object of $\mathscr{C}_{\mathrm{car}}$ over $U\to T$.

(2) It follows immediately from the arguments above.

(3) Consider the category $\mathscr{D}$ of morphisms of
$\mathscr{C}_{\mathrm{car}}$,

(9.8.5)

such that $K^{\prime}$ is a finite field extension of $K$. Similarly, this
category is also cofiltered with limit of diagrams of schemes
$(\mathop{\mathrm{Spec}}(L)\to\mathop{\mathrm{Spec}}(\mathcal{O}_{L}))\to(\mathop{\mathrm{Spec}}(\overline{K})\to\mathop{\mathrm{Spec}}(\mathcal{O}_{\overline{K}}))$.
It suffices to show that the full subcategory of $\mathscr{D}$ formed by
strictly semi-stable objects is initial. For any object $U\to T$ of
$\mathscr{C}_{\mathrm{car}}$, by de Jong’s alteration theorem [dJ96, 6.5],
there exists a proper surjective and generically finite morphism
$T^{\prime}\to T$ of integral schemes such that
$U^{\prime}=U\times_{T}T^{\prime}\to T^{\prime}$ is strictly semi-stable over
$\mathop{\mathrm{Spec}}(K^{\prime})\to\mathop{\mathrm{Spec}}(\mathcal{O}_{K^{\prime}})$
for a finite field extension $K\to K^{\prime}$. Since $L$ is algebraically
closed, the dominant morphism $\mathop{\mathrm{Spec}}(L)\to U$ lifts to a
dominant morphism $\mathop{\mathrm{Spec}}(L)\to U^{\prime}$ (9.6), which
further extends to a lifting $\mathop{\mathrm{Spec}}(\mathcal{O}_{L})\to
T^{\prime}$ of $\mathop{\mathrm{Spec}}(\mathcal{O}_{L})\to T$ by the valuative
criterion. After replacing $T^{\prime}$ by an affine open neighborhood of the
image of the closed point of $\mathop{\mathrm{Spec}}(\mathcal{O}_{L})$, we
obtain a strictly semi-stable object of $\mathscr{D}$ over $(U\to
T)\to(\mathop{\mathrm{Spec}}(K)\to\mathop{\mathrm{Spec}}(\mathcal{O}_{K}))$,
which completes the proof. ∎

######  Theorem 9.9 ([ILO14, X 3.5, 3.7]).

Let $K$ be a complete discrete valuation field with valuation ring
$\mathcal{O}_{K}$, $(Y\to X)\to(U\to T)$ a morphism of dominant open
immersions over
$\mathop{\mathrm{Spec}}(K)\to\mathop{\mathrm{Spec}}(\mathcal{O}_{K})$ between
irreducible $\mathcal{O}_{K}$-schemes of finite type such that $X\to T$ is
proper surjective. Then, there exists a commutative diagram of dominant open
immersions between irreducible $\mathcal{O}_{K}$-schemes of finite type

(9.9.5)

satisfying the following conditions:

  1. (i)

We have $Y^{\prime}=\beta^{-1}(Y)\cap f^{\prime-1}(U^{\prime})$, i.e.
$Y^{\prime}\to X^{\prime}$ is Cartesian over $U^{\prime}\times_{U}Y\to
T^{\prime}\times_{T}X$ (cf. 7.1).

  2. (ii)

The morphism $(X^{\prime},\mathscr{M}_{Y^{\prime}\to
X^{\prime}})\to(T^{\prime},\mathscr{M}_{U^{\prime}\to T^{\prime}})$ induced by
$(f^{\prime\circ},f^{\prime})$ is a smooth and saturated morphism of fine,
saturated and regular log schemes.

  3. (iii)

The morphisms $\alpha$ and $\beta$ are proper surjective and generically
finite, and $f^{\prime}$ is projective surjective.

###### Proof.

We may assume that $T$ is nonempty. Recall that
$\mathop{\mathrm{Spec}}(\mathcal{O}_{K})$ is universally
$\mathbb{Q}$-resolvable ([ILO14, X.3.3]) by de Jong’s alteration theorem
[dJ96, 6.5]. Thus, $T$ is also universally $\mathbb{Q}$-resolvable by [ILO14,
X 3.5, 3.5.2] so that we can apply [ILO14, X.3.5] to the proper surjective
morphism $f$ and the nowhere dense closed subset $X\setminus Y$. Then, we
obtain a commutative diagram of schemes

(9.9.10)

and dense open subsets $U^{\prime}\subseteq T^{\prime}$,
$Y^{\prime}=\beta^{-1}(Y)\cap f^{\prime-1}(U^{\prime})\subseteq X^{\prime}$
such that $(X^{\prime},\mathscr{M}_{Y^{\prime}\to X^{\prime}})$ and
$(T^{\prime},\mathscr{M}_{U^{\prime}\to T^{\prime}})$ are fine, saturated and
regular, that $(X^{\prime},\mathscr{M}_{Y^{\prime}\to
X^{\prime}})\to(T^{\prime},\mathscr{M}_{U^{\prime}\to T^{\prime}})$ is smooth,
that $\alpha,\beta$ are proper surjective and generically finite morphisms
which map generic points to generic points, and that $f^{\prime}$ is
projective (since $f$ is proper, cf. [ILO14, X 3.1.6, 3.1.7]). Since $X$
(resp. $T$) is irreducible and $X^{\prime}$ (resp. $T^{\prime}$) is a disjoint
union of normal integral schemes (9.4), after firstly replacing $X^{\prime}$
by an irreducible component and then replacing $T^{\prime}$ by the irreducible
component under $X^{\prime}$, we may assume that $X^{\prime}$ and $T^{\prime}$
are irreducible. Then, $Y^{\prime}\to U^{\prime}$ is dominant (so that
$f^{\prime}$ is projective surjective), since it is smooth and $Y^{\prime}$ is
nonempty ([EGA IV2, 2.3.4]). We claim that $\alpha$ maps $U^{\prime}$ into
$U$. Indeed, if there exists a point $u\in U^{\prime}$ with $\alpha(u)\notin
U$, then $f^{\prime-1}(u)\cap Y^{\prime}=\emptyset$. However, endowing $u$
with the trivial log structure, the log scheme
$(u,\mathcal{O}_{u_{\mathrm{\acute{e}t}}}^{\times})$ is fine, saturated and
regular, and the fibred product in the category of fine and saturated log
schemes

(9.9.11)
$\displaystyle(u,\mathcal{O}_{u_{\mathrm{\acute{e}t}}}^{\times})\times_{(T^{\prime},\mathscr{M}_{U^{\prime}\to
T^{\prime}})}^{\mathrm{fs}}(X^{\prime},\mathscr{M}_{Y^{\prime}\to
X^{\prime}})$

is regular with underlying scheme $f^{\prime-1}(u)$ (9.4, 9.2). Thus,
$f^{\prime-1}(u)\cap Y^{\prime}$ is dense in $f^{\prime-1}(u)$, which
contradicts the assumption that $f^{\prime-1}(u)\cap Y^{\prime}=\emptyset$
since $f^{\prime}$ is surjective. Thus, we obtain a diagram (9.9.5) satisfying
all the conditions except the saturatedness of
$(X^{\prime},\mathscr{M}_{Y^{\prime}\to
X^{\prime}})\to(T^{\prime},\mathscr{M}_{U^{\prime}\to T^{\prime}})$.

To make $(X^{\prime},\mathscr{M}_{Y^{\prime}\to
X^{\prime}})\to(T^{\prime},\mathscr{M}_{U^{\prime}\to T^{\prime}})$ saturated,
we apply [ILO14, X.3.7] to the morphism $(f^{\prime\circ},f^{\prime})$. We
obtain a Cartesian morphism $(\gamma^{\circ},\gamma):(U^{\prime\prime}\to
T^{\prime\prime})\to(U^{\prime}\to T^{\prime})$ of dominant open immersions
such that $(T^{\prime\prime},\mathscr{M}_{U^{\prime\prime}\to
T^{\prime\prime}})$ is a fine, saturated and regular log scheme, that $\gamma$
is a proper surjective and generically finite morphism which maps generic
points of $T^{\prime\prime}$ to the generic point of $T^{\prime}$, and that
the fibred product in the category of fine and saturated log schemes

(9.9.12) $\displaystyle(T^{\prime\prime},\mathscr{M}_{U^{\prime\prime}\to
T^{\prime\prime}})\times^{\mathrm{fs}}_{(T^{\prime},\mathscr{M}_{U^{\prime}\to
T^{\prime}})}(X^{\prime},\mathscr{M}_{Y^{\prime}\to X^{\prime}})$

is saturated over $(T^{\prime\prime},\mathscr{M}_{U^{\prime\prime}\to
T^{\prime\prime}})$. The fibred product (9.9.12) is still smooth over
$(T^{\prime\prime},\mathscr{M}_{U^{\prime\prime}\to T^{\prime\prime}})$, and
thus it is regular (9.4). Let $X^{\prime\prime}$ be the underlying scheme of
it and let $Y^{\prime\prime}=(X^{\prime\prime})^{\mathrm{tr}}$. Then, the
fibred product (9.9.12) is isomorphic to
$(X^{\prime\prime},\mathscr{M}_{Y^{\prime\prime}\to X^{\prime\prime}})$ (9.4).
Thus, we obtain a commutative diagram of dominant open immersions of schemes

(9.9.17)

Notice that $Y^{\prime\prime}=U^{\prime\prime}\times_{U^{\prime}}Y^{\prime}$
and $X^{\prime\prime}\to T^{\prime\prime}\times_{T^{\prime}}X^{\prime}$ is
finite, and that $Y^{\prime\prime}\to X^{\prime\prime}$ is Cartesian over
$U^{\prime\prime}\times_{U^{\prime}}Y^{\prime}\to
T^{\prime\prime}\times_{T^{\prime}}X^{\prime}$ (9.2). Thus, we see that
$Y^{\prime\prime}\to X^{\prime\prime}$ is Cartesian over
$U^{\prime\prime}\times_{U}Y\to T^{\prime\prime}\times_{T}X$ and that
$f^{\prime\prime}$ is projective. Since $T^{\prime}$ (resp. $X^{\prime}$) is
irreducible and $T^{\prime\prime}$ (resp. $X^{\prime\prime}$) is a disjoint
union of normal integral schemes (9.4), after firstly replacing
$T^{\prime\prime}$ by an irreducible component and then replacing
$X^{\prime\prime}$ by an irreducible component on which the restriction of
$\delta^{\circ}$ is dominant, we may assume that $T^{\prime\prime}$ and
$X^{\prime\prime}$ are irreducible. In particular, $\delta$ is generically
finite and so is $\beta\circ\delta$ (9.6), and again $Y^{\prime\prime}\to
U^{\prime\prime}$ is dominant so that $f^{\prime\prime}$ is projective
surjective. ∎

######  Lemma 9.10.

Let $X$ be a scheme of finite type over a valuation ring $A$ of height $1$.
Then, the underlying topological space of $X$ is Noetherian.

###### Proof.

Let $\eta$ and $s$ be the generic point and closed point of
$\mathop{\mathrm{Spec}}(A)$ respectively. Then, the generic fibre $X_{\eta}$
and the special fibre $X_{s}$ are both Noetherian. As a union of $X_{\eta}$
and $X_{s}$, the underlying topological space of $X$ is also Noetherian
([Sta22, 0053]). ∎

######  Proposition 9.11.

With the notation in 9.7 and 9.8, let $Y\to X$ be a quasi-compact dominant
open immersion over
$\mathop{\mathrm{Spec}}(L)\to\mathop{\mathrm{Spec}}(\mathcal{O}_{L})$ such
that $X\to\mathop{\mathrm{Spec}}(\mathcal{O}_{L})$ is proper of finite
presentation. Then, there exists a proper surjective
$\mathcal{O}_{L}$-morphism of finite presentation $X^{\prime}\to X$, an index
$\lambda_{1}\in\Lambda$, and a directed inverse system of open immersions
$(Y^{\prime}_{\lambda}\to X^{\prime}_{\lambda})_{\lambda\geq\lambda_{1}}$ over
$(U_{\lambda}\to T_{\lambda})_{\lambda\geq\lambda_{1}}$ satisfying the
following conditions for each $\lambda\geq\lambda_{1}$:

  1. (i)

We have
$Y^{\prime}=Y\times_{X}X^{\prime}=\lim_{\lambda\geq\lambda_{1}}Y^{\prime}_{\lambda}$
and $X^{\prime}=\lim_{\lambda\geq\lambda_{1}}X^{\prime}_{\lambda}$.

  2. (ii)

The log scheme $(X^{\prime}_{\lambda},\mathscr{M}_{Y^{\prime}_{\lambda}\to
X^{\prime}_{\lambda}})$ is fine, saturated and regular.

  3. (iii)

The morphism $(X^{\prime}_{\lambda},\mathscr{M}_{Y^{\prime}_{\lambda}\to
X^{\prime}_{\lambda}})\to(T_{\lambda},\mathscr{M}_{U_{\lambda}\to
T_{\lambda}})$ is smooth and saturated, and $X^{\prime}_{\lambda}\to
T_{\lambda}$ is projective.

  4. (iv)

If moreover
$Y=\mathop{\mathrm{Spec}}(L)\times_{\mathop{\mathrm{Spec}}(\mathcal{O}_{L})}X$,
then we can require that
$Y^{\prime}_{\lambda}=U_{\lambda}\times_{T_{\lambda}}X^{\prime}_{\lambda}$.

###### Proof.

We follow closely the proof of [ALPT19, 5.2.19]. Since the underlying
topological space of $X$ is Noetherian by 9.10, each irreducible component $Z$
of $X$ admits a closed subscheme structure such that $Z\to X$ is of finite
presentation ([Sta22, 01PH]). After replacing $X$ by the disjoint union of its
irreducible components, we may assume that $X$ is irreducible. Then, the
generic fibre of $X\to\mathop{\mathrm{Spec}}(\mathcal{O}_{L})$ is also
irreducible as an open subset of $X$. Using [EGA IV3, 8.8.2, 8.10.5], there
exists an index $\lambda_{0}\in\Lambda$, a proper $T_{\lambda_{0}}$-scheme
$X_{\lambda_{0}}$, and an open subscheme $Y_{\lambda_{0}}$ of
$U_{\lambda_{0}}\times_{T_{\lambda_{0}}}X_{\lambda_{0}}$, such that
$X=\mathop{\mathrm{Spec}}(\mathcal{O}_{L})\times_{T_{\lambda_{0}}}X_{\lambda_{0}}$
and that $Y=\mathop{\mathrm{Spec}}(L)\times_{U_{\lambda_{0}}}Y_{\lambda_{0}}$.
Let $\eta$ denote the generic point of $X$, $\eta_{\lambda_{0}}$ the image of
$\eta$ under the morphism $X\to X_{\lambda_{0}}$, $Z_{\lambda_{0}}$ the scheme
theoretic closure of $\eta_{\lambda_{0}}$ in $X_{\lambda_{0}}$. Notice that
$\mathop{\mathrm{Spec}}(\mathcal{O}_{L})\times_{T_{\lambda_{0}}}Z_{\lambda_{0}}\to
X$ is a surjective finitely presented closed immersion. After replacing $X$ by
$\mathop{\mathrm{Spec}}(\mathcal{O}_{L})\times_{T_{\lambda_{0}}}Z_{\lambda_{0}}$
and replacing $X_{\lambda_{0}}$ by $Z_{\lambda_{0}}$, we may assume that $X\to
X_{\lambda_{0}}$ is a dominant morphism of irreducible schemes. Since
$T_{\lambda_{0}}$ is irreducible and $L$ is algebraically closed, the generic
fibre of $f:X_{\lambda_{0}}\to T_{\lambda_{0}}$ is geometrically irreducible.
In particular, if $\xi_{\lambda_{0}}$ (resp. $\eta_{\lambda_{0}}$) denotes the
generic point of $T_{\lambda_{0}}$ (resp. $X_{\lambda_{0}}$), then
$\eta=\mathop{\mathrm{Spec}}(L)\times_{\xi_{\lambda_{0}}}\eta_{\lambda_{0}}$
([EGA IV2, 4.5.9]). In the situation of (iv), we can moreover assume that
$Y_{\lambda_{0}}=U_{\lambda_{0}}\times_{T_{\lambda_{0}}}X_{\lambda_{0}}$.

By 9.9, there exists a commutative diagram of dominant open immersions of
irreducible schemes,

(9.11.5)

where $Y^{\prime}_{\lambda_{0}}\to X^{\prime}_{\lambda_{0}}$ is Cartesian over
$U^{\prime}_{\lambda_{0}}\times_{U_{\lambda_{0}}}Y_{\lambda_{0}}\to
T^{\prime}_{\lambda_{0}}\times_{T_{\lambda_{0}}}X_{\lambda_{0}}$, and where
$(X^{\prime}_{\lambda_{0}},\mathscr{M}_{Y^{\prime}_{\lambda_{0}}\to
X^{\prime}_{\lambda_{0}}})\to(T^{\prime}_{\lambda_{0}},\mathscr{M}_{U^{\prime}_{\lambda_{0}}\to
T^{\prime}_{\lambda_{0}}})$ is a smooth and saturated morphism of fine,
saturated and regular log schemes, and where $\alpha$ and $\beta$ are proper
surjective and generically finite, and where $f^{\prime}$ is projective
surjective. We take a dominant morphism
$\gamma^{\circ}:\mathop{\mathrm{Spec}}(L)\to U_{\lambda_{0}}^{\prime}$ which
lifts $\mathop{\mathrm{Spec}}(L)\to U_{\lambda_{0}}$ since $L$ is
algebraically closed and $\alpha$ is generically finite, the morphism
$\mathop{\mathrm{Spec}}(\mathcal{O}_{L})\to T_{\lambda_{0}}$ lifts to
$\gamma:\mathop{\mathrm{Spec}}(\mathcal{O}_{L})\to T^{\prime}_{\lambda_{0}}$
by the valuative criterion. We set
$Y^{\prime}=\mathop{\mathrm{Spec}}(L)\times_{U^{\prime}_{\lambda_{0}}}Y^{\prime}_{\lambda_{0}}$
and
$X^{\prime}=\mathop{\mathrm{Spec}}(\mathcal{O}_{L})\times_{T^{\prime}_{\lambda_{0}}}X^{\prime}_{\lambda_{0}}$.
It is clear that $Y^{\prime}\to X^{\prime}$ is Cartesian over $Y\to X$ by base
change. Let $\xi^{\prime}_{\lambda_{0}}$ (resp. $\eta^{\prime}_{\lambda_{0}}$)
be the generic points of $T^{\prime}_{\lambda_{0}}$ (resp.
$X^{\prime}_{\lambda_{0}}$). Since the generic fibre of $f$ is geometrically
irreducible,
$\xi^{\prime}_{\lambda_{0}}\times_{\xi_{\lambda_{0}}}\eta_{\lambda_{0}}$ is a
single point and $\eta^{\prime}_{\lambda_{0}}$ maps to it ([EGA IV2, 4.5.9]).
Since $\mathop{\mathrm{Spec}}(L)\times_{\xi_{\lambda_{0}}}\eta_{\lambda_{0}}$
is the generic point of $X$, we see that $X^{\prime}\to X$ is proper
surjective and of finite presentation. It remains to construct
$(Y^{\prime}_{\lambda}\to X^{\prime}_{\lambda})_{\lambda\geq\lambda_{1}}$.

