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A characterization of ramification groups
of local fields with imperfect residue fields

Takeshi Saito
Abstract

We give a characterization of ramification groups of local fields with imperfect residue fields, using those for local fields with perfect residue fields. As an application, we reprove an equality of ramification groups for abelian extensions defined in different ways.

Let KK be a henselian discrete valuation field. Let LL be a Galois extension and let G=Gal​(L/K)G={\rm Gal}(L/K) be the Galois group. In the classical case where the residue field FF of KK is perfect, the lower numbering filtration (Gi,cl)(G_{i,{\rm cl}}) indexed by integers i≧0i\geqq 0 is defined by Gi,cl=Ker​(G→Aut​(𝒪L/𝔪Li+1))G_{i,{\rm cl}}={\rm Ker}(G\to{\rm Aut}({\cal O}_{L}/{\mathfrak{m}}^{i+1}_{L})). Further, the upper numbering filtration (Gclr)(G^{r}_{\rm cl}) indexed by rational numbers r>0r>0 is defined using the Herbrand function to renumber the lower numbering [8, Chapitre IV, Section 3].

In the general case where the residue field need not be perfect, an upper numbering filtration (Gr)(G^{r}) indexed by rational numbers r>0r>0 is defined first in [1] using rigid geometry and later in [6] purely in the language of schemes. In the classical case where FF is perfect, they are related to each other by the relation Gr=Gclr−1G^{r}=G^{r-1}_{\rm cl}.

We give an axiomatic characterization of the filtration (Gr)(G^{r}). The axiom has two conditions. The first condition is the relation in the classical case above. The second condition is the compatibility with tangentially dominant extensions. A similar approach reducing to the classical case was proposed in [3].

For a discrete valuation field KK, the tangent space at an algebraic closure F¯\bar{F} of the residue field FF is defined as an F¯\bar{F}-vector space using the cotangent complex. In the classical case where the residue field is perfect, it is nothing but the scalar extension of the Zariski tangent space defined as the dual HomF​(𝔪K/𝔪K2,F¯){\rm Hom}_{F}({\mathfrak{m}}_{K}/{\mathfrak{m}}_{K}^{2},\bar{F}). An extension of discrete valuation fields is said to be tangentially dominant if the induced morphism on the tangent spaces is dominant (Definition 2.1.2). An unramified extension is tangentially dominant and a tangentially dominant extension has ramification index 1.

The uniqueness is a consequence of the existence of tangentially dominant extension with perfect residue field. The existence follows from the functorial properties of the filtration (Gr)(G^{r}).

For r>1r>1, the graded quotient Grr​G=Gr/Gr+{\rm Gr}^{r}G=G^{r}/G^{r+} is defined by Gr+=⋃s>rGsG^{r+}=\bigcup_{s>r}G^{s} and is an 𝐅p{\mathbf{F}}_{p}-vector space. A canonical injection

(0.1) Hom​(Grr​G,𝐅p)→HomF¯​(𝔪K¯r/𝔪K¯r+,H1​(LF¯/S)){\rm Hom}({\rm Gr}^{r}G,{\mathbf{F}}_{p})\to{\rm Hom}_{\bar{F}}({\mathfrak{m}}_{\bar{K}}^{r}/{\mathfrak{m}}_{\bar{K}}^{r+},H_{1}(L_{\bar{F}/S}))

is defined in [7, (4.20)], as a generalization of a non-logarithmic variant of the refined Swan conductor defined by Kato in [4]. We also give an axiomatic characterization of this morphism, similar to the characterization for GrG^{r} itself.

As an application of the characterizations, we give a new proof of the equality of two filtrations for abelian extensions in positive characteristic. By the Hasse-Arf theorem, the filtration (Gn)(G^{n}) defined in [1] is in fact indexed by integers n>1n>1 for abelian extensions. The other filtration is the filtration (GMan)(G^{n}_{\rm Ma}) defined by Matsuda in [5] as a modification of that defined by Kato in [4]. The equality was proved in [2] except for the smallest index n=2n=2 and the remaining case was proved by Yatagawa in [9]. The equality is proved by verifying that the filtration (GMan)(G^{n}_{\rm Ma}) satisfies the same axiom. We also prove that the injection (0.1) equals the morphism rsw′{\rm rsw}^{\prime} defined in [5] and [9], as a modification of the refined Swan conductor defined in [4].

A variant (Glogr)(G^{r}_{\log}) of the upper numbering filtration (Gr)(G^{r}) called the logarithmic upper numbering filtration is also defined in [1]. In the case where the ramification index eL/Ke_{L/K} is 1, the two filtrations are the same: Gr=GlogrG^{r}=G^{r}_{\log}. If K′K^{\prime} is a log smooth extension of KK and L′=L​K′L^{\prime}=LK^{\prime}, the canonical injection G′=Gal​(L′/K′)→G=Gal​(L/K)G^{\prime}={\rm Gal}(L^{\prime}/K^{\prime})\to G={\rm Gal}(L/K) is known to induce isomorphisms Glog′⁣e​r=GlogrG^{\prime er}_{\log}=G^{r}_{\log} for e=eK′/Ke=e_{K^{\prime}/K}. Further if the ramification index eK′/Ke_{K^{\prime}/K} is divisible by eL/Ke_{L/K}, we have eL′/K′=1e_{L^{\prime}/K^{\prime}}=1. Thus, a characterization of (Gr)(G^{r}) gives an indirect characterization (Glogr)(G^{r}_{\log}).

The author thanks the referee for careful reading and the suggestion to include comments on the logarithmic filtration. The research is partially supported by Grant-in-Aid (B) 19H01780.

1 Totally ramified case

Let KK be a henselian discrete valuation field. Let LL be a totally ramified Galois extension of KK and let G=Gal​(L/K)G={\rm Gal}(L/K) be the Galois group. For a rational number r>1r>1, the upper ramification group GrG^{r} defined in [1, Definition 3.4] equals the subgroup defined in [8, Chapitre IV, Section 3] denoted Gclr−1G^{r-1}_{\rm cl}, by [1, Proposition 3.7 (3)].

Assume that LL is wildly ramified and let r>1r>1 be the largest rational number such that the subgroup GrG^{r} of the wild inertia subgroup P⊂GP\subset G is non-trivial. Let EE be the residue field and e=eL/Ke=e_{L/K} be the ramification index. We give a description of the canonical injection

(1.1) Gr∨=Hom𝐅p​(Gr,𝐅p)→HomE​(𝔪Le​(r−1)/𝔪Le​(r−1)+1,E)G^{r\vee}={\rm Hom}_{{\mathbf{F}}_{p}}(G^{r},{\mathbf{F}}_{p})\to{\rm Hom}_{E}({\mathfrak{m}}_{L}^{e(r-1)}/{\mathfrak{m}}_{L}^{e(r-1)+1},E)

for the 𝐅p{\mathbf{F}}_{p}-vector space GrG^{r}, in the case where LL is totally ramified over KK. The injection (1.1) is a special case of (0.1).

