余弦图边理想的符号幂的正则性
Regularity of Symbolic Powers of Co-Chordal Edge Ideals
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中文总结 AI 辅助
该研究针对有限简单余弦图的边理想，证明其符号幂的正则性公式为reg I(G)^(s)=2s，还指出这类符号幂未必是分量wise线性的，归约与论证过程结合了Takayama公式、团树等方法。
中文摘要 AI 辅助
设G为至少含一条边的有限简单余弦图，I(G)为其边理想。在任意域上，我们证明对所有s≥1，I(G)的符号幂满足reg I(G)^(s)=2s，因此I(G)的每个符号幂都具有极小次数的自由分解。我们利用Takayama公式和团树将正则性问题归约为拓扑问题，再通过有限凸几何得出非平凡同调对应合适的叶子集，经加权计数论证得到所需正则性公式，最后证明余弦图边理想的符号幂未必是分量wise线性的。
英文摘要
Let $G$ be a finite simple co-chordal graph with at least one edge, and let $I(G)$ be its edge ideal. Over an arbitrary field, we prove that the symbolic powers of $I(G)$ satisfy $\operatorname{reg} I(G)^{(s)}=2s$ for every $s\ge1$. Thus every symbolic power of $I(G)$ has a degree resolution. Using Takayama's formula and clique trees, we reduce the regularity problem to a topological one. Finite convex geometry then shows that nontrivial homology implies a suitable set of leaves, and a weighted counting argument gives the required regularity formula. Finally, we show that symbolic powers of co-chordal edge ideals are not necessarily componentwise linear.
