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 $\mathop {\mathrm {Spec}} R$ where R is a strictly Henselian local ring. Let $f \colon X \to S$ be a morphism of schemes. Let $q \colon U \to S$ be a morphism from a strictly local scheme U. The closed point of U is denoted by u. Let $\eta \in U$ be a point. Let $\overline {\eta } \to U$ be an algebraic geometric point lying above $\eta $ , that is, it is a geometric point lying above $\eta $ such that the residue field $\kappa (\overline {\eta })$ is a separable closure of the residue field $\kappa (\eta )$ of $\eta $ . The strict localization of U at $\overline {\eta } \to U$ is denoted by $U_{(\overline {\eta })}$ . We have the following commutative diagram:

Let $\Lambda $ be a commutative ring. We have the following functor:

$$\begin{align*}R\Psi_{f_U, \overline{\eta}}:=i^{*}Rj_*j^* \colon D^{+}(X_U, \Lambda) \to D^{+}(X_u, \Lambda). \end{align*}$$

This functor is called the sliced nearby cycles functor in [Reference Illusie19]. For a complex $\mathcal {K} \in D^{+}(X_U, \Lambda )$ , we have an action of the absolute Galois group $\mathop {\mathrm {Gal}}\nolimits (\kappa (\overline {\eta })/\kappa (\eta ))$ on $R\Psi _{f_U, \overline {\eta }}(\mathcal {K})$ via the canonical isomorphism

$$\begin{align*}\mathop{\mathrm{Aut}}\nolimits(U_{(\overline{\eta})}/\mathop{\mathrm{Spec}} ({\mathcal O}_{U, \eta})) \cong \mathop{\mathrm{Gal}}\nolimits(\kappa(\overline{\eta})/\kappa(\eta)). \end{align*}$$

Let $q \colon V \to U$ be a local morphism of strictly local schemes over S, that is, a morphism over S which sends the closed point v of V to the closed point u of U. Let $\xi \in V$ be a point with image $\eta =q(\xi ) \in U$ . For an algebraic geometric point $\overline {\xi } \to V$ lying above $\xi $ , we have an algebraic geometric point $\overline {\eta } \to U$ lying above $\eta $ by taking the separable closure of $\kappa (\eta )$ in $\kappa (\overline {\xi })$ . We call $\overline {\eta } \to U$ the image of $\overline {\xi } \to V$ under the morphism q. We have the following commutative diagram:

where the vertical morphisms are induced by q. For a complex $\mathcal {K} \in D^{+}(X_U, \Lambda )$ , we have the following base change map:

$$\begin{align*}q^*R\Psi_{f_U, \overline{\eta}}(\mathcal{K}) \to R\Psi_{f_V, \overline{\xi}}(\mathcal{K}_V). \end{align*}$$

Remark 2.4. We can restate Definition 2.3 (1) in terms of vanishing topoi as follows. Let $f \colon X \to S$ be a morphism of schemes. Let

$$\begin{align*}X \overset{\leftarrow}{\times}_S S \end{align*}$$

be the vanishing topos, where the étale topos of a scheme X is also denoted by X by abuse of notation. See [Reference Illusie19] for the definition and basic properties of the vanishing topos $X \overset {\leftarrow }{\times }_S S$ . Let $\Lambda $ be a commutative ring. We have a morphism of topoi $\Psi _f \colon X \to X \overset {\leftarrow }{\times }_S S$ . The direct image functor

$$\begin{align*}R\Psi_f \colon D^{+}(X, \Lambda) \to D^{+}(X \overset{\leftarrow}{\times}_S S, \Lambda) \end{align*}$$

defined by $\Psi _f$ is called the nearby cycles functor. For a morphism $q \colon T \to S$ of schemes, we have a morphism of topoi $ {\overset {\leftarrow }{q}} \colon X_T \overset {\leftarrow }{\times }_T T \to X \overset {\leftarrow }{\times }_S S $ and a $2$ -commutative diagram

where $X_T \to X$ is the projection. For a complex $\mathcal {K} \in D^{+}(X, \Lambda )$ , we have the base change map

$$\begin{align*}c_{f, q}(\mathcal{K}) \colon ({\overset{\leftarrow}{q}})^*R\Psi_{f}(\mathcal{K}) \to R\Psi_{f_T}(\mathcal{K}_T). \end{align*}$$

For a morphism $f \colon X \to S$ of schemes and a complex $\mathcal {K} \in D^{+}(X, \Lambda )$ , the sliced nearby cycles complexes for f and $\mathcal {K}$ are compatible with any base change in the sense of Definition 2.3 (1) if and only if, for every morphism $q \colon T \to S$ of schemes, the base change map $ c_{f, q}(\mathcal {K}) $ is an isomorphism (see also the proof of [Reference Orgogozo25, Lemme 4.1]). This follows from the following descriptions of the stalks of the nearby cycles functor and the sliced nearby cycles functors.

Let $x \to X$ be a geometric point of X, and let $s \to S$ denote the composition $x \to X \to S$ . Let $t \to S$ be a geometric point with a specialization map $\alpha \colon t \to s$ , that is, an S-morphism $\alpha \colon S_{(t)} \to S_{(s)}$ , where $S_{(s)}$ (resp. $S_{(t)}$ ) is the strict localization of S at $s \to S$ (resp. $t \to S$ ). The triple $(x, t, \alpha )$ defines a point of the vanishing topos $X \overset {\leftarrow }{\times }_S S$ , and every point of $X \overset {\leftarrow }{\times }_S S$ is of this form (up to equivalence). The topos $X \overset {\leftarrow }{\times }_S S$ has enough points. For the stalk $R\Psi _f(\mathcal {K})_{(x, t, \alpha )}$ of $R\Psi _f(\mathcal {K})$ at $(x, t, \alpha )$ , we have an isomorphism

$$\begin{align*}R\Psi_f(\mathcal{K})_{(x, t, \alpha)} \cong R\Gamma(X_{(x)}\times_{S_{(s)}} S_{(t)}, \mathcal{K}) \end{align*}$$

(see [Reference Illusie19, (1.3.2)]). Here, the pullback of $\mathcal {K}$ to $X_{(x)}\times _{S_{(s)}} S_{(t)}$ is also denoted by $\mathcal {K}$ (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 $q \colon U \to S$ be a morphism from a strictly local scheme U and $\overline {\eta } \to U$ an algebraic geometric point. Let $x \to X_u$ be a geometric point of the special fibre $X_u$ of $X_U$ . Then, since the morphism $X_{U_{(\overline {\eta })}} \to X_U$ is quasicompact and quasiseparated, we have

(2.1) $$ \begin{align} R\Psi_{f_U, \overline{\eta}}(\mathcal{K}_U)_x \cong R\Gamma((X_U)_{(x)} \times_{U} U_{(\overline{\eta})}, \mathcal{K}_U). \end{align} $$

2.2 Main results on nearby cycles over general bases

A proper surjective morphism $f \colon X \to Y$ of Noetherian schemes is called an alteration if it sends every generic point of X to a generic point of Y and it is generically finite, that is, there exists a dense open subset $U \subset Y$ such that the restriction $f^{-1}(U) \to U$ is a finite morphism. If, furthermore, X and Y are integral schemes, then f is called an integral alteration. An alteration $f \colon X \to Y$ is called a modification if there exists a dense open subset $U \subset Y$ such that the restriction $f^{-1}(U) \to U$ is an isomorphism.

Let $f \colon X \to S$ be a morphism of finite type of excellent Noetherian schemes. In [Reference Orgogozo25], Orgogozo proved the following result:

To prove Theorem 1.2, we need a uniform refinement of Theorem 2.5. More precisely, we need a modification (or an alteration) $S' \to S$ such that, for every positive integer n invertible on S, the sliced nearby cycles complexes for $f_{S'}$ and the constant sheaf ${\mathbb Z}/n{\mathbb Z}$ are compatible with any base change.

In order to prove the existence of such a modification, we will use the methods developed in a recent paper [Reference Orgogozo26] of Orgogozo. In fact, by the same methods, we can also prove that there exists an alteration $S' \to S$ such that, for every positive integer n invertible on S, the sliced nearby cycles complexes for $f_{S'}$ and the constant sheaf ${\mathbb Z}/n{\mathbb Z}$ are unipotent in the sense of Definition 2.3 (2). Such an alteration is also needed in the proof of Theorem 1.2.

We need to recall the definition of a locally unipotent sheaf on a Noetherian scheme from [Reference Orgogozo26]. Let X be a Noetherian scheme. In this paper, we call a finite set $\mathfrak {X}= \{ X_{\alpha } \}_{\alpha }$ of locally closed subsets of X a stratification if we have $X=\coprod _{\alpha } X_{\alpha }$ (set theoretically).

Our main result on nearby cycles over general bases is as follows.

For future reference, we state the following corollary.

In fact, as in [Reference Orgogozo25], we can show a more precise result for the compatibility of the sliced nearby cycles functors with base change as a corollary of Theorem 2.8:

3.1 Nodal curves

In this subsection, we recall some results on nodal curves from [Reference de Jong7, Reference Orgogozo26]. Let $f \colon X \to S$ 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 fibre 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 $(X^{\circ }, S^{\circ })$ of dense open subsets $X^{\circ }$ and $S^{\circ }$ of X and S, respectively, if the following conditions are satisfied:

• • f is a nodal curve which is smooth over $S^{\circ }$ .

