Bounds on the Ratliff-Rush Index and the Castelnuovo-Mumford Regularity

Abstract

Let $(R, \mathfrak m)$ be a Cohen-Macaulay local ring of dimension $d \geq 2,$ and $I$ an $\mathfrak m$-primary ideal of $R.$ Denote $r_{J}(I)$ as the reduction number of $I$ with respect to a minimal reduction $J$ of $I,$ and $\rho(I)$ as the Ratliff-Rush index of $I$. We establish upper bounds on $\rho(I)$ in terms of Hilbert coefficients $e_{i}(I)$ for $0 \leq i \leq d+1,$ and $r_{J}(I).$ Suppose $\widetilde{I^{r_{J}(I)}} \neq I^{r_{J}(I)}.$ We prove that $\rho(I) \leq r_{J}(I)-1+(-1)^{d+1}(e_{d+1}(I)-\widetilde{e}_{d+1}(I)).$ When $d=2,$ we prove that $\rho(I) \leq r_{J}(I) -1 +(e_{2}(I)-1)e_{2}(I)-e_{3}(I).$ This established bound on $\rho(I)$ consequently leads to a bound on the Castelnuovo-Mumford regularity of the associated graded ring of $I.$ We also determine bound on $\rho(I)$ in two-dimensional Buchsbaum rings with positive depth.