Ramification theory for degree $p$ extensions of arbitrary valuation rings in mixed characteristic $(0,p)$

TL;DR

This paper extends ramification theory results for degree p extensions of valuation rings in mixed characteristic, including defect cases and higher rank valuations, generalizing previous work to more complex valuation settings.

Contribution

It provides new ramification theory results for degree p extensions of valuation rings in mixed characteristic, accommodating defect extensions and higher rank valuations, with a more general base field assumption.

Findings

Results valid for defect extensions with higher rank valuations

Generalization to equal characteristic cases

Applicable to henselian valuation rings in mixed characteristic

Abstract

We previously obtained a generalization and refinement of results about the ramification theory of Artin-Schreier extensions of discretely valued fields in characteristic $p$ with perfect residue fields to the case of fields with more general valuations and residue fields. As seen in VT16, the "defect" case gives rise to many interesting complications. In this paper, we present analogous results for degree $p$ extensions of arbitrary valuation rings in mixed characteristic $(0,p)$ in a more general setting. More specifically, the only assumption here is that the base field $K$ is henselian. In particular, these results are true for defect extensions even if the rank of the valuation is greater than $1$. A similar method also works in equal characteristic, generalizing the results of VT16.