\ref{lem:open-BC} therefore identifies the restriction of $\mathbf Rg_*^{\An}$ to each base affinoid with the direct image there. For proper $g$, we can apply the affinoid theorem on these charts. \begin{proposition}\label{prop:global-Grauert} For proper finitely presented $g:Z\to X$, the functor \[ \mathbf Rg_*^{\An}:\Cbox(Z^{\An}_{\cX})\longrightarrow\Cbox(\cX) \] preserves arbitrary colimits and carries $\Dpcbox(Z^{\An}_{\cX})$ into $\Dpcbox(\cX)$. \end{proposition} \begin{proof} Both assertions can be checked on rational base charts. Take $V=\Spa(R,R^+)$ and set $Z_R=Z\times_X\Spec R$. Its inverse image is $Z_R^{\An}$, and Lemma~\ref{lem:open-BC} identifies \[ (\mathbf Rg^{\An}_*K)|_V \simeq\mathbf R(g_R)^{\An}_*(K|_{Z_R^{\An}}). \] For pseudocoherent $K$, Theorem~\ref{thm:local-solid-grauert} applies to its restriction on the proper scheme $Z_R$ over strongly noetherian $R$. The direct image belongs to $\Cond_R(\Dpc(R))$. The affinoid criterion therefore gives $\mathbf Rg_*^{\An}K\in\Dpcbox(\cX)$. For a diagram $(K_a)$, consider the canonical map \[ \delta:\colim_a\mathbf Rg^{\An}_*K_a \longrightarrow\mathbf Rg^{\An}_*\!\left(\colim_aK_a\right). \] Restriction to $V$ preserves colimits, so open base change identifies $\delta|_V$ with \[ \colim_a\mathbf R(g_R)^{\An}_*(K_a|_{Z_R^{\An}}) \longrightarrow \mathbf R(g_R)^{\An}_*\!\left(\colim_a K_a|_{Z_R^{\An}}\right). \] Corollary~\ref{cor:local-grauert-colimits} makes this an equivalence. The cofiber of $\delta$ thus vanishes on an affinoid cover and hence globally. \end{proof} \subsection{Algebraic localization}\label{subsec:Cartier} On $\Spec A$, restriction to $D(t)$ is scalar extension to $A[t^{-1}]$, and its right adjoint is restriction of scalars. For a section of a line bundle, these local calculations glue to a sequence of twists. We use the same notation for line bundles and sections and their images under the comparison functors. \begin{lemma}\label{lem:affinoid-localization} Let $(R,R^+)$ be a complete Tate Huber pair and let $S/R$ be separated and finitely presented. For a section $s:\cO_S\to\cL$ of a line bundle, let $j:S_s\hookrightarrow S$ be its nonvanishing open. Then $j^{\Alg,*}$ has a right adjoint and, for every $K\in\Cbox(S^{\Alg})$, \[ \mathbf Rj_*^{\Alg}j^{\Alg,*}K \simeq\colim_{n\geq0}\bigl(K\otimes\cL^{\otimes n}\bigr), \] naturally in $K$, with transition maps induced by $s$. \end{lemma} \begin{proof} Choose a finite affine cover trivializing $\cL$. On an intersection $\Spec A$, write the section as $t$; its nonvanishing locus is $\Spec A[t^{-1}]$. Lemma~\ref{lem:affinoid-module-category} identifies restriction with scalar extension along $A\to A[t^{-1}]$, whose right adjoint is restriction of scalars. For a further restriction $A\to A'$, flatness of localization gives \[ A'[t^{-1}]\simeq A'\otimes_A^{\mathbf L}A[t^{-1}], \] so, for an $A[t^{-1}]$-module $M$, tensor associativity gives \[ A'\otimes_A^{\mathbf L}\operatorname{Res}_{A[t^{-1}]/A}M \simeq \operatorname{Res}_{A'[t^{-1}]/A'} \bigl(A'[t^{-1}]\otimes_{A[t^{-1}]}^{\mathbf L}M\bigr). \] This is the comparison defined by the scalar-extension adjoint pairs, and tensor associativity gives compatibility with successive restrictions. Lemma~\ref{lem:right-adjoint-descent}, applied to the Zariski presentation of \cite[Theorem~A.13]{LiHuerta}, glues the right adjoints to $\mathbf Rj_*^{\Alg}$. To compute the composite, send the $n$th term of \[ A\xrightarrow{t}A\xrightarrow{t}A\xrightarrow{t}\cdots \] to $A[t^{-1}]$ by $a\mapsto a/t^n$. Every fraction appears, and it is zero exactly when a later multiplication by a power of $t$ kills the numerator. Hence the colimit is $A[t^{-1}]$. Filtered colimits of modules are exact, so this also computes the derived colimit. Tensoring with $K$ gives \[ \mathbf Rj_*^{\Alg}j^{\Alg,*}K \simeq K\otimes_A^{\mathbf L}A[t^{-1}] \simeq\colim(K\xrightarrow{t}K\xrightarrow{t}\cdots). \] In the chosen trivialization, the transition on $K\otimes\cL^n$ is multiplication by $t$. Changing the basis multiplies $t$ by a unit and the $n$th basis by the corresponding power, identifying the sequences and their localization maps on overlaps. The formulas therefore glue. \end{proof} Using the relative module description, we obtain the same localization formula over $X$. \begin{lemma}\label{lem:Cartier-localization} Let $Z$ be a separated finitely presented $X$-scheme, let $s:\cO_Z\to\cL$ be a line-bundle section, and put $j:U=Z_s\hookrightarrow Z$. Then $j^{\Alg,*}$ has a right adjoint and, naturally for $K\in\Cbox(Z^{\Alg}_{\cX})$, \[ \mathbf Rj_*^{\Alg}j^{\Alg,*}K \simeq\colim_{n\geq0}\bigl(K\otimes\cL^{\otimes n}\bigr), \] with transitions induced by $s$. For the canonical section of an effective Cartier divisor $D$, the terms are $K\otimes\cO_Z(nD)$. \end{lemma} \begin{proof} Choose a finite affine cover of $Z$ subordinate to affines of $X$ and trivializing $\cL$. On an intersection $\Spec A$, choose a basis $e$ and write $s=te$. Then $U\cap\Spec A=\Spec A[t^{-1}]$. By Lemma~\ref{lem:affine-module-category}, restriction is scalar extension to $A[t^{-1}]$ and its right adjoint is restriction of scalars. On a further intersection with ring $A'$, flatness of localization gives \[ A'[t^{-1}]\simeq A'\otimes_A^{\mathbf L}A[t^{-1}]. \] For an $A[t^{-1}]$-module $M$, we therefore compute \[ A'\otimes_A^{\mathbf L}\operatorname{Res}_{A[t^{-1}]/A}M \simeq\operatorname{Res}_{A'[t^{-1}]/A'} \bigl(A'[t^{-1}]\otimes_{A[t^{-1}]}^{\mathbf L}M\bigr). \] The displayed map is the comparison defined by the scalar-extension adjunctions. Tensor associativity makes the comparisons agree under further restriction, so Lemmas~\ref{lem:alg-zariski-descent} and~\ref{lem:right-adjoint-descent} glue the right adjoints to $\mathbf Rj_*^{\Alg}$. The maps from the $n$th term to $A[t^{-1}]$ given by $a\mapsto a/t^n$ identify \[ A[t^{-1}]\simeq\colim(A\xrightarrow{t}A\xrightarrow{t}\cdots). \] Every fraction occurs. Two fractions $a/t^n$ and $b/t^m$ agree exactly when $t^{k-n}a=t^{k-m}b$ for some $k\geq\max(m,n)$, which is precisely equality at a later stage of the sequence. Exactness of filtered colimits makes the same sequence a derived colimit, and tensoring gives \begin{align*} \mathbf Rj_*^{\Alg}j^{\Alg,*}K &\simeq K\otimes_A^{\mathbf L}A[t^{-1}]\\ &\simeq\colim(K\xrightarrow{t}K\xrightarrow{t}\cdots). \end{align*} Using the bases $e^{\otimes n}$, multiplication by $s$ on the twists is multiplication by $t$. Thus the last line is the sequence of twists in the statement. To check independence of the basis, write $e'=ue$. The coordinate of $s$ becomes $u^{-1}t$, while the $n$th basis changes by $u^n$. These changes identify the sequences and their localization maps, so the local equivalences glue. For an effective Cartier divisor with $\cO_Z(-D)=(t)$, the basis $t^{-1}$ of $\cO_Z(D)$ writes the canonical section as $1=t\cdot t^{-1}$. Its coordinate is therefore $t$, so the sequence consists of the twists $K\otimes\cO_Z(nD)$ with the stated transition maps \cite[proof of Theorem~A.18]{LiHuerta}. \end{proof} To apply the localization formula to a quasi-compact open, we first realize it as the complement of a Cartier divisor on a blowup. We begin with the noetherian case. \begin{lemma}\label{lem:blowup-boundary} Let $U\subset Y$ be a quasi-compact open in a noetherian scheme. There is a coherent ideal $\mathcal J$ cutting out $Y\setminus U$ whose blowup $p:\widetilde Y\to Y$ is projective, finitely presented, and an isomorphism over $U$, with \[ \mathcal J\cO_{\widetilde Y}=\cO_{\widetilde Y}(-D) \] for an effective Cartier divisor $D$ with complement $U$. If $Y/X$ is proper and finitely presented, so is $\widetilde Y/X$. \end{lemma} \begin{proof} Choose a coherent ideal $\mathcal J$ whose vanishing locus is $Y\setminus U$, and form \[ \widetilde Y=\operatorname{Proj}_Y\!\left(\bigoplus_{n\geq0}\mathcal J^n\right). \] Because the Rees algebra is generated in degree one by the finite ideal $\mathcal J$, this morphism is projective; noetherianity gives finite presentation. Over $U$, the ideal is the unit ideal, so the blowup is $\operatorname{Proj}_U\cO_U[T]=U$. On $\Spec A$, write $\mathcal J=(a_1,\ldots,a_r)$. The blowup charts have rings $A[\mathcal J/a_i]\subset A[a_i^{-1}]$, generated by the fractions $a_j/a_i$. Since $a_j=a_i(a_j/a_i)$, the extended ideal is $(a_i)$. Multiplication by $a_i$ is injective in this subring, so these ideals define effective Cartier divisors and glue to $\mathcal J\cO_{\widetilde Y}$ \cite[Section~31.33, especially Lemma~31.33.4]{Stacks}. The resulting divisor has support equal to the inverse image of $Y\setminus U$, so its complement is the copy of $U$. The final assertions about properness and finite presentation over $X$ follow by composition. \end{proof} For a quasi-compact quasi-separated scheme, descend the closed complement to a noetherian model and pull back the resulting modification. The line bundle and its section pull back with it. \begin{lemma}\label{lem:boundary-modification} Let $S$ be a quasi-compact quasi-separated scheme and let $U\subset S$ be quasi-compact open. There are a projective finitely presented $p:\widetilde S\to S$, an isomorphism over $U$, and a line-bundle section $s:\cO_{\widetilde S}\to\cL$ with nonvanishing locus $p^{-1}(U)\simeq U$. \end{lemma} \begin{proof} Choose a finite-type quasicoherent ideal $\mathcal J$ defining $S\setminus U$ \cite[Tag~01PH]{Stacks}. By the approximation in \cite[Tag~0EVA]{Stacks}, write $S$ as an inverse limit of noetherian schemes with affine transition maps and descend $V(\mathcal J)$ to one stage $S_0$. Let $\mathcal J_0$ be the ideal of that closed subscheme. Then $U_0=S_0\setminus V(\mathcal J_0)$ pulls back to $U$. The blowup $\widetilde S_0\to S_0$ of $\mathcal J_0$ is projective and finitely presented, is an isomorphism over $U_0$, and has effective Cartier boundary $D_0$ \cite[Section~31.33]{Stacks}. The canonical section of $\cO_{\widetilde S_0}(D_0)$ has nonvanishing locus $U_0$. Pull back the morphism, line bundle, and section to $S$. Projectivity, finite presentation, and the isomorphism over $U$ are preserved. Locally the nonvanishing locus is a principal open $D(t)$ and therefore commutes with base change. Thus the pulled-back section has nonvanishing locus $p^{-1}(U)\simeq U$. \end{proof} \subsection{Comparison with Huber analytification}\label{subsec:affinoid-Huber-comparison} We next compare the bounded charts with those of Huber analytification. Identifying their rings and analytic structures will give an affinoid description of the pseudocoherent subcategory. The assignment $\Spa(A,A^+)\mapsto\AnSpec(A,A^+)^\square$ sends rational covers to steady-localization covers and glues to a functor $(-)^\square$ from analytic adic spaces to analytic stacks \cite[A.19]{LiHuerta}. \begin{lemma}\label{lem:analytic-stack-open-gluing} For an open cover $(Z_i)_{i\in I}$ of a quasiseparated analytic adic space $Z$, put $Z_{i_0\ldots i_n}=Z_{i_0}\cap\cdots\cap Z_{i_n}$. The canonical map \[ \left|[n]\longmapsto\coprod_{(i_0,\ldots,i_n)\in I^{n+1}} Z_{i_0\ldots i_n}^{\square}\right| \xrightarrow{\sim}Z^{\square} \] is an equivalence. In particular, an increasing affinoid cover $(Z_n)$ with rational inclusions satisfies $\colim_n Z_n^\square\simeq Z^\square$. Every quasi-compact affinoid $W\subset Z$ then lies in some $Z_N$ and has a finite cover by common rational subdomains of $W$ and $Z_N$. \end{lemma} \begin{proof} The functor $(-)^\square$ already commutes with rational gluing \cite[A.19]{LiHuerta}. To handle an arbitrary open cover, choose affinoid charts in each $Z_i$ and regard them also as charts of $Z$. Quasiseparatedness makes their intersections quasi-compact, so we may refine those intersections by finitely many rational affinoids. Repeat this on the multiple intersections of the \v Cech diagram. The two covers give a bisimplicial diagram. Realizing in the rational direction first gives the stacks $Z_{i_0\ldots i_n}^\square$ and then the source of the displayed map. Reversing the order gives the affinoid presentation of $Z^\square$. Since the two colimits commute, the map is an equivalence. Common refinements identify the construction for different choices of charts. For an increasing cover, the rational inclusions give monomorphisms $Z_n^\square\to Z_{n+1}^\square$ \cite[Proposition~2.3.2]{RodriguezCamargo}. Their colimit is their union in the sheaf topos, and the first assertion identifies that union with $Z^\square$. Finally, $(W\cap Z_n)$ is an increasing open cover of the quasi-compact space $W$. A finite subcover is contained in one member, so $W\subset Z_N$ for some $N$. Rational subdomains of $Z_N$ contained in $W$ cover $W$; after pullback to $W$ they are still rational. A finite selection gives the common cover in the statement. \end{proof} Write $\alpha_T^{\An}=\jmath_T^*\alpha_T^{\Alg}$, where $\jmath_T:T^{\An}\to T^{\Alg}$ is the analytic-locus map. We compare the bounded charts and record the resulting formula for $\alpha_T^{\An}$ on a Huber affinoid. \begin{proposition}\label{prop:analytic-Huber-comparison} Let $V=\Spa(R,R^+)$ be a strongly noetherian complete Tate affinoid, and let $T/R$ be separated and finitely presented. There is a canonical equivalence over $V$, \[ \vartheta_{T,V}:T^{\An}\xrightarrow{\sim}(T^{\ad/V})^\square, \] natural in $T$ and coherently compatible with Zariski restriction and rational base change. Under $\vartheta_{T,V}$, an object belongs to $\Dpcbox$ exactly when its restriction to every strongly noetherian affinoid $W=\Spa(C,C^+)$ belongs to $\Cond_C(\Dpc(C))$. If $W\subset c_{T/V}^{-1}(U)$ for an affine $U=\Spec A\subset T$, then for $M\in\Dqc(T)$, \[ \alpha_T^{\An}(M)|_{W^\square} \simeq\Cond_C\bigl(C\otimes_A^{\mathbf L}\mathbf R\Gamma(U,M|_U)\bigr). \] \end{proposition} \begin{proof} We follow the bounded-chart comparison of \cite[proof of Theorem~A.19]{LiHuerta}. For $T=\Spec B$, choose a presentation \[ B=R[T_1,\ldots,T_m]/I. \] Let $b_i$ be the image of $T_i$, and choose a topologically nilpotent unit $\varpi\in R^+$. The inequalities $|\varpi^nb_i|\leq1$ define Huber affinoids $W_n$. Relative affine space is covered by the corresponding polydiscs, and the equations of $I$ cut out $T^{\ad/V}$ in them \cite[Proposition~3.8]{Huber1994} \cite[Examples~6.4--6.5]{Zavyalov}. Hence \[ T^{\ad/V}=\bigcup_{n\geq0}W_n. \] Inside $W_{n+1}$, the subspace $W_n$ is defined by $|\varpi^{n+1}b_i|\leq|\varpi|$. Since $\varpi$ is a unit, the inclusions form an increasing system of rational opens. The coordinate rings are \begin{align*} C_{0,n}&=R\langle\varpi^nT_1,\ldots,\varpi^nT_m\rangle,\\ C_n&=C_{0,n}/IC_{0,n}. \end{align*} Set $C_{0,n}^+=R^+\langle\varpi^nT_1,\ldots,\varpi^nT_m\rangle$, and let $C_n^+$ be the integral closure of its image in $C_n$. To write the equations in bounded coordinates, put $U_i=\varpi^nT_i$. They then generate \[ I_n=(f(\varpi^{-n}U_1,\ldots,\varpi^{-n}U_m):f\in I). \] Thus $C_n=R\langle\mathbf U\rangle/I_nR\langle\mathbf U\rangle$. The restriction from level $n+1$ to level $n$ is the substitution $U_i\mapsto\varpi U_i$. Put $P=R[T_1,\ldots,T_m]$. For a finite ideal $J\subset P$, pull $\widetilde J\hookrightarrow\cO_{\mathbf A_R^m}$ back to the polydisc. Flatness of analytification preserves the injection \cite[Lemma~6.7]{Zavyalov}. A finite presentation and the affinoid coherent-sheaf theorem \cite[Theorem~4.2]{Zavyalov} identify its map on sections with \[ J\otimes_PC_{0,n}\lhook\joinrel\longrightarrow C_{0,n}. \] The ideal criterion proves flatness \cite[Tag~00HD]{Stacks}. Hence \[ C_{0,n}\otimes_P^{\mathbf L}B\simeq C_n[0] \] in the ordinary derived category. The ideal $IC_{0,n}$ is finite since $C_{0,n}$ is noetherian. The finite-module completeness and open mapping theorems show that it is closed and that $C_n$ is complete for the quotient topology \cite[Theorem~1.2.9 and Corollaries~1.2.11--1.2.12]{KedlayaLiu}. Hence $W_n=\Spa(C_n,C_n^+)$. Let $B_n^{\mathrm{bd}}$ be the first bounded analytic ring, denoted by an underline in \cite[A.15]{LiHuerta}. For the polynomial algebra it is $(C_{0,n},C_{0,n}^+)^\square$ \cite[Lemma~A.8]{LiHuerta}. On passing to the quotient, the induced analytic structure has underlying condensed ring, before completion, \[ E_n=C_{0,n}\otimes_P^{\mathbf L}B, \] where $C_{0,n}$ has its topological condensed structure and $P,B$ are discrete \cite[proof of Proposition~A.16]{LiHuerta} \cite[Propositions~12.8 and~12.12]{ScholzeAnalytic}. Choose a bounded-above finite-free $P$-resolution $F^\bullet\to B$. Discrete condensification is exact \cite[Lemma~5.7]{Andreychev}, so this remains a free condensed resolution. It is K-flat, meaning that tensoring with an acyclic complex remains acyclic, and hence computes $E_n$. Tensoring with $\underline{C_{0,n}}$ gives the condensification of the ordinary complex $C_{0,n}\otimes_PF^\bullet$. The latter is quasi-isomorphic to $C_n[0]$ by flatness. Applying condensification to this quasi-isomorphism and identifying the finite module $C_n$ with its complete quotient topology gives \[ E_n\xrightarrow{\sim}C_n[0] \] \cite[Theorem~5.9 and Lemma~5.23]{Andreychev}. This is the canonical quotient map, so it is also an equivalence of the underlying animated condensed rings. The analytic structure on $C_n$ is induced from $(C_{0,n},C_{0,n}^+)^\square$, so completeness is tested after restriction to $C_{0,n}$, or equivalently on the discrete plus ring by \cite[Proposition~3.29]{Andreychev}. Factor its map as \[ C_{0,n}^+\twoheadrightarrow D_n^+:=\operatorname{im}(C_{0,n}^+\to C_n) \longrightarrow C_n^+. \] Both maps are integral; the first is a quotient map. The restriction criterion and invariance under integral extensions \cite[Propositions~3.21--3.22]{Andreychev} therefore identify all three completeness conditions with completeness for $(C_n,C_n^+)^\square$. The unit $C_n$ is complete \cite[Lemma~3.24]{Andreychev}, so analytic completion leaves the underlying ring unchanged. The two analytic structures have the same ring and complete connective