An Increasing normalized depth function

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.