Annihilators of Ideals of Exterior Algebras

The Orlik-Solomon algebra A of a matroid is isomorphic to the quotient of an exterior algebra E by a defining ideal I. We find an explicit presentation of the annihilator ideal of I or, equivalently, the E-module dual to A. As an application of that we provide a necessary, combinatorial condition for the algebra A to be quadratic. We show that this is stronger than matroid being line-closed thereby resolving (negatively) a conjecture by Falk. We also show that our condition is not sufficient for the quadraticity.