After replacing $T^{\prime}_{\lambda_{0}}$ by an affine open neighborhood of
the image of the closed point of $\mathop{\mathrm{Spec}}(\mathcal{O}_{L})$,
lemma 9.8 implies that there exists an index $\lambda_{1}\geq\lambda_{0}$ such
that the transition morphism $(U_{\lambda_{1}}\to
T_{\lambda_{1}})\to(U_{\lambda_{0}}\to T_{\lambda_{0}})$ factors through
$(U^{\prime}_{\lambda_{0}}\to T^{\prime}_{\lambda_{0}})$. For each index
$\lambda\geq\lambda_{1}$, consider the fibred product in the category of fine
and saturated log schemes

(9.11.6)
$\displaystyle(X^{\prime}_{\lambda},\mathscr{M}_{Y^{\prime}_{\lambda}\to
X^{\prime}_{\lambda}})=(T_{\lambda},\mathscr{M}_{U_{\lambda}\to
T_{\lambda}})\times_{(T^{\prime}_{\lambda_{0}},\mathscr{M}_{U^{\prime}_{\lambda_{0}}\to
T^{\prime}_{\lambda_{0}}})}^{\mathrm{fs}}(X^{\prime}_{\lambda_{0}},\mathscr{M}_{Y^{\prime}_{\lambda_{0}}\to
X^{\prime}_{\lambda_{0}}}),$

which is a fine, saturated and regular log scheme smooth and saturated over
$(T_{\lambda},\mathscr{M}_{U_{\lambda}\to T_{\lambda}})$ (9.2, 9.4). Moreover,
we have
$Y^{\prime}_{\lambda}=U_{\lambda}\times_{U^{\prime}_{\lambda_{0}}}Y^{\prime}_{\lambda_{0}}$,
$X^{\prime}_{\lambda}=T_{\lambda}\times_{T^{\prime}_{\lambda_{0}}}X^{\prime}_{\lambda_{0}}$,
and in the situation of (iv),
$Y^{\prime}_{\lambda}=U_{\lambda}\times_{T_{\lambda}}X^{\prime}_{\lambda}$ by
base change. Therefore, $(Y^{\prime}_{\lambda}\to
X^{\prime}_{\lambda})_{\lambda\geq\lambda_{1}}$ meets our requirements. ∎

##  10\. Faltings’ Main $p$-adic Comparison Theorem: the Absolute Case

######  Lemma 10.1.

Let $Y$ be a coherent scheme, $V$ a finite étale $Y$-scheme. Then, there
exists a finite étale surjective morphism $Y^{\prime}\to Y$ such that
$Y^{\prime}\times_{Y}V$ is isomorphic to a finite disjoint union of
$Y^{\prime}$.

###### Proof.

If $Y$ is connected, let $\overline{y}$ be a geometric point of $Y$,
$\pi_{1}(Y,\overline{y})$ the fundamental group of $Y$ with base point
$\overline{y}$. Then, $Y_{\mathrm{f\acute{e}t}}$ is equivalent to the category
of finite $\pi_{1}(Y,\overline{y})$-sets so that the lemma holds ([Sta22,
0BND]).

In general, for any connected component $Z$ of $Y$, let
$(Y_{\lambda})_{\lambda\in\Lambda_{Z}}$ be the directed inverse system of all
open and closed subschemes of $Y$ which contain $Z$ and whose transition
morphisms are inclusions. Notice that
$\lim_{\lambda\in\Lambda_{Z}}Y_{\lambda}$ is a closed subscheme of $Y$ with
underlying topological space $Z$ by [Sta22, 04PL] and [EGA IV3, 8.2.9]. We
endow $Z$ with the closed subscheme structure of
$\lim_{\lambda\in\Lambda_{Z}}Y_{\lambda}$. The first paragraph shows that
there exists a finite étale surjective morphism $Z^{\prime}\to Z$ such that
$Z^{\prime}\times_{Y}V=\coprod_{i=1}^{r}Z^{\prime}$. Using [EGA IV3, 8.8.2,
8.10.5] and [EGA IV4, 17.7.8], there exists an index
$\lambda_{0}\in\Lambda_{Z}$, a finite étale surjective morphism
$Y^{\prime}_{\lambda_{0}}\to Y_{\lambda_{0}}$ and an isomorphism
$Y^{\prime}_{\lambda_{0}}\times_{Y}V=\coprod_{i=1}^{r}Y^{\prime}_{\lambda_{0}}$.
Notice that $Y^{\prime}_{\lambda_{0}}$ is also finite étale over $Y$. Since
$Z$ is an arbitrary connected component of $Y$, the conclusion follows from
the quasi-compactness of $Y$. ∎

######  Lemma 10.2.

Let $Y$ be a coherent scheme, $\rho:Y_{\mathrm{\acute{e}t}}\to
Y_{\mathrm{f\acute{e}t}}$ the morphism of sites defined by the inclusion
functor. Then, the functor
$\rho^{-1}:\widetilde{Y_{\mathrm{f\acute{e}t}}}\to\widetilde{Y_{\mathrm{\acute{e}t}}}$
of the associated topoi induces an equivalence
$\rho^{-1}:\mathbf{LocSys}(Y_{\mathrm{f\acute{e}t}})\to\mathbf{LocSys}(Y_{\mathrm{\acute{e}t}})$
between the categories of finite locally constant abelian sheaves with quasi-
inverse $\rho_{*}$.

###### Proof.

Since any finite locally constant sheaf on $Y_{\mathrm{\acute{e}t}}$ (resp.
$Y_{\mathrm{f\acute{e}t}}$) is representable by a finite étale $Y$-scheme by
faithfully flat descent (cf. [Sta22, 03RV]), the Yoneda embeddings induce a
commutative diagram

(10.2.5)

where the horizontal arrows are fully faithful. In particular, $\rho^{-1}$ is
fully faithful. For a finite locally constant abelian sheaf $\mathbb{F}$ on
$Y_{\mathrm{\acute{e}t}}$, let $V$ be a finite étale $Y$-scheme representing
$\mathbb{F}$ and let $h_{V}^{\mathrm{\acute{e}t}}$ (resp.
$h_{V}^{\mathrm{f\acute{e}t}}$) be the representable sheaf of $V$ on
$Y_{\mathrm{\acute{e}t}}$ (resp. $Y_{\mathrm{f\acute{e}t}}$). We have
$\mathbb{F}=h_{V}^{\mathrm{\acute{e}t}}=\rho^{-1}h_{V}^{\mathrm{f\acute{e}t}}$
([Sta22, 04D3]). By 10.1, $h_{V}^{\mathrm{f\acute{e}t}}$ is finite locally
constant. It is clear that the adjunction morphism
$h_{V}^{\mathrm{f\acute{e}t}}\to\rho_{*}h_{V}^{\mathrm{\acute{e}t}}$ is an
isomorphism, which shows that $h_{V}^{\mathrm{f\acute{e}t}}$ is an abelian
sheaf. Thus, $\rho^{-1}$ is essentially surjective. Moreover, the argument
also shows that $\rho_{*}$ induces a functor
$\rho_{*}:\mathbf{LocSys}(Y_{\mathrm{\acute{e}t}})\to\mathbf{LocSys}(Y_{\mathrm{f\acute{e}t}})$
which is a quasi-inverse of $\rho^{-1}$. ∎

######  Proposition 10.3.

With the notation in 7.8, the functors between the categories of finite
locally constant abelian sheaves

(10.3.1)
$\displaystyle\mathbf{LocSys}(Y_{\mathrm{f\acute{e}t}})\stackrel{{\scriptstyle\beta^{-1}}}{{\longrightarrow}}\mathbf{LocSys}(\mathbf{E}_{Y\to
X}^{\mathrm{\acute{e}t}})\stackrel{{\scriptstyle\psi^{-1}}}{{\longrightarrow}}\mathbf{LocSys}(Y_{\mathrm{\acute{e}t}})$

are equivalences with quasi-inverses $\beta_{*}$ and $\psi_{*}$ respectively.

###### Proof.

Notice that for any finite locally constant abelian sheaf $\mathbb{G}$ on
$Y_{\mathrm{f\acute{e}t}}$, the canonical morphism
$\beta^{-1}\mathbb{G}\to\psi_{*}\rho^{-1}\mathbb{G}$, which is induced by the
adjunction $\mathrm{id}\to\psi_{*}\psi^{-1}$ and by the identity
$\psi^{-1}\beta^{-1}=\rho^{-1}$, is an isomorphism by 10.2 and the proof of
[AGT16, VI.6.3.(iii)]. For a finite locally constant abelian sheaf
$\mathbb{F}$ over $Y_{\mathrm{\acute{e}t}}$, we write
$\mathbb{F}=\rho^{-1}\mathbb{G}$ by 10.2. Then,
$\mathbb{F}=\psi^{-1}\beta^{-1}\mathbb{G}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\psi^{-1}\psi_{*}\rho^{-1}\mathbb{G}=\psi^{-1}\psi_{*}\mathbb{F}$,
whose inverse is the adjunction map $\psi^{-1}\psi_{*}\mathbb{F}\to\mathbb{F}$
since the composition of
$\psi^{-1}(\beta^{-1}\mathbb{G})\to\psi^{-1}(\psi_{*}\psi^{-1})(\beta^{-1}\mathbb{G})=(\psi^{-1}\psi_{*})\psi^{-1}(\beta^{-1}\mathbb{G})\to\psi^{-1}(\beta^{-1}\mathbb{G})$
is the identity. On the other hand, for a finite locally constant abelian
sheaf $\mathbb{L}$ over $\mathbf{E}_{Y\to X}^{\mathrm{\acute{e}t}}$, we claim
that $\mathbb{L}\to\psi_{*}\psi^{-1}\mathbb{L}$ is an isomorphism. The problem
is local on $\mathbf{E}_{Y\to X}^{\mathrm{\acute{e}t}}$. Thus, we may assume
that $\mathbb{L}$ is the constant sheaf with value $L$ where $L$ is a finite
abelian group. Let $\underline{L}$ be the constant sheaf with value $L$ on
$Y_{\mathrm{f\acute{e}t}}$. Then, $\mathbb{L}=\beta^{-1}\underline{L}$, and
the isomorphism
$\mathbb{L}=\beta^{-1}\underline{L}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\psi_{*}\rho^{-1}\underline{L}=\psi_{*}\psi^{-1}\mathbb{L}$
coincides with the adjunction map $\mathbb{L}\to\psi_{*}\psi^{-1}\mathbb{L}$.
Therefore, $\psi^{-1}:\mathbf{LocSys}(\mathbf{E}_{Y\to
X}^{\mathrm{\acute{e}t}})\to\mathbf{LocSys}(Y_{\mathrm{\acute{e}t}})$ is an
equivalence with quasi-inverse $\psi_{*}$. The conclusion follows from 10.2. ∎

######  10.4.

Let $f:(Y^{\prime}\to X^{\prime})\to(Y\to X)$ be a morphism of morphisms
between coherent schemes over
$\mathop{\mathrm{Spec}}(\mathbb{Q}_{p})\to\mathop{\mathrm{Spec}}(\mathbb{Z}_{p})$.
The base change by $f$ induces a commutative diagram of sites

(10.4.5)

Let $\mathbb{F}^{\prime}$ be a finite locally constant abelian sheaf on
$Y^{\prime}_{\mathrm{\acute{e}t}}$. Remark that the sheaf
$\overline{\mathscr{B}}$ on $\mathbf{E}_{Y\to X}^{\mathrm{\acute{e}t}}$ is
flat over $\mathbb{Z}$. Consider the natural morphisms in the derived category
$\mathbf{D}(\overline{\mathscr{B}}\textrm{-}\mathbf{Mod}_{\mathbf{E}_{Y\to
X}^{\mathrm{\acute{e}t}}})$,

(10.4.8)

where $\alpha_{1}$ is induced by the canonical morphism
$\psi^{\prime}_{*}\mathbb{F}^{\prime}\to\mathrm{R}\psi^{\prime}_{*}\mathbb{F}^{\prime}$,
and $\alpha_{2}$ is the canonical morphism.

######  10.5.

We keep the notation in 10.4 and assume that $X$ is the spectrum of an
absolutely integrally closed valuation ring $A$ and that $Y$ is a quasi-
compact open subscheme of $X$. Applying the functor $\mathrm{R}\Gamma(Y\to
X,-)$ on (10.4.8), we obtain the natural morphisms in the derived category
$\mathbf{D}(A\textrm{-}\mathbf{Mod})$ by 7.9,

(10.5.3)

######  Definition 10.6 ([AG20, 4.8.13, 5.7.3]).

Under the assumptions in 10.4 (resp. 10.5) and with the same notation, if
$\alpha_{1}$ is an isomorphism (for instance, if the canonical morphism
$\psi^{\prime}_{*}\mathbb{F}^{\prime}\to\mathrm{R}\psi^{\prime}_{*}\mathbb{F}^{\prime}$
is an isomorphism), then we call the canonical morphism

(10.6.1)
$\displaystyle\alpha_{2}\circ\alpha_{1}^{-1}:(\mathrm{R}\psi_{*}\mathrm{R}f_{\mathrm{\acute{e}t}*}\mathbb{F}^{\prime})\otimes^{\mathrm{L}}_{\mathbb{Z}}\overline{\mathscr{B}}\longrightarrow\mathrm{R}f_{\mathbf{E}*}(\psi^{\prime}_{*}\mathbb{F}^{\prime}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}^{\prime})$
(10.6.2) (resp.
$\displaystyle\alpha_{2}\circ\alpha_{1}^{-1}:\mathrm{R}\Gamma(Y^{\prime}_{\mathrm{\acute{e}t}},\mathbb{F}^{\prime})\otimes^{\mathrm{L}}_{\mathbb{Z}}A\longrightarrow\mathrm{R}\Gamma(\mathbf{E}_{Y^{\prime}\to
X^{\prime}}^{\mathrm{\acute{e}t}},\psi^{\prime}_{*}\mathbb{F}^{\prime}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}^{\prime})\textrm{)}$

the _relative_ (resp. _absolute_) _Faltings’ comparison morphism_ associated
to $f:(Y^{\prime}\to X^{\prime})\to(Y\to X)$ and $\mathbb{F}^{\prime}$. In
this case, we say that _Faltings’ comparison morphisms exist_.

######  Theorem 10.7 ([Ach17, Cor.6.9], cf. [AG20, 4.4.2]).

Let $\mathcal{O}_{K}$ be a strictly Henselian discrete valuation ring with
fraction field $K$ of characteristic $0$ and residue field of characteristic
$p$. We fix an algebraic closure $\overline{K}$ of $K$. Let $X$ be an
$\mathcal{O}_{K}$-scheme of finite type, $\mathbb{F}$ a finite locally
constant abelian sheaf on $X_{\overline{K},\mathrm{\acute{e}t}}$,
$\psi:X_{\overline{K},\mathrm{\acute{e}t}}\to\mathbf{E}_{X_{\overline{K}}\to
X}^{\mathrm{\acute{e}t}}$ the morphism of sites defined in 7.8. Then, the
canonical morphism $\psi_{*}\mathbb{F}\to\mathrm{R}\psi_{*}\mathbb{F}$ is an
isomorphism.

######  Corollary 10.8.

Let $\mathcal{O}_{K}$ be a strictly Henselian discrete valuation ring with
fraction field $K$ of characteristic $0$ and residue field of characteristic
$p$. We fix an algebraic closure $\overline{K}$ of $K$. Let $X$ be a coherent
$\mathcal{O}_{\overline{K}}$-scheme,
$Y=\mathop{\mathrm{Spec}}(\overline{K})\times_{\mathop{\mathrm{Spec}}(\mathcal{O}_{\overline{K}})}X$,
$\mathbb{F}$ a finite locally constant abelian sheaf on
$Y_{\mathrm{\acute{e}t}}$, $\psi:Y_{\mathrm{\acute{e}t}}\to\mathbf{E}_{Y\to
X}^{\mathrm{\acute{e}t}}$ the morphism of sites defined in 7.8. Then, the
canonical morphism $\psi_{*}\mathbb{F}\to\mathrm{R}\psi_{*}\mathbb{F}$ is an
isomorphism.

We emphasize that we don’t need any finiteness condition of $X$ over
$\mathcal{O}_{\overline{K}}$ in 10.8. In fact, one can replace
$\mathcal{O}_{\overline{K}}$ by $\overline{\mathbb{Z}_{p}}$ without loss of
generality, where $\overline{\mathbb{Z}_{p}}$ is the integral closure of
$\mathbb{Z}_{p}$ in an algebraic closure of $\mathbb{Q}_{p}$. We keep working
over $\mathcal{O}_{\overline{K}}$ only for the continuation of our usage of
notation.

###### Proof of 10.8.

We take a directed inverse system
$(X_{\lambda}\to\mathop{\mathrm{Spec}}(\mathcal{O}_{K_{\lambda}}))_{\lambda\in\Lambda}$
of morphisms of finite type of schemes by Noetherian approximation, such that
$K_{\lambda}$ is a finite field extension of $K$ and
$\overline{K}=\bigcup_{\lambda\in\Lambda}K_{\lambda}$, and that the transition
morphisms $X_{\lambda^{\prime}}\to X_{\lambda}$ are affine and
$X=\lim_{\lambda\in\Lambda}X_{\lambda}$ (cf. [Sta22, 09MV]). For each
$\lambda\in\Lambda$, we set
$Y_{\lambda}=\mathop{\mathrm{Spec}}(\overline{K})\times_{\mathop{\mathrm{Spec}}(\mathcal{O}_{K_{\lambda}})}X_{\lambda}$.
Notice that $Y=\lim Y_{\lambda}$. Then, there exists an index
$\lambda_{0}\in\Lambda$ and a finite locally constant abelian sheaf
$\mathbb{F}_{\lambda_{0}}$ on $Y_{\lambda_{0},\mathrm{\acute{e}t}}$ such that
$\mathbb{F}$ is the pullback of $\mathbb{F}_{\lambda_{0}}$ by
$Y_{\mathrm{\acute{e}t}}\to Y_{\lambda_{0},\mathrm{\acute{e}t}}$ (cf. [Sta22,
09YU]). Let $\mathbb{F}_{\lambda}$ be the pullback of
$\mathbb{F}_{\lambda_{0}}$ by $Y_{\lambda,\mathrm{\acute{e}t}}\to
Y_{\lambda_{0},\mathrm{\acute{e}t}}$ for each $\lambda\geq\lambda_{0}$. Notice
that $\mathcal{O}_{K_{\lambda}}$ also satisfies the conditions in 10.7. Let
$\psi_{\lambda}:Y_{\lambda,\mathrm{\acute{e}t}}\to\mathbf{E}_{Y_{\lambda}\to
X_{\lambda}}^{\mathrm{\acute{e}t}}$ be the morphism of sites defined in 7.8,
$\varphi_{\lambda}:\mathbf{E}_{Y\to
X}^{\mathrm{\acute{e}t}}\to\mathbf{E}_{Y_{\lambda}\to
X_{\lambda}}^{\mathrm{\acute{e}t}}$ the morphism of sites defined by the
transition morphism. Then, we have
$\mathrm{R}^{q}\psi_{\lambda*}\mathbb{F}_{\lambda}=0$ for each integer $q>0$
by 10.7, and moreover

(10.8.1)
$\displaystyle\mathrm{R}^{q}\psi_{*}\mathbb{F}=\mathop{\mathrm{colim}}_{\lambda\geq\lambda_{0}}\varphi_{\lambda}^{-1}\mathrm{R}^{q}\psi_{\lambda*}\mathbb{F}_{\lambda}=0$

by 7.12, [SGA 4${}_{\text{\rm II}}$, VII.5.6] and [SGA 4${}_{\text{\rm II}}$,
VI.8.7.3] whose conditions are satisfied because each object in each concerned
site is quasi-compact. ∎

######  Lemma 10.9.

With the notation in 10.4, let $\mathbb{F}$ be a finite locally constant
abelian sheaf on $Y_{\mathrm{\acute{e}t}}$. Then, the canonical morphism
$f_{\mathbf{E}}^{-1}\psi_{*}\mathbb{F}\to\psi^{\prime}_{*}f_{\mathrm{\acute{e}t}}^{-1}\mathbb{F}$
is an isomorphism.

