Proof. We first show that we have a quasi-isomorphism:

(2.6) $$ \begin{align} f_Y: \mathrm{R} \Gamma_{\mathrm{\acute{e}t} }(Y, \mathbf{Z}_{\ell}) \xrightarrow{\sim} \mathrm{R} \Gamma_{\mathrm{pro\acute{e}t} }(Y, \mathbf{Z}_{\ell}), \end{align} $$

for any smooth rigid analytic variety Y. For that, choose a covering of Y by an increasing sequence of quasi-compact opens $\{U_i\}_{i\in \mathbf {N}}$ . Since

$$ \begin{align*}\mathrm{R} \Gamma_{*}(Y, \mathbf{Z}_{\ell})\simeq \mathrm{R}\lim_i\mathrm{R}\Gamma_{*}(U_i,\mathbf{Z}_{\ell}), \end{align*} $$

it suffices to show the quasi-isomorphism (2.6) for a quasi-compact Y.

Assume thus that Y is quasi-compact. Then we have

$$\begin{align*}\mathrm{R} \Gamma_{\mathrm{\acute{e}t} }(Y, \mathbf{Z}_{\ell})= \mathrm{R} \lim_n \mathrm{R} \Gamma_{\mathrm{\acute{e}t} }(Y, \mathbf{Z}/\ell^n) \xrightarrow{\sim} \mathrm{R} \lim_n \mathrm{R} \Gamma_{\mathrm{pro\acute{e}t} }(Y, \mathbf{Z}/\ell^n) \end{align*}$$

using [Reference Scholze28, Cor. 3.17] . Since the functors $\mathrm {R} \Gamma $ and $\mathrm {R} \lim $ commute, we have

$$\begin{align*}\mathrm{R} \lim_n \mathrm{R} \Gamma_{\mathrm{pro\acute{e}t} }(Y, \mathbf{Z}/\ell^n) \xleftarrow{\sim} \mathrm{R} \Gamma_{\mathrm{pro\acute{e}t} }(Y, \mathrm{R} \lim_n \mathbf{Z}/\ell^n) \xrightarrow{\sim} \mathrm{R} \Gamma_{\mathrm{pro\acute{e}t} }(Y, \mathbf{Z}_{\ell}), \end{align*}$$

where the last quasi-isomorphism follows from [Reference Scholze28, Prop. 8.2]. Hence, we get the quasi-isomorphism (2.6), as wanted.

By the definitions of $\mathrm {R}\Gamma _{\mathrm {pro\acute {e}t} ,c}(X, \mathbf {Z}_{\ell })$ and $\mathrm {R}\Gamma _{\mathrm {\acute {e}t} ,c, {\mathrm naive}}(X, \mathbf {Z}_{\ell })$ , this induces a quasi-isomorphism:

$$\begin{align*}\mathrm{R}\Gamma_{\mathrm{pro\acute{e}t} ,c}(X, \mathbf{Z}_{\ell}) \xleftarrow[f]{\sim} \mathrm{R}\Gamma_{\mathrm{\acute{e}t} ,c, {\mathrm naive}}(X, \mathbf{Z}_{\ell}) \end{align*}$$

and composing with the natural morphism $\mathrm {R}\Gamma _{\mathrm {\acute {e}t} ,c, {\mathrm naive}}(X, \mathbf {Z}_{\ell }) \to \mathrm {R}\Gamma _{\mathrm {\acute {e}t} ,c}(X, \mathbf {Z}_{\ell })$ , we obtain the morphism from the second claim of the lemma.

Proof. We will recall here the proof from [Reference Jannsen21, Prop. 1.1]. For all s, we choose a monomorphism $\mathscr {F}_s \to \mathscr {J}_s$ of $\mathscr {F}_s$ into an injective object of the category of sheaves of $\mathbf {Z}/\ell ^s$ -modules and we define the projective system $(\mathscr {I}_n)_n$ as in the statement of the lemma. Since the $\mathscr {J}_s$ ’s are injective, $(\mathscr {J}_s)_{s\in \mathbf {N}}$ is an injective object of the product category

$$\begin{align*}C_{\ell}:= \prod_{s \in \mathbf{N}} \mathrm{mod}(X_{\mathrm{\acute{e}t} }-\mathbf{Z}/\ell^s) \end{align*}$$

and $(\mathscr {F}_s)_s \to (\mathscr {J}_s)_s$ is a monomorphism in that category. The functor

$$\begin{align*}P : C_{\ell} \to \mathrm{mod}(X_{\mathrm{\acute{e}t} }-\mathbf{Z}_{\ell}^{\bullet}),\quad (\mathscr{A}_s)_s \mapsto (\prod_{s=1}^n \mathscr{A}_s)_n \end{align*}$$

admits left adjoint given by the forgetful functor

$$\begin{align*}V : \mathrm{mod}(X_{\mathrm{\acute{e}t} }-\mathbf{Z}_{\ell}^{\bullet}) \to C_{\ell}. \end{align*}$$

Since V is exact and faithful, the functor P preserves monomorphisms and injective objects. It follows that $(\mathscr {I}_n)_n:=P((\mathscr {J}_s)_s)$ is an injective object and we have a monomorphism $(\mathscr {F}_n)_n\to (\mathscr {I}_n)_n$ .

Conversely, for all $m\in \mathbf {N}$ , the functor

$$\begin{align*}V_m : \mathrm{mod}(X_{\mathrm{\acute{e}t} }-\mathbf{Z}_{\ell}^{\bullet}) \to \mathrm{mod}(X_{\mathrm{\acute{e}t} }-\mathbf{Z}/\ell^m),\quad (\mathscr{A}_n)_n \mapsto \mathscr{A}_m \end{align*}$$

also admits an exact faithful left adjoint:

$$\begin{align*}U_m : \mathrm{mod}(X_{\mathrm{\acute{e}t} }-\mathbf{Z}/\ell^m) \to \mathrm{mod}(X_{\mathrm{\acute{e}t} }-\mathbf{Z}_{\ell}^{\bullet}), \text{ where } U_m((\mathscr{A}_n)_n)=\begin{cases} 0 & \text{ if } n < m, \\ \mathscr{A}_m & \text{ if } n \ge m. \end{cases} \end{align*}$$

It follows that $V_n$ preserves monomorphisms and injectives. Hence, we have obtained that $\mathscr {F}_n \to \mathscr {I}_n= V_n((\mathscr {I}_m)_m)$ is a monomorphism of $\mathscr {F}_n$ into an injective sheaf $\mathscr {I}_n$ , as wanted.

