H^{q}(\overline{U}^{/S}\times_{S}W,\operatorname{\mathscr{F}})$

and

$H^{q}(({X_{W}},\overline{|U_{W}|}\cap
f_{W}^{-1}(w)),\operatorname{\mathscr{F}})\simeq
H^{q}({f^{\prime}_{W}}^{-1}(w),\operatorname{\mathscr{F}}).$

This is sufficient to conclude the proof. ∎

######  Proposition 6.12.

In Situation 6.2, if in addition we assume

  1. (1)

The morphism $f$ is tight and $\mathcal{O}_{X}(X)$ is a Tate ring.

  2. (2)

Let $Y$ be the pseudo-adic space $(X,|Y|)$ where

$|Y|:=\\{x\in{X}||e_{i}(x)|<|e_{0}(x)|\mbox{ for }1\leq i\leq n\\}$

and $e_{0},\ldots,e_{n}\in\mathcal{O}_{{X}}({X})$ are such that
$\mathcal{O}_{{X}}({X})=\sum_{i}e_{i}\mathcal{O}_{{X}}({X})$. Let $i\colon
Y\hookrightarrow X$ be the natural closed embedding. The sheaf
$\operatorname{\mathscr{F}}$ is of the form $i_{*}(G_{Y})$ where $G_{Y}$ is
the constant sheaf on $Y_{\text{\'{e}t}}$ associated to an abelian torsion
group $G$.

Then (14) is an isomorphism.

Note that in what follows, when there is no ambiguity, we simplify notation
and write $G$ for the constant sheaf $G_{X}$ on $X_{\text{\'{e}t}}$ for $X$ a
pseudo-adic space or scheme.

###### Proof.

Recall that

$[R^{n}f_{*}(\operatorname{\mathscr{F}})]_{\overline{s}}=\varinjlim_{(W,\overline{w})}H^{n}(X_{W},\operatorname{\mathscr{F}})$

where the colimit runs over étale neighbourhoods of $(S,\overline{s})$ and we
must show that the natural morphism

$\varinjlim_{(W,\overline{w})}H^{n}(X_{W},\operatorname{\mathscr{F}})\to\varinjlim_{(W,\overline{w})}H^{n}(f_{W}^{-1}(w),\operatorname{\mathscr{F}})$

is an isomorphism. Note that we can restrict to when the colimits above run
over affine étale neighbourhoods of $(S,\overline{s})$. By [14, Tag 02GT], we
can further restrict so that the colimits are over standard étale
neighbourhoods of $(S,\overline{s})$. Let $\mathscr{J}$ denote the category of
standard étale neighbourhoods of $(S,\overline{s})$ and let
$(W,\overline{w})\in\mathscr{J}$. Observe that by Lemma 6.7, $X_{W}$ is
affinoid. For the remainder of the proof, all limits under consideration will
run over pairs $(W,\overline{w})\in\mathscr{J}$.

######  Lemma 6.13.

  1. (1)

We have a well defined pseudo-adic space $Y_{W}:=Y\times_{X}X_{W}$ such that
the morphism $i_{W}\colon Y_{W}\to X_{W}$ induced by base change is a closed
embedding. If for every $0\leq i\leq n$, $e^{\prime}_{i}$ denotes the image of
$e_{i}$ for the morphism $\mathcal{O}_{X}(X)\to\mathcal{O}_{X_{W}}(X_{W})$
then $|Y_{W}|=\\{x\in X_{W}||e^{\prime}_{i}(x)|<|e^{\prime}_{0}(x)|\mbox{ for
}1\leq i\leq n\\}.$

  2. (2)

We have that $i_{W_{*}}(G)=\operatorname{\mathscr{F}}_{|X_{W}}$.

  3. (3)

If $\epsilon_{W}$ denotes the closed embedding
$(X_{W},(f_{W}^{-1}(w)\cap|Y_{W}|))\hookrightarrow f_{W}^{-1}(w)$ of pseudo-
adic spaces then

$\operatorname{\mathscr{F}}_{|f_{W}^{-1}(w)}=\epsilon_{W*}(G).$

###### Proof.

The morphism $i\colon Y\to X$ is locally of finite type. By [8, Lemma
1.10.6(a)], $Y_{W}$ exists and from the proof of loc.cit., it must be of the
form described in part (1) of the lemma. Similarly,
$(X_{W},(f_{W}^{-1}(w)\cap|Y_{W}|))=f_{W}^{-1}(w)\times_{X}Y$. By [8, Lemma
1.10.17(i)], $i$ is proper. Parts (2) and (3) follow from [8, Proposition
4.1.2(b)]. ∎

We first reduce to when $e_{0}=1$. Let

$U:=\\{x\in{X}||e_{i}(x)|\leq|e_{0}(x)|\mbox{ for }i=1,\ldots,n\\}.$

Then $U$ is an affinoid open subset of $X$. Observe that it suffices to show
that the natural morphism

$\varinjlim_{(W,\overline{w})\in\mathscr{J}}H^{n}((X_{W},\overline{|U_{W}|}),\operatorname{\mathscr{F}})\to\varinjlim_{(W,\overline{w})\in\mathscr{J}}H^{n}((X_{W},\overline{|U_{W}|}\cap
f_{W}^{-1}(w)),\operatorname{\mathscr{F}})$

is an isomorphism where $U_{W}:=U\times_{S}W$. Indeed, by Lemma 6.13(1,2),
$i_{W}\colon Y_{W}\hookrightarrow X_{W}$ is a closed embedding and
$\operatorname{\mathscr{F}}_{|X_{W}}=i_{W*}(G)$. Hence,

(15) $\displaystyle H^{n}(X_{W},\operatorname{\mathscr{F}})$
$\displaystyle=H^{n}(Y_{W},G)$
$\displaystyle=H^{n}((X_{W},\overline{|U_{W}|}),\operatorname{\mathscr{F}}).$

Similarly, by Lemma 6.13(3), we deduce that

(16) $\displaystyle H^{n}(f_{W}^{-1}(w),\operatorname{\mathscr{F}})$
$\displaystyle=H^{n}((X_{W},|Y_{W}|\cap f_{W}^{-1}(w)),G)$
$\displaystyle=H^{n}((X_{W},\overline{|U_{W}|}\cap
f_{W}^{-1}(w)),\operatorname{\mathscr{F}}).$

By Lemma 6.5, $f$ is partially proper and by Remark 3.7, $f_{|U}$ is good. We
are hence in the context of Situation 6.9. By Lemma 6.11, it suffices to
verify the proposition after replacing $X$ with $\overline{U}^{/S}$. Since
$\mathcal{O}_{X}(U)=\mathcal{O}_{\overline{U}^{/S}}(\overline{U}^{/S})$,
$e_{0}$ is invertible on $\overline{U}^{/S}$ and after replacing $e_{i}$ with
$e_{i}/e_{0}$ for every $1\leq i\leq n$, we can suppose $e_{0}=1$ in the
definition of $|Y|$.

