2.1 We fix a prime number p throughout this paper. 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, Theorem 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 [SGA 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 $\mathbb {V}$ [SGA 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 [SGA 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 [SGA 4_I , I.1.3], and by $\widehat {C} \to \widetilde {C}$ , $\mathcal {F} \mapsto \mathcal {F}^{\mathrm {a}}$ the sheafification functor [SGA 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. [SGA 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 $u:D\to C$ [SGA 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 from that in [SGA 4_I ] (see [Sta23, 0CMZ]).

3.1 Firstly, we recall the definition of the Faltings site associated with a morphism of coherent schemes $Y\to X$ (see [Reference HeHe23, 7.7]). Let $\mathbf {E}_{Y\to X}^{\mathrm {\acute {e}t}}$ be the category of morphisms of coherent schemes $V\to U$ over $Y\to X$ , i.e., commutative diagrams

(3.1.1)

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

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

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 [Reference HeHe23, 7.6]).

3.2 Let $Y \to X$ be a morphism of coherent schemes. The natural left exact and continuous functors

(3.2.1) $$ \begin{align} \psi^+ &: \mathbf{E}_{Y\to X}^{\mathrm{\acute{e}t}} \longrightarrow Y_{\mathrm{\acute{e}t}},\ (V \to U) \longmapsto V, \end{align} $$

(3.2.2) $$ \begin{align} \beta^+ &: Y_{\mathrm{f\acute{e}t}} \longrightarrow \mathbf{E}_{Y\to X}^{\mathrm{\acute{e}t}},\ V \longmapsto (V \to X), \end{align} $$

induce a natural commutative diagram of sites (2.5),

(3.2.3)

where $\rho : Y_{\mathrm {\acute {e}t}} \to Y_{\mathrm {f\acute {e}t}}$ is defined by the inclusion functor (see [Reference HeHe23, 7.8]).

3.3 Recall that 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 $\mathop {\mathrm {Spec}}(B)\to T$ [Reference HeHe23, 3.1(1)]. Let $\mathbf {Sch}^{\mathrm {coh}}$ be the category of coherent schemes. We endow it with the topology generated by the pretopology formed by families of morphisms $\{S_i \to S\}_{i \in I}$ with I finite such that $\coprod _{i \in I} S_i \to S$ is a v-covering, and we denote the corresponding site by $\mathbf {Sch}^{\mathrm {coh}}_{\mathrm {v}}$ [Reference HeHe23, 3.3]. For any object S of $\mathbf {Sch}^{\mathrm {coh}}_{\mathrm {v}}$ , we denote by $(\mathbf {Sch}^{\mathrm {coh}}_{/S})_{\mathrm {v}}$ the localization of the v-site $\mathbf {Sch}^{\mathrm {coh}}_{\mathrm {v}}$ at S. The cohomological descent for étale cohomology can be stated as follows:

3.6 Let $Y \to X$ be a morphism of coherent schemes such that $Y \to X^Y$ is an open immersion, where $X^Y$ denotes the integral closure of X in Y (2.2). The natural left exact and continuous functors

(3.6.1) $$ \begin{align} \Psi^+:\mathbf{I}_{Y \to X^Y}&\longrightarrow (\mathbf{Sch}^{\mathrm{coh}}_{/Y})_{\mathrm{v}},\ Z\longmapsto Y\times_{X^Y}Z, \end{align} $$

(3.6.2) $$ \begin{align} \varepsilon^+:\mathbf{E}_{Y \to X}^{\mathrm{\acute{e}t}}&\longrightarrow \mathbf{I}_{Y \to X^Y},\ (V\to U)\longmapsto U^V, \end{align} $$

induce natural morphisms of sites (see [Reference HeHe23, 3.26, 8.6])

(3.6.3)

Combining with the morphisms of sites defined in 3.2 and 3.4, we obtain the following natural commutative diagram of sites:

(3.6.4)

Moreover, $\varepsilon ^+$ actually defines a morphism of ringed sites (see [Reference HeHe23, 8.6])

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

Proof The statement for $\mathbf {E}_{Y \to X^Y}^{\mathrm {\acute {e}t}}$ is proved in [Reference HeHe23, 7.9]. The same arguments work for $\mathbf {I}_{Y \to X^Y}$ .

4.1 We only consider logarithmic structures in étale topology. More precisely, let X be a scheme, let $X_{\mathrm {\acute {e}t}}$ be the étale site of X, let $\mathcal {O}_{X_{\mathrm {\acute {e}t}}}$ be the structure sheaf on $X_{\mathrm {\acute {e}t}}$ , and let $\mathcal {O}_{X_{\mathrm {\acute {e}t}}}^\times $ be the subsheaf of units of $\mathcal {O}_{X_{\mathrm {\acute {e}t}}}$ . A logarithmic structure on X is a homomorphism of sheaves of monoids $\alpha :\mathscr {M} \to \mathcal {O}_{X_{\mathrm {\acute {e}t}}}$ on $X_{\mathrm {\acute {e}t}}$ which induces an isomorphism $\alpha ^{-1}(\mathcal {O}_{X_{\mathrm {\acute {e}t}}}^\times ) \stackrel {\sim }{\longrightarrow } \mathcal {O}_{X_{\mathrm {\acute {e}t}}}^\times $ . We denote by $(X,\mathscr {M})$ the associated logarithmic scheme (cf. [Reference AchingerAGT16, II.5.11]).

4.2 Let $(X,\mathscr {M})$ be a coherent log scheme (cf. [Reference AchingerAGT16, II.5.15]). Then, there is a maximal open subscheme $X^{\mathrm {tr}}$ of X on which $\mathscr {M}$ is trivial, and moreover it is functorial in $(X,\mathscr {M})$ [Reference OgusOgu18, III.1.2.8]. Let $(X,\mathscr {M}) \to (S,\mathscr {L}) \leftarrow (Y,\mathscr {N})$ be a diagram of fine and saturated log schemes (cf. [Reference AchingerAGT16, II.5.15]). Then, the fibered product is representable in the category of fine and saturated log schemes by $(Z,\mathscr {P})=(X,\mathscr {M}) \times _{(S,\mathscr {L})}^{\mathrm {fs}} (Y,\mathscr {N})$ . We remark that $Z^{\mathrm {tr}}= X^{\mathrm {tr}} \times _{S^{\mathrm {tr}}} Y^{\mathrm {tr}}$ , that $Z \to X \times _S Y$ is finite, and that $Z^{\mathrm {tr}} \to Z$ is Cartesian over $X^{\mathrm {tr}} \times _{S^{\mathrm {tr}}} Y^{\mathrm {tr}} \to X \times _S Y$ [Reference OgusOgu18, III 2.1.2, 2.1.6]. Moreover, if $X^{\mathrm {tr}}=X$ , then $Z=X \times _S Y$ [Reference OgusOgu18, III.2.1.3].

4.3 For an open immersion $j:Y \to X$ , we denote by $j_{\mathrm {\acute {e}t}}: Y_{\mathrm {\acute {e}t}} \to X_{\mathrm {\acute {e}t}}$ the morphism of their étale sites defined by the base change by j. Let $\mathscr {M}_{Y \to X}$ be the preimage of $j_{\mathrm {\acute {e}t}*} \mathcal {O}_{Y_{\mathrm {\acute {e}t}}}^\times $ under the natural map $\mathcal {O}_{X_{\mathrm {\acute {e}t}}}\to j_{\mathrm {\acute {e}t}*}\mathcal {O}_{Y_{\mathrm {\acute {e}t}}}$ , and we endow X with the logarithmic structure $\mathscr {M}_{Y \to X} \to \mathcal {O}_{X_{\mathrm {\acute {e}t}}}$ , which is called the compactifying log structure associated with the open immersion j [Reference OgusOgu18, III.1.6.1]. Sometimes we write $\mathscr {M}_{Y \to X}$ as $\mathscr {M}_{X}$ if Y is clear in the context.

4.4 Let $(X,\mathscr {M})$ be a fine and saturated log scheme which is regular ([Reference KatoKat94, 2.1], [Reference NiziołNiz06, 2.3]). Then, X is locally Noetherian and normal, and $X^{\mathrm {tr}}$ is regular and dense in X [Reference KatoKat94, 4.1]. Moreover, there is a natural isomorphism $\mathscr {M} \stackrel {\sim }{\longrightarrow } \mathscr {M}_{X^{\mathrm {tr}} \to X}$ ([Reference KatoKat94, 11.6], [Reference NiziołNiz06, 2.6]). We remark that if X is a regular scheme with a strict normal crossings divisor D, then $(X,\mathscr {M}_{X\setminus D \to X})$ is fine, saturated, and regular [Reference OgusOgu18, III.1.11.9].

Let $f:(X,\mathscr {M}) \to (S,\mathscr {L})$ be a smooth (resp. saturated) morphism of fine and saturated log schemes (cf. [Reference AchingerAGT16, II 5.25, 5.18]). Then, f remains smooth (resp. saturated) under the base change in the category of fine and saturated log schemes ([Reference OgusOgu18, IV.3.1.2, IV.3.1.11], resp. [Reference OgusOgu18, III.2.5.3]). We remark that if f is smooth, then $f^{\mathrm {tr}}:X^{\mathrm {tr}}\to S^{\mathrm {tr}}$ is a smooth morphism of schemes. If, moreover, $(S,\mathscr {L})$ is regular, then $(X,\mathscr {M})$ is also regular [Reference OgusOgu18, IV.3.5.3]. We also remark that if f is saturated, then for any fibered product in the category of fine and saturated log schemes $(Z,\mathscr {P})=(X,\mathscr {M}) \times _{(S,\mathscr {L})}^{\mathrm {fs}} (Y,\mathscr {N})$ , we have $Z=X\times _S Y$ [Reference TsujiTsu19, II.2.13].

4.5 Let K be a complete discrete valuation field with valuation ring $\mathcal {O}_K$ , let k be the residue field of $\mathcal {O}_K$ , and let $\pi $ be a uniformizer of $\mathcal {O}_K$ . We set $\eta =\mathop {\mathrm {Spec}}(K)$ , $S= \mathop {\mathrm {Spec}}(\mathcal {O}_K)$ and $s=\mathop {\mathrm {Spec}}(k)$ . Then, $(S,\mathscr {M}_{\eta \to S})$ is fine, saturated, and regular, since $\mathbb {N} \to \Gamma (S,\mathscr {M}_{\eta \to S})$ sending $1$ to $\pi $ forms a chart of $(S,\mathscr {M}_{\eta \to S})$ (cf. [Reference AchingerAGT16, II.5.13, II.6.1]). Recall that an open immersion $Y \to X$ of quasi-compact and separated schemes over $\eta \to S$ is strictly semi-stable [Reference de JongdJ96, 6.3] if and only if the following conditions are satisfied ([Reference de JongdJ96, 6.4], [Reference GrothendieckEGA IV₄ , 17.5.3]):

1. (i) For each point x of the generic fiber $X_\eta $ , there is an open neighborhood $U\subseteq X_\eta $ of x and a smooth K-morphism

   (4.5.1) $$ \begin{align} f:U\longrightarrow \mathop{\mathrm{Spec}}(K[s_1,\dots,s_m]) \end{align} $$
   such that f maps x to the point associated with the maximal ideal $(s_1,\dots ,s_m)$ and that $U\setminus Y$ is the inverse image of the closed subset defined by $s_1\dots s_m=0$ .

2. (ii) For each point x of the special fiber $X_s$ , there is an open neighborhood $U\subseteq X$ of x and a smooth $\mathcal {O}_K$ -morphism

   (4.5.2) $$ \begin{align} f:U\longrightarrow \mathop{\mathrm{Spec}}(\mathcal{O}_K[t_1,\dots,t_n, s_1,\dots,s_m]/(\pi-t_1\dots t_n)) \end{align} $$
   such that f maps x to the point associated with the maximal ideal $(t_1,\dots ,t_n, s_1,\dots ,s_m)$ and that $U\setminus Y$ is the inverse image of the closed subset defined by $t_1\dots t_n\cdot s_1\dots s_m=0$ .

