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Cohomological Descent for Faltings Ringed Topos

Published online by Cambridge University Press:  02 April 2024

Tongmu He*
Affiliation:
Institut des Hautes Études Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette, France
*

Abstract

Faltings ringed topos, the keystone of Faltings’ approach to p-adic Hodge theory for a smooth variety over a local field, relies on the choice of an integral model, and its good properties depend on the (logarithmic) smoothness of this model. 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, that is, without any smoothness assumption. An essential ingredient of our proof is a variation of Bhatt–Scholze’s arc-descent of perfectoid rings.

Type
Number Theory
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2024. Published by Cambridge University Press

1 Introduction

1.1. Faltings ringed topos, the keystone of Faltings’ approach to p-adic Hodge theory, was originally introduced by Faltings in his proof of the Hodge–Tate decomposition [Reference FaltingsFal88, Reference FaltingsFal02] and since then became a fundamental tool in p-adic Hodge theory, in particular for p-adic comparison theorems and the p-adic Simpson correspondence (see [Reference FaltingsFal05, Reference Abbes, Gros and TsujiAGT16]). Faltings topos builds on an integral model of the p-adic variety, which has both benefits and limitations. On the one hand, Faltings’ approach uses only standard techniques from scheme theory and seems appropriate for cohomology with integral coefficients. But on the other hand, the (log-)smoothness of the integral model seems necessary for good properties of Faltings topos.

1.2. The goal of this work is to get rid of the (log-)smoothness assumption on integral models for Faltings’ approach to p-adic Hodge theory. For this, we establish a cohomological descent for Faltings ringed topos along a proper hypercovering of integral models, which allows us to descend important results for Faltings topos associated to nice models to Faltings topos associated to general models using alterations of de Jong et al. In order to avoid the complexity raised by proper hypercoverings, we introduce a variant of Faltings site in v-topology (where both faithfully flat morphisms and proper surjective morphisms are coverings), that we call the v-site of integrally closed schemes. We show that both p-adic étale cohomology and the cohomology of Faltings ringed topos can be computed by this new site (see 3.27, 8.12), which automatically implies the cohomological descent for Faltings ringed topos along proper hypercoverings. For this purpose, we prove a variation of Bhatt–Scholze’s arc-descent of perfectoid rings [Reference Bhatt and ScholzeBS22]. More precisely, we prove an almost arc-descent of almost perfectoid algebras (see 5.35), which couldn’t be obtained directly from their results but by adjusting their proof.

1.3. Our cohomological descent result has interesting applications in p-adic Hodge theory. Firstly, I extend in [Reference HeHe21] Faltings’ main p-adic comparison theorem, both in the absolute and the relative cases, to general integral models without any smoothness condition. Notably, the relative comparison takes place in our v-site of integrally closed schemes and remains valid for torsion abelian étale coefficients (not necessarily finite locally constant). Secondly, I deduce in [Reference HeHe21] an explicit local version of the relative Hodge–Tate filtration from the global version constructed by Abbes–Gros. Thirdly, Xu [Reference XuXu22] recently deduced from our cohomological descent a descent result for the p-adic Simpson correspondence. Finally, we would like to mention that our v-site of integrally closed schemes is a scheme theoretic analogue of the v-site of an adic space introduced by Scholze and that our cohomological descent is an analogue of the cohomological descent from the v-topos to the pro-étale topos of an adic space established by Scholze [Reference ScholzeSch17]. The advantage of our v-site is that it remains in the framework of algebraic geometry and uses only scheme theoretic arguments. Moreover, it may lead to an explicit comparison between Faltings and Scholze’s approaches to p-adic Hodge theory.

1.4. In order to state our cohomological descent result, we recall now the definition of the Faltings site associated to a morphism of coherent schemes $Y\to X$ (see 7.7), where ‘coherent’ stands for ‘quasi-compact and quasi-separated’. Let $\mathbf {E}_{Y\to X}^{\mathrm {\acute {e}t}}$ be the category of morphisms of coherent schemes $V\to U$ over $Y\to X$ , that is, commutative diagrams

(1.4.1)

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

(1.4.2) $$ \begin{align} \overline{\mathscr{B}}(V \to U) = \Gamma(U^V , \mathcal{O}_{U^V}), \end{align} $$

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

1.5. 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 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 (see 3.6). We will describe the cohomological descent for $\overline {\mathscr {B}}$ using a new site built from the v-topology defined as follows:

Definition 1.6 (see 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 $\mathbf {I}_{S^{\circ } \to S}$ .

