Hilbert Coefficients and Regularity of Binomial Edge Ideals

TL;DR

This paper investigates the relationship between Hilbert coefficients and regularity in binomial edge ideals, providing a reduction technique and demonstrating that these invariants can vary independently.

Contribution

It introduces a vertex degree condition to simplify Hilbert coefficient computation and shows that regularity and Hilbert coefficients are not inherently linked.

Findings

Hilbert coefficients remain unchanged under certain vertex removals.

For any regularity and Hilbert coefficient values, a corresponding graph exists.

Regularity and Hilbert coefficients can vary independently in binomial edge ideals.

Abstract

Let $G$ be a simple graph on $n$ vertices, and let $J_G$ denotes the corresponding binomial edge ideal in $S=\mathbb{K}[x_1,\ldots,x_n,y_1,\ldots,y_n]$, where $\mathbb{K}$ is a field. We show that if a vertex satisfies a certain degree condition, then some Hilbert coefficients remain unchanged upon its removal, thereby providing a reduction technique for computing Hilbert coefficients. As an application, for any $i\geq 0$ and a pair $(r,s)$ with $r\geq 2, s\in \mathbb{Z}$, we show that there always exists a graph $G$ such that $\mathrm{reg}(S/J_G)=r$ and $e_i(S/J_G)=s$, where $\mathrm{reg}(S/J_G)$ and $e_i(R/J_G)$ denote the Castelnuovo-Mumford regularity and the $i$-th Hilbert coefficient of $S/J_G$, respectively. In particular, this demonstrates that there is no inherent relationship between the regularity and the Hilbert coefficients for the class of binomial edge ideals.