modules; uniqueness \cite[Proposition~12.20]{ScholzeAnalytic} gives \begin{align*} B_n^{\mathrm{bd}}&\simeq(C_n,C_n^+)^\square,\\ \AnSpec B_n^{\mathrm{bd}}&\simeq W_n^\square. \end{align*} This also identifies the transition maps: on both sides they are the identity on $B$ together with $U_i\mapsto\varpi U_i$ between consecutive levels. Put $\underline{\mathcal U}_n=\AnSpec B_n^{\mathrm{bd}}$. The definition of the analytic locus also uses the second chart system $\mathcal U_n$ from \cite[A.15]{LiHuerta}. The two systems are related by maps \[ \underline{\mathcal U}_n\longrightarrow\mathcal U_n \longrightarrow\underline{\mathcal U}_{n+1} \] \cite[A.11]{LiHuerta}. The composites are the respective transition maps. On colimits they shift the index by one, hence act as the identity. Lemma~\ref{lem:analytic-stack-open-gluing} gives \[ T^{\An}\simeq\colim_n\underline{\mathcal U}_n \simeq\colim_nW_n^\square\simeq(T^{\ad/V})^\square. \] This defines $\vartheta_{T,V}$ for affine $T$. To compare the pseudocoherent subcategories, we also need the behavior of nuclearity under the maps between the two chart systems. Scalar extension along these maps preserves compact objects, since it sends measure generators to the corresponding target generators. It is monoidal and colimit-preserving. A class $\mathbf1\to P^\vee\otimes Q$ defining a trace-class map is carried, via the natural map to the dual of the image, to a class defining its scalar extension. Hence scalar extension preserves trace-class transitions, basic nuclear objects, and their colimits \cite[Propositions~2.9 and~2.11, Definition~5.32, and Proposition~5.33]{Andreychev} \cite[Definition~8.5 and Theorem~8.6]{ClausenScholze}. Let $W=\Spa(C,C^+)\subset c_{T/V}^{-1}(U)$ with $U=\Spec A$, and put $K=\mathbf R\Gamma(U,M|_U)$. By Lemma~\ref{lem:analytic-stack-open-gluing}, $W$ lies in some $W_N$ and has a finite cover $W_j=\Spa(C_j,C_j^+)$ rational in both. On $W_N$, comparison is scalar extension followed by condensification. Restricting to $W_j$ and to the algebraic open $U$ gives \[ \alpha_T^{\An}(M)|_{W_j^\square} \simeq\Cond_{C_j}(C_j\otimes_A^{\mathbf L}K) \] \cite[Lemma~5.10]{Andreychev}. On the other hand, restricting $\Cond_C(C\otimes_A^{\mathbf L}K)$ to $W_j$ gives the same object, because \[ C_j\otimes_C^{\mathbf L}(C\otimes_A^{\mathbf L}K) \simeq C_j\otimes_A^{\mathbf L}K. \] These identifications agree on rational intersections by tensor associativity. Descent therefore gives the claimed formula on $W$. Now suppose $K\in\Dpcbox(T^{\An})$. By definition, its restriction is pseudocompact on $\underline{\mathcal U}_N$ and nuclear on $\mathcal U_N$. Pulling back along $\underline{\mathcal U}_N\to\mathcal U_N$ makes it nuclear on $W_N^\square=\underline{\mathcal U}_N$ as well. For a Huber affinoid $W\subset W_N$, restrict to the common rational cover above. There the objects are both nuclear and pseudocompact. Nuclearity descends by \cite[Theorem~5.42]{Andreychev}, and pseudocompactness by \cite[Proposition~A.5]{LiHuerta} and \cite[Proposition~2.3.2]{RodriguezCamargo}. Thus $K|_{W^\square}\in\Cond_C(\Dpc(C))$ by \cite[A.15]{LiHuerta}. Conversely, suppose ordinary pseudocoherent representatives exist on all Huber affinoids. In particular, on $W_n^\square=\underline{\mathcal U}_n$ the object is nuclear and pseudocompact. The first chart system then gives pseudocompactness on $T^{\An}$ \cite[A.15 and Proposition~A.5]{LiHuerta}. Pullback along $\mathcal U_n\to\underline{\mathcal U}_{n+1}$ gives nuclearity on the second system and hence on $T^{\An}$ \cite[A.15 and Proposition~A.4]{LiHuerta}. This proves that the two descriptions of the pseudocoherent subcategory agree; see also \cite[Theorems~5.42, 5.44, and~5.50]{Andreychev}. Changing the presentation or $\varpi$ gives the common bounded refinements of Lemma~\ref{lem:bounded-chart-base-change}. On these refinements, the comparisons are defined by the same ring maps and scalar extensions. They are therefore independent of the choices and commute with morphisms and rational base change; tensor associativity gives the compatibilities for successive restrictions. Finally, let $T$ be separated and choose a finite affine cover with affine intersections. Huber analytification glues the local adic spaces, and Lemma~\ref{lem:analytic-stack-open-gluing} glues their associated stacks. The same intersections present $T^{\An}$ \cite[Theorem~A.13]{LiHuerta}, so the affine equivalences glue to $\vartheta_{T,V}$. The formula for complexes and the nuclearity and pseudocompactness conditions can all be checked on this cover. Their comparison, together with the functoriality and base-change identifications, therefore descends to $T$. \end{proof} On these affinoid representatives, pullback and tensor product are computed by ordinary derived tensor products. \begin{lemma}\label{lem:affinoid-pc-stability} Let $(R,R^+)$ be a strongly noetherian complete Tate Huber pair and let $h:S'\to S$ be a morphism of separated finitely presented $R$-schemes. For $K\in\Dpcbox(S^{\An})$ and $L\in\Dpcbox((S')^{\An})$, \[ h^{\An,*}K\otimes L\in\Dpcbox((S')^{\An}). \] These pullbacks are compatible with composition. \end{lemma} \begin{proof} Put $V=\Spa(R,R^+)$. Proposition~\ref{prop:analytic-Huber-comparison} lets us work on Huber affinoids. Given a strongly noetherian affinoid $W'\subset(S')^{\ad/V}$, choose a finite rational cover $W'_\nu=\Spa(C'_\nu,C_\nu^{\prime+})$ whose members map into affinoids $W_\nu=\Spa(C_\nu,C_\nu^+)$ of $S^{\ad/V}$. On these charts, choose ordinary pseudocoherent representatives $P_\nu,Q_\nu$ such that \begin{align*} K|_{W_\nu^\square}&\simeq\Cond_{C_\nu}(P_\nu),\\ L|_{(W'_\nu)^\square}&\simeq\Cond_{C'_\nu}(Q_\nu). \end{align*} Compatibility of condensification with pullback and tensor product gives \[ (h^{\An,*}K\otimes L)|_{(W'_\nu)^\square} \simeq\Cond_{C'_\nu} \bigl((C'_\nu\otimes_{C_\nu}^{\mathbf L}P_\nu) \otimes_{C'_\nu}^{\mathbf L}Q_\nu\bigr). \] Choose bounded-above finite-free resolutions $P_\nu^\bullet,Q_\nu^\bullet$ with upper bounds $a,b$. Scalar extension of the first remains finite free in each degree, and their tensor product has degree-$d$ term \[ \bigoplus_{d-b\leq i\leq a} (C'_\nu\otimes_{C_\nu}P_\nu^i)\otimes_{C'_\nu}Q_\nu^{d-i}. \] Only finitely many summands occur, all finite free, and the term is zero for $d>a+b$. This complex computes the derived tensor product and is a bounded-above finite-free resolution. Thus the tensor product is pseudocoherent \cite[Tags~0650 and~0DJE]{Stacks}. Rational descent gives the assertion on $W'$ and hence on $(S')^{\An}$. Setting $L$ equal to the unit gives preservation under pullback; setting $h$ equal to the identity gives closure under tensor product. Compatibility with composition is the associativity of scalar extension \cite[Lemma~5.10]{Andreychev}. \end{proof} \begin{lemma}\label{lem:pc-stability} Let $h:Z'\to Z$ be a morphism of separated finitely presented $X$-schemes. For $K\in\Dpcbox(Z^{\An}_{\cX})$ and $L\in\Dpcbox((Z')^{\An}_{\cX})$, \[ h^{\An,*}K\otimes L\in\Dpcbox((Z')^{\An}_{\cX}). \] \end{lemma} \begin{proof} This is local on the analytic base. Choose a rational-affinoid $V=\Spa(R,R^+)\subset\cX$ and let $h_R:Z'_R\to Z_R$ be the base change of $h$. On this chart, the complex in the statement becomes \[ h_R^{\An,*}(K|_{Z_R^{\An}})\otimes L|_{(Z'_R)^{\An}}. \] Both restrictions are pseudocoherent by Proposition~\ref{prop:relative-pc-descent}. The schemes $Z_R,Z'_R$ are separated and finitely presented over the strongly noetherian ring $R$, so Lemma~\ref{lem:affinoid-pc-stability} applies to this tensor product. These are restrictions of the given global complex, with its overlap maps. Proposition~\ref{prop:relative-pc-descent} therefore proves the global assertion. As before, taking the unit or the identity gives pullback preservation and tensor closure. Proposition~\ref{prop:forward-pc} also allows $L$ to be the image of an algebraic pseudocoherent complex. \end{proof} For proper $T/R$, a finite Huber affinoid cover gives an fpqc family of ordinary coordinate spectra. \begin{lemma}\label{lem:local-Huber-fpqc} Let $V=\Spa(R,R^+)$ be a strongly noetherian complete Tate affinoid and let $T/R$ be proper and finitely presented. There is a finite strongly noetherian affinoid cover $T^{\ad/V}=\bigcup_i W_i$, with $W_i=\Spa(B_i,B_i^+)\subset c_{T/V}^{-1}(U_i)$ for affine opens $U_i\subset T$, such that \[ \coprod_i\Spec B_i\longrightarrow T \] is fpqc. \end{lemma} \begin{proof} The proper adic space $T^{\ad/V}$ is quasi-compact \cite[Remark~6.3]{Zavyalov}. Choose a finite strongly noetherian affinoid cover $W_i=\Spa(B_i,B_i^+)$ subordinate to the inverse images of affine opens $U_i=\Spec A_i$ of $T$. For a finite ideal $J\subset A_i$, flatness of analytification gives \[ c_{T/V}^*\widetilde J|_{W_i}\lhook\joinrel\longrightarrow\cO_{W_i}, \] by \cite[Lemma~6.7]{Zavyalov}. Choose a finite presentation $A_i^r\to A_i^s\to J\to0$. After pullback it is the same matrix over $B_i$, so the coherent-sheaf theorem \cite[Theorem~4.2]{Zavyalov} identifies the source with the sheaf associated to $J\otimes_{A_i}B_i$. Taking sections gives an injection $J\otimes_{A_i}B_i\to B_i$. The ideal criterion now proves flatness \cite[Tag~00HD]{Stacks}. For surjectivity, start with a closed point $x\in T$. The skyscraper $\kappa(x)$ is nonzero. By Proposition~\ref{prop:affinoid-full-faithfulness} and the proper identification $T^{\An}\simeq T^{\Alg}$, its image $\alpha_T^{\An}\kappa(x)$ is also nonzero. It must therefore be nonzero on some $W_i^\square$. Proposition~\ref{prop:analytic-Huber-comparison} identifies that restriction with \[ \Cond_{B_i}\bigl(B_i\otimes_{A_i}^{\mathbf L} \Gamma(U_i,\kappa(x)|_{U_i})\bigr). \] Nonvanishing first forces $x\in U_i$. Flatness then identifies the ordinary complex with the nonzero ring $B_i\otimes_{A_i}\kappa(x)$ in degree zero. A prime of this ring gives a point of $\Spec B_i$ over $x$. Every point $y\in T$ has a closed specialization $x$, since $T$ is noetherian. Choose a lift of $x$ to $\Spec B_i$. Then $y\in U_i$ and $\mathfrak p_y\subset\mathfrak p_x$. Flat going-down lifts $\mathfrak p_y$ to a prime below the chosen prime over $x$. Thus the family covers every point of $T$. Each $\Spec B_i\to T$ is affine, as it factors through an affine open of the separated scheme $T$. Their finite disjoint union is therefore quasi-compact, flat, and surjective, hence fpqc. \end{proof} Combining localization with proper direct image gives the strongly noetherian case of \cite[Theorem~A.18]{LiHuerta}. We construct the preimage on affine opens and test its pseudocoherence on the preceding fpqc family. \begin{proposition}\label{prop:affinoid-pc-GAGA} Let $(R,R^+)$ be a strongly noetherian complete Tate Huber pair and let $S/R$ be proper and finitely presented. Then \[ \alpha_S^{\An}:\Dpc(S)\xrightarrow{\sim}\Dpcbox(S^{\An}). \] \end{proposition} \begin{proof} Analytification preserves pseudocoherence by the bounded-chart calculation \cite[A.15 and the paragraph preceding Theorem~A.18]{LiHuerta}. Properness identifies the two realizations \cite[Theorem~A.13]{LiHuerta}, and Proposition~\ref{prop:affinoid-full-faithfulness} gives full faithfulness. Let $C\in\Dpcbox(S^{\An})$. We construct its algebraic preimage on affine opens. Choose $j:U=\Spec A\hookrightarrow S$, with structural map $q:U\to\Spec R$. By Lemma~\ref{lem:boundary-modification}, there is a proper finitely presented $p:\widetilde S\to S$, unchanged over $U$, and a section $s$ of a line bundle $\cL$ whose nonvanishing locus is $U$. Write $\widetilde j:U\hookrightarrow\widetilde S$, $\widetilde\pi:\widetilde S\to\Spec R$, and $\widetilde C=p^{\An,*}C$. The scheme $\widetilde S$ is proper, so its algebraic and analytic realizations agree. Lemma~\ref{lem:affinoid-pc-stability} makes each twist $\widetilde C\otimes\cL^n$ pseudocoherent. Localization expresses the restriction to $U$ in terms of these twists: \[ \mathbf R\widetilde j_*^{\Alg}\widetilde j^{\Alg,*}\widetilde C \simeq\colim_n(\widetilde C\otimes\cL^n). \] Apply $\mathbf R\widetilde\pi_*$. By Theorem~\ref{thm:local-solid-grauert}, each \[ E_n=\mathbf R\widetilde\pi_*(\widetilde C\otimes\cL^n) \] is the condensification of a pseudocoherent $R$-complex, and Corollary~\ref{cor:local-grauert-colimits} permits passage of the colimit through this direct image. Full faithfulness of $\Cond_R$ and Lemma~\ref{lem:functor-descent} lift the sequence $E_\bullet$, with its transition maps, to $M_\bullet$ in $\Dpc(R)$. Put $M=\colim_nM_n$ in $\mathrm D(R)$. Colimit preservation gives \[ \mathbf R\widetilde\pi_*\mathbf R\widetilde j_*^{\Alg} \widetilde j^{\Alg,*}\widetilde C \simeq\colim_nE_n\simeq\Cond_R(M). \] Because $p$ is the identity over $U$, the restricted complex is $j^{\Alg,*}C$. Also $\widetilde\pi\circ\widetilde j=q$, so the composite direct image is right adjoint to $q^{\Alg,*}$ and is therefore $\mathbf Rq_*^{\Alg}$. Thus $\Cond_R(M)$ is the underlying complete $R$-module complex of $j^{\Alg,*}C$. Transfer the $\Cond_R(A)$-action of Lemma~\ref{lem:affinoid-module-category} to $\Cond_R(M)$. Strong monoidality identifies it with a map $\Cond_R(A\otimes_R^{\mathbf L}M)\to\Cond_R(M)$. Full faithfulness lifts this map, and Lemma~\ref{lem:module-image} lifts its unit, associativity, and higher relations. The resulting $A$-complex $N_U$ realizes $j^{\Alg,*}C$. Carry out the construction on a finite affine cover $(U_a)$. The restrictions of $C$ identify the analytic images on overlaps. Full faithfulness lifts those identifications and the homotopies on higher intersections. Zariski descent therefore gives $N\in\Dqc(S)$ with $\alpha_S^{\An}(N)\simeq C$. To prove that $N$ is pseudocoherent, use the family $s_i:\Spec B_i\to S$ from Lemma~\ref{lem:local-Huber-fpqc}, with corresponding affinoids $W_i$, and put $N_i=\mathbf Ls_i^*N$. The affinoid comparison gives \[ \Cond_{B_i}(N_i)\simeq C|_{W_i^\square} \simeq\Cond_{B_i}(P_i) \] for some $P_i\in\Dpc(B_i)$. Full faithfulness gives $N_i\simeq P_i$, so $N$ is pseudocoherent after the fpqc pullback. Faithfully flat descent \cite[Tags~066D and~068R]{Stacks} gives $N\in\Dpc(S)$ and proves essential surjectivity. \end{proof} \section{Pseudocoherent GAGA}\label{sec:stable-proof} Let $Y\to X$ be proper. By Section~\ref{sec:relative}, the comparison is already fully faithful. We prove essential surjectivity by constructing preimages on affine opens of $Y$ and gluing them. The boundary formula recovers their direct images on $X$, and lifting the algebra actions recovers the complexes on the opens. \subsection{Faithfully flat detection of pseudocoherence}\label{subsec:reflection} Before constructing the preimages, we show that they will be pseudocoherent. After pullback to the fpqc family on $X$, affinoid GAGA gives a pseudocoherent preimage. Full faithfulness identifies it with the pullback of the given complex, so pseudocoherence descends. \begin{proposition}\label{prop:reflection} Let $f:Y\to X$ be proper and let $j:U\hookrightarrow Y$ be quasi-compact open. For $C\in\Dpcbox(Y^{\An}_{\cX})$ and $N\in\Dqc(U)$, an equivalence \[ \alpha_U(N)\simeq j^{\Alg,*}C \] implies $N\in\Dpc(U)$, using the proper identification $Y^{\An}_{\cX}\simeq Y^{\Alg}_{\cX}$. \end{proposition} \begin{proof} We may check pseudocoherence locally on $U$. Restrict the given equivalence to a finite affine cover, using pullback compatibility of $\alpha$. It is therefore enough to treat $U=\Spec B$. Since $X$ has affine diagonal, the structural map $q:U\to X$ is affine. Choose the finite fpqc family from Lemma~\ref{lem:finite-fpqc-charts} and form \begin{align*} Y_i&=Y\times_X\Spec R_i,\\ U_i&=U\times_X\Spec R_i. \end{align*} The map $q$ is affine, so $U_i=\Spec B_i$. Their disjoint union is $\Spec B'$, where $B'=\prod_iB_i$. It is the base change of an fpqc family, hence $B\to B'$ is faithfully flat. Put $N_i=N\otimes_B^{\mathbf L}B_i$ and $C_i=C|_{Y_i^{\An}}$. Since $Y_i$ is proper over the strongly noetherian ring $R_i$, Proposition~\ref{prop:affinoid-pc-GAGA} gives $M_i\in\Dpc(Y_i)$ with $\alpha_{Y_i}^{\An}(M_i)\simeq C_i$. Rational base change identifies the image of $N_i$ with $C_i|_{U_i^{\Alg}}$, which is also the image of $M_i|_{U_i}$. Full faithfulness from Proposition~\ref{prop:affinoid-full-faithfulness} gives \[ N_i\simeq M_i|_{U_i}. \] Each $N_i$ is therefore pseudocoherent. Combining the members of the finite family gives \[ N\otimes_B^{\mathbf L}B'\simeq\prod_iN_i. \] Under $\mathrm D(B')\simeq\prod_i\mathrm D(B_i)$, this is a finite family of pseudocoherent complexes. Their upper bounds have a common maximum, so the product is pseudocoherent over $B'$. Faithfully flat descent \cite[Tag~068R]{Stacks} now gives $N\in\Dpc(B)$. \end{proof} \subsection{Boundary pushforward and affine algebraization}\label{subsec:local-algebraization} We express direct image from an open as a colimit of proper direct images on a blowup, following \cite[Theorem~13.10]{ClausenScholze} and \cite[Theorem~A.18]{LiHuerta}. GAGA on $X$ then algebraizes the terms together with their transition maps. Let $f:Y\to X$ be proper, let $j:U\hookrightarrow Y$ be quasi-compact open, and put $q=f\circ j$. Lemma~\ref{lem:blowup-boundary} gives a blowup $p:\widetilde Y\to Y$, unchanged over $U$, with effective Cartier boundary $D$. Write $\widetilde f=f\circ p$. Both $Y$ and $\widetilde Y$ are proper over $X$, so their algebraic and analytic realizations agree. For $C\in\Cbox(Y^{\An}_{\cX})$, put $\widetilde C=(p^{\An})^*C$ and \[ E_n(C)=\mathbf R\widetilde f_*^{\An} \bigl(\widetilde C\otimes\cO_{\widetilde Y}(nD)\bigr) \] for $n\geq0$. The canonical section of $\cO(D)$ induces the maps $E_n(C)\to E_{n+1}(C)$. \begin{theorem}\label{thm:boundary-pushforward} With this notation, for every $C\in\Cbox(Y^{\An}_{\cX})$ there is a canonical equivalence, natural in $C$, \[ \mathbf Rq_*^{\Alg}j^{\Alg,*}C \simeq\colim_{n\geq0} \mathbf R\widetilde f_*^{\An} \bigl(\widetilde C\otimes\cO_{\widetilde Y}(nD)\bigr). \] If $C\in\Dpcbox(Y^{\An}_{\cX})$, then $E_n(C)\in\Dpcbox(\cX)$ and the sequence lifts functorially to $M_\bullet(C)\in\operatorname{Fun}(\mathbf N,\Dpc(X))$, with $\alpha_XM_\bullet(C)\simeq E_\bullet(C)$. The lift with its comparison is unique up to contractible choice. Its colimit $M_U(C)\in\Dqc(X)$ satisfies \[ \alpha_X(M_U(C))\simeq\mathbf Rq_*^{\Alg}j^{\Alg,*}C \] and is canonically independent of the blowup, with this comparison. \end{theorem} \begin{proof} Let $\widetilde j:U\hookrightarrow\widetilde Y$ be the open immersion, and take $C\in\Cbox(Y^{\An}_{\cX})$. We use the identifications of the proper algebraic and analytic realizations from Proposition~\ref{prop:relative-construction}. The canonical section of $\cO_{\widetilde Y}(D)$ has nonvanishing locus $U$. Lemma~\ref{lem:Cartier-localization} therefore gives \[ \mathbf R\widetilde j_*^{\Alg}\widetilde j^{\Alg,*}\widetilde C \simeq\colim_n \bigl(\widetilde C\otimes\cO_{\widetilde Y}(nD)\bigr). \] Since $p$ is an isomorphism over $U$, we have $\widetilde j^{\Alg,*}\widetilde C\simeq j^{\Alg,*}C$. The identity $\widetilde f\circ\widetilde j=q$ identifies the composite pullback with $q^{\Alg,*}$. For $A\in\Cbox(U^{\Alg}_{\cX})$ and $B\in\Cbox(\cX)$, adjunction gives \begin{align*} \Map_{\cX}(B,\mathbf R\widetilde f_*^{\An}\mathbf R\widetilde j_*^{\Alg}A) &\simeq\Map_{U^{\Alg}_{\cX}} (\widetilde j^{\Alg,*}(\widetilde f^{\An})^*B,A)\\ &\simeq\Map_{U^{\Alg}_{\cX}}(q^{\Alg,*}B,A)\\ &\simeq\Map_{\cX}(B,\mathbf Rq_*^{\Alg}A). \end{align*} By Yoneda, this identifies $\mathbf R\widetilde f_*^{\An}\mathbf R\widetilde j_*^{\Alg}$ with $\mathbf Rq_*^{\Alg}$. Applying $\mathbf R\widetilde f_*^{\An}$ to the localization formula and using its colimit preservation from Proposition~\ref{prop:global-Grauert}, we obtain \begin{align*} \mathbf Rq_*^{\Alg}j^{\Alg,*}C &\simeq\mathbf R\widetilde f_*^{\An} \mathbf R\widetilde j_*^{\Alg}\widetilde j^{\Alg,*}\widetilde C\\ &\simeq\mathbf R\widetilde f_*^{\An} \left(\colim_n\widetilde C\otimes\cO_{\widetilde Y}(nD)\right)\\ &\simeq\colim_n E_n(C). \end{align*} Naturality follows from that of the localization map and the adjunctions. Now suppose $C$ is pseudocoherent. Lemma~\ref{lem:pc-stability} and Proposition~\ref{prop:forward-pc} show that \[ \widetilde C\otimes\cO_{\widetilde Y}(nD) \in\Dpcbox(\widetilde Y^{\An}_{\cX}). \] Proposition~\ref{prop:global-Grauert} gives $E_n(C)\in\Dpcbox(\cX)$ for every $n$. Applying an inverse of the equivalence in Proposition~\ref{prop:base-pc-GAGA} to $E_\bullet(C)$ gives a sequence $M_\bullet(C)$ with $\alpha_XM_\bullet(C)\simeq E_\bullet(C)$. Lemma~\ref{lem:functor-descent} ensures that this equivalence includes the transition maps and their composition homotopies. The sequence is functorial in $C$ and, together with its comparison, unique up to contractible choice. Set $M_U(C)=\colim_nM_n(C)$ in $\Dqc(X)$. Colimit preservation gives \[ \alpha_X(M_U(C))\simeq\colim_n E_n(C) \simeq\mathbf Rq_*^{\Alg}j^{\Alg,*}C. \] A second blowup gives a preimage of the same right-hand side. Full faithfulness lifts the specified equivalence between their images and all composition homotopies. The colimit with its comparison is therefore independent of the blowup. \end{proof} When $q:U\to X$ is affine, $j^{\Alg,*}C$ is determined by its direct image together with the $\alpha_X(q_*\cO_U)$-action. The preceding theorem algebraizes the underlying complex; full faithfulness now lets us lift the action. \begin{proposition}\label{prop:local-algebraization} Let $f:Y\to X$ be proper and let $j:U\hookrightarrow Y$ be any affine open. For every $C\in\Dpcbox(Y^{\An}_{\cX})$, there is $N_U\in\Dpc(U)$ with \[ \alpha_U(N_U)\simeq j^{\Alg,*}C \] in $\Cbox(U^{\Alg}_{\cX})$. \end{proposition} \begin{proof} Since $U$ is affine and $X$ has affine diagonal, $q=f\circ j:U\to X$ is affine. By Theorem~\ref{thm:boundary-pushforward}, there is $M\in\Dqc(X)$ with \[ \mathbf Rq_*^{\Alg}j^{\Alg,*}C\simeq\alpha_X(M). \] Put $\cA=q_*\cO_U$. By Lemma~\ref{lem:affine-module-category}, $j^{\Alg,*}C$ is an $\alpha_X(\cA)$-module whose underlying object is the displayed direct image. Transferring its action along this equivalence gives \[ \alpha_X(\cA)\otimes\alpha_X(M)\longrightarrow\alpha_X(M). \] Strong monoidality identifies the source with $\alpha_X(\cA\otimes M)$, so full faithfulness lifts the action to $\cA\otimes M\to M$. Lemma~\ref{lem:module-image} lifts the unit, associativity, and higher homotopies. This defines $N_U\in\operatorname{Mod}_{\cA}(\Dqc(X))\simeq\Dqc(U)$ with image $j^{\Alg,*}C$. Proposition~\ref{prop:reflection} then gives $N_U\in\Dpc(U)$. \end{proof} \subsection{The stable equivalence and proper pushforward}\label{subsec:stable-equivalence} We assemble the affine algebraizations into a global equivalence and then compare proper direct images. The algebraic direct images lie in the required category by the following finiteness lemma. \begin{lemma}\label{lem:proper-Dminus} Let $h:S\to T$ be a proper morphism between noetherian schemes. Then \(\mathbf Rh_*:\mathrm D^-_{\mathrm{coh}}(S) \longrightarrow \mathrm D^-_{\mathrm{coh}}(T)\). \end{lemma} \begin{proof} Choose a finite affine cover $(T_a)$ of $T$ and cover each $S_a=h^{-1}(T_a)$ by $r_a$ affine opens. Since $S_a$ is separated, their intersections are affine. The alternating \v Cech complex therefore computes quasicoherent cohomology on $S_a$ in degrees $0,\ldots,r_a-1$. This bound also holds after replacing $T_a$ by an affine open inside it: the covering opens and their intersections pull back to affine schemes, since they are fiber products of affines over $T_a$. Thus the same calculation bounds the sheaves $R^ih_*$. Set $d=\max_a(r_a-1)$. For a complex in degrees at most $q$, sections on an affine intersection are given by its ordinary module complex. The \v Cech degree adds at most $d$, so \[ \mathbf Rh_*\bigl(\Dqc(S)^{\leq q}\bigr) \subset\Dqc(T)^{\leq q+d}. \] Let $M\in\mathrm D^-_{\mathrm{coh}}(S)$ with $M\in\mathrm D^{\leq N}$. Its direct image lies in degrees at most $N+d$. For coherence in degree $n$, use the triangle \[ \tau_{\leq n-d-1}M\longrightarrow M\longrightarrow\tau_{\geq n-d}M. \] The direct image of the first term lies in degrees at most $n-1$. The long exact cohomology sequence therefore identifies \[ H^n(\mathbf Rh_*M) \simeq H^n\bigl(\mathbf Rh_*\tau_{\geq n-d}M\bigr). \] The complex $\tau_{\geq n-d}M$ has coherent cohomology in the finite interval $[n-d,N]$. Its Postnikov triangles express it as successive extensions of these shifted coherent sheaves. Proper direct image has coherent cohomology on each such sheaf, and coherence is preserved by exact triangles on the noetherian target \cite[Tags~0BCF and~02O5]{Stacks}. Hence the right side is coherent. Since $n$ was arbitrary, $\mathbf Rh_*M$ is bounded above with coherent cohomology. \end{proof} We can now prove the bootstrapping theorem. \begin{theorem}\label{thm:stable-main} Under the hypotheses of Section~\ref{subsec:setup}, let $f:Y\to X$ be proper. The functor \[ \alpha_Y:\Dqc(Y)\longrightarrow\Cbox(Y^{\An}_{\cX}) \] is fully faithful, colimit-preserving, and strong symmetric monoidal. It takes perfect complexes to compact objects and induces a symmetric monoidal equivalence \[ \Dpc(Y)\xrightarrow{\sim}\Dpcbox(Y^{\An}_{\cX}). \] For every $M\in\Dqc(Y)$, the canonical comparisons are equivalences: \begin{align*} \alpha_X(\mathbf Rf_*M)&\xrightarrow{\sim} \mathbf Rf_*^{\An}\alpha_Y(M),\\ \mathbf R\Gamma(Y,M)&\xrightarrow{\sim} \mathbf R\Gamma_\square(Y^{\An}_{\cX},\alpha_YM). \end{align*} \end{theorem} \begin{proof} Use properness to identify $Y^{\An}_{\cX}$ with $Y^{\Alg}_{\cX}$. Proposition~\ref{prop:relative-full-faithfulness} gives full faithfulness, colimit preservation, and the monoidal structure on $\alpha_Y$, while Proposition~\ref{prop:forward-pc} shows that it preserves pseudocoherence. We begin with essential surjectivity. Let $C\in\Dpcbox(Y^{\An}_{\cX})$ and choose a finite affine cover $(U_a)$ of $Y$. On each member, Proposition~\ref{prop:local-algebraization} supplies a preimage $N_a\in\Dpc(U_a)$ and an equivalence \[ e_a:\alpha_{U_a}(N_a)\xrightarrow{\sim}C|_{U_a^{\Alg}}. \] On $U_{ab}=U_a\cap U_b$, the composite of $e_a$ with the inverse of $e_b$ gives \[ \alpha_{U_{ab}}(N_a|_{U_{ab}}) \xrightarrow{\sim} \alpha_{U_{ab}}(N_b|_{U_{ab}}). \] Full faithfulness lifts this to an equivalence $\theta_{ab}$ of algebraic complexes. On a triple intersection, the comparisons with $C$ give a homotopy between the images of $\theta_{bc}\theta_{ab}$ and $\theta_{ac}$. The same mapping-space equivalence lifts that homotopy. Applying Lemma~\ref{lem:functor-descent} to the full \v Cech nerve recovers all higher compatibilities as well. These pseudocoherent descent data glue to $N\in\Dpc(Y)$. The comparisons $e_a$ descend at the same time, giving $\alpha_Y(N)\simeq C$ and proving essential surjectivity. Locally, the tensor product of two bounded-above finite-free resolutions has finitely many finite free summands in each degree, so $\Dpc(Y)$ is closed under tensor products \cite[Tag~0DJE]{Stacks}. The analytic subcategory is tensor-closed by Lemma~\ref{lem:pc-stability}. The strong symmetric monoidal structure on $\alpha_Y$ therefore restricts to the equivalence. For $M\in\Dqc(Y)$, apply $\alpha_Y$ to the counit $\mathbf Lf^*\mathbf Rf_*M\to M$. Compatibility with pullback gives \[ (f^{\An})^*\alpha_X(\mathbf Rf_*M) \simeq\alpha_Y(\mathbf Lf^*\mathbf Rf_*M) \longrightarrow\alpha_Y(M). \] By the adjoint pair $(f^{\An})^*\dashv\mathbf Rf_*^{\An}$, this defines \[ \gamma_M:\alpha_X(\mathbf Rf_*M) \longrightarrow E:=\mathbf Rf_*^{\An}\alpha_Y(M). \] Suppose first that $M$ is pseudocoherent. Lemma~\ref{lem:proper-Dminus} gives $\mathbf Rf_*M\in\Dpc(X)$, while Propositions~\ref{prop:forward-pc} and~\ref{prop:global-Grauert} give $E\in\Dpcbox(\cX)$. Choose $N\in\Dpc(X)$ with $E\simeq\alpha_X(N)$ by base GAGA. Full faithfulness lifts $\gamma_M$ to $u_M:\mathbf Rf_*M\to N$. For $P\in\Perf(X)$, the two adjunctions give \begin{align*} \Map_X(P,\mathbf Rf_*M)&\simeq\Map_Y(f^*P,M),\\ \Map_X(P,N)&\simeq\Map_{\cX}(\alpha_XP,E)\\ &\simeq\Map_{Y^{\An}_{\cX}}((f^{\An})^*\alpha_XP,\alpha_YM)\\ &\simeq\Map_{Y^{\An}_{\cX}}(\alpha_Y(f^*P),\alpha_YM)\\ &\simeq\Map_Y(f^*P,M). \end{align*} Under these identifications, $\Map_X(P,u_M)$ is the identity. Indeed, a map $P\to\mathbf Rf_*M$ corresponds to $f^*P\to M$. Applying $\alpha_Y$ and analytic adjunction gives its composite with $\gamma_M$, by the definition of $\gamma_M$ from the counit. The same calculation applies to the full mapping spectra. It follows that the cofiber $Q$ of $u_M$ satisfies $\Map_X(P,Q)=0$ for every perfect $P$. Perfect complexes generate $\Dqc(X)$, so $Q=0$. Both $u_M$ and $\gamma_M$ are therefore equivalences. For arbitrary $M$, compare the two functors \begin{align*} M&\longmapsto\alpha_X(\mathbf Rf_*M),\\ M&\longmapsto\mathbf Rf_*^{\An}\alpha_Y(M). \end{align*} Both functors preserve small colimits. On the analytic side, apply Proposition~\ref{prop:global-Grauert} and colimit preservation of $\alpha_Y$. On the algebraic side, $\mathbf Rf_*$ preserves direct sums for quasi-compact quasi-separated $f$ \cite[Tag~08DZ]{Stacks}. As it is exact, it preserves all small colimits, and so does its composite with $\alpha_X$. Write $M\simeq\colim_iP_i$ as a filtered colimit of perfect complexes \cite[Tags~09IS and~09M0]{Stacks}. Naturality of the counit identifies $\gamma_M$ with the composite \[ \alpha_X(\mathbf Rf_*M) \simeq\colim_i\alpha_X(\mathbf Rf_*P_i) \xrightarrow{\colim_i\gamma_{P_i}} \colim_i\mathbf Rf_*^{\An}\alpha_Y(P_i) \simeq\mathbf Rf_*^{\An}\alpha_Y(M). \] Every $\gamma_{P_i}$ is an equivalence by the pseudocoherent case. Taking their colimit proves the comparison for $M$. For $P\in\Perf(Y)$ and a filtered diagram $(N_i)$ in $\Cbox(Y^{\An}_{\cX})$, duality gives \[ \Map(\alpha_YP,N_i) \simeq\mathbf R\Gamma_\square \bigl(Y^{\An}_{\cX},\alpha_Y(P^\vee)\otimes N_i\bigr). \] Compute these global sections by first applying $\mathbf Rf_*^{\An}$ and then $\mathbf R\Gamma_\square(\cX,-)$. Tensor product and proper direct image preserve colimits, and the last functor preserves filtered colimits by Lemma~\ref{lem:compact-unit-base}. Consequently, \[ \colim_i\Map(\alpha_YP,N_i) \simeq\Map(\alpha_YP,\colim_iN_i), \] proving compactness of $\alpha_YP$. Finally, the cohomology comparison follows directly from full faithfulness, since $\alpha_Y$ sends the structure sheaf to the tensor unit: \[ \mathbf R\Gamma(Y,M)=\Map_Y(\cO_Y,M) \simeq\Map_{Y^{\An}_{\cX}}(\cO,\alpha_YM) =\mathbf R\Gamma(Y^{\An}_{\cX},\alpha_YM). \] This holds for every $M\in\Dqc(Y)$ and completes the proof. \end{proof} \section{Coherent sheaves and classical GAGA}\label{sec:classical} We pass from the solid comparison to ordinary Huber GAGA. On an affinoid, sheafification and condensification apply to the same finite modules and bounded complexes. Gluing these identifications gives the coherent and bounded-derived comparisons. We identify the resulting functors with ordinary pullback, then restrict to vector bundles and perfect complexes using duality and Tor-amplitude. On coherent hearts, tensor product means the ordinary tensor product, equivalently $H^0$ of derived tensor product. \subsection{The transported \texorpdfstring{$t$}{t}-structure}\label{subsec:pc-t} Let $T/X$ be proper and finitely presented, and put $\mathcal P_T=\Dpcbox(T^{\An}_{\cX})$. Transport the $t$-structure along the equivalence $\alpha_T:\Dpc(T)\simeq\mathcal P_T$ of Theorem~\ref{thm:stable-main}. Thus the degree conditions on $\alpha_T(M)$ are those on $M$. Write $\mathcal P_T^\heartsuit$ for the heart and $\mathcal P_T^b$ for the bounded subcategory. Restriction to a base affinoid preserves these degree conditions. \begin{lemma}\label{lem:transported-local} For $V=\Spa(R,R^+)\in\operatorname{RatAff}_X(\cX)$ and $T_R=T\times_X\Spec R$, restriction \[ \mathcal P_T\longrightarrow\Dpcbox(T_R^{\An}) \] is $t$-exact for the structures transported from $\Dpc(T)$ and $\Dpc(T_R)$. These structures are also preserved by rational localization of $V$. \end{lemma} \begin{proof} Take $M\in\Dpc(T)$ and write $M_R=\mathbf L(T_R\to T)^*M$. The base-change construction gives \[ \rho_V\alpha_T(M)\simeq\alpha_{T_R}^{\An}(M_R). \] On an affine $\Spec B\subset T$ over $\Spec A\subset X$, this pullback replaces $B$ by $B\otimes_A R$. Lemma~\ref{lem:base-flat} makes $A\to R$ flat, and hence $B\to B\otimes_A R$ flat as well. Thus \[ \mathcal H^j(M_R)\simeq (T_R\to T)^*\mathcal H^j(M). \] Thus pullback preserves both degree conditions and commutes with truncation, proving $t$-exactness for the transported structures. For rational $V'\subset V$, the same argument applies to the flat map $R\to R'$ and the pullback $T_{R'}\to T_R$. \end{proof} \subsection{Bounded and coherent comparison}\label{subsec:bounded-bridge} We first glue complexes with a fixed lower cohomological bound, then impose coherence and an upper bound. \begin{lemma}\label{lem:derived-open-descent} Let $(Z_i)_{i\in I}$ cover a ringed space $(Z,\cO_Z)$, and put $Z_{i_0\ldots i_n}=Z_{i_0}\cap\cdots\cap Z_{i_n}$. For every integer $a$, restriction gives \[ \mathrm D^{\geq a}(\cO_Z)\xrightarrow{\sim} \lim_{[n]\in\Delta}\prod_{(i_0,\ldots,i_n)\in I^{n+1}} \mathrm D^{\geq a}(\cO_{Z_{i_0\ldots i_n}}), \] including descent of the inherited mapping spectra. \end{lemma} \begin{proof} Let $H\mathcal O_Z$ denote the Eilenberg--MacLane sheaf of rings: its degree-zero homotopy sheaf is $\mathcal O_Z$, and its other homotopy sheaves vanish. We first work in \[ \operatorname{Mod}_{H\mathcal O_Z}(\operatorname{Shv}(Z,\mathrm{Sp})) \] where open descent follows by gluing sheaves of spectra together with their module actions and homotopies. Equivalently, formation of module categories commutes with the limits of symmetric monoidal categories in the descent diagram \cite[Section~4.8.5]{LurieHA}. Cohomology sheaves commute with restriction, so vanishing below $a$ is local. Descent therefore restricts to the given lower bound. For the topos of $Z$, \cite[Proposition~2.1.8]{LurieDAGVIII} identifies bounded-below complexes of $\mathcal O_Z$-modules with the corresponding sheaves of $H\mathcal O_Z$-module spectra. This identification commutes with open restriction. Applying it on each intersection, with the bound $\geq a$ in our cohomological convention, gives the asserted equivalence. The limit also computes mapping spaces. For any fixed shift, choose a lower bound containing the two complexes and that shift, and apply the same assertion. This identifies all homotopy groups of the inherited mapping spectra, proving the final claim. \end{proof} We apply this lower-bounded descent to complexes with coherent cohomology in a fixed interval. For a finite cover, a common interval can be chosen from the local bounds. \begin{lemma}\label{lem:finite-bounded-coherent-descent} Let $(Z,\cO_Z)$ be a ringed space with coherent structure sheaf. For a finite open cover $(Z_i)_{i\in I}$, put $Z_{i_0\ldots i_n}=Z_{i_0}\cap\cdots\cap Z_{i_n}$. Then \[ \Dbcoh(Z)\xrightarrow{\sim} \lim_{[n]\in\Delta}\prod_{(i_0,\ldots,i_n)}\Dbcoh(Z_{i_0\ldots i_n}). \] For a fixed cohomological interval $[a,b]$, the same assertion holds for an arbitrary open cover. \end{lemma} \begin{proof} First fix an interval $[a,b]$; the cover may then be arbitrary. Lemma~\ref{lem:derived-open-descent} glues the datum to $K\in\mathrm D^{\geq a}(\mathcal O_Z)$. For $j>b$, the sheaf $\mathcal H^j(K)$ is zero on every $Z_i$ and hence zero. In degrees $a\leq j\leq b$, its restrictions are coherent, so it is coherent. Thus the descended object belongs to $\Dbcoh(Z)^{[a,b]}$. The lower-bounded descent lemma also identifies the mapping spaces. Now let the cover be finite, without fixing an interval in advance. Choose $[a_i,b_i]$ for each local complex $K_i$ and set $a=\min_i a_i$, $b=\max_i b_i$. The higher \v Cech components are restrictions of the $K_i$, so the entire datum has bounds $[a,b]$. The fixed-interval assertion therefore gives a bounded