A formula for the *-core of an ideal

Abstract

Expanding on the 2010 work of Fouli and Vassilev, we determine a formula for the ∗ - core *\textrm {-}\textrm {core} of an ideal in two different settings: (1) in a Cohen–Macaulay local ring of characteristic p > 0 p>0 , with 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 p>0 , with perfect residue field and m \mathfrak {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.