Proof. This is trivial. ◻

Proof. We treat the case where $X^{\prime }=\text{Spa}\,A^{\prime }$ and $X=\text{Spa}\,A$ are affinoid, which is all we will need later. We can assume that they are reduced and that $f$ is surjective. If $i:Z\rightarrow X$ is Zariski-closed and nowhere dense, then $\text{dim}\,Z<\text{dim}\,X$; setting $Z^{\prime }=Z\times _{X}X^{\prime }$ and writing $f^{\prime }:Z^{\prime }\rightarrow Z$ and $i^{\prime }:Z^{\prime }\rightarrow X^{\prime }$ for the evident morphisms, we can assume that $i^{\ast }f_{\ast }\mathscr{F}\cong f_{\ast }^{\prime }{i^{\prime }}^{\ast }\mathscr{F}$ is Zariski-constructible by induction on $\text{dim}\,X$. By dévissage, it now suffices to find a dense Zariski-open subset $j:U\rightarrow X$ such that $j^{\ast }f_{\ast }\mathscr{F}$ is locally constant. To do this, choose a dense Zariski-open subset $V\subset X^{\prime }$ such that $\mathscr{F}|_{V}$ is locally constant. Then $W=X\smallsetminus f(X^{\prime }\smallsetminus V)$ is a dense Zariski-open subset of $X$, and $\mathscr{F}$ is locally constant after pullback along the open immersion $W^{\prime }=W\times _{X}X^{\prime }\rightarrow X^{\prime }$. If $\text{char}(K)=0$, we now conclude by taking $U$ to be any dense Zariski-open subset contained in $W$ such that $U^{\prime }=U\times _{X}X^{\prime }\rightarrow U$ is finite étale; if $\text{char}(K)=p$, we instead choose $U$ so that $U^{\prime }\rightarrow U$ factors as the composition of a universal homeomorphism followed by a finite étale map. (For the existence of such a $U$, look at the map of schemes $\text{Spec}\,A^{\prime }\rightarrow \text{Spec}\,A$; this morphism has the desired structure over all generic points of the target, and these structures then spread out over a dense Zariski-open subset of $\text{Spec}\,A$. One then concludes by analytifying.)◻

Proof. For (i), by induction on $\text{dim}\,X$ and dévissage, it suffices to find some dense Zariski-open subset $j:U\rightarrow X$ such that all three sheaves are locally constant after restriction to $U$. By assumption, we can choose $U$ such that two of the three sheaves have this property. Looking at the exact sequence $0\rightarrow \mathscr{F}|_{U}\rightarrow \mathscr{G}|_{U}\rightarrow \mathscr{H}|_{U}\rightarrow 0$, [Reference HuberHub96, Lemma 2.7.3] implies that all three sheaves are constructible (in the sense of [Reference HuberHub96]). By [Reference HuberHub96, Lemma 2.7.11], it now suffices to check that all three sheaves are overconvergent. But if $\overline{x}{\rightsquigarrow}\overline{y}$ is any specialization of geometric points, this follows immediately by applying the snake lemma to the diagram

since by assumption two of the three vertical arrows are isomorphisms.

For (ii), we first show that $\text{coker}\,f$ is Zariski-constructible. As in the argument for (i), it suffices to find some dense Zariski-open subset $j:U\rightarrow X$ such that $\text{coker}\,f|_{U}$ is locally constant. Choose $U$ such that $\mathscr{F}|_{U}$ and $\mathscr{G}|_{U}$ are locally constant, so $\text{coker}\,f|_{U}$ is constructible. Then for $\overline{x}{\rightsquigarrow}\overline{y}$ any specialization of geometric points of $U$, the first two vertical maps in the diagram

are isomorphisms by assumption, so the map $(\text{coker}\,f)_{\overline{y}}\rightarrow (\text{coker}\,f)_{\overline{x}}$ is an isomorphism by the five lemma. Thus, $\text{coker}\,f|_{U}$ is overconvergent and constructible and hence locally constant by [Reference HuberHub96, Lemma 2.7.11].

The remaining parts of (ii) now follow from the exact sequences $0\rightarrow \text{im}f\rightarrow \mathscr{G}\rightarrow \text{coker}\,f\rightarrow 0$ and $0\rightarrow \ker f\rightarrow \mathscr{F}\rightarrow \text{im}f\rightarrow 0$ by successive applications of (i).

For (iii), we can assume (possibly after a shift) that $F$ and $H$ are Zariski-constructible. The result then follows from parts (i) and (ii), by looking at the exact sequences

$$\begin{eqnarray}0\rightarrow \text{coker}\,(\mathscr{H}^{n-1}(H)\rightarrow \mathscr{H}^{n}(F))\rightarrow \mathscr{H}^{n}(G)\rightarrow \text{ker}(\mathscr{H}^{n}(H)\rightarrow \mathscr{H}^{n+1}(F))\rightarrow 0.\end{eqnarray}$$

◻

Proof. Any idempotent in ${\mathcal{O}}_{X}(X\smallsetminus Z)$ is power-bounded, so this is immediate from the previous theorem.◻

Proof. We immediately reduce to the case where $X$ is connected. Let $Z\subset X$ be as in the definition of a cover, and let $Y_{i}$ be any connected component of $Y$, so then $Y_{i}\cap \unicode[STIX]{x1D70B}^{-1}(Z)$ is closed and nowhere-dense in $Y_{i}$ and $Y_{i}\smallsetminus Y_{i}\cap \unicode[STIX]{x1D70B}^{-1}(Z)\rightarrow X\smallsetminus Z$ is finite étale. Then

$$\begin{eqnarray}\text{im}(Y_{i}\smallsetminus Y_{i}\cap \unicode[STIX]{x1D70B}^{-1}(Z)\rightarrow X\smallsetminus Z)\end{eqnarray}$$

is a nonempty open and closed subset of $X\smallsetminus Z$, so it coincides with $X\smallsetminus Z$ by the previous corollary. In particular, $\unicode[STIX]{x1D70B}(Y_{i})$ contains a dense subset of $X$. On the other hand, $\unicode[STIX]{x1D70B}(Y_{i})$ is a closed analytic subset of $X$ since $\unicode[STIX]{x1D70B}$ is finite. Therefore, $\unicode[STIX]{x1D70B}(Y_{i})=X$.

For the second claim, note that if $V$ is a closed analytic subset of a connected normal space $X$, then $V\subsetneq X$ if and only if $V$ is nowhere-dense if and only if $\text{dim}\,V<\text{dim}\,X$. Since

$$\begin{eqnarray}\text{dim}\,\unicode[STIX]{x1D70B}^{-1}(V)\cap Y_{i}=\text{dim}\,V<\text{dim}\,X=\text{dim}\,Y_{i}\end{eqnarray}$$

for any irreducible component $Y_{i}$ of $Y$, this gives the claim.◻

Proof. If $\unicode[STIX]{x1D70B}:Y\rightarrow X$ is any cover and $U\subset X$ is any open affinoid, then $\unicode[STIX]{x1D70B}^{-1}(U)$ is affinoid as well, and $\unicode[STIX]{x1D70B}^{-1}(Z\cap U)$ is nowhere-dense in $U$ by the previous proposition. But then ${\mathcal{O}}_{Y}(\unicode[STIX]{x1D70B}^{-1}(U))\cong {\mathcal{O}}_{Y}^{+}(\unicode[STIX]{x1D70B}^{-1}(U\smallsetminus U\cap Z))[\frac{1}{\unicode[STIX]{x1D71B}}]$ by Theorem 2.6, so ${\mathcal{O}}_{Y}(U)$ only depends on $Y\times _{X}(X\smallsetminus Z)$. This immediately gives the result.◻

Proof. This follows from a careful reading of [Reference KiehlKie67b, Theorem 1.18] (cf. also [Reference MitsuiMit09, Theorem 2.11]). ◻

Proof of Theorem 2.10.

We can assume that $X=\text{Spa}\,A$ is an affinoid rigid space, so $Z=\text{Spa}\,B$ is also affinoid, and we get a corresponding closed immersion of schemes ${\mathcal{Z}}=\text{Spec}\,B\rightarrow {\mathcal{X}}=\text{Spec}\,A$. These are quasi-excellent schemes over $\mathbf{Q}$, so, according to [Reference TemkinTem17, Theorem 1.11], we can find a projective birational morphism $f:{\mathcal{X}}^{\prime }\rightarrow {\mathcal{X}}$ such that ${\mathcal{X}}^{\prime }$ is regular and $f^{-1}({\mathcal{Z}})^{\text{red}}$ is a strict normal crossings divisor, and such that $f$ is an isomorphism away from ${\mathcal{Z}}\cup {\mathcal{X}}^{\text{sing}}$. Analytifying, we get a proper morphism of rigid spaces $g:X^{\prime }\rightarrow X$ with $X^{\prime }$ smooth such that $g^{-1}(Z)^{\text{red}}$ is a strict normal crossings divisor.

