Ramification of Local Fields

with Imperfect Residue Fields II

Dedicated to Kazuya Kato

on the occasion of his 50th birthday

Ahmed Abbes and Takeshi Saito

Received: October 10, 2002 Revised: August 6, 2003

Abstract. In [1], a filtration by ramification groups and its logarithmic version are defined on the absolute Galois group of a complete discrete valuation field without assuming that the residue field is perfect. In this paper, we study the graded pieces of these filtrations and show that they are abelian except possibly in the absolutely unramified and non-logarithmic case.

2000 Mathematics Subject Classification: primary 11S15, secondary 14G22 Keywords and Phrases: local fields, wild ramification, log structure, affinoid variety.

In the previous paper [1], a filtration by ramification groups and its logarithmic version are defined on the absolute Galois group G_{K} of a complete discrete valuation field K without assuming that the residue field is perfect. In this paper, we study the graded pieces of these filtrations and show that they are abelian except possibly in the absolutely unramified and non-logarithmic case. Let G_{K}^{j} (j > 0, ∈ ℚ) denote the decreasing filtration by ramification groups and G_{K,log}^{j} (j > 0, ∈ ℚ) be its logarithmic variant. We put G_{K}^{j+} = \overline{⋃_{j′>j} G_{K}^{j′}}

and G_{K,log}^{j+} = \overline{⋃_{j′>j} G_{K,log}^{j′}}. In [1], we show that the wild inertia subgroup P ⊂ G_{K} is equal to G_{K}^{1+} = G_{K,log}^{0+}. The main result is the following.

Theorem 1 Let K be a complete discrete valuation field.

1. (see Theorem 2.15) Assume either K has equal characteristics p > 0 or K has mixed characteristic and p is not a prime element. Then, for a rational number j > 1, the graded piece Gr^{j}G_{K} = G_{K}^{j}/G_{K}^{j+} is abelian and is a subgroup of the center of the pro-p-group G_{K}^{1+}/G_{K}^{j+}.

2. (see Theorem 5.12) For a rational number j > 0, the graded piece Gr_{log}^{j}G_{K} = G_{K,log}^{j}/G_{K,log}^{j+} is abelian and is a subgroup of the center of the pro-p-group G_{K,log}^{0+}/G_{K,log}^{j+}.

The idea of the proof of 1 is the following. Under some finiteness assumption, denoted by (F), we define a functor X̄^{j} from the category of finite étale Kalgebras with ramification bounded by j+ to the category of finite étale schemes over a certain tangent space Θ^{j} with continuous semi-linear action of G_{K}. For a finite Galois extension L of K with ramification bounded by j+, the image X̄^{j}(L) has two mutually commuting actions of G = Gal(L/K) and G_{K}. The arithmetic action of G_{K} comes from the definition of the functor X̄^{j} and the geometric action of G is defined by functoriality. Using these two commuting actions, we prove the assertion. The assumption that p is not a prime element is necessary in the construction of the functor X̄^{j}. In Section 1, for a rational number j > 0 and a smooth embedding of a finite flat O_{K}-algebra, we define its j-th tubular neighborhood as an affinoid variety. We also define its j-th twisted reduced normal cone. We recall the definition of the filtration by ramification groups in Section 2.1 using the notions introduced in Section 1. In the equal characteristic case, under the assumption (F), we define a functor X̄^{j} mentioned above in Section 2.2 using j-th tubular neighborhoods. In the mixed characteristic case, we give a similar but subtler construction using the twisted normal cones, assuming further that the residue characteristic p is not a prime element of K in Section 2.3. Then, we prove Theorem 2.15 in Section 2.4. We also define a canonical surjection π_{1}^{ab}(Θ^{j}) → Gr^{j}G_{K} under the assumption (F). After some preparations on generalities of log structures in Section 3, we study a logarithmic analogue in Sections 4 and 5. We define a canonical surjection π_{1}^{ab}(Θ_{log}^{j}) → Gr_{log}^{j}G_{K} under the assumption (F) and prove the logarithmic part, Theorem 5.12, of the main result in Section 5.2. Among other results, we compare the construction with the logarithmic construction given in [1] in Lemma 4.10. We also prove in Corollary 4.12 a logarithmic version of [1] Theorem 7.2 (see also Corollary 1.16). In Section 6, assuming the residue field is perfect, we show that the surjection π_{1}^{ab}(Θ_{log}^{j}) → Gr_{log}^{j}G_{K} induces an isomorphism π_{1}^{ab,gp}(Θ_{log}^{j}) → Gr_{log}^{j}G_{K} where π_{1}^{ab,gp}(Θ_{log}^{j}) denotes the quotient classifying the étale isogenies to Θ_{log}^{j} regarded as an algebraic group. When one of the authors (T.S.) started studing mathematics, Kazuya Kato, who was his adviser, suggested to read [13] and to study how to generalize it when the residue field is no longer assumed perfect. This paper is a partial answer to his suggestion. The authors are very happy to dedicate this paper to him for his 51st anniversary.

Notation. Let K be a complete discrete valuation field, O_{K} be its valuation ring and F be its residue field of characteristic p > 0. Let K̄ be a separable closure of K, OK̄ be the integral closure of O_{K} in K̄, F̄ be the residue field of OK̄, and G_{K} = Gal(K̄/K) be the Galois group of K̄ over K. Let π be a uniformizer of O_{K} and ord be the valuation of K normalized by ordπ = 1. We denote also by ord the unique extension of ord to K̄.

1 Tubular neighborhoods for finite flat algebras

For a semi-local ring R, let 𝔪_{R} denote the radical of R. We say that an O_{K}- algebra R is formally of finite type over O_{K} if R is semi-local, 𝔪_{R}-adically complete, Noetherian and the quotient R/𝔪_{R} is finite over F. An O_{K}-algebra R formally of finite type over O_{K} is formally smooth over O_{K} if and only if its factors are formally smooth. We say that an O_{K}-algebra R is topologically of finite type over O_{K} if R is π-adically complete, Noetherian and the quotient R/πR is of finite type over F. For an O_{K}-algebra R formally of finite type over O_{K}, we put Ω̂_{R/O_{K}} = \underleftarrow{lim}_{n} Ω_{(R/𝔪_{R}^{n})/O_{K}}. For an O_{K}-algebra R topologically of finite type over O_{K}, we put Ω̂_{R/O_{K}} = \underleftarrow{lim}_{n} Ω_{(R/π^{n}R)/O_{K}}. Here and in the following, Ω denotes the module of differential 1-forms. For a surjection R → R′ of rings, its formal completion is defined to be the projective limit R^{∧} = \underleftarrow{lim}_{n} R/(Ker(R → R′))^{n}. In this section, A will denote a finite flat O_{K}-algebra.

1.1 Embeddings of finite flat algebras

Definition 1.1 1. Let A be a finite flat O_{K}-algebra and A be an O_{K}-algebra formally of finite type and formally smooth over O_{K}. We say that a surjection A → A of O_{K}-algebras is an embedding if it induces an isomorphism A/𝔪_{A} → A/𝔪_{A}.

2. We define Emb_{O_{K}} to be the category whose objects and morphisms are as follows. An object of Emb_{O_{K}} is a triple (A → A) where:

• A is a finite flat O_{K}-algebra.

• A is an O_{K}-algebra formally of finite type and formally smooth over O_{K}.

• A → A is an embedding.

A morphism (f, f) : (A → A) → (B → B) of Emb_{O_{K}} is a pair of O_{K}- homomorphisms f : A → B and f : A → B such that the diagram

A ⟶ A
f↓  ↓^{f}
B ⟶ B

is commutative.

3. For a finite flat O_{K}-algebra A, let Emb_{O_{K}} (A) be the subcategory of Emb_{O_{K}} whose objects are of the form (A → A) and morphisms are of the form (id_{A}, f).

4. We say that a morphism (f, f) : (A → A) → (B → B) of Emb_{O_{K}} is finite flat if f : A → B is finite and flat and if the map B ⊗_{A} A → B is an isomorphism.

If (A → A) is an embedding, the A-module Ω̂_{A/O_{K}} is locally free of finite rank.

Lemma 1.2 1. For a finite flat O_{K}-algebra A, the category Emb_{O_{K}} (A) is nonempty.

2. For a morphism f : A → B of finite flat O_{K}-algebras and for embeddings (A → A) and (B → B), there exists a morphism (f, f) : (A → A) → (B → B) lifting f.

3. For a morphism f : A → B of finite flat O_{K}-algebras, the following conditions are equivalent.

(1) The map f : A → B is flat and locally of complete intersection.

(2) Their exists a finite flat morphism (f, f) : (A → A) → (B → B) of embeddings.

Proof. 1. Take a finite system of generators t_{1}, …, t_{n} of A over O_{K} and define a surjection O_{K}[T_{1}, …, T_{n}] → A by T_{i} ↦ t_{i}. Then the formal completion A → A of O_{K}[T_{1}, …, T_{n}] → A, where A = \underleftarrow{lim}_{m} O_{K}[T_{1}, …, T_{n}]/(Ker(O_{K}[T_{1}, …, T_{n}] → A))^{m}, is an embedding.

2. Since A is formally smooth over O_{K} and B = \underleftarrow{lim}_{n} B/I^{n} where I = Ker(B → B), the assertion follows.

3. (1)⇒(2). We may assume A and B are local. By 1 and 2, there exists a morphism (f, f) : (A → A) → (B → B) lifting f. Replacing B → B by the projective limit \underleftarrow{lim}_{n}(A/𝔪_{A}^{n} ⊗_{O_{K}} B/𝔪_{B}^{n})^{∧} → B/𝔪_{B}^{n} of the formal completion (A/𝔪_{A}^{n} ⊗_{O_{K}} B/𝔪_{B}^{n})^{∧} → B/𝔪_{B}^{n} of the surjections A/𝔪_{A}^{n} ⊗_{O_{K}} B/𝔪_{B}^{n} → B/𝔪_{B}^{n}, we may assume that the map A → B is formally smooth. Since A → B is locally of complete intersection, the kernel of the surjection B ⊗_{A} A → B is generated by a regular sequence (t_{1}, …, t_{n}). Take a lifting (t̃_{1}, … t̃_{n}) in B and define a map A[[T_{1}, …, T_{n}]] → B by T_{i} ↦ t_{i}. We consider an embedding A[[T_{1}, …, T_{n}]] → A defined by the composition A[[T_{1}, …, T_{n}]] → A → A sending T_{i} to 0. Replacing A by A[[T_{1}, …, T_{n}]], we obtain a map (A → A) → (B → B) such that the map B ⊗_{A} A → B is an isomorphism and dim A = dim B. By Nakayama’s lemma, the map A → B is finite. Hence the map A → B is flat by EGA Chap 0_{IV} Corollaire (17.3.5) (ii). (2)⇒(1). Since A and B are regular, B is locally of complete intersection over A. Since B is flat over A, B is also flat and locally of complete intersection over A. □ The base change of an embedding by an extension of complete discrete valuation fields is defined as follows.

Lemma 1.3 Let K′ be a complete discrete valuation field and K → K′ be a morphism of fields inducing a local homomorphism O_{K} → O_{K′}. Let (A → A) be an object of Emb_{O_{K}}. We define A ˆ⊗_{O_{K}} O_{K′} to be the projective limit \underleftarrow{lim}_{n}(A/𝔪_{A}^{n} ⊗_{O_{K}} O_{K′}). Then the O_{K′}-algebra A ˆ⊗_{O_{K}} O_{K′} is formally of finite type and formally smooth over O_{K′}. The natural surjection A ˆ⊗_{O_{K}} O_{K′} → A ˆ⊗_{O_{K}} O_{K′} defines an object (A ˆ⊗_{O_{K}} O_{K′} → A ⊗_{O_{K}} O_{K′}) of Emb_{O_{K′}}.

Proof. The O_{K}-algebra A is finite over the power series ring O_{K}[[T_{1}, …, T_{n}]] for some n ≥ 0. Hence the O_{K′}-algebra A ˆ⊗_{O_{K}} O_{K′} is finite over O_{K′}[[T_{1}, …, T_{n}]] and is formally of finite type over O_{K′}. The formal smoothness is clear from the definition. The rest is clear. □. For an object (A → A) of Emb_{O_{K}}, we let the object (A ˆ⊗_{O_{K}} O_{K′} → A⊗_{O_{K}} O_{K′}) of Emb_{O_{K′}} defined in Lemma 1.3 denoted by (A → A) ˆ⊗_{O_{K}} O_{K′}. By sending (A → A) to (A → A) ˆ⊗_{O_{K}} O_{K′}, we obtain a functor ˆ⊗_{O_{K}} O_{K′} : Emb_{O_{K}} → Emb_{O_{K′}}. If K′ is a finite extension of K, we have A ˆ⊗_{O_{K}} O_{K′} = A ⊗_{O_{K}} O_{K′}.

1.2 Tubular neighborhoods for embbedings

Let (A → A) be an object of Emb_{O_{K}} and I be the kernel of the surjection A → A. Mimicing [3] Chapter 7, for a pair of positive integers m, n > 0, we define an O_{K}-algebra A^{m/n} topologically of finite type as follows. Let A[I^{n}/π^{m}] be the subring of A ⊗_{O_{K}} K generated by A and the elements f/π^{m} for f ∈ I^{n} and let A^{m/n} be its π-adic completion. For two pairs of positive integers m, n and m′, n′, if m′ is a multiple of m and if m′/n′ ≤ m/n, we have an inclusion A[I^{n′}/π^{m′}] ⊂ A[I^{n}/π^{m}]. It induces a continuous homomorphism A^{m′/n′} → A^{m/n}. Then we have the following.

Lemma 1.4 Let (A → A) be an object of Emb_{O_{K}} and m, n > 0 be a pair of positive integers. Then,

1. The O_{K}-algebra A^{m/n} is topologically of finite type over O_{K}. The tensor product A_{K}^{m/n} = A^{m/n} ⊗_{O_{K}} K is an affinoid algebra over K.

2. The map A → A^{m/n} is continuous with respect to the 𝔪_{A}-adic topology on A and the π-adic topology on A^{m/n}.

3. Let m′, n′ be another pair of positive integers and assume that m′ is a multiple of m and j′ = m′/n′ ≤ j = m/n. Then, by the map X^{m/n} = Sp A_{K}^{m/n} → X^{m′/n′} = Sp A_{K}^{m′/n′} induced by the inclusion A[I^{n′}/π^{m′}] ⊂ A[I^{n}/π^{m}], the affinoid variety X^{m/n} is identified with a rational subdomain of X^{m′/n′}.

4. The affinoid variety X^{m/n} = Sp A_{K}^{m/n} depends only on the ratio j = m/n.

The proof is similar to that of [3] Lemma 7.1.2. Proof. 1. Since the O_{K}-algebra A^{m/n} is π-adically complete, it is sufficient to show that the quotient A[I^{n}/π^{m}]/(π) is of finite type over F. Since it is finitely generated over A/(π, I^{n}) and A/(π, I) = A/(π) is finite over F, the assertion follows.

2. Since A/π = A/(π, I) is of finite length, a power of 𝔪_{A} is in (π^{m}, I^{n}). Since the image of (π^{m}, I^{n}) in A^{m/n} is in π^{m}A^{m/n}, the assertion follows.

3. Take a system of generators f_{1}, …, f_{N} of I^{n} and define a surjection A[I^{n′}/π^{m′}][T_{1}, …, T_{N}]/(π^{m}T_{i}−f_{i}) → A[I^{n}/π^{m}] by sending T_{i} to f_{i}/π^{m}. Since it induces an isomorphism after tensoring with K, its kernel is annihilated by a power of π. Hence it induces an isomorphism A_{K}^{m′/n′}⟨T_{1}, …, T_{N}⟩/(π^{m}T_{i} − f_{i}, i = 1, …, N) → A_{K}^{m/n}.

4. Further assume m/n = m′/n′ and put k = m′/m. Let f_{1}, …, f_{N} ∈ I^{n} be a system of generators of I^{n} as above. Then A[I^{n}/π^{m}] is generated by (f_{1}/π^{m})^{k_{1}} ⋯ (f_{N}/π^{m})^{k_{N}} , 0 ≤ k_{i} < k as an A[I^{n′}/π^{m′}]-module. Hence the cokernel of the inclusion A^{m′/n′} → A^{m/n} is annihilated by a power of π and the assertion follows. □ If A = O_{K}[[T_{1}, …, T_{N}]] and I = (T_{1}, …, T_{N}), the ring A^{m/1} is isomorphic to the π-adic completion of O_{K}[T_{1}/π^{m}, …, T_{N}/π^{m}] and is denoted by O_{K}⟨T_{1}/π^{m}, …, T_{N}/π^{m}⟩. By Lemma 1.4.4, the integral closure A^{j} of A^{m/n} in the affinoid algebra A^{m/n} ⊗_{O_{K}} K depends only on j = m/n.

Definition 1.5 Let (A → A) be an object of Emb_{O_{K}} and j > 0 be a rational number. We define A^{j} to be the integral closure of A^{m/n} for j = m/n in the affinoid algebra A^{m/n}⊗_{O_{K}} K and define the j-th tubular neighborhood X^{j}(A → A) to be the affinoid variety Sp A_{K}^{j}.

In the case A = O_{K}[[T_{1}, …, T_{n}]] and the map A → A = O_{K} is defined by sending T_{i} to 0, the affinoid variety X^{j}(A → A) is the n-dimensional polydisk D(0, π^{j})^{n} of center 0 and of radius π^{j}. For each positive rational number j > 0, the construction attaching the j-th tubular neighboorhood X^{j}(A → A) to an object (A → A) of Emb_{O_{K}} defines a functor

X^{j} : Emb_{O_{K}} → (Affinoid/K)

to the category of affinoid varieties over K. For j′ ≤ j, we have a natural morphism X^{j} → X^{j′} of functors. A finite flat morphism of embeddings induces a finite flat morphism of affinoid varieties.

Lemma 1.6 Let j > 0 be a positive rational number and (A → A) → (B → B) be a finite and flat morphism in Emb_{O_{K}}. Then, the induced map f^{j} : X^{j}(B → B) → X^{j}(A → A) is a finite flat map of affinoid varieties.

Proof. Let I and J = IB be the kernels of the surjections A → A and B → B. Since the map A → B is flat, it induces isomorphisms B ⊗_{A} A[I^{n}/π^{m}] → B[J^{n}/π^{m}] and B ⊗_{A} A_{K}^{j} → B_{K}^{j}. The assertion follows from this immediately. □ For an extension K′ of complete discrete valuation field K, the construction of j-th tubular neighborhoods commutes with the base change. More precisely, we have the following. Let K′ be a complete discrete valuation field and K → K′ be a morphism of fields inducing a local homomorphism O_{K} → O_{K′}. Then by sending an affinoid variety Sp A_{K} over K to the affinoid variety Sp A_{K} ˆ⊗_{K}K′ over K′, we obtain a functor ˆ⊗_{K}K′ : (Affinoid/K) → (Affinoid/K′) (see [2] 9.3.6). Let e be the ramification index e_{K′/K} and j > 0 be a positive rational number. Then the canonical map A → A ˆ⊗_{O_{K}} O_{K′} induces an isomorphism X^{j}(A → A) ˆ⊗_{K}K′ → X^{ej}((A → A) ˆ⊗_{O_{K}} O_{K′}) of affinoid varieties over K′. In other words, we have a commutative diagram of functors

X^{j} : Emb_{O_{K}} ⟶ (Affinoid/K)
ˆ⊗_{O_{K}} O_{K′} ↓  ↓ ˆ⊗_{K}^{K′}
X^{ej}: Emb_{O_{K}} ⟶ (Affinoid/K′).

Lemma 1.7 For a rational number j > 0, the affinoid algebra A_{K}^{j} is smooth over K.

Proof. By the commutative diagram above, it is sufficient to show that there is a finite separable extension K′ of K such that the base change X^{j}(A → A) ⊗_{K} K′ = X^{j}(A ⊗_{O_{K}} O_{K′} → A ⊗_{O_{K}} O_{K′}) is smooth over K′. Replacing K by K′ and separating the factors of A, we may assume A/𝔪_{A} = F. Then we also have A/𝔪_{A} = F and an isomorphism O_{K}[[T_{1}, …, T_{n}]] → A. We define an object (A → O_{K}) of Emb_{O_{K}} by sending T_{i} ∈ A to 0. Let I and I′ be the kernel of A → A and A → O_{K} respectively and put j = m/n. Since A/(π^{m}, I^{n}) is of finite length, there is an integer n′ > 0 such that I^{′n′} ⊂ (π^{m}, I^{n}). Then we have an inclusion A[I^{′n′}/π^{m}] → A[I^{n}/π^{m}] and hence a map X^{m/n}(A → A) → X^{m/n′}(A → O_{K}). By the similar argument as in the proof of Lemma 1.4.3, the affinoid variety X^{m/n}(A → A) is identified with a rational subdomain of X^{m/n′}(A → O_{K}). Since the affinoid variety X^{m/n′}(A → O_{K}) is a polydisk, the assertion follows. □ By Lemma 1.7, the j-th tubular neighborhoods in fact define a functor

X^{j} : Emb_{O_{K}} ⟶ (smoooth Affinoid/K)

to the category of smooth affinoid varieties over K. Also by Lemma 1.7, Ω̂_{A^{j}/O_{K}} ⊗ K is a locally free A_{K}^{j}-module. An idea behind the definition of the j-th tubular neighborhood is the following description of the valued points. Let (A → A) be an object of Emb_{O_{K}} and j > 0 be a rational number. Let A_{K}^{j} be the affinoid algebra defining the affinoid variety X^{j}(A → A) and let X^{j}(A → A)(K̄) be the set of K̄-valued points. Since a continuous homomorphism A_{K}^{j} → K̄ is determined by the induced map A → OK̄, we have a natural injection X^{j}(A → A)(K̄) → Hom_{cont.O_{K}-alg}(A, OK̄). The surjection A → A induces an injection

(1.8.0)  Hom_{O_{K}-alg}(A, OK̄) ⟶ X^{j}(A → A)(K̄).

For a rational number j > 0, let 𝔪^{j} denote the ideal 𝔪^{j} = {x ∈ K̄; ordx ≥ j}. We naturally identify the set Hom_{O_{K}-alg}(A, OK̄/𝔪^{j}) of O_{K}-algebra homomorphisms with a subset of the set Hom_{cont.O_{K}-alg}(A, OK̄/𝔪^{j}) of continuous O_{K}- algebra homomorphisms.

Lemma 1.8 Let (A → A) be an object of Emb_{O_{K}} and j > 0 be a rational number. Then by the injection X^{j}(A → A)(K̄) → Hom_{cont.O_{K}-alg}(A, OK̄) above, the set X^{j}(A → A)(K̄) is identified with the inverse image of the subset Hom_{O_{K}-alg}(A, OK̄/𝔪^{j}) by the projection Hom_{cont.O_{K}-alg}(A, OK̄) → Hom_{cont.O_{K}-alg}(A, OK̄/𝔪^{j}). In other words, we have a cartesian diagram

(1.8.1)

X^{j}(A → A)(K̄)  ⟶ Hom_{cont.O_{K}-alg}(A, OK̄)
↓  ↓
Hom_{O_{K}-alg}(A, OK̄/𝔪^{j}) ⟶ Hom_{cont.O_{K}-alg}(A, OK̄/𝔪^{j}).

The arrows are compatible with the natural G_{K}-action.

Proof. Let j = m/n. By the definition of A^{m/n}, a continuous morphism A → OK̄ is extended to A_{K}^{j} → K̄, if and only if the image of I^{n} is contained in the ideal (π^{m}). Hence the assertion follows. □ For an affinoid variety X over K, let π_{0}(XK̄) denote the set \underleftarrow{lim}_{K′/K} π_{0}(X_{K′}) of geometric connected components, where K′ runs over finite extensions of K in K̄. The set π_{0}(XK̄) is finite and carries a natural continuous right action of the absolute Galois group G_{K}. To get a left action, we let σ ∈ G_{K} act on XK̄ by σ^{−1}. The natural map X^{j}(A → A)(K̄) → π_{0}(XK̄) is compatible with this left G_{K}-action. Let G_{K}-(Finite Sets) denote the category of finite sets with a continuous left action of G_{K} and let (Finite Flat/O_{K}) be the category of finite flat O_{K}-algebras. Then, for a rational number j > 0, we obtain a sequence of functors

(Finite Flat/O_{K}) ⟵ Emb_{O_{K}} ⟶^{X^{j}}
(smooth Affinoid/K) ⟶^{X↦π_{0}(XK̄)} G_{K}-(Finite Sets).

We show that the composition Emb_{O_{K}} → G_{K}-(Finite Sets) induces a functor (Finite Flat/O_{K}) → G_{K}-(Finite Sets).

Lemma 1.9 Let j > 0 be a positive rational number.

1. Let (A → A) be an embedding. Then, the map X^{j}(A → A)(K̄) → Hom_{O_{K}-alg}(A, OK̄/𝔪^{j}) (1.8.1) induces a surjection

(1.9.1)  Hom_{O_{K}-alg}(A, OK̄/𝔪^{j}) ⟶ π_{0}(X^{j}(A → A)K̄).

2. Let (A → A) and (A′ → A) be embeddings. Then, there exists a unique bijection π_{0}(X^{j}(A → A)K̄) → π_{0}(X^{j}(A′ → A)K̄) such that the diagram

(1.9.2)

Hom_{O_{K}-alg}(A, OK̄/𝔪^{j}) ⟶ π_{0}(X^{j}(A → A)K̄)
∥  ↓
Hom_{O_{K}-alg}(A, OK̄/𝔪^{j}) ⟶ π_{0}(X^{j}(A′ → A)K̄)

is commutative.

3. Let (f, f) : (A → A) → (B → B) be a morphism of Emb_{O_{K}}. Then, the induced map π_{0}(X^{j}(B → B)K̄) → π_{0}(X^{j}(A → A)K̄) depends only on f.

4. Let (f, f) : (A → O_{K}) → (B → B) be a finite flat morphism of Emb_{O_{K}}. Then the map (1.8.0) induces a surjection

(1.9.3)  Hom_{O_{K}-alg}(B, OK̄) ⟶ π_{0}(X^{j}(B → B)K̄).
Proof.  1.  The fibers of the map Hom_{cont.O_{K}-alg}(A, OK̄) →

Hom_{cont.O_{K}-alg}(A, OK̄/𝔪^{j}) are K̄-valued points of polydisks. Hence the surjection X^{j}(A → A)(K̄) → Hom_{cont.O_{K}-alg}(A, OK̄/𝔪^{j}) induces a surjection Hom_{cont.O_{K}-alg}(A, OK̄/𝔪^{j}) → π_{0}(X^{j}(A → A)K̄) by Lemma 1.8.

2. By 1 and Lemma 1.2.2, there exists a unique surjection π_{0}(X^{j}(A → A)K̄) → π_{0}(X^{j}(A′ → A)K̄) such that the diagram (1.9.2) is commutative. Switching A → A and A′ → A, we obtain the assertion.

3. In the commutative diagram

Hom_{cont.O_{K}-alg}(B, OK̄/𝔪^{j}) ⟶ π_{0}(X^{j}(B → B)K̄)
f^{∗}↓  ↓
Hom_{cont.O_{K}-alg}(A, OK̄/𝔪^{j}) ⟶ π_{0}(X^{j}(A → A)K̄),

the horizontal arrows are surjective by 1. Hence the assertion follows.

4. The map f^{j} : X^{j}(B → B) → X^{j}(A → O_{K}) is finite and flat by Lemma 1.6. Let y : X^{j}(A → O_{K})(K̄) be the point corresponding to the map A → O_{K}. Then the fiber (f^{j})^{−1}(y) is identified with the set Hom_{O_{K}-alg}(B, OK̄). Since X^{j}(A → O_{K})K̄ is isomorphic to a disk and is connected, the assertion follows. □ For a rational number j > 0 and a finite flat O_{K}-algebra A, we put

Ψ^{j}(A) = \underleftarrow{lim}_{(A→A)∈Emb_{O_{K}} (A)} π_{0}(X^{j}(A → A)K̄).

By Lemmas 1.2.1 and 1.9.2, the projective system in the right is constant. Further by Lemma 1.9.3, we obtain a functor

Ψ^{j} : (Finite Flat/O_{K}) ⟶ G_{K}-(Finite Sets)

sending a finite flat O_{K}-algebra A to Ψ^{j}(A). Let Ψ : (Finite Flat/O_{K}) → G_{K}-(Finite Sets) be the functor defined by Ψ(A) = Hom_{O_{K}-alg}(A, K̄). Then, the map (1.9.1) induces a map Ψ → Ψ^{j} of functors.

1.3 Stable normalized integral models and their closed fibers

We briefly recall the stable normalized integral model of an affinoid variety and its closed fiber (cf. [1] Section 4). It is based on the finiteness theorem of Grauert-Remmert.

Theorem 1.10 (Finiteness theorem of Grauert-Remmert, [1] Theorem 4.2) Let A be an O_{K}-algebra topologically of finite type. Assume that the generic fiber A_{K} = A ⊗_{O_{K}} K is geometrically reduced. Then,

1. There exists a finite separable extension K′ of K such that the geometric closed fiber A_{O_{K′}} ⊗_{O_{K′}} F̄ of the integral closure A_{O_{K′}} of A in A ⊗_{O_{K}} K′ is reduced.

2. Assume further that A is flat over O_{K} and that the geometric closed fiber A⊗_{O_{K}} F̄ is reduced. Let K′ be an extension of complete discrete valuation field over K and π′ be a prime element of K′. Then the π′-adic completion of the base change A ⊗_{O_{K}} O_{K′} is integrally closed in A ⊗_{O_{K}} K′.

