On Coefficient ideals

TL;DR

This paper investigates the relationship between coefficient ideals, Ratliff-Rush closures, and the properties of associated graded rings in Cohen-Macaulay local rings, establishing conditions under which these ideals coincide.

Contribution

It provides new criteria linking local cohomology conditions of the associated graded ring to the equality of coefficient ideals and Ratliff-Rush closures.

Findings

If certain local cohomology modules satisfy dimension bounds, coefficient ideals equal Ratliff-Rush closures.

For generalized Cohen-Macaulay associated graded rings, the first coefficient ideal equals the Ratliff-Rush closure.

The $S_2$-ification of the Rees algebra equals the direct sum of Ratliff-Rush closures under specified conditions.

Abstract

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d \geq 2$ with infinite residue field and let $I$ be an $\mathfrak{m}$-primary ideal. For $0 \leq i \leq d$ let $I_i$ be the $i^{th}$-coefficient ideal of $I$. Also let $\widetilde{I} = I_d$ denote the Ratliff-Rush closure of $A$. Let $G = G_I(A)$ be the associated graded ring of $I$. We show that if $\dim H^j_{G_+}(G)^\vee \leq j -1$ for $1 \leq j \leq i \leq d-1$ then $(I^n)_{d-i} = \widetilde{I^n}$ for all $n \geq 1$. In particular if $G$ is generalized Cohen-Macaulay then $(I^n)_1 = \widetilde{I^n}$ for all $n \geq 1$. As a consequence we get that if $A$ is an analytically unramified domain with $G$ generalized Cohen-Macaulay, then the $S_2$-ification of the Rees algebra $ A[It]$ is $\bigoplus_{n \geq 0} \widetilde{I^n}$.