Lower central series and free resolutions of hyperplane arrangements
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- by Henry K. Schenck and Alexander I. Suciu PDF
- Trans. Amer. Math. Soc. 354 (2002), 3409-3433 Request permission
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, $b’_{ij}=\dim \operatorname {Tor}^E_i(A,\Bbbk )_{j}$, 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.References
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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
- MR Author ID: 621581
- Email: schenck@math.tamu.edu
- Alexander I. Suciu
- Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
- MR Author ID: 168600
- ORCID: 0000-0002-5060-7754
- Email: alexsuciu@neu.edu
- 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 - © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 3409-3433
- MSC (2000): Primary 16E05, 20F14, 52C35; Secondary 16S37
- DOI: https://doi.org/10.1090/S0002-9947-02-03021-0
- MathSciNet review: 1911506