We begin with a description of extensions of vector spaces over a field of characteristic p>0p>0 by 𝐅p{\mathbf{F}}_{p}-vector spaces.

Lemma 1.1.

Let FF be a field of characteristic p>0p>0.

1. Let G⊂FG\subset F be a finite subgroup of the additive group. Then, the polynomial

(1.2) a1=∏σ∈G(X−σ)∏σ∈G,σ≠0(−σ)a_{1}=\dfrac{\prod_{\sigma\in G}(X-\sigma)}{\prod_{\sigma\in G,\,\sigma\neq 0}(-\sigma)}

∈F​[X]\in F[X] is a unique additive separable polynomial such that the coefficient of degree 11 is 11 and that the sequence

(1.3) 0→G→𝐆a→a1𝐆a→0\begin{CD}0@>{}>{}>G@>{}>{}>{\mathbf{G}}_{a}@>{a_{1}}>{}>{\mathbf{G}}_{a}@>{}>{}>0\end{CD}

is exact.

2. ([7, Proposition 2.1.6 (2)⇒\Rightarrow(3)]) Let EE be an FF-vector space of finite dimension and let 0→G→H→E→00\to G\to H\to E\to 0 be an extension of EE by an 𝐅p{\mathbf{F}}_{p}-vector space GG of finite dimension, as smooth group schemes over FF. Define a morphism

(1.4) [H]:G∨=Hom𝐅p​(G,𝐅p)→Ext​(E,𝐅p)=E∨=HomF​(E,F)[H]\colon G^{\vee}={\rm Hom}_{{\mathbf{F}}_{p}}(G,{\mathbf{F}}_{p})\to{\rm Ext}(E,{\mathbf{F}}_{p})=E^{\vee}={\rm Hom}_{F}(E,F)

by sending a character χ:G→𝐅p\chi\colon G\to{\mathbf{F}}_{p} to the linear form f:E→Ff\colon E\to F such that there exists a commutative diagram

0→G→H→E→0χ↓↓↓f0→𝐅p→𝐆a→xp−x𝐆a→0.\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>G@>{}>{}>H@>{}>{}>E@>{}>{}>0\\ @V{\chi}V{}V@V{}V{}V@V{}V{f}V\\ 0@>{}>{}>{\mathbf{F}}_{p}@>{}>{}>{\mathbf{G}}_{a}@>{x^{p}-x}>{}>{\mathbf{G}}_{a}@>{}>{}>0.\end{CD}

If HH is connected, then the morphism [H]:G∨→E∨[H]\colon G^{\vee}\to E^{\vee} is an injection.

Proof.

1. By [7, Lemma 2.1.5], a=∏σ∈G(X−σ)∈F​[X]a=\prod_{\sigma\in G}(X-\sigma)\in F[X] is an additive separable polynomial such that (1.3) with a1a_{1} replaced by aa is exact. Since the coefficient in aa of degree 11 is ∏σ∈G,σ≠0(−σ)\prod_{\sigma\in G,\,\sigma\neq 0}(-\sigma), the assertion follows. ∎

Let KK be a henselian discrete valuation field and LL be a totally ramified Galois extension of degree ee of Galois group GG. Let α∈L\alpha\in L be a uniformizer and let E=FE=F denote the residue field. The minimal polynomial f∈𝒪K​[X]f\in{\cal O}_{K}[X] is an Eisenstein polynomial and the constant term π=f​(0)\pi=f(0) is a uniformizer of KK. We define a closed immersion T=Spec​𝒪L→Q=Spec​𝒪K​[X]T={\rm Spec}\,{\cal O}_{L}\to Q={\rm Spec}\,{\cal O}_{K}[X] by sending XX to α\alpha. For a rational number r>1r>1 such that e​r∈𝐙er\in{\mathbf{Z}}, define a dilatation

QT[r]=Spec​𝒪L​[X]​[fαe​r]→QT=Spec​𝒪L​[X].Q^{[r]}_{T}={\rm Spec}\,{\cal O}_{L}[X]\Bigl{[}\dfrac{f}{\alpha^{er}}\Bigr{]}\to Q_{T}={\rm Spec}\,{\cal O}_{L}[X].

The generator ff of the kernel I=Ker​(𝒪K​[X]→𝒪L)I={\rm Ker}({\cal O}_{K}[X]\to{\cal O}_{L}) defines a basis over 𝒪L{\cal O}_{L} of the conormal module NT/Q=I/I2N_{T/Q}=I/I^{2} and αe​r\alpha^{er} defines a basis of the EE-vector space 𝔪Le​r/𝔪Le​r+1{\mathfrak{m}}_{L}^{er}/{\mathfrak{m}}_{L}^{er+1}. As subspaces of NE/Q=J/J2N_{E/Q}=J/J^{2} for J=Ker​(𝒪K​[X]→E)=(X,f)=(X,π)J={\rm Ker}({\cal O}_{K}[X]\to E)=(X,f)=(X,\pi), we have an equality

(1.5) NT/Q⊗𝒪LE=𝔪K/𝔪K2N_{T/Q}\otimes_{{\cal O}_{L}}E={\mathfrak{m}}_{K}/{\mathfrak{m}}_{K}^{2}

since ff is an Eisenstein polynomial. The basis ff of NT/Q⊗𝒪LEN_{T/Q}\otimes_{{\cal O}_{L}}E corresponds to the uniformizer π=f​(0)∈𝔪K/𝔪K2\pi=f(0)\in{\mathfrak{m}}_{K}/{\mathfrak{m}}_{K}^{2}.