Proof. By definition, $\mathrm {R} \Gamma _{\mathrm {\acute {e}t} , c, \mathrm {Hu}}(X,-)$ is equal to the functor $\mathrm {R} (\Gamma _c \circ \pi _*)$ . Take a resolution $(\mathscr {F}_n)_n \to (\mathscr {I}_n)^{\bullet }_n$ given by Remark 2.12. The functor $\pi _*$ is given by $(\mathscr {F}_n)_n \mapsto \lim _n \mathscr {F}_n$ , hence, we have

$$\begin{align*}\pi_*((\mathscr{I}^{\bullet}_n)_n)= \prod_{s \in \mathbf{N}} \mathscr{J}^{\bullet}_s. \end{align*}$$

This yields

$$ \begin{align*}\mathrm{R} (\Gamma_c \circ \pi_*)(({\mathscr{F}}_n)_n)\simeq \Gamma_c(X_{\mathrm{\acute{e}t} },\prod_{s \in \mathbf{N}} \mathscr{J}^{\bullet}_s). \end{align*} $$

But, by [Reference Huber19, (5.2.6)], every term of the complex $\prod _{s \in \mathbf {N}} \mathscr {J}^{\bullet }_n$ is $\Gamma _!$ -acyclic, hence, we have

$$ \begin{align*}(\mathrm{R} \Gamma_{!} \circ \mathrm{R} \pi_*)(({\mathscr{F}}_n)_n)\simeq\mathrm{R} \Gamma_{!}(\prod_{s \in \mathbf{N}} \mathscr{J}^{\bullet}_s) \simeq \Gamma_{!}(\prod_{s \in \mathbf{N}} \mathscr{J}^{\bullet}_s)= \Gamma_{c}(X_{\mathrm{\acute{e}t} }, \prod_{s \in \mathbf{N}} \mathscr{J}^{\bullet}_s). \end{align*} $$

We have obtained that $\mathrm {R} (\Gamma _! \circ \pi _*)\simeq \mathrm {R} \Gamma _{!} \circ \mathrm {R} \pi _*$ , as wanted.

Proof. As in the proof of the previous proposition, take an injective resolution $(\mathscr {F}_n)_n \to (\mathscr {I}_n)^{\bullet }_n$ given by Remark 2.12. We have $\pi _*((\mathscr {I}_n)^{\bullet }_n)= \prod _{s \in \mathbf {N}} \mathscr {J}^{\bullet }_s$ . But each term of the complex $\prod _{s \in \mathbf {N}} \mathscr {J}^{\bullet }_s$ is $\Gamma $ -acyclic and $\Gamma $ commutes with countable products, hence, we get

$$\begin{align*}\mathrm{R} \Gamma_{\mathrm{\acute{e}t} }(X, \mathrm{R} \pi_*(\mathscr{F}_n))\simeq \prod_{s \in \mathbf{N}} \Gamma_{\mathrm{\acute{e}t} }(X, \mathscr{J}^{\bullet}_s). \end{align*}$$

Remark 2.12 also gives us, for all n, an injective resolution $\mathscr {F}_n \to \mathscr {I}^{\bullet }_n$ of $\mathscr {F}_n$ . Hence,

$$\begin{align*}\mathrm{R} \lim_n \mathrm{R} \Gamma_{\mathrm{\acute{e}t} }(X, \mathscr{F}_n)\simeq \mathrm{R} \lim_n \Gamma_{\mathrm{\acute{e}t} }(X, \mathscr{I}^{\bullet}_n)\simeq \prod_{s \in \mathbf{N}} \Gamma_{\mathrm{\acute{e}t} }(X, \mathscr{J}^{\bullet}_s), \end{align*}$$

which proves our lemma.

Proof. If s is in $\Gamma _c(X_{\mathrm {\acute {e}t} }, \mathscr {F}),$ then, by definition (see (2.9)), there exists an admissible open covering $(Z, V)$ , with Z quasi-compact open, such that $s|_V=0$ . Since $V\supset X\setminus Z$ , we have $s|_{X \setminus Z}=0$ as well.

Conversely, let s be such that $s|_{X \setminus Z}=0$ , for Z admissible open and quasi-compact. Let $X= \bigcup _{i \in I} X_i$ be an admissible covering of X with $X_i$ open affinoid and such that, for all i, there exists $\tilde {X}_i$ an open affinoid of X with $X_i \Subset \tilde {X}_i$ . Since Z is quasi-compact, modulo reindexing, we may assume that $Z \subset X_1 \cup X_2 \cup \dots \cup X_n$ , for a constant n. Set $V:=X \setminus (X_1 \cup \dots \cup X_n)$ and $\tilde {V}:= \tilde {X}_1 \cup \dots \cup \tilde {X}_n$ . Then $\{V, \tilde {V} \}$ is an admissible open covering of X and $s|_{V}=0$ .

Take a formal model $\tilde {\mathfrak {V}}$ of $\tilde {V}$ and the opens $\mathfrak {X}_i$ and $\tilde {\mathfrak {X}}_i$ of $\tilde {\mathfrak {V}}$ with $\mathfrak {X}_i \subset \tilde {\mathfrak {X}}_i$ and $X_i= {\mathrm sp}^{-1}(\mathfrak {X}_i)$ , $\tilde {X}_i= { \mathrm sp}^{-1}(\tilde {\mathfrak {X}}_i)$ (where ${\mathrm sp} : \tilde {V} \to \tilde {\mathfrak {V}}$ is the specialization map). If we write $\bar {\mathfrak {X}}_i$ for the closure of $\mathfrak {X}_i$ in $\tilde {\mathfrak {X}}_i$ , then $\bar {\mathfrak {X}}_i$ is proper over ${\mathscr O}_L$ (by [Reference Lütkebohmert25, Lemma 2.5]). Let now $\mathfrak {Z}:= \bar {\mathfrak {X}}_1 \cup \dots \cup \bar {\mathfrak {X}}_n$ . Then $\mathfrak {Z}$ is proper over ${\mathscr O}_L$ and $\mathrm {sp}^{-1}(\tilde {\mathfrak {V}} \setminus \mathfrak {Z}) \subset (X \setminus Z)$ hence $s|_{ \mathrm {sp}^{-1}(\mathfrak {X} \setminus \mathfrak {Z})}=0$ . We have proved that $s \in \Gamma _c(X_{\mathrm {\acute {e}t} }, \mathscr {F})$ .

Proof. It suffices to check that, for all quasi-compact opens Z, we have a distinguished triangle:

$$\begin{align*}\mathrm{R} \Gamma_Z(X_{\mathrm{\acute{e}t} }, \mathscr{F}) \to \mathrm{R} \Gamma_{\mathrm{\acute{e}t} }(X, \mathscr{F}) \to \mathrm{R} \Gamma_{\mathrm{\acute{e}t} }(X\setminus Z, \mathscr{F}), \end{align*}$$

where $\mathrm {R} \Gamma _Z(X_{\mathrm {\acute {e}t} }, -)$ is the right derived functor of $\mathscr {F} \mapsto \Gamma _Z(X_{\mathrm {\acute {e}t} }, \mathscr {F})$ .

To prove that, take the injective resolution $\mathscr {F}\to \mathscr {I}^{\bullet }(\mathscr {F}) $ defined as follows. For $n=0$ , we set $\mathscr {I}^{-1}(\mathscr {F}) =\mathscr {F}$ and $\mathscr {I}^0(\mathscr {F})(U):= \prod _{x \in U} \widetilde {\mathscr {F}}_x$ , where the product is taken over the geometric points x of X such that $x \in U$ and $\widetilde {\mathscr {F}}_x$ is the injective envelope of $\mathscr {F}_x$ . We define then by induction $\mathscr {I}^{n+1}(\mathscr {F}):=\mathscr {I}^0(\mathscr {I}^{n}(\mathscr {F})/\mathscr {I}^{n-1}(\mathscr {F}))$ . We have that $\Gamma _{\mathrm {\acute {e}t} }(X, \mathscr {I}^n({\mathscr {F}})) \to \Gamma _{\mathrm {\acute {e}t} }(X \setminus Z, \mathscr {I}^n({\mathscr {F}}))$ is surjective, for all $n \ge 0$ . It follows that the sequence of complexes

$$ \begin{align*}0\to \Gamma_{Z}(X_{\mathrm{\acute{e}t} }, \mathscr{I}^{\bullet}({\mathscr{F}}))\to \Gamma_{\mathrm{\acute{e}t} }(X, \mathscr{I}^{\bullet}({\mathscr{F}})) \to \Gamma_{\mathrm{\acute{e}t} }(X \setminus Z, \mathscr{I}^{\bullet}({\mathscr{F}}))\to 0 \end{align*} $$

is exact, as wanted.

