arXiv 1711.020022017-11-06math.AC

单项理想的正则性与$h$ -多项式

Regularity and $h$-polynomials of monomial ideals

Takayuki Hibi, Kazunori Matsuda

备注：: 11 pages

摘要：

设$S = K[x_1, \ldots, x_n]$表示域$K$上的$n$变量的多项式环，其中每个$°x_i = 1$和$I \subset S$是$S$的齐次理想，其中$\dim S/I = d$。$S/I$的希尔伯特级数的形式为$h_{S/I}(λ)/(1 - λ)^d$，其中带有$h_s \neq 0$的$h_{S/I}(λ) = h_0 + h_1λ+ h_2λ^2 + \cdots + h_sλ^s$是$S/I$的$h$ -多项式。已知，当$S/I$是Cohen-Macaulay时，有$\reg(S/I) = °h_{S/I}(λ)$，其中$\reg(S/I)$是$S/I$的（Castelnuovo-Mumford）正则性。本文给出了任意整数$r$和$s$的$r \geq 1$和$s \geq 1$，构造了$S = K[x_1, \ldots, x_n]$和$n \gg 0$的单项式理想$I$的$\reg(S/I) = r$和$°h_{S/I}(λ) = s$.进一步，我们给出了具有$\reg(S/I) = °h_{S/I}(λ)$的Cameron-Walker图的一类边理想$I \subset S$，其中$S/I$不是Cohen-Macaulay.

英文摘要：

Let $S = K[x_1, \ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $°x_i = 1$ and $I \subset S$ a homogeneous ideal of $S$ with $\dim S/I = d$. The Hilbert series of $S/I$ is of the form $h_{S/I}(λ)/(1 - λ)^d$, where $h_{S/I}(λ) = h_0 + h_1λ+ h_2λ^2 + \cdots + h_sλ^s$ with $h_s \neq 0$ is the $h$-polynomial of $S/I$. It is known that, when $S/I$ is Cohen--Macaulay, one has $\reg(S/I) = °h_{S/I}(λ)$, where $\reg(S/I)$ is the (Castelnuovo--Mumford) regularity of $S/I$. In the present paper, given arbitrary integers $r$ and $s$ with $r \geq 1$ and $s \geq 1$, a monomial ideal $I$ of $S = K[x_1, \ldots, x_n]$ with $n \gg 0$ for which $\reg(S/I) = r$ and $°h_{S/I}(λ) = s$ will be constructed. Furthermore, we give a class of edge ideals $I \subset S$ of Cameron--Walker graphs with $\reg(S/I) = °h_{S/I}(λ)$ for which $S/I$ is not Cohen--Macaulay.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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