Let $(A,A^{+}):=(\mathcal{O}_{X}(X),\mathcal{O}^{+}_{X}(X))$. We begin by
simplifying the left hand side of (14). By Lemma 6.7, $X_{W}$ is affinoid and
let
$(A_{W},A_{W}^{+}):=(\mathcal{O}_{X_{W}}(X_{W}),\mathcal{O}^{+}_{X_{W}}(X_{W}))$.
Associated to the special subset $Y_{W}\subseteq X_{W}$, we define the algebra
$A_{W}(Y_{W})$ as follows. Let $A^{+}_{W}(Y_{W})$ denote the henselization of
$A^{+}_{W}[e_{1},\ldots,e_{n}]$ with respect to the ideal generated by
$\\{e_{1},\ldots,e_{n}\\}\cup A_{W}^{\circ\circ}$ where we abuse notation and
use $e_{i}$ to denote the image of $e_{i}$ for the morphism $A\to A_{W}$. We
define
$A_{W}(Y_{W}):=A^{+}_{W}(Y_{W})\otimes_{A^{+}_{W}[e_{1},\ldots,e_{n}]}A_{W}$.
By (15) and [8, Theorem 3.2.1], we have a natural isomorphism

(17) $H^{n}(X_{W},\operatorname{\mathscr{F}})\simeq
H^{n}(\mathrm{Spec}(A_{W}(Y_{W})),G).$

Hence we have that

$\displaystyle\varinjlim_{(W,\overline{w})}H^{n}(X_{W},\operatorname{\mathscr{F}})$
$\displaystyle\overset{(i)}{\simeq}\varinjlim_{(W,\overline{w})}H^{n}(\mathrm{Spec}(A_{W}(Y_{W})),G)$
$\displaystyle\overset{(ii)}{=}H^{n}(\varprojlim_{(W,\overline{w})}\mathrm{Spec}(A_{W}(Y_{W})),G)$
(18)
$\displaystyle\overset{(iii)}{=}H^{n}(\mathrm{Spec}(\varinjlim_{(W,\overline{w})}A_{W}(Y_{W})),G)$

where the colimit and limit runs over all standard étale neighbourhoods of
$(S,\overline{s})$. Observe that (i) is a consequence of the naturality of the
isomorphism (17). Note that (ii) follows from [14, Tag 09YQ] and (iii) is due
to [14, Tag 01YW].

We now focus on the right hand side of (14). Observe that by Lemma 6.7, the
morphism $f_{W}$ is tight. Then by Lemma 6.8, $f_{W}^{-1}(w)$ is a pro-special
subset of $X_{W}$. Furthermore, if $R$ and $R_{W}$ are such that
$S=\mathrm{Spec}(R)$ and $W=\mathrm{Spec}(R_{W})$, and $w\in W$ corresponds to
a maximal ideal $\mathfrak{p}_{W}$ in $R_{W}$ then we have the following
diagram

${A_{W}^{+}}$${R_{W}}$${A^{+}_{W}/\mathfrak{m}_{\mathcal{O}^{+}_{X_{W}}}(X_{W}).}$$\scriptstyle{d_{W}}$$\scriptstyle{e_{W}}$

We abuse notation and write $\mathfrak{p}_{W}$ for
$e_{W}(\mathfrak{p}_{W})\subset
A_{W}^{+}/\mathfrak{m}_{\mathcal{O}_{X_{W}}^{+}}(X_{W})$. As in the proof of
Lemma 6.8,

$f_{W}^{-1}(w)=\left\\{x\in X_{W}\text{ }\lvert\text{ }\lvert
a(x)\lvert<1\text{ }\forall a\in d_{W}^{-1}(\mathfrak{p}_{W}\cdot
A_{W}^{+}/\mathfrak{m}_{\mathcal{O}_{X_{W}}^{+}}(X_{W}))\right\\}.$

Let $I_{W}$ denote the ideal generated by $d_{W}^{-1}(\mathfrak{p}_{W}\cdot
A_{W}^{+}/\mathfrak{m}_{\mathcal{O}_{X_{W}}^{+}}(X_{W}))\bigcup\\{e_{1},\ldots,e_{n}\\}$
in $A^{+}_{W}[e_{1},\ldots,e_{n}]$ and let ${A^{+}_{W}}^{h(I_{W})}$ denote the
henselization of $A^{+}_{W}[e_{1},\ldots,e_{n}]$ along $I_{W}$.

Define

$A_{W}^{h(I_{W})}:={A^{+}_{W}}^{h(I_{W})}\otimes_{A^{+}_{W}[e_{1},\ldots,e_{n}]}A_{W}.$

Note that

$A_{W}^{h(I_{W})}={A^{+}_{W}}^{h(I_{W})}\otimes_{A^{+}[e_{1},\ldots,e_{n}]}A.$

The equality above is a consequence of the fact that $A$ is Tate and hence
$A=A^{+}[1/\pi]$ for some pseudo-uniformizer $\pi\in A$. By (16) and [8,
Theorem 3.2.1], we have a natural isomorphism

$H^{n}(f_{W}^{-1}(w),\operatorname{\mathscr{F}})\simeq
H^{n}(\mathrm{Spec}(A_{W}^{h(I_{W})}),G).$

By a similar calculation as made for (18),

$\displaystyle\varinjlim_{(W,\overline{w})}H^{n}(f_{W}^{-1}(w),\operatorname{\mathscr{F}})$
$\displaystyle\simeq\varinjlim_{(W,\overline{w})}H^{n}(\mathrm{Spec}(A_{W}^{h(I_{W})}),G)$
$\displaystyle=H^{n}(\varprojlim_{(W,\overline{w})}\mathrm{Spec}(A_{W}^{h(I_{W})}),G)$
(19)
$\displaystyle=H^{n}(\mathrm{Spec}(\varinjlim_{(W,\overline{w})}A_{W}^{h(I_{W})}),G).$

We claim that

$\varinjlim_{(W,\overline{w})}A_{W}^{h(I_{W})}=\varinjlim_{(W,\overline{w})}A_{W}(Y_{W}).$

Let $J_{W}\subset A^{+}_{W}(Y_{W})$ denote the ideal generated by the image of
$I_{W}$ for the map $A^{+}_{W}[e_{1},\ldots,e_{n}]\to A^{+}_{W}(Y_{W})$. A
diagram chase implies that if $(V,\overline{v})$ is a standard étale
neighbourhood of $(S,\overline{s})$ such that the map
$(V,\overline{v})\to(S,\overline{s})$ factors through
$(W,\overline{w})\to(S,\overline{s})$ then

$J_{W}\cdot A^{+}_{V}(Y_{V})\subseteq J_{V}$

where $J_{W}$ on the left is the image of the ideal $J_{W}\subset
A^{+}_{W}(Y_{W})$ for the morphism $A^{+}_{W}(Y_{W})\to A^{+}_{V}(Y_{V})$.

We set $B:=\varinjlim_{(W,\overline{w})}A^{+}_{W}(Y_{W})$. Let $J\subset B$
denote the ideal $\varinjlim_{(W,\overline{w})}J_{W}$. Let
$J^{\prime}_{W}\subset A^{+}_{W}(Y_{W})$ be the ideal generated by the image
of $\mathfrak{m}_{\mathcal{O}^{+}_{X_{W}}}(X_{W})\cup\\{e_{1},\ldots,e_{n}\\}$
for the morphism $A^{+}_{W}[e_{1},\ldots,e_{n}]\to A^{+}_{W}(Y_{W})$ and set
$J^{\prime}:=\varinjlim_{(W,\overline{w})}J^{\prime}_{W}\subset B$. Finally,
let $J^{\prime\prime}_{W}\subset A^{+}_{W}(Y_{W})$ be the ideal generated by
the image of $A_{W}^{\circ\circ}\cup\\{e_{1},\ldots,e_{n}\\}$ for the morphism
$A^{+}_{W}[e_{1},\ldots,e_{n}]\to A^{+}_{W}(Y_{W})$ and set
$J^{\prime\prime}:=\varinjlim_{(W,\overline{w})}J^{\prime\prime}_{W}\subset
B$.