###### Proof.

The base change morphism
$f_{\mathbf{E}}^{-1}\psi_{*}\mathbb{F}\to\psi^{\prime}_{*}f_{\mathrm{\acute{e}t}}^{-1}\mathbb{F}$
is the composition of the adjunction morphisms ([SGA 4${}_{\text{\rm III}}$,
XVII.2.1.3])

(10.9.1) $\displaystyle
f_{\mathbf{E}}^{-1}\psi_{*}\mathbb{F}\to\psi^{\prime}_{*}\psi^{\prime-1}(f_{\mathbf{E}}^{-1}\psi_{*}\mathbb{F})=\psi^{\prime}_{*}f_{\mathrm{\acute{e}t}}^{-1}(\psi^{-1}\psi_{*}\mathbb{F})\to\psi^{\prime}_{*}f_{\mathrm{\acute{e}t}}^{-1}\mathbb{F}$

which are both isomorphisms by 10.3. ∎

######  10.10.

Let $K$ be a complete discrete valuation field of characteristic $0$ with
valuation ring $\mathcal{O}_{K}$ whose residue field $k$ is algebraically
closed (a condition required by [AG20, 4.1.3, 5.1.3]) of characteristic $p>0$,
$\overline{K}$ an algebraic closure of $K$, $\mathcal{O}_{\overline{K}}$ the
integral closure of $\mathcal{O}_{K}$ in $\overline{K}$,
$\eta=\mathop{\mathrm{Spec}}(K)$,
$\overline{\eta}=\mathop{\mathrm{Spec}}(\overline{K})$,
$S=\mathop{\mathrm{Spec}}(\mathcal{O}_{K})$,
$\overline{S}=\mathop{\mathrm{Spec}}(\mathcal{O}_{\overline{K}})$,
$s=\mathop{\mathrm{Spec}}(k)$. Remark that $\overline{K}$ is a pre-perfectoid
field with valuation ring $\mathcal{O}_{\overline{K}}$ so we are also in the
situation 8.7.

######  10.11.

With the notation in 10.10, let $X$ be an $S$-scheme, $Y$ an open subscheme of
the generic fibre $X_{\eta}$. We simply denote by $\mathscr{M}_{X}$ the
compactifying log structure $\mathscr{M}_{X_{\eta}\to X}$ (9.3). Following
[AGT16, III.4.7], we say that $Y\to X$ is _adequate_ over $\eta\to S$ if the
following conditions are satisfied:

  1. (i)

$X$ is of finite type over $S$;

  2. (ii)

Any point of the special fibre $X_{s}$ admits an étale neighborhood $U$ such
that $U_{\eta}\to\eta$ is smooth and that $U_{\eta}\setminus Y$ is the support
of a strict normal crossings divisor on $U_{\eta}$;

  3. (iii)

$(X,\mathscr{M}_{Y\to X})$ is a fine log scheme and the structure morphism
$(X,\mathscr{M}_{Y\to X})\to(S,\mathscr{M}_{S})$ is smooth and saturated.

In this case, $(X,\mathscr{M}_{Y\to X})\to(S,\mathscr{M}_{S})$ is adequate in
the sense of [AGT16, III.4.7]. We remark that for any adequate
$(S,\mathscr{M}_{S})$-log scheme $(X,\mathscr{M})$, $X^{\mathrm{tr}}\to X$ is
adequate over $\eta\to S$ and
$(X,\mathscr{M})=(X,\mathscr{M}_{X^{\mathrm{tr}}\to X})$ (cf. 9.4, 9.5). Note
that if $Y\to X$ is semi-stable over $\eta\to S$ then it is adequate (cf.
9.5).

######  10.12.

We recall the statement of Faltings’ main $p$-adic comparison theorem
following Abbes-Gros [AG20]. We take the notation and assumptions in 10.10.
Firstly, recall that for any adequate open immersion of schemes $X^{\circ}\to
X$ over $\eta\to S$ and any finite locally constant abelian sheaf $\mathbb{F}$
on $X^{\circ}_{\overline{\eta},\mathrm{\acute{e}t}}$, the canonical morphism
$\psi_{*}\mathbb{F}\to\mathrm{R}\psi_{*}\mathbb{F}$ is an isomorphism, where
$\psi:X^{\circ}_{\overline{\eta},\mathrm{\acute{e}t}}\to\mathbf{E}_{X_{\overline{\eta}}^{\circ}\to
X}^{\mathrm{\acute{e}t}}$ is the morphism of sites defined in 7.8 ([AG20,
4.4.2]).

Let $(X^{\prime\triangleright}\to X^{\prime})\to(X^{\circ}\to X)$ be a
morphism of adequate open immersions of schemes over $\eta\to S$ such that
$X^{\prime}\to X$ is projective and that the induced morphism
$(X^{\prime},\mathscr{M}_{X^{\prime\triangleright}\to
X^{\prime}})\to(X,\mathscr{M}_{X^{\circ}\to X})$ is smooth and saturated. Let
$Y^{\prime}=\overline{\eta}\times_{\eta}X^{\prime\triangleright}$,
$Y=\overline{\eta}\times_{\eta}X^{\circ}$, $f:(Y^{\prime}\to
X^{\prime})\to(Y\to X)$ the natural morphism, $\mathbb{F}^{\prime}$ a finite
locally constant abelian sheaf on $Y^{\prime}_{\mathrm{\acute{e}t}}$. By the
first paragraph, we have the relative Faltings’ comparison morphism associated
to $f$ and $\mathbb{F}^{\prime}$ (10.6.1),

(10.12.1)
$\displaystyle(\mathrm{R}\psi_{*}\mathrm{R}f_{\mathrm{\acute{e}t}*}\mathbb{F}^{\prime})\otimes^{\mathrm{L}}_{\mathbb{Z}}\overline{\mathscr{B}}\longrightarrow\mathrm{R}f_{\mathbf{E}*}(\psi^{\prime}_{*}\mathbb{F}^{\prime}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}^{\prime}).$

Remark that under our assumption, the sheaf
$\mathrm{R}^{q}f_{\mathrm{\acute{e}t}*}\mathbb{F}^{\prime}$ on
$Y_{\mathrm{\acute{e}t}}$ is finite locally constant for each integer $q$
([AG20, 2.2.25]).

######  Theorem 10.13 ([Fal02, Thm.6, page 266], [AG20, 5.7.3]).

Under the assumptions in 10.12 and with the same notation, the relative
Faltings’ comparison morphism associated to $f$ and $\mathbb{F}^{\prime}$ is
an almost isomorphism in the derived category
$\mathbf{D}(\mathcal{O}_{\overline{K}}\textrm{-}\mathbf{Mod}_{\mathbf{E}_{Y\to
X}^{\mathrm{\acute{e}t}}})$ (5.7), and it induces an almost isomorphism

(10.13.1)
$\displaystyle(\psi_{*}\mathrm{R}^{q}f_{\mathrm{\acute{e}t}*}\mathbb{F}^{\prime})\otimes_{\mathbb{Z}}\overline{\mathscr{B}}\longrightarrow\mathrm{R}^{q}f_{\mathbf{E}*}(\psi^{\prime}_{*}\mathbb{F}^{\prime}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}^{\prime})$

of $\mathcal{O}_{\overline{K}}$-modules for each integer $q$.

######  Proposition 10.14.

With the natation in 10.10, let $A$ be an absolutely integrally closed
valuation ring of height $1$ extension of $\mathcal{O}_{\overline{K}}$, $X$ a
proper $A$-scheme of finite presentation,
$Y=\mathop{\mathrm{Spec}}(A[1/p])\times_{\mathop{\mathrm{Spec}}(A)}X$,
$\mathbb{F}$ a finite locally constant abelian sheaf on
$Y_{\mathrm{\acute{e}t}}$. Then, there exists a proper surjective morphism
$X^{\prime}\to X$ of finite presentation such that the relative and absolute
Faltings’ comparison morphisms associated to $f^{\prime}:(Y^{\prime}\to
X^{\prime})\to(\mathop{\mathrm{Spec}}(A[1/p])\to\mathop{\mathrm{Spec}}(A))$
and $\mathbb{F}^{\prime}$ (which exist by 10.8) are almost isomorphisms, where
$Y^{\prime}=Y\times_{X}X^{\prime}$ and $\mathbb{F}^{\prime}$ is the pullback
of $\mathbb{F}$ on $Y^{\prime}_{\mathrm{\acute{e}t}}$.

###### Proof.

Since the underlying topological space of $X$ is Noetherian by 9.10, each
irreducible component $Z$ of $X$ admits a closed subscheme structure such that
$Z\to X$ is of finite presentation ([Sta22, 01PH]). After replacing $X$ by the
disjoint union of its irreducible components, we may assume that $X$ is
irreducible. If $Y$ is empty, then we take $X^{\prime}=X$ and thus the
relative (resp. absolute) Faltings’ comparison morphism associated to
$f^{\prime}$ and $\mathbb{F}^{\prime}$ is an isomorphism between zero objects.
If $Y$ is not empty, then we are in the situation of 9.11.(iv) by taking
$\mathcal{O}_{L}=A$. With the notation in 9.11, we check that the morphism
$X^{\prime}\to X$ meets our requirements. We set
$\eta_{\lambda}=\mathop{\mathrm{Spec}}(K_{\lambda})$,
$S_{\lambda}=\mathop{\mathrm{Spec}}(\mathcal{O}_{K_{\lambda}})$,
$T_{\lambda,\overline{\eta_{\lambda}}}=\overline{\eta}\times_{\eta_{\lambda}}U_{\lambda}$,
$X^{\prime}_{\lambda,\overline{\eta_{\lambda}}}=\overline{\eta}\times_{\eta_{\lambda}}Y^{\prime}_{\lambda}$,
and denote by
$f^{\prime}_{\lambda}:(X^{\prime}_{\lambda,\overline{\eta_{\lambda}}}\to
X^{\prime}_{\lambda})\to(T_{\lambda,\overline{\eta_{\lambda}}}\to
T_{\lambda})$ the natural morphism. We obtain a commutative diagram

(10.14.9)

Firstly notice that the site $Y^{\prime}_{\mathrm{\acute{e}t}}$ (resp.
$\mathop{\mathrm{Spec}}(A[1/p])_{\mathrm{\acute{e}t}}$) is the limit of the
sites $X^{\prime}_{\lambda,\overline{\eta_{\lambda}},\mathrm{\acute{e}t}}$
(resp. $T_{\lambda,\overline{\eta_{\lambda}},\mathrm{\acute{e}t}}$) and the
site $\mathbf{E}_{Y^{\prime}\to X^{\prime}}^{\mathrm{\acute{e}t}}$ (resp.
$\mathbf{E}_{\mathop{\mathrm{Spec}}(A[1/p])\to\mathop{\mathrm{Spec}}(A)}^{\mathrm{\acute{e}t}}$)
is the limit of the sites
$\mathbf{E}_{X^{\prime}_{\lambda,\overline{\eta_{\lambda}}}\to
X^{\prime}_{\lambda}}^{\mathrm{\acute{e}t}}$ (resp.
$\mathbf{E}_{T_{\lambda,\overline{\eta_{\lambda}}}\to
T_{\lambda}}^{\mathrm{\acute{e}t}}$) ([SGA 4${}_{\text{\rm II}}$, VII.5.6] and
7.12). There exists an index $\lambda_{0}\in\Lambda$ and a finite locally
constant abelian sheaf $\mathbb{F}^{\prime}_{\lambda_{0}}$ on
$X^{\prime}_{\lambda_{0},\overline{\eta_{\lambda_{0}}},\mathrm{\acute{e}t}}$
such that $\mathbb{F}^{\prime}$ is the pullback of
$\mathbb{F}^{\prime}_{\lambda_{0}}$ by $Y^{\prime}_{\mathrm{\acute{e}t}}\to
X^{\prime}_{\lambda_{0},\overline{\eta_{\lambda_{0}}},\mathrm{\acute{e}t}}$
(cf. [Sta22, 09YU]). Let $\mathbb{F}^{\prime}_{\lambda}$ be the pullback of
$\mathbb{F}^{\prime}_{\lambda_{0}}$ by
$X^{\prime}_{\lambda,\overline{\eta_{\lambda}},\mathrm{\acute{e}t}}\to
X^{\prime}_{\lambda_{0},\overline{\eta_{\lambda_{0}}},\mathrm{\acute{e}t}}$
for each $\lambda\geq\lambda_{0}$. We also have
$\overline{\mathscr{B}}^{\prime}=\mathop{\mathrm{colim}}g_{\lambda,\mathbf{E}}^{-1}\overline{\mathscr{B}}^{\prime}$
(resp.
$\overline{\mathscr{B}}=\mathop{\mathrm{colim}}h_{\lambda,\mathbf{E}}^{-1}\overline{\mathscr{B}}$)
by 7.12. According to [SGA 4${}_{\text{\rm II}}$, VI.8.7.3], whose conditions
are satisfied because each object in each concerned site is quasi-compact,
there are canonical isomorphisms for each integer $q$,

(10.14.10) $\displaystyle(\mathrm{R}^{q}(\psi\circ
f^{\prime}_{\mathrm{\acute{e}t}})_{*}\mathbb{F}^{\prime})\otimes_{\mathbb{Z}}\overline{\mathscr{B}}$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathop{\mathrm{colim}}h_{\lambda,\mathbf{E}}^{-1}((\mathrm{R}^{q}(\psi_{\lambda}\circ
f^{\prime}_{\lambda,\mathrm{\acute{e}t}})_{*}\mathbb{F}^{\prime}_{\lambda})\otimes_{\mathbb{Z}}\overline{\mathscr{B}}),$
(10.14.11)
$\displaystyle\mathrm{R}^{q}f^{\prime}_{\mathbf{E}*}(\psi^{\prime}_{*}\mathbb{F}^{\prime}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}^{\prime})$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathop{\mathrm{colim}}h_{\lambda,\mathbf{E}}^{-1}\mathrm{R}^{q}f^{\prime}_{\lambda,\mathbf{E}*}(\psi^{\prime}_{\lambda*}\mathbb{F}^{\prime}_{\lambda}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}^{\prime}).$

On the other hand,
$(X^{\prime}_{\lambda},\mathscr{M}_{X^{\prime}_{\lambda}})\to(T_{\lambda},\mathscr{M}_{T_{\lambda}})$
is a smooth and saturated morphism of adequate
$(S_{\lambda},\mathscr{M}_{S_{\lambda}})$-log schemes with
$X^{\prime}_{\lambda}\to T_{\lambda}$ projective for each $\lambda\in\Lambda$
by construction. Thus, we are in the situation of 10.13, which implies that
the relative Faltings’ comparison morphism associated to
$f^{\prime}_{\lambda}$ and $\mathbb{F}^{\prime}_{\lambda}$,

(10.14.12) $\displaystyle(\mathrm{R}^{q}(\psi_{\lambda}\circ
f^{\prime}_{\lambda,\mathrm{\acute{e}t}})_{*}\mathbb{F}^{\prime}_{\lambda})\otimes_{\mathbb{Z}}\overline{\mathscr{B}}\longrightarrow\mathrm{R}^{q}f^{\prime}_{\lambda,\mathbf{E}*}(\psi^{\prime}_{\lambda*}\mathbb{F}^{\prime}_{\lambda}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}^{\prime})$

is an almost isomorphism for each $\lambda\geq\lambda_{0}$. Combining with
(10.14.10) and (10.14.11), we see that the relative Faltings’ comparison
morphism associated to $f^{\prime}$ and $\mathbb{F}^{\prime}$,

(10.14.13)
$\displaystyle\mathrm{R}\psi_{*}(\mathrm{R}f^{\prime}_{\mathrm{\acute{e}t}*}\mathbb{F}^{\prime})\otimes_{\mathbb{Z}}^{\mathrm{L}}\overline{\mathscr{B}}\longrightarrow\mathrm{R}f^{\prime}_{\mathbf{E}*}(\psi^{\prime}_{*}\mathbb{F}^{\prime}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}^{\prime}),$

is an almost isomorphism (and thus so is the absolute one). ∎

######  Corollary 10.15.

Under the assumptions in 10.14 and with the same notation, there exists a
proper hypercovering $X_{\bullet}\to X$ of coherent schemes ([Sta22, 0DHI])
such that for each degree $n$, the relative and absolute Faltings’ comparison
morphisms associated to $f_{n}:(Y_{n}\to
X_{n})\to(\mathop{\mathrm{Spec}}(A[1/p])\to\mathop{\mathrm{Spec}}(A))$ and
$\mathbb{F}_{n}$ (which exist by 10.8) are almost isomorphisms, where
$Y_{n}=Y\times_{X}X_{n}$ and $\mathbb{F}_{n}$ is the pullback of $\mathbb{F}$
by the natural morphism $Y_{n,\mathrm{\acute{e}t}}\to
Y_{\mathrm{\acute{e}t}}$. In particular, $Y_{\bullet}\to Y$ is a proper
hypercovering and $X_{\bullet}^{Y_{\bullet}}\to X^{Y}$ is a hypercovering in
$\mathbf{I}_{\overline{\eta}\to\overline{S}}$.

###### Proof.

Let $\mathscr{C}$ be the category of proper $A$-schemes of finite presentation
endowed with the pretopology formed by families of morphisms
$\\{f_{i}:X_{i}\to X\\}_{i\in I}$ with $I$ finite and $X=\bigcup_{i\in
I}f_{i}(X_{i})$. Consider the functor
$u^{+}:\mathscr{C}\to\mathbf{I}_{\mathop{\mathrm{Spec}}(A[1/p])\to\mathop{\mathrm{Spec}}(A)}$
sending $X$ to $X^{Y}$ where
$Y=\mathop{\mathrm{Spec}}(A[1/p])\times_{\mathop{\mathrm{Spec}}(A)}X$. It is
well-defined by 3.19.(4) and commutes with fibred products by 3.21 and
continuous by 3.15. Lemma 10.14 allows us to take a hypercovering
$X_{\bullet}\to X$ in $\mathscr{C}$ meeting our requirement by [Sta22, 094K
and 0DB1]. We see that $Y_{\bullet}\to Y$ is a proper hypercovering and that
$X_{\bullet}^{Y_{\bullet}}\to X^{Y}$ is a hypercovering in
$\mathbf{I}_{\overline{\eta}\to\overline{S}}$ by the properties of $u^{+}$
([Sta22, 0DAY]). ∎

######  Lemma 10.16.

Let $\overline{\mathbb{Z}_{p}}$ be the integral closure of $\mathbb{Z}_{p}$ in
an algebraic closure of $\mathbb{Q}_{p}$, $A$ a
$\overline{\mathbb{Z}_{p}}$-algebra which is an absolutely integrally closed
valuation ring, $X$ a proper $A$-scheme of finite presentation,
$Y=\mathop{\mathrm{Spec}}(A[1/p])\times_{\mathop{\mathrm{Spec}}(A)}X$,
$\mathbb{F}$ a finite locally constant abelian sheaf on
$Y_{\mathrm{\acute{e}t}}$. Let
$A^{\prime}=((A/\cap_{n>0}p^{n}A)_{\sqrt{pA}})^{\wedge}$ ($p$-adic
completion), $X^{\prime}=X_{A^{\prime}}$, $Y^{\prime}=Y_{A^{\prime}}$,
$\mathbb{F}^{\prime}$ the pullback of $\mathbb{F}$ on
$Y^{\prime}_{\mathrm{\acute{e}t}}$. Then, the following statements are
equivalent:

  1. (1)

The absolute Faltings’ comparison morphism associated to $f:(Y\to
X)\to(\mathop{\mathrm{Spec}}(A[1/p])\to\mathop{\mathrm{Spec}}(A))$ and
$\mathbb{F}$ (which exists by 10.8) is an almost isomorphism.

