  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,
$\phi_k$, of the lower central series quotients of $\pi_1(M)$ can be computed
from the Betti numbers, $b_{ii}=\dim_{\k} \Tor^A_i(\k,\k)_i$, 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, $b'_{ij}=\dim_{\k} \Tor^E_i(A,\k)_j$, 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 iff 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 $i \ge 2$. 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 $K_4$ subgraphs
in the graph.