We say that an open immersion $Y \to X$ of quasi-compact and separated schemes over $\eta \to S$ is semi-stable if for any point x of X there is an étale neighborhood U of x such that $Y\times _X U\to U$ is strictly semi-stable. In this case, $(X,\mathscr {M}_{Y \to X})$ is a fine, saturated, and regular log scheme smooth and saturated over $(S,\mathscr {M}_{\eta \to S})$ , since for any point x of X there is an étale neighborhood U of x such that there exists a chart for the morphism $(U,\mathscr {M}_{Y\times _X U \to U})\to (S,\mathscr {M}_{\eta \to S})$ subordinate to the morphism $\mathbb {N} \to \mathbb {N}^n\oplus \mathbb {N}^m$ sending $1$ to $(1,\dots ,1,0,\dots ,0)$ such that the induced morphism $U\to S\times _{\mathbb {A}_{\mathbb {N}}}\mathbb {A}_{\mathbb {N}^n\oplus \mathbb {N}^m}$ is smooth (cf. [Reference OgusOgu18, II.2.4.1, IV.3.1.18]).

4.6 Recall that a morphism of schemes $f:X\to S$ is called generically finite if there exists a dense open subscheme U of S such that $f^{-1}(U)\to U$ is finite. We remark that for a morphism $f:X\to S$ of finite type between Noetherian schemes which maps generic points to generic points, f is generically finite if and only if the residue field of any generic point $\eta $ of X is a finite field extension of the residue field of $f(\eta )$ [Reference Illusie, Laszlo and OrgogozoILO14, II.1.1.7].

4.7 Let K be a complete discrete valuation field with valuation ring $\mathcal {O}_K$ , let L be an algebraically closed valuation field of height $1$ extension of K with valuation ring $\mathcal {O}_L$ , and let $\overline {K}$ be the algebraic closure of K in L.

Consider the category $\mathscr {C}$ of open immersions between integral affine schemes $U\to T$ over $\mathop {\mathrm {Spec}}(K)\to \mathop {\mathrm {Spec}}(\mathcal {O}_K)$ under $\mathop {\mathrm {Spec}}(L)\to \mathop {\mathrm {Spec}}(\mathcal {O}_L)$ such that T is of finite type over $\mathcal {O}_K$ and that $\mathop {\mathrm {Spec}}(L)\to U$ is dominant. Let $\mathscr {C}_{\mathrm {car}}$ be the full subcategory of $\mathscr {C}$ formed by those objects $U \to T$ Cartesian over $\mathop {\mathrm {Spec}}(K) \to \mathop {\mathrm {Spec}}(\mathcal {O}_K)$ .

(4.7.1)

We note that the objects of $\mathscr {C}$ are of the form $(U=\mathop {\mathrm {Spec}}(B) \to T=\mathop {\mathrm {Spec}}(A)),$ where A (resp. B) is a finitely generated $\mathcal {O}_K$ -subalgebra of $\mathcal {O}_L$ (resp. K-subalgebra of L) with $A\subseteq B$ such that $\mathop {\mathrm {Spec}}(B) \to \mathop {\mathrm {Spec}}(A)$ is an open immersion.

Proof (1) For a diagram $(U_1\to T_1)\to (U_0 \to T_0) \leftarrow (U_2 \to T_2)$ in $\mathscr {C}$ , let T be the scheme theoretic image of $\mathop {\mathrm {Spec}}(L) \to T_1\times _{T_0} T_2$ , and let U be the intersection of $U_1\times _{U_0} U_2$ with T. It is clear that T is of finite type over $\mathcal {O}_K$ as $\mathcal {O}_K$ is Noetherian, that U and T are integral and affine, that $\mathop {\mathrm {Spec}}(L) \to U$ is dominant, and that $\mathop {\mathrm {Spec}}(L)\to T$ factors through $\mathop {\mathrm {Spec}}(\mathcal {O}_L)$ . Thus, $U \to T$ is an object of $\mathscr {C}$ , which shows that $\mathscr {C}$ is cofiltered. For an object $(U=\mathop {\mathrm {Spec}}(B)\to T=\mathop {\mathrm {Spec}}(A))$ of $\mathscr {C}$ , we write $\mathcal {O}_L$ as a filtered union of finitely generated A-subalgebras $A_i$ . Let $\pi $ be a uniformizer of K. Notice that $L=\mathcal {O}_L[1/\pi ]=\mathop {\mathrm {colim}} A_i[1/\pi ]$ and that $\mathrm {Hom}_{K\textrm {-}\mathbf {Alg}}(B,L)=\mathop {\mathrm {colim}} \mathrm {Hom}_{K\textrm {-}\mathbf {Alg}}(B,A_i[1/\pi ])$ by [Reference GrothendieckEGA IV₃ , 8.14.2.2]. Thus, there exists an index i such that $\mathop {\mathrm {Spec}}(A_i[1/\pi ]) \to \mathop {\mathrm {Spec}}(A_i)$ is an object of $\mathscr {C}_{\mathrm {car}}$ over $U \to T$ .

(2) It follows immediately from the arguments above.

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

(4.8.1)

such that $K'$ is a finite field extension of K. Similarly, this category is also cofiltered with limit of diagrams of schemes $(\mathop {\mathrm {Spec}}(L) \to \mathop {\mathrm {Spec}}(\mathcal {O}_L)) \to (\mathop {\mathrm {Spec}}(\overline {K}) \to \mathop {\mathrm {Spec}}(\mathcal {O}_{\overline {K}}))$ . It suffices to show that the full subcategory of $\mathscr {D}$ formed by strictly semi-stable objects is initial. For any object $U \to T$ of $\mathscr {C}_{\mathrm {car}}$ , by de Jong’s alteration theorem [Reference de JongdJ96, 6.5], there exists a proper surjective and generically finite morphism $T' \to T$ of integral schemes such that $U'=U\times _T T' \to T'$ is strictly semi-stable over $\mathop {\mathrm {Spec}}(K') \to \mathop {\mathrm {Spec}}(\mathcal {O}_{K'})$ for a finite field extension $K\to K'$ . Since L is algebraically closed, the dominant morphism $\mathop {\mathrm {Spec}}(L)\to U$ lifts to a dominant morphism $\mathop {\mathrm {Spec}}(L)\to U'$ (4.6), which further extends to a lifting $\mathop {\mathrm {Spec}}(\mathcal {O}_L) \to T'$ of $\mathop {\mathrm {Spec}}(\mathcal {O}_L) \to T$ by the valuative criterion. After replacing $T'$ by an affine open neighborhood of the image of the closed point of $\mathop {\mathrm {Spec}}(\mathcal {O}_L)$ , we obtain a strictly semi-stable object of $\mathscr {D}$ over $(U \to T)\to (\mathop {\mathrm {Spec}}(K) \to \mathop {\mathrm {Spec}}(\mathcal {O}_K))$ , which completes the proof.

Proof We may assume that T is nonempty. Recall that $\mathop {\mathrm {Spec}}(\mathcal {O}_K)$ is universally $\mathbb {Q}$ -resolvable [Reference Illusie, Laszlo and OrgogozoILO14, X.3.3] by de Jong’s alteration theorem [Reference de JongdJ96, 6.5]. Thus, T is also universally $\mathbb {Q}$ -resolvable by [Reference Illusie, Laszlo and OrgogozoILO14, X 3.5, 3.5.2] so that we can apply [Reference Illusie, Laszlo and OrgogozoILO14, X.3.5] to the proper surjective morphism f and the nowhere dense closed subset $X\setminus Y$ . Then, we obtain a commutative diagram of schemes

(4.9.2)

and dense open subsets $U'{\kern-1pt}\subseteq{\kern-1pt} T'$ , $Y'{\kern-1pt}={\kern-1pt}\beta ^{-1}(Y){\kern-1pt}\cap{\kern-1pt} f^{{\prime }-1}(U'){\kern-1pt}\subseteq{\kern-1pt} X'$ such that $(X',\mathscr {M}_{Y'\to X'})$ and $(T',\mathscr {M}_{U'\to T'})$ are fine, saturated, and regular, that $(X',\mathscr {M}_{Y'\to X'})\to (T',\mathscr {M}_{U'\to T'})$ is smooth, that $\alpha , \beta $ are proper surjective and generically finite morphisms which map generic points to generic points, and that $f'$ is projective (since f is proper, cf. [Reference Illusie, Laszlo and OrgogozoILO14, X 3.1.6, 3.1.7]). Since X (resp. T) is irreducible and $X'$ (resp. $T'$ ) is a disjoint union of normal integral schemes (4.4), after firstly replacing $X'$ by an irreducible component and then replacing $T'$ by the irreducible component under $X'$ , we may assume that $X'$ and $T'$ are irreducible. Then, $Y'\to U'$ is dominant (so that $f'$ is projective surjective), since it is smooth and $Y'$ is nonempty [Reference GrothendieckEGA IV₂ , 2.3.4]. We claim that $\alpha $ maps $U'$ into U. Indeed, if there exists a point $u\in U'$ with $\alpha (u)\notin U$ , then $f^{{\prime }-1}(u)\cap Y'=\emptyset $ . However, endowing u with the trivial log structure, the log scheme $(u,\mathcal {O}_{u_{\mathrm {\acute {e}t}}}^\times )$ is fine, saturated, and regular, and the fibered product in the category of fine and saturated log schemes

(4.9.3) $$ \begin{align} (u,\mathcal{O}_{u_{\mathrm{\acute{e}t}}}^\times)\times_{(T',\mathscr{M}_{U' \to T'})}^{\mathrm{fs}} (X',\mathscr{M}_{Y' \to X'}) \end{align} $$

is regular with underlying scheme $f^{{\prime }-1}(u)$ (4.4, 4.2). Thus, $f^{{\prime }-1}(u)\cap Y'$ is dense in $f^{{\prime }-1}(u)$ , which contradicts the assumption that $f^{{\prime }-1}(u)\cap Y'=\emptyset $ since $f'$ is surjective. Thus, we obtain a diagram (4.9.1) satisfying all the conditions except the saturatedness of $(X',\mathscr {M}_{Y'\to X'})\to (T',\mathscr {M}_{U'\to T'})$ .

To make $(X',\mathscr {M}_{Y'\to X'})\to (T',\mathscr {M}_{U'\to T'})$ saturated, we apply [Reference Illusie, Laszlo and OrgogozoILO14, X.3.7] to the morphism $(f^{{\prime }\circ }, f')$ . We obtain a Cartesian morphism $(\gamma ^{\circ }, \gamma ):(U" \to T")\to (U' \to T') $ of dominant open immersions such that $(T",\mathscr {M}_{U" \to T"})$ is a fine, saturated, and regular log scheme, that $\gamma $ is a proper surjective and generically finite morphism which maps generic points of $T"$ to the generic point of $T'$ , and that the fibered product in the category of fine and saturated log schemes

(4.9.4) $$ \begin{align} (T",\mathscr{M}_{U" \to T"})\times^{\mathrm{fs}}_{(T',\mathscr{M}_{U' \to T'})} (X',\mathscr{M}_{Y' \to X'}) \end{align} $$

is saturated over $(T",\mathscr {M}_{U" \to T"})$ . The fibered product (4.9.4) is still smooth over $(T",\mathscr {M}_{U" \to T"})$ , and thus it is regular (4.4). Let $X"$ be the underlying scheme of it, and let $Y"=(X")^{\mathrm {tr}}$ . Then, the fibered product (4.9.4) is isomorphic to $(X",\mathscr {M}_{Y"\to X"})$ (4.4). Thus, we obtain a commutative diagram of dominant open immersions of schemes:

(4.9.5)

Notice that $Y"= U" \times _{U'} Y'$ and $X" \to T" \times _{T'} X'$ is finite, and that $Y"\to X"$ is Cartesian over $U" \times _{U'} Y' \to T" \times _{T'} X'$ (4.2). Thus, we see that $Y"\to X"$ is Cartesian over $U" \times _{U} Y \to T" \times _{T} X$ and that $f"$ is projective. Since $T'$ (resp. $X'$ ) is irreducible and $T"$ (resp. $X"$ ) is a disjoint union of normal integral schemes (4.4), after firstly replacing $T"$ by an irreducible component and then replacing $X"$ by an irreducible component on which the restriction of $\delta ^{\circ }$ is dominant, we may assume that $T"$ and $X"$ are irreducible. In particular, $\delta $ is generically finite and so is $\beta \circ \delta $ (4.6), and again $Y"\to U"$ is dominant so that $f"$ is projective surjective.

