Equality of Two Non-Logarithmic Ramification

Filtrations of Abelianized Galois Group

in Positive Characteristic

Yuri Yatagawa

Received: Septeber 7, 2016 Revised: April 17, 2017

Communicated by Thomas Geisser

Abstract. We prove the equality of two non-logarithmic ramification filtrations defined by Matsuda and Abbes-Saito of the abelianized absolute Galois group of a complete discrete valuation field in positive characteristic. We compute the refined Swan conductor and the characteristic form of a character of the fundamental group of a smooth separated scheme over a perfect field of positive characteristic by using sheaves of Witt vectors.

2010 Mathematics Subject Classification: primary 11S15, secondary 14G22 Keywords and Phrases: local field, ramification filtration, characteristic form, Witt vector.

Introduction

Let K be a complete discrete valuation field with residue field F_{K} and G_{K} = Gal(K^{sep}/K) the absolute Galois group of K. In [Se], the definition of (upper numbering) ramification filtration of G_{K} is given in the case where F_{K} is perfect. In the general residue field case, Abbes-Saito ([AS1]) have given definitions of two ramification filtrations of G_{K} geometrically, one is logarithmic and the other is non-logarithmic. In Saito’s recent work ([Sa1], [Sa2]) on characteristic cycle of a constructible sheaf, the non-logarithmic filtration in equal characteristic plays important roles to give an example of characteristic cycle.

Assume that K is of positive characteristic. Let H^{1}(K, Q/Z) be the character group of G_{K}. In this case, Matsuda ([M]) has defined a non-logarithmic ramification filtration of H^{1}(K, Q/Z) as a non-logarithmic variant of Brylinski-Kato’s logarithmic filtration ([B], [K1]) using Witt vectors. In this paper, we prove that the abelianization of Abbes-Saito’s non-logarithmic filtration {G_{K}^{r}}_{r∈Q_{≥1}} is the same as Matsuda’s filtration {fil′_{m}H^{1}(K, Q/Z)}_{m∈Z_{≥1}} by taking dual, which enable us to compute abelianized Abbes-Saito’s filtration by using Witt vectors. This is stated as follows and proved in Section 3:

Theorem 0.1. Let m ≥ 1 be an integer and r a rational number such that m ≤ r < m + 1. For χ ∈ H^{1}(K, Q/Z), the following are equivalent:

(i) χ ∈ fil′_{m}H^{1}(K, Q/Z).

(ii) χ(G_{K}^{m+}) = 0.

(iii) χ(G_{K}^{r+}) = 0.

For m > 2, Theorem 0.1 has been proved by Abbes-Saito ([AS3]). The proof goes similarly as the proof by Abbes-Saito (loc. cit.). The proof in this paper relies on the characteristic form defined by Saito ([Sa1]) even in the exceptional case where p = 2 and an explicit computation of the characteristic form. Let X be a smooth separated scheme over a perfect field of positive characteristic and U = X − D the complement of a divisor D on X with simple normal crossings. The characteristic form of a character of the abelianized fundamental group π_{1}^{ab}(U) is an element of the restriction to a radicial covering of a sub divisor Z of D of a differential module of X. We compute the characteristic form using sheaves of Witt vectors. By taking X and D so that the local field at a generic point of D is K and using the injections defined by the characteristic form from the graded quotients of {fil′_{m}H^{1}(K, Q/Z)}_{m∈Z_{≥1}} and the modules of characters of the graded quotients of {G_{K}^{r}}_{r∈Q_{≥1}}, we obtain the proof of Theorem 0.1. This paper consists of three sections. In Section 1, we recall Kato and Matsuda’s ramification theories in positive characteristic. We give some complements to these theories to compute the refined Swan conductor ([K1]) and the characteristic form for a character of the fundamental group of a smooth separated scheme over a perfect field of positive characteristic in terms of sheaves of Witt vectors. In Section 2, we recall Abbes-Saito’s non-logarithmic ramification theory in positive characteristic in terms of schemes over a perfect field. We recall the definition of the characteristic form defined by Saito and show that this characteristic form is computed with sheaves of Witt vectors. Section 3 is devoted to prove Theorem 0.1. This paper is a refinement of a part of the author’s thesis at University of Tokyo. The author would like to express her sincere gratitude to her supervisor Takeshi Saito for suggesting her to refine the computation of characteristic form using sheaves of Witt vectors, reading the manuscript carefully, and giving a lot of advice on the manuscript. The research was partially supported by the Program for Leading Graduate Schools, MEXT, Japan and JSPS KAKENHI Grant Number 15J03851.

Contents

1 Kato and Matsuda’s ramification theories and complements919 1.1 Local theory: logarithmic case . . . . . . . . . . . . . . . . . . . 919 1.2 Local theory: non-logarithmic case . . . . . . . . . . . . . . . . 924 1.3 Sheafification: logarithmic case . . . . . . . . . . . . . . . . . . 930 1.4 Sheafification: non-logarithmic case . . . . . . . . . . . . . . . . 937

2 Abbes-Saito’s ramification theory and Witt vectors 942 2.1 Abbes-Saito’s ramification theory . . . . . . . . . . . . . . . . . 942 2.2 Valuation of Witt vectors . . . . . . . . . . . . . . . . . . . . . 944 2.3 Calculation of characteristic forms . . . . . . . . . . . . . . . . 948

3 Equality of ramification filtrations 950

1 Kato and Matsuda’s ramification theories and complements

1.1 Local theory: logarithmic case

We recall Kato’s ramification theory ([K1], [K2]) and prove some properties of graded quotients of some filtrations for the proof of Proposition 1.29 in Subsection 1.3. Let K be a complete discrete valuation field of characteristic p > 0. We regard H_{ét}^{1}(K, Z/nZ) as a subgroup of H_{ét}^{1}(K, Q/Z) = \underrightarrow{lim}_{n} H_{ét}^{1}(K, Z/nZ). Let W_{s}(K) be the Witt ring of K of length s ≥ 0. By definition, W_{0}(K) = 0 and W_{1}(K) = K. We write

F : W_{s}(K) → W_{s}(K); (a_{s−1}, ⋯, a_{0}) ↦ (a_{s−1}^{p}, ⋯, a_{0}^{p})

for the Frobenius. By the Artin-Schreier-Witt theory, we have the exact sequence

0 → W_{s}(F_{p}) → W_{s}(K) ⟶^{F −1} W_{s}(K) → H^{1}(K, Z/p^{s}Z) → 0.  (1.1)

We define

δ_{s} : W_{s}(K) → H^{1}(K, Q/Z)  (1.2)

to be the composition

W_{s}(K) → H^{1}(K, Z/p^{s}Z) → H^{1}(K, Q/Z),

where the first arrow is the fourth morphism in (1.1). Let O_{K} be the valuation ring of K and F_{K} the residue field of K. We write G_{K} for the absolute Galois group of K.

Definition 1.1 ([K1, Definition (3.1)]). Let s ≥ 0 be an integer.

(i) Let a = (a_{s−1}, …, a_{0}) be an element of W_{s}(K). We define ord_{K}(a) by ord_{K}(a) = min_{0≤i≤s−1}{p^{i} ord_{K}(a_{i})}.

(ii) We define an increasing filtration {fil_{n}W_{s}(K)}_{n∈Z} of W_{s}(K) by

fil_{n}W_{s}(K) = {a ∈ W_{s}(K) | ord_{K}(a) ≥ −n}.  (1.3)

The filtration {fil_{n}W_{s}(K)}_{n∈Z} in Definition 1.1 is first defined by Brylinski ([B, Proposition 1]) and fil_{n}W_{s}(K) is a submodule of W_{s}(K) for n ∈ Z (loc. cit.). Let n ≥ 0 be an integer and put s′ = ord_{p}(n). Suppose that s′ < s. Let V denote the Verschiebung

V : W_{s}(K) → W_{s+1}(K); (a_{s−1}, ⋯, a_{0}) ↦ (0, a_{s−1}, ⋯, a_{0}).

Since (a_{s−1}, …, a_{0}) = (a_{s−1}, …, a_{s′+1}, 0, …, 0)+V^{s−s′−1}(a_{s′}, …, a_{0}), we have

fil_{n}W_{s}(K) = fil_{n−1}W_{s}(K) + V^{s−s′−1}fil_{n}W_{s′+1}(K).  (1.4)

Definition 1.2 ([K1, Corollary (2.5), Theorem (3.2) (1)]). Let δ_{s} be as in (1.2).

(i) We define an increasing filtration {fil_{n}H^{1}(K, Z/p^{s}Z)}_{n∈Z_{≥0}} of H^{1}(K, Z/p^{s}Z) by

fil_{n}H^{1}(K, Z/p^{s}Z) = δ_{s}(fil_{n}W_{s}(K)).

(ii) We define an increasing filtration {fil_{n}H^{1}(K, Q/Z)}_{n∈Z_{≥0}} of H^{1}(K, Q/Z) by

fil_{n}H^{1}(K, Q/Z) = H^{1}(K, Q/Z){p′} + ⋃_{s≥1} δ_{s}(fil_{n}W_{s}(K)),  (1.5)

where H^{1}(K, Q/Z){p′} denotes the prime-to-p part of H^{1}(K, Q/Z).

Definition 1.3 ([K1, Definition (2.2)]). Let χ be an element of H^{1}(K, Q/Z). We define the Swan conductor sw(χ) of χ by sw(χ) = min{n ∈ Z_{≥0} | χ ∈ fil_{n}H^{1}(K, Q/Z)}.

We recall the definition of refined Swan conductor of χ ∈ H^{1}(K, Q/Z) given by Kato ([K2, (3.4.2)]). Let Ω_{K}^{1} be the differential module of K over K^{p} ⊂ K.

Definition 1.4. We define an increasing filtration {fil_{n}Ω_{K}^{1}}_{n∈Z_{≥0}} of Ω_{K}^{1} by

fil_{n}Ω_{K}^{1} = {(αdπ/π + β)/π^{n} | α ∈ O_{K}, β ∈ Ω_{O_{K}}^{1} } = 𝔪^{−n}Ω_{O_{K}}^{1} (log), (1.6)

where π is a uniformizer of K and 𝔪 is the maximal ideal of O_{K}.

We consider the morphism

−F^{s−1}d : W_{s}(K) → Ω_{K}^{1}; (a_{s−1}, ⋯, a_{0}) ↦ − ∑_{i=0}^{s−1} a_{i}^{p^{i}−1}da_{i}.  (1.7)

The morphism −F^{s−1}d (1.7) satisfies −F^{s−1}d(fil_{n}W_{s}(K)) ⊂ fil_{n}Ω_{K}^{1}. We put gr_{n} = fil_{n}/fil_{n−1} for n ∈ Z_{≥1}. Then, for n ∈ Z_{≥1}, the morphism (1.7) induces

φ_{s}^{(n)} : gr_{n}W_{s}(K) → gr_{n}Ω_{K}^{1}.

Let δ_{s}^{(n)} : gr_{n}W_{s}(K) → gr_{n}H^{1}(K, Q/Z) denote the morphism induced by δ_{s} (1.2) for n ∈ Z_{≥1}. For n ∈ Z_{≥1}, there exists a unique injection ϕ^{(n)} : gr_{n}H^{1}(K, Q/Z) → gr_{n}Ω_{K}^{1} such that the diagram

\xymatrix{gr_{n}W_{s}(K) \ar[rr]^{φ_{s}^{(n)}} \ar[dr]_{δ_{s}^{(n)}} &  & gr_{n}Ω_{K}^{1} \\  & gr_{n}H^{1}(K, Q/Z) \ar[ur]_{ϕ^{(n)}}}  (1.8)

is commutative for any s ∈ Z_{≥0} by [M, Remark 3.2.12], or [AS3, §10] for more detail. We note that gr_{n}Ω_{K}^{1} ≃ 𝔪^{−n}Ω_{O_{K}}^{1} (log) ⊗_{O_{K}} F_{K} is a vector space over F_{K}.

Definition 1.5 ([K2, (3.4.2)], [M, Remark 3.2.12], see also [AS3, Définition 10.16]). Let χ be an element of H^{1}(K, Q/Z). We put n = sw(χ). If n ≥ 1, then we define the refined Swan conductor rsw(χ) of χ to be the image of χ by ϕ^{(n)} in (1.8).

In the rest of this subsection, we prove some properties of graded quotients of filtrations. For q ∈ R, let [q] denote the integer n such that q − 1 < n ≤ q.

Lemma 1.6. Let m and r ≥ 0 be integers.

(i) [m/p^{r}] = [(m − 1)/p^{r}] + 1 if m ∈ p^{r}Z and [m/p^{r}] = [(m − 1)/p^{r}] if m ∉ p^{r}Z.

(ii) [[m/p^{r}]/p] = [m/p^{r+1}] = [[m/p]/p^{r}].

Proof. (i) We put m = p^{r}q + a, where q, a ∈ Z and 0 ≤ a < p^{r}. Then [m/p^{r}] = q. Further [(m − 1)/p^{r}] = q + [(a − 1)/p^{r}]. Since [(a − 1)/p^{r}] = −1 if a = 0 and [(a − 1)/p^{r}] = 0 if 0 < a < p^{r}, the assertion holds. (ii) We put m = p^{r+1}q′ + a′, where q′, a′ ∈ Z and 0 ≤ a′ < p^{r+1}. Then [m/p^{r}] = pq′ + [a′/p^{r}] and 0 ≤ [a′/p^{r}] < p. Further [m/p] = p^{r}q′ + [a′/p] and 0 ≤ [a′/p] < p^{r}. Hence we have [[m/p^{r}]/p] = q′ = [m/p^{r+1}] and [[m/p]/p^{r}] = q′ = [m/p^{r+1}].

Lemma 1.7. Let a be an element of W_{s}(K).

(i) ord_{K}(F (a)) = p · ord_{K}(a).

(ii) ord_{K}((F − 1)(a)) = p · ord_{K}(a) if ord_{K}(a) < 0 and ord_{K}((F − 1)(a)) ≥ 0 if ord_{K}(a) ≥ 0.

(iii) For an integer n ≥ 0, we have F^{−1}(fil_{n}W_{s}(K)) = (F −1)^{−1}(fil_{n}W_{s}(K)) = fil_{[n/p]}W_{s}(K).

Proof. (i) We put a = (a_{s−1}, …, a_{0}). Since F (a) = (a_{s−1}^{p}, …, a_{0}^{p}), the assertion holds. (ii) Suppose that ord_{K}(a) ≥ 0. Then, since both a and F (a) belong to fil_{0}W_{s}(K), we have (F −1)(a) ∈ fil_{0}W_{s}(K). Hence we have ord_{K}((F −1)(a)) ≥ 0 by (1.3). Suppose that ord_{K}(a) < 0. We put ord_{K}(a) = −n. Since both a and F (a) belong to fil_{pn}W_{s}(K), we have (F − 1)(a) ∈ fil_{pn}W_{s}(K). Since ord_{K}(F (a)) = −pn < ord_{K}(a) = −n, we have (F − 1)(a) ∉ fil_{pn−1}W_{s}(K). Hence we have ord_{K}((F − 1)(a)) = −pn. (iii) By (i), we have F (a) ∈ fil_{n}W_{s}(K) if and only if ord_{K}(a) ≥ −n/p for a ∈ W_{s}(K). Hence we have F^{−1}(fil_{n}W_{s}(K)) = fil_{[n/p]}W_{s}(K). By (ii), we have (F − 1)^{−1}(fil_{n}W_{s}(K)) = fil_{[n/p]}W_{s}(K) similarly.

Let n ≥ 1 be an integer. By Lemma 1.7 (iii), the Frobenius F: W_{s}(K) → W_{s}(K) induces the injection

F̄: fil_{[n/p]}W_{s}(K)/fil_{[(n−1)/p]}W_{s}(K) → gr_{n}W_{s}(K).  (1.9)

By Lemma 1.6 (i), the domain of (1.9) is equal to gr_{n/p}W_{s}(K) if n ∈ pZ and it is 0 if n ∉ pZ. By Lemma 1.7 (iii), the morphism F − 1: W_{s}(K) → W_{s}(K) induces the injection

\overline{F − 1}: fil_{[n/p]}W_{s}(K)/fil_{[(n−1)/p]}W_{s}(K) → gr_{n}W_{s}(K).  (1.10)

Since [n/p] < n if n ≥ 1, the morphisms (1.9) and (1.10) are the same.

Lemma 1.8 (cf. [K1, Theorem (3.2), Corollary (3.3)]). Let n ≥ 1 be an integer. Then we have the exact sequence

0 → fil_{[n/p]}W_{s}(K)/fil_{[(n−1)/p]}W_{s}(K) ⟶^{F̄} gr_{n}W_{s}(K) ⟶^{φ_{s}^{(n)}} gr_{n}Ω_{K}^{1},

where fil_{[n/p]}W_{s}(K)/fil_{[(n−1)/p]}W_{s}(K) is gr_{n/p}W_{s}(K) if n ∈ pZ and 0 if n ∉ pZ.

