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.

27 pages, minor modifications

References in corpus (3)

• Sheaf Cohomology and Free Resolutions over Exterior Algebras
• Lower central series and free resolutions of hyperplane arrangements
• Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence

Cited by in corpus (5)

• Broken circuit complexes and hyperplane arrangements
• Flawlessness of $h$-vectors of broken circuit complexes
• Exterior depth and exterior generic annihilator numbers
• Broken circuit complexes of series-parallel networks
• Note on resonance varieties