Proof. $\mathrm {R} \Gamma _{!}(X,-)$ is the right derived functor of the functor $\mathscr {F} \mapsto \Gamma _c(X_{\mathrm {\acute {e}t} }, \mathscr {F})$ , and hence, it is equal to $\mathrm {R} F$ by Lemma 2.15. The result follows now directly from Proposition 2.16.

Proof. Recall that for all primes $\ell $ (including $\ell =p$ ), we have

(2.21) $$ \begin{align} H^i_{\mathrm{\acute{e}t} } ({\mathbb A}_C^1, \mathbf{Z}/ \ell^n ) \simeq \begin{cases} \mathbf{Z}/ \ell^n & \text{ if } i=0, \\ 0 & \text{ if } i \neq 0. \end{cases} \end{align} $$

For $j \in \mathbf {N}$ , consider the closed discs

$$\begin{align*}D_j:=\{z \in {\mathbb A}^1_C, |z| \le p^j\} \end{align*}$$

and the associated distinguished triangle (1):

$$\begin{align*}\mathrm{R} \Gamma_{\mathrm{\acute{e}t} ,c}({\mathbb A}_C^1, \mathbf{Z}_{\ell}) \to \mathrm{R} \Gamma_{\mathrm{\acute{e}t} } ({\mathbb A}_C^1, \mathbf{Z}_{\ell}) \to \mathrm{colim} _{j} \mathrm{R} \Gamma_{\mathrm{\acute{e}t} } ({\mathbb A}_C^1 \setminus D_j, \mathbf{Z}_{\ell}). \end{align*}$$

Write $C_j:=\{z \in {\mathbb A}^1_C, |z| \ge p^j\}$ , then

$$\begin{align*}\mathrm{colim} _{j} \mathrm{R}\Gamma_{\mathrm{\acute{e}t} }({\mathbb A}^1_C \setminus D_j, \mathbf{Z}_{\ell}(1))\simeq \mathrm{colim} _{j} \mathrm{R}\Gamma_{\mathrm{\acute{e}t} }(C_j, \mathbf{Z}_{\ell}(1)). \end{align*}$$

Consider $E_j:=\{z \in \mathbb {P}^1, |z| \ge p^j\}$ so that $C_j={\mathbb A}^1_C \cap E_j$ . The Kummer sequence gives

$$\begin{align*}H^i_{\mathrm{\acute{e}t} } (E_j, \mathbf{Z}/ \ell^n(1)) \simeq \begin{cases} \mathbf{Z}/ \ell^n\mathbf{Z} & \text{if } i=0, \\ C\{(p^jz)^{-1}\}^{*}/C\{(p^jz)^{-1}\}^{*\ell^n} &\text{if } i=1, \\ 0 & \text{otherwise.} \end{cases} \end{align*}$$

Using the Mayer–Vietoris sequence for $\mathbb {P}^1_C=E_j \cup {\mathbb A}^1_C$ , we get

$$\begin{align*}H^i_{\mathrm{\acute{e}t} } (C_j, \mathbf{Z}/ \ell^n(1)) \simeq \begin{cases} \mathbf{Z}/ \ell^n\mathbf{Z} & \text{if } i=0, \\ C\{(p^jz)^{-1}\}^{*}/C\{(p^jz)^{-1}\}^{*\ell^n} \oplus \mathbf{Z}/\ell^n &\text{if } i=1, \\ 0 & \text{otherwise.} \end{cases} \end{align*}$$

If $\ell \neq p$ , then $C\{(p^jz)^{-1}\}^{*}$ is $\ell $ -divisible, and we get

$$\begin{align*}H^i_{\mathrm{\acute{e}t} } (C_j, \mathbf{Z}_{\ell}(1)) \simeq \begin{cases} \mathbf{Z}_{\ell} & \text{if } i=0, \\ \mathbf{Z}_{\ell} &\text{if } i=1, \\ 0 & \text{otherwise} \end{cases} \quad \text{and} \quad H^i_{\mathrm{\acute{e}t} } (C_j, \mathbf{Z}_{p}(1)) \simeq \begin{cases} \mathbf{Z}_{p} & \text{if } i=0, \\ C\{(p^jz)^{-1}\}^{*\wedge}\oplus \mathbf{Z}_{p} &\text{if } i=1, \\ 0 & \text{otherwise. } \end{cases} \end{align*}$$

Using (2.21) and the isomorphism, via the logarithm (see [Reference Colmez, Dospinescu, Hauseux and Nizioł9, Example A.2]), $ \mathrm {colim} _jC\{(p^jz)^{-1}\}^{*\wedge }\simeq ({\mathscr O}_{{\mathbb P}^1,\infty }/C), $ this yields (2.20), as wanted.

Proof. This follows from a result in general topology [Reference Fujiwara and Kato17, Chapter 0, Corollary 2.2.27, and Proposition 2.4.4] combined with the fact that the underlying topological spaces of rigid spaces (in Fujiwara–Kato’s sense) of type $V_{\mathbf {R}}$ are reflexive valuative spaces [Reference Fujiwara and Kato17, Chapter II, Theorem 8.2.19]. Since the result is less difficult than having to move between different formalisms, we give a proof below.

Let $U'$ be the interior of $\overline {U}$ in X. Trivially, $U\subseteq U'$ and we need to show the reverse inclusion. The underlying topological space of X is a locally spectral space, so by [Reference Fujiwara and Kato17, Chapter 0, Corollary 2.2.27], the closure $\overline {U}$ consists of all specializations of points of U inside X. Therefore, U and $U'$ have the same closed points, in particular the same classical points.

Let $V\subseteq U'$ be an affinoid open. As $U\hookrightarrow X$ is quasi-compact, the map $U\cap V\hookrightarrow V$ is an open immersion of qcqs rigid spaces which is surjective on classical points. It follows that $U\cap V = V$ , and since V was arbitrary, that $U \subseteq U'$ .

Proof. Let $\mathfrak {U}\subseteq \mathfrak {V}$ be an open immersion of admissible formal schemes over $\mathscr {O}_L$ which is a formal model of $U\subseteq V$ , which exists because U and V are qcqs. The assumption that $U\Subset V$ translates into the statement that $Z:=\overline {\mathfrak {U}_0}$ (the closure of the special fiber of $\mathfrak {U}$ inside the special fiber of $\mathfrak {V}$ ) is proper over k. Let $U'\subseteq V$ be the Berthelot tube of Z, that is, the interior of its preimage $\mathrm {sp}^{-1}(Z)$ under the specialization map $\mathrm {sp}\colon V\to \mathfrak {V}_0$ . Then $U'$ is a partially proper open of V containing U by construction.

Lemma 2.26 above implies that the quasi-compact open $W = \mathrm {sp}^{-1}(\mathfrak {V}_0\setminus Z)$ is the interior of its closure, so that $\mathrm {sp}^{-1}(Z)$ is the closure of its interior. This in turn implies that the interior $(V\setminus U')^\circ $ of $V\setminus U'$ equals W and hence is quasi-compact.