Proof. As in the proof of [AS3, Proposition 10.7], the morphism φ_{s}^{(n)} factors through

gr_{n}H^{1}(K, Z/p^{s}Z) ≃ fil_{n}W_{s}(K)/((F − 1)(W_{s}(K)) ∩ fil_{n}W_{s}(K) + fil_{n−1}W_{s}(K)).

Since this factorization defines the injection ϕ^{(n)} in (1.8) by [AS3, Proposition 10.14] and since the morphism F̄ (1.9) is equal to the morphism \overline{F − 1} (1.10), the assertion holds.

Definition 1.9. Let s ≥ 0 and r ≥ 0 be integers. We define an increasing filtration {fil_{n}^{(r)}W_{s}(K)}_{n∈Z_{≥0}} of W_{s}(K) by

fil_{n}^{(r)}W_{s}(K) = {a ∈ W_{s}(K) | ord_{K}(a) ≥ −n/p^{r}} = fil_{[n/p^{r}]}W_{s}(K). (1.11)

By (1.11), we have fil_{n}^{(0)}W_{s}(K) = fil_{n}W_{s}(K) for n ∈ Z_{≥0}. For integers 0 ≤ t ≤ s, let pr_{t} denote the projection

pr_{t}: W_{s}(K) → W_{t}(K) ; (a_{s−1}, …, a_{0}) ↦ (a_{s−1}, …, a_{s−t}).  (1.12)

We put gr_{n}^{(r)} = fil_{n}^{(r)}/fil_{n−1}^{(r)} for r ∈ Z_{≥0} and n ∈ Z_{≥1}.

Lemma 1.10. Let r ≥ 0 and 0 ≤ t ≤ s be integers. Let pr_{t}: W_{s}(K) → W_{t}(K) be as in (1.12). Let n ≥ 0 be an integer.

(i) pr_{t}(fil_{n}W_{s}(K)) = fil_{n}^{(s−t)}W_{t}(K).

(ii) (F − 1)^{−1}(fil_{n}^{(r)}W_{s}(K)) = fil_{[n/p]}^{(r)}W_{s}(K).

Proof. (i) By (1.3), we have pr_{t}(fil_{n}W_{s}(K)) = fil_{[n/p^{s−t}]}W_{t}(K). Hence the assertion holds by (1.11). (ii) By Lemma 1.7 (iii) and (1.11), we have (F − 1)^{−1}(fil_{n}^{(r)}W_{s}(K)) = fil_{[[n/p^{r}]/p]}W_{s}(K). By Lemma 1.6 (ii) and (1.11), the assertion holds.

Let n ≥ 0 and 0 ≤ t ≤ s be integers. Since pr_{t}(fil_{n}W_{s}(K)) = fil_{n}^{(s−t)}W_{t}(K) by Lemma 1.10 (i), we have the exact sequence

0 → fil_{n}W_{s−t}(K) ⟶^{V^{t}} fil_{n}W_{s}(K) ⟶^{pr_{t}} fil_{n}^{(s−t)}W_{t}(K) → 0.  (1.13)

Lemma 1.11. Let n ≥ 1 be an integer. Then the exact sequence (1.13) induces the exact sequence

0 → gr_{n}W_{s−t}(K) ⟶^{V̄^{t}} gr_{n}W_{s}(K) ⟶^{\overline{pr}_{t}} gr_{n}^{(s−t)}W_{t}(K) → 0,

where gr_{n}^{(s−t)}W_{t}(K) is equal to gr_{n/p^{s−t}}W_{t}(K) if n ∈ p^{s−t}Z and 0 if n ∉ p^{s−t}Z.

Proof. We consider the commutative diagram

\xymatrix{0 \ar[r] & fil_{n−1}W_{s−t}(K) \ar[r]^{V^{t}} \ar[d] & fil_{n−1}W_{s}(K) \ar[r]^{^{pr^{t}}} \ar[d] & fil_{n−1}^{(s−t)}W_{t}(K) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & fil_{n}W_{s−t}(K) \ar[r]^{V^{t}} & fil_{n}W_{s}(K) \ar[r]^{_{pr_{t}}} & fil_{n}^{(s−t)}W_{t}(K) \ar[r] & 0,} (1.14)

where the horizontal lines are exact and the vertical arrows are inclusions. By applying the snake lemma to (1.14), we obtain the exact sequence which we have desired. The last supplement to gr_{n}^{(s−t)}W_{t}(K) follows by Lemma 1.6 (i) and (1.11).

1.2 Local theory: non-logarithmic case

We recall a non-logarithmic variant, given by Matsuda ([M]), of Kato’s logarithmic ramification theory recalled in Subsection 1.1, and we consider the exceptional case of Matsuda’s theory. We also consider the graded quotients of filtrations. We keep the notation in Subsection 1.1.

Definition 1.12 (cf. [M, 3.1]). We define an increasing filtration {fil′_{m}W_{s}(K)}_{m∈Z_{≥1}} of W_{s}(K) by

fil′_{m}W_{s}(K) = fil_{m−1}W_{s}(K) + V^{s−s′}fil_{m}W_{s′}(K).  (1.15)

Here s′ = min{ord_{p}(m), s}.

The definition of {fil′_{m}W_{s}(K)}_{m∈Z_{≥1}} in Definition 1.12 is shifted by 1 from Matsuda’s definition ([M, 3.1]). Since fil_{n}W_{s}(K) is a submodule of W_{s}(K) for n ∈ Z, the subset fil′_{m}W_{s}(K) is a submodule of W_{s}(K) for m ∈ Z_{≥1}. By (1.15), we have

fil_{m−1}W_{s}(K) ⊂ fil′_{m}W_{s}(K) ⊂ fil_{m}W_{s}(K)  (1.16)

for m ∈ Z_{≥1}. Since min{ord_{p}(1), s} = 0 for s ∈ Z_{≥0}, we have

fil_{0}W_{s}(K) = fil′_{1}W_{s}(K).  (1.17)

Definition 1.13 (cf. [M, Definition 3.1.1]). Let δ_{s} be as in (1.2).

(i) We define an increasing filtration {fil′_{m}H^{1}(K, Z/p^{s}Z)}_{m∈Z_{≥1}} of H^{1}(K, Z/p^{s}Z) by

fil′_{m}H^{1}(K, Z/p^{s}Z) = δ_{s}(fil′_{m}W_{s}(K)).

(ii) We define an increasing filtration {fil′_{m}H^{1}(K, Q/Z)}_{m∈Z_{≥1}} of H^{1}(K, Q/Z) by

fil′_{m}H^{1}(K, Q/Z) = H^{1}(K, Q/Z){p′} + ⋃_{s≥1} δ_{s}(fil′_{m}W_{s}(K)), (1.18)

where H^{1}(K, Q/Z){p′} denotes the prime-to-p part of H^{1}(K, Q/Z).

By (1.16), we have

fil_{m−1}H^{1}(K, Q/Z) ⊂ fil′_{m}H^{1}(K, Q/Z) ⊂ fil_{m}H^{1}(K, Q/Z)  (1.19)

for m ∈ Z_{≥1}. By (1.17), we have fil_{0}H^{1}(K, Q/Z) = fil′_{1}H^{1}(K, Q/Z).

Definition 1.14 (cf. [M, Definition 3.2.5]). Let χ be an element of H^{1}(K, Q/Z). We define the total dimension dt(χ) of χ by dt(χ) = min{m ∈ Z_{≥1} | χ ∈ fil′_{m}H^{1}(K, Q/Z)}.

Definition 1.15. We define an increasing filtration {fil′_{m}Ω_{K}^{1}}_{m∈Z_{≥1}} of Ω_{K}^{1} by

fil′_{m}Ω_{K}^{1} = {γ/π^{m} | γ ∈ Ω_{O_{K}}^{1} } = 𝔪^{−m}Ω_{O_{K}}^{1} ,

where π is a uniformizer of K and 𝔪 is the maximal ideal of O_{K}.

Since 𝔪Ω_{O_{K}}^{1} (log) ⊂ Ω_{O_{K}}^{1} ⊂ Ω_{O_{K}}^{1} (log), we have

fil_{m−1}Ω_{K}^{1} ⊂ fil′_{m}Ω_{K}^{1} ⊂ fil_{m}Ω_{K}^{1}  (1.20)

for m ∈ Z_{≥1}. We consider the morphism (1.7). The morphism (1.7) satisfies −F^{s−1}d(fil′_{m}W_{s}(K)) ⊂ fil′_{m}Ω_{K}^{1} for m ∈ Z_{≥1}. We put gr′_{m} = fil′_{m}/fil′_{m−1} for m ∈ Z_{≥2}. Then, for m ∈ Z_{≥2}, the morphism (1.7) induces

φ′_{s}^{(m)} : gr′_{m}W_{s}(K) → gr′_{m}Ω_{K}^{1}.  (1.21)

Let δ_{s}^{′(m)} : gr′_{m}W_{s}(K) → gr′_{m}H^{1}(K, Q/Z) denote the morphism induced by δ_{s} (1.2) for m ∈ Z_{≥2}. If (p, m) ≠ (2, 2), there exists a unique injection ϕ^{′(m)} : gr′_{m}H^{1}(K, Q/Z) → gr′_{m}Ω_{K}^{1} such that the diagram

\xymatrix{gr′_{m}W_{s}(K) \ar[rr]^{φ_{s}^{′(m)}} \ar[dr]_{δ_{s}^{′(m)}} &  & gr′_{m}Ω_{K}^{1} \\  & gr′_{m}H^{1}(K, Q/Z) \ar@{-->}[ur]_{ϕ^{′(m)}}}  (1.22)

is commutative for any s ∈ Z_{≥0} by [M, Proposition 3.2.3]. We note that gr′_{m}Ω_{K}^{1} ≃ 𝔪^{−m}Ω_{O_{K}}^{1} ⊗_{O_{K}} F_{K} is a vector space over F_{K}. We consider the exceptional case where (p, m) = (2, 2).

Lemma 1.16. Let s ≥ 1 be an integer. Assume that p = 2. Then V^{s−1} : K → W_{s}(K) induces an isomorphism gr′_{2}K → gr′_{2}W_{s}(K).

Proof. Since p = 2, we have s′ = min{ord_{p}(2), s} = 1. Hence we have

fil′_{2}W_{s}(K) = fil_{1}W_{s}(K) + V^{s−1}fil_{2}K
= fil′_{1}W_{s}(K) + V^{s−1}fil_{2}K

by applying (1.15) for the first equality and (1.4) and (1.17) for the second equality. Since fil_{2}K = fil′_{2}K by (1.15), the assertion holds.

Proposition 1.17. Assume that p = 2. Let F_{K}^{1/2} ⊂ F̄_{K} denote the subfield of an algebraic closure F̄_{K} of F_{K} consisting of the square roots of F_{K}.

(i) There exists a unique morphism

φ̃_{s}^{′(2)} : gr′_{2}W_{s}(K) → gr′_{2}Ω_{K}^{1} ⊗_{F_{K}} F_{K}^{1/2}

such that φ̃_{s}^{′(2)}(ā) = −da_{0} + \sqrt{\overline{π^{2}a_{0}}}dπ/π^{2} for every ā ∈ gr′_{2}W_{s}(K) whose lift in fil′_{2}W_{s}(K) is a = (0, …, 0, a_{0}) and for every uniformizer π ∈ K. Here \sqrt{\overline{π^{2}a_{0}}} ∈ F_{K}^{1/2} denotes the square root of the image \overline{π^{2}a_{0}} of π^{2}a_{0} in F_{K}.

(ii) There exists a unique injection ϕ̃^{′(2)} : gr′_{2}H^{1}(K, Q/Z) → gr′_{2}Ω_{K}^{1} ⊗_{F_{K}} F_{K}^{1/2} such that the following diagram is commutative for every s ≥ 0:

\xymatrix{gr′_{2}W_{s}(K) \ar[rr]^{φ̃_{s}^{′(2)}} \ar[dr]_{δ_{s}^{′(2)}} &  & gr′_{2}Ω_{K}^{1} ⊗_{F_{K}} F_{K}^{1/2} \\  & gr′_{2}H^{1}(K, Q/Z). \ar[ur]_{ϕ̃^{′(2)}}}  (1.23)

Proof. By Lemma 1.16, we may assume that s = 1.

(i) Let a be an element of fil′_{2}K and π a uniformizer of K. Since p = 2, we have fil′_{2}K = fil_{2}K by (1.15). Hence we have π^{2}a ∈ O_{K} by (1.3). Since −d(fil′_{2}K) ⊂ fil′_{2}Ω_{K}^{1}, we have −da + \sqrt{\overline{π^{2}a}}dπ/π^{2} ∈ gr′_{2}Ω_{K}^{1} ⊗_{F_{K}} F_{K}^{1/2}. If a ∈ fil′_{1}K, we have a ∈ O_{K} by (1.3) and (1.17). Since −d(fil′_{1}K) ⊂ fil′_{1}Ω_{K}^{1}, we have −da + \sqrt{\overline{π^{2}a}}dπ/π^{2} = 0 in gr′_{2}Ω_{K}^{1} ⊗_{F_{K}} F_{K}^{1/2}. For a, b ∈ fil′_{2}K, we have

\sqrt{\overline{π^{2}(a + b)}} = \sqrt{\overline{π^{2}a}} + \sqrt{\overline{π^{2}b}}, since p = 2.

We prove that \sqrt{\overline{π^{2}a}}dπ/π^{2} is independent of the choice of a uniformizer π of K. Let u ∈ O_{K}^{×} be a unit. Then, in gr′_{2}Ω_{K}^{1} ⊗_{F_{K}} F_{K}^{1/2}, we have

\sqrt{\overline{(uπ)^{2}a}}d(uπ)/(uπ)^{2} = u\sqrt{\overline{π^{2}a}}udπ/(uπ)^{2} = \sqrt{\overline{π^{2}a}}dπ/π^{2}.

Hence the assertion holds. (ii) Since p = 2 and fil′_{2}K = fil_{2}K, we have fil′_{2}K ∩ (F − 1)(K) = (F − 1)(fil_{1}K) by Lemma 1.7 (iii). Hence it is sufficient to prove that Ker φ̃_{1}^{′(2)} is the image of (F − 1)(fil_{1}K) in gr′_{2}K. Let a be an element of fil_{1}K. By (1.3), we may put a = a′/π, where a′ ∈ O_{K}. Then we have

φ̃_{1}^{′(2)}(ā^{2} − ā) = −ā′dπ/π^{2} + \sqrt{ā′^{2}}dπ/π^{2} = 0.  (1.24)

Conversely, let a ∈ fil′_{2}K be a lift of an element of Ker φ̃_{1}^{′(2)}. Since fil′_{2}K = fil_{2}K, we can put a = a′/π^{2}, where a′ ∈ O_{K}, by (1.3). Suppose that ord_{K}(a′) > 0, that is a ∈ fil_{1}W_{s}(K). Since φ̃_{1}^{′(2)}(ā) = −(a′π^{−1})dπ/π^{2} = 0, we have a′π^{−1} = 0 in F_{K}. Hence a ∈ fil_{0}K = fil′_{1}K, that is ā = 0 in gr′_{2}K. Assume that a′ ∈ O_{K}^{×} is a unit. Since we have

φ̃_{1}^{′(2)}(ā) = −da + \sqrt{ā′}dπ/π^{2} = 0,  (1.25)

we have \sqrt{ā′} ∈ F_{K}. Hence there exist a unit a′′ ∈ O_{K}^{×} and an element b ∈ fil_{1}K such that a = (F − 1)(a′′/π) + b. By (1.24) and (1.25), we have φ̃_{1}^{′(2)}(b̄) = 0. Hence we have b ∈ fil′_{1}K by the case where ord_{K}(a′) > 0, which is proved above. Therefore ā ∈ gr′_{2}K is the image of an element of (F − 1)(fil_{1}K).

Let m ≥ 2 be an integer. By abuse of notation, we write

ϕ^{′(m)} : gr′_{m}H^{1}(K, Q/Z) → gr′_{m}Ω_{K}^{1} ⊗_{F_{K}} F_{K}^{1/p}  (1.26)

for the composition of ϕ^{′(m)} in (1.22) and the inclusion gr′_{m}Ω_{K}^{1} → gr′_{m}Ω_{K}^{1} ⊗_{F_{K}} F_{K}^{1/p} if (p, m) ≠ (2, 2) and ϕ̃^{′(2)} in Proposition 1.17 (ii) if (p, m) = (2, 2).

Definition 1.18. Let χ be an element of H^{1}(K, Q/Z). We put m = dt(χ) and assume that m ≥ 2. We define the characteristic form char(χ) ∈ gr′_{m}Ω_{K}^{1} ⊗_{F_{K}} F_{K}^{1/p} of χ to be the image of χ by ϕ^{′(m)} (1.26).

