Arrangements and Cohomology

To a matroid M is associated a graded commutative algebra A A M , the OrlikSolomon algebra of M. Motivated by its role in the construction of generalized hypergeometric functions, we study the cohomology H A dω of A M with coboundary map dω given by multiplication by a fixed element ω of A1. 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 dω as a function of ω. We describe the set R 1 A ω H1 A M dω 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 non-local components, and that combinatorial properties of R 1 A are often sufficient to distinguish non-isomorphic rank three Orlik-Solomon algebras.