Uniform local constancy of \'etale cohomology of rigid analytic varieties

We prove some $\ell$-independence results on local constancy of \'etale cohomology of rigid analytic varieties. As a result, we show that a closed subscheme of a proper scheme over an algebraically closed complete non-archimedean field has a small open neighborhood in the analytic topology such that, for every prime number $\ell$ different from the residue characteristic, the closed subscheme and the open neighborhood have the same \'etale cohomology with $\mathbb Z/\ell \mathbb Z$-coefficients. The existence of such an open neighborhood for each $\ell$ was proved by Huber. A key ingredient in the proof is a uniform refinement of a theorem of Orgogozo on the compatibility of the nearby cycles over general bases with base change.


Introduction
Let K be an algebraically closed complete non-archimedean field whose topology is given by a valuation | · | : K → R ≥0 of rank 1. Let O = K • be the ring of integers of K.
In this paper, we study local constancy ofétale cohomology of rigid analytic varieties over K, or more precisely, of adic spaces of finite type over Spa(K, O).

1.1.
A main result. The theory ofétale cohomology for adic spaces was developed by Huber; see [11]. Huber obtained several finiteness results onétale cohomology of adic spaces in a series of papers [12,13,15]. Let us recall one of the main results of [13]; see [13,Theorem 3.6] for a more precise statement. Theorem 1.1 (Huber [13,Theorem 3.6]). We assume that K is of characteristic zero. Let X be a separated adic space of finite type over Spa(K, O) and Z a closed adic subspace of X. Let n be a positive integer invertible in O. Then there exists an open subset V of X containing Z such that the restriction map H i (V, Z/nZ) → H i (Z, Z/nZ) onétale cohomology groups is an isomorphism for every integer i. Moreover, we can assume that V is quasi-compact.
It is a natural question to ask whether we can take an open subset V as in Theorem 1.1 independent of n. In the present paper, we answer this question in the affirmative for adic spaces which are arising from schemes of finite type over O.
More precisely, we will prove the following theorem. For a scheme X of finite type over O, let X denote the ̟-adic formal completion of X , where ̟ ∈ K × is an element with |̟| < 1. The Raynaud generic fiber of X is denoted by ( X ) rig in this section, which is an adic space of finite type over Spa(K, O). (It is denoted by d( X ) in [11] and in the main body of this paper.) Theorem 1.2 (Theorem 4.9). Let Z ֒→ X be a closed immersion of separated schemes of finite type over O. We have a closed embedding ( Z) rig ֒→ ( X ) rig . Then there exists an open subset V of ( X ) rig containing ( Z) rig such that, for every positive integer n invertible in O, the restriction map onétale cohomology groups is an isomorphism for every integer i. Moreover, we can assume that V is quasi-compact.
A more precise statement is given in Theorem 4.9. In this paper, we will use de Jong's alterations in several ways. This is the main reason why we restrict ourselves to the case where adic spaces are arising from schemes of finite type over O. We remark that, in our case, we need not impose any conditions on the characteristic of K. We will also prove an analogous statement forétale cohomology with compact support; see Theorem 4.8. Remark 1.3. In [26], Scholze proved the weight-monodromy conjecture for a projective smooth variety X over a non-archimedean local field L of mixed characteristic (0, p) which is a set-theoretic complete intersection in a projective smooth toric variety, by reduction to the function field case proved by Deligne. In the proof, Scholze used Theorem 1.1 to construct, for a fixed prime number ℓ = p, a projective smooth variety Y over a function field of characteristic p and an appropriate mapping frométale cohomology with Z/ℓZcoefficients of X to that of Y . The initial motivation for the present study is, following the method of Scholze, to prove that an analogue of the weight-monodromy conjecture holds forétale cohomology with Z/ℓZ-coefficients of such a variety X for all but finitely many ℓ = p by reduction to an ultraproduct variant of Weil II established by Cadoret [4]. For this, we shall use Theorem 1.2 instead of Theorem 1.1. The details will appear in a forthcoming paper [18].
1.2. Local constancy of higher direct images with proper support. For the proof of Theorem 1.2, we need to investigate local constancy of higher direct images with proper support for morphisms of adic spaces. Before stating our results on higher direct images with proper support, let us give an outline of the proof of Theorem 1.1.
Sketch of the proof of Theorem 1.1. We assume that K is of characteristic zero. Let Z ֒→ X be as in Theorem 1.1, which we may assume to be of finite presentation. By considering the blow-up of X along Z, we may assume further that the closed subscheme Z is defined by one global function f ∈ O X (X ). Let The inverse image (f rig ) −1 (0) of the origin 0 ∈ B(1) is the closed subspace ( Z) rig .
We fix a positive integer n invertible in O. We want to take an open subset V in Theorem 1.1 as the inverse image V = (f rig ) −1 (B(ǫ)) of the disc B(ǫ) ⊂ B(1) of radius ǫ centered at 0 for a small ǫ ∈ |K × |. Such a subset is called a tubular neighborhood of ( Z) rig . For this, we have to computeétale cohomology with Z/nZ-coefficients of (f rig ) −1 (B(ǫ)) for a small ǫ ∈ |K × |. By the Leray spectral sequence for f rig , it suffices to compute the cohomology group H i (B(ǫ), R j f rig * Z/nZ) for all i, j. The key steps are as follows.
In our case, the problem is to show that ǫ 0 and ǫ 1 in the above argument can be taken independent of n. To overcome this problem, by de Jong's alterations and by cohomological descent, we reduce to the case where there exists an element ǫ ∈ |K × | with ǫ ≤ 1 such that the restriction of f rig is smooth. In this case, we will analyze the higher direct image sheaf with proper support R j f rig ! Z/nZ on B(1), which is defined in [11,Definition 5.4.4]. An important fact is that, since f rig is smooth over B(ǫ)\{0}, the restriction (R j f rig ! Z/nZ)| B(ǫ)\{0} is a constructibleétale sheaf of Z/nZ-modules (in the sense of [11,Definition 2.7.2]) for every positive integer n invertible in O by [11,Theorem 6.2.2].
Our main result on local constancy of higher direct images with proper support is as follows. We do not suppose that K is of characteristic zero. For elements a, b ∈ |K × | with a < b ≤ 1, let B(a, b) ⊂ B(1) be the annulus with inner radius a and outer radius b centered at 0. Theorem 1.4 (Proposition 6.6 and Theorem 6.10). Let f : X → Spec O[T ] be a separated morphism of finite presentation. We assume that there exists an element ǫ ∈ |K × | with ǫ ≤ 1 such that the induced morphism is smooth over B(ǫ)\{0}. Then there exists an element ǫ 0 ∈ |K × | with ǫ 0 ≤ ǫ such that, for every positive integer n invertible in O, the following two assertions hold: (1) The restriction (R i f rig ! Z/nZ)| B(ǫ 0 )\{0} is a locally constant Z/nZ-sheaf of finite type for every i. Under the assumptions of Theorem 1.4, the same results hold for the higher direct image sheaf R i f rig * Z/nZ by Poincaré duality [11,Corollary 7.5.5], which will imply Theorem 1.2.
A key ingredient in the proof of Theorem 1.4 is the following uniform refinement of a theorem of Orgogozo [22,Théorème 2.1] on the compatibility of the sliced nearby cycles functors with base change. We also obtain a result on uniform unipotency of the sliced nearby cycles functors. See Section 2.1 for the definition of the sliced nearby cycles functors and see Definition 2.2 for the terminology used in the following theorem. Theorem 1.5 (Corollary 3.17). Let S be a Noetherian excellent scheme and g : Y → S a separated morphism of finite type. There exists an alteration S ′ → S such that, for every positive integer n invertible on S, the following assertions hold: (1) The sliced nearby cycles complexes for the base change g S ′ : Y S ′ → S ′ of g and the constant sheaf Z/nZ are compatible with any base change.
(2) The sliced nearby cycles complexes for g S ′ : Y S ′ → S ′ and the constant sheaf Z/nZ are unipotent.
Theorem 1.5 is a corollary of a more general result (Theorem 2.7), which may be of independent interest. The proof of Theorem 1.5 is quite similar to that of [23, Théorème 3.1.1]. In the proof, de Jong's alteration plays an important role.
By using a comparison theorem of Huber [11,Theorem 5.7.8], we will deduce Theorem 1.4 from Theorem 1.5. Roughly speaking, Theorem 1.4 (1) can be deduced from Theorem 1.5 (1) by considering a specialization map from an adic space of finite type over Spa(K, O) to its reduction; see Section 5.3 and Section 6.2 for details. Theorem 1.4 (2) can be deduced from Theorem 1.5 (2) and a study of the discriminant function δ h : [0, ∞) → R ≥0 associated with a finite Galoisétale covering h : Y → B(1)\{0} defined in [19,24,20]. See Section 6.1 and Appendix A for details.
1.3. The organization of this paper. This paper is organized as follows. In Section 2, we first recall the definition of the sliced nearby cycles functors. Then we formulate our main result (Theorem 2.7) on the sliced nearby cycles functors. In Section 3, we prove Theorem 2.7.
In Section 4, we recall the definition of tubular neighborhoods, and then we state our main results (Theorem 4.8 and Theorem 4.9) onétale cohomology of tubular neighborhoods. In Section 5, we recall a comparison theorem of Huber and use it to study the relation between higher direct images with proper support for morphisms of adic spaces and the sliced nearby cycles functors. In Section 6, we prove Theorem 1.4 in a slightly more general setting. In Section 7, we prove Theorem 4.8 and Theorem 4.9 (and hence Theorem 1.2) by using Theorem 1.4.
Finally, in Appendix A, we prove two theorems (Theorem 6.2 and Theorem 6.3) on finiteétale coverings of annuli in the unit disc, which are essentially proved in [19,24,20].

Nearby cycles over general bases
In this section, we formulate our main results on nearby cycles over general bases. We will use the following notation. Let f : X → S be a morphism of schemes. For a morphism T → S of schemes, the base change X × S T of X is denoted by X T and the base change of f is denoted by f T : X T → T . For a commutative ring Λ, let D + (X, Λ) be the derived category of bounded below complexes ofétale sheaves of Λ-modules on X. For a complex K ∈ D + (X, Λ), the pull-back of K to X T is denoted by K T . We often call anétale sheaf on X simply a sheaf on X.
2.1. Sliced nearby cycles functor. In this paper, a scheme is called a strictly local scheme if it is isomorphic to an affine scheme Spec R where R is a strictly Henselian local ring. Let f : X → S be a morphism of schemes. Let q : U → S be a morphism from a strictly local scheme U. The closed point of U is denoted by u. Let η ∈ U be a point. Let η → U be an algebraic geometric point lying above η, i.e. it is a geometric point lying above η such that the residue field κ(η) is a separable closure of the residue field κ(η) of η. The strict localization of U at η → U is denoted by U (η) . We have the following commutative diagram: Let Λ be a commutative ring. We have the following functor: This functor is called the sliced nearby cycles functor in [16]. For a complex K ∈ D + (X U , Λ), we have an action of the absolute Galois group Gal(κ(η)/κ(η)) on RΨ f U ,η (K) via the canonical isomorphism Aut(U (η) / Spec(O U,η )) ∼ = Gal(κ(η)/κ(η)).
Let q : V → U be a local morphism of strictly local schemes over S, i.e. a morphism over S which sends the closed point v of V to the closed point u of U. Let ξ ∈ V be a point with image η = q(ξ) ∈ U. For an algebraic geometric point ξ → V lying above ξ, we have an algebraic geometric point η → U lying above η by taking the separable closure of κ(η) in κ(ξ). We call η → U the image of ξ → V under the morphism q. We have the following commutative diagram: where the vertical morphisms are induced by q. For a complex K ∈ D + (X U , Λ), we have the following base change map: We will use the following terminology.