1.7. 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

(1.7.1)

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

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

Our cohomological descent for Faltings ringed topos is stated as follows:

Theorem 1.8 (see 8.14).

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

(1.8.1) $$ \begin{align} \mathbb{L}\otimes_{\mathbb{Z}} \overline{\mathscr{B}}\longrightarrow \mathrm{R}\varepsilon_*(\varepsilon^{-1}\mathbb{L}\otimes_{\mathbb{Z}}{\mathscr{O}}) \end{align} $$

is an almost isomorphism, that is, the cohomology groups of its cone are killed by $p^r$ for any rational number $r>0$ (see 5.7).

Corollary 1.9 (see 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

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

is an almost isomorphism.

The key ingredient of our proof of 1.8 is the almost descent of almost perfectoid algebras in arc-topology (a topology finer than the v-topology) (see 5.35). The analogue in characteristic p of 1.8 is Gabber’s computation of the cohomology of the structural sheaf in h-topology (see Section 4). Theorem 1.8 allows us to descend important results for Faltings sites associated to nice models to Faltings sites associated to general models. On the other hand, one important step of its proof, which has its own interests, is a characterization of ‘acyclic objects’ for Faltings ringed site in terms of almost perfectoid algebras. This result holds in the open case, that is, the complement of a normal crossings divisor in the generic fibre, using Abhyankar’s lemma (see 8.24).

1.10. The paper is structured as follows. In Section 3, we establish the foundation of the v-site $\mathbf {I}_{S^{\circ } \to S}$ of integrally closed schemes, where Proposition 3.27 proves that the étale cohomology of $S^{\circ }$ can be computed by this v-site. Sections 4 and 5 are devoted to a detailed proof of the almost arc-descent for almost perfectoid algebras. 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 include some preliminaries about Faltings sites and we introduce a pro-version of Faltings site to evaluate the structural sheaf on the spectrums of pre-perfectoid algebras. Finally, we prove our cohomological descent results in Section 8.

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 ([Sta23, 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 ([Reference BourbakiBou06, 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 ([Sta23, 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 [Reference Artin, Grothendieck and VerdierSGA 4II , 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 ([Sta23, 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 $\mathbb {V}$ ([Reference Artin, Grothendieck and VerdierSGA 4I , 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 ([Reference Artin, Grothendieck and VerdierSGA 4I , 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 embedding ([Reference Artin, Grothendieck and VerdierSGA 4I , I.1.3]) and by $\widehat {C} \to \widetilde {C}$ , $\mathcal {F} \mapsto \mathcal {F}^{\mathrm {a}}$ the sheafification functor ([Reference Artin, Grothendieck and VerdierSGA 4I , II.3.4]).

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 $u_{\mathrm {p}}$ [Sta23, 00VC] and a right adjoint ${}_{\mathrm {p}}u$ [Sta23, 00XF] (cf. [Reference Artin, Grothendieck and VerdierSGA 4I , I.5]). So we have a sequence of adjoint functors

(2.5.1) $$ \begin{align} u_{\mathrm{p}},\ u^{\mathrm{p}},\ {}_{\mathrm{p}}u. \end{align} $$

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. If finite limits are representable in C and D and if $u^+$ is left exact and continuous, then $u^+$ gives a morphism of sites $u:D\to C$ ([Reference Artin, Grothendieck and VerdierSGA 4I , IV.4.9.2]) and we also denote by

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

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 sites $u:(D,\mathcal {O}_D)\to (C,\mathcal {O}_C)$ , then we denote by $u^{*}=\mathcal {O}_D\otimes _{u^{-1}\mathcal {O}_C}u^{-1}$ the pullback functor of modules. We remark that the notation here, adopted by [Sta23], is slightly different with that in [Reference Artin, Grothendieck and VerdierSGA 4I ] (see [Sta23, 0CMZ]).

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. [Sta23, 0ETN])

    (3.1.1)
  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 (3.1.1) (cf. [Reference Bhatt and MathewBM21, 1.2], [Reference Cesnavicius and ScholzeCS19, 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. [Sta23, 0ETS]).

