Abstract
The purpose of this expository article is to survey some results and applications of free resolutions related to hyperplane arrangements. We include some computational examples and open problems.
Chapter PDF
Similar content being viewed by others
Mathematics Subject Classification (2000)
Keywords
References
David J. Anick, On the homology of associative algebras, Trans. Amer. Math. Soc. 296 (1986), no. 2, 641–659. MR846601 (87i:16046)
Annetta Aramova, Luchezar L. Avramov, and Jürgen Herzog, Resolutions of monomial ideals and cohomology over exterior algebras, Trans. Amer. Math. Soc. 352 (2000), no. 2, 579–594. MR1603874 (2000c:13021)
Annetta Aramova, Jürgen Herzog, and Takayuki Hibi, Gotzmann theorems for exterior algebras and combinatorics, J. Algebra 191 (1997), no. 1, 174–211. MR1444495 (98c:13025)
Luchezar L. Avramov, Local algebra and rational homotopy, Algebraic homotopy and local algebra (Luminy, 1982), Astérisque, vol. 113, Soc. Math. France, Paris, 1984, pp. 15-43. MR749041 (85j:55021)
Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473–527. MR1322847 (96k:17010)
Anders Björner and Günter M. Ziegler, Broken circuit complexes: factorizations and generalizations, J. Combin. Theory Ser. B 51 (1991), no. 1, 96–126. MR1088629 (92b:52027)
N. Bourbaki, Éléments de mathématique. Algébre. Chapitre 10. Algébre homologique, Springer Verlag, Berlin, 2007, Reprint of the 1980 original [Masson, Paris; MR0610795]. MR2327161
Egbert Brieskorn, Sur les groupes de tresses [d’aprés V. I. Arnol’d], Séminaire Bourbaki, 24éme année (1971/1972), Exp. No. 401, Springer, Berlin, 1973, pp. 21–44. Lecture Notes in Math., Vol. 317. MR0422674 (54 #10660)
Kuo-Tsai Chen, Integration in free groups, Ann. of Math. (2) 54 (1951), 147–162. MR0042414 (13,105c)
-, Extension of C ∞ function algebra by integrals and Malcev completion of π 1, Advances in Math. 23 (1977), no. 2, 181–210. MR0458461 (56 #16664)
Graham Denham and Henry K. Schenck, A spectral sequence stratification of cohomology jump loci, preprint.
Graham Denham and Alexander I. Suciu, On the homotopy Lie algebra of an arrangement, Michigan Math. J. 54 (2006), no. 2, 319–340. MR2252762 (2007f:17039)
Alexandru Dimca, Singularities and topology of hypersurfaces, Universitext, Springer Verlag, New York, 1992. MR1194180 (94b:32058)
Alexandru Dimca and Sergey Yuzvinsky, Lectures on Orlik-Solomon algebras, this volume, 83–110.
John A. Eagon and Victor Reiner, Resolutions of Stanley-Reisner rings and Alexander duality, J. Pure Appl. Algebra 130 (1998), no. 3, 265–275. MR1633767 (99h:13017)
David Eisenbud, The geometry of syzygies, Graduate Texts in Mathematics, vol. 229, Springer Verlag, New York, 2005, A second course in commutative algebra and algebraic geometry. MR2103875 (2005h:13021)
David Eisenbud, Gunnar Fløystad, and Frank-Olaf Schreyer, Sheaf cohomology and free resolutions over exterior algebras, Trans. Amer. Math. Soc. 355 (2003), no. 11, 4397–4426 (electronic). MR1990756 (2004f:14031)
David Eisenbud, Sorin Popescu, and Sergey Yuzvinsky, Hyperplane arrangement cohomology and monomials in the exterior algebra, Trans. Amer. Math. Soc. 355 (2003), no. 11, 4365–4383 (electronic). MR1986506 (2004g:52036)
Michael Falk, Geometry and combinatorics of resonant weights, this volume, 155–176.
Michael Falk and Richard Randell, The lower central series of a fiber-type arrangement, Invent. Math. 82 (1985), no. 1, 77–88. MR808110 (87c:32015b)
-, On the homotopy theory of arrangements. II, Arrangements—Tokyo 1998, Adv. Stud. Pure Math., vol. 27, Kinokuniya, Tokyo, 2000, pp. 93–125. MR1796895 (2002b:32044)
R. Fox and L. Neuwirth, The braid groups, Math. Scand. 10 (1962), 119–126. MR0150755 (27 #742)
R. Fröberg, Koszul algebras, Advances in commutative ring theory (Fez, 1997), Lecture Notes in Pure and Appl. Math., vol. 205, Dekker, New York, 1999, pp. 337–350. MR1767430 (2001i:16046)
R. Fröberg and C. Löfwall, Koszul homology and Lie algebras with application to generic forms and points, Homology Homotopy Appl. 4 (2002), no. 2, part 2, 227–258 (electronic), The Roos Festschrift volume, 2. MR1918511 (2003g:13018)
Daniel R. Grayson and Michael E. Stillman, Macaulay 2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/.