Definition 2.1. Let G be a group and X a scheme. We say that a sheaf F of Λ-modules on X with a G-action is G-unipotent if F has a finite filtration which is stable by the action of G such that the action of G on each successive quotient is trivial. We say that a complex K ∈ D + (X, Λ) with a G-action is G-unipotent if its cohomology sheaves are G-unipotent.
Definition 2.2. Let f : X → S be a morphism of schemes. Let Λ be a commutative ring and K ∈ D + (X, Λ) a complex.
(1) We say that the sliced nearby cycles complexes for f and K are compatible with any base change (or simply that the nearby cycles for f and K are compatible with any base change) if for every local morphism q : V → U of strictly local schemes over S and every algebraic geometric point ξ → V with image η → U, the base change map is an isomorphism. (2) We say that the sliced nearby cycles complexes for f and K are unipotent (or simply that the nearby cycles for f and K are unipotent) if for every morphism q : U → S from a strictly local scheme U, a point η ∈ U, and an algebraic geometric point η → U lying above η, the complex RΨ f U ,η (K U ) is Gal(κ(η)/κ(η))unipotent.
Remark 2.3. We can restate Definition 2.2 (1) in terms of vanishing topoi as follows.
Let f : X → S be a morphism of schemes. Let be the vanishing topos, where theétale topos of a scheme X is also denoted by X by abuse of notation. See [17, Exposé XI] and [16] for the definition and basic properties of the vanishing topos X ← × S S. Let Λ be a commutative ring. We have a morphism of defined by Ψ f is called the nearby cycles functor. For a morphism q : T → S of schemes, we have a morphism of topoi where X T → X is the projection. For a complex K ∈ D + (X, Λ), we have the base change map For a morphism f : X → S of schemes and a complex K ∈ D + (X, Λ), the sliced nearby cycles complexes for f and K are compatible with any base change in the sense of Definition 2.2 (1) if and only if, for every morphism q : T → S of schemes, the base change map c f,q (K) is an isomorphism. This follows from the following descriptions of the stalks of the nearby cycles functor and the sliced nearby cycles functors. Let x → X be a geometric point of X and let s → S denote the composition x → X → S. Let t → S be a geometric point with a specialization map α : t → s, i.e. an S-morphism α : S (t) → S (s) , where S (s) (resp. S (t) ) is the strict localization of S at s → S (resp. t → S). The triple (x, t, α) defines a point of the vanishing topos X ← × S S and every point of X ← × S S is of this form (up to equivalence). The topos X ← × S S has enough points. For the stalk RΨ f (K) (x,t,α) of RΨ f (K) at (x, t, α), we have an isomorphism see [16, (1.3.2)]. Here the pull-back of K to X (x) × S (s) S (t) is also denoted by K and we will use this notation in this paper when there is no possibility of confusion.
We have a similar description of the stalks of the sliced nearby cycles functors. More precisely, let q : U → S be a morphism from a strictly local scheme U and η → U an algebraic geometric point. Let x → X u be a geometric point of the special fiber X u of X U . Then, since the morphism X U (η) → X U is quasi-compact and quasi-separated, we have To prove Theorem 1.2, we need a uniform refinement of Theorem 2.4. More precisely, we need a modification (or an alteration) S ′ → S such that, for every positive integer n invertible on S, the sliced nearby cycles complexes for f S ′ and the constant sheaf Z/nZ are compatible with any base change, under the additional assumption that S is excellent.
In order to prove the existence of such a modification, we will use the methods developed in a recent paper [23] of Orgogozo. In fact, by the same methods, we can also prove that there exists an alteration S ′ → S such that, for every positive integer n invertible on S, the sliced nearby cycles complexes for f S ′ and the constant sheaf Z/nZ are unipotent in the sense of Definition 2.2 (2). Such an alteration is also needed in the proof of Theorem 1.2.
To formulate our results, we need to recall the definition of a locally unipotent sheaf on a Noetherian scheme from [23]. Let X be a Noetherian scheme. In this paper, we call a finite set X = {X α } α of locally closed subsets of X a stratification if we have X = α X α (set-theoretically). We endow each X α with the reduced subscheme structure.
Definition 2.5 (Orgogozo [23, 1.2.1]). Let X be a Noetherian scheme and X a stratification of X. We say that an abelian sheaf F on X is locally unipotent along X if, for every morphism q : U → X from a strictly local scheme U and every X α ∈ X, the pull-back of F to U × X X α has a finite filtration whose successive quotients are constant sheaves.
Remark 2.6. If a constructible abelian sheaf F on a Noetherian scheme X is locally unipotent along a stratification X, then it is constructible along X, i.e. for every X α ∈ X, the pull-back of F to X α is locally constant. (See [23, 1.2.2].) The main result on nearby cycles over general bases is as follows.
Theorem 2.7. Let S be a Noetherian excellent scheme. Let f : X → S be a proper morphism. Let X be a stratification of X. Then there exists an alteration S ′ → S such that, for every positive integer n invertible on S and every complex K ∈ D + (X, Z/nZ) whose cohomology sheaves are constructible sheaves of Z/nZ-modules and are locally unipotent along X, the following two assertions hold.
(1) The sliced nearby cycles complexes for f S ′ : X S ′ → S ′ and K S ′ are compatible with any base change.
(2) The sliced nearby cycles complexes for f S ′ : X S ′ → S ′ and K S ′ are unipotent.
In fact, as in [22], we can show a more precise result for the compatibility of the sliced nearby cycles functors with base change as a corollary of Theorem 2.7: Corollary 2.8. Under the assumptions of Theorem 2.7, there exists a modification S ′ → S such that, for every positive integer n invertible on S and every complex K ∈ D + (X, Z/nZ) whose cohomology sheaves are constructible sheaves of Z/nZ-modules and are locally unipotent along X, the sliced nearby cycles complexes for f S ′ : X S ′ → S ′ and K S ′ are compatible with any base change.
3. Proof of Theorem 2.7 3.1. Nodal curves. In this subsection, we recall some results on nodal curves from [6,23]. Let f : X → S be a morphism of Noetherian schemes. We say that f is a nodal curve if it is a flat projective morphism such that every geometric fiber of f is a connected reduced curve having at most ordinary double points as singularities. We say that f is a nodal curve adapted to a pair (X • , S • ) of dense open subsets X • and S • of X and S, respectively, if the following conditions are satisfied: • f is a nodal curve which is smooth over S • .
• There is a closed subscheme D of X which isétale over S and is contained in the smooth locus of f . Moreover we have f −1 (S • ) ∩ (X\D) = X • . The following proposition will be used in the proof of Theorem 2.7, which is one of the main reasons why we introduce the notion of locally unipotent sheaves. . Let S be a Noetherian scheme and f : X → S a nodal curve adapted to a pair (X • , S • ) of dense open subsets X • and S • of X and S, respectively. Let u : X • ֒→ X denote the open immersion. Assume that S • is normal. Then, for every positive integer n invertible on S and every locally constant constructible sheaf L of Z/nZ-modules on X • such that u ! L is locally unipotent along the stratification X = {X • , X\X • } of X, the sheaf is locally unipotent along the stratification S = {S • , S\S • } of S for every i. Remark 3.2. The proof of Theorem 2.7 is inspired by that of Proposition 3.1. In fact, we can show that, with the notation of Proposition 3.1, the nearby cycles for f and u ! L are compatible with any base change and unipotent. Since we will not use this fact in the proof of Theorem 2.7, we omit the proof of it.
We say that a morphism f : X → S of Noetherian integral schemes is a pluri nodal curve adapted to a dense open subset X • ⊂ X if there are an integer d ≥ 0, a sequence of morphisms of Noetherian integral schemes, and dense open subsets If d = 0, by convention, it means that X = S and f is the identity map.
The following theorem of de Jong plays an important role in the proof of Theorem 2.7. Theorem 3.3 (de Jong [6,Theorem 5.9]). Let f : X → S be a proper surjective morphism of Noetherian excellent integral schemes. Let X • ⊂ X be a dense open subset. We assume that the geometric generic fiber of f is irreducible. Then there is the following commutative diagram: where the vertical maps are integral alterations and f ′ is a pluri nodal curve adapted to a dense open subset X •• 0 ⊂ X 0 which is contained in the inverse image of X • ⊂ X. Proof. See [6,Theorem 5.9] and the proof of [6,Theorem 5.10]. We note that if the dimension of the generic fiber of f is zero, then f is an integral alteration. Hence we can take S ′ as X and take f ′ as the identity map on X in this case.
3.2. Preliminary lemmas. We shall give two lemmas, which will be used in the proof of Theorem 2.7.
We will need the following terminology.
Definition 3.4. Let f : X → S be a morphism of schemes. Let Λ be a commutative ring and K ∈ D + (X, Λ) a complex. Let ρ be an integer.
(1) We say that the sliced nearby cycles complexes for f and K are ρ-compatible with any base change (or simply that the nearby cycles for f and K are ρ-compatible with any base change) if for every local morphism q : V → U of strictly local schemes over S and every algebraic geometric point ξ → V with image η → U, we have τ ≤ρ ∆ = 0 for the cone ∆ of the base change map: (2) We say that the sliced nearby cycles complexes for f and K are ρ-unipotent (or simply that the nearby cycles for f and K are ρ-unipotent) if for every morphism q : U → S from a strictly local scheme U, a point η ∈ U, and an algebraic geometric point η → U lying above η, the complex is Gal(κ(η)/κ(η))-unipotent.
Lemma 3.5. Let f : X → Z and g : Z → S be morphisms of schemes. Let h := g • f denote the composition. Let K ∈ D + (X, Z/nZ) be a complex.
(1) Assume that g is a closed immersion. If the nearby cycles for f and K are ρcompatible with any base change (resp. are ρ-unipotent), then so are the nearby cycles for h and K.
(2) Assume that f is a closed immersion. If the nearby cycles for h and K are ρcompatible with any base change (resp. are ρ-unipotent), then so are the nearby cycles for g and f * K. Proof.
(1) Let q : U → S be a morphism from a strictly local scheme U and η → U an algebraic geometric point. If the image of η in S is not contained in Z, then we have RΨ h U ,η (K U ) = 0. If the image of η in S is contained in Z, then U ′ := Z × S U is a strictly local scheme and η induces an algebraic geometric point The assertion follows from this description.
(2) Let q : U → S and η → U be as above. Let u ∈ U be the closed point. Then we Since (f u ) * is exact, the assertion follows from this isomorphism.
As in [22], we need some results on cohomological descent. See [SGA 4 II, Exposé Vbis] and [7, Section 5] for the terminology used here. Let f : Y → X be a morphism of schemes. Let β : Y • := cosq 0 (Y /X) → X be the augmented simplicial object in the category of schemes defined as in [7, (5.1.4)], so Y m is the (m + 1)-times fiber product Y × X · · · × X Y for m ≥ 0. We can associate to theétale topoi of Y m (m ≥ 0) a topos (Y • ) ∼ ; see [7, (5 . Moreover, as in [7, (5.1.11)], we have a morphism of topoi The assertion (2) can be proved by the same arguments as in the proof of [22,Lemme 4.1]. We shall give a sketch here. Let q : U → S be a morphism from a strictly local scheme U and η → U an algebraic geometric point with image η ∈ U. Let u ∈ U be the closed point. We have the following diagram: The pull-back of the complex for every m ≥ 0. Thus we have the following spectral sequence: .) The assertion follows from this spectral sequence since the sheaf is Gal(κ(η)/κ(η))-unipotent if k + l ≤ ρ by our assumption.
3.3. Proof of Theorem 2.7. In this subsection, we prove Theorem 2.7. Let us stress that the proof is heavily inspired by the methods of [22,23].
In this section, we use the following terminology.
Definition 3.7. Let S be a Noetherian scheme and f : X → S a morphism of finite type. Let ρ be an integer.
