The reduction number of an ideal and the local cohomology of the associated graded ring

Abstract:

Let $(R,m)$ be a local ring and $I$ an $m$-primary ideal. A result of Trung shows that if the local cohomology of $g{r_I}(R)$ satisfies certain conditions, then the reduction number of $I$ is independent of the minimal reduction chosen. These conditions consist of $t = \dim R - \operatorname {grade} \;g{r_I}{(R)^ + }$ inequalities. We show that if $R$ is Cohen-Macaulay, then one of these inequalities is always satisied, while another can often be easily checked. Applications are then given in two-dimensional Cohen-Macaulay rings. For instance, we show that if the Hilbert function of $I$ equals the Hilbert polynomial of $I$ for all integers greater than $1$, then the reduction number is independent of the choice of minimal reduction.