Supersolvability–Koszulness conjecture for Orlik–Solomon algebras

Let $M$ be a matroid, or let $\mathcal{A}$ be a hyperplane arrangement, and let $A(M)$ , respectively $A(\mathcal{A})$ , denote its Orlik–Solomon algebra. A matroid or arrangement is supersolvable if its underlying lattice has the corresponding supersolvability property, and an Orlik–Solomon algebra is Koszul if it is a Koszul algebra. Supersolvability–Koszulness conjecture. A matroid (an arrangement) is supersolvable if and only if its Orlik–Solomon algebra is Koszul. The source states that this conjecture is verified in the paper for matroids whose Stanley–Reisner ideal of the broken circuit complex is a complete intersection; its general status is not established in the supplied text.

The conjecture, recorded in the 2012 literature, asks whether supersolvability of a matroid or arrangement is exactly equivalent to Koszulness of its Orlik–Solomon algebra. The general converse was explicitly open in 2018.

Known results

Supersolvable arrangements have Koszul Orlik–Solomon algebras (Shelton–Yuzvinsky, 1997).
The equivalence holds for ordered matroids with pairwise disjoint minimal broken circuits.
It holds for cones over Dirichlet arrangements.
Complete-intersection hypotheses yield further restricted equivalences for matroids and arrangements.

2026 non-supersolvable-arrangement paper

A 2026 arXiv paper reports that the standard Orlik–Solomon presentation has a quadratic Gröbner basis exactly in the supersolvable case, and that this implies Koszulness. The supplied record does not give a theorem establishing Koszulness for a non-supersolvable example, so it provides no confirmed counterexample or general resolution.

Current status (as of September 2026): Restricted classes are settled, but the general supersolvability–Koszulness equivalence remains open.

Supersolvability–Koszulness conjecture for Orlik–Solomon algebras

Primary source

Progress summary

Solutions 0