Computing the core of ideals in arbitrary characteristic

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 $1\le j\le \ell -g$. 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.

13 pages, revised. To appear in the Journal of Algebra

References in corpus (2)

• A formula for the core of an ideal
• The core of ideals in arbitrary characteristic