Let A be an O_{K}-algebra topologically of finite type such that A_{K} is smooth. If a finite separable extension K′ satisfies the condition in Theorem 1.10.1, we say that the integral closure A_{O_{K′}} of A in A_{K′} is a stable normalized integral model of the affinoid variety X_{K} = Sp A_{K} and that the stable normalized integral model is defined over K′. The geometric closed fiber X̄ = Spec A_{O_{K′}} ⊗_{O_{K′}} F̄ of a stable normalized integral model is independent of the choice of an extension K′ over which a stable normalized integral model is defined, by Theorem 1.10.2. Hence, the scheme X̄ carries a natural continuous action of the absolute Galois group G_{K} = Gal(K̄/K) compatible with its action on F̄. The construction above defines a functor as follows. Let G_{K}-(Aff/F̄) denote the category of affine schemes of finite type over F̄ with a semi-linear continuous action of the absolute Galois group G_{K}. More precisely, an object is an affine scheme Y over F̄ with an action of G_{K} compatible with the action of G_{K} on F̄ satisfying the following property: There exist a finite Galois extension K′ of K in K̄, an affine scheme Y_{K′} of finite type over the residue field F′ of K′, an action of Gal(K′/K) on Y_{K′} compatible with the action of Gal(K′/K) on F′ and a G_{K}-equivariant isomorphism Y_{K′} ⊗_{F′} F̄ → Y. Then Theorem 1.10 implies that the geometric closed fiber of a stable normalized integral model defines a functor

(smooth Affinoid/K) → G_{K}-(Aff/F̄) : X ↦ X̄.

Corollary 1.11 Let A be an O_{K}-algebra topologically of finite type such that the generic fiber A_{K} is geometrically reduced as in Theorem 1.10. Let X_{K} = Sp A_{K} be the affinoid variety and XF̄ be the geometric closed fiber of the stable normalized integral model. Then the natural map π_{0}(XF̄) → π_{0}(XK̄) is a bijection.

Proof. Replacing A by its image in A_{K}, we may assume A is flat over O_{K}. Let K′ be a finite separable extension of K in K̄ such that the stable normalized integral model A_{O_{K′}} is defined over K′. Then since A_{O_{K′}} is π-adically complete, the canonical maps π_{0}(SpecA_{O_{K′}}) → π_{0}(Spec(A_{O_{K′}} ⊗_{O_{K′}} F′)) is bijective. Since the idempotents of A_{K′} are in A_{O_{K′}}, the canonical maps π_{0}(SpecA_{O_{K′}}) → π_{0}(SpecA_{K′}) is also bijective. By taking the limit, we obtain the assertion. □

By Corollary 1.11, the functor (smooth Affinoid/K) → G_{K}-(Finite Sets) sending a smooth affinoid variety X to π_{0}(XK̄) may be also regarded as the composition of the functors

(smooth Affinoid/K) ⟶^{X↦X̄} G_{K}-(Aff/F̄) ⟶^{π_{0}} G_{K}-(Finite Sets).

Lemma 1.12 Let j > 0 be a positive rational number and (f, f) : (A → O_{K}) → (B → B) be a finite flat morphism of Emb_{O_{K}}. Let f^{j} : X^{j}(B → B) → X^{j}(A → O_{K}) be the induced map and f̄^{j} : X̄^{j}(B → B) → X̄^{j}(A → O_{K}) be its reduction. Let y ∈ X^{j}(A → O_{K})(K̄) be the point corresponding to A → A = O_{K} → K̄ and ȳ ∈ X̄^{j}(A → O_{K}) be its specialization. Then the surjections (f^{j})^{−1}(y) = Hom_{O_{K}-alg}(B, OK̄) → π_{0}(X^{j′}(B → B)K̄) (1.9.3) and the specialization map (f^{j})^{−1}(y) → (f̄^{j})^{−1}(ȳ) induces a bijection

(1.12.1)  \underrightarrow{lim}_{j′>j} π_{0}(X^{j′}(B → B)K̄) ⟶ (f̄^{j})^{−1}(ȳ).

Proof. The map (f^{j})^{−1}(y) → π_{0}(X^{j′}(B → B)K̄) is a surjection of finite sets by Lemma 1.9.4. Hence there exists a rational number j′ > j such that the surjection π_{0}(X^{j′}(B → B)K̄) → \underrightarrow{lim}_{j′′>j} π_{0}(X^{j′′}(B → B)K̄) is a bijection.

Let K′ be a finite separable extension such that the surjection π_{0}(X^{j′}(B → B)K̄) → π_{0}(X^{j′}(B → B)_{K′}) is a bijection and that the stable normalized integral models B_{O_{K′}}^{j} of X^{j}(B → B) is defined over K′. Enlarging K′ further if necessary, we assume that e′j is an integer where e′ = e_{K′/K} is the ramification index. Then the integral model A_{O_{K′}}^{j} of X^{j}(A → O_{K}) is also defined over K′. If O_{K}[[T_{1}, …, T_{n}]] → A is an isomorphism such that the kernel of A → O_{K} is generated by T_{1}, …, T_{n} and π′ is a prime element of K′, it induces an isomorphism O_{K′}⟨T_{1}/π^{′e′j}, …, T_{n}/π^{′e′j}⟩ → A_{O_{K′}}^{j}. Let A_{O_{K′}}^{j} → O_{K′} be the map induced by A → O_{K} and A_{O_{K′}}^{j} be the formal completion respect to the surjection A_{O_{K′}}^{j} → O_{K′}. If O_{K}[[T_{1}, …, T_{n}]] → A is an isomorphism as above, it induces an isomorphism O_{K′}[[T_{1}/π^{′e′j}, …, T_{n}/π^{′e′j}]] → A_{O_{K′}}^{j}. We put B_{O_{K′}}^{j} = B_{O_{K′}}^{j} ⊗_{A_{O_{K′}}^{j}} A_{O_{K′}}^{j}. The ring B_{O_{K′}}^{j} is finite over A_{O_{K′}}^{j} since B_{O_{K′}}^{j}

is finite over A_{O_{K′}}^{j}. Enlarging K′ further if necessary, we assume that the canonical map (f̄^{j})^{−1}(ȳ) → π_{0}(Spec B_{O_{K′}}^{j}) is a bijection. We show that the surjection π_{0}(X^{j′}(B → B)_{K′}) → π_{0}(Spec B_{O_{K′}}^{j}) is a bijec-

tion. For a rational number j′ > 0, let A_{K}^{j′} and B_{K}^{j′} denote the affinoid Kalgebras defining X^{j′}(A → O_{K}) and X^{j′}(B → B). We have B_{K}^{j′} = B ⊗_{A} A_{K}^{j′}. Since π_{0}(X^{j′}(B → B)K̄) → \underrightarrow{lim}_{j′′>j} π_{0}(X^{j′′}(B → B)K̄) is a bijection, the in-

jection Bj′′ ; K̄ → Bj′ ; K̄ induce a bijection of idempotents for j < j′′ < j′. Since π_{0}(X^{j′}(B → B)K̄) → π_{0}(X^{j′}(B → B)_{K′}) is a bijection, the idempotents of Bj′ ; K̄ are in B_{K′}^{j′}. Hence, for j < j′′ < j′, the map B_{K′}^{j′′} → B_{K′}^{j′} induces a bijection of idempotents for j < j′′ < j′. Therefore, the map B_{O_{K′}}^{j} → B_{K′}^{j′} induces a bijection of idempotents by [3] 7.3.6 Proposition. Thus, the map π_{0}(X^{j′}(B → B)_{K′}) → π_{0}(Spec B_{O_{K′}}^{j}) is a bijection as required. □

For later use in the proof of the commutativity in the logarithmic case, we give a more formal description of the functor (smooth Affinoid/K) → G_{K}-(Aff/F̄) : X ↦ X̄. For this purpose, we introduce a category \underrightarrow{lim}_{K′/K}(Aff/F′) and

an equivalence \underrightarrow{lim}_{K′/K}(Aff/F′) → G_{K}-(Aff/F̄) of categories. More generally, we define a category \underrightarrow{lim}_{K′/K} V(K′) in the following setting. Suppose we are given a category V(K′) for each finite separable extension K′ of K and a functor f^{∗} : V(K′′) → V(K′) for each morphism f : K′ → K′′ of finite separable extension of K satisfying (f ∘ g)^{∗} = g^{∗} ∘ f^{∗} and id_{K′}^{∗} = id_{V(K′)}. In the application here, we will take V(K′) to be (Aff/F′) for the residue field F′. In Section 4, we will take V(K′) to be Emb_{O_{K′}}. We say that a full subcategory C of the category (Ext/K) of finite separable extensions in K̄ is cofinal if C is non empty and a finite extension K′′ of an extension K′ in C is also in C. We define \underrightarrow{lim}_{K′/K} V(K′) to be the category whose objects and morphisms are as follows. An object of \underrightarrow{lim}_{K′/K} V(K′) is a system ((X_{K′})_{K′∈ob(C)}, (φ_{f})_{f:K′→K′′∈mor(C)}) where C is some cofinal full subcategory of (Ext/K), X_{K′} is an object of V(K′) for each object K′ in C and φ_{f} : X_{K′′} → f^{∗}(X_{K′}) is an isomorphism in V(K′′) for each morphism f : K′ → K′′ in C satisfying φ_{f∘f′} = f^{∗}(φ_{f′}) ∘ φ_{f} for morphisms f′ : K′ → K′′ and f : K′′ → K′′′ in C. For objects X = ((X_{K′})_{K′∈ob(C)}, (φ_{f})_{f:K′→K′′∈mor(C)}) and Y = ((Y_{K′})_{K′∈ob(C′)}, (ψ_{f})_{f:K′→K′′∈mor(C′)}) of the category \underrightarrow{lim}_{K′/K} V(K′), a morphism g : X → Y is a system (g_{K′})_{K′∈ob(C′′)}, where C′′ is some cofinal full subcategory of C ∩ C′ and g_{K′} : X_{K′} → Y_{K′} is a morphism in V(K′) such that the diagram

X_{K′′}  ⟶^{g_{K′}} Y_{K′′}
φ_{f} ↓  ↓^{ψ_{f}}
f^{∗}X_{K′} ⟶^{g_{K′′}} f^{∗}Y_{K′}

is commutative for each morphism f : K′ → K′′ in C′′. Applying the general construction above, we define a category \underrightarrow{lim}_{K′/K}(Aff/F′). An equivalence \underrightarrow{lim}_{K′/K}(Aff/F′) → G_{K}-(Aff/F̄) of categories is defined as follows. Let X = ((X_{K′})_{K′∈ob(C)}, (f^{∗})_{f:K′→K′′∈mor(C)}) be an object of \underrightarrow{lim}_{K′/K}(Aff/F′). Let CK̄ be the category

of finite extensions of K in K̄ which are in C. Then, XK̄ = \underleftarrow{lim}_{K′∈CK̄} X_{K′}

is an affine scheme over F̄ and has a natural continuous semi-linear action of the Galois group G_{K}. By sending X to XK̄, we obtain a functor \underrightarrow{lim}_{K′/K}(Aff/F′) → G_{K}-(Aff/F̄). We can easily verify that this functor gives an equivalence of categories. The reduced geometic closed fiber defines a functor (smooth Affinoid/K) → \underrightarrow{lim}_{K′/K}(Aff/F′) as follows. Let X be a smooth affinoid variety over K. Let C_{X} be the full subcategory of (Ext/K) consisting of finite extensions K′ such that a stable normalized integral model A_{O_{K′}} is defined over K′. By Theorem 1.10.1, the subcategory C_{X} is cofinal. Further, by Theorem 1.10.2, the system X̄ = (Spec A_{O_{K′}} ⊗_{O_{K′}} F′)_{K′∈obC_{X}} defines an object of \underrightarrow{lim}_{K′/K}(Aff/F′). Thus, by

sending X to X̄, we obtain a functor (smooth Affinoid/K) → \underrightarrow{lim}_{K′/K}(Aff/F̄′). By taking the composition with the equivalence of categories, we recover the functor (smooth Affinoid/K) → G_{K}-(Aff/F̄).

1.4 Twisted normal cones

Let (A → A) be an object in Emb_{O_{K}} and j > 0 be a positive rational number. We define X̄^{j}(A → A) to be the geometric closed fiber of the stable normalized integral model of X^{j}(A → A). We will also define a twisted normal cone C̄^{j}(A → A) as a scheme over A_{F̄,red} = (A ⊗_{O_{K}} F̄)_{red} and a canonical map X̄^{j}(A → A) → C̄^{j}(A → A). Let I be the kernel of the surjection A → A. Then the normal cone C_{A/A} of Spec A in Spec A is defined to be the spectrum of the graded A-algebra ⨁_{n=0}^{∞} I^{n}/I^{n+1}. We say that a surjection R → R′ of Noetherian rings is regular if the immersion SpecR′ → SpecR is a regular immersion. If the surjection A → A is regular, the conormal sheaf N_{A/A} = I/I^{2} is locally free and the normal cone C_{A/A} is equal to the normal bundle, namely the covariant vector bundle over SpecA defined by the locally free A-module Hom_{A}(N_{A/A}, A). For a rational number j, let 𝔪^{j} be the fractional ideal 𝔪^{j} = {x ∈ OK̄; ord(x) ≥ j} and put N^{j} = 𝔪^{j} ⊗_{OK̄} F̄.

Definition 1.13 Let (A → A) be an object of Emb_{O_{K}} and j > 0 be a rational number. We define the j-th twisted normal cone C̄^{j}(A → A) to be the reduced part

(Spec ⨁_{n=0}^{∞}(I^{n}/I^{n+1} ⊗_{O_{K}} N^{−jn}))_{red}

of the spectrum of the A ⊗_{O_{K}} F̄-algebra ⨁_{n=0}^{∞}(I^{n}/I^{n+1} ⊗_{O_{K}} N^{−jn}).

It is a reduced affine scheme over Spec A_{F̄,red} non-canonically isomorphic to the reduced part of the base change C_{A/A} ⊗_{O_{K}} F̄. It has a natural continuous semi-linear action of G_{K} via N^{−jn}. The restriction to the wild inertia subgroup P is trivial and the G_{K}-action induces an action of the tame quotient G_{K}^{tame} = G_{K}/P. If the surjection A → A is regular, the scheme C̄^{j}(A → A) is the covariant vector bundle over Spec A_{F̄,red} defined by the A_{F̄,red}-module (Hom_{A}(I/I^{2}, A) ⊗_{O_{K}} N^{j}) ⊗_{A⊗_{O_{K}} F̄} A_{F̄,red}. A canonical map X̄^{j}(A → A) → C̄^{j}(A → A) is defined as follows. Let K′ be a finite separable extension of K such that the stable normalized integral model A_{O_{K′}}^{j} is defined over K′ and that the product je with the ramification index e = e_{K′/K} is an integer. Then, we have a natural ring homomorphism

⨁_{n≥0} I^{n} ⊗_{O_{K}} 𝔪_{K′}^{−jen} ⟶ A_{O_{K′}}^{j} : f ⊗ a ↦ af.

Since IA_{O_{K′}}^{j} ⊂ 𝔪_{K′}^{je}A_{O_{K′}}^{j}, it induces a map ⨁_{n} I^{n}/I^{n+1} ⊗_{O_{K}} 𝔪_{K′}^{−jen} → A_{O_{K′}}^{j} /𝔪_{K′}A_{O_{K′}}^{j}. Let F′ be the residue field of K′. Then by extending the scalar, we obtain a map ⨁_{n=0}^{∞}(I^{n}/I^{n+1} ⊗_{O_{K}} N^{−jn}) → A_{O_{K′}}^{j} /𝔪_{K′}A_{O_{K′}}^{j} ⊗_{F′} F̄. By the assumption that A_{O_{K′}}^{j} is a stable normalized integral model, we have X̄^{j}(A → A) = Spec (A_{O_{K′}}^{j} /𝔪_{K′}A_{O_{K′}}^{j} ⊗_{F′} F̄). Since X̄^{j}(A → A) is a reduced scheme over F̄, we obtain a map X̄^{j}(A → A) → C̄^{j}(A → A) of schemes over F̄. For a positive rational number j > 0, the constructions above define a functor C̄^{j} : Emb_{O_{K}} → G_{K}-(Aff/F̄) and a morphism of functors X̄^{j} → C̄^{j}.

Lemma 1.14 Let (A → A) be an object of Emb_{O_{K}} and j > 0 be a rational number. Then, we have the following.

1. The canonical map X̄^{j}(A → A) → C̄^{j}(A → A) is finite.

2. Let (A → A) → (B → B) be a morphism in Emb_{O_{K}}. Then, the canonical maps form a commutative diagram

X̄^{j}(B → B) ⟶ C̄^{j}(B → B) ⟶ Spec B_{F̄,red}
↓  ↓  ↓
X̄^{j}(A → A) ⟶ C̄^{j}(A → A) ⟶ Spec A_{F̄,red}.

If the morphism (A → A) → (B → B) is finite flat, then the right square in the commutative diagram is cartesian.

3. Assume A = O_{K}. Then the surjection A → A is regular and the canonical map N_{A/A} → Ω̂_{A/O_{K}} ⊗_{A} A is an isomorphism. The twisted normal cone C̄^{j}(A → A) is equal to the F̄-vector space HomF̄(Ω̂_{A/O_{K}} ⊗_{A} F̄, N^{j}). The canonical map X̄^{j}(A → A) → C̄^{j}(A → A) is an isomorphism.

Proof. 1. Let K′ be a finite extension such that the stable normalized integral model A_{O_{K′}}^{j} is defined. Let A′ denote the π′-adic completion of the image of the map ⨁_{n≥0} I^{n} ⊗_{O_{K}} 𝔪_{K′}^{−jen} → A ⊗_{O_{K}} K′. Then by the definition and by Lemma 1.3, A_{O_{K′}}^{j} is the integral closure of A′ in A′_{K}. Hence A_{O_{K′}}^{j} /m_{K′}A_{O_{K′}}^{j} is finite over ⨁_{n} I^{n}/I^{n+1} ⊗_{O_{K}} 𝔪_{K′}^{−jen}. Thus the assertion follows.

2. Clear from the definitions.

3. If A = O_{K}, there is an isomorphism O_{K}[[T_{1}, …, T_{n}]] → A for some n such that the composition O_{K}[[T_{1}, …, T_{n}]] → A maps T_{i} to 0. Then the assertions are clear. □

1.5 Étale covering of tubular neighborhoods

Let A and B be the integer rings of finite étale K-algebras. For a finite flat morphism (A → A) → (B → B) of embeddings, we study conditions for the induced finite morphism X^{j}(A → A) → X^{j}(B → B) to be étale. Let X = Sp B_{K} and Y = Sp A_{K} be geometrically reduced affinoid varieties and A and B be the maximum integral models. Then a finite map f : X → Y of affinoid varieties is uniquelly extended to a finite map A → B of integral models.

Proposition 1.15 Let A and B = O_{L} be the integer rings of finite separable extensions of K and (A → A) → (B → B) be a finite flat morphism of embeddings. Let j > 1 be a rational number, π_{L} a prime element of L and e = ordπ_{L} be the ramification index.

1. ([1] Proposition 7.3) Assume A = O_{K}. Suppose that, for each j′ > j, there exists a finite separable extension K′ of K such that the base change X^{j′}(B → B)_{K′} is isomorphic to the disjoint union of finitely many copies of X^{j′}(A → A)_{K′} as an affinoid variety over X^{j′}(A → A). Then there is an integer 0 ≤ n < ej such that π_{L}^{n} annihilates Ω_{B/A}.

2. ([1] Proposition 7.5) If there is an integer 0 ≤ n < ej such that π_{L}^{n} annihilates Ω_{B/A}, then the finite flat map X^{j}(B → B) → X^{j}(A → A) is étale.

Corollary 1.16 ([1] Theorem 7.2) Let A = O_{K} and let B be the integer ring of a finite étale K-algebra. Let (A → A) → (B → B) be a finite flat morphism of embeddings. Let j > 1 be a rational number. Suppose that, for each j′ > j, there exists a finite separable extension K′ of K such that the base change X^{j′}(B → B)_{K′} is isomorphic to the disjoint union of finitely many copies of X^{j′}(A → A)_{K′} as in Proposition 1.15.1. Let I be the kernel of the surjection B → B and let N_{B/B} be the B-module I/I^{2}. Then, we have the following.

1. The finite map X^{j}(B → B) → X^{j}(A → A) is étale and is extended to a finite étale map of stable normalized integral models.

2. The finite map X̄^{j}(B → B) → X̄^{j}(A → A) is étale.

3. The twisted normal cone C̄^{j}(B → B) is canonically isomorphic to the covariant vector bundle defined by the B_{F̄,red}-module (Hom_{B}(N_{B/B}, B) ⊗_{O_{K}} N^{j}) ⊗_{BF̄} B_{F̄,red} and the finite map X̄^{j}(B → B) → C̄^{j}(B → B) is étale.

Though these statements except Corollary 1.16.3 are proved in [1] Section 7, we present here slightly modified proofs in order to compare with the proofs of the corresponding statements in the logarithmic setting given in Section 4.3. To prove Proposition 1.15, we use the following.

Lemma 1.17 Let A = O_{L} be the integer ring of a finite separable extension L, A → A be an embedding and let M be an A-module of finite type. Let j > 1 be a rational number and K′ be a finite separable extension of K such that the stable normalized integral model A_{O_{K′}}^{j} of X^{j}(A → A) is defined over K′. Let e and e′ be the ramification indices of L and of K′ over K and π_{L} and π′ be prime elements of L and K′. Assume that e′/e and e′j are integers. Then, the following conditions are equivalent.

(1) There exists an integer 0 ≤ n < ej such that the A-module M = M ⊗_{A} A is annihilated by π_{L}^{n}.

(2) The A_{O_{K′}}^{j} -module M^{j} = M ⊗_{A} A_{O_{K′}}^{j} is annihilated by π^{′e′j−1}.

Proof of Lemma 1.17. The image of an element in the kernel I of the surjection A → A in A_{O_{K′}}^{j} is divisible by π^{′e′j}. Hence we have a commutative diagram

A ⟶  A_{O_{K′}}^{j}
↓  ↓
A ⟶ A_{O_{K′}}^{j} /(π^{′e′j}).

We show that the ideals of A_{O_{K′}}^{j} /(π^{′e′j}) generated by the image of π_{L} ∈ A and by the image of π^{′e′/e} ∈ A_{O_{K′}}^{j} are equal. Take a lifting a ∈ A of π_{L} ∈ A. Then, the image of a^{e} is a unit times π and hence is a unit times π^{′e′} in A_{O_{K′}}^{j} /(π^{′e′j}). Since A_{O_{K′}}^{j} is π′-adically complete, we have a^{e} = uπ^{′e′} + vπ^{′e′j} for some u ∈ A_{O_{K′}}^{j×} and v ∈ A_{O_{K′}}^{j}. Since j > 1 and A_{O_{K′}}^{j} is π′-adically complete, we have (a/π^{′e′/e})^{e} = u + vπ^{′e′(j−1)} is a unit in A_{O_{K′}}^{j}. Since A_{O_{K′}}^{j} is normal, we have a/π^{′e′/e} ∈ A_{O_{K′}}^{j×} and the claim follows. Assume that the A-module M is isomorphic to A^{r} ⊕ ⨁_{i=1}^{s} A/(π_{L}^{n_{i}}) for integers 0 < n_{1} ≤ … ≤ n_{s}. Then, by the commutative diagram above and by the equality (π_{L}) = (π^{′e′/e}) of the ideals of A_{O_{K′}}^{j} /(π^{′e′j}) proved above, the A_{O_{K′}}^{j} /(π^{′e′j})-module M^{j}/π^{′e′j}M^{j} is isomorphic to (A_{O_{K′}}^{j} /(π^{′e′j}))^{r} ⊕ ⨁_{i=1}^{s} A_{O_{K′}}^{j} /(π^{′ min(e′j,e′n_{i}/e)}). The condition (1) is clearly equivalent to that r = 0 and n_{s} < ej. We see that the condition (2) is also equivalent to this condition by taking the localization at a prime ideal A_{O_{K′}}^{j} of height 1 containing π′. □ Proof of Proposition 1.15. 1. Since A = O_{K}, there is an isomorphism O_{K}[[T_{1}, …, T_{n}]] → A such that the composition O_{K}[[T_{1}, …, T_{n}]] → A maps T_{i} to 0. For j > 0, the affinoid variety X^{j}(A → A) is a polydisk. By the proof of Lemma 1.7, there exist a finite separable extension K′ of K of ramification index e′, an embedding (B ⊗_{O_{K}} O_{K′} → B′) in Emb_{O_{K′}} isomorphic to the embbedding (O_{K′}[[S_{1}, …, S_{n}]]^{N} → O_{K′}^{N}) sending S_{i} to 0 for some N > 0, a positive rational number ϵ < j and an open immersion X^{j}(B → B) ⊗_{K} K′ → X^{e′ϵ}(B ⊗_{O_{K}} O_{K′} → B′) as a rational subdomain. The affinoid variety X^{e′ϵ}(B ⊗_{O_{K}} O_{K′} → B′) is the disjoint union of finitely many copies of polydisks. Enlarging K′ if necessary, we may assume that e′j and e′ϵ are integers. We may further assume that there is a rational number j < j′ < j + ϵ such that e′j′ is an integer, that the stable normalized integral models B_{O_{K′}}^{j′} and B_{O_{K′}}^{′e′ϵ} of X^{j′}(B → B) and of X^{e′ϵ}(B ⊗_{O_{K}} O_{K′} → B′) are

Ramification of Local Fields 21

defined over K′ and that X^{j′}(B → B)_{K′} is isomorphic to the disjoint union of copies of X^{j′}(A → A)_{K′}. Since e′j′ is an integer, the stable normalized integral model A_{O_{K′}}^{j′} of X^{j′}(A → A) is also defined over K′. Then we have a commutative diagram

A  ⟶  A_{O_{K′}}^{j′}
↓  ↓
B ⟶ B_{O_{K′}}^{′e′ϵ} ⟶ B_{O_{K′}}^{j′} .

We consider the modules Ω̂_{A/O_{K}} = \underleftarrow{lim}_{n} Ω_{(A/𝔪_{A}^{n})/O_{K})}, Ω̂_{A_{O_{K′}}^{j′} /O_{K′}} = \underleftarrow{lim}_{n} Ω_{(A_{O_{K′}}^{j′} /π^{′n}A_{O_{K′}}^{j′})/O_{K′}} etc as defined in the beginning of Section 1.1. By Lemma 1.4.2, we have a commutative diagram

B_{O_{K′}}^{j′} ⊗_{A} Ω̂_{A/O_{K}}  ⟶  B_{O_{K′}}^{j′} ⊗_{A_{O_{K′}}^{j′}} Ω̂_{A_{O_{K′}}^{j′} /O_{K′}}
↓  ↓
B_{O_{K′}}^{j′} ⊗_{B} Ω̂_{B/O_{K}} ⟶ B_{O_{K′}}^{j′} ⊗_{B_{O_{K′}}^{′e′ϵ}} Ω̂_{B_{O_{K′}}^{′e′ϵ} /O_{K′}} ⟶  Ω̂_{B_{O_{K′}}^{j′} /O_{K′}} .

We show that the five B_{O_{K′}}^{j′} -modules are free of rank n and that the five maps are injective. We also show that by identifying the modules with their images in Ω̂_{B_{O_{K′}}^{j′} /O_{K′}}, we have an inclusion π^{′e′j′}B_{O_{K′}}^{j′} ⊗_{B} Ω̂_{B/O_{K}} ⊂ π^{′e′ϵ}B_{O_{K′}}^{j′} ⊗_{A} Ω̂_{A/O_{K}} of submodules of Ω̂_{B_{O_{K′}}^{j′} /O_{K′}}. By the assumption on the covering X^{j′}(B → B)_{K′} → X^{j′}(A → A)_{K′}, the A_{O_{K′}}^{j′} -algebra B_{O_{K′}}^{j′} is isomorphic to the product of finitely many copies of A_{O_{K′}}^{j′}. Hence the right vertical map B_{O_{K′}}^{j′} ⊗_{A_{O_{K′}}^{j′}} Ω̂_{A_{O_{K′}}^{j′} /O_{K′}} → Ω̂_{B_{O_{K′}}^{j′} /O_{K′}} is an isomorphism. The isomorphism O_{K}[[T_{1}, …, T_{n}]] → A in the beginning of the proof induces an isomorphism O_{K′}⟨T_{1}/π^{′e′j}, …, T_{n}/π^{′e′j}⟩ → A_{O_{K′}}^{j′} and we see that the Amodule Ω̂_{A/O_{K}} and the A_{O_{K′}}^{j′} -module Ω̂_{A_{O_{K′}}^{j′} /O_{K′}} are free of rank n. Hence Ω̂_{B_{O_{K′}}^{j′} /O_{K′}} is also a free B_{O_{K′}}^{j′} -module of rank n. Further by the canonical maps A_{O_{K′}}^{j′} ⊗_{A} Ω̂_{A/O_{K}} → Ω̂_{A_{O_{K′}}^{j′} /O_{K′}}, the module A_{O_{K′}}^{j′} ⊗_{A} Ω̂_{A/O_{K}} is identified with the submodule π^{′e′j′} Ω̂_{A_{O_{K′}}^{j′} /O_{K′}}. Similarly, the B-module Ω̂_{B/O_{K}} and the B_{O_{K′}}^{′e′ϵ} - module Ω̂_{B_{O_{K′}}^{′e′ϵ} /O_{K′}} are free of rank n and B_{O_{K′}}^{′e′ϵ} ⊗_{B} Ω̂_{B/O_{K}} is identified with the submodule π^{′e′ϵ}Ω̂_{B_{O_{K′}}^{′e′ϵ} /O_{K′}}. Since X^{j}(B → B) ⊗_{K} K′ is a rational subdomain of X^{e′ϵ}(B⊗_{O_{K}} O_{K′} → B′), the map B_{O_{K′}}^{j′} ⊗_{B_{O_{K′}}^{′e′ϵ}} Ω̂_{B_{O_{K′}}^{′e′ϵ} /O_{K′}} → Ω̂_{B_{O_{K′}}^{j′} /O_{K′}} is an

Documenta Mathematica · Extra Volume Kato (2003) 5–72

injection. Thus, we obtain an inclusion π^{′e′j′}B_{O_{K′}}^{j′} ⊗_{B} Ω̂_{B/O_{K}} ⊂ π^{′e′ϵ}B_{O_{K′}}^{j′} ⊗_{A} Ω̂_{A/O_{K}} as submodules of Ω̂_{B_{O_{K′}}^{j′} /O_{K′}}.

Thus the B_{O_{K′}}^{j′} -module B_{O_{K′}}^{j′} ⊗_{B} Ω_{B/A} = Coker(B_{O_{K′}}^{j′} ⊗_{A} Ω̂_{A/O_{K}} → B_{O_{K′}}^{j′} ⊗_{B} Ω̂_{B/O_{K}}) is annihilated by π^{′e′(j′−ϵ)}. Since 0 < j − ϵ < j′ − ϵ < j, it suffices to apply Lemma 1.17 (2)⇒(1).

