Annals of Combinatorics 1 (1997) 135-157Arrangements and Cohomology Michael Falk Department of Mathematics, Northern Arizona University, Flagstaff, AZ
86011, USA
Received January 28, 1997 AMS Subject Classification: 52B30, 05B35, 13D03, 33C50 Abstract. To a matroid M is associated a graded commutative algebra A=A(M), the Orlik-Solomon algebra of M. Motivated by its role in the construction of generalized hypergeometric functions, we study the cohomology H^{*}(A,d_{ω}) of A(M) with coboundary map d_{ω} given by multiplication by a fixed element ω of A^{1}. Using a description of decomposable relations in A, we construct new examples of "resonant" values of ω, and give a precise calculation of H^{1}(A,d_{ω}) as a function of ω We describe the set R_{1}(A)={ω | H^{1}(A(M),d_{ω})\ne 0}, and use it as a tool in the classification of Orlik-Solomon algebras, with applications to the topology of complex hyperplane complements. We show that R_{1}(A)is a complete invariant of the quadratic closure of A, and show under various hypotheses that one can reconstruct the matroid M, or at least its Tutte polynomial, from the variety R_{1}(A). We demonstrate with several examples that R_{1} is easily calculable, may contain nonlocal components, and that combinatorial properties of R_{1}(A) are often sufficient to distinguish nonisomorphic rank three Orlik-Solomon algebras. Keywords: arrangement, matroid, Orlik-Solomon algebra, Tutte polynomial, hypergeometric function, local system References 1. A. Björner, Topological methods, In: Handbook of Combinatorics, Elsevier, Amsterdam, 1995, pp. 1819–1872. 2. J.E. Blackburn, H.H. Crapo, and D.A. Higgs, A catalogue of combinatorial geometries, Math.~Computation 27 (1973) 155–166. 3. T. Brylawski, Hyperplane reconstruction of the Tutte polynomial of a geometric lattice, Discrete Math. 35 (1981) 25–28. 4. D. Cohen and A. Suciu, The Chen groups of the pure braid group, In: Proceedings of the Cech Centennial Homotopy Theory Conference, B. Cenkl and H. Miller, Eds., Contemporary Mathematics, Vol. 181, Providence, Amer. Math. Soc., 1995, pp. 45–64. 5. C. Eschenbrenner and M. Falk, Tutte polynomials and Orlik-Solomon algebras, preprint. 6. M. Falk, The minimal model of the complement of an arrangement of hyperplanes, Trans. Amer. Math. Soc. 309 (1988) 543–556. 7. M. Falk, The cohomology and fundamental group of a hyperplane complement, In: Singularities, R. Randell, Ed., Contemporary Mathematics, Vol. 90, Providence, Amer. Math. Soc., 1989, pp. 55–72. 8. M. Falk, On the algebra associated with a geometric lattice, Adv. Math. 80 (1989) 152–163. 9. M. Falk, Homotopy types of line arrangements, Invent. Math. 111 (1993) 139–150. 10. M. Falk and R. Randell, On the homotopy theory of arrangements, In: Complex Analytic Singularities, Advanced Studies in Mathematics, Vol. 8, North Holland, 1987, pp. 101–124. 11. I.M. Gelfand and A.V. Zelevinsky, Algebraic and combinatorial aspects of the general theory of hypergeometric functions, Functional Analysis and Appl. 20 (1986) 183–197. 12. E. Viehweg, H. Esnault, and V. Schechtman, Cohomology of local systems on the complement of hyperplanes, Invent. Math. 109 (1992) 557–561. Erratum: ibid. 112 (1993) 447. 13. P. Orlik and L. Solomon, Topology and combinatorics of complements of hyperplanes, Invent. Math. 56 (1980) 167–189. 14. P. Orlik and H. Terao, Arrangements of Hyperplanes, Springer-Verlag, Berlin Heidelberg New York, 1992. 15. J.G. Oxley, Matroid Theory, Oxford University Press, Oxford New York Tokyo, 1992. 16. G. Rybnikov, Private correspondence. 17. G. Rybnikov, On the fundamental group of a complex hyperplane arrangement, DIMACS Technical Reports 13 (1994). 18. V.V. Schechtman, H.~Terao, and A.N. Varchenko, Cohomology of local systems and Kac-Kazhdan condition for singular vectors, J. Pure Appl. Algebra 100 (1995) 93–102. 19. V.V. Schechtman and A.N. Varchenko, Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106 (1991) 139–194. 20. B. Shelton and S. Yuzvinsky, Koszul algebras from graphs and hyperplane arrangements, J. London Math. Soc., to appear. 21. D.J.A. Welsh, Matroid Theory, Academic Press, London, 1976. 22. N. White, Ed., Theory of Matroids, Cambridge University Press, Cambridge, 1986. 23. N. White, Ed., Matroid Applications, Cambridge University Press, Cambridge, 1992. 24. S. Yuzvinsky, Cohomology of Brieskorn-Orlik-Solomon algebras, Comm. Algebra 23 (1995) 5339–5354. 25. G.M. Ziegler, Private communication, 1988. |