tcb@breakable

# Cohomological Descent for Faltings’ $p$-adic Hodge Theory and Applications

Tongmu He  Tongmu He, Institut des Hautes Études Scientifiques, 35 route de
Chartres, 91440 Bures-sur-Yvette, France<EMAIL_ADDRESS>
###### Abstract.

Faltings’ approach in $p$-adic Hodge theory can be schematically divided into
two main steps: firstly, a local reduction of the computation of the $p$-adic
étale cohomology of a smooth variety over a $p$-adic local field to a Galois
cohomology computation and then, the establishment of a link between the
latter and differential forms. These relations are organized through Faltings
ringed topos whose definition relies on the choice of an integral model of the
variety, and whose good properties depend on the (logarithmic) smoothness of
this model. Scholze’s generalization for rigid analytic varieties has the
advantage of depending only on the variety (i.e. the generic fibre). Inspired
by Deligne’s approach to classical Hodge theory for singular varieties, we
establish a cohomological descent result for the structural sheaf of Faltings
topos, which makes it possible to extend Faltings’ approach to any integral
model, i.e. without any smoothness assumption. An essential ingredient of our
proof is a descent result of perfectoid algebras in the arc-topology due to
Bhatt and Scholze.

As an application of our cohomological descent, using a variant of de Jong’s
alteration theorem for morphisms of schemes due to Gabber-Illusie-Temkin, we
generalize Faltings’ main $p$-adic comparison theorem to any proper and
finitely presented morphism of coherent schemes over an absolute integral
closure of $\mathbb{Z}_{p}$ (without any assumption of smoothness) for torsion
étale sheaves (not necessarily finite locally constant). As a second
application, we prove a local version of the relative Hodge-Tate filtration as
a consequence of the global version constructed by Abbes-Gros.

00footnotetext: _2020 Mathematics Subject Classification_ 14F30 (primary).
Keywords: cohomological descent, Faltings topos, v-topology, $p$-adic Hodge
theory, comparison.

###### Contents

  1. 1 Introduction
  2. 2 Notation and Conventions
  3. 3 The v-site of Integrally Closed Schemes
  4. 4 The arc-Descent of Perfect Algebras
  5. 5 Almost Pre-perfectoid Algebras
  6. 6 Brief Review on Covanishing Fibred Sites
  7. 7 Faltings Ringed Sites
  8. 8 Cohomological Descent of the Structural Sheaves
  9. 9 Complements on Logarithmic Geometry
  10. 10 Faltings’ Main $p$-adic Comparison Theorem: the Absolute Case
  11. 11 Faltings’ Main $p$-adic Comparison Theorem: the Relative Case for More General Coefficients
  12. 12 A Local Version of the Relative Hodge-Tate Filtration

##  1\. Introduction

######  1.1.

Faltings and Scholze’s approaches to $p$-adic Hodge theory share several
similarities. The most recent approach, that of Scholze, generalizes Faltings’
main techniques from schemes to adic spaces. Nevertheless, beyond the
analogies, there is no thread connecting the two. The main difficulty stems
from the difference between the nature of their keystones, namely the Faltings
topos for Faltings’ approach and the pro-étale topos of an adic space for
Scholze’s approach. Faltings’ approach has the advantage of only using schemes
and their classical étale topoi. But it depends on the choice of an integral
model of the $p$-adic variety, which intervenes in the very definition of
Faltings topos and whose (log-)smoothness seems necessary for the good
properties of this one. On the other hand, Scholze’s approach which uses adic
spaces and their pro-étale topoi, does not depend on any integral model.

The initial goal of this work is to make Faltings’ approach “free of integral
models”. For this, we establish a cohomological descent result for Faltings
ringed topos. Along the way, we introduce a variant for the v-topology which
satisfies good cohomological descent properties and which can be regarded as a
scheme theoretic analogue of the v-topos of an adic space. In particular, we
establish a cohomological descent result from this topos to Faltings topos. It
is an analogue of the cohomological descent from the v-topos to the pro-étale
topos of an adic space established by Scholze [Sch21]. We give two
applications of our cohomological descent result. Firstly, we extend Faltings’
main $p$-adic comparison theorem (which we refer to as “Faltings’ comparison
theorem” for short in the rest of the introduction), both in the absolute and
the relative cases, to general integral models without any smoothness
condition. Faltings’ comparison theorem was generalized by Scholze to rigid
analytic varieties, first in the smooth case and then in the general case. Our
application is an analogue of this last generalization. Even in the smooth
case, Faltings’ comparison theorem and Scholze’s generalization cannot be
directly deduced from each other. Secondly, we prove a local version of the
relative Hodge-Tate filtration as a consequence of the global version
constructed by Abbes and Gros [AG20] and our cohomological descent result. We
would like to mention a third interesting application of our result to the
$p$-adic Simpson correspondence given by Xu [Xu22].

######  1.2.

Faltings’ proof of the Hodge-Tate decomposition illustrates his approach in
$p$-adic Hodge theory and the role of his ringed topos. Let $K$ be a complete
discrete valuation field of characteristic $0$ with algebraically closed
residue field of characteristic $p>0$. We fix an algebraic closure
$\overline{K}$ of $K$ and denote by $\widehat{\overline{K}}$ the $p$-adic
completion of $\overline{K}$. For a proper smooth $K$-scheme $X$, Tate
conjectured that there is a canonical
$G_{K}=\mathrm{Gal}(\overline{K}/K)$-equivariant decomposition, now called the
Hodge-Tate decomposition ([Tat67, Remark, page 180]),

(1.2.1) $\displaystyle
H^{n}_{\mathrm{\acute{e}t}}(X_{\overline{K}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}}\widehat{\overline{K}}=\bigoplus_{0\leq
q\leq n}H^{q}(X,\Omega_{X/K}^{n-q})\otimes_{K}\widehat{\overline{K}}(q-n),$

where $\widehat{\overline{K}}(q-n)$ is the $(q-n)$-th Tate twist of
$\widehat{\overline{K}}$. This conjecture was settled by Faltings [Fal88,
Fal02] and Tsuji [Tsu99, Tsu02] independently, and had been generalized to
rigid analytic settings by Scholze [Sch13a]. There is also a version for non-
proper smooth varieties showed by Faltings. Let $X^{\circ}$ be an open subset
of $X$ whose complement is a normal crossings divisor $D$. Then, there is a
canonical $G_{K}$-equivariant decomposition

(1.2.2) $\displaystyle
H^{n}_{\mathrm{\acute{e}t}}(X^{\circ}_{\overline{K}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}}\widehat{\overline{K}}=\bigoplus_{0\leq
q\leq n}H^{q}(X,\Omega_{X/K}^{n-q}(\log
D))\otimes_{K}\widehat{\overline{K}}(q-n).$

######  1.3.

One of the applications of our main result in this article is a generalization
of the Hodge-Tate decomposition to the relative case. Let
$(f,g):(X^{\prime\triangleright}\to X^{\prime})\to(X^{\circ}\to X)$ be a
morphism of open immersions of coherent schemes over
$\mathop{\mathrm{Spec}}(K)\to\mathop{\mathrm{Spec}}(\mathcal{O}_{K})$
(“coherent” stands for “quasi-compact and quasi-separated”). We assume that
the following conditions hold:

  1. (1)

The associated log schemes
$(X^{\prime},\mathscr{M}_{X^{\prime}}),(X,\mathscr{M}_{X})$ endowed with
compactifying log structures are adequate (a technical condition which holds
if the open immersions $X^{\prime\triangleright}\to X^{\prime},X^{\circ}\to X$
are semi-stable over $\mathcal{O}_{K}$, cf. 10.11).

  2. (2)

The morphism of log schemes
$(X^{\prime},\mathscr{M}_{X^{\prime}})\to(X,\mathscr{M}_{X})$ is smooth and
saturated.

  3. (3)

The morphism of schemes $g:X^{\prime}\to X$ is projective.

  4. (4)

The scheme $X=\mathop{\mathrm{Spec}}(R)$ is affine and there exist finitely
many nonzero divisors $f_{1},\dots,f_{r}$ of $R[1/p]$ such that the divisor
$D=\sum_{i=1}^{r}\mathrm{div}(f_{i})$ on $X_{K}$ has support $X_{K}\setminus
X^{\circ}_{K}$ and that at each strict henselization of $X_{K}$ those elements
$f_{i}$ contained in the maximal ideal form a subset of a regular system of
parameters (in particular, $D$ is a normal crossings divisor on $X_{K}$).

For any coherent $X^{\circ}_{K}$-scheme $Y$, we define a pro-finite étale
$Y$-scheme

(1.3.1) $\displaystyle Y_{\infty}$
$\displaystyle=\lim_{n}Y[T_{1},\dots,T_{r}]/(T_{1}^{n}-f_{1},\dots,T_{r}^{n}-f_{r}).$

######  Theorem 1.4 (cf. 12.7 and 8.24).

Under the assumptions in 1.3, let $U$ be an affine scheme pro-étale over $X$
and let $V$ be a pro-finite étale $U^{\circ}_{\overline{K},\infty}$-scheme
(where $U^{\circ}=X^{\circ}\times_{X}U$) satisfying the following conditions:

  1. (1)

The integral closure of $U$ in $V$ is the spectrum of an
$\mathcal{O}_{\overline{K}}$-algebra $A$ which is almost pre-perfectoid in the
sense of 5.19.

  2. (2)

For any integers $n\geq 0$ and $k\geq 0$, the pullback
$(\mathrm{R}^{n}f_{\mathrm{\acute{e}t}*}\mathbb{Z}/p^{k}\mathbb{Z})|_{V_{\mathrm{\acute{e}t}}}$
is a constant sheaf.

Let $\overline{x}$ be a geometric point of $V$. Then, for any integer $n\geq
0$, there is a canonical finite decreasing filtration
$(\mathrm{fil}^{q})_{q\in\mathbb{Z}}$ on
$H^{n}_{\mathrm{\acute{e}t}}(X^{\prime\triangleright}_{\overline{x}},\mathbb{Z}_{p})\otimes_{\mathbb{Z}_{p}}\widehat{A}[1/p]$
and a canonical isomorphism for each $q\in\mathbb{Z}$,

(1.4.1)
$\displaystyle\mathrm{gr}^{q}(H^{n}_{\mathrm{\acute{e}t}}(X^{\prime\triangleright}_{\overline{x}},\mathbb{Z}_{p})\otimes_{\mathbb{Z}_{p}}\widehat{A}[\frac{1}{p}])\cong
H^{q}(X^{\prime},\Omega^{n-q}_{(X^{\prime},\mathscr{M}_{X^{\prime}})/(X,\mathscr{M}_{X})})\otimes_{R}\widehat{A}[\frac{1}{p}](q-n),$

where $\mathrm{gr}^{q}$ denotes the graded piece
$\mathrm{fil}^{q}/\mathrm{fil}^{q+1}$. Moreover, if $U^{\circ}$ and $V$ are
connected and if the function field of $V$ is a Galois extension of that of
$U^{\circ}$ with Galois group $\Gamma$, then the filtration
$(\mathrm{fil}^{q})_{q\in\mathbb{Z}}$ and the isomorphisms (1.4.1) are
$\Gamma$-equivariant.

######  Remark 1.5.

The objects $V\to U$ satisfying the conditions in 1.4 form a topological
generating family of the pro-étale Faltings site of
$X^{\circ}_{\overline{K}}\to X$ (see the proof of 8.24).

######  1.6.

This local _relative Hodge-Tate filtration_ stems from the global relative
Hodge-Tate filtration constructed by Abbes-Gros [AG20]. Their filtration takes
place on the Faltings topos associated to $X^{\circ}_{\overline{K}}\to X$. In
the first version of their work, they asked for an explicit local version.
Scholze and Caraiani [CS17] constructed independently a relative Hodge-Tate
filtration for proper smooth morphisms of smooth adic spaces, and Scholze
announced that he can give a local version, answering the question of Abbes-
Gros. Our construction is obtained by applying our cohomological descent
result for Faltings ringed topos to the global relative Hodge-Tate filtration
of Abbes-Gros. In a new version of their manuscript, Abbes-Gros gave a third
construction of the local Hodge-Tate filtration in a slightly more restrictive
framework, using a cohomological descent result which is a special case of
ours.

######  1.7.

Faltings ringed topos plays a central role in the proof of the Hodge-Tate
decomposition. Let $X^{\circ}\to X$ be an open immersion of coherent schemes
over $\mathop{\mathrm{Spec}}(K)\to\mathop{\mathrm{Spec}}(\mathcal{O}_{K})$
such that the associated log scheme $(X,\mathscr{M}_{X})$ endowed with
compactifying log structure is adequate. We set $Y=X_{\overline{K}}$. The
Faltings ringed site $(\mathbf{E}_{Y\to
X}^{\mathrm{\acute{e}t}},\overline{\mathscr{B}})$ was constructed by Faltings
and developed by Abbes-Gros [AGT16, VI]. Faltings designed it as a bridge
between the $p$-adic étale cohomology of $Y$ and differential forms of $X$.
Concretely, these links are established through natural morphisms of sites

(1.7.1) $\displaystyle
Y_{\mathrm{\acute{e}t}}\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}\mathbf{E}_{Y\to
X}^{\mathrm{\acute{e}t}}\stackrel{{\scriptstyle\sigma}}{{\longrightarrow}}X_{\mathrm{\acute{e}t}}$

which satisfy the following properties:

  1. (1)

(Faltings’ comparison theorem, [Fal02, Thm.8, page 223], [AG20, 4.8.13]).
Assume that $X$ is proper over $\mathcal{O}_{K}$. For any finite locally
constant abelian sheaf $\mathbb{F}$ on $Y_{\mathrm{\acute{e}t}}$, there exists
a canonical morphism

(1.7.2)
$\displaystyle\mathrm{R}\Gamma(Y_{\mathrm{\acute{e}t}},\mathbb{F})\otimes^{\mathrm{L}}_{\mathbb{Z}}\mathcal{O}_{\overline{K}}\longrightarrow\mathrm{R}\Gamma(\mathbf{E}_{Y\to
X}^{\mathrm{\acute{e}t}},\psi_{*}\mathbb{F}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}),$

which is an almost isomorphism, that is, the cohomology groups of its cone are
killed by $p^{r}$ for any rational number $r>0$.

  2. (2)

(Faltings’ computation of Galois cohomology, [AG20, 6.3.8]). There exists a
canonical homomorphism of
$\mathcal{O}_{X}\otimes_{\mathcal{O}_{K}}\mathcal{O}_{\overline{K}}$-modules

(1.7.3)
$\displaystyle\Omega^{q}_{(X,\mathscr{M}_{X})/(S,\mathscr{M}_{S})}\otimes_{\mathcal{O}_{K}}\mathcal{O}_{\overline{K}}/p^{n}\mathcal{O}_{\overline{K}}\longrightarrow\mathrm{R}^{q}\sigma_{*}(\overline{\mathscr{B}}/p^{n}\overline{\mathscr{B}})$

whose kernel and cokernel are killed by $p^{r}$ for any rational number
$r>\frac{2\dim(Y)+1}{p-1}$.

Observing that
$\mathbb{Z}/p^{n}\mathbb{Z}=\psi_{*}(\mathbb{Z}/p^{n}\mathbb{Z})$, Faltings
deduced the Hodge-Tate decomposition from the degeneration and splitting of
the Cartan-Leray spectral sequence for the composed functor
$\mathrm{R}\Gamma(X_{\mathrm{\acute{e}t}},-)\circ\mathrm{R}\sigma_{*}$, later
named the Hodge-Tate spectral sequence by Scholze. Using de Jong’s alteration
theorem, one can deduce the Hodge-Tate decomposition for a general proper
smooth $K$-scheme by reducing to the case where it admits a semi-stable model
(cf. [Tsu02, A5]).

######  1.8.

In order to state our cohomological descent result, we recall now the
definition of the Faltings site associated to a morphism of coherent schemes
$Y\to X$ (cf. 7.7). Let $\mathbf{E}_{Y\to X}^{\mathrm{\acute{e}t}}$ be the
category of morphisms of coherent schemes $V\to U$ over $Y\to X$, i.e.
commutative diagrams

(1.8.5)

such that $U$ is étale over $X$ and that $V$ is finite étale over
$Y\times_{X}U$. We endow $\mathbf{E}_{Y\to X}^{\mathrm{\acute{e}t}}$ with the
topology generated by the following types of families of morphisms

  1. (v)

$\\{(V_{m}\to U)\to(V\to U)\\}_{m\in M}$, where $M$ is a finite set and
$\coprod_{m\in M}V_{m}\to V$ is surjective;

  2. (c)

$\\{(V\times_{U}{U_{n}}\to U_{n})\to(V\to U)\\}_{n\in N}$, where $N$ is a
finite set and $\coprod_{n\in N}U_{n}\to U$ is surjective.

Consider the presheaf $\overline{\mathscr{B}}$ on $\mathbf{E}_{Y\to
X}^{\mathrm{\acute{e}t}}$ defined by

(1.8.6) $\displaystyle\overline{\mathscr{B}}(V\to
U)=\Gamma(U^{V},\mathcal{O}_{U^{V}}),$

where $U^{V}$ is the integral closure of $U$ in $V$. It is indeed a sheaf of
rings, the structural sheaf of $\mathbf{E}_{Y\to X}^{\mathrm{\acute{e}t}}$
(cf. 7.6).

######  1.9.

Recall that the cohomological descent of étale cohomology along proper
hypercoverings can be generalized as follows: for a coherent $S$-scheme, we
endow the category of coherent $S$-schemes $\mathbf{Sch}^{\mathrm{coh}}_{/S}$
with Voevodsky’s _h-topology_ which is generated by étale coverings and proper
surjective morphisms of finite presentation. Then, for any torsion abelian
sheaf $\mathcal{F}$ on $S_{\mathrm{\acute{e}t}}$, denoting by
$a:(\mathbf{Sch}^{\mathrm{coh}}_{/S})_{\mathrm{h}}\to S_{\mathrm{\acute{e}t}}$
the natural morphism of sites, the adjunction morphism
$\mathcal{F}\to\mathrm{R}a_{*}a^{-1}\mathcal{F}$ is an isomorphism.

This result remains true for a finer topology, the _v-topology_. A morphism of
coherent schemes $T\to S$ is called a v-covering if for any morphism
$\mathop{\mathrm{Spec}}(A)\to S$ with $A$ a valuation ring, there exists an
extension of valuation rings $A\to B$ and a lifting
$\mathop{\mathrm{Spec}}(B)\to T$. In fact, a v-covering is a limit of
h-coverings (cf. 3.6). We will describe the cohomological descent for
$\overline{\mathscr{B}}$ using a new site built from the v-topology, which can
be regarded as a scheme theoretic analogue of the v-site of adic spaces (cf.
[Sch21, 8.1, 14.1, 15.5]).

######  Definition 1.10 (cf. 3.23).

Let $S^{\circ}\to S$ be an open immersion of coherent schemes such that $S$ is
integrally closed in $S^{\circ}$. We define a site $\mathbf{I}_{S^{\circ}\to
S}$ as follows:

  1. (1)

The underlying category is formed by coherent $S$-schemes $T$ which are
integrally closed in $S^{\circ}\times_{S}T$.

  2. (2)

The topology is generated by covering families $\\{T_{i}\to T\\}_{i\in I}$ in
the v-topology.

We call $\mathbf{I}_{S^{\circ}\to S}$ the _v-site of $S^{\circ}$-integrally
closed coherent $S$-schemes_, and we call the sheaf $\mathscr{O}$ on
$\mathbf{I}_{S^{\circ}\to S}$ associated to the presheaf
$T\mapsto\Gamma(T,\mathcal{O}_{T})$ the structural sheaf of
$\mathbf{I}_{S^{\circ}\to S}$.

######  1.11.

Let $p$ be a prime number, $\overline{\mathbb{Z}_{p}}$ the integral closure of
$\mathbb{Z}_{p}$ in an algebraic closure $\overline{\mathbb{Q}_{p}}$ of
$\mathbb{Q}_{p}$. We take
$S^{\circ}=\mathop{\mathrm{Spec}}(\overline{\mathbb{Q}_{p}})$ and
$S=\mathop{\mathrm{Spec}}(\overline{\mathbb{Z}_{p}})$. Consider a diagram of
coherent schemes

(1.11.5)

where $X^{Y}$ is the integral closure of $X$ in $Y$ and the square is
Cartesian (we don’t impose any condition on the regularity or finiteness of
$Y$ or $X$). The functor $\varepsilon^{+}:\mathbf{E}_{Y\to
X}^{\mathrm{\acute{e}t}}\to\mathbf{I}_{Y\to X^{Y}}$ sending $V\to U$ to
$U^{V}$ defines a natural morphism of ringed sites

(1.11.6) $\displaystyle\varepsilon:(\mathbf{I}_{Y\to
X^{Y}},\mathscr{O})\longrightarrow(\mathbf{E}_{Y\to
X}^{\mathrm{\acute{e}t}},\overline{\mathscr{B}}).$

Our cohomological descent results are stated as follows, which can be regarded
as a scheme theoretic analogue of the cohomological descent result for the
pro-étale site of an adic space (cf. [Sch21, 8.8, 14.7, 15.5]):

######  Theorem 1.12 (Cohomological descent for Faltings ringed sites, cf.
8.14).

For any finite locally constant abelian sheaf $\mathbb{L}$ on
$\mathbf{E}_{Y\to X}^{\mathrm{\acute{e}t}}$, the canonical morphism

(1.12.1)
$\displaystyle\mathbb{L}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}\longrightarrow\mathrm{R}\varepsilon_{*}(\varepsilon^{-1}\mathbb{L}\otimes_{\mathbb{Z}}\mathscr{O})$

is an almost isomorphism.