Finally, we claim that

$$\begin{align*}(X\setminus U')^\circ = (X\setminus V)\cup (V\setminus U')^\circ. \end{align*}$$

Suppose we know this to hold, then $X\setminus V\subseteq (X\setminus U')^\circ $ implies that $X = V\cup (X\setminus U')^\circ $ . Moreover, we can check that $(X\setminus U')^\circ $ is retrocompact on the open cover $X = V \cup (X\setminus U')^\circ $ , and we are done.

It remains to show that if $x\in X\setminus V$ belongs to the closure of V, then it lies in the closure of the quasi-compact open $W = (V\setminus U')^\circ $ . As taking the closure of a qc open amounts to adding specializations (Lemma 2.26 again), there exists a point $y\in V$ specializing to x, and x belongs to the closure of W if and only if y does. But if y is not in the closure of W, it belongs to the interior of $X\setminus W$ , which equals $U'$ . Since $U'$ is partially proper, it is closed under specialization in X (by the valuative criterion), and hence $x\in U'$ , a contradiction.

Proof. Since X is partially proper, it can be written as the filtered colimit of a family of quasi-compact opens $\{U_i\}_{i\in I}$ such that, for every $i\in I,$ there exists a $j\in I$ with $j>i$ such that $U_i\Subset U_j$ . Pick such a j for every i. Applying Lemma 2.27 to $U_i\Subset U_j\subseteq X$ , we obtain an open subset $U^{\prime }_i$ with $U_i\subset U^{\prime }_i\subset U_j$ which satisfies (1)–(3). It is easy to see that the resulting family of subsets (possibly no longer indexed by I) is cofiltering, and trivially covers X.

Proof. Claim (1) is clear. For claim (2), write X as the filtering colimit of opens $U_i\subseteq X$ ( $i\in I$ ) as in Corollary 2.28. Since the family is cofinal with the family of quasi-compact opens of X, we have

$$\begin{align*}P(\partial X) = \varinjlim_{i\in I} \, P((X\setminus U_i)^\circ). \end{align*}$$

It is therefore enough to show that the map $P(X)\to P((X\setminus U)^\circ )$ is an isomorphism for every U in the family. To this end, let $V\subseteq X$ be a quasi-compact open containing U. Thus, X can be covered by the quasi-compact open V and the non-quasi-compact open $(X\setminus U)^\circ $ whose intersection W is quasi-compact by retrocompactness of $(X\setminus U)^\circ $ . We have a morphism of distinguished triangles

By (1), we have $P(V) = 0 = P(W)$ , and it follows that $P(X)\to P((X\setminus U)^\circ )$ is a quasi-isomorphism.

Proof of Lemma 2.23 (and therefore also of Theorem 2.22).

For every quasi-compact open $U\subset X$ , let $U'\subset U$ denote its interior, which is a partially proper open. Since X is partially proper, the subsets $U'$ (as U varies over quasi-compact opens U) are cofinal with the subsets U (every U is contained in $V'$ for some V), so that we have

$$\begin{align*}\mathrm{R}\Gamma_{\mathrm{pro\acute{e}t} ,c}(\partial X, \mathbf{Q}_{\ell}) = \varinjlim_U \mathrm{R}\Gamma_{\mathrm{pro\acute{e}t} }(X\setminus U', \mathbf{Q}_{\ell}) \end{align*}$$

and, similarly,

$$\begin{align*}\mathrm{R}\Gamma_{\mathrm{pro\acute{e}t} ,c}(\partial X, \mathbf{Z}_{\ell})_{\mathbf{Q}_{\ell}} = \varinjlim_U \mathrm{R}\Gamma_{\mathrm{pro\acute{e}t} }(X\setminus U', \mathbf{Z}_{\ell})_{\mathbf{Q}_{\ell}}. \end{align*}$$

By Lemma 2.30, the map $P(X)\to P(\partial X)$ is a quasi-isomorphism. Therefore, the square

is Cartesian. Flipping the square, we deduce that $\mathrm {R}\Gamma _{\mathrm {pro\acute {e}t} ,c}(X, \mathbf {Z}_{\ell })_{\mathbf {Q}_{\ell }} \to \mathrm {R}\Gamma _{\mathrm {pro\acute {e}t} ,c}(X, \mathbf {Q}_{\ell })$ is a quasi-isomorphism as well, as wanted.

Proof. For the first claim, it suffices to show that, for any open covering $X=X_1\cup X_2$ by dagger affinoids, we have the Mayer–Vietoris distinguished triangle

$$ \begin{align*}\mathrm{R}\Gamma^{\natural}_{\mathrm{pro\acute{e}t} ,c}(X,\mathbf{Q}_{\ell}(r))\leftarrow \mathrm{R}\Gamma^{\natural}_{\mathrm{pro\acute{e}t} ,c}(X_1,\mathbf{Q}_{\ell}(r))\oplus \mathrm{R}\Gamma^{\natural}_{\mathrm{pro\acute{e}t} ,c}(X_2,\mathbf{Q}_{\ell}(r))\leftarrow \mathrm{R}\Gamma^{\natural}_{\mathrm{pro\acute{e}t} ,c}(X_1\cap X_2,\mathbf{Q}_{\ell}(r)). \end{align*} $$

Choosing a presentation $\{ X_h \}$ of X, an $h\in \mathbf {N}$ , and using excision, we can write this triangle as

$$ \begin{align*}\mathrm{R}\Gamma_{\mathrm{pro\acute{e}t} ,\widehat{X}}(X_h,\mathbf{Q}_{\ell}(r))\leftarrow \mathrm{R}\Gamma_{\mathrm{pro\acute{e}t} ,\widehat{X}_1}(X_h,\mathbf{Q}_{\ell}(r))\oplus \mathrm{R}\Gamma_{\mathrm{pro\acute{e}t} ,\widehat{X}_2}(X_h,\mathbf{Q}_{\ell}(r))\leftarrow \mathrm{R}\Gamma_{\mathrm{pro\acute{e}t} ,\widehat{X_1\cap X_2}}(X_h,\mathbf{Q}_{\ell}(r)), \end{align*} $$

which is known to be distinguished.

Let us now prove claim (2). If X is Stein, take $\{ U_n \}$ an increasing covering of X by admissible dagger affinoids. We have a commutative diagram:

where the first row is the distinguished triangle (1) and the second one—the distinguished triangle (2.39). (We note that there is no circular reasoning here: the proof of Proposition 2.37 uses only the first claim of this lemma.) The second and third vertical arrows are quasi-isomorphisms by the usual comparison results between overconvergent cohomologies and their rigid analytic analogs (see [Reference Colmez and Nizioł14, Proposition 3.17]). We obtain that the first vertical map (defined by the diagram) is a quasi-isomorphism as well.

Since a general partially proper variety X is locally Stein, we can conclude by analytic codescent.

