Séminaire Lotharingien de Combinatoire, 95B.48 (2026), 12 pp.
Christin Bibby and Peter Ramsey
The Orlik-Solomon Algebra of a Locally Geometric Poset
Abstract.
Motivated by the intersection data of hyperplane arrangements and their generalizations, we describe a matroidal framework for locally geometric posets.
Using this framework, we construct an Orlik-Solomon algebra associated to a locally geometric poset.
This construction generalizes that of geometric semilattices, where in the realizable case the Orlik-Solomon algebra arises as the cohomology ring of a hyperplane arrangement complement.
In our context, the Orlik-Solomon algebra introduced here is the "combinatorial" contribution to the cohomology of a toric or abelian arrangement complement, which interacts with a "topological" contribution from the cohomology of the ambient space.
Using Gröbner basis theory, we establish a no-broken-circuit basis analogous to that of the classical Orlik-Solomon algebra,
which provides the tools necessary for a deletion-contraction formula, and deeper understanding of the characteristic polynomial.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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