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 4_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 ([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 4_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 ([Reference Artin, Grothendieck and VerdierSGA 4_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 embedding ([Reference Artin, Grothendieck and VerdierSGA 4_I , 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 4_I , 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 4_I , 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 4_I , 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 4_I ] (see [Sta23, 0CMZ]).

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 4_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 $\mathbf {Sch}^{\mathrm {coh}}$ , 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 4_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 $H^q_{\tau }(X, \mathcal {F})$ .

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

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 4_I , 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.

Proof. Notice that the presheaf $\mathcal {F}$ is a filtered colimit of representable presheaves by [Reference Artin, Grothendieck and VerdierSGA 4_I , 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].

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 $(\mathbf {Sch}^{\mathrm {fp}}_{/X})_{\mathrm {h}}$ .

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.

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.

Proof. (1) is proved in [Sta23, 092T], and (2) follows from the proof of [Reference Bhatt and ScholzeBS17, 6.2].

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.

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.

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 $\mathop {\mathrm {Spec}}(W/\mathfrak {p}) \to \mathop {\mathrm {Spec}}(V_{\lambda }) \to \mathop {\mathrm {Spec}}(A)$ .

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).

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 $R=\mathop {\mathrm {colim}} R_{\lambda }$ .

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

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

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.

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 $\mathcal {F}$ .

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

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 4_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 [Reference Artin, Grothendieck and VerdierSGA 4_II , VII.5.8].

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

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]).

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.

Proof. This is a result of Gabber; see [Reference Bhatt, Schwede and TakagiBST17, 3.3] or [Sta23, 0EWU], and 4.4.

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.

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]).

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]).

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).

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.

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 $ .

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 $R/\pi _1R\to R$ the natural morphism $(\mathcal {O}_K/\pi )\otimes _{\mathrm {Frob},(\mathcal {O}_K/\pi )} R\to R$ induced by the Frobenius (cf. [Reference Gabber and RameroGR03, 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

(5.6.2)

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. [Reference Abbes and GrosAG20, 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 ([Sta23, 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 ([Sta23, 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 $\mathcal {S}^{-1}\mathbf {D}(\mathcal {O}_K\textrm {-}\mathbf {Mod}_C)$ ([Sta23, 05RI]). The natural functor

(5.7.1) $$ \begin{align} \mathcal{S}^{-1}\mathbf{D}(\mathcal{O}_K\textrm{-}\mathbf{Mod}_C) \longrightarrow \mathbf{D}(\mathcal{O}_K^{\mathrm{al}}\textrm{-}\mathbf{Mod}_C) \end{align} $$

is an equivalence by [Sta23, 06XM] and Equation (5.5.5) (cf. [Reference Gabber and RameroGR03, 2.4.9]).

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 $R = \bigcap _{n\geq 0} \pi _n^{-1}R$ .

Proof. There are natural maps

(5.9.2) $$ \begin{align} \mathrm{Hom}_{\mathcal{O}_K\textrm{-}\mathbf{Alg}} (R, V) \to \mathrm{Hom}_{\mathcal{O}_K^{\mathrm{al}}\textrm{-}\mathbf{Alg}}(R^{\mathrm{al}}, V^{\mathrm{al}}) \stackrel{\sim}{\longrightarrow} \mathrm{Hom}_{\mathcal{O}_K\textrm{-}\mathbf{Alg}}(R, (V^{\mathrm{al}})_*) \stackrel{\sim}{\longrightarrow} \mathrm{Hom}_{\mathcal{O}_K\textrm{-}\mathbf{Alg}}(R, V), \end{align} $$

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.

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) $$ \begin{align} \mathrm{Hom}_{\mathcal{O}_K^{\mathrm{al}}\textrm{-}\mathbf{Mod}}(M\otimes_{\mathcal{O}_K^{\mathrm{al}}} L_1, L_2) = \mathrm{Hom}_{\mathcal{O}_K^{\mathrm{al}}\textrm{-}\mathbf{Mod}}(L_1,\mathrm{Hom}_{\mathcal{O}_K\textrm{-}\mathbf{Mod}} (M_*, L_{2*})^{\mathrm{al}}) \end{align} $$

by Equations (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) $$ \begin{align} 0 \longrightarrow \mathcal{O}_K^{\mathrm{al}} \stackrel{\cdot \pi}{\longrightarrow} \mathcal{O}_K^{\mathrm{al}} \longrightarrow (\mathcal{O}_K/\pi\mathcal{O}_K)^{\mathrm{al}} \longrightarrow 0, \end{align} $$

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.

Proof. We follow the proof of [Reference ScholzeSch12, 5.3]. Applying the left exact functor $\alpha _*$ to the exact sequence

(5.13.2)

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 Equations (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 $b\equiv (\epsilon /\pi _n)\cdot f(\pi _n)\ \mod \epsilon x M_*$ 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, $\pi _n\cdot (b/\epsilon ) \equiv f(\pi _n)\ \mod x M_*$ 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 $\alpha ^{*}$ ) ([Sta23, 0596]).

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.