Proof. Let X and $\{U_n\}$ be as in the statement of the proposition. For each n, let $\{U_{n,h} \}_h$ be a presentation of $U_n$ . Then,

$$ \begin{align*} \mathrm{R}\Gamma_{\mathrm{pro\acute{e}t} ,c}(X,\mathbf{Q}_{\ell}(r)) & \xleftarrow{\sim} \mathrm{colim} _n \mathrm{R}\Gamma_{\mathrm{pro\acute{e}t} ,c}(U_n,\mathbf{Q}_{\ell}(r)) \xleftarrow{\sim} \mathrm{colim} _n \mathrm{R}\Gamma^{\natural}_{\mathrm{pro\acute{e}t} ,c}(U_n,\mathbf{Q}_{\ell}(r)) \\ & \xleftarrow{\sim} \mathrm{colim} _n \mathrm{R}\Gamma^{\mathrm{GK}}_{\mathrm{pro\acute{e}t} ,c}(U_n,\mathbf{Q}_{\ell}(r)), \end{align*} $$

where the first quasi-isomorphism comes from the fact that ${\mathscr {A}}_{\mathrm {pro\acute {e}t} ,c}$ is a cosheaf, the second one is the local–global compatibility from Lemma 2.34, and the third one is the one from Proposition 2.36. For $h \in \mathbf {N}$ , from the excision (2.1), we have

$$\begin{align*}\mathrm{R}\Gamma_{\mathrm{pro\acute{e}t} ,c}^{\mathrm{GK}}(U_n,\mathbf{Q}_{\ell}(r))= \mathrm{R}\Gamma_{\mathrm{pro\acute{e}t} , \widehat{U}_n}(U_{n,h},\mathbf{Q}_{\ell}(r)) \xleftarrow{\sim} \mathrm{R}\Gamma_{\mathrm{pro\acute{e}t} , \widehat{U}_n}(\widehat{X},\mathbf{Q}_{\ell}(r)). \end{align*}$$

We conclude the proof using the quasi-isomorphisms $\mathrm {R}\Gamma _{\mathrm {pro\acute {e}t} }(\widehat {X},\mathbf {Q}_{\ell }(r))\xleftarrow {\sim }\mathrm {R}\Gamma _{\mathrm {pro\acute {e}t} }(X,\mathbf {Q}_{\ell }(r))$ and $\mathrm {R}\Gamma _{\mathrm {pro\acute {e}t} }(\widehat {X}\setminus \widehat {U}_n,\mathbf {Q}_{\ell }(r))\xleftarrow {\sim }\mathrm {R}\Gamma _{\mathrm {pro\acute {e}t} }(X \setminus U_n,\mathbf {Q}_{\ell }(r))$ .

Proof. The proposition follows from formula (2.33) and the cosheaf condition for ${\mathscr {A}}_{\mathrm {pro\acute {e}t} ,c}(r)$ .

Proof. We can work étale locally. Consider the maps:

(3.4)

for ${\mathscr {X}}$ the base change to ${\mathscr {O}}_C$ of a semistable model ${\mathscr {X}}_{{\mathscr {O}}_L}$ over a finite extension L of K, and where we have defined

The first arrow in (3.4) is a quasi-isomorphism (use the fact that the ring ${\mathbf B}_{\mathrm {dR} }^+$ is the Hodge completed crystalline cohomology of the base ring ${\mathscr {O}}_{\overline {K}}$ ). For the second map, we know that, by the Künneth formula, we have quasi-isomorphisms:

$$\begin{align*}\mathrm{R}\Gamma_{\mathrm{cr} }({\mathscr{X}}_{{\mathscr{O}}_L,n}/{\mathscr{O}}_{L,n}^{\times}) \otimes^{\mathrm{L}}_{{\mathscr{O}}_L} \mathrm{R}\Gamma_{\mathrm{cr} }({\mathscr{O}}_{\overline{K},n}^{\times}/{\mathscr{O}}_{L,n}^{\times}) \xrightarrow[\sim]{\cup} \mathrm{R}\Gamma_{\mathrm{cr} }({\mathscr{X}}_n/{\mathscr{O}}_{L,n}^{\times}), \quad \text{ for } n \ge 1. \end{align*}$$

By taking the derived inverse limit over $n\ge 1$ , we get a quasi-isomorphisms:

$$\begin{align*}\mathrm{R}\Gamma_{\mathrm{cr} }({\mathscr{X}}_{{\mathscr{O}}_L}/{\mathscr{O}}_{L}^{\times}) \widehat{\otimes}^{\mathrm{L}}_{{\mathscr{O}}_L} \mathrm{R}\Gamma_{\mathrm{cr} }({\mathscr{O}}_{\overline{K}}^{\times}/{\mathscr{O}}_{L}^{\times}) \xrightarrow[\sim]{\cup} \mathrm{R}\Gamma_{\mathrm{cr} }({\mathscr{X}}/{\mathscr{O}}_{L}^{\times}).\end{align*}$$

We then use [Reference Bosco5, Lemma A.9] to identify the left-hand side with the solid tensor product:

Inverting p, we obtain that the second map in (3.4) is a quasi-isomorphism. The third arrow in (3.4) is given by the base change from ${\mathscr {O}}_F^{\times }$ to ${\mathscr {O}}_L^{\times }$ and it is a quasi-isomorphism by [Reference Nekovář and Nizioł26, Corollary 2.4]. By taking the derived limit $\mathrm {R}\lim _r \iota _{\mathrm {BK},r}$ , we get a quasi-isomorphism:

To conclude, it suffices to show that we have a quasi-isomorphism:

But this can be done as in [Reference Colmez and Nizioł15, Lemma 3.17], by writing the explicit definition of the filtration and arguing degreewise.

Proof. The proof from [Reference Colmez and Nizioł14, Proposition 3.11] still works in the solid setting: working étale locally, we use the $\iota _{\mathrm {BK}}$ from Lemma 3.3 to reduce to a similar statement for the de Rham cohomology instead of the crystalline one.

Proof. The proposition follows from Proposition 3.6, applying it to the smooth varieties X and $X \setminus Z$ , for Z quasi-compact. As usual, we use that the solid tensor product commutes with colimits for the first claim. The second and third points come from the fact that, by definition, we have distinguished triangles in ${\mathscr D}({{\mathbf B}_{\mathrm {dR} ,\Box }^+}) $ and in ${\mathscr D}(C_{\Box })$ , respectively:

$$ \begin{align*} F^r \mathrm{R}\Gamma_{\mathrm{dR} ,c}(X/ {\mathbf B}_{\mathrm{dR} }^+) & \to F^r \mathrm{R}\Gamma_{\mathrm{dR} }(X/ {\mathbf B}_{\mathrm{dR} }^+) \to F^r \mathrm{R}\Gamma_{\mathrm{dR} }(\partial X/ {\mathbf B}_{\mathrm{dR} }^+), \\ F^r \mathrm{R}\Gamma_{\mathrm{dR} ,c}(X)& \to F^r \mathrm{R}\Gamma_{\mathrm{dR} }(X) \to F^r \mathrm{R}\Gamma_{\mathrm{dR} }(\partial X), \quad \text{ for } r \ge 0.\\[-36pt] \end{align*} $$

Proof. This follows from the fact that, for $0 \le j\le d$ , the differential $\tilde {d}_{j+d}$ of the complex $\mathrm {DR} _c(X,r)$ factorizes as

where the first arrow is surjective with kernel .

Proof. It suffices to use Lemma 3.14 and the isomorphisms (3.15).