  2. (2)

The absolute Faltings’ comparison morphism associated to
$f^{\prime}:(Y^{\prime}\to
X^{\prime})\to(\mathop{\mathrm{Spec}}(A^{\prime}[1/p])\to\mathop{\mathrm{Spec}}(A^{\prime}))$
and $\mathbb{F}^{\prime}$ (which exists by 10.8) is an almost isomorphism.

###### Proof.

If $p$ is zero (resp. invertible) in $A$, then the absolute Faltings’
comparison morphisms are both isomorphisms between zero objects, since $Y$ and
$Y^{\prime}$ are empty (resp. the abelian sheaves $\mathbb{F}$ and
$\mathbb{F}^{\prime}$ are zero after inverting $p$). Thus, we may assume that
$p$ is a nonzero element of the maximal ideal of $A$. Notice that
$\cap_{n>0}p^{n}A$ is the maximal prime ideal of $A$ not containing $p$ and
that $\sqrt{pA}$ is the minimal prime ideal of $A$ containing $p$ (2.1). Thus,
$(A/\cap_{n>0}p^{n}A)_{\sqrt{pA}}$ is an absolutely integrally closed
valuation ring of height $1$ extension of $\overline{\mathbb{Z}_{p}}$ (2.1)
and thus so is its $p$-adic completion $A^{\prime}$.

We denote by $u:(Y^{\prime}\to X^{\prime})\to(Y\to X)$ the natural morphism.
We have $\mathbb{F}^{\prime}=u_{\mathrm{\acute{e}t}}^{-1}\mathbb{F}$. The
natural morphisms in (10.5.3) induce a commutative diagram

(10.16.5)

where $\gamma_{1}$ is induced by the canonical morphism
$\mathbb{F}\to\mathrm{R}u_{\mathrm{\acute{e}t}*}u_{\mathrm{\acute{e}t}}^{-1}\mathbb{F}$,
and $\gamma_{2}$ (resp. $\gamma_{3}$) is induced by the composition of
$\psi_{*}\mathbb{F}\to\mathrm{R}u_{\mathbf{E}*}u_{\mathbf{E}}^{-1}\psi_{*}\mathbb{F}\to\mathrm{R}u_{\mathbf{E}*}\psi^{\prime}_{*}u_{\mathrm{\acute{e}t}}^{-1}\mathbb{F}$
(resp. and by the canonical morphism
$\overline{\mathscr{B}}\to\mathrm{R}u_{\mathbf{E}*}\overline{\mathscr{B}}^{\prime}$).
Since $\alpha_{1}$ and $\alpha_{1}^{\prime}$ are isomorphisms by 10.8, it
suffices to show that $\gamma_{1}$ and $\gamma_{3}$ are almost isomorphisms.

Since $A/\cap_{n>0}p^{n}A\to(A/\cap_{n>0}p^{n}A)_{\sqrt{pA}}$ is injective
whose cokernel is killed by $\sqrt{pA}$ (4.7), the morphism $A\to A^{\prime}$
induces an almost isomorphism $A/p^{n}A\to A^{\prime}/p^{n}A^{\prime}$ for
each $n$. Then, for any torsion abelian group $M$, the natural morphism
$M\otimes_{\mathbb{Z}}A\to M\otimes_{\mathbb{Z}}A^{\prime}$ is an almost
isomorphism. Therefore, $\gamma_{1}$ is an almost isomorphism by the proper
base change theorem over the strictly Henselian local ring $A[1/p]$ ([SGA
4${}_{\text{\rm III}}$, XII 5.5, 5.4]). For $\gamma_{3}$, it suffices to show
that the canonical morphism
$\psi_{*}\mathbb{F}\otimes\overline{\mathscr{B}}\to\mathrm{R}u_{\mathbf{E}*}(\psi^{\prime}_{*}\mathbb{F}^{\prime}\otimes\overline{\mathscr{B}}^{\prime})$
is an almost isomorphism. The problem is local on $\mathbf{E}_{Y\to
X}^{\mathrm{\acute{e}t}}$, thus we may assume that $\psi_{*}\mathbb{F}$ is the
constant sheaf with value $\mathbb{Z}/p^{n}\mathbb{Z}$ by 10.3. Then,
$\psi^{\prime}_{*}\mathbb{F}^{\prime}$ is also the constant sheaf with value
$\mathbb{Z}/p^{n}\mathbb{Z}$ by 10.9. Let $V\to U$ be an object of
$\mathbf{E}_{Y\to X}^{\mathrm{pro\acute{e}t}}$ such that
$U^{V}=\mathop{\mathrm{Spec}}(R)$ is the spectrum of an
$\overline{\mathbb{Z}_{p}}$-algebra $R$ which is almost pre-perfectoid. Since
the almost isomorphisms $R/p^{n}\to(R\otimes_{A}A^{\prime})/p^{n}$ ($n\geq 1$)
induces an almost isomorphism of the $p$-adic completions $\widehat{R}\to
R\widehat{\otimes}_{A}A^{\prime}$, the $\overline{\mathbb{Z}_{p}}$-algebra
$R\otimes_{A}A^{\prime}$ is still almost pre-perfectoid (5.19). The pullback
of $V\to U$ in $\mathbf{E}_{Y^{\prime}\to
X^{\prime}}^{\mathrm{pro\acute{e}t}}$ is the object $V_{A^{\prime}}\to
U_{A^{\prime}}$ and $U_{A^{\prime}}^{V_{A^{\prime}}}$ is the spectrum of the
integral closure $R^{\prime}$ of $R\otimes_{A}A^{\prime}$ in
$R\otimes_{A}A^{\prime}[1/p]$. Since $R\otimes_{A}A^{\prime}$ is almost pre-
perfectoid, $R^{\prime}$ is also almost pre-perfectoid and the morphism
$(R\otimes_{A}A^{\prime})/p^{n}\to R^{\prime}/p^{n}$ is an almost isomorphism
by 5.30. Therefore, the morphism
$\overline{\mathscr{B}}/p^{n}\overline{\mathscr{B}}\to\mathrm{R}u_{\mathbf{E}*}(\overline{\mathscr{B}}^{\prime}/p^{n}\overline{\mathscr{B}}^{\prime})$
is an almost isomorphism by 7.32, 8.11 and 8.12. ∎

######  Theorem 10.17.

Let $\overline{\mathbb{Z}_{p}}$ be the integral closure of $\mathbb{Z}_{p}$ in
an algebraic closure of $\mathbb{Q}_{p}$, $A$ a
$\overline{\mathbb{Z}_{p}}$-algebra which is an absolutely integrally closed
valuation ring, $X$ a proper $A$-scheme of finite presentation,
$Y=\mathop{\mathrm{Spec}}(A[1/p])\times_{\mathop{\mathrm{Spec}}(A)}X$,
$\mathbb{F}$ a finite locally constant abelian sheaf on
$Y_{\mathrm{\acute{e}t}}$. Then, the absolute Faltings’ comparison morphism
associated to $f:(Y\to
X)\to(\mathop{\mathrm{Spec}}(A[1/p])\to\mathop{\mathrm{Spec}}(A))$ and
$\mathbb{F}$ (10.6.2) (which exists by 10.8),

(10.17.1)
$\displaystyle\mathrm{R}\Gamma(Y_{\mathrm{\acute{e}t}},\mathbb{F})\otimes^{\mathrm{L}}_{\mathbb{Z}}A\longrightarrow\mathrm{R}\Gamma(\mathbf{E}_{Y\to
X}^{\mathrm{\acute{e}t}},\psi_{*}\mathbb{F}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}),$

is an almost isomorphism in
$\mathbf{D}(\overline{\mathbb{Z}_{p}}\textrm{-}\mathbf{Mod})$ (5.7).

###### Proof.

Let $K$ be the $p$-adic completion of the maximal unramified extension of
$\mathbb{Q}_{p}$. By 10.16, we may assume that $A$ is a valuation ring of
height $1$ extension of $\mathcal{O}_{\overline{K}}$. Let $X_{\bullet}\to X$
be the proper hypercovering of coherent schemes constructed in 10.15. For each
degree $n$ the canonical morphisms (10.6.2)

(10.17.2)
$\displaystyle\mathrm{R}\Gamma(Y_{n,\mathrm{\acute{e}t}},\mathbb{F}_{n})\otimes_{\mathbb{Z}}^{\mathrm{L}}A\longleftarrow\mathrm{R}\Gamma(\mathbf{E}_{Y_{n}\to
X_{n}}^{\mathrm{\acute{e}t}},\psi_{n*}\mathbb{F}_{n})\otimes_{\mathbb{Z}}^{\mathrm{L}}A\longrightarrow\mathrm{R}\Gamma(\mathbf{E}_{Y_{n}\to
X_{n}}^{\mathrm{\acute{e}t}},\psi_{n*}\mathbb{F}_{n}\otimes_{\mathbb{Z}}\overline{\mathscr{B}})$

are an isomorphism and an almost isomorphism, where $\mathbb{F}_{n}$ is the
pullback of $\mathbb{F}$ by the natural morphism $Y_{n,\mathrm{\acute{e}t}}\to
Y_{\mathrm{\acute{e}t}}$. Consider the commutative diagram

(10.17.7)

where $\mathbb{F}_{\bullet}=(\mathbb{F}_{n})_{[n]\in\mathrm{Ob}(\Delta)}$ with
the notation in 6.5. By the functorial spectral sequence of simplicial sites
([Sta22, 09WJ]), we deduce from (10.17.2) that $\alpha_{1\bullet}$ is an
isomorphism and $\alpha_{2\bullet}$ is an almost isomorphism. Since
$\alpha_{1}$ is an isomorphism by 10.8, it remains to show that the left
vertical arrow is an isomorphism and the right vertical arrow is an almost
isomorphism.

We denote by $a:\mathbf{E}_{Y_{\bullet}\to
X_{\bullet}}^{\mathrm{\acute{e}t}}\to\mathbf{E}_{Y\to
X}^{\mathrm{\acute{e}t}}$ the augmentation of simplicial site and by
$a_{n}:\mathbf{E}_{Y_{n}\to X_{n}}^{\mathrm{\acute{e}t}}\to\mathbf{E}_{Y\to
X}^{\mathrm{\acute{e}t}}$ the natural morphism of sites. Notice that
$a^{-1}\psi_{*}\mathbb{F}=(a_{n}^{-1}\psi_{*}\mathbb{F})_{[n]\in\mathrm{Ob}(\Delta)}=(\psi_{n*}\mathbb{F}_{n})_{[n]\in\mathrm{Ob}(\Delta)}=\psi_{\bullet*}\mathbb{F}_{\bullet}$
by 10.9 ([Sta22, 0D70]). Since $X_{\bullet}^{Y_{\bullet}}\to X^{Y}$ forms a
hypercovering in $\mathbf{I}_{\overline{\eta}\to\overline{S}}$, the right
vertical arrow is an almost isomorphism by 10.3 and 8.18. Finally, the left
vertical arrow is an isomorphism by the cohomological descent for étale
cohomology [Sta22, 0DHL]. ∎

##  11\. Faltings’ Main $p$-adic Comparison Theorem: the Relative Case for
More General Coefficients

######  11.1.

Let $Y\to X$ be a morphism of coherent schemes such that $Y\to X^{Y}$ is an
open immersion. We obtain from 3.26, 8.6 and 7.8 a commutative diagram of
sites

(11.1.5)

where $a:(\mathbf{Sch}^{\mathrm{coh}}_{/Y})_{\mathrm{v}}\to
Y_{\mathrm{\acute{e}t}}$ and $\rho:Y_{\mathrm{\acute{e}t}}\to
Y_{\mathrm{f\acute{e}t}}$ are defined by the inclusion functors.

######  Lemma 11.2.

With the notation in 11.1, for any finite locally constant abelian sheaf
$\mathbb{F}$ on $Y_{\mathrm{\acute{e}t}}$, the canonical morphism
$\varepsilon^{-1}\psi_{*}\mathbb{F}\to\Psi_{*}a^{-1}\mathbb{F}$ is an
isomorphism.

###### Proof.

The base change morphism
$\varepsilon^{-1}\psi_{*}\mathbb{F}\to\Psi_{*}a^{-1}\mathbb{F}$ is the
composition of the adjunction morphisms ([SGA 4${}_{\text{\rm III}}$,
XVII.2.1.3])

(11.2.1)
$\displaystyle\varepsilon^{-1}\psi_{*}\mathbb{F}\to\Psi_{*}\Psi^{-1}(\varepsilon^{-1}\psi_{*}\mathbb{F})=\Psi_{*}a^{-1}(\psi^{-1}\psi_{*}\mathbb{F})\to\Psi_{*}a^{-1}\mathbb{F}$

which are both isomorphisms by 3.27.(2) and 10.3. ∎

######  11.3.

We fix an algebraic closure $\overline{\mathbb{Q}_{p}}$ of the $p$-adic number
field $\mathbb{Q}_{p}$ and we denote by $\overline{\mathbb{Z}_{p}}$ the
integral closure of $\mathbb{Z}_{p}$ in $\overline{\mathbb{Q}_{p}}$. We set
$\eta=\mathop{\mathrm{Spec}}(\mathbb{Q}_{p})$,
$\overline{\eta}=\mathop{\mathrm{Spec}}(\overline{\mathbb{Q}_{p}})$,
$S=\mathop{\mathrm{Spec}}(\mathbb{Z}_{p})$,
$\overline{S}=\mathop{\mathrm{Spec}}(\overline{\mathbb{Z}_{p}})$. Remark that
$\overline{\mathbb{Q}_{p}}$ is a pre-perfectoid field with valuation ring
$\overline{\mathbb{Z}_{p}}$ so we are also in the situation 8.7. Let
$f:(Y^{\prime}\to X^{\prime})\to(Y\to X)$ be a Cartesian morphism of morphisms
of coherent schemes with a Cartesian morphism $(Y\to
X^{Y})\to(\overline{\eta}\to\overline{S})$ (then, $Y^{\prime}\to X^{\prime
Y^{\prime}}$ is Cartesian over $\overline{\eta}\to\overline{S}$ by 3.19.(4)).
Thus, $X^{Y}$ and $X^{\prime Y^{\prime}}$ are objects of
$\mathbf{I}_{\overline{\eta}\to\overline{S}}$. Consider the following
commutative diagram of sites associated to $f$.

(11.3.5)

######  11.4.

Following 11.3, let $g:(\widetilde{Y}\to\widetilde{X})\to(Y\to X)$ be a
morphisms of coherent schemes such that
$\widetilde{Y}\to\widetilde{X}^{\widetilde{Y}}$ is also Cartesian over
$\overline{\eta}\to\overline{S}$. We denote by
$g^{\prime}:(\widetilde{Y^{\prime}}\to\widetilde{X^{\prime}})\to(Y^{\prime}\to
X^{\prime})$ the base change of $g$ by $f$, and denote by
$\widetilde{f}:(\widetilde{Y^{\prime}}\to\widetilde{X^{\prime}})\to(\widetilde{Y}\to\widetilde{X})$
the natural morphism which is Cartesian by base change. Thus,
$\widetilde{X}^{\widetilde{Y}}$ and
$\widetilde{X^{\prime}}^{\widetilde{Y^{\prime}}}$ are also objects of
$\mathbf{I}_{\overline{\eta}\to\overline{S}}$. We write the diagram (11.3.5)
associated $\widetilde{f}$ equipping all labels with tildes.

######  Lemma 11.5.

With the notation in 11.3 and 11.4, let $\mathbb{F}^{\prime}$ be a finite
locally constant abelian sheaf on $Y^{\prime}_{\mathrm{\acute{e}t}}$ and we
set $\mathscr{F}^{\prime}=\Psi^{\prime}_{*}a^{\prime-1}\mathbb{F}^{\prime}$.
Let $\widetilde{X}$ be an object of $\mathbf{I}_{Y\to X^{Y}}$,
$\widetilde{Y}=\overline{\eta}\times_{\overline{S}}\widetilde{X}$,
$\widetilde{\mathbb{F}^{\prime}}=g^{\prime-1}_{\mathrm{\acute{e}t}}\mathbb{F}^{\prime}$,
$q$ an integer.

  1. (1)

The sheaf $\mathrm{R}^{q}f_{\mathbf{I}*}\mathscr{F}^{\prime}$ on
$\mathbf{I}_{Y\to X^{Y}}$ is canonically isomorphic to the sheaf associated to
the presheaf $\widetilde{X}\mapsto
H^{q}_{\mathrm{\acute{e}t}}(\widetilde{Y^{\prime}},\widetilde{\mathbb{F}^{\prime}})$.

  2. (2)

The sheaf
$\mathrm{R}^{q}f_{\mathbf{I}*}(\mathscr{F}^{\prime}\otimes_{\mathbb{Z}}\mathscr{O}^{\prime})$
on $\mathbf{I}_{Y\to X^{Y}}$ is canonically almost isomorphic to the sheaf
associated to the presheaf $\widetilde{X}\mapsto
H^{q}(\mathbf{E}_{\widetilde{Y^{\prime}}\to\widetilde{X^{\prime}}}^{\mathrm{\acute{e}t}},\widetilde{\psi^{\prime}}_{*}\widetilde{\mathbb{F}^{\prime}}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}^{\prime})$.

  3. (3)

The canonical morphism
$(\mathrm{R}^{q}f_{\mathbf{I}*}\mathscr{F}^{\prime})\otimes_{\mathbb{Z}}\mathscr{O}\to(\mathrm{R}^{q}f_{\mathbf{I}*}\mathscr{F}^{\prime}\otimes_{\mathbb{Z}}\mathscr{O}^{\prime})$
is compatible with the canonical morphisms
$H^{q}_{\mathrm{\acute{e}t}}(\widetilde{Y^{\prime}},\widetilde{\mathbb{F}^{\prime}})\otimes_{\mathbb{Z}}R\stackrel{{\scriptstyle\alpha_{1}}}{{\longleftarrow}}H^{q}(\mathbf{E}_{\widetilde{Y^{\prime}}\to\widetilde{X^{\prime}}}^{\mathrm{\acute{e}t}},\widetilde{\psi^{\prime}}_{*}\widetilde{\mathbb{F}^{\prime}})\otimes_{\mathbb{Z}}R\stackrel{{\scriptstyle\alpha_{2}}}{{\longrightarrow}}H^{q}(\mathbf{E}_{\widetilde{Y^{\prime}}\to\widetilde{X^{\prime}}}^{\mathrm{\acute{e}t}},\widetilde{\psi^{\prime}}_{*}\widetilde{\mathbb{F}^{\prime}}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}^{\prime})$,
where $R=\overline{\mathscr{B}}(\widetilde{Y}\to\widetilde{X})$ (cf.10.5.3).

###### Proof.

Let $\widetilde{\mathscr{F}^{\prime}}$ be the restriction of
$\mathscr{F}^{\prime}$ on
$\mathbf{I}_{\widetilde{Y^{\prime}}\to\widetilde{X^{\prime}}^{\widetilde{Y^{\prime}}}}$.
We have
$\widetilde{\mathscr{F}^{\prime}}=\widetilde{\Psi^{\prime}}_{*}\widetilde{a^{\prime}}^{-1}\widetilde{\mathbb{F}^{\prime}}$.
We set
$\widetilde{\mathbb{L}^{\prime}}=\widetilde{\psi^{\prime}}_{*}\widetilde{\mathbb{F}^{\prime}}$
which is a finite locally constant abelian sheaf on
$\mathbf{E}_{\widetilde{Y^{\prime}}\to\widetilde{X^{\prime}}}^{\mathrm{\acute{e}t}}$
by 10.3. Notice that the canonical morphisms
$\widetilde{\psi^{\prime}}^{-1}\widetilde{\mathbb{L}^{\prime}}\to\widetilde{\mathbb{F}^{\prime}}$
and
$\widetilde{\varepsilon^{\prime}}^{-1}\widetilde{\mathbb{L}^{\prime}}\to\widetilde{\mathscr{F}^{\prime}}$
are isomorphisms by 10.3 and 11.2 respectively.

