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

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归一化深度函数的行为

设$I\subset S=K[x_1,\dots,x_n]$为无平方项理想，$K$为场。$I$的$k$无平方幂$I^{[k]}$是由所有属于$I^k$的无平方单项式生成的$S$的单项式理想。最大的整数$k$使得$I^{[k]}\ne(0)$称为$I$的单项等级，用$\nu(I)$表示。设$d_k$为属于$I^{[k]}$的单项式的最小次。然后，$\text{depth}(S/I^{[k]})\ge d_k-1$为所有$1\le k\le\nu(I)$。归一化深度函数$I$定义为$g_{I}(k)=\text{depth}(S/I^{[k]})-(d_k-1)$, $1\le k\le\nu(I)$。可以预期$g_I(k)$对于任何$I$都是一个不增加的函数。本文研究了$g_{I}(k)$在单项式理想的各种运算下的行为。我们的主要结果表征了所有的弦图$G$，对于$G$的边理想$I(G)$，我们有$g_{I(G)}(1)=0$。它们都是弦图$G$，它们的互补图$G^c$是连通的，并且有一个切顶点。作为一个深远的应用，对于给定的整数$1\le s

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