Upper Ramification Groups for Arbitrary Valuation Rings

TL;DR

This paper extends ramification theory to arbitrary Henselian valuation rings by expressing their integral closures as unions of complete intersection subrings, thus generalizing previous theories and addressing defect extensions.

Contribution

It introduces a new ramification framework for Henselian valuation rings as limits of existing theories, including defect extensions not previously covered.

Findings

Integral closure as union of complete intersection subrings

Generalization of Abbes-Saito ramification theory

Analysis of defect extensions

Abstract

T. Saito established a ramification theory for ring extensions locally of complete intersection. We show that for a Henselian valuation ring $A$ with field of fractions $K$ and for a finite Galois extension $L$ of $K$, the integral closure $B$ of $A$ in $L$ is a filtered union of subrings of $B$ which are of complete intersection over $A$. By this, we can obtain a ramification theory of Henselian valuation rings as the limit of the ramification theory of Saito. Our theory generalizes the ramification theory of complete discrete valuation rings of Abbes-Saito. We study "defect extensions" which are not treated in these previous works.