Proof. To prove the second point, we may work locally on $\vert Y \vert = \vert Y^{\diamondsuit } \vert $ and assume that $Y=\operatorname {\mathrm {Spa}}(B,B^+)$ is affinoid. We have the following adjunction, left to the reader

$$\begin{align*}\operatorname{\mathrm{Hom}}_{\operatorname{\mathrm{Spd}}(\mathbb{Q}_p)}(X^{\diamondsuit},\operatorname{\mathrm{Spd}}(B,B^+)) = \operatorname{\mathrm{Hom}}_{\operatorname{\mathrm{cts}}}((B,B^+),(\mathcal{O}_v(X^{\diamondsuit}),\mathcal{O}_v^{+}(X^{\diamondsuit}))).\end{align*}$$

It thus remains to show that the natural map

$$\begin{align*}\mathcal{O}^+(X) \rightarrow \mathcal{O}_v^+(X^{\diamondsuit})\end{align*}$$

is an isomorphism, so that it suffices to show the first point. The isomorphism $\nu _*(\mathcal {O}_{X_v}^+/p^n) = \mathcal {O}_{X_{\operatorname {\mathrm {\acute {e}t}}}}^+/p^n$ is [Reference Mann and Werner41, Prop. 2.8, Prop. 2.13] when X is a rigid space and [Reference Heuer24, Corollary 2.15] when X is sousperfectoid. The isomorphism $\nu _*\mathcal {O}_{X_v}^+ = \mathcal {O}_{X_{\operatorname {\mathrm {\acute {e}t}}}}^+$ follows from $\nu _*\mathcal {O}_{X_v} = \mathcal {O}_{X_{\operatorname {\mathrm {\acute {e}t}}}}$ . Finally, to show the last isomorphism, it is enough to show the isomorphism

$$\begin{align*}\mathcal{O}_{X_{v}}^+ \xrightarrow{\cong }\varprojlim_n \mathcal{O}_{X_{v}}^+/p^n.\end{align*}$$

This can be checked on affinoid perfectoids Y. Injectivity is clear. For surjectivity, given a collection $(s_n)_{n\geq 1}$ of compatible sections $s_n \in \mathcal {O}_v^+/p^n(Y)$ , let $c_n\in H_v^1(Y,\mathcal {O}_v^+)$ be the obstruction class to lifting $s_n$ to a section in $\mathcal {O}^+(Y)/p^n$ . Then $c_n = pc_{n+1}$ . Since $H_v^1(Y,\mathcal {O}_v^+)$ is almost zero, by [Reference Scholze47, Prop. 8.8], it is in particular annihilated by p. Hence $c_n$ vanishes and $(s_n)_n$ is in the image of the above map, as required. This concludes the proof.

Proof. The claim is local on $X_{\operatorname {\mathrm {\acute {e}t}}}$ . Therefore, we may assume that $X=\operatorname {\mathrm {Spa}}(A,A^+)$ , $X'=\operatorname {\mathrm {Spa}}(B,B^+)$ , and f is standard-smooth, that is, a composition of finite-étale maps, rational open immersions, and projections $\mathbb {B}_S^1 \rightarrow S$ . We may further assume that $X'=\mathbb {B}_X^1$ , so that $B=A\langle T \rangle $ . Fix a pro-étale presentation $X^{\diamondsuit } = Y/R$ for perfectoids Y and $R\subseteq Y\times Y$ . By Proposition 2.4.1, we have left exact sequences, for each $n\geq 1$

Upon adjoining a polynomial variable, taking the inverse limit over n and inverting p, we obtain the left exact sequence

By a $5$ -lemma argument, to show that

$$\begin{align*}\mathcal{O}_{\operatorname{\mathrm{\acute{e}t}}}(\mathbb{B}_X^1) \cong \mathcal{O}_v(\mathbb{B}_X^{1,\diamondsuit}), \end{align*}$$

it is enough to show that

$$\begin{align*}\mathcal{O}_{\operatorname{\mathrm{\acute{e}t}}}(\mathbb{B}_Y^1) \cong \mathcal{O}_v(\mathbb{B}_Y^{1,\diamondsuit}), \quad \mathcal{O}_{\operatorname{\mathrm{\acute{e}t}}}(\mathbb{B}_R^1) \cong \mathcal{O}_v(\mathbb{B}_R^{1,\diamondsuit}). \end{align*}$$

Hence, we may assume that X is affinoid perfectoid. In that case, X is diamantine, in the sense of [Reference Hansen and Kedlaya30, Def. 11.1], so that the result follows from [Reference Hansen and Kedlaya30, Thm. 11.18].

Proof. Let us write $\mathcal {F} = \varinjlim _{i\in I} \mathcal {F}_i$ where I is cofiltered and the $\mathcal {F}_i$ are Zariski-constructible. As $\pi \colon X \rightarrow S$ is qcqs, we have, by [48, Tag 0739]

$$\begin{align*}R^n\pi_{\operatorname{\mathrm{\acute{e}t}},*}\mathcal{F} = \varinjlim_i R^n\pi_{\operatorname{\mathrm{\acute{e}t}},*}\mathcal{F}_i. \end{align*}$$

From this, we reduce to the case where $\mathcal {F} = \mathcal {F}_i$ is Zariski-constructible. Let $X_0,S_0,Y_0$ denote the respective base change along $\operatorname {\mathrm {Spa}}(K,\mathcal {O}_K) \rightarrow \operatorname {\mathrm {Spa}}(K,K^+)$ . By [Reference Huber31, Prop. 8.2.3(ii)], the sheaves $R^n\pi _{\operatorname {\mathrm {\acute {e}t}},*}\mathcal {F}$ are overconvergent. Furthermore, their formations are easily seen to commute with base change along $S_0 \subseteq S$ . Hence, we may assume that $K^+=\mathcal {O}_K$ , in which case this is [Reference Bhatt and Hansen4, Thm. 3.10, Thm. 3.15].

Proof. Let $\mathbb {B} \rightarrow X$ denote the vector bundle underlying the locally free sheaf B. The norm map $\operatorname {\mathrm {Nr}}\colon B \rightarrow \mathcal {O}_X$ induces a map of X-spaces $N\colon \mathbb {B} \rightarrow \mathbb {G}_{a,X}$ and $B^{\times }$ is easily seen to be represented by the open subspace $N^{-1}(\mathbb {G}_{m,X})$ . Let $f\colon X' \rightarrow X$ denote the finite flat morphism associated with B. It is clear that

$$\begin{align*}f_{\operatorname{\mathrm{\acute{E}t}},*}^{\operatorname{\mathrm{rig}}}\mathbb{G}_{m,X'} = B^{\times},\end{align*}$$

where

$$\begin{align*}f_{\operatorname{\mathrm{\acute{E}t}}}^{\operatorname{\mathrm{rig}}}\colon \operatorname{\mathrm{Sm}}_{/X',\operatorname{\mathrm{\acute{e}t}}} \rightarrow \operatorname{\mathrm{Sm}}_{/X,\operatorname{\mathrm{\acute{e}t}}} \end{align*}$$

is the natural map of sites. By Theorem 2.12.1, $B^{\times }[p^m] = f_{\operatorname {\mathrm {\acute {E}t}},*}^{\operatorname {\mathrm {rig}}}\mu _{p^m}$ is Zariski-constructible. Moreover, $f_{\operatorname {\mathrm {\acute {E}t}},*}^{\operatorname {\mathrm {rig}}}$ is exact, by [Reference Huber31, Cor. 2.6.6], such that

$$\begin{align*}R^1f_{\operatorname{\mathrm{\acute{E}t}},*}^{\operatorname{\mathrm{rig}}}\mu_{p^m}=0.\end{align*}$$

It follows that $[p]\colon B^{\times } \rightarrow B^{\times }$ is surjective, and it is also étale by Remark 2.18. In particular $B^{\times }$ is locally p-divisible, as required.