(1) Let X be a stratification of X. We say that an alteration S ′ → S is ρ-adapted to the pair (f, X) if, for every positive integer n invertible on S and every constructible sheaf F of Z/nZ-modules on X which is locally unipotent along X, the nearby cycles for f S ′ : X S ′ → S ′ and F S ′ are ρ-compatible with any base change and ρ-unipotent. (2) Let u : U ֒→ X be an open immersion. We say that an alteration S ′ → S is ρ-adapted to the pair (f, U) if, for every positive integer n invertible on S and every locally constant constructible sheaf L of Z/nZ-modules on U such that u ! L is locally unipotent along the stratification {U, X\U}, the nearby cycles for f S ′ : X S ′ → S ′ and (u ! L) S ′ are ρ-compatible with any base change and ρunipotent.
Let S be a Noetherian excellent integral scheme. Let ρ and d be two integers. We shall consider the following statement P(S, ρ, d): P(S, ρ, d): Let T → S be an integral alteration and f : Y → T a proper morphism such that the dimension of the generic fiber of f is less than or equal to d. Let Y be a stratification of Y . Then there exists an alteration T ′ → T which is ρ-adapted to (f, Y) in the sense of Definition 3.7 (1).
(1) P(S, −2, d) holds trivially for every Noetherian excellent integral scheme S and every integer d.
(2) For an integral scheme T and a proper morphism f : Y → T , the condition that the dimension of the generic fiber is less than or equal to −1 means that f is not surjective. The statement P(S, ρ, −1) is not trivial.
Lemma 3.9. To prove Theorem 2.7, it is enough to prove that statement P(S, ρ, d) holds for every triple (S, ρ, d), where S is a Noetherian excellent integral scheme, and ρ and d are integers.
Proof. Let S be a Noetherian scheme and f : X → S a morphism of finite type. Let N be the supremum of dimensions of fibers of f . Let q : U → S be a morphism from a strictly local scheme U and η → U an algebraic geometric point. Then, for every sheaf F of Z/nZ-modules on X, where n is a positive integer, we have for i > 2N by [22,Proposition 3.1]; see also Remark 2.3. By using this fact, the assertion can be proved by standard arguments.
We will prove P(S, ρ, d) by induction on the triples (S, ρ, d). For two Noetherian excellent integral schemes S and S ′ , we denote if S ′ is isomorphic to a proper closed subscheme of an integral alteration of S. For a Noetherian excellent integral scheme S and an integer ρ, we also consider the following statements.
• P( * ≺ S, ρ, * ): The statement P(S ′ , ρ, d ′ ) holds for every Noetherian excellent integral scheme S ′ with S ′ ≺ S and every integer d ′ . We begin with the following lemma. Proof. This lemma can be proved by the same arguments as in [22,Section 4.2] by using [23, Proposition 1.6.2] instead of [22,Lemme 4.3]. We recall the arguments for the reader's convenience.
We assume that P( * ≺ S, ρ, * ) holds. Let T → S be an integral alteration and let f : Y → T be a proper morphism. We assume that f is not surjective. Let Y be a stratification of Y . Let Z := f (Y ) be the schematic image of f . We write g : Y → Z for the induced morphism. By applying P( * ≺ S, ρ, * ) to each irreducible component of Z, we can find an alteration Z ′ → Z which is ρ-adapted to (g, Y). By [23, Proposition 1.6.2], there is an alteration T ′ → T such that every irreducible component of Z T ′ := Z × T T ′ endowed with the reduced closed subscheme structure has a Z-morphism to Z ′ .
Let F be a constructible sheaf of Z/nZ-modules on Y which is locally unipotent along Y, where n is a positive integer invertible on T . We shall show that the nearby cycles for f T ′ and F T ′ are ρ-compatible with any base change and ρ-unipotent. By Lemma 3.5 (1), it suffices to show that the same properties hold for the nearby cycles for g T ′ : Y T ′ → Z T ′ and F T ′ . We put where {Z α } α∈Θ is the set of the irreducible components of Z T ′ , and put β : By the constructions of Z ′ and T ′ , and by Lemma 3.5 (1), we see that the nearby cycles for g m and F m are ρ-compatible with any base change and ρ-unipotent for every m ≥ 0, where we write F m := β * m F T ′ and g m := g T ′ • β m . By Lemma 3.6, it follows that the nearby cycles for g T ′ : Y T ′ → Z T ′ and F T ′ are ρ-compatible with any base change and ρ-unipotent.
Our next task is to show the following lemma.
The proof of Lemma 3.11 is divided into two steps. The first step is to prove the following lemma.
Lemma 3.12. We assume that P(S, ρ, d − 1), P(S, ρ − 1, * ), and P( * ≺ S, ρ, * ) hold. Under this assumption, to prove P(S, ρ, d), it suffices to prove the following statement P nd (S, ρ, d): P nd (S, ρ, d): Let T → S be an integral alteration and f : Y → T a pluri nodal curve adapted to a dense open subset Y • ⊂ Y such that the dimension of the generic fiber of f is less than or equal to d. Then there is an alteration T ′ → T which is ρ-adapted to (f, Y • ) in the sense of Definition 3.7 (2).
Proof. We assume that P nd (S, ρ, d) holds. Let T → S be an integral alteration and f : Y → T a proper morphism such that the dimension of the generic fiber of f is less than or equal to d. Let Y be a stratification of Y . We want to prove that there is an alteration Step 1. It suffices to prove the following claim (I): . Indeed, by replacing Y by a stratification refining it, we may assume that Y is a good stratification in the sense of [23, Section 1.1] (it is called a bonne stratification in French). Then every sheaf F of Z/nZ-modules on Y which is constructible along Y has a finite filtration such that each successive quotient is of the form u ! L where u : Y α ֒→ Y is an immersion for some Y α ∈ Y and L is a locally constant constructible sheaf of Z/nZ-modules on Y α ; see [23,Proposition 1.1.4]. If furthermore F is locally unipotent along Y, then so is every successive quotient of this filtration. Since Y consists of finitely many locally closed subsets, by using Lemma 3.5 (2), we see that it suffices to prove the claim (I).
Step 2. To prove the claim (I), we may assume that Y is integral, the morphism f is surjective, and the geometric generic fiber of f is irreducible.
Indeed, there is a field L which is a finite extension of the function field of T such that, every irreducible component of Y × T Spec L is geometrically irreducible. Let T ′ → T be the normalization of T in L. We put where {Y α } α∈Θ is the set of the irreducible components of Y T ′ . By P(S, ρ − 1, * ) and Lemma 3.6, it suffices to show that there is an alteration It is enough to show that the same assertion holds after restricting to each component Y α . By P( * ≺ S, ρ, * ) and Lemma 3.10, we may assume that Y α → T ′ is surjective. Then, by the construction of T ′ , the geometric generic fiber of Y α → T ′ is irreducible. This completes the proof of our claim.
Step 3. We may assume that Y • is non-empty. We claim that we may assume further Indeed, by Theorem 3.3, there is the following commutative diagram: where the vertical maps are integral alterations and f ′ is a pluri nodal curve adapted to a dense open subset Then the natural morphism W 0 → Y T ′ is a proper surjective morphism. By the same arguments as in the proof of the previous step, we see that it suffices to prove that there is an alteration Step 4. Finally, we complete the proof of the claim (I).
Let n be a positive integer invertible on T and L a locally constant constructible sheaf of Z/nZ-modules on The proof of Lemma 3.12 is now complete.
Proof. Let u : Y • ֒→ Y denote the open immersion. Let T → S be an integral alteration and f : Y → T a pluri nodal curve adapted to a dense open subset Y • ⊂ Y such that the dimension of the generic fiber of f is less than or equal to d. If f is an isomorphism, then there is nothing to prove. Hence we may assume that f is not an isomorphism, and hence there are a factorization and a dense open subset X • ⊂ X such that h : Y → X is a nodal curve adapted to the pair (Y • , X • ) and g : X → T is a pluri nodal curve adapted to X • . Since P(S, ρ, d − 1) holds, we may assume that the identity map T → T is ρ-adapted to the following two pairs . By replacing T with its normalization, we may assume that T is normal.
We claim that the identity map T → T is ρ-adapted to (f, Y • ). The proof is divided into two parts. First, we prove the assertion after restricting to the smooth locus Y ′ ⊂ Y of h. Then, we prove our claim by using the results on the smooth locus Y ′ .
Then the following assertions hold: (1) The nearby cycles for a : Y ′ → T and F are ρ-compatible with any base change.
(2) The nearby cycles for a : Y ′ → T and F are ρ-unipotent.
Proof. (1) We fix a local morphism q : V → U of strictly local schemes over T and an algebraic geometric point ξ → V with image η → U. In the following, for a morphism φ : Z → T and a complex K ∈ D + (Z, Z/nZ), the cone of the base change map is denoted by ∆(φ, K). For a morphism φ : Z → W of T -schemes and a T -scheme T ′ , the base change Z T ′ → W T ′ is often denoted by the same letter φ when there is no possibility of confusion. We want to show τ ≤ρ ∆(a, F ) = 0. It suffices to prove that τ ≤ρ ∆(a, F ) x = 0 at every on the stalks induced by the base change map can be identified with the pull-back map is denoted by the same letter u, it suffices to prove that τ ≤ρ ∆(a, Λ) = 0 and τ ≤ρ ∆(a, v * Λ) = 0.
It follows from the assumption on T that the nearby cycles for a • v and the constant sheaf Λ are ρ-compatible with any base change. Hence we have τ ≤ρ ∆(a, v * Λ) = 0 by Lemma 3.5 (2). By the assumption on T again, the nearby cycles for g and the constant sheaf Λ are ρ-compatible with any base change. Since the composition b : Y ′ ֒→ Y → X is smooth, we have ∆(a, Λ) ∼ = b * ∆(g, Λ) by the smooth base change theorem. Hence we obtain that (2) Let q : U → T be a morphism from a strictly local scheme U, a point η ∈ U, and an algebraic geometric point η → U lying above η. Let s ∈ U be the closed point. We want to show that the complex We first claim that, for every i ≤ ρ, the sheaf R i Ψ a U ,η (F U ) is constructible. Since we have already shown that the nearby cycles for a and F are ρ-compatible with any base change, we may assume that U is the strict localization of T at s → T , in particular, we may assume that U is Noetherian. Then, by using [EGA II, Proposition 7.1.9], we may assume that U is the spectrum of strictly Henselian discrete valuation ring, and in this case, the claim follows from [ By the exact sequence 0 → u ! Λ → Λ → v * Λ → 0, it suffices to prove that the nearby cycles for a and the sheaf v * Λ (resp. the constant sheaf Λ) are ρ-unipotent. By using the assumption on T , we conclude by the same argument as in the proof of (1). Claim 3.15. Let n be a positive integer invertible on T and L a locally constant constructible sheaf L of Z/nZ-modules on Y • such that F := u ! L is locally unipotent along the stratification Y = {Y • , Y \Y • }. Then the following assertions hold: (1) The nearby cycles for f and F are ρ-compatible with any base change.
(2) The nearby cycles for f and F are ρ-unipotent.
