Hyperplane arrangements and linear strands in resolutions

Peeva, Irena

doi:10.1090/S0002-9947-02-03128-8

Hyperplane arrangements and linear strands in resolutions
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by Irena Peeva

Trans. Amer. Math. Soc. 355 (2003), 609-618

DOI: https://doi.org/10.1090/S0002-9947-02-03128-8

Published electronically: September 6, 2002

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Abstract:

The cohomology ring of the complement of a central complex hyperplane arrangement is the well-studied Orlik-Solomon algebra. The homotopy group of the complement is interesting, complicated, and few results are known about it. We study the ranks for the lower central series of such a homotopy group via the linear strand of the minimal free resolution of the field $\mathbf {C}$ over the Orlik-Solomon algebra.

References

Annetta Aramova, Jürgen Herzog, and Takayuki Hibi, Gotzmann theorems for exterior algebras and combinatorics, J. Algebra 191 (1997), no. 1, 174–211. MR 1444495, DOI 10.1006/jabr.1996.6903
Anders Björner, The homology and shellability of matroids and geometric lattices, Matroid applications, Encyclopedia Math. Appl., vol. 40, Cambridge Univ. Press, Cambridge, 1992, pp. 226–283. MR 1165544, DOI 10.1017/CBO9780511662041.008
Anders Björner and Günter M. Ziegler, Broken circuit complexes: factorizations and generalizations, J. Combin. Theory Ser. B 51 (1991), no. 1, 96–126. MR 1088629, DOI 10.1016/0095-8956(91)90008-8
Paul H. Edelman and Victor Reiner, Free hyperplane arrangements between $A_{n-1}$ and $B_n$, Math. Z. 215 (1994), no. 3, 347–365. MR 1262522, DOI 10.1007/BF02571719
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
Michael Falk, The minimal model of the complement of an arrangement of hyperplanes, Trans. Amer. Math. Soc. 309 (1988), no. 2, 543–556. MR 929668, DOI 10.1090/S0002-9947-1988-0929668-7
Ralph Fröberg, Determination of a class of Poincaré series, Math. Scand. 37 (1975), no. 1, 29–39. MR 404254, DOI 10.7146/math.scand.a-11585
Toshitake Kohno, Série de Poincaré-Koszul associée aux groupes de tresses pures, Invent. Math. 82 (1985), no. 1, 57–75 (French). MR 808109, DOI 10.1007/BF01394779
Toshitake Kohno, Série de Poincaré-Koszul associée aux groupes de tresses pures, Invent. Math. 82 (1985), no. 1, 57–75 (French). MR 808109, DOI 10.1007/BF01394779
Peter Orlik and Hiroaki Terao, Arrangements of hyperplanes, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin, 1992. MR 1217488, DOI 10.1007/978-3-662-02772-1
Brad Shelton and Sergey Yuzvinsky, Koszul algebras from graphs and hyperplane arrangements, J. London Math. Soc. (2) 56 (1997), no. 3, 477–490. MR 1610447, DOI 10.1112/S0024610797005553
Günter M. Ziegler, Matroid representations and free arrangements, Trans. Amer. Math. Soc. 320 (1990), no. 2, 525–541. MR 986703, DOI 10.1090/S0002-9947-1990-0986703-7