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

Lemma 3.2. Let $Z\stackrel {g}{\longrightarrow }Y \stackrel {f}{\longrightarrow }X$ 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 [Reference Artin, Grothendieck and VerdierSGA 4I , 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 $\mathbf {Sch}^{\mathrm {coh}}$ , that is, 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 ([Reference Artin, Grothendieck and VerdierSGA 4I , 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 $Y' \to Y$ of finite presentation and a finite affine open covering $Y' = \bigcup _{r=1}^n Y^{\prime }_r$ 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 $\psi _{\lambda '\lambda }: X_{\lambda '} \to X_{\lambda }$ and $\phi _{\lambda '\lambda }: Y_{\lambda '} \to Y_{\lambda }$ 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 and is particularly an arc-covering.

  5. (5) Let $\pi '$ be another element of $\Gamma (Y,\mathcal {O}_Y)$ which divides $\pi $ . If f is a $\pi $ -complete arc-covering, then it is a $\pi '$ -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 [Sta23, 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 [Reference GrothendieckEGA 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 $Y^{\prime }_r\to Y' \to Y$ 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 Equation (3.1.1) 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 $W' = (W/\mathfrak {p})_{\mathfrak {q}}$ over V is an extension of valuation rings of height $\leq 1$ . Since $\pi \in \mathfrak {m}$ and $W'$ is of height $\leq 1$ , the $\pi $ -adic completion $\widehat {W'}$ is still a valuation ring extension of V of height $\leq 1$ (see [Reference BourbakiBou06, VI.§5.3, Prop.5]), which proves (4) (as arc-coverings are just $0$ -complete arc-coverings).

(5) Since a $\pi '$ -adically complete valuation ring V is also $\pi $ -adically complete ([Sta23, 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 $\pi '$ -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 $(\mathbf {Sch}^{\mathrm {fp}}_{/X})_{\mathrm {h}}\stackrel {\xi ^+}{\longrightarrow }(\mathbf {Sch}^{\mathrm {coh}}_{/X})_{\mathrm {h}}\stackrel {\zeta ^+}{\longrightarrow }(\mathbf {Sch}^{\mathrm {coh}}_{/X})_{\mathrm {v}}$ define morphisms of sites (2.5)

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

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 ([Sta23, 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 ([Sta23, 09MV]). By [Reference GrothendieckEGA 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 $\lambda '=(\alpha ',\beta ')$ , $\lambda =(\alpha ,\beta )$ , the set $\mathrm {Hom}_{\Lambda ^{\mathrm {op}}}(\lambda ',\lambda )$ is

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

  2. (ii) empty, if else.

The composition of morphisms $(g_i:Y_{i\alpha "\beta "}\to Y_{i\alpha '\beta "})_{i\in I}$ with $(f_i:Y_{i\alpha '\beta '}\to Y_{i\alpha \beta '})_{i\in I}$ in $\Lambda ^{\mathrm {op}}$ is $(g_i\circ f^{\prime }_i: Y_{i\alpha "\beta "}\to Y_{i\alpha \beta "})$ , where $f^{\prime }_i$ is the base change of $f_i$ by the transition morphism $Y_{\beta "}\to Y_{\beta '}$ . We see that $\Lambda ^{\mathrm {op}}$ is cofiltered by [Reference GrothendieckEGA 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 ([Reference Artin, Grothendieck and VerdierSGA 4I , I.8.1.3]). After replacing $\Lambda $ by a directed set ([Sta23, 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 [Reference Artin, Grothendieck and VerdierSGA 4I , I.3.4]

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

Thus, we may assume that $\mathcal {F}$ is representable by a finitely presented X-scheme $Y'$ since the section functor $\Gamma (Y,-)$ commutes with colimits of presheaves ([Sta23, 00VB]). Then, we have

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

where the first equality follows from [Sta23, 04D2], and the third equality follows from [Reference GrothendieckEGA 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) $$ \begin{align} H^q_{\tau}(Y, \nu^{-1}\mathcal{F})=\mathop{\mathrm{colim}} H^q((\mathbf{Sch}^{\mathrm{fp}}_{/Y_{\lambda}})_{\mathrm{h}},\mathcal{F}). \end{align} $$

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 Equation (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 Equation (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

(3.8.2)