(1) It follows from the canonical isomorphisms

(11.5.1) $\displaystyle
H^{q}(I_{\widetilde{Y^{\prime}}\to\widetilde{X^{\prime}}^{\widetilde{Y^{\prime}}}},\widetilde{\varepsilon^{\prime}}^{-1}\widetilde{\mathbb{L}^{\prime}})\stackrel{{\scriptstyle\gamma_{1}}}{{\longrightarrow}}H^{q}_{\mathrm{v}}(\widetilde{Y^{\prime}},\widetilde{\Psi^{\prime}}^{-1}\widetilde{\varepsilon^{\prime}}^{-1}\widetilde{\mathbb{L}^{\prime}})=H^{q}_{\mathrm{v}}(\widetilde{Y^{\prime}},\widetilde{a^{\prime}}^{-1}\widetilde{\psi^{\prime}}^{-1}\widetilde{\mathbb{L}^{\prime}})\stackrel{{\scriptstyle\gamma_{2}}}{{\longleftarrow}}H^{q}_{\mathrm{\acute{e}t}}(\widetilde{Y^{\prime}},\widetilde{\psi^{\prime}}^{-1}\widetilde{\mathbb{L}^{\prime}}),$

where $\gamma_{1}$ is induced by the canonical isomorphism
$\widetilde{\varepsilon^{\prime}}^{-1}\widetilde{\mathbb{L}^{\prime}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{R}\widetilde{\Psi^{\prime}}_{*}\widetilde{\Psi^{\prime}}^{-1}\widetilde{\varepsilon^{\prime}}^{-1}\widetilde{\mathbb{L}^{\prime}}$
(3.27.(2)), and $\gamma_{2}$ is induced by the canonical isomorphism
$\widetilde{\psi^{\prime}}^{-1}\widetilde{\mathbb{L}^{\prime}}\to\mathrm{R}\widetilde{a^{\prime}}_{*}\widetilde{a^{\prime}}^{-1}\widetilde{\psi^{\prime}}^{-1}\widetilde{\mathbb{L}^{\prime}}$
(3.9).

(2) It follows from the canonical almost isomorphism

(11.5.2)
$\displaystyle\gamma_{3}:H^{q}(\mathbf{E}_{\widetilde{Y^{\prime}}\to\widetilde{X^{\prime}}}^{\mathrm{\acute{e}t}},\widetilde{\mathbb{L}^{\prime}}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}^{\prime})\longrightarrow
H^{q}(I_{\widetilde{Y^{\prime}}\to\widetilde{X^{\prime}}^{\widetilde{Y^{\prime}}}},\widetilde{\varepsilon^{\prime}}^{-1}\widetilde{\mathbb{L}^{\prime}}\otimes\mathscr{O}^{\prime})$

which is induced by the canonical almost isomorphism
$\widetilde{\mathbb{L}^{\prime}}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}^{\prime}\to\mathrm{R}\widetilde{\varepsilon^{\prime}}_{*}(\widetilde{\varepsilon^{\prime}}^{-1}\widetilde{\mathbb{L}^{\prime}}\otimes\mathscr{O}^{\prime})$
(8.14).

(3) Consider the following diagram

(11.5.7)

where the unlabelled vertical arrow is induced by the canonical morphism
$\widetilde{\mathbb{L}^{\prime}}\to\mathrm{R}\widetilde{\varepsilon^{\prime}}_{*}\widetilde{\varepsilon^{\prime}}^{-1}\widetilde{\mathbb{L}^{\prime}}$,
and the unlabelled horizontal arrow is the canonical morphism which induces
$(\mathrm{R}^{q}f_{\mathbf{I}*}\mathscr{F}^{\prime})\otimes_{\mathbb{Z}}\mathscr{O}\to\mathrm{R}^{q}f_{\mathbf{I}*}(\mathscr{F}^{\prime}\otimes_{\mathbb{Z}}\mathscr{O}^{\prime})$
on $\mathbf{I}_{Y\to X^{Y}}$ by sheafification. It is clear that the diagram
(11.5.7) is commutative, which completes the proof. ∎

######  11.6.

We remark that 11.5 gives a new definition of the relative (resp. absolute)
Faltings’ comparison morphism without using 10.8. Following 11.3, let
$\mathbb{F}^{\prime}$ be a finite locally constant abelian sheaf on
$Y^{\prime}_{\mathrm{\acute{e}t}}$ and we set
$\mathscr{F}^{\prime}=\Psi^{\prime}_{*}a^{\prime-1}\mathbb{F}^{\prime}$. We
set $\mathbb{L}^{\prime}=\psi^{\prime}_{*}\mathbb{F}^{\prime}$, which is a
finite locally constant abelian sheaf on $\mathbf{E}_{Y^{\prime}\to
X^{\prime}}^{\mathrm{\acute{e}t}}$ by 10.3. Remark that the canonical
morphisms $\psi^{\prime-1}\mathbb{L}^{\prime}\to\mathbb{F}^{\prime}$ and
$\varepsilon^{\prime-1}\mathbb{L}^{\prime}\to\mathscr{F}^{\prime}$ are
isomorphisms by 10.3 and 11.2 respectively. We also remark that
$\overline{\mathscr{B}}$, $\mathscr{O}$ are flat over $\mathbb{Z}$. The
canonical morphisms in the derived category
$\mathbf{D}(\overline{\mathscr{B}}\textrm{-}\mathbf{Mod}_{\mathbf{E}_{Y\to
X}^{\mathrm{\acute{e}t}}})$ (cf. 10.4.8),

(11.6.3)

fit into the following commutative diagram

(11.6.8)

  1. (1)

The morphism $\alpha_{3}$ is induced by the canonical isomorphism
$\psi^{\prime-1}\mathbb{L}^{\prime}\to\mathrm{R}a^{\prime}_{*}a^{\prime-1}(\psi^{\prime-1}\mathbb{L}^{\prime})$
by 3.9, and thus $\alpha_{3}$ is an isomorphism.

  2. (2)

The morphism $\alpha_{5}$ is induced by the canonical isomorphism
$\varepsilon^{\prime-1}\mathbb{L}^{\prime}\to\mathrm{R}\Psi^{\prime}_{*}\Psi^{\prime-1}\varepsilon^{\prime-1}\mathbb{L}^{\prime}$
by 3.27, and thus $\alpha_{5}$ is an isomorphism.

  3. (3)

The unlabelled arrow is induced by the canonical morphism
$\mathbb{L}\to\mathrm{R}\varepsilon^{\prime}_{*}\varepsilon^{\prime-1}\mathbb{L}$.

  4. (4)

The morphism $\alpha_{4}$ is induced by the canonical almost isomorphism
$\mathbb{L}^{\prime}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}^{\prime}\to\mathrm{R}\varepsilon^{\prime}_{*}(\varepsilon^{\prime-1}\mathbb{L}^{\prime}\otimes_{\mathbb{Z}}\mathscr{O}^{\prime})$
by 8.14, and thus $\alpha_{4}$ is an almost isomorphism.

  5. (5)

The morphism $\alpha_{6}$ is the composition of

(11.6.9)
$\displaystyle\mathrm{R}\varepsilon_{*}(\mathrm{R}f_{\mathbf{I}*}\varepsilon^{\prime-1}\mathbb{L}^{\prime})\otimes_{\mathbb{Z}}^{\mathrm{L}}\overline{\mathscr{B}}$
$\displaystyle\longrightarrow\mathrm{R}\varepsilon_{*}((\mathrm{R}f_{\mathbf{I}*}\varepsilon^{\prime-1}\mathbb{L}^{\prime})\otimes_{\mathbb{Z}}^{\mathrm{L}}\mathscr{O})$
(11.6.10) $\displaystyle\textrm{ with
}\mathrm{R}\varepsilon_{*}((\mathrm{R}f_{\mathbf{I}*}\varepsilon^{\prime-1}\mathbb{L}^{\prime})\otimes_{\mathbb{Z}}^{\mathrm{L}}\mathscr{O})$
$\displaystyle\longrightarrow\mathrm{R}\varepsilon_{*}\mathrm{R}f_{\mathbf{I}*}(\varepsilon^{\prime-1}\mathbb{L}^{\prime}\otimes_{\mathbb{Z}}\mathscr{O}^{\prime}).$

In conclusion, the arrows $\alpha_{3}$, $\alpha_{5}$, $\alpha_{6}$ and
$\alpha_{4}$ induce an arrow

(11.6.11)
$\displaystyle\alpha_{4}^{-1}\circ\alpha_{6}\circ\alpha_{5}^{-1}\circ\alpha_{3}:\mathrm{R}\psi_{*}(\mathrm{R}f_{\mathrm{\acute{e}t}*}\mathbb{F}^{\prime})\otimes_{\mathbb{Z}}^{\mathrm{L}}\overline{\mathscr{B}}\longrightarrow\mathrm{R}f_{\mathbf{E}*}(\psi^{\prime}_{*}\mathbb{F}^{\prime}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}^{\prime})$

in the derived category of almost $\overline{\mathbb{Z}_{p}}$-modules on
$\mathbf{E}_{Y\to X}^{\mathrm{\acute{e}t}}$ (5.7). Remark that we don’t assume
that $\alpha_{1}$ is an isomorphism here. We also call (11.6.11) the _relative
Faltings’ comparison morphism_. Indeed, if $\alpha_{1}$ is an isomorphism,
then the relative Faltings’ comparison morphism (10.6.1) induces (11.6.11) in
$\mathbf{D}(\overline{\mathbb{Z}_{p}}^{\mathrm{al}}\textrm{-}\mathbf{Mod})$
due to the commutativity of the diagram (11.6.8).

If $X$ is the spectrum of an absolutely integrally closed valuation ring $A$
and if $Y=\overline{\eta}\times_{\overline{S}}X$, then applying the functor
$\mathrm{R}\Gamma(Y\to X,-)$ on (11.6.8) we obtain the natural morphisms in
the derived category $\mathbf{D}(A\textrm{-}\mathbf{Mod})$ by 7.9,

(11.6.16)

The arrows $\alpha_{3}$, $\alpha_{5}$, $\alpha_{6}$ and $\alpha_{4}$ induce an
arrow

(11.6.17)
$\displaystyle\alpha_{4}^{-1}\circ\alpha_{6}\circ\alpha_{5}^{-1}\circ\alpha_{3}:\mathrm{R}\Gamma(Y^{\prime}_{\mathrm{\acute{e}t}},\mathbb{F}^{\prime})\otimes_{\mathbb{Z}}^{\mathrm{L}}A\longrightarrow\mathrm{R}\Gamma(\mathbf{E}_{Y^{\prime}\to
X^{\prime}}^{\mathrm{\acute{e}t}},\psi^{\prime}_{*}\mathbb{F}^{\prime}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}^{\prime})$

in the derived category
$\mathbf{D}(\overline{\mathbb{Z}_{p}}^{\mathrm{al}}\textrm{-}\mathbf{Mod})$ of
almost $\overline{\mathbb{Z}_{p}}$-modules (5.7). We also call (11.6.17) the
_absolute Faltings’ comparison morphism_.

######  Lemma 11.7.

With the notation in 11.3, let $\mathbb{F}^{\prime}$ be a finite locally
constant abelian sheaf on $Y^{\prime}_{\mathrm{\acute{e}t}}$ and we set
$\mathscr{F}^{\prime}=\Psi^{\prime}_{*}a^{\prime-1}\mathbb{F}^{\prime}$.
Assume that $X^{\prime}\to X$ is proper of finite presentation. Then, the
canonical morphism

(11.7.1)
$\displaystyle(\mathrm{R}f_{\mathbf{I}*}\mathscr{F}^{\prime})\otimes_{\mathbb{Z}}^{\mathrm{L}}\mathscr{O}\longrightarrow\mathrm{R}f_{\mathbf{I}*}(\mathscr{F}^{\prime}\otimes_{\mathbb{Z}}\mathscr{O}^{\prime})$

is an almost isomorphism.

###### Proof.

Following 11.5, consider the following presheaves on $\mathbf{I}_{Y\to X^{Y}}$
for each integer $q$:

(11.7.2) $\displaystyle\mathcal{H}_{1}^{q}$
$\displaystyle:\widetilde{X}\longmapsto
H^{q}_{\mathrm{\acute{e}t}}(\widetilde{Y^{\prime}},\widetilde{\mathbb{F}^{\prime}})\otimes_{\mathbb{Z}}\overline{\mathscr{B}}(\widetilde{Y}\to\widetilde{X}),$
(11.7.3) $\displaystyle\mathcal{H}_{2}^{q}$
$\displaystyle:\widetilde{X}\longmapsto
H^{q}(\mathbf{E}_{\widetilde{Y^{\prime}}\to\widetilde{X^{\prime}}}^{\mathrm{\acute{e}t}},\widetilde{\psi^{\prime}}_{*}\widetilde{\mathbb{F}^{\prime}})\otimes_{\mathbb{Z}}\overline{\mathscr{B}}(\widetilde{Y}\to\widetilde{X}),$
(11.7.4) $\displaystyle\mathcal{H}_{3}^{q}$
$\displaystyle:\widetilde{X}\longmapsto
H^{q}(\mathbf{E}_{\widetilde{Y^{\prime}}\to\widetilde{X^{\prime}}}^{\mathrm{\acute{e}t}},\widetilde{\psi^{\prime}}_{*}\widetilde{\mathbb{F}^{\prime}}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}^{\prime}),$

They satisfy the limit-preserving condition 3.25.(ii) by 7.12, [SGA
4${}_{\text{\rm II}}$, VII.5.6] and [SGA 4${}_{\text{\rm II}}$, VI 8.5.9,
8.7.3]. Moreover, if $\widetilde{X}=\mathop{\mathrm{Spec}}(A)$ where $A$ is an
absolutely integrally closed valuation ring with $p$ nonzero in $A$, then the
canonical morphisms

(11.7.5)
$\displaystyle\mathcal{H}_{1}^{q}(\mathop{\mathrm{Spec}}(A))\leftarrow\mathcal{H}_{2}^{q}(\mathop{\mathrm{Spec}}(A))\to\mathcal{H}_{3}^{q}(\mathop{\mathrm{Spec}}(A))$

are an isomorphism and an almost isomorphism by 10.17. Thus, the canonical
morphisms
$\mathcal{H}_{1}^{q}\leftarrow\mathcal{H}_{2}^{q}\to\mathcal{H}_{3}^{q}$
induce an isomorphism and an almost isomorphism of their sheafifications by
3.25. The conclusion follows from 11.5. ∎

######  Lemma 11.8.

Let $Y\to X$ be an open immersion of coherent schemes, $Y^{\prime}\to Y$ a
finite morphism of finite presentation. Then, there exists a finite morphism
$X^{\prime}\to X$ of finite presentation whose base change by $Y\to X$ is
$Y^{\prime}\to Y$.

###### Proof.

Firstly, assume that $X$ is Noetherian. We have $Y^{\prime}=Y\times_{X}X^{Y}$
by 3.19.(4). We write $X^{Y}=\mathop{\mathrm{Spec}}_{X}(\mathcal{A})$ where
$\mathcal{A}$ is an integral quasi-coherent $\mathcal{O}_{X}$-algebra on $X$,
and we write $\mathcal{A}$ as a filtered colimit of its finite quasi-coherent
$\mathcal{O}_{X}$-subalgebras
$\mathcal{A}=\mathop{\mathrm{colim}}\mathcal{A}_{\alpha}$ ([Sta22, 0817]). Let
$\mathcal{B}_{\alpha}$ be the restriction of $\mathcal{A}_{\alpha}$ to $Y$.
Then, $\mathcal{B}=\mathop{\mathrm{colim}}\mathcal{B}_{\alpha}$ is a filtered
colimit of finite quasi-coherent $\mathcal{O}_{Y}$-algebras with injective
transition morphisms. Since
$Y^{\prime}=\mathop{\mathrm{Spec}}_{Y}(\mathcal{B})$ is finite over $Y$, there
exists an index $\alpha_{0}$ such that
$Y^{\prime}=\mathop{\mathrm{Spec}}_{Y}(\mathcal{B}_{\alpha_{0}})$. Therefore,
$X^{\prime}=\mathop{\mathrm{Spec}}_{X}(\mathcal{A}_{\alpha_{0}})$ meets our
requirements.

In general, we write $X$ as a cofiltered limit of coherent schemes of finite
type over $\mathbb{Z}$ with affine transition morphisms
$X=\lim_{\lambda\in\Lambda}X_{\lambda}$ ([Sta22, 01ZA]). Since $Y\to X$ is an
open immersion of finite presentation, using [EGA IV3, 8.8.2, 8.10.5] there
exists an index $\lambda_{0}\in\Lambda$, an open immersion $Y_{\lambda_{0}}\to
X_{\lambda_{0}}$ and a finite morphism $Y^{\prime}_{\lambda_{0}}\to
Y_{\lambda_{0}}$ such that the base change of the morphisms
$Y^{\prime}_{\lambda_{0}}\to Y_{\lambda_{0}}\to X_{\lambda_{0}}$ by $X\to
X_{\lambda_{0}}$ are the morphisms $Y^{\prime}\to Y\to X$. By the first
paragraph, there exists a finite morphism $X^{\prime}_{\lambda_{0}}\to
X_{\lambda_{0}}$ of finite presentation such that
$Y^{\prime}_{\lambda_{0}}=Y_{\lambda_{0}}\times_{X_{\lambda_{0}}}X^{\prime}_{\lambda_{0}}$.
We see that the base change $X^{\prime}\to X$ of $X^{\prime}_{\lambda_{0}}\to
X_{\lambda_{0}}$ by $X\to X_{\lambda_{0}}$ meets our requirements. ∎

######  Lemma 11.9.

With the notation in 11.3, let $g:Y^{\prime\prime}\to Y^{\prime}$ be a finite
morphism of finite presentation, $\mathbb{F}^{\prime\prime}$ a finite locally
constant abelian sheaf on $Y^{\prime\prime}_{\mathrm{\acute{e}t}}$ and we set
$\mathscr{F}^{\prime}=\Psi^{\prime}_{*}a^{\prime-1}(g_{\mathrm{\acute{e}t}*}\mathbb{F}^{\prime\prime})$.
Assume that $X^{\prime}\to X$ is proper of finite presentation. Then, the
canonical morphism

(11.9.1)
$\displaystyle(\mathrm{R}f_{\mathbf{I}*}\mathscr{F}^{\prime})\otimes_{\mathbb{Z}}^{\mathrm{L}}\mathscr{O}\longrightarrow\mathrm{R}f_{\mathbf{I}*}(\mathscr{F}^{\prime}\otimes_{\mathbb{Z}}\mathscr{O}^{\prime})$

is an almost isomorphism.

