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Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence


Authors: Henry K. Schenck and Alexander I. Suciu
Journal: Trans. Amer. Math. Soc. 358 (2006), 2269-2289
MSC (2000): Primary 16E05, 52C35; Secondary 13D07, 20F14
DOI: https://doi.org/10.1090/S0002-9947-05-03853-5
Published electronically: December 20, 2005
MathSciNet review: 2197444
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Abstract: If $ \mathcal A$ is a complex hyperplane arrangement, with complement $ X$, we show that the Chen ranks of $ G=\pi_1(X)$ are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring $ A=H^*(X,\Bbbk)$, viewed as a module over the exterior algebra $ E$ on $ \mathcal A$:

$\displaystyle \theta_k(G) = \dim_{\Bbbk}\operatorname{Tor}^E_{k-1}(A,\Bbbk)_k,$   for $ k\ge 2$$\displaystyle , $

where $ \Bbbk$ is a field of characteristic 0. The Chen ranks conjecture asserts that, for $ k$ sufficiently large, $ \theta_k(G) =(k-1) \sum_{r\ge 1} h_r \binom{r+k-1}{k}$, where $ h_r$ is the number of $ r$-dimensional components of the projective resonance variety $ \mathcal R^{1}(\mathcal A)$. Our earlier work on the resolution of $ A$ over $ E$ and the above equality yield a proof of the conjecture for graphic arrangements. Using results on the geometry of $ \mathcal R ^{1}(\mathcal A)$ and a localization argument, we establish the inequality

$\displaystyle \theta_k(G) \ge (k-1) \sum_{r\ge 1} h_r \binom{r+k-1}{k},$   for $ k\gg 0$$\displaystyle , $

for arbitrary $ \mathcal A$. Finally, we show that there is a polynomial $ \mathrm{P}(t)$ of degree equal to the dimension of $ \mathcal R^1(\mathcal A)$, such that $ \theta_k(G) = \mathrm{P}(k)$, for all $ k\gg 0$.


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Additional Information

Henry K. Schenck
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: schenck@math.tamu.edu

Alexander I. Suciu
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
Email: a.suciu@neu.edu

DOI: https://doi.org/10.1090/S0002-9947-05-03853-5
Received by editor(s): January 31, 2004
Received by editor(s) in revised form: August 17, 2004
Published electronically: December 20, 2005
Additional Notes: Both authors were supported by NSF Collaborative Research grant DMS 03-11142; the first author was also supported by NSA grant MDA 904-03-1-0006 and ATP grant 010366-0103.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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