Proof Let $\eta $ and s be the generic point and closed point of $\mathop {\mathrm {Spec}}(A),$ respectively. Then, the generic fiber $X_\eta $ and the special fiber $X_s$ are both Noetherian. As a union of $X_\eta $ and $X_s$ , the underlying topological space of X is also Noetherian [Sta23, 0053].

Proof We follow closely the proof of [Reference Adiprasito, Liu, Pak and TemkinALPT19, 5.2.19]. Since the underlying topological space of X is Noetherian by 4.10, each irreducible component Z of X admits a closed subscheme structure such that $Z\to X$ is of finite presentation [Sta23, 01PH]. After replacing X by the disjoint union of its irreducible components, we may assume that X is irreducible. Then, the generic fiber of $X\to \mathop {\mathrm {Spec}}(\mathcal {O}_L)$ is also irreducible as an open subset of X. Using [Reference GrothendieckEGA IV₃ , 8.8.2, 8.10.5], there exists an index $\lambda _0 \in \Lambda $ , a proper $T_{\lambda _0}$ -scheme $X_{\lambda _0}$ , and an open subscheme $Y_{\lambda _0}$ of $U_{\lambda _0}\times _{T_{\lambda _0}} X_{\lambda _0}$ , such that $X=\mathop {\mathrm {Spec}}(\mathcal {O}_L) \times _{T_{\lambda _0}} X_{\lambda _0}$ and that $Y=\mathop {\mathrm {Spec}}(L) \times _{U_{\lambda _0}} Y_{\lambda _0}$ . Let $\eta $ denote the generic point of X, $\eta _{\lambda _0}$ the image of $\eta $ under the morphism $X\to X_{\lambda _0}$ , $Z_{\lambda _0}$ the scheme theoretic closure of $\eta _{\lambda _0}$ in $X_{\lambda _0}$ . Notice that $\mathop {\mathrm {Spec}}(\mathcal {O}_L)\times _{T_{\lambda _0}} Z_{\lambda _0}\to X$ is a surjective finitely presented closed immersion. After replacing X by $\mathop {\mathrm {Spec}}(\mathcal {O}_L)\times _{T_{\lambda _0}} Z_{\lambda _0}$ and replacing $X_{\lambda _0}$ by $Z_{\lambda _0}$ , we may assume that $X\to X_{\lambda _0}$ is a dominant morphism of irreducible schemes. Since $T_{\lambda _0}$ is irreducible and L is algebraically closed, the generic fiber of $f:X_{\lambda _0}\to T_{\lambda _0}$ is geometrically irreducible. In particular, if $\xi _{\lambda _0}$ (resp. $\eta _{\lambda _0}$ ) denotes the generic point of $T_{\lambda _0}$ (resp. $X_{\lambda _0}$ ), then $\eta =\mathop {\mathrm {Spec}}(L)\times _{\xi _{\lambda _0}}\eta _{\lambda _0}$ [Reference GrothendieckEGA IV₂ , 4.5.9]. In the situation of (iv), we can moreover assume that $Y_{\lambda _0}= U_{\lambda _0} \times _{T_{\lambda _0}} X_{\lambda _0}$ .

By 4.9, there exists a commutative diagram of dominant open immersions of irreducible schemes,

(4.11.1)

where $Y^{\prime }_{\lambda _0}\to X^{\prime }_{\lambda _0}$ is Cartesian over $U^{\prime }_{\lambda _0}\times _{U_{\lambda _0}}Y_{\lambda _0} \to T^{\prime }_{\lambda _0}\times _{T_{\lambda _0}} X_{\lambda _0}$ , and where $(X^{\prime }_{\lambda _0},\mathscr {M}_{Y^{\prime }_{\lambda _0}\to X^{\prime }_{\lambda _0}})\to (T^{\prime }_{\lambda _0},\mathscr {M}_{U^{\prime }_{\lambda _0}\to T^{\prime }_{\lambda _0}})$ is a smooth and saturated morphism of fine, saturated, and regular log schemes, and where $\alpha $ and $\beta $ are proper surjective and generically finite, and where $f'$ is projective surjective. We take a dominant morphism $\gamma ^{\circ }:\mathop {\mathrm {Spec}}(L)\to U_{\lambda _0}'$ which lifts $\mathop {\mathrm {Spec}}(L)\to U_{\lambda _0}$ since L is algebraically closed and $\alpha $ is generically finite, the morphism $\mathop {\mathrm {Spec}}(\mathcal {O}_L) \to T_{\lambda _0}$ lifts to $\gamma :\mathop {\mathrm {Spec}}(\mathcal {O}_L) \to T^{\prime }_{\lambda _0}$ by the valuative criterion. We set $Y'=\mathop {\mathrm {Spec}}(L)\times _{U^{\prime }_{\lambda _0}}Y^{\prime }_{\lambda _0}$ and $X'=\mathop {\mathrm {Spec}}(\mathcal {O}_L)\times _{T^{\prime }_{\lambda _0}} X^{\prime }_{\lambda _0}$ . It is clear that $Y'\to X'$ is Cartesian over $Y \to X$ by base change. Let $\xi ^{\prime }_{\lambda _0}$ (resp. $\eta ^{\prime }_{\lambda _0}$ ) be the generic point of $T^{\prime }_{\lambda _0}$ (resp. $X^{\prime }_{\lambda _0}$ ). Since the generic fiber of f is geometrically irreducible, $\xi ^{\prime }_{\lambda _0}\times _{\xi _{\lambda _0}}\eta _{\lambda _0}$ is a single point and $\eta ^{\prime }_{\lambda _0}$ maps to it [Reference GrothendieckEGA IV₂ , 4.5.9]. Since $\mathop {\mathrm {Spec}}(L)\times _{\xi _{\lambda _0}}\eta _{\lambda _0}$ is the generic point of X, we see that $X' \to X$ is proper surjective and of finite presentation. It remains to construct $(Y^{\prime }_\lambda \to X^{\prime }_\lambda )_{\lambda \geq \lambda _1}$ .

After replacing $T^{\prime }_{\lambda _0}$ by an affine open neighborhood of the image of the closed point of $\mathop {\mathrm {Spec}}(\mathcal {O}_L)$ , Lemma 4.8 implies that there exists an index $\lambda _1 \geq \lambda _0$ such that the transition morphism $(U_{\lambda _1}\to T_{\lambda _1}) \to (U_{\lambda _0}\to T_{\lambda _0})$ factors through $(U^{\prime }_{\lambda _0}\to T^{\prime }_{\lambda _0})$ . For each index $\lambda \geq \lambda _1$ , consider the fibered product in the category of fine and saturated log schemes

(4.11.2) $$ \begin{align} (X^{\prime}_\lambda,\mathscr{M}_{Y^{\prime}_\lambda \to X^{\prime}_\lambda})=(T_{\lambda},\mathscr{M}_{U_{\lambda}\to T_{\lambda}}) \times_{(T^{\prime}_{\lambda_0},\mathscr{M}_{U^{\prime}_{\lambda_0}\to T^{\prime}_{\lambda_0}})}^{\mathrm{fs}} (X^{\prime}_{\lambda_0},\mathscr{M}_{Y^{\prime}_{\lambda_0}\to X^{\prime}_{\lambda_0}}), \end{align} $$

which is a fine, saturated, and regular log scheme smooth and saturated over $(T_{\lambda },\mathscr {M}_{U_{\lambda }\to T_{\lambda }})$ (4.2, 4.4). Moreover, we have $Y^{\prime }_\lambda = U_\lambda \times _{U^{\prime }_{\lambda _0}} Y^{\prime }_{\lambda _0}$ , $X^{\prime }_\lambda = T_\lambda \times _{T^{\prime }_{\lambda _0}} X^{\prime }_{\lambda _0}$ , and in the situation of (iv), $Y^{\prime }_\lambda =U_\lambda \times _{T_\lambda }X^{\prime }_\lambda $ by base change. Therefore, $(Y^{\prime }_\lambda \to X^{\prime }_\lambda )_{\lambda \geq \lambda _1}$ meets our requirements.

Proof If Y is connected, let $\overline {y}$ be a geometric point of Y, $\pi _1(Y,\overline {y})$ the fundamental group of Y with base point $\overline {y}$ . Then, $Y_{\mathrm {f\acute {e}t}}$ is equivalent to the category of finite $\pi _1(Y,\overline {y})$ -sets so that the lemma holds [Sta23, 0BND].

In general, for any connected component Z of Y, let $(Y_\lambda )_{\lambda \in \Lambda _Z}$ be the directed inverse system of all open and closed subschemes of Y which contain Z and whose transition morphisms are inclusions. Notice that $\lim _{\lambda \in \Lambda _Z} Y_\lambda $ is a closed subscheme of Y with underlying topological space Z by [Sta23, 04PL] and [Reference GrothendieckEGA IV₃ , 8.2.9]. We endow Z with the closed subscheme structure of $\lim _{\lambda \in \Lambda _Z} Y_\lambda $ . The first paragraph shows that there exists a finite étale surjective morphism $Z'\to Z$ such that $Z'\times _Y V=\coprod _{i=1}^r Z'$ . Using [Reference GrothendieckEGA IV₃ , 8.8.2, 8.10.5] and [Reference GrothendieckEGA IV₄ , 17.7.8], there exists an index $\lambda _0\in \Lambda _Z$ , a finite étale surjective morphism $Y^{\prime }_{\lambda _0}\to Y_{\lambda _0}$ and an isomorphism $Y^{\prime }_{\lambda _0}\times _Y V = \coprod _{i=1}^r Y^{\prime }_{\lambda _0}$ . Notice that $Y^{\prime }_{\lambda _0}$ is also finite étale over Y. Since Z is an arbitrary connected component of Y, the conclusion follows from the quasi-compactness of Y.

Proof Since any finite locally constant sheaf on $Y_{\mathrm {\acute {e}t}}$ (resp. $Y_{\mathrm {f\acute {e}t}}$ ) is representable by a finite étale Y-scheme by faithfully flat descent (cf. [Sta23, 03RV]), the Yoneda embeddings induce a commutative diagram:

(5.2.1)

where the horizontal arrows are fully faithful. In particular, $\rho ^{-1}$ is fully faithful. For a finite locally constant abelian sheaf $\mathbb {F}$ on $Y_{\mathrm {\acute {e}t}}$ , let V be a finite étale Y-scheme representing $\mathbb {F}$ , and let $h_V^{\mathrm {\acute {e}t}}$ (resp. $h_V^{\mathrm {f\acute {e}t}}$ ) be the representable sheaf associated with V on $Y_{\mathrm {\acute {e}t}}$ (resp. $Y_{\mathrm {f\acute {e}t}}$ ) (see 2.4). We have $\mathbb {F}=h_V^{\mathrm {\acute {e}t}}=\rho ^{-1}h_V^{\mathrm {f\acute {e}t}}$ [Sta23, 04D3]. By 5.1, $h_V^{\mathrm {f\acute {e}t}}$ is finite locally constant. It is clear that the adjunction morphism $h_V^{\mathrm {f\acute {e}t}}\to \rho _*h_V^{\mathrm {\acute {e}t}}$ is an isomorphism, which shows that $h_V^{\mathrm {f\acute {e}t}}$ is an abelian sheaf. Thus, $\rho ^{-1}$ is essentially surjective. Moreover, the argument also shows that $\rho _*$ induces a functor $\rho _*:\mathbf {LocSys}(Y_{\mathrm {\acute {e}t}}) \to \mathbf {LocSys}(Y_{\mathrm {f\acute {e}t}})$ which is a quasi-inverse of $\rho ^{-1}$ .

