# zbMATH — the first resource for mathematics

Commutative algebras for arrangements. (English) Zbl 0801.05019
Let $$V$$ be a vector space of dimension $$l$$ over some field $$\mathbb{K}$$. A hyperplane $$H$$ is a vector subspace of codimension one. An arrangement $$\mathcal A$$ is a finite collection of hyperplanes in $$V$$. Let $M({\mathcal A})= V- \bigcup_{H\in {\mathcal A}} H$ be the complement of the hyperplanes. Let $$V^*$$ be the dual space of $$V$$. Each hyperplane $$H\in {\mathcal A}$$ is the kernel of a linear form $$\alpha_ H\in V^*$$, defined up to a constant. The product $Q({\mathcal A})= \prod_{H\in {\mathcal A}} \alpha_ H$ is called a defining polynomial of $$\mathcal A$$. A subarrangement $${\mathcal B}\subseteq {\mathcal A}$$ is called independent if $$\bigcap_{H\in {\mathcal B}}H$$ has codimension $$|{\mathcal B}|$$, the cardinality of $$\mathcal B$$. In a special lecture at the Japan Mathematical Society in 1992, Aomoto sugested the study of the graded $$\mathbb{K}$$-vector space $\text{AO}({\mathcal A})= \sum_{{\mathcal B}} \mathbb{K} Q({\mathcal B})^{-1},\quad{\mathcal B}\text{ independent}.$ It appears as the top cohomology group of a certain ‘twisted’ de Rham chain complex. When $$\mathbb{K}= \mathbb{R}$$, he conjectured that the dimension of $$\text{AO}({\mathcal A})$$ is equal to the number of connected components (chambers) of $$M({\mathcal A})$$, which he proved for generic arrangements. In this paper, we prove this conjecture in general.
Reviewer: H.Terao

##### MSC:
 05B35 Combinatorial aspects of matroids and geometric lattices 13C15 Dimension theory, depth, related commutative rings (catenary, etc.) 51M20 Polyhedra and polytopes; regular figures, division of spaces
Full Text:
##### References:
 [1] DOI: 10.1007/BF02483913 · Zbl 0484.06014 · doi:10.1007/BF02483913 [2] Sügaku 45 pp 208– (1993) [3] Lecture Notes in Math. 317 pp 21– (1973) [4] DOI: 10.1007/BF01078470 · Zbl 0619.33004 · doi:10.1007/BF01078470 [5] Grundlehren der math. Wiss. 300 (1992) [6] DOI: 10.1007/BF01392549 · Zbl 0432.14016 · doi:10.1007/BF01392549 [7] Amer. Math. Soc. 90 pp 147– (1989) [8] Memoirs Amer. Math. Soc. 154 (1975)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.