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On the algebra associated with a geometric lattice. (English) Zbl 0733.05026
Summary: Let L be a geometric lattice. Following P. Orlik and L. Solomon [Invent. Math. 56, 167-189 (1980; Zbl 0432.14016)], we associated with L a graded commutative algebra A(L). We introduce a new invariant \(\psi\) of the algebra A(L) which suffices to distinguish algebras for which all other known invariants coincide. This result is applied to the study of arrangements of complex hyperplanes, with L being the intersection lattice. In this case A(L) is isomorphic to the cohomology algebra of the associated hyperplane complement. The goal is to find examples of arrangements with non-isomorphic lattices but homotopy equivalent complements. The invariant introduced here effectively narrows the list of candidates. Nevertheless, we exhibit combinatorially inequivalent arrangements for which all known invariants, including \(\psi\), coincide.

MSC:
05B35 Combinatorial aspects of matroids and geometric lattices
06C10 Semimodular lattices, geometric lattices
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[1] Falk, M, The cohomology and fundamental group of a hyperplane complement, (), 55-72
[2] Falk, M; Randell, R, The lower central series of a fiber-type arrangement, Invent. math., 82, 77-88, (1985) · Zbl 0574.55010
[3] Falk, M; Randell, R, On the homotopy theory of arrangements, complex analytic singularities, Adv. stud. pure math., 8, 101-124, (1986)
[4] Orlik, P; Solomon, L, Combinatorics and topology of complements of hyperplanes, Invent. math., 56, 167-189, (1980) · Zbl 0432.14016
[5] Terao, H, On modular elements and topological fibration, Adv. in math., 62, 135-154, (1986) · Zbl 0612.05019
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