Proof. We start by showing that there exists a finite locally free $\mathcal {O}_X^+$ -submodule $E \subseteq \operatorname {\mathrm {Lie}}(\mathcal {G})$ such that $\operatorname {\mathrm {Lie}}(\mathcal {G}) = E[\frac {1}{p}]$ . For this, we view X as a coherent rigid space in the sense of [Reference Abbes1, Def. 4.1.6] and thus we find a formal model ${\mathfrak {X}}$ of X topologically of finite presentation over $\operatorname {\mathrm {Spf}}(\mathcal {O}_K)$ . By [Reference Abbes1, Thm. 4.8.18.(ii)], the $\mathcal {O}_X$ -module $\operatorname {\mathrm {Lie}}(\mathcal {G})$ admits a coherent formal model $\mathcal {F}$ on ${\mathfrak {X}}$ . By Raynaud–Gruson’s flattening, see [Reference Abbes1, Thm. 5.8.1], up to replacing ${\mathfrak {X}}$ by an admissible formal blowup, we may assume that $\mathcal {F}$ is locally free. Then $E =\mathcal {F} \otimes _{\mathcal {O}_{{\mathfrak {X}}}} \mathcal {O}_X^+$ is as required.

Next, we consider the induced open subgroup

$$\begin{align*}{\mathfrak{h}} = E\otimes_{\mathcal{O}_X^+} \mathbb{G}_a^+ \subseteq {\mathfrak{g}}.\end{align*}$$

By [Reference Heuer and Xu33, Prop. 3.2.1], we can find an open cover of X by affinoid opens $X_i \subseteq X$ and open subgroups $V_i \subseteq {\mathfrak {g}}_{X_i}$ such that the logarithm restricts to an isomorphism $\log _{\mathcal {G}}^{-1}(V_i) \xrightarrow {\cong } V_i$ . By [Reference Heuer and Xu33, Lemma 3.1.2], there exists integers $n_i\geq 0$ such that $p^{n_i}{\mathfrak {h}}_{X_i} \subseteq V_i$ for each i. As X is quasi-compact, we can replace the open cover $\{X_i\}_i$ by a finite subcover, and we find that is as required, for $n \gg 0$ .

Proof. Let $G'= G \times _{\operatorname {\mathrm {Spa}}(K,K^+)}\operatorname {\mathrm {Spa}}(K,\mathcal {O}_K)$ be the associated locally p-divisible group over $(K,\mathcal {O}_K)$ . By [Reference Heuer, Werner and Zhang32, Thm. 6.12], we get an exponential, functorial in $G'$

$$\begin{align*}\operatorname{\mathrm{Exp}}_{G'}\colon \operatorname{\mathrm{Lie}}(G) \rightarrow \widehat{G}'(K,\mathcal{O}_K) = \widehat{G}(K,\mathcal{O}_K).\end{align*}$$

It remains to show that $\widehat {G}(K,\mathcal {O}_K) = \widehat {G}(K,K^+)$ . By the logarithm sequence (2.2), it is enough to show that $G[p^{\infty }](K,\mathcal {O}_K) = G[p^{\infty }](K,K^+)$ . This follows from the fact that $G[p^{\infty }]$ is ind-Zariski-constructible and hence overconvergent, by Remark 2.11.

Proof. Let C be a completed algebraic closure of K and let $C^+$ denote the completion of the integral closure of $K^+$ in C. We denote by $X_C$ (resp. $E_C$ ) the base change of X (resp. E) from K to C. By [Reference Heuer28, Cor. 3.10], we have

$$\begin{align*}H_v^n(X_C,E_C) = H_v^n(X,E)\otimes_K C\, , \quad H^i(X_C,\widetilde{\Omega}_{X_C}^j \otimes E_C) = H^i(X,\widetilde{\Omega}_{X}^j \otimes E)\otimes_K C.\end{align*}$$

As the degeneration can be deduced from a dimension count, we may assume without loss that K is algebraically closed. By the projection formula, we have

$$\begin{align*}R\nu_*E = R\nu_*\mathcal{O}_X \otimes_{\mathcal{O}_X}E.\end{align*}$$

Now by [Reference Guo20, Prop. 7.2.5], any lift $\mathbb {X}$ of X to ${\mathrm {B_{dR}^+}}/\xi ^2$ induces a splitting in the derived category $D(X_{\operatorname {\mathrm {\acute {e}t}}})$ , natural in $\mathbb {X}$

$$\begin{align*}R\nu_*\mathcal{O}_X = \bigoplus_{i\geq 0} \widetilde{\Omega}_X^i[-i].\end{align*}$$

There is thus a trivial Cartan–Eilenberg resolution of the complex $R\nu _*E$ given by

$$\begin{align*}J^{\bullet,\bullet} = \bigoplus_{i\geq 0} J_i^{\bullet}[-i],\end{align*}$$

where $J_i$ is an injective resolution of $\widetilde {\Omega }_X^i \otimes E$ in $D(X_{\operatorname {\mathrm {\acute {e}t}}})$ . It follows that the Hodge–Tate spectral sequence for E decomposes as the direct sum of the spectral sequence associated to the $1$ -column bicomplexes $J_i^{\bullet }$ , and this decomposition is natural in $\mathbb {X}$ . The result follows.

Proof. For the first point, see [Reference Scholze and Weinstein49, Prop. 2.4.5]. The second point then follows from [Reference Scholze47, Prop. 11.23].

Proof. This is proven in [Reference Heuer24, Prop. 2.14, Lemma 2.20] for $X=\operatorname {\mathrm {Spa}}(K,K^+)$ , but the proof extends without change to the general case.

Proof of Proposition 3.3

By Proposition 3.12, the sheaf $\overline {\mathcal {G}} = \mathcal {G}/\widehat {\mathcal {G}}$ has the approximation property. It follows from Proposition 3.11.3 that $R\nu _*\overline {\mathcal {G}} =\overline {\mathcal {G}}$ , such that

$$\begin{align*}R^j\nu_*\mathcal{G} = R^j\nu_*\widehat{\mathcal{G}}\, , \quad \forall j>0.\end{align*}$$

Moreover, recall that the logarithm defines an exact sequence

The étale X-group $\mathcal {G}[p^{\infty }]$ also satisfies the approximation property, by Example 3.10.2. Hence we also have $R\nu _*\mathcal {G}[p^{\infty }] =\mathcal {G}[p^{\infty }]$ and

$$\begin{align*}R^j\nu_*\widehat{\mathcal{G}}= R^j\nu_*{\mathfrak{g}}\, , \quad \forall j>0.\end{align*}$$

Finally, we have

$$\begin{align*}R^j\nu_*{\mathfrak{g}} = R^j\nu_*\mathbb{G}_a \otimes {\mathfrak{g}} = \widetilde{\Omega}_X^j \otimes {\mathfrak{g}}\, , \quad \forall j\geq 0,\end{align*}$$

where we use the projection formula for the first equality.

Proof of Proposition 3.12

• ○ $\underline{(1) \Rightarrow (2)\colon} $

  Let $\mathcal {F}$ have the approximation property. We may assume that X is affinoid. By arguing as in [Reference Fargues and Scholze17, Prop. III.1.3], we find that $\mathcal {F}$ is a small v-sheaf. Let $T \rightarrow \mathcal {F}$ be a surjection from a perfectoid space T and fix an open cover of T by affinoid perfectoids $T_i$ . We may write by [Reference Heuer22, Prop. 3.17]

  $$\begin{align*}T_i \approx \varprojlim_j U_{ij} \rightarrow X,\end{align*}$$
  where each $U_{ij}$ is affinoid smooth over X. Applying the approximation property, we find that $T_i \rightarrow \mathcal {F}$ factors through for some $j=j(i)$ . Then we obtain a surjection of v-sheaves $q\colon V = \coprod _{i} V_i \rightarrow \mathcal {F}$ from a smooth X-space. It remains to show that $R = V\times _{\mathcal {F}} V \subseteq V\times _X V$ is an open subspace, for any such map q. Let $y\in R$ with its associated geometric point $\overline {y}\colon \operatorname {\mathrm {Spa}}(C,C^+) \rightarrow V\times _X V$ . We may write
  $$\begin{align*}\operatorname{\mathrm{Spa}}(C,C^+) \approx \varprojlim_k W_k,\end{align*}$$
  where the limit ranges over all affinoid spaces $W_k$ étale over $V\times _X V$ together with a factorization
  

  Then the two following compositions

  
  are equal. By the approximation property, we deduce that already the compositions
  
  agree for $k\gg 0$ . This shows that the map $W_k \rightarrow V \times _X V$ factors through $R \subseteq V \times _X V$ and its image is an open neighborhood of $y \in V \times _X V$ contained in R. We deduce that R is open in $V \times _X V$ , as required. This shows the first implication and the last part of the statement.

