One-dimensional rings of finite F-representation type

TL;DR

This paper investigates conditions under which one-dimensional rings in prime characteristic have finite F-representation type, providing new examples and counterexamples based on residue field properties.

Contribution

It establishes criteria for finite F-representation type in one-dimensional domains and presents new examples of rings with or without this property.

Findings

Complete local or graded one-dimensional domains with algebraically closed or finite residue fields have finite F-representation type.

Examples of one-dimensional domains without finite F-representation type are provided for perfect residue fields.

Some higher-dimensional rings also exhibit finite F-representation type.

Abstract

We prove that a complete local or graded one-dimensional domain of prime characteristic has finite F-representation type if its residue field is algebraically closed or finite, and present examples of a complete local or graded one-dimensional domain which does not have finite F-representation type with a perfect residue field. We also present some examples of higher dimensional rings of finite F-representation type.