By (1.22) and Proposition 1.17, we need F_{K}^{1/p} only in the case where p = 2 and χ ∈ fil′_{2}H^{1}(K, Q/Z) − fil_{1}H^{1}(K, Q/Z). In the rest of this subsection, we prepare some lemmas for the proof of Proposition 1.29.

Definition 1.19. Let s ≥ 0 and r ≥ 0 be integers. We put r′ = min{ord_{p}(m), s + r} and s′′ = max{0, r′ − r}. We define increasing filtrations {fil_{m}^{′(r)}W_{s}(K)}_{m∈Z_{≥1}} and {fil_{m}^{′′(r)}W_{s}(K)}_{m∈Z_{≥1}} of W_{s}(K) by

fil_{m}^{′(r)}W_{s}(K) = fil_{m−1}^{(r)}W_{s}(K) + V^{s−s′′}fil_{m}^{(r)}W_{s′′}(K),  (1.27)
fil_{m}^{′′(r)}W_{s}(K) = fil_{[(m−1)/p]}^{(r)}W_{s}(K) + V^{s−s′′}fil_{[m/p]}^{(r)}W_{s′′}(K).  (1.28)

If r = 0, then we simply write fil′′_{m}W_{s}(K) for fil_{m}^{′′(0)}W_{s}(K).

If r = 0, since s′′ = s′ = min{ord_{p}(m), s}, we have fil_{m}^{′(0)}W_{s}(K) = fil′_{m}W_{s}(K). Further we have

fil′′_{m}W_{s}(K) = fil_{[(m−1)/p]}W_{s}(K) + V^{s−s′}fil_{[m/p]}W_{s′}(K).  (1.29)

Lemma 1.20. Let r ≥ 0 and 0 ≤ t ≤ s be integers. Let pr_{t}: W_{s}(K) → W_{t}(K) be as in (1.12). Let m ≥ 1 be an integer.

(i) pr_{t}(fil′_{m}W_{s}(K)) = fil_{m}^{′(s−t)}W_{t}(K).

(ii) We have the exact sequence

0 → fil′_{m}W_{s−t}(K) ⟶^{V^{t}} fil′_{m}W_{s}(K) ⟶^{pr_{t}} fil_{m}^{′(s−t)}W_{t}(K) → 0. (1.30)

(iii) pr_{t}(fil′′_{m}W_{s}(K)) = fil_{m}^{′′(s−t)}W_{t}(K).

(iv) We have the exact sequence

0 → fil′′_{m}W_{s−t}(K) ⟶^{V^{t}} fil′′_{m}W_{s}(K) ⟶^{pr_{t}} fil_{m}^{′′(s−t)}W_{t}(K) → 0. (1.31)

(v) fil_{m}^{′′(r)}W_{s}(K) = (F − 1)^{−1}(fil_{m}^{′(r)}W_{s}(K)). Especially, fil′′_{m}W_{s}(K) = (F − 1)^{−1}(fil′_{m}W_{s}(K)).

Proof. We put s′ = min{ord_{p}(m), s}, r′ = min{ord_{p}(m), s + r}, and s′′ = max{0, r′ − r}.

(i) By (1.27), we have fil_{m}^{′(s−t)}W_{t}(K) = fil_{m−1}^{(s−t)}W_{t}(K) if t ≤ s − s′ and fil_{m}^{′(s−t)}W_{t}(K) = fil_{m−1}^{(s−t)}W_{t}(K) + V^{s−s′}fil_{m}^{(s−t)}W_{t−s+s′}(K) if t > s − s′. By Lemma 1.10 (i), we have pr_{t}(fil_{m−1}W_{s}(K)) = fil_{m−1}^{(s−t)}W_{t}(K) and, if t > s − s′, we have pr_{t}(V^{s−s′}fil_{m}W_{s′}(K)) = V^{s−s′}fil_{m}^{(s−t)}W_{t−s+s′}(K). Hence the assertion holds by (1.15). (ii) The assertion holds by (1.15) and (i). (iii) The assertion holds similarly as the proof of (i) by (1.28) and (1.29). (iv) The assertion holds by (1.29) and (iii).

(v) Since V^{s−s′′} and pr_{s−s′′} commute with F − 1, the morphisms V^{s−s′′}: W_{s′′}(K) → W_{s}(K) and pr_{s−s′′}: W_{s}(K) → W_{s−s′′}(K) induce V^{s−s′′}: (F − 1)^{−1}(fil_{m}^{(r)}W_{s′′}(K)) → (F − 1)^{−1}(fil_{m}^{′(r)}W_{s}(K)) and pr_{s−s′′}: (F − 1)^{−1}(fil_{m}^{′(r)}W_{s}(K)) → (F − 1)^{−1}(fil_{m−1}^{(r+s′′)}W_{s−s′′}(K)) respectively. We prove that fil_{m}^{′′(r)}W_{s}(K) ⊂ (F − 1)^{−1}(fil_{m}^{′(r)}W_{s}(K)). By (1.11) and (1.28), we have fil_{m}^{′′(r)}W_{s}(K) = fil_{[[(m−1)/p]/p^{r}]}W_{s}(K) + V^{s−s′′}fil_{[[m/p]/p^{r}]}W_{s′′}(K). By (1.11) and (1.27), we have fil_{m}^{′(r)}W_{s}(K) = fil_{[(m−1)/p^{r}]}W_{s}(K) + V^{s−s′′}fil_{[m/p^{r}]}W_{s′′}(K). Hence, by Lemma 1.6 (ii) and Lemma 1.7 (iii), we have fil_{m}^{′′(r)}W_{s}(K) ⊂ (F − 1)^{−1}(fil_{m}^{′(r)}W_{s}(K)). We put A_{n} = fil_{n}^{(r)}W_{s′′}(K) and B_{n} = fil_{n}^{(r+s′′)}W_{s−s′′}(K) for n ∈ Z_{≥0}. We also put C_{n} = fil_{n}^{′(r)}W_{s}(K) and D_{n} = fil_{n}^{′′(r)}W_{s}(K) for n ∈ Z_{≥1}. We consider the commutative diagram

\xymatrix{A_{[m/p]} \ar[r]^{V^{s−s′′}} \ar[d] & D_{m} \ar[r]^{^{pr^{s−s′′}}} \ar[d] & B_{[(m−1)/p]} \ar[r] \ar[d] & 0 \\ (F − 1)^{−1}(A_{m}) \ar[r]^{V^{s−s′′}} & (F − 1)^{−1}(C_{m}) \ar[r]^{_{pr_{s−s′′}}} & (F − 1)^{−1}(B_{m−1}),}

where the left and right vertical arrows are the identities by Lemma 1.10 (ii), the middle vertical arrow is the inclusion, and the lower horizontal line is exact. Since the upper horizontal line is exact by Lemma 1.10 (i) and (1.28), the assertion holds by applying the snake lemma.

Corollary 1.21. Let m ≥ 2 and 0 ≤ t ≤ s be integers.

(i) The exact sequence (1.30) induces the exact sequence

0 → gr′_{m}W_{s−t}(K) ⟶^{V̄^{t}} gr′_{m}W_{s}(K) ⟶^{\overline{pr}_{t}} gr_{m}^{′(s−t)}W_{t}(K) → 0.

(ii) The exact sequence (1.31) induces the exact sequence

0 → gr′′_{m}W_{s−t}(K) ⟶^{V̄^{t}} gr′′_{m}W_{s}(K) ⟶^{\overline{pr}_{t}} gr_{m}^{′′(s−t)}W_{t}(K) → 0.

Proof. The assertion holds similarly as the proof of Lemma 1.11.

Let m ≥ 2 be an integer. By abuse of notation, let

φ_{s}^{′(m)} : gr′_{m}W_{s}(K) → gr′_{m}Ω_{K}^{1} ⊗_{F_{K}} F_{K}^{1/p}

be the composition of φ_{s}^{′(m)} (1.21) and the inclusion gr′_{m}Ω_{K}^{1} → gr′_{m}Ω_{K}^{1}⊗_{F_{K}} F_{K}^{1/p} if (p, m) ≠ (2, 2) and φ̃_{s}^{′(2)} in Proposition 1.17 (i) if (p, m) = (2, 2). Let r ≥ 0 be an integer. By Lemma 1.20 (v), the morphism F − 1: W_{s}(K) → W_{s}(K) induces the injection

\overline{F − 1}: gr_{m}^{′′(r)}W_{s}(K) → gr_{m}^{′(r)}W_{s}(K).

Especially, the morphism F − 1 induces the injection

\overline{F − 1}: gr′′_{m}W_{s}(K) → gr′_{m}W_{s}(K).

Lemma 1.22 (cf. [M, Proposition 3.2.1, Proposition 3.2.3]). Let m ≥ 2 be an integer. Then we have the exact sequence

0 → gr′′_{m}W_{s}(K) ⟶^{\overline{F −1}} gr′_{m}W_{s}(K) ⟶^{φ_{s}^{′(m)}} gr′_{m}Ω_{K}^{1} ⊗_{F} F^{1/p}.

Proof. As in the proof of [M, Proposition 3.2.1] and Proposition 1.17 (ii), the morphism φ_{s}^{′(m)} factors through

gr′_{m}H^{1}(K, Z/p^{s}Z) ≃ fil′_{m}W_{s}(K)/((F −1)(W_{s}(K))∩fil′_{m}W_{s}(K)+fil′_{m−1}W_{s}(K)).

Since this factorization defines the injection ϕ^{′(m)} by [M, Proposition 3.2.3] and Proposition 1.17 (ii), the assertion holds.

Lemma 1.23. Let m ≥ 1 and r ≥ 0 be integers.

(i) fil_{m}^{′(r)}K = fil_{m/p^{r}}K if m ∈ p^{r+1}Z and fil_{m}^{′(r)}K = fil_{[(m−1)/p^{r}]}K if m ∉ p^{r+1}Z.

(ii) fil_{m}^{′′(r)}K = fil_{[m/p^{r+1}]}K.

Proof. (i) By (1.27), we have fil_{m}^{′(r)}K = fil_{m}^{(r)}K if m ∈ p^{r+1}Z and fil_{m}^{′(r)}K = fil_{m−1}^{(r)}K if m ∉ p^{r+1}Z. Hence the assertion holds by (1.11). (ii) By Lemma 1.20 (v), we have fil_{m}^{′′(r)}K = (F − 1)^{−1}(fil_{m}^{′(r)}K). By (i) and Lemma 1.7 (iii), we have fil_{m}^{′′(r)}K = fil_{m/p^{r+1}}W_{s}(K) if m ∈ p^{r+1}Z and fil_{m}^{′′(r)}K = fil_{[[(m−1)/p^{r}]/p]}W_{s}(K) if m ∉ p^{r+1}Z. Hence the assertion holds by Lemma 1.6.

Corollary 1.24. Let m ≥ 2 and r ≥ 0 be integers.

(i) Assume that r ≥ 1. Then gr_{m}^{′(r)}K = gr_{[m/p^{r}]}K if m ∈ p^{r+1}Z or ord_{p}(m−

1) = r, and gr_{m}^{′(r)}K = 0 otherwise.

(ii) gr_{m}^{′′(r)}K = gr_{m/p^{r+1}}K if m ∈ p^{r+1}Z, and gr_{m}^{′′(r)}K = 0 if m ∉ p^{r+1}Z.

Proof. (i) Assume that m ∈ p^{r+1}Z. Since r ≥ 1, we have m − 1 ∉ p^{r}Z. Hence gr_{m}^{′(r)}K = fil_{[m/p^{r}]}K/fil_{[(m−2)/p^{r}]}K by Lemma 1.23 (i). By Lemma 1.6 (i), the assertion holds in this case. Assume that m ∉ p^{r+1}Z. By Lemma 1.23 (i), we have gr_{m}^{′(r)}K = fil_{[(m−1)/p^{r}]}K/fil_{[(m−2)/p^{r}]}K if m−1 ∉ p^{r+1}Z and gr_{m}^{′(r)}K = 0 if m−1 ∈ p^{r+1}Z. Suppose that m − 1 ∉ p^{r+1}Z. By Lemma 1.6 (i), we have gr_{m}^{′(r)}K = gr_{[(m−1)/p^{r}]}K if m − 1 ∈ p^{r}Z and gr_{m}^{′(r)}K = 0 if m − 1 ∉ p^{r}Z. If m − 1 ∈ p^{r}Z, then we have m ∉ p^{r}Z, since r ≥ 1. Hence the assertion holds by Lemma 1.6 (i). (ii) By Lemma 1.23 (ii), we have gr_{m}^{′′(r)}K = fil_{[m/p^{r+1}]}K/fil_{[(m−1)/p^{r+1}]}K. Hence the assertion holds by Lemma 1.6 (i).

We note that if r = 0 and if m ∈ pZ then gr_{m}^{′(r)}K = gr′_{m}K = fil_{m}K/fil_{m−2}K.

1.3 Sheafification: logarithmic case

Let X be a smooth separated scheme over a perfect field k of characteristic p > 0. Let D be a divisor on X with simple normal crossings and {D_{i}}_{i∈I} the irreducible components of D. The generic point of D_{i} is denoted by 𝔭_{i} for i ∈ I. We put U = X − D and let j: U → X be the canonical open immersion. For i ∈ I, let O_{K_{i}} denote the completion Ô_{X,𝔭_{i}} of the local ring O_{X,𝔭_{i}} at 𝔭_{i} and K_{i} the fractional field of O_{K_{i}} called local field at 𝔭_{i}. Let ϵ: X_{ét} → X_{Zar} be the canonical mapping from the étale site of X to the Zariski site of X. We use the same notation j_{∗} for the push-forward of both étale sheaves and Zariski sheaves. We consider the exact sequence

0 → W_{s}(F_{p}) → W_{s}(O_{U_{ét}}) ⟶^{F −1} W_{s}(O_{U_{ét}}) → 0

of étale sheaves on U for s ∈ Z_{≥0}. Since R^{1}(ϵ ∘ j)_{∗}W_{s}(O_{U_{ét}}) = 0, we have an exact sequence

0 → j_{∗}W_{s}(F_{p}) → j_{∗}W_{s}(O_{U}) ⟶^{F −1} j_{∗}W_{s}(O_{U}) → R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z → 0 (1.32)

We write

δ_{s}: j_{∗}W_{s}(O_{U}) → R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z  (1.33)

for the fourth morphism in (1.32). Let V be an open subset of X. Since we have the spectral sequence E_{2}^{p,q} = H_{Zar}^{p}(V, R^{q}(ϵ ∘ j)_{∗}Z/p^{s}Z) ⇒ H_{ét}^{p+q}(U ∩ V, Z/p^{s}Z) and E_{2}^{1,0} = E_{2}^{2,0} = 0, the canonical morphism

H_{ét}^{1}(U ∩ V, Z/p^{s}Z) → Γ(V, R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z)

is an isomorphism. By the exact sequence (1.32), the morphism δ_{s} (1.33) induces an isomorphism

j_{∗}W_{s}(O_{U})/(F − 1)j_{∗}W_{s}(O_{U}) → R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z.

If D_{i} ∩ V ≠ ∅ and if a ∈ Γ(U ∩ V, W_{s}(O_{U})), let a|_{K_{i}} denote the image of a by

Γ(U ∩ V, W_{s}(O_{U})) → W_{s}(K_{i}).

Similarly, if D_{i} ∩ V ≠ ∅ and if χ ∈ H_{ét}^{1}(U ∩ V, Z/p^{s}Z), let χ|_{K_{i}} denote the image of χ by

H_{ét}^{1}(U ∩ V, Z/p^{s}Z) → H^{1}(K_{i}, Z/p^{s}Z).

Definition 1.25. Let R = ∑_{i∈I} n_{i}D_{i}, where n_{i} ∈ Z_{≥0} for i ∈ I, and let j_{i}: Spec K_{i} → X denote the canonical morphism for i ∈ I.

(i) We define a subsheaf fil_{R}j_{∗}W_{s}(O_{U}) of Zariski sheaf j_{∗}W_{s}(O_{U}) to be the pull-back of ⨁_{i∈I} j_{i∗}fil_{n_{i}}W_{s}(K_{i}) by the morphism j_{∗}W_{s}(O_{U}) → ⨁_{i∈I} j_{i∗}W_{s}(K_{i}).

(ii) We define a subsheaf fil_{R}R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z of R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z to be the image of fil_{R}j_{∗}W_{s}(O_{U}) by δ_{s} (1.33).

(iii) We define a subsheaf fil_{R}j_{∗}Ω_{U}^{1} of j_{∗}Ω_{U}^{1} to be Ω_{X}^{1}(log D)(R).

We consider the morphism

−F^{s−1}d: j_{∗}W_{s}(O_{U}) → j_{∗}Ω_{U}^{1} ; (a_{s−1}, …, a_{0}) ↦ − ∑_{i=0}^{s−1} a_{i}^{p^{i}−1}da_{i}. (1.34)

Let R = ∑_{i∈I} n_{i}D_{i}, where n_{i} ∈ Z_{≥0} for i ∈ I. Then (1.34) induces the morphism

fil_{R}j_{∗}W_{s}(O_{U}) → fil_{R}j_{∗}Ω_{U}^{1}.

