Equivariant Cohomology and Conditional Oriented Matroids

We give a cohomological interpretation of the Heaviside filtration on the Varchenko–Gelfand ring of a pair $({\mathcal{A}},{\mathcal{K}})$, where ${\mathcal{A}}$ is a real hyperplane arrangement and ${\mathcal{K}}$ is a convex open subset of the ambient vector space. This builds on work of the first author, who studied the filtration from a purely algebraic perspective, as well as work of Moseley, who gave a cohomological interpretation in the special case where ${\mathcal{K}}$ is the ambient vector space. We also define the Gelfand–Rybnikov ring of a conditional oriented matroid, which simultaneously generalizes the Gelfand–Rybnikov ring of an oriented matroid and the aforementioned Varchenko–Gelfand ring of a pair. We give purely combinatorial presentations of the ring, its associated graded, and its Rees algebra.