$F$-nilpotent rings and permanence properties

TL;DR

This paper investigates classes of singularities called $F$-nilpotent and related types, analyzing their behavior under ring maps and establishing conditions for the openness of certain prime loci in Noetherian rings of prime characteristic.

Contribution

It introduces and studies the properties of $F$-nilpotent and related singularity classes, proving the openness of loci of primes with these properties in specific ring settings.

Findings

Loci of $F$-nilpotent primes are open in the Zariski topology.

Loci of weakly $F$-nilpotent primes are open under certain conditions.

The paper characterizes the behavior of these classes under faithfully flat local ring maps.

Abstract

We explore the singularity classes $F$-nilpotent, weakly $F$-nilpotent, and generalized weakly $F$-nilpotent under faithfully flat local ring maps. As an application, we show that the loci of primes in a Noetherian ring of prime characteristic which define either weakly $F$-nilpotent or $F$-nilpotent local rings are open with respect to the Zariski topology whenever $R$ is $F$-finite or essentially of finite type over an excellent local ring.