######  Corollary 1.13 (cf. 8.18).

For any proper hypercovering $X_{\bullet}\to X$, if
$a:\mathbf{E}_{Y_{\bullet}\to
X_{\bullet}}^{\mathrm{\acute{e}t}}\to\mathbf{E}_{Y\to
X}^{\mathrm{\acute{e}t}}$ denotes the augmentation of simplicial site where
$Y_{\bullet}=Y\times_{X}X_{\bullet}$, then the canonical morphism

(1.13.1)
$\displaystyle\mathbb{L}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}\longrightarrow\mathrm{R}a_{*}(a^{-1}\mathbb{L}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}_{\bullet})$

is an almost isomorphism.

The key ingredient of our proof of 1.12 is the descent of perfectoid algebras
in the arc-topology (a topology finer than the v-topology) due to Bhatt-
Scholze [BS19, 8.10] (cf. 5.35). The analogue in characteristic $p$ of 1.12 is
Gabber’s computation of the cohomology of the structural sheaf in the
h-topology (cf. 4). Theorem 1.12 allows us to descend important results for
Faltings sites with nice models to Faltings sites associated to general
models. On the other hand, its proof shows how to compute the cohomologies of
Faltings ringed sites locally. Using Abhyankar’s lemma, one can treat the open
case which in the generic fibre is the complement of a normal crossings
divisor.

######  Corollary 1.14 (cf. 8.24).

Under the assumptions in 1.4 and with the same notation, for any integer
$n>0$, the canonical morphism

(1.14.1) $\displaystyle
A/p^{n}A\longrightarrow\mathrm{R}\Gamma(\mathbf{E}_{V\to
U}^{\mathrm{\acute{e}t}},\overline{\mathscr{B}}/p^{n}\overline{\mathscr{B}})$

is an almost isomorphism.

Thus, we apply the derived functor $\mathrm{R}\Gamma(\mathbf{E}_{V\to
U}^{\mathrm{\acute{e}t}},-)$ to the global relative Hodge-Tate filtration
defined on the Faltings ringed site by Abbes-Gros, and then we obtain the
local version 1.4.

######  1.15.

On the other hand, we use 1.12 to generalize Faltings’ comparison theorem in
the absolute case. Let $A$ be a valuation ring extension of
$\overline{\mathbb{Z}_{p}}$ with algebraically closed fraction field. Consider
a Cartesian square of coherent schemes

(1.15.5)

######  Theorem 1.16 (Faltings’ comparison theorem in the absolute case, cf.
10.17).

Assume that $X$ is proper of finite presentation over $A$. Then, for any
finite locally constant abelian sheaf $\mathbb{F}$ on
$Y_{\mathrm{\acute{e}t}}$, there exists a canonical morphism

(1.16.1)
$\displaystyle\mathrm{R}\Gamma(Y_{\mathrm{\acute{e}t}},\mathbb{F})\otimes^{\mathrm{L}}_{\mathbb{Z}}A\longrightarrow\mathrm{R}\Gamma(\mathbf{E}_{Y\to
X}^{\mathrm{\acute{e}t}},\psi_{*}\mathbb{F}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}),$

which is an almost isomorphism.

We remark that the natural morphism
$\psi:Y_{\mathrm{\acute{e}t}}\to\mathbf{E}_{Y\to X}^{\mathrm{\acute{e}t}}$
induces an equivalence of the categories of finite locally constant abelian
sheaves on $Y_{\mathrm{\acute{e}t}}$ and $\mathbf{E}_{Y\to
X}^{\mathrm{\acute{e}t}}$ (cf. 10.3),

(1.16.4)

As a continuation of the work of Abbes-Gros, the canonical morphism (1.16.1)
(refered as _Faltings’ comparison morphism_) is constructed using the
acyclicity of $\psi$ for $\mathbb{F}$, i.e.
$\psi_{*}\mathbb{F}=\mathrm{R}\psi_{*}\mathbb{F}$ (so that
$\mathrm{R}\Gamma(Y_{\mathrm{\acute{e}t}},\mathbb{F})=\mathrm{R}\Gamma(\mathbf{E}_{Y\to
X}^{\mathrm{\acute{e}t}},\psi_{*}\mathbb{F})$), which is a consequence of
Achinger’s result on $K(\pi,1)$-schemes (cf. 10.6 and 10.8). We also propose a
new way to construct Faltings’ comparison morphism in the derived category of
almost modules using our cohomological descent result 1.12, which avoids using
the acyclicity of $\psi$. Indeed, there are natural morphisms of sites

(1.16.9)

and $\Psi$ is acyclic for any torsion abelian sheaf $\mathcal{F}$ on
$Y_{\mathrm{\acute{e}t}}$, i.e.
$\Psi_{*}(a^{-1}\mathcal{F})=\mathrm{R}\Psi_{*}(a^{-1}\mathcal{F})$, which
allows more general coefficients and whose proof is much easier than that of
$\psi$ (cf. 3.27). We remark that this new construction won’t give us a “real
morphism” (1.16.1) but a canonical morphism in the derived category of almost
modules (cf. 11.6).

We briefly explain the strategy for proving 1.16:

  1. (1)

Firstly, we use de Jong-Gabber-Illusie-Temkin’s alteration theorem for
morphisms of schemes [ILO14, X.3] to obtain a proper surjective morphism of
finite presentation $X^{\prime}\to X$ such that the morphism
$X^{\prime}\to\mathop{\mathrm{Spec}}(A)$ is the cofiltered limit of a system
of “nice” morphisms $X^{\prime}_{\lambda}\to T_{\lambda}$ of “nice” models
over $\mathcal{O}_{K_{\lambda}}$, where $K_{\lambda}$ is a finite extension of
$\mathbb{Q}_{p}$ (cf. 9.11).

  2. (2)

Then, we can apply Faltings’ comparison theorem in the relative case to the
“nice” morphisms $X^{\prime}_{\lambda}\to T_{\lambda}$ (formulated by Faltings
[Fal02, Thm.6, page 266] and proved by Abbes-Gros [AG20, 5.7.3], cf. 10.13).
By a limit argument, we get the comparison theorem for $X^{\prime}$.

  3. (3)

Finally, using our cohomological descent result 1.13, we deduce the comparison
theorem for $X$.

######  1.17.

The site $\mathbf{I}_{Y\to X^{Y}}$ is also appropriate to globalize Faltings’
comparison theorem. Consider a Cartesian square of coherent schemes

(1.17.5)

where $Y\to X$ is Cartesian over
$\mathop{\mathrm{Spec}}(\overline{\mathbb{Q}_{p}})\to\mathop{\mathrm{Spec}}(\overline{\mathbb{Z}_{p}})$.
In particular, there is a natural morphism of ringed sites by the
functoriality of (1.16.9),

(1.17.6) $\displaystyle f_{\mathbf{I}}:(\mathbf{I}_{Y^{\prime}\to X^{\prime
Y^{\prime}}},\mathscr{O}^{\prime})\longrightarrow(\mathbf{I}_{Y\to
X^{Y}},\mathscr{O}).$

######  Theorem 1.18 (cf. 11.11).

Assume that $X^{\prime}\to X$ is proper of finite presentation. Let
$\mathcal{F}^{\prime}$ be a torsion abelian sheaf on
$Y^{\prime}_{\mathrm{\acute{e}t}}$ and
$\mathscr{F}^{\prime}=\Psi^{\prime}_{*}a^{\prime-1}\mathcal{F}^{\prime}$ (cf.
(1.16.9)). Then, the canonical morphism

(1.18.1)
$\displaystyle(\mathrm{R}f_{\mathbf{I}*}\mathscr{F}^{\prime})\otimes_{\mathbb{Z}}^{\mathrm{L}}\mathscr{O}\longrightarrow\mathrm{R}f_{\mathbf{I}*}(\mathscr{F}^{\prime}\otimes_{\mathbb{Z}}\mathscr{O}^{\prime})$

is an almost isomorphism.

We remark that if $\mathcal{F}^{\prime}=\mathbb{Z}/p^{n}\mathbb{Z}$ then
$\mathscr{F}^{\prime}=\mathbb{Z}/p^{n}\mathbb{Z}$ (cf. 3.27), and that
$\mathrm{R}^{q}f_{\mathbf{I}*}\mathscr{F}^{\prime}$ is the sheafification of
étale cohomologies of $Y^{\prime}$ over $Y$ with coefficient
$\mathcal{F}^{\prime}$ in the v-topology (cf. 11.12). Very roughly speaking,
objects of $\mathbf{I}_{Y\to X^{Y}}$ are “locally” the spectrums of valuation
rings, and the “stalks” of (1.18.1) are Faltings’ comparison morphisms
(1.16.1) when $\mathcal{F}^{\prime}$ is finite locally constant (cf. 11.5).
Theorem 1.18 can be regarded as a scheme theoretical analogue of Scholze’s
comparison theorem for $p$-adic étale cohomology of a morphism of rigid
analytic varieties [Sch13b, 3.13].

Finally, we generalize Faltings’ comparison theorem in the relative case using
1.12 and 1.18.

######  Theorem 1.19 (Faltings’ comparison theorem in the relative case, cf.
11.13 and 11.14).

Assume that $Y^{\prime}\to Y$ is smooth and that $X^{\prime}\to X$ is proper
of finite presentation. Then, for any finite locally constant abelian sheaf
$\mathbb{F}^{\prime}$ on $Y^{\prime}_{\mathrm{\acute{e}t}}$, there exists a
canonical morphism

(1.19.3)

which is an almost isomorphism, and where
$f_{\mathrm{\acute{e}t}}:Y^{\prime}_{\mathrm{\acute{e}t}}\to
Y_{\mathrm{\acute{e}t}}$ and $f_{\mathbf{E}}:\mathbf{E}_{Y^{\prime}\to
X^{\prime}}^{\mathrm{\acute{e}t}}\to\mathbf{E}_{Y\to X}^{\mathrm{\acute{e}t}}$
are the natural morphisms of sites. In particular, there exists a canonical
morphism

(1.19.4)
$\displaystyle(\psi_{*}\mathrm{R}^{q}f_{\mathrm{\acute{e}t}*}\mathbb{F}^{\prime})\otimes_{\mathbb{Z}}\overline{\mathscr{B}}\longrightarrow\mathrm{R}^{q}f_{\mathbf{E}*}(\psi^{\prime}_{*}\mathbb{F}^{\prime}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}^{\prime}),$

which is an almost isomorphism, for any integer $q$.

######  1.20.

The paper is structured as follows. In section 3, we establish the foundation
of the site $\mathbf{I}_{S^{\circ}\to S}$, where proposition 3.27 discussing
the cohomological properties of
$\Psi:(\mathbf{Sch}^{\mathrm{coh}}_{/S^{\circ}})_{\mathrm{v}}\to\mathbf{I}_{S^{\circ}\to
S}$ is the key to our new construction of Faltings’ comparison morphism (cf.
11.6). Sections 4 and 5 are devoted to a detailed proof of the arc-descent for
perfectoid algebras following Bhatt-Scholze [BS19, 8.10]. Since we use the
language of schemes, the terminology “pre-perfectoid” is introduced for those
algebras whose $p$-adic completions are perfectoid. In sections 6 and 7, we
review the definition and some basic properties of Faltings sites and we
introduce a pro-version of Faltings site to evaluate the structural sheaf on
the spectrums of pre-perfectoid algebras. Then, we prove our cohomological
descent result in section 8. In section 9, we review de Jong-Gabber-Illusie-
Temkin’s alteration theorem and apply it to schemes over a valuation ring of
height $1$. Section 10 is devoted to proving our generalization of Faltings’
comparison theorem in the absolute case. In section 11, we give a new
construction of Faltings’ comparison morphism and our generalization of
Faltings’ comparison theorem in the relative case. Finally, we deduce from the
global relative Hodge-Tate filtration an explicit local version in section 12.

### Acknowledgements

This work is part of my thesis prepared at Université Paris-Saclay and
Institut des Hautes Études Scientifiques. I would like to express my sincere
gratitude to my doctoral supervisor, Ahmed Abbes, for his guidance to this
project, his thorough review of this work and his plenty of helpful
suggestions on both research and writing.

##  2\. Notation and Conventions

######  2.1.

We fix a prime number $p$ throughout this paper. For any monoid $M$, we denote
by $\mathrm{Frob}:M\to M$ the map sending an element $x$ to $x^{p}$ and we
call it the _Frobenius_ of $M$. For a ring $R$, we denote by $R^{\times}$ the
group of units of $R$. A ring $R$ is called _absolutely integrally closed_ if
any monic polynomial $f\in R[T]$ has a root in $R$ ([Sta22, 0DCK]). We remark
that quotients, localizations and products of absolutely integrally closed
rings are still absolutely integrally closed.

Recall that a valuation ring is a domain $V$ such that for any element $x$ in
its fraction field, if $x\notin V$ then $x^{-1}\in V$. The family of ideals of
$V$ is totally ordered by the inclusion relation ([Bou06, VI.§1.2, Thm.1]). In
particular, a radical ideal of $V$ is a prime ideal. Moreover, any quotient of
$V$ by a prime ideal and any localization of $V$ are still valuations rings
([Sta22, 088Y]). We remark that $V$ is normal, and that $V$ is absolutely
integrally closed if and only if its fraction field is algebraically closed.
An _extension of valuation rings_ is an injective and local homomorphism of
valuation rings.

######  2.2.

Following [SGA 4${}_{\text{\rm II}}$, VI.1.22], a _coherent_ scheme (resp.
morphism of schemes) stands for a quasi-compact and quasi-separated scheme
(resp. morphism of schemes). For a coherent morphism $Y\to X$ of schemes, we
denote by $X^{Y}$ the integral closure of $X$ in $Y$ ([Sta22, 0BAK]). For an
$X$-scheme $Z$, we say that $Z$ is _$Y$ -integrally closed_ if
$Z=Z^{Y\times_{X}Z}$.

######  2.3.

Throughout this paper, we fix two universes $\mathbb{U}$ and $\mathbb{V}$ such
that the set of natural numbers $\mathbb{N}$ is an element of $\mathbb{U}$ and
that $\mathbb{U}$ is an element of $\mathbb{V}$ ([SGA 4${}_{\text{\rm I}}$,
I.0]). In most cases, we won’t emphasize this set theoretical issue. Unless
stated otherwise, we only consider $\mathbb{U}$-small schemes and we denote by
$\mathbf{Sch}$ the category of $\mathbb{U}$-small schemes, which is a
$\mathbb{V}$-small category.

######  2.4.

Let $C$ be a category. We denote by $\widehat{C}$ the category of presheaves
of $\mathbb{V}$-small sets on $C$. If $C$ is a $\mathbb{V}$-site ([SGA
4${}_{\text{\rm I}}$, II.3.0.2]), we denote by $\widetilde{C}$ the topos of
sheaves of $\mathbb{V}$-small sets on $C$. We denote by
$h^{C}:C\to\widehat{C}$, $x\mapsto h_{x}^{C}$ the Yoneda embbeding ([SGA
4${}_{\text{\rm I}}$, I.1.3]), and by $\widehat{C}\to\widetilde{C}$,
$\mathcal{F}\mapsto\mathcal{F}^{\mathrm{a}}$ the sheafification functor ([SGA
4${}_{\text{\rm I}}$, II.3.4]). Unless stated otherwise, a site in this paper
stands for a site where _all finite limits are representable_.

######  2.5.

Let $u^{+}:C\to D$ be a functor of categories. We denote by
$u^{\mathrm{p}}:\widehat{D}\to\widehat{C}$ the functor that associates to a
presheaf $\mathcal{G}$ of $\mathbb{V}$-small sets on $D$ the presheaf
$u^{\mathrm{p}}\mathcal{G}=\mathcal{G}\circ u^{+}$. If $C$ is
$\mathbb{V}$-small and $D$ is a $\mathbb{V}$-category, then $u^{\mathrm{p}}$
admits a left adjoint $u_{\mathrm{p}}$ [Sta22, 00VC] and a right adjoint
${}_{\mathrm{p}}u$ [Sta22, 00XF] (cf. [SGA 4${}_{\text{\rm I}}$, I.5]). So we
have a sequence of adjoint functors

(2.5.1) $\displaystyle u_{\mathrm{p}},\ u^{\mathrm{p}},\ {}_{\mathrm{p}}u.$

If moreover $C$ and $D$ are $\mathbb{V}$-sites, then we denote by
$u_{\mathrm{s}},u^{\mathrm{s}},{}_{\mathrm{s}}u$ the functors of the topoi
$\widetilde{C}$ and $\widetilde{D}$ of sheaves of $\mathbb{V}$-small sets
induced by composing the sheafification functor with the functors
$u_{\mathrm{p}},u^{\mathrm{p}},{}_{\mathrm{p}}u$ respectively. As we only
consider finite complete sites, we say that the functor $u^{+}$ gives a
_morphism of sites_ , if $u^{+}$ is left exact and preserves covering families
([SGA 4${}_{\text{\rm I}}$, IV.4.9.2]). Then, we denote by

(2.5.2) $\displaystyle u=(u^{-1},u_{*}):\widetilde{D}\to\widetilde{C}$

the associated morphism of topoi, where $u^{-1}=u_{\mathrm{s}}$ and
$u_{*}=u^{\mathrm{s}}=u^{\mathrm{p}}|_{\widetilde{D}}$. If moreover $u$ is a
morphism of ringed topoi, then we denote by
$u^{*}=\mathcal{O}_{\widetilde{D}}\otimes_{u^{-1}\mathcal{O}_{\widetilde{C}}}u^{-1}$
the pullback functor of modules. We remark that the notation here, adopted by
[Sta22], is slightly different with that in [SGA 4${}_{\text{\rm I}}$] (cf.
[Sta22, 0CMZ]).

######  2.6.

Let $C$ be an additive category. The _catgory of objects of $C$ up to isogeny_
(cf. [AGT16, III.6]) is the category $C_{\mathbb{Q}}$ with a functor $F:C\to
C_{\mathbb{Q}}$ (called the localization functor) such that
$\mathrm{Ob}(C_{\mathbb{Q}})=\mathrm{Ob}(C)$ and that

(2.6.1)
$\displaystyle\mathrm{Hom}_{C_{\mathbb{Q}}}(M_{\mathbb{Q}},N_{\mathbb{Q}})=\mathrm{Hom}_{C}(M,N)\otimes_{\mathbb{Z}}\mathbb{Q},$

where we denote by $M_{\mathbb{Q}}$ the image of an object $M$ of $C$ via $F$
in $C_{\mathbb{Q}}$.

For a ringed site $(C,\mathcal{O})$, we denote by
$\mathcal{O}\textrm{-}\mathbf{Mod}_{\mathbb{Q}}$ the category of
$\mathcal{O}$-modules up to isogeny, whose objects are called
_$\mathcal{O}_{\mathbb{Q}}$ -modules_. It is an abelian category and the
localization functor
$\mathcal{O}\textrm{-}\mathbf{Mod}\to\mathcal{O}\textrm{-}\mathbf{Mod}_{\mathbb{Q}}$
sends injective objects to injective objects. We remark that if $\mathcal{O}$
is a $\mathbb{Q}$-module, then
$\mathcal{O}\textrm{-}\mathbf{Mod}\to\mathcal{O}\textrm{-}\mathbf{Mod}_{\mathbb{Q}}$
is an equivalence. A morphism of ringed sites
$u:(C,\mathcal{O})\to(C^{\prime},\mathcal{O}^{\prime})$ induces a pair of
adjoint functors

(2.6.2) $\displaystyle
u^{*}:\mathcal{O}^{\prime}\textrm{-}\mathbf{Mod}_{\mathbb{Q}}\to\mathcal{O}\textrm{-}\mathbf{Mod}_{\mathbb{Q}},\
u_{*}:\mathcal{O}\textrm{-}\mathbf{Mod}_{\mathbb{Q}}\to\mathcal{O}^{\prime}\textrm{-}\mathbf{Mod}_{\mathbb{Q}}.$

The derived functor $\mathrm{R}u_{*}$ commutes with the localization functor.

##  3\. The v-site of Integrally Closed Schemes

######  Definition 3.1.

Let $X\to Y$ be a quasi-compact morphism of schemes.

  1. (1)

We say that $X\to Y$ is a _v-covering_ , if for any valuation ring $V$ and any
morphism $\mathop{\mathrm{Spec}}(V)\to Y$, there exists an extension of
valuation rings $V\to W$ (2.1) and a commutative diagram (cf. [Sta22, 0ETN])

(3.1.5)

  2. (2)

Let $\pi$ be an element of $\Gamma(Y,\mathcal{O}_{Y})$. We say that $X\to Y$
is an _arc-covering_ (resp. _$\pi$ -complete arc-covering_), if for any
valuation ring (resp. $\pi$-adically complete valuation ring) $V$ of height
$\leq 1$ and any morphism $\mathop{\mathrm{Spec}}(V)\to Y$, there exists an
extension of valuation rings (resp. $\pi$-adically complete valuation rings)
$V\to W$ of height $\leq 1$ and a commutative diagram (3.1.5) (cf. [BM20,
1.2], [CS19, 2.2.1]).

  3. (3)

We say that $X\to Y$ is an _h-covering_ , if it is a v-covering and locally of
finite presentation (cf. [Sta22, 0ETS]).

We note that an arc-covering is simply a $0$-complete arc-covering.

######  Lemma 3.2.

Let $Z\stackrel{{\scriptstyle g}}{{\longrightarrow}}Y\stackrel{{\scriptstyle
f}}{{\longrightarrow}}X$ be quasi-compact morphisms of schemes,
$\pi\in\Gamma(X,\mathcal{O}_{X})$, $\tau\in\\{$h, v, $\pi$-complete arc$\\}$.