Proof Notice that for any finite locally constant abelian sheaf $\mathbb {G}$ on $Y_{\mathrm {f\acute {e}t}}$ , the canonical morphism $\beta ^{-1}\mathbb {G}\to \psi _*\rho ^{-1}\mathbb {G}$ , which is induced by the adjunction $\mathrm {id} \to \psi _*\psi ^{-1}$ and by the identity $\psi ^{-1}\beta ^{-1}=\rho ^{-1}$ , is an isomorphism by 5.2 and the proof of [Reference AchingerAGT16, VI.6.3(iii)]. For a finite locally constant abelian sheaf $\mathbb {F}$ over $Y_{\mathrm {\acute {e}t}}$ , we write $\mathbb {F}=\rho ^{-1}\mathbb {G}$ by 5.2. Then, $\mathbb {F}=\psi ^{-1}\beta ^{-1}\mathbb {G}\stackrel {\sim }{\longrightarrow }\psi ^{-1}\psi _*\rho ^{-1}\mathbb {G}=\psi ^{-1}\psi _*\mathbb {F}$ , whose inverse is the adjunction map $\psi ^{-1}\psi _*\mathbb {F} \to \mathbb {F}$ since the composition of $\psi ^{-1}(\beta ^{-1}\mathbb {G}) \to \psi ^{-1}(\psi _*\psi ^{-1})(\beta ^{-1}\mathbb {G})=(\psi ^{-1}\psi _*)\psi ^{-1}(\beta ^{-1}\mathbb {G})\to \psi ^{-1}(\beta ^{-1}\mathbb {G})$ is the identity. On the other hand, for a finite locally constant abelian sheaf $\mathbb {L}$ over $\mathbf {E}_{Y \to X}^{\mathrm {\acute {e}t}}$ , we claim that $\mathbb {L} \to \psi _*\psi ^{-1}\mathbb {L}$ is an isomorphism. The problem is local on $\mathbf {E}_{Y \to X}^{\mathrm {\acute {e}t}}$ . Thus, we may assume that $\mathbb {L}$ is the constant sheaf with value L, where L is a finite abelian group. Let $\underline {L}$ be the constant sheaf with value L on $Y_{\mathrm {f\acute {e}t}}$ . Then, $\mathbb {L}=\beta ^{-1}\underline {L}$ , and the isomorphism $\mathbb {L}=\beta ^{-1}\underline {L}\stackrel {\sim }{\longrightarrow } \psi _*\rho ^{-1}\underline {L}=\psi _*\psi ^{-1}\mathbb {L}$ coincides with the adjunction map $\mathbb {L} \to \psi _*\psi ^{-1}\mathbb {L}$ . Therefore, $\psi ^{-1}:\mathbf {LocSys}(\mathbf {E}_{Y \to X}^{\mathrm {\acute {e}t}}) \to \mathbf {LocSys}(Y_{\mathrm {\acute {e}t}}) $ is an equivalence with quasi-inverse $\psi _*$ . The conclusion follows from 5.2.

5.4 Let $f:(Y' \to X')\to (Y \to X)$ be a morphism of morphisms between coherent schemes over $\mathop {\mathrm {Spec}}(\mathbb {Q}_p)\to \mathop {\mathrm {Spec}}(\mathbb {Z}_p)$ . The base change by f induces a commutative diagram of sites (see 3.2):

(5.4.1)

Let $\mathbb {F}'$ be a finite locally constant abelian sheaf on $Y^{\prime }_{\mathrm {\acute {e}t}}$ . Remark that the sheaf $\overline {\mathscr {B}}$ on $\mathbf {E}_{Y \to X}^{\mathrm {\acute {e}t}}$ is flat over $\mathbb {Z}$ . Consider the natural morphisms in the derived category $\mathbf {D}(\overline {\mathscr {B}}\textrm {-}\mathbf {Mod}_{\mathbf {E}_{Y \to X}^{\mathrm {\acute {e}t}}})$ ,

(5.4.2)

where $\alpha _1$ is induced by the canonical morphism $\psi ^{\prime }_*\mathbb {F}' \to \mathrm {R}\psi ^{\prime }_*\mathbb {F}'$ , and $\alpha _2$ is the canonical morphism.

5.5 We keep the notation in 5.4 and assume that X is the spectrum of an absolutely integrally closed valuation ring A and that Y is a quasi-compact open subscheme of X. Then, the associated topos of $\mathbf {E}_{Y \to X}^{\mathrm {\acute {e}t}}$ is local (3.7). By applying the functor $\mathrm {R}\Gamma (Y \to X,-)$ on (5.4.2), we obtain the natural morphisms in the derived category $\mathbf {D}(A\textrm {-}\mathbf {Mod})$ ,

(5.5.1)

Proof of 5.8

We take a directed inverse system $(X_\lambda \to \mathop {\mathrm {Spec}}(\mathcal {O}_{K_\lambda }))_{\lambda \in \Lambda }$ of morphisms of finite type of schemes by Noetherian approximation, such that $K_\lambda $ is a finite field extension of K and $\overline {K}=\bigcup _{\lambda \in \Lambda }K_\lambda $ , and that the transition morphisms $X_{\lambda '} \to X_\lambda $ are affine and $X=\lim _{\lambda \in \Lambda } X_\lambda $ (cf. [Sta23, 09MV]). For each $\lambda \in \Lambda $ , we set $Y_\lambda =\mathop {\mathrm {Spec}}(\overline {K}) \times _{\mathop {\mathrm {Spec}}(\mathcal {O}_{K_\lambda })}X_\lambda $ . Notice that $Y=\lim Y_\lambda $ . Then, there exists an index $\lambda _0\in \Lambda $ and a finite locally constant abelian sheaf $\mathbb {F}_{\lambda _0}$ on $Y_{\lambda _0,\mathrm {\acute {e}t}}$ such that $\mathbb {F}$ is the pullback of $\mathbb {F}_{\lambda _0}$ by $Y_{\mathrm {\acute {e}t}} \to Y_{\lambda _0,\mathrm {\acute {e}t}}$ (cf. [Sta23, 09YU]). Let $\mathbb {F}_\lambda $ be the pullback of $\mathbb {F}_{\lambda _0}$ by $Y_{\lambda ,\mathrm {\acute {e}t}} \to Y_{\lambda _0,\mathrm {\acute {e}t}}$ for each $\lambda \geq \lambda _0$ . Notice that $\mathcal {O}_{K_\lambda }$ also satisfies the conditions in 5.7. Let $\psi _\lambda :Y_{\lambda ,\mathrm {\acute {e}t}} \to \mathbf {E}_{Y_\lambda \to X_\lambda }^{\mathrm {\acute {e}t}}$ be the morphism of sites defined in 3.2, and let $\varphi _\lambda : \mathbf {E}_{Y \to X}^{\mathrm {\acute {e}t}}\to \mathbf {E}_{Y_\lambda \to X_\lambda }^{\mathrm {\acute {e}t}}$ be the morphism of sites defined by the transition morphism. Then, we have $\mathrm {R}^q\psi _{\lambda *}\mathbb {F}_\lambda =0$ for each integer $q>0$ by 5.7, and moreover,

(5.8.1) $$ \begin{align} \mathrm{R}^q\psi_* \mathbb{F}=\mathop{\mathrm{colim}}_{\lambda\geq \lambda_0} \varphi_\lambda^{-1} \mathrm{R}^q\psi_{\lambda*}\mathbb{F}_\lambda=0 \end{align} $$

by [Reference HeHe23, 7.12], [SGA 4_II , VII.5.6], and [SGA 4_II , VI.8.7.3] whose conditions are satisfied because each object in each concerned site is quasi-compact.

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

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

which are both isomorphisms by 5.3.

5.10 Let K be a complete discrete valuation field of characteristic $0$ with valuation ring $\mathcal {O}_K$ whose residue field k is algebraically closed (a condition required by [Reference Abbes and GrosAG20, 4.1.3, 5.1.3]) of characteristic $p>0$ , and let $\overline {K}$ be an algebraic closure of K, $\mathcal {O}_{\overline {K}}$ the integral closure of $\mathcal {O}_K$ in $\overline {K}$ , $\eta = \mathop {\mathrm {Spec}} (K)$ , $\overline {\eta }=\mathop {\mathrm {Spec}}(\overline {K})$ , $S=\mathop {\mathrm {Spec}} (\mathcal {O}_K)$ , $\overline {S}=\mathop {\mathrm {Spec}}(\mathcal {O}_{\overline {K}})$ , $s=\mathop {\mathrm {Spec}} (k)$ . Remark that $\overline {K}$ is a pre-perfectoid field with valuation ring $\mathcal {O}_{\overline {K}}$ so we are also in the situation of 3.9.

5.11 With the notation in 5.10, let X be an S-scheme, and let Y be an open subscheme of the generic fiber $X_\eta $ . We simply denote by $\mathscr {M}_X$ the compactifying log structure $\mathscr {M}_{X_\eta \to X}$ (4.3). Following [Reference AchingerAGT16, III.4.7], we say that $Y\to X$ is adequate over $\eta \to S$ if the following conditions are satisfied:

1. (i) X is of finite type over S.

2. (ii) Any point of the special fiber $X_s$ admits an étale neighborhood U such that $U_\eta \to \eta $ is smooth and that $U_\eta \setminus Y$ is the support of a strict normal crossings divisor on $U_\eta $ .

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

In this case, $(X,\mathscr {M}_{Y \to X}) \to (S,\mathscr {M}_{S})$ is adequate in the sense of [Reference AchingerAGT16, III.4.7]. We remark that for any adequate $(S,\mathscr {M}_{S})$ -log scheme $(X,\mathscr {M})$ , $X^{\mathrm {tr}}\to X$ is adequate over $\eta \to S$ and $(X,\mathscr {M})=(X,\mathscr {M}_{X^{\mathrm {tr}}\to X})$ (see 4.4 and 4.5). Note that if $Y \to X$ is semi-stable over $\eta \to S$ then it is adequate (see 4.5).

5.12 We recall the statement of Faltings’ main p-adic comparison theorem following Abbes and Gros [Reference Abbes and GrosAG20]. We take the notation and assumptions in 5.10. Firstly, recall that for any adequate open immersion of schemes $X^\circ \to X$ over $\eta \to S$ and any finite locally constant abelian sheaf $\mathbb {F}$ on $X^\circ _{\overline {\eta },\mathrm {\acute {e}t}}$ , the canonical morphism $\psi _*\mathbb {F} \to \mathrm {R}\psi _*\mathbb {F}$ is an isomorphism, where $\psi :X^\circ _{\overline {\eta },\mathrm {\acute {e}t}} \to \mathbf {E}_{X_{\overline {\eta }}^\circ \to X}^{\mathrm {\acute {e}t}}$ is the morphism of sites defined in 3.2 [Reference Abbes and GrosAG20, 4.4.2].

Let $(X^{{\prime }\triangleright }\to X') \to (X^\circ \to X)$ be a morphism of adequate open immersions of schemes over $\eta \to S$ such that $X' \to X$ is projective and that the induced morphism $(X',\mathscr {M}_{X^{{\prime }\triangleright } \to X'})\to (X,\mathscr {M}_{X^\circ \to X})$ is smooth and saturated. Let $Y'=\overline {\eta }\times _{\eta } X^{{\prime }\triangleright }$ , $Y=\overline {\eta }\times _{\eta } X^\circ $ , $f:(Y' \to X') \to (Y\to X)$ be the natural morphism, and let $\mathbb {F}'$ be a finite locally constant abelian sheaf on $Y^{\prime }_{\mathrm {\acute {e}t}}$ . By the first paragraph, we have the relative Faltings’ comparison morphism associated with f and $\mathbb {F}'$ (5.6.1),

(5.12.1) $$ \begin{align} (\mathrm{R}\psi_*\mathrm{R} f_{\mathrm{\acute{e}t} *} \mathbb{F}')\otimes^{\mathrm{L}}_{\mathbb{Z}}\overline{\mathscr{B}} \longrightarrow \mathrm{R} f_{\mathbf{E} *} (\psi^{\prime}_*\mathbb{F}'\otimes_{\mathbb{Z}}\overline{\mathscr{B}}'). \end{align} $$

Remark that under our assumption, the sheaf $\mathrm {R}^q f_{\mathrm {\acute {e}t} *} \mathbb {F}'$ on $Y_{\mathrm {\acute {e}t}}$ is finite locally constant for each integer q [Reference Abbes and GrosAG20, 2.2.25].