• • There is a closed subscheme D of X which is étale over S and is contained in the smooth locus of f. Moreover, we have $f^{-1}(S^{\circ }) \cap (X \backslash D)=X^{\circ }$ .

The following proposition will be used in the proof of Theorem 2.8, which is one of the main reasons why we introduce the notion of locally unipotent sheaves.

Proposition 3.1 (Orgogozo [Reference Orgogozo26, Proposition 2.3.1])

Let S be a Noetherian scheme and $f \colon X \to S$ a nodal curve adapted to a pair $(X^{\circ }, S^{\circ })$ of dense open subsets $X^{\circ }$ and $S^{\circ }$ of X and S, respectively. Let $u \colon X^{\circ } \hookrightarrow X$ denote the open immersion. Assume that $S^{\circ }$ is normal. Then, for every positive integer n invertible on S and every locally constant constructible sheaf $\mathcal {L}$ of ${\mathbb Z}/n{\mathbb Z}$ -modules on $X^{\circ }$ such that $u_{!}\mathcal {L}$ is locally unipotent along the stratification $\mathfrak {X}=\{ X^{\circ }, X \backslash X^{\circ } \}$ of X, the sheaf

$$\begin{align*}R^{i}f_*(u_{!}\mathcal{L}) \end{align*}$$

is locally unipotent along the stratification $\mathfrak {S}=\{ S^{\circ }, S \backslash S^{\circ } \}$ of S for every i.

We say that a morphism $f \colon X \to S$ of Noetherian integral schemes is a pluri nodal curve adapted to a dense open subset $X^{\circ } \subset X$ if there are an integer $d \geq 0$ , a sequence

$$\begin{align*}(X=X_d \underset{f_d}{\to} X_{d-1} \to \cdots \to X_1 \underset{f_1}{\to} X_0=S) \end{align*}$$

of morphisms of Noetherian integral schemes and dense open subsets $X^{\circ }_i \subset X_i$ for every $0 \leq i \leq d$ with $X^{\circ }_d=X^{\circ }$ such that $f_i \colon X_i \to X_{i-1}$ is a nodal curve adapted to the pair $(X^{\circ }_i, X^{\circ }_{i-1})$ for every $ 1 \leq i \leq d$ . 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 2.8.

Theorem 3.3 (de Jong [Reference de Jong7, Theorem 5.9])

Let $f \colon X \to S$ be a proper surjective morphism of excellent Noetherian integral schemes. Let $X^{\circ } \subset X$ be a dense open subset. We assume that the geometric generic fibre 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 $X^{\circ \circ }_0 \subset X_0$ which is contained in the inverse image of $X^{\circ } \subset X$ .

3.2 Preliminary lemmas

We shall give two lemmas, which will be used in the proof of Theorem 2.8.

We will need the following terminology.

As in [Reference Orgogozo25], we need some results on cohomological descent (see [Reference Artin, Grothendieck and Verdier1, Exposé Vbis] and [Reference Deligne8, Section 5] for the terminology used here). Let $f \colon Y \to X$ be a morphism of schemes. Let

$$\begin{align*}\beta \colon Y_{\bullet}:=\mathop{\mathrm{cosq}}\nolimits_0(Y/X) \to X \end{align*}$$

be the augmented simplicial object in the category of schemes defined as in [Reference Deligne8, (5.1.4)], so $Y_m$ is the $(m+1)$ -times fibre product $Y \times _X \cdots \times _X Y$ for $m \geq 0$ . We can associate to the étale topoi of $Y_m$ ( $m \geq 0$ ) a topos $(Y_{\bullet })^{\sim }$ (see [Reference Deligne8, (5.1.6)–(5.1.8)]). Moreover, as in [Reference Deligne8, (5.1.11)], we have a morphism of topoi

$$\begin{align*}(\beta_*, \beta^*) \colon (Y_{\bullet})^{\sim} \to X^{\sim}_{\mathrm{\acute{e}t}} \end{align*}$$

from $(Y_{\bullet })^{\sim }$ to the étale topos $X^{\sim }_{\mathrm {\acute {e}t}}$ of X.

Proof. The assertion (1) is [Reference Orgogozo25, Lemme 4.1] (see also Remark 2.4). Although it is stated for constant sheaves, the same proof works for sheaves of ${\mathbb Z}/n{\mathbb Z}$ -modules (or more generally, for torsion abelian sheaves).

The assertion (2) can be proved by the same arguments as in the proof of [Reference Orgogozo25, Lemme 4.1]. We shall give a sketch here. Let $q \colon U \to S$ be a morphism from a strictly local scheme U and $\overline {\eta } \to U$ an algebraic geometric point with image $\eta \in U$ . Let $u \in U$ be the closed point. We have the following diagram:

where $\beta \colon (Y_{\bullet })_U \to X_U$ is the base change of $\beta $ , etc. By [Reference Artin, Grothendieck and Verdier1, Exposé Vbis, Proposition 4.3.2], the morphism $\beta _0 \colon Y \to X$ is universally of cohomological descent, and, hence, we have $\mathcal {F}_U \cong R\beta _*\beta ^*\mathcal {F}_U$ . Using this isomorphism and the proper base change theorem, we obtain

$$\begin{align*}R\Psi_{f_{U}, \overline{\eta}}(\mathcal{F}_{U}) \cong R\beta_*(i_{\bullet})^*R(j_{\bullet})_*(j_{\bullet})^*\beta^*\mathcal{F}_U. \end{align*}$$

The pullback of the complex

$$\begin{align*}(i_{\bullet})^*R(j_{\bullet})_*(j_{\bullet})^*\beta^*\mathcal{F}_U \end{align*}$$

to $(Y_m)_u$ is isomorphic to $R\Psi _{(f_m)_{U}, \overline {\eta }}((\mathcal {F}_m)_{U})$ for every $m \geq 0$ . Thus, we have the following spectral sequence:

$$\begin{align*}E^{k, l}_{1}=R^l (\beta_k)_*R\Psi_{(f_k)_{U}, \overline{\eta}}((\mathcal{F}_k)_{U}) \Rightarrow R^{k+l}\Psi_{f_{U}, \overline{\eta}}(\mathcal{F}_{U}) \end{align*}$$

(see [Reference Deligne8, (5.2.7.1)]). The assertion follows from this spectral sequence since our assumption implies that the sheaf

$$\begin{align*}R^l (\beta_k)_*R\Psi_{(f_k)_{U}, \overline{\eta}}((\mathcal{F}_k)_{U}) \cong R^l (\beta_k)_* \tau_{\leq l}R\Psi_{(f_k)_{U}, \overline{\eta}}((\mathcal{F}_k)_{U}) \end{align*}$$

is $\mathop {\mathrm {Gal}}\nolimits (\kappa (\overline {\eta })/\kappa (\eta ))$ -unipotent if $k+l \leq \rho $ .

3.3 Proof of Theorem 2.8

In this subsection, we prove Theorem 2.8. Our main technique is a combination of the methods of [Reference Orgogozo25] and [Reference Orgogozo26].

In this section, we use the following terminology.

Let S be an excellent Noetherian integral scheme. Let $\rho $ and d be two integers. We shall consider the following statement $\textbf {P}(S, \rho , d)$ :

• $\textbf {P}(S, \rho , d)$ : For every integral alteration $T \to S$ , for every proper morphism $f \colon Y \to T$ such that the dimension of the generic fibre of f is less than or equal to d and for any stratification $\mathfrak {Y}$ of Y, there exists an alteration $T' \to T$ which is $\rho $ -adapted to $(f, \mathfrak {Y})$ in the sense of Definition 3.6 (1).

We will prove $\textbf {P}(S, \rho , d)$ by induction on the triples $(S, \rho , d)$ . For two excellent Noetherian integral schemes S and $S'$ , we denote

$$\begin{align*}S' \prec S \end{align*}$$

if $S'$ is isomorphic to a proper closed subscheme of an integral alteration of S. For an excellent Noetherian integral scheme S and an integer $\rho $ , we also consider the following statements.

• • $\textbf {P}(S, \rho , *)$ : The statement $\textbf {P}(S, \rho , d')$ holds for every integer $d'$ .

• • $\textbf {P}(* \prec S, \rho , *)$ : The statement $\textbf {P}(S', \rho , d')$ holds for every excellent Noetherian integral scheme $S'$ with $S' \prec S$ and every integer $d'$ .

We begin with the following lemma.

Our next task is to show the following proposition, which is the most difficult part.

The proof of Proposition 3.10 is divided into two steps. The first step is to reduce to the case of pluri nodal curves:

Next, we prove $\textbf {P}_{\text {nd}}(S, \rho , d)$ in Lemma 3.11 under the assumptions:

Proof. Let $T \to S$ be an integral alteration and $f \colon Y \to T$ a pluri nodal curve adapted to a dense open subset $Y^{\circ } \subset Y$ such that the dimension of the generic fibre of f is less than or equal to d. Let $u \colon Y^{\circ } \hookrightarrow Y$ denote the open immersion. 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 factorisation

and a dense open subset $X^{\circ } \subset X$ such that $h \colon Y \to X$ is a nodal curve adapted to the pair $(Y^{\circ }, X^{\circ })$ and $g \colon X \to T$ is a pluri nodal curve adapted to $X^{\circ }$ . Since $\textbf {P}(S, \rho , d-1)$ holds, we may assume that the identity map $T \to T$ is $\rho $ -adapted to the following two pairs

$$\begin{align*}(Y \backslash Y^{\circ} \to T, \{ Y \backslash Y^{\circ} \}) \quad \text{and} \quad (g, \{ X^{\circ}, X \backslash X^{\circ} \}). \end{align*}$$

By replacing T with its normalization, we may assume that T is normal.