Suppose now that we are given a finite étale cover $Y\rightarrow X\smallsetminus Z$. Base changing along $g$, we get a finite étale cover of $X^{\prime }\smallsetminus g^{-1}(Z)$, which then extends to a cover $h:Y^{\prime }\rightarrow X^{\prime }$ by Theorem 2.11. Now, since $g\circ h$ is proper, the sheaf $(g\circ h)_{\ast }{\mathcal{O}}_{Y^{\prime }}$ defines a sheaf of coherent ${\mathcal{O}}_{X}$-algebras by [Reference KiehlKie67a]. Taking the normalization of the affinoid space associated with the global sections of this sheaf, we get a normal affinoid $Y^{\prime \prime }$ together with a finite map $Y^{\prime \prime }\rightarrow X$ and a canonical isomorphism $Y^{\prime \prime }|_{(X\smallsetminus Z)^{\text{sm}}}\cong Y|_{(X\smallsetminus Z)^{\text{sm}}}$. The cover ${\tilde{Y}}\rightarrow X$ we seek can then be defined as the Zariski closure of $Y^{\prime \prime }|_{(X\smallsetminus Z)^{\text{sm}}}$ in $Y^{\prime \prime }$; note that this is just a union of irreducible components of $Y^{\prime \prime }$, so it is still normal, and it is easy to check that ${\tilde{Y}}$ is a cover of $X$. Finally, since ${\tilde{Y}}$ and $Y$ are canonically isomorphic after restriction to $(X\smallsetminus Z)^{\text{sm}}$, the full faithfulness argument shows that this isomorphism extends to an isomorphism ${\tilde{Y}}|_{X\smallsetminus Z}\cong Y$, since $(X\smallsetminus Z)^{\text{sm}}$ is a dense Zariski-open subset of $X\smallsetminus Z$. This concludes the proof.◻

Proof. Apply Theorem 2.10 with $Z=V\cup W$ to construct ${\tilde{Y}}\rightarrow X$ extending

$$\begin{eqnarray}Y\times _{(X\smallsetminus V)}(X\smallsetminus V\cup W)\rightarrow X\smallsetminus V\cup W\end{eqnarray}$$

and then use full faithfulness to deduce that ${\tilde{Y}}|_{X\smallsetminus V}\cong Y$.◻

Proof. The essential point is to construct the functor $U_{\text{f}\acute{\text{e}}\text{t}}\mapsto {\mathcal{U}}_{\text{f}\acute{\text{e}}\text{t}}$. For this, let $Y\rightarrow U$ be any finite étale cover. Letting $S^{\unicode[STIX]{x1D708}}$ and ${\mathcal{S}}^{\unicode[STIX]{x1D708}}$ denote the normalizations of $S$ and ${\mathcal{S}}$, the map $U\rightarrow S$ factors canonically over a dense Zariski-open immersion $U\rightarrow S^{\unicode[STIX]{x1D708}}$, and similarly for ${\mathcal{U}}$. By Theorem 2.10, $Y\rightarrow U$ extends to a cover ${\tilde{Y}}\rightarrow S^{\unicode[STIX]{x1D708}}$. Since ${\tilde{Y}}\rightarrow S^{\unicode[STIX]{x1D708}}$ is finite, ${\tilde{Y}}=\text{Spa}\,B$ for $B$ some module-finite $A^{\unicode[STIX]{x1D708}}$-algebra. Then ${\mathcal{Y}}=\text{Spec}\,B\times _{{\mathcal{S}}^{\unicode[STIX]{x1D708}}}{\mathcal{U}}\rightarrow {\mathcal{U}}$ is the desired algebraization of $Y\rightarrow U$.◻

Proof of Theorem 1.3.

Fix a nonarchimedean field $K$ and a coefficient ring $\unicode[STIX]{x1D6EC}=\mathbf{Z}/n\mathbf{Z}$ as in the theorem. In what follows, ‘sheaf’ is shorthand for ‘étale sheaf of $\unicode[STIX]{x1D6EC}$-modules’. For nonnegative integers $d,i$, consider the following statement.

Statement ${\mathcal{T}}_{d,i}$: ‘For all $K$-affinoids $X$ of dimension ${\leqslant}d$, all Zariski-constructible sheaves $\mathscr{F}$ on $X$, and all integers $j>i$, we have $H_{\acute{\text{e}}\text{t}}^{j}(X_{\widehat{\overline{K}}},\mathscr{F})=0$’.

We are trying to prove that ${\mathcal{T}}_{d,d}$ is true for all $d\geqslant 0$. The idea is to argue by ascending induction on $d$ and descending induction on $i$. More precisely, it clearly suffices to assume the truth of ${\mathcal{T}}_{d-1,d-1}$ and then show that ${\mathcal{T}}_{d,i+1}$ implies ${\mathcal{T}}_{d,i}$ for any $i\geqslant d$; as noted in the introduction, ${\mathcal{T}}_{d,2d}$ is true for any $d\geqslant 0$, which gives a starting place for the descending induction.

We break the details into several steps.

Letting $i:Z\rightarrow X$ denote the closed complement, this is immediate by looking at the long exact sequence

$$\begin{eqnarray}\cdots \rightarrow H_{\acute{\text{e}}\text{t}}^{i-1}(Z_{\widehat{\overline{K}}},i^{\ast }\mathscr{F})\rightarrow H_{\acute{\text{e}}\text{t}}^{i}(X_{\widehat{\overline{K}}},j_{!}j^{\ast }\mathscr{F})\rightarrow H_{\acute{\text{e}}\text{t}}^{i}(X_{\widehat{\overline{K}}},\mathscr{F})\rightarrow H_{\acute{\text{e}}\text{t}}^{i}(Z_{\widehat{\overline{K}}},i^{\ast }\mathscr{F})\rightarrow \cdots\end{eqnarray}$$

associated with the short exact sequence

$$\begin{eqnarray}0\rightarrow j_{!}j^{\ast }\mathscr{F}\rightarrow \mathscr{F}\rightarrow i_{\ast }i^{\ast }\mathscr{F}\rightarrow 0\end{eqnarray}$$

and then applying ${\mathcal{T}}_{d-1,d-1}$ to control the outer terms.

One direction is trivial. For the other direction, note that by Noether normalization for affinoids [Reference Bosch, Güntzer and RemmertBGR84, Corollary 6.1.2/2], any $d$-dimensional affinoid $X$ admits a finite map

$$\begin{eqnarray}\unicode[STIX]{x1D70F}:X\rightarrow \mathbf{B}^{d}=\text{Spa}\,K\langle T_{1},\ldots ,T_{d}\rangle ,\end{eqnarray}$$

and $\unicode[STIX]{x1D70F}_{\ast }=R\unicode[STIX]{x1D70F}_{\ast }$ preserves Zariski-constructibility by Proposition 2.3.

To prove this, suppose that we are given $X,\,U$, and $\mathscr{H}$ as in the statement. By definition, we can find a finite étale cover $\unicode[STIX]{x1D70B}:Y\rightarrow U$ such that $\unicode[STIX]{x1D70B}^{\ast }\mathscr{H}$ is constant, i.e. such that there exists a surjection $\unicode[STIX]{x1D6EC}_{Y}^{n}\rightarrow \unicode[STIX]{x1D70B}^{\ast }\mathscr{H}$ for some $n$; fix such a surjection. This is adjoint to a surjection $\unicode[STIX]{x1D70B}_{!}(\unicode[STIX]{x1D6EC}_{Y}^{n})\rightarrow \mathscr{H}$,Footnote ⁷ which extends by zero to a surjection $s:j_{!}\unicode[STIX]{x1D70B}_{!}(\unicode[STIX]{x1D6EC}_{Y}^{n})\rightarrow j_{!}\mathscr{H}$. We claim that the sheaf $\mathscr{G}=j_{!}\unicode[STIX]{x1D70B}_{!}(\unicode[STIX]{x1D6EC}_{Y}^{n})$ has the required properties. Zariski-constructibility is clear from the identification $\unicode[STIX]{x1D70B}_{!}=\unicode[STIX]{x1D70B}_{\ast }$ and Proposition 2.3. For the vanishing statement, we apply Theorem 1.6 to extend $Y\rightarrow U$ to a cover $\tilde{\unicode[STIX]{x1D70B}}:{\tilde{Y}}\rightarrow X$ sitting in a cartesian diagram

where ${\tilde{Y}}$ is a normal affinoid and $h$ is a dense Zariski-open immersion. By proper base change and the finiteness of $\tilde{\unicode[STIX]{x1D70B}}$, we get isomorphisms

$$\begin{eqnarray}\mathscr{G}=j_{!}\unicode[STIX]{x1D70B}_{!}(\unicode[STIX]{x1D6EC}_{Y}^{n})\cong \tilde{\unicode[STIX]{x1D70B}}_{!}h_{!}(\unicode[STIX]{x1D6EC}_{Y}^{n})\cong \tilde{\unicode[STIX]{x1D70B}}_{\ast }h_{!}(\unicode[STIX]{x1D6EC}_{Y}^{n}),\end{eqnarray}$$

so $H_{\acute{\text{e}}\text{t}}^{i}(X_{\widehat{\overline{K}}},\mathscr{G})\cong H_{\acute{\text{e}}\text{t}}^{i}({\tilde{Y}}_{\widehat{\overline{K}}},h_{!}\unicode[STIX]{x1D6EC}_{Y})^{\oplus n}$. $$Now, writing $i:V\rightarrow {\tilde{Y}}$ for the closed complement of $Y$, we get exact sequences

$$\begin{eqnarray}H_{\acute{\text{e}}\text{t}}^{i-1}(V_{\widehat{\overline{K}}},\unicode[STIX]{x1D6EC}_{V})\rightarrow H_{\acute{\text{e}}\text{t}}^{i}({\tilde{Y}}_{\widehat{\overline{K}}},h_{!}\unicode[STIX]{x1D6EC}_{Y})\rightarrow H_{\acute{\text{e}}\text{t}}^{i}({\tilde{Y}}_{\widehat{\overline{K}}},\unicode[STIX]{x1D6EC}_{{\tilde{Y}}})\end{eqnarray}$$

for all $i$. Examining this sequence for any fixed $i>d$, we see that the rightmost term vanishes by Theorem 1.4, while the leftmost term vanishes by the assumption that ${\mathcal{T}}_{d-1,d-1}$ holds.Footnote ⁸ Therefore, $H_{\acute{\text{e}}\text{t}}^{i}({\tilde{Y}}_{\widehat{\overline{K}}},h_{!}\unicode[STIX]{x1D6EC}_{Y})=0$ for $i>d$, as desired.

