# Coincidence of two Swan conductors of abelian characters

## TL;DR

This paper proves that two different definitions of the Swan conductor for abelian characters of the Galois group of a local field are equivalent, clarifying their relationship.

## Contribution

It establishes the equality of two previously distinct notions of Swan conductors for abelian characters, resolving a conceptual ambiguity.

## Key findings

- The two Swan conductors are shown to be equal.
- The result applies to abelian characters of local Galois groups.
- Clarifies the theoretical understanding of ramification invariants.

## Abstract

There are two ways to define the Swan conductor of an abelian character of the absolute Galois group of a complete discrete valuation field. We prove that these two Swan conductors coincide.