We claim that the identity map $T \to T$ is $\rho $ -adapted to $(f, Y^{\circ })$ . The proof is divided into two parts. First, we prove the assertion after restricting to the smooth locus $Y' \subset Y$ of h. Then, we prove our claim by using the results on the smooth locus $Y'$ .

Claim 3.13. Let $a \colon Y' \to T$ denote the restriction of f to $Y'$ . Let n be a positive integer invertible on T and $\mathcal {L}$ a locally constant constructible sheaf of ${\mathbb Z}/n{\mathbb Z}$ -modules on $Y^{\circ }$ such that $u_!\mathcal {L}$ is locally unipotent along the stratification $\mathfrak {Y}:=\{ Y^{\circ }, Y\backslash Y^{\circ } \}$ . Let $\mathcal {F}$ be the pullback of $u_!\mathcal {L}$ to $Y'$ . Then the following assertions hold:

1. (1) The nearby cycles for $a \colon Y' \to T$ and $\mathcal {F}$ are $\rho $ -compatible with any base change.

2. (2) The nearby cycles for $a \colon Y' \to T$ and $\mathcal {F}$ are $\rho $ -unipotent.

Proof. (1) We fix a local morphism $q \colon V \to U$ of strictly local schemes over T and an algebraic geometric point $\overline {\xi } \to V$ with image $\overline {\eta } \to U$ . In the following, for a morphism $\phi \colon Z \to T$ and a complex $\mathcal {K} \in D^+(Z, {\mathbb Z}/n{\mathbb Z})$ , the cone of the base change map

$$\begin{align*}q^*R\Psi_{\phi_U, \overline{\eta}}(\mathcal{K}_U) \to R\Psi_{\phi_V, \overline{\xi}}(\mathcal{K}_V) \end{align*}$$

is denoted by $\Delta (\phi , \mathcal {K})$ . For a morphism $\psi \colon Z \to W$ of T-schemes and a T-scheme $T'$ , the base change $Z_{T'} \to W_{T'}$ is often denoted by the same letter $\psi $ when there is no possibility of confusion.

We want to show $\tau _{\leq \rho } \Delta (a, \mathcal {F})=0$ . It suffices to prove that $\tau _{\leq \rho } \Delta (a, \mathcal {F})_{x}=0$ at every geometric point $x \to Y^{\prime }_s$ , where $s \in V$ is the closed point. The morphism

$$\begin{align*}(q^*R\Psi_{a_U, \overline{\eta}}(\mathcal{F}_U))_x \to R\Psi_{a_V, \overline{\xi}}(\mathcal{F}_V)_x \end{align*}$$

on the stalks can be identified with the pullback map

$$\begin{align*}R\Gamma((Y^{\prime}_U)_{(x)}\times_{U} U_{(\overline{\eta})}, u_!\mathcal{L}) \to R\Gamma((Y^{\prime}_V)_{(x)}\times_{V} V_{(\overline{\xi})}, u_!\mathcal{L}) \end{align*}$$

(see also (2.1) in Remark 2.4).

In order to show $\tau _{\leq \rho } \Delta (a, \mathcal {F})_{x}=0$ , we can assume that $\mathcal {L}$ is a constant sheaf on $Y^{\circ }$ . Indeed, let $\mathcal {L}'$ denote the pullback of $\mathcal {L}$ to $(Y^{\prime }_U)_{(x)} \times _Y Y^{\circ }$ . It has a finite filtration

$$\begin{align*}0=\mathcal{L}^{\prime}_0 \subset \dots \subset \mathcal{L}^{\prime}_i \subset \mathcal{L}^{\prime}_{i+1} \subset \dots \subset \mathcal{L}' \end{align*}$$

such that each successive quotient $\mathcal {L}^{\prime }_{i+1}/\mathcal {L}^{\prime }_i$ is a constant sheaf since $u_!\mathcal {L}$ is locally unipotent along $\mathfrak {Y}=\{ Y^{\circ }, Y\backslash Y^{\circ } \}$ . The pullback of $u_!\mathcal {L}$ to $(Y^{\prime }_U)_{(x)}$ is isomorphic to $u^{\prime }_!\mathcal {L}'$ , where $u' \colon (Y^{\prime }_U)_{(x)} \times _Y Y^{\circ } \hookrightarrow (Y^{\prime }_U)_{(x)}$ denotes the open immersion. Let $\Delta _i$ (resp. $\Delta _{i+1, i}$ ) denote the cone of the map

$$\begin{align*}R\Gamma((Y^{\prime}_U)_{(x)}\times_{U} U_{(\overline{\eta})}, u^{\prime}_!\mathcal{L}^{\prime}_{i}) \to R\Gamma((Y^{\prime}_V)_{(x)}\times_{V} V_{(\overline{\xi})}, u^{\prime}_!\mathcal{L}^{\prime}_{i}) \end{align*}$$

(resp.

$$\begin{align*}R\Gamma((Y^{\prime}_U)_{(x)}\times_{U} U_{(\overline{\eta})}, u^{\prime}_!(\mathcal{L}^{\prime}_{i+1}/\mathcal{L}^{\prime}_i)) \to R\Gamma((Y^{\prime}_V)_{(x)}\times_{V} V_{(\overline{\xi})}, u^{\prime}_!(\mathcal{L}^{\prime}_{i+1}/\mathcal{L}^{\prime}_i))). \end{align*}$$

We obtain a distinguished triangle $\Delta _i \to \Delta _{i+1} \to \Delta _{i+1, i} \to $ for each i. Thus, if we have shown $\tau _{\leq \rho } \Delta _{i+1, i}=0$ for every i, then we can prove $\tau _{\leq \rho } \Delta _{i}=0$ for every i inductively, which, in turn, implies $\tau _{\leq \rho } \Delta (a, \mathcal {F})_{x}=0$ . Since $\mathcal {L}^{\prime }_{i+1}/\mathcal {L}^{\prime }_i$ is a constant sheaf (and, hence, is the pullback of a constant sheaf on $Y^{\circ }$ ), this means that it suffices to treat the case where $\mathcal {L}$ is a constant sheaf.

Now we assume that $\mathcal {L}=\Lambda $ is a constant sheaf on $Y^{\circ }$ . Note that $Y^{\circ }$ is contained in $Y'$ . Since we have the following exact sequence of sheaves on $Y'$

$$\begin{align*}0 \to u_!\Lambda \to \Lambda \to v_*\Lambda \to 0, \end{align*}$$

where $v \colon Y' \backslash Y^{\circ } \hookrightarrow Y'$ is the closed immersion and we denote the open immersion $Y^{\circ } \hookrightarrow Y'$ by the same letter u, it suffices to prove that $\tau _{\leq \rho }\Delta (a, \Lambda )=0$ and $\tau _{\leq \rho }\Delta (a, v_*\Lambda )=0$ .

It follows from the assumption on T that the nearby cycles for $a \circ v$ and the constant sheaf $\Lambda $ are $\rho $ -compatible with any base change. Hence, we have $\tau _{\leq \rho }\Delta (a, v_*\Lambda )=0$ . By the assumption on T again, the nearby cycles for g and the constant sheaf $\Lambda $ are $\rho $ -compatible with any base change. Since the composition $b \colon Y' \hookrightarrow Y \to X$ is smooth, we have $ \Delta (a, \Lambda ) \cong b^*\Delta (g, \Lambda ) $ by the smooth base change theorem. Hence, we obtain that

$$\begin{align*}\tau_{\leq \rho}\Delta(a, \Lambda) \cong \tau_{\leq \rho}b^*\Delta(g, \Lambda) \cong b^*\tau_{\leq \rho}\Delta(g, \Lambda)=0. \end{align*}$$

(2) Let $q \colon U \to T$ be a morphism from a strictly local scheme U, a point $\eta \in U$ and an algebraic geometric point $\overline {\eta } \to U$ lying above $\eta $ . Let $s \in U$ be the closed point. We want to show that the complex

$$\begin{align*}\tau_{\leq \rho}R\Psi_{a_{U}, \overline{\eta}}(\mathcal{F}_U) \end{align*}$$

is $\mathop {\mathrm {Gal}}\nolimits (\kappa (\overline {\eta })/\kappa (\eta ))$ -unipotent.

We first claim that for every $i \leq \rho $ , the sheaf $R^i\Psi _{a_{U}, \overline {\eta }}(\mathcal {F}_U)$ is constructible. Since we have already shown that the nearby cycles for a and $\mathcal {F}$ are $\rho $ -compatible with any base change, we may assume that U is the strict localization of T at $s \to T$ , in particular, we may assume that U is Noetherian. Then, by using [Reference Grothendieck11, 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 [Reference Deligne9, Th. finitude, Théorème 3.2] (see also [Reference Orgogozo25, Section 8]). Now, it suffices to prove that, for every geometric point $x \to Y^{\prime }_s$ , the complex

$$\begin{align*}\tau_{\leq \rho}R\Psi_{a_U, \overline{\eta}}(\mathcal{F}_U)_x \cong \tau_{\leq \rho}R\Gamma((Y^{\prime}_U)_{(x)} \times_{U} U_{(\overline{\eta})}, u_!\mathcal{L}) \end{align*}$$

is $\mathop {\mathrm {Gal}}\nolimits (\kappa (\overline {\eta })/\kappa (\eta ))$ -unipotent (see [Reference Orgogozo26, Lemme 1.2.5]). Since the sheaf $u_!\mathcal {L}$ is locally unipotent along $\mathfrak {Y}=\{ Y^{\circ }, Y\backslash Y^{\circ } \}$ , we reduce to the case where $\mathcal {L}=\Lambda $ is a constant sheaf on $Y^{\circ }$ as in the proof of (1).