Fix $d$ and $i\geqslant d$ as in the statement, and assume that ${\mathcal{T}}_{d,i+1}$ is true. Let $X$ be a $d$-dimensional affinoid and let $\mathscr{F}$ be a Zariski-constructible sheaf on $X$. We need to show that $H_{\acute{\text{e}}\text{t}}^{i+1}(X_{\widehat{\overline{K}}},\mathscr{F})=0$. By Step two, we can assume that $X$ is normal (or even that $X$ is the $d$-dimensional affinoid ball). By Step one, it suffices to show that $H_{\acute{\text{e}}\text{t}}^{i+1}(X_{\widehat{\overline{K}}},j_{!}j^{\ast }\mathscr{F})=0$, where $j:U\rightarrow X$ is the inclusion of any dense Zariski-open subset. Fix a choice of such a $U$ with the property that $j^{\ast }\mathscr{F}$ is locally constant. By Step three, we can choose a Zariski-constructible sheaf $\mathscr{G}$ on $X$ and a surjection $s:\mathscr{G}\rightarrow j_{!}j^{\ast }\mathscr{F}$ such that $H_{\acute{\text{e}}\text{t}}^{n}(X_{\widehat{\overline{K}}},\mathscr{G})=0$ for all $n>d$. By Proposition 2.4, the sheaf $\mathscr{K}=\ker s$ is Zariski-constructible. Now, looking at the exact sequence

$$\begin{eqnarray}H_{\acute{\text{e}}\text{t}}^{i+1}(X_{\widehat{\overline{K}}},\mathscr{G})\rightarrow H_{\acute{\text{e}}\text{t}}^{i+1}(X_{\widehat{\overline{K}}},j_{!}j^{\ast }\mathscr{F})\rightarrow H_{\acute{\text{e}}\text{t}}^{i+2}(X_{\widehat{\overline{K}}},\mathscr{K}),\end{eqnarray}$$

we see that the leftmost term vanishes by the construction of $\mathscr{G}$, while the rightmost term vanishes by the induction hypothesis. Therefore,

$$\begin{eqnarray}H_{\acute{\text{e}}\text{t}}^{i+1}(X_{\widehat{\overline{K}}},j_{!}j^{\ast }\mathscr{F})=0,\end{eqnarray}$$

as desired. ◻

Proof. By Noether normalization for affinoids and [Reference Bosch, Güntzer and RemmertBGR84, Corollary 6.4.1/6], $A^{\circ }$ can be realized as a module-finite integral extension of ${\mathcal{O}}_{K}\langle T_{1},\ldots ,T_{n}\rangle$ with $n=\text{dim}\,A$. By a result of Valabrega (cf. [Reference ValabregaVal75, Proposition 7] and [Reference ValabregaVal76, Theorem 9]), the convergent power series ring ${\mathcal{O}}_{K}\langle T_{1},\ldots ,T_{n}\rangle$ is excellent for any complete discrete valuation ring ${\mathcal{O}}_{K}$. Since excellence propagates along finite-type ring maps and localizations, cf. [Sta17, Tag 07QU], we see that $A^{\circ }$ and any localization thereof are excellent. Now, by a result of Greco [Reference GrecoGre76, Corollary 5.6(iii)], the strict Henselization of any excellent local ring is excellent, which gives what we want.◻

Proof. This is a special case of Gabber’s affine Lefschetz theorem for quasi-excellent schemes; cf. [Reference Illusie, Laszlo and OrgogozoILO14, Corollaire XV.1.2.4]. ◻

Proof of Theorem 1.4.

Let $X=\text{Spa}\,A$ be a $K$-affinoid as in the theorem. After replacing $K$ by $\widehat{K^{\text{nr}}}$ and $X$ by $X_{\widehat{K^{\text{nr}}}}^{\text{red}}$, we can assume that $A$ is reduced and that $K$ has separably closed residue field $k$. By an easy induction we can also assume that $n=l$ is prime. For notational simplicity we give the remainder of the proof in the case where $\text{char}(k)=p>0$; the equal characteristic-zero case is only easier.

By e.g. [Reference BerkovichBer93, Corollary 2.4.6], $\text{Gal}_{\overline{K}/K}$ sits in a short exact sequence

$$\begin{eqnarray}1\rightarrow P\rightarrow \text{Gal}_{\overline{K}/K}\rightarrow T\simeq \mathop{\prod }_{q\neq p}\mathbf{Z}_{q}\rightarrow 1,\end{eqnarray}$$

where $P$ is pro-$p$. In particular, if $L\subset \overline{K}$ is any finite extension of $K$, then

$$\begin{eqnarray}H^{i}(\text{Gal}_{\overline{K}/L},\mathbf{Z}/l\mathbf{Z})\simeq \left\{\begin{array}{@{}ll@{}}\mathbf{Z}/l\mathbf{Z}\quad \quad & \text{if}\,i=0,1,\\ 0\quad \quad & \text{if}\,i>1.\end{array}\right.\end{eqnarray}$$

For any such $L$, look at the Cartan–Leray spectral sequence

$$\begin{eqnarray}E_{2}^{i,j}=H^{i}(\text{Gal}_{\overline{K}/L},H_{\acute{\text{e}}\text{t}}^{j}(X_{\widehat{\overline{K}}},\mathbf{Z}/l\mathbf{Z}))\Rightarrow H_{\acute{\text{e}}\text{t}}^{i+j}(X_{L},\mathbf{Z}/l\mathbf{Z}).\end{eqnarray}$$

The group $\bigoplus _{j\geqslant 0}H_{\acute{\text{e}}\text{t}}^{j}(X_{\widehat{\overline{K}}},\mathbf{Z}/l\mathbf{Z})$ is finite by vanishing in degrees ${>}2\,\text{dim}\,X$ together with [Reference BerkovichBer15, Theorem 1.1.1], so all the Galois actions in the $E_{2}$-page are trivial for any large enough $L$,Footnote ⁹ in which case we can rewrite the spectral sequence as

$$\begin{eqnarray}E_{2}^{i,j}=H^{i}(\text{Gal}_{\overline{K}/L},\mathbf{Z}/l\mathbf{Z})\otimes H_{\acute{\text{e}}\text{t}}^{j}(X_{\widehat{\overline{K}}},\mathbf{Z}/l\mathbf{Z})\Rightarrow H_{\acute{\text{e}}\text{t}}^{i+j}(X_{L},\mathbf{Z}/l\mathbf{Z}).\end{eqnarray}$$

Now if $d$ is the largest integer such that $H_{\acute{\text{e}}\text{t}}^{d}(X_{\widehat{\overline{K}}},\mathbf{Z}/l\mathbf{Z})\neq 0$, then the $E_{2}^{1,d}$ term survives the spectral sequence, so $H_{\acute{\text{e}}\text{t}}^{d+1}(X_{L},\mathbf{Z}/l\mathbf{Z})\neq 0$. Since the residue field of $L$ is still separably closed, it thus suffices to prove the following statement.

1. (†) For any reduced affinoid $X=\text{Spa}\,A$ over a complete discretely valued nonarchimedean field $K$ with separably closed residue field $k$ of characteristic $p>0$, we have $H_{\acute{\text{e}}\text{t}}^{i}(X,\mathbf{Z}/l\mathbf{Z})=0$ for all $i>1+\text{dim}\,X$ and all primes $l\neq p$.

