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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
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