By the exact sequence $ 0 \to u_!\Lambda \to \Lambda \to v_*\Lambda \to 0, $ it suffices to prove that the nearby cycles for a and $v_*\Lambda $ and the nearby cycles for a and $\Lambda $ are $\rho $ -unipotent. By using the assumption on T, we conclude by the same argument as in the proof of (1).

Claim 3.14. Let n be a positive integer invertible on T and $\mathcal {L}$ a locally constant constructible sheaf of ${\mathbb Z}/n{\mathbb Z}$ -modules on $Y^{\circ }$ such that $\mathcal {F}:=u_!\mathcal {L}$ is locally unipotent along the stratification $\mathfrak {Y}=\{ Y^{\circ }, Y\backslash Y^{\circ } \}$ . Then the following assertions hold:

1. (1) The nearby cycles for f and $\mathcal {F}$ are $\rho $ -compatible with any base change.

2. (2) The nearby cycles for f and $\mathcal {F}$ are $\rho $ -unipotent.

Proof. (1) We fix a local morphism $q \colon V \to U$ of strictly local schemes over T and an algebraic geometric point $\overline {\xi } \to V$ with image $\overline {\eta } \to U$ . We retain the notation of the proof of Claim 3.13 (1). We write $\Delta :=\Delta (f, \mathcal {F})$ . We want to show $\tau _{\leq \rho } \Delta =0$ . Let $c \colon Z \hookrightarrow Y$ be a closed immersion whose complement is the smooth locus $Y'$ of h. By Claim 3.13 (1), we have

$$\begin{align*}\tau_{\leq \rho} \Delta \cong c_*c^* \tau_{\leq \rho}\Delta, \end{align*}$$

and, hence, it suffices to show that $c^* \tau _{\leq \rho }\Delta =0$ . Since the composition $d \colon Z \to Y \to X$ is a finite morphism, it is enough to prove that

$$\begin{align*}d_*c^* \tau_{\leq \rho}\Delta=0. \end{align*}$$

By using $\tau _{\leq \rho } \Delta \cong c_*c^* \tau _{\leq \rho }\Delta $ , we obtain an isomorphism $d_*c^* \tau _{\leq \rho }\Delta \cong \tau _{\leq \rho }Rh_*\Delta $ . By the proper base change theorem, we have $Rh_*\Delta \cong \Delta (g, Rh_*\mathcal {F})$ . Note that $X^{\circ }$ is normal since T is normal. Hence, the cohomology sheaves of $Rh_*\mathcal {F}$ are locally unipotent along the stratification $\{ X^{\circ }, X \backslash X^{\circ } \}$ by Proposition 3.1. By the assumption on T, we have $ \tau _{\leq \rho }\Delta (g, R^{i}h_*\mathcal {F})=0 $ for every i. It follows that $ \tau _{\leq \rho }\Delta (g, Rh_*\mathcal {F})=0. $ This completes the proof of (1).

(2) Let $q \colon U \to T$ be a morphism from a strictly local scheme U, a point $\eta \in U$ and an algebraic geometric point $\overline {\eta } \to U$ lying above $\eta $ . We write $\mathcal {K}:= R\Psi _{f_U, \overline {\eta }}(\mathcal {F}_U)$ . Let $e \colon Y' \to Y$ denote the open immersion. We have the following distinguished triangle:

$$\begin{align*}e_!e^*\tau_{\leq \rho}\mathcal{K} \to \tau_{\leq \rho}\mathcal{K} \to c_*c^*\tau_{\leq \rho}\mathcal{K} \to. \end{align*}$$

By Claim 3.13 (2), it suffices to prove that $c^*\tau _{\leq \rho }\mathcal {K}$ is $\mathop {\mathrm {Gal}}\nolimits (\kappa (\overline {\eta })/\kappa (\eta ))$ -unipotent. Since d is a finite morphism, it suffices to prove that

$$\begin{align*}d_*c^*\tau_{\leq \rho}\mathcal{K} \end{align*}$$

is $\mathop {\mathrm {Gal}}\nolimits (\kappa (\overline {\eta })/\kappa (\eta ))$ -unipotent. We have the following distinguished triangle:

$$\begin{align*}Rh_*e_!e^*\tau_{\leq \rho}\mathcal{K} \to Rh_*\tau_{\leq \rho}\mathcal{K} \to d_*c^*\tau_{\leq \rho}\mathcal{K} \to. \end{align*}$$

Since the complex $Rh_*e_!e^*\tau _{\leq \rho }\mathcal {K}$ is $\mathop {\mathrm {Gal}}\nolimits (\kappa (\overline {\eta })/\kappa (\eta ))$ -unipotent by Claim 3.13 (2), it is enough to show that $ \tau _{\leq \rho }Rh_*\tau _{\leq \rho }\mathcal {K} \cong \tau _{\leq \rho }Rh_*\mathcal {K} $ is $\mathop {\mathrm {Gal}}\nolimits (\kappa (\overline {\eta })/\kappa (\eta ))$ -unipotent. By the proper base change theorem, we have

$$\begin{align*}Rh_*\mathcal{K} \cong R\Psi_{g_{U}, \overline{\eta}}((Rh_*\mathcal{F})_U). \end{align*}$$

As above, by Proposition 3.1 and the assumption on T, it follows that the complex $\tau _{\leq \rho }R\Psi _{g_{U}, \overline {\eta }}((Rh_*\mathcal {F})_U)$ is $\mathop {\mathrm {Gal}}\nolimits (\kappa (\overline {\eta })/\kappa (\eta ))$ -unipotent, whence, (2).

The proof of Lemma 3.12 is complete.

Now Proposition 3.10 follows from Lemma 3.11 and Lemma 3.12. Finally, we prove the following proposition which completes the proof Theorem 2.8.

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 [Reference Huber12, Reference Huber13, Reference Huber14]. We shall recall the definitions very roughly. We will use the terminology in [Reference Huber14, 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

$$\begin{align*}X=(X, {\mathcal O}_X, \{ v_{x} \}_{x \in X}), \end{align*}$$

where X is a topological space, ${\mathcal O}_X$ is a sheaf of topological rings on the topological space X and $v_x$ is an equivalence class of valuations of the stalk ${\mathcal O}_{X, x}$ at $x \in X$ which is locally isomorphic to the affinoid adic space $\mathop {\mathrm {Spa}}\nolimits (A, A^+)$ associated with an affinoid ring $(A, A^+)$ (see [Reference Huber14, 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 $\mathop {\mathrm {Spa}}\nolimits (A, A^+)$ associated with an affinoid ring $(A, A^+)$ such that A is a strongly Noetherian Tate ring. So we can use the results in [Reference Huber14] (see [Reference Huber14, (1.1.1)]). In particular, we only treat analytic adic spaces (see [Reference Huber14, Section 1.1] for the definition of an analytic adic space).

A pseudo-adic space is a pair

$$\begin{align*}(X, S), \end{align*}$$

where X is an adic space and S is a subset of X satisfying certain conditions (see [Reference Huber14, Definition 1.10.3]). If X is an adic space and $S \subset X$ 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 $f \colon (X, S) \to (X', S')$ of pseudo-adic spaces is by definition a morphism $f \colon X \to X'$ of adic spaces with $f(S) \subset S'$ .

We have a functor $X \mapsto (X, X)$ 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, that is, it is a topological field whose topology is induced by a valuation $ \vert \cdot \vert \colon K \to {\mathbb R}_{\geq 0}$ of rank $1$ . We assume that K is complete. Let ${\mathcal O}=K^{\circ }$ be the valuation ring of $\vert \cdot \vert $ . We call ${\mathcal O}$ the ring of integers of K. Let $\varpi \in K^{\times }$ be an element with $\vert \varpi \vert < 1$ . Let $\mathcal {X}$ be a scheme of finite type over ${\mathcal O}$ . The $\varpi $ -adic formal completion of $\mathcal {X}$ is denoted by $\widehat {\mathcal {X}}$ or $\mathcal {X}^{\wedge }$ . Following [Reference Huber14, Section 1.9], the Raynaud generic fibre of $\widehat {\mathcal {X}}$ is denoted by $d(\widehat {\mathcal {X}})$ , which is an adic space of finite type over $\mathop {\mathrm {Spa}}\nolimits (K, {\mathcal O})$ . In particular, $d(\widehat {\mathcal {X}})$ is quasicompact. For example, we have

$$\begin{align*}d((\mathop{\mathrm{Spec}} {\mathcal O}[T])^{\wedge})=\mathop{\mathrm{Spa}}\nolimits (K\langle T \rangle, {\mathcal O} \langle T \rangle)=:\mathbb{B}(1). \end{align*}$$

We often identify $d((\mathop {\mathrm {Spec}} {\mathcal O}[T])^{\wedge })$ with $\mathbb {B}(1)$ . For a morphism $f \colon \mathcal {Y} \to \mathcal {X}$ of schemes of finite type over ${\mathcal O}$ , the induced morphism $d(\widehat {\mathcal {Y}}) \to d(\widehat {\mathcal {X}})$ is denoted by $d(f)$ (rather than $d(\widehat {f})$ ).