Proof. We can work étale locally. Let ${\mathscr {X}}$ be the base change of a formal model ${\mathscr {X}}_{{\mathscr {O}}_L}$ defined over L a finite extension of K. We have a quasi-isomorphism:

$$\begin{align*}\mathrm{R}\Gamma_{\mathrm{cr} }({\mathscr{X}}_0/{\mathscr{O}}_{F_L,n}^0)\otimes^{\mathrm{L}}_{{\mathscr{O}}_{F_L,n}} {\mathscr{O}}_{\breve{F},n} \stackrel{\sim}{\to} \mathrm{R}\Gamma_{\mathrm{cr} }({\mathscr{X}}_0/{\mathscr{O}}_{\breve{F},n}), \quad \text{ for } n \ge 1, \end{align*}$$

where ${\mathscr {O}}_{F_L}:=W(k_L)$ . By taking the derived inverse limit over n and inverting p, we get a quasi-ismorphism:

$$\begin{align*}\mathrm{R}\Gamma_{\mathrm{HK} }({\mathscr{X}}_0) \widehat{\otimes}^{\mathrm{L}}_{{F_L}} {\breve{F}} \stackrel{\sim}{\to} \mathrm{R}\Gamma_{\mathrm{HK} , \breve{F}}({\mathscr{X}}_0),\end{align*}$$

where $\widehat {\otimes }^{\mathrm {L}}$ is the derived completion. Since $ \mathrm {R}\Gamma _{\mathrm {HK} }({\mathscr {X}}_0) $ is built from Banach spaces, we can pass to the solid tensor product.

Proof. We can work étale locally. The local quasi-isomorphisms are constructed in [Reference Colmez and Nizioł15] and the construction goes through in the condensed setting. Recall that the main point is to show that we have a quasi-isomorphism:

Proof. The proposition follows immediately from Proposition 3.18 applied to $Y=X, X\setminus Z$ , for Z quasi-compact.

Proof. The proof of this lemma is analogous to the one of Lemma 2.34.

Proof. To prove the point (1)(a) of the proposition, it suffices to argue locally, then apply the analogous quasi-ismorphism obtained in the rigid analytic case (3.7) to the varieties $X_h$ and $X_h \setminus \widehat {X}$ and we conclude using that the solid tensor product commutes with colimits.

The arguments are similar for the distinguished triangles (1)(b) (using (3.8)) and (1)(c) (using (3.9)), and for the quasi-isomorphism (2) (using (3.3) and the previous remark for the compatibility with the filtrations).

Proof. We can work locally. Let X be a smooth dagger affinoid over C. Then,

where $F_L=\mathrm {Frac}(W(k_L))$ . The first quasi-isomorphism is obtained by commutativity of the solid tensor product with colimits, and the second one is the map $\beta $ from Lemma 3.17.

Proof. It suffices to argue locally. Let X be a smooth dagger affinoid over L. Then,

The first quasi-isomorphism is obtained by commutativity of the solid tensor product with colimits, and the second one is the Hyodo–Kato morphism from Proposition 3.18.

Proof. We start with the proof for the Hyodo–Kato cohomology. The proof is the same as [Reference Colmez and Nizioł15, Proposition 3.5]. It suffices to show that, for any analytic dagger affinoid hypercovering $U_{\bullet }$ of X, the map

(4.17) $$ \begin{align} \mathrm{R}\Gamma^{\natural}_{\mathrm{HK} ,c}(X) \leftarrow \mathrm{R}\Gamma^{\natural}_{\mathrm{HK} ,c}(U_{\bullet}) \end{align} $$

is a quasi-isomorphism. Consider the following commutative diagram (where $F_L=F^{\mathrm {nr}}, \breve {F}$ and $*=-, \breve {F}$ ) in ${\mathscr {D}}(C_{\Box })$ :

The vertical arrows are quasi-isomorphisms by (the proof of) Proposition 4.15 and the bottom horizontal one by (the proof of) Lemma 4.3. It follows that the upper horizontal arrow is a quasi-isomorphism as well.

It remains to check that this implies the quasi-isomorphism (4.17). For that, it is enough to show that if $K^{\bullet }$ is in ${\mathscr D}({F_{1,\Box }})$ , for $F_1=\breve {F}$ or a finite unramified extension of F, such that is acyclic then $K^{\bullet }$ itself is acyclic. To see that, writing $C \simeq F_1 \oplus W$ with W an $F_1$ -Banach space, we obtain

so $H^*(K^{\bullet })=0$ , and this concludes the proof of the first point.

The second point follows from the first one, and from Lemma 4.3, by the exact same argument as in [Reference Colmez and Nizioł15, Corollary 4.28].

Proof. The result follows from the long exact sequence (5.2) using that we have $H^i({\mathbb A}_L^d, \Omega ^d) = 0$ , for $i \neq 0$ , and

$$\begin{align*}H^i({\mathbb A}_L^d \setminus D_j, \Omega^d) \simeq \begin{cases} H^0({\mathbb A}_L^d, \Omega^d) &\text{ if } i=0 ,\\ \{ \sum_{ \nu>0} \frac{ a_{\nu}}{x^{\nu}} dx_1 \wedge \cdots \wedge dx_d \; | \; \lim_{|\nu| \to \infty} p^{-j| \nu |}|a_{\nu}|=0 \} &\text{ if } i=d-1, \\0 &\text{ if } i\neq 0, d-1. \end{cases} \end{align*}$$

To prove the latter formula, consider the admissible covering of ${\mathbb A}_L^d\setminus D_j$ by the open subsets $\{U_i\}_{1 \le i \le d}$ , where

$$\begin{align*}U_i:=\{x=(x_1, \ldots, x_d) \in {\mathbb A}^d_L \; | \; |x_i|> \frac{1}{p^j}\}, \quad \text{ for } i \in \{ 1, \ldots, d\} ,\end{align*}$$

and use the associated Čech cohomology.

Proof. The trace map is given by formula (5.4).

First, assume that ${\mathscr {F}}= {\mathscr {O}}_X$ . For $i \neq d$ , both terms of the product are zero, so the pairing is perfect. For $i=d$ , the pairing is given by

$$ \begin{align*} H_c^d(X, {\mathscr{O}}_X) & \times H^0(X,\Omega_X^{d}) \to L \\ (\sum_{\nu> 0} \frac{ a_{\nu}}{x^{\nu}} & ,\sum_{\nu \ge 0} b_{\nu} x^{\nu} dx) \mapsto \sum_{\nu\ge 0} a_{\nu+1} b_{\nu} \end{align*} $$

and it induces a perfect duality. This proves the proposition for ${\mathscr {F}}= {\mathscr {O}}_X$ and we can deduce from it the result, first, for ${\mathscr {F}}= {\mathscr {O}}_X^m$ and then for a general coherent sheaf.

Proof. The first claim follows from Lemma 5.6 using the fact that $i^{!}$ is right adjoint to $i_*$ .

To prove claim (2), note that the result is true for the usual cohomology (without compact support). We deduce the isomorphism for the compactly supported cohomology using that for each open affinoid V of X, we can find a finite set of open affinoids $Z_1, \ldots , Z_n$ in the affine space such that V is the intersection of the $Z_i$ ’s with X (see [Reference Chiarellotto6, Prop. 4.17]).

For the third claim, consider an injective resolution ${\mathscr {I}}_{\bullet }$ of $\Omega _{{\mathbb A}_L^N}^N$ . Then

$$\begin{align*}{\mathrm{Hom}} _{{\mathscr{O}}_{{\mathbb A}^N_L}}(i_*{\mathscr{F}}, {\mathscr{I}}^{\bullet}) \simeq {\mathrm{Hom}} _{{\mathscr{O}}_X}({\mathscr{F}}, i^{!}{\mathscr{I}}^{\bullet}). \end{align*}$$

We conclude using that the functor $i^!$ sends injectives to injectives and applying Lemma 5.6.

