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 _SM, the square-free module over a commutative polynomial ring 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.