Let R′ = ∑_{i∈I} n′_{i}D_{i}, where n′_{i} ∈ Z_{≥0} such that n′_{i} ≤ n_{i} for i ∈ I. Then we have fil_{R} ⊃ fil_{R′} and put gr_{R/R′} = fil_{R}/fil_{R′}. Then the morphism (1.34) induces the morphism

φ_{s}^{(R/R′)} : gr_{R/R′}j_{∗}W_{s}(O_{U}) → gr_{R/R′}j_{∗}Ω_{U}^{1}.  (1.35)

If R = R′ + D_{i} for some i ∈ I, then we simply write φ_{s}^{(R,i)} for φ_{s}^{(R,R′)} and gr_{R,i} for gr_{R/R′}. Let 0 ≤ t ≤ s be integers. We put [R/p^{j}] = ∑_{i∈I}[n_{i}/p^{j}]D_{i}. We consider the projection

pr_{t}: j_{∗}W_{s}(O_{U}) → j_{∗}W_{t}(O_{U}) ; (a_{s−1}, …, a_{0}) ↦ (a_{s−1}, …, a_{s−t}). (1.36)

Since we have pr_{t}(fil_{R}j_{∗}W_{s}(O_{U})) = fil_{[R/p^{s−t}]}j_{∗}W_{t}(O_{U}) by (1.11) and Lemma 1.10 (i), we have the exact sequence

0 → fil_{R}j_{∗}W_{s−t}(O_{U}) ⟶^{V^{t}} fil_{R}j_{∗}W_{s}(O_{U}) ⟶^{pr_{t}} fil_{[R/p^{s−t}]}j_{∗}W_{t}(O_{U}) → 0. (1.37)

Lemma 1.26. Let R = ∑_{i∈I} n_{i}D_{i} and R′ = ∑_{i∈I} n′_{i}D_{i}, where n_{i}, n′_{i} ∈ Z_{≥0} and n′_{i} ≤ n_{i} for every i ∈ I. Then the exact sequence (1.37) induces the exact sequence

0 → gr_{R/R′}j_{∗}W_{s−t}(O_{U}) ⟶^{V̄^{t}} gr_{R/R′}j_{∗}W_{s}(O_{U})  (1.38)

⟶^{\overline{pr}_{t}} gr_{[R/p^{s−t}]/[R′/p^{s−t}]}j_{∗}W_{t}(O_{U}) → 0.

Especially, if R = R′ + D_{i} for some i ∈ I, we have the exact sequence

0 → gr_{R,i}j_{∗}W_{s−t}(O_{U}) ⟶^{V̄^{t}} gr_{R,i}j_{∗}W_{s}(O_{U})

⟶^{\overline{pr}_{t}} gr_{[R/p^{s−t}]/[(R−D_{i})/p^{s−t}]}j_{∗}W_{t}(O_{U}) → 0.

Proof. The assertion holds similarly as the proof of Lemma 1.11. In fact, we consider the commutative diagram

\xymatrix{0 \ar[r] & fil_{R′}j_{∗}W_{s−t}(O_{U}) \ar[r]^{V^{t}} \ar[d] & fil_{R′}j_{∗}W_{s}(O_{U}) \ar[r]^{^{pr^{t}}} \ar[d] & fil_{[R′/p^{s−t}]}j_{∗}W_{t}(O_{U}) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & fil_{R}j_{∗}W_{s−t}(O_{U}) \ar[r]^{V^{t}} & fil_{R}j_{∗}W_{s}(O_{U}) \ar[r]^{_{pr_{t}}} & fil_{[R/p^{s−t}]}j_{∗}W_{t}(O_{U}) \ar[r] & 0,} (1.39)

where the horizontal lines are exact and the vertical arrows are inclusions. Then this diagram induces the sequence (1.38). By taking stalks of (1.39), the exactness of (1.38) follows.

Let R = ∑_{i∈I} n_{i}D_{i} and R′ = ∑_{i∈I} n′_{i}D_{i}, where n_{i}, n′_{i} ∈ Z_{≥0} and n′_{i} ≤ n_{i} for every i ∈ I. We consider the morphism

F̄: gr_{[R/p]/[R′/p]}j_{∗}W_{s}(O_{U}) → gr_{R/R′}j_{∗}W_{s}(O_{U})  (1.40)

induced by the Frobenius F: j_{∗}W_{s}(O_{U}) → j_{∗}W_{s}(O_{U}). Since F^{−1}(fil_{R}j_{∗}W_{s}(O_{U})) = fil_{[R/p]}j_{∗}W_{s}(O_{U}) by Lemma 1.7 (iii) and similarly for R′, the morphism (1.40) is injective.

We consider the morphism

\overline{F − 1}: gr_{[R/p]/[R′/p]}j_{∗}W_{s}(O_{U}) → gr_{R/R′}j_{∗}W_{s}(O_{U})  (1.41)

induced by F −1: j_{∗}W_{s}(O_{U}) → j_{∗}W_{s}(O_{U}). If R = R′ +D_{i} for some i ∈ I, then the morphisms (1.40) and (1.41) are the same, since [R/p] ≤ R′ with respect to product order.

Lemma 1.27. Let A be a smooth ring over k. Let t_{1}, …, t_{r} be elements of A such that (t_{1} ⋯ t_{r} = 0) is a divisor on Spec A with simple normal crossings whose irreducible components are {(t_{i} = 0)}_{i=1}^{r}. Let a be an element of Frac A. Assume that a^{p}t_{1}^{n_{1}} ⋯ t_{r}^{n_{r}} ∈ A, where n_{1}, …, n_{r} are integers such that 0 ≤ n_{i} < p for i = 1, …, r. Then we have a ∈ A.

Proof. Since a^{p}t_{1}^{n_{1}} ⋯ t_{r}^{n_{r}} ∈ A, the valuation of a^{p}t_{1}^{n_{1}} ⋯ t_{r}^{n_{r}} in A_{(t_{i})} is nonnegative for i = 1, …, r. Since the normalized valuation of a^{p} in Frac A_{(t_{i})} for i = 1, …, r is divided by p and 0 ≤ n_{i} < p for i = 1, …, r, the valuation of a in Frac A_{(t_{i})} for i = 1, …, r is non-negative. Since A is factorial, we have A[1/t_{1} ⋯ t_{r}] ∩ ⋂_{i=1}^{r} A_{(t_{i})} = A. Hence the assertion holds.

Lemma 1.28. Let F, G, and H be sheaves of abelian groups on X and let F_{i}, G_{i}, and H_{i} be subsheaves of F, G, and H respectively for i = 1, 2, 3. Assume that F_{3} = F_{1} ∩ F_{2}, H_{3} = H_{1} ∩ H_{2}, and that G_{3} ⊂ G_{1} ∩ G_{2}. If we have an exact sequence 0 → F → G → H → 0 and if this exact sequence induces the exact sequence 0 → F_{i} → G_{i} → H_{i} → 0 for i = 1, 2, 3, then we have G_{3} = G_{1} ∩ G_{2}.

Proof. We consider the commutative diagram

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & F_{3} \ar[r] \ar[d] & G_{3} \ar[r] \ar[d] & H_{3} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & F_{1} ⊕ F_{2} \ar[r] \ar[d] & G_{1} ⊕ G_{2} \ar[r] \ar[d] & H_{1} ⊕ H_{2} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & F \ar[r] & G \ar[r] & H \ar[r] & 0,}  (1.42)

where the bottom vertical arrows are defined by the difference. Since F_{3} = F_{1} ∩ F_{2} and H_{3} = H_{1} ∩ H_{2}, the left and right vertical columns are exact. By applying the snake lemma to the lower two lines, we have G_{3} = G_{1} ∩ G_{2}.

Proposition 1.29. Let R = ∑_{i∈I} n_{i}D_{i}, where n_{i} ∈ Z_{≥0} for i ∈ I. Let s ≥ 0 be an integer and let i be an element of I such that n_{i} ≥ 1. We put R′ = R−D_{i}. Then we have the exact sequence

0 → gr_{[R/p]/[R′/p]}j_{∗}W_{s}(O_{U}) ⟶^{F̄} gr_{R,i}j_{∗}W_{s}(O_{U}) ⟶^{φ_{s}^{(R,i)}} gr_{R,i}j_{∗}Ω_{U}^{1},

where gr_{[R/p]/[R′/p]}j_{∗}W_{s}(O_{U}) is gr_{[R/p],i}j_{∗}W_{s}(O_{U}) if n_{i} ∈ pZ and 0 if n_{i} ∉ pZ.

Proof. We may assume that s ≥ 1, I = {1, …, r}, and that i = 1. Let j_{1} : Spec K_{1} → X be the canonical morphism. We consider the commutative diagram

\xymatrix{0 \ar[r] & gr_{[R/p]/[R′/p]}j_{∗}W_{s}(O_{U}) \ar[r]^{F̄} \ar[d] & gr_{R,1}j_{∗}W_{s}(O_{U}) \ar[r]^{φ_{s}^{(R,1)}} \ar[d] & gr_{R,1}j_{∗}Ω_{U}^{1} \ar[d] \\ 0 \ar[r] & j_{1∗}(gr_{[n_{1}/p]/[(n_{1}−1)/p]}W_{s}(K_{1})) \ar[r]^{F̄} & j_{1∗}gr_{n_{1}}W_{s}(K_{1}) \ar[r]^{φ_{s}^{(n_{1})}} & j_{1∗}gr_{n_{1}}Ω_{K_{1}}^{1},} (1.43)

where we put gr_{[n_{1}/p]/[(n_{1}−1)/p]} = fil_{[n_{1}/p]}/fil_{[(n_{1}−1)/p]} and the vertical arrows are inclusions. Since the lower line is exact by Lemma 1.8, it is sufficient to prove that the left square in (1.43) is cartesian. If n_{1} ∉ pZ, then the assertion holds since gr_{[R/p]/[R′/p]}j_{∗}W_{s}(O_{U}) = 0 and gr_{[n_{1}/p]/[(n_{1}−1)/p]}W_{s}(K_{1}) = 0 by Lemma 1.6 (i). Assume that n_{1} ∈ pZ. Then we have gr_{[R/p]/[R′/p]}j_{∗}W_{s}(O_{U}) = gr_{[R/p],1}j_{∗}W_{s}(O_{U}) and gr_{[n_{1}/p]/[(n_{1}−1)/p]}W_{s}(K_{1}) = gr_{n_{1}/p}W_{s}(K_{1}) by Lemma 1.6 (i). We prove the assertion by induction on s. Suppose that s = 1. Since the assertion is local, we may assume that X = Spec A is affine and that D_{i} = (t_{i} = 0) for i ∈ I, where t_{i} ∈ A for i ∈ I. Further we may assume that the invertible O_{D_{1}}-modules gr_{R,1}j_{∗}O_{U} and gr_{[R/p],1}j_{∗}O_{U} are gererated by c_{0} = 1/t_{1}^{n_{1}} ⋯ t_{r}^{n_{r}}

and c_{1} = 1/t_{1}^{n_{1}/p}t_{2}^{m′_{2}} ⋯ t_{r}^{m′_{r}} respectively, where m′_{i} = [n_{i}/p] for i ∈ I − {1}. Let k(D_{1}) denote the functional field of D_{1}. We identify gr_{n_{1}}K_{1} with k(D_{1}) · c_{0} and gr_{n_{1}/p}K_{1} with k(D_{1}) · c_{1}. Let ā be an element of k(D_{1}) such that F̄(āc_{1}) = ā^{p}c_{1}^{p} ∈ gr_{R,1}j_{∗}O_{U}. Since (ā^{p}c_{1}^{p}/c_{0}) · c_{0} ∈ gr_{R,1}j_{∗}O_{U} = O_{D_{1}} · c_{0}, we have ā^{p}c_{1}^{p}/c_{0} ∈ O_{D_{1}}. Since c_{1}^{p}/c_{0} =

t_{2}^{n_{2}−pm′_{2}} ⋯ t_{r}^{n_{r}−pm′_{r}} and 0 ≤ n_{i} − pm′_{i} < p for i ∈ I − {1}, we have ā ∈ O_{D_{1}} by Lemma 1.27. Hence we have āc_{1} ∈ O_{D_{1}} · c_{1} = gr_{[R/p],1}j_{∗}O_{U}. Hence the assertion holds if s = 1. If s > 1, we put F = j_{1∗}gr_{n_{1}}W_{s−1}(K_{1}), F_{1} = gr_{R,1}j_{∗}W_{s−1}(O_{U}), F_{2} = j_{1∗}gr_{n_{1}/p}W_{s−1}(K_{1}), and F_{3} = gr_{[R/p],1}j_{∗}W_{s−1}(O_{U}). Since the canonical morphisms F_{1} → F and F_{3} → F_{2} are injective and both F̄: F_{3} → F_{1} and F̄: F_{2} → F are injective, we may identify F_{i} with a subsheaf of F for i = 1, 2, 3. We also put G = j_{1∗}gr_{n_{1}}W_{s}(K_{1}), G_{1} = gr_{R,1}j_{∗}W_{s}(O_{U}), G_{2} = j_{1∗}gr_{n_{1}/p}W_{s}(K_{1}),

and G_{3} = gr_{[R/p],1}j_{∗}W_{s}(O_{U}). We further put H = j_{1∗}(gr_{n}^{(s−1)}K_{1}), H_{1} =

gr_{[R/p^{s−1}]/[R′/p^{s−1}]}j_{∗}O_{U}, H_{2} = j_{1∗}(gr_{n_{1}/p}^{(s−1)}K_{1}), and H_{3} = gr_{[R/p^{s}]/[R′/p^{s}]}j_{∗}O_{U}. Similarly as F_{i}, we may identify G_{i} and H_{i} with subsheaves of G and H respectively for i = 1, 2, 3. By the induction hypothesis, we have F_{3} = F_{1} ∩ F_{2}. If n_{1} ∉ p^{s}Z, then H_{2} = H_{3} = 0 by Lemma 1.6 (i) and (1.11). If n_{1} ∈ p^{s}Z, then we have H_{3} = H_{1} ∩ H_{2} by Lemma 1.6 (i), (1.11), and the induction hypothesis. By the commutativity of (1.43), we have G_{3} ⊂ G_{1} ∩ G_{2}. Since exact sequences in Lemma 1.11 and Lemma 1.26 in the case where t = 1 are compatible with the inclusions of sheaves above, the assertion holds by Lemma 1.28.

Lemma 1.30. Let f: F → G be a surjection of sheaves of abelian groups on X. Let g: G → H be a morphism of sheaves of abelian groups on X. We put Γ = (Z_{≥0})^{r}, where r > 0 is an integer, and let 1_{i} ∈ Γ be the element whose i-th component is 1 and the others are 0 for i = 1, …, r. Let {fil_{n}F}_{n∈Γ} and {fil_{n}H}_{n∈Γ} be increasing filtrations of F and H respectively with respect to product order. Assume that ⋃_{n∈Γ} fil_{n}F = F and ⋃_{n∈Γ} fil_{n}H = H. We put fil_{n}G = f(fil_{n}F) for n ∈ Γ, which define an increasing filtration of G. If g(fil_{n}G) ⊂ fil_{n}H for every n ∈ Γ and if the morphism fil_{n+1_{i}}G/fil_{n}G → fil_{n+1_{i}}H/fil_{n}H induced by g is injective for every n ∈ Γ and i = 1, …, r, then we have fil_{n}G = g^{−1}(fil_{n}H) for every n ∈ Γ.

Proof. Let n ∈ Γ be an element. We prove that the morphism G/fil_{n}G → H/fil_{n}H is injective. Since F = ⋃_{n∈Γ} fil_{n}F and f is surjective, we have G = ⋃_{n∈Γ} fil_{n}G and hence G/fil_{n}G = \underrightarrow{lim}_{n′} fil_{n′}G/fil_{n}G, where n′ runs through the elements of Γ greater than n with respect to product order. Since H = ⋃_{n∈Γ} fil_{n}H, we have H/fil_{n}H = \underrightarrow{lim}_{n′} fil_{n′}H/fil_{n}H, where n′ runs through the elements of Γ greater than n. Hence it is sufficient to prove that fil_{n′}G/fil_{n}G → fil_{n′}H/fil_{n}H is injective for every n′ ∈ Γ such that n′ ≥ n. We prove this assertion by induction on n′. If n′ = n, the assertion holds since fil_{n′}G/fil_{n}G = 0 and fil_{n′}H/fil_{n}H = 0. For n′ > n, take i such that n′ − 1_{i} ≥ n. We consider the commutative diagram

\xymatrix{0 \ar[r] & fil_{n′−1_{i}}G/fil_{n}G \ar[r] \ar[d] & fil_{n′}G/fil_{n}G \ar[r] \ar[d] & fil_{n′}G/fil_{n′−1_{i}}G \ar[r] \ar[d] & 0 \\ 0 \ar[r] & fil_{n′−1_{i}}H/fil_{n}H \ar[r] & fil_{n′}H/fil_{n}H \ar[r] & fil_{n′}H/fil_{n′−1_{i}}H \ar[r] & 0,}

where the horizontal lines are exact. By the induction hypothesis, the left vertical arrow is injective. Since the right vertical arrow is injective, the middle vertical arrow is injective. Hence the assertion holds.

