An Increasing normalized depth function

TL;DR

This paper investigates the normalized depth function of squarefree monomial ideals, providing a counterexample to a conjecture that it is nonincreasing, and shows that the difference between its values can be arbitrarily large.

Contribution

The paper presents the first counterexample to the conjecture that the normalized depth function is nonincreasing for all squarefree monomial ideals.

Findings

Counterexample disproves the conjecture.

The difference g_I(2)-g_I(1) can be arbitrarily large.

Normalized depth function behavior is more complex than previously thought.

Abstract

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. Assume that $I$ is a squarefree monomial ideal of $S$. For every integer $k\geq 1$, we denote the $k$-th squarefree power of $I$ by $I^{[k]}$. The normalized depth function of $I$ is defined as $g_I(k)={\rm depth}(S/I^{[k]})-(d_k-1)$, where $d_k$ denotes the minimum degree of monomials belonging to $I^{[k]}$. Erey, Herzog, Hibi and Saeedi Madani conjectured that for any squarefree monomial ideal $I$, the function $g_I(k)$ is nonincreasing. In this short note, we provide a counterexample for this conjecture. Our example in fact shows that $g_I(2)-g_I(1)$ can be arbitrarily large.