######  Lemma 6.14.

  1. (1)

The pair $(B,J^{\prime})$ is henselian.

  2. (2)

The pair $(B/J^{\prime},J/J^{\prime})$ is henselian.

  3. (3)

The pair $(B,J)$ is henselian.

###### Proof.

Let us verify part (1) of the lemma. By construction, $A^{+}_{W}(Y_{W})$ is
henselian along $J^{\prime\prime}_{W}$. Using [14, Tag 0FWT], we deduce that
$B$ is henselian along $J^{\prime\prime}$. We check that the pair
$(B/J^{\prime\prime},J^{\prime}/J^{\prime\prime})$ is henselian. Since
filtered colimits are exact,

$(B/J^{\prime\prime},J^{\prime}/J^{\prime\prime})=(\varinjlim_{(W,\overline{w})}A^{+}_{W}(Y_{W})/J^{\prime\prime}_{W},\varinjlim_{(W,\overline{w})}J^{\prime}_{W}/J^{\prime\prime}_{W}).$

Since $A^{+}_{W}(Y_{W})$ is the henselization of
$A^{+}_{W}[e_{1},\ldots,e_{n}]$ along the ideal $\langle
A_{W}^{\circ\circ}\cup\\{e_{1},\ldots,e_{n}\\}\rangle$, we get that

(20) $A^{+}_{W}(Y_{W})/J^{\prime\prime}_{W}=A^{+}_{W}/A^{\circ\circ}_{W}.$

Furthermore,
$J^{\prime}_{W}/J^{\prime\prime}_{W}=\mathfrak{m}_{\mathcal{O}^{+}_{X_{W}}}(X_{W})/A^{\circ\circ}_{W}$.
Hence,

$(B/J^{\prime\prime},J^{\prime}/J^{\prime\prime})=(\varinjlim_{(W,\overline{w})}A^{+}_{W}/A^{\circ\circ}_{W},\varinjlim_{(W,\overline{w})}\mathfrak{m}_{\mathcal{O}^{+}_{X_{W}}}(X_{W})/A^{\circ\circ}_{W}).$

By Corollary 2.16 and [14, Tag 0DYD and Tag 0FWT], we deduce that
$(B/J^{\prime\prime},J^{\prime}/J^{\prime\prime})$ is a henselian pair. Since
$(B,J^{\prime\prime})$ is a henselian pair, applying [14, Tag 0DYD] once again
implies that $(B,J^{\prime})$ is henselian.

We now verify part (2) of the lemma. Since filtered colimits are exact,

(21)
$B/J^{\prime}=\varinjlim_{(W,\overline{w})}A^{+}_{W}(Y_{W})/J^{\prime}_{W}.$

Similarly,

$\varinjlim_{(W,\overline{w})}A^{+}_{W}/\varinjlim_{(W,\overline{w})}\mathfrak{m}_{\mathcal{O}^{+}_{X_{W}}}(X_{W})=\varinjlim_{(W,\overline{w})}A^{+}_{W}/\mathfrak{m}_{\mathcal{O}^{+}_{X_{W}}}(X_{W}).$

We have the following commutative diagram.

${\varinjlim_{(W,\overline{w})}A^{+}_{W}}$${B}$${\varinjlim_{(W,\overline{w})}R_{W}}$${\varinjlim_{(W,\overline{w})}A^{+}_{W}/\mathfrak{m}_{\mathcal{O}^{+}_{X_{W}}}(X_{W})}$${\varinjlim_{(W,\overline{w})}A^{+}_{W}(Y_{W})/J^{\prime}_{W}.}$$\scriptstyle{\varinjlim
d_{W}}$$\scriptstyle{\varinjlim e_{W}}$

The equality in the above diagram is because by construction,
$A^{+}_{W}(Y_{W})/J^{\prime}_{W}=A^{+}_{W}/\mathfrak{m}_{\mathcal{O}^{+}_{X_{W}}}(X_{W}).$

Observe that $\varinjlim_{(W,\overline{w})}R_{W}$ is a strictly henselian
local ring whose maximal ideal is
$\mathfrak{p}_{0}:=\varinjlim_{(W,\overline{w})}\mathfrak{p}_{W}$. By Lemma
6.7, the morphism

$e_{W}\colon R_{W}\to A^{+}_{W}/\mathfrak{m}_{\mathcal{O}^{+}_{X_{W}}}(X_{W})$

is integral and hence by the equality in the diagram, the induced map

$R_{W}\to A^{+}_{W}(Y_{W})/J^{\prime}_{W}$

is integral as well. By [14, Tag 09XK],
$\varinjlim_{(W,\overline{w})}A^{+}_{W}(Y_{W})/J^{\prime}_{W}$ is henselian
along the ideal
$\mathfrak{p}_{0}\cdot\varinjlim_{(W,\overline{w})}A^{+}_{W}(Y_{W})/J^{\prime}_{W}=\varinjlim_{(W,\overline{w})}\mathfrak{p}_{W}\cdot
A^{+}_{W}(Y_{W})/J^{\prime}_{W}$. Using that filtered colimits are exact, we
deduce that

$J/J^{\prime}=\varinjlim_{(W,\overline{w})}J_{W}/J^{\prime}_{W}.$

A diagram chase shows that $J_{W}/J^{\prime}_{W}=\mathfrak{p}_{W}\cdot
A_{W}^{+}(Y_{W})/J^{\prime}_{W}.$ The equality (21) now implies that
$B/J^{\prime}$ is henselian along the ideal $J/J^{\prime}$.

Part (3) of the lemma is a direct consequence of Parts (1)-(2) and [14, Tag
0DYD]. ∎

Note that

$\varinjlim_{(W,\overline{w})}A_{W}^{h(I_{W})}=(\varinjlim_{(W,\overline{w})}{A_{W}^{+}}^{h(I_{W})})\otimes_{A^{+}[e_{1},\ldots,e_{n}]}A.$

and similarly,

$\varinjlim_{(W,\overline{w})}A_{W}(Y_{W})=(\varinjlim_{(W,\overline{w})}{A_{W}^{+}}(Y_{W}))\otimes_{A^{+}[e_{1},\ldots,e_{n}]}A.$

Hence, to show that
$\varinjlim_{(W,\overline{w})}A_{W}^{h(I_{W})}\simeq\varinjlim_{(W,\overline{w})}A_{W}(Y_{W})$,
it suffices to verify that
$\varinjlim_{(W,\overline{w})}{A^{+}_{W}}^{h(I_{W})}\simeq\varinjlim_{(W,\overline{w})}A^{+}_{W}(Y_{W})$.

