Witt vectors and separably closed fields with higher derivations

TL;DR

This paper proposes a revised axiomatization for the theory of separably closed fields with higher derivations, aligning more closely with original ideas and utilizing Witt group-based iterativity conditions.

Contribution

It introduces a new axiomatization that corrects previous formulations by Messmer and Wood, using Witt group-based higher derivations.

Findings

Provides a consistent axiomatization for separably closed fields with higher derivations.

Aligns the axioms more closely with original conceptual frameworks.

Uses Witt group-based iterativity conditions for derivations.

Abstract

The main scope of this short paper is to provide a modification of the axioms given by Messmer and Wood for the theory of separably closed fields of positive characteristic and finite imperfectness degree. The original axioms failed to meet natural expectations, and therefore a new axiomatization was given (i.e. Ziegler's one), but the new axioms do not follow the initial idea of Messmer and Wood. Therefore, we aim to give a correct axiomatization which is more similar to the original one and which, as the original axioms,involves only one Hasse-Schmidt derivation, this time based on the iterativity conditions corresponding to the Witt group.