A comparison of the regularity of certain classes of monomial ideals and their integral closures

TL;DR

This paper proves K"{u}ronya and Pintye's conjecture that the regularity of the integral closure of certain monomial ideals does not exceed that of the original ideal, advancing understanding of their algebraic properties.

Contribution

It establishes the conjecture for specific classes of monomial ideals, providing new insights into their regularity behavior.

Findings

Proved the conjecture for certain monomial ideals.

Showed regularity of integral closures is bounded by original ideals.

Enhanced understanding of algebraic regularity in monomial ideals.

Abstract

Let $S = \mathsf{k}[x_1, \ldots, x_n]$, $I$ be an ideal of $S$, and $\bar{I}$ denote its integral closure. A conjecture of K\"{u}ronya and Pintye states that for any homogeneous ideal $I$ of $S$, the inequality $\operatorname{reg}(\bar{I}) \leq \operatorname{reg}(I)$ holds, where $\operatorname{reg}(\_)$ denotes the Castelnuovo-Mumford regularity. In this article, we prove the conjecture for certain classes of monomial ideals.