To complete the proof, we apply [2, Remark 3.19] with
$S=\varinjlim_{(W,\overline{w})}A_{W}^{+}[e_{1},\ldots,e_{n}]$,
$I=\varinjlim_{(W,\overline{w})}I_{W}$ and $\tilde{S}=B$. Note that by [14,
Tag 0A04], $B$ is the henselization of
$\varinjlim_{(W,\overline{w})}A^{+}_{W}[e_{1},\ldots,e_{n}]$ along the ideal
$\varinjlim_{(W,\overline{w})}\langle
A_{W}^{\circ\circ}\cup\\{e_{1},\ldots,e_{n}\\}\rangle$. It follows that $B$ is
a filtered colimit of étale $S$-algebras. Since $(B,J)$ is henselian by Lemma
6.14(3), we can apply [2, Remark 3.19] to get that $(B,J)$ is the
henselization of
$(\varinjlim_{(W,\overline{w})}{A_{W}^{+}}[e_{1},\ldots,e_{n}],\varinjlim_{(W,\overline{w})}I_{W})$.
By [14, Tag 0A04], $\varinjlim_{(W,\overline{w})}{A^{+}_{W}}^{h(I_{W})}$ is
the henselization of
$\varinjlim_{(W,\overline{w})}A^{+}_{W}[e_{1},\ldots,e_{n}]$ along
$\varinjlim_{(W,\overline{w})}I_{W}$. It follows that the natural morphism
$B\to\varinjlim_{(W,\overline{w})}{A^{+}_{W}}^{h(I_{W})}$ is an isomorphism.
Equations (18), (19) and the naturality of the isomorphism in [8, Theorem
3.2.1] allow us to conclude the proof.

∎

######  Lemma 6.15.

Let $\mathscr{W}$ be a cofiltered category. Let $\mathcal{A}$ be the category
of pseudo-adic spaces. Let $p\colon\mathscr{W}\to\mathcal{A}$ and
$q\colon\mathscr{W}\to\mathcal{A}$ be functors such for any $W\in\mathscr{W}$,
$\underline{q(W)}=\underline{p(W)}$ and $|q(W)|\subseteq|p(W)|$. Suppose
$\mathscr{W}$ has a final object $0$ and let $X_{0}:=p(0)$. Let
$\operatorname{\mathscr{F}}_{0}$ be a sheaf of abelian groups on
$X_{0,\text{\'{e}t}}$ and for every $W\in\mathscr{W}$, let
$\operatorname{\mathscr{F}}_{W}$ denote the pullback of
$\operatorname{\mathscr{F}}_{0}$ to $p(W)$. Assume that we have a finite index
set $\mathscr{I}$ such that $|X_{0}|=\bigcup_{j\in\mathscr{I}}U_{j}$ where for
every $j\in\mathscr{I}$, $U_{j}$ is a subspace of $|X_{0}|$. For every
$j\in\mathscr{I}$, let $U_{j,W}$ be the preimage of $U_{j}$ for the morphism
$|p(W)|\to|X_{0}|$. Then the natural map

$\varinjlim_{W\in\mathscr{W}}H^{n}(p(W),\operatorname{\mathscr{F}}_{W})\to\varinjlim_{W\in\mathscr{W}}H^{n}(q(W),\operatorname{\mathscr{F}}_{W})$

is an isomorphism if for every non-empty subset $J\subseteq\mathscr{I}$, the
natural morphism

(22) $\varinjlim_{W\in\mathscr{W}}H^{n}(\bigcap_{j\in
J}\overline{U_{j,W}},\operatorname{\mathscr{F}}_{W})\to\varinjlim_{W\in\mathscr{W}}H^{n}(\bigcap_{j\in
J}\overline{U_{j,W}}\cap q(W),\operatorname{\mathscr{F}}_{W}).$

is an isomorphism where by $\overline{U_{j,W}}$, we mean the closure of
$U_{j,W}$ in $|p(W)|$.

###### Proof.

By [8, Corollary 2.6.10], we have a spectral sequence

$\prod_{\alpha\in\mathscr{I}^{r+1}}H^{s}(\bigcap_{0\leq i\leq
r}\overline{U_{\alpha_{i},W}},\operatorname{\mathscr{F}}_{W})\implies
H^{r+s}(p(W),\operatorname{\mathscr{F}}_{W})$

where $\alpha=(\alpha_{i})_{0\leq i\leq r}$ and for every $i$,
$\alpha_{i}\in\mathscr{I}$. Note that $\mathscr{I}^{r+1}$ is a finite set.

Since $\varinjlim_{W\in\mathscr{W}}$ is a filtered colimit, it is exact by
[14, Tag 04B0]. Note that finite products are coproducts in the category of
abelian groups. It follows that we have a spectral sequence

$\prod_{\alpha\in\mathscr{I}^{r+1}}\varinjlim_{W\in\mathscr{W}}H^{s}(\bigcap_{0\leq
i\leq
r}\overline{U_{\alpha_{i},W}},\operatorname{\mathscr{F}}_{W})\implies\varinjlim_{W\in\mathscr{W}}H^{r+s}(p(W),\operatorname{\mathscr{F}}_{W}).$

Similar arguments then show that we have a spectral sequence

$\prod_{\alpha\in\mathscr{I}^{r+1}}\varinjlim_{W\in\mathscr{W}}H^{s}(\bigcap_{0\leq
i\leq r}\overline{U_{\alpha_{i},W}}\cap
q(W),\operatorname{\mathscr{F}}_{W})\implies\varinjlim_{W\in\mathscr{W}}H^{r+s}(q(W),\operatorname{\mathscr{F}}_{W}).$

Note that the natural morphisms

$\prod_{\alpha\in\mathscr{I}^{r+1}}\varinjlim_{W\in\mathscr{W}}H^{s}(\bigcap_{0\leq
i\leq
r}\overline{U_{\alpha_{i},W}},\operatorname{\mathscr{F}}_{W})\to\prod_{\alpha\in\mathscr{I}^{r+1}}\varinjlim_{W\in\mathscr{W}}H^{s}(\bigcap_{0\leq
i\leq r}\overline{U_{\alpha_{i},W}}\cap q(W),\operatorname{\mathscr{F}}_{W})$

are compatible with the differentials and hence they induce the natural map

$\varinjlim_{W\in\mathscr{W}}H^{r+s}(p(W),\operatorname{\mathscr{F}}_{W})\to\varinjlim_{W\in\mathscr{W}}H^{r+s}(q(W),\operatorname{\mathscr{F}}).$

The hypothesis (22) shows that this is sufficient to conclude the proof. ∎

###### Proof.

(of Theorem 6.3) We can reduce to when $S$ is an affine scheme. Indeed, this
is a consequence of Lemma 4.1 and Remark 3.20. Let $\mathscr{J}$ denote the
category of standard étale neighbourhoods of $(S,\overline{s})$ and let
$(W,\overline{w})\in\mathscr{J}$. Note that the category $\mathscr{J}$ is
cofiltered. Observe that it suffices to prove (14) when the colimit on the
right runs over $\mathscr{J}$.

We proceed as in [8, Proposition 4.4.3]. We first reduce to the case when $X$
is affinoid and $\mathcal{O}_{X}(X)$ is a Tate ring. Since $f$ is a quasi-
compact morphism and $S$ is quasi-compact, we get that $X$ is quasi-compact.
Let $\\{U_{i}\\}_{i\in I}$ be a finite covering of $X$ by affinoid open
subsets such that for every $i\in I$, the morphism $U_{i}\to S$ is good. By
Remark 3.7, we can assume in addition that $\mathcal{O}_{U_{i}}(U_{i})$ is a
Tate ring. By Lemma 3.8, $\\{U_{i,W}:=U_{i}\times_{S}W\\}_{i\in I}$ is a
finite covering of $X_{W}$ by affinoid open subsets.

