Lower central series and free resolutions of hyperplane arrangements期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

按检索

Lower central series and free resolutions of hyperplane arrangements

Authors:	Henry K Schenck Alexander I Suciu

Institution:	Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138 ; Department of Mathematics, Northeastern University, Boston, Massachusetts 02115

Abstract:	If is the complement of a hyperplane arrangement, and $A=H^*(M,\Bbbk)$ is the cohomology ring of over a field $\Bbbk$ of characteristic , then the ranks, $\phi_k$ , of the lower central series quotients of $\pi_1(M)$ can be computed from the Betti numbers, $b_{ii}=\dim \operatorname{Tor}^A_i(\Bbbk,\Bbbk)_i$ , of the linear strand in a minimal free resolution of $\Bbbk$ over . We use the Cartan-Eilenberg change of rings spectral sequence to relate these numbers to the graded Betti numbers, $b'_{ij}=\dim \Tor^E_i(A,\Bbbk)_{j}$ , of a minimal resolution of over the exterior algebra . From this analysis, we recover a formula of Falk for $\phi_3$ , and obtain a new formula for $\phi_4$ . The exact sequence of low-degree terms in the spectral sequence allows us to answer a question of Falk on graphic arrangements, and also shows that for these arrangements, the algebra is Koszul if and only if the arrangement is supersolvable. We also give combinatorial lower bounds on the Betti numbers, $b'_{i,i+1}$ , of the linear strand of the free resolution of over ; if the lower bound is attained for , then it is attained for all $i \ge 2$ . For such arrangements, we compute the entire linear strand of the resolution, and we prove that all components of the first resonance variety of are local. For graphic arrangements (which do not attain the lower bound, unless they have no braid subarrangements), we show that $b'_{i,i+1}$ is determined by the number of triangles and subgraphs in the graph.

Keywords:	Lower central series free resolution hyperplane arrangement change of rings spectral sequence Koszul algebra linear strand graphic arrangement

	点击此处可从《Transactions of the American Mathematical Society》浏览原始摘要信息
	点击此处可从《Transactions of the American Mathematical Society》下载免费的PDF全文

设为首页 | 免责声明 | 关于勤云 | 加入收藏