A generalization of Nakai's theorem on locally finite iterative higher derivations

TL;DR

This paper extends Nakai's theorem on locally finite iterative higher derivations to non-algebraically closed fields, leading to a broader cancellation theorem for certain finitely generated domains.

Contribution

It generalizes Nakai's structure theorem to arbitrary fields and derives a new cancellation theorem for two-dimensional UFDs over such fields.

Findings

Generalization of Nakai's theorem to non-algebraically closed fields.

Establishment of a cancellation theorem for certain $k$-domains.

Application to domains with trivial extensions and UFD properties.

Abstract

Let $k$ be a field of arbitrary characteristic. Nakai (1978) proved a structure theorem for $k$-domains admitting a nontrivial locally finite iterative higher derivation when $k$ is algebraically closed. In this paper, we generalize Nakai's theorem to cover the case where $k$ is not algebraically closed. As a consequence, we obtain a cancellation theorem of the following form: Let $A$ and $A'$ be finitely generated $k$-domains with $A[x]\simeq _kA'[x]$. If $A$ and $\bar{k}\otimes _kA$ are UFDs and $\mathop{\rm trans.deg}\nolimits_kA=2$, then we have $A\simeq _kA'$. This generalizes the cancellation theorem of Crachiola (2009).