Homological properties of Orlik–Solomon algebras

Abstract

The Orlik–Solomon algebra of a matroid can be considered as a quotient ring over the exterior algebra E. At first, we study homological properties of E-modules as e.g., complexity, depth and regularity. In particular, we consider modules with linear injective resolutions. We apply our results to Orlik–Solomon algebras of matroids and give formulas for the complexity, depth and regularity of such rings in terms of invariants of the matroid. Moreover, we characterize those matroids whose Orlik–Solomon ideal has a linear projective resolution and compute in these cases the Betti numbers of the ideal.