Fix a uniformizer $\unicode[STIX]{x1D71B}\in {\mathcal{O}}_{K}$. Set ${\mathcal{X}}=\text{Spec}\,A^{\circ }$ and ${\mathcal{X}}_{s}=\text{Spec}\,A^{\circ }/\unicode[STIX]{x1D71B}$, so ${\mathcal{X}}_{s}$ is an affine variety over $k$. As in [Reference HuberHub96, § 3.5] or [Reference BerkovichBer94], there is a natural map of sites $\unicode[STIX]{x1D706}:X_{\acute{\text{e}}\text{t}}\rightarrow {\mathcal{X}}_{s,\acute{\text{e}}\text{t}}$, corresponding to the natural functor

$$\begin{eqnarray}\displaystyle {\mathcal{X}}_{s,\acute{\text{e}}\text{t}} & \rightarrow & \displaystyle X_{\acute{\text{e}}\text{t}}\nonumber\\ \displaystyle {\mathcal{U}}/{\mathcal{X}}_{s} & \mapsto & \displaystyle \unicode[STIX]{x1D702}({\mathcal{U}})/X\nonumber\end{eqnarray}$$

given by (uniquely) deforming an étale map ${\mathcal{U}}\rightarrow {\mathcal{X}}_{s}$ to a $\unicode[STIX]{x1D71B}$-adic formal scheme étale over $\text{Spf}\,A^{\circ }$ and then passing to rigid generic fibers. (We follow Huber’s notation in writing $\unicode[STIX]{x1D706}$ – Berkovich denotes this map by $\unicode[STIX]{x1D6E9}$.) For any abelian étale sheaf $\mathscr{F}$ on $X$, the derived pushforward along $\unicode[STIX]{x1D706}$ gives rise to the so-called nearby cycle sheaves $R^{j}\unicode[STIX]{x1D706}_{\ast }\mathscr{F}$ on ${\mathcal{X}}_{s,\acute{\text{e}}\text{t}}$, which can be calculated as the sheafifications of the presheaves ${\mathcal{U}}\mapsto H_{\acute{\text{e}}\text{t}}^{j}(\unicode[STIX]{x1D702}({\mathcal{U}}),\mathscr{F})$, and there is a spectral sequence

$$\begin{eqnarray}H_{\acute{\text{e}}\text{t}}^{i}({\mathcal{X}}_{s},R^{j}\unicode[STIX]{x1D706}_{\ast }\mathscr{F})\Rightarrow H_{\acute{\text{e}}\text{t}}^{i+j}(X,\mathscr{F});\end{eqnarray}$$

cf. [Reference BerkovichBer94, Proposition 4.1 and Corollary 4.2(iii)]. Taking $\mathscr{F}=\mathbf{Z}/l\mathbf{Z}$, we see that to prove $(\dagger )$ it is enough to show that $H_{\acute{\text{e}}\text{t}}^{i}({\mathcal{X}}_{s},R^{j}\unicode[STIX]{x1D706}_{\ast }\mathbf{Z}/l\mathbf{Z})=0$ for any $j\geqslant 0$ and any $i>1+\text{dim}\,X-j$. By the strong form of the Artin–Grothendieck vanishing theorem recalled above, we are reduced to proving that if $x\in {\mathcal{X}}_{s}$ is any point such that

$$\begin{eqnarray}(R^{j}\unicode[STIX]{x1D706}_{\ast }\mathbf{Z}/l\mathbf{Z})_{\overline{x}}\neq 0,\end{eqnarray}$$

then $\text{tr.deg}\,k(x)/k\leqslant 1+\text{dim}\,X-j$.

We check this by a direct computation. So, let $x\in {\mathcal{X}}_{s}$ be any point and let $\mathfrak{p}_{x}\subset A^{\circ }$ be the associated prime ideal. Crucially, we have a ‘purely algebraic’ description of the stalk $(R^{j}\unicode[STIX]{x1D706}_{\ast }\mathbf{Z}/l\mathbf{Z})_{\overline{x}}$: letting ${\mathcal{O}}_{{\mathcal{X}},\overline{x}}$ denote the strict Henselization of ${\mathcal{O}}_{{\mathcal{X}},x}=(A^{\circ })_{\mathfrak{p}_{x}}$ as usual, then

(*)$$\begin{eqnarray}(R^{j}\unicode[STIX]{x1D706}_{\ast }\mathbf{Z}/l\mathbf{Z})_{\overline{x}}\cong H_{\acute{\text{e}}\text{t}}^{j}(\text{Spec}\,{\mathcal{O}}_{{\mathcal{X}},\overline{x}}[{\textstyle \frac{1}{\unicode[STIX]{x1D71B}}}],\mathbf{Z}/l\mathbf{Z}).\end{eqnarray}$$

This is a special case of [Reference HuberHub96, Theorem 3.5.10], and it is remarkable that we have a description like this which does not involve taking some completion. By Lemma 3.1, ${\mathcal{O}}_{{\mathcal{X}},\overline{x}}$ is excellent, so Theorem 3.2 implies that $H_{\acute{\text{e}}\text{t}}^{j}(U,\mathbf{Z}/l\mathbf{Z})=0$ for any open affine subscheme $U\subset \text{Spec}\,{\mathcal{O}}_{{\mathcal{X}},\overline{x}}$ and any $j>\text{dim}\,{\mathcal{O}}_{{\mathcal{X}},\overline{x}}$. In particular, taking $U=\text{Spec}\,{\mathcal{O}}_{{\mathcal{X}},\overline{x}}[\frac{1}{\unicode[STIX]{x1D71B}}]$ and applying Huber’s formula ($\ast$) above, we see that if $j$ is an integer such that $(R^{j}\unicode[STIX]{x1D706}_{\ast }\mathbf{Z}/l\mathbf{Z})_{\overline{x}}\neq 0$, then necessarily

$$\begin{eqnarray}j\leqslant \text{dim}\,{\mathcal{O}}_{{\mathcal{X}},\overline{x}}=\text{dim}\,{\mathcal{O}}_{{\mathcal{X}},x}=\text{ht}\,\mathfrak{p}_{x},\end{eqnarray}$$

where the first equality follows from e.g. [Sta17, Tag 06LK]. Writing $R=A^{\circ }/\unicode[STIX]{x1D71B}$ and $\overline{\mathfrak{p}_{x}}=\mathfrak{p}_{x}/\unicode[STIX]{x1D71B}\subset R$, we then have

$$\begin{eqnarray}\displaystyle j+\text{tr.deg}\,k(x)/k & {\leqslant} & \displaystyle \text{ht}\,\mathfrak{p}_{x}+\text{tr.deg}\,k(x)/k\nonumber\\ \displaystyle & = & \displaystyle 1+\text{ht}\,\overline{\mathfrak{p}_{x}}+\text{tr}.\text{deg}\,k(x)/k\nonumber\\ \displaystyle & = & \displaystyle 1+\text{dim}\,R_{\overline{\mathfrak{p}_{x}}}+\text{dim}\,R/\overline{\mathfrak{p}_{x}}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle 1+\text{dim}\,R\nonumber\\ \displaystyle & {\leqslant} & \displaystyle 1+\text{dim}\,X.\nonumber\end{eqnarray}$$

Here the second line follows from the fact that $\unicode[STIX]{x1D71B}\in \mathfrak{p}_{x}$ is part of a system of parameters of $(A^{\circ })_{\mathfrak{p}_{x}}$, so $\text{ht}\,\overline{\mathfrak{p}_{x}}=\text{dim}\,R_{\overline{\mathfrak{p}_{x}}}=\text{dim}\,(A^{\circ })_{\mathfrak{p}_{x}}-1$; the third line is immediate from the equality $\text{tr}.\text{deg}\,k(x)/k=\text{dim}\,R/\overline{\mathfrak{p}_{x}}$, which is a standard fact about domains of finite type over a field; the fourth line is trivial; and the fifth line follows from the fact that $R$ is module-finite over $k[T_{1},\ldots ,T_{n}]$ with $n=\text{dim}\,X$. But then

$$\begin{eqnarray}\text{tr.deg}\,k(x)/k\leqslant 1+\text{dim}\,X-j,\end{eqnarray}$$

as desired. ◻

Proof. By [Reference HuberHub96, Lemma 3.9.2], we have a short exact sequence

$$\begin{eqnarray}0\rightarrow \underset{\leftarrow n}{\text{lim}^{1}}H_{\acute{\text{e}}\text{t}}^{i-1}(U_{n,\widehat{\overline{K}}},\mathscr{F})\rightarrow H_{\acute{\text{e}}\text{t}}^{i}(X_{\widehat{\overline{K}}},\mathscr{F})\rightarrow \underset{\leftarrow n}{\text{lim}}\,H_{\acute{\text{e}}\text{t}}^{i}(U_{n,\widehat{\overline{K}}},\mathscr{F})\rightarrow 0.\end{eqnarray}$$

But the groups $H_{\acute{\text{e}}\text{t}}^{j}(U_{i,\widehat{\overline{K}}},\mathscr{F})$ are finite, so the $\text{lim}^{1}$ term vanishes, and the result now follows from Theorem 1.3.◻

Proof of Theorem 1.9.

First, observe that all functors involved in the statement of the theorem commute with filtered colimits: for $H_{\acute{\text{e}}\text{t}}^{n}({\mathcal{S}},-)$ this is standard, for $H_{\acute{\text{e}}\text{t}}^{n}(S,-)$ this follows from [Reference HuberHub96, Lemma 2.3.13], and for $\unicode[STIX]{x1D707}_{S}^{\ast }$ it is trivial (because $\unicode[STIX]{x1D707}_{S}^{\ast }$ is a left adjoint). Writing $\mathscr{F}$ as the filtered colimit of its $m$-torsion subsheaves, we therefore reduce to the case where $\mathscr{F}$ is killed by some integer $m\geqslant 1$. Since ${\mathcal{S}}$ is qcqs, we can write any sheaf of $\mathbf{Z}/m\mathbf{Z}$-modules on ${\mathcal{S}}_{\acute{\text{e}}\text{t}}$ as a filtered colimit of constructible sheaves of $\mathbf{Z}/m\mathbf{Z}$-modules, cf. [Sta17, Tag 03SA], which reduces us further to the case where $\mathscr{F}$ is a constructible sheaf of $\mathbf{Z}/m\mathbf{Z}$-modules.