By sending SS to f/αe​rf/\alpha^{er}, we define an isomorphism 𝒪L​[X,S]/(f−αe​r​S)→𝒪L​[X]​[f/αe​r]{\cal O}_{L}[X,S]/(f-\alpha^{er}S)\to{\cal O}_{L}[X][f/\alpha^{er}]. Since ff is an Eisenstein polynomial, the reduced closed fiber QE[r]=Spec​(𝒪L​[X]​[f/αe​r]⊗𝒪LE)redQ^{[r]}_{E}={\rm Spec}\,({\cal O}_{L}[X][f/\alpha^{er}]\otimes_{{\cal O}_{L}}E)_{\rm red} is identified with Spec​E​[S]{\rm Spec}\,E[S]. By this identification and (1.5), we define an isomorphism

QE[r]→HomE​(𝔪Le​r/𝔪Le​r+1,NT/Q⊗𝒪LE)∨→\displaystyle Q^{[r]}_{E}\to{\rm Hom}_{E}({\mathfrak{m}}_{L}^{er}/{\mathfrak{m}}_{L}^{er+1},N_{T/Q}\otimes_{{\cal O}_{L}}E)^{\vee}\to HomE​(𝔪Le​r/𝔪Le​r+1,𝔪K/𝔪K2)∨\displaystyle\,{\rm Hom}_{E}({\mathfrak{m}}_{L}^{er}/{\mathfrak{m}}_{L}^{er+1},{\mathfrak{m}}_{K}/{\mathfrak{m}}_{K}^{2})^{\vee}
(1.6) =𝔪Le​(r−1)/𝔪Le​(r−1)+1\displaystyle={\mathfrak{m}}_{L}^{e(r-1)}/{\mathfrak{m}}_{L}^{e(r-1)+1}

of smooth group schemes of dimension 1 over EE.

Let QT(r)→QT[r]Q^{(r)}_{T}\to Q^{[r]}_{T} be the normalization and define a section T→QT(r)T\to Q^{(r)}_{T} to be the unique lifting of the section T→QTT\to Q_{T} defined by sending XX to α\alpha. Let QE(r)Q^{(r)}_{E} denote the reduced part of the closed fiber QT(r)×TSpec​EQ^{(r)}_{T}\times_{T}{\rm Spec}\,E and let QE(r)⁣∘⊂QE(r)Q^{(r)\circ}_{E}\subset Q^{(r)}_{E} denote the connected component containing the image of the closed point of TT by the section T→QT(r)T\to Q^{(r)}_{T}.

Proposition 1.2.

Let KK be a henselian discrete valuation field with residue field FF of characteristic p>0p>0. Let LL be a totally ramified Galois extension of degree n=en=e with residue field E=FE=F and let G=Gal​(L/K)G={\rm Gal}(L/K) be the Galois group. Let α∈L\alpha\in L be a uniformizer and let f∈𝒪K​[X]f\in{\cal O}_{K}[X] be the minimal polynomial. Decompose f=∏i=1n(X−αi)f=\prod_{i=1}^{n}(X-\alpha_{i}) so that αn=α\alpha_{n}=\alpha and ordL​(αi−αn){\rm ord}_{L}(\alpha_{i}-\alpha_{n}) is increasing in ii.

1. Let r>1r>1 be the largest rational number such that Gr≠1G^{r}\neq 1. Then, we have

(1.7) e​r=ordL​f′​(α)+ordL​(αn−1−αn).er={\rm ord}_{L}f^{\prime}(\alpha)+{\rm ord}_{L}(\alpha_{n-1}-\alpha_{n}).

Define an injection β:Gr→𝐆a\beta\colon G^{r}\to{\mathbf{G}}_{a} by β​(σ)≡σ​(α)−ααn−1−αnmod𝔪L\beta(\sigma)\equiv\dfrac{\sigma(\alpha)-\alpha}{\alpha_{n-1}-\alpha_{n}}\bmod{\mathfrak{m}}_{L} and an additive polynomial b1∈E​[X]b_{1}\in E[X] by b1=∏σ∈Gr(X−β​(σ))/∏σ∈Gr,σ≠1(−β​(σ))b_{1}=\prod_{\sigma\in G^{r}}(X-\beta(\sigma))/\prod_{\sigma\in G^{r},\sigma\neq 1}(-\beta(\sigma)). Define an isomorphism 𝐆a→𝔪Le​(r−1)/𝔪Le​(r−1)+1{\mathbf{G}}_{a}\to{\mathfrak{m}}_{L}^{e(r-1)}/{\mathfrak{m}}_{L}^{e(r-1)+1} by f′​(α)​(αn−1−αn)/f​(0)∈𝔪Le​(r−1)f^{\prime}(\alpha)(\alpha_{n-1}-\alpha_{n})/f(0)\in{\mathfrak{m}}_{L}^{e(r-1)} and identify QE[r]Q^{[r]}_{E} with 𝔪Le​(r−1)/𝔪Le​(r−1)+1{\mathfrak{m}}_{L}^{e(r-1)}/{\mathfrak{m}}_{L}^{e(r-1)+1} by the isomorphism (1.6). Then, there exists an isomorphism

(1.8) 0→Gr→QE(r)⁣∘→QE[r]→0∥↑↑f′​(α)​(αn−1−αn)/f​(0)0→Gr→β𝐆a→b1𝐆a→0\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>G^{r}@>{}>{}>Q^{(r)\circ}_{E}@>{}>{}>Q^{[r]}_{E}@>{}>{}>0\\ \Big{\|}@A{}A{}A@A{}A{f^{\prime}(\alpha)(\alpha_{n-1}-\alpha_{n})/f(0)}A\\ 0@>{}>{}>G^{r}@>{\beta}>{}>{\mathbf{G}}_{a}@>{b_{1}}>{}>{\mathbf{G}}_{a}@>{}>{}>0\end{CD}

of exact sequences.

2. Let i>0i>0 be the largest integer such that Gi,cl=Ker​(G→Aut​(𝒪L/𝔪Li+1))≠1G_{i,{\rm cl}}={\rm Ker}(G\to{\rm Aut}({\cal O}_{L}/{\mathfrak{m}}_{L}^{i+1}))\neq 1. Then, we have

(1.9) i=ordL​(αn−1−αn)−1.i={\rm ord}_{L}(\alpha_{n-1}-\alpha_{n})-1.

Let K⊂M⊂LK\subset M\subset L be the intermediate extension corresponding to Gi,cl⊂GG_{i,{\rm cl}}\subset G and let ULi=1+𝔪Li⊂L×U^{i}_{L}=1+{\mathfrak{m}}^{i}_{L}\subset L^{\times} and UMi=1+𝔪Mi⊂M×U^{i}_{M}=1+{\mathfrak{m}}^{i}_{M}\subset M^{\times} be the multiplicative subgroups. Let Ni:ULi/ULi+1→UMi/UMi+1N^{i}\colon U^{i}_{L}/U^{i+1}_{L}\to U^{i}_{M}/U^{i+1}_{M} denote the morphism induced by the norm NL/M:L×→M×N_{L/M}\colon L^{\times}\to M^{\times} and Ti:ULi/ULi+1=𝔪Li/𝔪Li+1→UMi/UMi+1=𝔪Mi/𝔪Mi+1T^{i}\colon U^{i}_{L}/U^{i+1}_{L}={\mathfrak{m}}^{i}_{L}/{\mathfrak{m}}^{i+1}_{L}\to U^{i}_{M}/U^{i+1}_{M}={\mathfrak{m}}^{i}_{M}/{\mathfrak{m}}^{i+1}_{M} be the isomorphism induced by the trace TrL/M:L→M{\rm Tr}_{L/M}\colon L\to M. Define an isomorphism 𝐆a→ULi/ULi+1{\mathbf{G}}_{a}\to U^{i}_{L}/U^{i+1}_{L} by sending 11 to the class of αn−1/αn∈ULi\alpha_{n-1}/\alpha_{n}\in U^{i}_{L}. Then, the diagram