• ○ $\underline{(2) \Rightarrow (1)\colon}$

  Fix an affinoid perfectoid tilde-limit $Y\approx \varprojlim _i Y_i$ of good affinoid adic spaces. We first show that the natural map

  $$\begin{align*}\phi\colon \varinjlim_i \mathcal{F}(Y_i) \rightarrow \mathcal{F}(Y).\end{align*}$$
  is injective. Let $i_0\in I$ and $\overline {s},\overline {t}\in \mathcal {F}(Y_{i_0})$ be such that $\overline {s}|_{Y}=\overline {t}|_{Y}$ . We need to show that already $\overline {s}|_{Y_i}=\overline {t}|_{Y_i}$ for $i\gg i_0$ . This may be checked after taking a standard-étale cover of $Y_{i_0}$ (replacing $Y_i$ and Y by their respective pullbacks along this cover). Hence we may assume that the given sections admit lifts $s,t\in V(Y_{i_0})$ . By assumption, the following diagram commutes
  
  so that $(s,t)|_{Y}$ has image in $R \subseteq V\times _X V$ . By Lemma 3.13 below, we deduce that already $(s,t)|_{Y_i}$ maps in R, for $i\gg i_0$ , such that $\overline {s}|_{Y_i} = \overline {t}|_{Y_i}$ . This shows injectivity.

  We now show that $\phi $ is surjective. Let $\overline {s} \in \mathcal {F}(Y)$ . Assume that $\overline {s}$ is in the image of $\varphi $ after replacing Y by an étale cover, then we can conclude. Indeed, let $Y' \rightarrow Y$ be a quasi-compact étale cover, corresponding to an étale cover $Y_{i_0}' \rightarrow Y_{i_0}$ for some $i_0\in I$ , and let $\overline {s_i'} \in \mathcal {F}(Y_i')$ be a section for $i\geq i_0$ mapping to $\overline {s}|_{Y'}$ . We claim that the section $\overline {s_i'}$ descends to a section $\overline {s_i} \in \mathcal {F}(Y_i)$ (which is then automatically mapped to $\overline {s}$ ), up to increasing i: For this, it is enough to show that $p_1^*\overline {s_i'} = p_2^*\overline {s_i'} \in \mathcal {F}(Y_i' \times _{Y_i} Y_i')$ : This holds after pullback to $Y' \times _Y Y'$ , hence it holds after increasing i, by the injectivity step.

  Next we reduce to the case where $Y=\operatorname {\mathrm {Spa}}(C,C^+)$ for a complete algebraically closed field C. Let $y\in \vert Y \vert $ and let $\{ (U_j,u_j) \}_j$ denote the cofiltered inverse system of quasi-compact étale maps $U_j \rightarrow Y$ together with a lift $u_j \in \vert U_j \vert $ of y. Then the corresponding geometric point $\operatorname {\mathrm {Spa}}(C(y),C(y)^+) \rightarrow Y$ satisfies $\operatorname {\mathrm {Spa}}(C(y),C(y)^+) \approx \varprojlim _j U_j$ . Consider the cofiltered category consisting of triplets $(i,j,U_{ij}\rightarrow Y_i)$ , where $U_{ij}\rightarrow Y_i$ is an étale map with $U_j = U_{ij}\times _{Y_i}Y$ . Then it follows from [Reference Heuer22, Lemma 3.13] (or alternatively Lemma 3.19.2 below) that

  $$\begin{align*}U_j \approx \varprojlim_{i} U_{ij}. \end{align*}$$

  By sequential approximation, we deduce that also

  $$\begin{align*}\operatorname{\mathrm{Spa}}(C(y),C(y)^+) \approx \varprojlim_{i,j} U_{ij}.\end{align*}$$

  Suppose that $\overline {s}(y)\in \mathcal {F}(C(y),C(y)^+)$ coincides with $\overline {s}_{ij}(y)$ for some $\overline {s}_{ij}\in \mathcal {F}(U_{ij})$ . Then $\overline {s}|_{U_j}$ and $\overline {s}_{ij}|_{U_j}$ map to the same element of $\mathcal {F}(C(y),C(y)^+)$ . By the injectivity step, these two sections are equal, up to increasing j. Hence, if surjectivity of $\phi $ holds for points, for each $y\in \vert Y \vert $ , the restriction of $\overline {s}$ to some étale neighborhood of y is in the image of $\phi $ , which concludes, by the previous reduction step.

  From these two reduction steps, we can without loss assume that $Y=\operatorname {\mathrm {Spa}}(C,C^+)$ , $\overline {s}$ admits a lift $s\in V(Y)$ , X is affinoid and V is standard-smooth over X. Hence, we may write $V\rightarrow X$ as a composition

  $$\begin{align*}V \xrightarrow{g} B \rightarrow X,\end{align*}$$
  where g is standard-étale, $B=\mathbb {B}_X^n$ and the last map is the canonical projection. We write
  $$\begin{align*}\mathcal{O}(B) = \mathcal{O}(X)\langle T_1, \ldots, T_n\rangle.\end{align*}$$

  Here, $T_1,\ldots T_n$ are Tate variables and we abbreviate these by $\underline {T}$ . Let

  $$\begin{align*}\underline{t} = g(s)\in B(Y) \cong (C^+)^{\oplus n}.\end{align*}$$

  By definition of tilde-limits

  $$\begin{align*}\theta\colon \varinjlim_i \mathcal{O}^+(Y_i) \rightarrow \mathcal{O}^+(Y)=C^+ \end{align*}$$
  has dense image. Hence we may find $\underline {t_i} \in \mathcal {O}^+(Y_i)^{\oplus n}$ such that $\theta (\underline {t_i}) \cong \underline {t} \, (\, \operatorname {\mathrm {mod}} \varpi ^N)$ , where $\varpi \in K^+$ is a pseudo-uniformizer and $N\gg 0$ is a large integer. We write $\underline {t'} = \theta (\underline {t_i}) \in (C^+)^{\oplus n}$ .

  It remains to construct a lift $s' \in V(Y)$ of $\underline {t'} \in B(Y)$ that is R-equivalent to the original section $s \in V(Y)$ . Note that $s'$ will automatically live in finite level, that is, arise from a section $s_i \in V(Y_i)$ that lifts $\underline {t_i} \in B(Y_i)$ , up to increasing i. This follows from the fact that $V \rightarrow B$ is étale, such that by [Reference Scholze47, Prop. 14.9],

  $$\begin{align*}\operatorname{\mathrm{Hom}}_B(Y,V) = \varinjlim_i \operatorname{\mathrm{Hom}}_B(Y_i,V).\end{align*}$$

  Let $U \subseteq R \subseteq V \times _X V$ be a rational open subset containing the image of $(s,s)\colon Y \rightarrow R$ . We form the following pullback diagram

  

  A lift $s'$ as above corresponds to a diagonal dotted arrow in the above diagram and thus a section to the étale map $Y' \rightarrow Y$ . Let $ W \subseteq (\operatorname {\mathrm {id}}\times g)(U)$ be a rational open neighborhood of the image of $(s,\underline {t})\colon Y \rightarrow V \times _X B$ in the open image $(\operatorname {\mathrm {id}}\times g)(U) \subseteq V\times _X B$ . We write $W=U(\tfrac {F_1,\ldots ,F_r}{G})$ for $F_j,G\in \mathcal {O}^+(V\times _{X}B)$ and we let $f_j,g$ (resp. $f_j', g'$ ) denote their pullbacks in $\mathcal {O}^+(Y)=C^+$ under the map $(s,\underline {t})$ (resp. the map $(s,\underline {t'})$ ). By assumption, $(s,\underline {t})(Y) \subseteq W$ such that

  $$\begin{align*}\vert f_j\vert \leq \vert g \vert \neq 0, \,\, \forall j=1, \ldots r,\end{align*}$$
  where $\vert \cdot \vert $ denotes the valuation on C. By construction of $\underline {t'}$ , we have that $f_j\equiv f_j' \,(\, \operatorname {\mathrm {mod}} \varpi ^N)$ and $g \equiv g' \,(\, \operatorname {\mathrm {mod}} \varpi ^N)$ . Hence, as soon as N is big enough,
  $$\begin{align*}\vert f_j'\vert = \vert f_j\vert \leq \vert g \vert= \vert g' \vert, \, \,\forall j=1, \ldots r.\end{align*}$$

  It follows that also $(s,\underline {t'})\colon Y \rightarrow V\times _X B$ has image in the open subspace $W \subseteq V\times _X B$ . After passing to pullbacks, we deduce that the étale map $Y' \rightarrow Y$ is surjective. Since $Y=\operatorname {\mathrm {Spa}}(C,C^+)$ is strictly totally disconnected, the map $Y' \rightarrow Y$ thus admits a section, as required. This concludes the proof.