(2) Let q : U → T be a morphism from a strictly local scheme U, a point η ∈ U, and an algebraic geometric point η → U lying above η. We write K := RΨ f U ,η (F U ). Let e : Y ′ → Y denote the open immersion. We have the following distinguished triangle: By Claim 3.14 (2), it suffices to prove that c * τ ≤ρ K is Gal(κ(η)/κ(η))-unipotent. Since d is a finite morphism, it suffices to prove that is Gal(κ(η)/κ(η))-unipotent. We have the following distinguished triangle: Since the complex Rh * e ! e * τ ≤ρ K is Gal(κ(η)/κ(η))-unipotent by Claim 3.14 (2), it is enough to show that τ ≤ρ Rh * τ ≤ρ K ∼ = τ ≤ρ Rh * K is Gal(κ(η)/κ(η))-unipotent. By the proper base change theorem, we have As above, by Proposition 3.1 and the assumption on T , it follows that the complex The proof of Lemma 3.13 is complete. Now Lemma 3.11 follows from Lemma 3.12 and Lemma 3.13. Finally, we prove the following proposition which completes the proof Theorem 2.7. Proof. We assume that P(S, ρ, d) does not hold. Then, by Lemma 3.10 and Lemma 3.11, we can find infinitely many triples {(S n , ρ n , d n )} n∈Z ≥0 with the following properties: (1) P(S n , ρ n , d n ) does not hold for every n ∈ Z ≥0 .
For future reference, we state the following immediate consequence of Theorem 2.7 as a corollary.
Corollary 3.17. Let S be a Noetherian excellent scheme and f : X → S a separated morphism of finite type. There exists an alteration S ′ → S such that, for every positive integer n invertible on S, the sliced nearby cycles complexes for f S ′ : X S ′ → S ′ and the constant sheaf Z/nZ are compatible with any base change and are unipotent Proof. The morphism f has a factorization f = g • u where u : X ֒→ P is an open immersion and g : P → S is a proper morphism. Let q : U → S be a morphism from a strictly local scheme U and η → U an algebraic geometric point. Let u ∈ U be the closed point. Then the restriction of RΨ g U ,η (Z/nZ) to X u is isomorphic to RΨ f U ,η (Z/nZ). Thus, by applying Theorem 2.7 to g : P → S and the stratification {P } of P , we obtain the desired conclusion.

Tubular neighborhoods and main results
In this section, we will state our main results onétale cohomology of tubular neighborhoods.
4.1. Adic spaces and pseudo-adic spaces. In this paper, we will freely use the theory of adic spaces and pseudo-adic spaces developed by Huber. Our basis references are [9,10,11]. We shall recall the definitions very roughly. We will use the terminology in [11, Section 1.1], such as a valuation of a ring, an affinoid ring, a Tate ring, or a strongly Noetherian Tate ring.
An adic space is by definition a triple where X is a topological space, O X is a sheaf of topological rings on the topological space X, and v x is an equivalence class of valuations of the stalk O X,x at x ∈ X which is locally isomorphic to the affinoid adic space Spa(A, A + ) associated with an affinoid ring (A, A + ); see [11, Section 1.1] for details. In this paper, unless stated otherwise, we assume that every adic space is locally isomorphic to the affinoid adic space Spa(A, A + ) associated with an affinoid ring (A, A + ) such that A is a strongly Noetherian Tate ring. So we can use the results in [11]; see [11, (1.1.1)]. In particular, we only treat analytic adic spaces; see [11, Section 1.1] for the definition of an analytic adic space.
A pseudo-adic space is a pair (X, S) where X is an adic space and S is a subset of X satisfying certain conditions; see [11,Definition 1.10.3]. If X is an adic space and S ⊂ X is a locally closed subset, then (X, S) is a pseudo-adic space. Almost all pseudo-adic spaces which appear in this paper are of this form. A morphism f : (X, S) → (X ′ , S ′ ) of pseudo-adic spaces is by definition a morphism f : X → X ′ of adic spaces with f (S) ⊂ S ′ .
We have a functor X → (X, X) from the category of adic spaces to the category of pseudo-adic spaces. We often consider an adic space as a pseudo-adic space via this functor.
A typical example of an adic space is the following. Let K be a non-archimedean field, i.e. it is a topological field whose topology is induced by a valuation | · | : K → R ≥0 of rank 1. We assume that K is complete. Let O = K • be the valuation ring of | · |. We call O the ring of integers of K. Let ̟ ∈ K × be an element with |̟| < 1. Let X be a scheme of finite type over O. The ̟-adic formal completion of X is denoted by X or X ∧ . Following [11, Section 1.9], the Raynaud generic fiber of X is denoted by d( X ), which is an adic space of finite type over Spa(K, O). In particular d( X ) is quasi-compact. For example, we have Important examples of pseudo-adic spaces for us are tubular neighborhoods of adic spaces. In the next subsection, we will define them in the case where adic spaces are arising from schemes of finite type over O.

Tubular neighborhoods. Let
X be an open subset and g ∈ O X (U) an element. Following [11], for a point x ∈ U, we write |g(x)| := v x (g). (Strictly speaking, we implicitly choose a valuation from the equivalence class v x .) As in the previous subsection, let K be a complete non-archimedean field with ring of integers O.
Proposition 4.1. Let X be a scheme of finite type over O and Z ֒→ X a closed immersion of finite presentation. Let ǫ ∈ |K × | be an element.
(1) There exist subsets satisfying the following properties; for any affine open subset U ⊂ X and any set where ̟ ∈ K × is an element with ǫ = |̟| and the element of O d( U ) (d( U)) arising from g i is denoted by the same letter. Moreover, they are characterized by the above properties.
Proof. (1) Let ̟ ∈ K × be an element with ǫ = |̟|. Let U ⊂ X be an affine open subset. It suffices to show that the subsets be another set of such elements. Then, for every i, we have g i = Σ 1≤j≤r s ij h j for some elements {s ij } ⊂ O U (U). Since we have |s ij (x)| ≤ 1 for every x ∈ d( U) and every s ij , the assertion follows.
(2) We may assume that X is affine. The subset T (Z, ǫ) is a rational subset of the affinoid adic space d( X ), and hence it is open and quasi-compact. The subset S(Z, ǫ) is the complement of the union of finitely many rational subsets. It follows that S(Z, ǫ) is closed and constructible.
(3) We may assume that X and Y are affine. Then the assertion follows from the descriptions given in (1).
The subsets T (Z, ǫ) and S(Z, ǫ) in Proposition 4.1 are called an open tubular neighborhood and a closed tubular neighborhood of d( Z) in d( X ), respectively. For an element ǫ ∈ |K × |, we also consider the following subsets: This is a quasi-compact open subset of d( X ).
For a locally closed subset S of an adic space X, the pseudo-adic space (X, S) is often denoted by S for simplicity. For example, the pseudo-adic spaces (d( X ), S(Z, ǫ)) and (d( X ), T (Z, ǫ)) are denoted by S(Z, ǫ) and T (Z, ǫ), respectively. , we can also define tubular neighborhoods of d(Z ) in d(X ) in the same way. However, we will always work with algebraizable formal schemes of finite type over O in this paper.
We end this subsection with the following lemma.
Proof. We may assume that X is affine. Then the underlying topological space of d( X ) is a spectral space. We have Hence the intersection is empty. In the constructible topology, the subsets T (Z, ǫ) and d( X )\W are closed, Main results on tubular neighborhoods. In this subsection, let K be an algebraically closed complete non-archimedean field with ring of integers O.
To state the main results on tubular neighborhoods, we needétale cohomology and etale cohomology with proper support of pseudo-adic spaces. See [11,Section 2.3] for definition of theétale site of a pseudo-adic space. As shown in [11,Proposition 2.3.7], for an adic space X and an open subset U ⊂ X, theétale topos of the adic space U is naturally equivalent to theétale topos of the pseudo-adic space (X, U). For a commutative ring Λ, let D + (X, Λ) denote the derived category of bounded below complexes of etale sheaves of Λ-modules on a pseudo-adic space X.
Let f : X → Y be a morphism of analytic pseudo-adic spaces. We assume that f is separated, locally of finite type, and taut. (See [11,Definition 5.1.2] for the definitions of a taut pseudo-adic space and a taut morphism of pseudo-adic spaces. For example, if f is separated and quasi-compact, then f is taut.) For such a morphism f , the direct image functor with proper support is defined in [11,Definition 5.4.4], where Λ is a torsion commutative ring. Moreover, if Y = Spa(K, O), we obtain for a complex K ∈ D + (X, Λ) the cohomology group with proper support H i c (X, K). Let us recall the following results due to Huber in our setting. (1) There exists an element ǫ 0 ∈ |K × | such that, for every ǫ ∈ |K × | with ǫ ≤ ǫ 0 , the following natural maps are isomorphisms for every i: We assume further that F is locally constant. Then there exists an element ǫ 0 ∈ |K × | such that, for every ǫ ∈ |K × | with ǫ ≤ ǫ 0 , the restriction maps onétale cohomology groups are isomorphisms for every i.   The main objective of this paper is to prove uniform variants of Theorem 4.5 for constant sheaves. The main result onétale cohomology groups with proper support of tubular neighborhoods is as follows.
Theorem 4.8. Let K be an algebraically closed complete non-archimedean field with ring of integers O. Let X be a separated scheme of finite type over O and Z ֒→ X a closed immersion of finite presentation. Then there exists an element ǫ 0 ∈ |K × | such that, for every ǫ ∈ |K × | with ǫ ≤ ǫ 0 and for every positive integer n invertible in O, the following natural maps are isomorphisms for every i: The main result onétale cohomology groups of tubular neighborhoods is as follows.   The proofs of Theorem 4.8 and Theorem 4.9 will be given in Section 7. In the rest of this section, we will restate Theorem 4.9 for proper schemes over K.
Let L ⊂ K be a subfield of K which is a complete non-archimedean field with the induced topology. Let O L be the ring of integers of L. For a scheme X of finite type over L, the adic space associated with X is denoted by Corollary 4.11. Let X be a proper scheme over L and Z ֒→ X a closed immersion. We have a closed immersion Z ad ֒→ X ad of adic spaces over Spa(L, O L ). Then, there is a quasi-compact open subset V of X ad containing Z ad such that, for every positive integer n invertible in O, the restriction map is an isomorphism for every i.
Proof. There exist a proper scheme X over Spec O L and a closed immersion Z ֒→ X such that the base change of it to Spec L is isomorphic to the closed immersion Z ֒→ X by Nagata's compactification theorem; see [8, Chapter II, Theorem F.1.1] for example. As in Remark 4.10, we may assume that Z ֒→ X is of finite presentation. Let . By [11, Proposition 1.9.6], we have d( Z) = Z ad and d( X ) = X ad . Therefore, the assertion follows from Theorem 4.9.

5.Étale cohomology with proper support of adic spaces and nearby cycles
In this section, we study the relation between the compatibility of the sliced nearby cycles functors with base change and the bijectivity of specialization maps on stalks of higher direct image sheaves with proper support for adic spaces by using a comparison theorem of Huber [11, Theorem 5.7.8].
5.1. Analytic adic spaces associated with formal schemes. In this subsection, we recall the functor d(−) from a certain category of formal schemes to the category of analytic adic spaces defined in [11, Section 1.9].
Following [11], for a commutative ring A and an element s ∈ A, let A(s/s) denote the localization A[1/s] equipped with the structure of a Tate ring such that the image A 0 of the map A → A[1/s] is a ring of definition and sA 0 is an ideal of definition. We record the following well known results.
Lemma 5.1. Let A be a commutative ring endowed with the ̟-adic topology for an element ̟ ∈ A satisfying the following two properties: (i) A is ̟-adically complete, i.e. the following natural map is an isomorphism: (ii) Let A X 1 , . . . , X n be the ̟-adic completion of A[X 1 , . . . , X n ], called the restricted formal power series ring. Then A X 1 , . . . , X n [1/̟] is Noetherian for every n ≥ 0. Then the following assertions hold: (1) For every ideal I ⊂ A X 1 , . . . , X n , the quotient A X 1 , . . . , X n /I is ̟-adically complete.
(2) Let B be an A-algebra such that the ̟-adic completion B of B is isomorphic to A X 1 , . . . , X n . Let I ⊂ B be an ideal. Then, the ̟-adic completion B/I of B/I is isomorphic to B/I B.  (1). The rest of the proposition is an immediate consequence of (1). We will sketch the proof for the reader's convenience.
