Bounds for the reduction number of primary ideal in dimension three

TL;DR

This paper establishes bounds on the reduction number of primary ideals in Cohen-Macaulay local rings of dimension three, relating it to Hilbert coefficients and depth conditions of the associated graded ring.

Contribution

It provides new upper bounds for the reduction number of primary ideals in dimension three, depending on Hilbert coefficients and depth conditions, extending previous results.

Findings

Bound: r_J(I) ≤ e_1(I) - e_0(I) + λ(R/I) + 1 + (e_2(I) - 1)e_2(I) - e_3(I) for depth G(I) ≥ d-3.

Bound: r_J(I) ≤ e_1(I) - e_0(I) + λ(R/I) + t when d=3 and depth G(I^t) > 0.

Results connect reduction number bounds with Hilbert coefficients and depth conditions.

Abstract

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d\geq 3$ and $I$ an $\mathfrak{m}$-primary ideal of $R$. Let $r_J(I)$ be the reduction number of $I$ with respect to a minimal reduction $J$ of $I$. Suppose depth $G(I)\geq d-3$. We prove that $r_J(I)\leq e_1(I)-e_0(I)+\lambda(R/I)+1+(e_2(I)-1)e_2(I)-e_3(I)$, where $e_i(I)$ are Hilbert coefficients. Suppose $d=3$ and depth $G(I^t)>0$ for some $t\geq 1$. Then we prove that $r_J(I)\leq e_1(I)-e_0(I)+\lambda(R/I)+t$.