Proposition 1.31. Let R = ∑_{i∈I} n_{i}D_{i}, where n_{i} ∈ Z_{≥0} for i ∈ I. Let j_{i}: Spec K_{i} → X be the canonical morphism for i ∈ I.

(i) The subsheaf fil_{R}R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z is equal to the pull-back of ⨁_{i∈I} j_{i∗}fil_{n_{i}}H^{1}(K_{i}, Q/Z) by the morphism R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z → ⨁_{i∈I} j_{i∗}H^{1}(K_{i}, Q/Z).

(ii) Let R′ = ∑_{i∈I} n′_{i}D_{i}, where n′_{i} ∈ Z_{≥0} and n_{i} − 1 ≤ n′_{i} ≤ n_{i} for i ∈ I. Then there exists a unique injection ϕ_{s}^{(R/R′)} : gr_{R/R′}R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z → gr_{R/R′}j_{∗}Ω_{U}^{1} such that the following diagram is commutative:

\xymatrix{gr_{R/R′}j_{∗}W_{s}(O_{U}) \ar[dr]_{δ_{s}^{(R/R′)}} \ar[rr]^{φ_{s}^{(R/R′)}} &  & gr_{R/R′}j_{∗}Ω_{U}^{1} \\  & gr_{R/R′}R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z. \ar[ur]_{ϕ_{s}^{(R/R′)}}} (1.44)

Proof. Let i be an element of I such that n_{i} ≥ 1. Since the kernel of δ_{s}^{(R,i)} is the image of \overline{F − 1} (1.41) and the morphisms F̄ (1.40) and \overline{F − 1} (1.41) are the same, the kernel of δ_{s}^{(R,i)} is equal to the kernel of φ_{s}^{(R,i)} by Proposition 1.29. Since δ_{s}^{(R,i)} is surjective, there exists a unique injection ϕ_{s}^{(R,i)} : gr_{R,i}R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z → gr_{R,i}j_{∗}Ω_{U}^{1} such that the diagram (1.44) for R′ = R − D_{i} is commutative.

(i) Let i be an element of I such that n_{i} ≥ 1. We consider the commutative diagram

\xymatrix{gr_{R,i}R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z \ar[r] \ar[d]_{ϕ_{s}^{(R,i)}} & j_{i∗}gr_{n_{i}}H^{1}(K_{i}, Q/Z) \ar[d]^{ϕ^{(n_{i})}} \\ gr_{R,i}j_{∗}Ω_{U}^{1} \ar[r] & j_{i∗}gr_{n_{i}}Ω_{K_{i}}^{1},}

where the lower horizontal arrow is the inclusion and ϕ^{(n_{i})} is as in (1.8). Since the left vertical arrow is injective as proved above, the upper horizontal arrow is injective. Hence the assertion holds by applying Lemma 1.30 to the case where F = j_{∗}W_{s}(O_{U}), G = R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z, and H = ⨁_{i∈I} j_{i∗}H^{1}(K_{i}, Q/Z). (ii) Let J be the subset of I consisting of i ∈ I such that n′_{i} ≠ n_{i}. We consider the commutative diagram

\xymatrix{gr_{R/R′}j_{∗}W_{s}(O_{U}) \ar[rr]^{φ_{s}^{(R/R′)}} \ar[d]_{δ_{s}^{(R/R′)}} &  & gr_{R/R′}j_{∗}Ω_{U}^{1} \ar[d] \\ gr_{R/R′}R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z \ar[r] & ⨁_{i∈J} j_{i∗}gr_{n_{i}}H^{1}(K_{i}, Q/Z) \ar[r]^{⊕ϕ^{(n_{i})}} & ⨁_{i∈J} j_{i∗}gr_{n_{i}}Ω_{K_{i}}^{1},}

where ϕ^{(n_{i})} is as in (1.8) for i ∈ J. By (i), the left lower horizontal arrow is injective. Since gr_{n_{i}}Ω_{K_{i}}^{1} is the stalk of gr_{R/R′}j_{∗}Ω_{U}^{1} at the generic point of D_{i} for i ∈ J, the kernel of the canonical morphism fil_{R}j_{∗}Ω_{U}^{1} → ⨁_{i∈J} j_{i∗}gr_{n_{i}}Ω_{K_{i}}^{1} is the intersection of fil_{R−D_{i}}j_{∗}Ω_{U}^{1} for i ∈ J. Hence the right vertical arrow is injective. Since the right lower horizontal arrow is injective, the kernel of φ_{s}^{(R/R′)} is equal to that of δ_{s}^{(R/R′)}. Since δ_{s}^{(R/R′)} is surjective, the assertion holds.

Definition 1.32. Let χ be an element of H_{ét}^{1}(U, Q/Z). We define the Swan conductor divisor R_{χ} of χ by R_{χ} = ∑_{i∈I} sw(χ|_{K_{i}})D_{i}.

Definition 1.33. Let χ be an element of H_{ét}^{1}(U, Q/Z). Assume that sw(χ|_{K_{i}}) > 0 for some i ∈ I. Let p^{s} be the order of the p-part of χ. We put Z = Supp(R_{χ}). We define the refined Swan conductor rsw(χ) of χ to be the image of the p-part of χ by the composition

Γ(X, fil_{R_{χ}}R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z) → Γ(X, gr_{R_{χ}/(R_{χ}−Z)}R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z)

⟶^{ϕ_{s}^{(Rχ/(Rχ−Z))}(X)} Γ(X, gr_{R_{χ}/(R_{χ}−Z)}j_{∗}Ω_{U}^{1}) = Γ(Z, Ω_{X}^{1}(log D)(R_{χ}) ⊗_{O_{X}} O_{Z}).

By the construction of ϕ_{s}^{(R_{χ}/(R_{χ}−Z))}, the germ rsw(χ)_{𝔭_{i}} of rsw(χ) at the generic point 𝔭_{i} of D_{i} contained in Z is equal to rsw(χ|_{K_{i}}). This implies that rsw(χ) in Definition 1.33 is none other than the refined Swan conductor of χ in the sense of [K2, (3.4.2)].

1.4 Sheafification: non-logarithmic case

We recall the definition of the radicial covering S^{1/p} of a scheme S over a perfect field k of characteristic p > 0. We consider the commutative diagram

\xymatrix{S^{1/p} \ar[d] \ar[r] & S \ar[r]^{F_{S}} \ar[d] & S \ar[d] \\ Spec k \ar[r]_{F_{k}^{−1}} & Spec k \ar[r]_{F_{k}} & Spec k,}

where the left square is the base change over k by the inverse F_{k}^{−1} of F_{k}. The symbols F_{S} and F_{k} denote the absolute Frobenius of S and Spec k respectively. We define the radicial covering S^{1/p} → S as the composition of morphisms in the upper line. We keep the notation in Subsection 1.3.

Definition 1.34. Let R = ∑_{i∈I} n_{i}D_{i}, where n_{i} ∈ Z_{≥1} for i ∈ I, and let j_{i}: Spec K_{i} → X denote the canonical morphism for i ∈ I. Let r ≥ 0 be an integer.

(i) We define subsheaves fil_{R}^{′(r)}j_{∗}W_{s}(O_{U}) and fil_{R}^{′′(r)}j_{∗}W_{s}(O_{U}) of Zariski sheaf j_{∗}W_{s}(O_{U}) to be the pull-back of ⨁_{i∈I} j_{i∗}fil_{n_{i}}^{′(r)}W_{s}(K_{i}) and ⨁_{i∈I} j_{i∗}fil_{n_{i}}^{′′(r)}W_{s}(K_{i}) by the morphism j_{∗}W_{s}(O_{U}) → ⨁_{i∈I} j_{i∗}W_{s}(K_{i}) respectively.

If r = 0, then we simply write fil′_{R}j_{∗}W_{s}(O_{U}) and fil′′_{R}j_{∗}W_{s}(O_{U}) for fil_{R}^{′(0)}j_{∗}W_{s}(O_{U}) and fil_{R}^{′′(0)}j_{∗}W_{s}(O_{U}) respectively.

(ii) We define a subsheaf fil′_{R}R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z of R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z to be the image of fil′_{R}j_{∗}W_{s}(O_{U}) by δ_{s} (1.33).

(iii) We define a subsheaf fil′_{R}j_{∗}Ω_{U}^{1} of j_{∗}Ω_{U}^{1} to be Ω_{X}^{1}(R).

Similarly as in the logarithmic case, we consider the morphism −F^{s−1}d (1.34). Let R = ∑_{i∈I} n_{i}D_{i}, where n_{i} ∈ Z_{≥1} for i ∈ I. Then −F^{s−1}d (1.34) induces the morphism

fil′_{R}j_{∗}W_{s}(O_{U}) → fil′_{R}j_{∗}Ω_{U}^{1}.

For R′ = ∑_{i∈I} n′_{i}D_{i}, where n′_{i} ∈ Z_{≥1} such that n′_{i} ≤ n_{i} for i ∈ I, we put gr′_{R/R′} = fil′_{R}/fil′_{R′}. Then the morphism −F^{s−1}d (1.34) induces the morphism

φ_{s}^{′(R/R′)} : gr′_{R/R′}j_{∗}W_{s}(O_{U}) → gr′_{R/R′}j_{∗}Ω_{U}^{1}.  (1.45)

For R′ = ∑_{i∈I} n′_{i}D_{i}, where n′_{i} ∈ Z_{≥1} such that n_{i} − 1 ≤ n′_{i} ≤ n_{i} for i ∈ I, we put D^{(R/R′)} = R − R′ ⊂ D. If p ≠ 2 or there is no i ∈ I such that (n_{i}, n′_{i}) = (2, 1), let φ̃_{s}^{′(R/R′)} : gr′_{R/R′}j_{∗}W_{s}(O_{U}) → gr′_{R/R′}j_{∗}Ω_{U}^{1} ⊗_{O_{D^{(R/R′)}}} O_{D^{(R/R′)^{1/2}}} be the composition

gr′_{R/R′}j_{∗}W_{s}(O_{U}) ⟶^{φ_{s}^{′(R/R′)}} gr′_{R/R′}j_{∗}Ω_{U}^{1} → gr′_{R/R′}j_{∗}Ω_{U}^{1} ⊗_{O_{D^{(R/R′)}}} O_{D^{(R/R′)^{1/2}}} .

Otherwise, as in the proof of Proposition 1.17 (i), there exists a unique morphism

φ̃_{s}^{′(R/R′)} : gr′_{R/R′}j_{∗}W_{s}(O_{U}) → gr′_{R/R′}j_{∗}Ω_{U}^{1} ⊗_{O_{D^{(R/R′)}}} O_{D^{(R/R′)^{1/p}}}  (1.46)

such that locally φ̃_{s}^{′(R/R′)}(ā) = − ∑_{i=0}^{s−1} a_{i}^{p^{i}−1}da_{i} + ∑_{(n_{i},n′_{i})=(2,1)} \sqrt{\overline{t_{i}^{2}a_{0}}}dt_{i}/t_{i}^{2} for every ā ∈ gr′_{R/R′}j_{∗}W_{s}(O_{U}) whose lift is a = (a_{s−1}, …, a_{0}) ∈ fil′_{R}j_{∗}W_{s}(O_{U}) and for every local equation t_{i} of D_{i} for i ∈ I such that (n_{i}, n′_{i}) = (2, 1). If R = R′ + D_{i} for some i ∈ I, then we simply write gr′_{R,i} for gr′_{R/R′}, φ̃^{′(R,i)}

for φ̃^{′(R/R′)}, and similarly for gr′′_{R/R′}, gr_{R/R′}^{′(r)}, and gr_{R/R′}^{′′(r)}.

Lemma 1.35. Let R = ∑_{i∈I} n_{i}D_{i}, where n_{i} ∈ Z_{≥1} for i ∈ I, and let r ≥ 0 be an integer. Then we have fil_{R}^{′′(r)}j_{∗}W_{s}(O_{U}) = (F − 1)^{−1}(fil_{R}^{′(r)}j_{∗}W_{s}(O_{U})). Especially, we have fil′′_{R}j_{∗}W_{s}(O_{U}) = (F − 1)^{−1}(fil′_{R}j_{∗}W_{s}(O_{U})).

Proof. Let j_{i}: Spec K_{i} → X be the canonical morphism for i ∈ I. Since F − 1 is compatible with the canonical morphism j_{∗}W_{s}(O_{U}) → ⨁_{i∈I} j_{i∗}W_{s}(K_{i}), the assertions hold by Lemma 1.20 (v).

Lemma 1.36. Let r ≥ 0 be an integer. Let R = ∑_{i∈I} n_{i}D_{i} and R′ = ∑_{i∈I} n′_{i}D_{i}, where n_{i}, n′_{i} ∈ Z_{≥1} such that n′_{i} = n_{i}/p^{r} if n_{i} ∈ p^{r+1}Z and n′_{i} = [(n_{i} − 1)/p^{r}] if n_{i} ∉ p^{r+1}Z for every i ∈ I.

(i) fil_{R}^{′(r)}j_{∗}O_{U} = fil_{R′}j_{∗}O_{U}.

(ii) fil_{R}^{′′(r)}j_{∗}O_{U} = fil_{[R/p^{r+1}]}j_{∗}O_{U}.

Proof. The assertions hold by Lemma 1.23.

Corollary 1.37. Let the notation be as in Lemma 1.36. Let i be an element of I such that n_{i} ≥ 2.

(i) Assume that r ≥ 1. Then gr_{R,i}^{′(r)}j_{∗}O_{U} = gr_{R′,i}j_{∗}O_{U} if n_{i} ∈ p^{r+1}Z or

ord_{p}(n_{i} − 1) = r, and gr_{R,i}^{′(r)}j_{∗}O_{U} = 0 otherwise.

(ii) gr_{R,i}^{′′(r)}j_{∗}O_{U} = gr_{[R/p^{r+1}],i}j_{∗}O_{U} = gr_{[R′/p],i}j_{∗}O_{U} if n_{i} ∈ p^{r+1}Z, and

gr_{R,i}^{′′(r)}j_{∗}O_{U} = 0 if n_{i} ∉ p^{r+1}Z.

Proof. Since [R/p^{r+1}] = [R′/p] by Lemma 1.6, the assertions hold by Corollary 1.24 and Lemma 1.36.

Let R = ∑_{i∈I} n_{i}D_{i} and R′ = ∑_{i∈I} n′_{i}D_{i}, where n_{i}, n′_{i} ∈ Z_{≥1} and n′_{i} ≤ n_{i} for every i ∈ I. Let 0 ≤ t ≤ s be integers. Since we have pr_{t}(fil′_{R}j_{∗}W_{s}(O_{U})) = fil_{R}^{′(s−t)}j_{∗}W_{t}(O_{U}) by Lemma 1.20 (i), we have the exact sequence

0 → fil′_{R}j_{∗}W_{s−t}(O_{U}) ⟶^{V^{t}} fil′_{R}j_{∗}W_{s}(O_{U}) ⟶^{pr_{t}} fil_{R}^{′(s−t)}j_{∗}W_{t}(O_{U}) → 0. (1.47)

Similarly, since pr_{t}(fil′′_{R}j_{∗}W_{s}(O_{U})) is fil_{R}^{′′(s−t)}j_{∗}W_{t}(O_{U}) by Lemma 1.20 (iii), we have the exact sequence

0 → fil′′_{R}j_{∗}W_{s−t}(O_{U}) ⟶^{V^{t}} fil′′_{R}j_{∗}W_{s}(O_{U}) ⟶^{pr_{t}} fil_{R}^{′′(s−t)}j_{∗}W_{t}(O_{U}) → 0. (1.48)

Lemma 1.38. Let R = ∑_{i∈I} n_{i}D_{i} and R′ = ∑_{i∈I} n′_{i}D_{i}, where n_{i}, n′_{i} ∈ Z_{≥1} and n_{i} − 1 ≤ n′_{i} ≤ n_{i} for every i ∈ I. Let 0 ≤ t ≤ s be integers.

