Homological properties of Orlik-Solomon algebras

TL;DR

This paper investigates the homological properties of Orlik-Solomon algebras associated with matroids, providing formulas for invariants like complexity, depth, and regularity, and characterizing matroids with linear projective resolutions.

Contribution

It offers new formulas and characterizations for the homological invariants of Orlik-Solomon algebras in relation to matroid properties.

Findings

Formulas for complexity, depth, and regularity of Orlik-Solomon algebras.

Characterization of matroids with linear projective resolutions.

Computation of Betti numbers for specific cases.

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.