On matroids and Orlik-Solomon algebras
Suppose \(\mathcal A\) is a finite set of hyperplanes in a fixed vector space \(V\) over a field \(\mathbb K,\) with intersection poset \(L\) and Orlik-Solomon algebra \(A({\mathcal A}).\) If \(\mathcal A\) is a central arrangement, then \(L\) is a geometric lattice, and is the lattice of flats of a simple matroid with ground set \(\mathcal A.\) In general the intersection poset \(L\) is a geometric semi-lattice, as introduced by \textit{M. L. Wachs} and \textit{J. W. Walker} [Order 2, 367--385 (1986; Zbl 0589.06005)]. There is a simple process, called coning, by which the arrangement \(\mathcal A\) is replaced by a central arrangement \(c{\mathcal A}\) in \(V \oplus {\mathbb K}\) that includes an additional hyperplane (``at infinity'') \(V \oplus 0.\) Central arrangements can be deconed once a hyperplane is specified to be the hyperplane at infinity. This process is reflected in the intersection posets: every geometric semi-lattice is a subposet of a geometric lattice with one additional atom \(a,\) as the set of elements not greater than \(a\). The category of geometric semi-lattices is equivalent to the category of pointed geometric lattices. On the algebraic side, the Orlik-Solomon algebras fit into a split short exact sequence \(0 \to A({\mathcal A}) \to A(c{\mathcal A}) \to A({\mathcal A})[-1] \to 0.\) This sequence is extracted from an acyclic chain complex \((A(c{\mathcal A}),\partial).\) In the paper under review this construction is carried out at the more fundamental level of matroids. The author gives an axiomatic definition of a ``quasi-matroid,'' to model the intrinsic combinatorics of non-central arrangements. A quasi-matroid consists of a simplicial complex \(\mathcal E\) and a matroid subcomplex \(\mathcal I,\) subject to a single axiom. An arrangement \(\mathcal A\) determines a quasi-matroid, with \({\mathcal E}\) recording the sets of hyperplanes with nonempty intersection, and \(\mathcal I\) consisting of the independent sets of hyperplanes. One can define a notion of flat, and the poset of flats of a quasi-matroid is a geometric semi-lattice. Each quasi-matroid \(Q\) can be coned to result in a matroid \(M\) with one additional point. Strong maps are defined, and the category of quasi-matroids is equivalent to the category of pointed matroids. There is a direct sum of quasi-matroids which corresponds to the sum in the category of pointed matroids, namely parallel connection. The Orlik-Solomon algebra of a quasi-matroid is defined as for arrangements. It is shown that \(A(Q)\cong \partial A(M)\) where \(M\) is the cone of \(Q\), whence there is a split short exact sequence \(0\to A(Q) \to A(M) \to A(Q) \to 0.\) It is shown that \(A(Q_1) \cong A(Q_2)\) if and only if there is an isomorphism of the chain complexes \((A(M_1),\partial)\) and \((A(M_2),\partial).\) The Orlik-Solomon algebra of a pointed matroid is independent of base point, and \(A(Q_1\oplus Q_2) \cong A(Q_1)\otimes A(Q_2).\) These observations (at the level of pointed matroids) were exploited by \textit{C. J. Eschenbrenner} and the reviewer in [J. Algebr. Comb. 10, 189-199 (1999; Zbl 0955.52010)] to show the existence of matroids with isomorphic Orlik-Solomon algebras but different Tutte polynomials.
scientific article; zbMATH DE number 2121815