Proof. Let us take an open cover $T=\bigcup _j T_j$ by open spatial subdiamonds and $Z_{i_0} = \bigcup _k Z_{i_0,k}$ an open cover by spatial subdiamonds subdividing the preimage of the $T_j$ . By quasi-compactness of $Z_{i_0}$ , we may assume that there are finitely many k. Then letting $Z_{i,k},Z_k$ denote the inverse image of $Z_{i_0,k}$ in $Z_i, Z$ , we obtain again inverse limits $Z_k = \varprojlim _{i \geq i_0} Z_{i,k}$ . Replacing Z by $Z_k$ , we may assume that T is spatial. Let us take an open cover $U= \bigcup _{j} U_{j}$ by spatial subdiamonds. Again by quasi-compactness, finitely many $U_j$ cover the image of $Z \rightarrow U$ . Hence we may assume that U is quasi-compact and hence spatial. For $i\geq i_0$ , the preimage $V_i \subseteq Z_i$ of U under the composition $Z_i \rightarrow Z_{i_0} \xrightarrow {t} T$ are quasi-compact and $\varprojlim _{i\geq i_0} V_i = Z$ . By [48, Tag 0A2W], it follows that $V_i=Z_i$ for $i\gg i_0$ , as required.

Proof. The second point is [Reference Heuer22, Lemma 3.13]. In our case, it also follows from the first point if we redefine and . Let us now prove the first point. We claim that under our assumptions, the fiber products of diamonds $X\times _S Y_i$ are representable by good affinoid adic spaces, namely the adic spectrum of

$$\begin{align*}(\mathcal{O}(X),\mathcal{O}^+(X))\, \underset{(\mathcal{O}(S),\mathcal{O}^+(S))}{\widehat{\otimes}}(\mathcal{O}(Y_i),\mathcal{O}^+(Y_i)),\end{align*}$$

and similarly for Y. By assumption, the map $\mathcal {O}(S) \rightarrow \mathcal {O}(X)$ is a composition of rational open immersions, finite étale maps and adjunction of Tate variables. Then so is $\mathcal {O}(Y_i) \rightarrow \mathcal {O}(X)\, \widehat {\otimes }_{\mathcal {O}(S)} \,\mathcal {O}(Y_i)$ . It follows from Proposition 2.5 that the completed tensor product of Huber pairs above yields a good adic space, which then represents the diamond $X\times _S Y_i$ .

Next, we want to show that the image of the map

$$\begin{align*}\varinjlim_i \mathcal{O}(X)\, \widehat{\otimes}_{\mathcal{O}(S)} \,\mathcal{O}(Y_i) \rightarrow \mathcal{O}(X) \,\widehat{\otimes}_{\mathcal{O}(S)}\, \mathcal{O}(Y)\end{align*}$$

is dense in its target. It is enough to show that the image of

$$\begin{align*}\varinjlim_i \mathcal{O}(X) \otimes_{\mathcal{O}(S)} \mathcal{O}(Y_i) = \mathcal{O}(X)\otimes_{\mathcal{O}(S)}\varinjlim_i \mathcal{O}(Y_i)\end{align*}$$

is dense. This image is immediately seen to be dense in $\mathcal {O}(X)\otimes _{\mathcal {O}(S)} \mathcal {O}(Y)$ , which concludes.

It remains to show the condition on topological spaces. The diamond functor commutes with fiber products and, on the level of sheaves, fiber products commute with inverse limits. It follows that

$$\begin{align*}(X\times_S Y)^{\diamondsuit} = \varprojlim_i (X\times_S Y_i)^{\diamondsuit}.\end{align*}$$

The topological spaces of an analytic adic space and its diamondification are identified [Reference Scholze47, Lemma 15.6]. On the level of diamonds, taking underlying topological spaces commutes with the inverse limit [Reference Scholze47, Lemma 11.22], which concludes.

Proof of Proposition 3.18

We start with the case $n=0$ .

Claim 3.20. In the situation of Proposition 3.18, assume furthermore that $U_{i_0}$ is standard-étale over $X'\times _{S'} Y_{i_0}$ where $X' \subseteq X$ and $S'\subseteq S$ are affinoid open subspaces such that all $Y,Y_i$ are defined over $S'$ and the restriction $\pi \colon X' \rightarrow S'$ is standard-smooth. Then we have an isomorphism

$$\begin{align*}\underset{i\geq i_0}{\varinjlim}\, \mathcal{F}(U_i) \xrightarrow{\cong} \mathcal{F}(U).\end{align*}$$

Proof. By Lemma 3.19.1-2, we find that

$$\begin{align*}U \approx \varprojlim_{i\geq i_0} U_i.\end{align*}$$

The claim now follows from Proposition 3.11.2.

Since standard-étale maps form a basis of the étale site of $X\times _S Y$ , it follows that we have an isomorphism of sheaves

$$\begin{align*}\mathcal{F} = \varinjlim_{i\geq i_0} q_i^{*}q_{i,*}\mathcal{F}\end{align*}$$

on the site , where $q_i\colon X\times _S Y \rightarrow X\times _S Y_i$ denotes the projection and we denote the restriction of $\mathcal {F}$ to the site $(X \times _S Y)_{\operatorname {\mathrm {\acute {e}t}},\operatorname {\mathrm {qcqs}}}$ by the same symbol. The statement now follows from [48, Tag 09YP] in the abelian case, and the nonabelian case is deduced similarly.

Proof. Let $Y\approx \varprojlim _i Y_i$ be an affinoid perfectoid tilde-limit of good affinoid adic spaces over S. Let $V\rightarrow Y$ be a qcqs étale map, corresponding to a qcqs étale maps $V_i \rightarrow Y_i$ for $i\gg 0$ . By Proposition 3.18, we have

$$\begin{align*}H_{\operatorname{\mathrm{\acute{e}t}}}^n(X\times_S V,\mathcal{F}) = \varinjlim_{i} H_{\operatorname{\mathrm{\acute{e}t}}}^n(X\times_S V_i,\mathcal{F}).\end{align*}$$

Sheafifying on Y yields

$$\begin{align*}R^n\pi_{\operatorname{\mathrm{\acute{E}t}},*}\mathcal{F}(Y) = \varinjlim_{i} R^n\pi_{\operatorname{\mathrm{\acute{E}t}},*}\mathcal{F}(Y_i).\end{align*}$$

This proves that $R^n\pi _{\operatorname {\mathrm {\acute {E}t}},*}\mathcal {F}$ has the approximation property and is, in particular, a v-sheaf. Considering now $R^n\pi _{v,*}\mathcal {F}$ , which is the v-sheafification of the presheaf taking $Y \rightarrow S$ to $H_{v}^n(X\times _S Y,\mathcal {F})$ . By Proposition 3.11.3, we have

$$\begin{align*}H_{v}^n(X\times_S Y,\mathcal{F}) = H_{\operatorname{\mathrm{\acute{e}t}}}^n(X\times_S Y,\mathcal{F}).\end{align*}$$

We deduce that

$$\begin{align*}R^n\pi_{v,*}\mathcal{F} = R^n\pi_{\operatorname{\mathrm{\acute{E}t}},*}\mathcal{F}.\end{align*}$$

Hence the first two points are proven. For the last point, we argue as in the proof of [Reference Heuer22, Cor. 3.16]. Namely, observe that the two sheaves under consideration are the étale sheafification of the following presheaves. The left-hand side comes from the presheaf

$$\begin{align*}Y \in \operatorname{\mathrm{Perf}}_S \mapsto \underset{Y \rightarrow Y_i}{\varinjlim}H_{\operatorname{\mathrm{\acute{e}t}}}^n(X\times_S Y_i,\mathcal{F}),\end{align*}$$

where the colimit runs over all factorization of $Y\rightarrow S$ into an adic space $Y_i$ smooth over S. Here we use that so that it is enough to sheafify over Y. The right-hand side comes from the presheaf

$$\begin{align*}Y \in \operatorname{\mathrm{Perf}}_S \mapsto H_{\operatorname{\mathrm{\acute{e}t}}}^n(X\times_S Y,\mathcal{F}).\end{align*}$$

By [Reference Heuer22, Prop. 3.17], we may write

$$\begin{align*}Y\approx \underset{Y \rightarrow Y_i}{\varprojlim}\, Y_i,\end{align*}$$

where the limit ranges over all morphisms $Y \rightarrow Y_i$ into affinoid adic spaces smooth over S. It now follows from Proposition 3.18 that these two presheaves are isomorphic.