(1.10) 0→Gi,cl→σ↦σ​(α)/αULi/ULi+1→(Ti)−1∘NiULi/ULi+1→0∥↑αn−1/αn↑αn−1/αn0→Gi,cl→β𝐆a→b1𝐆a→0\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>G_{i,{\rm cl}}@>{\sigma\mapsto\sigma(\alpha)/\alpha}>{}>U^{i}_{L}/U^{i+1}_{L}@>{(T^{i})^{-1}\circ N^{i}}>{}>U^{i}_{L}/U^{i+1}_{L}@>{}>{}>0\\ \Big{\|}@A{}A{\alpha_{n-1}/\alpha_{n}}A@A{}A{\alpha_{n-1}/\alpha_{n}}A\\ 0@>{}>{}>G_{i,{\rm cl}}@>{\beta}>{}>{\mathbf{G}}_{a}@>{b_{1}}>{}>{\mathbf{G}}_{a}@>{}>{}>0\end{CD}

is an isomorphism of exact sequences.

Proof.

1. We have (1.7) by [7, Lemma 3.3.1.5].

We have a commutative diagram (1.8) with b1b_{1} and f′​(α)​(αn−1−αn)f^{\prime}(\alpha)(\alpha_{n-1}-\alpha_{n}) replaced by b=∏σ∈Gr(X−β​(σ))b=\prod_{\sigma\in G^{r}}(X-\beta(\sigma)) and c=∏i=1m(αn−αi)⋅(αn−1−αn)n−mc=\prod_{i=1}^{m}(\alpha_{n}-\alpha_{i})\cdot(\alpha_{n-1}-\alpha_{n})^{n-m} for m=#​G−#​Grm=\#G-\#G^{r} by [7, Lemma 3.3.1.1], since the canonical isomorphism NT/Q→NE/S⊗EN_{T/Q}\to N_{E/S}\otimes E maps ff to f​(0)f(0). Since b=∏σ∈Gr,σ≠1(−β​(σ))⋅b1b=\prod_{\sigma\in G^{r},\sigma\neq 1}(-\beta(\sigma))\cdot b_{1} and c=∏σ∈Gr,σ≠1(−β​(σ))⋅f′​(α)​(αn−1−αn)c=\prod_{\sigma\in G^{r},\sigma\neq 1}(-\beta(\sigma))\cdot f^{\prime}(\alpha)(\alpha_{n-1}-\alpha_{n}), we obtain (1.8).

2. Since 𝒪L=𝒪K​[α]{\cal O}_{L}={\cal O}_{K}[\alpha] and ordL​(αi−αn){\rm ord}_{L}(\alpha_{i}-\alpha_{n}) is increasing, the equality (1.9) follows from the definition of Gi,clG_{i,{\rm cl}}.

By [8, Chapitre V, Proposition 8, Section 6], the morphism (Ti)−1∘Ni:ULi/ULi+1→ULi/ULi+1(T^{i})^{-1}\circ N^{i}\colon U^{i}_{L}/U^{i+1}_{L}\to U^{i}_{L}/U^{i+1}_{L} is defined by a separable additive polynomial such that the coefficient of degree 1 is 1 and the upper line of (1.10) is exact. Since σ​(α)/α=1+(σ​(α)−αn)/(αn−1−αn)⋅(αn−1/αn−1)\sigma(\alpha)/\alpha=1+(\sigma(\alpha)-\alpha_{n})/(\alpha_{n-1}-\alpha_{n})\cdot(\alpha_{n-1}/\alpha_{n}-1), the left square is commutative. Since the left square is commutative, the right square is also commutative by the uniqueness of b1b_{1}. ∎

Corollary 1.3.

1. We have

(1.11) e​r=ordL​f′​(α)+(i+1).er={\rm ord}_{L}f^{\prime}(\alpha)+(i+1).

2. There exists an isomorphism

(1.12) 0→Gr→QE(r)⁣∘→𝔪Le​(r−1)/𝔪Le​(r−1)+1→0∥↑↑f′​(α)⋅αn/f​(0)0→Gi,cl→ULi/ULi+1→(Ti)−1∘NiULi/ULi+1→0\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>G^{r}@>{}>{}>Q^{(r)\circ}_{E}@>{}>{}>{\mathfrak{m}}_{L}^{e(r-1)}/{\mathfrak{m}}_{L}^{e(r-1)+1}@>{}>{}>0\\ \Big{\|}@A{}A{}A@A{}A{f^{\prime}(\alpha)\cdot\alpha_{n}/f(0)}A\\ 0@>{}>{}>G_{i,{\rm cl}}@>{}>{}>U^{i}_{L}/U^{i+1}_{L}@>{(T^{i})^{-1}\circ N^{i}}>{}>U^{i}_{L}/U^{i+1}_{L}@>{}>{}>0\end{CD}

of exact sequences.

Proof.

1. The equality (1.11) follows from (1.7) and (1.9).

2. Combining (1.8) and (1.10), we obtain (1.12). ∎

By Lemma 1.1.2, the extension in the upper line of (1.12) defines a canonical injection

(1.13) Gr∨=Hom𝐅p​(Gr,𝐅p)→HomE​(𝔪Le​(r−1)/𝔪Le​(r−1)+1,E).G^{r\vee}={\rm Hom}_{{\mathbf{F}}_{p}}(G^{r},{\mathbf{F}}_{p})\to{\rm Hom}_{E}({\mathfrak{m}}_{L}^{e(r-1)}/{\mathfrak{m}}_{L}^{e(r-1)+1},E).

