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

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.

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

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.

Paper Structure

This paper contains 3 sections, 11 theorems, 21 equations.

Table of Contents

  1. Totally ramified case
  2. Tangent spaces and a characterization of ramification groups
  3. Abelian extensions

Key Result

Lemma 1.1

Let $F$ be a field of characteristic $p>0$. 1. Let $G\subset F$ be a finite subgroup of the additive group. Then, the polynomial $\in F[X]$ is a unique additive separable polynomial such that the coefficient of degree $1$ is $1$ and that the sequence \begin{CD} 0@>>> G @>>> {\mathbf G}_a @>{a_1}>> {\mathbf G}_a @>>>0 \end{CD}is exact. 2. (red) Let $E$ be an $F$-vector space of finite dimension an

Theorems & Definitions (12)

  • Lemma 1.1
  • Proposition 1.2
  • Corollary 1.3
  • Theorem 1.4: CL
  • Definition 2.1: red
  • Proposition 2.2: red
  • Theorem 2.3
  • Theorem 2.4
  • Proposition 2.5: red
  • Theorem 3.1
  • ...and 2 more