Proof of Proposition 3.16

1. 1. Let us show that

   $$\begin{align*}R^n\pi_{\operatorname{\mathrm{\acute{E}t}},*}^{\diamondsuit}{\mathfrak{g}} = (R^n\pi_{\operatorname{\mathrm{\acute{e}t}},*}{\mathfrak{g}}) \otimes_{\mathcal{O}_{S_{\operatorname{\mathrm{\acute{e}t}}}}} \mathbb{G}_a.\end{align*}$$

   When $S=\operatorname {\mathrm {Spa}}(K,K^+)$ , it follows from [Reference Heuer28, Cor. 3.10], which says that for any affinoid perfectoid Y over K

   $$\begin{align*}H_{\operatorname{\mathrm{\acute{e}t}}}^n(X\times_K Y,{\mathfrak{g}}) = H_{\operatorname{\mathrm{\acute{e}t}}}^n(X,{\mathfrak{g}}) \otimes_K \mathcal{O}(Y)\end{align*}$$

   If K is perfectoid, loc. cit. also shows that

   $$\begin{align*}H_{v}^n(X\times_K Y,{\mathfrak{g}}) = H_{v}^n(X,{\mathfrak{g}}) \otimes_K \mathcal{O}(Y),\end{align*}$$
   which yields
   $$\begin{align*}R^n\pi_{v,*}^{\diamondsuit}{\mathfrak{g}} = H_v^n(X,{\mathfrak{g}}) \otimes_K \mathbb{G}_a.\end{align*}$$

   Assume now that S is arbitrary and that ${\mathfrak {g}}$ is the pullback of a vector bundle E on S. By the projection formula, we may assume that ${\mathfrak {g}}=\mathbb {G}_a$ . It now follows from the fact that $R^n\pi _{\operatorname {\mathrm {an}},*}\mathcal {O}_X$ is an analytic vector bundle whose formation commutes with base change along any perfectoid space or rigid space $Y \rightarrow S$ [Reference Heuer28, Thm. 5.7.3]. Note that this proves the $4$ th point for $\mathcal {G}={\mathfrak {g}}$ .

2. 2. By Corollary 3.21.2-3, we have, for any $1\leq m \leq \infty $

   $$\begin{align*}R^n\pi_{v,*}^{\diamondsuit}\mathcal{G}[p^{m}] = R^n\pi_{\operatorname{\mathrm{\acute{E}t}},*}^{\diamondsuit}\mathcal{G}[p^{m}] = (R^n\pi_{\operatorname{\mathrm{\acute{E}t}},*}^{\operatorname{\mathrm{rig}}}\mathcal{G}[p^{m}])^{\diamondsuit}.\end{align*}$$

   By proper base change (Theorem 2.12.2, the sheaf $R^n\pi _{\operatorname {\mathrm {\acute {E}t}},*}^{\operatorname {\mathrm {rig}}}\mathcal {G}[p^{m}]$ arises from the ind-Zariski-constructible sheaf $R^n\pi _{\operatorname {\mathrm {\acute {e}t}},*}\mathcal {G}[p^{m}]$ via pullback along $\operatorname {\mathrm {Sm}}_{/S,\operatorname {\mathrm {\acute {e}t}}} \rightarrow S_{\operatorname {\mathrm {\acute {e}t}}}$ , which shows the first part of the statement. If $S=\operatorname {\mathrm {Spa}}(K,K^+)$ , we conclude that the ind-Zariski-constructible sheaf $R^n\pi _{\operatorname {\mathrm {\acute {e}t}},*}\mathcal {G}[p^{m}]$ is representable by an étale-locally constant rigid group. Assume now that $[p]\colon \widehat {\mathcal {G}} \rightarrow \widehat {\mathcal {G}}$ is finite étale surjective. For $m< \infty $ , the group $\mathcal {G}[p^m] \rightarrow S$ is finite étale, such that $R^n\pi _{\operatorname {\mathrm {\acute {e}t}},*}\mathcal {G}[p^{m}]$ is a local system, by [Reference Scholze and Weinstein50, Thm. 10.5.1], and is thus easily seen to be representable by a finite étale S-group. By a standard limit argument (see the proof of [Reference Heuer28, Thm. 4.4]), we deduce that $\mathbb {M}^n = R^n\pi _{v,*}^{\diamondsuit }T_p\mathcal {G}$ is a lisse $\mathbb {Z}_p$ -sheaf on $S_v$ , where $T_p\mathcal {G} = \varprojlim _m \mathcal {G}[p^m]$ is the Tate module. It remains to show that $R^n\pi _{\operatorname {\mathrm {\acute {E}t}},*}^{\diamondsuit }\mathcal {G}[p^{\infty }]$ is representable by an étale S-group. By writing $\mathcal {G}[p^{\infty }] = T_p\mathcal {G}[\tfrac {1}{p}]/T_p\mathcal {G}$ , we obtain a short exact sequence of v-sheaves over S

   
   where $\mathbb {M}_{\operatorname {\mathrm {tor}}}^{n+1}$ is the torsion subsheaf of $\mathbb {M}^{n+1}$ , a finite locally free abelian sheaf. Hence it is enough to show that $\mathbb {M}^n[\tfrac {1}{p}]/\mathbb {M}^n$ is representable by a separated étale S-group. By [Reference Scholze47, Prop. 10.11.(iv)], this may be checked v-locally. Hence we may assume that S is strictly totally disconnected perfectoid space and $\mathbb {M}^n \cong \underline {M}$ for a finitely generated $\mathbb {Z}_p$ -module M, in which case it is clear.

3. 3. Applying $R\pi _{\tau ,*}^{\diamondsuit }$ to the logarithm exact sequence (2.2) yields a long exact sequence of sheaves on $\operatorname {\mathrm {Perf}}_{S,\tau }$

   

   We claim that $\delta $ vanishes, for all $n\geq 0$ . For this, we use the following argument that we learned in [Reference Heuer and Xu33, Prop. 6.1.15]. Assume first that $\tau $ is the étale topology. By the first two points, the source of $\delta $ is a vector bundle while its target comes from pullback from an ind-Zariski-constructible sheaf on $S_{\operatorname {\mathrm {\acute {e}t}}}$ . Hence, by [Reference Huber31, Prop. 2.6.1], the vanishing of $\delta $ may be checked on geometric fibers. We may thus assume that $S=\operatorname {\mathrm {Spa}}(C,C^+)$ for an algebraically closed field C, in which case

   $$\begin{align*}R^{n}\pi_{\operatorname{\mathrm{\acute{E}t}},*}^{\diamondsuit}{\mathfrak{g}} \cong \mathbb{G}_a^{\oplus r}, \quad R^{n+1}\pi_{\operatorname{\mathrm{\acute{E}t}},*}^{\diamondsuit}\mathcal{G}[p^{\infty}] = \underline{\Lambda},\end{align*}$$
   for a discrete group $\Lambda $ . Now there is no nonzero maps of groups from the connected rigid group $\mathbb {G}_a^{\oplus r}$ to a discrete group. This shows that $\delta =0$ as required. When $\tau =v$ , the vanishing of $\delta $ can be shown after pulling back along a pro-étale cover by a perfectoid space $\widetilde {S} \rightarrow S$ . Then the above argument applies verbatim over $\widetilde {S}$ , where $R^{n}\pi _{v,*}^{\diamondsuit }{\mathfrak {g}}$ is representable by an étale vector bundle, since étale and v-vector bundles on a perfectoid space agree [Reference Kedlaya and Liu36, Thm. 3.5.8].