coherent global complex. Conversely, restriction preserves coherence and bounds. For two global bounded coherent complexes, choose a common lower bound and apply Lemma~\ref{lem:derived-open-descent} to their mapping spaces. This proves full faithfulness. \end{proof} The corresponding description on an open basis is as follows. \begin{lemma}\label{lem:category-basis-descent} Let $S$ be quasi-compact and quasiseparated, and let $\mathcal B$ be a quasi-compact open basis whose finite intersections have finite $\mathcal B$-covers. If a presheaf $\mathcal F$ of $\infty$-categories on the quasi-compact opens satisfies finite \v Cech descent, including mapping spaces, then \[ \mathcal F(S)\xrightarrow{\sim} \operatorname{Cart}_{V\in\mathcal B}\mathcal F(V). \] \end{lemma} \begin{proof} Let $(K_V)_{V\in\mathcal B}$ be a cartesian family, and choose a finite basis cover $S=V_1\cup\cdots\cup V_r$. Cover $V_i\cap V_j$ by finitely many basis opens $W$. Cartesianness gives \[ K_{V_i}|_W\xrightarrow{\sim}K_W \xleftarrow{\sim}K_{V_j}|_W. \] Hence the two restrictions agree on each $W$. Their comparisons agree on smaller basis opens, again by cartesianness, so mapping-space descent glues an equivalence over $V_i\cap V_j$. Repeating the argument on triple and higher intersections glues the composition homotopies and their compatibilities. Finite descent gives $K\in\mathcal F(S)$. For any $V\in\mathcal B$, choose a finite basis refinement of the cover $(V\cap V_i)$. On this refinement, $K|_V$ and $K_V$ have the same descent data, so they are equivalent. The same mapping-space descent glues compatible maps and homotopies, proving full faithfulness. Passing to a common refinement compares different choices of cover, so the equivalence is the restriction functor in the statement. \end{proof} For bounded coherent complexes, the cartesian identifications carry the bounds on a finite basis cover to every member of the basis. \begin{lemma}\label{lem:bounded-coherent-descent} Let $(Z,\cO_Z)$ have coherent structure sheaf, let $S$ be a quasi-compact quasi-separated space, and let $p:Z\to S$ be continuous. Let $\mathcal B$ be a basis of quasi-compact opens whose finite intersections have finite $\mathcal B$-covers. Put $Z_V=p^{-1}(V)$. Restriction induces an equivalence \[ \Dbcoh(Z) \xrightarrow{\ \sim\ } \operatorname{Cart}_{V\in\mathcal B}\bigl(\Dbcoh(Z_V)\bigr). \] \end{lemma} \begin{proof} Let $(K_V)_{V\in\mathcal B}$ be a cartesian family in $\Dbcoh(Z_V)$. Choose a finite basis cover $S=\bigcup_iV_i$ and an interval $[a,b]$ containing the bounds of all $K_{V_i}$. For any $V\in\mathcal B$, quasi-compactness and quasiseparatedness give a finite refinement \[ V=\bigcup_{i,\nu}W_{i\nu} \] with $W_{i\nu}\in\mathcal B$ and $W_{i\nu}\subset V\cap V_i$. On each $W_{i\nu}$, cartesianness identifies the restrictions of $K_V$ and $K_{V_i}$. The latter has bounds $[a,b]$. Since cohomology vanishing is local, so does $K_V$. For quasi-compact $U\subset S$, put \[ \mathcal F_{[a,b]}(U)=\Dbcoh(Z_U)^{[a,b]}. \] Lemma~\ref{lem:finite-bounded-coherent-descent} gives finite descent for these categories and their mapping spaces. The basis satisfies the finite-refinement hypothesis of Lemma~\ref{lem:category-basis-descent}, so that lemma glues the family to a bounded coherent complex on $Z$. Conversely, a global bounded coherent complex restricts to such a family. Choosing a common interval for two global objects and applying the same mapping-space descent proves full faithfulness. \end{proof} We can now glue the affinoid identifications between sheafification and condensification. Let $Z$ be a quasi-compact quasiseparated locally strongly noetherian analytic adic space. An object of $\Cbox(Z^\square)$ belongs to $\Dpcbox(Z^\square)$ when, on every strongly noetherian affinoid $W=\Spa(A,A^+)$, it lies in $\Cond_A(\Dpc(A))$. We use the subscript $\mathrm{ch}$ for the $t$-structure defined by the degrees of these ordinary representatives. \begin{proposition}\label{prop:adic-solid-bounded} The ordinary affinoid $t$-structures descend to $\Dpcbox(Z^\square)$. Condensification gives canonical exact equivalences \begin{align*} \Coh(Z)&\xrightarrow{\sim}\Dpcbox(Z^\square)_{\mathrm{ch}}^\heartsuit,\\ \Dbcoh(Z)&\xrightarrow{\sim}\Dpcbox(Z^\square)_{\mathrm{ch}}^b, \end{align*} compatible with affinoid restriction. The first is symmetric monoidal. \end{proposition} \begin{proof} On a strongly noetherian affinoid $W=\Spa(A,A^+)$, the coherent-sheaf theorem gives an exact equivalence \[ \operatorname{Mod}_{\mathrm{fg}}(A)\xrightarrow{\sim}\Coh(W) \] and vanishing of higher coherent cohomology \cite[Theorem~4.2]{Zavyalov}. Write $\mathrm D^b_{\mathrm{fg}}(A)$ for complexes with finite cohomology in a finite interval. Since $A$ is noetherian, these are the objects coming from the bounded derived category of finite $A$-modules. Derived sheafification and sections form the adjoint pair \[ (-)^\sim:\mathrm D^b_{\mathrm{fg}}(A) \rightleftarrows\Dbcoh(W):\mathbf R\Gamma(W,-). \] For $K\in\Dbcoh(W)$, consider the spectral sequence \[ H^p(W,\mathcal H^q(K))\Longrightarrow H^{p+q}\mathbf R\Gamma(W,K). \] Coherent acyclicity makes the spectral sequence degenerate, with \[ H^q\mathbf R\Gamma(W,K)\simeq\Gamma(W,\mathcal H^q(K)), \] a finite $A$-module. The counit $(\mathbf R\Gamma(W,K))^\sim\to K$ induces in degree $q$ the coherent-sheaf isomorphism \[ \widetilde{\Gamma(W,\mathcal H^q(K))}\xrightarrow{\sim}\mathcal H^q(K). \] The counit is therefore a quasi-isomorphism. For $M\in\mathrm D^b_{\mathrm{fg}}(A)$, the same calculation shows that the unit $M\to\mathbf R\Gamma(W,\widetilde M)$ is an equivalence. We have proved \[ \Dbcoh(W)\simeq\mathrm D^b_{\mathrm{fg}}(A). \] Condensification identifies $\Dpc(A)$ with the nuclear and pseudocompact subcategory \cite[Theorem~5.9, Proposition~5.22, and Theorem~5.50]{Andreychev} \cite[A.15]{LiHuerta}. Its heart and bounded part are represented by finite modules and $\mathrm D^b_{\mathrm{fg}}(A)$, respectively. Sheafification of these same representatives gives the ordinary categories. For rational $A\to A'$, both constructions restrict by $M\mapsto A'\otimes_A^{\mathbf L}M$ \cite[Lemma~5.10]{Andreychev}. Flatness makes this commute with truncation. The truncation triangles therefore glue, and the limit of the local mapping spaces gives orthogonality, defining the asserted $t$-structure. Choose a finite affinoid cover of $Z$. Coherent sheaves descend on it, and bounded coherent complexes descend by Lemma~\ref{lem:bounded-coherent-descent}. For solid complexes, apply Proposition~\ref{prop:analytic-stack-Cbox-descent} and \cite[Theorems~5.3 and~5.44]{Andreychev}. Taking limits of the compatible affinoid equivalences gives both global comparisons. On hearts, the tensor comparisons are those of ordinary finite modules and hence give a symmetric monoidal equivalence. The construction commutes with affinoid restriction. \end{proof} Over a Tate affinoid, the fpqc family identifies the degree conditions on an algebraic preimage with those on its ordinary analytic representatives. \begin{proposition}\label{prop:local-bounded-bridge} Let $V=\Spa(R,R^+)$ be a strongly noetherian complete Tate affinoid and let $T/R$ be proper and finitely presented. Under Proposition~\ref{prop:analytic-Huber-comparison}, the transported and chartwise $t$-structures on $\Dpcbox(T^{\An})$ agree. There are canonical exact equivalences \begin{align*} \beta^\heartsuit_{T,V}:\Dpcbox(T^{\An})^\heartsuit &\xrightarrow{\sim}\Coh(T^{\ad/V}),\\ \beta^b_{T,V}:\Dpcbox(T^{\An})^b &\xrightarrow{\sim}\Dbcoh(T^{\ad/V}), \end{align*} with $\beta^\heartsuit_{T,V}$ symmetric monoidal. Their composites with $\alpha_T^{\An}$ are $c_{T/V}^*$ and $\mathbf Lc_{T/V}^*$, respectively; hence these ordinary pullbacks are equivalences as well. \end{proposition} \begin{proof} Choose the fpqc family of Lemma~\ref{lem:local-Huber-fpqc}, with affinoids $W_i=\Spa(B_i,B_i^+)$ lying over $U_i=\Spec A_i$. For $C\in\Dpcbox(T^{\An})$, take $M\in\Dpc(T)$ with $\alpha_T^{\An}(M)\simeq C$ by Proposition~\ref{prop:affinoid-pc-GAGA}. Proposition~\ref{prop:analytic-Huber-comparison} gives \[ C|_{W_i^\square} \simeq\Cond_{B_i}\bigl(B_i\otimes_{A_i}^{\mathbf L}\mathbf R\Gamma(U_i,M)\bigr). \] The ring maps $A_i\to B_i$ are flat, so \[ H^j\bigl(B_i\otimes_{A_i}^{\mathbf L}\mathbf R\Gamma(U_i,M)\bigr) \simeq B_i\otimes_{A_i}\Gamma(U_i,\mathcal H^j(M)). \] In particular, vanishing of $\mathcal H^j(M)$ implies vanishing of the degree-$j$ cohomology of every representative. Conversely, if these cohomology modules vanish for every $i$, then $\mathcal H^j(M)$ vanishes on the fpqc family and hence on $T$. Applying this in positive and negative degrees identifies the two $t$-structures. Since the cover is finite, the same argument compares bounded objects with a common interval. Proposition~\ref{prop:adic-solid-bounded}, transported through $\vartheta_{T,V}$, gives $\beta^\heartsuit_{T,V}$ and $\beta^b_{T,V}$. Their composites with $\alpha_T^{\An}$ sheafify the displayed scalar extension on each $W_i$. These are the formulas for $c_{T/V}^*$ on coherent sheaves and $\mathbf Lc_{T/V}^*$ on bounded complexes, and they agree on overlaps. Equality of the $t$-structures gives exactness; ordinary tensor products of finite modules give the tensor comparison on hearts. Composing with GAGA proves the final assertions. \end{proof} These equivalences commute with rational restriction. \begin{lemma}\label{lem:beta-pseudonatural} The functors $\beta^\heartsuit_{T,V}$ and $\beta^b_{T,V}$ commute naturally with restriction along rational localizations $V'\to V$, coherently under composition. \end{lemma} \begin{proof} For an object in the domain of $\beta^\heartsuit_{T,V}$ or $\beta^b_{T,V}$, write its restriction to $W=\Spa(C,C^+)$ as $\Cond_C(P)$. Here $P$ is a finite module in degree zero for the heart, and $P\in\mathrm D^b_{\mathrm{fg}}(C)$ for the bounded category. In either case, $\beta$ sheafifies $P$. After restricting to $V'$, choose $W'=\Spa(C',C'^+)$ mapping into $W$. Pullback compatibility in Proposition~\ref{prop:analytic-Huber-comparison} gives the representative \[ \Cond_{C'}(C'\otimes_C^{\mathbf L}P). \] Both applying $\beta$ to this representative and restricting the sheafification of $P$ to $W'$ give the sheafification of $C'\otimes_C^{\mathbf L}P$. This identification is natural in $P$. For a further restriction with ring $C''$, the comparison is \[ C''\otimes_{C'}^{\mathbf L}(C'\otimes_C^{\mathbf L}P) \simeq C''\otimes_C^{\mathbf L}P. \] Associativity identifies the comparisons for successive restrictions, together with their homotopies. The formulas also agree on common affinoid refinements, so they descend to the relative spaces. Proposition~\ref{prop:local-bounded-bridge} identifies them with the restriction comparisons for $c^*$ and $\mathbf Lc^*$. \end{proof} The rational-restriction identifications let us glue the local equivalences over $\cX$. We identify their source categories using the fpqc family. \begin{proposition}\label{prop:pc-ordinary} For proper $T/X$, the local comparisons glue to canonical exact equivalences \begin{align*} \beta_T^\heartsuit:\mathcal P_T^\heartsuit &\xrightarrow{\sim}\Coh(T^{\ad}_{\cX}),\\ \beta_T^b:\mathcal P_T^b &\xrightarrow{\sim}\Dbcoh(T^{\ad}_{\cX}). \end{align*} The first is symmetric monoidal. Their composites with $\alpha_T$ are $c_T^*$ and $\mathbf Lc_T^*$, respectively. \end{proposition} \begin{proof} The functors $\beta_{T,V}$ form a compatible diagram by Lemma~\ref{lem:beta-pseudonatural}. We identify its limit on hearts and bounded subcategories. Write $C\simeq\alpha_T(M)$ with $M\in\Dpc(T)$. By definition, $C$ lies in $\mathcal P_T^\heartsuit$ precisely when $\mathcal H^j(M)=0$ for $j\ne0$. Consider the fpqc family \[ T\times_X\coprod_i\Spec R_i\longrightarrow T \] Flat pullback commutes with cohomology, and faithful flatness detects its vanishing. Thus $M$ is concentrated in degree zero if and only if all its pullbacks are. Equivalently, the analytic restrictions of $C$ lie in the local transported hearts. Proposition~\ref{prop:relative-pc-descent} therefore identifies $\mathcal P_T^\heartsuit$ with their limit. On the ordinary side, coherent sheaves descend on the corresponding cover of $T^{\ad}_{\cX}$. Taking the limit of the functors $\beta^\heartsuit_{T,V}$ gives the heart equivalence. For bounded objects, choose one interval $[a,b]$ containing the bounds on the finite base cover. Every rational chart has a finite refinement by intersections with this cover. Cartesianness gives the same bounds on that refinement, and therefore on the chart itself, as in Lemma~\ref{lem:bounded-coherent-descent}. Pseudocoherent descent gives a global object $C\simeq\alpha_T(M)$. The cohomology of $M$ outside $[a,b]$ vanishes on the fpqc family, hence on $T$. Thus $C\in\mathcal P_T^b$. Conversely, a bounded algebraic preimage has bounded restrictions. This identifies the bounded source with the limit of the local bounded sources. For the ordinary target, \[ \Dbcoh(T^{\ad}_{\cX}) \simeq\operatorname{Cart}_{V\in\operatorname{RatAff}_X(\cX)} \Dbcoh(T_R^{\ad/V}) \] by Lemma~\ref{lem:bounded-coherent-descent}. The limit of the $\beta^b_{T,V}$ consequently gives $\beta_T^b$. Locally, the composites with $\alpha_T$ are scalar extension followed by sheafification, exactly as in $c_T^*$ and $\mathbf Lc_T^*$. Lemma~\ref{lem:beta-pseudonatural} shows that these identifications agree under restriction, so they identify the global functors. Exactness follows from the local degree conditions. The tensor comparisons for finite modules agree on overlaps as well, giving the symmetric monoidal equivalence on hearts. \end{proof} The restrictions to vector bundles and perfect complexes follow from duality and finite Tor-amplitude. \begin{corollary}\label{cor:coherent-bounded-gaga} For proper $T/X$, analytification gives exact equivalences \begin{align*} c_T^*:\Coh(T)&\xrightarrow{\sim}\Coh(T^{\ad}_{\cX}),\\ \mathbf Lc_T^*:\Dbcoh(T)&\xrightarrow{\sim}\Dbcoh(T^{\ad}_{\cX}). \end{align*} The first is symmetric monoidal, and these restrict to equivalences on vector bundles and perfect complexes. \end{corollary} \begin{proof} Theorem~\ref{thm:stable-main} and Proposition~\ref{prop:pc-ordinary} give the coherent and bounded-derived equivalences. The proposition identifies the composites with $c_T^*$ and $\mathbf Lc_T^*$, respectively. A coherent sheaf is a vector bundle exactly when it is dualizable for ordinary tensor product. Indeed, locally a duality datum writes the identity of a finite module as $m\mapsto\sum_i\phi_i(m)m_i$, exhibiting it as a summand of a finite free module. A finite projective module, conversely, has its usual dual and evaluation. The tensor equivalence preserves and reflects these data, and hence identifies vector bundles. Derived pullback preserves perfect complexes because it preserves bounded complexes of finite projectives. For the converse, first work over a base affinoid and let $M\in\Dbcoh(T_R)$ have perfect analytification. The analytic space is proper and therefore quasi-compact. Choose a finite affinoid cover on which this perfect complex has finite Tor-amplitude, and choose $[a,b]$ containing all the resulting intervals. For any $N\in\Coh(T_R)$, flatness gives \[ c_{T_R/V}^*H^j(M\otimes^{\mathbf L}N) \simeq H^j(\mathbf Lc_{T_R/V}^*M\otimes^{\mathbf L}c_{T_R/V}^*N). \] The right side vanishes for $j\notin[a,b]$, with bounds independent of $N$. The algebraic cohomology sheaf on the left is coherent: locally, tensor a bounded-above finite-free resolution of $M$ with the finite module representing $N$, and use noetherianity for its kernels and quotients. Coherent GAGA detects zero objects, so $H^j(M\otimes^{\mathbf L}N)=0$ outside the same interval. Let $U=\Spec A\subset T_R$ and let $Q$ be a finite $A$-module. Extend its associated sheaf to $N\in\Coh(T_R)$ using \cite[Tag~0G41]{Stacks}. Restricting the preceding vanishing to $U$ gives \[ H^j\bigl(\mathbf R\Gamma(U,M|_U)\otimes_A^{\mathbf L}Q\bigr)=0 \] for $j\notin[a,b]$. Every $A$-module is a filtered union of finite submodules, and derived tensor product preserves this colimit. Exactness of filtered colimits extends the vanishing to every $A$-module. Thus $M|_U$ has Tor-amplitude in $[a,b]$; being pseudocoherent, it is perfect by \cite[Tag~0658]{Stacks}. Finally, let $M\in\Dbcoh(T)$ have perfect analytification. Pull it back along the finite fpqc family on $X$. The affinoid argument makes each pullback perfect on $T_{R_i}$, and perfectness descends faithfully flatly. Hence $M$ itself is perfect, proving the remaining equivalence. \end{proof} \subsection{Ordinary proper pushforward}\label{subsec:pushforward-comparison} We compare ordinary proper direct images on the rational opens of the base, using open base change. \begin{lemma}\label{lem:ordinary-open-BC} Let \[ \begin{CD} Z' @>{j'}>> Z\\ @V{h'}VV @VV{h}V\\ V' @>{j}>> V \end{CD} \] be a cartesian square of ringed spaces in which $j$ is an open immersion. Then the canonical morphism \(j^*\mathbf Rh_*\longrightarrow \mathbf Rh'_*j'^*\) is an equivalence on derived categories of modules. \end{lemma} \begin{proof} For an open immersion $j$, extension by zero is an exact left adjoint $j_!\dashv j^*$. On stalks it is the given sheaf inside the open and zero outside. We first check \[ \mathbf Lh^*j_!\simeq j'_!\mathbf L(h')^*. \] Choose a K-flat resolution on $V'$ and extend it by zero. The extension is still K-flat, since its tensor product with an acyclic complex is acyclic on every stalk. At a point of $Z'$, the two displayed functors give the same scalar extension; outside $Z'$, they both give zero. This proves the equivalence. For $A\in\mathrm D(\mathcal O_Z)$ and $B\in\mathrm D(\mathcal O_{V'})$, adjunction gives \begin{align*} \Map_{V'}(B,j^*\mathbf Rh_*A) &\simeq\Map_V(j_!B,\mathbf Rh_*A)\\ &\simeq\Map_Z(\mathbf Lh^*j_!B,A)\displaybreak[3]\\ &\simeq\Map_Z(j'_!