A Leray model for the Orlik-Solomon algebra

TL;DR

This paper introduces a combinatorial generalization of Leray models for hyperplane arrangement complements, creating a new algebraic framework that interpolates between known structures and extends to non-realizable matroids.

Contribution

It constructs a new bigraded differential-graded algebra for matroids, generalizing the Morgan model and connecting Chow rings with Orlik-Solomon algebras.

Findings

The algebra is quasi-isomorphic to the classical Orlik-Solomon algebra.

A monomial basis is constructed via a Gröbner basis.

The model interpolates between Chow rings and Orlik-Solomon algebras.

Abstract

We construct a combinatorial generalization of the Leray models for hyperplane arrangement complements. Given a matroid and some combinatorial blowup data, we give a presentation for a bigraded (commutative) differential-graded algebra. If the matroid is realizable over $\mathbb{C}$, this is the familiar Morgan model for a hyperplane arrangement complement, embedded in a blowup of projective space. In general, we obtain a cdga that interpolates between the Chow ring of a matroid and the Orlik-Solomon algebra. Our construction can also be expressed in terms of sheaves on combinatorial blowups of geometric lattices. As a key technical device, we construct a monomial basis via a Gr\"obner basis for the ideal of relations. Combining these ingredients, we show that our algebra is quasi-isomorphic to the classical Orlik-Solomon algebra of the matroid.