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Transactions of the American Mathematical Society

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Lower central series and free resolutions of hyperplane arrangements

Authors: Henry K. Schenck and Alexander I. Suciu
Journal: Trans. Amer. Math. Soc. 354 (2002), 3409-3433
MSC (2000): Primary 16E05, 20F14, 52C35; Secondary 16S37
Published electronically: May 8, 2002
MathSciNet review: 1911506
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Abstract: If $M$ is the complement of a hyperplane arrangement, and $A=H^*(M,\Bbbk)$is the cohomology ring of $M$ over a field $\Bbbk$ of characteristic $0$, 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 $A$. 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 $A$ over the exterior algebra $E$.

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 $A$ 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 $A$ over $E$; if the lower bound is attained for $i=2$, 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 $A$ 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 $K_4$ subgraphs in the graph.

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Additional Information

Henry K. Schenck
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Address at time of publication: Department of Mathematics, Texas A&M University, College Station, Texas 77843

Alexander I. Suciu
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115

Keywords: Lower central series, free resolution, hyperplane arrangement, change of rings spectral sequence, Koszul algebra, linear strand, graphic arrangement
Received by editor(s): August 22, 2001
Received by editor(s) in revised form: September 19, 2001
Published electronically: May 8, 2002
Additional Notes: The first author was partially supported by an NSF postdoctoral research fellowship
The second author was partially supported by NSF grant DMS-0105342
Article copyright: © Copyright 2002 American Mathematical Society

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