Free arrangements and relation spaces
type: Item
instance of: scholarly article
title: Free arrangements and relation spaces
zbMATH Open document ID: 0802.52010
DOI: 10.1007/BF02574365
author: Hiroaki Terao
author: Keith Brandt
published in: Discrete \& Computational Geometry
publication date: 1994-08-10T00:00:00Z
full work available at URL: https://eudml.org/doc/131317
review text: For \(\{x_ 1, \dots, x_ n\}\) a basis of the dual vector \(n\)-space \((V^ n)^*\) and \(S\) a symmetric algebra of \((V^ n)^*\), isomorphic to the polynomial algebra \(\mathbb{K} [x_ 1, \dots, x_ n]\), each hyperplane \(H \subset V^ n\) has a defining form \(\alpha_ H = a_ 1x_ 1 + \cdots + a_ nx_ n\) with \(\text{ker} (\alpha_ H) = H\) (unique up to a constant). Thus, an arrangement \(A\) of hyperplanes in \(V^ n\) can be described by the product of such forms. So let \(Q = Q(A) = \prod_{H \in A} \alpha_ H\), and let \(D(A)\) be the \(S\)-module consisting of all deviations \(\Theta : S \to S\) such that \(\Theta (Q)\) is a multiple of \(Q\). If \(D(A)\) is a free \(S\)-module, then the arrangement \(A\) is said to be free. On the other hand, \(A\) is formal if all linear dependencies among the defining forms of the hyperplanes of \(A\) are generated by dependencies corresponding to the 2-codimensional subspaces. \textit{S. Yuzvinsky} [Trans. Am. Math. Soc. 335, No. 1, 231-244 (1993; Zbl 0768.05019)] proved that free arrangements are formal. In the present paper it is shown that if \(A\) is free, then \(A\) is even \(k\)-formal (with \(2 \leq k \leq r - 1\), where \(r\) is the codimension of the common intersection of all the hyperplanes from \(A)\). For this assertion, the more general \(k\)-formal arrangements are introduced, where, as starting point, \(A\) is 2-formal if it is formal. (The authors construct also a 3- formal arrangement which is not 2-formal, to distinguish the notion ``\(k\)-formal'' from the notion ``formal'').
reviewed by: Horst Martini
Mathematics Subject Classification ID: 52C35
zbMATH DE Number: 617324
zbMATH Keywords: free arrangements
zbMATH Keywords: formal arrangements
zbMATH Keywords: \(S\)-modules
zbMATH Keywords: polynomial algebra
MaRDI profile type: Publication
cites work: Q3775655
cites work: Q3676243
cites work: Q4003206
cites work: A formula for the characteristic polynomial of an arrangement
cites work: A Family of Regular Maps of Type {6, 6}
cites work: Cohomology of Local Sheaves on Arrangement Lattices
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description: scientific article; zbMATH DE number 617324