Line-closed matroids, quadratic algebras, and formal arrangements

Abstract

Let $G$ be a matroid on ground set \A. The Orlik-Solomon algebra $A(G)$ is the quotient of the exterior algebra \E on \A by the ideal \I generated by circuit boundaries. The quadratic closure $\bar{A}(G)$ of $A(G)$ is the quotient of \E by the ideal generated by the degree-two component of \I. We introduce the notion of \nbb set in $G$, determined by a linear order on \A, and show that the corresponding monomials are linearly independent in the quadratic closure $\bar{A}(G)$. As a consequence, $A(G)$ is a quadratic algebra only if $G$ is line-closed. An example of S.~Yuzvinsky proves the converse false. 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 $K(\pi,1)$ 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 $K(\pi,1)$, or for \A to be free.