(2) By (1) (1), the ring A 0 is ̟-adically complete, and hence A(̟/̟) is complete. It is clear from the definitions that Let N be the kernel of the surjection B := A[X 1 , . . . , X n ] → A 0 [X 1 , . . . , X n ]. By using (2), we have the following exact sequence: For the sake of completeness, we include a proof of the following result on the compatibility of the functor d(−) with fiber products.
Proposition 5.2. Let f : X → Y be a morphism locally of finite type of formal schemes in C. Let Z → Y be an adic morphism of formal schemes in C. Then the morphism induced by the universal property of the fiber product is an isomorphism.
Proof. First, we note that the fiber product d( We may assume that X = Spf A, Y = Spf B, and Z = Spf C are affine, where B and C satisfy the conditions in Lemma 5.1 for some element ̟ ∈ B and for its image in C, respectively. We may assume further that A is of the form B X 1 , . . . , X n /I. We A valuation ring R is called a microbial valuation ring if the field of fractions L of R admits a topologically nilpotent unit ̟ with respect to the valuation topology; see [11,Definition 1.1.4]. We equip R with the valuation topology unless explicitly mentioned otherwise. In this case, the element ̟ is contained in R, the ideal ̟R is an ideal of definition of R, and we have L = R[1/̟]. The completion R of R is also a microbial valuation ring.
Let R be a complete microbial valuation ring. It is well known that R X 1 , . . . , X n [1/̟] ∼ = L X 1 , . . . , X n is Noetherian for every n ≥ 0; see [2, 5.2.6, Theorem 1]. A formal scheme X locally of finite type over Spf R is in the category C.

5.2.Étale cohomology with proper support of adic spaces and nearby cycles.
We shall recall a comparison theorem of Huber. To formulate his result, we need some preparations. Let R be a microbial valuation ring with field of fractions L. We assume that R is a strictly Henselian local ring. Let ̟ be a topologically nilpotent unit in L. Let L be a separable closure of L and let R be the valuation ring of L which extends R.
We will use the following notation. For a scheme X over R, we write X := X × Spec R Spec R and X ′ := X × Spec R Spec R/̟R.
Let η ∈ Spec R and η ∈ Spec R be the generic points. We define X η := X × Spec R η and X η := X × Spec R η.
The ̟-adic formal completion of a scheme (or a ring) X over R is denoted by X . Let s ∈ Spec R be the closed point and X s the special fiber of X . We will use the same notation for morphisms of schemes over R when there is no possibility of confusion. We write  Let Λ be a torsion commutative ring. Let F be anétale sheaf of Λ-modules on X . Let F a denote the pull-back of F by the composition

Recall that we have the direct image functor with proper support
for d(f ) by [11,Definition 5.4.4]. We define R n d(f ) ! F a := H n (Rd(f ) ! F a ). We will describe the stalk at the geometric point ξ : t → S in terms of the sliced nearby cycles functor relative to f . Recall that we defined the sliced nearby cycles functor RΨ f,η := i * Rj * j * : D + (X , Λ) → D + (X s , Λ) in Section 2. Here we fix the notation by the following commutative diagram: Now we can state the following result due to Huber: Proof. We recall the construction of the isomorphism since it is a key ingredient in this paper and the construction will make the compatibility of it with specialization maps clear; see the proof of Proposition 5.5.
First, we recall the following fact. Let F be the pull-back of F to X . By [11, (1) in the proof of Proposition 4.2.4], we see that the base change map There is a factorization f = g • u where u : X ֒→ P is an open immersion and g : P → Spec R is a proper morphism by Nagata's compactification theorem. By using the valuation criterion [11, Corollary 1.3.9], we see that the morphism d(g) : d( P) → S is proper; see also the proof of [21, Lemma 3.5].
Let q : Spec R → Spec R denote the morphism induced by the inclusion R ⊂ R. The base change of it will be denoted by the same letter q when there is no possibility of confusion. We have the following Cartesian diagram: By [11, Proposition 2.5.13 i) and Proposition 2.6.1], we have Therefore, in view of (5.1), we reduce to the case where R = R. The construction given below shows that the desired isomorphism is Gal(L/L)-equivariant. Let λ : d( X )é t → (X ′ )é t denote the morphism of sites defined by sending anétale morphism h : Y → X ′ to d( Y ) → d( X ) where Y → X is anétale morphism of formal schemes lifting h; see [11,Lemma 3.5.1]. Similarly, we have a morphism λ : d( P)é t → (P ′ )é t of sites. By applying [11,Corollary 3.5.11 ii)] to the following diagram Moreover, by applying [11, Theorem 3.5.13] to the following diagram we have an isomorphism Rλ * F a ∼ = i ′ * Rj * j * F . So we have Together with (5.2), we obtain the following isomorphism

The proper base change theorem for schemes implies that
. This isomorphism completes the construction of the desired isomorphism.

5.3.
Specialization maps on the stalks of Rd(f ) ! . In this subsection, we work over a complete non-archimedean field K with ring of integers O for simplicity. We fix a topologically nilpotent unit ̟ in K.
For an adic space X over Spa(K, O), we will use the following notation. For a point x ∈ X, let k(x) be the residue field of the local ring O X,x and k(x) + the valuation ring corresponding to the valuation v x . We note that k(x) + is a microbial valuation ring and the image of ̟, also denoted by ̟, is a topologically nilpotent unit in k(x). For a geometric point ξ of X, let Supp(ξ) ∈ X denote the support of it.
We recall strict localizations of analytic adic spaces. Let ξ : s → X be a geometric point. The strict localization X(ξ) of X at ξ is defined in [11, Section 2.5.11]. It is an adic space over X with an X-morphism s → X(ξ). We write x := Supp(ξ). By [11, Proposition 2.5.13], the strict localization X(ξ) is isomorphic to Spa(k(x), k(x) + ) over X, where k(x) is a separable closure of k(x) and k(x) + is a valuation ring extending k(x) + . A specialization morphism ξ 1 → ξ 2 of geometric points of X is by definition a morphism X(ξ 1 ) → X(ξ 2 ) over X, and such a morphism exists if and only if we have Supp(ξ 2 ) ∈ {Supp(ξ 1 )}.
Let F be an abelianétale sheaf on X. A specialization morphism ξ 1 → ξ 2 of geometric points of X induces a mapping F ξ 2 → F ξ 1 on the stalks in the usual way; see [11, (2.5.16)].
Definition 5.4. Let X be an adic space over Spa(K, O) (or more generally an analytic adic space). Let F be an abelianétale sheaf on X. For a subset W ⊂ X, we say that F is overconvergent on W if, for every specialization morphism ξ 1 → ξ 2 of geometric points of X whose supports are contained in W , the induced map F ξ 2 → F ξ 1 is bijective.
Let X be a scheme of finite type over Spec O. We write X K := X × Spec O Spec K. For anétale sheaf F on X , let F a denote the pull-back of F by the composition Let Λ be a torsion commutative ring.
Proposition 5.5. Let X and Y be separated schemes of finite type over O. Let f : X → Y be a morphism over O. Let F be anétale sheaf of Λ-modules on X . We assume that the sliced nearby cycles complexes for f and F are compatible with any base change; see Definition 2.2 (1). Let s ∈ Y be a point. We consider the inverse image λ −1 (s) under the specialization map Then, the sheaf R n d(f ) ! F a is overconvergent on λ −1 (s) for every n.
Proof. Let ξ 1 → ξ 2 be a specialization morphism of geometric points of d( Y) whose supports are contained in λ −1 (s). We write y m := Supp(ξ m ) (m = 1, 2). Let k(y m ) be a separable closure of k(y m ) and let k(y m ) + be a valuation ring extending k(y m ) + . We identify d( Y)(ξ m ) with Spa(k(y m ), k(y m ) + ). Let R m be the completion of k(y m ) + and we put U m := Spec R m . The morphism Spa(k(y m ), k(y m ) By the assumption, we have q m (s m ) = s for the closed point s m ∈ U m , where the image of s ∈ Y in Y is denoted by the same letter. Let s → Y be an algebraic geometric point lying above s and let U = Spec R be the strict localization of Y at s. There are local Y-morphisms q m : U m → U (m = 1, 2) such that the following diagram commutes: We remark that r is not a local morphism if y 1 = y 2 . Let η m be the generic point of U m . Then we have r(η 1 ) = η 2 . We write η := q 1 (η 1 ) = q 2 (η 2 ). Let η → U denote the algebraic geometric point which is the image of η 2 . We fix the notation by the following commutative diagrams: There is a factorization f = g•u where u : X ֒→ P is an open immersion and g : P → Y is a proper morphism by Nagata's compactification theorem. Let Via these isomorphisms, the map ( We have the following commutative diagram: By the proper base change theorem, the map φ m is identified with the map For both m = 1 and m = 2, this map is an isomorphism by our assumption that the sliced nearby cycles complexes for f and F are compatible with any base change. Thus φ is also an isomorphism. The proof of the proposition is complete.

Local constancy of higher direct images with proper support for generically smooth morphisms
In this section, we study local constancy of higher direct images with proper support for generically smooth morphisms of adic spaces whose target is one-dimensional. We will formulate and prove the results not only for constant sheaves, but also for non-constant sheaves satisfying certain conditions related to the sliced nearby cycles functors.
Throughout this section, we fix an algebraically closed complete non-archimedean field K with ring of integers O.
6.1. Tame sheaves on annuli. In this subsection, we recall two theorems on finiteétale coverings on annuli and the punctured disc, which are essentially proved in [19,24,20]. We do not impose any conditions on the characteristic of K. Since we can not directly apply some results there and some results are only stated in the case where the base field is of characteristic zero, we give proofs of the theorems in Appendix A.
To state the two theorems, we need some preparations. Recall that we defined B(1) = Spa(K T , O T ). Let B(1) * := B(1)\{0} be the punctured disc, where 0 ∈ B(1) is the K-rational point corresponding to 0 ∈ K. It is an adic space locally of finite type over Spa(K, O). We fix a valuation | · | : K → R ≥0 of rank 1 such that the topology of K is induced by it. For elements a, b ∈ |K × | with a ≤ b ≤ 1, we define  B(a, b) for a, b ∈ |K × | and some m with a ≤ b ≤ 1.) In this paper, we use the following notion of tameness forétale sheaves on onedimensional smooth adic spaces over Spa(K, O). Definition 6.1. Let X be a one-dimensional smooth adic space over Spa (K, O). Let x ∈ X be a point which has a proper generalization in X, i.e. there exists a point x ′ ∈ X with x ∈ {x ′ } and x = x ′ . Let k(x) ∧h+ be the Henselization of the completion of the valuation ring k(x) + of x. Let L(x) be a separable closure of the field of fractions k(x) ∧h of k(x) ∧h+ . It induces a geometric point x → X with support x. For anétale sheaf F on X, we say that F is tame at x ∈ X if the action of Gal(L(x)/k(x) ∧h ) on the stalk F x at the geometric point x factors through a finite group G such that ♯ G is invertible in O, where ♯ G denotes the cardinality of G.  [20,Theorem 4.11], [24,Theorem 2.4.3]). Let f : X → B(1) * be a finiteétale morphism of adic spaces. There exists an element ǫ ∈ |K × | with ǫ ≤ 1 such that, for all a, b ∈ |K × | with a < b ≤ ǫ, we have If K is of characteristic zero, then we can take such an element ǫ ∈ |K × | so that the restriction of f to every component B(c i , d i ) appearing in the above decomposition is a Kummer covering.
