Regularity of symbolic powers of square-free monomial ideals

TL;DR

This paper investigates the regularity of symbolic powers of square-free monomial ideals, providing bounds related to the combinatorial structure of the underlying simplicial complex or graph.

Contribution

It establishes new upper bounds on the regularity of symbolic powers for square-free monomial ideals, including sharp bounds and specific results for edge ideals of graphs.

Findings

Bound on regularity: (n-1)+b for Stanley-Reisner ideals.

Linear upper bound for edge ideals: 2n + match(G)-1.

Sharpness of bounds demonstrated for all n.

Abstract

We study the regularity of symbolic powers of square-free monomial ideals. We prove that if $I = I_\Delta$ is the Stanley-Reisner ideal of a simplicial complex $\Delta$, then $\reg(I^{(n)}) \leqslant \delta(n-1) +b$ for all $n\geqslant 1$, where $\delta = \lim\limits_{n\to\infty} \reg(I^{(n)})/n$, and $b = \max\{\reg(I_\Gamma) \mid \Gamma \text{ is a subcomplex of } \Delta \text{ with } \F(\Gamma) \subseteq \F(\Delta)\}$. This bound is sharp for any $n$. When $I = I(G)$ is the edge ideal of a simple graph $G$, we obtain a general linear upper bound $\reg(I^{(n)}) \leqslant 2n + \ordmatch(G)-1$, where $\ordmatch(G)$ is the ordered matching number of $G$.