Proof Since the underlying topological space of X is Noetherian by 4.10, each irreducible component Z of X admits a closed subscheme structure such that ${Z\to X}$ is of finite presentation [Sta23, 01PH]. After replacing X by the disjoint union of its irreducible components, we may assume that X is irreducible. If Y is empty, then we take $X'=X$ and thus the relative (resp. absolute) Faltings’ comparison morphism associated with $f'$ and $\mathbb {F}'$ is an isomorphism between zero objects. If Y is not empty, then we are in the situation of 4.11(iv) by taking $\mathcal {O}_L=A$ . With the notation in 4.11, we check that the morphism $X' \to X$ meets our requirements. We set $\eta _\lambda =\mathop {\mathrm {Spec}}(K_\lambda )$ , $S_\lambda =\mathop {\mathrm {Spec}}(\mathcal {O}_{K_\lambda })$ , $T_{\lambda ,\overline {\eta _\lambda }} =\overline {\eta }\times _{\eta _\lambda } U_\lambda $ , $X^{\prime }_{\lambda ,\overline {\eta _\lambda }}=\overline {\eta }\times _{\eta _\lambda } Y^{\prime }_\lambda $ , and denote by $f^{\prime }_\lambda :(X^{\prime }_{\lambda ,\overline {\eta _\lambda }} \to X^{\prime }_\lambda ) \to (T_{\lambda ,\overline {\eta _\lambda }} \to T_\lambda )$ the natural morphism. We obtain a commutative diagram:

(5.14.1)

Firstly, notice that the site $Y^{\prime }_{\mathrm {\acute {e}t}}$ (resp. $\mathop {\mathrm {Spec}}(A[1/p])_{\mathrm {\acute {e}t}}$ ) is the limit of the sites $X^{\prime }_{\lambda ,\overline {\eta _\lambda },\mathrm {\acute {e}t}}$ (resp. $T_{\lambda ,\overline {\eta _\lambda },\mathrm {\acute {e}t}}$ ) and the site $\mathbf {E}_{Y' \to X'}^{\mathrm {\acute {e}t}}$ (resp. $\mathbf {E}_{\mathop {\mathrm {Spec}}(A[1/p]) \to \mathop {\mathrm {Spec}}(A)}^{\mathrm {\acute {e}t}}$ ) is the limit of the sites $\mathbf {E}_{X^{\prime }_{\lambda ,\overline {\eta _\lambda }} \to X^{\prime }_\lambda }^{\mathrm {\acute {e}t}}$ (resp. $\mathbf {E}_{T_{\lambda ,\overline {\eta _\lambda }} \to T_\lambda }^{\mathrm {\acute {e}t}}$ ) ([SGA 4_II , VII.5.6] and [Reference HeHe23, 7.12]). There exists an index $\lambda _0\in \Lambda $ and a finite locally constant abelian sheaf $\mathbb {F}^{\prime }_{\lambda _0}$ on $X^{\prime }_{\lambda _0,\overline {\eta _{\lambda _0}},\mathrm {\acute {e}t}}$ such that $\mathbb {F}'$ is the pullback of $\mathbb {F}^{\prime }_{\lambda _0}$ by $Y^{\prime }_{\mathrm {\acute {e}t}} \to X^{\prime }_{\lambda _0,\overline {\eta _{\lambda _0}},\mathrm {\acute {e}t}}$ (cf. [Sta23, 09YU]). Let $\mathbb {F}^{\prime }_\lambda $ be the pullback of $\mathbb {F}^{\prime }_{\lambda _0}$ by $X^{\prime }_{\lambda ,\overline {\eta _{\lambda }},\mathrm {\acute {e}t}} \to X^{\prime }_{\lambda _0,\overline {\eta _{\lambda _0}},\mathrm {\acute {e}t}}$ for each $\lambda \geq \lambda _0$ . We also have $\overline {\mathscr {B}}'=\mathop {\mathrm {colim}} g_{\lambda ,\mathbf {E}}^{-1}\overline {\mathscr {B}}'$ (resp. $\overline {\mathscr {B}}=\mathop {\mathrm {colim}} h_{\lambda ,\mathbf {E}}^{-1}\overline {\mathscr {B}}$ ) by [Reference HeHe23, 7.12]. According to [SGA 4_II , VI.8.7.3], whose conditions are satisfied because each object in each concerned site is quasi-compact, there are canonical isomorphisms for each integer q,

(5.14.2) $$ \begin{align} (\mathrm{R}^q (\psi\circ f^{\prime}_{\mathrm{\acute{e}t}})_*\mathbb{F}') \otimes_{\mathbb{Z}} \overline{\mathscr{B}} &\stackrel{\sim}{\longrightarrow} \mathop{\mathrm{colim}} h_{\lambda,\mathbf{E}}^{-1}((\mathrm{R}^q (\psi_{\lambda}\circ f^{\prime}_{\lambda,\mathrm{\acute{e}t}})_*\mathbb{F}^{\prime}_{\lambda}) \otimes_{\mathbb{Z}}\overline{\mathscr{B}}), \end{align} $$

(5.14.3) $$ \begin{align} \mathrm{R}^q f^{\prime}_{\mathbf{E} *}(\psi^{\prime}_*\mathbb{F}'\otimes_{\mathbb{Z}} \overline{\mathscr{B}}') & \stackrel{\sim}{\longrightarrow} \mathop{\mathrm{colim}} h_{\lambda,\mathbf{E}}^{-1} \mathrm{R}^q f^{\prime}_{\lambda, \mathbf{E} *}(\psi^{\prime}_{\lambda *}\mathbb{F}^{\prime}_{\lambda}\otimes_{\mathbb{Z}} \overline{\mathscr{B}}'). \end{align} $$

On the other hand, $(X^{\prime }_\lambda ,\mathscr {M}_{X^{\prime }_\lambda }) \to (T_\lambda ,\mathscr {M}_{T_\lambda })$ is a smooth and saturated morphism of adequate $(S_\lambda ,\mathscr {M}_{S_\lambda })$ -log schemes with $X^{\prime }_\lambda \to T_\lambda $ projective for each $\lambda \in \Lambda $ by construction. Thus, we are in the situation of 5.13, which implies that the relative Faltings’ comparison morphism associated with $f^{\prime }_\lambda $ and $\mathbb {F}^{\prime }_\lambda $ ,

(5.14.4) $$ \begin{align} (\mathrm{R}^q (\psi_{\lambda}\circ f^{\prime}_{\lambda,\mathrm{\acute{e}t}})_*\mathbb{F}^{\prime}_{\lambda}) \otimes_{\mathbb{Z}}\overline{\mathscr{B}}\longrightarrow \mathrm{R}^q f^{\prime}_{\lambda, \mathbf{E} *}(\psi^{\prime}_{\lambda*}\mathbb{F}^{\prime}_{\lambda}\otimes_{\mathbb{Z}} \overline{\mathscr{B}}') \end{align} $$

is an almost isomorphism for each $\lambda \geq \lambda _0$ . Combining with (5.14.2) and (5.14.3), we see that the relative Faltings’ comparison morphism associated with $f'$ and $\mathbb {F}'$ ,

(5.14.5) $$ \begin{align} \mathrm{R}\psi_*(\mathrm{R} f^{\prime}_{\mathrm{\acute{e}t} *}\mathbb{F}') \otimes_{\mathbb{Z}}^{\mathrm{L}} \overline{\mathscr{B}} \longrightarrow\mathrm{R} f^{\prime}_{\mathbf{E} *}(\psi^{\prime}_*\mathbb{F}'\otimes_{\mathbb{Z}} \overline{\mathscr{B}}'), \end{align} $$

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

Proof Let $\mathscr {C}$ be the category of proper A-schemes of finite presentation endowed with the pretopology formed by families of morphisms $\{f_i:X_i \to X\}_{i \in I}$ with I finite and $X=\bigcup _{i \in I} f_i(X_i)$ . Consider the functor $u^+:\mathscr {C} \to \mathbf {I}_{\mathop {\mathrm {Spec}}(A[1/p]) \to \mathop {\mathrm {Spec}}(A)}$ sending X to $X^{Y}$ , where $Y=\mathop {\mathrm {Spec}}(A[1/p])\times _{\mathop {\mathrm {Spec}}(A)} X$ . It is well-defined by [Reference HeHe23, 3.19(4)] and commutes with fibered products by [Reference HeHe23, 3.21] and is continuous by [Reference HeHe23, 3.15]. Proposition 5.14 allows us to take a hypercovering $X_\bullet \to X$ in $\mathscr {C}$ meeting our requirement by [Sta23, 094K and 0DB1]. We see that $Y_\bullet \to Y$ is a proper hypercovering and that $X_\bullet ^{Y_\bullet }\to X^Y$ is a hypercovering in $\mathbf {I}_{\overline {\eta }\to \overline {S}}$ by the properties of $u^+$ [Sta23, 0DAY].

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

We denote by $u:(Y' \to X') \to (Y \to X)$ the natural morphism. We have $\mathbb {F}'=u_{\mathrm {\acute {e}t}}^{-1}\mathbb {F}$ . The natural morphisms in (5.5.1) induce a commutative diagram:

(5.16.1)

where $\gamma _1$ is induced by the canonical morphism $\mathbb {F} \to \mathrm {R} u_{\mathrm {\acute {e}t} *}u_{\mathrm {\acute {e}t}}^{-1}\mathbb {F}$ , and $\gamma _2$ (resp. $\gamma _3$ ) is induced by the composition of $\psi _*\mathbb {F} \to \mathrm {R} u_{\mathbf {E} *} u_{\mathbf {E}}^{-1}\psi _*\mathbb {F} \to \mathrm {R} u_{\mathbf {E} *}\psi ^{\prime }_*u_{\mathrm {\acute {e}t}}^{-1}\mathbb {F}$ (resp. and by the canonical morphism $\overline {\mathscr {B}} \to \mathrm {R} u_{\mathbf {E} *}\overline {\mathscr {B}}'$ ). Since $\alpha _1$ and $\alpha _1'$ are isomorphisms by 5.8, it suffices to show that $\gamma _1$ and $\gamma _3$ are almost isomorphisms.

Since $A/\cap _{n>0}p^nA\to (A/\cap _{n>0}p^nA)_{\sqrt {pA}}$ is injective whose cokernel is killed by $\sqrt {pA}$ [Reference HeHe23, 4.7], the morphism $A \to A'$ induces an almost isomorphism $A/p^nA \to A'/p^nA'$ for each n. Then, for any torsion abelian group M, the natural morphism $M\otimes _{\mathbb {Z}} A \to M \otimes _{\mathbb {Z}} A'$ is an almost isomorphism. Therefore, $\gamma _1$ is an almost isomorphism by the proper base change theorem over the strictly Henselian local ring $A[1/p]$ [SGA 4_III , XII 5.5, 5.4]. For $\gamma _3$ , it suffices to show that the canonical morphism $\psi _*\mathbb {F}\otimes \overline {\mathscr {B}} \to \mathrm {R} u_{\mathbf {E}*} (\psi ^{\prime }_*\mathbb {F}'\otimes \overline {\mathscr {B}}')$ is an almost isomorphism. The problem is local on $\mathbf {E}_{Y \to X}^{\mathrm {\acute {e}t}}$ , thus we may assume that $\psi _*\mathbb {F}$ is the constant sheaf with value $\mathbb {Z}/p^n\mathbb {Z}$ by 5.3. Then, $\psi ^{\prime }_*\mathbb {F}'$ is also the constant sheaf with value $\mathbb {Z}/p^n\mathbb {Z}$ by 5.9. Thus, it remains to show that $\overline {\mathscr {B}}/p^n\overline {\mathscr {B}} \to \mathrm {R} u_{\mathbf {E}*} (\overline {\mathscr {B}}'/p^n\overline {\mathscr {B}}')$ is an almost isomorphism. Consider the following commutative diagram of ringed sites (see 3.6):

(5.16.2)

By the cohomological descent for Faltings ringed topos (3.9), it remains to show that $\mathscr {O}/p^n\mathscr {O} \to \mathrm {R} u_{\mathbf {E}*} (\mathscr {O}'/p^n\mathscr {O}')$ is an almost isomorphism. Let W be an object of $\mathbf {I}_{Y \to X^Y}$ such that W is the spectrum of a $\overline {\mathbb {Z}_p}$ -algebra R which is almost pre-perfectoid [Reference HeHe23, 5.19]. Since the almost isomorphisms $R/p^n \to (R\otimes _A A')/p^n$ ( $n\geq 1$ ) induces an almost isomorphism of the p-adic completions $\widehat {R}\to R\widehat {\otimes }_A A'$ , the $\overline {\mathbb {Z}_p}$ -algebra $R\otimes _A A'$ is still almost pre-perfectoid [Reference HeHe23, 5.19]. The pullback of W in $\mathbf {I}_{Y' \to X^{{\prime }Y'}}$ is the spectrum of the integral closure $R'$ of $R\otimes _A A'$ in $R\otimes _A A'[1/p]$ [Reference HeHe23, (3.21.1)]. Since $R\otimes _A A'$ is almost pre-perfectoid, $R'$ is also almost pre-perfectoid and the morphism $(R\otimes _A A')/p^n \to R'/p^n$ is an almost isomorphism by [Reference HeHe23, 5.30]. Recall that such W forms a topological generating family of $\mathbf {I}_{Y \to X^Y}$ [Reference HeHe23, 8.10]. Therefore, the morphism $\mathscr {O}/p^n\mathscr {O} \to \mathrm {R} u_{\mathbf {E}*} (\mathscr {O}'/p^n\mathscr {O}')$ is an almost isomorphism by [Reference HeHe23, 8.11].

