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arXiv 2203.01710math.AC
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Cohen-Macaulay加权定向边理想及其Alexander对偶

Cohen-Macaulay Weighted Oriented Edge Ideals and its Alexander Dual

Kamalesh Saha, Indranath Sengupta

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26 pages
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摘要：

在Reed-Muller型码的背景下，研究了带有底层图$G$的加权定向图$D_{G}$的边理想$I(D_{G})$.我们推广了Villarreal为单图的边理想给出的$I(D_{G})$的Cohen-Macaulay构造.我们使用这个结构来分类所有的Cohen-Macaulay加权定向边理想，其基础图是一个循环。我们证明了Pitones等人（2019）提出的关于$I(D_{G})$的Cohen-Macaulayness的猜想对于$I(D_{C_{n}})$成立，其中$C_{n}$表示长度为$n$的循环。Miller将无平方因子的单项理想的Alexander对偶理想推广到任意单项理想，并在此基础上研究了$I(D_{G})$的Alexander对偶及其成为Cohen-Macaulay的条件.

英文摘要：

The study of the edge ideal $I(D_{G})$ of a weighted oriented graph $D_{G}$ with underlying graph $G$ started in the context of Reed-Muller type codes. We generalize a Cohen-Macaulay construction for $I(D_{G})$, which Villarreal gave for edge ideals of simple graphs. We use this construction to classify all the Cohen-Macaulay weighted oriented edge ideals, whose underlying graph is a cycle. We show that the conjecture on Cohen-Macaulayness of $I(D_{G})$, proposed by Pitones et al. (2019), holds for $I(D_{C_{n}})$, where $C_{n}$ denotes the cycle of length $n$. Miller generalized the concept of Alexander dual ideals of square-free monomial ideals to arbitrary monomial ideals, and in that direction, we study the Alexander dual of $I(D_{G})$ and its conditions to be Cohen-Macaulay.

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