By Lemma 6.15, we deduce that to verify (14), it suffices to prove that for
every non-empty subset $J\subseteq I$ and every $q\in\mathbb{N}$, the natural
morphism

$\varinjlim_{(W,\overline{w})\in\mathscr{J}}H^{q}(({X_{W}},\bigcap_{j\in
J}\overline{|U_{j,W}|}),\operatorname{\mathscr{F}})\xrightarrow{}\varinjlim_{(W,\overline{w})\in\mathscr{J}}H^{q}(({X_{W}},\bigcap_{j\in
J}\overline{|U_{j,W}|}\cap f_{W}^{-1}(w)),\operatorname{\mathscr{F}})$

is an isomorphism.

Let $J\subseteq I$ be non-empty. Let $\\{V_{i}\\}_{i\in I_{J}}$ be a finite
affinoid cover of $\bigcap_{j\in J}U_{j}$. Hence,
$\\{\overline{|V_{i,W}|}\\}_{i\in I_{J}}$ is a finite cover of $\bigcap_{j\in
J}\overline{|U_{j,W}|}$ because $\overline{\bigcap_{j\in
J}|U_{j,W}|}=\bigcap_{j\in J}\overline{|U_{j,W}|}$. This is a consequence of
[8, Lemma 1.1.10(i)]. Then, as before, it suffices to show that for any
$J^{\prime}\subseteq I_{J}$, the natural morphism

$\varinjlim_{(W,\overline{w})\in\mathscr{J}}H^{q}(({X_{W}},\bigcap_{j^{\prime}\in
J^{\prime}}\overline{|V_{j^{\prime},W}|}),\operatorname{\mathscr{F}})\xrightarrow{}\varinjlim_{(W,\overline{w})\in\mathscr{J}}H^{q}((X_{W},\bigcap_{j^{\prime}\in
J^{\prime}}\overline{|V_{j^{\prime},W}|}\cap
f_{W}^{-1}(w)),\operatorname{\mathscr{F}})$

is an isomorphism.

We choose an index $j_{0}\in J$. Let $J^{\prime}\subseteq I_{J}$. Observe that
for every $j^{\prime}\in J^{\prime}$, $V_{j^{\prime}}\subseteq U_{j_{0}}$. It
follows that $\bigcap_{j^{\prime}\in J^{\prime}}V_{j^{\prime}}$ is an affinoid
space. Furthermore, by Remark 3.7, the composition $\bigcap_{j^{\prime}\in
J^{\prime}}V_{j^{\prime}}\to U_{j_{0}}\to S$ is good. In summary, it suffices
to prove that the natural map

$\varinjlim_{(W,\overline{w})\in\mathscr{J}}H^{q}(({X_{W}},\overline{|U_{W}|}),\operatorname{\mathscr{F}})\to\varinjlim_{(W,\overline{w})\in\mathscr{J}}H^{q}(({X_{W}},\overline{|U_{W}|}\cap
f_{W}^{-1}(s)),\operatorname{\mathscr{F}})$

is an isomorphism where $U$ is an affinoid open subset of $X$ and $U\to S$ is
good. In addition, we have that $\mathcal{O}_{U}(U)$ is a Tate ring.

Observe that $f_{|U}$ is good. We are hence in the context of Situation 6.9.
By Lemma 6.11, we can suppose $X$ is affinoid and is the universal
compactification of $U\to S$ with $\mathcal{O}_{U}(U)$ Tate. By Lemma 3.31(3),
$f\colon X\to S$ is tight and $\mathcal{O}_{X}(X)=\mathcal{O}_{U}(U)$ is Tate.

Suppose
$\operatorname{\mathscr{F}}\simeq\varinjlim_{i}\operatorname{\mathscr{F}}_{i}$
is a filtered colimit of torsion abelian sheaves. By Lemma 3.10 and [11,
Exposé VI, Theorem 5.1], we deduce that

$(R^{n}f_{*}\operatorname{\mathscr{F}})_{\overline{s}}\simeq\varinjlim_{i}((R^{n}f_{*}\operatorname{\mathscr{F}}_{i})_{\overline{s}}).$

By Lemma 3.8, $X_{W}$ is affinoid. Since $w$ is a closed point,
$f_{W}^{-1}(w)$ is a closed subspace of $X_{W}$. Observe that $f_{W}^{-1}(w)$
is a closed subspace of a spectral space. It is hence quasi-separated and
quasi-compact. By [8, Lemma 2.3.13(i)],

$H^{n}(f_{W}^{-1}(w),\operatorname{\mathscr{F}})\simeq\varinjlim_{i}H^{n}(f_{W}^{-1}(w),\operatorname{\mathscr{F}}_{i}).$

Just as in (8) of the proof of [8, Proposition 4.4.3], we can reduce to when
$\operatorname{\mathscr{F}}=p_{*}G$ where $p\colon Y\to X$ is a finite
morphism of pseudo-adic spaces and $G$ is a constant sheaf of
$\mathbb{Z}/m\mathbb{Z}$-modules of finite type for some $m\in\mathbb{N}$.

######  Lemma 6.16.

We can reduce to the case where there exists
$e_{0},\ldots,e_{n}\in\mathcal{O}_{\underline{Y}}(\underline{Y})$ such that
$\mathcal{O}_{\underline{Y}}(\underline{Y})=\sum_{i}e_{i}\mathcal{O}_{\underline{Y}}(\underline{Y})$
and $|Y|=\\{y\in\underline{Y}||e_{i}(y)|<|e_{0}(y)|\mbox{ for
}i=1,\ldots,n\\}$.

###### Proof.

Since the morphism $p$ is finite, $|Y|$ is closed in $\underline{Y}$. Hence
$|Y|=\bigcap_{i\in\mathscr{I}}C_{i}$, where $C_{i}\subset\underline{Y}$ is a
closed constructible set where we use the notion of constructible set as
introduced in [6, Chapitre 0, Définition 2.3.10]. We claim that we can reduce
to when $|Y|$ is closed and constructible. Note that Indeed, for every
$(W,\overline{w})\in\mathscr{J}$, let $Y_{W}:=Y\times_{X}X_{W}$ where we take
the fibre product in the category of pseudo-adic spaces. We then have that
$|Y_{W}|=\bigcap_{i\in\mathscr{I}}C_{i,W}$ where for every $i\in\mathscr{I}$,
$C_{i,W}$ is the preimage of $C_{i}$ in $\underline{Y_{W}}$ for the projection
$\underline{Y_{W}}\to\underline{Y}$. Since $p$ is finite, we deduce that to
show (14), it suffices to show that for every $q\in\mathbb{N}$, the natural
morphism

$\varinjlim_{(W,\overline{w})\in\mathscr{J}}H^{q}((\underline{Y_{W}},\bigcap_{i\in\mathscr{I}}C_{i,W}),G)\xrightarrow{}\varinjlim_{(W,\overline{w})\in\mathscr{J}}H^{q}((\underline{Y_{W}},\bigcap_{i\in\mathscr{I}}C_{i,W}\cap
f_{W}^{-1}(w)),G)$

is an isomorphism. Since colimits commute with colimits, we deduce from [8,
Corollary 2.4.6] that it suffices to verify that for every finite subset
$J\subseteq\mathscr{I}$ and $q\in\mathbb{N}$, the natural morphism

$\varinjlim_{(W,\overline{w})\in\mathscr{J}}H^{q}((\underline{Y_{W}},\bigcap_{j\in
J}C_{j,W}),G)\xrightarrow{}\varinjlim_{(W,\overline{w})\in\mathscr{J}}H^{q}((\underline{Y_{W}},\bigcap_{j\in
J}C_{j,W}\cap f_{W}^{-1}(w)),G)$

is an isomorphism. We may hence suppose that $|Y|$ is closed and
constructible. By [8, Lemma 3.1.10(ii)] and Lemma 6.15, we can conclude a
proof of the lemma. ∎

Note that since $X$ is affinoid, $\underline{Y}$ is affinoid as well. Let
$Y^{\prime}:=\underline{Y}$. Let $i$ denote the closed embedding
$Y\hookrightarrow Y^{\prime}$ and set $\mathscr{G}:=i_{*}(G)$. Let
$p^{\prime}$ denote the finite morphism $Y^{\prime}\to X$.

