Operators coming from ring schemes

TL;DR

This paper introduces a new class of operators called $$-operators derived from coordinate $$-algebra schemes, generalizing existing operators like endomorphisms and derivations, and explores their model-theoretic properties.

Contribution

It defines coordinate $$-algebra schemes and $$-operators, classifies these schemes over perfect fields, and analyzes the model theory of fields equipped with such operators.

Findings

Classified coordinate $$-algebra schemes for perfect fields.

Extended the framework of $$-operators to include Frobenius-related operators.

Studied model-theoretic properties of fields with $$-operators.

Abstract

We introduce the notion of a coordinate $\mathbf{k}$-algebra scheme and the corresponding notion of a $\mathcal{B}$-operator. This class of operators includes endomorphisms and derivations of the Frobenius map, and it also generalizes the operators related to $\mathcal{D}$-rings from [15]. We classify the (coordinate) $\mathbf{k}$-algebra schemes for a perfect field $\mathbf{k}$ and we also discuss the model-theoretic properties of fields with $\mathcal{B}$-operators.