A formula for the *-core of an ideal

TL;DR

This paper derives a formula for the *-core of an ideal in specific algebraic settings, expanding previous work and highlighting conditions where the formula applies or fails.

Contribution

It provides a new explicit formula for the *-core of an ideal in Cohen--Macaulay and normal local rings under certain conditions, extending prior research.

Findings

Formula for *-core in Cohen--Macaulay rings with test ideal of depth ≥ 2

Formula for *-core in normal local domains with high Frobenius powers

Examples showing failure of the formula outside specified hypotheses

Abstract

Expanding on the work of Fouli and Vassilev \cite{FV}, we determine a formula for the *-$\rm{core}$ of an ideal in two different settings: (1) in a Cohen--Macaulay local ring of characteristic $p>0$, perfect residue field and test ideal of depth at least two, where the ideal has a minimal *-reduction that is a parameter ideal and (2) in a normal local domain of characteristic $p>0$, perfect residue field and $\m$-primary test ideal, where the ideal is a sufficiently high Frobenius power of an ideal. We also exhibit some examples where our formula fails if our hypotheses are not met.