旗复形的第一至第七个Betti数的域独立性
Field independence of the first seven Betti numbers of flag complexes
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中文总结 AI 辅助
本文证明旗复形第七个Betti数域独立，通过更强的拓扑结果τ(d)≥d+10，并推广到边理想的前七个Betti数。
中文摘要 AI 辅助
2006年，Katzman证明了旗复形的Stanley-Reisner环的前六个Betti数与域无关。他还发现了具有十一个顶点的旗复形，其第八个Betti数依赖于域，并询问第七个Betti数是否总是与域无关。我们通过证明一个更强的纯拓扑结果来肯定地回答这个问题。设τ(d)为具有d阶约化整同调群有挠的旗复形的最少顶点数。我们证明对于每个d≥0，有τ(d)≥d+10。这个下界导出了第七个Betti数的域独立性。等价地，将我们的结果与Katzman的结果结合，对于每个有限简单图G，边理想I(G)的前七个Betti数都是与域无关的。
英文摘要
In 2006, Katzman showed that the first six Betti numbers of the Stanley--Reisner ring of a flag complex are field independent. He also found flag complexes on eleven vertices whose eighth Betti number depends on the field, and asked whether the seventh is always field independent. We answer this affirmatively by proving a stronger, purely topological result. Let $τ(d)$ be the least number of vertices of a flag complex whose $d$-th reduced integral homology has torsion. We prove that $τ(d)\geq d+10$ for every $d\ge0$. This bound yields the field independence of the seventh Betti number. Equivalently, combining our result with Katzman's, for every finite simple graph $G$, the first seven Betti numbers of the edge ideal $I(G)$ are field independent.
发表机构
- Chennai Mathematical Institute（金奈数学研究所）
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