Resonance Varieties and Free Resolutions Over an Exterior Algebra
Abstract
If E is an exterior algebra on a finite dimensional vector space and M is a graded Emodule, 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 Mis 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.
Degree
Ph.D.
Advisors
Walther, Purdue University.
Subject Area
Mathematics
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