  1. (1)

If $f$ is a $\tau$-covering, then any base change of $f$ is also a
$\tau$-covering.

  2. (2)

If $f$ and $g$ are $\tau$-coverings, then $f\circ g$ is also a
$\tau$-covering.

  3. (3)

If $f\circ g$ is a $\tau$-covering (and if $f$ is locally of finite
presentation when $\tau=\textrm{h}$), then $f$ is also a $\tau$-covering.

###### Proof.

It follows directly from the definitions. ∎

######  3.3.

Let $\mathbf{Sch}^{\mathrm{coh}}$ be the category of coherent
$\mathbb{U}$-small schemes, $\tau\in\\{\textrm{h, v, arc}\\}$. We endow
$\mathbf{Sch}^{\mathrm{coh}}$ with the $\tau$-topology generated by the
pretopology formed by families of morphisms $\\{X_{i}\to X\\}_{i\in I}$ with
$I$ finite such that $\coprod_{i\in I}X_{i}\to X$ is a $\tau$-covering, and we
denote the corresponding site by $\mathbf{Sch}^{\mathrm{coh}}_{\tau}$. It is
clear that a morphism $Y\to X$ (which is locally of finite presentation if
$\tau=\textrm{h}$) is a $\tau$-covering if and only if $\\{Y\to X\\}$ is a
covering family in $\mathbf{Sch}^{\mathrm{coh}}_{\tau}$ by 3.2 and [SGA
4${}_{\text{\rm I}}$, II.1.4].

For any coherent $\mathbb{U}$-small scheme $X$, we endow the category
$\mathbf{Sch}^{\mathrm{coh}}_{/X}$ of objects of $\mathbf{Sch}^{\mathrm{coh}}$
over $X$ with the topology induced by the $\tau$-topology of
$\mathbf{Sch}^{\mathrm{coh}}$, i.e. the topology generated by the pretopology
formed by families of $X$-morphisms $\\{Y_{i}\to Y\\}_{i\in I}$ with $I$
finite such that $\coprod_{i\in I}Y_{i}\to Y$ is a $\tau$-covering ([SGA
4${}_{\text{\rm I}}$, III.5.2]). For any sheaf $\mathcal{F}$ of
$\mathbb{V}$-small abelian groups on the site
$(\mathbf{Sch}^{\mathrm{coh}}_{/X})_{\tau}$, we denote its $q$-th cohomology
by $H^{q}_{\tau}(X,\mathcal{F})$.

######  Lemma 3.4.

Let $f:X\to Y$ be a quasi-compact morphism of schemes,
$\pi\in\Gamma(Y,\mathcal{O}_{Y})$.

  1. (1)

If $f$ is proper surjective or faithfully flat, then $f$ is a v-covering.

  2. (2)

If $f$ is an h-covering and $Y$ is affine, then there exists a proper
surjective morphism $Y^{\prime}\to Y$ of finite presentation and a finite
affine open covering $Y^{\prime}=\bigcup_{r=1}^{n}Y^{\prime}_{r}$ such that
$Y^{\prime}_{r}\to Y$ factors through $f$ for each $r$.

  3. (3)

If $f$ is an h-covering and if there exists a directed inverse system
$(f_{\lambda}:X_{\lambda}\to Y_{\lambda})_{\lambda\in\Lambda}$ of finitely
presented morphisms of coherent schemes with affine transition morphisms
$\psi_{\lambda^{\prime}\lambda}:X_{\lambda^{\prime}}\to X_{\lambda}$ and
$\phi_{\lambda^{\prime}\lambda}:Y_{\lambda^{\prime}}\to Y_{\lambda}$ such that
$X=\lim X_{\lambda}$, $Y=\lim Y_{\lambda}$ and that $f_{\lambda}$ is the base
change of $f_{\lambda_{0}}$ by $\phi_{\lambda\lambda_{0}}$ for some index
$\lambda_{0}\in\Lambda$ and any $\lambda\geq\lambda_{0}$, then there exists an
index $\lambda_{1}\geq\lambda_{0}$ such that $f_{\lambda}$ is an h-covering
for any $\lambda\geq\lambda_{1}$.

  4. (4)

If $f$ is a v-covering, then it is a $\pi$-complete arc-covering.

  5. (5)

Let $\pi^{\prime}$ be another element of $\Gamma(Y,\mathcal{O}_{Y})$ which
divides $\pi$. If $f$ is a $\pi$-complete arc-covering, then it is a
$\pi^{\prime}$-complete arc-covering.

  6. (6)

If $\mathop{\mathrm{Spec}}(B)\to\mathop{\mathrm{Spec}}(A)$ is a $\pi$-complete
arc-covering, then the morphism
$\mathop{\mathrm{Spec}}(\widehat{B})\to\mathop{\mathrm{Spec}}(\widehat{A})$
between the spectrums of their $\pi$-adic completions is also a $\pi$-complete
arc-covering.

###### Proof.

(1), (2) are proved in [Sta22, 0ETK, 0ETU] respectively.

(3) To show that one can take $\lambda_{1}\geq\lambda_{0}$ such that
$f_{\lambda_{1}}$ is an h-covering, we may assume that $Y_{\lambda_{0}}$ is
affine by replacing it by a finite affine open covering by 3.2 and (1). Thus,
applying (2) to the h-covering $f$ and using [EGA IV3, 8.8.2, 8.10.5], there
exists an index $\lambda_{1}\geq\lambda_{0}$, a proper surjective morphism
$Y^{\prime}_{\lambda_{1}}\to Y_{\lambda_{1}}$ and a finite affine open
covering $Y^{\prime}_{\lambda_{1}}=\bigcup_{r=1}^{n}Y^{\prime}_{r\lambda_{1}}$
such that the morphisms $Y^{\prime}_{r}\to Y^{\prime}\to Y$ are the base
changes of the morphisms $Y^{\prime}_{r\lambda_{1}}\to
Y^{\prime}_{\lambda_{1}}\to Y_{\lambda_{1}}$ by the transition morphism $Y\to
Y_{\lambda_{1}}$, and that $Y^{\prime}_{r\lambda_{1}}\to Y_{\lambda_{1}}$
factors through $X_{\lambda_{1}}$. This shows that $f_{\lambda_{1}}$ is an
h-covering by 3.2 and (1).

(4) With the notation in (3.1.5), if $V$ is a $\pi$-adically complete
valuation ring of height $\leq 1$ with maximal ideal $\mathfrak{m}$, then
since the family of prime ideals of $W$ is totally ordered by the inclusion
relation (2.1), we take the maximal prime ideal $\mathfrak{p}\subseteq W$ over
$0\subseteq V$ and the minimal prime ideal $\mathfrak{q}\subseteq W$ over
$\mathfrak{m}\subseteq V$. Then, $\mathfrak{p}\subseteq\mathfrak{q}$ and
$W^{\prime}=(W/\mathfrak{p})_{\mathfrak{q}}$ over $V$ is an extension of
valuation rings of height $\leq 1$. Since $\pi\in\mathfrak{m}$ and
$W^{\prime}$ is of height $\leq 1$, the $\pi$-adic completion
$\widehat{W^{\prime}}$ is still a valuation ring extension of $V$ of height
$\leq 1$ (cf. [Bou06, VI.§5.3, Prop.5]), which proves (4).

(5) Since a $\pi^{\prime}$-adically complete valuation ring $V$ is also
$\pi$-adically complete ([Sta22, 090T]), there exists a lifting
$\mathop{\mathrm{Spec}}(W)\to X$ for any morphism
$\mathop{\mathrm{Spec}}(V)\to Y$. After replacing $W$ by its
$\pi^{\prime}$-adic completion, the conclusion follows.

(6) Let $V$ be a $\pi$-adically complete valuation ring of height $\leq 1$.
Given a morphism $\widehat{A}\to V$, there exists a lifting $B\to W$ where
$V\to W$ is an extension of $\pi$-adically complete valuation rings of height
$\leq 1$. It is clear that $B\to W$ factors through $\widehat{B}$, which
proves (6). ∎

######  3.5.

Let $X$ be a coherent scheme, $\mathbf{Sch}^{\mathrm{fp}}_{/X}$ the full
subcategory of $\mathbf{Sch}^{\mathrm{coh}}_{/X}$ formed by finitely presented
$X$-schemes. We endow it with the topology generated by the pretopology formed
by families of morphisms $\\{Y_{i}\to Y\\}_{i\in I}$ with $I$ finite such that
$\coprod_{i\in I}Y_{i}\to Y$ is an h-covering, and we denote the corresponding
site by $(\mathbf{Sch}^{\mathrm{fp}}_{/X})_{\mathrm{h}}$. It is clear that
this topology coincides with the topologies induced from
$(\mathbf{Sch}^{\mathrm{coh}}_{/X})_{\mathrm{v}}$ and from
$(\mathbf{Sch}^{\mathrm{coh}}_{/X})_{\mathrm{h}}$. The inclusion functors
$(\mathbf{Sch}^{\mathrm{fp}}_{/X})_{\mathrm{h}}\stackrel{{\scriptstyle\xi^{+}}}{{\longrightarrow}}(\mathbf{Sch}^{\mathrm{coh}}_{/X})_{\mathrm{h}}\stackrel{{\scriptstyle\zeta^{+}}}{{\longrightarrow}}(\mathbf{Sch}^{\mathrm{coh}}_{/X})_{\mathrm{v}}$
define morphisms of sites (2.5)

(3.5.1)
$\displaystyle(\mathbf{Sch}^{\mathrm{coh}}_{/X})_{\mathrm{v}}\stackrel{{\scriptstyle\zeta}}{{\longrightarrow}}(\mathbf{Sch}^{\mathrm{coh}}_{/X})_{\mathrm{h}}\stackrel{{\scriptstyle\xi}}{{\longrightarrow}}(\mathbf{Sch}^{\mathrm{fp}}_{/X})_{\mathrm{h}}.$

######  Lemma 3.6.

Let $X$ be a coherent scheme. Then, for any covering family
$\mathfrak{U}=\\{Y_{i}\to Y\\}_{i\in I}$ in
$(\mathbf{Sch}^{\mathrm{coh}}_{/X})_{\mathrm{v}}$ with $I$ finite,

  1. (i)

there exists a directed inverse system $(Y_{\lambda})_{\lambda\in\Lambda}$ of
finitely presented $X$-schemes with affine transition morphisms such that
$Y=\lim Y_{\lambda}$, and

  2. (ii)

for each $i\in I$, there exists a directed inverse system
$(Y_{i\lambda})_{\lambda\in\Lambda}$ of finitely presented $X$-schemes with
affine transition morphisms over the inverse system
$(Y_{\lambda})_{\lambda\in\Lambda}$ such that $Y_{i}=\lim Y_{i\lambda}$, and

  3. (iii)

for each $\lambda\in\Lambda$, the family
$\mathfrak{U}_{\lambda}=\\{Y_{i\lambda}\to Y_{\lambda}\\}_{i\in I}$ is a
covering in $(\mathbf{Sch}^{\mathrm{fp}}_{/X})_{\mathrm{h}}$.

###### Proof.

We take a directed set $A$ such that for each $i\in I$, we can write $Y_{i}$
as a cofiltered limit of finitely presented $Y$-schemes $Y_{i}=\lim_{\alpha\in
A}Y_{i\alpha}$ with affine transition morphisms ([Sta22, 09MV]). We see that
$\coprod_{i\in I}Y_{i\alpha}\to Y$ is an h-covering for each $\alpha\in A$ by
3.2.

We write $Y$ as a cofiltered limit of finitely presented $X$-schemes
$Y=\lim_{\beta\in B}Y_{\beta}$ with affine transition morphisms ([Sta22,
09MV]). By [EGA IV3, 8.8.2, 8.10.5] and 3.4.(3), for each $\alpha\in A$, there
exists an index $\beta_{\alpha}\in B$ such that the morphism $Y_{i\alpha}\to
Y$ is the base change of a finitely presented morphism
$Y_{i\alpha\beta_{\alpha}}\to Y_{\beta_{\alpha}}$ by the transition morphism
$Y\to Y_{\beta_{\alpha}}$ for each $i\in I$, and that $\coprod_{i\in
I}Y_{i\alpha\beta_{\alpha}}\to Y_{\beta_{\alpha}}$ is an h-covering. For each
$\beta\geq\beta_{\alpha}$, let $Y_{i\alpha\beta}$ be the base change of
$Y_{i\alpha\beta_{\alpha}}$ by $Y_{\beta}\to Y_{\beta_{\alpha}}$.

We define a category $\Lambda^{\mathrm{op}}$, whose set of objects is
$\\{(\alpha,\beta)\in A\times B\ |\ \beta\geq\beta_{\alpha}\\}$, and for any
two objects $\lambda^{\prime}=(\alpha^{\prime},\beta^{\prime})$,
$\lambda=(\alpha,\beta)$, the set
$\mathrm{Hom}_{\Lambda^{\mathrm{op}}}(\lambda^{\prime},\lambda)$ is

  1. (i)

the subset of $\prod_{i\in
I}\mathrm{Hom}_{Y_{\beta^{\prime}}}(Y_{i\alpha^{\prime}\beta^{\prime}},Y_{i\alpha\beta^{\prime}})$
formed by elements $f=(f_{i})_{i\in I}$ such that for each $i\in I$,
$f_{i}:Y_{i\alpha^{\prime}\beta^{\prime}}\to Y_{i\alpha\beta^{\prime}}$ is
affine and the base change of $f_{i}$ by $Y\to Y_{\beta^{\prime}}$ is the
transition morphism $Y_{i\alpha^{\prime}}\to Y_{i\alpha}$, if
$\alpha^{\prime}\geq\alpha$ and $\beta^{\prime}\geq\beta$;

  2. (ii)

empty, if else.

The composition of morphisms
$(g_{i}:Y_{i\alpha^{\prime\prime}\beta^{\prime\prime}}\to
Y_{i\alpha^{\prime}\beta^{\prime\prime}})_{i\in I}$ with
$(f_{i}:Y_{i\alpha^{\prime}\beta^{\prime}}\to Y_{i\alpha\beta^{\prime}})_{i\in
I}$ in $\Lambda^{\mathrm{op}}$ is $(g_{i}\circ
f^{\prime}_{i}:Y_{i\alpha^{\prime\prime}\beta^{\prime\prime}}\to
Y_{i\alpha\beta^{\prime\prime}})$, where $f^{\prime}_{i}$ is the base change
of $f_{i}$ by the transition morphism $Y_{\beta^{\prime\prime}}\to
Y_{\beta^{\prime}}$. We see that $\Lambda^{\mathrm{op}}$ is cofiltered by [EGA
IV3, 8.8.2]. Let $\Lambda$ be the opposite category of
$\Lambda^{\mathrm{op}}$. For each $i\in I$ and
$\lambda=(\alpha,\beta)\in\Lambda$, we set $Y_{\lambda}=Y_{\beta}$ and
$Y_{i\lambda}=Y_{i\alpha\beta}$. It is clear that the natural functors
$\Lambda\to A$ and $\Lambda\to B$ are cofinal ([SGA 4${}_{\text{\rm I}}$,
I.8.1.3]). After replacing $\Lambda$ by a directed set ([Sta22, 0032]), the
families $\mathfrak{U}_{\lambda}=\\{Y_{i\lambda}\to Y_{\lambda}\\}_{i\in I}$
satisfy the required conditions. ∎

######  Lemma 3.7.

With the notation in 3.5, let $\mathcal{F}$ be a presheaf on
$(\mathbf{Sch}^{\mathrm{fp}}_{/X})_{\mathrm{h}}$, $(Y_{\lambda})$ a directed
inverse system of finitely presented $X$-schemes with affine transition
morphisms, $Y=\lim Y_{\lambda}$. Then, we have
$\nu_{\mathrm{p}}\mathcal{F}(Y)=\mathop{\mathrm{colim}}\mathcal{F}(Y_{\lambda})$,
where $\nu^{+}=\xi^{+}$ (resp. $\nu^{+}=\zeta^{+}\circ\xi^{+}$).

###### Proof.

Notice that the presheaf $\mathcal{F}$ is a filtered colimit of representable
presheaves by [SGA 4${}_{\text{\rm I}}$, I.3.4]

(3.7.1)
$\displaystyle\mathcal{F}=\mathop{\mathrm{colim}}_{Y^{\prime}\in(\mathbf{Sch}^{\mathrm{fp}}_{/X})_{/\mathcal{F}}}h_{Y^{\prime}}.$

Thus, we may assume that $\mathcal{F}$ is representable by a finitely
presented $X$-scheme $Y^{\prime}$ since the section functor $\Gamma(Y,-)$
commutes with colimits of presheaves ([Sta22, 00VB]). Then, we have

(3.7.2)
$\displaystyle\nu_{\mathrm{p}}h_{Y^{\prime}}(Y)=h_{\nu^{+}(Y^{\prime})}(Y)=\mathrm{Hom}_{X}(Y,Y^{\prime})=\mathop{\mathrm{colim}}\mathrm{Hom}_{X}(Y_{\lambda},Y^{\prime})=\mathop{\mathrm{colim}}h_{Y^{\prime}}(Y_{\lambda})$

where the first equality follows from [Sta22, 04D2], and the third equality
follows from [EGA IV3, 8.14.2]. ∎

######  Proposition 3.8.

With the notation in 3.5, let $\mathcal{F}$ be an abelian sheaf on
$(\mathbf{Sch}^{\mathrm{fp}}_{/X})_{\mathrm{h}}$, $(Y_{\lambda})$ a directed
inverse system of finitely presented $X$-schemes with affine transition
morphisms, $Y=\lim Y_{\lambda}$. Let $\tau=\mathrm{h}$ and $\nu^{+}=\xi^{+}$
(resp. $\tau=\mathrm{v}$ and $\nu^{+}=\zeta^{+}\circ\xi^{+}$). Then, for any
integer $q$, we have

(3.8.1) $\displaystyle
H^{q}_{\tau}(Y,\nu^{-1}\mathcal{F})=\mathop{\mathrm{colim}}H^{q}((\mathbf{Sch}^{\mathrm{fp}}_{/Y_{\lambda}})_{\mathrm{h}},\mathcal{F}).$

In particular, the canonical morphism
$\mathcal{F}\longrightarrow\mathrm{R}\nu_{*}\nu^{-1}\mathcal{F}$ is an
isomorphism.

###### Proof.

For the second assertion, the sheaf $\mathrm{R}^{q}\nu_{*}\nu^{-1}\mathcal{F}$
is the sheaf associated to the presheaf $Y\mapsto
H^{q}_{\tau}(Y,\nu^{-1}\mathcal{F})=H^{q}((\mathbf{Sch}^{\mathrm{fp}}_{/Y})_{\mathrm{h}},\mathcal{F})$
by the first assertion, which is $\mathcal{F}$ if $q=0$ and vanishes
otherwise.

We claim that it suffices to show that (3.8.1) holds for any injective abelian
sheaf $\mathcal{F}=\mathcal{I}$ on
$(\mathbf{Sch}^{\mathrm{fp}}_{/X})_{\mathrm{h}}$. Indeed, if so, then we prove
by induction on $q$ that (3.8.1) holds in general. The case where $q\leq-1$ is
trivial. We set $H^{q}_{1}(\mathcal{F})=H^{q}_{\tau}(Y,\nu^{-1}\mathcal{F})$
and
$H^{q}_{2}(\mathcal{F})=\mathop{\mathrm{colim}}H^{q}((\mathbf{Sch}^{\mathrm{fp}}_{/Y_{\lambda}})_{\mathrm{h}},\mathcal{F})$.
We embed an abelian sheaf $\mathcal{F}$ to an injective abelian sheaf
$\mathcal{I}$. Consider the exact sequence
$0\to\mathcal{F}\to\mathcal{I}\to\mathcal{G}\to 0$ and the morphism of long
exact sequences

(3.8.6)

If (3.8.1) holds for any abelian sheaf $\mathcal{F}$ for degree $q-1$, then
$\gamma_{1}$, $\gamma_{2}$, $\gamma_{4}$ are isomorphisms and thus
$\gamma_{3}$ is injective by the 5-lemma ([Sta22, 05QA]). Thus, $\gamma_{5}$
is also injective since $\mathcal{F}$ is an arbitrary abelian sheaf. Then, we
see that $\gamma_{3}$ is an isomorphism, which completes the induction
procedure.