If Equation (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 ([Sta23, 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 $\nu _{\mathrm {p}}\mathcal {I}$

(3.8.3) $$ \begin{align} \nu_{\mathrm{p}}\mathcal{I}(Y) \to \prod_{i \in I}\nu_{\mathrm{p}}\mathcal{I}(Y_i) \to \prod_{i,j \in I} \nu_{\mathrm{p}}\mathcal{I}(Y_i \times_{Y} Y_j)\to \cdots \end{align} $$

is exact. Admitting this claim, we see that $\nu _{\mathrm {p}}\mathcal {I}$ is indeed a sheaf, that is, $\nu ^{-1}\mathcal {I}=\nu _{\mathrm {p}}\mathcal {I}$ , and the vanishing of higher Čech cohomologies implies that $H^q_{\tau }(Y,\nu ^{-1}\mathcal {I})=0$ for $q>0$ by 3.6 ([Sta23, 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 (3.8.3) is the filtered colimit of the augmented Čech complexes

(3.8.4) $$ \begin{align} \mathcal{I}(Y_{\lambda}) \to \prod_{i \in I}\mathcal{I}(Y_{i\lambda}) \to \prod_{i,j \in I} \mathcal{I}(Y_{i\lambda} \times_{Y_{\lambda}} Y_{j\lambda})\to \cdots, \end{align} $$

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

Notice that the morphism $\mathcal {F} \to \mathrm {R} (\mu \circ \xi )_*(\mu \circ \xi )^{-1}\mathcal {F}$ is an isomorphism by [Sta23, 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

(3.10.1)

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

(3.10.2) $$ \begin{align} a_Y^{-1}\mathrm{R} f_{\mathrm{\acute{e}t} *} \mathcal{F} \longrightarrow \mathrm{R} f_{\mathrm{v}*} a_X^{-1}\mathcal{F} \end{align} $$

is an isomorphism.

Proof. Consider the commutative diagram

(3.10.3)

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 [Sta23, 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 $Y'$ be a coherent Y-scheme, and we set $g:X'=Y'\times _Y X \to X$ . 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 $Y' \mapsto H^q_{\mathrm {h}}(X', b_{X'}^{-1}g^{-1}_{\mathrm {\acute {e}t}}\mathcal {F})=H^q_{\mathrm {\acute {e}t}}(X', g^{-1}_{\mathrm {\acute {e}t}}\mathcal {F})$ by [Sta23, 0EWH], and $\mathrm {R}^q f_{\mathrm {v}*} a_X^{-1}\mathcal {F}$ is the sheaf associated to the presheaf $Y'\mapsto H^q_{\mathrm {v}}(X', a_{X'}^{-1}g^{-1}_{\mathrm {\acute {e}t}}\mathcal {F})=H^q_{\mathrm {\acute {e}t}}(X', g^{-1}_{\mathrm {\acute {e}t}}\mathcal {F})$ 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 [Sta23, 092T], and (2) follows from the proof of [Reference Bhatt and ScholzeBS17, 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 [Reference GrothendieckEGA 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 ([Reference GrothendieckEGA II, 7.1.2]). We denote by $\kappa (y)$ the residue field of the point y. Let $V'$ be a valuation ring extension of V with fraction field $\kappa (y)$ , and let W be the preimage of $V'$ 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 $W/\mathfrak {p} = V'$ . 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) ([Sta23, 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

(3.15.1)

Assume the following conditions hold:

  1. (1) $Y \to Z$ is dominant and $Y' \to Y \times _X X'$ is surjective.

  2. (2) $Z \to X$ is separated, $Z'\to Z$ is quasi-compact and $Z' \to X'$ is integral.

  3. (3) For any valuation ring W and any morphism $\mathop {\mathrm {Spec}}(W) \to X$ such that the generic point of $\mathop {\mathrm {Spec}}(W)$ lies over Y, there exists an extension of valuation rings $W\to W'$ and a commutative diagram

    (3.15.2)

Then, $Z' \to Z$ is a v-covering.

