A locally F-finite Noetherian domain that is not F-finite

TL;DR

This paper constructs a specific example of a Noetherian ring in prime characteristic where the Frobenius morphism is locally finite but not finite, illustrating a nuanced distinction in algebraic properties.

Contribution

It provides the first known example of a Noetherian ring with a locally finite Frobenius morphism that is not globally finite, clarifying the relationship between these properties.

Findings

Frobenius morphism can be locally finite without being finite

Constructs a concrete example based on Nagata's work

Highlights a subtlety in the structure of Noetherian rings in characteristic p

Abstract

Using an old example of Nagata, we construct a Noetherian ring of prime characteristic p, whose Frobenius morphism is locally finite, but not finite.