For an injective abelian sheaf $\mathcal{I}$ on
$(\mathbf{Sch}^{\mathrm{fp}}_{/X})_{\mathrm{h}}$, we claim that for any
covering family $\mathfrak{U}=\\{(Y_{i}\to Y)\\}_{i\in I}$ in
$(\mathbf{Sch}^{\mathrm{coh}}_{/X})_{\tau}$ with $I$ finite, the augmented
Čech complex associated to the presheaf $\nu_{\mathrm{p}}\mathcal{I}$

(3.8.7) $\displaystyle\nu_{\mathrm{p}}\mathcal{I}(Y)\to\prod_{i\in
I}\nu_{\mathrm{p}}\mathcal{I}(Y_{i})\to\prod_{i,j\in
I}\nu_{\mathrm{p}}\mathcal{I}(Y_{i}\times_{Y}Y_{j})\to\cdots$

is exact. Admitting this claim, we see that $\nu_{\mathrm{p}}\mathcal{I}$ is
indeed a sheaf, i.e. $\nu^{-1}\mathcal{I}=\nu_{\mathrm{p}}\mathcal{I}$, and
the vanishing of higher Čech cohomologies implies that
$H^{q}_{\tau}(Y,\nu^{-1}\mathcal{I})=0$ for $q>0$ by 3.6 ([Sta22, 03F9]),
which completes the proof together with 3.7. For the claim, we take the
covering families $\mathfrak{U}_{\lambda}=\\{Y_{i\lambda}\to
Y_{\lambda}\\}_{i\in I}$ in $(\mathbf{Sch}^{\mathrm{fp}}_{/X})_{\mathrm{h}}$
constructed by 3.6. By 3.7, the sequence (3.8.7) is the filtered colimit of
the augmented Čech complexes

(3.8.8) $\displaystyle\mathcal{I}(Y_{\lambda})\to\prod_{i\in
I}\mathcal{I}(Y_{i\lambda})\to\prod_{i,j\in
I}\mathcal{I}(Y_{i\lambda}\times_{Y_{\lambda}}Y_{j\lambda})\to\cdots,$

which are exact since $\mathcal{I}$ is an injective abelian sheaf on
$(\mathbf{Sch}^{\mathrm{fp}}_{/X})_{\mathrm{h}}$. ∎

######  Corollary 3.9.

Let $X$ be a coherent scheme, $\mathcal{F}$ a torsion abelian sheaf on the
site $X_{\mathrm{\acute{e}t}}$ formed by coherent étale $X$-schemes endowed
with the étale topology, $a:(\mathbf{Sch}^{\mathrm{coh}}_{/X})_{\mathrm{v}}\to
X_{\mathrm{\acute{e}t}}$ the morphism of sites defined by the inclusion
functor. Then, the canonical morphism
$\mathcal{F}\to\mathrm{R}a_{*}a^{-1}\mathcal{F}$ is an isomorphism.

###### Proof.

Consider the morphisms of sites defined by inclusion functors

(3.9.1)
$\displaystyle(\mathbf{Sch}^{\mathrm{coh}}_{/X})_{\mathrm{v}}\stackrel{{\scriptstyle\zeta}}{{\longrightarrow}}(\mathbf{Sch}^{\mathrm{coh}}_{/X})_{\mathrm{h}}\stackrel{{\scriptstyle\xi}}{{\longrightarrow}}(\mathbf{Sch}^{\mathrm{fp}}_{/X})_{\mathrm{h}}\stackrel{{\scriptstyle\mu}}{{\longrightarrow}}X_{\mathrm{\acute{e}t}}.$

Notice that the morphism
$\mathcal{F}\to\mathrm{R}(\mu\circ\xi)_{*}(\mu\circ\xi)^{-1}\mathcal{F}$ is an
isomorphism by [Sta22, 0EWG]. Hence,
$\mathcal{F}\to\mathrm{R}\mu_{*}\mu^{-1}\mathcal{F}$ is an isomorphism by 3.8,
and thus so is $\mathcal{F}\to\mathrm{R}a_{*}a^{-1}\mathcal{F}$ by 3.8. ∎

######  Corollary 3.10.

Let $f:X\to Y$ be a proper morphism of coherent schemes, $\mathcal{F}$ a
torsion abelian sheaf on $X_{\mathrm{\acute{e}t}}$. Consider the commutative
diagram

(3.10.5)

where $f_{\mathrm{v}}$ and $f_{\mathrm{\acute{e}t}}$ are defined by the base
change by $f$. Then, the canonical morphism

(3.10.6) $\displaystyle
a_{Y}^{-1}\mathrm{R}f_{\mathrm{\acute{e}t}*}\mathcal{F}\longrightarrow\mathrm{R}f_{\mathrm{v}*}a_{X}^{-1}\mathcal{F}$

is an isomorphism.

###### Proof.

Consider the commutative diagram

(3.10.11)

The canonical morphism
$b_{Y}^{-1}\mathrm{R}f_{\mathrm{\acute{e}t}*}\mathcal{F}\longrightarrow\mathrm{R}f_{\mathrm{h}*}b_{X}^{-1}\mathcal{F}$
is an isomorphism by [Sta22, 0EWF]. It remains to show that the canonical
morphism
$\zeta_{Y}^{-1}\mathrm{R}f_{\mathrm{h}*}b_{X}^{-1}\mathcal{F}\longrightarrow\mathrm{R}f_{\mathrm{v}*}a_{X}^{-1}\mathcal{F}$
is an isomorphism. Let $Y^{\prime}$ be a coherent $Y$-scheme and we set
$g:X^{\prime}=Y^{\prime}\times_{Y}X\to X$. For each integer $q$,
$\zeta_{Y}^{-1}\mathrm{R}^{q}f_{\mathrm{h}*}b_{X}^{-1}\mathcal{F}$ is the
sheaf associated to the presheaf $Y^{\prime}\mapsto
H^{q}_{\mathrm{h}}(X^{\prime},b_{X^{\prime}}^{-1}g^{-1}_{\mathrm{\acute{e}t}}\mathcal{F})=H^{q}_{\mathrm{\acute{e}t}}(X^{\prime},g^{-1}_{\mathrm{\acute{e}t}}\mathcal{F})$
by [Sta22, 0EWH], and $\mathrm{R}^{q}f_{\mathrm{v}*}a_{X}^{-1}\mathcal{F}$ is
the sheaf associated to the presheaf $Y^{\prime}\mapsto
H^{q}_{\mathrm{v}}(X^{\prime},a_{X^{\prime}}^{-1}g^{-1}_{\mathrm{\acute{e}t}}\mathcal{F})=H^{q}_{\mathrm{\acute{e}t}}(X^{\prime},g^{-1}_{\mathrm{\acute{e}t}}\mathcal{F})$
by 3.9. ∎

######  Lemma 3.11.

Let $A$ be a product of (resp. absolutely integrally closed) valuation rings
(2.1).

  1. (1)

Any finitely generated ideal of $A$ is principal.

  2. (2)

Any connected component of $\mathop{\mathrm{Spec}}(A)$ with the reduced closed
subscheme structure is isomorphic to the spectrum of a (resp. absolutely
integrally closed) valuation ring.

###### Proof.

(1) is proved in [Sta22, 092T], and (2) follows from the proof of [BS17, 6.2].
∎

######  Lemma 3.12.

Let $X$ be a $\mathbb{U}$-small scheme, $y\rightsquigarrow x$ a specialization
of points of $X$. Then, there exists a $\mathbb{U}$-small family
$\\{f_{\lambda}:\mathop{\mathrm{Spec}}(V_{\lambda})\to
X\\}_{\lambda\in\Lambda_{y\rightsquigarrow x}}$ of morphisms of schemes such
that

  1. (i)

the ring $V_{\lambda}$ is a $\mathbb{U}$-small (resp. absolutely integrally
closed) valuation ring, and that

  2. (ii)

the morphism $f_{\lambda}$ maps the generic point and closed point of
$\mathop{\mathrm{Spec}}(V_{\lambda})$ to $y$ and $x$ respectively, and that

  3. (iii)

for any morphism of schemes $f:\mathop{\mathrm{Spec}}(V)\to X$ where $V$ is a
(resp. absolutely integrally closed) valuation ring such that $f$ maps the
generic point and closed point of $V$ to $y$ and $x$ respectively, there
exists an element $\lambda\in\Lambda_{y\rightsquigarrow x}$ such that $f$
factors through $f_{\lambda}$ and that $V_{\lambda}\to V$ is an extension of
valuation rings.

###### Proof.

Let $K_{y}$ be the residue field $\kappa(y)$ of $y$ (resp. an algebraic
closure of $\kappa(y)$). Let $\mathfrak{p}_{y}$ be the prime ideal of the
local ring $\mathcal{O}_{X,x}$ corresponding to the point $y$, and let
$\\{V_{\lambda}\\}_{\lambda\in\Lambda_{y\rightsquigarrow x}}$ be the set of
all valuation rings with fraction field $K_{y}$ which contain
$\mathcal{O}_{X,x}/\mathfrak{p}_{y}$ such that the injective homomorphism
$\mathcal{O}_{X,x}/\mathfrak{p}_{y}\to V_{\lambda}$ is local. The smallness of
$\Lambda_{y\rightsquigarrow x}$ and $V_{\lambda}$ is clear, and the inclusion
$\mathcal{O}_{X,x}/\mathfrak{p}_{y}\to V_{\lambda}$ induces a morphism
$f_{\lambda}:\mathop{\mathrm{Spec}}(V_{\lambda})\to X$ satisfying (ii). It
remains to check (iii). The morphism $f$ induces an injective and local
homomorphism $\mathcal{O}_{X,x}/\mathfrak{p}_{y}\to V$. Notice that
$\mathcal{O}_{X,x}/\mathfrak{p}_{y}\to\mathrm{Frac}(V)$ factors through
$K_{y}$ and that $K_{y}\cap V$ is a valuation ring with fraction field
$K_{y}$. It is clear that $K_{y}\cap V\to V$ is local and injective, which
shows that $K_{y}\cap V$ belongs to the set
$\\{V_{\lambda}\\}_{\lambda\in\Lambda_{y\rightsquigarrow x}}$ constructed
before. ∎

######  Lemma 3.13.

Let $f:\mathop{\mathrm{Spec}}(V)\to X$ be a morphism of schemes where $V$ is a
valuation ring. We denote by $a$ and $b$ the closed point and generic point of
$\mathop{\mathrm{Spec}}(V)$ respectively. If $c\in X$ is a generalization of
$f(b)$, then there exists an absolutely integrally closed valuation ring $W$,
a prime ideal $\mathfrak{p}$ of $W$, and a morphism
$g:\mathop{\mathrm{Spec}}(W)\to X$ satisfying the following conditions:

  1. (i)

If $z$, $y$, $x$ denote respectively the generic point, the point
$\mathfrak{p}$ and the closed point of $\mathop{\mathrm{Spec}}(W)$, then
$g(z)=c$, $g(y)=f(b)$ and $g(x)=f(a)$.

  2. (ii)

The induced morphism $\mathop{\mathrm{Spec}}(W/\mathfrak{p})\to X$ factors
through $f$, and the induced morphism $V\to W/\mathfrak{p}$ is an extension of
valuation rings.

###### Proof.

According to [EGA II, 7.1.4], there exists an absolutely integrally closed
valuation ring $U$ and a morphism $\mathop{\mathrm{Spec}}(U)\to X$ which maps
the generic point $z$ and the closed point $y$ of $\mathop{\mathrm{Spec}}(U)$
to $c$ and $f(b)$ respectively. After extending $U$, we may assume that the
morphism $y\to f(b)$ factors through $b$ ([EGA II, 7.1.2]). We denote by
$\kappa(y)$ the residue field of the point $y$. Let $V^{\prime}$ be a
valuation ring extension of $V$ with fraction field $\kappa(y)$, and let $W$
be the preimage of $V^{\prime}$ by the surjection $U\to\kappa(y)$. Then, the
maximal ideal $\mathfrak{p}=\mathrm{Ker}(U\to\kappa(y))$ of $U$ is a prime
ideal of $W$, and $W/\mathfrak{p}=V^{\prime}$. We claim that $W$ is an
absolutely integrally closed valuation ring such that $W_{\mathfrak{p}}=U$.
Indeed, firstly note that the fraction fields of $U$ and $W$ are equal as
$\mathfrak{p}\subseteq W$. Let $\gamma$ be an element of
$\mathrm{Frac}(W)\setminus W$. If $\gamma\in U$, then $\gamma^{-1}\in
W\setminus\mathfrak{p}$ by definition since $\gamma^{-1}\in
U\setminus\mathfrak{p}$ and $V$ is a valuation ring, and then $\gamma\in
W_{\mathfrak{p}}$. If $\gamma\notin U$, then $\gamma^{-1}\in\mathfrak{p}$
since $U$ is a valuation ring, and then $\gamma\notin W_{\mathfrak{p}}$. Thus,
we have proved the claim, which shows that $W$ satisfies the required
conditions. ∎

######  Proposition 3.14.

Let $X$ be a coherent $\mathbb{U}$-small scheme, $X^{\circ}$ a quasi-compact
dense open subset of $X$. Then, there exists a $\mathbb{U}$-small product $A$
of absolutely integrally closed $\mathbb{U}$-small valuation rings and a
v-covering $\mathop{\mathrm{Spec}}(A)\to X$ such that
$\mathop{\mathrm{Spec}}(A)$ is $X^{\circ}$-integrally closed (2.2).

###### Proof.

After replacing $X$ by a finite affine open covering, we may assume that
$X=\mathop{\mathrm{Spec}}(R)$. For a specialization ${y\rightsquigarrow x}$ of
points of $X$, let $\\{R\to
V_{\lambda}\\}_{\lambda\in\Lambda_{y\rightsquigarrow x}}$ be the
$\mathbb{U}$-small set constructed in 3.12. Let $\Lambda=\coprod_{y\in
X^{\circ}}\Lambda_{y\rightsquigarrow x}$ where ${y\rightsquigarrow x}$ runs
through all specializations in $X$ such that $y\in X^{\circ}$. We take
$A=\prod_{\lambda\in\Lambda}V_{\lambda}$ and $R\to A$ the natural
homomorphism. As a quasi-compact open subscheme of
$\mathop{\mathrm{Spec}}(A)$, $X^{\circ}\times_{X}\mathop{\mathrm{Spec}}(A)$ is
the spectrum of $A[1/\pi]$ for an element
$\pi=(\pi_{\lambda})_{\lambda\in\Lambda}\in A$ by 3.11.(1) ([Sta22, 01PH]).
Notice that $\pi_{\lambda}\neq 0$ for any $\lambda\in\Lambda$. We see that $A$
is integrally closed in $A[1/\pi]$. It remains to check that
$\mathop{\mathrm{Spec}}(A)\to X$ is a v-covering. For any morphism
$f:\mathop{\mathrm{Spec}}(V)\to X$ where $V$ is a valuation ring, by 3.13,
there exists an absolutely integrally closed valuation ring $W$, a prime ideal
$\mathfrak{p}$ of $W$ and a morphism $g:\mathop{\mathrm{Spec}}(W)\to X$ such
that $g$ maps the generic point of $W$ into $X^{\circ}$ and that
$W/\mathfrak{p}$ is a valuation ring extension of $V$. By construction, there
exists $\lambda\in\Lambda$ such that $g$ factors through
$\mathop{\mathrm{Spec}}(V_{\lambda})\to X$. We see that $f$ lifts to the
composition of
$\mathop{\mathrm{Spec}}(W/\mathfrak{p})\to\mathop{\mathrm{Spec}}(V_{\lambda})\to\mathop{\mathrm{Spec}}(A)$.
∎

######  Proposition 3.15.

Consider a commutative diagram of schemes

(3.15.5)

where $Z^{\prime}\to Z$ and $X^{\prime}\to X$ are quasi-compact. Assume that
$Y^{\prime}\to Y\times_{X}X^{\prime}$ is surjective, $Y\to Z$ is dominant,
$Z\to X$ is separated and $Z^{\prime}\to X^{\prime}$ is integral. If
$X^{\prime}\to X$ is a v-covering, then $Z^{\prime}\to Z$ is also a
v-covering.

###### Proof.

Notice that $Z^{\prime}\to Z\times_{X}X^{\prime}$ is still integral as $Z\to
X$ is separated. After replacing $X^{\prime}\to X$ by
$Z\times_{X}X^{\prime}\to Z$, we may assume that $Z=X$. Let
$\mathop{\mathrm{Spec}}(V)\to Z$ be a morphism of schemes where $V$ is a
valuation ring. Since $Y\to Z$ is dominant, by 3.13, there exists a morphism
$\mathop{\mathrm{Spec}}(W)\to Z$ where $W$ is an absolutely integrally closed
valuation ring, a prime ideal $\mathfrak{p}$ of $W$ such that $W/\mathfrak{p}$
is a valuation ring extension of $V$ and that the generic point $\xi$ of
$\mathop{\mathrm{Spec}}(W)$ is over the image of $Y\to Z$. After extending $W$
([Sta22, 00IA]), we may assume that there exists a lifting $\xi\to Y$ of
$\xi\to Z$. The morphism $\mathop{\mathrm{Spec}}(W)\to Z=X$ admits a lifting
$\mathop{\mathrm{Spec}}(W^{\prime})\to X^{\prime}$ where $W\to W^{\prime}$ is
an extension of valuation rings. We claim that after extending $W^{\prime}$,
$\mathop{\mathrm{Spec}}(W^{\prime})\to X^{\prime}$ factors through
$Z^{\prime}$. Indeed, if $\xi^{\prime}$ denotes the generic point of
$\mathop{\mathrm{Spec}}(W^{\prime})$, as $Y^{\prime}\to Y\times_{X}X^{\prime}$
is surjective, after extending $W^{\prime}$, we may assume that there exists
an $X^{\prime}$-morphism $\xi^{\prime}\to Y^{\prime}$ which is over $\xi\to
Y$. Since $\mathop{\mathrm{Spec}}(W^{\prime})$ is integrally closed in
$\xi^{\prime}$ and $Z^{\prime}$ is integral over $X^{\prime}$, the morphism
$\mathop{\mathrm{Spec}}(W^{\prime})\to X^{\prime}$ factors through
$Z^{\prime}$ ([Sta22, 035I]). Finally, let
$\mathfrak{q}\in\mathop{\mathrm{Spec}}(W^{\prime})$ which lies over
$\mathfrak{p}\in\mathop{\mathrm{Spec}}(W)$, then we get a lifting
$\mathop{\mathrm{Spec}}(W^{\prime}/\mathfrak{q})\to Z^{\prime}$ of
$\mathop{\mathrm{Spec}}(V)\to Z$, which shows that $Z^{\prime}\to Z$ is a
v-covering. ∎

######  3.16.

Let $S^{\circ}\to S$ be an open immersion of coherent schemes such that $S$ is
$S^{\circ}$-integrally closed (2.2). For any $S$-scheme $X$, we set
$X^{\circ}=S^{\circ}\times_{S}X$. We denote by $\mathbf{I}_{S^{\circ}\to S}$
the category formed by coherent $S$-schemes which are $S^{\circ}$-integrally
closed. Note that any $S^{\circ}$-integrally closed coherent $S$-scheme $X$ is
also $X^{\circ}$-integrally closed by definition. It is clear that the
category $(\mathbf{I}_{S^{\circ}\to S})_{/X}$ of objects of
$\mathbf{I}_{S^{\circ}\to S}$ over $X$ is canonically equivalent to the
category $\mathbf{I}_{X^{\circ}\to X}$.

######  Lemma 3.17 ([Sta22, 03GV]).

Let $Y\to X$ be a coherent morphism of schemes, $X^{\prime}\to X$ a smooth
morphism of schemes, $Y^{\prime}=Y\times_{X}X^{\prime}$. Then, we have
$X^{\prime Y^{\prime}}=X^{Y}\times_{X}X^{\prime}$.

######  Lemma 3.18.

Let $(Y_{\lambda}\to X_{\lambda})_{\lambda\in\Lambda}$ be a directed inverse
system of morphisms of coherent schemes with affine transition morphisms
$Y_{\lambda^{\prime}}\to Y_{\lambda}$ and $X_{\lambda^{\prime}}\to
X_{\lambda}$ ($\lambda^{\prime}\geq\lambda$). We set $Y=\lim Y_{\lambda}$ and
$X=\lim X_{\lambda}$. Then, $(X_{\lambda}^{Y_{\lambda}})_{\lambda\in\Lambda}$
is a directed inverse system of coherent schemes with affine transition
morphisms and we have $X^{Y}=\lim X_{\lambda}^{Y_{\lambda}}$.

###### Proof.

We fix an index $\lambda_{0}\in\Lambda$. After replacing $X_{\lambda_{0}}$ by
an affine open covering, we may assume that $X_{\lambda_{0}}$ is affine
(3.17). We write $X_{\lambda}=\mathop{\mathrm{Spec}}(A_{\lambda})$ and
$B_{\lambda}=\Gamma(Y_{\lambda},\mathcal{O}_{Y_{\lambda}})$ for each
$\lambda\geq\lambda_{0}$, and we set $A=\mathop{\mathrm{colim}}A_{\lambda}$
and $B=\mathop{\mathrm{colim}}B_{\lambda}$. Then, we have
$X=\mathop{\mathrm{Spec}}(A)$ and $B=\Gamma(Y,\mathcal{O}_{Y})$ ([Sta22,
009F]). Let $R_{\lambda}$ (resp. $R$) be the integral closure of $A_{\lambda}$
in $B_{\lambda}$ (resp. $A$ in $B$). By definition, we have
$X_{\lambda}^{Y_{\lambda}}=\mathop{\mathrm{Spec}}(R_{\lambda})$ and
$X^{Y}=\mathop{\mathrm{Spec}}(R)$. The conclusion follows from the fact that
$R=\mathop{\mathrm{colim}}R_{\lambda}$. ∎

######  Lemma 3.19.

Let $S^{\circ}\to S$ be an open immersion of coherent schemes.

  1. (1)

If $X$ is an $S^{\circ}$-integrally closed coherent $S$-scheme, then the open
subscheme $X^{\circ}$ is scheme theoretically dense in $X$.

  2. (2)

If $X$ is an $S^{\circ}$-integrally closed coherent $S$-scheme and
$X^{\prime}$ is a coherent smooth $X$-scheme, then $X^{\prime}$ is also
$S^{\circ}$-integrally closed.

  3. (3)

If $(X_{\lambda})_{\lambda\in\Lambda}$ is a directed inverse system of
$S^{\circ}$-integrally closed coherent $S$-scheme with affine transition
morphisms, then $X=\lim_{\lambda\in\Lambda}X_{\lambda}$ is also
$S^{\circ}$-integrally closed.

  4. (4)

If $Y\to X$ is a morphism of coherent schemes over $S^{\circ}\to S$ such that
$Y$ is integral over $X^{\circ}$, then the integral closure $X^{Y}$ is
$S^{\circ}$-integrally closed with $(X^{Y})^{\circ}=Y$.