Proof. Let X be a smooth Stein variety. Our theorem follows from the duality for the affine space (Proposition 5.5) using the isomorphisms from Corollary 5.7:

$$ \begin{align*} H_c^i(X, {\mathscr{F}}) \xrightarrow{\sim} H_c^i({\mathbb A}_L^N, i_*{\mathscr{F}}) \quad \text{ and }\quad & \mathrm{ Ext}_{{\mathscr{O}}_X}^{d-i}({\mathscr{F}}, \Omega^d_X) \xrightarrow{\sim} \mathrm{Ext}_{{\mathscr{O}}_{{\mathbb A}_L^N}}^{N-i}(i_*{\mathscr{F}}, \Omega^N_{{\mathbb A}^N_L}), \\ H^i(X, {\mathscr{F}}) \xrightarrow{\sim} H^i({\mathbb A}_L^N, i_*{\mathscr{F}}) \quad \text{ and }\quad & \mathrm{ Ext}_{{\mathscr{O}}_X,c}^{d-i}({\mathscr{F}}, \Omega^d_X) \xrightarrow{\sim} \mathrm{Ext}_{{\mathscr{O}}_{{\mathbb A}_L^N},c}^{N-i}(i_*{\mathscr{F}}, \Omega^N_{{\mathbb A}^N_L}).\\[-36pt] \end{align*} $$

Proof. The proof is similar to the one of Lemma 5.3: if we write $X_{\delta }:=\mathrm {Sp}(L[\delta ^{-1}X_1, \ldots , \delta ^{-1}X_d])$ , for $\delta>1$ , then we can use the long exact sequence:

$$\begin{align*}\dots \to \mathrm{colim} _{\delta} H^{i-1}(X_{\delta} \setminus \widehat{X}, \Omega^d) \to H^i_c(X, \Omega^d) \to \mathrm{colim} _{\delta} H^i(X_{\delta}, \Omega^d) \to \mathrm{colim} _{\delta} H^{i-1}(X_{\delta}\setminus \widehat{X}, \Omega^d) \to \cdots \end{align*}$$

to compute $H^i_c(X, \Omega ^d)$ (using that $H^i(X_{\delta }, \Omega ^d)=0$ for $i\neq 0$ ).

Proof. The trace map is given by formula (5.13). The proof of the duality is the same as the one of Proposition 5.5: it is enough to prove the result for ${\mathscr {F}}={\mathscr {O}}_X$ , and in that case, the pairing can be written explicitly.

Proof. The proof can be reduced to the Stein case, that is, to Lemma 5.8.

Proof. In the case X is Stein, the trace map $\mathrm {tr}_{\mathrm {dR} }: H^{2d}_{\mathrm {dR} ,c}(X)\to L$ can be induced by the coherent trace map $\mathrm {tr}_{\mathrm {coh}}: H^d_c(X, \Omega _X^d) \to L$ from Corollary 5.18, noting that $\mathrm {tr}_{\mathrm {coh}}(\partial )=0$ , where $\partial : H^{d}_c(X, \Omega _X^{d-1}) \to H^d_c(X, \Omega _X^d)$ (see [Reference Beyer4, Lemma 2.1.4]). Its derived version $\mathrm {tr}_{\mathrm {dR} }: \mathrm {R}\Gamma _{\mathrm {dR} ,c}(X)[2d]\to L$ is defined as the composition

In the case of general partially proper X, we reduce to the Stein case as in the proof of Corollary 5.18.

In the case X is Stein, claim (2) follows immediately from the sequence representing $\mathrm {R}\Gamma _{\mathrm {dR} }(X)$ :

(5.24) $$ \begin{align} {\mathscr O}(X)\to \Omega^1(X)\to\dots\to\Omega^d(X), \end{align} $$

the sequence (3.13) representing $\mathrm {R}\Gamma _{\mathrm {dR} ,c}(X)$ , and derived Serre duality (5.19). For a general partially proper X, we cover it with Stein varieties $\{X_i\}, i\in I$

and consider the associated Čech hypercovering $Y_{\scriptscriptstyle \bullet }$ of X. Then we compute in ${\mathscr {D}}(L_{\Box })$

$$ \begin{align*} {\mathrm{R}\Gamma}_{\mathrm{dR} }(X)& \simeq\mathrm{R}\lim_{i} {\mathrm{R}\Gamma}_{\mathrm{dR} }(Y_{i})\simeq\mathrm{R}\lim_{i}\mathrm{R}{\mathrm{Hom}} _{L_{\Box}}({\mathrm{R}\Gamma}_{\mathrm{dR} ,c}(Y_{i})[2d],L)\\ & \simeq\mathrm{R}{\mathrm{Hom}} _{L_{\Box}}(\mathrm{colim} _{i}{\mathrm{R}\Gamma}_{\mathrm{dR} ,c}(Y_{i})[2d],L)\simeq\mathrm{R}{\mathrm{Hom}} _{L_{\Box}}({\mathrm{R}\Gamma}_{\mathrm{dR} ,c}(X)[2d],L), \end{align*} $$

as wanted. Here, the second quasi-isomorphism follows from the duality for Stein varieties.

The first isomorphism in claim (3) follows from claim (2) by taking cohomology. We use here the fact that $\mathrm {Ext} ^j_{L_{\Box }}(H^{m}_{\mathrm {dR} ,c}(X),L)=0$ , for $j\geq 1$ , because $H^{m}_{\mathrm {dR} ,c}(X)$ is of compact type (in fact, a countable direct sum of finite rank vector spaces). The second isomorphism follows from the computation:

$$ \begin{align*} {\mathrm{Hom}} _{L_{\Box}} (H^{2d-i}_{\mathrm{dR} }(X),L)\stackrel{\sim}{\leftarrow} {\mathrm{Hom}} _{L_{\Box}} ({\mathrm{Hom}} _{L_{\Box}} (H^{i}_{\mathrm{dR} ,c}(X),L),L)\stackrel{\sim}{\to} H^{i}_{\mathrm{dR} ,c}(X), \end{align*} $$

where the first isomorphism was just proven and the second one follows from reflexivity of $H^{i}_{\mathrm {dR} ,c}(X)$ .

Proof. This follows immediately from the definition of the de Rham trace map and an analogous claim for coherent traces (see Lemma 5.20).

Proof. Assume first that X is Stein. We compute

$$ \begin{align*} \mathrm{R}{\mathrm{Hom}} _{L_{\Box}}(F^{r^{\prime}}\mathrm{R}\Gamma_{\mathrm{dR} ,c}(X),L) & \simeq (H^d_c(X,\Omega^{r^{\prime}})^*\leftarrow H^d_c(X,\Omega^{r^{\prime}+1})^*\leftarrow \dots \leftarrow H^d_c(X,\Omega^d)^*)[-d-r^{\prime}]\\ & \simeq {\mathscr O}(X)\to\Omega^1(X)\to\dots\to \Omega^{r}(X)\\ &\simeq \mathrm{R}\Gamma_{\mathrm{dR} }(X)/F^{r+1}. \end{align*} $$

We wrote here $(-)^*$ for ${\mathrm {Hom}} _{L_{\Box }}(-,L)$ and used the fact that

$$ \begin{align*}{\mathrm{Hom}} _{L_{\Box}}(H^d_c(X,\Omega^{i}),L)\stackrel{\sim}{\to}\mathrm{R}{\mathrm{Hom}} _{L_{\Box}}(H^d_c(X,\Omega^{i}),L)\end{align*} $$

because the cohomology groups with compact support are of compact type. The second quasi-isomorphism follows from Corollary 5.18.