###### Proof.

There exists a Cartesian morphism $g:(Y^{\prime\prime}\to
X^{\prime\prime})\to(Y^{\prime}\to X^{Y}\times_{X}X^{\prime})$ of open
immersions of coherent schemes such that $X^{\prime\prime}\to
X^{Y}\times_{X}X^{\prime}$ is finite and of finite presentation by 11.8.
Consider the diagram (11.3.5) associated to $g$:

(11.9.6)

We set
$\mathscr{G}^{\prime\prime}=\Psi^{\prime\prime}_{*}a^{\prime\prime-1}\mathbb{F}^{\prime\prime}$.
The base change morphism $a^{\prime-1}g_{\mathrm{\acute{e}t}*}\to
g_{\mathrm{v}*}a^{\prime\prime-1}$ induces a canonical isomorphism
$\mathscr{F}^{\prime}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}g_{\mathbf{I}*}\mathscr{G}^{\prime\prime}$
by 3.10. Moreover, the canonical morphism
$g_{\mathbf{I}*}\mathscr{G}^{\prime\prime}\to\mathrm{R}g_{\mathbf{I}*}\mathscr{G}^{\prime\prime}$
is an isomorphism by 11.5.(1) and 3.25, since $g:Y^{\prime\prime}\to
Y^{\prime}$ is finite ([SGA 4${}_{\text{\rm II}}$, VIII.5.6]). By applying
11.7 to $g$ and $\mathbb{F}^{\prime\prime}$, the canonical morphism

(11.9.7)
$\displaystyle(\mathrm{R}g_{\mathbf{I}*}\mathscr{G}^{\prime\prime})\otimes_{\mathbb{Z}}^{\mathrm{L}}\mathscr{O}^{\prime}\longrightarrow\mathrm{R}g_{\mathbf{I}*}(\mathscr{G}^{\prime\prime}\otimes_{\mathbb{Z}}\mathscr{O}^{\prime\prime})$

is an almost isomorphism. Let $h$ be the composition of $(Y^{\prime\prime}\to
X^{\prime\prime})\to(Y^{\prime}\to X^{Y}\times_{X}X^{\prime})\to(Y\to X^{Y})$.
Note that $X^{\prime\prime}\to X^{Y}$ is also proper of finite presentation.
By applying 11.7 to $h$ and $\mathbb{F}^{\prime\prime}$, the canonical
morphism

(11.9.8)
$\displaystyle(\mathrm{R}h_{\mathbf{I}*}\mathscr{G}^{\prime\prime})\otimes_{\mathbb{Z}}^{\mathrm{L}}\mathscr{O}\longrightarrow\mathrm{R}h_{\mathbf{I}*}(\mathscr{G}^{\prime\prime}\otimes_{\mathbb{Z}}\mathscr{O}^{\prime\prime})$

is an almost isomorphism. It is clear that $h_{\mathbf{I}}=f_{\mathbf{I}}\circ
g_{\mathbf{I}}$. The conclusion follows from the canonical isomorphism
$\mathscr{F}^{\prime}\to\mathrm{R}g_{\mathbf{I}*}\mathscr{G}^{\prime\prime}$
and the canonical almost isomorphisms (11.9.7) and (11.9.8). ∎

######  Lemma 11.10.

With the notation in 11.3, let $\mathcal{F}^{\prime}$ be a constructible
abelian sheaf on $Y^{\prime}_{\mathrm{\acute{e}t}}$ and we set
$\mathscr{F}^{\prime}=\Psi^{\prime}_{*}a^{\prime-1}\mathcal{F}^{\prime}$.
Assume that $X^{\prime}\to X$ is proper of finite presentation. Then, the
canonical morphism

(11.10.1)
$\displaystyle(\mathrm{R}f_{\mathbf{I}*}\mathscr{F}^{\prime})\otimes_{\mathbb{Z}}^{\mathrm{L}}\mathscr{O}\longrightarrow\mathrm{R}f_{\mathbf{I}*}(\mathscr{F}^{\prime}\otimes_{\mathbb{Z}}\mathscr{O}^{\prime})$

is an almost isomorphism.

###### Proof.

We prove by induction on an integer $q$ that the canonical morphism
$(\mathrm{R}^{q}f_{\mathbf{I}*}\mathscr{F}^{\prime})\otimes_{\mathbb{Z}}\mathscr{O}\to\mathrm{R}^{q}f_{\mathbf{I}*}(\mathscr{F}^{\prime}\otimes_{\mathbb{Z}}\mathscr{O}^{\prime})$
is an almost isomorphism. It holds trivially for each $q\leq-1$. Notice that
there exists a finite morphism $g:Y^{\prime\prime}\to Y^{\prime}$ of finite
presentation, a finite locally constant abelian sheaf
$\mathbb{F}^{\prime\prime}$ on $Y^{\prime\prime}_{\mathrm{\acute{e}t}}$ and an
injective morphism $\mathcal{F}^{\prime}\to
g_{\mathrm{\acute{e}t}*}\mathbb{F}^{\prime\prime}$ by [Sta22, 09Z7] (cf. [SGA
4${}_{\text{\rm III}}$, IX.2.14]). Let $\mathcal{G}^{\prime}$ be the quotient
of $\mathcal{F}^{\prime}\to
g_{\mathrm{\acute{e}t}*}\mathbb{F}^{\prime\prime}$, which is also a
constructible abelian sheaf on $Y^{\prime}_{\mathrm{\acute{e}t}}$ since
$g_{\mathrm{\acute{e}t}*}\mathbb{F}^{\prime\prime}$ is so ([Sta22, 095R,
03RZ]). The exact sequence $0\to\mathcal{F}^{\prime}\to
g_{\mathrm{\acute{e}t}*}\mathbb{F}^{\prime\prime}\to\mathcal{G}^{\prime}\to 0$
induces an exact sequence by 3.27.(1),

(11.10.4)

We set
$\mathscr{H}^{\prime}=\Psi^{\prime}_{*}a^{\prime-1}(g_{\mathrm{\acute{e}t}*}\mathbb{F}^{\prime\prime})$
and $\mathscr{G}^{\prime}=\Psi^{\prime}_{*}a^{\prime-1}\mathcal{G}^{\prime}$.
Then, we obtain a morphism of long exact sequences

(11.10.9)

Notice that $\gamma_{1}$ and $\gamma_{2}$ are almost isomorphisms by
induction, and that $\gamma_{4}$ is an almost isomorphism by 11.9. Thus,
applying the 5-lemma ([Sta22, 05QA]) in the abelian category of almost
$\overline{\mathbb{Z}_{p}}$-modules over $\mathbf{I}_{Y\to X^{Y}}$, we see
that $\gamma_{3}$ is almost injective. Since $\mathcal{F}^{\prime}$ is an
arbitrary constructible abelian sheaf, the morphism $\gamma_{5}$ is also
almost injective. Thus, $\gamma_{3}$ is an almost isomorphism. ∎

######  Theorem 11.11.

With the notation in 11.3, let $\mathcal{F}^{\prime}$ be a torsion abelian
sheaf on $Y^{\prime}_{\mathrm{\acute{e}t}}$ and we set
$\mathscr{F}^{\prime}=\Psi^{\prime}_{*}a^{\prime-1}\mathcal{F}^{\prime}$.
Assume that $X^{\prime}\to X$ is proper of finite presentation. Then, the
canonical morphism

(11.11.1)
$\displaystyle(\mathrm{R}f_{\mathbf{I}*}\mathscr{F}^{\prime})\otimes_{\mathbb{Z}}^{\mathrm{L}}\mathscr{O}\longrightarrow\mathrm{R}f_{\mathbf{I}*}(\mathscr{F}^{\prime}\otimes_{\mathbb{Z}}\mathscr{O}^{\prime})$

is an almost isomorphism in the derived category
$\mathbf{D}(\overline{\mathbb{Z}_{p}}\textrm{-}\mathbf{Mod}_{\mathbf{I}_{Y\to
X^{Y}}})$ (5.7).

###### Proof.

We write $\mathcal{F}^{\prime}$ as a filtered colimit of constructible abelian
sheaves
$\mathcal{F}^{\prime}=\mathop{\mathrm{colim}}_{\lambda\in\Lambda}\mathcal{F}^{\prime}_{\lambda}$
([Sta22, 03SA], cf. [SGA 4${}_{\text{\rm III}}$, IX.2.7.2]). We set
$\mathscr{F}^{\prime}_{\lambda}=\Psi^{\prime}_{*}a^{\prime-1}\mathcal{F}_{\lambda}$.
We have
$\mathscr{F}^{\prime}=\mathop{\mathrm{colim}}_{\lambda\in\Lambda}\mathscr{F}^{\prime}_{\lambda}$
by [SGA 4${}_{\text{\rm II}}$, VI.5.1] whose conditions are satisfied since
each object in each concerned site is quasi-compact. Moreover, for each
integer $q$, we have

(11.11.2)
$\displaystyle(\mathrm{R}^{q}f_{\mathbf{I}*}\mathscr{F}^{\prime})\otimes_{\mathbb{Z}}\mathscr{O}=$
$\displaystyle\mathop{\mathrm{colim}}_{\lambda\in\Lambda}(\mathrm{R}^{q}f_{\mathbf{I}*}\mathscr{F}^{\prime}_{\lambda})\otimes_{\mathbb{Z}}\mathscr{O},$
(11.11.3)
$\displaystyle\mathrm{R}^{q}f_{\mathbf{I}*}(\mathscr{F}^{\prime}\otimes_{\mathbb{Z}}\mathscr{O}^{\prime})=$
$\displaystyle\mathop{\mathrm{colim}}_{\lambda\in\Lambda}\mathrm{R}^{q}f_{\mathbf{I}*}(\mathscr{F}^{\prime}_{\lambda}\otimes_{\mathbb{Z}}\mathscr{O}^{\prime}).$

The conclusion follows from 11.10. ∎

######  Lemma 11.12.

With the notation in 11.3 and 11.4, let $\mathcal{F}^{\prime}$ be a torsion
abelian sheaf on $Y^{\prime}_{\mathrm{\acute{e}t}}$,
$\mathcal{H}=\mathrm{R}f_{\mathrm{\acute{e}t}*}\mathcal{F}^{\prime}$, and we
set $\mathscr{F}^{\prime}=\Psi^{\prime}_{*}a^{\prime-1}\mathcal{F}^{\prime}$,
$\mathscr{H}=\mathrm{R}\Psi_{*}a^{-1}\mathcal{H}$. Let $\widetilde{X}$ be an
object of $\mathbf{I}_{Y\to X^{Y}}$,
$\widetilde{Y}=\overline{\eta}\times_{\overline{S}}\widetilde{X}$,
$\widetilde{\mathcal{F}^{\prime}}=g^{\prime-1}_{\mathrm{\acute{e}t}}\mathcal{F}^{\prime}$.

  1. (1)

The sheaf $\mathrm{R}^{q}f_{\mathbf{I}*}\mathscr{F}^{\prime}$ is canonically
isomorphic to the presheaf $\widetilde{X}\mapsto
H^{q}_{\mathrm{\acute{e}t}}(\widetilde{Y^{\prime}},\widetilde{\mathcal{F}^{\prime}})$
for each integer $q$.

  2. (2)

If $Y^{\prime}\to Y$ is proper, then there exists a canonical isomorphism
$\mathscr{H}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{R}f_{\mathbf{I}*}\mathscr{F}^{\prime}$.

###### Proof.

Note that the canonical morphism
$\mathscr{F}^{\prime}\to\mathrm{R}\Psi^{\prime}_{*}a^{\prime-1}\mathcal{F}^{\prime}$
is an isomorphism by 3.27.(1). Thus,
$\mathrm{R}f_{\mathbf{I}*}\mathscr{F}^{\prime}=\mathrm{R}(\Psi\circ
f_{\mathrm{v}})_{*}a^{\prime-1}\mathcal{F}^{\prime}$, whose $q$-th cohomology
is the sheaf associated to the presheaf $\widetilde{X}\mapsto
H^{q}_{\mathrm{v}}(\widetilde{Y^{\prime}},\widetilde{a^{\prime}}^{-1}\widetilde{\mathcal{F}^{\prime}})=H^{q}_{\mathrm{\acute{e}t}}(\widetilde{Y^{\prime}},\widetilde{\mathcal{F}^{\prime}})$
by 3.9, and thus (1) follows. If $Y^{\prime}\to Y$ is proper, then the base
change morphism
$a^{-1}\mathrm{R}f_{\mathrm{\acute{e}t}*}\to\mathrm{R}f_{\mathrm{v}*}a^{\prime-1}$
induces an isomorphism
$a^{-1}\mathcal{H}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{R}f_{\mathrm{v}*}a^{\prime-1}\mathcal{F}^{\prime}$
by 3.10, and thus (2) follows. ∎

######  Theorem 11.13.

With the notation in 11.3, let $\mathbb{F}^{\prime}$ be a finite locally
constant abelian sheaf on $Y^{\prime}_{\mathrm{\acute{e}t}}$. Assume that

  1. (i)

the morphism $X^{\prime}\to X$ is proper of finite presentation, and that

  2. (ii)

the sheaf $\mathrm{R}^{q}f_{\mathrm{\acute{e}t}*}\mathbb{F}^{\prime}$ is
finite locally constant for each integer $q$ and nonzero for finitely many
$q$, and that

  3. (iii)

we have $\mathrm{R}^{q}\psi_{*}\mathbb{H}=0$ (resp.
$\mathrm{R}^{q}\psi^{\prime}_{*}\mathbb{H}=0$) for any finite locally constant
abelian sheaf $\mathbb{H}$ on $Y_{\mathrm{\acute{e}t}}$ (resp.
$Y^{\prime}_{\mathrm{\acute{e}t}}$) and any integer $q>0$.

Then, the relative Faltings’ comparison morphism associated to $f$ and
$\mathbb{F}^{\prime}$ (10.6.1) (which exists by (iii)) is an almost
isomorphism in the derived category
$\mathbf{D}(\overline{\mathbb{Z}_{p}}\textrm{-}\mathbf{Mod}_{\mathbf{E}_{Y\to
X}^{\mathrm{\acute{e}t}}})$ (5.7), and it induces an almost isomorphism

(11.13.1)
$\displaystyle(\psi_{*}\mathrm{R}^{q}f_{\mathrm{\acute{e}t}*}\mathbb{F}^{\prime})\otimes_{\mathbb{Z}}\overline{\mathscr{B}}\longrightarrow\mathrm{R}^{q}f_{\mathbf{E}*}(\psi^{\prime}_{*}\mathbb{F}^{\prime}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}^{\prime})$

of $\overline{\mathbb{Z}_{p}}$-modules for each integer $q$.

###### Proof.

We follow the discussion of 11.6 and set
$\mathscr{F}^{\prime}=\Psi^{\prime}_{*}a^{\prime-1}\mathbb{F}^{\prime}$. The
canonical morphism (11.6.10)

(11.13.2)
$\displaystyle\mathrm{R}\varepsilon_{*}((\mathrm{R}f_{\mathbf{I}*}\mathscr{F}^{\prime})\otimes_{\mathbb{Z}}^{\mathrm{L}}\mathscr{O})\longrightarrow\mathrm{R}\varepsilon_{*}\mathrm{R}f_{\mathbf{I}*}(\mathscr{F}^{\prime}\otimes_{\mathbb{Z}}\mathscr{O}^{\prime})$

is an almost isomorphism by 11.7. It remains to show that the canonical
morphism (11.6.9)

(11.13.3)
$\displaystyle\mathrm{R}\varepsilon_{*}(\mathrm{R}f_{\mathbf{I}*}\mathscr{F}^{\prime})\otimes_{\mathbb{Z}}^{\mathrm{L}}\overline{\mathscr{B}}\longrightarrow\mathrm{R}\varepsilon_{*}((\mathrm{R}f_{\mathbf{I}*}\mathscr{F}^{\prime})\otimes_{\mathbb{Z}}^{\mathrm{L}}\mathscr{O})$

is also an almost isomorphism. With the notation in 11.12 by taking
$\mathcal{F}^{\prime}=\mathbb{F}^{\prime}$, the complex $\mathcal{H}$ is a
bounded complex whose cohomologies $H^{q}(\mathcal{H})$ are finite locally
constant abelian sheaves by condition (ii). Consider the commutative diagram
(11.1.5),

(11.13.8)

We set $\mathcal{L}=\mathrm{R}\psi_{*}\mathcal{H}$. Then,
$H^{q}(\mathcal{L})=\psi_{*}H^{q}(\mathcal{H})$ by Cartan-Leray spectral
sequence and condition (iii). Hence, $\mathcal{L}$ is a bounded complex of
abelian sheaves whose cohomologies are finite locally constant by 10.3 so that
the canonical morphism

(11.13.9)
$\displaystyle\mathcal{L}\otimes_{\mathbb{Z}}^{\mathrm{L}}\overline{\mathscr{B}}\longrightarrow\mathrm{R}\varepsilon_{*}(\varepsilon^{-1}\mathcal{L}\otimes_{\mathbb{Z}}^{\mathrm{L}}\mathscr{O})$

is an almost isomorphism by 8.15.

On the other hand, $H^{q}(\mathscr{H})=\Psi_{*}a^{-1}H^{q}(\mathcal{H})$ by
Cartan-Leray spectral sequence and 3.27.(1). Thus, the base change morphism
$\varepsilon^{-1}\mathrm{R}\psi_{*}\to\mathrm{R}\Psi_{*}a^{-1}$ induces an
isomorphism
$\varepsilon^{-1}\mathcal{L}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathscr{H}$
by 11.2. Moreover, the canonical morphism
$\mathcal{L}\to\mathrm{R}\varepsilon_{*}\varepsilon^{-1}\mathcal{L}=\mathrm{R}\varepsilon_{*}\mathscr{H}=\mathrm{R}\psi_{*}\mathrm{R}a_{*}a^{-1}\mathcal{H}$
is an isomorphism by 3.9. Thus, the canonical morphism

(11.13.10)
$\displaystyle(\mathrm{R}\varepsilon_{*}\varepsilon^{-1}\mathcal{L})\otimes_{\mathbb{Z}}^{\mathrm{L}}\overline{\mathscr{B}}\longrightarrow\mathrm{R}\varepsilon_{*}(\varepsilon^{-1}\mathcal{L}\otimes_{\mathbb{Z}}\mathscr{O})$

is an almost isomorphism by (11.13.9). In conclusion, (11.13.3) is an almost
isomorphism by (11.13.10) and by the canonical isomorphisms
$\varepsilon^{-1}\mathcal{L}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathscr{H}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{R}f_{\mathbf{I}*}\mathscr{F}^{\prime}$.
∎

######  Remark 11.14.

We give two concrete situations where the conditions in 11.13 are satisfied:

  1. (1)

Let $\overline{\mathbb{Z}_{p}}$ be the integral closure of $\mathbb{Z}_{p}$ in
an algebraic closure $\overline{\mathbb{Q}_{p}}$ of $\mathbb{Q}_{p}$,
$X^{\prime}\to X$ a proper and finitely presented morphism of coherent
$\overline{\mathbb{Z}_{p}}$-schemes, $Y^{\prime}\to Y$ the base change of
$X^{\prime}\to X$ by
$\mathop{\mathrm{Spec}}(\overline{\mathbb{Q}_{p}})\to\mathop{\mathrm{Spec}}(\overline{\mathbb{Z}_{p}})$.
Assume that $Y^{\prime}\to Y$ is smooth. Then, the condition (ii) is
guaranteed by [SGA 4${}_{\text{\rm III}}$, XVI.2.2 and XVII.5.2.8.1], and the
condition (iii) is guaranteed by 10.8.

  2. (2)

Let $\mathcal{O}_{K}$ be a strictly Henselian discrete valuation ring with
fraction field $K$ of characteristic $0$ and residue field of characteristic
$p$, $\overline{K}$ an algebraic closure of $K$, $X^{\prime}\to X$ a proper
morphism of $\mathcal{O}_{K}$-schemes of finite type, $Y^{\prime}\to Y$ the
base change of $X^{\prime}\to X$ by
$\mathop{\mathrm{Spec}}(\overline{K})\to\mathop{\mathrm{Spec}}(\mathcal{O}_{K})$.
Assume that $Y^{\prime}\to Y$ is smooth. Then, the condition (ii) is
guaranteed by [SGA 4${}_{\text{\rm III}}$, XVI.2.2 and XVII.5.2.8.1], and the
condition (iii) is guaranteed by 10.7.

