Abstract: For a complex hyperplane arrangement, the cohomology ring of the complement depends only on the combinatorics of the arrangement, and this cohomology ring has an explicit expression as the quotient of an exterior algebra E. One may study such rings by studying their resonance varieties, collections of points corresponding to (nontrivial) zero divisors. Viewing these cohomology rings as modules over E, the Chen Ranks Theorem expresses some of the graded Betti numbers of the ring in terms of the first resonance variety. Inspired by this result, I will define a collection of varieties consisting of the points in the resonance varieties that "contribute to the Betti numbers," and state a theorem I proved computing these for all square-free E-modules. I will conclude with an application of the result to cohomology rings of hyperplane arrangements.