Proof. Notice that $Z'\to Z\times _X X'$ is still integral as $Z \to X$ is separated. After replacing $X' \to X$ by $Z\times _X X' \to Z$ , 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 ([Sta23, 00IA]), we may assume that there exists a lifting $\xi \to Y$ of $\xi \to Z$ . Thus, by assumption (3.15), the morphism $\mathop {\mathrm {Spec}}(W) \to Z = X $ admits a lifting $\mathop {\mathrm {Spec}}(W') \to X'$ , where $W\to W'$ is an extension of valuation rings. We claim that after extending $W'$ , $\mathop {\mathrm {Spec}}(W') \to X'$ factors through $Z'$ . Indeed, if $\xi '$ denotes the generic point of $\mathop {\mathrm {Spec}}(W')$ , as $Y' \to Y \times _XX'$ is surjective, after extending $W'$ , we may assume that there exists an $X'$ -morphism $\xi ' \to Y'$ which is over $\xi \to Y$ . Since $\mathop {\mathrm {Spec}}(W')$ is integrally closed in $\xi '$ and $Z'$ is integral over $X'$ , the morphism $\mathop {\mathrm {Spec}}(W') \to X'$ factors through $Z'$ ([Sta23, 035I]). Finally, let $\mathfrak {q} \in \mathop {\mathrm {Spec}}(W')$ which lies over $\mathfrak {p} \in \mathop {\mathrm {Spec}}(W)$ , then we get a lifting $\mathop {\mathrm {Spec}}(W'/\mathfrak {q}) \to Z'$ of $\mathop {\mathrm {Spec}}(V) \to Z$ , which shows that $Z' \to Z$ is a v-covering (as we assume that $Z'\to Z$ is quasi-compact, cf. 3.1).

Definition 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 [Sta23, 03GV].

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

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 $Y_{\lambda '}\to Y_{\lambda }$ and $X_{\lambda '}\to X_{\lambda }$ ( $\lambda '\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 $B=\Gamma (Y, \mathcal {O}_{Y})$ ([Sta23, 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 $X'$ is a coherent smooth X-scheme, then $X'$ 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 [Sta23, 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) $$ \begin{align} \Phi^+: \mathbf{I}_{S^{\circ} \to S} \longrightarrow \mathbf{Sch}^{\mathrm{coh}}_{/S},\ X \longmapsto X, \end{align} $$

admits a right adjoint

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

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) $$ \begin{align} \Psi^+: \mathbf{I}_{S^{\circ} \to S} \longrightarrow \mathbf{Sch}^{\mathrm{coh}}_{/S^{\circ}},\ X \longmapsto X^{\circ}, \end{align} $$

admits a left adjoint

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

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) $$ \begin{align} X_1 \overline{\times}_{X_0} X_2 = (X_1 \times_{X_0} X_2)^{X_1^{\circ} \times_{X_0^{\circ}}X_2^{\circ}}. \end{align} $$

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 $X' \to X$ of $\mathbf {I}_{S^{\circ } \to S}$ , the base change $\{X_i^{\prime } \to X'\}_{i \in I}$ also lies in $\mathscr {C}$ , where $Z_i = X_i \times _X X'$ and $X_i^{\prime }=Z_i^{Z_i^{\circ }}$ by 3.21. Applying 3.15 to the following diagram

(3.22.1)

we deduce that $\coprod _{i \in I} X_i^{\prime } \to X'$ 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 $f:X'\to X$ 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 $h_{X'}^{\mathrm {a}} \to h_{X}^{\mathrm {a}}$ is an isomorphism.

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

(3.24.1) $$ \begin{align} \mathcal{F}(X) \to \mathcal{F}(X') \rightrightarrows \mathcal{F}(X'\overline{\times}_X X')\to \mathcal{F}(X'). \end{align} $$

The right arrow is injective as $X' \to X'\overline {\times }_X X'$ 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.

Assume moreover that

  1. (iii) $\mathcal {F}_i$ sends finite coproducts to finite products for $i=1,2$ .

Then, for any product A of absolutely integrally closed valuation rings such that $\mathop {\mathrm {Spec}}(A)$ is an $S^{\circ }$ -integrally closed S-scheme, the map $\mathcal {F}_1(\mathop {\mathrm {Spec}}(A))\to \mathcal {F}_2(\mathop {\mathrm {Spec}}(A))$ is bijective.

