Kummer-Artin-Schreier-Witt Theory

TL;DR

This paper develops a new method to lift Artin--Schreier--Witt isogenies from characteristic p to zero, using Kummer classes and Witt vectors, advancing the understanding of Galois covers in algebraic geometry.

Contribution

It introduces a novel technique linking Kummer classes to Witt vectors, enabling explicit lifting of isogenies across characteristics.

Findings

Constructs explicit lifts of isogenies over concrete base rings.

Establishes a framework connecting Kummer classes with Witt vectors.

Sets the stage for applications to inseparable extensions and Swan conductors.

Abstract

We study the problem of lifting the Artin--Schreier--Witt isogeny from characteristic $p>0$ to characteristic $0$, which is central to the lifting problem for Galois covers of algebraic schemes in positive characteristic. We introduce a new technique that associates a Kummer class, representing a tamely ramified cyclic extension, to a Witt vector via Matsuda's Kummer--Artin--Schreier--Witt theory. This viewpoint leads to an explicit construction of a lift of the isogeny over a concrete base ring. Our results lay the groundwork for further applications, including the study of inseparable extensions and Kato's refined Swan conductor.