赋权有向图的同调移位理想
Homological shift ideals of weighted oriented graphs
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中文总结 AI 辅助
该论文研究赋权有向图的同调移位理想，证明顶点可分裂时$HS_1(I(D))$有线性商，推导树的对应刻画，给出结构障碍，关联基础简单图的同调线性商性质。
中文摘要 AI 辅助
本文研究与赋权有向图关联的边理想的同调移位理想。对赋权有向图$D$，记$HS_k(I(D))$为其边理想$I(D)$的第$k$个同调移位理想。证明若$I(D)$是顶点可分裂的，则$HS_1(I(D))$有线性商；还证明对所有$k\text{≥}1$，$\sqrt{HS_k(I(D))}=HS_k(I(G))$，其中$G$是$D$的基础简单图。若$I(D)$有同调线性商，则$I(G)$也有同调线性商；此外，给出$HS_1(I(D))$有线性商的结构障碍。当$D$是树时，建立如下刻画：对所有$k\text{≥}0$，$HS_k(I(D))$有线性商当且仅当$D$不含$D_i$（$i=1,2,5,6,8$）。
英文摘要
In this paper, we study the homological shift ideals of edge ideals associated with weighted oriented graphs. For a weighted oriented graph $D$, let $HS_k(I(D))$ denote the $k^{th}$ homological shift ideal of its edge ideal $I(D)$. If $D$ is vertex-splittable, then we characterize that $HS_1(I(D))$ has linear quotients if and only if $D_6$, $D_7$, and $D_8$ are not induced subgraphs of $D$. Furthermore, we show that if $I(D)$ has linear quotients, then $\sqrt{HS_k(I(D))} = HS_k(I(G))$, for all $k\geq 1$, where $G$ is the underlying simple graph of $D$. We show that if $I(D)$ has homological linear quotients, then $I(G)$ also has homological linear quotients. If $D$ is a tree, then we establish the following characterization: \begin{align*} HS_k(I(D)) \text{ has linear quotients for all } k\geq 0 \iff ~ &G~ \text{is} \text{ a star graph or a broom graph} \\ &\text{ and }~ D \text{ is $D_i$-free, for } i=1,2,5,6,8. \end{align*}