Important examples of pseudo-adic spaces for us are tubular neighbourhoods of adic spaces. In the next subsection, we will define them in the case where adic spaces arise from schemes of finite type over ${\mathcal O}$ .

4.2 Tubular neighbourhoods

Let $X=(X, {\mathcal O}_X, \{ v_{x} \}_{x \in X})$ be an adic space. Let $U \subset X$ be an open subset and $g \in {\mathcal O}_X(U)$ an element. Following [Reference Huber14], for a point $x \in U$ , we write $\vert g(x) \vert :=v_x(g)$ (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 ${\mathcal O}$ .

Proof. (1) Let $\varpi \in K^{\times }$ be an element with $\epsilon = \vert \varpi \vert $ . Let $\mathcal {U} \subset \mathcal {X}$ be an affine open subset. It suffices to show that the subsets

$$\begin{align*}\{ x \in d(\widehat{\mathcal{U}}) \, \vert \, \vert g_i(x) \vert < \epsilon \ \text{for every } 1 \leq i \leq q \} \end{align*}$$

and

$$\begin{align*}\{ x \in d(\widehat{\mathcal{U}}) \, \vert \, \vert g_i(x) \vert \leq \epsilon \ \text{for every } 1 \leq i \leq q \} \end{align*}$$

are independent of the choice of a set $\{ g_1, \dotsc , g_q \} \subset {\mathcal O}_{\mathcal {U}}(\mathcal {U})$ of elements defining the closed subscheme $\mathcal {Z} \cap \mathcal {U}$ of $\mathcal {U}$ . Let $\{ h_1, \dotsc , h_r\} \subset {\mathcal O}_{\mathcal {U}}(\mathcal {U})$ be another set of such elements. Then, for every i, we have

$$\begin{align*}g_i = \Sigma_{1 \leq j \leq r} s_{ij} h_j \end{align*}$$

for some elements $\{ s_{ij} \} \subset {\mathcal O}_{\mathcal {U}}(\mathcal {U})$ . Since we have $\vert s_{ij}(x) \vert \leq 1$ for every $x \in d(\widehat {\mathcal {U}})$ and every $s_{ij}$ , the assertion follows.

(2) We may assume that $\mathcal {X}$ is affine. The subset $T(\mathcal {Z}, \epsilon )$ is a rational subset of the affinoid adic space $d(\widehat {\mathcal {X}})$ , and, hence, it is open and quasicompact. The subset $S(\mathcal {Z}, \epsilon )$ is the complement of the union of finitely many rational subsets. It follows that $S(\mathcal {Z}, \epsilon )$ is closed and constructible.

(3) We may assume that $\mathcal {X}$ and $\mathcal {Y}$ are affine. Then the assertion follows from the descriptions given in (1).

The subsets $T(\mathcal {Z}, \epsilon )$ and $S(\mathcal {Z}, \epsilon )$ in Proposition 4.1 are called an open tubular neighbourhood and a closed tubular neighbourhood of $d(\widehat {\mathcal {Z}})$ in $d(\widehat {\mathcal {X}})$ , respectively. For an element $\epsilon \in \vert K^{\times } \vert $ , we also consider the following subsets:

$$\begin{align*}Q(\mathcal{Z}, \epsilon):=d(\widehat{\mathcal{X}}) \backslash S(\mathcal{Z}, \epsilon). \end{align*}$$

This is a quasicompact open subset of $d(\widehat {\mathcal {X}})$ .

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 $(d(\widehat {\mathcal {X}}), S(\mathcal {Z}, \epsilon ))$ and $(d(\widehat {\mathcal {X}}), T(\mathcal {Z}, \epsilon ))$ are denoted by $S(\mathcal {Z}, \epsilon )$ and $T(\mathcal {Z}, \epsilon )$ , respectively.

We end this subsection with the following lemma.

Proof. We may assume that $\mathcal {X}$ is affine. Then the underlying topological space of $d(\widehat {\mathcal {X}})$ is a spectral space. We have

$$\begin{align*}d(\widehat{\mathcal{Z}})=\bigcap_{\epsilon \in \vert K^{\times} \vert} T(\mathcal{Z}, \epsilon). \end{align*}$$

Hence, the intersection

$$\begin{align*}\bigcap_{\epsilon \in \vert K^{\times} \vert} T(\mathcal{Z}, \epsilon) \cap (d(\widehat{\mathcal{X}}) \backslash W) \end{align*}$$

is empty. In the constructible topology, the subsets $T(\mathcal {Z}, \epsilon )$ and $d(\widehat {\mathcal {X}}) \backslash W$ are closed and $d(\widehat {\mathcal {X}})$ is quasicompact. It follows that there is an element $\epsilon \in \vert K^{\times } \vert $ such that the intersection $T(\mathcal {Z}, \epsilon ) \cap d(\widehat {\mathcal {X}}) \backslash W$ is empty, whence $T(\mathcal {Z}, \epsilon ) \subset W$ .

4.3 Main results on tubular neighbourhoods

In this subsection, let K be an algebraically closed complete non-archimedean field with ring of integers ${\mathcal O}$ .

To state our main results on tubular neighbourhoods, we need étale cohomology and étale cohomology with proper support of pseudo-adic spaces (see [Reference Huber14, Section 2.3] for definition of the étale site of a pseudo-adic space). As shown in [Reference Huber14, Proposition 2.3.7], for an adic space X and an open subset $U \subset X$ , the étale topos of the adic space U is naturally equivalent to the étale topos of the pseudo-adic space $(X, U)$ . For a commutative ring $\Lambda $ , let $D^+(X, \Lambda )$ denote the derived category of bounded below complexes of étale sheaves of $\Lambda $ -modules on a pseudo-adic space X.

Let $f \colon X \to Y$ be a morphism of analytic pseudo-adic spaces. We assume that f is separated, locally of finite type and taut (see [Reference Huber14, 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 quasicompact, then f is taut). For such a morphism f, the direct image functor with proper support

$$\begin{align*}Rf_! \colon D^+(X, \Lambda) \to D^+(Y, \Lambda) \end{align*}$$

is defined in [Reference Huber14, Definition 5.4.4], where $\Lambda $ is a torsion commutative ring. Moreover, if $Y=\mathop {\mathrm {Spa}}\nolimits (K, {\mathcal O})$ , we obtain for a complex $\mathcal {K} \in D^+(X, \Lambda )$ the cohomology group with proper support

$$\begin{align*}H^i_c(X, \mathcal{K}). \end{align*}$$

Let us recall the following results due to Huber in our setting.

The main objective of this paper is to prove uniform variants of Theorem 4.5 for constant sheaves. Our main result on étale cohomology groups with proper support of tubular neighbourhoods is as follows.

Our main result on étale cohomology groups of tubular neighbourhoods is as follows.

Theorem 4.9. Let K be an algebraically closed complete non-archimedean field with ring of integers ${\mathcal O}$ . Let $\mathcal {X}$ be a separated scheme of finite type over ${\mathcal O}$ and $\mathcal {Z} \hookrightarrow \mathcal {X}$ a closed immersion of finite presentation. Then there exists an element $\epsilon _0 \in \vert K^{\times } \vert $ such that, for every $\epsilon \in \vert K^{\times } \vert $ with $\epsilon \leq \epsilon _0$ and for every positive integer n invertible in ${\mathcal O}$ , the restriction maps

$$\begin{align*}H^i(T(\mathcal{Z}, \epsilon), {\mathbb Z}/n{\mathbb Z}) \overset{\cong}{\to} H^i(S(\mathcal{Z}, \epsilon), {\mathbb Z}/n{\mathbb Z}) \overset{\cong}{\to} H^i(S(\mathcal{Z}, \epsilon)^{\circ}, {\mathbb Z}/n{\mathbb Z}) \overset{\cong}{\to} H^i(d(\widehat{\mathcal{Z}}), {\mathbb Z}/n{\mathbb Z}) \end{align*}$$

are isomorphisms for every i.

Theorem 1.2 follows from Theorem 4.9 (see also the following remark).

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 $L \subset K$ be a subfield of K which is a complete non-archimedean field with the induced topology. Let ${\mathcal O}_L$ 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

$$\begin{align*}X^{\mathrm{ad}}:= X \times_{\mathop{\mathrm{Spec}} L} \mathop{\mathrm{Spa}}\nolimits(L, {\mathcal O}_L) \end{align*}$$

(see [Reference Huber13, Proposition 3.8]). For an adic space Y locally of finite type over $\mathop {\mathrm {Spa}}\nolimits (L, {\mathcal O}_L)$ , we denote by

$$\begin{align*}Y_{K}:=Y \times_{\mathop{\mathrm{Spa}}\nolimits(L, {\mathcal O}_L)} \mathop{\mathrm{Spa}}\nolimits(K, {\mathcal O}) \end{align*}$$

the base change of Y to $\mathop {\mathrm {Spa}}\nolimits (K, {\mathcal O})$ , which exists by [Reference Huber14, Proposition 1.2.2].

Corollary 4.11. Let X be a proper scheme over L and $Z \hookrightarrow X$ a closed immersion. We have a closed immersion $Z^{\mathrm {ad}} \hookrightarrow X^{\mathrm {ad}}$ of adic spaces over $\mathop {\mathrm {Spa}}\nolimits (L, {\mathcal O}_L)$ . Then, there is a quasicompact open subset V of $X^{\mathrm {ad}}$ containing $Z^{\mathrm {ad}}$ such that, for every positive integer n invertible in ${\mathcal O}$ , the restriction map

$$\begin{align*}H^i(V_K, {\mathbb Z}/n{\mathbb Z}) \to H^i((Z^{\mathrm{ad}})_K, {\mathbb Z}/n{\mathbb Z}) \end{align*}$$

is an isomorphism for every i.