2. Let K′ be a finite separable extension such that e′j is an integer and the stable normalized integral models A_{O_{K′}}^{j} and B_{O_{K′}}^{j} are defined over K′. By the proof of Lemma 1.9.2, we have B_{O_{K′}}^{j} ⊗_{O_{K′}} K′ = B ⊗_{A} A_{O_{K′}}^{j} ⊗_{O_{K′}} K′ and the map A_{O_{K′}}^{j} ⊗_{O_{K′}} K′ → B_{O_{K′}}^{j} ⊗_{O_{K′}} K′ is finite flat. By Lemma 1.17 (1)⇒(2), the B_{O_{K′}}^{j} -module B_{O_{K′}}^{j} ⊗_{B} Ω_{B/A} is annihilated by π^{′n′} for an integer 0 ≤ n′ < e′j. Hence the map A_{O_{K′}}^{j} ⊗_{O_{K′}} K′ → B_{O_{K′}}^{j} ⊗_{O_{K′}} K′ is étale. □ Proof of Corollary 1.16. 1. It follows from Proposition 1.15 that the map X^{j}(B → B) → X^{j}(A → A) is finite étale. By Lemma 1.12, the fiber (f̄^{j})^{−1}(ȳ) has the same cardinality as the degree of the map X^{j}(B → B) → X^{j}(A → A) in the notation there. Hence the finite map X^{j}(B → B)_{O_{K′}} → X^{j}(A → A)_{O_{K′}} of the normalized integral models is étale at a point of X^{j}(A → A)_{O_{K′}} in the closed fiber. Since X^{j}(A → A)_{O_{K′}} is a regular Noetherian scheme, the assertion follows by the purity of branch locus.

2. Clear from 1.

3. Since the surjection B → B is regular, the twisted normal cone C̄^{j}(B → B) is canonically isomorphic to the covariant vector bundle defined by the B_{F̄,red}- module (Hom_{B}(I/I^{2}, B) ⊗_{O_{K}} N^{j}) ⊗_{BF̄} B_{F̄,red}. We consider the commutative diagram

X̄^{j}(B → B) ⟶ C̄^{j}(B → B) ⟶ Spec B_{F̄,red}
↓  ↓  ↓
X̄^{j}(A → O_{K}) ⟶ C̄^{j}(A → O_{K}) ⟶ Spec F̄

in Lemma 1.14.2. Since the map (A → A) → (B → B) is finite and flat, the right square is cartesian. Hence the middle vertical arrow is étale. Since A = O_{K}, the lower left horizontal arrow is an isomorphism by Lemma 1.14.3. By 2, the left vertial arrow is finite étale. Thus the assertion is proved. □

2 Filtration by ramification groups: the non-logarithmic case

2.1 Construction

In this subsection, we rephrase the definition of the filtration by ramification groups given in the previous paper [1] by using the construction in Section 1. The main purpose is to emphasize the parallelism between the non-logarithmic construction recalled here and the logarithmic construction to be recalled in Section 5.1.

Let Φ : (Finite Étale/K) → G_{K}-(Finite Sets) denote the fiber functor sending a finite étale K-algebra L to the finite set Φ(L) = Hom_{K-alg}(L, K̄) with the continuous G_{K}-action. For a rational number j > 0, we define a functor Φ^{j} : (Finite Étale/K) → G_{K}-(Finite Sets) as the composition of the functor (Finite Étale/K) → (Finite Flat/O_{K}) sending a finite étale K-algebra L to the integral closure O_{L} of O_{K} in L and the functor Ψ^{j} : (Finite Flat/O_{K}) → G_{K}-(Finite Sets) defined at the end of Section 1.2. The map (1.9.3) defines a surjection Φ → Φ^{j} of functors. In [1], we define the filtration by ramification groups on G_{K} by using the family of surjections (Φ → Φ^{j})_{j>0,∈ℚ} of functors. The filtration by the ramification groups G_{K}^{j} ⊂ G_{K}, j > 0, ∈ ℚ is characterized by the condition that the canonical map Φ(L) → Φ^{j}(L) induces a bijection Φ(L)/G_{K}^{j} → Φ^{j}(L) for each finite étale algebra L over K. The functor Φ^{j} is defined by the commutativity of the diagram

\xymatrix{(Finite Étale/K) \ar[r]^{Φ^{j}} \ar[d] & G_{K}-(Finite Sets) \\ (Finite Flat/O_{K}) \ar[ur]_{Ψ^{j}} & G_{K}-(Aff/F̄) \ar[u]_{π_{0}} \\ Emb_{O_{K}} \ar[r]_{X^{j}} \ar[u] & (smooth Affinoid/K) \ar[u]_{X↦X̄}}

We briefly recall how the other arrows in the diagram are defined. The forgetful functor Emb_{O_{K}} → (Finite Flat/O_{K}) sends (A → A) to A. The functor X^{j} : Emb_{O_{K}} → (smooth Affinoid/K) is defined by the j-th tubular neighborhood. The functor (smooth Affinoid/K) → G_{K}-(Aff/F̄) sends X to the geometric closed fiber X̄ of the stable normalized integral model. The functor π_{0} : G_{K}-(Aff/F̄) → G_{K}-(Finite Sets) is defined by the set of connected components. They induce a functor Ψ^{j} : (Finite Flat/O_{K}) → G_{K}-(Finite Sets) by Lemma 1.9. The functor Φ^{j} is defined as the composition of Ψ^{j} with the functor (Finite Étale/K) → (Finite Flat/O_{K}) sending a finite étale algebra L over K to the integral closure O_{L} in L of O_{K}. More concretely, we have

Φ^{j}(L) = \underleftarrow{lim}_{(A→O_{L})∈Emb_{O_{K}} (O_{L})}π_{0}(X̄^{j}(A → O_{L}))

for a finite étale K-algebra L. For a rational number j ≥ 0, we define a functor Φ^{j+} : (Finite Étale/K) →→ G_{K}-(Finite Sets) by Φ^{j+}(L) = \underrightarrow{lim}_{j′>j} Φ^{j′}(L) for a finite étale K-algebra L. We

define a closed normal subgroup G_{K}^{j+} to be \overline{∪_{j′>j}G_{K}^{j′}}. Then we have Φ^{j+}(L) = Φ(L)/G_{K}^{j+}. The finite set Φ^{j+}(L) has the following geometric description.

Lemma 2.1 Let B be the integer ring of a finite étale algebra L over K and j > 0 be a rational number. Let (f, f) : (A → O_{K}) → (B → B) be a finite flat morphism of embeddings. Let f^{j} : X^{j}(B → B) → X^{j}(A → O_{K}) and f̄^{j} : X̄^{j}(B → B) → X̄^{j}(A → O_{K}) be the canonical maps. Let 0 ∈ X^{j}(A → O_{K}) be the point corresponding to the map A → O_{K} and ¯0 ∈ X̄^{j}(A → O_{K}) be its specialization. Then the maps (1.8.0), (1.12.1) and the specialization map form a commutative diagram

(2.1.1)  \begin{matrix}Φ(L) & ⟶ & Φ^{j+}(L) & ⟶ & Φ^{j}(L) \\ ↓ & & ↓ & & ↓ \\ (f^{j})^{−1}(0) & ⟶ & (f̄^{j})^{−1}(0) & ⟶ & π_{0}(X̄^{j}(B → B)) \end{matrix}

and the vertical arrows are bijections.

Proof. Since the map (A → O_{K}) → (B → B) is finite flat, the map B → B induces an isomorphism B_{K}^{j} ⊗_{A_{K}^{j}} K → L. Hence we obtain a bijection Φ(L) = Hom_{K-alg}(L, K̄) → (f^{j})^{−1}(0). By Lemma 1.12 and the definition of Φ^{j+}(L), we have a bijection Φ^{j+}(L) → (f̄^{j})^{−1}(0). The bijection Φ^{j}(L) → π_{0}(X̄^{j}(B → B)) is clear from the definition of Φ^{j}(L). The commutativity is clear. □ For a finite étale algebra L over K and a rational number j > 0, we say that the ramification of L is bounded by j if the canonical map Φ(L) → Φ^{j}(L) is a bijection. Let A = O_{K} and let B = O_{L} be the integer ring of a finite étale K-algebra L and (A → A) → (B → B) be a finite flat morphism of embeddings. Then, since the map X^{j}(B → B) → X^{j}(A → A) is finite flat of degree [L : K], the ramification of L is bounded by j if and only if there exists a finite separable extension K′ of K such that the affinoid variety X^{j}(B → B)_{K′} is isomorphic to the disjoint union of finitely many copies of X^{j}(A → A)_{K′} over X^{j}(A → A)_{K′}. We say that the ramification of L is bounded by j+ if the ramification of L is bounded by every rational number j′ > j. The ramification of L is bounded by j+ if and only if the canonical map Φ(L) → Φ^{j+}(L) is a bijection.

Lemma 2.2 Let K → K′ be a map of complete discrete valuation fields inducing a local homomorphism O_{K} → O_{K′} of integer rings. Assume that a prime element of K goes to a prime element of K′ and that the residue field F′ of K′ is a separable extension of the residue field F of K. Then, for a rational number j > 0, the map G_{K′} → G_{K} induces a surjection G_{K′}^{j} → G_{K}^{j}.

Proof. Let A be the integer ring of a finite étale K-algebra L and (A → A) be an object of Emb_{O_{K}}. By the assumption, the tensor product A ⊗_{O_{K}} O_{K′} is the integer ring of L ⊗_{K} K′. By the isomorphism X^{j}(A → A) ˆ⊗_{K}K′ → X^{j}(A ˆ⊗_{O_{K}} O_{K′} → A ⊗_{O_{K}} O_{K′}) in Section 1.2 and Theorem 1.10, the natural map Φ^{j}(L ⊗_{K} K′) → Φ^{j}(L) is a bijection. Hence the assertion follows. □ Example. Let K = 𝔽_{p}(x, y)((π)) and put L = K[t]/(t^{p} − t − \frac{x}{π^{p^{2}}} ), M = L[t_{1}, t_{2}]/(t_{1}^{p} − t_{1} − \frac{x}{π^{p^{3}}} , t_{2}^{p} − t_{2} − \frac{y}{π^{p^{3}}} ) and G = Gal(M/K) ≃ 𝔽_{p}^{3}. Then we have G^{j} = G for j ≤ p^{2}, G^{j} = H = Gal(M/L) ≃ 𝔽_{p}^{2} for p^{2} < j ≤ p^{3} and G^{j} = 1 for p^{3} < j.

We put z = π^{p}t. Then we have O_{L} = O_{K}[z]/(z^{p} − π^{p(p−1)}z − x) and L = 𝔽_{p}(z, y)((π)). By putting s = t_{1} − \frac{z}{π^{p^{2}}} , we also have M = L[s, t_{2}]/(s^{p} − s −

\frac{z(−1+π^{p(p−1)^{2}} )}{π^{p(p^{2}−p+1)}}, t_{2}^{p} − t_{2} − \frac{x}{π^{p^{3}}} ). We put M_{1} = L(s) ⊂ M. Then we have H^{j} = H for j ≤ p(p^{2} − p + 1), H^{j} = Gal(M/M_{1}) ≃ 𝔽_{p} for p(p^{2} − p + 1) < j ≤ p^{3} and H^{j} = 1 for p^{3} < j. This example shows that the filtration on the subgroup H induced from the filtration by ramification groups on G is not the filtration by ramification groups on H even after renumbering. It also shows that the “lower numbering” filtration is not equal to the upper numbering filtration defined here even after renumbering.

2.2 Functoriality of the closed fibers of tubular neighborhoods: An equal characteristic case

For a positive rational number j > 0, let (Finite Étale/K)^{≤j+} denote the full subcategory of (Finite Étale/K) consisting of étale K-algebras whose ramification is bounded by j+. In this subsection and the following one, we assume the following condition (F) is satisfied.

(F) There exists a perfect subfield F_{0} of F such that F is finitely generated over F_{0}.

Further assuming that p is not a uniformizer of K, we will define a twisted tangent space Θ^{j} and show that the functor X̄^{j} : Emb_{O_{K}} → G_{K}-(Aff/F̄) induces a functor

X̄^{j} : (Finite Étale/K)^{≤j+} → G_{K}-(Finite Étale/Θ^{j}).

In this subsection, we study the easier case where K is of characteristic p. Let F_{0} be a perfect subfield of F such that F is finitely generated over F_{0}. We assume K is of characteristic p. Then, F_{0} is naturally identified with a subfield of K. We first define a functor

(Finite Étale/K) → Emb_{O_{K}} .

In this subsection, A denotes the integer ring of a finite étale K-algebra.

Lemma 2.3 Let A be the integer ring of a finite étale K-algebra.

1. Let (A/𝔪_{A}^{n} ⊗_{F_{0}} O_{K})^{∧} denote the formal completion of A/𝔪_{A}^{n} ⊗_{F_{0}} O_{K} of the surjection A/𝔪_{A}^{n} ⊗_{F_{0}} O_{K} → A/𝔪_{A}^{n} sending a ⊗ b to ab. Then the projective limit

(A ˆ⊗_{F_{0}}O_{K})^{∧} = \underleftarrow{lim}_{n}(A/𝔪_{A}^{n} ⊗_{F_{0}} O_{K})^{∧}

is an O_{K}-algebra formally of finite type and formally smooth over O_{K}.

2. Let (A ˆ⊗_{F_{0}}O_{K})^{∧} → A be the limit of the surjections (A/𝔪_{A}^{n} ⊗_{F_{0}} O_{K})^{∧} → A/𝔪_{A}^{n}. Then ((A ˆ⊗_{F_{0}}O_{K})^{∧} → A) is an object of Emb_{O_{K}}.

3. Let A → B be a morphism of the integer rings of finite étale K-algebras. Then it induces a finite flat morphism ((A ˆ⊗_{F_{0}}O_{K})^{∧} → A) → ((B ˆ⊗_{F_{0}}O_{K})^{∧} → B) of Emb_{O_{K}}.

Proof. 1. We may assume A is local. Let E be the residue field of A and take a transcendental basis (t̄_{1}, …, t̄_{m}) of E over the perfect subfield F_{0} such that E is a finite separable extension of F_{0}(t̄_{1}, …, t̄_{m}). Take a lifting (t_{1}, …, t_{m}) in A of (t̄_{1}, …, t̄_{m}) and a prime element t_{0} ∈ A. We define a map F_{0}[T_{0}, …, T_{m}] → A by sending T_{i} to t_{i}. Then A is finite étale over the completion of the local ring of F_{0}[T_{0}, …, T_{m}] at the prime ideal (T_{0}). Hence there exist an étale scheme X over 𝔸_{F_{0}}^{m+1}, a point ξ of X above (T_{0}) and an F_{0}-isomorphism φ : Ô_{X,ξ} → A. Let i : Spec A → X ⊗_{F_{0}} O_{K} be the map defined by φ and O_{K} → A. Then (A ˆ⊗_{F_{0}}O_{K})^{∧} is isomorphic to the coordinate ring of the formal completion of X ⊗_{F_{0}} O_{K} along the closed immersion i : Spec A → X ⊗_{F_{0}} O_{K}. Hence (A ˆ⊗_{F_{0}}O_{K})^{∧} is formally of finite type and formally smooth over O_{K}.

2. Since the map (A ˆ⊗_{F_{0}}O_{K})^{∧} → A is surjective, the assertion follows from 1.

3. Since (B ˆ⊗_{F_{0}}O_{K})^{∧} = B ⊗_{A} (A ˆ⊗_{F_{0}}O_{K})^{∧}, the assertion follows. □ Thus, we obtain a functor (Finite Étale/K) → Emb_{O_{K}} sending a finite étale K-algebra L to ((O_{L} ˆ⊗_{F_{0}}O_{K})^{∧} → O_{L}). For a rational number j > 0, we have a sequence of functors

(Finite Étale/K) ⟶ Emb_{O_{K}} ⟶
(smooth Affinoid/K) ⟶ G_{K}-(Aff/F̄).

We also let X̄^{j} denote the composite functor (Finite Étale/K) → G_{K}-(Aff/F̄). For a finite étale K-algebra L, we have

X̄^{j}(L) = X̄^{j}((O_{L} ˆ⊗_{F_{0}}O_{K})^{∧} → O_{L}).

We define an object Θ^{j} of G_{K}-(Aff/F̄) to be the F̄-vector space Θ^{j} = Hom_{F} (Ω̂_{O_{K}/F_{0}} ⊗_{O_{K}} F, N^{j}) regarded as an affine scheme over F̄ with a natural G_{K}-action. Let G_{K}-(Finite Étale/Θ^{j}) denote the subcategory of G_{K}-(Aff/F̄) whose objects are finite étale schemes over Θ^{j} and morphisms are over Θ^{j}.

Lemma 2.4 For a rational number j > 1, the functor X̄^{j} : (Finite Étale/K) → G_{K}-(Aff/F̄) induces a functor X̄^{j} : (Finite Étale/K)^{≤j+} → G_{K}-(Finite Étale/Θ^{j}).

Proof. The canonical map Ω̂_{O_{K}/F_{0}} ⊗_{O_{K}} (O_{K} ˆ⊗_{F_{0}}O_{K})^{∧} → Ω̂_{(O_{K} ˆ⊗_{F_{0}} O_{K})^{∧}/O_{K}} is an isomorphism by the definition of (O_{K} ˆ⊗_{F_{0}}O_{K})^{∧}. Hence, we obtain isomorphisms X̄^{j}(K) → C̄^{j}((O_{K} ˆ⊗_{F_{0}}O_{K})^{∧} → O_{K}) → Θ^{j} by Lemma 1.14.3. We identify X̄^{j}(K) with Θ^{j} by this isomorphism. Let L be a finite étale Kalgebras whose ramification is bounded by j+. Then, by Corollary 1.16, the map X̄^{j}(L) → X̄^{j}(K) = Θ^{j} is finite and étale. Thus the assertion is proved. □ The construction in this subsection is independent of the choice of perfect subfield F_{0} ⊂ F by the following Lemma.

Lemma 2.5 Let K be a complete discrete valuation field of characteristic p > 0 satisfying the condition (F). Let F_{0} and F′_{0} be perfect subfields of F such that F is finitely generated over F_{0} and F′_{0}.

1. There exists a perfect subfield F′′_{0} of F containing F_{0} and F′_{0}.

2. Assume F_{0} ⊂ F′_{0}. Then F′_{0} is a finite separable extension of F_{0}. For the integer ring A of a finite étale algebra over K, the canonical map (A ˆ⊗_{F_{0}}O_{K})^{∧} → (A ˆ⊗_{F′_{0}}O_{K})^{∧} is an isomorphism.

Proof. 1. The maximum perfect subfield ⋂_{n} F^{p^{n}} of F contains F_{0} and F′_{0} as subfields.

2. Since F′_{0} is a perfect subfield of a finitely generated field F over F_{0}, it is a finite extension of F_{0}. Since the canonical map (A ˆ⊗_{F_{0}}O_{K})^{∧} → (A ˆ⊗_{F′_{0}}O_{K})^{∧} is finite étale and the induced map (A ˆ⊗_{F_{0}}O_{K})^{∧}/𝔪_{(A ˆ⊗_{F_{0}} O_{K})^{∧}} → (A ˆ⊗_{F′_{0}}O_{K})^{∧}/𝔪_{(A ˆ⊗_{F′_{0}} O_{K})^{∧}} is an isomorphism, the assertion follows. □

2.3 Functoriality of the closed fibers of tubular neighborhoods: A mixed characteristic case

In this subsection, we keep the assumption:

(F) There exists a perfect subfield F_{0} of F such that F is finitely generated over F_{0}.

We do not assume that the characterisic of K is p. Under the assumption (F), there exists a subfield K_{0} of K such that O_{K_{0}} = O_{K} ∩K_{0} is a complete discrete valuation ring with residue field F_{0}. If K is of characteristic 0, the fraction field K_{0} of the ring of the Witt vectors W (F_{0}) = O_{K_{0}} regarded as a subfield of K satisfies the conditions. If K is of characteristic p, we naturally identify F_{0} as a subfield of K and the subfield F_{0}((t)) for any non-zero element t ∈ 𝔪_{K} satisfies the conditions. In this subsection, we take a subfield K_{0} of K such that O_{K_{0}} = O_{K} ∩ K_{0} is a complete discrete valuation ring with residue field F_{0}. Here, we do not define a functor (Finite Étale/K) → Emb_{O_{K}}. Instead, we introduce a new category Emb_{K,O_{K_{0}}} and a functor

Emb_{K,O_{K_{0}}} → Emb_{O_{K}} .

In this subsection, A denotes the integer ring of a finite étale K-algebra and π_{0} denotes a prime element of the subfield K_{0} ⊂ K. For a complete Noetherian local O_{K_{0}}-algebra R formally smooth over O_{K_{0}}, we define its relative dimension over O_{K_{0}} to be the sum tr.deg(E/k)+dim_{E} 𝔪_{R}/(π_{0}, 𝔪_{R}^{2}) of the transcendental degree of E = R/𝔪_{R} over k and the dimension dim_{E} 𝔪_{R}/(π_{0}, 𝔪_{R}^{2}).

Definition 2.6 Let K be a complete discrete valuation field and K_{0} be a subfield of K such that O_{K_{0}} = O_{K} ∩ K_{0} is a complete discrete valuation ring with perfect residue field F_{0} and that F is finitely generated over F_{0}.

1. We define Emb_{K,O_{K_{0}}} to be the category whose objects and morphisms are as follows. An object of Emb_{K,O_{K_{0}}} is a triple (A_{0} → A) where:

• A is the integer ring of a finite étale K-algebra.

• A_{0} is a complete semi-local Noetherian O_{K_{0}}-algebra formally smooth of relative dimension tr.deg(F/F_{0}) + 1 over O_{K_{0}}.

• A_{0} → A is a regular surjection of codimension 1 of O_{K_{0}}-algebras inducing an isomorphism A_{0}/𝔪_{A_{0}} → A/𝔪_{A}.

A morphism (f, f) : (A_{0} → A) → (B_{0} → B) is a pair of an O_{K}-homomorphism f : A → B and an O_{K_{0}}-homomorphism f : A_{0} → B_{0} such that the diagram

A_{0} ⟶ A
f↓  ↓^{f}
B_{0} ⟶ B

is commutative.

2. For the integer ring A of a finite étale K-algebra, we define Emb_{K,O_{K_{0}}} (A) to be the subcategory of Emb_{K,O_{K_{0}}} whose objects are of the form (A_{0} → A) and morphisms are of the form (id_{A}, f).

3. We say that a morphism (A_{0} → A) → (B_{0} → B) is finite flat if A_{0} → B_{0} is finite flat and the map B_{0} ⊗_{A_{0}} A → B is an isomorphism.

Lemma 2.7 1. If A is the integer ring of a finite étale K-algebra, then the category Emb_{K,O_{K_{0}}} (A) is non-empty.

2. Let (A_{0} → A) and (B_{0} → B) be objects of Emb_{K,O_{K_{0}}} and A → B be an O_{K}- homomorphism. Then there exists a homomorphism (A_{0} → A) → (B_{0} → B) in Emb_{K,O_{K_{0}}} extending A → B.

3. Let (A_{0} → A) → (B_{0} → B) be a morphism of Emb_{K,O_{K_{0}}} . If a prime element π_{0} of K_{0} is not a prime element of any factor of A, then the map (A_{0} → A) → (B_{0} → B) is finite and flat.

Proof. 1. We may assume A is local. Take a transcendental basis (t̄_{1}, …, t̄_{m}) of the residue field E of A over k such that E is a finite separable extension of k(t̄_{1}, …, t̄_{m}). Take a lifting (t_{1}, …, t_{m}) in O_{K} of (t̄_{1}, …, t̄_{m}) and a prime element t_{0} of A. Then A is unramified over the completion of the local ring of O_{K_{0}}[T_{0}, …, T_{m}] at the prime ideal (π_{0}, T_{0}) by the map sending T_{i} to t_{i}. Hence there are an étale scheme X over 𝔸_{O_{K_{0}}}^{m+1}, a point ξ of X above (π_{0}, T_{0}) and a

regular surjection φ : Ô_{X,ξ} → A of codimension 1. Let A_{0} be the O_{K_{0}}-algebra Ô_{X,ξ}. Then (A_{0} → A) is an object of Emb_{K,O_{K_{0}}} .

2. Since A_{0} is formally smooth over O_{K_{0}}, it follows from that B_{0} is the formal completion of itself with respect to the surjection B_{0} → B.

3. We may assume A and B are local. We show that the map B_{0} ⊗_{A_{0}} A → B is an isomorphism. Let f be a generator of the kernel of A_{0} → A and consider the class of f in 𝔪_{A_{0}}/𝔪_{A_{0}}^{2}. We show that the image of the class of f in 𝔪_{B_{0}}/𝔪_{B_{0}}^{2} is not 0. Let t_{0} ∈ A_{0} and t′_{0} ∈ B_{0} be liftings of prime elements of A and B respectively. By the assumption that π_{0} is not a prime element, the surjection Ω̂_{A_{0}/O_{K_{0}}} → Ω̂_{A/O_{K_{0}}} induces an isomorphism Ω̂_{A_{0}/O_{K_{0}}} ⊗_{A_{0}} A/𝔪_{A} → Ω̂_{A/O_{K_{0}}} ⊗_{A} A/𝔪_{A}. Hence the image of dt_{0} is a basis of the kernel of Ω̂_{A_{0}/O_{K_{0}}} ⊗_{A_{0}} A/𝔪_{A} → Ω_{(A/𝔪_{A})/k}. Therefore, (π_{0}, t_{0}) is a basis of 𝔪_{A_{0}}/𝔪_{A_{0}}^{2}. Further, by the assumption that π_{0} is not a prime element, the kernel of the map 𝔪_{A_{0}}/𝔪_{A_{0}}^{2} → 𝔪_{A}/𝔪_{A}^{2} is generated by the class of π_{0}. Hence the class of f is a non-zero multiplie of the class of π_{0}. Similarly (π_{0}, t′_{0}) is a basis of 𝔪_{B_{0}}/𝔪_{B_{0}}^{2}. Thus the image of f in 𝔪_{B_{0}}/𝔪_{B_{0}}^{2} is not zero as is claimed. Hence the kernel of B_{0} → B is also generated by the image of f and the map B_{0} ⊗_{A_{0}} A → B is an isomorphism. Since B is finite over A, B_{0} is also finite over A_{0} by Nakayama’s lemma. Since dim A_{0} = dim B_{0} = 2, the assertion follows by EGA Chap 0_{IV} Corollaire (17.3.5) (ii). □

Corollary 2.8 Let A be the integer ring of a finite étale K-algebra. If a prime element π_{0} of K_{0} is not a prime element of any factor of A, then every morphism of Emb_{K,O_{K_{0}}} (A) is an isomorphism.

Proof. If (A_{0} → A) → (A′_{0} → A) is a map, the map A_{0} → A′_{0} is finite flat of degree 1 by Lemma 2.7.3. Hence it is an isomorphism. □ We define a functor Emb_{K,O_{K_{0}}} → Emb_{O_{K}}.

Lemma 2.9 Let (A_{0} → A) be an object of Emb_{K,O_{K_{0}}} .

1. Let (A_{0}/𝔪_{A_{0}}^{n} ⊗_{O_{K_{0}}} O_{K})^{∧} denote the formal completion of A_{0}/𝔪_{A_{0}}^{n} ⊗_{O_{K_{0}}} O_{K} of the surjection A_{0}/𝔪_{A_{0}}^{n} ⊗_{O_{K_{0}}} O_{K} → A/𝔪_{A}^{n} sending a ⊗ b to ab. Then the projective limit

(A_{0} ˆ⊗_{O_{K_{0}}} O_{K})^{∧} = \underleftarrow{lim}_{n}(A_{0}/𝔪_{A_{0}}^{n} ⊗_{O_{K_{0}}} O_{K})^{∧}

is an O_{K}-algebra formally of finite type and formally smooth over O_{K}.

2. Let (A_{0} ˆ⊗_{O_{K_{0}}} O_{K})^{∧} → A be the limit of the surjections (A_{0}/𝔪_{A_{0}}^{n} ⊗_{O_{K_{0}}} O_{K})^{∧} → A/𝔪_{A}^{n}. Then ((A_{0} ˆ⊗_{O_{K_{0}}} O_{K})^{∧} → A) is an object of Emb_{O_{K}}.

3. Let (A_{0} → A) → (B_{0} → B) be a morphism of Emb_{K,O_{K_{0}}} . Then it induces a morphism ((A_{0} ˆ⊗_{O_{K_{0}}} O_{K})^{∧} → A) → ((B_{0} ˆ⊗_{O_{K_{0}}} O_{K})^{∧} → B) of Emb_{O_{K}}.

Proof. 1. We may assume A and hence A_{0} are local. Let E be the residue field of A and take a transcendental basis (t̄_{1}, …, t̄_{m}) of E over k such that E is a finite separable extension of k(t̄_{1}, …, t̄_{m}). Take a lifting (t_{1}, …, t_{m}) in A_{0} of (t̄_{1}, …, t̄_{m}). By our assumption, the quotient ring A_{0}/π_{0}A_{0} is a regular local ring of dimension 1 and hence is a discrete valuation ring. Take a lifting t_{0} ∈ A_{0} of a prime element of A_{0}/π_{0}A_{0}. We define a map O_{K_{0}}[T_{0}, …, T_{m}] → A_{0} by sending T_{i} to t_{i}. Then A_{0} is finite étale over the completion of the local ring of O_{K_{0}}[T_{0}, …, T_{m}] at the prime ideal (T_{0}, π_{0}). Hence there exist an étale scheme X over 𝔸_{O_{K_{0}}}^{m+1}, a point ξ of X above (T_{0}, π_{0}) and a O_{K_{0}}-isomorphism φ : Ô_{X,ξ} → A_{0}. Let i : Spec A → X ⊗_{O_{K_{0}}} O_{K} be the map defined by φ and O_{K} → A. Then (A_{0} ˆ⊗_{O_{K_{0}}} O_{K})^{∧} is isomorphic to the coordinate ring of the formal completion of X ⊗_{O_{K_{0}}} O_{K} along the closed immersion i : Spec A → X ⊗_{O_{K_{0}}} O_{K}. Hence (A_{0} ˆ⊗_{O_{K_{0}}} O_{K})^{∧} is formally of finite type and formally smooth over O_{K}.