Theorem 6.3. Let a, b ∈ |K × | be elements with a < b ≤ 1. Let F be a locally constant etale sheaf with finite stalks on B(a, b). We assume that the sheaf F is tame at every x ∈ B(a, b) having a proper generalization in B(a, b), in the sense of Definition 6.1. Let t ∈ |K × | be an element with a/b < t 2 < 1. Then the restriction F | B(a/t,tb) of F to B(a/t, tb) is trivialized by a Kummer covering ϕ m of degree m, i.e. the pull-back ϕ * m (F | B(a/t,tb) ) is a constant sheaf. Moreover, we can assume that the degree m is invertible in O.
We prove Theorem 6.2 and Theorem 6.3 in Appendix A.
Remark 6.4. If K is of characteristic zero, then Theorem 6.2 is known as the p-adic Riemann existence theorem of Lütkebohmert [19].
6.2. Local constancy of higher direct images with proper support. As in Section 5, we use the following notation. Let X be a scheme of finite type over O. We write X K := X × Spec O Spec K. For anétale sheaf F on X , we denote by F a the pull-back of F by the composition d( X )é t a → (X K )é t → Xé t . (See Section 5.2 for the morphism a : d( X )é t → (X K )é t .) Let us introduce the following slightly technical definition.
Definition 6.5. We consider the following diagram: • Y = Spec A is an integral affine scheme of finite type over O such that Y K is one-dimensional and smooth over K, • f : X → Y is a separated morphism of finite type, and • π : Z → Y is a proper surjective morphism such that Z is an integral scheme whose generic fiber Z K is smooth over K, and the base change π K : Z K → Y K is a finite morphism.
Let n be a positive integer invertible in O and F a sheaf of Z/nZ-modules on X . We say that F is adapted to the pair (f, π) if the following conditions are satisfied: (  (2) and (3). Here we retain the notation of Section 2. For example f Z denotes the base change f Z : X × Y Z → Z of f and F Z denotes the pull-back of the sheaf F to X × Y Z.
For the proofs of Theorem 4.8 and Theorem 4.9, we need the following proposition, which is a consequence of Theorem 2.7. Proposition 6.6. Let Y = Spec A be an integral affine scheme of finite type over O such that Y K is one-dimensional and smooth over K. Let f : X → Y be a separated morphism of finite presentation. Then, there exists a proper surjective morphism π : Z → Y as in Definition 6.5 such that, for every positive integer n invertible in O, the constant sheaf Z/nZ on X is adapted to (f, π).
Proof. Let p ≥ 0 be the characteristic of the residue field of O. Let Z (p) be the localization of Z at the prime ideal (p). We may find a finitely generated Z (p) -subalgebra A 0 of A and a separated morphism f 0 : X 0 → Spec A 0 of finite type such that the base change X 0 × Spec A 0 Y is isomorphic to X over Y. By applying Corollary 3.17 to f 0 , we find an alteration π 0 : Z 0 → Spec A 0 satisfying the properties stated there. The base change to Y is generically finite, proper, and surjective. By restricting π ′ to an irreducible component Z ′′ of Z ′ dominating Y, we obtain a morphism π ′′ : Z ′′ → Y. The scheme Z ′′ K is one-dimensional since π ′′ is generically finite. It follows that π ′′ K : Z ′′ K → Y K is finite. Let h : Z → Z ′′ K be the normalization of Z ′′ K . There exists a proper surjective morphism Z → Z ′′ such that Z is integral and the base change Z K → Z ′′ K is isomorphic to h. Then we define π as the composition π : Z → Z ′′ → Y.
By the construction, the constant sheaf Z/nZ is adapted to (f, π) for every positive integer n invertible in O.
By using the results in Section 5, we prove the following proposition: Proposition 6.7. Let f : X → Y and π : Z → Y be morphisms as in Definition 6.5. We have the following diagram: Then the following assertions hold: (1) Let F be anétale sheaf of Z/nZ-modules on X adapted to (f, π) with n ∈ O × .
For every i, the sheaf is tame at every z ∈ d( Z) having a proper generalization in d( Z), in the sense of Definition 6.1.
containing y such that, for everyétale sheaf F of Z/nZ-modules on X adapted to (f, π) with n ∈ O × , the sheaf d(π) Proof. For anétale sheaf F of Z/nZ-modules on X with n ∈ O × , the pull-back of F a by d( X Z ) → d( X ) is isomorphic to (F Z ) a , and hence, by using the base change theorem [11,Theorem 5.4.6] for Rd(f ) ! , we have (1) This is an immediate consequence of Theorem 5.3. Indeed, let z ∈ d( Z) be an element having a proper generalization in d( Z). Let R := k(z) ∧h+ be the Henselization of the completion of the valuation ring k(z) + of z. By [14,Corollary 5.4], the residue field of R is algebraically closed. We write U := Spec R. Let L be the field of fractions of R. The composite is induced by a natural morphism q : U → Z of schemes over O. Let L be a separable closure of L, which induces a geometric point t → Spa(L, R) and a geometric point z → d( Z) in the usual way.
Let F be anétale sheaf of Z/nZ-modules on X adapted to (f, π) with n ∈ O × . By applying Theorem 5.3 to f U : X U → U, we have Gal(L/L)-equivariant isomorphisms , where s ∈ U is the closed point and η = Spec L → U is the algebraic geometric point. By [11,Corollary 5.4.8 and Proposition 6.2.1 i)], the left hand side is a finitely generated Z/nZ-module. Moreover, the action of Gal(L/L) on it factors through a finite group.
Since the complex RΨ f U ,η (F U ) is Gal(L/L)-unipotent and the integer n is invertible in O, it follows that the action of Gal(L/L) on the right hand side factors through a finite group G such that ♯ G is invertible in O. This proves (1).
(2) Since π : Z → Y is proper, by [11, Proposition 1.9.6], we have d( for every j; see Lemma 4.3. Since d(π) is a finite morphism, there is an open neighbor- By using Proposition 5.5, we see that, for everyétale sheaf F of Z/nZ-modules on X adapted to (f, π) with n ∈ O × , the sheaf We need the following finiteness result due to Huber. The main result of this section is the following theorem.
Theorem 6.10. Let f : X → Y and π : Z → Y be morphisms as in Definition 6.5. We assume that there is an open disc V ⊂ d( Y) such that is smooth over V (1) * . Then there exists an element ǫ 0 ∈ |K × | with ǫ 0 ≤ 1 such that, for everyétale sheaf F of Z/nZ-modules on X adapted to (f, π) with n ∈ O × , the following two assertions hold: (1) The restriction is a constant sheaf associated with a finitely generated Z/nZ-module for every i. If K is of characteristic zero, then we can take g as a Kummer covering. (The morphism h possibly depends on F .) Proof. Clearly, the first assertion (1) follows from the second assertion (2). We shall prove (2).
Step 1. We may assume that, for the dominant morphism π : Z → Y, the separable closure k(Y) sep of the function field k(Y) of Y in the function field of Z is Galois over k(Y).
Indeed, there is a finite surjective morphism Z ′ → Z K from an integral scheme Z ′ which is smooth over K such that the separable closure of the function field k(Y) of Y in the function field of Z ′ is Galois over k(Y). There exists a proper surjective morphism Z ′ → Z such that Z ′ is integral and the base change Z ′ K → Z K is isomorphic to Z ′ → Z K . We define π ′ as the composition π ′ : Z ′ → Z → Y. If a sheaf F is adapted to (f, π), then it is also adapted to (f, π ′ ). Thus it suffices to prove Theorem 6.10 for (f, π ′ ).
Then the induced morphism Z K → W is finite, radicial, and surjective and there is a dense open subset U ⊂ Y K over which W → Y K is a finite Galoisétale morphism. Let denote the structure morphism. The morphism d(π) can be written as the composition of finite morphisms Let ǫ 1 ∈ |K × | be an element with is finite andétale. By Theorem 6.2, there exists an element ǫ 2 ∈ |K × | with ǫ 2 ≤ ǫ 1 such that, for all a, b ∈ |K × | with a < b ≤ ǫ 2 , we have for some elements c j , d j ∈ |K × | with c j < d j ≤ 1 (1 ≤ j ≤ N). If K is of characteristic zero, then we can take such ǫ 2 ∈ |K × | so that the restriction V (a, b) of g to every component B(c j , d j ) appearing in the above decomposition is a Kummer covering. By Proposition 6.7 (2), there exists an element ǫ 3 ∈ |K × | with ǫ 3 ≤ ǫ 2 such that, for everyétale sheaf F of Z/nZ-modules on X adapted to (f, π) with n ∈ O × , the sheaf d(π) * R i d(f ) ! F a is overconvergent on d(π) −1 (V (ǫ 3 )) for every i. Let t ∈ |K × | be an element with t < 1. Then we put ǫ 0 := tǫ 3 .
Step 3. We shall show that ǫ 0 satisfies the condition. Indeed, let F be anétale sheaf of Z/nZ-modules on X adapted to (f, π) with n ∈ O × . Let a, b ∈ |K × | be elements with a < b ≤ ǫ 0 . We have g −1 (V (ta, b/t) ≤ N). We take a component B (c 1 , d 1 ) of the decomposition. The restriction of g is denoted by the same letter. By the construction, it is a finite Galoisétale morphism.
We remark that, since the morphism Z K → W is finite, radicial, and surjective, it follows that α : d( Z) → W ′ is a homeomorphism and, for every z ∈ d( Z), the extension k(α(z)) ∧ → k(z) ∧ of the completions of the residue fields is a finite purely inseparable extension, and hence the extension k(α(z)) ∧h → k(z) ∧h of the Henselizations of these fields is also a finite purely inseparable extension. Since on B(c 1 , d 1 ) is constructible by Theorem 6.8. By the construction, it is overconvergent on B(c 1 , d 1 ). Therefore, by [11,Lemma 2.7.11], the sheaf G is locally constant. Moreover, by Proposition 6.7 (1) and the remark above, the sheaf G is tame at every x ∈ B(c 1 , d 1 ) having a proper generalization in B(c 1 , d 1 ). We have g −1 (V (a, b) Hence, by Theorem 6.3, we conclude that the restriction of G to g −1 (V (a, b) For an element ǫ ∈ |K × |, we define This is a closed constructible subset of B(1). For later use, we record the following results. (2) Let F be a locally constant constructible sheaf of Z/nZ-modules on B(1) * . Assume that for all a, b ∈ |K × | with a < b ≤ 1, there exists a finiteétale morphism h : B(c, d) → B(a, b) such that h is a composition of finite Galoisétale morphisms and the pull-back h * (F | B(a,b) ) is a constant sheaf. Then we have Proof. (1) After possibly changing the coordinate function of B(c, d), we may assume that h −1 (B(a, a)) = B(c, c) and h −1 (B(b, b)) = B(d, d). Then, we have We shall show the first equality of (a). By the spectral sequence in [12, 4.2 ii)] and the fact that h is a composition of finite Galoisétale morphisms, it is enough to show that, for a constant sheaf M on B(d)\B(c) associated with a finitely generated Z/nZ-module, we have for every i We shall prove (b). By theČech-to-cohomology spectral sequences and by the fact that h is a composition of finite Galoisétale morphisms, it is enough to show that, for a constant sheaf M on B(c, d) associated with a finitely generated Z/nZ-module, the restriction map In this section, we shall prove Theorem 4.8 and Theorem 4.9. Let K be an algebraically closed complete non-archimedean field with ring of integers O.
The main part of the proofs of Theorem 4.8 and Theorem 4.9 is contained in the following lemma.
Lemma 7.1. Let X be a separated scheme of finite type over O and Z ֒→ X a closed immersion defined by one global section f ∈ O X (X ). Let f : X → Spec O[T ] be the morphism defined by T → f , which is also denoted by f . We assume that there is an element ǫ 1 ∈ |K × | with ǫ 1 ≤ 1 such that is smooth over B(ǫ 1 ) * = B(ǫ 1 )\{0}. Then, there exists an element ǫ 0 ∈ |K × | with ǫ 0 ≤ ǫ 1 , such that the following assertions hold for every ǫ ∈ |K × | with ǫ ≤ ǫ 0 , every positive integer n invertible in O, and every integer i.