Proof Let K be the p-adic completion of the maximal unramified extension of $\mathbb {Q}_p$ . By 5.16, we may assume that A is a valuation ring of height $1$ extension of $\mathcal {O}_{\overline {K}}$ . Let $X_\bullet \to X$ be the proper hypercovering of coherent schemes constructed in 5.15. For each degree n, the canonical morphisms (5.6.2),

(5.17.2) $$ \begin{align} \mathrm{R}\Gamma(Y_{n,\mathrm{\acute{e}t}}, \mathbb{F}_n) \otimes_{\mathbb{Z}}^{\mathrm{L}} A \longleftarrow \mathrm{R}\Gamma(\mathbf{E}_{Y_n \to X_n}^{\mathrm{\acute{e}t}},\psi_{n*}\mathbb{F}_n) \otimes_{\mathbb{Z}}^{\mathrm{L}} A \longrightarrow \mathrm{R}\Gamma(\mathbf{E}_{Y_n \to X_n}^{\mathrm{\acute{e}t}},\psi_{n*}\mathbb{F}_n\otimes_{\mathbb{Z}} \overline{\mathscr{B}}) \end{align} $$

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

(5.17.3)

where $\mathbb {F}_\bullet =(\mathbb {F}_n)_{[n]\in \mathrm {Ob}(\Delta )}$ (see [Reference HeHe23, 6.5] for the notation). By the functorial spectral sequence of simplicial sites [Sta23, 09WJ], we deduce from (5.17.2) that $\alpha _{1\bullet }$ is an isomorphism and $\alpha _{2\bullet }$ is an almost isomorphism. Since $\alpha _1$ is an isomorphism by 5.8, it remains to show that the left vertical arrow is an isomorphism and the right vertical arrow is an almost isomorphism.

We denote by $b:\mathbf {E}_{Y_{\bullet } \to X_\bullet }^{\mathrm {\acute {e}t}}\to \mathbf {E}_{Y \to X}^{\mathrm {\acute {e}t}}$ the augmentation of simplicial site and by $b_n:\mathbf {E}_{Y_{n} \to X_n}^{\mathrm {\acute {e}t}}\to \mathbf {E}_{Y \to X}^{\mathrm {\acute {e}t}}$ the natural morphism of sites. Notice that $b^{-1}\psi _*\mathbb {F}=(b_n^{-1}\psi _*\mathbb {F})_{[n]\in \mathrm {Ob}(\Delta )}=(\psi _{n *}\mathbb {F}_n)_{[n]\in \mathrm {Ob}(\Delta )}=\psi _{\bullet *}\mathbb {F}_\bullet $ by 5.9 [Sta23, 0D70]. Since $X_\bullet ^{Y_\bullet } \to X^Y$ forms a hypercovering in $\mathbf {I}_{\overline {\eta }\to \overline {S}}$ , the right vertical arrow is an almost isomorphism by 5.3 and 3.10. Finally, the left vertical arrow is an isomorphism by the cohomological descent for étale cohomology [Sta23, 0DHL].

6.1 Let $Y \to X$ be a morphism of coherent schemes such that $Y \to X^Y$ is an open immersion. Recall that there is a natural commutative diagram of sites (see 3.6):

(6.1.1)

Proof The base change morphism $\varepsilon ^{-1}\psi _*\mathbb {F}\to \Psi _*a^{-1}\mathbb {F}$ is the composition of the adjunction morphisms [SGA 4_III , XVII.2.1.3]

(6.2.1) $$ \begin{align} \varepsilon^{-1}\psi_*\mathbb{F}\to \Psi_*\Psi^{-1}(\varepsilon^{-1}\psi_*\mathbb{F})=\Psi_*a^{-1}(\psi^{-1}\psi_*\mathbb{F})\to \Psi_*a^{-1}\mathbb{F} \end{align} $$

which are both isomorphisms by 3.8(2) and 5.3.

6.3 We fix an algebraic closure $\overline {\mathbb {Q}_p}$ of $\mathbb {Q}_p$ and we denote by $\overline {\mathbb {Z}_p}$ the integral closure of $\mathbb {Z}_p$ in $\overline {\mathbb {Q}_p}$ . We set $\eta = \mathop {\mathrm {Spec}} (\mathbb {Q}_p)$ , $\overline {\eta }=\mathop {\mathrm {Spec}}(\overline {\mathbb {Q}_p})$ , $S=\mathop {\mathrm {Spec}} (\mathbb {Z}_p)$ , $\overline {S}=\mathop {\mathrm {Spec}}(\overline {\mathbb {Z}_p})$ . Remark that $\overline {\mathbb {Q}_p}$ is a pre-perfectoid field with valuation ring $\overline {\mathbb {Z}_p}$ so we are also in the situation of 3.9. Let $f:(Y'\to X')\to (Y\to X)$ be a Cartesian morphism of morphisms of coherent schemes with a Cartesian morphism $(Y\to X^Y)\to (\overline {\eta } \to \overline {S})$ (then, $Y'\to X^{{\prime }Y'}$ is Cartesian over $\overline {\eta } \to \overline {S}$ by [Reference HeHe23, 3.19(4)]). Thus, $X^Y$ and $X^{{\prime }Y'}$ are objects of $\mathbf {I}_{\overline {\eta } \to \overline {S}}$ . Consider the following commutative diagram of sites associated with f (see 3.6):

(6.3.1)

6.4 Following 6.3, let $g: (\widetilde {Y}\to \widetilde {X})\to (Y\to X)$ be a morphism of coherent schemes such that $\widetilde {Y}{\kern-1pt}\to{\kern-1pt} \widetilde {X}^{\widetilde {Y}}$ is also Cartesian over $\overline {\eta } {\kern-1pt}\to{\kern-1pt} \overline {S}$ . We denote by $g':(\widetilde {Y'}{\kern-1pt}\to{\kern-1pt} \widetilde {X'}){\kern-1pt}\to (Y'\to X')$ the base change of g by f, and denote by $\widetilde {f}: (\widetilde {Y'}\to \widetilde {X'})\to (\widetilde {Y}\to \widetilde {X})$ the natural morphism which is Cartesian by base change. Thus, $\widetilde {X}^{\widetilde {Y}}$ and $\widetilde {X'}^{\widetilde {Y'}}$ are also objects of $\mathbf {I}_{\overline {\eta } \to \overline {S}}$ . We write the diagram (6.3.1) associated with $\widetilde {f}$ by equipping all labels with tildes.

Proof Let $\widetilde {\mathscr {F}'}$ be the restriction of $\mathscr {F}'$ on $\mathbf {I}_{\widetilde {Y'}\to \widetilde {X'}^{\widetilde {Y'}}}$ . We have $\widetilde {\mathscr {F}'}=\widetilde {\Psi '}_*\widetilde {a'}^{-1}\widetilde {\mathbb {F}'}$ . We set $\widetilde {\mathbb {L}'}=\widetilde {\psi '}_*\widetilde {\mathbb {F}'}$ which is a finite locally constant abelian sheaf on $\mathbf {E}_{\widetilde {Y'} \to \widetilde {X'}}^{\mathrm {\acute {e}t}}$ by 5.3. Notice that the canonical morphisms $\widetilde {\psi '}^{-1}\widetilde {\mathbb {L}'}\to \widetilde {\mathbb {F}'}$ and $\widetilde {\varepsilon '}^{-1}\widetilde {\mathbb {L}'} \to \widetilde {\mathscr {F}'}$ are isomorphisms by 5.3 and 6.2, respectively.

(1) It follows from the canonical isomorphisms

(6.5.1) $$ \begin{align} H^q(\mathbf{I}_{\widetilde{Y'}\to \widetilde{X'}^{\widetilde{Y'}}},\widetilde{\varepsilon'}^{-1}\widetilde{\mathbb{L}'})\stackrel{\gamma_1}{\longrightarrow} H^q_{\mathrm{v}}(\widetilde{Y'},\widetilde{\Psi'}^{-1}\widetilde{\varepsilon'}^{-1}\widetilde{\mathbb{L}'})= H^q_{\mathrm{v}}(\widetilde{Y'},\widetilde{a'}^{-1}\widetilde{\psi'}^{-1}\widetilde{\mathbb{L}'}) \stackrel{\gamma_2}{\longleftarrow} H^q_{\mathrm{\acute{e}t}}(\widetilde{Y'},\widetilde{\psi'}^{-1}\widetilde{\mathbb{L}'}), \end{align} $$

where $\gamma _1$ is induced by the canonical isomorphism $\widetilde {\varepsilon '}^{-1}\widetilde {\mathbb {L}'}\stackrel {\sim }{\longrightarrow } \mathrm {R}\widetilde {\Psi '}_*\widetilde {\Psi '}^{-1}\widetilde {\varepsilon '}^{-1}\widetilde {\mathbb {L}'}$ (3.8(2)), and $\gamma _2$ is induced by the canonical isomorphism $\widetilde {\psi '}^{-1}\widetilde {\mathbb {L}'}\to \mathrm {R}\widetilde {a'}_*\widetilde {a'}^{-1}\widetilde {\psi '}^{-1}\widetilde {\mathbb {L}'}$ (3.4).

(2) It follows from the canonical almost isomorphism

(6.5.2) $$ \begin{align} \gamma_3: H^q(\mathbf{E}_{\widetilde{Y'}\to \widetilde{X'}}^{\mathrm{\acute{e}t}},\widetilde{\mathbb{L}'}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}')\longrightarrow H^q(\mathbf{I}_{\widetilde{Y'}\to \widetilde{X'}^{\widetilde{Y'}}},\widetilde{\varepsilon'}^{-1}\widetilde{\mathbb{L}'}\otimes \mathscr{O}'), \end{align} $$

which is induced by the canonical almost isomorphism $\widetilde {\mathbb {L}'}{\kern-1pt}\otimes _{\mathbb {Z}}\overline {\mathscr {B}}'{\kern-1pt}\to{\kern-1pt} \mathrm {R}\widetilde {\varepsilon '}_*(\widetilde {\varepsilon '}^{-1}\widetilde {\mathbb {L}'}{\kern-1pt}\otimes{\kern-1pt} \mathscr {O}')$ (3.9).

(3) Consider the following diagram:

(6.5.3)

where the unlabeled vertical arrow is induced by the canonical morphism $\widetilde {\mathbb {L}'}\to \mathrm {R}\widetilde {\varepsilon '}_*\widetilde {\varepsilon '}^{-1}\widetilde {\mathbb {L}'}$ , and the unlabeled horizontal arrow is the canonical morphism which induces $(\mathrm {R}^q f_{\mathbf {I} *}\mathscr {F}')\otimes _{\mathbb {Z}}\mathscr {O}\to \mathrm {R}^q f_{\mathbf {I} *}(\mathscr {F}'\otimes _{\mathbb {Z}}\mathscr {O}')$ on $\mathbf {I}_{Y \to X^{Y}}$ by sheafification. It is clear that the diagram (6.5.3) is commutative, which completes the proof.