2. Since the map (A_{0} ˆ⊗_{O_{K_{0}}} O_{K})^{∧} → A is surjective, the assertion follows from 1.

3. Clear. □ In the rest of this subsection, we put A = (A_{0} ˆ⊗_{O_{K_{0}}} O_{K})^{∧} for an object (A_{0} → A) of Emb_{K,O_{K_{0}}} . By Lemma 2.9, we obtain a functor Emb_{K,O_{K_{0}}} → Emb_{O_{K}} sending (A_{0} → A) to (A → A). For a rational number j > 0, we have a sequence of functors

Emb_{K,O_{K_{0}}} ⟶ Emb_{O_{K}} ⟶^{X^{j}} (smooth Affinoid/K) ⟶ G_{K}-(Aff/F̄).

We also let X̄^{j} denote the composite functor Emb_{K,O_{K_{0}}} → G_{K}-(Aff/F̄). For an object (A_{0} → A) of Emb_{K,O_{K_{0}}} , we have X̄^{j}(A_{0} → A) = X̄^{j}((A_{0} ˆ⊗_{O_{K_{0}}} O_{K})^{∧} → A). We study the dependence of the construction on the choice of a subfield K_{0} ⊂ K, assuming the characteristic of K is 0.

Lemma 2.10 Let K be a complete discrete valuation field of mixed characteristic satisfying the condition (F). Let K_{0} and K′_{0} be subfields of K such that O_{K_{0}} = O_{K} ∩K_{0} and O_{K′_{0}} = O_{K} ∩K′_{0} are complete discrete valuation rings with perfect residue field F_{0} and F′_{0} and that F is finitely generated over F_{0} and F′_{0}.

1. There exists a subfield K′′_{0} of K such that O_{K′′_{0}} = O_{K} ∩ K′′_{0} is a complete discrete valuation ring with perfect residue field and that K′′_{0} contains K_{0} and K′_{0} as subfields.

2. Assume K_{0} ⊂ K′_{0}. Then K′_{0} is a finite extension of K_{0}. For an object (A_{0} → A) of Emb_{K,O_{K_{0}}} , the formal completion A′_{0} → A of the surjection A_{0} ⊗_{O_{K_{0}}} O_{K′_{0}} → A defines an object (A′_{0} → A) of Emb_{K,O_{K′_{0}}} . Further, we have a canonical isomorphism ((A′_{0} ˆ⊗_{O_{K′_{0}}} O_{K})^{∧} → A) → ((A_{0} ˆ⊗_{O_{K_{0}}} O_{K})^{∧} → A) in Emb_{O_{K}}.

Proof. 1. By Lemma 2.5, we may assume the residue fields F_{0} and F′_{0} are the maximum perfect subfields of F. Then both of K_{0} and K′_{0} are finite over the fraction field of W (F_{0}) regarded as a subfield of K. Hence it is sufficient to take the composition field.

2. By Lemma 2.5.2, the extension K′_{0} is finite over K_{0}. The rest is clear from the construction. □ If K is of characteristic p, the construction in this subsection is related to that in the last subsection as follows. Let K_{0} be a subfield of K such that O_{K_{0}} = O_{K} ∩ K_{0} is a complete discrete valuation ring with perfect residue field F_{0} and that F is finitely generated over F_{0}. Then, if π_{0} is a prime element of K_{0}, we have an isomorphism F_{0}((t)) → K_{0} sending t to π_{0}. For the integer ring A of a finite étale algebra over K, let (A ˆ⊗_{F_{0}}O_{K_{0}})^{∧} denote the projective limit of the formal completions (A/𝔪_{A}^{n} ⊗_{F_{0}} O_{K_{0}})^{∧} of the surjections A/𝔪_{A}^{n} ⊗_{F_{0}} O_{K_{0}} → A/𝔪_{A}^{n}. The surjection (A ˆ⊗_{F_{0}}O_{K_{0}})^{∧} → A defines an object ((A ˆ⊗_{F_{0}}O_{K_{0}})^{∧} → A) of Emb_{K,O_{K_{0}}} . Further, we have a canonical isomorphism ((A ˆ⊗_{F_{0}}O_{K_{0}})^{∧} ˆ⊗_{O_{K_{0}}} O_{K})^{∧} → A) → ((A ˆ⊗_{F_{0}}O_{K})^{∧} → A) in Emb_{O_{K}}. In order to define a functor similar to the functor (Finite Étale/K)^{≤j+} → (Finite Étale/Θ^{j}) in Section 2.2, we assume that π_{0} is not a prime element of K in the rest of this subsection. Note that if p is not a prime element of K and if the condition (F) is satisfied, there exists a subfield K_{0} ⊂ K with residue field F_{0} such that a prime element of K_{0} is not a prime element of K. We compute the twisted normal cone C̄^{j}((A_{0} ˆ⊗_{O_{K_{0}}} O_{K})^{∧} → A) for an object (A_{0} → A) of Emb_{K,O_{K_{0}}} . Let N_{A/A} = I/I^{2} be the conormal module where I is the kernel of the surjection A → A. We put Ω̂_{O_{K}/O_{K_{0}}} = \underleftarrow{lim}_{n} Ω_{(O_{K}/𝔪_{K}^{n})/O_{K_{0}}} and let Ω̃_{F} be the F -vector space Ω̂_{O_{K}/O_{K_{0}}} ⊗_{O_{K}} F. Similarly, we put Ω̂_{A/A_{0}} = \underleftarrow{lim}_{n} Ω_{(A/𝔪_{A}^{n})/A_{0}}. We also consider the canonical maps N_{A/A} → Ω̂_{A/A_{0}} ⊗_{A} A and Ω̂_{O_{K}/O_{K_{0}}} ⊗_{O_{K}} A → Ω̂_{A/A_{0}} ⊗_{A} A.

Lemma 2.11 Assume π_{0} is not a prime element of K and let m be the transcendental dimension of F over k. Let (A_{0} → A) be an object of Emb_{K,O_{K_{0}}} . Then,

1. The dimension of the F -vector space Ω̃_{F} is m + 1.

2. The map N_{A/A} → Ω̂_{A/A_{0}} ⊗_{A} A is a surjection and the map Ω̂_{O_{K}/O_{K_{0}}} ⊗_{O_{K}} A → Ω̂_{A/A_{0}} ⊗_{A} A is an isomorphism. They induce an isomorphism N_{A/A} ⊗_{A} A/𝔪_{A} → Ω̃_{F} ⊗_{F} A/𝔪_{A}.

3. Let (A_{0} → A) → (B_{0} → B) be a morphism of Emb_{K,O_{K_{0}}} and put B = (B_{0} ˆ⊗_{O_{K_{0}}} O_{K})^{∧}. Then, the diagram

N_{A/A} ⊗_{A} A/𝔪_{A} ⟶ Ω̃_{F} ⊗_{F} A/𝔪_{A}
↓  ↓
N_{B/B} ⊗_{B} B/𝔪_{B} ⟶ Ω̃_{F} ⊗_{F} B/𝔪_{B}

is commutative.

Proof. 1. By the assumption that π_{0} is not a prime element of K, we have an exact sequence 0 → 𝔪_{K}/𝔪_{K}^{2} → Ω̃_{F} → Ω_{F/k} → 0. Since the F -vector space Ω_{F/k} is of dimension m, the assertion follows.

2. Since the cokernel of the map N_{A/A} → Ω̂_{A/A_{0}} ⊗_{A} A is Ω_{A/A_{0}} = 0, it is a surjection. By the definition of A, the map Ω̂_{O_{K}/O_{K_{0}}} ⊗_{O_{K}} A → Ω_{A/A_{0}} is an isomorphism. Hence the map Ω̂_{O_{K}/O_{K_{0}}} ⊗_{O_{K}} A → Ω̂_{A/A_{0}} ⊗_{A} A is also an isomorphism. Then the codimension of the regular surjection A → A is m + 1 and hence N_{A/A} is free of rank m+1. Since the induced map N_{A/A}⊗_{A}A/𝔪_{A} → Ω̃_{F} ⊗_{F} A/𝔪_{A} is a surjection of free A/𝔪_{A}-modules of rank m + 1, it is an isomorphism.

3. By the assumption that π_{0} is not a prime element of K, every map in Emb_{K,O_{K_{0}}} is finite flat by Lemma 2.7.3. Hence the assertion follows. □

For a rational number j > 0, let Θ^{j} be the F̄-vector space Θ^{j} = Hom_{F} (Ω̃_{F} , N^{j}) regarded as an affine scheme over F̄.

Corollary 2.12 Assume that π_{0} is not a prime element of K. Let (A_{0} → A) be an object of Emb_{K,O_{K_{0}}} and let (A → A) be the image in Emb_{K,O_{K_{0}}} . Let j > 0 be a rational number.

1. The isomorphism in Lemma 2.11.2 induces an isomorphism C̄^{j}(A → A) → Θ^{j} ⊗F̄ A_{F̄,red}.

2. Let (A_{0} → A) → (B_{0} → B) be a morphism of Emb_{K,O_{K_{0}}} . Then the diagram

X̄^{j}(B → B) ⟶ C̄^{j}(B → B) ⟶ Θ^{j} ⊗F̄ B_{F̄,red}
↓  ↓  ↓
X̄^{j}(A → A) ⟶ C̄^{j}(A → A) ⟶ Θ^{j} ⊗F̄ A_{F̄,red}

is commutative.

3. If the ramification of A ⊗_{O_{K}} K is bounded by j+ and j > 1, then the composition X̄^{j}(A → A) → C̄^{j}(A → A) → Θ^{j} ⊗F̄ A_{F̄,red} → Θ^{j} is finite and étale.

Proof. 1. Since the surjection A → A is regular, the assertion follows from the isomorphism in Lemma 2.11.2.

2. The left square is commutative by the construction. The commutativity of the right square is a consequence of Lemma 2.11.3.

3. By Lemma 2.7, there exist an embedding (A′_{0} → O_{K}) in Emb_{K,O_{K_{0}}} (O_{K}) and a finite flat morphism (A′_{0} → O_{K}) → (A_{0} → A). Since the ramification is bounded by j+, the finite map X̄^{j}(A → A) → C̄^{j}(A → A) is étale by Corollary 1.16.3. Since A_{F̄,red} is étale over F̄, the assertion follows from 1 and 2. □. For a rational number j > 0, we regard Θ^{j} as an object of G_{K}-(Aff/F̄) with the natural G_{K}-action. Let G_{K}-(Finite Étale/Θ^{j}) denote the subcategory of G_{K}-(Aff/F̄) whose objects are finite étale schemes over Θ^{j} and morphisms are over Θ^{j}. Let Emb_{K,O_{K_{0}}}^{≤j+} denote the full subcategory of Emb_{K,O_{K_{0}}} consisting of the objects (A_{0} → A) such that the ramifications of A ⊗_{O_{K}} K are bounded by j+. By Corollary 2.12, the functor X̄^{j} : Emb_{K,O_{K_{0}}} → G_{K}-(Aff/F̄) induces a functor X̄^{j} : Emb_{K,O_{K_{0}}}^{≤j+} → G_{K}-(Finite Étale/Θ^{j}).

We show that the functor X̄^{j} : Emb_{K,O_{K_{0}}}^{≤j+} → G_{K}-(Finite Étale/Θ^{j}) further

induces a functor (Finite Étale/K)^{≤j+} → G_{K}-(Finite Étale/Θ^{j}).

Lemma 2.13 Assume π_{0} is not a prime element of K. Let (f, f), (g, g) : (A_{0} → A) → (B_{0} → B) be maps in Emb_{K,O_{K_{0}}} and j > 1 be a rational number. If the ramifications of A ⊗_{O_{K}} K and B ⊗_{O_{K}} K are bounded by j+ and if f = g, then the induced maps

(f, f)_{∗}, (g, g)_{∗} : X̄^{j}(A_{0} → A) ⟶ X̄^{j}(B_{0} → B)

are equal.

Proof. By Corollary 2.12, the schemes X̄^{j}(A_{0} → A) and X̄^{j}(B_{0} → B) are finite étale over Θ^{j} and the maps (f, f)_{∗}, (g, g)_{∗} : X̄^{j}(A_{0} → A) → X̄^{j}(B_{0} → B) are maps over Θ^{j}. Hence they are determined by the restrictions on the inverse images of a point. The inverse images of the origin 0 ∈ Θ^{j} are canonically identified with the sets Hom_{O_{K}} (A, K̄) and Hom_{O_{K}} (B, K̄) respectively by Lemma 2.1. Hence the assertion follows. □

Corollary 2.14 Assume π_{0} is not a prime element of K. Let j > 1 be a rational number.

1. Let L be a finite étale K-algebra with ramification bounded by j+. Then the system X̄^{j}(A_{0} → O_{L}) parametrized by the objects (A_{0} → O_{L}) of Emb_{K,O_{K_{0}}} (O_{L}) is constant and the limit

X̄^{j}(L) = \underleftarrow{lim}_{(A_{0}→O_{L})∈Emb_{K,O_{K_{0}}} (O_{L})}X̄^{j}(A_{0} → O_{L})

is a finite étale scheme over Θ^{j}.

2. The functor X̄^{j} : Emb_{K,O_{K_{0}}}^{≤j+} → G_{K}-(Finite Étale/Θ^{j}) induces a functor

X̄^{j} : (Finite Étale/K)^{≤j+} ⟶ G_{K}-(Finite Étale/Θ^{j}).

Proof. 1. By Corollary 2.8 and by the assumption that π_{0} is not a prime element, every map in Emb_{K,O_{K_{0}}} (O_{L}) induces an isomorphism. By Lemma 2.7.1, the category Emb_{K,O_{K_{0}}} (O_{L}) is connected. To see that the system is constant, it suffices to apply Lemma 2.13 for f = g = id_{O_{L}}. The map X̄^{j}(L) → Θ^{j} is finite étale by Corollary 2.12.3.

2. It is also an immediate consequence of Lemma 2.13. □ By Lemma 2.10 and the canonical isomorphism Ω̂_{O_{K}/O_{K_{0}}} ⊗_{O_{K}} F → Ω̂_{O_{K}/O_{K′_{0}}} ⊗_{O_{K}} F, the functor X̄^{j} : (Finite Étale/K)^{≤j+} →

G_{K}-(Finite Étale/Θ^{j}) is independent of the choice of subfield K_{0} if the characteristic of K is 0. If the characteristic of K is p, it is the same as that defined in Section 2.2.

2.4 Proof of commutativity

Now we are ready to prove the main result. For an integer m prime to p, let I_{m} be the unique open subgroup of the inertia subgroup I ⊂ G_{K} of index m.

Theorem 2.15 Let K be a complete discrete valuation field. Let j > 1 be a rational number and m be the prime-to-p part of the denominator of j. Assume either K has equal characteristics p > 0 or K has mixed characteristic and p is not a prime element. Then we have the following.

1. The graded piece Gr^{j}G_{K} = G_{K}^{j}/G_{K}^{j+} is abelian.

2. The commutator [I_{m}, G_{K}^{j}] is a subgroup of G_{K}^{j+}. In particular, Gr^{j}G_{K} is a subgroup of the center of the pro-p-group G_{K}^{1+}/G_{K}^{j+}.

Proof. We first prove the case where the condition (F) There exists a perfect subfield F_{0} of F such that F is finitely generated over F_{0}. is satisfied. We use the functor X̄^{j} : (Finite Étale/K)^{≤j+} → G_{K}-(Finite Étale/Θ^{j}) defined in Sections 2.2 and 2.3. Let L be a finite Galois extension of K of ramification bounded by j+ and put G = Gal(L/K). To prove 1, it is sufficient to show that G^{j} is commutative. By the definition of the functor, the image X̄^{j}(L) is a finite étale covering of Θ^{j} with a left action of G_{K}. We call this action of G_{K} on X̄^{j}(L) the arithmetic action. On the other hand, by functoriality, we have a right action of G on X̄^{j}(L), which commutes with the arithmetic action of G_{K}. We call this action of G on X̄^{j}(L) the geometric action. We identify the inverse image in X̄^{j}(L) of the origin of Θ^{j} with Φ(L) as in Lemma 2.1. The arithmetic action of σ ∈ G_{K} on Φ(L) = Hom_{K}(L, K̄) is given by f ↦ σ ∘ f and the geometric action of τ ∈ G is given by f ↦ f ∘ τ. Hence Φ(L) is a G-torsor and the étale covering X̄^{j}(L) is also a G-torsor over Θ^{j}. The stabilizer in G_{K} of each connected component of X̄^{j}(L) with respect to the arithmeric action is equal to G_{K}^{j} since Φ^{j}(L) is identified with π_{0}(X̄^{j}(L)). Take a connected component X̄^{j}(L)_{0} of X̄^{j}(L). Then, the stabilizer of the intersection X̄^{j}(L)_{0} ∩ Φ(L) in G, with respect to the geometric action, is equal to G^{j}. Hence the stabilizer of the component X̄^{j}(L)_{0} in G, with respect to the geometric action, is also equal to G^{j} and X̄^{j}(L)_{0} is a connected G^{j}-torsor over Θ^{j}. Therefore the map G^{j} → Aut(X̄^{j}(L)_{0}/Θ^{j}) is an isomorphism. On the other hand, by the assumption that j > 1, the group G_{K}^{j} is a subgroup of the wild inertia subgroup G_{K}^{1+} = P. Hence the restriction to G_{K}^{j} of the arithmetic action on Θ^{j} is trivial and we get a map G_{K}^{j} → Aut(X̄^{j}(L)_{0}/Θ^{j}). Since G_{K}^{j} acts on the intersection X̄^{j}(L)_{0} ∩ Φ(L) transitively, the map G_{K}^{j} → Aut(X̄^{j}(L)_{0}/Θ^{j}) is surjective. Since the geometric action of G^{j} and the arithmetic action of G_{K}^{j} on X̄^{j}(L)_{0} are commutative to each other, the group G^{j} ≃ Aut(X̄^{j}(L)_{0}/Θ^{j}) is commutative. Thus assertion 1 is proved in this case. We prove assertion 2 assuming the condition (F). We define a canonical map π_{1}^{ab}(Θ^{j}) → Gr^{j}G_{K} as follows. By 1, the image of the functor X̄^{j} : (Finite Étale/K)^{≤j+} → G_{K}-(Finite Étale/Θ^{j}) is in the full subcategory consisting of abelian coverings. Taking the Galois groups, we obtain a map π_{1}^{ab}(Θ^{j}) → G_{K}/G_{K}^{j+} inducing a surjection

π_{1}^{ab}(Θ^{j}) ⟶ Gr^{j}G_{K}.

The canonical map π_{1}^{ab}(Θ^{j}) → Gr^{j}G_{K} is compatible with the actions of G_{K}. The action of G_{K} on π_{1}^{ab}(Θ^{j}) is induced by that on Θ^{j} and the action on Gr^{j}G_{K} is by conjugation. Since the subgroup I_{m} acts trivially on Θ^{j}, it also acts trivially on π_{1}^{ab}(Θ^{j}). Hence, assertion 2 follows in this case by the compatibility of the surjection π_{1}^{ab}(Θ^{j}) → Gr^{j}G_{K} with the G_{K}-action. To reduce the general case to the special case proved above, we show the following Lemma.

Lemma 2.16 Let K be a complete discrete valuation field and K_{0} be a subfield of K such that O_{K_{0}} = O_{K} ∩ K is a complete discrete valuation ring with perfect residue field F_{0}. Then there exist a filtered family of subextensions K_{μ} ⊂ K, μ ∈ M of K_{0} satisfying the following conditions: For each μ ∈ M, the intersection O_{K_{μ}} = O_{K} ∩ K_{μ} is a complete discrete valuation ring and the residue field F_{μ} is finitely generated over F_{0}, the residue field F is a separable extension of F_{μ} and a prime element of K_{μ} is a prime element of K. The residue field F is equal to the union \underrightarrow{lim}_{μ∈M} F_{μ}.

Proof. Let π_{0} be a prime element of K_{0}. Take a transcendental basis (t̄_{λ})_{λ∈Λ} of F over F_{0} such that F is separable over F_{0}(t̄_{λ}, λ ∈ Λ). We take liftings t_{λ} ∈ O_{K}, λ ∈ Λ of t̄_{λ}. For a finite subset σ ⊂ Λ, let K_{0,σ} be the fraction field of the completion of the local ring at the prime ideal (π_{0}) of the ring O_{K_{0}}[T_{λ}, λ ∈ σ] and regard it as a subfield of K. Let K_{0,μ} ⊂ K, μ ∈ M_{0} be the family of finite unramified subextensions of K_{0,σ}, σ ⊂ Λ. Let K′_{0} be the completion of the union \underrightarrow{lim}_{μ∈M_{0}} K_{0,μ}. Then K is a finite totally ramified extension of K′_{0}. Hence there is an index μ_{0} ∈ M_{0} and a finite totally ramified extension K_{μ_{0}} of K_{0,μ_{0}} such that K is the composite of K_{μ_{0}} and K′_{0}. We put M = {μ ∈ M_{0} : K_{0,μ_{0}} ⊂ K_{0,μ}}. Then the family K_{μ} = K_{μ_{0}}K_{0,μ}, μ ∈ M satisfies the conditions. □ We complete the proof of Theorem. It is sufficient to show assertion 2. Let F_{0} = ⋂_{n} F^{p^{n}} be the maximum perfect subfield of the residue field F. If the characteristic of K is positive, we take a element π_{0} ∈ 𝔪_{K}^{2}, ≠ 0 of K and put K_{0} = F_{0}((π_{0})) ⊂ K. If the characteristic of K is 0, let K_{0} be the fraction field of W (F_{0}) and regard it as a subfield of K. By the assumption that p is not a prime element of K, a prime element of K_{0} is not a prime element of K. Let K_{μ}, μ ∈ M be a family of subfields of K as in Lemma 2.16. Since K_{0} is a subfield of K_{μ} satisfying the condition (F) and a prime element of K_{0} is not a prime element of K_{μ}, we have [I_{m,K_{μ}}, G_{K_{μ}}^{j}] ⊂ G_{K_{μ}}^{j+} for μ ∈ M. Since K′ = \underleftarrow{lim}_{μ∈M} K_{μ} is a Henselian discrete valuation field and K is the completion of K′, the canonical maps G_{K} → G_{K′} → \underleftarrow{lim}_{μ∈M} G_{K_{μ}} are isomorphisms. It induces an isomorphism I_{m,K} → \underleftarrow{lim}_{μ} I_{m.K_{μ}}. By Lemma 2.2 and by the assumption that the residue field F is separable over F_{μ} and a prime element of K_{μ} is a prime element of K, the map G_{K}^{j} → G_{K_{μ}}^{j} is surjective. Hence we have isomorphisms G_{K}^{j} → \underleftarrow{lim}_{μ∈M} G_{K_{μ}}^{j} and G_{K}^{j+} → \underleftarrow{lim}_{μ∈M} G_{K_{μ}}^{j+}.

By taking the limit of [I_{m,K_{μ}}, G_{K_{μ}}^{j}] ⊂ G_{K_{μ}}^{j+}, we obtain [I_{m,K}, G_{K}^{j}] ⊂ G_{K}^{j+}. □

3 Some generalities on log structures

To study the logarithmic filtration in later sections, we recall and establish some generalities on log structures. More systematic account of a part is given in [10] Section 4. For the basic definitions on log schemes, we refer to [6]. In this paper, a log structure M_{X} → O_{X} on a scheme X means a Zariski fs-log structure. We prepare some basic terminologies on log schemes. We call a pair (X, P) of a log scheme X and a chart P on X a charted log scheme. For charted log schemes (X, P) and (S, N), we call a pair (f, φ) of a map f : X → S of log schemes and a map N → P of fs-monoid a map (X, P) → (S, N) of charted log schemes if the diagram

\begin{matrix}N & ⟶ & Γ(S, M_{S}) \\ ↓ & & ↓ \\ P & ⟶ & Γ(X, M_{X}) \end{matrix}  (1)

is commutative. For an fs-monoid P, we regard Spec ℤ[P] as a log scheme with the log structure defined by the chart P → ℤ[P]. For maps X → S and Y → S of log schemes, let X ×_{S}^{log} Y denote the fibered product in the category of fs-log schemes. If S = Spec A, X = Spec B and Y = Spec C are affine, N → A, P → B and Q → C are charts and if (f, φ) : (X, P) → (S, N) and (g, ψ) : (Y, Q) → (S, N) are morphisms of charted log schemes, we have X ×_{S}^{log} Y = Spec B⊗_{A}^{log} C where B ⊗_{A}^{log} C = (B ⊗_{A} C) ⊗_{ℤ[P +Q]} ⊗ℤ[P +_{N}^{sat} Q] and P +_{N}^{sat} Q is the saturation of the image of P + Q in the fibered sum P^{gp} ⊕_{N^{gp}} Q^{gp} = Coker(φ − ψ : N^{gp} → P^{gp} ⊕ Q^{gp}).

Definition 3.1 Let X → S be a morphism of log schemes.

1. (cf. [7], [11] Theorem 4.6 (iv)) We say that X → S is log flat if the following conditions are satisfied: For each x, there exist a commutative diagram

\begin{matrix}U & ⟶ & V \\ ↓ & & ↓ \\ X & ⟶ & S \end{matrix}

of log schemes, charts P on U and N on V and morphism (U, P) → (V, N) of charted log schemes such that the underlying map U → X is a flat surjection to an open neighborhood of x, the underlying map V → S is flat, the map N → P is injective and the underlying map U → V ⊗_{ℤ[N]} ℤ[P] is flat.

2. We say that X → S is log locally of complete intersection if the following conditions are satisfied: For each x, there exist a commutative diagram

\begin{matrix}U & ⟶ & V \\ ↓ & & ↓ \\ X & ⟶ & S \end{matrix}

of log schemes such that U is an open neighborhood of x, the map V → S is log smooth and U → V is an exact and regular immersion.

3. We say that X → S is log syntomic if it is log flat and log locally of complete intersection.

For the log syntomic morphisms, the definition here is slightly different from that in [9] (2.5). We introduce the new definition because it is a special case of the general definition due to Illusie and Olsson [5], [11] Definition 4.1 by Lemma 3.3 below. An equivalent statement of Lemma 3.2 in the resp. cases is proved in [6], and in the log flat case in [11] Theorem 4.6. Another proof is given in [10] Section 4.4.

Lemma 3.2 (cf. [11] Theorem 4.6) For a morphism X → S of log schemes, the following conditions are equivalent.

(1) The map X → S is log flat (resp. log smooth, log étale).

(2) Let

X′ ⟶ S′
↓  ↓
X ⟶ S

be a commutative diagram of log schemes such that X′ → X ×_{S}^{log} S′ is log étale and X′ → S′ is strict. Then the underlying map X′ → S′ is flat (resp. smooth, étale).

Lemma 3.3 For a morphism X → S of log schemes, the following conditions are equivalent.

(1) The map X → S is log syntomic.

(2) Let

X′ ⟶ S′
↓  ↓
X ⟶ S

be a commutative diagram of log schemes such that X′ → X ×_{S}^{log} S′ is log étale and X′ → S′ is strict. Then the underlying map X′ → S′ is flat and locally of complete intersection.

To deduce Lemma 3.3 from Lemma 3.2, we introduce some basic constructions on log schemes.

Lemma 3.4 Let f : X → S be a morphism of log schemes and x ∈ X. Then there exist charts P and N on open neighborhoods U of x and V ⊃ f(U) of s = f(x) and a morphism (U, P) → (V, N) of charted log schemes such that the map Spec ℤ[P] → Spec ℤ[N] is log smooth.

Proof. We put M̄_{S} = M_{S}/O_{S}^{×}, M̄_{X} = M_{X}/O_{X}^{×}, N = M̄_{S,s}, P_{0} = M̄_{X,x} and let N → P_{0} be the canonical map. We take charts N → Γ(V, M_{V}), P_{0} → Γ(U, M_{U}) on open neighborhoods lifting the identities. We define an fs-monoid P to be the inverse image of P_{0} by the map P_{0}^{gp}⊕N^{gp} → P_{0}^{gp} sending (m, n) to m+f(n). Then, shrinking U if necessary, we find a unique map P → Γ(U, M_{X}) extending the composition P_{0} + N → Γ(X, M_{X}) + Γ(S, M_{S}) → Γ(X, M_{X}). Thus, we obtain a morphism (U, P) → (V, N) of charted log schemes. Since the map N^{gp} → P^{gp} is an isomorphism to a direct summand, the map Spec ℤ[P] → Spec ℤ[N] is log smooth. □ For a morphism f : N → P of fs-monoids, we define an fs-monoid (P +_{N} P)^{∼} to be the inverse image of P by the map P^{gp} ⊕_{N^{gp}} P^{gp} → P^{gp} sending (m, m′) to m + m′.

Lemma 3.5 Let N → P be a map of fs-monoids and let (P +_{N} P)^{∼} ⊂ P^{gp}⊕_{N^{gp}} P^{gp} be as above. Then,

1. The map P × (P^{gp}/N^{gp}) → (P +_{N} P)^{∼} sending (m, m′) to (m + m′, −m′) is an isomorphism.

2. The ring homomorphism ℤ[P] → ℤ[(P +_{N} P)^{∼}] induced by the map P → (P +_{N} P)^{∼} of monoids sending m to (m, 0) is faithfully flat.

3. The map P + P + (P^{gp}/N^{gp}) → (P +_{N} P)^{∼} sending (m, m′, m′′) → (m + m′′, m′ − m′′) induces an isomorphism ℤ[P × P × (P^{gp}/N^{gp})]/((m, 0, 0) − (0, m, m); m ∈ P) → ℤ[(P +_{N} P)^{∼}] of rings.

Proof. 1. The inverse (P +_{N} P)^{∼} → P × (P^{gp}/N^{gp}) is given by (m, m′) → (m + m′, −m′). 2 and 3. Clear from 1. □

Corollary 3.6 Let (X, P) → (S, N) be a morphism of charted log schemes and put S′ = S ⊗_{ℤ[N]} ℤ[P] and X′ = X ⊗_{ℤ[P]} ℤ[(P +_{N} P)^{∼}]. Then the map X′ → S′ is strict, the map X′ → X ×_{S}^{log} S′ is log étale and X′ → X is faithfully flat.

