Classification of plethories in characteristic zero

TL;DR

This paper classifies plethories over fields of characteristic zero, showing they are all linear and free on a bialgebra, and extends related results in ring scheme theory.

Contribution

It provides a complete classification of plethories in characteristic zero and extends known results in the theory of ring schemes.

Findings

All plethories over characteristic zero fields are linear.

Plethories with trivial Verschiebung are classified over perfect fields of non-zero characteristic.

The paper extends existing results in ring scheme theory.

Abstract

We classify plethories over fields of characteristic zero, thus answering a question of Borger-Wieland and Bergman-Hausknecht. All plethories over characteristic zero fields are linear, in the sense that they are free plethories on a bialgebra. For the proof we need some facts from the theory of ring schemes where we extend previously known results. We also classify plethories with trivial Verschiebung over a perfect field of non-zero characteristic and indicate future work.