###### Proof.

(1), (2), (3) follow from [Sta22, 035I], 3.17 and 3.18 respectively. For (4),
$(X^{Y})^{\circ}=X^{\circ}\times_{X}X^{Y}$ is the integral closure of
$X^{\circ}$ in $X^{\circ}\times_{X}Y=Y$ by 3.17, which is $Y$ itself. ∎

######  3.20.

We take the notation in 3.16. The inclusion functor

(3.20.1) $\displaystyle\Phi^{+}:\mathbf{I}_{S^{\circ}\to
S}\longrightarrow\mathbf{Sch}^{\mathrm{coh}}_{/S},\ X\longmapsto X,$

admits a right adjoint

(3.20.2)
$\displaystyle\sigma^{+}:\mathbf{Sch}^{\mathrm{coh}}_{/S}\longrightarrow\mathbf{I}_{S^{\circ}\to
S},\ X\longmapsto\overline{X}=X^{X^{\circ}}.$

Indeed, $\sigma^{+}$ is well-defined by 3.19.(4), and the adjointness follows
from the functoriality of taking integral closures. We remark that
$\overline{X}^{\circ}=X^{\circ}$. On the other hand, the functor

(3.20.3) $\displaystyle\Psi^{+}:\mathbf{I}_{S^{\circ}\to
S}\longrightarrow\mathbf{Sch}^{\mathrm{coh}}_{/S^{\circ}},\ X\longmapsto
X^{\circ},$

admits a left adjoint

(3.20.4)
$\displaystyle\alpha^{+}:\mathbf{Sch}^{\mathrm{coh}}_{/S^{\circ}}\longrightarrow\mathbf{I}_{S^{\circ}\to
S},\ Y\longmapsto Y.$

######  Lemma 3.21.

With the notation in 3.16, let $\varphi:I\to\mathbf{I}_{S^{\circ}\to S}$ be a
functor sending $i$ to $X_{i}$. If $X=\lim X_{i}$ represents the limit of
$\varphi$ in the category of coherent $S$-schemes, then the integral closure
$\overline{X}=X^{X^{\circ}}$ represents the limit of $\varphi$ in
$\mathbf{I}_{S^{\circ}\to S}$ with $\overline{X}^{\circ}=X^{\circ}$.

###### Proof.

It follows directly from the adjoint pair $(\Phi^{+},\sigma^{+})$ (3.20). ∎

It follows from 3.21 that for a diagram $X_{1}\to X_{0}\leftarrow X_{2}$ in
$\mathbf{I}_{S^{\circ}\to S}$, the fibred product is representable by

(3.21.1) $\displaystyle
X_{1}\overline{\times}_{X_{0}}X_{2}=(X_{1}\times_{X_{0}}X_{2})^{X_{1}^{\circ}\times_{X_{0}^{\circ}}X_{2}^{\circ}}.$

######  Proposition 3.22.

With the notation in 3.16, let $\mathscr{C}$ be the set of families of
morphisms $\\{X_{i}\to X\\}_{i\in I}$ of $\mathbf{I}_{S^{\circ}\to S}$ with
$I$ finite such that $\coprod_{i\in I}X_{i}\to X$ is a v-covering. Then,
$\mathscr{C}$ forms a pretopology of $\mathbf{I}_{S^{\circ}\to S}$.

###### Proof.

Let $\\{X_{i}\to X\\}_{i\in I}$ be an element of $\mathscr{C}$. Firstly, we
check that for a morphism $X^{\prime}\to X$ of $\mathbf{I}_{S^{\circ}\to S}$,
the base change $\\{X_{i}^{\prime}\to X^{\prime}\\}_{i\in I}$ also lies in
$\mathscr{C}$, where $Z_{i}=X_{i}\times_{X}X^{\prime}$ and
$X_{i}^{\prime}=Z_{i}^{Z_{i}^{\circ}}$ by 3.21. Applying 3.15 to the following
diagram

(3.22.5)

we deduce that $\coprod_{i\in I}X_{i}^{\prime}\to X^{\prime}$ is also a
v-covering, which shows the stability of $\mathscr{C}$ under base change.

For each $i\in I$, let $\\{X_{ij}\to X_{i}\\}_{j\in J_{i}}$ be an element of
$\mathscr{C}$. We need to show that the composition $\\{X_{ij}\to X\\}_{i\in
I,j\in J_{i}}$ also lies in $\mathscr{C}$. This follows immediately from the
stability of v-coverings under composition. We conclude that $\mathscr{C}$
forms a pretopology. ∎

######  Definition 3.23.

With the notation in 3.16, we endow the category $\mathbf{I}_{S^{\circ}\to S}$
with the topology generated by the pretopology defined in 3.22, and we call
$\mathbf{I}_{S^{\circ}\to S}$ the _v-site of $S^{\circ}$-integrally closed
coherent $S$-schemes_.

By definition, any object in $\mathbf{I}_{S^{\circ}\to S}$ is quasi-compact.
Let $\mathscr{O}$ be the sheaf on $\mathbf{I}_{S^{\circ}\to S}$ associated to
the presheaf $X\mapsto\Gamma(X,\mathcal{O}_{X})$. We call $\mathscr{O}$ the
structural sheaf of $\mathbf{I}_{S^{\circ}\to S}$.

######  Proposition 3.24.

With the notation in 3.16, let $f:X^{\prime}\to X$ be a covering in
$\mathbf{I}_{S^{\circ}\to S}$ such that $f$ is separated and that the diagonal
morphism $X^{\prime\circ}\to X^{\prime\circ}\times_{X^{\circ}}X^{\prime\circ}$
is surjective. Then, the morphism of representable sheaves
$h_{X^{\prime}}^{\mathrm{a}}\to h_{X}^{\mathrm{a}}$ is an isomorphism.

###### Proof.

We need to show that for any sheaf $\mathcal{F}$ on $\mathbf{I}_{S^{\circ}\to
S}$, $\mathcal{F}(X)\to\mathcal{F}(X^{\prime})$ is an isomorphism. Since the
composition of $X^{\prime\circ}\to
X^{\prime\circ}\times_{X^{\circ}}X^{\prime\circ}\to
X^{\prime}\overline{\times}_{X}X^{\prime}$ factors through the closed
immersion $X^{\prime}\to X^{\prime}\overline{\times}_{X}X^{\prime}$ (as $f$ is
separated), the closed immersion $X^{\prime}\to
X^{\prime}\overline{\times}_{X}X^{\prime}$ is surjective (3.19.(1)). Consider
the following sequence

(3.24.1)
$\displaystyle\mathcal{F}(X)\to\mathcal{F}(X^{\prime})\rightrightarrows\mathcal{F}(X^{\prime}\overline{\times}_{X}X^{\prime})\to\mathcal{F}(X^{\prime}).$

The right arrow is injective as $X^{\prime}\to
X^{\prime}\overline{\times}_{X}X^{\prime}$ is a v-covering. Thus, the middle
two arrows are actually the same. Thus, the first arrow is an isomorphism by
the sheaf property of $\mathcal{F}$. ∎

######  Proposition 3.25.

With the notation in 3.16, let $\alpha:\mathcal{F}_{1}\to\mathcal{F}_{2}$ be a
morphism of presheaves on $\mathbf{I}_{S^{\circ}\to S}$. Assume that

  1. (i)

the morphism
$\mathcal{F}_{1}(\mathop{\mathrm{Spec}}(V))\to\mathcal{F}_{2}(\mathop{\mathrm{Spec}}(V))$
is an isomorphism for any $S^{\circ}$-integrally closed $S$-scheme
$\mathop{\mathrm{Spec}}(V)$ where $V$ is an absolutely integrally closed
valuation ring, and that

  2. (ii)

for any directed inverse system of $S^{\circ}$-integrally closed affine
schemes $(\mathop{\mathrm{Spec}}(A_{\lambda}))_{\lambda\in\Lambda}$ over $S$
the natural morphism
$\mathop{\mathrm{colim}}\mathcal{F}_{i}(\mathop{\mathrm{Spec}}(A_{\lambda}))\to\mathcal{F}_{i}(\mathop{\mathrm{Spec}}(\mathop{\mathrm{colim}}A_{\lambda}))$
is an isomorphism for $i=1,2$ (cf. 3.19.(3)).

Then, the morphism of the associated sheaves
$\mathcal{F}_{1}^{\mathrm{a}}\to\mathcal{F}_{2}^{\mathrm{a}}$ is an
isomorphism.

###### Proof.

Let $A$ be a product of absolutely integrally closed valuation rings such that
$X=\mathop{\mathrm{Spec}}(A)$ is an $S^{\circ}$-integrally closed $S$-scheme.
Let $\mathop{\mathrm{Spec}}(V)$ be a connected component of
$\mathop{\mathrm{Spec}}(A)$ with the reduced closed subscheme structure. Then,
$V$ is an absolutely integrally closed valuation ring by 3.11.(2), and
$\mathop{\mathrm{Spec}}(V)$ is also $S^{\circ}$-integrally closed since it has
nonempty intersection with the dense open subset $X^{\circ}$ of $X$. Notice
that each connected component of an affine scheme is the intersection of some
open and closed subsets ([Sta22, 04PP]). Moreover, since $A$ is reduced, we
have $V=\mathop{\mathrm{colim}}A^{\prime}$, where the colimit is taken over
all the open and closed subschemes
$X^{\prime}=\mathop{\mathrm{Spec}}(A^{\prime})$ of $X$ which contain
$\mathop{\mathrm{Spec}}(V)$. By assumption, we have an isomorphism

(3.25.1)
$\displaystyle\mathop{\mathrm{colim}}\mathcal{F}_{1}(X^{\prime})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathop{\mathrm{colim}}\mathcal{F}_{2}(X^{\prime}).$

For two elements $\xi_{1},\xi_{1}^{\prime}\in\mathcal{F}_{1}(X)$ with
$\alpha_{X}(\xi_{1})=\alpha_{X}(\xi_{1}^{\prime})$ in $\mathcal{F}_{2}(X)$, by
(3.25.1) and a limit argument, there exists a finite disjoin union
$X=\coprod_{i=1}^{r}X_{i}^{\prime}$ such that the images of $\xi_{1}$ and
$\xi_{1}^{\prime}$ in $\mathcal{F}_{1}(X_{i}^{\prime})$ are the same.
Therefore, $\mathcal{F}_{1}^{\mathrm{a}}\to\mathcal{F}_{2}^{\mathrm{a}}$ is
injective by 3.14. On the other hand, for an element
$\xi_{2}\in\mathcal{F}_{2}(X)$, by (3.25.1) and a limit argument, there exists
a finite disjoin union $X=\coprod_{i=1}^{r}X_{i}^{\prime}$ such that there
exists an element $\xi_{1,i}\in\mathcal{F}_{1}(X_{i}^{\prime})$ for each $i$
such that the image of $\xi_{2}$ in $\mathcal{F}_{2}(X_{i}^{\prime})$ is equal
to $\alpha_{X_{i}^{\prime}}(\xi_{1,i})$. Therefore,
$\mathcal{F}_{1}^{\mathrm{a}}\to\mathcal{F}_{2}^{\mathrm{a}}$ is surjective by
3.14. ∎

######  3.26.

We take the notation in 3.16. Endowing $\mathbf{Sch}^{\mathrm{coh}}$ with the
v-topology (3.3), we see that the functors $\sigma^{+}$ and $\Psi^{+}$ defined
in 3.20 are left exact (as they have left adjoints) and continuous by 3.15 and
3.22. Therefore, they define morphisms of sites (2.5)

(3.26.1)
$\displaystyle(\mathbf{Sch}^{\mathrm{coh}}_{/S^{\circ}})_{\mathrm{v}}\stackrel{{\scriptstyle\Psi}}{{\longrightarrow}}\mathbf{I}_{S^{\circ}\to
S}\stackrel{{\scriptstyle\sigma}}{{\longrightarrow}}(\mathbf{Sch}^{\mathrm{coh}}_{/S})_{\mathrm{v}}.$

######  Proposition 3.27.

With the notation in 3.26, let
$a:(\mathbf{Sch}^{\mathrm{coh}}_{/S^{\circ}})_{\mathrm{v}}\to
S^{\circ}_{\mathrm{\acute{e}t}}$ be the morphism of site defined by the
inclusion functor (3.9).

  1. (1)

For any torsion abelian sheaf $\mathcal{F}$ on
$S^{\circ}_{\mathrm{\acute{e}t}}$, the canonical morphism
$\Psi_{*}(a^{-1}\mathcal{F})\to\mathrm{R}\Psi_{*}(a^{-1}\mathcal{F})$ is an
isomorphism.

  2. (2)

For any locally constant torsion abelian sheaf $\mathbb{L}$ on
$\mathbf{I}_{S^{\circ}\to S}$, the canonical morphism
$\mathbb{L}\to\mathrm{R}\Psi_{*}\Psi^{-1}\mathbb{L}$ is an isomorphism.

###### Proof.

(1) For each integer $q$, the sheaf
$\mathrm{R}^{q}\Psi_{*}(a^{-1}\mathcal{F})$ is the sheaf associated to the
presheaf $X\mapsto
H^{q}_{\mathrm{v}}(X^{\circ},a^{-1}\mathcal{F})=H^{q}_{\mathrm{\acute{e}t}}(X^{\circ},f^{-1}_{\mathrm{\acute{e}t}}\mathcal{F})$
by 3.9, where $f_{\mathrm{\acute{e}t}}:X^{\circ}_{\mathrm{\acute{e}t}}\to
S^{\circ}_{\mathrm{\acute{e}t}}$ is the natural morphism. If $X$ is the
spectrum of a nonzero absolutely integrally closed valuation ring $V$, then
$X^{\circ}=\mathop{\mathrm{Spec}}(V[1/\pi])$ for a nonzero element $\pi\in V$
by 3.11.(1) and 3.19.(1), which is also the spectrum of an absolutely
integrally closed valuation ring (2.1). In this case,
$H^{q}_{\mathrm{\acute{e}t}}(X^{\circ},f^{-1}_{\mathrm{\acute{e}t}}\mathcal{F})=0$
for $q>0$, which proves (1) by 3.25 and [SGA 4${}_{\text{\rm II}}$, VII.5.8].

(2) The problem is local on $\mathbf{I}_{S^{\circ}\to S}$. We may assume that
$\mathbb{L}$ is the constant sheaf with value $L$. Then,
$\mathrm{R}^{q}\Psi_{*}\Psi^{-1}\mathbb{L}=0$ for $q>0$ by applying (1) on the
constant sheaf with value $L$ on $S^{\circ}_{\mathrm{\acute{e}t}}$. For $q=0$,
notice that $\mathbb{L}$ is also the sheaf associated to the presheaf
$X\mapsto H^{0}_{\mathrm{\acute{e}t}}(X,L)$, while
$\Psi_{*}\Psi^{-1}\mathbb{L}$ is the sheaf $X\mapsto
H^{0}_{\mathrm{\acute{e}t}}(X^{\circ},L)$ by the discussion in (1). If $X$ is
the spectrum of a nonzero absolutely integrally closed valuation ring, then so
is $X^{\circ}$ and so that
$H^{0}_{\mathrm{\acute{e}t}}(X,L)=H^{0}_{\mathrm{\acute{e}t}}(X^{\circ},L)=L$.
The conclusion follows from 3.25 and [SGA 4${}_{\text{\rm II}}$, VII.5.8]. ∎

##  4\. The arc-Descent of Perfect Algebras

######  Definition 4.1.

For any $\mathbb{F}_{p}$-algebra $R$, we denote by $R_{\mathrm{perf}}$ the
filtered colimit

(4.1.1) $\displaystyle
R_{\mathrm{perf}}=\mathop{\mathrm{colim}}_{\mathrm{Frob}}R$

indexed by $(\mathbb{N},\leq)$, where the transition map associated to
$i\leq(i+1)$ is the Frobenius of $R$.

It is clear that the endo-functor of the category of
$\mathbb{F}_{p}$-algebras, $R\mapsto R_{\mathrm{perf}}$, commutes with
colimits.

######  4.2.

We define a presheaf $\mathcal{O}_{\mathrm{perf}}$ on the category
$\mathbf{Sch}^{\mathrm{coh}}_{\mathbb{F}_{p}}$ of coherent $\mathbb{U}$-small
$\mathbb{F}_{p}$-schemes $X$ by

(4.2.1)
$\displaystyle\mathcal{O}_{\mathrm{perf}}(X)=\Gamma(X,\mathcal{O}_{X})_{\mathrm{perf}}.$

For any morphism $\mathop{\mathrm{Spec}}(B)\to\mathop{\mathrm{Spec}}(A)$ of
affine $\mathbb{F}_{p}$-schemes, we consider the augmented Čech complex of the
presheaf $\mathcal{O}_{\mathrm{perf}}$,

(4.2.2) $\displaystyle 0\to A_{\mathrm{perf}}\to B_{\mathrm{perf}}\to
B_{\mathrm{perf}}\otimes_{A_{\mathrm{perf}}}B_{\mathrm{perf}}\to\cdots.$

######  Lemma 4.3 ([Sta22, 0EWT]).

The presheaf $\mathcal{O}_{\mathrm{perf}}$ is a sheaf on
$\mathbf{Sch}^{\mathrm{coh}}_{\mathbb{F}_{p}}$ with respect to the fppf
topology ([Sta22, 021L]). Moreover, for any coherent $\mathbb{F}_{p}$-scheme
$X$ and any integer $q$,

(4.3.1) $\displaystyle
H^{q}_{\mathrm{fppf}}(X,\mathcal{O}_{\mathrm{perf}})=\mathop{\mathrm{colim}}_{\mathrm{Frob}}H^{q}(X,\mathcal{O}_{X}).$

###### Proof.

Firstly, we remark that for any integer $q$, the functor
$H^{q}_{\mathrm{fppf}}(X,-)$ commutes with filtered colimit of abelian sheaves
on $(\mathbf{Sch}^{\mathrm{coh}}_{/X})_{\mathrm{fppf}}$ for any coherent
scheme $X$ ([Sta22, 0739]). Since the presheaf $\mathcal{O}$ sending $X$ to
$\Gamma(X,\mathcal{O}_{X})$ on $\mathbf{Sch}^{\mathrm{coh}}_{\mathbb{F}_{p}}$
is an fppf-sheaf, we have
$H^{0}_{\mathrm{fppf}}(X,\mathcal{O}_{\mathrm{perf}}^{\mathrm{a}})=\mathop{\mathrm{colim}}_{\mathrm{Frob}}H^{0}_{\mathrm{fppf}}(X,\mathcal{O})=\mathcal{O}_{\mathrm{perf}}(X)$.
Thus, $\mathcal{O}_{\mathrm{perf}}$ is an fppf-sheaf. Moreover,
$H^{q}_{\mathrm{fppf}}(X,\mathcal{O}_{\mathrm{perf}})=\mathop{\mathrm{colim}}_{\mathrm{Frob}}H^{q}_{\mathrm{fppf}}(X,\mathcal{O})=\mathop{\mathrm{colim}}_{\mathrm{Frob}}H^{q}(X,\mathcal{O}_{X})$
by faithfully flat descent ([Sta22, 03DW]). ∎

######  Lemma 4.4.

Let $\tau\in\\{\textrm{fppf, h, v, arc}\\}$. The following propositions are
equivalent:

  1. (1)

The presheaf $\mathcal{O}_{\mathrm{perf}}$ on
$\mathbf{Sch}^{\mathrm{coh}}_{\mathbb{F}_{p}}$ is a $\tau$-sheaf and
$H^{q}_{\tau}(X,\mathcal{O}_{\mathrm{perf}})=\mathop{\mathrm{colim}}_{\mathrm{Frob}}H^{q}(X,\mathcal{O}_{X})$
for any coherent $\mathbb{F}_{p}$-scheme $X$ and any integer $q$.

  2. (2)

For any $\tau$-covering
$\mathop{\mathrm{Spec}}(B)\to\mathop{\mathrm{Spec}}(A)$ of affine
$\mathbb{F}_{p}$-schemes, the augmented Čech complex (4.2.2) is exact.

###### Proof.

For an affine scheme $X=\mathop{\mathrm{Spec}}(A)$, $H^{q}(X,\mathcal{O}_{X})$
vanishes for $q>0$ and $H^{0}(X,\mathcal{O}_{X})=A$. For (1) $\Rightarrow$
(2), the exactness of (4.2.2) follows from the Čech-cohomology-to-cohomology
spectral sequence associated to the $\tau$-covering
$\mathop{\mathrm{Spec}}(B)\to\mathop{\mathrm{Spec}}(A)$ [Sta22, 03AZ].
Therefore, (1) and (2) hold for $\tau=\textrm{fppf}$ by 4.3. Conversely, the
exactness of (4.2.2) shows the sheaf property for any $\tau$-covering of an
affine scheme by affine schemes, which implies the fppf-sheaf
$\mathcal{O}_{\mathrm{perf}}$ is a $\tau$-sheaf (cf. [Sta22, 0ETM]). The
vanishing of higher Čech cohomologies implies that
$H^{q}_{\tau}(X,\mathcal{O}_{\mathrm{perf}})=0$ for any affine
$\mathbb{F}_{p}$-scheme $X$ and any $q>0$ ([Sta22, 03F9]). Therefore, for a
coherent $\mathbb{F}_{p}$-scheme $X$,
$H^{q}_{\tau}(X,\mathcal{O}_{\mathrm{perf}})$ can be computed by the hyper-
Čech cohomology of a hypercovering of $X$ formed by affine open subschemes
([Sta22, 01GY]). In particular, we have
$H^{q}_{\tau}(X,\mathcal{O}_{\mathrm{perf}})=H^{q}_{\mathrm{fppf}}(X,\mathcal{O}_{\mathrm{perf}})$
for any integer $q$, which completes the proof by 4.3. ∎

######  Lemma 4.5 (Gabber).

The augmented Čech complex (4.2.2) is exact for any h-covering
$\mathop{\mathrm{Spec}}(B)\to\mathop{\mathrm{Spec}}(A)$ of affine
$\mathbb{F}_{p}$-schemes.