Next, by repeated application of [Reference Artin, Grothendieck and VerdierSGA4, Proposition IX.2.14(ii)], we may choose a resolution

$$\begin{eqnarray}\mathscr{F}\simeq [\mathscr{G}^{0}\rightarrow \mathscr{G}^{1}\rightarrow \mathscr{G}^{2}\rightarrow \cdots \,],\end{eqnarray}$$

where each sheaf $\mathscr{G}^{i}$ is isomorphic to a finite direct sum of sheaves of the form $\unicode[STIX]{x1D70B}_{\ast }\text{}\underline{M}$, where $\unicode[STIX]{x1D70B}:{\mathcal{S}}^{\prime }\rightarrow {\mathcal{S}}$ is a finite morphism and $M$ is a finite $\mathbf{Z}/n\mathbf{Z}$-module. By Lemma 4.1, it then suffices to prove that $H_{\acute{\text{e}}\text{t}}^{n}({\mathcal{S}},\unicode[STIX]{x1D70B}_{\ast }\text{}\underline{M})\rightarrow H_{\acute{\text{e}}\text{t}}^{n}(S,\unicode[STIX]{x1D707}_{S}^{\ast }\unicode[STIX]{x1D70B}_{\ast }\text{}\underline{M})$ is an isomorphism for any such $\unicode[STIX]{x1D70B}$ and $M$, and any $n$. Using the exactness of $\unicode[STIX]{x1D70B}_{\ast }$ and $\unicode[STIX]{x1D70B}_{\ast }^{\text{an}}$ and their compatibility with $\unicode[STIX]{x1D707}$-pullback, one immediately reduces further to the case of constant sheaves, which is exactly [Reference HuberHub96, Corollary 3.2.3].◻

Proof. Filter $\mathscr{F}^{\bullet }$ by the subcomplexes $F^{i}\mathscr{F}^{\bullet }=[\mathscr{F}^{i}\rightarrow \mathscr{F}^{i+1}\rightarrow \cdots \,]$. By [Sta17, 0BKI], we can choose a filtered complex $\mathscr{G}^{\bullet }$ of abelian sheaves on ${\mathcal{C}}$ together with a filtered quasi-isomorphism $\unicode[STIX]{x1D6FC}:\mathscr{F}^{\bullet }\rightarrow \mathscr{G}^{\bullet }$ such that (for all $j$) $\mathscr{G}^{\bullet }$ and $\text{gr}^{j}\mathscr{G}^{\bullet }$ are K-injective complexes of injective sheaves and

$$\begin{eqnarray}\text{gr}^{j}\unicode[STIX]{x1D6FC}:\text{gr}^{j}\mathscr{F}^{\bullet }=\mathscr{F}^{j}[-j]\rightarrow \text{gr}^{j}\mathscr{G}^{\bullet }\end{eqnarray}$$

is a quasi-isomorphism. Unwinding definitions, one checks that the upper spectral sequence in question coincides with the spectral sequence associated with the filtered complex $\unicode[STIX]{x1D6E4}({\mathcal{C}},\mathscr{G}^{\bullet })$.

Applying [Sta17, 0BKI] again, we can choose a filtered complex $\mathscr{H}^{\bullet }$ of abelian sheaves on ${\mathcal{D}}$ together with a filtered quasi-isomorphism $\unicode[STIX]{x1D6FD}:f^{-1}\mathscr{G}^{\bullet }\rightarrow \mathscr{H}^{\bullet }$ such that (for all $j$) $\mathscr{H}^{\bullet }$ and $\text{gr}^{j}\mathscr{H}^{\bullet }$ are K-injective complexes of injective sheaves and $\text{gr}^{j}\unicode[STIX]{x1D6FD}$ is a quasi-isomorphism. Again, one checks directly that the lower spectral sequence identifies with the spectral sequence associated with the filtered complex $\unicode[STIX]{x1D6E4}({\mathcal{D}},\mathscr{H}^{\bullet })$. By construction, we have filtered maps of filtered complexes

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}({\mathcal{C}},\mathscr{G}^{\bullet })\rightarrow \unicode[STIX]{x1D6E4}({\mathcal{D}},f^{-1}\mathscr{G}^{\bullet })\rightarrow \unicode[STIX]{x1D6E4}({\mathcal{D}},\mathscr{H}^{\bullet }),\end{eqnarray}$$

where the left-hand map exists by general nonsense and the right-hand map is induced by $\unicode[STIX]{x1D6FD}$. The composition of these maps induces the desired map of spectral sequences.◻

Proof. Cover ${\mathcal{X}}$ by open affines ${\mathcal{X}}_{i},0\leqslant i\leqslant n$; for any $0\leqslant i_{1}<\cdots <i_{p}\leqslant n$, set ${\mathcal{X}}_{i_{1}<\cdots <i_{p}}=\bigcap _{1\leqslant j\leqslant p}{\mathcal{X}}_{i_{j}}$, so each ${\mathcal{X}}_{i_{1}<\cdots <i_{p}}$ is a separated open subscheme of ${\mathcal{X}}$. Let $f_{i_{1}<\cdots <i_{p}}:{\mathcal{X}}_{i_{1}<\cdots <i_{p}}\rightarrow {\mathcal{Y}}$ be the evident morphism. Looking at the Čech spectral sequence

$$\begin{eqnarray}E_{1}^{p,q}=\bigoplus _{i_{1}<\cdots <i_{p}}R^{q}(f_{i_{1}<\cdots <i_{p}})_{\ast }F|_{{\mathcal{X}}_{i_{1}<\cdots <i_{p}}}\Rightarrow R^{p+q}f_{\ast }F,\end{eqnarray}$$

it suffices to show that each $R(f_{i_{1}<\cdots <i_{p}})_{\ast }$ sends $D_{c}^{b}({\mathcal{X}}_{i_{1}<\cdots <i_{p}},\unicode[STIX]{x1D6EC})$ into $D_{c}^{b}({\mathcal{Y}},\unicode[STIX]{x1D6EC})$. In particular, it suffices to consider the case where ${\mathcal{X}}$ is separated. Since $f$ is then separated, by Nagata’s compactification theorem we may choose a factorization $f=\overline{f}\circ j$, where $\overline{f}:{\mathcal{X}}^{\prime }\rightarrow {\mathcal{Y}}$ is proper and $j:{\mathcal{X}}\rightarrow {\mathcal{X}}^{\prime }$ is an open immersion. Combining [Reference Illusie, Laszlo and OrgogozoILO14, Theorem XIII.1.1.1] and [Reference Illusie, Laszlo and OrgogozoILO14, Proposition XVII.7.6.7], we see that $Rj_{\ast }$ sends $D_{c}^{b}({\mathcal{X}},\unicode[STIX]{x1D6EC})$ into $D_{c}^{b}({\mathcal{X}}^{\prime },\unicode[STIX]{x1D6EC})$. By [Reference Artin, Grothendieck and VerdierSGA4, Theorem XIV.1.1], $R\overline{f}_{\ast }$ sends $D_{c}^{b}({\mathcal{X}}^{\prime },\unicode[STIX]{x1D6EC})$ into $D_{c}^{b}({\mathcal{Y}},\unicode[STIX]{x1D6EC})$. Since $Rf_{\ast }\cong R\overline{f}_{\ast }Rj_{\ast }$, this gives the result.◻

Proof. Part (i) is a special case of the composability of base change maps; cf. [Sta17, Tag 0E46]. For (ii), consider the composition

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{Z}^{\ast }R(g\circ h)_{\ast }F\cong \unicode[STIX]{x1D707}_{Z}^{\ast }Rg_{\ast }Rh_{\ast }F\rightarrow Rg_{\ast }^{\text{an}}\unicode[STIX]{x1D707}_{Y}^{\ast }Rh_{\ast }F\rightarrow Rg_{\ast }^{\text{an}}Rh_{\ast }^{\text{an}}\unicode[STIX]{x1D707}_{X}^{\ast }F\cong R(g\circ h)_{\ast }^{\text{an}}\unicode[STIX]{x1D707}_{X}^{\ast }F,\end{eqnarray}$$

where the first arrow is $\text{comp}(g,Rh_{\ast }F)$ and the second arrow is $Rg_{\ast }^{\text{an}}\text{comp}(h,F)$. By the composability of base change maps, this composition is the comparison map $\text{comp}(g\circ h,F)$. By our assumptions, the second arrow and the composition of both arrows are isomorphisms. We then conclude by observing that if $A\overset{s}{\rightarrow }B\overset{t}{\rightarrow }C$ is any pair of morphisms in any category such that $t$ and $t\circ s$ are isomorphisms, then $s$ is an isomorphism as well.

