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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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##### References:
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