(i) The exact sequence (1.47) induces the exact sequence

\xymatrix{0 \ar[r] & gr′_{R/R′}j_{∗}W_{s−t}(O_{U}) \ar[r]^{V̄^{t}} & gr′_{R/R′}j_{∗}W_{s}(O_{U}) \ar[r]^{^{\overline{pr}^{t}}} & gr_{R/R′}^{′(s−t)}j_{∗}W_{t}(O_{U}) \ar[r] & 0.}

(ii) The exact sequence (1.48) induces the exact sequence

\xymatrix{0 \ar[r] & gr′′_{R/R′}j_{∗}W_{s−t}(O_{U}) \ar[r]^{V̄^{t}} & gr′′_{R/R′}j_{∗}W_{s}(O_{U}) \ar[r]^{^{\overline{pr}^{t}}} & gr_{R/R′}^{′′(s−t)}j_{∗}W_{t}(O_{U}) \ar[r] & 0.}

Proof. The assertions hold similarly as the proof of Lemma 1.26.

Let r ≥ 0 be an integer. By Lemma 1.35, the morphism F − 1: j_{∗}W_{s}(O_{U}) → j_{∗}W_{s}(O_{U}) induces the injection

\overline{F − 1}: gr_{R/R′}^{′′(r)}j_{∗}W_{s}(O_{U}) → gr_{R/R′}^{′(r)}j_{∗}W_{s}(O_{U}).

Especially, the morphism F − 1 induces the injection

\overline{F − 1}: gr′′_{R/R′}j_{∗}W_{s}(O_{U}) → gr′_{R/R′}j_{∗}W_{s}(O_{U}).

Lemma 1.39. Let R = ∑_{i∈I} n_{i}D_{i}, where n_{i} ∈ Z_{≥1} for i ∈ I. Let s ≥ 0 be an integer and let i be an element of I such that n_{i} ≥ 2. Then we have the exact sequence

0 → gr′′_{R,i}j_{∗}W_{s}(O_{U}) ⟶^{\overline{F −1}} gr′_{R,i}j_{∗}W_{s}(O_{U}) ⟶^{φ̃_{s}^{′(R,i)}} gr′_{R,i}j_{∗}Ω_{U}^{1} ⊗_{O_{D_{i}}} O_{D_{i}^{1/p}}.

Proof. We may assume that s ≥ 1, I = {1, …, r}, and that i = 1. Let j_{1} : Spec K_{1} → X be the canonical morphism. We consider the commutative diagram

\xymatrix{0 \ar[r] & gr′′_{R,1}j_{∗}W_{s}(O_{U}) \ar[r]^{\overline{F −1}} \ar[d] & gr′_{R,1}j_{∗}W_{s}(O_{U}) \ar[r]^{φ_{s}^{′(R,1)}} \ar[d] & gr′_{R,1}j_{∗}Ω_{U}^{1} ⊗_{O_{D_{1}}} O_{D_{1}^{1/p}} \ar[d] \\ 0 \ar[r] & j_{1∗}gr′′_{n_{1}}W_{s}(K_{1}) \ar[r]^{\overline{F −1}} & j_{1∗}gr′_{n_{1}}W_{s}(K_{1}) \ar[r]^{φ_{s}^{′(n_{1})}} & j_{1∗}(gr′_{n_{1}}Ω_{K_{1}}^{1} ⊗_{F_{K_{1}}} F_{K_{1}}^{1/p}),} (1.49)

where F_{K_{1}} denotes the residue field of K_{1} and the vertical arrows are canonical injections. By Lemma 1.22, the lower horizontal line is exact. Hence it is sufficient to prove that the left square in (1.49) is cartesian. We prove the assertion by induction on s. Suppose that s = 1. If n_{1} ∉ pZ, then we have gr′′_{n_{1}}W_{s}(K_{1}) = 0 and gr′′_{R,1}j_{∗}O_{U} = 0 by Corollary 1.24 (ii) and Corollary 1.37 (ii). Hence the assertion holds in this case. Assume that n_{1} ∈ pZ. By (1.15), we have gr′_{n_{1}}K_{1} = fil_{n_{1}}K_{1}/fil_{n_{1}−2}K_{1}. By Corollary 1.24 (ii), we have gr′′_{n_{1}}K_{1} = gr_{n_{1}/p}K_{1}. Since the assertion is a local property, we may assume that X = Spec A is affine and that D_{i} = (t_{i} = 0) for i ∈ I, where t_{i} ∈ A for i ∈ I. Further we may assume that the invertible O_{2D_{1}}- module gr′_{R,1}j_{∗}O_{U} is generated by c_{0} = 1/t_{1}^{n_{1}} ⋯ t_{r}^{n_{r}}, and that the invertible

O_{D_{1}}-module gr′′_{R,1}j_{∗}O_{U} is generated by c_{1} = 1/t_{1}^{n_{1}/p}t_{2}^{m′_{2}} ⋯ t_{r}^{m′_{r}}, where m′_{i} = [n_{i}/p] for i ∈ I −{1}. Let R(2D_{1}) denote the stalk of O_{2D_{1}} at the generic point of 2D_{1} and let k(D_{1}) denote the functional field of D_{1}. Then we may identify gr′_{n_{1}}K_{1} with R(2D_{1}) · c_{0} and gr′′_{n_{1}}K_{1} with k(D_{1}) · c_{1}. Let ā be an element of k(D_{1}) such that \overline{(F − 1)}(āc_{1}) ∈ gr′_{R,1}j_{∗}O_{U}. Since we have \overline{(F − 1)}(āc_{1}) = ((ā^{p}c_{1}^{p} − āc_{1})/c_{0}) · c_{0} ∈ gr′_{R,1}j_{∗}O_{U} = O_{2D_{1}} · c_{0}, we

have (ā^{p}c_{1}^{p} − āc_{1})/c_{0} ∈ O_{2D_{1}}. Since c_{1}/c_{0} = t_{1}^{n_{1}−n_{1}/p}t_{2}^{n_{2}−m′_{2}} ⋯ t_{r}^{n_{r}−m′_{r}} and n_{1} − n_{1}/p ≥ 1, we have (ā^{p}c_{1}^{p} − āc_{1})/c_{0} = ā^{p}c_{1}^{p}/c_{0} in O_{D_{1}}. Since c_{1}^{p}/c_{0} = t_{2}^{n_{2}−pm′_{2}} ⋯ t_{r}^{n_{r}−pm′_{r}} and 0 ≤ n_{i} − pm′_{i} < p for i ∈ I − {1}, we have ā ∈ O_{D_{1}} by Lemma 1.27. Hence we have āc_{1} ∈ O_{D_{1}} · c_{1} = gr′′_{R,1}j_{∗}O_{U}. Thus the assertion holds if s = 1. If s > 1, we put F = j_{1∗}gr′_{n_{1}}W_{s−1}(K_{1}), F_{1} = gr′_{R,1}j_{∗}W_{s−1}(O_{U}), F_{2} = j_{1∗}gr′′_{n_{1}}W_{s−1}(K_{1}), and F_{3} = gr′′_{R,1}j_{∗}W_{s−1}(O_{U}). Since the canonical morphisms F_{1} → F and F_{3} → F_{2} are injective and both \overline{F − 1}: F_{3} → F_{1} and \overline{F − 1}: F_{2} → F are injective, we may identify F_{i} with a subsheaf of F for i = 1, 2, 3. We also put G = j_{1∗}gr′_{n_{1}}W_{s}(K_{1}), G_{1} = gr′_{R,1}j_{∗}W_{s}(O_{U}), G_{2} = j_{1∗}gr′′_{n_{1}}W_{s}(K_{1}), and G_{3} = gr′′_{R,1}j_{∗}W_{s}(O_{U}). Further we put H = j_{1∗}gr_{n_{1}}^{′(s−1)}K_{1}, H_{1} = gr_{R,1}^{′(s−1)}j_{∗}O_{U},

H_{2} = j_{1∗}gr_{n_{1}}^{′′(s−1)}K_{1}, and H_{3} = gr_{R,1}^{′′(s−1)}j_{∗}O_{U}. Similarly as F_{i}, we may identify G_{i} and H_{i} with subsheaves of G and H respectively for i = 1, 2, 3. By the induction hypothesis, we have F_{3} = F_{1} ∩ F_{2}. If n_{1} ∉ p^{s}Z, then we have H_{2} = H_{3} = 0 by Corollary 1.24 (ii) and Corollary 1.37 (ii). If n_{1} ∈ p^{s}Z, then we have H_{3} = H_{1} ∩ H_{2} by Corollary 1.24, Corollary 1.37, and the case where s = 1 in the proof of Proposition 1.29. By the commutativity of (1.49), we have G_{3} ⊂ G_{1} ∩ G_{2}. Since exact sequences in Corollary 1.21 and Lemma 1.38 in the case where t = 1 are compatible with the inclusions of sheaves above, the assertion holds by Lemma 1.28.

Proposition 1.40. Let R = ∑_{i∈I} n_{i}D_{i}, where n_{i} ∈ Z_{≥1} for i ∈ I. Let j_{i}: Spec K_{i} → X be the canonical morphism for i ∈ I.

(i) The subsheaf fil′_{R}R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z is equal to the pull-back of ⨁_{i∈I} j_{i∗}fil′_{n_{i}}H^{1}(K_{i}, Q/Z) by the morphism R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z → ⨁_{i∈I} j_{i∗}H^{1}(K_{i}, Q/Z).

(ii) Let R′ = ∑_{i∈I} n′_{i}D_{i}, where n′_{i} ∈ Z_{≥1} such that n_{i} −1 ≤ n′_{i} ≤ n_{i} for i ∈ I. Then there exists a unique injection ϕ_{s}^{′(R/R′)} : gr′_{R/R′}R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z → (gr′_{R/R′}j_{∗}Ω_{U}^{1})|_{D^{(R/R′)1/p}} = gr′_{R/R′}j_{∗}Ω_{U}^{1} ⊗_{O_{D^{(R/R′)}}} O_{D^{(R/R′)1/p}} such that the following diagram is commutative:

\xymatrix{gr′_{R/R′}j_{∗}W_{s}(O_{U}) \ar@{-->}[dr]_{δ_{s}^{′(R/R′)}} \ar[rr]^{φ̃_{s}^{′(R/R′)}} &  & (gr′_{R/R′}j_{∗}Ω_{U}^{1})|_{D^{(R/R′)1/p}} \\  & gr′_{R/R′}R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z. \ar[ur]_{ϕ_{s}^{′(R/R′)}}}

(1.50)

Proof. Let i be an element of I such that n_{i} ≥ 2. By Lemma 1.39, the kernel of δ_{s}^{′(R,i)} is equal to the kernel of φ̃_{s}^{′(R,i)}. Since δ_{s}^{′(R,i)} is surjective, there exists a unique injection ϕ_{s}^{′(R,i)} : gr′_{R,i}R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z → gr′_{R,i}j_{∗}Ω_{U}^{1} ⊗_{O_{D_{i}}} O_{D_{i}^{1/p}} such that the diagram (1.50) for R′ = R − D_{i} is commutative.

(i) Let i be an element of I such that n_{i} ≥ 2. We consider the commutative diagram

\xymatrix{gr′_{R,i}R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z \ar[r] \ar[d]_{ϕ_{s}^{′(R,i)}} & j_{i∗}gr_{n_{i}}H^{1}(K_{i}, Q/Z) \ar[d]^{ϕ^{′(n_{i})}} \\ gr′_{R,i}j_{∗}Ω_{U}^{1} ⊗_{O_{D_{i}}} O_{D_{i}^{1/p}} \ar[r] & j_{i∗}(gr′_{n_{i}}Ω_{K_{i}}^{1} ⊗_{F_{K_{i}}} F_{K_{i}}^{1/p}),}

where F_{K_{i}} is the residue field of K_{i}, the lower horizontal arrow is the inclusion, and ϕ^{′(n_{i})} is as in (1.26). Since the left vertical arrow is injective as proved above, the upper horizontal arrow is injective. Hence the assertion holds by Lemma 1.30 similarly as the proof of Proposition 1.31 (i). (ii) Let J be the subset of I consisting of i ∈ I such that n′_{i} ≠ n_{i}. Since gr′_{n_{i}}j_{i∗}Ω_{K_{i}}^{1} ⊗_{F_{K_{i}}} F_{K_{i}}^{1/p} is the stalk of gr′_{R/R′}j_{∗}Ω_{U}^{1} ⊗_{O_{D^{(R/R′)}}} O_{D^{(R/R′)1/p}} at the

generic point of D_{i}^{1/p} for i ∈ J, the assertion holds similarly as the proof of Proposition 1.31 (ii).

Definition 1.41. Let χ be an element of H_{ét}^{1}(U, Q/Z). We define the total dimension divisor R′_{χ} of χ by R′_{χ} = ∑_{i∈I} dt(χ|_{K_{i}})D_{i}.

We note that we have Supp(R′_{χ} − D) = Supp(R_{χ}) by (1.17).

Definition 1.42. Let χ be an element of H_{ét}^{1}(U, Q/Z). Assume that dt(χ|_{K_{i}}) > 1 for some i ∈ I. Let p^{s} be the order of the p-part of χ. We put Z = Supp(R′_{χ} − D). We define the characteristic form char(χ) of χ to be the image of the p-part of χ by the composition

Γ(X, fil′_{R_{χ}}R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z) → Γ(X, gr′_{R′_{χ}/(R′_{χ}−Z)}R^{1}(ϵ ∘ j)_{∗}Z/p^{s}Z)

⟶^{ϕ_{s}^{′(R′_{χ}/(R′_{χ}−Z))}(X)} Γ(X, gr′_{R′_{χ}/(R′_{χ}−Z)}j_{∗}Ω_{U}^{1} ⊗_{O_{Z}} O_{Z^{1/p}})

= Γ(Z^{1/p}, Ω^{1}(R′_{χ}) ⊗_{O_{X}} O_{Z^{1/p}}).

2 Abbes-Saito’s ramification theory and Witt vectors

2.1 Abbes-Saito’s ramification theory

We briefly recall Abbes-Saito’s non-logarithmic ramification theory ([Sa1, Section 2, Subsection 3.1]).

Definition 2.1 ([Sa1, Definition 1.12]). Let P be a scheme. Let D be a Cartier divisor on P and X a closed subscheme of P. We define the dilatation P^{(D·X)} of P with respect to (D, X) to be the complement of the proper transform of D in the blow-up of X along D ∩ X.

Let X be a smooth separated scheme over a perfect field k of characteristic p > 0. Let D be a divisor on X with simple normal crossings and {D_{i}}_{i∈I} the irreducible components of D. We put U = X − D. Let R = ∑_{i∈I} r_{i}D_{i} be a linear combination with integral coefficients r_{i} ≥ 1 for every i ∈ I. Let Z be the support of R − D. We put P = X ×_{k} X. Let Δ: X → P be the diagonal and pr_{i}: P → X the i-th projection for i = 1, 2. We identify D ⊂ X with closed subschemes of P by the diagonal. We put P^{(D)} = ⋂_{i=1}^{2} P^{(pr_{i}^{∗}D·X)}, where the intersection is taken in the blow-up of P along D ⊂ P. Let D_{i}^{(D)} be the inverse image of D_{i} by P^{(D)} → P. Then D^{(D)} = ∑_{i∈I} D_{i}^{(D)} is a divisor on P^{(D)} with simple normal crossings. The diagonal Δ is canonically lifted to the closed immersion X → P^{(D)} and we identify X with a closed subscheme of P^{(D)} by the lift. We define P^{(R)} to be the dilatation of P^{(D)} with respect to (∑_{i∈I}(r_{i} − 1)D_{i}^{(D)}, X). Let T^{(R)} ⊂ D^{(R)} be the inverse image of Z ⊂ D by P^{(R)} → P. Then the complement P^{(R)} − D^{(R)} is U ×_{k} U ([Sa1, Lemma 2.4.3]) and T^{(R)} is T X(−R) ×_{X} Z ([Sa1, Corollary 2.9]), where T X = Spec S^{•}Ω_{X}^{1} denotes the tangent bundle of X and T X(−R) = Spec S^{•}Ω_{X}^{1}(R). Let G be a finite group and V → U a G-torsor. We consider the open immersion U ×_{k} U = P^{(R)} − D^{(R)} → P^{(R)}. The quotient (V ×_{k} V)/ΔG of V ×_{k} V by the diagonal action of G is finite étale over U ×_{k} U. Let Q^{(R)} be the normalization of P^{(R)} in the finite étale covering (V ×_{k} V)/ΔG → U ×_{k} U. Then the canonical lift X → P^{(R)} of the diagonal is canonically lifted to X → Q^{(R)}.

Definition 2.2 ([Sa1, Definition 2.12]). Let V → U be a G-torsor for a finite group G and R = ∑_{i∈I} r_{i}D_{i} a linear combination with integral coefficients r_{i} ≥ 1 for every i ∈ I.

(i) We say that the ramification of V over U at a point x on D is bounded by R+ if the finite morphism Q^{(R)} → P^{(R)} is étale on a neighborhood of the image of x by the lift X → Q^{(R)}.

(ii) We say that the ramification of V over U along D is bounded by R+ if the finite morphism Q^{(R)} → P^{(R)} is étale on a neighborhood of the image of the lift X → Q^{(R)}.

