On Coefficient ideals

Abstract

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d\ge 2$ with infinite residue field and let $I$ be an $\mathfrak{m}$-primary ideal. For $0\le i\le d$ let $I_i$ be the $i^{th}$-coefficient ideal of $I$. Also let $\tilde{I}=I_d$ denote the Ratliff-Rush closure of $A$. Let $G=G_I(A)$ be the associated graded ring of $I$. We show that if $\dim H_{G_{+}}^j(G{)}^{\vee }\le j-1$ for $1\le j\le i\le d-1$ then $(I^n{)}_{d-i}=\widetilde{I^n}$ for all $n\ge 1$. In particular if $G$ is generalized Cohen-Macaulay then $(I^n{)}_1=\widetilde{I^n}$ for all $n\ge 1$. 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 ${⨁}_{n\ge 0}\widetilde{I^n}$.