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arXiv 2012.13725math.ACmath.CO
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Cohen-Macaulay单纯复形的Stanley-Reisner环上的不变量不等式

Inequalities of invariants on Stanley-Reisner rings of Cohen-Macaulay simplicial complexes

Akihiro Higashitani, Hiroju Kanno, Kazunori Matsuda

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9 pages
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摘要：

本文的目的是研究维数为$d - 1$的Cohen-Macaulay单纯复形的Stanley-Reisner环的一些代数不变量.本文证明了不等式$d \leq \mathrm{reg}(Δ) \cdot \mathrm{type}(Δ)$对任何满足$Δ=\mathrm{core}(Δ)$的$(d-1)$维Cohen-Macaulay单纯复形$Δ$成立，其中$\mathrm{reg}(Δ)$（分别为. $\mathrm{type}(Δ)$）表示Castelnuovo-Mumford正则性（分别为Cohen-Macaulay型）的Stanley-Reisner环$\Bbbk[Δ]$。此外，对于任意给定的整数$d,r,t$，满足$r,t \geq 2$和$r \leq d \leq rt$，我们构造了一个Cohen-Macaulay单纯复形$Δ(G)$作为图$G$的独立复形，使得$\dim(Δ(G))=d-1$，$\mathrm{reg}(Δ(G))=r$和$\mathrm{type}(Δ(G))=t$ .

英文摘要：

The goal of the present paper is the study of some algebraic invariants of Stanley-Reisner rings of Cohen-Macaulay simplicial complexes of dimension $d - 1$. We prove that the inequality $d \leq \mathrm{reg}(Δ) \cdot \mathrm{type}(Δ)$ holds for any $(d-1)$-dimensional Cohen-Macaulay simplicial complex $Δ$ satisfying $Δ=\mathrm{core}(Δ)$, where $\mathrm{reg}(Δ)$ (resp. $\mathrm{type}(Δ)$) denotes the Castelnuovo-Mumford regularity (resp. Cohen-Macaulay type) of the Stanley-Reisner ring $\Bbbk[Δ]$. Moreover, for any given integers $d,r,t$ satisfying $r,t \geq 2$ and $r \leq d \leq rt$, we construct a Cohen-Macaulay simplicial complex $Δ(G)$ as an independent complex of a graph $G$ such that $\dim(Δ(G))=d-1$, $\mathrm{reg}(Δ(G))=r$ and $\mathrm{type}(Δ(G))=t$.

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