Part (iii) is an easy exercise, using the fact that the three maps $\text{comp}(h,F_{i})$ define a morphism between the evident distinguished triangles.◻

Proof. First, we show that if $\text{comp}(j)$ is an isomorphism for all open immersions, then $\text{comp}(f)$ is an isomorphism for all separated maps. For this, let $f:{\mathcal{X}}\rightarrow {\mathcal{Y}}$ be as in Theorem 1.8 and suppose that $f$ is separated. We can argue locally on ${\mathcal{Y}}$, so we can assume that ${\mathcal{Y}}$ is affine. By Nagata’s compactification theorem, we may choose a factorization $f=\overline{f}\circ j$, where $\overline{f}:{\mathcal{X}}^{\prime }\rightarrow {\mathcal{Y}}$ is proper and $j:{\mathcal{X}}\rightarrow {\mathcal{X}}^{\prime }$ is an open immersion. Then $\text{comp}(j)$ is an isomorphism by assumption, and $\text{comp}(\overline{f})$ is an isomorphism by Berkovich and Huber’s proof of the proper case. We then conclude by part (i) of the previous proposition.

For the general case, we can assume that ${\mathcal{Y}}$ is affine, so ${\mathcal{X}}$ is quasicompact. We now argue by induction on the minimal number $s({\mathcal{X}})$ of separated open subschemes required to cover ${\mathcal{X}}$. Choose an open cover ${\mathcal{X}}={\mathcal{U}}\cup {\mathcal{V}}$, where ${\mathcal{U}}$ is separated and $s({\mathcal{V}})=s({\mathcal{X}})-1$. Note that ${\mathcal{U}}\cap {\mathcal{V}}$ is separated. Let $j_{{\mathcal{U}}}:{\mathcal{U}}\rightarrow {\mathcal{X}}$, $j_{{\mathcal{V}}}:{\mathcal{V}}\rightarrow {\mathcal{X}}$, and $j_{{\mathcal{U}}\cap {\mathcal{V}}}:{\mathcal{U}}\cap {\mathcal{V}}\rightarrow {\mathcal{X}}$ be the evident open inclusions, so we get a functorial distinguished triangle

$$\begin{eqnarray}F\rightarrow Rj_{{\mathcal{U}}\ast }j_{{\mathcal{U}}}^{\ast }F\oplus Rj_{{\mathcal{V}}\ast }j_{{\mathcal{V}}}^{\ast }F\rightarrow Rj_{{\mathcal{U}}\cap {\mathcal{V}}\ast }j_{{\mathcal{U}}\cap {\mathcal{V}}}^{\ast }F\rightarrow\end{eqnarray}$$

for any $F\in D_{c}^{+}({\mathcal{X}},\unicode[STIX]{x1D6EC})$. Applying $Rf_{\ast }$, the comparison maps $\text{comp}(f,-)$ fit together into a morphism

of distinguished triangles. For each $\bullet \in \{{\mathcal{U}},{\mathcal{V}},{\mathcal{U}}\cap {\mathcal{V}}\}$, the maps $\text{comp}(j_{\bullet })$ are isomorphisms by assumption; by induction on $s$, the maps $\text{comp}(f\circ j_{\bullet })$ are isomorphisms as well. Applying part (ii) of the previous proposition with $g=f$ and $h=j_{\bullet }$, we see that the maps $(2)$ and $(3)$ are both isomorphisms. By the 2-out-of-3 property, the morphism $(1)$ is therefore an isomorphism, as desired.◻

Proof. By the previous proposition, it suffices to deduce under these assumptions that $\text{comp}(j,F)$ is an isomorphism for any open immersion $j:{\mathcal{X}}\rightarrow {\mathcal{Y}}$ and any $F\in D_{c}^{+}({\mathcal{X}},\unicode[STIX]{x1D6EC})$. We can clearly assume that ${\mathcal{Y}}$ is affine and that $j$ has dense image. We can also assume that $F=\mathscr{F}$ is a constructible sheaf (placed in degree zero). As in the proof of Theorem 1.9, we may choose an isomorphism

$$\begin{eqnarray}\mathscr{F}\simeq [\mathscr{G}^{0}\rightarrow \mathscr{G}^{1}\rightarrow \mathscr{G}^{2}\rightarrow \cdots \,],\end{eqnarray}$$

where each sheaf $\mathscr{G}^{i}$ is isomorphic to a finite direct sum of sheaves of the form $\unicode[STIX]{x1D70B}_{\ast }\text{}\underline{M}$, where $\unicode[STIX]{x1D70B}:{\mathcal{U}}\rightarrow {\mathcal{X}}$ is a finite morphism and $\text{}\underline{M}$ is a constant constructible sheaf of $\unicode[STIX]{x1D6EC}$-modules on ${\mathcal{U}}$. By an easy variant of the spectral sequence argument used in the proof of Theorem 1.9, it suffices to prove that $\text{comp}(j,\mathscr{G}^{i})$ is an isomorphism for all $i$. In particular, it suffices to prove that $\text{comp}(j,\unicode[STIX]{x1D70B}_{\ast }\text{}\underline{M})$ is an isomorphism for any finite morphism $\unicode[STIX]{x1D70B}:{\mathcal{U}}\rightarrow {\mathcal{X}}$ and any constant constructible sheaf $\text{}\underline{M}$ as above.

Fix such a $\unicode[STIX]{x1D70B}$ and $\text{}\underline{M}$. By Zariski’s main theorem, we can choose a commutative diagram

where $h$ is an open immersion and $\unicode[STIX]{x1D70F}$ is finite. In what follows, we freely use the facts that $\unicode[STIX]{x1D70B}_{\ast }=R\unicode[STIX]{x1D70B}_{\ast }$ and $\unicode[STIX]{x1D70F}=R\unicode[STIX]{x1D70F}_{\ast }$, and likewise for $\unicode[STIX]{x1D70B}^{\text{an}}$ and $\unicode[STIX]{x1D70F}^{\text{an}}$; we also freely use the fact that $\text{comp}(\unicode[STIX]{x1D70B})$ is an isomorphism for any finite morphism $\unicode[STIX]{x1D70B}$. Consider the sequence of canonical isomorphisms

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{Y}^{\ast }Rj_{\ast }\unicode[STIX]{x1D70B}_{\ast }\text{}\underline{M} & \cong & \displaystyle \unicode[STIX]{x1D707}_{Y}^{\ast }R(j\circ \unicode[STIX]{x1D70B})_{\ast }\text{}\underline{M}\nonumber\\ \displaystyle & \cong & \displaystyle \unicode[STIX]{x1D707}_{Y}^{\ast }R(\unicode[STIX]{x1D70F}\circ h)_{\ast }\text{}\underline{M}\nonumber\\ \displaystyle & \cong & \displaystyle \unicode[STIX]{x1D707}_{Y}^{\ast }\unicode[STIX]{x1D70F}_{\ast }Rh_{\ast }\text{}\underline{M}\nonumber\\ \displaystyle & \cong & \displaystyle \unicode[STIX]{x1D70F}_{\ast }^{\text{an}}\unicode[STIX]{x1D707}_{V}^{\ast }Rh_{\ast }\text{}\underline{M}\nonumber\\ \displaystyle & \overset{(\ast )}{\cong } & \displaystyle \unicode[STIX]{x1D70F}_{\ast }^{\text{an}}Rh_{\ast }^{\text{an}}\unicode[STIX]{x1D707}_{U}^{\ast }\text{}\underline{M}\nonumber\\ \displaystyle & \cong & \displaystyle Rj_{\ast }^{\text{an}}\unicode[STIX]{x1D70B}_{\ast }^{\text{an}}\unicode[STIX]{x1D707}_{U}^{\ast }\text{}\underline{M}\nonumber\\ \displaystyle & \cong & \displaystyle Rj_{\ast }^{\text{an}}\unicode[STIX]{x1D707}_{X}^{\ast }\unicode[STIX]{x1D70B}_{\ast }\text{}\underline{M},\nonumber\end{eqnarray}$$

where the starred isomorphism is the map $\unicode[STIX]{x1D70F}_{\ast }^{\text{an}}\text{comp}(h,\text{}\underline{M})$. One checks directly that the composition of these maps coincides with $\text{comp}(j,\unicode[STIX]{x1D70B}_{\ast }\text{}\underline{M})$, so the latter is an isomorphism, as desired.◻

Proof. By the previous proposition, it suffices to show that the hypothesis of the proposition implies that $\text{comp}(j,F)$ is an isomorphism for any open immersion $j:{\mathcal{X}}\rightarrow {\mathcal{Y}}$ of finite-type ${\mathcal{S}}$-schemes and any constant constructible sheaf $\text{}\underline{M}$ on ${\mathcal{X}}$.

