  We show that if X is the complement of a complex hyperplane arrangement, then
the homology of X has linear free resolution as a module over the exterior
algebra on the first cohomology of X. We study invariants of X that can be
deduced from this resolution. A key ingredient is a result of Aramova, Avramov,
and Herzog [2000] on resolutions of monomial ideals in the exterior algebra. We
give a new conceptual proof of this result.