Lemma 2.3. Let V → U be a G-torsor for a finite group G and R = ∑_{i∈I} r_{i}D_{i} a linear combination with integral coefficients r_{i} ≥ 1 for every i ∈ I. Let 𝔭_{i} be the generic point of D_{i} for i ∈ I. Then the following are equivalent:

(i) The ramification of V over U at 𝔭_{i} is bounded by R+ for every i ∈ I.

(ii) The ramification of V over U along D is bounded by R+.

Proof. Since Q^{(R)} → P^{(R)} is an isomorphism outside of the inverse image of D, the assertion holds by the purity of Zariski-Nagata.

In [Sa1], the notion of the bound of ramification of V over U is defined for R = ∑_{i∈I} r_{i}D_{i} of rational coefficients r_{i} ≥ 1. The next proposition relates the ramification of G-torsor to the ramification of local field.

Proposition 2.4 ([Sa1, Proposition 2.27]). Assume that D is irreducible. Let K be the local field at the generic point 𝔭 of D. Let {G_{K}^{r}}_{r∈Q_{>0}} be the ramification filtration of the absolute Galois group G_{K} of K ([AS1, Definition 3.4]).

Let r ≥ 1 be a rational number and let G_{K}^{r+} = \overline{⋃_{s>r} G_{K}^{s}} denote the closure of

the union of G_{K}^{s} for s > r. For a G-torsor V → U for a finite group G, the following are equivalent:

(i) The ramification of V over U at 𝔭 is bounded by rD+.

(ii) G_{K}^{r+} acts trivially on the finite étale K-algebra L = Γ(V ×_{U} K, O_{V ×_{U} K}).

We note that the filtration {G_{K}^{r}}_{r∈Q_{>0}} is decreasing. We recall the characteristic form defined in [Sa1, Subsection 2.4]. Let W^{(R)} be the largest open subscheme of Q^{(R)} étale over P^{(R)}. We define a scheme E^{(R)} over T^{(R)} to be the fiber product T^{(R)} ×_{P^{(R)}} W^{(R)}. Then there is a unique open sub group scheme E^{(R)0} of a smooth group scheme E^{(R)} over Z such that for every x ∈ Z the fiber E^{(R)0} ×_{Z} x is the connected component of E^{(R)} ×_{Z} x containing the unit section ([Sa1, Proposition 2.16]). Further E^{(R)0} is étale over T^{(R)}. Assume that the ramification of V over U along D is bounded by R+. Then, we say that the ramification of V over U along D is non-degenerate at the multiplicity R if the étale morphism E^{(R)0} → T^{(R)} is finite. We note that this condition is satisfied if we remove a sufficiently large closed subscheme of X of codimension ≥ 2. Assume that the ramification of V over U along D is non-degenerate at the multiplicity R. Then the exact sequence 0 → G̃^{(R)} → E^{(R)0} → T^{(R)} → 0 defines a closed immersion G̃^{(R)∨} → T^{(R)∨} of commutative group schemes to the dual vector bundle defined over Z^{1/p^{n}}, where n ≥ 0 is an integer.

Definition 2.5 ([Sa1, Definition 2.19]). Let V → U be a G-torsor for a finite group G. Assume that the ramification of V over U along D is bounded by R+ and non-degenerate at the multiplicity R. We define the characteristic form Char_{R}(V/U) to be the morphism G̃^{(R)∨} → T^{(R)∨} = (T^{∗}X ×_{X} Z)(R) over Z^{1/p^{n}} for a sufficiently large integer n ≥ 0.

Proposition 2.6 (cf. [Sa1, Corollary 2.28.2]). Let the notation be as in Proposition 2.4. Let O_{K} be the valuation ring of K and F_{K} the residue field of

K. We put N^{(r)} = 𝔪r ; K̄/𝔪r+ ; K̄, where 𝔪r ; K̄ = {a ∈ K̄ | ord_{K}(a) ≥ r} and

𝔪r+ ; K̄ = {a ∈ K̄ | ord_{K}(a) > r}. Let r > 1 be a rational number. Assume that the ramification of V over U along D is bounded by R+ and non-degenerate at the multiplicity rD. Then the following are equivalent:

(i) The characteristic form Char_{rD}(V/U) defines the non-zero mapping by taking the stalk at the generic point of D.

(ii) G_{K}^{r} acts non-trivially on L.

Proof. The assertion holds by [Sa1, Corollary 2.28.2] and its proof.

2.2 Valuation of Witt vectors

We keep the notation in Subsection 2.1. In this subsection, we assume that X is a smooth affine scheme Spec A over k and that D is an irreducible divisor defined by π ∈ A. We put U = Spec B and R = rD, where r ≥ 1 is an integer. Let J ⊂ A be the kernel of the multiplication A ⊗_{k} A → A. Following the construction of P^{(R)} recalled in the previous section, we have

P^{(R)} = Spec(A ⊗_{k} A)[J/(π^{r} ⊗ 1), ((1 ⊗ π)/(π ⊗ 1))^{−1}].

The divisor D^{(R)} is defined by π ⊗ 1. We put P^{(R)} = Spec A^{(r)}. Let Â denote the completion of the local ring O_{X,𝔭} at the generic point 𝔭 of D and Â^{(r)} the completion of the local ring O_{P^{(R)},𝔮} at the generic point 𝔮 of D^{(R)} respectively. Let u: Â → Â^{(r)} and v: Â → Â^{(r)} be the morphisms induced by the first and second projections P → X respectively. We put K = Frac Â and L^{(r)} = Frac Â^{(r)}.

Lemma 2.7. Let F_{K} be the residue field of K. Let a = a′π^{n} ∈ K be an element, where n is an integer and a′ ∈ Â^{×} is a unit. Let r ≥ 1 be an integer.

(i) If n = 0 and if r = 1, then we have ord_{L^{(r)}}(v(a)/u(a)) = 0.

(ii) If n ∉ pZ or r = 1, then ord_{L^{(r)}}(v(a)/u(a) − 1) = r − 1.

(iii) If n ∈ pZ and if r > 1, then ord_{L^{(r)}}(v(a)/u(a) − 1) ≥ r. Further if a′ is not a p-power in F_{K}, the equality holds.

Proof. We put w = (v(π) − u(π))/u(π)^{r} and w′ = (v(a′) − u(a′))/u(π)^{r}. Then we have v(π)/u(π) = 1 + u(π)^{r−1}w and v(a′)/u(a′) = 1 + u(a′)^{−1}u(π)^{r}w′. Hence we have

v(a)/u(a) − 1  (2.1)

= \begin{cases}(1 + u(a′)^{−1}u(π)^{r}w′)(1 + u(π)^{r−1}w)^{n} − 1 & (n ≥ 0) \\ (1 + u(π)^{r−1}w)^{n} ((1 + u(a′)^{−1}u(π)^{r}w′) − (1 + u(π)^{r−1}w)^{−n}) & (n < 0). \end{cases}

Suppose that n = 0 and r = 1. Then we have v(a)/u(a) = 1 + u(a′)^{−1}u(π)w′. Since u(π) = π ⊗ 1 is a uniformizer of Â^{(r)}, the assertion (i) holds. Suppose that n ∉ pZ. Then we have ord_{L^{(r)}}(v(a)/u(a) − 1) = r − 1. Assume that n ∈ pZ. Suppose that n = 0. Then we have ord_{L^{(r)}}(v(a)/u(a) − 1) ≥ r, and the equality holds if w′ is a unit in Â^{(r)}. Suppose that n ≠ 0. We put n = p^{s′}n′, where s′ = ord_{p}(n) ≥ 1. Then we have ord_{L^{(r)}}(v(a)/u(a) − 1) ≥ min{r, p^{s′}(r − 1)}. If r = 1, then we have r > p^{s′}(r − 1) = 0 = r − 1. Since w ∈ Â^{(r)×} is a unit, the assertion holds if r = 1. If r > 1, then p^{s′}(r − 1) ≥ r. Further the equality holds only if (p, r, s′) = (2, 2, 1). Hence we have ord_{L^{(r)}}(v(a)/u(a) − 1) ≥ r. Further, if (p, r, s′) ≠ (2, 2, 1) and if w′ is a unit in Â^{(r)}, the equality holds. If (p, r, s′) = (2, 2, 1), then we have ord_{L^{(r)}}(v(a)/u(a) − 1) = r if and only if u(a′)^{−1}w′ ≠ n′w^{p}. Assume that a is not a p-power in F_{K}. Then the elements π and a′ are pindependent over K^{p}. Hence the images in Â^{(r)}/u(π)Â^{(r)} of w and w′ form a part of a basis of the F_{K}-vector space π^{−r}Ω_{A}^{1} ⊗_{A} F_{K}, since T^{(R)} is T X(−R)×_{X} D. Hence w′ is a unit in Â^{(r)} and u(a′)^{−1}w′ ≠ n′w^{p}. Thus the assertions (ii) and (iii) follow.

Let s ≥ 0 be an integer and put Z[T, S]_{d} = Z[T_{d}, …, T_{s−1}, S_{d}, …, S_{s−1}] for an integer d such that 0 ≤ d ≤ s − 1. We define polynomials Q_{d}(T, S) ∈ Z[T, S]_{d}[1/p] for 0 ≤ d ≤ s − 1 inductively by the relation

∑_{i=d}^{s−1} p^{s−1−i}(T_{i}(1 + S_{i}))^{p^{i−d}} = ∑_{i=d}^{s−1} p^{s−1−i}T_{i}^{p^{i−d}} + ∑_{i=d}^{s−1} p^{s−1−i}Q_{i}^{p^{i−d}}.  (2.2)

It is well-known in the theory of Witt vectors that Q_{d} is an element of Z[T, S]_{d}. For elements x = (x_{s−1}, … x_{0}) and y = (y_{s−1}, …, y_{0}) of W_{s}(A) for a ring A, we put x′ = (x′_{s−1}, …, x′_{0}), where x′_{i} = x_{i}(1 + y_{i}) for i = 0, …, s − 1. Then we have

x′ − x = (Q_{s−1}(x, y), Q_{s−2}(x, y), …, Q_{0}(x, y)).  (2.3)

Lemma 2.8 (cf. [AS3, Lemma 12.2]). Let the notation be as above.

(i) Q_{d}(T, S) belongs to the ideal of Z[T, S]_{d} generated by (S_{i})_{d≤i≤s−1} for d = 0, …, s − 1.

(ii) Q_{d}(T, S) − ∑_{i=d}^{s−1} T_{i}^{p^{i−d}}S_{i} belongs to the ideal of Z[T, S]_{d} generated by (S_{i}S_{j})_{d≤i,j≤s−1} for d = 0, …, s − 1.

(iii) If we replace T_{i} by T_{i}^{p^{s−1−i}} in Q_{d}(T, S), the polynomial Q_{d}(T, S) is homogeneous of degree p^{s−1−d} as a polynomial of multi-value T for 0 ≤ d ≤ s − 1.

Proof. The assertions (i) and (ii) are the same as (i) and (ii) in [AS3, Lemma 12.2] respectively. We prove (iii) by induction on d. If d = s − 1, we have Q_{s−1} = T_{s−1}S_{s−1}. Hence the assertion holds. If d < s − 1, we have

Q_{d} = p^{d−s+1} (∑_{i=d}^{s−1} p^{s−1−i}T_{i}^{p^{i−d}} ((1 + S_{i})^{p^{i−d}} − 1) − ∑_{i=d+1}^{s−1} p^{s−1−i}Q_{i}^{p^{i−d}}) .

By the induction hypothesis, the polynomial Q_{i}(T, S) is homogeneous of degree p^{s−1−i} for T for d + 1 ≤ i ≤ s − 1 with T_{j} replaced by T_{j}^{p^{s−1−j}} for i ≤ j ≤ s − 1. Hence Q_{i}(T, S)^{p^{i−d}} is homogeneous of degree p^{s−1−d} for T for d + 1 ≤ i ≤ s − 1 with the same replacement of T_{j} for i ≤ j ≤ s − 1. Hence the assertion holds.

Lemma 2.9. Let a = (a_{s−1}, …, a_{0}) be an element of W_{s}(K) and put b = (b_{s−1}, …, b_{0}) ∈ W_{s}(L^{(r)}), where b_{i} = v(a_{i})/u(a_{i}) − 1 if a_{i} ≠ 0 and b_{i} = 0 if a_{i} = 0 for 0 ≤ i ≤ s − 1. Let m ≥ 1 be an integer and assume that a ∈ fil′_{m}W_{s}(K). Let r ≥ 1 be an integer.

(i) If (m, r) = (1, 1), then p^{i} ord_{L^{(r)}}(Q_{d}(u(a), b)) ≥ −m + 1 for every 0 ≤ d ≤ s − 1.

(ii) If r > 1, then p^{i} ord_{L^{(r)}}(Q_{d}(u(a), b)) > −m + r for every 0 < d ≤ s − 1, and ord_{L^{(r)}}(Q_{0}(u(a), b)) ≥ −m + r.

Proof. We put s′ = min{ord_{p}(m), s}. Let a′ = (a′_{s−1}, …, a′_{0}) be an element of W_{s}(K) such that a′_{i} = 0 if p^{i} ord_{K}(a_{i}) = −m and a′_{i} = a_{i} if p^{i} ord_{K}(a_{i}) ≥ −(m − 1) for 0 ≤ i ≤ s − 1. We note that if s′ ≤ i ≤ s − 1 then a′_{i} = a_{i} by (1.15). Let a′′ = (a′′_{s′−1}, …, a′′_{0}) be an element of W_{s′}(K) such that a′′_{i} = 0 if p^{i} ord_{K}(a_{i}) ≥ −(m − 1) and a′′_{i} = a_{i} if p^{i} ord_{K}(a_{i}) = −m for 0 ≤ i ≤ s′ − 1. Then we have a = a′ + V^{s−s′}(a′′). Let b′ ∈ W_{s}(L^{(r)}) and b′′ ∈ W_{s′}(L^{(r)}) be the elements defined from a′ and a′′ respectively similarly as b defined from a. Since we have Q(u(a), b) = (Q_{s−1}(u(a), b), …, Q_{0}(u(a), b)) = v(a) − u(a) and similarly for a′ and a′′ by (2.3), we have Q(u(a), b) = Q(u(a′), b′) + V^{s−s′}(Q(u(a′′), b′′)). Since fil_{n}W_{s}(L^{(r)}) is a submodule of W_{s}(L^{(r)}) for n ∈ Z, the assertions for a hold if the assertions for a′ and a′′ hold. Hence we prove the assertions for a′ and a′′. By the definitions of a′ and a′′, we have ord_{L^{(r)}}(u(a′_{i})) ≥ −(m − 1)/p^{i} for 0 ≤ i ≤ s − 1 and ord_{L^{(r)}}(u(a′′_{i})) ≥ −m/p^{i} for 0 ≤ i ≤ s′ − 1. If r > 1, then we have ord_{L^{(r)}}(b′_{i}) ≥ r − 1 for 0 ≤ i ≤ s − 1 and ord_{L^{(r)}}(b′′_{i}) ≥ r for 0 ≤ i ≤ s′ − 1 by Lemma 2.7 (ii) and (iii). If (m, r) = (1, 1), then we have s′ = 0 and ord_{L^{(r)}}(b′_{i}) ≥ r − 1 for 0 ≤ i ≤ s − 1 by Lemma 2.7 (ii). Hence, by Lemma 2.8 (i) and (iii), we have

p^{d} ord_{L^{(r)}}(Q_{d}(u(a′), b′)) ≥ −(m − 1) + p^{d}(r − 1) ≥ −m + r.  (2.4)

Further we have

p^{d} ord_{L^{(r)}}(Q_{d}(u(a′′), b′′)) ≥ −m + p^{d}r ≥ −m + r.  (2.5)

If r > 1, then the equality holds in the right inequality in (2.4) only if d = 0 and so in (2.5). Hence the assertions hold.

Lemma 2.10. Let the notation be as in Lemma 2.9. Let m ≥ 2 be an integer and assume that a ∈ fil′_{m}W_{s}(K). Then we have ord_{L^{(m)}}(Q_{0}(u(a), b) − ∑_{i=0}^{s−1} u(a_{i})^{p^{i}}b_{i}) > 0.

Proof. We put s′ = min{ord_{p}(m), s}. Let a′ = (a′_{s−1}, …, a′_{0}) and a′′ = (a′′_{s′−1}, …, a′′_{0}) be as in the proof of Lemma 2.9. We have a = a′ + V^{s−s′}(a′′). Let b′ ∈ W_{s}(L^{(m)}) and b′′ ∈ W_{s′}(L^{(m)}) be the elements defined from a′ and a′′ respectively similarly as b defined from a. Since Q(u(a), b) = Q(u(a′), b′) + V^{s−s′}Q(u(a′′), b′′) as in the proof of Lemma 2.9 and ∑_{i=0}^{s−1} u(a_{i})^{p^{i}}b_{i} = ∑_{i=0}^{s−1} u(a′_{i})^{p^{i}}b′_{i} + ∑_{i=0}^{s′−1} u(a′′_{i})^{p^{i}}b′′_{i}, it is sufficient to prove the assertion for a′ and a′′. As in the proof of Lemma 2.9, we have ord_{L^{(m)}}(u(a′_{i})) ≥ −(m − 1)/p^{i} for 0 ≤ i ≤ s − 1 and ord_{L^{(m)}}(u(a′′_{i})) ≥ −m/p^{i} for 0 ≤ i ≤ s′ − 1. Further we have ord_{L^{(m)}}(b′_{i}) ≥ m − 1 for 0 ≤ i ≤ s − 1 and ord_{L^{(m)}}(b′′_{i}) ≥ m for 0 ≤ i ≤ s′ − 1.

