Cohen-Macaulay seminormalizations of unions of linear subspaces
type: Item
instance of: scholarly article
title: Cohen-Macaulay seminormalizations of unions of linear subspaces
zbMATH Open document ID: 0705.13011
DOI: 10.1016/0021-8693(90)90139-F
author: Sergey Yuzvinzky
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: Let \(R\) be the coordinate ring of a finite union of subspaces of an affine space and \(S\) be its seminormalization. If \(X\) is the corresponding intersection poset and \(\mathcal A\) the sheaf of coordinate rings on \(X\), then the author shows that \(S\) is Cohen-Macaulay if and only if \(\mathcal A\) is ``gradual'' and for every \(x\in X\), \(\tilde H^i(X^x,\mathcal A)=0\) for \(i< \mathrm{rk}(x)-2\), where \(X^x:=\{y\in X\mid y<x\}\). The main idea is to use a result of Orecchia that expresses \(S\) as \(\Gamma(\mathcal A)\), then to estimate the depth of \(S\) by constructing a resolution by modules of known depth. The result generalizes an equivalent version of Reisner's theorem on the Cohen-Macaulay property of face rings of simplicial posets.
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13F45
Mathematics Subject Classification ID: 14A05
zbMATH DE Number: 4156609
zbMATH Keywords: unions of linear subspaces
zbMATH Keywords: coordinate ring
zbMATH Keywords: seminormalization
zbMATH Keywords: intersection poset
zbMATH Keywords: Cohen-Macaulay
zbMATH Keywords: face rings
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4156609