We first deduce Theorem 4.8 and Theorem 4.9 from Lemma 7.1. We will prove Lemma 7.1 in Section 7.3.
We say that f : Y → X is a morphism of cohomological descent for torsion abelianétale sheaves if for every torsion abelianétale sheaf F on X, the natural morphism F → Rβ * β * F in the derived category D + (X ∼ et ) is an isomorphism. See [SGA 4 II, Exposé Vbis, Section 2] for details.
As a consequence of the proper base change theorem for analytic pseudo-adic spaces [11,Theorem 4.4.1], we have the following proposition. We formulate it in the generality we need. Proposition 7.2. Let f : Y → X be a morphism of analytic adic spaces which is proper, of finite type, and surjective. Then for every morphism Z → X of analytic pseudo-adic spaces, the base change f : Y × X Z → Z is of cohomological descent for torsion abeliań etale sheaves.
Proof. First, we note that the fiber product Y × X Z → Z exists by [11,Proposition 1.10.6]. By the proper base change theorem for analytic pseudo-adic spaces [11,Theorem 4.4.1 (b)], it suffices to prove that, for every geometric point S → X, the base change Y × X S → S is of cohomological descent for torsion abelianétale sheaves. It is enough to show that there exists a section S → Y × X S; see [SGA 4 II, Exposé Vbis, Proposition 3.3.1] for example. The existence of a section can be easily proved in our case: By the properness of f and [11, Corollary 1.3.9], we may assume that S is of rank 1. Then it is well known.
For future reference, we deduce the following corollaries from Proposition 7.2. Corollary 7.3. Let X be a separated scheme of finite type over O. Let β 0 : Y → X be a proper surjective morphism. We put β : Y • = cosq 0 (Y/X ) → X . Let Z ⊂ d( X ) be a taut locally closed subset. Then we have the following spectral sequence: Proof. By Nagata's compactification theorem, there exists a proper scheme P over Spec O with a dense open immersion u : X ֒→ P over Spec O. Moreover, there is the following Cartesian diagram of schemes: where u ′ is an open immersion and β ′ 0 is a proper surjective morphism. The morphism d(β ′ 0 ) : d( Q) → d( P) is proper, of finite type, and surjective; see [21,Lemma 3.5] (although the base field is assumed to be a discrete valuation field in [21], the same proof works). We put β ′ : Q • = cosq 0 (Q/P) → P. We have a taut locally closed embedding j : Z → d( P). Let j m : Z m → d( Q m ) be the pull-back of j by d(β ′ m ). Then we have  Proof. The assertion can be proved by a similar method used in [22,Lemma 4.1]. We shall give a brief sketch here. Let β ′ : W • = cosq 0 (W 0 /W ) → W (resp. β ′′ : Z • = cosq 0 (Z 0 /Z) → Z) be the base change of β to W (resp. to Z). The morphism i induces a morphism i • : (Z • ) ∼ → (W • ) ∼ of topoi. For the sheaf Z/nZ on W , we have Z/nZ ∼ = Rβ ′ * β ′ * Z/nZ by Proposition 7.2, and we obtain isomorphisms [11,Theorem 4.4.1 (b)] and by a spectral sequence as in [7, (5.2.7.1)] (see also [27,Tag 0D7A]). Thus, by applying Rβ ′ * to the following distinguished triangle we have the following distinguished triangle   Indeed, let X be a separated scheme of finite type over O and Z ֒→ X a closed immersion of finite presentation. By applying [11,Remark 5.5.11 iv)] to the following diagram we have the following long exact sequence: We note that S(Z, ǫ)\d( Z) = (d( X )\d( Z))\Q(Z, ǫ). Hence we have a similar spectral sequence for the diagram Moreover, for elements ǫ, ǫ ′ ∈ |K × | with ǫ ≤ ǫ ′ , we have a similar spectral sequence for the diagram By [11,Proposition 5.5.8], there exists an integer N, which is independent of n and ǫ, ǫ ′ ∈ |K × |, such that we have H i c (S(Z, ǫ)\d( Z), Z/nZ) = 0 and H i c (T (Z, ǫ ′ )\T (Z, ǫ), Z/nZ) = 0 for every i ≥ N. Our claim follows from these results.
Step 2. We suppose that P c (i) holds for every i. We will prove P ′ c (i) by induction on i. The assertion holds trivially for i = −1. We assume that P ′ c (i 0 − 1) holds. First, we claim that, to prove P ′ c (i 0 ), we may assume that X is integral. We may assume that X is flat over O. Then every irreducible component of X dominates Spec O, and hence X has finitely many irreducible components. Let X ′ be the disjoint union of the irreducible components of X . Then X ′ → X is proper and surjective. By P ′ c (i 0 − 1) and Corollary 7.3, it suffices to prove P ′ c (i 0 ) for X ′ and Z × X X ′ . By considering each component of X ′ separately, our claim follows.
Step 3. We assume that X is integral. We may assume further that Z is not equal to X . Let Y → X be the blow-up of X along Z, which is proper and surjective. By P ′ c (i 0 − 1) and Corollary 7.3, it suffices to prove P ′ c (i 0 ) for Y and Z × X Y. Consequently, to prove P ′ c (i 0 ), we may assume further that Z ֒→ X is locally defined by one function. Finally, let X = α∈I U α be a finite affine covering such that Z ∩ U α ֒→ U α is defined by one global section in O X (U α ) for every α ∈ I. We have the following spectral sequence by [11,Remark 5.5.12 iii)]: Here we write U J := ∩ α∈J U α and Z J := Z × X U J ֒→ U J . We have a similar spectral sequence for T (Z, ǫ ′ )\T (Z, ǫ). Since P c (i) holds for every i by hypothesis, it follows that P ′ c (i 0 ) holds for X and Z ֒→ X .
Lemma 7.6. To prove Theorem 4.9, it suffices to prove the following statement P(i) for every integer i.

P(i):
Let X be a separated scheme of finite type over O and Z ֒→ X a closed immersion defined by one global section f ∈ O X (X ). We consider the following distinguished triangles: Then there exists an element ǫ 0 ∈ |K × |, such that for every ǫ ∈ |K × | with ǫ ≤ ǫ 0 and every positive integer n invertible in O, we have τ ≤i K m (ǫ, n) = 0 for every m ∈ {1, 2, 3}.
Proof. Let X be a separated scheme of finite type over O and Z ֒→ X a closed immersion of finite presentation. By [11,Corollary 2.8.3], there exists an integer N, which is independent of ǫ and n, such that cohomology groups H i (T (Z, ǫ), Z/nZ), H i (S(Z, ǫ), Z/nZ), and H i (d( Z), ǫ), Z/nZ) vanish for every i ≥ N. Let t ∈ |K × | be an element with t < 1. Then we have see [12,Lemma 1.3]. Therefore, by [11, Lemma 3.9.2 i)], the same holds for the cohomology group H i (S(Z, ǫ) • , Z/nZ).
We can prove the assertion by the same argument as in the proof of Lemma 7.5 by using the results remarked above instead of [11,Proposition 5.5.8] and by using Corollary 7.4 instead of Corollary 7.3.
We will now prove the desired statement: Proof. We suppose that Lemma 7.1 holds. By Lemma 7.5 and Lemma 7.6, it suffices to prove that P c (i) and P(i) hold for every i. Let us show the assertions by induction on i. The assertions P c (−1) and P(−2) hold trivially. Assume that P c (i − 1) and P(i − 1) hold. Let X be a separated scheme of finite type over O and Z ֒→ X a closed immersion defined by one global section f ∈ O X (X ). We shall show that P c (i) and P(i) hold for X and Z. As in the proof of Lemma 7.5, we may assume that X is integral.
First, we prove the assertions in the case where K is of characteristic zero. By [5,Theorem 4.1], there is an integral alteration Y → X K such that Y is smooth over K. By Nagata's compactification theorem, there is a proper surjective morphism Y → X such that Y K ∼ = Y over X K and Y is integral. By the induction hypothesis, Corollary 7.3, and Corollary 7.4, it suffices to prove P c (i) and P(i) for Y and Z × X Y. (As we have already seen in the proof of Corollary 7.3, the morphism d( Y) → d( X ) is proper and surjective.) Therefore, we may assume that X K is smooth over K. Let be the morphism defined by T → f . Since K is of characteristic zero, there is an open dense subset W ⊂ Spec K[T ] such that f K is smooth over W . It follows from [11,Proposition 1.9.6] that there exists an open subset V ⊂ B(1) whose complement consists of finitely many K-rational points of B(1) such that d(f ) : d( X ) → B(1) is smooth over V . Thus P c (i) and P(i) hold for X and Z since we suppose that Lemma 7.1 holds.
Let us now suppose that K is of characteristic p > 0. As above, let f : X → Spec O[T ] be the morphism defined by T → f . If f is not dominant, then P c (i) and P(i) hold trivially for X and Z. Thus we may assume that f is dominant. By applying [5,Theorem 4.1] to the underlying reduced subscheme of X × Spec O[T ] Spec K(T 1/p ∞ ), where K(T 1/p ∞ ) is the perfection of K(T ), we find an alteration such that Y is integral and smooth over K(T 1/p N ) for some integer N ≥ 0. By Nagata's compactification theorem, there is a proper surjective morphism whose base change to Spec K(T 1/p N ) is isomorphic to g K . As above, it suffices to prove P c (i) and P(i) for Y and Z × X Y.
Let f ′ be the image of for every ǫ ∈ |K × |. Thus, it suffices to prove that P c (i) and P(i) hold for Y and Z ′ ; see [11,Proposition 2.3.7]. By the construction, the generic fiber of the morphism Y → Spec O[T ] defined by T → f ′ is smooth over K(T ). Therefore, as in the case of characteristic zero, Lemma 7.1 implies P c (i) and P(i) for Y and Z ′ .
The proof of Lemma 7.7 is complete.

7.3.
Proof of the key case. In this subsection, we prove Lemma 7.1 and finish the proofs of Theorem 4.8 and Theorem 4.9.
Proof of Lemma 7.1. We may assume that X is flat over Spec O. Then X is of finite presentation over Spec O by [25, Première partie, Corollaire 3.4.7]. By Proposition 6.6 and Theorem 6.10, there exists an element ǫ 0 ∈ |K × | with ǫ 0 ≤ ǫ 1 such that, for all a, b ∈ |K × | with a < b ≤ ǫ 0 and every positive integer n invertible in O × , there exists a finiteétale morphism h : B(c, d) → B(a, b) such that h is a composition of finite Galoisétale morphisms and the pull-back is a constant sheaf associated with a finitely generated Z/nZ-module for every i. We shall show that ǫ 0 satisfies the desired properties. Let n be a positive integer invertible in O and ǫ ∈ |K × | an element with ǫ ≤ ǫ 0 .
(2) We have Be the Leray spectral spectral sequences, it suffices to prove that the restriction map is an isomorphism for all i, j. Let ǫ ′ ∈ |K × | be an element with ǫ ′ < ǫ. Then {B(ǫ ′ , ǫ), D(ǫ) • } is an open covering of B(ǫ). By theČech-to-cohomology spectral sequences, it is enough to prove that the restriction map is an isomorphism for all i, j.
The inverse image d(f ) −1 (B(ǫ ′ , ǫ)) has finitely many connected components. It is enough to show that, for every connected component W ⊂ d(f ) −1 (B(ǫ ′ , ǫ)) and the restriction g : W → B(ǫ ′ , ǫ) of d(f ), the restriction map is an isomorphism for all i, j. The morphism g is of pure dimension N for some integer N ≥ 0. (See [11,Section 1.8] for the definition of the dimension of a morphism of adic spaces.) Since g is smooth and R i g ! Z/nZ is a locally constant constructible sheaf of Z/nZ-modules, Poincaré duality [11, Corollary 7.5.5] implies that for every j, where (N) denotes the Tate twist and () ∨ denotes the Z/nZ-dual. (Here we use the fact that Z/nZ is an injective Z/nZ-module.) The right hand side satisfies the assumption of Lemma 6.11 (1), and hence the assertion follows from the lemma.