Assume that the residue field of FF is perfect and let LL be a Galois extension of KK. Let Kur⊂LK^{\rm ur}\subset L denote the maximum unramified extension corresponding to the inertia subgroup I⊂GI\subset G. For a rational number r>1r>1, we apply the construction of (1.13) to the totally ramified extension M⊂LM\subset L of KurK^{\rm ur} corresponding to Gr+=⋃s>rGs⊂I⊂GG^{r+}=\bigcup_{s>r}G^{s}\subset I\subset G and to Hr=Grr​G=Gr/Gr+⊂H=Gal​(M/Kur)=I/Gr+H^{r}={\rm Gr}^{r}G=G^{r}/G^{r+}\subset H={\rm Gal}(M/K^{\rm ur})=I/G^{r+}. Let e′=eM/Ke^{\prime}=e_{M/K} be the ramification index and E′⊂EE^{\prime}\subset E be the residue field of MM. We obtain an injection

(1.14) (Grr​G)∨=Hom𝐅p​(Grr​G,𝐅p)→\displaystyle({\rm Gr}^{r}G)^{\vee}={\rm Hom}_{{\mathbf{F}}_{p}}({\rm Gr}^{r}G,{\mathbf{F}}_{p})\to HomE′​(𝔪Me′​(r−1)/𝔪Me′​(r−1)+1,E′)\displaystyle\,{\rm Hom}_{E^{\prime}}({\mathfrak{m}}_{M}^{e^{\prime}(r-1)}/{\mathfrak{m}}_{M}^{e^{\prime}(r-1)+1},E^{\prime})
⊂HomE​(𝔪Le​(r−1)/𝔪Le​(r−1)+1,E).\displaystyle\subset{\rm Hom}_{E}({\mathfrak{m}}_{L}^{e(r-1)}/{\mathfrak{m}}_{L}^{e(r-1)+1},E).

For abelian extensions, we have the Hasse-Arf theorem.

Theorem 1.4 ([8, Chapitre V, Section 7, Théorème 1]).

Let KK be a henselian discrete valuation field with perfect residue field and let LL be a finite abelian extension of KK. Let n≧1n\geqq 1 be an integer and rr be a rational number satisfying n<r≦n+1n<r\leqq n+1. Then, we have Gr=Gn+1G^{r}=G^{n+1}.

2 Tangent spaces and a characterization of ramification groups

Definition 2.1 ([7, Definition 1.1.8]).

Let KK be a discrete valuation field, S=Spec​𝒪KS={\rm Spec}\,{\cal O}_{K} and FF be the residue field.

1. For an extension EE of FF, let LE/SL_{E/S} denote the cotangent complex and we call the spectrum

(2.1) ΘK,E=Spec​S​(H1​(LE/S))\Theta_{K,E}={\rm Spec}\,S(H_{1}(L_{E/S}))

of the symmetric algebra over EE the tangent space of SS at EE.

2. If 𝒪K→𝒪K′{\cal O}_{K}\to{\cal O}_{K^{\prime}} is a faithfully flat morphism of discrete valuation rings, we say that K′K^{\prime} is an extension of discrete valuation fields of KK. We say that an extension K′K^{\prime} of discrete valuation fields of KK is tangentially dominant if, for a morphism F¯→F¯′\bar{F}\to\bar{F}^{\prime} of algebraic closures of the residue fields, the morphism

S​(H1​(LF¯/S))→S​(H1​(LF¯′/S′))S(H_{1}(L_{\bar{F}/S}))\to S(H_{1}(L_{\bar{F}^{\prime}/S^{\prime}}))

is an injection.

The morphism

(2.2) 𝔪K/𝔪K2⊗FF¯=H1​(LF/S)⊗FF¯→H1​(LF¯/S){\mathfrak{m}}_{K}/{\mathfrak{m}}_{K}^{2}\otimes_{F}\bar{F}=H_{1}(L_{F/S})\otimes_{F}\bar{F}\to H_{1}(L_{\bar{F}/S})

defined by the functoriality of cotangent complexes is an injection by [7, Proposition 1.1.3.1]. The injection (2.2) is an isomorphism if FF is perfect. The distinguished triangle LS/𝐙⊗𝒪SLF¯→LF¯/𝐙→LF¯/S→L_{S/{\mathbf{Z}}}\otimes^{L}_{{\cal O}_{S}}\bar{F}\to L_{\bar{F}/{\mathbf{Z}}}\to L_{\bar{F}/S}\to defines a canonical surjection

(2.3) H1​(LF¯/S)→Ω𝒪K1⊗𝒪KF¯H_{1}(L_{\bar{F}/S})\to\Omega^{1}_{{\cal O}_{K}}\otimes_{{\cal O}_{K}}\bar{F}

[7, Proposition 1.1.7.3] such that the composition with (2.2) is induced by d:𝔪K/𝔪K2→Ω𝒪K1⊗𝒪KFd\colon{\mathfrak{m}}_{K}/{\mathfrak{m}}_{K}^{2}\to\Omega^{1}_{{\cal O}_{K}}\otimes_{{\cal O}_{K}}F. If KK is of characteristic p>0p>0, (2.3) is an isomorphism by [7, Proposition 1.1.7.3]. If K′K^{\prime} is a tangentially dominant extension of KK, the morphism H1​(LF¯/S)→H1​(LF¯′/S′)H_{1}(L_{\bar{F}/S})\to H_{1}(L_{\bar{F}^{\prime}/S^{\prime}}) is an injection.

Proposition 2.2 ([7, Proposition 1.1.10]).

Let K→K′K\to K^{\prime} be an extension of discrete valuation fields. We consider the following conditions:

(1) The ramification index eK′/Ke_{K^{\prime}/K} is 11 and F′=𝒪K′/𝔪K′F^{\prime}={\cal O}_{K^{\prime}}/{\mathfrak{m}}_{K^{\prime}} is a separable extension of F=𝒪K/𝔪KF={\cal O}_{K}/{\mathfrak{m}}_{K}.

(2) The extension K′K^{\prime} is tangentially dominant over KK.

(3) The ramification index eK′/Ke_{K^{\prime}/K} is 11.

Then, we have the implications (1)⇒\Rightarrow(2)⇒\Rightarrow(3).

Theorem 2.3.

Let r>1r>1 be a rational number. For finite Galois extensions LL of henselian discrete valuation fields KK, there exists a unique way to define a normal subgroup GrG^{r} of the Galois group G=Gal​(L/K)G={\rm Gal}(L/K) satisfying the following conditions:

(1) If the residue field of KK is perfect, then Gr=Gclr−1G^{r}=G^{r-1}_{\rm cl}.

(2) Let K′K^{\prime} be a tangentially dominant extension of KK. Then the natural injection G′=Gal​(L′/K′)→GG^{\prime}={\rm Gal}(L^{\prime}/K^{\prime})\to G for L′=L​K′L^{\prime}=LK^{\prime} induces an isomorphism G′⁣r→GrG^{\prime r}\to G^{r}.

