A note on Ratliff-Rush filtration, reduction number and postulation number of $\mathfrak m$-primary ideals

TL;DR

This paper investigates the relationships between reduction number, postulation number, and the Ratliff-Rush filtration of m-primary ideals in Cohen-Macaulay local rings, extending previous results and providing new bounds and conditions.

Contribution

It generalizes Marley’s result on Hilbert functions by relaxing depth conditions and explores the interplay between these invariants and the Ratliff-Rush filtration.

Findings

For dimension 2, specific bounds relate n(I) and rd(I).

For higher dimensions, conditions on integrally closed ideals imply bounds on rd(I).

If the Hilbert polynomial equals the Hilbert function at some point, they coincide thereafter.

Abstract

Let $(R,\mathfrak m)$ be a Cohen-Macaulay local ring of dimension $d\geq 2$ and $I$ an $\mathfrak m$-primary ideal. Let rd$(I)$ be the reduction number of $I$ and n$(I)$ the postulation number. We prove that for $d=2,$ if n$(I)=\rho(I)-1,$ then rd$(I) \leq$n$(I)+2$ and if n$(I)\neq \rho(I)-1,$ then rd$(I)\geq$n$(I)+2.$ For $d \geq 3$, if $I$ is integrally closed, depth gr$(I) = d-2$ and n$(I)=-(d-3).$ Then we prove that rd$(I)\geq$n$(I)+d$. Our main result is to generalize a result of T. Marley on the relation between the Hilbert-Samuel function and the Hilbert-Samuel polynomial by relaxing the condition on the depth of the associated graded ring with the good behaviour of the Ratliff-Rush filtration with respect to $I$ mod a superficial element. From this result, it follows that for a Cohen-Macaulay ring of dimension $d\geq2$, if $P_{I}(k)=H_{I}(k)$ for some $k \geq \rho(I)$, then $P_{I}(n)=H_{I}(n)$ for all $n \geq k.$