Toshitake Kohno, On the holonomy Lie algebra and the nilpotent completion of the fundamental group of the complement of hypersurfaces, Nagoya Math. J. 92 (1983), 21–37. MR726138 (85d:14032)
-, Série de Poincaré-Koszul associée aux groupes de tresses pures, Invent. Math. 82 (1985), no. 1, 57–75. MR808109 (87c:32015a)
Clas Löfwall, On the subalgebra generated by the one-dimensional elements in the Yoneda Ext-algebra, Algebra, algebraic topology and their interactions (Stockholm, 1983), Lecture Notes in Math., vol. 1183, Springer, Berlin, 1986, pp. 291–338. MR846457 (88f:16030)
W. S. Massey, Completion of link modules, Duke Math. J. 47 (1980), no. 2, 399–420. MR575904 (81g:57004)
John W. Milnor and John C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–264. MR0174052 (30 #4259)
P. Orlik and H. Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften, no. 300, Springer Verlag, 1992.
Stefan Papadima and Alexander I. Suciu, Chen Lie algebras, Int. Math. Res. Not. (2004), no. 21, 1057–1086. MR2037049 (2004m:17043)
-, Algebraic invariants for right-angled Artin groups, Math. Ann. 334 (2006), no. 3, 533–555. MR2207874 (2006k:20078)
-, When does the associated graded Lie algebra of an arrangement group decompose?, Comment. Math. Helv. 81 (2006), no. 4, 859–875. MR2271225 (2007h:52028)
Stefan Papadima and Sergey Yuzvinsky, On rational K[π, 1]_spaces and Koszul algebras, J. Pure Appl. Algebra 144 (1999), no. 2, 157–167. MR1731434 (2000k:55017)
Irena Peeva, Hyperplane arrangements and linear strands in resolutions, Trans. Amer. Math. Soc. 355 (2003), no. 2, 609–618 (electronic). MR1932716 (2004c:13020)
Alexander Polishchuk and Leonid Positselski, Quadratic algebras, University Lecture Series, vol. 37, American Mathematical Society, Providence, RI, 2005. MR2177131 (2006f:16043)
J. Scott Provan and Louis J. Billera, Decompositions of simplicial complexes related to diameters of convex polyhedra, Math. Oper. Res. 5 (1980), no. 4, 576–594. MR593648 (82c:52010)
Jan-Erik Roos, The homotopy Lie algebra of a complex hyperplane arrangement is not necessarily finitely presented, Experimental Mathematics 17 (2008), no. 2, 129–143.
Jan-Erik Roos and Bernd Sturmfels, A toric ring with irrational Poincaré-Betti series, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 2, 141–146. MR1646972 (2000e:14042)
Henry K. Schenck and Alexander I. Suciu, Lower central series and free resolutions of hyperplane arrangements, Trans. Amer. Math. Soc. 354 (2002), no. 9, 3409–3433 (electronic). MR1911506 (2003k:52022)
-, Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence, Trans. Amer. Math. Soc. 358 (2006), no. 5, 2269–2289 (electronic). MR2197444 (2007a:52026)
Brad Shelton and Sergey Yuzvinsky, Koszul algebras from graphs and hyperplane arrangements, J. London Math. Soc. (2) 56 (1997), no. 3, 477–490. MR1610447 (99c:16044)
Alexander I. Suciu, Fundamental groups of line arrangements: enumerative aspects, Advances in algebraic geometry motivated by physics (Lowell, MA, 2000), Contemp. Math., vol. 276, Amer. Math. Soc., Providence, RI, 2001, pp. 43–79. MR1837109 (2002k:14029)
Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 269–331 (1978). MR0646078 (58 #31119)
Hiroaki Terao, Modular elements of lattices and topological fibration, Adv. in Math. 62 (1986), no. 2, 135–154. MR865835 (88b:32032)
S. Yuzvinsky, Orlik-Solomon algebras in algebra and topology, Uspekhi Mat. Nauk 56 (2001), no. 2(338), 87–166. MR1859708 (2002i:14047)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Denham, G. (2009). Homological Aspects of Hyperplane Arrangements. In: El Zein, F., Suciu, A.I., Tosun, M., Uludağ, A.M., Yuzvinsky, S. (eds) Arrangements, Local Systems and Singularities. Progress in Mathematics, vol 283. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0209-9_2
Download citation
DOI: https://doi.org/10.1007/978-3-0346-0209-9_2
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0346-0208-2
Online ISBN: 978-3-0346-0209-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)