4. 4. To show that the natural map

   $$\begin{align*}(R^{n}\pi_{\operatorname{\mathrm{\acute{E}t}},*}^{\operatorname{\mathrm{rig}}}\mathcal{G})^{\diamondsuit} \rightarrow R^{n}\pi_{\operatorname{\mathrm{\acute{E}t}},*}^{\diamondsuit}\mathcal{G}\end{align*}$$
   is an isomorphism, it suffices, by the $5$ -Lemma and Corollary 3.21.3 applied to the sheaf $\overline {\mathcal {G}} = \mathcal {G}/\widehat {\mathcal {G}}$ , to show the statement with $\mathcal {G}$ replaced by $\widehat {\mathcal {G}}$ . Applying again Corollary 3.21.3 to the sheaf $\mathcal {G}[p^{\infty }]$ , it is enough to show the statement for ${\mathfrak {g}}$ , which we have already seen in the proof of the first point. This completes the proof.

Proof. We have a commutative diagram

By the usual comparison of cohomology between small and big étale sites, it is equivalent to work with $X_{\operatorname {\mathrm {\acute {e}t}}}$ or $\operatorname {\mathrm {Smd}}_{X,\operatorname {\mathrm {\acute {e}t}}}$ to compute $E_{2}^{ij}$ . By construction of the Leray sequence, there is a collection of isomorphisms

$$\begin{align*}E_0^{ij} =\Gamma_{\operatorname{\mathrm{\acute{e}t}}}(X,I^{i,j}) = \pi_{\operatorname{\mathrm{\acute{E}t}},*}(I^{i,j})(K) = \mathbf{E}_0^{ij}(K),\end{align*}$$

compatible with differentials, where $ I^{\bullet ,\bullet }$ is Cartan–Eilenberg resolution of $R\mu _*\mathcal {G}$ . These isomorphisms are immediately seen to induce maps

$$ \begin{align*} E_r^{ij} \rightarrow \mathbf{E}_r^{ij}(K)\, , \quad \forall \, 0\leq r \leq \infty, \end{align*} $$

that coincide with étale sheafification for $r=2$ . Moreover, if K is assumed to be algebraically closed, taking $(K,K^+)$ -rational points is an exact functor, such that the comparison maps $E_r^{ij} \rightarrow \mathbf {E}_r^{ij}(K)$ are isomorphisms, for each $0\leq r \leq \infty $ . This concludes the proof.

Proof. For $Z \in \operatorname {\mathrm {Smd}}_{X}$ , the isomorphism $\operatorname {\mathrm {HTlog}}_Z$ on $Z_{\operatorname {\mathrm {\acute {e}t}}}$ of Proposition 3.3 is natural in Z. This allows us to glue these isomorphisms to an isomorphism of sheaves on $\operatorname {\mathrm {Smd}}_{X,\operatorname {\mathrm {\acute {e}t}}}$ , as required.

Proof. Let us take an affinoid perfectoid $Y\xrightarrow {g} S$ and form the following cartesian diagram of adic spaces

The sheaf

$$\begin{align*}R^i\pi_{\operatorname{\mathrm{\acute{E}t}},*}(R^j\mu_*\mathcal{G}) = R^i\pi_{\operatorname{\mathrm{\acute{E}t}},*}(\widetilde{\Omega}^j \otimes \operatorname{\mathrm{Lie}}(\mathcal{G})) \end{align*}$$

is the étale sheafification of the presheaf taking $Y \in \operatorname {\mathrm {Perf}}_S$ to

$$\begin{align*}H_{\operatorname{\mathrm{\acute{e}t}}}^i(X\times_S Y, \widetilde{\Omega}_{X \times_S Y}^j \otimes\operatorname{\mathrm{Lie}}(\mathcal{G})) = H_{\operatorname{\mathrm{\acute{e}t}}}^i(X\times_S Y, g^{\prime*}\widetilde{\Omega}_{X/S}^j \otimes \operatorname{\mathrm{Lie}}(\mathcal{G})),\end{align*}$$

where we use Example 2.9. This shows that

$$\begin{align*}R^i\pi_{\operatorname{\mathrm{\acute{E}t}},*}(R^j\mu_*\mathcal{G}) = R^i\pi_{\operatorname{\mathrm{\acute{E}t}},*}(\widetilde{\Omega}_{X/S}^j \otimes \operatorname{\mathrm{Lie}}(\mathcal{G})) \end{align*}$$

and it remains to show that this sheaf is a vector bundle. In the case where $S=\operatorname {\mathrm {Spa}}(K,K^+)$ , we have by [Reference Heuer28, Cor. 3.10.1],

$$\begin{align*}H_{\operatorname{\mathrm{\acute{e}t}}}^i(X\times_K Y, g^{\prime*}\widetilde{\Omega}_{X}^j \otimes \operatorname{\mathrm{Lie}}(\mathcal{G})) = H_{\operatorname{\mathrm{\acute{e}t}}}^i(X,\widetilde{\Omega}_{X}^j \otimes \operatorname{\mathrm{Lie}}(\mathcal{G})) \otimes_K \mathcal{O}(Y). \end{align*}$$

This shows that

$$\begin{align*}R^i\pi_{\operatorname{\mathrm{\acute{E}t}},*}(\widetilde{\Omega}_{X/S}^j \otimes \operatorname{\mathrm{Lie}}(\mathcal{G})) = H_{\operatorname{\mathrm{\acute{e}t}}}^i(X,\widetilde{\Omega}_{X}^j \otimes \operatorname{\mathrm{Lie}}(\mathcal{G})) \otimes_K \mathbb{G}_a.\end{align*}$$

Assume now that we are in the second case, so that $\operatorname {\mathrm {Lie}}(\mathcal {G}) = \pi ^*E$ for some vector bundle E on S. By the projection formula and the above observations, it is enough to show that $R^i\pi _{\operatorname {\mathrm {\acute {E}t}},*}\widetilde {\Omega }_{X/S}^j$ is a vector bundle. It follows from the fact that $R^i\pi _{\operatorname {\mathrm {an}},*}\widetilde {\Omega }_{X/S}^j$ is a vector bundle whose formation commutes with pullbacks along perfectoid spaces $Y \rightarrow S$ [Reference Heuer28, Thm. 5.7.3].

Proof. This follows from Theorem 3.29 in the case $S=\operatorname {\mathrm {Spa}}(K,K^+)$ and the fact that the comparison maps (3.9) are isomorphisms, as K is assumed to be algebraically closed.

Proof. Let us first show that the abutment satisfies

$$\begin{align*}R^n(\mu_*\pi_{v,*}){\mathfrak{g}}=\mu_*{\mathbf{Bun}}_{{\mathfrak{g}},v}^{n}.\end{align*}$$

By a Leray sequence argument, it is enough to show that

$$\begin{align*}R^m\mu_*({\mathbf{Bun}}_{{\mathfrak{g}},v}^{n})=0, \quad \forall m>0.\end{align*}$$

Let $Y \in \operatorname {\mathrm {Perf}}_S$ be an affinoid perfectoid space and let $\nu \colon Y_v \rightarrow Y_{\operatorname {\mathrm {\acute {e}t}}}$ be the natural map. It is enough to show that

$$\begin{align*}R^m\nu_*({\mathbf{Bun}}_{{\mathfrak{g}},v}^{n})=0.\end{align*}$$

We know by Proposition 3.16.1 that $E = {\mathbf {Bun}}_{{\mathfrak {g}},v}^{n}$ is represented by a v-vector bundle, but étale and v-vector bundles on Y agree by [Reference Kedlaya and Liu36, Thm. 5.3.8]. Therefore Proposition 3.3 applies and we find that

$$\begin{align*}R^m\nu_*E = \widetilde{\Omega}_Y^m \otimes E =0, \end{align*}$$

as required.