Proof. There exist a proper scheme $\mathcal {X}$ over $\mathop {\mathrm {Spec}} {\mathcal O}_L$ and a closed immersion $\mathcal {Z} \hookrightarrow \mathcal {X}$ such that the base change of it to $\mathop {\mathrm {Spec}} L$ is isomorphic to the closed immersion $Z \hookrightarrow X$ by Nagata’s compactification theorem (see [Reference Fujiwara and Kato10, Chapter II, Theorem F.1.1] for example). As in Remark 4.10, we may assume that $\mathcal {Z} \hookrightarrow \mathcal {X}$ is of finite presentation. Let

$$\begin{align*}\overline{\mathcal{X}}:=\mathcal{X} \times_{\mathop{\mathrm{Spec}} {\mathcal O}_L} \mathop{\mathrm{Spec}} {\mathcal O} \quad \text{and} \quad \overline{\mathcal{Z}}:=\mathcal{Z} \times_{\mathop{\mathrm{Spec}} {\mathcal O}_L} \mathop{\mathrm{Spec}} {\mathcal O} \end{align*}$$

denote the fibre products. We have $ d(\widehat {\overline {\mathcal {Z}}}) \cong d(\widehat {\mathcal {Z}})_K $ and $ d(\widehat {\overline {\mathcal {X}}}) \cong d(\widehat {\mathcal {X}})_K. $ For an element $\epsilon \in \vert L^{\times } \vert $ , we have $T(\mathcal {Z}, \epsilon )_K=T(\overline {\mathcal {Z}}, \epsilon )$ in $d(\widehat {\overline {\mathcal {X}}})$ . By [Reference Huber14, Proposition 1.9.6], we have $d(\widehat {\mathcal {Z}}) = Z^{\mathrm {ad}}$ and $d(\widehat {\mathcal {X}}) = X^{\mathrm {ad}}$ . Therefore, the assertion follows from Theorem 4.9.

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 [Reference Huber14, Section 1.9].

Following [Reference Huber14], for a commutative ring A and an element $s \in A$ , let

$$\begin{align*}A(s/s) \end{align*}$$

denote the localization $A[1/s]$ equipped with the structure of a Tate ring such that the image $A_0$ of the map $A \to A[1/s]$ is a ring of definition and $s A_0$ is an ideal of definition.

We record the following well known results.

Proof. See [Reference Fujiwara and Kato10, Chapter 0, Proposition 8.4.4] for (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), the ring $\widehat {B}/I\widehat {B}$ is $\varpi $ -adically complete. Hence, we have

$$\begin{align*}\widehat{B/I}={\mathop{\varprojlim}\limits_{n}} (B/I)/\varpi^n \cong {\mathop{\varprojlim}\limits_{n}} (\widehat{B}/I\widehat{B})/\varpi^n \cong \widehat{B}/I\widehat{B}. \end{align*}$$

(3) Let $A_0$ be the image of the map $A \to A[1/\varpi ]$ . By (1), the ring $A_0$ is $\varpi $ -adically complete, and, hence, $A(\varpi /\varpi )$ is complete. It is clear from the definitions that

$$\begin{align*}A_0 \langle Y_1, \dotsc, Y_m \rangle[1/\varpi] \cong A(\varpi/\varpi)\langle Y_1, \dotsc, Y_m \rangle. \end{align*}$$

Let N be the kernel of the surjection $B:=A[X_1, \dotsc , X_n] \to A_0[X_1, \dotsc , X_n]$ . By using (2), we have the following exact sequence:

$$\begin{align*}N\otimes_{B} \widehat{B} \to \widehat{B} \to A_0 \langle Y_1, \dotsc, Y_m \rangle \to 0. \end{align*}$$

Since $N[1/\varpi ]=0$ , we have $(N\otimes _{B} \widehat {B})[1/\varpi ]=0$ , and, hence,

$$\begin{align*}\widehat{B}[1/\varpi] \cong A_0 \langle Y_1, \dotsc, Y_m \rangle[1/\varpi]. \end{align*}$$

This completes the proof of (3).

Let $\mathcal {C}$ be the category whose objects are formal schemes which are locally isomorphic to $\mathop {\mathrm {Spf}} A$ for an adic ring A with an ideal of definition $\varpi A$ such that the pair $(A, \varpi )$ satisfies the conditions in Lemma 5.1. The morphisms in $\mathcal {C}$ are the adic morphisms. A formal scheme in $\mathcal {C}$ satisfies the condition (S) in [Reference Huber14, Section 1.9] by Lemma 5.1 (3). In [Reference Huber14, Proposition 1.9.1], Huber defined a functor

$$\begin{align*}d(-) \end{align*}$$

from $\mathcal {C}$ to the category of analytic adic spaces. For a formal scheme $\mathscr {X}$ in $\mathcal {C}$ , the adic space $d(\mathscr {X})$ is equipped with a morphism of ringed spaces

$$\begin{align*}\lambda \colon d(\mathscr{X}) \to \mathscr{X}. \end{align*}$$

This map is called a specialization map. If A and $\varpi \in A$ satisfy the conditions in Lemma 5.1, then we have

$$\begin{align*}d(\mathop{\mathrm{Spf}} A)=\mathop{\mathrm{Spa}}\nolimits (A(\varpi/\varpi), A^{+}), \end{align*}$$

where $A^+$ is the integral closure of A in $A(\varpi /\varpi )=A[1/\varpi ]$ . The map $\lambda \colon d(\mathop {\mathrm {Spf}} A) \to \mathop {\mathrm {Spf}} A$ sends $x \in d(\mathop {\mathrm {Spf}} A)$ to the prime ideal $ \{ a \in A \, \vert \, \vert a (x) \vert < 1 \} \subset A. $ If $f \colon \mathscr {X} \to \mathscr {Y}$ is an adic morphism of formal schemes in $\mathcal {C}$ , then the induced morphism $d(f) \colon d(\mathscr {X}) \to d(\mathscr {Y})$ fits into the following commutative diagram:

For the sake of completeness, we include a proof of the following result on the compatibility of the functor $d(-)$ with fibre products.

Proposition 5.2. Let $f \colon \mathscr {X} \to \mathscr {Y}$ be a morphism locally of finite type of formal schemes in $\mathcal {C}$ . Let $\mathscr {Z} \to \mathscr {Y}$ be an adic morphism of formal schemes in $\mathcal {C}$ . Then the morphism

$$\begin{align*}d(\mathscr{X} \times_{\mathscr{Y}} \mathscr{Z}) \to d(\mathscr{X}) \times_{d(\mathscr{Y})} d(\mathscr{Z}) \end{align*}$$

induced by the universal property of the fibre product is an isomorphism.

Proof. First, we note that the fibre product $ d(\mathscr {X}) \times _{d(\mathscr {Y})} d(\mathscr {Z}) $ exists by [Reference Huber14, Proposition 1.2.2] since $d(f) \colon d(\mathscr {X}) \to d(\mathscr {Y})$ is locally of finite type.

We may assume that $\mathscr {X}=\mathop {\mathrm {Spf}} A$ , $\mathscr {Y}= \mathop {\mathrm {Spf}} B$ and $\mathscr {Z}= \mathop {\mathrm {Spf}} C$ are affine, where B and C satisfy the conditions in Lemma 5.1 for some element $\varpi \in B$ and for its image in C, respectively. We may assume further that A is of the form $B\langle X_1, \dotsc , X_n \rangle /I$ . We write $ D:=A \otimes _B C$ . The source of the morphism in question is isomorphic to

$$\begin{align*}\mathop{\mathrm{Spa}}\nolimits ((\widehat{D})(\varpi/\varpi), E^{+}), \end{align*}$$

where $\widehat {D}$ is the $\varpi $ -adic completion of D and $E^{+}$ is the integral closure of $\widehat {D}$ in $(\widehat {D})[1/\varpi ]$ . On the other hand, the target of the morphism in question is isomorphic to

$$\begin{align*}\mathop{\mathrm{Spa}}\nolimits (D(\varpi/\varpi), F^+), \end{align*}$$

where $F^+$ is the integral closure of D in $D[1/\varpi ]$ . Let $D_0$ be the image of the map $D \to D[1/\varpi ]$ . Clearly, the completion of $D(\varpi /\varpi )$ is isomorphic to $(\widehat {D}_0)(\varpi /\varpi )$ . By a similar argument as in the proof of Lemma 5.1 (3), we have $(\widehat {D})(\varpi /\varpi ) \cong (\widehat {D}_0)(\varpi /\varpi )$ . This completes the proof of the proposition since the adic spaces associated with an affinoid ring and its completion are naturally isomorphic (see [Reference Huber13, Lemma 1.5]).

A valuation ring R is called a microbial valuation ring if the field of fractions L of R admits a topologically nilpotent unit $\varpi $ with respect to the valuation topology (see [Reference Huber14, Definition 1.1.4]). We equip R with the valuation topology unless explicitly mentioned otherwise. In this case, the element $\varpi $ is contained in R, the ideal $\varpi R$ is an ideal of definition of R and we have $L=R[1/\varpi ]$ . The completion $ \widehat {R} $ of R is also a microbial valuation ring.