##  12\. A Local Version of the Relative Hodge-Tate Filtration

######  12.1.

Let $K$ be a complete discrete valuation field of characteristic $0$ with
valuation ring $\mathcal{O}_{K}$ whose residue field is algebraically closed
(a condition required by [AG20, 4.1.3, 5.1.3]) of characteristic $p>0$,
$\overline{K}$ an algebraic closure of $K$, $\mathcal{O}_{\overline{K}}$ the
integral closure of $\mathcal{O}_{K}$ in $\overline{K}$,
$\eta=\mathop{\mathrm{Spec}}(K)$,
$\overline{\eta}=\mathop{\mathrm{Spec}}(\overline{K})$,
$S=\mathop{\mathrm{Spec}}(\mathcal{O}_{K})$,
$\overline{S}=\mathop{\mathrm{Spec}}(\mathcal{O}_{\overline{K}})$. Let
$(f,g):(X^{\prime\triangleright}\to X^{\prime})\to(X^{\circ}\to X)$ be a
morphism of adequate open immersions of schemes (10.11) over $\eta\to S$ such
that $g:X^{\prime}\to X$ is projective and that the induced morphism
$(X^{\prime},\mathscr{M}_{X^{\prime\triangleright}\to
X^{\prime}})\to(X,\mathscr{M}_{X^{\circ}\to X})$ is smooth and saturated. We
are in fact in the situation 10.12 but with a slightly different notation.

Consider the morphisms of sites defined in 7.34:

(12.1.3)

We consider $\breve{\sigma}$ as a morphism of ringed sites
$\breve{\sigma}:(\mathbf{E}_{X^{\circ}_{\overline{\eta}}\to
X}^{\mathrm{\acute{e}t},\mathbb{N}},\breve{\overline{\mathscr{B}}})\to(X^{\mathbb{N}}_{\mathrm{\acute{e}t}},\breve{\mathcal{O}}_{X_{\mathrm{\acute{e}t}}})$,
and $\breve{f}_{\overline{\eta},\mathrm{\acute{e}t}}$ as a morphism of ringed
sites
$\breve{f}_{\overline{\eta},\mathrm{\acute{e}t}}:(X^{\prime\triangleright,\mathbb{N}}_{\overline{\eta},\mathrm{\acute{e}t}},\breve{\mathbb{Z}}_{p})\to(X^{\circ,\mathbb{N}}_{\overline{\eta},\mathrm{\acute{e}t}},\breve{\mathbb{Z}}_{p})$,
where $\breve{\mathbb{Z}}_{p}=(\mathbb{Z}/p^{n}\mathbb{Z})_{n\geq 0}$.

######  Theorem 12.2 ([AG20, 6.7.5, 6.7.10, 6.7.13]).

Under the assumptions in 12.1 and with the same notation, for each integer
$n\geq 0$, there is a canonical $G_{K}$-equivariant finite decreasing
filtration $(\mathrm{Fil}^{q})_{q\in\mathbb{Z}}$ on the
$\breve{\overline{\mathscr{B}}}_{\mathbb{Q}}$-module
$\breve{\psi}_{*}(\mathrm{R}^{n}\breve{f}_{\overline{\eta},\mathrm{\acute{e}t}*}(\breve{\mathbb{Z}}_{p}))\otimes_{\mathbb{Z}_{p}}\breve{\overline{\mathscr{B}}}_{\mathbb{Q}}$
(cf. 2.6) and a canonical $G_{K}$-equivariant isomorphism for each
$q\in\mathbb{Z}$,

(12.2.1)
$\displaystyle\mathrm{Gr}^{q}(\breve{\psi}_{*}(\mathrm{R}^{n}\breve{f}_{\overline{\eta},\mathrm{\acute{e}t}*}(\breve{\mathbb{Z}}_{p}))\otimes_{\breve{\mathbb{Z}}_{p}}\breve{\overline{\mathscr{B}}}_{\mathbb{Q}})\cong\breve{\sigma}^{*}(\mathrm{R}^{q}g_{*}(\Omega^{n-q}_{(X^{\prime},\mathscr{M}_{X^{\prime}})/(X,\mathscr{M}_{X})})\otimes_{\mathcal{O}_{X_{\mathrm{\acute{e}t}}}}\breve{\mathcal{O}}_{X_{\mathrm{\acute{e}t}}}(q-n))_{\mathbb{Q}},$

where $\mathrm{Gr}^{q}$ denotes the graded piece
$\mathrm{Fil}^{q}/\mathrm{Fil}^{q+1}$.

We call this filtration the _relative Hodge-Tate filtration_ of the morphism
$(f,g):(X^{\prime\triangleright}\to X^{\prime})\to(X^{\circ}\to X)$.

######  Remark 12.3.

We keep the notation and assumptions in 12.1 and 12.2.

  1. (1)

If we set
$\mathrm{R}^{n}\breve{f}_{\overline{\eta},\mathrm{\acute{e}t}*}(\breve{\mathbb{Z}}_{p})=\breve{\mathbb{L}}^{(n)}=(\mathbb{L}_{k}^{(n)})_{k\geq
0}$ where
$\mathbb{L}_{k}^{(n)}=\mathrm{R}^{n}f_{\overline{\eta},\mathrm{\acute{e}t}*}(\mathbb{Z}/p^{k}\mathbb{Z})$,
then $\mathbb{L}_{k}^{(n)}$ is a finite locally constant abelian sheaf on
$X^{\circ}_{\overline{\eta},\mathrm{\acute{e}t}}$ ([AG20, 2.2.25]), and the
inverse system $(\mathbb{L}_{k}^{(n)})_{k\geq 0}$ is Artin-Rees $p$-adic
([Fu15, 10.1.18.(iii)]).

  2. (2)

The $\mathcal{O}_{X}$-module
$\mathcal{M}^{q,n-q}=\mathrm{R}^{q}g_{*}(\Omega^{n-q}_{(X^{\prime},\mathscr{M}_{X^{\prime}})/(X,\mathscr{M}_{X})})$
is coherent and its restriction to $X_{\eta}$ is locally free ([IKN05, 7.2],
cf. the proof of [AG20, 6.7.13]).

######  12.4.

Under the assumptions in 12.1, 12.2, 12.3 and with the same notation, assume
further that $X=\mathop{\mathrm{Spec}}(R)$ is affine. We remark that $R$ is
$p$-torsion free ([AGT16, II.6.3.(ii)]). Let $V\to U$ be an object of
$\mathbf{E}_{X^{\circ}_{\overline{\eta}}\to X}^{\mathrm{pro\acute{e}t}}$
satisfying the following conditions:

  1. (1)

The morphism $V\to U$ is Faltings acyclic (cf. 8.1, 8.24).

  2. (2)

For any integers $n\geq 0$ and $k\geq 0$, the pullback
$\mathbb{L}_{k}^{(n)}|_{V_{\mathrm{\acute{e}t}}}$ is constant with value
$H^{n}_{\mathrm{\acute{e}t},k}$ .

We denote by $A$ the $\mathcal{O}_{K}$-algebra $\overline{\mathscr{B}}(V\to
U)$ (i.e. $U^{V}=\mathop{\mathrm{Spec}}(A)$), and we set
$H^{n}_{\mathrm{\acute{e}t}}=\lim_{k\to\infty}H^{n}_{\mathrm{\acute{e}t},k}$.

######  Remark 12.5.

Let $\overline{x}$ be a geometric point of $V$. Then, there is a natural
isomorphism $H^{n}_{\mathrm{\acute{e}t},k}\cong
H^{n}_{\mathrm{\acute{e}t}}(X^{\prime\triangleright}_{\overline{x}},\mathbb{Z}/p^{k}\mathbb{Z})$
([AG20, 2.2.25]). We remark that
$H^{n}_{\mathrm{\acute{e}t}}(X^{\prime\triangleright}_{\overline{x}},\mathbb{Z}_{p})=\lim_{k\to\infty}H^{n}_{\mathrm{\acute{e}t}}(X^{\prime\triangleright}_{\overline{x}},\mathbb{Z}/p^{k}\mathbb{Z})$
is a finitely generated $\mathbb{Z}_{p}$-module (thus so is
$H^{n}_{\mathrm{\acute{e}t}}$), and that the morphism of inverse systems of
abelian groups

(12.5.1)
$\displaystyle(H^{n}_{\mathrm{\acute{e}t}}(X^{\prime\triangleright}_{\overline{x}},\mathbb{Z}_{p})/p^{k}H^{n}_{\mathrm{\acute{e}t}}(X^{\prime\triangleright}_{\overline{x}},\mathbb{Z}_{p}))_{k\geq
0}\longrightarrow(H^{n}_{\mathrm{\acute{e}t},k})_{k\geq 0}$

is an Artin-Rees isomorphism, by which we mean its kernel and cokernel are
Artin-Rees zero (cf. 12.3.(1) and [Fu15, 10.1.4]).

######  12.6.

Following 12.4, notice that $U^{\circ}=X^{\circ}\times_{X}U$ is affine and
geometrically normal over $K$ and that $V$ is affine and normal, since
$X^{\circ}$ is smooth over $K$. We assume further that the following
conditions hold:

  1. (3)

The scheme $V$ is integral and lies over a connected component
$U^{\circ}_{\star}$ of $U^{\circ}$.

  2. (4)

The function field $\mathcal{L}$ of $V$ is a Galois extension of the function
field $\mathcal{K}$ of $U^{\circ}_{\star}$ with Galois group $\Gamma$.

Let $U^{\circ}_{\overline{\eta},\star}$ be the connected component of
$U^{\circ}_{\overline{\eta}}$ over which $V$ lies. Its function field is the
composite $\overline{K}\mathcal{K}$ of $\overline{K}$ and $\mathcal{K}$ in
$\mathcal{L}$, which is Galois over $\mathcal{K}$ whose Galois group
identifies with the closed subgroup $G_{L}$ of $G_{K}$ where $L$ is the
algebraic closure of $K$ in $\mathcal{K}$. We denote by $\Delta$ the Galois
group of $\mathcal{L}$ over $\overline{K}\mathcal{K}$. It is clear that
$G_{L}=\Gamma/\Delta$.

(12.6.7)

Since $V$ is the integral closure of $U^{\circ}_{\star}$ in $\mathcal{L}$, the
canonical homomorphism of groups

(12.6.8)
$\displaystyle\mathrm{Aut}_{U^{\circ}_{\star}}(V)^{\mathrm{op}}\longrightarrow\mathrm{Aut}_{\mathcal{K}}(\mathcal{L})=\Gamma$

is an isomorphism. In particular, $\Gamma$ acts naturally on $V$ on the right.
For $\gamma\in\Gamma$ with image $u\in G_{K}$, we denote by $f_{\gamma}:V\to
V$ the right action of $\gamma$ on $V$, and for any $K$-scheme $Z$, we denote
by $f_{u}:Z_{\overline{\eta}}\to Z_{\overline{\eta}}$ the base change of the
automorphism of $\overline{\eta}$ induced by $u$. There is a commutative
diagram

(12.6.13)

The natural isomorphism (induced by the base change)

(12.6.14) $\displaystyle
f_{\gamma,\mathrm{\acute{e}t}}^{-1}(\mathbb{L}_{k}^{(n)}|_{V_{\mathrm{\acute{e}t}}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathbb{L}_{k}^{(n)}|_{V_{\mathrm{\acute{e}t}}}$

defines a natural action of $\Gamma$ on $H^{n}_{\mathrm{\acute{e}t},k}$, and
thus an action on
$H^{n}_{\mathrm{\acute{e}t}}=\lim_{k}H^{n}_{\mathrm{\acute{e}t},k}$. On the
other hand, $\Gamma$ also acts naturally on $A$ as $\mathop{\mathrm{Spec}}(A)$
is the integral closure of $U$ in $\mathcal{L}$, and thus acts on the Tate
twist $A(i)$ via the map $\Gamma\to G_{K}$.

######  Theorem 12.7.

Under the assumptions in 12.4 and with the same notation, for each integer
$n\geq 0$, there is a canonical finite decreasing filtration
$(\mathrm{fil}^{q})_{q\in\mathbb{Z}}$ on
$H^{n}_{\mathrm{\acute{e}t}}\otimes_{\mathbb{Z}_{p}}\widehat{A}[1/p]$ and a
canonical isomorphism for each $q\in\mathbb{Z}$,

(12.7.1)
$\displaystyle\mathrm{gr}^{q}(H^{n}_{\mathrm{\acute{e}t}}\otimes_{\mathbb{Z}_{p}}\widehat{A}[\frac{1}{p}])\cong
H^{q}(X^{\prime},\Omega^{n-q}_{(X^{\prime},\mathscr{M}_{X^{\prime}})/(X,\mathscr{M}_{X})})\otimes_{R}\widehat{A}[\frac{1}{p}](q-n),$

where $\mathrm{gr}^{q}$ denotes the graded piece
$\mathrm{fil}^{q}/\mathrm{fil}^{q+1}$. Moreover, under the assumptions in 12.6
and with the same notation, the filtration
$(\mathrm{fil}^{q})_{q\in\mathbb{Z}}$ and the isomorphisms (12.7.1) are
$\Gamma$-equivariant.

###### Proof.

We set $Y=X^{\circ}_{\overline{\eta}}$. We start from the filtration of 12.2.
Consider the natural morphism of ringed sites (7.34.2)
$\breve{\nu}:(\mathbf{E}_{Y\to
X}^{\mathrm{pro\acute{e}t},\mathbb{N}},\breve{\overline{\mathscr{B}}})\to(\mathbf{E}_{Y\to
X}^{\mathrm{\acute{e}t},\mathbb{N}},\breve{\overline{\mathscr{B}}})$. We
obtain a filtration $\breve{\nu}^{*}\mathrm{Fil}^{q}$ on
$\breve{\nu}^{*}(\breve{\psi}_{*}\breve{\mathbb{L}}^{(n)}\otimes_{\breve{\mathbb{Z}}_{p}}\breve{\overline{\mathscr{B}}}_{\mathbb{Q}})$
with graded pieces
$\breve{\nu}^{*}\mathrm{Gr}^{q}=\breve{\nu}^{*}\breve{\sigma}^{*}\breve{\mathcal{M}}^{q,n-q}(q-n)_{\mathbb{Q}}$
(as $\breve{\nu}^{*}=\breve{\nu}^{-1}$ is exact, cf. (7.34.3)). We apply the
derived functor $\mathrm{R}\Gamma(\mathbf{E}_{V\to
U}^{\mathrm{pro\acute{e}t},\mathbb{N}},-)$ to these modules.

Consider the commutative diagram

(12.7.6)

Since the canonical morphism
$j^{-1}\psi_{*}\mathbb{L}^{(n)}_{k}\to\psi_{*}j^{-1}\mathbb{L}^{(n)}_{k}$ is
an isomorphism by 10.9, $j^{-1}\psi_{*}\mathbb{L}^{(n)}_{k}$ is constant with
value $H^{n}_{\mathrm{\acute{e}t},k}$ by 10.3. Thus, the restriction of
$\breve{\nu}^{-1}\breve{\psi}_{*}\breve{\mathbb{L}}^{(n)}$ to
$\mathbf{E}_{V\to U}^{\mathrm{pro\acute{e}t},\mathbb{N}}$ is the inverse
system of constant sheaves $(H^{n}_{\mathrm{\acute{e}t},k})_{k\geq 0}$.
Therefore, the canonical morphism

(12.7.7) $\displaystyle
H^{n}_{\mathrm{\acute{e}t},k}\otimes_{\mathbb{Z}_{p}}A\longrightarrow\mathrm{R}\Gamma(\mathbf{E}_{V\to
U}^{\mathrm{pro\acute{e}t}},\nu^{*}(\psi_{*}\mathbb{L}_{k}^{(n)}\otimes_{\mathbb{Z}_{p}}\overline{\mathscr{B}}))$

is an almost isomorphism, since $V\to U$ is Faltings acyclic and
$H^{n}_{\mathrm{\acute{e}t},k}$ is a finite abelian group. Notice that for any
integer $q\geq 0$ there exists a canonical exact sequence ([AGT16, VI.7.10])

(12.7.8) $\displaystyle
0\to\mathrm{R}^{1}\lim_{k\to\infty}H^{q-1}(\mathbf{E}^{\mathrm{pro\acute{e}t}}_{V\to
U},\nu^{*}(\psi_{*}\mathbb{L}_{k}^{(n)}\otimes_{\mathbb{Z}_{p}}\overline{\mathscr{B}}))\to$
$\displaystyle H^{q}(\mathbf{E}^{\mathrm{pro\acute{e}t},\mathbb{N}}_{V\to
U},\breve{\nu}^{*}(\breve{\psi}_{*}\breve{\mathbb{L}}^{(n)}\otimes_{\breve{\mathbb{Z}}_{p}}\breve{\overline{\mathscr{B}}}))$
$\displaystyle\to\lim_{k\to\infty}H^{q}(\mathbf{E}^{\mathrm{pro\acute{e}t}}_{V\to
U},\nu^{*}(\psi_{*}\mathbb{L}_{k}^{(n)}\otimes_{\mathbb{Z}_{p}}\overline{\mathscr{B}}))\to
0.$

Since the inverse system
$(H^{n}_{\mathrm{\acute{e}t},k}\otimes_{\mathbb{Z}_{p}}A)_{k\geq 0}$ is Artin-
Rees $p$-adic,
$\mathrm{R}^{1}\lim_{k\to\infty}H^{0}(\mathbf{E}^{\mathrm{pro\acute{e}t}}_{V\to
U},\nu^{*}(\psi_{*}\mathbb{L}_{k}^{(n)}\otimes_{\mathbb{Z}_{p}}\overline{\mathscr{B}}))$
is almost zero by the almost isomorphism (12.7.7). Moreover, we deduce that
$\lim_{k\to\infty}(H^{n}_{\mathrm{\acute{e}t},k}\otimes_{\mathbb{Z}_{p}}A)\to
H^{0}(\mathbf{E}^{\mathrm{pro\acute{e}t},\mathbb{N}}_{V\to
U},\breve{\nu}^{*}(\breve{\psi}_{*}\breve{\mathbb{L}}^{(n)}\otimes_{\breve{\mathbb{Z}}_{p}}\breve{\overline{\mathscr{B}}}))$
is an almost isomorphism and that
$H^{q}(\mathbf{E}^{\mathrm{pro\acute{e}t},\mathbb{N}}_{V\to
U},\breve{\nu}^{*}(\breve{\psi}_{*}\breve{\mathbb{L}}^{(n)}\otimes_{\breve{\mathbb{Z}}_{p}}\breve{\overline{\mathscr{B}}}))$
is almost zero for $q>0$. Since $H^{n}_{\mathrm{\acute{e}t}}$ is a finitely
generated $\mathbb{Z}_{p}$-module (12.5), we have
$\lim_{k\to\infty}(H^{n}_{\mathrm{\acute{e}t},k}\otimes_{\mathbb{Z}_{p}}A)=H^{n}_{\mathrm{\acute{e}t}}\otimes_{\mathbb{Z}_{p}}\widehat{A}$.
Therefore, the canonical morphism

(12.7.9) $\displaystyle
H^{n}_{\mathrm{\acute{e}t}}\otimes_{\mathbb{Z}_{p}}\widehat{A}\longrightarrow\mathrm{R}\Gamma(\mathbf{E}_{V\to
U}^{\mathrm{pro\acute{e}t},\mathbb{N}},\breve{\nu}^{*}(\breve{\psi}_{*}\breve{\mathbb{L}}^{(n)}\otimes_{\breve{\mathbb{Z}}_{p}}\breve{\overline{\mathscr{B}}}))$

is an almost isomorphism. Inverting $p$, we obtain a canonical isomorphism

(12.7.10) $\displaystyle
H^{n}_{\mathrm{\acute{e}t}}\otimes_{\mathbb{Z}_{p}}\widehat{A}[\frac{1}{p}]\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{R}\Gamma(\mathbf{E}_{V\to
U}^{\mathrm{pro\acute{e}t},\mathbb{N}},\breve{\nu}^{*}(\breve{\psi}_{*}\breve{\mathbb{L}}^{(n)}\otimes_{\breve{\mathbb{Z}}_{p}}\breve{\overline{\mathscr{B}}}))[\frac{1}{p}].$

By taking $H^{0}(\mathbf{E}_{V\to
U}^{\mathrm{pro\acute{e}t},\mathbb{N}},\breve{\nu}^{*}\mathrm{Fil}^{q})$, the
filtration $\breve{\nu}^{*}\mathrm{Fil}^{q}$ on
$\breve{\nu}^{*}(\breve{\psi}_{*}\breve{\mathbb{L}}^{(n)}\otimes_{\breve{\mathbb{Z}}_{p}}\breve{\overline{\mathscr{B}}}_{\mathbb{Q}})$
induces a canonical filtration $\mathrm{fil}^{q}$ on
$H^{n}_{\mathrm{\acute{e}t}}(X^{\prime\triangleright}_{\overline{x}},\mathbb{Z}_{p})\otimes_{\mathbb{Z}_{p}}\widehat{A}[1/p]$
(cf. 2.6).

