Ramification of local fields with imperfect residue fields III

Ramification of Local Fields with Imperfect Residue Fields II

In [1], a filtration by ramification groups and its logarithmic version are defined on the absolute Galois group of a complete discrete valuation field without assuming that the residue field is perfect. In this paper, we study the graded pieces of these filtrations and show that they are abelian except possibly in the absolutely unramified and non-logarithmic case. 2000 Mathematics Subject Cla...

Ramification of local fields with imperfect residue fields I

Let K be a complete discrete valuation field, and let G be the Galois group of a separable closure Ω. Classically the ramification filtration of G is defined in the case where the residue field of K is perfect ([5], Chapter IV). In this paper, we define without any assumption on the residue field, two ramification filtrations of G and study some of their properties. Our first filtration, (G)a∈Q...

Ramification of local fields with imperfect residue fields

We define two decreasing filtrations by ramification groups on the absolute Galois group of a complete discrete valuation field whose residue field may not be perfect. In the classical case where the residue field is perfect, we recover the classical upper numbering filtration. The definition uses rigid geometry and log-structures. We also establish some of their properties.

Ramification Theory for Local Fields with Imperfect Residue Fields

Notation 1.1. Let l/k be a finite Galois extension of complete discretely valued fields. Let Ok, Ol, πk, πl, k̄, and l̄ be rings of integers, uniformizers, and residue fields, respectively. For an element a ∈ Ol, we use ā to denote its reduction in l̄. Let G = Gl/k be the Galois group. Use vl(·) to denote the valuation on l so that vl(πl) = 1. We call e = vl(πk) the näıve ramification degree; it i...

Ramification of Higher Local Fields