arXiv 0912.40972009-12-21math.ACmath.CO

$k$ -Cohen-Macaulay复形的推广

A generalization of $k$-Cohen-Macaulay complexes

Hassan Haghighi, Rahim Zaare-Nahandi, Siamak Yassemi

备注：: 8 pages

摘要：

对于正整数$k$和非负整数$t$，引入了一类单纯复形，记为$k$ - ${\rm CM}_t$.这个类推广了单纯复形的两个概念：$k$ -Cohen-Macaulay和$k$ -Buchsbaum。类似于Cohen-Macaulay和Buchsbaum复形，我们给出了${\rm CM}_t(=$ 1- ${\rm CM}_t)$复形的一些特征，即它的链的某些同调的消失，以及复形及其穿孔空间的几何实现的某些相对奇异同调的消失.证明了复形是$k$ - ${\rm CM}_t$当且仅当其非空面的链环是$k$ - ${\rm CM}_{t-1}$ .证明了对整数$s\le d$，$(d-1)$维$k$ - ${\rm CM}_t$复形的$(d-s-1)$ -骨架是$(k+s)$ - ${\rm CM}_t$ .这个结果推广了Hibi关于Cohen-Macaulay复形的结果和Miyazaki关于Buchsbaum复形的结果。

英文摘要：

For a positive integer $k$ and a non-negative integer $t$ a class of simplicial complexes, to be denoted by $k$-${\rm CM}_t$, is introduced. This class generalizes two notions for simplicial complexes: being $k$-Cohen-Macaulay and $k$-Buchsbaum. In analogy with the Cohen-Macaulay and Buchsbaum complexes, we give some characterizations of ${\rm CM}_t(=$1-${\rm CM}_t)$ complexes, in terms of vanishing of some homologies of its links and, in terms of vanishing of some relative singular homologies of the geometric realization of the complex and its punctured space. We show that a complex is $k$-${\rm CM}_t$ if and only if the links of its nonempty faces are $k$-${\rm CM}_{t-1}$. We prove that for an integer $s\le d$, the $(d-s-1)$-skeleton of a $(d-1)$-dimensional $k$-${\rm CM}_t$ complex is $(k+s)$-${\rm CM}_t$. This result generalizes Hibi's result for Cohen-Macaulay complexes and Miyazaki's result for Buchsbaum complexes.

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