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 of characteristic 0, then the ranks, φk, of the lower central series quotients of π1(M) can be computed from the Betti numbers, bii = dimTor A i (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 = dimTori (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 sub-arrangements), we show that b′i,i+1 is determined by the number of triangles and K4 subgraphs in the graph.