We claim that it suffices to prove the theorem for the specialization morphism
$f\circ p^{\prime}\colon Y^{\prime}\to S$ and the sheaf $\mathscr{G}$. Indeed,
observe firstly that by [8, Lemma 1.10.17(i)] the morphisms $i$ and
$p^{\prime}$ are proper. Hence,

(23)
$\operatorname{\mathscr{F}}_{|X_{W}}\xrightarrow{\sim}p^{\prime}_{W*}(\mathscr{G}_{|Y^{\prime}_{W}})$

where $p^{\prime}_{W}\colon Y^{\prime}_{W}\to X_{W}$ is the morphism induced
by base change. This implies an isomorphism

(24) $\displaystyle
H^{n}(X_{W},\operatorname{\mathscr{F}}_{|X_{W}})\xrightarrow{\sim}H^{n}(Y^{\prime}_{W},\mathscr{G}_{|Y^{\prime}_{W}}).$

On the other hand, we have a sequence of ismorphisms,

$\displaystyle\operatorname{\mathscr{F}}_{|f_{W}^{-1}(w)}$
$\displaystyle\overset{(i)}{\to}(p^{\prime}_{W*}(\mathscr{G}_{|Y^{\prime}_{W}}))_{|f_{W}^{-1}(w)}$
(25)
$\displaystyle\overset{(ii)}{\to}p^{\prime}_{0W*}\mathscr{G}_{|(f_{W}\circ
p^{\prime}_{W})^{-1}(w)}$

where $p^{\prime}_{0W}$ denotes the finite morphism $(f_{W}\circ
p^{\prime}_{W})^{-1}(w)\to f_{W}^{-1}(w)$. The isomorphism (i) is due to
equation (23). Note that $p^{\prime}_{W}$ is finite and hence proper. Hence,
[8, Theorem 4.1.2(b)] implies the isomorphism (ii). Equation (6) gives an
isomorphism

(26) $\displaystyle
H^{n}(f_{W}^{-1}(w),\operatorname{\mathscr{F}})\xrightarrow{\sim}H^{n}((f_{W}\circ
p^{\prime}_{W})^{-1}(w),\mathscr{G}).$

One deduces the claim using equations (24) and (26).

By Lemma 6.6, since $Y^{\prime}$ is finite over $X$ and $X$ is a universal
compactification, the composition $Y^{\prime}\to X\to S$ is tight. We now
apply Proposition 6.12 to conclude a proof of the theorem. ∎

######  Corollary 6.17.

Let $f\colon X\to S$ be a proper specialization morphism where $S$ is strictly
local i.e. $S=\mathrm{Spec}(R)$ where $R$ is a strictly henselian local ring.
Let $s\in S$ denote the closed point and $X_{s}:=f^{-1}(s)$ be the fibre over
$s$. Let $\operatorname{\mathscr{F}}$ be a torsion abelian sheaf. We then have
that, for every $n\in\mathbb{N}$,

$\displaystyle[R^{n}f_{*}(\operatorname{\mathscr{F}})]_{{s}}\simeq
H^{n}(X_{{s}},\operatorname{\mathscr{F}}_{|X_{{s}}}).$

###### Proof.

This is a direct consequence of Theorem 6.3. ∎

######  Corollary 6.18.

Let $f\colon X\to S$ be a proper specialization morphism. Let $s\in S$ be a
closed point and $\overline{s}\to S$ be a geometric point over $s$. Let
$\operatorname{\mathscr{F}}$ be a complex in
$\mathcal{D}^{+}(X_{\text{\'{e}t}},\mathbb{Z})$ whose cohomology sheaves are
torsion. For every $n\in\mathbb{Z}$, we have a natural isomorphism

$\displaystyle[R^{n}f_{*}(\operatorname{\mathscr{F}})]_{\overline{s}}\to\varinjlim_{(W,\overline{w})}R^{n}\Gamma(f_{W}^{-1}(w),\operatorname{\mathscr{F}})$

where the colimit on the right runs over all étale neighbourhoods
$(W,\overline{w})$ of $(S,\overline{s})$, $w\in W$ is the image of
$\overline{w}$, $X_{W}:=X\times_{S}W$ and $f_{W}\colon X_{W}\to W$ is the
morphism induced by base change.

###### Proof.

Our proof is similar to that of [14, Tag 0DDE]. Let $(W,\overline{w})$ be an
étale neighbourhood of $(S,\overline{s})$. Observe that we have spectral
sequences

$R^{p}f_{*}H^{q}(\operatorname{\mathscr{F}})\implies
R^{p+q}f_{*}(\operatorname{\mathscr{F}})$

and

$H^{p}(f_{W}^{-1}(w),H^{q}(\operatorname{\mathscr{F}}))\implies
R^{p+q}\Gamma(f_{W}^{-1}(w),\operatorname{\mathscr{F}}).$

Since $\varinjlim_{(W,\overline{w})}$ is a filtered colimit, it is exact by
[14, Tag 04B0]. Moreover, the functor of taking the stalk at $\overline{s}$ is
exact as well. It follows that we have spectral sequences

$[R^{p}f_{*}H^{q}(\operatorname{\mathscr{F}})]_{\overline{s}}\implies[R^{p+q}f_{*}(\operatorname{\mathscr{F}})]_{\overline{s}}$

and

$\varinjlim_{(W,\overline{w})}H^{p}(f_{W}^{-1}(w),H^{q}(\operatorname{\mathscr{F}}))\implies\varinjlim_{(W,\overline{w})}R^{p+q}\Gamma(f_{W}^{-1}(w),\operatorname{\mathscr{F}}).$

We now apply Theorem 6.3 to conclude the proof. ∎

##  7\. An application

Recall from §4 that we fixed a torsion ring $A$. As an application of our work
we prove the following theorem concerning type (S) formal schemes (cf. [8,
§1.9]). We remind the reader that given a type (S) formal scheme
$\mathfrak{X}$, there is the associated reduced scheme $\mathfrak{X}_{s}$ and
the analytic adic generic fiber $\mathfrak{X}_{\eta}$ (cf. the paragraph
immediately preceding Definition 2.17). Moreover there is the specialization
morphism $\lambda_{\mathfrak{X}}\colon\mathfrak{X}_{\eta}\to\mathfrak{X}_{s}$
and by Lemma 2.29, there is no ambiguity whether one considers the pushforward
along (3) or (4).

