Line-closed matroids, quadratic algebras, and formal arrangements

Let G be a matroid on ground set A. The Orlik-Solomon algebra A(G) is the quotient of the exterior algebra ℬ on A by the ideal ℐ generated by circuit boundaries. The quadratic closure A(G) of A(G) is the quotient of ℬ by the ideal generated by the degree-two component of ℐ. We introduce the notion of the nbb set in G, determined by a linear order on A, and show that the corresponding monomials are linearly independent in the quadratic closure A(G). As a consequence, A(G) is a quadratic algebra only if G is line-closed. An example of S. Yuzvinsky proves the converse false. [G. Denham and S. Yuzvinsky, Adv. in Appl. Math. 28, 2002, doi:10.1006/aama.2001.0779]. These results generalize to the degree r closure of A(G). The motivation for studying line-closed matroids grew out of the study of formal arrangements. This is a geometric condition necessary for A to be free and for the complement M of A to be a K(π, 1) space. Formality of A is also necessary for A(G) to be a quadratic algebra. We clarify the relationship between formality, line-closure, and other matroidal conditions related to formality. We give examples to show that line-closure of G is not necessary or sufficient for M to be a K(π, 1) or for A to be free.