Annals of Combinatorics 1 (1997) 135-158 Annals of (original) (raw)

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, do,) ofA (M) with coboundary map do, given by multiplication by a fixed element o) ofA l. Using a description of decomposable relations in A, we construct new examples of "resonant" values of co, and give a precise calculation of H 1 (A, do,) as a function ofo). We describe the set Rt(A) = {co [ Hl(A(M),do,) ~ 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 R1 (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! (,4). We demonstrate with several examples that Rz is easily calculable, may contain nonlocal components, and that combinatorial properties of Rl (A) are often sufficient to distinguish nonisomorphic rank three Orlik-Solomon algebras.