Proof. The map X′ → X ×_{S}^{log} S′ is log étale by the definition of (P +_{N} P)^{∼}. The map X′ → S′ is strict by Lemma 3.5.1. The map X′ → X is faithfully flat by Lemma 3.5.2. □ Proof of Lemma 3.3. Since the assertion is local on X, we may assume there exist a log smooth scheme Y over S, an exact closed immersion X → Y over S and a morphism (Y, P) → (S, N) of charted log schemes as in Lemma 3.4. We put S_{1} = S ⊗_{ℤ[N]}^{log} ℤ[P], Y_{1} = Y ⊗_{ℤ[P]}^{log} ℤ[(P +_{N} P)^{∼}] and X_{1} = X ×_{Y}^{log} Y_{1}. We show (1)⇒(2). We assume X → S is log syntomic. We consider the diagram in (2). Since the question is local on X′, we may assume there exist a log étale scheme Y′ over Y ×_{S} S′ and an isomorphism X′ → X ×_{Y}^{log} Y′. Shrinking Y′, we may assume that the map Y′ → S′ is strict. Hence by Lemma 3.2, the underlying map Y′ → S′ is smooth. It is sufficient to show that the closed immersion X′ → Y′ is a regular immersion. We consider a commutative diagram

\xymatrix{ &  &  & X′_{1} \ar[dlll] \ar[r] \ar[d] & Y′_{1} \ar[dlll] \ar[d] \ar[dr] \\ X′ \ar[r] \ar[d] & Y′ \ar[d] \ar[dr] &  & X_{1} ×_{S}^{log} S′ \ar[dlll] \ar[r] \ar[d] & Y_{1} ×_{S}^{log} S′ \ar[dlll] \ar[r] \ar[d] & S′_{1} \ar[dlll] \ar[d] \\ X ×_{S}^{log} S′ \ar[r] \ar[d] & Y ×_{S}^{log} S′ \ar[r] \ar[d] & S′ \ar[d] & X_{1} \ar[dlll] \ar[r] & Y_{1} \ar[dlll] \ar[r] & S_{1} \ar[dlll] \\ X \ar[r] & Y \ar[r] & S}

by putting S′_{1} = S_{1} ×_{S}^{log} S′, Y′_{1} = Y_{1} ×_{Y}^{log} Y′ and X′_{1} = X_{1} ×_{X}^{log} X′. Since Y → S is log smooth, Y_{1} → S_{1} is strict and Y_{1} → Y ×_{S}^{log} S_{1} is log étale, the underlying map Y_{1} → S_{1} is smooth by Lemma 3.2. Similarly, since X → S is log flat, X_{1} → S_{1} is strict and X_{1} → X ×_{S}^{log} S_{1} is log étale, the underlying map X_{1} → S_{1} is flat by Lemma 3.2. Since Y_{1} → Y is flat by Lemma 3.5.2 and X → Y is a regular immersion, the immersion X_{1} → Y_{1} is a regular immersion. Thus X_{1} → S_{1} is flat and locally of complete intersection. Since the maps X_{1} → Y_{1} → S_{1} are strict, the underlying map X_{1} ×_{S}^{log} S′ → S′_{1} is flat and locally of complete intersection and the immersion X_{1} ×_{S}^{log} S′ → Y_{1} ×_{S}^{log} S′ is a regular immersion by EGA IV Propositions (19.3.9)(ii) and (19.3.7). Since Y′_{1} → Y_{1} ×_{S}^{log} S′ is a base change of Y′ → Y ×_{S}^{log} S′, the map Y′_{1} → Y_{1} ×_{S}^{log} S′ is log étale. Since it is strict, the underlying map Y′_{1} → Y_{1} ×_{S}^{log} S′ is étale by Lemma 3.2. Since X′_{1} → Y′_{1} is the base change of the regular immersion X_{1} ×_{S}^{log} S′ → Y_{1} ×_{S}^{log} S′ by the étale map Y′_{1} → Y_{1} ×_{S}^{log} S′, it is also a regular immersion. Since the regular immersion X′_{1} → Y′_{1} is also the base change of the immersion X′ → Y′ by the faithfully flat and strict map Y_{1} → Y, the immersion X′ → Y′ is a regular immersion as required. We show (2)⇒(1). We assume the condition (2) is satisfied. It is sufficient to show that the exact closed immersion X → Y is a regular immersion. By (2), the underlying map X_{1} → S_{1} is flat and locally of complete intersection and the underlying map Y_{1} → S_{1} is smooth. Hence the immersion X_{1} → Y_{1} is a regular immersion by EGA IV Proposition (19.3.7). Since the regular immersion X_{1} → Y_{1} is the base change of the immersion X → Y by the strict and faithfully flat map Y_{1} → Y, the immersion X → Y is a regular immersion as required. □

Corollary 3.7 (cf. [11] Corollary 4.12) Let f : X → S and S′ → S be morphisms of log schemes and let f′ : X′ = X ×_{S}^{log} S′ → S′ be the log base change. Then, if f : X → S is log flat (resp. log syntomic), the base change f′ : X′ → S′ is also log flat (resp. log syntomic).

Proof. Clear from Lemmas 3.2 and 3.3. □

Lemma 3.8 Let X → S be a log scheme over S log locally of complete intersection, Y → S be a log smooth log scheme over S and X → Y be an exact closed immersion over S. Then,

1. The immersion X → Y is a regular immersion.

2. Let Y′ → S be another log smooth log scheme over S and X → Y′ be an exact closed regular immersion over S. Let n and n′ be the relative dimensions of Y and of Y′ over S and r and r′ be the codimensions of the regular immersions X → Y and of X → Y′ respectively. Then we have n − r = n′ − r′.

Proof. 1. Since the assertion is local, we may assume there is an exact regular closed immersion X → Y′ into a log smooth scheme Y′ over S. By the same argument as in the proof of Lemma 3.4, we may assume that there exist a commutative diagram

(X, P) ⟶ (Y, P)
↓  ↓
(Y′, P) ⟶ (S, N)

of charted log schemes. We define an fs-monoid (P +_{N} P)^{∼} ⊂ P^{gp} ⊕_{N^{gp}} P^{gp} as above and put Y′′ = (Y ×_{S}^{log} Y′) ⊗_{ℤ[P +P]}^{log} ℤ[(P +_{N} P)^{∼}]. Then the projections Y′′ → Y and Y′′ → Y′ are log smooth and strict and hence are smooth. Since the immersion X → Y′ is a regular immersion, the immersion X → Y′′ is a regular immersion. Since the map Y′′ → Y is also smooth, the immersion X → Y is also a regular immersion by EGA IV Proposition (19.1.5)(iv)b)⇒a) applied to the immersions X ×_{Y}^{log} Y′′ → Y′′ and X → X ×_{Y}^{log} Y′′ and by loc.cit (ii). Hence the assertion follows.

2. In the notation above, the relative dimensions of Y′′ over Y and Y′ are n′ and n respectively. Hence the assertion follows. □ If X → Y is an exact regular immersion of codimension r, and Y is log smooth over S of relative dimension n, we say that X → S is of relative dimension n − r.

Lemma 3.9 Let X and S be log regular schemes and f : X → S be a morphism of finite type. Then f : X → S is log locally of complete intersection.

Proof. Since the assertion is local, we may assume there is a morphism (X, P) → (S, N) of charted log schemes as in Lemma 3.4. The map S′ = S ⊗_{ℤ[N]}^{log} ℤ[P] → S is log smooth and the map X → S is factorized as X → S′ → S where X → S′ is strict. Hence by replacing S by S′, we may assume X → S is strict. Further replacing S by a smooth scheme over S, we may assume X → S is an exact immersion. It is sufficient to show that the immersion X → S is a regular immersion. Since the question is local, we may assume S = Spec A and X = Spec B are local. We put P = M̄_{S,s} and take a chart α : P = M̄_{S,s} → A. We put Ā = A/α(P − {1}) and B̄ = B ⊗_{A} Ā. Since Ā → B̄ is a surjection of regular local rings, the kernel is generated by a regular sequence (t̄_{1}, …, t̄_{r}) of Ā. We take a lifting (t_{1}, …, t_{r}) in the maximal ideal 𝔪_{A}. We show that A_{i} = A/(t_{1}, …, t_{i}) is log regular of dimension dim A − i and that (t_{1}, …, t_{i}) is a regular sequence. By induction on i = 1, …, r, it is sufficient to show the case i = 1. Since t_{1} ≠ 0 and A is normal, we have dim A_{1} = dim A − 1. On the other hand, we have dim Ā_{1} + rank P^{gp} = dim Ā − 1 + rank P^{gp}. Hence, we have dim A_{1} = dim Ā_{1} + rank P^{gp} and A_{1} is log regular. Thus by induction, A_{r} is log regular of dimension dim A − r and (t_{1}, …, t_{r}) is a regular sequence. Since dim B = dim B̄ + rank P^{gp} = dim Ā − r + rank P^{gp} = dim A_{r} and A_{r} is normal, the surjection A_{r} → B is an isomorphism. Hence the immersion X → S is a regular immersion of codimension r. □ Let f : X → S be a map of log schemes such that the map of underlying schemes is locally of finite presentation and x ∈ X. We put s = f(x), X_{s} = X ⊗_{κ(s)} κ(x) and define

dim_{x}^{log} f^{−1}(f(x)) = = dim O_{X_{s},x}/(α(M_{X,x} − O_{X,x}^{×})) + tr.deg κ(x)/κ(s) + rank M̄_{X,x}^{gp}/M̄_{S,s}^{gp}.

Lemma 3.10 Let f : X → S be a morphism of log schemes such that the map of underlying schemes is of finite presentation.

1. Let (X, P) → (S, N) be a morphism of charted log schemes and let x ∈ X. Regard x as a log scheme with the log structure defined by the chart P. We put X′_{x} = (X ×_{S} x) ⊗_{ℤ[P +P]} ℤ[(P +_{N} P)^{∼}] and let x → X′_{x} be the section defined by x → X and the map (P +_{N} P)^{∼} → P → κ(x). Then, we have an equality

dim_{x}^{log} f^{−1}(f(x)) = dim O_{X′_{x},x}.

2. If X → S is log flat, the function dim_{x}^{log} f^{−1}(f(x)) is a locally constant function of x ∈ X.

3. Assume X → S is log locally of complete intersection of relative dimension d. If we have an equality dim_{x}^{log} f^{−1}(f(x)) = d for all x ∈ X, the map X → S is log flat and hence log syntomic.

Proof. 1. By Lemma 3.5.3, X′_{x} is the closed subscheme of (X_{s} ⊗_{κ(s)} κ(x)) ⊗_{ℤ} ℤ[P^{′gp}/N^{gp}] defined by the ideal I generated by (α(m) ⊗ 1) − (1 ⊗ α_{x}(m)) · (m) for m ∈ P. The ideal I is generated by α(m) ⊗ 1 for m ∈ P \Ker(P → M̄_{X,x}) and (m) − (1 ⊗ α_{x}(m))^{−1}(α(m) ⊗ 1) for m ∈ Ker(P → M̄_{X,x}). Hence X′_{x} is the closed subscheme of (X_{s} ⊗_{κ(s)} κ(x)) ⊗_{ℤ} ℤ[M̄_{X,x}^{gp}/M̄_{S,s}^{gp}] defined by the ideal J generated by α(m)⊗1 for m ∈ P \Ker(P → M̄_{X,x}). Thus the assertion follows.

2. Let S′ = S ⊗_{ℤ[N]}^{log} ℤ[P], X′ = X ⊗_{ℤ[P]}^{log} ℤ[(P +_{N} P)^{∼}] and f′ : X′ → S′

be the map. Since the map X′ → X ×_{S}^{log} S′ is log étale, and the composition X′ → S′ is strict, the underlying map X′ → S′ is flat. Hence the function dim_{x′} f^{′−1}(f′(x′)) = dim O_{X′_{f′(x′)},x′} is locally constant on x′ ∈ X′.

The function dim_{x}^{log} f^{−1}(f(x)) is the pull-back of the locally constant function dim_{x′} f^{′−1}(f′(x′)) by the section X → X′ induced by the map (P +_{N} P)^{∼} → P. Thus the assertion is proved.

3. Since the question is local, we may further assume that there is an exact regular immersion X → Y to a log scheme Y log smooth over S. Let n be the relative dimension of Y over S and r = n − d be the codimension of the regular immersion X → Y. We put Y′ = Y ⊗_{ℤ[P′]}^{log} ℤ[(P′ +_{N} P′)^{∼}]. Then we

have X′ = X ×_{Y}^{log} Y′. Since X′ → X is faithfully flat by Lemma 3.5.2, it is sufficient to show that the map X′ → S′ is flat. Since Y′ → Y is flat, the immersion X′ → Y′ is regular of codimension r. The map Y′ → S′ is smooth of relative dimension n. Hence the strict map X′ → S′ is locally of complete intersection of relative dimension d. By the assumption and the computation above, each fiber of X′ → S′ has dimension d. Hence by EGA IV Théorème (11.3.8) d)⇒a), X′ → S′ is flat.

Corollary 3.11 Let f : X → S be a finite morphism of log regular schemes. Assume dim X = dim S and f^{∗}M̄_{S}^{gp} ⊗ ℚ → M̄_{X}^{gp} ⊗ ℚ is surjective. Then X is log flat and hence log syntomic over S.

Proof. By Lemma 3.9, the map f : X → S is log locally of complete intersection. Further, by the assumption that X → S is finite and dim X = dim S, the map X → S has relative dimension 0. Since dim_{x}^{log} f^{−1}(f(x)) = 0 for all x ∈ X, it is sufficient to apply Lemma 3.10 □ For a ring A, we call a Zariski fs-log structure on X = Spec A a log structure on A. We call a ring with a log structure a log ring. If A is a local ring, a log structure on A is defined by a chart P → A. We say that a map A → B of log rings is a surjection if the underlying ring homomorphism A → B is surjective and the map f^{∗}M_{Y} → M_{X} is surjective where f : X = Spec B → Y = Spec A denotes the corresponding map of log schemes and M_{X} and M_{Y} denote the log structures. We say that a surjection A → B of log rings is an exact surjection if the log structure M_{X} is the pull-back log structure of M_{Y}. We say that a surjection A → B is regular if the immersion Spec B → Spec A of the underlying schemes is a regular immersion. For a map A → B of log rings, let Ω_{B/A}(log / log) denote the module of logarithmic differential forms, denoted by ω_{B/A} in [6]. If A and B are local and N and P denote the stalks of the log structures at the closed points, we have

Ω_{B/A}(log / log) = (Ω_{B/A} ⊕ (B ⊗_{ℤ} (P^{gp}/N^{gp}))) /(dm − m ⊗ m : m ∈ P).

We study formally log smooth maps of complete local Noetherian log rings.

Definition 3.12 (cf. [11] Definition 4.4) Let A and B be complete local Noetherian rings with log structures and f : A → B a morphism of log rings such that the underlying ring homomorphism is local.

1. We say f : A → B is formally log smooth (resp. formally log étale) if, for a nilpotent exact surjection R → R′ of discrete log A-algebras and a continuous homomorphism B → R′ of log A-algebras, there exists a (resp. a unique) continuous homomorphism B → R of log A-algebras lifting B → R′.

2. We put Ω̂_{B/A}(log / log) = \underleftarrow{lim}_{n} Ω_{(B/𝔪_{B}^{n})/A}(log / log).

Lemma 3.13 Let A and B be complete local Noetherian rings with log structures and f : A → B a morphism of log rings such that the underlying ring homomorphism is local. Assume that the residue field of B is finitely generated over the residue field of A. Then, the following conditions are equivalent.

(1) B is formally log smooth over A.

(2) There exist a log smooth scheme X over A, a point x of X over the closed point of Spec A and an étale local homomorphism B → Ô_{X,x} over A.

Proof. It is clear that (2) implies (1). The implication (1)⇒(2) is proved similarly as in the proof of [6] (3.5.1)⇒(3.5.2). □

Corollary 3.14 Let A → B be as in Lemma and assume A → B is log smooth.

1. The B-module Ω̂_{B/A}(log / log) is free of finite rank.

2. If A is log regular (cf. [8] Definition (2.1)), then B is also log regular.

Proof. 1. It follows from Lemma 3.13 (1)⇒(2) and [6] Proposition (3.10).

2. It follows from Lemma 3.13 (1)⇒(2) and [8] Theorem (8.2). □

4 Tubular neighborhoods for finite flat and log flat log algebras

In the rest of the paper, the integer ring O_{K} is considered as a log ring with its canonical log structure defined by the chart ℕ → O_{K} sending 1 ∈ ℕ to a prime element. The letter A denotes a finite flat and log flat log O_{K}-algebra such that the log structure on A_{K} is trivial. For a finite étale algebra L over K, its integer ring O_{L} is considered as a log O_{K}-algebra with its canonical log structure defined by taking the product of the canononical log structures on its factors. The log O_{K}-algebra O_{L} is log flat by Corollary 3.11. Hence it is finite flat and log flat and the log structure on L is trivial.

4.1 Log embeddings

Definition 4.1 1. Let A be a finite flat and log flat log O_{K}-algebra such that the log structure on A_{K} is trivial. Let A be a log O_{K}-algebra formally of finite type and formally log smooth over O_{K}. We say that an exact surjection A → A of log O_{K}-algebras is a log embedding if it induces an isomorphism A/𝔪_{A} → A/𝔪_{A}.

2. We define Emb_{O_{K}}^{log} to be the category whose objects and morphisms are as follows. An object of Emb_{O_{K}}^{log} is a triple (A → A) where:

• A is a finite flat and log flat log O_{K}-algebra such that the log structure on A_{K} is trivial.

• A is a log O_{K}-algebra formally of finite type and formally log smooth over O_{K}.

• A → A is a log embedding.

A morphism (f, f) : (A → A) → (B → B) is a pair of homomorphisms f : A → B and f : A → B of log O_{K}-algebras such that the diagram

A ⟶ A
f↓  ↓^{f}
B ⟶ B

of log O_{K}-algebra homomorphisms is commutative.

3. For a finite flat and log flat log O_{K}-algebra A such that the log structure on A_{K} is trivial, let Emb_{O_{K}}^{log} (A) be the subcategory of Emb_{O_{K}}^{log} whose objects are of the form (A → A) and morphisms are of the form (id_{A}, f).

4. We say that a morphism (f, f) : (A → A) → (B → B) of Emb_{O_{K}} is finite flat and log flat if A → B is finite flat and log flat and the map B ⊗_{A}^{log} A → B is an isomorphism of log O_{K}-algebras.

5. We say that a log embedding A → A is strict if the maps O_{K} → A and O_{K} → A of log rings are strict.

For a complete semi-local Noetherian log O_{K}-algebra R such that R/𝔪_{R} is finite over F, we put Ω̂_{R/O_{K}} (log / log) = \underleftarrow{lim}_{n} Ω_{(R/𝔪_{R}^{n})/O_{K}} (log / log). If (A → A) is a log embedding, the A-module Ω̂_{A/O_{K}} (log / log) is locally free of finite rank. If (A → A) is a strict object of Emb_{O_{K}}^{log} , by forgetting the log structures, we obtain an object (A → A)^{∘} of Emb_{O_{K}}. For an object (A → A) of Emb_{O_{K}}, by putting the pull-back log structures on A and A from that on O_{K}, we obtain an object (A → A)^{log} of Emb_{O_{K}}^{log} . Thus, we obtain an equivalence of categories between Emb_{O_{K}} and the full subcategory of Emb_{O_{K}}^{log} consisting of strict objects.

Lemma 4.2 Let A be a finite flat and log flat log O_{K}-algebra such that the log structure on A_{K} is trivial. We put X = Spec A and S = Spec O_{K}.

1. For a closed point x of X = Spec A, the stalk M̄_{X,x} of the sheaf M̄_{X} = M_{X}/O_{X}^{×} is isomorphic to ℕ and the map M̄_{S,s} = ℕ → M̄_{X,x} = ℕ is the multiplication by an integer e_{x} ≥ 1.

2. Let (A → A) be a log embedding. Then, the ring A is regular and the reduced closed fiber (A ⊗_{O_{K}} F)_{red} is a regular divisor. The log ring A is log regular and the log structure is defined by the reduced closed fiber (A⊗_{O_{K}} F)_{red}.

3. A log embedding (A → A) is strict if and only if the map O_{K} → A is strict.

Proof. 1. Clear from Lemma 3.10.1.

2. We may assume A is local and the log structure is defined by a chart ℕ → A. Since A is formally log smooth over O_{K}, it is log regular by Corollary 3.14.2. Since the stalks of M̄ are either ℕ or 0, the ring A is regular and the image t ∈ A of 1 ∈ ℕ defines a regular divisor. Since π/t^{e_{x}} ∈ A^{×}, the assertion follows.

3. We may assume A is local. Assume the map O_{K} → A is strict. Then, in the notation of the proof of 2, we have e_{x} = 1 and π/t ∈ A^{×}. Hence the map O_{K} → A is strict. The only if part is obvious. □ To prove the logarithmic version Lemma 4.5 below of Lemma 1.2, we make another definition.

Definition 4.3 1. Let A be a finite flat and log flat log O_{K}-algebra such that the log structure on A_{K} is trivial. Let A be a log O_{K}-algebra formally of finite type, formally smooth and formally log smooth over O_{K}. We say that a surjection A → A of log O_{K}-algebra is a log pre-embedding if it induces an isomorphism A/𝔪_{A} → A/𝔪_{A} of underlying F -algebras.

2. We define preEmb_{O_{K}}^{log} to be the category whose objects and morphisms are as follows. An object of Emb_{O_{K}}^{log} is a triple (A → A) where:

• A is a finite flat and log flat log O_{K}-algebra such that the log structure on A_{K} is trivial.

• A is a log O_{K}-algebra formally of finite type, formally smooth and formally log smooth over O_{K}.

• A → A is a log pre-embedding.

A morphism (f, f) : (A → A) → (B → B) is a pair of log O_{K}-homomorphism f : A → B and f : A → B such that the diagram

A ⟶ A
f↓  ↓^{f}
B ⟶ B

is commutative.

3. For a finite flat and log flat log O_{K}-algebra A such that the log structure on A_{K} is trivial, let preEmb_{O_{K}}^{log} (A) be the subcategory of preEmb_{O_{K}}^{log} whose objects are of the form (A → A) and morphisms are of the form (id_{A}, f).

A log pre-embedding (A → A) is an embedding together with log structures on A and on A such that the log ring A is formally log smooth, that the log ring A is log flat and the log structure on A_{K} is trivial and that the map A → A is a surjection of log O_{K}-algebras. Hence, by forgetting the log structures, we obtain a functor preEmb_{O_{K}}^{log} → Emb_{O_{K}}. We also define a functor preEmb_{O_{K}}^{log} → Emb_{O_{K}}^{log} . For an object (A → A) of preEmb_{O_{K}}^{log} , we attach a log embedding (A^{∼} → A) as follows. First, we consider the case where A is local. Assume the log structure of A is defined by a chart P → A. Let P → ℕ be the map P → M̄_{X,x} = ℕ where x is the closed point of X = Spec A and we identify M̄_{X,x} = ℕ by the unique isomorphism. Let P^{∼} be the inverse image of ℕ by the induced map P^{gp} → M̄_{X,x}^{gp} = ℤ. The map P → A → A is extended uniquely to a map P^{∼} → A. We define A^{∼} to be the formal completion of the surjection A ⊗_{ℤ[P]} ℤ[P^{∼}] → A induced by P^{∼} → A. Let A^{∼} → A be the canonical map. The log ring A^{∼} and the homomorphism A^{∼} → A are independent of the choice of the chart P → A upto a unique isomorphism. In general, we define A^{∼} and A^{∼} → A by taking the product. By the construction, the canonical map A → A^{∼} is formally log étale.

Lemma 4.4 Let A be a finite flat and log flat log O_{K}-algebra such that the log structure on A_{K} is trivial.

1. The category preEmb_{O_{K}}^{log} (A) is non-empty.

2. Let (A → A) be an object of preEmb_{O_{K}}^{log} and define A^{∼} and A^{∼} → A as above. Then (A^{∼} → A) is an object of Emb_{O_{K}}^{log} .

Proof. 1. We may assume A is local. Take a system of generators t_{1}, …, t_{n} of A over O_{K} and a chart ℕ → A. Let t_{0} ∈ A be the image of 1 ∈ ℕ. We define a surjection O_{K}[T_{0}, …, T_{n}] → A by sending T_{i} to t_{i} and a log structure on O_{K}[T_{0}, …, T_{n}] by the chart ℕ^{2} → O_{K}[T_{0}, …, T_{n}] sending (1, 0) and (0, 1) ∈ ℕ^{2} to T_{0} and π. Then its formal completion A → A is a log pre-embedding.

2. By the definition, the O_{K}-algebra A^{∼} is formally of finite type over O_{K} and the surjection A^{∼} → A is exact. Since the map A → A^{∼} is formally log étale, the log O_{K}-algebra A^{∼} is formally log smooth over O_{K}. Hence the assertion follows. □ By Lemma 4.4.2, we obtain a functor preEmb_{O_{K}}^{log} → Emb_{O_{K}}^{log} .

Lemma 4.5 1. For a finite flat and log flat log O_{K}-algebra A such that the log structure on A_{K} is trivial, the category Emb_{O_{K}}^{log} (A) is non-empty.

2. For a morphism f : A → B of finite flat and log flat log O_{K}-algebras such that the log structures on A_{K} and B_{K} are trivial and for objects (A → A) and (B → B) of Emb_{O_{K}}^{log} , there exists a morphism (f, f) : (A → A) → (B → B) lifting f.

3. For a morphism f : A → B of finite flat and log flat log O_{K}-algebras such that the log structures on A_{K} and B_{K} are trivial, the following conditions are equivalent.

(1) The map f : A → B is log syntomic.

(2) Their exists a finite flat and log flat morphism (f, f) : (A → A) → (B → B) of log embeddings.

Proof. 1. Clear from Lemma 4.4.

2. Since A is formally log smooth, B = \underleftarrow{lim}_{n} B/I^{n} where I = Ker(B → B) and the surjection B/I^{n} → B is exact, the assertion follows.

3. (1)⇒(2). We may assume A and B are local. We take log embeddings A → A and B → B. We define a log embedding B′ → B by applying an argument similar to the proof of Lemma 4.4.2 to \underleftarrow{lim}_{n}(A/𝔪_{A}^{n} ⊗_{O_{K}}^{log} B/𝔪_{B}^{n})^{∧} → B. Replacing B → B by B′ → B, we may assume that there is a map (A → A) → (B → B) such that A → B is formally log smooth. Since A → B is log syntomic, the exact surjection B ⊗_{A}^{log} A → B is regular by Lemma 3.8.1 and the kernel is generated by a regular sequence (t_{1}, …, t_{n}). Take a lifting (t̃_{1}, … t̃_{n}) in B and define a map A[[T_{1}, …, T_{n}]] → B by sending T_{i} to t_{i}. We consider A[[T_{1}, …, T_{n}]] as a log ring with the pull-back log structure by the map A → A[[T_{1}, …, T_{n}]]. Then the composition A[[T_{1}, …, T_{n}]] → A → A sending T_{i} to 0 defines a log embedding. Replacing A by A[[T_{1}, …, T_{n}]], we obtain a map (A → A) → (B → B) such that the map B ⊗_{A}^{log} A → B is an isomorphism and that dim A = dim B. By Nakayama’s lemma, the map A → B is finite. Since A and B are regular, the map A → B is flat by EGA Chap 0_{IV} Corollaire (17.3.5) (ii). Further by Corollary 3.11, it is log syntomic. (2)⇒(1). Since A and B are log regular and have the same dimension, B is log syntomic over A by Corollary 3.11. Hence B is also log syntomic over A by Lemma 3.7.2. □ The base change of a log embedding by an extension of complete discrete valuation fields is defined as follows.

Lemma 4.6 Let K′ be a complete discrete valuation field and K → K′ be a morphism of fields inducing a local homomorphism O_{K} → O_{K′}. Let (A → A) be an object of Emb_{O_{K}}^{log} . We define A ˆ⊗_{O_{K}}^{log} O_{K′} to be the projec-

tive limit \underleftarrow{lim}_{n}(A/𝔪_{A}^{n} ⊗_{O_{K}}^{log} O_{K′}). Then the log O_{K′}-algebra A ˆ⊗_{O_{K}}^{log} O_{K′} is formally of finite type and formally log smooth over O_{K′}. The natural surjection A ˆ⊗_{O_{K}}^{log} O_{K′} → A ˆ⊗_{O_{K}}^{log} O_{K′} defines an object (A ˆ⊗_{O_{K}}^{log} O_{K′} → A ⊗_{O_{K}}^{log} O_{K′}) of Emb_{O_{K′}}^{log} .

Proof. Since A ˆ⊗_{O_{K}}^{log} O_{K′} is finite over A ˆ⊗_{O_{K}} O_{K′}, it is formally of finite type over O_{K′}. The formal log smoothness is clear from the definition. The rest is clear. □. Thus we obtain a functor ˆ⊗_{O_{K}}^{log} O_{K′} : Emb_{O_{K}}^{log} → Emb_{O_{K′}}^{log} . If K′′ is an extension of complete discrete valuation fields of K′, the composition Emb_{O_{K}}^{log} →

Emb_{O_{K′}}^{log} → Emb_{O_{K′′}}^{log} is the same as ˆ⊗_{O_{K}}^{log} O_{K′′} : Emb_{O_{K}}^{log} → Emb_{O_{K′′}}^{log} . If K′ is a

finite extension, we have A ⊗_{O_{K}}^{log} O_{K′} = A ˆ⊗_{O_{K}}^{log} O_{K′}. If (A → A) is strict, we have (A → A) ⊗_{O_{K}}^{log} O_{K′} = ((A → A)^{∘} ⊗_{O_{K}} O_{K′})^{log}. Similarly as for \underrightarrow{lim}_{K′/K}(Aff/F′) defined in Section 1.3, we define a category

\underrightarrow{lim}_{K′/K} Emb_{O_{K′}}. We define a functor Emb_{O_{K}}^{log} → \underrightarrow{lim}_{K′/K} Emb_{O_{K′}} as follows.

Lemma 4.7 Let A be a finite flat and log flat log O_{K}-algebra. Let e = e_{A/O_{K}} denote the least common multiple of e_{x} in Lemma 4.2.1 for the closed points x in X = Spec A. Let K′ be a finite separable extension of K of ramification index e_{K′/K}. If e_{K′/K} is divisible by e_{A/O_{K}}, then the log tensor product A_{O_{K′}} = A ⊗_{O_{K}}^{log} O_{K′} is strict over O_{K′}.