(4) Similarly as above, it suffices to prove that the restriction map is an isomorphism for all i, j. Let ǫ ′ ∈ |K × | be an element with ǫ ′ < ǫ. As in the proof of (2), it suffices to prove that the restriction map is an isomorphism for all i, j. By the proof of (3), the sheaf F j | D(ǫ)∩B(ǫ ′ ,ǫ) is a locally constant constructible sheaf of Z/nZ-modules. Hence the assertion follows from the proof of [13,Lemma 2.5]. (In [13], the characteristic of the base field is always assumed to be zero. However [13, Lemma 2.5] holds in positive characteristic without changing the proof.) The proof of Lemma 7.1 is complete. In this appendix, we prove Theorem 6.2 and Theorem 6.3. We retain the notation of Section 6.1. In particular, we fix an algebraically closed complete non-archimedean field K with ring of integers O. We will follow the methods given in Ramero's paper [24].
Following [24], we will use the following notation in this appendix. Recall that for a morphism of finite type Spa ( is a finitely generated algebra over κ. We note that the ideal mA • coincides with the set of topologically nilpotent elements of A. In particular, the ring A ∼ is reduced. A.1. Open annuli in the unit disc. We recall some basic properties of open annuli in the unit disc B(1) = Spa(K T ).
Let a, b ∈ |K × | be elements with a ≤ b ≤ 1. Recall that we defined where ̟ a , ̟ b ∈ K × are elements such that a = |̟ a | and b = |̟ b |. We have as an adic space over B (1). The adic space B(a, b) is isomorphic to B(a/b, 1) as an adic space over Spa(K). We write A(a, b) := O B(1) (B(a, b)). We will focus on the following points of the unit disc B(1). Let r ∈ |K × | be an element with r ≤ 1.
• Let η(r) ♭ : be the Gauss norm of radius r centered at 0. The corresponding point η(r) ♭ ∈ B(1) is denoted by the same letter. • Let δ be an infinite cyclic group with generator δ. We equip |K × | × δ with a total order such that (s, δ m ) < (t, δ n ) ⇐⇒ s < t, or s = t and m > n.
We recall the following example from [24], which is useful to study finiteétale coverings of B(a, b).
Example A.2 ([24, Example 2.1.12]). We assume that a < b. Let Ψ : B(a, b) = Spa (A(a, b)) → B(1) = Spa(K S ) be the morphism over Spa(K) defined by the following homomorphism The homomorphism ψ makes A(a, b) • into a free O S -module of rank 2.
Remark A.3. In the rest of this section, we shall study finiteétale coverings of B(a, b). We recall the following fact from [11, Example 1.6.6 ii)], which we will use freely: Let X be an affinoid adic space of finite type over Spa(K, O). Let Y → X be a finiteétale morphism of adic spaces. Then Y is affinoid and the induced morphism Spec O Y (Y ) → Spec O X (X) of schemes is finite andétale. This construction gives an equivalence of categories between the category of adic spaces which are finite andétale over X and the category of schemes which are finite andétale over Spec O X (X).
In the rest of this subsection, we give two lemmas about the connected components of a finiteétale covering of B(a, b).
Lemma A.4. We assume that a < b. Let f : X → B(a, b) be a finiteétale morphism of adic spaces. For every t ∈ |K × | with a/b < t 2 < 1, there exists an element s 0 ∈ |K × | with t < s 0 ≤ 1 such that every connected component of f −1 (B(a/s 0 , s 0 b)) remains connected after restricting to B(a/s, sb) for every s ∈ |K × | with t < s ≤ s 0 .
Proof. The number of the connected components of f −1 (B(a/s, sb)) increases with decreasing s and it is bounded above by the degree of f (i.e. the rank of O X (X) as an A(a, b)-module) for every s ∈ |K × | with a/b < s 2 ≤ 1. The assertion follows from these properties.
Lemma A.5. We assume that a < b. Let f : X → B(a, b) be a finiteétale morphism of adic spaces. We write B := O X (X) and consider the composition where ψ is the homomorphism defined in Example A.2 and the second homomorphism is the one induced by f . Let {q 1 , . . . , q n } ⊂ Spec B • be the set of the prime ideals of B • lying above the maximal ideal mO S + SO S ⊂ O S . Then, for every t ∈ |K × | with a/b < t 2 < 1, the adic space f −1 (B(a/t, tb)) has at least n connected components.
Proof. This lemma is proved in the proof of [24,Theorem 2.4.3]. We recall the arguments for the reader's convenience.
We define g as the composition Let t ∈ |K × | be an element with a/b < t 2 < 1. We define B(t) := {x ∈ B(1) | |S(x)| ≤ t}. Since Ψ −1 (B(t)) = B(a/t, tb), it is enough to show that g  B(t)). This proves our claim since Spa(B i ) is non-empty for every 1 ≤ i ≤ n.
Proof. The first assertion is a consequence of [24, (2.4.4) in the proof of Theorem 2.4.3].
The assumption that the characteristic of the base field is zero in loc. cit. is not needed here. Moreover, the morphism f need not be Galois. The second assertion is claimed in [24,Remark 2.4.8] (at least when K is of characteristic zero) without proof. Indeed, the hard parts of the proof were already done in [24]. We shall explain how to use the results in loc. cit. to deduce the second assertion.
We deduce the following result from Proposition A.7, which is used in the proof of Theorem 6.2 (in the case where K is of positive characteristic).
Proposition A.8. Let f : X → B(a, b) be a finiteétale morphism of adic spaces. We assume that the discriminant function δ f is linear. Let t ∈ |K × | be an element with a/b < t 2 < 1. Then we have for some elements c i ∈ |K × | with c i < 1 (1 ≤ i ≤ n).
Proof. By Lemma A.4, without loss of generality, we may assume that every connected component of X remains connected after restricting to B(a/s, sb) for every s ∈ |K × | with t < s ≤ 1. Let X 1 , . . . , X m be the connected components of X and let f i : X i → B(a, b) be the restriction of f . By Theorem A.6, each discriminant function δ f i associated with f i is a continuous, piecewise linear, and convex function. Since δ f = m i=1 δ f i , it follows that δ f i is linear for every i. Thus we may further assume that X is connected.
Thus we may consider f −1 (B(a/s, sb)) as a connected affinoid open subset of B(1).
We write X t := f −1 (B(a/t, tb)). Let X c t be the closure of X t in B(1), which is contained in g −1 (D(1) • ). In view of Example A.1 (3), to prove the assertion, it suffices to prove that the complement X c t \X t consists of exactly two points. The map f induces a map f ′ : X c t \X t → B(a/t, tb) c \B(a/t, tb) = {η(a/t), η(tb) ′ }. We prove that f ′ is bijective. Since f is surjective and specializing by [11,Lemma 1.4.5 ii)], it follows that the map f ′ is surjective. To show that the map f ′ is injective, it suffices to prove the following claim: Claim A. 9. The inverse images f −1 (η(a/t)) and f −1 (η(tb) ′ ) both consist of one point.
Proof. Recall that we assume that X remains connected after restricting to B(a/s, sb) for every s ∈ |K × | with t < s ≤ 1. Thus, by Lemma A.5 and Proposition A.7, the inverse images f −1 (η(a/s) ′ ) and f −1 (η(sb)) both consist of one point for every s ∈ |K × | with t < s ≤ 1. This fact implies that is connected for every s 1 , s 2 ∈ |K × | with t ≤ s 1 < s 2 ≤ 1. (Indeed, if it is not connected, then there exist at least two points mapped to η(s 2 b).) By applying Lemma A.5 and Proposition A.7 to f −1 (B(tb, b)) → B(tb, b), we see that f −1 (η(tb) ′ ) consists of one point. The same arguments show that f −1 (η(a/t)) consists of one point.
The proof of Proposition A.8 is complete.
Remark A.10. Here we prove Proposition A.7 and Proposition A.8 in the context of adic spaces as in [24]. It is probably possible to prove these results by using the methods of [19,20].
We now give a proof of Theorem 6.2.
Proof of Theorem 6.2. Let f : X → B(1) * be a finiteétale morphism. Clearly, the discriminant functions on open annuli constructed in Theorem A.6 can be glued to a continuous, piecewise linear, and convex function Moreover, the slopes of δ f are integers. By [24, Lemma 2.1.10], the function δ f is bounded above by some positive real number (depending only on the degree of f ). It follows that there exists an element ǫ 0 ∈ |K × | with ǫ 0 ≤ 1 such that the restriction of δ f to [− log ǫ 0 , ∞) is constant. Let t ∈ |K × | be an element with t < 1. We put ǫ := tǫ 0 . Then, for elements a, b ∈ |K × | with a < b ≤ ǫ, we have  A(a, b)) be the opposite of the group of B(a, b)-automorphisms on X, or equivalently, the group of A(a, b)-automorphisms of B. We assume that f is Galois, i.e. A(a, b) coincides with the ring B G of G-invariants. (This is equivalent to saying that the finiteétale morphism Spec B → Spec A(a, b) of schemes is Galois; see Remark A.3.) In this case, we call G the Galois group of f .
We assume that X is connected. Let r ∈ |K × | be an element with a < r ≤ b and let x ∈ f −1 (η(r)) be an element. Let Stab x := {g ∈ G | g(x) = x} be the stabilizer of x. Let k(x) ∧h+ (resp. k(r) ∧h+ ) be the Henselization of the completion of the valuation ring k(x) + (resp. k(η(r)) + ). Let k(x) ∧h and k(r) ∧h be the fields of fractions of k(x) ∧h+ and k(r) ∧h+ , respectively. Then by [14, 5.5] the extension of fields k(r) ∧h → k(x) ∧h is finite and Galois, and we have a natural isomorphism Stab x ∼ = → Gal(k(x) ∧h /k(r) ∧h ).
In [14], Huber defined higher ramification subgroups and the Swan character of the Galois group Gal(k(x) ∧h /k(r) ∧h ). In [24], Ramero investigated the relation between the discriminant functions and the Swan characters. We are interested in the case where all higher ramification subgroups and the Swan character of Gal(k(x) ∧h /k(r) ∧h ) are trivial. All we need is the following lemma: Lemma A.11 ([24, Lemma 3.3.10]). Let f : X → B(a, b) be a finite Galoisétale morphism such that X is connected. We assume that ♯ Stab x is invertible in O for every r ∈ |K × | with a < r ≤ b and every x ∈ f −1 (η(r)). Then the discriminant function Proof. This follows from the second equality of [24,Lemma 3.3.10]. Indeed, under the assumption, we have Sw ♮ x = 0 for the Swan character Sw ♮ x attached to x ∈ f −1 (η(r)) defined in [24,Section 3.3].
Proof of Theorem 6.3. In fact, we will show that if a locally constantétale sheaf F with finite stalks on B(a, b) is tame at η(r) ∈ B(a, b) for every r ∈ |K × | with a < r ≤ b, then, for every t ∈ |K × | with a/b < t 2 < 1, there exists an integer m invertible in O such that the restriction F | B(a/t,tb) is trivialized by a Kummer covering ϕ m .
There is a finite Galoisétale morphism f : X → B(a, b) such that X is connected and f * F is a constant sheaf. Let G be the Galois group of f . By replacing X by a quotient of it by a subgroup of G (this makes sense by Remark A.3), we may assume that the induced homomorphism ρ : G → Aut(Γ(X, f * F ))