We now prove the degeneration of the sequence. Let us first assume that $S=\operatorname {\mathrm {Spa}}(K,K^+)$ . In that case, by [Reference Heuer28, Cor. 3.10.1], we further have, for any perfectoid space Y

$$\begin{align*}H_{v}^i(X\times_K Y, {\mathfrak{g}}) = H_{v}^i(X,{\mathfrak{g}}) \otimes_K \mathcal{O}(Y). \end{align*}$$

This shows that

$$\begin{align*}R^n\pi_{v,*}{\mathfrak{g}} = H_{v}^n(X,{\mathfrak{g}}) \otimes_K \mathbb{G}_a.\end{align*}$$

The geometric Hodge–Tate spectral sequence thus has the following shape

$$ \begin{align*} \mathbf{E}_2^{ij}({\mathfrak{g}}) = H_{\operatorname{\mathrm{\acute{e}t}}}^i(X,\widetilde{\Omega}_{X/S}^j\otimes_{\mathcal{O}_X} {\mathfrak{g}})\otimes_K \mathbb{G}_a \Rightarrow H_v^{i+j}(X,{\mathfrak{g}}) \otimes_K \mathbb{G}_a, \end{align*} $$

and its degeneration can clearly be checked on K-points, which is Proposition 3.4. Let us now assume that S is arbitrary and that ${\mathfrak {g}}$ comes from a vector bundle on S. In that case the statement follows from the projection formula and the degeneration of the relative Hodge–Tate spectral sequence for $\mathcal {O}_X$ [Reference Heuer28, Cor. 5.12].

Proof of Theorem 3.29

We have morphisms of X-groups and hence of v-sheaves

$$\begin{align*}\mathcal{G} \supseteq \widehat{\mathcal{G}} \xrightarrow{\log} {\mathfrak{g}}.\end{align*}$$

These induce morphisms between the $E_2$ -pages of their respective geometric Hodge–Tate spectral sequences. Moreover, these maps induce isomorphisms

$$\begin{align*}R^j\mu_*\mathcal{G} \xleftarrow{\cong} R^j\mu_*\widehat{\mathcal{G}} \xrightarrow{\cong} R^j\mu_*{\mathfrak{g}}, \end{align*}$$

for all $j\geq 1$ , by Lemma 3.27. These induce isomorphisms between the respective terms $\mathbf {E}_2^{ij}$ , for $j\geq 1$ and any i. As a consequence, since the geometric Hodge–Tate spectral sequence for ${\mathfrak {g}}$ degenerates, by Proposition 3.31, we deduce that $\mathbf {d}_2^{ij}\colon \mathbf {E}_2^{ij}(\mathcal {G}) \rightarrow \mathbf {E}_2^{i+2,j-1}(\mathcal {G})$ is the zero map, for all $j\geq 2$ and all i. It remains to show that the differential

$$\begin{align*}\mathbf{d}_2^{i,1}\colon R^i\pi_{\operatorname{\mathrm{\acute{E}t}},*}(\widetilde{\Omega}_{X/S}^1\otimes\operatorname{\mathrm{Lie}}(\mathcal{G})) \rightarrow {\mathbf{Bun}}_{\mathcal{G},\operatorname{\mathrm{\acute{e}t}}}^{i+2}\end{align*}$$

vanishes, for all $i\geq 0$ . We have a commutative diagram

so that it is enough to show that $\mathbf {d}=\mathbf {d}_2^{i,1}(\widehat {\mathcal {G}}) =0$ . From the above diagram, we have a factorization

As in the proof of Proposition 3.16.3, we deduce that, as the target of $\mathbf {d}'$ is an ind-Zariski-constructible sheaf on $S_{\operatorname {\mathrm {\acute {e}t}}}$ (Proposition 3.16.3 and its source is fiberwise connected (Proposition 3.28), it must be the zero map. Hence $\mathbf {d}=\mathbf {d}_2^{i,2}(\widehat {\mathcal {G}})$ vanishes, as required.

Proof. The last equality in the statement follows from the first part, using the Leray sequence computing $R^{n}(\mu _*\pi _{v,*})\mathcal {G}$ . Let’s prove the first part, starting with the case $\mathcal {G}=\widehat {\mathcal {G}}$ . Let Y be any affinoid perfectoid space over S and consider the canonical map $\nu \colon Y_v \rightarrow Y_{\operatorname {\mathrm {\acute {e}t}}}$ . We need to show that

$$\begin{align*}R^m\nu_*({\mathbf{Bun}}_{\widehat{\mathcal{G}},v}^{n}) = 0, \quad \forall m>0.\end{align*}$$

From the short exact sequence (3.6), it is enough to show that

$$\begin{align*}R^m\nu_*({\mathbf{Bun}}_{\mathcal{G}[p^{\infty}],v}^{n}) = R^m\nu_*({\mathbf{Bun}}_{{\mathfrak{g}},v}^{n}) = 0.\end{align*}$$

By Corollary 3.21.1-2, the sheaf ${\mathbf {Bun}}_{\widehat {\mathcal {G}}[p^{\infty }],v}^{n}$ has the approximation property, hence it follows from Proposition 3.11.3 that the left term vanishes. The vanishing of the right term was already shown in the proof of Proposition 3.31.

We now treat the case of general $\mathcal {G}$ . For this, we need the following intermediary result.

Let us assume the lemma. We have an exact sequence of v-sheaves on $\operatorname {\mathrm {Perf}}_{S,v}$

Those sheaves and the above exact sequence admit a natural extension to $\operatorname {\mathrm {Adic}}_{S,v}$ . By Corollary 3.21, the sheaves ${\mathbf {Bun}}_{\overline {\mathcal {G}},v}^n$ satisfy the approximation property. We claim that the sheaves $\operatorname {\mathrm {Ker}}(\delta _n)$ and $\operatorname {\mathrm {Im}}(\delta _n)$ also satisfy the approximation property. Indeed, consider the composition $V\rightarrow {\mathbf {Bun}}_{\mathcal {G},v}^n \rightarrow {\mathbf {Bun}}_{\overline {\mathcal {G}},v}^n$ , where $V \rightarrow {\mathbf {Bun}}_{\mathcal {G},v}^n$ is as in Lemma 3.33. Then by Proposition 3.12, the diamond

$$\begin{align*}R = V \times_{{\mathbf{Bun}}_{\overline{\mathcal{G}},v}^n} V \end{align*}$$

is open in $V \times _S V$ , such that the étale quotient $V/^{\operatorname {\mathrm {\acute {e}t}}}R$ has the approximation property. In particular, it coincides with its v-sheafification $V/^{v}R = \operatorname {\mathrm {Ker}}(\delta _n)$ . We now consider the short exact sequence of étale sheaves

where the first and second sheaves satisfy the approximation property. It is then easily seen that the third sheaf $\operatorname {\mathrm {Im}}_{\operatorname {\mathrm {\acute {e}t}}}(\delta _n)$ automatically satisfies the approximation property as well. In particular, $\operatorname {\mathrm {Im}}_{\operatorname {\mathrm {\acute {e}t}}}(\delta _n)$ is a v-sheaf and thus the above sequence is also right-exact in the v-topology.

We now consider the exact sequence of v-sheaves

Each sheaf $\mathcal {F}$ on the left or right of ${\mathbf {Bun}}_{\mathcal {G},v}^{n}$ in the above sequence satisfies $R^m\mu _*\mathcal {F} = 0$ , by the case $\mathcal {G}=\widehat {\mathcal {G}}$ and Proposition 3.11.3 respectively. By decomposing the above sequence into short exact sequences, we finally deduce that

$$\begin{align*}R^m\mu_*({\mathbf{Bun}}_{\mathcal{G},v}^{n}) = 0,\end{align*}$$

which concludes the proof.