Let R be a complete microbial valuation ring. It is well known that

$$\begin{align*}R\langle X_1, \dotsc, X_n \rangle[1/\varpi] \cong L \langle X_1, \dotsc, X_n \rangle \end{align*}$$

is Noetherian for every $n \geq 0$ (see [Reference Bosch, Güntzer and Remmert3, 5.2.6, Theorem 1]). A formal scheme $\mathscr {X}$ locally of finite type over $\mathop {\mathrm {Spf}} R$ is in the category $\mathcal {C}$ .

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 $\varpi $ be a topologically nilpotent unit in L. Let $\overline {L}$ be a separable closure of L, and let $\overline {R}$ be the valuation ring of $\overline {L}$ which extends R.

We will use the following notation. Let $\eta \in \mathop {\mathrm {Spec}} R$ and $\overline {\eta } \in \mathop {\mathrm {Spec}} \overline {R}$ be the generic points. For a scheme $\mathcal {X}$ over R, we write

$$\begin{align*}\mathcal{X}_{\eta}:=\mathcal{X} \times_{\mathop{\mathrm{Spec}} R} \eta \quad \text{and} \quad \mathcal{X}_{\overline{\eta}}:=\mathcal{X} \times_{\mathop{\mathrm{Spec}} R} \overline{\eta}. \end{align*}$$

The $\varpi $ -adic formal completion of a scheme (or a ring) $\mathcal {X}$ over R is denoted by $\widehat {\mathcal {X}}$ . Let $s \in \mathop {\mathrm {Spec}} R$ be the closed point and $\mathcal {X}_s$ the special fibre of $\mathcal {X}$ .

We write

$$\begin{align*}S:=\mathop{\mathrm{Spa}}\nolimits(L, R)= d(\mathop{\mathrm{Spf}} \widehat{R}) \quad \text{and} \quad \overline{S}:=\mathop{\mathrm{Spa}}\nolimits(\overline{L}, \overline{R}) = d(\mathop{\mathrm{Spf}} \widehat{\overline{R}}). \end{align*}$$

Let $t \in S$ and $\overline {t} \in \overline {S}$ be the closed points corresponding to the valuation rings R and $\overline {R}$ , respectively. The pseudo-adic space $ (\overline {S}, \{ \overline {t} \}) $ is also denoted by $\overline {t}$ . The natural morphism $\xi \colon \overline {t} \to S$ is a geometric point with support $t \in S$ in the sense of [Reference Huber14, Definition 2.5.1].

Let $f \colon \mathcal {X} \to \mathop {\mathrm {Spec}} R$ be a separated morphism of finite type of schemes. The induced morphism

$$\begin{align*}d(f) \colon d(\widehat{\mathcal{X}}) \to S \end{align*}$$

is separated and of finite type; the separatedness can be checked, for example, by using Proposition 5.2 (we often write $d(f)$ instead of $d(\widehat {f})$ ). There is a natural morphism $d(\widehat {\mathcal {X}}) \to \mathcal {X}_{\eta }$ of locally ringed spaces (see [Reference Huber14, (1.9.4)]). An étale morphism $Y \to \mathcal {X}_{\eta }$ defines an adic space $d(\widehat {\mathcal {X}}) \times _{\mathcal {X}_{\eta }} Y$ , which is étale over $d(\widehat {\mathcal {X}})$ (see [Reference Huber13, Proposition 3.8] and [Reference Huber14, Corollary 1.7.3 i)]). In this way, we get a morphism of étale sites

$$\begin{align*}a \colon d(\widehat{\mathcal{X}})_{\mathrm{\acute{e}t}} \to (\mathcal{X}_{\eta})_{\mathrm{\acute{e}t}}. \end{align*}$$

Let $\Lambda $ be a torsion commutative ring. Let $\mathcal {F}$ be an étale sheaf of $\Lambda $ -modules on $\mathcal {X}$ . Let $ \mathcal {F}^a $ denote the pullback of $\mathcal {F}$ by the composition

$$\begin{align*}d(\widehat{\mathcal{X}})_{\mathrm{\acute{e}t}} \overset{a}{\to} (\mathcal{X}_{\eta})_{\mathrm{\acute{e}t}} \to \mathcal{X}_{\mathrm{\acute{e}t}}. \end{align*}$$

Recall that we have the direct image functor with proper support

$$\begin{align*}Rd(f)_! \colon D^+(d(\widehat{\mathcal{X}}), \Lambda) \to D^+(S, \Lambda) \end{align*}$$

for $d(f)$ by [Reference Huber14, Definition 5.4.4]. We define $ R^nd(f)_!\mathcal {F}^a:=H^n(Rd(f)_!\mathcal {F}^a). $ We will describe the stalk

$$\begin{align*}(R^nd(f)_!\mathcal{F}^a)_{\overline{t}}:=\Gamma(\overline{t}, \xi^*R^nd(f)_!\mathcal{F}^a) \end{align*}$$

at the geometric point $\xi \colon \overline {t} \to S$ in terms of the sliced nearby cycles functor relative to f. We defined the sliced nearby cycles functor

$$\begin{align*}R\Psi_{f, \overline{\eta}}:=i^{*}Rj_*j^* \colon D^{+}(\mathcal{X}, \Lambda) \to D^{+}(\mathcal{X}_s, \Lambda) \end{align*}$$

in Section 2. Here, we fix the notation by the following commutative diagram:

Now we can state the following result due to Huber:

Theorem 5.3 (Huber [Reference Huber14, Theorem 5.7.8])

There is an isomorphism

$$\begin{align*}(R^nd(f)_!\mathcal{F}^a)_{\overline{t}} \cong H^n_c(\mathcal{X}_s, R\Psi_{f, \overline{\eta}}(\mathcal{F})) \end{align*}$$

for every n. This isomorphism is compatible with the natural actions of $\mathop {\mathrm {Gal}}\nolimits (\overline {L}/L)$ on both sides.

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 ${\mathcal O}$ for simplicity. We fix a topologically nilpotent unit $\varpi $ in K.

For an adic space X over $\mathop {\mathrm {Spa}}\nolimits (K, {\mathcal O})$ , we will use the following notation. For a point $x \in X$ , let $k(x)$ be the residue field of the local ring ${\mathcal O}_{X, x}$ 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 $\varpi $ , also denoted by $\varpi $ , is a topologically nilpotent unit in $k(x)$ . For a geometric point $\xi $ of X, let $\mathop {\mathrm {Supp}}(\xi ) \in X$ denote the support of it.

We recall strict localizations of analytic adic spaces. Let $ \xi \colon s \to X $ be a geometric point. The strict localization

$$\begin{align*}X(\xi) \end{align*}$$

of X at $\xi $ is defined in [Reference Huber14, Section 2.5.11]. It is an adic space over X with an X-morphism $s \to X(\xi )$ . We write $x:=\mathop {\mathrm {Supp}}(\xi )$ . By [Reference Huber14, Proposition 2.5.13], the strict localization $X(\xi )$ is isomorphic to

$$\begin{align*}\mathop{\mathrm{Spa}}\nolimits(\overline{k}(x), \overline{k}(x)^+) \end{align*}$$

over X, where $\overline {k}(x)$ is a separable closure of $k(x)$ and $\overline {k}(x)^+$ is a valuation ring extending $k(x)^+$ . A specialization morphism $\xi _1 \to \xi _2$ of geometric points of X is by definition a morphism $X(\xi _1) \to X(\xi _2)$ over X, and such a morphism exists if and only if we have

$$\begin{align*}\mathop{\mathrm{Supp}}(\xi_2) \in \overline{\{ \mathop{\mathrm{Supp}}(\xi_1) \}}. \end{align*}$$

Let $\mathcal {F}$ be an abelian étale sheaf on X. A specialization morphism $\xi _1 \to \xi _2$ of geometric points of X induces a mapping

$$\begin{align*}\mathcal{F}_{\xi_2} \to \mathcal{F}_{\xi_1} \end{align*}$$

on the stalks in the usual way (see [Reference Huber14, (2.5.16)]).

Let $\mathcal {X}$ be a scheme of finite type over $\mathop {\mathrm {Spec}} {\mathcal O}$ . We write $ \mathcal {X}_{K}:=\mathcal {X} \times _{\mathop {\mathrm {Spec}} {\mathcal O}} \mathop {\mathrm {Spec}} K. $ For an étale sheaf $\mathcal {F}$ on $\mathcal {X}$ , let $ \mathcal {F}^a $ denote the pullback of $\mathcal {F}$ by the composition

$$\begin{align*}d(\widehat{\mathcal{X}})_{\mathrm{\acute{e}t}} \overset{a}{\to} (\mathcal{X}_{K})_{\mathrm{\acute{e}t}} \to \mathcal{X}_{\mathrm{\acute{e}t}}. \end{align*}$$

Let $\Lambda $ be a torsion commutative ring.

Proposition 5.5. Let $\mathcal {X}$ and $\mathcal {Y}$ be separated schemes of finite type over ${\mathcal O}$ . Let $f \colon \mathcal {X} \to \mathcal {Y}$ be a morphism over ${\mathcal O}$ . Let $\mathcal {F}$ be an étale sheaf of $\Lambda $ -modules on $\mathcal {X}$ . We assume that the sliced nearby cycles complexes for f and $\mathcal {F}$ are compatible with any base change (see Definition 2.3 (1)). Let $s \in \widehat {\mathcal {Y}}$ be a point. We consider the inverse image $\lambda ^{-1}(s)$ under the specialization map

$$\begin{align*}\lambda \colon d(\widehat{\mathcal{Y}}) \to \widehat{\mathcal{Y}}. \end{align*}$$

Then, the sheaf $R^nd(f)_!\mathcal {F}^a$ is overconvergent on $\lambda ^{-1}(s)$ for every n.