######  Theorem 7.1.

Let $\mathfrak{f}\colon\mathfrak{X}\to\mathfrak{Y}$ be a finite type and
separated morphism of type (S) formal schemes. Suppose in addition that
$\mathfrak{Y}$ is quasi-compact and quasi-separated. Then there is a canonical
equivalence

$\displaystyle R^{+}\mathfrak{f}_{s!}\circ
R^{+}\lambda_{\mathfrak{X}*}\xrightarrow{\sim}R^{+}\lambda_{\mathfrak{Y}*}\circ
R^{+}\mathfrak{f}_{\eta!}$

of functors $D^{+}(\mathfrak{X}_{\eta},A)\to D^{+}(\mathfrak{Y}_{s},A)$.

The idea for proving Theorem 7.1 is simple. One takes the compactification of
the morphism $\mathfrak{f}_{\eta}$ in the sense of [8, Definition 5.1.8] and
reduces Theorem 7.1 to the case of an open immersion, where the result is
known by Corollary 3.5.11(ii) in loc.cit. However the compactification of
$\mathfrak{f}_{\eta}$ may no longer be of finite type (in particular may not
possess a formal model). Thus the strategy is use the enlarged class of
specialization morphisms we have constructed and their compactifications (cf.
Theorem 3.30). First we need some preparation.

######  Situation 7.2.

_Let_

${X}$${Y}$${S}$${T}$$\scriptstyle{j_{1}}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{j_{2}}$

_be a commutative diagram where $\alpha$ and $\beta$ are proper specialization
morphisms, and $j_{1}$ and $j_{2}$ are open embeddings._

######  Lemma 7.3.

We work within the context of Situation 7.2. There is a natural equivalence

$j_{2!}j^{*}_{2}R^{+}\beta_{*}j_{1!}\to R^{+}\beta_{*}j_{1!}.$

###### Proof.

By adjointness, we have a natural morphism

$\Phi\colon j_{2!}j^{*}_{2}R^{+}\beta_{*}j_{1!}\to R^{+}\beta_{*}j_{1!}.$

Observe that it suffices to show that if
$\operatorname{\mathscr{F}}\in\mathcal{D}^{+}(X_{\text{\'{e}t}},A)$, $t\in
T\smallsetminus S$ and $n\in\mathbb{Z}$ then
$[R^{n}\beta_{*}j_{1!}(\operatorname{\mathscr{F}})]_{\overline{t}}=0$. By
Corollary 6.18, for every $n\in\mathbb{Z}$, we have an isomorphism

(27)
$[R^{n}\beta_{*}j_{1!}(\operatorname{\mathscr{F}})]_{\overline{t}}\xrightarrow{\sim}\varinjlim_{(W,\overline{w})}R^{n}\Gamma(\beta_{W}^{-1}(w),j_{1!}(\operatorname{\mathscr{F}}))$

where the colimit on the right runs over all étale neighbourhoods
$(W,\overline{w})$ of $(T,\overline{t})$. Let $(W,\overline{w})$ be an étale
neighbourhood of $(T,\overline{t})$ and let $X_{W}$ be the preimage of $X$ for
the morphism $Y_{W}\to Y$. If $j_{1W}$ denotes the open embedding
$X_{W}\hookrightarrow Y_{W}$ then by [8, Theorem 5.2.2(iv)], we have an
isomorphism

$(j_{1!}(\operatorname{\mathscr{F}}))_{|Y_{W}}\xrightarrow{\sim}j_{1W!}(\operatorname{\mathscr{F}}_{|X_{W}}).$

Since $\beta_{W}^{-1}(w)\cap X_{W}=\emptyset$, we see that
$R^{n}\Gamma(\beta_{W}^{-1}(w),j_{1!}(\operatorname{\mathscr{F}}))=0$. This
concludes the proof. ∎

######  Corollary 7.4.

We work within the context of Situation 7.2.

  1. (1)

Let $I$ be an injective $A$-module on $X_{\text{\'{e}t}}$. Then, for every
$n\geq 1$,

$R^{n}\beta_{*}j_{1!}(I)=0.$

  2. (2)

There is a natural equivalence

$R^{+}\beta_{*}\circ j_{1!}\xrightarrow{\sim}j_{2!}\circ R^{+}\alpha_{*}.$

###### Proof.

We verify part (1) as follows. Let $Z$ be the fibre product $Y\times_{T}S$,
which exists by Proposition 2.19. Consider the commutative diagram coming from
the universal property of fibre products

${X}$${Z}$${Y}$${S}$${T.}$$\scriptstyle{j_{1}}$$\scriptstyle{j_{3}}$$\scriptstyle{\alpha}$$\scriptstyle{j_{4}}$$\scriptstyle{\gamma}$$\scriptstyle{\beta}$$\scriptstyle{j_{2}}$

Note that since $j_{1}$ and $j_{4}$ are étale, so is $j_{3}$ by [8,
Proposition 1.6.7(iii)]. By Remark 3.20, since $\beta$ is proper, $\gamma$ is
proper. Since $\alpha$ is proper as well, it follows that $j_{3}$ is both
quasi-compact and separated (by similar arguments appearing in the proof of
Lemma 5.2).

Then

$\displaystyle j_{2}^{*}\circ R^{+}\beta_{*}\circ j_{1!}$
$\displaystyle\overset{(i)}{=}R^{+}\gamma_{*}\circ j_{4}^{*}\circ j_{1!}$
$\displaystyle\overset{(ii)}{=}R^{+}\gamma_{*}\circ j_{3!}$ (28)
$\displaystyle\overset{(iii)}{=}R^{+}\alpha_{*}$

where (i) follows from Lemma 4.1, (ii) follows from $j_{1}=j_{4}\circ j_{3}$
and (iii) follows from Lemma 5.3 after noting that $j_{3}$ is quasi-compact
because $\alpha$ is quasi-compact and $\gamma$ is quasi-separated. Hence, we
get that for every $n\geq 1$,

$j_{2}^{*}R^{n}\beta_{*}j_{1!}(I)=R^{n}\alpha_{*}(I)=0.$

It remains to show that $R^{n}\beta_{*}j_{1!}(I)$ vanishes outside of $S$.
This is an immediate consequence of Lemma 7.3. Hence,
$R^{n}\beta_{*}j_{1!}(I)=0$ everywhere.

We now verify part (2). Recall that we have a natural morphism $j_{!}\to
j_{*}$ which implies a natural map $\beta_{*}j_{1!}\to\beta_{*}j_{1*}$. Since
$\beta\circ j_{1}=j_{2}\circ\alpha$, $\beta_{*}j_{1*}=j_{2*}\alpha_{*}$ and
hence we have a natural morphism

$\beta_{*}j_{1!}\to j_{2*}\alpha_{*}.$

Applying $j_{2!}j^{*}_{2}$, we get a morphism

$j_{2!}j^{*}_{2}\beta_{*}j_{1!}\to j_{2!}j^{*}_{2}j_{2*}\alpha_{*}.$

By Lemma 7.3, we get a natural morphism

$\epsilon\colon\beta_{*}j_{1!}\to j_{2!}\alpha_{*}.$

We claim that $\epsilon$ is an equivalence. Indeed, observe that Equation (7)
shows that this is true over $S$ while Lemma 7.3 shows that this true outside
of $S$. Part (1) of the lemma implies that
$R^{+}(\beta_{*}j_{1!})=R^{+}\beta_{*}j_{1!}$. Since $\epsilon$ is a natural
equivalence, we have that the induced morphism

$R^{+}\beta_{*}j_{1!}\to j_{2!}R^{+}\alpha_{*}$

is an isomorphism.

