Artin-Schreier-Witt lifts of purely inseparable extensions

TL;DR

This paper investigates the existence and explicit construction of Artin-Schreier-Witt lifts for purely inseparable extensions over discrete valued fields of positive characteristic, revealing new possibilities for wild ramification and field extensions.

Contribution

It proves the existence of cyclic lifts for purely inseparable modular extensions and constructs Artin-Schreier-Witt lifts of any finite degree, extending ramification theory in positive characteristic.

Findings

Cyclic lifts of purely inseparable modular extensions always exist.

Explicit construction methods for Artin-Schreier-Witt lifts are provided.

There is no cap on wild ramification index in the studied setting.

Abstract

Given a discrete valued field $K$ of positive characteristic, we study the cyclic lifting problem of purely inseparable extensions of the residue field. We prove that unlike the mixed characteristic case, cyclic lifts of any finite purely inseparable modular extension exist and show how to explicitly construct them. Moreover, given such a residual extension, we prove the existence of Artin-Schreier-Witt lifts of any finite degree. This follows from a more general construction based on the notion of $\mathcal{G}$-weaves and $\mathcal{G}$-cyclic extensions where $\mathcal{G}$ is an arbitrary gene over $K$. In particular, this gives an affirmative answer to a question in \cite{ramification_survey} as well as implies that there is no cap on the wild ramification index unlike the mixed characteristic case. We also show some interesting applications such as constructing cyclic lifts of fields that are isomorphic to the tensor product of a purely inseparable modular extension and an Artin-Schreier-Witt extension. Finally, we prove a structure theorem of Artin-Schreier-Witt extensions over a discrete valued fields which restricts the ramification type of intermediate field extensions complementing the above results.