Annals of Combinatorics 1 (1997) 135-157

Arrangements 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,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 H1(A,dω) as a function of ω We describe the set R1(A)={ω | H1(A(M),dω)\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 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 R1(A). We demonstrate with several examples that R1 is easily calculable, may contain nonlocal components, and that combinatorial properties of R1(A) are often sufficient to distinguish nonisomorphic rank three Orlik-Solomon algebras.

Keywords: arrangement, matroid, Orlik-Solomon algebra, Tutte polynomial, hypergeometric function, local system


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