F-coherent rings with applications to tight closure theory

TL;DR

This paper introduces the class of F-coherent rings in positive characteristic, explores their properties, and examines their implications for tight closure theory and related F-singularity classes.

Contribution

It defines F-coherent rings via perfect closures and investigates their properties and applications in tight closure and F-singularity theories.

Findings

F-coherent rings are characterized by their perfect closures being coherent.

Relationships between F-coherent, F-pure, F-regular, and F-injective rings are established.

The coherent property influences the behavior of tight closure in perfect rings.

Abstract

The aim of this paper is to introduce a new class of Noetherian rings of positive characteristic in terms of perfect closures and study their basic properties. If the perfect closure of a Noetherian ring is coherent, we call it an $F$-coherent ring. Some interesting applications are given in connection with tight closure theory. In particular, we discuss relationships between $F$-coherent rings and $F$-pure, $F$-regular, and $F$-injective rings. The final section discusses how the coherent property effects the behavior of tight closure for general perfect rings.