\mathbf L(h')^*B,A)\\ &\simeq\Map_{Z'}(\mathbf L(h')^*B,j'^*A)\\ &\simeq\Map_{V'}(B,\mathbf Rh'_*j'^*A). \end{align*} The middle equivalence is the one just proved; the others use extension by zero and pullback--direct-image adjunction. Tracing a map through the units and counits gives the canonical base-change morphism. It is an equivalence after mapping from every $B$, so Yoneda proves the claim. \end{proof} Over a Tate affinoid, full faithfulness compares global sections by mapping from the structure sheaf. Applying this comparison on rational opens identifies the proper direct images. \begin{proposition}\label{prop:affinoid-ordinary-pushforward} Let $V=\Spa(R,R^+)$ be a strongly noetherian complete Tate affinoid and let $h:Z\to\Spec R$ be proper and finitely presented. For $M\in\Dbcoh(Z)$, the canonical map \[ \mathbf Lc_V^*\mathbf Rh_*M\xrightarrow{\sim} \mathbf Rh_*^{\ad}\mathbf Lc_{Z/V}^*M \] is an equivalence in $\Dbcoh(V)$, natural in $M$ and coherently compatible with rational localization. \end{proposition} \begin{proof} Apply analytification to the algebraic counit. Compatibility with pullback gives \[ \mathbf L(h^{\ad})^*\mathbf Lc_V^*\mathbf Rh_*M \simeq\mathbf Lc_{Z/V}^*\mathbf Lh^*\mathbf Rh_*M \longrightarrow\mathbf Lc_{Z/V}^*M. \] Its adjoint under $\mathbf L(h^{\ad})^*\dashv\mathbf Rh_*^{\ad}$ is the map in the statement. Take a rational open $V'=\Spa(R',R'^+)\subset V$, and put $Z_{R'}=Z\times_R\Spec R'$ and $M_{R'}=M\otimes_R^{\mathbf L}R'$. Flat base change and coherent affinoid acyclicity identify sections of the source on $V'$ with $\mathbf R\Gamma(Z_{R'},M_{R'})$. By Lemma~\ref{lem:ordinary-open-BC}, sections of the target are computed on the inverse-image analytic open. The resulting map is \[ \mathbf R\Gamma(Z_{R'},M_{R'}) \longrightarrow \mathbf R\Gamma(Z_{R'}^{\ad/V'},\mathbf Lc^*M_{R'}). \] By Proposition~\ref{prop:local-bounded-bridge}, analytification over $R'$ is fully faithful on bounded coherent complexes and preserves the structure sheaf. We therefore have an equivalence \[ \Map_{Z_{R'}}(\cO,M_{R'}) \xrightarrow{\mathbf Lc^*} \Map_{Z_{R'}^{\ad/V'}}(\cO,\mathbf Lc^*M_{R'}). \] Its source and target are the section complexes above, and it sends $s:\cO\to M_{R'}$ to $\mathbf Lc^*(s)$. After flat base change, the canonical comparison is defined by applying $\mathbf Lc^*$ to \[ \mathbf Lh_{R'}^*\mathbf R(h_{R'})_*M_{R'}\longrightarrow M_{R'}, \] and taking the adjoint. It therefore induces the same map on sections and is an equivalence on every rational $V'$. Let $Q$ be the cofiber. Choose $a$ with $M\in\mathrm D^{\geq a}(Z)$. Flat analytification preserves this bound, and derived direct image is left $t$-exact, so both terms of the comparison lie in degrees at least $a$. Hence $Q\in\mathrm D^{\geq a-1}(V)$. Lemma~\ref{lem:derived-open-descent} identifies it with a sheaf of module spectra whose values on the rational basis are \[ \mathbf R\Gamma(V',Q|_{V'})=0. \] These values determine the sheaf, so $Q=0$. Algebraic proper direct image and flat pullback make the source bounded coherent; the equivalent target is therefore bounded coherent as well. Every restriction comparison comes from the same counit. Tensor associativity and naturality of the units and counits identify the composites, together with their higher homotopies. This proves compatibility with rational localization. \end{proof} For the base itself, Proposition~\ref{cor:global-Huber} identifies $X^{\ad}_{\cX}=\cX$ and $c_X=\lambda$. Applying the affinoid comparison on its charts gives the global direct-image comparison. \begin{proposition}\label{prop:ordinary-pushforward} For proper finitely presented $f:Y\to X$ and $M\in\Dbcoh(Y)$, the canonical map \[ \mathbf Lc_X^*\mathbf Rf_*M\xrightarrow{\sim} \mathbf Rf_*^{\ad}\mathbf Lc_Y^*M \] is an equivalence in $\Dbcoh(\cX)$. \end{proposition} \begin{proof} As in the affinoid case, apply $\mathbf Lc_Y^*$ to the algebraic counit. Pullback compatibility gives \[ \mathbf L(f^{\ad})^*\mathbf Lc_X^*\mathbf Rf_*M \simeq\mathbf Lc_Y^*\mathbf Lf^*\mathbf Rf_*M \longrightarrow\mathbf Lc_Y^*M. \] Take its adjoint to define the global comparison, and restrict to a rational-affinoid chart. Choose $V=\Spa(R,R^+)\subset\lambda^{-1}(W)$ with $W=\Spec A$, and put $Y_R=Y\times_X\Spec R$ and $M_R=\mathbf L(Y_R\to Y)^*M$. Lemma~\ref{lem:base-flat} makes $A\to R$ flat. Flat base change, Proposition~\ref{prop:affinoid-ordinary-pushforward}, and open base change then give \begin{align*} (\mathbf Lc_X^*\mathbf Rf_*M)|_V &\simeq\mathbf Lc_V^*\mathbf R(f_R)_*M_R\\ &\simeq\mathbf R(f_R^{\ad})_*\mathbf Lc_{Y_R/V}^*M_R\\ &\simeq(\mathbf Rf^{\ad}_*\mathbf Lc_Y^*M)|_V. \end{align*} The first map is obtained by flat base change of the counit, the second by the affinoid construction from that counit, and the last by open restriction. Their composite is therefore the restriction of our global map. On a finite cover $(V_i)$, the target is bounded coherent by the affinoid comparison. Taking the minimum of the lower bounds and maximum of the upper bounds gives a common interval. Coherence and cohomology vanishing are local, so the target lies in $\Dbcoh(\cX)$. The cofiber vanishes on every $V_i$, hence globally. Thus the comparison is an equivalence in $\Dbcoh(\cX)$. \end{proof} \Needspace{4\baselineskip} \subsection{Classical GAGA}\label{subsec:classical-proof} The category equivalences and direct-image comparison give classical GAGA. Global cohomology follows by applying the base comparison to the proper direct image. \begin{theorem}\label{thm:classical-main} Under the hypotheses of Section~\ref{subsec:setup}, let $f:Y\to X$ be proper. Ordinary analytification $c_Y:Y^{\ad}_{\cX}\to Y$ induces canonical equivalences \begin{align*} \Coh(Y)&\xrightarrow{\sim}\Coh(Y^{\ad}_{\cX}),\\ \Dbcoh(Y)&\xrightarrow{\sim}\Dbcoh(Y^{\ad}_{\cX}),\\ \Perf(Y)&\xrightarrow{\sim}\Perf(Y^{\ad}_{\cX}),\\ \Vect(Y)&\xrightarrow{\sim}\Vect(Y^{\ad}_{\cX}). \end{align*} The coherent equivalence is exact and symmetric monoidal; the bounded-derived equivalence is exact. Under Theorem~\ref{thm:stable-main}, they identify the transported heart and bounded part of $\Dpcbox(Y^{\An}_{\cX})$ with the ordinary coherent and bounded coherent categories, through $c_Y^*$ and $\mathbf Lc_Y^*$. For $\mathcal F\in\Coh(Y)$ and $i\geq0$, the canonical maps are isomorphisms: \begin{align*} (R^if_*\mathcal F)^{\ad}&\xrightarrow{\sim}R^if_*^{\ad}(\mathcal F^{\ad}),\\ H^i(Y,\mathcal F)&\xrightarrow{\sim}H^i(Y^{\ad}_{\cX},\mathcal F^{\ad}). \end{align*} \end{theorem} \begin{proof} By Proposition~\ref{prop:pc-ordinary}, the transported heart and bounded part are the ordinary coherent and bounded coherent categories, and the functors are $c_Y^*$ and $\mathbf Lc_Y^*$. Corollary~\ref{cor:coherent-bounded-gaga} gives the restrictions to vector bundles and perfect complexes, so all four equivalences follow. For $\mathcal F\in\Coh(Y)$, apply Proposition~\ref{prop:ordinary-pushforward} to $\mathcal F[0]$. Flatness identifies degree-$i$ cohomology of the source with $c_X^*(R^if_*\mathcal F)$, while degree-$i$ cohomology of the target is $R^if_*^{\ad}(\mathcal F^{\ad})$. The resulting isomorphism is the asserted canonical direct-image map. The complex $\mathbf Rf_*\mathcal F$ is bounded coherent: proper direct image gives coherence, Lemma~\ref{lem:proper-Dminus} gives an upper bound, and negative cohomology vanishes. Applying the base comparison yields \begin{align*} \mathbf R\Gamma(Y,\mathcal F) &\simeq\mathbf R\Gamma(X,\mathbf Rf_*\mathcal F)\\ &\simeq\mathbf R\Gamma_{\ad}(\cX,(\mathbf Rf_*\mathcal F)^{\ad})\\ &\simeq\mathbf R\Gamma_{\ad}(\cX,\mathbf Rf_*^{\ad}\mathcal F^{\ad})\\ &\simeq\mathbf R\Gamma_{\ad}(Y^{\ad}_{\cX},\mathcal F^{\ad}). \end{align*} The first and last maps compose direct images. The second is Lemma~\ref{lem:solid-ordinary-cohomology}, and the third is the proper-direct-image comparison. Each is canonical, so taking degree-$i$ cohomology gives the required isomorphism. \end{proof} \subsection{Changing the proper base}\label{subsec:proper-base-change} A proper $X$-scheme, together with its analytification, is another base to which the theorem applies. To compare direct images after this change of base, we first identify the successive analytifications. \begin{lemma}\label{lem:analytification-transitivity} Let $Z\to X$ be proper, put $\mathcal Z=Z^{\ad}_{\cX}$, and let $Y\to Z$ be separated and finitely presented. There is a canonical isomorphism \[ Y^{\ad}_{\mathcal Z}\xrightarrow{\sim}Y^{\ad}_{\cX}, \] compatible with the maps to $Y$ and $\mathcal Z$. It induces an equivalence $Y^{\An}_{\mathcal Z}\simeq Y^{\An}_{\cX}$ identifying the comparison functors $\alpha_Y^{\An}$. \end{lemma} \begin{proof} A map from an analytic test space $H$ to $Y^{\ad}_{\cX}$ is a compatible pair $H\to\cX$, $H\to Y$, where the latter is a morphism of locally ringed spaces. Composing with $Y\to Z$ gives the pair defining a unique map $H\to\mathcal Z$, compatible with $H\to Y$ over $Z$. Conversely, such a pair over $\mathcal Z$ gives one over $\cX$. These constructions are inverse and natural in $H$. Huber's mapping property \cite[Proposition~3.8]{Huber1994}, applied on affinoids and glued as in Proposition~\ref{cor:global-Huber}, therefore gives the displayed isomorphism. For a further change of base, both composites are specified by this same mapping property. Since $Y$ is separated and finitely presented over $X$, Proposition~\ref{prop:analytic-Huber-comparison} applies over both locally strongly noetherian analytic bases. It identifies the analytic loci $Y^{\An}_{\mathcal Z}$ and $Y^{\An}_{\cX}$ with the stack associated to this adic space. Their comparison functors agree as well: if $W=\Spa(D,D^+)$ maps into $U=\Spec A\subset Y$, both send $M\in\Dqc(Y)$ to \[ \Cond_D\bigl(D\otimes_A^{\mathbf L}\mathbf R\Gamma(U,M|_U)\bigr) \] on $W$. The formulas agree under restriction by tensor associativity and hence identify the global functors. \end{proof} \begin{corollary}\label{cor:proper-relative-pushforward} Let $Y,Z$ be proper $X$-schemes and let $g:Y\to Z$ be an $X$-morphism. For every $M\in\Dqc(Y)$, the canonical map \[ \alpha_Z(\mathbf Rg_*M) \xrightarrow{\sim}\mathbf Rg_*^{\An}\alpha_Y(M) \] is an equivalence. For $M\in\Dbcoh(Y)$, the canonical map \[ \mathbf Lc_Z^*\mathbf Rg_*M \xrightarrow{\sim}\mathbf Rg_*^{\ad}\mathbf Lc_Y^*M \] is an equivalence in $\Dbcoh(Z^{\ad}_{\cX})$. Both comparisons commute with composition of such morphisms. \end{corollary} \begin{proof} First observe that $g$ is proper. Its graph in $Y\times_XZ$ is closed because $Z\to X$ is separated, and the projection to $Z$ is a base change of the proper map $Y\to X$. The pair $(Z,\mathcal Z)$, with $\mathcal Z=Z^{\ad}_{\cX}$, satisfies Section~\ref{subsec:setup}. Indeed, $Z$ is noetherian and has affine diagonal by the factorization in Section~\ref{sec:base}. Its analytification is locally strongly noetherian and proper over the quasi-compact quasi-separated space $\cX$, hence is quasi-compact and quasi-separated. Theorem~\ref{thm:classical-main} supplies exact coherent GAGA and the canonical coherent-cohomology comparison. Apply Theorem~\ref{thm:stable-main} and Proposition~\ref{prop:ordinary-pushforward} to $g$ over $(Z,\mathcal Z)$. Lemma~\ref{lem:analytification-transitivity} identifies their spaces and functors with the ones in the statement. Both constructions use the algebraic counit, so the resulting maps are the stated canonical maps. For successive morphisms, the composite counit is the counit of the composite adjoint pair. Hence the direct and iterated comparisons agree, proving compatibility with composition. \end{proof} \section{The Fargues--Fontaine curve}\label{sec:ff} For the Fargues--Fontaine curve, coherent GAGA and cohomology comparison provide the input to the bootstrapping theorem. We recall the schematic and adic constructions and their comparison map \cite{FarguesFontaine,FarguesScholze}, then give the affinoid cover needed to apply it. \subsection{The adic and schematic curves}\label{subsec:curve-constructions} Let $E$ be a nonarchimedean local field with residue field $\mathbf F_q$, and let $F$ be a complete perfectoid field of characteristic $p$ containing $\mathbf F_q$. Write $\cO_E$ and $\cO_F$ for their valuation rings, and choose a uniformizer $\varpi_E$ of $E$ and a nonzero topologically nilpotent element $\varpi_F\in\cO_F$. In both constructions $F$ has characteristic $p$; it is the characteristic of $E$ that distinguishes the equal- and mixed-characteristic cases. The starting ring is \[ A_{\inf}= \begin{cases} W(\cO_F)\widehat\otimes_{W(\mathbf F_q)}\cO_E, &\operatorname{char}E=0,\\ \cO_F[[\varpi_E]],&E=\mathbf F_q((\varpi_E)). \end{cases} \] Here $W$ denotes $p$-typical Witt vectors, and the tensor product is completed $\varpi_E$-adically. In either case, $A_{\inf}/\varpi_E\simeq\cO_F$. For $x\in\cO_F$, write $[x]$ for its Teichm\"uller lift; in equal characteristic, this is just the constant series $x$. Give $A_{\inf}$ the $(\varpi_E,[\varpi_F])$-adic topology. The $q$-power Frobenius on $F$ then induces an automorphism $\varphi$ satisfying \begin{align*} \varphi([x])&=[x^q],\\ \varphi(\varpi_E)&=\varpi_E. \end{align*} Thus in equal characteristic, Frobenius acts on the coefficients and fixes the variable. In mixed characteristic, the same formulas describe the ramified Witt-vector construction \cite[Section~II.1.1]{FarguesScholze}. To obtain the adic curve, first consider the open period-ring space \[ \mathcal Y=\Spa(A_{\inf},A_{\inf}) \setminus V(\varpi_E[\varpi_F]). \] Both $\varpi_E$ and $[\varpi_F]$ have nonzero value on $\mathcal Y$. Inverting $\varpi_E$ makes $\mathcal Y$ an analytic adic space over $E$. In equal characteristic, it is the punctured open unit disc over $F$ with coordinate $\varpi_E$ \cite[Section~II.1.1 and the proof of Proposition~II.2.2]{FarguesScholze}. For a rank-one point $x$, define its radius by \[ \operatorname{rad}(x)= \frac{\log|[\varpi_F](x)|}{\log|\varpi_E(x)|}. \] For any other point, use its rank-one generalization in the formula. The identities for Frobenius give $\operatorname{rad}(\varphi x)=q\operatorname{rad}(x)$. Thus an annulus of sufficiently small multiplicative width does not meet any of its nontrivial Frobenius translates. The action is free and totally discontinuous, so we can form the quotient \[ \cX=X^{\ad}_{F,E}=\mathcal Y/\varphi^{\mathbf Z}, \] with quotient map $\pi:\mathcal Y\to\cX$ \cite[Definition~II.1.15 and Proposition~II.1.16]{FarguesScholze}. Annuli disjoint from their nontrivial translates give affinoid charts on this adic curve. For the schematic construction, put $B=\Gamma(\mathcal Y,\cO_{\mathcal Y})$ and form the graded $E$-algebra of Frobenius eigenvectors \begin{gather*} P=\bigoplus_{n\geq0}P_n,\\ P_n=B^{\varphi=\varpi_E^n}. \end{gather*} Here $P_n=\{b\in B:\varphi(b)=\varpi_E^n b\}$. If $a\in P_m$ and $b\in P_n$, then $\varphi(ab)=\varpi_E^{m+n}ab$, so multiplication respects the grading. Define \[ X=X^{\mathrm{sch}}_{F,E}=\operatorname{Proj}(P). \] The graded algebra $P$ is also a section ring on the adic curve. Descending the trivial line on $\mathcal Y$ with semilinear Frobenius $\varpi_E^{-1}\varphi$ gives $\cO_{\cX}(1)$. Sections of its $n$th tensor power are functions $b$ satisfying $\varphi(b)=\varpi_E^n b$, so $P_n=H^0(\cX,\cO_{\cX}(n))$ \cite[Section~II.2 and Proposition~II.2.7]{FarguesScholze}. To describe the comparison map, let $s\in P_d$ be homogeneous of positive degree. The corresponding standard affine is \[ D_+(s)=\Spec\bigl(P[s^{-1}]_0\bigr), \] where the subscript means degree zero. An element of this ring is a fraction $a/s^k$ with $a\in P_{kd}$. Wherever $s$ is nonzero, we compute \[ \varphi(a/s^k)= \frac{\varpi_E^{kd}a}{(\varpi_E^d s)^k}=a/s^k. \] Thus the fraction descends to an analytic function on the nonvanishing open $\cX_s$, giving $\cX_s\to D_+(s)$. These opens cover $\cX$ because $\cO_{\cX}(d)$ is globally generated for sufficiently large $d$ \cite[Theorem~II.2.6]{FarguesScholze}. On $\cX_s\cap\cX_t$, both maps are given by localization through $D_+(st)$. They glue to \[ \lambda:\cX\longrightarrow X \] as a morphism of locally ringed spaces \cite[Proposition~II.2.7 and its proof]{FarguesScholze}. This is the map used in Section~\ref{subsec:base-comparison}. For nonzero homogeneous $s$ of positive degree, the ring $P[s^{-1}]_0$ is a Dedekind domain, and the annular rings are strongly noetherian \cite[Theorem~4.7.3]{KedlayaLiu}. In particular, $X$ is a noetherian regular separated curve. The known GAGA theorem for $\lambda$ gives \begin{align*} \lambda^*:\Coh(X)&\xrightarrow{\sim}\Coh(\cX),\\ H^i(X,\mathcal F)&\xrightarrow{\sim} H^i(\cX,\lambda^*\mathcal F) \end{align*} for coherent $\mathcal F$ and $i\geq0$ \cite[Theorems~4.7.4--4.7.5]{KedlayaLiu}. The following affinoid cover verifies quasi-compactness and quasiseparatedness of $\cX$. \subsection{A two-affinoid cover of the curve}\label{subsec:MV} We use the cover of \cite[Remark~3.1.9]{KedlayaNotes}. It is obtained from two annuli disjoint from their nontrivial Frobenius translates; their intersections with one another's translates determine the overlap. For an interval $I=[a,b]\subset(0,\infty)$ with rational endpoints, the corresponding rational affinoid in $\mathcal Y$ is \[ \mathcal Y_I =\bigl\{x\in\mathcal Y: |\varpi_E(x)|^b\leq |[\varpi_F](x)|\leq |\varpi_E(x)|^a \bigr\} \subset \operatorname{rad}^{-1}(I), \] with $\operatorname{rad}\circ\varphi=q\,\operatorname{rad}$ \cite[Definition~II.1.15 and Proposition~II.1.16]{FarguesScholze}. Substituting $\varphi(\varpi_E)=\varpi_E$ and $\varphi([\varpi_F])=[\varpi_F]^q$ in the inequalities gives $\varphi(\mathcal Y_I)=\mathcal Y_{qI}$. \begin{lemma}\label{lem:annulus-calculus} For rational intervals $I=[a,b]$ and $J=[c,d]$ in $(0,\infty)$, \[ \mathcal Y_I\cap\mathcal Y_J = \begin{cases} \mathcal Y_{[\max(a,c),\,\min(b,d)]},&\max(a,c)\leq\min(b,d),\\ \varnothing,&\max(a,c)>\min(b,d). \end{cases} \] For every $n\in\mathbf Z$, we also have $\varphi^n(\mathcal Y_I)=\mathcal Y_{q^nI}$. \end{lemma} \begin{proof} At a