6.6 We remark that 6.5 gives a new definition of the relative (resp. absolute) Faltings’ comparison morphism without using 5.8. Following 6.3, let $\mathbb {F}'$ be a finite locally constant abelian sheaf on $Y^{\prime }_{\mathrm {\acute {e}t}}$ and we set $\mathscr {F}'=\Psi ^{\prime }_*a^{{\prime }-1}\mathbb {F}'$ . We set $\mathbb {L}'=\psi ^{\prime }_*\mathbb {F}'$ , which is a finite locally constant abelian sheaf on $\mathbf {E}_{Y'\to X'}^{\mathrm {\acute {e}t}}$ by 5.3. Remark that the canonical morphisms $\psi ^{{\prime }-1}\mathbb {L}'\to \mathbb {F}'$ and $\varepsilon ^{{\prime }-1}\mathbb {L}'\to \mathscr {F}'$ are isomorphisms by 5.3 and 6.2, respectively. We also remark that $\overline {\mathscr {B}}$ , $\mathscr {O}$ are flat over $\mathbb {Z}$ . The canonical morphisms in the derived category $\mathbf {D}(\overline {\mathscr {B}}\textrm {-}\mathbf {Mod}_{\mathbf {E}_{Y \to X}^{\mathrm {\acute {e}t}}})$ (cf. (5.4.2)),

(6.6.1)

fit into the following commutative diagram:

(6.6.2)

1. (1) The morphism $\alpha _3$ is induced by the canonical isomorphism $\psi ^{{\prime }-1}\mathbb {L}'\to \mathrm {R} a^{\prime }_*a^{{\prime }-1} (\psi ^{{\prime }-1}\mathbb {L}')$ by 3.4, and thus $\alpha _3$ is an isomorphism.

2. (2) The morphism $\alpha _5$ is induced by the canonical isomorphism $\varepsilon ^{{\prime }-1}\mathbb {L}' \to \mathrm {R} \Psi ^{\prime }_*\Psi ^{{\prime }-1}\varepsilon ^{{\prime }-1}\mathbb {L}'$ by 3.8(2), and thus $\alpha _5$ is an isomorphism.

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

4. (4) The morphism $\alpha _4$ is induced by the canonical almost isomorphism $\mathbb {L}'\otimes _{\mathbb {Z}} \overline {\mathscr {B}}' \to \mathrm {R} \varepsilon ^{\prime }_* (\varepsilon ^{{\prime }-1}\mathbb {L}'\otimes _{\mathbb {Z}} \mathscr {O}')$ by 3.9, and thus $\alpha _4$ is an almost isomorphism.

5. (5) The morphism $\alpha _6$ is the composition of

   (6.6.3) $$ \begin{align} \mathrm{R} \varepsilon_*(\mathrm{R} f_{\mathbf{I} *}\varepsilon^{{\prime}-1}\mathbb{L}') \otimes_{\mathbb{Z}}^{\mathrm{L}} \overline{\mathscr{B}} &\longrightarrow \mathrm{R} \varepsilon_*((\mathrm{R} f_{\mathbf{I} *}\varepsilon^{{\prime}-1}\mathbb{L}') \otimes_{\mathbb{Z}}^{\mathrm{L}} \mathscr{O}) \end{align} $$
   (6.6.4) $$ \begin{align} \textrm{ with } \mathrm{R} \varepsilon_*((\mathrm{R} f_{\mathbf{I} *}\varepsilon^{{\prime}-1}\mathbb{L}') \otimes_{\mathbb{Z}}^{\mathrm{L}} \mathscr{O}) &\longrightarrow \mathrm{R} \varepsilon_* \mathrm{R} f_{\mathbf{I} *}(\varepsilon^{{\prime}-1}\mathbb{L}'\otimes_{\mathbb{Z}} \mathscr{O}'). \end{align} $$

In conclusion, the arrows $\alpha _3$ , $\alpha _5$ , $\alpha _6$ and $\alpha _4$ induce an arrow

(6.6.5) $$ \begin{align} \alpha_4^{-1}\circ\alpha_6\circ \alpha_5^{-1}\circ\alpha_3: \mathrm{R}\psi_*(\mathrm{R} f_{\mathrm{\acute{e}t} *}\mathbb{F}') \otimes_{\mathbb{Z}}^{\mathrm{L}} \overline{\mathscr{B}} \longrightarrow \mathrm{R} f_{\mathbf{E} *}(\psi^{\prime}_*\mathbb{F}'\otimes_{\mathbb{Z}}\overline{\mathscr{B}}') \end{align} $$

in the derived category of almost $\overline {\mathbb {Z}_p}$ -modules on $\mathbf {E}_{Y \to X}^{\mathrm {\acute {e}t}}$ [Reference HeHe23, 5.7]. Remark that we don’t assume that $\alpha _1$ is an isomorphism here. We also call (6.6.5) the relative Faltings’ comparison morphism. Indeed, if $\alpha _1$ is an isomorphism, then the relative Faltings’ comparison morphism (5.6.1) induces (6.6.5) in $\mathbf {D}(\overline {\mathbb {Z}_p}^{\mathrm {al}}\textrm {-}\mathbf {Mod})$ (the derived category of almost $\overline {\mathbb {Z}_p}$ -modules, see [Reference HeHe23, 5.7]) due to the commutativity of the diagram (6.6.2).

If X is the spectrum of an absolutely integrally closed valuation ring A and if $Y=\overline {\eta }\times _{\overline {S}} X$ , then applying the functor $\mathrm {R}\Gamma (Y \to X,-)$ on (6.6.2) we obtain the natural morphisms in the derived category $\mathbf {D}(A\textrm {-}\mathbf {Mod})$ by 3.7 making the following diagram commutative.

(6.6.6)

The arrows $\alpha _3$ , $\alpha _5$ , $\alpha _6,$ and $\alpha _4$ induce an arrow

(6.6.7) $$ \begin{align} \alpha_4^{-1}\circ\alpha_6\circ \alpha_5^{-1}\circ\alpha_3: \mathrm{R}\Gamma(Y^{\prime}_{\mathrm{\acute{e}t}}, \mathbb{F}') \otimes_{\mathbb{Z}}^{\mathrm{L}} A \longrightarrow \mathrm{R}\Gamma(\mathbf{E}_{Y' \to X'}^{\mathrm{\acute{e}t}},\psi^{\prime}_*\mathbb{F}'\otimes_{\mathbb{Z}} \overline{\mathscr{B}}') \end{align} $$

in the derived category $\mathbf {D}(\overline {\mathbb {Z}_p}^{\mathrm {al}}\textrm {-}\mathbf {Mod})$ of almost $\overline {\mathbb {Z}_p}$ -modules [Reference HeHe23, 5.7]. We also call (6.6.7) the absolute Faltings’ comparison morphism.

Proof Following 6.5, consider the following presheaves on $\mathbf {I}_{Y \to X^{Y}}$ for each integer q:

(6.7.2) $$ \begin{align} \mathcal{H}_1^q&: \widetilde{X}\longmapsto H^q_{\mathrm{\acute{e}t}}(\widetilde{Y'},\widetilde{\mathbb{F}'})\otimes_{\mathbb{Z}} \overline{\mathscr{B}}(\widetilde{Y}\to \widetilde{X}), \end{align} $$

(6.7.3) $$ \begin{align} \mathcal{H}_2^q&: \widetilde{X} \longmapsto H^q(\mathbf{E}_{\widetilde{Y'}\to \widetilde{X'}}^{\mathrm{\acute{e}t}},\widetilde{\psi'}_*\widetilde{\mathbb{F}'})\otimes_{\mathbb{Z}} \overline{\mathscr{B}}(\widetilde{Y}\to \widetilde{X}), \end{align} $$

(6.7.4) $$ \begin{align} \mathcal{H}_3^q&: \widetilde{X} \longmapsto H^q(\mathbf{E}_{\widetilde{Y'}\to \widetilde{X'}}^{\mathrm{\acute{e}t}},\widetilde{\psi'}_*\widetilde{\mathbb{F}'}\otimes_{\mathbb{Z}}\overline{\mathscr{B}}'). \end{align} $$

They satisfy the limit-preserving condition [Reference HeHe23, 3.25(ii)] by [Reference HeHe23, 7.12], [SGA 4_II , VII.5.6], and [SGA 4_II , VI 8.5.9, 8.7.3]. Moreover, if $\widetilde {X}=\mathop {\mathrm {Spec}}(A),$ where A is an absolutely integrally closed valuation ring with p nonzero in A, then the canonical morphisms

(6.7.5) $$ \begin{align} \mathcal{H}_1^q(\mathop{\mathrm{Spec}}(A))\leftarrow \mathcal{H}_2^q(\mathop{\mathrm{Spec}}(A)) \to \mathcal{H}_3^q(\mathop{\mathrm{Spec}}(A)) \end{align} $$

are, respectively, an isomorphism and an almost isomorphism by 5.17. Thus, the canonical morphisms $\mathcal {H}_1^q\leftarrow \mathcal {H}_2^q \to \mathcal {H}_3^q$ induce an isomorphism and an almost isomorphism of their sheafifications by [Reference HeHe23, 3.25]. The conclusion follows from 6.5.

Proof Firstly, assume that X is Noetherian. We have $Y'=Y\times _X X^Y$ by [Reference HeHe23, 3.19(4)]. We write $X^Y=\mathop {\mathrm {Spec}}_X(\mathcal {A})$ where $\mathcal {A}$ is an integral quasi-coherent $\mathcal {O}_X$ -algebra on X, and we write $\mathcal {A}$ as a filtered colimit of its finite quasi-coherent $\mathcal {O}_X$ -subalgebras $\mathcal {A}=\mathop {\mathrm {colim}} \mathcal {A}_\alpha $ [Sta23, 0817]. Let $\mathcal {B}_\alpha $ be the restriction of $\mathcal {A}_\alpha $ to Y. Then, $\mathcal {B}=\mathop {\mathrm {colim}} \mathcal {B}_\alpha $ is a filtered colimit of finite quasi-coherent $\mathcal {O}_Y$ -algebras with injective transition morphisms. Since $Y'=\mathop {\mathrm {Spec}}_Y(\mathcal {B})$ is finite over Y, there exists an index $\alpha _0$ such that $Y'=\mathop {\mathrm {Spec}}_Y(\mathcal {B}_{\alpha _0})$ . Therefore, $X'=\mathop {\mathrm {Spec}}_X(\mathcal {A}_{\alpha _0})$ meets our requirements.

In general, we write X as a cofiltered limit of coherent schemes of finite type over $\mathbb {Z}$ with affine transition morphisms $X=\lim _{\lambda \in \Lambda } X_\lambda $ [Sta23, 01ZA]. Since $Y\to X$ is an open immersion of finite presentation, using [Reference GrothendieckEGA IV₃ , 8.8.2, 8.10.5] there exists an index $\lambda _0\in \Lambda $ , an open immersion $Y_{\lambda _0}\to X_{\lambda _0}$ and a finite morphism $Y^{\prime }_{\lambda _0}\to Y_{\lambda _0}$ such that the base change of the morphisms $Y^{\prime }_{\lambda _0}\to Y_{\lambda _0} \to X_{\lambda _0}$ by $X\to X_{\lambda _0}$ are the morphisms $Y'\to Y \to X$ . By the first paragraph, there exists a finite morphism $X^{\prime }_{\lambda _0}\to X_{\lambda _0}$ of finite presentation such that $Y^{\prime }_{\lambda _0}=Y_{\lambda _0}\times _{X_{\lambda _0}} X^{\prime }_{\lambda _0}$ . We see that the base change $X'\to X$ of $X^{\prime }_{\lambda _0}\to X_{\lambda _0}$ by $X\to X_{\lambda _0}$ meets our requirements.