###### Proof.

This is a result of Gabber, cf. [BST17, 3.3] or [Sta22, 0EWU], and 4.4. ∎

######  Lemma 4.6 ([BS17, 4.1]).

The augmented Čech complex (4.2.2) is exact for any v-covering
$\mathop{\mathrm{Spec}}(B)\to\mathop{\mathrm{Spec}}(A)$ of affine
$\mathbb{F}_{p}$-schemes.

###### Proof.

We write $B$ as a filtered colimit of finitely presented $A$-algebras
$B=\mathop{\mathrm{colim}}B_{\lambda}$. Then,
$\mathop{\mathrm{Spec}}(B_{\lambda})\to\mathop{\mathrm{Spec}}(A)$ is an
h-covering for each $\lambda$ by 3.2. Notice that
$B_{\mathrm{perf}}=\mathop{\mathrm{colim}}B_{\lambda,\mathrm{perf}}$, then the
conclusion follows from applying 4.5 on
$\mathop{\mathrm{Spec}}(B_{\lambda})\to\mathop{\mathrm{Spec}}(A)$ and taking
colimit. ∎

######  Lemma 4.7 ([BS17, 6.3]).

For any valuation ring $V$ and any prime ideal $\mathfrak{p}$ of $V$, the
sequence

(4.7.1) $\displaystyle 0\longrightarrow
V\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}V/\mathfrak{p}\oplus
V_{\mathfrak{p}}\stackrel{{\scriptstyle\beta}}{{\longrightarrow}}V_{\mathfrak{p}}/\mathfrak{p}V_{\mathfrak{p}}\longrightarrow
0$

is exact, where $\alpha(a)=(a,a)$ and $\beta(a,b)=a-b$. If moreover $V$ is a
perfect $\mathbb{F}_{p}$-algebra, then for any perfect $V$-algebra $R$, the
base change of (4.7.1) by $V\to R$,

(4.7.2) $\displaystyle 0\longrightarrow R\longrightarrow R/\mathfrak{p}R\oplus
R_{\mathfrak{p}}\longrightarrow
R_{\mathfrak{p}}/\mathfrak{p}R_{\mathfrak{p}}\longrightarrow 0$

is exact.

###### Proof.

The sequence (4.7.1) is exact if and only if
$\mathfrak{p}=\mathfrak{p}V_{\mathfrak{p}}$. Let $a\in\mathfrak{p}$ and $s\in
V\setminus\mathfrak{p}$. Since $\mathfrak{p}$ is an ideal, $s/a\notin V$, thus
$a/s\in V$ as $V$ is a valuation ring. Moreover, we must have
$a/s\in\mathfrak{p}$ as $\mathfrak{p}$ is a prime ideal. This shows the
equality $\mathfrak{p}=\mathfrak{p}V_{\mathfrak{p}}$.

The second assertion follows directly from the fact that
$\mathop{\mathrm{Tor}}_{q}^{A}(B,C)=0$ for any $q>0$ and any diagram
$B\leftarrow A\to C$ of perfect $\mathbb{F}_{p}$-algebras ([BS17, 3.16]). ∎

######  Lemma 4.8 ([BM20, 4.8]).

The augmented Čech complex (4.2.2) is exact for any arc-covering
$\mathop{\mathrm{Spec}}(B)\to\mathop{\mathrm{Spec}}(A)$ of affine
$\mathbb{F}_{p}$-schemes with $A$ a valuation ring.

###### Proof.

We follow the proof of Bhatt-Mathew [BM20, 4.8]. Let
$B=\mathop{\mathrm{colim}}B_{\lambda}$ be a filtered colimit of finitely
presented $A$-algebras. Then,
$\mathop{\mathrm{Spec}}(B_{\lambda})\to\mathop{\mathrm{Spec}}(A)$ is also an
arc-covering by 3.2. Thus, we may assume that $B$ is a finitely presented
$A$-algebra.

An interval $I=[\mathfrak{p},\mathfrak{q}]$ of a valuation ring $A$ is a pair
of prime ideals $\mathfrak{p}\subseteq\mathfrak{q}$ of $A$. We denote by
$A_{I}=(A/\mathfrak{p})_{\mathfrak{q}}$. The set $\mathcal{I}$ of intervals of
$A$ is partially ordered under inclusion. Let $\mathcal{P}$ be the subset
consisting of intervals $I$ such that the lemma holds for
$\mathop{\mathrm{Spec}}(B\otimes_{A}A_{I})\to\mathop{\mathrm{Spec}}(A_{I})$.
It suffices to show that $\mathcal{P}=\mathcal{I}$.

  1. (1)

If the valuation ring $A_{I}$ is of height $\leq 1$, we claim that
$\mathop{\mathrm{Spec}}(B\otimes_{A}A_{I})\to\mathop{\mathrm{Spec}}(A_{I})$ is
automatically a v-covering. Indeed, there is an extension of valuation rings
$A_{I}\to V$ of height $\leq 1$ which factors through $B\otimes_{A}A_{I}$. As
$A_{I}\to V$ is faithfully flat,
$\mathop{\mathrm{Spec}}(B\otimes_{A}A_{I})\to\mathop{\mathrm{Spec}}(A_{I})$ is
a v-covering by 3.2 and 3.4.(1). Therefore, $I\in\mathcal{P}$ by 4.6.

  2. (2)

For any interval $J\subseteq I$, if $I\in\mathcal{P}$ then $J\in\mathcal{P}$.
Indeed, applying $\otimes_{\mathbb{F}_{p}}(A_{J})_{\mathrm{perf}}$ to the
exact sequence (4.2.2) for
$\mathop{\mathrm{Spec}}(B\otimes_{A}A_{I})\to\mathop{\mathrm{Spec}}(A_{I})$,
we still get an exact sequence by the Tor-independence of perfect
$\mathbb{F}_{p}$-algebras ([BS17, 3.16]).

  3. (3)

If $\mathfrak{p}\subseteq A$ is not maximal, then there exists
$\mathfrak{q}\supsetneq\mathfrak{p}$ with
$I=[\mathfrak{p},\mathfrak{q}]\in\mathcal{P}$. Indeed, if there is no such $I$
with the height of $A_{I}$ no more than $1$, then
$\mathfrak{p}=\bigcap_{\mathfrak{q}\supsetneq\mathfrak{p}}\mathfrak{q}$, and
thus,

(4.8.1) $\displaystyle
A_{\mathfrak{p}}/\mathfrak{p}A_{\mathfrak{p}}=\mathop{\mathrm{colim}}_{I=[\mathfrak{p},\mathfrak{q}],\mathfrak{q}\supsetneq\mathfrak{p}}A_{I}.$

Since
$\mathop{\mathrm{Spec}}(B\otimes_{A}{A_{\mathfrak{p}}/\mathfrak{p}A_{\mathfrak{p}}})\to\mathop{\mathrm{Spec}}(A_{\mathfrak{p}}/\mathfrak{p}A_{\mathfrak{p}})$
is an h-covering as $A_{\mathfrak{p}}/\mathfrak{p}A_{\mathfrak{p}}$ is a field
(and we have assumed that $B$ is of finite presentation over $A$), there
exists an interval $I$ in the above colimit, such that
$\mathop{\mathrm{Spec}}(B\otimes_{A}A_{I})\to\mathop{\mathrm{Spec}}(A_{I})$ is
also an h-covering by 3.4.(3). Therefore, this $I$ lies in $\mathcal{P}$ by
4.6.

  4. (4)

If $\mathfrak{p}\subseteq A$ is nonzero, then there exists
$\mathfrak{q}\subsetneq\mathfrak{p}$ with
$I=[\mathfrak{q},\mathfrak{p}]\in\mathcal{P}$. This is similar to (3).

  5. (5)

If $I,J\in\mathcal{P}$ are overlapping, then $I\cup J\in\mathcal{P}$. Indeed,
by (2) and replacing $A$ by $A_{I\cup J}$, we may assume that
$I=[0,\mathfrak{p}]$, $J=[\mathfrak{p},\mathfrak{m}]$ with $\mathfrak{m}$ the
maximal ideal. In particular, $A_{I}=A_{\mathfrak{p}}$,
$A_{J}=A/\mathfrak{p}$, and $A_{I\cap
J}=A_{\mathfrak{p}}/\mathfrak{p}A_{\mathfrak{p}}$. Since for each
$R=\otimes_{A_{\mathrm{perf}}}^{n}B_{\mathrm{perf}}$ we have the short exact
sequence (4.7.2), we get $I\cup J\in\mathcal{P}$.

In general, by Zorn’s lemma, the above five properties of $\mathcal{P}$
guarantee that $\mathcal{P}=\mathcal{I}$ (cf. [BM20, 4.7]). ∎

######  Lemma 4.9 (cf. [BM20, 3.30]).

The augmented Čech complex (4.2.2) is exact for any arc-covering
$\mathop{\mathrm{Spec}}(B)\to\mathop{\mathrm{Spec}}(A)$ of affine
$\mathbb{F}_{p}$-schemes with $A$ a product of valuation rings.

###### Proof.

We follow closely the proof of 3.25. Let $\mathop{\mathrm{Spec}}(V)$ be a
connected component of $\mathop{\mathrm{Spec}}(A)$ with the reduced closed
subscheme structure. Then, $V$ is a valuation ring by 3.11.(2). By 4.8, the
augmented Čech complex

(4.9.1) $\displaystyle 0\to
V_{\mathrm{perf}}\to(B\otimes_{A}V)_{\mathrm{perf}}\to(B\otimes_{A}V)_{\mathrm{perf}}\otimes_{V_{\mathrm{perf}}}(B\otimes_{A}V)_{\mathrm{perf}}\to\cdots$

is exact. Notice that each connected component of an affine scheme is the
intersection of some open and closed subsets ([Sta22, 04PP]). Moreover, since
$A$ is reduced, we have $V=\mathop{\mathrm{colim}}A^{\prime}$, where the
colimit is taken over all the open and closed subschemes
$\mathop{\mathrm{Spec}}(A^{\prime})$ which contain
$\mathop{\mathrm{Spec}}(V)$.

Therefore, by a limit argument, for an element
$f\in\otimes^{n}_{A_{\mathrm{perf}}}B_{\mathrm{perf}}$ which maps to zero in
$\otimes^{n+1}_{A_{\mathrm{perf}}}B_{\mathrm{perf}}$, as
$\mathop{\mathrm{Spec}}(A)$ is quasi-compact, we can decompose
$\mathop{\mathrm{Spec}}(A)$ into a finite disjoint union
$\coprod_{i=1}^{N}\mathop{\mathrm{Spec}}(A_{i})$ such that there exists
$g_{i}\in\otimes^{n-1}_{A_{i,\mathrm{perf}}}(B\otimes_{A}{A_{i}})_{\mathrm{perf}}$
which maps to the image $f_{i}$ of $f$ in
$\otimes^{n}_{A_{i,\mathrm{perf}}}(B\otimes_{A}{A_{i}})_{\mathrm{perf}}$.
Since we have

(4.9.2)
$\displaystyle\otimes^{n}_{A_{\mathrm{perf}}}B_{\mathrm{perf}}=\prod_{i=1}^{N}\otimes^{n}_{A_{i,\mathrm{perf}}}(B\otimes_{A}{A_{i}})_{\mathrm{perf}},$

the element $g=(g_{i})_{i=1}^{N}$ maps to $f$, which shows the exactness of
(4.2.2). ∎

######  Proposition 4.10 ([BS19, 8.10]).

Let $\tau\in\\{\textrm{fppf, h, v, arc}\\}$.

  1. (1)

The presheaf $\mathcal{O}_{\mathrm{perf}}$ is a $\tau$-sheaf over
$\mathbf{Sch}^{\mathrm{coh}}_{\mathbb{F}_{p}}$, and for any coherent
$\mathbb{F}_{p}$-scheme $X$ and any integer $q$,

(4.10.1) $\displaystyle
H^{q}_{\tau}(X,\mathcal{O}_{\mathrm{perf}})=\mathop{\mathrm{colim}}_{\mathrm{Frob}}H^{q}(X,\mathcal{O}_{X}).$

  2. (2)

For any $\tau$-covering
$\mathop{\mathrm{Spec}}(B)\to\mathop{\mathrm{Spec}}(A)$ of affine
$\mathbb{F}_{p}$-schemes, the augmented Čech complex

(4.10.2) $\displaystyle 0\to A_{\mathrm{perf}}\to B_{\mathrm{perf}}\to
B_{\mathrm{perf}}\otimes_{A_{\mathrm{perf}}}B_{\mathrm{perf}}\to\cdots$

is exact.

###### Proof.

We follow closely the proof of Bhatt-Scholze [BS19, 8.10]. (1) and (2) are
equivalent by 4.4, and they hold for $\tau\in\\{\textrm{fppf, h, v}\\}$ by
4.3, 4.5 and 4.6. In particular,

(4.10.3) $\displaystyle
H^{0}_{\mathrm{v}}(\mathop{\mathrm{Spec}}(A),\mathcal{O}_{\mathrm{perf}})=A_{\mathrm{perf}}\textrm{
and
}H^{q}_{\mathrm{v}}(\mathop{\mathrm{Spec}}(A),\mathcal{O}_{\mathrm{perf}})=0,\
\forall q>0.$

We take a hypercovering in the v-topology
$\mathop{\mathrm{Spec}}(A_{\bullet})\to\mathop{\mathrm{Spec}}(A)$ such that
$A_{n}$ is a product of valuation rings for each degree $n$ by 3.14 and
[Sta22, 094K and 0DB1]. The associated sequence

(4.10.4) $\displaystyle 0\to A_{\mathrm{perf}}\to A_{0,\mathrm{perf}}\to
A_{1,\mathrm{perf}}\to\cdots$

is exact by the hyper-Čech-cohomology-to-cohomology spectral sequence [Sta22,
01GY].

Consider the double complex $(A_{i}^{j})$ where the $i$-th row
$A_{i}^{\bullet}$ is the base change of (4.10.2) by $A_{\mathrm{perf}}\to
A_{i,\mathrm{perf}}$, i.e. the augmented Čech complex (4.2.2) associated to
$\mathop{\mathrm{Spec}}(B\otimes_{A}{A_{i}})\to\mathop{\mathrm{Spec}}(A_{i})$
(we set $A_{-1}=A$). On the other hand, the $j$-th column $A_{\bullet}^{j}$ is
the associated sequence (4.10.4) to the hypercovering
$\mathop{\mathrm{Spec}}({A_{\bullet}}\otimes_{A}(\otimes_{A}^{j}B))\to\mathop{\mathrm{Spec}}(\otimes_{A}^{j}B)$,
which is exact by the previous discussion. Since
$A_{-1}^{\bullet}\to\mathrm{Tot}(A_{i}^{j})_{i\geq 0}^{j\geq 0}$ is a quasi-
isomorphism ([Sta22, 0133]), for the exactness of the $(-1)$-row
$A_{-1}^{\bullet}$, we only need to show the exactness of the $i$-th row
$A_{i}^{\bullet}$ for any $i\geq 0$ but this has been proved in 4.9 thanks to
our choice of the hypercovering, which completes the proof. ∎

##  5\. Almost Pre-perfectoid Algebras

######  Definition 5.1.

  1. (1)

A _pre-perfectoid field_ $K$ is a valuation field whose valuation ring
$\mathcal{O}_{K}$ is non-discrete, of height $1$ and of residue characteristic
$p$, and such that the Frobenius map on $\mathcal{O}_{K}/p\mathcal{O}_{K}$ is
surjective.

  2. (2)

A _perfectoid field_ $K$ is a pre-perfectoid field which is complete for the
topology defined by its valuation (cf. [Sch12, 3.1]).

  3. (3)

A _pseudo-uniformizer_ $\pi$ of a pre-perfectoid field $K$ is a nonzero
element of the maximal ideal $\mathfrak{m}_{K}$ of $\mathcal{O}_{K}$.

A _morphism of pre-perfectoid fields_ $K\to L$ is a homomorphism of fields
which induces an extension of valuation rings
$\mathcal{O}_{K}\to\mathcal{O}_{L}$.

######  Lemma 5.2.

Let $K$ be a pre-perfectoid field with a pseudo-uniformizer $\pi$. Then, the
fraction field $\widehat{K}$ of the $\pi$-adic completion of $\mathcal{O}_{K}$
is a perfectoid field.

###### Proof.

The $\pi$-adic completion $\widehat{\mathcal{O}_{K}}$ of $\mathcal{O}_{K}$ is
still a non-discrete valuation ring of height $1$ with residue characteristic
$p$ (cf. [Bou06, VI.§5.3, Prop.5]). If $p\neq 0$, then it is canonically
isomorphic to the $p$-adic completion of $\mathcal{O}_{K}$, so that there is a
canonical isomorphism
$\mathcal{O}_{K}/p\mathcal{O}_{K}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\widehat{\mathcal{O}_{K}}/p\widehat{\mathcal{O}_{K}}$,
from which we see that $\widehat{K}$ is a perfectoid field. If $p=0$, then the
Frobenius induces a surjection $\mathcal{O}_{K}\to\mathcal{O}_{K}$ if and only
if $\mathcal{O}_{K}$ is perfect. Thus, $\widehat{\mathcal{O}_{K}}$ is also
perfect, and we see that $\widehat{K}$ is a perfectoid field. ∎

######  5.3.

Let $K$ be a pre-perfectoid field. There is a unique (up to scalar) ordered
group homomorphism $v_{K}:K^{\times}\to\mathbb{R}$ such that
$v_{K}^{-1}(0)=\mathcal{O}_{K}^{\times}$, where the group structure on
$\mathbb{R}$ is given by the addition. In particular,
$\mathcal{O}_{K}\setminus 0=v_{K}^{-1}(\mathbb{R}_{\geq 0})$ and
$\mathfrak{m}_{K}\setminus 0=v_{K}^{-1}(\mathbb{R}_{>0})$ (cf. [Bou06, VI.§4.5
Prop.7] and [Bou07, V.§2 Prop.1, Rem.2]). The non-discrete assumption on
$\mathcal{O}_{K}$ implies that the image
$v_{K}(K^{\times})\subseteq\mathbb{R}$ is dense. We set $v_{K}(0)=+\infty$.

######  Lemma 5.4 ([Sch12, 3.2]).

Let $K$ be a pre-perfectoid field. Then, for any pseudo-uniformizer $\pi$ of
$K$, there exists $\pi_{n}\in\mathfrak{m}_{K}$ for each integer $n\geq 0$ such
that $\pi_{0}=\pi$ and $\pi_{n}=u_{n}\cdot\pi_{n+1}^{p}$ for some unit
$u_{n}\in\mathcal{O}_{K}^{\times}$, and $\mathfrak{m}_{K}$ is generated by
$\\{\pi_{n}\\}_{n\geq 0}$.

###### Proof.

If $v_{K}(\pi)<v_{K}(p)$, since the Frobenius is surjective on
$\mathcal{O}_{K}/p$, there exists $\pi_{1}\in\mathcal{O}_{K}$ such that
$v_{K}(\pi-\pi_{1}^{p})\geq v_{K}(p)$. Then, $v_{K}(\pi)=v_{K}(\pi_{1}^{p})$
and thus $\pi=u\cdot\pi_{1}^{p}$ with $u\in\mathcal{O}_{K}^{\times}$. In
general, since $v_{K}(K^{\times})\subseteq\mathbb{R}$ is dense, any pseudo-
uniformizer $\pi$ is a finite product of pseudo-uniformizers whose valuation
values are strictly less than $v_{K}(p)$, from which we get a $p$-th root
$\pi_{1}$ of $\pi$ up to a unit. Since $\pi_{1}$ is also a pseudo-uniformizer,
we get $\pi_{n}$ inductively. As $v_{K}(\pi_{n})$ tends to zero when $n$ tends
to infinity, $\mathfrak{m}_{K}$ is generated by $\\{\pi_{n}\\}_{n\geq 0}$. ∎

######  5.5.

Let $K$ be a pre-perfectoid field. We briefly review almost algebra over
$(\mathcal{O}_{K},\mathfrak{m}_{K})$ for which we mainly refer to [AG20, 2.6],
[AGT16, V] and [GR03]. Remark that
$\mathfrak{m}_{K}\otimes_{\mathcal{O}_{K}}\mathfrak{m}_{K}\cong\mathfrak{m}_{K}^{2}=\mathfrak{m}_{K}$
is flat over $\mathcal{O}_{K}$.