point $x$, write $u=|\varpi_E(x)|$ and $v=|[\varpi_F](x)|$. To belong to both annuli, we need $u^b\leq v\leq u^a$ and $u^d\leq v\leq u^c$. Since $0\min(b,d)$, the inequalities cannot hold. Otherwise they define exactly the claimed rational subspace, with its analytic structure. Frobenius carries the interval $[a,b]$ to $[qa,qb]$ \cite[Definition~II.1.15 and Proposition~II.1.16]{FarguesScholze}. Applying it repeatedly, or applying its inverse, gives the formula for every $n\in\mathbf Z$. \end{proof} An open disjoint from its nontrivial translates descends to an open of the quotient. \begin{lemma}\label{lem:quotient-open} Let $U\subset\mathcal Y$ be an open analytic subspace satisfying \(U\cap\varphi^n(U)=\varnothing\) for every \(n\in\mathbf Z\setminus\{0\}\). Then the restriction $\pi|_U:U\to\cX$ is an open immersion. \end{lemma} \begin{proof} The quotient is formed from the free, totally discontinuous $\varphi^{\mathbf Z}$-action of \cite[Definition~II.1.15 and Proposition~II.1.16]{FarguesScholze}. Consider the saturation \[ U^{\mathrm{sat}}=\bigcup_{n\in\mathbf Z}\varphi^n(U). \] Its translates are pairwise disjoint: applying $\varphi^{-m}$ turns an intersection of two distinct translates into $U\cap\varphi^{n-m}(U)$, which is empty by hypothesis. We have $\pi^{-1}(\pi(U))=U^{\mathrm{sat}}$, so $\pi(U)$ is open. Every orbit in this saturation meets $U$ exactly once. The same argument for an open subset of $U$ shows that $|U|\to\pi(U)$ is an open bijection, hence a homeomorphism. Powers of $\varphi$ identify each component of $U^{\mathrm{sat}}$ with $U$, preserving the structure sheaf and valuations. The quotient therefore has the analytic structure of $U$, proving the open-immersion assertion. \end{proof} Choose one annulus crossing the ends of a fundamental interval and another contained in its interior. \begin{proposition}\label{prop:annular-MV} Choose rational $1b'$. Only $n=0,1$ remain. Lemma~\ref{lem:annulus-calculus} gives \begin{align*} \mathcal Y_{I_1}\cap\mathcal Y_{I_0}&=\mathcal Y_{[a',a]},\\ \mathcal Y_{I_1}\cap\varphi(\mathcal Y_{I_0})&=\mathcal Y_{[b,b']}. \end{align*} These pieces are disjoint because $a{\Cond_R}>> \Dsquare(R,R^+)\\ @V{(-)_{\mathrm{disc}}}VV @VV{\mathrm{inclusion}}V\\ \Dcond(R) @>{\underline R\otimes_{R_{\mathrm{disc}}}^{\mathbf L}-}>> \mathrm D(\operatorname{Mod}^{\mathrm{cond}}_{\underline R}). \end{CD} \] The left arrow assigns the discrete topology; the top arrow also extends scalars to $\underline R$. The right arrow includes the full subcategory of complete modules without changing their underlying condensed modules. For a finite free module, the formula reads $\Cond_R(R^n)=\underline R^{\,n}$, and a map is still given by the same matrix. An idempotent matrix then gives the corresponding construction for a finite projective summand. More generally, a bounded-above finite-free resolution, when available, computes derived scalar extension term by term. Condensification is fully faithful, preserves exact triangles and colimits, and is strong symmetric monoidal \cite[Theorem~5.9 and Lemma~5.10]{Andreychev}. Its essential image consists of the \emph{relatively discrete} objects. Thus $\underline R=\Cond_R(R)$ is relatively discrete: the adjective records its origin in $\mathrm D(R)$ rather than a choice of discrete topology. For strongly noetherian $R$, a finite module $M$ has a complete quotient topology independent of the choice of finite free presentation. Choose a presentation \[ R^a\longrightarrow R^b\longrightarrow M\longrightarrow0. \] The same matrix over $\underline R$ has cokernel $\underline M$, by completeness of this quotient topology \cite[Lemma~5.23]{Andreychev}. Consequently, a finite-free resolution computes a pseudocoherent object on either side using the same differentials and the same degree conditions on its ordinary representative. For the infinite module $R[T]=\bigoplus_{n\geq0}RT^n$, colimit preservation gives \[ \Cond_R(R[T])\simeq\bigoplus_{n\geq0}\underline R\,T^n. \] The right side is the condensed polynomial module. By contrast, imposing a bound on $T$ in the analytic-ring construction gives the closed unit disc, whose ring $\underline{R\langle T\rangle}$ contains restricted power series. For rational localization $(R,R^+)\to(R',R'^+)$, \[ \Cond_R(M)|_{\Spa(R',R'^+)} \simeq\Cond_{R'}(R'\otimes_R^{\mathbf L}M). \] This is the scalar-extension compatibility used throughout the proof. For strongly noetherian rings, $R\to R'$ is also flat, so restriction commutes with cohomology and truncation \cite[Lemma~5.10]{Andreychev}. We also use discrete integral pairs $(B,B^+)$. Their complete modules lie in $\Dcond(B)$, since the underlying ring is discrete. Lemmas~\ref{lem:integral-induced-modules} and~\ref{lem:integral-base-change} describe these categories and their Tor-independent base changes. The algebraic realization below combines such a discrete pair with a topological one. \subsection{Analytic stacks and descent}\label{app:analytic-stacks} The analytic affine associated with $\mathcal A$ is denoted by $\AnSpec\mathcal A$. The construction is contravariant, with quasicoherent category $\mathsf D(\mathcal A)$. A map of analytic rings induces pullback by extension of scalars and direct image by restriction of scalars. A map $\mathcal A\to\mathcal B$ is a \emph{localization} if restriction $\mathsf D(\mathcal B)\to\mathsf D(\mathcal A)$ is fully faithful. It is \emph{steady} if extension and restriction of scalars satisfy the following base-change condition. Let $f:\cU\to\cZ$ be the associated map of analytic affines. For every map $g:\cV\to\cZ$ from an analytic affine, the canonical map \[ f^*\mathbf Rg_*\xrightarrow{\sim}\mathbf Rg'_*f'^* \] must be an equivalence of functors from $\Cbox(\cV)$ to $\Cbox(\cU)$, in the cartesian square \[ \begin{CD} \cU' @>{g'}>> \cU\\ @V{f'}VV @VV{f}V\\ \cV @>{g}>> \cZ, \end{CD} \] This is the characterization of steadiness in \cite[Proposition~12.14]{ScholzeAnalytic}. A \emph{steady localization} satisfies both conditions. Steady maps are stable under composition and base change \cite[Definitions~12.13 and~12.16 and Proposition~12.15]{ScholzeAnalytic}. An \emph{analytic stack} is a sheaf of spaces for the topology generated by finite jointly conservative families of steady localizations. Joint conservativity means that a map is an equivalence if its pullback to every member is an equivalence; for these exact functors, it is enough to test zero objects. Rational covers of our Huber affinoids give examples of these covers \cite[A.2]{LiHuerta}. We define quasicoherent complexes on a stack by specifying them on all its analytic affines: \[ \Cbox(\cZ)= \lim_{(\AnSpec\mathcal A\to\cZ)^{\mathrm{op}}}\mathsf D(\mathcal A). \] An object assigns a complete module to each analytic affine over $\cZ$, with pullback equivalences and unit, composition, and higher homotopies; morphisms are compatible families of module maps. For the Huber-pair stacks considered here, these are the \emph{solid quasicoherent complexes} \cite[Proposition~5.16]{ScholzeSix}. They are defined using complete modules on analytic affines, rather than arbitrary ordinary sheaves of modules on a topological space. For $V=\Spa(R,R^+)$, put $V^\square=\AnSpec(R,R^+)^\square$. Then \[ \Cbox(V^\square)=\Dsquare(R,R^+). \] Gluing this assignment on an analytic adic space $Z$ gives the analytic stack $Z^\square$. In the main text we abbreviate $\Cbox(Z^\square)$ to $\Cbox(Z)$. We will identify the subcategory corresponding to ordinary coherent sheaves in Section~\ref{app:finiteness}. Let $\mathcal U_0\to\cZ$ be an effective epimorphism, with \v Cech nerve $\mathcal U_\bullet$. Descent gives \[ \Cbox(\cZ)\simeq\lim_{[n]\in\Delta}\Cbox(\mathcal U_n). \] Here $\mathcal U_n$ is the $(n+1)$-fold fiber product over $\cZ$, and effective epimorphism means $|\mathcal U_\bullet|\simeq\cZ$. The coproduct of an open cover is an example. For two opens, descent gives a complex on each open and an equivalence on their intersection; for a larger cover, it also records the compatibility homotopies on higher intersections. On an affinoid basis, we can express the same gluing data as \emph{cartesian sections}. An object of $\operatorname{Cart}_V\Dpc(R_V)$ consists of ordinary complexes $M_V\in\Dpc(R_V)$ and equivalences \[ R_{V'}\otimes_{R_V}^{\mathbf L}M_V\xrightarrow{\sim}M_{V'} \] for $V'\subset V$, compatible with successive restrictions. The pullback identifications and their compatibility homotopies are included in the cartesian section. For a finite open cover, Lemma~\ref{lem:finite-Cech-descent} replaces the full \v Cech nerve by the finite poset of nonempty intersections. Filtered colimits commute with the resulting finite limit of mapping spectra, giving Proposition~\ref{prop:finite-descent}. Derived global sections are \[ \mathbf R\Gamma_\square(\cZ,K) =\Map_{\Cbox(\cZ)}(\cO_{\cZ},K). \] On an affine, this is derived evaluation at $*$. On a stack, descent computes it as a limit of local section complexes. Pullback $f^*$ is given by derived scalar extension on charts, and $\mathbf Rf_*$ is its right adjoint. For the morphisms of Section~\ref{sec:grauert}, there is also an exceptional direct image $\mathbf Rf_!$ with right adjoint $f^!$, called exceptional pullback. Properness identifies $\mathbf Rf_!$ with $\mathbf Rf_*$, making proper direct image a left adjoint as well \cite[A.2 and A.16--A.17]{LiHuerta}. \subsection{The three realizations and the comparison functors}\label{app:three-realizations} Let $(R,R^+)$ be a complete Tate pair, let $B$ be a finitely presented $R$-algebra, and let $B^+$ be the integral closure of the image of $R^+$ in $B$. We describe the algebraic realization, analytic locus, and Huber analytification of $\Spec B$. For the algebraic realization, form the analytic ring \[ B_R^{\mathrm{alg}} =(R,R^+)^\square \mathbin{\otimes^{\mathrm{AnRing}}_{(R_{\mathrm{disc}},R^+_{\mathrm{disc}})^\square}} (B_{\mathrm{disc}},B^+_{\mathrm{disc}})^\square. \] The subscript pair and the second factor use discrete rings; only the first factor carries the topology of $R$. Lemma~\ref{lem:affinoid-module-category} identifies its modules as \[ \Cbox((\Spec B)^{\Alg}) \simeq\operatorname{Mod}_{\Cond_R(B)}\bigl(\Dsquare(R,R^+)\bigr) \] Thus we have a complete condensed $\underline R$-complex together with a $\Cond_R(B)$-action. When $B=R[T]$, giving the action means giving an endomorphism in $\Dsquare(R,R^+)$. For a quotient of a polynomial algebra, one also specifies the relations and their derived compatibilities. Direct image forgets the $\Cond_R(B)$-action and retains the underlying object of $\Dsquare(R,R^+)$. To algebraize the module, we first recover an ordinary $R$-complex and then lift the $B$-action by full faithfulness. To form the analytic locus, choose generators $b_1,\ldots,b_m$ and a topologically nilpotent unit $\varpi$. The bounded charts $|\varpi^nb_i|\leq1$ increase with $n$, and their union is $(\Spec B)^{\An}$. For example, if $B=R[T]$ with $R$ strongly noetherian, the $n$th chart is the disc of radius $|\varpi|^{-n}$ with ring $R\langle\varpi^nT\rangle$. For a quotient, impose the defining equations on these polydiscs. The construction is independent of the chosen generators and extends by Zariski gluing to separated finitely presented schemes \cite[A.9--A.13]{LiHuerta}. The Huber analytification $T^{\ad/V}$ over $V=\Spa(R,R^+)$ is instead an adic space. We associate to it the analytic stack $(T^{\ad/V})^\square$. Proposition~\ref{prop:analytic-Huber-comparison} identifies \[ T^{\An}\simeq(T^{\ad/V})^\square \] for strongly noetherian $R$, by identifying bounded chart rings and their complete modules. For proper $T/R$, the valuative criterion also gives $T^{\An}\simeq T^{\Alg}$. The first equivalence compares solid complexes with ordinary adic sheaves; the second permits algebraic localization of an analytic complex. Over $(X,\cX)$, perform these constructions on each $V=\Spa(R,R^+)\subset\lambda^{-1}(W)$ with $W=\Spec A$, and then glue. This gives $T^{\Alg}_{\cX}$, $T^{\An}_{\cX}$, and $T^{\ad}_{\cX}$. The comparison on the base is given by \[ \alpha_X(M)|_V =\Cond_R\bigl(R\otimes_A^{\mathbf L}\mathbf R\Gamma(W,M|_W)\bigr). \] This is scalar extension of sections on $W$, followed by condensification. Its compatibility on overlaps is proved in Section~\ref{subsec:base-comparison}. For affine $q:U\to X$, put $\cA=q_*\cO_U$. The two module descriptions are \begin{align*} \Dqc(U)&\simeq\operatorname{Mod}_{\cA}(\Dqc(X)),\\ \Cbox(U^{\Alg}_{\cX})&\simeq \operatorname{Mod}_{\alpha_X\cA}(\Cbox(\cX)). \end{align*} Apply $\alpha_X$ to the algebra, the complex, and the action to define $\alpha_U$. Zariski descent then defines $\alpha_T$ for general $T$. An \emph{algebraization} is a preimage equipped with its comparison equivalence. We call the forward functor the comparison functor, or analytification when its target is the analytic locus. \subsection{Finiteness conditions and the GAGA subcategories}\label{app:finiteness} Over an ordinary ring, a perfect complex has a bounded resolution by finite projective modules. A pseudocoherent complex has a resolution of the same kind that is only required to be bounded above. On schemes these are local conditions, and for noetherian $T$ we have \[ \Dpc(T)=\mathrm D^-_{\mathrm{coh}}(T). \] A complex $P\in\mathrm D(B)$ has Tor-amplitude in $[a,b]$ when $H^j(P\otimes_B^{\mathbf L}N)=0$ for every $B$-module $N$ and every $j\notin[a,b]$. On schemes, test this locally. Pseudocoherence together with finite Tor-amplitude gives perfectness. Derived pullback and tensor product preserve pseudocoherence \cite[Tags~064Q, 0650, 0658, and~0DJE]{Stacks}. For example, take $B=k[\epsilon]/(\epsilon^2)$. The module $k$ has the resolution \[ \cdots\xrightarrow{\epsilon}B\xrightarrow{\epsilon}B \xrightarrow{\epsilon}B\longrightarrow k\longrightarrow0. \] The image and kernel of multiplication by $\epsilon$ are both $(\epsilon)$, so the displayed complex is a resolution by finite free modules. It makes $k$ pseudocoherent. After tensoring with $k$, however, every differential is zero, giving $\operatorname{Tor}_i^B(k,k)=k$ for all $i\geq0$. Hence $k$ is not perfect. There is an analogous finiteness condition in a derived category generated by compact projective modules: an object is \emph{pseudocompact} if it has a bounded-above resolution by compact projectives. We use this condition for nonanimated Huber analytic rings. It can be tested by maps into families with a common lower cohomological bound: for every such family $(N_a)$, \[ \bigoplus_a\Map(P,N_a) \longrightarrow\Map\!\left(P,\bigoplus_aN_a\right) \] is an equivalence \cite[Definition~9.9]{ClausenScholze}. In each total degree, only finitely many terms of a bounded-above projective resolution contribute against targets with a common lower bound. Compactness of those terms gives the direct-sum comparison. A compact $P$ satisfies the analogous comparison for arbitrary filtered colimits, without a common degree bound. For a Huber analytic ring $\mathcal A=(A,\mathcal M)$, a complete module $N$ is \emph{nuclear} when \[ \bigl(\mathcal M[S]^\vee\otimes_{\mathcal A}^{\mathbf L}N\bigr)(*) \longrightarrow N(S) \] is an equivalence for every extremally disconnected $S$ \cite[Definition~5.30]{Andreychev}. Here $\mathcal M[S]^\vee$ is the internal dual and $N(S)$ means derived evaluation. For finite $S$, the formula is the tensor--Hom comparison for a finite sum of units. Nuclearity requires the same comparison after evaluation for infinite profinite sets. It is a condition on these families, not a bound on the dimension of $N(*)$. For compact $P,Q$, a map $P\to Q$ is \emph{trace class} when a map $\mathbf1\to P^\vee\otimes Q$ induces it by evaluation. It therefore factors as \[ P\longrightarrow P\otimes P^\vee\otimes Q \longrightarrow Q. \] A \emph{basic nuclear object} is a sequential colimit of compact objects whose transition maps are trace class. Nuclear objects are closed under colimits and admit presentations by basic nuclear objects \cite[Definitions~5.31--5.32 and Proposition~5.33]{Andreychev}. In Section~\ref{sec:relative}, we use these presentations to prove that scalar extension on bounded charts preserves nuclearity. The link with ordinary pseudocoherence is the following comparison: \[ \Cond_R(\Dpc(R)) =\{N\in\Dsquare(R,R^+): N\text{ is nuclear and pseudocompact}\}. \] This is the relative-discreteness theorem \cite[Proposition~5.22 and Theorem~5.50]{Andreychev} \cite[A.15]{LiHuerta}. The compact-projective resolution condition is called pseudocoherence in \cite[Section~5.2]{Andreychev} for nonanimated Huber rings. We call it pseudocompactness here to distinguish it from ordinary pseudocoherence in $\Dpc(R)$. On a strongly noetherian affinoid, the ordinary representatives give \begin{align*} \Cond_R(\Dpc(R))&=\Dpcbox(V),\\ \Cond_R(\mathrm D^b_{\mathrm{fg}}(R)) &=\Dpcbox(V)^b,\\ \Cond_R(\operatorname{Mod}_{\mathrm{fg}}(R)) &=\Dpcbox(V)^\heartsuit. \end{align*} The subscript $\mathrm{fg}$ means finitely generated, and $\mathrm D^b_{\mathrm{fg}}(R)$ requires finite cohomology in a finite interval. The last two lines use the degrees of the ordinary representatives. Finite projective modules give vector bundles, while perfect complexes give dualizable complete modules \cite[Proposition~5.12 and Corollary~5.51.1]{Andreychev}. For relative realizations, the two bounded-chart systems of \cite[A.15]{LiHuerta} test nuclearity and pseudocompactness and thus define $\Dpcbox$. Proposition~\ref{prop:analytic-Huber-comparison} gives an equivalent description: on each Huber affinoid $W=\Spa(C,C^+)$, the object is $\Cond_C(P_W)$ for some $P_W\in\Dpc(C)$. Rational restriction is ordinary derived scalar extension of these representatives. Since it is flat, it commutes with cohomology and truncation, allowing the local $t$-structures to glue. Sheafification and condensification of a finite $C$-module give its coherent sheaf and complete condensed module, respectively. The bounded comparisons use the same representatives with a common interval on a finite cover. Section~\ref{sec:classical} identifies these comparisons with $c_T^*$ and $\mathbf Lc_T^*$. \section{Descent}\label{sec:preliminaries} We collect the descent arguments used in the main text. We first pass from analytic affines to stacks, then treat the lifting of diagrams and module actions and the gluing of right adjoints. Finite open descent gives the compactness statements. The appendix ends with the integral and bounded-chart calculations underlying the affine module description. For ordinary limits, adjunctions, and Kan extensions, see \cite[Chapters~III--V and~X]{MacLane}. The arguments below take place in $\infty$-categories, for which we use \cite{LurieHTT,LurieHA}. We retain the notation of Appendix~\ref{app:analytic-foundations} and the analytic-ring constructions of \cite[Section~12]{ScholzeAnalytic} and \cite[A.1--A.3]{LiHuerta}. \subsection{Descent and fully faithful functors}\label{subsec:categorical-preliminaries} Put $\mathcal C=\operatorname{AnRing}^{\mathrm{op}}$, with the topology $\mathcal D$ generated by finite jointly conservative families of steady localizations. Complete modules on analytic affines extend to compatible families on $\operatorname{Shv}_{\mathcal D}(\mathcal C)$. The resulting functor takes colimits of stacks to limits of module categories, as we now prove. In particular, this applies to the simplicial presentations of algebraic realizations in \cite[Theorem~A.13]{LiHuerta}. \begin{proposition}\label{prop:analytic-stack-Cbox-descent} Let $a:\cU_0\to\cZ$ be an effective epimorphism of analytic stacks with \v Cech nerve $\cU_\bullet$. Pullback gives an equivalence in symmetric monoidal presentable stable $\infty$-categories \[ \Cbox(\cZ)\xrightarrow{\sim}\lim_{[n]\in\Delta}\Cbox(\cU_n). \] More generally, for every small diagram $(\cZ_i)_{i\in I}$, \[ \Cbox\!