If A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G = π1(X) are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring A = H(X, k), viewed as a module over the exterior algebra E on A: θk(G) = dimk Tor E k−1(A, k)k , for k ≥ 2, where k is a field of characteristic 0. The Chen ranks conjecture asserts ...