The algebra of glued polynomials on an arrangement of hyperplanes
type: Item
instance of: scholarly article
title: The algebra of glued polynomials on an arrangement of hyperplanes
zbMATH Open document ID: 0817.13010
DOI: 10.1016/0022-4049(94)90065-5
author: Christopher O'Connor
author: Sergey Yuzvinsky
published in: Journal of Pure and Applied Algebra
publication date: 1994-12-08T00:00:00Z
review text: Let \(F\) be an infinite field and \({\mathcal A} = \{H_ 1, \dots, H_ n\}\) a nonempty set (arrangement) of hyperplanes in an \(F\)-vector space \(V\) of dimension \(l\). Let \(S\) be the symmetric algebra of \(V^*\). Associated to \({\mathcal A}\) are two graded algebras which are graded \(S\)-modules: The coordinate ring \(R\) of the affine variety \(\bigcup^ n_{i=1} H_ i\), the second one \(^ +R\) is the pull-back of the system of the coordinate rings \(R(X)\) of various intersections \(X\) of the hyperplanes and the natural projections \(\rho_{X,Y} : R(X) \to R(Y)\). In fact \(^ +R\) is the seminormalisation of \(R\). The study of the Hilbert-Poincaré series of \(^ +R\) constitutes a central point in this paper. As a consequence the authors give examples of arrangements with non Cohen-Macaulay \(^ +R\) (of course \(R\) is Cohen-Macaulay). They suggest that all arrangements of a given dimension and of a given number of hyperplanes can be broken in several nontrivial classes depending on the depth of \(^ +R\).
Mathematics Subject Classification ID: 13F45
Mathematics Subject Classification ID: 14J70
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 14N10
zbMATH DE Number: 681811
zbMATH Keywords: seminormal ring
zbMATH Keywords: arrangement of hyperplanes
zbMATH Keywords: Hilbert-Poincaré series
zbMATH Keywords: depth
reviewed by: Marcel Morales
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)90065-5
OpenAlex ID: W2086926515
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description: scientific article; zbMATH DE number 681811