Kato's conductor and generic residual perfection

TL;DR

This paper proves that Kato's Swan conductor remains unchanged when pulled back to the generic residual perfection of a complete discrete valuation ring, extending the applicability of the conductor concept to imperfect residue fields.

Contribution

It establishes the invariance of the non-logarithmic Kato's Swan conductor under the generic residual perfection, broadening its theoretical framework.

Findings

Kato's Swan conductor is invariant under generic residual perfection.

The non-logarithmic variant of Kato's conductor extends to imperfect residue fields.

The result links conductors in different residue field contexts.

Abstract

Let A be a complete discrete valuation ring with possibly imperfect residue field, and let $\chi$ be a one-dimensional Galois representation over A. I show that the non-logarithmic variant of Kato's Swan conductor is the same for $\chi$ and the pullback of $\chi$ to the generic residual perfection of A. This implies the conductor from "Conductors and the moduli of residual perfection" (math.NT/0112305) extends the non-logarithmic variant of Kato's.