\left(\colim_{i\in I}^{\operatorname{Shv}_{\mathcal D}}\cZ_i\right) \xrightarrow{\sim}\lim_{i\in I^{\mathrm{op}}}\Cbox(\cZ_i). \] For $A\in\operatorname{CAlg}(\Cbox(\cZ))$, we also have \[ \operatorname{Mod}_A(\Cbox(\cZ))\xrightarrow{\sim} \lim_{[n]\in\Delta}\operatorname{Mod}_{A|_{\cU_n}}(\Cbox(\cU_n)). \] Colimits in these limits are computed componentwise. In particular, $\Cbox(|\cV_\bullet|)\simeq\lim_{[n]\in\Delta}\Cbox(\cV_n)$ for every simplicial analytic stack. \end{proposition} \begin{proof} Let $D_0$ denote the functor on analytic affines \begin{align*} \mathcal C^{\mathrm{op}}&\longrightarrow \operatorname{CAlg}(\operatorname{Pr}^L_{\mathrm{st}}),\\ \AnSpec(A,M)&\longmapsto\mathrm D(A,M). \end{align*} It satisfies descent for the generating covers \cite[A.2]{LiHuerta} \cite[Proposition~12.18]{ScholzeAnalytic}. For a presheaf $F$, define \[ \overline D(F)=\lim_{(U\to F)\in(\mathcal C_{/F})^{\mathrm{op}}}D_0(U). \] An object of this right Kan extension is a compatible family of complete modules on analytic affines over $F$. Since every presheaf is a colimit of representables, the universal property of the presheaf category shows that $\overline D$ carries colimits to limits \cite[Proposition~5.16]{ScholzeSix}. Let $L$ denote sheafification. We claim that $\overline D(F)\simeq\overline D(LF)$. Affine descent makes $\overline D$ invert every covering-sieve inclusion. These inclusions generate the sheafification equivalences under two-out-of-three, retracts, pushouts, and transfinite compositions \cite[Section~6.2.2]{LurieHTT}. The maps inverted by $\overline D$ are closed under these operations: pushouts and transfinite compositions become pullbacks and inverse limits of equivalences, while two-out-of-three and retracts follow from the same properties for equivalences. Thus $\overline D$ factors through sheafification, giving the functor $\Cbox$ on stacks. Since a colimit of sheaves is the sheafification of the presheaf colimit, \begin{align*} \Cbox\!\left(\colim_i^{\operatorname{Shv}}\cZ_i\right) &\simeq\overline D\!\left(L\colim_i^{\operatorname{PSh}}\cZ_i\right)\\ &\simeq\overline D\!\left(\colim_i^{\operatorname{PSh}}\cZ_i\right)\\ &\simeq\lim_{i\in I^{\mathrm{op}}}\overline D(\cZ_i)\\ &\simeq\lim_{i\in I^{\mathrm{op}}}\Cbox(\cZ_i). \end{align*} This proves the colimit formula, including the assertion for geometric realizations. For a generating cover $(U_i\to U)$, let $S\subset y(U)$ be the sieve of maps factoring through some $U_i$. The coproduct map factors as \[ \coprod_i y(U_i)\longrightarrow S\longrightarrow y(U) \] with the first arrow an effective epimorphism and the second a monomorphism \cite[Lemma~6.2.3.18]{LurieHTT}. Sheafification preserves finite limits and colimits, hence both the \v Cech nerve and its geometric realization. It therefore preserves effective epimorphisms. Since $S$ is covering, $LS\simeq Ly(U)$, and \[ \coprod_iLy(U_i)\longrightarrow Ly(U) \] is an effective epimorphism \cite[Lemma~6.2.3.19]{LurieHTT}. For an open cover, its image is the union of the opens, namely the whole target. The argument remains valid after base change. An effective epimorphism satisfies $|\cU_\bullet|\simeq\cZ$ \cite[Corollary~6.2.3.5]{LurieHTT}. Applying the colimit formula gives descent, with the overlap equivalences and all their higher homotopies. Finally, pullback preserves colimits, so taking a colimit componentwise retains the pullback identifications and computes the colimit in the limit category. Tensor products are componentwise too. An algebra or module structure is therefore a compatible collection of local actions and their homotopies; formation of these structures commutes with limits \cite[Sections~3.2.2 and~4.2.3]{LurieHA}. Applying this to $A$ gives the module-category equivalence. \end{proof} We next record how full faithfulness behaves under descent, both for diagrams and for algebra actions. \begin{lemma}\label{lem:functor-descent} Let $F:\mathcal C\to\mathcal D$ be a natural transformation of diagrams $I\to\operatorname{Cat}_\infty$, with $I$ small. Componentwise full faithfulness or equivalence passes to $\lim_I F$. In the fully faithful case, its essential image is detected componentwise. Natural transformations pass to limits. If the component categories admit small colimits and the transition functors preserve them, colimits in the limits are computed componentwise; if the $F_i$ also preserve them, so does $\lim_I F$. For fully faithful $H:\mathcal A\to\mathcal B$ and small $J$, the functor $\operatorname{Fun}(J,\mathcal A)\to\operatorname{Fun}(J,\mathcal B)$ is fully faithful, with essential image the diagrams valued in the essential image of $H$. \end{lemma} \begin{proof} An object of the source limit consists of $x_i\in\mathcal C_i$, transition equivalences, and their homotopies. Applying $F_i$ gives the corresponding family in the target. For two such objects, maps are computed by \[ \Map_{\lim_I\mathcal C_i}(x,y) \simeq\lim_{i\in I}\Map_{\mathcal C_i}(x_i,y_i). \] The transition maps on the right use the given identifications for $x$ and $y$. Full faithfulness of each $F_i$ identifies this diagram with the one in $\mathcal D$, so taking limits proves full faithfulness. If the $F_i$ are equivalences, the resulting map of limits is an equivalence as well. To identify the essential image, let $\mathcal E_i\subset\mathcal D_i$ be the full essential image of $F_i$. Naturality preserves these subcategories and gives a pointwise equivalence $\mathcal C\simeq\mathcal E$. A compatible family in $\mathcal D$ whose terms lie in the $\mathcal E_i$ is already a family in $\mathcal E$: fullness includes its maps and homotopies. It therefore lifts through \[ \lim_I\mathcal C_i\xrightarrow{\sim}\lim_I\mathcal E_i. \] This proves the essential-image assertion. Natural transformations act on the component families. For colimits, the transition functors preserve the componentwise colimits and hence their gluing identifications. Commuting the limits of local mapping spaces verifies the universal property. The same calculation shows that $\lim_I F$ preserves colimits whenever all the $F_i$ do. For the last assertion, identify $\mathcal A$ with its full essential image $\mathcal E\subset\mathcal B$. A diagram is valued in $\mathcal E$ exactly when all its objects are: its arrows and homotopies then lie there by fullness. The same is true of natural transformations. Hence \[ \operatorname{Fun}(J,\mathcal A)\simeq\operatorname{Fun}(J,\mathcal E) \subset\operatorname{Fun}(J,\mathcal B) \] is the asserted full subcategory. \end{proof} A fully faithful strong monoidal functor lifts a module structure whenever it lifts the underlying object. Write $\operatorname{LMod}_A$ for left modules over an associative algebra $A$. When $A$ is commutative and the category is symmetric monoidal, this is our usual $\operatorname{Mod}_A$. \begin{lemma}\label{lem:module-image} Let $F:\mathcal C\to\mathcal D$ be fully faithful and strong monoidal between monoidal $\infty$-categories, and let $A$ be an associative algebra in $\mathcal C$. Then \[ \operatorname{LMod}_A(\mathcal C)\longrightarrow \operatorname{LMod}_{F(A)}(\mathcal D) \] is fully faithful, with essential image detected on underlying objects. \end{lemma} \begin{proof} Let $N$ be an $F(A)$-module with underlying object identified with $F(M)$. Strong monoidality and full faithfulness identify the mapping spaces for its action: \[ \Map_{\mathcal C}(A\otimes M,M) \simeq\Map_{\mathcal D}(F(A)\otimes F(M),F(M)). \] The action therefore has a preimage $\mu:A\otimes M\to M$. Associativity compares \begin{align*} A\otimes A\otimes M&\xrightarrow{m_A\otimes1}A\otimes M \xrightarrow{\mu}M,\\ A\otimes A\otimes M&\xrightarrow{1\otimes\mu}A\otimes M \xrightarrow{\mu}M. \end{align*} The action on $N$ provides a homotopy between their images. Full faithfulness on $\Map(A\otimes A\otimes M,M)$ lifts that homotopy. For the unit, the image of $M\to A\otimes M\to M$ is homotopic to the identity, and the same argument lifts this homotopy as well. For the higher relations, use the equivalences \[ \Map_{\mathcal C}(A^{\otimes r}\otimes M,M) \simeq\Map_{\mathcal D}(F(A)^{\otimes r}\otimes F(M),F(M)). \] Compatibility with composition and insertion of the unit identifies all the operations and homotopies of the module structure. These are encoded by the two-colored module operad $\mathrm{LM}^{\otimes}$, whose colors specify an algebra and its module; here the algebra is fixed to be $A$ \cite[Sections~2.2.1 and~4.2.1]{LurieHA}. The lifted data therefore make $M$ an $A$-module with image $N$. The same argument applies to module maps. Full faithfulness lifts the underlying map $F(M)\to F(M')$ and its linearity homotopy, and the mapping-space equivalences for all tensor powers lift the remaining relations and homotopies between maps. Thus the functor on modules is fully faithful. Conversely, every module in its image has underlying object in the image of $F$. \end{proof} The right adjoints in the localization argument are restriction-of-scalars functors on affines. They glue when the comparisons defined by their units and counits are invertible. \begin{lemma}\label{lem:right-adjoint-descent} Let $F:\mathcal C\to\mathcal D$ be a natural transformation of diagrams $I\to\operatorname{Cat}_\infty$, with $I$ small, and suppose $F_i\dashv G_i$. For each $\theta:i\to j$, let $c_\theta,d_\theta$ be the transition functors. If each comparison map obtained by adjunction \[ c_\theta G_i\longrightarrow G_jd_\theta \] is invertible, these maps assemble the $G_i$ into a natural transformation and give a right adjoint to $\lim_I F$, computed componentwise. \end{lemma} \begin{proof} For $\theta:i\to j$, the given natural transformation supplies $d_\theta F_i\simeq F_jc_\theta$. Let $\eta_j$ be the unit of $F_j\dashv G_j$ and $\varepsilon_i$ the counit of $F_i\dashv G_i$. The comparison of right adjoints is \begin{align*} \beta_\theta:c_\theta G_i &\xrightarrow{\eta_j}G_jF_jc_\theta G_i\\ &\simeq G_jd_\theta F_iG_i \xrightarrow{G_jd_\theta\varepsilon_i}G_jd_\theta. \end{align*} This is an equivalence by hypothesis. For a compatible family $y=(y_i)$, set $(Gy)_i=G_iy_i$ and equip it with transition maps \[ c_\theta(G_iy_i)\xrightarrow{\beta_\theta} G_j(d_\theta y_i)\xrightarrow{\sim}G_jy_j. \] For two successive arrows, the intermediate unit and counit cancel by the triangle identity, leaving the comparison for the composite. Naturality gives the higher homotopies. Thus the displayed maps make $Gy$ a compatible family. Applying the $G_i$ to the component maps defines $G$ on morphisms. By construction, the local adjunctions respect these transition maps. Therefore \begin{align*} \Map_{\lim_I\mathcal D_i}(Fx,y) &\simeq\lim_{i\in I}\Map_{\mathcal D_i}(F_ix_i,y_i)\\ &\simeq\lim_{i\in I}\Map_{\mathcal C_i}(x_i,G_iy_i)\\ &\simeq\Map_{\lim_I\mathcal C_i}(x,Gy). \end{align*} These equivalences are natural in $x$ and $y$, proving that the componentwise functor $G$ is right adjoint to the functor on limits. \end{proof} \subsection{Finite open-cover descent}\label{subsec:finite-open-preliminaries} A finite open cover admits a finite descent diagram, indexed by its nonempty intersections. We use this description to test compactness locally. Let $\cZ=\bigcup_{i=1}^r\cU_i$. Put $\cU_I=\bigcap_{i\in I}\cU_i$ for nonempty $I\subset\{1,\ldots,r\}$, and let $P_r$ be the poset of these subsets ordered by inclusion. Write $K_I=K|_{\cU_I}$. An inclusion $I\subseteq J$ induces restriction from $\cU_I$ to $\cU_J$. \begin{lemma}\label{lem:binary-MV}\label{lem:finite-Cech-descent} Restriction gives natural equivalences \begin{align*} \Cbox(\cZ)&\xrightarrow{\sim}\lim_{I\in P_r}\Cbox(\cU_I),\\ \Map_{\cZ}(K,L)&\xrightarrow{\sim} \lim_{I\in P_r}\Map_{\cU_I}(K_I,L_I) \end{align*} for $K,L\in\Cbox(\cZ)$. \end{lemma} \begin{proof} Put $Q=P_r^{\mathrm{op}}$. We first prove the union formula \[ \colim_{I\in Q}\cU_I\xrightarrow{\sim}\cZ. \] The map $\coprod_i\cU_i\to\cZ$ is an effective epimorphism by Proposition~\ref{prop:analytic-stack-Cbox-descent}, so we may check the formula after pulling back to each $\cU_k$. Pullback commutes with colimits and replaces $\cU_I$ by $\cU_I\times_{\cZ}\cU_k=\cU_{I\cup\{k\}}$. Let $Q_k\subset Q$ consist of the subsets containing $k$. The functor $r_k(I)=I\cup\{k\}$ is cofinal. To check this, fix $J\in Q_k$. A map $J\to r_k(I)$ means $I\subseteq J$, so its indexing category is the reverse-inclusion poset of nonempty subsets of $J$. It has initial object $J$, and \cite[Theorem~4.1.3.1]{LurieHTT} applies. Since $\{k\}$ is terminal in $Q_k$, we obtain \[ \colim_{I\in Q}\cU_{I\cup\{k\}} \simeq\colim_{J\in Q_k}\cU_J \simeq\cU_k. \] This proves the union formula on the cover and hence globally. Proposition~\ref{prop:analytic-stack-Cbox-descent} turns it into the limit formula for module categories. Mapping spaces in this limit are limits of the local mapping spaces; applying the same calculation to shifts gives the statement for mapping spectra. For two opens, the limit is a homotopy pullback over the intersection, hence in spectra the fiber of the difference of restrictions. Taking $K=\cO_{\cZ}$ gives Mayer--Vietoris for global sections. \end{proof} The nerve of $P_r$ has only finitely many nondegenerate simplices: they are the strictly increasing chains of nonempty subsets. Consequently, the limits above are finite homotopy limits and commute with filtered colimits of spectra. \begin{proposition}\label{prop:finite-descent} Let $\cZ=\bigcup_{i=1}^r\cU_i$ be a finite open cover of an analytic stack, and put $\cU_I=\bigcap_{i\in I}\cU_i$ for nonempty $I$. Then: \begin{enumerate}[label=\textup{(\roman*)}] \item $\mathbf R\Gamma(\cZ,-)$ preserves filtered colimits if the unit of each $\Cbox(\cU_I)$ is compact; \item $K\in\Cbox(\cZ)$ is compact if every $K|_{\cU_I}$ is compact. \end{enumerate} The condition in \textup{(i)} holds when the intersections are finite disjoint unions of affines $\AnSpec(R,R^+)^\square$ for complete Huber pairs. \end{proposition} \begin{proof} Suppose each $K_I$ is compact, and let $(L_a)$ be a filtered diagram. Restriction preserves colimits, so the finite descent formula gives \begin{align*} \Map_{\cZ}(K,\colim_aL_a) &\simeq\lim_{I\in P_r} \Map_{\cU_I}(K_I,\colim_aL_{a,I})\\ &\simeq\lim_{I\in P_r}\colim_a \Map_{\cU_I}(K_I,L_{a,I})\\ &\simeq\colim_a\lim_{I\in P_r} \Map_{\cU_I}(K_I,L_{a,I})\\ &\simeq\colim_a\Map_{\cZ}(K,L_a). \end{align*} The second equivalence uses local compactness, and the third commutes a filtered colimit with a finite homotopy limit. Thus $K$ is compact, proving (ii). For (i), apply (ii) to $\cO_{\cZ}$. In the affine case $\AnSpec(R,R^+)^\square$, the unit is the measure module associated with the one-point set and is compact \cite[Theorem~3.28 and Proposition~2.11]{Andreychev}. A finite disjoint union gives a finite product of mapping spectra, so compactness persists. Empty intersections contribute zero. This proves the final assertion. \end{proof} \subsection{Integral subrings and bounded charts}\label{subsec:analytic-ring-preliminaries} Fix a complete Tate Huber pair $(R,R^+)$. For any $R$-algebra $B$, let $B^+$ be the integral closure of the image of $R^+$. Define its algebraic realization by \[ B_R^{\mathrm{alg}} =(R,R^+)^\square \mathbin{\otimes^{\mathrm{AnRing}}_{(R_{\mathrm{disc}},R^+_{\mathrm{disc}})^\square}} (B_{\mathrm{disc}},B^+_{\mathrm{disc}})^\square. \] Write $(\Spec B)^{\Alg}=\AnSpec B_R^{\mathrm{alg}}$. When $B$ is finitely presented, this is the construction of \cite[A.9]{LiHuerta}. Its analytic locus is the union of the bounded charts associated with a pseudo-uniformizer and finitely many generators. In that case the constructions extend by Zariski gluing and agree for proper schemes \cite[A.9--A.13]{LiHuerta}. We begin with the discrete pairs. The ring $B$ determines the module action, while $B^+$ determines which of those modules are complete. \begin{lemma}\label{lem:integral-induced-modules} Let $R\to B$ be a ring map, let $R^+\subset R$ be integrally closed, and let $B^+$ be the integral closure of the image of $R^+$ in $B$. Put \begin{align*} \mathcal R_0&=(R_{\mathrm{disc}},R^+_{\mathrm{disc}})^\square,\\ \mathcal B_0&=(B_{\mathrm{disc}},B^+_{\mathrm{disc}})^\square. \end{align*} Then $B$ is $\mathcal R_0$-complete, its induced analytic structure is $\mathcal B_0$, and restriction gives a canonical symmetric monoidal equivalence \[ \mathsf D(\mathcal B_0)\simeq\operatorname{Mod}_B\bigl(\mathsf D(\mathcal R_0)\bigr). \] Both sides consist of condensed $B$-complexes complete over $\mathcal R_0$. These identifications are functorial in maps of integral pairs. \end{lemma} \begin{proof} Let $N$ be a condensed $B$-complex. Put $C^+=\operatorname{im}(R^+\to B)$ and factor the map of plus rings as \[ R^+\twoheadrightarrow C^+\longrightarrow B^+. \] Both arrows are integral, the first being a quotient map. Completeness may be tested on the plus ring \cite[Proposition~3.21]{Andreychev}, and integral extension does not change this condition \cite[Proposition~3.22]{Andreychev}. We therefore have \begin{align*} N\text{ is }\mathcal B_0\text{-complete} &\Longleftrightarrow N|_{B^+}\text{ is }(B^+_{\mathrm{disc}})^\square\text{-complete}\\ &\Longleftrightarrow N|_{C^+}\text{ is }(C^+_{\mathrm{disc}})^\square\text{-complete}\\ &\Longleftrightarrow N|_{R^+}\text{ is }(R^+_{\mathrm{disc}})^\square\text{-complete}\\ &\Longleftrightarrow N|_R\text{ is }\mathcal R_0\text{-complete}. \end{align*} In other words, $N$ is $\mathcal B_0$-complete exactly when its restriction to $R$ is $\mathcal R_0$-complete. Apply this criterion to $B[0]$. It is $\mathcal B_0$-complete by \cite[Lemma~3.24]{Andreychev}, and hence $\mathcal R_0$-complete. The usual multiplication and unit therefore make $B$ an algebra in $\mathsf D(\mathcal R_0)$. Let $\mathcal B_{\mathrm{ind}}$ be the analytic structure induced along $R\to B$. By \cite[Proposition~12.8]{ScholzeAnalytic}, its module description is \[ \mathsf D(\mathcal B_{\mathrm{ind}}) \simeq\operatorname{Mod}_B(\mathsf D(\mathcal R_0)) \] \cite[Lemma~A.1 and A.3]{LiHuerta}. Its underlying ring is the $\mathcal R_0$-completion of $B$, hence $B$ itself. Its modules are precisely the condensed $B$-complexes whose restrictions are $\mathcal R_0$-complete, as in the displayed chain. The two analytic structures have the same underlying ring and complete connective modules. By uniqueness \cite[Proposition~12.20]{ScholzeAnalytic}, \[ \mathcal B_{\mathrm{ind}}\simeq\mathcal B_0. \] The equivalence identifies the module categories and their forgetful functors. On both sides, tensor product is tensor product over $B$ followed by the same analytic completion, so the equivalence is symmetric monoidal. All these identifications use the given ring maps and restriction functors. The left adjoint to the inclusion of complete connective modules is unique up to contractible choice. This compares the constructions for a map of integral pairs and supplies the composition homotopies for successive maps. \end{proof} For a Tor-independent base change, the derived tensor product is the ordinary ring $B\otimes_RR'$. We identify its analytic structure by comparing the complete modules over this ring. \begin{lemma}\label{lem:integral-base-change} Let $R\to R'$ be a ring map carrying an integrally closed subring $R^+\subset R$ into an integrally closed subring $R'^+\subset R'$. Let $B$ be an $R$-algebra with $\operatorname{Tor}_i^R(B,R')=0$ for $i>0$, and put $B'=B\otimes_RR'$. Write $B^+,B'^+$ for the integral closures of their images in $B,B'$, respectively. There is a canonical equivalence of complete analytic rings \[ (R'_{\mathrm{disc}},R'^+_{\mathrm{disc}})^\square \mathbin{\otimes^{\mathrm{AnRing}}_{(R_{\mathrm{disc}},R^+_{\mathrm{disc}})^\square}} (B_{\mathrm{disc}},B^+_{\mathrm{disc}})^\square \simeq (B'_{\mathrm{disc}},B'^+_{\mathrm{disc}})^\square. \] It is compatible with composition under the same Tor-vanishing for successive base changes, in particular for chains of flat maps. \end{lemma} \begin{proof} Write $\mathcal R_0,\mathcal B_0,\mathcal R'_0,\mathcal B'_0$ for the analytic rings of the four discrete pairs, and set \[ \mathcal P=\mathcal R'_0\otimes^{\mathrm{AnRing}}_{\mathcal R_0}\mathcal B_0. \] A monic equation for $b\in B^+$ remains monic over the image of $R'^+$ in $B'$, so $b$ maps into $B'^+$. This map of pairs and the map from $(R',R'^+)$ define $\mathcal P\to\mathcal B'_0$ by the pushout property. We compute its underlying ring before analytic completion: \[ D=B_{\mathrm{disc}}\otimes^{\mathbf L}_{R_{\mathrm{disc}}}R'_{\mathrm{disc}}. \] Discrete condensification commutes with derived tensor product, so the Tor-vanishing assumption gives \[ D\simeq(B\otimes_RR')_{\mathrm{disc}}[0]=B'_{\mathrm{disc}}[0]. \] By \cite[Proposition~12.12]{ScholzeAnalytic}, a connective condensed $B'$-complex $N$ is complete for the pushout precisely when $N|_{R'}$ is $\mathcal R'_0$-complete and $N|_B$ is $\mathcal B_0$-complete. The first condition implies the second by restriction to $R$ and Lemma~\ref{lem:integral-induced-modules}. Applying the same lemma to $R'\to B'$ gives \begin{align*} N\text{ is }\mathcal P\text{-complete} &\Longleftrightarrow N|_{R'}\text{ is }\mathcal R'_0\text{-complete}\\ &\Longleftrightarrow N\text{ is }\mathcal B'_0\text{-complete}. \end{align*} The unit $B'[0]$ is complete by Lemma~\ref{lem:integral-induced-modules}, so analytic completion leaves $D\simeq B'_{\mathrm{disc}}$ unchanged. The source and target have the same ring and complete connective modules; \cite[Proposition~12.20]{ScholzeAnalytic} therefore makes the comparison an equivalence. For $R'\to R''$ with $\operatorname{Tor}_i^{R'}(B',R'')=0$ for $i>0$, use \[ (B\otimes_R^{\mathbf L}R')\otimes_{R'}^{\mathbf L}R'' \simeq B\otimes_R^{\mathbf L}R''. \] Tor-vanishing identifies this derived tensor product with $B\otimes_RR''$. Both analytic pushouts impose completeness over $(R''_{\mathrm{disc}},R''^+_{\mathrm{disc}})^\square$, so the displayed associativity map compares them. Uniqueness of analytic completion gives the compatibilities for longer chains. \end{proof} Rational localizations of strongly noetherian Tate pairs are flat \cite[Lemma~2.2.6]{KedlayaLiu}, so the preceding calculation applies to the base changes in the main text. The bounded charts also commute with Tor-independent base change. In the following lemma the realizations and fiber products are taken in analytic stacks; for $B=R[T]$, its charts impose the bounds $|T|\leq|\varpi|^{-n}$. \begin{lemma}\label{lem:bounded-chart-base-change} Let $(R,R^+)\to(R',R'^+)$ be a morphism of complete Tate Huber pairs, and let $B$ be finitely presented over $R$ with $\operatorname{Tor}_i^R(B,R')=0$ for $i>0$. Choose a topologically nilpotent unit $\varpi\in R$ and generators $\mathbf b=(b_1,\ldots,b_m)$, and put $B'=B\otimes_RR'$. Use primes for their images after base change. Then: \begin{enumerate}[label=\textup{(\roman*)}] \item There is a canonical equivalence \[ B_R^{\mathrm{alg}} \mathbin{\otimes^{\mathrm{AnRing}}_{(R,R^+)^\square}} (R',R'^+)^\square\simeq(B')_{R'}^{\mathrm{alg}}. \] \item The chart $|\varpi^nb_i|\leq1$ base changes to $|{\varpi'}^nb'_i|\leq1$, compatibly with the transitions for $n\geq0$. \item Their increasing union is $(\Spec B)^{\An}$, and \[ (\Spec B)^{\An}\times_{\AnSpec(R,R^+)^\square} \AnSpec(R',R'^+)^\square \simeq(\Spec B')^{\An}. \] This is functorial in affine schemes, coherent under successive base changes satisfying the same Tor-vanishing condition, and independent of $\varpi$ and $\mathbf b$. \end{enumerate} \end{lemma} \begin{proof} We compute the algebraic realization using Lemma~\ref{lem:integral-base-change}. Put \begin{align*} \mathcal R_0&=(R_{\mathrm{disc}},R^+_{\mathrm{disc}})^\square,\\ \mathcal B_0&=(B_{\mathrm{disc}},B^+_{\mathrm{disc}})^\square, \end{align*} and $\mathcal R=(R,R^+)^\square$, with primes after base change. By \cite[A.9]{LiHuerta}, \[ B_R^{\mathrm{alg}}=\mathcal R\otimes^{\mathrm{AnRing}}_{\mathcal R_0}\mathcal B_0. \] Reassociating the pushouts and applying Lemma~\ref{lem:integral-base-change} gives \begin{align*} B_R^{\mathrm{alg}}\otimes^{\mathrm{AnRing}}_{\mathcal R}\mathcal R' &\simeq\mathcal R'\otimes^{\mathrm{AnRing}}_{\mathcal R'_0} \left(\mathcal R'_0\otimes^{\mathrm{AnRing}}_{\mathcal R_0}\mathcal B_0\right)\\ &\simeq\mathcal R'\otimes^{\mathrm{AnRing}}_{\mathcal R'_0}\mathcal B'_0\\ &=(B')_{R'}^{\mathrm{alg}}. \end{align*} Here Lemma~\ref{lem:integral-base-change} computes the inner pushout in the first line. This proves (i). To include the bounds at level $n$, form the pushout \[ \begin{CD} (\mathbf Z[T_1,\ldots,T_m],\mathbf Z)^\square @>>>\mathbf Z[T_1,\ldots,T_m]^\square\\ @V{T_i\mapsto\varpi^nb_i}VV @VVV\\ B_R^{\mathrm{alg}}@>>>B^{\mathrm{univ}}_{|\varpi^n\mathbf b|\leq1}. \end{CD} \] The upper map imposes integral bounds on the variables. Thus a map from the lower right ring is a map from $B_R^{\mathrm{alg}}$ for which the chosen elements satisfy those bounds. The upper map is a steady localization \cite[A.9]{LiHuerta}; its base change therefore defines a subobject of $\AnSpec B_R^{\mathrm{alg}}$, with a contractible space of factorizations whenever one exists. The upper row does not depend on $R$. After base change to $\mathcal R'$, the lower left ring becomes $(B')_{R'}^{\mathrm{alg}}$ and the bounded elements become ${\varpi'}^nb'_i$. The pushout consequently gives \[ B^{\mathrm{univ}}_{|\varpi^n\mathbf b|\leq1} \otimes^{\mathrm{AnRing}}_{\mathcal R}\mathcal R' \simeq(B')^{\mathrm{univ}}_{|{\varpi'}^n\mathbf b'|\leq1}. \] The transition to level $n+1$ uses \[ \varpi^{n+1}b_i=\varpi(\varpi^nb_i). \] Since $\varpi\in R^+$, a bound at level $n$ implies the bound at level $n+1$. Thus the charts increase, with their ring maps going in the opposite direction. The same identity holds over $R'$, so the comparison respects these transitions. This proves (ii). Separate bounds on the $m$ generators give an $\mathbf N^m$-indexed system. Each tuple is dominated by its largest entry, so the diagonal $\mathbf N\to\mathbf N^m$ is cofinal. By \cite[A.11--A.15]{LiHuerta}, \[ (\Spec B)^{\An} \simeq\colim_{n\geq0}\AnSpec B^{\mathrm{univ}}_{|\varpi^n\mathbf b|\leq1}. \] Pullback preserves this colimit, so taking unions proves (iii). Successive Tor-independent base changes reassociate the same pushouts, compatibly with the transitions in $n$. To compare the choices of generators and pseudo-uniformizer, write \[ U_{\varpi,n}(\mathbf b)=\AnSpec B^{\mathrm{univ}}_{|\varpi^n\mathbf b|\leq1}. \] Every topologically nilpotent element lies in $R^+$: a sufficiently large power lies in this open subring, so integral closedness gives the claim. We also use the universal bounded-locus maps for sums and products from \cite[A.9 and the proof of Proposition~A.12\textup{(2)}]{LiHuerta}, which turn the following polynomial identities into maps of bounded charts. For two generating tuples, enlarge both to their union $\mathbf c$. It is enough to compare one tuple $\mathbf b$ with this union. Write each added generator as \[ c_j=\sum_\nu a_{j\nu}\mathbf b^\nu. \] Fix a level $n$. Only finitely many coefficients occur, so openness of $R^+$ and topological nilpotence of $\varpi$ let us choose $m\geq n$ such that \[ \varpi^{m-n|\nu|}a_{j\nu}\in R^+ \] for every coefficient. On $U_{\varpi,n}(\mathbf b)$, this choice gives \[ \varpi^mc_j =\sum_\nu \bigl(\varpi^{m-n|\nu|}a_{j\nu}\bigr)(\varpi^n\mathbf b)^\nu. \] Each factor on the right is bounded, so the sum and product maps give the required bound on $\varpi^mc_j$. The original generators remain bounded at level $m$ because $m\geq n$. Hence \[ U_{\varpi,n}(\mathbf b)\subset U_{\varpi,m}(\mathbf c). \] Forgetting the added bounds gives the reverse comparison \[ U_{\varpi,n}(\mathbf c)\subset U_{\varpi,n}(\mathbf b). \] We may choose $m$ nondecreasing with $n$. The two comparisons then identify the unions of the chart systems. Now change the topologically nilpotent unit to $\widetilde\varpi$. Choose $r,s>0$ with \begin{align*} \widetilde\varpi^r/\varpi&\in R^+,\\ \varpi^s/\widetilde\varpi&\in R^+. \end{align*} Such choices exist because the corresponding sequences of quotients tend to zero. On $U_{\varpi,n}(\mathbf c)$, the identity \[ \widetilde\varpi^{rn}c_i =\left(\frac{\widetilde\varpi^r}{\varpi}\right)^n(\varpi^nc_i) \] expresses the new coordinate as a product of bounded elements. Reversing the roles of the two units gives \begin{align*} U_{\varpi,n}(\mathbf c)&\subset U_{\widetilde\varpi,rn}(\mathbf c),\\ U_{\widetilde\varpi,n}(\mathbf c)&\subset U_{\varpi,sn}(\mathbf c). \end{align*} Thus these unions agree as well. On their common refinements the base-change map is still obtained from the same pushout, so it is independent of both choices. For an $R$-algebra map $B\to C$, express the images of the $b_i$ in generators of $C$. The same coefficient bound sends each chart for $C$ into a chart for $B$; taking unions gives the map of analytic loci. Enlarging bounds compares choices. Contractibility of factorizations through localization subobjects supplies compatibility with identities, composition, the stated base changes, and higher homotopies. \end{proof} \subsection{Comparison over a Tate affinoid}\label{subsec:affinoid-full-faithfulness} The module description used in the algebraization argument follows by applying condensification to an algebra and its modules. Recall the formula \[ \Cond_R(M)=\underline R\otimes^{\mathbf L}_{R_{\mathrm{disc}}}M_{\mathrm{disc}} \] for $M\in\mathrm D(R)$, with values in complete modules \cite[Theorem~5.9]{Andreychev}. \begin{lemma}\label{lem:affinoid-module-category} Let $(R,R^+)$ be a complete Tate Huber pair and let $A$ be an $R$-algebra. There is a canonical symmetric monoidal equivalence \[ \Cbox((\Spec A)^{\Alg})\simeq \operatorname{Mod}_{\Cond_R(A)}\bigl(\Dsquare(R,R^+)\bigr). \] For the structural map $q_A^{\Alg}$, $\mathbf Rq_{A,*}^{\Alg}$ is the forgetful functor. The comparison $\alpha_A^{\Alg}$ is induced by $\Cond_R$ on modules and is fully faithful, with essential image detected on underlying objects in $\Cond_R(\mathrm D(R))$. \end{lemma} \begin{proof} Let $A^+$ be the integral closure of the image of $R^+$ in $A$. Put \begin{align*} \mathcal R_0&=(R_{\mathrm{disc}},R^+_{\mathrm{disc}})^\square,\\ \mathcal A_0&=(A_{\mathrm{disc}},A^+_{\mathrm{disc}})^\square. \end{align*} Set $\mathcal R=(R,R^+)^\square$. By the definition above, \[ A_R^{\mathrm{alg}}=\mathcal R\otimes^{\mathrm{AnRing}}_{\mathcal R_0}\mathcal A_0. \] Pass to complete-module categories and apply Lemma~\ref{lem:integral-induced-modules}: \begin{align*} \mathsf D(A_R^{\mathrm{alg}}) &\simeq\mathsf D(\mathcal R)\otimes_{\mathsf D(\mathcal R_0)}\mathsf D(\mathcal A_0)\\ &\simeq\mathsf D(\mathcal R)\otimes_{\mathsf D(\mathcal R_0)} \operatorname{Mod}_A(\mathsf D(\mathcal R_0))\\ &\simeq\operatorname{Mod}_{\Cond_R(A)}(\Dsquare(R,R^+)). \end{align*} The first equivalence is the pushout formula \cite[Lemma~A.1]{LiHuerta}; the second tests completeness after restriction to $\mathcal R_0$. The last extends the module category to $\mathsf D(\mathcal R)$ \cite[Section~4.8.5]{LurieHA}, carrying $A$ to $\Cond_R(A)$. All three respect tensor products, so the equivalence is symmetric monoidal. Under this equivalence, structural pullback is the free-module functor \[ K\longmapsto\Cond_R(A)\otimes K. \] Maps from this free module to $N$ are maps from $K$ to its underlying object in $\Dsquare(R,R^+)$. Thus the right adjoint $\mathbf Rq_{A,*}^{\Alg}$ forgets the $\Cond_R(A)$-action. An ordinary $A$-complex is an $A$-module in $\mathrm D(R)$. Condensifying the object and its action defines the comparison. Since $\Cond_R$ is fully faithful, exact, and colimit-preserving \cite[Theorem~5.9 and Lemma~5.10]{Andreychev}, Lemma~\ref{lem:module-image} gives full faithfulness on modules. If $N$ has underlying object $\Cond_R(M)$, the same lemma lifts its action to $A\otimes_R^{\mathbf L}M\to M$, including the unit, associativity, and higher relations. Hence $N$ comes from an ordinary $A$-complex. Conversely, an object in the image has underlying complex in $\Cond_R(\mathrm D(R))$ by construction. \end{proof} Zariski descent glues the affine comparisons, with their scalar-extension identifications. \begin{proposition}\label{prop:affinoid-full-faithfulness} Let $(R,R^+)$ be a complete Tate Huber pair and let $S$ be a separated finitely presented $R$-scheme. Scalar extension over the Tate base induces a fully faithful colimit-preserving strong symmetric monoidal functor \(\alpha_S^{\Alg}:\Dqc(S)\longrightarrow\Cbox(S^{\Alg})\). It is functorial in $S$ and compatible with pullback. \end{proposition} \begin{proof} Choose a finite affine cover $(S_a)$ of $S$. Separatedness makes all multiple intersections affine, so Lemma~\ref{lem:affinoid-module-category} gives a fully faithful comparison on each. For $\Spec A'\subset\Spec A$, restriction is $M\mapsto A'\otimes_A^{\mathbf L}M$. Condensification commutes with scalar extension \cite[Lemma~5.10]{Andreychev}, identifying its image with analytic restriction. For successive inclusions, the comparisons are \[ A''\otimes_{A'}^{\mathbf L}(A'\otimes_A^{\mathbf L}M) \simeq A''\otimes_A^{\mathbf L}M. \] Associativity includes the higher compatibilities, so these functors form a map of the full \v Cech diagrams. Zariski descent computes $\Dqc(S)$ as the ordinary limit. The presentation in \cite[Theorem~A.13]{LiHuerta} and Proposition~\ref{prop:analytic-stack-Cbox-descent} compute $\Cbox(S^{\Alg})$ as the analytic limit. Taking limits defines $\alpha_S^{\Alg}$, and Lemma~\ref{lem:functor-descent} gives full faithfulness and colimit preservation. The affine tensor and unit comparisons agree under restriction, so they descend to the strong symmetric monoidal structure. Common affine refinements identify different covers. For $S'\to S$, choose source affines mapping into target affines. Pullback compatibility there is the scalar-extension formula above; it commutes with restriction and hence glues. 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