Lower central series and free resolutions of hyperplane arrangements

If M is the complement of a hyperplane arrangement, and A = H*(M, k) is the cohomology ring of M over a field k of characteristic 0, then the ranks, phi(k), of the lower central series quotients of pi(1)(M) can be computed from the Betti numbers, b(ii) = dim Tor(i)(A) (k,k)(i), of the linear strand in a minimal free resolution of k over A. We use the Cartan-Eilenberg change of rings spectral sequence to relate these numbers to the graded Betti numbers, b(ij)' = dimTor(i)(E)(A, k)(j), of a minimal resolution of A over the exterior algebra E. From this analysis, we recover a formula of Falk for phi(3), and obtain a new formula for phi(4). The exact sequence of low-degree terms in the spectral sequence allows us to answer a question of Falk on graphic arrangements, and also shows that for these arrangements, the algebra A is Koszul if and only if the arrangement is supersolvable. We also give combinatorial lower bounds on the Betti numbers, b(i,i+1)', of the linear strand of the free resolution of A over E; if the lower bound is attained for i = 2, then it is attained for all i greater than or equal to 2. For such arrangements, we compute the entire linear strand of the resolution, and we prove that all components of the first resonance variety of A are local. For graphic arrangements ( which do not attain the lower bound, unless they have no braid subarrangements), we show that b(i,i+1)' is determined by the number of triangles and K-4 subgraphs in the graph.