On the other hand, recall that $\mathcal{M}^{q,n-q}$ is the coherent
$\mathcal{O}_{X}$-module associated to the coherent $R$-module
$M^{q,n-q}=H^{q}(X^{\prime},\Omega^{n-q}_{(X^{\prime},\mathscr{M}_{X^{\prime}})/(X,\mathscr{M}_{X})})$
and that $M^{q,n-q}[1/p]$ is a projective $R[1/p]$-module (12.3.(2)). By 8.4,
we see that the canonical morphism

(12.7.11) $\displaystyle
H^{q}(X^{\prime},\Omega^{n-q}_{(X^{\prime},\mathscr{M}_{X^{\prime}})/(X,\mathscr{M}_{X})})\otimes_{R}\widehat{A}[\frac{1}{p}]\longrightarrow\mathrm{R}\Gamma(\mathbf{E}_{V\to
U}^{\mathrm{pro\acute{e}t},\mathbb{N}},\breve{\nu}^{*}\breve{\sigma}^{*}\breve{\mathcal{M}}^{q,n-q})[\frac{1}{p}]$

is an isomorphism. Thus, the canonical isomorphisms for the graded pieces
$\breve{\nu}^{*}\mathrm{Gr}^{q}=\breve{\nu}^{*}\breve{\sigma}^{*}\breve{\mathcal{M}}^{q,n-q}(q-n)_{\mathbb{Q}}$
induce canonical isomorphisms (12.7.1). This completes the proof of the first
statement.

For the $\Gamma$-equivariance, let $\gamma\in\Gamma$ with image $u\in G_{K}$.
We obtain from 12.6 a commutative diagram

(12.7.16)

where the vertical arrows are the same pro-étale morphism. It induces a
commutative diagram of fibred ringed sites over $\mathbb{N}$,

(12.7.21)

where the vertical morphisms induce the same localization morphism of the
associated topoi ([AGT16, III.7.9]). Recall that the $G_{K}$-actions on the
$\breve{\overline{\mathscr{B}}}$-modules
$\breve{\nu}^{*}(\breve{\psi}_{*}\breve{\mathbb{L}}^{(n)}\otimes_{\breve{\mathbb{Z}}_{p}}\breve{\overline{\mathscr{B}}})$,
$\breve{\nu}^{*}\breve{\sigma}^{*}\breve{\mathcal{M}}^{q,n-q}(q-n)$ on
$\mathbf{E}^{\mathrm{pro\acute{e}t},\mathbb{N}}_{Y\to X}$ define isomorphisms

(12.7.22)
$\displaystyle\breve{\nu}^{*}(\breve{\psi}_{*}\breve{\mathbb{L}}^{(n)}\otimes_{\breve{\mathbb{Z}}_{p}}\breve{\overline{\mathscr{B}}})$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}f_{u}^{-1}(\breve{\nu}^{*}(\breve{\psi}_{*}\breve{\mathbb{L}}^{(n)}\otimes_{\breve{\mathbb{Z}}_{p}}\breve{\overline{\mathscr{B}}})),$
(12.7.23)
$\displaystyle\breve{\nu}^{*}\breve{\sigma}^{*}\breve{\mathcal{M}}^{q,n-q}(q-n)$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}f_{u}^{-1}(\breve{\nu}^{*}\breve{\sigma}^{*}\breve{\mathcal{M}}^{q,n-q}(q-n)).$

By 12.2, up to isogenies, they are respectively compatible with the relative
Hodge-Tate filtration $\breve{\nu}^{*}\mathrm{Fil}^{q}$ and compatible with
the canonical isomorphisms for the graded pieces
$\breve{\nu}^{*}\mathrm{Gr}^{q}$. Passing to localization by $j^{-1}$, we
obtain from the isomorphisms (12.7.22) and (12.7.23) the isomorphisms

(12.7.24) $\displaystyle
j^{-1}(\breve{\nu}^{*}(\breve{\psi}_{*}\breve{\mathbb{L}}^{(n)}\otimes_{\breve{\mathbb{Z}}_{p}}\breve{\overline{\mathscr{B}}}))$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}j^{-1}f_{u}^{-1}(\breve{\nu}^{*}(\breve{\psi}_{*}\breve{\mathbb{L}}^{(n)}\otimes_{\breve{\mathbb{Z}}_{p}}\breve{\overline{\mathscr{B}}}))\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}f_{\gamma}^{-1}j^{-1}(\breve{\nu}^{*}(\breve{\psi}_{*}\breve{\mathbb{L}}^{(n)}\otimes_{\breve{\mathbb{Z}}_{p}}\breve{\overline{\mathscr{B}}})),$
(12.7.25) $\displaystyle
j^{-1}(\breve{\nu}^{*}\breve{\sigma}^{*}\breve{\mathcal{M}}^{q,n-q}(q-n))$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}j^{-1}f_{u}^{-1}(\breve{\nu}^{*}\breve{\sigma}^{*}\breve{\mathcal{M}}^{q,n-q}(q-n))\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}f_{\gamma}^{-1}j^{-1}(\breve{\nu}^{*}\breve{\sigma}^{*}\breve{\mathcal{M}}^{q,n-q}(q-n)).$

Applying the derived functor $\mathrm{R}\Gamma(\mathbf{E}_{V\to
U}^{\mathrm{pro\acute{e}t},\mathbb{N}},-)[1/p]$ and combining with the
canonical isomorphisms (12.7.10) and (12.7.11), we obtain automorphisms of
$H^{n}_{\mathrm{\acute{e}t}}\otimes_{\mathbb{Z}_{p}}\widehat{A}[1/p]$ and of
$H^{q}(X^{\prime},\Omega^{n-q}_{(X^{\prime},\mathscr{M}_{X^{\prime}})/(X,\mathscr{M}_{X})})\otimes_{R}\widehat{A}[1/p](q-n)$,
which are exactly the semi-linear actions of $\gamma\in\Gamma$ on these
$\widehat{A}[1/p]$-modules defined in 12.6 by going through the definitions.
Thus, we see that the actions of $\Gamma$ are compatible with the relative
Hodge-Tate filtration and the canonical isomorphisms for the graded pieces,
which completes the proof. ∎

######  Remark 12.8.

The arguments for 12.7 does not make use of the assumption that the residue
field of $K$ is algebraically closed.

######  Remark 12.9.

In 12.6, we take $U$ to be an étale neighborhood of a point of the special
fibre of $X$ which is affine and admits an adequate chart in the sense of
[AGT16, III.4.4] (cf. [AGT16, III.4.7]), and take $V$ to be the inverse limit
of the normalized universal cover $(V_{i})$ of $U^{\circ}_{\star}$ at
$\overline{x}$ (cf. [SGA 1, V.7], [AGT16, VI.9.8]). We set
$U^{V_{i}}=\mathop{\mathrm{Spec}}(R_{i})$ and
$\overline{R}=\mathop{\mathrm{colim}}(R_{i})$. Then,
$\Gamma=\pi_{1}(U^{\circ}_{\star},\overline{x})$ and $A=\overline{R}$. We
obtain from the adequate chart finitely many nonzero divisors
$f_{1},\dots,f_{r}$ of $\Gamma(U_{\eta},\mathcal{O}_{U_{\eta}})$ such that the
divisor $D=\sum_{i=1}^{r}\mathrm{div}(f_{i})$ has support $U_{\eta}\setminus
U^{\circ}$ and that at each strict henselization of $U_{\eta}$ those elements
$f_{i}$ contained in the maximal ideal form a subset of a regular system of
parameters (cf. [AG20, 4.2.2.(ii)]). Then, $A$ is almost pre-perfectoid and
admits compatible $n$-th power roots of $f_{i}$ ([AGT16, II.9.10]). Hence,
$V\to U$ is Faltings acyclic by 8.24, and thus Theorem 12.7 holds in this
setting, which gives a local version of the relative Hodge-Tate filtration
answering the question of Abbes-Gros raised in the first version of [AG20]
(cf. [AG20, 1.2.3]).

## References

  * [Abb10] Ahmed Abbes.  Éléments de géométrie rigide. Volume I, volume 286 of Progress in Mathematics.  Birkhäuser/Springer Basel AG, Basel, 2010.  Construction et étude géométrique des espaces rigides. [Construction and geometric study of rigid spaces], With a preface by Michel Raynaud.
  * [Ach15] Piotr Achinger.  $K(\pi,1)$-neighborhoods and comparison theorems.  Compos. Math., 151(10):1945–1964, 2015.
  * [Ach17] Piotr Achinger.  Wild ramification and $K(\pi,1)$ spaces.  Invent. Math., 210(2):453–499, 2017.
  * [AG20] Ahmed Abbes and Michel Gros.  Les suites spectrales de Hodge-Tate.  https://arxiv.org/abs/2003.04714v2, 2020.
  * [AGT16] Ahmed Abbes, Michel Gros, and Takeshi Tsuji.  The $p$-adic Simpson correspondence, volume 193 of Annals of Mathematics Studies.  Princeton University Press, Princeton, NJ, 2016.
  * [ALPT19] Karim Adiprasito, Gaku Liu, Igor Pak, and Michael Temkin.  Log smoothness and polystability over valuation rings.  https://arxiv.org/abs/1806.09168v3, 2019.
  * [And18] Yves André.  Le lemme d’Abhyankar perfectoide.  Publ. Math. Inst. Hautes Études Sci., 127:1–70, 2018.
  * [Bei12] Alexander Beilinson.  $p$-adic periods and derived de Rham cohomology.  J. Amer. Math. Soc., 25(3):715–738, 2012.
  * [BM20] Bhargav Bhatt and Akhil Mathew.  The arc-topology.  https://arxiv.org/abs/1807.04725v4, 2020.
  * [Bou06] Nicolas Bourbaki.  Algèbre commutative. Chapitres 5 à 7. Springer-Verlag, Berlin Heidelberg, 2006.  Edition originale publiée par Herman, Paris, 1975.
  * [Bou07] Nicolas Bourbaki.  Topologie générale. Chapitres 5 à 7. Springer-Verlag, Berlin Heidelberg, 2007.  Edition originale publiée par Herrman, Paris, 1974.
  * [BS15] Bhargav Bhatt and Peter Scholze.  The pro-étale topology for schemes.  Astérisque, (369):99–201, 2015.
  * [BS17] Bhargav Bhatt and Peter Scholze.  Projectivity of the Witt vector affine Grassmannian.  Invent. Math., 209(2):329–423, 2017.
  * [BS19] Bhargav Bhatt and Peter Scholze.  Prisms and prismatic cohomology.  https://arxiv.org/abs/1905.08229v4, 2019.
  * [BST17] Bhargav Bhatt, Karl Schwede, and Shunsuke Takagi.  The weak ordinarity conjecture and $F$-singularities.  In Higher dimensional algebraic geometry—in honour of Professor Yujiro Kawamata’s sixtieth birthday, volume 74 of Adv. Stud. Pure Math., pages 11–39. Math. Soc. Japan, Tokyo, 2017.
  * [CS17] Ana Caraiani and Peter Scholze.  On the generic part of the cohomology of compact unitary Shimura varieties.  Ann. of Math. (2), 186(3):649–766, 2017.
  * [CS19] Kestutis Cesnavicius and Peter Scholze.  Purity for flat cohomology.  https://arxiv.org/abs/1912.10932v2, 2019.
  * [DB81] Philippe Du Bois.  Complexe de de Rham filtré d’une variété singulière.  Bull. Soc. Math. France, 109(1):41–81, 1981.
  * [Del74] Pierre Deligne.  Théorie de Hodge. III.  Inst. Hautes Études Sci. Publ. Math., (44):5–77, 1974.
  * [dJ96] Aise Johan de Jong.  Smoothness, semi-stability and alterations.  Inst. Hautes Études Sci. Publ. Math., (83):51–93, 1996.
  * [dJ97] Aise Johan de Jong.  Families of curves and alterations.  Ann. Inst. Fourier (Grenoble), 47(2):599–621, 1997.
  * [EGA II] Alexander Grothendieck.  Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes.  Inst. Hautes Études Sci. Publ. Math., (8):222, 1961.
  * [EGA IV2] Alexander Grothendieck.  Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II.  Inst. Hautes Études Sci. Publ. Math., (24):231, 1965.
  * [EGA IV3] Alexander Grothendieck.  Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III.  Inst. Hautes Études Sci. Publ. Math., (28):255, 1966.
  * [EGA IV4] Alexander Grothendieck.  Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV.  Inst. Hautes Études Sci. Publ. Math., (32):361, 1967.
  * [Fal88] Gerd Faltings.  $p$-adic Hodge theory.  J. Amer. Math. Soc., 1(1):255–299, 1988.
  * [Fal02] Gerd Faltings.  Almost étale extensions.  Astérisque, (279):185–270, 2002.  Cohomologies $p$-adiques et applications arithmétiques, II.
  * [Fu15] Lei Fu.  Etale cohomology theory, volume 14 of Nankai Tracts in Mathematics.  World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, revised edition, 2015.
  * [GR03] Ofer Gabber and Lorenzo Ramero.  Almost ring theory, volume 1800 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 2003.
  * [GR04] Ofer Gabber and Lorenzo Ramero.  Foundations for almost ring theory – release 7.5.  https://arxiv.org/abs/math/0409584v13, 2004.
  * [HJ14] Annette Huber and Clemens Jörder.  Differential forms in the h-topology.  Algebr. Geom., 1(4):449–478, 2014.
  * [IKN05] Luc Illusie, Kazuya Kato, and Chikara Nakayama.  Quasi-unipotent logarithmic Riemann-Hilbert correspondences.  J. Math. Sci. Univ. Tokyo, 12(1):1–66, 2005.
  * [ILO14] Luc Illusie, Yves Laszlo, and Fabrice Orgogozo, editors.  Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents.  Société Mathématique de France, Paris, 2014.  Séminaire à l’École Polytechnique 2006–2008. [Seminar of the Polytechnic School 2006–2008], With the collaboration of Frédéric Déglise, Alban Moreau, Vincent Pilloni, Michel Raynaud, Joël Riou, Benoît Stroh, Michael Temkin and Weizhe Zheng, Astérisque No. 363-364 (2014) (2014).
  * [Kat89] Kazuya Kato.  Logarithmic structures of Fontaine-Illusie.  In Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), pages 191–224. Johns Hopkins Univ. Press, Baltimore, MD, 1989.
  * [Kat94] Kazuya Kato.  Toric singularities.  Amer. J. Math., 116(5):1073–1099, 1994.
  * [Ker16] Moritz Kerz.  Transfinite limits in topos theory.  Theory Appl. Categ., 31:Paper No. 7, 175–200, 2016.
  * [Niz06] Wiesława Nizioł.  Toric singularities: log-blow-ups and global resolutions.  J. Algebraic Geom., 15(1):1–29, 2006.
  * [Ogu18] Arthur Ogus.  Lectures on logarithmic algebraic geometry, volume 178 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 2018.
  * [Ryd10] David Rydh.  Submersions and effective descent of étale morphisms.  Bull. Soc. Math. France, 138(2):181–230, 2010.
  * [Sch12] Peter Scholze.  Perfectoid spaces.  Publ. Math. Inst. Hautes Études Sci., 116:245–313, 2012.
  * [Sch13a] Peter Scholze.  $p$-adic Hodge theory for rigid-analytic varieties.  Forum Math. Pi, 1:e1, 77, 2013.
  * [Sch13b] Peter Scholze.  Perfectoid spaces: a survey.  In Current developments in mathematics 2012, pages 193–227. Int. Press, Somerville, MA, 2013.
  * [Sch16] Peter Scholze.  $p$-adic Hodge theory for rigid-analytic varieties—corrigendum [MR3090230].  Forum Math. Pi, 4:e6, 4, 2016.
  * [Sch21] Peter Scholze.  Etale cohomology of diamonds.  https://arxiv.org/abs/1709.07343v2, 2021.
  * [SGA 1] Revêtements étales et groupe fondamental (SGA 1), volume 3 of Documents Mathématiques (Paris) [Mathematical Documents (Paris)].  Société Mathématique de France, Paris, 2003.  Séminaire de géométrie algébrique du Bois Marie 1960–61. [Algebraic Geometry Seminar of Bois Marie 1960-61], Directed by A. Grothendieck, With two papers by M. Raynaud, Updated and annotated reprint of the 1971 original [Lecture Notes in Math., 224, Springer, Berlin; MR0354651 (50 #7129)].
  * [SGA 4${}_{\text{\rm I}}$] Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos.  Lecture Notes in Mathematics, Vol. 269. Springer-Verlag, Berlin-New York, 1972.  Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat.
  * [SGA 4${}_{\text{\rm II}}$] Théorie des topos et cohomologie étale des schémas. Tome 2.  Lecture Notes in Mathematics, Vol. 270. Springer-Verlag, Berlin-New York, 1972.  Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat.
  * [SGA 4${}_{\text{\rm III}}$] Théorie des topos et cohomologie étale des schémas. Tome 3.  Lecture Notes in Mathematics, Vol. 305. Springer-Verlag, Berlin-New York, 1973.  Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat.
  * [Sta22] The Stacks project authors.  The stacks project.  https://stacks.math.columbia.edu, 2022.
  * [Tat67] John Tate.  $p$-divisible groups.  In Proc. Conf. Local Fields (Driebergen, 1966), pages 158–183. Springer, Berlin, 1967.
  * [Tsu99] Takeshi Tsuji.  $p$-adic étale cohomology and crystalline cohomology in the semi-stable reduction case.  Invent. Math., 137(2):233–411, 1999.
  * [Tsu02] Takeshi Tsuji.  Semi-stable conjecture of Fontaine-Jannsen: a survey.  Number 279, pages 323–370. 2002.  Cohomologies $p$-adiques et applications arithmétiques, II.
  * [Tsu19] Takeshi Tsuji.  Saturated morphisms of logarithmic schemes.  Tunis. J. Math., 1(2):185–220, 2019.
  * [Voe96] Vladimir Voevodsky.  Homology of schemes.  Selecta Math. (N.S.), 2(1):111–153, 1996.
  * [Xu22] Daxin Xu.  Parallel transport for Higgs bundles over $p$-adic curves.  https://arxiv.org/abs/2201.06697v1, 2022.