Proof There exists a Cartesian morphism $g:(Y"\to X")\to (Y'\to X^{Y}\times _X X')$ of open immersions of coherent schemes such that $X" \to X^{Y}\times _X X'$ is finite and of finite presentation by 6.8. Consider the diagram (6.3.1) associated with g:

(6.9.2)

We set $\mathscr {G}"=\Psi ^{{\prime }\prime }_*a^{{\prime }{\prime }-1}\mathbb {F}"$ . As $g:Y"\to Y'$ is finite, the base change morphism $a^{{\prime }-1}g_{\mathrm {\acute {e}t} *}\to g_{\mathrm {v}*}a^{{\prime }{\prime }-1}$ induces a canonical isomorphism $\mathscr {F}'\stackrel {\sim }{\longrightarrow } g_{\mathbf {I}*}\mathscr {G}"$ by [Reference HeHe23, 3.10]. Moreover, the canonical morphism $g_{\mathbf {I} *}\mathscr {G}" \to \mathrm {R} g_{\mathbf {I} *} \mathscr {G}"$ is an isomorphism by 6.5(1) and [Reference HeHe23, 3.25], since $g:Y"\to Y'$ is finite [SGA 4_II , VIII.5.6]. By applying 6.7 to g and $\mathbb {F}"$ , the canonical morphism

(6.9.3) $$ \begin{align} (\mathrm{R} g_{\mathbf{I} *}\mathscr{G}")\otimes_{\mathbb{Z}}^{\mathrm{L}} \mathscr{O}' \longrightarrow \mathrm{R} g_{\mathbf{I} *}(\mathscr{G}"\otimes_{\mathbb{Z}} \mathscr{O}") \end{align} $$

is an almost isomorphism. Let h be the composition of $(Y"\to X")\to (Y'\to X^{Y} \times _X X')\to (Y\to X^Y)$ . Note that $X"\to X^Y$ is also proper of finite presentation. By applying 6.7 to h and $\mathbb {F}"$ , the canonical morphism

(6.9.4) $$ \begin{align} (\mathrm{R} h_{\mathbf{I} *}\mathscr{G}")\otimes_{\mathbb{Z}}^{\mathrm{L}} \mathscr{O} \longrightarrow \mathrm{R} h_{\mathbf{I} *}(\mathscr{G}"\otimes_{\mathbb{Z}} \mathscr{O}") \end{align} $$

is an almost isomorphism. It is clear that $h_{\mathbf {I}}=f_{\mathbf {I}}\circ g_{\mathbf {I}}$ . The conclusion follows from the canonical isomorphism $\mathscr {F}'\to \mathrm {R} g_{\mathbf {I} *}\mathscr {G}"$ and the canonical almost isomorphisms (6.9.3) and (6.9.4).

Proof We prove by induction on an integer q that the canonical morphism $(\mathrm {R}^q f_{\mathbf {I} *}\mathscr {F}')\otimes _{\mathbb {Z}} \mathscr {O} \to \mathrm {R}^q f_{\mathbf {I} *}(\mathscr {F}'\otimes _{\mathbb {Z}} \mathscr {O}')$ is an almost isomorphism. It holds trivially for each $q\leq -1$ . Notice that there exists a finite morphism $g:Y"\to Y'$ of finite presentation, a finite locally constant abelian sheaf $\mathbb {F}"$ on $Y^{{\prime }\prime }_{\mathrm {\acute {e}t}}$ and an injective morphism $\mathcal {F}'\to g_{\mathrm {\acute {e}t}*}\mathbb {F}"$ by [Sta23, 09Z7] (cf. [SGA 4_III , IX.2.14]). Let $\mathcal {G}'$ be the quotient of $\mathcal {F}'\to g_{\mathrm {\acute {e}t}*}\mathbb {F}"$ , which is also a constructible abelian sheaf on $Y^{\prime }_{\mathrm {\acute {e}t}}$ since $g_{\mathrm {\acute {e}t}*}\mathbb {F}"$ is so [Sta23, 095R, 03RZ]. The exact sequence $0\to \mathcal {F}' \to g_{\mathrm {\acute {e}t}*}\mathbb {F}" \to \mathcal {G}'\to 0$ induces an exact sequence by 3.8(1),

(6.10.2)

We set $\mathscr {H}'=\Psi ^{\prime }_*a^{{\prime }-1}(g_{\mathrm {\acute {e}t}*}\mathbb {F}")$ and $\mathscr {G}'=\Psi ^{\prime }_*a^{{\prime }-1}\mathcal {G}'$ . Then, we obtain a morphism of long exact sequences:

(6.10.3)

Notice that $\gamma _1$ and $\gamma _2$ are almost isomorphisms by induction, and that $\gamma _4$ is an almost isomorphism by 6.9. Thus, applying the 5-lemma [Sta23, 05QA] in the abelian category of almost $\overline {\mathbb {Z}_p}$ -modules over $\mathbf {I}_{Y \to X^{Y}}$ , we see that $\gamma _3$ is almost injective. Since $\mathcal {F}'$ is an arbitrary constructible abelian sheaf, the morphism $\gamma _5$ is also almost injective. Thus, $\gamma _3$ is an almost isomorphism.

Proof We write $\mathcal {F}'$ as a filtered colimit of constructible abelian sheaves $\mathcal {F}'=\mathop {\mathrm {colim}}_{\lambda \in \Lambda }\mathcal {F}^{\prime }_\lambda $ ([Sta23, 03SA], cf. [SGA 4_III , IX.2.7.2]). We set $\mathscr {F}^{\prime }_\lambda =\Psi ^{\prime }_*a^{{\prime }-1}\mathcal {F}_\lambda $ . We have $\mathscr {F}'=\mathop {\mathrm {colim}}_{\lambda \in \Lambda }\mathscr {F}^{\prime }_\lambda $ by [SGA 4_II , VI.5.1] whose conditions are satisfied since each object in each concerned site is quasi-compact. Moreover, for each integer q, we have

(6.11.2) $$ \begin{align} (\mathrm{R}^q f_{\mathbf{I} *}\mathscr{F}')\otimes_{\mathbb{Z}} \mathscr{O}=&\mathop{\mathrm{colim}}_{\lambda\in \Lambda} (\mathrm{R}^q f_{\mathbf{I} *}\mathscr{F}^{\prime}_\lambda)\otimes_{\mathbb{Z}} \mathscr{O}, \end{align} $$

(6.11.3) $$ \begin{align} \mathrm{R}^q f_{\mathbf{I} *}(\mathscr{F}'\otimes_{\mathbb{Z}} \mathscr{O}')=& \mathop{\mathrm{colim}}_{\lambda\in \Lambda} \mathrm{R}^q f_{\mathbf{I} *}(\mathscr{F}^{\prime}_\lambda\otimes_{\mathbb{Z}} \mathscr{O}'). \end{align} $$

The conclusion follows from 6.10.

Proof The first assertion follows from the canonical isomorphisms $\mathrm {R}\Gamma (Y_{\mathrm {\acute {e}t}},\mathcal {F})\stackrel {\sim }{\longrightarrow } \mathrm {R}\Gamma ((\mathbf {Sch}^{\mathrm {coh}}_{/Y})_{\mathrm {v}},a^{-1}\mathcal {F})=\mathrm {R}\Gamma (\mathbf {I}_{Y \to X^Y},\mathrm {R}\Psi _*a^{-1}\mathcal {F}) \stackrel {\sim }{\longleftarrow }\mathrm {R}\Gamma (\mathbf {I}_{Y \to X^Y},\Psi _*a^{-1}\mathcal {F})$ by 3.4 and 3.8(1) (cf. 6.13(1)). The second assertion follows from applying 6.11 to the morphism $X\to \mathop {\mathrm {Spec}}(A)$ and from the fact that the associated topos of $\mathbf {I}_{\mathop {\mathrm {Spec}}(A[1/p])\to \mathop {\mathrm {Spec}}(A)}$ is local (3.7).

Proof Note that the canonical morphism $\mathscr {F}'\to \mathrm {R}\Psi ^{\prime }_*a^{{\prime }-1}\mathcal {F}'$ is an isomorphism by 3.8(1). Thus, $\mathrm {R} f_{\mathbf {I} *}\mathscr {F}'=\mathrm {R}(\Psi \circ f_{\mathrm {v}})_*a^{{\prime }-1}\mathcal {F}'$ , whose qth cohomology is the sheaf associated with the presheaf $\widetilde {X}\mapsto H^q_{\mathrm {v}}(\widetilde {Y'}, \widetilde {a'}^{-1}\widetilde {\mathcal {F}'})=H^q_{\mathrm {\acute {e}t}}(\widetilde {Y'}, \widetilde {\mathcal {F}'})$ by 3.4, and thus (1) follows. If $Y'\to Y$ is proper, then the base change morphism $a^{-1}\mathrm {R} f_{\mathrm {\acute {e}t} *}\to \mathrm {R} f_{\mathrm {v}*}a^{{\prime }-1}$ induces an isomorphism $a^{-1} \mathcal {H}\stackrel {\sim }{\longrightarrow } \mathrm {R} f_{\mathrm {v}*}a^{{\prime }-1}\mathcal {F}'$ by [Reference HeHe23, 3.10], and thus (2) follows.

Proof We follow the discussion of 6.6 and set $\mathscr {F}'=\Psi ^{\prime }_*a^{{\prime }-1}\mathbb {F}'$ . The canonical morphism (6.6.4)

(6.14.2) $$ \begin{align} \mathrm{R} \varepsilon_*((\mathrm{R} f_{\mathbf{I} *}\mathscr{F}') \otimes_{\mathbb{Z}}^{\mathrm{L}} \mathscr{O}) \longrightarrow \mathrm{R} \varepsilon_* \mathrm{R} f_{\mathbf{I} *}(\mathscr{F}'\otimes_{\mathbb{Z}} \mathscr{O}') \end{align} $$

is an almost isomorphism by 6.7. It remains to show that the canonical morphism (6.6.3)

(6.14.3) $$ \begin{align} \mathrm{R} \varepsilon_*(\mathrm{R} f_{\mathbf{I} *}\mathscr{F}') \otimes_{\mathbb{Z}}^{\mathrm{L}} \overline{\mathscr{B}} \longrightarrow \mathrm{R} \varepsilon_*((\mathrm{R} f_{\mathbf{I} *}\mathscr{F}') \otimes_{\mathbb{Z}}^{\mathrm{L}} \mathscr{O}) \end{align} $$

is also an almost isomorphism. With the notation in 6.13 by taking $\mathcal {F}'=\mathbb {F}'$ , the complex $\mathcal {H}$ is a bounded complex whose cohomologies $H^q(\mathcal {H})$ are finite locally constant abelian sheaves by condition (ii). Consider the commutative diagram (3.6.4):

(6.14.4)

We set $\mathcal {L}=\mathrm {R}\psi _*\mathcal {H}$ . Then, $H^q(\mathcal {L})=\psi _*H^q(\mathcal {H})$ by Cartan–Leray spectral sequence and condition (iii). Hence, $\mathcal {L}$ is a bounded complex of abelian sheaves whose cohomologies are finite locally constant by 5.3 so that the canonical morphism

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

is an almost isomorphism by 3.9 (cf. [Reference HeHe23, 8.15]).

On the other hand, $H^q(\mathscr {H})=\Psi _*a^{-1}H^q(\mathcal {H})$ by Cartan–Leray spectral sequence and 3.8(1). Thus, the base change morphism $\varepsilon ^{-1}\mathrm {R}\psi _*\to \mathrm {R}\Psi _*a^{-1}$ induces an isomorphism $\varepsilon ^{-1}\mathcal {L}\stackrel {\sim }{\longrightarrow } \mathscr {H}$ by 6.2. Moreover, the canonical morphism $\mathcal {L}\to \mathrm {R}\varepsilon _*\varepsilon ^{-1}\mathcal {L}=\mathrm {R}\varepsilon _*\mathscr {H}=\mathrm {R}\psi _*\mathrm {R} a_*a^{-1}\mathcal {H}$ is an isomorphism by 3.4. Thus, the canonical morphism

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

is an almost isomorphism by (6.14.5). In conclusion, (6.14.3) is an almost isomorphism by (6.14.6) and by the canonical isomorphisms $\varepsilon ^{-1}\mathcal {L}\stackrel {\sim }{\longrightarrow } \mathscr {H}\stackrel {\sim }{\longrightarrow } \mathrm {R} f_{\mathbf {I} *}\mathscr {F}'$ .