Hence, by Lemma 2.8 (ii) and (iii), we have

ord_{L^{(m)}}(Q_{0}(u(a′), b′) − ∑_{i=0}^{s−1} u(a′_{i})^{p^{i}}b′_{i}) ≥ −(m − 1) + 2(m − 1) = m − 1 > 0.

Further we have

ord_{L^{(m)}}(Q_{0}(u(a′′), b′′) − ∑_{i=0}^{s′−1} u(a′′_{i})^{p^{i}}b′′_{i}) ≥ −m + 2m = m > 0.

Hence the assertion holds.

2.3 Calculation of characteristic forms

Let X be a smooth separated scheme over a perfect field k of characteristic p > 0. Let D be a divisor on X with simple normal crossings and {D_{i}}_{i∈I} the irreducible components of D. We put U = X − D and let j: U → X denote the canonical open immersion. Let K_{i} be the local field at the generic point of D_{i} for i ∈ I and let O_{K_{i}} be the valuation ring of K_{i} for i ∈ I. Let χ be an element of H_{ét}^{1}(U, Q/Z). In this subsection, we prove the equality of the characteristic form char(χ) of χ and the characteristic form Char_{R}(V/U) of the Galois torsor V → U corresponding to χ. Let p_{i}: P^{(R)} → X be the morphism induced by the i-th projection for i = 1, 2. Let u: p_{1}^{−1}O_{X} → O_{P^{(R)}} and v: p_{2}^{−1}O_{X} → O_{P^{(R)}} be the canonical morphisms of sheaves on P^{(R)} by abuse of notation. Let L_{i}^{(R)} be the fractional field of the completion of the local ring O_{P^{(R)},𝔮_{i}}, where R = ∑_{i∈I} r_{i}D_{i} is a linear combination with integer coefficients r_{i} ≥ 1 for every i ∈ I and 𝔮_{i} is the generic point of the pull-back D_{i}^{(R)} of D_{i}^{(D)} by P^{(R)} → P^{(D)}. If D = D_{1} is irreducible, then we simply write L^{(r_{1})} for L_{1}^{(R)} as in the previous section. We first consider the tamely ramified case.

Lemma 2.11. Assume that the order n of χ is prime to p and regard χ as an element of H_{ét}^{1}(U, Z/nZ). We put G = Z/nZ. Let V → U be the G-torsor corresponding to χ. Let R = ∑_{i∈I} r_{i}D_{i} be a linear combination with integral coefficients r_{i} ≥ 1 for every i ∈ I.

(i) The ramification of V over U along D is bounded by D+.

(ii) The characteristic form Char_{R}(V/U) is the zero mapping.

Proof. (i) By Lemma 2.3, we may assume that D = D_{1} is irreducible. Since the assertion is local, we may assume that X = Spec A is affine and D is defined by an element of A. Since G_{K_{1}}^{1} is the inertia group of G_{K_{1}} ([AS1, Proposition 3.7 (1)]), we may assume that k contains a primitive n-th root of unity by Lemma 2.3 and Proposition 2.4. Since ord_{L^{(r_{1})}}(v(a)/u(a)) = 0 for every unit a ∈ O_{K_{1}}^{×} by Lemma 2.7 (i), the assertion holds.

(ii) Let Z be the support of R−D. By (i) and Proposition 2.4, the ramification group G_{K_{i}}^{r_{i}+} acts trivially on L_{i} = Γ(V ×_{U} K_{i}, O_{V ×_{U} K_{i}}) for D_{i} contained in Z. By Proposition 2.6, the stalk of the characteristic form Char_{R}(V/U) at the generic point of D_{i} defines the zero mapping for D_{i} contained in Z. Hence the assertion holds.

By Lemma 2.11, the bound of the ramification of the Galois torsor V → U corresponding to χ and its characteristic form Char_{R}(V/U) does not depend on the prime-to-p-part of χ, that is, they are dependent only on the p-part of χ.

Proposition 2.12. Assume that the order of χ is p^{s} and regard χ as an element of H_{ét}^{1}(U, Z/p^{s}Z). We put G = Z/p^{s}Z. Let V → U be the G-torsor corresponding to χ.

(i) The ramification of V over U along D is bounded by R′_{χ}+, where R′_{χ} is the total dimension divisor of χ (Definition 1.41).

(ii) Assume that R′_{χ} ≠ D and put Z = Supp(R′_{χ} − D). Then the scheme E^{(R′_{χ})} → T^{(R′_{χ})} = T X(−R′_{χ}) ×_{X} Z is defined by the Artin-Schreier equation t^{p} − t = char(χ).

Proof. We put m_{i} = dt(χ|_{K_{i}}) for i ∈ I. Let a = (a_{s−1}, …, a_{0}) ∈ fil′_{R′_{χ}}j_{∗}W_{s}(O_{U}) be an element whose image by δ_{s} (1.33) is χ. Then V ×_{k} V/ΔG → U ×_{k} U is the G-torsor defined by the Artin-Schreier-Witt equation (F − 1)(t) = v(a) − u(a).

(i) By Lemma 2.3, we may assume that D is irreducible. Since the assertion is local, we may assume that X = Spec A is affine and that D is defined by an element of A. By (2.3) and Lemma 2.9, the difference v(a) − u(a) is a regular function on P^{(R′_{χ})}. Hence the assertion holds. (ii) By (i), (2.3), Lemma 2.9 (ii), and Lemma 2.10, the scheme E^{(R′_{χ})} → T^{(R′_{χ})} is the G-torsor defined by the Artin-Schreier equation t^{p} − t = ∑_{j=0}^{s−1} u(a_{j})^{p^{j}−1}(v(a_{j}) − u(a_{j})). We put n_{ij} = ord_{K_{i}}(a_{j}) for i ∈ I and 0 ≤ j ≤ s − 1. As calculating in the proof of Lemma 2.7, we have the following on a neighborhood of the generic point of D_{i}^{(R′_{χ})} for i ∈ I such that m_{i} > 1:

(a) If n_{ij} ∉ pZ, we have u(a_{j})^{p^{j}−1}(v(a_{j}) − u(a_{j})) = n_{ij}u(a_{j})^{p^{j}}u(t_{i})^{m_{i}−1}w_{i};

(b) If n_{ij} ∈ pZ and if (p, m_{i}, ord_{p}(n_{ij})) ≠ (2, 2, 1), we have u(a_{j})^{p^{j}−1}(v(a_{j}) − u(a_{j})) = u(a_{j})^{p^{j}}u(a′_{j})^{−1}u(t_{i})^{m_{i}}w′_{ij};

(c) If (p, m_{i}, ord_{p}(n_{ij})) = (2, 2, 1), we have u(a_{j})^{p^{j}−1}(v(a_{j}) − u(a_{j})) = u(a_{j})^{p^{j}}(u(a′_{j})^{−1}u(t_{i})^{2}w′_{ij} + (n_{ij}/2)u(t_{i})^{2}w_{i}^{2}),

where t_{i} is a local equation of D_{i}, a′_{j} = a_{j}/t_{i}^{n_{ij}}, w_{i} = (v(t_{i}) − u(t_{i}))/u(t_{i})^{m_{i}}, and w′_{ij} = (v(a′_{j}) − u(a′_{j}))/u(t_{i})^{m_{i}} for every j = 0, …, s − 1. Since a ∈ fil′_{R′_{χ}}j_{∗}W_{s}(O_{U}), we have p^{j} ord_{L_{i}^{(m_{i})}}(a_{j}) ≥ −(m_{i} − 1) if n_{ij} ∉ pZ and

p^{j} ord_{L_{i}^{(m_{i})}}(a_{j}) ≥ −m_{i} if n_{ij} ∈ pZ. If (p, m_{i}, ord_{p}(n_{ij}), p^{j}n_{ij}) = (2, 2, 1, −2), we have (p, j, n_{ij}) = (2, 0, −2). Hence the assertion holds by identifying w_{i} and w′_{ij} with dt_{i}/t_{i}^{m_{i}} and da′_{j}/t_{i}^{m_{i}} respectively.

Corollary 2.13. Let V → U be the Galois torsor corresponding to χ. Assume that the ramification of V over U along D is non-degenerate at the multiplicity R′_{χ}.

(i) The image of the generator 1 ∈ G̃^{(R′_{χ})∨} = F_{p} by Char_{R′_{χ}}(V/U) is equal to char(χ).

(ii) Assume that D = D_{1} is irreducible and that dt(χ|_{K_{1}}) > 1. Then the ramification of V over U at the generic point of D is not bounded by rD+ for any rational number r such that 1 ≤ r < dt(χ|_{K_{1}}).

Proof. (i) The assertion holds by Lemma 2.11 and Proposition 2.12 (ii). (ii) We put K = K_{1}. Assume that G_{K}^{r+} acts trivially on L = Γ(V ×_{U}K, O_{V ×_{U} K}) for a rational number r such that 1 ≤ r < dt(χ|_{K}). Then, by (i) and Proposition 2.6, the stalk char(χ|_{K}) of char(χ) at the generic point of D must be 0. However char(χ) is non-zero. Hence the assertion holds by Proposition 2.4.

3 Equality of ramification filtrations

Let K be a complete discrete valuation field of characteristic p > 0 and F_{K} the residue field. Let G_{K} be the absolute Galois group of K. We show that the abelianization of Abbes-Saito’s filtration {G_{K}^{r}}_{r∈Q>0} ([AS1, Definition 3.4]) is the same as {fil′_{m}H^{1}(K, Q/Z)}_{m∈Z_{≥1}} (Definition 1.2) by taking dual. If m > 2, then it has been proved by Abbes-Saito ([AS3, Théorème 9.10]).

Theorem 3.1. Let χ be an element of H^{1}(K, Q/Z). Let m ≥ 1 be an integer. Let r be a rational number such that m ≤ r < m+1. If F_{K} is finitely generated over a perfect subfield k ⊂ F_{K}, then the following are equivalent:

(i) χ ∈ fil′_{m}H^{1}(K, Q/Z).

(ii) χ(G_{K}^{m+}) = 0.

(iii) χ(G_{K}^{r+}) = 0.

Proof. Since G_{K}^{1+} is a pro-p-subgroup of G_{K} ([AS1, Proposition 3.7.1]), we may assume that the order of χ is a power of p. Let p^{s} be the order of χ and regard χ as an element of H_{ét}^{1}(U, Z/p^{s}Z). We put G = Z/p^{s}Z. As in [AS3, 6.1], we take a smooth affine connected scheme X over k and a smooth irreducible divisor D on X such that the completion Ô_{X,𝔭} of the local ring O_{X,𝔭} at the generic point 𝔭 of D is isomorphic to O_{K}. By shrinking X if necessary, we take a G-torsor V → U = X − D corresponding to a character of π_{1}^{ab}(U) whose restriction to G_{K} is χ.

By Proposition 2.12 (i) and Corollary 2.13 (ii), the ramification of V over U at the generic point of D is bounded by rD+ for a rational number r ≥ 1 if and only if r ≥ dt(χ). Further, by Proposition 2.4, the former condition is equivalent to that G_{K}^{r+} acts trivially on L = Γ(V ×_{U} K, O_{V ×_{U} K}). Hence χ(G_{K}^{r+}) = 0 if and only if r ≥ dt(χ). Since the condition (i) holds if and only if m ≥ dt(χ), the equivalence of (i) and (ii) follows. Since m ≤ r, the condition (ii) deduces the condition (iii). Suppose that the condition (iii) holds. Since r ≥ dt(χ), we have m = [r] ≥ dt(χ). Hence the condition (ii) holds.

Proof of Theorem 0.1. We may identify K with F_{K}((π)) by taking a uniformizer of K. Let K_{h} = Frac(F_{K}[π]_{(π)}^{h}) be the fractional field of the henselization of the localization F_{K}[π]_{(π)} of F_{K}[π] at the prime ideal (π). Since the completion of K_{h} is K, the canonical morphisms G_{K} → G_{K_{h}} and H^{1}(K_{h}, Q/Z) → H^{1}(K, Q/Z) are isomorphisms. Let k be a perfect subfield of F_{K} and take a separating transcendental basis S of F_{K} over k. For a finite subextension E of F_{K} over k(S′), where S′ is a finite set of S, let K_{E,h} denote the fractional field of the henselization of the local ring E[π]_{(π)}. Since F_{K} = \underrightarrow{lim} E, we may identify K_{h} with the inductive limit \underrightarrow{lim} K_{E,h} and H^{1}(K_{h}, Q/Z) with \underrightarrow{lim} H^{1}(K_{E,h}, Q/Z), where E runs through such subfields of F_{K}. Let χ be an element of H^{1}(K, Q/Z). Take a subfield E of F_{K} such that E is a subextension of F_{K} over k(S′) for a finite subset S′ ⊂ S and that χ ∈ H^{1}(K_{E,h}, Q/Z). Let K_{E} denote the completion of K_{E,h}. We identify H^{1}(K_{E}, Q/Z) with H^{1}(K_{E,h}, Q/Z) and χ ∈ H^{1}(K_{E,h}, Q/Z) with an element of H^{1}(K_{E}, Q/Z). We prove that each condition in Theorem 3.1 holds for K if and only if it holds for K_{E}. Let dt_{K}(χ) and dt_{K_{E}}(χ) denote the total dimension of χ as an element of H^{1}(K, Q/Z) and H^{1}(K_{E}, Q/Z) respectively. We put dt_{K}(χ) = n and dt_{K_{E}}(χ) = n′ and prove that n = n′. Since fil′_{m}W_{s}(K_{E}) ⊂ fil′_{m}W_{s}(K) for every integer m ≥ 1, we have fil′_{m}H_{1}(K_{E}, Q/Z) ⊂ fil′_{m}H^{1}(K, Q/Z). Hence we have 1 ≤ n ≤ n′, which proves that n = 1 if n′ = 1. Assume that n′ > 1. Take an element ā of gr′_{n′}W_{s}(E(π)) whose image in gr′_{n′}H^{1}(K_{E}, Q/Z) is χ. Let char_{K}(χ) and char_{K_{E}}(χ) denote the characteristic form of χ as an element of H^{1}(K, Q/Z) and H^{1}(K_{E}, Q/Z) respectively. Let O_{K} and O_{K_{E}} denote the valuation rings of K and K_{E} respectively. Since F_{K} is separable over E, we have an injection Ω_{E[π]_{(π)}}^{1} → Ω_{F_{K}[π]_{(π)}}^{1}. This injection induces the injection Ω_{O_{K_{E}}}^{1} → Ω_{O_{K}}^{1} , and further the injection gr′_{n′}Ω_{K_{E}}^{1} →

gr′_{n′}Ω_{K}^{1}. Hence the canonical morphism gr′_{n′}Ω_{K_{E}}^{1} ⊗_{F_{K}} F_{K}^{1/p} → gr′_{n′}Ω_{K}^{1} ⊗_{F_{K}} F_{K}^{1/p} is injective. Since char_{K_{E}}(χ) ≠ 0, the image of char_{K_{E}}(χ) in gr′_{n′}Ω_{K}^{1} ⊗_{F_{K}} F_{K}^{1/p} is not 0. This implies that char_{K}(χ) is the image of char_{K_{E}}(χ) in gr′_{n′}Ω_{K}^{1} ⊗_{F_{K}} F_{K}^{1/p}. Hence we have n = n′. Since the condition (i) in Theorem 3.1 holds for K if and only if m ≥ n and similarly for K_{E}, the condition (i) in Theorem 3.1 for K is equivalent to that for K_{E}.

Let r ≥ 1 be a rational number. Since K is an extension of K_{E} of ramification index 1 and the extension of residue fields is separable, by applying [AS2, Lemma 2.2], the canonical morphism G_{K} → G_{K_{E}} induces the surjection G_{K}^{s} → G_{K_{E}}^{s} for every s ∈ Q_{≥1}. Hence we have χ(G_{K}^{r+}) = 0 if and only if χ(G_{K_{E}}^{r+}) = 0, which proves the assertions for conditions (ii) and (iii) in Theorem 3.1.

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Yuri Yatagawa

Fakultät für Mathematik Universität Regensburg Universitätsstraße 31 93040 Regensburg Germany yuri.yatagawa@mathematik.uniregensburg.de