Graded annihilators and uniformly $F$-compatible ideals

TL;DR

This paper investigates the relationships between certain radical ideals in F-pure local rings of prime characteristic, establishing equalities among sets linked to tight closure and proposing a generalization of the splitting prime concept.

Contribution

It proves the equality of two sets of radical ideals in F-pure rings and introduces a generalized notion of the splitting prime.

Findings

Two sets of radical ideals are shown to be equal.

Connections established between these ideals and tight closure test ideals.

A new generalization of the splitting prime is proposed.

Abstract

Let $R$ be a commutative (Noetherian) local ring of prime characteristic $p$ that is $F$-pure. This paper is concerned with comparison of three finite sets of radical ideals of $R$, one of which is only defined in the case when $R$ is $F$-finite (that is, is finitely generated when viewed as a module over itself via the Frobenius homomorphism). Two of the afore-mentioned three sets have links to tight closure, via test ideals. Among the aims of the paper are a proof that two of the sets are equal, and a proposal for a generalization of I. M. Aberbach's and F. Enescu's splitting prime.