Proof. We may assume A is local. We put P = N′ = ℕ × ℤ and define maps ℕ → P and ℕ → N′ by sending 1 ∈ ℕ to (e_{A/O_{K}} , 1) and to (e_{O_{K′} /O_{K}} , 1) respectively. There exist morphisms of charts (ℕ → O_{K}) → (P → A) and (ℕ → O_{K}) → (N′ → O_{K′}). Since e_{A/O_{K}} divides e_{O_{K′} /O_{K}}, the saturation P +_{ℕ}^{sat} N′ is isomorphic to ℕ × (ℤ/e_{A/O_{K}} ℤ) × ℤ^{2} and the composition ℕ ⊂ N′ → P +_{ℕ}^{sat} N′ → ℕ is the identity. Hence A ⊗_{O_{K}}^{log} O_{K′} is strict over O_{K′}. □ Let (A → A) be an object of Emb_{O_{K}}^{log} and define e = e_{A/K} as in Lemma 4.7. Let C_{e} be the full subcategory of the category (Ext/K) of finite separable extensions of K consisting of the extensions with ramification index divisible by e. If K′ is a finite separable extension in C_{e}, then by Lemmas 4.7 and 4.2.3, the base change (A ⊗_{O_{K}}^{log} O_{K′} → A ⊗_{O_{K}}^{log} O_{K′}) is strict and defines an object (A ⊗_{O_{K}}^{log} O_{K′} → A ⊗_{O_{K}}^{log} O_{K′})^{∘} of Emb_{O_{K′}}. We consider a system consisting of (A ⊗_{O_{K}}^{log} O_{K′} → A ⊗_{O_{K}}^{log} O_{K′})^{∘} for extensions K′ in C_{e} and isomorphisms (A ⊗_{O_{K}}^{log} O_{K′} → A ⊗_{O_{K}}^{log} O_{K′})^{∘} ⊗_{O_{K′}} O_{K′′} → (A ⊗_{O_{K}}^{log} O_{K′′} → A ⊗_{O_{K}}^{log} O_{K′′})^{∘} for K-morphisms K′ → K′′ of extensions in C_{e}. Then it defines an object of \underrightarrow{lim}_{K′/K} Emb_{O_{K′}}. Thus we obtain a functor Emb_{O_{K′}}^{log} → \underrightarrow{lim}_{K′/K} Emb_{O_{K′}}.

4.2 Tubular neighborhoods for log embeddings

For a rational number j > 0, a functor X^{j} : \underrightarrow{lim}_{K′/K} Emb_{O_{K′}} →

\underrightarrow{lim}_{K′/K}(smooth Affinoid/K′) is defined as the limit of the functors X^{je_{K′/K}} : Emb_{O_{K′}} → (smooth Affinoid/K′) defined in Section 1.2. We define a functor \underrightarrow{lim}_{K′/K}(smooth Affinoid/K′) → \underrightarrow{lim}_{K′/K}(Aff/F′) as follows. Let (X_{K′})_{K′∈obC} be an object of (smooth Affinoid/K′). Then the extensions K′ in C such that the stable normalized integral model A_{O_{K′}} is defined over K′ form a cofinal full subcategory C′ by Theorem 1.10. For an extension K′ in C′, let X̄_{F′} denote the affine scheme A_{O_{K′}} ⊗_{O_{K′}} F′ over the residue field F′ of K′. By sending (X_{K′})_{K′∈obC} to (X̄_{F′})_{K′∈obC′}, we obtain a functor \underrightarrow{lim}_{K′/K}(smooth Affinoid/K′) → \underrightarrow{lim}_{K′/K}(Aff/F′). Thus, we have a sequence of functors

Emb_{O_{K}}^{log} ⟶ \underrightarrow{lim}_{K′/K} Emb_{O_{K}} ⟶ \underrightarrow{lim}_{K′/K}(smooth Affinoid/K′)
⟶ \underrightarrow{lim}_{K′/K}(Aff/F′) ⟶ G_{K}-(Aff/F̄).

The compositions X_{log}^{j} : Emb_{O_{K}}^{log} → \underrightarrow{lim}_{K′/K}(smooth Affinoid/K′) and X̄_{log}^{j} :

Emb_{O_{K}}^{log} → G_{K}-(Aff/F̄) are more concretely described as follows. For an object (A → A) of Emb_{O_{K}}^{log} and a finite separable extension K′ such that the ramification index e′ = e_{K′/K} is divisible by the integer e_{A/O_{K}} in Lemma 4.7, the base change (A ˆ⊗_{O_{K}}^{log} O_{K′} → A ⊗_{O_{K}}^{log} O_{K′}) is strict and we define an affinoid variety X_{log}^{j}(A → A)_{K′} over K′ by

X_{log}^{j}(A → A)_{K′} = X^{e′j}((A ˆ⊗_{O_{K}}^{log} O_{K′} → A ⊗_{O_{K}}^{log} O_{K′})^{∘}).

The composite functors X_{log}^{j} : Emb_{O_{K}}^{log} → \underrightarrow{lim}_{K′/K}(smooth Affinoid/K′) sends

an object (A → A) of Emb_{O_{K}}^{log} to the system X_{log}^{j}(A → A) = (X_{log}^{j}(A → A)_{K′})_{K′} where K′ runs over finite separable extensions such that the ramification index e′ = e_{K′/K} is divisible by the integer e_{A/O_{K}}. By Lemma 1.8 and the universality of ⊗^{log}, we obtain a cartesian diagram

X_{log}^{j}(A → A)(K̄)  ⟶ Hom_{cont.logO_{K}-alg}(A, OK̄)
↓  ↓
Hom_{logO_{K}-alg}(A, OK̄/𝔪^{j}) ⟶ Hom_{cont.logO_{K}-alg}(A, OK̄/𝔪^{j}).

Here OK̄/𝔪^{j} denotes the limit O_{K′}/𝔪^{je_{K′/K}} of fs-log rings where K′ runs finite extensions in K̄ such that je_{K′/K} is an integer. Similarly as in Section 1.2, the surjection X_{log}^{j}(A → A)(K̄) → π_{0}(X_{log}^{j}(A → A))K̄ induces a surjection

(4.2.1)  Hom_{cont.logO_{K}-alg}(A, OK̄/𝔪^{j}) ⟶ π_{0}(X_{log}^{j}(A → A))K̄.

The map A → A also induces a map

(4.2.2)  Hom_{logO_{K}-alg}(A, OK̄) ⟶ X_{log}^{j}(A → A)(K̄).

Similarly as Lemma 1.9.4, if (f, f) : (A → O_{K}) → (B → B) is a finite flat and log flat morphism of Emb_{O_{K}}^{log} , the map (4.2.2) induces a surjection

(4.2.3)  Hom_{log O_{K}-alg}(B, OK̄) ⟶ π_{0}(X_{log}^{j}(B → B)K̄).

Let (Finite Flat and Log Flat/O_{K}) denote the category of finite flat and log flat log O_{K}-algebras A such that the log structure on A⊗_{O_{K}} K is trivial. We define functors Ψ_{log} and Ψ_{log}^{j} : (Finite Flat and Log Flat/O_{K}) → G_{K}-(Finite Sets) for a rational number j > 0 as in Section 1.2 by sending a finite flat and log flat log O_{K}-algebra A such that the log structure on A ⊗_{O_{K}} K to the set Ψ_{log}(A) = Hom_{O_{K}}^{log} (A, OK̄) and to the set

Ψ_{log}^{j}(A) = \underleftarrow{lim}_{(A→A)∈Emb_{O_{K}}^{log} (A)} π_{0}(X_{log}^{j}(A → A)K̄)

respectively. As in Section 1.2, the surjection (4.2.1) implies that the projective system in the right hand side is constant. Further it induces a map Ψ_{log} → Ψ_{log}^{j} of functors. Similarly, for an object (A → A) of Emb_{O_{K}}^{log} and a finite separable extension K′ such that the ramification index e′ = e_{K′/K} is divisible by the integer e_{A/O_{K}} and that a stable normalized integral model A_{O_{K′}}^{j} of X_{log}^{j}(A → A)_{K′} is defined over K′, an affine scheme X̄_{log}^{j}(A → A)_{K′} over the residue field F′ of K′ is defined as the closed fiber Spec(A_{O_{K′}}^{j} ⊗_{O_{K′}} F′). The system X̄_{log}^{j}(A → A) = (X̄_{log}^{j}(A → A)_{K′})_{K′} defines an object of \underrightarrow{lim}_{K′/K}(Aff/F′).

By identifying the category \underrightarrow{lim}_{K′/K}(Aff/F′) with G_{K}-(Aff/F̄), we obtain the

composite functor X̄_{log}^{j} : Emb_{O_{K}}^{log} → G_{K}-(Aff/F̄). For j > 0, the functor Ψ_{log}^{j} : (Finite Flat and Log Flat/O_{K}) → G_{K}-(Finite Sets) is induced by the composition of the functors

Emb_{O_{K}}^{log} ⟶^{X̄_{log}^{j}} G_{K}-(Aff/F̄) ⟶^{π_{0}} G_{K}-(Finite Sets).

We also have a functor C̄_{log}^{j} : Emb_{O_{K}}^{log} → G_{K}-(Aff/F̄) and a map of functors X̄_{log}^{j} → C̄_{log}^{j}. Let (A → A) be an object of Emb_{O_{K}}^{log} and j > 0 be a rational number. Let K′ be a finite separable extension of K such that the ramification index e′ = e_{K′/K} is divisible by e_{A/O_{K}} and by the denominator of j and that ((A ⊗_{O_{K}}^{log} O′_{K}) ⊗_{O_{K′}} F′)_{red} is étale over F′. Let I be the kernel of A ⊗_{O_{K}}^{log} O′_{K} → A ⊗_{O_{K}}^{log} O′_{K} and we put

C̄_{log}^{j}(A → A)_{K′} = Spec (⨁_{n=0}^{∞} I^{n}/I^{n+1} ⊗_{O_{K′}} 𝔪_{K′}^{e′jn}/m_{K′}^{e′jn+1})_{red.}

Then the system (C̄_{log}^{j}(A → A)_{K′})_{K′} defines an object \underrightarrow{lim}_{K′/K}(Aff/F′) and

hence an object C̄_{log}^{j}(A → A) of G_{K}-(Aff/F̄). It is a scheme over ((A ⊗_{O_{K}}^{log} O_{K′})⊗_{O_{K′}} F̄)_{red} for K′ as above. In the following, we put A_{log F̄,red} = ((A⊗_{O_{K}}^{log} O_{K′}) ⊗_{O_{K′}} F̄)_{red} = (A ⊗_{O_{K}}^{log} F̄)_{red}. In the right hand side, F̄ is regarded as the limit of an fs-log ring with the chart ℚ_{≥0} → F̄ sending positive rational numbers to 0. We study relations between X^{j} and X_{log}^{j}. Let (A → A) be an object of Emb_{O_{K}} and (B → B) be an object of Emb_{O_{K}}^{log} . Let (A → A)^{log} be the object of Emb_{O_{K}}^{log} defined by the pull-back log structures. An O_{K}-algebra homomorphism A → B can be lifted to a morphism (A → A)^{log} → (B → B) of Emb_{O_{K}}^{log} by Lemma 4.2. For a rational number j > 0, a morphism (A → A)^{log} → (B → B) of Emb_{O_{K}}^{log} induces a morphism X_{log}^{j}(B → B) → X_{log}^{j}((A → A)^{log}) = X^{j}(A → A) of affinoid varieties. Let (A → A) be a log pre-embedding. We have an embedding (A → A)^{∘}, a log embedding (A^{∼} → A) and a canonical map ((A → A)^{∘})^{log} → (A^{∼} → A) of log embeddings by the construction in Lemma 4.4.2. For a rational number j > 0, we have affinoid varieties X^{j}((A → A)^{∘}) and X_{log}^{j}(A^{∼} → A) and a map of affinoid varieties X_{log}^{j}(A^{∼} → A) → X^{j}((A → A)^{∘}).

Lemma 4.8 Let (A → A) be an object of preEmb_{O_{K}}^{log} and j > 0 be a positive integers.

1. The canonical map X_{log}^{j}(A^{∼} → A) → X^{j}((A → A)^{∘}) is an open immersion and X_{log}^{j}(A^{∼} → A) is identified with a rational subdomain.

2. Assume A is local and put S = Spec O_{K}, X = Spec A and X = Spec A and let s and x be the closed points of S and of X. We put P = M̄_{X,x} and identify M̄_{X,x} and M̄_{S,s} with ℕ. Let e = e_{A/O_{K}} be the image of 1 ∈ M̄_{S,s} = ℕ by the composition M̄_{S,s} → M̄_{X,x} → M̄_{X,x} = ℕ as in Lemma 4.7. Let m_{1}, …, m_{n} be a system of generators of the monoid P and e_{1}, …, e_{n} be their images by P → ℕ = M̄_{X,x}. Let j′ ≥ j + max_{i} e_{i}/e be a rational number strictly greater than 1. Then we have an open immersion X^{j′}((A → A)^{∘}) → X_{log}^{j}(A^{∼} → A) of rational subdomains X^{j}((A → A)^{∘}).

Proof. 1. We may assume A is local. We use the notation in 2. Let I be the kernel of the surjection A → A and J be the kernel of the surjection A^{∼} → A. By renumbering the indices if necessary, we may assume e_{1} = 1. We take a chart φ : P → A and put t_{i} = φ(m_{i}) ∈ A. We define a monoid P^{∼} as in Lemma 4.4.2 and φ̃ : P^{∼} → A^{∼} be the extension. The monoid P^{∼} is generated by P and (m_{i}m_{1}^{−e_{i}})^{±1}, i = 2, …, n. Hence the ring A^{∼} is the completion of the subring generated by φ̃(m_{i}m_{1}^{−e_{i}})^{±1} over A. For i = 2, …, n, take liftings u_{i} ∈ A^{×} of the image of φ̃(m_{i}m_{1}^{−e_{i}}) in A^{×}. Then, the ideal J is generated by the image of I and φ̃(m_{i}m_{1}^{−e_{i}}) − u_{i}, i = 2, …, n. Hence X_{log}^{j}(A^{∼} → A) is the rational subdomain X^{j}((A → A)^{∘}) defined by the conditions ord(t_{i}t_{1}^{−e_{i}} − u_{i}) ≥ j for i = 2, …, n.

2. Similarly as in the proof of Lemma 1.17, we have ord t_{1} = 1/e on X^{j′}((A → A)^{∘}) by the assumption j′ > 1. Since t_{i} − u_{i}t_{1}^{−e_{i}} ∈ I for i = 2, …, n, we have ord(t_{i} − u_{i}t_{1}^{e_{i}}) ≥ j′ ≥ j + e_{i}/e on X^{j′}((A → A)^{∘}). Hence the assertion follows. □

Corollary 4.9 Let (A → A) be a log pre-embedding constucted in the proof of Lemma 4.4.1. Then, for a rational number j > 0, we have open immersions

X^{j+1}((A → A)^{∘}) ⟶ X_{log}^{j}(A^{∼} → A) ⟶ X^{j}((A → A)^{∘})

of rational subdomains.

Proof. The log structure on A is defined by a chart ℕ^{2} → A and we have e_{1} = 1 and e_{2} = e_{L/K} for m_{1} = (1, 0) and m_{2} = (0, 1) in the notation of Lemma 4.8.2. Hence the assertion follows. □ The affinoid varieties X_{log}^{j}(A → A) and Y_{Z,P}^{j} defined in [1] Section 3.2 are related as follows. Let L be a finite separable extension of K and A = O_{L} be the integer ring. Let Z = (z_{i})_{i∈I} be a finite system of generators of O_{L} over O_{K} and P ⊂ I be a subset such that z_{i} is a prime element of O_{L} for some i ∈ P and z_{i} is not zero for any i ∈ P. We recall a description of Y_{Z,P}^{j} for a rational number j > 0. We put e_{i} = ord_{L}z_{i} and e = e_{L/K} and let π be a prime element of K. Let I_{Z} be the kernel of the surjection O_{K}[T_{i}; i ∈ I] → A sending T_{i} to z_{i} and (f_{1}, …, f_{m}) be a finite set of generators of I_{Z}. For i ∈ P and (i, j) ∈ P^{2}, we take polynomials g_{i}, h_{i,j} ∈ O_{K}[T_{i}; i ∈ I] such that the images in O_{L} are u_{i} = z_{i}^{e}/π^{e_{i}} and u_{i,j} = z_{j}^{e_{i}}/z_{i}^{e_{j}}. If z_{ι} is a prime element for ι ∈ P, then we have

Y_{Z,P}^{j} (K̄) = \{(x_{i})_{i∈I} ∈ OI ; K̄ |\begin{matrix}ordf_{l}(x_{i}) ≥ j & for 1 ≤ l ≤ m \\ ord(x_{ι}^{e}/π^{e_{ι}} − g_{ι}(x_{i})) ≥ j \\ ord(x_{k}^{e_{ι}}/x_{ι}^{e_{k}} − h_{k,ι}(x_{i})) ≥ j & for k ∈ P \end{matrix}\}

by [1] Lemma 3.9 (2). Furthermore, for (x_{i})_{i∈I} ∈ Y_{Z,P}^{j} (K̄), we have x_{i}/x_{ι}^{e_{i}} ∈

O× ; K̄ for i ∈ P.

We define a log structure on O_{K}[T_{i}, i ∈ I] by the chart M = ℕ × ℕ^{P} → O_{K}[T_{i}, i ∈ I] sending (1, 0) to π and (0, f_{i}) to T_{i} where f_{i} ∈ ℕ^{P} is the i-th standard basis. Let A be the formal completion of the surjection O_{K}[T_{i}, i ∈ I] → A sending T_{i} to z_{i}.

Lemma 4.10 Let the notation be as above. Then (A → A) is a log preembedding and the affinoid variety X_{log}^{j}(A^{∼} → A)K̄ defined by the log embedding (A^{∼} → A) is the same as Y_{Z,P}^{j} defined in [1] Section 3.2.

Proof. It is clear that (A → A) is a log pre-embedding. We describe the log O_{K}-algebra A^{∼}. As in Lemma 4.4.2, let P^{∼} ⊂ P^{gp} = ℤ × ℤ^{P} be the inverse image of ℕ by the map ℤ × ℤ^{P} → ℤ sending T_{0} = (1, 0) to e and the standard basis T_{i} of ℤ^{P} to e_{i} for i ∈ P. We consider a chart ℕ → O_{K} and a map of monoids ℕ → P^{∼} sending 1 ∈ ℕ to a prime element π ∈ O_{K} and to T_{0} ∈ P^{∼}. We put A_{I,P} = O_{K} ⊗_{ℤ[ℕ]} ℤ[P^{∼}][T_{i}, i ∈ I − P] and define a log structure by the chart P^{∼} → A_{I,P}. Then, A^{∼} is identified with the formal completion of the natural surjection A_{I,P} → A. Let K′ be a finite separable extension of K containing L as a subfield. We compute the log tensor product A_{I,P} ⊗_{O_{K}}^{log} O_{K′}. By choosing a numbering, we assume P = {1, …, r} ⊂ I = {1, …, m} and z_{r} is a prime element. Let T_{i}, i = 0, …, r be the standard basis of P = ℕ × ℕ^{P} and put U_{i} = T_{i}T_{r}^{−e_{i}} for i = 1, …, r − 1 and U_{0} = T_{0}T_{r}^{−e}. Then the monoid P^{∼} is generated by U_{i}^{±1}, i = 0, …, r − 1 and T_{r} and is isomorphic to ℤ^{r} × ℕ. Let N′ be the monoid ℕ × ℤ with the map ℕ → ℕ × ℤ sending 1 ∈ ℕ to (e′, 1). Let π′ be a prime element of K′ and e′ = e_{K′/K} be the ramification index and define a unit u′ of O_{K′} by π = u′π^{′e′}. We consider a chart N′ → O_{K′} sending U′ = (0, 1) to u′ and T′ = (1, 0) to π′. By the assumption L ⊂ K′, ē = e′/e is an integer and the saturation P^{∼} +_{ℕ}^{sat} N′ is generated by U_{i}^{±1}, i = 1, …, r−1, V^{±1}, U^{′±1} and T′ where V = T_{r}T^{′−ē} and is isomorphic to ℤ^{r+1}×ℕ. Hence A_{I,P} ⊗_{O_{K}}^{log} O_{K′} = O_{K′} ⊗_{ℤ[N′]} ℤ[P^{∼} +_{ℕ}^{sat} N′][T_{r+1}, …, T_{m}] is isomorphic to O_{K′}[U_{1}^{±1}, …, U_{r−1}^{±1}, T_{r+1}, …, T_{m}, V^{±1}]. The log structure is the pull-back of that on O_{K′}. The base change A ˆ⊗_{O_{K}}^{log} O_{K′} is the formal completion of the surjection A_{I,P} ⊗_{O_{K}}^{log} O_{K′} → O_{L} ⊗_{O_{K}}^{log} O_{K′}. We claim that the kernel of the surjection A_{I,P} ⊗_{O_{K}}^{log} O_{K′} → O_{L} ⊗_{O_{K}}^{log} O_{K′} is generated by I_{Z} and U_{0} − π/z_{r}^{e}, U_{i} − z_{i}/z_{r}^{e_{i}}, i = 1, …, r.

The kernel Ker(A ˆ⊗_{O_{K}}^{log} O_{K′} → O_{L} ⊗_{O_{K}}^{log} O_{K′}) is generated by Ker(A_{I,P} → O_{L}) since the surjection A_{I,P} → O_{L} is exact. Since P^{∼} is generated by U_{0} = T_{0}T_{r}^{−e}, U_{1}, …, U_{r−1} and P, the ring A_{I,P} is also generated by U_{0}, U_{1}, …, U_{r−1} over O_{K}[T_{1}, …, T_{m}]. Hence, Ker(A_{I,P} → O_{L}) is generated by I_{Z} and U_{0} − π/z_{r}^{e}, U_{i} − z_{i}/z_{r}^{e_{i}}, i = 1, …, r and the claim is proved.

For an element (u_{1}, …, u_{r−1}, v, x_{r+1}, …, x_{m}) ∈ O×r ; K̄ ×Om−r ; K̄, we put x_{r} = vπ^{′ē}

and x_{i} = u_{i}x_{r}^{e_{i}} for i = 1, …, r − 1. Then, the underlying set of X_{log}^{j}(A → A)K̄ is

\begin{cases}(u_{1}, …, u_{r−1}, v, x_{r+1}, …, x_{m}) \\ ∈ O×r ; K̄ × Om−r ; K̄ \end{cases} |\begin{matrix}ordf_{l}(x_{i}) ≥ j for 1 ≤ l ≤ m \\ ord(v^{e}/u′ − g_{r}(x_{i})) ≥ j \\ ord(u_{k} − h_{k}(x_{i})) ≥ j for k = 1, …, r \end{matrix} \} .

Hence the map X_{log}^{j}(A → A)K̄ → Y_{Z,P}^{j} sending (u_{1}, …, u_{r−1}, v, x_{r+1}, …, x_{m}) to (x_{1}, …, x_{m}) is an isomorphism. □

4.3 Étale covering of log tubular neighborhoods

Let A and B be the integer rings of finite étale K-algebras. For a finite flat and log flat morphism (A → A) → (B → B) of log embeddings, we study conditions for the induced finite morphism X_{log}^{j}(A → A) → X_{log}^{j}(B → B) to be étale.

Proposition 4.11 Let A and B = O_{L} be the integer rings of finite separable extensions of K and (A → A) → (B → B) be a finite flat and log flat morphism of log embeddings. Let j > 0 be a rational number, π_{L} a prime element of L and e = ordπ_{L} be the ramification index.

1. Assume A = O_{K}. Suppose that, for each j′ > j, there exists a finite separable extension K′ of K such that X_{log}^{j′}(B → B)_{K′} is isomorphic to the

disjoint union of finitely many copies of X_{log}^{j′}(A → A)_{K′} as an affinoid variety

over X_{log}^{j′}(A → A)_{K′}. Then there is an integer 0 ≤ n < ej such that π_{L}^{n} annihilates Ω_{B/A}(log / log).

2. If there is an integer 0 ≤ n < ej such that π_{L}^{n} annihilates Ω_{B/A}(log / log), then the finite flat map X_{log}^{j}(B → B) → X_{log}^{j}(A → A) is étale.

Corollary 4.12 Let A = O_{K} and let B be the integer ring of a finite étale K-algebra and (A → A) → (B → B) be a finite flat and log flat morphism of log embeddings. Let j > 0 be a rational number. Suppose that, for each j′ > j, there exists a finite separable extension K′ of K such that X_{log}^{j′}(B → B)_{K′} is

isomorphic to the disjoint union of finitely many copies of X_{log}^{j′}(A → A)_{K′} as in Proposition 4.11.1. Let I be the kernel of the surjection B → B and let N_{B/B} be the B-module I/I^{2}. Then we have the following.

1. The finite map X_{log}^{j}(B → B) → X_{log}^{j}(A → A) is étale and is extended to a finite étale map of stable normalized integral models.

2. The finite map X̄_{log}^{j}(B → B) → X̄_{log}^{j}(A → A) is étale.

3. The twisted normal cone C̄_{log}^{j}(B → B) is canonically isomorphic to the covariant vector bundle defined by the B_{F̄,red}-module (Hom_{B}(N_{B/B}, B) ⊗_{O_{K}} N^{j}) ⊗_{BF̄} B_{log F̄,red} and the finite map X̄_{log}^{j}(B → B) → C̄_{log}^{j}(B → B) is étale.

To prove Proposition 4.11, we use the following.

Lemma 4.13 Let A = O_{L} be the integer ring of a finite separable extension L, A → A be a log embedding and let M be an A-module of finite type. Let j > 0 be a rational number and K′ be a finite separable extension of K such that the map O_{K′} → A⊗_{O_{K}}^{log} O_{K′} is strict and the stable normalized integral model A_{O_{K′}}^{j} of X_{log}^{j}(A → A) is defined over K′. Let e and e′ be the ramification indices of L and of K′ over K and π_{L} and π′ be prime elements of L and K′. Assume that e′/e and e′j are integers. Then the following conditions are equivalent.

(1) There exists an integer 0 ≤ n < ej such that the A-module M = M ⊗_{A} A is annihilated by π_{L}^{n}.

(2) The A_{O_{K′}}^{j} -module M^{j} = M ⊗_{A} A_{O_{K′}}^{j} is annihilated by π^{′e′j−1}.

Proof of Lemma 4.13. The proof is similar to that of Lemma 1.17. The image of an element in the kernel I of the surjection A ⊗_{O_{K}}^{log} O_{K′} → A ⊗_{O_{K}}^{log} O_{K′} in A_{O_{K′}}^{j} is divisible by π^{′e′j}. Hence we have a commutative diagram

A ⟶  A_{O_{K′}}^{j}
↓  ↓
A ⟶ A_{O_{K′}}^{j} /(π^{′e′j})

of log rings. The image of π_{L} ∈ A is a unit times π^{′e′/e} in A_{O_{K′}}^{j} /(π^{′e′j}). The rest of the proof is the same as that of Lemma 1.17. Proof of Proposition 4.11. Proof is similar to that of Proposition 1.15.

1. For j > 0, the affinoid variety X_{log}^{j}(A → A) is a polydisk. By the proof of Lemma 1.7, there exist a finite separable extension K′ of K of ramification index e′, an embedding (B ⊗_{O_{K}}^{log} O_{K′} → B′) in Emb_{O_{K′}} isomorphic to (O_{K′}[[T_{1}, …, T_{n}]]^{N} → O_{K′}^{N}) for some N > 0, a positive rational number ϵ < j and an open immersion X_{log}^{j}(B → B)_{K′} → X^{e′ϵ}((B ⊗_{O_{K}}^{log} O_{K′} → B′)^{∘}) as a rational subdomain. The affinoid variety X^{e′ϵ}((B ⊗_{O_{K}}^{log} O_{K′} → B′)^{∘}) is the disjoint union of finitely many copies of polydisks. Enlarging K′ if necessary, we may assume that e′j and e′ϵ are integers. We may further assume that there is a rational number j < j′ < j + ϵ such that e′j′ is an integer, that the stable normalized integral models B_{O_{K′}}^{j′} and B_{O_{K′}}^{′e′ϵ} of X_{log}^{j′}(B → B)_{K′} and

of X^{e′ϵ}((B ⊗_{O_{K}}^{log} O_{K′} → B′)^{∘})_{K′} are defined over K′ and X_{log}^{j′}(B → B)_{K′} is

isomorphic to the disjoint union of copies of X_{log}^{j′}(A → A)_{K′}. Since e′j′ is an

integer, the stable normalized integral model A_{O_{K′}}^{j′} of X_{log}^{j′}(A → A) is also defined over K′. We have a commutative diagram

A  ⟶  A_{O_{K′}}^{j′}
↓  ↓
B ⟶ B_{O_{K′}}^{′e′ϵ} ⟶ B_{O_{K′}}^{j′} .

We consider the modules Ω̂_{A/O_{K}} (log / log) = \underleftarrow{lim}_{n} Ω_{(A/𝔪_{A}^{n})/O_{K})}(log / log), Ω̂_{A_{O_{K′}}^{j′} /O_{K′}} = \underleftarrow{lim}_{n} Ω_{(A_{O_{K′}}^{j′} /π^{′n}A_{O_{K′}}^{j′})/O_{K′}} etc. Since A is strict over O_{K} and B ⊗_{O_{K}}^{log} O_{K′} is strict over O_{K′}, the canonical maps Ω̂_{A/O_{K}} (log / log) → Ω̂_{A/O_{K}} and (B ⊗_{O_{K}}^{log} O_{K′}) ⊗_{B} Ω̂_{B/O_{K}} (log / log) → Ω̂_{(B⊗_{O_{K}}^{log} O_{K′})/O_{K′}} are isomorphisms. Thus, as in the proof of Proposition 1.15, we have a commutative diagram

[figure: B_{O_{K′}}^{j′} ⊗_{A} Ω̂_{A/O_{K}} (log / log); B_{O_{K′}}^{j′} ⊗_{A_{O_{K′}}^{j′}} Ω̂_{A_{O_{K′}}^{j′} /O_{K′}}; B_{O_{K′}}^{j′} ⊗_{B} Ω̂_{B/O_{K}} (log / log); B_{O_{K′}}^{j′} ⊗_{B_{O_{K′}}^{′e′ϵ}} Ω̂_{B_{O_{K′}}^{′e′ϵ} /O_{K′}}; Ω̂_{B_{O_{K′}}^{j′} /O_{K′}} .]

