arXiv 1701.047342017-01-17math.AC

关于扩张单纯复形的刻面理想

On the facet ideal of an expanded simplicial complex

Somayeh Moradi, Rahim Rahmati-Asghar

备注：: To appear in: Bull. Iranian Math. Soc. arXiv admin note: substantial text overlap with arXiv:1511.04676

摘要：

对于单纯复形$Δ$，展开函子对$Δ$的组合性质及其Stanley-Reisner环的代数性质的影响已在以前的一些文章中得到研究.在本文中，我们考虑刻面理想$I(Δ)$和它的亚历山大对偶，我们表示为$J_Δ$，看看如何扩展函子改变这些理想的代数性质。证明了对于任意的扩张$Δ^α$，理想$J_Δ$和理想$J_{Δ^α}$具有相同的总贝蒂数，它们的Cohen-Macaulay性是等价的，这意味着理想$I(Δ)$和理想$I(Δ^α)$的Cohen-Macaulay性是相等的.此外，还比较了$I(Δ)$和$I(Δ^α)$的投射维数。在图$G$的续集中，一些性质是等价的$G$和它的扩展，并为Cohen-Macaulay（分别）。序列Cohen-Macaulay和可壳）图$G$，我们给出了从$G$中增加或删除一个顶点的条件，使得剩下的图仍然是Cohen-Macaulay（分别为.顺序为Cohen-Macaulay和可壳化）。

英文摘要：

For a simplicial complex $Δ$, the affect of the expansion functor on combinatorial properties of $Δ$ and algebraic properties of its Stanley-Reisner ring has been studied in some previous papers. In this paper, we consider the facet ideal $I(Δ)$ and its Alexander dual which we denote by $J_Δ$ to see how the expansion functor alter the algebraic properties of these ideals. It is shown that for any expansion $Δ^α$ the ideals $J_Δ$ and $J_{Δ^α}$ have the same total Betti numbers and their Cohen-Macaulayness are equivalent, which implies that the regularities of the ideals $I(Δ)$ and $I(Δ^α)$ are equal. Moreover, the projective dimensions of $I(Δ)$ and $I(Δ^α)$ are compared. In the sequel for a graph $G$, some properties that are equivalent in $G$ and its expansions are presented and for a Cohen-Macaulay (resp. sequentially Cohen-Macaulay and shellable) graph $G$, we give some conditions for adding or removing a vertex from $G$, so that the remaining graph is still Cohen-Macaulay (resp. sequentially Cohen-Macaulay and shellable).

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英文摘要：

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