An $\mathcal{O}_{K}$-module $M$ is called _almost zero_ if
$\mathfrak{m}_{K}M=0$. A morphism of $\mathcal{O}_{K}$-modules $M\to N$ is
called an _almost isomorphism_ if its kernel and cokernel are almost zero. Let
$\mathscr{N}$ be the full subcategory of the category
$\mathcal{O}_{K}\textrm{-}\mathbf{Mod}$ of $\mathcal{O}_{K}$-modules formed by
almost zero objects. It is clear that $\mathscr{N}$ is a Serre subcategory of
$\mathcal{O}_{K}\textrm{-}\mathbf{Mod}$ ([Sta22, 02MO]). Let $\mathcal{S}$ be
the set of almost isomorphisms in $\mathcal{O}_{K}\textrm{-}\mathbf{Mod}$.
Since $\mathscr{N}$ is a Serre subcategory, $\mathcal{S}$ is a multiplicative
system, and moreover the quotient abelian category
$\mathcal{O}_{K}\textrm{-}\mathbf{Mod}/\mathscr{N}$ is representable by the
localized category $\mathcal{S}^{-1}\mathcal{O}_{K}\textrm{-}\mathbf{Mod}$
(cf. [Sta22, 02MS]). We denote
$\mathcal{S}^{-1}\mathcal{O}_{K}\textrm{-}\mathbf{Mod}$ by
$\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Mod}$, whose objects are
called _almost $\mathcal{O}_{K}$-modules_ or simply
_$\mathcal{O}_{K}^{\mathrm{al}}$ -modules_ (cf. [AG20, 2.6.2]). We denote by

(5.5.1)
$\displaystyle\alpha^{*}:\mathcal{O}_{K}\textrm{-}\mathbf{Mod}\longrightarrow\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Mod},\
M\longmapsto M^{\mathrm{al}}$

the localization functor. It induces an $\mathcal{O}_{K}$-linear structure on
$\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Mod}$. For any two
$\mathcal{O}_{K}$-modules $M$ and $N$, we have a natural
$\mathcal{O}_{K}$-linear isomorphism ([AG20, 2.6.7.1])

(5.5.2)
$\displaystyle\mathrm{Hom}_{\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Mod}}(M^{\mathrm{al}},N^{\mathrm{al}})=\mathrm{Hom}_{\mathcal{O}_{K}\textrm{-}\mathbf{Mod}}(\mathfrak{m}_{K}\otimes_{\mathcal{O}_{K}}M,N).$

The localization functor $\alpha^{*}$ admits a right adjoint

(5.5.3)
$\displaystyle\alpha_{*}:\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Mod}\longrightarrow\mathcal{O}_{K}\textrm{-}\mathbf{Mod},\
M\longmapsto
M_{*}=\mathrm{Hom}_{\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Mod}}(\mathcal{O}_{K}^{\mathrm{al}},M),$

and a left adjoint

(5.5.4)
$\displaystyle\alpha_{!}:\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Mod}\longrightarrow\mathcal{O}_{K}\textrm{-}\mathbf{Mod},\
M\longmapsto M_{!}=\mathfrak{m}_{K}\otimes_{\mathcal{O}_{K}}M_{*}.$

Moreover, the natural morphisms

(5.5.5)
$\displaystyle(M_{*})^{\mathrm{al}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}M,\
M\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}(M_{!})^{\mathrm{al}}$

are isomorphisms for any $\mathcal{O}_{K}^{\mathrm{al}}$-module $M$ (cf.
[AG20, 2.6.8]). In particular, for any functor
$\varphi:I\to\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Mod}$ sending $i$
to $M_{i}$, the colimit and limit of $\varphi$ are representable by

(5.5.6)
$\displaystyle\mathop{\mathrm{colim}}M_{i}=(\mathop{\mathrm{colim}}M_{i*})^{\mathrm{al}},\
\lim M_{i}=(\lim M_{i*})^{\mathrm{al}}.$

The tensor product in $\mathcal{O}_{K}\textrm{-}\mathbf{Mod}$ induces a tensor
product in $\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Mod}$ by

(5.5.7) $\displaystyle
M^{\mathrm{al}}\otimes_{\mathcal{O}_{K}^{\mathrm{al}}}N^{\mathrm{al}}=(M\otimes_{\mathcal{O}_{K}}N)^{\mathrm{al}}$

making $\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Mod}$ an abelian tensor
category ([AG20, 2.6.4]). We denote by
$\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Alg}$ the category of
commutative unitary monoids in
$\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Mod}$ induced by the tensor
structure, whose objects are called _almost $\mathcal{O}_{K}$-algebras_ or
simply _$\mathcal{O}_{K}^{\mathrm{al}}$ -algebras_ (cf. [AG20, 2.6.11]).
Notice that $R^{\mathrm{al}}$ (resp. $R_{*}$) admits a canonical algebra
structure for any $\mathcal{O}_{K}$-algebra (resp.
$\mathcal{O}_{K}^{\mathrm{al}}$-algebra) $R$. Moreover, $\alpha^{*}$ and
$\alpha_{*}$ induce adjoint functors between
$\mathcal{O}_{K}\textrm{-}\mathbf{Alg}$ and
$\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Alg}$ ([AG20, 2.6.12]).
Combining with (5.5.5) and (5.5.6), we see that for any functor
$\varphi:I\to\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Alg}$ sending $i$
to $R_{i}$, the colimit and limit of $\varphi$ are representable by (cf.
[GR03, 2.2.16])

(5.5.8)
$\displaystyle\mathop{\mathrm{colim}}R_{i}=(\mathop{\mathrm{colim}}R_{i*})^{\mathrm{al}},\
\lim R_{i}=(\lim R_{i*})^{\mathrm{al}}.$

In particular, for any diagram $B\leftarrow A\to C$ of
$\mathcal{O}_{K}^{\mathrm{al}}$-algebras, we denote its colimit by

(5.5.9) $\displaystyle
B\otimes_{A}C=(B_{*}\otimes_{A_{*}}C_{*})^{\mathrm{al}},$

which is clearly compatible with the tensor products of modules. We remark
that $\alpha^{*}$ commutes with arbitrary colimits (resp. limits), since it
has a right adjoint $\alpha_{*}$ (resp. since the forgetful functor
$\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Alg}\to\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Mod}$
and the localization functor
$\alpha^{*}:\mathcal{O}_{K}\textrm{-}\mathbf{Mod}\to\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Mod}$
commute with arbitrary limits).

######  5.6.

For an element $a$ of $\mathcal{O}_{K}$, we denote by
$(\mathcal{O}_{K}/a\mathcal{O}_{K})^{\mathrm{al}}\textrm{-}\mathbf{Mod}$ the
full subcategory of $\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Mod}$
formed by the objects on which the morphism induced by multiplication by $a$
is zero. Notice that for an
$(\mathcal{O}_{K}/a\mathcal{O}_{K})^{\mathrm{al}}$-module $M$, $M_{*}$ is an
$\mathcal{O}_{K}/a\mathcal{O}_{K}$-module. Thus, the localization functor
$\alpha^{*}$ induces an essentially surjective exact functor
$(\mathcal{O}_{K}/a\mathcal{O}_{K})\textrm{-}\mathbf{Mod}\to(\mathcal{O}_{K}/a\mathcal{O}_{K})^{\mathrm{al}}\textrm{-}\mathbf{Mod}$,
which identifies the latter with the quotient abelian category
$(\mathcal{O}_{K}/a\mathcal{O}_{K})\textrm{-}\mathbf{Mod}/\mathscr{N}\cap(\mathcal{O}_{K}/a\mathcal{O}_{K})\textrm{-}\mathbf{Mod}$.

Let $\pi$ be a pseudo-uniformizer of $K$ dividing $p$ with a $p$-th root
$\pi_{1}$ up to a unit. The Frobenius on $\mathcal{O}_{K}/\pi\mathcal{O}_{K}$
induces an isomorphism
$\mathcal{O}_{K}/\pi_{1}\mathcal{O}_{K}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathcal{O}_{K}/\pi\mathcal{O}_{K}$.
The Frobenius on $(\mathcal{O}_{K}/\pi)$-algebras and the localization functor
$\alpha^{*}$ induce a natural transformation from the base change functor
$(\mathcal{O}_{K}/\pi)^{\mathrm{al}}\textrm{-}\mathbf{Alg}\to(\mathcal{O}_{K}/\pi)^{\mathrm{al}}\textrm{-}\mathbf{Alg}$,
$R\mapsto(\mathcal{O}_{K}/\pi)\otimes_{\mathrm{Frob},(\mathcal{O}_{K}/\pi)}R$
to the identity functor.

(5.6.3)

For an $(\mathcal{O}_{K}/\pi)^{\mathrm{al}}$-algebra $R$, we usually identify
the $(\mathcal{O}_{K}/\pi_{1})^{\mathrm{al}}$-algebra $R/\pi_{1}R$ with the
$(\mathcal{O}_{K}/\pi)^{\mathrm{al}}$-algebra
$(\mathcal{O}_{K}/\pi)\otimes_{\mathrm{Frob},(\mathcal{O}_{K}/\pi)}R$, and we
denote by $R/\pi_{1}R\to R$ the natural morphism
$(\mathcal{O}_{K}/\pi)\otimes_{\mathrm{Frob},(\mathcal{O}_{K}/\pi)}R\to R$
induced by the Frobenius (cf. [GR03, 3.5.6]). Moreover, the natural
transformations induced by Frobenius for
$(\mathcal{O}_{K}/\pi)\textrm{-}\mathbf{Alg}$ and
$(\mathcal{O}_{K}/\pi)^{\mathrm{al}}\textrm{-}\mathbf{Alg}$ are also
compatible with the functor $\alpha_{*}$. Indeed, it follows from the fact
that for any $(\mathcal{O}_{K}/\pi)$-algebra $R$, the composition of

(5.6.6)

is the relative Frobenius on
$(R^{\mathrm{al}})_{*}=\mathrm{Hom}_{\mathcal{O}_{K}\textrm{-}\mathbf{Mod}}(\mathfrak{m}_{K},R)$.

######  5.7.

Let $C$ be a site. A presheaf $\mathcal{F}$ of $\mathcal{O}_{K}$-modules on
$C$ is called _almost zero_ if $\mathcal{F}(U)$ is almost zero for any object
$U$ of $C$. A morphism of presheaves $\mathcal{F}\to\mathcal{G}$ of
$\mathcal{O}_{K}$-modules on $C$ is called an _almost isomorphism_ if
$\mathcal{F}(U)\to\mathcal{G}(U)$ is an almost isomorphism for any object $U$
of $C$ (cf. [AG20, 2.6.23]). Let $\mathscr{N}$ be the full subcategory of the
category $\mathcal{O}_{K}\textrm{-}\mathbf{Mod}_{C}$ of sheaves of
$\mathcal{O}_{K}$-modules on $C$ formed by almost zero objects. Similarly,
$\mathscr{N}$ is a Serre subcategory of
$\mathcal{O}_{K}\textrm{-}\mathbf{Mod}_{C}$. Let
$\mathbf{D}_{\mathscr{N}}(\mathcal{O}_{K}\textrm{-}\mathbf{Mod}_{C})$ be the
full subcategory of the derived category
$\mathbf{D}(\mathcal{O}_{K}\textrm{-}\mathbf{Mod}_{C})$ formed by complexes
with almost zero cohomologies. It is a strictly full saturated triangulated
subcategory ([Sta22, 06UQ]). We also say that the objects of
$\mathbf{D}_{\mathscr{N}}(\mathcal{O}_{K}\textrm{-}\mathbf{Mod}_{C})$ are
_almost zero_. Let $\mathcal{S}$ be the set of arrows in
$\mathbf{D}(\mathcal{O}_{K}\textrm{-}\mathbf{Mod}_{C})$ which induce almost
isomorphisms on cohomologies. We also call the elements of $\mathcal{S}$
_almost isomorphisms_. Then, $\mathcal{S}$ is a saturated multiplicative
system ([Sta22, 05RG]), and moreover the quotient triangulated category
$\mathbf{D}(\mathcal{O}_{K}\textrm{-}\mathbf{Mod}_{C})/\mathbf{D}_{\mathscr{N}}(\mathcal{O}_{K}\textrm{-}\mathbf{Mod}_{C})$
is representable by the localized triangulated category
$\mathcal{S}^{-1}\mathbf{D}(\mathcal{O}_{K}\textrm{-}\mathbf{Mod}_{C})$
([Sta22, 05RI]). The natural functor

(5.7.1)
$\displaystyle\mathcal{S}^{-1}\mathbf{D}(\mathcal{O}_{K}\textrm{-}\mathbf{Mod}_{C})\longrightarrow\mathbf{D}(\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Mod}_{C})$

is an equivalence by [Sta22, 06XM] and (5.5.5) (cf. [GR03, 2.4.9]).

######  Lemma 5.8.

Let $K$ be a pre-perfectoid field with a pseudo-uniformizer $\pi$, $M$ a flat
$\mathcal{O}_{K}$-module. We fix a system of $p^{n}$-th roots
$(\pi_{n})_{n\geq 0}$ of $\pi$ up to units (5.4), then the map

(5.8.1) $\displaystyle\bigcap_{n\geq
0}\pi_{n}^{-1}M\to(M^{\mathrm{al}})_{*}=\mathrm{Hom}_{\mathcal{O}_{K}\textrm{-}\mathbf{Mod}}(\mathfrak{m}_{K},M),\
a\mapsto(x\mapsto xa)$

where $\pi_{n}^{-1}M\subseteq M[1/\pi]$, is an isomorphism of
$\mathcal{O}_{K}$-modules. Moreover, for an extension of valuation rings
$\mathcal{O}_{K}\to R$ of height $1$, we have $R=\bigcap_{n\geq
0}\pi_{n}^{-1}R$ and the above isomorphism coincides with the unit map
$R\to(R^{\mathrm{al}})_{*}$.

###### Proof.

Since $\mathfrak{m}_{K}$ is generated by $\\{\pi_{n}\\}_{n\geq 0}$, any
$\mathcal{O}_{K}$-linear morphism $f:\mathfrak{m}_{K}\to M$ is determined by
its values $f(\pi_{n})\in M$. Notice that $(\pi/\pi_{n})\cdot
f(\pi_{n})=f(\pi)$ and $M$ is $\pi$-torsion free, so that $f$ must be given by
the multiplication by an element $a=f(\pi)/\pi\in M[1/\pi]$. It is clear that
such a multiplication sends $\mathfrak{m}_{K}$ to $M$ if and only if
$a\in\bigcap_{n\geq 0}\pi_{n}^{-1}M$, which shows the first assertion. If
$\mathcal{O}_{K}\to R$ is an extension of valuation rings of height $1$, then
we directly deduce from the valuation map $v:R[1/\pi]\setminus 0\to\mathbb{R}$
(5.3) the equality $R=\bigcap_{n\geq 0}\pi_{n}^{-1}R$. ∎

######  Lemma 5.9.

Let $K$ be a pre-perfectoid field, $R$ an $\mathcal{O}_{K}$-algebra,
$\mathcal{O}_{K}\to V$ an extension of valuation rings of height $1$. Then,
the canonical map

(5.9.1)
$\displaystyle\mathrm{Hom}_{\mathcal{O}_{K}\textrm{-}\mathbf{Alg}}(R,V)\longrightarrow\mathrm{Hom}_{\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Alg}}(R^{\mathrm{al}},V^{\mathrm{al}})$

is bijective.

###### Proof.

There are natural maps

(5.9.2)
$\displaystyle\mathrm{Hom}_{\mathcal{O}_{K}\textrm{-}\mathbf{Alg}}(R,V)\to\mathrm{Hom}_{\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Alg}}(R^{\mathrm{al}},V^{\mathrm{al}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{Hom}_{\mathcal{O}_{K}\textrm{-}\mathbf{Alg}}(R,(V^{\mathrm{al}})_{*})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{Hom}_{\mathcal{O}_{K}\textrm{-}\mathbf{Alg}}(R,V),$

where the middle isomorphism is given by adjunction and the last isomorphism
is induced by the inverse of the unit map $V\to(V^{\mathrm{al}})_{*}$ by 5.8.
The composition is the identity map, which completes the proof. ∎

######  Definition 5.10.

Let $K$ be a pre-perfectoid field. We say that an
$\mathcal{O}_{K}^{\mathrm{al}}$-module $M$ (resp. an $\mathcal{O}_{K}$-module
$M$) is _flat_ (resp. _almost flat_) if the functor
$\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Mod}\to\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Mod}$
given by tensoring with $M$ is exact (resp. $M^{\mathrm{al}}$ is flat).

######  Remark 5.11.

In general, one can define the flatness of a morphism of
$\mathcal{O}_{K}^{\mathrm{al}}$-algebras (cf. [GR03, 3.1.1.(i)]). We say that
a morphism of $\mathcal{O}_{K}$-algebras $A\to B$ is _almost flat_ if
$A^{\mathrm{al}}\to B^{\mathrm{al}}$ is flat.

######  Lemma 5.12.

Let $K$ be a pre-perfectoid field with a pseudo-uniformizer $\pi$. Then, an
$\mathcal{O}_{K}^{\mathrm{al}}$-module $M$ is flat if and only if $M_{*}$ is
$\pi$-torsion free. In particular, an $\mathcal{O}_{K}$-module $N$ is almost
flat if and only if the submodule of $\pi$-torsion elements of $N$ is almost
zero.

###### Proof.

First of all, for any $\mathcal{O}_{K}^{\mathrm{al}}$-modules $L_{1}$ and
$L_{2}$, we have a canonical isomorphism

(5.12.1)
$\displaystyle\mathrm{Hom}_{\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Mod}}(M\otimes_{\mathcal{O}_{K}^{\mathrm{al}}}L_{1},L_{2})=\mathrm{Hom}_{\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Mod}}(L_{1},\mathrm{Hom}_{\mathcal{O}_{K}\textrm{-}\mathbf{Mod}}(M_{*},L_{2*})^{\mathrm{al}})$

by (5.5.2), (5.5.5) and (5.5.7). Therefore, the functor defined by tensoring
with $M$ admits a right adjoint, and thus it is right exact. Consider the
sequence

(5.12.2) $\displaystyle
0\longrightarrow\mathcal{O}_{K}^{\mathrm{al}}\stackrel{{\scriptstyle\cdot\pi}}{{\longrightarrow}}\mathcal{O}_{K}^{\mathrm{al}}\longrightarrow(\mathcal{O}_{K}/\pi\mathcal{O}_{K})^{\mathrm{al}}\longrightarrow
0,$

which is exact since the localization functor $\alpha^{*}$ is exact. If $M$ is
flat, tensoring the above sequence with $M$ and applying $\alpha_{*}$, we
deduce that $M_{*}$ is $\pi$-torsion free since $\alpha_{*}$ is left exact (as
a right adjoint to $\alpha^{*}$). Conversely, if $M_{*}$ is $\pi$-torsion
free, then it is flat over $\mathcal{O}_{K}$. For any injective morphism
$L_{1}\to L_{2}$ of $\mathcal{O}_{K}^{\mathrm{al}}$-modules, $L_{1*}\to
L_{2*}$ is also injective, and it remains injective after tensoring with
$M_{*}$. Therefore, $L_{1}\to L_{2}$ also remains injective after tensoring
with $M$ since $\alpha^{*}$ is exact. This shows that $M$ is flat.

The second assertion follows from the almost isomorphism
$N\to(N^{\mathrm{al}})_{*}$ and the fact that
$(N^{\mathrm{al}})_{*}=\mathrm{Hom}_{\mathcal{O}_{K}\textrm{-}\mathbf{Mod}}(\mathfrak{m}_{K},N)$
has no nonzero almost zero element. ∎

######  Lemma 5.13.

Let $K$ be a pre-perfectoid field with a pseudo-uniformizer $\pi$, $M$ a flat
$\mathcal{O}_{K}^{\mathrm{al}}$-module, $x$ an element of $\mathcal{O}_{K}$.
Then, the canonical morphism $M_{*}/xM_{*}\to(M/xM)_{*}$ is injective, and for
any $\epsilon\in\mathfrak{m}_{K}$, the image of
$\varphi_{\epsilon}:(M/\epsilon xM)_{*}\to(M/xM)_{*}$ is $M_{*}/xM_{*}$. In
particular, the canonical morphism

(5.13.1)
$\displaystyle\varprojlim_{n}M_{*}/\pi^{n}M_{*}\longrightarrow(\varprojlim_{n}M/\pi^{n}M)_{*}$

is an isomorphism of $\mathcal{O}_{K}$-modules.

###### Proof.

We follow the proof of [Sch12, 5.3]. Applying the left exact functor
$\alpha_{*}$ to the exact sequence

(5.13.4)

we see that $M_{*}/xM_{*}\to(M/xM)_{*}$ is injective.

To show that the image of $\varphi_{\epsilon}$ is $M_{*}/xM_{*}$, it suffices
to show that $\varphi_{\epsilon}$ factors through $M_{*}/xM_{*}$. We identify
$(M/xM)_{*}$ with
$\mathrm{Hom}_{\mathcal{O}_{K}\textrm{-}\mathbf{Mod}}(\mathfrak{m}_{K},M_{*}/xM_{*})$
by (5.5.5) and (5.5.2) so that $M_{*}/xM_{*}$ identifies with the subset
consisting of the $\mathcal{O}_{K}$-morphisms $\mathfrak{m}_{K}\to
M_{*}/xM_{*}$ sending $y$ to $ya$ for some element $a\in M_{*}/xM_{*}$. For an
$\mathcal{O}_{K}$-morphism $f:\mathfrak{m}_{K}\to M_{*}/\epsilon xM_{*}$, let
$b$ be an element of $M_{*}$ which lifts $f(\epsilon)$. Notice that $M_{*}$ is
$\pi$-torsion free by 5.12. With notation in 5.8, we have
$b\equiv(\epsilon/\pi_{n})\cdot f(\pi_{n})\ \mod\epsilon xM_{*}$ for $n$ big
enough so that the element $b/\epsilon\in M_{*}[1/\pi]$ lies in
$\bigcap_{n\geq 0}\pi_{n}^{-1}M_{*}=M_{*}$. Moreover,
$\pi_{n}\cdot(b/\epsilon)\equiv f(\pi_{n})\ \mod xM_{*}$ for $n$ big enough.
As $\varphi_{\epsilon}(f)$ is determined by its values on $\pi_{n}$ for $n$
big enough, it follows that $\varphi_{\epsilon}(f)=a$, where $a$ is the image
of $b/\epsilon$ in $M_{*}/xM_{*}$.

