# Resonance Varieties and Free Resolutions Over an Exterior Algebra

If

*E*is an exterior algebra on a finite dimensional vector space and*M*is a graded*E*-module, the relationship between the resonance varieties of*M*and the minimal free resolution of*M*is studied. In the context of Orlik–Solomon algebras, we give a condition under which elements of the second resonance variety can be obtained. We show that the resonance varieties of a general*M*are invariant under taking syzygy modules up to a shift. As corollary, it is shown that certain points in the resonance varieties of*M*can be determined from minimal syzygies of a special form, and we define syzygetic resonance varieties to be the subvarieties consisting of such points. The (depth one) syzygetic resonance varieties of a square-free module*M*over*E*are shown to be subspace arrangements whose components correspond to graded shifts in the minimal free resolution of*, the square-free module over a commutative polynomial ring*_{S}M*S*corresponding to*M*. Using this, a lower bound for the graded Betti numbers of the square-free module*M*is given. As another application, it is shown that the minimality of certain syzygies of Orlik–Solomon algebras yield linear subspaces of their (syzygetic) resonance varieties and lower bounds for their graded Betti numbers.