For a separable closure K¯\bar{K} of KK, extend the normalized discrete valuation ordK{\rm ord}_{K} to K¯\bar{K}. For a rational number rr, set 𝔪K¯r={x∈K¯∣ordK​x≧r}⊃𝔪K¯r+={x∈K¯∣ordK​x>r}{\mathfrak{m}}_{\bar{K}}^{r}=\{x\in\bar{K}\mid{\rm ord}_{K}x\geqq r\}\supset{\mathfrak{m}}_{\bar{K}}^{r+}=\{x\in\bar{K}\mid{\rm ord}_{K}x>r\}. The quotient 𝔪K¯r/𝔪K¯r+{\mathfrak{m}}_{\bar{K}}^{r}/{\mathfrak{m}}_{\bar{K}}^{r+} is a vector space of dimension 1 over the residue field F¯\bar{F}. For r>1r>1, define Gr+=⋃s>rGsG^{r+}=\bigcup_{s>r}G^{s} and Grr​G=Gr/Gr+{\rm Gr}^{r}G=G^{r}/G^{r+}.

Theorem 2.4.

Let r>1r>1 be a rational number. For finite Galois extensions LL of henselian discrete valuation fields KK, for morphisms L→K¯L\to\bar{K} to separable closures over KK and for the residue field F¯\bar{F} of K¯\bar{K}, there exists a unique way to define an injection

(2.4) Hom​(Grr​G,𝐅p)→HomF¯​(𝔪K¯r/𝔪K¯r+,H1​(LF¯/S)).{\rm Hom}({\rm Gr}^{r}G,{\mathbf{F}}_{p})\to{\rm Hom}_{\bar{F}}({\mathfrak{m}}_{\bar{K}}^{r}/{\mathfrak{m}}_{\bar{K}}^{r+},H_{1}(L_{\bar{F}/S})).

satisfying the following conditions:

(1) Assume that the residue field of KK is perfect. Let EE be the residue field of LL, e=eL/Ke=e_{L/K} be the ramification index and identify HomE​(𝔪Le​(r−1)/𝔪Le​(r−1)+1,E){\rm Hom}_{E}({\mathfrak{m}}_{L}^{e(r-1)}/{\mathfrak{m}}_{L}^{e(r-1)+1},E) with a subgroup of HomF¯​(𝔪K¯r/𝔪K¯r+,H1​(LF¯/S)){\rm Hom}_{\bar{F}}({\mathfrak{m}}_{\bar{K}}^{r}/{\mathfrak{m}}_{\bar{K}}^{r+},H_{1}(L_{\bar{F}/S})) by the injection 𝔪K/𝔪K2→H1​(LF¯/S){\mathfrak{m}}_{K}/{\mathfrak{m}}_{K}^{2}\to H_{1}(L_{\bar{F}/S}) (2.2). Then, the diagram

(2.5) Hom​(Grr​G,𝐅p)→HomF¯​(𝔪K¯r/𝔪K¯r+,H1​(LF¯/S))∥↑Hom​(Grr​G,𝐅p)→(1.13)HomE​(𝔪Le​(r−1)/𝔪Le​(r−1)+1,E)\begin{CD}{\rm Hom}({\rm Gr}^{r}G,{\mathbf{F}}_{p})@>{}>{}>{\rm Hom}_{\bar{F}}({\mathfrak{m}}_{\bar{K}}^{r}/{\mathfrak{m}}_{\bar{K}}^{r+},H_{1}(L_{\bar{F}/S}))\\ \Big{\|}@A{}A{}A\\ {\rm Hom}({\rm Gr}^{r}G,{\mathbf{F}}_{p})@>{\rm(\ref{eqchcl})}>{}>{\rm Hom}_{E}({\mathfrak{m}}_{L}^{e(r-1)}/{\mathfrak{m}}_{L}^{e(r-1)+1},E)\end{CD}

is commutative.

(2) Let K′K^{\prime} be a tangentially dominant extension of KK, let K¯→K¯′\bar{K}\to\bar{K}^{\prime} be a morphism of separable closures extending L→L′=L​K′L\to L^{\prime}=LK^{\prime} and let F¯→F¯′\bar{F}\to\bar{F}^{\prime} be the morphism of residue fields. Then, for the natural injection G′=Gal​(L′/K′)→GG^{\prime}={\rm Gal}(L^{\prime}/K^{\prime})\to G, the diagram

(2.6) Hom​(Grr​G,𝐅p)→HomF¯​(𝔪K¯r/𝔪K¯r+,H1​(LF¯/S))↓↓Hom​(Grr​G′,𝐅p)→HomF¯′​(𝔪K¯′r/𝔪K¯′r+,H1​(LF¯′/S′))\begin{CD}{\rm Hom}({\rm Gr}^{r}G,{\mathbf{F}}_{p})@>{}>{}>{\rm Hom}_{\bar{F}}({\mathfrak{m}}_{\bar{K}}^{r}/{\mathfrak{m}}_{\bar{K}}^{r+},H_{1}(L_{\bar{F}/S}))\\ @V{}V{}V@V{}V{}V\\ {\rm Hom}({\rm Gr}^{r}G^{\prime},{\mathbf{F}}_{p})@>{}>{}>{\rm Hom}_{\bar{F}^{\prime}}({\mathfrak{m}}_{\bar{K}^{\prime}}^{r}/{\mathfrak{m}}_{\bar{K}^{\prime}}^{r+},H_{1}(L_{\bar{F}^{\prime}/S^{\prime}}))\end{CD}

is commutative.

The uniqueness is a consequence of the following existence of a tangentially dominant extension with perfect residue field.

Proposition 2.5 ([7, Proposition 1.1.12]).

Let KK be a discrete valuation field. Then, there exists a tangentially dominant extension K′K^{\prime} of KK such that the residue field F′F^{\prime} is perfect.

Proof of Theorem 2.3.

We show the uniqueness. By Proposition 2.5, there exists a tangentially dominant extension K′K^{\prime} of KK with perfect residue field. Let G′=Gal​(L′/K′)→GG^{\prime}={\rm Gal}(L^{\prime}/K^{\prime})\to G be the natural injection for L′=L​K′L^{\prime}=LK^{\prime}. Then, by the conditions (1) and (2), the subgroup Gr⊂GG^{r}\subset G is the image of Gcl′⁣r−1⊂G′G^{\prime r-1}_{\rm cl}\subset G^{\prime}.

To show the existence, it suffices to prove that the subgroup Gr⊂GG^{r}\subset G defined in [1] satisfies the conditions (1) and (2). The equality Gr=Gclr−1G^{r}=G^{r-1}_{\rm cl} is proved in [1, Proposition 3.7 (3)]. The condition (2) is satisfied by [7, Proposition 4.2.4 (1)]. ∎

Proof of Theorem 2.4.