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 ([Sta23, 04PP]). Moreover, since A is reduced, we have $V = \mathop {\mathrm {colim}} A'$ , where the colimit is taken over all the open and closed subschemes $X'=\mathop {\mathrm {Spec}}(A')$ of X which contain $\mathop {\mathrm {Spec}}(V)$ . By assumptions (i) and (ii), we have an isomorphism

(3.25.1) $$ \begin{align} \mathop{\mathrm{colim}} \mathcal{F}_1(X') \stackrel{\sim}{\longrightarrow} \mathop{\mathrm{colim}} \mathcal{F}_2(X'). \end{align} $$

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 Equation (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 each $\mathcal {F}_1(X_i^{\prime })$ are the same. Therefore, $\mathcal {F}_1^{\mathrm {a}}\to \mathcal {F}_2^{\mathrm {a}}$ is injective by 3.14. Moreover, under the assumption (iii), we have $\xi _1=\xi _1^{\prime }$ in $\mathcal {F}_1(X)=\prod _{i=1}^r\mathcal {F}_1(X_i^{\prime })$ (so that $\mathcal {F}_1(X)\to \mathcal {F}_2(X)$ is injective).

On the other hand, for an element $\xi _2 \in \mathcal {F}_2(X)$ , by Equation (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. Moreover, under the assumption (iii), let $\xi _1$ be the section $(\xi _{1,i})_{1\leq i\leq r}\in \mathcal {F}_1(X)=\prod _{i=1}^r\mathcal {F}_1(X_i^{\prime })$ . Then, $\alpha _X(\xi _1)=\xi _2\in \mathcal {F}_2(X)=\prod _{i=1}^r\mathcal {F}_2(X_i^{\prime })$ (so that $\mathcal {F}_1(X)\to \mathcal {F}_2(X)$ is surjective).

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) $$ \begin{align} (\mathbf{Sch}^{\mathrm{coh}}_{/S^{\circ}})_{\mathrm{v}}\stackrel{\Psi}{\longrightarrow}\mathbf{I}_{S^{\circ} \to S} \stackrel{\sigma}{\longrightarrow} (\mathbf{Sch}^{\mathrm{coh}}_{/S})_{\mathrm{v}}. \end{align} $$

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 [Reference Artin, Grothendieck and VerdierSGA 4II , 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 [Reference Artin, Grothendieck and VerdierSGA 4II , 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) $$ \begin{align} R_{\mathrm{perf}} = \mathop{\mathrm{colim}}_{\mathrm{Frob}} R \end{align} $$

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) $$ \begin{align} \mathcal{O}_{\mathrm{perf}}(X) = \Gamma(X,\mathcal{O}_X)_{\mathrm{perf}}. \end{align} $$

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) $$ \begin{align} 0 \to A_{\mathrm{perf}} \to B_{\mathrm{perf}} \to B_{\mathrm{perf}} \otimes_{A_{\mathrm{perf}}} B_{\mathrm{perf}} \to \cdots. \end{align} $$

Lemma 4.3 [Sta23, 0EWT].

The presheaf $\mathcal {O}_{\mathrm {perf}}$ is a sheaf on $\mathbf {Sch}^{\mathrm {coh}}_{\mathbb {F}_p}$ with respect to the fppf topology ([Sta23, 021L]). Moreover, for any coherent $\mathbb {F}_p$ -scheme X and any integer q,

(4.3.1) $$ \begin{align} H^q_{\mathrm{fppf}}(X, \mathcal{O}_{\mathrm{perf}}) = \mathop{\mathrm{colim}}_{\mathrm{Frob}} H^q(X,\mathcal{O}_X). \end{align} $$

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 ([Sta23, 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 ([Sta23, 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 Equation (4.2.2) follows from the Čech-cohomology-to-cohomology spectral sequence associated to the $\tau $ -covering $\mathop {\mathrm {Spec}}(B) \to \mathop {\mathrm {Spec}}(A)$ [Sta23, 03AZ]. Therefore, (1) and (2) hold for $\tau =\textrm {fppf}$ by 4.3. Conversely, the exactness of Equation (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. [Sta23, 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 $q>0$ ([Sta23, 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 ([Sta23, 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; see [Reference Bhatt, Schwede and TakagiBST17, 3.3] or [Sta23, 0EWU], and 4.4.

Lemma 4.6 [Reference Bhatt and ScholzeBS17, 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 [Reference Bhatt and ScholzeBS17, 6.3].

