Behaviour of the normalized depth function

Abstract

Let $I\subset S=K[x_1,\dots ,x_n]$ be a squarefree monomial ideal, $K$ a field. The $k$th squarefree power $I^{[k]}$ of $I$ is the monomial ideal of $S$ generated by all squarefree monomials belonging to $I^k$. The biggest integer $k$ such that $I^{[k]}\ne (0)$ is called the monomial grade of $I$ and it is denoted by $\nu (I)$. Let $d_k$ be the minimum degree of the monomials belonging to $I^{[k]}$. Then, $\text{depth}(S/I^{[k]})\ge d_k-1$ for all $1\le k\le \nu (I)$. The normalized depth function of $I$ is defined as $g_I(k)=\text{depth}(S/I^{[k]})-(d_k-1)$, $1\le k\le \nu (I)$. It is expected that $g_I(k)$ is a non-increasing function for any $I$. In this article we study the behaviour of $g_I(k)$ under various operations on monomial ideals. Our main result characterizes all cochordal graphs $G$ such that for the edge ideal $I(G)$ of $G$ we have $g_{I(G)}(1)=0$. They are precisely all cochordal graphs $G$ whose complementary graph $G^c$ is connected and has a cut vertex. As a far-reaching application, for given integers $1\le s<m$ we construct a graph $G$ such that $\nu (I(G))=m$ and $g_{I(G)}(k)=0$ if and only if $k=s+1,\dots ,m$. Finally, we show that any non-increasing function of non-negative integers is the normalized depth function of some squarefree monomial ideal.

This is the final version of our paper accepted for pubblication in the Electronic Journal of Combinatorics. Some minor changes and typos fixed, bybliography updated