Fix such a choice of $j$ and $\text{}\underline{M}$. We can clearly assume that ${\mathcal{Y}}$ is affine. By the topological invariant of the étale site, we can also assume that ${\mathcal{Y}}$ and ${\mathcal{X}}$ are reduced. Choose a dense open regular subscheme $g:{\mathcal{U}}\rightarrow {\mathcal{X}}$ with closed complement $i:{\mathcal{Z}}\rightarrow {\mathcal{X}}$, so we get a distinguished triangle

$$\begin{eqnarray}i_{\ast }i^{!}\text{}\underline{M}\rightarrow \text{}\underline{M}\rightarrow Rg_{\ast }g^{\ast }\text{}\underline{M}\rightarrow .\end{eqnarray}$$

By induction on $\text{dim}\,\text{supp}$, we can assume that $\text{comp}(j,i_{\ast }i^{!}\text{}\underline{M})$ is an isomorphism. By the 2-out-of-3 property, it now suffices to prove that $\text{comp}(j,Rg_{\ast }g^{\ast }\text{}\underline{M})$ is an isomorphism. By another application of Proposition 4.3(ii), we observe that the latter map is an isomorphism if both $\text{comp}(g,g^{\ast }\text{}\underline{M})$ and $\text{comp}(j\circ g,g^{\ast }\text{}\underline{M})$ are isomorphisms. In particular, we are reduced to showing that $\text{comp}(h,\text{}\underline{M})$ is an isomorphism for any open immersion $h:{\mathcal{X}}\rightarrow {\mathcal{Y}}$ of reduced finite-type ${\mathcal{S}}$-schemes with ${\mathcal{X}}$ regular.

Since ${\mathcal{Y}}$ is regular, by applying the results of [Reference TemkinTem17] we can choose a commutative diagram

where $\tilde{h}$ is a dense open embedding of regular finite-type ${\mathcal{S}}$-schemes and $f$ is a proper birational morphism. In particular, $\text{comp}(f)$ is an isomorphism. By the hypotheses of the proposition, $\text{comp}(\tilde{h},\text{}\underline{M})$ is an isomorphism as well. Arguing as in the proof of Proposition 4.3(ii), one deduces that $\text{comp}(h,\text{}\underline{M})$ is an isomorphism.◻

Proof. This follows exactly as in the final stages of the proof of [Reference Artin, Grothendieck and VerdierSGA4, Theorem XVI.4.1], with the appeal to results in [Reference Artin, Grothendieck and VerdierSGA4, § XVI.3] replaced by Huber’s cohomological purity theorem [Reference HuberHub96, Theorem 3.9.1]. ◻

Proof of Theorem 1.10.

For the first part of (i), we compute that

$$\begin{eqnarray}\displaystyle R\unicode[STIX]{x1D6E4}({\mathcal{X}},F) & \overset{(1)}{\cong } & \displaystyle R\unicode[STIX]{x1D6E4}({\mathcal{S}},Rf_{\ast }F)\nonumber\\ \displaystyle & \overset{(2)}{\cong } & \displaystyle R\unicode[STIX]{x1D6E4}(S,\unicode[STIX]{x1D707}_{S}^{\ast }Rf_{\ast }F)\nonumber\\ \displaystyle & \overset{(3)}{\cong } & \displaystyle R\unicode[STIX]{x1D6E4}(S,Rf_{\ast }^{\text{an}}\unicode[STIX]{x1D707}_{X}^{\ast }F)\nonumber\\ \displaystyle & \overset{(4)}{\cong } & \displaystyle R\unicode[STIX]{x1D6E4}(X,\unicode[STIX]{x1D707}_{X}^{\ast }F),\nonumber\end{eqnarray}$$

where (1) and (4) are the evident Leray isomorphisms, (2) follows from Theorem 1.9, and (3) follows from Theorem 1.8. For the second part of (i), it suffices to assume that $F=\mathscr{F}[0]$ is a sheaf. In this case, we need to prove that $\mathscr{F}\overset{{\sim}}{\rightarrow }\unicode[STIX]{x1D707}_{X\ast }\unicode[STIX]{x1D707}_{X}^{\ast }\mathscr{F}$ and that $R^{i}\unicode[STIX]{x1D707}_{X\ast }\unicode[STIX]{x1D707}_{X}^{\ast }\mathscr{F}=0$ for all $i\geqslant 1$. By definition, $R^{i}\unicode[STIX]{x1D707}_{X\ast }\unicode[STIX]{x1D707}_{X}^{\ast }\mathscr{F}$ is the sheafification of the presheaf sending any finite-type ${\mathcal{S}}$-scheme ${\mathcal{U}}$ with an étale map $g:{\mathcal{U}}\rightarrow {\mathcal{X}}$ to $H_{\acute{\text{e}}\text{t}}^{i}(U,g^{\text{an}\ast }\unicode[STIX]{x1D707}_{X}^{\ast }\mathscr{F})$, where of course $U={\mathcal{U}}^{\text{an}}$ is the analytification of ${\mathcal{U}}$. But then

$$\begin{eqnarray}H_{\acute{\text{e}}\text{t}}^{i}(U,g^{\text{an}\ast }\unicode[STIX]{x1D707}_{X}^{\ast }\mathscr{F})\cong H_{\acute{\text{e}}\text{t}}^{i}(U,\unicode[STIX]{x1D707}_{U}^{\ast }g^{\ast }\mathscr{F})\cong H_{\acute{\text{e}}\text{t}}^{i}({\mathcal{U}},g^{\ast }\mathscr{F}),\end{eqnarray}$$

where the second isomorphism follows from the first part of (i), so the result follows from the locality of cohomology (cf. [Sta17, Tag 01FW]).

For (ii), note that

$$\begin{eqnarray}\displaystyle \text{Hom}_{D(X,\unicode[STIX]{x1D6EC})}(\unicode[STIX]{x1D707}_{X}^{\ast }G,\unicode[STIX]{x1D707}_{X}^{\ast }F) & \cong & \displaystyle \text{Hom}_{D({\mathcal{X}},\unicode[STIX]{x1D6EC})}(G,R\unicode[STIX]{x1D707}_{X\ast }\unicode[STIX]{x1D707}_{X}^{\ast }F)\nonumber\\ \displaystyle & \cong & \displaystyle \text{Hom}_{D({\mathcal{X}},\unicode[STIX]{x1D6EC})}(G,F),\nonumber\end{eqnarray}$$

where the first isomorphism follows from the adjointness of $\unicode[STIX]{x1D707}_{X}^{\ast }$ and $R\unicode[STIX]{x1D707}_{X\ast }$, and the second isomorphism follows from part (i).◻

Proof of Theorem 1.7.

Full faithfulness is immediate from Theorem 1.10(ii) together with the natural identification $\text{Hom}_{{\mathcal{A}}}(M,N)=\text{Hom}_{D({\mathcal{A}})}(M,N)$, which holds for any abelian category ${\mathcal{A}}$.

For essential surjectivity, we argue as follows. By the topological invariance of étale sites, we can assume that $A$ is reduced. Fix $\mathscr{F}\in \text{Sh}_{zc}(S,\unicode[STIX]{x1D6EC})$. Pick a dense Zariski-open subset $j:U\rightarrow S$ such that $U$ is normal and $j^{\ast }\mathscr{F}$ is locally constant, and let $i:Z\rightarrow S$ be the inclusion of the closed complement. Let $j^{\text{alg}}:{\mathcal{U}}\rightarrow {\mathcal{S}}$ and $i^{\text{alg}}:{\mathcal{Z}}\rightarrow {\mathcal{S}}$ be the associated open and closed subschemes of ${\mathcal{S}}$. By induction on $\text{dim}\,\text{supp}\,\mathscr{F}$, we can assume that $i_{\ast }i^{\ast }\mathscr{F}\simeq \unicode[STIX]{x1D707}_{S}^{\ast }{\mathcal{G}}$ for some constructible sheaf ${\mathcal{G}}$. By Corollary 2.16, analytification induces an equivalence of categories ${\mathcal{U}}_{\text{f}\acute{\text{e}}\text{t}}\cong U_{\text{f}\acute{\text{e}}\text{t}}$. Since (on individual connected components of $U$) $j^{\ast }\mathscr{F}$ corresponds to a representation of $\unicode[STIX]{x1D70B}_{1}(U,\overline{x})\cong \unicode[STIX]{x1D70B}_{1}({\mathcal{U}},\overline{x})$, we immediately see that $j^{\ast }\mathscr{F}\simeq \unicode[STIX]{x1D707}_{U}^{\ast }{\mathcal{F}}^{\prime }$ for some locally constant constructible sheaf ${\mathcal{F}}^{\prime }$ on ${\mathcal{U}}$, so then

$$\begin{eqnarray}j_{!}j^{\ast }\mathscr{F}\simeq j_{!}\unicode[STIX]{x1D707}_{U}^{\ast }{\mathcal{F}}^{\prime }\cong \unicode[STIX]{x1D707}_{S}^{\ast }j_{!}^{\text{alg}}{\mathcal{F}}^{\prime }=\unicode[STIX]{x1D707}_{S}^{\ast }{\mathcal{F}},\end{eqnarray}$$

where we have set ${\mathcal{F}}=j_{!}^{\text{alg}}{\mathcal{F}}^{\prime }$. Thus, $\mathscr{F}$ sits in a short exact sequence

$$\begin{eqnarray}0\rightarrow \unicode[STIX]{x1D707}_{S}^{\ast }{\mathcal{F}}\rightarrow \mathscr{F}\rightarrow \unicode[STIX]{x1D707}_{S}^{\ast }{\mathcal{G}}\rightarrow 0.\end{eqnarray}$$

To conclude, recall that for any abelian category ${\mathcal{A}}$, there is a canonical identification $\text{Ext}_{{\mathcal{A}}}^{i}(B,A)=\text{Hom}_{D({\mathcal{A}})}(B,A[i])$ for any objects $A,B$ in ${\mathcal{A}}$; cf. [Sta17, Tag 06XP]. Combining this observation with Theorem 1.10(ii), we conclude that $\unicode[STIX]{x1D707}_{S}^{\ast }$ induces a bijection

$$\begin{eqnarray}\text{Ext}_{\text{Sh}({\mathcal{S}},\unicode[STIX]{x1D6EC})}^{1}({\mathcal{G}},{\mathcal{F}})\rightarrow \text{Ext}_{\text{Sh}(S,\unicode[STIX]{x1D6EC})}^{1}(\unicode[STIX]{x1D707}_{S}^{\ast }{\mathcal{G}},\unicode[STIX]{x1D707}_{S}^{\ast }{\mathcal{F}})\end{eqnarray}$$

on Yoneda Ext groups. Therefore, $\mathscr{F}\simeq \unicode[STIX]{x1D707}_{S}^{\ast }{\mathcal{H}}$ for some constructible sheaf ${\mathcal{H}}$ on ${\mathcal{S}}$, as desired.◻

Proof of Theorem 1.11.