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

(4.7.1) $$ \begin{align} 0 \longrightarrow V \stackrel{\alpha}{\longrightarrow} V/\mathfrak{p} \oplus V_{\mathfrak{p}} \stackrel{\beta}{\longrightarrow} V_{\mathfrak{p}}/\mathfrak{p}V_{\mathfrak{p}} \longrightarrow 0 \end{align} $$

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 Equation (4.7.1) by $V\to R$ ,

(4.7.2) $$ \begin{align} 0 \longrightarrow R \longrightarrow R/\mathfrak{p}R \oplus R_{\mathfrak{p}} \longrightarrow R_{\mathfrak{p}}/\mathfrak{p}R_{\mathfrak{p}} \longrightarrow 0 \end{align} $$

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 ([Reference Bhatt and ScholzeBS17, 3.16]).

Lemma 4.8 [Reference Bhatt and MathewBM21, 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 [Reference Bhatt and MathewBM21, 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 ([Reference Bhatt and ScholzeBS17, 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) $$ \begin{align} A_{\mathfrak{p}}/\mathfrak{p}A_{\mathfrak{p}} = \mathop{\mathrm{colim}}_{I=[\mathfrak{p}, \mathfrak{q}],\mathfrak{q} \supsetneq \mathfrak{p}} A_I. \end{align} $$
    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 $\mathcal {P} = \mathcal {I}$ (see [Reference Bhatt and MathewBM21, 4.7]).

Lemma 4.9 (cf. [Reference Bhatt and MathewBM21, 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) $$ \begin{align} 0 \to V_{\mathrm{perf}} \to (B\otimes_A V)_{\mathrm{perf}} \to (B\otimes_A V)_{\mathrm{perf}}\otimes_{V_{\mathrm{perf}}}(B\otimes_A V)_{\mathrm{perf}} \to \cdots \end{align} $$

is exact. Notice that each connected component of an affine scheme is the intersection of some open and closed subsets ([Sta23, 04PP]). Moreover, since A is reduced, we have $V = \mathop {\mathrm {colim}} A'$ , where the colimit is taken over all the open and closed subschemes $\mathop {\mathrm {Spec}}(A')$ 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) $$ \begin{align} \otimes^{n}_{A_{\mathrm{perf}}}B_{\mathrm{perf}} = \prod_{i=1}^N \otimes^{n}_{A_{i,\mathrm{perf}}}(B \otimes_A {A_i})_{\mathrm{perf}}, \end{align} $$

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

Proposition 4.10 [Reference Bhatt and ScholzeBS22, 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) $$ \begin{align} H^q_{\tau}(X, \mathcal{O}_{\mathrm{perf}}) = \mathop{\mathrm{colim}}_{\mathrm{Frob}} H^q(X,\mathcal{O}_X). \end{align} $$
  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) $$ \begin{align} 0 \to A_{\mathrm{perf}} \to B_{\mathrm{perf}} \to B_{\mathrm{perf}} \otimes_{A_{\mathrm{perf}}} B_{\mathrm{perf}} \to \cdots \end{align} $$
    is exact.

Proof. We follow closely the proof of Bhatt–Scholze [Reference Bhatt and ScholzeBS22, 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) $$ \begin{align} H^0_{\mathrm{v}}(\mathop{\mathrm{Spec}}(A),\mathcal{O}_{\mathrm{perf}}) = A_{\mathrm{perf}} \textrm{ and } H^q_{\mathrm{v}}(\mathop{\mathrm{Spec}}(A),\mathcal{O}_{\mathrm{perf}}) = 0, \ \forall q>0. \end{align} $$

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 [Sta23, 094K and 0DB1]. The associated sequence

(4.10.4) $$ \begin{align} 0 \to A_{\mathrm{perf}} \to A_{0,\mathrm{perf}} \to A_{1,\mathrm{perf}} \to \cdots \end{align} $$

is exact by the hyper-Čech-cohomology-to-cohomology spectral sequence [Sta23, 01GY].

