论文标题
甚至是FC型的Artin组的Sigma不变性
On the Sigma invariants of even Artin groups of FC-type
论文作者
论文摘要
在本文中，我们研究了即使是FC型Artin组的Sigma不变式，从而扩展了一些已知的右角Artin组结果。特别是，我们定义了一个条件，我们称之为图$γ$的强同源$ n $ link条件，并证明它为字符$χ：a_γ\ to \ mathbb {z} $提供足够的条件，以满足满足$ [χ] \inς^n（a_γ，\ mathbb {z}）$。这意味着内核$ a^χ_γ= \kerχ$属于$ fp_n $。也证明了同型对应物。讨论了相反的部分结果。 我们还为$ h_n（a^χ_γ; \ mathbb {f}）$的自由部分提供了一个通用公式，作为$ \ mathbb {f} [t^{\ pm 1}] $模块，并具有由$χ$引起的自然动作。这给出了$ h_n（a^χ_γ; \ mathbb {f}）$的表征是$ \ mathbb {f} $的有限维矢量空间。 在最后一个版本中，我们在引理4.3的证明中纠正了一个问题，也纠正了第3.3款末尾的评论。
In this paper we study Sigma invariants of even Artin groups of FC-type, extending some known results for right-angled Artin groups. In particular, we define a condition that we call the strong homological $n$-link condition for a graph $Γ$ and prove that it gives a sufficient condition for a character $χ:A_Γ\to \mathbb{Z}$ to satisfy $[χ]\inΣ^n(A_Γ,\mathbb{Z})$. This implies that the kernel $A^χ_Γ=\ker χ$ is of type $FP_n$. The homotopy counterpart is also proved. Partial results on the converse are discussed. We also provide a general formula for the free part of $H_n(A^χ_Γ;\mathbb{F})$ as an $\mathbb{F}[t^{\pm 1}]$-module with the natural action induced by $χ$. This gives a characterization of when $H_n(A^χ_Γ;\mathbb{F})$ is a finite dimensional vector space over $\mathbb{F}$. In the last version we correct a problem in the proof of Lemma 4.3 and also a remark at the end of subsection 3.3.
