来自非超可解构形的Koszul Orlik–Solomon代数
Koszul Orlik--Solomon Algebras from Non-supersolvable Arrangements
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中文总结 AI 辅助
该研究否定了所有Koszul Orlik–Solomon代数都来自超可解构形的猜想，通过构造秩≥3、在ℚ上可实现的不可约非超可解构形，得到了对应的Koszul Orlik–Solomon代数。
中文摘要 AI 辅助
复超平面构形补集的上同调环由其Orlik–Solomon代数给出。已知当且仅当交格是超可解时，Orlik–Solomon代数的定义理想在标准表示下具有二次Gröbner基，此类代数自动为Koszul。1997年，Shelton和Yuzvinsky提出问题：是否所有Koszul Orlik–Solomon代数都来自超可解构形？我们通过三种相关构造否定该问题，这些构造可生成非超可解构形，其Orlik–Solomon代数为Koszul，且构形可选不可约、在ℚ上可实现，秩≥3。我们的构造基于Falk和Proudfoot的结果，对其进行了强化和推广。在三种情况中的两种，我们利用Ziegler关于平行连接超可解性结果的修正形式证明非超可解性。我们还构造了来自非超可解构形的Koszul Orlik–Terao代数。
英文摘要
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 $\mathbb{Q}$, and of any rank $\geq 3$. 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.
发表机构
- Princeton University（普林斯顿大学）
- Iowa State University（爱荷华州立大学）
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