Consider the double complex $(A_i^j)$ where the i-th row $A_{i}^{\bullet }$ is the base change of Equation (4.10.2) by $A_{\mathrm {perf}} \to A_{i,\mathrm {perf}}$ , that is, 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 ([Sta23, 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 nondiscrete, 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. [Reference ScholzeSch12, 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 nondiscrete valuation ring of height $1$ with residue characteristic p (see [Reference BourbakiBou06, VI.§5.3, Prop.5]). If $p\neq 0$ in $\mathcal {O}_K$ , then it is canonically isomorphic to the p-adic completion of $\mathcal {O}_K$ so that there is a canonical isomorphism $\mathcal {O}_K/p\mathcal {O}_K \stackrel {\sim }{\longrightarrow } \widehat {\mathcal {O}_K}/p\widehat {\mathcal {O}_K}$ , from which we see that $\widehat {K}$ is a perfectoid field. If $p=0$ in $\mathcal {O}_K$ , 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 $\mathfrak {m}_K\setminus 0= v_K^{-1}(\mathbb {R}_{> 0})$ (see [Reference BourbakiBou06, VI.§4.5 Prop.7] and [Reference BourbakiBou07, V.§2 Prop.1, Rem.2]). The nondiscrete 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 [Reference ScholzeSch12, 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 $\{\pi _n\}_{n\geq 0}$ . We call $\pi _{n+1}$ a p-th root of $\pi _n$ up to a unit for simplicity.

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 [Reference Abbes and GrosAG20, 2.6], [Reference Abbes, Gros and TsujiAGT16, V] and [Reference Gabber and RameroGR03]. 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 $\mathcal {O}_K\textrm {-}\mathbf {Mod}$ ([Sta23, 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. [Sta23, 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. [Reference Abbes and GrosAG20, 2.6.2]). We denote by

(5.5.1) $$ \begin{align} \alpha^{*}: \mathcal{O}_K\textrm{-}\mathbf{Mod} \longrightarrow \mathcal{O}_K^{\mathrm{al}}\textrm{-}\mathbf{Mod},\ M \longmapsto M^{\mathrm{al}} \end{align} $$

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 ([Reference Abbes and GrosAG20, 2.6.7.1])

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

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

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

and a left adjoint

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

Moreover, the natural morphisms

(5.5.5) $$ \begin{align} (M_*)^{\mathrm{al}} \stackrel{\sim}{\longrightarrow} M,\ M \stackrel{\sim}{\longrightarrow} (M_!)^{\mathrm{al}} \end{align} $$

are isomorphisms for any $\mathcal {O}_K^{\mathrm {al}}$ -module M (cf. [Reference Abbes and GrosAG20, 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) $$ \begin{align} \mathop{\mathrm{colim}} M_i = (\mathop{\mathrm{colim}} M_{i*})^{\mathrm{al}},\ \lim M_i = (\lim M_{i*})^{\mathrm{al}}. \end{align} $$

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) $$ \begin{align} M^{\mathrm{al}} \otimes_{\mathcal{O}_K^{\mathrm{al}}} N^{\mathrm{al}} = (M\otimes_{\mathcal{O}_K} N)^{\mathrm{al}} \end{align} $$

making $\mathcal {O}_K^{\mathrm {al}}\textrm {-}\mathbf {Mod}$ an abelian tensor category ([Reference Abbes and GrosAG20, 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. [Reference Abbes and GrosAG20, 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 $\mathcal {O}_K^{\mathrm {al}}\textrm {-}\mathbf {Alg}$ ([Reference Abbes and GrosAG20, 2.6.12]). Combining with Equations (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. [Reference Gabber and RameroGR03, 2.2.16])

(5.5.8) $$ \begin{align} \mathop{\mathrm{colim}} R_i = (\mathop{\mathrm{colim}} R_{i*})^{\mathrm{al}},\ \lim R_i = (\lim R_{i*})^{\mathrm{al}}. \end{align} $$

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) $$ \begin{align} B\otimes_A C = (B_* \otimes_{A_*} C_*)^{\mathrm{al}}, \end{align} $$

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 (5.4). The Frobenius on $\mathcal {O}_K/\pi \mathcal {O}_K$ induces an isomorphism $\mathcal {O}_K/\pi _1\mathcal {O}_K \stackrel {\sim }{\longrightarrow } \mathcal {O}_K/\pi \mathcal {O}_K$ . 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.

(5.6.1)

For an $(\mathcal {O}_K/\pi )^{\mathrm {al}}$ -algebra R, we usually identify the $(\mathcal {O}_K/\pi _1)^{\mathrm {al}}$ -algebra $R/\pi _1R$ 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