The case of X partially proper follows from the Stein case by a limit/colimit argument as in the proof of Theorem 5.23.

Proof. Assume first that X is Stein. This follows from the computation:

(5.28) $$ \begin{align}\begin{aligned} \mathrm{R}{\mathrm{Hom}} _{{\mathbf B}^+_{\mathrm{dR} ,\Box}}(F^{r^{\prime}}& \mathrm{R}\Gamma_{\mathrm{dR} ,c}(X_C/{\mathbf B}^+_{\mathrm{dR} }),F^s{\mathbf B}^+_{\mathrm{dR} }) \simeq {\mathrm{Hom}} _{{\mathbf B}^+_{\mathrm{dR} ,\Box}}(F^{r^{\prime}}\mathrm{R}\Gamma_{\mathrm{dR} ,c}(X_C/{\mathbf B}^+_{\mathrm{dR} }),F^s{\mathbf B}^+_{\mathrm{dR} })\\& \simeq (H^d_c(X,\Omega^d)^*\otimes^{\Box}_K t^{r}{\mathbf B}^+_{\mathrm{dR} }\to \dots \to H^d_c(X,{\mathscr O})^*\otimes^{\Box}_K t^{r-d}{\mathbf B}^+_{\mathrm{dR} })[-d],\\& \simeq {\mathscr O}(X)\otimes^{\Box}_K t^r{\mathbf B}^+_{\mathrm{dR} }\to\Omega^1(X)\otimes^{\Box}_K t^{r-1}{\mathbf B}^+_{\mathrm{dR} }\to\dots\to \Omega^d(X)\otimes^{\Box}_K t^{r-d}{\mathbf B}^+_{\mathrm{dR} }\\&\simeq F^r\mathrm{R}\Gamma_{\mathrm{dR} }(X_C/{\mathbf B}^+_{\mathrm{dR} }).\end{aligned} \end{align} $$

We used here the quasi-isomorphisms

$$ \begin{align*} \mathrm{R}{\mathrm{Hom}} _{{\mathbf B}^+_{\mathrm{dR} ,\Box}}(H^d_c(X,\Omega^i)\otimes^{\mathrm{L}_{\Box}}_K t^{r^{\prime}-i}{\mathbf B}^+_{\mathrm{dR} },F^s{\mathbf B}^+_{\mathrm{dR} }) \simeq H^d_c(X,\Omega^i)^*\otimes^{\Box}_K t^{r-i}{\mathbf B}^+_{\mathrm{dR} }, \end{align*} $$

which hold because we have

$$ \begin{align*} \mathrm{R}{\mathrm{Hom}} _{{\mathbf B}^+_{\mathrm{dR} ,\Box}}(H^d_c(X,\Omega^i)\otimes^{\mathrm{L}_{\Box}}_K t^{r^{\prime}-i}{\mathbf B}^+_{\mathrm{dR} },t^s{\mathbf B}^+_{\mathrm{dR} }) & \simeq \mathrm{R}{\mathrm{Hom}} _{K_{\Box}}(H^d_c(X,\Omega^i),t^{s-r^{\prime}+i}{\mathbf B}^+_{\mathrm{dR} })\\ & \simeq \mathrm{R}{\mathrm{Hom}} _{K_{\Box}}(H^d_c(X,\Omega^i),K)\otimes_K^{\mathrm{L}_{\Box}}F^{s-r^{\prime}+i}{\mathbf B}^+_{\mathrm{dR} }\\ & \simeq H^d_c(X,\Omega^i)^*\otimes^{\Box}_K t^{r-d+i}{\mathbf B}^+_{\mathrm{dR} }. \end{align*} $$

Here, the first quasi-isomorphism is by adjunction; the second quasi-isomorphism follows from the fact that $H^d_c(X,\Omega ^i)$ is of compact type, ${\mathbf B}^+_{\mathrm {dR} }$ is Fréchet, and we have [Reference Rodrigues Jacinto and Rodriguez Camargo27, Theorem 3.40].

The case of X partially proper follows from the Stein case by a limit/colimit argument as in the proof of Theorem 5.23.

Proof. In the case X is affinoid, the trace map $\mathrm {tr}_{\mathrm {dR} }: H^{2d}_{\mathrm {dR} ,c}(X)\to L$ can be induced by the coherent trace map $\mathrm {tr}_{\mathrm {coh}}: H^d_c(X, \Omega _X^d) \to L$ from Corollary 5.21, noting that $\mathrm {tr}_{\mathrm {coh}}(\partial )=0$ , where $\partial : H^{d}_c(X, \Omega _X^{d-1}) \to H^d_c(X, \Omega _X^d)$ (reduce to the Stein case and use [Reference Beyer4, Lemma 2.1.4]). Its derived version $\mathrm {tr}_{\mathrm {dR} }: \mathrm {R}\Gamma _{\mathrm {dR} ,c}(X)[2d]\to L$ is defined as the composition

In the case of general smooth X, we reduce to the affinoid case as in the proof of Corollary 5.21.

In the case X is affinoid, the first part of claim (2) follows immediately from the sequence representing $\mathrm {R}\Gamma _{\mathrm {dR} }(X)$ :

$$ \begin{align*}{\mathscr O}(X)\to \Omega^1(X)\to\dots\to\Omega^d(X), \end{align*} $$

the analog of the sequence (3.13) representing $\mathrm {R}\Gamma _{\mathrm {dR} ,c}(X)$ , and derived Serre duality (5.19). For a general quasi-compact X, we cover it with dagger affinoid varieties $\{X_i\}, i\in I$ , and consider the associated Čech hypercovering $Y_{\scriptscriptstyle \bullet }$ of X. Then we compute in ${\mathscr {D}}(L_{\Box })$

$$ \begin{align*} {\mathrm{R}\Gamma}_{\mathrm{dR} }(X)& \simeq\mathrm{R}\lim_{i} {\mathrm{R}\Gamma}_{\mathrm{dR} }(Y_{i})\simeq\mathrm{R}\lim_{i}\mathrm{R}{\mathrm{Hom}} _{L_{\Box}}({\mathrm{R}\Gamma}_{\mathrm{dR} ,c}(Y_{i})[2d],L)\\ & \simeq\mathrm{R}{\mathrm{Hom}} _{L_{\Box}}(\mathrm{colim} _{i}{\mathrm{R}\Gamma}_{\mathrm{dR} ,c}(Y_{i})[2d],L)\simeq\mathrm{R}{\mathrm{Hom}} _{L_{\Box}}({\mathrm{R}\Gamma}_{\mathrm{dR} ,c}(X)[2d],L), \end{align*} $$

as wanted. Here, the second quasi-isomorphism follows from the duality for affinoid varieties. The second part of claim (2) follows now from the first part since ${\mathrm {R}\Gamma }_{\mathrm {dR} ,c}(X)$ is reflexive (the de Rham cohomologies $H^i_{\mathrm {dR} ,c}(X)$ are of finite rank): we have