For part (i), let $j^{\text{alg}}:{\mathcal{U}}\rightarrow {\mathcal{S}}$ be the evident open immersion with analytification $j$. By Theorem 1.7, we can write $\mathscr{F}\simeq \unicode[STIX]{x1D707}_{U}^{\ast }{\mathcal{F}}$ for some ${\mathcal{F}}\in \text{Sh}_{c}({\mathcal{U}},\unicode[STIX]{x1D6EC})$. (Precisely, Theorem 1.7 produces a sheaf ${\mathcal{G}}\in \text{Sh}_{c}({\mathcal{S}},\unicode[STIX]{x1D6EC})$ with $j_{!}\mathscr{F}\simeq \unicode[STIX]{x1D707}_{S}^{\ast }{\mathcal{G}}$; we then take ${\mathcal{F}}=j^{\text{alg}\ast }{\mathcal{G}}$.) By Proposition 4.2, $Rj_{\ast }^{\text{alg}}{\mathcal{F}}$ is constructible, so

$$\begin{eqnarray}Rj_{\ast }\mathscr{F}\cong Rj_{\ast }\unicode[STIX]{x1D707}_{U}^{\ast }{\mathcal{F}}\cong \unicode[STIX]{x1D707}_{S}^{\ast }Rj_{\ast }^{\text{alg}}{\mathcal{F}}\end{eqnarray}$$

is Zariski-constructible by Theorem 1.8.

For part (ii), the spectral sequence

$$\begin{eqnarray}E_{p,q}^{2}=\bigoplus _{i+j=q}\text{Tor}_{p}^{\unicode[STIX]{x1D6EC}}(\mathscr{H}^{-i}(\mathscr{F}),\mathscr{H}^{-j}(\mathscr{G}))\Rightarrow \mathscr{H}^{-p-q}(\mathscr{F}\otimes _{\unicode[STIX]{x1D6EC}}^{\mathbf{L}}\mathscr{G})\end{eqnarray}$$

shows that any given cohomology sheaf $\mathscr{H}^{n}(\mathscr{F}\otimes _{\unicode[STIX]{x1D6EC}}^{\mathbf{L}}\mathscr{G})$ is unchanged if we replace $\mathscr{F}$ by $\unicode[STIX]{x1D70F}^{{\geqslant}-n}\mathscr{F}$ for some large $n\gg 0$. This reduces us to the case where $\mathscr{F}$ is bounded, and then (by an easy induction using the two-out-of-three property) to the case where $\mathscr{F}$ is a sheaf. Repeating this argument, we can also assume that $\mathscr{G}$ is a sheaf. Going to a stratification $S=\coprod S_{i}$ such that $\mathscr{F}$ and $\mathscr{G}$ are locally constant on each stratum, we reduce to the case where both sheaves are locally constant, which is easy.

For part (iii), note that $\mathscr{H}^{n}(\mathbf{D}_{S}\mathscr{F})$ is the sheafification of the presheaf sending any quasicompact separated étale map $j:U\rightarrow S$ to the module

$$\begin{eqnarray}H^{n}(R\mathscr{H}\text{om}_{\unicode[STIX]{x1D6EC}}(R\unicode[STIX]{x1D6E4}_{c}(U,j^{\ast }\mathscr{F}),\unicode[STIX]{x1D6EC})).\end{eqnarray}$$

Our assumptions on $K$ and $\unicode[STIX]{x1D6EC}$ guarantee that $R\unicode[STIX]{x1D6E4}_{c}(U,-)$ sends $D^{[a,b]}(U,\unicode[STIX]{x1D6EC})$ into $D^{[a,b+N]}(\unicode[STIX]{x1D6EC})$ for some $N$ depending only on the pair $(S,\unicode[STIX]{x1D6EC})$, and that $R\mathscr{H}\text{om}_{\unicode[STIX]{x1D6EC}}(-,\unicode[STIX]{x1D6EC})$ sends $D^{[a,b]}(\unicode[STIX]{x1D6EC})$ into $D^{[-b,-a+\text{dim}\,\unicode[STIX]{x1D6EC}]}(\unicode[STIX]{x1D6EC})$. This shows that $\mathbf{D}_{S}$ exchanges the boundedness conditions as claimed, and that any given cohomology sheaf $\mathscr{H}^{n}(\mathbf{D}_{S}\mathscr{F})$ depends only on the truncation $\unicode[STIX]{x1D70F}^{{\leqslant}b}\unicode[STIX]{x1D70F}^{{\geqslant}a}\mathscr{F}$ for some large $a\ll 0\ll b$. For the Zariski-constructibility of $\mathbf{D}_{S}\mathscr{F}$, we can therefore assume that $\mathscr{F}$ is bounded, and then that $\mathscr{F}$ is a sheaf. Pick a smooth dense Zariski-open $j:U\rightarrow S$ with closed (affinoid) complement $i:Z\rightarrow X$, so $j^{\ast }\unicode[STIX]{x1D714}_{S}=\unicode[STIX]{x1D714}_{U}$ is locally constant up to a shift and twist. Shrinking $U$ further, if necessary, we can also assume that $j^{\ast }\mathscr{F}$ is locally constant, so

$$\begin{eqnarray}j^{\ast }\mathbf{D}_{S}\mathscr{F}\cong \mathbf{D}_{U}j^{\ast }\mathscr{F}=R\mathscr{H}\text{om}_{U}(j^{\ast }\mathscr{F},\unicode[STIX]{x1D714}_{U})\end{eqnarray}$$

is locally constant constructible, up to a shift and twist. Dualizing the distinguished triangle

$$\begin{eqnarray}j_{!}j^{\ast }\mathscr{F}\rightarrow \mathscr{F}\rightarrow i_{\ast }i^{\ast }\mathscr{F}\rightarrow\end{eqnarray}$$

and using the habitual isomorphisms $\mathbf{D}_{S}j_{!}\cong Rj_{\ast }\mathbf{D}_{U}$ and $\mathbf{D}_{S}i_{\ast }i^{\ast }\cong i_{\ast }\mathbf{D}_{Z}i^{\ast }$, we get a distinguished triangle

$$\begin{eqnarray}i_{\ast }\mathbf{D}_{Z}i^{\ast }\mathscr{F}\rightarrow \mathbf{D}_{S}\mathscr{F}\rightarrow Rj_{\ast }\mathbf{D}_{U}j^{\ast }\mathscr{F}\rightarrow .\end{eqnarray}$$

By induction on $\text{dim}\,S$, the first term in this triangle is Zariski-constructible. Since $\mathbf{D}_{U}j^{\ast }\mathscr{F}$ is locally constant, the third term is Zariski-constructible by part (i). The result now follows from the two-out-of-three property. Finally, for the reflexivity of $\mathscr{F}$, we can again assume that $\mathscr{F}$ is a sheaf, in which case the desired result is [Reference HansenHan17, Theorem B.1.2].

For part (iv), note that we have isomorphisms

$$\begin{eqnarray}\displaystyle R\mathscr{H}\text{om}_{S}(\mathscr{F},\mathscr{G}) & \cong & \displaystyle R\mathscr{H}\text{om}_{S}(\mathscr{F},\mathbf{D}_{S}\mathbf{D}_{S}\mathscr{G})\nonumber\\ \displaystyle & \cong & \displaystyle \mathbf{D}_{S}(\mathscr{F}\otimes _{\unicode[STIX]{x1D6EC}}^{\mathbf{L}}\mathbf{D}_{S}\mathscr{G}),\nonumber\end{eqnarray}$$

where the first comes from the reflexivity of $\mathscr{G}$ proved in (iii), and the second is an easy exercise in tensor–hom adjunction. The result now follows from parts (ii) and (iii).

For part (v), the habitual isomorphism $Rf^{!}\mathbf{D}_{S}\cong \mathbf{D}_{T}f^{\ast }$ together with the reflexivity of $\mathscr{F}$ gives isomorphisms

$$\begin{eqnarray}Rf^{!}\mathscr{F}\cong Rf^{!}\mathbf{D}_{S}\mathbf{D}_{S}\mathscr{F}\cong \mathbf{D}_{T}f^{\ast }\mathbf{D}_{S}\mathscr{F},\end{eqnarray}$$

so the result now follows from part (iii). ◻