∎

######  Corollary 7.5.

Let $\alpha\colon X\to S$ be a separated and finite type specialization
morphism. Let $g\colon S\to T$ be a separated and finite type morphism of
schemes. Then $g\circ\alpha$ is a separated and finite type specialization
morphism. Moreover, if $T$ is quasi-compact and quasi-separated, there is a
natural equivalence

$R^{+}(g\circ\alpha)_{!}\simeq R^{+}g_{!}\circ R^{+}\alpha_{!}$

of functors
$\mathcal{D}^{+}(X_{\text{\'{e}t}},A)\to\mathcal{D}^{+}(T_{\text{\'{e}t}},A)$.

###### Proof.

The strategy is very similar to the proof of Lemma 5.3. The fact that the
composition $g\circ\alpha$ is a separated and finite type specialization
morphism follows from Proposition 3.22. For the second part consider the
following diagram

${X}$${\overline{X}^{/S}}$${Z}$${S}$${U}$${T}$$\scriptstyle{j_{1}}$$\scriptstyle{\alpha}$$\scriptstyle{\beta_{1}}$$\scriptstyle{j_{2}}$$\scriptstyle{\beta_{2}}$$\scriptstyle{j_{3}}$$\scriptstyle{g}$$\scriptstyle{h}$

where the top left triangle is the universal compactification of $\alpha$ via
Theorem 3.30, the bottom right triangle is a compactification of $g$ via [3,
Theorem 4.1]. Let us explain the top right square. The specialization morphism
$j_{3}\circ\beta_{1}\colon\overline{X}^{/S}\to U$ is separated and of finite
type by Proposition 3.22. Thus again by Theorem 3.30, it admits a universal
compactification which we denote by the triple $(Z,\beta_{2},j_{2})$. In
particular all the horizontal arrows (in the diagram) are quasi-compact open
embeddings and the vertical arrows are proper. Therefore the composition
$j_{2}\circ j_{1}$ is a quasi-compact open embedding of analytic adic spaces
and by Proposition 3.22, the composition $h\circ\beta_{2}$ is a proper
specialization morphism. Thus the outer triangle is a compactification of
$g\circ\alpha$.

Then

$R^{+}g_{!}\circ R^{+}\alpha_{!}=R^{+}h_{*}\circ j_{3!}\circ
R^{+}\beta_{1*}\circ j_{1!}$

and

$\displaystyle R^{+}(g\circ\alpha)_{!}$
$\displaystyle\overset{(i)}{=}R^{+}(h\circ\beta_{2})_{*}\circ(j_{2}\circ
j_{1})_{!}$ $\displaystyle\overset{(ii)}{=}R^{+}h_{*}\circ
R^{+}\beta_{2*}\circ j_{2!}\circ j_{1!}$
$\displaystyle\overset{(iii)}{=}R^{+}h_{*}\circ j_{3!}\circ
R^{+}\beta_{1*}\circ j_{1!}$

where (i) follows from Lemma 5.2, (ii) follows from Proposition 2.28 and [8,
Theorem 5.4.3], and (iii) follows from Corollary 7.4. This completes the
proof. ∎

We now arrive at the promised application of the theory developed thus far.

###### Proof.

(of Theorem 7.1) We have a commutative diagram

${\mathfrak{X}_{\eta}}$${\mathfrak{X}_{s}}$${\mathfrak{Y}_{\eta}}$${\mathfrak{Y}_{s}}$$\scriptstyle{\lambda_{\mathfrak{X}}}$$\scriptstyle{\mathfrak{f}_{\eta}}$$\scriptstyle{\mathfrak{f}_{s}}$$\scriptstyle{\lambda_{\mathfrak{Y}}}$

where we remind the reader that $\lambda_{\mathfrak{X}}$ (resp.
$\lambda_{\mathfrak{Y}}$) are the specialization morphisms, in the sense of
Definition 2.17 induced by the morphisms of locally ringed spaces
$(\mathfrak{X}_{\eta},\mathcal{O}_{\mathfrak{X}_{\eta}}^{+})\to(\mathfrak{X},\mathcal{O}_{\mathfrak{X}})$
(resp.
$(\mathfrak{Y}_{\eta},\mathcal{O}_{\mathfrak{Y}_{\eta}}^{+})\to(\mathfrak{Y},\mathcal{O}_{\mathfrak{Y}})$),
cf. §2. The composition
$\mathfrak{f}_{s}\circ\lambda_{\mathfrak{X}}=\lambda_{\mathfrak{Y}}\circ\mathfrak{f}_{\eta}$
is a separated and finite type specialization morphism (because both
$\lambda_{\mathfrak{X}}$ and $\lambda_{\mathfrak{Y}}$ are proper by
Proposition 3.24, and both $\mathfrak{f}_{s}$ and $\mathfrak{f}_{\eta}$ are
separated and of finite type by assumption on $\mathfrak{f}$). Thus by Theorem
3.30 it admits a (universal) compactification. By Definition 5.1/Lemma 5.2, we
have a well defined
$R^{+}(\mathfrak{f}_{s}\circ\lambda_{\mathfrak{X}})_{!}=R^{+}(\lambda_{\mathfrak{Y}}\circ\mathfrak{f}_{\eta})_{!}$
right derived lower shriek functor. We compute

$\displaystyle R^{+}(\mathfrak{f}_{s}\circ\lambda_{\mathfrak{X}})_{!}$
$\displaystyle\overset{(i)}{=}R^{+}\mathfrak{f}_{s!}\circ
R^{+}\lambda_{\mathfrak{X}!}$
$\displaystyle\overset{(ii)}{=}R^{+}\mathfrak{f}_{s!}\circ
R^{+}\lambda_{\mathfrak{X}*}$

where (i) follows from Corollary 7.5, (ii) follows from the fact that
$\lambda_{\mathfrak{X}}$ is proper (cf. Proposition 3.24).

Similarly one obtains
$R^{+}(\lambda_{\mathfrak{Y}}\circ\mathfrak{f}_{\eta})_{!}=R^{+}\lambda_{\mathfrak{Y}*}\circ
R^{+}\mathfrak{f}_{\eta!}$ where one uses Lemma 5.3 in place of Corollary 7.5.
The result now follows. ∎

We end this section with the following remark.

######  Remark 7.6.

_Let $\alpha\colon X\to S$ be a separated, taut and locally of finite type
specialization morphism. Then as in Definition 5.1, one can define a functor
$R^{+}\alpha_{!}$ with the expected properties. Moreover if $X$ satisfies some
finiteness conditions (e.g. $X$ is locally of finite type over an
algebraically closed valued field $k$), then the functor_

$R^{+}\alpha_{!}\colon\mathcal{D}^{+}(X_{\text{\'{e}t}},A)\to\mathcal{D}^{+}(S_{\text{\'{e}t}},A)$

_admits a right adjoint functor_

$R^{+}\alpha^{!}\colon\mathcal{D}^{+}(S_{\text{\'{e}t}},A)\to\mathcal{D}^{+}(X_{\text{\'{e}t}},A).$

_This places[5, Corollary 4.3(1)] into a larger framework._

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Ildar Gaisin
Graduate School of Mathematical Sciences, The University of Tokyo,
3-8-1 Komaba, Meguro, Tokyo, 153-0041, Japan.
email<EMAIL_ADDRESS>

John Welliaveetil
Kavli Institute for the Physics and Mathematics of the Universe, The
University of Tokyo,
5-1-5 Kashiwanoha Kashiwa, 277-8583, Japan
email<EMAIL_ADDRESS>