Proof. Let $\xi _1 \to \xi _2$ be a specialization morphism of geometric points of $d(\widehat {\mathcal {Y}})$ whose supports are contained in $\lambda ^{-1}(s)$ . We write $y_m:= \mathop {\mathrm {Supp}}(\xi _m)$ ( $m=1, 2$ ). Let $\overline {k}(y_m)$ be a separable closure of $k(y_m)$ , and let $\overline {k}(y_m)^+$ be a valuation ring extending $k(y_m)^+$ . We identify $d(\widehat {\mathcal {Y}})(\xi _m)$ with $\mathop {\mathrm {Spa}}\nolimits (\overline {k}(y_m), \overline {k}(y_m)^+)$ . Let $R_m$ be the completion of $\overline {k}(y_m)^+$ , and we put $U_m:= \mathop {\mathrm {Spec}} R_m$ . The morphism $\mathop {\mathrm {Spa}}\nolimits (\overline {k}(y_m), \overline {k}(y_m)^+) \to d(\widehat {\mathcal {Y}})$ induces a natural morphism

$$\begin{align*}q_m \colon U_m \to \mathcal{Y} \end{align*}$$

over $\mathop {\mathrm {Spec}} {\mathcal O}$ , and the specialization morphism $d(\widehat {\mathcal {Y}})(\xi _1) \to d(\widehat {\mathcal {Y}})(\xi _2)$ induces a natural $\mathcal {Y}$ -morphism

$$\begin{align*}r \colon U_1 \to U_2. \end{align*}$$

By the assumption, we have $ q_m(s_m)=s $ for the closed point $s_m \in U_m$ , where the image of $s \in \widehat {\mathcal {Y}}$ in $\mathcal {Y}$ is denoted by the same letter. Let $\overline {s} \to \mathcal {Y}$ be an algebraic geometric point lying above s, and let U be the strict localization of $\mathcal {Y}$ at $\overline {s}$ . There are local $\mathcal {Y}$ -morphisms $\widetilde {q}_m \colon U_m \to U \ (m=1, 2)$ such that the following diagram commutes:

We remark that r is not a local morphism if $y_1 \neq y_2$ . Let $\eta _m$ be the generic point of $U_m$ . Then we have $r(\eta _1)=\eta _2$ . We write $\eta :=\widetilde {q}_1(\eta _1)=\widetilde {q}_2(\eta _2)$ . Let $\overline {\eta } \to U$ denote the algebraic geometric point which is the image of $\eta _2$ .

Let $\mathcal {F}_{m}$ be the pullback of $\mathcal {F}$ to $\mathcal {X} \times _{\mathcal {Y}} U_m$ and $\mathcal {F}_U$ the pullback of $\mathcal {F}$ to $\mathcal {X} \times _{\mathcal {Y}} U$ . For $m=1, 2$ , we have

$$\begin{align*}(R^nd(f)_!\mathcal{F}^a)_{\xi_m} \cong H^n_c(\mathcal{X} \times_{\mathcal{Y}} s_m, R\Psi_{f_{U_m}, \eta_m} (\mathcal{F}_m)) \end{align*}$$

by Theorem 5.3. By our assumption that the sliced nearby cycles complexes for f and $\mathcal {F}$ are compatible with any base change, the map

$$\begin{align*}H^n_c(\mathcal{X} \times_{\mathcal{Y}} \overline{s}, R\Psi_{f_U, \overline{\eta}}(\mathcal{F}_U)) \to H^n_c(\mathcal{X} \times_{\mathcal{Y}} s_m, R\Psi_{f_{U_m}, \eta_m} (\mathcal{F}_m)) \end{align*}$$

is bijective for both $m=1$ and $m=2$ . Hence, the proposition follows from the following commutative diagram:

The commutativity of the diagram can be verified by tracing the construction of the isomorphism in Theorem 5.3.

6.1 Tame sheaves on annuli

In this subsection, we recall two theorems on finite étale coverings on annuli and the punctured disc, which basically follow from the results in [Reference Lütkebohmert22, Reference Lütkebohmert and Schmechta23, Reference Ramero27]. 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 $\mathbb {B}(1)=\mathop {\mathrm {Spa}}\nolimits (K \langle T \rangle , {\mathcal O} \langle T \rangle )$ . Let

$$\begin{align*}\mathbb{B}(1)^{*}:=\mathbb{B}(1) \backslash \{ 0 \} \end{align*}$$

be the punctured disc, where $0 \in \mathbb {B}(1)$ is the K-rational point corresponding to $0 \in K$ . It is an adic space locally of finite type over $\mathop {\mathrm {Spa}}\nolimits (K, {\mathcal O})$ . We fix a valuation $ \vert \cdot \vert \colon K \to {\mathbb R}_{\geq 0}$ of rank $1$ such that the topology of K is induced by it. For elements $a, b \in \vert K^{\times } \vert $ with $a \leq b \leq 1$ , we define

$$ \begin{align*} \mathbb{B}(a, b)&:= \{ x \in \mathbb{B}(1) \, \vert \, a \leq \vert T(x) \vert \leq b \} \\ &:= \{ x \in \mathbb{B}(1) \, \vert \, \vert \varpi_a(x) \vert \leq \vert T(x) \vert \leq \vert \varpi_b(x) \vert \}, \end{align*} $$

which is called an open annulus. Here, $\varpi _a, \varpi _b \in K^{\times }$ are elements such that $a=\vert \varpi _a \vert $ and $b=\vert \varpi _b \vert $ . It is a rational subset of $\mathbb {B}(1)$ , and, hence, it is an affinoid open subspace of $\mathbb {B}(1)$ .

Let m be a positive integer invertible in K. The finite étale morphism $ \varphi _m \colon \mathbb {B}(1)^{*} \to \mathbb {B}(1)^{*} $ defined by $T \mapsto T^m$ is called a Kummer covering of degree m. For elements $a, b \in \vert K^{\times } \vert $ with $a \leq b \leq 1$ , the restriction

$$\begin{align*}\varphi_m \colon \mathbb{B}(a^{1/m}, b^{1/m}) \to \mathbb{B}(a, b) \end{align*}$$

of $\varphi _m$ is also called a Kummer covering of degree m (we also call a morphism of affinoid adic spaces of finite type over $\mathop {\mathrm {Spa}}\nolimits (K, {\mathcal O})$ a Kummer covering if it is isomorphic to $\varphi _m \colon \mathbb {B}(a^{1/m}, b^{1/m}) \to \mathbb {B}(a, b)$ for $a, b \in \vert K^{\times } \vert $ and some m with $a \leq b \leq 1$ ).

In this paper, we use the following notion of tameness for étale sheaves on one-dimensional smooth adic spaces over $\mathop {\mathrm {Spa}}\nolimits (K, {\mathcal O})$ .

Definition 6.1. Let X be a one-dimensional smooth adic space over $\mathop {\mathrm {Spa}}\nolimits (K, {\mathcal O})$ . Let $x \in X$ be a point which has a proper generalization in X, that is, there exists a point $x' \in X$ with $x \in \overline {\{ x' \}}$ and $x \neq x'$ . Let

$$\begin{align*}k(x)^{\wedge h+} \end{align*}$$

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 $k(x)^{\wedge h}$ of $k(x)^{\wedge h+}$ . It induces a geometric point $\overline {x} \to X$ with support x. For an étale sheaf $\mathcal {F}$ on X, we say that $\mathcal {F}$ is tame at $x \in X$ if the action of

$$\begin{align*}\mathop{\mathrm{Gal}}\nolimits(L(x)/k(x)^{\wedge h}) \end{align*}$$

on the stalk $ \mathcal {F}_{\overline {x}} $ at the geometric point $\overline {x}$ factors through a finite group G such that $\sharp \, G$ is invertible in ${\mathcal O}$ , where $\sharp \, G$ denotes the cardinality of G.

Now we can formulate the results.

Theorem 6.2 ([Reference Lütkebohmert22, Theorem 2.2], [Reference Lütkebohmert and Schmechta23, Theorem 4.11], [Reference Ramero27, Theorem 2.4.3])

Let $f \colon X \to \mathbb {B}(1)^{*}$ be a finite étale morphism of adic spaces. There exists an element $\epsilon \in \vert K^{\times } \vert $ with $\epsilon \leq 1$ such that, for all $a, b \in \vert K^{\times } \vert $ with $a < b \leq \epsilon $ , we have

$$\begin{align*}f^{-1}(\mathbb{B}(a, b)) \cong \coprod^{n}_{i=1} \mathbb{B}(c_i, d_i) \end{align*}$$

for some elements $c_i, d_i \in \vert K^{\times } \vert $ with $c_i < d_i \leq 1$ ( $1 \leq i \leq n$ ). If K is of characteristic zero, after replacing $\epsilon \in \vert K^{\times } \vert $ by a smaller one, the restriction

$$\begin{align*}\mathbb{B}(c_i, d_i) \to \mathbb{B}(a, b) \end{align*}$$

of f to every component $\mathbb {B}(c_i, d_i)$ appearing in the above decomposition becomes a Kummer covering.