We show the uniqueness. If the residue field is perfect, the morphism (2.4) is uniquely determined by the commutative diagram (2.5) since its right vertical arrow is an injection induced by the injection (2.2). In general, by Proposition 2.5, there exists a tangentially dominant extension K′K^{\prime} of KK with perfect residue field. Then, the morphism (2.4) is uniquely determined by the commutative diagram (2.6) since its right vertical arrow is an injection.

To show the existence, it suffices to prove that the morphism [7, (4.20)] satisfies the conditions (1) and (2). Assume that the residue field is perfect. To show the commutative diagram (2.5), we may assume that Gr+=1G^{r+}=1 and Grr​G=Gr{\rm Gr}^{r}G=G^{r} by the construction of the morphisms. Then, since the construction of (1.13) is a special case of [7, (4.20)], the condition (1) is satisfied. The condition (2) follows from [7, (4.19)]. ∎

3 Abelian extensions

Theorem 3.1.

Let r>1r>1 be a rational number.

1. For finite abelian extensions LL of henselian discrete valuation fields KK, there exists a unique way to define a normal subgroup GrG^{r} of the Galois group G=Gal​(L/K)G={\rm Gal}(L/K) satisfying the following conditions:

(1) If the residue field of KK is perfect, then Gr=Gclr−1G^{r}=G^{r-1}_{\rm cl}.

(2) Let K′K^{\prime} be a tangentially dominant extension of KK. Then the natural injection G′=Gal​(L′/K′)→GG^{\prime}={\rm Gal}(L^{\prime}/K^{\prime})\to G for L′=L​K′L^{\prime}=LK^{\prime} induces an isomorphism G′⁣r→GrG^{\prime r}\to G^{r}.

2. Let LL be a finite abelian extension of a henselian discrete valuation field KK and let n≧1n\geqq 1 be the integer satisfying n<r≦n+1n<r\leqq n+1. Then, we have Gr=Gn+1G^{r}=G^{n+1}.

Theorem 3.2.

Let n>1n>1 be an integer. For finite abelian extensions LL of henselian discrete valuation fields KK, for morphisms L→K¯L\to\bar{K} to separable closures over KK and for the residue fields F¯\bar{F} of K¯\bar{K}, there exists a unique way to define an injection

(3.1) Hom​(Gn/Gn+1,𝐅p)→HomF​(𝔪Kn/𝔪Kn+1,H1​(LF¯/S)).{\rm Hom}(G^{n}/G^{n+1},{\mathbf{F}}_{p})\to{\rm Hom}_{F}({\mathfrak{m}}_{K}^{n}/{\mathfrak{m}}_{K}^{n+1},H_{1}(L_{\bar{F}/S})).

satisfying the following conditions:

(1) Assume that the residue field of KK is perfect and let the notation be as in Theorem 2.4 (1). Then the diagram

(3.2) Hom​(Gn/Gn+1,𝐅p)→HomF​(𝔪Kn/𝔪Kn+1,H1​(LF¯/S))∥↑Hom​(Gcln−1/Gcln,𝐅p)→HomF​(𝔪Kn−1/𝔪Kn,E)\begin{CD}{\rm Hom}(G^{n}/G^{n+1},{\mathbf{F}}_{p})@>{}>{}>{\rm Hom}_{F}({\mathfrak{m}}_{K}^{n}/{\mathfrak{m}}_{K}^{n+1},H_{1}(L_{\bar{F}/S}))\\ \Big{\|}@A{}A{}A\\ {\rm Hom}(G^{n-1}_{\rm cl}/G^{n}_{\rm cl},{\mathbf{F}}_{p})@>{}>{}>{\rm Hom}_{F}({\mathfrak{m}}_{K}^{n-1}/{\mathfrak{m}}_{K}^{n},E)\end{CD}

is commutative.

(2) Let K′K^{\prime} be a tangentially dominant extension of KK and let the notation be as in Theorem 2.4 (2). Then, the diagram

(3.3) Hom​(Gn/Gn+1,𝐅p)→HomF​(𝔪Kn/𝔪Kn+1,H1​(LF¯/S))↓↓Hom​(G′⁣n/G′⁣n+1,𝐅p)→HomF′​(𝔪K′n/𝔪K′n+1,H1​(LF¯′/S′))\begin{CD}{\rm Hom}(G^{n}/G^{n+1},{\mathbf{F}}_{p})@>{}>{}>{\rm Hom}_{F}({\mathfrak{m}}_{K}^{n}/{\mathfrak{m}}_{K}^{n+1},H_{1}(L_{\bar{F}/S}))\\ @V{}V{}V@V{}V{}V\\ {\rm Hom}(G^{\prime n}/G^{\prime n+1},{\mathbf{F}}_{p})@>{}>{}>{\rm Hom}_{F^{\prime}}({\mathfrak{m}}_{K^{\prime}}^{n}/{\mathfrak{m}}_{K^{\prime}}^{n+1},H_{1}(L_{\bar{F}^{\prime}/S^{\prime}}))\end{CD}

is commutative.

Proof of Theorem 3.1.

1. is proved in the same way as Theorem 2.3.

2. By 1, this follows from the Hasse-Arf theorem Theorem 1.4. ∎

Proof of Theorem 3.2.

This is a special case of Theorem 2.4. ∎

Assume that KK is a henselian discrete valuation field of equal characteristic p>0p>0 and let LL be a finite abelian extension. Then, by the Hasse-Arf theorem Theorem 3.1.2 and by the isomorphism H1​(LF¯/S)→Ω𝒪K1⊗𝒪KF¯H_{1}(L_{\bar{F}/S})\to\Omega^{1}_{{\cal O}_{K}}\otimes_{{\cal O}_{K}}\bar{F} (2.3), for an integer n>0n>0, the injection (3.1) defines an injection

(3.4) Hom​(Gn/Gn+1,𝐅p)→HomF​(𝔪Kn/𝔪Kn+1,Ω𝒪K1⊗𝒪KF¯).{\rm Hom}(G^{n}/G^{n+1},{\mathbf{F}}_{p})\to{\rm Hom}_{F}({\mathfrak{m}}_{K}^{n}/{\mathfrak{m}}_{K}^{n+1},\Omega^{1}_{{\cal O}_{K}}\otimes_{{\cal O}_{K}}\bar{F}).