We show that the modules are free B_{O_{K′}}^{j′} -modules of rank n, the maps are injective and that we have an inclusion π^{′e′j′}B_{O_{K′}}^{j′} ⊗_{B} Ω̂_{B/O_{K}} (log / log) ⊂ π^{′e′ϵ}B_{O_{K′}}^{j′} ⊗_{A} Ω̂_{A/O_{K}} (log / log) as submodules of Ω̂_{B_{O_{K′}}^{j′} /O_{K′}}. By the assumption on the covering X_{log}^{j′}(B → B)_{K′} → X_{log}^{j′}(A → A)_{K′}, the A_{O_{K′}}^{j′} - algebra B_{O_{K′}}^{j′} is isomorphic to the product of finitely many copies of A_{O_{K′}}^{j′}. Hence the right vertical map B_{O_{K′}}^{j′} ⊗_{A_{O_{K′}}^{j′}} Ω̂_{A_{O_{K′}}^{j′} /O_{K′}} → Ω̂_{B_{O_{K′}}^{j′} /O_{K′}} is an isomorphism. Similarly as in the proof of Proposition 1.15.1, by the canonical map A_{O_{K′}}^{j′} ⊗_{A} Ω̂_{A/O_{K}} (log / log) → Ω̂_{A_{O_{K′}}^{j′} /O_{K′}}, the module A_{O_{K′}}^{j′} ⊗_{A} Ω̂_{A/O_{K}} (log / log) is identified with the submodules π^{′e′j′} Ω̂_{A_{O_{K′}}^{j′} /O_{K′}} of the free module Ω̂_{A_{O_{K′}}^{j′} /O_{K′}}. Also by B_{O_{K′}}^{′e′ϵ} ⊗_{B} Ω̂_{B/O_{K}} (log / log) → Ω̂_{B_{O_{K′}}^{′e′ϵ} /O_{K′}}, the module B_{O_{K′}}^{′e′ϵ} ⊗_{B} Ω̂_{B/O_{K}} (log / log) is identified with the submodule π^{′e′ϵ}Ω̂_{B_{O_{K′}}^{′e′ϵ} /O_{K′}} of the free module Ω̂_{B_{O_{K′}}^{′e′ϵ} /O_{K′}}. Hence we obtain an inclusion π^{′e′j′}B_{O_{K′}}^{j′} ⊗_{B} Ω̂_{B/O_{K}} (log / log) ⊂ π^{′e′ϵ}B_{O_{K′}}^{j′} ⊗_{A} Ω̂_{A/O_{K}} (log / log) as submodules of Ω̂_{B_{O_{K′}}^{j′} /O_{K′}}. Thus the B_{O_{K′}}^{j′} -module B_{O_{K′}}^{j′} ⊗_{B} Ω_{B/A}(log / log) = Coker(B_{O_{K′}}^{j′} ⊗_{A} Ω̂_{A/O_{K}} (log / log) → B_{O_{K′}}^{j′} ⊗_{B} Ω̂_{B/O_{K}} (log / log)) is annihilated by π^{′e′(j′−ϵ)}.

Since 0 < j − ϵ < j′ − ϵ < j, applying Lemma 4.13 (2)⇒(1), the assertion is proved.

2. Let K′ be a finite separable extension such that e′j is an integer, that B ⊗_{O_{K}}^{log} O_{K′} is strict over O_{K′} and that the stable normalized integral models A_{O_{K′}}^{j} and B_{O_{K′}}^{j} are defined over K′. By Lemma 4.13 (1)⇒(2), the B_{O_{K′}}^{j} - module B_{O_{K′}}^{j} ⊗_{B} Ω_{B/A}(log / log) is annihilated by π^{′n′} for an integer n′ < e′j. The rest of proof is the same as that of Proposition 1.15.2. □ Proof of Corollary 4.12. The same as that of Corollary 1.16. □

5 Filtration by ramification groups: the logarithmic case

5.1 Construction

In this subsection, we rephrase the definition of the logarithmic filtration by ramification groups given in the previous paper [1] by using the preceeding constructions. Let Φ : (Finite Étale/K) → G_{K}-(Finite Sets) be the fiber functor as in Section 2.1. For a rational number j > 0, we define a functor Φ_{log}^{j} : (Finite Étale/K) → G_{K}-(Finite Sets) as the composition of the functor (Finite Étale/K) → (Finite Flat and log Flat/O_{K}) sending a finite étale Kalgebra L to the integral closure O_{L} of O_{K} in L with the standard log structure and the functor Ψ_{log}^{j} : (Finite Flat and log Flat/O_{K}) → G_{K}-(Finite Sets) defined in Section 4.2. The map (4.2.3) defines a surjection Φ → Φ_{log}^{j} of functors. In [1], we define the logarithmic filtration by ramification groups on G_{K} by using the family of surjections (Φ → Φ_{log}^{j})_{j>0,∈ℚ} of functors. The filtration by the log ramification groups G_{K,log}^{j} ⊂ G_{K}, j > 0, ∈ ℚ is characterized by the condition that the canonical map Φ(L) → Φ_{log}^{j}(L) induces a bijection Φ(L)/G_{K,log}^{j} → Φ_{log}^{j}(L) for each finite étale algebra L over K. The functor Φ_{log}^{j} is defined by the commutativity of the diagram

\xymatrix{(Finite Étale/K) \ar[r]^{Φ_{log}^{j}} \ar[d] & G_{K}-(Finite Sets) \\ (Finite Flat and Log Flat/O_{K}) \ar[ur]_{Ψ_{log}^{j}} & G_{K}-(Aff/F̄) \ar[u]_{π_{0}} \\ Emb_{O_{K}}^{log} \ar[d]_{( ⊗^{log}O_{K′})_{K′}} \ar[u] & _{lim_{⟶}_{K′/K}(Aff/F′)} \ar[u] \\ lim_{⟶}_{K′/K}Emb_{O_{K′}} \ar[r]_{(X^{e_{K′/K} j})_{K′}} & _{lim_{⟶}_{K′/K}(smooth Affinoid/K′)} \ar[u]_{(X_{K′})_{K′} ↦(X̄_{F′})_{K′}}}

Documenta Mathematica · Extra Volume Kato (2003) 5–72

We briefly recall how the other arrows in the diagram are defined. The forgetful functor Emb_{O_{K}}^{log} → (Finite Flat and Log Flat/O_{K}) sends (A → A) to A. The functor Emb_{O_{K}}^{log} → \underrightarrow{lim}_{K′/K} Emb_{O_{K′}} sends a log embedding to the system of strict base changes. The functor \underrightarrow{lim}_{K′/K} Emb_{O_{K}} → \underrightarrow{lim}_{K′/K}(smooth Affinoid/K′) is defined by the system of tubular neighborhoods. The functor \underrightarrow{lim}_{K′/K}(smooth Affinoid/K′) → \underrightarrow{lim}_{K′/K}(Aff/F′) is defined by the closed fiber of the stable normalized integral models. The functor \underrightarrow{lim}_{K′/K}(Aff/F′) → G_{K}-(Aff/F̄) is the equivalence of category defined in Section 1.3. The functor π_{0} is defined by the set of connecteds components. They induce a functor Ψ_{log}^{j} : (Finite Flat and log flat/O_{K}) → G_{K}-(Finite Sets). The functor Φ_{log}^{j} is defined as the composition of Ψ_{log}^{j} with the functor sending a finite étale algebra L to the integral closure O_{L} in L of O_{K} with the canonical log structure. More concretely, we have

Φ_{log}^{j}(L) =

\underleftarrow{lim}_{(A→O_{L})∈Emb_{O_{K}}^{log} (O_{L})}π_{0}( \underrightarrow{lim}_{K′/K} X̄^{e_{K′/K}j}((A ⊗_{O_{K}}^{log} O_{K′} → O_{L} ⊗_{O_{K}}^{log} O_{K′})^{∘}))

for a finite étale K-algebra L. This definition agrees with that given in [1] by Lemma 4.10. For a rational number j ≥ 0, we define a functor Φ_{log}^{j+} : (Finite Étale/K) →→

G_{K}-(Finite Sets) by Φ_{log}^{j+}(L) = \underrightarrow{lim}_{j′>j} Φ_{log}^{j′}(L) for a finite étale K-algebra L.

We define a closed normal subgroup G_{K,log}^{j+} to be \overline{∪_{j′>j}G_{K}^{j′}}. Then we have Φ_{log}^{j+}(L) = Φ(L)/G_{K,log}^{j+}. Similarly as Lemma 2.1, the finite set Φ_{log}^{j+}(L) has the following geometric description.

Lemma 5.1 Let B be the integer ring with the standard log structure of a finite étale algebra L over K and j > 0 be a rational number. Let (f, f) : (A → O_{K}) → (B → B) be a finite flat and log flat morphism of embeddings. Let f^{j} : X_{log}^{j}(B → B) → X_{log}^{j}(A → O_{K}) and f̄^{j} : X̄_{log}^{j}(B → B) → X̄_{log}^{j}(A → O_{K}) be the canonical maps. Let 0 ∈ X_{log}^{j}(A → O_{K}) be the point corresponding to the map A → O_{K} and ¯0 ∈ X̄_{log}^{j}(A → O_{K}) be its specialization. Then the maps (1.8.0), (1.12.1) and the specialization map form a commutative diagram

(5.1.1)

Φ(L) ⟶ Φ_{log}^{j+}(L) ⟶  Φ_{log}^{j}(L)
↓  ↓  ↓
(f^{j})^{−1}(0) ⟶ (f̄^{j})^{−1}(0) ⟶ π_{0}(X̄_{log}^{j}(B → B))

and the vertical arrows are bijections.

For a finite étale algebra L over K and a rational number j > 0, we say that the log ramification of L is bounded by j if the canonical map Φ(L) → Φ_{log}^{j}(L) is a bijection. Let A = O_{K} and let B = O_{L} be the integer ring of a finite étale K-algebra L and (A → A) → (B → B) be a finite flat and log flat morphism of log embeddings. Then, since the map X_{log}^{j}(B → B) → X_{log}^{j}(A → A) is finite flat of degree [L : K], the ramification of L is bounded by j if and only if there exists a finite separable extension K′ of K such that the affinoid variety X_{log}^{j}(B → B)_{K′} is isomorphic to the disjoint union of finitely many copies of X_{log}^{j}(A → A)_{K′} over X_{log}^{j}(A → A)_{K′}. We say that the log ramification of L is bounded by j+ if the log ramification of L is bounded by every rational number j′ > j. The log ramification of L is bounded by j+ if and only if the canonical map Φ(L) → Φ_{log}^{j+}(L) is a bijection.

Lemma 5.2 Let K → K′ be a map of complete discrete valuation fields inducing a local homomorphism O_{K} → O_{K′} of integer rings. Assume that the ramification index e = e_{K′/K} is prime to p and that the residue field F′ of K′ is a separable extension of the residue field F of K. Then, for a rational number j > 0, the map G_{K′} → G_{K} induces a surjection G_{log,K′}^{ej} → G_{log,K}^{j}.

Proof. Let A be the integer ring of a finite étale K-algebra L and (A → A) be an object of Emb_{O_{K}}. By the assumption, the log tensor product A ⊗_{O_{K}}^{log} O_{K′} is the integer ring of L ⊗_{K} K′. The rest is the same as the proof of Lemma 2.2. □ The two filtrations by ramification groups are related as follows.

Lemma 5.3 Let K be a complete discrete valuation field and j > 0 be a rational number. Then, we have inclusions G_{K}^{j} ⊃ G_{K,log}^{j} ⊃ G_{K}^{j+1}.

Proof. By Corollary 4.9, there are natural morphisms Φ^{j+1} → Φ_{log}^{j} → Φ^{j} of functors. Hence the assertion follows. □

5.2 Functoriality of the closed fibers of log tubular neighborhoods

For a positive rational number j > 0, let (Finite Étale/K)_{log}^{≤j+} denote the full subcategory of (Finite Étale/K) consisting of étale K-algebras whose log ramification is bounded by j+. At the end of the section, we prove Theorem 5.12. As in the proof of Theorem 2.15, we reduce it to the case where the condition

(F) There exists a perfect subfield F_{0} of F such that F is finitely generated over F_{0}.

is satisfied. Assuming the condition (F), we define a twisted tangent space Θ_{log}^{j} and show that the functor X̄_{log}^{j} : Emb_{O_{K}}^{log} → G_{K}-(Aff/F̄) induces a functor

X̄_{log}^{j} : (Finite Étale/K)_{log}^{≤j+} → G_{K}-(Finite Étale/Θ_{log}^{j}).

In this subsection, L denotes a finite étale K-algebra and A = O_{L} denotes the integer ring with the canonical log structure. We assume that the condition (F) is satisfied. Let K_{0} be a subfield of K such that O_{K_{0}} = O_{K} ∩ K_{0} is a complete discrete valuation ring with perfect residue field F_{0} and F is finitely generated over F_{0} as in Section 2.3. Let π_{0} denote a prime element of O_{K_{0}}. We consider O_{K_{0}} as a log ring with the trivial log structure. We introduce a new category Emb_{K,O_{K_{0}}}^{log} and a functor

Emb_{K,O_{K_{0}}}^{log} → Emb_{O_{K}}^{log} similarly as in Section 2.3.

Definition 5.4 Let K be a complete discrete valuation field and K_{0} be a subfield of K such that O_{K_{0}} = O_{K} ∩ K_{0} is a complete discrete valuation ring with perfect residue field F_{0} and that F is finitely generated over F_{0}. We put m = tr.deg(F/F_{0}). We consider O_{K_{0}} as a log ring with the trivial log structure.

1. We define Emb_{K,O_{K_{0}}}^{log} to be the category whose objects and morphisms are

as follows. An object of Emb_{K,O_{K_{0}}}^{log} is a triple (A_{0} → A) where:

• A is the integer ring of a finite étale K-algebra with the canonical log structure.

• A_{0} is a complete semi-local Noetherian log O_{K_{0}}-algebras formally smooth and formally log smooth of relative dimension m + 1 = tr.deg(F/F_{0}) + 1 over O_{K_{0}}.

• A_{0} → A is an exact and regular surjection of codimension 1 of log O_{K_{0}}- algebras and induces an isomorphism A_{0}/𝔪_{A_{0}} → A/𝔪_{A} of underlying F_{0}-algebras.

A morphism (f, f) : (A_{0} → A) → (B_{0} → B) is a pair of a log O_{K}- homomorphism f : A → B and a log O_{K_{0}}-homomorphism f : A_{0} → B_{0} such that the diagram

A_{0} ⟶ A
f↓  ↓^{f}
B_{0} ⟶ B

is commutative.

2. For the integer ring A of a finite étale K-algebra, we define Emb_{K,O_{K_{0}}}^{log} (A)

to be the subcategory of Emb_{K,O_{K_{0}}}^{log} whose objects are of the form (A_{0} → A) and morphisms are of the form (id_{A}, f).

3. We say that a morphism (A_{0} → A) → (B_{0} → B) is finite flat and log flat if A_{0} → B_{0} is finite flat and log flat and the canonical map B_{0} ⊗_{A_{0}}^{log} A → B is an isomorphism.

An object (A_{0} → A) of Emb_{K,O_{K_{0}}}^{log} is an object (A_{0} → A) of Emb_{K,O_{K_{0}}} together with a log strucure on A_{0} such that the log ring A_{0} is formally log smooth over O_{K_{0}} and that the surjection A_{0} → A is exact.

Lemma 5.5 1. Let A be the integer ring of a finite étale K-algebra with the canonical log structure. Then, the category Emb_{K,O_{K_{0}}}^{log} (A) is non-empty.

2. Let (A_{0} → A) and (B_{0} → B) be objects of Emb_{K,O_{K_{0}}}^{log} and A → B be an O_{K}- homomorphism. Then there exists a homomorphism (A_{0} → A) → (B_{0} → B) in Emb_{K,O_{K_{0}}}^{log} extending A → B.

3. Every morphism in Emb_{K,O_{K_{0}}}^{log} is finite flat and log flat.

Proof. 1. We may assume A is local. Take a transcendental basis (t̄_{1}, …, t̄_{m}) of the residue field E of A over F_{0} such that E is a finite separable extension of F_{0}(t̄_{1}, …, t̄_{m}). Take a lifting (t_{1}, …, t_{m}) in A of (t̄_{1}, …, t̄_{m}) and prime elements t_{0} of A and π_{0} of O_{K_{0}}. Then A is unramified over the completion of the local ring of O_{K_{0}}[T_{0}, …, T_{m}] at the prime ideal (π_{0}, T_{0}) by the map defined by sending T_{i} to t_{i}. Hence there are an étale scheme X over 𝔸_{O_{K_{0}}}^{m+1}, a point ξ

of X above (π_{0}, T_{0}) and a regular immersion φ : Ô_{X,ξ} → A of codimension 1. Let A_{0} be the O_{K_{0}}-algebra Ô_{X,ξ} with the log structure defined by the chart ℕ → A_{0} sending 1 ∈ ℕ to T_{0}. Then (A_{0} → A) is an object of Emb_{K,O_{K_{0}}}^{log} .

2. Since A_{0} is formally log smooth over O_{K_{0}}, it follows from that B_{0} is the formal completion of itself with respect to the surjection B_{0} → B.

3. We may assume A and B are local. We show that the map B_{0} ⊗_{A_{0}}^{log} A → B is an isomorphism. Let f be a generator of the kernel of A_{0} → A. It is sufficient to show that the image of f in 𝔪_{B_{0}}/𝔪_{B_{0}}^{2} is not 0. We take charts ℕ → A_{0} and ℕ → B_{0} and let t_{0} ∈ A_{0} and t′_{0} ∈ B_{0} be the images of 1 ∈ ℕ. The charts ℕ → A_{0} and ℕ → B_{0} induces isomorphisms ℕ → M_{Y,y} and ℕ → M_{X,x} where y and x are the closed points of the log schemes Y = Spec A_{0} and X = Spec X_{0}. The map ℕ = M_{Y,y} → ℕ = M_{X,x} is the multiplication by the ramification index e of B ⊗_{O_{K}} K over A ⊗_{O_{K}} K. Since dt_{0} is in the kernel of the surjection Ω̂_{A_{0}/O_{K_{0}}} ⊗_{A_{0}} A/𝔪_{A} → Ω_{(A/𝔪_{A})/F_{0}} and is non-zero, (π_{0}, t_{0}) is a basis of 𝔪_{A_{0}}/𝔪_{A_{0}}^{2}. We put f = aπ_{0} + bt_{0} in 𝔪_{A_{0}}/𝔪_{A_{0}}^{2} for some element a, b in the residue field E of A. Since the surjection A_{0} → A is regular of codimension 1, either of a and b is not 0. Since the image of t_{0} is a basis of 𝔪_{A}/𝔪_{A}^{2} and the image of f is 0, we have a ≠ 0. Similarly (π_{0}, t′_{0}) is a basis of 𝔪_{B_{0}}/𝔪_{B_{0}}^{2}. Since the map ℕ = M_{Y,y} → ℕ = M_{X,x} is the multiplication by the ramification index e, the image of t_{0} is a unit times t_{0}^{′e}. Hence the image of f in 𝔪_{B_{0}}/𝔪_{B_{0}}^{2} is not zero. Thus the map B_{0} ⊗_{A_{0}}^{log} A → B is an isomorphism. Since B is finite over A, B_{0} is also finite over A_{0} by Nakayama’s lemma. Since dim A_{0} = dim B_{0} = 2 the assertion follows by Corollary 3.11. □

Corollary 5.6 Every morphism in Emb_{K,O_{K_{0}}}^{log} (A) is an isomorphism.

Proof. If (A_{0} → A) → (A′_{0} → A) is a map, the map A_{0} → A′_{0} is finite flat of degree 1 and is an isomorphism. □ We define a functor Emb_{K,O_{K_{0}}}^{log} → Emb_{O_{K}}^{log} as follows. Let (A_{0} → A) be an

object of Emb_{K,O_{K_{0}}}^{log} . We define an embedding ((A_{0} ˆ⊗_{O_{K_{0}}} O_{K})^{∧} → A) by regarding (A_{0} → A) as an object of Emb_{K,O_{K_{0}}} . Since the underlying ring of A/𝔪_{A}^{n} ⊗_{O_{K_{0}}}^{log} O_{K}/𝔪_{K}^{n} is A/𝔪_{A}^{n} ⊗_{O_{K_{0}}} O_{K}/𝔪_{K}^{n}, we define a log structure

on (A_{0} ˆ⊗_{O_{K_{0}}} O_{K})^{∧} as the limit of those on A/𝔪_{A}^{n} ⊗_{O_{K_{0}}}^{log} O_{K}/𝔪_{K}^{n}. We let

(A_{0} ˆ⊗_{O_{K_{0}}}^{log} O_{K})^{∧} denote the log ring (A_{0} ˆ⊗_{O_{K_{0}}} O_{K})^{∧} with this log structure.

Lemma 5.7 Let (A_{0} → A) be an object of Emb_{K,O_{K_{0}}}^{log} . Then,

((A_{0} ˆ⊗_{O_{K_{0}}}^{log} O_{K})^{∧} → A) is a log pre-embedding and hence ((A_{0} ˆ⊗_{O_{K_{0}}}^{log} O_{K})^{∧∼} → A) is a log embedding.

Proof. By the construction, the log O_{K}-algebra (A_{0} ˆ⊗_{O_{K_{0}}}^{log} O_{K})^{∧} is formally log

smooth and ((A_{0} ˆ⊗_{O_{K_{0}}}^{log} O_{K})^{∧} → A) is a log pre-embedding. The rest follows from Lemma 4.4.2. □ In the following, we put A = (A_{0} ˆ⊗_{O_{K_{0}}} O_{K})^{∧∼}. We obtain a functor Emb_{K,O_{K_{0}}}^{log} → Emb_{O_{K}}^{log} sending (A_{0} → A) to (A → A) = ((A_{0} ˆ⊗_{O_{K_{0}}} O_{K})^{∧∼} → A) by Lemma 5.7. For a rational number j > 0, we have a sequence of functors

Emb_{K,O_{K_{0}}}^{log} ⟶ Emb_{O_{K}}^{log} ⟶^{X_{log}^{j}}
\underrightarrow{lim}_{K′/K}(smooth Affinoid/K′) ⟶ G_{K}-(Aff/F̄).

We also let X̄_{log}^{j} denote the composite functor Emb_{K,O_{K_{0}}}^{log} → G_{K}-(Aff/F̄).

Thus, for an object (A_{0} → A) of Emb_{K,O_{K_{0}}}^{log} , we have X̄_{log}^{j}(A_{0} → A) =

X̄_{log}^{j}((A_{0} ˆ⊗_{O_{K_{0}}} O_{K})^{∧∼} → A). For a rational number j > 0, the composition

Emb_{K,O_{K_{0}}}^{log} ⟶ Emb_{O_{K}}^{log} ⟶^{C̄_{log}^{j}} G_{K}-(Aff/F̄).

defines a functor C̄_{log}^{j} : Emb_{K,O_{K_{0}}}^{log} → G_{K}-(Aff/F̄). We compute the twisted

normal cone C̄_{log}^{j}(A → A) for an object (A_{0} → A) of Emb_{K,O_{K_{0}}}^{log} and A =

(A_{0} ˆ⊗_{O_{K_{0}}} O_{K})^{∧∼}. It is a scheme over (A_{log} F̄)_{red} = (A ⊗_{O_{K}}^{log} F̄)_{red}. Let N_{A/A} = I/I^{2} be the conormal module where I is the kernel of the surjection A → A. We put Ω̂_{O_{K}/O_{K_{0}}} (log) = \underleftarrow{lim}_{n} Ω_{(O_{K}/𝔪_{K}^{n})/O_{K_{0}}} (log) with respect to the canonical log structure on O_{K} and the trivial log structure on O_{K_{0}}. Similarly, we put Ω̂_{A/A_{0}}(log / log) = \underleftarrow{lim}_{n} Ω_{(A/𝔪_{A}^{n})/A_{0}}(log / log). Since the map A → A_{0} is strict, we have Ω̂_{A/A_{0}}(log / log) = Ω̂_{A/A_{0}}. Let Ω_{F} (log) be the F -vector space Ω_{F/F_{0}}(log) with respect to the trivial log structure on F_{0} and the log structure on F defined by the chart ℕ → F sending 1 ∈ ℕ to 0. The canonical map Ω̂_{O_{K}/O_{K_{0}}} (log)⊗_{O_{K}} F → Ω_{F} (log) is an isomorphism. We have an exact sequence 0 → Ω_{F/F_{0}} → Ω_{F/F_{0}}(log) →^{res} F → 0. We have canonical maps N_{A/A} → Ω̂_{A/A_{0}} ⊗_{A} A and Ω̂_{O_{K}/O_{K_{0}}} (log) ⊗_{O_{K}} A → Ω̂_{A/A_{0}}(log / log) ⊗_{A} A. Similarly as Lemma 2.11, we have the following.

Lemma 5.8 Let (A_{0} → A) be an object of Emb_{K,O_{K_{0}}}^{log} .

1. If m is the transcendental dimension of F over F_{0}, the dimension of the F -vector space Ω_{F} (log) is m + 1.

2. The map N_{A/A} → Ω_{A/A_{0}} ⊗_{A} A is a surjection and the map Ω_{O_{K}/O_{K_{0}}} (log) ⊗_{O_{K}} A → Ω_{A/A_{0}} ⊗_{A} A is an isomorphism. They induce an isomorphism N_{A/A} ⊗_{A} A/𝔪_{A} → Ω_{F} (log) ⊗_{F} A/𝔪_{A}.

3. Let (A_{0} → A) → (B_{0} → B) be a morphism of Emb_{K,O_{K_{0}}}^{log} and put B = (B_{0} ˆ⊗_{O_{K_{0}}} O_{K})^{∧∼}. Then, the diagram

N_{A/A} ⊗_{A} A/𝔪_{A} ⟶ Ω_{F} (log) ⊗_{F} A/𝔪_{A}
↓  ↓
N_{B/B} ⊗_{B} B/𝔪_{B} ⟶ Ω_{F} (log) ⊗_{F} B/𝔪_{B}

is commutative.

For a rational number j > 0, let Θ_{log}^{j} be the F̄-vector space Hom_{F} (Ω_{F} (log), N^{j}) regarded as an affine scheme over F̄. Similarly as Corollary 2.12, we have the following.

Corollary 5.9 Let (A_{0} → A) be an object of Emb_{K,O_{K_{0}}}^{log} and let (A → A) be

its image in Emb_{O_{K_{0}}}^{log} . Let j > 0 be a rational number.

1. Let C̄_{log}^{j}(A → A) be the twisted normal cone. The isomorphism in Lemma 5.8.2 induces an isomorphism C̄_{log}^{j}(A → A) → Θ_{log}^{j} ⊗F̄ (A_{log} F̄)_{red}.

2. Let (A_{0} → A) → (B_{0} → B) be a morphism of Emb_{K,O_{K_{0}}}^{log} . Then the diagram

X̄_{log}^{j}(B → B) ⟶ C̄_{log}^{j}(B → B) ⟶ Θ_{log}^{j} ⊗F̄ (B_{log} F̄)_{red}
↓  ↓  ↓
X̄_{log}^{j}(A → A) ⟶ C̄_{log}^{j}(A → A) ⟶ Θ_{log}^{j} ⊗F̄ (A_{log} F̄)_{red}

is commutative.

3. If the ramification of A ⊗_{O_{K}} K is bounded by j+, then the composition X̄_{log}^{j}(A → A) → C̄_{log}^{j}(A → A) → Θ_{log}^{j} is finite and étale.

For a rational number j > 0, we regard Θ_{log}^{j} as an object of G_{K}-(Aff/F̄) with the natural G_{K}-action. Let G_{K}-(Finite Étale/Θ_{log}^{j}) denote the subcategory of G_{K}-(Aff/F̄) whose objects are finite étale schemes over Θ_{log}^{j} and morphisms are over Θ_{log}^{j}. Let Emb_{K,O_{K_{0}}}^{log,≤j+} denote the full subcategory of Emb_{K,O_{K_{0}}}^{log} consisting of the objects (A_{0} → A) such that the log ramifications of A⊗_{O_{K}} K are bounded by j+. By Corollary 5.9, the functor X̄_{log}^{j} : Emb_{K,O_{K_{0}}}^{log} → G_{K}-(Aff/F̄) induces

a functor X̄_{log}^{j} : Emb_{K,O_{K_{0}}}^{log,≤j+} → G_{K}-(Finite Étale/Θ_{log}^{j}).

The functor X̄_{log}^{j} : Emb_{K,O_{K_{0}}}^{log,≤j+} → G_{K}-(Finite Étale/Θ_{log}^{j}) further induces a

functor X̄_{log}^{j} : (Finite Étale/K)_{log}^{≤j+} → G_{K}-(Finite Étale/T_{log}^{j}). In fact, similarly as Lemma 2.13 and Corollary 2.14, we have the following.

Lemma 5.10 Let f : A → B be a map over O_{K} and let (f, f), (g, g) : (A_{0} → A) → (B_{0} → B) be maps in Emb_{K,O_{K_{0}}}^{log} . If f = g, then the induced maps

(f, f)_{∗}, (g, g)_{∗} : X̄_{log}^{j}(A_{0} → A) ⟶ X̄_{log}^{j}(B_{0} → B)

are equal.

Corollary 5.11 Let j > 0 be a rational number.

1. Let L be a finite étale K-algebra L such that the log ramification is bounded by j+. Then the system X̄_{log}^{j}(A_{0} → O_{L}) parametrized by the objects (A_{0} → O_{L}) of Emb_{K,O_{K_{0}}}^{log} (O_{L}) is constant and the limit

X̄_{log}^{j}(L) = \underleftarrow{lim}_{(A_{0}→O_{L})∈Emb_{K,O_{K_{0}}}^{log} (O_{L})}X̄_{log}^{j}(A_{0} → O_{L})

is a finite étale scheme over Θ_{log}^{j}.