Proof of Lemma 3.33

By Theorem 3.29, we have a descending filtration

$$\begin{align*}0={\mathbf{Fil}}^{n+1} \subseteq {\mathbf{Fil}}^n \subseteq \ldots \subseteq {\mathbf{Fil}}^0= R^n(\mu_*\pi_{v,*})\mathcal{G}\end{align*}$$

by étale sheaves on $\operatorname {\mathrm {Perf}}_{S}$ with ${\mathbf {Fil}}^n = {\mathbf {Bun}}_{\mathcal {G},\operatorname {\mathrm {\acute {e}t}}}^n$ and short exact sequences, for $0\leq i < n$

where $\mathbf {V}_{i,n} = R^i\pi _{\operatorname {\mathrm {\acute {E}t}},*}(\widetilde {\Omega }_{X/S}^{n-i}\otimes \operatorname {\mathrm {Lie}}(\mathcal {G}))$ . By passing to the v-sheafification $\mu ^*$ , we obtain a filtration by v-sheaves on $\operatorname {\mathrm {Perf}}_S$

$$\begin{align*}\mu^*{\mathbf{Fil}}^{\bullet} \subseteq \mu^*R^n(\mu_*\pi_{v,*})\mathcal{G} = {\mathbf{Bun}}_{\mathcal{G},v}^n,\end{align*}$$

with short exact sequences

Here, we use the fact that $\mathbf {V}_{i,n}$ is a vector bundle and thus is already a v-sheaf. Observe that the sequence is already exact for the étale topology, using that the surjection of étale sheaves ${\mathbf {Fil}}^i \rightarrow \mathbf {V}_{i,n}$ factors through $\mu ^*{\mathbf {Fil}}^i$ . By Lemma 3.34 below, the map $\mu ^*{\mathbf {Fil}}^i \rightarrow \mathbf {V}_{i,n}$ admits sections étale locally on $\mathbf {V}_{i,n}$ . Hence, by arguing inductively, it is enough to show that each $\mu ^*{\mathbf {Fil}}^i$ admits a surjection of v-sheaves from a smooth S-space, and we reduce to $i=n$ . For this, it is enough to show that ${\mathbf {Fil}}^n = {\mathbf {Bun}}_{\mathcal {G},\operatorname {\mathrm {\acute {e}t}}}^n$ admits a surjection of étale sheaves from a smooth S-space. By Proposition 3.16.4, it is enough to show that the sheaf ${\mathbf {Bun}}_{\mathcal {G},\operatorname {\mathrm {\acute {e}t}}}^{n,\operatorname {\mathrm {rig}}}$ on $\operatorname {\mathrm {Sm}}_{/S,\operatorname {\mathrm {\acute {e}t}}}$ admits a surjection from a smooth S-space. This follows from the co-Yoneda Lemma, since the site $\operatorname {\mathrm {Sm}}_{/S,\operatorname {\mathrm {\acute {e}t}}}$ is essentially small.

Proof. We may assume that Y is affinoid. By [Reference Scholze47, Lemma 15.3], there exists a v-cover

$$\begin{align*}\widetilde{Y}\approx \varprojlim_j Y_j \rightarrow Y\end{align*}$$

where $\widetilde {Y}$ is an affinoid perfectoid space, $Y_j$ are affinoid, finite étale over Y and each $\widetilde {Y} \rightarrow Y_j$ is surjective. Since $\mathcal {F} \rightarrow \mathcal {G}$ is surjective for the étale topology on $\operatorname {\mathrm {Perf}}_S$ , we may find a qcqs étale cover $\widetilde {Z} \rightarrow \widetilde {Y}$ and a map $\widetilde {s}$ fitting in the following commutative diagram of sheaves on $\operatorname {\mathrm {Perf}}_S$

By Lemma 3.7.2, $\widetilde {Z}$ arises via pullback from an étale cover $Z_j\rightarrow Y_j$ . Using that $\widetilde {Y} \rightarrow Y_j$ is surjective, it is easy to see that the map $\widetilde {f}$ descends to a map $f_j\colon Z_j \rightarrow \mathcal {F}$ satisfying the required properties.

Proof. Let $v\colon Y \rightarrow \mathbf {V}$ be any v-cover from a perfectoid space. By assumption, there exists a v-cover $Y' \rightarrow Y$ and a section $g\in \mathcal {G}(Y')$ such that $f(g)=v|_{Y'}$ . Then $\mathcal {G} \times _{\mathbf {V}} Y' \cong \Gamma \times _S Y'$ is represented by an étale separated perfectoid space over $Y'$ . By [Reference Scholze47, Prop. 10.11(iv)], $\mathcal {G} \rightarrow \mathbf {V}$ is étale. We deduce, using [Reference Scholze47, Lemma 15.6], that $\mathcal {G}$ is represented by an object of the small étale site $\mathbf {V}_{\operatorname {\mathrm {\acute {e}t}}}$ of $\mathbf {V}$ considered as an adic space. This shows that $\mathcal {G}$ is represented by a smooth S-group.

Next, observe that the composition

$$\begin{align*}h\colon \mathcal{G} \xrightarrow{f} \mathbf{V} \xrightarrow{D(f)^{-1}} {\mathfrak{g}}\end{align*}$$

is an étale morphism with $D(h) = \operatorname {\mathrm {id}}$ . By the unicity part of Lemma 2.17.1, we deduce that $h|_{\widehat {\mathcal {G}}}=\log _{\mathcal {G}}$ .

Let us now show that $\mathcal {G}=\widehat {\mathcal {G}}$ . This may be checked locally, such that we may assume that S is affinoid and $\mathbf {V}$ is a trivial vector bundle. Let Y be an affinoid perfectoid space over S and let $g \in \mathcal {G}(Y)$ . It suffices to show that $[p^n]g \in \widehat {\mathcal {G}}(Y)$ for some $n\geq 0$ , since the cokernel $\overline {\mathcal {G}}$ of the inclusion $\widehat {\mathcal {G}} \hookrightarrow \mathcal {G}$ is p-torsionfree, by [Reference Heuer25, Lemma 2.11.2]. Let $U \cong \mathbb {B}^n \subseteq \mathbf {V}$ denote an open subspace isomorphic to a closed ball where $\log _{\mathcal {G}}$ , and hence also f has a section, as in Lemma 2.26, see Remark 2.27. We have $f([p^n]g)=p^nf(g) \in U$ for large enough n. Note that we have $f^{-1}(U) \cong \Gamma \times U$ as rigid groups, such that

$$\begin{align*}\widehat{f^{-1}(U)} = \widehat{U} \times \widehat{\Gamma} = U \times \Gamma = f^{-1}(U). \end{align*}$$

Since $[p^n]g \in \widehat {f^{-1}(U)} \subset \widehat {\mathcal {G}}$ , this proves that $\mathcal {G}=\widehat {\mathcal {G}}$ , as required. In particular, $\log _{\mathcal {G}} =h$ is étale and surjective, such that $\mathcal {G}$ is locally p-divisible.

For the last part of the statement, assume that $[p]\colon \Gamma \rightarrow \Gamma $ is surjective and finite étale. As $[p]\colon \mathbf {V} \rightarrow \mathbf {V}$ is an isomorphism because $\mathbf {V}$ is a vector bundle, the assumption implies that $[p]\colon \mathcal {G} \rightarrow \mathcal {G}$ is also a surjection with the kernel $\mathcal {G}[p] = \Gamma [p]$ being finite étale over S, which concludes the proof.

Proof. Let us prove the first part. We apply Lemma 3.38 to the exact sequence (3.6). This is possible, since ${\mathbf {Bun}}_{\widehat {\mathcal {G}},\operatorname {\mathrm {\acute {e}t}} }$ is a v-sheaf, by Remark 3.35. In the case $S=\operatorname {\mathrm {Spa}}(K,K^+)$ and $\tau =v$ , Lemma 3.38 also applies to the exact sequence (3.6), as ${\mathbf {Bun}}_{{\mathfrak {g}},v}^n = H_v^n(X,{\mathfrak {g}}) \otimes \mathbb {G}_a$ is an étale vector bundle, by Proposition 3.16.1. This proves the first point. The second point follows from Proposition 2.29.

Proof of Theorem 3.37

We have a commutative diagram

Proposition 3.39 provides the splitting $s_{\exp }$ and hence yields a splitting $s_n$ of the left vertical map. For general $i<n$ , we have a commutative diagram

By descending induction on i, we have a section $s_{i+1}$ splitting the left vertical map. Since the left square above is a pushout square, we obtain a section $s_i$ of the middle vertical map. By Proposition 3.4, the lift $\mathbb {X}$ induces a splitting $s_{\mathbb {X}}$ of the Hodge–Tate spectral sequence with ${\mathfrak {g}}$ -coefficients. We may precompose $s_i$ with $s_{\mathbb {X}}$ and postcompose it with the natural map

$$\begin{align*}\operatorname{\mathrm{Fil}}^iH_v^n(X,\widehat{\mathcal{G}}) \rightarrow \operatorname{\mathrm{Fil}}^iH_v^n(X,\mathcal{G}).\end{align*}$$

This yields a map $H^i(X,\widetilde {\Omega }_X^{n-i} \otimes \operatorname {\mathrm {Lie}}(\mathcal {G})) \rightarrow \operatorname {\mathrm {Fil}}^iH_v^n(X,\mathcal {G})$ that splits the Hodge–Tate spectral sequence with $\mathcal {G}$ -coefficients, as required.