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, φ_k, of the lower central series quotients of π₁ (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 = dim Tor_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 φ₃, and obtain a new formula for φ₄. 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 ≥ 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 K₄ subgraphs in the graph.