Koszul Orlik--Solomon Algebras from Non-supersolvable Arrangements
Abstract
The cohomology ring of the complement of a complex hyperplane arrangement is given by its Orlik--Solomon algebra. It is known that the defining ideal of the Orlik--Solomon algebra has a quadratic Gröbner basis in the standard presentation if and only if the intersection lattice is supersolvable; such algebras are automatically Koszul. In 1997, Shelton and Yuzvinsky posed the question as to whether all Koszul Orlik--Solomon algebras arise from supersolvable arrangements. We answer this question negatively using three related constructions that produce non-supersolvable arrangements whose Orlik--Solomon algebras are Koszul. Moreover, these arrangements may be chosen to be irreducible, realizable over ℚ, and of any rank ≥ 3. Our constructions rely on a result of Falk and Proudfoot which we strengthen and generalize. In two of the three cases, we show non-supersolvability using a corrected form of a result of Ziegler regarding supersolvability of parallel connections. We also construct Koszul Orlik--Terao algebras coming from non-supersolvable arrangements.
Related
- The Orlik--Solomon algebra of a locally geometric poset and cohomology of complex abelian arrangements
- Orlik-Solomon algebras determine the Tutte polynomials of matroids of rank at most three
- Generalized Weight Polynomials of Codes through Flats and Orlik-Solomon Algebras of Matroids
- Monodromy of supersolvable toric arrangements
- Koszul duality for finite-dimensional absolutely Koszul algebras
BibTeX
@misc{le2026koszul,
title = {{Koszul Orlik--Solomon Algebras from Non-supersolvable Arrangements}},
author = {Tuong Le and Chayim Lowen and Jason McCullough},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03836},
eprint = {2609.03836},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03836},
}
RIS
TY - UNPB
TI - Koszul Orlik--Solomon Algebras from Non-supersolvable Arrangements
AU - Tuong Le
AU - Chayim Lowen
AU - Jason McCullough
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03836
UR - https://arxiv.org/abs/2609.03836
AB - The cohomology ring of the complement of a complex hyperplane arrangement is given by its Orlik--Solomon algebra. It is known that the defining ideal of the Orlik--Solomon algebra has a quadratic Gröbner basis in the standard presentation if and only if the intersection lattice is supersolvable; such algebras are automatically Koszul. In 1997, Shelton and Yuzvinsky posed the question as to whether all Koszul Orlik--Solomon algebras arise from supersolvable arrangements. We answer this question negatively using three related constructions that produce non-supersolvable arrangements whose Orlik--Solomon algebras are Koszul. Moreover, these arrangements may be chosen to be irreducible, realizable over ℚ, and of any rank ≥ 3. Our constructions rely on a result of Falk and Proudfoot which we strengthen and generalize. In two of the three cases, we show non-supersolvability using a corrected form of a result of Ziegler regarding supersolvability of parallel connections. We also construct Koszul Orlik--Terao algebras coming from non-supersolvable arrangements.
ER -