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.

21 pages, 6 figures. To appear in Advances in Applied Mathematics. Text has been shortened and substantially revised to clarify status of problem and several other important points. Theorem 2.4 has been made more precise and Corollary 2.20 has been strengthened