论文标题
归一化深度功能的行为
Behaviour of the normalized depth function
论文作者
论文摘要
令$ i \ subset s = k [x_1，\ dots，x_n] $为平方英尺的单一理想，$ k $ a字段。 $ k $ th squarefree $ i^{[k]} $ $ i $是所有属于$ i^k $的所有SquareFree Monomials生成的$ S $的单一理想。最大的整数$ k $，使得$ i^{[k]} \ ne（0）$称为$ i $的单一等级，并用$ν（i）$表示。令$ d_k $为属于$ i^{[k]} $的单一元素的最低度。然后，$ \ text {depth}（s/i^{[k]}）\ ge d_k-1 $ for las $ 1 \ le K \ lev（i）$。 $ i $的归一化深度函数定义为$ g_ {i}（k）= \ text {depth}（s/i^{[k]}） - （d_k-1）$，$ 1 \ le k \ le k \ lev（i）$。预计$ g_i（k）$是任何$ i $的非进攻功能。在本文中，我们研究了$ g_ {i}（k）$在单一理想的各种操作下的行为。我们的主要结果是所有cochordal图$ g $的特征，使得边缘理想$ i（g）$ g $ of $ g $我们有$ g_ {i（g）}（1）（1）= 0 $。它们恰恰是所有的cochordal图$ g $，其互补图$ g^c $已连接并具有切割顶点。作为一个深远的应用程序，对于给定的整数$ 1 \ le s <m $，我们构造一个图$ g $，使得$ν（i（g））= m $和$ g_ $ $ g_ {i（g）}（k）}（k）= 0 $ if，并且仅当$ k = s+s+s+1，\ dots，m $。最后，我们表明，非阴性整数的任何非进攻函数都是某些无方形单体理想的归一化深度函数。
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 $ν(I)$. 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 $1\le k\leν(I)$. The normalized depth function of $I$ is defined as $g_{I}(k)=\text{depth}(S/I^{[k]})-(d_k-1)$, $1\le k\leν(I)$. 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 $ν(I(G))=m$ 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.