Finally, the previous result implies that the inverse system
$((M/\pi^{n}M)_{*})_{n\geq 1}$ is Mittag-Leffler so that the “in particular”
part follows immediately from the fact that $\alpha_{*}$ commutes with
arbitrary limits (as a right adjoint to $\alpha^{*}$) ([Sta22, 0596]). ∎

######  Definition 5.14.

Let $K$ be a pre-perfectoid field. For any $\mathcal{O}_{K}$-algebra $R$, we
define a perfect ring $R^{\flat}$ as the projective limit

(5.14.1) $\displaystyle R^{\flat}=\varprojlim_{\mathrm{Frob}}R/pR$

indexed by $(\mathbb{N},\leq)$, where transition map associated to
$i\leq(i+1)$ is the Frobenius on $R/pR$. We call $R^{\flat}$ the _tilt_ of
$R$.

######  Lemma 5.15 ([Sch12, 3.4]).

Let $K$ be a perfectoid field with a pseudo-uniformizer $\pi$ dividing $p$.

  1. (1)

The projection induces an isomorphism of multiplicative monoids

(5.15.1)
$\displaystyle\varprojlim_{\mathrm{Frob}}\mathcal{O}_{K}\longrightarrow\varprojlim_{\mathrm{Frob}}\mathcal{O}_{K}/\pi\mathcal{O}_{K}.$

In particular, the right hand side is canonically isomorphic to
$(\mathcal{O}_{K})^{\flat}$ as a ring.

  2. (2)

We denote by

(5.15.2)
$\displaystyle\sharp:(\mathcal{O}_{K})^{\flat}\longrightarrow\mathcal{O}_{K},\
x\mapsto x^{\sharp},$

the composition of the inverse of (5.15.1) and the projection onto the first
component. Then $v_{K}\circ\sharp:(\mathcal{O}_{K})^{\flat}\setminus
0\to\mathbb{R}_{\geq 0}$ defines a valuation of height $1$ on
$(\mathcal{O}_{K})^{\flat}$.

  3. (3)

The fraction field $K^{\flat}$ of $(\mathcal{O}_{K})^{\flat}$ is a perfectoid
field of characteristic $p$ and the element

(5.15.3)
$\displaystyle\pi^{\flat}=(\cdots,\pi_{1}^{1/p^{2}},\pi_{1}^{1/p},\pi_{1},0)\in(\mathcal{O}_{K})^{\flat}$

is a pseudo-uniformizer of $K^{\flat}$, where $\pi=u\cdot\pi_{1}^{p}$ with
$\pi_{1}\in\mathfrak{m}_{K}$ and $u\in\mathcal{O}_{K}^{\times}$.

  4. (4)

We have $\mathcal{O}_{K^{\flat}}=(\mathcal{O}_{K})^{\flat}$, and there is a
canonical isomorphism

(5.15.4)
$\displaystyle\mathcal{O}_{K^{\flat}}/\pi^{\flat}\mathcal{O}_{K^{\flat}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathcal{O}_{K}/\pi\mathcal{O}_{K}$

induced by (1) and the projection onto the first component.

######  5.16.

We see that the tilt defines a functor
$\mathcal{O}_{K}\textrm{-}\mathbf{Alg}\to\mathcal{O}_{K^{\flat}}\textrm{-}\mathbf{Alg}$,
$R\mapsto R^{\flat}$, which preserves almost zero objects and almost
isomorphisms. For an $\mathcal{O}_{K}^{\mathrm{al}}$-algebra $R$, we set
$R^{\flat}=((R_{*})^{\flat})^{\mathrm{al}}$ and call it the _tilt_ of $R$,
which induces a functor
$\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Alg}\to\mathcal{O}_{K^{\flat}}^{\mathrm{al}}\textrm{-}\mathbf{Alg}$,
$R\mapsto R^{\flat}$. Note that the tilt functor commutes with the
localization functor $\alpha^{*}$ up to a canonical isomorphism, and commutes
with the functor $\alpha_{*}$ up to a canonical almost isomorphism.

######  Definition 5.17 ([Sch12, 5.1]).

Let $K$ be a perfectoid field, $\pi$ a pseudo-uniformizer of $K$ dividing $p$
with a $p$-th root $\pi_{1}$ up to a unit.

  1. (1)

A _perfectoid $\mathcal{O}_{K}^{\mathrm{al}}$-algebra_ is an
$\mathcal{O}_{K}^{\mathrm{al}}$-algebra $R$ such that

    1. (i)

$R$ is flat over $\mathcal{O}_{K}^{\mathrm{al}}$;

    2. (ii)

the Frobenius of $R/\pi R$ induces an isomorphism $R/\pi_{1}R\to R/\pi R$ of
$\mathcal{O}_{K}^{\mathrm{al}}$-algebras (5.6);

    3. (iii)

the canonical morphism $R\to\varprojlim_{n}R/\pi^{n}R$ is an isomorphism in
$\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Alg}$.

We denote by $\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Perf}$ the full
subcategory of $\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Alg}$ formed by
perfectoid $\mathcal{O}_{K}^{\mathrm{al}}$-algebras.

  2. (2)

A _perfectoid $(\mathcal{O}_{K}/\pi)^{\mathrm{al}}$-algebra_ is a flat
$(\mathcal{O}_{K}/\pi)^{\mathrm{al}}$-algebra $R$ such that the Frobenius map
induces an isomorphism
$R/\pi_{1}R\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}R$. We denote by
$(\mathcal{O}_{K}/\pi)^{\mathrm{al}}\textrm{-}\mathbf{Perf}$ the full
subcategory of $(\mathcal{O}_{K}/\pi)^{\mathrm{al}}\textrm{-}\mathbf{Alg}$
formed by perfectoid $(\mathcal{O}_{K}/\pi)^{\mathrm{al}}$-algebras.

######  Lemma 5.18.

Let $K$ be a pre-perfectoid field, $\pi$ a pseudo-uniformizer of $K$ dividing
$p$ with a $p$-th root $\pi_{1}$ up to a unit. Then, for an
$\mathcal{O}_{K}$-algebra $R$, the following conditions are equivalent:

  1. (1)

The almost algebra $\widehat{R}^{\mathrm{al}}$ associated to the $\pi$-adic
completion $\widehat{R}$ of $R$ is a perfectoid
$\mathcal{O}_{\widehat{K}}^{\mathrm{al}}$-algebra.

  2. (2)

The $\mathcal{O}_{\widehat{K}}$-module $\widehat{R}$ is almost flat, and the
Frobenius of $R/\pi R$ induces an almost isomorphism $R/\pi_{1}R\to R/\pi R$.

###### Proof.

We have seen that $\widehat{K}$ is a perfectoid field in 5.2 and $\pi$ is
obviously a pseudo-uniformizer of $\widehat{K}$. Since the localization
functor
$\alpha^{*}:\mathcal{O}_{K}\textrm{-}\mathbf{Alg}\to\mathcal{O}_{K}^{\mathrm{al}}\textrm{-}\mathbf{Alg}$
commutes with arbitrary limits and colimits (5.5), we have a canonical
isomorphism
$\widehat{R}^{\mathrm{al}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\varprojlim_{n}\widehat{R}^{\mathrm{al}}/\pi^{n}\widehat{R}^{\mathrm{al}}$.
Thus, the third condition in 5.17.(1) holds for $\widehat{R}^{\mathrm{al}}$.
Since there are canonical isomorphisms

(5.18.1) $\displaystyle
R/\pi_{1}R\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\widehat{R}/\pi_{1}\widehat{R},\
R/\pi
R\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\widehat{R}/\pi\widehat{R},$

the conditions (1) and (2) are clearly equivalent. ∎

######  Definition 5.19.

Let $K$ be a pre-perfectoid field, $\pi$ a pseudo-uniformizer of $K$ dividing
$p$ with a $p$-th root $\pi_{1}$ up to a unit. We say that an
$\mathcal{O}_{K}$-algebra is _almost pre-perfectoid_ if it satisfies the
equivalent conditions in 5.18.

We remark that in 5.19, if a morphism of $\mathcal{O}_{K}$-algebras $R\to
R^{\prime}$ induces an almost isomorphism $R/\pi^{n}R\to
R^{\prime}/\pi^{n}R^{\prime}$ for each $n\geq 1$, then the morphism of the
$\pi$-adic completions $\widehat{R}\to\widehat{R^{\prime}}$ is an almost
isomorphism since $\alpha^{*}$ commutes with limits. In particular, $R$ is
almost pre-perfectoid if and only if $R^{\prime}$ is almost pre-perfectoid.

######  Lemma 5.20.

Let $K$ be a pre-perfectoid field with a pseudo-uniformizer $\pi$, $R$ an
$\mathcal{O}_{K}$-algebra. If $R$ is almost flat (resp. flat) over
$\mathcal{O}_{K}$, then the $\pi$-adic completion $\widehat{R}$ is almost flat
(resp. flat) over $\mathcal{O}_{\widehat{K}}$.

###### Proof.

For any integer $n>0$, there is a canonical isomorphism

(5.20.1) $\displaystyle
R/\pi^{n}R\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\widehat{R}/\pi^{n}\widehat{R}.$

Let $x\in\widehat{R}$ be a $\pi$-torsion element. Since any $\pi$-torsion
element of $R$ is almost zero (resp. zero) by 5.12, for any
$\epsilon\in\mathfrak{m}_{K}$ (resp. $\epsilon=1$), the image of $\epsilon x$
in $\widehat{R}/\pi^{n}\widehat{R}$ lies in
$\pi^{n-1}\widehat{R}/\pi^{n}\widehat{R}$. Therefore, $\epsilon
x\in\bigcap_{n>0}\pi^{n-1}\widehat{R}=0$, which amounts to say that
$\widehat{R}$ is almost flat (resp. flat) over $\mathcal{O}_{\widehat{K}}$. ∎

######  Lemma 5.21.

Let $K$ be a pre-perfectoid field, $\pi$ a pseudo-uniformizer of $K$ dividing
$p$ with a $p$-th root $\pi_{1}$ up to a unit, $R$ a flat
$\mathcal{O}_{K}$-algebra. Then, the following conditions are equivalent:

  1. (1)

The Frobenius induces an injection $R/\pi_{1}R\to R/\pi R$.

  2. (2)

For any $x\in R[1/\pi]$, if $x^{p}\in R$, then $x\in R$.

###### Proof.

We follow the proof of [Sch12, 5.7]. Assume first that $R/\pi_{1}R\to R/\pi R$
is injective. Let $x\in R[1/\pi]$ with $x^{p}\in R$, $k$ the minimal natural
number such that $y=\pi_{1}^{k}x\in R$. If $k\geq 1$, then
$y^{p}=\pi_{1}^{pk}x^{p}\in\pi R$. Therefore, $y\in\pi_{1}R$ by the
injectivity of the Frobenius. However, as $R$ is $\pi$-torsion free, we have
$y^{\prime}=y/\pi_{1}=\pi_{1}^{k-1}x\in R$ which contradicts the minimality of
$k$.

Conversely, for any $x\in R$ with $x^{p}\in\pi R$, we have $(x/\pi_{1})^{p}\in
R$. Thus, $x/\pi_{1}\in R$ by assumption, i.e. $x\in\pi_{1}R$, which implies
the injectivity of the Frobenius. ∎

######  Lemma 5.22.

Let $K$ be a pre-perfectoid field, $\pi$ a pseudo-uniformizer of $K$ dividing
$p$ with a $p$-th root $\pi_{1}$ up to a unit, $R$ an
$\mathcal{O}_{K}$-algebra which is almost flat. Then, the following conditions
are equivalent:

  1. (1)

The Frobenius induces an almost injection (resp. almost isomorphism)
$R/\pi_{1}R\to R/\pi R$.

  2. (2)

The Frobenius induces an injection (resp. isomorphism)
$(R^{\mathrm{al}})_{*}/\pi_{1}(R^{\mathrm{al}})_{*}\to(R^{\mathrm{al}})_{*}/\pi(R^{\mathrm{al}})_{*}$.

###### Proof.

We follow the proof of [Sch12, 5.6]. Notice that the Frobenius is compatible
with the functors $\alpha^{*}$ and $\alpha_{*}$ (5.6). (2) $\Rightarrow$ (1)
follows from the almost isomorphism $R\to(R^{\mathrm{al}})_{*}$. The
“injection” part of (1) $\Rightarrow$ (2) follows from the inclusions (5.13)

(5.22.1)
$\displaystyle(R^{\mathrm{al}})_{*}/\pi_{1}(R^{\mathrm{al}})_{*}\subseteq((R/\pi_{1}R)^{\mathrm{al}})_{*},\
(R^{\mathrm{al}})_{*}/\pi(R^{\mathrm{al}})_{*}\subseteq((R/\pi
R)^{\mathrm{al}})_{*}.$

For the “isomorphism” part of (1) $\Rightarrow$ (2), notice that
$(R^{\mathrm{al}})_{*}/\pi_{1}(R^{\mathrm{al}})_{*}\to(R^{\mathrm{al}})_{*}/\pi(R^{\mathrm{al}})_{*}$
is almost surjective. Let $\pi_{2}$ be a $p$-th root of $\pi_{1}$ up to a unit
(5.4). Then, for an element $x$ of $(R^{\mathrm{al}})_{*}$, there exist
elements $y$ and $x^{\prime}$ of $(R^{\mathrm{al}})_{*}$ such that
$\pi_{2}^{p}x=y^{p}+\pi_{2}^{p^{2}}x^{\prime}$. Thus, $x=y^{\prime
p}+\pi_{2}^{p^{2}-p}x^{\prime}$ where
$y^{\prime}=y/\pi_{2}\in(R^{\mathrm{al}})_{*}[1/\pi]$ (as
$(R^{\mathrm{al}})_{*}$ is flat over $\mathcal{O}_{K}$ by 5.12). In fact,
$y^{\prime}$ lies in $(R^{\mathrm{al}})_{*}$ by 5.21 and the “injection” part
of (1) $\Rightarrow$ (2). By applying this process to $x^{\prime}$, there
exist elements $y^{\prime\prime}$ and $x^{\prime\prime}$ of
$(R^{\mathrm{al}})_{*}$ such that $x^{\prime}=y^{\prime\prime
p}+\pi_{2}^{p^{2}-p}x^{\prime\prime}$. In conclusion, we have $x=y^{\prime
p}+\pi_{2}^{p^{2}-p}(y^{\prime\prime
p}+\pi_{2}^{p^{2}-p}x^{\prime\prime})\equiv(y^{\prime}+\pi_{2}^{p-1}y^{\prime\prime})^{p}\
\mod\ \pi(R^{\mathrm{al}})_{*}$, which shows the surjectivity of
$(R^{\mathrm{al}})_{*}/\pi_{1}(R^{\mathrm{al}})_{*}\to(R^{\mathrm{al}})_{*}/\pi(R^{\mathrm{al}})_{*}$.
∎

######  Lemma 5.23.

Let $K$ be a pre-perfectoid field, $R$ an almost flat
$\mathcal{O}_{K}$-algebra, $\pi,\pi^{\prime}$ pseudo-uniformizers dividing $p$
with $p$-th roots $\pi_{1},\pi_{1}^{\prime}$ respectively up to units. Then,
the following conditions are equivalent:

  1. (1)

The Frobenius induces an almost injection (resp. almost surjection)
$R/\pi_{1}R\to R/\pi R$.

  2. (2)

The Frobenius induces an almost injection (resp. almost surjection)
$R/\pi_{1}^{\prime}R\to R/\pi^{\prime}R$.

In particular, the definitions 5.17.(1) and 5.19 do not depend on the choice
of the pseudo-uniformizer.

###### Proof.

Notice that $(R^{\mathrm{al}})_{*}$ is flat over $\mathcal{O}_{K}$ by 5.12.
The “injection” part follows from 5.21 and 5.22. For the “surjection” part, we
assume that $R/\pi_{1}R\to R/\pi R$ is almost surjective. Let
$\epsilon\in\mathfrak{m}_{K}$. We can take a pseudo-uniformizer
$\widetilde{\pi}$ of $K$ dividing $p$ with
$\widetilde{\pi}_{1}^{p}=\widetilde{\pi}$ and
$v_{K}(\pi)/3<v_{K}(\widetilde{\pi})<v_{K}(\pi)/2$. For any $x\in R$, by the
almost surjectivity, we have $\epsilon x=y^{p}+\widetilde{\pi}^{2}z$ for some
$y,z\in R$. We also have $\widetilde{\pi}z=v^{p}+\pi w$ for some $v,w\in R$,
then $\epsilon x=y^{p}+\widetilde{\pi}v^{p}+\widetilde{\pi}\pi w$. Since
$y^{p}+\widetilde{\pi}v^{p}\equiv(y+\widetilde{\pi}_{1}v)^{p}\mod pR$,
$R^{\prime}/\pi_{1}^{\prime}R\to R/\pi^{\prime}R$ is almost surjective for any
pseudo-uniformizer $\pi^{\prime}$ dividing $p$ with
$v_{K}(\pi^{\prime})<4v_{K}(\pi)/3$. By induction, we see that
$R^{\prime}/\pi_{1}^{\prime}R\to R/\pi^{\prime}R$ is almost surjective in
general. ∎

######  Proposition 5.24.

Let $K$ be a pre-perfectoid field of characteristic $p$ with a pseudo-
uniformizer $\pi$, $R$ an $\mathcal{O}_{K}$-algebra, $\widehat{R}$ the
$\pi$-adic completion of $R$. Then, $R$ is almost pre-perfectoid if and only
if $(\widehat{R}^{\mathrm{al}})_{*}$ is perfect.

###### Proof.

Note that $\mathcal{O}_{K}$ is perfect by definition. If $R$ is almost pre-
perfectoid, then $\widehat{R}$ is almost flat so that
$(\widehat{R}^{\mathrm{al}})_{*}$ is $\pi$-adically complete by taking
$M=\widehat{R}^{\mathrm{al}}$ in 5.13. Moreover, the Frobenius induces an
isomorphism
$(\widehat{R}^{\mathrm{al}})_{*}/\pi^{n}(\widehat{R}^{\mathrm{al}})_{*}\to(\widehat{R}^{\mathrm{al}})_{*}/\pi^{pn}(\widehat{R}^{\mathrm{al}})_{*}$
for any integer $n\geq 1$ by 5.22 and 5.23, which implies that
$(\widehat{R}^{\mathrm{al}})_{*}$ is perfect. Conversely, assume that
$(\widehat{R}^{\mathrm{al}})_{*}$ is perfect. For any $\pi$-torsion element
$f\in(\widehat{R}^{\mathrm{al}})_{*}$, we have $\pi^{1/p^{n}}f=0$ for any
integer $n\geq 0$, which shows that $\widehat{R}$ is almost flat by 5.12.
Moreover, it is clear that the Frobenius induces an isomorphism
$(\widehat{R}^{\mathrm{al}})_{*}/\pi(\widehat{R}^{\mathrm{al}})_{*}\to(\widehat{R}^{\mathrm{al}})_{*}/\pi^{p}(\widehat{R}^{\mathrm{al}})_{*}$,
which shows that $R$ is almost pre-perfectoid by 5.22 and 5.23. ∎

######  Proposition 5.25.

Let $K$ be a pre-perfectoid field with a pseudo-uniformizer $\pi$, $R$ an
$\mathcal{O}_{K}$-algebra which is almost flat, $R^{\prime}$ the integral
closure of $R$ in $R[1/\pi]$. If the Frobenius induces an almost injection
$R/\pi_{1}R\to R/\pi R$, then $R\to R^{\prime}$ is an almost isomorphism.

###### Proof.

Since $R\to(R^{\mathrm{al}})_{*}$ is an almost isomorphism, we may replace $R$
by $(R^{\mathrm{al}})_{*}$ so that we may assume that
$R=(R^{\mathrm{al}})_{*}$, $R\subseteq R[1/\pi]$ by 5.12 and for any $x\in
R[1/\pi]$ such that $x^{p}\in R$, then $x\in R$ by 5.21 and 5.22. It suffices
to show that $R$ is integrally closed in $R[1/\pi]$. Suppose that $x\in
R[1/\pi]$ is integral over $R$. There is an integer $N>0$ such that $x^{r}$ is
an $R$-linear combination of $1,x,\dots,x^{N}$ for any $r>0$. Therefore, there
exists an integer $k>0$ such that $\pi^{k}x^{r}\in R$ for any $r>0$. Taking
$r=p^{n}$, we get $x\in\bigcap_{n\geq 0}\pi_{n}^{-1}R=(R^{\mathrm{al}})_{*}=R$
by 5.8, which completes our proof. ∎

######  Lemma 5.26.

Let $R$ be a ring, $\pi$ a nonzero divisor of $R$, $\widehat{R}$ the
$\pi$-adic completion of $R$, $\varphi:R[1/\pi]\to\widehat{R}[1/\pi]$ the
canonical morphism. Then, $\varphi^{-1}(\pi^{n}\widehat{R})=\pi^{n}R$ for any
integer $n$.

###### Proof.

Remark that $\widehat{R}$ is also $\pi$-torsion free by 5.20. For an element
$x/\pi^{k}\in R[1/\pi]$ (where $x\in R$, $k\geq 0$) such that
$\varphi(x/\pi^{k})=\pi^{n}y$ for some $y\in\widehat{R}$. After enlarging $k$,
we may assume that $k+n>0$. Thus, we deduce from the canonical isomorphism
$R/\pi^{k+n}R\to\widehat{R}/\pi^{k+n}\widehat{R}$ that $x\in\pi^{k+n}R$, which