2. The functor X̄_{log}^{j} : Emb_{K,O_{K_{0}}}^{log,≤j+} → G_{K}-(Finite Étale/Θ_{log}^{j}) induces a functor

X̄_{log}^{j} : (Finite Étale/K)_{log}^{≤j+} → G_{K}-(Finite Étale/Θ_{log}^{j}).

Using the functor X̄_{log}^{j} : (Finite Étale/K)^{≤j+} → G_{K}-(Finite Étale/Θ^{j}) defined under the condition (F), we obtain the following theorem by the same argument as the proof of Theorem 2.15.

Theorem 5.12 Let K be a complete discrete valuation field and let j > 0 be a rational number. Let m be the prime-to-p part of the denominator of j and I_{m} be the subgroup of the inertia group I ⊂ G_{K} of index m. Then we have the following.

1. The graded piece Gr^{j}G_{K} = G_{K,log}^{j}/G_{K,log}^{j+} is abelian.

2. The commutator [I_{m}, G_{K,log}^{j}] is a subgroup of G_{K,log}^{j+}. In particular, Gr_{log}^{j}G_{K} is a subgroup of the center of the pro-p-group G_{K,log}^{0+}/G_{K,log}^{j+}.

Similarly as in the proof of Theorem 2.15, assuming the condition (F), we obtain a canonical surjection

(5.12.1)  π_{1}^{ab}(Θ_{log}^{j}) ⟶ Gr_{log}^{j}G_{K}.

The canonical surjections π_{1}^{ab}(Θ_{log}^{j}) → Gr_{log}^{j}G_{K} and π_{1}^{ab}(Θ^{j}) → Gr^{j}G_{K} are related as follows. The exact sequences 0 → N → Ω̃_{F} → Ω_{F} → 0 and 0 → Ω_{F} → Ω_{F} (log) → F → 0 induces canonical maps Θ_{log}^{j} → Θ^{j} and Θ^{j+1} → Θ_{log}^{j}.

Lemma 5.13 Assume that the condition (F) is satisfied and that p is not a prime element of K. Then, for a rational number j > 0, we have a commutative diagram

π_{1}^{ab}(Θ^{j+1}) ⟶ π_{1}^{ab}(Θ_{log}^{j}) ⟶ π_{1}^{ab}(Θ^{j})
↓  ↓  ↓
Gr^{j+1}G_{K} ⟶ Gr_{log}^{j}G_{K} ⟶ Gr^{j}G_{K}.

Proof. We show the commutativity of the left square. Let L be a finite separable extension of K such that the log ramification is bounded by j+ and A be the integer ring of L. By Lemma 5.3, the ramification of L is bounded by (j + 1)+. Let (A_{0} → A) be an object of Emb_{K,O_{K_{0}}}^{log} . By Lemma 5.7, the surjection

A = A_{0} ⊗_{O_{K_{0}}}^{log} O_{K} → A defines a log pre-embedding (A_{0} ⊗_{O_{K_{0}}}^{log} O_{K} → A). By forgetting the log structure, we obtain an embedding (A → A)^{∘}. By applying Lemma 4.4, we obtain a log embedding (A^{∼} → A). Then, by Lemma 4.8 we have an open immersion X^{j+1}((A → A)^{∘}) → X_{log}^{j}(A^{∼} → A) of affinoid subdomains of X^{j}((A → A)^{∘}). It induces a map X̄^{j+1}(L) → X̄_{log}^{j}(L). By the functoriality, we obtain a commutative diagram

X̄^{j+1}(L) ⟶  X̄_{log}^{j}(L)
↓  ↓
Θ^{j+1} =X̄^{j+1}(K) ⟶ Θ_{log}^{j} =X̄_{log}^{j}(K).

From this diagram, we deduce the commutativity of the left square. The proof for the right square is similar and omitted. □

6 The perfect residue field case

6.1 The Newton polygon of a polynomial

We recall the notion of Newton polygons and establish some properties. We say that a function l : [0, n] → ℝ ∪ {∞} is convex if for every 0 ≤ x ≤ y ≤ n, the graph of l is below the line segment connecting (x, l(x)) and (y, l(y)). If at least one of l(x) and l(y) is ∞, we define the line segment connecting (x, l(x)) and (y, l(y)) to be the union {(z, ∞)|x < z < y} ∪ {(x, l(x)), (y, l(y))}. For a polynomial h(T) = ∑_{i=0}^{n} b_{i}T^{n−i} ∈ K̄[T] of degree ≤ n, we define its Newton polygon to be the graph of the maximum convex function l_{h} : [0, n] → ℝ ∪ {∞} satisfying l_{h}(i) ≤ ord b_{i}. If b_{0} = 1, the Newton polygon of h and the solutions of the equation h(T) = 0 are related as follows. Let z_{1}, …, z_{n} be the solution of h(T) = ∏_{i=1}^{n}(T − z_{i}) = 0 and assume ordz_{i} is increasing in i. Then, since b_{i} = (−1)^{i} ∑_{1≤k_{1}<...<k_{i}≤n} z_{k_{1}} ⋯ z_{k_{i}}, the slope of l_{h} on the interval (i − 1, i) is equal to ordz_{i}. If l(x) = ∞, we define the slope of l at x to be ∞.

Lemma 6.1 Let f(T) = ∑_{i=0}^{n} a_{i}T^{n−i} ∈ O_{K}[T] be a polynomial of degree n and z be an element of K̄^{×} such that ordz = \frac{1}{n}. We assume a_{0} = 1 and orda_{i} ≥ 1 for 1 ≤ i < n. We put

h(T) = \frac{f(z(T + 1)) − f(z)}{z^{n}} = ∑_{i=0}^{n−1} b_{i}T^{n−i} ∈ K̄[T]

and let l_{h} : [0, n] → ℝ ∪ {∞} be the function defining the Newton polygon of h(T). Then, for an integer 0 < i < n, the equality l_{h}(i) = ordb_{i} implies i = n − p^{k} for some integer k ≥ 0.

Proof. For an integer 0 ≤ r ≤ n, we put f_{r}(T) = a_{n−r}T^{r}, h_{r}(T) = (f_{r}(z(T + 1)) − f_{r}(z))/z^{n} = a_{n−r}z^{−(n−r)}((T + 1)^{r} − 1) and let l_{r} : [0, n] → ℝ ∪ {∞} denote the function defining the Newton polygon of h_{r}(T). We have

h(T) = ∑_{r=1}^{n} h_{r}(T) = ∑_{i=1}^{n} (∑_{r=i}^{n} a_{n−r}z^{−(n−r)}\binom{r}{i}) T^{i}.

Since ordz = \frac{1}{n}, we have ord b_{n−i} = min_{i≤r≤n}(ord a_{n−r}z^{−(n−r)}\binom{r}{i}). Hence l_{h} is the maximum convex function satisfying l_{h} ≤ l_{r} for 1 ≤ r ≤ n. We compute the function l_{r} for 1 ≤ r ≤ n. We have h_{r}(T) = a_{n−r}z^{−(n−r)} ∑_{i=1}^{r} \binom{r}{i}T^{i}. For an integer 0 < i ≤ p^{k}|r, we have

ord\binom{r}{i} = ord \frac{r}{i} + ∑_{j=1}^{i−1} ord \frac{r − j}{j} = ord \frac{r}{i} ≥ ord \frac{r}{p^{k}} .

The equality holds only for i = p^{k}. Hence, l_{r} is the maximum convex function satisfying

l_{r}(i) = (ord a_{n−r}−1)+\frac{r}{n}+\begin{cases}0 & if i = n − r \\ ord \frac{r}{p^{k}} & if i = n − p^{k} for an integer 0 ≤ k ≤ ord_{p}r. \end{cases}

Thus, for an integer i satisfying n − p^{ord_{p}r} ≤ i ≤ n, the equality l_{h}(i) = l_{r}(i) implies i = n − p^{k} for an integer 1 ≤ k ≤ ord_{p}r. It also follows that we have 0 = l_{h}(0) < l_{r}(n − r) = l_{r}(n − p^{ord_{p}r}) for 1 ≤ r < n. Hence the equality l_{h}(i) = l_{r}(i) implies i ≥ n − p^{ord_{p}r}. Thus the assertion is proved. □ For a polynomial h(T) ∈ K̄[T], ≠ 0, let ord h(T) denote the minimum of the valuations of the coefficients. For a rational number u, let π^{u} denote an element of K̄^{×} satisfying ordπ^{u} = u. We define a function φ_{h} : [0, ∞) → [0, ∞) by φ_{h}(u) = ord h(π^{u}T). The function φ_{h} is continuous, convex and piecewise linear.

Lemma 6.2 Let h(T) = ∑_{i=0}^{n} b_{i}T^{n−i} = ∏_{i=1}^{n}(T − z_{i}) ∈ K̄[T] be a monic polynomial of degree n. Let l_{h} : [0, n] → ℝ ∪ {∞} be the function defining the Newton polygon of h(T) and φ_{h} : [0, ∞) → [0, ∞) be the function φ_{h}(u) = ord h(π^{u}T) defined above. Then,

1. The minimum value of the function l_{h}(t) + (n − t)u on t ∈ [0, n] is equal to φ_{h}(u).

2. We have an equality

φ_{h}(u) = ∑_{i=1}^{n} min(u, ord z_{i}).

3. If the coefficient of T^{r} in \overline{(\frac{h(π^{u}T)}{π^{φ_{h}(u)}})} ∈ F̄[T] is not zero, then the function

l_{h}(t) + (n − t)u attains the minimum value at t = r and we have l_{h}(r) = ordb_{r}.

Proof. 1. Since the function l_{h}(t) + (n − t)u defines the Newton polygon of h(π^{u}T), the assertion follows.

2. We put s_{i} = ord z_{i}. Let t_{0} ∈ [0, n] be the minimum where the function l_{h}(t) + (n − t)u takes the minimum value. Then t_{0} is the maximum such that the function l_{h}(t) + (n − t)u is strictly decreasing on [0, t_{0}]. Hence t_{0} is the cardinality of the set {i|s_{i} < u} and the minimum value of l_{h}(t) + (n − t)u is given by

l_{h}(t_{0}) + (n − t_{0})u = ∑_{s_{i}<u} s_{i} + ∑_{s_{i}≥u} u = ∑_{i=1}^{n} min(s_{i}, u).

Thus the assertion follows from 1.

3. The coefficient of T^{r} in \overline{h(π^{u}T)/π^{φ_{h}(u)}} ∈ F̄[T] is not zero if and only if the value of the function defining the Newton polygon of h(π^{u}T)/π^{φ_{h}(u)} at r is zero and l_{h}(r) = ordb_{r}. Hence the assertion follows from 1. □

6.2 The structure of graded pieces

In this subsection, we assume that the residue field F is perfect. Since the residue map Ω_{F} (log) → F is an isomorphism in this case, we have an isomorphism Θ_{log}^{j} → N^{j} of F̄-vector spaces of dimension 1. Let π_{1}^{ab,gp}(N^{j}) denote the quotient of π_{1}^{ab}(N^{j}) classifying the étale isogenies to the algebraic group N^{j}.

Proposition 6.3 Let K be a complete discrete valuation field with perfect residue field and j > 0 be a positive rational number. Then,

1. ([1] Propositions 3.7 (3) and 3.15 (4)) We have G_{log,K}^{j} = G_{K}^{j+1}. If p is not a prime element of K, the horizontal arrows in the diagram of Lemma 5.13 are isomorphism.

2. The canonical surjection π_{1}^{ab}(N^{j}) → Gr_{log}^{j}G_{K} (5.12.1) induces an isomorphism π_{1}^{ab,gp}(N^{j}) → Gr_{log}^{j}G_{K}.

Contrary to the proof given in [12], we give a proof without using the “lower numbering” filtration or local class field theory. Before starting proof, we introduce some notations. Let L be a finite separable extension of K and π_{L} be a prime element of L. Let K_{1} be the maximum unramified extension of K in L and let f(T) ∈ O_{K_{1}}[T] be the minimal polynomial of π_{L} over K_{1}. Since, L is totally ramified over K_{1}, the polynomial f(T) is an Eisenstein polynomial. We put n = [L : K_{1}] = deg f. We put A = O_{L} and K_{0} = K and define an object (A → A) of Emb_{K,O_{K_{0}}}^{log} as follows. We define a log structure on O_{K_{1}}[T] by the chart ℕ → O_{K_{1}}[T] sending 1 to T. We define a log O_{K_{0}}-algebra A = O_{K_{1}}[[T]] to be the formal completion of the surjection O_{K_{1}}[T] → O_{L} sending T to π_{L} with the induced log structure. Then the surjection A → A defines an object (A → A) of Emb_{K,O_{K_{0}}}^{log} .

By Lemma 5.7, it defines a log pre-embedding (A ⊗_{O_{K_{0}}}^{log} O_{K} → A). The log

ring A ⊗_{O_{K_{0}}}^{log} O_{K} is the ring A itself with the log structure defined by the chart ℕ^{2} → A sending (1, 0) to T and (0, 1) to a prime element π of O_{K}. By forgetting the log structure, we obtain an embedding (A → A)^{∘}. By applying Lemma 4.4, we obtain a log embedding (A^{∼} → A). The log ring A^{∼} is identified with the formal completion of the surjection O_{K}[T, U^{±1}]/(T^{n} − Uπ) → A of log O_{K}-algebras sending T to π_{L} and U to π_{L}^{n}/π ∈ A^{×} with log structure defined by the chart ℕ → O_{K_{1}}[T, U^{±1}]/(T^{n} − Uπ) sending 1 to T. Let K′ be a finite separable extension of K containing the conjugates of K_{1} over K and an element z of ord z = 1/n. Then, the log tensor product A^{∼} ⊗_{O_{K}}^{log} O_{K′} is further identified with the formal completion of the surjection O_{K_{1}} ⊗_{O_{K}} O_{K′}[W^{±1}] = ∏_{σ:K_{1}→K′} O_{K′}[W^{±1}] → A ⊗_{O_{K}}^{log} O_{K′} of strict log O_{K′}-algebras sending W to (π_{L}⊗1)/(1⊗z). With this identification, the canonical map A^{∼} → A^{∼}⊗_{O_{K}}^{log} O_{K′} sends T to (1⊗z)W and U to ((1⊗z)^{n}/π)·W^{n}. Further, we identify the affinoid variety X_{log}^{j}(A^{∼} → A)_{K′} as an affinoid subdomain of ∐_{σ:K_{1}→K′} SpK′⟨W^{±1}⟩. Similarly as for (A → A), by taking a prime element π of O_{K}, we define an object (B → O_{K}) of Emb_{K,O_{K_{0}}}^{log} as the formal completion of the surjection O_{K_{0}}[S] → O_{K} sending S to π. By Lemma 4.4, the log ring B^{∼} is identified with the formal completion of the surjection O_{K}[V^{±1}] → A of strict log O_{K}- algebras sending V to 1. With this identification, the canonical map B → B^{∼} sends S to πV. Further, we identify the affinoid variety X_{log}^{j}(B^{∼} → O_{K})_{K} with the subdisk D(1, π^{j}) ⊂ SpK⟨V^{±1}⟩. We define a map (B → O_{K}) → (A → A) of Emb_{K,O_{K_{0}}}^{log} as follows. Since f(T) is an Eisenstein polynomial of degree n, g(T) = (T^{n} − f(T))/π is in O_{K_{1}}[T] and its image is invertible in A. By sending S to T^{n}g(T)^{−1}, we obtain a map (B → O_{K}) → (A → A) of Emb_{K,O_{K_{0}}}^{log} . The Herbrand functions φ and ψ : [0, ∞) → [0, ∞) are defined as follows (cf. [4] Appendix). We put h(T) = f(π_{L}(T +1))/π_{L}^{n} and define φ to be the function φ_{h} in Lemma 6.2. The function φ is strictly increasing, continuous and piecewise linear. We define ψ : [0, ∞) → [0, ∞) to be the inverse φ^{−1}. The function ψ is also strictly increasing, continuous and piecewise linear. For an embedding σ : K_{1} → K̄ over K, let f^{σ}(T) ∈ OK̄[T] denote the image of f(T) by σ. For w ∈ K̄ and a rational number u > 0, let D(w, π^{u}) denote the disk with center w and radius π^{u}.

Lemma 6.4 Let the notation be as above.

1. The open immersion X^{j+1}((A → A)^{∘}) ⊂ X_{log}^{j}(A^{∼} → A) in Corollary 4.9 is an isomorphism.

2. As affinoid subdomains of ∐_{σ:K_{1}→K′} SpK′⟨W^{±1}⟩, we have an equality

X_{log}^{j}(A^{∼} → A) = ∐_{σ:K_{1}→K′} ⋃_{f^{σ}(z_{i}^{σ})=0} D(\frac{z_{i}^{σ}}{z} , π^{ψ(j)}).  (2)

The log ramification of L is bounded by j if and only if ψ(j) is larger than the slope s_{n−1} of the Newton polygon of h on the interval (n − 2, n − 1).

3. Let σ : K_{1} → K̄ be an embedding and z_{i}^{σ} ∈ OK̄ be a solution of f^{σ}(T) = 0. We put

h_{i}^{σ}(T) = −\frac{f^{σ}(z(π^{ψ(j)}T + \frac{z_{i}^{σ}}{z}))}{π^{j}f^{σ}(0)}.

Then we have h_{i}^{σ} ∈ OK̄[T]. Let h̄_{i}^{σ} ∈ F̄[T] be the reduction and let h̄_{i}^{σ} : 𝔸^{1} → 𝔸^{1} be the map defined by the polynomial h̄_{i}^{σ}. Then the isomorphisms ×π^{ψ(j)}+ \frac{z_{i}^{σ}}{z} : D(0, 1) → D(\frac{z_{i}^{σ}}{z} , π^{ψ(j)}) ⊂ X_{log}^{j}(A^{∼} → A) and ×π^{j}+1 : D(0, 1) → D(1, π^{j}) induce a commutative diagrams

𝔸^{1} ⟶ \overline{D(\frac{z_{i}^{σ}}{z} , π^{ψ(j)})} ⊂ X̄_{log}^{j}(A^{∼} → A)
h̄_{i}^{σ} ↓  ↓
𝔸^{1} ⟶ \overline{D(1, π^{j})} = X̄_{log}^{j}(B^{∼} → O_{K})= N^{j}.

Proof. 1. As in Lemma 4.8, we identify X_{log}^{j}(A^{∼} → A) and X^{j+1}((A → A)^{∘}) as affinoid subdomains of X^{j}((A → A)^{∘}). The kernels of the surjections A → A and A^{∼} → A are generated by f(T) and U^{−1} − g(T) = f(T)/π respectively. Hence, the affinoid subdomains X_{log}^{j}(A^{∼} → A) and X^{j+1}((A → A)^{∘}) of X^{j}((A → A)^{∘}) are defined by the conditions ord f(x)/π ≥ j and by ord f(x) ≥ j + 1 respectively. Hence the assertion follows.

2. Since the kernel of surjection A^{∼} → A is generated by f(T)/π, the kernel of surjection A^{∼} ⊗_{O_{K}}^{log} O_{K′} → A ⊗_{O_{K}}^{log} O_{K′} is generated by (z^{n}/π) · (f(zW)/z). Hence we have

X_{log}^{j}(A^{∼} → A)(K̄) = ∐_{σ:K_{1}→K′}{w ∈ OK̄|ord f^{σ}(zw)/z^{n} ≥ j}

We fix an embedding σ : K_{1} → K̄ and drop σ in the notation. For i = 1, …, n, we put U_{i} = {w|ord(w − z_{i}/z) ≥ ord(w − z_{k}/z) for k = 1, …, n}. By the equality above, to prove (2), it is sufficient to show

{w ∈ OK̄|ord f(zw)/z^{n} ≥ j} ∩ U_{i} ⊂ D(z_{i}/z, π^{ψ(j)}) ⊂ {w ∈ OK̄|ord f(zw)/z^{n} ≥ j}

for each i. Let w ∈ OK̄. We put I_{1} = {k : ord(w − z_{i}/z) > ord(w − z_{k}/z)} and I_{2} = {k : ord(w − z_{i}/z) ≤ ord(w − z_{k}/z)}. For i ∈ I_{1}, we have ord(w − z_{k}/z) = ord(z_{k} − z_{i})/z < ord(w − z_{i}/z) and, for i ∈ I_{2}, we have ord(w − z_{k}/z) ≥ ord(w −z_{i}/z) with the equality if x ∈ U_{i}. Since f(zW)/z^{n} = ∏_{k=1}^{n}(W −z_{k}/z), we have an inequality

ord\frac{f(zw)}{z^{n}} = ∑_{k=1}^{n} ord(w − \frac{z_{k}}{z}) ≥

≥ ∑_{k∈I_{1}} ord(\frac{z_{k}}{z} − \frac{z_{i}}{z}) + ∑_{k∈I_{2}} ord(w − \frac{z_{i}}{z}) = φ(ord(w − \frac{z_{i}}{z})).

We have an equality if x ∈ U_{i}. Thus the equality (2) is proved. The last assertion follows from the equality (2) and s_{n−1} = max_{i≠k} ord (z_{i}/z − z_{k}/z).

3. We show h_{i}^{σ}(T) ∈ OK̄[T]. We extend σ : K_{1} → K̄ to σ_{i} : L → K̄ by sending π_{L} to z_{i}^{σ} and put u = ψ(j). Then we have h_{i}^{σ}(T) = −h^{σ_{i}}(π^{u} · (z/z_{i}) · T)/π^{φ(u)}f(0). Since z/z_{i} and f(0)/z^{n} are units, we have h_{i}^{σ}(T) ∈ OK̄[T] by the definition of φ(u). We show the commutativity of the diagram. Since B → A sends S to T^{n}g(T)^{−1}, the induced map B^{∼} → A^{∼} ⊗_{O_{K}}^{log} O_{K′} sends V to

\frac{T^{n}}{π · g(T)} = \frac{f(T)}{π · g(T)} + 1 = \frac{f((1 ⊗ z)W)}{π · g((1 ⊗ z)W)} + 1.

We fix σ : K_{1} → K and we drop σ in the notation. We define a map D(z_{i}/z, π^{ψ(j)}) → D(1, π^{j}) by sending w to (f(zw)/(πg(zw))) + 1. Then, we have a commutative diagram

D(\frac{z_{i}}{z} , π^{ψ(j)}) ⟶^{⊂} X_{log}^{j}(A^{∼} → A)
↓  ↓
D(1, π^{j})  X_{log}^{j}(B^{∼} → O_{K}).

The polynomial g(zW) is congruent to the constant −f(0)/π modulo the maximal ideal. Hence, by substituting W = π^{ψ(j)}T + z_{i}/z, we get the assertion. □ Proof of Proposition 6.3. 1. The equality G_{K,log}^{j} = G_{K}^{j+1} follows from Lemma 6.4.1. The rest is clear.

2. First we show that the map π_{1}^{ab}(N^{j}) → Gr_{log}^{j}G_{K} factors the quotient π_{1}^{ab,gp}(N^{j}). By Lemma 6.4.3, it is sufficient to show that the map h̄_{i}^{σ} : 𝔸^{1} → 𝔸^{1} is an isogeny. In other words, it is enough to show that if the coefficient of T^{r} in h̄_{i}^{σ} is non-zero, then r is a power of p. Since h_{i}^{σ}(T) = −h^{σ_{i}}(π^{u} · (z/z_{i}) · T)/π^{φ(u)}f(0) and z/z_{i} and f(0)/z^{n} are units, the coefficient of T^{r} in h̄_{i}^{σ} is non-zero if and only if the coefficient of T^{r} in \overline{h(π^{u}T)/π^{φ(u)}} is non-zero. Let l_{h} be the function defining the Newton polygon of h. We apply Lemma 6.2 to h(T) = f(π_{L}(T + 1))/π_{L}^{n} = ∑_{i=0}^{n−1} b_{i}T^{n−i}. Then, if the coefficient of T^{r} in \overline{h(π^{u}T)/π^{φ(u)}} is non-zero, we have l_{h}(r) = ord b_{r}. Since ord z = 1/n, we may apply Lemma 6.1 to the polynomial h(T). Thus the equality l_{h}(r) = ord b_{r} implies that r is a power of p as required. We show that the surjection π_{1}^{ab,gp}(N^{j}) → Gr_{log}^{j}G_{K} is an isomorphism. By Lemma 5.2, we may replace K by the completion of a maximum unramified extension and assume the residue field F is algebraically closed. To show the isomorphism, it is sufficient to construct every étale isogeny of degree p to N^{j} from a finite separable extension of K. Recall that every étale isogeny of degree p to N^{j} is obtained by pulling-back the isogeny 𝔸^{1} → 𝔸^{1} defined by the polynomial T^{p} − T by an isomorphism N^{j} → 𝔸^{1}. We show the following Lemma.

Lemma 6.5 Let n, m, l ≥ 1 be integers such that m ≤ n and pl ≤ n, m and l are prime to p and that p^{2}|n. Let π be a prime element of O_{K} and a, b be element of O_{K}. We put m′ = n · ord a + m and l′ = n · ord b + pl and assume pl′ < m′ < pl′ + n · ord p and pl′ < n · ord p · ord_{p}(n/p^{2}). Let f(T) be the Eisenstein polynomial

f(T) = T^{n} − π(aT^{m} − bT^{pl} + 1)

and let z = π_{L} be the image of T in L = K[T]/f(T). We put

j = \frac{p}{p − 1} · \frac{m′ − l′}{n}  and  π^{j} = maz^{m} (\frac{maz^{m}}{bz^{pl}} )^{\frac{1}{p−1}} .

Then,

1. The log ramification of the extension L = K[T]/(f(T)) is bounded by j+.

2. We define a map (B^{∼} → O_{K}) → (A^{∼} → O_{L}) as above and consider X_{log}^{j}(A^{∼} → A)_{L} as an affinoid subdomain of SpO_{L}⟨W ⟩ by taking K′ = L and z to be the image of T. Let 𝔸^{1} → 𝔸^{1} be the map defined by the polynomial T^{p}−T. Then, D(1, π^{ψ(j)}) is a connected component of X_{log}^{j}(A^{∼} → A). Further, the isomorphism ×π^{j} + 1 : D(0, 1) → D(1, π^{j}) induce a commutative diagram

𝔸^{1} ⟶ \overline{D(1, π^{ψ(j)})} ⊂ X̄_{log}^{j}(A^{∼} → A)
↓  ↓
𝔸^{1} ⟶ \overline{D(1, π^{j})} = X̄_{log}^{j}(B^{∼} → O_{K})= N^{j}.

Proof. 1. We put h(T) = f(z(T + 1))/z^{n} and let l_{h} : [0, n] → ℝ ∪ {∞} be the function defining the Newton polygon of h(T). Let l_{1} : [0, n] → ℝ ∪ {∞} be the linear function characterized by l_{1}(n − 1) = m′/n and l_{1}(n − p) = pl′/n. We claim that we have an equality l_{h} = l_{1} on and only on the interval (n − p, n − 1). By Lemma 6.1, it is sufficient to show l_{h}(n − 1) = m′/n, l_{h}(n − p) = pl′/n and l_{h}(n − p^{2}) > l_{1}(n − p^{2}). By the proof of Lemma 6.1, we have l_{h}(n − 1) = min(ord n, ord maz^{m}, ord plbz^{pl}) = min(ord p · ord_{p}n, m′/n, ord p + pl′/n). By the assumptions, we have m′ < n · ord p + pl′ < n · ord p · ord_{p}n and l_{h}(n − 1) = m′/n. Similarly, we have l_{h}(n − p) = min(ord \binom{n}{p}, m′/n, ord \binom{pl}{p}bz^{pl}) = min(ord p · ord_{p}(n/p), m′/n, pl′/n) = pl′/n and l_{h}(n − p^{2}) ≥ min(ord \binom{n}{p^{2}}, m′/n, pl′/n) = pl′/n ≥ l_{1}(n − p) > l_{1}(n − p^{2}). Thus the claim is proved. By Lemma 6.4.2, it is sufficient to show that the slope s_{n−1} of l_{h} on the interval (n − 2, n − 1) is ψ(j). By the claim above, we have s_{n−1} = (l_{h}(n − 1) − l_{h}(n − p))/(p−1) and φ(s_{n−1}) = l_{h}(n−1)+s_{n−1} = (p·l_{h}(n−1)−l_{h}(n−p))/(p−1) = p(m′ − l′)/(p − 1)n = j. Thus the assertion follows.

2. In Lemma 6.4.3, we put π^{ψ(j)} = (maz^{m}/bz^{pl})^{1/(p−1)} and π^{j} = maz^{m}π^{ψ(j)}. Then we have

−\frac{f(z(π^{ψ(j)}T + 1))}{π^{j}f(0)} ≡ −\frac{−\binom{pl}{p}bz^{pl}π^{pψ(j)}T^{p} + maz^{m}π^{ψ(j)}T}{π^{j}} ≡ T^{p} − T.

Hence the assertion follows. □ We complete the proof of Proposition 6.3.2. By Lemma 6.5, it is sufficient to show the following: For every rational number j > 0, there exist integers n, m′, l′ > 0 satisfying the conditions in Lemma 6.5 and, for every non-zero element x of N^{j}, there exist a, b ∈ O_{K} such that ord a is the integral part of m′/n, ord b is the integral part of pl′/n and x ≡ maz^{m}(maz^{m}/bz^{pl})^{1/(p−1)}. First, we prove the claim for j. Assume p is odd (resp. even). Let n > 0 be an integer such that n(p − 1)j/p (resp. n(p − 1)j/2p) and n/p^{2} are integers and (p − 1)j/p ∈ [(p + 1)/n, (p − 1)n/p^{2} · ord p · ord_{p}(n/p^{2}) − (p + 1)/n]. Then there exist integers l′, m′ such that (p − 1)j/p = (m′ − l′)/n, l′ and m′ are prime to p, pl′ < m′ < pl′ + n · ord p and pl′ < n · ord p · ord_{p}(n/p^{2}). Thus the claim is proved for j. Since we may multiply a an arbitrary unit, the claim for x is clear. Hence the assertion is proved. □

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Ahmed Abbes LAGA, Institut Galilée Université Paris-Nord 93430 Villetaneuse, France abbes@math.univ-paris13.fr Takeshi Saito Department of Mathematical Sciences University of Tokyo Tokyo 153-8914 Japan t-saito@ms.u-tokyo.ac.jp