A decreasing filtration (GMan)(G^{n}_{\rm Ma}) indexed by integers n>0n>0 is defined in [5, Definition 3.1.1] as a non-logarithmic modification of a filtration (GKan)(G^{n}_{\rm Ka}) defined in [4, Definition (2.1)]. Further, for an integer n>0n>0, a canonical morphism

(3.5) rsw′:Hom​(GMan/GMan+1,𝐅p)→HomF​(𝔪Kn/𝔪Kn+1,Ω𝒪K1⊗𝒪KF¯){\rm rsw}^{\prime}\colon{\rm Hom}(G^{n}_{\rm Ma}/G_{\rm Ma}^{n+1},{\mathbf{F}}_{p})\to{\rm Hom}_{F}({\mathfrak{m}}_{K}^{n}/{\mathfrak{m}}_{K}^{n+1},\Omega^{1}_{{\cal O}_{K}}\otimes_{{\cal O}_{K}}\bar{F})

is defined in [5, Definition 3.2.5] except the case p=2p=2, n=2n=2 and in [9, Definition 1.18] in the exceptional case p=2p=2, n=2n=2, as a modification of the refined Swan conductor defined in [4, Corollary (5.2)].

As an application, we give a new proof of the equalities of the two filtrations and the two morphisms, different from that in [2] and [9].

Corollary 3.3.

Let LL be an abelian extension of a henselian discrete valuation field KK of equal characteristic p>0p>0 and let n>1n>1 be an integer.

1.([2, Théorème 9.10 (i)] for n>1n>1, [9, Theorem 3.1] for nn general) We have an equality Gn=GManG^{n}=G^{n}_{\rm Ma} of subgroups of GG.

2.([2, Théorème 9.10 (ii)] for n>1n>1, [9, Corollary 2.13] for nn general) The injection (3.4) is the same as rsw′{\rm rsw}^{\prime} (3.5).

The following proof is by the reduction to the logarithmic variant [2, Théorème 9.11] in the classical case where the residue field is perfect.

Proof.

It suffices to show that the filtration (GMan)(G^{n}_{\rm Ma}) and the morphism rsw′{\rm rsw}^{\prime} satisfy the conditions in Theorems 3.1 and 3.2.

We show that the conditions (1) are satisfied. Assume that the residue field FF is perfect. Then, we have Gn=Gcln−1G^{n}=G^{n-1}_{\rm cl} and GMan=GKan−1G^{n}_{\rm Ma}=G^{n-1}_{\rm Ka}. Since GKan−1=Gcln−1G^{n-1}_{\rm Ka}=G^{n-1}_{\rm cl} in this case by [2, Théorème 9.11 (i)], the condition (1) in Theorem 3.1 is satisfied.

We identify Ω𝒪K1⊗𝒪KF\Omega^{1}_{{\cal O}_{K}}\otimes_{{\cal O}_{K}}F with 𝔪K/𝔪K2{\mathfrak{m}}_{K}/{\mathfrak{m}}_{K}^{2} by d:𝔪K/𝔪K2→Ω𝒪K1⊗𝒪KFd\colon{\mathfrak{m}}_{K}/{\mathfrak{m}}_{K}^{2}\to\Omega^{1}_{{\cal O}_{K}}\otimes_{{\cal O}_{K}}F and HomF(𝔪Kn/𝔪Kn+1,{\rm Hom}_{F}({\mathfrak{m}}_{K}^{n}/{\mathfrak{m}}_{K}^{n+1}, Ω𝒪K1⊗𝒪KF¯)\Omega^{1}_{{\cal O}_{K}}\otimes_{{\cal O}_{K}}\bar{F}) with HomF​(𝔪Kn−1/𝔪Kn,F¯){\rm Hom}_{F}({\mathfrak{m}}_{K}^{n-1}/{\mathfrak{m}}_{K}^{n},\bar{F}) by the induced isomorphism. Then, the morphism (3.5) is identified with the morphism

(3.6) rsw:Hom​(Gcln−1/Gcln,𝐅p)→HomF​(𝔪Kn−1/𝔪Kn,F){\rm rsw}\colon{\rm Hom}(G^{n-1}_{\rm cl}/G^{n}_{\rm cl},{\mathbf{F}}_{p})\to{\rm Hom}_{F}({\mathfrak{m}}_{K}^{n-1}/{\mathfrak{m}}_{K}^{n},F)

defined in [4, Corollary (5.2)]. Since the morphism (3.6) equals (3.4) by [2, Théorème 9.11 (ii)], the condition (1) in Theorem 3.2 is satisfied.

We show that the conditions (2) are satisfied. For an extension K′K^{\prime} of henselian discrete valuation field of ramification index 11, the diagram

(3.7) Hom​(GMan/GMan+1,𝐅p)→rsw′HomF​(𝔪Kn/𝔪Kn+1,Ω𝒪K1⊗𝒪KF¯)↓↓Hom​(GMa′⁣n/GMa′⁣n+1,𝐅p)→rsw′HomF′​(𝔪K′n/𝔪K′n+1,Ω𝒪K′1⊗𝒪K′F¯′)\begin{CD}{\rm Hom}(G^{n}_{\rm Ma}/G^{n+1}_{\rm Ma},{\mathbf{F}}_{p})@>{{\rm rsw}^{\prime}}>{}>{\rm Hom}_{F}({\mathfrak{m}}_{K}^{n}/{\mathfrak{m}}_{K}^{n+1},\Omega^{1}_{{\cal O}_{K}}\otimes_{{\cal O}_{K}}\bar{F})\\ @V{}V{}V@V{}V{}V\\ {\rm Hom}(G^{\prime n}_{\rm Ma}/G^{\prime n+1}_{\rm Ma},{\mathbf{F}}_{p})@>{{\rm rsw}^{\prime}}>{}>{\rm Hom}_{F^{\prime}}({\mathfrak{m}}_{K^{\prime}}^{n}/{\mathfrak{m}}_{K^{\prime}}^{n+1},\Omega^{1}_{{\cal O}_{K^{\prime}}}\otimes_{{\cal O}_{K^{\prime}}}\bar{F}^{\prime})\end{CD}

is commutative. Hence the condition (2) in Theorem 3.2 is satisfied.

If K′K^{\prime} is tangentially dominant over KK, then the morphism Ω𝒪K1⊗𝒪KF¯→Ω𝒪K′1⊗𝒪K′F¯′\Omega^{1}_{{\cal O}_{K}}\otimes_{{\cal O}_{K}}\bar{F}\to\Omega^{1}_{{\cal O}_{K^{\prime}}}\otimes_{{\cal O}_{K^{\prime}}}\bar{F}^{\prime} is an injection. Hence by the commutative diagram (3.7), the morphism G′⁣n/G′⁣n+1→Gn/Gn+1G^{\prime n}/G^{\prime n+1}\to G^{n}/G^{n+1} is a surjection. By the descending induction on nn, the condition (2) in Theorem 3.1 is satisfied. ∎

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