Computing the core of ideals in arbitrary characteristic

TL;DR

This paper investigates the computation of the core of ideals in local Gorenstein rings of arbitrary characteristic, providing a counterexample to a known formula for higher analytic spreads and proposing a new formula.

Contribution

It extends the understanding of ideal cores beyond the analytic spread one case, offering a counterexample and a new formula for higher spreads.

Findings

Counterexample to the core formula for higher analytic spreads

Proposed a new formula for the core of ideals with higher analytic spread

Clarified conditions under which the core formula holds

Abstract

Let $R$ be a local Gorenstein ring with infinite residue field of arbitrary characteristic. Let $I$ be an $R$--ideal with $g=\height I >0$, analytic spread $\ell$, and let $J$ be a minimal reduction of $I$. We further assume that $I$ satisfies $G_{\ell}$ and ${\depth} R/I^j \geq \dim R/I -j+1$ for $1 \leq j \leq \ell-g$. The question we are interested in is whether $\core{I}=J^{n+1}:\ds \sum_{b \in I} (J,b)^n$ for $n \gg 0$. In the case of analytic spread one Polini and Ulrich show that this is true with even weaker assumptions (\cite[Theorem 3.4]{PU}). We give a negative answer to this question for higher analytic spreads and suggest a formula for the core of such ideals.